id	sid	tid	token	lemma	pos
iajs-3051	1	1	ihjpas	ihjpas	PROPN
iajs-3051	1	2	.	.	PUNCT
iajs-3051	2	1	36	36	NUM
iajs-3051	2	2	(	(	PUNCT
iajs-3051	2	3	3	3	NUM
iajs-3051	2	4	)	)	PUNCT
iajs-3051	2	5	2023	2023	NUM
iajs-3051	2	6	398	398	NUM
iajs-3051	2	7	this	this	DET
iajs-3051	2	8	work	work	NOUN
iajs-3051	2	9	is	be	AUX
iajs-3051	2	10	licensed	license	VERB
iajs-3051	2	11	under	under	ADP
iajs-3051	2	12	a	a	DET
iajs-3051	2	13	creative	creative	ADJ
iajs-3051	2	14	commons	common	NOUN
iajs-3051	2	15	attribution	attribution	NOUN
iajs-3051	2	16	4.0	4.0	NUM
iajs-3051	2	17	international	international	ADJ
iajs-3051	2	18	license	license	NOUN
iajs-3051	2	19	*	*	PUNCT
iajs-3051	2	20	corresponding	correspond	VERB
iajs-3051	2	21	author	author	NOUN
iajs-3051	2	22	:	:	PUNCT
iajs-3051	2	23	stabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	stabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-3051	2	24	abstract	abstract	ADJ
iajs-3051	2	25	the	the	DET
iajs-3051	2	26	structure	structure	NOUN
iajs-3051	2	27	of	of	ADP
iajs-3051	2	28	this	this	DET
iajs-3051	2	29	paper	paper	NOUN
iajs-3051	2	30	includes	include	VERB
iajs-3051	2	31	introduction	introduction	NOUN
iajs-3051	2	32	the	the	DET
iajs-3051	2	33	definition	definition	NOUN
iajs-3051	2	34	of	of	ADP
iajs-3051	2	35	the	the	DET
iajs-3051	2	36	nano	nano	ADJ
iajs-3051	2	37	topological	topological	ADJ
iajs-3051	2	38	space	space	NOUN
iajs-3051	2	39	,	,	PUNCT
iajs-3051	2	40	which	which	PRON
iajs-3051	2	41	was	be	AUX
iajs-3051	2	42	defined	define	VERB
iajs-3051	2	43	by	by	ADP
iajs-3051	2	44	m.	m.	NOUN
iajs-3051	2	45	l.	l.	PROPN
iajs-3051	2	46	thivagar	thivagar	PROPN
iajs-3051	2	47	,	,	PUNCT
iajs-3051	2	48	who	who	PRON
iajs-3051	2	49	defined	define	VERB
iajs-3051	2	50	the	the	DET
iajs-3051	2	51	lower	low	ADJ
iajs-3051	2	52	approximation	approximation	NOUN
iajs-3051	2	53	of	of	ADP
iajs-3051	2	54	g	g	PROPN
iajs-3051	2	55	and	and	CCONJ
iajs-3051	2	56	the	the	DET
iajs-3051	2	57	upper	upper	ADJ
iajs-3051	2	58	approximation	approximation	NOUN
iajs-3051	2	59	of	of	ADP
iajs-3051	2	60	g	g	NOUN
iajs-3051	2	61	,	,	PUNCT
iajs-3051	2	62	as	as	ADV
iajs-3051	2	63	well	well	ADV
iajs-3051	2	64	as	as	ADP
iajs-3051	2	65	defined	define	VERB
iajs-3051	2	66	the	the	DET
iajs-3051	2	67	boundary	boundary	ADJ
iajs-3051	2	68	region	region	NOUN
iajs-3051	2	69	of	of	ADP
iajs-3051	2	70	g	g	PROPN
iajs-3051	2	71	and	and	CCONJ
iajs-3051	2	72	some	some	DET
iajs-3051	2	73	other	other	ADJ
iajs-3051	2	74	important	important	ADJ
iajs-3051	2	75	definitions	definition	NOUN
iajs-3051	2	76	that	that	PRON
iajs-3051	2	77	were	be	AUX
iajs-3051	2	78	mentioned	mention	VERB
iajs-3051	2	79	in	in	ADP
iajs-3051	2	80	this	this	DET
iajs-3051	2	81	paper	paper	NOUN
iajs-3051	2	82	with	with	ADP
iajs-3051	2	83	giving	give	VERB
iajs-3051	2	84	some	some	DET
iajs-3051	2	85	theories	theory	NOUN
iajs-3051	2	86	on	on	ADP
iajs-3051	2	87	this	this	DET
iajs-3051	2	88	subject	subject	NOUN
iajs-3051	2	89	.	.	PUNCT
iajs-3051	3	1	some	some	DET
iajs-3051	3	2	examples	example	NOUN
iajs-3051	3	3	of	of	ADP
iajs-3051	3	4	defining	define	VERB
iajs-3051	3	5	nano	nano	NOUN
iajs-3051	3	6	perfect	perfect	ADJ
iajs-3051	3	7	mappings	mapping	NOUN
iajs-3051	3	8	are	be	AUX
iajs-3051	3	9	presented	present	VERB
iajs-3051	3	10	along	along	ADP
iajs-3051	3	11	with	with	ADP
iajs-3051	3	12	some	some	DET
iajs-3051	3	13	basic	basic	ADJ
iajs-3051	3	14	theories	theory	NOUN
iajs-3051	3	15	.	.	PUNCT
iajs-3051	4	1	also	also	ADV
iajs-3051	4	2	,	,	PUNCT
iajs-3051	4	3	some	some	DET
iajs-3051	4	4	basic	basic	ADJ
iajs-3051	4	5	definitions	definition	NOUN
iajs-3051	4	6	were	be	AUX
iajs-3051	4	7	presented	present	VERB
iajs-3051	4	8	that	that	PRON
iajs-3051	4	9	form	form	VERB
iajs-3051	4	10	the	the	DET
iajs-3051	4	11	focus	focus	NOUN
iajs-3051	4	12	of	of	ADP
iajs-3051	4	13	this	this	DET
iajs-3051	4	14	paper	paper	NOUN
iajs-3051	4	15	,	,	PUNCT
iajs-3051	4	16	including	include	VERB
iajs-3051	4	17	the	the	DET
iajs-3051	4	18	definition	definition	NOUN
iajs-3051	4	19	of	of	ADP
iajs-3051	4	20	nano	nano	ADJ
iajs-3051	4	21	pseudometrizable	pseudometrizable	ADJ
iajs-3051	4	22	space	space	NOUN
iajs-3051	4	23	,	,	PUNCT
iajs-3051	4	24	the	the	DET
iajs-3051	4	25	definition	definition	NOUN
iajs-3051	4	26	of	of	ADP
iajs-3051	4	27	nano	nano	NOUN
iajs-3051	4	28	compactly	compactly	ADV
iajs-3051	4	29	generated	generate	VERB
iajs-3051	4	30	space	space	NOUN
iajs-3051	4	31	,	,	PUNCT
iajs-3051	4	32	and	and	CCONJ
iajs-3051	4	33	the	the	DET
iajs-3051	4	34	definition	definition	NOUN
iajs-3051	4	35	of	of	ADP
iajs-3051	4	36	completely	completely	ADV
iajs-3051	4	37	nano	nano	ADJ
iajs-3051	4	38	para	para	ADJ
iajs-3051	4	39	-	-	PUNCT
iajs-3051	4	40	compact	compact	ADJ
iajs-3051	4	41	.	.	PUNCT
iajs-3051	5	1	in	in	ADP
iajs-3051	5	2	this	this	DET
iajs-3051	5	3	paper	paper	NOUN
iajs-3051	5	4	,	,	PUNCT
iajs-3051	5	5	we	we	PRON
iajs-3051	5	6	presented	present	VERB
iajs-3051	5	7	images	image	NOUN
iajs-3051	5	8	of	of	ADP
iajs-3051	5	9	nano	nano	VERB
iajs-3051	5	10	perfect	perfect	ADJ
iajs-3051	5	11	mappings	mapping	NOUN
iajs-3051	5	12	with	with	ADP
iajs-3051	5	13	some	some	DET
iajs-3051	5	14	definitions	definition	NOUN
iajs-3051	5	15	and	and	CCONJ
iajs-3051	5	16	important	important	ADJ
iajs-3051	5	17	evidence	evidence	NOUN
iajs-3051	5	18	related	relate	VERB
iajs-3051	5	19	to	to	ADP
iajs-3051	5	20	them	they	PRON
iajs-3051	5	21	,	,	PUNCT
iajs-3051	5	22	then	then	ADV
iajs-3051	5	23	we	we	PRON
iajs-3051	5	24	presented	present	VERB
iajs-3051	5	25	inverse	inverse	NOUN
iajs-3051	5	26	images	image	NOUN
iajs-3051	5	27	of	of	ADP
iajs-3051	5	28	nano	nano	VERB
iajs-3051	5	29	perfect	perfect	ADJ
iajs-3051	5	30	mappings	mapping	NOUN
iajs-3051	5	31	with	with	ADP
iajs-3051	5	32	related	related	ADJ
iajs-3051	5	33	theories	theory	NOUN
iajs-3051	5	34	.	.	PUNCT
iajs-3051	6	1	keywords	keyword	NOUN
iajs-3051	6	2	:	:	PUNCT
iajs-3051	6	3	nano	nano	NOUN
iajs-3051	6	4	topological	topological	ADJ
iajs-3051	6	5	space	space	NOUN
iajs-3051	6	6	,	,	PUNCT
iajs-3051	6	7	nano	nano	ADJ
iajs-3051	6	8	continuous	continuous	ADJ
iajs-3051	6	9	mappings	mapping	NOUN
iajs-3051	6	10	,	,	PUNCT
iajs-3051	6	11	nano	nano	ADJ
iajs-3051	6	12	compact	compact	ADJ
iajs-3051	6	13	space	space	NOUN
iajs-3051	6	14	,	,	PUNCT
iajs-3051	6	15	nano	nano	NOUN
iajs-3051	6	16	perfect	perfect	ADJ
iajs-3051	6	17	mappings	mapping	NOUN
iajs-3051	6	18	.	.	PUNCT
iajs-3051	7	1	1	1	X
iajs-3051	7	2	.	.	X
iajs-3051	7	3	introduction	introduction	NOUN
iajs-3051	7	4	[	[	X
iajs-3051	7	5	1	1	NUM
iajs-3051	7	6	]	]	PUNCT
iajs-3051	7	7	introduced	introduce	VERB
iajs-3051	7	8	the	the	DET
iajs-3051	7	9	concept	concept	NOUN
iajs-3051	7	10	of	of	ADP
iajs-3051	7	11	nano	nano	NOUN
iajs-3051	7	12	topological	topological	ADJ
iajs-3051	7	13	spaces	space	NOUN
iajs-3051	7	14	with	with	ADP
iajs-3051	7	15	respect	respect	NOUN
iajs-3051	7	16	to	to	ADP
iajs-3051	7	17	a	a	DET
iajs-3051	7	18	subset	subset	NOUN
iajs-3051	7	19	g	g	NOUN
iajs-3051	7	20	of	of	ADP
iajs-3051	7	21	a	a	DET
iajs-3051	7	22	universe	universe	ADJ
iajs-3051	7	23	u	u	NOUN
iajs-3051	7	24	,	,	PUNCT
iajs-3051	7	25	which	which	PRON
iajs-3051	7	26	known	know	VERB
iajs-3051	7	27	as	as	ADP
iajs-3051	7	28	terms	term	NOUN
iajs-3051	7	29	of	of	ADP
iajs-3051	7	30	approximations	approximation	NOUN
iajs-3051	7	31	and	and	CCONJ
iajs-3051	7	32	boundary	boundary	ADJ
iajs-3051	7	33	region	region	NOUN
iajs-3051	7	34	of	of	ADP
iajs-3051	7	35	a	a	DET
iajs-3051	7	36	subset	subset	NOUN
iajs-3051	7	37	of	of	ADP
iajs-3051	7	38	an	an	DET
iajs-3051	7	39	universe	universe	NOUN
iajs-3051	7	40	using	use	VERB
iajs-3051	7	41	an	an	DET
iajs-3051	7	42	equivalence	equivalence	NOUN
iajs-3051	7	43	relation	relation	NOUN
iajs-3051	7	44	on	on	ADP
iajs-3051	7	45	it	it	PRON
iajs-3051	7	46	and	and	CCONJ
iajs-3051	7	47	also	also	ADV
iajs-3051	7	48	known	know	VERB
iajs-3051	7	49	as	as	ADP
iajs-3051	7	50	nano	nano	NOUN
iajs-3051	7	51	-	-	PUNCT
iajs-3051	7	52	closed	closed	ADJ
iajs-3051	7	53	-	-	PUNCT
iajs-3051	7	54	sets	set	NOUN
iajs-3051	7	55	,	,	PUNCT
iajs-3051	7	56	nano	nano	NOUN
iajs-3051	7	57	-	-	ADJ
iajs-3051	7	58	interior	interior	ADJ
iajs-3051	7	59	and	and	CCONJ
iajs-3051	7	60	nano	nano	NOUN
iajs-3051	7	61	-	-	PUNCT
iajs-3051	7	62	closure	closure	NOUN
iajs-3051	7	63	.	.	PUNCT
iajs-3051	8	1	[	[	X
iajs-3051	8	2	2	2	X
iajs-3051	8	3	]	]	PUNCT
iajs-3051	8	4	introduced	introduce	VERB
iajs-3051	8	5	fibrewise	fibrewise	ADV
iajs-3051	8	6	ij	ij	ADJ
iajs-3051	8	7	-	-	ADJ
iajs-3051	8	8	perfect	perfect	ADJ
iajs-3051	8	9	bitopological	bitopological	ADJ
iajs-3051	8	10	spaces	space	NOUN
iajs-3051	8	11	and	and	CCONJ
iajs-3051	8	12	[	[	X
iajs-3051	8	13	3	3	NUM
iajs-3051	8	14	]	]	PUNCT
iajs-3051	8	15	introduced	introduce	VERB
iajs-3051	8	16	weak	weak	ADJ
iajs-3051	8	17	and	and	CCONJ
iajs-3051	8	18	strong	strong	ADJ
iajs-3051	8	19	forms	form	NOUN
iajs-3051	8	20	of	of	ADP
iajs-3051	8	21	ωperfect	ωperfect	NOUN
iajs-3051	8	22	mappings	mapping	NOUN
iajs-3051	8	23	.	.	PUNCT
iajs-3051	9	1	[	[	X
iajs-3051	9	2	4	4	X
iajs-3051	9	3	]	]	PUNCT
iajs-3051	9	4	introduced	introduce	VERB
iajs-3051	9	5	r𝛼-compactness	r𝛼-compactness	NOUN
iajs-3051	9	6	on	on	ADP
iajs-3051	9	7	bitopological	bitopological	ADJ
iajs-3051	9	8	spaces	space	NOUN
iajs-3051	9	9	.	.	PUNCT
iajs-3051	10	1	[	[	X
iajs-3051	10	2	5	5	NUM
iajs-3051	10	3	]	]	PUNCT
iajs-3051	10	4	introduced	introduce	VERB
iajs-3051	10	5	on	on	ADP
iajs-3051	10	6	cohomology	cohomology	NOUN
iajs-3051	10	7	groups	group	NOUN
iajs-3051	10	8	of	of	ADP
iajs-3051	10	9	four	four	NUM
iajs-3051	10	10	-	-	PUNCT
iajs-3051	10	11	dimensional	dimensional	ADJ
iajs-3051	10	12	nilpotent	nilpotent	ADJ
iajs-3051	10	13	associative	associative	NOUN
iajs-3051	10	14	algebras	algebra	NOUN
iajs-3051	10	15	.	.	PUNCT
iajs-3051	11	1	[	[	X
iajs-3051	11	2	6,7,8,9,10	6,7,8,9,10	NOUN
iajs-3051	11	3	]	]	PUNCT
iajs-3051	11	4	introduce	introduce	VERB
iajs-3051	11	5	some	some	DET
iajs-3051	11	6	basic	basic	ADJ
iajs-3051	11	7	concepts	concept	NOUN
iajs-3051	11	8	that	that	PRON
iajs-3051	11	9	helped	help	VERB
iajs-3051	11	10	us	we	PRON
iajs-3051	11	11	build	build	VERB
iajs-3051	11	12	this	this	DET
iajs-3051	11	13	work	work	NOUN
iajs-3051	11	14	.	.	PUNCT
iajs-3051	12	1	[	[	X
iajs-3051	12	2	11,12	11,12	NUM
iajs-3051	12	3	,	,	PUNCT
iajs-3051	12	4	13,14	13,14	NUM
iajs-3051	12	5	]	]	PUNCT
iajs-3051	12	6	introduce	introduce	VERB
iajs-3051	12	7	some	some	DET
iajs-3051	12	8	types	type	NOUN
iajs-3051	12	9	of	of	ADP
iajs-3051	12	10	mappings	mapping	NOUN
iajs-3051	12	11	in	in	ADP
iajs-3051	12	12	bitopological	bitopological	ADJ
iajs-3051	12	13	spaces	space	NOUN
iajs-3051	12	14	and	and	CCONJ
iajs-3051	12	15	soft	soft	ADJ
iajs-3051	12	16	simply	simply	ADV
iajs-3051	12	17	compact	compact	ADJ
iajs-3051	12	18	spaces	space	NOUN
iajs-3051	12	19	and	and	CCONJ
iajs-3051	12	20	other	other	ADJ
iajs-3051	12	21	topics	topic	NOUN
iajs-3051	12	22	related	relate	VERB
iajs-3051	12	23	to	to	ADP
iajs-3051	12	24	the	the	DET
iajs-3051	12	25	doi.org/10.30526/36.3.3051	doi.org/10.30526/36.3.3051	ADJ
iajs-3051	12	26	article	article	NOUN
iajs-3051	12	27	history	history	NOUN
iajs-3051	12	28	:	:	PUNCT
iajs-3051	12	29	received	receive	VERB
iajs-3051	12	30	30	30	NUM
iajs-3051	12	31	september	september	PROPN
iajs-3051	12	32	2022	2022	NUM
iajs-3051	12	33	,	,	PUNCT
iajs-3051	12	34	accepted	accept	VERB
iajs-3051	12	35	20	20	NUM
iajs-3051	12	36	february	february	NOUN
iajs-3051	12	37	2022	2022	NUM
iajs-3051	12	38	,	,	PUNCT
iajs-3051	12	39	published	publish	VERB
iajs-3051	12	40	in	in	ADP
iajs-3051	12	41	july	july	PROPN
iajs-3051	12	42	2023	2023	NUM
iajs-3051	12	43	.	.	PUNCT
iajs-3051	13	1	ibn	ibn	PROPN
iajs-3051	13	2	al	al	PROPN
iajs-3051	13	3	-	-	PUNCT
iajs-3051	13	4	haitham	haitham	PROPN
iajs-3051	13	5	journal	journal	PROPN
iajs-3051	13	6	for	for	ADP
iajs-3051	13	7	pure	pure	ADJ
iajs-3051	13	8	and	and	CCONJ
iajs-3051	13	9	applied	applied	ADJ
iajs-3051	13	10	sciences	sciences	PROPN
iajs-3051	13	11	journal	journal	PROPN
iajs-3051	13	12	homepage	homepage	NOUN
iajs-3051	13	13	:	:	PUNCT
iajs-3051	13	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	VERB
iajs-3051	13	15	some	some	DET
iajs-3051	13	16	results	result	NOUN
iajs-3051	13	17	on	on	ADP
iajs-3051	13	18	nano	nano	NOUN
iajs-3051	13	19	perfect	perfect	ADJ
iajs-3051	13	20	mappings	mapping	NOUN
iajs-3051	13	21	e.	e.	PROPN
iajs-3051	13	22	a.	a.	PROPN
iajs-3051	13	23	mohsin	mohsin	PROPN
iajs-3051	13	24	department	department	PROPN
iajs-3051	13	25	of	of	ADP
iajs-3051	13	26	mathematics	mathematics	PROPN
iajs-3051	13	27	,	,	PUNCT
iajs-3051	13	28	college	college	NOUN
iajs-3051	13	29	of	of	ADP
iajs-3051	13	30	education	education	NOUN
iajs-3051	13	31	for	for	ADP
iajs-3051	13	32	pure	pure	ADJ
iajs-3051	13	33	science	science	NOUN
iajs-3051	13	34	ibn	ibn	PROPN
iajs-3051	13	35	al	al	PROPN
iajs-3051	13	36	-	-	PUNCT
iajs-3051	13	37	haytham	haytham	PROPN
iajs-3051	13	38	,	,	PUNCT
iajs-3051	13	39	university	university	NOUN
iajs-3051	13	40	of	of	ADP
iajs-3051	13	41	baghdad	baghdad	PROPN
iajs-3051	13	42	,	,	PUNCT
iajs-3051	13	43	baghdad	baghdad	PROPN
iajs-3051	13	44	,	,	PUNCT
iajs-3051	13	45	iraq	iraq	PROPN
iajs-3051	13	46	.	.	PUNCT
iajs-3051	14	1	ihcoedu.uobaghdad.edu.iq@istabraq.abd2103	ihcoedu.uobaghdad.edu.iq@istabraq.abd2103	X
iajs-3051	14	2	m	m	VERB
iajs-3051	14	3	y.	y.	PROPN
iajs-3051	14	4	y.	y.	PROPN
iajs-3051	14	5	y𝐨us𝐢f	y𝐨us𝐢f	PROPN
iajs-3051	14	6	department	department	PROPN
iajs-3051	14	7	of	of	ADP
iajs-3051	14	8	mathematics	mathematics	PROPN
iajs-3051	14	9	,	,	PUNCT
iajs-3051	14	10	college	college	NOUN
iajs-3051	14	11	of	of	ADP
iajs-3051	14	12	education	education	NOUN
iajs-3051	14	13	for	for	ADP
iajs-3051	14	14	pure	pure	ADJ
iajs-3051	14	15	science	science	NOUN
iajs-3051	14	16	ibn	ibn	PROPN
iajs-3051	14	17	al	al	PROPN
iajs-3051	14	18	-	-	PUNCT
iajs-3051	14	19	haytham	haytham	PROPN
iajs-3051	14	20	,	,	PUNCT
iajs-3051	14	21	university	university	NOUN
iajs-3051	14	22	of	of	ADP
iajs-3051	14	23	baghdad	baghdad	PROPN
iajs-3051	14	24	,	,	PUNCT
iajs-3051	14	25	baghdad	baghdad	PROPN
iajs-3051	14	26	,	,	PUNCT
iajs-3051	14	27	iraq	iraq	PROPN
iajs-3051	14	28	.	.	PUNCT
iajs-3051	15	1	yousif.y.y@ihcoedu.uobaghdad.edu.iq	yousif.y.y@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-3051	15	2	m.	m.	PROPN
iajs-3051	15	3	el	el	PROPN
iajs-3051	15	4	sayed	say	VERB
iajs-3051	15	5	najran	najran	PROPN
iajs-3051	15	6	university	university	PROPN
iajs-3051	15	7	,	,	PUNCT
iajs-3051	15	8	college	college	NOUN
iajs-3051	15	9	of	of	ADP
iajs-3051	15	10	science	science	NOUN
iajs-3051	15	11	and	and	CCONJ
iajs-3051	15	12	arts	art	NOUN
iajs-3051	15	13	,	,	PUNCT
iajs-3051	15	14	department	department	NOUN
iajs-3051	15	15	of	of	ADP
iajs-3051	15	16	mathematics	mathematic	NOUN
iajs-3051	15	17	,	,	PUNCT
iajs-3051	15	18	66445	66445	NUM
iajs-3051	15	19	,	,	PUNCT
iajs-3051	15	20	saudi	saudi	PROPN
iajs-3051	15	21	arabia	arabia	PROPN
iajs-3051	15	22	.	.	PUNCT
iajs-3051	16	1	mebadris@nu.edu.sa	mebadris@nu.edu.sa	PROPN
iajs-3051	16	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3051	16	3	mailto:stabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	mailto:stabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-3051	16	4	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3051	16	5	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3051	16	6	mailto:istabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	mailto:istabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-3051	16	7	mailto:istabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	mailto:istabraq.abd2103m@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-3051	16	8	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3051	16	9	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3051	16	10	mailto:yousif.y.y@ihcoedu.uobaghdad.edu.iq	mailto:yousif.y.y@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-3051	16	11	mailto:yousif.y.y@ihcoedu.uobaghdad.edu.iq	mailto:yousif.y.y@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-3051	16	12	mailto:mebadris@nu.edu.sa	mailto:mebadris@nu.edu.sa	PROPN
iajs-3051	16	13	mailto:mebadris@nu.edu.sa	mailto:mebadris@nu.edu.sa	PROPN
iajs-3051	16	14	ihjpas	ihjpa	VERB
iajs-3051	16	15	.	.	PUNCT
iajs-3051	17	1	36	36	NUM
iajs-3051	17	2	(	(	PUNCT
iajs-3051	17	3	3	3	NUM
iajs-3051	17	4	)	)	PUNCT
iajs-3051	17	5	2023	2023	NUM
iajs-3051	17	6	399	399	NUM
iajs-3051	17	7	mentioned	mention	VERB
iajs-3051	17	8	sources	source	NOUN
iajs-3051	17	9	.	.	PUNCT
iajs-3051	18	1	in	in	ADP
iajs-3051	18	2	this	this	DET
iajs-3051	18	3	paper	paper	NOUN
iajs-3051	18	4	,	,	PUNCT
iajs-3051	18	5	we	we	PRON
iajs-3051	18	6	introduce	introduce	VERB
iajs-3051	18	7	a	a	DET
iajs-3051	18	8	nano	nano	NOUN
iajs-3051	18	9	perfect	perfect	ADJ
iajs-3051	18	10	mappings	mapping	NOUN
iajs-3051	18	11	and	and	CCONJ
iajs-3051	18	12	several	several	ADJ
iajs-3051	18	13	related	relate	VERB
iajs-3051	18	14	theorems	theorem	NOUN
iajs-3051	18	15	.	.	PUNCT
iajs-3051	19	1	definition	definition	NOUN
iajs-3051	19	2	1.1	1.1	NUM
iajs-3051	19	3	:	:	PUNCT
iajs-3051	20	1	[	[	X
iajs-3051	20	2	15	15	NUM
iajs-3051	20	3	]	]	PUNCT
iajs-3051	20	4	let	let	VERB
iajs-3051	20	5	u	u	PRON
iajs-3051	20	6	be	be	AUX
iajs-3051	20	7	a	a	DET
iajs-3051	20	8	non	non	X
iajs-3051	20	9	empty	empty	ADJ
iajs-3051	20	10	finite	finite	ADJ
iajs-3051	20	11	set	set	NOUN
iajs-3051	20	12	of	of	ADP
iajs-3051	20	13	objects	object	NOUN
iajs-3051	20	14	is	be	AUX
iajs-3051	20	15	said	say	VERB
iajs-3051	20	16	to	to	PART
iajs-3051	20	17	be	be	AUX
iajs-3051	20	18	the	the	DET
iajs-3051	20	19	universe	universe	NOUN
iajs-3051	20	20	and	and	CCONJ
iajs-3051	20	21	r	r	NOUN
iajs-3051	20	22	be	be	VERB
iajs-3051	20	23	an	an	DET
iajs-3051	20	24	equivalence	equivalence	NOUN
iajs-3051	20	25	relation	relation	NOUN
iajs-3051	20	26	on	on	ADP
iajs-3051	20	27	u	u	NOUN
iajs-3051	20	28	is	be	AUX
iajs-3051	20	29	called	call	VERB
iajs-3051	20	30	indiscernibility	indiscernibility	NOUN
iajs-3051	20	31	relation	relation	NOUN
iajs-3051	20	32	.	.	PUNCT
iajs-3051	21	1	then	then	ADV
iajs-3051	21	2	u	u	NOUN
iajs-3051	21	3	is	be	AUX
iajs-3051	21	4	divided	divide	VERB
iajs-3051	21	5	into	into	ADP
iajs-3051	21	6	disjoint	disjoint	ADJ
iajs-3051	21	7	equivalence	equivalence	NOUN
iajs-3051	21	8	classes	class	NOUN
iajs-3051	21	9	.	.	PUNCT
iajs-3051	22	1	components	component	NOUN
iajs-3051	22	2	belonging	belong	VERB
iajs-3051	22	3	to	to	ADP
iajs-3051	22	4	the	the	DET
iajs-3051	22	5	same	same	ADJ
iajs-3051	22	6	equivalence	equivalence	NOUN
iajs-3051	22	7	class	class	NOUN
iajs-3051	22	8	are	be	AUX
iajs-3051	22	9	called	call	VERB
iajs-3051	22	10	the	the	DET
iajs-3051	22	11	indiscernible	indiscernible	ADJ
iajs-3051	22	12	with	with	ADP
iajs-3051	22	13	one	one	NUM
iajs-3051	22	14	another	another	DET
iajs-3051	22	15	.	.	PUNCT
iajs-3051	23	1	the	the	DET
iajs-3051	23	2	spouse	spouse	NOUN
iajs-3051	23	3	(	(	PUNCT
iajs-3051	23	4	u	u	NOUN
iajs-3051	23	5	,	,	PUNCT
iajs-3051	23	6	r	r	NOUN
iajs-3051	23	7	)	)	PUNCT
iajs-3051	23	8	is	be	AUX
iajs-3051	23	9	called	call	VERB
iajs-3051	23	10	the	the	DET
iajs-3051	23	11	approximation	approximation	NOUN
iajs-3051	23	12	-	-	PUNCT
iajs-3051	23	13	space	space	NOUN
iajs-3051	23	14	.	.	PUNCT
iajs-3051	24	1	let	let	VERB
iajs-3051	24	2	g	g	PROPN
iajs-3051	24	3	⊆	⊆	NUM
iajs-3051	24	4	u.	u.	NOUN
iajs-3051	24	5	then	then	ADV
iajs-3051	24	6	,	,	PUNCT
iajs-3051	24	7	a	a	X
iajs-3051	24	8	)	)	PUNCT
iajs-3051	24	9	the	the	DET
iajs-3051	24	10	lower	low	ADJ
iajs-3051	24	11	approximation	approximation	NOUN
iajs-3051	24	12	of	of	ADP
iajs-3051	24	13	g	g	NOUN
iajs-3051	24	14	with	with	ADP
iajs-3051	24	15	reference	reference	NOUN
iajs-3051	24	16	to	to	ADP
iajs-3051	24	17	r	r	NOUN
iajs-3051	24	18	is	be	AUX
iajs-3051	24	19	the	the	DET
iajs-3051	24	20	set	set	NOUN
iajs-3051	24	21	of	of	ADP
iajs-3051	24	22	all	all	DET
iajs-3051	24	23	objects	object	NOUN
iajs-3051	24	24	which	which	PRON
iajs-3051	24	25	can	can	AUX
iajs-3051	24	26	be	be	AUX
iajs-3051	24	27	categorized	categorize	VERB
iajs-3051	24	28	as	as	ADP
iajs-3051	24	29	g	g	NOUN
iajs-3051	24	30	with	with	ADP
iajs-3051	24	31	reference	reference	NOUN
iajs-3051	24	32	to	to	ADP
iajs-3051	24	33	r	r	NOUN
iajs-3051	24	34	and	and	CCONJ
iajs-3051	24	35	is	be	AUX
iajs-3051	24	36	symboly	symboly	ADJ
iajs-3051	24	37	by	by	ADP
iajs-3051	24	38	lor(g	lor(g	PROPN
iajs-3051	24	39	)	)	PUNCT
iajs-3051	24	40	.	.	PUNCT
iajs-3051	25	1	that	that	PRON
iajs-3051	25	2	is	be	AUX
iajs-3051	25	3	,	,	PUNCT
iajs-3051	25	4	lor(g	lor(g	X
iajs-3051	25	5	)	)	PUNCT
iajs-3051	25	6	=	=	SYM
iajs-3051	26	1	⋃{r(g	⋃{r(g	NOUN
iajs-3051	26	2	):	):	PUNCT
iajs-3051	26	3	r(g	r(g	NUM
iajs-3051	26	4	)	)	PUNCT
iajs-3051	26	5	⊆	⊆	NUM
iajs-3051	26	6	g	g	NOUN
iajs-3051	26	7	,	,	PUNCT
iajs-3051	26	8	g	g	PROPN
iajs-3051	26	9	∈	∈	PROPN
iajs-3051	26	10	u	u	NOUN
iajs-3051	26	11	}	}	PUNCT
iajs-3051	26	12	where	where	SCONJ
iajs-3051	26	13	r(g	r(g	NUM
iajs-3051	26	14	)	)	PUNCT
iajs-3051	26	15	indicate	indicate	VERB
iajs-3051	26	16	the	the	DET
iajs-3051	26	17	equivalence	equivalence	NOUN
iajs-3051	26	18	class	class	NOUN
iajs-3051	26	19	determined	determine	VERB
iajs-3051	26	20	by	by	ADP
iajs-3051	26	21	g	g	PROPN
iajs-3051	26	22	∈	∈	PROPN
iajs-3051	26	23	u.	u.	PROPN
iajs-3051	26	24	b	b	PROPN
iajs-3051	26	25	)	)	PUNCT
iajs-3051	26	26	the	the	DET
iajs-3051	26	27	upper	upper	ADJ
iajs-3051	26	28	approximation	approximation	NOUN
iajs-3051	26	29	of	of	ADP
iajs-3051	26	30	g	g	NOUN
iajs-3051	26	31	with	with	ADP
iajs-3051	26	32	reference	reference	NOUN
iajs-3051	26	33	to	to	ADP
iajs-3051	26	34	r	r	NOUN
iajs-3051	26	35	is	be	AUX
iajs-3051	26	36	the	the	DET
iajs-3051	26	37	set	set	NOUN
iajs-3051	26	38	of	of	ADP
iajs-3051	26	39	all	all	DET
iajs-3051	26	40	objects	object	NOUN
iajs-3051	26	41	which	which	PRON
iajs-3051	26	42	can	can	AUX
iajs-3051	26	43	be	be	AUX
iajs-3051	26	44	possibly	possibly	ADV
iajs-3051	26	45	classified	classify	VERB
iajs-3051	26	46	as	as	ADP
iajs-3051	26	47	g	g	PROPN
iajs-3051	26	48	with	with	ADP
iajs-3051	26	49	reference	reference	NOUN
iajs-3051	26	50	to	to	ADP
iajs-3051	26	51	r	r	NOUN
iajs-3051	26	52	and	and	CCONJ
iajs-3051	26	53	is	be	AUX
iajs-3051	26	54	symboly	symboly	ADJ
iajs-3051	26	55	by	by	ADP
iajs-3051	26	56	upr(g	upr(g	PROPN
iajs-3051	26	57	)	)	PUNCT
iajs-3051	26	58	.	.	PUNCT
iajs-3051	27	1	that	that	PRON
iajs-3051	27	2	is	be	AUX
iajs-3051	27	3	,	,	PUNCT
iajs-3051	27	4	upr(g	upr(g	PROPN
iajs-3051	27	5	)	)	PUNCT
iajs-3051	27	6	=	=	SYM
iajs-3051	28	1	⋃{r(g	⋃{r(g	NOUN
iajs-3051	28	2	):	):	PUNCT
iajs-3051	28	3	r(g	r(g	NUM
iajs-3051	28	4	)	)	PUNCT
iajs-3051	28	5	⋂	⋂	PROPN
iajs-3051	28	6	g	g	PROPN
iajs-3051	28	7	≠	≠	PROPN
iajs-3051	28	8	∅	∅	NOUN
iajs-3051	28	9	,	,	PUNCT
iajs-3051	28	10	g	g	PROPN
iajs-3051	28	11	∈	∈	PROPN
iajs-3051	28	12	u	u	NOUN
iajs-3051	28	13	}	}	PUNCT
iajs-3051	28	14	.	.	PUNCT
iajs-3051	29	1	c	c	X
iajs-3051	29	2	)	)	PUNCT
iajs-3051	29	3	the	the	DET
iajs-3051	29	4	boundary	boundary	ADJ
iajs-3051	29	5	region	region	NOUN
iajs-3051	29	6	of	of	ADP
iajs-3051	29	7	g	g	PROPN
iajs-3051	29	8	with	with	ADP
iajs-3051	29	9	reference	reference	NOUN
iajs-3051	29	10	to	to	ADP
iajs-3051	29	11	r	r	NOUN
iajs-3051	29	12	is	be	AUX
iajs-3051	29	13	the	the	DET
iajs-3051	29	14	set	set	NOUN
iajs-3051	29	15	of	of	ADP
iajs-3051	29	16	all	all	DET
iajs-3051	29	17	objects	object	NOUN
iajs-3051	29	18	which	which	PRON
iajs-3051	29	19	can	can	AUX
iajs-3051	29	20	be	be	AUX
iajs-3051	29	21	classified	classify	VERB
iajs-3051	29	22	neither	neither	CCONJ
iajs-3051	29	23	as	as	ADP
iajs-3051	29	24	g	g	NOUN
iajs-3051	29	25	nor	nor	CCONJ
iajs-3051	29	26	as	as	ADV
iajs-3051	29	27	not	not	PART
iajs-3051	29	28	/g	/g	PUNCT
iajs-3051	29	29	with	with	ADP
iajs-3051	29	30	reference	reference	NOUN
iajs-3051	29	31	to	to	ADP
iajs-3051	29	32	r	r	NOUN
iajs-3051	29	33	and	and	CCONJ
iajs-3051	29	34	is	be	AUX
iajs-3051	29	35	symboly	symboly	ADJ
iajs-3051	29	36	by	by	ADP
iajs-3051	29	37	bor(g	bor(g	PROPN
iajs-3051	29	38	)	)	PUNCT
iajs-3051	29	39	.	.	PUNCT
iajs-3051	30	1	bor(g	bor(g	X
iajs-3051	30	2	)	)	PUNCT
iajs-3051	30	3	=	=	SYM
iajs-3051	30	4	upr(g	upr(g	PROPN
iajs-3051	30	5	)	)	PUNCT
iajs-3051	30	6	/	/	SYM
iajs-3051	30	7	lor(g	lor(g	NOUN
iajs-3051	30	8	)	)	PUNCT
iajs-3051	30	9	.	.	PUNCT
iajs-3051	31	1	property	property	NOUN
iajs-3051	31	2	1.2	1.2	NUM
iajs-3051	31	3	:	:	PUNCT
iajs-3051	32	1	[	[	X
iajs-3051	32	2	16	16	NUM
iajs-3051	32	3	]	]	X
iajs-3051	32	4	if	if	SCONJ
iajs-3051	32	5	(	(	PUNCT
iajs-3051	32	6	u	u	NOUN
iajs-3051	32	7	,	,	PUNCT
iajs-3051	32	8	r	r	NOUN
iajs-3051	32	9	)	)	PUNCT
iajs-3051	32	10	is	be	AUX
iajs-3051	32	11	an	an	DET
iajs-3051	32	12	approximation	approximation	NOUN
iajs-3051	32	13	space	space	NOUN
iajs-3051	32	14	and	and	CCONJ
iajs-3051	32	15	g	g	NOUN
iajs-3051	32	16	,	,	PUNCT
iajs-3051	32	17	h	h	NOUN
iajs-3051	32	18	⊆	⊆	NUM
iajs-3051	32	19	u	u	NOUN
iajs-3051	32	20	,	,	PUNCT
iajs-3051	32	21	then	then	ADV
iajs-3051	32	22	a	a	X
iajs-3051	32	23	)	)	PUNCT
iajs-3051	32	24	lor(g	lor(g	PROPN
iajs-3051	32	25	)	)	PUNCT
iajs-3051	32	26	⊆	⊆	NUM
iajs-3051	32	27	g	g	PROPN
iajs-3051	32	28	⊆	⊆	NUM
iajs-3051	32	29	upr(g	upr(g	PROPN
iajs-3051	32	30	)	)	PUNCT
iajs-3051	32	31	.	.	PUNCT
iajs-3051	33	1	b	b	X
iajs-3051	33	2	)	)	PUNCT
iajs-3051	33	3	lor(∅	lor(∅	NOUN
iajs-3051	33	4	)	)	PUNCT
iajs-3051	33	5	=	=	SYM
iajs-3051	33	6	upr(∅	upr(∅	NOUN
iajs-3051	33	7	)	)	PUNCT
iajs-3051	33	8	=	=	PUNCT
iajs-3051	34	1	∅.	∅.	PRON
iajs-3051	34	2	c	c	X
iajs-3051	34	3	)	)	PUNCT
iajs-3051	34	4	lor(u	lor(u	PROPN
iajs-3051	34	5	)	)	PUNCT
iajs-3051	34	6	=	=	SYM
iajs-3051	34	7	upr(u	upr(u	PROPN
iajs-3051	34	8	)	)	PUNCT
iajs-3051	34	9	=	=	PUNCT
iajs-3051	35	1	u.	u.	NOUN
iajs-3051	35	2	d	d	NOUN
iajs-3051	35	3	)	)	PUNCT
iajs-3051	35	4	upr(g⋃h	upr(g⋃h	PUNCT
iajs-3051	35	5	)	)	PUNCT
iajs-3051	35	6	=	=	SYM
iajs-3051	35	7	upr(g	upr(g	PROPN
iajs-3051	35	8	)	)	PUNCT
iajs-3051	35	9	⋃	⋃	ADP
iajs-3051	35	10	upr(h	upr(h	NOUN
iajs-3051	35	11	)	)	PUNCT
iajs-3051	35	12	.	.	PUNCT
iajs-3051	36	1	e	e	X
iajs-3051	36	2	)	)	PUNCT
iajs-3051	36	3	upr(g⋂	upr(g⋂	PROPN
iajs-3051	36	4	h	h	NOUN
iajs-3051	36	5	)	)	PUNCT
iajs-3051	36	6	⊆	⊆	NUM
iajs-3051	36	7	upr(g	upr(g	PROPN
iajs-3051	36	8	)	)	PUNCT
iajs-3051	36	9	⋂	⋂	PROPN
iajs-3051	36	10	upr(h	upr(h	PROPN
iajs-3051	36	11	)	)	PUNCT
iajs-3051	36	12	.	.	PUNCT
iajs-3051	37	1	f	f	X
iajs-3051	37	2	)	)	PUNCT
iajs-3051	37	3	lor(g⋃h	lor(g⋃h	PROPN
iajs-3051	37	4	)	)	PUNCT
iajs-3051	37	5	⊇	⊇	PROPN
iajs-3051	37	6	lor(g	lor(g	PROPN
iajs-3051	37	7	)	)	PUNCT
iajs-3051	37	8	⋃	⋃	PROPN
iajs-3051	37	9	lor(h	lor(h	PROPN
iajs-3051	37	10	)	)	PUNCT
iajs-3051	37	11	.	.	PUNCT
iajs-3051	38	1	g	g	NOUN
iajs-3051	38	2	)	)	PUNCT
iajs-3051	38	3	lor(g⋂h	lor(g⋂h	PROPN
iajs-3051	38	4	)	)	PUNCT
iajs-3051	39	1	=	=	SYM
iajs-3051	39	2	lor(g	lor(g	X
iajs-3051	39	3	)	)	PUNCT
iajs-3051	39	4	⋂	⋂	PROPN
iajs-3051	39	5	lor(h	lor(h	PROPN
iajs-3051	39	6	)	)	PUNCT
iajs-3051	39	7	.	.	PUNCT
iajs-3051	40	1	h	h	X
iajs-3051	40	2	)	)	PUNCT
iajs-3051	40	3	lor(g	lor(g	PROPN
iajs-3051	40	4	)	)	PUNCT
iajs-3051	40	5	⊆	⊆	NUM
iajs-3051	40	6	lor(h	lor(h	PROPN
iajs-3051	40	7	)	)	PUNCT
iajs-3051	40	8	and	and	CCONJ
iajs-3051	40	9	upr(g	upr(g	PROPN
iajs-3051	40	10	)	)	PUNCT
iajs-3051	41	1	⊆	⊆	NUM
iajs-3051	41	2	upr(h	upr(h	NOUN
iajs-3051	41	3	)	)	PUNCT
iajs-3051	41	4	whenever	whenever	SCONJ
iajs-3051	41	5	g⊆h	g⊆h	PROPN
iajs-3051	41	6	.	.	PUNCT
iajs-3051	42	1	i	i	PRON
iajs-3051	42	2	)	)	PUNCT
iajs-3051	42	3	upr(gc	upr(gc	X
iajs-3051	42	4	)	)	PUNCT
iajs-3051	42	5	=[	=[	NOUN
iajs-3051	42	6	lor(g)]𝐶	lor(g)]𝐶	NOUN
iajs-3051	42	7	and	and	CCONJ
iajs-3051	42	8	lor(gc	lor(gc	NOUN
iajs-3051	42	9	)	)	PUNCT
iajs-3051	43	1	=	=	NOUN
iajs-3051	44	1	[	[	X
iajs-3051	44	2	upr(g)]c	upr(g)]c	X
iajs-3051	44	3	.	.	PUNCT
iajs-3051	44	4	j	j	PROPN
iajs-3051	44	5	)	)	PUNCT
iajs-3051	45	1	upr[upr(g	upr[upr(g	PROPN
iajs-3051	45	2	)	)	PUNCT
iajs-3051	45	3	]	]	PUNCT
iajs-3051	46	1	=	=	PUNCT
iajs-3051	46	2	lor[upr(g	lor[upr(g	PROPN
iajs-3051	46	3	)	)	PUNCT
iajs-3051	46	4	]	]	PUNCT
iajs-3051	47	1	=	=	PUNCT
iajs-3051	47	2	upr(g	upr(g	PROPN
iajs-3051	47	3	)	)	PUNCT
iajs-3051	47	4	.	.	PUNCT
iajs-3051	48	1	k	k	X
iajs-3051	48	2	)	)	PUNCT
iajs-3051	48	3	lor[lor(g	lor[lor(g	PROPN
iajs-3051	48	4	)	)	PUNCT
iajs-3051	48	5	]	]	PUNCT
iajs-3051	49	1	=	=	PUNCT
iajs-3051	49	2	upr[lor(g	upr[lor(g	PROPN
iajs-3051	49	3	)	)	PUNCT
iajs-3051	49	4	]	]	PUNCT
iajs-3051	50	1	=	=	PUNCT
iajs-3051	50	2	lor(g	lor(g	NOUN
iajs-3051	50	3	)	)	PUNCT
iajs-3051	50	4	.	.	PUNCT
iajs-3051	51	1	definition	definition	NOUN
iajs-3051	51	2	1.3	1.3	NUM
iajs-3051	51	3	:	:	PUNCT
iajs-3051	52	1	[	[	X
iajs-3051	52	2	16	16	NUM
iajs-3051	52	3	]	]	X
iajs-3051	52	4	if	if	SCONJ
iajs-3051	52	5	τr(g	τr(g	PUNCT
iajs-3051	52	6	)	)	PUNCT
iajs-3051	52	7	is	be	AUX
iajs-3051	52	8	the	the	DET
iajs-3051	52	9	nano-top-sp.on	nano-top-sp.on	NUM
iajs-3051	52	10	u	u	NOUN
iajs-3051	52	11	with	with	ADP
iajs-3051	52	12	respect	respect	NOUN
iajs-3051	52	13	to	to	ADP
iajs-3051	52	14	g	g	NOUN
iajs-3051	52	15	,	,	PUNCT
iajs-3051	52	16	then	then	ADV
iajs-3051	52	17	the	the	DET
iajs-3051	52	18	set	set	NOUN
iajs-3051	52	19	b	b	PROPN
iajs-3051	52	20	=	=	SYM
iajs-3051	52	21	{	{	PUNCT
iajs-3051	52	22	u	u	PROPN
iajs-3051	52	23	,	,	PUNCT
iajs-3051	52	24	lor(g	lor(g	PROPN
iajs-3051	52	25	)	)	PUNCT
iajs-3051	52	26	,	,	PUNCT
iajs-3051	52	27	bor(g	bor(g	PROPN
iajs-3051	52	28	)	)	PUNCT
iajs-3051	52	29	}	}	PUNCT
iajs-3051	52	30	is	be	AUX
iajs-3051	52	31	the	the	DET
iajs-3051	52	32	nano	nano	NOUN
iajs-3051	52	33	-	-	PUNCT
iajs-3051	52	34	basis	basis	NOUN
iajs-3051	52	35	for	for	ADP
iajs-3051	52	36	τr(g	τr(g	PUNCT
iajs-3051	52	37	)	)	PUNCT
iajs-3051	52	38	.	.	PUNCT
iajs-3051	53	1	definition	definition	NOUN
iajs-3051	53	2	1.4	1.4	NUM
iajs-3051	53	3	:	:	PUNCT
iajs-3051	54	1	[	[	X
iajs-3051	54	2	17	17	NUM
iajs-3051	54	3	]	]	PUNCT
iajs-3051	54	4	let	let	VERB
iajs-3051	54	5	u	u	PRON
iajs-3051	54	6	be	be	AUX
iajs-3051	54	7	the	the	DET
iajs-3051	54	8	universe	universe	NOUN
iajs-3051	54	9	,	,	PUNCT
iajs-3051	54	10	r	r	NOUN
iajs-3051	54	11	be	be	VERB
iajs-3051	54	12	an	an	DET
iajs-3051	54	13	equivalence	equivalence	NOUN
iajs-3051	54	14	relation	relation	NOUN
iajs-3051	54	15	on	on	ADP
iajs-3051	54	16	u	u	NOUN
iajs-3051	54	17	and	and	CCONJ
iajs-3051	54	18	τr(g	τr(g	PUNCT
iajs-3051	54	19	)	)	PUNCT
iajs-3051	55	1	=	=	PRON
iajs-3051	55	2	{	{	PUNCT
iajs-3051	55	3	u	u	NOUN
iajs-3051	55	4	,	,	PUNCT
iajs-3051	55	5	∅	∅	NOUN
iajs-3051	55	6	,	,	PUNCT
iajs-3051	55	7	lor(g	lor(g	PROPN
iajs-3051	55	8	)	)	PUNCT
iajs-3051	55	9	,	,	PUNCT
iajs-3051	55	10	upr(g	upr(g	PROPN
iajs-3051	55	11	)	)	PUNCT
iajs-3051	55	12	,	,	PUNCT
iajs-3051	55	13	bor(g	bor(g	PROPN
iajs-3051	55	14	)	)	PUNCT
iajs-3051	55	15	}	}	PUNCT
iajs-3051	55	16	,	,	PUNCT
iajs-3051	55	17	where	where	SCONJ
iajs-3051	55	18	g	g	PROPN
iajs-3051	55	19	⊆	⊆	NUM
iajs-3051	55	20	u.	u.	NOUN
iajs-3051	55	21	then	then	ADV
iajs-3051	55	22	τr(g	τr(g	PUNCT
iajs-3051	55	23	)	)	PUNCT
iajs-3051	55	24	it	it	PRON
iajs-3051	55	25	achieves	achieve	VERB
iajs-3051	55	26	the	the	DET
iajs-3051	55	27	following	follow	VERB
iajs-3051	55	28	axioms	axiom	NOUN
iajs-3051	55	29	:	:	PUNCT
iajs-3051	55	30	a	a	X
iajs-3051	55	31	)	)	PUNCT
iajs-3051	55	32	u	u	NOUN
iajs-3051	55	33	and	and	CCONJ
iajs-3051	55	34	∅	∅	NOUN
iajs-3051	55	35	∈	∈	PROPN
iajs-3051	55	36	τr(g	τr(g	PUNCT
iajs-3051	55	37	)	)	PUNCT
iajs-3051	55	38	.	.	PUNCT
iajs-3051	56	1	b	b	X
iajs-3051	56	2	)	)	PUNCT
iajs-3051	56	3	the	the	DET
iajs-3051	56	4	union	union	NOUN
iajs-3051	56	5	of	of	ADP
iajs-3051	56	6	the	the	DET
iajs-3051	56	7	components	component	NOUN
iajs-3051	56	8	of	of	ADP
iajs-3051	56	9	any	any	DET
iajs-3051	56	10	subset	subset	NOUN
iajs-3051	56	11	of	of	ADP
iajs-3051	56	12	τr(g	τr(g	PUNCT
iajs-3051	56	13	)	)	PUNCT
iajs-3051	56	14	is	be	AUX
iajs-3051	56	15	in	in	ADP
iajs-3051	56	16	τr(g	τr(g	PUNCT
iajs-3051	56	17	)	)	PUNCT
iajs-3051	56	18	.	.	PUNCT
iajs-3051	57	1	c	c	X
iajs-3051	57	2	)	)	PUNCT
iajs-3051	57	3	the	the	DET
iajs-3051	57	4	intersection	intersection	NOUN
iajs-3051	57	5	of	of	ADP
iajs-3051	57	6	the	the	DET
iajs-3051	57	7	components	component	NOUN
iajs-3051	57	8	of	of	ADP
iajs-3051	57	9	any	any	DET
iajs-3051	57	10	finite	finite	NOUN
iajs-3051	57	11	subset	subset	NOUN
iajs-3051	57	12	of	of	ADP
iajs-3051	57	13	τr(g	τr(g	PUNCT
iajs-3051	57	14	)	)	PUNCT
iajs-3051	57	15	is	be	AUX
iajs-3051	57	16	in	in	ADP
iajs-3051	57	17	τr(g	τr(g	PUNCT
iajs-3051	57	18	)	)	PUNCT
iajs-3051	57	19	.	.	PUNCT
iajs-3051	58	1	therefore	therefore	ADV
iajs-3051	58	2	,	,	PUNCT
iajs-3051	58	3	τr(g	τr(g	PUNCT
iajs-3051	58	4	)	)	PUNCT
iajs-3051	58	5	is	be	AUX
iajs-3051	58	6	a	a	DET
iajs-3051	58	7	topology	topology	NOUN
iajs-3051	58	8	on	on	ADP
iajs-3051	58	9	u	u	NOUN
iajs-3051	58	10	called	call	VERB
iajs-3051	58	11	the	the	DET
iajs-3051	58	12	nano	nano	NOUN
iajs-3051	58	13	topology	topology	NOUN
iajs-3051	58	14	(	(	PUNCT
iajs-3051	58	15	denoted	denote	VERB
iajs-3051	58	16	by	by	ADP
iajs-3051	58	17	nano	nano	NOUN
iajs-3051	58	18	-	-	PUNCT
iajs-3051	58	19	top	top	NOUN
iajs-3051	58	20	.	.	PUNCT
iajs-3051	58	21	)	)	PUNCT
iajs-3051	59	1	on	on	ADP
iajs-3051	59	2	u	u	NOUN
iajs-3051	59	3	with	with	ADP
iajs-3051	59	4	reference	reference	NOUN
iajs-3051	59	5	to	to	ADP
iajs-3051	59	6	g.	g.	PROPN
iajs-3051	59	7	we	we	PRON
iajs-3051	59	8	call	call	VERB
iajs-3051	59	9	(	(	PUNCT
iajs-3051	59	10	u	u	NOUN
iajs-3051	59	11	,	,	PUNCT
iajs-3051	59	12	τr(g	τr(g	PUNCT
iajs-3051	59	13	)	)	PUNCT
iajs-3051	59	14	)	)	PUNCT
iajs-3051	60	1	as	as	ADP
iajs-3051	60	2	the	the	DET
iajs-3051	60	3	nano	nano	NOUN
iajs-3051	60	4	topological	topological	ADJ
iajs-3051	60	5	space	space	NOUN
iajs-3051	60	6	(	(	PUNCT
iajs-3051	60	7	denoted	denote	VERB
iajs-3051	60	8	by	by	ADP
iajs-3051	60	9	nano	nano	NOUN
iajs-3051	60	10	-	-	PUNCT
iajs-3051	60	11	top	top	NOUN
iajs-3051	60	12	-	-	PUNCT
iajs-3051	60	13	sp	sp	NOUN
iajs-3051	60	14	.	.	PUNCT
iajs-3051	60	15	)	)	PUNCT
iajs-3051	61	1	the	the	DET
iajs-3051	61	2	components	component	NOUN
iajs-3051	61	3	of	of	ADP
iajs-3051	61	4	τr(g	τr(g	PUNCT
iajs-3051	61	5	)	)	PUNCT
iajs-3051	61	6	are	be	AUX
iajs-3051	61	7	said	say	VERB
iajs-3051	61	8	to	to	PART
iajs-3051	61	9	be	be	AUX
iajs-3051	61	10	nano	nano	ADJ
iajs-3051	61	11	open	open	ADJ
iajs-3051	61	12	sets	set	NOUN
iajs-3051	61	13	(	(	PUNCT
iajs-3051	61	14	denoted	denote	VERB
iajs-3051	61	15	by	by	ADP
iajs-3051	61	16	nano	nano	NOUN
iajs-3051	61	17	-	-	PUNCT
iajs-3051	61	18	ope	ope	NOUN
iajs-3051	61	19	-	-	PUNCT
iajs-3051	61	20	sets	set	NOUN
iajs-3051	61	21	)	)	PUNCT
iajs-3051	61	22	.	.	PUNCT
iajs-3051	62	1	the	the	DET
iajs-3051	62	2	complement	complement	NOUN
iajs-3051	62	3	of	of	ADP
iajs-3051	62	4	a	a	DET
iajs-3051	62	5	nano	nano	NOUN
iajs-3051	62	6	-	-	PUNCT
iajs-3051	62	7	ope	ope	NOUN
iajs-3051	62	8	-	-	PUNCT
iajs-3051	62	9	set	set	NOUN
iajs-3051	62	10	is	be	AUX
iajs-3051	62	11	called	call	VERB
iajs-3051	62	12	a	a	DET
iajs-3051	62	13	nano	nano	NOUN
iajs-3051	62	14	closed	close	VERB
iajs-3051	62	15	set(denoted	set(denote	VERB
iajs-3051	62	16	by	by	ADP
iajs-3051	62	17	nano	nano	NOUN
iajs-3051	62	18	-	-	PUNCT
iajs-3051	62	19	clos	clo	NOUN
iajs-3051	62	20	-	-	PUNCT
iajs-3051	62	21	set	set	NOUN
iajs-3051	62	22	)	)	PUNCT
iajs-3051	62	23	.	.	PUNCT
iajs-3051	63	1	ihjpas	ihjpas	PROPN
iajs-3051	63	2	.	.	PUNCT
iajs-3051	64	1	36	36	NUM
iajs-3051	64	2	(	(	PUNCT
iajs-3051	64	3	3	3	NUM
iajs-3051	64	4	)	)	PUNCT
iajs-3051	64	5	2023	2023	NUM
iajs-3051	64	6	400	400	NUM
iajs-3051	64	7	definition	definition	NOUN
iajs-3051	64	8	1.5	1.5	NUM
iajs-3051	64	9	:	:	PUNCT
iajs-3051	65	1	[	[	X
iajs-3051	65	2	18	18	NUM
iajs-3051	65	3	]	]	X
iajs-3051	65	4	let	let	VERB
iajs-3051	65	5	(	(	PUNCT
iajs-3051	65	6	u	u	NOUN
iajs-3051	65	7	,	,	PUNCT
iajs-3051	65	8	τr(g	τr(g	PUNCT
iajs-3051	65	9	)	)	PUNCT
iajs-3051	65	10	)	)	PUNCT
iajs-3051	65	11	and	and	CCONJ
iajs-3051	65	12	(	(	PUNCT
iajs-3051	65	13	v	v	NOUN
iajs-3051	65	14	,	,	PUNCT
iajs-3051	65	15	τr`(h	τr`(h	NOUN
iajs-3051	65	16	)	)	PUNCT
iajs-3051	65	17	)	)	PUNCT
iajs-3051	65	18	be	be	AUX
iajs-3051	65	19	nano	nano	NOUN
iajs-3051	65	20	-	-	PUNCT
iajs-3051	65	21	top	top	NOUN
iajs-3051	65	22	-	-	PUNCT
iajs-3051	65	23	sp	sp	NOUN
iajs-3051	65	24	.	.	PUNCT
iajs-3051	66	1	then	then	ADV
iajs-3051	66	2	a	a	DET
iajs-3051	66	3	mapping	mapping	NOUN
iajs-3051	66	4	(	(	PUNCT
iajs-3051	66	5	dented	dent	VERB
iajs-3051	66	6	by	by	ADP
iajs-3051	66	7	map	map	NOUN
iajs-3051	66	8	.	.	PUNCT
iajs-3051	66	9	)	)	PUNCT
iajs-3051	67	1	f	f	NOUN
iajs-3051	67	2	:	:	PUNCT
iajs-3051	67	3	(	(	PUNCT
iajs-3051	67	4	u	u	NOUN
iajs-3051	67	5	,	,	PUNCT
iajs-3051	67	6	τr(g	τr(g	PUNCT
iajs-3051	67	7	)	)	PUNCT
iajs-3051	67	8	)	)	PUNCT
iajs-3051	67	9	⟶	⟶	NOUN
iajs-3051	67	10	(	(	PUNCT
iajs-3051	67	11	v	v	NOUN
iajs-3051	67	12	,	,	PUNCT
iajs-3051	67	13	τr`(h	τr`(h	NOUN
iajs-3051	67	14	)	)	PUNCT
iajs-3051	67	15	)	)	PUNCT
iajs-3051	67	16	is	be	AUX
iajs-3051	67	17	nano	nano	NOUN
iajs-3051	67	18	continuous	continuous	ADJ
iajs-3051	67	19	(	(	PUNCT
iajs-3051	67	20	denoted	denote	VERB
iajs-3051	67	21	by	by	ADP
iajs-3051	67	22	nano	nano	NOUN
iajs-3051	67	23	-	-	PUNCT
iajs-3051	67	24	cont	cont	NOUN
iajs-3051	67	25	.	.	PUNCT
iajs-3051	67	26	)	)	PUNCT
iajs-3051	68	1	on	on	ADP
iajs-3051	68	2	u	u	PRON
iajs-3051	68	3	if	if	SCONJ
iajs-3051	68	4	the	the	DET
iajs-3051	68	5	inverse	inverse	ADJ
iajs-3051	68	6	image	image	NOUN
iajs-3051	68	7	of	of	ADP
iajs-3051	68	8	each	each	DET
iajs-3051	68	9	nano	nano	NOUN
iajs-3051	68	10	-	-	PUNCT
iajs-3051	68	11	ope	ope	NOUN
iajs-3051	68	12	-	-	PUNCT
iajs-3051	68	13	set	set	NOUN
iajs-3051	68	14	in	in	ADP
iajs-3051	68	15	v	v	NOUN
iajs-3051	68	16	is	be	AUX
iajs-3051	68	17	nano	nano	NOUN
iajs-3051	68	18	-	-	PUNCT
iajs-3051	68	19	ope	ope	NOUN
iajs-3051	68	20	.	.	PUNCT
iajs-3051	69	1	in	in	ADP
iajs-3051	69	2	u.	u.	PROPN
iajs-3051	69	3	definition	definition	NOUN
iajs-3051	69	4	1.6	1.6	NUM
iajs-3051	69	5	:	:	PUNCT
iajs-3051	69	6	[	[	X
iajs-3051	69	7	18	18	NUM
iajs-3051	69	8	]	]	PUNCT
iajs-3051	69	9	a	a	DET
iajs-3051	69	10	function	function	NOUN
iajs-3051	69	11	f	f	NOUN
iajs-3051	69	12	:	:	PUNCT
iajs-3051	69	13	(	(	PUNCT
iajs-3051	69	14	u	u	NOUN
iajs-3051	69	15	,	,	PUNCT
iajs-3051	69	16	τr(g	τr(g	PUNCT
iajs-3051	69	17	)	)	PUNCT
iajs-3051	69	18	)	)	PUNCT
iajs-3051	69	19	⟶	⟶	NOUN
iajs-3051	69	20	(	(	PUNCT
iajs-3051	69	21	v	v	NOUN
iajs-3051	69	22	,	,	PUNCT
iajs-3051	69	23	τr`(h	τr`(h	NOUN
iajs-3051	69	24	)	)	PUNCT
iajs-3051	69	25	)	)	PUNCT
iajs-3051	69	26	is	be	AUX
iajs-3051	69	27	a	a	DET
iajs-3051	69	28	nano	nano	NOUN
iajs-3051	69	29	-	-	PUNCT
iajs-3051	69	30	clos	clo	NOUN
iajs-3051	69	31	.	.	PUNCT
iajs-3051	70	1	mapping	mapping	NOUN
iajs-3051	70	2	if	if	SCONJ
iajs-3051	70	3	the	the	DET
iajs-3051	70	4	image	image	NOUN
iajs-3051	70	5	of	of	ADP
iajs-3051	70	6	each	each	DET
iajs-3051	70	7	nano	nano	NOUN
iajs-3051	70	8	-	-	PUNCT
iajs-3051	70	9	clos	clo	NOUN
iajs-3051	70	10	-	-	PUNCT
iajs-3051	70	11	set	set	NOUN
iajs-3051	70	12	in	in	ADP
iajs-3051	70	13	u	u	NOUN
iajs-3051	70	14	is	be	AUX
iajs-3051	70	15	nano	nano	NOUN
iajs-3051	70	16	-	-	PUNCT
iajs-3051	70	17	clos	clo	NOUN
iajs-3051	70	18	.	.	PUNCT
iajs-3051	71	1	in	in	ADP
iajs-3051	71	2	v.	v.	ADP
iajs-3051	71	3	definition	definition	NOUN
iajs-3051	71	4	1.7	1.7	NUM
iajs-3051	71	5	:	:	PUNCT
iajs-3051	71	6	[	[	X
iajs-3051	71	7	13	13	NUM
iajs-3051	71	8	]	]	PUNCT
iajs-3051	71	9	let	let	VERB
iajs-3051	71	10	(	(	PUNCT
iajs-3051	71	11	g	g	NOUN
iajs-3051	71	12	,	,	PUNCT
iajs-3051	71	13	τr(g	τr(g	PUNCT
iajs-3051	71	14	)	)	PUNCT
iajs-3051	71	15	)	)	PUNCT
iajs-3051	72	1	be	be	AUX
iajs-3051	72	2	a	a	DET
iajs-3051	72	3	nano-top-sp.and	nano-top-sp.and	NOUN
iajs-3051	72	4	a	a	DET
iajs-3051	72	5	⊆	⊆	NUM
iajs-3051	72	6	g.the	g.the	DET
iajs-3051	72	7	nano	nano	NOUN
iajs-3051	72	8	-	-	PUNCT
iajs-3051	72	9	closure	closure	NOUN
iajs-3051	72	10	of	of	ADP
iajs-3051	72	11	a	a	DET
iajs-3051	72	12	set	set	NOUN
iajs-3051	72	13	a	a	PRON
iajs-3051	72	14	is	be	AUX
iajs-3051	72	15	symboly	symboly	ADJ
iajs-3051	72	16	by	by	ADP
iajs-3051	72	17	nci(a	nci(a	PROPN
iajs-3051	72	18	)	)	PUNCT
iajs-3051	73	1	is	be	AUX
iajs-3051	73	2	the	the	DET
iajs-3051	73	3	intersection	intersection	NOUN
iajs-3051	73	4	of	of	ADP
iajs-3051	73	5	all	all	DET
iajs-3051	73	6	nano	nano	NOUN
iajs-3051	73	7	-	-	PUNCT
iajs-3051	73	8	clos	clo	NOUN
iajs-3051	73	9	-	-	PUNCT
iajs-3051	73	10	sets	set	NOUN
iajs-3051	73	11	containing	contain	VERB
iajs-3051	73	12	a	a	DET
iajs-3051	73	13	or	or	CCONJ
iajs-3051	73	14	all	all	PRON
iajs-3051	73	15	clos	clo	NOUN
iajs-3051	73	16	-	-	PUNCT
iajs-3051	73	17	super	super	NOUN
iajs-3051	73	18	-	-	NOUN
iajs-3051	73	19	sets	set	NOUN
iajs-3051	73	20	of	of	ADP
iajs-3051	73	21	a	a	PRON
iajs-3051	73	22	;	;	PUNCT
iajs-3051	73	23	i.e.	i.e.	X
iajs-3051	73	24	,the	,the	PUNCT
iajs-3051	73	25	smallest	small	ADJ
iajs-3051	73	26	clos	clos	NOUN
iajs-3051	73	27	-	-	PUNCT
iajs-3051	73	28	set	set	VERB
iajs-3051	73	29	containing	contain	VERB
iajs-3051	73	30	a.	a.	NOUN
iajs-3051	73	31	definition	definition	NOUN
iajs-3051	73	32	1.8	1.8	NUM
iajs-3051	73	33	:	:	PUNCT
iajs-3051	74	1	[	[	X
iajs-3051	74	2	18	18	NUM
iajs-3051	74	3	]	]	PUNCT
iajs-3051	74	4	a	a	DET
iajs-3051	74	5	function	function	NOUN
iajs-3051	74	6	f:(u	f:(u	PROPN
iajs-3051	74	7	,	,	PUNCT
iajs-3051	74	8	τr(g))⟶(v	τr(g))⟶(v	X
iajs-3051	74	9	,	,	PUNCT
iajs-3051	74	10	τr`(h	τr`(h	PROPN
iajs-3051	74	11	)	)	PUNCT
iajs-3051	74	12	)	)	PUNCT
iajs-3051	74	13	is	be	AUX
iajs-3051	74	14	called	call	VERB
iajs-3051	74	15	a	a	DET
iajs-3051	74	16	nano	nano	NOUN
iajs-3051	74	17	homeomorphism	homeomorphism	NOUN
iajs-3051	74	18	if	if	SCONJ
iajs-3051	74	19	:	:	PUNCT
iajs-3051	74	20	a	a	X
iajs-3051	74	21	)	)	PUNCT
iajs-3051	74	22	f	f	PROPN
iajs-3051	74	23	is	be	AUX
iajs-3051	74	24	1	1	NUM
iajs-3051	74	25	-	-	SYM
iajs-3051	74	26	1	1	NUM
iajs-3051	74	27	and	and	CCONJ
iajs-3051	74	28	onto	onto	ADP
iajs-3051	74	29	.	.	PUNCT
iajs-3051	75	1	b	b	X
iajs-3051	75	2	)	)	PUNCT
iajs-3051	75	3	f	f	PROPN
iajs-3051	75	4	is	be	AUX
iajs-3051	75	5	nano	nano	NOUN
iajs-3051	75	6	-	-	PUNCT
iajs-3051	75	7	cont	cont	NOUN
iajs-3051	75	8	.	.	PUNCT
iajs-3051	76	1	c	c	X
iajs-3051	76	2	)	)	PUNCT
iajs-3051	76	3	f	f	PROPN
iajs-3051	76	4	is	be	AUX
iajs-3051	76	5	nano	nano	NOUN
iajs-3051	76	6	-	-	PUNCT
iajs-3051	76	7	ope	ope	NOUN
iajs-3051	76	8	.	.	PUNCT
iajs-3051	77	1	definition	definition	NOUN
iajs-3051	77	2	1.9	1.9	NUM
iajs-3051	77	3	:	:	PUNCT
iajs-3051	78	1	[	[	X
iajs-3051	78	2	19	19	NUM
iajs-3051	78	3	]	]	PUNCT
iajs-3051	78	4	a	a	DET
iajs-3051	78	5	set	set	NOUN
iajs-3051	78	6	{	{	PUNCT
iajs-3051	78	7	ai	ai	NOUN
iajs-3051	78	8	:	:	PUNCT
iajs-3051	78	9	i	i	PROPN
iajs-3051	78	10	∈	∈	VERB
iajs-3051	78	11	i	i	PRON
iajs-3051	78	12	}	}	PUNCT
iajs-3051	78	13	of	of	ADP
iajs-3051	78	14	nano	nano	NOUN
iajs-3051	78	15	-	-	PUNCT
iajs-3051	78	16	ope	ope	NOUN
iajs-3051	78	17	-	-	PUNCT
iajs-3051	78	18	sets	set	NOUN
iajs-3051	78	19	in	in	ADP
iajs-3051	78	20	a	a	DET
iajs-3051	78	21	nano	nano	NOUN
iajs-3051	78	22	-	-	PUNCT
iajs-3051	78	23	top	top	NOUN
iajs-3051	78	24	-	-	PUNCT
iajs-3051	78	25	sp	sp	NOUN
iajs-3051	78	26	.	.	PUNCT
iajs-3051	79	1	(	(	PUNCT
iajs-3051	79	2	(	(	PUNCT
iajs-3051	79	3	u	u	NOUN
iajs-3051	79	4	,	,	PUNCT
iajs-3051	79	5	τr(g	τr(g	PUNCT
iajs-3051	79	6	)	)	PUNCT
iajs-3051	79	7	)	)	PUNCT
iajs-3051	79	8	is	be	AUX
iajs-3051	79	9	said	say	VERB
iajs-3051	79	10	to	to	PART
iajs-3051	79	11	be	be	AUX
iajs-3051	79	12	nano	nano	NOUN
iajs-3051	79	13	-	-	PUNCT
iajs-3051	79	14	ope	ope	NOUN
iajs-3051	79	15	-	-	PUNCT
iajs-3051	79	16	cover	cover	NOUN
iajs-3051	79	17	of	of	ADP
iajs-3051	79	18	a	a	DET
iajs-3051	79	19	subset	subset	NOUN
iajs-3051	79	20	b	b	NOUN
iajs-3051	79	21	of	of	ADP
iajs-3051	79	22	u	u	PRON
iajs-3051	79	23	if	if	SCONJ
iajs-3051	79	24	b	b	PROPN
iajs-3051	79	25	⊂	⊂	PROPN
iajs-3051	79	26	{	{	PUNCT
iajs-3051	79	27	ai	ai	VERB
iajs-3051	79	28	:	:	PUNCT
iajs-3051	79	29	i	i	PRON
iajs-3051	79	30	∈	∈	VERB
iajs-3051	79	31	i	i	PRON
iajs-3051	79	32	}	}	PUNCT
iajs-3051	79	33	holds	hold	VERB
iajs-3051	79	34	.	.	PUNCT
iajs-3051	80	1	definition	definition	NOUN
iajs-3051	80	2	1.10	1.10	NUM
iajs-3051	80	3	:	:	PUNCT
iajs-3051	81	1	[	[	X
iajs-3051	81	2	19	19	NUM
iajs-3051	81	3	]	]	PUNCT
iajs-3051	81	4	a	a	DET
iajs-3051	81	5	space	space	NOUN
iajs-3051	81	6	(	(	PUNCT
iajs-3051	81	7	u	u	NOUN
iajs-3051	81	8	,	,	PUNCT
iajs-3051	81	9	τrʹ(g	τrʹ(g	PROPN
iajs-3051	81	10	)	)	PUNCT
iajs-3051	81	11	)	)	PUNCT
iajs-3051	81	12	is	be	AUX
iajs-3051	81	13	called	call	VERB
iajs-3051	81	14	nano	nano	NOUN
iajs-3051	81	15	hausdroff	hausdroff	NOUN
iajs-3051	81	16	space	space	NOUN
iajs-3051	81	17	(	(	PUNCT
iajs-3051	81	18	denoted	denote	VERB
iajs-3051	81	19	by	by	ADP
iajs-3051	81	20	nano	nano	NOUN
iajs-3051	81	21	-	-	PUNCT
iajs-3051	81	22	hausdsp	hausdsp	NOUN
iajs-3051	81	23	.	.	PUNCT
iajs-3051	81	24	)	)	PUNCT
iajs-3051	82	1	if	if	SCONJ
iajs-3051	82	2	whenever	whenever	SCONJ
iajs-3051	82	3	g	g	PROPN
iajs-3051	82	4	and	and	CCONJ
iajs-3051	82	5	h	h	NOUN
iajs-3051	82	6	are	be	AUX
iajs-3051	82	7	distinct	distinct	ADJ
iajs-3051	82	8	points	point	NOUN
iajs-3051	82	9	of	of	ADP
iajs-3051	82	10	(	(	PUNCT
iajs-3051	82	11	u	u	NOUN
iajs-3051	82	12	,	,	PUNCT
iajs-3051	82	13	τr(g	τr(g	PUNCT
iajs-3051	82	14	)	)	PUNCT
iajs-3051	82	15	)	)	PUNCT
iajs-3051	82	16	,	,	PUNCT
iajs-3051	82	17	find	find	VERB
iajs-3051	82	18	disjoint	disjoint	ADJ
iajs-3051	82	19	nano	nano	NOUN
iajs-3051	82	20	-	-	PUNCT
iajs-3051	82	21	ope	ope	NOUN
iajs-3051	82	22	-	-	PUNCT
iajs-3051	82	23	sets	set	NOUN
iajs-3051	82	24	a	a	PRON
iajs-3051	82	25	and	and	CCONJ
iajs-3051	82	26	b	b	NOUN
iajs-3051	82	27	such	such	ADJ
iajs-3051	82	28	as	as	ADP
iajs-3051	82	29	g	g	PROPN
iajs-3051	82	30	∈	∈	PROPN
iajs-3051	82	31	a	a	PRON
iajs-3051	82	32	and	and	CCONJ
iajs-3051	82	33	h	h	PROPN
iajs-3051	82	34	∈	∈	PROPN
iajs-3051	82	35	b.	b.	PROPN
iajs-3051	82	36	definition	definition	NOUN
iajs-3051	82	37	1.11	1.11	NUM
iajs-3051	82	38	:	:	PUNCT
iajs-3051	83	1	[	[	X
iajs-3051	83	2	19	19	NUM
iajs-3051	83	3	]	]	X
iajs-3051	83	4	let	let	NOUN
iajs-3051	83	5	(	(	PUNCT
iajs-3051	83	6	u	u	NOUN
iajs-3051	83	7	,	,	PUNCT
iajs-3051	83	8	τr(g	τr(g	PUNCT
iajs-3051	83	9	)	)	PUNCT
iajs-3051	83	10	)	)	PUNCT
iajs-3051	83	11	be	be	AUX
iajs-3051	83	12	a	a	DET
iajs-3051	83	13	nano	nano	NOUN
iajs-3051	83	14	-	-	PUNCT
iajs-3051	83	15	top	top	NOUN
iajs-3051	83	16	-	-	PUNCT
iajs-3051	83	17	sp	sp	NOUN
iajs-3051	83	18	.	.	NOUN
iajs-3051	84	1	and	and	CCONJ
iajs-3051	84	2	w	w	PROPN
iajs-3051	84	3	be	be	AUX
iajs-3051	84	4	a	a	DET
iajs-3051	84	5	subspace	subspace	NOUN
iajs-3051	84	6	of	of	ADP
iajs-3051	84	7	g.	g.	PROPN
iajs-3051	84	8	we	we	PRON
iajs-3051	84	9	said	say	VERB
iajs-3051	84	10	to	to	PART
iajs-3051	84	11	be	be	AUX
iajs-3051	84	12	space	space	NOUN
iajs-3051	84	13	w	w	NOUN
iajs-3051	84	14	is	be	AUX
iajs-3051	84	15	nano	nano	NOUN
iajs-3051	84	16	-	-	PUNCT
iajs-3051	84	17	comp	comp	NOUN
iajs-3051	84	18	-	-	PUNCT
iajs-3051	84	19	sp	sp	NOUN
iajs-3051	84	20	.	.	PUNCT
iajs-3051	85	1	iff	iff	VERB
iajs-3051	85	2	each	each	DET
iajs-3051	85	3	open	open	ADJ
iajs-3051	85	4	-	-	PUNCT
iajs-3051	85	5	cover	cover	NOUN
iajs-3051	85	6	from	from	ADP
iajs-3051	85	7	g	g	PROPN
iajs-3051	85	8	cover	cover	NOUN
iajs-3051	85	9	w	w	NOUN
iajs-3051	85	10	has	have	VERB
iajs-3051	85	11	a	a	DET
iajs-3051	85	12	finite	finite	ADJ
iajs-3051	85	13	-	-	ADJ
iajs-3051	85	14	sub	sub	ADJ
iajs-3051	85	15	-	-	NOUN
iajs-3051	85	16	cover	cover	NOUN
iajs-3051	85	17	.	.	PUNCT
iajs-3051	86	1	definition	definition	NOUN
iajs-3051	86	2	1.12	1.12	NUM
iajs-3051	86	3	:	:	PUNCT
iajs-3051	87	1	[	[	X
iajs-3051	87	2	18	18	NUM
iajs-3051	87	3	]	]	X
iajs-3051	87	4	let	let	VERB
iajs-3051	87	5	(	(	PUNCT
iajs-3051	87	6	g	g	NOUN
iajs-3051	87	7	,	,	PUNCT
iajs-3051	87	8	τr(g	τr(g	PUNCT
iajs-3051	87	9	)	)	PUNCT
iajs-3051	87	10	)	)	PUNCT
iajs-3051	87	11	be	be	AUX
iajs-3051	87	12	anano	anano	ADJ
iajs-3051	87	13	-	-	PUNCT
iajs-3051	87	14	top	top	NOUN
iajs-3051	87	15	-	-	PUNCT
iajs-3051	87	16	sp	sp	NOUN
iajs-3051	87	17	.	.	NOUN
iajs-3051	87	18	,	,	PUNCT
iajs-3051	87	19	and	and	CCONJ
iajs-3051	87	20	let	let	VERB
iajs-3051	87	21	{	{	PUNCT
iajs-3051	87	22	ui	ui	NOUN
iajs-3051	87	23	:	:	PUNCT
iajs-3051	87	24	ui	ui	PROPN
iajs-3051	88	1	⊂	⊂	PROPN
iajs-3051	88	2	g}i∈λ	g}i∈λ	AUX
iajs-3051	88	3	be	be	AUX
iajs-3051	88	4	a	a	DET
iajs-3051	88	5	open	open	ADJ
iajs-3051	88	6	-	-	PUNCT
iajs-3051	88	7	cover	cover	NOUN
iajs-3051	88	8	of	of	ADP
iajs-3051	88	9	g	g	PROPN
iajs-3051	88	10	is	be	AUX
iajs-3051	88	11	called	call	VERB
iajs-3051	88	12	locally	locally	ADV
iajs-3051	88	13	finite	finite	ADJ
iajs-3051	88	14	-	-	ADJ
iajs-3051	88	15	cover	cover	NOUN
iajs-3051	88	16	if	if	SCONJ
iajs-3051	88	17	for	for	ADP
iajs-3051	88	18	all	all	DET
iajs-3051	88	19	point	point	NOUN
iajs-3051	88	20	g	g	PROPN
iajs-3051	88	21	∈	∈	PROPN
iajs-3051	88	22	g	g	NOUN
iajs-3051	88	23	,	,	PUNCT
iajs-3051	88	24	find	find	VERB
iajs-3051	88	25	a	a	DET
iajs-3051	88	26	nbd	nbd	PROPN
iajs-3051	88	27	ug	ug	ADP
iajs-3051	88	28	of	of	ADP
iajs-3051	88	29	g	g	PROPN
iajs-3051	88	30	⊂	⊂	PROPN
iajs-3051	88	31	ug	ug	ADP
iajs-3051	88	32	such	such	ADJ
iajs-3051	88	33	as	as	SCONJ
iajs-3051	88	34	it	it	PRON
iajs-3051	88	35	intersects	intersect	VERB
iajs-3051	88	36	only	only	ADV
iajs-3051	88	37	limited	limit	VERB
iajs-3051	88	38	many	many	ADJ
iajs-3051	88	39	components	component	NOUN
iajs-3051	88	40	of	of	ADP
iajs-3051	88	41	the	the	DET
iajs-3051	88	42	cover	cover	NOUN
iajs-3051	88	43	,	,	PUNCT
iajs-3051	88	44	hence	hence	ADV
iajs-3051	88	45	such	such	ADJ
iajs-3051	88	46	as	as	ADP
iajs-3051	88	47	ug	ug	ADP
iajs-3051	88	48	⋂	⋂	PROPN
iajs-3051	88	49	ui	ui	PROPN
iajs-3051	88	50	≠	≠	PROPN
iajs-3051	88	51	∅	∅	NOUN
iajs-3051	88	52	,	,	PUNCT
iajs-3051	88	53	for	for	ADP
iajs-3051	88	54	only	only	ADV
iajs-3051	88	55	a	a	DET
iajs-3051	88	56	finite	finite	ADJ
iajs-3051	88	57	number	number	NOUN
iajs-3051	88	58	of	of	ADP
iajs-3051	88	59	i	i	PROPN
iajs-3051	88	60	∈	∈	PROPN
iajs-3051	88	61	λ	λ	PROPN
iajs-3051	88	62	.	.	PUNCT
iajs-3051	88	63	definition	definition	NOUN
iajs-3051	88	64	1.13	1.13	NUM
iajs-3051	88	65	:	:	PUNCT
iajs-3051	89	1	[	[	X
iajs-3051	89	2	20	20	NUM
iajs-3051	89	3	]	]	PUNCT
iajs-3051	89	4	a	a	DET
iajs-3051	89	5	top	top	ADV
iajs-3051	89	6	-	-	PUNCT
iajs-3051	89	7	sp.(g	sp.(g	PROPN
iajs-3051	89	8	,	,	PUNCT
iajs-3051	89	9	τ	τ	PROPN
iajs-3051	89	10	)	)	PUNCT
iajs-3051	89	11	and	and	CCONJ
iajs-3051	89	12	the	the	DET
iajs-3051	89	13	open	open	ADJ
iajs-3051	89	14	-	-	PUNCT
iajs-3051	89	15	cover	cover	NOUN
iajs-3051	89	16	{	{	PUNCT
iajs-3051	89	17	ui	ui	NOUN
iajs-3051	89	18	:	:	PUNCT
iajs-3051	89	19	ui	ui	PROPN
iajs-3051	89	20	⊂	⊂	PROPN
iajs-3051	89	21	g}i∈i	g}i∈i	PROPN
iajs-3051	89	22	is	be	AUX
iajs-3051	89	23	called	call	VERB
iajs-3051	89	24	refinement	refinement	NOUN
iajs-3051	89	25	if	if	SCONJ
iajs-3051	89	26	a	a	DET
iajs-3051	89	27	set	set	NOUN
iajs-3051	89	28	of	of	ADP
iajs-3051	89	29	open	open	ADJ
iajs-3051	89	30	subsets	subset	NOUN
iajs-3051	89	31	{	{	PUNCT
iajs-3051	89	32	vj	vj	PROPN
iajs-3051	89	33	:	:	PUNCT
iajs-3051	89	34	vj	vj	PROPN
iajs-3051	89	35	⊂	⊂	PROPN
iajs-3051	89	36	g	g	PROPN
iajs-3051	89	37	}	}	PUNCT
iajs-3051	89	38	i∈i	i∈i	ADJ
iajs-3051	89	39	which	which	PRON
iajs-3051	89	40	is	be	AUX
iajs-3051	89	41	still	still	ADV
iajs-3051	89	42	an	an	DET
iajs-3051	89	43	open	open	ADJ
iajs-3051	89	44	-	-	PUNCT
iajs-3051	89	45	cover	cover	NOUN
iajs-3051	89	46	in	in	ADP
iajs-3051	89	47	itself	itself	PRON
iajs-3051	89	48	and	and	CCONJ
iajs-3051	89	49	such	such	ADJ
iajs-3051	89	50	as	as	ADP
iajs-3051	89	51	for	for	ADP
iajs-3051	89	52	each	each	DET
iajs-3051	89	53	j	j	PROPN
iajs-3051	89	54	∈	∈	PROPN
iajs-3051	89	55	j	j	PROPN
iajs-3051	89	56	,	,	PUNCT
iajs-3051	89	57	find	find	VERB
iajs-3051	89	58	an	an	DET
iajs-3051	89	59	i	i	NOUN
iajs-3051	89	60	∈	∈	PROPN
iajs-3051	89	61	i	i	PRON
iajs-3051	89	62	with	with	ADP
iajs-3051	89	63	vj	vj	PROPN
iajs-3051	89	64	⊂	⊂	PROPN
iajs-3051	89	65	ui	ui	PROPN
iajs-3051	89	66	.	.	PROPN
iajs-3051	90	1	definition	definition	NOUN
iajs-3051	90	2	1.14	1.14	NUM
iajs-3051	90	3	:	:	PUNCT
iajs-3051	91	1	[	[	X
iajs-3051	91	2	21	21	NUM
iajs-3051	91	3	]	]	X
iajs-3051	91	4	a	a	DET
iajs-3051	91	5	nano	nano	NOUN
iajs-3051	91	6	-	-	PUNCT
iajs-3051	91	7	top	top	NOUN
iajs-3051	91	8	-	-	PUNCT
iajs-3051	91	9	sp.(g	sp.(g	ADJ
iajs-3051	91	10	,	,	PUNCT
iajs-3051	91	11	𝜏𝑅(𝐺	𝜏𝑅(𝐺	ADJ
iajs-3051	91	12	)	)	PUNCT
iajs-3051	91	13	)	)	PUNCT
iajs-3051	91	14	is	be	AUX
iajs-3051	91	15	called	call	VERB
iajs-3051	91	16	nano	nano	ADJ
iajs-3051	91	17	regular	regular	ADJ
iajs-3051	91	18	space	space	NOUN
iajs-3051	91	19	(	(	PUNCT
iajs-3051	91	20	denoted	denote	VERB
iajs-3051	91	21	by	by	ADP
iajs-3051	91	22	nanoregu	nanoregu	NOUN
iajs-3051	91	23	-	-	PUNCT
iajs-3051	91	24	sp	sp	NOUN
iajs-3051	91	25	.	.	PUNCT
iajs-3051	91	26	)	)	PUNCT
iajs-3051	92	1	iff	iff	NOUN
iajs-3051	92	2	for	for	ADP
iajs-3051	92	3	all	all	DET
iajs-3051	92	4	nano	nano	NOUN
iajs-3051	92	5	closed	close	VERB
iajs-3051	92	6	set	set	NOUN
iajs-3051	92	7	(	(	PUNCT
iajs-3051	92	8	denoted	denote	VERB
iajs-3051	92	9	by	by	ADP
iajs-3051	92	10	nano	nano	NOUN
iajs-3051	92	11	-	-	PUNCT
iajs-3051	92	12	clos	clo	NOUN
iajs-3051	92	13	-	-	PUNCT
iajs-3051	92	14	set	set	NOUN
iajs-3051	92	15	)	)	PUNCT
iajs-3051	93	1	f	f	PROPN
iajs-3051	94	1	⊂	⊂	PROPN
iajs-3051	94	2	g	g	PROPN
iajs-3051	94	3	,	,	PUNCT
iajs-3051	94	4	and	and	CCONJ
iajs-3051	94	5	all	all	DET
iajs-3051	94	6	point	point	NOUN
iajs-3051	94	7	g	g	PROPN
iajs-3051	94	8	∉	∉	PROPN
iajs-3051	94	9	f	f	PROPN
iajs-3051	94	10	find	find	VERB
iajs-3051	94	11	nano	nano	NOUN
iajs-3051	94	12	open	open	ADJ
iajs-3051	94	13	sets(denoted	sets(denote	VERB
iajs-3051	94	14	by	by	ADP
iajs-3051	94	15	nano	nano	NOUN
iajs-3051	94	16	-	-	PUNCT
iajs-3051	94	17	ope	ope	NOUN
iajs-3051	94	18	-	-	PUNCT
iajs-3051	94	19	sets	set	NOUN
iajs-3051	94	20	)	)	PUNCT
iajs-3051	94	21	u	u	NOUN
iajs-3051	94	22	and	and	CCONJ
iajs-3051	94	23	v	v	ADP
iajs-3051	94	24	such	such	ADJ
iajs-3051	94	25	as	as	ADP
iajs-3051	94	26	g	g	PROPN
iajs-3051	94	27	∈	∈	PROPN
iajs-3051	94	28	u	u	PROPN
iajs-3051	94	29	,	,	PUNCT
iajs-3051	94	30	f	f	PROPN
iajs-3051	94	31	⊂	⊂	PROPN
iajs-3051	94	32	v	v	PROPN
iajs-3051	94	33	and	and	CCONJ
iajs-3051	94	34	u⋂v	u⋂v	NOUN
iajs-3051	94	35	=	=	PUNCT
iajs-3051	94	36	∅.	∅.	ADP
iajs-3051	94	37	definition	definition	NOUN
iajs-3051	94	38	1.15	1.15	NUM
iajs-3051	94	39	:	:	PUNCT
iajs-3051	95	1	[	[	X
iajs-3051	95	2	21	21	NUM
iajs-3051	95	3	]	]	PUNCT
iajs-3051	95	4	a	a	DET
iajs-3051	95	5	anao	anao	NOUN
iajs-3051	95	6	-	-	PUNCT
iajs-3051	95	7	top	top	NOUN
iajs-3051	95	8	-	-	PUNCT
iajs-3051	95	9	sp.(u	sp.(u	NOUN
iajs-3051	95	10	,	,	PUNCT
iajs-3051	95	11	τr(g	τr(g	PUNCT
iajs-3051	95	12	)	)	PUNCT
iajs-3051	95	13	)	)	PUNCT
iajs-3051	95	14	is	be	AUX
iajs-3051	95	15	called	call	VERB
iajs-3051	95	16	nano	nano	ADJ
iajs-3051	95	17	-	-	PUNCT
iajs-3051	95	18	normal	normal	ADJ
iajs-3051	95	19	-	-	PUNCT
iajs-3051	95	20	space	space	NOUN
iajs-3051	95	21	(	(	PUNCT
iajs-3051	95	22	denoted	denote	VERB
iajs-3051	95	23	by	by	ADP
iajs-3051	95	24	nanonorm	nanonorm	NOUN
iajs-3051	95	25	-	-	PUNCT
iajs-3051	95	26	sp	sp	NOUN
iajs-3051	95	27	.	.	PUNCT
iajs-3051	95	28	)	)	PUNCT
iajs-3051	96	1	if	if	SCONJ
iajs-3051	96	2	for	for	ADP
iajs-3051	96	3	any	any	DET
iajs-3051	96	4	pair	pair	NOUN
iajs-3051	96	5	of	of	ADP
iajs-3051	96	6	disjoint	disjoint	ADJ
iajs-3051	96	7	nano	nano	NOUN
iajs-3051	96	8	-	-	PUNCT
iajs-3051	96	9	clos	clo	NOUN
iajs-3051	96	10	-	-	PUNCT
iajs-3051	96	11	sets	set	NOUN
iajs-3051	96	12	a	a	PRON
iajs-3051	96	13	and	and	CCONJ
iajs-3051	96	14	b	b	NOUN
iajs-3051	96	15	of	of	ADP
iajs-3051	96	16	u	u	PRON
iajs-3051	96	17	find	find	VERB
iajs-3051	96	18	nano	nano	NOUN
iajs-3051	96	19	-	-	PUNCT
iajs-3051	96	20	ope	ope	NOUN
iajs-3051	96	21	-	-	PUNCT
iajs-3051	96	22	sets	set	NOUN
iajs-3051	96	23	v	v	NOUN
iajs-3051	96	24	and	and	CCONJ
iajs-3051	96	25	w	w	NOUN
iajs-3051	96	26	of	of	ADP
iajs-3051	96	27	u	u	PRON
iajs-3051	96	28	such	such	ADJ
iajs-3051	96	29	as	as	ADP
iajs-3051	96	30	a	a	DET
iajs-3051	96	31	⊆	⊆	NUM
iajs-3051	96	32	v	v	NOUN
iajs-3051	96	33	and	and	CCONJ
iajs-3051	96	34	b	b	NOUN
iajs-3051	96	35	⊆	⊆	NUM
iajs-3051	96	36	w.	w.	NOUN
iajs-3051	96	37	definition	definition	NOUN
iajs-3051	96	38	1.16	1.16	NUM
iajs-3051	96	39	:	:	PUNCT
iajs-3051	96	40	a	a	DET
iajs-3051	96	41	nano	nano	NOUN
iajs-3051	96	42	-	-	PUNCT
iajs-3051	96	43	top	top	NOUN
iajs-3051	96	44	-	-	PUNCT
iajs-3051	96	45	sp.(g	sp.(g	ADJ
iajs-3051	96	46	,	,	PUNCT
iajs-3051	96	47	𝜏𝑅(𝐺	𝜏𝑅(𝐺	ADJ
iajs-3051	96	48	)	)	PUNCT
iajs-3051	96	49	)	)	PUNCT
iajs-3051	96	50	is	be	AUX
iajs-3051	96	51	said	say	VERB
iajs-3051	96	52	to	to	PART
iajs-3051	96	53	be	be	AUX
iajs-3051	96	54	nano	nano	ADJ
iajs-3051	96	55	paracompact	paracompact	ADJ
iajs-3051	96	56	space	space	NOUN
iajs-3051	96	57	(	(	PUNCT
iajs-3051	96	58	denoted	denote	VERB
iajs-3051	96	59	by	by	ADP
iajs-3051	96	60	nano	nano	NOUN
iajs-3051	96	61	-	-	PUNCT
iajs-3051	96	62	para	para	ADJ
iajs-3051	96	63	-	-	PUNCT
iajs-3051	96	64	comp	comp	NOUN
iajs-3051	96	65	-	-	PUNCT
iajs-3051	96	66	sp	sp	NOUN
iajs-3051	96	67	.	.	PUNCT
iajs-3051	96	68	)	)	PUNCT
iajs-3051	97	1	if	if	SCONJ
iajs-3051	97	2	it	it	PRON
iajs-3051	97	3	satisfies	satisfy	VERB
iajs-3051	97	4	for	for	ADP
iajs-3051	97	5	each	each	DET
iajs-3051	97	6	nano	nano	NOUN
iajs-3051	97	7	-	-	PUNCT
iajs-3051	97	8	ope	ope	NOUN
iajs-3051	97	9	-	-	PUNCT
iajs-3051	97	10	cover	cover	NOUN
iajs-3051	97	11	has	have	VERB
iajs-3051	97	12	a	a	DET
iajs-3051	97	13	locally	locally	ADV
iajs-3051	97	14	finite	finite	ADJ
iajs-3051	97	15	-	-	ADJ
iajs-3051	97	16	nano	nano	ADJ
iajs-3051	97	17	-	-	PUNCT
iajs-3051	97	18	open	open	ADJ
iajs-3051	97	19	refinement	refinement	NOUN
iajs-3051	97	20	.	.	PUNCT
iajs-3051	98	1	definition	definition	NOUN
iajs-3051	98	2	1.17	1.17	NUM
iajs-3051	98	3	:	:	PUNCT
iajs-3051	98	4	a	a	DET
iajs-3051	98	5	nano	nano	NOUN
iajs-3051	98	6	-	-	PUNCT
iajs-3051	98	7	top	top	NOUN
iajs-3051	98	8	-	-	PUNCT
iajs-3051	98	9	sp.(g	sp.(g	ADJ
iajs-3051	98	10	,	,	PUNCT
iajs-3051	98	11	𝜏𝑅(𝐺	𝜏𝑅(𝐺	ADJ
iajs-3051	98	12	)	)	PUNCT
iajs-3051	98	13	)	)	PUNCT
iajs-3051	98	14	is	be	AUX
iajs-3051	98	15	said	say	VERB
iajs-3051	98	16	to	to	PART
iajs-3051	98	17	be	be	AUX
iajs-3051	98	18	nano	nano	ADJ
iajs-3051	98	19	metrizable	metrizable	ADJ
iajs-3051	98	20	-	-	PUNCT
iajs-3051	98	21	space	space	NOUN
iajs-3051	98	22	if	if	SCONJ
iajs-3051	98	23	there	there	PRON
iajs-3051	98	24	is	be	VERB
iajs-3051	98	25	a	a	DET
iajs-3051	98	26	metric	metric	ADJ
iajs-3051	98	27	d	d	NOUN
iajs-3051	98	28	:	:	PUNCT
iajs-3051	98	29	g×g	g×g	PROPN
iajs-3051	98	30	⟶	⟶	NOUN
iajs-3051	98	31	[	[	X
iajs-3051	98	32	0,∞	0,∞	X
iajs-3051	98	33	]	]	X
iajs-3051	98	34	,	,	PUNCT
iajs-3051	98	35	such	such	ADJ
iajs-3051	98	36	as	as	ADP
iajs-3051	98	37	the	the	DET
iajs-3051	98	38	nano	nano	NOUN
iajs-3051	98	39	topology	topology	NOUN
iajs-3051	98	40	induced	induce	VERB
iajs-3051	98	41	by	by	ADP
iajs-3051	98	42	d	d	PROPN
iajs-3051	98	43	is	be	AUX
iajs-3051	98	44	τ	τ	PROPN
iajs-3051	98	45	,	,	PUNCT
iajs-3051	98	46	and	and	CCONJ
iajs-3051	98	47	is	be	AUX
iajs-3051	98	48	nano	nano	NOUN
iajs-3051	98	49	homeomorphic	homeomorphic	ADJ
iajs-3051	98	50	to	to	ADP
iajs-3051	98	51	a	a	DET
iajs-3051	98	52	metric	metric	ADJ
iajs-3051	98	53	-	-	PUNCT
iajs-3051	98	54	space	space	NOUN
iajs-3051	98	55	.	.	PUNCT
iajs-3051	99	1	2	2	X
iajs-3051	99	2	.	.	NUM
iajs-3051	99	3	images	image	NOUN
iajs-3051	99	4	of	of	ADP
iajs-3051	99	5	nano	nano	VERB
iajs-3051	99	6	perfect	perfect	ADJ
iajs-3051	99	7	mappings	mapping	NOUN
iajs-3051	99	8	.	.	PUNCT
iajs-3051	100	1	we	we	PRON
iajs-3051	100	2	study	study	VERB
iajs-3051	100	3	the	the	DET
iajs-3051	100	4	images	image	NOUN
iajs-3051	100	5	of	of	ADP
iajs-3051	100	6	nano	nano	VERB
iajs-3051	100	7	perfect	perfect	ADJ
iajs-3051	100	8	mappings	mapping	NOUN
iajs-3051	100	9	and	and	CCONJ
iajs-3051	100	10	some	some	DET
iajs-3051	100	11	theories	theory	NOUN
iajs-3051	100	12	related	relate	VERB
iajs-3051	100	13	to	to	ADP
iajs-3051	100	14	the	the	DET
iajs-3051	100	15	topic	topic	NOUN
iajs-3051	100	16	.	.	PUNCT
iajs-3051	101	1	definition	definition	NOUN
iajs-3051	101	2	2.1	2.1	NUM
iajs-3051	101	3	:	:	PUNCT
iajs-3051	101	4	a	a	DET
iajs-3051	101	5	map	map	NOUN
iajs-3051	101	6	f	f	X
iajs-3051	101	7	:	:	PUNCT
iajs-3051	101	8	g	g	PROPN
iajs-3051	101	9	⟶	⟶	NOUN
iajs-3051	101	10	h	h	NOUN
iajs-3051	101	11	is	be	AUX
iajs-3051	101	12	said	say	VERB
iajs-3051	101	13	to	to	PART
iajs-3051	101	14	be	be	AUX
iajs-3051	101	15	nano	nano	NOUN
iajs-3051	101	16	perfect	perfect	ADJ
iajs-3051	101	17	(	(	PUNCT
iajs-3051	101	18	dented	dent	VERB
iajs-3051	101	19	by	by	ADP
iajs-3051	101	20	nano	nano	NOUN
iajs-3051	101	21	-	-	PUNCT
iajs-3051	101	22	perf	perf	NOUN
iajs-3051	101	23	.	.	PUNCT
iajs-3051	101	24	)	)	PUNCT
iajs-3051	102	1	if	if	SCONJ
iajs-3051	102	2	it	it	PRON
iajs-3051	102	3	is	be	AUX
iajs-3051	102	4	nano	nano	NOUN
iajs-3051	102	5	continuous	continuous	ADJ
iajs-3051	102	6	(	(	PUNCT
iajs-3051	102	7	denoted	denote	VERB
iajs-3051	102	8	by	by	ADP
iajs-3051	102	9	nano	nano	NOUN
iajs-3051	102	10	-	-	PUNCT
iajs-3051	102	11	cont	cont	NOUN
iajs-3051	102	12	.	.	PUNCT
iajs-3051	102	13	)	)	PUNCT
iajs-3051	102	14	,	,	PUNCT
iajs-3051	102	15	nano	nano	NOUN
iajs-3051	102	16	closed	close	VERB
iajs-3051	102	17	(	(	PUNCT
iajs-3051	102	18	denoted	denote	VERB
iajs-3051	102	19	by	by	ADP
iajs-3051	102	20	nano	nano	NOUN
iajs-3051	102	21	-	-	PUNCT
iajs-3051	102	22	clos	clo	NOUN
iajs-3051	102	23	.	.	PUNCT
iajs-3051	102	24	)	)	PUNCT
iajs-3051	102	25	,	,	PUNCT
iajs-3051	102	26	and	and	CCONJ
iajs-3051	102	27	for	for	ADP
iajs-3051	102	28	each	each	DET
iajs-3051	102	29	h	h	NOUN
iajs-3051	102	30	∈	∈	PROPN
iajs-3051	102	31	h	h	NOUN
iajs-3051	102	32	,	,	PUNCT
iajs-3051	102	33	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	102	34	)	)	PUNCT
iajs-3051	102	35	is	be	AUX
iajs-3051	102	36	nano	nano	ADJ
iajs-3051	102	37	compact	compact	ADJ
iajs-3051	102	38	(	(	PUNCT
iajs-3051	102	39	denoted	denote	VERB
iajs-3051	102	40	by	by	ADP
iajs-3051	102	41	nano	nano	NOUN
iajs-3051	102	42	-	-	PUNCT
iajs-3051	102	43	comp	comp	NOUN
iajs-3051	102	44	.	.	PUNCT
iajs-3051	102	45	)	)	PUNCT
iajs-3051	102	46	.	.	PUNCT
iajs-3051	103	1	ihjpas	ihjpas	PROPN
iajs-3051	103	2	.	.	PUNCT
iajs-3051	104	1	36	36	NUM
iajs-3051	104	2	(	(	PUNCT
iajs-3051	104	3	3	3	NUM
iajs-3051	104	4	)	)	PUNCT
iajs-3051	104	5	2023	2023	NUM
iajs-3051	104	6	401	401	NUM
iajs-3051	104	7	we	we	PRON
iajs-3051	104	8	show	show	VERB
iajs-3051	104	9	by	by	ADP
iajs-3051	104	10	the	the	DET
iajs-3051	104	11	next	next	ADJ
iajs-3051	104	12	examples	example	NOUN
iajs-3051	104	13	:	:	PUNCT
iajs-3051	104	14	example	example	NOUN
iajs-3051	104	15	2.2	2.2	NUM
iajs-3051	104	16	:	:	PUNCT
iajs-3051	104	17	let	let	VERB
iajs-3051	104	18	f	f	PRON
iajs-3051	104	19	:	:	PUNCT
iajs-3051	104	20	u	u	NOUN
iajs-3051	104	21	⟶	⟶	NOUN
iajs-3051	104	22	v	v	AUX
iajs-3051	104	23	be	be	AUX
iajs-3051	104	24	a	a	DET
iajs-3051	104	25	map	map	NOUN
iajs-3051	104	26	such	such	ADJ
iajs-3051	104	27	as	as	ADP
iajs-3051	104	28	u=	u=	PROPN
iajs-3051	104	29	{	{	PUNCT
iajs-3051	104	30	a	a	PRON
iajs-3051	104	31	,	,	PUNCT
iajs-3051	104	32	b	b	NOUN
iajs-3051	104	33	,	,	PUNCT
iajs-3051	104	34	c	c	NOUN
iajs-3051	104	35	,	,	PUNCT
iajs-3051	104	36	d	d	NOUN
iajs-3051	104	37	}	}	PUNCT
iajs-3051	104	38	with	with	ADP
iajs-3051	104	39	u	u	NOUN
iajs-3051	104	40	r⁄	r⁄	NOUN
iajs-3051	104	41	=	=	SYM
iajs-3051	104	42	{	{	PUNCT
iajs-3051	104	43	{	{	PUNCT
iajs-3051	104	44	a},{b	a},{b	NOUN
iajs-3051	104	45	}	}	PUNCT
iajs-3051	104	46	,	,	PUNCT
iajs-3051	104	47	{	{	PUNCT
iajs-3051	104	48	c	c	X
iajs-3051	104	49	,	,	PUNCT
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iajs-3051	105	6	,	,	PUNCT
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iajs-3051	105	12	τr(g	τr(g	PUNCT
iajs-3051	105	13	)	)	PUNCT
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iajs-3051	106	4	,	,	PUNCT
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iajs-3051	106	17	.	.	PUNCT
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iajs-3051	107	3	{	{	PUNCT
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iajs-3051	107	5	,	,	PUNCT
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iajs-3051	107	20	,	,	PUNCT
iajs-3051	107	21	{	{	PUNCT
iajs-3051	107	22	z	z	NOUN
iajs-3051	107	23	,	,	PUNCT
iajs-3051	107	24	w	w	NOUN
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iajs-3051	107	26	}	}	PUNCT
iajs-3051	107	27	and	and	CCONJ
iajs-3051	107	28	h=	h=	X
iajs-3051	107	29	{	{	PUNCT
iajs-3051	107	30	x	x	X
iajs-3051	107	31	,	,	PUNCT
iajs-3051	107	32	z	z	NOUN
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iajs-3051	107	34	⊂	⊂	PROPN
iajs-3051	107	35	v	v	NOUN
iajs-3051	107	36	.	.	PUNCT
iajs-3051	108	1	then	then	ADV
iajs-3051	108	2	τrʹ(h	τrʹ(h	X
iajs-3051	108	3	)	)	PUNCT
iajs-3051	109	1	=	=	PRON
iajs-3051	109	2	{	{	PUNCT
iajs-3051	109	3	v	v	NOUN
iajs-3051	109	4	,	,	PUNCT
iajs-3051	109	5	∅	∅	NOUN
iajs-3051	109	6	,	,	PUNCT
iajs-3051	109	7	{	{	PUNCT
iajs-3051	109	8	x},{x	x},{x	ADJ
iajs-3051	109	9	,	,	PUNCT
iajs-3051	109	10	z	z	NOUN
iajs-3051	109	11	,	,	PUNCT
iajs-3051	109	12	w},{z	w},{z	NOUN
iajs-3051	109	13	,	,	PUNCT
iajs-3051	109	14	w	w	NOUN
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iajs-3051	109	16	}	}	PUNCT
iajs-3051	109	17	.	.	PUNCT
iajs-3051	110	1	such	such	ADJ
iajs-3051	110	2	as	as	ADP
iajs-3051	110	3	f(a	f(a	NOUN
iajs-3051	110	4	)	)	PUNCT
iajs-3051	110	5	=	=	SYM
iajs-3051	110	6	x	x	SYM
iajs-3051	110	7	,	,	PUNCT
iajs-3051	110	8	f(b	f(b	PROPN
iajs-3051	110	9	)	)	PUNCT
iajs-3051	110	10	=	=	SYM
iajs-3051	110	11	y	y	PROPN
iajs-3051	110	12	,	,	PUNCT
iajs-3051	110	13	f(c	f(c	PROPN
iajs-3051	110	14	)	)	PUNCT
iajs-3051	110	15	=	=	SYM
iajs-3051	110	16	z	z	NOUN
iajs-3051	110	17	,	,	PUNCT
iajs-3051	110	18	f(d	f(d	PROPN
iajs-3051	110	19	)	)	PUNCT
iajs-3051	110	20	=	=	SYM
iajs-3051	110	21	w	w	PROPN
iajs-3051	110	22	.	.	PUNCT
iajs-3051	111	1	then	then	ADV
iajs-3051	111	2	f	f	PROPN
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iajs-3051	111	4	nano	nano	NOUN
iajs-3051	111	5	-	-	PUNCT
iajs-3051	111	6	perf	perf	NOUN
iajs-3051	111	7	-	-	PUNCT
iajs-3051	111	8	map	map	NOUN
iajs-3051	111	9	.	.	PUNCT
iajs-3051	112	1	example	example	NOUN
iajs-3051	112	2	2.3	2.3	NUM
iajs-3051	112	3	:	:	PUNCT
iajs-3051	112	4	let	let	VERB
iajs-3051	112	5	u	u	PRON
iajs-3051	112	6	=	=	VERB
iajs-3051	112	7	{	{	PUNCT
iajs-3051	112	8	a	a	PRON
iajs-3051	112	9	,	,	PUNCT
iajs-3051	112	10	b	b	NOUN
iajs-3051	112	11	,	,	PUNCT
iajs-3051	112	12	c	c	X
iajs-3051	112	13	,	,	PUNCT
iajs-3051	112	14	d	d	NOUN
iajs-3051	112	15	}	}	PUNCT
iajs-3051	112	16	with	with	ADP
iajs-3051	112	17	u	u	NOUN
iajs-3051	112	18	r⁄	r⁄	NOUN
iajs-3051	112	19	=	=	SYM
iajs-3051	112	20	{	{	PUNCT
iajs-3051	112	21	{	{	PUNCT
iajs-3051	112	22	b},{c},{a	b},{c},{a	PROPN
iajs-3051	112	23	,	,	PUNCT
iajs-3051	112	24	d	d	NOUN
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iajs-3051	112	26	}	}	PUNCT
iajs-3051	112	27	.	.	PUNCT
iajs-3051	113	1	let	let	VERB
iajs-3051	113	2	g	g	NOUN
iajs-3051	113	3	=	=	VERB
iajs-3051	113	4	{	{	PUNCT
iajs-3051	113	5	a	a	PRON
iajs-3051	113	6	,	,	PUNCT
iajs-3051	113	7	b	b	NOUN
iajs-3051	113	8	}	}	PUNCT
iajs-3051	113	9	⊂	⊂	PROPN
iajs-3051	113	10	u	u	PROPN
iajs-3051	113	11	.	.	PUNCT
iajs-3051	114	1	then	then	ADV
iajs-3051	114	2	τr(g	τr(g	PUNCT
iajs-3051	114	3	)	)	PUNCT
iajs-3051	115	1	=	=	PRON
iajs-3051	115	2	{	{	PUNCT
iajs-3051	115	3	u	u	NOUN
iajs-3051	115	4	,	,	PUNCT
iajs-3051	115	5	∅	∅	NOUN
iajs-3051	115	6	,	,	PUNCT
iajs-3051	115	7	{	{	PUNCT
iajs-3051	115	8	b},{a	b},{a	ADV
iajs-3051	115	9	,	,	PUNCT
iajs-3051	115	10	b	b	NOUN
iajs-3051	115	11	,	,	PUNCT
iajs-3051	115	12	d},{a	d},{a	ADV
iajs-3051	115	13	,	,	PUNCT
iajs-3051	115	14	d	d	NOUN
iajs-3051	115	15	}	}	PUNCT
iajs-3051	115	16	}	}	PUNCT
iajs-3051	115	17	.	.	PUNCT
iajs-3051	116	1	let	let	VERB
iajs-3051	116	2	v	v	VERB
iajs-3051	116	3	=	=	SYM
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iajs-3051	116	5	1	1	NUM
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iajs-3051	116	7	2	2	NUM
iajs-3051	116	8	,	,	PUNCT
iajs-3051	116	9	3	3	NUM
iajs-3051	116	10	,	,	PUNCT
iajs-3051	116	11	4	4	NUM
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iajs-3051	116	13	with	with	ADP
iajs-3051	116	14	v	v	ADP
iajs-3051	116	15	rʹ⁄	rʹ⁄	ADJ
iajs-3051	116	16	=	=	PRON
iajs-3051	116	17	{	{	PUNCT
iajs-3051	116	18	{	{	PUNCT
iajs-3051	116	19	1},{2},{3},{4	1},{2},{3},{4	NOUN
iajs-3051	116	20	}	}	PUNCT
iajs-3051	116	21	}	}	PUNCT
iajs-3051	116	22	and	and	CCONJ
iajs-3051	116	23	h	h	NOUN
iajs-3051	116	24	=	=	SYM
iajs-3051	116	25	{	{	PUNCT
iajs-3051	116	26	1	1	NUM
iajs-3051	116	27	,	,	PUNCT
iajs-3051	116	28	3	3	NUM
iajs-3051	116	29	,	,	PUNCT
iajs-3051	116	30	4	4	NUM
iajs-3051	116	31	}	}	PUNCT
iajs-3051	116	32	⊂	⊂	PROPN
iajs-3051	116	33	v	v	NOUN
iajs-3051	116	34	.	.	PUNCT
iajs-3051	117	1	then	then	ADV
iajs-3051	117	2	τrʹ(h	τrʹ(h	X
iajs-3051	117	3	)	)	PUNCT
iajs-3051	118	1	=	=	PRON
iajs-3051	118	2	{	{	PUNCT
iajs-3051	118	3	v	v	NOUN
iajs-3051	118	4	,	,	PUNCT
iajs-3051	118	5	∅	∅	NOUN
iajs-3051	118	6	,	,	PUNCT
iajs-3051	118	7	{	{	PUNCT
iajs-3051	118	8	1	1	NUM
iajs-3051	118	9	,	,	PUNCT
iajs-3051	118	10	3	3	NUM
iajs-3051	118	11	,	,	PUNCT
iajs-3051	118	12	4	4	NUM
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iajs-3051	118	15	}	}	PUNCT
iajs-3051	118	16	.	.	PUNCT
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iajs-3051	119	2	f	f	X
iajs-3051	120	1	:	:	PUNCT
iajs-3051	120	2	u	u	NOUN
iajs-3051	120	3	⟶	⟶	NOUN
iajs-3051	120	4	v	v	NOUN
iajs-3051	120	5	such	such	ADJ
iajs-3051	120	6	as	as	ADP
iajs-3051	120	7	f(a	f(a	NOUN
iajs-3051	120	8	)	)	PUNCT
iajs-3051	120	9	=	=	SYM
iajs-3051	120	10	1	1	NUM
iajs-3051	120	11	,	,	PUNCT
iajs-3051	120	12	f(b	f(b	PROPN
iajs-3051	120	13	)	)	PUNCT
iajs-3051	120	14	=	=	SYM
iajs-3051	120	15	2	2	NUM
iajs-3051	120	16	,	,	PUNCT
iajs-3051	120	17	f(c	f(c	PROPN
iajs-3051	120	18	)	)	PUNCT
iajs-3051	120	19	=	=	SYM
iajs-3051	120	20	3	3	NUM
iajs-3051	120	21	,	,	PUNCT
iajs-3051	120	22	f(4	f(4	PROPN
iajs-3051	120	23	)	)	PUNCT
iajs-3051	120	24	=	=	NOUN
iajs-3051	121	1	4	4	NUM
iajs-3051	121	2	then	then	ADV
iajs-3051	121	3	f	f	PROPN
iajs-3051	121	4	is	be	AUX
iajs-3051	121	5	not	not	PART
iajs-3051	121	6	nano	nano	NOUN
iajs-3051	121	7	-	-	PUNCT
iajs-3051	121	8	perf	perf	NOUN
iajs-3051	121	9	-	-	PUNCT
iajs-3051	121	10	map	map	NOUN
iajs-3051	121	11	.	.	PUNCT
iajs-3051	122	1	theorem	theorem	VERB
iajs-3051	122	2	2.4	2.4	NUM
iajs-3051	122	3	:	:	PUNCT
iajs-3051	122	4	let	let	VERB
iajs-3051	122	5	f	f	PRON
iajs-3051	122	6	:	:	PUNCT
iajs-3051	122	7	g	g	PROPN
iajs-3051	122	8	⟶	⟶	PROPN
iajs-3051	122	9	h	h	NOUN
iajs-3051	122	10	be	be	AUX
iajs-3051	122	11	a	a	DET
iajs-3051	122	12	nano	nano	NOUN
iajs-3051	122	13	-	-	PUNCT
iajs-3051	122	14	perf	perf	NOUN
iajs-3051	122	15	-	-	PUNCT
iajs-3051	122	16	map	map	NOUN
iajs-3051	122	17	.	.	PUNCT
iajs-3051	123	1	of	of	ADP
iajs-3051	123	2	g	g	PROPN
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iajs-3051	123	4	h	h	NOUN
iajs-3051	123	5	.	.	PUNCT
iajs-3051	124	1	if	if	SCONJ
iajs-3051	124	2	the	the	DET
iajs-3051	124	3	weight	weight	NOUN
iajs-3051	124	4	of	of	ADP
iajs-3051	124	5	g	g	PROPN
iajs-3051	124	6	is	be	AUX
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iajs-3051	124	8	,	,	PUNCT
iajs-3051	124	9	then	then	ADV
iajs-3051	124	10	w(h	w(h	PROPN
iajs-3051	124	11	)	)	PUNCT
iajs-3051	124	12	≤	≤	NOUN
iajs-3051	124	13	w(g	w(g	PROPN
iajs-3051	124	14	)	)	PUNCT
iajs-3051	124	15	.	.	PUNCT
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iajs-3051	125	2	:	:	PUNCT
iajs-3051	125	3	let	let	VERB
iajs-3051	125	4	v	v	NUM
iajs-3051	125	5	∽	∽	NOUN
iajs-3051	125	6	be	be	AUX
iajs-3051	125	7	a	a	DET
iajs-3051	125	8	nano	nano	NOUN
iajs-3051	125	9	-	-	PUNCT
iajs-3051	125	10	base	base	NOUN
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iajs-3051	125	12	the	the	DET
iajs-3051	125	13	nano	nano	NOUN
iajs-3051	125	14	-	-	PUNCT
iajs-3051	125	15	topology	topology	NOUN
iajs-3051	125	16	of	of	ADP
iajs-3051	125	17	g	g	NOUN
iajs-3051	125	18	such	such	ADJ
iajs-3051	125	19	as	as	ADP
iajs-3051	125	20	|v	|v	PROPN
iajs-3051	125	21	∽	∽	NUM
iajs-3051	125	22	|	|	NOUN
iajs-3051	125	23	=	=	SYM
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iajs-3051	125	25	)	)	PUNCT
iajs-3051	125	26	and	and	CCONJ
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iajs-3051	125	28	w	w	PRON
iajs-3051	125	29	∽	∽	NOUN
iajs-3051	125	30	consist	consist	NOUN
iajs-3051	125	31	of	of	ADP
iajs-3051	125	32	every	every	DET
iajs-3051	125	33	sets	set	NOUN
iajs-3051	125	34	of	of	ADP
iajs-3051	125	35	the	the	DET
iajs-3051	125	36	form	form	NOUN
iajs-3051	125	37	⋃	⋃	PUNCT
iajs-3051	125	38	bb∈b⏟	bb∈b⏟	PROPN
iajs-3051	125	39	,	,	PUNCT
iajs-3051	125	40	where	where	SCONJ
iajs-3051	125	41	b	b	X
iajs-3051	125	42	∽	∽	NOUN
iajs-3051	125	43	is	be	AUX
iajs-3051	125	44	a	a	DET
iajs-3051	125	45	finite	finite	NOUN
iajs-3051	125	46	subset	subset	NOUN
iajs-3051	125	47	of	of	ADP
iajs-3051	125	48	v	v	NOUN
iajs-3051	125	49	∽	∽	NUM
iajs-3051	125	50	.	.	PUNCT
iajs-3051	126	1	if	if	SCONJ
iajs-3051	126	2	w	w	PROPN
iajs-3051	126	3	∈	∈	PROPN
iajs-3051	126	4	w	w	PROPN
iajs-3051	126	5	̴	̴	PROPN
iajs-3051	126	6	,	,	PUNCT
iajs-3051	126	7	let	let	VERB
iajs-3051	126	8	tw	tw	NOUN
iajs-3051	126	9	=	=	NOUN
iajs-3051	126	10	h\f(g\w	h\f(g\w	NOUN
iajs-3051	126	11	)	)	PUNCT
iajs-3051	126	12	and	and	CCONJ
iajs-3051	126	13	let	let	VERB
iajs-3051	126	14	t	t	PROPN
iajs-3051	126	15	be	be	AUX
iajs-3051	126	16	the	the	DET
iajs-3051	126	17	set	set	NOUN
iajs-3051	126	18	of	of	ADP
iajs-3051	126	19	nano	nano	NOUN
iajs-3051	126	20	-	-	PUNCT
iajs-3051	126	21	ope	ope	NOUN
iajs-3051	126	22	-	-	PUNCT
iajs-3051	126	23	sets	set	NOUN
iajs-3051	126	24	of	of	ADP
iajs-3051	126	25	h	h	NOUN
iajs-3051	126	26	of	of	ADP
iajs-3051	126	27	the	the	DET
iajs-3051	126	28	form	form	NOUN
iajs-3051	126	29	tw	tw	NOUN
iajs-3051	126	30	for	for	ADP
iajs-3051	126	31	some	some	DET
iajs-3051	126	32	w	w	NOUN
iajs-3051	126	33	in	in	ADP
iajs-3051	126	34	w	w	PROPN
iajs-3051	126	35	̴	̴	PROPN
iajs-3051	126	36	.	.	PUNCT
iajs-3051	127	1	so	so	ADV
iajs-3051	127	2	w(g	w(g	PROPN
iajs-3051	127	3	)	)	PUNCT
iajs-3051	127	4	is	be	AUX
iajs-3051	127	5	infinite	infinite	ADJ
iajs-3051	127	6	,	,	PUNCT
iajs-3051	127	7	|	|	ADV
iajs-3051	127	8	w	w	PROPN
iajs-3051	127	9	̴	̴	PROPN
iajs-3051	127	10	|	|	PROPN
iajs-3051	127	11	=	=	SYM
iajs-3051	127	12	w(g	w(g	PROPN
iajs-3051	127	13	)	)	PUNCT
iajs-3051	127	14	,	,	PUNCT
iajs-3051	127	15	so	so	SCONJ
iajs-3051	127	16	that	that	SCONJ
iajs-3051	127	17	the	the	DET
iajs-3051	127	18	cardinality	cardinality	NOUN
iajs-3051	127	19	of	of	ADP
iajs-3051	127	20	the	the	DET
iajs-3051	127	21	set	set	NOUN
iajs-3051	127	22	t	t	PROPN
iajs-3051	127	23	∽	∽	NUM
iajs-3051	127	24	of	of	ADP
iajs-3051	127	25	nano	nano	NOUN
iajs-3051	127	26	-	-	PUNCT
iajs-3051	127	27	ope	ope	NOUN
iajs-3051	127	28	-	-	PUNCT
iajs-3051	127	29	sets	set	NOUN
iajs-3051	127	30	of	of	ADP
iajs-3051	127	31	h	h	NOUN
iajs-3051	127	32	does	do	AUX
iajs-3051	127	33	not	not	PART
iajs-3051	127	34	exceed	exceed	VERB
iajs-3051	127	35	w(g	w(g	PROPN
iajs-3051	127	36	)	)	PUNCT
iajs-3051	127	37	.	.	PUNCT
iajs-3051	128	1	furthermore	furthermore	ADV
iajs-3051	128	2	t	t	PROPN
iajs-3051	128	3	∽	∽	NOUN
iajs-3051	128	4	is	be	AUX
iajs-3051	128	5	a	a	DET
iajs-3051	128	6	nano	nano	NOUN
iajs-3051	128	7	-	-	PUNCT
iajs-3051	128	8	base	base	NOUN
iajs-3051	128	9	for	for	ADP
iajs-3051	128	10	the	the	DET
iajs-3051	128	11	nano	nano	NOUN
iajs-3051	128	12	-	-	PUNCT
iajs-3051	128	13	topology	topology	NOUN
iajs-3051	128	14	of	of	ADP
iajs-3051	128	15	h.	h.	PROPN
iajs-3051	128	16	for	for	SCONJ
iajs-3051	128	17	let	let	VERB
iajs-3051	128	18	j	j	PROPN
iajs-3051	128	19	be	be	AUX
iajs-3051	128	20	an	an	DET
iajs-3051	128	21	nano	nano	NOUN
iajs-3051	128	22	-	-	PUNCT
iajs-3051	128	23	ope	ope	NOUN
iajs-3051	128	24	-	-	PUNCT
iajs-3051	128	25	set	set	NOUN
iajs-3051	128	26	of	of	ADP
iajs-3051	128	27	h	h	NOUN
iajs-3051	128	28	and	and	CCONJ
iajs-3051	128	29	let	let	VERB
iajs-3051	128	30	h	h	PRON
iajs-3051	128	31	be	be	AUX
iajs-3051	128	32	a	a	DET
iajs-3051	128	33	point	point	NOUN
iajs-3051	128	34	of	of	ADP
iajs-3051	128	35	j.	j.	PROPN
iajs-3051	128	36	since	since	SCONJ
iajs-3051	128	37	𝑓−1	𝑓−1	PROPN
iajs-3051	128	38	(	(	PUNCT
iajs-3051	128	39	h	h	NOUN
iajs-3051	128	40	)	)	PUNCT
iajs-3051	128	41	⊂	⊂	PROPN
iajs-3051	128	42	𝑓−1(j	𝑓−1(j	PROPN
iajs-3051	128	43	)	)	PUNCT
iajs-3051	128	44	and	and	CCONJ
iajs-3051	128	45	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	128	46	)	)	PUNCT
iajs-3051	128	47	is	be	AUX
iajs-3051	128	48	nano	nano	NOUN
iajs-3051	128	49	-	-	NOUN
iajs-3051	128	50	comp	comp	NOUN
iajs-3051	128	51	.	.	PUNCT
iajs-3051	128	52	,	,	PUNCT
iajs-3051	128	53	find	find	VERB
iajs-3051	128	54	w	w	NOUN
iajs-3051	128	55	of	of	ADP
iajs-3051	128	56	w	w	NOUN
iajs-3051	128	57	∽	∽	NOUN
iajs-3051	128	58	such	such	ADJ
iajs-3051	128	59	as	as	ADP
iajs-3051	128	60	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	128	61	)	)	PUNCT
iajs-3051	129	1	⊂	⊂	PROPN
iajs-3051	129	2	w	w	PROPN
iajs-3051	129	3	⊂	⊂	PROPN
iajs-3051	129	4	𝑓−1(j	𝑓−1(j	PROPN
iajs-3051	129	5	)	)	PUNCT
iajs-3051	129	6	.	.	PUNCT
iajs-3051	130	1	then	then	ADV
iajs-3051	130	2	h	h	PROPN
iajs-3051	130	3	∈	∈	PROPN
iajs-3051	130	4	tw	tw	VERB
iajs-3051	131	1	⊂	⊂	PROPN
iajs-3051	131	2	j.	j.	PROPN
iajs-3051	132	1	it	it	PRON
iajs-3051	132	2	follows	follow	VERB
iajs-3051	132	3	that	that	SCONJ
iajs-3051	132	4	w(h	w(h	PROPN
iajs-3051	132	5	)	)	PUNCT
iajs-3051	132	6	≤	≤	NOUN
iajs-3051	132	7	w(g	w(g	PROPN
iajs-3051	132	8	)	)	PUNCT
iajs-3051	132	9	.	.	PUNCT
iajs-3051	133	1	theorem	theorem	VERB
iajs-3051	133	2	2.5	2.5	NUM
iajs-3051	133	3	:	:	PUNCT
iajs-3051	133	4	the	the	DET
iajs-3051	133	5	class	class	NOUN
iajs-3051	133	6	of	of	ADP
iajs-3051	133	7	nano	nano	NOUN
iajs-3051	133	8	-	-	PUNCT
iajs-3051	133	9	ti	ti	NOUN
iajs-3051	133	10	-	-	NOUN
iajs-3051	133	11	spaces	space	NOUN
iajs-3051	133	12	(	(	PUNCT
iajs-3051	133	13	denoted	denote	VERB
iajs-3051	133	14	by	by	ADP
iajs-3051	133	15	nano	nano	NOUN
iajs-3051	133	16	-	-	PUNCT
iajs-3051	133	17	ti	ti	NOUN
iajs-3051	133	18	-	-	NOUN
iajs-3051	133	19	sp	sp	NOUN
iajs-3051	133	20	.	.	PUNCT
iajs-3051	133	21	)	)	PUNCT
iajs-3051	133	22	is	be	AUX
iajs-3051	133	23	invariant	invariant	ADJ
iajs-3051	133	24	under	under	ADP
iajs-3051	133	25	nano	nano	NOUN
iajs-3051	133	26	-	-	PUNCT
iajs-3051	133	27	perfmaps	perfmap	NOUN
iajs-3051	133	28	for	for	ADP
iajs-3051	133	29	i=1,2,3,4	i=1,2,3,4	PROPN
iajs-3051	133	30	.	.	PUNCT
iajs-3051	134	1	proof	proof	NOUN
iajs-3051	134	2	:	:	PUNCT
iajs-3051	134	3	let	let	VERB
iajs-3051	134	4	f	f	PRON
iajs-3051	134	5	:	:	PUNCT
iajs-3051	134	6	g	g	PROPN
iajs-3051	134	7	⟶	⟶	PROPN
iajs-3051	134	8	h	h	NOUN
iajs-3051	134	9	be	be	AUX
iajs-3051	134	10	a	a	DET
iajs-3051	134	11	nano	nano	NOUN
iajs-3051	134	12	-	-	PUNCT
iajs-3051	134	13	perf	perf	NOUN
iajs-3051	134	14	.	.	PUNCT
iajs-3051	135	1	onto	onto	ADP
iajs-3051	135	2	map	map	NOUN
iajs-3051	135	3	:	:	PUNCT
iajs-3051	135	4	(	(	PUNCT
iajs-3051	135	5	a	a	X
iajs-3051	135	6	)	)	PUNCT
iajs-3051	135	7	if	if	SCONJ
iajs-3051	135	8	g	g	PROPN
iajs-3051	135	9	is	be	AUX
iajs-3051	135	10	a	a	DET
iajs-3051	135	11	nano	nano	NOUN
iajs-3051	135	12	-	-	PUNCT
iajs-3051	135	13	t1	t1	NUM
iajs-3051	135	14	-	-	PUNCT
iajs-3051	135	15	sp	sp	NOUN
iajs-3051	135	16	.	.	PUNCT
iajs-3051	135	17	then	then	ADV
iajs-3051	135	18	all	all	DET
iajs-3051	135	19	point	point	NOUN
iajs-3051	135	20	{	{	PUNCT
iajs-3051	135	21	g	g	NOUN
iajs-3051	135	22	}	}	PUNCT
iajs-3051	135	23	is	be	AUX
iajs-3051	135	24	a	a	DET
iajs-3051	135	25	nano	nano	NOUN
iajs-3051	135	26	-	-	PUNCT
iajs-3051	135	27	clos	clo	NOUN
iajs-3051	135	28	-	-	PUNCT
iajs-3051	135	29	set	set	NOUN
iajs-3051	135	30	.	.	PUNCT
iajs-3051	136	1	so	so	ADV
iajs-3051	136	2	let	let	VERB
iajs-3051	136	3	h	h	PRON
iajs-3051	136	4	∈	∈	PROPN
iajs-3051	136	5	h	h	NOUN
iajs-3051	136	6	,	,	PUNCT
iajs-3051	136	7	then	then	ADV
iajs-3051	136	8	find	find	VERB
iajs-3051	136	9	g	g	ADP
iajs-3051	136	10	∈	∈	PROPN
iajs-3051	136	11	g	g	NOUN
iajs-3051	136	12	such	such	ADJ
iajs-3051	136	13	as	as	ADP
iajs-3051	136	14	f(g	f(g	NOUN
iajs-3051	136	15	)	)	PUNCT
iajs-3051	136	16	=	=	SYM
iajs-3051	137	1	h.	h.	NOUN
iajs-3051	138	1	but	but	CCONJ
iajs-3051	138	2	f	f	PROPN
iajs-3051	138	3	is	be	AUX
iajs-3051	138	4	nano	nano	NOUN
iajs-3051	138	5	-	-	PUNCT
iajs-3051	138	6	clos	clo	NOUN
iajs-3051	138	7	.	.	PUNCT
iajs-3051	139	1	and	and	CCONJ
iajs-3051	139	2	{	{	PUNCT
iajs-3051	139	3	g	g	NOUN
iajs-3051	139	4	}	}	PUNCT
iajs-3051	139	5	is	be	AUX
iajs-3051	139	6	nano	nano	NOUN
iajs-3051	139	7	-	-	PUNCT
iajs-3051	139	8	clos	clo	NOUN
iajs-3051	139	9	.	.	PUNCT
iajs-3051	140	1	;	;	PUNCT
iajs-3051	140	2	hence	hence	ADV
iajs-3051	140	3	{	{	PUNCT
iajs-3051	140	4	h	h	NOUN
iajs-3051	140	5	}	}	PUNCT
iajs-3051	140	6	is	be	AUX
iajs-3051	140	7	nano	nano	NOUN
iajs-3051	140	8	-	-	PUNCT
iajs-3051	140	9	clos	clo	NOUN
iajs-3051	140	10	.	.	PUNCT
iajs-3051	141	1	in	in	ADP
iajs-3051	141	2	h.	h.	PROPN
iajs-3051	141	3	(	(	PUNCT
iajs-3051	141	4	b	b	X
iajs-3051	141	5	)	)	PUNCT
iajs-3051	141	6	let	let	VERB
iajs-3051	141	7	g	g	NOUN
iajs-3051	141	8	be	be	AUX
iajs-3051	141	9	a	a	DET
iajs-3051	141	10	nano	nano	NOUN
iajs-3051	141	11	-	-	PUNCT
iajs-3051	141	12	t2	t2	NOUN
iajs-3051	141	13	-	-	PUNCT
iajs-3051	141	14	sp	sp	NOUN
iajs-3051	141	15	.	.	PUNCT
iajs-3051	142	1	let	let	VERB
iajs-3051	142	2	h1	h1	PROPN
iajs-3051	142	3	,	,	PUNCT
iajs-3051	142	4	h2	h2	PROPN
iajs-3051	142	5	be	be	AUX
iajs-3051	142	6	a	a	DET
iajs-3051	142	7	pair	pair	NOUN
iajs-3051	142	8	of	of	ADP
iajs-3051	142	9	distinct	distinct	ADJ
iajs-3051	142	10	points	point	NOUN
iajs-3051	142	11	of	of	ADP
iajs-3051	142	12	h.	h.	NOUN
iajs-3051	142	13	the	the	DET
iajs-3051	142	14	inverse	inverse	NOUN
iajs-3051	142	15	images	image	NOUN
iajs-3051	142	16	𝑓−1(h1	𝑓−1(h1	PRON
iajs-3051	142	17	)	)	PUNCT
iajs-3051	142	18	and	and	CCONJ
iajs-3051	142	19	𝑓−1(h2	𝑓−1(h2	NOUN
iajs-3051	142	20	)	)	PUNCT
iajs-3051	142	21	are	be	AUX
iajs-3051	142	22	nano-comp.and	nano-comp.and	PRON
iajs-3051	142	23	disjoint	disjoint	NOUN
iajs-3051	142	24	subsets	subset	NOUN
iajs-3051	142	25	of	of	ADP
iajs-3051	142	26	g	g	NOUN
iajs-3051	142	27	,	,	PUNCT
iajs-3051	142	28	so	so	ADV
iajs-3051	142	29	find	find	VERB
iajs-3051	142	30	nano	nano	NOUN
iajs-3051	142	31	-	-	PUNCT
iajs-3051	142	32	ope	ope	NOUN
iajs-3051	142	33	-	-	PUNCT
iajs-3051	142	34	sets	set	NOUN
iajs-3051	142	35	u	u	NOUN
iajs-3051	142	36	,	,	PUNCT
iajs-3051	142	37	v	v	ADP
iajs-3051	142	38	⊂	⊂	PROPN
iajs-3051	142	39	g	g	PROPN
iajs-3051	142	40	such	such	ADJ
iajs-3051	142	41	as	as	ADP
iajs-3051	142	42	𝑓−1(h1	𝑓−1(h1	PROPN
iajs-3051	142	43	)	)	PUNCT
iajs-3051	143	1	⊂	⊂	PROPN
iajs-3051	143	2	u	u	NOUN
iajs-3051	143	3	,	,	PUNCT
iajs-3051	143	4	𝑓−1(h2	𝑓−1(h2	NOUN
iajs-3051	143	5	)	)	PUNCT
iajs-3051	143	6	⊂	⊂	PROPN
iajs-3051	143	7	v	v	NOUN
iajs-3051	143	8	and	and	CCONJ
iajs-3051	143	9	u⋂v=∅.	u⋂v=∅.	NUM
iajs-3051	143	10	so	so	SCONJ
iajs-3051	143	11	f	f	PROPN
iajs-3051	143	12	is	be	AUX
iajs-3051	143	13	nano	nano	NOUN
iajs-3051	143	14	-	-	PUNCT
iajs-3051	143	15	clos	clo	NOUN
iajs-3051	143	16	.	.	PUNCT
iajs-3051	143	17	,	,	PUNCT
iajs-3051	143	18	the	the	PRON
iajs-3051	143	19	sets	set	VERB
iajs-3051	143	20	a	a	DET
iajs-3051	143	21	=	=	NOUN
iajs-3051	143	22	h\f(g\u	h\f(g\u	NOUN
iajs-3051	143	23	)	)	PUNCT
iajs-3051	143	24	and	and	CCONJ
iajs-3051	143	25	b	b	X
iajs-3051	143	26	=	=	NOUN
iajs-3051	143	27	h\f(g\v	h\f(g\v	NOUN
iajs-3051	143	28	)	)	PUNCT
iajs-3051	143	29	are	be	AUX
iajs-3051	143	30	nano	nano	NOUN
iajs-3051	143	31	-	-	PUNCT
iajs-3051	143	32	ope	ope	NOUN
iajs-3051	143	33	.	.	PUNCT
iajs-3051	144	1	in	in	ADP
iajs-3051	144	2	h	h	NOUN
iajs-3051	144	3	such	such	ADJ
iajs-3051	144	4	as	as	ADP
iajs-3051	144	5	𝑓−1(h1	𝑓−1(h1	PROPN
iajs-3051	144	6	)	)	PUNCT
iajs-3051	144	7	⊂	⊂	PROPN
iajs-3051	144	8	𝑓−1(a	𝑓−1(a	PROPN
iajs-3051	144	9	)	)	PUNCT
iajs-3051	145	1	⊂	⊂	PROPN
iajs-3051	145	2	u	u	NOUN
iajs-3051	145	3	and	and	CCONJ
iajs-3051	145	4	𝑓−1(h2	𝑓−1(h2	NOUN
iajs-3051	145	5	)	)	PUNCT
iajs-3051	145	6	⊂	⊂	PROPN
iajs-3051	145	7	𝑓−1(b	𝑓−1(b	PROPN
iajs-3051	145	8	)	)	PUNCT
iajs-3051	146	1	⊂	⊂	PROPN
iajs-3051	146	2	v	v	PROPN
iajs-3051	146	3	,	,	PUNCT
iajs-3051	146	4	a	a	DET
iajs-3051	146	5	and	and	CCONJ
iajs-3051	146	6	b	b	NOUN
iajs-3051	146	7	are	be	AUX
iajs-3051	146	8	nano	nano	NOUN
iajs-3051	146	9	-	-	PUNCT
iajs-3051	146	10	nbds	nbds	NOUN
iajs-3051	146	11	of	of	ADP
iajs-3051	146	12	h1	h1	NOUN
iajs-3051	146	13	and	and	CCONJ
iajs-3051	146	14	h2	h2	NOUN
iajs-3051	146	15	respectively	respectively	ADV
iajs-3051	146	16	.	.	PUNCT
iajs-3051	147	1	moreover	moreover	ADV
iajs-3051	147	2	a⋂	a⋂	NOUN
iajs-3051	147	3	b	b	PROPN
iajs-3051	147	4	=[	=[	NUM
iajs-3051	147	5	h\f(g\u	h\f(g\u	PROPN
iajs-3051	147	6	)	)	PUNCT
iajs-3051	147	7	]	]	PUNCT
iajs-3051	148	1	⋂	⋂	PROPN
iajs-3051	149	1	[	[	X
iajs-3051	149	2	h\f(g\v	h\f(g\v	NOUN
iajs-3051	149	3	)	)	PUNCT
iajs-3051	149	4	]	]	PUNCT
iajs-3051	150	1	=	=	PUNCT
iajs-3051	150	2	h\[f(g\u	h\[f(g\u	PROPN
iajs-3051	150	3	)	)	PUNCT
iajs-3051	150	4	⋃	⋃	NOUN
iajs-3051	150	5	f(g\v	f(g\v	NOUN
iajs-3051	150	6	)	)	PUNCT
iajs-3051	150	7	]	]	PUNCT
iajs-3051	151	1	=	=	PUNCT
iajs-3051	151	2	h\f[g\(u⋂v	h\f[g\(u⋂v	NOUN
iajs-3051	151	3	)	)	PUNCT
iajs-3051	151	4	]	]	PUNCT
iajs-3051	152	1	=	=	X
iajs-3051	152	2	h\f(g	h\f(g	NOUN
iajs-3051	152	3	)	)	PUNCT
iajs-3051	152	4	=	=	NOUN
iajs-3051	152	5	∅	∅	NOUN
iajs-3051	152	6	so	so	ADV
iajs-3051	152	7	h	h	NOUN
iajs-3051	152	8	is	be	AUX
iajs-3051	152	9	a	a	DET
iajs-3051	152	10	nano	nano	NOUN
iajs-3051	152	11	-	-	PUNCT
iajs-3051	152	12	t2	t2	NOUN
iajs-3051	152	13	-	-	PUNCT
iajs-3051	152	14	sp	sp	NOUN
iajs-3051	152	15	.	.	PUNCT
iajs-3051	153	1	(	(	PUNCT
iajs-3051	153	2	c	c	X
iajs-3051	153	3	)	)	PUNCT
iajs-3051	153	4	let	let	VERB
iajs-3051	153	5	g	g	NOUN
iajs-3051	153	6	be	be	AUX
iajs-3051	153	7	a	a	DET
iajs-3051	153	8	nano	nano	NOUN
iajs-3051	153	9	-	-	PUNCT
iajs-3051	153	10	t3	t3	NOUN
iajs-3051	153	11	-	-	PUNCT
iajs-3051	153	12	sp	sp	NOUN
iajs-3051	153	13	.	.	PUNCT
iajs-3051	154	1	let	let	VERB
iajs-3051	154	2	h	h	PRON
iajs-3051	154	3	∈	∈	PROPN
iajs-3051	154	4	h	h	NOUN
iajs-3051	154	5	and	and	CCONJ
iajs-3051	154	6	f	f	PROPN
iajs-3051	154	7	be	be	AUX
iajs-3051	154	8	a	a	DET
iajs-3051	154	9	nano	nano	NOUN
iajs-3051	154	10	-	-	PUNCT
iajs-3051	154	11	clos	clo	NOUN
iajs-3051	154	12	-	-	PUNCT
iajs-3051	154	13	set	set	NOUN
iajs-3051	154	14	in	in	ADP
iajs-3051	154	15	h	h	NOUN
iajs-3051	154	16	such	such	ADJ
iajs-3051	154	17	as	as	ADP
iajs-3051	154	18	h	h	PROPN
iajs-3051	154	19	∈	∈	PROPN
iajs-3051	154	20	h	h	NOUN
iajs-3051	155	1	so	so	ADV
iajs-3051	155	2	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	155	3	)	)	PUNCT
iajs-3051	155	4	is	be	AUX
iajs-3051	155	5	nano	nano	NOUN
iajs-3051	155	6	-	-	PUNCT
iajs-3051	155	7	comp	comp	NOUN
iajs-3051	155	8	-	-	PUNCT
iajs-3051	155	9	sub	sub	NOUN
iajs-3051	155	10	-	-	ADJ
iajs-3051	155	11	set	set	NOUN
iajs-3051	155	12	of	of	ADP
iajs-3051	155	13	g	g	PROPN
iajs-3051	155	14	and	and	CCONJ
iajs-3051	155	15	𝑓−1(f	𝑓−1(f	PROPN
iajs-3051	155	16	)	)	PUNCT
iajs-3051	155	17	is	be	AUX
iajs-3051	155	18	a	a	DET
iajs-3051	155	19	nano	nano	NOUN
iajs-3051	155	20	-	-	PUNCT
iajs-3051	155	21	clos	clo	NOUN
iajs-3051	155	22	-	-	PUNCT
iajs-3051	155	23	sub	sub	NOUN
iajs-3051	155	24	-	-	ADJ
iajs-3051	155	25	set	set	NOUN
iajs-3051	155	26	of	of	ADP
iajs-3051	155	27	g	g	PROPN
iajs-3051	155	28	and	and	CCONJ
iajs-3051	155	29	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	155	30	)	)	PUNCT
iajs-3051	155	31	⋂𝑓−1(f	⋂𝑓−1(f	NUM
iajs-3051	155	32	)	)	PUNCT
iajs-3051	156	1	=	=	NOUN
iajs-3051	156	2	∅.	∅.	NOUN
iajs-3051	156	3	so	so	ADV
iajs-3051	156	4	find	find	VERB
iajs-3051	156	5	u	u	NOUN
iajs-3051	156	6	,	,	PUNCT
iajs-3051	156	7	v	v	ADP
iajs-3051	156	8	nano	nano	NOUN
iajs-3051	156	9	-	-	PUNCT
iajs-3051	156	10	ope	ope	NOUN
iajs-3051	156	11	-	-	PUNCT
iajs-3051	156	12	sub	sub	NOUN
iajs-3051	156	13	-	-	NOUN
iajs-3051	156	14	sets	set	NOUN
iajs-3051	156	15	of	of	ADP
iajs-3051	156	16	g	g	NOUN
iajs-3051	156	17	such	such	ADJ
iajs-3051	156	18	as	as	ADP
iajs-3051	156	19	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	156	20	)	)	PUNCT
iajs-3051	156	21	⊂	⊂	PROPN
iajs-3051	156	22	u	u	PROPN
iajs-3051	156	23	and	and	CCONJ
iajs-3051	156	24	𝑓−1(f	𝑓−1(f	NUM
iajs-3051	156	25	)	)	PUNCT
iajs-3051	157	1	⊂	⊂	PROPN
iajs-3051	157	2	v	v	NOUN
iajs-3051	157	3	with	with	ADP
iajs-3051	157	4	u⋂v	u⋂v	NOUN
iajs-3051	157	5	=	=	PUNCT
iajs-3051	157	6	∅.	∅.	PROPN
iajs-3051	157	7	so	so	ADV
iajs-3051	157	8	f	f	PROPN
iajs-3051	157	9	is	be	AUX
iajs-3051	157	10	nanoclos	nanoclo	NOUN
iajs-3051	157	11	.	.	PUNCT
iajs-3051	157	12	,	,	PUNCT
iajs-3051	157	13	find	find	VERB
iajs-3051	157	14	a	a	DET
iajs-3051	157	15	,	,	PUNCT
iajs-3051	157	16	b	b	NOUN
iajs-3051	157	17	nano	nano	VERB
iajs-3051	157	18	-	-	PUNCT
iajs-3051	157	19	ope	ope	NOUN
iajs-3051	157	20	-	-	PUNCT
iajs-3051	157	21	sub	sub	NOUN
iajs-3051	157	22	-	-	NOUN
iajs-3051	157	23	sets	set	NOUN
iajs-3051	157	24	of	of	ADP
iajs-3051	157	25	h	h	NOUN
iajs-3051	157	26	such	such	ADJ
iajs-3051	157	27	as	as	ADP
iajs-3051	157	28	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	157	29	)	)	PUNCT
iajs-3051	157	30	⊂	⊂	PROPN
iajs-3051	157	31	𝑓−1(a	𝑓−1(a	PROPN
iajs-3051	157	32	)	)	PUNCT
iajs-3051	158	1	⊂	⊂	PROPN
iajs-3051	158	2	u	u	PROPN
iajs-3051	158	3	and	and	CCONJ
iajs-3051	158	4	𝑓−1(f	𝑓−1(f	NUM
iajs-3051	158	5	)	)	PUNCT
iajs-3051	158	6	⊂	⊂	PROPN
iajs-3051	158	7	𝑓−1(b	𝑓−1(b	PROPN
iajs-3051	158	8	)	)	PUNCT
iajs-3051	159	1	⊂	⊂	PROPN
iajs-3051	160	1	v.	v.	CCONJ
iajs-3051	160	2	clearly	clearly	ADV
iajs-3051	160	3	,	,	PUNCT
iajs-3051	160	4	h	h	PROPN
iajs-3051	160	5	∈	∈	PROPN
iajs-3051	160	6	a	a	PRON
iajs-3051	160	7	,	,	PUNCT
iajs-3051	160	8	f	f	PROPN
iajs-3051	160	9	⊂	⊂	PROPN
iajs-3051	160	10	b	b	PROPN
iajs-3051	160	11	and	and	CCONJ
iajs-3051	160	12	a⋂b	a⋂b	PROPN
iajs-3051	160	13	=	=	SYM
iajs-3051	160	14	∅	∅	NOUN
iajs-3051	160	15	so	so	ADV
iajs-3051	160	16	h	h	PROPN
iajs-3051	160	17	is	be	AUX
iajs-3051	160	18	a	a	DET
iajs-3051	160	19	nano	nano	NOUN
iajs-3051	160	20	-	-	PUNCT
iajs-3051	160	21	t3	t3	NOUN
iajs-3051	160	22	-	-	PUNCT
iajs-3051	160	23	sp	sp	NOUN
iajs-3051	160	24	.	.	PUNCT
iajs-3051	161	1	(	(	PUNCT
iajs-3051	161	2	d	d	X
iajs-3051	161	3	)	)	PUNCT
iajs-3051	161	4	let	let	VERB
iajs-3051	161	5	g	g	NOUN
iajs-3051	161	6	be	be	AUX
iajs-3051	161	7	a	a	DET
iajs-3051	161	8	nano	nano	NOUN
iajs-3051	161	9	-	-	PUNCT
iajs-3051	161	10	t4	t4	PROPN
iajs-3051	161	11	-	-	PUNCT
iajs-3051	161	12	sp	sp	PROPN
iajs-3051	161	13	.	.	PUNCT
iajs-3051	162	1	let	let	VERB
iajs-3051	162	2	f1	f1	NOUN
iajs-3051	162	3	,	,	PUNCT
iajs-3051	162	4	f2	f2	PROPN
iajs-3051	162	5	be	be	VERB
iajs-3051	162	6	distinct	distinct	ADJ
iajs-3051	162	7	nano	nano	NOUN
iajs-3051	162	8	-	-	PUNCT
iajs-3051	162	9	clos	clo	NOUN
iajs-3051	162	10	.	.	PUNCT
iajs-3051	163	1	subsets	subset	NOUN
iajs-3051	163	2	of	of	ADP
iajs-3051	163	3	h.	h.	PROPN
iajs-3051	163	4	so	so	ADV
iajs-3051	163	5	𝑓−1(f1	𝑓−1(f1	ADV
iajs-3051	163	6	)	)	PUNCT
iajs-3051	163	7	and	and	CCONJ
iajs-3051	163	8	𝑓−1	𝑓−1	NUM
iajs-3051	163	9	(	(	PUNCT
iajs-3051	163	10	f2	f2	PROPN
iajs-3051	163	11	)	)	PUNCT
iajs-3051	163	12	are	be	AUX
iajs-3051	163	13	nano	nano	NOUN
iajs-3051	163	14	-	-	PUNCT
iajs-3051	163	15	clos	clo	NOUN
iajs-3051	163	16	-	-	PUNCT
iajs-3051	163	17	sub	sub	NOUN
iajs-3051	163	18	-	-	NOUN
iajs-3051	163	19	sets	set	NOUN
iajs-3051	163	20	of	of	ADP
iajs-3051	163	21	g	g	NOUN
iajs-3051	163	22	,	,	PUNCT
iajs-3051	163	23	find	find	VERB
iajs-3051	163	24	u	u	NOUN
iajs-3051	163	25	,	,	PUNCT
iajs-3051	163	26	v	v	ADP
iajs-3051	163	27	nano	nano	NOUN
iajs-3051	163	28	-	-	PUNCT
iajs-3051	163	29	ope	ope	NOUN
iajs-3051	163	30	-	-	PUNCT
iajs-3051	163	31	sub	sub	NOUN
iajs-3051	163	32	-	-	NOUN
iajs-3051	163	33	sets	set	NOUN
iajs-3051	163	34	of	of	ADP
iajs-3051	163	35	g	g	PROPN
iajs-3051	163	36	such	such	ADJ
iajs-3051	163	37	as𝑓−1(f1	as𝑓−1(f1	NOUN
iajs-3051	163	38	)	)	PUNCT
iajs-3051	164	1	⊂	⊂	PROPN
iajs-3051	164	2	u	u	NOUN
iajs-3051	164	3	,	,	PUNCT
iajs-3051	164	4	𝑓−1(f2	𝑓−1(f2	NOUN
iajs-3051	164	5	)	)	PUNCT
iajs-3051	165	1	⊂	⊂	PROPN
iajs-3051	165	2	v.	v.	CCONJ
iajs-3051	165	3	since	since	SCONJ
iajs-3051	165	4	f	f	PROPN
iajs-3051	165	5	is	be	AUX
iajs-3051	165	6	nano	nano	NOUN
iajs-3051	165	7	-	-	PUNCT
iajs-3051	165	8	clos	clo	NOUN
iajs-3051	165	9	.	.	PUNCT
iajs-3051	165	10	,	,	PUNCT
iajs-3051	165	11	find	find	VERB
iajs-3051	165	12	nano	nano	NOUN
iajs-3051	165	13	-	-	PUNCT
iajs-3051	165	14	ope	ope	NOUN
iajs-3051	165	15	-	-	PUNCT
iajs-3051	165	16	sub	sub	NOUN
iajs-3051	165	17	-	-	NOUN
iajs-3051	165	18	sets	set	NOUN
iajs-3051	165	19	a	a	DET
iajs-3051	165	20	,	,	PUNCT
iajs-3051	165	21	b	b	PROPN
iajs-3051	165	22	of	of	ADP
iajs-3051	165	23	h	h	PROPN
iajs-3051	165	24	such	such	ADJ
iajs-3051	165	25	as𝑓−1(f1	as𝑓−1(f1	NOUN
iajs-3051	165	26	)	)	PUNCT
iajs-3051	166	1	⊂	⊂	PROPN
iajs-3051	166	2	𝑓−1(a	𝑓−1(a	PROPN
iajs-3051	166	3	)	)	PUNCT
iajs-3051	167	1	⊂	⊂	PROPN
iajs-3051	167	2	u	u	NOUN
iajs-3051	167	3	and	and	CCONJ
iajs-3051	167	4	𝑓−1(f2	𝑓−1(f2	NOUN
iajs-3051	167	5	)	)	PUNCT
iajs-3051	167	6	⊂	⊂	PROPN
iajs-3051	167	7	𝑓−1(b	𝑓−1(b	PROPN
iajs-3051	167	8	)	)	PUNCT
iajs-3051	168	1	⊂	⊂	PROPN
iajs-3051	168	2	v.	v.	CCONJ
iajs-3051	168	3	clearly	clearly	ADV
iajs-3051	168	4	,	,	PUNCT
iajs-3051	168	5	f1	f1	PROPN
iajs-3051	168	6	⊂	⊂	PROPN
iajs-3051	168	7	a	a	PROPN
iajs-3051	168	8	,	,	PUNCT
iajs-3051	168	9	f2	f2	PROPN
iajs-3051	168	10	⊂	⊂	PROPN
iajs-3051	168	11	b	b	PROPN
iajs-3051	168	12	and	and	CCONJ
iajs-3051	168	13	a	a	DET
iajs-3051	168	14	,	,	PUNCT
iajs-3051	168	15	b	b	NOUN
iajs-3051	168	16	are	be	AUX
iajs-3051	168	17	disjoint	disjoint	ADJ
iajs-3051	168	18	,	,	PUNCT
iajs-3051	168	19	hence	hence	ADV
iajs-3051	168	20	h	h	NOUN
iajs-3051	168	21	is	be	AUX
iajs-3051	168	22	a	a	DET
iajs-3051	168	23	nano	nano	NOUN
iajs-3051	168	24	-	-	PUNCT
iajs-3051	168	25	t4	t4	PROPN
iajs-3051	168	26	-	-	PUNCT
iajs-3051	168	27	sp	sp	PROPN
iajs-3051	168	28	.	.	PROPN
iajs-3051	168	29	ihjpas	ihjpas	PROPN
iajs-3051	168	30	.	.	PUNCT
iajs-3051	169	1	36	36	NUM
iajs-3051	169	2	(	(	PUNCT
iajs-3051	169	3	3	3	NUM
iajs-3051	169	4	)	)	PUNCT
iajs-3051	169	5	2023	2023	NUM
iajs-3051	169	6	402	402	NUM
iajs-3051	169	7	theorem	theorem	VERB
iajs-3051	169	8	2.6	2.6	NUM
iajs-3051	169	9	:	:	PUNCT
iajs-3051	169	10	nano	nano	NOUN
iajs-3051	169	11	-	-	PUNCT
iajs-3051	169	12	compactness	compactness	NOUN
iajs-3051	169	13	is	be	AUX
iajs-3051	169	14	invariant	invariant	ADJ
iajs-3051	169	15	under	under	ADP
iajs-3051	169	16	nano	nano	NOUN
iajs-3051	169	17	-	-	PUNCT
iajs-3051	169	18	perf	perf	NOUN
iajs-3051	169	19	-	-	PUNCT
iajs-3051	169	20	maps	map	NOUN
iajs-3051	169	21	.	.	PUNCT
iajs-3051	170	1	proof	proof	NOUN
iajs-3051	170	2	:	:	PUNCT
iajs-3051	170	3	let	let	VERB
iajs-3051	170	4	f	f	PRON
iajs-3051	170	5	:	:	PUNCT
iajs-3051	170	6	g	g	PROPN
iajs-3051	170	7	⟶	⟶	PROPN
iajs-3051	170	8	h	h	NOUN
iajs-3051	170	9	be	be	AUX
iajs-3051	170	10	a	a	DET
iajs-3051	170	11	nano	nano	NOUN
iajs-3051	170	12	-	-	PUNCT
iajs-3051	170	13	perf	perf	NOUN
iajs-3051	170	14	-	-	PUNCT
iajs-3051	170	15	map	map	NOUN
iajs-3051	170	16	from	from	ADP
iajs-3051	170	17	a	a	DET
iajs-3051	170	18	nano	nano	NOUN
iajs-3051	170	19	-	-	PUNCT
iajs-3051	170	20	comp	comp	NOUN
iajs-3051	170	21	-	-	PUNCT
iajs-3051	170	22	sp	sp	NOUN
iajs-3051	170	23	.	.	NOUN
iajs-3051	171	1	g	g	NOUN
iajs-3051	171	2	onto	onto	ADP
iajs-3051	171	3	a	a	DET
iajs-3051	171	4	space	space	NOUN
iajs-3051	171	5	h.	h.	NOUN
iajs-3051	171	6	let	let	VERB
iajs-3051	171	7	u	u	PRON
iajs-3051	171	8	∽	∽	NOUN
iajs-3051	171	9	=	=	SYM
iajs-3051	171	10	{	{	PUNCT
iajs-3051	171	11	uα	uα	X
iajs-3051	171	12	:	:	PUNCT
iajs-3051	171	13	α	α	PROPN
iajs-3051	171	14	∈	∈	PROPN
iajs-3051	171	15	λ	λ	PROPN
iajs-3051	171	16	}	}	PUNCT
iajs-3051	171	17	be	be	AUX
iajs-3051	171	18	any	any	DET
iajs-3051	171	19	nano	nano	NOUN
iajs-3051	171	20	-	-	PUNCT
iajs-3051	171	21	ope	ope	NOUN
iajs-3051	171	22	-	-	PUNCT
iajs-3051	171	23	cover	cover	NOUN
iajs-3051	171	24	of	of	ADP
iajs-3051	171	25	h.	h.	PROPN
iajs-3051	171	26	then	then	ADV
iajs-3051	171	27	𝑓−1	𝑓−1	NUM
iajs-3051	171	28	(	(	PUNCT
iajs-3051	171	29	u	u	NOUN
iajs-3051	171	30	∽	∽	PROPN
iajs-3051	171	31	)	)	PUNCT
iajs-3051	171	32	is	be	AUX
iajs-3051	171	33	an	an	DET
iajs-3051	171	34	nano	nano	NOUN
iajs-3051	171	35	-	-	PUNCT
iajs-3051	171	36	ope	ope	NOUN
iajs-3051	171	37	-	-	PUNCT
iajs-3051	171	38	cover	cover	NOUN
iajs-3051	171	39	of	of	ADP
iajs-3051	171	40	g.	g.	PROPN
iajs-3051	171	41	so	so	ADV
iajs-3051	171	42	find	find	VERB
iajs-3051	171	43	a	a	DET
iajs-3051	171	44	finitesub	finitesub	NOUN
iajs-3051	171	45	-	-	ADJ
iajs-3051	171	46	cover	cover	NOUN
iajs-3051	171	47	{	{	PUNCT
iajs-3051	171	48	𝑓−1(uαi	𝑓−1(uαi	NOUN
iajs-3051	171	49	)	)	PUNCT
iajs-3051	171	50	}	}	PUNCT
iajs-3051	171	51	i=1	i=1	PROPN
iajs-3051	171	52	n	n	PROPN
iajs-3051	171	53	for	for	ADP
iajs-3051	171	54	g.	g.	PROPN
iajs-3051	171	55	then	then	ADV
iajs-3051	171	56	{	{	PUNCT
iajs-3051	171	57	uαi	uαi	PROPN
iajs-3051	171	58	}	}	PUNCT
iajs-3051	171	59	i=1	i=1	PROPN
iajs-3051	171	60	n	n	PRON
iajs-3051	171	61	is	be	AUX
iajs-3051	171	62	a	a	DET
iajs-3051	171	63	finite	finite	ADJ
iajs-3051	171	64	-	-	PUNCT
iajs-3051	171	65	sub	sub	NOUN
iajs-3051	171	66	-	-	NOUN
iajs-3051	171	67	cover	cover	NOUN
iajs-3051	171	68	of	of	ADP
iajs-3051	171	69	h	h	NOUN
iajs-3051	171	70	,	,	PUNCT
iajs-3051	171	71	hence	hence	ADV
iajs-3051	171	72	h	h	NOUN
iajs-3051	171	73	is	be	AUX
iajs-3051	171	74	nano	nano	NOUN
iajs-3051	171	75	-	-	PUNCT
iajs-3051	171	76	comp	comp	NOUN
iajs-3051	171	77	.	.	PUNCT
iajs-3051	172	1	lemma	lemma	PROPN
iajs-3051	172	2	2.7	2.7	NUM
iajs-3051	172	3	:	:	PUNCT
iajs-3051	172	4	for	for	ADP
iajs-3051	172	5	each	each	DET
iajs-3051	172	6	nano	nano	NOUN
iajs-3051	172	7	-	-	PUNCT
iajs-3051	172	8	comp	comp	NOUN
iajs-3051	172	9	-	-	PUNCT
iajs-3051	172	10	sub	sub	NOUN
iajs-3051	172	11	-	-	NOUN
iajs-3051	172	12	sp	sp	NOUN
iajs-3051	172	13	.	.	PUNCT
iajs-3051	173	1	a	a	PRON
iajs-3051	173	2	of	of	ADP
iajs-3051	173	3	a	a	DET
iajs-3051	173	4	locally	locally	ADV
iajs-3051	173	5	nano	nano	NOUN
iajs-3051	173	6	-	-	PUNCT
iajs-3051	173	7	comp	comp	NOUN
iajs-3051	173	8	-	-	PUNCT
iajs-3051	173	9	sp	sp	NOUN
iajs-3051	173	10	.	.	NOUN
iajs-3051	173	11	g	g	PROPN
iajs-3051	173	12	and	and	CCONJ
iajs-3051	173	13	each	each	DET
iajs-3051	173	14	nano	nano	NOUN
iajs-3051	173	15	-	-	PUNCT
iajs-3051	173	16	ope	ope	NOUN
iajs-3051	173	17	-	-	PUNCT
iajs-3051	173	18	set	set	VERB
iajs-3051	173	19	v	v	NOUN
iajs-3051	173	20	that	that	PRON
iajs-3051	173	21	contains	contain	VERB
iajs-3051	173	22	a	a	DET
iajs-3051	173	23	find	find	NOUN
iajs-3051	173	24	an	an	DET
iajs-3051	173	25	nano	nano	NOUN
iajs-3051	173	26	-	-	PUNCT
iajs-3051	173	27	ope	ope	NOUN
iajs-3051	173	28	-	-	PUNCT
iajs-3051	173	29	set	set	VERB
iajs-3051	173	30	u	u	NOUN
iajs-3051	173	31	such	such	ADJ
iajs-3051	173	32	as	as	ADP
iajs-3051	173	33	a	a	DET
iajs-3051	173	34	⊂	⊂	X
iajs-3051	173	35	u	u	X
iajs-3051	173	36	⊂	⊂	PROPN
iajs-3051	173	37	ncl(u	ncl(u	PROPN
iajs-3051	173	38	)	)	PUNCT
iajs-3051	173	39	⊂	⊂	PROPN
iajs-3051	173	40	v	v	NOUN
iajs-3051	173	41	and	and	CCONJ
iajs-3051	173	42	ncl(u	ncl(u	PROPN
iajs-3051	173	43	)	)	PUNCT
iajs-3051	173	44	is	be	AUX
iajs-3051	173	45	nano	nano	NOUN
iajs-3051	173	46	-	-	PUNCT
iajs-3051	173	47	comp	comp	NOUN
iajs-3051	173	48	.	.	PUNCT
iajs-3051	174	1	proof	proof	NOUN
iajs-3051	174	2	:	:	PUNCT
iajs-3051	174	3	for	for	SCONJ
iajs-3051	174	4	each	each	DET
iajs-3051	174	5	g	g	PROPN
iajs-3051	174	6	∈	∈	PROPN
iajs-3051	174	7	a	a	DET
iajs-3051	174	8	take	take	VERB
iajs-3051	174	9	a	a	DET
iajs-3051	174	10	nano	nano	NOUN
iajs-3051	174	11	-	-	PUNCT
iajs-3051	174	12	nbd	nbd	PROPN
iajs-3051	174	13	vg	vg	NOUN
iajs-3051	174	14	of	of	ADP
iajs-3051	174	15	the	the	DET
iajs-3051	174	16	point	point	NOUN
iajs-3051	174	17	g	g	ADP
iajs-3051	174	18	such	such	ADJ
iajs-3051	174	19	as	as	ADP
iajs-3051	174	20	ncl(𝑉g	ncl(𝑉g	PROPN
iajs-3051	174	21	)	)	PUNCT
iajs-3051	174	22	⊂	⊂	PROPN
iajs-3051	174	23	v	v	NOUN
iajs-3051	174	24	and	and	CCONJ
iajs-3051	174	25	a	a	DET
iajs-3051	174	26	nano	nano	NOUN
iajs-3051	174	27	-	-	PUNCT
iajs-3051	174	28	nbd	nbd	PROPN
iajs-3051	174	29	wg	wg	PROPN
iajs-3051	174	30	of	of	ADP
iajs-3051	174	31	g	g	NOUN
iajs-3051	174	32	such	such	ADJ
iajs-3051	174	33	as	as	ADP
iajs-3051	174	34	ncl(𝑊g	ncl(𝑊g	PROPN
iajs-3051	174	35	)	)	PUNCT
iajs-3051	174	36	is	be	AUX
iajs-3051	174	37	nano	nano	NOUN
iajs-3051	174	38	-	-	NOUN
iajs-3051	174	39	comp	comp	NOUN
iajs-3051	174	40	.	.	PUNCT
iajs-3051	175	1	the	the	DET
iajs-3051	175	2	set	set	NOUN
iajs-3051	175	3	ncl(ug	ncl(ug	NOUN
iajs-3051	175	4	)	)	PUNCT
iajs-3051	175	5	,	,	PUNCT
iajs-3051	175	6	where	where	SCONJ
iajs-3051	175	7	ug=	ug=	PROPN
iajs-3051	175	8	vg⋂wg	vg⋂wg	PROPN
iajs-3051	175	9	is	be	AUX
iajs-3051	175	10	nano	nano	NOUN
iajs-3051	175	11	-	-	NOUN
iajs-3051	175	12	comp	comp	NOUN
iajs-3051	175	13	.	.	PUNCT
iajs-3051	176	1	because	because	SCONJ
iajs-3051	176	2	it	it	PRON
iajs-3051	176	3	is	be	AUX
iajs-3051	176	4	a	a	DET
iajs-3051	176	5	nano	nano	NOUN
iajs-3051	176	6	-	-	PUNCT
iajs-3051	176	7	clos	clo	NOUN
iajs-3051	176	8	-	-	PUNCT
iajs-3051	176	9	sub	sub	NOUN
iajs-3051	176	10	-	-	ADJ
iajs-3051	176	11	set	set	NOUN
iajs-3051	176	12	of	of	ADP
iajs-3051	176	13	the	the	DET
iajs-3051	176	14	nano	nano	NOUN
iajs-3051	176	15	-	-	PUNCT
iajs-3051	176	16	comp	comp	NOUN
iajs-3051	176	17	-	-	PUNCT
iajs-3051	176	18	sp	sp	NOUN
iajs-3051	176	19	.	.	PUNCT
iajs-3051	176	20	ncl(wg	ncl(wg	PROPN
iajs-3051	176	21	)	)	PUNCT
iajs-3051	176	22	.	.	PUNCT
iajs-3051	177	1	so	so	ADV
iajs-3051	177	2	find	find	VERB
iajs-3051	177	3	a	a	DET
iajs-3051	177	4	finite	finite	ADJ
iajs-3051	177	5	set	set	NOUN
iajs-3051	177	6	{	{	PUNCT
iajs-3051	177	7	g1	g1	PROPN
iajs-3051	177	8	,	,	PUNCT
iajs-3051	177	9	g2	g2	PROPN
iajs-3051	177	10	,	,	PUNCT
iajs-3051	177	11	…	…	PUNCT
iajs-3051	177	12	,	,	PUNCT
iajs-3051	177	13	gk	gk	PROPN
iajs-3051	177	14	}	}	PUNCT
iajs-3051	177	15	⊂	⊂	PROPN
iajs-3051	177	16	a	a	PRON
iajs-3051	177	17	such	such	ADJ
iajs-3051	177	18	as	as	ADP
iajs-3051	177	19	a	a	DET
iajs-3051	177	20	⊂	⊂	X
iajs-3051	177	21	u	u	NOUN
iajs-3051	177	22	=	=	NOUN
iajs-3051	177	23	ug1	ug1	PROPN
iajs-3051	177	24	⋃	⋃	PROPN
iajs-3051	177	25	ug2	ug2	NOUN
iajs-3051	177	26	⋃	⋃	NOUN
iajs-3051	177	27	…	…	PUNCT
iajs-3051	178	1	⋃	⋃	PROPN
iajs-3051	178	2	ugk	ugk	NOUN
iajs-3051	178	3	.	.	PUNCT
iajs-3051	179	1	the	the	DET
iajs-3051	179	2	set	set	NOUN
iajs-3051	179	3	ncl(u	ncl(u	PROPN
iajs-3051	179	4	)	)	PUNCT
iajs-3051	179	5	=	=	SYM
iajs-3051	179	6	ncl(ug1	ncl(ug1	PROPN
iajs-3051	179	7	)	)	PUNCT
iajs-3051	180	1	⋃	⋃	VERB
iajs-3051	180	2	ncl(𝑈𝑔2	ncl(𝑈𝑔2	NOUN
iajs-3051	180	3	)	)	PUNCT
iajs-3051	180	4	⋃	⋃	ADP
iajs-3051	180	5	…	…	PUNCT
iajs-3051	180	6	⋃	⋃	NOUN
iajs-3051	180	7	ncl(ugk	ncl(ugk	NOUN
iajs-3051	180	8	)	)	PUNCT
iajs-3051	180	9	is	be	AUX
iajs-3051	180	10	nano	nano	NOUN
iajs-3051	180	11	-	-	NOUN
iajs-3051	180	12	comp	comp	NOUN
iajs-3051	180	13	.	.	PUNCT
iajs-3051	181	1	and	and	CCONJ
iajs-3051	181	2	clearly	clearly	ADV
iajs-3051	181	3	we	we	PRON
iajs-3051	181	4	have	have	VERB
iajs-3051	181	5	ncl(u	ncl(u	PROPN
iajs-3051	181	6	)	)	PUNCT
iajs-3051	181	7	⊂	⊂	PROPN
iajs-3051	181	8	ncl(vg1	ncl(vg1	PROPN
iajs-3051	181	9	)	)	PUNCT
iajs-3051	182	1	⋃	⋃	ADP
iajs-3051	182	2	ncl(vg2	ncl(vg2	NUM
iajs-3051	182	3	)	)	PUNCT
iajs-3051	182	4	⋃	⋃	ADP
iajs-3051	182	5	…	…	PUNCT
iajs-3051	182	6	⋃	⋃	NOUN
iajs-3051	182	7	ncl(vgk	ncl(vgk	NOUN
iajs-3051	182	8	)	)	PUNCT
iajs-3051	183	1	⊂	⊂	PROPN
iajs-3051	183	2	v.	v.	CCONJ
iajs-3051	183	3	theorem	theorem	VERB
iajs-3051	183	4	2.8	2.8	NUM
iajs-3051	183	5	:	:	PUNCT
iajs-3051	183	6	locally	locally	ADV
iajs-3051	183	7	nano	nano	NOUN
iajs-3051	183	8	-	-	PUNCT
iajs-3051	183	9	compactness	compactness	NOUN
iajs-3051	183	10	is	be	AUX
iajs-3051	183	11	invariant	invariant	ADJ
iajs-3051	183	12	under	under	ADP
iajs-3051	183	13	nano	nano	NOUN
iajs-3051	183	14	-	-	PUNCT
iajs-3051	183	15	perf	perf	NOUN
iajs-3051	183	16	-	-	PUNCT
iajs-3051	183	17	maps	map	NOUN
iajs-3051	183	18	.	.	PUNCT
iajs-3051	184	1	proof	proof	NOUN
iajs-3051	184	2	:	:	PUNCT
iajs-3051	184	3	let	let	VERB
iajs-3051	184	4	f	f	PRON
iajs-3051	184	5	:	:	PUNCT
iajs-3051	184	6	g	g	PROPN
iajs-3051	184	7	⟶	⟶	PROPN
iajs-3051	184	8	h	h	NOUN
iajs-3051	184	9	be	be	AUX
iajs-3051	184	10	a	a	DET
iajs-3051	184	11	nano	nano	NOUN
iajs-3051	184	12	-	-	PUNCT
iajs-3051	184	13	perf	perf	NOUN
iajs-3051	184	14	-	-	PUNCT
iajs-3051	184	15	map	map	NOUN
iajs-3051	184	16	of	of	ADP
iajs-3051	184	17	a	a	DET
iajs-3051	184	18	locally	locally	ADV
iajs-3051	184	19	nano	nano	NOUN
iajs-3051	184	20	-	-	PUNCT
iajs-3051	184	21	comp	comp	NOUN
iajs-3051	184	22	-	-	PUNCT
iajs-3051	184	23	sp	sp	NOUN
iajs-3051	184	24	.	.	NOUN
iajs-3051	185	1	g	g	NOUN
iajs-3051	185	2	onto	onto	ADP
iajs-3051	185	3	a	a	DET
iajs-3051	185	4	space	space	NOUN
iajs-3051	185	5	h.	h.	NOUN
iajs-3051	185	6	by	by	ADP
iajs-3051	185	7	lemma	lemma	PROPN
iajs-3051	185	8	2.7	2.7	NUM
iajs-3051	185	9	,	,	PUNCT
iajs-3051	185	10	for	for	ADP
iajs-3051	185	11	each	each	DET
iajs-3051	185	12	h	h	NOUN
iajs-3051	185	13	∈	∈	PROPN
iajs-3051	185	14	h	h	NOUN
iajs-3051	185	15	find	find	VERB
iajs-3051	185	16	an	an	DET
iajs-3051	185	17	nano	nano	NOUN
iajs-3051	185	18	-	-	PUNCT
iajs-3051	185	19	ope	ope	NOUN
iajs-3051	185	20	-	-	PUNCT
iajs-3051	185	21	set	set	NOUN
iajs-3051	185	22	v	v	ADP
iajs-3051	185	23	⊂	⊂	PROPN
iajs-3051	185	24	g	g	PROPN
iajs-3051	185	25	such	such	ADJ
iajs-3051	185	26	as	as	ADP
iajs-3051	185	27	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	185	28	)	)	PUNCT
iajs-3051	186	1	⊂	⊂	PROPN
iajs-3051	186	2	v	v	PROPN
iajs-3051	186	3	and	and	CCONJ
iajs-3051	186	4	ncl(v	ncl(v	PROPN
iajs-3051	186	5	)	)	PUNCT
iajs-3051	186	6	is	be	AUX
iajs-3051	186	7	nano	nano	NOUN
iajs-3051	186	8	-	-	NOUN
iajs-3051	186	9	comp	comp	NOUN
iajs-3051	186	10	.	.	PUNCT
iajs-3051	187	1	the	the	DET
iajs-3051	187	2	set	set	NOUN
iajs-3051	187	3	w	w	NOUN
iajs-3051	187	4	=	=	SYM
iajs-3051	187	5	h\f(g\v	h\f(g\v	NOUN
iajs-3051	187	6	)	)	PUNCT
iajs-3051	187	7	is	be	AUX
iajs-3051	187	8	a	a	DET
iajs-3051	187	9	nbd	nbd	PROPN
iajs-3051	187	10	of	of	ADP
iajs-3051	187	11	h	h	PROPN
iajs-3051	187	12	and	and	CCONJ
iajs-3051	187	13	since	since	SCONJ
iajs-3051	187	14	w	w	NOUN
iajs-3051	187	15	=	=	SYM
iajs-3051	187	16	h\f(g\v	h\f(g\v	NOUN
iajs-3051	187	17	)	)	PUNCT
iajs-3051	187	18	⊂	⊂	PROPN
iajs-3051	187	19	h\(g\ncl(v	h\(g\ncl(v	NOUN
iajs-3051	187	20	)	)	PUNCT
iajs-3051	187	21	)	)	PUNCT
iajs-3051	188	1	⊂	⊂	PROPN
iajs-3051	188	2	f(ncl(v	f(ncl(v	NOUN
iajs-3051	188	3	)	)	PUNCT
iajs-3051	188	4	)	)	PUNCT
iajs-3051	188	5	,	,	PUNCT
iajs-3051	188	6	the	the	DET
iajs-3051	188	7	nano	nano	NOUN
iajs-3051	188	8	-	-	PUNCT
iajs-3051	188	9	closure	closure	NOUN
iajs-3051	188	10	ncl(w	ncl(w	NOUN
iajs-3051	188	11	)	)	PUNCT
iajs-3051	188	12	is	be	AUX
iajs-3051	188	13	a	a	DET
iajs-3051	188	14	nano-comp-sub-sp.of	nano-comp-sub-sp.of	PROPN
iajs-3051	188	15	h.	h.	PROPN
iajs-3051	188	16	theorem	theorem	VERB
iajs-3051	188	17	2.9	2.9	NUM
iajs-3051	188	18	:	:	PUNCT
iajs-3051	188	19	a	a	DET
iajs-3051	188	20	nano	nano	NOUN
iajs-3051	188	21	-	-	PUNCT
iajs-3051	188	22	cont	cont	NOUN
iajs-3051	188	23	.	.	PUNCT
iajs-3051	189	1	map	map	NOUN
iajs-3051	190	1	f	f	X
iajs-3051	190	2	:	:	PUNCT
iajs-3051	190	3	g	g	PROPN
iajs-3051	190	4	⟶	⟶	NOUN
iajs-3051	190	5	h	h	NOUN
iajs-3051	190	6	is	be	AUX
iajs-3051	190	7	nano	nano	NOUN
iajs-3051	190	8	-	-	PUNCT
iajs-3051	190	9	clos	clo	NOUN
iajs-3051	190	10	.	.	PUNCT
iajs-3051	191	1	if	if	SCONJ
iajs-3051	191	2	and	and	CCONJ
iajs-3051	191	3	only	only	ADV
iajs-3051	191	4	if	if	SCONJ
iajs-3051	191	5	for	for	ADP
iajs-3051	191	6	each	each	DET
iajs-3051	191	7	point	point	NOUN
iajs-3051	191	8	h	h	NOUN
iajs-3051	192	1	∈	∈	PROPN
iajs-3051	193	1	h	h	NOUN
iajs-3051	193	2	and	and	CCONJ
iajs-3051	193	3	each	each	DET
iajs-3051	193	4	nano	nano	NOUN
iajs-3051	193	5	-	-	PUNCT
iajs-3051	193	6	ope	ope	NOUN
iajs-3051	193	7	-	-	PUNCT
iajs-3051	193	8	set	set	VERB
iajs-3051	193	9	u	u	NOUN
iajs-3051	193	10	⊂	⊂	PROPN
iajs-3051	193	11	g	g	PROPN
iajs-3051	193	12	such	such	ADJ
iajs-3051	193	13	as	as	ADP
iajs-3051	193	14	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	193	15	)	)	PUNCT
iajs-3051	194	1	⊂	⊂	PROPN
iajs-3051	194	2	u	u	NOUN
iajs-3051	194	3	,	,	PUNCT
iajs-3051	194	4	then	then	ADV
iajs-3051	194	5	find	find	VERB
iajs-3051	194	6	an	an	DET
iajs-3051	194	7	nano	nano	NOUN
iajs-3051	194	8	-	-	PUNCT
iajs-3051	194	9	ope	ope	NOUN
iajs-3051	194	10	-	-	PUNCT
iajs-3051	194	11	set	set	VERB
iajs-3051	194	12	v	v	NOUN
iajs-3051	194	13	of	of	ADP
iajs-3051	194	14	h	h	NOUN
iajs-3051	194	15	such	such	ADJ
iajs-3051	194	16	as	as	ADP
iajs-3051	194	17	h	h	PROPN
iajs-3051	194	18	∈	∈	PROPN
iajs-3051	194	19	v	v	NOUN
iajs-3051	194	20	and	and	CCONJ
iajs-3051	194	21	𝑓−1(v	𝑓−1(v	NOUN
iajs-3051	194	22	)	)	PUNCT
iajs-3051	195	1	⊂	⊂	PROPN
iajs-3051	195	2	u.	u.	PROPN
iajs-3051	195	3	proof	proof	NOUN
iajs-3051	195	4	:	:	PUNCT
iajs-3051	195	5	(	(	PUNCT
iajs-3051	195	6	⇒	⇒	NOUN
iajs-3051	195	7	)	)	PUNCT
iajs-3051	195	8	let	let	VERB
iajs-3051	195	9	f	f	PRON
iajs-3051	195	10	be	be	AUX
iajs-3051	195	11	a	a	DET
iajs-3051	195	12	nano	nano	NOUN
iajs-3051	195	13	-	-	PUNCT
iajs-3051	195	14	cont	cont	NOUN
iajs-3051	195	15	.	.	PUNCT
iajs-3051	196	1	map	map	NOUN
iajs-3051	197	1	f	f	NOUN
iajs-3051	197	2	:	:	PUNCT
iajs-3051	197	3	g⟶	g⟶	NOUN
iajs-3051	197	4	h	h	NOUN
iajs-3051	197	5	is	be	AUX
iajs-3051	197	6	nano	nano	NOUN
iajs-3051	197	7	-	-	PUNCT
iajs-3051	197	8	clos	clo	NOUN
iajs-3051	197	9	.	.	PUNCT
iajs-3051	198	1	and	and	CCONJ
iajs-3051	198	2	g	g	PROPN
iajs-3051	198	3	∈	∈	PROPN
iajs-3051	198	4	h.	h.	PROPN
iajs-3051	198	5	let	let	VERB
iajs-3051	198	6	u	u	PRON
iajs-3051	198	7	be	be	AUX
iajs-3051	198	8	an	an	DET
iajs-3051	198	9	nano	nano	NOUN
iajs-3051	198	10	-	-	PUNCT
iajs-3051	198	11	opeset	opeset	NOUN
iajs-3051	198	12	in	in	ADP
iajs-3051	198	13	g	g	PROPN
iajs-3051	198	14	,	,	PUNCT
iajs-3051	198	15	such	such	ADJ
iajs-3051	198	16	as	as	ADP
iajs-3051	198	17	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	198	18	)	)	PUNCT
iajs-3051	199	1	⊂	⊂	PROPN
iajs-3051	199	2	u	u	PROPN
iajs-3051	199	3	,	,	PUNCT
iajs-3051	199	4	so	so	SCONJ
iajs-3051	199	5	g	g	PROPN
iajs-3051	199	6	\	\	NOUN
iajs-3051	199	7	u	u	NOUN
iajs-3051	199	8	is	be	AUX
iajs-3051	199	9	nano	nano	NOUN
iajs-3051	199	10	-	-	PUNCT
iajs-3051	199	11	clos	clo	NOUN
iajs-3051	199	12	.	.	PUNCT
iajs-3051	200	1	in	in	ADP
iajs-3051	200	2	g.	g.	PROPN
iajs-3051	200	3	this	this	PRON
iajs-3051	200	4	implies	imply	VERB
iajs-3051	200	5	f(g	f(g	PROPN
iajs-3051	200	6	\	\	NOUN
iajs-3051	200	7	u	u	NOUN
iajs-3051	200	8	)	)	PUNCT
iajs-3051	200	9	is	be	AUX
iajs-3051	200	10	nano	nano	NOUN
iajs-3051	200	11	-	-	PUNCT
iajs-3051	200	12	clos	clo	NOUN
iajs-3051	200	13	.	.	PUNCT
iajs-3051	201	1	in	in	ADP
iajs-3051	201	2	h.	h.	PROPN
iajs-3051	201	3	let	let	VERB
iajs-3051	201	4	v	v	NOUN
iajs-3051	201	5	=	=	SYM
iajs-3051	201	6	h	h	NOUN
iajs-3051	201	7	\	\	PROPN
iajs-3051	201	8	f(g	f(g	PROPN
iajs-3051	201	9	\	\	NOUN
iajs-3051	201	10	u	u	NOUN
iajs-3051	201	11	)	)	PUNCT
iajs-3051	201	12	,	,	PUNCT
iajs-3051	201	13	then	then	ADV
iajs-3051	201	14	v	v	NOUN
iajs-3051	201	15	is	be	AUX
iajs-3051	201	16	an	an	DET
iajs-3051	201	17	nano	nano	NOUN
iajs-3051	201	18	-	-	PUNCT
iajs-3051	201	19	ope	ope	NOUN
iajs-3051	201	20	-	-	PUNCT
iajs-3051	201	21	sub	sub	NOUN
iajs-3051	201	22	-	-	ADJ
iajs-3051	201	23	set	set	NOUN
iajs-3051	201	24	of	of	ADP
iajs-3051	201	25	h	h	NOUN
iajs-3051	201	26	such	such	ADJ
iajs-3051	201	27	as	as	ADP
iajs-3051	201	28	h	h	PROPN
iajs-3051	201	29	∈	∈	PROPN
iajs-3051	201	30	v	v	NOUN
iajs-3051	201	31	and	and	CCONJ
iajs-3051	201	32	𝑓−1(v	𝑓−1(v	NOUN
iajs-3051	201	33	)	)	PUNCT
iajs-3051	202	1	=	=	SYM
iajs-3051	202	2	g	g	ADP
iajs-3051	202	3	\	\	NOUN
iajs-3051	202	4	𝑓−1(f(g	𝑓−1(f(g	ADJ
iajs-3051	202	5	\	\	NOUN
iajs-3051	202	6	u	u	NOUN
iajs-3051	202	7	)	)	PUNCT
iajs-3051	202	8	)	)	PUNCT
iajs-3051	203	1	⊂	⊂	PROPN
iajs-3051	203	2	u.	u.	PROPN
iajs-3051	203	3	(	(	PUNCT
iajs-3051	203	4	⇐	⇐	ADJ
iajs-3051	203	5	)	)	PUNCT
iajs-3051	203	6	assume	assume	VERB
iajs-3051	203	7	that	that	SCONJ
iajs-3051	203	8	the	the	DET
iajs-3051	203	9	assumption	assumption	NOUN
iajs-3051	203	10	holds	hold	VERB
iajs-3051	203	11	.	.	PUNCT
iajs-3051	204	1	let	let	VERB
iajs-3051	204	2	f	f	PRON
iajs-3051	204	3	be	be	AUX
iajs-3051	204	4	a	a	DET
iajs-3051	204	5	nano	nano	NOUN
iajs-3051	204	6	-	-	PUNCT
iajs-3051	204	7	clos	clo	NOUN
iajs-3051	204	8	-	-	PUNCT
iajs-3051	204	9	sub	sub	NOUN
iajs-3051	204	10	-	-	ADJ
iajs-3051	204	11	set	set	NOUN
iajs-3051	204	12	of	of	ADP
iajs-3051	204	13	g	g	NOUN
iajs-3051	204	14	and	and	CCONJ
iajs-3051	204	15	h	h	NOUN
iajs-3051	204	16	∈	∈	PROPN
iajs-3051	204	17	h	h	NOUN
iajs-3051	204	18	\	\	PROPN
iajs-3051	204	19	f(f	f(f	PROPN
iajs-3051	204	20	)	)	PUNCT
iajs-3051	204	21	and	and	CCONJ
iajs-3051	204	22	each	each	DET
iajs-3051	204	23	nano	nano	NOUN
iajs-3051	204	24	-	-	PUNCT
iajs-3051	204	25	ope	ope	NOUN
iajs-3051	204	26	-	-	PUNCT
iajs-3051	204	27	set	set	VERB
iajs-3051	204	28	u	u	NOUN
iajs-3051	204	29	⊂	⊂	PROPN
iajs-3051	204	30	g	g	PROPN
iajs-3051	204	31	such	such	ADJ
iajs-3051	204	32	as	as	ADP
iajs-3051	204	33	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	204	34	)	)	PUNCT
iajs-3051	205	1	⊂	⊂	PROPN
iajs-3051	205	2	u	u	PRON
iajs-3051	205	3	find	find	VERB
iajs-3051	205	4	an	an	DET
iajs-3051	205	5	nano	nano	NOUN
iajs-3051	205	6	-	-	PUNCT
iajs-3051	205	7	ope	ope	NOUN
iajs-3051	205	8	-	-	PUNCT
iajs-3051	205	9	set	set	VERB
iajs-3051	205	10	v	v	NOUN
iajs-3051	205	11	of	of	ADP
iajs-3051	205	12	h	h	NOUN
iajs-3051	205	13	such	such	ADJ
iajs-3051	205	14	as	as	ADP
iajs-3051	205	15	h	h	PROPN
iajs-3051	205	16	∈	∈	PROPN
iajs-3051	205	17	v	v	NOUN
iajs-3051	205	18	and	and	CCONJ
iajs-3051	205	19	𝑓−1(v	𝑓−1(v	NOUN
iajs-3051	205	20	)	)	PUNCT
iajs-3051	206	1	⊂	⊂	PROPN
iajs-3051	206	2	u.	u.	VERB
iajs-3051	206	3	it	it	PRON
iajs-3051	206	4	is	be	AUX
iajs-3051	206	5	easy	easy	ADJ
iajs-3051	206	6	to	to	PART
iajs-3051	206	7	show	show	VERB
iajs-3051	206	8	that	that	SCONJ
iajs-3051	206	9	v	v	X
iajs-3051	206	10	⊂	⊂	PROPN
iajs-3051	206	11	h	h	NOUN
iajs-3051	206	12	\	\	PROPN
iajs-3051	206	13	f(f	f(f	PROPN
iajs-3051	206	14	)	)	PUNCT
iajs-3051	206	15	.	.	PUNCT
iajs-3051	207	1	hence	hence	ADV
iajs-3051	207	2	h	h	NOUN
iajs-3051	207	3	\	\	PROPN
iajs-3051	207	4	f(f	f(f	PROPN
iajs-3051	207	5	)	)	PUNCT
iajs-3051	207	6	is	be	AUX
iajs-3051	207	7	nano	nano	NOUN
iajs-3051	207	8	-	-	PUNCT
iajs-3051	207	9	ope	ope	NOUN
iajs-3051	207	10	.	.	PUNCT
iajs-3051	208	1	in	in	ADP
iajs-3051	208	2	h	h	NOUN
iajs-3051	208	3	and	and	CCONJ
iajs-3051	208	4	this	this	PRON
iajs-3051	208	5	implies	imply	VERB
iajs-3051	208	6	f(f	f(f	PROPN
iajs-3051	208	7	)	)	PUNCT
iajs-3051	208	8	is	be	AUX
iajs-3051	208	9	nano	nano	NOUN
iajs-3051	208	10	-	-	PUNCT
iajs-3051	208	11	clos	clo	NOUN
iajs-3051	208	12	.	.	PUNCT
iajs-3051	209	1	in	in	ADP
iajs-3051	209	2	h.	h.	PROPN
iajs-3051	209	3	lemma	lemma	PROPN
iajs-3051	209	4	2.10	2.10	NUM
iajs-3051	209	5	:	:	PUNCT
iajs-3051	209	6	if	if	SCONJ
iajs-3051	209	7	f	f	X
iajs-3051	209	8	:	:	PUNCT
iajs-3051	209	9	g	g	PROPN
iajs-3051	209	10	⟶	⟶	NOUN
iajs-3051	209	11	h	h	NOUN
iajs-3051	209	12	is	be	AUX
iajs-3051	209	13	a	a	DET
iajs-3051	209	14	nano	nano	NOUN
iajs-3051	209	15	-	-	PUNCT
iajs-3051	209	16	perf	perf	NOUN
iajs-3051	209	17	.	.	PUNCT
iajs-3051	210	1	onto	onto	ADP
iajs-3051	210	2	map	map	NOUN
iajs-3051	210	3	and	and	CCONJ
iajs-3051	210	4	{	{	PUNCT
iajs-3051	210	5	aα}α∈λ	aα}α∈λ	ADV
iajs-3051	210	6	is	be	AUX
iajs-3051	210	7	a	a	DET
iajs-3051	210	8	locally	locally	ADV
iajs-3051	210	9	finite	finite	ADJ
iajs-3051	210	10	family	family	NOUN
iajs-3051	210	11	of	of	ADP
iajs-3051	210	12	subsets	subset	NOUN
iajs-3051	210	13	of	of	ADP
iajs-3051	210	14	g	g	NOUN
iajs-3051	210	15	,	,	PUNCT
iajs-3051	210	16	then	then	ADV
iajs-3051	210	17	{	{	PUNCT
iajs-3051	210	18	𝑓(aα)}α∈λ	𝑓(aα)}α∈λ	PROPN
iajs-3051	210	19	is	be	AUX
iajs-3051	210	20	a	a	DET
iajs-3051	210	21	locally	locally	ADV
iajs-3051	210	22	finite	finite	ADJ
iajs-3051	210	23	family	family	NOUN
iajs-3051	210	24	of	of	ADP
iajs-3051	210	25	subsets	subset	NOUN
iajs-3051	210	26	of	of	ADP
iajs-3051	210	27	h.	h.	PROPN
iajs-3051	210	28	proof	proof	NOUN
iajs-3051	210	29	:	:	PUNCT
iajs-3051	210	30	let	let	VERB
iajs-3051	210	31	h	h	NOUN
iajs-3051	210	32	be	be	AUX
iajs-3051	210	33	a	a	DET
iajs-3051	210	34	point	point	NOUN
iajs-3051	210	35	of	of	ADP
iajs-3051	210	36	h.	h.	NOUN
iajs-3051	210	37	if	if	SCONJ
iajs-3051	210	38	g	g	PROPN
iajs-3051	210	39	∈	∈	PROPN
iajs-3051	210	40	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	210	41	)	)	PUNCT
iajs-3051	210	42	,	,	PUNCT
iajs-3051	210	43	then	then	ADV
iajs-3051	210	44	find	find	VERB
iajs-3051	210	45	an	an	DET
iajs-3051	210	46	nano	nano	NOUN
iajs-3051	210	47	-	-	PUNCT
iajs-3051	210	48	ope	ope	NOUN
iajs-3051	210	49	-	-	PUNCT
iajs-3051	210	50	set	set	VERB
iajs-3051	210	51	vg	vg	NOUN
iajs-3051	210	52	of	of	ADP
iajs-3051	210	53	g	g	NOUN
iajs-3051	210	54	such	such	ADJ
iajs-3051	210	55	as	as	ADP
iajs-3051	210	56	g	g	PROPN
iajs-3051	210	57	∈	∈	PROPN
iajs-3051	210	58	vg	vg	NOUN
iajs-3051	211	1	and	and	CCONJ
iajs-3051	211	2	λg	λg	NOUN
iajs-3051	212	1	=	=	SYM
iajs-3051	212	2	{	{	PUNCT
iajs-3051	212	3	α	α	NOUN
iajs-3051	212	4	∈	∈	PROPN
iajs-3051	212	5	λ	λ	X
iajs-3051	212	6	:	:	PUNCT
iajs-3051	212	7	vg⋂aα	vg⋂aα	PROPN
iajs-3051	212	8	≠	≠	PROPN
iajs-3051	212	9	∅	∅	NOUN
iajs-3051	212	10	}	}	PUNCT
iajs-3051	212	11	is	be	AUX
iajs-3051	212	12	finite	finite	ADJ
iajs-3051	212	13	.	.	PUNCT
iajs-3051	213	1	since	since	SCONJ
iajs-3051	213	2	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	213	3	)	)	PUNCT
iajs-3051	213	4	is	be	AUX
iajs-3051	213	5	nano	nano	NOUN
iajs-3051	213	6	-	-	NOUN
iajs-3051	213	7	comp	comp	NOUN
iajs-3051	213	8	.	.	PUNCT
iajs-3051	214	1	,	,	PUNCT
iajs-3051	214	2	find	find	VERB
iajs-3051	214	3	a	a	DET
iajs-3051	214	4	finite	finite	NOUN
iajs-3051	214	5	subset	subset	NOUN
iajs-3051	214	6	b	b	PROPN
iajs-3051	214	7	of	of	ADP
iajs-3051	214	8	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	214	9	)	)	PUNCT
iajs-3051	214	10	such	such	ADJ
iajs-3051	214	11	as	as	ADP
iajs-3051	214	12	𝑓−1	𝑓−1	NUM
iajs-3051	214	13	(	(	PUNCT
iajs-3051	214	14	h	h	NOUN
iajs-3051	214	15	)	)	PUNCT
iajs-3051	214	16	⊂	⊂	PROPN
iajs-3051	214	17	⋃g∈b	⋃g∈b	NOUN
iajs-3051	214	18	vg	vg	NOUN
iajs-3051	214	19	.	.	PUNCT
iajs-3051	215	1	by	by	ADP
iajs-3051	215	2	theorem	theorem	NOUN
iajs-3051	215	3	2.9	2.9	NUM
iajs-3051	215	4	,	,	PUNCT
iajs-3051	215	5	find	find	VERB
iajs-3051	215	6	an	an	DET
iajs-3051	215	7	nano	nano	NOUN
iajs-3051	215	8	-	-	PUNCT
iajs-3051	215	9	ope	ope	NOUN
iajs-3051	215	10	-	-	PUNCT
iajs-3051	215	11	set	set	VERB
iajs-3051	215	12	w	w	NOUN
iajs-3051	215	13	of	of	ADP
iajs-3051	215	14	h	h	NOUN
iajs-3051	215	15	such	such	ADJ
iajs-3051	215	16	as	as	ADP
iajs-3051	215	17	h	h	PROPN
iajs-3051	215	18	∈	∈	PROPN
iajs-3051	215	19	w	w	PROPN
iajs-3051	215	20	and	and	CCONJ
iajs-3051	215	21	𝑓−1(w	𝑓−1(w	ADJ
iajs-3051	215	22	)	)	PUNCT
iajs-3051	215	23	⊂	⊂	PROPN
iajs-3051	215	24	⋃g∈bvg	⋃g∈bvg	NOUN
iajs-3051	215	25	.	.	PUNCT
iajs-3051	216	1	if	if	SCONJ
iajs-3051	216	2	w⋂f(aα	w⋂f(aα	PROPN
iajs-3051	216	3	)	)	PUNCT
iajs-3051	216	4	≠	≠	PROPN
iajs-3051	216	5	∅	∅	NOUN
iajs-3051	216	6	,	,	PUNCT
iajs-3051	216	7	then	then	ADV
iajs-3051	216	8	𝑓−1(w	𝑓−1(w	ADJ
iajs-3051	216	9	)	)	PUNCT
iajs-3051	216	10	⋂aα	⋂aα	PROPN
iajs-3051	216	11	≠	≠	PROPN
iajs-3051	216	12	∅	∅	NOUN
iajs-3051	216	13	,	,	PUNCT
iajs-3051	216	14	so	so	SCONJ
iajs-3051	216	15	that	that	SCONJ
iajs-3051	216	16	α	α	PROPN
iajs-3051	216	17	∈	∈	PROPN
iajs-3051	216	18	m=	m=	ADJ
iajs-3051	216	19	⋃g∈b	⋃g∈b	PROPN
iajs-3051	216	20	.	.	PUNCT
iajs-3051	217	1	λg	λg	PROPN
iajs-3051	217	2	since	since	SCONJ
iajs-3051	217	3	m	m	PROPN
iajs-3051	217	4	is	be	AUX
iajs-3051	217	5	a	a	DET
iajs-3051	217	6	finite	finite	NOUN
iajs-3051	217	7	subset	subset	NOUN
iajs-3051	217	8	of	of	ADP
iajs-3051	217	9	λ	λ	PROPN
iajs-3051	217	10	,	,	PUNCT
iajs-3051	217	11	it	it	PRON
iajs-3051	217	12	follows	follow	VERB
iajs-3051	217	13	that	that	SCONJ
iajs-3051	217	14	the	the	DET
iajs-3051	217	15	family	family	NOUN
iajs-3051	217	16	{	{	PUNCT
iajs-3051	217	17	𝑓(aα)}𝛼∈λ	𝑓(aα)}𝛼∈λ	PROPN
iajs-3051	217	18	is	be	AUX
iajs-3051	217	19	locally	locally	ADV
iajs-3051	217	20	finite	finite	ADJ
iajs-3051	217	21	.	.	PUNCT
iajs-3051	218	1	theorem	theorem	VERB
iajs-3051	218	2	2.11	2.11	NUM
iajs-3051	218	3	:	:	PUNCT
iajs-3051	218	4	for	for	SCONJ
iajs-3051	218	5	each	each	DET
iajs-3051	218	6	point	point	NOUN
iajs-3051	218	7	-	-	PUNCT
iajs-3051	218	8	finite	finite	ADJ
iajs-3051	218	9	nano	nano	NOUN
iajs-3051	218	10	-	-	PUNCT
iajs-3051	218	11	ope	ope	NOUN
iajs-3051	218	12	-	-	PUNCT
iajs-3051	218	13	cover	cover	NOUN
iajs-3051	218	14	{	{	PUNCT
iajs-3051	218	15	us}s∈s	us}s∈s	NOUN
iajs-3051	218	16	of	of	ADP
iajs-3051	218	17	a	a	DET
iajs-3051	218	18	nano	nano	NOUN
iajs-3051	218	19	-	-	PUNCT
iajs-3051	218	20	normal	normal	ADJ
iajs-3051	218	21	space	space	NOUN
iajs-3051	218	22	g	g	PROPN
iajs-3051	218	23	find	find	VERB
iajs-3051	218	24	an	an	DET
iajs-3051	218	25	nano	nano	NOUN
iajs-3051	218	26	-	-	PUNCT
iajs-3051	218	27	open	open	ADJ
iajs-3051	218	28	-	-	PUNCT
iajs-3051	218	29	cover	cover	NOUN
iajs-3051	218	30	{	{	PUNCT
iajs-3051	218	31	vs}s∈s	vs}s∈s	NOUN
iajs-3051	218	32	of	of	ADP
iajs-3051	218	33	g	g	NOUN
iajs-3051	218	34	such	such	ADJ
iajs-3051	218	35	as	as	ADP
iajs-3051	218	36	ncl(vs	ncl(vs	NOUN
iajs-3051	218	37	)	)	PUNCT
iajs-3051	218	38	⊂	⊂	PROPN
iajs-3051	218	39	us	us	PROPN
iajs-3051	218	40	for	for	SCONJ
iajs-3051	218	41	each	each	DET
iajs-3051	218	42	s	s	PROPN
iajs-3051	218	43	∈	∈	PROPN
iajs-3051	218	44	s.	s.	PROPN
iajs-3051	218	45	theorem	theorem	VERB
iajs-3051	218	46	2.12	2.12	NUM
iajs-3051	218	47	:	:	PUNCT
iajs-3051	218	48	if	if	SCONJ
iajs-3051	218	49	each	each	DET
iajs-3051	218	50	nano	nano	NOUN
iajs-3051	218	51	-	-	PUNCT
iajs-3051	218	52	ope	ope	NOUN
iajs-3051	218	53	-	-	PUNCT
iajs-3051	218	54	cover	cover	NOUN
iajs-3051	218	55	of	of	ADP
iajs-3051	218	56	a	a	DET
iajs-3051	218	57	top	top	ADJ
iajs-3051	218	58	-	-	PUNCT
iajs-3051	218	59	sp	sp	NOUN
iajs-3051	218	60	.	.	PUNCT
iajs-3051	219	1	g	g	PROPN
iajs-3051	219	2	has	have	VERB
iajs-3051	219	3	a	a	DET
iajs-3051	219	4	locally	locally	ADV
iajs-3051	219	5	finite	finite	ADJ
iajs-3051	219	6	nano	nano	NOUN
iajs-3051	219	7	-	-	PUNCT
iajs-3051	219	8	clos	clo	NOUN
iajs-3051	219	9	.	.	PUNCT
iajs-3051	220	1	refinement	refinement	NOUN
iajs-3051	220	2	,	,	PUNCT
iajs-3051	220	3	then	then	ADV
iajs-3051	220	4	each	each	DET
iajs-3051	220	5	nano	nano	NOUN
iajs-3051	220	6	-	-	PUNCT
iajs-3051	220	7	ope	ope	NOUN
iajs-3051	220	8	-	-	PUNCT
iajs-3051	220	9	cover	cover	NOUN
iajs-3051	220	10	of	of	ADP
iajs-3051	220	11	g	g	PROPN
iajs-3051	220	12	has	have	VERB
iajs-3051	220	13	also	also	ADV
iajs-3051	220	14	a	a	DET
iajs-3051	220	15	locally	locally	ADV
iajs-3051	220	16	finite	finite	ADJ
iajs-3051	220	17	nano	nano	NOUN
iajs-3051	220	18	-	-	PUNCT
iajs-3051	220	19	ope	ope	NOUN
iajs-3051	220	20	.	.	PUNCT
iajs-3051	221	1	refinement	refinement	NOUN
iajs-3051	221	2	.	.	PUNCT
iajs-3051	222	1	theorem	theorem	VERB
iajs-3051	222	2	2.13	2.13	NUM
iajs-3051	222	3	:	:	PUNCT
iajs-3051	222	4	nano	nano	NOUN
iajs-3051	222	5	-	-	PUNCT
iajs-3051	222	6	para	para	NOUN
iajs-3051	222	7	-	-	PUNCT
iajs-3051	222	8	compactness	compactness	NOUN
iajs-3051	222	9	is	be	AUX
iajs-3051	222	10	invariant	invariant	ADJ
iajs-3051	222	11	under	under	ADP
iajs-3051	222	12	nano	nano	NOUN
iajs-3051	222	13	-	-	PUNCT
iajs-3051	222	14	perf	perf	NOUN
iajs-3051	222	15	-	-	PUNCT
iajs-3051	222	16	maps	map	NOUN
iajs-3051	222	17	.	.	PUNCT
iajs-3051	223	1	ihjpas	ihjpas	PROPN
iajs-3051	223	2	.	.	PUNCT
iajs-3051	224	1	36	36	NUM
iajs-3051	224	2	(	(	PUNCT
iajs-3051	224	3	3	3	NUM
iajs-3051	224	4	)	)	PUNCT
iajs-3051	224	5	2023	2023	NUM
iajs-3051	224	6	403	403	NUM
iajs-3051	224	7	proof	proof	NOUN
iajs-3051	224	8	:	:	PUNCT
iajs-3051	224	9	let	let	VERB
iajs-3051	224	10	f	f	PRON
iajs-3051	224	11	:	:	PUNCT
iajs-3051	224	12	g	g	PROPN
iajs-3051	224	13	⟶	⟶	PROPN
iajs-3051	224	14	h	h	NOUN
iajs-3051	224	15	be	be	AUX
iajs-3051	224	16	a	a	DET
iajs-3051	224	17	nano	nano	NOUN
iajs-3051	224	18	-	-	PUNCT
iajs-3051	224	19	perf	perf	NOUN
iajs-3051	224	20	-	-	PUNCT
iajs-3051	224	21	map	map	NOUN
iajs-3051	224	22	from	from	ADP
iajs-3051	224	23	a	a	DET
iajs-3051	224	24	nano	nano	NOUN
iajs-3051	224	25	-	-	PUNCT
iajs-3051	224	26	para	para	ADJ
iajs-3051	224	27	-	-	PUNCT
iajs-3051	224	28	comp	comp	NOUN
iajs-3051	224	29	-	-	PUNCT
iajs-3051	224	30	sp	sp	NOUN
iajs-3051	224	31	.	.	NOUN
iajs-3051	225	1	g	g	NOUN
iajs-3051	225	2	onto	onto	ADP
iajs-3051	225	3	a	a	DET
iajs-3051	225	4	nano	nano	NOUN
iajs-3051	225	5	-	-	PUNCT
iajs-3051	225	6	top	top	NOUN
iajs-3051	225	7	-	-	PUNCT
iajs-3051	225	8	sp	sp	NOUN
iajs-3051	225	9	.	.	PUNCT
iajs-3051	226	1	h.	h.	PROPN
iajs-3051	226	2	let	let	VERB
iajs-3051	226	3	v	v	NUM
iajs-3051	226	4	∽	∽	NOUN
iajs-3051	226	5	be	be	AUX
iajs-3051	226	6	any	any	DET
iajs-3051	226	7	nano	nano	NOUN
iajs-3051	226	8	-	-	PUNCT
iajs-3051	226	9	ope	ope	NOUN
iajs-3051	226	10	-	-	PUNCT
iajs-3051	226	11	cover	cover	NOUN
iajs-3051	226	12	of	of	ADP
iajs-3051	226	13	the	the	DET
iajs-3051	226	14	space	space	NOUN
iajs-3051	226	15	h.	h.	PROPN
iajs-3051	227	1	so	so	ADV
iajs-3051	227	2	g	g	PROPN
iajs-3051	227	3	is	be	AUX
iajs-3051	227	4	nano	nano	NOUN
iajs-3051	227	5	-	-	PUNCT
iajs-3051	227	6	para	para	NOUN
iajs-3051	227	7	-	-	PUNCT
iajs-3051	227	8	comp	comp	NOUN
iajs-3051	227	9	.	.	PUNCT
iajs-3051	227	10	,	,	PUNCT
iajs-3051	227	11	{	{	PUNCT
iajs-3051	227	12	𝑓−1(v)}v∈v⏟	𝑓−1(v)}v∈v⏟	PROPN
iajs-3051	227	13	has	have	VERB
iajs-3051	227	14	a	a	DET
iajs-3051	227	15	locally	locally	ADV
iajs-3051	227	16	finite	finite	ADJ
iajs-3051	227	17	nano	nano	NOUN
iajs-3051	227	18	-	-	PUNCT
iajs-3051	227	19	ope	ope	NOUN
iajs-3051	227	20	.	.	PUNCT
iajs-3051	228	1	improvement	improvement	NOUN
iajs-3051	228	2	{	{	PUNCT
iajs-3051	228	3	us}s∈s	us}s∈s	PROPN
iajs-3051	228	4	.	.	PROPN
iajs-3051	228	5	therefore	therefore	ADV
iajs-3051	228	6	g	g	PROPN
iajs-3051	228	7	is	be	AUX
iajs-3051	228	8	norm	norm	NOUN
iajs-3051	228	9	-	-	PUNCT
iajs-3051	228	10	sp	sp	NOUN
iajs-3051	228	11	.	.	PROPN
iajs-3051	228	12	,	,	PUNCT
iajs-3051	228	13	therefore	therefore	ADV
iajs-3051	228	14	by	by	ADP
iajs-3051	228	15	theorem	theorem	NOUN
iajs-3051	228	16	2.11,find	2.11,find	NUM
iajs-3051	228	17	a	a	DET
iajs-3051	228	18	nano	nano	NOUN
iajs-3051	228	19	-	-	PUNCT
iajs-3051	228	20	clos	clo	NOUN
iajs-3051	228	21	-	-	PUNCT
iajs-3051	228	22	cover	cover	NOUN
iajs-3051	228	23	f	f	NOUN
iajs-3051	228	24	∼	∼	NOUN
iajs-3051	228	25	=	=	SYM
iajs-3051	228	26	{	{	PUNCT
iajs-3051	228	27	fs}s∈s	fs}s∈s	X
iajs-3051	228	28	of	of	ADP
iajs-3051	228	29	g	g	NOUN
iajs-3051	228	30	such	such	ADJ
iajs-3051	228	31	as	as	ADP
iajs-3051	228	32	fs	fs	ADP
iajs-3051	228	33	⊂	⊂	PROPN
iajs-3051	228	34	us	us	PROPN
iajs-3051	228	35	for	for	ADP
iajs-3051	228	36	each	each	DET
iajs-3051	228	37	s	s	PROPN
iajs-3051	228	38	∈	∈	PROPN
iajs-3051	228	39	s.	s.	PROPN
iajs-3051	228	40	since	since	SCONJ
iajs-3051	228	41	f	f	PROPN
iajs-3051	228	42	∼	∼	NOUN
iajs-3051	228	43	is	be	AUX
iajs-3051	228	44	locally	locally	ADV
iajs-3051	228	45	finite	finite	ADJ
iajs-3051	228	46	and	and	CCONJ
iajs-3051	228	47	nano	nano	NOUN
iajs-3051	228	48	-	-	PUNCT
iajs-3051	228	49	clos	clo	NOUN
iajs-3051	228	50	.	.	PUNCT
iajs-3051	229	1	,	,	PUNCT
iajs-3051	229	2	it	it	PRON
iajs-3051	229	3	follows	follow	VERB
iajs-3051	229	4	that	that	SCONJ
iajs-3051	229	5	{	{	PUNCT
iajs-3051	229	6	𝑓(fs)}s∈s	𝑓(fs)}s∈s	NOUN
iajs-3051	229	7	is	be	AUX
iajs-3051	229	8	a	a	DET
iajs-3051	229	9	locally	locally	ADV
iajs-3051	229	10	finite	finite	ADJ
iajs-3051	229	11	nano-clos.improvement	nano-clos.improvement	NOUN
iajs-3051	229	12	of	of	ADP
iajs-3051	229	13	the	the	DET
iajs-3051	229	14	cover	cover	NOUN
iajs-3051	229	15	v	v	NOUN
iajs-3051	229	16	∼	∼	NOUN
iajs-3051	229	17	.	.	PUNCT
iajs-3051	230	1	hence	hence	ADV
iajs-3051	230	2	by	by	ADP
iajs-3051	230	3	theorem	theorem	NOUN
iajs-3051	230	4	2.12	2.12	NUM
iajs-3051	230	5	,	,	PUNCT
iajs-3051	230	6	v	v	ADP
iajs-3051	230	7	∼	∼	NOUN
iajs-3051	230	8	has	have	VERB
iajs-3051	230	9	an	an	DET
iajs-3051	230	10	nano	nano	NOUN
iajs-3051	230	11	-	-	PUNCT
iajs-3051	230	12	ope	ope	NOUN
iajs-3051	230	13	.	.	PUNCT
iajs-3051	231	1	locally	locally	ADV
iajs-3051	231	2	finite	finite	VERB
iajs-3051	231	3	improvement	improvement	NOUN
iajs-3051	231	4	and	and	CCONJ
iajs-3051	231	5	hence	hence	ADV
iajs-3051	231	6	h	h	NOUN
iajs-3051	231	7	is	be	AUX
iajs-3051	231	8	nano	nano	NOUN
iajs-3051	231	9	-	-	PUNCT
iajs-3051	231	10	paracomp	paracomp	NOUN
iajs-3051	231	11	.	.	PUNCT
iajs-3051	232	1	definition	definition	NOUN
iajs-3051	232	2	2.14	2.14	NUM
iajs-3051	232	3	:	:	PUNCT
iajs-3051	233	1	[	[	X
iajs-3051	233	2	15	15	NUM
iajs-3051	233	3	]	]	PUNCT
iajs-3051	233	4	let	let	VERB
iajs-3051	233	5	a	a	PRON
iajs-3051	233	6	~	~	PUNCT
iajs-3051	233	7	=	=	SYM
iajs-3051	233	8	{	{	PUNCT
iajs-3051	233	9	as}s∈s	as}s∈s	NOUN
iajs-3051	233	10	be	be	AUX
iajs-3051	233	11	a	a	DET
iajs-3051	233	12	cover	cover	NOUN
iajs-3051	233	13	of	of	ADP
iajs-3051	233	14	a	a	DET
iajs-3051	233	15	set	set	NOUN
iajs-3051	233	16	g	g	NOUN
iajs-3051	233	17	;	;	PUNCT
iajs-3051	233	18	the	the	DET
iajs-3051	233	19	star	star	NOUN
iajs-3051	233	20	of	of	ADP
iajs-3051	233	21	a	a	DET
iajs-3051	233	22	set	set	NOUN
iajs-3051	233	23	m	m	PROPN
iajs-3051	233	24	⊂	⊂	PROPN
iajs-3051	233	25	g	g	NOUN
iajs-3051	233	26	with	with	ADP
iajs-3051	233	27	reference	reference	NOUN
iajs-3051	233	28	to	to	ADP
iajs-3051	233	29	a	a	DET
iajs-3051	233	30	∽	∽	NOUN
iajs-3051	233	31	is	be	AUX
iajs-3051	233	32	the	the	DET
iajs-3051	233	33	set	set	NOUN
iajs-3051	233	34	st(m	st(m	NOUN
iajs-3051	233	35	,	,	PUNCT
iajs-3051	233	36	a	a	PRON
iajs-3051	233	37	)	)	PUNCT
iajs-3051	233	38	=	=	SYM
iajs-3051	233	39	⋃	⋃	NOUN
iajs-3051	233	40	{	{	PUNCT
iajs-3051	233	41	as	as	ADP
iajs-3051	233	42	:	:	PUNCT
iajs-3051	233	43	m⋂as	m⋂as	NOUN
iajs-3051	233	44	≠	≠	PROPN
iajs-3051	233	45	∅	∅	NOUN
iajs-3051	233	46	}	}	PUNCT
iajs-3051	233	47	.	.	PUNCT
iajs-3051	234	1	the	the	DET
iajs-3051	234	2	star	star	NOUN
iajs-3051	234	3	of	of	ADP
iajs-3051	234	4	one	one	NUM
iajs-3051	234	5	̴	̴	NOUN
iajs-3051	234	6	point	point	NOUN
iajs-3051	234	7	set	set	NOUN
iajs-3051	234	8	{	{	PUNCT
iajs-3051	234	9	g	g	NOUN
iajs-3051	234	10	}	}	PUNCT
iajs-3051	234	11	with	with	ADP
iajs-3051	234	12	reference	reference	NOUN
iajs-3051	234	13	to	to	ADP
iajs-3051	234	14	a	a	DET
iajs-3051	234	15	cover	cover	NOUN
iajs-3051	234	16	a	a	PRON
iajs-3051	234	17	~	~	PUNCT
iajs-3051	234	18	is	be	AUX
iajs-3051	234	19	said	say	VERB
iajs-3051	234	20	to	to	PART
iajs-3051	234	21	be	be	AUX
iajs-3051	234	22	the	the	DET
iajs-3051	234	23	star	star	NOUN
iajs-3051	234	24	of	of	ADP
iajs-3051	234	25	the	the	DET
iajs-3051	234	26	point	point	NOUN
iajs-3051	234	27	g	g	NOUN
iajs-3051	234	28	with	with	ADP
iajs-3051	234	29	reference	reference	NOUN
iajs-3051	234	30	to	to	ADP
iajs-3051	234	31	a	a	PRON
iajs-3051	234	32	~	~	PUNCT
iajs-3051	234	33	and	and	CCONJ
iajs-3051	234	34	is	be	AUX
iajs-3051	234	35	symboly	symboly	ADJ
iajs-3051	234	36	by	by	ADP
iajs-3051	234	37	st(g	st(g	NOUN
iajs-3051	234	38	,	,	PUNCT
iajs-3051	234	39	a	a	PRON
iajs-3051	234	40	~	~	PUNCT
iajs-3051	234	41	)	)	PUNCT
iajs-3051	234	42	.	.	PUNCT
iajs-3051	235	1	definition	definition	NOUN
iajs-3051	235	2	2.15	2.15	NUM
iajs-3051	235	3	:	:	PUNCT
iajs-3051	235	4	a	a	DET
iajs-3051	235	5	nano	nano	NOUN
iajs-3051	235	6	-	-	PUNCT
iajs-3051	235	7	pseudometrizable	pseudometrizable	ADJ
iajs-3051	235	8	-	-	PUNCT
iajs-3051	235	9	space	space	NOUN
iajs-3051	235	10	is	be	AUX
iajs-3051	235	11	a	a	DET
iajs-3051	235	12	pair	pair	NOUN
iajs-3051	235	13	(	(	PUNCT
iajs-3051	235	14	g	g	NOUN
iajs-3051	235	15	,	,	PUNCT
iajs-3051	235	16	d	d	NOUN
iajs-3051	235	17	)	)	PUNCT
iajs-3051	235	18	where	where	SCONJ
iajs-3051	235	19	g	g	PROPN
iajs-3051	235	20	is	be	AUX
iajs-3051	235	21	a	a	DET
iajs-3051	235	22	set	set	NOUN
iajs-3051	235	23	and	and	CCONJ
iajs-3051	235	24	d	d	NOUN
iajs-3051	235	25	is	be	AUX
iajs-3051	235	26	a	a	DET
iajs-3051	235	27	function	function	NOUN
iajs-3051	235	28	d	d	NOUN
iajs-3051	235	29	:	:	PUNCT
iajs-3051	235	30	g×g⟶[0,∞	g×g⟶[0,∞	NOUN
iajs-3051	235	31	]	]	PUNCT
iajs-3051	235	32	,	,	PUNCT
iajs-3051	235	33	is	be	AUX
iajs-3051	235	34	said	say	VERB
iajs-3051	235	35	to	to	PART
iajs-3051	235	36	be	be	AUX
iajs-3051	235	37	nano	nano	NOUN
iajs-3051	235	38	-	-	PUNCT
iajs-3051	235	39	pseudometrizable	pseudometrizable	ADJ
iajs-3051	235	40	,	,	PUNCT
iajs-3051	235	41	that	that	PRON
iajs-3051	235	42	satisfies	satisfy	VERB
iajs-3051	235	43	the	the	DET
iajs-3051	235	44	following	follow	VERB
iajs-3051	235	45	properties	property	NOUN
iajs-3051	235	46	for	for	ADP
iajs-3051	235	47	all	all	DET
iajs-3051	235	48	g	g	PROPN
iajs-3051	235	49	,	,	PUNCT
iajs-3051	235	50	h	h	NOUN
iajs-3051	235	51	,	,	PUNCT
iajs-3051	235	52	i	i	PRON
iajs-3051	235	53	in	in	ADP
iajs-3051	235	54	g	g	NOUN
iajs-3051	235	55	:	:	PUNCT
iajs-3051	235	56	a	a	X
iajs-3051	235	57	)	)	PUNCT
iajs-3051	235	58	d(g	d(g	PROPN
iajs-3051	235	59	,	,	PUNCT
iajs-3051	235	60	h	h	NOUN
iajs-3051	235	61	)	)	PUNCT
iajs-3051	236	1	=	=	SYM
iajs-3051	236	2	0	0	PUNCT
iajs-3051	237	1	if	if	SCONJ
iajs-3051	237	2	and	and	CCONJ
iajs-3051	237	3	only	only	ADV
iajs-3051	237	4	if	if	SCONJ
iajs-3051	237	5	g	g	PROPN
iajs-3051	237	6	=	=	PROPN
iajs-3051	237	7	h.	h.	PROPN
iajs-3051	237	8	b	b	PROPN
iajs-3051	237	9	)	)	PUNCT
iajs-3051	237	10	d(g	d(g	PROPN
iajs-3051	237	11	,	,	PUNCT
iajs-3051	237	12	h	h	NOUN
iajs-3051	237	13	)	)	PUNCT
iajs-3051	237	14	=	=	PUNCT
iajs-3051	237	15	d(h	d(h	PROPN
iajs-3051	237	16	,	,	PUNCT
iajs-3051	237	17	g	g	NOUN
iajs-3051	237	18	)	)	PUNCT
iajs-3051	237	19	.	.	PUNCT
iajs-3051	238	1	c	c	X
iajs-3051	238	2	)	)	PUNCT
iajs-3051	238	3	d(g	d(g	PROPN
iajs-3051	238	4	,	,	PUNCT
iajs-3051	238	5	h	h	NOUN
iajs-3051	238	6	)	)	PUNCT
iajs-3051	238	7	≤	≤	NOUN
iajs-3051	239	1	d(g	d(g	PROPN
iajs-3051	239	2	,	,	PUNCT
iajs-3051	239	3	i	i	NOUN
iajs-3051	239	4	)	)	PUNCT
iajs-3051	240	1	+	+	CCONJ
iajs-3051	240	2	d(i	d(i	PROPN
iajs-3051	240	3	,	,	PUNCT
iajs-3051	240	4	h	h	NOUN
iajs-3051	240	5	)	)	PUNCT
iajs-3051	240	6	.	.	PUNCT
iajs-3051	241	1	lemma	lemma	PROPN
iajs-3051	241	2	2.16	2.16	NUM
iajs-3051	241	3	:	:	PUNCT
iajs-3051	241	4	a	a	DET
iajs-3051	241	5	space	space	NOUN
iajs-3051	241	6	g	g	NOUN
iajs-3051	241	7	is	be	AUX
iajs-3051	241	8	nano	nano	NOUN
iajs-3051	241	9	-	-	ADJ
iajs-3051	241	10	pseudometrizable	pseudometrizable	ADJ
iajs-3051	241	11	if	if	SCONJ
iajs-3051	241	12	and	and	CCONJ
iajs-3051	241	13	only	only	ADV
iajs-3051	241	14	if	if	SCONJ
iajs-3051	241	15	find	find	VERB
iajs-3051	241	16	a	a	DET
iajs-3051	241	17	sequence	sequence	NOUN
iajs-3051	241	18	{	{	PUNCT
iajs-3051	241	19	f	f	PROPN
iajs-3051	241	20	~n	~n	NUM
iajs-3051	241	21	}	}	PUNCT
iajs-3051	241	22	n∈n	n∈n	NOUN
iajs-3051	241	23	of	of	ADP
iajs-3051	241	24	locally	locally	ADV
iajs-3051	241	25	finite	finite	VERB
iajs-3051	241	26	nano	nano	NOUN
iajs-3051	241	27	-	-	PUNCT
iajs-3051	241	28	clos	clo	NOUN
iajs-3051	241	29	-	-	PUNCT
iajs-3051	241	30	covering	covering	NOUN
iajs-3051	241	31	of	of	ADP
iajs-3051	241	32	g	g	NOUN
iajs-3051	241	33	such	such	ADJ
iajs-3051	241	34	as	as	ADP
iajs-3051	241	35	for	for	ADP
iajs-3051	241	36	all	all	DET
iajs-3051	241	37	point	point	NOUN
iajs-3051	241	38	g	g	NOUN
iajs-3051	241	39	of	of	ADP
iajs-3051	241	40	g	g	PROPN
iajs-3051	241	41	and	and	CCONJ
iajs-3051	241	42	all	all	DET
iajs-3051	241	43	nano	nano	NOUN
iajs-3051	241	44	-	-	PUNCT
iajs-3051	241	45	ope	ope	NOUN
iajs-3051	241	46	-	-	PUNCT
iajs-3051	241	47	set	set	VERB
iajs-3051	241	48	u	u	NOUN
iajs-3051	241	49	such	such	ADJ
iajs-3051	241	50	as	as	ADP
iajs-3051	241	51	g	g	PROPN
iajs-3051	241	52	∈	∈	NOUN
iajs-3051	241	53	u	u	NOUN
iajs-3051	241	54	find	find	VERB
iajs-3051	241	55	an	an	DET
iajs-3051	241	56	integer	integer	NOUN
iajs-3051	241	57	n	n	CCONJ
iajs-3051	241	58	such	such	ADJ
iajs-3051	241	59	as	as	ADP
iajs-3051	241	60	st(g	st(g	NOUN
iajs-3051	241	61	,	,	PUNCT
iajs-3051	241	62	f	f	X
iajs-3051	241	63	~n	~n	NUM
iajs-3051	241	64	)	)	PUNCT
iajs-3051	242	1	⊂	⊂	PROPN
iajs-3051	242	2	u.	u.	PROPN
iajs-3051	242	3	theorem	theorem	VERB
iajs-3051	242	4	2.17	2.17	NUM
iajs-3051	242	5	:	:	PUNCT
iajs-3051	242	6	if	if	SCONJ
iajs-3051	242	7	f	f	X
iajs-3051	242	8	:	:	PUNCT
iajs-3051	242	9	g	g	PROPN
iajs-3051	242	10	⟶	⟶	NOUN
iajs-3051	242	11	h	h	NOUN
iajs-3051	242	12	is	be	AUX
iajs-3051	242	13	a	a	DET
iajs-3051	242	14	nano	nano	NOUN
iajs-3051	242	15	-	-	PUNCT
iajs-3051	242	16	perf	perf	NOUN
iajs-3051	242	17	.	.	PUNCT
iajs-3051	243	1	onto	onto	ADP
iajs-3051	243	2	map	map	NOUN
iajs-3051	243	3	.	.	PUNCT
iajs-3051	244	1	and	and	CCONJ
iajs-3051	244	2	g	g	PROPN
iajs-3051	244	3	is	be	AUX
iajs-3051	244	4	nano	nano	NOUN
iajs-3051	244	5	-	-	PUNCT
iajs-3051	244	6	pseudometizable	pseudometizable	NOUN
iajs-3051	244	7	-	-	PUNCT
iajs-3051	244	8	space	space	NOUN
iajs-3051	244	9	,	,	PUNCT
iajs-3051	244	10	then	then	ADV
iajs-3051	244	11	h	h	NOUN
iajs-3051	244	12	is	be	AUX
iajs-3051	244	13	a	a	DET
iajs-3051	244	14	nano	nano	NOUN
iajs-3051	244	15	-	-	PUNCT
iajs-3051	244	16	pseudometizable	pseudometizable	NOUN
iajs-3051	244	17	-	-	PUNCT
iajs-3051	244	18	space	space	NOUN
iajs-3051	244	19	.	.	PUNCT
iajs-3051	245	1	proof	proof	NOUN
iajs-3051	245	2	:	:	PUNCT
iajs-3051	245	3	choose	choose	VERB
iajs-3051	245	4	a	a	DET
iajs-3051	245	5	nano	nano	NOUN
iajs-3051	245	6	-	-	PUNCT
iajs-3051	245	7	pseudometric	pseudometric	ADJ
iajs-3051	245	8	d	d	NOUN
iajs-3051	245	9	on	on	ADP
iajs-3051	245	10	g	g	NOUN
iajs-3051	245	11	which	which	PRON
iajs-3051	245	12	induces	induce	VERB
iajs-3051	245	13	the	the	DET
iajs-3051	245	14	nano	nano	NOUN
iajs-3051	245	15	-	-	PUNCT
iajs-3051	245	16	topology	topology	NOUN
iajs-3051	245	17	of	of	ADP
iajs-3051	245	18	g.	g.	PROPN
iajs-3051	245	19	for	for	ADP
iajs-3051	245	20	all	all	DET
iajs-3051	245	21	positive	positive	ADJ
iajs-3051	245	22	integer	integer	NOUN
iajs-3051	245	23	n	n	NOUN
iajs-3051	245	24	,	,	PUNCT
iajs-3051	245	25	let	let	VERB
iajs-3051	245	26	e	e	X
iajs-3051	245	27	~𝑛	~𝑛	PUNCT
iajs-3051	245	28	be	be	AUX
iajs-3051	245	29	a	a	DET
iajs-3051	245	30	locally	locally	ADV
iajs-3051	245	31	finite	finite	ADJ
iajs-3051	245	32	nano	nano	NOUN
iajs-3051	245	33	-	-	PUNCT
iajs-3051	245	34	clos	clo	NOUN
iajs-3051	245	35	.	.	PUNCT
iajs-3051	246	1	improvement	improvement	NOUN
iajs-3051	246	2	of	of	ADP
iajs-3051	246	3	the	the	DET
iajs-3051	246	4	covering	covering	NOUN
iajs-3051	246	5	{	{	PUNCT
iajs-3051	246	6	b1	b1	NOUN
iajs-3051	246	7	2n⁄	2n⁄	NUM
iajs-3051	246	8	(	(	PUNCT
iajs-3051	246	9	g	g	NOUN
iajs-3051	246	10	)	)	PUNCT
iajs-3051	246	11	}	}	PUNCT
iajs-3051	246	12	g∈g	g∈g	NOUN
iajs-3051	246	13	of	of	ADP
iajs-3051	246	14	g.	g.	PROPN
iajs-3051	246	15	then	then	ADV
iajs-3051	246	16	by	by	ADP
iajs-3051	246	17	lemma	lemma	PROPN
iajs-3051	246	18	2.10	2.10	NUM
iajs-3051	246	19	,	,	PUNCT
iajs-3051	246	20	f	f	X
iajs-3051	246	21	~n	~n	NUM
iajs-3051	246	22	=	=	SYM
iajs-3051	246	23	{	{	PUNCT
iajs-3051	246	24	f(e	f(e	NOUN
iajs-3051	246	25	)	)	PUNCT
iajs-3051	246	26	:	:	PUNCT
iajs-3051	247	1	e	e	X
iajs-3051	247	2	∈	∈	PROPN
iajs-3051	247	3	e	e	X
iajs-3051	247	4	~n	~n	NUM
iajs-3051	247	5	}	}	PUNCT
iajs-3051	247	6	is	be	AUX
iajs-3051	247	7	a	a	DET
iajs-3051	247	8	locally	locally	ADV
iajs-3051	247	9	finite	finite	ADJ
iajs-3051	247	10	nano	nano	NOUN
iajs-3051	247	11	-	-	PUNCT
iajs-3051	247	12	clos	clo	NOUN
iajs-3051	247	13	-	-	PUNCT
iajs-3051	247	14	covering	covering	NOUN
iajs-3051	247	15	of	of	ADP
iajs-3051	247	16	h	h	NOUN
iajs-3051	247	17	for	for	ADP
iajs-3051	247	18	all	all	DET
iajs-3051	247	19	positive	positive	ADJ
iajs-3051	247	20	integer	integer	NOUN
iajs-3051	247	21	n.	n.	NOUN
iajs-3051	247	22	we	we	PRON
iajs-3051	247	23	shall	shall	AUX
iajs-3051	247	24	complete	complete	VERB
iajs-3051	247	25	the	the	DET
iajs-3051	247	26	proof	proof	NOUN
iajs-3051	247	27	by	by	ADP
iajs-3051	247	28	showing	show	VERB
iajs-3051	247	29	that	that	SCONJ
iajs-3051	247	30	the	the	DET
iajs-3051	247	31	sequence	sequence	NOUN
iajs-3051	247	32	{	{	PUNCT
iajs-3051	247	33	f	f	PROPN
iajs-3051	247	34	~n	~n	NUM
iajs-3051	247	35	}	}	PUNCT
iajs-3051	247	36	n∈n	n∈n	NOUN
iajs-3051	247	37	satisfies	satisfy	VERB
iajs-3051	247	38	the	the	DET
iajs-3051	247	39	condition	condition	NOUN
iajs-3051	247	40	of	of	ADP
iajs-3051	247	41	lemma	lemma	PROPN
iajs-3051	247	42	2.16	2.16	NUM
iajs-3051	247	43	.	.	PUNCT
iajs-3051	248	1	let	let	VERB
iajs-3051	248	2	j	j	PROPN
iajs-3051	248	3	be	be	AUX
iajs-3051	248	4	an	an	DET
iajs-3051	248	5	nano	nano	NOUN
iajs-3051	248	6	-	-	PUNCT
iajs-3051	248	7	ope	ope	NOUN
iajs-3051	248	8	-	-	PUNCT
iajs-3051	248	9	set	set	NOUN
iajs-3051	248	10	of	of	ADP
iajs-3051	248	11	h	h	NOUN
iajs-3051	248	12	and	and	CCONJ
iajs-3051	248	13	let	let	VERB
iajs-3051	248	14	h	h	PRON
iajs-3051	248	15	be	be	AUX
iajs-3051	248	16	a	a	DET
iajs-3051	248	17	point	point	NOUN
iajs-3051	248	18	of	of	ADP
iajs-3051	248	19	j.	j.	PROPN
iajs-3051	248	20	then	then	ADV
iajs-3051	248	21	𝑓−1	𝑓−1	NUM
iajs-3051	248	22	(	(	PUNCT
iajs-3051	248	23	h	h	NOUN
iajs-3051	248	24	)	)	PUNCT
iajs-3051	248	25	is	be	AUX
iajs-3051	248	26	nano	nano	NOUN
iajs-3051	248	27	-	-	PUNCT
iajs-3051	248	28	comp	comp	NOUN
iajs-3051	248	29	.	.	PUNCT
iajs-3051	249	1	and	and	CCONJ
iajs-3051	249	2	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	249	3	)	)	PUNCT
iajs-3051	250	1	⊂	⊂	PROPN
iajs-3051	250	2	𝑓−1(j	𝑓−1(j	PROPN
iajs-3051	250	3	)	)	PUNCT
iajs-3051	250	4	.	.	PUNCT
iajs-3051	251	1	the	the	DET
iajs-3051	251	2	nano	nano	NOUN
iajs-3051	251	3	-	-	PUNCT
iajs-3051	251	4	cont	cont	NOUN
iajs-3051	251	5	.	.	PUNCT
iajs-3051	252	1	real	real	ADJ
iajs-3051	252	2	value	value	NOUN
iajs-3051	252	3	function	function	NOUN
iajs-3051	252	4	∅	∅	NOUN
iajs-3051	252	5	know	know	VERB
iajs-3051	252	6	in	in	ADP
iajs-3051	252	7	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	252	8	)	)	PUNCT
iajs-3051	252	9	,	,	PUNCT
iajs-3051	252	10	by	by	ADP
iajs-3051	252	11	putting	put	VERB
iajs-3051	252	12	∅(g	∅(g	NOUN
iajs-3051	252	13	)	)	PUNCT
iajs-3051	253	1	=	=	SYM
iajs-3051	253	2	d(g	d(g	PROPN
iajs-3051	253	3	,	,	PUNCT
iajs-3051	253	4	g\𝑓−1(j	g\𝑓−1(j	NOUN
iajs-3051	253	5	)	)	PUNCT
iajs-3051	253	6	)	)	PUNCT
iajs-3051	254	1	if	if	SCONJ
iajs-3051	254	2	g	g	PROPN
iajs-3051	254	3	∈	∈	PROPN
iajs-3051	254	4	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	254	5	)	)	PUNCT
iajs-3051	254	6	,	,	PUNCT
iajs-3051	254	7	is	be	AUX
iajs-3051	254	8	bounded	bound	VERB
iajs-3051	254	9	and	and	CCONJ
iajs-3051	254	10	attains	attain	VERB
iajs-3051	254	11	its	its	PRON
iajs-3051	254	12	bounds	bound	NOUN
iajs-3051	254	13	.	.	PUNCT
iajs-3051	255	1	since	since	SCONJ
iajs-3051	255	2	∅(g	∅(g	PROPN
iajs-3051	255	3	)	)	PUNCT
iajs-3051	255	4	>	>	PUNCT
iajs-3051	255	5	0	0	PUNCT
iajs-3051	255	6	if	if	SCONJ
iajs-3051	255	7	g	g	PROPN
iajs-3051	255	8	∈	∈	PROPN
iajs-3051	255	9	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	255	10	)	)	PUNCT
iajs-3051	255	11	,	,	PUNCT
iajs-3051	255	12	it	it	PRON
iajs-3051	255	13	follows	follow	VERB
iajs-3051	255	14	that	that	PRON
iajs-3051	255	15	find	find	VERB
iajs-3051	255	16	a	a	DET
iajs-3051	255	17	point	point	NOUN
iajs-3051	255	18	go	go	VERB
iajs-3051	255	19	∈	∈	PROPN
iajs-3051	255	20	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	255	21	)	)	PUNCT
iajs-3051	255	22	such	such	ADJ
iajs-3051	255	23	as	as	ADP
iajs-3051	255	24	∅	∅	NOUN
iajs-3051	255	25	(	(	PUNCT
iajs-3051	255	26	g	g	NOUN
iajs-3051	255	27	)	)	PUNCT
iajs-3051	255	28	≥	≥	NOUN
iajs-3051	255	29	∅(g0	∅(g0	PROPN
iajs-3051	255	30	)	)	PUNCT
iajs-3051	255	31	>	>	X
iajs-3051	255	32	0	0	PUNCT
iajs-3051	256	1	if	if	SCONJ
iajs-3051	256	2	g	g	PROPN
iajs-3051	256	3	∈	∈	PROPN
iajs-3051	256	4	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	256	5	)	)	PUNCT
iajs-3051	256	6	.	.	PUNCT
iajs-3051	257	1	thus	thus	ADV
iajs-3051	257	2	find	find	VERB
iajs-3051	257	3	a	a	DET
iajs-3051	257	4	positive	positive	ADJ
iajs-3051	257	5	integer	integer	NOUN
iajs-3051	257	6	n	n	CCONJ
iajs-3051	257	7	such	such	ADJ
iajs-3051	257	8	as	as	ADP
iajs-3051	257	9	d(g	d(g	PROPN
iajs-3051	257	10	,	,	PUNCT
iajs-3051	257	11	g\f(j	g\f(j	NOUN
iajs-3051	257	12	)	)	PUNCT
iajs-3051	257	13	)	)	PUNCT
iajs-3051	257	14	≥	≥	AUX
iajs-3051	257	15	1	1	NUM
iajs-3051	257	16	2⁄	2⁄	NUM
iajs-3051	257	17	𝑛	𝑛	NOUN
iajs-3051	257	18	if	if	SCONJ
iajs-3051	257	19	g	g	PROPN
iajs-3051	257	20	∈	∈	PROPN
iajs-3051	257	21	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	257	22	)	)	PUNCT
iajs-3051	257	23	.	.	PUNCT
iajs-3051	258	1	suppose	suppose	VERB
iajs-3051	258	2	that	that	SCONJ
iajs-3051	258	3	f	f	PROPN
iajs-3051	258	4	f	f	X
iajs-3051	258	5	~n+1	~n+1	NOUN
iajs-3051	258	6	and	and	CCONJ
iajs-3051	258	7	h	h	PROPN
iajs-3051	258	8	∈	∈	PROPN
iajs-3051	258	9	f.	f.	PROPN
iajs-3051	258	10	then	then	ADV
iajs-3051	258	11	f	f	PROPN
iajs-3051	258	12	=	=	SYM
iajs-3051	258	13	f(e	f(e	PROPN
iajs-3051	258	14	)	)	PUNCT
iajs-3051	258	15	,	,	PUNCT
iajs-3051	258	16	where	where	SCONJ
iajs-3051	258	17	e	e	X
iajs-3051	258	18	∈	∈	PROPN
iajs-3051	258	19	e	e	X
iajs-3051	258	20	∼n+1	∼n+1	PROPN
iajs-3051	258	21	and	and	CCONJ
iajs-3051	258	22	e⋂𝑓−1(h	e⋂𝑓−1(h	PROPN
iajs-3051	258	23	)	)	PUNCT
iajs-3051	258	24	≠	≠	NOUN
iajs-3051	258	25	∅.	∅.	AUX
iajs-3051	258	26	let	let	VERB
iajs-3051	258	27	g	g	NOUN
iajs-3051	258	28	be	be	AUX
iajs-3051	258	29	a	a	DET
iajs-3051	258	30	point	point	NOUN
iajs-3051	258	31	of	of	ADP
iajs-3051	258	32	e⋂𝑓−1(h	e⋂𝑓−1(h	NOUN
iajs-3051	258	33	)	)	PUNCT
iajs-3051	258	34	and	and	CCONJ
iajs-3051	258	35	suppose	suppose	VERB
iajs-3051	258	36	find	find	VERB
iajs-3051	258	37	g2	g2	PROPN
iajs-3051	258	38	∈	∈	PROPN
iajs-3051	258	39	g	g	PROPN
iajs-3051	258	40	such	such	ADJ
iajs-3051	258	41	as	as	ADP
iajs-3051	258	42	g2	g2	PROPN
iajs-3051	258	43	∈	∈	PROPN
iajs-3051	258	44	e⋂(g\𝑓−1(j	e⋂(g\𝑓−1(j	PROPN
iajs-3051	258	45	)	)	PUNCT
iajs-3051	258	46	)	)	PUNCT
iajs-3051	258	47	.	.	PUNCT
iajs-3051	259	1	since	since	SCONJ
iajs-3051	259	2	there	there	PRON
iajs-3051	259	3	g1	g1	VERB
iajs-3051	259	4	∈	∈	PROPN
iajs-3051	259	5	g	g	PROPN
iajs-3051	259	6	such	such	ADJ
iajs-3051	259	7	that	that	SCONJ
iajs-3051	259	8	e	e	PROPN
iajs-3051	259	9	⊂	⊂	PROPN
iajs-3051	259	10	b1	b1	VERB
iajs-3051	259	11	2⁄	2⁄	NUM
iajs-3051	259	12	n+1(g1	n+1(g1	ADJ
iajs-3051	259	13	)	)	PUNCT
iajs-3051	259	14	,	,	PUNCT
iajs-3051	259	15	it	it	PRON
iajs-3051	259	16	follows	follow	VERB
iajs-3051	259	17	that	that	SCONJ
iajs-3051	259	18	d(g	d(g	PROPN
iajs-3051	259	19	,	,	PUNCT
iajs-3051	259	20	g2	g2	PROPN
iajs-3051	259	21	)	)	PUNCT
iajs-3051	259	22	≥	≥	NOUN
iajs-3051	259	23	d(g	d(g	PROPN
iajs-3051	259	24	,	,	PUNCT
iajs-3051	259	25	g1	g1	PROPN
iajs-3051	259	26	)	)	PUNCT
iajs-3051	260	1	+	+	CCONJ
iajs-3051	260	2	d(g1	d(g1	NOUN
iajs-3051	260	3	,	,	PUNCT
iajs-3051	260	4	g2	g2	PROPN
iajs-3051	260	5	)	)	PUNCT
iajs-3051	261	1	<	<	X
iajs-3051	261	2	1	1	NUM
iajs-3051	261	3	2⁄	2⁄	NUM
iajs-3051	261	4	𝑛	𝑛	NOUN
iajs-3051	261	5	.	.	PUNCT
iajs-3051	262	1	which	which	PRON
iajs-3051	262	2	is	be	AUX
iajs-3051	262	3	a	a	DET
iajs-3051	262	4	inconsistency	inconsistency	NOUN
iajs-3051	262	5	.	.	PUNCT
iajs-3051	263	1	hence	hence	ADV
iajs-3051	263	2	e	e	PROPN
iajs-3051	263	3	⊂	⊂	PROPN
iajs-3051	263	4	𝑓−1(j	𝑓−1(j	PROPN
iajs-3051	263	5	)	)	PUNCT
iajs-3051	264	1	so	so	SCONJ
iajs-3051	264	2	that	that	SCONJ
iajs-3051	264	3	f	f	PROPN
iajs-3051	264	4	⊂	⊂	PROPN
iajs-3051	264	5	j.	j.	PROPN
iajs-3051	264	6	thus	thus	ADV
iajs-3051	264	7	st(h	st(h	PROPN
iajs-3051	264	8	,	,	PUNCT
iajs-3051	264	9	f	f	PROPN
iajs-3051	264	10	∼n	∼n	PROPN
iajs-3051	264	11	)	)	PUNCT
iajs-3051	265	1	⊂	⊂	PROPN
iajs-3051	265	2	j.	j.	PROPN
iajs-3051	265	3	corollary	corollary	PROPN
iajs-3051	265	4	2.18	2.18	NUM
iajs-3051	265	5	:	:	PUNCT
iajs-3051	265	6	if	if	SCONJ
iajs-3051	265	7	f	f	X
iajs-3051	265	8	:	:	PUNCT
iajs-3051	265	9	g	g	PROPN
iajs-3051	265	10	⟶	⟶	NOUN
iajs-3051	265	11	h	h	NOUN
iajs-3051	265	12	is	be	AUX
iajs-3051	265	13	a	a	DET
iajs-3051	265	14	nano	nano	NOUN
iajs-3051	265	15	-	-	PUNCT
iajs-3051	265	16	perf	perf	NOUN
iajs-3051	265	17	.	.	PUNCT
iajs-3051	266	1	onto	onto	ADP
iajs-3051	266	2	map	map	NOUN
iajs-3051	266	3	and	and	CCONJ
iajs-3051	266	4	g	g	NOUN
iajs-3051	266	5	is	be	AUX
iajs-3051	266	6	a	a	DET
iajs-3051	266	7	nano	nano	NOUN
iajs-3051	266	8	-	-	PUNCT
iajs-3051	266	9	metrizable	metrizable	ADJ
iajs-3051	266	10	-	-	PUNCT
iajs-3051	266	11	space	space	NOUN
iajs-3051	266	12	,	,	PUNCT
iajs-3051	266	13	then	then	ADV
iajs-3051	266	14	h	h	NOUN
iajs-3051	266	15	is	be	AUX
iajs-3051	266	16	a	a	DET
iajs-3051	266	17	nano	nano	NOUN
iajs-3051	266	18	-	-	PUNCT
iajs-3051	266	19	metrizable	metrizable	ADJ
iajs-3051	266	20	-	-	PUNCT
iajs-3051	266	21	space	space	NOUN
iajs-3051	266	22	.	.	PUNCT
iajs-3051	267	1	defintion	defintion	NOUN
iajs-3051	267	2	2.19	2.19	NUM
iajs-3051	267	3	:	:	PUNCT
iajs-3051	267	4	in	in	ADP
iajs-3051	267	5	a	a	DET
iajs-3051	267	6	nano	nano	NOUN
iajs-3051	267	7	-	-	PUNCT
iajs-3051	267	8	top	top	NOUN
iajs-3051	267	9	-	-	PUNCT
iajs-3051	267	10	sp	sp	NOUN
iajs-3051	267	11	.	.	PUNCT
iajs-3051	268	1	g	g	PROPN
iajs-3051	268	2	is	be	AUX
iajs-3051	268	3	a	a	DET
iajs-3051	268	4	nano	nano	NOUN
iajs-3051	268	5	-	-	PUNCT
iajs-3051	268	6	compactly	compactly	ADV
iajs-3051	268	7	-	-	PUNCT
iajs-3051	268	8	generated	generate	VERB
iajs-3051	268	9	-	-	PUNCT
iajs-3051	268	10	sp	sp	NOUN
iajs-3051	268	11	.	.	PUNCT
iajs-3051	269	1	(	(	PUNCT
iajs-3051	269	2	denoted	denote	VERB
iajs-3051	269	3	by	by	ADP
iajs-3051	269	4	nano	nano	NOUN
iajs-3051	269	5	kspace	kspace	NOUN
iajs-3051	269	6	)	)	PUNCT
iajs-3051	269	7	in	in	ADP
iajs-3051	269	8	which	which	PRON
iajs-3051	269	9	a	a	DET
iajs-3051	269	10	subset	subset	NOUN
iajs-3051	269	11	is	be	AUX
iajs-3051	269	12	nano	nano	NOUN
iajs-3051	269	13	-	-	PUNCT
iajs-3051	269	14	clos	clo	NOUN
iajs-3051	269	15	.	.	PUNCT
iajs-3051	270	1	if	if	SCONJ
iajs-3051	270	2	it	it	PRON
iajs-3051	270	3	is	be	AUX
iajs-3051	270	4	intersection	intersection	NOUN
iajs-3051	270	5	with	with	ADP
iajs-3051	270	6	any	any	DET
iajs-3051	270	7	nano	nano	NOUN
iajs-3051	270	8	-	-	PUNCT
iajs-3051	270	9	comp	comp	NOUN
iajs-3051	270	10	-	-	PUNCT
iajs-3051	270	11	sub	sub	NOUN
iajs-3051	270	12	-	-	ADJ
iajs-3051	270	13	set	set	ADJ
iajs-3051	270	14	is	be	AUX
iajs-3051	270	15	clos	clo	NOUN
iajs-3051	270	16	.	.	PUNCT
iajs-3051	271	1	ihjpas	ihjpas	PROPN
iajs-3051	271	2	.	.	PUNCT
iajs-3051	272	1	36	36	NUM
iajs-3051	272	2	(	(	PUNCT
iajs-3051	272	3	3	3	NUM
iajs-3051	272	4	)	)	PUNCT
iajs-3051	272	5	2023	2023	NUM
iajs-3051	272	6	404	404	NUM
iajs-3051	272	7	lemma	lemma	PROPN
iajs-3051	272	8	2.20	2.20	NUM
iajs-3051	272	9	:	:	PUNCT
iajs-3051	272	10	a	a	DET
iajs-3051	272	11	nano	nano	NOUN
iajs-3051	272	12	-	-	PUNCT
iajs-3051	272	13	hausd	hausd	NOUN
iajs-3051	272	14	-	-	PUNCT
iajs-3051	272	15	sp	sp	NOUN
iajs-3051	272	16	.	.	PUNCT
iajs-3051	273	1	g	g	PROPN
iajs-3051	273	2	is	be	AUX
iajs-3051	273	3	a	a	DET
iajs-3051	273	4	nano	nano	ADJ
iajs-3051	273	5	k	k	NOUN
iajs-3051	273	6	-	-	NOUN
iajs-3051	273	7	space	space	NOUN
iajs-3051	273	8	if	if	SCONJ
iajs-3051	273	9	and	and	CCONJ
iajs-3051	273	10	only	only	ADV
iajs-3051	273	11	if	if	SCONJ
iajs-3051	273	12	for	for	ADP
iajs-3051	273	13	all	all	DET
iajs-3051	273	14	a	a	DET
iajs-3051	273	15	⊂	⊂	PROPN
iajs-3051	273	16	g	g	NOUN
iajs-3051	273	17	,	,	PUNCT
iajs-3051	273	18	the	the	DET
iajs-3051	273	19	set	set	NOUN
iajs-3051	273	20	a	a	PRON
iajs-3051	273	21	is	be	AUX
iajs-3051	273	22	nano	nano	NOUN
iajs-3051	273	23	-	-	PUNCT
iajs-3051	273	24	clos	clo	NOUN
iajs-3051	273	25	.	.	PUNCT
iajs-3051	274	1	in	in	ADP
iajs-3051	274	2	g	g	NOUN
iajs-3051	274	3	on	on	ADP
iajs-3051	274	4	condition	condition	NOUN
iajs-3051	274	5	the	the	DET
iajs-3051	274	6	intersection	intersection	NOUN
iajs-3051	274	7	of	of	ADP
iajs-3051	274	8	a	a	PRON
iajs-3051	274	9	with	with	ADP
iajs-3051	274	10	any	any	DET
iajs-3051	274	11	nano	nano	NOUN
iajs-3051	274	12	-	-	PUNCT
iajs-3051	274	13	comp	comp	NOUN
iajs-3051	274	14	-	-	PUNCT
iajs-3051	274	15	sub	sub	NOUN
iajs-3051	274	16	-	-	NOUN
iajs-3051	274	17	sp	sp	NOUN
iajs-3051	274	18	.	.	PUNCT
iajs-3051	274	19	f	f	PROPN
iajs-3051	274	20	of	of	ADP
iajs-3051	274	21	the	the	DET
iajs-3051	274	22	space	space	NOUN
iajs-3051	274	23	g	g	NOUN
iajs-3051	274	24	is	be	AUX
iajs-3051	274	25	nano	nano	NOUN
iajs-3051	274	26	-	-	PUNCT
iajs-3051	274	27	clos	clo	NOUN
iajs-3051	274	28	.	.	PUNCT
iajs-3051	275	1	in	in	ADP
iajs-3051	275	2	f.	f.	PROPN
iajs-3051	275	3	theorem	theorem	VERB
iajs-3051	275	4	2.21	2.21	NUM
iajs-3051	275	5	:	:	PUNCT
iajs-3051	275	6	if	if	SCONJ
iajs-3051	275	7	there	there	PRON
iajs-3051	275	8	exists	exist	VERB
iajs-3051	275	9	a	a	DET
iajs-3051	275	10	nano	nano	NOUN
iajs-3051	275	11	-	-	PUNCT
iajs-3051	275	12	perf	perf	NOUN
iajs-3051	275	13	-	-	PUNCT
iajs-3051	275	14	map	map	NOUN
iajs-3051	275	15	f	f	NOUN
iajs-3051	275	16	:	:	PUNCT
iajs-3051	275	17	g	g	PROPN
iajs-3051	275	18	⟶	⟶	NOUN
iajs-3051	275	19	h	h	NOUN
iajs-3051	275	20	of	of	ADP
iajs-3051	275	21	a	a	DET
iajs-3051	275	22	nano	nano	ADJ
iajs-3051	275	23	k	k	NOUN
iajs-3051	275	24	-	-	NOUN
iajs-3051	275	25	sp	sp	NOUN
iajs-3051	275	26	.	.	PUNCT
iajs-3051	276	1	g	g	NOUN
iajs-3051	276	2	onto	onto	ADP
iajs-3051	276	3	a	a	DET
iajs-3051	276	4	top	top	ADJ
iajs-3051	276	5	-	-	PUNCT
iajs-3051	276	6	sp	sp	NOUN
iajs-3051	276	7	.	.	NOUN
iajs-3051	276	8	h	h	NOUN
iajs-3051	276	9	,	,	PUNCT
iajs-3051	276	10	then	then	ADV
iajs-3051	276	11	h	h	NOUN
iajs-3051	276	12	is	be	AUX
iajs-3051	276	13	a	a	DET
iajs-3051	276	14	nano	nano	ADJ
iajs-3051	276	15	k	k	NOUN
iajs-3051	276	16	-	-	PUNCT
iajs-3051	276	17	sp	sp	NOUN
iajs-3051	276	18	.	.	PUNCT
iajs-3051	276	19	proof	proof	NOUN
iajs-3051	276	20	:	:	PUNCT
iajs-3051	276	21	let	let	VERB
iajs-3051	276	22	a	a	DET
iajs-3051	276	23	⊂	⊂	PROPN
iajs-3051	276	24	h	h	NOUN
iajs-3051	276	25	such	such	ADJ
iajs-3051	276	26	as	as	ADP
iajs-3051	276	27	a⋂f	a⋂f	NOUN
iajs-3051	276	28	is	be	AUX
iajs-3051	276	29	nano	nano	NOUN
iajs-3051	276	30	-	-	PUNCT
iajs-3051	276	31	clos	clo	NOUN
iajs-3051	276	32	.	.	PUNCT
iajs-3051	277	1	in	in	ADP
iajs-3051	277	2	f	f	PROPN
iajs-3051	277	3	for	for	ADP
iajs-3051	277	4	each	each	DET
iajs-3051	277	5	nano	nano	NOUN
iajs-3051	277	6	-	-	PUNCT
iajs-3051	277	7	comp	comp	NOUN
iajs-3051	277	8	-	-	PUNCT
iajs-3051	277	9	sub	sub	NOUN
iajs-3051	277	10	-	-	ADJ
iajs-3051	277	11	set	set	ADJ
iajs-3051	277	12	f	f	PROPN
iajs-3051	277	13	of	of	ADP
iajs-3051	277	14	h.	h.	PROPN
iajs-3051	277	15	it	it	PRON
iajs-3051	277	16	suffices	suffice	VERB
iajs-3051	277	17	to	to	PART
iajs-3051	277	18	show	show	VERB
iajs-3051	277	19	that	that	SCONJ
iajs-3051	277	20	a	a	PRON
iajs-3051	277	21	is	be	AUX
iajs-3051	277	22	nano	nano	NOUN
iajs-3051	277	23	-	-	PUNCT
iajs-3051	277	24	clos	clo	NOUN
iajs-3051	277	25	.	.	PUNCT
iajs-3051	278	1	in	in	ADP
iajs-3051	278	2	h.	h.	PROPN
iajs-3051	278	3	let	let	VERB
iajs-3051	278	4	k	k	PRON
iajs-3051	278	5	be	be	AUX
iajs-3051	278	6	a	a	DET
iajs-3051	278	7	nano	nano	NOUN
iajs-3051	278	8	-	-	PUNCT
iajs-3051	278	9	comp	comp	NOUN
iajs-3051	278	10	-	-	PUNCT
iajs-3051	278	11	sub	sub	NOUN
iajs-3051	278	12	-	-	ADJ
iajs-3051	278	13	set	set	NOUN
iajs-3051	278	14	of	of	ADP
iajs-3051	278	15	g	g	NOUN
iajs-3051	278	16	,	,	PUNCT
iajs-3051	278	17	so	so	ADV
iajs-3051	278	18	f(k	f(k	VERB
iajs-3051	278	19	)	)	PUNCT
iajs-3051	278	20	is	be	AUX
iajs-3051	278	21	nano	nano	NOUN
iajs-3051	278	22	-	-	NOUN
iajs-3051	278	23	comp	comp	NOUN
iajs-3051	278	24	.	.	PUNCT
iajs-3051	279	1	in	in	ADP
iajs-3051	279	2	h	h	NOUN
iajs-3051	279	3	,	,	PUNCT
iajs-3051	279	4	hence	hence	ADV
iajs-3051	279	5	a⋂f(k	a⋂f(k	VERB
iajs-3051	279	6	)	)	PUNCT
iajs-3051	279	7	is	be	AUX
iajs-3051	279	8	nano	nano	NOUN
iajs-3051	279	9	-	-	PUNCT
iajs-3051	279	10	clos	clo	NOUN
iajs-3051	279	11	.	.	PUNCT
iajs-3051	280	1	in	in	ADP
iajs-3051	280	2	f(k	f(k	PROPN
iajs-3051	280	3	)	)	PUNCT
iajs-3051	280	4	,	,	PUNCT
iajs-3051	280	5	so	so	ADV
iajs-3051	280	6	𝑓−1(a)⋂k	𝑓−1(a)⋂k	PROPN
iajs-3051	280	7	is	be	AUX
iajs-3051	280	8	nano	nano	NOUN
iajs-3051	280	9	-	-	PUNCT
iajs-3051	280	10	clos	clo	NOUN
iajs-3051	280	11	.	.	PUNCT
iajs-3051	281	1	in	in	ADP
iajs-3051	281	2	k.	k.	PROPN
iajs-3051	282	1	but	but	CCONJ
iajs-3051	282	2	g	g	PROPN
iajs-3051	282	3	is	be	AUX
iajs-3051	282	4	a	a	DET
iajs-3051	282	5	nano	nano	ADJ
iajs-3051	282	6	k	k	NOUN
iajs-3051	282	7	-	-	NOUN
iajs-3051	282	8	sp	sp	NOUN
iajs-3051	282	9	.	.	PROPN
iajs-3051	282	10	,	,	PUNCT
iajs-3051	282	11	so	so	ADV
iajs-3051	282	12	𝑓−1(a	𝑓−1(a	PROPN
iajs-3051	282	13	)	)	PUNCT
iajs-3051	282	14	is	be	AUX
iajs-3051	282	15	nano	nano	NOUN
iajs-3051	282	16	-	-	PUNCT
iajs-3051	282	17	clos	clo	NOUN
iajs-3051	282	18	.	.	PUNCT
iajs-3051	283	1	in	in	ADP
iajs-3051	283	2	g.	g.	PROPN
iajs-3051	283	3	thus	thus	ADV
iajs-3051	283	4	𝑓𝑓−1(a	𝑓𝑓−1(a	VERB
iajs-3051	283	5	)	)	PUNCT
iajs-3051	283	6	=	=	PUNCT
iajs-3051	284	1	a	a	PRON
iajs-3051	284	2	is	be	AUX
iajs-3051	284	3	nano	nano	NOUN
iajs-3051	284	4	-	-	PUNCT
iajs-3051	284	5	clos	clo	NOUN
iajs-3051	284	6	.	.	PUNCT
iajs-3051	285	1	in	in	ADP
iajs-3051	285	2	h	h	NOUN
iajs-3051	285	3	,	,	PUNCT
iajs-3051	285	4	hence	hence	ADV
iajs-3051	285	5	h	h	NOUN
iajs-3051	285	6	is	be	AUX
iajs-3051	285	7	a	a	DET
iajs-3051	285	8	nano	nano	ADJ
iajs-3051	285	9	k	k	NOUN
iajs-3051	285	10	-	-	PUNCT
iajs-3051	285	11	sp	sp	NOUN
iajs-3051	285	12	.	.	NOUN
iajs-3051	285	13	3	3	NUM
iajs-3051	285	14	.	.	PUNCT
iajs-3051	285	15	inverse	inverse	NOUN
iajs-3051	285	16	images	image	NOUN
iajs-3051	285	17	of	of	ADP
iajs-3051	285	18	nana	nana	PROPN
iajs-3051	285	19	perfect	perfect	ADJ
iajs-3051	285	20	mappings	mapping	NOUN
iajs-3051	285	21	.	.	PUNCT
iajs-3051	286	1	we	we	PRON
iajs-3051	286	2	study	study	VERB
iajs-3051	286	3	the	the	DET
iajs-3051	286	4	inverse	inverse	NOUN
iajs-3051	286	5	images	image	NOUN
iajs-3051	286	6	of	of	ADP
iajs-3051	286	7	nano	nano	VERB
iajs-3051	286	8	perfect	perfect	ADJ
iajs-3051	286	9	mappings	mapping	NOUN
iajs-3051	286	10	and	and	CCONJ
iajs-3051	286	11	several	several	ADJ
iajs-3051	286	12	related	relate	VERB
iajs-3051	286	13	theorems	theorem	NOUN
iajs-3051	286	14	.	.	PUNCT
iajs-3051	287	1	theorem	theorem	VERB
iajs-3051	287	2	3.1	3.1	NUM
iajs-3051	287	3	:	:	PUNCT
iajs-3051	287	4	nano	nano	NOUN
iajs-3051	287	5	-	-	PUNCT
iajs-3051	287	6	regularity	regularity	NOUN
iajs-3051	287	7	is	be	AUX
iajs-3051	287	8	an	an	DET
iajs-3051	287	9	inverse	inverse	NOUN
iajs-3051	287	10	of	of	ADP
iajs-3051	287	11	nano	nano	NOUN
iajs-3051	287	12	-	-	PUNCT
iajs-3051	287	13	perf	perf	NOUN
iajs-3051	287	14	-	-	PUNCT
iajs-3051	287	15	maps	map	NOUN
iajs-3051	287	16	.	.	PUNCT
iajs-3051	288	1	proof	proof	NOUN
iajs-3051	288	2	:	:	PUNCT
iajs-3051	288	3	let	let	VERB
iajs-3051	288	4	f	f	PRON
iajs-3051	288	5	:	:	PUNCT
iajs-3051	288	6	g	g	PROPN
iajs-3051	288	7	⟶	⟶	PROPN
iajs-3051	288	8	h	h	NOUN
iajs-3051	288	9	be	be	AUX
iajs-3051	288	10	a	a	DET
iajs-3051	288	11	nano	nano	NOUN
iajs-3051	288	12	-	-	PUNCT
iajs-3051	288	13	perf	perf	NOUN
iajs-3051	288	14	-	-	PUNCT
iajs-3051	288	15	map	map	NOUN
iajs-3051	288	16	onto	onto	ADP
iajs-3051	288	17	a	a	DET
iajs-3051	288	18	nano	nano	NOUN
iajs-3051	288	19	-	-	PUNCT
iajs-3051	288	20	regu	regu	NOUN
iajs-3051	288	21	-	-	PUNCT
iajs-3051	288	22	sp	sp	NOUN
iajs-3051	288	23	.	.	PUNCT
iajs-3051	289	1	h.	h.	PROPN
iajs-3051	289	2	take	take	VERB
iajs-3051	289	3	a	a	DET
iajs-3051	289	4	point	point	NOUN
iajs-3051	289	5	g	g	ADP
iajs-3051	289	6	∈	∈	PROPN
iajs-3051	289	7	g	g	PROPN
iajs-3051	289	8	and	and	CCONJ
iajs-3051	289	9	a	a	DET
iajs-3051	289	10	nanoclos	nanoclo	NOUN
iajs-3051	289	11	-	-	PUNCT
iajs-3051	289	12	set	set	VERB
iajs-3051	289	13	f	f	PROPN
iajs-3051	289	14	⊂	⊂	PROPN
iajs-3051	289	15	g	g	PROPN
iajs-3051	289	16	such	such	ADJ
iajs-3051	289	17	as	as	ADP
iajs-3051	289	18	g	g	PROPN
iajs-3051	289	19	∉	∉	PROPN
iajs-3051	289	20	f.	f.	PROPN
iajs-3051	289	21	the	the	DET
iajs-3051	289	22	set	set	NOUN
iajs-3051	289	23	f⋂𝑓−1(f(g	f⋂𝑓−1(f(g	NOUN
iajs-3051	289	24	)	)	PUNCT
iajs-3051	289	25	)	)	PUNCT
iajs-3051	290	1	is	be	AUX
iajs-3051	290	2	nano	nano	NOUN
iajs-3051	290	3	-	-	NOUN
iajs-3051	290	4	comp	comp	NOUN
iajs-3051	290	5	.	.	PUNCT
iajs-3051	291	1	and	and	CCONJ
iajs-3051	291	2	does	do	AUX
iajs-3051	291	3	not	not	PART
iajs-3051	291	4	contain	contain	VERB
iajs-3051	291	5	g	g	NOUN
iajs-3051	291	6	,	,	PUNCT
iajs-3051	291	7	since	since	SCONJ
iajs-3051	291	8	g	g	PROPN
iajs-3051	291	9	is	be	AUX
iajs-3051	291	10	nano	nano	NOUN
iajs-3051	291	11	-	-	PUNCT
iajs-3051	291	12	hausd	hausd	NOUN
iajs-3051	291	13	-	-	PUNCT
iajs-3051	291	14	sp	sp	NOUN
iajs-3051	291	15	.	.	NOUN
iajs-3051	291	16	find	find	VERB
iajs-3051	291	17	disjoint	disjoint	ADJ
iajs-3051	291	18	nano	nano	NOUN
iajs-3051	291	19	-	-	PUNCT
iajs-3051	291	20	ope	ope	NOUN
iajs-3051	291	21	-	-	PUNCT
iajs-3051	291	22	sets	set	NOUN
iajs-3051	291	23	u1	u1	NOUN
iajs-3051	291	24	,	,	PUNCT
iajs-3051	291	25	v1	v1	PROPN
iajs-3051	291	26	⊂	⊂	PROPN
iajs-3051	291	27	g	g	PROPN
iajs-3051	291	28	such	such	ADJ
iajs-3051	291	29	as	as	ADP
iajs-3051	291	30	g	g	PROPN
iajs-3051	291	31	∈	∈	PROPN
iajs-3051	291	32	u1	u1	NOUN
iajs-3051	291	33	and	and	CCONJ
iajs-3051	291	34	f⋂𝑓−1(f(g	f⋂𝑓−1(f(g	NOUN
iajs-3051	291	35	)	)	PUNCT
iajs-3051	291	36	)	)	PUNCT
iajs-3051	292	1	⊂	⊂	PROPN
iajs-3051	292	2	v1	v1	PROPN
iajs-3051	292	3	.	.	PUNCT
iajs-3051	293	1	the	the	DET
iajs-3051	293	2	set	set	NOUN
iajs-3051	293	3	f(f\v1	f(f\v1	ADJ
iajs-3051	293	4	)	)	PUNCT
iajs-3051	293	5	is	be	AUX
iajs-3051	293	6	nano	nano	NOUN
iajs-3051	293	7	-	-	PUNCT
iajs-3051	293	8	clos	clo	NOUN
iajs-3051	293	9	.	.	PUNCT
iajs-3051	294	1	in	in	ADP
iajs-3051	294	2	h	h	NOUN
iajs-3051	294	3	and	and	CCONJ
iajs-3051	294	4	does	do	AUX
iajs-3051	294	5	not	not	PART
iajs-3051	294	6	contain	contain	VERB
iajs-3051	294	7	f(g	f(g	NOUN
iajs-3051	294	8	)	)	PUNCT
iajs-3051	294	9	,	,	PUNCT
iajs-3051	294	10	hence	hence	ADV
iajs-3051	294	11	by	by	ADP
iajs-3051	294	12	nano	nano	NOUN
iajs-3051	294	13	-	-	PUNCT
iajs-3051	294	14	regularity	regularity	NOUN
iajs-3051	294	15	of	of	ADP
iajs-3051	294	16	h	h	NOUN
iajs-3051	294	17	,	,	PUNCT
iajs-3051	294	18	find	find	VERB
iajs-3051	294	19	disjoint	disjoint	ADJ
iajs-3051	294	20	nano	nano	NOUN
iajs-3051	294	21	-	-	PUNCT
iajs-3051	294	22	ope	ope	NOUN
iajs-3051	294	23	-	-	PUNCT
iajs-3051	294	24	sets	set	NOUN
iajs-3051	294	25	u2	u2	NOUN
iajs-3051	294	26	,	,	PUNCT
iajs-3051	294	27	v2	v2	PROPN
iajs-3051	294	28	⊂	⊂	PROPN
iajs-3051	294	29	h	h	NOUN
iajs-3051	294	30	such	such	ADJ
iajs-3051	294	31	as	as	ADP
iajs-3051	294	32	f(g	f(g	NOUN
iajs-3051	294	33	)	)	PUNCT
iajs-3051	294	34	∈	∈	PROPN
iajs-3051	294	35	u2	u2	NOUN
iajs-3051	294	36	and	and	CCONJ
iajs-3051	294	37	f(f\v1	f(f\v1	ADJ
iajs-3051	294	38	)	)	PUNCT
iajs-3051	295	1	⊂	⊂	PROPN
iajs-3051	295	2	v2	v2	PROPN
iajs-3051	295	3	.	.	PUNCT
iajs-3051	296	1	the	the	DET
iajs-3051	296	2	sets	set	NOUN
iajs-3051	296	3	u	u	NOUN
iajs-3051	296	4	=	=	PROPN
iajs-3051	296	5	u1⋂𝑓−1(u2	u1⋂𝑓−1(u2	PROPN
iajs-3051	296	6	)	)	PUNCT
iajs-3051	296	7	and	and	CCONJ
iajs-3051	296	8	v	v	NOUN
iajs-3051	296	9	=	=	SYM
iajs-3051	296	10	v1⋂𝑓−1(v2	v1⋂𝑓−1(v2	NOUN
iajs-3051	296	11	)	)	PUNCT
iajs-3051	296	12	are	be	AUX
iajs-3051	296	13	nano	nano	NOUN
iajs-3051	296	14	-	-	PUNCT
iajs-3051	296	15	ope	ope	NOUN
iajs-3051	296	16	.	.	PUNCT
iajs-3051	297	1	in	in	ADP
iajs-3051	297	2	g	g	PROPN
iajs-3051	297	3	,	,	PUNCT
iajs-3051	297	4	disjoint	disjoint	NOUN
iajs-3051	297	5	,	,	PUNCT
iajs-3051	297	6	and	and	CCONJ
iajs-3051	297	7	contain	contain	VERB
iajs-3051	297	8	the	the	DET
iajs-3051	297	9	point	point	NOUN
iajs-3051	297	10	g	g	NOUN
iajs-3051	297	11	and	and	CCONJ
iajs-3051	297	12	the	the	DET
iajs-3051	297	13	set	set	NOUN
iajs-3051	297	14	f	f	NOUN
iajs-3051	297	15	alternately	alternately	ADV
iajs-3051	297	16	.	.	PUNCT
iajs-3051	298	1	remark	remark	VERB
iajs-3051	298	2	3.2	3.2	NUM
iajs-3051	298	3	:	:	PUNCT
iajs-3051	298	4	the	the	DET
iajs-3051	298	5	remaining	remain	VERB
iajs-3051	298	6	axioms	axiom	NOUN
iajs-3051	298	7	of	of	ADP
iajs-3051	298	8	separation	separation	NOUN
iajs-3051	298	9	are	be	AUX
iajs-3051	298	10	not	not	PART
iajs-3051	298	11	inverse	inverse	ADJ
iajs-3051	298	12	invariants	invariant	NOUN
iajs-3051	298	13	of	of	ADP
iajs-3051	298	14	nano	nano	NOUN
iajs-3051	298	15	-	-	PUNCT
iajs-3051	298	16	perf	perf	NOUN
iajs-3051	298	17	-	-	PUNCT
iajs-3051	298	18	maps	map	NOUN
iajs-3051	298	19	.	.	PUNCT
iajs-3051	299	1	also	also	ADV
iajs-3051	299	2	complete	complete	VERB
iajs-3051	299	3	nano	nano	NOUN
iajs-3051	299	4	-	-	PUNCT
iajs-3051	299	5	regularity	regularity	NOUN
iajs-3051	299	6	is	be	AUX
iajs-3051	299	7	not	not	PART
iajs-3051	299	8	inverse	inverse	ADJ
iajs-3051	299	9	invariant	invariant	NOUN
iajs-3051	299	10	of	of	ADP
iajs-3051	299	11	nano	nano	NOUN
iajs-3051	299	12	-	-	PUNCT
iajs-3051	299	13	perf	perf	NOUN
iajs-3051	299	14	-	-	PUNCT
iajs-3051	299	15	maps	map	NOUN
iajs-3051	299	16	.	.	PUNCT
iajs-3051	300	1	see	see	VERB
iajs-3051	300	2	[	[	X
iajs-3051	300	3	8	8	NUM
iajs-3051	300	4	]	]	X
iajs-3051	300	5	exercis	exercis	PROPN
iajs-3051	300	6	3.10.c	3.10.c	NUM
iajs-3051	300	7	(	(	PUNCT
iajs-3051	300	8	c	c	NOUN
iajs-3051	300	9	)	)	PUNCT
iajs-3051	300	10	,	,	PUNCT
iajs-3051	300	11	problem	problem	NOUN
iajs-3051	300	12	3.12.20	3.12.20	NUM
iajs-3051	300	13	(	(	PUNCT
iajs-3051	300	14	e	e	NOUN
iajs-3051	300	15	)	)	PUNCT
iajs-3051	300	16	,	,	PUNCT
iajs-3051	300	17	example	example	NOUN
iajs-3051	300	18	5.1.40	5.1.40	NUM
iajs-3051	300	19	,	,	PUNCT
iajs-3051	300	20	and	and	CCONJ
iajs-3051	300	21	the	the	DET
iajs-3051	300	22	book	book	NOUN
iajs-3051	300	23	.	.	PUNCT
iajs-3051	301	1	theorem	theorem	VERB
iajs-3051	301	2	3.3	3.3	NUM
iajs-3051	301	3	:	:	PUNCT
iajs-3051	301	4	if	if	SCONJ
iajs-3051	301	5	f	f	X
iajs-3051	301	6	:	:	PUNCT
iajs-3051	301	7	g	g	PROPN
iajs-3051	301	8	⟶	⟶	NOUN
iajs-3051	301	9	h	h	NOUN
iajs-3051	301	10	is	be	AUX
iajs-3051	301	11	a	a	DET
iajs-3051	301	12	nano	nano	NOUN
iajs-3051	301	13	-	-	PUNCT
iajs-3051	301	14	perf	perf	NOUN
iajs-3051	301	15	-	-	PUNCT
iajs-3051	301	16	map	map	NOUN
iajs-3051	301	17	,	,	PUNCT
iajs-3051	301	18	then	then	ADV
iajs-3051	301	19	for	for	ADP
iajs-3051	301	20	each	each	DET
iajs-3051	301	21	nano	nano	NOUN
iajs-3051	301	22	-	-	PUNCT
iajs-3051	301	23	comp	comp	NOUN
iajs-3051	301	24	-	-	PUNCT
iajs-3051	301	25	sub	sub	NOUN
iajs-3051	301	26	-	-	NOUN
iajs-3051	301	27	sp	sp	NOUN
iajs-3051	301	28	.	.	PUNCT
iajs-3051	302	1	r	r	NOUN
iajs-3051	302	2	⊂	⊂	PROPN
iajs-3051	302	3	h	h	NOUN
iajs-3051	302	4	the	the	DET
iajs-3051	302	5	inverse	inverse	NOUN
iajs-3051	302	6	image	image	NOUN
iajs-3051	302	7	𝑓−1(r	𝑓−1(r	NOUN
iajs-3051	302	8	)	)	PUNCT
iajs-3051	302	9	is	be	AUX
iajs-3051	302	10	nano	nano	NOUN
iajs-3051	302	11	-	-	PUNCT
iajs-3051	302	12	comp	comp	NOUN
iajs-3051	302	13	.	.	PUNCT
iajs-3051	303	1	proof	proof	NOUN
iajs-3051	303	2	:	:	PUNCT
iajs-3051	303	3	let	let	VERB
iajs-3051	303	4	ṵ	ṵ	NOUN
iajs-3051	303	5	=	=	PUNCT
iajs-3051	303	6	{	{	PUNCT
iajs-3051	303	7	uα	uα	X
iajs-3051	303	8	:	:	PUNCT
iajs-3051	303	9	α	α	PROPN
iajs-3051	303	10	∈	∈	PROPN
iajs-3051	303	11	λ	λ	X
iajs-3051	303	12	}	}	PUNCT
iajs-3051	303	13	be	be	AUX
iajs-3051	303	14	a	a	DET
iajs-3051	303	15	family	family	NOUN
iajs-3051	303	16	of	of	ADP
iajs-3051	303	17	nano	nano	NOUN
iajs-3051	303	18	-	-	PUNCT
iajs-3051	303	19	ope	ope	NOUN
iajs-3051	303	20	-	-	PUNCT
iajs-3051	303	21	sets	set	NOUN
iajs-3051	303	22	of	of	ADP
iajs-3051	303	23	g	g	NOUN
iajs-3051	303	24	such	such	ADJ
iajs-3051	303	25	as	as	ADP
iajs-3051	303	26	𝑓−1(r	𝑓−1(r	NOUN
iajs-3051	303	27	)	)	PUNCT
iajs-3051	304	1	⊂	⊂	PROPN
iajs-3051	304	2	∪α∈λ	∪α∈λ	PROPN
iajs-3051	304	3	uα	uα	PROPN
iajs-3051	304	4	.	.	PUNCT
iajs-3051	305	1	if	if	SCONJ
iajs-3051	305	2	h	h	NOUN
iajs-3051	305	3	∈r	∈r	NOUN
iajs-3051	305	4	,	,	PUNCT
iajs-3051	305	5	then	then	ADV
iajs-3051	305	6	find	find	VERB
iajs-3051	305	7	a	a	DET
iajs-3051	305	8	finite	finite	NOUN
iajs-3051	305	9	subset	subset	NOUN
iajs-3051	305	10	m(h	m(h	NOUN
iajs-3051	305	11	)	)	PUNCT
iajs-3051	305	12	of	of	ADP
iajs-3051	305	13	λ	λ	NOUN
iajs-3051	305	14	such	such	ADJ
iajs-3051	305	15	as	as	ADP
iajs-3051	305	16	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	305	17	)	)	PUNCT
iajs-3051	305	18	⊂	⊂	PROPN
iajs-3051	305	19	∪α∈m(h	∪α∈m(h	NOUN
iajs-3051	305	20	)	)	PUNCT
iajs-3051	305	21	uα	uα	PROPN
iajs-3051	305	22	.	.	PUNCT
iajs-3051	306	1	since	since	SCONJ
iajs-3051	306	2	f	f	PROPN
iajs-3051	306	3	is	be	AUX
iajs-3051	306	4	a	a	DET
iajs-3051	306	5	nano-clos.map	nano-clos.map	NOUN
iajs-3051	306	6	,	,	PUNCT
iajs-3051	306	7	by	by	ADP
iajs-3051	306	8	theorem	theorem	VERB
iajs-3051	306	9	2.6	2.6	NUM
iajs-3051	306	10	find	find	VERB
iajs-3051	306	11	an	an	DET
iajs-3051	306	12	nano	nano	NOUN
iajs-3051	306	13	-	-	PUNCT
iajs-3051	306	14	ope	ope	NOUN
iajs-3051	306	15	-	-	PUNCT
iajs-3051	306	16	set	set	VERB
iajs-3051	306	17	vh	vh	NOUN
iajs-3051	306	18	of	of	ADP
iajs-3051	306	19	h	h	NOUN
iajs-3051	306	20	such	such	ADJ
iajs-3051	306	21	as	as	ADP
iajs-3051	306	22	h	h	PROPN
iajs-3051	306	23	∈	∈	PROPN
iajs-3051	306	24	vh	vh	PROPN
iajs-3051	306	25	and	and	CCONJ
iajs-3051	306	26	𝑓−1(vh	𝑓−1(vh	PROPN
iajs-3051	306	27	)	)	PUNCT
iajs-3051	307	1	⊂	⊂	PROPN
iajs-3051	307	2	∪α∈m(h	∪α∈m(h	NOUN
iajs-3051	307	3	)	)	PUNCT
iajs-3051	307	4	uα	uα	PROPN
iajs-3051	307	5	.	.	PUNCT
iajs-3051	308	1	since	since	SCONJ
iajs-3051	308	2	r	r	NOUN
iajs-3051	308	3	is	be	AUX
iajs-3051	308	4	nano	nano	NOUN
iajs-3051	308	5	-	-	NOUN
iajs-3051	308	6	comp	comp	NOUN
iajs-3051	308	7	.	.	PUNCT
iajs-3051	308	8	,	,	PUNCT
iajs-3051	308	9	find	find	VERB
iajs-3051	308	10	a	a	DET
iajs-3051	308	11	finite	finite	NOUN
iajs-3051	308	12	subset	subset	NOUN
iajs-3051	308	13	b	b	PROPN
iajs-3051	308	14	of	of	ADP
iajs-3051	308	15	r	r	NOUN
iajs-3051	308	16	such	such	ADJ
iajs-3051	308	17	as	as	ADP
iajs-3051	308	18	r	r	PROPN
iajs-3051	308	19	⊂	⊂	X
iajs-3051	308	20	∪h∈b	∪h∈b	NUM
iajs-3051	308	21	vh	vh	NOUN
iajs-3051	308	22	.	.	PUNCT
iajs-3051	309	1	hence	hence	ADV
iajs-3051	309	2	𝑓−1(r	𝑓−1(r	ADV
iajs-3051	309	3	)	)	PUNCT
iajs-3051	310	1	⊂	⊂	X
iajs-3051	310	2	∪h∈b	∪h∈b	NUM
iajs-3051	310	3	𝑓−1(vh	𝑓−1(vh	PROPN
iajs-3051	310	4	)	)	PUNCT
iajs-3051	310	5	⊂	⊂	PRON
iajs-3051	310	6	∪h∈b	∪h∈b	VERB
iajs-3051	310	7	∪α∈m(h	∪α∈m(h	NOUN
iajs-3051	310	8	)	)	PUNCT
iajs-3051	310	9	uα	uα	NOUN
iajs-3051	310	10	.	.	PUNCT
iajs-3051	311	1	thus	thus	ADV
iajs-3051	311	2	if	if	SCONJ
iajs-3051	311	3	m	m	NOUN
iajs-3051	311	4	=	=	VERB
iajs-3051	311	5	∪h∈b	∪h∈b	NUM
iajs-3051	311	6	m(h	m(h	NUM
iajs-3051	311	7	)	)	PUNCT
iajs-3051	311	8	,	,	PUNCT
iajs-3051	311	9	then	then	ADV
iajs-3051	311	10	m	m	VERB
iajs-3051	311	11	is	be	AUX
iajs-3051	311	12	a	a	DET
iajs-3051	311	13	finite	finite	NOUN
iajs-3051	311	14	subset	subset	NOUN
iajs-3051	311	15	of	of	ADP
iajs-3051	311	16	λ	λ	PROPN
iajs-3051	311	17	and	and	CCONJ
iajs-3051	311	18	𝑓−1(r	𝑓−1(r	NOUN
iajs-3051	311	19	)	)	PUNCT
iajs-3051	312	1	⊂	⊂	PROPN
iajs-3051	312	2	∪α∈m	∪α∈m	VERB
iajs-3051	312	3	uα	uα	PROPN
iajs-3051	312	4	.	.	PUNCT
iajs-3051	313	1	thus	thus	ADV
iajs-3051	313	2	𝑓−1(r	𝑓−1(r	NOUN
iajs-3051	313	3	)	)	PUNCT
iajs-3051	313	4	is	be	AUX
iajs-3051	313	5	nano	nano	NOUN
iajs-3051	313	6	-	-	PUNCT
iajs-3051	313	7	comp	comp	NOUN
iajs-3051	313	8	.	.	PUNCT
iajs-3051	314	1	theorem	theorem	VERB
iajs-3051	314	2	3.4	3.4	NUM
iajs-3051	314	3	:	:	PUNCT
iajs-3051	314	4	nano	nano	NOUN
iajs-3051	314	5	-	-	PUNCT
iajs-3051	314	6	compactness	compactness	NOUN
iajs-3051	314	7	and	and	CCONJ
iajs-3051	314	8	local	local	ADJ
iajs-3051	314	9	nano	nano	NOUN
iajs-3051	314	10	-	-	PUNCT
iajs-3051	314	11	compactness	compactness	NOUN
iajs-3051	314	12	are	be	AUX
iajs-3051	314	13	inverse	inverse	NOUN
iajs-3051	314	14	invariants	invariant	NOUN
iajs-3051	314	15	of	of	ADP
iajs-3051	314	16	nanoperf	nanoperf	NOUN
iajs-3051	314	17	-	-	PUNCT
iajs-3051	314	18	maps	map	NOUN
iajs-3051	314	19	.	.	PUNCT
iajs-3051	315	1	proof	proof	NOUN
iajs-3051	315	2	:	:	PUNCT
iajs-3051	315	3	the	the	DET
iajs-3051	315	4	inverse	inverse	NOUN
iajs-3051	315	5	invariance	invariance	NOUN
iajs-3051	315	6	of	of	ADP
iajs-3051	315	7	nano	nano	NOUN
iajs-3051	315	8	-	-	PUNCT
iajs-3051	315	9	compactness	compactness	NOUN
iajs-3051	315	10	follows	follow	VERB
iajs-3051	315	11	directly	directly	ADV
iajs-3051	315	12	from	from	ADP
iajs-3051	315	13	theorem	theorem	ADJ
iajs-3051	315	14	3.3	3.3	NUM
iajs-3051	315	15	.	.	PUNCT
iajs-3051	316	1	if	if	SCONJ
iajs-3051	316	2	f	f	PROPN
iajs-3051	316	3	:	:	PUNCT
iajs-3051	316	4	g	g	PROPN
iajs-3051	316	5	⟶	⟶	NOUN
iajs-3051	316	6	h	h	NOUN
iajs-3051	316	7	is	be	AUX
iajs-3051	316	8	a	a	DET
iajs-3051	316	9	nano	nano	NOUN
iajs-3051	316	10	-	-	PUNCT
iajs-3051	316	11	perf	perf	NOUN
iajs-3051	316	12	-	-	PUNCT
iajs-3051	316	13	map	map	NOUN
iajs-3051	316	14	and	and	CCONJ
iajs-3051	316	15	h	h	NOUN
iajs-3051	316	16	is	be	AUX
iajs-3051	316	17	a	a	DET
iajs-3051	316	18	locally	locally	ADV
iajs-3051	316	19	nano	nano	NOUN
iajs-3051	316	20	-	-	PUNCT
iajs-3051	316	21	comp	comp	NOUN
iajs-3051	316	22	-	-	PUNCT
iajs-3051	316	23	sp	sp	NOUN
iajs-3051	316	24	.	.	PROPN
iajs-3051	316	25	,	,	PUNCT
iajs-3051	316	26	then	then	ADV
iajs-3051	316	27	for	for	ADP
iajs-3051	316	28	each	each	DET
iajs-3051	316	29	g	g	PROPN
iajs-3051	316	30	∈	∈	PROPN
iajs-3051	316	31	g	g	NOUN
iajs-3051	316	32	,	,	PUNCT
iajs-3051	316	33	find	find	VERB
iajs-3051	316	34	a	a	DET
iajs-3051	316	35	nbd	nbd	PROPN
iajs-3051	316	36	u	u	PROPN
iajs-3051	316	37	⊂	⊂	PROPN
iajs-3051	316	38	g	g	PROPN
iajs-3051	316	39	such	such	ADJ
iajs-3051	316	40	as	as	ADP
iajs-3051	316	41	f(u	f(u	PROPN
iajs-3051	316	42	)	)	PUNCT
iajs-3051	316	43	is	be	AUX
iajs-3051	316	44	contained	contain	VERB
iajs-3051	316	45	in	in	ADP
iajs-3051	316	46	a	a	DET
iajs-3051	316	47	nano	nano	NOUN
iajs-3051	316	48	-	-	PUNCT
iajs-3051	316	49	comp	comp	NOUN
iajs-3051	316	50	-	-	PUNCT
iajs-3051	316	51	sub	sub	NOUN
iajs-3051	316	52	-	-	NOUN
iajs-3051	316	53	sp	sp	NOUN
iajs-3051	316	54	.	.	PUNCT
iajs-3051	317	1	i	i	PRON
iajs-3051	317	2	of	of	ADP
iajs-3051	317	3	the	the	DET
iajs-3051	317	4	space	space	NOUN
iajs-3051	317	5	h.	h.	PROPN
iajs-3051	317	6	since	since	SCONJ
iajs-3051	317	7	f(ncl(u	f(ncl(u	PROPN
iajs-3051	317	8	)	)	PUNCT
iajs-3051	317	9	)	)	PUNCT
iajs-3051	318	1	⊂	⊂	PROPN
iajs-3051	318	2	ncl(f(u	ncl(f(u	ADJ
iajs-3051	318	3	)	)	PUNCT
iajs-3051	318	4	)	)	PUNCT
iajs-3051	319	1	⊂	⊂	PROPN
iajs-3051	319	2	r	r	X
iajs-3051	319	3	,	,	PUNCT
iajs-3051	319	4	the	the	DET
iajs-3051	319	5	set	set	NOUN
iajs-3051	319	6	ncl(u	ncl(u	PROPN
iajs-3051	319	7	)	)	PUNCT
iajs-3051	319	8	⊂	⊂	PROPN
iajs-3051	319	9	𝑓−1(r	𝑓−1(r	PROPN
iajs-3051	319	10	)	)	PUNCT
iajs-3051	319	11	is	be	AUX
iajs-3051	319	12	nano	nano	NOUN
iajs-3051	319	13	-	-	PUNCT
iajs-3051	319	14	comp	comp	NOUN
iajs-3051	319	15	.	.	PUNCT
iajs-3051	320	1	theorem	theorem	VERB
iajs-3051	320	2	3.5	3.5	NUM
iajs-3051	320	3	:	:	PUNCT
iajs-3051	320	4	nano	nano	NOUN
iajs-3051	320	5	-	-	PUNCT
iajs-3051	320	6	para	para	NOUN
iajs-3051	320	7	-	-	PUNCT
iajs-3051	320	8	compactness	compactness	NOUN
iajs-3051	320	9	is	be	AUX
iajs-3051	320	10	an	an	DET
iajs-3051	320	11	inverse	inverse	ADJ
iajs-3051	320	12	invariant	invariant	NOUN
iajs-3051	320	13	of	of	ADP
iajs-3051	320	14	nano	nano	NOUN
iajs-3051	320	15	-	-	PUNCT
iajs-3051	320	16	perf	perf	NOUN
iajs-3051	320	17	-	-	PUNCT
iajs-3051	320	18	maps	map	NOUN
iajs-3051	320	19	.	.	PUNCT
iajs-3051	321	1	proof	proof	NOUN
iajs-3051	321	2	:	:	PUNCT
iajs-3051	321	3	let	let	VERB
iajs-3051	321	4	f	f	PRON
iajs-3051	321	5	:	:	PUNCT
iajs-3051	321	6	g	g	PROPN
iajs-3051	321	7	⟶	⟶	PROPN
iajs-3051	321	8	h	h	NOUN
iajs-3051	321	9	be	be	AUX
iajs-3051	321	10	a	a	DET
iajs-3051	321	11	nano	nano	NOUN
iajs-3051	321	12	-	-	PUNCT
iajs-3051	321	13	perf	perf	NOUN
iajs-3051	321	14	-	-	PUNCT
iajs-3051	321	15	map	map	NOUN
iajs-3051	321	16	.	.	PUNCT
iajs-3051	322	1	onto	onto	ADP
iajs-3051	322	2	a	a	DET
iajs-3051	322	3	nano	nano	NOUN
iajs-3051	322	4	-	-	PUNCT
iajs-3051	322	5	para	para	ADJ
iajs-3051	322	6	-	-	PUNCT
iajs-3051	322	7	comp	comp	NOUN
iajs-3051	322	8	-	-	PUNCT
iajs-3051	322	9	sp.h	sp.h	NOUN
iajs-3051	322	10	.	.	PUNCT
iajs-3051	323	1	consider	consider	VERB
iajs-3051	323	2	an	an	DET
iajs-3051	323	3	nano	nano	NOUN
iajs-3051	323	4	-	-	PUNCT
iajs-3051	323	5	opecover	opecover	NOUN
iajs-3051	323	6	{	{	PUNCT
iajs-3051	323	7	us}s∈s	us}s∈s	PROPN
iajs-3051	323	8	of	of	ADP
iajs-3051	323	9	the	the	DET
iajs-3051	323	10	nano	nano	NOUN
iajs-3051	323	11	-	-	PUNCT
iajs-3051	323	12	space	space	NOUN
iajs-3051	323	13	g	g	NOUN
iajs-3051	323	14	and	and	CCONJ
iajs-3051	323	15	for	for	ADP
iajs-3051	323	16	each	each	DET
iajs-3051	323	17	h	h	NOUN
iajs-3051	323	18	∈	∈	NOUN
iajs-3051	323	19	h	h	NOUN
iajs-3051	323	20	choose	choose	VERB
iajs-3051	323	21	a	a	DET
iajs-3051	323	22	finite	finite	NOUN
iajs-3051	323	23	set	set	VERB
iajs-3051	323	24	s(h	s(h	PROPN
iajs-3051	323	25	)	)	PUNCT
iajs-3051	324	1	⊂	⊂	PRON
iajs-3051	324	2	s	s	VERB
iajs-3051	324	3	such	such	ADJ
iajs-3051	324	4	as	as	ADP
iajs-3051	324	5	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	324	6	)	)	PUNCT
iajs-3051	325	1	⊂	⊂	PROPN
iajs-3051	325	2	⋃	⋃	PROPN
iajs-3051	325	3	uss∈s(h	uss∈s(h	PROPN
iajs-3051	325	4	)	)	PUNCT
iajs-3051	325	5	.	.	PUNCT
iajs-3051	326	1	since	since	SCONJ
iajs-3051	326	2	f	f	PROPN
iajs-3051	326	3	is	be	AUX
iajs-3051	326	4	nano	nano	NOUN
iajs-3051	326	5	-	-	PUNCT
iajs-3051	326	6	clos	clo	NOUN
iajs-3051	326	7	.	.	PUNCT
iajs-3051	326	8	,	,	PUNCT
iajs-3051	326	9	by	by	ADP
iajs-3051	326	10	theorem	theorem	NOUN
iajs-3051	326	11	2.9	2.9	NUM
iajs-3051	326	12	,	,	PUNCT
iajs-3051	326	13	find	find	VERB
iajs-3051	326	14	a	a	DET
iajs-3051	326	15	nbd	nbd	PROPN
iajs-3051	326	16	𝑉ℎ	𝑉ℎ	PROPN
iajs-3051	326	17	of	of	ADP
iajs-3051	326	18	the	the	DET
iajs-3051	326	19	point	point	NOUN
iajs-3051	326	20	h	h	NOUN
iajs-3051	326	21	such	such	ADJ
iajs-3051	326	22	as	as	ADP
iajs-3051	326	23	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	326	24	)	)	PUNCT
iajs-3051	326	25	⊂	⊂	PROPN
iajs-3051	326	26	𝑓−1(vh	𝑓−1(vh	PROPN
iajs-3051	326	27	)	)	PUNCT
iajs-3051	326	28	⊂	⊂	PROPN
iajs-3051	326	29	⋃	⋃	PROPN
iajs-3051	326	30	uss∈s(h	uss∈s(h	PROPN
iajs-3051	326	31	)	)	PUNCT
iajs-3051	326	32	.	.	PUNCT
iajs-3051	327	1	the	the	DET
iajs-3051	327	2	nano	nano	NOUN
iajs-3051	327	3	-	-	PUNCT
iajs-3051	327	4	ope	ope	NOUN
iajs-3051	327	5	-	-	PUNCT
iajs-3051	327	6	cover	cover	NOUN
iajs-3051	327	7	{	{	PUNCT
iajs-3051	327	8	vh}h∈h	vh}h∈h	NOUN
iajs-3051	327	9	of	of	ADP
iajs-3051	327	10	h	h	NOUN
iajs-3051	327	11	has	have	VERB
iajs-3051	327	12	an	an	DET
iajs-3051	327	13	nano	nano	NOUN
iajs-3051	327	14	-	-	PUNCT
iajs-3051	327	15	ope	ope	NOUN
iajs-3051	327	16	.	.	PUNCT
iajs-3051	328	1	locally	locally	ADV
iajs-3051	328	2	finite	finite	VERB
iajs-3051	328	3	improvement	improvement	NOUN
iajs-3051	328	4	{	{	PUNCT
iajs-3051	328	5	wt}t∈t	wt}t∈t	PROPN
iajs-3051	328	6	.	.	PUNCT
iajs-3051	329	1	the	the	DET
iajs-3051	329	2	family	family	NOUN
iajs-3051	329	3	{	{	PUNCT
iajs-3051	329	4	𝑓−1(wt)}t∈t	𝑓−1(wt)}t∈t	PROPN
iajs-3051	329	5	is	be	AUX
iajs-3051	329	6	an	an	DET
iajs-3051	329	7	nano	nano	NOUN
iajs-3051	329	8	-	-	PUNCT
iajs-3051	329	9	ope	ope	NOUN
iajs-3051	329	10	.	.	PUNCT
iajs-3051	330	1	locally	locally	ADV
iajs-3051	330	2	finite	finite	ADJ
iajs-3051	330	3	-	-	NOUN
iajs-3051	330	4	cover	cover	NOUN
iajs-3051	330	5	of	of	ADP
iajs-3051	330	6	g	g	NOUN
iajs-3051	330	7	and	and	CCONJ
iajs-3051	330	8	for	for	ADP
iajs-3051	330	9	each	each	DET
iajs-3051	330	10	t	t	PROPN
iajs-3051	330	11	∈	∈	PROPN
iajs-3051	330	12	t	t	PROPN
iajs-3051	330	13	find	find	VERB
iajs-3051	330	14	a	a	DET
iajs-3051	330	15	ht	ht	PROPN
iajs-3051	330	16	∈	∈	PROPN
iajs-3051	330	17	h	h	NOUN
iajs-3051	330	18	satisfying	satisfy	VERB
iajs-3051	330	19	𝑓−1(wt	𝑓−1(wt	NOUN
iajs-3051	330	20	)	)	PUNCT
iajs-3051	330	21	⊂	⊂	PROPN
iajs-3051	330	22	𝑓−1(vht	𝑓−1(vht	ADV
iajs-3051	330	23	)	)	PUNCT
iajs-3051	331	1	⊂	⊂	PROPN
iajs-3051	331	2	⋃	⋃	VERB
iajs-3051	331	3	uss∈s(ht	uss∈s(ht	NOUN
iajs-3051	331	4	)	)	PUNCT
iajs-3051	331	5	.	.	PUNCT
iajs-3051	332	1	so	so	ADV
iajs-3051	332	2	the	the	DET
iajs-3051	332	3	family	family	NOUN
iajs-3051	332	4	{	{	PUNCT
iajs-3051	332	5	𝑓−1(wt	𝑓−1(wt	PROPN
iajs-3051	332	6	)	)	PUNCT
iajs-3051	332	7	⋂	⋂	PROPN
iajs-3051	332	8	us	we	PRON
iajs-3051	332	9	:	:	PUNCT
iajs-3051	332	10	t	t	PROPN
iajs-3051	332	11	∈	∈	PROPN
iajs-3051	332	12	t	t	PROPN
iajs-3051	332	13	and	and	CCONJ
iajs-3051	332	14	s	s	PROPN
iajs-3051	332	15	∈	∈	PROPN
iajs-3051	332	16	s(ht	s(ht	NOUN
iajs-3051	332	17	)	)	PUNCT
iajs-3051	332	18	}	}	PUNCT
iajs-3051	332	19	is	be	AUX
iajs-3051	332	20	an	an	DET
iajs-3051	332	21	nano	nano	NOUN
iajs-3051	332	22	-	-	PUNCT
iajs-3051	332	23	ope	ope	NOUN
iajs-3051	332	24	.	.	PUNCT
iajs-3051	333	1	locally	locally	ADV
iajs-3051	333	2	finite	finite	VERB
iajs-3051	333	3	improvement	improvement	NOUN
iajs-3051	333	4	of	of	ADP
iajs-3051	333	5	{	{	PUNCT
iajs-3051	333	6	us}s∈s	us}s∈s	PROPN
iajs-3051	333	7	.	.	PROPN
iajs-3051	333	8	ihjpas	ihjpas	PROPN
iajs-3051	333	9	.	.	PUNCT
iajs-3051	334	1	36	36	NUM
iajs-3051	334	2	(	(	PUNCT
iajs-3051	334	3	3	3	NUM
iajs-3051	334	4	)	)	PUNCT
iajs-3051	334	5	2023	2023	NUM
iajs-3051	334	6	405	405	NUM
iajs-3051	334	7	theorem	theorem	VERB
iajs-3051	334	8	3.6	3.6	NUM
iajs-3051	334	9	:	:	PUNCT
iajs-3051	334	10	the	the	DET
iajs-3051	334	11	cartesian	cartesian	ADJ
iajs-3051	334	12	product	product	NOUN
iajs-3051	334	13	g×h	g×h	PROPN
iajs-3051	334	14	of	of	ADP
iajs-3051	334	15	a	a	DET
iajs-3051	334	16	nano	nano	NOUN
iajs-3051	334	17	-	-	PUNCT
iajs-3051	334	18	para	para	ADJ
iajs-3051	334	19	-	-	PUNCT
iajs-3051	334	20	comp	comp	NOUN
iajs-3051	334	21	-	-	PUNCT
iajs-3051	334	22	sp	sp	NOUN
iajs-3051	334	23	.	.	NOUN
iajs-3051	334	24	g	g	PROPN
iajs-3051	334	25	and	and	CCONJ
iajs-3051	334	26	a	a	DET
iajs-3051	334	27	nano	nano	NOUN
iajs-3051	334	28	-	-	PUNCT
iajs-3051	334	29	comp	comp	NOUN
iajs-3051	334	30	-	-	PUNCT
iajs-3051	334	31	sp	sp	NOUN
iajs-3051	334	32	.	.	PUNCT
iajs-3051	335	1	h	h	PROPN
iajs-3051	335	2	is	be	AUX
iajs-3051	335	3	nano	nano	NOUN
iajs-3051	335	4	-	-	PUNCT
iajs-3051	335	5	para	para	NOUN
iajs-3051	335	6	-	-	PUNCT
iajs-3051	335	7	comp	comp	NOUN
iajs-3051	335	8	.	.	PUNCT
iajs-3051	336	1	proof	proof	NOUN
iajs-3051	336	2	:	:	PUNCT
iajs-3051	336	3	consider	consider	VERB
iajs-3051	336	4	the	the	DET
iajs-3051	336	5	projection	projection	NOUN
iajs-3051	336	6	p	p	X
iajs-3051	336	7	:	:	PUNCT
iajs-3051	336	8	g×h	g×h	VERB
iajs-3051	336	9	⟶	⟶	NOUN
iajs-3051	336	10	h.	h.	NOUN
iajs-3051	336	11	since	since	SCONJ
iajs-3051	336	12	h	h	PROPN
iajs-3051	336	13	is	be	AUX
iajs-3051	336	14	nano	nano	NOUN
iajs-3051	336	15	-	-	NOUN
iajs-3051	336	16	comp	comp	NOUN
iajs-3051	336	17	.	.	PUNCT
iajs-3051	337	1	,	,	PUNCT
iajs-3051	337	2	then	then	ADV
iajs-3051	337	3	p	p	PROPN
iajs-3051	337	4	is	be	AUX
iajs-3051	337	5	nano	nano	NOUN
iajs-3051	337	6	-	-	PUNCT
iajs-3051	337	7	perf	perf	NOUN
iajs-3051	337	8	.	.	PUNCT
iajs-3051	338	1	since	since	SCONJ
iajs-3051	338	2	h	h	PROPN
iajs-3051	338	3	is	be	AUX
iajs-3051	338	4	nano	nano	NOUN
iajs-3051	338	5	-	-	PUNCT
iajs-3051	338	6	para	para	NOUN
iajs-3051	338	7	-	-	PUNCT
iajs-3051	338	8	comp	comp	NOUN
iajs-3051	338	9	.	.	PUNCT
iajs-3051	338	10	,	,	PUNCT
iajs-3051	338	11	therefore	therefore	ADV
iajs-3051	338	12	by	by	ADP
iajs-3051	338	13	theorem	theorem	NOUN
iajs-3051	338	14	3.5	3.5	NUM
iajs-3051	338	15	,	,	PUNCT
iajs-3051	338	16	g×h	g×h	PROPN
iajs-3051	338	17	is	be	AUX
iajs-3051	338	18	nano	nano	NOUN
iajs-3051	338	19	-	-	PUNCT
iajs-3051	338	20	para	para	NOUN
iajs-3051	338	21	-	-	PUNCT
iajs-3051	338	22	comp	comp	NOUN
iajs-3051	338	23	.	.	PUNCT
iajs-3051	339	1	definition	definition	NOUN
iajs-3051	339	2	3.7	3.7	NUM
iajs-3051	339	3	:	:	PUNCT
iajs-3051	340	1	[	[	X
iajs-3051	340	2	22	22	NUM
iajs-3051	340	3	]	]	PUNCT
iajs-3051	340	4	a	a	DET
iajs-3051	340	5	family	family	NOUN
iajs-3051	340	6	{	{	PUNCT
iajs-3051	340	7	bα	bα	NOUN
iajs-3051	340	8	:	:	PUNCT
iajs-3051	340	9	α	α	PROPN
iajs-3051	340	10	∈	∈	PROPN
iajs-3051	340	11	γ	γ	X
iajs-3051	340	12	}	}	PUNCT
iajs-3051	340	13	of	of	ADP
iajs-3051	340	14	subsets	subset	NOUN
iajs-3051	340	15	of	of	ADP
iajs-3051	340	16	a	a	DET
iajs-3051	340	17	space	space	NOUN
iajs-3051	340	18	g	g	NOUN
iajs-3051	340	19	is	be	AUX
iajs-3051	340	20	said	say	VERB
iajs-3051	340	21	to	to	PART
iajs-3051	340	22	be	be	AUX
iajs-3051	340	23	a	a	DET
iajs-3051	340	24	weak	weak	ADJ
iajs-3051	340	25	improvement	improvement	NOUN
iajs-3051	340	26	of	of	ADP
iajs-3051	340	27	covering	cover	VERB
iajs-3051	340	28	u	u	NOUN
iajs-3051	340	29	∼	∼	NOUN
iajs-3051	340	30	of	of	ADP
iajs-3051	340	31	g	g	NOUN
iajs-3051	340	32	if	if	SCONJ
iajs-3051	340	33	find	find	VERB
iajs-3051	340	34	a	a	DET
iajs-3051	340	35	subset	subset	NOUN
iajs-3051	340	36	γ`	γ`	NOUN
iajs-3051	340	37	of	of	ADP
iajs-3051	340	38	γ	γ	PRON
iajs-3051	340	39	such	such	ADJ
iajs-3051	340	40	as	as	ADP
iajs-3051	340	41	{	{	PUNCT
iajs-3051	340	42	bα	bα	NOUN
iajs-3051	340	43	:	:	PUNCT
iajs-3051	340	44	α	α	PROPN
iajs-3051	340	45	∈	∈	PROPN
iajs-3051	340	46	γ`	γ`	NOUN
iajs-3051	340	47	}	}	PUNCT
iajs-3051	340	48	is	be	AUX
iajs-3051	340	49	a	a	DET
iajs-3051	340	50	covering	covering	NOUN
iajs-3051	340	51	of	of	ADP
iajs-3051	340	52	g	g	PROPN
iajs-3051	340	53	and	and	CCONJ
iajs-3051	340	54	a	a	DET
iajs-3051	340	55	improvement	improvement	NOUN
iajs-3051	340	56	of	of	ADP
iajs-3051	340	57	u	u	NOUN
iajs-3051	340	58	∼	∼	NOUN
iajs-3051	340	59	.	.	PUNCT
iajs-3051	341	1	definition	definition	NOUN
iajs-3051	341	2	3.8	3.8	NUM
iajs-3051	341	3	:	:	PUNCT
iajs-3051	341	4	a	a	DET
iajs-3051	341	5	nano	nano	NOUN
iajs-3051	341	6	-	-	PUNCT
iajs-3051	341	7	top	top	NOUN
iajs-3051	341	8	-	-	PUNCT
iajs-3051	341	9	sp	sp	NOUN
iajs-3051	341	10	.	.	PUNCT
iajs-3051	342	1	g	g	PROPN
iajs-3051	342	2	is	be	AUX
iajs-3051	342	3	called	call	VERB
iajs-3051	342	4	completely	completely	ADV
iajs-3051	342	5	nano	nano	NOUN
iajs-3051	342	6	-	-	PUNCT
iajs-3051	342	7	para	para	NOUN
iajs-3051	342	8	-	-	PUNCT
iajs-3051	342	9	comp	comp	NOUN
iajs-3051	342	10	.	.	PUNCT
iajs-3051	343	1	if	if	SCONJ
iajs-3051	343	2	all	all	DET
iajs-3051	343	3	nano	nano	NOUN
iajs-3051	343	4	-	-	PUNCT
iajs-3051	343	5	ope	ope	NOUN
iajs-3051	343	6	-	-	PUNCT
iajs-3051	343	7	covering	covering	NOUN
iajs-3051	343	8	of	of	ADP
iajs-3051	343	9	g	g	PROPN
iajs-3051	343	10	has	have	VERB
iajs-3051	343	11	a	a	DET
iajs-3051	343	12	weak	weak	ADJ
iajs-3051	343	13	improvement	improvement	NOUN
iajs-3051	343	14	of	of	ADP
iajs-3051	343	15	the	the	DET
iajs-3051	343	16	form	form	NOUN
iajs-3051	343	17	{	{	PUNCT
iajs-3051	343	18	vα	vα	PROPN
iajs-3051	343	19	:	:	PUNCT
iajs-3051	343	20	α	α	PROPN
iajs-3051	343	21	∈	∈	PROPN
iajs-3051	343	22	λ	λ	PROPN
iajs-3051	343	23	}	}	PUNCT
iajs-3051	343	24	where	where	SCONJ
iajs-3051	343	25	λ	λ	X
iajs-3051	343	26	=	=	VERB
iajs-3051	343	27	⋃n∈nλn	⋃n∈nλn	PRON
iajs-3051	343	28	and	and	CCONJ
iajs-3051	343	29	{	{	PUNCT
iajs-3051	343	30	v𝛼	v𝛼	PRON
iajs-3051	343	31	:	:	PUNCT
iajs-3051	343	32	α	α	PROPN
iajs-3051	343	33	∈	∈	PROPN
iajs-3051	343	34	λn	λn	PROPN
iajs-3051	343	35	}	}	PUNCT
iajs-3051	343	36	is	be	AUX
iajs-3051	343	37	a	a	DET
iajs-3051	343	38	star	star	NOUN
iajs-3051	343	39	finite	finite	ADJ
iajs-3051	343	40	nano	nano	NOUN
iajs-3051	343	41	-	-	PUNCT
iajs-3051	343	42	ope	ope	NOUN
iajs-3051	343	43	-	-	PUNCT
iajs-3051	343	44	covering	covering	NOUN
iajs-3051	343	45	of	of	ADP
iajs-3051	343	46	g	g	NOUN
iajs-3051	343	47	for	for	ADP
iajs-3051	343	48	all	all	DET
iajs-3051	343	49	n.	n.	NOUN
iajs-3051	343	50	theorem	theorem	VERB
iajs-3051	343	51	3.9	3.9	NUM
iajs-3051	343	52	:	:	PUNCT
iajs-3051	343	53	let	let	VERB
iajs-3051	343	54	f	f	PRON
iajs-3051	343	55	:	:	PUNCT
iajs-3051	343	56	g	g	PROPN
iajs-3051	343	57	⟶	⟶	PROPN
iajs-3051	343	58	h	h	NOUN
iajs-3051	343	59	be	be	AUX
iajs-3051	343	60	a	a	DET
iajs-3051	343	61	nano	nano	NOUN
iajs-3051	343	62	-	-	PUNCT
iajs-3051	343	63	perf	perf	NOUN
iajs-3051	343	64	.	.	PUNCT
iajs-3051	344	1	onto	onto	ADP
iajs-3051	344	2	map	map	NOUN
iajs-3051	344	3	.	.	PUNCT
iajs-3051	345	1	if	if	SCONJ
iajs-3051	345	2	h	h	NOUN
iajs-3051	345	3	is	be	AUX
iajs-3051	345	4	a	a	DET
iajs-3051	345	5	completely	completely	ADV
iajs-3051	345	6	nano	nano	NOUN
iajs-3051	345	7	-	-	PUNCT
iajs-3051	345	8	para	para	ADJ
iajs-3051	345	9	-	-	PUNCT
iajs-3051	345	10	comp	comp	NOUN
iajs-3051	345	11	-	-	PUNCT
iajs-3051	345	12	sp	sp	NOUN
iajs-3051	345	13	.	.	PROPN
iajs-3051	345	14	,	,	PUNCT
iajs-3051	345	15	then	then	ADV
iajs-3051	345	16	g	g	PROPN
iajs-3051	345	17	is	be	AUX
iajs-3051	345	18	a	a	DET
iajs-3051	345	19	completely	completely	ADV
iajs-3051	345	20	nano	nano	NOUN
iajs-3051	345	21	-	-	PUNCT
iajs-3051	345	22	para	para	ADJ
iajs-3051	345	23	-	-	PUNCT
iajs-3051	345	24	comp	comp	NOUN
iajs-3051	345	25	-	-	PUNCT
iajs-3051	345	26	sp	sp	NOUN
iajs-3051	345	27	.	.	PUNCT
iajs-3051	345	28	proof	proof	NOUN
iajs-3051	345	29	:	:	PUNCT
iajs-3051	345	30	let	let	VERB
iajs-3051	345	31	u	u	PRON
iajs-3051	345	32	~	~	PUNCT
iajs-3051	345	33	=	=	SYM
iajs-3051	345	34	{	{	PUNCT
iajs-3051	345	35	uα	uα	NOUN
iajs-3051	345	36	:	:	PUNCT
iajs-3051	345	37	α	α	PROPN
iajs-3051	345	38	∈	∈	PROPN
iajs-3051	345	39	λ	λ	AUX
iajs-3051	345	40	}	}	PUNCT
iajs-3051	345	41	be	be	AUX
iajs-3051	345	42	an	an	DET
iajs-3051	345	43	nano	nano	NOUN
iajs-3051	345	44	-	-	PUNCT
iajs-3051	345	45	ope	ope	NOUN
iajs-3051	345	46	-	-	PUNCT
iajs-3051	345	47	cover	cover	NOUN
iajs-3051	345	48	of	of	ADP
iajs-3051	345	49	g.	g.	PROPN
iajs-3051	345	50	if	if	SCONJ
iajs-3051	345	51	h	h	PROPN
iajs-3051	345	52	∈	∈	PROPN
iajs-3051	345	53	h	h	NOUN
iajs-3051	345	54	,	,	PUNCT
iajs-3051	345	55	find	find	VERB
iajs-3051	345	56	a	a	DET
iajs-3051	345	57	finite	finite	NOUN
iajs-3051	345	58	subset	subset	NOUN
iajs-3051	345	59	λ(h	λ(h	NOUN
iajs-3051	345	60	)	)	PUNCT
iajs-3051	345	61	of	of	ADP
iajs-3051	345	62	λ	λ	NOUN
iajs-3051	345	63	such	such	ADJ
iajs-3051	345	64	as	as	ADP
iajs-3051	345	65	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	345	66	)	)	PUNCT
iajs-3051	346	1	⊂	⊂	PROPN
iajs-3051	346	2	⋃α∈λ(h)uα	⋃α∈λ(h)uα	PROPN
iajs-3051	346	3	.	.	PUNCT
iajs-3051	347	1	so	so	ADV
iajs-3051	347	2	f	f	PROPN
iajs-3051	347	3	is	be	AUX
iajs-3051	347	4	a	a	DET
iajs-3051	347	5	nano	nano	NOUN
iajs-3051	347	6	-	-	PUNCT
iajs-3051	347	7	clos	clo	NOUN
iajs-3051	347	8	.	.	PUNCT
iajs-3051	347	9	map	map	NOUN
iajs-3051	347	10	,	,	PUNCT
iajs-3051	347	11	find	find	VERB
iajs-3051	347	12	an	an	DET
iajs-3051	347	13	nano	nano	NOUN
iajs-3051	347	14	-	-	PUNCT
iajs-3051	347	15	ope	ope	NOUN
iajs-3051	347	16	-	-	PUNCT
iajs-3051	347	17	set	set	VERB
iajs-3051	347	18	wh	wh	NOUN
iajs-3051	347	19	of	of	ADP
iajs-3051	347	20	h	h	NOUN
iajs-3051	347	21	such	such	ADJ
iajs-3051	347	22	as	as	ADP
iajs-3051	347	23	h	h	NOUN
iajs-3051	347	24	∈	∈	PROPN
iajs-3051	347	25	wh	wh	NOUN
iajs-3051	347	26	and	and	CCONJ
iajs-3051	347	27	𝑓−1(wh	𝑓−1(wh	PROPN
iajs-3051	347	28	)	)	PUNCT
iajs-3051	348	1	⊂	⊂	PROPN
iajs-3051	348	2	⋃𝛼∈λ(h)uα	⋃𝛼∈λ(h)uα	PROPN
iajs-3051	348	3	,	,	PUNCT
iajs-3051	348	4	and	and	CCONJ
iajs-3051	348	5	w	w	NOUN
iajs-3051	348	6	∼	∼	NOUN
iajs-3051	348	7	=	=	PUNCT
iajs-3051	348	8	{	{	PUNCT
iajs-3051	348	9	wh	wh	PROPN
iajs-3051	348	10	:	:	PUNCT
iajs-3051	348	11	h	h	PROPN
iajs-3051	348	12	∈	∈	PROPN
iajs-3051	348	13	h	h	PROPN
iajs-3051	348	14	}	}	PUNCT
iajs-3051	348	15	is	be	AUX
iajs-3051	348	16	an	an	DET
iajs-3051	348	17	nano	nano	NOUN
iajs-3051	348	18	-	-	PUNCT
iajs-3051	348	19	ope	ope	NOUN
iajs-3051	348	20	-	-	PUNCT
iajs-3051	348	21	covering	covering	NOUN
iajs-3051	348	22	of	of	ADP
iajs-3051	348	23	h.	h.	NOUN
iajs-3051	349	1	so	so	ADV
iajs-3051	349	2	h	h	PROPN
iajs-3051	349	3	is	be	AUX
iajs-3051	349	4	completely	completely	ADV
iajs-3051	349	5	nano	nano	VERB
iajs-3051	349	6	-	-	PUNCT
iajs-3051	349	7	para	para	NOUN
iajs-3051	349	8	-	-	PUNCT
iajs-3051	349	9	comp	comp	NOUN
iajs-3051	349	10	.	.	PUNCT
iajs-3051	349	11	,	,	PUNCT
iajs-3051	349	12	find	find	VERB
iajs-3051	349	13	a	a	DET
iajs-3051	349	14	family	family	NOUN
iajs-3051	349	15	a	a	DET
iajs-3051	349	16	{	{	PUNCT
iajs-3051	349	17	vβ	vβ	NOUN
iajs-3051	349	18	:	:	PUNCT
iajs-3051	349	19	β	β	NOUN
iajs-3051	349	20	∈	∈	PROPN
iajs-3051	349	21	γ	γ	X
iajs-3051	349	22	}	}	PUNCT
iajs-3051	349	23	of	of	ADP
iajs-3051	349	24	nano	nano	NOUN
iajs-3051	349	25	-	-	PUNCT
iajs-3051	349	26	ope	ope	NOUN
iajs-3051	349	27	-	-	PUNCT
iajs-3051	349	28	sets	set	NOUN
iajs-3051	349	29	of	of	ADP
iajs-3051	349	30	h	h	NOUN
iajs-3051	349	31	such	such	ADJ
iajs-3051	349	32	that	that	PRON
iajs-3051	349	33	:	:	PUNCT
iajs-3051	349	34	(	(	PUNCT
iajs-3051	349	35	a	a	X
iajs-3051	349	36	)	)	PUNCT
iajs-3051	349	37	γ	γ	NOUN
iajs-3051	349	38	=	=	SYM
iajs-3051	349	39	⋃n∈n	⋃n∈n	PROPN
iajs-3051	349	40	γ(n	γ(n	X
iajs-3051	349	41	)	)	PUNCT
iajs-3051	349	42	and	and	CCONJ
iajs-3051	349	43	the	the	DET
iajs-3051	349	44	family	family	NOUN
iajs-3051	349	45	{	{	PUNCT
iajs-3051	349	46	vβ	vβ	NOUN
iajs-3051	349	47	:	:	PUNCT
iajs-3051	349	48	β	β	X
iajs-3051	349	49	∈	∈	PROPN
iajs-3051	349	50	γ(n	γ(n	X
iajs-3051	349	51	)	)	PUNCT
iajs-3051	349	52	}	}	PUNCT
iajs-3051	349	53	is	be	AUX
iajs-3051	349	54	a	a	DET
iajs-3051	349	55	star	star	NOUN
iajs-3051	349	56	finite	finite	NOUN
iajs-3051	349	57	-	-	NOUN
iajs-3051	349	58	covering	covering	NOUN
iajs-3051	349	59	of	of	ADP
iajs-3051	349	60	h	h	NOUN
iajs-3051	349	61	for	for	ADP
iajs-3051	349	62	all	all	DET
iajs-3051	349	63	n	n	CCONJ
iajs-3051	349	64	,	,	PUNCT
iajs-3051	349	65	and	and	CCONJ
iajs-3051	349	66	(	(	PUNCT
iajs-3051	349	67	b	b	X
iajs-3051	349	68	)	)	PUNCT
iajs-3051	349	69	find	find	VERB
iajs-3051	349	70	a	a	DET
iajs-3051	349	71	subsets	subset	NOUN
iajs-3051	349	72	γ`	γ`	NOUN
iajs-3051	349	73	of	of	ADP
iajs-3051	349	74	γ	γ	PRON
iajs-3051	349	75	such	such	ADJ
iajs-3051	349	76	as	as	ADP
iajs-3051	349	77	{	{	PUNCT
iajs-3051	349	78	vβ	vβ	NOUN
iajs-3051	349	79	:	:	PUNCT
iajs-3051	349	80	β	β	NOUN
iajs-3051	349	81	∈	∈	PROPN
iajs-3051	349	82	γ`	γ`	NOUN
iajs-3051	349	83	}	}	PUNCT
iajs-3051	349	84	is	be	AUX
iajs-3051	349	85	a	a	DET
iajs-3051	349	86	improvement	improvement	NOUN
iajs-3051	349	87	of	of	ADP
iajs-3051	349	88	w	w	NOUN
iajs-3051	349	89	∽	∽	NOUN
iajs-3051	349	90	.	.	PUNCT
iajs-3051	350	1	if	if	SCONJ
iajs-3051	350	2	β	β	PROPN
iajs-3051	350	3	∈	∈	PROPN
iajs-3051	350	4	γ`	γ`	NOUN
iajs-3051	350	5	,	,	PUNCT
iajs-3051	350	6	choose	choose	VERB
iajs-3051	350	7	h(β	h(β	NOUN
iajs-3051	350	8	)	)	PUNCT
iajs-3051	350	9	in	in	ADP
iajs-3051	350	10	h	h	NOUN
iajs-3051	350	11	such	such	ADJ
iajs-3051	350	12	as	as	ADP
iajs-3051	350	13	vβ	vβ	X
iajs-3051	350	14	⊂	⊂	ADJ
iajs-3051	350	15	wh(β	wh(β	X
iajs-3051	350	16	)	)	PUNCT
iajs-3051	350	17	.	.	PUNCT
iajs-3051	351	1	let	let	VERB
iajs-3051	351	2	o	o	NOUN
iajs-3051	351	3	be	be	AUX
iajs-3051	351	4	the	the	DET
iajs-3051	351	5	family	family	NOUN
iajs-3051	351	6	of	of	ADP
iajs-3051	351	7	nano	nano	NOUN
iajs-3051	351	8	-	-	PUNCT
iajs-3051	351	9	ope	ope	NOUN
iajs-3051	351	10	-	-	PUNCT
iajs-3051	351	11	sets	set	NOUN
iajs-3051	351	12	of	of	ADP
iajs-3051	351	13	g	g	NOUN
iajs-3051	351	14	which	which	PRON
iajs-3051	351	15	consists	consist	VERB
iajs-3051	351	16	of	of	ADP
iajs-3051	351	17	each	each	DET
iajs-3051	351	18	sets	set	NOUN
iajs-3051	351	19	𝐹−1(vβ)⋂uα	𝐹−1(vβ)⋂uα	NUM
iajs-3051	351	20	,	,	PUNCT
iajs-3051	351	21	where	where	SCONJ
iajs-3051	351	22	β	β	X
iajs-3051	351	23	∈	∈	PROPN
iajs-3051	351	24	γ`and	γ`and	PROPN
iajs-3051	351	25	α	α	PROPN
iajs-3051	351	26	∈	∈	PROPN
iajs-3051	351	27	λ(h(β	λ(h(β	PROPN
iajs-3051	351	28	)	)	PUNCT
iajs-3051	351	29	)	)	PUNCT
iajs-3051	351	30	,	,	PUNCT
iajs-3051	351	31	together	together	ADV
iajs-3051	351	32	with	with	ADP
iajs-3051	351	33	each	each	DET
iajs-3051	351	34	sets	set	NOUN
iajs-3051	351	35	𝑓−1(vβ	𝑓−1(vβ	PROPN
iajs-3051	351	36	)	)	PUNCT
iajs-3051	351	37	,	,	PUNCT
iajs-3051	351	38	where	where	SCONJ
iajs-3051	351	39	β	β	X
iajs-3051	351	40	∈	∈	PROPN
iajs-3051	351	41	γ\γ`.	γ\γ`.	AUX
iajs-3051	351	42	then	then	ADV
iajs-3051	351	43	o	o	NOUN
iajs-3051	351	44	consists	consist	VERB
iajs-3051	351	45	of	of	ADP
iajs-3051	351	46	countably	countably	ADV
iajs-3051	351	47	many	many	ADJ
iajs-3051	351	48	star	star	NOUN
iajs-3051	351	49	finite	finite	VERB
iajs-3051	351	50	nano	nano	NOUN
iajs-3051	351	51	-	-	PUNCT
iajs-3051	351	52	ope	ope	NOUN
iajs-3051	351	53	-	-	PUNCT
iajs-3051	351	54	coverings	covering	NOUN
iajs-3051	351	55	of	of	ADP
iajs-3051	351	56	g	g	PROPN
iajs-3051	351	57	and	and	CCONJ
iajs-3051	351	58	o	o	PROPN
iajs-3051	351	59	is	be	AUX
iajs-3051	351	60	a	a	DET
iajs-3051	351	61	weak	weak	ADJ
iajs-3051	351	62	improvement	improvement	NOUN
iajs-3051	351	63	of	of	ADP
iajs-3051	351	64	u	u	NOUN
iajs-3051	351	65	∽	∽	NOUN
iajs-3051	351	66	since	since	SCONJ
iajs-3051	351	67	the	the	DET
iajs-3051	351	68	subfamily	subfamily	ADV
iajs-3051	351	69	{	{	PUNCT
iajs-3051	351	70	𝑓−1(vβ)⋂uα	𝑓−1(vβ)⋂uα	PROPN
iajs-3051	351	71	:	:	PUNCT
iajs-3051	351	72	α	α	PROPN
iajs-3051	351	73	∈	∈	PROPN
iajs-3051	351	74	λ(h(β	λ(h(β	PROPN
iajs-3051	351	75	)	)	PUNCT
iajs-3051	351	76	)	)	PUNCT
iajs-3051	351	77	,	,	PUNCT
iajs-3051	351	78	β	β	X
iajs-3051	351	79	∈	∈	PROPN
iajs-3051	351	80	γ`	γ`	NOUN
iajs-3051	351	81	}	}	PUNCT
iajs-3051	351	82	is	be	AUX
iajs-3051	351	83	a	a	DET
iajs-3051	351	84	covering	covering	NOUN
iajs-3051	351	85	of	of	ADP
iajs-3051	351	86	g	g	PROPN
iajs-3051	351	87	and	and	CCONJ
iajs-3051	351	88	a	a	DET
iajs-3051	351	89	improvement	improvement	NOUN
iajs-3051	351	90	of	of	ADP
iajs-3051	351	91	u.	u.	NOUN
iajs-3051	352	1	then	then	ADV
iajs-3051	352	2	if	if	SCONJ
iajs-3051	352	3	h	h	NOUN
iajs-3051	352	4	is	be	AUX
iajs-3051	352	5	completely	completely	ADV
iajs-3051	352	6	nano	nano	VERB
iajs-3051	352	7	-	-	PUNCT
iajs-3051	352	8	para	para	NOUN
iajs-3051	352	9	-	-	PUNCT
iajs-3051	352	10	comp	comp	NOUN
iajs-3051	352	11	.	.	PUNCT
iajs-3051	352	12	,	,	PUNCT
iajs-3051	352	13	then	then	ADV
iajs-3051	352	14	g	g	PROPN
iajs-3051	352	15	is	be	AUX
iajs-3051	352	16	completely	completely	ADV
iajs-3051	352	17	nano	nano	VERB
iajs-3051	352	18	-	-	PUNCT
iajs-3051	352	19	para	para	NOUN
iajs-3051	352	20	-	-	PUNCT
iajs-3051	352	21	comp	comp	NOUN
iajs-3051	352	22	.	.	PUNCT
iajs-3051	353	1	definition	definition	NOUN
iajs-3051	353	2	3.10	3.10	NUM
iajs-3051	353	3	:	:	PUNCT
iajs-3051	353	4	let	let	VERB
iajs-3051	353	5	g	g	PRON
iajs-3051	353	6	be	be	AUX
iajs-3051	353	7	a	a	DET
iajs-3051	353	8	nano-hausd-sp.and	nano-hausd-sp.and	PROPN
iajs-3051	353	9	let	let	VERB
iajs-3051	353	10	c	c	NOUN
iajs-3051	353	11	be	be	AUX
iajs-3051	353	12	a	a	DET
iajs-3051	353	13	family	family	NOUN
iajs-3051	353	14	consisting	consist	VERB
iajs-3051	353	15	of	of	ADP
iajs-3051	353	16	those	those	DET
iajs-3051	353	17	subsets	subset	NOUN
iajs-3051	353	18	of	of	ADP
iajs-3051	353	19	g	g	PROPN
iajs-3051	353	20	that	that	PRON
iajs-3051	353	21	have	have	VERB
iajs-3051	353	22	nano-clos.interssections	nano-clos.interssection	NOUN
iajs-3051	353	23	with	with	ADP
iajs-3051	353	24	all	all	DET
iajs-3051	353	25	nano-comp-sub-sps.of	nano-comp-sub-sps.of	PROPN
iajs-3051	353	26	g.	g.	PROPN
iajs-3051	353	27	the	the	DET
iajs-3051	353	28	set	set	VERB
iajs-3051	353	29	g	g	NOUN
iajs-3051	353	30	with	with	ADP
iajs-3051	353	31	the	the	DET
iajs-3051	353	32	nanotopology	nanotopology	NOUN
iajs-3051	353	33	generated	generate	VERB
iajs-3051	353	34	by	by	ADP
iajs-3051	353	35	the	the	DET
iajs-3051	353	36	family	family	NOUN
iajs-3051	353	37	c	c	PROPN
iajs-3051	353	38	of	of	ADP
iajs-3051	353	39	nano	nano	NOUN
iajs-3051	353	40	-	-	PUNCT
iajs-3051	353	41	clos	clo	NOUN
iajs-3051	353	42	-	-	PUNCT
iajs-3051	353	43	subsets	subset	NOUN
iajs-3051	353	44	will	will	AUX
iajs-3051	353	45	be	be	AUX
iajs-3051	353	46	symboly	symboly	ADJ
iajs-3051	353	47	by	by	ADP
iajs-3051	353	48	kg	kg	PROPN
iajs-3051	353	49	is	be	AUX
iajs-3051	353	50	nano	nano	NOUN
iajs-3051	353	51	-	-	PUNCT
iajs-3051	353	52	ope	ope	NOUN
iajs-3051	353	53	.	.	PUNCT
iajs-3051	354	1	if	if	SCONJ
iajs-3051	354	2	and	and	CCONJ
iajs-3051	354	3	only	only	ADV
iajs-3051	354	4	if	if	SCONJ
iajs-3051	354	5	its	its	PRON
iajs-3051	354	6	intersection	intersection	NOUN
iajs-3051	354	7	with	with	ADP
iajs-3051	354	8	any	any	DET
iajs-3051	354	9	nano	nano	NOUN
iajs-3051	354	10	-	-	PUNCT
iajs-3051	354	11	comp	comp	NOUN
iajs-3051	354	12	-	-	PUNCT
iajs-3051	354	13	sub	sub	NOUN
iajs-3051	354	14	-	-	NOUN
iajs-3051	354	15	sp	sp	NOUN
iajs-3051	354	16	.	.	PUNCT
iajs-3051	355	1	r	r	NOUN
iajs-3051	355	2	of	of	ADP
iajs-3051	355	3	the	the	DET
iajs-3051	355	4	space	space	NOUN
iajs-3051	355	5	g	g	NOUN
iajs-3051	355	6	is	be	AUX
iajs-3051	355	7	nano-ope.in	nano-ope.in	PROPN
iajs-3051	355	8	r.	r.	NOUN
iajs-3051	355	9	the	the	DET
iajs-3051	355	10	nano	nano	NOUN
iajs-3051	355	11	-	-	PUNCT
iajs-3051	355	12	topology	topology	NOUN
iajs-3051	355	13	of	of	ADP
iajs-3051	355	14	kg	kg	PROPN
iajs-3051	355	15	is	be	AUX
iajs-3051	355	16	finer	fine	ADJ
iajs-3051	355	17	than	than	ADP
iajs-3051	355	18	the	the	DET
iajs-3051	355	19	nano	nano	NOUN
iajs-3051	355	20	-	-	PUNCT
iajs-3051	355	21	topology	topology	NOUN
iajs-3051	355	22	of	of	ADP
iajs-3051	355	23	g.	g.	PROPN
iajs-3051	355	24	proposition	proposition	PROPN
iajs-3051	355	25	3.11	3.11	NUM
iajs-3051	355	26	:	:	PUNCT
iajs-3051	355	27	the	the	DET
iajs-3051	355	28	nano	nano	NOUN
iajs-3051	355	29	-	-	PUNCT
iajs-3051	355	30	spaces	space	NOUN
iajs-3051	355	31	kg	kg	NOUN
iajs-3051	355	32	and	and	CCONJ
iajs-3051	355	33	g	g	PROPN
iajs-3051	355	34	have	have	VERB
iajs-3051	355	35	the	the	DET
iajs-3051	355	36	same	same	ADJ
iajs-3051	355	37	nano	nano	NOUN
iajs-3051	355	38	-	-	PUNCT
iajs-3051	355	39	comp	comp	NOUN
iajs-3051	355	40	-	-	PUNCT
iajs-3051	355	41	sub	sub	NOUN
iajs-3051	355	42	-	-	NOUN
iajs-3051	355	43	sps	sps	NOUN
iajs-3051	355	44	.	.	PUNCT
iajs-3051	356	1	proposition	proposition	NOUN
iajs-3051	356	2	3.12	3.12	NUM
iajs-3051	356	3	:	:	PUNCT
iajs-3051	356	4	define	define	VERB
iajs-3051	356	5	for	for	ADP
iajs-3051	356	6	each	each	DET
iajs-3051	356	7	nano	nano	NOUN
iajs-3051	356	8	-	-	PUNCT
iajs-3051	356	9	cont	cont	NOUN
iajs-3051	356	10	.	.	PUNCT
iajs-3051	357	1	map	map	NOUN
iajs-3051	358	1	f	f	X
iajs-3051	358	2	:	:	PUNCT
iajs-3051	358	3	g	g	PROPN
iajs-3051	358	4	⟶	⟶	NOUN
iajs-3051	358	5	h	h	NOUN
iajs-3051	358	6	where	where	SCONJ
iajs-3051	358	7	g	g	NOUN
iajs-3051	358	8	and	and	CCONJ
iajs-3051	358	9	h	h	NOUN
iajs-3051	358	10	nano	nano	NOUN
iajs-3051	358	11	-	-	PUNCT
iajs-3051	358	12	hausd	hausd	NOUN
iajs-3051	358	13	-	-	PUNCT
iajs-3051	358	14	sps	sps	NOUN
iajs-3051	358	15	.	.	PUNCT
iajs-3051	358	16	,	,	PUNCT
iajs-3051	358	17	the	the	DET
iajs-3051	358	18	map	map	NOUN
iajs-3051	358	19	kf	kf	NOUN
iajs-3051	358	20	assigning	assign	VERB
iajs-3051	358	21	to	to	ADP
iajs-3051	358	22	g	g	PROPN
iajs-3051	358	23	∈	∈	PROPN
iajs-3051	358	24	kg	kg	NOUN
iajs-3051	358	25	the	the	DET
iajs-3051	358	26	point	point	NOUN
iajs-3051	358	27	f(g	f(g	NOUN
iajs-3051	358	28	)	)	PUNCT
iajs-3051	359	1	∈	∈	PROPN
iajs-3051	359	2	kh	kh	PROPN
iajs-3051	359	3	.	.	PUNCT
iajs-3051	360	1	and	and	CCONJ
iajs-3051	360	2	ǥg	ǥg	ADP
iajs-3051	360	3	:	:	PUNCT
iajs-3051	360	4	kg	kg	PROPN
iajs-3051	360	5	⟶	⟶	PROPN
iajs-3051	360	6	g	g	NOUN
iajs-3051	360	7	by	by	ADP
iajs-3051	360	8	ǥg(g	ǥg(g	NUM
iajs-3051	360	9	)	)	PUNCT
iajs-3051	360	10	=	=	SYM
iajs-3051	361	1	g.	g.	PROPN
iajs-3051	361	2	then	then	ADV
iajs-3051	361	3	:	:	PUNCT
iajs-3051	361	4	(	(	PUNCT
iajs-3051	361	5	a	a	X
iajs-3051	361	6	)	)	PUNCT
iajs-3051	361	7	ǥg	ǥg	ADP
iajs-3051	361	8	:	:	PUNCT
iajs-3051	361	9	kg	kg	PROPN
iajs-3051	361	10	⟶	⟶	PROPN
iajs-3051	361	11	g	g	PROPN
iajs-3051	361	12	is	be	AUX
iajs-3051	361	13	nano	nano	NOUN
iajs-3051	361	14	-	-	PUNCT
iajs-3051	361	15	cont	cont	NOUN
iajs-3051	361	16	.	.	PUNCT
iajs-3051	362	1	(	(	PUNCT
iajs-3051	362	2	b	b	X
iajs-3051	362	3	)	)	PUNCT
iajs-3051	362	4	kf	kf	NOUN
iajs-3051	362	5	:	:	PUNCT
iajs-3051	362	6	kg	kg	PROPN
iajs-3051	362	7	⟶	⟶	PROPN
iajs-3051	362	8	kh	kh	PROPN
iajs-3051	362	9	is	be	AUX
iajs-3051	362	10	nano	nano	NOUN
iajs-3051	362	11	-	-	PUNCT
iajs-3051	362	12	cont	cont	NOUN
iajs-3051	362	13	.	.	PUNCT
iajs-3051	363	1	(	(	PUNCT
iajs-3051	363	2	c	c	X
iajs-3051	363	3	)	)	PUNCT
iajs-3051	363	4	kf	kf	PROPN
iajs-3051	363	5	satisfies	satisfy	VERB
iajs-3051	363	6	the	the	DET
iajs-3051	363	7	equality	equality	NOUN
iajs-3051	363	8	foǥg	foǥg	NOUN
iajs-3051	363	9	=	=	SYM
iajs-3051	363	10	ǥhokf	ǥhokf	ADJ
iajs-3051	363	11	.	.	PUNCT
iajs-3051	364	1	theorem	theorem	VERB
iajs-3051	364	2	3.13	3.13	NUM
iajs-3051	364	3	:	:	PUNCT
iajs-3051	364	4	for	for	ADP
iajs-3051	364	5	a	a	DET
iajs-3051	364	6	nano	nano	NOUN
iajs-3051	364	7	-	-	PUNCT
iajs-3051	364	8	cont	cont	NOUN
iajs-3051	364	9	.	.	PUNCT
iajs-3051	365	1	map	map	NOUN
iajs-3051	366	1	f	f	X
iajs-3051	366	2	:	:	PUNCT
iajs-3051	366	3	g	g	PROPN
iajs-3051	366	4	⟶	⟶	NOUN
iajs-3051	366	5	h	h	NOUN
iajs-3051	366	6	of	of	ADP
iajs-3051	366	7	a	a	DET
iajs-3051	366	8	nano	nano	NOUN
iajs-3051	366	9	-	-	PUNCT
iajs-3051	366	10	hausd	hausd	NOUN
iajs-3051	366	11	-	-	PUNCT
iajs-3051	366	12	sp	sp	NOUN
iajs-3051	366	13	.	.	NOUN
iajs-3051	367	1	g	g	NOUN
iajs-3051	367	2	to	to	ADP
iajs-3051	367	3	a	a	DET
iajs-3051	367	4	nano	nano	ADJ
iajs-3051	367	5	k	k	NOUN
iajs-3051	367	6	-	-	NOUN
iajs-3051	367	7	sp	sp	NOUN
iajs-3051	367	8	.	.	PUNCT
iajs-3051	367	9	h	h	NOUN
iajs-3051	368	1	the	the	DET
iajs-3051	368	2	statement	statement	NOUN
iajs-3051	368	3	are	be	AUX
iajs-3051	368	4	equivalent	equivalent	ADJ
iajs-3051	368	5	:	:	PUNCT
iajs-3051	368	6	(	(	PUNCT
iajs-3051	368	7	a	a	X
iajs-3051	368	8	)	)	PUNCT
iajs-3051	368	9	the	the	DET
iajs-3051	368	10	map	map	NOUN
iajs-3051	368	11	.	.	PUNCT
iajs-3051	369	1	f	f	PROPN
iajs-3051	369	2	is	be	AUX
iajs-3051	369	3	nano	nano	NOUN
iajs-3051	369	4	-	-	PUNCT
iajs-3051	369	5	perf	perf	NOUN
iajs-3051	369	6	.	.	PUNCT
iajs-3051	370	1	(	(	PUNCT
iajs-3051	370	2	b	b	X
iajs-3051	370	3	)	)	PUNCT
iajs-3051	370	4	for	for	ADP
iajs-3051	370	5	each	each	DET
iajs-3051	370	6	nano	nano	NOUN
iajs-3051	370	7	-	-	PUNCT
iajs-3051	370	8	comp	comp	NOUN
iajs-3051	370	9	-	-	PUNCT
iajs-3051	370	10	sub	sub	NOUN
iajs-3051	370	11	-	-	NOUN
iajs-3051	370	12	sp	sp	NOUN
iajs-3051	370	13	.	.	PUNCT
iajs-3051	371	1	r	r	NOUN
iajs-3051	371	2	⊂	⊂	PROPN
iajs-3051	371	3	h	h	NOUN
iajs-3051	371	4	the	the	DET
iajs-3051	371	5	restriction	restriction	NOUN
iajs-3051	371	6	𝑓i	𝑓i	ADP
iajs-3051	371	7	:	:	PUNCT
iajs-3051	371	8	𝑓	𝑓	DET
iajs-3051	371	9	−1(r	−1(r	NOUN
iajs-3051	371	10	)	)	PUNCT
iajs-3051	371	11	⟶	⟶	NOUN
iajs-3051	371	12	r	r	NOUN
iajs-3051	371	13	is	be	AUX
iajs-3051	371	14	nano	nano	NOUN
iajs-3051	371	15	-	-	PUNCT
iajs-3051	371	16	perf	perf	NOUN
iajs-3051	371	17	.	.	PUNCT
iajs-3051	372	1	(	(	PUNCT
iajs-3051	372	2	c	c	X
iajs-3051	372	3	)	)	PUNCT
iajs-3051	372	4	for	for	ADP
iajs-3051	372	5	each	each	DET
iajs-3051	372	6	nano	nano	NOUN
iajs-3051	372	7	-	-	PUNCT
iajs-3051	372	8	comp	comp	NOUN
iajs-3051	372	9	-	-	PUNCT
iajs-3051	372	10	sub	sub	NOUN
iajs-3051	372	11	-	-	NOUN
iajs-3051	372	12	sp	sp	NOUN
iajs-3051	372	13	.	.	PUNCT
iajs-3051	373	1	r	r	NOUN
iajs-3051	373	2	⊂	⊂	PROPN
iajs-3051	373	3	h	h	NOUN
iajs-3051	373	4	the	the	DET
iajs-3051	373	5	inverse	inverse	NOUN
iajs-3051	373	6	image	image	NOUN
iajs-3051	373	7	𝑓−1(r	𝑓−1(r	NOUN
iajs-3051	373	8	)	)	PUNCT
iajs-3051	373	9	is	be	AUX
iajs-3051	373	10	nano	nano	NOUN
iajs-3051	373	11	-	-	PUNCT
iajs-3051	373	12	comp	comp	NOUN
iajs-3051	373	13	.	.	PUNCT
iajs-3051	374	1	proposition	proposition	NOUN
iajs-3051	374	2	3.14	3.14	NUM
iajs-3051	374	3	:	:	PUNCT
iajs-3051	374	4	if	if	SCONJ
iajs-3051	374	5	the	the	DET
iajs-3051	374	6	composition	composition	NOUN
iajs-3051	374	7	gof	gof	NOUN
iajs-3051	374	8	of	of	ADP
iajs-3051	374	9	nano	nano	NOUN
iajs-3051	374	10	-	-	PUNCT
iajs-3051	374	11	cont	cont	NOUN
iajs-3051	374	12	.	.	PUNCT
iajs-3051	375	1	map	map	NOUN
iajs-3051	376	1	f	f	X
iajs-3051	376	2	:	:	PUNCT
iajs-3051	376	3	g	g	PROPN
iajs-3051	376	4	⟶	⟶	PROPN
iajs-3051	376	5	h	h	NOUN
iajs-3051	376	6	and	and	CCONJ
iajs-3051	376	7	j	j	PROPN
iajs-3051	376	8	:	:	PUNCT
iajs-3051	376	9	h	h	NOUN
iajs-3051	376	10	⟶	⟶	PROPN
iajs-3051	376	11	r	r	NOUN
iajs-3051	376	12	,	,	PUNCT
iajs-3051	376	13	where	where	SCONJ
iajs-3051	376	14	h	h	NOUN
iajs-3051	376	15	is	be	AUX
iajs-3051	376	16	a	a	DET
iajs-3051	376	17	nano	nano	NOUN
iajs-3051	376	18	-	-	PUNCT
iajs-3051	376	19	hausd	hausd	NOUN
iajs-3051	376	20	-	-	PUNCT
iajs-3051	376	21	sp	sp	NOUN
iajs-3051	376	22	.	.	PROPN
iajs-3051	376	23	,	,	PUNCT
iajs-3051	376	24	is	be	AUX
iajs-3051	376	25	nano	nano	NOUN
iajs-3051	376	26	-	-	PUNCT
iajs-3051	376	27	perf	perf	NOUN
iajs-3051	376	28	.	.	PUNCT
iajs-3051	377	1	,	,	PUNCT
iajs-3051	377	2	then	then	ADV
iajs-3051	377	3	the	the	DET
iajs-3051	377	4	map	map	NOUN
iajs-3051	377	5	j	j	NOUN
iajs-3051	377	6	│	│	NOUN
iajs-3051	377	7	f(g	f(g	NOUN
iajs-3051	377	8	)	)	PUNCT
iajs-3051	377	9	and	and	CCONJ
iajs-3051	377	10	f	f	PROPN
iajs-3051	377	11	are	be	AUX
iajs-3051	377	12	nano	nano	NOUN
iajs-3051	377	13	-	-	PUNCT
iajs-3051	377	14	perf	perf	NOUN
iajs-3051	377	15	.	.	PUNCT
iajs-3051	378	1	ihjpas	ihjpas	PROPN
iajs-3051	378	2	.	.	PUNCT
iajs-3051	379	1	36	36	NUM
iajs-3051	379	2	(	(	PUNCT
iajs-3051	379	3	3	3	NUM
iajs-3051	379	4	)	)	PUNCT
iajs-3051	379	5	2023	2023	NUM
iajs-3051	379	6	406	406	NUM
iajs-3051	379	7	proof	proof	NOUN
iajs-3051	379	8	:	:	PUNCT
iajs-3051	379	9	(	(	PUNCT
iajs-3051	379	10	a	a	X
iajs-3051	379	11	)	)	PUNCT
iajs-3051	379	12	for	for	ADP
iajs-3051	379	13	each	each	DET
iajs-3051	379	14	point	point	NOUN
iajs-3051	379	15	r	r	NOUN
iajs-3051	379	16	∈	∈	NOUN
iajs-3051	379	17	r	r	NOUN
iajs-3051	379	18	the	the	DET
iajs-3051	379	19	fiber	fiber	NOUN
iajs-3051	379	20	(	(	PUNCT
iajs-3051	379	21	j	j	NOUN
iajs-3051	379	22	│	│	X
iajs-3051	379	23	f(g))−1	f(g))−1	NOUN
iajs-3051	379	24	(	(	PUNCT
iajs-3051	379	25	r	r	NOUN
iajs-3051	379	26	)	)	PUNCT
iajs-3051	379	27	=	=	NOUN
iajs-3051	379	28	f(g	f(g	NOUN
iajs-3051	379	29	)	)	PUNCT
iajs-3051	379	30	⋂	⋂	PROPN
iajs-3051	379	31	j−1(r	j−1(r	NOUN
iajs-3051	379	32	)	)	PUNCT
iajs-3051	380	1	=	=	SYM
iajs-3051	381	1	f	f	X
iajs-3051	381	2	(	(	PUNCT
iajs-3051	381	3	(	(	PUNCT
iajs-3051	381	4	jof)−1(r	jof)−1(r	NOUN
iajs-3051	381	5	)	)	PUNCT
iajs-3051	381	6	)	)	PUNCT
iajs-3051	381	7	is	be	AUX
iajs-3051	381	8	nanocomp	nanocomp	ADJ
iajs-3051	381	9	.	.	PUNCT
iajs-3051	381	10	,	,	PUNCT
iajs-3051	381	11	because	because	SCONJ
iajs-3051	381	12	the	the	DET
iajs-3051	381	13	fiber	fiber	NOUN
iajs-3051	381	14	(	(	PUNCT
iajs-3051	381	15	jof	jof	PROPN
iajs-3051	381	16	)	)	PUNCT
iajs-3051	381	17	−1(r	−1(r	NOUN
iajs-3051	381	18	)	)	PUNCT
iajs-3051	381	19	is	be	AUX
iajs-3051	381	20	nano	nano	NOUN
iajs-3051	381	21	-	-	PUNCT
iajs-3051	381	22	comp	comp	NOUN
iajs-3051	381	23	.	.	PUNCT
iajs-3051	382	1	any	any	DET
iajs-3051	382	2	nano	nano	NOUN
iajs-3051	382	3	-	-	PUNCT
iajs-3051	382	4	clos	clo	NOUN
iajs-3051	382	5	.	.	PUNCT
iajs-3051	383	1	subset	subset	NOUN
iajs-3051	383	2	of	of	ADP
iajs-3051	383	3	f(g	f(g	NOUN
iajs-3051	383	4	)	)	PUNCT
iajs-3051	383	5	is	be	AUX
iajs-3051	383	6	of	of	ADP
iajs-3051	383	7	the	the	DET
iajs-3051	383	8	form	form	NOUN
iajs-3051	383	9	a⋂f(g	a⋂f(g	ADV
iajs-3051	383	10	)	)	PUNCT
iajs-3051	383	11	,	,	PUNCT
iajs-3051	383	12	where	where	SCONJ
iajs-3051	383	13	a	a	PRON
iajs-3051	383	14	is	be	AUX
iajs-3051	383	15	nano	nano	NOUN
iajs-3051	383	16	-	-	PUNCT
iajs-3051	383	17	clos	clo	NOUN
iajs-3051	383	18	.	.	PUNCT
iajs-3051	384	1	in	in	ADP
iajs-3051	384	2	h.	h.	PROPN
iajs-3051	384	3	as	as	SCONJ
iajs-3051	384	4	the	the	DET
iajs-3051	384	5	inverse	inverse	NOUN
iajs-3051	384	6	image	image	NOUN
iajs-3051	384	7	𝑓−1(a	𝑓−1(a	PROPN
iajs-3051	384	8	)	)	PUNCT
iajs-3051	384	9	is	be	AUX
iajs-3051	384	10	nano	nano	NOUN
iajs-3051	384	11	-	-	PUNCT
iajs-3051	384	12	clos	clo	NOUN
iajs-3051	384	13	.	.	PUNCT
iajs-3051	385	1	in	in	ADP
iajs-3051	385	2	g	g	NOUN
iajs-3051	385	3	and	and	CCONJ
iajs-3051	385	4	jof	jof	PROPN
iajs-3051	385	5	is	be	AUX
iajs-3051	385	6	a	a	DET
iajs-3051	385	7	nano	nano	NOUN
iajs-3051	385	8	-	-	PUNCT
iajs-3051	385	9	clos	clo	NOUN
iajs-3051	385	10	.	.	PUNCT
iajs-3051	386	1	map	map	NOUN
iajs-3051	386	2	,	,	PUNCT
iajs-3051	386	3	the	the	DET
iajs-3051	386	4	set	set	NOUN
iajs-3051	386	5	(	(	PUNCT
iajs-3051	386	6	j	j	NOUN
iajs-3051	386	7	│	│	ADJ
iajs-3051	386	8	f(g	f(g	NOUN
iajs-3051	386	9	)	)	PUNCT
iajs-3051	386	10	)	)	PUNCT
iajs-3051	386	11	(	(	PUNCT
iajs-3051	386	12	a⋂f(g	a⋂f(g	NOUN
iajs-3051	386	13	)	)	PUNCT
iajs-3051	386	14	)	)	PUNCT
iajs-3051	387	1	=	=	PUNCT
iajs-3051	387	2	j(a⋂f(g	j(a⋂f(g	NOUN
iajs-3051	387	3	)	)	PUNCT
iajs-3051	387	4	)	)	PUNCT
iajs-3051	388	1	=	=	SYM
iajs-3051	388	2	jof	jof	NOUN
iajs-3051	388	3	(	(	PUNCT
iajs-3051	388	4	𝑓−1(a	𝑓−1(a	PROPN
iajs-3051	388	5	)	)	PUNCT
iajs-3051	388	6	)	)	PUNCT
iajs-3051	388	7	,	,	PUNCT
iajs-3051	388	8	is	be	AUX
iajs-3051	388	9	nano	nano	NOUN
iajs-3051	388	10	-	-	PUNCT
iajs-3051	388	11	clos	clo	NOUN
iajs-3051	388	12	.	.	PUNCT
iajs-3051	389	1	in	in	ADP
iajs-3051	389	2	r	r	NOUN
iajs-3051	389	3	,	,	PUNCT
iajs-3051	389	4	we	we	PRON
iajs-3051	389	5	have	have	AUX
iajs-3051	389	6	j	j	NOUN
iajs-3051	389	7	│	│	ADJ
iajs-3051	389	8	f(g	f(g	NOUN
iajs-3051	389	9	)	)	PUNCT
iajs-3051	389	10	is	be	AUX
iajs-3051	389	11	a	a	DET
iajs-3051	389	12	nano	nano	NOUN
iajs-3051	389	13	-	-	PUNCT
iajs-3051	389	14	clos	clo	NOUN
iajs-3051	389	15	.	.	PUNCT
iajs-3051	390	1	map	map	NOUN
iajs-3051	390	2	and	and	CCONJ
iajs-3051	390	3	thus	thus	ADV
iajs-3051	390	4	the	the	DET
iajs-3051	390	5	map	map	NOUN
iajs-3051	390	6	j	j	NOUN
iajs-3051	390	7	│	│	ADJ
iajs-3051	390	8	f(g	f(g	NOUN
iajs-3051	390	9	)	)	PUNCT
iajs-3051	390	10	is	be	AUX
iajs-3051	390	11	a	a	DET
iajs-3051	390	12	nano	nano	NOUN
iajs-3051	390	13	-	-	PUNCT
iajs-3051	390	14	clos	clo	NOUN
iajs-3051	390	15	.	.	PUNCT
iajs-3051	391	1	map	map	NOUN
iajs-3051	391	2	and	and	CCONJ
iajs-3051	391	3	thus	thus	ADV
iajs-3051	391	4	the	the	DET
iajs-3051	391	5	map	map	NOUN
iajs-3051	391	6	j	j	NOUN
iajs-3051	391	7	│	│	ADJ
iajs-3051	391	8	f(g	f(g	NOUN
iajs-3051	391	9	)	)	PUNCT
iajs-3051	391	10	is	be	AUX
iajs-3051	391	11	nano	nano	NOUN
iajs-3051	391	12	-	-	PUNCT
iajs-3051	391	13	perf	perf	NOUN
iajs-3051	391	14	.	.	PUNCT
iajs-3051	392	1	(	(	PUNCT
iajs-3051	392	2	b	b	X
iajs-3051	392	3	)	)	PUNCT
iajs-3051	392	4	for	for	ADP
iajs-3051	392	5	each	each	DET
iajs-3051	392	6	point	point	NOUN
iajs-3051	392	7	h	h	NOUN
iajs-3051	392	8	∈	∈	PROPN
iajs-3051	392	9	h	h	NOUN
iajs-3051	392	10	the	the	DET
iajs-3051	392	11	fiber	fiber	NOUN
iajs-3051	392	12	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	392	13	)	)	PUNCT
iajs-3051	393	1	=	=	PUNCT
iajs-3051	393	2	[	[	PUNCT
iajs-3051	393	3	(	(	PUNCT
iajs-3051	393	4	jof)−1(j(h)]⋂	jof)−1(j(h)]⋂	PROPN
iajs-3051	393	5	𝑓−1(h	𝑓−1(h	PROPN
iajs-3051	393	6	)	)	PUNCT
iajs-3051	393	7	is	be	AUX
iajs-3051	393	8	nano	nano	NOUN
iajs-3051	393	9	-	-	NOUN
iajs-3051	393	10	comp	comp	NOUN
iajs-3051	393	11	.	.	PUNCT
iajs-3051	394	1	to	to	PART
iajs-3051	394	2	conclude	conclude	VERB
iajs-3051	394	3	the	the	DET
iajs-3051	394	4	proof	proof	NOUN
iajs-3051	394	5	it	it	PRON
iajs-3051	394	6	suffices	suffice	VERB
iajs-3051	394	7	to	to	PART
iajs-3051	394	8	show	show	VERB
iajs-3051	394	9	that	that	SCONJ
iajs-3051	394	10	f	f	PROPN
iajs-3051	394	11	is	be	AUX
iajs-3051	394	12	nano	nano	NOUN
iajs-3051	394	13	-	-	PUNCT
iajs-3051	394	14	clos	clo	NOUN
iajs-3051	394	15	.	.	PUNCT
iajs-3051	395	1	for	for	ADP
iajs-3051	395	2	each	each	DET
iajs-3051	395	3	nano	nano	NOUN
iajs-3051	395	4	-	-	PUNCT
iajs-3051	395	5	clos	clo	NOUN
iajs-3051	395	6	-	-	PUNCT
iajs-3051	395	7	set	set	VERB
iajs-3051	395	8	f	f	PROPN
iajs-3051	395	9	⊂	⊂	PROPN
iajs-3051	395	10	g	g	ADP
iajs-3051	395	11	the	the	DET
iajs-3051	395	12	map	map	NOUN
iajs-3051	395	13	(	(	PUNCT
iajs-3051	395	14	jof)	jof)	NOUN
iajs-3051	395	15	│	│	ADJ
iajs-3051	395	16	f	f	PROPN
iajs-3051	395	17	is	be	AUX
iajs-3051	395	18	nano	nano	NOUN
iajs-3051	395	19	-	-	PUNCT
iajs-3051	395	20	perf	perf	NOUN
iajs-3051	395	21	.	.	PUNCT
iajs-3051	395	22	,	,	PUNCT
iajs-3051	395	23	so	so	SCONJ
iajs-3051	395	24	that	that	SCONJ
iajs-3051	395	25	by	by	ADP
iajs-3051	395	26	the	the	DET
iajs-3051	395	27	first	first	ADJ
iajs-3051	395	28	section	section	NOUN
iajs-3051	395	29	of	of	ADP
iajs-3051	395	30	us	we	PRON
iajs-3051	395	31	proof	proof	NOUN
iajs-3051	395	32	the	the	DET
iajs-3051	395	33	restriction	restriction	NOUN
iajs-3051	395	34	j	j	PROPN
iajs-3051	395	35	│	│	X
iajs-3051	395	36	f(f	f(f	PROPN
iajs-3051	395	37	)	)	PUNCT
iajs-3051	395	38	is	be	AUX
iajs-3051	395	39	nano	nano	NOUN
iajs-3051	395	40	-	-	PUNCT
iajs-3051	395	41	perf	perf	NOUN
iajs-3051	395	42	.	.	PUNCT
iajs-3051	396	1	;	;	PUNCT
iajs-3051	396	2	since	since	SCONJ
iajs-3051	396	3	the	the	DET
iajs-3051	396	4	latter	latter	ADJ
iajs-3051	396	5	map	map	NOUN
iajs-3051	396	6	ability	ability	NOUN
iajs-3051	396	7	be	be	AUX
iajs-3051	396	8	continuously	continuously	ADV
iajs-3051	396	9	extended	extend	VERB
iajs-3051	396	10	through	through	ADP
iajs-3051	396	11	ncl(f(f	ncl(f(f	NOUN
iajs-3051	396	12	)	)	PUNCT
iajs-3051	396	13	)	)	PUNCT
iajs-3051	396	14	,	,	PUNCT
iajs-3051	396	15	it	it	PRON
iajs-3051	396	16	follows	follow	VERB
iajs-3051	396	17	by	by	ADP
iajs-3051	396	18	lemma	lemma	PROPN
iajs-3051	396	19	2.11	2.11	NUM
iajs-3051	396	20	that	that	DET
iajs-3051	396	21	f(f	f(f	PROPN
iajs-3051	396	22	)	)	PUNCT
iajs-3051	396	23	=	=	SYM
iajs-3051	396	24	ncl(f(f	ncl(f(f	NOUN
iajs-3051	396	25	)	)	PUNCT
iajs-3051	396	26	)	)	PUNCT
iajs-3051	396	27	,	,	PUNCT
iajs-3051	396	28	and	and	CCONJ
iajs-3051	396	29	thus	thus	ADV
iajs-3051	396	30	f	f	PROPN
iajs-3051	396	31	is	be	AUX
iajs-3051	396	32	a	a	DET
iajs-3051	396	33	nano	nano	NOUN
iajs-3051	396	34	-	-	PUNCT
iajs-3051	396	35	clos	clo	NOUN
iajs-3051	396	36	.	.	PUNCT
iajs-3051	396	37	map	map	NOUN
iajs-3051	396	38	.	.	PUNCT
iajs-3051	397	1	theorem	theorem	VERB
iajs-3051	397	2	3.15	3.15	NUM
iajs-3051	397	3	:	:	PUNCT
iajs-3051	397	4	if	if	SCONJ
iajs-3051	397	5	find	find	VERB
iajs-3051	397	6	a	a	DET
iajs-3051	397	7	nano	nano	NOUN
iajs-3051	397	8	-	-	PUNCT
iajs-3051	397	9	perfmap	perfmap	ADJ
iajs-3051	397	10	f	f	X
iajs-3051	397	11	:	:	PUNCT
iajs-3051	397	12	g	g	PROPN
iajs-3051	397	13	⟶	⟶	NOUN
iajs-3051	397	14	h	h	NOUN
iajs-3051	397	15	of	of	ADP
iajs-3051	397	16	g	g	NOUN
iajs-3051	397	17	onto	onto	ADP
iajs-3051	397	18	a	a	DET
iajs-3051	397	19	nano	nano	ADJ
iajs-3051	397	20	k	k	NOUN
iajs-3051	397	21	-	-	PUNCT
iajs-3051	397	22	sp	sp	NOUN
iajs-3051	397	23	.	.	PROPN
iajs-3051	397	24	h	h	NOUN
iajs-3051	397	25	,	,	PUNCT
iajs-3051	397	26	then	then	ADV
iajs-3051	397	27	g	g	PROPN
iajs-3051	397	28	is	be	AUX
iajs-3051	397	29	a	a	DET
iajs-3051	397	30	nano	nano	ADJ
iajs-3051	397	31	ksp	ksp	PROPN
iajs-3051	397	32	.	.	PUNCT
iajs-3051	398	1	proof	proof	NOUN
iajs-3051	398	2	:	:	PUNCT
iajs-3051	398	3	let	let	VERB
iajs-3051	398	4	us	we	PRON
iajs-3051	398	5	consider	consider	VERB
iajs-3051	398	6	spaces	space	NOUN
iajs-3051	398	7	kg	kg	X
iajs-3051	398	8	and	and	CCONJ
iajs-3051	398	9	map	map	VERB
iajs-3051	398	10	kf	kf	INTJ
iajs-3051	398	11	:	:	PUNCT
iajs-3051	398	12	kg	kg	PROPN
iajs-3051	398	13	⟶	⟶	NOUN
iajs-3051	398	14	kh.since	kh.since	NOUN
iajs-3051	398	15	h	h	NOUN
iajs-3051	398	16	is	be	AUX
iajs-3051	398	17	a	a	DET
iajs-3051	398	18	nano	nano	ADJ
iajs-3051	398	19	k	k	NOUN
iajs-3051	398	20	-	-	NOUN
iajs-3051	398	21	sp	sp	NOUN
iajs-3051	398	22	.	.	PROPN
iajs-3051	398	23	,	,	PUNCT
iajs-3051	398	24	we	we	PRON
iajs-3051	398	25	have	have	VERB
iajs-3051	398	26	h	h	NOUN
iajs-3051	398	27	=	=	SYM
iajs-3051	398	28	kh	kh	PROPN
iajs-3051	398	29	and	and	CCONJ
iajs-3051	398	30	kf	kf	PROPN
iajs-3051	398	31	=	=	PROPN
iajs-3051	398	32	foǥg	foǥg	PROPN
iajs-3051	398	33	.	.	PUNCT
iajs-3051	399	1	from	from	ADP
iajs-3051	399	2	theorem	theorem	ADJ
iajs-3051	399	3	3.13	3.13	NUM
iajs-3051	399	4	,	,	PUNCT
iajs-3051	399	5	it	it	PRON
iajs-3051	399	6	follows	follow	VERB
iajs-3051	399	7	that	that	SCONJ
iajs-3051	399	8	kf	kf	PROPN
iajs-3051	399	9	is	be	AUX
iajs-3051	399	10	a	a	DET
iajs-3051	399	11	nano	nano	NOUN
iajs-3051	399	12	-	-	PUNCT
iajs-3051	399	13	perf	perf	NOUN
iajs-3051	399	14	-	-	PUNCT
iajs-3051	399	15	map	map	NOUN
iajs-3051	399	16	and	and	CCONJ
iajs-3051	399	17	this	this	PRON
iajs-3051	399	18	along	along	ADP
iajs-3051	399	19	with	with	ADP
iajs-3051	399	20	proposition	proposition	NOUN
iajs-3051	399	21	3.14	3.14	NUM
iajs-3051	399	22	,	,	PUNCT
iajs-3051	399	23	implies	imply	VERB
iajs-3051	399	24	that	that	SCONJ
iajs-3051	399	25	ǥg	ǥg	PROPN
iajs-3051	399	26	is	be	AUX
iajs-3051	399	27	nano	nano	NOUN
iajs-3051	399	28	-	-	PUNCT
iajs-3051	399	29	perf	perf	NOUN
iajs-3051	399	30	.	.	PUNCT
iajs-3051	400	1	as	as	SCONJ
iajs-3051	400	2	ǥg	ǥg	X
iajs-3051	400	3	is	be	VERB
iajs-3051	400	4	a	a	DET
iajs-3051	400	5	one	one	NUM
iajs-3051	400	6	-	-	PUNCT
iajs-3051	400	7	to	to	ADP
iajs-3051	400	8	-	-	PUNCT
iajs-3051	400	9	one	one	NUM
iajs-3051	400	10	map	map	NOUN
iajs-3051	400	11	,	,	PUNCT
iajs-3051	400	12	it	it	PRON
iajs-3051	400	13	is	be	AUX
iajs-3051	400	14	a	a	DET
iajs-3051	400	15	nanohomeomorphism	nanohomeomorphism	NOUN
iajs-3051	400	16	;	;	PUNCT
iajs-3051	400	17	therefore	therefore	ADV
iajs-3051	400	18	g	g	PROPN
iajs-3051	400	19	is	be	AUX
iajs-3051	400	20	a	a	DET
iajs-3051	400	21	nano	nano	ADJ
iajs-3051	400	22	k	k	NOUN
iajs-3051	400	23	-	-	PUNCT
iajs-3051	400	24	sp	sp	NOUN
iajs-3051	400	25	.	.	NOUN
iajs-3051	400	26	4	4	NUM
iajs-3051	400	27	.	.	X
iajs-3051	400	28	conclusion	conclusion	VERB
iajs-3051	400	29	the	the	DET
iajs-3051	400	30	main	main	ADJ
iajs-3051	400	31	purpose	purpose	NOUN
iajs-3051	400	32	of	of	ADP
iajs-3051	400	33	this	this	DET
iajs-3051	400	34	paper	paper	NOUN
iajs-3051	400	35	is	be	AUX
iajs-3051	400	36	to	to	PART
iajs-3051	400	37	present	present	VERB
iajs-3051	400	38	and	and	CCONJ
iajs-3051	400	39	study	study	VERB
iajs-3051	400	40	images	image	NOUN
iajs-3051	400	41	of	of	ADP
iajs-3051	400	42	nano	nano	VERB
iajs-3051	400	43	perfect	perfect	ADJ
iajs-3051	400	44	mappings	mapping	NOUN
iajs-3051	400	45	,	,	PUNCT
iajs-3051	400	46	as	as	ADV
iajs-3051	400	47	well	well	ADV
iajs-3051	400	48	as	as	ADP
iajs-3051	400	49	to	to	PART
iajs-3051	400	50	present	present	ADJ
iajs-3051	400	51	invers	inver	NOUN
iajs-3051	400	52	images	image	NOUN
iajs-3051	400	53	of	of	ADP
iajs-3051	400	54	nano	nano	VERB
iajs-3051	400	55	perfect	perfect	ADJ
iajs-3051	400	56	mappings	mapping	NOUN
iajs-3051	400	57	.	.	PUNCT
iajs-3051	401	1	references	reference	NOUN
iajs-3051	401	2	1	1	NUM
iajs-3051	401	3	.	.	PUNCT
iajs-3051	401	4	thivagar	thivagar	NOUN
iajs-3051	401	5	,	,	PUNCT
iajs-3051	401	6	m.	m.	NOUN
iajs-3051	401	7	l.	l.	PROPN
iajs-3051	401	8	;	;	PUNCT
iajs-3051	401	9	richard	richard	PROPN
iajs-3051	401	10	,	,	PUNCT
iajs-3051	401	11	c.	c.	PROPN
iajs-3051	401	12	on	on	ADP
iajs-3051	401	13	nano	nano	NOUN
iajs-3051	401	14	continuity	continuity	NOUN
iajs-3051	401	15	.	.	PUNCT
iajs-3051	402	1	mathematical	mathematical	ADJ
iajs-3051	402	2	theory	theory	NOUN
iajs-3051	402	3	and	and	CCONJ
iajs-3051	402	4	modelling	modelling	NOUN
iajs-3051	402	5	.	.	PUNCT
iajs-3051	403	1	2013	2013	NUM
iajs-3051	403	2	,	,	PUNCT
iajs-3051	403	3	3	3	NUM
iajs-3051	403	4	,	,	PUNCT
iajs-3051	403	5	7	7	NUM
iajs-3051	403	6	,	,	PUNCT
iajs-3051	403	7	32–37	32–37	NUM
iajs-3051	403	8	.	.	PUNCT
iajs-3051	404	1	2	2	NUM
iajs-3051	404	2	.	.	X
iajs-3051	404	3	yousif	yousif	PROPN
iajs-3051	404	4	,	,	PUNCT
iajs-3051	404	5	y.	y.	PROPN
iajs-3051	404	6	y.	y.	PROPN
iajs-3051	404	7	;	;	PUNCT
iajs-3051	404	8	hussain	hussain	PROPN
iajs-3051	404	9	,	,	PUNCT
iajs-3051	404	10	l.	l.	PROPN
iajs-3051	404	11	a.	a.	PROPN
iajs-3051	404	12	fibrewise	fibrewise	PROPN
iajs-3051	404	13	ij	ij	ADJ
iajs-3051	404	14	-	-	ADJ
iajs-3051	404	15	perfect	perfect	ADJ
iajs-3051	404	16	bitopological	bitopological	ADJ
iajs-3051	404	17	spaces	space	NOUN
iajs-3051	404	18	.	.	PUNCT
iajs-3051	405	1	journal	journal	PROPN
iajs-3051	405	2	of	of	ADP
iajs-3051	405	3	physics	physics	PROPN
iajs-3051	405	4	:	:	PUNCT
iajs-3051	405	5	conference	conference	NOUN
iajs-3051	405	6	series	series	NOUN
iajs-3051	405	7	,	,	PUNCT
iajs-3051	405	8	iop	iop	PROPN
iajs-3051	405	9	publishing	publishing	NOUN
iajs-3051	405	10	,	,	PUNCT
iajs-3051	405	11	ibn	ibn	PROPN
iajs-3051	405	12	al	al	PROPN
iajs-3051	405	13	-	-	PUNCT
iajs-3051	405	14	haitham	haitham	PROPN
iajs-3051	405	15	1^stinternational	1^stinternational	NUM
iajs-3051	405	16	scientific	scientific	ADJ
iajs-3051	405	17	conference	conference	NOUN
iajs-3051	405	18	.	.	PUNCT
iajs-3051	406	1	december	december	PROPN
iajs-3051	406	2	2017	2017	NUM
iajs-3051	406	3	,	,	PUNCT
iajs-3051	406	4	2018	2018	NUM
iajs-3051	406	5	,	,	PUNCT
iajs-3051	406	6	volume	volume	NOUN
iajs-3051	406	7	1003	1003	NUM
iajs-3051	406	8	,	,	PUNCT
iajs-3051	406	9	13	13	NUM
iajs-3051	406	10	-	-	SYM
iajs-3051	406	11	14	14	NUM
iajs-3051	406	12	,	,	PUNCT
iajs-3051	406	13	,	,	PUNCT
iajs-3051	406	14	doi	doi	X
iajs-3051	406	15	:	:	PUNCT
iajs-3051	406	16	10.1088/1742	10.1088/1742	NUM
iajs-3051	406	17	-	-	SYM
iajs-3051	406	18	6596/1003/1/012063	6596/1003/1/012063	NUM
iajs-3051	406	19	,	,	PUNCT
iajs-3051	406	20	pp	pp	X
iajs-3051	406	21	.	.	PUNCT
iajs-3051	407	1	112	112	NUM
iajs-3051	407	2	.	.	X
iajs-3051	408	1	3	3	X
iajs-3051	408	2	.	.	NOUN
iajs-3051	408	3	ashaea	ashaea	PROPN
iajs-3051	408	4	,	,	PUNCT
iajs-3051	408	5	g.	g.	PROPN
iajs-3051	408	6	s.	s.	PROPN
iajs-3051	408	7	;	;	PUNCT
iajs-3051	409	1	yousif	yousif	PROPN
iajs-3051	409	2	,	,	PUNCT
iajs-3051	409	3	y.y	y.y	PROPN
iajs-3051	409	4	.	.	PROPN
iajs-3051	409	5	weak	weak	ADJ
iajs-3051	409	6	and	and	CCONJ
iajs-3051	409	7	strong	strong	ADJ
iajs-3051	409	8	forms	form	NOUN
iajs-3051	409	9	of	of	ADP
iajs-3051	409	10	𝜔-perfect	𝜔-perfect	ADJ
iajs-3051	409	11	mappings	mapping	NOUN
iajs-3051	409	12	.	.	PUNCT
iajs-3051	410	1	iraqi	iraqi	ADJ
iajs-3051	410	2	journal	journal	PROPN
iajs-3051	410	3	of	of	ADP
iajs-3051	410	4	science	science	NOUN
iajs-3051	410	5	.	.	PUNCT
iajs-3051	411	1	2020	2020	NUM
iajs-3051	411	2	,	,	PUNCT
iajs-3051	411	3	special	special	ADJ
iajs-3051	411	4	issue	issue	NOUN
iajs-3051	411	5	,	,	PUNCT
iajs-3051	411	6	45	45	NUM
iajs-3051	411	7	-	-	SYM
iajs-3051	411	8	55	55	NUM
iajs-3051	411	9	.	.	PUNCT
iajs-3051	412	1	4	4	NUM
iajs-3051	412	2	.	.	X
iajs-3051	412	3	majeed	majeed	PROPN
iajs-3051	412	4	,	,	PUNCT
iajs-3051	412	5	r.	r.	PROPN
iajs-3051	412	6	n.	n.	PROPN
iajs-3051	412	7	rα	rα	PROPN
iajs-3051	412	8	-	-	PUNCT
iajs-3051	412	9	compactness	compactness	NOUN
iajs-3051	412	10	on	on	ADP
iajs-3051	412	11	bitopological	bitopological	ADJ
iajs-3051	412	12	spases	spase	NOUN
iajs-3051	412	13	.	.	PUNCT
iajs-3051	413	1	journal	journal	PROPN
iajs-3051	413	2	of	of	ADP
iajs-3051	413	3	al	al	PROPN
iajs-3051	413	4	-	-	PUNCT
iajs-3051	413	5	nahrain	nahrain	PROPN
iajs-3051	413	6	university	university	NOUN
iajs-3051	413	7	science	science	NOUN
iajs-3051	413	8	.	.	PUNCT
iajs-3051	414	1	2010	2010	NUM
iajs-3051	414	2	,	,	PUNCT
iajs-3051	414	3	vol.13	vol.13	NOUN
iajs-3051	414	4	(	(	PUNCT
iajs-3051	414	5	3	3	NUM
iajs-3051	414	6	)	)	PUNCT
iajs-3051	414	7	,	,	PUNCT
iajs-3051	414	8	september,134	september,134	PROPN
iajs-3051	414	9	-	-	PUNCT
iajs-3051	414	10	137	137	NUM
iajs-3051	414	11	.	.	PUNCT
iajs-3051	415	1	5	5	NUM
iajs-3051	415	2	.	.	PUNCT
iajs-3051	416	1	mohammed	mohammed	PROPN
iajs-3051	416	2	,	,	PUNCT
iajs-3051	416	3	n.	n.	PROPN
iajs-3051	416	4	f.	f.	PROPN
iajs-3051	416	5	;	;	PUNCT
iajs-3051	416	6	gasim	gasim	PROPN
iajs-3051	416	7	,	,	PUNCT
iajs-3051	416	8	s.	s.	PROPN
iajs-3051	416	9	g.	g.	PROPN
iajs-3051	416	10	;	;	PUNCT
iajs-3051	416	11	mohammed	mohammed	PROPN
iajs-3051	416	12	,	,	PUNCT
iajs-3051	416	13	a.	a.	PROPN
iajs-3051	416	14	s.	s.	PROPN
iajs-3051	416	15	on	on	ADP
iajs-3051	416	16	cohomology	cohomology	NOUN
iajs-3051	416	17	groups	group	NOUN
iajs-3051	416	18	of	of	ADP
iajs-3051	416	19	fourdimensional	fourdimensional	ADJ
iajs-3051	416	20	nilpotent	nilpotent	ADJ
iajs-3051	416	21	associative	associative	ADJ
iajs-3051	416	22	algebras	algebra	NOUN
iajs-3051	416	23	.	.	PUNCT
iajs-3051	416	24	baghdad	baghdad	PROPN
iajs-3051	416	25	science	science	PROPN
iajs-3051	416	26	journal	journal	PROPN
iajs-3051	416	27	.	.	PUNCT
iajs-3051	417	1	university	university	NOUN
iajs-3051	417	2	of	of	ADP
iajs-3051	417	3	baghdad	baghdad	PROPN
iajs-3051	417	4	–	–	PUNCT
iajs-3051	417	5	collage	collage	NOUN
iajs-3051	417	6	of	of	ADP
iajs-3051	417	7	science	science	NOUN
iajs-3051	417	8	for	for	ADP
iajs-3051	417	9	woman	woman	NOUN
iajs-3051	417	10	.	.	PUNCT
iajs-3051	418	1	published	publish	VERB
iajs-3051	418	2	online	online	ADV
iajs-3051	418	3	first	first	ADV
iajs-3051	418	4	:	:	PUNCT
iajs-3051	418	5	september	september	PROPN
iajs-3051	418	6	,	,	PUNCT
iajs-3051	418	7	doi	doi	PROPN
iajs-3051	418	8	:	:	PUNCT
iajs-3051	418	9	http://dx.doi.org/10.21123/bsj.2022.19.2.029	http://dx.doi.org/10.21123/bsj.2022.19.2.029	NOUN
iajs-3051	418	10	,	,	PUNCT
iajs-3051	418	11	2021	2021	NUM
iajs-3051	418	12	,	,	PUNCT
iajs-3051	418	13	vol	vol	NOUN
iajs-3051	418	14	.	.	PROPN
iajs-3051	418	15	19	19	NUM
iajs-3051	418	16	,	,	PUNCT
iajs-3051	418	17	no.2	no.2	PROPN
iajs-3051	418	18	,	,	PUNCT
iajs-3051	418	19	pp.329	pp.329	PROPN
iajs-3051	418	20	-	-	PUNCT
iajs-3051	418	21	335	335	NUM
iajs-3051	418	22	,	,	PUNCT
iajs-3051	418	23	202	202	NUM
iajs-3051	418	24	.	.	NOUN
iajs-3051	418	25	6	6	NUM
iajs-3051	418	26	.	.	X
iajs-3051	418	27	nasef	nasef	PROPN
iajs-3051	418	28	,	,	PUNCT
iajs-3051	418	29	a.	a.	NOUN
iajs-3051	418	30	a.	a.	PROPN
iajs-3051	418	31	;	;	PUNCT
iajs-3051	418	32	aggour	aggour	NOUN
iajs-3051	418	33	,	,	PUNCT
iajs-3051	418	34	a.	a.	NOUN
iajs-3051	418	35	i.	i.	PROPN
iajs-3051	418	36	;	;	PUNCT
iajs-3051	418	37	aggour	aggour	PROPN
iajs-3051	418	38	,	,	PUNCT
iajs-3051	418	39	s.	s.	PROPN
iajs-3051	418	40	m.	m.	NOUN
iajs-3051	418	41	on	on	ADP
iajs-3051	418	42	some	some	DET
iajs-3051	418	43	classes	class	NOUN
iajs-3051	418	44	of	of	ADP
iajs-3051	418	45	nearly	nearly	ADV
iajs-3051	418	46	open	open	ADJ
iajs-3051	418	47	sets	set	NOUN
iajs-3051	418	48	in	in	ADP
iajs-3051	418	49	nano	nano	ADJ
iajs-3051	418	50	topological	topological	ADJ
iajs-3051	418	51	spaces	space	NOUN
iajs-3051	418	52	.	.	PUNCT
iajs-3051	419	1	journal	journal	NOUN
iajs-3051	419	2	of	of	ADP
iajs-3051	419	3	the	the	DET
iajs-3051	419	4	egyptian	egyptian	PROPN
iajs-3051	419	5	mathematical	mathematical	ADJ
iajs-3051	419	6	society	society	NOUN
iajs-3051	419	7	.	.	PUNCT
iajs-3051	420	1	2016	2016	NUM
iajs-3051	420	2	,	,	PUNCT
iajs-3051	420	3	24,585	24,585	NUM
iajs-3051	420	4	-	-	SYM
iajs-3051	420	5	589	589	NUM
iajs-3051	420	6	.	.	PUNCT
iajs-3051	421	1	7	7	X
iajs-3051	421	2	.	.	X
iajs-3051	421	3	vadivel	vadivel	NOUN
iajs-3051	421	4	,	,	PUNCT
iajs-3051	421	5	a.	a.	NOUN
iajs-3051	421	6	;	;	PUNCT
iajs-3051	422	1	padma	padma	NOUN
iajs-3051	422	2	,	,	PUNCT
iajs-3051	422	3	a.	a.	NOUN
iajs-3051	422	4	;	;	PUNCT
iajs-3051	422	5	saraswathi	saraswathi	PROPN
iajs-3051	422	6	,	,	PUNCT
iajs-3051	422	7	m.	m.	NOUN
iajs-3051	422	8	;	;	PUNCT
iajs-3051	422	9	saravanakumar	saravanakumar	PROPN
iajs-3051	422	10	,	,	PUNCT
iajs-3051	422	11	g.	g.	PROPN
iajs-3051	422	12	nano	nano	NOUN
iajs-3051	422	13	continuous	continuous	ADJ
iajs-3051	422	14	mappings	mapping	NOUN
iajs-3051	422	15	via	via	ADP
iajs-3051	422	16	nano	nano	NOUN
iajs-3051	422	17	ℳ	ℳ	NOUN
iajs-3051	422	18	open	open	ADJ
iajs-3051	422	19	sets	set	NOUN
iajs-3051	422	20	.	.	PUNCT
iajs-3051	423	1	applications	application	NOUN
iajs-3051	423	2	and	and	CCONJ
iajs-3051	423	3	applied	apply	VERB
iajs-3051	423	4	mathematics	mathematic	NOUN
iajs-3051	423	5	:	:	PUNCT
iajs-3051	423	6	an	an	DET
iajs-3051	423	7	international	international	ADJ
iajs-3051	423	8	journal	journal	NOUN
iajs-3051	423	9	(	(	PUNCT
iajs-3051	423	10	aam	aam	PROPN
iajs-3051	423	11	)	)	PUNCT
iajs-3051	423	12	.	.	PUNCT
iajs-3051	424	1	december	december	PROPN
iajs-3051	424	2	2021	2021	NUM
iajs-3051	424	3	,	,	PUNCT
iajs-3051	424	4	issn	issn	PROPN
iajs-3051	424	5	:	:	PUNCT
iajs-3051	424	6	1932	1932	NUM
iajs-3051	424	7	-	-	SYM
iajs-3051	424	8	9466	9466	NUM
iajs-3051	424	9	,	,	PUNCT
iajs-3051	424	10	issue	issue	NOUN
iajs-3051	424	11	2	2	NUM
iajs-3051	424	12	,	,	PUNCT
iajs-3051	424	13	vol.16	vol.16	NOUN
iajs-3051	424	14	,	,	PUNCT
iajs-3051	424	15	1099	1099	NUM
iajs-3051	424	16	-	-	SYM
iajs-3051	424	17	1119	1119	NUM
iajs-3051	424	18	,	,	PUNCT
iajs-3051	424	19	.	.	PUNCT
iajs-3051	424	20	8	8	X
iajs-3051	424	21	.	.	X
iajs-3051	425	1	engelking	engelke	VERB
iajs-3051	425	2	,	,	PUNCT
iajs-3051	425	3	r.	r.	PROPN
iajs-3051	425	4	general	general	PROPN
iajs-3051	425	5	topology	topology	PROPN
iajs-3051	425	6	.	.	PUNCT
iajs-3051	426	1	heldermann	heldermann	PROPN
iajs-3051	426	2	.	.	PUNCT
iajs-3051	427	1	1989	1989	NUM
iajs-3051	427	2	,	,	PUNCT
iajs-3051	427	3	berlin	berlin	PROPN
iajs-3051	427	4	.	.	PUNCT
iajs-3051	428	1	http://dx.doi.org/10.21123/bsj.2022.19.2.029	http://dx.doi.org/10.21123/bsj.2022.19.2.029	PROPN
iajs-3051	428	2	ihjpas	ihjpa	VERB
iajs-3051	428	3	.	.	PUNCT
iajs-3051	429	1	36	36	NUM
iajs-3051	429	2	(	(	PUNCT
iajs-3051	429	3	3	3	NUM
iajs-3051	429	4	)	)	PUNCT
iajs-3051	429	5	2023	2023	NUM
iajs-3051	429	6	407	407	NUM
iajs-3051	429	7	9	9	NUM
iajs-3051	429	8	.	.	PUNCT
iajs-3051	430	1	mohammed	mohammed	PROPN
iajs-3051	430	2	,	,	PUNCT
iajs-3051	430	3	n.	n.	PROPN
iajs-3051	430	4	f.	f.	PROPN
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iajs-3051	430	8	subspaces	subspace	NOUN
iajs-3051	430	9	.	.	PUNCT
iajs-3051	431	1	baghdad	baghdad	PROPN
iajs-3051	431	2	science	science	PROPN
iajs-3051	431	3	journal	journal	PROPN
iajs-3051	431	4	.	.	PUNCT
iajs-3051	432	1	university	university	NOUN
iajs-3051	432	2	of	of	ADP
iajs-3051	432	3	baghdad	baghdad	PROPN
iajs-3051	432	4	–	–	PUNCT
iajs-3051	432	5	collage	collage	NOUN
iajs-3051	432	6	of	of	ADP
iajs-3051	432	7	science	science	NOUN
iajs-3051	432	8	for	for	ADP
iajs-3051	432	9	woman	woman	NOUN
iajs-3051	432	10	.	.	PUNCT
iajs-3051	433	1	2010	2010	NUM
iajs-3051	433	2	,	,	PUNCT
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iajs-3051	433	4	.	.	PROPN
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iajs-3051	433	6	,	,	PUNCT
iajs-3051	433	7	no.1	no.1	NUM
iajs-3051	433	8	,	,	PUNCT
iajs-3051	433	9	pp.174	pp.174	PROPN
iajs-3051	433	10	-	-	PUNCT
iajs-3051	433	11	179	179	NUM
iajs-3051	433	12	.	.	PUNCT
iajs-3051	434	1	10	10	NUM
iajs-3051	434	2	.	.	X
iajs-3051	435	1	majeed	majeed	PROPN
iajs-3051	435	2	,	,	PUNCT
iajs-3051	435	3	r.	r.	PROPN
iajs-3051	435	4	n.	n.	PROPN
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iajs-3051	435	6	on	on	ADP
iajs-3051	435	7	bitopological	bitopological	ADJ
iajs-3051	435	8	spases	spase	NOUN
iajs-3051	435	9	.	.	PUNCT
iajs-3051	436	1	journal	journal	PROPN
iajs-3051	436	2	of	of	ADP
iajs-3051	436	3	al	al	PROPN
iajs-3051	436	4	-	-	PUNCT
iajs-3051	436	5	qadisiyah	qadisiyah	NOUN
iajs-3051	436	6	for	for	ADP
iajs-3051	436	7	computer	computer	NOUN
iajs-3051	436	8	science	science	NOUN
iajs-3051	436	9	and	and	CCONJ
iajs-3051	436	10	mathematics	mathematic	NOUN
iajs-3051	436	11	.	.	PUNCT
iajs-3051	437	1	2011	2011	NUM
iajs-3051	437	2	,	,	PUNCT
iajs-3051	437	3	vol	vol	NOUN
iajs-3051	437	4	.	.	PUNCT
iajs-3051	437	5	3(1	3(1	NUM
iajs-3051	437	6	)	)	PUNCT
iajs-3051	437	7	,	,	PUNCT
iajs-3051	438	1	pp	pp	ADP
iajs-3051	438	2	.	.	PUNCT
iajs-3051	439	1	1	1	NUM
iajs-3051	439	2	-	-	SYM
iajs-3051	439	3	7	7	NUM
iajs-3051	439	4	.	.	NOUN
iajs-3051	439	5	11	11	NUM
iajs-3051	439	6	.	.	PUNCT
iajs-3051	439	7	thivagar	thivagar	NOUN
iajs-3051	439	8	,	,	PUNCT
iajs-3051	439	9	m.	m.	NOUN
iajs-3051	439	10	l.	l.	PROPN
iajs-3051	439	11	;	;	PUNCT
iajs-3051	439	12	richard	richard	PROPN
iajs-3051	439	13	,	,	PUNCT
iajs-3051	439	14	c.	c.	PROPN
iajs-3051	439	15	on	on	ADP
iajs-3051	439	16	nano	nano	NOUN
iajs-3051	439	17	continuity	continuity	NOUN
iajs-3051	439	18	.	.	PUNCT
iajs-3051	440	1	international	international	ADJ
iajs-3051	440	2	journal	journal	PROPN
iajs-3051	440	3	of	of	ADP
iajs-3051	440	4	mathematics	mathematics	PROPN
iajs-3051	440	5	and	and	CCONJ
iajs-3051	440	6	statistics	statistic	NOUN
iajs-3051	440	7	invention	invention	NOUN
iajs-3051	440	8	.	.	PUNCT
iajs-3051	441	1	2013	2013	NUM
iajs-3051	441	2	,	,	PUNCT
iajs-3051	441	3	1(1	1(1	NUM
iajs-3051	441	4	)	)	PUNCT
iajs-3051	441	5	,	,	PUNCT
iajs-3051	442	1	31–37	31–37	NUM
iajs-3051	442	2	.	.	PUNCT
iajs-3051	443	1	12	12	NUM
iajs-3051	443	2	.	.	PUNCT
iajs-3051	444	1	parimala	parimala	NOUN
iajs-3051	444	2	,	,	PUNCT
iajs-3051	444	3	m.	m.	NOUN
iajs-3051	444	4	;	;	PUNCT
iajs-3051	444	5	indirani	indirani	PROPN
iajs-3051	444	6	,	,	PUNCT
iajs-3051	444	7	c.	c.	PROPN
iajs-3051	444	8	;	;	PUNCT
iajs-3051	444	9	jafari	jafari	PROPN
iajs-3051	444	10	,	,	PUNCT
iajs-3051	444	11	s.	s.	PROPN
iajs-3051	444	12	on	on	ADP
iajs-3051	444	13	nano	nano	PROPN
iajs-3051	444	14	b	b	PROPN
iajs-3051	444	15	–	–	PUNCT
iajs-3051	444	16	open	open	ADJ
iajs-3051	444	17	sets	set	NOUN
iajs-3051	444	18	in	in	ADP
iajs-3051	444	19	nano	nano	ADJ
iajs-3051	444	20	topological	topological	ADJ
iajs-3051	444	21	spaces	space	NOUN
iajs-3051	444	22	.	.	PUNCT
iajs-3051	445	1	jordan	jordan	PROPN
iajs-3051	445	2	journal	journal	PROPN
iajs-3051	445	3	of	of	ADP
iajs-3051	445	4	mathematics	mathematics	PROPN
iajs-3051	445	5	and	and	CCONJ
iajs-3051	445	6	statistics	statistic	NOUN
iajs-3051	445	7	(	(	PUNCT
iajs-3051	445	8	jjms	jjms	PROPN
iajs-3051	445	9	)	)	PUNCT
iajs-3051	445	10	9(3	9(3	NUM
iajs-3051	445	11	)	)	PUNCT
iajs-3051	445	12	.	.	PUNCT
iajs-3051	446	1	2016	2016	NUM
iajs-3051	446	2	pp	pp	ADP
iajs-3051	446	3	173	173	NUM
iajs-3051	446	4	-	-	SYM
iajs-3051	446	5	184	184	NUM
iajs-3051	446	6	.	.	NOUN
iajs-3051	446	7	13	13	NUM
iajs-3051	446	8	.	.	X
iajs-3051	446	9	rajasekaran	rajasekaran	NOUN
iajs-3051	446	10	,	,	PUNCT
iajs-3051	446	11	i.	i.	NOUN
iajs-3051	446	12	;	;	PUNCT
iajs-3051	446	13	nethaji	nethaji	PROPN
iajs-3051	446	14	,	,	PUNCT
iajs-3051	446	15	o.	o.	ADJ
iajs-3051	446	16	simple	simple	ADJ
iajs-3051	446	17	forms	form	NOUN
iajs-3051	446	18	of	of	ADP
iajs-3051	446	19	nano	nano	NOUN
iajs-3051	446	20	open	open	ADJ
iajs-3051	446	21	sets	set	NOUN
iajs-3051	446	22	in	in	ADP
iajs-3051	446	23	an	an	DET
iajs-3051	446	24	ideal	ideal	ADJ
iajs-3051	446	25	nano	nano	NOUN
iajs-3051	446	26	topological	topological	ADJ
iajs-3051	446	27	spaces	space	NOUN
iajs-3051	446	28	.	.	PUNCT
iajs-3051	447	1	journal	journal	NOUN
iajs-3051	447	2	of	of	ADP
iajs-3051	447	3	new	new	ADJ
iajs-3051	447	4	theory	theory	NOUN
iajs-3051	447	5	.	.	PUNCT
iajs-3051	448	1	2018	2018	NUM
iajs-3051	448	2	,	,	PUNCT
iajs-3051	448	3	issn	issn	PROPN
iajs-3051	448	4	:	:	PUNCT
iajs-3051	448	5	2149	2149	NUM
iajs-3051	448	6	-	-	SYM
iajs-3051	448	7	1402	1402	NUM
iajs-3051	448	8	,	,	PUNCT
iajs-3051	448	9	number	number	NOUN
iajs-3051	448	10	:	:	PUNCT
iajs-3051	448	11	24	24	NUM
iajs-3051	448	12	,	,	PUNCT
iajs-3051	448	13	pages	page	NOUN
iajs-3051	448	14	:	:	PUNCT
iajs-3051	448	15	35	35	NUM
iajs-3051	448	16	-	-	SYM
iajs-3051	448	17	43	43	NUM
iajs-3051	448	18	.	.	PUNCT
iajs-3051	449	1	14	14	NUM
iajs-3051	449	2	.	.	PUNCT
iajs-3051	450	1	el	el	PROPN
iajs-3051	450	2	-	-	PUNCT
iajs-3051	450	3	atik	atik	PROPN
iajs-3051	450	4	,	,	PUNCT
iajs-3051	450	5	a.	a.	NOUN
iajs-3051	450	6	a.	a.	NOUN
iajs-3051	450	7	;	;	PUNCT
iajs-3051	451	1	hassan	hassan	PROPN
iajs-3051	451	2	,	,	PUNCT
iajs-3051	451	3	h.	h.	PROPN
iajs-3051	451	4	z.	z.	PROPN
iajs-3051	452	1	some	some	DET
iajs-3051	452	2	nano	nano	PROPN
iajs-3051	452	3	topological	topological	ADJ
iajs-3051	452	4	structures	structure	NOUN
iajs-3051	452	5	via	via	ADP
iajs-3051	452	6	ideals	ideal	NOUN
iajs-3051	452	7	and	and	CCONJ
iajs-3051	452	8	graphs	graph	NOUN
iajs-3051	452	9	.	.	PUNCT
iajs-3051	453	1	journal	journal	NOUN
iajs-3051	453	2	of	of	ADP
iajs-3051	453	3	the	the	DET
iajs-3051	453	4	egyptian	egyptian	PROPN
iajs-3051	453	5	mathematical	mathematical	PROPN
iajs-3051	453	6	society	society	NOUN
iajs-3051	453	7	,	,	PUNCT
iajs-3051	453	8	2020	2020	NUM
iajs-3051	453	9	.	.	PUNCT
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iajs-3051	454	2	.	.	PUNCT
iajs-3051	454	3	krishnaprakash	krishnaprakash	PROPN
iajs-3051	454	4	,	,	PUNCT
iajs-3051	454	5	s.	s.	PROPN
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iajs-3051	454	7	ramesh	ramesh	PROPN
iajs-3051	454	8	,	,	PUNCT
iajs-3051	454	9	r.	r.	PROPN
iajs-3051	454	10	;	;	PUNCT
iajs-3051	454	11	suresh	suresh	PROPN
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iajs-3051	454	13	r.	r.	PROPN
iajs-3051	454	14	nano	nano	PROPN
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iajs-3051	454	16	and	and	CCONJ
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iajs-3051	454	20	nano	nano	ADJ
iajs-3051	454	21	topological	topological	ADJ
iajs-3051	454	22	space	space	NOUN
iajs-3051	454	23	.	.	PUNCT
iajs-3051	455	1	international	international	ADJ
iajs-3051	455	2	journal	journal	NOUN
iajs-3051	455	3	of	of	ADP
iajs-3051	455	4	pure	pure	ADJ
iajs-3051	455	5	and	and	CCONJ
iajs-3051	455	6	applied	applied	ADJ
iajs-3051	455	7	mathematics	mathematic	NOUN
iajs-3051	455	8	.	.	PUNCT
iajs-3051	456	1	2018	2018	NUM
iajs-3051	456	2	,	,	PUNCT
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iajs-3051	456	4	.	.	PROPN
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iajs-3051	457	2	,	,	PUNCT
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iajs-3051	457	4	,	,	PUNCT
iajs-3051	457	5	pp	pp	ADJ
iajs-3051	457	6	.	.	PUNCT
iajs-3051	458	1	107	107	NUM
iajs-3051	458	2	–	–	PUNCT
iajs-3051	458	3	115	115	NUM
iajs-3051	458	4	.	.	PUNCT
iajs-3051	459	1	16	16	NUM
iajs-3051	459	2	.	.	X
iajs-3051	459	3	bourbaki	bourbaki	VERB
iajs-3051	459	4	,	,	PUNCT
iajs-3051	459	5	n	n	X
iajs-3051	459	6	.	.	PUNCT
iajs-3051	460	1	general	general	ADJ
iajs-3051	460	2	topology	topology	NOUN
iajs-3051	460	3	;	;	PUNCT
iajs-3051	460	4	part	part	NOUN
iajs-3051	460	5	ι	ι	PROPN
iajs-3051	460	6	.	.	PROPN
iajs-3051	460	7	addison	addison	PROPN
iajs-3051	460	8	wesley	wesley	PROPN
iajs-3051	460	9	.	.	PUNCT
iajs-3051	461	1	reading	reading	NOUN
iajs-3051	461	2	.	.	PUNCT
iajs-3051	462	1	mass	mass	PROPN
iajs-3051	462	2	,	,	PUNCT
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iajs-3051	462	4	.	.	PUNCT
iajs-3051	463	1	17	17	NUM
iajs-3051	463	2	.	.	PUNCT
iajs-3051	463	3	upadhya	upadhya	PROPN
iajs-3051	463	4	,	,	PUNCT
iajs-3051	463	5	a.	a.	NOUN
iajs-3051	463	6	c.	c.	PROPN
iajs-3051	463	7	;	;	PUNCT
iajs-3051	463	8	mamata	mamata	PROPN
iajs-3051	463	9	,	,	PUNCT
iajs-3051	463	10	m.	m.	PROPN
iajs-3051	463	11	k.	k.	PROPN
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iajs-3051	463	13	nano	nano	ADJ
iajs-3051	463	14	gp	gp	NOUN
iajs-3051	463	15	-	-	ADJ
iajs-3051	463	16	regular	regular	ADJ
iajs-3051	463	17	and	and	CCONJ
iajs-3051	463	18	nano	nano	ADJ
iajs-3051	463	19	gp	gp	NOUN
iajs-3051	463	20	-	-	ADJ
iajs-3051	463	21	normal	normal	ADJ
iajs-3051	463	22	spaces	space	NOUN
iajs-3051	463	23	.	.	PUNCT
iajs-3051	464	1	the	the	DET
iajs-3051	464	2	international	international	ADJ
iajs-3051	464	3	journal	journal	NOUN
iajs-3051	464	4	of	of	ADP
iajs-3051	464	5	analytical	analytical	ADJ
iajs-3051	464	6	and	and	CCONJ
iajs-3051	464	7	experimental	experimental	ADJ
iajs-3051	464	8	modal	modal	ADJ
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iajs-3051	464	10	.	.	PUNCT
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iajs-3051	467	2	,	,	PUNCT
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iajs-3051	467	4	,	,	PUNCT
iajs-3051	467	5	issn	issn	PROPN
iajs-3051	467	6	no:0886	no:0886	NOUN
iajs-3051	467	7	-	-	NOUN
iajs-3051	467	8	9367	9367	NUM
iajs-3051	467	9	,	,	PUNCT
iajs-3051	467	10	p:3770	p:3770	NOUN
iajs-3051	467	11	-	-	PUNCT
iajs-3051	467	12	3775	3775	NUM
iajs-3051	467	13	.	.	PUNCT
iajs-3051	468	1	18	18	NUM
iajs-3051	468	2	.	.	PUNCT
iajs-3051	468	3	pears	pear	NOUN
iajs-3051	468	4	,	,	PUNCT
iajs-3051	468	5	a.	a.	PROPN
iajs-3051	468	6	r.	r.	PROPN
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iajs-3051	468	10	general	general	ADJ
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iajs-3051	468	12	.	.	PUNCT
iajs-3051	469	1	cambridge	cambridge	PROPN
iajs-3051	469	2	university	university	PROPN
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iajs-3051	469	4	.	.	PUNCT
iajs-3051	470	1	london	london	PROPN
iajs-3051	470	2	.	.	PUNCT
iajs-3051	471	1	new	new	PROPN
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iajs-3051	471	3	,	,	PUNCT
iajs-3051	471	4	melbourne	melbourne	PROPN
iajs-3051	471	5	,	,	PUNCT
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iajs-3051	471	7	.	.	PUNCT
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iajs-3051	472	2	.	.	NUM
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iajs-3051	472	4	,	,	PUNCT
iajs-3051	472	5	g.	g.	PROPN
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iajs-3051	472	7	;	;	PUNCT
iajs-3051	472	8	yousif	yousif	PROPN
iajs-3051	472	9	,	,	PUNCT
iajs-3051	473	1	y.	y.	PROPN
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iajs-3051	473	10	.	.	PUNCT
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iajs-3051	475	4	–	–	PUNCT
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iajs-3051	475	6	of	of	ADP
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iajs-3051	475	8	for	for	ADP
iajs-3051	475	9	woman	woman	NOUN
iajs-3051	475	10	,	,	PUNCT
iajs-3051	475	11	published	publish	VERB
iajs-3051	475	12	online	online	ADV
iajs-3051	475	13	first	first	ADV
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iajs-3051	475	22	,	,	PUNCT
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iajs-3051	475	24	,	,	PUNCT
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iajs-3051	475	28	.	.	PUNCT
iajs-3051	476	1	20	20	NUM
iajs-3051	476	2	.	.	X
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iajs-3051	477	2	,	,	PUNCT
iajs-3051	477	3	s.	s.	PROPN
iajs-3051	477	4	;	;	PUNCT
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iajs-3051	477	6	,	,	PUNCT
iajs-3051	477	7	y.	y.	PROPN
iajs-3051	477	8	y.	y.	PROPN
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iajs-3051	477	13	.	.	PUNCT
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iajs-3051	478	5	.	.	PUNCT
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iajs-3051	479	2	of	of	ADP
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iajs-3051	479	8	of	of	ADP
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iajs-3051	479	11	at	at	ADP
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iajs-3051	479	14	–	–	PUNCT
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iajs-3051	479	19	in	in	ADP
iajs-3051	479	20	5	5	NUM
iajs-3051	479	21	-	-	SYM
iajs-3051	479	22	6	6	NUM
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iajs-3051	479	24	.	.	PROPN
iajs-3051	479	25	2020	2020	NUM
iajs-3051	479	26	,	,	PUNCT
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iajs-3051	479	28	/	/	SYM
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iajs-3051	479	30	,	,	PUNCT
iajs-3051	479	31	special	special	ADJ
iajs-3051	479	32	issue,108	issue,108	NOUN
iajs-3051	479	33	-	-	PUNCT
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iajs-3051	479	35	.	.	PUNCT
iajs-3051	480	1	21	21	NUM
iajs-3051	480	2	.	.	X
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iajs-3051	480	4	,	,	PUNCT
iajs-3051	480	5	r.b	r.b	PROPN
iajs-3051	480	6	.	.	PROPN
iajs-3051	480	7	;	;	PUNCT
iajs-3051	480	8	saeed	saeed	PROPN
iajs-3051	480	9	,	,	PUNCT
iajs-3051	480	10	s.	s.	PROPN
iajs-3051	480	11	nano	nano	PROPN
iajs-3051	480	12	ag	ag	PROPN
iajs-3051	480	13	-	-	PUNCT
iajs-3051	480	14	open	open	ADJ
iajs-3051	480	15	set	set	NOUN
iajs-3051	480	16	.	.	PUNCT
iajs-3051	481	1	journal	journal	PROPN
iajs-3051	481	2	of	of	ADP
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iajs-3051	481	4	:	:	PUNCT
iajs-3051	481	5	conference	conference	NOUN
iajs-3051	481	6	seriesthis	seriesthis	PROPN
iajs-3051	481	7	link	link	NOUN
iajs-3051	481	8	is	be	AUX
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iajs-3051	481	10	.	.	PUNCT
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iajs-3051	482	2	,	,	PUNCT
iajs-3051	482	3	1897(1	1897(1	NUM
iajs-3051	482	4	)	)	PUNCT
iajs-3051	482	5	,	,	PUNCT
iajs-3051	482	6	012031	012031	NUM
iajs-3051	482	7	.	.	PUNCT
iajs-3051	483	1	22	22	NUM
iajs-3051	483	2	.	.	X
iajs-3051	483	3	esmaeel	esmaeel	PROPN
iajs-3051	483	4	,	,	PUNCT
iajs-3051	483	5	r.	r.	PROPN
iajs-3051	483	6	b.	b.	PROPN
iajs-3051	483	7	;	;	PUNCT
iajs-3051	483	8	jassam	jassam	PROPN
iajs-3051	483	9	,	,	PUNCT
iajs-3051	483	10	a.	a.	NOUN
iajs-3051	483	11	a.	a.	NOUN
iajs-3051	483	12	on	on	ADP
iajs-3051	483	13	nano	nano	PROPN
iajs-3051	483	14	ḟ	ḟ	NOUN
iajs-3051	483	15	-	-	ADJ
iajs-3051	483	16	pre	pre	ADJ
iajs-3051	483	17	-	-	ADJ
iajs-3051	483	18	g	g	ADV
iajs-3051	483	19	-	-	PUNCT
iajs-3051	483	20	open	open	ADJ
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iajs-3051	483	22	.	.	PUNCT
iajs-3051	484	1	ibn	ibn	PROPN
iajs-3051	484	2	al	al	PROPN
iajs-3051	484	3	-	-	PUNCT
iajs-3051	484	4	haitham	haitham	PROPN
iajs-3051	484	5	journal	journal	PROPN
iajs-3051	484	6	for	for	ADP
iajs-3051	484	7	pure	pure	ADJ
iajs-3051	484	8	and	and	CCONJ
iajs-3051	484	9	applied	applied	ADJ
iajs-3051	484	10	science	science	NOUN
iajs-3051	484	11	.	.	PUNCT
iajs-3051	485	1	2020	2020	NUM
iajs-3051	485	2	,	,	PUNCT
iajs-3051	485	3	33(3),1	33(3),1	NUM
iajs-3051	485	4	-	-	SYM
iajs-3051	485	5	12	12	NUM
iajs-3051	485	6	.	.	PUNCT