id	sid	tid	token	lemma	pos
iajs-2958	1	1	ihjpas	ihjpas	PROPN
iajs-2958	1	2	.	.	PUNCT
iajs-2958	2	1	36(2)2023	36(2)2023	NUM
iajs-2958	2	2	306	306	NUM
iajs-2958	2	3	this	this	DET
iajs-2958	2	4	work	work	NOUN
iajs-2958	2	5	is	be	AUX
iajs-2958	2	6	licensed	license	VERB
iajs-2958	2	7	under	under	ADP
iajs-2958	2	8	a	a	DET
iajs-2958	2	9	creative	creative	ADJ
iajs-2958	2	10	commons	common	NOUN
iajs-2958	2	11	attribution	attribution	NOUN
iajs-2958	2	12	4.0	4.0	NUM
iajs-2958	2	13	international	international	ADJ
iajs-2958	2	14	license	license	NOUN
iajs-2958	2	15	.	.	PUNCT
iajs-2958	3	1	abstract	abstract	ADJ
iajs-2958	3	2	the	the	DET
iajs-2958	3	3	objective	objective	NOUN
iajs-2958	3	4	of	of	ADP
iajs-2958	3	5	this	this	DET
iajs-2958	3	6	paper	paper	NOUN
iajs-2958	3	7	is	be	AUX
iajs-2958	3	8	to	to	PART
iajs-2958	3	9	define	define	VERB
iajs-2958	3	10	and	and	CCONJ
iajs-2958	3	11	introduce	introduce	VERB
iajs-2958	3	12	a	a	DET
iajs-2958	3	13	new	new	ADJ
iajs-2958	3	14	type	type	NOUN
iajs-2958	3	15	of	of	ADP
iajs-2958	3	16	nano	nano	NOUN
iajs-2958	3	17	semi	semi	ADJ
iajs-2958	3	18	-	-	ADJ
iajs-2958	3	19	open	open	ADJ
iajs-2958	3	20	set	set	NOUN
iajs-2958	3	21	which	which	PRON
iajs-2958	3	22	called	call	VERB
iajs-2958	3	23	nano	nano	NOUN
iajs-2958	3	24	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	3	25	set	set	VERB
iajs-2958	3	26	as	as	ADP
iajs-2958	3	27	a	a	DET
iajs-2958	3	28	strong	strong	ADJ
iajs-2958	3	29	form	form	NOUN
iajs-2958	3	30	of	of	ADP
iajs-2958	3	31	nano	nano	NOUN
iajs-2958	3	32	semi	semi	ADJ
iajs-2958	3	33	-	-	ADJ
iajs-2958	3	34	open	open	ADJ
iajs-2958	3	35	set	set	NOUN
iajs-2958	3	36	which	which	PRON
iajs-2958	3	37	is	be	AUX
iajs-2958	3	38	related	relate	VERB
iajs-2958	3	39	to	to	ADP
iajs-2958	3	40	nano	nano	NOUN
iajs-2958	3	41	closed	close	VERB
iajs-2958	3	42	sets	set	NOUN
iajs-2958	3	43	in	in	ADP
iajs-2958	3	44	nano	nano	ADJ
iajs-2958	3	45	topological	topological	ADJ
iajs-2958	3	46	spaces	space	NOUN
iajs-2958	3	47	.	.	PUNCT
iajs-2958	4	1	in	in	ADP
iajs-2958	4	2	this	this	DET
iajs-2958	4	3	paper	paper	NOUN
iajs-2958	4	4	,	,	PUNCT
iajs-2958	4	5	we	we	PRON
iajs-2958	4	6	find	find	VERB
iajs-2958	4	7	all	all	DET
iajs-2958	4	8	forms	form	NOUN
iajs-2958	4	9	of	of	ADP
iajs-2958	4	10	the	the	DET
iajs-2958	4	11	family	family	NOUN
iajs-2958	4	12	of	of	ADP
iajs-2958	4	13	nano	nano	NOUN
iajs-2958	4	14	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	4	15	sets	set	VERB
iajs-2958	4	16	in	in	ADP
iajs-2958	4	17	term	term	NOUN
iajs-2958	4	18	of	of	ADP
iajs-2958	4	19	upper	upper	ADJ
iajs-2958	4	20	and	and	CCONJ
iajs-2958	4	21	lower	low	ADJ
iajs-2958	4	22	approximations	approximation	NOUN
iajs-2958	4	23	of	of	ADP
iajs-2958	4	24	sets	set	NOUN
iajs-2958	4	25	and	and	CCONJ
iajs-2958	4	26	we	we	PRON
iajs-2958	4	27	can	can	AUX
iajs-2958	4	28	easily	easily	ADV
iajs-2958	4	29	find	find	VERB
iajs-2958	4	30	nano	nano	NOUN
iajs-2958	4	31	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	4	32	sets	set	NOUN
iajs-2958	4	33	and	and	CCONJ
iajs-2958	4	34	they	they	PRON
iajs-2958	4	35	are	be	AUX
iajs-2958	4	36	a	a	DET
iajs-2958	4	37	gate	gate	NOUN
iajs-2958	4	38	to	to	ADP
iajs-2958	4	39	more	more	ADJ
iajs-2958	4	40	study	study	NOUN
iajs-2958	4	41	.	.	PUNCT
iajs-2958	5	1	several	several	ADJ
iajs-2958	5	2	types	type	NOUN
iajs-2958	5	3	of	of	ADP
iajs-2958	5	4	nano	nano	NOUN
iajs-2958	5	5	open	open	ADJ
iajs-2958	5	6	sets	set	NOUN
iajs-2958	5	7	are	be	AUX
iajs-2958	5	8	known	know	VERB
iajs-2958	5	9	,	,	PUNCT
iajs-2958	5	10	so	so	ADV
iajs-2958	5	11	we	we	PRON
iajs-2958	5	12	study	study	VERB
iajs-2958	5	13	relationship	relationship	NOUN
iajs-2958	5	14	between	between	ADP
iajs-2958	5	15	the	the	DET
iajs-2958	5	16	nano	nano	ADJ
iajs-2958	5	17	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	5	18	sets	set	VERB
iajs-2958	5	19	with	with	ADP
iajs-2958	5	20	the	the	DET
iajs-2958	5	21	other	other	ADJ
iajs-2958	5	22	known	know	VERB
iajs-2958	5	23	types	type	NOUN
iajs-2958	5	24	of	of	ADP
iajs-2958	5	25	nano	nano	NOUN
iajs-2958	5	26	open	open	ADJ
iajs-2958	5	27	sets	set	NOUN
iajs-2958	5	28	in	in	ADP
iajs-2958	5	29	nano	nano	ADJ
iajs-2958	5	30	topological	topological	ADJ
iajs-2958	5	31	spaces	space	NOUN
iajs-2958	5	32	.	.	PUNCT
iajs-2958	6	1	the	the	DET
iajs-2958	6	2	operators	operator	NOUN
iajs-2958	6	3	such	such	ADJ
iajs-2958	6	4	as	as	ADP
iajs-2958	6	5	nano	nano	NOUN
iajs-2958	6	6	𝑆𝐶-interior	𝑆𝐶-interior	NOUN
iajs-2958	6	7	and	and	CCONJ
iajs-2958	6	8	nano	nano	PROPN
iajs-2958	6	9	𝑆𝐶-closure	𝑆𝐶-closure	PROPN
iajs-2958	6	10	are	be	AUX
iajs-2958	6	11	the	the	DET
iajs-2958	6	12	part	part	NOUN
iajs-2958	6	13	of	of	ADP
iajs-2958	6	14	this	this	DET
iajs-2958	6	15	paper	paper	NOUN
iajs-2958	6	16	.	.	PUNCT
iajs-2958	7	1	keywords	keyword	NOUN
iajs-2958	7	2	:	:	PUNCT
iajs-2958	7	3	nano	nano	NOUN
iajs-2958	7	4	closed	close	VERB
iajs-2958	7	5	sets	set	NOUN
iajs-2958	7	6	,	,	PUNCT
iajs-2958	7	7	nano	nano	NOUN
iajs-2958	7	8	semi	semi	ADJ
iajs-2958	7	9	-	-	ADJ
iajs-2958	7	10	open	open	ADJ
iajs-2958	7	11	sets	set	NOUN
iajs-2958	7	12	,	,	PUNCT
iajs-2958	7	13	nano	nano	ADJ
iajs-2958	7	14	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	7	15	sets	set	VERB
iajs-2958	7	16	,	,	PUNCT
iajs-2958	7	17	nano	nano	PROPN
iajs-2958	7	18	𝑆𝐶-interior	𝑆𝐶-interior	PROPN
iajs-2958	7	19	,	,	PUNCT
iajs-2958	7	20	nano	nano	NOUN
iajs-2958	7	21	𝑆𝐶closure	𝑆𝐶closure	PROPN
iajs-2958	7	22	.	.	PUNCT
iajs-2958	8	1	1.introduction	1.introduction	NUM
iajs-2958	8	2	the	the	DET
iajs-2958	8	3	notion	notion	NOUN
iajs-2958	8	4	of	of	ADP
iajs-2958	8	5	nano	nano	NOUN
iajs-2958	8	6	topological	topological	ADJ
iajs-2958	8	7	space	space	NOUN
iajs-2958	8	8	(	(	PUNCT
iajs-2958	8	9	briefly	briefly	NOUN
iajs-2958	8	10	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	8	11	)	)	PUNCT
iajs-2958	8	12	introduced	introduce	VERB
iajs-2958	8	13	by	by	ADP
iajs-2958	8	14	thivagar	thivagar	NOUN
iajs-2958	8	15	and	and	CCONJ
iajs-2958	8	16	carmel	carmel	NOUN
iajs-2958	8	17	[	[	X
iajs-2958	8	18	1	1	NUM
iajs-2958	8	19	]	]	PUNCT
iajs-2958	8	20	with	with	ADP
iajs-2958	8	21	respect	respect	NOUN
iajs-2958	8	22	to	to	ADP
iajs-2958	8	23	a	a	DET
iajs-2958	8	24	subset	subset	ADJ
iajs-2958	8	25	𝑋	𝑋	NOUN
iajs-2958	8	26	of	of	ADP
iajs-2958	8	27	a	a	DET
iajs-2958	8	28	universe	universe	ADJ
iajs-2958	8	29	𝑈	𝑈	NOUN
iajs-2958	8	30	which	which	PRON
iajs-2958	8	31	is	be	AUX
iajs-2958	8	32	defined	define	VERB
iajs-2958	8	33	in	in	ADP
iajs-2958	8	34	terms	term	NOUN
iajs-2958	8	35	of	of	ADP
iajs-2958	8	36	lower	low	ADJ
iajs-2958	8	37	and	and	CCONJ
iajs-2958	8	38	upper	upper	ADJ
iajs-2958	8	39	approximations	approximation	NOUN
iajs-2958	8	40	.	.	PUNCT
iajs-2958	9	1	levine	levine	PROPN
iajs-2958	9	2	[	[	X
iajs-2958	9	3	2	2	NUM
iajs-2958	9	4	]	]	PUNCT
iajs-2958	9	5	introduced	introduce	VERB
iajs-2958	9	6	the	the	DET
iajs-2958	9	7	notions	notion	NOUN
iajs-2958	9	8	of	of	ADP
iajs-2958	9	9	semi	semi	ADJ
iajs-2958	9	10	-	-	ADJ
iajs-2958	9	11	open	open	ADJ
iajs-2958	9	12	.	.	PUNCT
iajs-2958	10	1	later	later	ADV
iajs-2958	10	2	,	,	PUNCT
iajs-2958	10	3	nano	nano	NOUN
iajs-2958	10	4	semi	semi	ADJ
iajs-2958	10	5	-	-	ADJ
iajs-2958	10	6	open	open	ADJ
iajs-2958	10	7	sets	set	NOUN
iajs-2958	10	8	introduced	introduce	VERB
iajs-2958	10	9	by	by	ADP
iajs-2958	10	10	thivagar	thivagar	NOUN
iajs-2958	10	11	carmel	carmel	NOUN
iajs-2958	11	1	[	[	X
iajs-2958	11	2	1	1	NUM
iajs-2958	11	3	]	]	PUNCT
iajs-2958	11	4	,	,	PUNCT
iajs-2958	11	5	also	also	ADV
iajs-2958	11	6	nano	nano	NOUN
iajs-2958	11	7	𝑆𝛽-open	𝑆𝛽-open	PUNCT
iajs-2958	11	8	sets	set	NOUN
iajs-2958	11	9	introduced	introduce	VERB
iajs-2958	11	10	by	by	ADP
iajs-2958	11	11	[	[	X
iajs-2958	11	12	4	4	NUM
iajs-2958	11	13	]	]	PUNCT
iajs-2958	11	14	,	,	PUNCT
iajs-2958	11	15	and	and	CCONJ
iajs-2958	11	16	more	more	ADV
iajs-2958	11	17	nano	nano	ADJ
iajs-2958	11	18	open	open	ADJ
iajs-2958	11	19	sets	set	NOUN
iajs-2958	11	20	defined	define	VERB
iajs-2958	11	21	in	in	ADP
iajs-2958	11	22	[	[	PUNCT
iajs-2958	11	23	5	5	NUM
iajs-2958	11	24	-	-	SYM
iajs-2958	11	25	7	7	NUM
iajs-2958	11	26	]	]	PUNCT
iajs-2958	11	27	.	.	PUNCT
iajs-2958	12	1	in	in	ADP
iajs-2958	12	2	this	this	DET
iajs-2958	12	3	paper	paper	NOUN
iajs-2958	12	4	,	,	PUNCT
iajs-2958	12	5	we	we	PRON
iajs-2958	12	6	introduce	introduce	VERB
iajs-2958	12	7	the	the	DET
iajs-2958	12	8	concept	concept	NOUN
iajs-2958	12	9	nano	nano	NOUN
iajs-2958	12	10	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	12	11	sets	set	VERB
iajs-2958	12	12	as	as	ADP
iajs-2958	12	13	a	a	DET
iajs-2958	12	14	strong	strong	ADJ
iajs-2958	12	15	form	form	NOUN
iajs-2958	12	16	of	of	ADP
iajs-2958	12	17	nano	nano	NOUN
iajs-2958	12	18	semi	semi	ADJ
iajs-2958	12	19	-	-	ADJ
iajs-2958	12	20	open	open	ADJ
iajs-2958	12	21	sets	set	NOUN
iajs-2958	12	22	,	,	PUNCT
iajs-2958	12	23	since	since	SCONJ
iajs-2958	12	24	every	every	DET
iajs-2958	12	25	nano	nano	ADJ
iajs-2958	12	26	𝑆𝐶-open	𝑆𝐶-open	NOUN
iajs-2958	12	27	(	(	PUNCT
iajs-2958	12	28	briefly	briefly	NOUN
iajs-2958	12	29	𝑛𝑆𝐶-oprn	𝑛𝑆𝐶-oprn	NOUN
iajs-2958	12	30	)	)	PUNCT
iajs-2958	12	31	sets	set	NOUN
iajs-2958	12	32	is	be	AUX
iajs-2958	12	33	nano	nano	ADJ
iajs-2958	12	34	semi	semi	ADJ
iajs-2958	12	35	-	-	ADJ
iajs-2958	12	36	open	open	ADJ
iajs-2958	12	37	sets	set	NOUN
iajs-2958	12	38	and	and	CCONJ
iajs-2958	12	39	the	the	DET
iajs-2958	12	40	relationship	relationship	NOUN
iajs-2958	12	41	with	with	ADP
iajs-2958	12	42	some	some	DET
iajs-2958	12	43	class	class	NOUN
iajs-2958	12	44	of	of	ADP
iajs-2958	12	45	nano	nano	NOUN
iajs-2958	12	46	near	near	ADP
iajs-2958	12	47	open	open	ADJ
iajs-2958	12	48	sets	set	NOUN
iajs-2958	12	49	.	.	PUNCT
iajs-2958	13	1	all	all	DET
iajs-2958	13	2	forms	form	NOUN
iajs-2958	13	3	of	of	ADP
iajs-2958	13	4	family	family	NOUN
iajs-2958	13	5	of	of	ADP
iajs-2958	13	6	nano	nano	NOUN
iajs-2958	13	7	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	13	8	sets	set	VERB
iajs-2958	13	9	under	under	ADP
iajs-2958	13	10	various	various	ADJ
iajs-2958	13	11	cases	case	NOUN
iajs-2958	13	12	of	of	ADP
iajs-2958	13	13	approximations	approximation	NOUN
iajs-2958	13	14	idea	idea	NOUN
iajs-2958	13	15	also	also	ADV
iajs-2958	13	16	derived	derive	VERB
iajs-2958	13	17	.	.	PUNCT
iajs-2958	14	1	also	also	ADV
iajs-2958	14	2	,	,	PUNCT
iajs-2958	14	3	operators	operator	NOUN
iajs-2958	14	4	such	such	ADJ
iajs-2958	14	5	as	as	ADP
iajs-2958	14	6	nano	nano	NOUN
iajs-2958	14	7	𝑆𝐶-interior	𝑆𝐶-interior	NOUN
iajs-2958	14	8	and	and	CCONJ
iajs-2958	14	9	nano	nano	PROPN
iajs-2958	14	10	𝑆𝐶-closure	𝑆𝐶-closure	PROPN
iajs-2958	14	11	are	be	AUX
iajs-2958	14	12	the	the	DET
iajs-2958	14	13	part	part	NOUN
iajs-2958	14	14	of	of	ADP
iajs-2958	14	15	this	this	DET
iajs-2958	14	16	paper	paper	NOUN
iajs-2958	14	17	.	.	PUNCT
iajs-2958	15	1	doi.org/10.30526/36.2.2958	doi.org/10.30526/36.2.2958	PROPN
iajs-2958	15	2	article	article	NOUN
iajs-2958	15	3	history	history	NOUN
iajs-2958	15	4	:	:	PUNCT
iajs-2958	15	5	received	receive	VERB
iajs-2958	15	6	14	14	NUM
iajs-2958	15	7	august	august	PROPN
iajs-2958	15	8	2022	2022	NUM
iajs-2958	15	9	,	,	PUNCT
iajs-2958	15	10	accepted	accept	VERB
iajs-2958	15	11	12	12	NUM
iajs-2958	15	12	september	september	PROPN
iajs-2958	15	13	2022	2022	NUM
iajs-2958	15	14	,	,	PUNCT
iajs-2958	15	15	published	publish	VERB
iajs-2958	15	16	in	in	ADP
iajs-2958	15	17	april	april	PROPN
iajs-2958	15	18	2023	2023	NUM
iajs-2958	15	19	.	.	PUNCT
iajs-2958	16	1	ibn	ibn	PROPN
iajs-2958	16	2	al	al	PROPN
iajs-2958	16	3	-	-	PUNCT
iajs-2958	16	4	haitham	haitham	PROPN
iajs-2958	16	5	journal	journal	PROPN
iajs-2958	16	6	for	for	ADP
iajs-2958	16	7	pure	pure	ADJ
iajs-2958	16	8	and	and	CCONJ
iajs-2958	16	9	applied	applied	ADJ
iajs-2958	16	10	sciences	sciences	PROPN
iajs-2958	16	11	journal	journal	PROPN
iajs-2958	16	12	homepage	homepage	NOUN
iajs-2958	16	13	:	:	PUNCT
iajs-2958	16	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	VERB
iajs-2958	16	15	nano	nano	NOUN
iajs-2958	16	16	𝑺𝑪-open	𝑺𝑪-open	NOUN
iajs-2958	16	17	sets	set	NOUN
iajs-2958	16	18	in	in	ADP
iajs-2958	16	19	nano	nano	NOUN
iajs-2958	16	20	topological	topological	ADJ
iajs-2958	16	21	spaces	space	NOUN
iajs-2958	16	22	nehmat	nehmat	PROPN
iajs-2958	16	23	k.	k.	PROPN
iajs-2958	16	24	ahmed	ahmed	PROPN
iajs-2958	16	25	department	department	PROPN
iajs-2958	16	26	of	of	ADP
iajs-2958	16	27	mathematics	mathematics	PROPN
iajs-2958	16	28	,	,	PUNCT
iajs-2958	16	29	college	college	NOUN
iajs-2958	16	30	of	of	ADP
iajs-2958	16	31	education	education	NOUN
iajs-2958	16	32	,	,	PUNCT
iajs-2958	16	33	salahaddin	salahaddin	VERB
iajs-2958	16	34	university	university	NOUN
iajs-2958	16	35	-	-	PUNCT
iajs-2958	16	36	erbil	erbil	PROPN
iajs-2958	16	37	,	,	PUNCT
iajs-2958	16	38	44001	44001	NUM
iajs-2958	16	39	,	,	PUNCT
iajs-2958	16	40	erbil	erbil	PROPN
iajs-2958	16	41	.	.	PUNCT
iajs-2958	17	1	nehmat.ahmed@su.edu.krd	nehmat.ahmed@su.edu.krd	NOUN
iajs-2958	17	2	osama	osama	NOUN
iajs-2958	17	3	t.	t.	PROPN
iajs-2958	17	4	pirbal	pirbal	PROPN
iajs-2958	17	5	department	department	PROPN
iajs-2958	17	6	of	of	ADP
iajs-2958	17	7	mathematics	mathematics	PROPN
iajs-2958	17	8	,	,	PUNCT
iajs-2958	17	9	college	college	NOUN
iajs-2958	17	10	of	of	ADP
iajs-2958	17	11	education	education	NOUN
iajs-2958	17	12	,	,	PUNCT
iajs-2958	17	13	salahaddin	salahaddin	VERB
iajs-2958	17	14	university	university	NOUN
iajs-2958	17	15	-	-	PUNCT
iajs-2958	17	16	erbil	erbil	PROPN
iajs-2958	17	17	,	,	PUNCT
iajs-2958	17	18	44001	44001	NUM
iajs-2958	17	19	,	,	PUNCT
iajs-2958	17	20	erbil	erbil	PROPN
iajs-2958	17	21	.	.	PUNCT
iajs-2958	18	1	osama.pirbal@su.edu.krd	osama.pirbal@su.edu.krd	PROPN
iajs-2958	18	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2958	18	3	mailto:nehmat.ahmed@su.edu.krd	mailto:nehmat.ahmed@su.edu.krd	NOUN
iajs-2958	18	4	mailto:osama.pirbal@su.edu.krd	mailto:osama.pirbal@su.edu.krd	PROPN
iajs-2958	18	5	ihjpas	ihjpa	VERB
iajs-2958	18	6	.	.	PUNCT
iajs-2958	19	1	36(2)2023	36(2)2023	NUM
iajs-2958	19	2	307	307	NUM
iajs-2958	19	3	2	2	NUM
iajs-2958	19	4	.	.	PUNCT
iajs-2958	19	5	preliminaries	preliminary	NOUN
iajs-2958	19	6	definition	definition	NOUN
iajs-2958	19	7	2.1	2.1	NUM
iajs-2958	19	8	.	.	PUNCT
iajs-2958	20	1	[	[	X
iajs-2958	20	2	8	8	NUM
iajs-2958	20	3	]	]	PUNCT
iajs-2958	20	4	let	let	VERB
iajs-2958	20	5	𝒲	𝒲	NOUN
iajs-2958	20	6	≠	≠	PROPN
iajs-2958	20	7	𝜙	𝜙	PRON
iajs-2958	20	8	denote	denote	VERB
iajs-2958	20	9	the	the	DET
iajs-2958	20	10	finite	finite	ADJ
iajs-2958	20	11	universe	universe	NOUN
iajs-2958	20	12	and	and	CCONJ
iajs-2958	20	13	the	the	DET
iajs-2958	20	14	equivalence	equivalence	NOUN
iajs-2958	20	15	relation	relation	NOUN
iajs-2958	20	16	𝑅	𝑅	PROPN
iajs-2958	20	17	on	on	ADP
iajs-2958	20	18	the	the	DET
iajs-2958	20	19	universe	universe	ADJ
iajs-2958	20	20	𝑊	𝑊	NOUN
iajs-2958	20	21	called	call	VERB
iajs-2958	20	22	the	the	DET
iajs-2958	20	23	indiscernibility	indiscernibility	NOUN
iajs-2958	20	24	relation	relation	NOUN
iajs-2958	20	25	.	.	PUNCT
iajs-2958	21	1	the	the	DET
iajs-2958	21	2	pair	pair	NOUN
iajs-2958	21	3	(	(	PUNCT
iajs-2958	21	4	𝒲	𝒲	PROPN
iajs-2958	21	5	,	,	PUNCT
iajs-2958	21	6	𝑅	𝑅	PROPN
iajs-2958	21	7	)	)	PUNCT
iajs-2958	21	8	is	be	AUX
iajs-2958	21	9	called	call	VERB
iajs-2958	21	10	the	the	DET
iajs-2958	21	11	approximation	approximation	NOUN
iajs-2958	21	12	space	space	NOUN
iajs-2958	21	13	.	.	PUNCT
iajs-2958	22	1	let	let	VERB
iajs-2958	22	2	𝑋	𝑋	PROPN
iajs-2958	22	3	⊆	⊆	NUM
iajs-2958	22	4	𝒲	𝒲	PROPN
iajs-2958	22	5	:	:	PUNCT
iajs-2958	22	6	i.	i.	NOUN
iajs-2958	22	7	the	the	DET
iajs-2958	22	8	lower	low	ADJ
iajs-2958	22	9	approximation	approximation	NOUN
iajs-2958	22	10	defined	define	VERB
iajs-2958	22	11	by	by	ADP
iajs-2958	22	12	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	22	13	)	)	PUNCT
iajs-2958	23	1	=	=	SYM
iajs-2958	23	2	⋃	⋃	NOUN
iajs-2958	23	3	{	{	PUNCT
iajs-2958	23	4	𝑅(𝑥	𝑅(𝑥	NOUN
iajs-2958	23	5	)	)	PUNCT
iajs-2958	23	6	;	;	PUNCT
iajs-2958	23	7	𝑅(𝑥	𝑅(𝑥	NUM
iajs-2958	23	8	)	)	PUNCT
iajs-2958	23	9	⊆	⊆	NUM
iajs-2958	23	10	𝑋}𝑥∈𝑈	𝑋}𝑥∈𝑈	NOUN
iajs-2958	23	11	,	,	PUNCT
iajs-2958	23	12	where	where	SCONJ
iajs-2958	23	13	𝑅(𝑥	𝑅(𝑥	VERB
iajs-2958	23	14	)	)	PUNCT
iajs-2958	23	15	stands	stand	VERB
iajs-2958	23	16	the	the	DET
iajs-2958	23	17	equivalence	equivalence	NOUN
iajs-2958	23	18	class	class	NOUN
iajs-2958	23	19	by	by	ADP
iajs-2958	23	20	𝑥.	𝑥.	PROPN
iajs-2958	23	21	ii	ii	PROPN
iajs-2958	23	22	.	.	PUNCT
iajs-2958	24	1	the	the	DET
iajs-2958	24	2	upper	upper	ADJ
iajs-2958	24	3	approximation	approximation	NOUN
iajs-2958	24	4	defined	define	VERB
iajs-2958	24	5	by	by	ADP
iajs-2958	24	6	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	24	7	)	)	PUNCT
iajs-2958	24	8	=	=	SYM
iajs-2958	24	9	⋃	⋃	NOUN
iajs-2958	24	10	{	{	PUNCT
iajs-2958	24	11	𝑅(𝑥	𝑅(𝑥	NOUN
iajs-2958	24	12	)	)	PUNCT
iajs-2958	24	13	;	;	PUNCT
iajs-2958	24	14	𝑅(𝑥)⋂𝑋	𝑅(𝑥)⋂𝑋	NOUN
iajs-2958	24	15	≠	≠	PROPN
iajs-2958	24	16	𝜙}𝑥∈𝑈	𝜙}𝑥∈𝑈	PROPN
iajs-2958	24	17	.	.	PUNCT
iajs-2958	25	1	iii	iii	X
iajs-2958	25	2	.	.	PUNCT
iajs-2958	26	1	the	the	DET
iajs-2958	26	2	boundary	boundary	ADJ
iajs-2958	26	3	region	region	NOUN
iajs-2958	26	4	defined	define	VERB
iajs-2958	26	5	by	by	ADP
iajs-2958	26	6	𝐵𝑅(𝑋	𝐵𝑅(𝑋	NOUN
iajs-2958	26	7	)	)	PUNCT
iajs-2958	26	8	=	=	SYM
iajs-2958	26	9	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	26	10	)	)	PUNCT
iajs-2958	26	11	−	−	PROPN
iajs-2958	26	12	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	26	13	)	)	PUNCT
iajs-2958	26	14	.	.	PUNCT
iajs-2958	27	1	definition	definition	NOUN
iajs-2958	27	2	2.2	2.2	NUM
iajs-2958	27	3	.	.	PUNCT
iajs-2958	28	1	[	[	X
iajs-2958	28	2	1	1	X
iajs-2958	28	3	]	]	X
iajs-2958	28	4	let	let	VERB
iajs-2958	28	5	𝒲	𝒲	NOUN
iajs-2958	28	6	denote	denote	VERB
iajs-2958	28	7	the	the	DET
iajs-2958	28	8	universe	universe	NOUN
iajs-2958	28	9	and	and	CCONJ
iajs-2958	28	10	r	r	NOUN
iajs-2958	28	11	be	be	VERB
iajs-2958	28	12	an	an	DET
iajs-2958	28	13	equivalence	equivalence	NOUN
iajs-2958	28	14	relation	relation	NOUN
iajs-2958	28	15	on	on	ADP
iajs-2958	28	16	𝑊	𝑊	PROPN
iajs-2958	28	17	and	and	CCONJ
iajs-2958	28	18	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	28	19	)	)	PUNCT
iajs-2958	28	20	=	=	PRON
iajs-2958	28	21	{	{	PUNCT
iajs-2958	28	22	𝜙	𝜙	NOUN
iajs-2958	28	23	,	,	PUNCT
iajs-2958	28	24	𝒲	𝒲	PROPN
iajs-2958	28	25	,	,	PUNCT
iajs-2958	28	26	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	28	27	)	)	PUNCT
iajs-2958	28	28	,	,	PUNCT
iajs-2958	28	29	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	28	30	)	)	PUNCT
iajs-2958	28	31	,	,	PUNCT
iajs-2958	28	32	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	28	33	)	)	PUNCT
iajs-2958	28	34	}	}	PUNCT
iajs-2958	28	35	where	where	SCONJ
iajs-2958	28	36	𝑋	𝑋	PROPN
iajs-2958	28	37	⊆	⊆	NUM
iajs-2958	28	38	𝒲.	𝒲.	PROPN
iajs-2958	28	39	then	then	ADV
iajs-2958	28	40	the	the	DET
iajs-2958	28	41	followings	following	NOUN
iajs-2958	28	42	axioms	axiom	VERB
iajs-2958	28	43	hold	hold	NOUN
iajs-2958	28	44	for	for	ADP
iajs-2958	28	45	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	28	46	):	):	PUNCT
iajs-2958	28	47	i.	i.	PROPN
iajs-2958	28	48	𝑊	𝑊	PROPN
iajs-2958	28	49	and	and	CCONJ
iajs-2958	28	50	𝜙	𝜙	PRON
iajs-2958	28	51	∈	∈	PROPN
iajs-2958	28	52	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	28	53	)	)	PUNCT
iajs-2958	28	54	ii	ii	PROPN
iajs-2958	28	55	.	.	PUNCT
iajs-2958	29	1	𝐴	𝐴	PROPN
iajs-2958	29	2	,	,	PUNCT
iajs-2958	29	3	𝐵	𝐵	PROPN
iajs-2958	29	4	∈	∈	PROPN
iajs-2958	29	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	29	6	)	)	PUNCT
iajs-2958	29	7	,	,	PUNCT
iajs-2958	29	8	then	then	ADV
iajs-2958	29	9	𝐴	𝐴	PROPN
iajs-2958	29	10	∪	∪	VERB
iajs-2958	29	11	𝐵𝜏𝑅(𝑋	𝐵𝜏𝑅(𝑋	PROPN
iajs-2958	29	12	)	)	PUNCT
iajs-2958	29	13	iii	iii	NOUN
iajs-2958	29	14	.	.	PUNCT
iajs-2958	30	1	the	the	DET
iajs-2958	30	2	intersection	intersection	NOUN
iajs-2958	30	3	of	of	ADP
iajs-2958	30	4	any	any	DET
iajs-2958	30	5	finite	finite	ADJ
iajs-2958	30	6	subcollection	subcollection	NOUN
iajs-2958	30	7	of	of	ADP
iajs-2958	30	8	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	30	9	)	)	PUNCT
iajs-2958	30	10	is	be	AUX
iajs-2958	30	11	in	in	ADP
iajs-2958	30	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	30	13	)	)	PUNCT
iajs-2958	30	14	.	.	PUNCT
iajs-2958	31	1	then	then	ADV
iajs-2958	31	2	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	31	3	)	)	PUNCT
iajs-2958	31	4	forms	form	VERB
iajs-2958	31	5	a	a	DET
iajs-2958	31	6	topology	topology	NOUN
iajs-2958	31	7	on	on	ADP
iajs-2958	31	8	𝒲	𝒲	NOUN
iajs-2958	31	9	and	and	CCONJ
iajs-2958	31	10	called	call	VERB
iajs-2958	31	11	nano	nano	NOUN
iajs-2958	31	12	topology	topology	NOUN
iajs-2958	31	13	on	on	ADP
iajs-2958	31	14	𝒲	𝒲	NOUN
iajs-2958	31	15	with	with	ADP
iajs-2958	31	16	respect	respect	NOUN
iajs-2958	31	17	to	to	ADP
iajs-2958	31	18	𝑋.	𝑋.	PROPN
iajs-2958	31	19	also	also	ADV
iajs-2958	31	20	(	(	PUNCT
iajs-2958	31	21	𝒲	𝒲	PROPN
iajs-2958	31	22	,	,	PUNCT
iajs-2958	31	23	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	31	24	)	)	PUNCT
iajs-2958	31	25	)	)	PUNCT
iajs-2958	31	26	is	be	AUX
iajs-2958	31	27	called	call	VERB
iajs-2958	31	28	the	the	DET
iajs-2958	31	29	𝑁𝑇𝑆	𝑁𝑇𝑆	NOUN
iajs-2958	31	30	and	and	CCONJ
iajs-2958	31	31	the	the	DET
iajs-2958	31	32	members	member	NOUN
iajs-2958	31	33	of	of	ADP
iajs-2958	31	34	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	31	35	)	)	PUNCT
iajs-2958	31	36	are	be	AUX
iajs-2958	31	37	called	call	VERB
iajs-2958	31	38	nano	nano	NOUN
iajs-2958	31	39	open	open	ADJ
iajs-2958	31	40	sets	set	NOUN
iajs-2958	31	41	.	.	PUNCT
iajs-2958	32	1	definition	definition	NOUN
iajs-2958	32	2	2.3	2.3	NUM
iajs-2958	32	3	.	.	PUNCT
iajs-2958	33	1	let	let	VERB
iajs-2958	33	2	(	(	PUNCT
iajs-2958	33	3	𝒲	𝒲	NOUN
iajs-2958	33	4	,	,	PUNCT
iajs-2958	33	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	33	6	)	)	PUNCT
iajs-2958	33	7	)	)	PUNCT
iajs-2958	33	8	be	be	AUX
iajs-2958	33	9	a	a	DET
iajs-2958	33	10	𝑁𝑇𝑆	𝑁𝑇𝑆	NOUN
iajs-2958	33	11	and	and	CCONJ
iajs-2958	33	12	𝐾	𝐾	PROPN
iajs-2958	33	13	⊆	⊆	PROPN
iajs-2958	33	14	𝒲.	𝒲.	PROPN
iajs-2958	33	15	the	the	DET
iajs-2958	33	16	set	set	NOUN
iajs-2958	33	17	𝐾	𝐾	PROPN
iajs-2958	33	18	is	be	AUX
iajs-2958	33	19	called	call	VERB
iajs-2958	33	20	nano	nano	NOUN
iajs-2958	33	21	:	:	PUNCT
iajs-2958	33	22	i.	i.	PROPN
iajs-2958	33	23	regular	regular	PROPN
iajs-2958	33	24	-	-	PUNCT
iajs-2958	33	25	open	open	NOUN
iajs-2958	33	26	[	[	X
iajs-2958	33	27	1	1	NUM
iajs-2958	33	28	]	]	PUNCT
iajs-2958	33	29	,	,	PUNCT
iajs-2958	33	30	if	if	SCONJ
iajs-2958	33	31	𝐾	𝐾	PROPN
iajs-2958	33	32	=	=	PUNCT
iajs-2958	33	33	𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝐾	𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝐾	PROPN
iajs-2958	33	34	)	)	PUNCT
iajs-2958	33	35	)	)	PUNCT
iajs-2958	33	36	.	.	PUNCT
iajs-2958	34	1	ii	ii	PROPN
iajs-2958	34	2	.	.	PUNCT
iajs-2958	35	1	𝛼-open	𝛼-open	PROPN
iajs-2958	36	1	[	[	X
iajs-2958	36	2	1	1	NUM
iajs-2958	36	3	]	]	PUNCT
iajs-2958	36	4	,	,	PUNCT
iajs-2958	36	5	if	if	SCONJ
iajs-2958	36	6	𝐾	𝐾	PROPN
iajs-2958	36	7	⊆	⊆	NUM
iajs-2958	36	8	𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝑛𝑖𝑛𝑡((𝐾	𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝑛𝑖𝑛𝑡((𝐾	NOUN
iajs-2958	36	9	)	)	PUNCT
iajs-2958	36	10	)	)	PUNCT
iajs-2958	36	11	)	)	PUNCT
iajs-2958	36	12	.	.	PUNCT
iajs-2958	37	1	iii	iii	X
iajs-2958	37	2	.	.	PUNCT
iajs-2958	38	1	semi	semi	ADJ
iajs-2958	38	2	-	-	ADJ
iajs-2958	38	3	open	open	ADJ
iajs-2958	38	4	[	[	X
iajs-2958	38	5	1	1	NUM
iajs-2958	38	6	]	]	PUNCT
iajs-2958	38	7	,	,	PUNCT
iajs-2958	38	8	if	if	SCONJ
iajs-2958	38	9	𝐾	𝐾	PROPN
iajs-2958	38	10	⊆	⊆	NUM
iajs-2958	38	11	𝑛𝑐𝑙(𝑛𝑖𝑛𝑡(𝐾	𝑛𝑐𝑙(𝑛𝑖𝑛𝑡(𝐾	PROPN
iajs-2958	38	12	)	)	PUNCT
iajs-2958	38	13	)	)	PUNCT
iajs-2958	38	14	.	.	PUNCT
iajs-2958	39	1	iv	iv	X
iajs-2958	39	2	.	.	PUNCT
iajs-2958	39	3	𝛽-open	𝛽-open	PROPN
iajs-2958	39	4	(	(	PUNCT
iajs-2958	39	5	nano	nano	NOUN
iajs-2958	39	6	semi	semi	ADJ
iajs-2958	39	7	pre	pre	ADJ
iajs-2958	39	8	-	-	ADJ
iajs-2958	39	9	open	open	ADJ
iajs-2958	39	10	)	)	PUNCT
iajs-2958	40	1	[	[	X
iajs-2958	40	2	3	3	NUM
iajs-2958	40	3	]	]	PUNCT
iajs-2958	40	4	,	,	PUNCT
iajs-2958	40	5	if	if	SCONJ
iajs-2958	40	6	𝐾	𝐾	PROPN
iajs-2958	40	7	⊆	⊆	NUM
iajs-2958	40	8	𝑛𝑐𝑙(𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝐾	𝑛𝑐𝑙(𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝐾	NOUN
iajs-2958	40	9	)	)	PUNCT
iajs-2958	40	10	)	)	PUNCT
iajs-2958	40	11	)	)	PUNCT
iajs-2958	40	12	.	.	PUNCT
iajs-2958	41	1	v.	v.	ADP
iajs-2958	41	2	𝜃-open	𝜃-open	NOUN
iajs-2958	42	1	[	[	X
iajs-2958	42	2	1	1	NUM
iajs-2958	42	3	]	]	PUNCT
iajs-2958	42	4	,	,	PUNCT
iajs-2958	42	5	if	if	SCONJ
iajs-2958	42	6	for	for	ADP
iajs-2958	42	7	each	each	DET
iajs-2958	42	8	𝑥	𝑥	PRON
iajs-2958	42	9	∈	∈	PROPN
iajs-2958	42	10	𝐾	𝐾	PROPN
iajs-2958	42	11	,	,	PUNCT
iajs-2958	42	12	there	there	PRON
iajs-2958	42	13	exists	exist	VERB
iajs-2958	42	14	a	a	DET
iajs-2958	42	15	nano	nano	NOUN
iajs-2958	42	16	open	open	ADJ
iajs-2958	42	17	set	set	NOUN
iajs-2958	42	18	𝐺	𝐺	PROPN
iajs-2958	42	19	such	such	ADJ
iajs-2958	42	20	that	that	SCONJ
iajs-2958	42	21	𝑥	𝑥	PROPN
iajs-2958	42	22	∈	∈	PROPN
iajs-2958	42	23	𝐺	𝐺	PROPN
iajs-2958	42	24	⊆	⊆	NUM
iajs-2958	42	25	𝑛𝑐𝑙(𝐾	𝑛𝑐𝑙(𝐾	NUM
iajs-2958	42	26	)	)	PUNCT
iajs-2958	42	27	⊆	⊆	NUM
iajs-2958	42	28	𝐾.	𝐾.	PROPN
iajs-2958	42	29	vi	vi	PROPN
iajs-2958	42	30	.	.	PUNCT
iajs-2958	43	1	𝑆𝛽-open	𝑆𝛽-open	X
iajs-2958	44	1	[	[	X
iajs-2958	44	2	4	4	NUM
iajs-2958	44	3	]	]	PUNCT
iajs-2958	44	4	,	,	PUNCT
iajs-2958	44	5	if	if	SCONJ
iajs-2958	44	6	𝐾	𝐾	PROPN
iajs-2958	44	7	is	be	AUX
iajs-2958	44	8	nano	nano	ADJ
iajs-2958	44	9	semi	semi	ADJ
iajs-2958	44	10	-	-	ADJ
iajs-2958	44	11	open	open	ADJ
iajs-2958	44	12	and	and	CCONJ
iajs-2958	44	13	the	the	DET
iajs-2958	44	14	union	union	NOUN
iajs-2958	44	15	of	of	ADP
iajs-2958	44	16	nano	nano	NOUN
iajs-2958	44	17	𝛽-closed	𝛽-close	VERB
iajs-2958	44	18	sets	set	NOUN
iajs-2958	44	19	.	.	PUNCT
iajs-2958	45	1	the	the	DET
iajs-2958	45	2	set	set	NOUN
iajs-2958	45	3	of	of	ADP
iajs-2958	45	4	all	all	DET
iajs-2958	45	5	nano	nano	VERB
iajs-2958	45	6	regular	regular	ADJ
iajs-2958	45	7	-	-	PUNCT
iajs-2958	45	8	open	open	ADJ
iajs-2958	45	9	(	(	PUNCT
iajs-2958	45	10	resp	resp	NOUN
iajs-2958	45	11	.	.	PUNCT
iajs-2958	46	1	nano	nano	PROPN
iajs-2958	46	2	𝛼-open	𝛼-open	PROPN
iajs-2958	46	3	,	,	PUNCT
iajs-2958	46	4	nano	nano	NOUN
iajs-2958	46	5	semi	semi	ADJ
iajs-2958	46	6	-	-	ADJ
iajs-2958	46	7	open	open	ADJ
iajs-2958	46	8	,	,	PUNCT
iajs-2958	46	9	nano	nano	NOUN
iajs-2958	46	10	𝛽-open	𝛽-open	NOUN
iajs-2958	46	11	,	,	PUNCT
iajs-2958	46	12	𝜃-open	𝜃-open	NOUN
iajs-2958	46	13	and	and	CCONJ
iajs-2958	46	14	nano	nano	NOUN
iajs-2958	46	15	𝑆𝛽-open	𝑆𝛽-open	NOUN
iajs-2958	46	16	)	)	PUNCT
iajs-2958	46	17	sets	set	NOUN
iajs-2958	46	18	denoted	denote	VERB
iajs-2958	46	19	by	by	ADP
iajs-2958	46	20	𝑛𝑅𝑂(𝒲	𝑛𝑅𝑂(𝒲	PROPN
iajs-2958	46	21	,	,	PUNCT
iajs-2958	46	22	𝑋	𝑋	PROPN
iajs-2958	46	23	)	)	PUNCT
iajs-2958	46	24	(	(	PUNCT
iajs-2958	46	25	resp	resp	NOUN
iajs-2958	46	26	.	.	PUNCT
iajs-2958	47	1	𝑛𝛼𝑂(𝒲	𝑛𝛼𝑂(𝒲	ADJ
iajs-2958	47	2	,	,	PUNCT
iajs-2958	47	3	𝑋	𝑋	NOUN
iajs-2958	47	4	)	)	PUNCT
iajs-2958	47	5	,	,	PUNCT
iajs-2958	47	6	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	47	7	,	,	PUNCT
iajs-2958	47	8	𝑋	𝑋	PROPN
iajs-2958	47	9	)	)	PUNCT
iajs-2958	47	10	,	,	PUNCT
iajs-2958	47	11	𝑛𝛽𝑂(𝒲	𝑛𝛽𝑂(𝒲	PROPN
iajs-2958	47	12	,	,	PUNCT
iajs-2958	47	13	𝑋	𝑋	PROPN
iajs-2958	47	14	)	)	PUNCT
iajs-2958	47	15	,	,	PUNCT
iajs-2958	47	16	𝑛𝜃𝑂(𝒲	𝑛𝜃𝑂(𝒲	ADJ
iajs-2958	47	17	,	,	PUNCT
iajs-2958	47	18	𝑋	𝑋	NOUN
iajs-2958	47	19	)	)	PUNCT
iajs-2958	47	20	and	and	CCONJ
iajs-2958	47	21	𝑛𝑆𝛽𝑂(𝒲	𝑛𝑆𝛽𝑂(𝒲	PROPN
iajs-2958	47	22	,	,	PUNCT
iajs-2958	47	23	𝑋	𝑋	PROPN
iajs-2958	47	24	)	)	PUNCT
iajs-2958	47	25	)	)	PUNCT
iajs-2958	47	26	.	.	PUNCT
iajs-2958	48	1	theorem	theorem	VERB
iajs-2958	48	2	2.4	2.4	NUM
iajs-2958	48	3	.	.	PUNCT
iajs-2958	49	1	[	[	X
iajs-2958	49	2	1	1	X
iajs-2958	49	3	]	]	PUNCT
iajs-2958	49	4	if	if	SCONJ
iajs-2958	49	5	𝐴	𝐴	PROPN
iajs-2958	49	6	,	,	PUNCT
iajs-2958	49	7	𝐵	𝐵	PROPN
iajs-2958	49	8	∈	∈	PROPN
iajs-2958	49	9	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	49	10	,	,	PUNCT
iajs-2958	49	11	𝑋	𝑋	PROPN
iajs-2958	49	12	)	)	PUNCT
iajs-2958	49	13	,	,	PUNCT
iajs-2958	49	14	then	then	ADV
iajs-2958	49	15	𝐴	𝐴	PROPN
iajs-2958	49	16	∪	∪	AUX
iajs-2958	49	17	𝐵	𝐵	PROPN
iajs-2958	49	18	∈	∈	PROPN
iajs-2958	49	19	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	49	20	,	,	PUNCT
iajs-2958	49	21	𝑋	𝑋	PROPN
iajs-2958	49	22	)	)	PUNCT
iajs-2958	49	23	.	.	PUNCT
iajs-2958	50	1	theorem	theorem	VERB
iajs-2958	50	2	2.5	2.5	NUM
iajs-2958	50	3	.	.	PUNCT
iajs-2958	51	1	[	[	X
iajs-2958	51	2	1	1	X
iajs-2958	51	3	]	]	X
iajs-2958	51	4	let	let	VERB
iajs-2958	51	5	(	(	PUNCT
iajs-2958	51	6	𝒲	𝒲	NOUN
iajs-2958	51	7	,	,	PUNCT
iajs-2958	51	8	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	51	9	)	)	PUNCT
iajs-2958	51	10	)	)	PUNCT
iajs-2958	51	11	be	be	AUX
iajs-2958	51	12	a	a	DET
iajs-2958	51	13	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	51	14	,	,	PUNCT
iajs-2958	51	15	then	then	ADV
iajs-2958	51	16	:	:	PUNCT
iajs-2958	51	17	i.	i.	NOUN
iajs-2958	51	18	if	if	SCONJ
iajs-2958	51	19	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	51	20	)	)	PUNCT
iajs-2958	51	21	=	=	SYM
iajs-2958	51	22	𝒲	𝒲	PROPN
iajs-2958	51	23	and	and	CCONJ
iajs-2958	51	24	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	51	25	)	)	PUNCT
iajs-2958	52	1	=	=	SYM
iajs-2958	52	2	𝜙	𝜙	NOUN
iajs-2958	52	3	⟹	⟹	NUM
iajs-2958	52	4	τ𝑅	τ𝑅	NOUN
iajs-2958	52	5	𝜃(𝑋	𝜃(𝑋	ADV
iajs-2958	52	6	)	)	PUNCT
iajs-2958	52	7	=	=	PRON
iajs-2958	52	8	{	{	PUNCT
iajs-2958	52	9	𝜙	𝜙	NOUN
iajs-2958	52	10	,	,	PUNCT
iajs-2958	52	11	𝒲	𝒲	PROPN
iajs-2958	52	12	}	}	PUNCT
iajs-2958	52	13	.	.	PUNCT
iajs-2958	53	1	ii	ii	PROPN
iajs-2958	53	2	.	.	PUNCT
iajs-2958	54	1	if	if	SCONJ
iajs-2958	54	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	54	3	)	)	PUNCT
iajs-2958	54	4	=	=	SYM
iajs-2958	54	5	𝒲	𝒲	PROPN
iajs-2958	54	6	and	and	CCONJ
iajs-2958	54	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	54	8	)	)	PUNCT
iajs-2958	55	1	≠	≠	PROPN
iajs-2958	55	2	𝜙	𝜙	NOUN
iajs-2958	55	3	⟹	⟹	PUNCT
iajs-2958	55	4	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	55	5	)	)	PUNCT
iajs-2958	56	1	=	=	SYM
iajs-2958	56	2	τ𝑅	τ𝑅	NOUN
iajs-2958	56	3	𝜃(𝑋	𝜃(𝑋	ADV
iajs-2958	56	4	)	)	PUNCT
iajs-2958	56	5	.	.	PUNCT
iajs-2958	57	1	iii	iii	X
iajs-2958	57	2	.	.	PUNCT
iajs-2958	58	1	if	if	SCONJ
iajs-2958	58	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	58	3	)	)	PUNCT
iajs-2958	58	4	=	=	SYM
iajs-2958	58	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	58	6	)	)	PUNCT
iajs-2958	58	7	≠	≠	PROPN
iajs-2958	58	8	𝒲	𝒲	PROPN
iajs-2958	58	9	⟹	⟹	PUNCT
iajs-2958	58	10	τ𝑅	τ𝑅	NOUN
iajs-2958	58	11	𝜃(𝑋	𝜃(𝑋	ADV
iajs-2958	58	12	)	)	PUNCT
iajs-2958	58	13	=	=	PRON
iajs-2958	58	14	{	{	PUNCT
iajs-2958	58	15	𝜙	𝜙	NOUN
iajs-2958	58	16	,	,	PUNCT
iajs-2958	58	17	𝒲	𝒲	PROPN
iajs-2958	58	18	}	}	PUNCT
iajs-2958	58	19	.	.	PUNCT
iajs-2958	59	1	iv	iv	X
iajs-2958	59	2	.	.	PUNCT
iajs-2958	60	1	if	if	SCONJ
iajs-2958	60	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	60	3	)	)	PUNCT
iajs-2958	60	4	≠	≠	PROPN
iajs-2958	60	5	𝒲	𝒲	PROPN
iajs-2958	60	6	,	,	PUNCT
iajs-2958	60	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	60	8	)	)	PUNCT
iajs-2958	60	9	=	=	SYM
iajs-2958	61	1	𝜙	𝜙	NOUN
iajs-2958	61	2	⟹	⟹	NUM
iajs-2958	61	3	τ𝑅	τ𝑅	NOUN
iajs-2958	61	4	𝜃(𝑋	𝜃(𝑋	ADV
iajs-2958	61	5	)	)	PUNCT
iajs-2958	61	6	=	=	PRON
iajs-2958	61	7	{	{	PUNCT
iajs-2958	61	8	𝜙	𝜙	NOUN
iajs-2958	61	9	,	,	PUNCT
iajs-2958	61	10	𝒲	𝒲	PROPN
iajs-2958	61	11	}	}	PUNCT
iajs-2958	61	12	.	.	PUNCT
iajs-2958	62	1	v.	v.	INTJ
iajs-2958	62	2	if	if	SCONJ
iajs-2958	62	3	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	62	4	)	)	PUNCT
iajs-2958	62	5	≠	≠	PROPN
iajs-2958	62	6	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	62	7	)	)	PUNCT
iajs-2958	62	8	where	where	SCONJ
iajs-2958	62	9	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	62	10	)	)	PUNCT
iajs-2958	62	11	≠	≠	PROPN
iajs-2958	62	12	𝒲	𝒲	PROPN
iajs-2958	62	13	and	and	CCONJ
iajs-2958	62	14	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	62	15	)	)	PUNCT
iajs-2958	62	16	≠	≠	PROPN
iajs-2958	62	17	𝜙	𝜙	PRON
iajs-2958	62	18	⟹	⟹	NUM
iajs-2958	62	19	τ𝑅	τ𝑅	NOUN
iajs-2958	62	20	𝜃(𝑋	𝜃(𝑋	ADV
iajs-2958	62	21	)	)	PUNCT
iajs-2958	62	22	=	=	PRON
iajs-2958	62	23	{	{	PUNCT
iajs-2958	62	24	𝜙	𝜙	NOUN
iajs-2958	62	25	,	,	PUNCT
iajs-2958	62	26	𝒲	𝒲	PROPN
iajs-2958	62	27	}	}	PUNCT
iajs-2958	62	28	.	.	PUNCT
iajs-2958	63	1	theorem	theorem	VERB
iajs-2958	63	2	2.6	2.6	NUM
iajs-2958	63	3	.	.	PUNCT
iajs-2958	64	1	[	[	X
iajs-2958	64	2	1	1	X
iajs-2958	64	3	]	]	X
iajs-2958	64	4	let	let	VERB
iajs-2958	64	5	(	(	PUNCT
iajs-2958	64	6	𝒲	𝒲	NOUN
iajs-2958	64	7	,	,	PUNCT
iajs-2958	64	8	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	64	9	)	)	PUNCT
iajs-2958	64	10	)	)	PUNCT
iajs-2958	64	11	be	be	AUX
iajs-2958	64	12	a	a	DET
iajs-2958	64	13	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	64	14	,	,	PUNCT
iajs-2958	64	15	then	then	ADV
iajs-2958	64	16	:	:	PUNCT
iajs-2958	64	17	i.	i.	NOUN
iajs-2958	64	18	if	if	SCONJ
iajs-2958	64	19	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	64	20	)	)	PUNCT
iajs-2958	64	21	=	=	SYM
iajs-2958	64	22	𝒲	𝒲	PROPN
iajs-2958	64	23	and	and	CCONJ
iajs-2958	64	24	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	64	25	)	)	PUNCT
iajs-2958	65	1	=	=	SYM
iajs-2958	65	2	𝜙	𝜙	X
iajs-2958	65	3	⟹	⟹	PUNCT
iajs-2958	65	4	𝑛𝑅𝑂(𝒲	𝑛𝑅𝑂(𝒲	PROPN
iajs-2958	65	5	,	,	PUNCT
iajs-2958	65	6	𝑋	𝑋	NOUN
iajs-2958	65	7	)	)	PUNCT
iajs-2958	65	8	=	=	PUNCT
iajs-2958	65	9	{	{	PUNCT
iajs-2958	65	10	𝜙	𝜙	NOUN
iajs-2958	65	11	,	,	PUNCT
iajs-2958	65	12	𝒲	𝒲	PROPN
iajs-2958	65	13	}	}	PUNCT
iajs-2958	65	14	.	.	PUNCT
iajs-2958	66	1	ii	ii	PROPN
iajs-2958	66	2	.	.	PUNCT
iajs-2958	67	1	if	if	SCONJ
iajs-2958	67	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	67	3	)	)	PUNCT
iajs-2958	67	4	=	=	SYM
iajs-2958	67	5	𝒲	𝒲	PROPN
iajs-2958	67	6	and	and	CCONJ
iajs-2958	67	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	67	8	)	)	PUNCT
iajs-2958	68	1	≠	≠	PROPN
iajs-2958	68	2	𝜙	𝜙	NOUN
iajs-2958	68	3	⟹	⟹	PUNCT
iajs-2958	68	4	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	68	5	)	)	PUNCT
iajs-2958	68	6	=	=	PUNCT
iajs-2958	69	1	𝑛𝑅𝑂(𝒲	𝑛𝑅𝑂(𝒲	PROPN
iajs-2958	69	2	,	,	PUNCT
iajs-2958	69	3	𝑋	𝑋	PROPN
iajs-2958	69	4	)	)	PUNCT
iajs-2958	69	5	.	.	PUNCT
iajs-2958	70	1	iii	iii	X
iajs-2958	70	2	.	.	PUNCT
iajs-2958	71	1	if	if	SCONJ
iajs-2958	71	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	71	3	)	)	PUNCT
iajs-2958	71	4	=	=	SYM
iajs-2958	71	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	71	6	)	)	PUNCT
iajs-2958	71	7	≠	≠	PROPN
iajs-2958	71	8	𝒲	𝒲	PROPN
iajs-2958	71	9	⟹	⟹	NUM
iajs-2958	71	10	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	71	11	)	)	PUNCT
iajs-2958	71	12	=	=	PUNCT
iajs-2958	72	1	𝑛𝑅𝑂(𝒲	𝑛𝑅𝑂(𝒲	PROPN
iajs-2958	72	2	,	,	PUNCT
iajs-2958	72	3	𝑋	𝑋	PROPN
iajs-2958	72	4	)	)	PUNCT
iajs-2958	72	5	.	.	PUNCT
iajs-2958	73	1	iv	iv	X
iajs-2958	73	2	.	.	PUNCT
iajs-2958	74	1	if	if	SCONJ
iajs-2958	74	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	74	3	)	)	PUNCT
iajs-2958	74	4	≠	≠	PROPN
iajs-2958	74	5	𝒲	𝒲	PROPN
iajs-2958	74	6	,	,	PUNCT
iajs-2958	74	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	74	8	)	)	PUNCT
iajs-2958	74	9	=	=	SYM
iajs-2958	74	10	𝜙	𝜙	X
iajs-2958	74	11	⟹	⟹	NUM
iajs-2958	74	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	VERB
iajs-2958	74	13	)	)	PUNCT
iajs-2958	74	14	=	=	PUNCT
iajs-2958	75	1	𝑛𝑅𝑂(𝒲	𝑛𝑅𝑂(𝒲	PROPN
iajs-2958	75	2	,	,	PUNCT
iajs-2958	75	3	𝑋	𝑋	PROPN
iajs-2958	75	4	)	)	PUNCT
iajs-2958	75	5	.	.	PUNCT
iajs-2958	76	1	ihjpas	ihjpas	PROPN
iajs-2958	76	2	.	.	PUNCT
iajs-2958	77	1	36(2)2023	36(2)2023	NUM
iajs-2958	77	2	308	308	NUM
iajs-2958	77	3	v.	v.	ADP
iajs-2958	77	4	if	if	SCONJ
iajs-2958	77	5	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	77	6	)	)	PUNCT
iajs-2958	77	7	≠	≠	PROPN
iajs-2958	77	8	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	77	9	)	)	PUNCT
iajs-2958	77	10	where	where	SCONJ
iajs-2958	77	11	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	77	12	)	)	PUNCT
iajs-2958	77	13	≠	≠	PROPN
iajs-2958	77	14	𝒲	𝒲	PROPN
iajs-2958	77	15	and	and	CCONJ
iajs-2958	77	16	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	77	17	)	)	PUNCT
iajs-2958	78	1	≠	≠	PROPN
iajs-2958	78	2	𝜙	𝜙	NOUN
iajs-2958	78	3	⟹	⟹	ADP
iajs-2958	78	4	𝑛𝑅𝑂(𝒲	𝑛𝑅𝑂(𝒲	PROPN
iajs-2958	78	5	,	,	PUNCT
iajs-2958	78	6	𝑋	𝑋	NOUN
iajs-2958	78	7	)	)	PUNCT
iajs-2958	78	8	=	=	PUNCT
iajs-2958	78	9	{	{	PUNCT
iajs-2958	78	10	𝜙	𝜙	NOUN
iajs-2958	78	11	,	,	PUNCT
iajs-2958	78	12	𝒲	𝒲	PROPN
iajs-2958	78	13	,	,	PUNCT
iajs-2958	78	14	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	78	15	)	)	PUNCT
iajs-2958	78	16	,	,	PUNCT
iajs-2958	78	17	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	78	18	)	)	PUNCT
iajs-2958	78	19	}	}	PUNCT
iajs-2958	78	20	.	.	PUNCT
iajs-2958	79	1	theorem	theorem	VERB
iajs-2958	79	2	2.7	2.7	NUM
iajs-2958	79	3	.	.	PUNCT
iajs-2958	80	1	[	[	X
iajs-2958	80	2	1	1	X
iajs-2958	80	3	]	]	X
iajs-2958	80	4	let	let	VERB
iajs-2958	80	5	(	(	PUNCT
iajs-2958	80	6	𝒲	𝒲	NOUN
iajs-2958	80	7	,	,	PUNCT
iajs-2958	80	8	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	80	9	)	)	PUNCT
iajs-2958	80	10	)	)	PUNCT
iajs-2958	80	11	be	be	AUX
iajs-2958	80	12	a	a	DET
iajs-2958	80	13	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	80	14	,	,	PUNCT
iajs-2958	80	15	then	then	ADV
iajs-2958	80	16	:	:	PUNCT
iajs-2958	80	17	i.	i.	NOUN
iajs-2958	80	18	if	if	SCONJ
iajs-2958	80	19	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	80	20	)	)	PUNCT
iajs-2958	80	21	=	=	SYM
iajs-2958	80	22	𝒲	𝒲	PROPN
iajs-2958	80	23	and	and	CCONJ
iajs-2958	80	24	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	80	25	)	)	PUNCT
iajs-2958	80	26	=	=	SYM
iajs-2958	80	27	𝜙	𝜙	X
iajs-2958	80	28	⟹	⟹	NUM
iajs-2958	80	29	𝑛𝛼𝑂(𝒲	𝑛𝛼𝑂(𝒲	ADJ
iajs-2958	80	30	,	,	PUNCT
iajs-2958	80	31	𝑋	𝑋	NOUN
iajs-2958	80	32	)	)	PUNCT
iajs-2958	80	33	=	=	PUNCT
iajs-2958	80	34	{	{	PUNCT
iajs-2958	80	35	𝜙	𝜙	NOUN
iajs-2958	80	36	,	,	PUNCT
iajs-2958	80	37	𝒲	𝒲	PROPN
iajs-2958	80	38	}	}	PUNCT
iajs-2958	80	39	.	.	PUNCT
iajs-2958	81	1	ii	ii	PROPN
iajs-2958	81	2	.	.	PUNCT
iajs-2958	82	1	if	if	SCONJ
iajs-2958	82	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	82	3	)	)	PUNCT
iajs-2958	82	4	=	=	SYM
iajs-2958	82	5	𝒲	𝒲	PROPN
iajs-2958	82	6	and	and	CCONJ
iajs-2958	82	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	82	8	)	)	PUNCT
iajs-2958	83	1	≠	≠	PROPN
iajs-2958	83	2	𝜙	𝜙	NOUN
iajs-2958	83	3	⟹	⟹	NUM
iajs-2958	83	4	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	83	5	)	)	PUNCT
iajs-2958	83	6	=	=	PUNCT
iajs-2958	83	7	𝑛𝛼𝑂(𝒲	𝑛𝛼𝑂(𝒲	PROPN
iajs-2958	83	8	,	,	PUNCT
iajs-2958	83	9	𝑋	𝑋	NOUN
iajs-2958	83	10	)	)	PUNCT
iajs-2958	83	11	.	.	PUNCT
iajs-2958	84	1	iii	iii	X
iajs-2958	84	2	.	.	PUNCT
iajs-2958	85	1	if	if	SCONJ
iajs-2958	85	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	85	3	)	)	PUNCT
iajs-2958	85	4	=	=	SYM
iajs-2958	85	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	85	6	)	)	PUNCT
iajs-2958	85	7	≠	≠	PROPN
iajs-2958	85	8	𝒲	𝒲	PROPN
iajs-2958	85	9	⟹	⟹	NUM
iajs-2958	85	10	𝜙	𝜙	NOUN
iajs-2958	85	11	and	and	CCONJ
iajs-2958	85	12	those	those	PRON
iajs-2958	85	13	sets	set	VERB
iajs-2958	85	14	𝐴	𝐴	PROPN
iajs-2958	85	15	for	for	ADP
iajs-2958	85	16	which	which	PRON
iajs-2958	85	17	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	85	18	)	)	PUNCT
iajs-2958	85	19	⊆	⊆	NUM
iajs-2958	85	20	𝐴	𝐴	PROPN
iajs-2958	85	21	are	be	AUX
iajs-2958	85	22	the	the	DET
iajs-2958	85	23	only	only	ADJ
iajs-2958	85	24	𝑛𝛼open	𝑛𝛼open	ADJ
iajs-2958	85	25	sets	set	NOUN
iajs-2958	85	26	in	in	ADP
iajs-2958	85	27	𝒲.	𝒲.	PROPN
iajs-2958	85	28	iv	iv	X
iajs-2958	85	29	.	.	PUNCT
iajs-2958	86	1	if	if	SCONJ
iajs-2958	86	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	86	3	)	)	PUNCT
iajs-2958	86	4	≠	≠	PROPN
iajs-2958	86	5	𝒲	𝒲	PROPN
iajs-2958	86	6	,	,	PUNCT
iajs-2958	86	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	86	8	)	)	PUNCT
iajs-2958	86	9	=	=	SYM
iajs-2958	87	1	𝜙	𝜙	PROPN
iajs-2958	87	2	⟹	⟹	VERB
iajs-2958	87	3	𝜙	𝜙	NOUN
iajs-2958	87	4	and	and	CCONJ
iajs-2958	87	5	those	those	PRON
iajs-2958	87	6	sets	set	VERB
iajs-2958	87	7	𝐴	𝐴	PROPN
iajs-2958	87	8	for	for	ADP
iajs-2958	87	9	which	which	PRON
iajs-2958	87	10	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	87	11	)	)	PUNCT
iajs-2958	87	12	⊆	⊆	NUM
iajs-2958	87	13	𝐴	𝐴	PROPN
iajs-2958	87	14	are	be	AUX
iajs-2958	87	15	the	the	DET
iajs-2958	87	16	only	only	ADJ
iajs-2958	87	17	𝑛𝛼-open	𝑛𝛼-open	ADJ
iajs-2958	87	18	sets	set	NOUN
iajs-2958	87	19	in	in	ADP
iajs-2958	87	20	𝒲.	𝒲.	PROPN
iajs-2958	87	21	v.	v.	ADP
iajs-2958	87	22	if	if	SCONJ
iajs-2958	87	23	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	87	24	)	)	PUNCT
iajs-2958	87	25	≠	≠	PROPN
iajs-2958	87	26	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	87	27	)	)	PUNCT
iajs-2958	87	28	where	where	SCONJ
iajs-2958	87	29	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	87	30	)	)	PUNCT
iajs-2958	87	31	≠	≠	PROPN
iajs-2958	87	32	𝒲	𝒲	PROPN
iajs-2958	87	33	and	and	CCONJ
iajs-2958	87	34	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	87	35	)	)	PUNCT
iajs-2958	88	1	≠	≠	PROPN
iajs-2958	88	2	𝜙	𝜙	VERB
iajs-2958	88	3	⟹	⟹	NUM
iajs-2958	88	4	𝜙	𝜙	PROPN
iajs-2958	88	5	,	,	PUNCT
iajs-2958	88	6	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	88	7	)	)	PUNCT
iajs-2958	88	8	,	,	PUNCT
iajs-2958	88	9	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	88	10	)	)	PUNCT
iajs-2958	88	11	,	,	PUNCT
iajs-2958	88	12	any	any	DET
iajs-2958	88	13	set	set	NOUN
iajs-2958	88	14	containing	contain	VERB
iajs-2958	88	15	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	88	16	)	)	PUNCT
iajs-2958	88	17	are	be	AUX
iajs-2958	88	18	the	the	DET
iajs-2958	88	19	only	only	ADJ
iajs-2958	88	20	𝑛𝛼-open	𝑛𝛼-open	ADJ
iajs-2958	88	21	sets	set	NOUN
iajs-2958	88	22	in	in	ADP
iajs-2958	88	23	𝒲.	𝒲.	PROPN
iajs-2958	88	24	theorem	theorem	VERB
iajs-2958	88	25	2.8	2.8	NUM
iajs-2958	88	26	.	.	PUNCT
iajs-2958	89	1	[	[	X
iajs-2958	89	2	1	1	X
iajs-2958	89	3	]	]	X
iajs-2958	89	4	let	let	VERB
iajs-2958	89	5	(	(	PUNCT
iajs-2958	89	6	𝒲	𝒲	NOUN
iajs-2958	89	7	,	,	PUNCT
iajs-2958	89	8	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	89	9	)	)	PUNCT
iajs-2958	89	10	)	)	PUNCT
iajs-2958	89	11	be	be	AUX
iajs-2958	89	12	a	a	DET
iajs-2958	89	13	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	89	14	,	,	PUNCT
iajs-2958	89	15	then	then	ADV
iajs-2958	89	16	:	:	PUNCT
iajs-2958	89	17	i.	i.	NOUN
iajs-2958	89	18	if	if	SCONJ
iajs-2958	89	19	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	89	20	)	)	PUNCT
iajs-2958	89	21	=	=	SYM
iajs-2958	89	22	𝒲	𝒲	PROPN
iajs-2958	89	23	and	and	CCONJ
iajs-2958	89	24	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	89	25	)	)	PUNCT
iajs-2958	89	26	=	=	SYM
iajs-2958	90	1	𝜙	𝜙	NOUN
iajs-2958	90	2	⟹	⟹	X
iajs-2958	90	3	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	90	4	,	,	PUNCT
iajs-2958	90	5	𝑋	𝑋	NOUN
iajs-2958	90	6	)	)	PUNCT
iajs-2958	90	7	=	=	PUNCT
iajs-2958	90	8	{	{	PUNCT
iajs-2958	90	9	𝜙	𝜙	NOUN
iajs-2958	90	10	,	,	PUNCT
iajs-2958	90	11	𝒲	𝒲	PROPN
iajs-2958	90	12	}	}	PUNCT
iajs-2958	90	13	.	.	PUNCT
iajs-2958	91	1	ii	ii	PROPN
iajs-2958	91	2	.	.	PUNCT
iajs-2958	92	1	if	if	SCONJ
iajs-2958	92	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	92	3	)	)	PUNCT
iajs-2958	92	4	=	=	SYM
iajs-2958	92	5	𝒲	𝒲	PROPN
iajs-2958	92	6	and	and	CCONJ
iajs-2958	92	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	92	8	)	)	PUNCT
iajs-2958	93	1	≠	≠	PROPN
iajs-2958	93	2	𝜙	𝜙	NOUN
iajs-2958	93	3	⟹	⟹	NUM
iajs-2958	93	4	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	93	5	)	)	PUNCT
iajs-2958	93	6	=	=	PUNCT
iajs-2958	94	1	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	94	2	,	,	PUNCT
iajs-2958	94	3	𝑋	𝑋	PROPN
iajs-2958	94	4	)	)	PUNCT
iajs-2958	94	5	.	.	PUNCT
iajs-2958	95	1	iii	iii	X
iajs-2958	95	2	.	.	PUNCT
iajs-2958	96	1	if	if	SCONJ
iajs-2958	96	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	96	3	)	)	PUNCT
iajs-2958	96	4	=	=	SYM
iajs-2958	96	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	96	6	)	)	PUNCT
iajs-2958	96	7	≠	≠	PROPN
iajs-2958	96	8	𝒲	𝒲	PROPN
iajs-2958	96	9	⟹	⟹	NUM
iajs-2958	96	10	𝜙	𝜙	NOUN
iajs-2958	96	11	and	and	CCONJ
iajs-2958	96	12	those	those	PRON
iajs-2958	96	13	sets	set	VERB
iajs-2958	96	14	𝐴	𝐴	PROPN
iajs-2958	96	15	for	for	ADP
iajs-2958	96	16	which	which	PRON
iajs-2958	96	17	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	96	18	)	)	PUNCT
iajs-2958	96	19	⊆	⊆	NUM
iajs-2958	96	20	𝐴	𝐴	PROPN
iajs-2958	96	21	are	be	AUX
iajs-2958	96	22	the	the	DET
iajs-2958	96	23	only	only	ADJ
iajs-2958	96	24	𝑛𝑆open	𝑛𝑆open	PROPN
iajs-2958	96	25	sets	set	NOUN
iajs-2958	96	26	in	in	ADP
iajs-2958	96	27	𝒲.	𝒲.	PROPN
iajs-2958	96	28	iv	iv	X
iajs-2958	96	29	.	.	PUNCT
iajs-2958	97	1	if	if	SCONJ
iajs-2958	97	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	97	3	)	)	PUNCT
iajs-2958	97	4	≠	≠	PROPN
iajs-2958	97	5	𝒲	𝒲	PROPN
iajs-2958	97	6	,	,	PUNCT
iajs-2958	97	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	97	8	)	)	PUNCT
iajs-2958	97	9	=	=	SYM
iajs-2958	98	1	𝜙	𝜙	PROPN
iajs-2958	98	2	⟹	⟹	VERB
iajs-2958	98	3	𝜙	𝜙	NOUN
iajs-2958	98	4	and	and	CCONJ
iajs-2958	98	5	those	those	PRON
iajs-2958	98	6	sets	set	VERB
iajs-2958	98	7	𝐴	𝐴	PROPN
iajs-2958	98	8	for	for	ADP
iajs-2958	98	9	which	which	PRON
iajs-2958	98	10	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	98	11	)	)	PUNCT
iajs-2958	98	12	⊆	⊆	NUM
iajs-2958	98	13	𝐴	𝐴	PROPN
iajs-2958	98	14	are	be	AUX
iajs-2958	98	15	the	the	DET
iajs-2958	98	16	only	only	ADJ
iajs-2958	98	17	𝑛𝑆-open	𝑛𝑆-open	NOUN
iajs-2958	98	18	sets	set	VERB
iajs-2958	98	19	in	in	ADP
iajs-2958	98	20	𝒲.	𝒲.	PROPN
iajs-2958	98	21	v.	v.	ADP
iajs-2958	98	22	if	if	SCONJ
iajs-2958	98	23	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	98	24	)	)	PUNCT
iajs-2958	98	25	≠	≠	PROPN
iajs-2958	98	26	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	98	27	)	)	PUNCT
iajs-2958	98	28	where	where	SCONJ
iajs-2958	98	29	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	98	30	)	)	PUNCT
iajs-2958	98	31	≠	≠	PROPN
iajs-2958	98	32	𝒲	𝒲	PROPN
iajs-2958	98	33	and	and	CCONJ
iajs-2958	98	34	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	98	35	)	)	PUNCT
iajs-2958	99	1	≠	≠	PROPN
iajs-2958	99	2	𝜙	𝜙	VERB
iajs-2958	99	3	⟹	⟹	NUM
iajs-2958	99	4	𝜙	𝜙	PROPN
iajs-2958	99	5	,	,	PUNCT
iajs-2958	99	6	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	99	7	)	)	PUNCT
iajs-2958	99	8	,	,	PUNCT
iajs-2958	99	9	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	99	10	)	)	PUNCT
iajs-2958	99	11	,	,	PUNCT
iajs-2958	99	12	any	any	DET
iajs-2958	99	13	set	set	NOUN
iajs-2958	99	14	containing	contain	VERB
iajs-2958	99	15	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	99	16	)	)	PUNCT
iajs-2958	99	17	,	,	PUNCT
iajs-2958	99	18	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	99	19	)	)	PUNCT
iajs-2958	99	20	∪	∪	ADP
iajs-2958	99	21	𝐵	𝐵	PROPN
iajs-2958	99	22	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2958	99	23	𝐵𝑅(𝑋	𝐵𝑅(𝑋	NOUN
iajs-2958	99	24	)	)	PUNCT
iajs-2958	99	25	∪	∪	NOUN
iajs-2958	99	26	𝐵	𝐵	NOUN
iajs-2958	99	27	where	where	SCONJ
iajs-2958	99	28	𝐵	𝐵	PROPN
iajs-2958	99	29	⊆	⊆	PROPN
iajs-2958	100	1	[	[	X
iajs-2958	100	2	𝑈𝑅(𝑋)]𝑐	𝑈𝑅(𝑋)]𝑐	NOUN
iajs-2958	100	3	are	be	AUX
iajs-2958	100	4	the	the	DET
iajs-2958	100	5	only	only	ADJ
iajs-2958	100	6	𝑛𝑆open	𝑛𝑆open	PROPN
iajs-2958	100	7	sets	set	NOUN
iajs-2958	100	8	in	in	ADP
iajs-2958	100	9	𝒲.	𝒲.	PROPN
iajs-2958	100	10	theorem	theorem	VERB
iajs-2958	100	11	2.9	2.9	NUM
iajs-2958	100	12	.	.	PUNCT
iajs-2958	101	1	[	[	X
iajs-2958	101	2	4	4	X
iajs-2958	101	3	]	]	X
iajs-2958	101	4	let	let	VERB
iajs-2958	101	5	(	(	PUNCT
iajs-2958	101	6	𝑊	𝑊	PROPN
iajs-2958	101	7	,	,	PUNCT
iajs-2958	101	8	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	101	9	)	)	PUNCT
iajs-2958	101	10	)	)	PUNCT
iajs-2958	101	11	be	be	AUX
iajs-2958	101	12	a	a	DET
iajs-2958	101	13	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	101	14	,	,	PUNCT
iajs-2958	101	15	then	then	ADV
iajs-2958	101	16	:	:	PUNCT
iajs-2958	101	17	i.	i.	NOUN
iajs-2958	101	18	if	if	SCONJ
iajs-2958	101	19	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	101	20	)	)	PUNCT
iajs-2958	101	21	=	=	SYM
iajs-2958	101	22	𝒲	𝒲	PROPN
iajs-2958	101	23	and	and	CCONJ
iajs-2958	101	24	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	101	25	)	)	PUNCT
iajs-2958	102	1	=	=	SYM
iajs-2958	103	1	𝜙	𝜙	X
iajs-2958	103	2	⟹	⟹	X
iajs-2958	103	3	𝑛𝑆𝛽𝑂(𝒲	𝑛𝑆𝛽𝑂(𝒲	PROPN
iajs-2958	103	4	,	,	PUNCT
iajs-2958	103	5	𝑋	𝑋	PROPN
iajs-2958	103	6	)	)	PUNCT
iajs-2958	103	7	=	=	PUNCT
iajs-2958	103	8	{	{	PUNCT
iajs-2958	103	9	𝜙	𝜙	NOUN
iajs-2958	103	10	,	,	PUNCT
iajs-2958	103	11	𝒲	𝒲	PROPN
iajs-2958	103	12	}	}	PUNCT
iajs-2958	103	13	.	.	PUNCT
iajs-2958	104	1	ii	ii	PROPN
iajs-2958	104	2	.	.	PUNCT
iajs-2958	105	1	if	if	SCONJ
iajs-2958	105	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	105	3	)	)	PUNCT
iajs-2958	105	4	=	=	SYM
iajs-2958	105	5	𝒲	𝒲	PROPN
iajs-2958	105	6	and	and	CCONJ
iajs-2958	105	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	105	8	)	)	PUNCT
iajs-2958	106	1	≠	≠	PROPN
iajs-2958	106	2	𝜙	𝜙	NOUN
iajs-2958	106	3	⟹	⟹	NUM
iajs-2958	106	4	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	106	5	)	)	PUNCT
iajs-2958	106	6	=	=	PUNCT
iajs-2958	107	1	𝑛𝑆𝛽𝑂(𝒲	𝑛𝑆𝛽𝑂(𝒲	PROPN
iajs-2958	107	2	,	,	PUNCT
iajs-2958	107	3	𝑋	𝑋	PROPN
iajs-2958	107	4	)	)	PUNCT
iajs-2958	107	5	.	.	PUNCT
iajs-2958	108	1	iii	iii	X
iajs-2958	108	2	.	.	PUNCT
iajs-2958	109	1	if	if	SCONJ
iajs-2958	109	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	109	3	)	)	PUNCT
iajs-2958	109	4	=	=	SYM
iajs-2958	109	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	109	6	)	)	PUNCT
iajs-2958	109	7	=	=	PRON
iajs-2958	109	8	{	{	PUNCT
iajs-2958	109	9	𝑥	𝑥	X
iajs-2958	109	10	}	}	PUNCT
iajs-2958	109	11	,	,	PUNCT
iajs-2958	109	12	𝑥	𝑥	DET
iajs-2958	109	13	∈	∈	PROPN
iajs-2958	109	14	𝒲	𝒲	PROPN
iajs-2958	109	15	,	,	PUNCT
iajs-2958	109	16	⟹	⟹	NUM
iajs-2958	109	17	𝜙	𝜙	NOUN
iajs-2958	109	18	and	and	CCONJ
iajs-2958	109	19	those	those	DET
iajs-2958	109	20	sets	set	VERB
iajs-2958	109	21	𝐴	𝐴	PROPN
iajs-2958	109	22	for	for	ADP
iajs-2958	109	23	which	which	PRON
iajs-2958	109	24	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	109	25	)	)	PUNCT
iajs-2958	109	26	⊆	⊆	NUM
iajs-2958	109	27	𝐴	𝐴	PROPN
iajs-2958	109	28	are	be	AUX
iajs-2958	109	29	the	the	DET
iajs-2958	109	30	only	only	ADJ
iajs-2958	109	31	𝑛𝑆𝛽-open	𝑛𝑆𝛽-open	NOUN
iajs-2958	109	32	sets	set	NOUN
iajs-2958	109	33	in	in	ADP
iajs-2958	109	34	𝒲.	𝒲.	PROPN
iajs-2958	109	35	iv	iv	X
iajs-2958	109	36	.	.	PUNCT
iajs-2958	110	1	if	if	SCONJ
iajs-2958	110	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	110	3	)	)	PUNCT
iajs-2958	110	4	=	=	SYM
iajs-2958	110	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	110	6	)	)	PUNCT
iajs-2958	110	7	≠	≠	PROPN
iajs-2958	110	8	𝒲	𝒲	PROPN
iajs-2958	110	9	and	and	CCONJ
iajs-2958	110	10	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	110	11	)	)	PUNCT
iajs-2958	110	12	containing	contain	VERB
iajs-2958	110	13	more	more	ADJ
iajs-2958	110	14	than	than	ADP
iajs-2958	110	15	one	one	NUM
iajs-2958	110	16	element	element	NOUN
iajs-2958	110	17	of	of	ADP
iajs-2958	110	18	𝑈	𝑈	PROPN
iajs-2958	110	19	⟹	⟹	PUNCT
iajs-2958	110	20	𝜙	𝜙	NOUN
iajs-2958	110	21	and	and	CCONJ
iajs-2958	110	22	those	those	PRON
iajs-2958	110	23	sets	set	VERB
iajs-2958	110	24	𝐴	𝐴	PROPN
iajs-2958	110	25	for	for	ADP
iajs-2958	110	26	which	which	PRON
iajs-2958	110	27	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	110	28	)	)	PUNCT
iajs-2958	110	29	⊆	⊆	NUM
iajs-2958	110	30	𝐴	𝐴	PROPN
iajs-2958	110	31	are	be	AUX
iajs-2958	110	32	the	the	DET
iajs-2958	110	33	only	only	ADJ
iajs-2958	110	34	𝑛𝑆𝛽-open	𝑛𝑆𝛽-open	NOUN
iajs-2958	110	35	sets	set	NOUN
iajs-2958	110	36	in	in	ADP
iajs-2958	110	37	𝒲.	𝒲.	PROPN
iajs-2958	110	38	v.	v.	ADP
iajs-2958	110	39	if	if	SCONJ
iajs-2958	110	40	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	110	41	)	)	PUNCT
iajs-2958	110	42	≠	≠	PROPN
iajs-2958	110	43	𝒲	𝒲	PROPN
iajs-2958	110	44	,	,	PUNCT
iajs-2958	110	45	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	110	46	)	)	PUNCT
iajs-2958	110	47	=	=	SYM
iajs-2958	110	48	𝜙	𝜙	NOUN
iajs-2958	110	49	and	and	CCONJ
iajs-2958	110	50	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	110	51	)	)	PUNCT
iajs-2958	110	52	containing	contain	VERB
iajs-2958	110	53	more	more	ADJ
iajs-2958	110	54	than	than	ADP
iajs-2958	110	55	one	one	NUM
iajs-2958	110	56	element	element	NOUN
iajs-2958	110	57	of	of	ADP
iajs-2958	110	58	𝒲	𝒲	NOUN
iajs-2958	110	59	⟹	⟹	NUM
iajs-2958	110	60	𝜙	𝜙	NOUN
iajs-2958	110	61	and	and	CCONJ
iajs-2958	110	62	those	those	PRON
iajs-2958	110	63	sets	set	VERB
iajs-2958	110	64	𝐴	𝐴	PROPN
iajs-2958	110	65	for	for	ADP
iajs-2958	110	66	which	which	PRON
iajs-2958	110	67	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	110	68	)	)	PUNCT
iajs-2958	110	69	⊆	⊆	NUM
iajs-2958	110	70	𝐴	𝐴	PROPN
iajs-2958	110	71	are	be	AUX
iajs-2958	110	72	the	the	DET
iajs-2958	110	73	only	only	ADJ
iajs-2958	110	74	𝑛𝑆𝛽-open	𝑛𝑆𝛽-open	NOUN
iajs-2958	110	75	sets	set	NOUN
iajs-2958	110	76	in	in	ADP
iajs-2958	110	77	𝒲.	𝒲.	PROPN
iajs-2958	110	78	vi	vi	PROPN
iajs-2958	110	79	.	.	PUNCT
iajs-2958	111	1	if	if	SCONJ
iajs-2958	111	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	111	3	)	)	PUNCT
iajs-2958	111	4	≠	≠	PROPN
iajs-2958	111	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	111	6	)	)	PUNCT
iajs-2958	111	7	where	where	SCONJ
iajs-2958	111	8	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	111	9	)	)	PUNCT
iajs-2958	111	10	≠	≠	PROPN
iajs-2958	111	11	𝒲	𝒲	PROPN
iajs-2958	111	12	and	and	CCONJ
iajs-2958	111	13	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	111	14	)	)	PUNCT
iajs-2958	111	15	≠	≠	PROPN
iajs-2958	111	16	𝜙	𝜙	VERB
iajs-2958	111	17	⟹	⟹	NUM
iajs-2958	111	18	𝜙	𝜙	PROPN
iajs-2958	111	19	,	,	PUNCT
iajs-2958	111	20	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	111	21	)	)	PUNCT
iajs-2958	111	22	,	,	PUNCT
iajs-2958	111	23	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	111	24	)	)	PUNCT
iajs-2958	111	25	,	,	PUNCT
iajs-2958	111	26	any	any	DET
iajs-2958	111	27	set	set	NOUN
iajs-2958	111	28	containing	contain	VERB
iajs-2958	111	29	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	111	30	)	)	PUNCT
iajs-2958	111	31	,	,	PUNCT
iajs-2958	111	32	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	111	33	)	)	PUNCT
iajs-2958	111	34	∪	∪	ADP
iajs-2958	111	35	𝐵	𝐵	PROPN
iajs-2958	111	36	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2958	111	37	𝐵𝑅(𝑋	𝐵𝑅(𝑋	NOUN
iajs-2958	111	38	)	)	PUNCT
iajs-2958	111	39	∪	∪	NOUN
iajs-2958	111	40	𝐵	𝐵	NOUN
iajs-2958	111	41	where	where	SCONJ
iajs-2958	111	42	𝐵	𝐵	PROPN
iajs-2958	111	43	⊆	⊆	PROPN
iajs-2958	112	1	[	[	X
iajs-2958	112	2	𝑈𝑅(𝑋)]𝑐	𝑈𝑅(𝑋)]𝑐	NOUN
iajs-2958	112	3	are	be	AUX
iajs-2958	112	4	the	the	DET
iajs-2958	112	5	only	only	ADJ
iajs-2958	112	6	𝑛𝑆𝛽open	𝑛𝑆𝛽open	PROPN
iajs-2958	112	7	sets	set	NOUN
iajs-2958	112	8	in	in	ADP
iajs-2958	112	9	𝒲.	𝒲.	PROPN
iajs-2958	112	10	3	3	X
iajs-2958	112	11	.	.	X
iajs-2958	112	12	nano	nano	NOUN
iajs-2958	112	13	𝑺𝑪-open	𝑺𝑪-open	PROPN
iajs-2958	112	14	sets	set	NOUN
iajs-2958	112	15	definition	definition	NOUN
iajs-2958	112	16	3.1	3.1	NUM
iajs-2958	112	17	.	.	PUNCT
iajs-2958	113	1	a	a	DET
iajs-2958	113	2	subset	subset	NOUN
iajs-2958	113	3	𝐴	𝐴	NOUN
iajs-2958	113	4	∈	∈	PROPN
iajs-2958	113	5	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	113	6	,	,	PUNCT
iajs-2958	113	7	𝑋	𝑋	PROPN
iajs-2958	113	8	)	)	PUNCT
iajs-2958	113	9	is	be	AUX
iajs-2958	113	10	said	say	VERB
iajs-2958	113	11	to	to	PART
iajs-2958	113	12	be	be	AUX
iajs-2958	113	13	nano	nano	ADJ
iajs-2958	113	14	𝑆𝐶-open	𝑆𝐶-open	NOUN
iajs-2958	113	15	(	(	PUNCT
iajs-2958	113	16	briefly	briefly	ADV
iajs-2958	113	17	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	113	18	)	)	PUNCT
iajs-2958	113	19	sets	set	NOUN
iajs-2958	113	20	in	in	ADP
iajs-2958	113	21	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	113	22	𝒲	𝒲	PROPN
iajs-2958	113	23	if	if	SCONJ
iajs-2958	113	24	for	for	ADP
iajs-2958	113	25	each	each	DET
iajs-2958	113	26	𝑥	𝑥	PRON
iajs-2958	113	27	∈	∈	PROPN
iajs-2958	113	28	𝐴	𝐴	PROPN
iajs-2958	113	29	,	,	PUNCT
iajs-2958	113	30	there	there	PRON
iajs-2958	113	31	exist	exist	VERB
iajs-2958	113	32	a	a	DET
iajs-2958	113	33	nano	nano	NOUN
iajs-2958	113	34	closed	close	VERB
iajs-2958	113	35	set	set	VERB
iajs-2958	113	36	𝐹	𝐹	PRON
iajs-2958	113	37	such	such	ADJ
iajs-2958	113	38	that	that	SCONJ
iajs-2958	113	39	𝑥	𝑥	PROPN
iajs-2958	113	40	∈	∈	PROPN
iajs-2958	113	41	𝐹	𝐹	PROPN
iajs-2958	113	42	⊆	⊆	NUM
iajs-2958	113	43	𝐴.	𝐴.	PROPN
iajs-2958	113	44	the	the	DET
iajs-2958	113	45	family	family	NOUN
iajs-2958	113	46	of	of	ADP
iajs-2958	113	47	all	all	DET
iajs-2958	113	48	nano	nano	ADJ
iajs-2958	113	49	𝑆𝐶-open	𝑆𝐶-open	PROPN
iajs-2958	113	50	subsets	subset	NOUN
iajs-2958	113	51	of	of	ADP
iajs-2958	113	52	a	a	DET
iajs-2958	113	53	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	113	54	𝒲	𝒲	NOUN
iajs-2958	113	55	denoted	denote	VERB
iajs-2958	113	56	by	by	ADP
iajs-2958	113	57	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	113	58	,	,	PUNCT
iajs-2958	113	59	𝑋	𝑋	PROPN
iajs-2958	113	60	)	)	PUNCT
iajs-2958	113	61	.	.	PUNCT
iajs-2958	114	1	definition	definition	NOUN
iajs-2958	114	2	3.2	3.2	NUM
iajs-2958	114	3	.	.	PUNCT
iajs-2958	115	1	the	the	DET
iajs-2958	115	2	complement	complement	NOUN
iajs-2958	115	3	of	of	ADP
iajs-2958	115	4	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	115	5	sets	set	NOUN
iajs-2958	115	6	in	in	ADP
iajs-2958	115	7	a	a	DET
iajs-2958	115	8	𝑁𝑇𝑆	𝑁𝑇𝑆	NOUN
iajs-2958	115	9	(	(	PUNCT
iajs-2958	115	10	𝒲	𝒲	PROPN
iajs-2958	115	11	,	,	PUNCT
iajs-2958	115	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	115	13	)	)	PUNCT
iajs-2958	115	14	)	)	PUNCT
iajs-2958	115	15	is	be	AUX
iajs-2958	115	16	said	say	VERB
iajs-2958	115	17	to	to	PART
iajs-2958	115	18	ne	ne	VERB
iajs-2958	115	19	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	115	20	sets	set	NOUN
iajs-2958	115	21	.	.	PUNCT
iajs-2958	116	1	the	the	DET
iajs-2958	116	2	family	family	NOUN
iajs-2958	116	3	of	of	ADP
iajs-2958	116	4	all	all	DET
iajs-2958	116	5	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	116	6	sets	set	NOUN
iajs-2958	116	7	denoted	denote	VERB
iajs-2958	116	8	by	by	ADP
iajs-2958	116	9	𝑛𝑆𝐶𝐶(𝒲	𝑛𝑆𝐶𝐶(𝒲	PROPN
iajs-2958	116	10	,	,	PUNCT
iajs-2958	116	11	𝑋	𝑋	PROPN
iajs-2958	116	12	)	)	PUNCT
iajs-2958	116	13	.	.	PUNCT
iajs-2958	117	1	remark	remark	VERB
iajs-2958	117	2	3.3	3.3	NUM
iajs-2958	117	3	.	.	PUNCT
iajs-2958	118	1	every	every	DET
iajs-2958	118	2	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	118	3	set	set	NOUN
iajs-2958	118	4	is	be	AUX
iajs-2958	118	5	𝑛𝑆-open	𝑛𝑆-open	NOUN
iajs-2958	118	6	set	set	VERB
iajs-2958	118	7	,	,	PUNCT
iajs-2958	118	8	but	but	CCONJ
iajs-2958	118	9	the	the	DET
iajs-2958	118	10	converse	converse	NOUN
iajs-2958	118	11	may	may	AUX
iajs-2958	118	12	not	not	PART
iajs-2958	118	13	be	be	AUX
iajs-2958	118	14	true	true	ADJ
iajs-2958	118	15	in	in	ADP
iajs-2958	118	16	general	general	ADJ
iajs-2958	118	17	,	,	PUNCT
iajs-2958	118	18	as	as	SCONJ
iajs-2958	118	19	it	it	PRON
iajs-2958	118	20	shown	show	VERB
iajs-2958	118	21	in	in	ADP
iajs-2958	118	22	the	the	DET
iajs-2958	118	23	next	next	ADJ
iajs-2958	118	24	example	example	NOUN
iajs-2958	118	25	.	.	PUNCT
iajs-2958	119	1	ihjpas	ihjpas	PROPN
iajs-2958	119	2	.	.	PUNCT
iajs-2958	120	1	36(2)2023	36(2)2023	NUM
iajs-2958	120	2	309	309	NUM
iajs-2958	120	3	example	example	NOUN
iajs-2958	120	4	3.4	3.4	NUM
iajs-2958	120	5	.	.	PUNCT
iajs-2958	121	1	let	let	VERB
iajs-2958	121	2	𝒲	𝒲	NOUN
iajs-2958	121	3	=	=	PUNCT
iajs-2958	121	4	{	{	PUNCT
iajs-2958	121	5	𝑎	𝑎	NOUN
iajs-2958	121	6	,	,	PUNCT
iajs-2958	121	7	𝑏	𝑏	NOUN
iajs-2958	121	8	,	,	PUNCT
iajs-2958	121	9	𝑐	𝑐	NOUN
iajs-2958	121	10	,	,	PUNCT
iajs-2958	121	11	𝑑	𝑑	NOUN
iajs-2958	121	12	}	}	PUNCT
iajs-2958	121	13	with	with	ADP
iajs-2958	121	14	𝒲	𝒲	PROPN
iajs-2958	121	15	𝑅⁄	𝑅⁄	PROPN
iajs-2958	121	16	=	=	PUNCT
iajs-2958	121	17	{	{	PUNCT
iajs-2958	121	18	{	{	PUNCT
iajs-2958	121	19	𝑎	𝑎	NOUN
iajs-2958	121	20	,	,	PUNCT
iajs-2958	121	21	𝑏	𝑏	NOUN
iajs-2958	121	22	}	}	PUNCT
iajs-2958	121	23	,	,	PUNCT
iajs-2958	121	24	{	{	PUNCT
iajs-2958	121	25	𝑐	𝑐	NOUN
iajs-2958	121	26	}	}	PUNCT
iajs-2958	121	27	,	,	PUNCT
iajs-2958	121	28	{	{	PUNCT
iajs-2958	121	29	𝑑	𝑑	NOUN
iajs-2958	121	30	}	}	PUNCT
iajs-2958	121	31	}	}	PUNCT
iajs-2958	121	32	and	and	CCONJ
iajs-2958	121	33	𝑋	𝑋	PROPN
iajs-2958	121	34	=	=	SYM
iajs-2958	121	35	{	{	PUNCT
iajs-2958	121	36	𝑏	𝑏	NOUN
iajs-2958	121	37	,	,	PUNCT
iajs-2958	121	38	𝑐	𝑐	NOUN
iajs-2958	121	39	}	}	PUNCT
iajs-2958	121	40	,	,	PUNCT
iajs-2958	121	41	then	then	ADV
iajs-2958	121	42	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	121	43	)	)	PUNCT
iajs-2958	121	44	=	=	PRON
iajs-2958	121	45	{	{	PUNCT
iajs-2958	121	46	𝜙	𝜙	NOUN
iajs-2958	121	47	,	,	PUNCT
iajs-2958	121	48	𝒲	𝒲	NOUN
iajs-2958	121	49	,	,	PUNCT
iajs-2958	121	50	{	{	PUNCT
iajs-2958	121	51	𝑎	𝑎	NOUN
iajs-2958	121	52	,	,	PUNCT
iajs-2958	121	53	𝑏	𝑏	NOUN
iajs-2958	121	54	,	,	PUNCT
iajs-2958	121	55	𝑐	𝑐	NOUN
iajs-2958	121	56	}	}	PUNCT
iajs-2958	121	57	,	,	PUNCT
iajs-2958	121	58	{	{	PUNCT
iajs-2958	121	59	𝑐	𝑐	NOUN
iajs-2958	121	60	}	}	PUNCT
iajs-2958	121	61	,	,	PUNCT
iajs-2958	121	62	{	{	PUNCT
iajs-2958	121	63	𝑎	𝑎	X
iajs-2958	121	64	,	,	PUNCT
iajs-2958	121	65	𝑏	𝑏	NOUN
iajs-2958	121	66	}	}	PUNCT
iajs-2958	121	67	}	}	PUNCT
iajs-2958	121	68	and	and	CCONJ
iajs-2958	121	69	[	[	X
iajs-2958	121	70	𝜏𝑅(𝑋)]𝐶	𝜏𝑅(𝑋)]𝐶	NOUN
iajs-2958	121	71	=	=	NOUN
iajs-2958	121	72	{	{	PUNCT
iajs-2958	121	73	𝜙	𝜙	NOUN
iajs-2958	121	74	,	,	PUNCT
iajs-2958	121	75	𝒲	𝒲	NOUN
iajs-2958	121	76	,	,	PUNCT
iajs-2958	121	77	{	{	PUNCT
iajs-2958	121	78	𝑑	𝑑	NOUN
iajs-2958	121	79	}	}	PUNCT
iajs-2958	121	80	,	,	PUNCT
iajs-2958	121	81	{	{	PUNCT
iajs-2958	121	82	𝑎	𝑎	X
iajs-2958	121	83	,	,	PUNCT
iajs-2958	121	84	𝑏	𝑏	NOUN
iajs-2958	121	85	,	,	PUNCT
iajs-2958	121	86	𝑑	𝑑	NOUN
iajs-2958	121	87	}	}	PUNCT
iajs-2958	121	88	,	,	PUNCT
iajs-2958	121	89	{	{	PUNCT
iajs-2958	121	90	𝑐	𝑐	NOUN
iajs-2958	121	91	,	,	PUNCT
iajs-2958	121	92	𝑑	𝑑	NOUN
iajs-2958	121	93	}	}	PUNCT
iajs-2958	121	94	}	}	PUNCT
iajs-2958	121	95	.	.	PUNCT
iajs-2958	122	1	the	the	DET
iajs-2958	122	2	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	122	3	,	,	PUNCT
iajs-2958	122	4	𝑋	𝑋	NOUN
iajs-2958	122	5	)	)	PUNCT
iajs-2958	122	6	=	=	PUNCT
iajs-2958	122	7	{	{	PUNCT
iajs-2958	122	8	𝜙	𝜙	NOUN
iajs-2958	122	9	,	,	PUNCT
iajs-2958	122	10	𝒲	𝒲	NOUN
iajs-2958	122	11	,	,	PUNCT
iajs-2958	122	12	{	{	PUNCT
iajs-2958	122	13	𝑎	𝑎	NOUN
iajs-2958	122	14	,	,	PUNCT
iajs-2958	122	15	𝑏	𝑏	NOUN
iajs-2958	122	16	,	,	PUNCT
iajs-2958	122	17	𝑐	𝑐	NOUN
iajs-2958	122	18	}	}	PUNCT
iajs-2958	122	19	,	,	PUNCT
iajs-2958	122	20	{	{	PUNCT
iajs-2958	122	21	𝑐	𝑐	NOUN
iajs-2958	122	22	}	}	PUNCT
iajs-2958	122	23	,	,	PUNCT
iajs-2958	122	24	{	{	PUNCT
iajs-2958	122	25	𝑎	𝑎	X
iajs-2958	122	26	,	,	PUNCT
iajs-2958	122	27	𝑏	𝑏	NOUN
iajs-2958	122	28	}	}	PUNCT
iajs-2958	122	29	,	,	PUNCT
iajs-2958	122	30	{	{	PUNCT
iajs-2958	122	31	𝑐	𝑐	NOUN
iajs-2958	122	32	,	,	PUNCT
iajs-2958	122	33	𝑑	𝑑	NOUN
iajs-2958	122	34	}	}	PUNCT
iajs-2958	122	35	,	,	PUNCT
iajs-2958	122	36	{	{	PUNCT
iajs-2958	122	37	𝑎	𝑎	X
iajs-2958	122	38	,	,	PUNCT
iajs-2958	122	39	𝑏	𝑏	NOUN
iajs-2958	122	40	,	,	PUNCT
iajs-2958	122	41	𝑑	𝑑	NOUN
iajs-2958	122	42	}	}	PUNCT
iajs-2958	122	43	}	}	PUNCT
iajs-2958	122	44	.	.	PUNCT
iajs-2958	123	1	then	then	ADV
iajs-2958	123	2	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	123	3	,	,	PUNCT
iajs-2958	123	4	𝑋	𝑋	PROPN
iajs-2958	123	5	)	)	PUNCT
iajs-2958	123	6	=	=	PUNCT
iajs-2958	123	7	{	{	PUNCT
iajs-2958	123	8	𝜙	𝜙	NOUN
iajs-2958	123	9	,	,	PUNCT
iajs-2958	123	10	𝒲	𝒲	NOUN
iajs-2958	123	11	,	,	PUNCT
iajs-2958	123	12	{	{	PUNCT
iajs-2958	123	13	𝑐	𝑐	NOUN
iajs-2958	123	14	,	,	PUNCT
iajs-2958	123	15	𝑑	𝑑	NOUN
iajs-2958	123	16	}	}	PUNCT
iajs-2958	123	17	,	,	PUNCT
iajs-2958	123	18	{	{	PUNCT
iajs-2958	123	19	𝑎	𝑎	X
iajs-2958	123	20	,	,	PUNCT
iajs-2958	123	21	𝑏	𝑏	NOUN
iajs-2958	123	22	,	,	PUNCT
iajs-2958	123	23	𝑑	𝑑	NOUN
iajs-2958	123	24	}	}	PUNCT
iajs-2958	123	25	}	}	PUNCT
iajs-2958	123	26	and	and	CCONJ
iajs-2958	123	27	it	it	PRON
iajs-2958	123	28	is	be	AUX
iajs-2958	123	29	clear	clear	ADJ
iajs-2958	123	30	that	that	SCONJ
iajs-2958	123	31	the	the	DET
iajs-2958	123	32	subset	subset	NOUN
iajs-2958	123	33	{	{	PUNCT
iajs-2958	123	34	𝑐	𝑐	NOUN
iajs-2958	123	35	}	}	PUNCT
iajs-2958	123	36	is	be	AUX
iajs-2958	123	37	𝑛𝑆-open	𝑛𝑆-open	ADJ
iajs-2958	123	38	but	but	CCONJ
iajs-2958	123	39	not	not	PART
iajs-2958	123	40	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	123	41	set	set	NOUN
iajs-2958	123	42	in	in	ADP
iajs-2958	123	43	𝒲.	𝒲.	PROPN
iajs-2958	123	44	proposition	proposition	NOUN
iajs-2958	123	45	3.5	3.5	NUM
iajs-2958	123	46	.	.	PUNCT
iajs-2958	124	1	a	a	DET
iajs-2958	124	2	subset	subset	ADJ
iajs-2958	124	3	𝐴	𝐴	PROPN
iajs-2958	124	4	of	of	ADP
iajs-2958	124	5	a	a	DET
iajs-2958	124	6	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	124	7	(	(	PUNCT
iajs-2958	124	8	𝒲	𝒲	PROPN
iajs-2958	124	9	,	,	PUNCT
iajs-2958	124	10	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	124	11	)	)	PUNCT
iajs-2958	124	12	)	)	PUNCT
iajs-2958	124	13	is	be	AUX
iajs-2958	124	14	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	124	15	if	if	SCONJ
iajs-2958	124	16	and	and	CCONJ
iajs-2958	124	17	only	only	ADV
iajs-2958	124	18	if	if	SCONJ
iajs-2958	124	19	𝐴	𝐴	PROPN
iajs-2958	124	20	is	be	AUX
iajs-2958	124	21	𝑛𝑆-open	𝑛𝑆-open	ADJ
iajs-2958	124	22	and	and	CCONJ
iajs-2958	124	23	the	the	DET
iajs-2958	124	24	union	union	NOUN
iajs-2958	124	25	of	of	ADP
iajs-2958	124	26	nano	nano	NOUN
iajs-2958	124	27	closed	close	VERB
iajs-2958	124	28	sets	set	NOUN
iajs-2958	124	29	.	.	PUNCT
iajs-2958	125	1	proof	proof	NOUN
iajs-2958	125	2	.	.	PUNCT
iajs-2958	126	1	obvious	obvious	ADJ
iajs-2958	126	2	.	.	PUNCT
iajs-2958	126	3	remark	remark	NOUN
iajs-2958	126	4	3.6	3.6	NUM
iajs-2958	126	5	.	.	PUNCT
iajs-2958	127	1	i.	i.	PROPN
iajs-2958	127	2	nano	nano	PROPN
iajs-2958	127	3	open	open	ADJ
iajs-2958	127	4	sets	set	NOUN
iajs-2958	127	5	and	and	CCONJ
iajs-2958	127	6	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	127	7	sets	set	NOUN
iajs-2958	127	8	are	be	AUX
iajs-2958	127	9	independent	independent	ADJ
iajs-2958	127	10	.	.	PUNCT
iajs-2958	128	1	in	in	ADP
iajs-2958	128	2	above	above	ADP
iajs-2958	128	3	example	example	NOUN
iajs-2958	128	4	,	,	PUNCT
iajs-2958	128	5	the	the	DET
iajs-2958	128	6	subset	subset	NOUN
iajs-2958	128	7	{	{	PUNCT
iajs-2958	128	8	𝑐	𝑐	NOUN
iajs-2958	128	9	,	,	PUNCT
iajs-2958	128	10	𝑑	𝑑	NOUN
iajs-2958	128	11	}	}	PUNCT
iajs-2958	128	12	is	be	AUX
iajs-2958	128	13	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	128	14	but	but	CCONJ
iajs-2958	128	15	not	not	PART
iajs-2958	128	16	nano	nano	VERB
iajs-2958	128	17	open	open	ADJ
iajs-2958	128	18	in	in	ADP
iajs-2958	128	19	𝑈	𝑈	PROPN
iajs-2958	128	20	,	,	PUNCT
iajs-2958	128	21	also	also	ADV
iajs-2958	128	22	,	,	PUNCT
iajs-2958	128	23	the	the	DET
iajs-2958	128	24	subset	subset	NOUN
iajs-2958	128	25	{	{	PUNCT
iajs-2958	128	26	𝑐	𝑐	X
iajs-2958	128	27	}	}	PUNCT
iajs-2958	128	28	is	be	AUX
iajs-2958	128	29	nano	nano	NOUN
iajs-2958	128	30	open	open	ADJ
iajs-2958	128	31	but	but	CCONJ
iajs-2958	128	32	not	not	PART
iajs-2958	128	33	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	128	34	set	set	VERB
iajs-2958	128	35	in	in	ADP
iajs-2958	128	36	𝑈.	𝑈.	PROPN
iajs-2958	128	37	ii	ii	PROPN
iajs-2958	128	38	.	.	PUNCT
iajs-2958	128	39	𝑛𝛼-open	𝑛𝛼-open	ADJ
iajs-2958	128	40	sets	set	NOUN
iajs-2958	128	41	and	and	CCONJ
iajs-2958	128	42	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	128	43	sets	set	NOUN
iajs-2958	128	44	are	be	AUX
iajs-2958	128	45	independent	independent	ADJ
iajs-2958	128	46	.	.	PUNCT
iajs-2958	129	1	in	in	ADP
iajs-2958	129	2	above	above	ADP
iajs-2958	129	3	example	example	NOUN
iajs-2958	129	4	,	,	PUNCT
iajs-2958	129	5	{	{	PUNCT
iajs-2958	129	6	𝑎	𝑎	X
iajs-2958	129	7	,	,	PUNCT
iajs-2958	129	8	𝑏	𝑏	NOUN
iajs-2958	129	9	,	,	PUNCT
iajs-2958	129	10	𝑐	𝑐	PRON
iajs-2958	129	11	}	}	PUNCT
iajs-2958	129	12	is	be	AUX
iajs-2958	129	13	𝑛𝛼-open	𝑛𝛼-open	ADJ
iajs-2958	129	14	set	set	ADJ
iajs-2958	129	15	but	but	CCONJ
iajs-2958	129	16	not	not	PART
iajs-2958	129	17	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	129	18	,	,	PUNCT
iajs-2958	129	19	also	also	ADV
iajs-2958	129	20	{	{	PUNCT
iajs-2958	129	21	𝑐	𝑐	NOUN
iajs-2958	129	22	,	,	PUNCT
iajs-2958	129	23	𝑑	𝑑	NOUN
iajs-2958	129	24	}	}	PUNCT
iajs-2958	129	25	is	be	AUX
iajs-2958	129	26	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	129	27	but	but	CCONJ
iajs-2958	129	28	not	not	PART
iajs-2958	129	29	𝑛𝛼-open	𝑛𝛼-open	ADJ
iajs-2958	129	30	.	.	PUNCT
iajs-2958	130	1	iii	iii	X
iajs-2958	130	2	.	.	PROPN
iajs-2958	130	3	𝑛𝑅-open	𝑛𝑅-open	PROPN
iajs-2958	130	4	sets	set	NOUN
iajs-2958	130	5	and	and	CCONJ
iajs-2958	130	6	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	130	7	sets	set	NOUN
iajs-2958	130	8	are	be	AUX
iajs-2958	130	9	independent	independent	ADJ
iajs-2958	130	10	.	.	PUNCT
iajs-2958	131	1	in	in	ADP
iajs-2958	131	2	above	above	ADP
iajs-2958	131	3	example	example	NOUN
iajs-2958	131	4	,	,	PUNCT
iajs-2958	131	5	{	{	PUNCT
iajs-2958	131	6	𝑎	𝑎	X
iajs-2958	131	7	,	,	PUNCT
iajs-2958	131	8	𝑏	𝑏	NOUN
iajs-2958	131	9	}	}	PUNCT
iajs-2958	131	10	is	be	AUX
iajs-2958	131	11	𝑛𝑅-open	𝑛𝑅-open	NOUN
iajs-2958	131	12	set	set	VERB
iajs-2958	131	13	but	but	CCONJ
iajs-2958	131	14	not	not	PART
iajs-2958	131	15	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	131	16	,	,	PUNCT
iajs-2958	131	17	also	also	ADV
iajs-2958	131	18	{	{	PUNCT
iajs-2958	131	19	𝑐	𝑐	NOUN
iajs-2958	131	20	,	,	PUNCT
iajs-2958	131	21	𝑑	𝑑	NOUN
iajs-2958	131	22	}	}	PUNCT
iajs-2958	131	23	is	be	AUX
iajs-2958	131	24	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	131	25	but	but	CCONJ
iajs-2958	131	26	not	not	PART
iajs-2958	131	27	𝑛𝑅-open	𝑛𝑅-open	PROPN
iajs-2958	131	28	.	.	PUNCT
iajs-2958	132	1	iv	iv	X
iajs-2958	132	2	.	.	PUNCT
iajs-2958	133	1	the	the	DET
iajs-2958	133	2	intersection	intersection	NOUN
iajs-2958	133	3	of	of	ADP
iajs-2958	133	4	tow	tow	NOUN
iajs-2958	133	5	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	133	6	sets	set	NOUN
iajs-2958	133	7	may	may	AUX
iajs-2958	133	8	not	not	PART
iajs-2958	133	9	be	be	AUX
iajs-2958	133	10	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	133	11	.	.	PUNCT
iajs-2958	134	1	in	in	ADP
iajs-2958	134	2	above	above	ADP
iajs-2958	134	3	example	example	NOUN
iajs-2958	134	4	,	,	PUNCT
iajs-2958	134	5	{	{	PUNCT
iajs-2958	134	6	𝑐	𝑐	NOUN
iajs-2958	134	7	,	,	PUNCT
iajs-2958	134	8	𝑑	𝑑	NOUN
iajs-2958	134	9	}	}	PUNCT
iajs-2958	134	10	and	and	CCONJ
iajs-2958	134	11	{	{	PUNCT
iajs-2958	134	12	𝑎	𝑎	NOUN
iajs-2958	134	13	,	,	PUNCT
iajs-2958	134	14	𝑏	𝑏	NOUN
iajs-2958	134	15	,	,	PUNCT
iajs-2958	134	16	𝑑	𝑑	NOUN
iajs-2958	134	17	}	}	PUNCT
iajs-2958	134	18	are	be	AUX
iajs-2958	134	19	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	134	20	but	but	CCONJ
iajs-2958	134	21	{	{	PUNCT
iajs-2958	134	22	𝑐	𝑐	NOUN
iajs-2958	134	23	,	,	PUNCT
iajs-2958	134	24	𝑑	𝑑	NOUN
iajs-2958	134	25	}	}	PUNCT
iajs-2958	134	26	∩	∩	ADJ
iajs-2958	134	27	{	{	PUNCT
iajs-2958	134	28	𝑎	𝑎	NOUN
iajs-2958	134	29	,	,	PUNCT
iajs-2958	134	30	𝑏	𝑏	NOUN
iajs-2958	134	31	,	,	PUNCT
iajs-2958	134	32	𝑑	𝑑	NOUN
iajs-2958	134	33	}	}	PUNCT
iajs-2958	134	34	=	=	SYM
iajs-2958	134	35	{	{	PUNCT
iajs-2958	134	36	𝑑	𝑑	NOUN
iajs-2958	134	37	}	}	PUNCT
iajs-2958	134	38	which	which	PRON
iajs-2958	134	39	is	be	AUX
iajs-2958	134	40	not	not	PART
iajs-2958	134	41	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	134	42	in	in	ADP
iajs-2958	134	43	𝑈.	𝑈.	PROPN
iajs-2958	134	44	so	so	SCONJ
iajs-2958	134	45	that	that	SCONJ
iajs-2958	134	46	,	,	PUNCT
iajs-2958	134	47	the	the	DET
iajs-2958	134	48	family	family	NOUN
iajs-2958	134	49	of	of	ADP
iajs-2958	134	50	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	134	51	sets	set	NOUN
iajs-2958	134	52	forma	forma	NOUN
iajs-2958	134	53	supra	supra	PROPN
iajs-2958	134	54	topology	topology	NOUN
iajs-2958	134	55	.	.	PUNCT
iajs-2958	135	1	proposition	proposition	NOUN
iajs-2958	135	2	3.7	3.7	NUM
iajs-2958	135	3	.	.	PUNCT
iajs-2958	136	1	let	let	VERB
iajs-2958	136	2	{	{	PUNCT
iajs-2958	136	3	𝐴𝛼	𝐴𝛼	VERB
iajs-2958	136	4	:	:	PUNCT
iajs-2958	136	5	𝛼	𝛼	PROPN
iajs-2958	136	6	∈	∈	PROPN
iajs-2958	136	7	δ	δ	PROPN
iajs-2958	136	8	}	}	PUNCT
iajs-2958	136	9	be	be	VERB
iajs-2958	136	10	a	a	DET
iajs-2958	136	11	collection	collection	NOUN
iajs-2958	136	12	of	of	ADP
iajs-2958	136	13	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	136	14	sets	set	NOUN
iajs-2958	136	15	in	in	ADP
iajs-2958	136	16	a	a	DET
iajs-2958	136	17	𝑁𝑇𝑆	𝑁𝑇𝑆	NOUN
iajs-2958	136	18	(	(	PUNCT
iajs-2958	136	19	𝒲	𝒲	PROPN
iajs-2958	136	20	,	,	PUNCT
iajs-2958	136	21	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	136	22	)	)	PUNCT
iajs-2958	136	23	)	)	PUNCT
iajs-2958	136	24	.	.	PUNCT
iajs-2958	137	1	then	then	ADV
iajs-2958	137	2	⋃{𝐴𝛼	⋃{𝐴𝛼	PROPN
iajs-2958	137	3	:	:	PUNCT
iajs-2958	137	4	𝛼	𝛼	PROPN
iajs-2958	137	5	∈	∈	PROPN
iajs-2958	137	6	δ	δ	PROPN
iajs-2958	137	7	}	}	PUNCT
iajs-2958	137	8	is	be	AUX
iajs-2958	137	9	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	137	10	.	.	PUNCT
iajs-2958	138	1	proof	proof	NOUN
iajs-2958	138	2	.	.	PUNCT
iajs-2958	139	1	let	let	VERB
iajs-2958	139	2	𝐴𝛼	𝐴𝛼	VERB
iajs-2958	139	3	be	be	AUX
iajs-2958	139	4	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	139	5	set	set	ADJ
iajs-2958	139	6	for	for	ADP
iajs-2958	139	7	each	each	DET
iajs-2958	139	8	𝛼	𝛼	NOUN
iajs-2958	139	9	,	,	PUNCT
iajs-2958	139	10	then	then	ADV
iajs-2958	139	11	𝐴𝛼	𝐴𝛼	PROPN
iajs-2958	139	12	is	be	AUX
iajs-2958	139	13	𝑛𝑆-open	𝑛𝑆-open	X
iajs-2958	139	14	and	and	CCONJ
iajs-2958	139	15	hence	hence	ADV
iajs-2958	139	16	by	by	ADP
iajs-2958	139	17	theorem	theorem	NOUN
iajs-2958	139	18	5	5	NUM
iajs-2958	139	19	,	,	PUNCT
iajs-2958	139	20	∪	∪	X
iajs-2958	139	21	{	{	PUNCT
iajs-2958	139	22	𝐴𝛼	𝐴𝛼	PROPN
iajs-2958	139	23	:	:	PUNCT
iajs-2958	139	24	𝛼	𝛼	PROPN
iajs-2958	139	25	∈	∈	PROPN
iajs-2958	139	26	δ	δ	PROPN
iajs-2958	139	27	}	}	PUNCT
iajs-2958	139	28	is	be	AUX
iajs-2958	139	29	𝑛𝑆-open	𝑛𝑆-open	ADJ
iajs-2958	139	30	.	.	PUNCT
iajs-2958	140	1	let	let	VERB
iajs-2958	140	2	𝑥	𝑥	PRON
iajs-2958	140	3	∈	∈	PROPN
iajs-2958	140	4	⋃{𝐴𝛼	⋃{𝐴𝛼	PROPN
iajs-2958	140	5	:	:	PUNCT
iajs-2958	140	6	𝛼	𝛼	PROPN
iajs-2958	140	7	∈	∈	PROPN
iajs-2958	140	8	δ	δ	PROPN
iajs-2958	140	9	}	}	PUNCT
iajs-2958	140	10	,	,	PUNCT
iajs-2958	140	11	there	there	PRON
iajs-2958	140	12	exist	exist	VERB
iajs-2958	140	13	𝛼	𝛼	DET
iajs-2958	140	14	∈	∈	PROPN
iajs-2958	140	15	δ	δ	NOUN
iajs-2958	140	16	such	such	ADJ
iajs-2958	140	17	that	that	SCONJ
iajs-2958	140	18	𝑥	𝑥	PROPN
iajs-2958	140	19	∈	∈	PROPN
iajs-2958	140	20	𝐴𝛼.	𝐴𝛼.	PROPN
iajs-2958	140	21	since	since	SCONJ
iajs-2958	140	22	𝐴𝛼	𝐴𝛼	PROPN
iajs-2958	140	23	is	be	AUX
iajs-2958	140	24	𝑛𝑆-open	𝑛𝑆-open	ADJ
iajs-2958	140	25	for	for	ADP
iajs-2958	140	26	each	each	DET
iajs-2958	140	27	𝛼	𝛼	NOUN
iajs-2958	140	28	,	,	PUNCT
iajs-2958	140	29	there	there	PRON
iajs-2958	140	30	exists	exist	VERB
iajs-2958	140	31	a	a	DET
iajs-2958	140	32	nano	nano	NOUN
iajs-2958	140	33	closed	close	VERB
iajs-2958	140	34	set	set	VERB
iajs-2958	140	35	𝐹	𝐹	PRON
iajs-2958	140	36	such	such	ADJ
iajs-2958	140	37	that	that	SCONJ
iajs-2958	140	38	𝑥	𝑥	PROPN
iajs-2958	140	39	∈	∈	NOUN
iajs-2958	140	40	𝐹	𝐹	PROPN
iajs-2958	140	41	⊆	⊆	NUM
iajs-2958	140	42	𝐴𝛼	𝐴𝛼	PROPN
iajs-2958	140	43	⊆	⊆	NUM
iajs-2958	140	44	⋃{𝐴𝛼	⋃{𝐴𝛼	PROPN
iajs-2958	140	45	:	:	PUNCT
iajs-2958	140	46	𝛼	𝛼	PROPN
iajs-2958	140	47	∈	∈	PROPN
iajs-2958	140	48	δ	δ	X
iajs-2958	140	49	}	}	PUNCT
iajs-2958	140	50	,	,	PUNCT
iajs-2958	140	51	so	so	ADV
iajs-2958	141	1	𝑥	𝑥	ADP
iajs-2958	141	2	∈	∈	PROPN
iajs-2958	141	3	𝐹	𝐹	PROPN
iajs-2958	141	4	⊆	⊆	X
iajs-2958	141	5	⋃{𝐴𝛼	⋃{𝐴𝛼	PROPN
iajs-2958	141	6	:	:	PUNCT
iajs-2958	141	7	𝛼	𝛼	PROPN
iajs-2958	141	8	∈	∈	PROPN
iajs-2958	141	9	δ	δ	PROPN
iajs-2958	141	10	}	}	PUNCT
iajs-2958	141	11	.	.	PUNCT
iajs-2958	142	1	therefore	therefore	ADV
iajs-2958	142	2	,	,	PUNCT
iajs-2958	142	3	∪	∪	X
iajs-2958	142	4	{	{	PUNCT
iajs-2958	142	5	𝐴𝛼	𝐴𝛼	PROPN
iajs-2958	142	6	:	:	PUNCT
iajs-2958	142	7	𝛼	𝛼	PROPN
iajs-2958	142	8	∈	∈	PROPN
iajs-2958	142	9	δ	δ	PROPN
iajs-2958	142	10	}	}	PUNCT
iajs-2958	142	11	is	be	AUX
iajs-2958	142	12	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	142	13	set	set	VERB
iajs-2958	142	14	.	.	PUNCT
iajs-2958	143	1	in	in	ADP
iajs-2958	143	2	the	the	DET
iajs-2958	143	3	following	follow	VERB
iajs-2958	143	4	results	result	NOUN
iajs-2958	143	5	,	,	PUNCT
iajs-2958	143	6	we	we	PRON
iajs-2958	143	7	study	study	VERB
iajs-2958	143	8	all	all	DET
iajs-2958	143	9	form	form	NOUN
iajs-2958	143	10	of	of	ADP
iajs-2958	143	11	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	143	12	sets	set	NOUN
iajs-2958	143	13	in	in	ADP
iajs-2958	143	14	term	term	NOUN
iajs-2958	143	15	of	of	ADP
iajs-2958	143	16	upper	upper	ADJ
iajs-2958	143	17	and	and	CCONJ
iajs-2958	143	18	lower	low	ADJ
iajs-2958	143	19	approximations	approximation	NOUN
iajs-2958	143	20	in	in	ADP
iajs-2958	143	21	𝑁𝑇𝑆.	𝑁𝑇𝑆.	X
iajs-2958	143	22	theorem	theorem	NOUN
iajs-2958	143	23	3.8	3.8	NUM
iajs-2958	143	24	.	.	PUNCT
iajs-2958	144	1	let	let	VERB
iajs-2958	144	2	(	(	PUNCT
iajs-2958	144	3	𝒲	𝒲	NOUN
iajs-2958	144	4	,	,	PUNCT
iajs-2958	144	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	144	6	)	)	PUNCT
iajs-2958	144	7	)	)	PUNCT
iajs-2958	145	1	be	be	AUX
iajs-2958	145	2	a	a	DET
iajs-2958	145	3	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	145	4	,	,	PUNCT
iajs-2958	145	5	then	then	ADV
iajs-2958	145	6	:	:	PUNCT
iajs-2958	145	7	i.	i.	NOUN
iajs-2958	145	8	if	if	SCONJ
iajs-2958	145	9	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	145	10	)	)	PUNCT
iajs-2958	145	11	=	=	SYM
iajs-2958	145	12	𝒲	𝒲	PROPN
iajs-2958	145	13	and	and	CCONJ
iajs-2958	145	14	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	145	15	)	)	PUNCT
iajs-2958	146	1	=	=	SYM
iajs-2958	146	2	𝜙	𝜙	NOUN
iajs-2958	146	3	,	,	PUNCT
iajs-2958	146	4	then	then	ADV
iajs-2958	146	5	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	146	6	,	,	PUNCT
iajs-2958	146	7	𝑋	𝑋	PROPN
iajs-2958	146	8	)	)	PUNCT
iajs-2958	146	9	=	=	PUNCT
iajs-2958	146	10	{	{	PUNCT
iajs-2958	146	11	𝜙	𝜙	NOUN
iajs-2958	146	12	,	,	PUNCT
iajs-2958	146	13	𝒲	𝒲	PROPN
iajs-2958	146	14	}	}	PUNCT
iajs-2958	146	15	.	.	PUNCT
iajs-2958	147	1	ii	ii	PROPN
iajs-2958	147	2	.	.	PUNCT
iajs-2958	148	1	if	if	SCONJ
iajs-2958	148	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	148	3	)	)	PUNCT
iajs-2958	148	4	=	=	SYM
iajs-2958	148	5	𝒲	𝒲	PROPN
iajs-2958	148	6	and	and	CCONJ
iajs-2958	148	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	148	8	)	)	PUNCT
iajs-2958	149	1	≠	≠	PROPN
iajs-2958	149	2	𝜙	𝜙	NOUN
iajs-2958	149	3	,	,	PUNCT
iajs-2958	149	4	then	then	ADV
iajs-2958	149	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NUM
iajs-2958	149	6	)	)	PUNCT
iajs-2958	149	7	=	=	PUNCT
iajs-2958	149	8	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	149	9	,	,	PUNCT
iajs-2958	149	10	𝑋	𝑋	PROPN
iajs-2958	149	11	)	)	PUNCT
iajs-2958	149	12	.	.	PUNCT
iajs-2958	150	1	iii	iii	X
iajs-2958	150	2	.	.	PUNCT
iajs-2958	151	1	if	if	SCONJ
iajs-2958	151	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	151	3	)	)	PUNCT
iajs-2958	151	4	=	=	SYM
iajs-2958	151	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	151	6	)	)	PUNCT
iajs-2958	151	7	≠	≠	PROPN
iajs-2958	151	8	𝒲	𝒲	PROPN
iajs-2958	151	9	,	,	PUNCT
iajs-2958	151	10	then	then	ADV
iajs-2958	151	11	𝑛𝑆𝐶𝑂(𝑊	𝑛𝑆𝐶𝑂(𝑊	PROPN
iajs-2958	151	12	,	,	PUNCT
iajs-2958	151	13	𝑋	𝑋	PROPN
iajs-2958	151	14	)	)	PUNCT
iajs-2958	151	15	=	=	PUNCT
iajs-2958	151	16	{	{	PUNCT
iajs-2958	151	17	𝜙	𝜙	NOUN
iajs-2958	151	18	,	,	PUNCT
iajs-2958	151	19	𝒲	𝒲	PROPN
iajs-2958	151	20	}	}	PUNCT
iajs-2958	151	21	.	.	PUNCT
iajs-2958	152	1	iv	iv	X
iajs-2958	152	2	.	.	PUNCT
iajs-2958	153	1	if	if	SCONJ
iajs-2958	153	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	153	3	)	)	PUNCT
iajs-2958	153	4	≠	≠	PROPN
iajs-2958	153	5	𝑊	𝑊	PROPN
iajs-2958	153	6	,	,	PUNCT
iajs-2958	153	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	153	8	)	)	PUNCT
iajs-2958	153	9	=	=	SYM
iajs-2958	154	1	𝜙	𝜙	NOUN
iajs-2958	154	2	,	,	PUNCT
iajs-2958	154	3	then	then	ADV
iajs-2958	154	4	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	154	5	,	,	PUNCT
iajs-2958	154	6	𝑋	𝑋	PROPN
iajs-2958	154	7	)	)	PUNCT
iajs-2958	154	8	=	=	PUNCT
iajs-2958	154	9	{	{	PUNCT
iajs-2958	154	10	𝜙	𝜙	NOUN
iajs-2958	154	11	,	,	PUNCT
iajs-2958	154	12	𝒲	𝒲	PROPN
iajs-2958	154	13	}	}	PUNCT
iajs-2958	154	14	.	.	PUNCT
iajs-2958	155	1	v.	v.	INTJ
iajs-2958	155	2	if	if	SCONJ
iajs-2958	155	3	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	155	4	)	)	PUNCT
iajs-2958	155	5	≠	≠	PROPN
iajs-2958	155	6	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	155	7	)	)	PUNCT
iajs-2958	155	8	where	where	SCONJ
iajs-2958	155	9	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	155	10	)	)	PUNCT
iajs-2958	155	11	≠	≠	PROPN
iajs-2958	155	12	𝒲	𝒲	PROPN
iajs-2958	155	13	and	and	CCONJ
iajs-2958	155	14	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	155	15	)	)	PUNCT
iajs-2958	155	16	≠	≠	PROPN
iajs-2958	155	17	𝜙	𝜙	NOUN
iajs-2958	155	18	,	,	PUNCT
iajs-2958	155	19	then	then	ADV
iajs-2958	155	20	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	155	21	,	,	PUNCT
iajs-2958	155	22	𝑋	𝑋	PROPN
iajs-2958	155	23	)	)	PUNCT
iajs-2958	155	24	=	=	PUNCT
iajs-2958	155	25	{	{	PUNCT
iajs-2958	155	26	𝜙	𝜙	NOUN
iajs-2958	155	27	,	,	PUNCT
iajs-2958	155	28	𝒲	𝒲	PROPN
iajs-2958	155	29	,	,	PUNCT
iajs-2958	155	30	[	[	NOUN
iajs-2958	155	31	𝐵𝑅(𝑋	𝐵𝑅(𝑋	NOUN
iajs-2958	155	32	)	)	PUNCT
iajs-2958	155	33	∪	∪	X
iajs-2958	155	34	𝐵	𝐵	PROPN
iajs-2958	155	35	]	]	PUNCT
iajs-2958	155	36	,	,	PUNCT
iajs-2958	155	37	[	[	X
iajs-2958	155	38	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	155	39	)	)	PUNCT
iajs-2958	155	40	∪	∪	X
iajs-2958	155	41	𝐵	𝐵	PROPN
iajs-2958	155	42	]	]	PUNCT
iajs-2958	155	43	}	}	PUNCT
iajs-2958	155	44	,	,	PUNCT
iajs-2958	155	45	where	where	SCONJ
iajs-2958	155	46	𝐵	𝐵	NOUN
iajs-2958	155	47	=	=	PUNCT
iajs-2958	156	1	[	[	X
iajs-2958	156	2	𝑈𝑅(𝑋)]𝐶.	𝑈𝑅(𝑋)]𝐶.	NOUN
iajs-2958	156	3	proof	proof	NOUN
iajs-2958	156	4	.	.	PUNCT
iajs-2958	157	1	i.	i.	PROPN
iajs-2958	157	2	follows	follow	VERB
iajs-2958	157	3	form	form	NOUN
iajs-2958	157	4	that	that	SCONJ
iajs-2958	157	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	157	6	)	)	PUNCT
iajs-2958	157	7	=	=	PUNCT
iajs-2958	158	1	[	[	X
iajs-2958	158	2	𝜏𝑅(𝑋)]𝐶	𝜏𝑅(𝑋)]𝐶	X
iajs-2958	158	3	=	=	SYM
iajs-2958	158	4	𝑛𝑆𝑂(𝑋	𝑛𝑆𝑂(𝑋	PROPN
iajs-2958	158	5	,	,	PUNCT
iajs-2958	158	6	𝒲	𝒲	NOUN
iajs-2958	158	7	)	)	PUNCT
iajs-2958	158	8	=	=	SYM
iajs-2958	158	9	{	{	PUNCT
iajs-2958	158	10	𝜙	𝜙	NOUN
iajs-2958	158	11	,	,	PUNCT
iajs-2958	158	12	𝒲	𝒲	PROPN
iajs-2958	158	13	}	}	PUNCT
iajs-2958	158	14	.	.	PUNCT
iajs-2958	159	1	ii	ii	PROPN
iajs-2958	159	2	.	.	PROPN
iajs-2958	159	3	suppose	suppose	VERB
iajs-2958	159	4	that	that	SCONJ
iajs-2958	159	5	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	159	6	)	)	PUNCT
iajs-2958	159	7	=	=	SYM
iajs-2958	159	8	𝒲	𝒲	PROPN
iajs-2958	159	9	and	and	CCONJ
iajs-2958	159	10	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	159	11	)	)	PUNCT
iajs-2958	159	12	≠	≠	PROPN
iajs-2958	159	13	𝜙	𝜙	NOUN
iajs-2958	159	14	,	,	PUNCT
iajs-2958	159	15	then	then	ADV
iajs-2958	159	16	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	159	17	)	)	PUNCT
iajs-2958	159	18	=	=	PRON
iajs-2958	159	19	{	{	PUNCT
iajs-2958	159	20	𝜙	𝜙	NOUN
iajs-2958	159	21	,	,	PUNCT
iajs-2958	159	22	𝒲	𝒲	PROPN
iajs-2958	159	23	,	,	PUNCT
iajs-2958	159	24	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	159	25	)	)	PUNCT
iajs-2958	159	26	,	,	PUNCT
iajs-2958	159	27	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	159	28	)	)	PUNCT
iajs-2958	159	29	}	}	PUNCT
iajs-2958	159	30	=	=	PUNCT
iajs-2958	160	1	[	[	X
iajs-2958	160	2	𝜏𝑅(𝑋)]𝐶.	𝜏𝑅(𝑋)]𝐶.	NOUN
iajs-2958	160	3	then	then	ADV
iajs-2958	160	4	by	by	ADP
iajs-2958	160	5	theorem	theorem	NOUN
iajs-2958	160	6	9	9	NUM
iajs-2958	160	7	,	,	PUNCT
iajs-2958	160	8	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	160	9	,	,	PUNCT
iajs-2958	160	10	𝑋	𝑋	NOUN
iajs-2958	160	11	)	)	PUNCT
iajs-2958	160	12	=	=	SYM
iajs-2958	160	13	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	160	14	)	)	PUNCT
iajs-2958	160	15	.	.	PUNCT
iajs-2958	161	1	therefore	therefore	ADV
iajs-2958	161	2	,	,	PUNCT
iajs-2958	161	3	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
iajs-2958	161	4	)	)	PUNCT
iajs-2958	161	5	=	=	PUNCT
iajs-2958	161	6	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	161	7	,	,	PUNCT
iajs-2958	161	8	𝑋	𝑋	PROPN
iajs-2958	161	9	)	)	PUNCT
iajs-2958	161	10	.	.	PUNCT
iajs-2958	162	1	iii	iii	X
iajs-2958	162	2	.	.	PUNCT
iajs-2958	162	3	let	let	VERB
iajs-2958	162	4	𝐴	𝐴	PROPN
iajs-2958	162	5	∈	∈	PROPN
iajs-2958	162	6	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	162	7	,	,	PUNCT
iajs-2958	162	8	𝑋	𝑋	PROPN
iajs-2958	162	9	)	)	PUNCT
iajs-2958	162	10	.	.	PUNCT
iajs-2958	163	1	by	by	ADP
iajs-2958	163	2	theorem	theorem	NOUN
iajs-2958	163	3	9	9	NUM
iajs-2958	163	4	,	,	PUNCT
iajs-2958	163	5	𝜙	𝜙	NOUN
iajs-2958	163	6	,	,	PUNCT
iajs-2958	163	7	𝒲	𝒲	NOUN
iajs-2958	163	8	and	and	CCONJ
iajs-2958	163	9	any	any	DET
iajs-2958	163	10	subset	subset	ADJ
iajs-2958	163	11	𝐴	𝐴	PROPN
iajs-2958	163	12	for	for	ADP
iajs-2958	163	13	which	which	PRON
iajs-2958	163	14	containing	contain	VERB
iajs-2958	163	15	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	163	16	)	)	PUNCT
iajs-2958	163	17	are	be	AUX
iajs-2958	163	18	the	the	DET
iajs-2958	163	19	only	only	ADJ
iajs-2958	163	20	𝑛𝑆-open	𝑛𝑆-open	NOUN
iajs-2958	163	21	sets	set	VERB
iajs-2958	163	22	in	in	ADP
iajs-2958	163	23	𝒲.	𝒲.	PROPN
iajs-2958	163	24	if	if	SCONJ
iajs-2958	163	25	𝐴	𝐴	PROPN
iajs-2958	163	26	=	=	PUNCT
iajs-2958	163	27	𝜙	𝜙	NOUN
iajs-2958	163	28	or	or	CCONJ
iajs-2958	163	29	𝐴	𝐴	PROPN
iajs-2958	163	30	=	=	SYM
iajs-2958	163	31	𝒲	𝒲	PROPN
iajs-2958	163	32	,	,	PUNCT
iajs-2958	163	33	the	the	DET
iajs-2958	163	34	result	result	NOUN
iajs-2958	163	35	is	be	AUX
iajs-2958	163	36	clear	clear	ADJ
iajs-2958	163	37	.	.	PUNCT
iajs-2958	164	1	let	let	VERB
iajs-2958	164	2	𝜙	𝜙	NOUN
iajs-2958	164	3	,	,	PUNCT
iajs-2958	164	4	𝒲	𝒲	PROPN
iajs-2958	164	5	≠	≠	PROPN
iajs-2958	164	6	𝐴	𝐴	PROPN
iajs-2958	164	7	∈	∈	PROPN
iajs-2958	164	8	𝑛𝑆𝑂(𝑈	𝑛𝑆𝑂(𝑈	PROPN
iajs-2958	164	9	,	,	PUNCT
iajs-2958	164	10	𝑋	𝑋	PROPN
iajs-2958	164	11	)	)	PUNCT
iajs-2958	164	12	,	,	PUNCT
iajs-2958	164	13	then	then	ADV
iajs-2958	164	14	𝐴	𝐴	PROPN
iajs-2958	164	15	containing	contain	VERB
iajs-2958	164	16	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	164	17	)	)	PUNCT
iajs-2958	164	18	,	,	PUNCT
iajs-2958	164	19	but	but	CCONJ
iajs-2958	164	20	since	since	SCONJ
iajs-2958	164	21	[	[	X
iajs-2958	164	22	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	164	23	is	be	AUX
iajs-2958	164	24	the	the	DET
iajs-2958	164	25	only	only	ADJ
iajs-2958	164	26	non	non	ADJ
iajs-2958	164	27	-	-	ADJ
iajs-2958	164	28	empty	empty	ADJ
iajs-2958	164	29	proper	proper	ADJ
iajs-2958	164	30	ihjpas	ihjpa	NOUN
iajs-2958	164	31	.	.	PUNCT
iajs-2958	165	1	36(2)2023	36(2)2023	NUM
iajs-2958	165	2	310	310	NUM
iajs-2958	165	3	nano	nano	NOUN
iajs-2958	165	4	closed	close	VERB
iajs-2958	165	5	set	set	VERB
iajs-2958	165	6	in	in	ADP
iajs-2958	165	7	𝒲	𝒲	NOUN
iajs-2958	165	8	and	and	CCONJ
iajs-2958	165	9	[	[	X
iajs-2958	165	10	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	X
iajs-2958	165	11	⊈	⊈	PROPN
iajs-2958	165	12	𝐴	𝐴	PROPN
iajs-2958	165	13	for	for	ADP
iajs-2958	165	14	every	every	DET
iajs-2958	165	15	𝑥	𝑥	PROPN
iajs-2958	165	16	∈	∈	PROPN
iajs-2958	165	17	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	165	18	)	)	PUNCT
iajs-2958	165	19	⊇	⊇	PROPN
iajs-2958	165	20	𝐴	𝐴	PROPN
iajs-2958	165	21	,	,	PUNCT
iajs-2958	165	22	hence	hence	ADV
iajs-2958	165	23	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	165	24	,	,	PUNCT
iajs-2958	165	25	𝑋	𝑋	PROPN
iajs-2958	165	26	)	)	PUNCT
iajs-2958	165	27	=	=	PUNCT
iajs-2958	165	28	{	{	PUNCT
iajs-2958	165	29	𝜙	𝜙	NOUN
iajs-2958	165	30	,	,	PUNCT
iajs-2958	165	31	𝒲	𝒲	PROPN
iajs-2958	165	32	}	}	PUNCT
iajs-2958	165	33	.	.	PUNCT
iajs-2958	166	1	iv	iv	X
iajs-2958	166	2	.	.	PUNCT
iajs-2958	167	1	the	the	DET
iajs-2958	167	2	proof	proof	NOUN
iajs-2958	167	3	is	be	AUX
iajs-2958	167	4	similar	similar	ADJ
iajs-2958	167	5	to	to	ADP
iajs-2958	167	6	part	part	NOUN
iajs-2958	167	7	(	(	PUNCT
iajs-2958	167	8	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	167	9	)	)	PUNCT
iajs-2958	167	10	.	.	PUNCT
iajs-2958	168	1	v.	v.	CCONJ
iajs-2958	168	2	let	let	VERB
iajs-2958	168	3	𝐴	𝐴	PROPN
iajs-2958	168	4	∈	∈	PROPN
iajs-2958	168	5	𝑛𝑆𝑂(𝒲	𝑛𝑆𝑂(𝒲	PROPN
iajs-2958	168	6	,	,	PUNCT
iajs-2958	168	7	𝑋	𝑋	PROPN
iajs-2958	168	8	)	)	PUNCT
iajs-2958	168	9	.	.	PUNCT
iajs-2958	169	1	by	by	ADP
iajs-2958	169	2	theorem	theorem	NOUN
iajs-2958	169	3	9	9	NUM
iajs-2958	169	4	,	,	PUNCT
iajs-2958	169	5	𝜙	𝜙	NOUN
iajs-2958	169	6	,	,	PUNCT
iajs-2958	169	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	169	8	)	)	PUNCT
iajs-2958	169	9	,	,	PUNCT
iajs-2958	169	10	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	169	11	)	)	PUNCT
iajs-2958	169	12	,	,	PUNCT
iajs-2958	169	13	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	169	14	)	)	PUNCT
iajs-2958	169	15	,	,	PUNCT
iajs-2958	169	16	any	any	DET
iajs-2958	169	17	set	set	NOUN
iajs-2958	169	18	𝐺	𝐺	PROPN
iajs-2958	169	19	⊆	⊆	NUM
iajs-2958	169	20	𝒲	𝒲	NOUN
iajs-2958	169	21	for	for	ADP
iajs-2958	169	22	which	which	PRON
iajs-2958	169	23	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	169	24	)	)	PUNCT
iajs-2958	169	25	⊆	⊆	NUM
iajs-2958	169	26	𝐺	𝐺	PROPN
iajs-2958	169	27	,	,	PUNCT
iajs-2958	169	28	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	169	29	)	)	PUNCT
iajs-2958	169	30	∪	∪	NOUN
iajs-2958	169	31	𝐵	𝐵	PROPN
iajs-2958	169	32	and	and	CCONJ
iajs-2958	169	33	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	169	34	)	)	PUNCT
iajs-2958	169	35	∪	∪	ADP
iajs-2958	169	36	𝐵	𝐵	PROPN
iajs-2958	169	37	are	be	AUX
iajs-2958	169	38	the	the	DET
iajs-2958	169	39	only	only	ADJ
iajs-2958	169	40	𝑛𝑆-open	𝑛𝑆-open	NOUN
iajs-2958	169	41	sets	set	NOUN
iajs-2958	169	42	in	in	ADP
iajs-2958	169	43	𝒲	𝒲	NOUN
iajs-2958	169	44	where	where	SCONJ
iajs-2958	169	45	𝐵	𝐵	PROPN
iajs-2958	169	46	⊆	⊆	NUM
iajs-2958	170	1	[	[	X
iajs-2958	170	2	𝑈𝑅(𝑋)]𝐶.	𝑈𝑅(𝑋)]𝐶.	NOUN
iajs-2958	170	3	it	it	PRON
iajs-2958	170	4	is	be	AUX
iajs-2958	170	5	clear	clear	ADJ
iajs-2958	170	6	𝜙	𝜙	NOUN
iajs-2958	170	7	and	and	CCONJ
iajs-2958	170	8	𝒲	𝒲	NOUN
iajs-2958	170	9	are	be	AUX
iajs-2958	170	10	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	170	11	sets	set	NOUN
iajs-2958	170	12	in	in	ADP
iajs-2958	170	13	𝒲.	𝒲.	PROPN
iajs-2958	170	14	if	if	SCONJ
iajs-2958	170	15	𝐴	𝐴	PROPN
iajs-2958	170	16	=	=	SYM
iajs-2958	170	17	𝐿𝑅(𝑋	𝐿𝑅(𝑋	PROPN
iajs-2958	170	18	)	)	PUNCT
iajs-2958	170	19	,	,	PUNCT
iajs-2958	170	20	then	then	ADV
iajs-2958	170	21	𝐴	𝐴	PROPN
iajs-2958	170	22	is	be	AUX
iajs-2958	170	23	not	not	PART
iajs-2958	170	24	𝑛𝑆𝐶open	𝑛𝑆𝐶open	X
iajs-2958	170	25	set	set	VERB
iajs-2958	170	26	,	,	PUNCT
iajs-2958	170	27	since	since	SCONJ
iajs-2958	170	28	every	every	DET
iajs-2958	170	29	non	non	ADJ
iajs-2958	170	30	-	-	ADJ
iajs-2958	170	31	empty	empty	ADJ
iajs-2958	170	32	proper	proper	ADJ
iajs-2958	170	33	nano	nano	NOUN
iajs-2958	170	34	closed	close	VERB
iajs-2958	170	35	set	set	NOUN
iajs-2958	170	36	containing	contain	VERB
iajs-2958	170	37	[	[	PUNCT
iajs-2958	170	38	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	170	39	and	and	CCONJ
iajs-2958	170	40	[	[	X
iajs-2958	170	41	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	X
iajs-2958	170	42	⊈	⊈	X
iajs-2958	170	43	𝐴.	𝐴.	NOUN
iajs-2958	170	44	if	if	SCONJ
iajs-2958	170	45	𝐴	𝐴	PROPN
iajs-2958	170	46	=	=	SYM
iajs-2958	170	47	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	170	48	)	)	PUNCT
iajs-2958	170	49	,	,	PUNCT
iajs-2958	170	50	then	then	ADV
iajs-2958	170	51	𝐴	𝐴	PROPN
iajs-2958	170	52	is	be	AUX
iajs-2958	170	53	not	not	PART
iajs-2958	170	54	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	170	55	set	set	NOUN
iajs-2958	170	56	,	,	PUNCT
iajs-2958	170	57	since	since	SCONJ
iajs-2958	170	58	every	every	DET
iajs-2958	170	59	non	non	ADJ
iajs-2958	170	60	-	-	ADJ
iajs-2958	170	61	empty	empty	ADJ
iajs-2958	170	62	proper	proper	ADJ
iajs-2958	170	63	nano	nano	NOUN
iajs-2958	170	64	closed	close	VERB
iajs-2958	170	65	set	set	NOUN
iajs-2958	170	66	containing	contain	VERB
iajs-2958	170	67	[	[	PUNCT
iajs-2958	170	68	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	170	69	and	and	CCONJ
iajs-2958	170	70	[	[	X
iajs-2958	170	71	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	X
iajs-2958	170	72	⊈	⊈	X
iajs-2958	170	73	𝐴.	𝐴.	NOUN
iajs-2958	170	74	if	if	SCONJ
iajs-2958	170	75	𝐴	𝐴	PROPN
iajs-2958	170	76	=	=	SYM
iajs-2958	170	77	𝑈𝑅(𝑋	𝑈𝑅(𝑋	PROPN
iajs-2958	170	78	)	)	PUNCT
iajs-2958	170	79	,	,	PUNCT
iajs-2958	170	80	then	then	ADV
iajs-2958	170	81	𝐴	𝐴	PROPN
iajs-2958	170	82	is	be	AUX
iajs-2958	170	83	not	not	PART
iajs-2958	170	84	𝑛𝑆𝐶open	𝑛𝑆𝐶open	X
iajs-2958	170	85	set	set	VERB
iajs-2958	170	86	,	,	PUNCT
iajs-2958	170	87	since	since	SCONJ
iajs-2958	170	88	every	every	DET
iajs-2958	170	89	non	non	ADJ
iajs-2958	170	90	-	-	ADJ
iajs-2958	170	91	empty	empty	ADJ
iajs-2958	170	92	proper	proper	ADJ
iajs-2958	170	93	nano	nano	NOUN
iajs-2958	170	94	closed	close	VERB
iajs-2958	170	95	set	set	NOUN
iajs-2958	170	96	containing	contain	VERB
iajs-2958	170	97	[	[	PUNCT
iajs-2958	170	98	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	170	99	and	and	CCONJ
iajs-2958	170	100	[	[	X
iajs-2958	170	101	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	X
iajs-2958	170	102	⊈	⊈	X
iajs-2958	170	103	𝐴.	𝐴.	NOUN
iajs-2958	170	104	if	if	SCONJ
iajs-2958	170	105	𝐴	𝐴	PROPN
iajs-2958	170	106	containing	contain	VERB
iajs-2958	170	107	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	170	108	)	)	PUNCT
iajs-2958	170	109	,	,	PUNCT
iajs-2958	170	110	then	then	ADV
iajs-2958	170	111	𝐴	𝐴	PROPN
iajs-2958	170	112	is	be	AUX
iajs-2958	170	113	not	not	PART
iajs-2958	170	114	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	170	115	set	set	NOUN
iajs-2958	170	116	,	,	PUNCT
iajs-2958	170	117	since	since	SCONJ
iajs-2958	170	118	every	every	DET
iajs-2958	170	119	non	non	ADJ
iajs-2958	170	120	-	-	ADJ
iajs-2958	170	121	empty	empty	ADJ
iajs-2958	170	122	proper	proper	ADJ
iajs-2958	170	123	nano	nano	NOUN
iajs-2958	170	124	closed	close	VERB
iajs-2958	170	125	set	set	NOUN
iajs-2958	170	126	containing	contain	VERB
iajs-2958	170	127	[	[	PUNCT
iajs-2958	170	128	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	170	129	and	and	CCONJ
iajs-2958	170	130	[	[	X
iajs-2958	170	131	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	X
iajs-2958	170	132	⊈	⊈	PROPN
iajs-2958	170	133	𝐴	𝐴	PROPN
iajs-2958	170	134	⊇	⊇	PROPN
iajs-2958	170	135	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
iajs-2958	170	136	)	)	PUNCT
iajs-2958	170	137	.	.	PUNCT
iajs-2958	171	1	if	if	SCONJ
iajs-2958	171	2	𝐴	𝐴	PROPN
iajs-2958	171	3	=	=	SYM
iajs-2958	171	4	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	171	5	)	)	PUNCT
iajs-2958	171	6	∪	∪	PROPN
iajs-2958	171	7	𝐵	𝐵	NOUN
iajs-2958	171	8	where	where	SCONJ
iajs-2958	171	9	𝐵	𝐵	PROPN
iajs-2958	171	10	⊂	⊂	PROPN
iajs-2958	171	11	[	[	X
iajs-2958	171	12	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	171	13	,	,	PUNCT
iajs-2958	171	14	then	then	ADV
iajs-2958	171	15	𝐴	𝐴	PROPN
iajs-2958	171	16	is	be	AUX
iajs-2958	171	17	not	not	PART
iajs-2958	171	18	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	171	19	set	set	NOUN
iajs-2958	171	20	,	,	PUNCT
iajs-2958	171	21	since	since	SCONJ
iajs-2958	171	22	every	every	DET
iajs-2958	171	23	non	non	ADJ
iajs-2958	171	24	-	-	ADJ
iajs-2958	171	25	empty	empty	ADJ
iajs-2958	171	26	proper	proper	ADJ
iajs-2958	171	27	nano	nano	NOUN
iajs-2958	171	28	closed	close	VERB
iajs-2958	171	29	set	set	NOUN
iajs-2958	171	30	containing	contain	VERB
iajs-2958	171	31	[	[	PUNCT
iajs-2958	171	32	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	171	33	and	and	CCONJ
iajs-2958	171	34	[	[	X
iajs-2958	171	35	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	X
iajs-2958	171	36	⊈	⊈	X
iajs-2958	171	37	𝐴.	𝐴.	NOUN
iajs-2958	171	38	if	if	SCONJ
iajs-2958	171	39	𝐴	𝐴	PROPN
iajs-2958	171	40	=	=	SYM
iajs-2958	171	41	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	171	42	)	)	PUNCT
iajs-2958	171	43	∪	∪	NOUN
iajs-2958	171	44	𝐵	𝐵	NOUN
iajs-2958	171	45	where	where	SCONJ
iajs-2958	171	46	𝐵	𝐵	PROPN
iajs-2958	171	47	⊂	⊂	PROPN
iajs-2958	171	48	[	[	X
iajs-2958	171	49	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	171	50	,	,	PUNCT
iajs-2958	171	51	similarly	similarly	ADV
iajs-2958	171	52	𝐴	𝐴	PROPN
iajs-2958	171	53	is	be	AUX
iajs-2958	171	54	not	not	PART
iajs-2958	171	55	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	171	56	set	set	NOUN
iajs-2958	171	57	.	.	PUNCT
iajs-2958	172	1	if	if	SCONJ
iajs-2958	172	2	𝐴	𝐴	PROPN
iajs-2958	172	3	=	=	SYM
iajs-2958	172	4	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	172	5	)	)	PUNCT
iajs-2958	172	6	∪	∪	NOUN
iajs-2958	172	7	𝐵	𝐵	NOUN
iajs-2958	172	8	where	where	SCONJ
iajs-2958	172	9	𝐵	𝐵	NOUN
iajs-2958	172	10	=	=	PUNCT
iajs-2958	173	1	[	[	X
iajs-2958	173	2	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	173	3	,	,	PUNCT
iajs-2958	173	4	then	then	ADV
iajs-2958	173	5	𝐴	𝐴	PROPN
iajs-2958	173	6	is	be	AUX
iajs-2958	173	7	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	173	8	set	set	NOUN
iajs-2958	173	9	,	,	PUNCT
iajs-2958	173	10	since	since	SCONJ
iajs-2958	173	11	every	every	DET
iajs-2958	173	12	non	non	ADJ
iajs-2958	173	13	-	-	ADJ
iajs-2958	173	14	empty	empty	ADJ
iajs-2958	173	15	proper	proper	ADJ
iajs-2958	173	16	nano	nano	NOUN
iajs-2958	173	17	closed	close	VERB
iajs-2958	173	18	set	set	NOUN
iajs-2958	173	19	containing	contain	VERB
iajs-2958	173	20	[	[	PUNCT
iajs-2958	173	21	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	173	22	and	and	CCONJ
iajs-2958	173	23	[	[	X
iajs-2958	173	24	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	173	25	⊆	⊆	NUM
iajs-2958	173	26	𝐴	𝐴	PROPN
iajs-2958	173	27	for	for	ADP
iajs-2958	173	28	every	every	DET
iajs-2958	173	29	𝑥	𝑥	PROPN
iajs-2958	173	30	∈	∈	PROPN
iajs-2958	173	31	𝐴.	𝐴.	NOUN
iajs-2958	173	32	if	if	SCONJ
iajs-2958	173	33	𝐴	𝐴	PROPN
iajs-2958	173	34	=	=	SYM
iajs-2958	173	35	𝐵𝑅(𝑋	𝐵𝑅(𝑋	PROPN
iajs-2958	173	36	)	)	PUNCT
iajs-2958	173	37	∪	∪	NOUN
iajs-2958	173	38	𝐵	𝐵	NOUN
iajs-2958	173	39	where	where	SCONJ
iajs-2958	173	40	𝐵	𝐵	NOUN
iajs-2958	173	41	=	=	PUNCT
iajs-2958	174	1	[	[	X
iajs-2958	174	2	𝑈𝑅(𝑋)]𝐶	𝑈𝑅(𝑋)]𝐶	NOUN
iajs-2958	174	3	,	,	PUNCT
iajs-2958	174	4	similarly	similarly	ADV
iajs-2958	174	5	𝐴	𝐴	PROPN
iajs-2958	174	6	is	be	AUX
iajs-2958	174	7	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	174	8	set	set	NOUN
iajs-2958	174	9	.	.	PUNCT
iajs-2958	175	1	therefore	therefore	ADV
iajs-2958	175	2	,	,	PUNCT
iajs-2958	175	3	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	175	4	,	,	PUNCT
iajs-2958	175	5	𝑋	𝑋	NOUN
iajs-2958	175	6	)	)	PUNCT
iajs-2958	175	7	=	=	PUNCT
iajs-2958	175	8	{	{	PUNCT
iajs-2958	175	9	𝜙	𝜙	NOUN
iajs-2958	175	10	,	,	PUNCT
iajs-2958	175	11	𝒲	𝒲	PROPN
iajs-2958	175	12	,	,	PUNCT
iajs-2958	175	13	[	[	NOUN
iajs-2958	175	14	𝐵𝑅(𝑋	𝐵𝑅(𝑋	NOUN
iajs-2958	175	15	)	)	PUNCT
iajs-2958	175	16	∪	∪	X
iajs-2958	175	17	𝐵	𝐵	PROPN
iajs-2958	175	18	]	]	PUNCT
iajs-2958	175	19	,	,	PUNCT
iajs-2958	175	20	[	[	X
iajs-2958	175	21	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
iajs-2958	175	22	)	)	PUNCT
iajs-2958	175	23	∪	∪	X
iajs-2958	175	24	𝐵	𝐵	PROPN
iajs-2958	175	25	]	]	PUNCT
iajs-2958	175	26	}	}	PUNCT
iajs-2958	175	27	,	,	PUNCT
iajs-2958	175	28	where	where	SCONJ
iajs-2958	175	29	𝐵	𝐵	NOUN
iajs-2958	175	30	=	=	PUNCT
iajs-2958	176	1	[	[	X
iajs-2958	176	2	𝑈𝑅(𝑋)]𝐶.	𝑈𝑅(𝑋)]𝐶.	NOUN
iajs-2958	176	3	proposition	proposition	NOUN
iajs-2958	176	4	3.9	3.9	NUM
iajs-2958	176	5	.	.	PUNCT
iajs-2958	177	1	let	let	VERB
iajs-2958	177	2	(	(	PUNCT
iajs-2958	177	3	𝒲	𝒲	NOUN
iajs-2958	177	4	,	,	PUNCT
iajs-2958	177	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	177	6	)	)	PUNCT
iajs-2958	177	7	)	)	PUNCT
iajs-2958	177	8	be	be	AUX
iajs-2958	177	9	a	a	DET
iajs-2958	177	10	𝑁𝑇𝑆	𝑁𝑇𝑆	NOUN
iajs-2958	177	11	and	and	CCONJ
iajs-2958	177	12	𝐾	𝐾	PROPN
iajs-2958	177	13	be	be	VERB
iajs-2958	177	14	any	any	DET
iajs-2958	177	15	subset	subset	NOUN
iajs-2958	177	16	of	of	ADP
iajs-2958	177	17	𝑈	𝑈	PROPN
iajs-2958	177	18	:	:	PUNCT
iajs-2958	177	19	i.	i.	NOUN
iajs-2958	177	20	if	if	SCONJ
iajs-2958	177	21	𝐾	𝐾	PROPN
iajs-2958	177	22	is	be	AUX
iajs-2958	177	23	𝑛𝜃-open	𝑛𝜃-open	ADJ
iajs-2958	177	24	set	set	NOUN
iajs-2958	177	25	,	,	PUNCT
iajs-2958	177	26	then	then	ADV
iajs-2958	177	27	𝐾	𝐾	PROPN
iajs-2958	177	28	is	be	AUX
iajs-2958	177	29	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	177	30	.	.	PUNCT
iajs-2958	178	1	ii	ii	X
iajs-2958	178	2	.	.	PUNCT
iajs-2958	179	1	if	if	SCONJ
iajs-2958	179	2	𝐾	𝐾	PROPN
iajs-2958	179	3	is	be	AUX
iajs-2958	179	4	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	179	5	set	set	NOUN
iajs-2958	179	6	,	,	PUNCT
iajs-2958	179	7	then	then	ADV
iajs-2958	179	8	𝐾	𝐾	PROPN
iajs-2958	179	9	is	be	AUX
iajs-2958	179	10	𝑛𝑆𝛽-open	𝑛𝑆𝛽-open	NOUN
iajs-2958	179	11	.	.	PUNCT
iajs-2958	180	1	iii	iii	X
iajs-2958	180	2	.	.	PUNCT
iajs-2958	181	1	if	if	SCONJ
iajs-2958	181	2	𝐾	𝐾	PROPN
iajs-2958	181	3	is	be	AUX
iajs-2958	181	4	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	181	5	set	set	NOUN
iajs-2958	181	6	,	,	PUNCT
iajs-2958	181	7	then	then	ADV
iajs-2958	181	8	𝐾	𝐾	PROPN
iajs-2958	181	9	is	be	AUX
iajs-2958	181	10	𝑛𝛽-open	𝑛𝛽-open	ADJ
iajs-2958	181	11	.	.	PUNCT
iajs-2958	182	1	iv	iv	X
iajs-2958	182	2	.	.	PUNCT
iajs-2958	183	1	if	if	SCONJ
iajs-2958	183	2	𝐾	𝐾	PROPN
iajs-2958	183	3	is	be	AUX
iajs-2958	183	4	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	183	5	set	set	NOUN
iajs-2958	183	6	,	,	PUNCT
iajs-2958	183	7	then	then	ADV
iajs-2958	183	8	𝐾	𝐾	PROPN
iajs-2958	183	9	is	be	AUX
iajs-2958	183	10	𝑛𝜆-open	𝑛𝜆-open	ADJ
iajs-2958	183	11	.	.	PUNCT
iajs-2958	184	1	v.	v.	INTJ
iajs-2958	184	2	if	if	SCONJ
iajs-2958	184	3	𝐾	𝐾	PROPN
iajs-2958	184	4	is	be	AUX
iajs-2958	184	5	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	184	6	set	set	NOUN
iajs-2958	184	7	,	,	PUNCT
iajs-2958	184	8	then	then	ADV
iajs-2958	184	9	𝐾	𝐾	PROPN
iajs-2958	184	10	is	be	AUX
iajs-2958	184	11	𝑛𝛿𝛽-open	𝑛𝛿𝛽-open	ADJ
iajs-2958	184	12	.	.	PUNCT
iajs-2958	185	1	proof	proof	NOUN
iajs-2958	185	2	.	.	PUNCT
iajs-2958	186	1	obvious	obvious	ADJ
iajs-2958	186	2	.	.	PUNCT
iajs-2958	187	1	4	4	X
iajs-2958	187	2	.	.	X
iajs-2958	187	3	nano	nano	NOUN
iajs-2958	187	4	𝑺𝑪-operators	𝑺𝑪-operators	PROPN
iajs-2958	187	5	definition	definition	NOUN
iajs-2958	187	6	4.1	4.1	NUM
iajs-2958	187	7	.	.	PUNCT
iajs-2958	188	1	a	a	DET
iajs-2958	188	2	subset	subset	NOUN
iajs-2958	188	3	𝑁	𝑁	NOUN
iajs-2958	188	4	of	of	ADP
iajs-2958	188	5	a	a	DET
iajs-2958	188	6	𝑁𝑇𝑆	𝑁𝑇𝑆	NOUN
iajs-2958	188	7	(	(	PUNCT
iajs-2958	188	8	𝒲	𝒲	PROPN
iajs-2958	188	9	,	,	PUNCT
iajs-2958	188	10	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	188	11	)	)	PUNCT
iajs-2958	188	12	)	)	PUNCT
iajs-2958	188	13	is	be	AUX
iajs-2958	188	14	said	say	VERB
iajs-2958	188	15	to	to	PART
iajs-2958	188	16	be	be	AUX
iajs-2958	188	17	a	a	DET
iajs-2958	188	18	𝑛𝑆𝐶-neighborhood	𝑛𝑆𝐶-neighborhood	NOUN
iajs-2958	188	19	of	of	ADP
iajs-2958	188	20	a	a	DET
iajs-2958	188	21	subset	subset	ADJ
iajs-2958	188	22	𝐴	𝐴	PROPN
iajs-2958	188	23	of	of	ADP
iajs-2958	188	24	𝑊	𝑊	PROPN
iajs-2958	188	25	,	,	PUNCT
iajs-2958	188	26	if	if	SCONJ
iajs-2958	188	27	there	there	PRON
iajs-2958	188	28	exists	exist	VERB
iajs-2958	188	29	a	a	DET
iajs-2958	188	30	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	188	31	set	set	NOUN
iajs-2958	188	32	𝐺	𝐺	PROPN
iajs-2958	188	33	such	such	ADJ
iajs-2958	188	34	that	that	SCONJ
iajs-2958	188	35	𝐴	𝐴	PROPN
iajs-2958	188	36	⊆	⊆	NUM
iajs-2958	188	37	𝐺	𝐺	PROPN
iajs-2958	188	38	⊆	⊆	NUM
iajs-2958	188	39	𝑁	𝑁	PROPN
iajs-2958	188	40	,	,	PUNCT
iajs-2958	188	41	and	and	CCONJ
iajs-2958	188	42	denoted	denote	VERB
iajs-2958	188	43	by	by	ADP
iajs-2958	188	44	𝑛𝑆𝐶-neighborhood	𝑛𝑆𝐶-neighborhood	NOUN
iajs-2958	188	45	.	.	PUNCT
iajs-2958	189	1	definition	definition	NOUN
iajs-2958	189	2	4.2	4.2	NUM
iajs-2958	189	3	.	.	PUNCT
iajs-2958	190	1	a	a	DET
iajs-2958	190	2	point	point	NOUN
iajs-2958	190	3	𝑥	𝑥	PRON
iajs-2958	190	4	∈	∈	NOUN
iajs-2958	190	5	𝒲	𝒲	PROPN
iajs-2958	190	6	is	be	AUX
iajs-2958	190	7	called	call	VERB
iajs-2958	190	8	a	a	DET
iajs-2958	190	9	𝑛𝑆𝐶-interior	𝑛𝑆𝐶-interior	PROPN
iajs-2958	190	10	point	point	NOUN
iajs-2958	190	11	of	of	ADP
iajs-2958	190	12	𝐴	𝐴	PROPN
iajs-2958	190	13	⊆	⊆	PROPN
iajs-2958	190	14	𝑊	𝑊	PROPN
iajs-2958	190	15	,	,	PUNCT
iajs-2958	190	16	if	if	SCONJ
iajs-2958	190	17	∃	∃	PROPN
iajs-2958	190	18	a	a	DET
iajs-2958	190	19	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	190	20	set	set	NOUN
iajs-2958	190	21	𝐺	𝐺	PROPN
iajs-2958	190	22	containing	contain	VERB
iajs-2958	190	23	𝑥	𝑥	PRON
iajs-2958	190	24	such	such	ADJ
iajs-2958	190	25	that	that	SCONJ
iajs-2958	190	26	𝑥	𝑥	PROPN
iajs-2958	190	27	∈	∈	PROPN
iajs-2958	190	28	𝐺	𝐺	NOUN
iajs-2958	190	29	⊆	⊆	NUM
iajs-2958	190	30	𝐴.	𝐴.	PROPN
iajs-2958	190	31	the	the	DET
iajs-2958	190	32	set	set	NOUN
iajs-2958	190	33	of	of	ADP
iajs-2958	190	34	all	all	DET
iajs-2958	190	35	𝑛𝑆𝐶-interior	𝑛𝑆𝐶-interior	ADJ
iajs-2958	190	36	points	point	NOUN
iajs-2958	190	37	of	of	ADP
iajs-2958	190	38	𝐴	𝐴	PROPN
iajs-2958	190	39	is	be	AUX
iajs-2958	190	40	called	call	VERB
iajs-2958	190	41	𝑛𝑆𝐶-interior	𝑛𝑆𝐶-interior	PROPN
iajs-2958	190	42	of	of	ADP
iajs-2958	190	43	𝐴	𝐴	PROPN
iajs-2958	190	44	and	and	CCONJ
iajs-2958	190	45	denoted	denote	VERB
iajs-2958	190	46	by	by	ADP
iajs-2958	190	47	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	190	48	)	)	PUNCT
iajs-2958	190	49	.	.	PUNCT
iajs-2958	191	1	theorem	theorem	VERB
iajs-2958	191	2	4.3	4.3	NUM
iajs-2958	191	3	.	.	PUNCT
iajs-2958	192	1	let	let	VERB
iajs-2958	192	2	𝐴	𝐴	PROPN
iajs-2958	192	3	be	be	AUX
iajs-2958	192	4	any	any	DET
iajs-2958	192	5	subset	subset	NOUN
iajs-2958	192	6	of	of	ADP
iajs-2958	192	7	a	a	DET
iajs-2958	192	8	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	192	9	(	(	PUNCT
iajs-2958	192	10	𝒲	𝒲	PROPN
iajs-2958	192	11	,	,	PUNCT
iajs-2958	192	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	192	13	)	)	PUNCT
iajs-2958	192	14	)	)	PUNCT
iajs-2958	192	15	.	.	PUNCT
iajs-2958	193	1	if	if	SCONJ
iajs-2958	193	2	a	a	DET
iajs-2958	193	3	point	point	NOUN
iajs-2958	193	4	𝑥	𝑥	X
iajs-2958	193	5	∈	∈	PROPN
iajs-2958	193	6	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	193	7	)	)	PUNCT
iajs-2958	193	8	,	,	PUNCT
iajs-2958	193	9	then	then	ADV
iajs-2958	193	10	∃	∃	PROPN
iajs-2958	193	11	a	a	DET
iajs-2958	193	12	nano	nano	NOUN
iajs-2958	193	13	closed	close	VERB
iajs-2958	193	14	set	set	VERB
iajs-2958	193	15	𝐹	𝐹	PROPN
iajs-2958	193	16	containing	contain	VERB
iajs-2958	193	17	𝑥	𝑥	PRON
iajs-2958	193	18	such	such	ADJ
iajs-2958	193	19	that	that	SCONJ
iajs-2958	193	20	𝐹	𝐹	PROPN
iajs-2958	193	21	⊆	⊆	NUM
iajs-2958	193	22	𝐴.	𝐴.	PROPN
iajs-2958	193	23	proof	proof	NOUN
iajs-2958	193	24	.	.	PUNCT
iajs-2958	194	1	suppose	suppose	VERB
iajs-2958	194	2	that	that	SCONJ
iajs-2958	194	3	𝑥	𝑥	PROPN
iajs-2958	194	4	∈	∈	PROPN
iajs-2958	194	5	𝑛𝑆𝐶𝑖𝑛𝑡	𝑛𝑆𝐶𝑖𝑛𝑡	PROPN
iajs-2958	194	6	𝐴	𝐴	PROPN
iajs-2958	194	7	,	,	PUNCT
iajs-2958	194	8	then	then	ADV
iajs-2958	194	9	∃	∃	PROPN
iajs-2958	194	10	a	a	DET
iajs-2958	194	11	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	194	12	set	set	NOUN
iajs-2958	194	13	𝐺	𝐺	PROPN
iajs-2958	194	14	containing	contain	VERB
iajs-2958	194	15	𝑥	𝑥	PROPN
iajs-2958	194	16	such	such	ADJ
iajs-2958	194	17	that	that	SCONJ
iajs-2958	194	18	𝐺	𝐺	PROPN
iajs-2958	194	19	⊆	⊆	NUM
iajs-2958	194	20	𝐴.	𝐴.	PROPN
iajs-2958	194	21	since	since	SCONJ
iajs-2958	194	22	𝐺	𝐺	PROPN
iajs-2958	194	23	∈	∈	PROPN
iajs-2958	194	24	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	194	25	,	,	PUNCT
iajs-2958	194	26	𝑋	𝑋	PROPN
iajs-2958	194	27	)	)	PUNCT
iajs-2958	194	28	,	,	PUNCT
iajs-2958	194	29	then	then	ADV
iajs-2958	194	30	∃	∃	PROPN
iajs-2958	194	31	a	a	DET
iajs-2958	194	32	nano	nano	NOUN
iajs-2958	194	33	closed	close	VERB
iajs-2958	194	34	set	set	VERB
iajs-2958	194	35	𝐹	𝐹	PROPN
iajs-2958	194	36	containing	contain	VERB
iajs-2958	194	37	𝑥	𝑥	PRON
iajs-2958	194	38	such	such	ADJ
iajs-2958	194	39	that	that	SCONJ
iajs-2958	194	40	𝐹	𝐹	PROPN
iajs-2958	194	41	⊆	⊆	NUM
iajs-2958	194	42	𝐺	𝐺	NOUN
iajs-2958	194	43	⊆	⊆	NUM
iajs-2958	194	44	𝐴.	𝐴.	NOUN
iajs-2958	194	45	hence	hence	ADV
iajs-2958	194	46	𝑥	𝑥	ADP
iajs-2958	194	47	∈	∈	NOUN
iajs-2958	194	48	𝐹	𝐹	PROPN
iajs-2958	194	49	⊆	⊆	NUM
iajs-2958	194	50	𝐴.	𝐴.	PROPN
iajs-2958	194	51	theorem	theorem	VERB
iajs-2958	194	52	4.4	4.4	NUM
iajs-2958	194	53	.	.	PUNCT
iajs-2958	195	1	let	let	VERB
iajs-2958	195	2	𝐴	𝐴	PROPN
iajs-2958	195	3	be	be	AUX
iajs-2958	195	4	any	any	DET
iajs-2958	195	5	subset	subset	NOUN
iajs-2958	195	6	of	of	ADP
iajs-2958	195	7	a	a	DET
iajs-2958	195	8	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	195	9	(	(	PUNCT
iajs-2958	195	10	𝒲	𝒲	PROPN
iajs-2958	195	11	,	,	PUNCT
iajs-2958	195	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	195	13	)	)	PUNCT
iajs-2958	195	14	)	)	PUNCT
iajs-2958	195	15	,	,	PUNCT
iajs-2958	195	16	then	then	ADV
iajs-2958	195	17	:	:	PUNCT
iajs-2958	195	18	i.	i.	PROPN
iajs-2958	195	19	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	195	20	)	)	PUNCT
iajs-2958	195	21	⊆	⊆	NUM
iajs-2958	195	22	𝐴.	𝐴.	PROPN
iajs-2958	195	23	ii	ii	PROPN
iajs-2958	195	24	.	.	PUNCT
iajs-2958	196	1	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	196	2	)	)	PUNCT
iajs-2958	197	1	=	=	SYM
iajs-2958	197	2	∪	∪	X
iajs-2958	197	3	{	{	PUNCT
iajs-2958	197	4	𝐺	𝐺	NOUN
iajs-2958	197	5	:	:	PUNCT
iajs-2958	197	6	𝐺	𝐺	PROPN
iajs-2958	197	7	𝑖𝑠	𝑖𝑠	NOUN
iajs-2958	197	8	𝑛𝑆𝐶open	𝑛𝑆𝐶open	X
iajs-2958	197	9	and	and	CCONJ
iajs-2958	197	10	𝐺	𝐺	PROPN
iajs-2958	197	11	⊆	⊆	NUM
iajs-2958	197	12	𝐴	𝐴	PROPN
iajs-2958	197	13	}	}	PUNCT
iajs-2958	197	14	iii	iii	PROPN
iajs-2958	197	15	.	.	PUNCT
iajs-2958	198	1	𝐴	𝐴	PROPN
iajs-2958	198	2	is	be	AUX
iajs-2958	198	3	𝑛𝑆𝐶open	𝑛𝑆𝐶open	X
iajs-2958	198	4	if	if	SCONJ
iajs-2958	198	5	and	and	CCONJ
iajs-2958	198	6	only	only	ADV
iajs-2958	198	7	if	if	SCONJ
iajs-2958	198	8	𝐴	𝐴	PROPN
iajs-2958	198	9	=	=	SYM
iajs-2958	198	10	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	198	11	)	)	PUNCT
iajs-2958	198	12	.	.	PUNCT
iajs-2958	199	1	ihjpas	ihjpas	PROPN
iajs-2958	199	2	.	.	PUNCT
iajs-2958	200	1	36(2)2023	36(2)2023	NUM
iajs-2958	200	2	311	311	NUM
iajs-2958	200	3	iv	iv	NOUN
iajs-2958	200	4	.	.	PUNCT
iajs-2958	201	1	𝑛𝑆𝐶𝑖𝑛𝑡	𝑛𝑆𝐶𝑖𝑛𝑡	PROPN
iajs-2958	201	2	(	(	PUNCT
iajs-2958	201	3	𝑛𝑆𝛽𝑖𝑛𝑡(𝐴	𝑛𝑆𝛽𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	201	4	)	)	PUNCT
iajs-2958	201	5	)	)	PUNCT
iajs-2958	202	1	=	=	PUNCT
iajs-2958	202	2	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	202	3	)	)	PUNCT
iajs-2958	202	4	.	.	PUNCT
iajs-2958	203	1	v.	v.	ADP
iajs-2958	203	2	𝑛𝑆𝐶𝑖𝑛𝑡(𝜙	𝑛𝑆𝐶𝑖𝑛𝑡(𝜙	NOUN
iajs-2958	203	3	)	)	PUNCT
iajs-2958	203	4	=	=	SYM
iajs-2958	203	5	𝜙	𝜙	NOUN
iajs-2958	203	6	and	and	CCONJ
iajs-2958	203	7	𝑛𝑆𝐶𝑖𝑛𝑡(𝒲	𝑛𝑆𝐶𝑖𝑛𝑡(𝒲	NOUN
iajs-2958	203	8	)	)	PUNCT
iajs-2958	203	9	=	=	SYM
iajs-2958	203	10	𝒲.	𝒲.	PROPN
iajs-2958	203	11	proof	proof	NOUN
iajs-2958	203	12	.	.	PUNCT
iajs-2958	204	1	i.	i.	PROPN
iajs-2958	204	2	follows	follow	VERB
iajs-2958	204	3	form	form	NOUN
iajs-2958	204	4	definition	definition	NOUN
iajs-2958	204	5	.	.	PUNCT
iajs-2958	205	1	ii	ii	PROPN
iajs-2958	205	2	.	.	PUNCT
iajs-2958	206	1	𝑥	𝑥	PROPN
iajs-2958	206	2	∈	∈	PROPN
iajs-2958	206	3	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	206	4	)	)	PUNCT
iajs-2958	206	5	,	,	PUNCT
iajs-2958	206	6	then	then	ADV
iajs-2958	206	7	𝐺	𝐺	PROPN
iajs-2958	206	8	⊆	⊆	NUM
iajs-2958	206	9	𝐴	𝐴	PROPN
iajs-2958	206	10	for	for	ADP
iajs-2958	206	11	some	some	DET
iajs-2958	206	12	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	206	13	set	set	NOUN
iajs-2958	206	14	𝐺	𝐺	PROPN
iajs-2958	206	15	such	such	ADJ
iajs-2958	206	16	that	that	SCONJ
iajs-2958	206	17	𝑥	𝑥	PROPN
iajs-2958	206	18	∈	∈	PROPN
iajs-2958	206	19	𝐺.	𝐺.	NOUN
iajs-2958	206	20	therefore	therefore	ADV
iajs-2958	206	21	,	,	PUNCT
iajs-2958	206	22	𝑥	𝑥	DET
iajs-2958	206	23	∈∪	∈∪	NOUN
iajs-2958	206	24	{	{	PUNCT
iajs-2958	206	25	𝐺	𝐺	NOUN
iajs-2958	206	26	:	:	PUNCT
iajs-2958	206	27	𝐺	𝐺	NOUN
iajs-2958	206	28	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2958	206	29	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	206	30	and	and	CCONJ
iajs-2958	206	31	𝑥	𝑥	PRON
iajs-2958	206	32	∈	∈	PROPN
iajs-2958	206	33	𝐺	𝐺	PROPN
iajs-2958	206	34	⊆	⊆	NUM
iajs-2958	206	35	𝐴	𝐴	PROPN
iajs-2958	206	36	}	}	PUNCT
iajs-2958	206	37	.	.	PUNCT
iajs-2958	207	1	if	if	SCONJ
iajs-2958	207	2	𝑥	𝑥	PRON
iajs-2958	207	3	∈∪	∈∪	VERB
iajs-2958	207	4	{	{	PUNCT
iajs-2958	207	5	𝐺	𝐺	NOUN
iajs-2958	207	6	:	:	PUNCT
iajs-2958	207	7	𝐺	𝐺	NOUN
iajs-2958	207	8	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2958	207	9	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	207	10	and	and	CCONJ
iajs-2958	207	11	𝑥	𝑥	DET
iajs-2958	207	12	∈	∈	PROPN
iajs-2958	207	13	𝐺	𝐺	PROPN
iajs-2958	207	14	⊆	⊆	NUM
iajs-2958	207	15	𝐴	𝐴	PROPN
iajs-2958	207	16	}	}	PUNCT
iajs-2958	207	17	,	,	PUNCT
iajs-2958	207	18	then	then	ADV
iajs-2958	207	19	𝑥	𝑥	PROPN
iajs-2958	207	20	∈	∈	PROPN
iajs-2958	207	21	𝐺	𝐺	PROPN
iajs-2958	207	22	for	for	ADP
iajs-2958	207	23	some	some	DET
iajs-2958	207	24	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	207	25	set	set	NOUN
iajs-2958	207	26	𝐺	𝐺	PROPN
iajs-2958	207	27	⊆	⊆	PROPN
iajs-2958	207	28	𝐴.	𝐴.	PROPN
iajs-2958	207	29	therefore	therefore	ADV
iajs-2958	207	30	,	,	PUNCT
iajs-2958	207	31	𝑥	𝑥	DET
iajs-2958	207	32	∈	∈	PROPN
iajs-2958	207	33	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	207	34	)	)	PUNCT
iajs-2958	207	35	.	.	PUNCT
iajs-2958	208	1	iii	iii	X
iajs-2958	208	2	.	.	PUNCT
iajs-2958	209	1	if	if	SCONJ
iajs-2958	209	2	𝐴	𝐴	PROPN
iajs-2958	209	3	is	be	AUX
iajs-2958	209	4	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	209	5	and	and	CCONJ
iajs-2958	209	6	𝑥	𝑥	DET
iajs-2958	209	7	∈	∈	PROPN
iajs-2958	209	8	𝐴	𝐴	PROPN
iajs-2958	209	9	,	,	PUNCT
iajs-2958	209	10	then	then	ADV
iajs-2958	209	11	𝑥	𝑥	DET
iajs-2958	209	12	∈∪	∈∪	NOUN
iajs-2958	209	13	{	{	PUNCT
iajs-2958	209	14	𝐺	𝐺	NOUN
iajs-2958	209	15	:	:	PUNCT
iajs-2958	209	16	𝐺	𝐺	NOUN
iajs-2958	209	17	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2958	209	18	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	209	19	and	and	CCONJ
iajs-2958	209	20	𝐺	𝐺	PROPN
iajs-2958	209	21	⊆	⊆	NUM
iajs-2958	209	22	𝐴	𝐴	PROPN
iajs-2958	209	23	}	}	PUNCT
iajs-2958	209	24	.	.	PUNCT
iajs-2958	210	1	that	that	PRON
iajs-2958	210	2	is	be	AUX
iajs-2958	210	3	,	,	PUNCT
iajs-2958	210	4	𝑥	𝑥	PROPN
iajs-2958	210	5	∈	∈	PROPN
iajs-2958	210	6	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	210	7	)	)	PUNCT
iajs-2958	210	8	,	,	PUNCT
iajs-2958	210	9	hence	hence	ADV
iajs-2958	210	10	𝐴	𝐴	PROPN
iajs-2958	210	11	⊆	⊆	NUM
iajs-2958	210	12	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	210	13	)	)	PUNCT
iajs-2958	210	14	,	,	PUNCT
iajs-2958	210	15	but	but	CCONJ
iajs-2958	210	16	since	since	SCONJ
iajs-2958	210	17	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	210	18	)	)	PUNCT
iajs-2958	210	19	⊆	⊆	NUM
iajs-2958	210	20	a.	a.	NOUN
iajs-2958	210	21	therefore	therefore	ADV
iajs-2958	210	22	,	,	PUNCT
iajs-2958	210	23	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	210	24	)	)	PUNCT
iajs-2958	210	25	=	=	SYM
iajs-2958	211	1	𝐴.	𝐴.	PROPN
iajs-2958	211	2	conversely	conversely	ADV
iajs-2958	211	3	,	,	PUNCT
iajs-2958	211	4	if	if	SCONJ
iajs-2958	211	5	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	211	6	)	)	PUNCT
iajs-2958	211	7	=	=	SYM
iajs-2958	211	8	a	a	PRON
iajs-2958	211	9	,	,	PUNCT
iajs-2958	211	10	then	then	ADV
iajs-2958	211	11	𝐴	𝐴	PROPN
iajs-2958	211	12	is	be	AUX
iajs-2958	211	13	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	211	14	in	in	ADP
iajs-2958	211	15	𝑈	𝑈	PROPN
iajs-2958	211	16	since	since	SCONJ
iajs-2958	211	17	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	211	18	)	)	PUNCT
iajs-2958	211	19	is	be	AUX
iajs-2958	211	20	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	211	21	.	.	PUNCT
iajs-2958	212	1	iv	iv	X
iajs-2958	212	2	.	.	PROPN
iajs-2958	212	3	follows	follow	VERB
iajs-2958	212	4	from	from	ADP
iajs-2958	212	5	part	part	NOUN
iajs-2958	212	6	(	(	PUNCT
iajs-2958	212	7	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	212	8	)	)	PUNCT
iajs-2958	212	9	.	.	PUNCT
iajs-2958	213	1	v.	v.	CCONJ
iajs-2958	213	2	since	since	SCONJ
iajs-2958	213	3	𝜙	𝜙	PROPN
iajs-2958	213	4	and	and	CCONJ
iajs-2958	213	5	𝒲	𝒲	NOUN
iajs-2958	213	6	are	be	AUX
iajs-2958	213	7	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	213	8	set	set	NOUN
iajs-2958	213	9	,	,	PUNCT
iajs-2958	213	10	then	then	ADV
iajs-2958	213	11	by	by	ADP
iajs-2958	213	12	part	part	NOUN
iajs-2958	213	13	(	(	PUNCT
iajs-2958	213	14	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	213	15	)	)	PUNCT
iajs-2958	213	16	,	,	PUNCT
iajs-2958	213	17	𝑛𝑆𝐶𝑖𝑛𝑡(𝜙	𝑛𝑆𝐶𝑖𝑛𝑡(𝜙	NOUN
iajs-2958	213	18	)	)	PUNCT
iajs-2958	213	19	=	=	SYM
iajs-2958	213	20	𝜙	𝜙	NOUN
iajs-2958	213	21	and	and	CCONJ
iajs-2958	213	22	𝑛𝑆𝐶𝑖𝑛𝑡(𝑈	𝑛𝑆𝐶𝑖𝑛𝑡(𝑈	ADJ
iajs-2958	213	23	)	)	PUNCT
iajs-2958	213	24	=	=	SYM
iajs-2958	213	25	𝒲.	𝒲.	PROPN
iajs-2958	213	26	theorem	theorem	VERB
iajs-2958	213	27	4.5	4.5	NUM
iajs-2958	213	28	.	.	PUNCT
iajs-2958	214	1	let	let	VERB
iajs-2958	214	2	𝐴	𝐴	PROPN
iajs-2958	214	3	and	and	CCONJ
iajs-2958	214	4	𝐵	𝐵	NOUN
iajs-2958	214	5	be	be	VERB
iajs-2958	214	6	any	any	DET
iajs-2958	214	7	two	two	NUM
iajs-2958	214	8	subset	subset	NOUN
iajs-2958	214	9	of	of	ADP
iajs-2958	214	10	a	a	DET
iajs-2958	214	11	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	214	12	(	(	PUNCT
iajs-2958	214	13	𝒲	𝒲	PROPN
iajs-2958	214	14	,	,	PUNCT
iajs-2958	214	15	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	214	16	)	)	PUNCT
iajs-2958	214	17	)	)	PUNCT
iajs-2958	214	18	,	,	PUNCT
iajs-2958	214	19	then	then	ADV
iajs-2958	214	20	:	:	PUNCT
iajs-2958	214	21	i.	i.	NOUN
iajs-2958	215	1	if	if	SCONJ
iajs-2958	215	2	𝐴	𝐴	PROPN
iajs-2958	215	3	⊆	⊆	NUM
iajs-2958	215	4	𝐵	𝐵	PROPN
iajs-2958	215	5	,	,	PUNCT
iajs-2958	215	6	then	then	ADV
iajs-2958	215	7	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	215	8	)	)	PUNCT
iajs-2958	215	9	⊆	⊆	NUM
iajs-2958	215	10	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	215	11	)	)	PUNCT
iajs-2958	215	12	.	.	PUNCT
iajs-2958	216	1	ii	ii	PROPN
iajs-2958	216	2	.	.	PUNCT
iajs-2958	217	1	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	217	2	)	)	PUNCT
iajs-2958	217	3	∪	∪	ADP
iajs-2958	217	4	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	217	5	)	)	PUNCT
iajs-2958	218	1	⊆	⊆	NUM
iajs-2958	218	2	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	218	3	∪	∪	ADJ
iajs-2958	218	4	𝐵	𝐵	NOUN
iajs-2958	218	5	)	)	PUNCT
iajs-2958	218	6	.	.	PUNCT
iajs-2958	219	1	iii	iii	X
iajs-2958	219	2	.	.	PUNCT
iajs-2958	219	3	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	219	4	∩	∩	ADJ
iajs-2958	219	5	𝐵	𝐵	NOUN
iajs-2958	219	6	)	)	PUNCT
iajs-2958	219	7	⊆	⊆	NUM
iajs-2958	219	8	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	219	9	)	)	PUNCT
iajs-2958	219	10	∩	∩	NOUN
iajs-2958	219	11	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	219	12	)	)	PUNCT
iajs-2958	219	13	.	.	PUNCT
iajs-2958	220	1	proof	proof	NOUN
iajs-2958	220	2	.	.	PUNCT
iajs-2958	221	1	i.	i.	NOUN
iajs-2958	221	2	if	if	SCONJ
iajs-2958	221	3	𝐴	𝐴	PROPN
iajs-2958	221	4	⊆	⊆	NUM
iajs-2958	221	5	𝐵	𝐵	NOUN
iajs-2958	221	6	and	and	CCONJ
iajs-2958	221	7	𝑥	𝑥	PRON
iajs-2958	221	8	∈	∈	NOUN
iajs-2958	221	9	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	221	10	)	)	PUNCT
iajs-2958	221	11	,	,	PUNCT
iajs-2958	221	12	then	then	ADV
iajs-2958	221	13	𝐺	𝐺	PROPN
iajs-2958	221	14	⊆	⊆	NUM
iajs-2958	221	15	𝐴	𝐴	PROPN
iajs-2958	221	16	for	for	ADP
iajs-2958	221	17	some	some	DET
iajs-2958	221	18	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	221	19	set	set	NOUN
iajs-2958	221	20	𝐺	𝐺	PROPN
iajs-2958	221	21	containing	contain	VERB
iajs-2958	221	22	𝑥.	𝑥.	ADV
iajs-2958	221	23	hence	hence	ADV
iajs-2958	221	24	𝐺	𝐺	PROPN
iajs-2958	221	25	⊆	⊆	NUM
iajs-2958	221	26	𝐴	𝐴	PROPN
iajs-2958	221	27	and	and	CCONJ
iajs-2958	221	28	𝑥	𝑥	PRON
iajs-2958	221	29	∈	∈	NOUN
iajs-2958	221	30	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	221	31	)	)	PUNCT
iajs-2958	221	32	.	.	PUNCT
iajs-2958	222	1	therefore	therefore	ADV
iajs-2958	222	2	,	,	PUNCT
iajs-2958	222	3	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	222	4	)	)	PUNCT
iajs-2958	222	5	⊆	⊆	NUM
iajs-2958	222	6	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	222	7	)	)	PUNCT
iajs-2958	222	8	.	.	PUNCT
iajs-2958	223	1	ii	ii	PROPN
iajs-2958	223	2	.	.	PUNCT
iajs-2958	224	1	since	since	ADV
iajs-2958	224	2	,	,	PUNCT
iajs-2958	224	3	𝐴	𝐴	PROPN
iajs-2958	224	4	⊆	⊆	NUM
iajs-2958	224	5	𝐴	𝐴	PROPN
iajs-2958	224	6	∪	∪	PROPN
iajs-2958	224	7	𝐵	𝐵	PROPN
iajs-2958	224	8	,	,	PUNCT
iajs-2958	224	9	by	by	ADP
iajs-2958	224	10	(	(	PUNCT
iajs-2958	224	11	𝑖	𝑖	X
iajs-2958	224	12	)	)	PUNCT
iajs-2958	224	13	,	,	PUNCT
iajs-2958	224	14	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	224	15	)	)	PUNCT
iajs-2958	224	16	⊆	⊆	NUM
iajs-2958	224	17	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	224	18	∪	∪	ADJ
iajs-2958	224	19	𝐵	𝐵	NOUN
iajs-2958	224	20	)	)	PUNCT
iajs-2958	224	21	.	.	PUNCT
iajs-2958	225	1	again	again	ADV
iajs-2958	225	2	since	since	SCONJ
iajs-2958	225	3	𝐵	𝐵	PROPN
iajs-2958	225	4	⊆	⊆	NUM
iajs-2958	225	5	𝐴	𝐴	PROPN
iajs-2958	225	6	∪	∪	NOUN
iajs-2958	225	7	𝐵	𝐵	PROPN
iajs-2958	225	8	,	,	PUNCT
iajs-2958	225	9	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	225	10	)	)	PUNCT
iajs-2958	225	11	⊆	⊆	NUM
iajs-2958	225	12	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	225	13	∪	∪	ADJ
iajs-2958	225	14	𝐵	𝐵	NOUN
iajs-2958	225	15	)	)	PUNCT
iajs-2958	225	16	.	.	PUNCT
iajs-2958	226	1	therefore	therefore	ADV
iajs-2958	226	2	,	,	PUNCT
iajs-2958	226	3	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	226	4	)	)	PUNCT
iajs-2958	226	5	∪	∪	ADP
iajs-2958	226	6	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	226	7	)	)	PUNCT
iajs-2958	226	8	⊆	⊆	NUM
iajs-2958	226	9	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	226	10	∪	∪	ADJ
iajs-2958	226	11	𝐵	𝐵	NOUN
iajs-2958	226	12	)	)	PUNCT
iajs-2958	226	13	.	.	PUNCT
iajs-2958	227	1	iii	iii	X
iajs-2958	227	2	.	.	PUNCT
iajs-2958	228	1	since	since	SCONJ
iajs-2958	228	2	𝐴	𝐴	PROPN
iajs-2958	228	3	∩	∩	NOUN
iajs-2958	228	4	𝐵	𝐵	PROPN
iajs-2958	228	5	⊆	⊆	NUM
iajs-2958	228	6	𝐴	𝐴	PROPN
iajs-2958	228	7	and	and	CCONJ
iajs-2958	228	8	𝐴	𝐴	PROPN
iajs-2958	228	9	∩	∩	NOUN
iajs-2958	228	10	𝐵	𝐵	PROPN
iajs-2958	228	11	⊆	⊆	NUM
iajs-2958	228	12	𝐵	𝐵	PROPN
iajs-2958	228	13	,	,	PUNCT
iajs-2958	228	14	by	by	ADP
iajs-2958	228	15	part	part	NOUN
iajs-2958	228	16	(	(	PUNCT
iajs-2958	228	17	𝑖	𝑖	NOUN
iajs-2958	228	18	)	)	PUNCT
iajs-2958	228	19	,	,	PUNCT
iajs-2958	228	20	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	228	21	∩	∩	ADJ
iajs-2958	228	22	𝐵	𝐵	NOUN
iajs-2958	228	23	)	)	PUNCT
iajs-2958	228	24	⊆	⊆	NUM
iajs-2958	228	25	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	228	26	)	)	PUNCT
iajs-2958	228	27	∩	∩	NOUN
iajs-2958	228	28	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	228	29	)	)	PUNCT
iajs-2958	228	30	.	.	PUNCT
iajs-2958	229	1	the	the	DET
iajs-2958	229	2	inclusion	inclusion	NOUN
iajs-2958	229	3	of	of	ADP
iajs-2958	229	4	parts	part	NOUN
iajs-2958	229	5	(	(	PUNCT
iajs-2958	229	6	𝑖𝑖	𝑖𝑖	X
iajs-2958	229	7	and	and	CCONJ
iajs-2958	229	8	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	229	9	)	)	PUNCT
iajs-2958	229	10	of	of	ADP
iajs-2958	229	11	above	above	ADP
iajs-2958	229	12	theorem	theorem	NOUN
iajs-2958	229	13	can	can	AUX
iajs-2958	229	14	not	not	PART
iajs-2958	229	15	be	be	AUX
iajs-2958	229	16	replaced	replace	VERB
iajs-2958	229	17	by	by	ADP
iajs-2958	229	18	equality	equality	NOUN
iajs-2958	229	19	in	in	ADP
iajs-2958	229	20	general	general	ADJ
iajs-2958	229	21	,	,	PUNCT
iajs-2958	229	22	as	as	SCONJ
iajs-2958	229	23	it	it	PRON
iajs-2958	229	24	shown	show	VERB
iajs-2958	229	25	in	in	ADP
iajs-2958	229	26	the	the	DET
iajs-2958	229	27	following	follow	VERB
iajs-2958	229	28	example	example	NOUN
iajs-2958	229	29	.	.	PUNCT
iajs-2958	230	1	example	example	NOUN
iajs-2958	230	2	4.6	4.6	NUM
iajs-2958	230	3	.	.	PUNCT
iajs-2958	231	1	let	let	VERB
iajs-2958	231	2	𝒲	𝒲	NOUN
iajs-2958	231	3	=	=	PUNCT
iajs-2958	231	4	{	{	PUNCT
iajs-2958	231	5	𝑎	𝑎	NOUN
iajs-2958	231	6	,	,	PUNCT
iajs-2958	231	7	𝑏	𝑏	NOUN
iajs-2958	231	8	,	,	PUNCT
iajs-2958	231	9	𝑐	𝑐	NOUN
iajs-2958	231	10	,	,	PUNCT
iajs-2958	231	11	𝑑	𝑑	NOUN
iajs-2958	231	12	}	}	PUNCT
iajs-2958	231	13	with	with	ADP
iajs-2958	231	14	𝒲	𝒲	PROPN
iajs-2958	231	15	𝑅⁄	𝑅⁄	PROPN
iajs-2958	231	16	=	=	PUNCT
iajs-2958	231	17	{	{	PUNCT
iajs-2958	231	18	{	{	PUNCT
iajs-2958	231	19	𝑎	𝑎	NOUN
iajs-2958	231	20	}	}	PUNCT
iajs-2958	231	21	,	,	PUNCT
iajs-2958	231	22	{	{	PUNCT
iajs-2958	231	23	𝑏	𝑏	NOUN
iajs-2958	231	24	,	,	PUNCT
iajs-2958	231	25	𝑐	𝑐	NOUN
iajs-2958	231	26	}	}	PUNCT
iajs-2958	231	27	,	,	PUNCT
iajs-2958	231	28	{	{	PUNCT
iajs-2958	231	29	𝑑	𝑑	NOUN
iajs-2958	231	30	}	}	PUNCT
iajs-2958	231	31	}	}	PUNCT
iajs-2958	231	32	and	and	CCONJ
iajs-2958	231	33	𝑋	𝑋	PROPN
iajs-2958	231	34	=	=	SYM
iajs-2958	231	35	{	{	PUNCT
iajs-2958	231	36	𝑎	𝑎	NOUN
iajs-2958	231	37	,	,	PUNCT
iajs-2958	231	38	𝑏	𝑏	NOUN
iajs-2958	231	39	}	}	PUNCT
iajs-2958	231	40	,	,	PUNCT
iajs-2958	231	41	then	then	ADV
iajs-2958	231	42	𝜏𝑅	𝜏𝑅	PROPN
iajs-2958	231	43	(	(	PUNCT
iajs-2958	231	44	𝑋	𝑋	NOUN
iajs-2958	231	45	)	)	PUNCT
iajs-2958	231	46	=	=	PUNCT
iajs-2958	231	47	{	{	PUNCT
iajs-2958	231	48	𝜙	𝜙	NOUN
iajs-2958	231	49	,	,	PUNCT
iajs-2958	231	50	𝒲	𝒲	NOUN
iajs-2958	231	51	,	,	PUNCT
iajs-2958	231	52	{	{	PUNCT
iajs-2958	231	53	𝑎	𝑎	NOUN
iajs-2958	231	54	}	}	PUNCT
iajs-2958	231	55	,	,	PUNCT
iajs-2958	231	56	{	{	PUNCT
iajs-2958	231	57	𝑏	𝑏	NOUN
iajs-2958	231	58	,	,	PUNCT
iajs-2958	231	59	𝑐	𝑐	NOUN
iajs-2958	231	60	}	}	PUNCT
iajs-2958	231	61	,	,	PUNCT
iajs-2958	231	62	{	{	PUNCT
iajs-2958	231	63	𝑎	𝑎	X
iajs-2958	231	64	,	,	PUNCT
iajs-2958	231	65	𝑏	𝑏	NOUN
iajs-2958	231	66	,	,	PUNCT
iajs-2958	231	67	𝑐	𝑐	NOUN
iajs-2958	231	68	}	}	PUNCT
iajs-2958	231	69	}	}	PUNCT
iajs-2958	231	70	and	and	CCONJ
iajs-2958	231	71	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	231	72	,	,	PUNCT
iajs-2958	231	73	𝑋	𝑋	PROPN
iajs-2958	231	74	)	)	PUNCT
iajs-2958	231	75	=	=	PUNCT
iajs-2958	231	76	{	{	PUNCT
iajs-2958	231	77	𝜙	𝜙	NOUN
iajs-2958	231	78	,	,	PUNCT
iajs-2958	231	79	𝒲	𝒲	NOUN
iajs-2958	231	80	,	,	PUNCT
iajs-2958	231	81	{	{	PUNCT
iajs-2958	231	82	𝑎	𝑎	NOUN
iajs-2958	231	83	,	,	PUNCT
iajs-2958	231	84	𝑑	𝑑	NOUN
iajs-2958	231	85	}	}	PUNCT
iajs-2958	231	86	,	,	PUNCT
iajs-2958	231	87	{	{	PUNCT
iajs-2958	231	88	𝑏	𝑏	NOUN
iajs-2958	231	89	,	,	PUNCT
iajs-2958	231	90	𝑐	𝑐	PROPN
iajs-2958	231	91	,	,	PUNCT
iajs-2958	231	92	𝑑	𝑑	NOUN
iajs-2958	231	93	}	}	PUNCT
iajs-2958	231	94	}	}	PUNCT
iajs-2958	231	95	.	.	PUNCT
iajs-2958	232	1	for	for	ADP
iajs-2958	232	2	part	part	NOUN
iajs-2958	232	3	(	(	PUNCT
iajs-2958	232	4	𝑖𝑖	𝑖𝑖	NOUN
iajs-2958	232	5	)	)	PUNCT
iajs-2958	232	6	,	,	PUNCT
iajs-2958	232	7	take	take	VERB
iajs-2958	232	8	𝐴	𝐴	NOUN
iajs-2958	232	9	=	=	SYM
iajs-2958	232	10	{	{	PUNCT
iajs-2958	232	11	𝑏	𝑏	NOUN
iajs-2958	232	12	,	,	PUNCT
iajs-2958	232	13	𝑑	𝑑	NOUN
iajs-2958	232	14	}	}	PUNCT
iajs-2958	232	15	and	and	CCONJ
iajs-2958	232	16	𝐵	𝐵	NOUN
iajs-2958	232	17	=	=	PUNCT
iajs-2958	232	18	{	{	PUNCT
iajs-2958	232	19	𝑐	𝑐	NOUN
iajs-2958	232	20	,	,	PUNCT
iajs-2958	232	21	𝑑	𝑑	NOUN
iajs-2958	232	22	}	}	PUNCT
iajs-2958	232	23	,	,	PUNCT
iajs-2958	232	24	then	then	ADV
iajs-2958	232	25	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	232	26	)	)	PUNCT
iajs-2958	232	27	∪	∪	ADP
iajs-2958	232	28	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	232	29	)	)	PUNCT
iajs-2958	232	30	=	=	SYM
iajs-2958	232	31	𝜙	𝜙	NOUN
iajs-2958	232	32	∪	∪	VERB
iajs-2958	232	33	𝜙	𝜙	NOUN
iajs-2958	232	34	=	=	SYM
iajs-2958	232	35	𝜙	𝜙	NOUN
iajs-2958	232	36	but	but	CCONJ
iajs-2958	232	37	𝑛𝑆𝐶𝑖𝑛𝑡	𝑛𝑆𝐶𝑖𝑛𝑡	PRON
iajs-2958	232	38	(	(	PUNCT
iajs-2958	232	39	𝐴	𝐴	PROPN
iajs-2958	232	40	∪	∪	NOUN
iajs-2958	232	41	𝐵	𝐵	NOUN
iajs-2958	232	42	)	)	PUNCT
iajs-2958	232	43	=	=	SYM
iajs-2958	232	44	{	{	PUNCT
iajs-2958	232	45	𝑏	𝑏	NOUN
iajs-2958	232	46	,	,	PUNCT
iajs-2958	232	47	𝑐	𝑐	PROPN
iajs-2958	232	48	,	,	PUNCT
iajs-2958	232	49	𝑑	𝑑	NOUN
iajs-2958	232	50	}	}	PUNCT
iajs-2958	232	51	.	.	PUNCT
iajs-2958	233	1	therefore	therefore	ADV
iajs-2958	233	2	,	,	PUNCT
iajs-2958	233	3	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	233	4	)	)	PUNCT
iajs-2958	233	5	∪	∪	ADP
iajs-2958	233	6	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	233	7	)	)	PUNCT
iajs-2958	233	8	≠	≠	PROPN
iajs-2958	233	9	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	233	10	∪	∪	ADJ
iajs-2958	233	11	𝐵	𝐵	NOUN
iajs-2958	233	12	)	)	PUNCT
iajs-2958	233	13	.	.	PUNCT
iajs-2958	234	1	for	for	ADP
iajs-2958	234	2	part	part	NOUN
iajs-2958	234	3	(	(	PUNCT
iajs-2958	234	4	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	234	5	)	)	PUNCT
iajs-2958	234	6	,	,	PUNCT
iajs-2958	234	7	take	take	VERB
iajs-2958	234	8	𝐴	𝐴	NOUN
iajs-2958	234	9	=	=	PUNCT
iajs-2958	234	10	{	{	PUNCT
iajs-2958	234	11	𝑎	𝑎	NOUN
iajs-2958	234	12	,	,	PUNCT
iajs-2958	234	13	𝑑	𝑑	NOUN
iajs-2958	234	14	}	}	PUNCT
iajs-2958	234	15	and	and	CCONJ
iajs-2958	234	16	𝐵	𝐵	NOUN
iajs-2958	234	17	=	=	PUNCT
iajs-2958	234	18	{	{	PUNCT
iajs-2958	234	19	𝑏	𝑏	NOUN
iajs-2958	234	20	,	,	PUNCT
iajs-2958	234	21	𝑐	𝑐	PROPN
iajs-2958	234	22	,	,	PUNCT
iajs-2958	234	23	𝑑	𝑑	NOUN
iajs-2958	234	24	}	}	PUNCT
iajs-2958	234	25	,	,	PUNCT
iajs-2958	234	26	then	then	ADV
iajs-2958	234	27	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	234	28	∩	∩	ADJ
iajs-2958	234	29	𝐵	𝐵	NOUN
iajs-2958	234	30	)	)	PUNCT
iajs-2958	234	31	=	=	SYM
iajs-2958	234	32	𝑛𝑆𝐶𝑖𝑛𝑡({𝑑	𝑛𝑆𝐶𝑖𝑛𝑡({𝑑	NOUN
iajs-2958	234	33	}	}	PUNCT
iajs-2958	234	34	)	)	PUNCT
iajs-2958	234	35	=	=	SYM
iajs-2958	234	36	𝜙	𝜙	NOUN
iajs-2958	234	37	,	,	PUNCT
iajs-2958	234	38	but	but	CCONJ
iajs-2958	234	39	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	234	40	)	)	PUNCT
iajs-2958	234	41	∩	∩	NOUN
iajs-2958	234	42	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	234	43	)	)	PUNCT
iajs-2958	234	44	=	=	SYM
iajs-2958	234	45	{	{	PUNCT
iajs-2958	234	46	𝑎	𝑎	NOUN
iajs-2958	234	47	,	,	PUNCT
iajs-2958	234	48	𝑑	𝑑	NOUN
iajs-2958	234	49	}	}	PUNCT
iajs-2958	234	50	∩	∩	NOUN
iajs-2958	234	51	{	{	PUNCT
iajs-2958	234	52	𝑏	𝑏	NOUN
iajs-2958	234	53	,	,	PUNCT
iajs-2958	234	54	𝑐	𝑐	PROPN
iajs-2958	234	55	,	,	PUNCT
iajs-2958	234	56	𝑑	𝑑	NOUN
iajs-2958	234	57	}	}	PUNCT
iajs-2958	234	58	=	=	SYM
iajs-2958	234	59	{	{	PUNCT
iajs-2958	234	60	𝑑	𝑑	NOUN
iajs-2958	234	61	}	}	PUNCT
iajs-2958	234	62	.	.	PUNCT
iajs-2958	235	1	therefore	therefore	ADV
iajs-2958	235	2	,	,	PUNCT
iajs-2958	235	3	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	X
iajs-2958	235	4	∩	∩	ADJ
iajs-2958	235	5	𝐵	𝐵	NOUN
iajs-2958	235	6	)	)	PUNCT
iajs-2958	235	7	≠	≠	PROPN
iajs-2958	235	8	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	235	9	)	)	PUNCT
iajs-2958	235	10	∩	∩	NOUN
iajs-2958	235	11	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	𝑛𝑆𝐶𝑖𝑛𝑡(𝐵	NOUN
iajs-2958	235	12	)	)	PUNCT
iajs-2958	235	13	.	.	PUNCT
iajs-2958	236	1	definition	definition	NOUN
iajs-2958	236	2	4.7	4.7	NUM
iajs-2958	236	3	.	.	PUNCT
iajs-2958	237	1	a	a	DET
iajs-2958	237	2	point	point	NOUN
iajs-2958	237	3	𝑥	𝑥	PRON
iajs-2958	237	4	∈	∈	PROPN
iajs-2958	237	5	𝒲	𝒲	NOUN
iajs-2958	237	6	of	of	ADP
iajs-2958	237	7	a	a	DET
iajs-2958	237	8	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	237	9	(	(	PUNCT
iajs-2958	237	10	𝒲	𝒲	PROPN
iajs-2958	237	11	,	,	PUNCT
iajs-2958	237	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	237	13	)	)	PUNCT
iajs-2958	237	14	)	)	PUNCT
iajs-2958	237	15	is	be	AUX
iajs-2958	237	16	said	say	VERB
iajs-2958	237	17	to	to	PART
iajs-2958	237	18	be	be	AUX
iajs-2958	237	19	𝑛𝑆𝐶-cluster	𝑛𝑆𝐶-cluster	NOUN
iajs-2958	237	20	point	point	NOUN
iajs-2958	237	21	of	of	ADP
iajs-2958	237	22	a	a	DET
iajs-2958	237	23	subset	subset	ADJ
iajs-2958	237	24	𝐴	𝐴	PROPN
iajs-2958	237	25	of	of	ADP
iajs-2958	237	26	𝑈	𝑈	PROPN
iajs-2958	237	27	,	,	PUNCT
iajs-2958	237	28	if	if	SCONJ
iajs-2958	237	29	𝐴	𝐴	PROPN
iajs-2958	237	30	∩	∩	ADJ
iajs-2958	237	31	𝐺	𝐺	NOUN
iajs-2958	237	32	≠	≠	PROPN
iajs-2958	237	33	𝜙	𝜙	NOUN
iajs-2958	237	34	for	for	ADP
iajs-2958	237	35	every	every	DET
iajs-2958	237	36	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	237	37	set	set	NOUN
iajs-2958	237	38	𝐺	𝐺	PROPN
iajs-2958	237	39	containing	contain	VERB
iajs-2958	237	40	𝑥.	𝑥.	ADJ
iajs-2958	237	41	definition	definition	NOUN
iajs-2958	237	42	4.8	4.8	NUM
iajs-2958	237	43	.	.	PUNCT
iajs-2958	238	1	the	the	DET
iajs-2958	238	2	set	set	NOUN
iajs-2958	238	3	of	of	ADP
iajs-2958	238	4	all	all	DET
iajs-2958	238	5	𝑛𝑆𝐶-cluster	𝑛𝑆𝐶-cluster	NOUN
iajs-2958	238	6	points	point	NOUN
iajs-2958	238	7	of	of	ADP
iajs-2958	238	8	a	a	DET
iajs-2958	238	9	subset	subset	ADJ
iajs-2958	238	10	𝐴	𝐴	NOUN
iajs-2958	238	11	of	of	ADP
iajs-2958	238	12	𝒲	𝒲	PROPN
iajs-2958	238	13	is	be	AUX
iajs-2958	238	14	said	say	VERB
iajs-2958	238	15	to	to	PART
iajs-2958	238	16	be	be	AUX
iajs-2958	238	17	𝑛𝑆𝐶-closure	𝑛𝑆𝐶-closure	NOUN
iajs-2958	238	18	of	of	ADP
iajs-2958	238	19	𝐴	𝐴	PROPN
iajs-2958	238	20	and	and	CCONJ
iajs-2958	238	21	denoted	denote	VERB
iajs-2958	238	22	by	by	ADP
iajs-2958	238	23	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	238	24	)	)	PUNCT
iajs-2958	238	25	.	.	PUNCT
iajs-2958	239	1	equivalently	equivalently	ADV
iajs-2958	239	2	,	,	PUNCT
iajs-2958	239	3	the	the	DET
iajs-2958	239	4	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	239	5	)	)	PUNCT
iajs-2958	239	6	is	be	AUX
iajs-2958	239	7	the	the	DET
iajs-2958	239	8	intersection	intersection	NOUN
iajs-2958	239	9	of	of	ADP
iajs-2958	239	10	all	all	DET
iajs-2958	239	11	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	239	12	sets	set	NOUN
iajs-2958	239	13	containing	contain	VERB
iajs-2958	239	14	𝐴.	𝐴.	PROPN
iajs-2958	239	15	ihjpas	ihjpa	NOUN
iajs-2958	239	16	.	.	PUNCT
iajs-2958	240	1	36(2)2023	36(2)2023	NUM
iajs-2958	240	2	312	312	NUM
iajs-2958	240	3	theorem	theorem	NOUN
iajs-2958	240	4	4.9	4.9	NUM
iajs-2958	240	5	.	.	PUNCT
iajs-2958	241	1	let	let	VERB
iajs-2958	241	2	𝐴	𝐴	PROPN
iajs-2958	241	3	be	be	AUX
iajs-2958	241	4	any	any	DET
iajs-2958	241	5	subset	subset	NOUN
iajs-2958	241	6	of	of	ADP
iajs-2958	241	7	a	a	DET
iajs-2958	241	8	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	241	9	(	(	PUNCT
iajs-2958	241	10	𝒲	𝒲	PROPN
iajs-2958	241	11	,	,	PUNCT
iajs-2958	241	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	241	13	)	)	PUNCT
iajs-2958	241	14	)	)	PUNCT
iajs-2958	241	15	.	.	PUNCT
iajs-2958	242	1	a	a	DET
iajs-2958	242	2	point	point	NOUN
iajs-2958	242	3	𝑥	𝑥	X
iajs-2958	242	4	∈	∈	NOUN
iajs-2958	242	5	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	242	6	)	)	PUNCT
iajs-2958	242	7	if	if	SCONJ
iajs-2958	242	8	and	and	CCONJ
iajs-2958	242	9	only	only	ADV
iajs-2958	242	10	if	if	SCONJ
iajs-2958	242	11	𝐴	𝐴	PROPN
iajs-2958	242	12	∩	∩	VERB
iajs-2958	242	13	𝐻	𝐻	PROPN
iajs-2958	242	14	≠	≠	PROPN
iajs-2958	242	15	𝜙	𝜙	NOUN
iajs-2958	242	16	for	for	ADP
iajs-2958	242	17	every	every	DET
iajs-2958	242	18	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	242	19	set	set	NOUN
iajs-2958	242	20	𝐻	𝐻	PROPN
iajs-2958	242	21	containing	contain	VERB
iajs-2958	242	22	𝑥.	𝑥.	ADJ
iajs-2958	242	23	proof	proof	NOUN
iajs-2958	242	24	.	.	PUNCT
iajs-2958	243	1	obvious	obvious	ADJ
iajs-2958	243	2	.	.	PUNCT
iajs-2958	244	1	corollary	corollary	ADJ
iajs-2958	244	2	4.10	4.10	NUM
iajs-2958	244	3	.	.	PUNCT
iajs-2958	245	1	for	for	ADP
iajs-2958	245	2	any	any	DET
iajs-2958	245	3	subset	subset	NOUN
iajs-2958	245	4	𝐴	𝐴	PROPN
iajs-2958	245	5	of	of	ADP
iajs-2958	245	6	a	a	DET
iajs-2958	245	7	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	245	8	(	(	PUNCT
iajs-2958	245	9	𝒲	𝒲	PROPN
iajs-2958	245	10	,	,	PUNCT
iajs-2958	245	11	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	245	12	)	)	PUNCT
iajs-2958	245	13	)	)	PUNCT
iajs-2958	245	14	,	,	PUNCT
iajs-2958	245	15	the	the	DET
iajs-2958	245	16	following	follow	VERB
iajs-2958	245	17	statements	statement	NOUN
iajs-2958	245	18	are	be	AUX
iajs-2958	245	19	true	true	ADJ
iajs-2958	245	20	.	.	PUNCT
iajs-2958	246	1	i.	i.	PROPN
iajs-2958	246	2	𝑛𝑆𝐶𝑐𝑙	𝑛𝑆𝐶𝑐𝑙	PROPN
iajs-2958	246	3	(	(	PUNCT
iajs-2958	246	4	𝒲	𝒲	PROPN
iajs-2958	246	5	−	−	PROPN
iajs-2958	246	6	𝐴	𝐴	PROPN
iajs-2958	246	7	)	)	PUNCT
iajs-2958	246	8	=	=	SYM
iajs-2958	246	9	𝒲−	𝒲−	X
iajs-2958	246	10	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	246	11	)	)	PUNCT
iajs-2958	246	12	.	.	PUNCT
iajs-2958	247	1	ii	ii	PROPN
iajs-2958	247	2	.	.	PUNCT
iajs-2958	247	3	𝑛𝑆𝐶𝑖𝑛𝑡	𝑛𝑆𝐶𝑖𝑛𝑡	PROPN
iajs-2958	247	4	(	(	PUNCT
iajs-2958	247	5	𝒲	𝒲	PROPN
iajs-2958	247	6	−	−	PROPN
iajs-2958	247	7	𝐴	𝐴	PROPN
iajs-2958	247	8	)	)	PUNCT
iajs-2958	248	1	=	=	PUNCT
iajs-2958	248	2	𝒲	𝒲	NOUN
iajs-2958	248	3	−	−	NOUN
iajs-2958	248	4	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	248	5	)	)	PUNCT
iajs-2958	248	6	.	.	PUNCT
iajs-2958	249	1	proof	proof	NOUN
iajs-2958	249	2	.	.	PUNCT
iajs-2958	250	1	i.	i.	PROPN
iajs-2958	250	2	let	let	VERB
iajs-2958	250	3	𝑥	𝑥	DET
iajs-2958	250	4	∈	∈	PROPN
iajs-2958	250	5	𝑛𝑆𝐶𝑐𝑙	𝑛𝑆𝐶𝑐𝑙	PROPN
iajs-2958	250	6	(	(	PUNCT
iajs-2958	250	7	𝒲	𝒲	PROPN
iajs-2958	250	8	−	−	PROPN
iajs-2958	250	9	𝐴	𝐴	PROPN
iajs-2958	250	10	)	)	PUNCT
iajs-2958	250	11	,	,	PUNCT
iajs-2958	250	12	then	then	ADV
iajs-2958	250	13	𝐺	𝐺	PROPN
iajs-2958	250	14	∩	∩	NOUN
iajs-2958	250	15	(	(	PUNCT
iajs-2958	250	16	𝒲	𝒲	PROPN
iajs-2958	250	17	−	−	PROPN
iajs-2958	250	18	𝐴	𝐴	PROPN
iajs-2958	250	19	)	)	PUNCT
iajs-2958	250	20	≠	≠	PROPN
iajs-2958	250	21	𝜙	𝜙	NOUN
iajs-2958	250	22	for	for	ADP
iajs-2958	250	23	any	any	DET
iajs-2958	250	24	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	250	25	set	set	NOUN
iajs-2958	250	26	𝐺	𝐺	PROPN
iajs-2958	250	27	containing	contain	VERB
iajs-2958	250	28	𝑥.	𝑥.	ADV
iajs-2958	250	29	therefore	therefore	ADV
iajs-2958	250	30	,	,	PUNCT
iajs-2958	250	31	𝐺	𝐺	PROPN
iajs-2958	250	32	⊈	⊈	PROPN
iajs-2958	250	33	𝐴	𝐴	PROPN
iajs-2958	250	34	where	where	SCONJ
iajs-2958	250	35	𝐺	𝐺	PROPN
iajs-2958	250	36	is	be	AUX
iajs-2958	250	37	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	250	38	set	set	NOUN
iajs-2958	250	39	containing	contain	VERB
iajs-2958	250	40	𝑥.	𝑥.	NOUN
iajs-2958	250	41	that	that	PRON
iajs-2958	250	42	is	be	AUX
iajs-2958	250	43	,	,	PUNCT
iajs-2958	250	44	𝑥	𝑥	PROPN
iajs-2958	250	45	∉	∉	PROPN
iajs-2958	250	46	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	250	47	)	)	PUNCT
iajs-2958	250	48	.	.	PUNCT
iajs-2958	251	1	therefore	therefore	ADV
iajs-2958	251	2	,	,	PUNCT
iajs-2958	251	3	𝑥	𝑥	DET
iajs-2958	251	4	∈	∈	PROPN
iajs-2958	251	5	𝒲	𝒲	PROPN
iajs-2958	251	6	−	−	NOUN
iajs-2958	251	7	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	251	8	)	)	PUNCT
iajs-2958	251	9	.	.	PUNCT
iajs-2958	252	1	thus	thus	ADV
iajs-2958	252	2	,	,	PUNCT
iajs-2958	252	3	𝑛𝑆𝐶𝑐𝑙(𝒲	𝑛𝑆𝐶𝑐𝑙(𝒲	X
iajs-2958	252	4	−	−	PROPN
iajs-2958	252	5	𝐴	𝐴	PROPN
iajs-2958	252	6	)	)	PUNCT
iajs-2958	252	7	⊆	⊆	NUM
iajs-2958	252	8	𝒲	𝒲	PROPN
iajs-2958	252	9	−	−	NOUN
iajs-2958	252	10	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	252	11	)	)	PUNCT
iajs-2958	252	12	.	.	PUNCT
iajs-2958	253	1	conversely	conversely	ADV
iajs-2958	253	2	,	,	PUNCT
iajs-2958	253	3	if	if	SCONJ
iajs-2958	253	4	𝑥	𝑥	PRON
iajs-2958	253	5	∈	∈	PROPN
iajs-2958	253	6	𝒲	𝒲	PROPN
iajs-2958	253	7	−	−	NOUN
iajs-2958	253	8	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	253	9	)	)	PUNCT
iajs-2958	253	10	,	,	PUNCT
iajs-2958	253	11	then	then	ADV
iajs-2958	253	12	𝑥	𝑥	PROPN
iajs-2958	253	13	∉	∉	PROPN
iajs-2958	253	14	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	PROPN
iajs-2958	253	15	)	)	PUNCT
iajs-2958	253	16	,	,	PUNCT
iajs-2958	253	17	and	and	CCONJ
iajs-2958	253	18	this	this	PRON
iajs-2958	253	19	mean	mean	ADJ
iajs-2958	253	20	𝐺	𝐺	PROPN
iajs-2958	253	21	⊈	⊈	PROPN
iajs-2958	253	22	𝐴	𝐴	PROPN
iajs-2958	253	23	for	for	ADP
iajs-2958	253	24	every	every	DET
iajs-2958	253	25	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	253	26	set	set	NOUN
iajs-2958	253	27	𝐺	𝐺	PROPN
iajs-2958	253	28	containing	contain	VERB
iajs-2958	253	29	𝑥.	𝑥.	ADV
iajs-2958	253	30	therefore	therefore	ADV
iajs-2958	253	31	,	,	PUNCT
iajs-2958	253	32	𝐺	𝐺	PROPN
iajs-2958	253	33	∩	∩	NOUN
iajs-2958	253	34	(	(	PUNCT
iajs-2958	253	35	𝒲	𝒲	PROPN
iajs-2958	253	36	−	−	PROPN
iajs-2958	253	37	𝐴	𝐴	PROPN
iajs-2958	253	38	)	)	PUNCT
iajs-2958	253	39	≠	≠	PROPN
iajs-2958	253	40	𝜙	𝜙	NOUN
iajs-2958	253	41	and	and	CCONJ
iajs-2958	253	42	so	so	ADV
iajs-2958	253	43	𝑥	𝑥	PRON
iajs-2958	253	44	∈	∈	PROPN
iajs-2958	253	45	𝑛𝑆𝐶𝑐𝑙(𝒲	𝑛𝑆𝐶𝑐𝑙(𝒲	X
iajs-2958	253	46	−	−	PROPN
iajs-2958	253	47	𝐴	𝐴	PROPN
iajs-2958	253	48	)	)	PUNCT
iajs-2958	253	49	.	.	PUNCT
iajs-2958	254	1	hence	hence	ADV
iajs-2958	254	2	,	,	PUNCT
iajs-2958	254	3	𝒲	𝒲	PROPN
iajs-2958	254	4	−	−	NOUN
iajs-2958	254	5	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	254	6	)	)	PUNCT
iajs-2958	254	7	⊆	⊆	NUM
iajs-2958	254	8	𝑛𝑆𝐶𝑐𝑙(𝒲	𝑛𝑆𝐶𝑐𝑙(𝒲	X
iajs-2958	254	9	−	−	PROPN
iajs-2958	254	10	𝐴	𝐴	PROPN
iajs-2958	254	11	)	)	PUNCT
iajs-2958	254	12	.	.	PUNCT
iajs-2958	255	1	hence	hence	ADV
iajs-2958	255	2	,	,	PUNCT
iajs-2958	255	3	𝑛𝑆𝐶𝑐𝑙(𝒲	𝑛𝑆𝐶𝑐𝑙(𝒲	X
iajs-2958	255	4	−	−	PROPN
iajs-2958	255	5	𝐴	𝐴	PROPN
iajs-2958	255	6	)	)	PUNCT
iajs-2958	255	7	=	=	PUNCT
iajs-2958	255	8	𝒲	𝒲	PROPN
iajs-2958	255	9	−	−	NOUN
iajs-2958	255	10	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	𝑛𝑆𝐶𝑖𝑛𝑡(𝐴	NOUN
iajs-2958	255	11	)	)	PUNCT
iajs-2958	255	12	.	.	PUNCT
iajs-2958	256	1	ii	ii	PROPN
iajs-2958	256	2	.	.	PUNCT
iajs-2958	257	1	the	the	DET
iajs-2958	257	2	proof	proof	NOUN
iajs-2958	257	3	is	be	AUX
iajs-2958	257	4	similar	similar	ADJ
iajs-2958	257	5	to	to	ADP
iajs-2958	257	6	part	part	NOUN
iajs-2958	257	7	(	(	PUNCT
iajs-2958	257	8	𝑖	𝑖	NOUN
iajs-2958	257	9	)	)	PUNCT
iajs-2958	257	10	.	.	PUNCT
iajs-2958	258	1	theorem	theorem	VERB
iajs-2958	258	2	4.11	4.11	NUM
iajs-2958	258	3	.	.	PUNCT
iajs-2958	259	1	for	for	ADP
iajs-2958	259	2	any	any	DET
iajs-2958	259	3	subset	subset	NOUN
iajs-2958	259	4	𝐴	𝐴	NOUN
iajs-2958	259	5	and	and	CCONJ
iajs-2958	259	6	𝐵	𝐵	NOUN
iajs-2958	259	7	of	of	ADP
iajs-2958	259	8	a	a	DET
iajs-2958	259	9	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	259	10	(	(	PUNCT
iajs-2958	259	11	𝒲	𝒲	PROPN
iajs-2958	259	12	,	,	PUNCT
iajs-2958	259	13	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	259	14	)	)	PUNCT
iajs-2958	259	15	)	)	PUNCT
iajs-2958	259	16	,	,	PUNCT
iajs-2958	259	17	the	the	DET
iajs-2958	259	18	following	follow	VERB
iajs-2958	259	19	statements	statement	NOUN
iajs-2958	259	20	are	be	AUX
iajs-2958	259	21	true	true	ADJ
iajs-2958	259	22	:	:	PUNCT
iajs-2958	259	23	i.	i.	NOUN
iajs-2958	259	24	if	if	SCONJ
iajs-2958	259	25	𝐴	𝐴	PROPN
iajs-2958	259	26	⊆	⊆	NUM
iajs-2958	259	27	𝐵	𝐵	PROPN
iajs-2958	259	28	,	,	PUNCT
iajs-2958	259	29	then	then	ADV
iajs-2958	259	30	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	259	31	)	)	PUNCT
iajs-2958	259	32	⊆	⊆	NUM
iajs-2958	259	33	𝑛𝑆𝐶𝑐𝑙(𝐵	𝑛𝑆𝐶𝑐𝑙(𝐵	NOUN
iajs-2958	259	34	)	)	PUNCT
iajs-2958	259	35	.	.	PUNCT
iajs-2958	260	1	ii	ii	PROPN
iajs-2958	260	2	.	.	PROPN
iajs-2958	260	3	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	260	4	)	)	PUNCT
iajs-2958	260	5	∪	∪	ADP
iajs-2958	260	6	𝑛𝑆𝐶𝑐𝑙(𝐵	𝑛𝑆𝐶𝑐𝑙(𝐵	PRON
iajs-2958	260	7	)	)	PUNCT
iajs-2958	260	8	⊆	⊆	NUM
iajs-2958	260	9	𝑛𝑆𝐶𝑐𝑙	𝑛𝑆𝐶𝑐𝑙	PROPN
iajs-2958	260	10	(	(	PUNCT
iajs-2958	260	11	𝐴	𝐴	PROPN
iajs-2958	260	12	∪	∪	PROPN
iajs-2958	260	13	𝐵	𝐵	PROPN
iajs-2958	260	14	)	)	PUNCT
iajs-2958	260	15	.	.	PUNCT
iajs-2958	261	1	iii	iii	X
iajs-2958	261	2	.	.	PROPN
iajs-2958	261	3	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	261	4	∩	∩	ADJ
iajs-2958	261	5	𝐵	𝐵	NOUN
iajs-2958	261	6	)	)	PUNCT
iajs-2958	261	7	⊆	⊆	NUM
iajs-2958	261	8	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	261	9	)	)	PUNCT
iajs-2958	261	10	∩	∩	NOUN
iajs-2958	261	11	𝑛𝑆𝐶𝑐𝑙(𝐵	𝑛𝑆𝐶𝑐𝑙(𝐵	X
iajs-2958	261	12	)	)	PUNCT
iajs-2958	261	13	.	.	PUNCT
iajs-2958	262	1	proof	proof	NOUN
iajs-2958	262	2	.	.	PUNCT
iajs-2958	263	1	i.	i.	NOUN
iajs-2958	263	2	if	if	SCONJ
iajs-2958	263	3	𝐴	𝐴	PROPN
iajs-2958	263	4	⊆	⊆	NUM
iajs-2958	263	5	𝐵	𝐵	NOUN
iajs-2958	263	6	and	and	CCONJ
iajs-2958	263	7	𝑥	𝑥	PRON
iajs-2958	263	8	∈	∈	NOUN
iajs-2958	263	9	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	263	10	)	)	PUNCT
iajs-2958	263	11	,	,	PUNCT
iajs-2958	263	12	then	then	ADV
iajs-2958	263	13	𝐺	𝐺	PROPN
iajs-2958	263	14	∩	∩	ADJ
iajs-2958	263	15	𝐴	𝐴	NOUN
iajs-2958	263	16	≠	≠	PROPN
iajs-2958	263	17	𝜙	𝜙	NOUN
iajs-2958	263	18	for	for	ADP
iajs-2958	263	19	every	every	DET
iajs-2958	263	20	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	263	21	set	set	NOUN
iajs-2958	263	22	𝐺	𝐺	PROPN
iajs-2958	263	23	containing	contain	VERB
iajs-2958	263	24	𝑥.since	𝑥.since	ADJ
iajs-2958	263	25	𝐺	𝐺	NOUN
iajs-2958	263	26	∩	∩	NOUN
iajs-2958	263	27	𝐴	𝐴	PROPN
iajs-2958	263	28	⊆	⊆	NUM
iajs-2958	263	29	𝐺	𝐺	PROPN
iajs-2958	263	30	∩	∩	ADJ
iajs-2958	263	31	𝐵	𝐵	PROPN
iajs-2958	263	32	,	,	PUNCT
iajs-2958	263	33	𝐺	𝐺	NOUN
iajs-2958	263	34	∩	∩	ADJ
iajs-2958	263	35	𝐵	𝐵	NOUN
iajs-2958	263	36	≠	≠	PROPN
iajs-2958	263	37	𝜙	𝜙	NOUN
iajs-2958	263	38	whenever	whenever	SCONJ
iajs-2958	263	39	𝐺	𝐺	PROPN
iajs-2958	263	40	is	be	AUX
iajs-2958	263	41	𝑛𝑆𝐶-open	𝑛𝑆𝐶-open	ADJ
iajs-2958	263	42	set	set	NOUN
iajs-2958	263	43	containing	contain	VERB
iajs-2958	263	44	𝑥.	𝑥.	ADV
iajs-2958	263	45	therefore	therefore	ADV
iajs-2958	263	46	,	,	PUNCT
iajs-2958	263	47	𝑥	𝑥	PRON
iajs-2958	263	48	∈	∈	PROPN
iajs-2958	263	49	𝑛𝑆𝐶𝑐𝑙(𝐵	𝑛𝑆𝐶𝑐𝑙(𝐵	PRON
iajs-2958	263	50	)	)	PUNCT
iajs-2958	263	51	.	.	PUNCT
iajs-2958	264	1	hence	hence	ADV
iajs-2958	264	2	,	,	PUNCT
iajs-2958	264	3	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	264	4	)	)	PUNCT
iajs-2958	264	5	⊆	⊆	NUM
iajs-2958	264	6	𝑛𝑆𝐶𝑐𝑙(𝐵	𝑛𝑆𝐶𝑐𝑙(𝐵	NOUN
iajs-2958	264	7	)	)	PUNCT
iajs-2958	264	8	.	.	PUNCT
iajs-2958	265	1	ii	ii	PROPN
iajs-2958	265	2	.	.	PUNCT
iajs-2958	266	1	since	since	SCONJ
iajs-2958	266	2	𝐴	𝐴	PROPN
iajs-2958	266	3	,	,	PUNCT
iajs-2958	266	4	𝐵	𝐵	PROPN
iajs-2958	266	5	⊆	⊆	NUM
iajs-2958	266	6	𝐴	𝐴	PROPN
iajs-2958	266	7	∪	∪	NOUN
iajs-2958	266	8	𝐵	𝐵	PROPN
iajs-2958	266	9	,	,	PUNCT
iajs-2958	266	10	then	then	ADV
iajs-2958	266	11	by	by	ADP
iajs-2958	266	12	part	part	NOUN
iajs-2958	266	13	(	(	PUNCT
iajs-2958	266	14	𝑖	𝑖	NOUN
iajs-2958	266	15	)	)	PUNCT
iajs-2958	266	16	,	,	PUNCT
iajs-2958	266	17	we	we	PRON
iajs-2958	266	18	get	get	VERB
iajs-2958	266	19	the	the	DET
iajs-2958	266	20	result	result	NOUN
iajs-2958	266	21	.	.	PUNCT
iajs-2958	267	1	iii	iii	X
iajs-2958	267	2	.	.	PUNCT
iajs-2958	268	1	since	since	SCONJ
iajs-2958	268	2	𝐴	𝐴	PROPN
iajs-2958	268	3	∩	∩	NOUN
iajs-2958	268	4	𝐵	𝐵	PROPN
iajs-2958	268	5	⊆	⊆	NUM
iajs-2958	268	6	𝐴	𝐴	PROPN
iajs-2958	268	7	,	,	PUNCT
iajs-2958	268	8	𝐵	𝐵	PROPN
iajs-2958	268	9	,	,	PUNCT
iajs-2958	268	10	then	then	ADV
iajs-2958	268	11	by	by	ADP
iajs-2958	268	12	part	part	NOUN
iajs-2958	268	13	(	(	PUNCT
iajs-2958	268	14	𝑖	𝑖	NOUN
iajs-2958	268	15	)	)	PUNCT
iajs-2958	268	16	,	,	PUNCT
iajs-2958	268	17	we	we	PRON
iajs-2958	268	18	get	get	VERB
iajs-2958	268	19	the	the	DET
iajs-2958	268	20	result	result	NOUN
iajs-2958	268	21	.	.	PUNCT
iajs-2958	269	1	the	the	DET
iajs-2958	269	2	inclusion	inclusion	NOUN
iajs-2958	269	3	in	in	ADP
iajs-2958	269	4	(	(	PUNCT
iajs-2958	269	5	𝑖𝑖	𝑖𝑖	X
iajs-2958	269	6	and	and	CCONJ
iajs-2958	269	7	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	269	8	)	)	PUNCT
iajs-2958	269	9	of	of	ADP
iajs-2958	269	10	above	above	ADP
iajs-2958	269	11	theorem	theorem	NOUN
iajs-2958	269	12	can	can	AUX
iajs-2958	269	13	not	not	PART
iajs-2958	269	14	be	be	AUX
iajs-2958	269	15	replaced	replace	VERB
iajs-2958	269	16	by	by	ADP
iajs-2958	269	17	quality	quality	NOUN
iajs-2958	269	18	in	in	ADP
iajs-2958	269	19	general	general	ADJ
iajs-2958	269	20	,	,	PUNCT
iajs-2958	269	21	as	as	SCONJ
iajs-2958	269	22	it	it	PRON
iajs-2958	269	23	shown	show	VERB
iajs-2958	269	24	in	in	ADP
iajs-2958	269	25	the	the	DET
iajs-2958	269	26	following	follow	VERB
iajs-2958	269	27	two	two	NUM
iajs-2958	269	28	examples	example	NOUN
iajs-2958	269	29	.	.	PUNCT
iajs-2958	270	1	example	example	NOUN
iajs-2958	270	2	4.12	4.12	NUM
iajs-2958	270	3	.	.	PUNCT
iajs-2958	271	1	let	let	VERB
iajs-2958	271	2	𝒲	𝒲	NOUN
iajs-2958	271	3	=	=	PUNCT
iajs-2958	271	4	{	{	PUNCT
iajs-2958	271	5	𝑎	𝑎	NOUN
iajs-2958	271	6	,	,	PUNCT
iajs-2958	271	7	𝑏	𝑏	NOUN
iajs-2958	271	8	,	,	PUNCT
iajs-2958	271	9	𝑐	𝑐	NOUN
iajs-2958	271	10	,	,	PUNCT
iajs-2958	271	11	𝑑	𝑑	NOUN
iajs-2958	271	12	}	}	PUNCT
iajs-2958	271	13	with	with	ADP
iajs-2958	271	14	𝒲	𝒲	PROPN
iajs-2958	271	15	𝑅⁄	𝑅⁄	PROPN
iajs-2958	271	16	=	=	PUNCT
iajs-2958	271	17	{	{	PUNCT
iajs-2958	271	18	{	{	PUNCT
iajs-2958	271	19	𝑎	𝑎	NOUN
iajs-2958	271	20	}	}	PUNCT
iajs-2958	271	21	,	,	PUNCT
iajs-2958	271	22	{	{	PUNCT
iajs-2958	271	23	𝑏	𝑏	NOUN
iajs-2958	271	24	,	,	PUNCT
iajs-2958	271	25	𝑐	𝑐	NOUN
iajs-2958	271	26	}	}	PUNCT
iajs-2958	271	27	,	,	PUNCT
iajs-2958	271	28	{	{	PUNCT
iajs-2958	271	29	𝑑	𝑑	NOUN
iajs-2958	271	30	}	}	PUNCT
iajs-2958	271	31	}	}	PUNCT
iajs-2958	271	32	and	and	CCONJ
iajs-2958	271	33	𝑋	𝑋	PROPN
iajs-2958	271	34	=	=	SYM
iajs-2958	271	35	{	{	PUNCT
iajs-2958	271	36	𝑎	𝑎	NOUN
iajs-2958	271	37	,	,	PUNCT
iajs-2958	271	38	𝑏	𝑏	NOUN
iajs-2958	271	39	}	}	PUNCT
iajs-2958	271	40	,	,	PUNCT
iajs-2958	271	41	then	then	ADV
iajs-2958	271	42	𝜏𝑅	𝜏𝑅	PROPN
iajs-2958	271	43	(	(	PUNCT
iajs-2958	271	44	𝑋	𝑋	NOUN
iajs-2958	271	45	)	)	PUNCT
iajs-2958	271	46	=	=	PUNCT
iajs-2958	271	47	{	{	PUNCT
iajs-2958	271	48	𝜙	𝜙	NOUN
iajs-2958	271	49	,	,	PUNCT
iajs-2958	271	50	𝒲	𝒲	NOUN
iajs-2958	271	51	,	,	PUNCT
iajs-2958	271	52	{	{	PUNCT
iajs-2958	271	53	𝑎	𝑎	NOUN
iajs-2958	271	54	}	}	PUNCT
iajs-2958	271	55	,	,	PUNCT
iajs-2958	271	56	{	{	PUNCT
iajs-2958	271	57	𝑏	𝑏	NOUN
iajs-2958	271	58	,	,	PUNCT
iajs-2958	271	59	𝑐	𝑐	NOUN
iajs-2958	271	60	}	}	PUNCT
iajs-2958	271	61	,	,	PUNCT
iajs-2958	271	62	{	{	PUNCT
iajs-2958	271	63	𝑎	𝑎	X
iajs-2958	271	64	,	,	PUNCT
iajs-2958	271	65	𝑏	𝑏	NOUN
iajs-2958	271	66	,	,	PUNCT
iajs-2958	271	67	𝑐	𝑐	NOUN
iajs-2958	271	68	}	}	PUNCT
iajs-2958	271	69	}	}	PUNCT
iajs-2958	271	70	,	,	PUNCT
iajs-2958	271	71	𝑛𝑆𝐶𝑂(𝒲	𝑛𝑆𝐶𝑂(𝒲	ADV
iajs-2958	271	72	,	,	PUNCT
iajs-2958	271	73	𝑋	𝑋	NOUN
iajs-2958	271	74	)	)	PUNCT
iajs-2958	271	75	=	=	PUNCT
iajs-2958	271	76	{	{	PUNCT
iajs-2958	271	77	𝜙	𝜙	NOUN
iajs-2958	271	78	,	,	PUNCT
iajs-2958	271	79	𝒲	𝒲	NOUN
iajs-2958	271	80	,	,	PUNCT
iajs-2958	271	81	{	{	PUNCT
iajs-2958	271	82	𝑎	𝑎	NOUN
iajs-2958	271	83	,	,	PUNCT
iajs-2958	271	84	𝑑	𝑑	NOUN
iajs-2958	271	85	}	}	PUNCT
iajs-2958	271	86	,	,	PUNCT
iajs-2958	271	87	{	{	PUNCT
iajs-2958	271	88	𝑏	𝑏	NOUN
iajs-2958	271	89	,	,	PUNCT
iajs-2958	271	90	𝑐	𝑐	PROPN
iajs-2958	271	91	,	,	PUNCT
iajs-2958	271	92	𝑑	𝑑	NOUN
iajs-2958	271	93	}	}	PUNCT
iajs-2958	271	94	}	}	PUNCT
iajs-2958	271	95	and	and	CCONJ
iajs-2958	271	96	𝑛𝑆𝐶𝐶(𝒲	𝑛𝑆𝐶𝐶(𝒲	PROPN
iajs-2958	271	97	,	,	PUNCT
iajs-2958	271	98	𝑋	𝑋	NOUN
iajs-2958	271	99	)	)	PUNCT
iajs-2958	271	100	=	=	PUNCT
iajs-2958	271	101	{	{	PUNCT
iajs-2958	271	102	𝜙	𝜙	NOUN
iajs-2958	271	103	,	,	PUNCT
iajs-2958	271	104	𝒲	𝒲	NOUN
iajs-2958	271	105	,	,	PUNCT
iajs-2958	271	106	{	{	PUNCT
iajs-2958	271	107	𝑏	𝑏	NOUN
iajs-2958	271	108	,	,	PUNCT
iajs-2958	271	109	𝑐	𝑐	NOUN
iajs-2958	271	110	}	}	PUNCT
iajs-2958	271	111	,	,	PUNCT
iajs-2958	271	112	{	{	PUNCT
iajs-2958	271	113	𝑎	𝑎	VERB
iajs-2958	271	114	}	}	PUNCT
iajs-2958	271	115	}	}	PUNCT
iajs-2958	271	116	.	.	PUNCT
iajs-2958	272	1	for	for	ADP
iajs-2958	272	2	part	part	NOUN
iajs-2958	272	3	(	(	PUNCT
iajs-2958	272	4	𝑖𝑖	𝑖𝑖	NOUN
iajs-2958	272	5	)	)	PUNCT
iajs-2958	272	6	,	,	PUNCT
iajs-2958	272	7	take	take	VERB
iajs-2958	272	8	𝐹	𝐹	PROPN
iajs-2958	272	9	=	=	PUNCT
iajs-2958	272	10	{	{	PUNCT
iajs-2958	272	11	𝑎	𝑎	NOUN
iajs-2958	272	12	,	,	PUNCT
iajs-2958	272	13	𝑑	𝑑	NOUN
iajs-2958	272	14	}	}	PUNCT
iajs-2958	272	15	and	and	CCONJ
iajs-2958	272	16	𝐸	𝐸	PROPN
iajs-2958	272	17	=	=	SYM
iajs-2958	272	18	{	{	PUNCT
iajs-2958	272	19	𝑎	𝑎	NOUN
iajs-2958	272	20	,	,	PUNCT
iajs-2958	272	21	𝑏	𝑏	NOUN
iajs-2958	272	22	}	}	PUNCT
iajs-2958	272	23	,	,	PUNCT
iajs-2958	272	24	then	then	ADV
iajs-2958	272	25	𝑛𝑆𝐶𝑐𝑙(𝐹	𝑛𝑆𝐶𝑐𝑙(𝐹	AUX
iajs-2958	272	26	∩	∩	ADJ
iajs-2958	272	27	𝐸	𝐸	PROPN
iajs-2958	272	28	)	)	PUNCT
iajs-2958	272	29	=	=	SYM
iajs-2958	272	30	𝑛𝑆𝐶𝑐𝑙({𝑎	𝑛𝑆𝐶𝑐𝑙({𝑎	PROPN
iajs-2958	272	31	}	}	PUNCT
iajs-2958	272	32	)	)	PUNCT
iajs-2958	273	1	=	=	PUNCT
iajs-2958	273	2	{	{	PUNCT
iajs-2958	273	3	𝑎	𝑎	NOUN
iajs-2958	273	4	}	}	PUNCT
iajs-2958	273	5	,	,	PUNCT
iajs-2958	273	6	but	but	CCONJ
iajs-2958	273	7	𝑛𝑆𝐶𝑐𝑙(𝐹	𝑛𝑆𝐶𝑐𝑙(𝐹	X
iajs-2958	273	8	)	)	PUNCT
iajs-2958	273	9	∩	∩	NOUN
iajs-2958	273	10	𝑛𝑆𝐶𝑐𝑙(𝐸	𝑛𝑆𝐶𝑐𝑙(𝐸	ADJ
iajs-2958	273	11	)	)	PUNCT
iajs-2958	274	1	=	=	SYM
iajs-2958	274	2	𝒲.	𝒲.	PROPN
iajs-2958	274	3	therefore	therefore	ADV
iajs-2958	274	4	,	,	PUNCT
iajs-2958	274	5	𝑛𝑆𝐶𝑐𝑙(𝐹	𝑛𝑆𝐶𝑐𝑙(𝐹	X
iajs-2958	274	6	∩	∩	ADJ
iajs-2958	274	7	𝐸	𝐸	PROPN
iajs-2958	274	8	)	)	PUNCT
iajs-2958	274	9	≠	≠	PROPN
iajs-2958	274	10	𝑛𝑆𝐶𝑐𝑙(𝐹	𝑛𝑆𝐶𝑐𝑙(𝐹	NUM
iajs-2958	274	11	)	)	PUNCT
iajs-2958	274	12	∩	∩	NOUN
iajs-2958	274	13	𝑛𝑆𝐶𝑐𝑙(𝐸	𝑛𝑆𝐶𝑐𝑙(𝐸	NOUN
iajs-2958	274	14	)	)	PUNCT
iajs-2958	274	15	.	.	PUNCT
iajs-2958	275	1	for	for	ADP
iajs-2958	275	2	part	part	NOUN
iajs-2958	275	3	(	(	PUNCT
iajs-2958	275	4	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	275	5	)	)	PUNCT
iajs-2958	275	6	,	,	PUNCT
iajs-2958	275	7	take	take	VERB
iajs-2958	275	8	𝐹	𝐹	PROPN
iajs-2958	275	9	=	=	PUNCT
iajs-2958	275	10	{	{	PUNCT
iajs-2958	275	11	𝑏	𝑏	NOUN
iajs-2958	275	12	,	,	PUNCT
iajs-2958	275	13	𝑐	𝑐	NOUN
iajs-2958	275	14	}	}	PUNCT
iajs-2958	275	15	and	and	CCONJ
iajs-2958	275	16	𝐸	𝐸	PROPN
iajs-2958	275	17	=	=	SYM
iajs-2958	275	18	{	{	PUNCT
iajs-2958	275	19	𝑎	𝑎	NOUN
iajs-2958	275	20	}	}	PUNCT
iajs-2958	275	21	,	,	PUNCT
iajs-2958	275	22	then	then	ADV
iajs-2958	275	23	𝑛𝑆𝐶𝑐𝑙({𝑏	𝑛𝑆𝐶𝑐𝑙({𝑏	ADJ
iajs-2958	275	24	,	,	PUNCT
iajs-2958	275	25	𝑐	𝑐	NOUN
iajs-2958	275	26	}	}	PUNCT
iajs-2958	275	27	∪	∪	X
iajs-2958	275	28	{	{	PUNCT
iajs-2958	275	29	𝑎	𝑎	NOUN
iajs-2958	275	30	}	}	PUNCT
iajs-2958	275	31	)	)	PUNCT
iajs-2958	275	32	=	=	SYM
iajs-2958	275	33	𝒲	𝒲	NOUN
iajs-2958	275	34	,	,	PUNCT
iajs-2958	275	35	but	but	CCONJ
iajs-2958	275	36	𝑛𝑆𝐶𝑐𝑙({𝑏	𝑛𝑆𝐶𝑐𝑙({𝑏	ADJ
iajs-2958	275	37	,	,	PUNCT
iajs-2958	275	38	𝑐	𝑐	NOUN
iajs-2958	275	39	}	}	PUNCT
iajs-2958	275	40	)	)	PUNCT
iajs-2958	275	41	∪	∪	ADP
iajs-2958	275	42	𝑛𝑆𝐶𝑐𝑙({𝑎	𝑛𝑆𝐶𝑐𝑙({𝑎	PROPN
iajs-2958	275	43	}	}	PUNCT
iajs-2958	275	44	)	)	PUNCT
iajs-2958	275	45	=	=	SYM
iajs-2958	275	46	{	{	PUNCT
iajs-2958	275	47	𝑎	𝑎	NOUN
iajs-2958	275	48	,	,	PUNCT
iajs-2958	275	49	𝑏	𝑏	NOUN
iajs-2958	275	50	,	,	PUNCT
iajs-2958	275	51	𝑐	𝑐	NOUN
iajs-2958	275	52	}	}	PUNCT
iajs-2958	275	53	.	.	PUNCT
iajs-2958	276	1	therefore	therefore	ADV
iajs-2958	276	2	,	,	PUNCT
iajs-2958	276	3	𝑛𝑆𝐶𝑐𝑙(𝐹	𝑛𝑆𝐶𝑐𝑙(𝐹	NOUN
iajs-2958	276	4	)	)	PUNCT
iajs-2958	276	5	∪	∪	ADP
iajs-2958	276	6	𝑛𝑆𝐶𝑐𝑙(𝐸	𝑛𝑆𝐶𝑐𝑙(𝐸	NOUN
iajs-2958	276	7	)	)	PUNCT
iajs-2958	276	8	≠	≠	PROPN
iajs-2958	276	9	𝑛𝑆𝐶𝑐𝑙	𝑛𝑆𝐶𝑐𝑙	PROPN
iajs-2958	276	10	(	(	PUNCT
iajs-2958	276	11	𝐹	𝐹	PROPN
iajs-2958	276	12	∪	∪	PROPN
iajs-2958	276	13	𝐸	𝐸	PROPN
iajs-2958	276	14	)	)	PUNCT
iajs-2958	276	15	.	.	PUNCT
iajs-2958	277	1	theorem	theorem	VERB
iajs-2958	277	2	4.13	4.13	NUM
iajs-2958	277	3	.	.	PUNCT
iajs-2958	278	1	let	let	VERB
iajs-2958	278	2	𝐴	𝐴	PROPN
iajs-2958	278	3	be	be	AUX
iajs-2958	278	4	any	any	DET
iajs-2958	278	5	subset	subset	NOUN
iajs-2958	278	6	of	of	ADP
iajs-2958	278	7	a	a	DET
iajs-2958	278	8	𝑁𝑇𝑆	𝑁𝑇𝑆	PROPN
iajs-2958	278	9	(	(	PUNCT
iajs-2958	278	10	𝒲	𝒲	PROPN
iajs-2958	278	11	,	,	PUNCT
iajs-2958	278	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
iajs-2958	278	13	)	)	PUNCT
iajs-2958	278	14	)	)	PUNCT
iajs-2958	278	15	,	,	PUNCT
iajs-2958	278	16	then	then	ADV
iajs-2958	278	17	the	the	DET
iajs-2958	278	18	following	following	ADJ
iajs-2958	278	19	statements	statement	NOUN
iajs-2958	278	20	are	be	AUX
iajs-2958	278	21	true	true	ADJ
iajs-2958	278	22	:	:	PUNCT
iajs-2958	278	23	i.	i.	NOUN
iajs-2958	278	24	𝑛𝑆𝐶𝑐𝑙(𝜙	𝑛𝑆𝐶𝑐𝑙(𝜙	NOUN
iajs-2958	278	25	)	)	PUNCT
iajs-2958	278	26	=	=	SYM
iajs-2958	278	27	𝜙	𝜙	NOUN
iajs-2958	278	28	and	and	CCONJ
iajs-2958	278	29	𝑛𝑆𝐶𝑐𝑙(𝒲	𝑛𝑆𝐶𝑐𝑙(𝒲	NUM
iajs-2958	278	30	)	)	PUNCT
iajs-2958	279	1	=	=	SYM
iajs-2958	279	2	𝒲.	𝒲.	PROPN
iajs-2958	279	3	ii	ii	PROPN
iajs-2958	279	4	.	.	PUNCT
iajs-2958	280	1	𝐴	𝐴	PROPN
iajs-2958	280	2	⊆	⊆	NUM
iajs-2958	280	3	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	280	4	)	)	PUNCT
iajs-2958	280	5	.	.	PUNCT
iajs-2958	281	1	iii	iii	X
iajs-2958	281	2	.	.	PUNCT
iajs-2958	282	1	𝐴	𝐴	PROPN
iajs-2958	282	2	∈	∈	PROPN
iajs-2958	282	3	𝑛𝑆𝐶𝐶(𝑊	𝑛𝑆𝐶𝐶(𝑊	PROPN
iajs-2958	282	4	,	,	PUNCT
iajs-2958	282	5	𝑋	𝑋	PROPN
iajs-2958	282	6	)	)	PUNCT
iajs-2958	282	7	if	if	SCONJ
iajs-2958	283	1	and	and	CCONJ
iajs-2958	283	2	only	only	ADV
iajs-2958	283	3	if	if	SCONJ
iajs-2958	283	4	𝐴	𝐴	PROPN
iajs-2958	283	5	=	=	PUNCT
iajs-2958	283	6	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	283	7	)	)	PUNCT
iajs-2958	283	8	.	.	PUNCT
iajs-2958	283	9	ihjpas	ihjpas	PROPN
iajs-2958	283	10	.	.	PUNCT
iajs-2958	284	1	36(2)2023	36(2)2023	NUM
iajs-2958	284	2	313	313	NUM
iajs-2958	284	3	iv	iv	NOUN
iajs-2958	284	4	.	.	PUNCT
iajs-2958	285	1	𝑛𝑆𝐶𝑐𝑙	𝑛𝑆𝐶𝑐𝑙	PROPN
iajs-2958	285	2	(	(	PUNCT
iajs-2958	285	3	𝑛𝑆𝛽𝑐𝑙(𝐴	𝑛𝑆𝛽𝑐𝑙(𝐴	PROPN
iajs-2958	285	4	)	)	PUNCT
iajs-2958	285	5	)	)	PUNCT
iajs-2958	286	1	=	=	PUNCT
iajs-2958	286	2	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	286	3	)	)	PUNCT
iajs-2958	286	4	proof	proof	NOUN
iajs-2958	286	5	.	.	PUNCT
iajs-2958	287	1	i.	i.	PROPN
iajs-2958	287	2	follows	follow	VERB
iajs-2958	287	3	form	form	VERB
iajs-2958	287	4	the	the	DET
iajs-2958	287	5	fact	fact	NOUN
iajs-2958	287	6	that	that	SCONJ
iajs-2958	287	7	𝜙	𝜙	PROPN
iajs-2958	287	8	and	and	CCONJ
iajs-2958	287	9	𝒲	𝒲	NOUN
iajs-2958	287	10	are	be	AUX
iajs-2958	287	11	𝑛𝑆𝛽-closed	𝑛𝑆𝛽-close	VERB
iajs-2958	287	12	set	set	VERB
iajs-2958	287	13	.	.	PUNCT
iajs-2958	288	1	ii	ii	PROPN
iajs-2958	288	2	.	.	PUNCT
iajs-2958	289	1	by	by	ADP
iajs-2958	289	2	definition	definition	NOUN
iajs-2958	289	3	of	of	ADP
iajs-2958	289	4	𝑛𝑆𝐶-closure	𝑛𝑆𝐶-closure	NOUN
iajs-2958	289	5	,	,	PUNCT
iajs-2958	289	6	𝐴	𝐴	PROPN
iajs-2958	289	7	⊆	⊆	NUM
iajs-2958	289	8	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	289	9	)	)	PUNCT
iajs-2958	289	10	.	.	PUNCT
iajs-2958	290	1	iii	iii	X
iajs-2958	290	2	.	.	PUNCT
iajs-2958	291	1	let	let	VERB
iajs-2958	291	2	𝐴	𝐴	PROPN
iajs-2958	291	3	is	be	AUX
iajs-2958	291	4	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	291	5	set	set	NOUN
iajs-2958	291	6	,	,	PUNCT
iajs-2958	291	7	then	then	ADV
iajs-2958	291	8	𝐴	𝐴	PROPN
iajs-2958	291	9	is	be	AUX
iajs-2958	291	10	smallest	smallest	ADV
iajs-2958	291	11	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	291	12	set	set	NOUN
iajs-2958	291	13	containing	contain	VERB
iajs-2958	291	14	itself	itself	PRON
iajs-2958	291	15	and	and	CCONJ
iajs-2958	291	16	hence	hence	ADV
iajs-2958	291	17	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	291	18	)	)	PUNCT
iajs-2958	291	19	=	=	SYM
iajs-2958	292	1	𝐴.	𝐴.	PROPN
iajs-2958	292	2	conversely	conversely	ADV
iajs-2958	292	3	,	,	PUNCT
iajs-2958	292	4	if	if	SCONJ
iajs-2958	292	5	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	292	6	)	)	PUNCT
iajs-2958	292	7	=	=	SYM
iajs-2958	292	8	𝐴	𝐴	PROPN
iajs-2958	292	9	,	,	PUNCT
iajs-2958	292	10	then	then	ADV
iajs-2958	292	11	𝐴	𝐴	PROPN
iajs-2958	292	12	is	be	AUX
iajs-2958	292	13	the	the	DET
iajs-2958	292	14	smallest	small	ADJ
iajs-2958	292	15	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	292	16	set	set	NOUN
iajs-2958	292	17	containing	contain	VERB
iajs-2958	292	18	itself	itself	PRON
iajs-2958	292	19	and	and	CCONJ
iajs-2958	292	20	hence	hence	ADV
iajs-2958	292	21	𝐴	𝐴	PROPN
iajs-2958	292	22	is	be	AUX
iajs-2958	292	23	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	292	24	set	set	VERB
iajs-2958	292	25	in	in	ADP
iajs-2958	292	26	𝑊.	𝑊.	PROPN
iajs-2958	292	27	iv	iv	NUM
iajs-2958	292	28	.	.	PUNCT
iajs-2958	293	1	since	since	SCONJ
iajs-2958	293	2	𝑛𝑆𝐶𝑐𝑙(𝐴	𝑛𝑆𝐶𝑐𝑙(𝐴	NOUN
iajs-2958	293	3	)	)	PUNCT
iajs-2958	293	4	is	be	AUX
iajs-2958	293	5	𝑛𝑆𝐶-closed	𝑛𝑆𝐶-close	VERB
iajs-2958	293	6	set	set	NOUN
iajs-2958	293	7	,	,	PUNCT
iajs-2958	293	8	then	then	ADV
iajs-2958	293	9	the	the	DET
iajs-2958	293	10	proof	proof	NOUN
iajs-2958	293	11	follows	follow	VERB
iajs-2958	293	12	from	from	ADP
iajs-2958	293	13	part	part	NOUN
iajs-2958	293	14	(	(	PUNCT
iajs-2958	293	15	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2958	293	16	)	)	PUNCT
iajs-2958	293	17	.	.	PUNCT
iajs-2958	294	1	references	reference	NOUN
iajs-2958	294	2	1	1	NUM
iajs-2958	294	3	.	.	PUNCT
iajs-2958	294	4	thivagar	thivagar	NOUN
iajs-2958	294	5	,	,	PUNCT
iajs-2958	294	6	m.	m.	NOUN
iajs-2958	294	7	l.	l.	PROPN
iajs-2958	294	8	;	;	PUNCT
iajs-2958	294	9	richard	richard	PROPN
iajs-2958	294	10	,	,	PUNCT
iajs-2958	294	11	c.	c.	PROPN
iajs-2958	294	12	on	on	ADP
iajs-2958	294	13	nano	nano	NOUN
iajs-2958	294	14	forms	form	NOUN
iajs-2958	294	15	of	of	ADP
iajs-2958	294	16	weakly	weakly	ADJ
iajs-2958	294	17	open	open	ADJ
iajs-2958	294	18	sets	set	NOUN
iajs-2958	294	19	.	.	PUNCT
iajs-2958	295	1	international	international	ADJ
iajs-2958	295	2	journal	journal	NOUN
iajs-2958	295	3	of	of	ADP
iajs-2958	295	4	mathematics	mathematics	PROPN
iajs-2958	295	5	and	and	CCONJ
iajs-2958	295	6	statistics	statistic	NOUN
iajs-2958	295	7	invention	invention	NOUN
iajs-2958	295	8	,	,	PUNCT
iajs-2958	295	9	2013	2013	NUM
iajs-2958	295	10	,	,	PUNCT
iajs-2958	295	11	1	1	NUM
iajs-2958	295	12	,	,	PUNCT
iajs-2958	295	13	31	31	NUM
iajs-2958	295	14	-	-	SYM
iajs-2958	295	15	37	37	NUM
iajs-2958	295	16	.	.	NOUN
iajs-2958	296	1	2	2	NUM
iajs-2958	296	2	.	.	X
iajs-2958	296	3	levine	levine	PROPN
iajs-2958	296	4	,	,	PUNCT
iajs-2958	296	5	n.	n.	PROPN
iajs-2958	296	6	semi	semi	ADJ
iajs-2958	296	7	-	-	ADJ
iajs-2958	296	8	open	open	ADJ
iajs-2958	296	9	sets	set	NOUN
iajs-2958	296	10	and	and	CCONJ
iajs-2958	296	11	semi	semi	ADJ
iajs-2958	296	12	-	-	NOUN
iajs-2958	296	13	continuity	continuity	NOUN
iajs-2958	296	14	in	in	ADP
iajs-2958	296	15	topological	topological	ADJ
iajs-2958	296	16	spaces	space	NOUN
iajs-2958	296	17	.	.	PUNCT
iajs-2958	297	1	the	the	DET
iajs-2958	297	2	american	american	PROPN
iajs-2958	297	3	mathematical	mathematical	PROPN
iajs-2958	297	4	monthly	monthly	ADV
iajs-2958	297	5	,	,	PUNCT
iajs-2958	297	6	1963	1963	NUM
iajs-2958	297	7	,	,	PUNCT
iajs-2958	297	8	70	70	NUM
iajs-2958	297	9	,	,	PUNCT
iajs-2958	297	10	36	36	NUM
iajs-2958	297	11	-	-	SYM
iajs-2958	297	12	41	41	NUM
iajs-2958	297	13	.	.	PUNCT
iajs-2958	298	1	3	3	X
iajs-2958	298	2	.	.	X
iajs-2958	298	3	revathy	revathy	PROPN
iajs-2958	298	4	,	,	PUNCT
iajs-2958	298	5	a.	a.	NOUN
iajs-2958	298	6	,	,	PUNCT
iajs-2958	298	7	ilango	ilango	NOUN
iajs-2958	298	8	,	,	PUNCT
iajs-2958	298	9	g.	g.	PROPN
iajs-2958	298	10	on	on	ADP
iajs-2958	298	11	nano	nano	NOUN
iajs-2958	298	12	β	β	NOUN
iajs-2958	298	13	-	-	ADJ
iajs-2958	298	14	open	open	ADJ
iajs-2958	298	15	sets	set	NOUN
iajs-2958	298	16	.	.	PUNCT
iajs-2958	299	1	int	int	NOUN
iajs-2958	299	2	.	.	PUNCT
iajs-2958	300	1	j.	j.	PROPN
iajs-2958	300	2	eng	eng	PROPN
iajs-2958	300	3	.	.	PROPN
iajs-2958	301	1	contemp	contemp	PROPN
iajs-2958	301	2	.	.	PUNCT
iajs-2958	302	1	math	math	NOUN
iajs-2958	302	2	.	.	PUNCT
iajs-2958	303	1	sci	sci	PROPN
iajs-2958	303	2	,	,	PUNCT
iajs-2958	303	3	2015	2015	NUM
iajs-2958	303	4	,	,	PUNCT
iajs-2958	303	5	1	1	NUM
iajs-2958	303	6	,	,	PUNCT
iajs-2958	303	7	1	1	NUM
iajs-2958	303	8	-	-	SYM
iajs-2958	303	9	6	6	NUM
iajs-2958	303	10	.	.	NOUN
iajs-2958	303	11	4	4	NUM
iajs-2958	303	12	.	.	X
iajs-2958	303	13	pirbal	pirbal	ADJ
iajs-2958	303	14	,	,	PUNCT
iajs-2958	303	15	o.	o.	PROPN
iajs-2958	303	16	t.	t.	PROPN
iajs-2958	303	17	;	;	PUNCT
iajs-2958	303	18	ahmed	ahme	VERB
iajs-2958	303	19	,	,	PUNCT
iajs-2958	303	20	n.	n.	PROPN
iajs-2958	303	21	k.	k.	PROPN
iajs-2958	303	22	on	on	ADP
iajs-2958	303	23	nano	nano	NOUN
iajs-2958	303	24	𝑆𝛽-open	𝑆𝛽-open	VERB
iajs-2958	303	25	sets	set	NOUN
iajs-2958	303	26	in	in	ADP
iajs-2958	303	27	nano	nano	ADJ
iajs-2958	303	28	topological	topological	ADJ
iajs-2958	303	29	spaces	space	NOUN
iajs-2958	303	30	.	.	PUNCT
iajs-2958	304	1	general	general	ADJ
iajs-2958	304	2	letters	letter	NOUN
iajs-2958	304	3	in	in	ADP
iajs-2958	304	4	mathematics	mathematic	NOUN
iajs-2958	304	5	,	,	PUNCT
iajs-2958	304	6	2022	2022	NUM
iajs-2958	304	7	,	,	PUNCT
iajs-2958	304	8	12	12	NUM
iajs-2958	304	9	,	,	PUNCT
iajs-2958	304	10	23	23	NUM
iajs-2958	304	11	-	-	SYM
iajs-2958	304	12	30	30	NUM
iajs-2958	304	13	.	.	PUNCT
iajs-2958	305	1	5	5	NUM
iajs-2958	305	2	.	.	X
iajs-2958	305	3	rajasekaran	rajasekaran	NOUN
iajs-2958	305	4	,	,	PUNCT
iajs-2958	305	5	i.	i.	NOUN
iajs-2958	305	6	;	;	PUNCT
iajs-2958	305	7	meharin	meharin	NOUN
iajs-2958	305	8	,	,	PUNCT
iajs-2958	305	9	m.	m.	NOUN
iajs-2958	305	10	;	;	PUNCT
iajs-2958	305	11	nethaji	nethaji	PROPN
iajs-2958	305	12	,	,	PUNCT
iajs-2958	305	13	o.	o.	NOUN
iajs-2958	305	14	on	on	ADP
iajs-2958	305	15	nano	nano	ADJ
iajs-2958	305	16	gβ	gβ	NOUN
iajs-2958	305	17	-	-	PUNCT
iajs-2958	305	18	closed	closed	ADJ
iajs-2958	305	19	sets	set	NOUN
iajs-2958	305	20	.	.	PUNCT
iajs-2958	306	1	2017	2017	NUM
iajs-2958	306	2	,	,	PUNCT
iajs-2958	306	3	5	5	NUM
iajs-2958	306	4	,	,	PUNCT
iajs-2958	306	5	377–382	377–382	NUM
iajs-2958	306	6	.	.	NOUN
iajs-2958	307	1	6	6	NUM
iajs-2958	307	2	.	.	X
iajs-2958	307	3	padma	padma	PROPN
iajs-2958	307	4	,	,	PUNCT
iajs-2958	307	5	a.	a.	NOUN
iajs-2958	307	6	;	;	PUNCT
iajs-2958	307	7	saraswathi	saraswathi	PROPN
iajs-2958	307	8	,	,	PUNCT
iajs-2958	307	9	m.	m.	NOUN
iajs-2958	307	10	;	;	PUNCT
iajs-2958	307	11	vadivel	vadivel	NOUN
iajs-2958	307	12	,	,	PUNCT
iajs-2958	307	13	a.	a.	NOUN
iajs-2958	307	14	;	;	PUNCT
iajs-2958	307	15	saravanakumar	saravanakumar	PROPN
iajs-2958	307	16	,	,	PUNCT
iajs-2958	307	17	g.	g.	PROPN
iajs-2958	307	18	new	new	ADJ
iajs-2958	307	19	notions	notion	NOUN
iajs-2958	307	20	of	of	ADP
iajs-2958	307	21	nano	nano	NOUN
iajs-2958	307	22	m	m	ADJ
iajs-2958	307	23	-	-	ADJ
iajs-2958	307	24	open	open	ADJ
iajs-2958	307	25	sets	set	NOUN
iajs-2958	307	26	.	.	PUNCT
iajs-2958	308	1	malaya	malaya	PROPN
iajs-2958	308	2	journal	journal	PROPN
iajs-2958	308	3	of	of	ADP
iajs-2958	308	4	matematik	matematik	PROPN
iajs-2958	308	5	,	,	PUNCT
iajs-2958	308	6	2019	2019	NUM
iajs-2958	308	7	,	,	PUNCT
iajs-2958	308	8	1	1	NUM
iajs-2958	308	9	,	,	PUNCT
iajs-2958	308	10	656	656	NUM
iajs-2958	308	11	-	-	SYM
iajs-2958	308	12	660	660	NUM
iajs-2958	308	13	.	.	NOUN
iajs-2958	309	1	7	7	X
iajs-2958	309	2	.	.	NUM
iajs-2958	309	3	rajasekaran	rajasekaran	NOUN
iajs-2958	309	4	,	,	PUNCT
iajs-2958	309	5	i.	i.	NOUN
iajs-2958	309	6	;	;	PUNCT
iajs-2958	309	7	meharin	meharin	NOUN
iajs-2958	309	8	,	,	PUNCT
iajs-2958	309	9	m.	m.	NOUN
iajs-2958	309	10	;	;	PUNCT
iajs-2958	309	11	nethaji	nethaji	PROPN
iajs-2958	309	12	,	,	PUNCT
iajs-2958	309	13	o.	o.	NOUN
iajs-2958	309	14	on	on	ADP
iajs-2958	309	15	new	new	ADJ
iajs-2958	309	16	classes	class	NOUN
iajs-2958	309	17	of	of	ADP
iajs-2958	309	18	some	some	DET
iajs-2958	309	19	nano	nano	ADJ
iajs-2958	309	20	open	open	ADJ
iajs-2958	309	21	sets	set	NOUN
iajs-2958	309	22	.	.	PUNCT
iajs-2958	310	1	international	international	ADJ
iajs-2958	310	2	journal	journal	NOUN
iajs-2958	310	3	of	of	ADP
iajs-2958	310	4	pure	pure	ADJ
iajs-2958	310	5	and	and	CCONJ
iajs-2958	310	6	applied	applied	ADJ
iajs-2958	310	7	mathematical	mathematical	ADJ
iajs-2958	310	8	sciences	science	NOUN
iajs-2958	310	9	,	,	PUNCT
iajs-2958	310	10	2017	2017	NUM
iajs-2958	310	11	,	,	PUNCT
iajs-2958	310	12	10	10	NUM
iajs-2958	310	13	,	,	PUNCT
iajs-2958	310	14	147	147	NUM
iajs-2958	310	15	-	-	SYM
iajs-2958	310	16	155	155	NUM
iajs-2958	310	17	.	.	NOUN
iajs-2958	310	18	8	8	NUM
iajs-2958	310	19	.	.	PUNCT
iajs-2958	311	1	pawlak	pawlak	ADJ
iajs-2958	311	2	,	,	PUNCT
iajs-2958	311	3	z.	z.	PROPN
iajs-2958	311	4	rough	rough	ADJ
iajs-2958	311	5	sets	set	NOUN
iajs-2958	311	6	.	.	PUNCT
iajs-2958	312	1	international	international	ADJ
iajs-2958	312	2	journal	journal	PROPN
iajs-2958	312	3	of	of	ADP
iajs-2958	312	4	computer	computer	PROPN
iajs-2958	312	5	&	&	CCONJ
iajs-2958	312	6	information	information	NOUN
iajs-2958	312	7	sciences	sciences	PROPN
iajs-2958	312	8	,	,	PUNCT
iajs-2958	312	9	1982	1982	NUM
iajs-2958	312	10	,	,	PUNCT
iajs-2958	312	11	11	11	NUM
iajs-2958	312	12	,	,	PUNCT
iajs-2958	312	13	341	341	NUM
iajs-2958	312	14	-	-	SYM
iajs-2958	312	15	356	356	NUM
iajs-2958	312	16	.	.	PUNCT