id	sid	tid	token	lemma	pos
ejpam-4465	1	1	european	european	PROPN
ejpam-4465	1	2	journal	journal	PROPN
ejpam-4465	1	3	of	of	ADP
ejpam-4465	1	4	pure	pure	ADJ
ejpam-4465	1	5	and	and	CCONJ
ejpam-4465	1	6	applied	apply	VERB
ejpam-4465	1	7	mathematics	mathematic	NOUN
ejpam-4465	1	8	vol	vol	NOUN
ejpam-4465	1	9	.	.	PROPN
ejpam-4465	2	1	15	15	NUM
ejpam-4465	2	2	,	,	PUNCT
ejpam-4465	2	3	no	no	INTJ
ejpam-4465	2	4	.	.	NOUN
ejpam-4465	2	5	3	3	NUM
ejpam-4465	2	6	,	,	PUNCT
ejpam-4465	2	7	2022	2022	NUM
ejpam-4465	2	8	,	,	PUNCT
ejpam-4465	2	9	1344	1344	NUM
ejpam-4465	2	10	-	-	SYM
ejpam-4465	2	11	1347	1347	NUM
ejpam-4465	2	12	issn	issn	PROPN
ejpam-4465	2	13	1307	1307	NUM
ejpam-4465	2	14	-	-	SYM
ejpam-4465	2	15	5543	5543	NUM
ejpam-4465	2	16	–	–	PUNCT
ejpam-4465	2	17	ejpam.com	ejpam.com	X
ejpam-4465	2	18	published	publish	VERB
ejpam-4465	2	19	by	by	ADP
ejpam-4465	2	20	new	new	PROPN
ejpam-4465	2	21	york	york	PROPN
ejpam-4465	2	22	business	business	PROPN
ejpam-4465	2	23	global	global	PROPN
ejpam-4465	2	24	a	a	DET
ejpam-4465	2	25	short	short	ADJ
ejpam-4465	2	26	introduction	introduction	NOUN
ejpam-4465	2	27	to	to	ADP
ejpam-4465	2	28	similarity	similarity	NOUN
ejpam-4465	2	29	via	via	ADP
ejpam-4465	2	30	ideals	ideal	NOUN
ejpam-4465	2	31	asli	asli	PROPN
ejpam-4465	2	32	guldurdek	guldurdek	PROPN
ejpam-4465	2	33	college	college	PROPN
ejpam-4465	2	34	of	of	ADP
ejpam-4465	2	35	engineering	engineering	NOUN
ejpam-4465	2	36	and	and	CCONJ
ejpam-4465	2	37	technology	technology	NOUN
ejpam-4465	2	38	,	,	PUNCT
ejpam-4465	2	39	american	american	PROPN
ejpam-4465	2	40	university	university	PROPN
ejpam-4465	2	41	of	of	ADP
ejpam-4465	2	42	the	the	DET
ejpam-4465	2	43	middle	middle	PROPN
ejpam-4465	2	44	east	east	PROPN
ejpam-4465	2	45	,	,	PUNCT
ejpam-4465	2	46	egaila	egaila	PROPN
ejpam-4465	2	47	54200	54200	NUM
ejpam-4465	2	48	,	,	PUNCT
ejpam-4465	2	49	kuwait	kuwait	PROPN
ejpam-4465	2	50	abstract	abstract	NOUN
ejpam-4465	2	51	.	.	PUNCT
ejpam-4465	3	1	in	in	ADP
ejpam-4465	3	2	this	this	DET
ejpam-4465	3	3	work	work	NOUN
ejpam-4465	3	4	,	,	PUNCT
ejpam-4465	3	5	we	we	PRON
ejpam-4465	3	6	introduce	introduce	VERB
ejpam-4465	3	7	the	the	DET
ejpam-4465	3	8	notion	notion	NOUN
ejpam-4465	3	9	of	of	ADP
ejpam-4465	3	10	similarity	similarity	NOUN
ejpam-4465	3	11	between	between	ADP
ejpam-4465	3	12	topologies	topology	NOUN
ejpam-4465	3	13	τ1	τ1	NOUN
ejpam-4465	3	14	and	and	CCONJ
ejpam-4465	3	15	τ2	τ2	NOUN
ejpam-4465	3	16	,	,	PUNCT
ejpam-4465	3	17	on	on	ADP
ejpam-4465	3	18	a	a	DET
ejpam-4465	3	19	set	set	NOUN
ejpam-4465	3	20	x	x	PUNCT
ejpam-4465	3	21	via	via	ADP
ejpam-4465	3	22	ideals	ideal	NOUN
ejpam-4465	3	23	.	.	PUNCT
ejpam-4465	4	1	then	then	ADV
ejpam-4465	4	2	,	,	PUNCT
ejpam-4465	4	3	we	we	PRON
ejpam-4465	4	4	give	give	VERB
ejpam-4465	4	5	some	some	DET
ejpam-4465	4	6	characterizations	characterization	NOUN
ejpam-4465	4	7	regarding	regard	VERB
ejpam-4465	4	8	this	this	DET
ejpam-4465	4	9	kind	kind	NOUN
ejpam-4465	4	10	of	of	ADP
ejpam-4465	4	11	similarity	similarity	NOUN
ejpam-4465	4	12	by	by	ADP
ejpam-4465	4	13	using	use	VERB
ejpam-4465	4	14	∗−dense	∗−dense	NOUN
ejpam-4465	4	15	,	,	PUNCT
ejpam-4465	4	16	and	and	CCONJ
ejpam-4465	4	17	i−dense	i−dense	NOUN
ejpam-4465	4	18	subsets	subset	NOUN
ejpam-4465	4	19	.	.	PUNCT
ejpam-4465	5	1	we	we	PRON
ejpam-4465	5	2	also	also	ADV
ejpam-4465	5	3	examine	examine	VERB
ejpam-4465	5	4	the	the	DET
ejpam-4465	5	5	preservation	preservation	NOUN
ejpam-4465	5	6	of	of	ADP
ejpam-4465	5	7	similarity	similarity	NOUN
ejpam-4465	5	8	with	with	ADP
ejpam-4465	5	9	respect	respect	NOUN
ejpam-4465	5	10	to	to	ADP
ejpam-4465	5	11	the	the	DET
ejpam-4465	5	12	topologies	topology	NOUN
ejpam-4465	5	13	τ∗1	τ∗1	ADP
ejpam-4465	5	14	and	and	CCONJ
ejpam-4465	5	15	τ∗2	τ∗2	NOUN
ejpam-4465	5	16	.	.	PUNCT
ejpam-4465	6	1	2020	2020	NUM
ejpam-4465	6	2	mathematics	mathematic	NOUN
ejpam-4465	6	3	subject	subject	NOUN
ejpam-4465	6	4	classifications	classification	NOUN
ejpam-4465	6	5	:	:	PUNCT
ejpam-4465	6	6	54a05	54a05	NUM
ejpam-4465	6	7	,	,	PUNCT
ejpam-4465	6	8	54a10	54a10	NUM
ejpam-4465	6	9	,	,	PUNCT
ejpam-4465	6	10	54d80	54d80	NUM
ejpam-4465	6	11	key	key	ADJ
ejpam-4465	6	12	words	word	NOUN
ejpam-4465	6	13	and	and	CCONJ
ejpam-4465	6	14	phrases	phrase	NOUN
ejpam-4465	6	15	:	:	PUNCT
ejpam-4465	6	16	ideal	ideal	ADJ
ejpam-4465	6	17	topological	topological	ADJ
ejpam-4465	6	18	space	space	NOUN
ejpam-4465	6	19	,	,	PUNCT
ejpam-4465	6	20	similarity	similarity	NOUN
ejpam-4465	6	21	1	1	NUM
ejpam-4465	6	22	.	.	PUNCT
ejpam-4465	7	1	introduction	introduction	NOUN
ejpam-4465	7	2	and	and	CCONJ
ejpam-4465	7	3	preliminaries	preliminary	NOUN
ejpam-4465	7	4	the	the	DET
ejpam-4465	7	5	idea	idea	NOUN
ejpam-4465	7	6	of	of	ADP
ejpam-4465	7	7	adding	add	VERB
ejpam-4465	7	8	the	the	DET
ejpam-4465	7	9	notion	notion	NOUN
ejpam-4465	7	10	of	of	ADP
ejpam-4465	7	11	ideal	ideal	NOUN
ejpam-4465	7	12	into	into	ADP
ejpam-4465	7	13	the	the	DET
ejpam-4465	7	14	topological	topological	ADJ
ejpam-4465	7	15	spaces	space	NOUN
ejpam-4465	7	16	started	start	VERB
ejpam-4465	7	17	with	with	ADP
ejpam-4465	7	18	the	the	DET
ejpam-4465	7	19	works	work	NOUN
ejpam-4465	7	20	of	of	ADP
ejpam-4465	7	21	kuratowski	kuratowski	NOUN
ejpam-4465	7	22	[	[	X
ejpam-4465	7	23	6	6	NUM
ejpam-4465	7	24	]	]	PUNCT
ejpam-4465	7	25	,	,	PUNCT
ejpam-4465	7	26	and	and	CCONJ
ejpam-4465	7	27	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-4465	7	28	[	[	X
ejpam-4465	7	29	7	7	NUM
ejpam-4465	7	30	]	]	PUNCT
ejpam-4465	7	31	.	.	PUNCT
ejpam-4465	8	1	after	after	ADP
ejpam-4465	8	2	that	that	PRON
ejpam-4465	8	3	the	the	DET
ejpam-4465	8	4	notion	notion	NOUN
ejpam-4465	8	5	of	of	ADP
ejpam-4465	8	6	ideal	ideal	ADJ
ejpam-4465	8	7	topological	topological	ADJ
ejpam-4465	8	8	space	space	NOUN
ejpam-4465	8	9	and	and	CCONJ
ejpam-4465	8	10	applications	application	NOUN
ejpam-4465	8	11	have	have	AUX
ejpam-4465	8	12	been	be	AUX
ejpam-4465	8	13	examined	examine	VERB
ejpam-4465	8	14	deeply	deeply	ADV
ejpam-4465	8	15	.	.	PUNCT
ejpam-4465	9	1	an	an	DET
ejpam-4465	9	2	ideal	ideal	NOUN
ejpam-4465	9	3	i	i	PRON
ejpam-4465	9	4	on	on	ADP
ejpam-4465	9	5	a	a	DET
ejpam-4465	9	6	set	set	NOUN
ejpam-4465	9	7	x	x	PUNCT
ejpam-4465	9	8	is	be	AUX
ejpam-4465	9	9	a	a	DET
ejpam-4465	9	10	nonempty	nonempty	ADJ
ejpam-4465	9	11	collection	collection	NOUN
ejpam-4465	9	12	of	of	ADP
ejpam-4465	9	13	subsets	subset	NOUN
ejpam-4465	9	14	of	of	ADP
ejpam-4465	9	15	x	x	PROPN
ejpam-4465	9	16	,	,	PUNCT
ejpam-4465	9	17	which	which	PRON
ejpam-4465	9	18	satisfies	satisfy	VERB
ejpam-4465	9	19	the	the	DET
ejpam-4465	9	20	following	follow	VERB
ejpam-4465	9	21	conditions	condition	NOUN
ejpam-4465	9	22	:	:	PUNCT
ejpam-4465	9	23	i.	i.	NOUN
ejpam-4465	9	24	if	if	SCONJ
ejpam-4465	9	25	a	a	DET
ejpam-4465	9	26	∈	∈	PROPN
ejpam-4465	9	27	i	i	X
ejpam-4465	9	28	,	,	PUNCT
ejpam-4465	9	29	and	and	CCONJ
ejpam-4465	10	1	b	b	X
ejpam-4465	10	2	⊂	⊂	PROPN
ejpam-4465	10	3	a	a	X
ejpam-4465	10	4	,	,	PUNCT
ejpam-4465	10	5	then	then	ADV
ejpam-4465	10	6	b	b	X
ejpam-4465	10	7	∈	∈	PROPN
ejpam-4465	10	8	i	i	PROPN
ejpam-4465	10	9	ii	ii	PROPN
ejpam-4465	10	10	.	.	PUNCT
ejpam-4465	11	1	if	if	SCONJ
ejpam-4465	11	2	a	a	PRON
ejpam-4465	11	3	,	,	PUNCT
ejpam-4465	11	4	b	b	X
ejpam-4465	11	5	∈	∈	PROPN
ejpam-4465	12	1	i	i	PRON
ejpam-4465	12	2	,	,	PUNCT
ejpam-4465	12	3	then	then	ADV
ejpam-4465	12	4	a	a	DET
ejpam-4465	12	5	∪b	∪b	X
ejpam-4465	12	6	∈	∈	PROPN
ejpam-4465	12	7	i.	i.	NOUN
ejpam-4465	12	8	we	we	PRON
ejpam-4465	12	9	denote	denote	VERB
ejpam-4465	12	10	a	a	DET
ejpam-4465	12	11	topological	topological	ADJ
ejpam-4465	12	12	space	space	NOUN
ejpam-4465	12	13	(	(	PUNCT
ejpam-4465	12	14	x	x	X
ejpam-4465	12	15	,	,	PUNCT
ejpam-4465	12	16	τ	τ	X
ejpam-4465	12	17	)	)	PUNCT
ejpam-4465	12	18	with	with	ADP
ejpam-4465	12	19	an	an	DET
ejpam-4465	12	20	ideal	ideal	NOUN
ejpam-4465	12	21	i	i	PRON
ejpam-4465	12	22	defined	define	VERB
ejpam-4465	12	23	on	on	ADP
ejpam-4465	12	24	x	x	PUNCT
ejpam-4465	12	25	by	by	ADP
ejpam-4465	12	26	(	(	PUNCT
ejpam-4465	12	27	x	x	NOUN
ejpam-4465	12	28	,	,	PUNCT
ejpam-4465	12	29	τ	τ	PROPN
ejpam-4465	12	30	,	,	PUNCT
ejpam-4465	12	31	i	i	PROPN
ejpam-4465	12	32	)	)	PUNCT
ejpam-4465	12	33	.	.	PUNCT
ejpam-4465	13	1	an	an	DET
ejpam-4465	13	2	ideal	ideal	NOUN
ejpam-4465	13	3	i	i	PRON
ejpam-4465	13	4	on	on	ADP
ejpam-4465	13	5	(	(	PUNCT
ejpam-4465	13	6	x	x	X
ejpam-4465	13	7	,	,	PUNCT
ejpam-4465	13	8	τ	τ	X
ejpam-4465	13	9	)	)	PUNCT
ejpam-4465	13	10	is	be	AUX
ejpam-4465	13	11	said	say	VERB
ejpam-4465	13	12	to	to	PART
ejpam-4465	13	13	be	be	AUX
ejpam-4465	13	14	τ	τ	NOUN
ejpam-4465	13	15	-codense	-codense	NOUN
ejpam-4465	13	16	if	if	SCONJ
ejpam-4465	13	17	i	i	PRON
ejpam-4465	13	18	∩	∩	NOUN
ejpam-4465	13	19	τ	τ	X
ejpam-4465	13	20	=	=	PUNCT
ejpam-4465	13	21	{	{	PUNCT
ejpam-4465	13	22	∅	∅	NOUN
ejpam-4465	13	23	}	}	PUNCT
ejpam-4465	13	24	.	.	PUNCT
ejpam-4465	14	1	on	on	ADP
ejpam-4465	14	2	the	the	DET
ejpam-4465	14	3	other	other	ADJ
ejpam-4465	14	4	hand	hand	NOUN
ejpam-4465	14	5	,	,	PUNCT
ejpam-4465	14	6	there	there	PRON
ejpam-4465	14	7	are	be	VERB
ejpam-4465	14	8	many	many	ADJ
ejpam-4465	14	9	papers	paper	NOUN
ejpam-4465	14	10	devoted	devote	VERB
ejpam-4465	14	11	to	to	ADP
ejpam-4465	14	12	constructing	construct	VERB
ejpam-4465	14	13	new	new	ADJ
ejpam-4465	14	14	topologies	topology	NOUN
ejpam-4465	14	15	via	via	ADP
ejpam-4465	14	16	ideals	ideal	NOUN
ejpam-4465	14	17	.	.	PUNCT
ejpam-4465	15	1	to	to	PART
ejpam-4465	15	2	do	do	VERB
ejpam-4465	15	3	that	that	PRON
ejpam-4465	15	4	,	,	PUNCT
ejpam-4465	15	5	firstly	firstly	ADV
ejpam-4465	15	6	an	an	DET
ejpam-4465	15	7	operator	operator	NOUN
ejpam-4465	15	8	called	call	VERB
ejpam-4465	15	9	the	the	DET
ejpam-4465	15	10	local	local	ADJ
ejpam-4465	15	11	function	function	NOUN
ejpam-4465	15	12	is	be	AUX
ejpam-4465	15	13	invented	invent	VERB
ejpam-4465	15	14	.	.	PUNCT
ejpam-4465	16	1	then	then	ADV
ejpam-4465	16	2	by	by	ADP
ejpam-4465	16	3	using	use	VERB
ejpam-4465	16	4	this	this	PRON
ejpam-4465	16	5	,	,	PUNCT
ejpam-4465	16	6	one	one	PRON
ejpam-4465	16	7	can	can	AUX
ejpam-4465	16	8	get	get	VERB
ejpam-4465	16	9	a	a	DET
ejpam-4465	16	10	kuratowski	kuratowski	ADJ
ejpam-4465	16	11	closure	closure	NOUN
ejpam-4465	16	12	operator	operator	NOUN
ejpam-4465	16	13	.	.	PUNCT
ejpam-4465	17	1	definition	definition	NOUN
ejpam-4465	17	2	1	1	NUM
ejpam-4465	17	3	(	(	PUNCT
ejpam-4465	17	4	[	[	X
ejpam-4465	17	5	6	6	NUM
ejpam-4465	17	6	]	]	PUNCT
ejpam-4465	17	7	)	)	PUNCT
ejpam-4465	17	8	.	.	PUNCT
ejpam-4465	18	1	let	let	VERB
ejpam-4465	18	2	(	(	PUNCT
ejpam-4465	18	3	x	x	NOUN
ejpam-4465	18	4	,	,	PUNCT
ejpam-4465	18	5	τ	τ	X
ejpam-4465	18	6	)	)	PUNCT
ejpam-4465	18	7	be	be	VERB
ejpam-4465	18	8	a	a	DET
ejpam-4465	18	9	topological	topological	ADJ
ejpam-4465	18	10	space	space	NOUN
ejpam-4465	18	11	,	,	PUNCT
ejpam-4465	18	12	and	and	CCONJ
ejpam-4465	18	13	i	i	PRON
ejpam-4465	18	14	be	be	VERB
ejpam-4465	18	15	an	an	DET
ejpam-4465	18	16	ideal	ideal	NOUN
ejpam-4465	18	17	on	on	ADP
ejpam-4465	18	18	x.	x.	NOUN
ejpam-4465	18	19	then	then	ADV
ejpam-4465	18	20	the	the	DET
ejpam-4465	18	21	local	local	ADJ
ejpam-4465	18	22	function	function	NOUN
ejpam-4465	18	23	a∗(i	a∗(i	PROPN
ejpam-4465	18	24	,	,	PUNCT
ejpam-4465	18	25	τ	τ	X
ejpam-4465	18	26	)	)	PUNCT
ejpam-4465	18	27	of	of	ADP
ejpam-4465	18	28	a	a	DET
ejpam-4465	18	29	⊂	⊂	X
ejpam-4465	18	30	x	x	X
ejpam-4465	18	31	is	be	AUX
ejpam-4465	18	32	defined	define	VERB
ejpam-4465	18	33	as	as	ADP
ejpam-4465	18	34	following	follow	VERB
ejpam-4465	18	35	:	:	PUNCT
ejpam-4465	18	36	a∗(i	a∗(i	PROPN
ejpam-4465	18	37	,	,	PUNCT
ejpam-4465	18	38	τ	τ	X
ejpam-4465	18	39	)	)	PUNCT
ejpam-4465	18	40	=	=	PRON
ejpam-4465	19	1	{	{	PUNCT
ejpam-4465	19	2	x	x	PUNCT
ejpam-4465	19	3	∈	∈	NOUN
ejpam-4465	19	4	x	x	X
ejpam-4465	19	5	|	|	ADV
ejpam-4465	19	6	u	u	X
ejpam-4465	20	1	∩a	∩a	PROPN
ejpam-4465	20	2	/∈	/∈	PUNCT
ejpam-4465	21	1	i	i	PRON
ejpam-4465	21	2	for	for	ADP
ejpam-4465	21	3	every	every	DET
ejpam-4465	21	4	u	u	PROPN
ejpam-4465	21	5	∈	∈	PROPN
ejpam-4465	21	6	τ(x	τ(x	NOUN
ejpam-4465	21	7	)	)	PUNCT
ejpam-4465	21	8	}	}	PUNCT
ejpam-4465	21	9	where	where	SCONJ
ejpam-4465	21	10	τ(x	τ(x	NOUN
ejpam-4465	21	11	)	)	PUNCT
ejpam-4465	21	12	=	=	PRON
ejpam-4465	21	13	{	{	PUNCT
ejpam-4465	21	14	u	u	X
ejpam-4465	21	15	∈	∈	PROPN
ejpam-4465	21	16	τ	τ	X
ejpam-4465	21	17	|	|	ADV
ejpam-4465	21	18	x	x	X
ejpam-4465	21	19	∈	∈	PROPN
ejpam-4465	21	20	u	u	NOUN
ejpam-4465	21	21	}	}	PUNCT
ejpam-4465	21	22	.	.	PUNCT
ejpam-4465	22	1	doi	doi	NOUN
ejpam-4465	22	2	:	:	PUNCT
ejpam-4465	22	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4465	https://doi.org/10.29020/nybg.ejpam.v15i3.4465	VERB
ejpam-4465	22	4	email	email	NOUN
ejpam-4465	22	5	address	address	NOUN
ejpam-4465	22	6	:	:	PUNCT
ejpam-4465	22	7	asli.guldurdek@aum.edu.kw	asli.guldurdek@aum.edu.kw	PROPN
ejpam-4465	22	8	(	(	PUNCT
ejpam-4465	22	9	asli	asli	PROPN
ejpam-4465	22	10	guldurdek	guldurdek	PROPN
ejpam-4465	22	11	)	)	PUNCT
ejpam-4465	22	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4465	22	13	1344	1344	NUM
ejpam-4465	23	1	©	©	PROPN
ejpam-4465	23	2	2022	2022	NUM
ejpam-4465	23	3	ejpam	ejpam	VERB
ejpam-4465	23	4	all	all	DET
ejpam-4465	23	5	rights	right	NOUN
ejpam-4465	23	6	reserved	reserve	VERB
ejpam-4465	23	7	.	.	PUNCT
ejpam-4465	24	1	a.	a.	PROPN
ejpam-4465	24	2	guldurdek	guldurdek	PROPN
ejpam-4465	24	3	/	/	SYM
ejpam-4465	24	4	eur	eur	PROPN
ejpam-4465	24	5	.	.	PUNCT
ejpam-4465	25	1	j.	j.	PROPN
ejpam-4465	25	2	pure	pure	PROPN
ejpam-4465	25	3	appl	appl	PROPN
ejpam-4465	25	4	.	.	PROPN
ejpam-4465	25	5	math	math	PROPN
ejpam-4465	25	6	,	,	PUNCT
ejpam-4465	25	7	15	15	NUM
ejpam-4465	25	8	(	(	PUNCT
ejpam-4465	25	9	3	3	NUM
ejpam-4465	25	10	)	)	PUNCT
ejpam-4465	25	11	(	(	PUNCT
ejpam-4465	25	12	2022	2022	NUM
ejpam-4465	25	13	)	)	PUNCT
ejpam-4465	25	14	,	,	PUNCT
ejpam-4465	25	15	1344	1344	NUM
ejpam-4465	25	16	-	-	SYM
ejpam-4465	25	17	1347	1347	NUM
ejpam-4465	25	18	1345	1345	NUM
ejpam-4465	25	19	one	one	NUM
ejpam-4465	25	20	can	can	AUX
ejpam-4465	25	21	see	see	VERB
ejpam-4465	25	22	that	that	PRON
ejpam-4465	25	23	,	,	PUNCT
ejpam-4465	25	24	(	(	PUNCT
ejpam-4465	25	25	.)∗	.)∗	X
ejpam-4465	25	26	:	:	PUNCT
ejpam-4465	25	27	p(x	p(x	PROPN
ejpam-4465	25	28	)	)	PUNCT
ejpam-4465	25	29	→	→	SYM
ejpam-4465	25	30	p(x	p(x	NOUN
ejpam-4465	25	31	)	)	PUNCT
ejpam-4465	25	32	satisfies	satisfy	VERB
ejpam-4465	25	33	the	the	DET
ejpam-4465	25	34	conditions	condition	NOUN
ejpam-4465	25	35	to	to	PART
ejpam-4465	25	36	make	make	VERB
ejpam-4465	25	37	c∗(a	c∗(a	NOUN
ejpam-4465	25	38	)	)	PUNCT
ejpam-4465	25	39	=	=	PUNCT
ejpam-4465	25	40	a	a	DET
ejpam-4465	25	41	∪a∗(i	∪a∗(i	NOUN
ejpam-4465	25	42	,	,	PUNCT
ejpam-4465	25	43	τ	τ	PROPN
ejpam-4465	25	44	)	)	PUNCT
ejpam-4465	25	45	a	a	DET
ejpam-4465	25	46	kuratowski	kuratowski	ADJ
ejpam-4465	25	47	closure	closure	NOUN
ejpam-4465	25	48	operator	operator	NOUN
ejpam-4465	25	49	.	.	PUNCT
ejpam-4465	26	1	definition	definition	NOUN
ejpam-4465	26	2	2	2	NUM
ejpam-4465	26	3	(	(	PUNCT
ejpam-4465	26	4	[	[	X
ejpam-4465	26	5	5	5	NUM
ejpam-4465	26	6	]	]	PUNCT
ejpam-4465	26	7	)	)	PUNCT
ejpam-4465	26	8	.	.	PUNCT
ejpam-4465	27	1	let	let	VERB
ejpam-4465	27	2	(	(	PUNCT
ejpam-4465	27	3	x	x	NOUN
ejpam-4465	27	4	,	,	PUNCT
ejpam-4465	27	5	τ	τ	X
ejpam-4465	27	6	)	)	PUNCT
ejpam-4465	27	7	be	be	VERB
ejpam-4465	27	8	a	a	DET
ejpam-4465	27	9	topological	topological	ADJ
ejpam-4465	27	10	space	space	NOUN
ejpam-4465	27	11	,	,	PUNCT
ejpam-4465	27	12	and	and	CCONJ
ejpam-4465	27	13	i	i	PRON
ejpam-4465	27	14	be	be	VERB
ejpam-4465	27	15	an	an	DET
ejpam-4465	27	16	ideal	ideal	NOUN
ejpam-4465	27	17	on	on	ADP
ejpam-4465	27	18	x.	x.	NOUN
ejpam-4465	27	19	since	since	SCONJ
ejpam-4465	27	20	c∗(a	c∗(a	PROPN
ejpam-4465	27	21	)	)	PUNCT
ejpam-4465	27	22	=	=	SYM
ejpam-4465	27	23	a∪a∗(i	a∪a∗(i	PROPN
ejpam-4465	27	24	,	,	PUNCT
ejpam-4465	27	25	τ	τ	X
ejpam-4465	27	26	)	)	PUNCT
ejpam-4465	27	27	is	be	AUX
ejpam-4465	27	28	a	a	DET
ejpam-4465	27	29	kuratowski	kuratowski	ADJ
ejpam-4465	27	30	closure	closure	NOUN
ejpam-4465	27	31	operator	operator	NOUN
ejpam-4465	27	32	,	,	PUNCT
ejpam-4465	27	33	generates	generate	VERB
ejpam-4465	27	34	a	a	DET
ejpam-4465	27	35	topology	topology	NOUN
ejpam-4465	27	36	τ∗(i	τ∗(i	PROPN
ejpam-4465	27	37	,	,	PUNCT
ejpam-4465	27	38	τ	τ	PROPN
ejpam-4465	27	39	)	)	PUNCT
ejpam-4465	27	40	on	on	ADP
ejpam-4465	27	41	x.	x.	NOUN
ejpam-4465	27	42	if	if	SCONJ
ejpam-4465	27	43	there	there	PRON
ejpam-4465	27	44	is	be	VERB
ejpam-4465	27	45	no	no	DET
ejpam-4465	27	46	chance	chance	NOUN
ejpam-4465	27	47	of	of	ADP
ejpam-4465	27	48	confusion	confusion	NOUN
ejpam-4465	27	49	this	this	DET
ejpam-4465	27	50	topological	topological	ADJ
ejpam-4465	27	51	space	space	NOUN
ejpam-4465	27	52	is	be	AUX
ejpam-4465	27	53	denoted	denote	VERB
ejpam-4465	27	54	as	as	ADP
ejpam-4465	27	55	(	(	PUNCT
ejpam-4465	27	56	x	x	NOUN
ejpam-4465	27	57	,	,	PUNCT
ejpam-4465	27	58	τ∗	τ∗	NOUN
ejpam-4465	27	59	)	)	PUNCT
ejpam-4465	27	60	.	.	PUNCT
ejpam-4465	28	1	now	now	ADV
ejpam-4465	28	2	we	we	PRON
ejpam-4465	28	3	recall	recall	VERB
ejpam-4465	28	4	some	some	DET
ejpam-4465	28	5	definitions	definition	NOUN
ejpam-4465	28	6	in	in	ADP
ejpam-4465	28	7	ideal	ideal	ADJ
ejpam-4465	28	8	topological	topological	ADJ
ejpam-4465	28	9	spaces	space	NOUN
ejpam-4465	28	10	,	,	PUNCT
ejpam-4465	28	11	which	which	PRON
ejpam-4465	28	12	are	be	AUX
ejpam-4465	28	13	crucial	crucial	ADJ
ejpam-4465	28	14	in	in	ADP
ejpam-4465	28	15	our	our	PRON
ejpam-4465	28	16	work	work	NOUN
ejpam-4465	28	17	.	.	PUNCT
ejpam-4465	29	1	definition	definition	NOUN
ejpam-4465	29	2	3	3	NUM
ejpam-4465	29	3	.	.	PUNCT
ejpam-4465	30	1	let	let	VERB
ejpam-4465	30	2	(	(	PUNCT
ejpam-4465	30	3	x	x	X
ejpam-4465	30	4	,	,	PUNCT
ejpam-4465	30	5	τ	τ	PROPN
ejpam-4465	30	6	,	,	PUNCT
ejpam-4465	30	7	i	i	PRON
ejpam-4465	30	8	)	)	PUNCT
ejpam-4465	30	9	be	be	VERB
ejpam-4465	30	10	an	an	DET
ejpam-4465	30	11	ideal	ideal	ADJ
ejpam-4465	30	12	topological	topological	ADJ
ejpam-4465	30	13	space	space	NOUN
ejpam-4465	30	14	and	and	CCONJ
ejpam-4465	30	15	a	a	DET
ejpam-4465	30	16	be	be	AUX
ejpam-4465	30	17	a	a	DET
ejpam-4465	30	18	subset	subset	NOUN
ejpam-4465	30	19	of	of	ADP
ejpam-4465	30	20	x.	x.	NOUN
ejpam-4465	30	21	we	we	PRON
ejpam-4465	30	22	say	say	VERB
ejpam-4465	30	23	that	that	SCONJ
ejpam-4465	30	24	a	a	PRON
ejpam-4465	30	25	is	be	AUX
ejpam-4465	30	26	i.	i.	NOUN
ejpam-4465	30	27	∗-dense	∗-dense	NOUN
ejpam-4465	31	1	[	[	X
ejpam-4465	31	2	4	4	X
ejpam-4465	31	3	]	]	X
ejpam-4465	31	4	if	if	SCONJ
ejpam-4465	31	5	c∗a	c∗a	PROPN
ejpam-4465	31	6	=	=	SYM
ejpam-4465	31	7	x	x	PROPN
ejpam-4465	31	8	,	,	PUNCT
ejpam-4465	31	9	ii	ii	PROPN
ejpam-4465	31	10	.	.	PUNCT
ejpam-4465	32	1	i	i	PRON
ejpam-4465	32	2	-	-	PUNCT
ejpam-4465	32	3	dense	dense	ADJ
ejpam-4465	32	4	[	[	X
ejpam-4465	32	5	3	3	NUM
ejpam-4465	32	6	]	]	X
ejpam-4465	32	7	if	if	SCONJ
ejpam-4465	32	8	a∗(i	a∗(i	PROPN
ejpam-4465	32	9	,	,	PUNCT
ejpam-4465	32	10	τ	τ	X
ejpam-4465	32	11	)	)	PUNCT
ejpam-4465	32	12	=	=	PUNCT
ejpam-4465	33	1	x.	x.	NOUN
ejpam-4465	33	2	it	it	PRON
ejpam-4465	33	3	is	be	AUX
ejpam-4465	33	4	easy	easy	ADJ
ejpam-4465	33	5	to	to	PART
ejpam-4465	33	6	show	show	VERB
ejpam-4465	33	7	that	that	SCONJ
ejpam-4465	33	8	τ	τ	PROPN
ejpam-4465	33	9	⊂	⊂	PROPN
ejpam-4465	33	10	τ∗.	τ∗.	AUX
ejpam-4465	33	11	also	also	ADV
ejpam-4465	33	12	note	note	VERB
ejpam-4465	33	13	that	that	SCONJ
ejpam-4465	33	14	,	,	PUNCT
ejpam-4465	33	15	if	if	SCONJ
ejpam-4465	33	16	i	i	PRON
ejpam-4465	33	17	=	=	SYM
ejpam-4465	33	18	{	{	PUNCT
ejpam-4465	33	19	∅	∅	NOUN
ejpam-4465	33	20	}	}	PUNCT
ejpam-4465	33	21	then	then	ADV
ejpam-4465	33	22	τ	τ	PROPN
ejpam-4465	33	23	=	=	SYM
ejpam-4465	33	24	τ∗	τ∗	NOUN
ejpam-4465	33	25	,	,	PUNCT
ejpam-4465	33	26	and	and	CCONJ
ejpam-4465	33	27	if	if	SCONJ
ejpam-4465	33	28	i	i	PRON
ejpam-4465	33	29	=	=	SYM
ejpam-4465	33	30	p(x	p(x	PROPN
ejpam-4465	33	31	)	)	PUNCT
ejpam-4465	33	32	then	then	ADV
ejpam-4465	33	33	a∗(p(x	a∗(p(x	PROPN
ejpam-4465	33	34	)	)	PUNCT
ejpam-4465	33	35	,	,	PUNCT
ejpam-4465	33	36	τ	τ	X
ejpam-4465	33	37	)	)	PUNCT
ejpam-4465	33	38	=	=	SYM
ejpam-4465	33	39	∅	∅	NOUN
ejpam-4465	33	40	which	which	PRON
ejpam-4465	33	41	implies	imply	VERB
ejpam-4465	33	42	τ∗	τ∗	ADJ
ejpam-4465	33	43	=	=	SYM
ejpam-4465	33	44	p(x	p(x	PROPN
ejpam-4465	33	45	)	)	PUNCT
ejpam-4465	33	46	.	.	PUNCT
ejpam-4465	34	1	carrying	carry	VERB
ejpam-4465	34	2	general	general	ADJ
ejpam-4465	34	3	topological	topological	ADJ
ejpam-4465	34	4	notions	notion	NOUN
ejpam-4465	34	5	into	into	ADP
ejpam-4465	34	6	the	the	DET
ejpam-4465	34	7	ideal	ideal	ADJ
ejpam-4465	34	8	topological	topological	ADJ
ejpam-4465	34	9	spaces	space	NOUN
ejpam-4465	34	10	is	be	AUX
ejpam-4465	34	11	a	a	DET
ejpam-4465	34	12	very	very	ADV
ejpam-4465	34	13	fruitful	fruitful	ADJ
ejpam-4465	34	14	,	,	PUNCT
ejpam-4465	34	15	and	and	CCONJ
ejpam-4465	34	16	generalizing	generalize	VERB
ejpam-4465	34	17	process	process	NOUN
ejpam-4465	34	18	.	.	PUNCT
ejpam-4465	35	1	to	to	ADP
ejpam-4465	35	2	this	this	DET
ejpam-4465	35	3	end	end	NOUN
ejpam-4465	35	4	we	we	PRON
ejpam-4465	35	5	turned	turn	VERB
ejpam-4465	35	6	our	our	PRON
ejpam-4465	35	7	attention	attention	NOUN
ejpam-4465	35	8	to	to	ADP
ejpam-4465	35	9	similarity	similarity	NOUN
ejpam-4465	35	10	between	between	ADP
ejpam-4465	35	11	the	the	DET
ejpam-4465	35	12	topologies	topology	NOUN
ejpam-4465	35	13	defined	define	VERB
ejpam-4465	35	14	on	on	ADP
ejpam-4465	35	15	the	the	DET
ejpam-4465	35	16	same	same	ADJ
ejpam-4465	35	17	set	set	NOUN
ejpam-4465	35	18	.	.	PUNCT
ejpam-4465	36	1	this	this	DET
ejpam-4465	36	2	topic	topic	NOUN
ejpam-4465	36	3	is	be	AUX
ejpam-4465	36	4	introduced	introduce	VERB
ejpam-4465	36	5	in	in	ADP
ejpam-4465	36	6	[	[	X
ejpam-4465	36	7	2	2	NUM
ejpam-4465	36	8	]	]	PUNCT
ejpam-4465	36	9	.	.	PUNCT
ejpam-4465	37	1	according	accord	VERB
ejpam-4465	37	2	to	to	ADP
ejpam-4465	37	3	bartoszewicz	bartoszewicz	NOUN
ejpam-4465	37	4	and	and	CCONJ
ejpam-4465	37	5	et	et	NOUN
ejpam-4465	37	6	al	al	PROPN
ejpam-4465	37	7	.	.	PUNCT
ejpam-4465	38	1	(	(	PUNCT
ejpam-4465	38	2	x	x	NOUN
ejpam-4465	38	3	,	,	PUNCT
ejpam-4465	38	4	τ1	τ1	NOUN
ejpam-4465	38	5	)	)	PUNCT
ejpam-4465	38	6	and	and	CCONJ
ejpam-4465	38	7	(	(	PUNCT
ejpam-4465	38	8	x	x	NOUN
ejpam-4465	38	9	,	,	PUNCT
ejpam-4465	38	10	τ2	τ2	ADJ
ejpam-4465	38	11	)	)	PUNCT
ejpam-4465	38	12	are	be	AUX
ejpam-4465	38	13	similar	similar	ADJ
ejpam-4465	38	14	if	if	SCONJ
ejpam-4465	38	15	the	the	DET
ejpam-4465	38	16	families	family	NOUN
ejpam-4465	38	17	of	of	ADP
ejpam-4465	38	18	sets	set	NOUN
ejpam-4465	38	19	which	which	PRON
ejpam-4465	38	20	have	have	AUX
ejpam-4465	38	21	nonempty	nonempty	VERB
ejpam-4465	38	22	interior	interior	ADJ
ejpam-4465	38	23	with	with	ADP
ejpam-4465	38	24	respect	respect	NOUN
ejpam-4465	38	25	to	to	ADP
ejpam-4465	38	26	τ1	τ1	NOUN
ejpam-4465	38	27	and	and	CCONJ
ejpam-4465	38	28	τ2	τ2	NOUN
ejpam-4465	38	29	coincide	coincide	NOUN
ejpam-4465	38	30	.	.	PUNCT
ejpam-4465	39	1	similarity	similarity	NOUN
ejpam-4465	39	2	between	between	ADP
ejpam-4465	39	3	topological	topological	ADJ
ejpam-4465	39	4	spaces	space	NOUN
ejpam-4465	39	5	is	be	AUX
ejpam-4465	39	6	denoted	denote	VERB
ejpam-4465	39	7	by	by	ADP
ejpam-4465	39	8	τ1	τ1	NOUN
ejpam-4465	39	9	∼	∼	NOUN
ejpam-4465	39	10	τ2	τ2	NOUN
ejpam-4465	39	11	.	.	PUNCT
ejpam-4465	40	1	in	in	ADP
ejpam-4465	40	2	[	[	X
ejpam-4465	40	3	2	2	NUM
ejpam-4465	40	4	]	]	PUNCT
ejpam-4465	40	5	,	,	PUNCT
ejpam-4465	40	6	besides	besides	SCONJ
ejpam-4465	40	7	some	some	DET
ejpam-4465	40	8	other	other	ADJ
ejpam-4465	40	9	characterizations	characterization	NOUN
ejpam-4465	40	10	,	,	PUNCT
ejpam-4465	40	11	it	it	PRON
ejpam-4465	40	12	is	be	AUX
ejpam-4465	40	13	shown	show	VERB
ejpam-4465	40	14	that	that	SCONJ
ejpam-4465	40	15	two	two	NUM
ejpam-4465	40	16	topologies	topology	NOUN
ejpam-4465	40	17	are	be	AUX
ejpam-4465	40	18	similar	similar	ADJ
ejpam-4465	40	19	if	if	SCONJ
ejpam-4465	40	20	and	and	CCONJ
ejpam-4465	40	21	only	only	ADV
ejpam-4465	40	22	if	if	SCONJ
ejpam-4465	40	23	the	the	DET
ejpam-4465	40	24	families	family	NOUN
ejpam-4465	40	25	of	of	ADP
ejpam-4465	40	26	dense	dense	ADJ
ejpam-4465	40	27	subsets	subset	NOUN
ejpam-4465	40	28	coincide	coincide	NOUN
ejpam-4465	40	29	,	,	PUNCT
ejpam-4465	40	30	or	or	CCONJ
ejpam-4465	40	31	τ1	τ1	ADP
ejpam-4465	40	32	\	\	NOUN
ejpam-4465	40	33	{	{	PUNCT
ejpam-4465	40	34	∅	∅	NOUN
ejpam-4465	40	35	}	}	PUNCT
ejpam-4465	40	36	and	and	CCONJ
ejpam-4465	40	37	τ2	τ2	PROPN
ejpam-4465	40	38	\	\	NOUN
ejpam-4465	40	39	{	{	PUNCT
ejpam-4465	40	40	∅	∅	NOUN
ejpam-4465	40	41	}	}	PUNCT
ejpam-4465	40	42	are	be	AUX
ejpam-4465	40	43	mutually	mutually	ADV
ejpam-4465	40	44	coinitial	coinitial	ADJ
ejpam-4465	40	45	[	[	X
ejpam-4465	40	46	1	1	NUM
ejpam-4465	40	47	]	]	PUNCT
ejpam-4465	40	48	.	.	PUNCT
ejpam-4465	41	1	that	that	PRON
ejpam-4465	41	2	is	be	AUX
ejpam-4465	41	3	for	for	ADP
ejpam-4465	41	4	all	all	PRON
ejpam-4465	41	5	u	u	PRON
ejpam-4465	41	6	∈	∈	PROPN
ejpam-4465	41	7	τ1	τ1	NOUN
ejpam-4465	41	8	\	\	NOUN
ejpam-4465	41	9	{	{	PUNCT
ejpam-4465	41	10	∅	∅	NOUN
ejpam-4465	41	11	}	}	PUNCT
ejpam-4465	41	12	there	there	PRON
ejpam-4465	41	13	exists	exist	VERB
ejpam-4465	41	14	v	v	ADP
ejpam-4465	41	15	∈	∈	PROPN
ejpam-4465	41	16	τ2	τ2	PROPN
ejpam-4465	41	17	\	\	NOUN
ejpam-4465	41	18	{	{	PUNCT
ejpam-4465	41	19	∅	∅	NOUN
ejpam-4465	41	20	}	}	PUNCT
ejpam-4465	42	1	such	such	ADJ
ejpam-4465	42	2	that	that	SCONJ
ejpam-4465	42	3	v	v	ADP
ejpam-4465	42	4	⊂	⊂	PROPN
ejpam-4465	42	5	u	u	NOUN
ejpam-4465	42	6	and	and	CCONJ
ejpam-4465	42	7	for	for	ADP
ejpam-4465	42	8	all	all	PRON
ejpam-4465	42	9	u	u	PRON
ejpam-4465	42	10	∈	∈	PROPN
ejpam-4465	42	11	τ2	τ2	PROPN
ejpam-4465	42	12	\	\	NOUN
ejpam-4465	42	13	{	{	PUNCT
ejpam-4465	42	14	∅	∅	NOUN
ejpam-4465	42	15	}	}	PUNCT
ejpam-4465	42	16	there	there	PRON
ejpam-4465	42	17	exists	exist	VERB
ejpam-4465	42	18	v	v	ADP
ejpam-4465	42	19	∈	∈	PROPN
ejpam-4465	42	20	τ1	τ1	NOUN
ejpam-4465	42	21	\	\	NOUN
ejpam-4465	42	22	{	{	PUNCT
ejpam-4465	42	23	∅	∅	NOUN
ejpam-4465	42	24	}	}	PUNCT
ejpam-4465	42	25	such	such	ADJ
ejpam-4465	42	26	that	that	PRON
ejpam-4465	42	27	v	v	ADP
ejpam-4465	42	28	⊂	⊂	PROPN
ejpam-4465	42	29	u	u	PROPN
ejpam-4465	42	30	.	.	PUNCT
ejpam-4465	43	1	in	in	ADP
ejpam-4465	43	2	this	this	DET
ejpam-4465	43	3	work	work	NOUN
ejpam-4465	43	4	we	we	PRON
ejpam-4465	43	5	first	first	ADV
ejpam-4465	43	6	define	define	VERB
ejpam-4465	43	7	similarity	similarity	NOUN
ejpam-4465	43	8	with	with	ADP
ejpam-4465	43	9	respect	respect	NOUN
ejpam-4465	43	10	to	to	ADP
ejpam-4465	43	11	an	an	DET
ejpam-4465	43	12	ideal	ideal	NOUN
ejpam-4465	43	13	,	,	PUNCT
ejpam-4465	43	14	and	and	CCONJ
ejpam-4465	43	15	then	then	ADV
ejpam-4465	43	16	give	give	VERB
ejpam-4465	43	17	some	some	DET
ejpam-4465	43	18	characterizations	characterization	NOUN
ejpam-4465	43	19	.	.	PUNCT
ejpam-4465	44	1	throughout	throughout	ADP
ejpam-4465	44	2	this	this	DET
ejpam-4465	44	3	work	work	NOUN
ejpam-4465	44	4	,	,	PUNCT
ejpam-4465	44	5	(	(	PUNCT
ejpam-4465	44	6	x	x	X
ejpam-4465	44	7	,	,	PUNCT
ejpam-4465	44	8	τ	τ	PROPN
ejpam-4465	44	9	)	)	PUNCT
ejpam-4465	44	10	,	,	PUNCT
ejpam-4465	44	11	cia	cia	PROPN
ejpam-4465	44	12	,	,	PUNCT
ejpam-4465	44	13	and	and	CCONJ
ejpam-4465	44	14	c∗ia	c∗ia	PROPN
ejpam-4465	44	15	will	will	AUX
ejpam-4465	44	16	denote	denote	VERB
ejpam-4465	44	17	the	the	DET
ejpam-4465	44	18	topological	topological	ADJ
ejpam-4465	44	19	space	space	NOUN
ejpam-4465	44	20	,	,	PUNCT
ejpam-4465	44	21	closure	closure	NOUN
ejpam-4465	44	22	in	in	ADP
ejpam-4465	44	23	τi	τi	NOUN
ejpam-4465	44	24	,	,	PUNCT
ejpam-4465	44	25	and	and	CCONJ
ejpam-4465	44	26	in	in	ADP
ejpam-4465	44	27	τ∗i	τ∗i	NUM
ejpam-4465	44	28	,	,	PUNCT
ejpam-4465	44	29	respectively	respectively	ADV
ejpam-4465	44	30	where	where	SCONJ
ejpam-4465	44	31	i	i	PRON
ejpam-4465	44	32	=	=	NOUN
ejpam-4465	44	33	1	1	NUM
ejpam-4465	44	34	,	,	PUNCT
ejpam-4465	44	35	2	2	NUM
ejpam-4465	44	36	.	.	X
ejpam-4465	44	37	2	2	X
ejpam-4465	44	38	.	.	X
ejpam-4465	44	39	main	main	ADJ
ejpam-4465	44	40	results	result	NOUN
ejpam-4465	44	41	definition	definition	NOUN
ejpam-4465	44	42	4	4	NUM
ejpam-4465	44	43	.	.	PUNCT
ejpam-4465	45	1	let	let	VERB
ejpam-4465	45	2	x	x	PRON
ejpam-4465	45	3	be	be	AUX
ejpam-4465	45	4	a	a	DET
ejpam-4465	45	5	set	set	NOUN
ejpam-4465	45	6	,	,	PUNCT
ejpam-4465	45	7	τ1	τ1	NOUN
ejpam-4465	45	8	,	,	PUNCT
ejpam-4465	45	9	τ2	τ2	NOUN
ejpam-4465	45	10	be	be	VERB
ejpam-4465	45	11	two	two	NUM
ejpam-4465	45	12	given	give	VERB
ejpam-4465	45	13	topologies	topology	NOUN
ejpam-4465	45	14	,	,	PUNCT
ejpam-4465	45	15	and	and	CCONJ
ejpam-4465	45	16	i	i	PRON
ejpam-4465	45	17	be	be	VERB
ejpam-4465	45	18	an	an	DET
ejpam-4465	45	19	ideal	ideal	NOUN
ejpam-4465	45	20	on	on	ADP
ejpam-4465	45	21	x.	x.	NOUN
ejpam-4465	46	1	we	we	PRON
ejpam-4465	46	2	say	say	VERB
ejpam-4465	46	3	that	that	SCONJ
ejpam-4465	46	4	τ1	τ1	NOUN
ejpam-4465	46	5	and	and	CCONJ
ejpam-4465	46	6	τ2	τ2	NOUN
ejpam-4465	46	7	are	be	AUX
ejpam-4465	46	8	similar	similar	ADJ
ejpam-4465	46	9	with	with	ADP
ejpam-4465	46	10	respect	respect	NOUN
ejpam-4465	46	11	to	to	ADP
ejpam-4465	46	12	i	i	PRON
ejpam-4465	46	13	or	or	CCONJ
ejpam-4465	46	14	i	i	PRON
ejpam-4465	46	15	-	-	PUNCT
ejpam-4465	46	16	similar	similar	ADJ
ejpam-4465	46	17	and	and	CCONJ
ejpam-4465	46	18	denote	denote	VERB
ejpam-4465	46	19	by	by	ADP
ejpam-4465	46	20	τ1	τ1	PROPN
ejpam-4465	46	21	∼i	∼i	PROPN
ejpam-4465	46	22	τ2	τ2	PROPN
ejpam-4465	46	23	if	if	SCONJ
ejpam-4465	46	24	,	,	PUNCT
ejpam-4465	46	25	for	for	ADP
ejpam-4465	46	26	every	every	DET
ejpam-4465	46	27	nonempty	nonempty	ADJ
ejpam-4465	46	28	u	u	NOUN
ejpam-4465	46	29	∈	∈	PROPN
ejpam-4465	46	30	τ1	τ1	NOUN
ejpam-4465	46	31	,	,	PUNCT
ejpam-4465	46	32	there	there	PRON
ejpam-4465	46	33	exists	exist	VERB
ejpam-4465	46	34	a	a	DET
ejpam-4465	46	35	nonempty	nonempty	ADJ
ejpam-4465	46	36	v	v	ADP
ejpam-4465	46	37	∈	∈	NOUN
ejpam-4465	46	38	τ2	τ2	NOUN
ejpam-4465	47	1	such	such	ADJ
ejpam-4465	47	2	that	that	PRON
ejpam-4465	47	3	v	v	NUM
ejpam-4465	47	4	\u	\u	NOUN
ejpam-4465	47	5	∈	∈	PROPN
ejpam-4465	48	1	i	i	PRON
ejpam-4465	48	2	,	,	PUNCT
ejpam-4465	48	3	and	and	CCONJ
ejpam-4465	48	4	for	for	ADP
ejpam-4465	48	5	every	every	DET
ejpam-4465	48	6	u	u	PROPN
ejpam-4465	48	7	∈	∈	PROPN
ejpam-4465	48	8	τ2	τ2	NOUN
ejpam-4465	48	9	,	,	PUNCT
ejpam-4465	48	10	there	there	PRON
ejpam-4465	48	11	exists	exist	VERB
ejpam-4465	48	12	a	a	DET
ejpam-4465	48	13	nonempty	nonempty	ADJ
ejpam-4465	48	14	v	v	ADP
ejpam-4465	48	15	∈	∈	NOUN
ejpam-4465	48	16	τ1	τ1	NOUN
ejpam-4465	48	17	such	such	ADJ
ejpam-4465	48	18	that	that	PRON
ejpam-4465	48	19	v	v	NOUN
ejpam-4465	48	20	\	\	NOUN
ejpam-4465	48	21	u	u	PROPN
ejpam-4465	48	22	∈	∈	PROPN
ejpam-4465	48	23	i.	i.	NOUN
ejpam-4465	48	24	it	it	PRON
ejpam-4465	48	25	is	be	AUX
ejpam-4465	48	26	clear	clear	ADJ
ejpam-4465	48	27	that	that	SCONJ
ejpam-4465	48	28	if	if	SCONJ
ejpam-4465	48	29	τ1	τ1	NOUN
ejpam-4465	48	30	and	and	CCONJ
ejpam-4465	48	31	τ2	τ2	NOUN
ejpam-4465	48	32	are	be	AUX
ejpam-4465	48	33	similar	similar	ADJ
ejpam-4465	48	34	topologies	topology	NOUN
ejpam-4465	48	35	,	,	PUNCT
ejpam-4465	48	36	then	then	ADV
ejpam-4465	48	37	they	they	PRON
ejpam-4465	48	38	are	be	AUX
ejpam-4465	48	39	similar	similar	ADJ
ejpam-4465	48	40	with	with	ADP
ejpam-4465	48	41	respect	respect	NOUN
ejpam-4465	48	42	to	to	ADP
ejpam-4465	48	43	any	any	DET
ejpam-4465	48	44	ideal	ideal	ADJ
ejpam-4465	48	45	i.	i.	NOUN
ejpam-4465	48	46	on	on	ADP
ejpam-4465	48	47	the	the	DET
ejpam-4465	48	48	other	other	ADJ
ejpam-4465	48	49	hand	hand	NOUN
ejpam-4465	48	50	the	the	DET
ejpam-4465	48	51	following	follow	VERB
ejpam-4465	48	52	example	example	NOUN
ejpam-4465	48	53	shows	show	VERB
ejpam-4465	48	54	that	that	SCONJ
ejpam-4465	48	55	ideal	ideal	ADJ
ejpam-4465	48	56	similarity	similarity	NOUN
ejpam-4465	48	57	does	do	AUX
ejpam-4465	48	58	not	not	PART
ejpam-4465	48	59	imply	imply	VERB
ejpam-4465	48	60	similarity	similarity	NOUN
ejpam-4465	48	61	between	between	ADP
ejpam-4465	48	62	topologies	topology	NOUN
ejpam-4465	48	63	.	.	PUNCT
ejpam-4465	49	1	example	example	NOUN
ejpam-4465	50	1	1	1	NUM
ejpam-4465	50	2	.	.	PUNCT
ejpam-4465	50	3	let	let	VERB
ejpam-4465	50	4	x	x	PUNCT
ejpam-4465	50	5	=	=	PRON
ejpam-4465	50	6	{	{	PUNCT
ejpam-4465	50	7	a	a	PRON
ejpam-4465	50	8	,	,	PUNCT
ejpam-4465	50	9	b	b	NOUN
ejpam-4465	50	10	,	,	PUNCT
ejpam-4465	50	11	c	c	NOUN
ejpam-4465	50	12	}	}	PUNCT
ejpam-4465	50	13	,	,	PUNCT
ejpam-4465	50	14	τ1	τ1	NOUN
ejpam-4465	50	15	=	=	SYM
ejpam-4465	50	16	{	{	PUNCT
ejpam-4465	50	17	∅	∅	NOUN
ejpam-4465	50	18	,	,	PUNCT
ejpam-4465	50	19	x	x	X
ejpam-4465	50	20	,	,	PUNCT
ejpam-4465	50	21	{	{	PUNCT
ejpam-4465	50	22	a	a	PRON
ejpam-4465	50	23	,	,	PUNCT
ejpam-4465	50	24	b	b	NOUN
ejpam-4465	50	25	}	}	PUNCT
ejpam-4465	50	26	}	}	PUNCT
ejpam-4465	50	27	,	,	PUNCT
ejpam-4465	50	28	τ2	τ2	NOUN
ejpam-4465	50	29	=	=	SYM
ejpam-4465	50	30	{	{	PUNCT
ejpam-4465	50	31	∅	∅	NOUN
ejpam-4465	50	32	,	,	PUNCT
ejpam-4465	50	33	x	x	X
ejpam-4465	50	34	,	,	PUNCT
ejpam-4465	50	35	{	{	PUNCT
ejpam-4465	50	36	b	b	NOUN
ejpam-4465	50	37	,	,	PUNCT
ejpam-4465	50	38	c	c	NOUN
ejpam-4465	50	39	}	}	PUNCT
ejpam-4465	50	40	}	}	PUNCT
ejpam-4465	50	41	,	,	PUNCT
ejpam-4465	50	42	and	and	CCONJ
ejpam-4465	50	43	i	i	PRON
ejpam-4465	50	44	=	=	PUNCT
ejpam-4465	50	45	{	{	PUNCT
ejpam-4465	50	46	∅	∅	NOUN
ejpam-4465	50	47	,	,	PUNCT
ejpam-4465	50	48	{	{	PUNCT
ejpam-4465	50	49	a	a	X
ejpam-4465	50	50	}	}	PUNCT
ejpam-4465	50	51	,	,	PUNCT
ejpam-4465	50	52	{	{	PUNCT
ejpam-4465	50	53	c	c	X
ejpam-4465	50	54	}	}	PUNCT
ejpam-4465	50	55	,	,	PUNCT
ejpam-4465	50	56	{	{	PUNCT
ejpam-4465	50	57	a	a	PRON
ejpam-4465	50	58	,	,	PUNCT
ejpam-4465	50	59	c	c	NOUN
ejpam-4465	50	60	}	}	PUNCT
ejpam-4465	50	61	}	}	PUNCT
ejpam-4465	50	62	.	.	PUNCT
ejpam-4465	51	1	τ1	τ1	NOUN
ejpam-4465	51	2	and	and	CCONJ
ejpam-4465	51	3	τ2	τ2	NOUN
ejpam-4465	51	4	are	be	AUX
ejpam-4465	51	5	i	i	NOUN
ejpam-4465	51	6	-	-	PUNCT
ejpam-4465	51	7	similar	similar	ADJ
ejpam-4465	51	8	,	,	PUNCT
ejpam-4465	51	9	but	but	CCONJ
ejpam-4465	51	10	not	not	PART
ejpam-4465	51	11	similar	similar	ADJ
ejpam-4465	51	12	.	.	PUNCT
ejpam-4465	52	1	also	also	ADV
ejpam-4465	52	2	,	,	PUNCT
ejpam-4465	52	3	if	if	SCONJ
ejpam-4465	52	4	i	i	PRON
ejpam-4465	52	5	=	=	SYM
ejpam-4465	52	6	{	{	PUNCT
ejpam-4465	52	7	∅	∅	NOUN
ejpam-4465	52	8	}	}	PUNCT
ejpam-4465	52	9	,	,	PUNCT
ejpam-4465	52	10	then	then	ADV
ejpam-4465	52	11	similarity	similarity	NOUN
ejpam-4465	52	12	is	be	AUX
ejpam-4465	52	13	equivalent	equivalent	ADJ
ejpam-4465	52	14	to	to	ADP
ejpam-4465	52	15	i	i	NOUN
ejpam-4465	52	16	-	-	NOUN
ejpam-4465	52	17	similarity	similarity	NOUN
ejpam-4465	52	18	.	.	PUNCT
ejpam-4465	53	1	what	what	PRON
ejpam-4465	53	2	is	be	AUX
ejpam-4465	53	3	more	more	ADJ
ejpam-4465	53	4	,	,	PUNCT
ejpam-4465	53	5	if	if	SCONJ
ejpam-4465	53	6	τ1	τ1	NOUN
ejpam-4465	53	7	and	and	CCONJ
ejpam-4465	53	8	τ2	τ2	NOUN
ejpam-4465	53	9	are	be	AUX
ejpam-4465	53	10	i	i	NOUN
ejpam-4465	53	11	-	-	PUNCT
ejpam-4465	53	12	similar	similar	ADJ
ejpam-4465	53	13	topologies	topology	NOUN
ejpam-4465	53	14	and	and	CCONJ
ejpam-4465	53	15	j	j	PROPN
ejpam-4465	53	16	is	be	AUX
ejpam-4465	53	17	an	an	DET
ejpam-4465	53	18	ideal	ideal	NOUN
ejpam-4465	53	19	with	with	ADP
ejpam-4465	53	20	i	i	PROPN
ejpam-4465	53	21	⊂	⊂	PROPN
ejpam-4465	53	22	j	j	PROPN
ejpam-4465	53	23	,	,	PUNCT
ejpam-4465	53	24	then	then	ADV
ejpam-4465	53	25	τ1	τ1	NOUN
ejpam-4465	53	26	and	and	CCONJ
ejpam-4465	53	27	τ2	τ2	NOUN
ejpam-4465	53	28	are	be	AUX
ejpam-4465	53	29	also	also	ADV
ejpam-4465	53	30	j	j	NOUN
ejpam-4465	53	31	-	-	PUNCT
ejpam-4465	53	32	similar	similar	ADJ
ejpam-4465	53	33	.	.	PUNCT
ejpam-4465	54	1	a.	a.	PROPN
ejpam-4465	54	2	guldurdek	guldurdek	PROPN
ejpam-4465	54	3	/	/	SYM
ejpam-4465	54	4	eur	eur	PROPN
ejpam-4465	54	5	.	.	PUNCT
ejpam-4465	55	1	j.	j.	PROPN
ejpam-4465	55	2	pure	pure	PROPN
ejpam-4465	55	3	appl	appl	PROPN
ejpam-4465	55	4	.	.	PROPN
ejpam-4465	55	5	math	math	PROPN
ejpam-4465	55	6	,	,	PUNCT
ejpam-4465	55	7	15	15	NUM
ejpam-4465	55	8	(	(	PUNCT
ejpam-4465	55	9	3	3	NUM
ejpam-4465	55	10	)	)	PUNCT
ejpam-4465	55	11	(	(	PUNCT
ejpam-4465	55	12	2022	2022	NUM
ejpam-4465	55	13	)	)	PUNCT
ejpam-4465	55	14	,	,	PUNCT
ejpam-4465	55	15	1344	1344	NUM
ejpam-4465	55	16	-	-	SYM
ejpam-4465	55	17	1347	1347	NUM
ejpam-4465	55	18	1346	1346	NUM
ejpam-4465	55	19	theorem	theorem	NOUN
ejpam-4465	55	20	1	1	NUM
ejpam-4465	55	21	.	.	PUNCT
ejpam-4465	56	1	let	let	VERB
ejpam-4465	56	2	x	x	PRON
ejpam-4465	56	3	be	be	AUX
ejpam-4465	56	4	a	a	DET
ejpam-4465	56	5	set	set	NOUN
ejpam-4465	56	6	,	,	PUNCT
ejpam-4465	56	7	τ1	τ1	NOUN
ejpam-4465	56	8	,	,	PUNCT
ejpam-4465	56	9	τ2	τ2	NOUN
ejpam-4465	56	10	topologies	topology	NOUN
ejpam-4465	56	11	on	on	ADP
ejpam-4465	56	12	x	x	NOUN
ejpam-4465	56	13	,	,	PUNCT
ejpam-4465	56	14	and	and	CCONJ
ejpam-4465	56	15	i	i	PRON
ejpam-4465	56	16	be	be	VERB
ejpam-4465	56	17	an	an	DET
ejpam-4465	56	18	ideal	ideal	NOUN
ejpam-4465	56	19	on	on	ADP
ejpam-4465	56	20	x.	x.	NOUN
ejpam-4465	56	21	then	then	ADV
ejpam-4465	56	22	τ1	τ1	NOUN
ejpam-4465	56	23	and	and	CCONJ
ejpam-4465	56	24	τ2	τ2	NOUN
ejpam-4465	56	25	are	be	AUX
ejpam-4465	56	26	similar	similar	ADJ
ejpam-4465	56	27	with	with	ADP
ejpam-4465	56	28	respect	respect	NOUN
ejpam-4465	56	29	to	to	ADP
ejpam-4465	56	30	i	i	PRON
ejpam-4465	56	31	if	if	SCONJ
ejpam-4465	56	32	and	and	CCONJ
ejpam-4465	56	33	only	only	ADV
ejpam-4465	56	34	if	if	SCONJ
ejpam-4465	56	35	∗-dense	∗-dense	ADJ
ejpam-4465	56	36	subsets	subset	NOUN
ejpam-4465	56	37	coincide	coincide	NOUN
ejpam-4465	56	38	.	.	PUNCT
ejpam-4465	57	1	proof	proof	NOUN
ejpam-4465	57	2	.	.	PUNCT
ejpam-4465	58	1	let	let	VERB
ejpam-4465	58	2	τ1	τ1	VERB
ejpam-4465	58	3	∼i	∼i	PROPN
ejpam-4465	58	4	τ2	τ2	PROPN
ejpam-4465	58	5	,	,	PUNCT
ejpam-4465	58	6	a	a	DET
ejpam-4465	58	7	subset	subset	NOUN
ejpam-4465	58	8	a	a	DET
ejpam-4465	58	9	⊂	⊂	PROPN
ejpam-4465	58	10	x	x	PUNCT
ejpam-4465	58	11	with	with	ADP
ejpam-4465	58	12	c∗1a	c∗1a	PROPN
ejpam-4465	58	13	=	=	SYM
ejpam-4465	58	14	x	x	NOUN
ejpam-4465	58	15	,	,	PUNCT
ejpam-4465	58	16	and	and	CCONJ
ejpam-4465	58	17	c∗2a	c∗2a	VERB
ejpam-4465	58	18	̸=	̸=	PROPN
ejpam-4465	58	19	x	x	PART
ejpam-4465	58	20	be	be	AUX
ejpam-4465	58	21	given	give	VERB
ejpam-4465	58	22	.	.	PUNCT
ejpam-4465	59	1	by	by	ADP
ejpam-4465	59	2	definition	definition	NOUN
ejpam-4465	59	3	,	,	PUNCT
ejpam-4465	59	4	there	there	PRON
ejpam-4465	59	5	exists	exist	VERB
ejpam-4465	59	6	an	an	DET
ejpam-4465	59	7	element	element	NOUN
ejpam-4465	59	8	x	x	PUNCT
ejpam-4465	59	9	in	in	ADP
ejpam-4465	59	10	x	x	PRON
ejpam-4465	59	11	,	,	PUNCT
ejpam-4465	59	12	so	so	SCONJ
ejpam-4465	59	13	that	that	SCONJ
ejpam-4465	59	14	x	x	X
ejpam-4465	59	15	/∈	/∈	PUNCT
ejpam-4465	59	16	a	a	PRON
ejpam-4465	59	17	and	and	CCONJ
ejpam-4465	59	18	x	x	ADJ
ejpam-4465	59	19	/∈	/∈	SYM
ejpam-4465	59	20	a∗(i	a∗(i	PROPN
ejpam-4465	59	21	,	,	PUNCT
ejpam-4465	59	22	τ2	τ2	NOUN
ejpam-4465	59	23	)	)	PUNCT
ejpam-4465	59	24	.	.	PUNCT
ejpam-4465	60	1	hence	hence	ADV
ejpam-4465	60	2	there	there	PRON
ejpam-4465	60	3	exists	exist	VERB
ejpam-4465	60	4	u	u	NOUN
ejpam-4465	60	5	∈	∈	PROPN
ejpam-4465	60	6	τ2(x	τ2(x	PUNCT
ejpam-4465	60	7	)	)	PUNCT
ejpam-4465	60	8	so	so	SCONJ
ejpam-4465	60	9	that	that	SCONJ
ejpam-4465	60	10	u	u	NOUN
ejpam-4465	60	11	∩a	∩a	PROPN
ejpam-4465	60	12	∈	∈	PROPN
ejpam-4465	60	13	i.	i.	NOUN
ejpam-4465	60	14	together	together	ADV
ejpam-4465	60	15	with	with	ADP
ejpam-4465	60	16	this	this	PRON
ejpam-4465	60	17	,	,	PUNCT
ejpam-4465	60	18	c∗1a	c∗1a	PROPN
ejpam-4465	60	19	=	=	NOUN
ejpam-4465	60	20	x	x	NOUN
ejpam-4465	60	21	implies	imply	VERB
ejpam-4465	60	22	x	x	PUNCT
ejpam-4465	60	23	∈	∈	PROPN
ejpam-4465	60	24	a∗(i	a∗(i	PROPN
ejpam-4465	60	25	,	,	PUNCT
ejpam-4465	60	26	τ1	τ1	NOUN
ejpam-4465	60	27	)	)	PUNCT
ejpam-4465	60	28	.	.	PUNCT
ejpam-4465	61	1	by	by	ADP
ejpam-4465	61	2	i	i	PROPN
ejpam-4465	61	3	-	-	PUNCT
ejpam-4465	61	4	similarity	similarity	NOUN
ejpam-4465	61	5	there	there	PRON
ejpam-4465	61	6	exists	exist	VERB
ejpam-4465	61	7	a	a	DET
ejpam-4465	61	8	set	set	NOUN
ejpam-4465	61	9	v	v	NUM
ejpam-4465	61	10	∈	∈	NOUN
ejpam-4465	61	11	τ1	τ1	NOUN
ejpam-4465	61	12	so	so	SCONJ
ejpam-4465	61	13	that	that	SCONJ
ejpam-4465	61	14	v	v	X
ejpam-4465	61	15	\	\	NOUN
ejpam-4465	61	16	u	u	PROPN
ejpam-4465	61	17	∈	∈	PROPN
ejpam-4465	61	18	i.	i.	NOUN
ejpam-4465	61	19	note	note	PROPN
ejpam-4465	61	20	also	also	ADV
ejpam-4465	61	21	that	that	SCONJ
ejpam-4465	61	22	a	a	DET
ejpam-4465	61	23	∩	∩	NOUN
ejpam-4465	61	24	v	v	ADP
ejpam-4465	61	25	̸=	̸=	PROPN
ejpam-4465	61	26	∅.	∅.	ADV
ejpam-4465	61	27	if	if	SCONJ
ejpam-4465	61	28	we	we	PRON
ejpam-4465	61	29	consider	consider	VERB
ejpam-4465	61	30	a	a	DET
ejpam-4465	61	31	∩	∩	ADJ
ejpam-4465	61	32	v	v	NOUN
ejpam-4465	61	33	,	,	PUNCT
ejpam-4465	61	34	we	we	PRON
ejpam-4465	61	35	see	see	VERB
ejpam-4465	61	36	that	that	SCONJ
ejpam-4465	61	37	a	a	DET
ejpam-4465	61	38	∩	∩	NOUN
ejpam-4465	61	39	v	v	X
ejpam-4465	61	40	∈	∈	NOUN
ejpam-4465	61	41	i	i	PRON
ejpam-4465	61	42	since	since	SCONJ
ejpam-4465	61	43	a	a	DET
ejpam-4465	61	44	∩	∩	ADJ
ejpam-4465	61	45	v	v	X
ejpam-4465	61	46	⊂	⊂	PROPN
ejpam-4465	61	47	(	(	PUNCT
ejpam-4465	61	48	u	u	NOUN
ejpam-4465	61	49	∩a	∩a	PROPN
ejpam-4465	61	50	)	)	PUNCT
ejpam-4465	61	51	∪	∪	NOUN
ejpam-4465	61	52	(	(	PUNCT
ejpam-4465	61	53	v	v	NOUN
ejpam-4465	61	54	∩	∩	NOUN
ejpam-4465	61	55	(	(	PUNCT
ejpam-4465	61	56	x	x	SYM
ejpam-4465	61	57	\	\	PROPN
ejpam-4465	61	58	u	u	NOUN
ejpam-4465	61	59	)	)	PUNCT
ejpam-4465	61	60	)	)	PUNCT
ejpam-4465	61	61	.	.	PUNCT
ejpam-4465	62	1	however	however	ADV
ejpam-4465	62	2	,	,	PUNCT
ejpam-4465	62	3	that	that	PRON
ejpam-4465	62	4	contradicts	contradict	VERB
ejpam-4465	62	5	the	the	DET
ejpam-4465	62	6	fact	fact	NOUN
ejpam-4465	62	7	that	that	SCONJ
ejpam-4465	62	8	c∗1a	c∗1a	PROPN
ejpam-4465	62	9	=	=	SYM
ejpam-4465	62	10	x.	x.	NOUN
ejpam-4465	62	11	on	on	ADP
ejpam-4465	62	12	the	the	DET
ejpam-4465	62	13	other	other	ADJ
ejpam-4465	62	14	hand	hand	NOUN
ejpam-4465	62	15	,	,	PUNCT
ejpam-4465	62	16	let	let	VERB
ejpam-4465	62	17	u	u	PRON
ejpam-4465	62	18	∈	∈	PROPN
ejpam-4465	62	19	τ1	τ1	NOUN
ejpam-4465	62	20	\	\	NOUN
ejpam-4465	62	21	{	{	PUNCT
ejpam-4465	62	22	∅	∅	NOUN
ejpam-4465	62	23	}	}	PUNCT
ejpam-4465	62	24	,	,	PUNCT
ejpam-4465	62	25	and	and	CCONJ
ejpam-4465	62	26	suppose	suppose	VERB
ejpam-4465	62	27	v	v	ADP
ejpam-4465	62	28	\	\	PROPN
ejpam-4465	62	29	u	u	NOUN
ejpam-4465	62	30	/∈	/∈	PUNCT
ejpam-4465	63	1	i	i	PRON
ejpam-4465	63	2	for	for	ADP
ejpam-4465	63	3	every	every	DET
ejpam-4465	63	4	v	v	NUM
ejpam-4465	63	5	∈	∈	PROPN
ejpam-4465	63	6	τ2	τ2	PROPN
ejpam-4465	63	7	\	\	NOUN
ejpam-4465	63	8	{	{	PUNCT
ejpam-4465	63	9	∅	∅	NOUN
ejpam-4465	63	10	}	}	PUNCT
ejpam-4465	63	11	.	.	PUNCT
ejpam-4465	64	1	so	so	ADV
ejpam-4465	64	2	,	,	PUNCT
ejpam-4465	64	3	x	x	SYM
ejpam-4465	64	4	\	\	NOUN
ejpam-4465	64	5	u	u	NOUN
ejpam-4465	64	6	is	be	AUX
ejpam-4465	64	7	∗-dense	∗-dense	NOUN
ejpam-4465	64	8	with	with	ADP
ejpam-4465	64	9	respect	respect	NOUN
ejpam-4465	64	10	to	to	ADP
ejpam-4465	64	11	τ2	τ2	NOUN
ejpam-4465	64	12	.	.	PUNCT
ejpam-4465	65	1	by	by	ADP
ejpam-4465	65	2	hypothesis	hypothesis	NOUN
ejpam-4465	65	3	x	x	X
ejpam-4465	65	4	\	\	NOUN
ejpam-4465	65	5	u	u	NOUN
ejpam-4465	65	6	is	be	AUX
ejpam-4465	65	7	also	also	ADV
ejpam-4465	65	8	∗-dense	∗-dense	NOUN
ejpam-4465	65	9	with	with	ADP
ejpam-4465	65	10	respect	respect	NOUN
ejpam-4465	65	11	to	to	ADP
ejpam-4465	65	12	τ1	τ1	NOUN
ejpam-4465	65	13	.	.	PUNCT
ejpam-4465	66	1	that	that	PRON
ejpam-4465	66	2	is	be	AUX
ejpam-4465	66	3	(	(	PUNCT
ejpam-4465	66	4	x	x	SYM
ejpam-4465	66	5	\	\	PROPN
ejpam-4465	66	6	u	u	NOUN
ejpam-4465	66	7	)	)	PUNCT
ejpam-4465	66	8	∪	∪	VERB
ejpam-4465	66	9	(	(	PUNCT
ejpam-4465	66	10	x	x	SYM
ejpam-4465	66	11	\	\	PROPN
ejpam-4465	66	12	u)∗(i	u)∗(i	PROPN
ejpam-4465	66	13	,	,	PUNCT
ejpam-4465	66	14	τ1	τ1	NOUN
ejpam-4465	66	15	)	)	PUNCT
ejpam-4465	66	16	=	=	SYM
ejpam-4465	67	1	x	x	NOUN
ejpam-4465	67	2	,	,	PUNCT
ejpam-4465	67	3	and	and	CCONJ
ejpam-4465	67	4	this	this	PRON
ejpam-4465	67	5	implies	imply	VERB
ejpam-4465	67	6	u	u	PROPN
ejpam-4465	67	7	⊂	⊂	PROPN
ejpam-4465	67	8	(	(	PUNCT
ejpam-4465	67	9	x	x	SYM
ejpam-4465	67	10	\	\	PROPN
ejpam-4465	67	11	u)∗(i	u)∗(i	PROPN
ejpam-4465	67	12	,	,	PUNCT
ejpam-4465	67	13	τ1	τ1	NOUN
ejpam-4465	67	14	)	)	PUNCT
ejpam-4465	67	15	.	.	PUNCT
ejpam-4465	68	1	as	as	ADP
ejpam-4465	68	2	a	a	DET
ejpam-4465	68	3	result	result	NOUN
ejpam-4465	68	4	we	we	PRON
ejpam-4465	68	5	have	have	VERB
ejpam-4465	68	6	the	the	DET
ejpam-4465	68	7	contradiction	contradiction	NOUN
ejpam-4465	68	8	:	:	PUNCT
ejpam-4465	68	9	u	u	NOUN
ejpam-4465	68	10	∩	∩	NOUN
ejpam-4465	68	11	(	(	PUNCT
ejpam-4465	68	12	x	x	SYM
ejpam-4465	68	13	\	\	PROPN
ejpam-4465	68	14	u	u	NOUN
ejpam-4465	68	15	)	)	PUNCT
ejpam-4465	68	16	/∈	/∈	PUNCT
ejpam-4465	69	1	i.	i.	PROPN
ejpam-4465	69	2	corollary	corollary	NOUN
ejpam-4465	69	3	1	1	X
ejpam-4465	69	4	.	.	PUNCT
ejpam-4465	70	1	let	let	VERB
ejpam-4465	70	2	x	x	PRON
ejpam-4465	70	3	be	be	AUX
ejpam-4465	70	4	a	a	DET
ejpam-4465	70	5	set	set	NOUN
ejpam-4465	70	6	,	,	PUNCT
ejpam-4465	70	7	τ1	τ1	NOUN
ejpam-4465	70	8	,	,	PUNCT
ejpam-4465	70	9	τ2	τ2	NOUN
ejpam-4465	70	10	topologies	topology	NOUN
ejpam-4465	70	11	on	on	ADP
ejpam-4465	70	12	x	x	NOUN
ejpam-4465	70	13	,	,	PUNCT
ejpam-4465	70	14	and	and	CCONJ
ejpam-4465	70	15	i	i	PRON
ejpam-4465	70	16	be	be	VERB
ejpam-4465	70	17	an	an	DET
ejpam-4465	70	18	ideal	ideal	NOUN
ejpam-4465	70	19	on	on	ADP
ejpam-4465	70	20	x.	x.	NOUN
ejpam-4465	70	21	then	then	ADV
ejpam-4465	70	22	τ∗1	τ∗1	NOUN
ejpam-4465	70	23	and	and	CCONJ
ejpam-4465	70	24	τ∗2	τ∗2	NOUN
ejpam-4465	70	25	are	be	AUX
ejpam-4465	70	26	similar	similar	ADJ
ejpam-4465	70	27	if	if	SCONJ
ejpam-4465	70	28	and	and	CCONJ
ejpam-4465	70	29	only	only	ADV
ejpam-4465	70	30	if	if	SCONJ
ejpam-4465	70	31	τ1	τ1	NOUN
ejpam-4465	70	32	and	and	CCONJ
ejpam-4465	70	33	τ2	τ2	NOUN
ejpam-4465	70	34	are	be	AUX
ejpam-4465	70	35	i	i	NOUN
ejpam-4465	70	36	-	-	PUNCT
ejpam-4465	70	37	similar	similar	ADJ
ejpam-4465	70	38	.	.	PUNCT
ejpam-4465	71	1	proof	proof	NOUN
ejpam-4465	71	2	.	.	PUNCT
ejpam-4465	72	1	by	by	ADP
ejpam-4465	72	2	previous	previous	ADJ
ejpam-4465	72	3	theorem	theorem	ADJ
ejpam-4465	72	4	,	,	PUNCT
ejpam-4465	72	5	τ1	τ1	NOUN
ejpam-4465	72	6	and	and	CCONJ
ejpam-4465	72	7	τ2	τ2	NOUN
ejpam-4465	72	8	are	be	AUX
ejpam-4465	72	9	i	i	NOUN
ejpam-4465	72	10	-	-	PUNCT
ejpam-4465	72	11	similar	similar	ADJ
ejpam-4465	72	12	if	if	SCONJ
ejpam-4465	73	1	and	and	CCONJ
ejpam-4465	73	2	only	only	ADV
ejpam-4465	73	3	if	if	SCONJ
ejpam-4465	73	4	∗-dense	∗-dense	ADJ
ejpam-4465	73	5	subsets	subset	NOUN
ejpam-4465	73	6	coincide	coincide	NOUN
ejpam-4465	73	7	,	,	PUNCT
ejpam-4465	73	8	and	and	CCONJ
ejpam-4465	73	9	by	by	ADP
ejpam-4465	73	10	[	[	X
ejpam-4465	73	11	2	2	NUM
ejpam-4465	73	12	]	]	PUNCT
ejpam-4465	73	13	,	,	PUNCT
ejpam-4465	73	14	theorem	theorem	VERB
ejpam-4465	73	15	2.2	2.2	NUM
ejpam-4465	73	16	,	,	PUNCT
ejpam-4465	73	17	this	this	PRON
ejpam-4465	73	18	is	be	AUX
ejpam-4465	73	19	true	true	ADJ
ejpam-4465	73	20	if	if	SCONJ
ejpam-4465	73	21	and	and	CCONJ
ejpam-4465	73	22	only	only	ADV
ejpam-4465	73	23	if	if	SCONJ
ejpam-4465	73	24	τ∗1	τ∗1	NOUN
ejpam-4465	73	25	and	and	CCONJ
ejpam-4465	73	26	τ∗2	τ∗2	NOUN
ejpam-4465	73	27	are	be	AUX
ejpam-4465	73	28	similar	similar	ADJ
ejpam-4465	73	29	topologies	topology	NOUN
ejpam-4465	73	30	.	.	PUNCT
ejpam-4465	74	1	we	we	PRON
ejpam-4465	74	2	have	have	VERB
ejpam-4465	74	3	another	another	DET
ejpam-4465	74	4	characterization	characterization	NOUN
ejpam-4465	74	5	for	for	ADP
ejpam-4465	74	6	i	i	PROPN
ejpam-4465	74	7	-	-	PUNCT
ejpam-4465	74	8	similarity	similarity	NOUN
ejpam-4465	74	9	.	.	PUNCT
ejpam-4465	75	1	theorem	theorem	NOUN
ejpam-4465	75	2	2	2	NUM
ejpam-4465	75	3	.	.	PUNCT
ejpam-4465	76	1	let	let	VERB
ejpam-4465	76	2	x	x	PRON
ejpam-4465	76	3	be	be	AUX
ejpam-4465	76	4	a	a	DET
ejpam-4465	76	5	set	set	NOUN
ejpam-4465	76	6	,	,	PUNCT
ejpam-4465	76	7	τ1	τ1	NOUN
ejpam-4465	76	8	,	,	PUNCT
ejpam-4465	76	9	τ2	τ2	NOUN
ejpam-4465	76	10	topologies	topology	NOUN
ejpam-4465	76	11	on	on	ADP
ejpam-4465	76	12	x	x	NOUN
ejpam-4465	76	13	,	,	PUNCT
ejpam-4465	76	14	and	and	CCONJ
ejpam-4465	76	15	i	i	PRON
ejpam-4465	76	16	be	be	VERB
ejpam-4465	76	17	an	an	DET
ejpam-4465	76	18	ideal	ideal	NOUN
ejpam-4465	76	19	on	on	ADP
ejpam-4465	76	20	x.	x.	NOUN
ejpam-4465	76	21	then	then	ADV
ejpam-4465	76	22	τ1	τ1	NOUN
ejpam-4465	76	23	and	and	CCONJ
ejpam-4465	76	24	τ2	τ2	NOUN
ejpam-4465	76	25	are	be	AUX
ejpam-4465	76	26	similar	similar	ADJ
ejpam-4465	76	27	with	with	ADP
ejpam-4465	76	28	respect	respect	NOUN
ejpam-4465	76	29	to	to	ADP
ejpam-4465	76	30	i	i	PRON
ejpam-4465	76	31	if	if	SCONJ
ejpam-4465	76	32	and	and	CCONJ
ejpam-4465	76	33	only	only	ADV
ejpam-4465	76	34	if	if	SCONJ
ejpam-4465	76	35	i	i	PRON
ejpam-4465	76	36	-	-	PUNCT
ejpam-4465	76	37	dense	dense	ADJ
ejpam-4465	76	38	subsets	subset	NOUN
ejpam-4465	76	39	coincide	coincide	NOUN
ejpam-4465	76	40	.	.	PUNCT
ejpam-4465	77	1	proof	proof	NOUN
ejpam-4465	77	2	.	.	PUNCT
ejpam-4465	78	1	let	let	VERB
ejpam-4465	78	2	τ1	τ1	VERB
ejpam-4465	78	3	∼i	∼i	PROPN
ejpam-4465	78	4	τ2	τ2	PROPN
ejpam-4465	78	5	,	,	PUNCT
ejpam-4465	78	6	a	a	DET
ejpam-4465	78	7	∗(i	∗(i	NOUN
ejpam-4465	78	8	,	,	PUNCT
ejpam-4465	78	9	τ1	τ1	NOUN
ejpam-4465	78	10	)	)	PUNCT
ejpam-4465	78	11	=	=	SYM
ejpam-4465	79	1	x	x	NOUN
ejpam-4465	79	2	,	,	PUNCT
ejpam-4465	79	3	and	and	CCONJ
ejpam-4465	79	4	a∗(i	a∗(i	PROPN
ejpam-4465	79	5	,	,	PUNCT
ejpam-4465	79	6	τ2	τ2	NOUN
ejpam-4465	79	7	)	)	PUNCT
ejpam-4465	79	8	̸=	̸=	PROPN
ejpam-4465	79	9	x.	x.	NOUN
ejpam-4465	79	10	then	then	ADV
ejpam-4465	79	11	,	,	PUNCT
ejpam-4465	79	12	there	there	PRON
ejpam-4465	79	13	exists	exist	VERB
ejpam-4465	79	14	an	an	DET
ejpam-4465	79	15	element	element	NOUN
ejpam-4465	79	16	x	x	PUNCT
ejpam-4465	79	17	in	in	ADP
ejpam-4465	79	18	x	x	PUNCT
ejpam-4465	79	19	so	so	SCONJ
ejpam-4465	79	20	that	that	SCONJ
ejpam-4465	79	21	x	x	SYM
ejpam-4465	79	22	/∈	/∈	SYM
ejpam-4465	79	23	a∗(i	a∗(i	PROPN
ejpam-4465	79	24	,	,	PUNCT
ejpam-4465	79	25	τ2	τ2	NOUN
ejpam-4465	79	26	)	)	PUNCT
ejpam-4465	79	27	.	.	PUNCT
ejpam-4465	80	1	that	that	PRON
ejpam-4465	80	2	is	be	AUX
ejpam-4465	80	3	,	,	PUNCT
ejpam-4465	80	4	for	for	ADP
ejpam-4465	80	5	a	a	DET
ejpam-4465	80	6	set	set	NOUN
ejpam-4465	80	7	u	u	NOUN
ejpam-4465	80	8	∈	∈	PROPN
ejpam-4465	80	9	τ2(x	τ2(x	PROPN
ejpam-4465	80	10	)	)	PUNCT
ejpam-4465	80	11	,	,	PUNCT
ejpam-4465	80	12	we	we	PRON
ejpam-4465	80	13	have	have	VERB
ejpam-4465	80	14	u	u	NOUN
ejpam-4465	80	15	∩a	∩a	PROPN
ejpam-4465	80	16	∈	∈	PROPN
ejpam-4465	80	17	i.	i.	NOUN
ejpam-4465	80	18	by	by	ADP
ejpam-4465	80	19	i	i	PROPN
ejpam-4465	80	20	-	-	PUNCT
ejpam-4465	80	21	similarity	similarity	NOUN
ejpam-4465	80	22	there	there	PRON
ejpam-4465	80	23	exists	exist	VERB
ejpam-4465	80	24	a	a	DET
ejpam-4465	80	25	subset	subset	NOUN
ejpam-4465	80	26	v	v	ADP
ejpam-4465	80	27	∈	∈	PROPN
ejpam-4465	80	28	τ1	τ1	NOUN
ejpam-4465	80	29	\	\	NOUN
ejpam-4465	80	30	{	{	PUNCT
ejpam-4465	80	31	∅	∅	NOUN
ejpam-4465	80	32	}	}	PUNCT
ejpam-4465	80	33	such	such	ADJ
ejpam-4465	80	34	that	that	SCONJ
ejpam-4465	80	35	v	v	NOUN
ejpam-4465	80	36	\	\	NOUN
ejpam-4465	80	37	u	u	NOUN
ejpam-4465	80	38	∈	∈	PROPN
ejpam-4465	80	39	i	i	PRON
ejpam-4465	80	40	which	which	PRON
ejpam-4465	80	41	implies	imply	VERB
ejpam-4465	80	42	v	v	ADP
ejpam-4465	80	43	∩	∩	NOUN
ejpam-4465	80	44	a	a	DET
ejpam-4465	80	45	∈	∈	PROPN
ejpam-4465	80	46	i.	i.	NOUN
ejpam-4465	80	47	that	that	PRON
ejpam-4465	80	48	contradicts	contradict	VERB
ejpam-4465	80	49	the	the	DET
ejpam-4465	80	50	fact	fact	NOUN
ejpam-4465	80	51	that	that	SCONJ
ejpam-4465	80	52	x	x	SYM
ejpam-4465	80	53	∈	∈	PROPN
ejpam-4465	80	54	a∗(i	a∗(i	PROPN
ejpam-4465	80	55	,	,	PUNCT
ejpam-4465	80	56	τ1	τ1	NOUN
ejpam-4465	80	57	)	)	PUNCT
ejpam-4465	80	58	=	=	PUNCT
ejpam-4465	81	1	x.	x.	NOUN
ejpam-4465	81	2	let	let	VERB
ejpam-4465	81	3	this	this	DET
ejpam-4465	81	4	time	time	NOUN
ejpam-4465	81	5	the	the	DET
ejpam-4465	81	6	families	family	NOUN
ejpam-4465	81	7	of	of	ADP
ejpam-4465	81	8	i	i	PRON
ejpam-4465	81	9	-	-	PUNCT
ejpam-4465	81	10	dense	dense	ADJ
ejpam-4465	81	11	subsets	subset	NOUN
ejpam-4465	81	12	coincide	coincide	NOUN
ejpam-4465	81	13	,	,	PUNCT
ejpam-4465	81	14	and	and	CCONJ
ejpam-4465	81	15	u	u	PRON
ejpam-4465	81	16	be	be	VERB
ejpam-4465	81	17	a	a	DET
ejpam-4465	81	18	set	set	NOUN
ejpam-4465	81	19	belonging	belong	VERB
ejpam-4465	81	20	to	to	ADP
ejpam-4465	81	21	τ1	τ1	NOUN
ejpam-4465	81	22	\	\	NOUN
ejpam-4465	81	23	{	{	PUNCT
ejpam-4465	81	24	∅	∅	NOUN
ejpam-4465	81	25	}	}	PUNCT
ejpam-4465	81	26	.	.	PUNCT
ejpam-4465	82	1	suppose	suppose	VERB
ejpam-4465	82	2	v	v	ADP
ejpam-4465	82	3	\	\	PROPN
ejpam-4465	82	4	u	u	NOUN
ejpam-4465	82	5	/∈	/∈	PUNCT
ejpam-4465	83	1	i	i	PRON
ejpam-4465	83	2	for	for	ADP
ejpam-4465	83	3	every	every	DET
ejpam-4465	83	4	v	v	NUM
ejpam-4465	83	5	∈	∈	PROPN
ejpam-4465	83	6	τ2	τ2	PROPN
ejpam-4465	83	7	\	\	NOUN
ejpam-4465	83	8	{	{	PUNCT
ejpam-4465	83	9	∅	∅	NOUN
ejpam-4465	83	10	}	}	PUNCT
ejpam-4465	83	11	.	.	PUNCT
ejpam-4465	84	1	then	then	ADV
ejpam-4465	84	2	v	v	X
ejpam-4465	84	3	\	\	PROPN
ejpam-4465	84	4	u	u	PROPN
ejpam-4465	84	5	̸=	̸=	PROPN
ejpam-4465	84	6	∅	∅	NOUN
ejpam-4465	84	7	,	,	PUNCT
ejpam-4465	84	8	and	and	CCONJ
ejpam-4465	84	9	hence	hence	ADV
ejpam-4465	84	10	x	x	X
ejpam-4465	84	11	\	\	NOUN
ejpam-4465	84	12	u	u	NOUN
ejpam-4465	84	13	is	be	AUX
ejpam-4465	84	14	i	i	PRON
ejpam-4465	84	15	-	-	PUNCT
ejpam-4465	84	16	dense	dense	ADJ
ejpam-4465	84	17	in	in	ADP
ejpam-4465	84	18	τ2	τ2	PROPN
ejpam-4465	84	19	.	.	PUNCT
ejpam-4465	85	1	by	by	ADP
ejpam-4465	85	2	hypothesis	hypothesis	NOUN
ejpam-4465	85	3	,	,	PUNCT
ejpam-4465	85	4	x	x	SYM
ejpam-4465	85	5	\	\	NOUN
ejpam-4465	85	6	u	u	NOUN
ejpam-4465	85	7	is	be	AUX
ejpam-4465	85	8	also	also	ADV
ejpam-4465	85	9	i	i	PRON
ejpam-4465	85	10	-	-	PUNCT
ejpam-4465	85	11	dense	dense	ADJ
ejpam-4465	85	12	in	in	ADP
ejpam-4465	85	13	τ1	τ1	NOUN
ejpam-4465	85	14	,	,	PUNCT
ejpam-4465	85	15	which	which	PRON
ejpam-4465	85	16	brings	bring	VERB
ejpam-4465	85	17	the	the	DET
ejpam-4465	85	18	contradiction	contradiction	NOUN
ejpam-4465	85	19	that	that	PRON
ejpam-4465	85	20	u	u	NOUN
ejpam-4465	85	21	∩	∩	NOUN
ejpam-4465	85	22	(	(	PUNCT
ejpam-4465	85	23	x	x	SYM
ejpam-4465	85	24	\	\	PROPN
ejpam-4465	85	25	u	u	NOUN
ejpam-4465	85	26	)	)	PUNCT
ejpam-4465	85	27	/∈	/∈	PUNCT
ejpam-4465	85	28	i.	i.	PROPN
ejpam-4465	85	29	lemma	lemma	PROPN
ejpam-4465	85	30	1	1	X
ejpam-4465	85	31	.	.	PUNCT
ejpam-4465	86	1	let	let	VERB
ejpam-4465	86	2	x	x	PRON
ejpam-4465	86	3	be	be	AUX
ejpam-4465	86	4	a	a	DET
ejpam-4465	86	5	set	set	NOUN
ejpam-4465	86	6	,	,	PUNCT
ejpam-4465	86	7	τ1	τ1	NOUN
ejpam-4465	86	8	,	,	PUNCT
ejpam-4465	86	9	τ2	τ2	NOUN
ejpam-4465	86	10	topologies	topology	NOUN
ejpam-4465	86	11	on	on	ADP
ejpam-4465	86	12	x	x	NOUN
ejpam-4465	86	13	,	,	PUNCT
ejpam-4465	86	14	and	and	CCONJ
ejpam-4465	86	15	i	i	PRON
ejpam-4465	86	16	be	be	VERB
ejpam-4465	86	17	an	an	DET
ejpam-4465	86	18	ideal	ideal	NOUN
ejpam-4465	86	19	on	on	ADP
ejpam-4465	86	20	x.	x.	NOUN
ejpam-4465	86	21	if	if	SCONJ
ejpam-4465	86	22	τ1	τ1	NOUN
ejpam-4465	86	23	and	and	CCONJ
ejpam-4465	86	24	τ2	τ2	NOUN
ejpam-4465	86	25	are	be	AUX
ejpam-4465	86	26	similar	similar	ADJ
ejpam-4465	86	27	with	with	ADP
ejpam-4465	86	28	respect	respect	NOUN
ejpam-4465	86	29	to	to	ADP
ejpam-4465	86	30	i	i	PRON
ejpam-4465	86	31	,	,	PUNCT
ejpam-4465	86	32	and	and	CCONJ
ejpam-4465	86	33	i	i	PRON
ejpam-4465	86	34	is	be	AUX
ejpam-4465	86	35	a	a	DET
ejpam-4465	86	36	τi	τi	ADJ
ejpam-4465	86	37	-	-	PUNCT
ejpam-4465	86	38	codense	codense	NOUN
ejpam-4465	86	39	ideal	ideal	NOUN
ejpam-4465	86	40	for	for	ADP
ejpam-4465	86	41	i	i	PROPN
ejpam-4465	86	42	=	=	NOUN
ejpam-4465	86	43	1	1	NUM
ejpam-4465	86	44	,	,	PUNCT
ejpam-4465	86	45	2	2	NUM
ejpam-4465	86	46	then	then	ADV
ejpam-4465	86	47	i	i	PRON
ejpam-4465	86	48	is	be	AUX
ejpam-4465	86	49	also	also	ADV
ejpam-4465	86	50	τj	τj	ADP
ejpam-4465	86	51	-	-	PUNCT
ejpam-4465	86	52	codense	codense	NOUN
ejpam-4465	86	53	ideal	ideal	NOUN
ejpam-4465	86	54	for	for	ADP
ejpam-4465	86	55	j	j	PROPN
ejpam-4465	86	56	=	=	SYM
ejpam-4465	86	57	3−	3−	NUM
ejpam-4465	86	58	i.	i.	NOUN
ejpam-4465	86	59	proof	proof	NOUN
ejpam-4465	86	60	.	.	PUNCT
ejpam-4465	87	1	without	without	ADP
ejpam-4465	87	2	loss	loss	NOUN
ejpam-4465	87	3	of	of	ADP
ejpam-4465	87	4	generality	generality	NOUN
ejpam-4465	87	5	,	,	PUNCT
ejpam-4465	87	6	assume	assume	VERB
ejpam-4465	87	7	i	i	PRON
ejpam-4465	87	8	be	be	VERB
ejpam-4465	87	9	a	a	DET
ejpam-4465	87	10	τ1	τ1	NOUN
ejpam-4465	87	11	-	-	PUNCT
ejpam-4465	87	12	codense	codense	NOUN
ejpam-4465	87	13	ideal	ideal	NOUN
ejpam-4465	87	14	,	,	PUNCT
ejpam-4465	87	15	and	and	CCONJ
ejpam-4465	87	16	let	let	VERB
ejpam-4465	87	17	u	u	PRON
ejpam-4465	87	18	∈	∈	PROPN
ejpam-4465	87	19	i∩τ2\{∅	i∩τ2\{∅	PROPN
ejpam-4465	87	20	}	}	PUNCT
ejpam-4465	87	21	.	.	PUNCT
ejpam-4465	88	1	by	by	ADP
ejpam-4465	88	2	i	i	PROPN
ejpam-4465	88	3	-	-	PUNCT
ejpam-4465	88	4	similarity	similarity	NOUN
ejpam-4465	88	5	,	,	PUNCT
ejpam-4465	88	6	there	there	PRON
ejpam-4465	88	7	exists	exist	VERB
ejpam-4465	88	8	a	a	DET
ejpam-4465	88	9	set	set	NOUN
ejpam-4465	88	10	v	v	NUM
ejpam-4465	88	11	∈	∈	NOUN
ejpam-4465	88	12	τ1	τ1	NOUN
ejpam-4465	88	13	\	\	NOUN
ejpam-4465	88	14	{	{	PUNCT
ejpam-4465	88	15	∅	∅	NOUN
ejpam-4465	88	16	}	}	PUNCT
ejpam-4465	88	17	so	so	SCONJ
ejpam-4465	88	18	that	that	SCONJ
ejpam-4465	88	19	v	v	X
ejpam-4465	88	20	\	\	NOUN
ejpam-4465	88	21	u	u	PROPN
ejpam-4465	88	22	∈	∈	PROPN
ejpam-4465	88	23	i.	i.	NOUN
ejpam-4465	88	24	however	however	ADV
ejpam-4465	88	25	these	these	PRON
ejpam-4465	88	26	imply	imply	VERB
ejpam-4465	88	27	v	v	ADP
ejpam-4465	88	28	∈	∈	PROPN
ejpam-4465	89	1	i	i	PRON
ejpam-4465	89	2	,	,	PUNCT
ejpam-4465	89	3	which	which	PRON
ejpam-4465	89	4	is	be	AUX
ejpam-4465	89	5	impossible	impossible	ADJ
ejpam-4465	89	6	since	since	SCONJ
ejpam-4465	89	7	i	i	PRON
ejpam-4465	89	8	∩	∩	NOUN
ejpam-4465	89	9	τ1	τ1	NOUN
ejpam-4465	89	10	=	=	SYM
ejpam-4465	89	11	{	{	PUNCT
ejpam-4465	89	12	∅	∅	NOUN
ejpam-4465	89	13	}	}	PUNCT
ejpam-4465	89	14	.	.	PUNCT
ejpam-4465	90	1	now	now	ADV
ejpam-4465	90	2	,	,	PUNCT
ejpam-4465	90	3	we	we	PRON
ejpam-4465	90	4	examine	examine	VERB
ejpam-4465	90	5	the	the	DET
ejpam-4465	90	6	similarity	similarity	NOUN
ejpam-4465	90	7	between	between	ADP
ejpam-4465	90	8	τ∗1	τ∗1	NOUN
ejpam-4465	90	9	and	and	CCONJ
ejpam-4465	90	10	τ∗2	τ∗2	NOUN
ejpam-4465	90	11	.	.	PUNCT
ejpam-4465	91	1	one	one	PRON
ejpam-4465	91	2	can	can	AUX
ejpam-4465	91	3	easily	easily	ADV
ejpam-4465	91	4	show	show	VERB
ejpam-4465	91	5	that	that	SCONJ
ejpam-4465	91	6	,	,	PUNCT
ejpam-4465	91	7	if	if	SCONJ
ejpam-4465	91	8	τ1	τ1	NOUN
ejpam-4465	91	9	and	and	CCONJ
ejpam-4465	91	10	τ2	τ2	NOUN
ejpam-4465	91	11	are	be	AUX
ejpam-4465	91	12	similar	similar	ADJ
ejpam-4465	91	13	topologies	topology	NOUN
ejpam-4465	91	14	,	,	PUNCT
ejpam-4465	91	15	then	then	ADV
ejpam-4465	91	16	τ∗1	τ∗1	ADV
ejpam-4465	91	17	,	,	PUNCT
ejpam-4465	91	18	and	and	CCONJ
ejpam-4465	91	19	τ∗2	τ∗2	NOUN
ejpam-4465	91	20	are	be	AUX
ejpam-4465	91	21	also	also	ADV
ejpam-4465	91	22	similar	similar	ADJ
ejpam-4465	91	23	.	.	PUNCT
ejpam-4465	92	1	on	on	ADP
ejpam-4465	92	2	the	the	DET
ejpam-4465	92	3	other	other	ADJ
ejpam-4465	92	4	hand	hand	NOUN
ejpam-4465	92	5	the	the	DET
ejpam-4465	92	6	converse	converse	NOUN
ejpam-4465	92	7	is	be	AUX
ejpam-4465	92	8	not	not	PART
ejpam-4465	92	9	true	true	ADJ
ejpam-4465	92	10	,	,	PUNCT
ejpam-4465	92	11	as	as	SCONJ
ejpam-4465	92	12	the	the	DET
ejpam-4465	92	13	following	follow	VERB
ejpam-4465	92	14	example	example	NOUN
ejpam-4465	92	15	shows	show	VERB
ejpam-4465	92	16	:	:	PUNCT
ejpam-4465	92	17	references	reference	NOUN
ejpam-4465	92	18	1347	1347	NUM
ejpam-4465	92	19	example	example	NOUN
ejpam-4465	92	20	2	2	NUM
ejpam-4465	92	21	.	.	PUNCT
ejpam-4465	93	1	let	let	VERB
ejpam-4465	93	2	us	we	PRON
ejpam-4465	93	3	reconsider	reconsider	VERB
ejpam-4465	93	4	the	the	DET
ejpam-4465	93	5	example	example	NOUN
ejpam-4465	93	6	1	1	NUM
ejpam-4465	93	7	.	.	PUNCT
ejpam-4465	94	1	the	the	DET
ejpam-4465	94	2	topologies	topology	NOUN
ejpam-4465	94	3	τ∗1	τ∗1	ADV
ejpam-4465	94	4	,	,	PUNCT
ejpam-4465	94	5	and	and	CCONJ
ejpam-4465	94	6	τ∗2	τ∗2	NOUN
ejpam-4465	94	7	satisfy	satisfy	VERB
ejpam-4465	94	8	:	:	PUNCT
ejpam-4465	94	9	τ∗1	τ∗1	ADP
ejpam-4465	94	10	=	=	SYM
ejpam-4465	94	11	{	{	PUNCT
ejpam-4465	94	12	x	x	NOUN
ejpam-4465	94	13	,	,	PUNCT
ejpam-4465	94	14	∅	∅	NOUN
ejpam-4465	94	15	,	,	PUNCT
ejpam-4465	94	16	{	{	PUNCT
ejpam-4465	94	17	b	b	NOUN
ejpam-4465	94	18	}	}	PUNCT
ejpam-4465	94	19	,	,	PUNCT
ejpam-4465	94	20	{	{	PUNCT
ejpam-4465	94	21	a	a	DET
ejpam-4465	94	22	,	,	PUNCT
ejpam-4465	94	23	b	b	NOUN
ejpam-4465	94	24	}	}	PUNCT
ejpam-4465	94	25	,	,	PUNCT
ejpam-4465	94	26	{	{	PUNCT
ejpam-4465	94	27	b	b	X
ejpam-4465	94	28	,	,	PUNCT
ejpam-4465	94	29	c	c	NOUN
ejpam-4465	94	30	}	}	PUNCT
ejpam-4465	94	31	}	}	PUNCT
ejpam-4465	94	32	=	=	NOUN
ejpam-4465	94	33	τ∗2	τ∗2	NOUN
ejpam-4465	94	34	,	,	PUNCT
ejpam-4465	94	35	are	be	AUX
ejpam-4465	94	36	the	the	DET
ejpam-4465	94	37	same	same	ADJ
ejpam-4465	94	38	.	.	PUNCT
ejpam-4465	95	1	however	however	ADV
ejpam-4465	95	2	,	,	PUNCT
ejpam-4465	95	3	as	as	SCONJ
ejpam-4465	95	4	we	we	PRON
ejpam-4465	95	5	mentioned	mention	VERB
ejpam-4465	95	6	,	,	PUNCT
ejpam-4465	95	7	τ1	τ1	NOUN
ejpam-4465	95	8	,	,	PUNCT
ejpam-4465	95	9	and	and	CCONJ
ejpam-4465	95	10	τ2	τ2	NOUN
ejpam-4465	95	11	are	be	AUX
ejpam-4465	95	12	not	not	PART
ejpam-4465	95	13	similar	similar	ADJ
ejpam-4465	95	14	.	.	PUNCT
ejpam-4465	96	1	note	note	VERB
ejpam-4465	96	2	that	that	SCONJ
ejpam-4465	96	3	,	,	PUNCT
ejpam-4465	96	4	the	the	DET
ejpam-4465	96	5	ideal	ideal	NOUN
ejpam-4465	96	6	of	of	ADP
ejpam-4465	96	7	the	the	DET
ejpam-4465	96	8	previous	previous	ADJ
ejpam-4465	96	9	example	example	NOUN
ejpam-4465	96	10	is	be	AUX
ejpam-4465	96	11	codense	codense	NOUN
ejpam-4465	96	12	with	with	ADP
ejpam-4465	96	13	respect	respect	NOUN
ejpam-4465	96	14	to	to	ADP
ejpam-4465	96	15	both	both	DET
ejpam-4465	96	16	topologies	topology	NOUN
ejpam-4465	96	17	,	,	PUNCT
ejpam-4465	96	18	but	but	CCONJ
ejpam-4465	96	19	this	this	PRON
ejpam-4465	96	20	is	be	AUX
ejpam-4465	96	21	not	not	PART
ejpam-4465	96	22	enough	enough	ADJ
ejpam-4465	96	23	to	to	PART
ejpam-4465	96	24	have	have	VERB
ejpam-4465	96	25	similarity	similarity	NOUN
ejpam-4465	96	26	between	between	ADP
ejpam-4465	96	27	τ1	τ1	NOUN
ejpam-4465	96	28	,	,	PUNCT
ejpam-4465	96	29	and	and	CCONJ
ejpam-4465	96	30	τ2	τ2	NOUN
ejpam-4465	96	31	.	.	PUNCT
ejpam-4465	97	1	this	this	PRON
ejpam-4465	97	2	motivates	motivate	VERB
ejpam-4465	97	3	the	the	DET
ejpam-4465	97	4	following	follow	VERB
ejpam-4465	97	5	question	question	NOUN
ejpam-4465	97	6	:	:	PUNCT
ejpam-4465	97	7	are	be	AUX
ejpam-4465	97	8	there	there	PRON
ejpam-4465	97	9	any	any	DET
ejpam-4465	97	10	conditions	condition	NOUN
ejpam-4465	97	11	can	can	AUX
ejpam-4465	97	12	be	be	AUX
ejpam-4465	97	13	added	add	VERB
ejpam-4465	97	14	to	to	ADP
ejpam-4465	97	15	an	an	DET
ejpam-4465	97	16	ideal	ideal	NOUN
ejpam-4465	97	17	i	i	PRON
ejpam-4465	97	18	for	for	ADP
ejpam-4465	97	19	carrying	carry	VERB
ejpam-4465	97	20	similarity	similarity	NOUN
ejpam-4465	97	21	between	between	ADP
ejpam-4465	97	22	τ∗1	τ∗1	NOUN
ejpam-4465	97	23	and	and	CCONJ
ejpam-4465	97	24	τ∗2	τ∗2	NOUN
ejpam-4465	97	25	to	to	ADP
ejpam-4465	97	26	the	the	DET
ejpam-4465	97	27	case	case	NOUN
ejpam-4465	97	28	of	of	ADP
ejpam-4465	97	29	τ1	τ1	NOUN
ejpam-4465	97	30	and	and	CCONJ
ejpam-4465	97	31	τ2	τ2	NOUN
ejpam-4465	97	32	.	.	PUNCT
ejpam-4465	98	1	3	3	X
ejpam-4465	98	2	.	.	X
ejpam-4465	98	3	conclusion	conclusion	NOUN
ejpam-4465	98	4	in	in	ADP
ejpam-4465	98	5	this	this	DET
ejpam-4465	98	6	work	work	NOUN
ejpam-4465	98	7	the	the	DET
ejpam-4465	98	8	notion	notion	NOUN
ejpam-4465	98	9	of	of	ADP
ejpam-4465	98	10	similarity	similarity	NOUN
ejpam-4465	98	11	between	between	ADP
ejpam-4465	98	12	topological	topological	ADJ
ejpam-4465	98	13	spaces	space	NOUN
ejpam-4465	98	14	,	,	PUNCT
ejpam-4465	98	15	is	be	AUX
ejpam-4465	98	16	blended	blend	VERB
ejpam-4465	98	17	with	with	ADP
ejpam-4465	98	18	ideals	ideal	NOUN
ejpam-4465	98	19	on	on	ADP
ejpam-4465	98	20	topological	topological	ADJ
ejpam-4465	98	21	spaces	space	NOUN
ejpam-4465	98	22	.	.	PUNCT
ejpam-4465	99	1	it	it	PRON
ejpam-4465	99	2	is	be	AUX
ejpam-4465	99	3	proved	prove	VERB
ejpam-4465	99	4	that	that	SCONJ
ejpam-4465	99	5	,	,	PUNCT
ejpam-4465	99	6	we	we	PRON
ejpam-4465	99	7	can	can	AUX
ejpam-4465	99	8	deduce	deduce	VERB
ejpam-4465	99	9	the	the	DET
ejpam-4465	99	10	similarity	similarity	NOUN
ejpam-4465	99	11	with	with	ADP
ejpam-4465	99	12	respect	respect	NOUN
ejpam-4465	99	13	to	to	ADP
ejpam-4465	99	14	an	an	DET
ejpam-4465	99	15	ideal	ideal	NOUN
ejpam-4465	99	16	i	i	PRON
ejpam-4465	99	17	,	,	PUNCT
ejpam-4465	99	18	under	under	ADP
ejpam-4465	99	19	the	the	DET
ejpam-4465	99	20	condition	condition	NOUN
ejpam-4465	99	21	of	of	ADP
ejpam-4465	99	22	coinciding	coincide	VERB
ejpam-4465	99	23	i	i	PRON
ejpam-4465	99	24	-	-	PUNCT
ejpam-4465	99	25	dense	dense	ADJ
ejpam-4465	99	26	subsets	subset	NOUN
ejpam-4465	99	27	,	,	PUNCT
ejpam-4465	99	28	and	and	CCONJ
ejpam-4465	99	29	the	the	DET
ejpam-4465	99	30	similarity	similarity	NOUN
ejpam-4465	99	31	between	between	ADP
ejpam-4465	99	32	τ∗1	τ∗1	NOUN
ejpam-4465	99	33	,	,	PUNCT
ejpam-4465	99	34	and	and	CCONJ
ejpam-4465	99	35	τ∗2	τ∗2	ADJ
ejpam-4465	99	36	topologies	topology	NOUN
ejpam-4465	99	37	is	be	AUX
ejpam-4465	99	38	equivalent	equivalent	ADJ
ejpam-4465	99	39	the	the	DET
ejpam-4465	99	40	i	i	NOUN
ejpam-4465	99	41	-	-	PUNCT
ejpam-4465	99	42	similarity	similarity	NOUN
ejpam-4465	99	43	between	between	ADP
ejpam-4465	99	44	τ1	τ1	NOUN
ejpam-4465	99	45	and	and	CCONJ
ejpam-4465	99	46	τ2	τ2	NOUN
ejpam-4465	99	47	topologies	topology	NOUN
ejpam-4465	99	48	.	.	PUNCT
ejpam-4465	100	1	however	however	ADV
ejpam-4465	100	2	,	,	PUNCT
ejpam-4465	100	3	the	the	DET
ejpam-4465	100	4	question	question	NOUN
ejpam-4465	100	5	asking	ask	VERB
ejpam-4465	100	6	if	if	SCONJ
ejpam-4465	100	7	there	there	PRON
ejpam-4465	100	8	are	be	VERB
ejpam-4465	100	9	any	any	DET
ejpam-4465	100	10	conditions	condition	NOUN
ejpam-4465	100	11	can	can	AUX
ejpam-4465	100	12	be	be	AUX
ejpam-4465	100	13	added	add	VERB
ejpam-4465	100	14	to	to	ADP
ejpam-4465	100	15	an	an	DET
ejpam-4465	100	16	ideal	ideal	NOUN
ejpam-4465	100	17	i	i	PRON
ejpam-4465	100	18	for	for	ADP
ejpam-4465	100	19	carrying	carry	VERB
ejpam-4465	100	20	similarity	similarity	NOUN
ejpam-4465	100	21	between	between	ADP
ejpam-4465	100	22	τ∗1	τ∗1	NOUN
ejpam-4465	100	23	and	and	CCONJ
ejpam-4465	100	24	τ∗2	τ∗2	NOUN
ejpam-4465	100	25	to	to	ADP
ejpam-4465	100	26	the	the	DET
ejpam-4465	100	27	spaces	space	NOUN
ejpam-4465	100	28	τ1	τ1	NOUN
ejpam-4465	100	29	and	and	CCONJ
ejpam-4465	100	30	τ2	τ2	NOUN
ejpam-4465	100	31	is	be	AUX
ejpam-4465	100	32	still	still	ADV
ejpam-4465	100	33	open	open	ADJ
ejpam-4465	100	34	for	for	ADP
ejpam-4465	100	35	a	a	DET
ejpam-4465	100	36	possible	possible	ADJ
ejpam-4465	100	37	future	future	ADJ
ejpam-4465	100	38	work	work	NOUN
ejpam-4465	100	39	.	.	PUNCT
ejpam-4465	101	1	acknowledgements	acknowledgement	VERB
ejpam-4465	101	2	the	the	DET
ejpam-4465	101	3	author	author	NOUN
ejpam-4465	101	4	thanks	thank	NOUN
ejpam-4465	101	5	the	the	DET
ejpam-4465	101	6	referees	referee	NOUN
ejpam-4465	101	7	for	for	ADP
ejpam-4465	101	8	their	their	PRON
ejpam-4465	101	9	useful	useful	ADJ
ejpam-4465	101	10	suggestions	suggestion	NOUN
ejpam-4465	101	11	which	which	PRON
ejpam-4465	101	12	were	be	AUX
ejpam-4465	101	13	very	very	ADV
ejpam-4465	101	14	helpful	helpful	ADJ
ejpam-4465	101	15	to	to	PART
ejpam-4465	101	16	improve	improve	VERB
ejpam-4465	101	17	this	this	DET
ejpam-4465	101	18	article	article	NOUN
ejpam-4465	101	19	.	.	PUNCT
ejpam-4465	102	1	references	reference	NOUN
ejpam-4465	102	2	[	[	X
ejpam-4465	102	3	1	1	NUM
ejpam-4465	102	4	]	]	PUNCT
ejpam-4465	102	5	m	m	NOUN
ejpam-4465	102	6	balcerzak	balcerzak	NOUN
ejpam-4465	102	7	,	,	PUNCT
ejpam-4465	102	8	a	a	DET
ejpam-4465	102	9	bartoszewicz	bartoszewicz	NOUN
ejpam-4465	102	10	,	,	PUNCT
ejpam-4465	102	11	j	j	PROPN
ejpam-4465	102	12	rzepecka	rzepecka	NOUN
ejpam-4465	102	13	,	,	PUNCT
ejpam-4465	102	14	and	and	CCONJ
ejpam-4465	102	15	s	s	VERB
ejpam-4465	102	16	wroński	wroński	NOUN
ejpam-4465	102	17	.	.	PUNCT
ejpam-4465	102	18	marczewski	marczewski	ADJ
ejpam-4465	102	19	fields	field	NOUN
ejpam-4465	102	20	and	and	CCONJ
ejpam-4465	102	21	ideals	ideal	NOUN
ejpam-4465	102	22	.	.	PUNCT
ejpam-4465	103	1	real	real	ADJ
ejpam-4465	103	2	anal	anal	PROPN
ejpam-4465	103	3	.	.	PUNCT
ejpam-4465	104	1	exch	exch	PROPN
ejpam-4465	104	2	.	.	PROPN
ejpam-4465	104	3	,	,	PUNCT
ejpam-4465	105	1	26:703–715	26:703–715	PROPN
ejpam-4465	105	2	,	,	PUNCT
ejpam-4465	105	3	2000	2000	NUM
ejpam-4465	105	4	.	.	PUNCT
ejpam-4465	106	1	[	[	X
ejpam-4465	106	2	2	2	X
ejpam-4465	106	3	]	]	PUNCT
ejpam-4465	106	4	a	a	DET
ejpam-4465	106	5	bartoszewicz	bartoszewicz	NOUN
ejpam-4465	106	6	,	,	PUNCT
ejpam-4465	106	7	m	m	PROPN
ejpam-4465	106	8	filipczak	filipczak	NOUN
ejpam-4465	106	9	,	,	PUNCT
ejpam-4465	106	10	a	a	DET
ejpam-4465	106	11	kowalski	kowalski	NOUN
ejpam-4465	106	12	,	,	PUNCT
ejpam-4465	106	13	and	and	CCONJ
ejpam-4465	106	14	m	m	PROPN
ejpam-4465	106	15	terepeta	terepeta	NOUN
ejpam-4465	106	16	.	.	PUNCT
ejpam-4465	107	1	on	on	ADP
ejpam-4465	107	2	similarity	similarity	NOUN
ejpam-4465	107	3	between	between	ADP
ejpam-4465	107	4	topologies	topology	NOUN
ejpam-4465	107	5	.	.	PUNCT
ejpam-4465	108	1	cent	cent	NOUN
ejpam-4465	108	2	.	.	PUNCT
ejpam-4465	109	1	eur	eur	PROPN
ejpam-4465	109	2	.	.	PUNCT
ejpam-4465	110	1	j.	j.	PROPN
ejpam-4465	110	2	math	math	PROPN
ejpam-4465	110	3	.	.	PROPN
ejpam-4465	110	4	,	,	PUNCT
ejpam-4465	110	5	12:603–610	12:603–610	NUM
ejpam-4465	110	6	,	,	PUNCT
ejpam-4465	110	7	2014	2014	NUM
ejpam-4465	110	8	.	.	PUNCT
ejpam-4465	111	1	[	[	X
ejpam-4465	111	2	3	3	X
ejpam-4465	111	3	]	]	X
ejpam-4465	111	4	j	j	PROPN
ejpam-4465	111	5	dontchev	dontchev	PROPN
ejpam-4465	111	6	,	,	PUNCT
ejpam-4465	111	7	m	m	NOUN
ejpam-4465	111	8	ganster	ganster	NOUN
ejpam-4465	111	9	,	,	PUNCT
ejpam-4465	111	10	and	and	CCONJ
ejpam-4465	111	11	d	d	NOUN
ejpam-4465	111	12	rose	rise	VERB
ejpam-4465	111	13	.	.	PUNCT
ejpam-4465	112	1	ideal	ideal	ADJ
ejpam-4465	112	2	resolvability	resolvability	NOUN
ejpam-4465	112	3	.	.	PUNCT
ejpam-4465	113	1	top	top	ADJ
ejpam-4465	113	2	.	.	PUNCT
ejpam-4465	113	3	appl	appl	PROPN
ejpam-4465	113	4	.	.	PROPN
ejpam-4465	113	5	,	,	PUNCT
ejpam-4465	113	6	93:1–16	93:1–16	NUM
ejpam-4465	113	7	,	,	PUNCT
ejpam-4465	113	8	1999	1999	NUM
ejpam-4465	113	9	.	.	PUNCT
ejpam-4465	114	1	[	[	X
ejpam-4465	114	2	4	4	NUM
ejpam-4465	114	3	]	]	X
ejpam-4465	114	4	e	e	X
ejpam-4465	114	5	hayashi	hayashi	PROPN
ejpam-4465	114	6	.	.	PUNCT
ejpam-4465	114	7	topologies	topology	NOUN
ejpam-4465	114	8	defined	define	VERB
ejpam-4465	114	9	by	by	ADP
ejpam-4465	114	10	local	local	ADJ
ejpam-4465	114	11	properties	property	NOUN
ejpam-4465	114	12	.	.	PUNCT
ejpam-4465	115	1	math	math	NOUN
ejpam-4465	115	2	.	.	PUNCT
ejpam-4465	116	1	ann	ann	PROPN
ejpam-4465	116	2	.	.	PROPN
ejpam-4465	116	3	,	,	PUNCT
ejpam-4465	116	4	156:205–215	156:205–215	NUM
ejpam-4465	116	5	,	,	PUNCT
ejpam-4465	116	6	1964	1964	NUM
ejpam-4465	116	7	.	.	PUNCT
ejpam-4465	117	1	[	[	X
ejpam-4465	117	2	5	5	NUM
ejpam-4465	117	3	]	]	X
ejpam-4465	117	4	d	d	NOUN
ejpam-4465	117	5	janković	janković	NOUN
ejpam-4465	117	6	and	and	CCONJ
ejpam-4465	117	7	t	t	PROPN
ejpam-4465	117	8	r	r	PROPN
ejpam-4465	117	9	hamlett	hamlett	PROPN
ejpam-4465	117	10	.	.	PUNCT
ejpam-4465	118	1	new	new	ADJ
ejpam-4465	118	2	topologies	topology	NOUN
ejpam-4465	118	3	from	from	ADP
ejpam-4465	118	4	old	old	ADJ
ejpam-4465	118	5	via	via	ADP
ejpam-4465	118	6	ideals	ideal	NOUN
ejpam-4465	118	7	.	.	PUNCT
ejpam-4465	119	1	amer	amer	PROPN
ejpam-4465	119	2	.	.	PUNCT
ejpam-4465	119	3	math	math	PROPN
ejpam-4465	119	4	.	.	PUNCT
ejpam-4465	120	1	mon	mon	PROPN
ejpam-4465	120	2	.	.	PROPN
ejpam-4465	120	3	,	,	PUNCT
ejpam-4465	120	4	97:295–310	97:295–310	PROPN
ejpam-4465	120	5	,	,	PUNCT
ejpam-4465	120	6	1990	1990	NUM
ejpam-4465	120	7	.	.	PUNCT
ejpam-4465	121	1	[	[	X
ejpam-4465	121	2	6	6	NUM
ejpam-4465	121	3	]	]	X
ejpam-4465	121	4	k	k	PROPN
ejpam-4465	121	5	kuratowski	kuratowski	PROPN
ejpam-4465	121	6	.	.	PUNCT
ejpam-4465	122	1	topology	topology	NOUN
ejpam-4465	122	2	.	.	PUNCT
ejpam-4465	123	1	academic	academic	ADJ
ejpam-4465	123	2	press	press	NOUN
ejpam-4465	123	3	,	,	PUNCT
ejpam-4465	123	4	new	new	PROPN
ejpam-4465	123	5	york	york	PROPN
ejpam-4465	123	6	,	,	PUNCT
ejpam-4465	123	7	ny	ny	PROPN
ejpam-4465	123	8	,	,	PUNCT
ejpam-4465	123	9	1966	1966	NUM
ejpam-4465	123	10	.	.	PUNCT
ejpam-4465	124	1	[	[	X
ejpam-4465	124	2	7	7	NUM
ejpam-4465	124	3	]	]	X
ejpam-4465	124	4	r	r	NOUN
ejpam-4465	124	5	vaidyanathaswamy	vaidyanathaswamy	NOUN
ejpam-4465	124	6	.	.	PUNCT
ejpam-4465	125	1	the	the	DET
ejpam-4465	125	2	localisation	localisation	NOUN
ejpam-4465	125	3	theory	theory	NOUN
ejpam-4465	125	4	in	in	ADP
ejpam-4465	125	5	the	the	DET
ejpam-4465	125	6	set	set	NOUN
ejpam-4465	125	7	topology	topology	NOUN
ejpam-4465	125	8	.	.	PUNCT
ejpam-4465	126	1	proc	proc	PROPN
ejpam-4465	126	2	.	.	PUNCT
ejpam-4465	127	1	indian	indian	PROPN
ejpam-4465	127	2	acad	acad	PROPN
ejpam-4465	127	3	.	.	PUNCT
ejpam-4465	128	1	sci	sci	PROPN
ejpam-4465	128	2	.	.	PROPN
ejpam-4465	128	3	,	,	PUNCT
ejpam-4465	128	4	20:51–61	20:51–61	NUM
ejpam-4465	128	5	,	,	PUNCT
ejpam-4465	128	6	1945	1945	NUM
ejpam-4465	128	7	.	.	PUNCT