id	sid	tid	token	lemma	pos
ma-246	1	1	2025	2025	NUM
ma-246	1	2	ada	ada	PROPN
ma-246	1	3	academica	academica	PROPN
ma-246	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-246	1	5	.	.	PUNCT
ma-246	2	1	j.	j.	PROPN
ma-246	2	2	math	math	PROPN
ma-246	2	3	.	.	PUNCT
ma-246	3	1	anal	anal	ADJ
ma-246	3	2	.	.	PUNCT
ma-246	4	1	5	5	NUM
ma-246	4	2	(	(	PUNCT
ma-246	4	3	2025	2025	NUM
ma-246	4	4	)	)	PUNCT
ma-246	5	1	2doi	2doi	NUM
ma-246	5	2	:	:	PUNCT
ma-246	5	3	10.28924	10.28924	NUM
ma-246	5	4	/	/	SYM
ma-246	5	5	ada	ada	PROPN
ma-246	5	6	/	/	SYM
ma-246	5	7	ma.5.2	ma.5.2	PROPN
ma-246	5	8	on	on	ADP
ma-246	5	9	η	η	ADJ
ma-246	5	10	-	-	ADJ
ma-246	5	11	local	local	ADJ
ma-246	5	12	functions	function	NOUN
ma-246	5	13	in	in	ADP
ma-246	5	14	ideal	ideal	ADJ
ma-246	5	15	topological	topological	ADJ
ma-246	5	16	spaces	space	NOUN
ma-246	5	17	junvon	junvon	PROPN
ma-246	5	18	a.	a.	PROPN
ma-246	5	19	almocera∗	almocera∗	PROPN
ma-246	5	20	,	,	PUNCT
ma-246	5	21	lezel	lezel	ADJ
ma-246	5	22	m.	m.	NOUN
ma-246	5	23	tutanes	tutane	VERB
ma-246	5	24	department	department	PROPN
ma-246	5	25	of	of	ADP
ma-246	5	26	mathematics	mathematics	PROPN
ma-246	5	27	,	,	PUNCT
ma-246	5	28	college	college	NOUN
ma-246	5	29	of	of	ADP
ma-246	5	30	arts	art	NOUN
ma-246	5	31	and	and	CCONJ
ma-246	5	32	sciences	sciences	PROPN
ma-246	5	33	,	,	PUNCT
ma-246	5	34	bukidnon	bukidnon	NOUN
ma-246	5	35	state	state	PROPN
ma-246	5	36	university	university	PROPN
ma-246	5	37	,	,	PUNCT
ma-246	5	38	malaybalay	malaybalay	NOUN
ma-246	5	39	city	city	NOUN
ma-246	5	40	,	,	PUNCT
ma-246	5	41	bukidnon	bukidnon	NOUN
ma-246	5	42	,	,	PUNCT
ma-246	5	43	philippines	philippine	NOUN
ma-246	5	44	1901103646@student.buksu.edu.ph	1901103646@student.buksu.edu.ph	PRON
ma-246	5	45	,	,	PUNCT
ma-246	5	46	lezeltutanes@buksu.edu.ph	lezeltutanes@buksu.edu.ph	NOUN
ma-246	5	47	∗correspondence	∗correspondence	NOUN
ma-246	5	48	:	:	PUNCT
ma-246	5	49	1901103646@student.buksu.edu.ph	1901103646@student.buksu.edu.ph	NUM
ma-246	5	50	abstract	abstract	ADJ
ma-246	5	51	.	.	PUNCT
ma-246	6	1	this	this	DET
ma-246	6	2	study	study	NOUN
ma-246	6	3	introduces	introduce	NOUN
ma-246	6	4	and	and	CCONJ
ma-246	6	5	investigates	investigate	VERB
ma-246	6	6	a	a	DET
ma-246	6	7	new	new	ADJ
ma-246	6	8	local	local	ADJ
ma-246	6	9	function	function	NOUN
ma-246	6	10	called	call	VERB
ma-246	6	11	η	η	ADJ
ma-246	6	12	-	-	ADJ
ma-246	6	13	local	local	ADJ
ma-246	6	14	function	function	NOUN
ma-246	6	15	in	in	ADP
ma-246	6	16	idealtopological	idealtopological	ADJ
ma-246	6	17	space	space	NOUN
ma-246	6	18	(	(	PUNCT
ma-246	6	19	x	x	X
ma-246	6	20	,	,	PUNCT
ma-246	6	21	τ	τ	PROPN
ma-246	6	22	,	,	PUNCT
ma-246	6	23	i	i	NOUN
ma-246	6	24	)	)	PUNCT
ma-246	6	25	by	by	ADP
ma-246	6	26	using	use	VERB
ma-246	6	27	the	the	DET
ma-246	6	28	notion	notion	NOUN
ma-246	6	29	of	of	ADP
ma-246	6	30	η	η	ADJ
ma-246	6	31	-	-	ADJ
ma-246	6	32	open	open	ADJ
ma-246	6	33	sets	set	NOUN
ma-246	6	34	in	in	ADP
ma-246	6	35	topological	topological	ADJ
ma-246	6	36	space	space	NOUN
ma-246	6	37	(	(	PUNCT
ma-246	6	38	x	x	X
ma-246	6	39	,	,	PUNCT
ma-246	6	40	τ	τ	PROPN
ma-246	6	41	)	)	PUNCT
ma-246	6	42	.	.	PUNCT
ma-246	7	1	theoperator	theoperator	NOUN
ma-246	7	2	(	(	PUNCT
ma-246	7	3	·	·	PUNCT
ma-246	7	4	)	)	PUNCT
ma-246	7	5	∗η	∗η	NOUN
ma-246	7	6	:	:	PUNCT
ma-246	7	7	p(x	p(x	PROPN
ma-246	7	8	)	)	PUNCT
ma-246	7	9	→	→	SYM
ma-246	7	10	p(x	p(x	PROPN
ma-246	7	11	)	)	PUNCT
ma-246	7	12	is	be	AUX
ma-246	7	13	defined	define	VERB
ma-246	7	14	as	as	ADP
ma-246	7	15	(	(	PUNCT
ma-246	7	16	·	·	PUNCT
ma-246	7	17	)	)	PUNCT
ma-246	7	18	∗η(a	∗η(a	NOUN
ma-246	7	19	)	)	PUNCT
ma-246	7	20	=	=	PUNCT
ma-246	8	1	a∗η	a∗η	PROPN
ma-246	8	2	=	=	PRON
ma-246	8	3	{	{	PUNCT
ma-246	8	4	x	x	PUNCT
ma-246	8	5	∈	∈	PROPN
ma-246	8	6	x	x	X
ma-246	8	7	:	:	PUNCT
ma-246	8	8	a	a	DET
ma-246	8	9	∩	∩	ADJ
ma-246	8	10	u	u	NOUN
ma-246	8	11	/∈	/∈	PUNCT
ma-246	9	1	i	i	PRON
ma-246	9	2	for	for	ADP
ma-246	9	3	every	every	DET
ma-246	9	4	u	u	PROPN
ma-246	9	5	∈	∈	PROPN
ma-246	9	6	η	η	PROPN
ma-246	9	7	-	-	PROPN
ma-246	9	8	o(x	o(x	PROPN
ma-246	9	9	)	)	PUNCT
ma-246	9	10	}	}	PUNCT
ma-246	9	11	for	for	ADP
ma-246	9	12	each	each	DET
ma-246	9	13	a	a	DET
ma-246	9	14	⊆	⊆	NUM
ma-246	9	15	x	x	SYM
ma-246	9	16	,	,	PUNCT
ma-246	9	17	where	where	SCONJ
ma-246	9	18	η	η	PROPN
ma-246	9	19	-	-	PROPN
ma-246	9	20	o(x	o(x	PROPN
ma-246	9	21	)	)	PUNCT
ma-246	9	22	is	be	AUX
ma-246	9	23	the	the	DET
ma-246	9	24	set	set	NOUN
ma-246	9	25	of	of	ADP
ma-246	9	26	all	all	DET
ma-246	9	27	η	η	ADJ
ma-246	9	28	-	-	ADJ
ma-246	9	29	open	open	ADJ
ma-246	9	30	subset	subset	NOUN
ma-246	9	31	of	of	ADP
ma-246	9	32	x	x	SYM
ma-246	9	33	containing	contain	VERB
ma-246	9	34	x	x	PUNCT
ma-246	9	35	.	.	PUNCT
ma-246	10	1	thisstudy	thisstudy	NOUN
ma-246	10	2	establishes	establish	VERB
ma-246	10	3	some	some	DET
ma-246	10	4	properties	property	NOUN
ma-246	10	5	of	of	ADP
ma-246	10	6	a∗η	a∗η	PROPN
ma-246	10	7	including	include	VERB
ma-246	10	8	its	its	PRON
ma-246	10	9	relationships	relationship	NOUN
ma-246	10	10	to	to	ADP
ma-246	10	11	the	the	DET
ma-246	10	12	local	local	ADJ
ma-246	10	13	function	function	NOUN
ma-246	10	14	and	and	CCONJ
ma-246	10	15	localfunction	localfunction	NOUN
ma-246	10	16	γ∗	γ∗	NOUN
ma-246	10	17	in	in	ADP
ma-246	10	18	ideal	ideal	ADJ
ma-246	10	19	topological	topological	ADJ
ma-246	10	20	space	space	NOUN
ma-246	10	21	(	(	PUNCT
ma-246	10	22	x	x	X
ma-246	10	23	,	,	PUNCT
ma-246	10	24	τ	τ	PROPN
ma-246	10	25	,	,	PUNCT
ma-246	10	26	i	i	PROPN
ma-246	10	27	)	)	PUNCT
ma-246	10	28	.	.	PUNCT
ma-246	11	1	this	this	DET
ma-246	11	2	study	study	NOUN
ma-246	11	3	also	also	ADV
ma-246	11	4	introduces	introduce	VERB
ma-246	11	5	a	a	DET
ma-246	11	6	new	new	ADJ
ma-246	11	7	type	type	NOUN
ma-246	11	8	of	of	ADP
ma-246	11	9	closurecalled	closurecalle	VERB
ma-246	11	10	the	the	DET
ma-246	11	11	η	η	ADJ
ma-246	11	12	-	-	ADJ
ma-246	11	13	local	local	ADJ
ma-246	11	14	closure	closure	NOUN
ma-246	11	15	in	in	ADP
ma-246	11	16	ideal	ideal	ADJ
ma-246	11	17	topological	topological	ADJ
ma-246	11	18	space	space	NOUN
ma-246	11	19	(	(	PUNCT
ma-246	11	20	x	x	X
ma-246	11	21	,	,	PUNCT
ma-246	11	22	τ	τ	PROPN
ma-246	11	23	,	,	PUNCT
ma-246	11	24	i	i	PROPN
ma-246	11	25	)	)	PUNCT
ma-246	11	26	which	which	PRON
ma-246	11	27	is	be	AUX
ma-246	11	28	denoted	denote	VERB
ma-246	11	29	by	by	ADP
ma-246	11	30	cl∗η(a	cl∗η(a	PROPN
ma-246	11	31	)	)	PUNCT
ma-246	11	32	for	for	ADP
ma-246	11	33	each	each	DET
ma-246	11	34	a	a	DET
ma-246	11	35	⊆	⊆	NUM
ma-246	11	36	x	x	SYM
ma-246	11	37	.	.	PUNCT
ma-246	12	1	furthermore	furthermore	ADV
ma-246	12	2	,	,	PUNCT
ma-246	12	3	this	this	DET
ma-246	12	4	study	study	NOUN
ma-246	12	5	establishes	establish	VERB
ma-246	12	6	some	some	DET
ma-246	12	7	properties	property	NOUN
ma-246	12	8	of	of	ADP
ma-246	12	9	the	the	DET
ma-246	12	10	η	η	ADJ
ma-246	12	11	-	-	ADJ
ma-246	12	12	local	local	ADJ
ma-246	12	13	closure	closure	NOUN
ma-246	12	14	.	.	PUNCT
ma-246	13	1	1	1	X
ma-246	13	2	.	.	X
ma-246	13	3	introduction	introduction	NOUN
ma-246	13	4	the	the	DET
ma-246	13	5	concept	concept	NOUN
ma-246	13	6	of	of	ADP
ma-246	13	7	ideal	ideal	ADJ
ma-246	13	8	topological	topological	ADJ
ma-246	13	9	spaces	space	NOUN
ma-246	13	10	was	be	AUX
ma-246	13	11	first	first	ADV
ma-246	13	12	studied	study	VERB
ma-246	13	13	by	by	ADP
ma-246	13	14	kuratowski	kuratowski	NOUN
ma-246	14	1	[	[	X
ma-246	14	2	4	4	NUM
ma-246	14	3	]	]	PUNCT
ma-246	14	4	and	and	CCONJ
ma-246	14	5	vaidyanathas	vaidyanatha	NOUN
ma-246	14	6	-	-	ADJ
ma-246	14	7	wamy	wamy	ADJ
ma-246	14	8	[	[	X
ma-246	14	9	9	9	NUM
ma-246	14	10	]	]	PUNCT
ma-246	14	11	.	.	PUNCT
ma-246	15	1	the	the	DET
ma-246	15	2	notion	notion	NOUN
ma-246	15	3	of	of	ADP
ma-246	15	4	topological	topological	ADJ
ma-246	15	5	spaces	space	NOUN
ma-246	15	6	with	with	ADP
ma-246	15	7	ideals	ideal	NOUN
ma-246	15	8	were	be	AUX
ma-246	15	9	investigated	investigate	VERB
ma-246	15	10	by	by	ADP
ma-246	15	11	jankovic	jankovic	PROPN
ma-246	16	1	[	[	X
ma-246	16	2	3	3	NUM
ma-246	16	3	]	]	PUNCT
ma-246	16	4	.	.	PUNCT
ma-246	17	1	thereafter	thereafter	ADV
ma-246	17	2	,	,	PUNCT
ma-246	17	3	the	the	DET
ma-246	17	4	study	study	NOUN
ma-246	17	5	of	of	ADP
ma-246	17	6	ideal	ideal	ADJ
ma-246	17	7	topological	topological	ADJ
ma-246	17	8	spaces	space	NOUN
ma-246	17	9	attracts	attract	VERB
ma-246	17	10	the	the	DET
ma-246	17	11	attention	attention	NOUN
ma-246	17	12	of	of	ADP
ma-246	17	13	many	many	ADJ
ma-246	17	14	topologists	topologist	NOUN
ma-246	17	15	.	.	PUNCT
ma-246	18	1	recently	recently	ADV
ma-246	18	2	,	,	PUNCT
ma-246	18	3	al	al	PROPN
ma-246	18	4	-	-	PUNCT
ma-246	18	5	omari	omari	PROPN
ma-246	18	6	[	[	X
ma-246	18	7	1	1	NUM
ma-246	18	8	]	]	PUNCT
ma-246	18	9	have	have	AUX
ma-246	18	10	introduced	introduce	VERB
ma-246	18	11	and	and	CCONJ
ma-246	18	12	investigated	investigate	VERB
ma-246	18	13	the	the	DET
ma-246	18	14	notion	notion	NOUN
ma-246	18	15	of	of	ADP
ma-246	18	16	local	local	ADJ
ma-246	18	17	function	function	NOUN
ma-246	18	18	γ∗	γ∗	NOUN
ma-246	18	19	in	in	ADP
ma-246	18	20	an	an	DET
ma-246	18	21	ideal	ideal	ADJ
ma-246	18	22	topologicalspace	topologicalspace	NOUN
ma-246	18	23	and	and	CCONJ
ma-246	18	24	showed	show	VERB
ma-246	18	25	that	that	SCONJ
ma-246	18	26	γ∗	γ∗	NOUN
ma-246	18	27	is	be	AUX
ma-246	18	28	equivalent	equivalent	ADJ
ma-246	18	29	to	to	ADP
ma-246	18	30	the	the	DET
ma-246	18	31	δ	δ	PROPN
ma-246	18	32	-	-	ADJ
ma-246	18	33	local	local	ADJ
ma-246	18	34	function	function	NOUN
ma-246	18	35	due	due	ADP
ma-246	18	36	to	to	PART
ma-246	18	37	hatir	hatir	PROPN
ma-246	18	38	et	et	PROPN
ma-246	18	39	al	al	PROPN
ma-246	18	40	.	.	PUNCT
ma-246	19	1	[	[	X
ma-246	19	2	2	2	NUM
ma-246	19	3	]	]	PUNCT
ma-246	19	4	.	.	PUNCT
ma-246	20	1	in	in	ADP
ma-246	20	2	this	this	DET
ma-246	20	3	paper	paper	NOUN
ma-246	20	4	,	,	PUNCT
ma-246	20	5	the	the	DET
ma-246	20	6	researcher	researcher	NOUN
ma-246	20	7	defined	define	VERB
ma-246	20	8	a	a	DET
ma-246	20	9	new	new	ADJ
ma-246	20	10	type	type	NOUN
ma-246	20	11	of	of	ADP
ma-246	20	12	local	local	ADJ
ma-246	20	13	function	function	NOUN
ma-246	20	14	called	call	VERB
ma-246	20	15	the	the	DET
ma-246	20	16	η	η	ADJ
ma-246	20	17	-	-	ADJ
ma-246	20	18	local	local	ADJ
ma-246	20	19	function	function	NOUN
ma-246	20	20	in	in	ADP
ma-246	20	21	ideal	ideal	ADJ
ma-246	20	22	topologicalspaces	topologicalspace	NOUN
ma-246	20	23	by	by	ADP
ma-246	20	24	using	use	VERB
ma-246	20	25	the	the	DET
ma-246	20	26	η	η	ADJ
ma-246	20	27	-	-	ADJ
ma-246	20	28	open	open	ADJ
ma-246	20	29	set	set	NOUN
ma-246	20	30	of	of	ADP
ma-246	20	31	subbulakshmi	subbulakshmi	NOUN
ma-246	20	32	[	[	X
ma-246	20	33	8	8	NUM
ma-246	20	34	]	]	PUNCT
ma-246	20	35	and	and	CCONJ
ma-246	20	36	established	establish	VERB
ma-246	20	37	some	some	PRON
ma-246	20	38	of	of	ADP
ma-246	20	39	its	its	PRON
ma-246	20	40	properties	property	NOUN
ma-246	20	41	,	,	PUNCT
ma-246	20	42	includingits	includingit	VERB
ma-246	20	43	relationship	relationship	NOUN
ma-246	20	44	to	to	ADP
ma-246	20	45	the	the	DET
ma-246	20	46	local	local	ADJ
ma-246	20	47	function	function	NOUN
ma-246	20	48	and	and	CCONJ
ma-246	20	49	local	local	ADJ
ma-246	20	50	function	function	NOUN
ma-246	20	51	γ∗	γ∗	NOUN
ma-246	20	52	in	in	ADP
ma-246	20	53	ideal	ideal	ADJ
ma-246	20	54	topological	topological	ADJ
ma-246	20	55	spaces	space	NOUN
ma-246	20	56	.	.	PUNCT
ma-246	21	1	subsequently	subsequently	ADV
ma-246	21	2	,	,	PUNCT
ma-246	21	3	the	the	DET
ma-246	21	4	η	η	PROPN
ma-246	21	5	-	-	ADJ
ma-246	21	6	local	local	ADJ
ma-246	21	7	closure	closure	NOUN
ma-246	21	8	has	have	AUX
ma-246	21	9	been	be	AUX
ma-246	21	10	defined	define	VERB
ma-246	21	11	,	,	PUNCT
ma-246	21	12	and	and	CCONJ
ma-246	21	13	some	some	PRON
ma-246	21	14	of	of	ADP
ma-246	21	15	the	the	DET
ma-246	21	16	properties	property	NOUN
ma-246	21	17	are	be	AUX
ma-246	21	18	established	establish	VERB
ma-246	21	19	.	.	PUNCT
ma-246	22	1	2	2	X
ma-246	22	2	.	.	NUM
ma-246	22	3	preliminaries	preliminary	NOUN
ma-246	22	4	throughout	throughout	ADP
ma-246	22	5	this	this	DET
ma-246	22	6	paper	paper	NOUN
ma-246	22	7	(	(	PUNCT
ma-246	22	8	x	x	X
ma-246	22	9	,	,	PUNCT
ma-246	22	10	τ	τ	X
ma-246	22	11	)	)	PUNCT
ma-246	22	12	and	and	CCONJ
ma-246	22	13	(	(	PUNCT
ma-246	22	14	x	x	X
ma-246	22	15	,	,	PUNCT
ma-246	22	16	τ	τ	PROPN
ma-246	22	17	,	,	PUNCT
ma-246	22	18	i	i	NOUN
ma-246	22	19	)	)	PUNCT
ma-246	22	20	denote	denote	VERB
ma-246	22	21	a	a	DET
ma-246	22	22	topological	topological	ADJ
ma-246	22	23	space	space	NOUN
ma-246	22	24	and	and	CCONJ
ma-246	22	25	an	an	DET
ma-246	22	26	ideal	ideal	ADJ
ma-246	22	27	topologicalspace	topologicalspace	NOUN
ma-246	22	28	,	,	PUNCT
ma-246	22	29	respectively	respectively	ADV
ma-246	22	30	.	.	PUNCT
ma-246	23	1	the	the	DET
ma-246	23	2	members	member	NOUN
ma-246	23	3	of	of	ADP
ma-246	23	4	τ	τ	PROPN
ma-246	23	5	are	be	AUX
ma-246	23	6	called	call	VERB
ma-246	23	7	open	open	ADJ
ma-246	23	8	sets	set	NOUN
ma-246	23	9	and	and	CCONJ
ma-246	23	10	their	their	PRON
ma-246	23	11	complement	complement	NOUN
ma-246	23	12	are	be	AUX
ma-246	23	13	calledclosed	calledclose	VERB
ma-246	23	14	sets	set	NOUN
ma-246	23	15	.	.	PUNCT
ma-246	24	1	for	for	ADP
ma-246	24	2	any	any	DET
ma-246	24	3	subset	subset	NOUN
ma-246	24	4	a	a	PRON
ma-246	24	5	of	of	ADP
ma-246	24	6	x	x	PRON
ma-246	24	7	,	,	PUNCT
ma-246	24	8	the	the	DET
ma-246	24	9	closure	closure	NOUN
ma-246	24	10	and	and	CCONJ
ma-246	24	11	interior	interior	NOUN
ma-246	24	12	of	of	ADP
ma-246	24	13	a	a	PRON
ma-246	24	14	are	be	AUX
ma-246	24	15	denoted	denote	VERB
ma-246	24	16	by	by	ADP
ma-246	24	17	cl(a	cl(a	NOUN
ma-246	24	18	)	)	PUNCT
ma-246	24	19	and	and	CCONJ
ma-246	24	20	received	receive	VERB
ma-246	24	21	:	:	PUNCT
ma-246	24	22	4	4	NUM
ma-246	24	23	jun	jun	PROPN
ma-246	24	24	2024	2024	NUM
ma-246	24	25	.	.	PUNCT
ma-246	25	1	key	key	ADJ
ma-246	25	2	words	word	NOUN
ma-246	25	3	and	and	CCONJ
ma-246	25	4	phrases	phrase	NOUN
ma-246	25	5	.	.	PUNCT
ma-246	26	1	ideal	ideal	ADJ
ma-246	26	2	topoloical	topoloical	ADJ
ma-246	26	3	space	space	NOUN
ma-246	26	4	,	,	PUNCT
ma-246	26	5	η	η	NOUN
ma-246	26	6	-	-	ADJ
ma-246	26	7	open	open	ADJ
ma-246	26	8	set	set	NOUN
ma-246	26	9	;	;	PUNCT
ma-246	26	10	η	η	ADJ
ma-246	26	11	-	-	ADJ
ma-246	26	12	local	local	ADJ
ma-246	26	13	function	function	NOUN
ma-246	26	14	;	;	PUNCT
ma-246	26	15	η	η	PROPN
ma-246	26	16	-	-	ADJ
ma-246	26	17	local	local	ADJ
ma-246	26	18	closure.1	closure.1	PROPN
ma-246	26	19	https://adac.ee	https://adac.ee	PROPN
ma-246	26	20	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	26	21	eur	eur	PROPN
ma-246	26	22	.	.	PUNCT
ma-246	27	1	j.	j.	PROPN
ma-246	27	2	math	math	PROPN
ma-246	27	3	.	.	PUNCT
ma-246	28	1	anal	anal	PROPN
ma-246	28	2	.	.	PUNCT
ma-246	29	1	10.28924	10.28924	NUM
ma-246	29	2	/	/	SYM
ma-246	29	3	ada	ada	PROPN
ma-246	29	4	/	/	SYM
ma-246	29	5	ma.5.2	ma.5.2	PROPN
ma-246	29	6	2	2	NUM
ma-246	29	7	int(a	int(a	NOUN
ma-246	29	8	)	)	PUNCT
ma-246	29	9	,	,	PUNCT
ma-246	29	10	respectively	respectively	ADV
ma-246	29	11	.	.	PUNCT
ma-246	30	1	a	a	DET
ma-246	30	2	subset	subset	NOUN
ma-246	30	3	a	a	PRON
ma-246	30	4	of	of	ADP
ma-246	30	5	a	a	DET
ma-246	30	6	space	space	NOUN
ma-246	30	7	(	(	PUNCT
ma-246	30	8	x	x	X
ma-246	30	9	,	,	PUNCT
ma-246	30	10	τ	τ	X
ma-246	30	11	)	)	PUNCT
ma-246	30	12	is	be	AUX
ma-246	30	13	said	say	VERB
ma-246	30	14	to	to	PART
ma-246	30	15	be	be	AUX
ma-246	30	16	η	η	NOUN
ma-246	30	17	-	-	ADJ
ma-246	30	18	open	open	ADJ
ma-246	30	19	(	(	PUNCT
ma-246	30	20	resp	resp	NOUN
ma-246	30	21	.	.	PUNCT
ma-246	31	1	η	η	VERB
ma-246	31	2	-	-	ADJ
ma-246	31	3	closed	closed	ADJ
ma-246	31	4	)	)	PUNCT
ma-246	32	1	[	[	X
ma-246	32	2	8	8	X
ma-246	32	3	]	]	X
ma-246	32	4	if	if	SCONJ
ma-246	32	5	a	a	DET
ma-246	32	6	⊆	⊆	NUM
ma-246	32	7	int(cl(int(a	int(cl(int(a	NOUN
ma-246	32	8	)	)	PUNCT
ma-246	32	9	)	)	PUNCT
ma-246	32	10	)	)	PUNCT
ma-246	32	11	∪	∪	ADP
ma-246	32	12	cl(int(a	cl(int(a	PROPN
ma-246	32	13	)	)	PUNCT
ma-246	32	14	)	)	PUNCT
ma-246	32	15	(	(	PUNCT
ma-246	32	16	resp	resp	NOUN
ma-246	32	17	.	.	PUNCT
ma-246	33	1	a	a	DET
ma-246	33	2	⊇	⊇	ADJ
ma-246	33	3	cl(int(cl(a	cl(int(cl(a	NOUN
ma-246	33	4	)	)	PUNCT
ma-246	33	5	)	)	PUNCT
ma-246	33	6	)	)	PUNCT
ma-246	34	1	∩	∩	PROPN
ma-246	34	2	int(cl(a	int(cl(a	PROPN
ma-246	34	3	)	)	PUNCT
ma-246	34	4	)	)	PUNCT
ma-246	34	5	)	)	PUNCT
ma-246	34	6	.	.	PUNCT
ma-246	35	1	the	the	DET
ma-246	35	2	η	η	NOUN
ma-246	35	3	-	-	NOUN
ma-246	35	4	closure	closure	NOUN
ma-246	35	5	of	of	ADP
ma-246	35	6	a	a	DET
ma-246	35	7	isdefined	isdefine	VERB
ma-246	35	8	by	by	ADP
ma-246	35	9	the	the	DET
ma-246	35	10	intersection	intersection	NOUN
ma-246	35	11	of	of	ADP
ma-246	35	12	all	all	DET
ma-246	35	13	η	η	ADJ
ma-246	35	14	-	-	ADJ
ma-246	35	15	closed	closed	ADJ
ma-246	35	16	sets	set	NOUN
ma-246	35	17	containing	contain	VERB
ma-246	35	18	the	the	DET
ma-246	35	19	set	set	NOUN
ma-246	35	20	a	a	PRON
ma-246	35	21	and	and	CCONJ
ma-246	35	22	it	it	PRON
ma-246	35	23	is	be	AUX
ma-246	35	24	denoted	denote	VERB
ma-246	35	25	by	by	ADP
ma-246	35	26	η	η	NOUN
ma-246	35	27	-	-	NOUN
ma-246	35	28	cl(a	cl(a	NUM
ma-246	35	29	)	)	PUNCT
ma-246	36	1	[	[	X
ma-246	36	2	8].a	8].a	PRON
ma-246	36	3	subset	subset	VERB
ma-246	36	4	a	a	PRON
ma-246	36	5	of	of	ADP
ma-246	36	6	a	a	DET
ma-246	36	7	space	space	NOUN
ma-246	36	8	(	(	PUNCT
ma-246	36	9	x	x	X
ma-246	36	10	,	,	PUNCT
ma-246	36	11	τ	τ	X
ma-246	36	12	)	)	PUNCT
ma-246	36	13	is	be	AUX
ma-246	36	14	said	say	VERB
ma-246	36	15	to	to	PART
ma-246	36	16	be	be	AUX
ma-246	36	17	semi	semi	ADJ
ma-246	36	18	-	-	ADJ
ma-246	36	19	open	open	ADJ
ma-246	36	20	[	[	X
ma-246	36	21	6	6	NUM
ma-246	36	22	]	]	PUNCT
ma-246	36	23	if	if	SCONJ
ma-246	36	24	a	a	DET
ma-246	36	25	⊆	⊆	NUM
ma-246	36	26	cl(int(a	cl(int(a	NOUN
ma-246	36	27	)	)	PUNCT
ma-246	36	28	)	)	PUNCT
ma-246	36	29	.	.	PUNCT
ma-246	37	1	a	a	DET
ma-246	37	2	subset	subset	NOUN
ma-246	37	3	a	a	PRON
ma-246	37	4	of	of	ADP
ma-246	37	5	a	a	DET
ma-246	37	6	space	space	NOUN
ma-246	37	7	(	(	PUNCT
ma-246	37	8	x	x	X
ma-246	37	9	,	,	PUNCT
ma-246	37	10	τ	τ	X
ma-246	37	11	)	)	PUNCT
ma-246	37	12	is	be	AUX
ma-246	37	13	said	say	VERB
ma-246	37	14	to	to	PART
ma-246	37	15	be	be	AUX
ma-246	37	16	regular	regular	ADJ
ma-246	37	17	-	-	PUNCT
ma-246	37	18	open	open	ADJ
ma-246	38	1	[	[	X
ma-246	38	2	7	7	NUM
ma-246	38	3	]	]	X
ma-246	38	4	if	if	SCONJ
ma-246	38	5	a	a	PRON
ma-246	38	6	=	=	X
ma-246	38	7	int(cl(a	int(cl(a	PROPN
ma-246	38	8	)	)	PUNCT
ma-246	38	9	)	)	PUNCT
ma-246	38	10	.	.	PUNCT
ma-246	39	1	the	the	DET
ma-246	39	2	familiy	familiy	NOUN
ma-246	39	3	of	of	ADP
ma-246	39	4	all	all	DET
ma-246	39	5	η	η	NOUN
ma-246	39	6	-	-	ADJ
ma-246	39	7	open	open	ADJ
ma-246	39	8	(	(	PUNCT
ma-246	39	9	resp	resp	NOUN
ma-246	39	10	.	.	PUNCT
ma-246	40	1	semi	semi	ADJ
ma-246	40	2	-	-	ADJ
ma-246	40	3	open	open	ADJ
ma-246	40	4	,	,	PUNCT
ma-246	40	5	regular	regular	ADJ
ma-246	40	6	open	open	ADJ
ma-246	40	7	)	)	PUNCT
ma-246	40	8	sets	set	NOUN
ma-246	40	9	in	in	ADP
ma-246	40	10	x	x	PROPN
ma-246	40	11	is	be	AUX
ma-246	40	12	denoted	denote	VERB
ma-246	40	13	by	by	ADP
ma-246	40	14	η	η	PROPN
ma-246	40	15	-	-	PROPN
ma-246	40	16	o(x	o(x	PROPN
ma-246	40	17	)	)	PUNCT
ma-246	40	18	(	(	PUNCT
ma-246	40	19	resp	resp	NOUN
ma-246	40	20	.	.	PUNCT
ma-246	40	21	so(x	so(x	NUM
ma-246	40	22	)	)	PUNCT
ma-246	40	23	,	,	PUNCT
ma-246	40	24	ro(x)).an	ro(x)).an	PROPN
ma-246	40	25	ideal	ideal	NOUN
ma-246	40	26	i	i	PRON
ma-246	41	1	[	[	X
ma-246	41	2	5	5	X
ma-246	41	3	]	]	PUNCT
ma-246	41	4	on	on	ADP
ma-246	41	5	a	a	DET
ma-246	41	6	topological	topological	ADJ
ma-246	41	7	spaces	space	NOUN
ma-246	41	8	(	(	PUNCT
ma-246	41	9	x	x	X
ma-246	41	10	,	,	PUNCT
ma-246	41	11	τ	τ	X
ma-246	41	12	)	)	PUNCT
ma-246	41	13	is	be	AUX
ma-246	41	14	a	a	DET
ma-246	41	15	nonempty	nonempty	ADJ
ma-246	41	16	collection	collection	NOUN
ma-246	41	17	of	of	ADP
ma-246	41	18	subsets	subset	NOUN
ma-246	41	19	of	of	ADP
ma-246	41	20	x	x	PRON
ma-246	41	21	,	,	PUNCT
ma-246	41	22	whichsatisfies	whichsatisfie	NOUN
ma-246	41	23	(	(	PUNCT
ma-246	41	24	i	i	NOUN
ma-246	41	25	)	)	PUNCT
ma-246	42	1	a	a	DET
ma-246	42	2	∈	∈	NOUN
ma-246	42	3	i	i	PRON
ma-246	42	4	and	and	CCONJ
ma-246	42	5	b	b	X
ma-246	42	6	∈	∈	PROPN
ma-246	42	7	i	i	PRON
ma-246	42	8	implies	imply	VERB
ma-246	42	9	a	a	DET
ma-246	42	10	∪	∪	X
ma-246	42	11	b	b	NOUN
ma-246	42	12	∈	∈	PROPN
ma-246	42	13	i	i	PRON
ma-246	42	14	;	;	PUNCT
ma-246	42	15	and	and	CCONJ
ma-246	42	16	(	(	PUNCT
ma-246	42	17	ii	ii	NOUN
ma-246	42	18	)	)	PUNCT
ma-246	43	1	a	a	PRON
ma-246	43	2	∈	∈	PROPN
ma-246	44	1	i	i	PRON
ma-246	44	2	and	and	CCONJ
ma-246	44	3	b	b	X
ma-246	44	4	⊆	⊆	NUM
ma-246	44	5	a	a	PRON
ma-246	44	6	implies	imply	VERB
ma-246	44	7	b	b	X
ma-246	44	8	∈	∈	ADJ
ma-246	44	9	i	i	PRON
ma-246	44	10	.	.	PUNCT
ma-246	45	1	then	then	ADV
ma-246	45	2	thetriplet	thetriplet	PROPN
ma-246	45	3	(	(	PUNCT
ma-246	45	4	x	x	PROPN
ma-246	45	5	,	,	PUNCT
ma-246	45	6	τ	τ	PROPN
ma-246	45	7	,	,	PUNCT
ma-246	45	8	i	i	PROPN
ma-246	45	9	)	)	PUNCT
ma-246	45	10	is	be	AUX
ma-246	45	11	called	call	VERB
ma-246	45	12	an	an	DET
ma-246	45	13	ideal	ideal	ADJ
ma-246	45	14	topological	topological	ADJ
ma-246	45	15	space	space	NOUN
ma-246	45	16	.	.	PUNCT
ma-246	46	1	if	if	SCONJ
ma-246	46	2	p(x	p(x	PROPN
ma-246	46	3	)	)	PUNCT
ma-246	46	4	is	be	AUX
ma-246	46	5	the	the	DET
ma-246	46	6	set	set	NOUN
ma-246	46	7	of	of	ADP
ma-246	46	8	all	all	DET
ma-246	46	9	subsets	subset	NOUN
ma-246	46	10	of	of	ADP
ma-246	46	11	x	x	PRON
ma-246	46	12	,	,	PUNCT
ma-246	46	13	a	a	DET
ma-246	46	14	setoperator	setoperator	NOUN
ma-246	46	15	(	(	PUNCT
ma-246	46	16	·	·	PUNCT
ma-246	46	17	)	)	PUNCT
ma-246	46	18	∗	∗	NOUN
ma-246	46	19	:	:	PUNCT
ma-246	46	20	p(x)→	p(x)→	X
ma-246	46	21	p(x	p(x	NOUN
ma-246	46	22	)	)	PUNCT
ma-246	46	23	called	call	VERB
ma-246	46	24	a	a	DET
ma-246	46	25	local	local	ADJ
ma-246	46	26	function	function	NOUN
ma-246	46	27	[	[	X
ma-246	46	28	3,9	3,9	NUM
ma-246	46	29	]	]	PUNCT
ma-246	46	30	of	of	ADP
ma-246	46	31	a	a	PRON
ma-246	46	32	with	with	ADP
ma-246	46	33	respect	respect	NOUN
ma-246	46	34	to	to	ADP
ma-246	46	35	τ	τ	PROPN
ma-246	46	36	and	and	CCONJ
ma-246	46	37	i	i	PRON
ma-246	46	38	is	be	AUX
ma-246	46	39	defined	define	VERB
ma-246	46	40	asfollows	asfollow	VERB
ma-246	46	41	:	:	PUNCT
ma-246	46	42	for	for	ADP
ma-246	46	43	a	a	DET
ma-246	46	44	⊆	⊆	NUM
ma-246	46	45	x	x	SYM
ma-246	46	46	,	,	PUNCT
ma-246	46	47	a∗(i	a∗(i	PROPN
ma-246	46	48	,	,	PUNCT
ma-246	46	49	τ	τ	X
ma-246	46	50	)	)	PUNCT
ma-246	46	51	=	=	PRON
ma-246	47	1	{	{	PUNCT
ma-246	47	2	x	x	PUNCT
ma-246	47	3	∈	∈	PROPN
ma-246	47	4	x	x	X
ma-246	47	5	:	:	PUNCT
ma-246	47	6	u∩a	u∩a	PROPN
ma-246	47	7	/∈	/∈	PUNCT
ma-246	48	1	i	i	PRON
ma-246	48	2	,	,	PUNCT
ma-246	48	3	for	for	ADP
ma-246	48	4	every	every	DET
ma-246	48	5	u	u	PROPN
ma-246	48	6	∈	∈	PROPN
ma-246	48	7	τ(x	τ(x	NOUN
ma-246	48	8	)	)	PUNCT
ma-246	48	9	}	}	PUNCT
ma-246	48	10	where	where	SCONJ
ma-246	48	11	τ(x	τ(x	NOUN
ma-246	48	12	)	)	PUNCT
ma-246	48	13	=	=	PRON
ma-246	48	14	{	{	PUNCT
ma-246	48	15	u	u	X
ma-246	48	16	∈	∈	PROPN
ma-246	48	17	τ	τ	X
ma-246	48	18	:	:	PUNCT
ma-246	48	19	x	x	SYM
ma-246	48	20	∈	∈	PROPN
ma-246	48	21	u	u	NOUN
ma-246	48	22	}	}	PUNCT
ma-246	48	23	.	.	PUNCT
ma-246	49	1	a∗(i	a∗(i	PROPN
ma-246	49	2	,	,	PUNCT
ma-246	49	3	τ	τ	X
ma-246	49	4	)	)	PUNCT
ma-246	49	5	can	can	AUX
ma-246	49	6	simply	simply	ADV
ma-246	49	7	be	be	AUX
ma-246	49	8	written	write	VERB
ma-246	49	9	as	as	ADP
ma-246	49	10	a∗.	a∗.	NOUN
ma-246	49	11	for	for	ADP
ma-246	49	12	every	every	DET
ma-246	49	13	ideal	ideal	ADJ
ma-246	49	14	topological	topological	ADJ
ma-246	49	15	space	space	NOUN
ma-246	49	16	,	,	PUNCT
ma-246	49	17	there	there	PRON
ma-246	49	18	exists	exist	VERB
ma-246	49	19	a	a	DET
ma-246	49	20	topology	topology	NOUN
ma-246	49	21	τ∗(i	τ∗(i	PROPN
ma-246	49	22	,	,	PUNCT
ma-246	49	23	τ	τ	PROPN
ma-246	49	24	)	)	PUNCT
ma-246	49	25	or	or	CCONJ
ma-246	49	26	briefly	briefly	ADV
ma-246	49	27	τ∗	τ∗	VERB
ma-246	50	1	[	[	X
ma-246	50	2	3	3	NUM
ma-246	50	3	]	]	X
ma-246	50	4	,	,	PUNCT
ma-246	50	5	finer	fine	ADJ
ma-246	50	6	than	than	ADP
ma-246	50	7	τ	τ	PROPN
ma-246	50	8	,	,	PUNCT
ma-246	50	9	generated	generate	VERB
ma-246	50	10	by	by	ADP
ma-246	50	11	the	the	DET
ma-246	50	12	b(i	b(i	PROPN
ma-246	50	13	,	,	PUNCT
ma-246	50	14	τ	τ	X
ma-246	50	15	)	)	PUNCT
ma-246	50	16	=	=	SYM
ma-246	50	17	{	{	PUNCT
ma-246	50	18	u	u	NOUN
ma-246	50	19	\	\	PROPN
ma-246	50	20	j	j	PROPN
ma-246	50	21	:	:	PUNCT
ma-246	50	22	u	u	PROPN
ma-246	50	23	∈	∈	PROPN
ma-246	50	24	τ	τ	X
ma-246	50	25	and	and	CCONJ
ma-246	50	26	j	j	PROPN
ma-246	50	27	∈	∈	PROPN
ma-246	50	28	i},however	i},however	PROPN
ma-246	50	29	,	,	PUNCT
ma-246	50	30	b(i	b(i	PROPN
ma-246	50	31	,	,	PUNCT
ma-246	50	32	τ	τ	X
ma-246	50	33	)	)	PUNCT
ma-246	50	34	is	be	AUX
ma-246	50	35	not	not	PART
ma-246	50	36	a	a	DET
ma-246	50	37	topology	topology	NOUN
ma-246	50	38	in	in	ADP
ma-246	50	39	general	general	ADJ
ma-246	50	40	.	.	PUNCT
ma-246	51	1	additionally	additionally	ADV
ma-246	51	2	,	,	PUNCT
ma-246	51	3	cl∗(a	cl∗(a	NOUN
ma-246	51	4	)	)	PUNCT
ma-246	51	5	=	=	SYM
ma-246	52	1	a∪a∗	a∪a∗	PROPN
ma-246	52	2	defines	define	VERB
ma-246	52	3	a	a	DET
ma-246	52	4	kuratowskiclosure	kuratowskiclosure	NOUN
ma-246	52	5	operator	operator	NOUN
ma-246	52	6	[	[	X
ma-246	52	7	5	5	NUM
ma-246	52	8	]	]	PUNCT
ma-246	52	9	for	for	ADP
ma-246	52	10	τ∗.	τ∗.	NOUN
ma-246	52	11	a	a	DET
ma-246	52	12	subset	subset	NOUN
ma-246	52	13	a	a	PRON
ma-246	52	14	of	of	ADP
ma-246	52	15	an	an	DET
ma-246	52	16	ideal	ideal	ADJ
ma-246	52	17	topological	topological	ADJ
ma-246	52	18	spaces	space	NOUN
ma-246	52	19	is	be	AUX
ma-246	52	20	τ∗-closed	τ∗-close	VERB
ma-246	52	21	set	set	ADJ
ma-246	52	22	or	or	CCONJ
ma-246	52	23	∗-closedset	∗-closedset	VERB
ma-246	52	24	[	[	X
ma-246	52	25	3	3	X
ma-246	52	26	]	]	PUNCT
ma-246	52	27	if	if	SCONJ
ma-246	52	28	a∗	a∗	PROPN
ma-246	52	29	⊆	⊆	NUM
ma-246	52	30	a.	a.	NOUN
ma-246	52	31	let	let	VERB
ma-246	52	32	(	(	PUNCT
ma-246	52	33	x	x	X
ma-246	52	34	,	,	PUNCT
ma-246	52	35	τ	τ	PROPN
ma-246	52	36	,	,	PUNCT
ma-246	52	37	i	i	PRON
ma-246	52	38	)	)	PUNCT
ma-246	52	39	be	be	VERB
ma-246	52	40	an	an	DET
ma-246	52	41	ideal	ideal	ADJ
ma-246	52	42	topological	topological	ADJ
ma-246	52	43	spaces	space	NOUN
ma-246	52	44	and	and	CCONJ
ma-246	52	45	a	a	DET
ma-246	52	46	be	be	AUX
ma-246	52	47	a	a	DET
ma-246	52	48	subset	subset	NOUN
ma-246	52	49	of	of	ADP
ma-246	52	50	x	x	X
ma-246	52	51	.	.	PUNCT
ma-246	53	1	then	then	ADV
ma-246	53	2	γ∗(a)(i	γ∗(a)(i	NUM
ma-246	53	3	,	,	PUNCT
ma-246	53	4	τ	τ	X
ma-246	53	5	)	)	PUNCT
ma-246	53	6	=	=	PRON
ma-246	53	7	{	{	PUNCT
ma-246	53	8	x	x	PUNCT
ma-246	53	9	∈	∈	PROPN
ma-246	53	10	x	x	X
ma-246	53	11	:	:	PUNCT
ma-246	53	12	a∩u	a∩u	PROPN
ma-246	53	13	/∈	/∈	PUNCT
ma-246	54	1	i	i	PRON
ma-246	54	2	,	,	PUNCT
ma-246	54	3	for	for	ADP
ma-246	54	4	every	every	DET
ma-246	54	5	u	u	PROPN
ma-246	54	6	∈	∈	PROPN
ma-246	54	7	ro(x	ro(x	PUNCT
ma-246	54	8	)	)	PUNCT
ma-246	54	9	}	}	PUNCT
ma-246	55	1	where	where	SCONJ
ma-246	55	2	ro(x	ro(x	VERB
ma-246	55	3	)	)	PUNCT
ma-246	55	4	=	=	SYM
ma-246	55	5	{	{	PUNCT
ma-246	55	6	u	u	NOUN
ma-246	55	7	∈	∈	PROPN
ma-246	55	8	ro(x	ro(x	PUNCT
ma-246	55	9	)	)	PUNCT
ma-246	55	10	:	:	PUNCT
ma-246	56	1	x	x	X
ma-246	56	2	∈	∈	X
ma-246	56	3	u	u	NOUN
ma-246	56	4	}	}	PUNCT
ma-246	56	5	.	.	PUNCT
ma-246	57	1	γ∗(a)(i	γ∗(a)(i	NOUN
ma-246	57	2	,	,	PUNCT
ma-246	57	3	τ	τ	X
ma-246	57	4	)	)	PUNCT
ma-246	57	5	can	can	AUX
ma-246	57	6	simply	simply	ADV
ma-246	57	7	be	be	AUX
ma-246	57	8	denoted	denote	VERB
ma-246	57	9	as	as	ADP
ma-246	57	10	γ∗(a	γ∗(a	NOUN
ma-246	57	11	)	)	PUNCT
ma-246	58	1	[	[	X
ma-246	58	2	1	1	NUM
ma-246	58	3	]	]	PUNCT
ma-246	58	4	.	.	PUNCT
ma-246	59	1	3	3	X
ma-246	59	2	.	.	X
ma-246	59	3	η	η	ADJ
ma-246	59	4	-	-	ADJ
ma-246	59	5	local	local	ADJ
ma-246	59	6	functions	function	NOUN
ma-246	59	7	definition	definition	NOUN
ma-246	59	8	1	1	NUM
ma-246	59	9	.	.	PUNCT
ma-246	60	1	let	let	VERB
ma-246	60	2	(	(	PUNCT
ma-246	60	3	x	x	X
ma-246	60	4	,	,	PUNCT
ma-246	60	5	τ	τ	PROPN
ma-246	60	6	,	,	PUNCT
ma-246	60	7	i	i	PRON
ma-246	60	8	)	)	PUNCT
ma-246	60	9	be	be	VERB
ma-246	60	10	an	an	DET
ma-246	60	11	ideal	ideal	ADJ
ma-246	60	12	topological	topological	ADJ
ma-246	60	13	space	space	NOUN
ma-246	60	14	.	.	PUNCT
ma-246	61	1	then	then	ADV
ma-246	61	2	the	the	DET
ma-246	61	3	operator	operator	NOUN
ma-246	61	4	(	(	PUNCT
ma-246	61	5	·	·	PUNCT
ma-246	61	6	)	)	PUNCT
ma-246	61	7	∗η	∗η	NOUN
ma-246	61	8	:	:	PUNCT
ma-246	61	9	p(x)→	p(x)→	X
ma-246	61	10	p(x	p(x	NOUN
ma-246	61	11	)	)	PUNCT
ma-246	61	12	is	be	AUX
ma-246	61	13	defined	define	VERB
ma-246	61	14	as	as	ADP
ma-246	61	15	for	for	ADP
ma-246	61	16	a	a	DET
ma-246	61	17	⊆	⊆	NUM
ma-246	61	18	x	x	SYM
ma-246	61	19	,	,	PUNCT
ma-246	61	20	a∗η	a∗η	PROPN
ma-246	61	21	(	(	PUNCT
ma-246	61	22	i	i	PROPN
ma-246	61	23	,	,	PUNCT
ma-246	61	24	η	η	PROPN
ma-246	61	25	-	-	PROPN
ma-246	61	26	o(x	o(x	PROPN
ma-246	61	27	)	)	PUNCT
ma-246	61	28	)	)	PUNCT
ma-246	62	1	=	=	PRON
ma-246	62	2	{	{	PUNCT
ma-246	62	3	x	x	PUNCT
ma-246	62	4	∈	∈	PROPN
ma-246	62	5	x	x	X
ma-246	62	6	:	:	PUNCT
ma-246	62	7	a	a	DET
ma-246	62	8	∩	∩	ADJ
ma-246	62	9	u	u	NOUN
ma-246	62	10	/∈	/∈	PUNCT
ma-246	63	1	i	i	PRON
ma-246	63	2	,	,	PUNCT
ma-246	63	3	for	for	ADP
ma-246	63	4	every	every	DET
ma-246	63	5	u	u	PROPN
ma-246	63	6	∈	∈	PROPN
ma-246	63	7	η	η	PROPN
ma-246	63	8	-	-	PROPN
ma-246	63	9	o(x	o(x	PROPN
ma-246	63	10	)	)	PUNCT
ma-246	63	11	}	}	PUNCT
ma-246	64	1	where	where	SCONJ
ma-246	64	2	η	η	PROPN
ma-246	64	3	-	-	ADJ
ma-246	64	4	o(x	o(x	PROPN
ma-246	64	5	)	)	PUNCT
ma-246	64	6	=	=	PRON
ma-246	64	7	{	{	PUNCT
ma-246	64	8	u	u	NOUN
ma-246	64	9	∈	∈	PROPN
ma-246	64	10	η	η	PROPN
ma-246	64	11	-	-	PROPN
ma-246	64	12	o(x	o(x	PROPN
ma-246	64	13	)	)	PUNCT
ma-246	64	14	:	:	PUNCT
ma-246	64	15	x	x	X
ma-246	64	16	∈	∈	X
ma-246	64	17	u	u	NOUN
ma-246	64	18	}	}	PUNCT
ma-246	64	19	is	be	AUX
ma-246	64	20	called	call	VERB
ma-246	64	21	the	the	DET
ma-246	64	22	η	η	ADJ
ma-246	64	23	-	-	ADJ
ma-246	64	24	local	local	ADJ
ma-246	64	25	f	f	PROPN
ma-246	64	26	unction	unction	NOUN
ma-246	64	27	of	of	ADP
ma-246	64	28	a	a	PRON
ma-246	64	29	with	with	ADP
ma-246	64	30	respect	respect	NOUN
ma-246	64	31	to	to	ADP
ma-246	64	32	i	i	PRON
ma-246	64	33	and	and	CCONJ
ma-246	64	34	η	η	PROPN
ma-246	64	35	-	-	PROPN
ma-246	64	36	o(x	o(x	PROPN
ma-246	64	37	)	)	PUNCT
ma-246	64	38	.	.	PUNCT
ma-246	65	1	a∗η	a∗η	PROPN
ma-246	65	2	(	(	PUNCT
ma-246	65	3	i	i	PROPN
ma-246	65	4	,	,	PUNCT
ma-246	65	5	η	η	PROPN
ma-246	65	6	-	-	PROPN
ma-246	65	7	o(x	o(x	PROPN
ma-246	65	8	)	)	PUNCT
ma-246	65	9	)	)	PUNCT
ma-246	65	10	can	can	AUX
ma-246	65	11	simply	simply	ADV
ma-246	65	12	be	be	AUX
ma-246	65	13	denoted	denote	VERB
ma-246	65	14	by	by	ADP
ma-246	65	15	a∗η	a∗η	PROPN
ma-246	65	16	.	.	PUNCT
ma-246	66	1	example	example	NOUN
ma-246	67	1	1	1	NUM
ma-246	67	2	.	.	PUNCT
ma-246	68	1	let	let	VERB
ma-246	68	2	(	(	PUNCT
ma-246	68	3	x	x	X
ma-246	68	4	,	,	PUNCT
ma-246	68	5	τ	τ	PROPN
ma-246	68	6	,	,	PUNCT
ma-246	68	7	i	i	PRON
ma-246	68	8	)	)	PUNCT
ma-246	68	9	be	be	VERB
ma-246	68	10	an	an	DET
ma-246	68	11	ideal	ideal	ADJ
ma-246	68	12	topological	topological	ADJ
ma-246	68	13	where	where	SCONJ
ma-246	68	14	x	x	X
ma-246	68	15	=	=	PRON
ma-246	68	16	{	{	PUNCT
ma-246	68	17	a	a	PRON
ma-246	68	18	,	,	PUNCT
ma-246	68	19	b	b	NOUN
ma-246	68	20	,	,	PUNCT
ma-246	68	21	c	c	NOUN
ma-246	68	22	,	,	PUNCT
ma-246	68	23	d	d	NOUN
ma-246	68	24	}	}	PUNCT
ma-246	68	25	,	,	PUNCT
ma-246	68	26	τ	τ	X
ma-246	68	27	=	=	PUNCT
ma-246	68	28	{	{	PUNCT
ma-246	68	29	x,∅	x,∅	PROPN
ma-246	68	30	,	,	PUNCT
ma-246	68	31	{	{	PUNCT
ma-246	68	32	a	a	X
ma-246	68	33	}	}	PUNCT
ma-246	68	34	,	,	PUNCT
ma-246	68	35	{	{	PUNCT
ma-246	68	36	b	b	NOUN
ma-246	68	37	}	}	PUNCT
ma-246	68	38	,	,	PUNCT
ma-246	68	39	{	{	PUNCT
ma-246	68	40	a	a	DET
ma-246	68	41	,	,	PUNCT
ma-246	68	42	b	b	NOUN
ma-246	68	43	}	}	PUNCT
ma-246	68	44	,	,	PUNCT
ma-246	68	45	{	{	PUNCT
ma-246	68	46	a	a	DET
ma-246	68	47	,	,	PUNCT
ma-246	68	48	b	b	NOUN
ma-246	68	49	,	,	PUNCT
ma-246	68	50	c	c	NOUN
ma-246	68	51	}	}	PUNCT
ma-246	68	52	,	,	PUNCT
ma-246	68	53	{	{	PUNCT
ma-246	68	54	a	a	DET
ma-246	68	55	,	,	PUNCT
ma-246	68	56	b	b	NOUN
ma-246	68	57	,	,	PUNCT
ma-246	68	58	d	d	NOUN
ma-246	68	59	}	}	PUNCT
ma-246	68	60	}	}	PUNCT
ma-246	68	61	,	,	PUNCT
ma-246	68	62	and	and	CCONJ
ma-246	68	63	i	i	PRON
ma-246	68	64	=	=	PUNCT
ma-246	68	65	{	{	PUNCT
ma-246	68	66	∅	∅	NOUN
ma-246	68	67	,	,	PUNCT
ma-246	68	68	{	{	PUNCT
ma-246	68	69	c	c	NOUN
ma-246	68	70	}	}	PUNCT
ma-246	68	71	,	,	PUNCT
ma-246	68	72	{	{	PUNCT
ma-246	68	73	d	d	X
ma-246	68	74	}	}	PUNCT
ma-246	68	75	,	,	PUNCT
ma-246	68	76	{	{	PUNCT
ma-246	68	77	c	c	X
ma-246	68	78	,	,	PUNCT
ma-246	68	79	d	d	NOUN
ma-246	68	80	}	}	PUNCT
ma-246	68	81	}	}	PUNCT
ma-246	68	82	.	.	PUNCT
ma-246	69	1	then	then	ADV
ma-246	69	2	η	η	PROPN
ma-246	69	3	-	-	PROPN
ma-246	69	4	o(x	o(x	PROPN
ma-246	69	5	)	)	PUNCT
ma-246	69	6	=	=	SYM
ma-246	69	7	{	{	PUNCT
ma-246	69	8	∅	∅	NOUN
ma-246	69	9	,	,	PUNCT
ma-246	69	10	x	x	X
ma-246	69	11	,	,	PUNCT
ma-246	69	12	{	{	PUNCT
ma-246	69	13	a	a	X
ma-246	69	14	}	}	PUNCT
ma-246	69	15	,	,	PUNCT
ma-246	69	16	{	{	PUNCT
ma-246	69	17	b	b	NOUN
ma-246	69	18	}	}	PUNCT
ma-246	69	19	,	,	PUNCT
ma-246	69	20	{	{	PUNCT
ma-246	69	21	a	a	DET
ma-246	69	22	,	,	PUNCT
ma-246	69	23	b	b	NOUN
ma-246	69	24	}	}	PUNCT
ma-246	69	25	,	,	PUNCT
ma-246	69	26	{	{	PUNCT
ma-246	69	27	a	a	X
ma-246	69	28	,	,	PUNCT
ma-246	69	29	c	c	NOUN
ma-246	69	30	}	}	PUNCT
ma-246	69	31	,	,	PUNCT
ma-246	69	32	{	{	PUNCT
ma-246	69	33	a	a	DET
ma-246	69	34	,	,	PUNCT
ma-246	69	35	d	d	NOUN
ma-246	69	36	}	}	PUNCT
ma-246	69	37	,	,	PUNCT
ma-246	69	38	{	{	PUNCT
ma-246	69	39	b	b	X
ma-246	69	40	,	,	PUNCT
ma-246	69	41	c	c	NOUN
ma-246	69	42	}	}	PUNCT
ma-246	69	43	,	,	PUNCT
ma-246	69	44	{	{	PUNCT
ma-246	69	45	b	b	X
ma-246	69	46	,	,	PUNCT
ma-246	69	47	d	d	NOUN
ma-246	69	48	}	}	PUNCT
ma-246	69	49	,	,	PUNCT
ma-246	69	50	{	{	PUNCT
ma-246	69	51	a	a	DET
ma-246	69	52	,	,	PUNCT
ma-246	69	53	b	b	NOUN
ma-246	69	54	,	,	PUNCT
ma-246	69	55	c	c	NOUN
ma-246	69	56	}	}	PUNCT
ma-246	69	57	,	,	PUNCT
ma-246	69	58	{	{	PUNCT
ma-246	69	59	a	a	PRON
ma-246	69	60	,	,	PUNCT
ma-246	69	61	c	c	NOUN
ma-246	69	62	,	,	PUNCT
ma-246	69	63	d	d	NOUN
ma-246	69	64	}	}	PUNCT
ma-246	69	65	,	,	PUNCT
ma-246	69	66	{	{	PUNCT
ma-246	69	67	a	a	DET
ma-246	69	68	,	,	PUNCT
ma-246	69	69	b	b	NOUN
ma-246	69	70	,	,	PUNCT
ma-246	69	71	d	d	NOUN
ma-246	69	72	}	}	PUNCT
ma-246	69	73	,	,	PUNCT
ma-246	69	74	{	{	PUNCT
ma-246	69	75	b	b	X
ma-246	69	76	,	,	PUNCT
ma-246	69	77	c	c	NOUN
ma-246	69	78	,	,	PUNCT
ma-246	69	79	d	d	NOUN
ma-246	69	80	}	}	PUNCT
ma-246	69	81	}	}	PUNCT
ma-246	69	82	.	.	PUNCT
ma-246	70	1	now	now	ADV
ma-246	70	2	,	,	PUNCT
ma-246	70	3	let	let	VERB
ma-246	70	4	a	a	DET
ma-246	70	5	=	=	X
ma-246	70	6	{	{	PUNCT
ma-246	70	7	a	a	PROPN
ma-246	70	8	,	,	PUNCT
ma-246	70	9	b	b	NOUN
ma-246	70	10	,	,	PUNCT
ma-246	70	11	d	d	NOUN
ma-246	70	12	}	}	PUNCT
ma-246	70	13	.	.	PUNCT
ma-246	71	1	then	then	ADV
ma-246	71	2	by	by	ADP
ma-246	71	3	definition	definition	NOUN
ma-246	71	4	1	1	NUM
ma-246	71	5	,	,	PUNCT
ma-246	71	6	a∗η	a∗η	PROPN
ma-246	71	7	=	=	SYM
ma-246	71	8	{	{	PUNCT
ma-246	71	9	a	a	PRON
ma-246	71	10	,	,	PUNCT
ma-246	71	11	b	b	NOUN
ma-246	71	12	,	,	PUNCT
ma-246	71	13	c	c	NOUN
ma-246	71	14	,	,	PUNCT
ma-246	71	15	d	d	NOUN
ma-246	71	16	}	}	PUNCT
ma-246	71	17	=	=	SYM
ma-246	71	18	x	x	X
ma-246	71	19	.	.	PUNCT
ma-246	72	1	theorem	theorem	NOUN
ma-246	72	2	1	1	X
ma-246	72	3	.	.	PUNCT
ma-246	73	1	let	let	VERB
ma-246	73	2	(	(	PUNCT
ma-246	73	3	x	x	X
ma-246	73	4	,	,	PUNCT
ma-246	73	5	τ	τ	PROPN
ma-246	73	6	,	,	PUNCT
ma-246	73	7	i	i	PRON
ma-246	73	8	)	)	PUNCT
ma-246	73	9	be	be	VERB
ma-246	73	10	an	an	DET
ma-246	73	11	ideal	ideal	ADJ
ma-246	73	12	topological	topological	ADJ
ma-246	73	13	space	space	NOUN
ma-246	73	14	and	and	CCONJ
ma-246	73	15	a	a	DET
ma-246	73	16	,	,	PUNCT
ma-246	73	17	b	b	PROPN
ma-246	73	18	be	be	AUX
ma-246	73	19	subsets	subset	NOUN
ma-246	73	20	of	of	ADP
ma-246	73	21	x	x	X
ma-246	73	22	.	.	PUNCT
ma-246	74	1	then	then	ADV
ma-246	74	2	for	for	ADP
ma-246	74	3	any	any	DET
ma-246	74	4	η	η	ADJ
ma-246	74	5	-	-	ADJ
ma-246	74	6	local	local	ADJ
ma-246	74	7	functions	function	NOUN
ma-246	74	8	,	,	PUNCT
ma-246	74	9	the	the	DET
ma-246	74	10	following	follow	VERB
ma-246	74	11	properties	property	NOUN
ma-246	74	12	hold	hold	VERB
ma-246	74	13	:	:	PUNCT
ma-246	74	14	(	(	PUNCT
ma-246	74	15	i	i	NOUN
ma-246	74	16	)	)	PUNCT
ma-246	74	17	if	if	SCONJ
ma-246	74	18	a	a	DET
ma-246	74	19	⊆	⊆	NUM
ma-246	74	20	b	b	NOUN
ma-246	74	21	,	,	PUNCT
ma-246	74	22	then	then	ADV
ma-246	74	23	a∗η	a∗η	PROPN
ma-246	74	24	⊆	⊆	NUM
ma-246	74	25	b∗η	b∗η	NUM
ma-246	74	26	;	;	PUNCT
ma-246	74	27	(	(	PUNCT
ma-246	74	28	ii	ii	NOUN
ma-246	74	29	)	)	PUNCT
ma-246	74	30	(	(	PUNCT
ma-246	74	31	a	a	DET
ma-246	74	32	∩	∩	NOUN
ma-246	74	33	b)∗η	b)∗η	VERB
ma-246	74	34	⊆	⊆	NUM
ma-246	74	35	a∗η	a∗η	PROPN
ma-246	74	36	∩	∩	NOUN
ma-246	74	37	b∗η	b∗η	PROPN
ma-246	74	38	;	;	PUNCT
ma-246	74	39	and	and	CCONJ
ma-246	74	40	(	(	PUNCT
ma-246	74	41	iii	iii	X
ma-246	74	42	)	)	PUNCT
ma-246	74	43	a∗η	a∗η	ADP
ma-246	74	44	∪	∪	ADP
ma-246	74	45	b∗η	b∗η	PROPN
ma-246	74	46	⊆	⊆	NUM
ma-246	74	47	(	(	PUNCT
ma-246	74	48	a	a	DET
ma-246	74	49	∪	∪	ADJ
ma-246	74	50	b)∗η	b)∗η	NOUN
ma-246	74	51	;	;	PUNCT
ma-246	74	52	(	(	PUNCT
ma-246	74	53	iv	iv	X
ma-246	74	54	)	)	PUNCT
ma-246	74	55	if	if	SCONJ
ma-246	74	56	a	a	DET
ma-246	74	57	=	=	NOUN
ma-246	74	58	∅	∅	NOUN
ma-246	74	59	,	,	PUNCT
ma-246	74	60	then	then	ADV
ma-246	74	61	a∗η	a∗η	PROPN
ma-246	74	62	=	=	SYM
ma-246	74	63	∅	∅	NOUN
ma-246	74	64	;	;	PUNCT
ma-246	74	65	(	(	PUNCT
ma-246	74	66	v	v	NOUN
ma-246	74	67	)	)	PUNCT
ma-246	74	68	if	if	SCONJ
ma-246	74	69	a∗η	a∗η	PROPN
ma-246	74	70	∩	∩	PROPN
ma-246	74	71	b	b	X
ma-246	74	72	/∈	/∈	PUNCT
ma-246	75	1	i	i	PRON
ma-246	75	2	,	,	PUNCT
ma-246	75	3	then	then	ADV
ma-246	75	4	a∗η	a∗η	PROPN
ma-246	75	5	∩	∩	PROPN
ma-246	75	6	b	b	PROPN
ma-246	75	7	6=	6=	NUM
ma-246	75	8	∅	∅	NOUN
ma-246	75	9	;	;	PUNCT
ma-246	75	10	(	(	PUNCT
ma-246	75	11	vi	vi	NOUN
ma-246	75	12	)	)	PUNCT
ma-246	75	13	(	(	PUNCT
ma-246	75	14	a∗η	a∗η	PROPN
ma-246	75	15	)	)	PUNCT
ma-246	75	16	∗	∗	PROPN
ma-246	75	17	η	η	PROPN
ma-246	75	18	⊆	⊆	NUM
ma-246	75	19	a∗η	a∗η	NUM
ma-246	75	20	;	;	PUNCT
ma-246	75	21	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	75	22	eur	eur	PROPN
ma-246	75	23	.	.	PUNCT
ma-246	76	1	j.	j.	PROPN
ma-246	76	2	math	math	PROPN
ma-246	76	3	.	.	PUNCT
ma-246	77	1	anal	anal	PROPN
ma-246	77	2	.	.	PUNCT
ma-246	78	1	10.28924	10.28924	NUM
ma-246	78	2	/	/	SYM
ma-246	78	3	ada	ada	PROPN
ma-246	78	4	/	/	SYM
ma-246	78	5	ma.5.2	ma.5.2	PROPN
ma-246	78	6	3	3	NUM
ma-246	78	7	(	(	PUNCT
ma-246	78	8	vii	vii	PROPN
ma-246	78	9	)	)	PUNCT
ma-246	78	10	if	if	SCONJ
ma-246	78	11	a	a	DET
ma-246	78	12	∈	∈	X
ma-246	78	13	i	i	PRON
ma-246	78	14	,	,	PUNCT
ma-246	78	15	then	then	ADV
ma-246	78	16	a∗η	a∗η	PROPN
ma-246	78	17	=	=	SYM
ma-246	78	18	∅	∅	NOUN
ma-246	78	19	;	;	PUNCT
ma-246	78	20	(	(	PUNCT
ma-246	78	21	iix	iix	PROPN
ma-246	78	22	)	)	PUNCT
ma-246	79	1	if	if	SCONJ
ma-246	79	2	i	i	PRON
ma-246	79	3	=	=	SYM
ma-246	79	4	{	{	PUNCT
ma-246	79	5	∅	∅	NOUN
ma-246	79	6	}	}	PUNCT
ma-246	79	7	,	,	PUNCT
ma-246	79	8	then	then	ADV
ma-246	79	9	a∗η	a∗η	PROPN
ma-246	79	10	=	=	SYM
ma-246	79	11	η	η	PROPN
ma-246	79	12	-	-	NOUN
ma-246	79	13	cl(a	cl(a	NUM
ma-246	79	14	)	)	PUNCT
ma-246	79	15	;	;	PUNCT
ma-246	79	16	(	(	PUNCT
ma-246	79	17	ix	ix	INTJ
ma-246	79	18	)	)	PUNCT
ma-246	79	19	if	if	SCONJ
ma-246	79	20	i	i	PRON
ma-246	79	21	=	=	SYM
ma-246	79	22	p(x	p(x	PROPN
ma-246	79	23	)	)	PUNCT
ma-246	79	24	,	,	PUNCT
ma-246	79	25	then	then	ADV
ma-246	79	26	a∗η	a∗η	PROPN
ma-246	79	27	=	=	SYM
ma-246	79	28	∅	∅	NOUN
ma-246	79	29	;	;	PUNCT
ma-246	79	30	and	and	CCONJ
ma-246	79	31	(	(	PUNCT
ma-246	79	32	x	x	X
ma-246	79	33	)	)	PUNCT
ma-246	79	34	a∗η	a∗η	PROPN
ma-246	79	35	⊆	⊆	NUM
ma-246	79	36	η	η	PROPN
ma-246	79	37	-	-	NOUN
ma-246	79	38	cl(a	cl(a	NUM
ma-246	79	39	)	)	PUNCT
ma-246	79	40	;	;	PUNCT
ma-246	79	41	proof	proof	NOUN
ma-246	79	42	.	.	PUNCT
ma-246	80	1	(	(	PUNCT
ma-246	80	2	i	i	NOUN
ma-246	80	3	)	)	PUNCT
ma-246	80	4	let	let	VERB
ma-246	80	5	a	a	DET
ma-246	80	6	,	,	PUNCT
ma-246	80	7	b	b	NOUN
ma-246	80	8	⊆	⊆	NUM
ma-246	80	9	x	x	PUNCT
ma-246	80	10	and	and	CCONJ
ma-246	80	11	a	a	DET
ma-246	80	12	⊆	⊆	NUM
ma-246	80	13	b.	b.	NOUN
ma-246	80	14	suppose	suppose	VERB
ma-246	80	15	x	x	X
ma-246	80	16	/∈	/∈	INTJ
ma-246	80	17	b∗η	b∗η	PROPN
ma-246	80	18	,	,	PUNCT
ma-246	80	19	then	then	ADV
ma-246	80	20	there	there	PRON
ma-246	80	21	exist	exist	VERB
ma-246	80	22	u	u	PROPN
ma-246	80	23	∈	∈	PROPN
ma-246	80	24	η	η	PROPN
ma-246	80	25	-	-	PROPN
ma-246	80	26	o(x	o(x	ADJ
ma-246	80	27	)	)	PUNCT
ma-246	80	28	such	such	ADJ
ma-246	80	29	that	that	SCONJ
ma-246	80	30	b	b	NOUN
ma-246	80	31	∩	∩	NOUN
ma-246	80	32	u	u	NOUN
ma-246	80	33	∈	∈	PROPN
ma-246	80	34	i	i	PRON
ma-246	80	35	.	.	PUNCT
ma-246	81	1	since	since	SCONJ
ma-246	81	2	a	a	DET
ma-246	81	3	⊆	⊆	NUM
ma-246	81	4	b	b	NOUN
ma-246	81	5	,	,	PUNCT
ma-246	81	6	a	a	DET
ma-246	81	7	∩	∩	ADJ
ma-246	81	8	u	u	NOUN
ma-246	81	9	⊆	⊆	PROPN
ma-246	81	10	b	b	NUM
ma-246	81	11	∩	∩	ADJ
ma-246	81	12	u	u	X
ma-246	81	13	∈	∈	PROPN
ma-246	81	14	i	i	PRON
ma-246	81	15	,	,	PUNCT
ma-246	81	16	by	by	ADP
ma-246	81	17	definition	definition	NOUN
ma-246	81	18	of	of	ADP
ma-246	81	19	ideal	ideal	NOUN
ma-246	81	20	,	,	PUNCT
ma-246	81	21	a	a	DET
ma-246	81	22	∩	∩	ADJ
ma-246	81	23	u	u	NOUN
ma-246	81	24	∈	∈	NOUN
ma-246	81	25	i	i	PRON
ma-246	81	26	.	.	PUNCT
ma-246	82	1	hence	hence	ADV
ma-246	82	2	,	,	PUNCT
ma-246	82	3	x	x	PROPN
ma-246	82	4	/∈	/∈	PUNCT
ma-246	82	5	a∗η	a∗η	PROPN
ma-246	82	6	.	.	PUNCT
ma-246	83	1	(	(	PUNCT
ma-246	83	2	ii	ii	NOUN
ma-246	83	3	)	)	PUNCT
ma-246	83	4	let	let	VERB
ma-246	83	5	a	a	DET
ma-246	83	6	,	,	PUNCT
ma-246	83	7	b	b	NOUN
ma-246	83	8	⊆	⊆	NUM
ma-246	83	9	x	x	X
ma-246	83	10	.	.	PUNCT
ma-246	84	1	since	since	SCONJ
ma-246	84	2	a	a	DET
ma-246	84	3	∩	∩	NOUN
ma-246	84	4	b	b	ADP
ma-246	84	5	⊆	⊆	NUM
ma-246	84	6	a	a	PRON
ma-246	84	7	and	and	CCONJ
ma-246	84	8	a	a	DET
ma-246	84	9	∩	∩	ADJ
ma-246	84	10	b	b	PROPN
ma-246	84	11	⊆	⊆	NUM
ma-246	84	12	b	b	NOUN
ma-246	84	13	,	,	PUNCT
ma-246	84	14	by	by	ADP
ma-246	84	15	theorem	theorem	NOUN
ma-246	84	16	1	1	NUM
ma-246	84	17	(	(	PUNCT
ma-246	84	18	i	i	NOUN
ma-246	84	19	)	)	PUNCT
ma-246	84	20	,	,	PUNCT
ma-246	84	21	(	(	PUNCT
ma-246	84	22	a	a	DET
ma-246	84	23	∩	∩	NOUN
ma-246	84	24	b)∗η	b)∗η	VERB
ma-246	84	25	⊆	⊆	NUM
ma-246	84	26	a∗η	a∗η	PROPN
ma-246	84	27	and	and	CCONJ
ma-246	84	28	(	(	PUNCT
ma-246	84	29	a	a	DET
ma-246	84	30	∩	∩	NOUN
ma-246	84	31	b)∗η	b)∗η	VERB
ma-246	84	32	⊆	⊆	NUM
ma-246	84	33	b∗η	b∗η	PROPN
ma-246	84	34	,	,	PUNCT
ma-246	84	35	respectively	respectively	ADV
ma-246	84	36	.	.	PUNCT
ma-246	85	1	hence	hence	ADV
ma-246	85	2	,	,	PUNCT
ma-246	85	3	(	(	PUNCT
ma-246	85	4	a	a	DET
ma-246	85	5	∩	∩	NOUN
ma-246	85	6	b)∗η	b)∗η	VERB
ma-246	85	7	⊆	⊆	NUM
ma-246	85	8	a∗η	a∗η	NOUN
ma-246	85	9	∩	∩	NOUN
ma-246	85	10	b∗η	b∗η	PROPN
ma-246	85	11	.	.	PUNCT
ma-246	86	1	(	(	PUNCT
ma-246	86	2	iii	iii	X
ma-246	86	3	)	)	PUNCT
ma-246	86	4	let	let	VERB
ma-246	86	5	a	a	DET
ma-246	86	6	,	,	PUNCT
ma-246	86	7	b	b	NOUN
ma-246	86	8	⊆	⊆	NUM
ma-246	86	9	x	x	X
ma-246	86	10	.	.	PUNCT
ma-246	87	1	since	since	SCONJ
ma-246	87	2	a	a	DET
ma-246	87	3	⊆	⊆	NUM
ma-246	87	4	a	a	DET
ma-246	87	5	∪	∪	NOUN
ma-246	87	6	b	b	NOUN
ma-246	87	7	and	and	CCONJ
ma-246	87	8	b	b	NOUN
ma-246	87	9	⊆	⊆	NUM
ma-246	87	10	a	a	DET
ma-246	87	11	∪	∪	ADJ
ma-246	87	12	b	b	NOUN
ma-246	87	13	,	,	PUNCT
ma-246	87	14	by	by	ADP
ma-246	87	15	theorem	theorem	NOUN
ma-246	87	16	1	1	NUM
ma-246	87	17	(	(	PUNCT
ma-246	87	18	i	i	NOUN
ma-246	87	19	)	)	PUNCT
ma-246	87	20	,	,	PUNCT
ma-246	87	21	a∗η	a∗η	PROPN
ma-246	87	22	⊆	⊆	NUM
ma-246	87	23	(	(	PUNCT
ma-246	87	24	a	a	DET
ma-246	87	25	∪	∪	ADJ
ma-246	87	26	b)∗η	b)∗η	NOUN
ma-246	87	27	and	and	CCONJ
ma-246	87	28	b∗η	b∗η	PROPN
ma-246	87	29	⊆	⊆	NUM
ma-246	87	30	(	(	PUNCT
ma-246	87	31	a	a	DET
ma-246	87	32	∪	∪	ADJ
ma-246	87	33	b)∗η	b)∗η	NOUN
ma-246	87	34	,	,	PUNCT
ma-246	87	35	respectively	respectively	ADV
ma-246	87	36	.	.	PUNCT
ma-246	88	1	hence	hence	ADV
ma-246	88	2	,	,	PUNCT
ma-246	88	3	a∗η	a∗η	ADV
ma-246	88	4	∪	∪	VERB
ma-246	88	5	b∗η	b∗η	PROPN
ma-246	88	6	⊆	⊆	NUM
ma-246	88	7	(	(	PUNCT
ma-246	88	8	a	a	DET
ma-246	88	9	∪	∪	ADJ
ma-246	88	10	b)∗η	b)∗η	NOUN
ma-246	88	11	.	.	PUNCT
ma-246	89	1	(	(	PUNCT
ma-246	89	2	iv	iv	X
ma-246	89	3	)	)	PUNCT
ma-246	89	4	let	let	VERB
ma-246	89	5	a	a	PRON
ma-246	89	6	=	=	PUNCT
ma-246	89	7	∅.	∅.	VERB
ma-246	89	8	suppose	suppose	VERB
ma-246	89	9	a∗η	a∗η	PROPN
ma-246	89	10	6=	6=	ADP
ma-246	89	11	∅.	∅.	VERB
ma-246	89	12	then	then	ADV
ma-246	89	13	there	there	PRON
ma-246	89	14	exists	exist	VERB
ma-246	89	15	x	x	X
ma-246	89	16	∈	∈	PROPN
ma-246	89	17	a∗η	a∗η	PROPN
ma-246	89	18	.	.	PUNCT
ma-246	90	1	it	it	PRON
ma-246	90	2	follows	follow	VERB
ma-246	90	3	that	that	SCONJ
ma-246	90	4	a	a	DET
ma-246	90	5	∩	∩	ADJ
ma-246	90	6	u	u	NOUN
ma-246	90	7	=	=	NOUN
ma-246	90	8	∅	∅	NOUN
ma-246	90	9	∩	∩	NOUN
ma-246	90	10	u	u	NOUN
ma-246	90	11	=	=	NOUN
ma-246	90	12	∅	∅	NOUN
ma-246	90	13	/∈	/∈	PUNCT
ma-246	91	1	i	i	PRON
ma-246	91	2	for	for	ADP
ma-246	91	3	every	every	DET
ma-246	91	4	u	u	PROPN
ma-246	91	5	∈	∈	PROPN
ma-246	91	6	η	η	PROPN
ma-246	91	7	-	-	PROPN
ma-246	91	8	o(x	o(x	PROPN
ma-246	91	9	)	)	PUNCT
ma-246	91	10	.	.	PUNCT
ma-246	92	1	since	since	SCONJ
ma-246	92	2	i	i	PRON
ma-246	92	3	is	be	AUX
ma-246	92	4	an	an	DET
ma-246	92	5	ideal	ideal	ADJ
ma-246	92	6	,	,	PUNCT
ma-246	92	7	∅	∅	NOUN
ma-246	92	8	∈	∈	PROPN
ma-246	92	9	i	i	PRON
ma-246	92	10	which	which	PRON
ma-246	92	11	is	be	AUX
ma-246	92	12	a	a	DET
ma-246	92	13	contradiction	contradiction	NOUN
ma-246	92	14	.	.	PUNCT
ma-246	93	1	(	(	PUNCT
ma-246	93	2	v	v	NOUN
ma-246	93	3	)	)	PUNCT
ma-246	93	4	let	let	VERB
ma-246	93	5	a∗η	a∗η	PROPN
ma-246	93	6	∩	∩	PROPN
ma-246	93	7	b	b	X
ma-246	93	8	/∈	/∈	PUNCT
ma-246	94	1	i	i	PRON
ma-246	94	2	.	.	PUNCT
ma-246	95	1	suppose	suppose	VERB
ma-246	95	2	a∗η	a∗η	PROPN
ma-246	95	3	∩	∩	PROPN
ma-246	95	4	b	b	X
ma-246	95	5	=	=	PUNCT
ma-246	95	6	∅.	∅.	NOUN
ma-246	95	7	note	note	VERB
ma-246	95	8	that	that	SCONJ
ma-246	95	9	by	by	ADP
ma-246	95	10	definition	definition	NOUN
ma-246	95	11	of	of	ADP
ma-246	95	12	ideal	ideal	ADJ
ma-246	95	13	,	,	PUNCT
ma-246	95	14	∅	∅	NOUN
ma-246	95	15	∈	∈	PROPN
ma-246	95	16	i	i	PRON
ma-246	95	17	for	for	ADP
ma-246	95	18	anyideal	anyideal	NOUN
ma-246	95	19	i	i	PRON
ma-246	95	20	.	.	PUNCT
ma-246	96	1	now	now	ADV
ma-246	96	2	,	,	PUNCT
ma-246	96	3	since	since	SCONJ
ma-246	96	4	a∗η	a∗η	PROPN
ma-246	96	5	∩	∩	PROPN
ma-246	96	6	b	b	NOUN
ma-246	96	7	=	=	SYM
ma-246	96	8	∅	∅	NOUN
ma-246	97	1	and	and	CCONJ
ma-246	97	2	i	i	PRON
ma-246	97	3	is	be	AUX
ma-246	97	4	an	an	DET
ma-246	97	5	ideal	ideal	ADJ
ma-246	97	6	,	,	PUNCT
ma-246	97	7	a∗η	a∗η	PROPN
ma-246	97	8	∩	∩	ADJ
ma-246	97	9	b	b	NOUN
ma-246	97	10	=	=	SYM
ma-246	97	11	∅	∅	NOUN
ma-246	97	12	∈	∈	PROPN
ma-246	97	13	i	i	PRON
ma-246	97	14	implies	imply	VERB
ma-246	97	15	a∗η	a∗η	PROPN
ma-246	97	16	∩	∩	PROPN
ma-246	97	17	b	b	X
ma-246	97	18	∈	∈	PROPN
ma-246	97	19	i	i	PRON
ma-246	97	20	,	,	PUNCT
ma-246	97	21	acontradiction	acontradiction	NOUN
ma-246	97	22	.	.	PUNCT
ma-246	98	1	(	(	PUNCT
ma-246	98	2	vi	vi	X
ma-246	98	3	)	)	PUNCT
ma-246	98	4	let	let	VERB
ma-246	98	5	x	x	X
ma-246	98	6	∈	∈	PROPN
ma-246	98	7	(	(	PUNCT
ma-246	98	8	a∗η)∗η	a∗η)∗η	ADJ
ma-246	98	9	.	.	PUNCT
ma-246	99	1	then	then	ADV
ma-246	99	2	,	,	PUNCT
ma-246	99	3	for	for	ADP
ma-246	99	4	every	every	DET
ma-246	99	5	u	u	PROPN
ma-246	99	6	∈	∈	PROPN
ma-246	99	7	η	η	PROPN
ma-246	99	8	-	-	PROPN
ma-246	99	9	o(x	o(x	PROPN
ma-246	99	10	)	)	PUNCT
ma-246	99	11	,	,	PUNCT
ma-246	99	12	u	u	NOUN
ma-246	99	13	∩	∩	PROPN
ma-246	99	14	a∗η	a∗η	PROPN
ma-246	99	15	/∈	/∈	PUNCT
ma-246	99	16	i	i	PRON
ma-246	99	17	and	and	CCONJ
ma-246	99	18	hence	hence	ADV
ma-246	99	19	,	,	PUNCT
ma-246	99	20	by	by	ADP
ma-246	99	21	(	(	PUNCT
ma-246	99	22	ii	ii	NOUN
ma-246	99	23	)	)	PUNCT
ma-246	99	24	,	,	PUNCT
ma-246	99	25	u	u	NOUN
ma-246	99	26	∩	∩	X
ma-246	99	27	a∗η	a∗η	PROPN
ma-246	99	28	6=	6=	SYM
ma-246	99	29	∅.now	∅.now	PROPN
ma-246	99	30	,	,	PUNCT
ma-246	99	31	let	let	VERB
ma-246	99	32	y	y	PROPN
ma-246	99	33	∈	∈	PROPN
ma-246	99	34	u	u	PROPN
ma-246	99	35	∩a∗η	∩a∗η	PROPN
ma-246	99	36	.	.	PUNCT
ma-246	100	1	then	then	ADV
ma-246	100	2	,	,	PUNCT
ma-246	100	3	u	u	PROPN
ma-246	100	4	∈	∈	PROPN
ma-246	100	5	η	η	PROPN
ma-246	100	6	-	-	PROPN
ma-246	100	7	o(y	o(y	PROPN
ma-246	100	8	)	)	PUNCT
ma-246	100	9	and	and	CCONJ
ma-246	100	10	y	y	PROPN
ma-246	100	11	∈	∈	PROPN
ma-246	100	12	a∗η	a∗η	PROPN
ma-246	100	13	.	.	PUNCT
ma-246	101	1	hence	hence	ADV
ma-246	101	2	,	,	PUNCT
ma-246	101	3	we	we	PRON
ma-246	101	4	have	have	VERB
ma-246	101	5	u	u	NOUN
ma-246	101	6	∩a	∩a	NOUN
ma-246	101	7	/∈	/∈	PUNCT
ma-246	102	1	i	i	PRON
ma-246	102	2	.	.	PUNCT
ma-246	103	1	note	note	VERB
ma-246	103	2	that	that	SCONJ
ma-246	103	3	u	u	PROPN
ma-246	103	4	∈	∈	PROPN
ma-246	103	5	η	η	PROPN
ma-246	103	6	-	-	PROPN
ma-246	103	7	o(x	o(x	ADJ
ma-246	103	8	)	)	PUNCT
ma-246	103	9	and	and	CCONJ
ma-246	103	10	u	u	PROPN
ma-246	103	11	∩	∩	NOUN
ma-246	103	12	a	a	X
ma-246	103	13	/∈	/∈	INTJ
ma-246	104	1	i	i	INTJ
ma-246	104	2	.	.	PUNCT
ma-246	105	1	it	it	PRON
ma-246	105	2	follows	follow	VERB
ma-246	105	3	that	that	SCONJ
ma-246	105	4	x	x	SYM
ma-246	105	5	∈	∈	PROPN
ma-246	105	6	a∗η	a∗η	PROPN
ma-246	105	7	.	.	PUNCT
ma-246	106	1	therefore	therefore	ADV
ma-246	106	2	,	,	PUNCT
ma-246	106	3	(	(	PUNCT
ma-246	106	4	a∗η)∗η	a∗η)∗η	NOUN
ma-246	106	5	⊆	⊆	NUM
ma-246	106	6	a∗η	a∗η	PROPN
ma-246	106	7	.	.	PUNCT
ma-246	107	1	(	(	PUNCT
ma-246	107	2	vii	vii	PROPN
ma-246	107	3	)	)	PUNCT
ma-246	107	4	let	let	VERB
ma-246	107	5	a	a	DET
ma-246	107	6	∈	∈	NOUN
ma-246	108	1	i	i	PRON
ma-246	108	2	.	.	PUNCT
ma-246	109	1	suppose	suppose	VERB
ma-246	109	2	a∗η	a∗η	PROPN
ma-246	109	3	6=	6=	ADP
ma-246	109	4	∅.	∅.	VERB
ma-246	109	5	then	then	ADV
ma-246	109	6	there	there	PRON
ma-246	109	7	exists	exist	VERB
ma-246	109	8	an	an	DET
ma-246	109	9	element	element	NOUN
ma-246	109	10	x	x	SYM
ma-246	109	11	∈	∈	PROPN
ma-246	109	12	a∗η	a∗η	PROPN
ma-246	109	13	.	.	PUNCT
ma-246	110	1	then	then	ADV
ma-246	110	2	a	a	DET
ma-246	110	3	∩	∩	ADJ
ma-246	110	4	u	u	NOUN
ma-246	110	5	/∈	/∈	PUNCT
ma-246	111	1	i	i	PRON
ma-246	111	2	forevery	forevery	VERB
ma-246	111	3	u	u	PROPN
ma-246	111	4	∈	∈	PROPN
ma-246	111	5	η	η	PROPN
ma-246	111	6	-	-	PROPN
ma-246	111	7	o(x	o(x	PROPN
ma-246	111	8	)	)	PUNCT
ma-246	111	9	.	.	PUNCT
ma-246	112	1	now	now	ADV
ma-246	112	2	,	,	PUNCT
ma-246	112	3	since	since	ADV
ma-246	112	4	,	,	PUNCT
ma-246	112	5	a	a	DET
ma-246	112	6	∩	∩	ADJ
ma-246	112	7	u	u	NOUN
ma-246	112	8	⊆	⊆	NUM
ma-246	112	9	a	a	DET
ma-246	112	10	∈	∈	NOUN
ma-246	113	1	i	i	PRON
ma-246	113	2	and	and	CCONJ
ma-246	113	3	i	i	PRON
ma-246	113	4	is	be	AUX
ma-246	113	5	an	an	DET
ma-246	113	6	ideal	ideal	NOUN
ma-246	113	7	,	,	PUNCT
ma-246	113	8	a	a	DET
ma-246	113	9	∩	∩	ADJ
ma-246	113	10	u	u	ADJ
ma-246	113	11	∈	∈	PROPN
ma-246	113	12	i	i	PRON
ma-246	113	13	which	which	PRON
ma-246	113	14	is	be	AUX
ma-246	113	15	acontradiction	acontradiction	NOUN
ma-246	113	16	.	.	PUNCT
ma-246	114	1	(	(	PUNCT
ma-246	114	2	iix	iix	PROPN
ma-246	114	3	)	)	PUNCT
ma-246	114	4	let	let	VERB
ma-246	114	5	i	i	PRON
ma-246	114	6	=	=	PUNCT
ma-246	114	7	{	{	PUNCT
ma-246	114	8	∅	∅	NOUN
ma-246	114	9	}	}	PUNCT
ma-246	114	10	.	.	PUNCT
ma-246	115	1	suppose	suppose	VERB
ma-246	115	2	that	that	SCONJ
ma-246	115	3	a∗η	a∗η	PROPN
ma-246	115	4	6=	6=	NUM
ma-246	115	5	η	η	PROPN
ma-246	115	6	-	-	NOUN
ma-246	115	7	cl(a	cl(a	NUM
ma-246	115	8	)	)	PUNCT
ma-246	115	9	.	.	PUNCT
ma-246	116	1	let	let	VERB
ma-246	116	2	η	η	NOUN
ma-246	116	3	-	-	ADJ
ma-246	116	4	cl(a	cl(a	NUM
ma-246	116	5	)	)	PUNCT
ma-246	116	6	⊂	⊂	PROPN
ma-246	116	7	a∗η	a∗η	PROPN
ma-246	116	8	,	,	PUNCT
ma-246	116	9	then	then	ADV
ma-246	116	10	there	there	PRON
ma-246	116	11	exists	exist	VERB
ma-246	116	12	an	an	DET
ma-246	116	13	element	element	NOUN
ma-246	116	14	x	x	SYM
ma-246	116	15	∈	∈	PROPN
ma-246	116	16	a∗η	a∗η	PROPN
ma-246	116	17	and	and	CCONJ
ma-246	116	18	x	x	PROPN
ma-246	116	19	/∈	/∈	PUNCT
ma-246	116	20	η	η	PROPN
ma-246	116	21	-	-	NOUN
ma-246	116	22	cl(a	cl(a	NUM
ma-246	116	23	)	)	PUNCT
ma-246	116	24	.	.	PUNCT
ma-246	117	1	it	it	PRON
ma-246	117	2	follows	follow	VERB
ma-246	117	3	that	that	SCONJ
ma-246	117	4	for	for	ADP
ma-246	117	5	every	every	DET
ma-246	117	6	a	a	DET
ma-246	117	7	⊆	⊆	NUM
ma-246	117	8	x	x	SYM
ma-246	117	9	,	,	PUNCT
ma-246	117	10	since	since	SCONJ
ma-246	117	11	x	x	PROPN
ma-246	117	12	∈	∈	PROPN
ma-246	117	13	a∗η	a∗η	PROPN
ma-246	117	14	,	,	PUNCT
ma-246	117	15	a∩u	a∩u	PROPN
ma-246	117	16	/∈	/∈	PUNCT
ma-246	118	1	i	i	PRON
ma-246	118	2	for	for	ADP
ma-246	118	3	every	every	DET
ma-246	118	4	u	u	PROPN
ma-246	118	5	∈	∈	PROPN
ma-246	118	6	η	η	PROPN
ma-246	118	7	-	-	PROPN
ma-246	118	8	o(x	o(x	PROPN
ma-246	118	9	)	)	PUNCT
ma-246	118	10	.	.	PUNCT
ma-246	119	1	since	since	SCONJ
ma-246	119	2	i	i	PRON
ma-246	119	3	=	=	SYM
ma-246	119	4	{	{	PUNCT
ma-246	119	5	∅	∅	NOUN
ma-246	119	6	}	}	PUNCT
ma-246	119	7	,	,	PUNCT
ma-246	119	8	a∩u	a∩u	PROPN
ma-246	119	9	6=	6=	NOUN
ma-246	119	10	∅	∅	NOUN
ma-246	119	11	for	for	ADP
ma-246	119	12	every	every	DET
ma-246	119	13	u	u	PROPN
ma-246	119	14	∈	∈	PROPN
ma-246	119	15	η	η	PROPN
ma-246	119	16	-	-	PROPN
ma-246	119	17	o(x	o(x	PROPN
ma-246	119	18	)	)	PUNCT
ma-246	119	19	.	.	PUNCT
ma-246	119	20	note	note	VERB
ma-246	119	21	that	that	SCONJ
ma-246	119	22	u	u	PROPN
ma-246	119	23	∈	∈	PROPN
ma-246	119	24	η	η	PROPN
ma-246	119	25	-	-	PROPN
ma-246	119	26	o(x	o(x	ADJ
ma-246	119	27	)	)	PUNCT
ma-246	119	28	means	mean	VERB
ma-246	119	29	x	x	SYM
ma-246	119	30	∈	∈	PROPN
ma-246	119	31	u	u	NOUN
ma-246	119	32	where	where	SCONJ
ma-246	119	33	u	u	NOUN
ma-246	119	34	is	be	AUX
ma-246	119	35	η	η	ADJ
ma-246	119	36	-	-	ADJ
ma-246	119	37	open	open	ADJ
ma-246	119	38	set	set	NOUN
ma-246	119	39	.	.	PUNCT
ma-246	120	1	since	since	SCONJ
ma-246	120	2	,	,	PUNCT
ma-246	120	3	x	x	PROPN
ma-246	120	4	/∈	/∈	PUNCT
ma-246	120	5	η	η	PROPN
ma-246	120	6	-	-	NOUN
ma-246	120	7	cl(a	cl(a	NUM
ma-246	120	8	)	)	PUNCT
ma-246	120	9	,	,	PUNCT
ma-246	120	10	x	x	PUNCT
ma-246	120	11	/∈	/∈	PUNCT
ma-246	120	12	⋂{k	⋂{k	VERB
ma-246	120	13	:	:	PUNCT
ma-246	120	14	k	k	X
ma-246	120	15	is	be	AUX
ma-246	120	16	η	η	NOUN
ma-246	120	17	-	-	ADJ
ma-246	120	18	closed	closed	ADJ
ma-246	120	19	and	and	CCONJ
ma-246	120	20	a	a	DET
ma-246	120	21	⊆	⊆	NUM
ma-246	120	22	k}.it	k}.it	NOUN
ma-246	120	23	follows	follow	VERB
ma-246	120	24	that	that	SCONJ
ma-246	121	1	x	x	PROPN
ma-246	121	2	/∈	/∈	PUNCT
ma-246	122	1	k	k	PROPN
ma-246	122	2	for	for	ADP
ma-246	122	3	some	some	DET
ma-246	122	4	η	η	NOUN
ma-246	122	5	-	-	ADJ
ma-246	122	6	closed	closed	ADJ
ma-246	122	7	set	set	NOUN
ma-246	122	8	k	k	PROPN
ma-246	123	1	such	such	ADJ
ma-246	123	2	that	that	SCONJ
ma-246	123	3	a	a	DET
ma-246	123	4	⊆	⊆	NUM
ma-246	123	5	k.	k.	NOUN
ma-246	123	6	hence	hence	ADV
ma-246	123	7	,	,	PUNCT
ma-246	123	8	x	x	PUNCT
ma-246	123	9	∈	∈	PROPN
ma-246	123	10	kc	kc	PROPN
ma-246	123	11	for	for	ADP
ma-246	123	12	some	some	DET
ma-246	123	13	η	η	NOUN
ma-246	123	14	-	-	ADJ
ma-246	123	15	open	open	ADJ
ma-246	123	16	set	set	NOUN
ma-246	123	17	kc	kc	PROPN
ma-246	123	18	such	such	ADJ
ma-246	123	19	that	that	SCONJ
ma-246	123	20	a	a	DET
ma-246	123	21	∩kc	∩kc	NOUN
ma-246	123	22	=	=	PUNCT
ma-246	123	23	∅.	∅.	NOUN
ma-246	123	24	it	it	PRON
ma-246	123	25	implies	imply	VERB
ma-246	123	26	that	that	SCONJ
ma-246	123	27	there	there	PRON
ma-246	123	28	exists	exist	VERB
ma-246	123	29	an	an	DET
ma-246	123	30	η	η	NOUN
ma-246	123	31	-	-	ADJ
ma-246	123	32	open	open	ADJ
ma-246	123	33	set	set	NOUN
ma-246	123	34	kc	kc	PROPN
ma-246	123	35	where	where	SCONJ
ma-246	123	36	x	x	SYM
ma-246	123	37	∈	∈	PROPN
ma-246	123	38	kc	kc	PROPN
ma-246	123	39	and	and	CCONJ
ma-246	123	40	a	a	DET
ma-246	123	41	∩kc	∩kc	NOUN
ma-246	123	42	=	=	NOUN
ma-246	123	43	∅	∅	NOUN
ma-246	123	44	,	,	PUNCT
ma-246	123	45	a	a	DET
ma-246	123	46	contradiction	contradiction	NOUN
ma-246	123	47	.	.	PUNCT
ma-246	124	1	(	(	PUNCT
ma-246	124	2	ix	ix	ADV
ma-246	124	3	)	)	PUNCT
ma-246	124	4	let	let	VERB
ma-246	124	5	i	i	PRON
ma-246	124	6	=	=	SYM
ma-246	124	7	p(x	p(x	PROPN
ma-246	124	8	)	)	PUNCT
ma-246	124	9	.	.	PUNCT
ma-246	125	1	note	note	VERB
ma-246	125	2	that	that	SCONJ
ma-246	125	3	a	a	DET
ma-246	125	4	⊆	⊆	NUM
ma-246	125	5	x	x	SYM
ma-246	125	6	,	,	PUNCT
ma-246	125	7	then	then	ADV
ma-246	125	8	a	a	DET
ma-246	125	9	∈	∈	PROPN
ma-246	125	10	p(x	p(x	NOUN
ma-246	125	11	)	)	PUNCT
ma-246	125	12	.	.	PUNCT
ma-246	126	1	since	since	SCONJ
ma-246	126	2	i	i	PRON
ma-246	126	3	=	=	SYM
ma-246	126	4	p(x	p(x	PROPN
ma-246	126	5	)	)	PUNCT
ma-246	126	6	,	,	PUNCT
ma-246	126	7	a	a	DET
ma-246	126	8	∈	∈	NOUN
ma-246	126	9	i	i	PRON
ma-246	126	10	.	.	PUNCT
ma-246	127	1	hence	hence	ADV
ma-246	127	2	,	,	PUNCT
ma-246	127	3	bytheorem	bytheorem	ADJ
ma-246	127	4	1	1	NUM
ma-246	127	5	(	(	PUNCT
ma-246	127	6	vii	vii	PROPN
ma-246	127	7	)	)	PUNCT
ma-246	127	8	,	,	PUNCT
ma-246	127	9	a∗η	a∗η	PROPN
ma-246	127	10	=	=	SYM
ma-246	127	11	∅.	∅.	X
ma-246	127	12	(	(	PUNCT
ma-246	127	13	x	x	X
ma-246	127	14	)	)	PUNCT
ma-246	127	15	let	let	VERB
ma-246	127	16	x	x	X
ma-246	127	17	/∈	/∈	PUNCT
ma-246	127	18	η	η	PROPN
ma-246	127	19	-	-	NOUN
ma-246	127	20	cl(a	cl(a	NUM
ma-246	127	21	)	)	PUNCT
ma-246	127	22	.	.	PUNCT
ma-246	128	1	then	then	ADV
ma-246	128	2	,	,	PUNCT
ma-246	128	3	x	x	PUNCT
ma-246	128	4	/∈	/∈	PUNCT
ma-246	128	5	⋂{k	⋂{k	VERB
ma-246	128	6	:	:	PUNCT
ma-246	128	7	k	k	X
ma-246	128	8	is	be	AUX
ma-246	128	9	η	η	NOUN
ma-246	128	10	-	-	ADJ
ma-246	128	11	closed	closed	ADJ
ma-246	128	12	and	and	CCONJ
ma-246	128	13	a	a	DET
ma-246	128	14	⊆	⊆	NUM
ma-246	128	15	k	k	NOUN
ma-246	128	16	}	}	PUNCT
ma-246	128	17	.	.	PUNCT
ma-246	129	1	it	it	PRON
ma-246	129	2	follows	follow	VERB
ma-246	129	3	that	that	PRON
ma-246	129	4	x	x	PROPN
ma-246	129	5	/∈	/∈	PUNCT
ma-246	130	1	k	k	PROPN
ma-246	130	2	forsome	forsome	PROPN
ma-246	131	1	η	η	PROPN
ma-246	131	2	-	-	ADJ
ma-246	131	3	closed	closed	ADJ
ma-246	131	4	set	set	NOUN
ma-246	131	5	k	k	PROPN
ma-246	131	6	such	such	ADJ
ma-246	131	7	that	that	SCONJ
ma-246	131	8	a	a	DET
ma-246	131	9	⊆	⊆	NUM
ma-246	131	10	k.	k.	NOUN
ma-246	131	11	hence	hence	ADV
ma-246	131	12	,	,	PUNCT
ma-246	131	13	x	x	PUNCT
ma-246	131	14	∈	∈	PROPN
ma-246	131	15	kc	kc	PROPN
ma-246	131	16	for	for	ADP
ma-246	131	17	some	some	DET
ma-246	131	18	η	η	NOUN
ma-246	131	19	-	-	ADJ
ma-246	131	20	open	open	ADJ
ma-246	131	21	set	set	NOUN
ma-246	131	22	kc	kc	PROPN
ma-246	131	23	such	such	ADJ
ma-246	131	24	that	that	SCONJ
ma-246	131	25	a	a	DET
ma-246	131	26	∩	∩	ADJ
ma-246	131	27	kc	kc	NOUN
ma-246	131	28	=	=	PUNCT
ma-246	131	29	∅.	∅.	NOUN
ma-246	131	30	it	it	PRON
ma-246	131	31	implies	imply	VERB
ma-246	131	32	that	that	SCONJ
ma-246	131	33	there	there	PRON
ma-246	131	34	exists	exist	VERB
ma-246	131	35	kc	kc	PROPN
ma-246	131	36	∈	∈	PROPN
ma-246	131	37	η	η	PROPN
ma-246	131	38	-	-	PROPN
ma-246	131	39	o(x	o(x	ADJ
ma-246	131	40	)	)	PUNCT
ma-246	132	1	such	such	ADJ
ma-246	132	2	that	that	SCONJ
ma-246	132	3	a	a	DET
ma-246	132	4	∩	∩	ADJ
ma-246	132	5	kc	kc	NOUN
ma-246	132	6	=	=	NOUN
ma-246	132	7	∅	∅	NOUN
ma-246	132	8	,	,	PUNCT
ma-246	132	9	and	and	CCONJ
ma-246	132	10	bydefinition	bydefinition	NOUN
ma-246	132	11	of	of	ADP
ma-246	132	12	ideal	ideal	ADJ
ma-246	132	13	,	,	PUNCT
ma-246	132	14	∅	∅	NOUN
ma-246	132	15	∈	∈	PROPN
ma-246	132	16	i	i	PRON
ma-246	132	17	for	for	ADP
ma-246	132	18	any	any	DET
ma-246	132	19	ideal	ideal	NOUN
ma-246	132	20	i	i	PRON
ma-246	132	21	.	.	PUNCT
ma-246	133	1	hence	hence	ADV
ma-246	133	2	,	,	PUNCT
ma-246	133	3	a	a	DET
ma-246	133	4	∩	∩	ADJ
ma-246	133	5	kc	kc	PROPN
ma-246	133	6	∈	∈	PROPN
ma-246	133	7	i	i	PRON
ma-246	133	8	for	for	ADP
ma-246	133	9	some	some	DET
ma-246	133	10	kc	kc	PROPN
ma-246	133	11	∈	∈	PROPN
ma-246	133	12	η	η	PROPN
ma-246	133	13	-	-	PROPN
ma-246	133	14	o(x	o(x	PROPN
ma-246	133	15	)	)	PUNCT
ma-246	133	16	.	.	PUNCT
ma-246	133	17	thisshows	thisshow	VERB
ma-246	133	18	that	that	PRON
ma-246	133	19	x	x	PUNCT
ma-246	133	20	/∈	/∈	PUNCT
ma-246	133	21	a∗η	a∗η	PROPN
ma-246	133	22	.	.	PUNCT
ma-246	134	1	�	�	PROPN
ma-246	134	2	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	134	3	eur	eur	PROPN
ma-246	134	4	.	.	PUNCT
ma-246	135	1	j.	j.	PROPN
ma-246	135	2	math	math	PROPN
ma-246	135	3	.	.	PUNCT
ma-246	136	1	anal	anal	PROPN
ma-246	136	2	.	.	PUNCT
ma-246	137	1	10.28924	10.28924	NUM
ma-246	137	2	/	/	SYM
ma-246	137	3	ada	ada	PROPN
ma-246	137	4	/	/	SYM
ma-246	137	5	ma.5.2	ma.5.2	PROPN
ma-246	137	6	4	4	NUM
ma-246	137	7	remark	remark	NOUN
ma-246	137	8	1	1	NUM
ma-246	137	9	.	.	PUNCT
ma-246	138	1	let	let	VERB
ma-246	138	2	(	(	PUNCT
ma-246	138	3	x	x	X
ma-246	138	4	,	,	PUNCT
ma-246	138	5	τ	τ	PROPN
ma-246	138	6	,	,	PUNCT
ma-246	138	7	i	i	PRON
ma-246	138	8	)	)	PUNCT
ma-246	138	9	be	be	VERB
ma-246	138	10	an	an	DET
ma-246	138	11	ideal	ideal	ADJ
ma-246	138	12	topological	topological	ADJ
ma-246	138	13	space	space	NOUN
ma-246	138	14	and	and	CCONJ
ma-246	138	15	a	a	DET
ma-246	138	16	be	be	AUX
ma-246	138	17	any	any	DET
ma-246	138	18	subset	subset	NOUN
ma-246	138	19	of	of	ADP
ma-246	138	20	x	x	X
ma-246	138	21	.	.	PUNCT
ma-246	139	1	then	then	ADV
ma-246	139	2	for	for	ADP
ma-246	139	3	any	any	DET
ma-246	139	4	η	η	ADJ
ma-246	139	5	-	-	ADJ
ma-246	139	6	local	local	ADJ
ma-246	139	7	functions	function	NOUN
ma-246	139	8	,	,	PUNCT
ma-246	139	9	the	the	DET
ma-246	139	10	following	follow	VERB
ma-246	139	11	properties	property	NOUN
ma-246	139	12	hold:(i	hold:(i	NOUN
ma-246	139	13	)	)	PUNCT
ma-246	140	1	the	the	DET
ma-246	140	2	reverse	reverse	ADJ
ma-246	140	3	inclusion	inclusion	NOUN
ma-246	140	4	of	of	ADP
ma-246	140	5	theorem	theorem	ADJ
ma-246	140	6	1	1	NUM
ma-246	140	7	(	(	PUNCT
ma-246	140	8	iii	iii	NOUN
ma-246	140	9	)	)	PUNCT
ma-246	140	10	need	need	AUX
ma-246	140	11	not	not	PART
ma-246	140	12	be	be	AUX
ma-246	140	13	true	true	ADJ
ma-246	140	14	in	in	ADP
ma-246	140	15	general.(ii	general.(ii	NOUN
ma-246	140	16	)	)	PUNCT
ma-246	140	17	neither	neither	CCONJ
ma-246	140	18	a	a	DET
ma-246	140	19	⊆	⊆	NUM
ma-246	140	20	a∗η	a∗η	NOUN
ma-246	140	21	nor	nor	CCONJ
ma-246	140	22	a∗η	a∗η	PROPN
ma-246	140	23	⊆	⊆	NUM
ma-246	140	24	a	a	PRON
ma-246	140	25	in	in	ADP
ma-246	140	26	general.(iii	general.(iii	NOUN
ma-246	140	27	)	)	PUNCT
ma-246	140	28	a∗η	a∗η	PROPN
ma-246	140	29	is	be	AUX
ma-246	140	30	an	an	DET
ma-246	140	31	η	η	NOUN
ma-246	140	32	-	-	ADJ
ma-246	140	33	closed	closed	ADJ
ma-246	140	34	set	set	VERB
ma-246	140	35	iff	iff	PROPN
ma-246	140	36	a∗η	a∗η	PROPN
ma-246	140	37	=	=	SYM
ma-246	140	38	η	η	PROPN
ma-246	140	39	-	-	PROPN
ma-246	140	40	cl(a∗η	cl(a∗η	NOUN
ma-246	140	41	)	)	PUNCT
ma-246	140	42	.	.	PUNCT
ma-246	141	1	in	in	ADP
ma-246	141	2	order	order	NOUN
ma-246	141	3	to	to	PART
ma-246	141	4	verify	verify	VERB
ma-246	141	5	remark	remark	NOUN
ma-246	141	6	1	1	NUM
ma-246	141	7	(	(	PUNCT
ma-246	141	8	i	i	NOUN
ma-246	141	9	)	)	PUNCT
ma-246	141	10	and	and	CCONJ
ma-246	141	11	(	(	PUNCT
ma-246	141	12	ii	ii	NOUN
ma-246	141	13	)	)	PUNCT
ma-246	141	14	,	,	PUNCT
ma-246	141	15	the	the	DET
ma-246	141	16	following	follow	VERB
ma-246	141	17	examples	example	NOUN
ma-246	141	18	are	be	AUX
ma-246	141	19	shown	show	VERB
ma-246	141	20	.	.	PUNCT
ma-246	142	1	example	example	NOUN
ma-246	142	2	2.(i	2.(i	NUM
ma-246	142	3	)	)	PUNCT
ma-246	142	4	consider	consider	VERB
ma-246	142	5	the	the	DET
ma-246	142	6	ideal	ideal	ADJ
ma-246	142	7	topological	topological	ADJ
ma-246	142	8	space	space	NOUN
ma-246	142	9	(	(	PUNCT
ma-246	142	10	x	x	X
ma-246	142	11	,	,	PUNCT
ma-246	142	12	τ	τ	PROPN
ma-246	142	13	,	,	PUNCT
ma-246	142	14	i	i	PROPN
ma-246	142	15	)	)	PUNCT
ma-246	142	16	,	,	PUNCT
ma-246	142	17	where	where	SCONJ
ma-246	142	18	x	x	X
ma-246	142	19	=	=	PRON
ma-246	142	20	{	{	PUNCT
ma-246	142	21	a	a	PRON
ma-246	142	22	,	,	PUNCT
ma-246	142	23	b	b	NOUN
ma-246	142	24	,	,	PUNCT
ma-246	142	25	c	c	NOUN
ma-246	142	26	,	,	PUNCT
ma-246	142	27	d	d	NOUN
ma-246	142	28	}	}	PUNCT
ma-246	142	29	,	,	PUNCT
ma-246	142	30	τ	τ	X
ma-246	142	31	=	=	PUNCT
ma-246	142	32	{	{	PUNCT
ma-246	142	33	x,∅	x,∅	PROPN
ma-246	142	34	,	,	PUNCT
ma-246	142	35	{	{	PUNCT
ma-246	142	36	b	b	X
ma-246	142	37	}	}	PUNCT
ma-246	142	38	,	,	PUNCT
ma-246	142	39	{	{	PUNCT
ma-246	142	40	c	c	X
ma-246	142	41	}	}	PUNCT
ma-246	142	42	,	,	PUNCT
ma-246	142	43	{	{	PUNCT
ma-246	142	44	b	b	X
ma-246	142	45	,	,	PUNCT
ma-246	142	46	c	c	NOUN
ma-246	142	47	}	}	PUNCT
ma-246	142	48	,	,	PUNCT
ma-246	142	49	{	{	PUNCT
ma-246	142	50	a	a	PRON
ma-246	142	51	,	,	PUNCT
ma-246	142	52	b	b	NOUN
ma-246	142	53	,	,	PUNCT
ma-246	142	54	c	c	NOUN
ma-246	142	55	}	}	PUNCT
ma-246	142	56	}	}	PUNCT
ma-246	142	57	,	,	PUNCT
ma-246	142	58	and	and	CCONJ
ma-246	142	59	i	i	PRON
ma-246	142	60	=	=	PUNCT
ma-246	142	61	{	{	PUNCT
ma-246	142	62	∅	∅	NOUN
ma-246	142	63	,	,	PUNCT
ma-246	142	64	{	{	PUNCT
ma-246	142	65	a	a	X
ma-246	142	66	}	}	PUNCT
ma-246	142	67	}	}	PUNCT
ma-246	142	68	.	.	PUNCT
ma-246	143	1	then	then	ADV
ma-246	143	2	the	the	DET
ma-246	143	3	η	η	PROPN
ma-246	143	4	-	-	ADJ
ma-246	143	5	open	open	ADJ
ma-246	143	6	sets	set	NOUN
ma-246	143	7	are	be	AUX
ma-246	143	8	∅	∅	NOUN
ma-246	143	9	,	,	PUNCT
ma-246	143	10	x	x	INTJ
ma-246	143	11	,	,	PUNCT
ma-246	143	12	{	{	PUNCT
ma-246	143	13	b	b	NOUN
ma-246	143	14	}	}	PUNCT
ma-246	143	15	,	,	PUNCT
ma-246	143	16	{	{	PUNCT
ma-246	143	17	c	c	X
ma-246	143	18	}	}	PUNCT
ma-246	143	19	,	,	PUNCT
ma-246	143	20	{	{	PUNCT
ma-246	143	21	a	a	DET
ma-246	143	22	,	,	PUNCT
ma-246	143	23	b	b	NOUN
ma-246	143	24	}	}	PUNCT
ma-246	143	25	,	,	PUNCT
ma-246	143	26	{	{	PUNCT
ma-246	143	27	a	a	X
ma-246	143	28	,	,	PUNCT
ma-246	143	29	c	c	NOUN
ma-246	143	30	}	}	PUNCT
ma-246	143	31	,	,	PUNCT
ma-246	143	32	{	{	PUNCT
ma-246	143	33	b	b	X
ma-246	143	34	,	,	PUNCT
ma-246	143	35	c	c	NOUN
ma-246	143	36	}	}	PUNCT
ma-246	143	37	,	,	PUNCT
ma-246	143	38	{	{	PUNCT
ma-246	143	39	b	b	X
ma-246	143	40	,	,	PUNCT
ma-246	143	41	d	d	NOUN
ma-246	143	42	}	}	PUNCT
ma-246	143	43	,	,	PUNCT
ma-246	143	44	{	{	PUNCT
ma-246	143	45	c	c	X
ma-246	143	46	,	,	PUNCT
ma-246	143	47	d	d	NOUN
ma-246	143	48	}	}	PUNCT
ma-246	143	49	,	,	PUNCT
ma-246	143	50	{	{	PUNCT
ma-246	143	51	a	a	DET
ma-246	143	52	,	,	PUNCT
ma-246	143	53	b	b	NOUN
ma-246	143	54	,	,	PUNCT
ma-246	143	55	c	c	NOUN
ma-246	143	56	}	}	PUNCT
ma-246	143	57	,	,	PUNCT
ma-246	143	58	{	{	PUNCT
ma-246	143	59	a	a	PRON
ma-246	143	60	,	,	PUNCT
ma-246	143	61	c	c	NOUN
ma-246	143	62	,	,	PUNCT
ma-246	143	63	d	d	NOUN
ma-246	143	64	}	}	PUNCT
ma-246	143	65	,	,	PUNCT
ma-246	143	66	{	{	PUNCT
ma-246	143	67	a	a	DET
ma-246	143	68	,	,	PUNCT
ma-246	143	69	b	b	NOUN
ma-246	143	70	,	,	PUNCT
ma-246	143	71	d	d	NOUN
ma-246	143	72	}	}	PUNCT
ma-246	143	73	,	,	PUNCT
ma-246	143	74	and	and	CCONJ
ma-246	143	75	{	{	PUNCT
ma-246	143	76	b	b	NOUN
ma-246	143	77	,	,	PUNCT
ma-246	143	78	c	c	NOUN
ma-246	143	79	,	,	PUNCT
ma-246	143	80	d	d	NOUN
ma-246	143	81	}	}	PUNCT
ma-246	143	82	.	.	PUNCT
ma-246	144	1	now	now	ADV
ma-246	144	2	,	,	PUNCT
ma-246	144	3	let	let	VERB
ma-246	144	4	a	a	PRON
ma-246	144	5	=	=	SYM
ma-246	144	6	{	{	PUNCT
ma-246	144	7	b	b	NOUN
ma-246	144	8	}	}	PUNCT
ma-246	144	9	and	and	CCONJ
ma-246	144	10	b	b	X
ma-246	144	11	=	=	PUNCT
ma-246	144	12	{	{	PUNCT
ma-246	144	13	c	c	NOUN
ma-246	144	14	}	}	PUNCT
ma-246	144	15	,	,	PUNCT
ma-246	144	16	then	then	ADV
ma-246	144	17	a	a	DET
ma-246	144	18	∪	∪	X
ma-246	144	19	b	b	NOUN
ma-246	144	20	=	=	SYM
ma-246	144	21	{	{	PUNCT
ma-246	144	22	b	b	NOUN
ma-246	144	23	,	,	PUNCT
ma-246	144	24	c	c	NOUN
ma-246	144	25	}	}	PUNCT
ma-246	144	26	.	.	PUNCT
ma-246	145	1	then	then	ADV
ma-246	145	2	by	by	ADP
ma-246	145	3	applying	apply	VERB
ma-246	145	4	definition	definition	NOUN
ma-246	145	5	1	1	NUM
ma-246	145	6	,	,	PUNCT
ma-246	145	7	a∗η	a∗η	PROPN
ma-246	145	8	=	=	SYM
ma-246	145	9	{	{	PUNCT
ma-246	145	10	b	b	NOUN
ma-246	145	11	}	}	PUNCT
ma-246	145	12	,	,	PUNCT
ma-246	145	13	b∗η	b∗η	PROPN
ma-246	145	14	=	=	PUNCT
ma-246	145	15	{	{	PUNCT
ma-246	145	16	c	c	NOUN
ma-246	145	17	}	}	PUNCT
ma-246	145	18	,	,	PUNCT
ma-246	145	19	and	and	CCONJ
ma-246	145	20	(	(	PUNCT
ma-246	145	21	a	a	DET
ma-246	145	22	∪	∪	ADJ
ma-246	145	23	b)∗η	b)∗η	X
ma-246	145	24	=	=	SYM
ma-246	145	25	x	x	X
ma-246	145	26	.	.	PUNCT
ma-246	146	1	observe	observe	VERB
ma-246	146	2	that	that	SCONJ
ma-246	146	3	(	(	PUNCT
ma-246	146	4	a	a	DET
ma-246	146	5	∪	∪	ADJ
ma-246	146	6	b)∗η	b)∗η	X
ma-246	146	7	=	=	SYM
ma-246	146	8	x	x	X
ma-246	146	9	and	and	CCONJ
ma-246	146	10	a∗η	a∗η	PROPN
ma-246	146	11	∪	∪	VERB
ma-246	146	12	b∗η	b∗η	PROPN
ma-246	146	13	=	=	SYM
ma-246	146	14	{	{	PUNCT
ma-246	146	15	b	b	PROPN
ma-246	146	16	,	,	PUNCT
ma-246	146	17	c	c	NOUN
ma-246	146	18	}	}	PUNCT
ma-246	146	19	.	.	PUNCT
ma-246	147	1	these	these	PRON
ma-246	147	2	shows	show	VERB
ma-246	147	3	that	that	SCONJ
ma-246	147	4	(	(	PUNCT
ma-246	147	5	a	a	DET
ma-246	147	6	∪	∪	ADJ
ma-246	147	7	b)∗η	b)∗η	X
ma-246	147	8	*	*	PUNCT
ma-246	147	9	a∗η	a∗η	PROPN
ma-246	147	10	∪	∪	ADP
ma-246	147	11	b∗η	b∗η	PROPN
ma-246	147	12	.(ii	.(ii	PROPN
ma-246	147	13	)	)	PUNCT
ma-246	147	14	consider	consider	VERB
ma-246	147	15	the	the	DET
ma-246	147	16	ideal	ideal	ADJ
ma-246	147	17	topological	topological	ADJ
ma-246	147	18	space	space	NOUN
ma-246	147	19	(	(	PUNCT
ma-246	147	20	x	x	X
ma-246	147	21	,	,	PUNCT
ma-246	147	22	τ	τ	PROPN
ma-246	147	23	,	,	PUNCT
ma-246	147	24	i	i	PROPN
ma-246	147	25	)	)	PUNCT
ma-246	147	26	,	,	PUNCT
ma-246	147	27	where	where	SCONJ
ma-246	147	28	x	x	X
ma-246	147	29	=	=	PRON
ma-246	147	30	{	{	PUNCT
ma-246	147	31	a	a	PRON
ma-246	147	32	,	,	PUNCT
ma-246	147	33	b	b	NOUN
ma-246	147	34	,	,	PUNCT
ma-246	147	35	c	c	NOUN
ma-246	147	36	,	,	PUNCT
ma-246	147	37	d	d	NOUN
ma-246	147	38	}	}	PUNCT
ma-246	147	39	,	,	PUNCT
ma-246	147	40	τ	τ	X
ma-246	147	41	=	=	PUNCT
ma-246	147	42	{	{	PUNCT
ma-246	147	43	x,∅	x,∅	PROPN
ma-246	147	44	,	,	PUNCT
ma-246	147	45	{	{	PUNCT
ma-246	147	46	b	b	X
ma-246	147	47	}	}	PUNCT
ma-246	147	48	,	,	PUNCT
ma-246	147	49	{	{	PUNCT
ma-246	147	50	c	c	X
ma-246	147	51	}	}	PUNCT
ma-246	147	52	,	,	PUNCT
ma-246	147	53	{	{	PUNCT
ma-246	147	54	b	b	X
ma-246	147	55	,	,	PUNCT
ma-246	147	56	c	c	NOUN
ma-246	147	57	,	,	PUNCT
ma-246	147	58	d	d	NOUN
ma-246	147	59	}	}	PUNCT
ma-246	147	60	}	}	PUNCT
ma-246	147	61	,	,	PUNCT
ma-246	147	62	and	and	CCONJ
ma-246	147	63	i	i	PRON
ma-246	147	64	=	=	PUNCT
ma-246	147	65	{	{	PUNCT
ma-246	147	66	∅	∅	NOUN
ma-246	147	67	,	,	PUNCT
ma-246	147	68	{	{	PUNCT
ma-246	147	69	c	c	NOUN
ma-246	147	70	}	}	PUNCT
ma-246	147	71	}	}	PUNCT
ma-246	147	72	.	.	PUNCT
ma-246	148	1	then	then	ADV
ma-246	148	2	the	the	DET
ma-246	148	3	η	η	PROPN
ma-246	148	4	-	-	ADJ
ma-246	148	5	open	open	ADJ
ma-246	148	6	sets	set	NOUN
ma-246	148	7	are	be	AUX
ma-246	148	8	∅	∅	NOUN
ma-246	148	9	,	,	PUNCT
ma-246	148	10	x	x	INTJ
ma-246	148	11	,	,	PUNCT
ma-246	148	12	{	{	PUNCT
ma-246	148	13	b	b	NOUN
ma-246	148	14	}	}	PUNCT
ma-246	148	15	,	,	PUNCT
ma-246	148	16	{	{	PUNCT
ma-246	148	17	c	c	X
ma-246	148	18	}	}	PUNCT
ma-246	148	19	,	,	PUNCT
ma-246	148	20	{	{	PUNCT
ma-246	148	21	a	a	DET
ma-246	148	22	,	,	PUNCT
ma-246	148	23	b	b	NOUN
ma-246	148	24	}	}	PUNCT
ma-246	148	25	,	,	PUNCT
ma-246	148	26	{	{	PUNCT
ma-246	148	27	b	b	X
ma-246	148	28	,	,	PUNCT
ma-246	148	29	c	c	NOUN
ma-246	148	30	}	}	PUNCT
ma-246	148	31	,	,	PUNCT
ma-246	148	32	{	{	PUNCT
ma-246	148	33	b	b	X
ma-246	148	34	,	,	PUNCT
ma-246	148	35	d	d	NOUN
ma-246	148	36	}	}	PUNCT
ma-246	148	37	,	,	PUNCT
ma-246	148	38	{	{	PUNCT
ma-246	148	39	a	a	DET
ma-246	148	40	,	,	PUNCT
ma-246	148	41	b	b	NOUN
ma-246	148	42	,	,	PUNCT
ma-246	148	43	c	c	NOUN
ma-246	148	44	}	}	PUNCT
ma-246	148	45	,	,	PUNCT
ma-246	148	46	{	{	PUNCT
ma-246	148	47	a	a	DET
ma-246	148	48	,	,	PUNCT
ma-246	148	49	b	b	NOUN
ma-246	148	50	,	,	PUNCT
ma-246	148	51	d	d	NOUN
ma-246	148	52	}	}	PUNCT
ma-246	148	53	,	,	PUNCT
ma-246	148	54	and	and	CCONJ
ma-246	148	55	{	{	PUNCT
ma-246	148	56	b	b	NOUN
ma-246	148	57	,	,	PUNCT
ma-246	148	58	c	c	NOUN
ma-246	148	59	,	,	PUNCT
ma-246	148	60	d	d	NOUN
ma-246	148	61	}	}	PUNCT
ma-246	148	62	.	.	PUNCT
ma-246	149	1	let	let	VERB
ma-246	149	2	a	a	PRON
ma-246	149	3	,	,	PUNCT
ma-246	149	4	b	b	X
ma-246	149	5	⊂	⊂	PROPN
ma-246	149	6	x	x	PUNCT
ma-246	149	7	where	where	SCONJ
ma-246	149	8	,	,	PUNCT
ma-246	149	9	a	a	PRON
ma-246	149	10	=	=	X
ma-246	149	11	{	{	PUNCT
ma-246	149	12	a	a	X
ma-246	149	13	,	,	PUNCT
ma-246	149	14	c	c	NOUN
ma-246	149	15	,	,	PUNCT
ma-246	149	16	d	d	NOUN
ma-246	149	17	}	}	PUNCT
ma-246	149	18	and	and	CCONJ
ma-246	149	19	b	b	X
ma-246	149	20	=	=	NOUN
ma-246	149	21	{	{	PUNCT
ma-246	149	22	a	a	PROPN
ma-246	149	23	,	,	PUNCT
ma-246	149	24	b	b	NOUN
ma-246	149	25	}	}	PUNCT
ma-246	149	26	.	.	PUNCT
ma-246	150	1	then	then	ADV
ma-246	150	2	by	by	ADP
ma-246	150	3	definition	definition	NOUN
ma-246	150	4	1	1	NUM
ma-246	150	5	,	,	PUNCT
ma-246	150	6	a∗η	a∗η	PROPN
ma-246	150	7	=	=	SYM
ma-246	150	8	{	{	PUNCT
ma-246	150	9	a	a	X
ma-246	150	10	,	,	PUNCT
ma-246	150	11	d	d	NOUN
ma-246	150	12	}	}	PUNCT
ma-246	150	13	and	and	CCONJ
ma-246	150	14	b∗η	b∗η	PROPN
ma-246	150	15	=	=	SYM
ma-246	150	16	{	{	PUNCT
ma-246	150	17	a	a	PRON
ma-246	150	18	,	,	PUNCT
ma-246	150	19	b	b	NOUN
ma-246	150	20	,	,	PUNCT
ma-246	150	21	d	d	NOUN
ma-246	150	22	}	}	PUNCT
ma-246	150	23	.	.	PUNCT
ma-246	150	24	obeserve	obeserve	VERB
ma-246	150	25	that	that	SCONJ
ma-246	150	26	a	a	DET
ma-246	150	27	*	*	PUNCT
ma-246	150	28	a∗η	a∗η	PROPN
ma-246	150	29	and	and	CCONJ
ma-246	150	30	b∗η	b∗η	PROPN
ma-246	150	31	*	*	PUNCT
ma-246	150	32	b.	b.	PROPN
ma-246	150	33	theorem	theorem	PROPN
ma-246	150	34	2	2	X
ma-246	150	35	.	.	PUNCT
ma-246	151	1	let	let	VERB
ma-246	151	2	(	(	PUNCT
ma-246	151	3	x	x	X
ma-246	151	4	,	,	PUNCT
ma-246	151	5	τ	τ	PROPN
ma-246	151	6	,	,	PUNCT
ma-246	151	7	i	i	PRON
ma-246	151	8	)	)	PUNCT
ma-246	151	9	be	be	VERB
ma-246	151	10	an	an	DET
ma-246	151	11	ideal	ideal	ADJ
ma-246	151	12	topological	topological	ADJ
ma-246	151	13	space	space	NOUN
ma-246	151	14	and	and	CCONJ
ma-246	151	15	a	a	DET
ma-246	151	16	,	,	PUNCT
ma-246	151	17	b	b	PROPN
ma-246	151	18	be	be	AUX
ma-246	151	19	subsets	subset	NOUN
ma-246	151	20	of	of	ADP
ma-246	151	21	x	x	X
ma-246	151	22	.	.	PUNCT
ma-246	152	1	then	then	ADV
ma-246	152	2	for	for	ADP
ma-246	152	3	any	any	DET
ma-246	152	4	η	η	ADJ
ma-246	152	5	-	-	ADJ
ma-246	152	6	local	local	ADJ
ma-246	152	7	functions	function	NOUN
ma-246	152	8	,	,	PUNCT
ma-246	152	9	the	the	DET
ma-246	152	10	following	follow	VERB
ma-246	152	11	properties	property	NOUN
ma-246	152	12	hold	hold	VERB
ma-246	152	13	:	:	PUNCT
ma-246	152	14	(	(	PUNCT
ma-246	152	15	i	i	NOUN
ma-246	152	16	)	)	PUNCT
ma-246	152	17	(	(	PUNCT
ma-246	152	18	a	a	DET
ma-246	152	19	\	\	PROPN
ma-246	152	20	b)∗η	b)∗η	X
ma-246	152	21	\	\	X
ma-246	152	22	b∗η	b∗η	PROPN
ma-246	152	23	⊆	⊆	NUM
ma-246	152	24	a∗η	a∗η	PROPN
ma-246	152	25	\	\	NOUN
ma-246	152	26	b∗η	b∗η	PROPN
ma-246	152	27	;	;	PUNCT
ma-246	152	28	(	(	PUNCT
ma-246	152	29	ii	ii	NOUN
ma-246	152	30	)	)	PUNCT
ma-246	152	31	if	if	SCONJ
ma-246	152	32	b	b	X
ma-246	152	33	∈	∈	PROPN
ma-246	153	1	i	i	PRON
ma-246	153	2	,	,	PUNCT
ma-246	153	3	then	then	ADV
ma-246	153	4	(	(	PUNCT
ma-246	153	5	a	a	DET
ma-246	153	6	∪	∪	ADJ
ma-246	153	7	b)∗η	b)∗η	X
ma-246	153	8	=	=	PUNCT
ma-246	153	9	a∗η	a∗η	PROPN
ma-246	153	10	=	=	SYM
ma-246	153	11	(	(	PUNCT
ma-246	153	12	a	a	DET
ma-246	153	13	\	\	PROPN
ma-246	153	14	b)∗η	b)∗η	NOUN
ma-246	153	15	;	;	PUNCT
ma-246	153	16	(	(	PUNCT
ma-246	153	17	iii	iii	X
ma-246	153	18	)	)	PUNCT
ma-246	153	19	(	(	PUNCT
ma-246	153	20	a	a	DET
ma-246	153	21	\	\	PROPN
ma-246	153	22	b)∗η	b)∗η	X
ma-246	153	23	∪	∪	ADJ
ma-246	153	24	(	(	PUNCT
ma-246	153	25	b	b	NOUN
ma-246	153	26	\	\	NOUN
ma-246	153	27	a)∗η	a)∗η	NOUN
ma-246	153	28	⊆	⊆	NUM
ma-246	153	29	(	(	PUNCT
ma-246	153	30	a	a	DET
ma-246	153	31	∪	∪	ADJ
ma-246	153	32	b)∗η	b)∗η	NOUN
ma-246	153	33	;	;	PUNCT
ma-246	153	34	(	(	PUNCT
ma-246	153	35	iv	iv	X
ma-246	153	36	)	)	PUNCT
ma-246	153	37	if	if	SCONJ
ma-246	153	38	u	u	PROPN
ma-246	153	39	⊆	⊆	NUM
ma-246	153	40	x	x	X
ma-246	153	41	,	,	PUNCT
ma-246	153	42	then	then	ADV
ma-246	153	43	u	u	NOUN
ma-246	153	44	∩	∩	NOUN
ma-246	153	45	(	(	PUNCT
ma-246	153	46	u	u	NOUN
ma-246	153	47	∩	∩	NOUN
ma-246	153	48	a)∗η	a)∗η	NOUN
ma-246	153	49	⊆	⊆	NUM
ma-246	153	50	u	u	NOUN
ma-246	153	51	∩	∩	NOUN
ma-246	153	52	a∗η	a∗η	PROPN
ma-246	153	53	;	;	PUNCT
ma-246	153	54	(	(	PUNCT
ma-246	153	55	v	v	NOUN
ma-246	153	56	)	)	PUNCT
ma-246	153	57	if	if	SCONJ
ma-246	153	58	u	u	PROPN
ma-246	153	59	∈	∈	PROPN
ma-246	154	1	i	i	PRON
ma-246	154	2	,	,	PUNCT
ma-246	154	3	then	then	ADV
ma-246	154	4	(	(	PUNCT
ma-246	154	5	a	a	DET
ma-246	154	6	∩	∩	NOUN
ma-246	154	7	u)∗η	u)∗η	NOUN
ma-246	154	8	=	=	SYM
ma-246	154	9	∅	∅	NOUN
ma-246	154	10	;	;	PUNCT
ma-246	154	11	(	(	PUNCT
ma-246	154	12	vi	vi	X
ma-246	154	13	)	)	PUNCT
ma-246	154	14	if	if	SCONJ
ma-246	154	15	a	a	PRON
ma-246	154	16	is	be	AUX
ma-246	154	17	an	an	DET
ma-246	154	18	η	η	ADJ
ma-246	154	19	-	-	ADJ
ma-246	154	20	closed	closed	ADJ
ma-246	154	21	set	set	NOUN
ma-246	154	22	,	,	PUNCT
ma-246	154	23	then	then	ADV
ma-246	154	24	a∗η	a∗η	PROPN
ma-246	154	25	⊆	⊆	NUM
ma-246	154	26	a	a	PRON
ma-246	154	27	;	;	PUNCT
ma-246	154	28	(	(	PUNCT
ma-246	154	29	vii	vii	PROPN
ma-246	154	30	)	)	PUNCT
ma-246	154	31	(	(	PUNCT
ma-246	154	32	a	a	DET
ma-246	154	33	∩	∩	X
ma-246	154	34	a∗η	a∗η	NUM
ma-246	154	35	)	)	PUNCT
ma-246	154	36	∗	∗	PROPN
ma-246	154	37	η	η	PROPN
ma-246	154	38	⊆	⊆	NUM
ma-246	154	39	a∗η	a∗η	NUM
ma-246	154	40	;	;	PUNCT
ma-246	154	41	(	(	PUNCT
ma-246	154	42	iix	iix	PROPN
ma-246	154	43	)	)	PUNCT
ma-246	154	44	if	if	SCONJ
ma-246	154	45	a	a	DET
ma-246	154	46	∪	∪	X
ma-246	154	47	b	b	X
ma-246	154	48	∈	∈	NOUN
ma-246	155	1	i	i	PRON
ma-246	155	2	,	,	PUNCT
ma-246	155	3	then	then	ADV
ma-246	155	4	(	(	PUNCT
ma-246	155	5	a	a	DET
ma-246	155	6	∪	∪	ADJ
ma-246	155	7	b)∗η	b)∗η	X
ma-246	155	8	=	=	SYM
ma-246	155	9	a∗η	a∗η	PUNCT
ma-246	155	10	∪	∪	X
ma-246	155	11	b∗η	b∗η	PROPN
ma-246	155	12	=	=	SYM
ma-246	155	13	∅.	∅.	X
ma-246	155	14	(	(	PUNCT
ma-246	155	15	ix	ix	ADJ
ma-246	155	16	)	)	PUNCT
ma-246	155	17	a∗η	a∗η	PROPN
ma-246	155	18	=	=	SYM
ma-246	155	19	η	η	PROPN
ma-246	155	20	-	-	PROPN
ma-246	155	21	cl(a∗η	cl(a∗η	NOUN
ma-246	155	22	)	)	PUNCT
ma-246	155	23	⊆	⊆	NUM
ma-246	155	24	η	η	PROPN
ma-246	155	25	-	-	NOUN
ma-246	155	26	cl(a	cl(a	NUM
ma-246	155	27	)	)	PUNCT
ma-246	155	28	and	and	CCONJ
ma-246	155	29	a∗η	a∗η	PROPN
ma-246	155	30	is	be	AUX
ma-246	155	31	an	an	DET
ma-246	155	32	η	η	ADJ
ma-246	155	33	-	-	ADJ
ma-246	155	34	closed	closed	ADJ
ma-246	155	35	set	set	NOUN
ma-246	155	36	;	;	PUNCT
ma-246	155	37	and	and	CCONJ
ma-246	155	38	(	(	PUNCT
ma-246	155	39	x	x	X
ma-246	155	40	)	)	PUNCT
ma-246	155	41	if	if	SCONJ
ma-246	155	42	a	a	DET
ma-246	155	43	⊆	⊆	NUM
ma-246	155	44	a∗η	a∗η	PROPN
ma-246	155	45	,	,	PUNCT
ma-246	155	46	then	then	ADV
ma-246	155	47	a∗η	a∗η	PROPN
ma-246	155	48	=	=	SYM
ma-246	155	49	η	η	PROPN
ma-246	155	50	-	-	PROPN
ma-246	155	51	cl(a∗η	cl(a∗η	NOUN
ma-246	155	52	)	)	PUNCT
ma-246	155	53	=	=	PUNCT
ma-246	155	54	η	η	PROPN
ma-246	155	55	-	-	NOUN
ma-246	155	56	cl(a	cl(a	NUM
ma-246	155	57	)	)	PUNCT
ma-246	155	58	.	.	PUNCT
ma-246	156	1	proof	proof	NOUN
ma-246	156	2	.	.	PUNCT
ma-246	157	1	(	(	PUNCT
ma-246	157	2	i	i	NOUN
ma-246	157	3	)	)	PUNCT
ma-246	157	4	let	let	VERB
ma-246	157	5	a	a	DET
ma-246	157	6	,	,	PUNCT
ma-246	157	7	b	b	NOUN
ma-246	157	8	⊆	⊆	NUM
ma-246	157	9	x	x	X
ma-246	157	10	.	.	PUNCT
ma-246	158	1	since	since	SCONJ
ma-246	158	2	a\b	a\b	ADP
ma-246	158	3	⊆	⊆	NUM
ma-246	158	4	a	a	PRON
ma-246	158	5	,	,	PUNCT
ma-246	158	6	by	by	ADP
ma-246	158	7	theorem	theorem	NOUN
ma-246	158	8	1	1	NUM
ma-246	158	9	(	(	PUNCT
ma-246	158	10	i	i	NOUN
ma-246	158	11	)	)	PUNCT
ma-246	158	12	,	,	PUNCT
ma-246	158	13	(	(	PUNCT
ma-246	158	14	a\b)∗η	a\b)∗η	NOUN
ma-246	158	15	⊆	⊆	NUM
ma-246	158	16	a∗η	a∗η	PROPN
ma-246	158	17	implies	imply	VERB
ma-246	158	18	that	that	SCONJ
ma-246	158	19	(	(	PUNCT
ma-246	158	20	a\b)∗η\b∗η	a\b)∗η\b∗η	PROPN
ma-246	158	21	⊆	⊆	NUM
ma-246	158	22	a∗η	a∗η	PROPN
ma-246	158	23	\	\	NOUN
ma-246	158	24	b∗η	b∗η	PROPN
ma-246	158	25	.	.	PUNCT
ma-246	159	1	(	(	PUNCT
ma-246	159	2	ii	ii	NOUN
ma-246	159	3	)	)	PUNCT
ma-246	159	4	let	let	VERB
ma-246	159	5	b	b	X
ma-246	159	6	∈	∈	PROPN
ma-246	160	1	i	i	PRON
ma-246	160	2	.	.	PUNCT
ma-246	161	1	suppose	suppose	VERB
ma-246	161	2	x	x	X
ma-246	161	3	∈	∈	PROPN
ma-246	161	4	(	(	PUNCT
ma-246	161	5	a∪b)∗η	a∪b)∗η	NOUN
ma-246	161	6	.	.	PUNCT
ma-246	162	1	then	then	ADV
ma-246	162	2	for	for	ADP
ma-246	162	3	every	every	DET
ma-246	162	4	u	u	PROPN
ma-246	162	5	∈	∈	PROPN
ma-246	162	6	η	η	PROPN
ma-246	162	7	-	-	PROPN
ma-246	162	8	o(x	o(x	PROPN
ma-246	162	9	)	)	PUNCT
ma-246	162	10	,	,	PUNCT
ma-246	162	11	(	(	PUNCT
ma-246	162	12	a∪b)∩	a∪b)∩	NOUN
ma-246	162	13	u	u	NOUN
ma-246	162	14	/∈	/∈	PUNCT
ma-246	163	1	i	i	PRON
ma-246	163	2	.	.	PUNCT
ma-246	164	1	note	note	VERB
ma-246	164	2	that	that	SCONJ
ma-246	164	3	(	(	PUNCT
ma-246	164	4	a∩u)∪(b∩u	a∩u)∪(b∩u	PROPN
ma-246	164	5	)	)	PUNCT
ma-246	164	6	=	=	SYM
ma-246	164	7	(	(	PUNCT
ma-246	165	1	a∪b)∩u	a∪b)∩u	PROPN
ma-246	165	2	/∈	/∈	VERB
ma-246	165	3	i	i	PRON
ma-246	165	4	implies	imply	VERB
ma-246	165	5	that	that	SCONJ
ma-246	165	6	(	(	PUNCT
ma-246	165	7	a∩u)∪(b∩u	a∩u)∪(b∩u	PROPN
ma-246	165	8	)	)	PUNCT
ma-246	165	9	/∈	/∈	PUNCT
ma-246	166	1	i	i	PRON
ma-246	166	2	.	.	PUNCT
ma-246	167	1	it	it	PRON
ma-246	167	2	shows	show	VERB
ma-246	167	3	that	that	PRON
ma-246	167	4	a∩u	a∩u	PROPN
ma-246	167	5	/∈	/∈	PUNCT
ma-246	168	1	i	i	PRON
ma-246	168	2	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	168	3	eur	eur	PROPN
ma-246	168	4	.	.	PUNCT
ma-246	169	1	j.	j.	PROPN
ma-246	169	2	math	math	PROPN
ma-246	169	3	.	.	PUNCT
ma-246	170	1	anal	anal	PROPN
ma-246	170	2	.	.	PUNCT
ma-246	171	1	10.28924	10.28924	NUM
ma-246	171	2	/	/	SYM
ma-246	171	3	ada	ada	PROPN
ma-246	171	4	/	/	SYM
ma-246	171	5	ma.5.2	ma.5.2	PROPN
ma-246	171	6	5or	5or	ADJ
ma-246	171	7	b∩u	b∩u	NOUN
ma-246	171	8	/∈	/∈	PUNCT
ma-246	172	1	i	i	PRON
ma-246	172	2	,	,	PUNCT
ma-246	172	3	or	or	CCONJ
ma-246	172	4	both	both	PRON
ma-246	172	5	,	,	PUNCT
ma-246	172	6	and	and	CCONJ
ma-246	172	7	as	as	ADP
ma-246	172	8	a	a	DET
ma-246	172	9	result	result	NOUN
ma-246	172	10	,	,	PUNCT
ma-246	172	11	x	x	SYM
ma-246	172	12	∈	∈	PROPN
ma-246	172	13	a∗η	a∗η	PROPN
ma-246	172	14	or	or	CCONJ
ma-246	172	15	x	x	SYM
ma-246	172	16	∈	∈	PROPN
ma-246	172	17	b∗η	b∗η	PROPN
ma-246	172	18	,	,	PUNCT
ma-246	172	19	or	or	CCONJ
ma-246	172	20	both	both	PRON
ma-246	172	21	.	.	PUNCT
ma-246	173	1	hence	hence	ADV
ma-246	173	2	,	,	PUNCT
ma-246	173	3	x	x	PROPN
ma-246	173	4	∈	∈	PROPN
ma-246	173	5	a∗η∪b∗η	a∗η∪b∗η	NOUN
ma-246	173	6	.	.	PUNCT
ma-246	174	1	now	now	ADV
ma-246	174	2	,	,	PUNCT
ma-246	174	3	note	note	VERB
ma-246	174	4	that	that	SCONJ
ma-246	174	5	b	b	X
ma-246	174	6	∈	∈	PROPN
ma-246	175	1	i	i	PRON
ma-246	175	2	,	,	PUNCT
ma-246	175	3	then	then	ADV
ma-246	175	4	by	by	ADP
ma-246	175	5	theorem	theorem	NOUN
ma-246	175	6	1	1	NUM
ma-246	175	7	(	(	PUNCT
ma-246	175	8	vii	vii	PROPN
ma-246	175	9	)	)	PUNCT
ma-246	175	10	,	,	PUNCT
ma-246	175	11	b∗η	b∗η	PROPN
ma-246	175	12	=	=	PUNCT
ma-246	175	13	∅.	∅.	ADP
ma-246	175	14	thus	thus	ADV
ma-246	175	15	,	,	PUNCT
ma-246	175	16	x	x	SYM
ma-246	175	17	∈	∈	PROPN
ma-246	175	18	a∗η	a∗η	PROPN
ma-246	175	19	∪b∗η	∪b∗η	PROPN
ma-246	175	20	=	=	SYM
ma-246	175	21	a∗η	a∗η	PROPN
ma-246	175	22	∪∅	∪∅	PROPN
ma-246	175	23	=	=	SYM
ma-246	175	24	a∗η	a∗η	PROPN
ma-246	175	25	.	.	PUNCT
ma-246	176	1	thisimplies	thisimplie	VERB
ma-246	176	2	that	that	SCONJ
ma-246	176	3	x	x	SYM
ma-246	176	4	∈	∈	PROPN
ma-246	176	5	a∗η	a∗η	PROPN
ma-246	176	6	.	.	PUNCT
ma-246	177	1	hence	hence	ADV
ma-246	177	2	,	,	PUNCT
ma-246	177	3	it	it	PRON
ma-246	177	4	shows	show	VERB
ma-246	177	5	that	that	SCONJ
ma-246	177	6	(	(	PUNCT
ma-246	177	7	a∪b)∗η	a∪b)∗η	NOUN
ma-246	177	8	⊆	⊆	NUM
ma-246	177	9	a∗η	a∗η	NOUN
ma-246	177	10	.	.	PUNCT
ma-246	178	1	in	in	ADP
ma-246	178	2	contrast	contrast	NOUN
ma-246	178	3	,	,	PUNCT
ma-246	178	4	since	since	SCONJ
ma-246	178	5	a	a	DET
ma-246	178	6	⊆	⊆	NUM
ma-246	178	7	a∪b	a∪b	NOUN
ma-246	178	8	,	,	PUNCT
ma-246	178	9	bytheorem	bytheorem	VERB
ma-246	178	10	1	1	NUM
ma-246	178	11	(	(	PUNCT
ma-246	178	12	i	i	NOUN
ma-246	178	13	)	)	PUNCT
ma-246	178	14	,	,	PUNCT
ma-246	178	15	it	it	PRON
ma-246	178	16	implies	imply	VERB
ma-246	178	17	that	that	SCONJ
ma-246	178	18	a∗η	a∗η	PROPN
ma-246	178	19	⊆	⊆	NUM
ma-246	178	20	(	(	PUNCT
ma-246	178	21	a	a	DET
ma-246	178	22	∪	∪	ADJ
ma-246	178	23	b)∗η	b)∗η	NOUN
ma-246	178	24	.	.	PUNCT
ma-246	179	1	consequently	consequently	ADV
ma-246	179	2	,	,	PUNCT
ma-246	179	3	as	as	ADP
ma-246	179	4	a	a	DET
ma-246	179	5	result	result	NOUN
ma-246	179	6	,	,	PUNCT
ma-246	179	7	a∗η	a∗η	PROPN
ma-246	179	8	=	=	SYM
ma-246	179	9	(	(	PUNCT
ma-246	179	10	a	a	DET
ma-246	179	11	∪	∪	ADJ
ma-246	179	12	b)∗η	b)∗η	NOUN
ma-246	179	13	.now	.now	NOUN
ma-246	179	14	,	,	PUNCT
ma-246	179	15	suppose	suppose	VERB
ma-246	179	16	that	that	SCONJ
ma-246	179	17	(	(	PUNCT
ma-246	179	18	a\b)∗η	a\b)∗η	NOUN
ma-246	179	19	6=	6=	NUM
ma-246	179	20	a∗η	a∗η	PROPN
ma-246	179	21	.	.	PUNCT
ma-246	180	1	let	let	VERB
ma-246	180	2	(	(	PUNCT
ma-246	180	3	a\b)∗η	a\b)∗η	NOUN
ma-246	180	4	⊂	⊂	PROPN
ma-246	180	5	a∗η	a∗η	PROPN
ma-246	180	6	.	.	PUNCT
ma-246	181	1	then	then	ADV
ma-246	181	2	there	there	PRON
ma-246	181	3	exists	exist	VERB
ma-246	181	4	an	an	DET
ma-246	181	5	element	element	NOUN
ma-246	181	6	x	x	SYM
ma-246	181	7	∈	∈	PROPN
ma-246	181	8	a∗ηsuch	a∗ηsuch	NOUN
ma-246	181	9	that	that	PRON
ma-246	181	10	x	x	X
ma-246	181	11	/∈	/∈	INTJ
ma-246	182	1	(	(	PUNCT
ma-246	182	2	a	a	DET
ma-246	182	3	\b)∗η	\b)∗η	PROPN
ma-246	182	4	.	.	PUNCT
ma-246	183	1	note	note	VERB
ma-246	183	2	that	that	SCONJ
ma-246	183	3	x	x	SYM
ma-246	183	4	∈	∈	PROPN
ma-246	183	5	a∗η	a∗η	PROPN
ma-246	183	6	implies	imply	VERB
ma-246	183	7	that	that	SCONJ
ma-246	183	8	for	for	ADP
ma-246	183	9	every	every	DET
ma-246	183	10	a	a	DET
ma-246	183	11	⊆	⊆	NUM
ma-246	183	12	x	x	SYM
ma-246	183	13	,	,	PUNCT
ma-246	183	14	a∩u	a∩u	PROPN
ma-246	183	15	/∈	/∈	PUNCT
ma-246	184	1	i	i	PRON
ma-246	184	2	for	for	ADP
ma-246	184	3	every	every	DET
ma-246	184	4	u	u	PROPN
ma-246	184	5	∈	∈	PROPN
ma-246	184	6	η	η	PROPN
ma-246	184	7	-	-	PROPN
ma-246	184	8	o(x	o(x	PROPN
ma-246	184	9	)	)	PUNCT
ma-246	184	10	.	.	PUNCT
ma-246	185	1	now	now	ADV
ma-246	185	2	,	,	PUNCT
ma-246	185	3	since	since	SCONJ
ma-246	185	4	x	x	PROPN
ma-246	185	5	/∈	/∈	INTJ
ma-246	185	6	(	(	PUNCT
ma-246	185	7	a	a	DET
ma-246	185	8	\b)∗η	\b)∗η	PROPN
ma-246	185	9	,	,	PUNCT
ma-246	185	10	there	there	PRON
ma-246	185	11	exists	exist	VERB
ma-246	185	12	u	u	PROPN
ma-246	185	13	∈	∈	PROPN
ma-246	185	14	η	η	PROPN
ma-246	185	15	-	-	PROPN
ma-246	185	16	o(x	o(x	ADJ
ma-246	185	17	)	)	PUNCT
ma-246	185	18	such	such	ADJ
ma-246	185	19	that	that	SCONJ
ma-246	185	20	(	(	PUNCT
ma-246	185	21	a	a	DET
ma-246	185	22	\b	\b	ADJ
ma-246	185	23	)	)	PUNCT
ma-246	185	24	∩	∩	NOUN
ma-246	185	25	u	u	X
ma-246	185	26	∈	∈	PROPN
ma-246	185	27	i	i	PRON
ma-246	185	28	.note	.note	VERB
ma-246	185	29	that	that	SCONJ
ma-246	185	30	(	(	PUNCT
ma-246	185	31	a	a	DET
ma-246	185	32	\	\	PROPN
ma-246	185	33	b	b	NOUN
ma-246	185	34	)	)	PUNCT
ma-246	185	35	∩	∩	ADJ
ma-246	185	36	u	u	NOUN
ma-246	185	37	=	=	X
ma-246	185	38	(	(	PUNCT
ma-246	185	39	a	a	DET
ma-246	185	40	∩	∩	ADJ
ma-246	185	41	u	u	NOUN
ma-246	185	42	)	)	PUNCT
ma-246	185	43	\	\	PUNCT
ma-246	185	44	(	(	PUNCT
ma-246	185	45	b	b	NUM
ma-246	185	46	∩	∩	ADJ
ma-246	185	47	u	u	NOUN
ma-246	185	48	)	)	PUNCT
ma-246	185	49	∈	∈	PROPN
ma-246	185	50	i	i	PROPN
ma-246	185	51	and	and	CCONJ
ma-246	185	52	b	b	PROPN
ma-246	185	53	∩	∩	X
ma-246	185	54	u	u	NOUN
ma-246	185	55	⊆	⊆	PROPN
ma-246	185	56	b	b	PROPN
ma-246	185	57	∈	∈	PROPN
ma-246	186	1	i	i	PRON
ma-246	186	2	.	.	PUNCT
ma-246	187	1	now	now	ADV
ma-246	187	2	,	,	PUNCT
ma-246	187	3	since	since	SCONJ
ma-246	187	4	i	i	PRON
ma-246	187	5	is	be	AUX
ma-246	187	6	anideal	anideal	NOUN
ma-246	187	7	,	,	PUNCT
ma-246	187	8	b	b	PROPN
ma-246	187	9	∩	∩	X
ma-246	187	10	u	u	NOUN
ma-246	187	11	∈	∈	PROPN
ma-246	187	12	i	i	PRON
ma-246	187	13	and	and	CCONJ
ma-246	187	14	[	[	X
ma-246	187	15	(	(	PUNCT
ma-246	187	16	a	a	DET
ma-246	187	17	∩	∩	ADJ
ma-246	187	18	u	u	NOUN
ma-246	187	19	)	)	PUNCT
ma-246	187	20	\	\	PUNCT
ma-246	188	1	(	(	PUNCT
ma-246	188	2	b	b	NUM
ma-246	188	3	∩	∩	ADJ
ma-246	188	4	u	u	NOUN
ma-246	188	5	)	)	PUNCT
ma-246	188	6	]	]	PUNCT
ma-246	188	7	∪	∪	X
ma-246	188	8	(	(	PUNCT
ma-246	188	9	b	b	NOUN
ma-246	188	10	∩	∩	ADJ
ma-246	188	11	u	u	NOUN
ma-246	188	12	)	)	PUNCT
ma-246	188	13	∈	∈	PROPN
ma-246	189	1	i	i	PRON
ma-246	189	2	,	,	PUNCT
ma-246	189	3	respectively	respectively	ADV
ma-246	189	4	.	.	PUNCT
ma-246	190	1	again	again	ADV
ma-246	190	2	,	,	PUNCT
ma-246	190	3	note	note	VERB
ma-246	190	4	that	that	SCONJ
ma-246	190	5	[	[	PUNCT
ma-246	190	6	(	(	PUNCT
ma-246	190	7	a∩u)\(b∩u	a∩u)\(b∩u	PROPN
ma-246	190	8	)	)	PUNCT
ma-246	190	9	]	]	PUNCT
ma-246	191	1	∪(b∩u	∪(b∩u	PROPN
ma-246	191	2	)	)	PUNCT
ma-246	191	3	=	=	PRON
ma-246	191	4	(	(	PUNCT
ma-246	191	5	a∩u)∪(b∩u	a∩u)∪(b∩u	PROPN
ma-246	191	6	)	)	PUNCT
ma-246	191	7	∈	∈	PROPN
ma-246	192	1	i	i	PRON
ma-246	192	2	.	.	PUNCT
ma-246	193	1	since	since	SCONJ
ma-246	193	2	(	(	PUNCT
ma-246	193	3	a∩u	a∩u	PROPN
ma-246	193	4	)	)	PUNCT
ma-246	193	5	⊆	⊆	NUM
ma-246	193	6	(	(	PUNCT
ma-246	193	7	a∩u)∪(b∩u	a∩u)∪(b∩u	PROPN
ma-246	193	8	)	)	PUNCT
ma-246	193	9	∈	∈	PROPN
ma-246	193	10	i	i	PRON
ma-246	193	11	,	,	PUNCT
ma-246	193	12	it	it	PRON
ma-246	193	13	implies	imply	VERB
ma-246	193	14	that	that	SCONJ
ma-246	193	15	a	a	DET
ma-246	193	16	∩	∩	ADJ
ma-246	193	17	u	u	NOUN
ma-246	193	18	∈	∈	PROPN
ma-246	193	19	i	i	PRON
ma-246	193	20	.	.	PUNCT
ma-246	194	1	so	so	ADV
ma-246	194	2	,	,	PUNCT
ma-246	194	3	there	there	PRON
ma-246	194	4	exists	exist	VERB
ma-246	194	5	u	u	PROPN
ma-246	194	6	∈	∈	PROPN
ma-246	194	7	η	η	PROPN
ma-246	194	8	-	-	PROPN
ma-246	194	9	o(x	o(x	ADJ
ma-246	194	10	)	)	PUNCT
ma-246	194	11	such	such	ADJ
ma-246	194	12	that	that	SCONJ
ma-246	194	13	a	a	DET
ma-246	194	14	∩	∩	ADJ
ma-246	194	15	u	u	NOUN
ma-246	194	16	∈	∈	PROPN
ma-246	194	17	i	i	PRON
ma-246	194	18	which	which	PRON
ma-246	194	19	is	be	AUX
ma-246	194	20	acontradiction	acontradiction	NOUN
ma-246	194	21	.	.	PUNCT
ma-246	195	1	(	(	PUNCT
ma-246	195	2	iii	iii	X
ma-246	195	3	)	)	PUNCT
ma-246	195	4	let	let	VERB
ma-246	195	5	a	a	DET
ma-246	195	6	,	,	PUNCT
ma-246	195	7	b	b	NOUN
ma-246	195	8	⊆	⊆	NUM
ma-246	195	9	x	x	SYM
ma-246	195	10	.	.	PUNCT
ma-246	196	1	note	note	VERB
ma-246	196	2	that	that	SCONJ
ma-246	196	3	a	a	DET
ma-246	196	4	\b	\b	NOUN
ma-246	196	5	⊆	⊆	NUM
ma-246	196	6	a	a	PRON
ma-246	196	7	and	and	CCONJ
ma-246	196	8	b	b	NOUN
ma-246	196	9	\a	\a	PRON
ma-246	196	10	⊆	⊆	NUM
ma-246	196	11	b.	b.	NOUN
ma-246	196	12	then	then	ADV
ma-246	196	13	by	by	ADP
ma-246	196	14	theorem	theorem	NOUN
ma-246	196	15	1	1	NUM
ma-246	196	16	(	(	PUNCT
ma-246	196	17	i	i	NOUN
ma-246	196	18	)	)	PUNCT
ma-246	196	19	,	,	PUNCT
ma-246	196	20	(	(	PUNCT
ma-246	196	21	a	a	DET
ma-246	196	22	\b)∗η	\b)∗η	ADP
ma-246	196	23	⊆	⊆	NUM
ma-246	196	24	a∗ηand	a∗ηand	SYM
ma-246	196	25	(	(	PUNCT
ma-246	196	26	b	b	NOUN
ma-246	196	27	\	\	PROPN
ma-246	196	28	a)∗η	a)∗η	NOUN
ma-246	196	29	⊆	⊆	NUM
ma-246	196	30	b∗η	b∗η	X
ma-246	196	31	,	,	PUNCT
ma-246	196	32	and	and	CCONJ
ma-246	196	33	so	so	ADV
ma-246	196	34	,	,	PUNCT
ma-246	196	35	(	(	PUNCT
ma-246	196	36	a	a	DET
ma-246	196	37	\	\	PROPN
ma-246	196	38	b)∗η	b)∗η	X
ma-246	196	39	∪	∪	ADJ
ma-246	196	40	(	(	PUNCT
ma-246	196	41	b	b	NOUN
ma-246	196	42	\	\	NOUN
ma-246	196	43	a)∗η	a)∗η	NOUN
ma-246	196	44	⊆	⊆	NUM
ma-246	196	45	a∗η	a∗η	PUNCT
ma-246	196	46	∪	∪	ADP
ma-246	196	47	b∗η	b∗η	PROPN
ma-246	196	48	.	.	PUNCT
ma-246	197	1	now	now	ADV
ma-246	197	2	,	,	PUNCT
ma-246	197	3	by	by	ADP
ma-246	197	4	theorem	theorem	NOUN
ma-246	197	5	1	1	NUM
ma-246	197	6	(	(	PUNCT
ma-246	197	7	iii	iii	NOUN
ma-246	197	8	)	)	PUNCT
ma-246	197	9	,	,	PUNCT
ma-246	197	10	a∗η	a∗η	PROPN
ma-246	197	11	∪	∪	VERB
ma-246	197	12	b∗η	b∗η	PROPN
ma-246	197	13	⊆	⊆	NUM
ma-246	197	14	(	(	PUNCT
ma-246	197	15	a	a	DET
ma-246	197	16	∪	∪	ADJ
ma-246	197	17	b)∗η	b)∗η	NOUN
ma-246	197	18	.	.	PUNCT
ma-246	198	1	hence	hence	ADV
ma-246	198	2	,	,	PUNCT
ma-246	198	3	(	(	PUNCT
ma-246	198	4	a	a	DET
ma-246	198	5	\	\	PROPN
ma-246	198	6	b)∗η	b)∗η	X
ma-246	198	7	∪	∪	ADJ
ma-246	198	8	(	(	PUNCT
ma-246	198	9	b	b	NOUN
ma-246	198	10	\	\	NOUN
ma-246	198	11	a)∗η	a)∗η	NOUN
ma-246	198	12	⊆	⊆	NUM
ma-246	198	13	(	(	PUNCT
ma-246	198	14	a	a	DET
ma-246	198	15	∪	∪	ADJ
ma-246	198	16	b)∗η	b)∗η	NOUN
ma-246	198	17	.	.	PUNCT
ma-246	199	1	(	(	PUNCT
ma-246	199	2	iv	iv	X
ma-246	199	3	)	)	PUNCT
ma-246	199	4	let	let	VERB
ma-246	199	5	u	u	PRON
ma-246	199	6	⊆	⊆	NUM
ma-246	199	7	x	x	SYM
ma-246	199	8	.	.	PUNCT
ma-246	200	1	since	since	SCONJ
ma-246	200	2	u	u	NOUN
ma-246	200	3	∩a	∩a	PROPN
ma-246	200	4	⊆	⊆	NUM
ma-246	200	5	a	a	PRON
ma-246	200	6	,	,	PUNCT
ma-246	200	7	by	by	ADP
ma-246	200	8	theorem	theorem	NOUN
ma-246	200	9	1	1	NUM
ma-246	200	10	(	(	PUNCT
ma-246	200	11	i	i	NOUN
ma-246	200	12	)	)	PUNCT
ma-246	200	13	,	,	PUNCT
ma-246	200	14	(	(	PUNCT
ma-246	200	15	u	u	NOUN
ma-246	200	16	∩a)∗η	∩a)∗η	ADP
ma-246	200	17	⊆	⊆	NUM
ma-246	200	18	a∗η	a∗η	PROPN
ma-246	200	19	,	,	PUNCT
ma-246	200	20	and	and	CCONJ
ma-246	200	21	hence	hence	ADV
ma-246	200	22	,	,	PUNCT
ma-246	200	23	u	u	PROPN
ma-246	200	24	∩	∩	NOUN
ma-246	200	25	(	(	PUNCT
ma-246	200	26	u	u	NOUN
ma-246	200	27	∩a)∗η	∩a)∗η	ADP
ma-246	200	28	⊆	⊆	NUM
ma-246	200	29	u	u	NOUN
ma-246	200	30	∩	∩	NOUN
ma-246	200	31	a∗η	a∗η	PROPN
ma-246	200	32	.	.	PUNCT
ma-246	201	1	(	(	PUNCT
ma-246	201	2	v	v	NOUN
ma-246	201	3	)	)	PUNCT
ma-246	201	4	let	let	VERB
ma-246	201	5	u	u	PRON
ma-246	201	6	∈	∈	PROPN
ma-246	202	1	i	i	PRON
ma-246	202	2	.	.	PUNCT
ma-246	203	1	since	since	SCONJ
ma-246	203	2	a	a	DET
ma-246	203	3	∩	∩	ADJ
ma-246	203	4	u	u	NOUN
ma-246	203	5	⊆	⊆	NUM
ma-246	203	6	u	u	NOUN
ma-246	203	7	∈	∈	NOUN
ma-246	204	1	i	i	PRON
ma-246	205	1	and	and	CCONJ
ma-246	205	2	i	i	PRON
ma-246	205	3	is	be	AUX
ma-246	205	4	an	an	DET
ma-246	205	5	ideal	ideal	NOUN
ma-246	205	6	,	,	PUNCT
ma-246	205	7	a	a	DET
ma-246	205	8	∩	∩	ADJ
ma-246	205	9	u	u	NOUN
ma-246	205	10	∈	∈	NOUN
ma-246	206	1	i	i	PRON
ma-246	206	2	.	.	PUNCT
ma-246	207	1	hence	hence	ADV
ma-246	207	2	,	,	PUNCT
ma-246	207	3	by	by	ADP
ma-246	207	4	theorem	theorem	NOUN
ma-246	207	5	1	1	NUM
ma-246	207	6	(	(	PUNCT
ma-246	207	7	vii	vii	PROPN
ma-246	207	8	)	)	PUNCT
ma-246	207	9	,	,	PUNCT
ma-246	207	10	(	(	PUNCT
ma-246	207	11	a	a	DET
ma-246	207	12	∩	∩	NOUN
ma-246	207	13	u)∗η	u)∗η	SCONJ
ma-246	207	14	=	=	SYM
ma-246	207	15	∅.	∅.	X
ma-246	207	16	(	(	PUNCT
ma-246	207	17	vi	vi	NOUN
ma-246	207	18	)	)	PUNCT
ma-246	207	19	let	let	VERB
ma-246	207	20	a	a	PRON
ma-246	207	21	be	be	AUX
ma-246	207	22	an	an	DET
ma-246	207	23	η	η	ADJ
ma-246	207	24	-	-	ADJ
ma-246	207	25	closed	closed	ADJ
ma-246	207	26	set	set	NOUN
ma-246	207	27	.	.	PUNCT
ma-246	208	1	then	then	ADV
ma-246	208	2	a	a	DET
ma-246	208	3	=	=	X
ma-246	208	4	η	η	NOUN
ma-246	208	5	-	-	NOUN
ma-246	208	6	cl(a	cl(a	NUM
ma-246	208	7	)	)	PUNCT
ma-246	208	8	.	.	PUNCT
ma-246	209	1	now	now	ADV
ma-246	209	2	,	,	PUNCT
ma-246	209	3	note	note	VERB
ma-246	209	4	that	that	SCONJ
ma-246	209	5	by	by	ADP
ma-246	209	6	theorem	theorem	NOUN
ma-246	209	7	1	1	NUM
ma-246	209	8	(	(	PUNCT
ma-246	209	9	x	x	NOUN
ma-246	209	10	)	)	PUNCT
ma-246	209	11	,	,	PUNCT
ma-246	209	12	a∗η	a∗η	PROPN
ma-246	209	13	⊆	⊆	NUM
ma-246	209	14	η	η	PROPN
ma-246	209	15	-	-	NOUN
ma-246	209	16	cl(a	cl(a	NUM
ma-246	209	17	)	)	PUNCT
ma-246	209	18	.	.	PUNCT
ma-246	210	1	hence	hence	ADV
ma-246	210	2	,	,	PUNCT
ma-246	210	3	a∗η	a∗η	PROPN
ma-246	210	4	⊆	⊆	NUM
ma-246	210	5	a.	a.	NOUN
ma-246	210	6	(	(	PUNCT
ma-246	210	7	vii	vii	PROPN
ma-246	210	8	)	)	PUNCT
ma-246	210	9	let	let	VERB
ma-246	210	10	a	a	DET
ma-246	210	11	⊆	⊆	NUM
ma-246	210	12	x	x	SYM
ma-246	210	13	.	.	PUNCT
ma-246	211	1	since	since	SCONJ
ma-246	211	2	a	a	DET
ma-246	211	3	∩	∩	NOUN
ma-246	211	4	a∗η	a∗η	PROPN
ma-246	211	5	⊆	⊆	NUM
ma-246	211	6	a∗η	a∗η	PROPN
ma-246	211	7	,	,	PUNCT
ma-246	211	8	by	by	ADP
ma-246	211	9	theorem	theorem	NOUN
ma-246	211	10	1	1	NUM
ma-246	211	11	(	(	PUNCT
ma-246	211	12	i	i	NOUN
ma-246	211	13	)	)	PUNCT
ma-246	211	14	,	,	PUNCT
ma-246	211	15	(	(	PUNCT
ma-246	211	16	a	a	DET
ma-246	211	17	∩	∩	NOUN
ma-246	211	18	a∗η)∗η	a∗η)∗η	CCONJ
ma-246	211	19	⊆	⊆	NUM
ma-246	211	20	(	(	PUNCT
ma-246	211	21	a∗η)∗η	a∗η)∗η	X
ma-246	211	22	.	.	PUNCT
ma-246	212	1	note	note	VERB
ma-246	212	2	that	that	PRON
ma-246	212	3	bytheorem	bytheorem	VERB
ma-246	212	4	1	1	NUM
ma-246	212	5	(	(	PUNCT
ma-246	212	6	vi	vi	NOUN
ma-246	212	7	)	)	PUNCT
ma-246	212	8	,	,	PUNCT
ma-246	212	9	(	(	PUNCT
ma-246	212	10	a∗η)∗η	a∗η)∗η	NOUN
ma-246	212	11	⊆	⊆	NUM
ma-246	212	12	a∗η	a∗η	PROPN
ma-246	212	13	.	.	PUNCT
ma-246	213	1	therefore	therefore	ADV
ma-246	213	2	,	,	PUNCT
ma-246	213	3	(	(	PUNCT
ma-246	213	4	a	a	DET
ma-246	213	5	∩	∩	NOUN
ma-246	213	6	a∗η)∗η	a∗η)∗η	VERB
ma-246	213	7	⊆	⊆	X
ma-246	213	8	a∗η	a∗η	PROPN
ma-246	213	9	.	.	PUNCT
ma-246	214	1	(	(	PUNCT
ma-246	214	2	iix	iix	PROPN
ma-246	214	3	)	)	PUNCT
ma-246	214	4	let	let	VERB
ma-246	214	5	a∪b	a∪b	NOUN
ma-246	214	6	∈	∈	PROPN
ma-246	215	1	i	i	PRON
ma-246	215	2	.	.	PUNCT
ma-246	216	1	since	since	SCONJ
ma-246	216	2	a∪b	a∪b	ADJ
ma-246	216	3	∈	∈	PROPN
ma-246	216	4	i	i	PRON
ma-246	216	5	and	and	CCONJ
ma-246	216	6	i	i	PRON
ma-246	216	7	is	be	AUX
ma-246	216	8	an	an	DET
ma-246	216	9	ideal	ideal	NOUN
ma-246	216	10	,	,	PUNCT
ma-246	217	1	a	a	DET
ma-246	217	2	∈	∈	NOUN
ma-246	217	3	i	i	PRON
ma-246	217	4	and	and	CCONJ
ma-246	217	5	b	b	X
ma-246	217	6	∈	∈	PROPN
ma-246	217	7	i	i	PRON
ma-246	217	8	.	.	PUNCT
ma-246	218	1	these	these	PRON
ma-246	218	2	imply	imply	VERB
ma-246	218	3	by	by	ADP
ma-246	218	4	theorem1	theorem1	PROPN
ma-246	218	5	(	(	PUNCT
ma-246	218	6	vii	vii	PROPN
ma-246	218	7	)	)	PUNCT
ma-246	218	8	,	,	PUNCT
ma-246	218	9	(	(	PUNCT
ma-246	218	10	a	a	DET
ma-246	218	11	∪	∪	ADJ
ma-246	218	12	b)∗η	b)∗η	ADJ
ma-246	218	13	=	=	SYM
ma-246	218	14	∅	∅	NOUN
ma-246	218	15	,	,	PUNCT
ma-246	218	16	a∗η	a∗η	PROPN
ma-246	218	17	=	=	SYM
ma-246	218	18	∅	∅	NOUN
ma-246	218	19	,	,	PUNCT
ma-246	218	20	and	and	CCONJ
ma-246	218	21	b∗η	b∗η	PROPN
ma-246	218	22	=	=	PUNCT
ma-246	218	23	∅.	∅.	VERB
ma-246	218	24	therefore	therefore	ADV
ma-246	218	25	,	,	PUNCT
ma-246	218	26	(	(	PUNCT
ma-246	218	27	a	a	DET
ma-246	218	28	∪	∪	ADJ
ma-246	218	29	b)∗η	b)∗η	X
ma-246	218	30	=	=	SYM
ma-246	218	31	a∗η	a∗η	PUNCT
ma-246	218	32	∪	∪	X
ma-246	218	33	b∗η	b∗η	PROPN
ma-246	218	34	=	=	SYM
ma-246	218	35	∅.	∅.	X
ma-246	218	36	(	(	PUNCT
ma-246	218	37	ix	ix	ADV
ma-246	218	38	)	)	PUNCT
ma-246	218	39	suppose	suppose	VERB
ma-246	218	40	that	that	SCONJ
ma-246	218	41	a∗η	a∗η	PROPN
ma-246	218	42	6=	6=	ADP
ma-246	218	43	η	η	PROPN
ma-246	218	44	-	-	PROPN
ma-246	218	45	cl(a∗η	cl(a∗η	NOUN
ma-246	218	46	)	)	PUNCT
ma-246	218	47	.	.	PUNCT
ma-246	219	1	let	let	VERB
ma-246	219	2	η	η	PROPN
ma-246	219	3	-	-	ADJ
ma-246	219	4	cl(a∗η	cl(a∗η	NOUN
ma-246	219	5	)	)	PUNCT
ma-246	219	6	⊂	⊂	PROPN
ma-246	219	7	a∗η	a∗η	PROPN
ma-246	219	8	.	.	PUNCT
ma-246	220	1	then	then	ADV
ma-246	220	2	there	there	PRON
ma-246	220	3	exists	exist	VERB
ma-246	220	4	an	an	DET
ma-246	220	5	element	element	NOUN
ma-246	220	6	x	x	SYM
ma-246	220	7	∈	∈	PROPN
ma-246	220	8	a∗ηsuch	a∗ηsuch	NOUN
ma-246	220	9	that	that	SCONJ
ma-246	220	10	x	x	PROPN
ma-246	220	11	/∈	/∈	PUNCT
ma-246	220	12	η	η	PROPN
ma-246	220	13	-	-	PROPN
ma-246	220	14	cl(a∗η	cl(a∗η	NOUN
ma-246	220	15	)	)	PUNCT
ma-246	220	16	.	.	PUNCT
ma-246	221	1	note	note	VERB
ma-246	221	2	that	that	SCONJ
ma-246	222	1	since	since	SCONJ
ma-246	222	2	x	x	PROPN
ma-246	222	3	∈	∈	PROPN
ma-246	222	4	a∗η	a∗η	PROPN
ma-246	222	5	,	,	PUNCT
ma-246	222	6	for	for	ADP
ma-246	222	7	every	every	DET
ma-246	222	8	u	u	PROPN
ma-246	222	9	∈	∈	PROPN
ma-246	222	10	η	η	PROPN
ma-246	222	11	-	-	PROPN
ma-246	222	12	o(x	o(x	PROPN
ma-246	222	13	)	)	PUNCT
ma-246	222	14	,	,	PUNCT
ma-246	222	15	a	a	DET
ma-246	222	16	∩	∩	ADJ
ma-246	222	17	u	u	NOUN
ma-246	222	18	/∈	/∈	PUNCT
ma-246	222	19	i	i	INTJ
ma-246	222	20	.	.	PUNCT
ma-246	223	1	now	now	ADV
ma-246	223	2	,	,	PUNCT
ma-246	223	3	note	note	VERB
ma-246	223	4	that	that	SCONJ
ma-246	223	5	x	x	X
ma-246	223	6	/∈	/∈	PUNCT
ma-246	223	7	η	η	PROPN
ma-246	223	8	-	-	PROPN
ma-246	223	9	cl(a∗η	cl(a∗η	NOUN
ma-246	223	10	)	)	PUNCT
ma-246	223	11	.	.	PUNCT
ma-246	224	1	then	then	ADV
ma-246	224	2	x	x	X
ma-246	224	3	/∈	/∈	PUNCT
ma-246	224	4	⋂{k	⋂{k	VERB
ma-246	224	5	:	:	PUNCT
ma-246	225	1	k	k	X
ma-246	225	2	is	be	AUX
ma-246	225	3	η	η	NOUN
ma-246	225	4	-	-	ADJ
ma-246	225	5	closed	closed	ADJ
ma-246	225	6	and	and	CCONJ
ma-246	225	7	a∗η	a∗η	PROPN
ma-246	225	8	⊆	⊆	NUM
ma-246	225	9	k	k	NOUN
ma-246	225	10	}	}	PUNCT
ma-246	225	11	.	.	PUNCT
ma-246	226	1	this	this	PRON
ma-246	226	2	shows	show	VERB
ma-246	226	3	that	that	SCONJ
ma-246	226	4	x	x	PROPN
ma-246	226	5	/∈	/∈	PUNCT
ma-246	226	6	k	k	PROPN
ma-246	226	7	for	for	ADP
ma-246	226	8	some	some	DET
ma-246	226	9	η	η	NOUN
ma-246	226	10	-	-	ADJ
ma-246	226	11	closed	closed	ADJ
ma-246	226	12	set	set	NOUN
ma-246	226	13	k	k	ADP
ma-246	226	14	such	such	ADJ
ma-246	226	15	that	that	SCONJ
ma-246	226	16	a∗η	a∗η	PROPN
ma-246	226	17	⊆	⊆	NUM
ma-246	226	18	k.	k.	NOUN
ma-246	226	19	this	this	PRON
ma-246	226	20	implies	imply	VERB
ma-246	226	21	that	that	SCONJ
ma-246	226	22	x	x	PUNCT
ma-246	226	23	∈	∈	PROPN
ma-246	226	24	kc	kc	PROPN
ma-246	226	25	for	for	ADP
ma-246	226	26	some	some	DET
ma-246	226	27	η	η	NOUN
ma-246	226	28	-	-	ADJ
ma-246	226	29	open	open	ADJ
ma-246	226	30	set	set	NOUN
ma-246	226	31	kc	kc	PROPN
ma-246	226	32	such	such	ADJ
ma-246	226	33	that	that	SCONJ
ma-246	226	34	kc	kc	PROPN
ma-246	226	35	∩	∩	PROPN
ma-246	226	36	a∗η	a∗η	PROPN
ma-246	226	37	=	=	SYM
ma-246	226	38	∅.	∅.	AUX
ma-246	226	39	note	note	VERB
ma-246	226	40	that	that	SCONJ
ma-246	226	41	x	x	X
ma-246	226	42	∈	∈	PROPN
ma-246	226	43	kc	kc	PROPN
ma-246	226	44	and	and	CCONJ
ma-246	226	45	kc	kc	PROPN
ma-246	226	46	∩	∩	PROPN
ma-246	226	47	a∗η	a∗η	PROPN
ma-246	226	48	=	=	SYM
ma-246	226	49	∅	∅	NOUN
ma-246	226	50	,	,	PUNCT
ma-246	226	51	then	then	ADV
ma-246	226	52	itfollows	itfollow	VERB
ma-246	226	53	that	that	SCONJ
ma-246	226	54	x	x	SYM
ma-246	226	55	/∈	/∈	PUNCT
ma-246	226	56	a∗η	a∗η	PROPN
ma-246	226	57	,	,	PUNCT
ma-246	226	58	and	and	CCONJ
ma-246	226	59	so	so	ADV
ma-246	226	60	,	,	PUNCT
ma-246	226	61	for	for	ADP
ma-246	226	62	some	some	DET
ma-246	226	63	kc	kc	PROPN
ma-246	226	64	∈	∈	PROPN
ma-246	226	65	η	η	PROPN
ma-246	226	66	-	-	PROPN
ma-246	226	67	o(x	o(x	PROPN
ma-246	226	68	)	)	PUNCT
ma-246	226	69	,	,	PUNCT
ma-246	226	70	a	a	DET
ma-246	226	71	∩	∩	ADJ
ma-246	226	72	kc	kc	PROPN
ma-246	226	73	∈	∈	PROPN
ma-246	226	74	i	i	PRON
ma-246	226	75	,	,	PUNCT
ma-246	226	76	which	which	PRON
ma-246	226	77	is	be	AUX
ma-246	226	78	a	a	PRON
ma-246	226	79	contradiction.consequently	contradiction.consequently	ADV
ma-246	226	80	,	,	PUNCT
ma-246	226	81	a∗η	a∗η	PROPN
ma-246	226	82	=	=	SYM
ma-246	226	83	η	η	PROPN
ma-246	226	84	-	-	PROPN
ma-246	226	85	cl(a∗η	cl(a∗η	NOUN
ma-246	226	86	)	)	PUNCT
ma-246	226	87	,	,	PUNCT
ma-246	226	88	then	then	ADV
ma-246	226	89	by	by	ADP
ma-246	226	90	remark	remark	NOUN
ma-246	226	91	1	1	NUM
ma-246	226	92	(	(	PUNCT
ma-246	226	93	iii	iii	NOUN
ma-246	226	94	)	)	PUNCT
ma-246	226	95	,	,	PUNCT
ma-246	226	96	a∗η	a∗η	PROPN
ma-246	226	97	is	be	AUX
ma-246	226	98	an	an	DET
ma-246	226	99	η	η	NOUN
ma-246	226	100	-	-	ADJ
ma-246	226	101	closed	closed	ADJ
ma-246	226	102	set	set	NOUN
ma-246	226	103	.	.	PUNCT
ma-246	227	1	now	now	ADV
ma-246	227	2	,	,	PUNCT
ma-246	227	3	notethat	notethat	PROPN
ma-246	227	4	a∗η	a∗η	PROPN
ma-246	227	5	=	=	SYM
ma-246	227	6	η	η	PROPN
ma-246	227	7	-	-	PROPN
ma-246	227	8	cl(a∗η	cl(a∗η	NOUN
ma-246	227	9	)	)	PUNCT
ma-246	227	10	and	and	CCONJ
ma-246	227	11	by	by	ADP
ma-246	227	12	theorem	theorem	NOUN
ma-246	227	13	1	1	NUM
ma-246	227	14	(	(	PUNCT
ma-246	227	15	x	x	NOUN
ma-246	227	16	)	)	PUNCT
ma-246	227	17	,	,	PUNCT
ma-246	227	18	hence	hence	ADV
ma-246	227	19	,	,	PUNCT
ma-246	227	20	a∗η	a∗η	PROPN
ma-246	227	21	=	=	SYM
ma-246	227	22	η	η	PROPN
ma-246	227	23	-	-	PROPN
ma-246	227	24	cl(a∗η	cl(a∗η	NOUN
ma-246	227	25	)	)	PUNCT
ma-246	227	26	⊆	⊆	NUM
ma-246	227	27	η	η	PROPN
ma-246	227	28	-	-	NOUN
ma-246	227	29	cl(a	cl(a	NUM
ma-246	227	30	)	)	PUNCT
ma-246	227	31	.	.	PUNCT
ma-246	228	1	(	(	PUNCT
ma-246	228	2	x	x	X
ma-246	228	3	)	)	PUNCT
ma-246	228	4	let	let	VERB
ma-246	228	5	a	a	DET
ma-246	228	6	⊆	⊆	NUM
ma-246	228	7	a∗η	a∗η	PROPN
ma-246	228	8	.	.	PUNCT
ma-246	229	1	suppose	suppose	VERB
ma-246	229	2	that	that	SCONJ
ma-246	229	3	x	x	PROPN
ma-246	229	4	∈	∈	PROPN
ma-246	229	5	η	η	PROPN
ma-246	229	6	-	-	NOUN
ma-246	229	7	cl(a	cl(a	NUM
ma-246	229	8	)	)	PUNCT
ma-246	229	9	.	.	PUNCT
ma-246	230	1	then	then	ADV
ma-246	230	2	x	x	PUNCT
ma-246	230	3	∈	∈	NOUN
ma-246	230	4	⋂{k	⋂{k	VERB
ma-246	230	5	:	:	PUNCT
ma-246	230	6	k	k	PROPN
ma-246	230	7	is	be	AUX
ma-246	230	8	η	η	NOUN
ma-246	230	9	-	-	ADJ
ma-246	230	10	closed	closed	ADJ
ma-246	230	11	and	and	CCONJ
ma-246	230	12	a	a	DET
ma-246	230	13	⊆	⊆	NUM
ma-246	230	14	k}.this	k}.this	PROPN
ma-246	230	15	shows	show	VERB
ma-246	230	16	that	that	SCONJ
ma-246	230	17	x	x	SYM
ma-246	230	18	∈	∈	PROPN
ma-246	230	19	k	k	PROPN
ma-246	230	20	for	for	ADP
ma-246	230	21	every	every	DET
ma-246	230	22	η	η	PROPN
ma-246	230	23	-	-	ADJ
ma-246	230	24	closed	closed	ADJ
ma-246	230	25	set	set	NOUN
ma-246	230	26	k	k	PROPN
ma-246	230	27	such	such	ADJ
ma-246	230	28	that	that	SCONJ
ma-246	230	29	a	a	DET
ma-246	230	30	⊆	⊆	NUM
ma-246	230	31	k.	k.	NOUN
ma-246	230	32	note	note	NOUN
ma-246	230	33	that	that	SCONJ
ma-246	230	34	by	by	ADP
ma-246	230	35	theorem2	theorem2	PROPN
ma-246	230	36	(	(	PUNCT
ma-246	230	37	ix	ix	PROPN
ma-246	230	38	)	)	PUNCT
ma-246	230	39	,	,	PUNCT
ma-246	230	40	a∗η	a∗η	PROPN
ma-246	230	41	is	be	AUX
ma-246	230	42	an	an	DET
ma-246	230	43	η	η	NOUN
ma-246	230	44	-	-	ADJ
ma-246	230	45	closed	closed	ADJ
ma-246	230	46	set	set	NOUN
ma-246	230	47	.	.	PUNCT
ma-246	231	1	now	now	ADV
ma-246	231	2	,	,	PUNCT
ma-246	231	3	note	note	VERB
ma-246	231	4	that	that	SCONJ
ma-246	231	5	since	since	SCONJ
ma-246	231	6	a	a	DET
ma-246	231	7	⊆	⊆	NUM
ma-246	231	8	a∗η	a∗η	PROPN
ma-246	231	9	and	and	CCONJ
ma-246	231	10	a∗η	a∗η	PROPN
ma-246	231	11	is	be	AUX
ma-246	231	12	an	an	DET
ma-246	231	13	η	η	ADJ
ma-246	231	14	-	-	ADJ
ma-246	231	15	closed	closed	ADJ
ma-246	231	16	set	set	NOUN
ma-246	231	17	,	,	PUNCT
ma-246	231	18	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	231	19	eur	eur	PROPN
ma-246	231	20	.	.	PUNCT
ma-246	232	1	j.	j.	PROPN
ma-246	232	2	math	math	PROPN
ma-246	232	3	.	.	PUNCT
ma-246	233	1	anal	anal	PROPN
ma-246	233	2	.	.	PUNCT
ma-246	234	1	10.28924	10.28924	NUM
ma-246	234	2	/	/	SYM
ma-246	234	3	ada	ada	PROPN
ma-246	234	4	/	/	SYM
ma-246	234	5	ma.5.2	ma.5.2	PROPN
ma-246	234	6	6	6	NUM
ma-246	234	7	a∗η	a∗η	PROPN
ma-246	234	8	∈	∈	PROPN
ma-246	234	9	{	{	PUNCT
ma-246	234	10	k	k	NOUN
ma-246	234	11	:	:	PUNCT
ma-246	234	12	k	k	PROPN
ma-246	234	13	is	be	AUX
ma-246	234	14	η	η	NOUN
ma-246	234	15	-	-	ADJ
ma-246	234	16	closed	closed	ADJ
ma-246	234	17	and	and	CCONJ
ma-246	234	18	a	a	DET
ma-246	234	19	⊆	⊆	NUM
ma-246	234	20	k	k	NOUN
ma-246	234	21	}	}	PUNCT
ma-246	234	22	.	.	PUNCT
ma-246	235	1	this	this	PRON
ma-246	235	2	implies	imply	VERB
ma-246	235	3	that	that	SCONJ
ma-246	235	4	x	x	SYM
ma-246	235	5	∈	∈	PROPN
ma-246	235	6	a∗η	a∗η	PROPN
ma-246	235	7	.	.	PUNCT
ma-246	236	1	hence	hence	ADV
ma-246	236	2	,	,	PUNCT
ma-246	236	3	η	η	PROPN
ma-246	236	4	-	-	NOUN
ma-246	236	5	cl(a	cl(a	NUM
ma-246	236	6	)	)	PUNCT
ma-246	236	7	⊆	⊆	NUM
ma-246	236	8	a∗η	a∗η	PROPN
ma-246	236	9	.	.	PUNCT
ma-246	237	1	asa	asa	PROPN
ma-246	237	2	result	result	NOUN
ma-246	237	3	,	,	PUNCT
ma-246	237	4	by	by	ADP
ma-246	237	5	theorem	theorem	NOUN
ma-246	237	6	1	1	NUM
ma-246	237	7	(	(	PUNCT
ma-246	237	8	x	x	NOUN
ma-246	237	9	)	)	PUNCT
ma-246	237	10	and	and	CCONJ
ma-246	237	11	2	2	NUM
ma-246	237	12	(	(	PUNCT
ma-246	237	13	ix	ix	PROPN
ma-246	237	14	)	)	PUNCT
ma-246	237	15	,	,	PUNCT
ma-246	237	16	a∗η	a∗η	PROPN
ma-246	237	17	=	=	SYM
ma-246	237	18	η	η	PROPN
ma-246	237	19	-	-	PROPN
ma-246	237	20	cl(a∗η	cl(a∗η	NOUN
ma-246	237	21	)	)	PUNCT
ma-246	237	22	=	=	PUNCT
ma-246	237	23	η	η	PROPN
ma-246	237	24	-	-	NOUN
ma-246	237	25	cl(a	cl(a	NUM
ma-246	237	26	)	)	PUNCT
ma-246	237	27	.	.	PUNCT
ma-246	238	1	�	�	PROPN
ma-246	238	2	theorem	theorem	VERB
ma-246	238	3	3	3	X
ma-246	238	4	.	.	PUNCT
ma-246	239	1	let	let	VERB
ma-246	239	2	(	(	PUNCT
ma-246	239	3	x	x	X
ma-246	239	4	,	,	PUNCT
ma-246	239	5	τ	τ	PROPN
ma-246	239	6	,	,	PUNCT
ma-246	239	7	i	i	PRON
ma-246	239	8	)	)	PUNCT
ma-246	239	9	be	be	VERB
ma-246	239	10	an	an	DET
ma-246	239	11	ideal	ideal	ADJ
ma-246	239	12	topological	topological	ADJ
ma-246	239	13	space	space	NOUN
ma-246	239	14	where	where	SCONJ
ma-246	239	15	η	η	PROPN
ma-246	239	16	-	-	PROPN
ma-246	239	17	o(x	o(x	PROPN
ma-246	239	18	)	)	PUNCT
ma-246	239	19	is	be	AUX
ma-246	239	20	closed	close	VERB
ma-246	239	21	under	under	ADP
ma-246	239	22	any	any	DET
ma-246	239	23	two	two	NUM
ma-246	239	24	intersections	intersection	NOUN
ma-246	239	25	.	.	PUNCT
ma-246	240	1	then	then	ADV
ma-246	240	2	for	for	ADP
ma-246	240	3	any	any	DET
ma-246	240	4	a	a	DET
ma-246	240	5	,	,	PUNCT
ma-246	240	6	b	b	NOUN
ma-246	240	7	subsets	subset	NOUN
ma-246	240	8	of	of	ADP
ma-246	240	9	x	x	PRON
ma-246	240	10	,	,	PUNCT
ma-246	240	11	the	the	DET
ma-246	240	12	following	follow	VERB
ma-246	240	13	properties	property	NOUN
ma-246	240	14	hold	hold	VERB
ma-246	240	15	:	:	PUNCT
ma-246	240	16	(	(	PUNCT
ma-246	240	17	i	i	NOUN
ma-246	240	18	)	)	PUNCT
ma-246	240	19	(	(	PUNCT
ma-246	240	20	a	a	DET
ma-246	240	21	∪	∪	ADJ
ma-246	240	22	b)∗η	b)∗η	X
ma-246	240	23	=	=	SYM
ma-246	240	24	a∗η	a∗η	PUNCT
ma-246	240	25	∪	∪	X
ma-246	240	26	b∗η	b∗η	PROPN
ma-246	240	27	;	;	PUNCT
ma-246	240	28	(	(	PUNCT
ma-246	240	29	ii	ii	NOUN
ma-246	240	30	)	)	PUNCT
ma-246	240	31	for	for	ADP
ma-246	240	32	u	u	PROPN
ma-246	240	33	∈	∈	PROPN
ma-246	240	34	η	η	PROPN
ma-246	240	35	-	-	PROPN
ma-246	240	36	o(x	o(x	PROPN
ma-246	240	37	)	)	PUNCT
ma-246	240	38	,	,	PUNCT
ma-246	240	39	u	u	NOUN
ma-246	240	40	∩	∩	NOUN
ma-246	240	41	a∗η	a∗η	PROPN
ma-246	240	42	=	=	SYM
ma-246	240	43	u	u	PROPN
ma-246	240	44	∩	∩	NOUN
ma-246	240	45	(	(	PUNCT
ma-246	240	46	u	u	NOUN
ma-246	240	47	∩	∩	NOUN
ma-246	240	48	a)∗η	a)∗η	NOUN
ma-246	240	49	⊆	⊆	NUM
ma-246	240	50	(	(	PUNCT
ma-246	240	51	u	u	NOUN
ma-246	240	52	∩	∩	NOUN
ma-246	240	53	a)∗η	a)∗η	NOUN
ma-246	240	54	;	;	PUNCT
ma-246	240	55	and	and	CCONJ
ma-246	240	56	(	(	PUNCT
ma-246	240	57	iii	iii	NOUN
ma-246	240	58	)	)	PUNCT
ma-246	240	59	a∗η	a∗η	ADP
ma-246	240	60	\	\	NOUN
ma-246	240	61	b∗η	b∗η	PROPN
ma-246	241	1	=	=	SYM
ma-246	241	2	(	(	PUNCT
ma-246	241	3	a	a	DET
ma-246	241	4	\	\	PROPN
ma-246	241	5	b)∗η	b)∗η	X
ma-246	241	6	\	\	NOUN
ma-246	241	7	b∗η	b∗η	PROPN
ma-246	241	8	⊆	⊆	NUM
ma-246	241	9	(	(	PUNCT
ma-246	241	10	a	a	DET
ma-246	241	11	\	\	PROPN
ma-246	241	12	b)∗η	b)∗η	NOUN
ma-246	241	13	.	.	PUNCT
ma-246	242	1	proof	proof	NOUN
ma-246	242	2	.	.	PUNCT
ma-246	243	1	(	(	PUNCT
ma-246	243	2	i	i	NOUN
ma-246	243	3	)	)	PUNCT
ma-246	243	4	let	let	VERB
ma-246	243	5	η	η	PROPN
ma-246	243	6	-	-	ADJ
ma-246	243	7	o(x	o(x	VERB
ma-246	243	8	)	)	PUNCT
ma-246	243	9	be	be	AUX
ma-246	243	10	closed	close	VERB
ma-246	243	11	under	under	ADP
ma-246	243	12	any	any	DET
ma-246	243	13	two	two	NUM
ma-246	243	14	intersections	intersection	NOUN
ma-246	243	15	.	.	PUNCT
ma-246	244	1	suppose	suppose	VERB
ma-246	244	2	that	that	SCONJ
ma-246	244	3	x	x	PROPN
ma-246	244	4	/∈	/∈	PROPN
ma-246	244	5	a∗η∪b∗η	a∗η∪b∗η	NOUN
ma-246	244	6	,	,	PUNCT
ma-246	244	7	then	then	ADV
ma-246	244	8	x	x	X
ma-246	244	9	/∈	/∈	PUNCT
ma-246	245	1	a∗ηand	a∗ηand	NOUN
ma-246	245	2	x	x	SYM
ma-246	245	3	/∈	/∈	PUNCT
ma-246	246	1	b∗η	b∗η	PUNCT
ma-246	246	2	implying	imply	VERB
ma-246	246	3	that	that	SCONJ
ma-246	246	4	there	there	PRON
ma-246	246	5	exist	exist	VERB
ma-246	246	6	u	u	NOUN
ma-246	246	7	,	,	PUNCT
ma-246	246	8	v	v	PROPN
ma-246	246	9	∈	∈	PROPN
ma-246	246	10	η	η	PROPN
ma-246	246	11	-	-	PROPN
ma-246	246	12	o(x	o(x	ADJ
ma-246	246	13	)	)	PUNCT
ma-246	246	14	such	such	ADJ
ma-246	246	15	that	that	SCONJ
ma-246	246	16	a	a	DET
ma-246	246	17	∩	∩	ADJ
ma-246	246	18	u	u	X
ma-246	246	19	∈	∈	PROPN
ma-246	246	20	i	i	PROPN
ma-246	246	21	and	and	CCONJ
ma-246	246	22	b	b	PROPN
ma-246	246	23	∩	∩	NOUN
ma-246	246	24	v	v	X
ma-246	246	25	∈	∈	X
ma-246	246	26	i	i	PRON
ma-246	246	27	.note	.note	VERB
ma-246	246	28	that	that	SCONJ
ma-246	246	29	a	a	DET
ma-246	246	30	∩	∩	ADJ
ma-246	246	31	u	u	X
ma-246	246	32	∈	∈	PROPN
ma-246	246	33	i	i	PRON
ma-246	246	34	,	,	PUNCT
ma-246	246	35	b	b	PROPN
ma-246	246	36	∩	∩	X
ma-246	246	37	v	v	X
ma-246	246	38	∈	∈	NOUN
ma-246	247	1	i	i	PRON
ma-246	247	2	,	,	PUNCT
ma-246	247	3	and	and	CCONJ
ma-246	247	4	i	i	PRON
ma-246	247	5	is	be	AUX
ma-246	247	6	an	an	DET
ma-246	247	7	ideal	ideal	NOUN
ma-246	247	8	.	.	PUNCT
ma-246	248	1	then	then	ADV
ma-246	248	2	(	(	PUNCT
ma-246	248	3	a	a	DET
ma-246	248	4	∩	∩	ADJ
ma-246	248	5	u	u	NOUN
ma-246	248	6	)	)	PUNCT
ma-246	248	7	∪	∪	ADV
ma-246	248	8	(	(	PUNCT
ma-246	248	9	b	b	PROPN
ma-246	248	10	∩	∩	ADJ
ma-246	248	11	v	v	X
ma-246	248	12	)	)	PUNCT
ma-246	248	13	∈	∈	PROPN
ma-246	249	1	i	i	PRON
ma-246	249	2	.	.	PUNCT
ma-246	250	1	since	since	SCONJ
ma-246	250	2	u	u	NOUN
ma-246	250	3	∩	∩	NOUN
ma-246	250	4	v	v	ADP
ma-246	250	5	⊆	⊆	NUM
ma-246	250	6	u	u	NOUN
ma-246	250	7	and	and	CCONJ
ma-246	250	8	u	u	NOUN
ma-246	250	9	∩	∩	NOUN
ma-246	250	10	v	v	ADP
ma-246	250	11	⊆	⊆	NUM
ma-246	250	12	v	v	NOUN
ma-246	250	13	,	,	PUNCT
ma-246	250	14	(	(	PUNCT
ma-246	250	15	a	a	DET
ma-246	250	16	∩	∩	ADJ
ma-246	250	17	u	u	NOUN
ma-246	250	18	)	)	PUNCT
ma-246	250	19	∪	∪	ADV
ma-246	250	20	(	(	PUNCT
ma-246	250	21	b	b	PROPN
ma-246	250	22	∩	∩	ADJ
ma-246	250	23	v	v	NOUN
ma-246	250	24	)	)	PUNCT
ma-246	250	25	⊇	⊇	NOUN
ma-246	250	26	[	[	PUNCT
ma-246	250	27	a	a	DET
ma-246	250	28	∩	∩	NOUN
ma-246	250	29	(	(	PUNCT
ma-246	250	30	u	u	PROPN
ma-246	250	31	∩	∩	NOUN
ma-246	250	32	v	v	NOUN
ma-246	250	33	)	)	PUNCT
ma-246	250	34	]	]	PUNCT
ma-246	250	35	∪	∪	ADP
ma-246	250	36	[	[	PUNCT
ma-246	250	37	b	b	NOUN
ma-246	250	38	∩	∩	NOUN
ma-246	250	39	(	(	PUNCT
ma-246	250	40	u	u	PROPN
ma-246	250	41	∩	∩	NOUN
ma-246	250	42	v	v	NOUN
ma-246	250	43	)	)	PUNCT
ma-246	250	44	]	]	PUNCT
ma-246	251	1	=	=	PUNCT
ma-246	251	2	(	(	PUNCT
ma-246	251	3	a	a	DET
ma-246	251	4	∪	∪	ADJ
ma-246	251	5	b	b	NOUN
ma-246	251	6	)	)	PUNCT
ma-246	251	7	∩	∩	NOUN
ma-246	251	8	(	(	PUNCT
ma-246	251	9	u	u	PROPN
ma-246	251	10	∩	∩	NOUN
ma-246	251	11	v	v	NOUN
ma-246	251	12	)	)	PUNCT
ma-246	251	13	.	.	PUNCT
ma-246	252	1	it	it	PRON
ma-246	252	2	implies	imply	VERB
ma-246	252	3	that	that	SCONJ
ma-246	252	4	(	(	PUNCT
ma-246	252	5	a	a	DET
ma-246	252	6	∪	∪	ADJ
ma-246	252	7	b	b	NOUN
ma-246	252	8	)	)	PUNCT
ma-246	252	9	∩	∩	NOUN
ma-246	252	10	(	(	PUNCT
ma-246	252	11	u	u	PROPN
ma-246	252	12	∩	∩	NOUN
ma-246	252	13	v	v	NOUN
ma-246	252	14	)	)	PUNCT
ma-246	252	15	⊆	⊆	NUM
ma-246	252	16	(	(	PUNCT
ma-246	252	17	a	a	DET
ma-246	252	18	∩	∩	ADJ
ma-246	252	19	u	u	NOUN
ma-246	252	20	)	)	PUNCT
ma-246	252	21	∪	∪	ADV
ma-246	252	22	(	(	PUNCT
ma-246	252	23	b	b	PROPN
ma-246	252	24	∩	∩	ADJ
ma-246	252	25	v	v	X
ma-246	252	26	)	)	PUNCT
ma-246	252	27	∈	∈	PROPN
ma-246	252	28	i	i	PRON
ma-246	252	29	.	.	PUNCT
ma-246	253	1	again	again	ADV
ma-246	253	2	,	,	PUNCT
ma-246	253	3	since	since	SCONJ
ma-246	253	4	i	i	PRON
ma-246	253	5	is	be	AUX
ma-246	253	6	an	an	DET
ma-246	253	7	ideal	ideal	NOUN
ma-246	253	8	,	,	PUNCT
ma-246	253	9	(	(	PUNCT
ma-246	253	10	a	a	DET
ma-246	253	11	∪	∪	ADJ
ma-246	253	12	b	b	NOUN
ma-246	253	13	)	)	PUNCT
ma-246	253	14	∩	∩	NOUN
ma-246	253	15	(	(	PUNCT
ma-246	253	16	u	u	PROPN
ma-246	253	17	∩	∩	NOUN
ma-246	253	18	v	v	X
ma-246	253	19	)	)	PUNCT
ma-246	253	20	∈	∈	PROPN
ma-246	254	1	i	i	PRON
ma-246	254	2	.	.	PUNCT
ma-246	255	1	now	now	ADV
ma-246	255	2	,	,	PUNCT
ma-246	255	3	note	note	VERB
ma-246	255	4	that	that	SCONJ
ma-246	255	5	by	by	ADP
ma-246	255	6	assumption	assumption	NOUN
ma-246	255	7	,	,	PUNCT
ma-246	255	8	η	η	PROPN
ma-246	255	9	-	-	PROPN
ma-246	255	10	o(x	o(x	ADJ
ma-246	255	11	)	)	PUNCT
ma-246	255	12	is	be	AUX
ma-246	255	13	closed	close	VERB
ma-246	255	14	under	under	ADP
ma-246	255	15	any	any	DET
ma-246	255	16	twointersections	twointersection	NOUN
ma-246	255	17	,	,	PUNCT
ma-246	255	18	and	and	CCONJ
ma-246	255	19	so	so	ADV
ma-246	255	20	,	,	PUNCT
ma-246	255	21	there	there	PRON
ma-246	255	22	exists	exist	VERB
ma-246	255	23	u	u	NOUN
ma-246	255	24	∩	∩	NOUN
ma-246	255	25	v	v	ADP
ma-246	255	26	∈	∈	PROPN
ma-246	255	27	η	η	PROPN
ma-246	255	28	-	-	PROPN
ma-246	255	29	o(x	o(x	ADJ
ma-246	255	30	)	)	PUNCT
ma-246	255	31	such	such	ADJ
ma-246	255	32	that	that	SCONJ
ma-246	255	33	(	(	PUNCT
ma-246	255	34	a	a	DET
ma-246	255	35	∪	∪	ADJ
ma-246	255	36	b	b	NOUN
ma-246	255	37	)	)	PUNCT
ma-246	255	38	∩	∩	NOUN
ma-246	255	39	(	(	PUNCT
ma-246	255	40	u	u	PROPN
ma-246	255	41	∩	∩	NOUN
ma-246	255	42	v	v	X
ma-246	255	43	)	)	PUNCT
ma-246	255	44	∈	∈	PROPN
ma-246	255	45	i	i	PRON
ma-246	255	46	.	.	PUNCT
ma-246	256	1	thisshows	thisshow	VERB
ma-246	256	2	that	that	PRON
ma-246	256	3	x	x	PUNCT
ma-246	256	4	/∈	/∈	INTJ
ma-246	257	1	(	(	PUNCT
ma-246	257	2	a	a	DET
ma-246	257	3	∪b)∗η	∪b)∗η	NOUN
ma-246	257	4	.	.	PUNCT
ma-246	258	1	hence	hence	ADV
ma-246	258	2	,	,	PUNCT
ma-246	258	3	(	(	PUNCT
ma-246	258	4	a	a	DET
ma-246	258	5	∪b)∗η	∪b)∗η	ADP
ma-246	258	6	⊆	⊆	NUM
ma-246	258	7	a∗η	a∗η	NOUN
ma-246	258	8	∪b∗η	∪b∗η	PROPN
ma-246	258	9	.	.	PUNCT
ma-246	259	1	now	now	ADV
ma-246	259	2	,	,	PUNCT
ma-246	259	3	by	by	ADP
ma-246	259	4	theorem	theorem	NOUN
ma-246	259	5	1	1	NUM
ma-246	259	6	(	(	PUNCT
ma-246	259	7	iii	iii	NOUN
ma-246	259	8	)	)	PUNCT
ma-246	259	9	,	,	PUNCT
ma-246	259	10	therefore	therefore	ADV
ma-246	259	11	,	,	PUNCT
ma-246	259	12	(	(	PUNCT
ma-246	259	13	a	a	DET
ma-246	259	14	∪	∪	ADJ
ma-246	259	15	b)∗η	b)∗η	X
ma-246	259	16	=	=	SYM
ma-246	259	17	a∗η	a∗η	PUNCT
ma-246	259	18	∪	∪	ADP
ma-246	259	19	b∗η	b∗η	PROPN
ma-246	259	20	.	.	PUNCT
ma-246	260	1	(	(	PUNCT
ma-246	260	2	ii	ii	NOUN
ma-246	260	3	)	)	PUNCT
ma-246	260	4	let	let	VERB
ma-246	260	5	η	η	PROPN
ma-246	260	6	-	-	ADJ
ma-246	260	7	o(x	o(x	VERB
ma-246	260	8	)	)	PUNCT
ma-246	260	9	be	be	AUX
ma-246	260	10	closed	close	VERB
ma-246	260	11	under	under	ADP
ma-246	260	12	any	any	DET
ma-246	260	13	two	two	NUM
ma-246	260	14	intersections	intersection	NOUN
ma-246	260	15	.	.	PUNCT
ma-246	261	1	for	for	ADP
ma-246	261	2	u	u	PROPN
ma-246	261	3	∈	∈	PROPN
ma-246	261	4	η	η	PROPN
ma-246	261	5	-	-	PROPN
ma-246	261	6	o(x	o(x	PROPN
ma-246	261	7	)	)	PUNCT
ma-246	261	8	,	,	PUNCT
ma-246	261	9	suppose	suppose	VERB
ma-246	261	10	that	that	SCONJ
ma-246	261	11	x	x	PUNCT
ma-246	261	12	∈	∈	PROPN
ma-246	261	13	u	u	NOUN
ma-246	261	14	∩a∗η	∩a∗η	PROPN
ma-246	261	15	.	.	PUNCT
ma-246	262	1	then	then	ADV
ma-246	262	2	x	x	SYM
ma-246	262	3	∈	∈	PROPN
ma-246	262	4	u	u	NOUN
ma-246	262	5	and	and	CCONJ
ma-246	262	6	x	x	SYM
ma-246	262	7	∈	∈	PROPN
ma-246	262	8	a∗η	a∗η	PROPN
ma-246	262	9	.	.	PUNCT
ma-246	263	1	to	to	PART
ma-246	263	2	show	show	VERB
ma-246	263	3	that	that	SCONJ
ma-246	263	4	x	x	SYM
ma-246	263	5	∈	∈	PROPN
ma-246	263	6	(	(	PUNCT
ma-246	263	7	u	u	NOUN
ma-246	263	8	∩a)∗η	∩a)∗η	PUNCT
ma-246	263	9	,	,	PUNCT
ma-246	263	10	let	let	VERB
ma-246	263	11	v	v	ADP
ma-246	263	12	∈	∈	PROPN
ma-246	263	13	η	η	PROPN
ma-246	263	14	-	-	PROPN
ma-246	263	15	o(x	o(x	PROPN
ma-246	263	16	)	)	PUNCT
ma-246	263	17	.	.	PUNCT
ma-246	264	1	since	since	SCONJ
ma-246	264	2	x	x	PROPN
ma-246	264	3	∈	∈	NOUN
ma-246	264	4	uand	uand	NOUN
ma-246	264	5	u	u	PROPN
ma-246	264	6	∈	∈	PROPN
ma-246	264	7	η	η	PROPN
ma-246	264	8	-	-	PROPN
ma-246	264	9	o(x	o(x	PROPN
ma-246	264	10	)	)	PUNCT
ma-246	264	11	,	,	PUNCT
ma-246	264	12	we	we	PRON
ma-246	264	13	can	can	AUX
ma-246	264	14	write	write	VERB
ma-246	264	15	it	it	PRON
ma-246	264	16	as	as	ADP
ma-246	264	17	u	u	PROPN
ma-246	264	18	∈	∈	PROPN
ma-246	264	19	η	η	PROPN
ma-246	264	20	-	-	PROPN
ma-246	264	21	o(x	o(x	PROPN
ma-246	264	22	)	)	PUNCT
ma-246	264	23	.	.	PUNCT
ma-246	265	1	hence	hence	ADV
ma-246	265	2	,	,	PUNCT
ma-246	265	3	by	by	ADP
ma-246	265	4	assumption	assumption	NOUN
ma-246	265	5	,	,	PUNCT
ma-246	265	6	u	u	PROPN
ma-246	265	7	∩	∩	PROPN
ma-246	265	8	v	v	ADP
ma-246	265	9	∈	∈	PROPN
ma-246	265	10	η	η	PROPN
ma-246	265	11	-	-	NOUN
ma-246	265	12	o(x).note	o(x).note	NOUN
ma-246	265	13	that	that	SCONJ
ma-246	265	14	since	since	SCONJ
ma-246	265	15	x	x	PROPN
ma-246	265	16	∈	∈	PROPN
ma-246	265	17	a∗η	a∗η	PROPN
ma-246	265	18	and	and	CCONJ
ma-246	265	19	u∩v	u∩v	PROPN
ma-246	265	20	∈	∈	PROPN
ma-246	265	21	η	η	PROPN
ma-246	265	22	-	-	PROPN
ma-246	265	23	o(x	o(x	PROPN
ma-246	265	24	)	)	PUNCT
ma-246	265	25	,	,	PUNCT
ma-246	265	26	then	then	ADV
ma-246	265	27	a∩(u∩v	a∩(u∩v	VERB
ma-246	265	28	)	)	PUNCT
ma-246	265	29	/∈	/∈	PUNCT
ma-246	266	1	i	i	PRON
ma-246	266	2	for	for	ADP
ma-246	266	3	every	every	DET
ma-246	266	4	u∩v	u∩v	PROPN
ma-246	266	5	∈	∈	PROPN
ma-246	266	6	η	η	PROPN
ma-246	266	7	-	-	PROPN
ma-246	266	8	o(x).now	o(x).now	PROPN
ma-246	266	9	,	,	PUNCT
ma-246	266	10	by	by	ADP
ma-246	266	11	associativity	associativity	NOUN
ma-246	266	12	and	and	CCONJ
ma-246	266	13	commutativity	commutativity	NOUN
ma-246	266	14	,	,	PUNCT
ma-246	266	15	a	a	DET
ma-246	266	16	∩	∩	NOUN
ma-246	266	17	(	(	PUNCT
ma-246	266	18	u	u	PROPN
ma-246	266	19	∩	∩	NOUN
ma-246	266	20	v	v	NOUN
ma-246	266	21	)	)	PUNCT
ma-246	266	22	=	=	SYM
ma-246	266	23	(	(	PUNCT
ma-246	266	24	a	a	DET
ma-246	266	25	∩	∩	ADJ
ma-246	266	26	u	u	NOUN
ma-246	266	27	)	)	PUNCT
ma-246	266	28	∩	∩	ADJ
ma-246	266	29	v	v	NOUN
ma-246	266	30	/∈	/∈	PUNCT
ma-246	267	1	i	i	PRON
ma-246	267	2	=	=	PUNCT
ma-246	267	3	(	(	PUNCT
ma-246	267	4	u	u	PROPN
ma-246	267	5	∩	∩	ADJ
ma-246	267	6	a	a	PRON
ma-246	267	7	)	)	PUNCT
ma-246	267	8	∩	∩	ADJ
ma-246	267	9	v	v	X
ma-246	267	10	/∈	/∈	PUNCT
ma-246	267	11	i.	i.	NOUN
ma-246	268	1	this	this	PRON
ma-246	268	2	shows	show	VERB
ma-246	268	3	that	that	SCONJ
ma-246	268	4	for	for	ADP
ma-246	268	5	every	every	DET
ma-246	268	6	v	v	PROPN
ma-246	268	7	∈	∈	PROPN
ma-246	268	8	η	η	PROPN
ma-246	268	9	-	-	PROPN
ma-246	268	10	o(x	o(x	PROPN
ma-246	268	11	)	)	PUNCT
ma-246	268	12	,	,	PUNCT
ma-246	268	13	(	(	PUNCT
ma-246	268	14	u∩a)∩v	u∩a)∩v	PROPN
ma-246	268	15	/∈	/∈	PUNCT
ma-246	269	1	i	i	INTJ
ma-246	269	2	.	.	PUNCT
ma-246	270	1	it	it	PRON
ma-246	270	2	implies	imply	VERB
ma-246	270	3	that	that	SCONJ
ma-246	270	4	x	x	SYM
ma-246	270	5	∈	∈	PROPN
ma-246	270	6	(	(	PUNCT
ma-246	270	7	u∩a)∗η	u∩a)∗η	NOUN
ma-246	270	8	.	.	PUNCT
ma-246	271	1	hence	hence	ADV
ma-246	271	2	,	,	PUNCT
ma-246	271	3	u	u	X
ma-246	271	4	∩a∗η	∩a∗η	NOUN
ma-246	271	5	⊆	⊆	NUM
ma-246	271	6	(	(	PUNCT
ma-246	271	7	u	u	NOUN
ma-246	271	8	∩a)∗η	∩a)∗η	PUNCT
ma-246	271	9	.	.	PUNCT
ma-246	272	1	now	now	ADV
ma-246	272	2	,	,	PUNCT
ma-246	272	3	note	note	VERB
ma-246	272	4	that	that	SCONJ
ma-246	272	5	u	u	PRON
ma-246	272	6	∩a∗η	∩a∗η	NOUN
ma-246	272	7	⊆	⊆	NUM
ma-246	272	8	(	(	PUNCT
ma-246	272	9	u	u	NOUN
ma-246	272	10	∩a)∗η	∩a)∗η	ADP
ma-246	272	11	,	,	PUNCT
ma-246	272	12	then	then	ADV
ma-246	272	13	u	u	NOUN
ma-246	272	14	∩	∩	NOUN
ma-246	272	15	(	(	PUNCT
ma-246	272	16	u	u	NOUN
ma-246	272	17	∩a∗η	∩a∗η	NOUN
ma-246	272	18	)	)	PUNCT
ma-246	273	1	⊆	⊆	NUM
ma-246	273	2	u	u	NOUN
ma-246	273	3	∩	∩	NOUN
ma-246	273	4	(	(	PUNCT
ma-246	273	5	u	u	NOUN
ma-246	273	6	∩a)∗η	∩a)∗η	ADP
ma-246	273	7	.since	.since	PUNCT
ma-246	273	8	u	u	PROPN
ma-246	273	9	∩	∩	NOUN
ma-246	273	10	(	(	PUNCT
ma-246	273	11	u	u	PROPN
ma-246	273	12	∩	∩	NOUN
ma-246	273	13	a∗η	a∗η	NUM
ma-246	273	14	)	)	PUNCT
ma-246	273	15	,	,	PUNCT
ma-246	273	16	by	by	ADP
ma-246	273	17	associativity	associativity	NOUN
ma-246	273	18	again	again	ADV
ma-246	273	19	,	,	PUNCT
ma-246	273	20	u	u	NOUN
ma-246	273	21	∩	∩	NOUN
ma-246	273	22	(	(	PUNCT
ma-246	273	23	u	u	PROPN
ma-246	273	24	∩	∩	NOUN
ma-246	273	25	a∗η	a∗η	NUM
ma-246	273	26	)	)	PUNCT
ma-246	273	27	=	=	SYM
ma-246	273	28	(	(	PUNCT
ma-246	273	29	u	u	PROPN
ma-246	273	30	∩	∩	ADJ
ma-246	273	31	u	u	NOUN
ma-246	273	32	)	)	PUNCT
ma-246	273	33	∩	∩	NOUN
ma-246	273	34	a∗η	a∗η	PROPN
ma-246	273	35	=	=	SYM
ma-246	273	36	u	u	NOUN
ma-246	273	37	∩	∩	NOUN
ma-246	273	38	a∗η.this	a∗η.this	PROPN
ma-246	273	39	implies	imply	VERB
ma-246	273	40	that	that	SCONJ
ma-246	273	41	u	u	PROPN
ma-246	273	42	∩	∩	NOUN
ma-246	273	43	a∗η	a∗η	PROPN
ma-246	273	44	⊆	⊆	NUM
ma-246	273	45	u	u	NOUN
ma-246	273	46	∩	∩	NOUN
ma-246	273	47	(	(	PUNCT
ma-246	273	48	u	u	NOUN
ma-246	273	49	∩	∩	NOUN
ma-246	273	50	a)∗η	a)∗η	NOUN
ma-246	273	51	.	.	PUNCT
ma-246	274	1	in	in	ADP
ma-246	274	2	contrast	contrast	NOUN
ma-246	274	3	,	,	PUNCT
ma-246	274	4	note	note	VERB
ma-246	274	5	that	that	SCONJ
ma-246	274	6	u	u	PROPN
ma-246	274	7	∩	∩	NOUN
ma-246	274	8	a	a	DET
ma-246	274	9	⊆	⊆	NUM
ma-246	274	10	a	a	PRON
ma-246	274	11	,	,	PUNCT
ma-246	274	12	then	then	ADV
ma-246	274	13	bytheorem	bytheorem	VERB
ma-246	274	14	1	1	NUM
ma-246	274	15	(	(	PUNCT
ma-246	274	16	i	i	NOUN
ma-246	274	17	)	)	PUNCT
ma-246	274	18	,	,	PUNCT
ma-246	274	19	(	(	PUNCT
ma-246	274	20	u	u	NOUN
ma-246	274	21	∩	∩	NOUN
ma-246	274	22	a)∗η	a)∗η	VERB
ma-246	274	23	⊆	⊆	NUM
ma-246	274	24	a∗η	a∗η	NOUN
ma-246	274	25	.	.	PUNCT
ma-246	275	1	thus	thus	ADV
ma-246	275	2	,	,	PUNCT
ma-246	275	3	u	u	NOUN
ma-246	275	4	∩	∩	NOUN
ma-246	275	5	(	(	PUNCT
ma-246	275	6	u	u	NOUN
ma-246	275	7	∩	∩	NOUN
ma-246	275	8	a)∗η	a)∗η	NOUN
ma-246	275	9	⊆	⊆	NUM
ma-246	275	10	u	u	NOUN
ma-246	275	11	∩	∩	NOUN
ma-246	275	12	a∗η	a∗η	PROPN
ma-246	275	13	.	.	PUNCT
ma-246	276	1	consequently	consequently	ADV
ma-246	276	2	,	,	PUNCT
ma-246	276	3	as	as	ADP
ma-246	276	4	a	a	DET
ma-246	276	5	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	276	6	eur	eur	NOUN
ma-246	276	7	.	.	PUNCT
ma-246	277	1	j.	j.	PROPN
ma-246	277	2	math	math	PROPN
ma-246	277	3	.	.	PUNCT
ma-246	278	1	anal	anal	PROPN
ma-246	278	2	.	.	PUNCT
ma-246	279	1	10.28924	10.28924	NUM
ma-246	279	2	/	/	SYM
ma-246	279	3	ada	ada	PROPN
ma-246	279	4	/	/	SYM
ma-246	279	5	ma.5.2	ma.5.2	PROPN
ma-246	279	6	7result	7result	NUM
ma-246	279	7	,	,	PUNCT
ma-246	279	8	u	u	NOUN
ma-246	279	9	∩	∩	NOUN
ma-246	279	10	a∗η	a∗η	PROPN
ma-246	279	11	=	=	SYM
ma-246	279	12	u	u	PROPN
ma-246	279	13	∩	∩	NOUN
ma-246	279	14	(	(	PUNCT
ma-246	279	15	u	u	NOUN
ma-246	279	16	∩	∩	NOUN
ma-246	279	17	a)∗η	a)∗η	NOUN
ma-246	279	18	.	.	PUNCT
ma-246	280	1	note	note	VERB
ma-246	280	2	that	that	SCONJ
ma-246	280	3	u	u	PROPN
ma-246	280	4	∩	∩	NOUN
ma-246	280	5	(	(	PUNCT
ma-246	280	6	u	u	NOUN
ma-246	280	7	∩	∩	NOUN
ma-246	280	8	a)∗η	a)∗η	NOUN
ma-246	280	9	⊆	⊆	NUM
ma-246	280	10	(	(	PUNCT
ma-246	280	11	u	u	NOUN
ma-246	280	12	∩	∩	NOUN
ma-246	280	13	a)∗η	a)∗η	NOUN
ma-246	280	14	.	.	PUNCT
ma-246	281	1	this	this	PRON
ma-246	281	2	shows	show	VERB
ma-246	281	3	that	that	SCONJ
ma-246	281	4	u	u	NOUN
ma-246	281	5	∩	∩	NOUN
ma-246	281	6	a∗η	a∗η	PROPN
ma-246	281	7	=	=	SYM
ma-246	281	8	u	u	PROPN
ma-246	281	9	∩	∩	NOUN
ma-246	281	10	(	(	PUNCT
ma-246	281	11	u	u	NOUN
ma-246	281	12	∩	∩	NOUN
ma-246	281	13	a)∗η	a)∗η	NOUN
ma-246	281	14	⊆	⊆	NUM
ma-246	281	15	(	(	PUNCT
ma-246	281	16	u	u	NOUN
ma-246	281	17	∩	∩	NOUN
ma-246	281	18	a)∗η	a)∗η	NOUN
ma-246	281	19	.	.	PUNCT
ma-246	282	1	(	(	PUNCT
ma-246	282	2	iii	iii	X
ma-246	282	3	)	)	PUNCT
ma-246	282	4	let	let	VERB
ma-246	282	5	a	a	DET
ma-246	282	6	,	,	PUNCT
ma-246	282	7	b	b	NOUN
ma-246	282	8	⊆	⊆	NUM
ma-246	282	9	x	x	SYM
ma-246	282	10	.	.	PUNCT
ma-246	283	1	note	note	VERB
ma-246	283	2	that	that	SCONJ
ma-246	283	3	a	a	PRON
ma-246	283	4	=	=	X
ma-246	283	5	(	(	PUNCT
ma-246	283	6	a	a	DET
ma-246	283	7	\	\	PROPN
ma-246	283	8	b	b	NOUN
ma-246	283	9	)	)	PUNCT
ma-246	283	10	∪	∪	NOUN
ma-246	283	11	(	(	PUNCT
ma-246	283	12	b	b	PROPN
ma-246	283	13	∩	∩	NOUN
ma-246	283	14	a	a	X
ma-246	283	15	)	)	PUNCT
ma-246	283	16	.	.	PUNCT
ma-246	284	1	thus	thus	ADV
ma-246	284	2	a∗η	a∗η	PROPN
ma-246	284	3	=	=	SYM
ma-246	284	4	[	[	PUNCT
ma-246	284	5	(	(	PUNCT
ma-246	284	6	a	a	DET
ma-246	284	7	\	\	PROPN
ma-246	284	8	b	b	NOUN
ma-246	284	9	)	)	PUNCT
ma-246	284	10	∪	∪	NOUN
ma-246	284	11	(	(	PUNCT
ma-246	284	12	b	b	PROPN
ma-246	284	13	∩	∩	NOUN
ma-246	284	14	a	a	X
ma-246	284	15	)	)	PUNCT
ma-246	284	16	]	]	PUNCT
ma-246	284	17	∗	∗	X
ma-246	284	18	η	η	PROPN
ma-246	284	19	.	.	PUNCT
ma-246	285	1	notethat	notethat	PROPN
ma-246	285	2	by	by	ADP
ma-246	285	3	assumption	assumption	NOUN
ma-246	285	4	,	,	PUNCT
ma-246	285	5	η	η	PROPN
ma-246	285	6	-	-	PROPN
ma-246	285	7	o(x	o(x	ADJ
ma-246	285	8	)	)	PUNCT
ma-246	285	9	is	be	AUX
ma-246	285	10	closed	close	VERB
ma-246	285	11	under	under	ADP
ma-246	285	12	any	any	DET
ma-246	285	13	two	two	NUM
ma-246	285	14	intersections	intersection	NOUN
ma-246	285	15	,	,	PUNCT
ma-246	285	16	then	then	ADV
ma-246	285	17	by	by	ADP
ma-246	285	18	theorem	theorem	NOUN
ma-246	285	19	3	3	NUM
ma-246	285	20	(	(	PUNCT
ma-246	285	21	i	i	NOUN
ma-246	285	22	)	)	PUNCT
ma-246	285	23	,	,	PUNCT
ma-246	285	24	a∗η	a∗η	PROPN
ma-246	285	25	=	=	PUNCT
ma-246	285	26	[	[	PUNCT
ma-246	285	27	(	(	PUNCT
ma-246	285	28	a	a	DET
ma-246	285	29	\	\	PROPN
ma-246	285	30	b	b	NOUN
ma-246	285	31	)	)	PUNCT
ma-246	285	32	∪	∪	NOUN
ma-246	285	33	(	(	PUNCT
ma-246	285	34	b	b	PROPN
ma-246	285	35	∩	∩	NOUN
ma-246	285	36	a	a	X
ma-246	285	37	)	)	PUNCT
ma-246	285	38	]	]	PUNCT
ma-246	285	39	∗	∗	X
ma-246	285	40	η	η	X
ma-246	285	41	=	=	PROPN
ma-246	285	42	(	(	PUNCT
ma-246	285	43	a	a	DET
ma-246	285	44	\	\	PROPN
ma-246	285	45	b)∗η	b)∗η	X
ma-246	285	46	∪	∪	ADJ
ma-246	285	47	(	(	PUNCT
ma-246	285	48	b	b	NOUN
ma-246	285	49	∩	∩	NOUN
ma-246	285	50	a)∗η.so	a)∗η.so	NOUN
ma-246	285	51	,	,	PUNCT
ma-246	285	52	a∗η	a∗η	PROPN
ma-246	285	53	=	=	SYM
ma-246	285	54	(	(	PUNCT
ma-246	285	55	a	a	DET
ma-246	285	56	\	\	PROPN
ma-246	285	57	b)∗η	b)∗η	X
ma-246	285	58	∪	∪	ADJ
ma-246	285	59	(	(	PUNCT
ma-246	285	60	b	b	NOUN
ma-246	285	61	∩	∩	ADJ
ma-246	285	62	a)∗η	a)∗η	NOUN
ma-246	285	63	.	.	PUNCT
ma-246	286	1	now	now	ADV
ma-246	286	2	,	,	PUNCT
ma-246	286	3	note	note	VERB
ma-246	286	4	that	that	SCONJ
ma-246	286	5	a∗η	a∗η	ADP
ma-246	286	6	\	\	NOUN
ma-246	286	7	b∗η	b∗η	PROPN
ma-246	286	8	=	=	SYM
ma-246	286	9	a∗η	a∗η	PROPN
ma-246	286	10	∩	∩	NOUN
ma-246	286	11	(	(	PUNCT
ma-246	286	12	b∗η	b∗η	PROPN
ma-246	286	13	)	)	PUNCT
ma-246	286	14	c	c	NOUN
ma-246	286	15	,	,	PUNCT
ma-246	286	16	and	and	CCONJ
ma-246	286	17	since	since	SCONJ
ma-246	286	18	a∗η	a∗η	PROPN
ma-246	286	19	=	=	SYM
ma-246	286	20	(	(	PUNCT
ma-246	286	21	a	a	DET
ma-246	286	22	\	\	PROPN
ma-246	286	23	b)∗η	b)∗η	X
ma-246	286	24	∪	∪	ADJ
ma-246	286	25	(	(	PUNCT
ma-246	286	26	b	b	NOUN
ma-246	286	27	∩	∩	ADJ
ma-246	286	28	a)∗η	a)∗η	NOUN
ma-246	286	29	,	,	PUNCT
ma-246	286	30	a∗η	a∗η	ADP
ma-246	286	31	\	\	X
ma-246	286	32	b∗η	b∗η	PROPN
ma-246	286	33	=	=	SYM
ma-246	286	34	a∗η	a∗η	PROPN
ma-246	286	35	∩	∩	NOUN
ma-246	286	36	(	(	PUNCT
ma-246	286	37	b∗η	b∗η	PROPN
ma-246	286	38	)	)	PUNCT
ma-246	286	39	c	c	X
ma-246	286	40	=	=	PUNCT
ma-246	287	1	[	[	PUNCT
ma-246	287	2	(	(	PUNCT
ma-246	287	3	a	a	DET
ma-246	287	4	\	\	PROPN
ma-246	287	5	b)∗η	b)∗η	X
ma-246	287	6	∪	∪	ADJ
ma-246	287	7	(	(	PUNCT
ma-246	287	8	b	b	NOUN
ma-246	287	9	∩	∩	ADJ
ma-246	287	10	a)∗η	a)∗η	NOUN
ma-246	287	11	]	]	PUNCT
ma-246	287	12	∩	∩	NOUN
ma-246	287	13	(	(	PUNCT
ma-246	287	14	b∗η	b∗η	PROPN
ma-246	287	15	)	)	PUNCT
ma-246	287	16	c	c	X
ma-246	287	17	=	=	PUNCT
ma-246	288	1	[	[	PUNCT
ma-246	288	2	(	(	PUNCT
ma-246	288	3	a	a	DET
ma-246	288	4	\	\	PROPN
ma-246	288	5	b)∗η	b)∗η	NOUN
ma-246	288	6	∩	∩	NOUN
ma-246	288	7	(	(	PUNCT
ma-246	288	8	b∗η	b∗η	PROPN
ma-246	288	9	)	)	PUNCT
ma-246	288	10	c	c	X
ma-246	288	11	]	]	X
ma-246	288	12	∪	∪	X
ma-246	288	13	[	[	X
ma-246	288	14	(	(	PUNCT
ma-246	288	15	b	b	NOUN
ma-246	288	16	∩	∩	ADJ
ma-246	288	17	a)∗η	a)∗η	NOUN
ma-246	288	18	∩	∩	NOUN
ma-246	288	19	(	(	PUNCT
ma-246	288	20	b∗η	b∗η	PROPN
ma-246	288	21	)	)	PUNCT
ma-246	288	22	c	c	X
ma-246	288	23	]	]	X
ma-246	289	1	=	=	X
ma-246	290	1	[	[	PUNCT
ma-246	290	2	(	(	PUNCT
ma-246	290	3	a	a	DET
ma-246	290	4	\	\	PROPN
ma-246	290	5	b)∗η	b)∗η	X
ma-246	290	6	\	\	X
ma-246	290	7	b∗η	b∗η	X
ma-246	290	8	]	]	PUNCT
ma-246	290	9	∪	∪	X
ma-246	290	10	[	[	PUNCT
ma-246	290	11	(	(	PUNCT
ma-246	290	12	b	b	NOUN
ma-246	290	13	∩	∩	ADJ
ma-246	290	14	a)∗η	a)∗η	NOUN
ma-246	290	15	\	\	X
ma-246	290	16	b∗η	b∗η	PROPN
ma-246	290	17	]	]	PUNCT
ma-246	290	18	.	.	PUNCT
ma-246	291	1	hence	hence	ADV
ma-246	291	2	,	,	PUNCT
ma-246	291	3	a∗η	a∗η	ADP
ma-246	291	4	\	\	X
ma-246	291	5	b∗η	b∗η	PROPN
ma-246	292	1	=	=	SYM
ma-246	293	1	[	[	PUNCT
ma-246	293	2	(	(	PUNCT
ma-246	293	3	a	a	DET
ma-246	293	4	\	\	PROPN
ma-246	293	5	b)∗η	b)∗η	X
ma-246	293	6	\	\	X
ma-246	293	7	b∗η	b∗η	X
ma-246	293	8	]	]	PUNCT
ma-246	293	9	∪	∪	X
ma-246	293	10	[	[	PUNCT
ma-246	293	11	(	(	PUNCT
ma-246	293	12	b	b	NOUN
ma-246	293	13	∩	∩	ADJ
ma-246	293	14	a)∗η	a)∗η	NOUN
ma-246	293	15	\	\	X
ma-246	293	16	b∗η	b∗η	PROPN
ma-246	293	17	]	]	PUNCT
ma-246	293	18	.	.	PUNCT
ma-246	294	1	note	note	VERB
ma-246	294	2	that	that	SCONJ
ma-246	294	3	b	b	NOUN
ma-246	294	4	∩	∩	NOUN
ma-246	294	5	a	a	DET
ma-246	294	6	⊆	⊆	NUM
ma-246	294	7	b	b	NOUN
ma-246	294	8	,	,	PUNCT
ma-246	294	9	then	then	ADV
ma-246	294	10	bytheorem	bytheorem	VERB
ma-246	294	11	1	1	NUM
ma-246	294	12	(	(	PUNCT
ma-246	294	13	i	i	NOUN
ma-246	294	14	)	)	PUNCT
ma-246	294	15	,	,	PUNCT
ma-246	294	16	(	(	PUNCT
ma-246	294	17	b	b	NOUN
ma-246	294	18	∩	∩	NOUN
ma-246	294	19	a)∗η	a)∗η	NOUN
ma-246	294	20	⊆	⊆	NUM
ma-246	294	21	b∗η	b∗η	PROPN
ma-246	294	22	implies	imply	VERB
ma-246	294	23	that	that	SCONJ
ma-246	294	24	(	(	PUNCT
ma-246	294	25	b	b	NOUN
ma-246	294	26	∩	∩	ADJ
ma-246	294	27	a)∗η	a)∗η	NOUN
ma-246	294	28	\	\	X
ma-246	294	29	b∗η	b∗η	PROPN
ma-246	294	30	=	=	PUNCT
ma-246	294	31	∅.	∅.	ADP
ma-246	294	32	now	now	ADV
ma-246	294	33	,	,	PUNCT
ma-246	294	34	since	since	SCONJ
ma-246	294	35	a∗η	a∗η	PROPN
ma-246	294	36	\	\	PROPN
ma-246	294	37	b∗η	b∗η	PROPN
ma-246	294	38	=[	=[	NOUN
ma-246	294	39	(	(	PUNCT
ma-246	294	40	a	a	DET
ma-246	294	41	\	\	PROPN
ma-246	294	42	b)∗η	b)∗η	X
ma-246	294	43	\	\	X
ma-246	294	44	b∗η	b∗η	X
ma-246	294	45	]	]	PUNCT
ma-246	294	46	∪	∪	X
ma-246	294	47	[	[	PUNCT
ma-246	294	48	(	(	PUNCT
ma-246	294	49	b	b	NOUN
ma-246	294	50	∩	∩	ADJ
ma-246	294	51	a)∗η	a)∗η	NOUN
ma-246	294	52	\	\	X
ma-246	294	53	b∗η	b∗η	PROPN
ma-246	294	54	]	]	PUNCT
ma-246	294	55	and	and	CCONJ
ma-246	294	56	(	(	PUNCT
ma-246	294	57	b	b	NOUN
ma-246	294	58	∩	∩	ADJ
ma-246	294	59	a)∗η	a)∗η	NOUN
ma-246	294	60	\	\	X
ma-246	294	61	b∗η	b∗η	PROPN
ma-246	294	62	=	=	SYM
ma-246	294	63	∅	∅	NOUN
ma-246	294	64	,	,	PUNCT
ma-246	294	65	it	it	PRON
ma-246	294	66	follows	follow	VERB
ma-246	294	67	that	that	SCONJ
ma-246	294	68	a∗η	a∗η	PUNCT
ma-246	294	69	\	\	X
ma-246	294	70	b∗η	b∗η	PROPN
ma-246	295	1	=	=	SYM
ma-246	296	1	[	[	PUNCT
ma-246	296	2	(	(	PUNCT
ma-246	296	3	a	a	DET
ma-246	296	4	\	\	PROPN
ma-246	296	5	b)∗η	b)∗η	X
ma-246	296	6	\	\	X
ma-246	296	7	b∗η	b∗η	X
ma-246	296	8	]	]	PUNCT
ma-246	296	9	∪	∪	X
ma-246	296	10	[	[	PUNCT
ma-246	296	11	(	(	PUNCT
ma-246	296	12	b	b	NOUN
ma-246	296	13	∩	∩	ADJ
ma-246	296	14	a)∗η	a)∗η	NOUN
ma-246	296	15	\	\	X
ma-246	296	16	b∗η	b∗η	X
ma-246	296	17	]	]	PUNCT
ma-246	296	18	=	=	PUNCT
ma-246	297	1	[	[	PUNCT
ma-246	297	2	(	(	PUNCT
ma-246	297	3	a	a	DET
ma-246	297	4	\	\	PROPN
ma-246	297	5	b)∗η	b)∗η	X
ma-246	297	6	\	\	X
ma-246	297	7	b∗η	b∗η	PROPN
ma-246	297	8	]	]	PUNCT
ma-246	297	9	∪∅	∪∅	PROPN
ma-246	297	10	=	=	SYM
ma-246	298	1	(	(	PUNCT
ma-246	298	2	a	a	DET
ma-246	298	3	\	\	PROPN
ma-246	298	4	b)∗η	b)∗η	X
ma-246	298	5	\	\	NOUN
ma-246	298	6	b∗η	b∗η	PROPN
ma-246	298	7	⊆	⊆	NUM
ma-246	298	8	(	(	PUNCT
ma-246	298	9	a	a	DET
ma-246	298	10	\	\	PROPN
ma-246	298	11	b)∗η.as	b)∗η.as	X
ma-246	298	12	a	a	DET
ma-246	298	13	result	result	NOUN
ma-246	298	14	,	,	PUNCT
ma-246	298	15	it	it	PRON
ma-246	298	16	shows	show	VERB
ma-246	298	17	that	that	SCONJ
ma-246	298	18	a∗η	a∗η	PUNCT
ma-246	298	19	\	\	X
ma-246	298	20	b∗η	b∗η	PROPN
ma-246	298	21	=	=	SYM
ma-246	298	22	(	(	PUNCT
ma-246	298	23	a	a	DET
ma-246	298	24	\	\	PROPN
ma-246	298	25	b)∗η	b)∗η	X
ma-246	298	26	\	\	NOUN
ma-246	298	27	b∗η	b∗η	PROPN
ma-246	298	28	⊆	⊆	NUM
ma-246	298	29	(	(	PUNCT
ma-246	298	30	a	a	DET
ma-246	298	31	\	\	PROPN
ma-246	298	32	b)∗η	b)∗η	NOUN
ma-246	298	33	.	.	PUNCT
ma-246	299	1	�	�	PROPN
ma-246	299	2	theorem	theorem	VERB
ma-246	299	3	4	4	NUM
ma-246	299	4	.	.	PUNCT
ma-246	300	1	let	let	VERB
ma-246	300	2	(	(	PUNCT
ma-246	300	3	x	x	X
ma-246	300	4	,	,	PUNCT
ma-246	300	5	τ	τ	PROPN
ma-246	300	6	,	,	PUNCT
ma-246	300	7	i	i	PRON
ma-246	300	8	)	)	PUNCT
ma-246	300	9	be	be	VERB
ma-246	300	10	an	an	DET
ma-246	300	11	ideal	ideal	ADJ
ma-246	300	12	topological	topological	ADJ
ma-246	300	13	space	space	NOUN
ma-246	300	14	and	and	CCONJ
ma-246	300	15	a	a	DET
ma-246	300	16	⊆	⊆	NUM
ma-246	300	17	x	x	SYM
ma-246	300	18	.	.	PUNCT
ma-246	301	1	then	then	ADV
ma-246	301	2	for	for	ADP
ma-246	301	3	any	any	DET
ma-246	301	4	η	η	ADJ
ma-246	301	5	-	-	ADJ
ma-246	301	6	local	local	ADJ
ma-246	301	7	function	function	NOUN
ma-246	301	8	,	,	PUNCT
ma-246	301	9	the	the	DET
ma-246	301	10	following	follow	VERB
ma-246	301	11	properties	property	NOUN
ma-246	301	12	hold	hold	VERB
ma-246	301	13	:	:	PUNCT
ma-246	301	14	(	(	PUNCT
ma-246	301	15	i	i	NOUN
ma-246	301	16	)	)	PUNCT
ma-246	301	17	a∗η	a∗η	PROPN
ma-246	301	18	⊆	⊆	NUM
ma-246	301	19	a∗	a∗	NOUN
ma-246	301	20	;	;	PUNCT
ma-246	301	21	(	(	PUNCT
ma-246	301	22	ii	ii	NOUN
ma-246	301	23	)	)	PUNCT
ma-246	301	24	a∗η	a∗η	PROPN
ma-246	301	25	⊆	⊆	NUM
ma-246	301	26	γ∗(a	γ∗(a	NOUN
ma-246	301	27	)	)	PUNCT
ma-246	301	28	;	;	PUNCT
ma-246	301	29	and	and	CCONJ
ma-246	301	30	(	(	PUNCT
ma-246	301	31	iii	iii	X
ma-246	301	32	)	)	PUNCT
ma-246	301	33	a∗	a∗	NOUN
ma-246	301	34	⊆	⊆	NUM
ma-246	301	35	γ∗(a	γ∗(a	NOUN
ma-246	301	36	)	)	PUNCT
ma-246	301	37	.	.	PUNCT
ma-246	302	1	proof	proof	NOUN
ma-246	302	2	.	.	PUNCT
ma-246	303	1	(	(	PUNCT
ma-246	303	2	i	i	NOUN
ma-246	303	3	)	)	PUNCT
ma-246	303	4	let	let	VERB
ma-246	303	5	x	x	SYM
ma-246	303	6	∈	∈	PROPN
ma-246	303	7	a∗η	a∗η	PROPN
ma-246	303	8	and	and	CCONJ
ma-246	303	9	u	u	PROPN
ma-246	303	10	∈	∈	PROPN
ma-246	303	11	τ(x	τ(x	NOUN
ma-246	303	12	)	)	PUNCT
ma-246	303	13	.	.	PUNCT
ma-246	304	1	since	since	SCONJ
ma-246	304	2	every	every	DET
ma-246	304	3	open	open	ADJ
ma-246	304	4	set	set	NOUN
ma-246	304	5	is	be	AUX
ma-246	304	6	η	η	ADJ
ma-246	304	7	-	-	ADJ
ma-246	304	8	open	open	ADJ
ma-246	304	9	set	set	NOUN
ma-246	304	10	,	,	PUNCT
ma-246	304	11	u	u	PROPN
ma-246	304	12	∈	∈	PROPN
ma-246	304	13	η	η	PROPN
ma-246	304	14	-	-	PROPN
ma-246	304	15	o(x	o(x	PROPN
ma-246	304	16	)	)	PUNCT
ma-246	304	17	.	.	PUNCT
ma-246	305	1	also	also	ADV
ma-246	305	2	,	,	PUNCT
ma-246	305	3	since	since	SCONJ
ma-246	305	4	x	x	PROPN
ma-246	305	5	∈	∈	PROPN
ma-246	305	6	a∗η	a∗η	PROPN
ma-246	305	7	and	and	CCONJ
ma-246	305	8	u	u	PROPN
ma-246	305	9	∈	∈	PROPN
ma-246	305	10	η	η	PROPN
ma-246	305	11	-	-	PROPN
ma-246	305	12	o(x	o(x	PROPN
ma-246	305	13	)	)	PUNCT
ma-246	305	14	,	,	PUNCT
ma-246	305	15	a	a	DET
ma-246	305	16	∩	∩	ADJ
ma-246	305	17	u	u	NOUN
ma-246	305	18	/∈	/∈	PUNCT
ma-246	305	19	i	i	INTJ
ma-246	305	20	.	.	PUNCT
ma-246	306	1	note	note	VERB
ma-246	306	2	that	that	SCONJ
ma-246	306	3	a	a	DET
ma-246	306	4	∩	∩	ADJ
ma-246	306	5	u	u	NOUN
ma-246	306	6	/∈	/∈	PUNCT
ma-246	307	1	i	i	PRON
ma-246	307	2	and	and	CCONJ
ma-246	307	3	u	u	NOUN
ma-246	307	4	∈	∈	PROPN
ma-246	307	5	τ(x	τ(x	NOUN
ma-246	307	6	)	)	PUNCT
ma-246	307	7	.	.	PUNCT
ma-246	308	1	hence	hence	ADV
ma-246	308	2	,	,	PUNCT
ma-246	308	3	a	a	DET
ma-246	308	4	∩	∩	ADJ
ma-246	308	5	u	u	NOUN
ma-246	308	6	/∈	/∈	PUNCT
ma-246	308	7	ifor	ifor	ADP
ma-246	308	8	every	every	DET
ma-246	308	9	u	u	PROPN
ma-246	308	10	∈	∈	PROPN
ma-246	308	11	τ(x	τ(x	NOUN
ma-246	308	12	)	)	PUNCT
ma-246	308	13	,	,	PUNCT
ma-246	308	14	and	and	CCONJ
ma-246	308	15	so	so	ADV
ma-246	308	16	,	,	PUNCT
ma-246	308	17	x	x	PROPN
ma-246	308	18	∈	∈	NOUN
ma-246	308	19	a∗.	a∗.	NOUN
ma-246	308	20	therefore	therefore	ADV
ma-246	308	21	,	,	PUNCT
ma-246	308	22	a∗η	a∗η	PROPN
ma-246	308	23	⊆	⊆	NUM
ma-246	308	24	a∗.	a∗.	NOUN
ma-246	308	25	(	(	PUNCT
ma-246	308	26	ii	ii	NOUN
ma-246	308	27	)	)	PUNCT
ma-246	308	28	let	let	VERB
ma-246	308	29	x	x	SYM
ma-246	308	30	∈	∈	PROPN
ma-246	308	31	a∗η	a∗η	PROPN
ma-246	308	32	and	and	CCONJ
ma-246	308	33	u	u	PROPN
ma-246	308	34	∈	∈	PROPN
ma-246	308	35	ro(x	ro(x	PUNCT
ma-246	308	36	)	)	PUNCT
ma-246	308	37	.	.	PUNCT
ma-246	309	1	since	since	SCONJ
ma-246	309	2	every	every	DET
ma-246	309	3	regular	regular	ADJ
ma-246	309	4	-	-	PUNCT
ma-246	309	5	open	open	ADJ
ma-246	309	6	set	set	NOUN
ma-246	309	7	is	be	AUX
ma-246	309	8	η	η	ADJ
ma-246	309	9	-	-	ADJ
ma-246	309	10	open	open	ADJ
ma-246	309	11	set	set	NOUN
ma-246	309	12	,	,	PUNCT
ma-246	309	13	u	u	PROPN
ma-246	309	14	∈	∈	PROPN
ma-246	309	15	η	η	PROPN
ma-246	309	16	-	-	PROPN
ma-246	309	17	o(x	o(x	PROPN
ma-246	309	18	)	)	PUNCT
ma-246	309	19	.	.	PUNCT
ma-246	310	1	also	also	ADV
ma-246	310	2	,	,	PUNCT
ma-246	310	3	since	since	SCONJ
ma-246	310	4	x	x	PROPN
ma-246	310	5	∈	∈	PROPN
ma-246	310	6	a∗η	a∗η	PROPN
ma-246	310	7	and	and	CCONJ
ma-246	310	8	u	u	PROPN
ma-246	310	9	∈	∈	PROPN
ma-246	310	10	η	η	PROPN
ma-246	310	11	-	-	PROPN
ma-246	310	12	o(x	o(x	PROPN
ma-246	310	13	)	)	PUNCT
ma-246	310	14	,	,	PUNCT
ma-246	310	15	a	a	DET
ma-246	310	16	∩	∩	ADJ
ma-246	310	17	u	u	NOUN
ma-246	310	18	/∈	/∈	PUNCT
ma-246	310	19	i	i	INTJ
ma-246	310	20	.	.	PUNCT
ma-246	311	1	hence	hence	ADV
ma-246	311	2	,	,	PUNCT
ma-246	311	3	a	a	DET
ma-246	311	4	∩	∩	ADJ
ma-246	311	5	u	u	NOUN
ma-246	311	6	/∈	/∈	PUNCT
ma-246	311	7	i	i	PRON
ma-246	311	8	for	for	ADP
ma-246	311	9	every	every	DET
ma-246	311	10	u	u	PROPN
ma-246	311	11	∈	∈	PROPN
ma-246	311	12	ro(x	ro(x	PUNCT
ma-246	311	13	)	)	PUNCT
ma-246	311	14	,	,	PUNCT
ma-246	311	15	and	and	CCONJ
ma-246	311	16	so	so	ADV
ma-246	311	17	,	,	PUNCT
ma-246	311	18	x	x	PUNCT
ma-246	311	19	∈	∈	PROPN
ma-246	311	20	γ∗(a	γ∗(a	PROPN
ma-246	311	21	)	)	PUNCT
ma-246	311	22	.	.	PUNCT
ma-246	312	1	therefore	therefore	ADV
ma-246	312	2	,	,	PUNCT
ma-246	312	3	a∗η	a∗η	PROPN
ma-246	312	4	⊆	⊆	NUM
ma-246	312	5	γ∗(a	γ∗(a	NOUN
ma-246	312	6	)	)	PUNCT
ma-246	312	7	.	.	PUNCT
ma-246	313	1	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	313	2	eur	eur	PROPN
ma-246	313	3	.	.	PUNCT
ma-246	314	1	j.	j.	PROPN
ma-246	314	2	math	math	PROPN
ma-246	314	3	.	.	PUNCT
ma-246	315	1	anal	anal	PROPN
ma-246	315	2	.	.	PUNCT
ma-246	316	1	10.28924	10.28924	NUM
ma-246	316	2	/	/	SYM
ma-246	316	3	ada	ada	PROPN
ma-246	316	4	/	/	SYM
ma-246	316	5	ma.5.2	ma.5.2	PROPN
ma-246	316	6	8	8	NUM
ma-246	316	7	(	(	PUNCT
ma-246	316	8	iii	iii	NOUN
ma-246	316	9	)	)	PUNCT
ma-246	316	10	let	let	VERB
ma-246	316	11	x	x	SYM
ma-246	316	12	∈	∈	PROPN
ma-246	316	13	a∗	a∗	NOUN
ma-246	316	14	and	and	CCONJ
ma-246	316	15	u	u	NOUN
ma-246	316	16	∈	∈	PROPN
ma-246	316	17	ro(x	ro(x	PUNCT
ma-246	316	18	)	)	PUNCT
ma-246	316	19	.	.	PUNCT
ma-246	317	1	since	since	SCONJ
ma-246	317	2	every	every	DET
ma-246	317	3	regular	regular	ADJ
ma-246	317	4	-	-	PUNCT
ma-246	317	5	open	open	ADJ
ma-246	317	6	set	set	NOUN
ma-246	317	7	is	be	AUX
ma-246	317	8	open	open	ADJ
ma-246	317	9	set	set	VERB
ma-246	317	10	,	,	PUNCT
ma-246	317	11	u	u	PROPN
ma-246	317	12	∈	∈	PROPN
ma-246	317	13	τ	τ	X
ma-246	317	14	.	.	PUNCT
ma-246	318	1	also	also	ADV
ma-246	318	2	,	,	PUNCT
ma-246	318	3	since	since	SCONJ
ma-246	318	4	x	x	PROPN
ma-246	318	5	∈	∈	PROPN
ma-246	318	6	a∗	a∗	NOUN
ma-246	318	7	and	and	CCONJ
ma-246	318	8	u	u	PROPN
ma-246	318	9	∈	∈	PROPN
ma-246	318	10	τ	τ	X
ma-246	318	11	,	,	PUNCT
ma-246	318	12	a	a	DET
ma-246	318	13	∩	∩	ADJ
ma-246	318	14	u	u	NOUN
ma-246	318	15	/∈	/∈	PUNCT
ma-246	319	1	i	i	INTJ
ma-246	319	2	.	.	PUNCT
ma-246	320	1	hence	hence	ADV
ma-246	320	2	,	,	PUNCT
ma-246	320	3	a	a	DET
ma-246	320	4	∩	∩	ADJ
ma-246	320	5	u	u	NOUN
ma-246	320	6	/∈	/∈	PUNCT
ma-246	320	7	i	i	PRON
ma-246	320	8	for	for	ADP
ma-246	320	9	every	every	DET
ma-246	320	10	u	u	PROPN
ma-246	320	11	∈	∈	PROPN
ma-246	320	12	ro(x	ro(x	PUNCT
ma-246	320	13	)	)	PUNCT
ma-246	320	14	,	,	PUNCT
ma-246	320	15	and	and	CCONJ
ma-246	320	16	so	so	ADV
ma-246	320	17	,	,	PUNCT
ma-246	320	18	x	x	SYM
ma-246	320	19	∈	∈	PROPN
ma-246	320	20	γ∗(a).therefore	γ∗(a).therefore	ADV
ma-246	320	21	,	,	PUNCT
ma-246	320	22	a∗	a∗	PROPN
ma-246	320	23	⊆	⊆	NUM
ma-246	320	24	γ∗(a	γ∗(a	NOUN
ma-246	320	25	)	)	PUNCT
ma-246	320	26	.	.	PUNCT
ma-246	321	1	�	�	PROPN
ma-246	321	2	remark	remark	VERB
ma-246	321	3	2	2	NUM
ma-246	321	4	.	.	PUNCT
ma-246	322	1	let	let	VERB
ma-246	322	2	(	(	PUNCT
ma-246	322	3	x	x	X
ma-246	322	4	,	,	PUNCT
ma-246	322	5	τ	τ	PROPN
ma-246	322	6	,	,	PUNCT
ma-246	322	7	i	i	PRON
ma-246	322	8	)	)	PUNCT
ma-246	322	9	be	be	VERB
ma-246	322	10	an	an	DET
ma-246	322	11	ideal	ideal	ADJ
ma-246	322	12	topological	topological	ADJ
ma-246	322	13	space	space	NOUN
ma-246	322	14	and	and	CCONJ
ma-246	322	15	a	a	DET
ma-246	322	16	⊆	⊆	NUM
ma-246	322	17	x	x	SYM
ma-246	322	18	.	.	PUNCT
ma-246	323	1	then	then	ADV
ma-246	323	2	for	for	ADP
ma-246	323	3	any	any	DET
ma-246	323	4	η	η	ADJ
ma-246	323	5	-	-	ADJ
ma-246	323	6	local	local	ADJ
ma-246	323	7	functions	function	NOUN
ma-246	323	8	,	,	PUNCT
ma-246	323	9	the	the	DET
ma-246	323	10	following	follow	VERB
ma-246	323	11	properties	property	NOUN
ma-246	323	12	hold	hold	VERB
ma-246	323	13	:	:	PUNCT
ma-246	323	14	(	(	PUNCT
ma-246	323	15	ii	ii	NOUN
ma-246	323	16	)	)	PUNCT
ma-246	323	17	a∗η	a∗η	PROPN
ma-246	323	18	⊆	⊆	NUM
ma-246	323	19	a∗	a∗	PROPN
ma-246	323	20	⊆	⊆	NUM
ma-246	323	21	γ∗(a);(ii	γ∗(a);(ii	NUM
ma-246	323	22	)	)	PUNCT
ma-246	323	23	if	if	SCONJ
ma-246	323	24	η	η	PROPN
ma-246	323	25	-	-	ADJ
ma-246	323	26	o(x	o(x	PROPN
ma-246	323	27	)	)	PUNCT
ma-246	323	28	=	=	SYM
ma-246	323	29	τ	τ	PROPN
ma-246	323	30	,	,	PUNCT
ma-246	323	31	then	then	ADV
ma-246	323	32	a∗η	a∗η	PROPN
ma-246	323	33	=	=	SYM
ma-246	323	34	a∗	a∗	X
ma-246	323	35	;	;	PUNCT
ma-246	323	36	and(iii	and(iii	NOUN
ma-246	323	37	)	)	PUNCT
ma-246	323	38	if	if	SCONJ
ma-246	323	39	η	η	PROPN
ma-246	323	40	-	-	ADJ
ma-246	323	41	o(x	o(x	ADJ
ma-246	323	42	)	)	PUNCT
ma-246	323	43	=	=	PUNCT
ma-246	323	44	ro(x	ro(x	X
ma-246	323	45	)	)	PUNCT
ma-246	323	46	,	,	PUNCT
ma-246	323	47	then	then	ADV
ma-246	323	48	a∗η	a∗η	PROPN
ma-246	323	49	=	=	SYM
ma-246	323	50	γ∗(a	γ∗(a	PROPN
ma-246	323	51	)	)	PUNCT
ma-246	323	52	theorem	theorem	NOUN
ma-246	323	53	5	5	NUM
ma-246	323	54	.	.	PUNCT
ma-246	324	1	let	let	VERB
ma-246	324	2	(	(	PUNCT
ma-246	324	3	x	x	NOUN
ma-246	324	4	,	,	PUNCT
ma-246	324	5	τ	τ	X
ma-246	324	6	)	)	PUNCT
ma-246	324	7	be	be	VERB
ma-246	324	8	a	a	DET
ma-246	324	9	topological	topological	ADJ
ma-246	324	10	space	space	NOUN
ma-246	324	11	with	with	ADP
ma-246	324	12	ideals	ideal	NOUN
ma-246	324	13	i1	i1	PROPN
ma-246	324	14	and	and	CCONJ
ma-246	324	15	i2	i2	PROPN
ma-246	324	16	on	on	ADP
ma-246	324	17	x	x	X
ma-246	324	18	and	and	CCONJ
ma-246	324	19	a	a	DET
ma-246	324	20	⊆	⊆	NUM
ma-246	324	21	x	x	SYM
ma-246	324	22	.	.	PUNCT
ma-246	325	1	then	then	ADV
ma-246	325	2	,	,	PUNCT
ma-246	325	3	for	for	ADP
ma-246	325	4	any	any	DET
ma-246	325	5	η	η	ADJ
ma-246	325	6	-	-	ADJ
ma-246	325	7	local	local	ADJ
ma-246	325	8	functions	function	NOUN
ma-246	325	9	,	,	PUNCT
ma-246	325	10	the	the	DET
ma-246	325	11	following	follow	VERB
ma-246	325	12	properties	property	NOUN
ma-246	325	13	hold	hold	VERB
ma-246	325	14	:	:	PUNCT
ma-246	325	15	(	(	PUNCT
ma-246	325	16	i	i	NOUN
ma-246	325	17	)	)	PUNCT
ma-246	325	18	if	if	SCONJ
ma-246	325	19	i1	i1	PROPN
ma-246	325	20	⊆	⊆	NUM
ma-246	325	21	i2	i2	PROPN
ma-246	325	22	,	,	PUNCT
ma-246	325	23	then	then	ADV
ma-246	325	24	a∗η	a∗η	PROPN
ma-246	325	25	(	(	PUNCT
ma-246	325	26	i2	i2	PROPN
ma-246	325	27	,	,	PUNCT
ma-246	325	28	η	η	PROPN
ma-246	325	29	-	-	PROPN
ma-246	325	30	o(x	o(x	PROPN
ma-246	325	31	)	)	PUNCT
ma-246	325	32	)	)	PUNCT
ma-246	326	1	⊆	⊆	X
ma-246	326	2	a∗η	a∗η	PROPN
ma-246	326	3	(	(	PUNCT
ma-246	326	4	i1	i1	PROPN
ma-246	326	5	,	,	PUNCT
ma-246	326	6	η	η	PROPN
ma-246	326	7	-	-	PROPN
ma-246	326	8	o(x	o(x	PROPN
ma-246	326	9	)	)	PUNCT
ma-246	326	10	)	)	PUNCT
ma-246	326	11	;	;	PUNCT
ma-246	326	12	and	and	CCONJ
ma-246	326	13	(	(	PUNCT
ma-246	326	14	ii	ii	NOUN
ma-246	326	15	)	)	PUNCT
ma-246	326	16	a∗η	a∗η	PROPN
ma-246	326	17	(	(	PUNCT
ma-246	326	18	(	(	PUNCT
ma-246	326	19	i1	i1	PROPN
ma-246	326	20	∩	∩	PROPN
ma-246	326	21	i2	i2	PROPN
ma-246	326	22	)	)	PUNCT
ma-246	326	23	,	,	PUNCT
ma-246	326	24	η	η	PROPN
ma-246	326	25	-	-	PROPN
ma-246	326	26	o(x	o(x	PROPN
ma-246	326	27	)	)	PUNCT
ma-246	326	28	)	)	PUNCT
ma-246	327	1	=	=	PUNCT
ma-246	327	2	a∗η	a∗η	PROPN
ma-246	327	3	(	(	PUNCT
ma-246	327	4	i1	i1	PROPN
ma-246	327	5	,	,	PUNCT
ma-246	327	6	η	η	PROPN
ma-246	327	7	-	-	PROPN
ma-246	327	8	o(x	o(x	PROPN
ma-246	327	9	)	)	PUNCT
ma-246	327	10	)	)	PUNCT
ma-246	327	11	∪	∪	ADP
ma-246	327	12	a∗η	a∗η	PROPN
ma-246	327	13	(	(	PUNCT
ma-246	327	14	i2	i2	PROPN
ma-246	327	15	,	,	PUNCT
ma-246	327	16	η	η	PROPN
ma-246	327	17	-	-	PROPN
ma-246	327	18	o(x	o(x	PROPN
ma-246	327	19	)	)	PUNCT
ma-246	327	20	)	)	PUNCT
ma-246	327	21	.	.	PUNCT
ma-246	328	1	proof	proof	NOUN
ma-246	328	2	.	.	PUNCT
ma-246	329	1	(	(	PUNCT
ma-246	329	2	i	i	NOUN
ma-246	329	3	)	)	PUNCT
ma-246	329	4	let	let	VERB
ma-246	329	5	i1	i1	PROPN
ma-246	329	6	⊆	⊆	NUM
ma-246	329	7	i2	i2	PROPN
ma-246	329	8	and	and	CCONJ
ma-246	329	9	x	x	PART
ma-246	329	10	∈	∈	PROPN
ma-246	329	11	a∗η(i2	a∗η(i2	NOUN
ma-246	329	12	,	,	PUNCT
ma-246	329	13	η	η	NOUN
ma-246	329	14	-	-	PROPN
ma-246	329	15	o(x	o(x	PROPN
ma-246	329	16	)	)	PUNCT
ma-246	329	17	)	)	PUNCT
ma-246	329	18	.	.	PUNCT
ma-246	330	1	then	then	ADV
ma-246	330	2	for	for	ADP
ma-246	330	3	every	every	DET
ma-246	330	4	u	u	PROPN
ma-246	330	5	∈	∈	PROPN
ma-246	330	6	η	η	PROPN
ma-246	330	7	-	-	PROPN
ma-246	330	8	o(x	o(x	PROPN
ma-246	330	9	)	)	PUNCT
ma-246	330	10	,	,	PUNCT
ma-246	330	11	a∩u	a∩u	PROPN
ma-246	330	12	/∈	/∈	PUNCT
ma-246	330	13	i2	i2	PROPN
ma-246	330	14	.	.	PUNCT
ma-246	331	1	since	since	SCONJ
ma-246	331	2	i1	i1	PROPN
ma-246	331	3	⊆	⊆	NUM
ma-246	331	4	i2	i2	PROPN
ma-246	331	5	,	,	PUNCT
ma-246	331	6	a	a	DET
ma-246	331	7	∩	∩	ADJ
ma-246	331	8	u	u	NOUN
ma-246	331	9	/∈	/∈	PROPN
ma-246	331	10	i1	i1	PROPN
ma-246	331	11	for	for	ADP
ma-246	331	12	every	every	DET
ma-246	331	13	u	u	PROPN
ma-246	331	14	∈	∈	PROPN
ma-246	331	15	η	η	PROPN
ma-246	331	16	-	-	PROPN
ma-246	331	17	o(x	o(x	PROPN
ma-246	331	18	)	)	PUNCT
ma-246	331	19	.	.	PUNCT
ma-246	332	1	hence	hence	ADV
ma-246	332	2	,	,	PUNCT
ma-246	332	3	a∗η(i2	a∗η(i2	PROPN
ma-246	332	4	,	,	PUNCT
ma-246	332	5	η	η	NOUN
ma-246	332	6	-	-	PROPN
ma-246	332	7	o(x	o(x	PROPN
ma-246	332	8	)	)	PUNCT
ma-246	332	9	)	)	PUNCT
ma-246	333	1	⊆	⊆	X
ma-246	333	2	a∗η	a∗η	PROPN
ma-246	333	3	(	(	PUNCT
ma-246	333	4	i1	i1	PROPN
ma-246	333	5	,	,	PUNCT
ma-246	333	6	η	η	PROPN
ma-246	333	7	-	-	PROPN
ma-246	333	8	o(x	o(x	PROPN
ma-246	333	9	)	)	PUNCT
ma-246	333	10	)	)	PUNCT
ma-246	333	11	.	.	PUNCT
ma-246	334	1	(	(	PUNCT
ma-246	334	2	ii	ii	NOUN
ma-246	334	3	)	)	PUNCT
ma-246	334	4	let	let	VERB
ma-246	334	5	i1	i1	PROPN
ma-246	334	6	and	and	CCONJ
ma-246	334	7	i2	i2	PROPN
ma-246	334	8	be	be	VERB
ma-246	334	9	ideals	ideal	NOUN
ma-246	334	10	on	on	ADP
ma-246	334	11	x	x	X
ma-246	334	12	.	.	PUNCT
ma-246	335	1	note	note	VERB
ma-246	335	2	that	that	SCONJ
ma-246	335	3	i1	i1	PROPN
ma-246	335	4	∩	∩	PROPN
ma-246	335	5	i2	i2	PROPN
ma-246	335	6	⊆	⊆	NUM
ma-246	335	7	i1	i1	PROPN
ma-246	335	8	and	and	CCONJ
ma-246	335	9	i1	i1	PROPN
ma-246	336	1	∩	∩	PROPN
ma-246	336	2	i2	i2	PROPN
ma-246	336	3	⊆	⊆	NUM
ma-246	336	4	i2	i2	NOUN
ma-246	336	5	.	.	PUNCT
ma-246	337	1	then	then	ADV
ma-246	337	2	by	by	ADP
ma-246	337	3	theorem	theorem	NOUN
ma-246	337	4	5	5	NUM
ma-246	337	5	(	(	PUNCT
ma-246	337	6	i	i	NOUN
ma-246	337	7	)	)	PUNCT
ma-246	337	8	,	,	PUNCT
ma-246	337	9	a∗η	a∗η	PROPN
ma-246	337	10	(	(	PUNCT
ma-246	337	11	i1	i1	PROPN
ma-246	337	12	,	,	PUNCT
ma-246	337	13	η	η	PROPN
ma-246	337	14	-	-	PROPN
ma-246	337	15	o(x	o(x	PROPN
ma-246	337	16	)	)	PUNCT
ma-246	337	17	)	)	PUNCT
ma-246	338	1	⊆	⊆	X
ma-246	338	2	a∗η	a∗η	PROPN
ma-246	338	3	(	(	PUNCT
ma-246	338	4	(	(	PUNCT
ma-246	338	5	i1	i1	PROPN
ma-246	338	6	∩	∩	PROPN
ma-246	338	7	i2	i2	PROPN
ma-246	338	8	)	)	PUNCT
ma-246	338	9	,	,	PUNCT
ma-246	338	10	η	η	PROPN
ma-246	338	11	-	-	PROPN
ma-246	338	12	o(x	o(x	PROPN
ma-246	338	13	)	)	PUNCT
ma-246	338	14	)	)	PUNCT
ma-246	338	15	and	and	CCONJ
ma-246	338	16	a∗η	a∗η	PROPN
ma-246	338	17	(	(	PUNCT
ma-246	338	18	i2	i2	PROPN
ma-246	338	19	,	,	PUNCT
ma-246	338	20	η	η	PROPN
ma-246	338	21	-	-	PROPN
ma-246	338	22	o(x	o(x	PROPN
ma-246	338	23	)	)	PUNCT
ma-246	338	24	)	)	PUNCT
ma-246	339	1	⊆	⊆	X
ma-246	339	2	a∗η	a∗η	PROPN
ma-246	339	3	(	(	PUNCT
ma-246	339	4	(	(	PUNCT
ma-246	339	5	i1	i1	PROPN
ma-246	339	6	∩	∩	PROPN
ma-246	339	7	i2	i2	PROPN
ma-246	339	8	)	)	PUNCT
ma-246	339	9	,	,	PUNCT
ma-246	339	10	η	η	PROPN
ma-246	339	11	-	-	PROPN
ma-246	339	12	o(x	o(x	PROPN
ma-246	339	13	)	)	PUNCT
ma-246	339	14	)	)	PUNCT
ma-246	339	15	,	,	PUNCT
ma-246	339	16	and	and	CCONJ
ma-246	339	17	hence	hence	ADV
ma-246	339	18	,	,	PUNCT
ma-246	339	19	a∗η	a∗η	PROPN
ma-246	339	20	(	(	PUNCT
ma-246	339	21	i1	i1	PROPN
ma-246	339	22	,	,	PUNCT
ma-246	339	23	η	η	PROPN
ma-246	339	24	-	-	PROPN
ma-246	339	25	o(x	o(x	PROPN
ma-246	339	26	)	)	PUNCT
ma-246	339	27	)	)	PUNCT
ma-246	339	28	∪	∪	ADP
ma-246	339	29	a∗η	a∗η	PROPN
ma-246	339	30	(	(	PUNCT
ma-246	339	31	i2	i2	PROPN
ma-246	339	32	,	,	PUNCT
ma-246	339	33	η	η	PROPN
ma-246	339	34	-	-	PROPN
ma-246	339	35	o(x	o(x	PROPN
ma-246	339	36	)	)	PUNCT
ma-246	339	37	)	)	PUNCT
ma-246	340	1	⊆	⊆	X
ma-246	340	2	a∗η	a∗η	PROPN
ma-246	340	3	(	(	PUNCT
ma-246	340	4	(	(	PUNCT
ma-246	340	5	i1	i1	PROPN
ma-246	340	6	∩	∩	PROPN
ma-246	340	7	i2	i2	PROPN
ma-246	340	8	)	)	PUNCT
ma-246	340	9	,	,	PUNCT
ma-246	340	10	η	η	PROPN
ma-246	340	11	-	-	PROPN
ma-246	340	12	o(x	o(x	PROPN
ma-246	340	13	)	)	PUNCT
ma-246	340	14	)	)	PUNCT
ma-246	340	15	.	.	PUNCT
ma-246	341	1	next	next	ADV
ma-246	341	2	,	,	PUNCT
ma-246	341	3	let	let	VERB
ma-246	341	4	x	x	X
ma-246	341	5	∈	∈	PROPN
ma-246	341	6	a∗η((i1∩i2	a∗η((i1∩i2	PROPN
ma-246	341	7	)	)	PUNCT
ma-246	341	8	,	,	PUNCT
ma-246	341	9	η	η	PROPN
ma-246	341	10	-	-	PROPN
ma-246	341	11	o(x	o(x	PROPN
ma-246	341	12	)	)	PUNCT
ma-246	341	13	)	)	PUNCT
ma-246	341	14	,	,	PUNCT
ma-246	341	15	then	then	ADV
ma-246	341	16	for	for	ADP
ma-246	341	17	every	every	DET
ma-246	341	18	u	u	PROPN
ma-246	341	19	∈	∈	PROPN
ma-246	341	20	η	η	PROPN
ma-246	341	21	-	-	PROPN
ma-246	341	22	o(x	o(x	PROPN
ma-246	341	23	)	)	PUNCT
ma-246	341	24	,	,	PUNCT
ma-246	341	25	a∩u	a∩u	PROPN
ma-246	341	26	/∈	/∈	PUNCT
ma-246	342	1	i1∩i2	i1∩i2	NOUN
ma-246	342	2	.	.	PUNCT
ma-246	343	1	this	this	DET
ma-246	343	2	impliesthat	impliesthat	PROPN
ma-246	343	3	a∩u	a∩u	PROPN
ma-246	343	4	/∈	/∈	PUNCT
ma-246	344	1	i1	i1	PROPN
ma-246	344	2	or	or	CCONJ
ma-246	344	3	a∩u	a∩u	PROPN
ma-246	344	4	/∈	/∈	PUNCT
ma-246	344	5	i2	i2	PROPN
ma-246	344	6	.	.	PUNCT
ma-246	345	1	this	this	PRON
ma-246	345	2	shows	show	VERB
ma-246	345	3	that	that	SCONJ
ma-246	345	4	x	x	PUNCT
ma-246	345	5	∈	∈	PROPN
ma-246	345	6	a∗η(i1	a∗η(i1	NOUN
ma-246	345	7	,	,	PUNCT
ma-246	345	8	η	η	PROPN
ma-246	345	9	-	-	PROPN
ma-246	345	10	o(x	o(x	PROPN
ma-246	345	11	)	)	PUNCT
ma-246	345	12	)	)	PUNCT
ma-246	345	13	or	or	CCONJ
ma-246	345	14	x	x	PUNCT
ma-246	345	15	∈	∈	PROPN
ma-246	345	16	a∗η(i2	a∗η(i2	NOUN
ma-246	345	17	,	,	PUNCT
ma-246	345	18	η	η	NOUN
ma-246	345	19	-	-	PROPN
ma-246	345	20	o(x	o(x	PROPN
ma-246	345	21	)	)	PUNCT
ma-246	345	22	)	)	PUNCT
ma-246	345	23	.hence	.hence	NOUN
ma-246	345	24	,	,	PUNCT
ma-246	345	25	x	x	SYM
ma-246	345	26	∈	∈	PROPN
ma-246	345	27	a∗η(i1	a∗η(i1	NOUN
ma-246	345	28	,	,	PUNCT
ma-246	345	29	η	η	PROPN
ma-246	345	30	-	-	PROPN
ma-246	345	31	o(x	o(x	PROPN
ma-246	345	32	)	)	PUNCT
ma-246	345	33	)	)	PUNCT
ma-246	345	34	∪	∪	ADP
ma-246	345	35	a∗η	a∗η	PROPN
ma-246	345	36	(	(	PUNCT
ma-246	345	37	i2	i2	PROPN
ma-246	345	38	,	,	PUNCT
ma-246	345	39	η	η	PROPN
ma-246	345	40	-	-	PROPN
ma-246	345	41	o(x	o(x	PROPN
ma-246	345	42	)	)	PUNCT
ma-246	345	43	)	)	PUNCT
ma-246	345	44	,	,	PUNCT
ma-246	345	45	and	and	CCONJ
ma-246	345	46	so	so	ADV
ma-246	345	47	,	,	PUNCT
ma-246	345	48	a∗η	a∗η	PROPN
ma-246	345	49	(	(	PUNCT
ma-246	345	50	(	(	PUNCT
ma-246	345	51	i1	i1	PROPN
ma-246	345	52	∩	∩	PROPN
ma-246	345	53	i2	i2	PROPN
ma-246	345	54	)	)	PUNCT
ma-246	345	55	,	,	PUNCT
ma-246	345	56	η	η	PROPN
ma-246	345	57	-	-	PROPN
ma-246	345	58	o(x	o(x	PROPN
ma-246	345	59	)	)	PUNCT
ma-246	345	60	)	)	PUNCT
ma-246	346	1	⊆	⊆	X
ma-246	346	2	a∗η	a∗η	PROPN
ma-246	346	3	(	(	PUNCT
ma-246	346	4	i1	i1	PROPN
ma-246	346	5	,	,	PUNCT
ma-246	346	6	η	η	PROPN
ma-246	346	7	-	-	PROPN
ma-246	346	8	o(x	o(x	PROPN
ma-246	346	9	)	)	PUNCT
ma-246	346	10	)	)	PUNCT
ma-246	346	11	∪	∪	ADP
ma-246	346	12	a∗η	a∗η	PROPN
ma-246	346	13	(	(	PUNCT
ma-246	346	14	i2	i2	PROPN
ma-246	346	15	,	,	PUNCT
ma-246	346	16	η	η	PROPN
ma-246	346	17	-	-	PROPN
ma-246	346	18	o(x	o(x	PROPN
ma-246	346	19	)	)	PUNCT
ma-246	346	20	)	)	PUNCT
ma-246	346	21	.	.	PUNCT
ma-246	347	1	as	as	ADP
ma-246	347	2	a	a	DET
ma-246	347	3	result	result	NOUN
ma-246	347	4	,	,	PUNCT
ma-246	347	5	thus	thus	ADV
ma-246	347	6	,	,	PUNCT
ma-246	347	7	a∗η	a∗η	PROPN
ma-246	347	8	(	(	PUNCT
ma-246	347	9	(	(	PUNCT
ma-246	347	10	i1	i1	PROPN
ma-246	347	11	∩	∩	PROPN
ma-246	347	12	i2	i2	PROPN
ma-246	347	13	)	)	PUNCT
ma-246	347	14	,	,	PUNCT
ma-246	347	15	η	η	PROPN
ma-246	347	16	-	-	PROPN
ma-246	347	17	o(x	o(x	PROPN
ma-246	347	18	)	)	PUNCT
ma-246	347	19	)	)	PUNCT
ma-246	348	1	=	=	PUNCT
ma-246	348	2	a∗η	a∗η	PROPN
ma-246	348	3	(	(	PUNCT
ma-246	348	4	i1	i1	PROPN
ma-246	348	5	,	,	PUNCT
ma-246	348	6	η	η	PROPN
ma-246	348	7	-	-	PROPN
ma-246	348	8	o(x	o(x	PROPN
ma-246	348	9	)	)	PUNCT
ma-246	348	10	)	)	PUNCT
ma-246	348	11	∪	∪	ADP
ma-246	348	12	a∗η	a∗η	PROPN
ma-246	348	13	(	(	PUNCT
ma-246	348	14	i2	i2	PROPN
ma-246	348	15	,	,	PUNCT
ma-246	348	16	η	η	PROPN
ma-246	348	17	-	-	PROPN
ma-246	348	18	o(x	o(x	PROPN
ma-246	348	19	)	)	PUNCT
ma-246	348	20	)	)	PUNCT
ma-246	348	21	.	.	PUNCT
ma-246	349	1	�	�	PROPN
ma-246	349	2	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	349	3	eur	eur	PROPN
ma-246	349	4	.	.	PUNCT
ma-246	350	1	j.	j.	PROPN
ma-246	350	2	math	math	PROPN
ma-246	350	3	.	.	PUNCT
ma-246	351	1	anal	anal	PROPN
ma-246	351	2	.	.	PUNCT
ma-246	352	1	10.28924	10.28924	NUM
ma-246	352	2	/	/	SYM
ma-246	352	3	ada	ada	PROPN
ma-246	352	4	/	/	SYM
ma-246	352	5	ma.5.2	ma.5.2	PROPN
ma-246	352	6	94	94	NUM
ma-246	352	7	.	.	PUNCT
ma-246	353	1	η	η	ADJ
ma-246	353	2	-	-	ADJ
ma-246	353	3	local	local	ADJ
ma-246	353	4	closure	closure	NOUN
ma-246	353	5	definition	definition	NOUN
ma-246	353	6	2	2	NUM
ma-246	353	7	.	.	PUNCT
ma-246	354	1	let	let	VERB
ma-246	354	2	(	(	PUNCT
ma-246	354	3	x	x	X
ma-246	354	4	,	,	PUNCT
ma-246	354	5	τ	τ	PROPN
ma-246	354	6	,	,	PUNCT
ma-246	354	7	i	i	PRON
ma-246	354	8	)	)	PUNCT
ma-246	354	9	be	be	VERB
ma-246	354	10	an	an	DET
ma-246	354	11	ideal	ideal	ADJ
ma-246	354	12	topological	topological	ADJ
ma-246	354	13	space	space	NOUN
ma-246	354	14	.	.	PUNCT
ma-246	355	1	the	the	DET
ma-246	355	2	η	η	ADJ
ma-246	355	3	-	-	ADJ
ma-246	355	4	local	local	ADJ
ma-246	355	5	closure	closure	NOUN
ma-246	355	6	of	of	ADP
ma-246	355	7	a	a	DET
ma-246	355	8	denoted	denote	VERB
ma-246	355	9	by	by	ADP
ma-246	355	10	cl∗η(a	cl∗η(a	PROPN
ma-246	355	11	)	)	PUNCT
ma-246	355	12	is	be	AUX
ma-246	355	13	defined	define	VERB
ma-246	355	14	by	by	ADP
ma-246	355	15	the	the	DET
ma-246	355	16	union	union	NOUN
ma-246	355	17	of	of	ADP
ma-246	355	18	a	a	PRON
ma-246	355	19	and	and	CCONJ
ma-246	355	20	the	the	DET
ma-246	355	21	η	η	ADJ
ma-246	355	22	-	-	ADJ
ma-246	355	23	local	local	ADJ
ma-246	355	24	function	function	NOUN
ma-246	355	25	of	of	ADP
ma-246	355	26	a	a	DET
ma-246	355	27	,	,	PUNCT
ma-246	355	28	i.e	i.e	PROPN
ma-246	355	29	,	,	PUNCT
ma-246	355	30	cl∗η(a	cl∗η(a	PROPN
ma-246	355	31	)	)	PUNCT
ma-246	355	32	=	=	PUNCT
ma-246	356	1	a	a	DET
ma-246	356	2	∪	∪	X
ma-246	356	3	a∗η	a∗η	NOUN
ma-246	356	4	for	for	ADP
ma-246	356	5	any	any	DET
ma-246	356	6	a	a	DET
ma-246	356	7	⊆	⊆	NUM
ma-246	356	8	x	x	SYM
ma-246	356	9	.	.	PUNCT
ma-246	356	10	example	example	NOUN
ma-246	357	1	3	3	X
ma-246	357	2	.	.	PUNCT
ma-246	358	1	let	let	VERB
ma-246	358	2	(	(	PUNCT
ma-246	358	3	x	x	X
ma-246	358	4	,	,	PUNCT
ma-246	358	5	τ	τ	PROPN
ma-246	358	6	,	,	PUNCT
ma-246	358	7	i	i	PRON
ma-246	358	8	)	)	PUNCT
ma-246	358	9	be	be	VERB
ma-246	358	10	an	an	DET
ma-246	358	11	ideal	ideal	ADJ
ma-246	358	12	topological	topological	ADJ
ma-246	358	13	space	space	NOUN
ma-246	358	14	where	where	SCONJ
ma-246	358	15	x	x	X
ma-246	358	16	=	=	PRON
ma-246	358	17	{	{	PUNCT
ma-246	358	18	a	a	PRON
ma-246	358	19	,	,	PUNCT
ma-246	358	20	b	b	NOUN
ma-246	358	21	,	,	PUNCT
ma-246	358	22	c	c	NOUN
ma-246	358	23	}	}	PUNCT
ma-246	358	24	,	,	PUNCT
ma-246	358	25	τ	τ	X
ma-246	358	26	=	=	PUNCT
ma-246	358	27	{	{	PUNCT
ma-246	358	28	∅	∅	NOUN
ma-246	358	29	,	,	PUNCT
ma-246	358	30	x	x	X
ma-246	358	31	,	,	PUNCT
ma-246	358	32	{	{	PUNCT
ma-246	358	33	a	a	X
ma-246	358	34	}	}	PUNCT
ma-246	358	35	,	,	PUNCT
ma-246	358	36	{	{	PUNCT
ma-246	358	37	b	b	NOUN
ma-246	358	38	}	}	PUNCT
ma-246	358	39	,	,	PUNCT
ma-246	358	40	{	{	PUNCT
ma-246	358	41	a	a	PRON
ma-246	358	42	,	,	PUNCT
ma-246	358	43	b	b	NOUN
ma-246	358	44	}	}	PUNCT
ma-246	358	45	}	}	PUNCT
ma-246	358	46	,	,	PUNCT
ma-246	358	47	and	and	CCONJ
ma-246	358	48	i	i	PRON
ma-246	358	49	=	=	PUNCT
ma-246	358	50	{	{	PUNCT
ma-246	358	51	∅	∅	NOUN
ma-246	358	52	,	,	PUNCT
ma-246	358	53	{	{	PUNCT
ma-246	358	54	b	b	NOUN
ma-246	358	55	}	}	PUNCT
ma-246	358	56	}	}	PUNCT
ma-246	358	57	.	.	PUNCT
ma-246	359	1	then	then	ADV
ma-246	359	2	the	the	DET
ma-246	359	3	η	η	ADJ
ma-246	359	4	-	-	ADJ
ma-246	359	5	open	open	ADJ
ma-246	359	6	sets	set	NOUN
ma-246	359	7	of	of	ADP
ma-246	359	8	x	x	SYM
ma-246	359	9	are	be	AUX
ma-246	359	10	∅	∅	NOUN
ma-246	359	11	,	,	PUNCT
ma-246	359	12	x	x	INTJ
ma-246	359	13	,	,	PUNCT
ma-246	359	14	{	{	PUNCT
ma-246	359	15	a	a	NOUN
ma-246	359	16	}	}	PUNCT
ma-246	359	17	,	,	PUNCT
ma-246	359	18	{	{	PUNCT
ma-246	359	19	b	b	NOUN
ma-246	359	20	}	}	PUNCT
ma-246	359	21	,	,	PUNCT
ma-246	359	22	{	{	PUNCT
ma-246	359	23	a	a	DET
ma-246	359	24	,	,	PUNCT
ma-246	359	25	b	b	NOUN
ma-246	359	26	}	}	PUNCT
ma-246	359	27	,	,	PUNCT
ma-246	359	28	{	{	PUNCT
ma-246	359	29	a	a	X
ma-246	359	30	,	,	PUNCT
ma-246	359	31	c	c	NOUN
ma-246	359	32	}	}	PUNCT
ma-246	359	33	,	,	PUNCT
ma-246	359	34	and	and	CCONJ
ma-246	359	35	{	{	PUNCT
ma-246	359	36	b	b	NOUN
ma-246	359	37	,	,	PUNCT
ma-246	359	38	c	c	NOUN
ma-246	359	39	}	}	PUNCT
ma-246	359	40	.	.	PUNCT
ma-246	360	1	let	let	VERB
ma-246	360	2	a	a	DET
ma-246	360	3	=	=	X
ma-246	360	4	{	{	PUNCT
ma-246	360	5	a	a	PROPN
ma-246	360	6	,	,	PUNCT
ma-246	360	7	b	b	NOUN
ma-246	360	8	}	}	PUNCT
ma-246	360	9	,	,	PUNCT
ma-246	360	10	then	then	ADV
ma-246	360	11	by	by	ADP
ma-246	360	12	definition	definition	NOUN
ma-246	360	13	1	1	NUM
ma-246	360	14	and	and	CCONJ
ma-246	360	15	2	2	NUM
ma-246	360	16	,	,	PUNCT
ma-246	360	17	a∗η	a∗η	PROPN
ma-246	360	18	=	=	SYM
ma-246	360	19	{	{	PUNCT
ma-246	360	20	a	a	NOUN
ma-246	360	21	}	}	PUNCT
ma-246	360	22	and	and	CCONJ
ma-246	360	23	cl∗η(a	cl∗η(a	PROPN
ma-246	360	24	)	)	PUNCT
ma-246	360	25	=	=	PRON
ma-246	360	26	{	{	PUNCT
ma-246	360	27	a	a	PRON
ma-246	360	28	,	,	PUNCT
ma-246	360	29	b	b	NOUN
ma-246	360	30	}	}	PUNCT
ma-246	360	31	∪	∪	ADJ
ma-246	360	32	{	{	PUNCT
ma-246	360	33	a	a	NOUN
ma-246	360	34	}	}	PUNCT
ma-246	360	35	=	=	SYM
ma-246	360	36	{	{	PUNCT
ma-246	360	37	a	a	DET
ma-246	360	38	,	,	PUNCT
ma-246	360	39	b	b	NOUN
ma-246	360	40	}	}	PUNCT
ma-246	360	41	,	,	PUNCT
ma-246	360	42	respectively	respectively	ADV
ma-246	360	43	.	.	PUNCT
ma-246	361	1	theorem	theorem	VERB
ma-246	361	2	6	6	NUM
ma-246	361	3	.	.	PUNCT
ma-246	362	1	let	let	VERB
ma-246	362	2	(	(	PUNCT
ma-246	362	3	x	x	X
ma-246	362	4	,	,	PUNCT
ma-246	362	5	τ	τ	PROPN
ma-246	362	6	,	,	PUNCT
ma-246	362	7	i	i	PRON
ma-246	362	8	)	)	PUNCT
ma-246	362	9	be	be	VERB
ma-246	362	10	an	an	DET
ma-246	362	11	ideal	ideal	ADJ
ma-246	362	12	topological	topological	ADJ
ma-246	362	13	space	space	NOUN
ma-246	362	14	and	and	CCONJ
ma-246	362	15	a	a	DET
ma-246	362	16	,	,	PUNCT
ma-246	362	17	b	b	NOUN
ma-246	362	18	⊆	⊆	NUM
ma-246	362	19	x	x	X
ma-246	362	20	.	.	PUNCT
ma-246	363	1	then	then	ADV
ma-246	363	2	the	the	DET
ma-246	363	3	following	follow	VERB
ma-246	363	4	properties	property	NOUN
ma-246	363	5	hold	hold	VERB
ma-246	363	6	:	:	PUNCT
ma-246	363	7	(	(	PUNCT
ma-246	363	8	i	i	NOUN
ma-246	363	9	)	)	PUNCT
ma-246	363	10	if	if	SCONJ
ma-246	363	11	a	a	DET
ma-246	363	12	⊆	⊆	NUM
ma-246	363	13	b	b	NOUN
ma-246	363	14	,	,	PUNCT
ma-246	363	15	then	then	ADV
ma-246	363	16	cl∗η(a	cl∗η(a	PROPN
ma-246	363	17	)	)	PUNCT
ma-246	363	18	⊆	⊆	NUM
ma-246	363	19	cl∗η(b	cl∗η(b	NUM
ma-246	363	20	)	)	PUNCT
ma-246	363	21	;	;	PUNCT
ma-246	363	22	(	(	PUNCT
ma-246	363	23	ii	ii	NOUN
ma-246	363	24	)	)	PUNCT
ma-246	363	25	cl∗η(a	cl∗η(a	PROPN
ma-246	363	26	∩	∩	ADJ
ma-246	363	27	b	b	X
ma-246	363	28	)	)	PUNCT
ma-246	363	29	⊆	⊆	NUM
ma-246	363	30	cl∗η(a	cl∗η(a	ADJ
ma-246	363	31	)	)	PUNCT
ma-246	363	32	∩	∩	NOUN
ma-246	363	33	cl∗η(b	cl∗η(b	NOUN
ma-246	363	34	)	)	PUNCT
ma-246	363	35	;	;	PUNCT
ma-246	363	36	(	(	PUNCT
ma-246	363	37	iii	iii	X
ma-246	363	38	)	)	PUNCT
ma-246	363	39	if	if	SCONJ
ma-246	363	40	a	a	PRON
ma-246	363	41	is	be	AUX
ma-246	363	42	an	an	DET
ma-246	363	43	η	η	ADJ
ma-246	363	44	-	-	ADJ
ma-246	363	45	closed	closed	ADJ
ma-246	363	46	set	set	NOUN
ma-246	363	47	,	,	PUNCT
ma-246	363	48	then	then	ADV
ma-246	363	49	cl∗η(a	cl∗η(a	PROPN
ma-246	363	50	)	)	PUNCT
ma-246	363	51	=	=	PUNCT
ma-246	363	52	η	η	PROPN
ma-246	363	53	-	-	NOUN
ma-246	363	54	cl(a	cl(a	NUM
ma-246	363	55	)	)	PUNCT
ma-246	363	56	;	;	PUNCT
ma-246	363	57	(	(	PUNCT
ma-246	363	58	iv	iv	X
ma-246	363	59	)	)	PUNCT
ma-246	363	60	if	if	SCONJ
ma-246	363	61	a	a	DET
ma-246	363	62	∈	∈	X
ma-246	364	1	i	i	PRON
ma-246	364	2	,	,	PUNCT
ma-246	364	3	then	then	ADV
ma-246	364	4	cl∗η(a	cl∗η(a	PROPN
ma-246	364	5	)	)	PUNCT
ma-246	364	6	=	=	SYM
ma-246	365	1	a	a	PRON
ma-246	365	2	;	;	PUNCT
ma-246	365	3	(	(	PUNCT
ma-246	365	4	v	v	NOUN
ma-246	365	5	)	)	PUNCT
ma-246	365	6	cl∗η(a∗η	cl∗η(a∗η	NOUN
ma-246	365	7	)	)	PUNCT
ma-246	365	8	=	=	PUNCT
ma-246	366	1	a∗η	a∗η	PROPN
ma-246	366	2	;	;	PUNCT
ma-246	366	3	(	(	PUNCT
ma-246	366	4	vi	vi	NOUN
ma-246	366	5	)	)	PUNCT
ma-246	366	6	cl∗η(a	cl∗η(a	PROPN
ma-246	366	7	)	)	PUNCT
ma-246	366	8	=	=	PUNCT
ma-246	367	1	η	η	PROPN
ma-246	367	2	-	-	NOUN
ma-246	367	3	cl(a	cl(a	NUM
ma-246	367	4	)	)	PUNCT
ma-246	367	5	;	;	PUNCT
ma-246	367	6	and	and	CCONJ
ma-246	367	7	(	(	PUNCT
ma-246	367	8	vii	vii	PROPN
ma-246	367	9	)	)	PUNCT
ma-246	367	10	(	(	PUNCT
ma-246	367	11	cl∗η(a	cl∗η(a	PROPN
ma-246	367	12	)	)	PUNCT
ma-246	367	13	)	)	PUNCT
ma-246	367	14	∗	∗	PROPN
ma-246	367	15	η	η	PROPN
ma-246	367	16	=	=	PUNCT
ma-246	367	17	a∗η	a∗η	PROPN
ma-246	367	18	.	.	PUNCT
ma-246	368	1	proof	proof	NOUN
ma-246	368	2	.	.	PUNCT
ma-246	369	1	(	(	PUNCT
ma-246	369	2	i	i	NOUN
ma-246	369	3	)	)	PUNCT
ma-246	369	4	let	let	VERB
ma-246	369	5	a	a	DET
ma-246	369	6	,	,	PUNCT
ma-246	369	7	b	b	NOUN
ma-246	369	8	⊆	⊆	NUM
ma-246	369	9	x	x	PUNCT
ma-246	369	10	and	and	CCONJ
ma-246	369	11	a	a	DET
ma-246	369	12	⊆	⊆	NUM
ma-246	369	13	b.	b.	NOUN
ma-246	369	14	by	by	ADP
ma-246	369	15	definition	definition	NOUN
ma-246	369	16	2	2	NUM
ma-246	369	17	,	,	PUNCT
ma-246	369	18	cl∗η(a	cl∗η(a	PROPN
ma-246	369	19	)	)	PUNCT
ma-246	370	1	=	=	PUNCT
ma-246	370	2	a	a	PRON
ma-246	370	3	∪	∪	ADJ
ma-246	370	4	a∗η	a∗η	NOUN
ma-246	370	5	and	and	CCONJ
ma-246	370	6	cl∗η(b	cl∗η(b	NUM
ma-246	370	7	)	)	PUNCT
ma-246	370	8	=	=	SYM
ma-246	370	9	b	b	X
ma-246	370	10	∪	∪	ADP
ma-246	370	11	b∗η	b∗η	PROPN
ma-246	370	12	.since	.since	NOUN
ma-246	370	13	a	a	DET
ma-246	370	14	⊆	⊆	NUM
ma-246	370	15	b	b	NOUN
ma-246	370	16	,	,	PUNCT
ma-246	370	17	by	by	ADP
ma-246	370	18	theorem	theorem	NOUN
ma-246	370	19	1	1	NUM
ma-246	370	20	(	(	PUNCT
ma-246	370	21	i	i	NOUN
ma-246	370	22	)	)	PUNCT
ma-246	370	23	,	,	PUNCT
ma-246	370	24	a∗η	a∗η	PROPN
ma-246	370	25	⊆	⊆	NUM
ma-246	370	26	b∗η	b∗η	PROPN
ma-246	370	27	.	.	PUNCT
ma-246	371	1	this	this	PRON
ma-246	371	2	shows	show	VERB
ma-246	371	3	that	that	SCONJ
ma-246	371	4	a	a	PRON
ma-246	371	5	∪	∪	ADJ
ma-246	371	6	a∗η	a∗η	PROPN
ma-246	371	7	⊆	⊆	NUM
ma-246	371	8	b	b	NOUN
ma-246	371	9	∪	∪	X
ma-246	371	10	b∗η	b∗η	PROPN
ma-246	371	11	,	,	PUNCT
ma-246	371	12	and	and	CCONJ
ma-246	371	13	hence	hence	ADV
ma-246	371	14	,	,	PUNCT
ma-246	371	15	cl∗η(a	cl∗η(a	PROPN
ma-246	371	16	)	)	PUNCT
ma-246	371	17	⊆	⊆	NUM
ma-246	371	18	cl∗η(b	cl∗η(b	NUM
ma-246	371	19	)	)	PUNCT
ma-246	371	20	.	.	PUNCT
ma-246	372	1	(	(	PUNCT
ma-246	372	2	ii	ii	NOUN
ma-246	372	3	)	)	PUNCT
ma-246	372	4	let	let	VERB
ma-246	372	5	a	a	DET
ma-246	372	6	,	,	PUNCT
ma-246	372	7	b	b	NOUN
ma-246	372	8	⊆	⊆	NUM
ma-246	372	9	x	x	X
ma-246	372	10	.	.	PUNCT
ma-246	373	1	since	since	SCONJ
ma-246	373	2	a	a	DET
ma-246	373	3	∩	∩	NOUN
ma-246	373	4	b	b	ADP
ma-246	373	5	⊆	⊆	NUM
ma-246	373	6	a	a	PRON
ma-246	373	7	and	and	CCONJ
ma-246	373	8	a	a	DET
ma-246	373	9	∩	∩	ADJ
ma-246	373	10	b	b	PROPN
ma-246	373	11	⊆	⊆	NUM
ma-246	373	12	b	b	NOUN
ma-246	373	13	,	,	PUNCT
ma-246	373	14	by	by	ADP
ma-246	373	15	theorem	theorem	NOUN
ma-246	373	16	6	6	NUM
ma-246	373	17	(	(	PUNCT
ma-246	373	18	i	i	NOUN
ma-246	373	19	)	)	PUNCT
ma-246	373	20	,	,	PUNCT
ma-246	373	21	cl∗η(a	cl∗η(a	PROPN
ma-246	373	22	∩	∩	ADJ
ma-246	373	23	b	b	X
ma-246	373	24	)	)	PUNCT
ma-246	373	25	⊆	⊆	NUM
ma-246	373	26	cl∗η(a)and	cl∗η(a)and	NOUN
ma-246	373	27	cl∗η(a	cl∗η(a	PROPN
ma-246	373	28	∩	∩	ADJ
ma-246	373	29	b	b	X
ma-246	373	30	)	)	PUNCT
ma-246	373	31	⊆	⊆	NUM
ma-246	373	32	cl∗η(b	cl∗η(b	NUM
ma-246	373	33	)	)	PUNCT
ma-246	373	34	.	.	PUNCT
ma-246	374	1	hence	hence	ADV
ma-246	374	2	,	,	PUNCT
ma-246	374	3	it	it	PRON
ma-246	374	4	implies	imply	VERB
ma-246	374	5	cl∗η(a	cl∗η(a	PROPN
ma-246	374	6	∩	∩	ADJ
ma-246	374	7	b	b	NOUN
ma-246	374	8	)	)	PUNCT
ma-246	374	9	⊆	⊆	NUM
ma-246	374	10	cl∗η(a	cl∗η(a	ADJ
ma-246	374	11	)	)	PUNCT
ma-246	374	12	∩	∩	NOUN
ma-246	374	13	cl∗η(b	cl∗η(b	NOUN
ma-246	374	14	)	)	PUNCT
ma-246	374	15	.	.	PUNCT
ma-246	375	1	(	(	PUNCT
ma-246	375	2	iii	iii	X
ma-246	375	3	)	)	PUNCT
ma-246	375	4	let	let	VERB
ma-246	375	5	a	a	PRON
ma-246	375	6	be	be	AUX
ma-246	375	7	an	an	DET
ma-246	375	8	η	η	ADJ
ma-246	375	9	-	-	ADJ
ma-246	375	10	closed	closed	ADJ
ma-246	375	11	set	set	NOUN
ma-246	375	12	,	,	PUNCT
ma-246	375	13	then	then	ADV
ma-246	375	14	a	a	DET
ma-246	375	15	=	=	X
ma-246	375	16	η	η	NOUN
ma-246	375	17	-	-	NOUN
ma-246	375	18	cl(a	cl(a	NUM
ma-246	375	19	)	)	PUNCT
ma-246	375	20	.	.	PUNCT
ma-246	376	1	now	now	ADV
ma-246	376	2	,	,	PUNCT
ma-246	376	3	suppose	suppose	VERB
ma-246	376	4	that	that	SCONJ
ma-246	376	5	x	x	SYM
ma-246	376	6	/∈	/∈	PUNCT
ma-246	376	7	a.	a.	NOUN
ma-246	377	1	it	it	PRON
ma-246	377	2	implies	imply	VERB
ma-246	377	3	that	that	SCONJ
ma-246	377	4	x	x	PROPN
ma-246	377	5	/∈	/∈	PUNCT
ma-246	377	6	η	η	PROPN
ma-246	377	7	-	-	NOUN
ma-246	377	8	cl(a	cl(a	NUM
ma-246	377	9	)	)	PUNCT
ma-246	377	10	,	,	PUNCT
ma-246	377	11	then	then	ADV
ma-246	377	12	x	x	PUNCT
ma-246	377	13	/∈	/∈	PUNCT
ma-246	377	14	⋂{k	⋂{k	VERB
ma-246	377	15	:	:	PUNCT
ma-246	378	1	k	k	X
ma-246	378	2	is	be	AUX
ma-246	378	3	η	η	NOUN
ma-246	378	4	-	-	ADJ
ma-246	378	5	closed	closed	ADJ
ma-246	378	6	and	and	CCONJ
ma-246	378	7	a	a	DET
ma-246	378	8	⊆	⊆	NUM
ma-246	378	9	k	k	NOUN
ma-246	378	10	}	}	PUNCT
ma-246	378	11	.	.	PUNCT
ma-246	379	1	it	it	PRON
ma-246	379	2	follows	follow	VERB
ma-246	379	3	that	that	SCONJ
ma-246	379	4	x	x	PROPN
ma-246	379	5	/∈	/∈	PUNCT
ma-246	380	1	k	k	PROPN
ma-246	380	2	for	for	ADP
ma-246	380	3	some	some	DET
ma-246	380	4	η	η	NOUN
ma-246	380	5	-	-	ADJ
ma-246	380	6	closed	closed	ADJ
ma-246	380	7	set	set	NOUN
ma-246	380	8	k	k	PROPN
ma-246	381	1	such	such	ADJ
ma-246	381	2	that	that	SCONJ
ma-246	381	3	a	a	DET
ma-246	381	4	⊆	⊆	NUM
ma-246	381	5	k.	k.	NOUN
ma-246	381	6	hence	hence	ADV
ma-246	381	7	,	,	PUNCT
ma-246	381	8	x	x	PUNCT
ma-246	381	9	∈	∈	PROPN
ma-246	381	10	kc	kc	PROPN
ma-246	381	11	for	for	ADP
ma-246	381	12	some	some	DET
ma-246	381	13	η	η	NOUN
ma-246	381	14	-	-	ADJ
ma-246	381	15	open	open	ADJ
ma-246	381	16	set	set	NOUN
ma-246	381	17	kc	kc	PROPN
ma-246	381	18	such	such	ADJ
ma-246	381	19	that	that	SCONJ
ma-246	381	20	a	a	DET
ma-246	381	21	∩	∩	ADJ
ma-246	381	22	kc	kc	NOUN
ma-246	381	23	=	=	PUNCT
ma-246	381	24	∅.	∅.	NOUN
ma-246	381	25	it	it	PRON
ma-246	381	26	implies	imply	VERB
ma-246	381	27	that	that	SCONJ
ma-246	381	28	there	there	PRON
ma-246	381	29	exists	exist	VERB
ma-246	381	30	kc	kc	PROPN
ma-246	381	31	∈	∈	PROPN
ma-246	381	32	η	η	PROPN
ma-246	381	33	-	-	PROPN
ma-246	381	34	o(x	o(x	ADJ
ma-246	381	35	)	)	PUNCT
ma-246	382	1	such	such	ADJ
ma-246	382	2	that	that	SCONJ
ma-246	382	3	a	a	DET
ma-246	382	4	∩	∩	ADJ
ma-246	382	5	kc	kc	NOUN
ma-246	382	6	=	=	NOUN
ma-246	382	7	∅	∅	NOUN
ma-246	382	8	,	,	PUNCT
ma-246	382	9	and	and	CCONJ
ma-246	382	10	bydefinition	bydefinition	NOUN
ma-246	382	11	of	of	ADP
ma-246	382	12	ideal	ideal	ADJ
ma-246	382	13	,	,	PUNCT
ma-246	382	14	∅	∅	NOUN
ma-246	382	15	∈	∈	PROPN
ma-246	382	16	i	i	PRON
ma-246	382	17	for	for	ADP
ma-246	382	18	any	any	DET
ma-246	382	19	ideal	ideal	NOUN
ma-246	382	20	i	i	PRON
ma-246	382	21	.	.	PUNCT
ma-246	383	1	hence	hence	ADV
ma-246	383	2	,	,	PUNCT
ma-246	383	3	a	a	DET
ma-246	383	4	∩	∩	ADJ
ma-246	383	5	kc	kc	PROPN
ma-246	383	6	∈	∈	PROPN
ma-246	383	7	i	i	PRON
ma-246	383	8	for	for	ADP
ma-246	383	9	some	some	DET
ma-246	383	10	kc	kc	PROPN
ma-246	383	11	∈	∈	PROPN
ma-246	383	12	η	η	PROPN
ma-246	383	13	-	-	PROPN
ma-246	383	14	o(x	o(x	PROPN
ma-246	383	15	)	)	PUNCT
ma-246	383	16	.	.	PUNCT
ma-246	383	17	thisshows	thisshow	VERB
ma-246	383	18	that	that	PRON
ma-246	383	19	x	x	PUNCT
ma-246	383	20	/∈	/∈	PUNCT
ma-246	383	21	a∗η	a∗η	PROPN
ma-246	383	22	,	,	PUNCT
ma-246	383	23	and	and	CCONJ
ma-246	383	24	hence	hence	ADV
ma-246	383	25	,	,	PUNCT
ma-246	383	26	a∗η	a∗η	PROPN
ma-246	383	27	⊆	⊆	NUM
ma-246	383	28	a.	a.	NOUN
ma-246	383	29	it	it	PRON
ma-246	383	30	follows	follow	VERB
ma-246	383	31	that	that	SCONJ
ma-246	383	32	cl∗η(a	cl∗η(a	PROPN
ma-246	383	33	)	)	PUNCT
ma-246	384	1	=	=	PUNCT
ma-246	384	2	a	a	DET
ma-246	384	3	∪	∪	X
ma-246	384	4	a∗η	a∗η	PROPN
ma-246	384	5	=	=	SYM
ma-246	384	6	a.	a.	NOUN
ma-246	384	7	note	note	NOUN
ma-246	384	8	that	that	SCONJ
ma-246	384	9	a	a	DET
ma-246	384	10	=	=	PUNCT
ma-246	384	11	η	η	NOUN
ma-246	384	12	-	-	NOUN
ma-246	384	13	cl(a	cl(a	NUM
ma-246	384	14	)	)	PUNCT
ma-246	384	15	.	.	PUNCT
ma-246	385	1	therefore	therefore	ADV
ma-246	385	2	,	,	PUNCT
ma-246	385	3	cl∗η(a	cl∗η(a	PROPN
ma-246	385	4	)	)	PUNCT
ma-246	385	5	=	=	PUNCT
ma-246	385	6	η	η	PROPN
ma-246	385	7	-	-	NOUN
ma-246	385	8	cl(a	cl(a	NUM
ma-246	385	9	)	)	PUNCT
ma-246	385	10	.	.	PUNCT
ma-246	386	1	(	(	PUNCT
ma-246	386	2	iv	iv	X
ma-246	386	3	)	)	PUNCT
ma-246	386	4	let	let	VERB
ma-246	386	5	a	a	DET
ma-246	386	6	∈	∈	NOUN
ma-246	387	1	i	i	PRON
ma-246	387	2	.	.	PUNCT
ma-246	388	1	then	then	ADV
ma-246	388	2	by	by	ADP
ma-246	388	3	definition	definition	NOUN
ma-246	388	4	2	2	NUM
ma-246	388	5	and	and	CCONJ
ma-246	388	6	theorem	theorem	VERB
ma-246	388	7	1	1	NUM
ma-246	388	8	(	(	PUNCT
ma-246	388	9	vii	vii	PROPN
ma-246	388	10	)	)	PUNCT
ma-246	388	11	,	,	PUNCT
ma-246	388	12	cl∗η(a	cl∗η(a	PROPN
ma-246	388	13	)	)	PUNCT
ma-246	388	14	=	=	PUNCT
ma-246	389	1	a	a	DET
ma-246	389	2	∪	∪	X
ma-246	389	3	a∗η	a∗η	PROPN
ma-246	389	4	=	=	PUNCT
ma-246	389	5	a	a	DET
ma-246	389	6	∪	∪	ADJ
ma-246	389	7	∅	∅	NOUN
ma-246	389	8	=	=	PUNCT
ma-246	389	9	a.consequently	a.consequently	ADV
ma-246	389	10	,	,	PUNCT
ma-246	389	11	cl∗η(a	cl∗η(a	PROPN
ma-246	389	12	)	)	PUNCT
ma-246	389	13	=	=	SYM
ma-246	389	14	a.	a.	NOUN
ma-246	389	15	(	(	PUNCT
ma-246	389	16	v	v	NOUN
ma-246	389	17	)	)	PUNCT
ma-246	389	18	let	let	VERB
ma-246	389	19	a	a	DET
ma-246	389	20	⊆	⊆	NUM
ma-246	389	21	x	x	SYM
ma-246	389	22	.	.	PUNCT
ma-246	390	1	then	then	ADV
ma-246	390	2	by	by	ADP
ma-246	390	3	definition	definition	NOUN
ma-246	390	4	2	2	NUM
ma-246	390	5	and	and	CCONJ
ma-246	390	6	theorem	theorem	VERB
ma-246	390	7	1	1	NUM
ma-246	390	8	(	(	PUNCT
ma-246	390	9	vi	vi	NOUN
ma-246	390	10	)	)	PUNCT
ma-246	390	11	,	,	PUNCT
ma-246	390	12	cl∗η(a∗η	cl∗η(a∗η	PROPN
ma-246	390	13	)	)	PUNCT
ma-246	390	14	=	=	PUNCT
ma-246	390	15	a∗η	a∗η	PROPN
ma-246	390	16	∪	∪	X
ma-246	390	17	(	(	PUNCT
ma-246	390	18	a∗η	a∗η	PROPN
ma-246	390	19	)	)	PUNCT
ma-246	390	20	∗	∗	PROPN
ma-246	390	21	η	η	PROPN
ma-246	390	22	=	=	PROPN
ma-246	390	23	a∗η	a∗η	PROPN
ma-246	390	24	.	.	PUNCT
ma-246	390	25	itfollows	itfollow	NOUN
ma-246	390	26	that	that	SCONJ
ma-246	390	27	,	,	PUNCT
ma-246	390	28	cl∗η(a∗η	cl∗η(a∗η	PROPN
ma-246	390	29	)	)	PUNCT
ma-246	390	30	=	=	PUNCT
ma-246	391	1	a∗η	a∗η	PROPN
ma-246	391	2	.	.	PUNCT
ma-246	392	1	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	392	2	eur	eur	PROPN
ma-246	392	3	.	.	PUNCT
ma-246	393	1	j.	j.	PROPN
ma-246	393	2	math	math	PROPN
ma-246	393	3	.	.	PUNCT
ma-246	394	1	anal	anal	PROPN
ma-246	394	2	.	.	PUNCT
ma-246	395	1	10.28924	10.28924	NUM
ma-246	395	2	/	/	SYM
ma-246	395	3	ada	ada	PROPN
ma-246	395	4	/	/	SYM
ma-246	395	5	ma.5.2	ma.5.2	PROPN
ma-246	395	6	10	10	NUM
ma-246	395	7	(	(	PUNCT
ma-246	395	8	vi	vi	NOUN
ma-246	395	9	)	)	PUNCT
ma-246	395	10	let	let	VERB
ma-246	395	11	a	a	DET
ma-246	395	12	⊆	⊆	NUM
ma-246	395	13	x	x	SYM
ma-246	395	14	.	.	PUNCT
ma-246	395	15	suppose	suppose	VERB
ma-246	395	16	that	that	SCONJ
ma-246	395	17	cl∗η(a	cl∗η(a	PROPN
ma-246	395	18	)	)	PUNCT
ma-246	395	19	6=	6=	ADP
ma-246	395	20	η	η	PROPN
ma-246	395	21	-	-	NOUN
ma-246	395	22	cl(a	cl(a	NUM
ma-246	395	23	)	)	PUNCT
ma-246	395	24	.	.	PUNCT
ma-246	396	1	let	let	VERB
ma-246	396	2	η	η	NOUN
ma-246	396	3	-	-	ADJ
ma-246	396	4	cl(a	cl(a	NUM
ma-246	396	5	)	)	PUNCT
ma-246	396	6	⊂	⊂	PROPN
ma-246	396	7	cl∗η(a	cl∗η(a	PROPN
ma-246	396	8	)	)	PUNCT
ma-246	396	9	.	.	PUNCT
ma-246	397	1	then	then	ADV
ma-246	397	2	there	there	ADV
ma-246	397	3	existsan	existsan	PROPN
ma-246	397	4	element	element	NOUN
ma-246	397	5	x	x	SYM
ma-246	397	6	∈	∈	PROPN
ma-246	397	7	cl∗η(a	cl∗η(a	PROPN
ma-246	397	8	)	)	PUNCT
ma-246	398	1	such	such	ADJ
ma-246	398	2	that	that	SCONJ
ma-246	398	3	x	x	PROPN
ma-246	398	4	/∈	/∈	PUNCT
ma-246	398	5	η	η	PROPN
ma-246	398	6	-	-	NOUN
ma-246	398	7	cl(a	cl(a	NUM
ma-246	398	8	)	)	PUNCT
ma-246	398	9	.	.	PUNCT
ma-246	399	1	note	note	VERB
ma-246	399	2	that	that	SCONJ
ma-246	399	3	since	since	SCONJ
ma-246	399	4	x	x	PROPN
ma-246	399	5	∈	∈	PROPN
ma-246	399	6	cl∗η(a	cl∗η(a	PROPN
ma-246	399	7	)	)	PUNCT
ma-246	399	8	,	,	PUNCT
ma-246	399	9	by	by	ADP
ma-246	399	10	definition2	definition2	NOUN
ma-246	399	11	,	,	PUNCT
ma-246	399	12	x	x	X
ma-246	399	13	∈	∈	PROPN
ma-246	399	14	a	a	DET
ma-246	399	15	∪	∪	NOUN
ma-246	399	16	a∗η	a∗η	PROPN
ma-246	399	17	implies	imply	VERB
ma-246	399	18	that	that	SCONJ
ma-246	399	19	x	x	SYM
ma-246	399	20	∈	∈	PROPN
ma-246	399	21	a	a	DET
ma-246	399	22	or	or	CCONJ
ma-246	399	23	x	x	SYM
ma-246	399	24	∈	∈	PROPN
ma-246	399	25	a∗η	a∗η	PROPN
ma-246	399	26	,	,	PUNCT
ma-246	399	27	or	or	CCONJ
ma-246	399	28	both	both	PRON
ma-246	399	29	.	.	PUNCT
ma-246	400	1	suppose	suppose	VERB
ma-246	400	2	x	x	SYM
ma-246	400	3	∈	∈	PROPN
ma-246	400	4	a∗η	a∗η	PROPN
ma-246	400	5	.	.	PUNCT
ma-246	401	1	then	then	ADV
ma-246	401	2	for	for	ADP
ma-246	401	3	every	every	DET
ma-246	401	4	u	u	PROPN
ma-246	401	5	∈	∈	PROPN
ma-246	401	6	η	η	PROPN
ma-246	401	7	-	-	PROPN
ma-246	401	8	o(x	o(x	PROPN
ma-246	401	9	)	)	PUNCT
ma-246	401	10	,	,	PUNCT
ma-246	401	11	a	a	DET
ma-246	401	12	∩	∩	ADJ
ma-246	401	13	u	u	NOUN
ma-246	401	14	/∈	/∈	PUNCT
ma-246	402	1	i	i	INTJ
ma-246	402	2	.	.	PUNCT
ma-246	403	1	now	now	ADV
ma-246	403	2	,	,	PUNCT
ma-246	403	3	since	since	SCONJ
ma-246	403	4	x	x	PROPN
ma-246	403	5	/∈	/∈	PROPN
ma-246	403	6	η	η	PROPN
ma-246	403	7	-	-	NOUN
ma-246	403	8	cl(a	cl(a	NUM
ma-246	403	9	)	)	PUNCT
ma-246	403	10	,	,	PUNCT
ma-246	403	11	x	x	PUNCT
ma-246	403	12	/∈	/∈	PUNCT
ma-246	403	13	⋂{k	⋂{k	VERB
ma-246	403	14	:	:	PUNCT
ma-246	404	1	k	k	X
ma-246	404	2	is	be	AUX
ma-246	404	3	η	η	NOUN
ma-246	404	4	-	-	ADJ
ma-246	404	5	closed	closed	ADJ
ma-246	404	6	and	and	CCONJ
ma-246	404	7	a	a	DET
ma-246	404	8	⊆	⊆	NUM
ma-246	404	9	k}.it	k}.it	NOUN
ma-246	404	10	follows	follow	VERB
ma-246	404	11	that	that	SCONJ
ma-246	404	12	x	x	PROPN
ma-246	404	13	/∈	/∈	PUNCT
ma-246	404	14	k	k	PROPN
ma-246	404	15	for	for	ADP
ma-246	404	16	some	some	DET
ma-246	404	17	η	η	NOUN
ma-246	404	18	-	-	ADJ
ma-246	404	19	closed	closed	ADJ
ma-246	404	20	set	set	NOUN
ma-246	404	21	k	k	PROPN
ma-246	404	22	such	such	ADJ
ma-246	404	23	that	that	SCONJ
ma-246	404	24	a	a	DET
ma-246	404	25	⊆	⊆	NUM
ma-246	404	26	k.	k.	NOUN
ma-246	404	27	hence	hence	ADV
ma-246	404	28	,	,	PUNCT
ma-246	404	29	x	x	PUNCT
ma-246	404	30	∈	∈	PROPN
ma-246	404	31	kc	kc	PROPN
ma-246	404	32	for	for	ADP
ma-246	404	33	some	some	DET
ma-246	404	34	η	η	NOUN
ma-246	404	35	-	-	ADJ
ma-246	404	36	open	open	ADJ
ma-246	404	37	set	set	NOUN
ma-246	404	38	kc	kc	PROPN
ma-246	404	39	such	such	ADJ
ma-246	404	40	that	that	SCONJ
ma-246	404	41	a	a	DET
ma-246	404	42	∩kc	∩kc	NOUN
ma-246	404	43	=	=	NOUN
ma-246	404	44	∅	∅	NOUN
ma-246	404	45	,	,	PUNCT
ma-246	404	46	and	and	CCONJ
ma-246	404	47	by	by	ADP
ma-246	404	48	definition	definition	NOUN
ma-246	404	49	of	of	ADP
ma-246	404	50	an	an	DET
ma-246	404	51	ideal	ideal	ADJ
ma-246	404	52	,	,	PUNCT
ma-246	404	53	∅	∅	NOUN
ma-246	404	54	∈	∈	PROPN
ma-246	404	55	i	i	PRON
ma-246	404	56	for	for	ADP
ma-246	404	57	any	any	DET
ma-246	404	58	ideal	ideal	NOUN
ma-246	404	59	i	i	PRON
ma-246	404	60	.it	.it	PUNCT
ma-246	404	61	implies	imply	VERB
ma-246	404	62	that	that	SCONJ
ma-246	404	63	there	there	PRON
ma-246	404	64	exists	exist	VERB
ma-246	404	65	kc	kc	PROPN
ma-246	404	66	∈	∈	PROPN
ma-246	404	67	η	η	PROPN
ma-246	404	68	-	-	PROPN
ma-246	404	69	o(x	o(x	ADJ
ma-246	404	70	)	)	PUNCT
ma-246	404	71	such	such	ADJ
ma-246	404	72	that	that	SCONJ
ma-246	404	73	a	a	DET
ma-246	404	74	∩	∩	ADJ
ma-246	404	75	kc	kc	PROPN
ma-246	404	76	∈	∈	PROPN
ma-246	404	77	i	i	PRON
ma-246	404	78	,	,	PUNCT
ma-246	404	79	and	and	CCONJ
ma-246	404	80	hence	hence	ADV
ma-246	404	81	,	,	PUNCT
ma-246	404	82	x	x	X
ma-246	404	83	/∈	/∈	PUNCT
ma-246	404	84	a∗η	a∗η	PROPN
ma-246	404	85	.	.	PUNCT
ma-246	405	1	also	also	ADV
ma-246	405	2	,	,	PUNCT
ma-246	405	3	note	note	VERB
ma-246	405	4	that	that	SCONJ
ma-246	405	5	since	since	SCONJ
ma-246	405	6	x	x	PROPN
ma-246	405	7	∈	∈	PROPN
ma-246	405	8	kc	kc	PROPN
ma-246	405	9	and	and	CCONJ
ma-246	405	10	a	a	DET
ma-246	405	11	∩	∩	ADJ
ma-246	405	12	kc	kc	NOUN
ma-246	405	13	=	=	VERB
ma-246	405	14	∅	∅	NOUN
ma-246	405	15	,	,	PUNCT
ma-246	405	16	x	x	PUNCT
ma-246	405	17	/∈	/∈	PUNCT
ma-246	405	18	a.	a.	NOUN
ma-246	406	1	this	this	PRON
ma-246	406	2	shows	show	VERB
ma-246	406	3	that	that	SCONJ
ma-246	406	4	x	x	SYM
ma-246	406	5	/∈	/∈	PUNCT
ma-246	406	6	a	a	DET
ma-246	406	7	and	and	CCONJ
ma-246	406	8	x	x	NOUN
ma-246	406	9	/∈	/∈	PUNCT
ma-246	406	10	a∗η	a∗η	PROPN
ma-246	406	11	,	,	PUNCT
ma-246	406	12	acontradiction	acontradiction	NOUN
ma-246	406	13	.	.	PUNCT
ma-246	407	1	(	(	PUNCT
ma-246	407	2	vii	vii	PROPN
ma-246	407	3	)	)	PUNCT
ma-246	407	4	let	let	VERB
ma-246	407	5	a	a	DET
ma-246	407	6	⊆	⊆	NUM
ma-246	407	7	x	x	SYM
ma-246	407	8	.	.	PUNCT
ma-246	408	1	then	then	ADV
ma-246	408	2	by	by	ADP
ma-246	408	3	definition	definition	NOUN
ma-246	408	4	2	2	NUM
ma-246	408	5	and	and	CCONJ
ma-246	408	6	theorem	theorem	VERB
ma-246	408	7	1	1	NUM
ma-246	408	8	(	(	PUNCT
ma-246	408	9	iii	iii	NOUN
ma-246	408	10	)	)	PUNCT
ma-246	408	11	,	,	PUNCT
ma-246	408	12	(	(	PUNCT
ma-246	408	13	cl∗η(a	cl∗η(a	PROPN
ma-246	408	14	)	)	PUNCT
ma-246	408	15	)	)	PUNCT
ma-246	408	16	∗	∗	NOUN
ma-246	408	17	η	η	X
ma-246	408	18	=	=	PROPN
ma-246	408	19	(	(	PUNCT
ma-246	408	20	a∪a∗η	a∪a∗η	ADV
ma-246	408	21	)	)	PUNCT
ma-246	408	22	∗	∗	NOUN
ma-246	408	23	η	η	PROPN
ma-246	408	24	⊇	⊇	PROPN
ma-246	408	25	a∗η∪	a∗η∪	PROPN
ma-246	408	26	(	(	PUNCT
ma-246	408	27	a∗η	a∗η	PROPN
ma-246	408	28	)	)	PUNCT
ma-246	408	29	∗	∗	PROPN
ma-246	408	30	η	η	PROPN
ma-246	408	31	.note	.note	X
ma-246	408	32	that	that	SCONJ
ma-246	408	33	by	by	ADP
ma-246	408	34	theorem	theorem	NOUN
ma-246	408	35	1	1	NUM
ma-246	408	36	(	(	PUNCT
ma-246	408	37	vi	vi	NOUN
ma-246	408	38	)	)	PUNCT
ma-246	408	39	,	,	PUNCT
ma-246	408	40	(	(	PUNCT
ma-246	408	41	a∗η)∗η	a∗η)∗η	NOUN
ma-246	408	42	⊆	⊆	NUM
ma-246	408	43	a∗η	a∗η	PROPN
ma-246	408	44	,	,	PUNCT
ma-246	408	45	then	then	ADV
ma-246	408	46	a∗η	a∗η	PUNCT
ma-246	408	47	∪	∪	X
ma-246	408	48	(	(	PUNCT
ma-246	408	49	a∗η)∗η	a∗η)∗η	NOUN
ma-246	408	50	=	=	PUNCT
ma-246	408	51	a∗η	a∗η	PROPN
ma-246	408	52	.	.	PUNCT
ma-246	409	1	it	it	PRON
ma-246	409	2	implies	imply	VERB
ma-246	409	3	that	that	SCONJ
ma-246	409	4	a∗η	a∗η	PROPN
ma-246	409	5	⊆	⊆	SYM
ma-246	409	6	(	(	PUNCT
ma-246	409	7	cl∗η(a	cl∗η(a	PROPN
ma-246	409	8	)	)	PUNCT
ma-246	409	9	)	)	PUNCT
ma-246	409	10	∗	∗	PROPN
ma-246	409	11	η	η	PROPN
ma-246	409	12	.	.	PUNCT
ma-246	410	1	now	now	ADV
ma-246	410	2	,	,	PUNCT
ma-246	410	3	let	let	VERB
ma-246	410	4	x	x	X
ma-246	410	5	∈	∈	PROPN
ma-246	410	6	(	(	PUNCT
ma-246	410	7	cl∗η(a	cl∗η(a	PROPN
ma-246	410	8	)	)	PUNCT
ma-246	410	9	)	)	PUNCT
ma-246	410	10	∗	∗	PROPN
ma-246	410	11	η	η	PROPN
ma-246	410	12	.	.	PUNCT
ma-246	411	1	then	then	ADV
ma-246	411	2	for	for	ADP
ma-246	411	3	every	every	DET
ma-246	411	4	u	u	PROPN
ma-246	411	5	∈	∈	PROPN
ma-246	411	6	η	η	PROPN
ma-246	411	7	-	-	PROPN
ma-246	411	8	o(x	o(x	PROPN
ma-246	411	9	)	)	PUNCT
ma-246	411	10	,	,	PUNCT
ma-246	411	11	cl∗η(a	cl∗η(a	PROPN
ma-246	411	12	)	)	PUNCT
ma-246	411	13	∩	∩	NOUN
ma-246	411	14	u	u	NOUN
ma-246	411	15	/∈	/∈	PUNCT
ma-246	412	1	i	i	INTJ
ma-246	412	2	.	.	PUNCT
ma-246	413	1	now	now	ADV
ma-246	413	2	,	,	PUNCT
ma-246	413	3	bydefinition	bydefinition	NOUN
ma-246	413	4	2	2	NUM
ma-246	413	5	,	,	PUNCT
ma-246	413	6	cl∗η(a)∩u	cl∗η(a)∩u	NOUN
ma-246	413	7	=	=	PUNCT
ma-246	413	8	(	(	PUNCT
ma-246	413	9	a∪a∗η)∩u	a∪a∗η)∩u	PROPN
ma-246	413	10	/∈	/∈	INTJ
ma-246	414	1	i	i	PRON
ma-246	414	2	=	=	SYM
ma-246	414	3	(	(	PUNCT
ma-246	414	4	a∩u)∪(a∗η∩u	a∩u)∪(a∗η∩u	PROPN
ma-246	414	5	)	)	PUNCT
ma-246	414	6	/∈	/∈	PUNCT
ma-246	415	1	i	i	PRON
ma-246	415	2	.	.	PUNCT
ma-246	416	1	it	it	PRON
ma-246	416	2	implies	imply	VERB
ma-246	416	3	that	that	PRON
ma-246	416	4	a∩u	a∩u	PROPN
ma-246	416	5	/∈	/∈	PUNCT
ma-246	416	6	ior	ior	PROPN
ma-246	416	7	a∗η∩u	a∗η∩u	PROPN
ma-246	416	8	/∈	/∈	PUNCT
ma-246	417	1	i	i	PRON
ma-246	417	2	,	,	PUNCT
ma-246	417	3	or	or	CCONJ
ma-246	417	4	both	both	PRON
ma-246	417	5	,	,	PUNCT
ma-246	417	6	and	and	CCONJ
ma-246	417	7	so	so	ADV
ma-246	417	8	,	,	PUNCT
ma-246	417	9	x	x	SYM
ma-246	417	10	∈	∈	PROPN
ma-246	417	11	a∗η	a∗η	PROPN
ma-246	417	12	or	or	CCONJ
ma-246	417	13	x	x	SYM
ma-246	417	14	∈	∈	PROPN
ma-246	417	15	(	(	PUNCT
ma-246	417	16	a∗η)∗η	a∗η)∗η	NOUN
ma-246	417	17	,	,	PUNCT
ma-246	417	18	or	or	CCONJ
ma-246	417	19	both	both	PRON
ma-246	417	20	.	.	PUNCT
ma-246	418	1	it	it	PRON
ma-246	418	2	follows	follow	VERB
ma-246	418	3	that	that	SCONJ
ma-246	418	4	x	x	PUNCT
ma-246	418	5	∈	∈	NOUN
ma-246	418	6	a∗η∪(a∗η)∗η	a∗η∪(a∗η)∗η	NOUN
ma-246	418	7	.note	.note	PUNCT
ma-246	418	8	that	that	SCONJ
ma-246	418	9	a∗η	a∗η	PROPN
ma-246	418	10	∪	∪	ADJ
ma-246	418	11	(	(	PUNCT
ma-246	418	12	a∗η)∗η	a∗η)∗η	NOUN
ma-246	418	13	=	=	SYM
ma-246	418	14	a∗η	a∗η	PROPN
ma-246	418	15	,	,	PUNCT
ma-246	418	16	and	and	CCONJ
ma-246	418	17	so	so	ADV
ma-246	418	18	,	,	PUNCT
ma-246	418	19	x	x	SYM
ma-246	418	20	∈	∈	PROPN
ma-246	418	21	a∗η	a∗η	PROPN
ma-246	418	22	.	.	PUNCT
ma-246	419	1	consequently	consequently	ADV
ma-246	419	2	,	,	PUNCT
ma-246	419	3	(	(	PUNCT
ma-246	419	4	cl∗η(a	cl∗η(a	PROPN
ma-246	419	5	)	)	PUNCT
ma-246	419	6	)	)	PUNCT
ma-246	419	7	∗	∗	PROPN
ma-246	419	8	η	η	PROPN
ma-246	419	9	⊆	⊆	NUM
ma-246	419	10	a∗η	a∗η	PROPN
ma-246	419	11	.	.	PUNCT
ma-246	420	1	thus	thus	ADV
ma-246	420	2	,	,	PUNCT
ma-246	420	3	(	(	PUNCT
ma-246	420	4	cl∗η(a	cl∗η(a	PROPN
ma-246	420	5	)	)	PUNCT
ma-246	420	6	)	)	PUNCT
ma-246	421	1	∗	∗	PROPN
ma-246	421	2	η	η	PROPN
ma-246	421	3	=	=	PROPN
ma-246	421	4	a∗η	a∗η	PROPN
ma-246	421	5	.	.	PUNCT
ma-246	422	1	�	�	PROPN
ma-246	422	2	theorem	theorem	VERB
ma-246	422	3	7	7	NUM
ma-246	422	4	.	.	PUNCT
ma-246	423	1	let	let	VERB
ma-246	423	2	(	(	PUNCT
ma-246	423	3	x	x	X
ma-246	423	4	,	,	PUNCT
ma-246	423	5	τ	τ	PROPN
ma-246	423	6	,	,	PUNCT
ma-246	423	7	i	i	PRON
ma-246	423	8	)	)	PUNCT
ma-246	423	9	be	be	VERB
ma-246	423	10	an	an	DET
ma-246	423	11	ideal	ideal	ADJ
ma-246	423	12	topological	topological	ADJ
ma-246	423	13	space	space	NOUN
ma-246	423	14	and	and	CCONJ
ma-246	423	15	a	a	DET
ma-246	423	16	,	,	PUNCT
ma-246	423	17	b	b	NOUN
ma-246	423	18	⊆	⊆	NUM
ma-246	423	19	x	x	X
ma-246	423	20	.	.	PUNCT
ma-246	424	1	then	then	ADV
ma-246	424	2	the	the	DET
ma-246	424	3	following	follow	VERB
ma-246	424	4	properties	property	NOUN
ma-246	424	5	hold	hold	VERB
ma-246	424	6	:	:	PUNCT
ma-246	424	7	(	(	PUNCT
ma-246	424	8	i	i	NOUN
ma-246	424	9	)	)	PUNCT
ma-246	424	10	a	a	DET
ma-246	424	11	⊆	⊆	NUM
ma-246	424	12	cl∗η(a	cl∗η(a	NUM
ma-246	424	13	)	)	PUNCT
ma-246	424	14	and	and	CCONJ
ma-246	424	15	a∗η	a∗η	PROPN
ma-246	424	16	⊆	⊆	NUM
ma-246	424	17	cl∗η(a	cl∗η(a	PROPN
ma-246	424	18	)	)	PUNCT
ma-246	424	19	;	;	PUNCT
ma-246	424	20	(	(	PUNCT
ma-246	424	21	ii	ii	NOUN
ma-246	424	22	)	)	PUNCT
ma-246	424	23	cl∗η(∅	cl∗η(∅	NOUN
ma-246	424	24	)	)	PUNCT
ma-246	424	25	=	=	SYM
ma-246	424	26	∅	∅	NOUN
ma-246	424	27	and	and	CCONJ
ma-246	424	28	cl∗η(x	cl∗η(x	NOUN
ma-246	424	29	)	)	PUNCT
ma-246	424	30	=	=	SYM
ma-246	424	31	x	x	X
ma-246	424	32	;	;	PUNCT
ma-246	424	33	(	(	PUNCT
ma-246	424	34	iii	iii	NOUN
ma-246	424	35	)	)	PUNCT
ma-246	424	36	cl∗η(a	cl∗η(a	PROPN
ma-246	424	37	)	)	PUNCT
ma-246	424	38	∪	∪	ADP
ma-246	424	39	cl∗η(b	cl∗η(b	NOUN
ma-246	424	40	)	)	PUNCT
ma-246	424	41	⊆	⊆	NUM
ma-246	424	42	cl∗η(a	cl∗η(a	PROPN
ma-246	424	43	∪	∪	ADJ
ma-246	424	44	b	b	NOUN
ma-246	424	45	)	)	PUNCT
ma-246	424	46	;	;	PUNCT
ma-246	424	47	and	and	CCONJ
ma-246	424	48	(	(	PUNCT
ma-246	424	49	iv	iv	X
ma-246	424	50	)	)	PUNCT
ma-246	424	51	(	(	PUNCT
ma-246	424	52	cl∗η(a	cl∗η(a	PROPN
ma-246	424	53	)	)	PUNCT
ma-246	424	54	)	)	PUNCT
ma-246	424	55	∗	∗	PROPN
ma-246	424	56	η	η	PROPN
ma-246	424	57	⊆	⊆	NUM
ma-246	424	58	cl∗η(a	cl∗η(a	PROPN
ma-246	424	59	)	)	PUNCT
ma-246	424	60	=	=	SYM
ma-246	425	1	cl∗η	cl∗η	PROPN
ma-246	425	2	(	(	PUNCT
ma-246	425	3	cl∗η(a	cl∗η(a	PROPN
ma-246	425	4	)	)	PUNCT
ma-246	425	5	)	)	PUNCT
ma-246	425	6	.	.	PUNCT
ma-246	426	1	proof	proof	NOUN
ma-246	426	2	.	.	PUNCT
ma-246	427	1	(	(	PUNCT
ma-246	427	2	i	i	NOUN
ma-246	427	3	)	)	PUNCT
ma-246	427	4	let	let	VERB
ma-246	427	5	a	a	DET
ma-246	427	6	⊆	⊆	NUM
ma-246	427	7	x	x	SYM
ma-246	427	8	.	.	PUNCT
ma-246	428	1	note	note	VERB
ma-246	428	2	that	that	SCONJ
ma-246	428	3	a	a	DET
ma-246	428	4	⊆	⊆	NUM
ma-246	428	5	a	a	DET
ma-246	428	6	∪	∪	NOUN
ma-246	428	7	a∗η	a∗η	PROPN
ma-246	428	8	.	.	PUNCT
ma-246	429	1	then	then	ADV
ma-246	429	2	by	by	ADP
ma-246	429	3	definition	definition	NOUN
ma-246	429	4	2	2	NUM
ma-246	429	5	,	,	PUNCT
ma-246	429	6	a	a	DET
ma-246	429	7	⊆	⊆	NUM
ma-246	429	8	cl∗η(a	cl∗η(a	NOUN
ma-246	429	9	)	)	PUNCT
ma-246	429	10	.	.	PUNCT
ma-246	430	1	next	next	ADV
ma-246	430	2	,	,	PUNCT
ma-246	430	3	note	note	VERB
ma-246	430	4	that	that	SCONJ
ma-246	430	5	a∗η	a∗η	PROPN
ma-246	430	6	⊆	⊆	NUM
ma-246	430	7	a	a	DET
ma-246	430	8	∪	∪	ADJ
ma-246	430	9	a∗η	a∗η	PUNCT
ma-246	430	10	,	,	PUNCT
ma-246	430	11	by	by	ADP
ma-246	430	12	definition	definition	NOUN
ma-246	430	13	2	2	NUM
ma-246	430	14	again	again	ADV
ma-246	430	15	,	,	PUNCT
ma-246	430	16	it	it	PRON
ma-246	430	17	implies	imply	VERB
ma-246	430	18	that	that	SCONJ
ma-246	430	19	a∗η	a∗η	PROPN
ma-246	430	20	⊆	⊆	NUM
ma-246	430	21	cl∗η(a	cl∗η(a	NUM
ma-246	430	22	)	)	PUNCT
ma-246	430	23	.	.	PUNCT
ma-246	431	1	(	(	PUNCT
ma-246	431	2	ii	ii	NOUN
ma-246	431	3	)	)	PUNCT
ma-246	431	4	by	by	ADP
ma-246	431	5	definition	definition	NOUN
ma-246	431	6	2	2	NUM
ma-246	431	7	and	and	CCONJ
ma-246	431	8	theorem	theorem	VERB
ma-246	431	9	1	1	NUM
ma-246	431	10	(	(	PUNCT
ma-246	431	11	iv	iv	NUM
ma-246	431	12	)	)	PUNCT
ma-246	431	13	,	,	PUNCT
ma-246	431	14	cl∗η(∅	cl∗η(∅	NOUN
ma-246	431	15	)	)	PUNCT
ma-246	431	16	=	=	NOUN
ma-246	431	17	∅	∅	NOUN
ma-246	431	18	∪	∪	X
ma-246	431	19	(	(	PUNCT
ma-246	431	20	∅)∗η	∅)∗η	NOUN
ma-246	431	21	=	=	SYM
ma-246	431	22	∅	∅	NOUN
ma-246	431	23	∪∅	∪∅	X
ma-246	431	24	=	=	PUNCT
ma-246	431	25	∅.	∅.	ADP
ma-246	431	26	next	next	ADV
ma-246	431	27	,	,	PUNCT
ma-246	431	28	note	note	VERB
ma-246	431	29	that	that	SCONJ
ma-246	431	30	x	x	PRON
ma-246	431	31	is	be	AUX
ma-246	431	32	auniversal	auniversal	ADJ
ma-246	431	33	set	set	NOUN
ma-246	431	34	,	,	PUNCT
ma-246	431	35	then	then	ADV
ma-246	431	36	(	(	PUNCT
ma-246	431	37	x)∗η	x)∗η	PROPN
ma-246	431	38	⊆	⊆	NUM
ma-246	431	39	x	x	X
ma-246	431	40	.	.	PUNCT
ma-246	432	1	hence	hence	ADV
ma-246	432	2	,	,	PUNCT
ma-246	432	3	by	by	ADP
ma-246	432	4	definition	definition	NOUN
ma-246	432	5	2	2	NUM
ma-246	432	6	,	,	PUNCT
ma-246	432	7	cl∗η(x	cl∗η(x	NOUN
ma-246	432	8	)	)	PUNCT
ma-246	432	9	=	=	SYM
ma-246	432	10	x	x	SYM
ma-246	432	11	∪	∪	X
ma-246	432	12	(	(	PUNCT
ma-246	432	13	x)∗η	x)∗η	NOUN
ma-246	432	14	=	=	PUNCT
ma-246	432	15	x	x	X
ma-246	432	16	.	.	PUNCT
ma-246	433	1	(	(	PUNCT
ma-246	433	2	iii	iii	X
ma-246	433	3	)	)	PUNCT
ma-246	433	4	let	let	VERB
ma-246	433	5	a	a	DET
ma-246	433	6	,	,	PUNCT
ma-246	433	7	b	b	NOUN
ma-246	433	8	⊆	⊆	NUM
ma-246	433	9	x	x	X
ma-246	433	10	.	.	PUNCT
ma-246	434	1	by	by	ADP
ma-246	434	2	definition	definition	NOUN
ma-246	434	3	2	2	NUM
ma-246	434	4	and	and	CCONJ
ma-246	434	5	thoerem	thoerem	ADJ
ma-246	434	6	1	1	NUM
ma-246	434	7	(	(	PUNCT
ma-246	434	8	iii	iii	NOUN
ma-246	434	9	)	)	PUNCT
ma-246	434	10	,	,	PUNCT
ma-246	434	11	cl∗η(a	cl∗η(a	PROPN
ma-246	434	12	∪	∪	ADP
ma-246	434	13	b	b	NOUN
ma-246	434	14	)	)	PUNCT
ma-246	434	15	=	=	SYM
ma-246	434	16	(	(	PUNCT
ma-246	434	17	a	a	DET
ma-246	434	18	∪	∪	ADJ
ma-246	434	19	b	b	NOUN
ma-246	434	20	)	)	PUNCT
ma-246	434	21	∪	∪	NOUN
ma-246	434	22	(	(	PUNCT
ma-246	434	23	a	a	DET
ma-246	434	24	∪	∪	ADJ
ma-246	434	25	b)∗η	b)∗η	ADJ
ma-246	434	26	⊇	⊇	NOUN
ma-246	434	27	(	(	PUNCT
ma-246	434	28	a	a	DET
ma-246	434	29	∪	∪	ADJ
ma-246	434	30	b	b	NOUN
ma-246	434	31	)	)	PUNCT
ma-246	434	32	∪	∪	NOUN
ma-246	434	33	(	(	PUNCT
ma-246	434	34	a∗η	a∗η	ADV
ma-246	434	35	∪	∪	X
ma-246	434	36	b∗η	b∗η	PROPN
ma-246	434	37	)	)	PUNCT
ma-246	434	38	=	=	PUNCT
ma-246	434	39	(	(	PUNCT
ma-246	434	40	a	a	DET
ma-246	434	41	∪	∪	ADJ
ma-246	434	42	a∗η	a∗η	NUM
ma-246	434	43	)	)	PUNCT
ma-246	434	44	∪	∪	NOUN
ma-246	434	45	(	(	PUNCT
ma-246	434	46	b	b	NOUN
ma-246	434	47	∪	∪	X
ma-246	434	48	b∗η	b∗η	PROPN
ma-246	434	49	)	)	PUNCT
ma-246	434	50	=	=	SYM
ma-246	434	51	cl∗η(a	cl∗η(a	PROPN
ma-246	434	52	)	)	PUNCT
ma-246	434	53	∪	∪	ADP
ma-246	434	54	cl∗η(b	cl∗η(b	NOUN
ma-246	434	55	)	)	PUNCT
ma-246	434	56	.	.	PUNCT
ma-246	435	1	this	this	PRON
ma-246	435	2	shows	show	VERB
ma-246	435	3	that	that	SCONJ
ma-246	435	4	cl∗η(a	cl∗η(a	PROPN
ma-246	435	5	)	)	PUNCT
ma-246	435	6	∪	∪	ADP
ma-246	435	7	cl∗η(b	cl∗η(b	NOUN
ma-246	435	8	)	)	PUNCT
ma-246	435	9	⊆	⊆	NUM
ma-246	435	10	cl∗η(a	cl∗η(a	PROPN
ma-246	435	11	∪	∪	ADJ
ma-246	435	12	b	b	NOUN
ma-246	435	13	)	)	PUNCT
ma-246	435	14	.	.	PUNCT
ma-246	436	1	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	436	2	eur	eur	PROPN
ma-246	436	3	.	.	PUNCT
ma-246	437	1	j.	j.	PROPN
ma-246	437	2	math	math	PROPN
ma-246	437	3	.	.	PUNCT
ma-246	438	1	anal	anal	PROPN
ma-246	438	2	.	.	PUNCT
ma-246	439	1	10.28924	10.28924	NUM
ma-246	439	2	/	/	SYM
ma-246	439	3	ada	ada	PROPN
ma-246	439	4	/	/	SYM
ma-246	439	5	ma.5.2	ma.5.2	PROPN
ma-246	439	6	11	11	NUM
ma-246	439	7	(	(	PUNCT
ma-246	439	8	iv	iv	X
ma-246	439	9	)	)	PUNCT
ma-246	439	10	let	let	VERB
ma-246	439	11	a	a	DET
ma-246	439	12	⊆	⊆	NUM
ma-246	439	13	x	x	SYM
ma-246	439	14	.	.	PUNCT
ma-246	440	1	note	note	VERB
ma-246	440	2	that	that	SCONJ
ma-246	440	3	by	by	ADP
ma-246	440	4	theorem	theorem	NOUN
ma-246	440	5	6	6	NUM
ma-246	440	6	(	(	PUNCT
ma-246	440	7	vii	vii	PROPN
ma-246	440	8	)	)	PUNCT
ma-246	440	9	,	,	PUNCT
ma-246	440	10	(	(	PUNCT
ma-246	440	11	cl∗η(a	cl∗η(a	PROPN
ma-246	440	12	)	)	PUNCT
ma-246	440	13	)	)	PUNCT
ma-246	440	14	∗	∗	PROPN
ma-246	440	15	η	η	PROPN
ma-246	440	16	=	=	PUNCT
ma-246	440	17	a∗η	a∗η	PROPN
ma-246	440	18	,	,	PUNCT
ma-246	440	19	and	and	CCONJ
ma-246	440	20	by	by	ADP
ma-246	440	21	theorem	theorem	NOUN
ma-246	440	22	7	7	NUM
ma-246	440	23	(	(	PUNCT
ma-246	440	24	i	i	NOUN
ma-246	440	25	)	)	PUNCT
ma-246	440	26	,	,	PUNCT
ma-246	440	27	a∗η	a∗η	PROPN
ma-246	440	28	⊆	⊆	NUM
ma-246	440	29	cl∗η(a).hence	cl∗η(a).hence	NOUN
ma-246	440	30	,	,	PUNCT
ma-246	440	31	it	it	PRON
ma-246	440	32	shows	show	VERB
ma-246	440	33	that	that	SCONJ
ma-246	440	34	(	(	PUNCT
ma-246	440	35	cl∗η(a	cl∗η(a	PROPN
ma-246	440	36	)	)	PUNCT
ma-246	440	37	)	)	PUNCT
ma-246	440	38	∗	∗	PROPN
ma-246	440	39	η	η	PROPN
ma-246	440	40	⊆	⊆	NUM
ma-246	440	41	cl∗η(a	cl∗η(a	PROPN
ma-246	440	42	)	)	PUNCT
ma-246	440	43	.	.	PUNCT
ma-246	441	1	next	next	ADV
ma-246	441	2	,	,	PUNCT
ma-246	441	3	by	by	ADP
ma-246	441	4	definition	definition	NOUN
ma-246	441	5	2	2	NUM
ma-246	441	6	,	,	PUNCT
ma-246	441	7	cl∗η(cl∗η(a	cl∗η(cl∗η(a	ADJ
ma-246	441	8	)	)	PUNCT
ma-246	441	9	)	)	PUNCT
ma-246	442	1	=	=	SYM
ma-246	442	2	cl∗η(a	cl∗η(a	PROPN
ma-246	442	3	)	)	PUNCT
ma-246	442	4	∪	∪	PROPN
ma-246	442	5	(	(	PUNCT
ma-246	442	6	cl∗η(a	cl∗η(a	PROPN
ma-246	442	7	)	)	PUNCT
ma-246	442	8	)	)	PUNCT
ma-246	442	9	∗	∗	PROPN
ma-246	442	10	η	η	PROPN
ma-246	442	11	.	.	PUNCT
ma-246	443	1	note	note	VERB
ma-246	443	2	that	that	SCONJ
ma-246	443	3	since	since	SCONJ
ma-246	443	4	(	(	PUNCT
ma-246	443	5	cl∗η(a	cl∗η(a	PROPN
ma-246	443	6	)	)	PUNCT
ma-246	443	7	)	)	PUNCT
ma-246	443	8	∗	∗	PROPN
ma-246	443	9	η	η	PROPN
ma-246	443	10	⊆	⊆	NUM
ma-246	443	11	cl∗η(a	cl∗η(a	PROPN
ma-246	443	12	)	)	PUNCT
ma-246	443	13	,	,	PUNCT
ma-246	443	14	cl∗η	cl∗η	PROPN
ma-246	443	15	(	(	PUNCT
ma-246	443	16	cl∗η(a	cl∗η(a	PROPN
ma-246	443	17	)	)	PUNCT
ma-246	443	18	)	)	PUNCT
ma-246	444	1	=	=	SYM
ma-246	444	2	cl∗η(a	cl∗η(a	PROPN
ma-246	444	3	)	)	PUNCT
ma-246	444	4	∪	∪	NOUN
ma-246	444	5	(	(	PUNCT
ma-246	444	6	cl∗η(a	cl∗η(a	PROPN
ma-246	444	7	)	)	PUNCT
ma-246	444	8	)	)	PUNCT
ma-246	444	9	∗	∗	PROPN
ma-246	444	10	η	η	NOUN
ma-246	444	11	=	=	PROPN
ma-246	444	12	cl∗η(a	cl∗η(a	PROPN
ma-246	444	13	)	)	PUNCT
ma-246	444	14	.	.	PUNCT
ma-246	445	1	it	it	PRON
ma-246	445	2	follows	follow	VERB
ma-246	445	3	that	that	SCONJ
ma-246	445	4	,	,	PUNCT
ma-246	445	5	cl∗η(a	cl∗η(a	PROPN
ma-246	445	6	)	)	PUNCT
ma-246	445	7	=	=	SYM
ma-246	446	1	cl∗η	cl∗η	PROPN
ma-246	446	2	(	(	PUNCT
ma-246	446	3	cl∗η(a	cl∗η(a	PROPN
ma-246	446	4	)	)	PUNCT
ma-246	446	5	)	)	PUNCT
ma-246	446	6	.	.	PUNCT
ma-246	447	1	�	�	PROPN
ma-246	447	2	remark	remark	VERB
ma-246	447	3	3	3	NUM
ma-246	447	4	.	.	PUNCT
ma-246	448	1	the	the	DET
ma-246	448	2	reverse	reverse	ADJ
ma-246	448	3	inclusion	inclusion	NOUN
ma-246	448	4	of	of	ADP
ma-246	448	5	theorem	theorem	ADJ
ma-246	448	6	7	7	NUM
ma-246	448	7	(	(	PUNCT
ma-246	448	8	iii	iii	NOUN
ma-246	448	9	)	)	PUNCT
ma-246	448	10	need	need	AUX
ma-246	448	11	not	not	PART
ma-246	448	12	be	be	AUX
ma-246	448	13	true	true	ADJ
ma-246	448	14	in	in	ADP
ma-246	448	15	general	general	ADJ
ma-246	448	16	as	as	SCONJ
ma-246	448	17	shown	show	VERB
ma-246	448	18	from	from	ADP
ma-246	448	19	the	the	DET
ma-246	448	20	following	follow	VERB
ma-246	448	21	example	example	NOUN
ma-246	448	22	.	.	PUNCT
ma-246	449	1	example	example	NOUN
ma-246	450	1	4	4	NUM
ma-246	450	2	.	.	PUNCT
ma-246	451	1	let	let	VERB
ma-246	451	2	(	(	PUNCT
ma-246	451	3	x	x	X
ma-246	451	4	,	,	PUNCT
ma-246	451	5	τ	τ	PROPN
ma-246	451	6	,	,	PUNCT
ma-246	451	7	i	i	PRON
ma-246	451	8	)	)	PUNCT
ma-246	451	9	be	be	VERB
ma-246	451	10	an	an	DET
ma-246	451	11	ideal	ideal	ADJ
ma-246	451	12	topological	topological	ADJ
ma-246	451	13	space	space	NOUN
ma-246	451	14	where	where	SCONJ
ma-246	451	15	x	x	X
ma-246	451	16	=	=	PRON
ma-246	451	17	{	{	PUNCT
ma-246	451	18	a	a	PRON
ma-246	451	19	,	,	PUNCT
ma-246	451	20	b	b	NOUN
ma-246	451	21	,	,	PUNCT
ma-246	451	22	c	c	NOUN
ma-246	451	23	,	,	PUNCT
ma-246	451	24	d	d	NOUN
ma-246	451	25	}	}	PUNCT
ma-246	451	26	,	,	PUNCT
ma-246	451	27	τ	τ	X
ma-246	451	28	=	=	PUNCT
ma-246	451	29	{	{	PUNCT
ma-246	451	30	∅	∅	NOUN
ma-246	451	31	,	,	PUNCT
ma-246	451	32	x	x	X
ma-246	451	33	,	,	PUNCT
ma-246	451	34	{	{	PUNCT
ma-246	451	35	c	c	NOUN
ma-246	451	36	}	}	PUNCT
ma-246	451	37	,	,	PUNCT
ma-246	451	38	{	{	PUNCT
ma-246	451	39	d	d	X
ma-246	451	40	}	}	PUNCT
ma-246	451	41	,	,	PUNCT
ma-246	451	42	{	{	PUNCT
ma-246	451	43	c	c	X
ma-246	451	44	,	,	PUNCT
ma-246	451	45	d	d	NOUN
ma-246	451	46	}	}	PUNCT
ma-246	451	47	}	}	PUNCT
ma-246	451	48	,	,	PUNCT
ma-246	451	49	and	and	CCONJ
ma-246	451	50	i	i	PRON
ma-246	451	51	=	=	NOUN
ma-246	451	52	{	{	PUNCT
ma-246	451	53	∅	∅	NOUN
ma-246	451	54	,	,	PUNCT
ma-246	451	55	{	{	PUNCT
ma-246	451	56	a	a	X
ma-246	451	57	}	}	PUNCT
ma-246	451	58	,	,	PUNCT
ma-246	451	59	{	{	PUNCT
ma-246	451	60	b	b	NOUN
ma-246	451	61	}	}	PUNCT
ma-246	451	62	,	,	PUNCT
ma-246	451	63	{	{	PUNCT
ma-246	451	64	a	a	DET
ma-246	451	65	,	,	PUNCT
ma-246	451	66	b	b	NOUN
ma-246	451	67	}	}	PUNCT
ma-246	451	68	}	}	PUNCT
ma-246	451	69	.	.	PUNCT
ma-246	452	1	then	then	ADV
ma-246	452	2	the	the	DET
ma-246	452	3	η	η	ADJ
ma-246	452	4	-	-	ADJ
ma-246	452	5	open	open	ADJ
ma-246	452	6	sets	set	NOUN
ma-246	452	7	of	of	ADP
ma-246	452	8	x	x	SYM
ma-246	452	9	are	be	AUX
ma-246	452	10	∅	∅	NOUN
ma-246	452	11	,	,	PUNCT
ma-246	452	12	x	x	INTJ
ma-246	452	13	,	,	PUNCT
ma-246	452	14	{	{	PUNCT
ma-246	452	15	c	c	NOUN
ma-246	452	16	}	}	PUNCT
ma-246	452	17	,	,	PUNCT
ma-246	452	18	{	{	PUNCT
ma-246	452	19	d	d	X
ma-246	452	20	}	}	PUNCT
ma-246	452	21	,	,	PUNCT
ma-246	452	22	{	{	PUNCT
ma-246	452	23	a	a	X
ma-246	452	24	,	,	PUNCT
ma-246	452	25	c	c	NOUN
ma-246	452	26	}	}	PUNCT
ma-246	452	27	,	,	PUNCT
ma-246	452	28	{	{	PUNCT
ma-246	452	29	a	a	PRON
ma-246	452	30	,	,	PUNCT
ma-246	452	31	d	d	NOUN
ma-246	452	32	}	}	PUNCT
ma-246	452	33	,	,	PUNCT
ma-246	452	34	{	{	PUNCT
ma-246	452	35	b	b	X
ma-246	452	36	,	,	PUNCT
ma-246	452	37	c	c	NOUN
ma-246	452	38	}	}	PUNCT
ma-246	452	39	,	,	PUNCT
ma-246	452	40	{	{	PUNCT
ma-246	452	41	b	b	X
ma-246	452	42	,	,	PUNCT
ma-246	452	43	d	d	NOUN
ma-246	452	44	}	}	PUNCT
ma-246	452	45	,	,	PUNCT
ma-246	452	46	{	{	PUNCT
ma-246	452	47	c	c	X
ma-246	452	48	,	,	PUNCT
ma-246	452	49	d	d	NOUN
ma-246	452	50	}	}	PUNCT
ma-246	452	51	,	,	PUNCT
ma-246	452	52	{	{	PUNCT
ma-246	452	53	a	a	DET
ma-246	452	54	,	,	PUNCT
ma-246	452	55	b	b	NOUN
ma-246	452	56	,	,	PUNCT
ma-246	452	57	c	c	NOUN
ma-246	452	58	}	}	PUNCT
ma-246	452	59	,	,	PUNCT
ma-246	452	60	{	{	PUNCT
ma-246	452	61	a	a	PRON
ma-246	452	62	,	,	PUNCT
ma-246	452	63	c	c	NOUN
ma-246	452	64	,	,	PUNCT
ma-246	452	65	d	d	NOUN
ma-246	452	66	}	}	PUNCT
ma-246	452	67	,	,	PUNCT
ma-246	452	68	{	{	PUNCT
ma-246	452	69	a	a	DET
ma-246	452	70	,	,	PUNCT
ma-246	452	71	b	b	NOUN
ma-246	452	72	,	,	PUNCT
ma-246	452	73	d	d	NOUN
ma-246	452	74	}	}	PUNCT
ma-246	452	75	,	,	PUNCT
ma-246	452	76	and	and	CCONJ
ma-246	452	77	{	{	PUNCT
ma-246	452	78	b	b	NOUN
ma-246	452	79	,	,	PUNCT
ma-246	452	80	c	c	NOUN
ma-246	452	81	,	,	PUNCT
ma-246	452	82	d	d	NOUN
ma-246	452	83	}	}	PUNCT
ma-246	452	84	.	.	PUNCT
ma-246	453	1	let	let	VERB
ma-246	453	2	a	a	DET
ma-246	453	3	=	=	X
ma-246	453	4	{	{	PUNCT
ma-246	453	5	c	c	NOUN
ma-246	453	6	}	}	PUNCT
ma-246	453	7	and	and	CCONJ
ma-246	453	8	b	b	X
ma-246	453	9	=	=	SYM
ma-246	453	10	{	{	PUNCT
ma-246	453	11	d	d	NOUN
ma-246	453	12	}	}	PUNCT
ma-246	453	13	such	such	ADJ
ma-246	453	14	that	that	DET
ma-246	453	15	a∪b	a∪b	NOUN
ma-246	453	16	=	=	X
ma-246	453	17	{	{	PUNCT
ma-246	453	18	c	c	NOUN
ma-246	453	19	,	,	PUNCT
ma-246	453	20	d	d	NOUN
ma-246	453	21	}	}	PUNCT
ma-246	453	22	,	,	PUNCT
ma-246	453	23	then	then	ADV
ma-246	453	24	by	by	ADP
ma-246	453	25	definition	definition	NOUN
ma-246	453	26	1	1	NUM
ma-246	453	27	,	,	PUNCT
ma-246	453	28	a∗η	a∗η	PROPN
ma-246	453	29	=	=	SYM
ma-246	453	30	{	{	PUNCT
ma-246	453	31	c	c	NOUN
ma-246	453	32	}	}	PUNCT
ma-246	453	33	,	,	PUNCT
ma-246	453	34	b∗η	b∗η	PROPN
ma-246	453	35	=	=	PUNCT
ma-246	453	36	{	{	PUNCT
ma-246	453	37	d	d	NOUN
ma-246	453	38	}	}	PUNCT
ma-246	453	39	,	,	PUNCT
ma-246	453	40	and	and	CCONJ
ma-246	453	41	(	(	PUNCT
ma-246	453	42	a∪b)∗η	a∪b)∗η	NOUN
ma-246	453	43	=	=	SYM
ma-246	453	44	x	x	X
ma-246	453	45	.	.	PUNCT
ma-246	454	1	now	now	ADV
ma-246	454	2	,	,	PUNCT
ma-246	454	3	by	by	ADP
ma-246	454	4	definition	definition	NOUN
ma-246	454	5	2	2	NUM
ma-246	454	6	,	,	PUNCT
ma-246	454	7	cl∗η(a	cl∗η(a	PROPN
ma-246	454	8	)	)	PUNCT
ma-246	454	9	=	=	PRON
ma-246	454	10	{	{	PUNCT
ma-246	454	11	c	c	NOUN
ma-246	454	12	}	}	PUNCT
ma-246	454	13	,	,	PUNCT
ma-246	454	14	cl∗η(b	cl∗η(b	NOUN
ma-246	454	15	)	)	PUNCT
ma-246	454	16	=	=	PRON
ma-246	455	1	{	{	PUNCT
ma-246	455	2	d	d	NOUN
ma-246	455	3	}	}	PUNCT
ma-246	455	4	,	,	PUNCT
ma-246	455	5	and	and	CCONJ
ma-246	455	6	cl∗η(a	cl∗η(a	PROPN
ma-246	455	7	∪	∪	ADJ
ma-246	455	8	b	b	NOUN
ma-246	455	9	)	)	PUNCT
ma-246	455	10	=	=	SYM
ma-246	456	1	x	x	X
ma-246	456	2	.	.	PUNCT
ma-246	457	1	observe	observe	VERB
ma-246	457	2	that	that	SCONJ
ma-246	457	3	cl∗η(a	cl∗η(a	PROPN
ma-246	457	4	∪	∪	ADP
ma-246	457	5	b	b	NOUN
ma-246	457	6	)	)	PUNCT
ma-246	457	7	=	=	SYM
ma-246	457	8	x	x	PROPN
ma-246	457	9	and	and	CCONJ
ma-246	457	10	cl∗η(a	cl∗η(a	PROPN
ma-246	457	11	)	)	PUNCT
ma-246	457	12	∪	∪	ADP
ma-246	457	13	cl∗η(b	cl∗η(b	NOUN
ma-246	457	14	)	)	PUNCT
ma-246	457	15	=	=	PRON
ma-246	458	1	{	{	PUNCT
ma-246	458	2	c	c	NOUN
ma-246	458	3	,	,	PUNCT
ma-246	458	4	d	d	NOUN
ma-246	458	5	}	}	PUNCT
ma-246	458	6	.	.	PUNCT
ma-246	459	1	these	these	PRON
ma-246	459	2	shows	show	VERB
ma-246	459	3	that	that	SCONJ
ma-246	459	4	cl∗η(a	cl∗η(a	PROPN
ma-246	459	5	∪	∪	ADP
ma-246	459	6	b	b	NOUN
ma-246	459	7	)	)	PUNCT
ma-246	459	8	*	*	PUNCT
ma-246	459	9	cl∗η(a	cl∗η(a	PROPN
ma-246	459	10	)	)	PUNCT
ma-246	459	11	∪	∪	ADP
ma-246	459	12	cl∗η(b	cl∗η(b	NOUN
ma-246	459	13	)	)	PUNCT
ma-246	459	14	.	.	PUNCT
ma-246	460	1	hence	hence	ADV
ma-246	460	2	,	,	PUNCT
ma-246	460	3	the	the	DET
ma-246	460	4	above	above	ADJ
ma-246	460	5	assertion	assertion	NOUN
ma-246	460	6	has	have	AUX
ma-246	460	7	been	be	AUX
ma-246	460	8	verified	verify	VERB
ma-246	460	9	.	.	PUNCT
ma-246	461	1	theorem	theorem	ADJ
ma-246	461	2	8	8	NUM
ma-246	461	3	.	.	PUNCT
ma-246	462	1	let	let	VERB
ma-246	462	2	(	(	PUNCT
ma-246	462	3	x	x	X
ma-246	462	4	,	,	PUNCT
ma-246	462	5	τ	τ	PROPN
ma-246	462	6	,	,	PUNCT
ma-246	462	7	i	i	PRON
ma-246	462	8	)	)	PUNCT
ma-246	462	9	be	be	VERB
ma-246	462	10	an	an	DET
ma-246	462	11	ideal	ideal	ADJ
ma-246	462	12	topological	topological	ADJ
ma-246	462	13	space	space	NOUN
ma-246	462	14	and	and	CCONJ
ma-246	462	15	a	a	DET
ma-246	462	16	⊆	⊆	NUM
ma-246	462	17	x	x	SYM
ma-246	462	18	.	.	PUNCT
ma-246	463	1	then	then	ADV
ma-246	463	2	cl∗η(a	cl∗η(a	PROPN
ma-246	463	3	)	)	PUNCT
ma-246	463	4	⊆	⊆	NUM
ma-246	463	5	cl∗(a	cl∗(a	NOUN
ma-246	463	6	)	)	PUNCT
ma-246	463	7	.	.	PUNCT
ma-246	464	1	proof	proof	NOUN
ma-246	464	2	.	.	PUNCT
ma-246	465	1	let	let	VERB
ma-246	465	2	a	a	DET
ma-246	465	3	⊆	⊆	NUM
ma-246	465	4	x	x	X
ma-246	465	5	.	.	PUNCT
ma-246	466	1	by	by	ADP
ma-246	466	2	definition	definition	NOUN
ma-246	466	3	2	2	NUM
ma-246	466	4	and	and	CCONJ
ma-246	466	5	kuratowski	kuratowski	ADJ
ma-246	466	6	closure	closure	NOUN
ma-246	466	7	operator	operator	NOUN
ma-246	466	8	,	,	PUNCT
ma-246	466	9	cl∗η(a	cl∗η(a	PROPN
ma-246	466	10	)	)	PUNCT
ma-246	466	11	=	=	PUNCT
ma-246	467	1	a	a	PRON
ma-246	467	2	∪	∪	X
ma-246	467	3	a∗η	a∗η	PROPN
ma-246	467	4	and	and	CCONJ
ma-246	467	5	cl∗(a	cl∗(a	NOUN
ma-246	467	6	)	)	PUNCT
ma-246	467	7	=	=	PUNCT
ma-246	467	8	a	a	DET
ma-246	467	9	∪	∪	ADJ
ma-246	467	10	a∗	a∗	NOUN
ma-246	467	11	,	,	PUNCT
ma-246	467	12	respectively	respectively	ADV
ma-246	467	13	.	.	PUNCT
ma-246	468	1	since	since	SCONJ
ma-246	468	2	by	by	ADP
ma-246	468	3	theorem	theorem	NOUN
ma-246	468	4	4	4	NUM
ma-246	468	5	(	(	PUNCT
ma-246	468	6	i	i	NOUN
ma-246	468	7	)	)	PUNCT
ma-246	468	8	,	,	PUNCT
ma-246	468	9	a∗η	a∗η	PROPN
ma-246	468	10	⊆	⊆	NUM
ma-246	468	11	a∗	a∗	NOUN
ma-246	468	12	,	,	PUNCT
ma-246	468	13	a	a	DET
ma-246	468	14	∪	∪	ADJ
ma-246	468	15	a∗η	a∗η	PROPN
ma-246	468	16	⊆	⊆	NUM
ma-246	468	17	a	a	DET
ma-246	468	18	∪	∪	ADJ
ma-246	468	19	a∗.	a∗.	NOUN
ma-246	468	20	hence	hence	ADV
ma-246	468	21	,	,	PUNCT
ma-246	468	22	cl∗η(a	cl∗η(a	PROPN
ma-246	468	23	)	)	PUNCT
ma-246	468	24	⊆	⊆	NUM
ma-246	468	25	cl∗(a	cl∗(a	NOUN
ma-246	468	26	)	)	PUNCT
ma-246	468	27	.	.	PUNCT
ma-246	469	1	�	�	PROPN
ma-246	469	2	theorem	theorem	VERB
ma-246	469	3	9	9	NUM
ma-246	469	4	.	.	PUNCT
ma-246	470	1	let	let	VERB
ma-246	470	2	(	(	PUNCT
ma-246	470	3	x	x	X
ma-246	470	4	,	,	PUNCT
ma-246	470	5	τ	τ	PROPN
ma-246	470	6	,	,	PUNCT
ma-246	470	7	i	i	PRON
ma-246	470	8	)	)	PUNCT
ma-246	470	9	be	be	VERB
ma-246	470	10	an	an	DET
ma-246	470	11	ideal	ideal	ADJ
ma-246	470	12	topological	topological	ADJ
ma-246	470	13	space	space	NOUN
ma-246	470	14	and	and	CCONJ
ma-246	470	15	a	a	DET
ma-246	470	16	be	be	AUX
ma-246	470	17	any	any	DET
ma-246	470	18	subset	subset	NOUN
ma-246	470	19	of	of	ADP
ma-246	470	20	x	x	X
ma-246	470	21	.	.	PUNCT
ma-246	471	1	then	then	ADV
ma-246	471	2	a	a	PRON
ma-246	471	3	is	be	AUX
ma-246	471	4	an	an	DET
ma-246	471	5	η	η	NOUN
ma-246	471	6	-	-	ADJ
ma-246	471	7	closed	closed	ADJ
ma-246	471	8	set	set	NOUN
ma-246	471	9	iff	iff	PROPN
ma-246	471	10	a	a	DET
ma-246	471	11	=	=	PUNCT
ma-246	471	12	cl∗η(a	cl∗η(a	PROPN
ma-246	471	13	)	)	PUNCT
ma-246	471	14	.	.	PUNCT
ma-246	472	1	proof	proof	NOUN
ma-246	472	2	.	.	PUNCT
ma-246	473	1	let	let	VERB
ma-246	473	2	a	a	PRON
ma-246	473	3	be	be	AUX
ma-246	473	4	an	an	DET
ma-246	473	5	η	η	ADJ
ma-246	473	6	-	-	ADJ
ma-246	473	7	closed	closed	ADJ
ma-246	473	8	set	set	NOUN
ma-246	473	9	.	.	PUNCT
ma-246	474	1	then	then	ADV
ma-246	474	2	a	a	DET
ma-246	474	3	=	=	X
ma-246	474	4	η	η	NOUN
ma-246	474	5	-	-	NOUN
ma-246	474	6	cl(a	cl(a	NUM
ma-246	474	7	)	)	PUNCT
ma-246	474	8	and	and	CCONJ
ma-246	474	9	by	by	ADP
ma-246	474	10	theroem	theroem	NOUN
ma-246	474	11	6	6	NUM
ma-246	474	12	(	(	PUNCT
ma-246	474	13	vi	vi	NOUN
ma-246	474	14	)	)	PUNCT
ma-246	474	15	,	,	PUNCT
ma-246	474	16	cl∗η(a	cl∗η(a	PROPN
ma-246	474	17	)	)	PUNCT
ma-246	474	18	=	=	SYM
ma-246	474	19	η	η	PROPN
ma-246	474	20	-	-	PUNCT
ma-246	474	21	cl(a),respectively	cl(a),respectively	ADV
ma-246	474	22	.	.	PUNCT
ma-246	475	1	note	note	VERB
ma-246	475	2	that	that	SCONJ
ma-246	475	3	a	a	DET
ma-246	475	4	=	=	PUNCT
ma-246	475	5	η	η	NOUN
ma-246	475	6	-	-	NOUN
ma-246	475	7	cl(a	cl(a	NUM
ma-246	475	8	)	)	PUNCT
ma-246	475	9	and	and	CCONJ
ma-246	475	10	η	η	NOUN
ma-246	475	11	-	-	NOUN
ma-246	475	12	cl(a	cl(a	NUM
ma-246	475	13	)	)	PUNCT
ma-246	475	14	=	=	SYM
ma-246	476	1	cl∗η(a	cl∗η(a	PROPN
ma-246	476	2	)	)	PUNCT
ma-246	476	3	,	,	PUNCT
ma-246	476	4	then	then	ADV
ma-246	476	5	by	by	ADP
ma-246	476	6	transitive	transitive	ADJ
ma-246	476	7	property	property	NOUN
ma-246	476	8	,	,	PUNCT
ma-246	476	9	it	it	PRON
ma-246	476	10	impliesthat	impliesthat	VERB
ma-246	476	11	a	a	DET
ma-246	476	12	=	=	NOUN
ma-246	476	13	cl∗η(a	cl∗η(a	PROPN
ma-246	476	14	)	)	PUNCT
ma-246	476	15	.	.	PUNCT
ma-246	477	1	now	now	ADV
ma-246	477	2	,	,	PUNCT
ma-246	477	3	on	on	ADP
ma-246	477	4	the	the	DET
ma-246	477	5	other	other	ADJ
ma-246	477	6	hand	hand	NOUN
ma-246	477	7	,	,	PUNCT
ma-246	477	8	let	let	VERB
ma-246	477	9	a	a	DET
ma-246	477	10	=	=	NOUN
ma-246	477	11	cl∗η(a	cl∗η(a	PROPN
ma-246	477	12	)	)	PUNCT
ma-246	477	13	.	.	PUNCT
ma-246	478	1	note	note	VERB
ma-246	478	2	that	that	SCONJ
ma-246	478	3	by	by	ADP
ma-246	478	4	theorem	theorem	NOUN
ma-246	478	5	6	6	NUM
ma-246	478	6	(	(	PUNCT
ma-246	478	7	vi	vi	NOUN
ma-246	478	8	)	)	PUNCT
ma-246	478	9	,	,	PUNCT
ma-246	478	10	cl∗η(a	cl∗η(a	PROPN
ma-246	478	11	)	)	PUNCT
ma-246	478	12	=	=	PUNCT
ma-246	478	13	η	η	PROPN
ma-246	478	14	-	-	NOUN
ma-246	478	15	cl(a	cl(a	NUM
ma-246	478	16	)	)	PUNCT
ma-246	478	17	.	.	PUNCT
ma-246	479	1	now	now	ADV
ma-246	479	2	that	that	SCONJ
ma-246	479	3	a	a	DET
ma-246	479	4	=	=	SYM
ma-246	479	5	cl∗η(a	cl∗η(a	PROPN
ma-246	479	6	)	)	PUNCT
ma-246	479	7	and	and	CCONJ
ma-246	479	8	cl∗η(a	cl∗η(a	PROPN
ma-246	479	9	)	)	PUNCT
ma-246	479	10	=	=	PUNCT
ma-246	479	11	η	η	PROPN
ma-246	479	12	-	-	NOUN
ma-246	479	13	cl(a	cl(a	NUM
ma-246	479	14	)	)	PUNCT
ma-246	479	15	,	,	PUNCT
ma-246	479	16	by	by	ADP
ma-246	479	17	transitive	transitive	ADJ
ma-246	479	18	property	property	NOUN
ma-246	479	19	again	again	ADV
ma-246	479	20	,	,	PUNCT
ma-246	479	21	a	a	DET
ma-246	479	22	=	=	X
ma-246	479	23	η	η	NOUN
ma-246	479	24	-	-	NOUN
ma-246	479	25	cl(a	cl(a	NUM
ma-246	479	26	)	)	PUNCT
ma-246	479	27	.	.	PUNCT
ma-246	480	1	therefore	therefore	ADV
ma-246	480	2	,	,	PUNCT
ma-246	480	3	a	a	PRON
ma-246	480	4	is	be	AUX
ma-246	480	5	an	an	DET
ma-246	480	6	η	η	ADJ
ma-246	480	7	-	-	ADJ
ma-246	480	8	closed	closed	ADJ
ma-246	480	9	set	set	NOUN
ma-246	480	10	.	.	PUNCT
ma-246	481	1	�	�	PROPN
ma-246	481	2	note	note	VERB
ma-246	481	3	that	that	SCONJ
ma-246	481	4	theorem	theorem	VERB
ma-246	481	5	7	7	NUM
ma-246	481	6	(	(	PUNCT
ma-246	481	7	i	i	NOUN
ma-246	481	8	)	)	PUNCT
ma-246	481	9	,	,	PUNCT
ma-246	481	10	(	(	PUNCT
ma-246	481	11	ii	ii	NOUN
ma-246	481	12	)	)	PUNCT
ma-246	481	13	,	,	PUNCT
ma-246	481	14	and	and	CCONJ
ma-246	481	15	(	(	PUNCT
ma-246	481	16	iv	iv	X
ma-246	481	17	)	)	PUNCT
ma-246	481	18	satisfy	satisfy	NOUN
ma-246	481	19	three	three	NUM
ma-246	481	20	of	of	ADP
ma-246	481	21	the	the	DET
ma-246	481	22	kuratowski	kuratowski	ADJ
ma-246	481	23	closure	closure	NOUN
ma-246	481	24	axioms	axiom	NOUN
ma-246	481	25	.	.	PUNCT
ma-246	482	1	however	however	ADV
ma-246	482	2	,	,	PUNCT
ma-246	482	3	theorem	theorem	VERB
ma-246	482	4	7	7	NUM
ma-246	482	5	(	(	PUNCT
ma-246	482	6	iii	iii	NOUN
ma-246	482	7	)	)	PUNCT
ma-246	482	8	did	do	AUX
ma-246	482	9	not	not	PART
ma-246	482	10	satisfy	satisfy	VERB
ma-246	482	11	one	one	NUM
ma-246	482	12	of	of	ADP
ma-246	482	13	the	the	DET
ma-246	482	14	kuratowski	kuratowski	ADJ
ma-246	482	15	closure	closure	NOUN
ma-246	482	16	axioms	axiom	NOUN
ma-246	482	17	because	because	SCONJ
ma-246	482	18	it	it	PRON
ma-246	482	19	is	be	AUX
ma-246	482	20	an	an	DET
ma-246	482	21	inclusionproperty	inclusionproperty	NOUN
ma-246	482	22	.	.	PUNCT
ma-246	483	1	as	as	ADP
ma-246	483	2	a	a	DET
ma-246	483	3	result	result	NOUN
ma-246	483	4	,	,	PUNCT
ma-246	483	5	the	the	DET
ma-246	483	6	following	follow	VERB
ma-246	483	7	remark	remark	NOUN
ma-246	483	8	is	be	AUX
ma-246	483	9	obtained	obtain	VERB
ma-246	483	10	.	.	PUNCT
ma-246	484	1	remark	remark	PROPN
ma-246	484	2	4	4	NUM
ma-246	484	3	.	.	PUNCT
ma-246	485	1	the	the	DET
ma-246	485	2	η	η	ADJ
ma-246	485	3	-	-	ADJ
ma-246	485	4	local	local	ADJ
ma-246	485	5	closure	closure	NOUN
ma-246	485	6	,	,	PUNCT
ma-246	485	7	i.e	i.e	PRON
ma-246	485	8	,	,	PUNCT
ma-246	485	9	cl∗η	cl∗η	PROPN
ma-246	485	10	,	,	PUNCT
ma-246	485	11	need	need	AUX
ma-246	485	12	not	not	PART
ma-246	485	13	be	be	AUX
ma-246	485	14	a	a	DET
ma-246	485	15	kuratowski	kuratowski	ADJ
ma-246	485	16	closure	closure	NOUN
ma-246	485	17	operator	operator	NOUN
ma-246	485	18	with	with	ADP
ma-246	485	19	respect	respect	NOUN
ma-246	485	20	to	to	ADP
ma-246	485	21	η	η	PROPN
ma-246	485	22	in	in	ADP
ma-246	485	23	general	general	NOUN
ma-246	485	24	.	.	PUNCT
ma-246	486	1	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	486	2	eur	eur	PROPN
ma-246	486	3	.	.	PUNCT
ma-246	487	1	j.	j.	PROPN
ma-246	487	2	math	math	PROPN
ma-246	487	3	.	.	PUNCT
ma-246	488	1	anal	anal	PROPN
ma-246	488	2	.	.	PUNCT
ma-246	489	1	10.28924	10.28924	NUM
ma-246	489	2	/	/	SYM
ma-246	489	3	ada	ada	PROPN
ma-246	489	4	/	/	SYM
ma-246	489	5	ma.5.2	ma.5.2	PROPN
ma-246	489	6	12	12	NUM
ma-246	489	7	theorem	theorem	NOUN
ma-246	489	8	10	10	NUM
ma-246	489	9	.	.	PUNCT
ma-246	490	1	let	let	VERB
ma-246	490	2	(	(	PUNCT
ma-246	490	3	x	x	X
ma-246	490	4	,	,	PUNCT
ma-246	490	5	τ	τ	PROPN
ma-246	490	6	,	,	PUNCT
ma-246	490	7	i	i	PRON
ma-246	490	8	)	)	PUNCT
ma-246	490	9	be	be	VERB
ma-246	490	10	an	an	DET
ma-246	490	11	ideal	ideal	ADJ
ma-246	490	12	topological	topological	ADJ
ma-246	490	13	space	space	NOUN
ma-246	490	14	where	where	SCONJ
ma-246	490	15	η	η	PROPN
ma-246	490	16	-	-	PROPN
ma-246	490	17	o(x	o(x	PROPN
ma-246	490	18	)	)	PUNCT
ma-246	490	19	is	be	AUX
ma-246	490	20	closed	close	VERB
ma-246	490	21	under	under	ADP
ma-246	490	22	any	any	DET
ma-246	490	23	two	two	NUM
ma-246	490	24	intersections	intersection	NOUN
ma-246	490	25	and	and	CCONJ
ma-246	490	26	a	a	DET
ma-246	490	27	,	,	PUNCT
ma-246	490	28	b	b	NOUN
ma-246	490	29	⊆	⊆	NUM
ma-246	490	30	x	x	X
ma-246	490	31	.	.	PUNCT
ma-246	491	1	then	then	ADV
ma-246	491	2	cl∗η(a	cl∗η(a	PROPN
ma-246	491	3	∪	∪	ADJ
ma-246	491	4	b	b	NOUN
ma-246	491	5	)	)	PUNCT
ma-246	491	6	=	=	SYM
ma-246	491	7	cl∗η(a	cl∗η(a	PROPN
ma-246	491	8	)	)	PUNCT
ma-246	491	9	∪	∪	ADP
ma-246	491	10	cl∗η(b	cl∗η(b	NOUN
ma-246	491	11	)	)	PUNCT
ma-246	491	12	.	.	PUNCT
ma-246	492	1	proof	proof	NOUN
ma-246	492	2	.	.	PUNCT
ma-246	493	1	let	let	VERB
ma-246	493	2	η	η	PROPN
ma-246	493	3	-	-	ADJ
ma-246	493	4	o(x	o(x	VERB
ma-246	493	5	)	)	PUNCT
ma-246	493	6	be	be	AUX
ma-246	493	7	closed	close	VERB
ma-246	493	8	under	under	ADP
ma-246	493	9	any	any	DET
ma-246	493	10	two	two	NUM
ma-246	493	11	intersections	intersection	NOUN
ma-246	493	12	.	.	PUNCT
ma-246	494	1	then	then	ADV
ma-246	494	2	by	by	ADP
ma-246	494	3	definition	definition	NOUN
ma-246	494	4	2	2	NUM
ma-246	494	5	and	and	CCONJ
ma-246	494	6	theorem	theorem	VERB
ma-246	494	7	3	3	NUM
ma-246	494	8	(	(	PUNCT
ma-246	494	9	i),it	i),it	PROPN
ma-246	494	10	follows	follow	VERB
ma-246	494	11	that	that	SCONJ
ma-246	494	12	cl∗η(a	cl∗η(a	PROPN
ma-246	494	13	∪	∪	ADP
ma-246	494	14	b	b	NOUN
ma-246	494	15	)	)	PUNCT
ma-246	494	16	=	=	SYM
ma-246	494	17	(	(	PUNCT
ma-246	494	18	a	a	DET
ma-246	494	19	∪	∪	ADJ
ma-246	494	20	b	b	NOUN
ma-246	494	21	)	)	PUNCT
ma-246	494	22	∪	∪	NOUN
ma-246	494	23	(	(	PUNCT
ma-246	494	24	a	a	DET
ma-246	494	25	∪	∪	ADJ
ma-246	494	26	b)∗η	b)∗η	X
ma-246	494	27	=	=	SYM
ma-246	494	28	(	(	PUNCT
ma-246	494	29	a	a	DET
ma-246	494	30	∪	∪	ADJ
ma-246	494	31	b	b	NOUN
ma-246	494	32	)	)	PUNCT
ma-246	494	33	∪	∪	NOUN
ma-246	494	34	(	(	PUNCT
ma-246	494	35	a∗η	a∗η	ADV
ma-246	494	36	∪	∪	X
ma-246	494	37	b∗η	b∗η	PROPN
ma-246	494	38	)	)	PUNCT
ma-246	495	1	=	=	PUNCT
ma-246	495	2	(	(	PUNCT
ma-246	495	3	a	a	DET
ma-246	495	4	∪	∪	ADJ
ma-246	495	5	a∗η	a∗η	NUM
ma-246	495	6	)	)	PUNCT
ma-246	495	7	∪	∪	NOUN
ma-246	495	8	(	(	PUNCT
ma-246	495	9	b	b	NOUN
ma-246	495	10	∪	∪	X
ma-246	495	11	b∗η	b∗η	PROPN
ma-246	495	12	)	)	PUNCT
ma-246	495	13	=	=	SYM
ma-246	495	14	cl∗η(a	cl∗η(a	PROPN
ma-246	495	15	)	)	PUNCT
ma-246	495	16	∪	∪	ADP
ma-246	495	17	cl∗η(b	cl∗η(b	NOUN
ma-246	495	18	)	)	PUNCT
ma-246	495	19	.	.	PUNCT
ma-246	496	1	hence	hence	ADV
ma-246	496	2	,	,	PUNCT
ma-246	496	3	cl∗η(a	cl∗η(a	ADJ
ma-246	496	4	∪	∪	ADP
ma-246	496	5	b	b	NOUN
ma-246	496	6	)	)	PUNCT
ma-246	496	7	=	=	SYM
ma-246	496	8	cl∗η(a	cl∗η(a	PROPN
ma-246	496	9	)	)	PUNCT
ma-246	496	10	∪	∪	ADP
ma-246	496	11	cl∗η(b	cl∗η(b	NOUN
ma-246	496	12	)	)	PUNCT
ma-246	496	13	.	.	PUNCT
ma-246	497	1	�	�	PROPN
ma-246	497	2	note	note	VERB
ma-246	497	3	that	that	SCONJ
ma-246	497	4	theorem	theorem	VERB
ma-246	497	5	7	7	NUM
ma-246	497	6	(	(	PUNCT
ma-246	497	7	i	i	NOUN
ma-246	497	8	)	)	PUNCT
ma-246	497	9	,	,	PUNCT
ma-246	497	10	(	(	PUNCT
ma-246	497	11	ii	ii	NOUN
ma-246	497	12	)	)	PUNCT
ma-246	497	13	,	,	PUNCT
ma-246	497	14	(	(	PUNCT
ma-246	497	15	iv	iv	X
ma-246	497	16	)	)	PUNCT
ma-246	497	17	and	and	CCONJ
ma-246	497	18	theorem	theorem	VERB
ma-246	497	19	10	10	NUM
ma-246	497	20	using	use	VERB
ma-246	497	21	the	the	DET
ma-246	497	22	condition	condition	NOUN
ma-246	497	23	,	,	PUNCT
ma-246	497	24	for	for	ADP
ma-246	497	25	any	any	DET
ma-246	497	26	ideal	ideal	ADJ
ma-246	497	27	topologicalspaces	topologicalspace	NOUN
ma-246	497	28	(	(	PUNCT
ma-246	497	29	x	x	X
ma-246	497	30	,	,	PUNCT
ma-246	497	31	τ	τ	PROPN
ma-246	497	32	,	,	PUNCT
ma-246	497	33	i	i	PROPN
ma-246	497	34	)	)	PUNCT
ma-246	497	35	where	where	SCONJ
ma-246	497	36	η	η	PROPN
ma-246	497	37	-	-	PROPN
ma-246	497	38	o(x	o(x	PROPN
ma-246	497	39	)	)	PUNCT
ma-246	497	40	is	be	AUX
ma-246	497	41	closed	close	VERB
ma-246	497	42	under	under	ADP
ma-246	497	43	any	any	DET
ma-246	497	44	two	two	NUM
ma-246	497	45	intersections	intersection	NOUN
ma-246	497	46	,	,	PUNCT
ma-246	497	47	satisfy	satisfy	VERB
ma-246	497	48	the	the	DET
ma-246	497	49	kuratowski	kuratowski	PROPN
ma-246	497	50	closureaxioms	closureaxioms	PROPN
ma-246	497	51	.	.	PUNCT
ma-246	498	1	as	as	ADP
ma-246	498	2	a	a	DET
ma-246	498	3	result	result	NOUN
ma-246	498	4	,	,	PUNCT
ma-246	498	5	the	the	DET
ma-246	498	6	following	follow	VERB
ma-246	498	7	remark	remark	NOUN
ma-246	498	8	is	be	AUX
ma-246	498	9	obtained	obtain	VERB
ma-246	498	10	remark	remark	NOUN
ma-246	498	11	5	5	NUM
ma-246	498	12	.	.	PUNCT
ma-246	499	1	let	let	VERB
ma-246	499	2	(	(	PUNCT
ma-246	499	3	x	x	X
ma-246	499	4	,	,	PUNCT
ma-246	499	5	τ	τ	PROPN
ma-246	499	6	,	,	PUNCT
ma-246	499	7	i	i	PRON
ma-246	499	8	)	)	PUNCT
ma-246	499	9	be	be	VERB
ma-246	499	10	an	an	DET
ma-246	499	11	ideal	ideal	ADJ
ma-246	499	12	topological	topological	ADJ
ma-246	499	13	space	space	NOUN
ma-246	499	14	where	where	SCONJ
ma-246	499	15	η	η	PROPN
ma-246	499	16	-	-	PROPN
ma-246	499	17	o(x	o(x	PROPN
ma-246	499	18	)	)	PUNCT
ma-246	499	19	is	be	AUX
ma-246	499	20	closed	close	VERB
ma-246	499	21	under	under	ADP
ma-246	499	22	any	any	DET
ma-246	499	23	two	two	NUM
ma-246	499	24	intersections	intersection	NOUN
ma-246	499	25	,	,	PUNCT
ma-246	499	26	the	the	DET
ma-246	499	27	η	η	PROPN
ma-246	499	28	-	-	ADJ
ma-246	499	29	local	local	ADJ
ma-246	499	30	closure	closure	NOUN
ma-246	499	31	,	,	PUNCT
ma-246	499	32	i.e	i.e	PRON
ma-246	499	33	,	,	PUNCT
ma-246	499	34	cl∗η	cl∗η	PROPN
ma-246	499	35	,	,	PUNCT
ma-246	499	36	is	be	AUX
ma-246	499	37	a	a	DET
ma-246	499	38	kuratowski	kuratowski	ADJ
ma-246	499	39	closure	closure	NOUN
ma-246	499	40	operator	operator	NOUN
ma-246	499	41	(	(	PUNCT
ma-246	499	42	or	or	CCONJ
ma-246	499	43	almocera	almocera	NOUN
ma-246	499	44	closure	closure	NOUN
ma-246	499	45	operator	operator	NOUN
ma-246	499	46	)	)	PUNCT
ma-246	499	47	with	with	ADP
ma-246	499	48	respect	respect	NOUN
ma-246	499	49	to	to	ADP
ma-246	499	50	η	η	PROPN
ma-246	499	51	.	.	PROPN
ma-246	499	52	note	note	VERB
ma-246	499	53	that	that	SCONJ
ma-246	499	54	by	by	ADP
ma-246	499	55	remark	remark	NOUN
ma-246	499	56	5	5	NUM
ma-246	499	57	,	,	PUNCT
ma-246	499	58	cl∗η	cl∗η	PROPN
ma-246	499	59	is	be	AUX
ma-246	499	60	a	a	DET
ma-246	499	61	kuratowski	kuratowski	ADJ
ma-246	499	62	closure	closure	NOUN
ma-246	499	63	operator	operator	NOUN
ma-246	499	64	(	(	PUNCT
ma-246	499	65	or	or	CCONJ
ma-246	499	66	almocera	almocera	NOUN
ma-246	499	67	closure	closure	NOUN
ma-246	499	68	operator)with	operator)with	ADP
ma-246	499	69	respect	respect	NOUN
ma-246	499	70	to	to	ADP
ma-246	499	71	η	η	PROPN
ma-246	499	72	for	for	ADP
ma-246	499	73	any	any	DET
ma-246	499	74	ideal	ideal	ADJ
ma-246	499	75	topological	topological	ADJ
ma-246	499	76	space	space	NOUN
ma-246	499	77	(	(	PUNCT
ma-246	499	78	x	x	X
ma-246	499	79	,	,	PUNCT
ma-246	499	80	τ	τ	PROPN
ma-246	499	81	,	,	PUNCT
ma-246	499	82	i	i	PROPN
ma-246	499	83	)	)	PUNCT
ma-246	499	84	where	where	SCONJ
ma-246	499	85	η	η	PROPN
ma-246	499	86	-	-	PROPN
ma-246	499	87	o(x	o(x	PROPN
ma-246	499	88	)	)	PUNCT
ma-246	499	89	is	be	AUX
ma-246	499	90	closed	close	VERB
ma-246	499	91	under	under	ADP
ma-246	499	92	any	any	DET
ma-246	499	93	twointersections	twointersection	NOUN
ma-246	499	94	.	.	PUNCT
ma-246	500	1	now	now	ADV
ma-246	500	2	,	,	PUNCT
ma-246	500	3	let	let	VERB
ma-246	500	4	a	a	PRON
ma-246	500	5	be	be	AUX
ma-246	500	6	a	a	DET
ma-246	500	7	τ∗η	τ∗η	ADV
ma-246	500	8	-	-	PUNCT
ma-246	500	9	closed	closed	ADJ
ma-246	500	10	set	set	VERB
ma-246	500	11	iff	iff	PROPN
ma-246	500	12	a∗η	a∗η	PROPN
ma-246	500	13	⊆	⊆	NUM
ma-246	500	14	a	a	PRON
ma-246	500	15	in	in	ADP
ma-246	500	16	any	any	DET
ma-246	500	17	ideal	ideal	ADJ
ma-246	500	18	topological	topological	ADJ
ma-246	500	19	space	space	NOUN
ma-246	500	20	(	(	PUNCT
ma-246	500	21	x	x	X
ma-246	500	22	,	,	PUNCT
ma-246	500	23	τ	τ	PROPN
ma-246	500	24	,	,	PUNCT
ma-246	500	25	i)where	i)where	X
ma-246	501	1	η	η	PROPN
ma-246	501	2	-	-	PROPN
ma-246	501	3	o(x	o(x	ADJ
ma-246	501	4	)	)	PUNCT
ma-246	501	5	is	be	AUX
ma-246	501	6	closed	close	VERB
ma-246	501	7	under	under	ADP
ma-246	501	8	any	any	DET
ma-246	501	9	two	two	NUM
ma-246	501	10	intersections	intersection	NOUN
ma-246	501	11	.	.	PUNCT
ma-246	502	1	then	then	ADV
ma-246	502	2	the	the	DET
ma-246	502	3	following	follow	VERB
ma-246	502	4	lemma	lemma	PROPN
ma-246	502	5	is	be	AUX
ma-246	502	6	obtained	obtain	VERB
ma-246	502	7	.	.	PUNCT
ma-246	503	1	lemma	lemma	PROPN
ma-246	503	2	1	1	NUM
ma-246	503	3	.	.	PUNCT
ma-246	504	1	let	let	VERB
ma-246	504	2	a	a	PRON
ma-246	504	3	be	be	AUX
ma-246	504	4	a	a	DET
ma-246	504	5	τ∗η	τ∗η	ADV
ma-246	504	6	-	-	PUNCT
ma-246	504	7	closed	closed	ADJ
ma-246	504	8	set	set	VERB
ma-246	504	9	iff	iff	PROPN
ma-246	504	10	a∗η	a∗η	PROPN
ma-246	504	11	⊆	⊆	NUM
ma-246	504	12	a	a	PRON
ma-246	504	13	in	in	ADP
ma-246	504	14	any	any	DET
ma-246	504	15	ideal	ideal	ADJ
ma-246	504	16	topological	topological	ADJ
ma-246	504	17	space	space	NOUN
ma-246	504	18	(	(	PUNCT
ma-246	504	19	x	x	X
ma-246	504	20	,	,	PUNCT
ma-246	504	21	τ	τ	PROPN
ma-246	504	22	,	,	PUNCT
ma-246	504	23	i	i	PROPN
ma-246	504	24	)	)	PUNCT
ma-246	504	25	where	where	SCONJ
ma-246	504	26	η	η	PROPN
ma-246	504	27	-	-	PROPN
ma-246	504	28	o(x	o(x	PROPN
ma-246	504	29	)	)	PUNCT
ma-246	504	30	is	be	AUX
ma-246	504	31	closed	close	VERB
ma-246	504	32	under	under	ADP
ma-246	504	33	any	any	DET
ma-246	504	34	two	two	NUM
ma-246	504	35	intersections	intersection	NOUN
ma-246	504	36	.	.	PUNCT
ma-246	505	1	then	then	ADV
ma-246	505	2	a	a	PRON
ma-246	505	3	is	be	AUX
ma-246	505	4	τ∗η	τ∗η	PUNCT
ma-246	505	5	-	-	PUNCT
ma-246	505	6	closed	closed	ADJ
ma-246	505	7	set	set	ADJ
ma-246	505	8	iff	iff	PROPN
ma-246	505	9	cl∗η(a	cl∗η(a	PROPN
ma-246	505	10	)	)	PUNCT
ma-246	506	1	=	=	NOUN
ma-246	506	2	a.	a.	NOUN
ma-246	506	3	proof	proof	NOUN
ma-246	506	4	.	.	PUNCT
ma-246	507	1	let	let	VERB
ma-246	507	2	a	a	PRON
ma-246	507	3	be	be	AUX
ma-246	507	4	a	a	DET
ma-246	507	5	τ∗η	τ∗η	ADV
ma-246	507	6	-	-	PUNCT
ma-246	507	7	closed	closed	ADJ
ma-246	507	8	in	in	ADP
ma-246	507	9	(	(	PUNCT
ma-246	507	10	x	x	NOUN
ma-246	507	11	,	,	PUNCT
ma-246	507	12	τ	τ	PROPN
ma-246	507	13	,	,	PUNCT
ma-246	507	14	i	i	PROPN
ma-246	507	15	)	)	PUNCT
ma-246	507	16	where	where	SCONJ
ma-246	507	17	η	η	PROPN
ma-246	507	18	-	-	PROPN
ma-246	507	19	o(x	o(x	PROPN
ma-246	507	20	)	)	PUNCT
ma-246	507	21	is	be	AUX
ma-246	507	22	closed	close	VERB
ma-246	507	23	under	under	ADP
ma-246	507	24	any	any	DET
ma-246	507	25	two	two	NUM
ma-246	507	26	intersections	intersection	NOUN
ma-246	507	27	.	.	PUNCT
ma-246	508	1	now	now	ADV
ma-246	508	2	,	,	PUNCT
ma-246	508	3	since	since	SCONJ
ma-246	508	4	a	a	PRON
ma-246	508	5	is	be	AUX
ma-246	508	6	a	a	DET
ma-246	508	7	τ∗η	τ∗η	ADV
ma-246	508	8	-	-	ADJ
ma-246	508	9	closed	closed	ADJ
ma-246	508	10	,	,	PUNCT
ma-246	508	11	by	by	ADP
ma-246	508	12	assumption	assumption	NOUN
ma-246	508	13	,	,	PUNCT
ma-246	508	14	a∗η	a∗η	PROPN
ma-246	508	15	⊆	⊆	NUM
ma-246	508	16	a.	a.	NOUN
ma-246	508	17	it	it	PRON
ma-246	508	18	follows	follow	VERB
ma-246	508	19	that	that	SCONJ
ma-246	508	20	a	a	DET
ma-246	508	21	∪	∪	ADJ
ma-246	508	22	a∗η	a∗η	NOUN
ma-246	508	23	=	=	SYM
ma-246	508	24	a.	a.	NOUN
ma-246	508	25	now	now	ADV
ma-246	508	26	,	,	PUNCT
ma-246	508	27	by	by	ADP
ma-246	508	28	definiton2	definiton2	NOUN
ma-246	508	29	,	,	PUNCT
ma-246	508	30	cl∗η(a	cl∗η(a	PROPN
ma-246	508	31	)	)	PUNCT
ma-246	508	32	=	=	PUNCT
ma-246	509	1	a	a	DET
ma-246	509	2	∪	∪	X
ma-246	509	3	a∗η	a∗η	PROPN
ma-246	509	4	=	=	SYM
ma-246	509	5	a.	a.	NOUN
ma-246	509	6	therefore	therefore	ADV
ma-246	509	7	,	,	PUNCT
ma-246	509	8	cl∗η	cl∗η	PROPN
ma-246	509	9	=	=	SYM
ma-246	509	10	a.	a.	NOUN
ma-246	509	11	on	on	ADP
ma-246	509	12	the	the	DET
ma-246	509	13	other	other	ADJ
ma-246	509	14	hand	hand	NOUN
ma-246	509	15	,	,	PUNCT
ma-246	509	16	let	let	VERB
ma-246	509	17	cl∗η(a	cl∗η(a	PROPN
ma-246	509	18	)	)	PUNCT
ma-246	510	1	=	=	PUNCT
ma-246	510	2	a.	a.	NOUN
ma-246	510	3	now	now	ADV
ma-246	510	4	,	,	PUNCT
ma-246	510	5	bydefiniton	bydefiniton	VERB
ma-246	510	6	2	2	NUM
ma-246	510	7	,	,	PUNCT
ma-246	510	8	cl∗η(a	cl∗η(a	PROPN
ma-246	510	9	)	)	PUNCT
ma-246	510	10	=	=	PUNCT
ma-246	511	1	a	a	DET
ma-246	511	2	∪	∪	X
ma-246	511	3	a∗η	a∗η	PROPN
ma-246	511	4	=	=	SYM
ma-246	511	5	a.	a.	NOUN
ma-246	511	6	so	so	ADV
ma-246	511	7	,	,	PUNCT
ma-246	511	8	a	a	DET
ma-246	511	9	∪	∪	ADJ
ma-246	511	10	a∗η	a∗η	PROPN
ma-246	511	11	=	=	SYM
ma-246	511	12	a	a	PRON
ma-246	511	13	implies	imply	VERB
ma-246	511	14	a∗η	a∗η	PROPN
ma-246	511	15	⊆	⊆	NUM
ma-246	511	16	a	a	PRON
ma-246	511	17	,	,	PUNCT
ma-246	511	18	and	and	CCONJ
ma-246	511	19	so	so	ADV
ma-246	511	20	,	,	PUNCT
ma-246	511	21	by	by	ADP
ma-246	511	22	assumption	assumption	NOUN
ma-246	511	23	,	,	PUNCT
ma-246	511	24	a	a	PRON
ma-246	511	25	is	be	AUX
ma-246	511	26	τ∗η	τ∗η	PUNCT
ma-246	511	27	-	-	ADJ
ma-246	511	28	closed	closed	ADJ
ma-246	511	29	.	.	PUNCT
ma-246	512	1	�	�	PROPN
ma-246	512	2	theorem	theorem	VERB
ma-246	512	3	11	11	NUM
ma-246	512	4	.	.	PUNCT
ma-246	513	1	let	let	VERB
ma-246	513	2	(	(	PUNCT
ma-246	513	3	x	x	X
ma-246	513	4	,	,	PUNCT
ma-246	513	5	τ	τ	PROPN
ma-246	513	6	,	,	PUNCT
ma-246	513	7	i	i	PRON
ma-246	513	8	)	)	PUNCT
ma-246	513	9	be	be	VERB
ma-246	513	10	an	an	DET
ma-246	513	11	ideal	ideal	ADJ
ma-246	513	12	topological	topological	ADJ
ma-246	513	13	space	space	NOUN
ma-246	513	14	where	where	SCONJ
ma-246	513	15	η	η	PROPN
ma-246	513	16	-	-	PROPN
ma-246	513	17	o(x	o(x	PROPN
ma-246	513	18	)	)	PUNCT
ma-246	513	19	is	be	AUX
ma-246	513	20	closed	close	VERB
ma-246	513	21	under	under	ADP
ma-246	513	22	any	any	DET
ma-246	513	23	two	two	NUM
ma-246	513	24	intersections	intersection	NOUN
ma-246	513	25	.	.	PUNCT
ma-246	514	1	let	let	VERB
ma-246	514	2	τ∗η	τ∗η	PUNCT
ma-246	514	3	=	=	SYM
ma-246	514	4	{	{	PUNCT
ma-246	514	5	j	j	NOUN
ma-246	514	6	⊆	⊆	NUM
ma-246	514	7	x	x	SYM
ma-246	514	8	:	:	PUNCT
ma-246	514	9	cl∗η(jc	cl∗η(jc	VERB
ma-246	514	10	)	)	PUNCT
ma-246	515	1	=	=	SYM
ma-246	515	2	jc	jc	PROPN
ma-246	515	3	}	}	PUNCT
ma-246	515	4	.	.	PUNCT
ma-246	516	1	then	then	ADV
ma-246	516	2	τ∗η	τ∗η	PUNCT
ma-246	516	3	is	be	AUX
ma-246	516	4	a	a	DET
ma-246	516	5	topology	topology	NOUN
ma-246	516	6	for	for	ADP
ma-246	516	7	x	x	SYM
ma-246	516	8	such	such	ADJ
ma-246	516	9	that	that	DET
ma-246	516	10	τ∗	τ∗	NOUN
ma-246	516	11	⊆	⊆	NUM
ma-246	516	12	τ∗η	τ∗η	PUNCT
ma-246	516	13	and	and	CCONJ
ma-246	516	14	η	η	PROPN
ma-246	516	15	-	-	PROPN
ma-246	516	16	o(x	o(x	ADJ
ma-246	516	17	)	)	PUNCT
ma-246	516	18	⊆	⊆	NUM
ma-246	516	19	τ∗η	τ∗η	PUNCT
ma-246	516	20	.	.	PUNCT
ma-246	517	1	proof	proof	NOUN
ma-246	517	2	.	.	PUNCT
ma-246	518	1	let	let	VERB
ma-246	518	2	η	η	PROPN
ma-246	518	3	-	-	ADJ
ma-246	518	4	o(x	o(x	VERB
ma-246	518	5	)	)	PUNCT
ma-246	518	6	be	be	AUX
ma-246	518	7	closed	close	VERB
ma-246	518	8	under	under	ADP
ma-246	518	9	any	any	DET
ma-246	518	10	two	two	NUM
ma-246	518	11	intersections	intersection	NOUN
ma-246	518	12	.	.	PUNCT
ma-246	519	1	note	note	VERB
ma-246	519	2	that	that	SCONJ
ma-246	519	3	by	by	ADP
ma-246	519	4	remark	remark	NOUN
ma-246	519	5	5	5	NUM
ma-246	519	6	,	,	PUNCT
ma-246	519	7	cl∗η	cl∗η	PROPN
ma-246	519	8	is	be	AUX
ma-246	519	9	akuratowski	akuratowski	ADJ
ma-246	519	10	closure	closure	NOUN
ma-246	519	11	operator	operator	NOUN
ma-246	519	12	with	with	ADP
ma-246	519	13	respect	respect	NOUN
ma-246	519	14	to	to	ADP
ma-246	519	15	η	η	PROPN
ma-246	519	16	.	.	PROPN
ma-246	519	17	therefore	therefore	ADV
ma-246	519	18	,	,	PUNCT
ma-246	519	19	τ∗η	τ∗η	PUNCT
ma-246	519	20	is	be	AUX
ma-246	519	21	a	a	DET
ma-246	519	22	topology	topology	NOUN
ma-246	519	23	generated	generate	VERB
ma-246	519	24	by	by	ADP
ma-246	519	25	cl∗η	cl∗η	PROPN
ma-246	519	26	.	.	PUNCT
ma-246	520	1	now	now	ADV
ma-246	520	2	,	,	PUNCT
ma-246	520	3	to	to	PART
ma-246	520	4	show	show	VERB
ma-246	520	5	that	that	SCONJ
ma-246	520	6	τ∗	τ∗	NOUN
ma-246	520	7	⊆	⊆	NUM
ma-246	520	8	τ∗η	τ∗η	PUNCT
ma-246	520	9	,	,	PUNCT
ma-246	520	10	let	let	VERB
ma-246	520	11	a	a	PRON
ma-246	520	12	be	be	AUX
ma-246	520	13	a	a	DET
ma-246	520	14	τ∗-open	τ∗-open	NOUN
ma-246	520	15	.	.	PUNCT
ma-246	521	1	then	then	ADV
ma-246	521	2	ac	ac	PROPN
ma-246	521	3	is	be	AUX
ma-246	521	4	a	a	DET
ma-246	521	5	τ∗-closed	τ∗-close	VERB
ma-246	521	6	.	.	PUNCT
ma-246	522	1	then	then	ADV
ma-246	522	2	by	by	ADP
ma-246	522	3	definition	definition	NOUN
ma-246	522	4	of	of	ADP
ma-246	522	5	τ∗-closed	τ∗-closed	PROPN
ma-246	522	6	,	,	PUNCT
ma-246	522	7	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	PROPN
ma-246	522	8	eur	eur	PROPN
ma-246	522	9	.	.	PUNCT
ma-246	523	1	j.	j.	PROPN
ma-246	523	2	math	math	PROPN
ma-246	523	3	.	.	PUNCT
ma-246	524	1	anal	anal	PROPN
ma-246	524	2	.	.	PUNCT
ma-246	525	1	10.28924	10.28924	NUM
ma-246	525	2	/	/	SYM
ma-246	525	3	ada	ada	PROPN
ma-246	525	4	/	/	SYM
ma-246	525	5	ma.5.2	ma.5.2	PROPN
ma-246	525	6	13	13	NUM
ma-246	525	7	(	(	PUNCT
ma-246	525	8	ac	ac	PROPN
ma-246	525	9	)	)	PUNCT
ma-246	525	10	∗	∗	NOUN
ma-246	525	11	⊆	⊆	NUM
ma-246	525	12	ac	ac	NOUN
ma-246	525	13	.	.	PUNCT
ma-246	526	1	hence	hence	ADV
ma-246	526	2	,	,	PUNCT
ma-246	526	3	cl∗(ac	cl∗(ac	PROPN
ma-246	526	4	)	)	PUNCT
ma-246	527	1	=	=	PUNCT
ma-246	527	2	ac	ac	ADP
ma-246	527	3	∪	∪	ADV
ma-246	527	4	(	(	PUNCT
ma-246	527	5	ac	ac	ADJ
ma-246	527	6	)	)	PUNCT
ma-246	527	7	∗	∗	NOUN
ma-246	527	8	=	=	PUNCT
ma-246	527	9	ac	ac	PROPN
ma-246	527	10	implies	imply	VERB
ma-246	527	11	that	that	SCONJ
ma-246	527	12	cl∗(ac	cl∗(ac	X
ma-246	527	13	)	)	PUNCT
ma-246	528	1	=	=	PUNCT
ma-246	528	2	ac	ac	PROPN
ma-246	528	3	.	.	PUNCT
ma-246	529	1	then	then	ADV
ma-246	529	2	by	by	ADP
ma-246	529	3	theorem8	theorem8	PROPN
ma-246	529	4	,	,	PUNCT
ma-246	529	5	cl∗η(ac	cl∗η(ac	PRON
ma-246	529	6	)	)	PUNCT
ma-246	529	7	⊆	⊆	NUM
ma-246	529	8	ac	ac	PROPN
ma-246	529	9	.	.	PUNCT
ma-246	530	1	now	now	ADV
ma-246	530	2	,	,	PUNCT
ma-246	530	3	since	since	SCONJ
ma-246	530	4	cl∗η(ac	cl∗η(ac	PRON
ma-246	530	5	)	)	PUNCT
ma-246	530	6	⊆	⊆	NUM
ma-246	530	7	ac	ac	NOUN
ma-246	530	8	,	,	PUNCT
ma-246	530	9	by	by	ADP
ma-246	530	10	theorem	theorem	NOUN
ma-246	530	11	7	7	NUM
ma-246	530	12	(	(	PUNCT
ma-246	530	13	i	i	NOUN
ma-246	530	14	)	)	PUNCT
ma-246	530	15	,	,	PUNCT
ma-246	530	16	cl∗η(ac	cl∗η(ac	X
ma-246	530	17	)	)	PUNCT
ma-246	530	18	=	=	SYM
ma-246	530	19	ac	ac	PROPN
ma-246	530	20	.	.	PUNCT
ma-246	531	1	hence	hence	ADV
ma-246	531	2	,	,	PUNCT
ma-246	531	3	by	by	ADP
ma-246	531	4	lemma1	lemma1	PROPN
ma-246	531	5	,	,	PUNCT
ma-246	531	6	ac	ac	PROPN
ma-246	531	7	is	be	AUX
ma-246	531	8	a	a	DET
ma-246	531	9	τ∗η	τ∗η	ADV
ma-246	531	10	-	-	PUNCT
ma-246	531	11	closed	closed	ADJ
ma-246	531	12	,	,	PUNCT
ma-246	531	13	and	and	CCONJ
ma-246	531	14	so	so	ADV
ma-246	531	15	,	,	PUNCT
ma-246	531	16	a	a	PRON
ma-246	531	17	is	be	AUX
ma-246	531	18	a	a	DET
ma-246	531	19	τ∗η	τ∗η	ADV
ma-246	531	20	-	-	ADJ
ma-246	531	21	open	open	ADJ
ma-246	531	22	.	.	PUNCT
ma-246	532	1	as	as	ADP
ma-246	532	2	a	a	DET
ma-246	532	3	result	result	NOUN
ma-246	532	4	,	,	PUNCT
ma-246	532	5	thus	thus	ADV
ma-246	532	6	,	,	PUNCT
ma-246	532	7	τ∗	τ∗	ADJ
ma-246	532	8	⊆	⊆	NUM
ma-246	532	9	τ∗η	τ∗η	PUNCT
ma-246	532	10	.	.	PUNCT
ma-246	533	1	next	next	ADV
ma-246	533	2	,	,	PUNCT
ma-246	533	3	to	to	PART
ma-246	533	4	show	show	VERB
ma-246	533	5	that	that	SCONJ
ma-246	533	6	η	η	PROPN
ma-246	533	7	-	-	ADJ
ma-246	533	8	o(x	o(x	ADJ
ma-246	533	9	)	)	PUNCT
ma-246	533	10	⊆	⊆	NUM
ma-246	533	11	τ∗η	τ∗η	PUNCT
ma-246	533	12	,	,	PUNCT
ma-246	533	13	let	let	VERB
ma-246	533	14	a	a	PRON
ma-246	533	15	be	be	AUX
ma-246	533	16	an	an	DET
ma-246	533	17	η	η	NOUN
ma-246	533	18	-	-	ADJ
ma-246	533	19	open	open	ADJ
ma-246	533	20	.	.	PUNCT
ma-246	534	1	then	then	ADV
ma-246	534	2	ac	ac	PROPN
ma-246	534	3	is	be	AUX
ma-246	534	4	an	an	DET
ma-246	534	5	η	η	NOUN
ma-246	534	6	-	-	ADJ
ma-246	534	7	closed	closed	ADJ
ma-246	534	8	and	and	CCONJ
ma-246	534	9	by	by	ADP
ma-246	534	10	theorem	theorem	NOUN
ma-246	534	11	9	9	NUM
ma-246	534	12	,	,	PUNCT
ma-246	534	13	ac	ac	PROPN
ma-246	534	14	=	=	SYM
ma-246	534	15	cl∗η(ac	cl∗η(ac	PROPN
ma-246	534	16	)	)	PUNCT
ma-246	534	17	.	.	PUNCT
ma-246	535	1	itfollows	itfollow	NOUN
ma-246	535	2	that	that	SCONJ
ma-246	535	3	by	by	ADP
ma-246	535	4	lemma	lemma	PROPN
ma-246	535	5	1	1	NUM
ma-246	535	6	,	,	PUNCT
ma-246	535	7	ac	ac	PROPN
ma-246	535	8	is	be	AUX
ma-246	535	9	a	a	DET
ma-246	535	10	τ∗η	τ∗η	ADV
ma-246	535	11	-	-	PUNCT
ma-246	535	12	closed	closed	ADJ
ma-246	535	13	implies	imply	VERB
ma-246	535	14	that	that	SCONJ
ma-246	535	15	a	a	PRON
ma-246	535	16	is	be	AUX
ma-246	535	17	a	a	DET
ma-246	535	18	τ∗η	τ∗η	ADV
ma-246	535	19	-	-	ADJ
ma-246	535	20	open	open	ADJ
ma-246	535	21	.	.	PUNCT
ma-246	536	1	hence	hence	ADV
ma-246	536	2	,	,	PUNCT
ma-246	536	3	η	η	PROPN
ma-246	536	4	-	-	PROPN
ma-246	536	5	o(x	o(x	ADJ
ma-246	536	6	)	)	PUNCT
ma-246	536	7	⊆	⊆	NUM
ma-246	536	8	τ∗η	τ∗η	PUNCT
ma-246	536	9	.	.	PUNCT
ma-246	537	1	�	�	PROPN
ma-246	537	2	5	5	NUM
ma-246	537	3	.	.	PUNCT
ma-246	537	4	conclusion	conclusion	VERB
ma-246	537	5	the	the	DET
ma-246	537	6	concept	concept	NOUN
ma-246	537	7	of	of	ADP
ma-246	537	8	the	the	DET
ma-246	537	9	η	η	ADJ
ma-246	537	10	-	-	ADJ
ma-246	537	11	local	local	ADJ
ma-246	537	12	function	function	NOUN
ma-246	537	13	and	and	CCONJ
ma-246	537	14	the	the	DET
ma-246	537	15	closure	closure	NOUN
ma-246	537	16	cl∗η	cl∗η	PROPN
ma-246	537	17	has	have	AUX
ma-246	537	18	been	be	AUX
ma-246	537	19	introduced	introduce	VERB
ma-246	537	20	and	and	CCONJ
ma-246	537	21	demonstratedthrough	demonstratedthrough	ADJ
ma-246	537	22	illustrative	illustrative	ADJ
ma-246	537	23	examples	example	NOUN
ma-246	537	24	.	.	PUNCT
ma-246	538	1	additionally	additionally	ADV
ma-246	538	2	,	,	PUNCT
ma-246	538	3	certain	certain	ADJ
ma-246	538	4	properties	property	NOUN
ma-246	538	5	have	have	AUX
ma-246	538	6	been	be	AUX
ma-246	538	7	studied	study	VERB
ma-246	538	8	and	and	CCONJ
ma-246	538	9	explored	explore	VERB
ma-246	538	10	.	.	PUNCT
ma-246	539	1	itcan	itcan	PROPN
ma-246	539	2	be	be	AUX
ma-246	539	3	concluded	conclude	VERB
ma-246	539	4	that	that	SCONJ
ma-246	539	5	the	the	DET
ma-246	539	6	closure	closure	NOUN
ma-246	539	7	cl∗η	cl∗η	PROPN
ma-246	539	8	can	can	AUX
ma-246	539	9	only	only	ADV
ma-246	539	10	be	be	AUX
ma-246	539	11	a	a	DET
ma-246	539	12	kuratowski	kuratowski	ADJ
ma-246	539	13	closure	closure	NOUN
ma-246	539	14	operator	operator	NOUN
ma-246	539	15	(	(	PUNCT
ma-246	539	16	almocera	almocera	NOUN
ma-246	539	17	closureoperator	closureoperator	NOUN
ma-246	539	18	)	)	PUNCT
ma-246	539	19	if	if	SCONJ
ma-246	539	20	η	η	PROPN
ma-246	539	21	-	-	PROPN
ma-246	539	22	o(x	o(x	ADJ
ma-246	539	23	)	)	PUNCT
ma-246	539	24	is	be	AUX
ma-246	539	25	closed	close	VERB
ma-246	539	26	under	under	ADP
ma-246	539	27	two	two	NUM
ma-246	539	28	intersections	intersection	NOUN
ma-246	539	29	.	.	PUNCT
ma-246	540	1	under	under	ADP
ma-246	540	2	this	this	DET
ma-246	540	3	condition	condition	NOUN
ma-246	540	4	,	,	PUNCT
ma-246	540	5	τ∗η	τ∗η	PUNCT
ma-246	540	6	can	can	AUX
ma-246	540	7	form	form	VERB
ma-246	540	8	a	a	DET
ma-246	540	9	topology	topology	NOUN
ma-246	540	10	,	,	PUNCT
ma-246	540	11	making	make	VERB
ma-246	540	12	τ∗η	τ∗η	PUNCT
ma-246	540	13	a	a	DET
ma-246	540	14	more	more	ADV
ma-246	540	15	generalized	generalized	ADJ
ma-246	540	16	version	version	NOUN
ma-246	540	17	of	of	ADP
ma-246	540	18	τ∗	τ∗	NOUN
ma-246	540	19	and	and	CCONJ
ma-246	540	20	η	η	PROPN
ma-246	540	21	-	-	PROPN
ma-246	540	22	o(x	o(x	PROPN
ma-246	540	23	)	)	PUNCT
ma-246	540	24	.	.	PUNCT
ma-246	541	1	references	reference	NOUN
ma-246	541	2	[	[	X
ma-246	541	3	1	1	NUM
ma-246	541	4	]	]	PUNCT
ma-246	541	5	a.	a.	PROPN
ma-246	541	6	al	al	PROPN
ma-246	541	7	-	-	PUNCT
ma-246	541	8	omari	omari	PROPN
ma-246	541	9	,	,	PUNCT
ma-246	541	10	t.	t.	PROPN
ma-246	541	11	noiri	noiri	PROPN
ma-246	541	12	,	,	PUNCT
ma-246	541	13	local	local	ADJ
ma-246	541	14	function	function	NOUN
ma-246	541	15	γ∗	γ∗	NOUN
ma-246	541	16	in	in	ADP
ma-246	541	17	ideal	ideal	ADJ
ma-246	541	18	topological	topological	ADJ
ma-246	541	19	spaces	space	NOUN
ma-246	541	20	,	,	PUNCT
ma-246	541	21	sci	sci	PROPN
ma-246	541	22	.	.	PROPN
ma-246	541	23	stud	stud	PROPN
ma-246	541	24	.	.	PUNCT
ma-246	542	1	res	re	NOUN
ma-246	542	2	.	.	PUNCT
ma-246	542	3	ser	ser	PROPN
ma-246	542	4	.	.	PROPN
ma-246	542	5	math	math	PROPN
ma-246	542	6	.	.	PUNCT
ma-246	543	1	inform	inform	NOUN
ma-246	543	2	.	.	PUNCT
ma-246	544	1	26	26	NUM
ma-246	544	2	(	(	PUNCT
ma-246	544	3	1	1	NUM
ma-246	544	4	)	)	PUNCT
ma-246	544	5	(	(	PUNCT
ma-246	544	6	2016)5	2016)5	NUM
ma-246	544	7	-	-	SYM
ma-246	544	8	16.[2	16.[2	NUM
ma-246	544	9	]	]	X
ma-246	544	10	e.	e.	PROPN
ma-246	544	11	hatir	hatir	PROPN
ma-246	544	12	,	,	PUNCT
ma-246	544	13	a.	a.	PROPN
ma-246	544	14	al	al	PROPN
ma-246	544	15	-	-	PUNCT
ma-246	544	16	omari	omari	PROPN
ma-246	544	17	,	,	PUNCT
ma-246	544	18	s.	s.	PROPN
ma-246	544	19	jafari	jafari	PROPN
ma-246	544	20	,	,	PUNCT
ma-246	544	21	δ	δ	PROPN
ma-246	544	22	-	-	ADJ
ma-246	544	23	local	local	ADJ
ma-246	544	24	functions	function	NOUN
ma-246	544	25	and	and	CCONJ
ma-246	544	26	its	its	PRON
ma-246	544	27	properties	property	NOUN
ma-246	544	28	in	in	ADP
ma-246	544	29	ideal	ideal	ADJ
ma-246	544	30	topological	topological	ADJ
ma-246	544	31	spaces	space	NOUN
ma-246	544	32	,	,	PUNCT
ma-246	544	33	fasciculi	fasciculi	PROPN
ma-246	544	34	math	math	PROPN
ma-246	544	35	.	.	PUNCT
ma-246	545	1	53(2014	53(2014	PROPN
ma-246	545	2	)	)	PUNCT
ma-246	545	3	53	53	NUM
ma-246	545	4	-	-	PUNCT
ma-246	545	5	64.[3	64.[3	NUM
ma-246	545	6	]	]	PUNCT
ma-246	545	7	d.	d.	PROPN
ma-246	545	8	jankovic	jankovic	PROPN
ma-246	545	9	,	,	PUNCT
ma-246	545	10	t.r	t.r	PROPN
ma-246	545	11	.	.	PROPN
ma-246	545	12	hamlett	hamlett	PROPN
ma-246	545	13	,	,	PUNCT
ma-246	545	14	new	new	ADJ
ma-246	545	15	topologies	topology	NOUN
ma-246	545	16	from	from	ADP
ma-246	545	17	old	old	ADJ
ma-246	545	18	via	via	ADP
ma-246	545	19	ideals	ideal	NOUN
ma-246	545	20	,	,	PUNCT
ma-246	545	21	amer	amer	PROPN
ma-246	545	22	.	.	PROPN
ma-246	545	23	math	math	PROPN
ma-246	545	24	.	.	PUNCT
ma-246	546	1	monthly	monthly	ADJ
ma-246	546	2	97	97	NUM
ma-246	546	3	(	(	PUNCT
ma-246	546	4	4	4	NUM
ma-246	546	5	)	)	PUNCT
ma-246	546	6	(	(	PUNCT
ma-246	546	7	1990	1990	NUM
ma-246	546	8	)	)	PUNCT
ma-246	546	9	295	295	NUM
ma-246	546	10	-	-	SYM
ma-246	546	11	310.[4	310.[4	NUM
ma-246	546	12	]	]	PUNCT
ma-246	546	13	k.	k.	PROPN
ma-246	546	14	kuratowski	kuratowski	PROPN
ma-246	546	15	,	,	PUNCT
ma-246	546	16	topology	topology	PROPN
ma-246	546	17	i	i	PRON
ma-246	546	18	,	,	PUNCT
ma-246	546	19	warszawa	warszawa	PROPN
ma-246	546	20	,	,	PUNCT
ma-246	546	21	1933.[5	1933.[5	NUM
ma-246	546	22	]	]	X
ma-246	546	23	k.	k.	PROPN
ma-246	546	24	kuratowski	kuratowski	PROPN
ma-246	546	25	,	,	PUNCT
ma-246	546	26	topology	topology	NOUN
ma-246	546	27	,	,	PUNCT
ma-246	546	28	academic	academic	ADJ
ma-246	546	29	press	press	NOUN
ma-246	546	30	,	,	PUNCT
ma-246	546	31	new	new	PROPN
ma-246	546	32	york	york	PROPN
ma-246	546	33	,	,	PUNCT
ma-246	546	34	1966.[6	1966.[6	NUM
ma-246	546	35	]	]	X
ma-246	546	36	n.	n.	PROPN
ma-246	546	37	levine	levine	PROPN
ma-246	546	38	,	,	PUNCT
ma-246	546	39	semi	semi	ADJ
ma-246	546	40	-	-	ADJ
ma-246	546	41	open	open	ADJ
ma-246	546	42	sets	set	NOUN
ma-246	546	43	and	and	CCONJ
ma-246	546	44	semi	semi	ADJ
ma-246	546	45	-	-	NOUN
ma-246	546	46	continuity	continuity	NOUN
ma-246	546	47	in	in	ADP
ma-246	546	48	topological	topological	ADJ
ma-246	546	49	spaces	space	NOUN
ma-246	546	50	,	,	PUNCT
ma-246	546	51	amer	amer	PROPN
ma-246	546	52	.	.	PROPN
ma-246	546	53	math	math	PROPN
ma-246	546	54	.	.	PUNCT
ma-246	547	1	monthly	monthly	ADJ
ma-246	547	2	70	70	NUM
ma-246	547	3	(	(	PUNCT
ma-246	547	4	1963	1963	NUM
ma-246	547	5	)	)	PUNCT
ma-246	547	6	36	36	NUM
ma-246	547	7	-	-	SYM
ma-246	547	8	41.[7	41.[7	NUM
ma-246	547	9	]	]	X
ma-246	547	10	p.l	p.l	PROPN
ma-246	547	11	.	.	PROPN
ma-246	547	12	powar	powar	PROPN
ma-246	547	13	,	,	PUNCT
ma-246	547	14	k.	k.	PROPN
ma-246	547	15	rajak	rajak	PROPN
ma-246	547	16	,	,	PUNCT
ma-246	547	17	some	some	DET
ma-246	547	18	new	new	ADJ
ma-246	547	19	concepts	concept	NOUN
ma-246	547	20	of	of	ADP
ma-246	547	21	continuity	continuity	NOUN
ma-246	547	22	in	in	ADP
ma-246	547	23	generalized	generalized	ADJ
ma-246	547	24	topological	topological	ADJ
ma-246	547	25	space	space	NOUN
ma-246	547	26	,	,	PUNCT
ma-246	547	27	int	int	NOUN
ma-246	547	28	.	.	PUNCT
ma-246	548	1	j.	j.	PROPN
ma-246	548	2	com	com	PROPN
ma-246	548	3	.	.	PUNCT
ma-246	548	4	appl	appl	PROPN
ma-246	548	5	.	.	PUNCT
ma-246	549	1	38	38	NUM
ma-246	549	2	(	(	PUNCT
ma-246	549	3	5)(2012	5)(2012	NUM
ma-246	549	4	)	)	PUNCT
ma-246	549	5	12	12	NUM
ma-246	549	6	-	-	SYM
ma-246	549	7	17.[8	17.[8	PROPN
ma-246	549	8	]	]	X
ma-246	549	9	d.	d.	PROPN
ma-246	549	10	subbulakshmi	subbulakshmi	PROPN
ma-246	549	11	,	,	PUNCT
ma-246	549	12	k.	k.	PROPN
ma-246	549	13	sumathi	sumathi	PROPN
ma-246	549	14	,	,	PUNCT
ma-246	549	15	k.	k.	PROPN
ma-246	549	16	indiran	indiran	PROPN
ma-246	549	17	,	,	PUNCT
ma-246	549	18	η	η	ADJ
ma-246	549	19	-	-	ADJ
ma-246	549	20	open	open	ADJ
ma-246	549	21	sets	set	NOUN
ma-246	549	22	in	in	ADP
ma-246	549	23	topological	topological	ADJ
ma-246	549	24	,	,	PUNCT
ma-246	549	25	int	int	NOUN
ma-246	549	26	.	.	PUNCT
ma-246	550	1	j.	j.	PROPN
ma-246	550	2	innov	innov	PROPN
ma-246	550	3	.	.	PUNCT
ma-246	551	1	techno	techno	PROPN
ma-246	551	2	.	.	PUNCT
ma-246	552	1	explor	explor	PROPN
ma-246	552	2	.	.	PUNCT
ma-246	553	1	eng	eng	PROPN
ma-246	553	2	.	.	PROPN
ma-246	553	3	8	8	NUM
ma-246	553	4	(	(	PUNCT
ma-246	553	5	10s)(2019	10s)(2019	NUM
ma-246	553	6	)	)	PUNCT
ma-246	553	7	,	,	PUNCT
ma-246	553	8	276	276	NUM
ma-246	553	9	-	-	SYM
ma-246	553	10	282.[9	282.[9	NUM
ma-246	553	11	]	]	PUNCT
ma-246	553	12	r.	r.	NOUN
ma-246	553	13	vaidyanathaswamy	vaidyanathaswamy	PROPN
ma-246	553	14	,	,	PUNCT
ma-246	553	15	the	the	DET
ma-246	553	16	localization	localization	NOUN
ma-246	553	17	theory	theory	NOUN
ma-246	553	18	in	in	ADP
ma-246	553	19	set	set	NOUN
ma-246	553	20	-	-	PUNCT
ma-246	553	21	topology	topology	NOUN
ma-246	553	22	,	,	PUNCT
ma-246	553	23	proc	proc	NOUN
ma-246	553	24	.	.	PUNCT
ma-246	554	1	indian	indian	PROPN
ma-246	554	2	acad	acad	PROPN
ma-246	554	3	.	.	PUNCT
ma-246	555	1	sci	sci	PROPN
ma-246	555	2	.	.	PUNCT
ma-246	555	3	sect	sect	NOUN
ma-246	555	4	.	.	PUNCT
ma-246	556	1	20	20	NUM
ma-246	556	2	(	(	PUNCT
ma-246	556	3	1944	1944	NUM
ma-246	556	4	)	)	PUNCT
ma-246	556	5	51	51	NUM
ma-246	556	6	-	-	SYM
ma-246	556	7	61	61	NUM
ma-246	556	8	.	.	PUNCT
ma-246	557	1	https://doi.org/10.28924/ada/ma.5.2	https://doi.org/10.28924/ada/ma.5.2	NUM
ma-246	557	2	1	1	NUM
ma-246	557	3	.	.	PUNCT
ma-246	557	4	introduction	introduction	NOUN
ma-246	557	5	2	2	NUM
ma-246	557	6	.	.	PUNCT
ma-246	557	7	preliminaries	preliminary	NOUN
ma-246	557	8	3	3	NUM
ma-246	557	9	.	.	X
ma-246	557	10	-local	-local	ADJ
ma-246	557	11	functions	function	NOUN
ma-246	557	12	4	4	NUM
ma-246	557	13	.	.	PUNCT
ma-246	558	1	-local	-local	ADJ
ma-246	558	2	closure	closure	NOUN
ma-246	558	3	5	5	NUM
ma-246	558	4	.	.	PUNCT
ma-246	558	5	conclusion	conclusion	NOUN
ma-246	558	6	references	reference	NOUN