id	sid	tid	token	lemma	pos
ejpam-237	1	1	1_hatir.dvi	1_hatir.dvi	NUM
ejpam-237	1	2	european	european	ADJ
ejpam-237	1	3	journal	journal	PROPN
ejpam-237	1	4	of	of	ADP
ejpam-237	1	5	pure	pure	ADJ
ejpam-237	1	6	and	and	CCONJ
ejpam-237	1	7	applied	apply	VERB
ejpam-237	1	8	mathematics	mathematic	NOUN
ejpam-237	1	9	vol	vol	NOUN
ejpam-237	1	10	.	.	PROPN
ejpam-237	2	1	2	2	NUM
ejpam-237	2	2	,	,	PUNCT
ejpam-237	2	3	no	no	INTJ
ejpam-237	2	4	.	.	NOUN
ejpam-237	2	5	2	2	NUM
ejpam-237	2	6	,	,	PUNCT
ejpam-237	2	7	2009	2009	NUM
ejpam-237	2	8	,	,	PUNCT
ejpam-237	2	9	(	(	PUNCT
ejpam-237	2	10	172	172	NUM
ejpam-237	2	11	-	-	SYM
ejpam-237	2	12	181	181	NUM
ejpam-237	2	13	)	)	PUNCT
ejpam-237	2	14	issn	issn	PROPN
ejpam-237	2	15	1307	1307	NUM
ejpam-237	2	16	-	-	SYM
ejpam-237	2	17	5543	5543	NUM
ejpam-237	2	18	–	–	PUNCT
ejpam-237	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-237	2	20	on	on	ADP
ejpam-237	2	21	hausdorff	hausdorff	NOUN
ejpam-237	2	22	spaces	space	NOUN
ejpam-237	2	23	via	via	ADP
ejpam-237	2	24	ideals	ideal	NOUN
ejpam-237	2	25	and	and	CCONJ
ejpam-237	2	26	semi	semi	ADJ
ejpam-237	2	27	-	-	ADJ
ejpam-237	2	28	i	i	NOUN
ejpam-237	2	29	-	-	PUNCT
ejpam-237	2	30	irresolute	irresolute	ADJ
ejpam-237	2	31	functions	function	NOUN
ejpam-237	2	32	e.	e.	PROPN
ejpam-237	2	33	hatir1∗	hatir1∗	PROPN
ejpam-237	2	34	and	and	CCONJ
ejpam-237	2	35	t.	t.	NOUN
ejpam-237	2	36	noiri2	noiri2	PROPN
ejpam-237	2	37	1	1	NUM
ejpam-237	2	38	selçuk	selçuk	NOUN
ejpam-237	2	39	üniversitesi	üniversitesi	NOUN
ejpam-237	2	40	,	,	PUNCT
ejpam-237	2	41	eğitim	eğitim	NOUN
ejpam-237	2	42	fakültesi	fakültesi	NOUN
ejpam-237	2	43	,	,	PUNCT
ejpam-237	2	44	42090	42090	NUM
ejpam-237	2	45	,	,	PUNCT
ejpam-237	2	46	meram	meram	NOUN
ejpam-237	2	47	-	-	PUNCT
ejpam-237	2	48	konya	konya	PROPN
ejpam-237	2	49	,	,	PUNCT
ejpam-237	2	50	turkey	turkey	PROPN
ejpam-237	2	51	2	2	NUM
ejpam-237	2	52	2949	2949	NUM
ejpam-237	2	53	shiokita	shiokita	PROPN
ejpam-237	2	54	-	-	PUNCT
ejpam-237	2	55	shi	shi	PROPN
ejpam-237	2	56	,	,	PUNCT
ejpam-237	2	57	kumamoto	kumamoto	PROPN
ejpam-237	2	58	-	-	PUNCT
ejpam-237	2	59	ken	ken	PROPN
ejpam-237	2	60	869	869	NUM
ejpam-237	2	61	-	-	SYM
ejpam-237	2	62	5142	5142	NUM
ejpam-237	2	63	,	,	PUNCT
ejpam-237	2	64	japan	japan	PROPN
ejpam-237	2	65	abstract	abstract	PROPN
ejpam-237	2	66	.	.	PUNCT
ejpam-237	3	1	we	we	PRON
ejpam-237	3	2	introduce	introduce	VERB
ejpam-237	3	3	the	the	DET
ejpam-237	3	4	notion	notion	NOUN
ejpam-237	3	5	of	of	ADP
ejpam-237	3	6	semi	semi	ADJ
ejpam-237	3	7	-	-	ADJ
ejpam-237	3	8	i	i	ADJ
ejpam-237	3	9	-	-	PUNCT
ejpam-237	3	10	hausdorff	hausdorff	NOUN
ejpam-237	3	11	spaces	space	NOUN
ejpam-237	3	12	which	which	PRON
ejpam-237	3	13	is	be	AUX
ejpam-237	3	14	weaker	weak	ADJ
ejpam-237	3	15	than	than	ADP
ejpam-237	3	16	hausdorff	hausdorff	NOUN
ejpam-237	3	17	spaces	space	NOUN
ejpam-237	3	18	and	and	CCONJ
ejpam-237	3	19	independent	independent	ADJ
ejpam-237	3	20	both	both	CCONJ
ejpam-237	3	21	i	i	PROPN
ejpam-237	3	22	-	-	PUNCT
ejpam-237	3	23	hausdorff	hausdorff	NOUN
ejpam-237	3	24	and	and	CCONJ
ejpam-237	3	25	quasi	quasi	ADJ
ejpam-237	3	26	-	-	ADJ
ejpam-237	3	27	i	i	NOUN
ejpam-237	3	28	-	-	PUNCT
ejpam-237	3	29	hausdorff	hausdorff	NOUN
ejpam-237	3	30	.	.	PUNCT
ejpam-237	4	1	ams	am	NOUN
ejpam-237	4	2	subject	subject	ADJ
ejpam-237	4	3	classifications	classification	NOUN
ejpam-237	4	4	:	:	PUNCT
ejpam-237	4	5	primary	primary	ADJ
ejpam-237	4	6	54c08	54c08	NOUN
ejpam-237	4	7	,	,	PUNCT
ejpam-237	4	8	54h05	54h05	NUM
ejpam-237	4	9	;	;	PUNCT
ejpam-237	4	10	secondary	secondary	ADJ
ejpam-237	4	11	54c10	54c10	NUM
ejpam-237	4	12	key	key	ADJ
ejpam-237	4	13	words	word	NOUN
ejpam-237	4	14	:	:	PUNCT
ejpam-237	4	15	i	i	NOUN
ejpam-237	4	16	-	-	PUNCT
ejpam-237	4	17	hausdorff	hausdorff	NOUN
ejpam-237	4	18	,	,	PUNCT
ejpam-237	4	19	quasi	quasi	ADJ
ejpam-237	4	20	-	-	ADJ
ejpam-237	4	21	i	i	NOUN
ejpam-237	4	22	-	-	PUNCT
ejpam-237	4	23	hausdorff	hausdorff	NOUN
ejpam-237	4	24	,	,	PUNCT
ejpam-237	4	25	semi	semi	ADJ
ejpam-237	4	26	-	-	ADJ
ejpam-237	4	27	i	i	NOUN
ejpam-237	4	28	-	-	PUNCT
ejpam-237	4	29	hausdorff	hausdorff	NOUN
ejpam-237	4	30	,	,	PUNCT
ejpam-237	4	31	semi	semi	ADJ
ejpam-237	4	32	-	-	ADJ
ejpam-237	4	33	i	i	NOUN
ejpam-237	4	34	-	-	PUNCT
ejpam-237	4	35	irresolute	irresolute	ADJ
ejpam-237	4	36	,	,	PUNCT
ejpam-237	4	37	semi	semi	ADJ
ejpam-237	4	38	-	-	ADJ
ejpam-237	4	39	i	i	PRON
ejpam-237	4	40	-	-	PUNCT
ejpam-237	4	41	open	open	ADJ
ejpam-237	4	42	set	set	NOUN
ejpam-237	4	43	.	.	PUNCT
ejpam-237	5	1	1	1	X
ejpam-237	5	2	.	.	X
ejpam-237	5	3	introduction	introduction	NOUN
ejpam-237	5	4	in	in	ADP
ejpam-237	5	5	[	[	X
ejpam-237	5	6	4	4	NUM
ejpam-237	5	7	]	]	PUNCT
ejpam-237	5	8	,	,	PUNCT
ejpam-237	5	9	dontchev	dontchev	PROPN
ejpam-237	5	10	has	have	AUX
ejpam-237	5	11	introduced	introduce	VERB
ejpam-237	5	12	and	and	CCONJ
ejpam-237	5	13	studied	study	VERB
ejpam-237	5	14	i	i	PROPN
ejpam-237	5	15	-	-	PUNCT
ejpam-237	5	16	hausdorff	hausdorff	NOUN
ejpam-237	5	17	spaces	space	NOUN
ejpam-237	5	18	.	.	PUNCT
ejpam-237	6	1	in	in	ADP
ejpam-237	6	2	[	[	X
ejpam-237	6	3	13	13	NUM
ejpam-237	6	4	]	]	PUNCT
ejpam-237	6	5	,	,	PUNCT
ejpam-237	6	6	nasef	nasef	PROPN
ejpam-237	6	7	has	have	AUX
ejpam-237	6	8	improved	improve	VERB
ejpam-237	6	9	i	i	PROPN
ejpam-237	6	10	-	-	PUNCT
ejpam-237	6	11	hausdorff	hausdorff	NOUN
ejpam-237	6	12	spaces	space	NOUN
ejpam-237	6	13	and	and	CCONJ
ejpam-237	6	14	defined	define	VERB
ejpam-237	6	15	quasi	quasi	ADJ
ejpam-237	6	16	-	-	ADJ
ejpam-237	6	17	i	i	NOUN
ejpam-237	6	18	-	-	PUNCT
ejpam-237	6	19	hausdorff	hausdorff	NOUN
ejpam-237	6	20	spaces	space	NOUN
ejpam-237	6	21	.	.	PUNCT
ejpam-237	7	1	in	in	ADP
ejpam-237	7	2	[	[	X
ejpam-237	7	3	5	5	NUM
ejpam-237	7	4	]	]	PUNCT
ejpam-237	7	5	,	,	PUNCT
ejpam-237	7	6	the	the	DET
ejpam-237	7	7	∗corresponding	∗corresponde	VERB
ejpam-237	7	8	author	author	NOUN
ejpam-237	7	9	.	.	PUNCT
ejpam-237	8	1	email	email	NOUN
ejpam-237	8	2	addresses	address	NOUN
ejpam-237	8	3	:	:	PUNCT
ejpam-237	8	4	hatir10	hatir10	X
ejpam-237	8	5	�	�	PROPN
ejpam-237	8	6	yahoo	yahoo	PROPN
ejpam-237	8	7	.	.	PUNCT
ejpam-237	9	1	om	om	PROPN
ejpam-237	9	2	(	(	PUNCT
ejpam-237	9	3	e.	e.	PROPN
ejpam-237	9	4	hatir	hatir	PROPN
ejpam-237	9	5	)	)	PUNCT
ejpam-237	9	6	,	,	PUNCT
ejpam-237	9	7	t.noiri	t.noiri	ADV
ejpam-237	9	8	�	�	NOUN
ejpam-237	9	9	nifty	nifty	ADJ
ejpam-237	9	10	.	.	PUNCT
ejpam-237	10	1	om	om	PROPN
ejpam-237	10	2	(	(	PUNCT
ejpam-237	10	3	t.	t.	PROPN
ejpam-237	10	4	noiri	noiri	PROPN
ejpam-237	10	5	)	)	PUNCT
ejpam-237	10	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-237	11	1	172	172	NUM
ejpam-237	11	2	c	c	X
ejpam-237	11	3	©	©	PROPN
ejpam-237	11	4	2009	2009	NUM
ejpam-237	11	5	ejpam	ejpam	NOUN
ejpam-237	11	6	all	all	DET
ejpam-237	11	7	rights	right	NOUN
ejpam-237	11	8	reserved	reserve	VERB
ejpam-237	11	9	.	.	PUNCT
ejpam-237	12	1	e.	e.	PROPN
ejpam-237	12	2	hatir	hatir	PROPN
ejpam-237	12	3	and	and	CCONJ
ejpam-237	12	4	t.	t.	PROPN
ejpam-237	12	5	noiri	noiri	PROPN
ejpam-237	12	6	/	/	SYM
ejpam-237	12	7	eur	eur	PROPN
ejpam-237	12	8	.	.	PUNCT
ejpam-237	13	1	j.	j.	PROPN
ejpam-237	13	2	pure	pure	PROPN
ejpam-237	13	3	appl	appl	PROPN
ejpam-237	13	4	.	.	PROPN
ejpam-237	13	5	math	math	PROPN
ejpam-237	13	6	,	,	PUNCT
ejpam-237	13	7	2	2	NUM
ejpam-237	13	8	(	(	PUNCT
ejpam-237	13	9	2009	2009	NUM
ejpam-237	13	10	)	)	PUNCT
ejpam-237	13	11	,	,	PUNCT
ejpam-237	13	12	(	(	PUNCT
ejpam-237	13	13	172	172	NUM
ejpam-237	13	14	-	-	SYM
ejpam-237	13	15	181	181	NUM
ejpam-237	13	16	)	)	PUNCT
ejpam-237	13	17	173	173	NUM
ejpam-237	13	18	present	present	ADJ
ejpam-237	13	19	authors	author	NOUN
ejpam-237	13	20	defined	define	VERB
ejpam-237	13	21	the	the	DET
ejpam-237	13	22	notion	notion	NOUN
ejpam-237	13	23	of	of	ADP
ejpam-237	13	24	semi	semi	ADJ
ejpam-237	13	25	-	-	ADJ
ejpam-237	13	26	open	open	ADJ
ejpam-237	13	27	sets	set	NOUN
ejpam-237	13	28	via	via	ADP
ejpam-237	13	29	ideals	ideal	NOUN
ejpam-237	13	30	to	to	PART
ejpam-237	13	31	obtain	obtain	VERB
ejpam-237	13	32	decomposition	decomposition	NOUN
ejpam-237	13	33	of	of	ADP
ejpam-237	13	34	continuity	continuity	NOUN
ejpam-237	13	35	.	.	PUNCT
ejpam-237	14	1	in	in	ADP
ejpam-237	14	2	the	the	DET
ejpam-237	14	3	present	present	ADJ
ejpam-237	14	4	paper	paper	NOUN
ejpam-237	14	5	,	,	PUNCT
ejpam-237	14	6	we	we	PRON
ejpam-237	14	7	introduce	introduce	VERB
ejpam-237	14	8	the	the	DET
ejpam-237	14	9	notion	notion	NOUN
ejpam-237	14	10	of	of	ADP
ejpam-237	14	11	semi	semi	ADJ
ejpam-237	14	12	-	-	ADJ
ejpam-237	14	13	i	i	ADJ
ejpam-237	14	14	-	-	PUNCT
ejpam-237	14	15	hausdorff	hausdorff	NOUN
ejpam-237	14	16	spaces	space	NOUN
ejpam-237	14	17	which	which	PRON
ejpam-237	14	18	is	be	AUX
ejpam-237	14	19	weaker	weak	ADJ
ejpam-237	14	20	than	than	ADP
ejpam-237	14	21	hausdorff	hausdorff	NOUN
ejpam-237	14	22	spaces	space	NOUN
ejpam-237	14	23	and	and	CCONJ
ejpam-237	14	24	independent	independent	ADJ
ejpam-237	14	25	both	both	CCONJ
ejpam-237	14	26	i	i	PROPN
ejpam-237	14	27	-	-	PUNCT
ejpam-237	14	28	hausdorff	hausdorff	NOUN
ejpam-237	14	29	and	and	CCONJ
ejpam-237	14	30	quasi	quasi	ADJ
ejpam-237	14	31	-	-	ADJ
ejpam-237	14	32	i	i	NOUN
ejpam-237	14	33	-	-	PUNCT
ejpam-237	14	34	hausdorff	hausdorff	NOUN
ejpam-237	14	35	spaces	space	NOUN
ejpam-237	14	36	.	.	PUNCT
ejpam-237	15	1	using	use	VERB
ejpam-237	15	2	semi	semi	ADJ
ejpam-237	15	3	-	-	ADJ
ejpam-237	15	4	i	i	NOUN
ejpam-237	15	5	-	-	NOUN
ejpam-237	15	6	irresolute	irresolute	ADJ
ejpam-237	15	7	[	[	X
ejpam-237	15	8	6	6	NUM
ejpam-237	15	9	]	]	PUNCT
ejpam-237	15	10	functions	function	NOUN
ejpam-237	15	11	,	,	PUNCT
ejpam-237	15	12	we	we	PRON
ejpam-237	15	13	also	also	ADV
ejpam-237	15	14	investigate	investigate	VERB
ejpam-237	15	15	its	its	PRON
ejpam-237	15	16	relation	relation	NOUN
ejpam-237	15	17	with	with	ADP
ejpam-237	15	18	semi	semi	ADJ
ejpam-237	15	19	-	-	ADJ
ejpam-237	15	20	i	i	ADJ
ejpam-237	15	21	-	-	PUNCT
ejpam-237	15	22	hausdorff	hausdorff	NOUN
ejpam-237	15	23	spaces	space	NOUN
ejpam-237	15	24	.	.	PUNCT
ejpam-237	16	1	2	2	X
ejpam-237	16	2	.	.	NUM
ejpam-237	16	3	preliminaries	preliminary	NOUN
ejpam-237	16	4	throughout	throughout	ADP
ejpam-237	16	5	this	this	DET
ejpam-237	16	6	paper	paper	NOUN
ejpam-237	16	7	,	,	PUNCT
ejpam-237	16	8	(	(	PUNCT
ejpam-237	16	9	x	x	X
ejpam-237	16	10	,	,	PUNCT
ejpam-237	16	11	τ	τ	X
ejpam-237	16	12	)	)	PUNCT
ejpam-237	16	13	(	(	PUNCT
ejpam-237	16	14	simply	simply	ADV
ejpam-237	16	15	x	x	X
ejpam-237	16	16	)	)	PUNCT
ejpam-237	16	17	denotes	denote	VERB
ejpam-237	16	18	a	a	DET
ejpam-237	16	19	topological	topological	ADJ
ejpam-237	16	20	space	space	NOUN
ejpam-237	16	21	on	on	ADP
ejpam-237	16	22	which	which	PRON
ejpam-237	16	23	no	no	DET
ejpam-237	16	24	separation	separation	NOUN
ejpam-237	16	25	axiom	axiom	NOUN
ejpam-237	16	26	is	be	AUX
ejpam-237	16	27	assumed	assume	VERB
ejpam-237	16	28	unless	unless	SCONJ
ejpam-237	16	29	explicitly	explicitly	ADV
ejpam-237	16	30	stated	state	VERB
ejpam-237	16	31	.	.	PUNCT
ejpam-237	17	1	for	for	ADP
ejpam-237	17	2	a	a	DET
ejpam-237	17	3	subset	subset	NOUN
ejpam-237	17	4	a	a	PRON
ejpam-237	17	5	of	of	ADP
ejpam-237	17	6	a	a	DET
ejpam-237	17	7	topological	topological	ADJ
ejpam-237	17	8	space	space	NOUN
ejpam-237	17	9	x	x	SYM
ejpam-237	17	10	,	,	PUNCT
ejpam-237	17	11	the	the	DET
ejpam-237	17	12	closure	closure	NOUN
ejpam-237	17	13	and	and	CCONJ
ejpam-237	17	14	the	the	DET
ejpam-237	17	15	interior	interior	NOUN
ejpam-237	17	16	of	of	ADP
ejpam-237	17	17	a	a	DET
ejpam-237	17	18	in	in	NOUN
ejpam-237	17	19	x	x	X
ejpam-237	17	20	are	be	AUX
ejpam-237	17	21	denoted	denote	VERB
ejpam-237	17	22	by	by	ADP
ejpam-237	17	23	cl(a	cl(a	NOUN
ejpam-237	17	24	)	)	PUNCT
ejpam-237	17	25	and	and	CCONJ
ejpam-237	17	26	int(a	int(a	PROPN
ejpam-237	17	27	)	)	PUNCT
ejpam-237	17	28	,	,	PUNCT
ejpam-237	17	29	respectively	respectively	ADV
ejpam-237	17	30	.	.	PUNCT
ejpam-237	18	1	a	a	DET
ejpam-237	18	2	nonempty	nonempty	ADJ
ejpam-237	18	3	collection	collection	NOUN
ejpam-237	18	4	i	i	PRON
ejpam-237	18	5	of	of	ADP
ejpam-237	18	6	subsets	subset	NOUN
ejpam-237	18	7	on	on	ADP
ejpam-237	18	8	a	a	DET
ejpam-237	18	9	topological	topological	ADJ
ejpam-237	18	10	space	space	NOUN
ejpam-237	18	11	(	(	PUNCT
ejpam-237	18	12	x	x	X
ejpam-237	18	13	,	,	PUNCT
ejpam-237	18	14	τ	τ	X
ejpam-237	18	15	)	)	PUNCT
ejpam-237	18	16	is	be	AUX
ejpam-237	18	17	called	call	VERB
ejpam-237	18	18	a	a	DET
ejpam-237	18	19	topological	topological	ADJ
ejpam-237	18	20	ideal	ideal	NOUN
ejpam-237	18	21	on	on	ADP
ejpam-237	18	22	(	(	PUNCT
ejpam-237	18	23	x	x	INTJ
ejpam-237	18	24	,	,	PUNCT
ejpam-237	18	25	τ	τ	X
ejpam-237	18	26	)	)	PUNCT
ejpam-237	18	27	if	if	SCONJ
ejpam-237	18	28	it	it	PRON
ejpam-237	18	29	satisfies	satisfy	VERB
ejpam-237	18	30	the	the	DET
ejpam-237	18	31	following	follow	VERB
ejpam-237	18	32	two	two	NUM
ejpam-237	18	33	conditions	condition	NOUN
ejpam-237	18	34	:	:	PUNCT
ejpam-237	18	35	(	(	PUNCT
ejpam-237	18	36	1	1	X
ejpam-237	18	37	)	)	PUNCT
ejpam-237	18	38	if	if	SCONJ
ejpam-237	18	39	a	a	DET
ejpam-237	18	40	∈	∈	X
ejpam-237	18	41	i	i	PRON
ejpam-237	18	42	and	and	CCONJ
ejpam-237	18	43	b	b	PROPN
ejpam-237	18	44	⊂	⊂	PROPN
ejpam-237	18	45	a	a	PROPN
ejpam-237	18	46	,	,	PUNCT
ejpam-237	18	47	then	then	ADV
ejpam-237	18	48	b	b	X
ejpam-237	18	49	∈	∈	PROPN
ejpam-237	18	50	i	i	PRON
ejpam-237	18	51	(	(	PUNCT
ejpam-237	18	52	heredity	heredity	NOUN
ejpam-237	18	53	)	)	PUNCT
ejpam-237	18	54	;	;	PUNCT
ejpam-237	18	55	(	(	PUNCT
ejpam-237	18	56	2	2	X
ejpam-237	18	57	)	)	PUNCT
ejpam-237	18	58	if	if	SCONJ
ejpam-237	18	59	a	a	DET
ejpam-237	18	60	∈	∈	X
ejpam-237	18	61	i	i	PRON
ejpam-237	18	62	and	and	CCONJ
ejpam-237	18	63	b	b	X
ejpam-237	18	64	∈	∈	PROPN
ejpam-237	18	65	i	i	PRON
ejpam-237	18	66	,	,	PUNCT
ejpam-237	18	67	then	then	ADV
ejpam-237	18	68	a∪	a∪	PROPN
ejpam-237	18	69	b	b	PROPN
ejpam-237	18	70	∈	∈	PROPN
ejpam-237	19	1	i	i	PRON
ejpam-237	19	2	(	(	PUNCT
ejpam-237	19	3	finite	finite	PROPN
ejpam-237	19	4	additivity	additivity	NOUN
ejpam-237	19	5	)	)	PUNCT
ejpam-237	19	6	.	.	PUNCT
ejpam-237	20	1	if	if	SCONJ
ejpam-237	20	2	i	i	PRON
ejpam-237	20	3	is	be	AUX
ejpam-237	20	4	a	a	DET
ejpam-237	20	5	proper	proper	ADJ
ejpam-237	20	6	ideal	ideal	NOUN
ejpam-237	20	7	,	,	PUNCT
ejpam-237	20	8	that	that	ADV
ejpam-237	20	9	is	is	ADV
ejpam-237	20	10	,	,	PUNCT
ejpam-237	20	11	x	x	PROPN
ejpam-237	20	12	/∈	/∈	PUNCT
ejpam-237	21	1	i	i	PRON
ejpam-237	21	2	,	,	PUNCT
ejpam-237	21	3	then	then	ADV
ejpam-237	21	4	{	{	PUNCT
ejpam-237	21	5	a	a	X
ejpam-237	21	6	:	:	PUNCT
ejpam-237	21	7	x	x	SYM
ejpam-237	21	8	−	−	PROPN
ejpam-237	21	9	a∈	a∈	PROPN
ejpam-237	21	10	i	i	PRON
ejpam-237	21	11	}	}	PUNCT
ejpam-237	21	12	is	be	AUX
ejpam-237	21	13	a	a	DET
ejpam-237	21	14	filter	filter	NOUN
ejpam-237	21	15	,	,	PUNCT
ejpam-237	21	16	hence	hence	ADV
ejpam-237	21	17	proper	proper	ADJ
ejpam-237	21	18	ideals	ideal	NOUN
ejpam-237	21	19	are	be	AUX
ejpam-237	21	20	sometimes	sometimes	ADV
ejpam-237	21	21	called	call	VERB
ejpam-237	21	22	dual	dual	ADJ
ejpam-237	21	23	filters	filter	NOUN
ejpam-237	21	24	.	.	PUNCT
ejpam-237	22	1	by	by	ADP
ejpam-237	22	2	(	(	PUNCT
ejpam-237	22	3	x	x	INTJ
ejpam-237	22	4	,	,	PUNCT
ejpam-237	22	5	τ	τ	PROPN
ejpam-237	22	6	,	,	PUNCT
ejpam-237	22	7	i	i	PROPN
ejpam-237	22	8	)	)	PUNCT
ejpam-237	22	9	,	,	PUNCT
ejpam-237	22	10	we	we	PRON
ejpam-237	22	11	will	will	AUX
ejpam-237	22	12	denote	denote	VERB
ejpam-237	22	13	an	an	DET
ejpam-237	22	14	ideal	ideal	ADJ
ejpam-237	22	15	topological	topological	ADJ
ejpam-237	22	16	space	space	NOUN
ejpam-237	22	17	which	which	PRON
ejpam-237	22	18	means	mean	VERB
ejpam-237	22	19	a	a	DET
ejpam-237	22	20	topological	topological	ADJ
ejpam-237	22	21	space	space	NOUN
ejpam-237	22	22	(	(	PUNCT
ejpam-237	22	23	x	x	X
ejpam-237	22	24	,	,	PUNCT
ejpam-237	22	25	τ	τ	PROPN
ejpam-237	22	26	)	)	PUNCT
ejpam-237	22	27	with	with	ADP
ejpam-237	22	28	an	an	DET
ejpam-237	22	29	ideal	ideal	ADJ
ejpam-237	22	30	i	i	PRON
ejpam-237	22	31	on	on	ADP
ejpam-237	22	32	x	x	X
ejpam-237	22	33	.	.	PUNCT
ejpam-237	23	1	no	no	DET
ejpam-237	23	2	separation	separation	NOUN
ejpam-237	23	3	property	property	NOUN
ejpam-237	23	4	is	be	AUX
ejpam-237	23	5	assumed	assume	VERB
ejpam-237	23	6	on	on	ADP
ejpam-237	23	7	x	x	X
ejpam-237	23	8	.	.	PUNCT
ejpam-237	24	1	for	for	ADP
ejpam-237	24	2	a	a	DET
ejpam-237	24	3	space	space	NOUN
ejpam-237	24	4	(	(	PUNCT
ejpam-237	24	5	x	x	X
ejpam-237	24	6	,	,	PUNCT
ejpam-237	24	7	τ	τ	PROPN
ejpam-237	24	8	,	,	PUNCT
ejpam-237	24	9	i	i	PROPN
ejpam-237	24	10	)	)	PUNCT
ejpam-237	24	11	and	and	CCONJ
ejpam-237	24	12	a	a	DET
ejpam-237	24	13	subset	subset	NOUN
ejpam-237	24	14	a	a	PRON
ejpam-237	24	15	of	of	ADP
ejpam-237	24	16	x	x	SYM
ejpam-237	24	17	,	,	PUNCT
ejpam-237	24	18	a∗(i	a∗(i	PROPN
ejpam-237	24	19	)	)	PUNCT
ejpam-237	24	20	=	=	SYM
ejpam-237	24	21	�	�	PROPN
ejpam-237	24	22	x	x	SYM
ejpam-237	24	23	∈	∈	PROPN
ejpam-237	24	24	x	x	X
ejpam-237	24	25	:	:	PUNCT
ejpam-237	24	26	u	u	NOUN
ejpam-237	24	27	∩	∩	NOUN
ejpam-237	24	28	a	a	X
ejpam-237	24	29	/∈	/∈	PUNCT
ejpam-237	24	30	i	i	PRON
ejpam-237	24	31	for	for	ADP
ejpam-237	24	32	each	each	DET
ejpam-237	24	33	neighborhood	neighborhood	NOUN
ejpam-237	24	34	u	u	NOUN
ejpam-237	24	35	of	of	ADP
ejpam-237	24	36	x	x	PRON
ejpam-237	24	37	is	be	AUX
ejpam-237	24	38	called	call	VERB
ejpam-237	24	39	the	the	DET
ejpam-237	24	40	local	local	ADJ
ejpam-237	24	41	function	function	NOUN
ejpam-237	24	42	of	of	ADP
ejpam-237	24	43	a	a	PRON
ejpam-237	24	44	with	with	ADP
ejpam-237	24	45	respect	respect	NOUN
ejpam-237	24	46	to	to	ADP
ejpam-237	24	47	i	i	PRON
ejpam-237	24	48	and	and	CCONJ
ejpam-237	24	49	τ	τ	X
ejpam-237	25	1	[	[	X
ejpam-237	25	2	7	7	X
ejpam-237	25	3	]	]	PUNCT
ejpam-237	25	4	.	.	PUNCT
ejpam-237	26	1	we	we	PRON
ejpam-237	26	2	simply	simply	ADV
ejpam-237	26	3	write	write	VERB
ejpam-237	26	4	a∗	a∗	PROPN
ejpam-237	26	5	instead	instead	ADV
ejpam-237	26	6	of	of	ADP
ejpam-237	26	7	a∗(i	a∗(i	PROPN
ejpam-237	26	8	)	)	PUNCT
ejpam-237	26	9	in	in	ADP
ejpam-237	26	10	case	case	NOUN
ejpam-237	26	11	there	there	PRON
ejpam-237	26	12	is	be	VERB
ejpam-237	26	13	no	no	DET
ejpam-237	26	14	chance	chance	NOUN
ejpam-237	26	15	for	for	ADP
ejpam-237	26	16	confusion	confusion	NOUN
ejpam-237	26	17	.	.	PUNCT
ejpam-237	27	1	the	the	DET
ejpam-237	27	2	simplest	simple	ADJ
ejpam-237	27	3	ideals	ideal	NOUN
ejpam-237	27	4	are	be	AUX
ejpam-237	27	5	{	{	PUNCT
ejpam-237	27	6	∅	∅	NOUN
ejpam-237	27	7	}	}	PUNCT
ejpam-237	27	8	and	and	CCONJ
ejpam-237	27	9	℘(x	℘(x	ADJ
ejpam-237	27	10	)	)	PUNCT
ejpam-237	27	11	which	which	PRON
ejpam-237	27	12	satisfy	satisfy	VERB
ejpam-237	27	13	{	{	PUNCT
ejpam-237	27	14	∅	∅	NOUN
ejpam-237	27	15	}	}	PUNCT
ejpam-237	27	16	⊂	⊂	PROPN
ejpam-237	28	1	i	i	PRON
ejpam-237	28	2	⊂	⊂	PROPN
ejpam-237	28	3	℘(x	℘(x	ADJ
ejpam-237	28	4	)	)	PUNCT
ejpam-237	28	5	,	,	PUNCT
ejpam-237	28	6	for	for	ADP
ejpam-237	28	7	any	any	DET
ejpam-237	28	8	ideal	ideal	NOUN
ejpam-237	29	1	i	i	PRON
ejpam-237	29	2	on	on	ADP
ejpam-237	29	3	x	x	X
ejpam-237	29	4	.	.	PUNCT
ejpam-237	29	5	note	note	VERB
ejpam-237	29	6	that	that	SCONJ
ejpam-237	29	7	cl∗(a	cl∗(a	NOUN
ejpam-237	29	8	)	)	PUNCT
ejpam-237	29	9	=	=	SYM
ejpam-237	30	1	a∪a∗	a∪a∗	PROPN
ejpam-237	30	2	defines	define	VERB
ejpam-237	30	3	a	a	DET
ejpam-237	30	4	kuratowski	kuratowski	ADJ
ejpam-237	30	5	closure	closure	NOUN
ejpam-237	30	6	operator	operator	NOUN
ejpam-237	30	7	for	for	ADP
ejpam-237	30	8	a	a	DET
ejpam-237	30	9	topology	topology	NOUN
ejpam-237	30	10	τ∗(i	τ∗(i	PROPN
ejpam-237	30	11	)	)	PUNCT
ejpam-237	30	12	(	(	PUNCT
ejpam-237	30	13	also	also	ADV
ejpam-237	30	14	denoted	denote	VERB
ejpam-237	30	15	by	by	ADP
ejpam-237	30	16	τ∗	τ∗	NOUN
ejpam-237	30	17	when	when	SCONJ
ejpam-237	30	18	there	there	PRON
ejpam-237	30	19	is	be	VERB
ejpam-237	30	20	no	no	DET
ejpam-237	30	21	chance	chance	NOUN
ejpam-237	30	22	for	for	ADP
ejpam-237	30	23	confusion	confusion	NOUN
ejpam-237	30	24	)	)	PUNCT
ejpam-237	30	25	finer	fine	ADJ
ejpam-237	30	26	than	than	ADP
ejpam-237	30	27	τ	τ	PROPN
ejpam-237	30	28	.	.	PUNCT
ejpam-237	30	29	definition	definition	NOUN
ejpam-237	30	30	2.1	2.1	NUM
ejpam-237	30	31	.	.	PUNCT
ejpam-237	31	1	a	a	DET
ejpam-237	31	2	subset	subset	NOUN
ejpam-237	31	3	a	a	PRON
ejpam-237	31	4	of	of	ADP
ejpam-237	31	5	an	an	DET
ejpam-237	31	6	ideal	ideal	ADJ
ejpam-237	31	7	topological	topological	ADJ
ejpam-237	31	8	space	space	NOUN
ejpam-237	31	9	(	(	PUNCT
ejpam-237	31	10	x	x	X
ejpam-237	31	11	,	,	PUNCT
ejpam-237	31	12	τ	τ	PROPN
ejpam-237	31	13	,	,	PUNCT
ejpam-237	31	14	i	i	PROPN
ejpam-237	31	15	)	)	PUNCT
ejpam-237	31	16	is	be	AUX
ejpam-237	31	17	said	say	VERB
ejpam-237	31	18	to	to	PART
ejpam-237	31	19	be	be	AUX
ejpam-237	31	20	semiopen	semiopen	ADJ
ejpam-237	31	21	[	[	X
ejpam-237	31	22	10	10	NUM
ejpam-237	31	23	]	]	PUNCT
ejpam-237	31	24	(	(	PUNCT
ejpam-237	31	25	resp	resp	NOUN
ejpam-237	31	26	.	.	PUNCT
ejpam-237	32	1	β	β	X
ejpam-237	33	1	−	−	PROPN
ejpam-237	33	2	open	open	ADJ
ejpam-237	33	3	[	[	X
ejpam-237	33	4	1	1	NUM
ejpam-237	33	5	]	]	PUNCT
ejpam-237	33	6	,	,	PUNCT
ejpam-237	33	7	semi	semi	ADJ
ejpam-237	33	8	-	-	ADJ
ejpam-237	33	9	i	i	PRON
ejpam-237	33	10	-	-	PUNCT
ejpam-237	33	11	open	open	ADJ
ejpam-237	33	12	[	[	X
ejpam-237	33	13	5	5	NUM
ejpam-237	33	14	]	]	PUNCT
ejpam-237	33	15	,	,	PUNCT
ejpam-237	33	16	i	i	PRON
ejpam-237	33	17	-	-	VERB
ejpam-237	33	18	open	open	ADJ
ejpam-237	34	1	[	[	X
ejpam-237	34	2	9	9	NUM
ejpam-237	34	3	]	]	PUNCT
ejpam-237	34	4	,	,	PUNCT
ejpam-237	34	5	quasi	quasi	ADJ
ejpam-237	34	6	-	-	ADJ
ejpam-237	34	7	i	i	PRON
ejpam-237	34	8	-	-	PUNCT
ejpam-237	34	9	open	open	ADJ
ejpam-237	34	10	[	[	X
ejpam-237	34	11	2	2	NUM
ejpam-237	34	12	]	]	PUNCT
ejpam-237	34	13	)	)	PUNCT
ejpam-237	34	14	if	if	SCONJ
ejpam-237	34	15	a	a	DET
ejpam-237	34	16	⊂	⊂	PROPN
ejpam-237	34	17	cl(int(a	cl(int(a	PROPN
ejpam-237	34	18	)	)	PUNCT
ejpam-237	34	19	)	)	PUNCT
ejpam-237	35	1	(	(	PUNCT
ejpam-237	35	2	resp	resp	NOUN
ejpam-237	35	3	.	.	PUNCT
ejpam-237	36	1	a	a	DET
ejpam-237	36	2	⊂	⊂	PROPN
ejpam-237	36	3	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-237	36	4	)	)	PUNCT
ejpam-237	36	5	)	)	PUNCT
ejpam-237	36	6	)	)	PUNCT
ejpam-237	36	7	,	,	PUNCT
ejpam-237	36	8	a	a	DET
ejpam-237	36	9	⊂	⊂	PROPN
ejpam-237	36	10	cl∗(int(a	cl∗(int(a	PROPN
ejpam-237	36	11	)	)	PUNCT
ejpam-237	36	12	)	)	PUNCT
ejpam-237	36	13	,	,	PUNCT
ejpam-237	36	14	a	a	DET
ejpam-237	36	15	⊂	⊂	PROPN
ejpam-237	36	16	int(a∗	int(a∗	PART
ejpam-237	36	17	)	)	PUNCT
ejpam-237	36	18	,	,	PUNCT
ejpam-237	36	19	a	a	DET
ejpam-237	36	20	⊂	⊂	PROPN
ejpam-237	36	21	e.	e.	PROPN
ejpam-237	36	22	hatir	hatir	PROPN
ejpam-237	36	23	and	and	CCONJ
ejpam-237	36	24	t.	t.	PROPN
ejpam-237	36	25	noiri	noiri	PROPN
ejpam-237	36	26	/	/	SYM
ejpam-237	36	27	eur	eur	PROPN
ejpam-237	36	28	.	.	PUNCT
ejpam-237	37	1	j.	j.	PROPN
ejpam-237	37	2	pure	pure	PROPN
ejpam-237	37	3	appl	appl	PROPN
ejpam-237	37	4	.	.	PROPN
ejpam-237	37	5	math	math	PROPN
ejpam-237	37	6	,	,	PUNCT
ejpam-237	37	7	2	2	NUM
ejpam-237	37	8	(	(	PUNCT
ejpam-237	37	9	2009	2009	NUM
ejpam-237	37	10	)	)	PUNCT
ejpam-237	37	11	,	,	PUNCT
ejpam-237	37	12	(	(	PUNCT
ejpam-237	37	13	172	172	NUM
ejpam-237	37	14	-	-	SYM
ejpam-237	37	15	181	181	NUM
ejpam-237	37	16	)	)	PUNCT
ejpam-237	37	17	174	174	NUM
ejpam-237	37	18	cl(int(a∗	cl(int(a∗	PROPN
ejpam-237	37	19	)	)	PUNCT
ejpam-237	37	20	)	)	PUNCT
ejpam-237	37	21	)	)	PUNCT
ejpam-237	37	22	.	.	PUNCT
ejpam-237	38	1	for	for	ADP
ejpam-237	38	2	a	a	DET
ejpam-237	38	3	subsets	subset	NOUN
ejpam-237	38	4	defined	define	VERB
ejpam-237	38	5	above	above	ADP
ejpam-237	38	6	,	,	PUNCT
ejpam-237	38	7	the	the	DET
ejpam-237	38	8	following	follow	VERB
ejpam-237	38	9	diagram	diagram	NOUN
ejpam-237	38	10	holds	hold	VERB
ejpam-237	38	11	:	:	PUNCT
ejpam-237	39	1	diagram	diagram	NOUN
ejpam-237	39	2	i	i	PRON
ejpam-237	39	3	open	open	VERB
ejpam-237	39	4	−→	−→	NOUN
ejpam-237	39	5	semi	semi	ADJ
ejpam-237	39	6	-	-	ADJ
ejpam-237	39	7	i	i	PRON
ejpam-237	39	8	-	-	PUNCT
ejpam-237	39	9	open	open	ADJ
ejpam-237	39	10	−→	−→	NOUN
ejpam-237	39	11	semi	semi	ADJ
ejpam-237	39	12	-	-	ADJ
ejpam-237	39	13	open	open	ADJ
ejpam-237	39	14	↓	↓	NOUN
ejpam-237	40	1	i	i	PRON
ejpam-237	40	2	−	−	PROPN
ejpam-237	40	3	open	open	VERB
ejpam-237	40	4	−→	−→	NOUN
ejpam-237	40	5	quasi	quasi	ADJ
ejpam-237	40	6	-	-	ADJ
ejpam-237	40	7	i	i	PRON
ejpam-237	40	8	-	-	PUNCT
ejpam-237	40	9	open	open	ADJ
ejpam-237	40	10	−→	−→	NOUN
ejpam-237	40	11	β	β	X
ejpam-237	40	12	−	−	NOUN
ejpam-237	40	13	open	open	ADJ
ejpam-237	40	14	definition	definition	NOUN
ejpam-237	40	15	2.2	2.2	NUM
ejpam-237	40	16	.	.	PUNCT
ejpam-237	41	1	a	a	DET
ejpam-237	41	2	space	space	NOUN
ejpam-237	41	3	(	(	PUNCT
ejpam-237	41	4	x	x	X
ejpam-237	41	5	,	,	PUNCT
ejpam-237	41	6	τ	τ	X
ejpam-237	41	7	)	)	PUNCT
ejpam-237	41	8	is	be	AUX
ejpam-237	41	9	said	say	VERB
ejpam-237	41	10	to	to	PART
ejpam-237	41	11	be	be	AUX
ejpam-237	41	12	semi	semi	ADJ
ejpam-237	41	13	-	-	ADJ
ejpam-237	41	14	hausdorff	hausdorff	ADJ
ejpam-237	41	15	[	[	X
ejpam-237	41	16	11	11	NUM
ejpam-237	41	17	]	]	PUNCT
ejpam-237	41	18	(	(	PUNCT
ejpam-237	41	19	resp	resp	NOUN
ejpam-237	41	20	.	.	PUNCT
ejpam-237	41	21	β−hausdorff	β−hausdorff	PUNCT
ejpam-237	42	1	[	[	X
ejpam-237	42	2	12	12	NUM
ejpam-237	42	3	]	]	SYM
ejpam-237	42	4	)	)	PUNCT
ejpam-237	42	5	if	if	SCONJ
ejpam-237	42	6	for	for	ADP
ejpam-237	42	7	every	every	DET
ejpam-237	42	8	two	two	NUM
ejpam-237	42	9	different	different	ADJ
ejpam-237	42	10	points	point	NOUN
ejpam-237	42	11	x	x	X
ejpam-237	42	12	,	,	PUNCT
ejpam-237	42	13	y	y	PROPN
ejpam-237	42	14	of	of	ADP
ejpam-237	42	15	x	x	SYM
ejpam-237	42	16	,	,	PUNCT
ejpam-237	42	17	there	there	PRON
ejpam-237	42	18	exist	exist	VERB
ejpam-237	42	19	disjoint	disjoint	NOUN
ejpam-237	42	20	semi	semi	ADJ
ejpam-237	42	21	-	-	ADJ
ejpam-237	42	22	open	open	ADJ
ejpam-237	42	23	(	(	PUNCT
ejpam-237	42	24	resp	resp	NOUN
ejpam-237	42	25	.	.	PUNCT
ejpam-237	43	1	β	β	NOUN
ejpam-237	43	2	−	−	ADP
ejpam-237	43	3	open	open	ADJ
ejpam-237	43	4	)	)	PUNCT
ejpam-237	43	5	sets	set	VERB
ejpam-237	43	6	u	u	NOUN
ejpam-237	43	7	,	,	PUNCT
ejpam-237	43	8	v	v	NOUN
ejpam-237	43	9	of	of	ADP
ejpam-237	43	10	x	x	INTJ
ejpam-237	43	11	such	such	ADJ
ejpam-237	43	12	that	that	SCONJ
ejpam-237	43	13	x	x	SYM
ejpam-237	43	14	∈	∈	PROPN
ejpam-237	43	15	u	u	NOUN
ejpam-237	43	16	and	and	CCONJ
ejpam-237	43	17	y	y	PROPN
ejpam-237	43	18	∈	∈	PROPN
ejpam-237	43	19	v.	v.	ADP
ejpam-237	43	20	definition	definition	NOUN
ejpam-237	43	21	2.3	2.3	NUM
ejpam-237	43	22	.	.	PUNCT
ejpam-237	44	1	an	an	DET
ejpam-237	44	2	ideal	ideal	ADJ
ejpam-237	44	3	topological	topological	ADJ
ejpam-237	44	4	space	space	NOUN
ejpam-237	44	5	(	(	PUNCT
ejpam-237	44	6	x	x	X
ejpam-237	44	7	,	,	PUNCT
ejpam-237	44	8	τ	τ	PROPN
ejpam-237	44	9	,	,	PUNCT
ejpam-237	44	10	i	i	PROPN
ejpam-237	44	11	)	)	PUNCT
ejpam-237	44	12	is	be	AUX
ejpam-237	44	13	called	call	VERB
ejpam-237	44	14	i	i	PRON
ejpam-237	44	15	-	-	PUNCT
ejpam-237	44	16	hausdorff	hausdorff	NOUN
ejpam-237	44	17	[	[	X
ejpam-237	44	18	4	4	NUM
ejpam-237	44	19	]	]	PUNCT
ejpam-237	44	20	(	(	PUNCT
ejpam-237	44	21	resp	resp	NOUN
ejpam-237	44	22	.	.	PUNCT
ejpam-237	45	1	quasi	quasi	ADJ
ejpam-237	45	2	-	-	PROPN
ejpam-237	45	3	i	i	NOUN
ejpam-237	45	4	-	-	PUNCT
ejpam-237	45	5	hausdorff	hausdorff	NOUN
ejpam-237	45	6	[	[	X
ejpam-237	45	7	13	13	NUM
ejpam-237	45	8	]	]	SYM
ejpam-237	45	9	)	)	PUNCT
ejpam-237	45	10	if	if	SCONJ
ejpam-237	45	11	for	for	ADP
ejpam-237	45	12	every	every	DET
ejpam-237	45	13	two	two	NUM
ejpam-237	45	14	different	different	ADJ
ejpam-237	45	15	points	point	NOUN
ejpam-237	45	16	x	x	X
ejpam-237	45	17	,	,	PUNCT
ejpam-237	45	18	y	y	PROPN
ejpam-237	45	19	of	of	ADP
ejpam-237	45	20	x	x	SYM
ejpam-237	45	21	,	,	PUNCT
ejpam-237	45	22	there	there	PRON
ejpam-237	45	23	exist	exist	VERB
ejpam-237	45	24	disjoint	disjoint	NOUN
ejpam-237	45	25	i	i	PRON
ejpam-237	45	26	-	-	PUNCT
ejpam-237	45	27	open	open	ADJ
ejpam-237	45	28	sets	set	NOUN
ejpam-237	45	29	(	(	PUNCT
ejpam-237	45	30	resp	resp	NOUN
ejpam-237	45	31	.	.	PUNCT
ejpam-237	46	1	quasi	quasi	ADJ
ejpam-237	46	2	-	-	PROPN
ejpam-237	46	3	i	i	PRON
ejpam-237	46	4	-	-	PUNCT
ejpam-237	46	5	open	open	ADJ
ejpam-237	46	6	)	)	PUNCT
ejpam-237	46	7	u	u	NOUN
ejpam-237	46	8	,	,	PUNCT
ejpam-237	46	9	v	v	NOUN
ejpam-237	46	10	of	of	ADP
ejpam-237	46	11	x	x	INTJ
ejpam-237	46	12	such	such	ADJ
ejpam-237	46	13	that	that	SCONJ
ejpam-237	46	14	x	x	SYM
ejpam-237	46	15	∈	∈	PROPN
ejpam-237	46	16	u	u	NOUN
ejpam-237	46	17	and	and	CCONJ
ejpam-237	46	18	y	y	PROPN
ejpam-237	46	19	∈	∈	PROPN
ejpam-237	46	20	v.	v.	ADP
ejpam-237	46	21	3	3	NUM
ejpam-237	46	22	.	.	PUNCT
ejpam-237	46	23	semi	semi	ADJ
ejpam-237	46	24	-	-	ADJ
ejpam-237	46	25	i	i	NOUN
ejpam-237	46	26	-	-	PUNCT
ejpam-237	46	27	hausdorff	hausdorff	NOUN
ejpam-237	46	28	spaces	space	NOUN
ejpam-237	46	29	definition	definition	NOUN
ejpam-237	46	30	3.1	3.1	NUM
ejpam-237	46	31	.	.	PUNCT
ejpam-237	47	1	an	an	DET
ejpam-237	47	2	ideal	ideal	ADJ
ejpam-237	47	3	topological	topological	ADJ
ejpam-237	47	4	space	space	NOUN
ejpam-237	47	5	(	(	PUNCT
ejpam-237	47	6	x	x	X
ejpam-237	47	7	,	,	PUNCT
ejpam-237	47	8	τ	τ	PROPN
ejpam-237	47	9	,	,	PUNCT
ejpam-237	47	10	i	i	PROPN
ejpam-237	47	11	)	)	PUNCT
ejpam-237	47	12	is	be	AUX
ejpam-237	47	13	called	call	VERB
ejpam-237	47	14	semi	semi	ADJ
ejpam-237	47	15	-	-	ADJ
ejpam-237	47	16	i	i	NOUN
ejpam-237	47	17	-	-	PUNCT
ejpam-237	47	18	hausdorff	hausdorff	NOUN
ejpam-237	47	19	if	if	SCONJ
ejpam-237	47	20	for	for	ADP
ejpam-237	47	21	each	each	DET
ejpam-237	47	22	two	two	NUM
ejpam-237	47	23	distinct	distinct	ADJ
ejpam-237	47	24	points	point	NOUN
ejpam-237	47	25	x	x	PUNCT
ejpam-237	47	26	6=	6=	NUM
ejpam-237	47	27	y	y	PROPN
ejpam-237	47	28	,	,	PUNCT
ejpam-237	47	29	there	there	PRON
ejpam-237	47	30	exist	exist	VERB
ejpam-237	47	31	semi	semi	ADJ
ejpam-237	47	32	-	-	ADJ
ejpam-237	47	33	i	i	PRON
ejpam-237	47	34	-	-	PUNCT
ejpam-237	47	35	open	open	ADJ
ejpam-237	47	36	sets	set	VERB
ejpam-237	47	37	u	u	NOUN
ejpam-237	47	38	and	and	CCONJ
ejpam-237	47	39	v	v	ADP
ejpam-237	47	40	containig	containig	PROPN
ejpam-237	47	41	x	x	PUNCT
ejpam-237	47	42	and	and	CCONJ
ejpam-237	47	43	y	y	PROPN
ejpam-237	47	44	,	,	PUNCT
ejpam-237	47	45	respectively	respectively	ADV
ejpam-237	47	46	such	such	ADJ
ejpam-237	47	47	that	that	SCONJ
ejpam-237	47	48	u	u	PROPN
ejpam-237	47	49	∩	∩	NOUN
ejpam-237	47	50	v	v	ADP
ejpam-237	47	51	=	=	PUNCT
ejpam-237	47	52	∅.	∅.	NOUN
ejpam-237	47	53	then	then	ADV
ejpam-237	47	54	the	the	DET
ejpam-237	47	55	points	point	NOUN
ejpam-237	47	56	x	x	PUNCT
ejpam-237	47	57	and	and	CCONJ
ejpam-237	47	58	y	y	PROPN
ejpam-237	47	59	are	be	AUX
ejpam-237	47	60	said	say	VERB
ejpam-237	47	61	to	to	PART
ejpam-237	47	62	be	be	AUX
ejpam-237	47	63	semi	semi	ADV
ejpam-237	47	64	−	−	PROPN
ejpam-237	48	1	i	i	PRON
ejpam-237	48	2	−	−	PROPN
ejpam-237	48	3	separated	separate	VERB
ejpam-237	48	4	.	.	PUNCT
ejpam-237	49	1	theorem	theorem	VERB
ejpam-237	49	2	3.1	3.1	NUM
ejpam-237	49	3	.	.	PUNCT
ejpam-237	50	1	for	for	ADP
ejpam-237	50	2	an	an	DET
ejpam-237	50	3	ideal	ideal	ADJ
ejpam-237	50	4	topological	topological	ADJ
ejpam-237	50	5	space	space	NOUN
ejpam-237	50	6	(	(	PUNCT
ejpam-237	50	7	x	x	X
ejpam-237	50	8	,	,	PUNCT
ejpam-237	50	9	τ	τ	PROPN
ejpam-237	50	10	,	,	PUNCT
ejpam-237	50	11	i	i	PROPN
ejpam-237	50	12	)	)	PUNCT
ejpam-237	50	13	,	,	PUNCT
ejpam-237	50	14	the	the	DET
ejpam-237	50	15	following	follow	VERB
ejpam-237	50	16	statements	statement	NOUN
ejpam-237	50	17	hold	hold	VERB
ejpam-237	50	18	:	:	PUNCT
ejpam-237	51	1	1	1	X
ejpam-237	51	2	.	.	X
ejpam-237	52	1	every	every	DET
ejpam-237	52	2	hausdorff	hausdorff	NOUN
ejpam-237	52	3	space	space	NOUN
ejpam-237	52	4	is	be	AUX
ejpam-237	52	5	semi	semi	ADJ
ejpam-237	52	6	-	-	ADJ
ejpam-237	52	7	i	i	NOUN
ejpam-237	52	8	-	-	PUNCT
ejpam-237	52	9	hausdorff	hausdorff	NOUN
ejpam-237	52	10	.	.	PUNCT
ejpam-237	53	1	2	2	NUM
ejpam-237	53	2	.	.	X
ejpam-237	53	3	every	every	DET
ejpam-237	53	4	semi	semi	ADJ
ejpam-237	53	5	-	-	ADJ
ejpam-237	53	6	i	i	ADJ
ejpam-237	53	7	-	-	PUNCT
ejpam-237	53	8	hausdorff	hausdorff	NOUN
ejpam-237	53	9	space	space	NOUN
ejpam-237	53	10	is	be	AUX
ejpam-237	53	11	semi	semi	ADJ
ejpam-237	53	12	-	-	ADJ
ejpam-237	53	13	hausdorff	hausdorff	ADJ
ejpam-237	53	14	.	.	PUNCT
ejpam-237	54	1	proof	proof	NOUN
ejpam-237	54	2	.	.	PUNCT
ejpam-237	55	1	this	this	PRON
ejpam-237	55	2	follows	follow	VERB
ejpam-237	55	3	from	from	ADP
ejpam-237	55	4	the	the	DET
ejpam-237	55	5	definition	definition	NOUN
ejpam-237	55	6	of	of	ADP
ejpam-237	55	7	semi	semi	ADJ
ejpam-237	55	8	-	-	ADJ
ejpam-237	55	9	i	i	ADJ
ejpam-237	55	10	-open	-open	NOUN
ejpam-237	55	11	sets	set	NOUN
ejpam-237	55	12	.	.	PUNCT
ejpam-237	56	1	e.	e.	PROPN
ejpam-237	56	2	hatir	hatir	PROPN
ejpam-237	56	3	and	and	CCONJ
ejpam-237	56	4	t.	t.	PROPN
ejpam-237	56	5	noiri	noiri	PROPN
ejpam-237	56	6	/	/	SYM
ejpam-237	56	7	eur	eur	PROPN
ejpam-237	56	8	.	.	PUNCT
ejpam-237	57	1	j.	j.	PROPN
ejpam-237	57	2	pure	pure	PROPN
ejpam-237	57	3	appl	appl	PROPN
ejpam-237	57	4	.	.	PROPN
ejpam-237	57	5	math	math	PROPN
ejpam-237	57	6	,	,	PUNCT
ejpam-237	57	7	2	2	NUM
ejpam-237	57	8	(	(	PUNCT
ejpam-237	57	9	2009	2009	NUM
ejpam-237	57	10	)	)	PUNCT
ejpam-237	57	11	,	,	PUNCT
ejpam-237	57	12	(	(	PUNCT
ejpam-237	57	13	172	172	NUM
ejpam-237	57	14	-	-	SYM
ejpam-237	57	15	181	181	NUM
ejpam-237	57	16	)	)	PUNCT
ejpam-237	57	17	175	175	NUM
ejpam-237	57	18	for	for	ADP
ejpam-237	57	19	ideal	ideal	ADJ
ejpam-237	57	20	topological	topological	ADJ
ejpam-237	57	21	spaces	space	NOUN
ejpam-237	57	22	,	,	PUNCT
ejpam-237	57	23	the	the	DET
ejpam-237	57	24	following	follow	VERB
ejpam-237	57	25	diagram	diagram	NOUN
ejpam-237	57	26	holds	hold	VERB
ejpam-237	57	27	:	:	PUNCT
ejpam-237	58	1	diagram	diagram	PROPN
ejpam-237	58	2	ii	ii	PROPN
ejpam-237	58	3	hausdorff	hausdorff	PROPN
ejpam-237	58	4	−→	−→	PROPN
ejpam-237	58	5	semi	semi	ADJ
ejpam-237	58	6	-	-	ADJ
ejpam-237	58	7	i	i	NOUN
ejpam-237	58	8	-	-	PUNCT
ejpam-237	58	9	hausdorff	hausdorff	NOUN
ejpam-237	58	10	−→	−→	NOUN
ejpam-237	58	11	semi	semi	ADJ
ejpam-237	58	12	-	-	ADJ
ejpam-237	58	13	hausdorff	hausdorff	ADJ
ejpam-237	58	14	↓	↓	PROPN
ejpam-237	58	15	i	i	PROPN
ejpam-237	58	16	-	-	PUNCT
ejpam-237	58	17	hausdorff	hausdorff	PROPN
ejpam-237	58	18	−→	−→	NOUN
ejpam-237	58	19	quasi	quasi	PROPN
ejpam-237	58	20	-	-	ADJ
ejpam-237	58	21	i	i	NOUN
ejpam-237	58	22	-	-	PUNCT
ejpam-237	58	23	hausdorff	hausdorff	NOUN
ejpam-237	58	24	−→	−→	NOUN
ejpam-237	58	25	β	β	PART
ejpam-237	58	26	−hausdorff	−hausdorff	NOUN
ejpam-237	58	27	remark	remark	VERB
ejpam-237	58	28	3.1	3.1	NUM
ejpam-237	58	29	.	.	PUNCT
ejpam-237	59	1	(	(	PUNCT
ejpam-237	59	2	1	1	X
ejpam-237	59	3	)	)	PUNCT
ejpam-237	59	4	it	it	PRON
ejpam-237	59	5	is	be	AUX
ejpam-237	59	6	shown	show	VERB
ejpam-237	59	7	in	in	ADP
ejpam-237	59	8	example	example	NOUN
ejpam-237	59	9	2.3	2.3	NUM
ejpam-237	59	10	and	and	CCONJ
ejpam-237	59	11	2.4	2.4	NUM
ejpam-237	59	12	of	of	ADP
ejpam-237	59	13	[	[	X
ejpam-237	59	14	4	4	X
ejpam-237	59	15	]	]	PUNCT
ejpam-237	59	16	that	that	PRON
ejpam-237	59	17	hausdorffness	hausdorffness	NOUN
ejpam-237	59	18	and	and	CCONJ
ejpam-237	59	19	i	i	PROPN
ejpam-237	59	20	-	-	PUNCT
ejpam-237	59	21	hausdorffness	hausdorffness	PROPN
ejpam-237	59	22	are	be	AUX
ejpam-237	59	23	independent	independent	ADJ
ejpam-237	59	24	of	of	ADP
ejpam-237	59	25	each	each	DET
ejpam-237	59	26	other	other	ADJ
ejpam-237	59	27	.	.	PUNCT
ejpam-237	60	1	(	(	PUNCT
ejpam-237	60	2	2	2	X
ejpam-237	60	3	)	)	PUNCT
ejpam-237	60	4	in	in	ADP
ejpam-237	60	5	the	the	DET
ejpam-237	60	6	following	following	ADJ
ejpam-237	60	7	examples	example	NOUN
ejpam-237	60	8	,	,	PUNCT
ejpam-237	60	9	it	it	PRON
ejpam-237	60	10	will	will	AUX
ejpam-237	60	11	be	be	AUX
ejpam-237	60	12	shown	show	VERB
ejpam-237	60	13	that	that	SCONJ
ejpam-237	60	14	semi	semi	ADJ
ejpam-237	60	15	-	-	ADJ
ejpam-237	60	16	i	i	PRON
ejpam-237	60	17	-	-	PUNCT
ejpam-237	60	18	hausdorffness	hausdorffness	NOUN
ejpam-237	60	19	is	be	AUX
ejpam-237	60	20	independent	independent	ADJ
ejpam-237	60	21	to	to	ADP
ejpam-237	60	22	quasi	quasi	ADJ
ejpam-237	60	23	-	-	PROPN
ejpam-237	60	24	i	i	NOUN
ejpam-237	60	25	-	-	PUNCT
ejpam-237	60	26	hausdorffness	hausdorffness	NOUN
ejpam-237	60	27	and	and	CCONJ
ejpam-237	60	28	to	to	ADP
ejpam-237	60	29	i	i	NOUN
ejpam-237	60	30	-	-	PUNCT
ejpam-237	60	31	hausdorffness	hausdorffness	NOUN
ejpam-237	60	32	.	.	PUNCT
ejpam-237	61	1	example	example	NOUN
ejpam-237	61	2	3.1	3.1	NUM
ejpam-237	61	3	.	.	PUNCT
ejpam-237	62	1	let	let	VERB
ejpam-237	62	2	x	x	PRON
ejpam-237	62	3	be	be	AUX
ejpam-237	62	4	the	the	DET
ejpam-237	62	5	real	real	ADJ
ejpam-237	62	6	line	line	NOUN
ejpam-237	62	7	with	with	ADP
ejpam-237	62	8	the	the	DET
ejpam-237	62	9	"	"	PUNCT
ejpam-237	62	10	rigth	rigth	NOUN
ejpam-237	62	11	-	-	PUNCT
ejpam-237	62	12	ray	ray	NOUN
ejpam-237	62	13	"	"	PUNCT
ejpam-237	62	14	topology	topology	NOUN
ejpam-237	62	15	τ	τ	PROPN
ejpam-237	62	16	that	that	PRON
ejpam-237	62	17	is	be	AUX
ejpam-237	62	18	the	the	DET
ejpam-237	62	19	nontrivial	nontrivial	ADJ
ejpam-237	62	20	open	open	ADJ
ejpam-237	62	21	sets	set	NOUN
ejpam-237	62	22	are	be	AUX
ejpam-237	62	23	the	the	DET
ejpam-237	62	24	form	form	NOUN
ejpam-237	62	25	(	(	PUNCT
ejpam-237	62	26	x,∞	x,∞	PROPN
ejpam-237	62	27	)	)	PUNCT
ejpam-237	62	28	,	,	PUNCT
ejpam-237	62	29	where	where	SCONJ
ejpam-237	62	30	x	x	PRON
ejpam-237	62	31	is	be	AUX
ejpam-237	62	32	any	any	DET
ejpam-237	62	33	real	real	ADJ
ejpam-237	62	34	number	number	NOUN
ejpam-237	62	35	.	.	PUNCT
ejpam-237	63	1	let	let	VERB
ejpam-237	63	2	i	i	PRON
ejpam-237	63	3	be	be	AUX
ejpam-237	63	4	the	the	DET
ejpam-237	63	5	ideal	ideal	NOUN
ejpam-237	63	6	of	of	ADP
ejpam-237	63	7	all	all	DET
ejpam-237	63	8	finite	finite	ADJ
ejpam-237	63	9	subsets	subset	NOUN
ejpam-237	63	10	of	of	ADP
ejpam-237	63	11	x.	x.	NOUN
ejpam-237	63	12	then	then	ADV
ejpam-237	63	13	the	the	DET
ejpam-237	63	14	ideal	ideal	ADJ
ejpam-237	63	15	topological	topological	ADJ
ejpam-237	63	16	space	space	NOUN
ejpam-237	63	17	(	(	PUNCT
ejpam-237	63	18	x	x	X
ejpam-237	63	19	,	,	PUNCT
ejpam-237	63	20	τ	τ	PROPN
ejpam-237	63	21	,	,	PUNCT
ejpam-237	63	22	i	i	PROPN
ejpam-237	63	23	)	)	PUNCT
ejpam-237	63	24	is	be	AUX
ejpam-237	63	25	an	an	DET
ejpam-237	63	26	i	i	NOUN
ejpam-237	63	27	-	-	PUNCT
ejpam-237	63	28	hausdorff	hausdorff	NOUN
ejpam-237	63	29	space	space	NOUN
ejpam-237	63	30	which	which	PRON
ejpam-237	63	31	is	be	AUX
ejpam-237	63	32	not	not	PART
ejpam-237	63	33	hausdorff	hausdorff	ADJ
ejpam-237	63	34	�	�	PROPN
ejpam-237	63	35	4	4	NUM
ejpam-237	63	36	,	,	PUNCT
ejpam-237	63	37	example	example	NOUN
ejpam-237	63	38	2.3	2.3	NUM
ejpam-237	63	39	�	�	NOUN
ejpam-237	63	40	.	.	PUNCT
ejpam-237	64	1	however	however	ADV
ejpam-237	64	2	,	,	PUNCT
ejpam-237	64	3	this	this	DET
ejpam-237	64	4	space	space	NOUN
ejpam-237	64	5	is	be	AUX
ejpam-237	64	6	not	not	PART
ejpam-237	64	7	even	even	ADV
ejpam-237	64	8	semi	semi	ADJ
ejpam-237	64	9	-	-	ADJ
ejpam-237	64	10	hausdorff	hausdorff	ADJ
ejpam-237	64	11	because	because	SCONJ
ejpam-237	64	12	,	,	PUNCT
ejpam-237	64	13	every	every	PRON
ejpam-237	64	14	nonempty	nonempty	ADJ
ejpam-237	64	15	semi	semi	ADJ
ejpam-237	64	16	-	-	ADJ
ejpam-237	64	17	open	open	ADJ
ejpam-237	64	18	set	set	NOUN
ejpam-237	64	19	has	have	VERB
ejpam-237	64	20	the	the	DET
ejpam-237	64	21	nonempty	nonempty	ADJ
ejpam-237	64	22	interior	interior	NOUN
ejpam-237	64	23	.	.	PUNCT
ejpam-237	65	1	example	example	NOUN
ejpam-237	65	2	3.2	3.2	NUM
ejpam-237	65	3	.	.	PUNCT
ejpam-237	66	1	let	let	VERB
ejpam-237	66	2	x	x	PUNCT
ejpam-237	66	3	=	=	PRON
ejpam-237	66	4	{	{	PUNCT
ejpam-237	66	5	a	a	DET
ejpam-237	66	6	,	,	PUNCT
ejpam-237	66	7	b	b	NOUN
ejpam-237	66	8	}	}	PUNCT
ejpam-237	66	9	,	,	PUNCT
ejpam-237	66	10	τ	τ	PROPN
ejpam-237	66	11	be	be	AUX
ejpam-237	66	12	the	the	DET
ejpam-237	66	13	discrete	discrete	ADJ
ejpam-237	66	14	topology	topology	NOUN
ejpam-237	66	15	on	on	ADP
ejpam-237	66	16	x	x	PUNCT
ejpam-237	66	17	and	and	CCONJ
ejpam-237	66	18	i	i	NOUN
ejpam-237	66	19	=	=	PUNCT
ejpam-237	66	20	℘(x	℘(x	ADJ
ejpam-237	66	21	)	)	PUNCT
ejpam-237	66	22	.	.	PUNCT
ejpam-237	67	1	then	then	ADV
ejpam-237	67	2	dontchev	dontchev	VERB
ejpam-237	67	3	[	[	X
ejpam-237	67	4	4	4	X
ejpam-237	67	5	]	]	PUNCT
ejpam-237	67	6	showed	show	VERB
ejpam-237	67	7	that	that	SCONJ
ejpam-237	67	8	the	the	DET
ejpam-237	67	9	space	space	NOUN
ejpam-237	67	10	is	be	AUX
ejpam-237	67	11	hausdorff	hausdorff	NOUN
ejpam-237	67	12	,	,	PUNCT
ejpam-237	67	13	but	but	CCONJ
ejpam-237	67	14	it	it	PRON
ejpam-237	67	15	is	be	AUX
ejpam-237	67	16	not	not	PART
ejpam-237	67	17	i	i	NOUN
ejpam-237	67	18	-	-	PUNCT
ejpam-237	67	19	hausdorff	hausdorff	NOUN
ejpam-237	67	20	.	.	PUNCT
ejpam-237	68	1	moreover	moreover	ADV
ejpam-237	68	2	,	,	PUNCT
ejpam-237	68	3	nasef	nasef	PROPN
ejpam-237	68	4	[	[	X
ejpam-237	68	5	13	13	NUM
ejpam-237	68	6	]	]	PUNCT
ejpam-237	68	7	showed	show	VERB
ejpam-237	68	8	that	that	SCONJ
ejpam-237	68	9	the	the	DET
ejpam-237	68	10	space	space	NOUN
ejpam-237	68	11	is	be	AUX
ejpam-237	68	12	not	not	PART
ejpam-237	68	13	even	even	ADV
ejpam-237	68	14	quasi	quasi	ADJ
ejpam-237	68	15	-	-	ADJ
ejpam-237	68	16	i	i	NOUN
ejpam-237	68	17	-	-	PUNCT
ejpam-237	68	18	hausdorff	hausdorff	PROPN
ejpam-237	68	19	.	.	PUNCT
ejpam-237	68	20	example	example	NOUN
ejpam-237	68	21	3.3	3.3	NUM
ejpam-237	68	22	.	.	PUNCT
ejpam-237	69	1	let	let	VERB
ejpam-237	69	2	x	x	PUNCT
ejpam-237	69	3	=	=	PRON
ejpam-237	69	4	{	{	PUNCT
ejpam-237	69	5	a	a	DET
ejpam-237	69	6	,	,	PUNCT
ejpam-237	69	7	b	b	NOUN
ejpam-237	69	8	,	,	PUNCT
ejpam-237	69	9	c	c	NOUN
ejpam-237	69	10	}	}	PUNCT
ejpam-237	69	11	,	,	PUNCT
ejpam-237	69	12	τ	τ	X
ejpam-237	69	13	=	=	PUNCT
ejpam-237	69	14	{	{	PUNCT
ejpam-237	69	15	∅	∅	NOUN
ejpam-237	69	16	,	,	PUNCT
ejpam-237	69	17	x	x	INTJ
ejpam-237	69	18	,	,	PUNCT
ejpam-237	69	19	{	{	PUNCT
ejpam-237	69	20	a	a	NOUN
ejpam-237	69	21	}	}	PUNCT
ejpam-237	69	22	,	,	PUNCT
ejpam-237	69	23	{	{	PUNCT
ejpam-237	69	24	b	b	NOUN
ejpam-237	69	25	}	}	PUNCT
ejpam-237	69	26	,	,	PUNCT
ejpam-237	69	27	{	{	PUNCT
ejpam-237	69	28	a	a	PRON
ejpam-237	69	29	,	,	PUNCT
ejpam-237	69	30	b	b	NOUN
ejpam-237	69	31	}	}	PUNCT
ejpam-237	69	32	}	}	PUNCT
ejpam-237	69	33	and	and	CCONJ
ejpam-237	69	34	i	i	PRON
ejpam-237	69	35	=	=	PUNCT
ejpam-237	69	36	{	{	PUNCT
ejpam-237	69	37	∅	∅	NOUN
ejpam-237	69	38	}	}	PUNCT
ejpam-237	69	39	.	.	PUNCT
ejpam-237	70	1	then	then	ADV
ejpam-237	70	2	(	(	PUNCT
ejpam-237	70	3	x	x	X
ejpam-237	70	4	,	,	PUNCT
ejpam-237	70	5	τ	τ	PROPN
ejpam-237	70	6	,	,	PUNCT
ejpam-237	70	7	i	i	PROPN
ejpam-237	70	8	)	)	PUNCT
ejpam-237	70	9	is	be	AUX
ejpam-237	70	10	a	a	DET
ejpam-237	70	11	semi	semi	ADJ
ejpam-237	70	12	-	-	ADJ
ejpam-237	70	13	i	i	ADJ
ejpam-237	70	14	-	-	PUNCT
ejpam-237	70	15	hausdorff	hausdorff	NOUN
ejpam-237	70	16	space	space	NOUN
ejpam-237	70	17	which	which	PRON
ejpam-237	70	18	is	be	AUX
ejpam-237	70	19	not	not	PART
ejpam-237	70	20	hausdorff	hausdorff	ADJ
ejpam-237	70	21	.	.	PUNCT
ejpam-237	71	1	if	if	SCONJ
ejpam-237	71	2	we	we	PRON
ejpam-237	71	3	take	take	VERB
ejpam-237	71	4	i	i	NOUN
ejpam-237	71	5	=	=	PUNCT
ejpam-237	71	6	℘(x	℘(x	ADJ
ejpam-237	71	7	)	)	PUNCT
ejpam-237	71	8	,	,	PUNCT
ejpam-237	71	9	then	then	ADV
ejpam-237	71	10	(	(	PUNCT
ejpam-237	71	11	x	x	X
ejpam-237	71	12	,	,	PUNCT
ejpam-237	71	13	τ	τ	PROPN
ejpam-237	71	14	,	,	PUNCT
ejpam-237	71	15	i	i	PROPN
ejpam-237	71	16	)	)	PUNCT
ejpam-237	71	17	is	be	AUX
ejpam-237	71	18	semi	semi	ADJ
ejpam-237	71	19	-	-	ADJ
ejpam-237	71	20	hausdorff	hausdorff	ADJ
ejpam-237	71	21	,	,	PUNCT
ejpam-237	71	22	but	but	CCONJ
ejpam-237	71	23	it	it	PRON
ejpam-237	71	24	is	be	AUX
ejpam-237	71	25	neither	neither	PRON
ejpam-237	71	26	semi	semi	ADJ
ejpam-237	71	27	-	-	ADJ
ejpam-237	71	28	i	i	NOUN
ejpam-237	71	29	-	-	PUNCT
ejpam-237	71	30	hausdorff	hausdorff	NOUN
ejpam-237	71	31	nor	nor	CCONJ
ejpam-237	71	32	quasi	quasi	ADJ
ejpam-237	71	33	-	-	ADJ
ejpam-237	71	34	i	i	NOUN
ejpam-237	71	35	-	-	PUNCT
ejpam-237	71	36	hausdorff	hausdorff	NOUN
ejpam-237	71	37	.	.	PUNCT
ejpam-237	72	1	theorem	theorem	PROPN
ejpam-237	72	2	3.2	3.2	NUM
ejpam-237	72	3	.	.	PUNCT
ejpam-237	73	1	let	let	VERB
ejpam-237	73	2	(	(	PUNCT
ejpam-237	73	3	x	x	X
ejpam-237	73	4	,	,	PUNCT
ejpam-237	73	5	τ	τ	PROPN
ejpam-237	73	6	,	,	PUNCT
ejpam-237	73	7	i	i	PRON
ejpam-237	73	8	)	)	PUNCT
ejpam-237	73	9	be	be	VERB
ejpam-237	73	10	an	an	DET
ejpam-237	73	11	ideal	ideal	ADJ
ejpam-237	73	12	topological	topological	ADJ
ejpam-237	73	13	space	space	NOUN
ejpam-237	73	14	.	.	PUNCT
ejpam-237	74	1	1	1	X
ejpam-237	74	2	.	.	X
ejpam-237	74	3	let	let	VERB
ejpam-237	74	4	i	i	PRON
ejpam-237	74	5	=	=	PUNCT
ejpam-237	74	6	{	{	PUNCT
ejpam-237	74	7	∅	∅	NOUN
ejpam-237	74	8	}	}	PUNCT
ejpam-237	74	9	.	.	PUNCT
ejpam-237	75	1	then	then	ADV
ejpam-237	75	2	(	(	PUNCT
ejpam-237	75	3	x	x	X
ejpam-237	75	4	,	,	PUNCT
ejpam-237	75	5	τ	τ	PROPN
ejpam-237	75	6	,	,	PUNCT
ejpam-237	75	7	i	i	PROPN
ejpam-237	75	8	)	)	PUNCT
ejpam-237	75	9	is	be	AUX
ejpam-237	75	10	semi	semi	ADJ
ejpam-237	75	11	-	-	ADJ
ejpam-237	75	12	i	i	NOUN
ejpam-237	75	13	-	-	PUNCT
ejpam-237	75	14	hausdorff	hausdorff	NOUN
ejpam-237	75	15	(	(	PUNCT
ejpam-237	75	16	resp	resp	NOUN
ejpam-237	75	17	.	.	PUNCT
ejpam-237	76	1	quasi	quasi	ADJ
ejpam-237	76	2	-	-	PROPN
ejpam-237	76	3	i	i	NOUN
ejpam-237	76	4	-	-	PUNCT
ejpam-237	76	5	hausdorff	hausdorff	NOUN
ejpam-237	76	6	)	)	PUNCT
ejpam-237	77	1	if	if	SCONJ
ejpam-237	77	2	and	and	CCONJ
ejpam-237	77	3	only	only	ADV
ejpam-237	77	4	if	if	SCONJ
ejpam-237	77	5	it	it	PRON
ejpam-237	77	6	is	be	AUX
ejpam-237	77	7	semi	semi	ADJ
ejpam-237	77	8	-	-	ADJ
ejpam-237	77	9	hausdorff	hausdorff	ADJ
ejpam-237	77	10	(	(	PUNCT
ejpam-237	77	11	resp	resp	NOUN
ejpam-237	77	12	.	.	PUNCT
ejpam-237	78	1	β−hausdorff	β−hausdorff	NOUN
ejpam-237	78	2	)	)	PUNCT
ejpam-237	78	3	.	.	PUNCT
ejpam-237	79	1	e.	e.	PROPN
ejpam-237	79	2	hatir	hatir	PROPN
ejpam-237	79	3	and	and	CCONJ
ejpam-237	79	4	t.	t.	PROPN
ejpam-237	79	5	noiri	noiri	PROPN
ejpam-237	79	6	/	/	SYM
ejpam-237	79	7	eur	eur	PROPN
ejpam-237	79	8	.	.	PUNCT
ejpam-237	80	1	j.	j.	PROPN
ejpam-237	80	2	pure	pure	PROPN
ejpam-237	80	3	appl	appl	PROPN
ejpam-237	80	4	.	.	PROPN
ejpam-237	80	5	math	math	PROPN
ejpam-237	80	6	,	,	PUNCT
ejpam-237	80	7	2	2	NUM
ejpam-237	80	8	(	(	PUNCT
ejpam-237	80	9	2009	2009	NUM
ejpam-237	80	10	)	)	PUNCT
ejpam-237	80	11	,	,	PUNCT
ejpam-237	80	12	(	(	PUNCT
ejpam-237	80	13	172	172	NUM
ejpam-237	80	14	-	-	SYM
ejpam-237	80	15	181	181	NUM
ejpam-237	80	16	)	)	PUNCT
ejpam-237	80	17	176	176	NUM
ejpam-237	80	18	2	2	NUM
ejpam-237	80	19	.	.	PUNCT
ejpam-237	81	1	let	let	VERB
ejpam-237	81	2	i	i	PRON
ejpam-237	81	3	=	=	PUNCT
ejpam-237	81	4	℘(x	℘(x	ADJ
ejpam-237	81	5	)	)	PUNCT
ejpam-237	81	6	.	.	PUNCT
ejpam-237	82	1	then	then	ADV
ejpam-237	82	2	(	(	PUNCT
ejpam-237	82	3	x	x	X
ejpam-237	82	4	,	,	PUNCT
ejpam-237	82	5	τ	τ	PROPN
ejpam-237	82	6	,	,	PUNCT
ejpam-237	82	7	i	i	PROPN
ejpam-237	82	8	)	)	PUNCT
ejpam-237	82	9	is	be	AUX
ejpam-237	82	10	hausdorff	hausdorff	NOUN
ejpam-237	82	11	if	if	SCONJ
ejpam-237	83	1	and	and	CCONJ
ejpam-237	83	2	only	only	ADV
ejpam-237	83	3	if	if	SCONJ
ejpam-237	83	4	it	it	PRON
ejpam-237	83	5	is	be	AUX
ejpam-237	83	6	semi	semi	ADJ
ejpam-237	83	7	-	-	ADJ
ejpam-237	83	8	i	i	NOUN
ejpam-237	83	9	-	-	PUNCT
ejpam-237	83	10	hausdorff	hausdorff	NOUN
ejpam-237	83	11	.	.	PUNCT
ejpam-237	84	1	proof	proof	NOUN
ejpam-237	84	2	.	.	PUNCT
ejpam-237	85	1	(	(	PUNCT
ejpam-237	85	2	1	1	X
ejpam-237	85	3	)	)	PUNCT
ejpam-237	85	4	let	let	VERB
ejpam-237	85	5	i	i	PRON
ejpam-237	85	6	=	=	PUNCT
ejpam-237	85	7	{	{	PUNCT
ejpam-237	85	8	∅	∅	NOUN
ejpam-237	85	9	}	}	PUNCT
ejpam-237	85	10	.	.	PUNCT
ejpam-237	86	1	then	then	ADV
ejpam-237	86	2	a∗	a∗	PROPN
ejpam-237	86	3	=	=	SYM
ejpam-237	86	4	cl(a	cl(a	X
ejpam-237	86	5	)	)	PUNCT
ejpam-237	86	6	and	and	CCONJ
ejpam-237	86	7	cl∗(a	cl∗(a	NOUN
ejpam-237	86	8	)	)	PUNCT
ejpam-237	86	9	=	=	SYM
ejpam-237	86	10	cl(a	cl(a	X
ejpam-237	86	11	)	)	PUNCT
ejpam-237	86	12	for	for	ADP
ejpam-237	86	13	every	every	DET
ejpam-237	86	14	subset	subset	NOUN
ejpam-237	86	15	a	a	PRON
ejpam-237	86	16	of	of	ADP
ejpam-237	86	17	x	x	X
ejpam-237	86	18	.	.	PUNCT
ejpam-237	87	1	therefore	therefore	ADV
ejpam-237	87	2	,	,	PUNCT
ejpam-237	87	3	we	we	PRON
ejpam-237	87	4	have	have	VERB
ejpam-237	87	5	sio(x	sio(x	PROPN
ejpam-237	87	6	,	,	PUNCT
ejpam-237	87	7	τ	τ	X
ejpam-237	87	8	)	)	PUNCT
ejpam-237	88	1	=	=	SYM
ejpam-237	88	2	so(x	so(x	NOUN
ejpam-237	88	3	,	,	PUNCT
ejpam-237	88	4	τ	τ	X
ejpam-237	88	5	)	)	PUNCT
ejpam-237	88	6	(	(	PUNCT
ejpam-237	88	7	resp	resp	NOUN
ejpam-237	88	8	.	.	PUNCT
ejpam-237	89	1	qio(x	qio(x	PROPN
ejpam-237	89	2	,	,	PUNCT
ejpam-237	89	3	τ	τ	PROPN
ejpam-237	89	4	)	)	PUNCT
ejpam-237	89	5	=	=	NOUN
ejpam-237	90	1	β(x	β(x	NOUN
ejpam-237	90	2	,	,	PUNCT
ejpam-237	90	3	τ	τ	PROPN
ejpam-237	90	4	)	)	PUNCT
ejpam-237	90	5	)	)	PUNCT
ejpam-237	90	6	and	and	CCONJ
ejpam-237	90	7	hence	hence	ADV
ejpam-237	90	8	(	(	PUNCT
ejpam-237	90	9	x	x	X
ejpam-237	90	10	,	,	PUNCT
ejpam-237	90	11	τ	τ	PROPN
ejpam-237	90	12	,	,	PUNCT
ejpam-237	90	13	i	i	PROPN
ejpam-237	90	14	)	)	PUNCT
ejpam-237	90	15	is	be	AUX
ejpam-237	90	16	semi	semi	ADJ
ejpam-237	90	17	-	-	ADJ
ejpam-237	90	18	i	i	NOUN
ejpam-237	90	19	-	-	PUNCT
ejpam-237	90	20	hausdorff	hausdorff	NOUN
ejpam-237	90	21	(	(	PUNCT
ejpam-237	90	22	resp	resp	NOUN
ejpam-237	90	23	.	.	PUNCT
ejpam-237	91	1	quasi	quasi	ADJ
ejpam-237	91	2	-	-	PROPN
ejpam-237	91	3	i	i	NOUN
ejpam-237	91	4	-	-	PUNCT
ejpam-237	91	5	hausdorff	hausdorff	NOUN
ejpam-237	91	6	)	)	PUNCT
ejpam-237	92	1	if	if	SCONJ
ejpam-237	92	2	and	and	CCONJ
ejpam-237	92	3	only	only	ADV
ejpam-237	92	4	if	if	SCONJ
ejpam-237	92	5	semihausdorff	semihausdorff	NOUN
ejpam-237	92	6	(	(	PUNCT
ejpam-237	92	7	resp	resp	NOUN
ejpam-237	92	8	.	.	PUNCT
ejpam-237	93	1	β−hausdorff	β−hausdorff	NOUN
ejpam-237	93	2	)	)	PUNCT
ejpam-237	93	3	,	,	PUNCT
ejpam-237	93	4	where	where	SCONJ
ejpam-237	93	5	qio(x	qio(x	PROPN
ejpam-237	93	6	,	,	PUNCT
ejpam-237	93	7	τ	τ	PROPN
ejpam-237	93	8	)	)	PUNCT
ejpam-237	93	9	denotes	denote	VERB
ejpam-237	93	10	the	the	DET
ejpam-237	93	11	set	set	NOUN
ejpam-237	93	12	of	of	ADP
ejpam-237	93	13	all	all	DET
ejpam-237	93	14	quasi	quasi	ADJ
ejpam-237	93	15	-	-	ADJ
ejpam-237	93	16	i	i	NOUN
ejpam-237	93	17	-	-	PUNCT
ejpam-237	93	18	open	open	ADJ
ejpam-237	93	19	sets	set	NOUN
ejpam-237	93	20	.	.	PUNCT
ejpam-237	94	1	(	(	PUNCT
ejpam-237	94	2	2	2	X
ejpam-237	94	3	)	)	PUNCT
ejpam-237	94	4	let	let	VERB
ejpam-237	94	5	i	i	PRON
ejpam-237	94	6	=	=	PUNCT
ejpam-237	94	7	℘(x	℘(x	ADJ
ejpam-237	94	8	)	)	PUNCT
ejpam-237	94	9	.	.	PUNCT
ejpam-237	95	1	then	then	ADV
ejpam-237	95	2	a∗	a∗	NOUN
ejpam-237	95	3	=	=	SYM
ejpam-237	95	4	∅	∅	NOUN
ejpam-237	95	5	and	and	CCONJ
ejpam-237	95	6	cl∗(a	cl∗(a	NOUN
ejpam-237	95	7	)	)	PUNCT
ejpam-237	95	8	=	=	SYM
ejpam-237	96	1	a	a	PRON
ejpam-237	96	2	for	for	ADP
ejpam-237	96	3	every	every	DET
ejpam-237	96	4	subset	subset	NOUN
ejpam-237	96	5	a	a	PRON
ejpam-237	96	6	of	of	ADP
ejpam-237	96	7	x	x	X
ejpam-237	96	8	.	.	PUNCT
ejpam-237	97	1	let	let	VERB
ejpam-237	97	2	a	a	DET
ejpam-237	97	3	∈	∈	ADJ
ejpam-237	97	4	sio(x	sio(x	NOUN
ejpam-237	97	5	,	,	PUNCT
ejpam-237	97	6	τ	τ	PROPN
ejpam-237	97	7	)	)	PUNCT
ejpam-237	97	8	,	,	PUNCT
ejpam-237	97	9	then	then	ADV
ejpam-237	97	10	a⊂	a⊂	VERB
ejpam-237	97	11	cl∗(int(a	cl∗(int(a	NOUN
ejpam-237	97	12	)	)	PUNCT
ejpam-237	97	13	)	)	PUNCT
ejpam-237	98	1	=	=	PUNCT
ejpam-237	98	2	int(a	int(a	NOUN
ejpam-237	98	3	)	)	PUNCT
ejpam-237	98	4	and	and	CCONJ
ejpam-237	98	5	hence	hence	ADV
ejpam-237	98	6	a	a	PRON
ejpam-237	98	7	is	be	AUX
ejpam-237	98	8	open	open	ADJ
ejpam-237	98	9	in	in	ADP
ejpam-237	98	10	(	(	PUNCT
ejpam-237	98	11	x	x	INTJ
ejpam-237	98	12	,	,	PUNCT
ejpam-237	98	13	τ	τ	PROPN
ejpam-237	98	14	)	)	PUNCT
ejpam-237	98	15	.	.	PUNCT
ejpam-237	99	1	therefore	therefore	ADV
ejpam-237	99	2	,	,	PUNCT
ejpam-237	99	3	(	(	PUNCT
ejpam-237	99	4	x	x	X
ejpam-237	99	5	,	,	PUNCT
ejpam-237	99	6	τ	τ	PROPN
ejpam-237	99	7	,	,	PUNCT
ejpam-237	99	8	i	i	PROPN
ejpam-237	99	9	)	)	PUNCT
ejpam-237	99	10	is	be	AUX
ejpam-237	99	11	hausdorff	hausdorff	NOUN
ejpam-237	99	12	if	if	SCONJ
ejpam-237	99	13	and	and	CCONJ
ejpam-237	99	14	only	only	ADV
ejpam-237	99	15	if	if	SCONJ
ejpam-237	99	16	it	it	PRON
ejpam-237	99	17	is	be	AUX
ejpam-237	99	18	semi	semi	ADJ
ejpam-237	99	19	-	-	ADJ
ejpam-237	99	20	i	i	NOUN
ejpam-237	99	21	-	-	PUNCT
ejpam-237	99	22	hausdorff	hausdorff	NOUN
ejpam-237	99	23	.	.	PUNCT
ejpam-237	100	1	definition	definition	NOUN
ejpam-237	100	2	3.2	3.2	NUM
ejpam-237	100	3	.	.	PUNCT
ejpam-237	101	1	an	an	DET
ejpam-237	101	2	ideal	ideal	ADJ
ejpam-237	101	3	topological	topological	ADJ
ejpam-237	101	4	space	space	NOUN
ejpam-237	101	5	(	(	PUNCT
ejpam-237	101	6	x	x	X
ejpam-237	101	7	,	,	PUNCT
ejpam-237	101	8	τ	τ	PROPN
ejpam-237	101	9	,	,	PUNCT
ejpam-237	101	10	i	i	PROPN
ejpam-237	101	11	)	)	PUNCT
ejpam-237	101	12	is	be	AUX
ejpam-237	101	13	called	call	VERB
ejpam-237	101	14	semi	semi	ADJ
ejpam-237	101	15	-	-	ADJ
ejpam-237	101	16	i	i	PRON
ejpam-237	101	17	-	-	PUNCT
ejpam-237	101	18	complete	complete	ADJ
ejpam-237	101	19	(	(	PUNCT
ejpam-237	101	20	resp	resp	NOUN
ejpam-237	101	21	.	.	PUNCT
ejpam-237	101	22	quasii	quasii	PROPN
ejpam-237	101	23	-	-	PUNCT
ejpam-237	101	24	complete	complete	ADJ
ejpam-237	101	25	[	[	X
ejpam-237	101	26	13	13	NUM
ejpam-237	101	27	]	]	SYM
ejpam-237	101	28	)	)	PUNCT
ejpam-237	101	29	if	if	SCONJ
ejpam-237	101	30	τ∗	τ∗	NOUN
ejpam-237	101	31	=	=	SYM
ejpam-237	101	32	sio(x	sio(x	PROPN
ejpam-237	101	33	,	,	PUNCT
ejpam-237	101	34	τ	τ	PROPN
ejpam-237	101	35	)	)	PUNCT
ejpam-237	101	36	(	(	PUNCT
ejpam-237	101	37	resp	resp	NOUN
ejpam-237	101	38	.	.	PUNCT
ejpam-237	102	1	τ∗	τ∗	NOUN
ejpam-237	102	2	=	=	SYM
ejpam-237	102	3	qio(x	qio(x	PROPN
ejpam-237	102	4	,	,	PUNCT
ejpam-237	102	5	τ	τ	PROPN
ejpam-237	102	6	)	)	PUNCT
ejpam-237	102	7	)	)	PUNCT
ejpam-237	102	8	,	,	PUNCT
ejpam-237	102	9	that	that	ADV
ejpam-237	102	10	is	is	ADV
ejpam-237	102	11	,	,	PUNCT
ejpam-237	102	12	a	a	DET
ejpam-237	102	13	subset	subset	NOUN
ejpam-237	102	14	a	a	PRON
ejpam-237	102	15	of	of	ADP
ejpam-237	102	16	x	x	NOUN
ejpam-237	102	17	is	be	AUX
ejpam-237	102	18	τ∗−	τ∗−	ADV
ejpam-237	102	19	open	open	ADJ
ejpam-237	102	20	if	if	SCONJ
ejpam-237	102	21	and	and	CCONJ
ejpam-237	102	22	only	only	ADV
ejpam-237	102	23	if	if	SCONJ
ejpam-237	102	24	it	it	PRON
ejpam-237	102	25	is	be	AUX
ejpam-237	102	26	semi	semi	ADJ
ejpam-237	102	27	-	-	ADJ
ejpam-237	102	28	i	i	PRON
ejpam-237	102	29	-	-	PUNCT
ejpam-237	102	30	open	open	ADJ
ejpam-237	102	31	(	(	PUNCT
ejpam-237	102	32	resp	resp	NOUN
ejpam-237	102	33	.	.	PUNCT
ejpam-237	103	1	quasi	quasi	ADJ
ejpam-237	103	2	-	-	PROPN
ejpam-237	103	3	i	i	PRON
ejpam-237	103	4	-	-	PUNCT
ejpam-237	103	5	open	open	ADJ
ejpam-237	103	6	)	)	PUNCT
ejpam-237	103	7	.	.	PUNCT
ejpam-237	104	1	theorem	theorem	VERB
ejpam-237	104	2	3.3	3.3	NUM
ejpam-237	104	3	.	.	PUNCT
ejpam-237	105	1	let	let	AUX
ejpam-237	105	2	(	(	PUNCT
ejpam-237	105	3	x	x	X
ejpam-237	105	4	,	,	PUNCT
ejpam-237	105	5	τ	τ	PROPN
ejpam-237	105	6	,	,	PUNCT
ejpam-237	105	7	in	in	ADV
ejpam-237	105	8	)	)	PUNCT
ejpam-237	105	9	be	be	AUX
ejpam-237	105	10	an	an	DET
ejpam-237	105	11	ideal	ideal	ADJ
ejpam-237	105	12	topological	topological	ADJ
ejpam-237	105	13	space	space	NOUN
ejpam-237	105	14	,	,	PUNCT
ejpam-237	105	15	where	where	SCONJ
ejpam-237	105	16	in	in	ADP
ejpam-237	105	17	is	be	AUX
ejpam-237	105	18	the	the	DET
ejpam-237	105	19	ideal	ideal	NOUN
ejpam-237	105	20	of	of	ADP
ejpam-237	105	21	the	the	DET
ejpam-237	105	22	nowhere	nowhere	ADV
ejpam-237	105	23	dense	dense	ADJ
ejpam-237	105	24	sets	set	NOUN
ejpam-237	105	25	of	of	ADP
ejpam-237	105	26	(	(	PUNCT
ejpam-237	105	27	x	x	INTJ
ejpam-237	105	28	,	,	PUNCT
ejpam-237	105	29	τ	τ	PROPN
ejpam-237	105	30	)	)	PUNCT
ejpam-237	105	31	.	.	PUNCT
ejpam-237	106	1	1	1	X
ejpam-237	106	2	.	.	X
ejpam-237	106	3	(	(	PUNCT
ejpam-237	106	4	x	x	X
ejpam-237	106	5	,	,	PUNCT
ejpam-237	106	6	τ	τ	PROPN
ejpam-237	106	7	,	,	PUNCT
ejpam-237	106	8	in	in	ADP
ejpam-237	106	9	)	)	PUNCT
ejpam-237	106	10	is	be	AUX
ejpam-237	106	11	semi	semi	ADJ
ejpam-237	106	12	-	-	ADJ
ejpam-237	106	13	i	i	NOUN
ejpam-237	106	14	-	-	PUNCT
ejpam-237	106	15	hausdorff	hausdorff	NOUN
ejpam-237	106	16	(	(	PUNCT
ejpam-237	106	17	resp	resp	NOUN
ejpam-237	106	18	.	.	PUNCT
ejpam-237	107	1	quasi	quasi	ADJ
ejpam-237	107	2	-	-	PROPN
ejpam-237	107	3	i	i	NOUN
ejpam-237	107	4	-	-	PUNCT
ejpam-237	107	5	hausdorff	hausdorff	NOUN
ejpam-237	107	6	)	)	PUNCT
ejpam-237	108	1	if	if	SCONJ
ejpam-237	108	2	and	and	CCONJ
ejpam-237	108	3	only	only	ADV
ejpam-237	108	4	if	if	SCONJ
ejpam-237	108	5	it	it	PRON
ejpam-237	108	6	is	be	AUX
ejpam-237	108	7	semi	semi	ADJ
ejpam-237	108	8	-	-	ADJ
ejpam-237	108	9	hausdorff	hausdorff	ADJ
ejpam-237	108	10	(	(	PUNCT
ejpam-237	108	11	resp	resp	NOUN
ejpam-237	108	12	.	.	PUNCT
ejpam-237	109	1	β−hausdorff	β−hausdorff	NOUN
ejpam-237	109	2	)	)	PUNCT
ejpam-237	109	3	.	.	PUNCT
ejpam-237	110	1	2	2	X
ejpam-237	110	2	.	.	X
ejpam-237	110	3	(	(	PUNCT
ejpam-237	110	4	x	x	X
ejpam-237	110	5	,	,	PUNCT
ejpam-237	110	6	τ	τ	PROPN
ejpam-237	110	7	,	,	PUNCT
ejpam-237	110	8	in	in	ADP
ejpam-237	110	9	)	)	PUNCT
ejpam-237	110	10	is	be	AUX
ejpam-237	110	11	semi	semi	ADJ
ejpam-237	110	12	-	-	ADJ
ejpam-237	110	13	hausdorff	hausdorff	ADJ
ejpam-237	110	14	and	and	CCONJ
ejpam-237	110	15	semi	semi	ADJ
ejpam-237	110	16	-	-	ADJ
ejpam-237	110	17	i	i	PRON
ejpam-237	110	18	-	-	PUNCT
ejpam-237	110	19	complete	complete	ADJ
ejpam-237	110	20	(	(	PUNCT
ejpam-237	110	21	resp	resp	NOUN
ejpam-237	110	22	.	.	PUNCT
ejpam-237	111	1	β−hausdorff	β−hausdorff	PUNCT
ejpam-237	111	2	and	and	CCONJ
ejpam-237	111	3	quasi	quasi	ADJ
ejpam-237	111	4	-	-	ADJ
ejpam-237	111	5	i	i	NOUN
ejpam-237	111	6	-	-	PUNCT
ejpam-237	111	7	complete	complete	ADJ
ejpam-237	111	8	)	)	PUNCT
ejpam-237	111	9	,	,	PUNCT
ejpam-237	111	10	then	then	ADV
ejpam-237	111	11	it	it	PRON
ejpam-237	111	12	is	be	AUX
ejpam-237	111	13	hausdorff	hausdorff	NOUN
ejpam-237	111	14	.	.	PUNCT
ejpam-237	112	1	proof	proof	NOUN
ejpam-237	112	2	.	.	PUNCT
ejpam-237	113	1	(	(	PUNCT
ejpam-237	113	2	1	1	X
ejpam-237	113	3	)	)	PUNCT
ejpam-237	113	4	since	since	SCONJ
ejpam-237	113	5	in	in	ADV
ejpam-237	113	6	is	be	AUX
ejpam-237	113	7	the	the	DET
ejpam-237	113	8	ideal	ideal	NOUN
ejpam-237	113	9	of	of	ADP
ejpam-237	113	10	nowhere	nowhere	PRON
ejpam-237	113	11	dense	dense	ADJ
ejpam-237	113	12	sets	set	NOUN
ejpam-237	113	13	of	of	ADP
ejpam-237	113	14	(	(	PUNCT
ejpam-237	113	15	x	x	INTJ
ejpam-237	113	16	,	,	PUNCT
ejpam-237	113	17	τ	τ	PROPN
ejpam-237	113	18	)	)	PUNCT
ejpam-237	113	19	,	,	PUNCT
ejpam-237	113	20	we	we	PRON
ejpam-237	113	21	have	have	VERB
ejpam-237	113	22	a∗	a∗	NOUN
ejpam-237	113	23	=	=	SYM
ejpam-237	113	24	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-237	113	25	)	)	PUNCT
ejpam-237	113	26	)	)	PUNCT
ejpam-237	113	27	)	)	PUNCT
ejpam-237	113	28	and	and	CCONJ
ejpam-237	113	29	hence	hence	ADV
ejpam-237	113	30	by	by	ADP
ejpam-237	113	31	example	example	NOUN
ejpam-237	113	32	2.10	2.10	NUM
ejpam-237	113	33	of	of	ADP
ejpam-237	113	34	[	[	X
ejpam-237	113	35	8	8	NUM
ejpam-237	113	36	]	]	SYM
ejpam-237	113	37	cl∗(a	cl∗(a	NOUN
ejpam-237	113	38	)	)	PUNCT
ejpam-237	113	39	=	=	SYM
ejpam-237	113	40	a∪cl(int(cl(a	a∪cl(int(cl(a	NUM
ejpam-237	113	41	)	)	PUNCT
ejpam-237	113	42	)	)	PUNCT
ejpam-237	113	43	)	)	PUNCT
ejpam-237	114	1	=	=	SYM
ejpam-237	114	2	αcl(a	αcl(a	NUM
ejpam-237	114	3	)	)	PUNCT
ejpam-237	114	4	,	,	PUNCT
ejpam-237	114	5	where	where	SCONJ
ejpam-237	114	6	αcl(a	αcl(a	NUM
ejpam-237	114	7	)	)	PUNCT
ejpam-237	114	8	denotes	denote	VERB
ejpam-237	114	9	the	the	PRON
ejpam-237	114	10	α−	α−	ADP
ejpam-237	114	11	closure	closure	NOUN
ejpam-237	114	12	of	of	ADP
ejpam-237	114	13	a.for	a.for	ADP
ejpam-237	114	14	every	every	DET
ejpam-237	114	15	subset	subset	NOUN
ejpam-237	114	16	a	a	PRON
ejpam-237	114	17	of	of	ADP
ejpam-237	114	18	x	x	PRON
ejpam-237	114	19	,	,	PUNCT
ejpam-237	114	20	cl∗(int(a	cl∗(int(a	PROPN
ejpam-237	114	21	)	)	PUNCT
ejpam-237	114	22	)	)	PUNCT
ejpam-237	115	1	=	=	PRON
ejpam-237	115	2	int(a)∪	int(a)∪	PROPN
ejpam-237	116	1	cl(int(cl(int(a	cl(int(cl(int(a	NOUN
ejpam-237	116	2	)	)	PUNCT
ejpam-237	116	3	)	)	PUNCT
ejpam-237	116	4	)	)	PUNCT
ejpam-237	116	5	)	)	PUNCT
ejpam-237	117	1	=	=	SYM
ejpam-237	117	2	int(a)∪	int(a)∪	PROPN
ejpam-237	117	3	cl(int(a	cl(int(a	PROPN
ejpam-237	117	4	)	)	PUNCT
ejpam-237	117	5	)	)	PUNCT
ejpam-237	118	1	=	=	SYM
ejpam-237	118	2	cl(int(a	cl(int(a	PROPN
ejpam-237	118	3	)	)	PUNCT
ejpam-237	118	4	)	)	PUNCT
ejpam-237	118	5	.	.	PUNCT
ejpam-237	119	1	therefore	therefore	ADV
ejpam-237	119	2	,	,	PUNCT
ejpam-237	119	3	a	a	DET
ejpam-237	119	4	∈	∈	PROPN
ejpam-237	119	5	sio(x	sio(x	NOUN
ejpam-237	119	6	,	,	PUNCT
ejpam-237	119	7	τ	τ	PROPN
ejpam-237	119	8	)	)	PUNCT
ejpam-237	119	9	if	if	SCONJ
ejpam-237	119	10	and	and	CCONJ
ejpam-237	119	11	only	only	ADV
ejpam-237	119	12	if	if	SCONJ
ejpam-237	119	13	a	a	DET
ejpam-237	119	14	∈	∈	PROPN
ejpam-237	119	15	so(x	so(x	NOUN
ejpam-237	119	16	,	,	PUNCT
ejpam-237	119	17	τ).by	τ).by	PROPN
ejpam-237	119	18	this	this	DET
ejpam-237	119	19	fact	fact	NOUN
ejpam-237	119	20	,	,	PUNCT
ejpam-237	119	21	it	it	PRON
ejpam-237	119	22	follows	follow	VERB
ejpam-237	119	23	that	that	SCONJ
ejpam-237	119	24	(	(	PUNCT
ejpam-237	119	25	x	x	X
ejpam-237	119	26	,	,	PUNCT
ejpam-237	119	27	τ	τ	PROPN
ejpam-237	119	28	,	,	PUNCT
ejpam-237	119	29	in	in	ADP
ejpam-237	119	30	)	)	PUNCT
ejpam-237	119	31	is	be	AUX
ejpam-237	119	32	semi	semi	ADJ
ejpam-237	119	33	-	-	ADJ
ejpam-237	119	34	i	i	NOUN
ejpam-237	119	35	-	-	PUNCT
ejpam-237	119	36	hausdorff	hausdorff	NOUN
ejpam-237	119	37	if	if	SCONJ
ejpam-237	120	1	and	and	CCONJ
ejpam-237	120	2	only	only	ADV
ejpam-237	120	3	if	if	SCONJ
ejpam-237	120	4	it	it	PRON
ejpam-237	120	5	is	be	AUX
ejpam-237	120	6	semi	semi	ADJ
ejpam-237	120	7	-	-	ADJ
ejpam-237	120	8	hausdorff	hausdorff	ADJ
ejpam-237	120	9	.	.	PUNCT
ejpam-237	121	1	on	on	ADP
ejpam-237	121	2	the	the	DET
ejpam-237	121	3	other	other	ADJ
ejpam-237	121	4	hand	hand	NOUN
ejpam-237	121	5	,	,	PUNCT
ejpam-237	121	6	e.	e.	PROPN
ejpam-237	121	7	hatir	hatir	PROPN
ejpam-237	121	8	and	and	CCONJ
ejpam-237	121	9	t.	t.	PROPN
ejpam-237	121	10	noiri	noiri	PROPN
ejpam-237	121	11	/	/	SYM
ejpam-237	121	12	eur	eur	PROPN
ejpam-237	121	13	.	.	PUNCT
ejpam-237	122	1	j.	j.	PROPN
ejpam-237	122	2	pure	pure	PROPN
ejpam-237	122	3	appl	appl	PROPN
ejpam-237	122	4	.	.	PROPN
ejpam-237	122	5	math	math	PROPN
ejpam-237	122	6	,	,	PUNCT
ejpam-237	122	7	2	2	NUM
ejpam-237	122	8	(	(	PUNCT
ejpam-237	122	9	2009	2009	NUM
ejpam-237	122	10	)	)	PUNCT
ejpam-237	122	11	,	,	PUNCT
ejpam-237	122	12	(	(	PUNCT
ejpam-237	122	13	172	172	NUM
ejpam-237	122	14	-	-	SYM
ejpam-237	122	15	181	181	NUM
ejpam-237	122	16	)	)	PUNCT
ejpam-237	122	17	177	177	NUM
ejpam-237	122	18	cl(int(a∗	cl(int(a∗	NOUN
ejpam-237	122	19	)	)	PUNCT
ejpam-237	122	20	)	)	PUNCT
ejpam-237	123	1	=	=	SYM
ejpam-237	123	2	cl(int(cl(int(cl(a	cl(int(cl(int(cl(a	ADJ
ejpam-237	123	3	)	)	PUNCT
ejpam-237	123	4	)	)	PUNCT
ejpam-237	123	5	)	)	PUNCT
ejpam-237	123	6	)	)	PUNCT
ejpam-237	123	7	)	)	PUNCT
ejpam-237	124	1	=	=	SYM
ejpam-237	124	2	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-237	124	3	)	)	PUNCT
ejpam-237	124	4	)	)	PUNCT
ejpam-237	124	5	)	)	PUNCT
ejpam-237	124	6	for	for	ADP
ejpam-237	124	7	every	every	DET
ejpam-237	124	8	subset	subset	NOUN
ejpam-237	124	9	a	a	PRON
ejpam-237	124	10	of	of	ADP
ejpam-237	124	11	x	x	X
ejpam-237	124	12	.	.	PUNCT
ejpam-237	125	1	therefore	therefore	ADV
ejpam-237	125	2	,	,	PUNCT
ejpam-237	125	3	a∈	a∈	PROPN
ejpam-237	125	4	qio(x	qio(x	PROPN
ejpam-237	125	5	,	,	PUNCT
ejpam-237	125	6	τ	τ	PROPN
ejpam-237	125	7	)	)	PUNCT
ejpam-237	126	1	if	if	SCONJ
ejpam-237	126	2	and	and	CCONJ
ejpam-237	126	3	only	only	ADV
ejpam-237	126	4	if	if	SCONJ
ejpam-237	126	5	a∈	a∈	PROPN
ejpam-237	126	6	β(x	β(x	PROPN
ejpam-237	126	7	,	,	PUNCT
ejpam-237	126	8	τ	τ	PROPN
ejpam-237	126	9	)	)	PUNCT
ejpam-237	126	10	.	.	PUNCT
ejpam-237	127	1	it	it	PRON
ejpam-237	127	2	follows	follow	VERB
ejpam-237	127	3	that	that	SCONJ
ejpam-237	127	4	(	(	PUNCT
ejpam-237	127	5	x	x	X
ejpam-237	127	6	,	,	PUNCT
ejpam-237	127	7	τ	τ	PROPN
ejpam-237	127	8	,	,	PUNCT
ejpam-237	127	9	in	in	ADV
ejpam-237	127	10	)	)	PUNCT
ejpam-237	127	11	is	be	AUX
ejpam-237	127	12	quasi	quasi	ADJ
ejpam-237	127	13	-	-	ADJ
ejpam-237	127	14	i	i	NOUN
ejpam-237	127	15	-	-	PUNCT
ejpam-237	127	16	hausdorff	hausdorff	NOUN
ejpam-237	127	17	if	if	SCONJ
ejpam-237	128	1	and	and	CCONJ
ejpam-237	128	2	only	only	ADV
ejpam-237	128	3	if	if	SCONJ
ejpam-237	128	4	it	it	PRON
ejpam-237	128	5	is	be	AUX
ejpam-237	128	6	β	β	NOUN
ejpam-237	128	7	−hausdor	−hausdor	PROPN
ejpam-237	128	8	f	f	PROPN
ejpam-237	128	9	f	f	PROPN
ejpam-237	128	10	.	.	PUNCT
ejpam-237	129	1	(	(	PUNCT
ejpam-237	129	2	2	2	X
ejpam-237	129	3	)	)	PUNCT
ejpam-237	129	4	let	let	AUX
ejpam-237	129	5	(	(	PUNCT
ejpam-237	129	6	x	x	X
ejpam-237	129	7	,	,	PUNCT
ejpam-237	129	8	τ	τ	PROPN
ejpam-237	129	9	,	,	PUNCT
ejpam-237	129	10	in	in	ADP
ejpam-237	129	11	)	)	PUNCT
ejpam-237	129	12	be	be	AUX
ejpam-237	129	13	semi	semi	ADJ
ejpam-237	129	14	-	-	ADJ
ejpam-237	129	15	hausdorff	hausdorff	ADJ
ejpam-237	129	16	and	and	CCONJ
ejpam-237	129	17	semi	semi	ADJ
ejpam-237	129	18	-	-	ADJ
ejpam-237	129	19	i	i	NOUN
ejpam-237	129	20	-	-	PUNCT
ejpam-237	129	21	complete	complete	ADJ
ejpam-237	129	22	.	.	PUNCT
ejpam-237	130	1	then	then	ADV
ejpam-237	130	2	a	a	DET
ejpam-237	130	3	∈	∈	PROPN
ejpam-237	130	4	sio(x	sio(x	NOUN
ejpam-237	130	5	,	,	PUNCT
ejpam-237	130	6	τ	τ	PROPN
ejpam-237	130	7	)	)	PUNCT
ejpam-237	130	8	if	if	SCONJ
ejpam-237	130	9	and	and	CCONJ
ejpam-237	130	10	only	only	ADV
ejpam-237	130	11	if	if	SCONJ
ejpam-237	130	12	a	a	DET
ejpam-237	130	13	∈	∈	PROPN
ejpam-237	130	14	τ∗	τ∗	NOUN
ejpam-237	131	1	if	if	SCONJ
ejpam-237	131	2	and	and	CCONJ
ejpam-237	131	3	only	only	ADV
ejpam-237	131	4	if	if	SCONJ
ejpam-237	131	5	a	a	PRON
ejpam-237	131	6	is	be	AUX
ejpam-237	131	7	α	α	PRON
ejpam-237	131	8	−	−	NOUN
ejpam-237	131	9	open	open	ADJ
ejpam-237	131	10	.	.	PUNCT
ejpam-237	132	1	by	by	ADP
ejpam-237	132	2	the	the	DET
ejpam-237	132	3	proof	proof	NOUN
ejpam-237	132	4	of	of	ADP
ejpam-237	132	5	(	(	PUNCT
ejpam-237	132	6	1	1	NUM
ejpam-237	132	7	)	)	PUNCT
ejpam-237	132	8	,	,	PUNCT
ejpam-237	132	9	so(x	so(x	X
ejpam-237	132	10	,	,	PUNCT
ejpam-237	132	11	τ	τ	X
ejpam-237	132	12	)	)	PUNCT
ejpam-237	132	13	=	=	SYM
ejpam-237	133	1	sio(x	sio(x	PROPN
ejpam-237	133	2	,	,	PUNCT
ejpam-237	133	3	τ	τ	PROPN
ejpam-237	133	4	)	)	PUNCT
ejpam-237	133	5	and	and	CCONJ
ejpam-237	133	6	hence	hence	ADV
ejpam-237	133	7	(	(	PUNCT
ejpam-237	133	8	x	x	X
ejpam-237	133	9	,	,	PUNCT
ejpam-237	133	10	τ	τ	PROPN
ejpam-237	133	11	,	,	PUNCT
ejpam-237	133	12	i	i	PROPN
ejpam-237	133	13	)	)	PUNCT
ejpam-237	133	14	is	be	AUX
ejpam-237	133	15	hausdorff	hausdorff	NOUN
ejpam-237	133	16	.	.	PUNCT
ejpam-237	134	1	the	the	DET
ejpam-237	134	2	another	another	DET
ejpam-237	134	3	result	result	NOUN
ejpam-237	134	4	is	be	AUX
ejpam-237	134	5	shown	show	VERB
ejpam-237	134	6	similarly	similarly	ADV
ejpam-237	134	7	.	.	PUNCT
ejpam-237	135	1	lemma	lemma	PROPN
ejpam-237	135	2	3.1	3.1	NUM
ejpam-237	135	3	.	.	PUNCT
ejpam-237	136	1	let	let	VERB
ejpam-237	136	2	i	i	PRON
ejpam-237	136	3	and	and	CCONJ
ejpam-237	136	4	j	j	PROPN
ejpam-237	136	5	be	be	VERB
ejpam-237	136	6	two	two	NUM
ejpam-237	136	7	ideals	ideal	NOUN
ejpam-237	136	8	on	on	ADP
ejpam-237	136	9	a	a	DET
ejpam-237	136	10	topological	topological	ADJ
ejpam-237	136	11	space	space	NOUN
ejpam-237	136	12	(	(	PUNCT
ejpam-237	136	13	x	x	X
ejpam-237	136	14	,	,	PUNCT
ejpam-237	136	15	τ	τ	PROPN
ejpam-237	136	16	)	)	PUNCT
ejpam-237	136	17	.	.	PUNCT
ejpam-237	137	1	if	if	SCONJ
ejpam-237	137	2	i	i	PRON
ejpam-237	137	3	⊂	⊂	PROPN
ejpam-237	137	4	j	j	PROPN
ejpam-237	137	5	,	,	PUNCT
ejpam-237	137	6	then	then	ADV
ejpam-237	137	7	the	the	DET
ejpam-237	137	8	following	follow	VERB
ejpam-237	137	9	properties	property	NOUN
ejpam-237	137	10	hold	hold	VERB
ejpam-237	137	11	:	:	PUNCT
ejpam-237	137	12	1	1	X
ejpam-237	137	13	.	.	X
ejpam-237	138	1	c	c	NOUN
ejpam-237	138	2	l∗j	l∗j	PUNCT
ejpam-237	138	3	(	(	PUNCT
ejpam-237	138	4	a)⊂	a)⊂	PROPN
ejpam-237	138	5	cl∗i	cl∗i	X
ejpam-237	138	6	(	(	PUNCT
ejpam-237	138	7	a	a	NOUN
ejpam-237	138	8	)	)	PUNCT
ejpam-237	138	9	for	for	ADP
ejpam-237	138	10	each	each	PRON
ejpam-237	138	11	subset	subset	VERB
ejpam-237	138	12	a	a	PRON
ejpam-237	138	13	of	of	ADP
ejpam-237	138	14	x	x	SYM
ejpam-237	138	15	,	,	PUNCT
ejpam-237	138	16	2	2	X
ejpam-237	138	17	.	.	PUNCT
ejpam-237	139	1	sio(x	sio(x	VERB
ejpam-237	139	2	,	,	PUNCT
ejpam-237	139	3	τ	τ	PROPN
ejpam-237	139	4	,	,	PUNCT
ejpam-237	139	5	j	j	PROPN
ejpam-237	139	6	)	)	PUNCT
ejpam-237	140	1	⊂	⊂	PROPN
ejpam-237	141	1	sio(x	sio(x	PROPN
ejpam-237	141	2	,	,	PUNCT
ejpam-237	141	3	τ	τ	PROPN
ejpam-237	141	4	,	,	PUNCT
ejpam-237	141	5	i	i	PROPN
ejpam-237	141	6	)	)	PUNCT
ejpam-237	141	7	.	.	PUNCT
ejpam-237	142	1	proof	proof	NOUN
ejpam-237	142	2	.	.	PUNCT
ejpam-237	143	1	(	(	PUNCT
ejpam-237	143	2	1	1	X
ejpam-237	143	3	)	)	PUNCT
ejpam-237	143	4	if	if	SCONJ
ejpam-237	143	5	i	i	PRON
ejpam-237	143	6	⊂	⊂	PROPN
ejpam-237	143	7	j	j	PROPN
ejpam-237	143	8	,	,	PUNCT
ejpam-237	143	9	then	then	ADV
ejpam-237	143	10	a∗(j	a∗(j	PROPN
ejpam-237	143	11	)	)	PUNCT
ejpam-237	144	1	⊂	⊂	PROPN
ejpam-237	144	2	a∗(i	a∗(i	PROPN
ejpam-237	144	3	)	)	PUNCT
ejpam-237	144	4	and	and	CCONJ
ejpam-237	144	5	cl∗j	cl∗j	ADV
ejpam-237	144	6	(	(	PUNCT
ejpam-237	144	7	a	a	X
ejpam-237	144	8	)	)	PUNCT
ejpam-237	144	9	=	=	SYM
ejpam-237	144	10	a∪a∗(j	a∪a∗(j	PROPN
ejpam-237	144	11	)	)	PUNCT
ejpam-237	144	12	⊂	⊂	PROPN
ejpam-237	144	13	a∪a∗(i	a∪a∗(i	PROPN
ejpam-237	144	14	)	)	PUNCT
ejpam-237	144	15	=	=	SYM
ejpam-237	144	16	cl∗i	cl∗i	X
ejpam-237	144	17	(	(	PUNCT
ejpam-237	144	18	a	a	NOUN
ejpam-237	144	19	)	)	PUNCT
ejpam-237	144	20	for	for	ADP
ejpam-237	144	21	each	each	PRON
ejpam-237	144	22	subset	subset	VERB
ejpam-237	144	23	a	a	PRON
ejpam-237	144	24	of	of	ADP
ejpam-237	144	25	x	x	X
ejpam-237	144	26	.	.	PUNCT
ejpam-237	145	1	(	(	PUNCT
ejpam-237	145	2	2	2	X
ejpam-237	145	3	)	)	PUNCT
ejpam-237	145	4	let	let	VERB
ejpam-237	145	5	a	a	DET
ejpam-237	145	6	∈	∈	ADJ
ejpam-237	145	7	sio(x	sio(x	NOUN
ejpam-237	145	8	,	,	PUNCT
ejpam-237	145	9	τ	τ	PROPN
ejpam-237	145	10	,	,	PUNCT
ejpam-237	145	11	j	j	PROPN
ejpam-237	145	12	)	)	PUNCT
ejpam-237	145	13	.	.	PUNCT
ejpam-237	146	1	then	then	ADV
ejpam-237	146	2	a	a	DET
ejpam-237	146	3	⊂	⊂	PROPN
ejpam-237	146	4	cl∗j	cl∗j	ADJ
ejpam-237	146	5	(	(	PUNCT
ejpam-237	146	6	int(a	int(a	NOUN
ejpam-237	146	7	)	)	PUNCT
ejpam-237	146	8	)	)	PUNCT
ejpam-237	147	1	⊂	⊂	PROPN
ejpam-237	147	2	cl∗i	cl∗i	PROPN
ejpam-237	147	3	(	(	PUNCT
ejpam-237	147	4	int(a	int(a	PROPN
ejpam-237	147	5	)	)	PUNCT
ejpam-237	147	6	)	)	PUNCT
ejpam-237	147	7	and	and	CCONJ
ejpam-237	147	8	hence	hence	ADV
ejpam-237	147	9	a	a	DET
ejpam-237	147	10	∈	∈	ADJ
ejpam-237	147	11	sio(x	sio(x	NOUN
ejpam-237	147	12	,	,	PUNCT
ejpam-237	147	13	τ	τ	PROPN
ejpam-237	147	14	,	,	PUNCT
ejpam-237	147	15	i	i	PROPN
ejpam-237	147	16	)	)	PUNCT
ejpam-237	147	17	.	.	PUNCT
ejpam-237	148	1	theorem	theorem	VERB
ejpam-237	148	2	3.4	3.4	NUM
ejpam-237	148	3	.	.	PUNCT
ejpam-237	149	1	let	let	VERB
ejpam-237	149	2	i	i	PRON
ejpam-237	149	3	and	and	CCONJ
ejpam-237	149	4	j	j	PROPN
ejpam-237	149	5	be	be	VERB
ejpam-237	149	6	two	two	NUM
ejpam-237	149	7	ideals	ideal	NOUN
ejpam-237	149	8	on	on	ADP
ejpam-237	149	9	a	a	DET
ejpam-237	149	10	topological	topological	ADJ
ejpam-237	149	11	space	space	NOUN
ejpam-237	149	12	(	(	PUNCT
ejpam-237	149	13	x	x	X
ejpam-237	149	14	,	,	PUNCT
ejpam-237	149	15	τ	τ	PROPN
ejpam-237	149	16	)	)	PUNCT
ejpam-237	149	17	and	and	CCONJ
ejpam-237	150	1	i	i	PRON
ejpam-237	150	2	⊂	⊂	PROPN
ejpam-237	150	3	j	j	PROPN
ejpam-237	150	4	.	.	PUNCT
ejpam-237	151	1	if	if	SCONJ
ejpam-237	151	2	(	(	PUNCT
ejpam-237	151	3	x	x	X
ejpam-237	151	4	,	,	PUNCT
ejpam-237	151	5	τ	τ	PROPN
ejpam-237	151	6	,	,	PUNCT
ejpam-237	151	7	j	j	NOUN
ejpam-237	151	8	)	)	PUNCT
ejpam-237	151	9	is	be	AUX
ejpam-237	151	10	semi	semi	ADJ
ejpam-237	151	11	-	-	ADJ
ejpam-237	151	12	i	i	NOUN
ejpam-237	151	13	-	-	PUNCT
ejpam-237	151	14	hausdorff	hausdorff	NOUN
ejpam-237	151	15	,	,	PUNCT
ejpam-237	151	16	then	then	ADV
ejpam-237	151	17	(	(	PUNCT
ejpam-237	151	18	x	x	X
ejpam-237	151	19	,	,	PUNCT
ejpam-237	151	20	τ	τ	PROPN
ejpam-237	151	21	,	,	PUNCT
ejpam-237	151	22	i	i	PROPN
ejpam-237	151	23	)	)	PUNCT
ejpam-237	151	24	is	be	AUX
ejpam-237	151	25	semi	semi	ADJ
ejpam-237	151	26	-	-	ADJ
ejpam-237	151	27	i	i	NOUN
ejpam-237	151	28	-	-	PUNCT
ejpam-237	151	29	hausdorff	hausdorff	NOUN
ejpam-237	151	30	.	.	PUNCT
ejpam-237	152	1	proof	proof	NOUN
ejpam-237	152	2	.	.	PUNCT
ejpam-237	153	1	this	this	PRON
ejpam-237	153	2	is	be	AUX
ejpam-237	153	3	an	an	DET
ejpam-237	153	4	immediate	immediate	ADJ
ejpam-237	153	5	consequence	consequence	NOUN
ejpam-237	153	6	of	of	ADP
ejpam-237	153	7	lemma	lemma	PROPN
ejpam-237	153	8	1	1	NUM
ejpam-237	153	9	a	a	DET
ejpam-237	153	10	semi	semi	ADJ
ejpam-237	153	11	-	-	ADJ
ejpam-237	153	12	i	i	PRON
ejpam-237	153	13	-	-	PUNCT
ejpam-237	153	14	open	open	ADJ
ejpam-237	153	15	subspace	subspace	NOUN
ejpam-237	153	16	of	of	ADP
ejpam-237	153	17	semi	semi	ADJ
ejpam-237	153	18	-	-	ADJ
ejpam-237	153	19	i	i	ADJ
ejpam-237	153	20	-	-	PUNCT
ejpam-237	153	21	hausdorff	hausdorff	NOUN
ejpam-237	153	22	space	space	NOUN
ejpam-237	153	23	need	need	AUX
ejpam-237	153	24	not	not	PART
ejpam-237	153	25	be	be	AUX
ejpam-237	153	26	semi	semi	ADJ
ejpam-237	153	27	-	-	ADJ
ejpam-237	153	28	i	i	NOUN
ejpam-237	153	29	-	-	PUNCT
ejpam-237	153	30	hausdorff	hausdorff	NOUN
ejpam-237	153	31	as	as	SCONJ
ejpam-237	153	32	shown	show	VERB
ejpam-237	153	33	in	in	ADP
ejpam-237	153	34	the	the	DET
ejpam-237	153	35	following	follow	VERB
ejpam-237	153	36	example	example	NOUN
ejpam-237	153	37	.	.	PUNCT
ejpam-237	154	1	example	example	NOUN
ejpam-237	154	2	3.4	3.4	NUM
ejpam-237	154	3	.	.	PUNCT
ejpam-237	155	1	let	let	VERB
ejpam-237	155	2	x	x	PUNCT
ejpam-237	155	3	=	=	PRON
ejpam-237	155	4	{	{	PUNCT
ejpam-237	155	5	a	a	DET
ejpam-237	155	6	,	,	PUNCT
ejpam-237	155	7	b	b	NOUN
ejpam-237	155	8	,	,	PUNCT
ejpam-237	155	9	c	c	NOUN
ejpam-237	155	10	}	}	PUNCT
ejpam-237	155	11	,	,	PUNCT
ejpam-237	155	12	τ	τ	X
ejpam-237	155	13	=	=	PUNCT
ejpam-237	155	14	{	{	PUNCT
ejpam-237	155	15	∅	∅	NOUN
ejpam-237	155	16	,	,	PUNCT
ejpam-237	155	17	x	x	INTJ
ejpam-237	155	18	,	,	PUNCT
ejpam-237	155	19	{	{	PUNCT
ejpam-237	155	20	a	a	NOUN
ejpam-237	155	21	}	}	PUNCT
ejpam-237	155	22	,	,	PUNCT
ejpam-237	155	23	{	{	PUNCT
ejpam-237	155	24	b	b	NOUN
ejpam-237	155	25	}	}	PUNCT
ejpam-237	155	26	,	,	PUNCT
ejpam-237	155	27	{	{	PUNCT
ejpam-237	155	28	a	a	PRON
ejpam-237	155	29	,	,	PUNCT
ejpam-237	155	30	b	b	NOUN
ejpam-237	155	31	}	}	PUNCT
ejpam-237	155	32	}	}	PUNCT
ejpam-237	155	33	and	and	CCONJ
ejpam-237	155	34	i	i	PRON
ejpam-237	155	35	=	=	PUNCT
ejpam-237	155	36	{	{	PUNCT
ejpam-237	155	37	∅	∅	NOUN
ejpam-237	155	38	}	}	PUNCT
ejpam-237	155	39	.	.	PUNCT
ejpam-237	156	1	then	then	ADV
ejpam-237	156	2	(	(	PUNCT
ejpam-237	156	3	x	x	X
ejpam-237	156	4	,	,	PUNCT
ejpam-237	156	5	τ	τ	PROPN
ejpam-237	156	6	,	,	PUNCT
ejpam-237	156	7	i	i	PROPN
ejpam-237	156	8	)	)	PUNCT
ejpam-237	156	9	is	be	AUX
ejpam-237	156	10	semi	semi	ADJ
ejpam-237	156	11	-	-	ADJ
ejpam-237	156	12	i	i	NOUN
ejpam-237	156	13	-	-	PUNCT
ejpam-237	156	14	hausdorff	hausdorff	NOUN
ejpam-237	156	15	.	.	PUNCT
ejpam-237	157	1	but	but	CCONJ
ejpam-237	157	2	,	,	PUNCT
ejpam-237	157	3	take	take	VERB
ejpam-237	157	4	a	a	DET
ejpam-237	157	5	=	=	X
ejpam-237	157	6	{	{	PUNCT
ejpam-237	157	7	a	a	NOUN
ejpam-237	157	8	,	,	PUNCT
ejpam-237	157	9	c	c	NOUN
ejpam-237	157	10	}	}	PUNCT
ejpam-237	157	11	∈	∈	PROPN
ejpam-237	157	12	sio(x	sio(x	NOUN
ejpam-237	157	13	,	,	PUNCT
ejpam-237	157	14	τ	τ	PROPN
ejpam-237	157	15	)	)	PUNCT
ejpam-237	157	16	,	,	PUNCT
ejpam-237	157	17	then	then	ADV
ejpam-237	157	18	(	(	PUNCT
ejpam-237	157	19	a	a	PRON
ejpam-237	157	20	,	,	PUNCT
ejpam-237	157	21	τ|a	τ|a	NUM
ejpam-237	157	22	,	,	PUNCT
ejpam-237	157	23	i|a	i|a	PUNCT
ejpam-237	157	24	)	)	PUNCT
ejpam-237	157	25	is	be	AUX
ejpam-237	157	26	not	not	PART
ejpam-237	157	27	semi	semi	ADJ
ejpam-237	157	28	-	-	ADJ
ejpam-237	157	29	i	i	NOUN
ejpam-237	157	30	-	-	PUNCT
ejpam-237	157	31	hausdorff	hausdorff	NOUN
ejpam-237	157	32	.	.	PUNCT
ejpam-237	158	1	lemma	lemma	PROPN
ejpam-237	158	2	3.2	3.2	NUM
ejpam-237	158	3	.	.	PUNCT
ejpam-237	159	1	(	(	PUNCT
ejpam-237	159	2	hatir	hatir	NOUN
ejpam-237	159	3	and	and	CCONJ
ejpam-237	159	4	noiri	noiri	ADV
ejpam-237	160	1	[	[	X
ejpam-237	160	2	5	5	NUM
ejpam-237	160	3	]	]	PUNCT
ejpam-237	160	4	)	)	PUNCT
ejpam-237	160	5	let	let	VERB
ejpam-237	160	6	(	(	PUNCT
ejpam-237	160	7	x	x	X
ejpam-237	160	8	,	,	PUNCT
ejpam-237	160	9	τ	τ	PROPN
ejpam-237	160	10	,	,	PUNCT
ejpam-237	160	11	i	i	PRON
ejpam-237	160	12	)	)	PUNCT
ejpam-237	160	13	be	be	VERB
ejpam-237	160	14	an	an	DET
ejpam-237	160	15	ideal	ideal	ADJ
ejpam-237	160	16	topological	topological	ADJ
ejpam-237	160	17	space	space	NOUN
ejpam-237	160	18	.	.	PUNCT
ejpam-237	161	1	if	if	SCONJ
ejpam-237	161	2	u	u	PROPN
ejpam-237	161	3	∈	∈	PROPN
ejpam-237	161	4	τ	τ	X
ejpam-237	161	5	and	and	CCONJ
ejpam-237	161	6	v	v	ADP
ejpam-237	161	7	∈	∈	PROPN
ejpam-237	162	1	sio(x	sio(x	NOUN
ejpam-237	162	2	,	,	PUNCT
ejpam-237	162	3	τ	τ	PROPN
ejpam-237	162	4	)	)	PUNCT
ejpam-237	162	5	,	,	PUNCT
ejpam-237	162	6	then	then	ADV
ejpam-237	162	7	u	u	NOUN
ejpam-237	162	8	∩	∩	PROPN
ejpam-237	162	9	v	v	ADP
ejpam-237	162	10	∈	∈	PROPN
ejpam-237	162	11	sio(u	sio(u	PROPN
ejpam-237	162	12	,	,	PUNCT
ejpam-237	162	13	τ|u	τ|u	NOUN
ejpam-237	162	14	,	,	PUNCT
ejpam-237	162	15	i|u	i|u	NOUN
ejpam-237	162	16	)	)	PUNCT
ejpam-237	162	17	.	.	PUNCT
ejpam-237	163	1	e.	e.	PROPN
ejpam-237	163	2	hatir	hatir	PROPN
ejpam-237	163	3	and	and	CCONJ
ejpam-237	163	4	t.	t.	PROPN
ejpam-237	163	5	noiri	noiri	PROPN
ejpam-237	163	6	/	/	SYM
ejpam-237	163	7	eur	eur	PROPN
ejpam-237	163	8	.	.	PUNCT
ejpam-237	164	1	j.	j.	PROPN
ejpam-237	164	2	pure	pure	PROPN
ejpam-237	164	3	appl	appl	PROPN
ejpam-237	164	4	.	.	PROPN
ejpam-237	164	5	math	math	PROPN
ejpam-237	164	6	,	,	PUNCT
ejpam-237	164	7	2	2	NUM
ejpam-237	164	8	(	(	PUNCT
ejpam-237	164	9	2009	2009	NUM
ejpam-237	164	10	)	)	PUNCT
ejpam-237	164	11	,	,	PUNCT
ejpam-237	164	12	(	(	PUNCT
ejpam-237	164	13	172	172	NUM
ejpam-237	164	14	-	-	SYM
ejpam-237	164	15	181	181	NUM
ejpam-237	164	16	)	)	PUNCT
ejpam-237	164	17	178	178	NUM
ejpam-237	164	18	theorem	theorem	NOUN
ejpam-237	164	19	3.5	3.5	NUM
ejpam-237	164	20	.	.	PUNCT
ejpam-237	165	1	let	let	VERB
ejpam-237	165	2	(	(	PUNCT
ejpam-237	165	3	x	x	X
ejpam-237	165	4	,	,	PUNCT
ejpam-237	165	5	τ	τ	PROPN
ejpam-237	165	6	,	,	PUNCT
ejpam-237	165	7	i	i	PRON
ejpam-237	165	8	)	)	PUNCT
ejpam-237	165	9	be	be	VERB
ejpam-237	165	10	a	a	DET
ejpam-237	165	11	semi	semi	ADJ
ejpam-237	165	12	-	-	ADJ
ejpam-237	165	13	i	i	ADJ
ejpam-237	165	14	-	-	PUNCT
ejpam-237	165	15	hausdorff	hausdorff	NOUN
ejpam-237	165	16	space	space	NOUN
ejpam-237	165	17	and	and	CCONJ
ejpam-237	165	18	a⊂	a⊂	PRON
ejpam-237	165	19	x	x	X
ejpam-237	165	20	.	.	PUNCT
ejpam-237	166	1	then	then	ADV
ejpam-237	166	2	if	if	SCONJ
ejpam-237	166	3	a	a	PRON
ejpam-237	166	4	is	be	AUX
ejpam-237	166	5	open	open	ADJ
ejpam-237	166	6	,	,	PUNCT
ejpam-237	166	7	then	then	ADV
ejpam-237	166	8	(	(	PUNCT
ejpam-237	166	9	a	a	PRON
ejpam-237	166	10	,	,	PUNCT
ejpam-237	166	11	τ|a	τ|a	NUM
ejpam-237	166	12	,	,	PUNCT
ejpam-237	166	13	i|a	i|a	PUNCT
ejpam-237	166	14	)	)	PUNCT
ejpam-237	166	15	semi	semi	ADJ
ejpam-237	166	16	-	-	ADJ
ejpam-237	166	17	i	i	NOUN
ejpam-237	166	18	-	-	PUNCT
ejpam-237	166	19	hausdorff	hausdorff	NOUN
ejpam-237	166	20	.	.	PUNCT
ejpam-237	167	1	proof	proof	NOUN
ejpam-237	167	2	.	.	PUNCT
ejpam-237	168	1	this	this	PRON
ejpam-237	168	2	follows	follow	VERB
ejpam-237	168	3	from	from	ADP
ejpam-237	168	4	lemma	lemma	PROPN
ejpam-237	168	5	2	2	NUM
ejpam-237	168	6	4	4	NUM
ejpam-237	168	7	.	.	PUNCT
ejpam-237	168	8	semi	semi	ADJ
ejpam-237	168	9	-	-	ADJ
ejpam-237	168	10	i	i	NOUN
ejpam-237	168	11	-	-	PUNCT
ejpam-237	168	12	irresolute	irresolute	ADJ
ejpam-237	168	13	functions	function	NOUN
ejpam-237	168	14	in	in	ADP
ejpam-237	168	15	this	this	DET
ejpam-237	168	16	section	section	NOUN
ejpam-237	168	17	,	,	PUNCT
ejpam-237	168	18	we	we	PRON
ejpam-237	168	19	investigate	investigate	VERB
ejpam-237	168	20	some	some	DET
ejpam-237	168	21	properties	property	NOUN
ejpam-237	168	22	of	of	ADP
ejpam-237	168	23	semi	semi	ADJ
ejpam-237	168	24	-	-	ADJ
ejpam-237	168	25	i	i	NOUN
ejpam-237	168	26	-	-	PUNCT
ejpam-237	168	27	irresolute	irresolute	ADJ
ejpam-237	168	28	functions	function	NOUN
ejpam-237	168	29	.	.	PUNCT
ejpam-237	169	1	first	first	ADV
ejpam-237	169	2	,	,	PUNCT
ejpam-237	169	3	we	we	PRON
ejpam-237	169	4	shall	shall	AUX
ejpam-237	169	5	recall	recall	VERB
ejpam-237	169	6	some	some	DET
ejpam-237	169	7	definition	definition	NOUN
ejpam-237	169	8	of	of	ADP
ejpam-237	169	9	functions	function	NOUN
ejpam-237	169	10	.	.	PUNCT
ejpam-237	170	1	definition	definition	NOUN
ejpam-237	170	2	4.1	4.1	NUM
ejpam-237	170	3	.	.	PUNCT
ejpam-237	171	1	a	a	DET
ejpam-237	171	2	function	function	NOUN
ejpam-237	171	3	f	f	NOUN
ejpam-237	171	4	:	:	PUNCT
ejpam-237	171	5	(	(	PUNCT
ejpam-237	171	6	x	x	X
ejpam-237	171	7	,	,	PUNCT
ejpam-237	171	8	τ	τ	PROPN
ejpam-237	171	9	,	,	PUNCT
ejpam-237	171	10	i	i	NOUN
ejpam-237	171	11	)	)	PUNCT
ejpam-237	171	12	−→	−→	NOUN
ejpam-237	171	13	(	(	PUNCT
ejpam-237	171	14	y	y	PROPN
ejpam-237	171	15	,	,	PUNCT
ejpam-237	171	16	σ	σ	PROPN
ejpam-237	171	17	,	,	PUNCT
ejpam-237	171	18	j	j	PROPN
ejpam-237	171	19	)	)	PUNCT
ejpam-237	171	20	is	be	AUX
ejpam-237	171	21	said	say	VERB
ejpam-237	171	22	to	to	PART
ejpam-237	171	23	be	be	AUX
ejpam-237	171	24	1	1	NUM
ejpam-237	171	25	.	.	PUNCT
ejpam-237	171	26	semi	semi	ADJ
ejpam-237	171	27	-	-	ADJ
ejpam-237	171	28	i	i	PRON
ejpam-237	171	29	-	-	ADJ
ejpam-237	171	30	continuous	continuous	ADJ
ejpam-237	171	31	[	[	X
ejpam-237	171	32	5	5	NUM
ejpam-237	171	33	]	]	X
ejpam-237	171	34	if	if	SCONJ
ejpam-237	171	35	for	for	ADP
ejpam-237	171	36	every	every	DET
ejpam-237	171	37	v	v	PROPN
ejpam-237	171	38	∈	∈	PROPN
ejpam-237	171	39	σ	σ	PROPN
ejpam-237	171	40	,	,	PUNCT
ejpam-237	171	41	f	f	PROPN
ejpam-237	171	42	−1(v	−1(v	PROPN
ejpam-237	171	43	)	)	PUNCT
ejpam-237	171	44	is	be	AUX
ejpam-237	171	45	semi	semi	ADJ
ejpam-237	171	46	-	-	ADJ
ejpam-237	171	47	i	i	PRON
ejpam-237	171	48	-	-	PUNCT
ejpam-237	171	49	open	open	ADJ
ejpam-237	171	50	set	set	NOUN
ejpam-237	171	51	,	,	PUNCT
ejpam-237	171	52	2	2	NUM
ejpam-237	171	53	.	.	PUNCT
ejpam-237	171	54	semi	semi	ADJ
ejpam-237	171	55	-	-	ADJ
ejpam-237	171	56	i	i	NOUN
ejpam-237	171	57	-	-	NOUN
ejpam-237	171	58	irresolute	irresolute	ADJ
ejpam-237	171	59	if	if	SCONJ
ejpam-237	171	60	for	for	ADP
ejpam-237	171	61	every	every	DET
ejpam-237	171	62	v	v	NOUN
ejpam-237	171	63	∈	∈	PROPN
ejpam-237	171	64	sjo(y	sjo(y	PROPN
ejpam-237	171	65	,	,	PUNCT
ejpam-237	171	66	σ	σ	PROPN
ejpam-237	171	67	)	)	PUNCT
ejpam-237	171	68	,	,	PUNCT
ejpam-237	172	1	f	f	PROPN
ejpam-237	172	2	−1(v	−1(v	PROPN
ejpam-237	172	3	)	)	PUNCT
ejpam-237	172	4	∈	∈	PROPN
ejpam-237	172	5	sio(x	sio(x	PROPN
ejpam-237	172	6	,	,	PUNCT
ejpam-237	172	7	τ	τ	PROPN
ejpam-237	172	8	)	)	PUNCT
ejpam-237	172	9	,	,	PUNCT
ejpam-237	172	10	3	3	X
ejpam-237	172	11	.	.	X
ejpam-237	172	12	irresolute	irresolute	PROPN
ejpam-237	173	1	[	[	X
ejpam-237	173	2	3	3	X
ejpam-237	173	3	]	]	X
ejpam-237	173	4	if	if	SCONJ
ejpam-237	173	5	for	for	ADP
ejpam-237	173	6	every	every	DET
ejpam-237	173	7	v	v	NOUN
ejpam-237	173	8	∈	∈	PROPN
ejpam-237	173	9	so(y	so(y	NOUN
ejpam-237	173	10	,	,	PUNCT
ejpam-237	173	11	σ	σ	PROPN
ejpam-237	173	12	)	)	PUNCT
ejpam-237	173	13	,	,	PUNCT
ejpam-237	173	14	f	f	PROPN
ejpam-237	173	15	−1(v	−1(v	PROPN
ejpam-237	173	16	)	)	PUNCT
ejpam-237	173	17	∈	∈	PROPN
ejpam-237	173	18	so(x	so(x	NOUN
ejpam-237	173	19	,	,	PUNCT
ejpam-237	173	20	τ	τ	PROPN
ejpam-237	173	21	)	)	PUNCT
ejpam-237	173	22	.	.	PUNCT
ejpam-237	174	1	remark	remark	VERB
ejpam-237	174	2	4.1	4.1	NUM
ejpam-237	174	3	.	.	PUNCT
ejpam-237	175	1	in	in	ADP
ejpam-237	175	2	[	[	X
ejpam-237	175	3	6	6	NUM
ejpam-237	175	4	]	]	PUNCT
ejpam-237	175	5	,	,	PUNCT
ejpam-237	175	6	the	the	DET
ejpam-237	175	7	present	present	ADJ
ejpam-237	175	8	authors	author	NOUN
ejpam-237	175	9	called	call	VERB
ejpam-237	175	10	semi	semi	ADJ
ejpam-237	175	11	-	-	ADJ
ejpam-237	175	12	i	i	NOUN
ejpam-237	175	13	-	-	PUNCT
ejpam-237	175	14	irresolute	irresolute	ADJ
ejpam-237	175	15	functions	function	NOUN
ejpam-237	175	16	i	i	PRON
ejpam-237	175	17	-	-	PUNCT
ejpam-237	175	18	irresolute	irresolute	PROPN
ejpam-237	175	19	.	.	PUNCT
ejpam-237	176	1	however	however	ADV
ejpam-237	176	2	,	,	PUNCT
ejpam-237	176	3	dontchev	dontchev	ADJ
ejpam-237	176	4	[	[	X
ejpam-237	176	5	4	4	X
ejpam-237	176	6	]	]	PUNCT
ejpam-237	176	7	defined	define	VERB
ejpam-237	176	8	a	a	DET
ejpam-237	176	9	function	function	NOUN
ejpam-237	176	10	f	f	NOUN
ejpam-237	176	11	:	:	PUNCT
ejpam-237	176	12	(	(	PUNCT
ejpam-237	176	13	x	x	X
ejpam-237	176	14	,	,	PUNCT
ejpam-237	176	15	τ	τ	PROPN
ejpam-237	176	16	,	,	PUNCT
ejpam-237	176	17	i	i	NOUN
ejpam-237	176	18	)	)	PUNCT
ejpam-237	176	19	−→	−→	NOUN
ejpam-237	176	20	(	(	PUNCT
ejpam-237	176	21	y	y	PROPN
ejpam-237	176	22	,	,	PUNCT
ejpam-237	176	23	σ	σ	PROPN
ejpam-237	176	24	,	,	PUNCT
ejpam-237	176	25	j	j	PROPN
ejpam-237	176	26	)	)	PUNCT
ejpam-237	176	27	to	to	PART
ejpam-237	176	28	be	be	AUX
ejpam-237	176	29	i	i	NOUN
ejpam-237	176	30	-	-	NOUN
ejpam-237	176	31	irresolute	irresolute	ADJ
ejpam-237	176	32	if	if	SCONJ
ejpam-237	176	33	f	f	PROPN
ejpam-237	176	34	−1(v	−1(v	PROPN
ejpam-237	176	35	)	)	PUNCT
ejpam-237	176	36	is	be	AUX
ejpam-237	176	37	i	i	PRON
ejpam-237	176	38	-	-	PUNCT
ejpam-237	176	39	open	open	ADJ
ejpam-237	176	40	in	in	ADP
ejpam-237	176	41	(	(	PUNCT
ejpam-237	176	42	x	x	INTJ
ejpam-237	176	43	,	,	PUNCT
ejpam-237	176	44	τ	τ	PROPN
ejpam-237	176	45	,	,	PUNCT
ejpam-237	176	46	i	i	NOUN
ejpam-237	176	47	)	)	PUNCT
ejpam-237	176	48	for	for	ADP
ejpam-237	176	49	every	every	DET
ejpam-237	176	50	i	i	NOUN
ejpam-237	176	51	-	-	PUNCT
ejpam-237	176	52	open	open	ADJ
ejpam-237	176	53	in	in	ADP
ejpam-237	176	54	(	(	PUNCT
ejpam-237	176	55	y	y	PROPN
ejpam-237	176	56	,	,	PUNCT
ejpam-237	176	57	σ	σ	PROPN
ejpam-237	176	58	,	,	PUNCT
ejpam-237	176	59	j	j	PROPN
ejpam-237	176	60	)	)	PUNCT
ejpam-237	176	61	.	.	PUNCT
ejpam-237	177	1	theorem	theorem	VERB
ejpam-237	177	2	4.1	4.1	NUM
ejpam-237	177	3	.	.	PUNCT
ejpam-237	178	1	for	for	ADP
ejpam-237	178	2	a	a	DET
ejpam-237	178	3	function	function	NOUN
ejpam-237	178	4	f	f	NOUN
ejpam-237	178	5	:	:	PUNCT
ejpam-237	178	6	(	(	PUNCT
ejpam-237	178	7	x	x	X
ejpam-237	178	8	,	,	PUNCT
ejpam-237	178	9	τ	τ	PROPN
ejpam-237	178	10	,	,	PUNCT
ejpam-237	178	11	i	i	NOUN
ejpam-237	178	12	)	)	PUNCT
ejpam-237	178	13	−→	−→	NOUN
ejpam-237	178	14	(	(	PUNCT
ejpam-237	178	15	y	y	PROPN
ejpam-237	178	16	,	,	PUNCT
ejpam-237	178	17	σ	σ	PROPN
ejpam-237	178	18	,	,	PUNCT
ejpam-237	178	19	j	j	PROPN
ejpam-237	178	20	)	)	PUNCT
ejpam-237	178	21	,	,	PUNCT
ejpam-237	178	22	the	the	DET
ejpam-237	178	23	following	follow	VERB
ejpam-237	178	24	properties	property	NOUN
ejpam-237	178	25	are	be	AUX
ejpam-237	178	26	equivalent	equivalent	ADJ
ejpam-237	178	27	;	;	PUNCT
ejpam-237	178	28	1	1	X
ejpam-237	178	29	.	.	X
ejpam-237	178	30	f	f	PROPN
ejpam-237	178	31	is	be	AUX
ejpam-237	178	32	semi	semi	ADJ
ejpam-237	178	33	-	-	ADJ
ejpam-237	178	34	i	i	NOUN
ejpam-237	178	35	-	-	PUNCT
ejpam-237	178	36	irresolute	irresolute	ADJ
ejpam-237	178	37	,	,	PUNCT
ejpam-237	178	38	2	2	NUM
ejpam-237	178	39	.	.	PUNCT
ejpam-237	179	1	the	the	DET
ejpam-237	179	2	inverse	inverse	ADJ
ejpam-237	179	3	image	image	NOUN
ejpam-237	179	4	of	of	ADP
ejpam-237	179	5	each	each	DET
ejpam-237	179	6	semi	semi	ADJ
ejpam-237	179	7	-	-	ADJ
ejpam-237	179	8	i	i	ADV
ejpam-237	179	9	-	-	PUNCT
ejpam-237	179	10	closed	close	VERB
ejpam-237	179	11	set	set	NOUN
ejpam-237	179	12	in	in	ADP
ejpam-237	179	13	(	(	PUNCT
ejpam-237	179	14	y	y	PROPN
ejpam-237	179	15	,	,	PUNCT
ejpam-237	179	16	σ	σ	PROPN
ejpam-237	179	17	,	,	PUNCT
ejpam-237	179	18	j	j	PROPN
ejpam-237	179	19	)	)	PUNCT
ejpam-237	179	20	is	be	AUX
ejpam-237	179	21	semi	semi	ADJ
ejpam-237	179	22	-	-	ADJ
ejpam-237	179	23	i	i	PRON
ejpam-237	179	24	-	-	PUNCT
ejpam-237	179	25	closed	closed	ADJ
ejpam-237	179	26	in	in	ADP
ejpam-237	179	27	(	(	PUNCT
ejpam-237	179	28	x	x	INTJ
ejpam-237	179	29	,	,	PUNCT
ejpam-237	179	30	τ	τ	PROPN
ejpam-237	179	31	,	,	PUNCT
ejpam-237	179	32	i	i	PROPN
ejpam-237	179	33	)	)	PUNCT
ejpam-237	179	34	,	,	PUNCT
ejpam-237	179	35	3	3	X
ejpam-237	179	36	.	.	X
ejpam-237	179	37	for	for	ADP
ejpam-237	179	38	each	each	DET
ejpam-237	179	39	x	x	SYM
ejpam-237	179	40	∈	∈	PROPN
ejpam-237	179	41	x	x	X
ejpam-237	179	42	and	and	CCONJ
ejpam-237	179	43	each	each	DET
ejpam-237	179	44	v	v	ADP
ejpam-237	179	45	∈	∈	PROPN
ejpam-237	179	46	sio(y	sio(y	PROPN
ejpam-237	179	47	,	,	PUNCT
ejpam-237	179	48	σ	σ	NOUN
ejpam-237	179	49	)	)	PUNCT
ejpam-237	179	50	containing	contain	VERB
ejpam-237	179	51	f	f	PROPN
ejpam-237	179	52	(	(	PUNCT
ejpam-237	179	53	x	x	NOUN
ejpam-237	179	54	)	)	PUNCT
ejpam-237	179	55	,	,	PUNCT
ejpam-237	179	56	there	there	PRON
ejpam-237	179	57	exists	exist	VERB
ejpam-237	179	58	u	u	PROPN
ejpam-237	179	59	∈	∈	PROPN
ejpam-237	179	60	sio(x	sio(x	PROPN
ejpam-237	179	61	,	,	PUNCT
ejpam-237	179	62	τ	τ	PROPN
ejpam-237	179	63	)	)	PUNCT
ejpam-237	179	64	containing	contain	VERB
ejpam-237	179	65	x	x	PUNCT
ejpam-237	179	66	such	such	ADJ
ejpam-237	179	67	that	that	SCONJ
ejpam-237	179	68	f	f	PROPN
ejpam-237	179	69	(	(	PUNCT
ejpam-237	179	70	u)⊂	u)⊂	NOUN
ejpam-237	179	71	v.	v.	ADP
ejpam-237	179	72	proof	proof	NOUN
ejpam-237	179	73	.	.	PUNCT
ejpam-237	180	1	the	the	DET
ejpam-237	180	2	proof	proof	NOUN
ejpam-237	180	3	is	be	AUX
ejpam-237	180	4	obvious	obvious	ADJ
ejpam-237	180	5	from	from	ADP
ejpam-237	180	6	the	the	DET
ejpam-237	180	7	fact	fact	NOUN
ejpam-237	180	8	that	that	SCONJ
ejpam-237	180	9	the	the	DET
ejpam-237	180	10	arbitrary	arbitrary	ADJ
ejpam-237	180	11	union	union	NOUN
ejpam-237	180	12	of	of	ADP
ejpam-237	180	13	semi	semi	ADJ
ejpam-237	180	14	-	-	ADJ
ejpam-237	180	15	i	i	PRON
ejpam-237	180	16	-	-	PUNCT
ejpam-237	180	17	open	open	ADJ
ejpam-237	180	18	sets	set	NOUN
ejpam-237	180	19	is	be	AUX
ejpam-237	180	20	semi	semi	ADJ
ejpam-237	180	21	-	-	ADJ
ejpam-237	180	22	i	i	PRON
ejpam-237	180	23	-	-	PUNCT
ejpam-237	180	24	open	open	ADJ
ejpam-237	180	25	[	[	X
ejpam-237	180	26	6	6	NUM
ejpam-237	180	27	,	,	PUNCT
ejpam-237	180	28	theorem	theorem	VERB
ejpam-237	180	29	3.4	3.4	NUM
ejpam-237	180	30	]	]	PUNCT
ejpam-237	180	31	.	.	PUNCT
ejpam-237	181	1	e.	e.	PROPN
ejpam-237	181	2	hatir	hatir	PROPN
ejpam-237	181	3	and	and	CCONJ
ejpam-237	181	4	t.	t.	PROPN
ejpam-237	181	5	noiri	noiri	PROPN
ejpam-237	181	6	/	/	SYM
ejpam-237	181	7	eur	eur	PROPN
ejpam-237	181	8	.	.	PUNCT
ejpam-237	182	1	j.	j.	PROPN
ejpam-237	182	2	pure	pure	PROPN
ejpam-237	182	3	appl	appl	PROPN
ejpam-237	182	4	.	.	PROPN
ejpam-237	182	5	math	math	PROPN
ejpam-237	182	6	,	,	PUNCT
ejpam-237	182	7	2	2	NUM
ejpam-237	182	8	(	(	PUNCT
ejpam-237	182	9	2009	2009	NUM
ejpam-237	182	10	)	)	PUNCT
ejpam-237	182	11	,	,	PUNCT
ejpam-237	182	12	(	(	PUNCT
ejpam-237	182	13	172	172	NUM
ejpam-237	182	14	-	-	SYM
ejpam-237	182	15	181	181	NUM
ejpam-237	182	16	)	)	PUNCT
ejpam-237	182	17	179	179	NUM
ejpam-237	182	18	remark	remark	NOUN
ejpam-237	182	19	4.2	4.2	NUM
ejpam-237	182	20	.	.	PUNCT
ejpam-237	183	1	irresolute	irresolute	ADJ
ejpam-237	183	2	functions	function	NOUN
ejpam-237	183	3	are	be	AUX
ejpam-237	183	4	not	not	PART
ejpam-237	183	5	in	in	ADP
ejpam-237	183	6	general	general	ADJ
ejpam-237	183	7	semi	semi	ADJ
ejpam-237	183	8	-	-	ADJ
ejpam-237	183	9	i	i	NOUN
ejpam-237	183	10	-	-	PUNCT
ejpam-237	183	11	irresolute	irresolute	ADJ
ejpam-237	183	12	as	as	SCONJ
ejpam-237	183	13	shown	show	VERB
ejpam-237	183	14	by	by	ADP
ejpam-237	183	15	the	the	DET
ejpam-237	183	16	following	follow	VERB
ejpam-237	183	17	example	example	NOUN
ejpam-237	183	18	.	.	PUNCT
ejpam-237	184	1	example	example	NOUN
ejpam-237	184	2	4.1	4.1	NUM
ejpam-237	184	3	.	.	PUNCT
ejpam-237	185	1	let	let	VERB
ejpam-237	185	2	x	x	PUNCT
ejpam-237	185	3	=	=	PRON
ejpam-237	185	4	{	{	PUNCT
ejpam-237	185	5	a	a	PRON
ejpam-237	185	6	,	,	PUNCT
ejpam-237	185	7	b	b	NOUN
ejpam-237	185	8	,	,	PUNCT
ejpam-237	185	9	c	c	NOUN
ejpam-237	185	10	}	}	PUNCT
ejpam-237	185	11	,	,	PUNCT
ejpam-237	185	12	τ=	τ=	X
ejpam-237	185	13	{	{	PUNCT
ejpam-237	185	14	∅	∅	NOUN
ejpam-237	185	15	,	,	PUNCT
ejpam-237	185	16	x	x	INTJ
ejpam-237	185	17	,	,	PUNCT
ejpam-237	185	18	{	{	PUNCT
ejpam-237	185	19	a	a	X
ejpam-237	185	20	}	}	PUNCT
ejpam-237	185	21	}	}	PUNCT
ejpam-237	185	22	and	and	CCONJ
ejpam-237	185	23	i	i	PRON
ejpam-237	185	24	=	=	PUNCT
ejpam-237	185	25	{	{	PUNCT
ejpam-237	185	26	∅	∅	NOUN
ejpam-237	185	27	,	,	PUNCT
ejpam-237	185	28	{	{	PUNCT
ejpam-237	185	29	a	a	X
ejpam-237	185	30	}	}	PUNCT
ejpam-237	185	31	}	}	PUNCT
ejpam-237	185	32	and	and	CCONJ
ejpam-237	185	33	j	j	PROPN
ejpam-237	185	34	=	=	SYM
ejpam-237	185	35	{	{	PUNCT
ejpam-237	185	36	∅	∅	NOUN
ejpam-237	185	37	}	}	PUNCT
ejpam-237	185	38	.	.	PUNCT
ejpam-237	186	1	then	then	ADV
ejpam-237	186	2	the	the	DET
ejpam-237	186	3	identity	identity	NOUN
ejpam-237	186	4	function	function	NOUN
ejpam-237	186	5	f	f	NOUN
ejpam-237	186	6	:	:	PUNCT
ejpam-237	186	7	(	(	PUNCT
ejpam-237	186	8	x	x	X
ejpam-237	186	9	,	,	PUNCT
ejpam-237	186	10	τ	τ	PROPN
ejpam-237	186	11	,	,	PUNCT
ejpam-237	186	12	i	i	NOUN
ejpam-237	186	13	)	)	PUNCT
ejpam-237	186	14	−→	−→	NOUN
ejpam-237	186	15	(	(	PUNCT
ejpam-237	186	16	x	x	PROPN
ejpam-237	186	17	,	,	PUNCT
ejpam-237	186	18	σ	σ	PROPN
ejpam-237	186	19	,	,	PUNCT
ejpam-237	186	20	j	j	PROPN
ejpam-237	186	21	)	)	PUNCT
ejpam-237	186	22	is	be	AUX
ejpam-237	186	23	irresolute	irresolute	ADJ
ejpam-237	186	24	,	,	PUNCT
ejpam-237	186	25	but	but	CCONJ
ejpam-237	186	26	it	it	PRON
ejpam-237	186	27	is	be	AUX
ejpam-237	186	28	not	not	PART
ejpam-237	186	29	semi	semi	ADJ
ejpam-237	186	30	-	-	ADJ
ejpam-237	186	31	i	i	NOUN
ejpam-237	186	32	-	-	PUNCT
ejpam-237	186	33	irresolute	irresolute	ADJ
ejpam-237	186	34	since	since	SCONJ
ejpam-237	186	35	{	{	PUNCT
ejpam-237	186	36	a	a	PRON
ejpam-237	186	37	,	,	PUNCT
ejpam-237	186	38	c	c	NOUN
ejpam-237	186	39	}	}	PUNCT
ejpam-237	186	40	∈	∈	PROPN
ejpam-237	186	41	sio(x	sio(x	PROPN
ejpam-237	186	42	,	,	PUNCT
ejpam-237	186	43	σ	σ	PROPN
ejpam-237	186	44	,	,	PUNCT
ejpam-237	186	45	j	j	PROPN
ejpam-237	186	46	)	)	PUNCT
ejpam-237	186	47	and	and	CCONJ
ejpam-237	186	48	f	f	PROPN
ejpam-237	186	49	−1({a	−1({a	PROPN
ejpam-237	186	50	,	,	PUNCT
ejpam-237	186	51	c	c	NOUN
ejpam-237	186	52	}	}	PUNCT
ejpam-237	186	53	)	)	PUNCT
ejpam-237	187	1	=	=	PRON
ejpam-237	187	2	{	{	PUNCT
ejpam-237	187	3	a	a	X
ejpam-237	187	4	,	,	PUNCT
ejpam-237	187	5	c	c	NOUN
ejpam-237	187	6	}	}	PUNCT
ejpam-237	187	7	/∈	/∈	PUNCT
ejpam-237	188	1	sio(x	sio(x	NOUN
ejpam-237	188	2	,	,	PUNCT
ejpam-237	188	3	τ	τ	PROPN
ejpam-237	188	4	,	,	PUNCT
ejpam-237	188	5	i	i	PROPN
ejpam-237	188	6	)	)	PUNCT
ejpam-237	188	7	.	.	PUNCT
ejpam-237	189	1	theorem	theorem	VERB
ejpam-237	189	2	4.2	4.2	NUM
ejpam-237	189	3	.	.	PUNCT
ejpam-237	190	1	let	let	VERB
ejpam-237	190	2	f	f	NOUN
ejpam-237	190	3	:	:	PUNCT
ejpam-237	190	4	(	(	PUNCT
ejpam-237	190	5	x	x	X
ejpam-237	190	6	,	,	PUNCT
ejpam-237	190	7	τ	τ	PROPN
ejpam-237	190	8	,	,	PUNCT
ejpam-237	190	9	i	i	NOUN
ejpam-237	190	10	)	)	PUNCT
ejpam-237	190	11	−→	−→	NOUN
ejpam-237	190	12	(	(	PUNCT
ejpam-237	190	13	y	y	PROPN
ejpam-237	190	14	,	,	PUNCT
ejpam-237	190	15	σ	σ	PROPN
ejpam-237	190	16	,	,	PUNCT
ejpam-237	190	17	j	j	PROPN
ejpam-237	190	18	)	)	PUNCT
ejpam-237	190	19	be	be	VERB
ejpam-237	190	20	a	a	DET
ejpam-237	190	21	function	function	NOUN
ejpam-237	190	22	,	,	PUNCT
ejpam-237	190	23	where	where	SCONJ
ejpam-237	190	24	i	i	PRON
ejpam-237	190	25	and	and	CCONJ
ejpam-237	190	26	j	j	PROPN
ejpam-237	190	27	are	be	AUX
ejpam-237	190	28	ideals	ideal	NOUN
ejpam-237	190	29	on	on	ADP
ejpam-237	190	30	y	y	PROPN
ejpam-237	190	31	,	,	PUNCT
ejpam-237	190	32	respectively	respectively	ADV
ejpam-237	190	33	.	.	PUNCT
ejpam-237	191	1	if	if	SCONJ
ejpam-237	191	2	i	i	PRON
ejpam-237	191	3	=	=	SYM
ejpam-237	191	4	j	j	PROPN
ejpam-237	191	5	=	=	SYM
ejpam-237	191	6	{	{	PUNCT
ejpam-237	191	7	∅	∅	NOUN
ejpam-237	191	8	}	}	PUNCT
ejpam-237	191	9	or	or	CCONJ
ejpam-237	191	10	in	in	ADP
ejpam-237	191	11	,	,	PUNCT
ejpam-237	191	12	then	then	ADV
ejpam-237	191	13	semi	semi	ADJ
ejpam-237	191	14	-	-	ADJ
ejpam-237	191	15	i	i	NOUN
ejpam-237	191	16	-	-	PUNCT
ejpam-237	191	17	irresoluteness	irresoluteness	PROPN
ejpam-237	191	18	and	and	CCONJ
ejpam-237	191	19	irresoluteness	irresoluteness	NOUN
ejpam-237	191	20	are	be	AUX
ejpam-237	191	21	equivalent	equivalent	ADJ
ejpam-237	191	22	.	.	PUNCT
ejpam-237	192	1	proof	proof	NOUN
ejpam-237	192	2	.	.	PUNCT
ejpam-237	193	1	this	this	PRON
ejpam-237	193	2	follows	follow	VERB
ejpam-237	193	3	from	from	ADP
ejpam-237	193	4	the	the	DET
ejpam-237	193	5	proofs	proof	NOUN
ejpam-237	193	6	of	of	ADP
ejpam-237	193	7	theorems	theorem	NOUN
ejpam-237	193	8	2(1	2(1	NUM
ejpam-237	193	9	)	)	PUNCT
ejpam-237	193	10	and	and	CCONJ
ejpam-237	193	11	3(1	3(1	NUM
ejpam-237	193	12	)	)	PUNCT
ejpam-237	193	13	.	.	PUNCT
ejpam-237	194	1	theorem	theorem	VERB
ejpam-237	194	2	4.3	4.3	NUM
ejpam-237	194	3	.	.	PUNCT
ejpam-237	195	1	let	let	VERB
ejpam-237	195	2	f	f	PRON
ejpam-237	195	3	be	be	AUX
ejpam-237	195	4	a	a	DET
ejpam-237	195	5	semi	semi	ADJ
ejpam-237	195	6	-	-	ADJ
ejpam-237	195	7	i	i	NOUN
ejpam-237	195	8	-	-	PUNCT
ejpam-237	195	9	irresolute	irresolute	ADJ
ejpam-237	195	10	injection	injection	NOUN
ejpam-237	195	11	from	from	ADP
ejpam-237	195	12	a	a	DET
ejpam-237	195	13	space	space	NOUN
ejpam-237	195	14	(	(	PUNCT
ejpam-237	195	15	x	x	X
ejpam-237	195	16	,	,	PUNCT
ejpam-237	195	17	τ	τ	PROPN
ejpam-237	195	18	,	,	PUNCT
ejpam-237	195	19	i	i	NOUN
ejpam-237	195	20	)	)	PUNCT
ejpam-237	195	21	into	into	ADP
ejpam-237	195	22	a	a	DET
ejpam-237	195	23	space	space	NOUN
ejpam-237	195	24	(	(	PUNCT
ejpam-237	195	25	y	y	PROPN
ejpam-237	195	26	,	,	PUNCT
ejpam-237	195	27	σ	σ	PROPN
ejpam-237	195	28	,	,	PUNCT
ejpam-237	195	29	j	j	PROPN
ejpam-237	195	30	)	)	PUNCT
ejpam-237	195	31	.	.	PUNCT
ejpam-237	196	1	if	if	SCONJ
ejpam-237	196	2	y	y	PROPN
ejpam-237	196	3	is	be	AUX
ejpam-237	196	4	semi	semi	ADJ
ejpam-237	196	5	-	-	ADJ
ejpam-237	196	6	i	i	NOUN
ejpam-237	196	7	-	-	PUNCT
ejpam-237	196	8	hausdorff	hausdorff	NOUN
ejpam-237	196	9	,	,	PUNCT
ejpam-237	196	10	then	then	ADV
ejpam-237	196	11	x	x	PUNCT
ejpam-237	196	12	is	be	AUX
ejpam-237	196	13	also	also	ADV
ejpam-237	196	14	semi	semi	ADJ
ejpam-237	196	15	-	-	ADJ
ejpam-237	196	16	i	i	NOUN
ejpam-237	196	17	-	-	PUNCT
ejpam-237	196	18	hausdorff	hausdorff	NOUN
ejpam-237	196	19	.	.	PUNCT
ejpam-237	197	1	proof	proof	NOUN
ejpam-237	197	2	.	.	PUNCT
ejpam-237	198	1	let	let	VERB
ejpam-237	198	2	x	x	PRON
ejpam-237	198	3	,	,	PUNCT
ejpam-237	198	4	y	y	PROPN
ejpam-237	198	5	∈	∈	PROPN
ejpam-237	198	6	x	x	X
ejpam-237	198	7	and	and	CCONJ
ejpam-237	198	8	x	x	SYM
ejpam-237	198	9	6=	6=	ADP
ejpam-237	198	10	y.	y.	NOUN
ejpam-237	198	11	then	then	ADV
ejpam-237	198	12	f	f	PROPN
ejpam-237	198	13	(	(	PUNCT
ejpam-237	198	14	x	x	X
ejpam-237	198	15	)	)	PUNCT
ejpam-237	198	16	6=	6=	ADP
ejpam-237	199	1	f	f	PROPN
ejpam-237	199	2	(	(	PUNCT
ejpam-237	199	3	y	y	NOUN
ejpam-237	199	4	)	)	PUNCT
ejpam-237	199	5	thus	thus	ADV
ejpam-237	199	6	f	f	X
ejpam-237	199	7	(	(	PUNCT
ejpam-237	199	8	x	x	NOUN
ejpam-237	199	9	)	)	PUNCT
ejpam-237	199	10	and	and	CCONJ
ejpam-237	199	11	f	f	PROPN
ejpam-237	199	12	(	(	PUNCT
ejpam-237	199	13	y	y	NOUN
ejpam-237	199	14	)	)	PUNCT
ejpam-237	199	15	are	be	AUX
ejpam-237	199	16	semi	semi	ADJ
ejpam-237	199	17	-	-	ADJ
ejpam-237	199	18	iseparated	iseparated	ADJ
ejpam-237	199	19	in	in	ADP
ejpam-237	199	20	y	y	PROPN
ejpam-237	199	21	by	by	ADP
ejpam-237	199	22	semi	semi	ADJ
ejpam-237	199	23	-	-	ADJ
ejpam-237	199	24	i	i	PRON
ejpam-237	199	25	-	-	PUNCT
ejpam-237	199	26	open	open	ADJ
ejpam-237	199	27	sets	set	VERB
ejpam-237	199	28	u	u	NOUN
ejpam-237	199	29	and	and	CCONJ
ejpam-237	199	30	v	v	NOUN
ejpam-237	199	31	,	,	PUNCT
ejpam-237	199	32	respectively	respectively	ADV
ejpam-237	199	33	.	.	PUNCT
ejpam-237	200	1	since	since	SCONJ
ejpam-237	200	2	f	f	PROPN
ejpam-237	200	3	is	be	AUX
ejpam-237	200	4	semi	semi	ADJ
ejpam-237	200	5	-	-	ADJ
ejpam-237	200	6	i	i	NOUN
ejpam-237	200	7	-	-	PUNCT
ejpam-237	200	8	irresolute	irresolute	PROPN
ejpam-237	200	9	,	,	PUNCT
ejpam-237	200	10	f	f	PROPN
ejpam-237	200	11	−1(u	−1(u	NOUN
ejpam-237	200	12	)	)	PUNCT
ejpam-237	200	13	and	and	CCONJ
ejpam-237	200	14	f	f	PROPN
ejpam-237	200	15	−1(v	−1(v	PROPN
ejpam-237	200	16	)	)	PUNCT
ejpam-237	200	17	are	be	AUX
ejpam-237	200	18	disjoint	disjoint	ADJ
ejpam-237	200	19	semi	semi	ADJ
ejpam-237	200	20	-	-	ADJ
ejpam-237	200	21	i	i	PRON
ejpam-237	200	22	-	-	PUNCT
ejpam-237	200	23	open	open	ADJ
ejpam-237	200	24	sets	set	NOUN
ejpam-237	200	25	containing	contain	VERB
ejpam-237	200	26	x	x	PROPN
ejpam-237	200	27	and	and	CCONJ
ejpam-237	200	28	y	y	PROPN
ejpam-237	200	29	,	,	PUNCT
ejpam-237	200	30	respectively	respectively	ADV
ejpam-237	200	31	.	.	PUNCT
ejpam-237	201	1	this	this	PRON
ejpam-237	201	2	shows	show	VERB
ejpam-237	201	3	that	that	SCONJ
ejpam-237	201	4	x	x	PRON
ejpam-237	201	5	is	be	AUX
ejpam-237	201	6	semi	semi	ADJ
ejpam-237	201	7	-	-	ADJ
ejpam-237	201	8	i	i	NOUN
ejpam-237	201	9	-	-	PUNCT
ejpam-237	201	10	hausdorff	hausdorff	NOUN
ejpam-237	201	11	.	.	PUNCT
ejpam-237	202	1	theorem	theorem	PROPN
ejpam-237	202	2	4.4	4.4	NUM
ejpam-237	202	3	.	.	PUNCT
ejpam-237	203	1	let	let	VERB
ejpam-237	203	2	(	(	PUNCT
ejpam-237	203	3	x	x	X
ejpam-237	203	4	,	,	PUNCT
ejpam-237	203	5	τ	τ	PROPN
ejpam-237	203	6	,	,	PUNCT
ejpam-237	203	7	i	i	PRON
ejpam-237	203	8	)	)	PUNCT
ejpam-237	203	9	be	be	VERB
ejpam-237	203	10	an	an	DET
ejpam-237	203	11	ideal	ideal	ADJ
ejpam-237	203	12	topological	topological	ADJ
ejpam-237	203	13	space	space	NOUN
ejpam-237	203	14	with	with	ADP
ejpam-237	203	15	the	the	DET
ejpam-237	203	16	following	follow	VERB
ejpam-237	203	17	property	property	NOUN
ejpam-237	203	18	;	;	PUNCT
ejpam-237	204	1	if	if	SCONJ
ejpam-237	204	2	x	x	PROPN
ejpam-237	204	3	6=	6=	PROPN
ejpam-237	204	4	y	y	PROPN
ejpam-237	204	5	,	,	PUNCT
ejpam-237	204	6	where	where	SCONJ
ejpam-237	204	7	x	x	X
ejpam-237	204	8	,	,	PUNCT
ejpam-237	204	9	y	y	PROPN
ejpam-237	204	10	∈	∈	PROPN
ejpam-237	204	11	x	x	INTJ
ejpam-237	204	12	,	,	PUNCT
ejpam-237	204	13	then	then	ADV
ejpam-237	204	14	there	there	PRON
ejpam-237	204	15	exist	exist	VERB
ejpam-237	204	16	a	a	DET
ejpam-237	204	17	hausdorff	hausdorff	NOUN
ejpam-237	204	18	space	space	NOUN
ejpam-237	204	19	(	(	PUNCT
ejpam-237	204	20	y	y	PROPN
ejpam-237	204	21	,	,	PUNCT
ejpam-237	204	22	σ	σ	PROPN
ejpam-237	204	23	)	)	PUNCT
ejpam-237	204	24	and	and	CCONJ
ejpam-237	204	25	a	a	DET
ejpam-237	204	26	semii	semii	NOUN
ejpam-237	204	27	-	-	PUNCT
ejpam-237	204	28	continuous	continuous	ADJ
ejpam-237	204	29	function	function	NOUN
ejpam-237	204	30	f	f	NOUN
ejpam-237	204	31	:	:	PUNCT
ejpam-237	204	32	(	(	PUNCT
ejpam-237	204	33	x	x	X
ejpam-237	204	34	,	,	PUNCT
ejpam-237	204	35	τ	τ	PROPN
ejpam-237	204	36	,	,	PUNCT
ejpam-237	204	37	i	i	NOUN
ejpam-237	204	38	)	)	PUNCT
ejpam-237	204	39	−→	−→	NOUN
ejpam-237	204	40	(	(	PUNCT
ejpam-237	204	41	y	y	PROPN
ejpam-237	204	42	,	,	PUNCT
ejpam-237	204	43	σ	σ	PROPN
ejpam-237	204	44	)	)	PUNCT
ejpam-237	204	45	such	such	ADJ
ejpam-237	204	46	that	that	SCONJ
ejpam-237	204	47	f	f	PROPN
ejpam-237	204	48	(	(	PUNCT
ejpam-237	204	49	x	x	X
ejpam-237	204	50	)	)	PUNCT
ejpam-237	204	51	6=	6=	ADP
ejpam-237	205	1	f	f	PROPN
ejpam-237	205	2	(	(	PUNCT
ejpam-237	205	3	y	y	PROPN
ejpam-237	205	4	)	)	PUNCT
ejpam-237	205	5	.	.	PUNCT
ejpam-237	206	1	then	then	ADV
ejpam-237	206	2	x	x	X
ejpam-237	206	3	is	be	AUX
ejpam-237	206	4	semi	semi	ADJ
ejpam-237	206	5	-	-	ADJ
ejpam-237	206	6	i	i	NOUN
ejpam-237	206	7	-	-	PUNCT
ejpam-237	206	8	hausdorff	hausdorff	NOUN
ejpam-237	206	9	.	.	PUNCT
ejpam-237	207	1	proof	proof	NOUN
ejpam-237	207	2	.	.	PUNCT
ejpam-237	208	1	the	the	DET
ejpam-237	208	2	proof	proof	NOUN
ejpam-237	208	3	is	be	AUX
ejpam-237	208	4	straightforward	straightforward	ADJ
ejpam-237	208	5	.	.	PUNCT
ejpam-237	209	1	theorem	theorem	VERB
ejpam-237	209	2	4.5	4.5	NUM
ejpam-237	209	3	.	.	PUNCT
ejpam-237	210	1	let	let	VERB
ejpam-237	210	2	f	f	NOUN
ejpam-237	210	3	:	:	PUNCT
ejpam-237	210	4	(	(	PUNCT
ejpam-237	210	5	x	x	X
ejpam-237	210	6	,	,	PUNCT
ejpam-237	210	7	τ	τ	PROPN
ejpam-237	210	8	,	,	PUNCT
ejpam-237	210	9	i	i	NOUN
ejpam-237	210	10	)	)	PUNCT
ejpam-237	210	11	−→	−→	NOUN
ejpam-237	210	12	(	(	PUNCT
ejpam-237	210	13	y	y	PROPN
ejpam-237	210	14	,	,	PUNCT
ejpam-237	210	15	σ	σ	PROPN
ejpam-237	210	16	,	,	PUNCT
ejpam-237	210	17	j	j	PROPN
ejpam-237	210	18	)	)	PUNCT
ejpam-237	210	19	be	be	VERB
ejpam-237	210	20	a	a	DET
ejpam-237	210	21	function	function	NOUN
ejpam-237	210	22	and	and	CCONJ
ejpam-237	210	23	v	v	ADP
ejpam-237	210	24	∈	∈	PROPN
ejpam-237	210	25	σ	σ	PROPN
ejpam-237	210	26	.	.	PUNCT
ejpam-237	211	1	then	then	ADV
ejpam-237	211	2	e.	e.	PROPN
ejpam-237	211	3	hatir	hatir	PROPN
ejpam-237	211	4	and	and	CCONJ
ejpam-237	211	5	t.	t.	PROPN
ejpam-237	211	6	noiri	noiri	PROPN
ejpam-237	211	7	/	/	SYM
ejpam-237	211	8	eur	eur	PROPN
ejpam-237	211	9	.	.	PUNCT
ejpam-237	212	1	j.	j.	PROPN
ejpam-237	212	2	pure	pure	PROPN
ejpam-237	212	3	appl	appl	PROPN
ejpam-237	212	4	.	.	PROPN
ejpam-237	212	5	math	math	PROPN
ejpam-237	212	6	,	,	PUNCT
ejpam-237	212	7	2	2	NUM
ejpam-237	212	8	(	(	PUNCT
ejpam-237	212	9	2009	2009	NUM
ejpam-237	212	10	)	)	PUNCT
ejpam-237	212	11	,	,	PUNCT
ejpam-237	212	12	(	(	PUNCT
ejpam-237	212	13	172	172	NUM
ejpam-237	212	14	-	-	SYM
ejpam-237	212	15	181	181	NUM
ejpam-237	212	16	)	)	PUNCT
ejpam-237	212	17	180	180	NUM
ejpam-237	212	18	f	f	NOUN
ejpam-237	212	19	−1(v	−1(v	PROPN
ejpam-237	212	20	∗)⊂	∗)⊂	PROPN
ejpam-237	212	21	(	(	PUNCT
ejpam-237	212	22	f	f	PROPN
ejpam-237	212	23	−1(v	−1(v	PROPN
ejpam-237	212	24	)	)	PUNCT
ejpam-237	212	25	)	)	PUNCT
ejpam-237	212	26	∗	∗	NOUN
ejpam-237	212	27	implies	imply	VERB
ejpam-237	212	28	f	f	X
ejpam-237	212	29	−1(cl∗(v	−1(cl∗(v	PROPN
ejpam-237	212	30	)	)	PUNCT
ejpam-237	212	31	)	)	PUNCT
ejpam-237	213	1	⊂	⊂	PROPN
ejpam-237	213	2	cl∗	cl∗	PROPN
ejpam-237	213	3	(	(	PUNCT
ejpam-237	213	4	f	f	PROPN
ejpam-237	213	5	−1(v	−1(v	PROPN
ejpam-237	213	6	)	)	PUNCT
ejpam-237	213	7	)	)	PUNCT
ejpam-237	213	8	.	.	PUNCT
ejpam-237	214	1	proof	proof	NOUN
ejpam-237	214	2	.	.	PUNCT
ejpam-237	215	1	f	f	PROPN
ejpam-237	215	2	−1(cl∗(v	−1(cl∗(v	PROPN
ejpam-237	215	3	)	)	PUNCT
ejpam-237	215	4	)	)	PUNCT
ejpam-237	216	1	=	=	PUNCT
ejpam-237	216	2	f	f	PROPN
ejpam-237	216	3	−1(v	−1(v	NOUN
ejpam-237	216	4	∪	∪	VERB
ejpam-237	216	5	v	v	ADP
ejpam-237	216	6	∗	∗	NOUN
ejpam-237	216	7	)	)	PUNCT
ejpam-237	217	1	=	=	SYM
ejpam-237	218	1	f	f	PROPN
ejpam-237	218	2	−1(v	−1(v	NOUN
ejpam-237	218	3	)	)	PUNCT
ejpam-237	218	4	∪	∪	ADP
ejpam-237	218	5	f	f	PROPN
ejpam-237	218	6	−1(v	−1(v	PROPN
ejpam-237	218	7	∗	∗	NOUN
ejpam-237	218	8	)	)	PUNCT
ejpam-237	219	1	⊂	⊂	PROPN
ejpam-237	219	2	f	f	PROPN
ejpam-237	219	3	−1(v	−1(v	X
ejpam-237	219	4	)	)	PUNCT
ejpam-237	219	5	∪	∪	ADV
ejpam-237	219	6	(	(	PUNCT
ejpam-237	219	7	f	f	PROPN
ejpam-237	219	8	−1(v	−1(v	PROPN
ejpam-237	219	9	)	)	PUNCT
ejpam-237	219	10	)	)	PUNCT
ejpam-237	220	1	∗	∗	NOUN
ejpam-237	220	2	=	=	SYM
ejpam-237	220	3	cl∗	cl∗	PROPN
ejpam-237	220	4	(	(	PUNCT
ejpam-237	220	5	f	f	PROPN
ejpam-237	220	6	−1(v	−1(v	PROPN
ejpam-237	220	7	)	)	PUNCT
ejpam-237	220	8	)	)	PUNCT
ejpam-237	220	9	.	.	PUNCT
ejpam-237	221	1	remark	remark	VERB
ejpam-237	221	2	4.3	4.3	NUM
ejpam-237	221	3	.	.	PUNCT
ejpam-237	222	1	the	the	DET
ejpam-237	222	2	converse	converse	NOUN
ejpam-237	222	3	of	of	ADP
ejpam-237	222	4	theorem	theorem	NOUN
ejpam-237	222	5	10	10	NUM
ejpam-237	222	6	is	be	AUX
ejpam-237	222	7	false	false	ADJ
ejpam-237	222	8	as	as	SCONJ
ejpam-237	222	9	shown	show	VERB
ejpam-237	222	10	by	by	ADP
ejpam-237	222	11	the	the	DET
ejpam-237	222	12	following	follow	VERB
ejpam-237	222	13	example	example	NOUN
ejpam-237	222	14	.	.	PUNCT
ejpam-237	223	1	example	example	NOUN
ejpam-237	223	2	4.2	4.2	NUM
ejpam-237	223	3	.	.	PUNCT
ejpam-237	224	1	let	let	VERB
ejpam-237	224	2	x	x	PUNCT
ejpam-237	224	3	=	=	PRON
ejpam-237	224	4	{	{	PUNCT
ejpam-237	224	5	a	a	PRON
ejpam-237	224	6	,	,	PUNCT
ejpam-237	224	7	b	b	NOUN
ejpam-237	224	8	,	,	PUNCT
ejpam-237	224	9	c	c	NOUN
ejpam-237	224	10	}	}	PUNCT
ejpam-237	224	11	,	,	PUNCT
ejpam-237	224	12	τ=	τ=	X
ejpam-237	224	13	{	{	PUNCT
ejpam-237	224	14	∅	∅	NOUN
ejpam-237	224	15	,	,	PUNCT
ejpam-237	224	16	x	x	INTJ
ejpam-237	224	17	,	,	PUNCT
ejpam-237	224	18	{	{	PUNCT
ejpam-237	224	19	a	a	NOUN
ejpam-237	224	20	}	}	PUNCT
ejpam-237	224	21	,	,	PUNCT
ejpam-237	224	22	{	{	PUNCT
ejpam-237	224	23	c	c	NOUN
ejpam-237	224	24	}	}	PUNCT
ejpam-237	224	25	,	,	PUNCT
ejpam-237	224	26	{	{	PUNCT
ejpam-237	224	27	a	a	X
ejpam-237	224	28	,	,	PUNCT
ejpam-237	224	29	c	c	NOUN
ejpam-237	224	30	}	}	PUNCT
ejpam-237	224	31	}	}	PUNCT
ejpam-237	224	32	and	and	CCONJ
ejpam-237	224	33	σ	σ	NUM
ejpam-237	224	34	=	=	SYM
ejpam-237	224	35	{	{	PUNCT
ejpam-237	224	36	∅	∅	NOUN
ejpam-237	224	37	,	,	PUNCT
ejpam-237	224	38	x	x	INTJ
ejpam-237	224	39	,	,	PUNCT
ejpam-237	224	40	{	{	PUNCT
ejpam-237	224	41	c	c	NOUN
ejpam-237	224	42	}	}	PUNCT
ejpam-237	224	43	,	,	PUNCT
ejpam-237	224	44	{	{	PUNCT
ejpam-237	224	45	a	a	DET
ejpam-237	224	46	,	,	PUNCT
ejpam-237	224	47	b	b	NOUN
ejpam-237	224	48	}	}	PUNCT
ejpam-237	224	49	}	}	PUNCT
ejpam-237	224	50	.	.	PUNCT
ejpam-237	225	1	let	let	VERB
ejpam-237	225	2	us	we	PRON
ejpam-237	225	3	take	take	VERB
ejpam-237	225	4	i	i	NOUN
ejpam-237	225	5	=	=	PUNCT
ejpam-237	225	6	℘(x	℘(x	ADJ
ejpam-237	225	7	)	)	PUNCT
ejpam-237	225	8	and	and	CCONJ
ejpam-237	225	9	j	j	X
ejpam-237	225	10	=	=	SYM
ejpam-237	225	11	{	{	PUNCT
ejpam-237	225	12	∅	∅	NOUN
ejpam-237	225	13	,	,	PUNCT
ejpam-237	225	14	{	{	PUNCT
ejpam-237	225	15	c	c	NOUN
ejpam-237	225	16	}	}	PUNCT
ejpam-237	225	17	}	}	PUNCT
ejpam-237	225	18	.	.	PUNCT
ejpam-237	226	1	the	the	DET
ejpam-237	226	2	define	define	VERB
ejpam-237	226	3	the	the	DET
ejpam-237	226	4	identity	identity	NOUN
ejpam-237	226	5	function	function	NOUN
ejpam-237	226	6	f	f	NOUN
ejpam-237	226	7	:	:	PUNCT
ejpam-237	226	8	(	(	PUNCT
ejpam-237	226	9	x	x	X
ejpam-237	226	10	,	,	PUNCT
ejpam-237	226	11	τ	τ	PROPN
ejpam-237	226	12	,	,	PUNCT
ejpam-237	226	13	i	i	NOUN
ejpam-237	226	14	)	)	PUNCT
ejpam-237	226	15	−→	−→	NOUN
ejpam-237	226	16	(	(	PUNCT
ejpam-237	226	17	x	x	PROPN
ejpam-237	226	18	,	,	PUNCT
ejpam-237	226	19	σ	σ	PROPN
ejpam-237	226	20	,	,	PUNCT
ejpam-237	226	21	j	j	PROPN
ejpam-237	226	22	)	)	PUNCT
ejpam-237	226	23	.	.	PUNCT
ejpam-237	227	1	for	for	ADP
ejpam-237	227	2	the	the	DET
ejpam-237	227	3	subset	subset	NOUN
ejpam-237	227	4	{	{	PUNCT
ejpam-237	227	5	a	a	PRON
ejpam-237	227	6	,	,	PUNCT
ejpam-237	227	7	b	b	NOUN
ejpam-237	227	8	}	}	PUNCT
ejpam-237	227	9	∈	∈	PROPN
ejpam-237	227	10	σ	σ	NOUN
ejpam-237	227	11	,	,	PUNCT
ejpam-237	227	12	we	we	PRON
ejpam-237	227	13	have	have	VERB
ejpam-237	227	14	(	(	PUNCT
ejpam-237	227	15	{	{	PUNCT
ejpam-237	227	16	a	a	NOUN
ejpam-237	227	17	,	,	PUNCT
ejpam-237	227	18	b})∗	b})∗	NOUN
ejpam-237	227	19	=	=	PUNCT
ejpam-237	227	20	{	{	PUNCT
ejpam-237	227	21	a	a	DET
ejpam-237	227	22	,	,	PUNCT
ejpam-237	227	23	b	b	NOUN
ejpam-237	227	24	}	}	PUNCT
ejpam-237	227	25	and	and	CCONJ
ejpam-237	227	26	(	(	PUNCT
ejpam-237	227	27	f	f	PROPN
ejpam-237	227	28	−1({a	−1({a	PROPN
ejpam-237	227	29	,	,	PUNCT
ejpam-237	227	30	b}))∗	b}))∗	NOUN
ejpam-237	227	31	=	=	SYM
ejpam-237	227	32	(	(	PUNCT
ejpam-237	227	33	{	{	PUNCT
ejpam-237	227	34	a	a	NOUN
ejpam-237	227	35	,	,	PUNCT
ejpam-237	227	36	b})∗	b})∗	NOUN
ejpam-237	227	37	=	=	NOUN
ejpam-237	227	38	∅	∅	NOUN
ejpam-237	227	39	and	and	CCONJ
ejpam-237	227	40	thus	thus	ADV
ejpam-237	227	41	f	f	PROPN
ejpam-237	227	42	−1(v	−1(v	PROPN
ejpam-237	227	43	∗	∗	PROPN
ejpam-237	227	44	)	)	PUNCT
ejpam-237	227	45	(	(	PUNCT
ejpam-237	227	46	f	f	PROPN
ejpam-237	227	47	−1(v	−1(v	PROPN
ejpam-237	227	48	)	)	PUNCT
ejpam-237	227	49	)	)	PUNCT
ejpam-237	227	50	∗	∗	NOUN
ejpam-237	227	51	for	for	ADP
ejpam-237	227	52	v	v	NOUN
ejpam-237	227	53	=	=	SYM
ejpam-237	227	54	{	{	PUNCT
ejpam-237	227	55	a	a	PRON
ejpam-237	227	56	,	,	PUNCT
ejpam-237	227	57	b	b	NOUN
ejpam-237	227	58	}	}	PUNCT
ejpam-237	227	59	∈	∈	PROPN
ejpam-237	227	60	σ	σ	PROPN
ejpam-237	227	61	.	.	PUNCT
ejpam-237	228	1	but	but	CCONJ
ejpam-237	228	2	f	f	PROPN
ejpam-237	228	3	−1(cl∗(v	−1(cl∗(v	PROPN
ejpam-237	228	4	)	)	PUNCT
ejpam-237	228	5	)	)	PUNCT
ejpam-237	229	1	⊂	⊂	PROPN
ejpam-237	229	2	cl∗	cl∗	PROPN
ejpam-237	229	3	(	(	PUNCT
ejpam-237	229	4	f	f	PROPN
ejpam-237	229	5	−1(v	−1(v	PROPN
ejpam-237	229	6	)	)	PUNCT
ejpam-237	229	7	)	)	PUNCT
ejpam-237	229	8	for	for	ADP
ejpam-237	229	9	every	every	DET
ejpam-237	229	10	v	v	PROPN
ejpam-237	229	11	∈	∈	PROPN
ejpam-237	229	12	σ	σ	PROPN
ejpam-237	229	13	.	.	PUNCT
ejpam-237	230	1	lemma	lemma	PROPN
ejpam-237	230	2	4.1	4.1	NUM
ejpam-237	230	3	.	.	PUNCT
ejpam-237	231	1	(	(	PUNCT
ejpam-237	231	2	hatir	hatir	NOUN
ejpam-237	231	3	and	and	CCONJ
ejpam-237	231	4	noiri	noiri	ADV
ejpam-237	232	1	[	[	X
ejpam-237	232	2	6	6	NUM
ejpam-237	232	3	]	]	PUNCT
ejpam-237	232	4	)	)	PUNCT
ejpam-237	232	5	let	let	VERB
ejpam-237	232	6	a	a	PRON
ejpam-237	232	7	and	and	CCONJ
ejpam-237	232	8	b	b	NOUN
ejpam-237	232	9	be	be	AUX
ejpam-237	232	10	subsets	subset	NOUN
ejpam-237	232	11	of	of	ADP
ejpam-237	232	12	an	an	DET
ejpam-237	232	13	ideal	ideal	ADJ
ejpam-237	232	14	topological	topological	ADJ
ejpam-237	232	15	space	space	NOUN
ejpam-237	232	16	(	(	PUNCT
ejpam-237	232	17	x	x	X
ejpam-237	232	18	,	,	PUNCT
ejpam-237	232	19	τ	τ	PROPN
ejpam-237	232	20	,	,	PUNCT
ejpam-237	232	21	i	i	PROPN
ejpam-237	232	22	)	)	PUNCT
ejpam-237	232	23	.	.	PUNCT
ejpam-237	233	1	then	then	ADV
ejpam-237	233	2	the	the	DET
ejpam-237	233	3	following	follow	VERB
ejpam-237	233	4	properties	property	NOUN
ejpam-237	233	5	hold	hold	VERB
ejpam-237	233	6	:	:	PUNCT
ejpam-237	234	1	1	1	X
ejpam-237	234	2	.	.	X
ejpam-237	234	3	a∈	a∈	PROPN
ejpam-237	234	4	sio(x	sio(x	PROPN
ejpam-237	234	5	,	,	PUNCT
ejpam-237	234	6	τ	τ	PROPN
ejpam-237	234	7	)	)	PUNCT
ejpam-237	234	8	if	if	SCONJ
ejpam-237	234	9	and	and	CCONJ
ejpam-237	234	10	only	only	ADV
ejpam-237	234	11	if	if	SCONJ
ejpam-237	234	12	there	there	PRON
ejpam-237	234	13	exists	exist	VERB
ejpam-237	234	14	u	u	PROPN
ejpam-237	234	15	∈	∈	PROPN
ejpam-237	234	16	τ	τ	X
ejpam-237	234	17	such	such	ADJ
ejpam-237	234	18	that	that	SCONJ
ejpam-237	234	19	u	u	PROPN
ejpam-237	234	20	⊂	⊂	X
ejpam-237	234	21	a⊂	a⊂	VERB
ejpam-237	234	22	cl∗(u	cl∗(u	NOUN
ejpam-237	234	23	)	)	PUNCT
ejpam-237	234	24	,	,	PUNCT
ejpam-237	234	25	2	2	X
ejpam-237	234	26	.	.	X
ejpam-237	235	1	if	if	SCONJ
ejpam-237	235	2	a∈	a∈	PROPN
ejpam-237	235	3	sio(x	sio(x	PROPN
ejpam-237	235	4	,	,	PUNCT
ejpam-237	235	5	τ	τ	PROPN
ejpam-237	235	6	)	)	PUNCT
ejpam-237	235	7	and	and	CCONJ
ejpam-237	235	8	a⊂	a⊂	VERB
ejpam-237	235	9	b	b	PROPN
ejpam-237	235	10	⊂	⊂	X
ejpam-237	235	11	cl∗(a	cl∗(a	NOUN
ejpam-237	235	12	)	)	PUNCT
ejpam-237	235	13	,	,	PUNCT
ejpam-237	235	14	then	then	ADV
ejpam-237	235	15	b	b	X
ejpam-237	235	16	∈	∈	PROPN
ejpam-237	235	17	sio(x	sio(x	NOUN
ejpam-237	235	18	,	,	PUNCT
ejpam-237	235	19	τ	τ	PROPN
ejpam-237	235	20	)	)	PUNCT
ejpam-237	236	1	.	.	PUNCT
ejpam-237	237	1	the	the	DET
ejpam-237	237	2	following	follow	VERB
ejpam-237	237	3	theorem	theorem	NOUN
ejpam-237	237	4	slightly	slightly	ADV
ejpam-237	237	5	improve	improve	VERB
ejpam-237	237	6	the	the	DET
ejpam-237	237	7	theorem	theorem	ADJ
ejpam-237	237	8	4.5	4.5	NUM
ejpam-237	237	9	in	in	ADP
ejpam-237	237	10	[	[	X
ejpam-237	237	11	6	6	NUM
ejpam-237	237	12	]	]	PUNCT
ejpam-237	237	13	which	which	PRON
ejpam-237	237	14	states	state	VERB
ejpam-237	237	15	that	that	SCONJ
ejpam-237	237	16	if	if	SCONJ
ejpam-237	237	17	f	f	X
ejpam-237	237	18	:	:	PUNCT
ejpam-237	237	19	(	(	PUNCT
ejpam-237	237	20	x	x	X
ejpam-237	237	21	,	,	PUNCT
ejpam-237	237	22	τ	τ	PROPN
ejpam-237	237	23	,	,	PUNCT
ejpam-237	237	24	i	i	NOUN
ejpam-237	237	25	)	)	PUNCT
ejpam-237	237	26	−→	−→	NOUN
ejpam-237	237	27	(	(	PUNCT
ejpam-237	237	28	y	y	PROPN
ejpam-237	237	29	,	,	PUNCT
ejpam-237	237	30	σ	σ	PROPN
ejpam-237	237	31	,	,	PUNCT
ejpam-237	237	32	j	j	PROPN
ejpam-237	237	33	)	)	PUNCT
ejpam-237	237	34	is	be	AUX
ejpam-237	237	35	semi	semi	ADJ
ejpam-237	237	36	-	-	ADJ
ejpam-237	237	37	i	i	PRON
ejpam-237	237	38	-	-	PUNCT
ejpam-237	237	39	continuous	continuous	ADJ
ejpam-237	237	40	and	and	CCONJ
ejpam-237	237	41	f	f	PROPN
ejpam-237	237	42	−1(v	−1(v	PROPN
ejpam-237	237	43	∗	∗	NOUN
ejpam-237	237	44	)	)	PUNCT
ejpam-237	238	1	⊂	⊂	PROPN
ejpam-237	238	2	(	(	PUNCT
ejpam-237	238	3	f	f	X
ejpam-237	238	4	−1(v	−1(v	PROPN
ejpam-237	238	5	)	)	PUNCT
ejpam-237	238	6	)	)	PUNCT
ejpam-237	238	7	∗	∗	NOUN
ejpam-237	238	8	for	for	ADP
ejpam-237	238	9	each	each	DET
ejpam-237	238	10	v	v	NUM
ejpam-237	238	11	∈	∈	PROPN
ejpam-237	238	12	σ	σ	PROPN
ejpam-237	238	13	,	,	PUNCT
ejpam-237	238	14	then	then	ADV
ejpam-237	238	15	f	f	PROPN
ejpam-237	238	16	is	be	AUX
ejpam-237	238	17	semi	semi	ADJ
ejpam-237	238	18	-	-	ADJ
ejpam-237	238	19	i	i	NOUN
ejpam-237	238	20	-	-	PUNCT
ejpam-237	238	21	irresolute	irresolute	NOUN
ejpam-237	238	22	.	.	PUNCT
ejpam-237	239	1	theorem	theorem	VERB
ejpam-237	239	2	4.6	4.6	NUM
ejpam-237	239	3	.	.	PUNCT
ejpam-237	240	1	if	if	SCONJ
ejpam-237	240	2	f	f	PROPN
ejpam-237	240	3	:	:	PUNCT
ejpam-237	240	4	(	(	PUNCT
ejpam-237	240	5	x	x	X
ejpam-237	240	6	,	,	PUNCT
ejpam-237	240	7	τ	τ	PROPN
ejpam-237	240	8	,	,	PUNCT
ejpam-237	240	9	i	i	NOUN
ejpam-237	240	10	)	)	PUNCT
ejpam-237	240	11	−→	−→	NOUN
ejpam-237	240	12	(	(	PUNCT
ejpam-237	240	13	y	y	PROPN
ejpam-237	240	14	,	,	PUNCT
ejpam-237	240	15	σ	σ	PROPN
ejpam-237	240	16	,	,	PUNCT
ejpam-237	240	17	j	j	PROPN
ejpam-237	240	18	)	)	PUNCT
ejpam-237	240	19	is	be	AUX
ejpam-237	240	20	semi	semi	ADJ
ejpam-237	240	21	-	-	ADJ
ejpam-237	240	22	i	i	PRON
ejpam-237	240	23	-	-	PUNCT
ejpam-237	240	24	continuous	continuous	ADJ
ejpam-237	240	25	and	and	CCONJ
ejpam-237	240	26	f	f	X
ejpam-237	240	27	−1(cl∗(v	−1(cl∗(v	PROPN
ejpam-237	240	28	)	)	PUNCT
ejpam-237	240	29	)	)	PUNCT
ejpam-237	241	1	⊂	⊂	PROPN
ejpam-237	241	2	cl∗	cl∗	PROPN
ejpam-237	241	3	(	(	PUNCT
ejpam-237	241	4	f	f	PROPN
ejpam-237	241	5	−1(v	−1(v	PROPN
ejpam-237	241	6	)	)	PUNCT
ejpam-237	241	7	)	)	PUNCT
ejpam-237	241	8	for	for	ADP
ejpam-237	241	9	each	each	DET
ejpam-237	241	10	v	v	NUM
ejpam-237	241	11	∈	∈	PROPN
ejpam-237	241	12	σ	σ	PROPN
ejpam-237	241	13	,	,	PUNCT
ejpam-237	241	14	then	then	ADV
ejpam-237	241	15	f	f	PROPN
ejpam-237	241	16	is	be	AUX
ejpam-237	241	17	semi	semi	ADJ
ejpam-237	241	18	-	-	ADJ
ejpam-237	241	19	i	i	NOUN
ejpam-237	241	20	-	-	PUNCT
ejpam-237	241	21	irresolute	irresolute	ADJ
ejpam-237	241	22	.	.	PUNCT
ejpam-237	242	1	proof	proof	NOUN
ejpam-237	242	2	.	.	PUNCT
ejpam-237	243	1	let	let	VERB
ejpam-237	243	2	b	b	X
ejpam-237	243	3	be	be	AUX
ejpam-237	243	4	any	any	DET
ejpam-237	243	5	semi	semi	ADJ
ejpam-237	243	6	-	-	ADJ
ejpam-237	243	7	i	i	PRON
ejpam-237	243	8	-	-	PUNCT
ejpam-237	243	9	open	open	ADJ
ejpam-237	243	10	set	set	NOUN
ejpam-237	243	11	of	of	ADP
ejpam-237	243	12	(	(	PUNCT
ejpam-237	243	13	y	y	PROPN
ejpam-237	243	14	,	,	PUNCT
ejpam-237	243	15	σ	σ	PROPN
ejpam-237	243	16	,	,	PUNCT
ejpam-237	243	17	j	j	PROPN
ejpam-237	243	18	)	)	PUNCT
ejpam-237	243	19	.	.	PUNCT
ejpam-237	244	1	by	by	ADP
ejpam-237	244	2	lemma	lemma	PROPN
ejpam-237	244	3	3	3	NUM
ejpam-237	244	4	,	,	PUNCT
ejpam-237	244	5	there	there	PRON
ejpam-237	244	6	exists	exist	VERB
ejpam-237	244	7	v	v	ADP
ejpam-237	244	8	∈	∈	PROPN
ejpam-237	244	9	σ	σ	NOUN
ejpam-237	244	10	such	such	ADJ
ejpam-237	244	11	that	that	DET
ejpam-237	244	12	v	v	ADP
ejpam-237	244	13	⊂	⊂	PROPN
ejpam-237	244	14	b	b	X
ejpam-237	244	15	⊂	⊂	PROPN
ejpam-237	244	16	cl∗(v	cl∗(v	PROPN
ejpam-237	244	17	)	)	PUNCT
ejpam-237	244	18	.	.	PUNCT
ejpam-237	245	1	therefore	therefore	ADV
ejpam-237	245	2	,	,	PUNCT
ejpam-237	245	3	we	we	PRON
ejpam-237	245	4	have	have	VERB
ejpam-237	245	5	f	f	PROPN
ejpam-237	245	6	−1(v	−1(v	PROPN
ejpam-237	245	7	)	)	PUNCT
ejpam-237	246	1	⊂	⊂	PROPN
ejpam-237	246	2	f	f	PROPN
ejpam-237	247	1	−1(b	−1(b	ADV
ejpam-237	247	2	)	)	PUNCT
ejpam-237	247	3	⊂	⊂	PROPN
ejpam-237	247	4	f	f	PROPN
ejpam-237	247	5	−1(cl∗(v	−1(cl∗(v	PROPN
ejpam-237	247	6	)	)	PUNCT
ejpam-237	247	7	)	)	PUNCT
ejpam-237	248	1	⊂	⊂	PROPN
ejpam-237	248	2	cl∗	cl∗	PROPN
ejpam-237	248	3	(	(	PUNCT
ejpam-237	248	4	f	f	PROPN
ejpam-237	248	5	−1(v	−1(v	PROPN
ejpam-237	248	6	)	)	PUNCT
ejpam-237	248	7	)	)	PUNCT
ejpam-237	248	8	.	.	PUNCT
ejpam-237	249	1	since	since	SCONJ
ejpam-237	249	2	f	f	PROPN
ejpam-237	249	3	is	be	AUX
ejpam-237	249	4	semi	semi	ADJ
ejpam-237	249	5	-	-	ADJ
ejpam-237	249	6	i	i	PRON
ejpam-237	249	7	-	-	PUNCT
ejpam-237	249	8	continuous	continuous	ADJ
ejpam-237	249	9	and	and	CCONJ
ejpam-237	249	10	v	v	ADP
ejpam-237	249	11	∈	∈	PROPN
ejpam-237	249	12	σ	σ	PROPN
ejpam-237	249	13	,	,	PUNCT
ejpam-237	249	14	f	f	PROPN
ejpam-237	249	15	−1(v	−1(v	PROPN
ejpam-237	249	16	)	)	PUNCT
ejpam-237	250	1	∈	∈	PROPN
ejpam-237	250	2	sio(x	sio(x	PROPN
ejpam-237	250	3	,	,	PUNCT
ejpam-237	250	4	τ	τ	PROPN
ejpam-237	250	5	)	)	PUNCT
ejpam-237	250	6	and	and	CCONJ
ejpam-237	250	7	hence	hence	ADV
ejpam-237	250	8	by	by	ADP
ejpam-237	250	9	lemma	lemma	PROPN
ejpam-237	250	10	3	3	NUM
ejpam-237	250	11	,	,	PUNCT
ejpam-237	250	12	f	f	PROPN
ejpam-237	250	13	−1(b	−1(b	NOUN
ejpam-237	250	14	)	)	PUNCT
ejpam-237	250	15	is	be	AUX
ejpam-237	250	16	semi	semi	ADJ
ejpam-237	250	17	-	-	ADJ
ejpam-237	250	18	i	i	PRON
ejpam-237	250	19	-	-	PUNCT
ejpam-237	250	20	open	open	ADJ
ejpam-237	250	21	in	in	ADP
ejpam-237	250	22	(	(	PUNCT
ejpam-237	250	23	x	x	INTJ
ejpam-237	250	24	,	,	PUNCT
ejpam-237	250	25	τ	τ	PROPN
ejpam-237	250	26	,	,	PUNCT
ejpam-237	250	27	i	i	PROPN
ejpam-237	250	28	)	)	PUNCT
ejpam-237	250	29	.	.	PUNCT
ejpam-237	251	1	this	this	PRON
ejpam-237	251	2	shows	show	VERB
ejpam-237	251	3	that	that	SCONJ
ejpam-237	251	4	f	f	PROPN
ejpam-237	251	5	is	be	AUX
ejpam-237	251	6	semi	semi	ADJ
ejpam-237	251	7	-	-	ADJ
ejpam-237	251	8	i	i	NOUN
ejpam-237	251	9	-	-	PUNCT
ejpam-237	251	10	irresolute	irresolute	NOUN
ejpam-237	251	11	.	.	PUNCT
ejpam-237	252	1	references	reference	NOUN
ejpam-237	252	2	181	181	NUM
ejpam-237	252	3	references	reference	NOUN
ejpam-237	252	4	[	[	X
ejpam-237	252	5	1	1	NUM
ejpam-237	252	6	]	]	PUNCT
ejpam-237	252	7	m.	m.	NOUN
ejpam-237	252	8	e.	e.	PROPN
ejpam-237	252	9	abd	abd	PROPN
ejpam-237	252	10	el	el	PROPN
ejpam-237	252	11	-	-	PROPN
ejpam-237	252	12	monsef	monsef	PROPN
ejpam-237	252	13	,	,	PUNCT
ejpam-237	252	14	s.	s.	PROPN
ejpam-237	252	15	n.	n.	PROPN
ejpam-237	252	16	el	el	PROPN
ejpam-237	252	17	-	-	PUNCT
ejpam-237	252	18	deeb	deeb	PROPN
ejpam-237	252	19	and	and	CCONJ
ejpam-237	252	20	r.	r.	PROPN
ejpam-237	252	21	a.	a.	PROPN
ejpam-237	252	22	mahmoud	mahmoud	PROPN
ejpam-237	252	23	,	,	PUNCT
ejpam-237	252	24	β	β	X
ejpam-237	252	25	−	−	NOUN
ejpam-237	252	26	open	open	ADJ
ejpam-237	252	27	sets	set	NOUN
ejpam-237	252	28	and	and	CCONJ
ejpam-237	252	29	β	β	X
ejpam-237	252	30	−	−	NOUN
ejpam-237	252	31	continuous	continuous	ADJ
ejpam-237	252	32	mappings	mapping	NOUN
ejpam-237	252	33	,	,	PUNCT
ejpam-237	252	34	bull	bull	NOUN
ejpam-237	252	35	.	.	PUNCT
ejpam-237	253	1	fac	fac	PROPN
ejpam-237	253	2	.	.	PUNCT
ejpam-237	254	1	sci	sci	PROPN
ejpam-237	254	2	.	.	PUNCT
ejpam-237	254	3	assiut	assiut	PROPN
ejpam-237	254	4	univ	univ	PROPN
ejpam-237	254	5	.	.	PROPN
ejpam-237	254	6	,	,	PUNCT
ejpam-237	254	7	12(1983	12(1983	NUM
ejpam-237	254	8	)	)	PUNCT
ejpam-237	254	9	,	,	PUNCT
ejpam-237	254	10	77	77	NUM
ejpam-237	254	11	-	-	SYM
ejpam-237	254	12	90	90	NUM
ejpam-237	254	13	.	.	PUNCT
ejpam-237	255	1	[	[	X
ejpam-237	255	2	2	2	NUM
ejpam-237	255	3	]	]	PUNCT
ejpam-237	255	4	m.	m.	NOUN
ejpam-237	255	5	e.	e.	PROPN
ejpam-237	255	6	abd	abd	PROPN
ejpam-237	256	1	el	el	PROPN
ejpam-237	256	2	-	-	PROPN
ejpam-237	256	3	monsef	monsef	PROPN
ejpam-237	256	4	,	,	PUNCT
ejpam-237	256	5	r.	r.	PROPN
ejpam-237	256	6	a.	a.	PROPN
ejpam-237	256	7	mahmoud	mahmoud	PROPN
ejpam-237	256	8	and	and	CCONJ
ejpam-237	256	9	a.	a.	NOUN
ejpam-237	256	10	a.	a.	NOUN
ejpam-237	256	11	nasef	nasef	PROPN
ejpam-237	256	12	,	,	PUNCT
ejpam-237	256	13	on	on	ADP
ejpam-237	256	14	quasi	quasi	ADJ
ejpam-237	256	15	-	-	ADJ
ejpam-237	256	16	i	i	NOUN
ejpam-237	256	17	-	-	PUNCT
ejpam-237	256	18	openness	openness	NOUN
ejpam-237	256	19	and	and	CCONJ
ejpam-237	256	20	quasii	quasii	ADJ
ejpam-237	256	21	-	-	PUNCT
ejpam-237	256	22	continuity	continuity	NOUN
ejpam-237	256	23	,	,	PUNCT
ejpam-237	256	24	tamkang	tamkang	PROPN
ejpam-237	256	25	j.	j.	PROPN
ejpam-237	256	26	math	math	PROPN
ejpam-237	256	27	.	.	PUNCT
ejpam-237	256	28	,	,	PUNCT
ejpam-237	256	29	31(2000	31(2000	NUM
ejpam-237	256	30	)	)	PUNCT
ejpam-237	256	31	,	,	PUNCT
ejpam-237	256	32	101	101	NUM
ejpam-237	256	33	-	-	SYM
ejpam-237	256	34	108	108	NUM
ejpam-237	256	35	.	.	PUNCT
ejpam-237	257	1	[	[	X
ejpam-237	257	2	3	3	X
ejpam-237	257	3	]	]	PUNCT
ejpam-237	257	4	s.	s.	PROPN
ejpam-237	257	5	g.	g.	PROPN
ejpam-237	257	6	crossley	crossley	PROPN
ejpam-237	257	7	and	and	CCONJ
ejpam-237	257	8	s.	s.	PROPN
ejpam-237	257	9	k.	k.	PROPN
ejpam-237	257	10	hildebrand	hildebrand	PROPN
ejpam-237	257	11	,	,	PUNCT
ejpam-237	257	12	semi	semi	ADJ
ejpam-237	257	13	-	-	ADJ
ejpam-237	257	14	topological	topological	ADJ
ejpam-237	257	15	properties	property	NOUN
ejpam-237	257	16	,	,	PUNCT
ejpam-237	257	17	fund	fund	NOUN
ejpam-237	257	18	.	.	PUNCT
ejpam-237	257	19	math	math	NOUN
ejpam-237	257	20	.	.	PUNCT
ejpam-237	257	21	,	,	PUNCT
ejpam-237	257	22	74(1972	74(1972	PROPN
ejpam-237	257	23	)	)	PUNCT
ejpam-237	257	24	,	,	PUNCT
ejpam-237	257	25	233	233	NUM
ejpam-237	257	26	-	-	SYM
ejpam-237	257	27	254	254	NUM
ejpam-237	257	28	.	.	PUNCT
ejpam-237	258	1	[	[	X
ejpam-237	258	2	4	4	X
ejpam-237	258	3	]	]	PUNCT
ejpam-237	258	4	j.	j.	PROPN
ejpam-237	258	5	dontchev	dontchev	PROPN
ejpam-237	258	6	,	,	PUNCT
ejpam-237	258	7	on	on	ADP
ejpam-237	258	8	hausdorff	hausdorff	NOUN
ejpam-237	258	9	spaces	space	NOUN
ejpam-237	258	10	via	via	ADP
ejpam-237	258	11	topological	topological	ADJ
ejpam-237	258	12	ideals	ideal	NOUN
ejpam-237	258	13	and	and	CCONJ
ejpam-237	258	14	i	i	NOUN
ejpam-237	258	15	-	-	PUNCT
ejpam-237	258	16	irresolute	irresolute	ADJ
ejpam-237	258	17	functions	function	NOUN
ejpam-237	258	18	,	,	PUNCT
ejpam-237	258	19	in	in	ADP
ejpam-237	258	20	papers	paper	NOUN
ejpam-237	258	21	on	on	ADP
ejpam-237	258	22	general	general	ADJ
ejpam-237	258	23	topology	topology	NOUN
ejpam-237	258	24	and	and	CCONJ
ejpam-237	258	25	applications	application	NOUN
ejpam-237	258	26	,	,	PUNCT
ejpam-237	258	27	annals	annal	NOUN
ejpam-237	258	28	of	of	ADP
ejpam-237	258	29	new	new	PROPN
ejpam-237	258	30	york	york	PROPN
ejpam-237	258	31	academy	academy	PROPN
ejpam-237	258	32	of	of	ADP
ejpam-237	258	33	sciences	sciences	PROPN
ejpam-237	258	34	,	,	PUNCT
ejpam-237	258	35	vol.767(1995	vol.767(1995	NUM
ejpam-237	258	36	)	)	PUNCT
ejpam-237	258	37	,	,	PUNCT
ejpam-237	258	38	28	28	NUM
ejpam-237	258	39	-	-	SYM
ejpam-237	258	40	38	38	NUM
ejpam-237	258	41	.	.	PUNCT
ejpam-237	259	1	[	[	X
ejpam-237	259	2	5	5	X
ejpam-237	259	3	]	]	PUNCT
ejpam-237	259	4	e.	e.	PROPN
ejpam-237	259	5	hatir	hatir	PROPN
ejpam-237	259	6	and	and	CCONJ
ejpam-237	259	7	t.	t.	PROPN
ejpam-237	259	8	noiri	noiri	PROPN
ejpam-237	259	9	,	,	PUNCT
ejpam-237	259	10	on	on	ADP
ejpam-237	259	11	decompositions	decomposition	NOUN
ejpam-237	259	12	of	of	ADP
ejpam-237	259	13	continuity	continuity	NOUN
ejpam-237	259	14	via	via	ADP
ejpam-237	259	15	idealization	idealization	NOUN
ejpam-237	259	16	,	,	PUNCT
ejpam-237	259	17	acta	acta	PROPN
ejpam-237	259	18	math	math	PROPN
ejpam-237	259	19	.	.	PUNCT
ejpam-237	260	1	hungar	hungar	PROPN
ejpam-237	260	2	.	.	PUNCT
ejpam-237	260	3	,	,	PUNCT
ejpam-237	260	4	96(4)(2002	96(4)(2002	NUM
ejpam-237	260	5	)	)	PUNCT
ejpam-237	260	6	,	,	PUNCT
ejpam-237	260	7	314	314	NUM
ejpam-237	260	8	-	-	SYM
ejpam-237	260	9	349	349	NUM
ejpam-237	260	10	.	.	PUNCT
ejpam-237	261	1	[	[	X
ejpam-237	261	2	6	6	X
ejpam-237	261	3	]	]	PUNCT
ejpam-237	261	4	e.	e.	PROPN
ejpam-237	261	5	hatir	hatir	PROPN
ejpam-237	261	6	and	and	CCONJ
ejpam-237	261	7	t.	t.	PROPN
ejpam-237	261	8	noiri	noiri	PROPN
ejpam-237	261	9	,	,	PUNCT
ejpam-237	261	10	on	on	ADP
ejpam-237	261	11	semi	semi	ADJ
ejpam-237	261	12	-	-	ADJ
ejpam-237	261	13	i	i	PRON
ejpam-237	261	14	-	-	PUNCT
ejpam-237	261	15	open	open	ADJ
ejpam-237	261	16	sets	set	NOUN
ejpam-237	261	17	and	and	CCONJ
ejpam-237	261	18	semi	semi	ADJ
ejpam-237	261	19	-	-	ADJ
ejpam-237	261	20	i	i	ADJ
ejpam-237	261	21	-	-	PUNCT
ejpam-237	261	22	continuous	continuous	ADJ
ejpam-237	261	23	functions	function	NOUN
ejpam-237	261	24	,	,	PUNCT
ejpam-237	261	25	acta	acta	PROPN
ejpam-237	261	26	math	math	PROPN
ejpam-237	261	27	.	.	PUNCT
ejpam-237	262	1	hungar	hungar	PROPN
ejpam-237	262	2	.	.	PUNCT
ejpam-237	263	1	,	,	PUNCT
ejpam-237	263	2	107(4)(2005	107(4)(2005	NUM
ejpam-237	263	3	)	)	PUNCT
ejpam-237	263	4	,	,	PUNCT
ejpam-237	263	5	345	345	NUM
ejpam-237	263	6	-	-	SYM
ejpam-237	263	7	353	353	NUM
ejpam-237	263	8	.	.	PUNCT
ejpam-237	264	1	[	[	X
ejpam-237	264	2	7	7	X
ejpam-237	264	3	]	]	X
ejpam-237	264	4	e.	e.	PROPN
ejpam-237	264	5	hayashi	hayashi	PROPN
ejpam-237	264	6	,	,	PUNCT
ejpam-237	264	7	topologies	topology	NOUN
ejpam-237	264	8	defined	define	VERB
ejpam-237	264	9	by	by	ADP
ejpam-237	264	10	local	local	ADJ
ejpam-237	264	11	properties	property	NOUN
ejpam-237	264	12	,	,	PUNCT
ejpam-237	264	13	math	math	NOUN
ejpam-237	264	14	.	.	PUNCT
ejpam-237	265	1	ann	ann	PROPN
ejpam-237	265	2	.	.	PROPN
ejpam-237	265	3	,	,	PUNCT
ejpam-237	265	4	156(1964	156(1964	NUM
ejpam-237	265	5	)	)	PUNCT
ejpam-237	265	6	,	,	PUNCT
ejpam-237	265	7	205	205	NUM
ejpam-237	265	8	-	-	SYM
ejpam-237	265	9	215	215	NUM
ejpam-237	265	10	.	.	PUNCT
ejpam-237	266	1	[	[	X
ejpam-237	266	2	8	8	NUM
ejpam-237	266	3	]	]	X
ejpam-237	266	4	d.	d.	PROPN
ejpam-237	266	5	janković	janković	PROPN
ejpam-237	266	6	and	and	CCONJ
ejpam-237	266	7	t.	t.	PROPN
ejpam-237	266	8	r.	r.	PROPN
ejpam-237	266	9	hamlett	hamlett	PROPN
ejpam-237	266	10	,	,	PUNCT
ejpam-237	266	11	new	new	ADJ
ejpam-237	266	12	topologies	topology	NOUN
ejpam-237	266	13	from	from	ADP
ejpam-237	266	14	old	old	ADJ
ejpam-237	266	15	via	via	ADP
ejpam-237	266	16	ideals	ideal	NOUN
ejpam-237	266	17	,	,	PUNCT
ejpam-237	266	18	amer	amer	PROPN
ejpam-237	266	19	.	.	PROPN
ejpam-237	266	20	math	math	PROPN
ejpam-237	266	21	.	.	PUNCT
ejpam-237	267	1	monthly	monthly	ADJ
ejpam-237	267	2	,	,	PUNCT
ejpam-237	267	3	97(1990	97(1990	NUM
ejpam-237	267	4	)	)	PUNCT
ejpam-237	267	5	,	,	PUNCT
ejpam-237	267	6	295	295	NUM
ejpam-237	267	7	-	-	SYM
ejpam-237	267	8	310	310	NUM
ejpam-237	267	9	.	.	PUNCT
ejpam-237	268	1	[	[	X
ejpam-237	268	2	9	9	NUM
ejpam-237	268	3	]	]	X
ejpam-237	268	4	d.	d.	PROPN
ejpam-237	268	5	janković	janković	PROPN
ejpam-237	268	6	and	and	CCONJ
ejpam-237	268	7	t.	t.	PROPN
ejpam-237	268	8	r.	r.	PROPN
ejpam-237	268	9	hamlett	hamlett	PROPN
ejpam-237	268	10	,	,	PUNCT
ejpam-237	268	11	compatible	compatible	ADJ
ejpam-237	268	12	extensions	extension	NOUN
ejpam-237	268	13	of	of	ADP
ejpam-237	268	14	ideals	ideal	NOUN
ejpam-237	268	15	,	,	PUNCT
ejpam-237	268	16	boll	boll	NOUN
ejpam-237	268	17	.	.	PUNCT
ejpam-237	269	1	un	un	PROPN
ejpam-237	269	2	.	.	PROPN
ejpam-237	269	3	mat	mat	PROPN
ejpam-237	269	4	.	.	PUNCT
ejpam-237	269	5	ital	ital	PROPN
ejpam-237	269	6	.	.	PROPN
ejpam-237	269	7	,	,	PUNCT
ejpam-237	269	8	(	(	PUNCT
ejpam-237	269	9	7)(6	7)(6	NUM
ejpam-237	269	10	-	-	PUNCT
ejpam-237	269	11	b)(1992	b)(1992	VERB
ejpam-237	269	12	)	)	PUNCT
ejpam-237	269	13	,	,	PUNCT
ejpam-237	269	14	453	453	NUM
ejpam-237	269	15	-	-	SYM
ejpam-237	269	16	465	465	NUM
ejpam-237	269	17	.	.	PUNCT
ejpam-237	270	1	[	[	X
ejpam-237	270	2	10	10	NUM
ejpam-237	270	3	]	]	X
ejpam-237	270	4	n.	n.	PROPN
ejpam-237	270	5	levine	levine	PROPN
ejpam-237	270	6	,	,	PUNCT
ejpam-237	270	7	semi	semi	ADJ
ejpam-237	270	8	-	-	ADJ
ejpam-237	270	9	open	open	ADJ
ejpam-237	270	10	sets	set	NOUN
ejpam-237	270	11	and	and	CCONJ
ejpam-237	270	12	semi	semi	ADJ
ejpam-237	270	13	-	-	NOUN
ejpam-237	270	14	continuity	continuity	NOUN
ejpam-237	270	15	in	in	ADP
ejpam-237	270	16	topological	topological	ADJ
ejpam-237	270	17	spaces	space	NOUN
ejpam-237	270	18	,	,	PUNCT
ejpam-237	270	19	amer	amer	PROPN
ejpam-237	270	20	.	.	PROPN
ejpam-237	270	21	math	math	PROPN
ejpam-237	270	22	.	.	PUNCT
ejpam-237	271	1	monthly	monthly	ADJ
ejpam-237	271	2	,	,	PUNCT
ejpam-237	271	3	70(1963	70(1963	NUM
ejpam-237	271	4	)	)	PUNCT
ejpam-237	271	5	,	,	PUNCT
ejpam-237	271	6	36	36	NUM
ejpam-237	271	7	-	-	SYM
ejpam-237	271	8	41	41	NUM
ejpam-237	271	9	.	.	PUNCT
ejpam-237	272	1	[	[	X
ejpam-237	272	2	11	11	NUM
ejpam-237	272	3	]	]	PUNCT
ejpam-237	272	4	s.	s.	PROPN
ejpam-237	272	5	n.	n.	PROPN
ejpam-237	272	6	maheshwari	maheshwari	PROPN
ejpam-237	272	7	and	and	CCONJ
ejpam-237	272	8	r.	r.	PROPN
ejpam-237	272	9	prasad	prasad	PROPN
ejpam-237	272	10	,	,	PUNCT
ejpam-237	272	11	some	some	DET
ejpam-237	272	12	new	new	ADJ
ejpam-237	272	13	separation	separation	NOUN
ejpam-237	272	14	axioms	axiom	VERB
ejpam-237	272	15	,	,	PUNCT
ejpam-237	272	16	ann	ann	PROPN
ejpam-237	272	17	.	.	PROPN
ejpam-237	272	18	soc	soc	PROPN
ejpam-237	272	19	.	.	PUNCT
ejpam-237	273	1	sci	sci	PROPN
ejpam-237	273	2	.	.	PROPN
ejpam-237	273	3	bruxelles	bruxelles	PROPN
ejpam-237	273	4	,	,	PUNCT
ejpam-237	273	5	89(1975	89(1975	NUM
ejpam-237	273	6	)	)	PUNCT
ejpam-237	273	7	,	,	PUNCT
ejpam-237	273	8	395	395	NUM
ejpam-237	273	9	-	-	SYM
ejpam-237	273	10	402	402	NUM
ejpam-237	273	11	.	.	PUNCT
ejpam-237	274	1	[	[	X
ejpam-237	274	2	12	12	NUM
ejpam-237	274	3	]	]	PUNCT
ejpam-237	274	4	r.	r.	PROPN
ejpam-237	274	5	a.	a.	PROPN
ejpam-237	274	6	mahmoud	mahmoud	PROPN
ejpam-237	274	7	and	and	CCONJ
ejpam-237	274	8	m.	m.	PROPN
ejpam-237	274	9	e.	e.	PROPN
ejpam-237	274	10	abd	abd	PROPN
ejpam-237	274	11	el	el	PROPN
ejpam-237	274	12	-	-	PROPN
ejpam-237	274	13	monsef	monsef	ADJ
ejpam-237	274	14	,	,	PUNCT
ejpam-237	274	15	β−	β−	PRON
ejpam-237	274	16	ir	ir	VERB
ejpam-237	274	17	resolute	resolute	ADJ
ejpam-237	274	18	and	and	CCONJ
ejpam-237	274	19	β−	β−	PRON
ejpam-237	274	20	topo	topo	NOUN
ejpam-237	274	21	log	log	NOUN
ejpam-237	274	22	ical	ical	PROPN
ejpam-237	274	23	invariant	invariant	PROPN
ejpam-237	274	24	,	,	PUNCT
ejpam-237	274	25	proc	proc	NOUN
ejpam-237	274	26	.	.	PUNCT
ejpam-237	275	1	pakistan	pakistan	PROPN
ejpam-237	275	2	acad	acad	PROPN
ejpam-237	275	3	.	.	PUNCT
ejpam-237	276	1	sci	sci	PROPN
ejpam-237	276	2	.	.	PROPN
ejpam-237	276	3	,	,	PUNCT
ejpam-237	276	4	27(1990	27(1990	NUM
ejpam-237	276	5	)	)	PUNCT
ejpam-237	276	6	,	,	PUNCT
ejpam-237	276	7	285	285	NUM
ejpam-237	276	8	-	-	SYM
ejpam-237	276	9	296	296	NUM
ejpam-237	276	10	.	.	PUNCT
ejpam-237	277	1	[	[	X
ejpam-237	277	2	13	13	NUM
ejpam-237	277	3	]	]	PUNCT
ejpam-237	277	4	a.	a.	NOUN
ejpam-237	277	5	a.	a.	NOUN
ejpam-237	277	6	nasef	nasef	PROPN
ejpam-237	277	7	,	,	PUNCT
ejpam-237	277	8	on	on	ADP
ejpam-237	277	9	hausdorff	hausdorff	NOUN
ejpam-237	277	10	spaces	space	NOUN
ejpam-237	277	11	via	via	ADP
ejpam-237	277	12	ideals	ideal	NOUN
ejpam-237	277	13	and	and	CCONJ
ejpam-237	277	14	quai	quai	NOUN
ejpam-237	277	15	-	-	PUNCT
ejpam-237	277	16	i	i	NOUN
ejpam-237	277	17	-	-	PUNCT
ejpam-237	277	18	irresolute	irresolute	ADJ
ejpam-237	277	19	functions	function	NOUN
ejpam-237	277	20	,	,	PUNCT
ejpam-237	277	21	chaos	chaos	NOUN
ejpam-237	277	22	solitons	soliton	NOUN
ejpam-237	277	23	and	and	CCONJ
ejpam-237	277	24	fractals	fractal	NOUN
ejpam-237	277	25	,	,	PUNCT
ejpam-237	277	26	14(2002	14(2002	NUM
ejpam-237	277	27	)	)	PUNCT
ejpam-237	277	28	,	,	PUNCT
ejpam-237	277	29	619	619	NUM
ejpam-237	277	30	-	-	SYM
ejpam-237	277	31	625	625	NUM
ejpam-237	277	32	.	.	PUNCT