id	sid	tid	token	lemma	pos
ejpam-842	1	1	4_842_khan.dvi	4_842_khan.dvi	NUM
ejpam-842	1	2	european	european	ADJ
ejpam-842	1	3	journal	journal	NOUN
ejpam-842	1	4	of	of	ADP
ejpam-842	1	5	pure	pure	ADJ
ejpam-842	1	6	and	and	CCONJ
ejpam-842	1	7	applied	apply	VERB
ejpam-842	1	8	mathematics	mathematic	NOUN
ejpam-842	1	9	vol	vol	NOUN
ejpam-842	1	10	.	.	PROPN
ejpam-842	1	11	4	4	NUM
ejpam-842	1	12	,	,	PUNCT
ejpam-842	1	13	no	no	INTJ
ejpam-842	1	14	.	.	NOUN
ejpam-842	1	15	3	3	NUM
ejpam-842	1	16	,	,	PUNCT
ejpam-842	1	17	2011	2011	NUM
ejpam-842	1	18	,	,	PUNCT
ejpam-842	1	19	237	237	NUM
ejpam-842	1	20	-	-	SYM
ejpam-842	1	21	243	243	NUM
ejpam-842	1	22	issn	issn	PROPN
ejpam-842	1	23	1307	1307	NUM
ejpam-842	1	24	-	-	SYM
ejpam-842	1	25	5543	5543	NUM
ejpam-842	1	26	–	–	PUNCT
ejpam-842	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-842	1	28	on	on	ADP
ejpam-842	1	29	is⋆g	is⋆g	ADJ
ejpam-842	1	30	-	-	PUNCT
ejpam-842	1	31	continuous	continuous	ADJ
ejpam-842	1	32	functions	function	NOUN
ejpam-842	1	33	in	in	ADP
ejpam-842	1	34	ideal	ideal	ADJ
ejpam-842	1	35	topological	topological	ADJ
ejpam-842	1	36	spaces	space	NOUN
ejpam-842	1	37	m.	m.	NOUN
ejpam-842	1	38	khan1,∗,̧	khan1,∗,̧	NOUN
ejpam-842	1	39	t.	t.	NOUN
ejpam-842	1	40	noiri	noiri	PROPN
ejpam-842	1	41	2	2	NUM
ejpam-842	1	42	1	1	NUM
ejpam-842	1	43	department	department	NOUN
ejpam-842	1	44	of	of	ADP
ejpam-842	1	45	mathematics	mathematic	NOUN
ejpam-842	1	46	,	,	PUNCT
ejpam-842	1	47	comsats	comsats	PROPN
ejpam-842	1	48	institute	institute	PROPN
ejpam-842	1	49	of	of	ADP
ejpam-842	1	50	information	information	NOUN
ejpam-842	1	51	technology	technology	NOUN
ejpam-842	1	52	,	,	PUNCT
ejpam-842	1	53	park	park	NOUN
ejpam-842	1	54	road	road	NOUN
ejpam-842	1	55	,	,	PUNCT
ejpam-842	1	56	islamabad	islamabad	PROPN
ejpam-842	1	57	,	,	PUNCT
ejpam-842	1	58	pakistan	pakistan	PROPN
ejpam-842	1	59	2	2	NUM
ejpam-842	1	60	2949	2949	NUM
ejpam-842	1	61	-	-	SYM
ejpam-842	1	62	1	1	NUM
ejpam-842	1	63	shiokita	shiokita	NOUN
ejpam-842	1	64	-	-	PUNCT
ejpam-842	1	65	cho	cho	ADJ
ejpam-842	1	66	,	,	PUNCT
ejpam-842	1	67	hinagu	hinagu	ADJ
ejpam-842	1	68	,	,	PUNCT
ejpam-842	1	69	yatsushiro	yatsushiro	PROPN
ejpam-842	1	70	-	-	PUNCT
ejpam-842	1	71	shi	shi	PROPN
ejpam-842	1	72	,	,	PUNCT
ejpam-842	1	73	kumamoto	kumamoto	PROPN
ejpam-842	1	74	-	-	PUNCT
ejpam-842	1	75	ken	ken	PROPN
ejpam-842	1	76	,	,	PUNCT
ejpam-842	1	77	869	869	NUM
ejpam-842	1	78	-	-	SYM
ejpam-842	1	79	5142	5142	NUM
ejpam-842	1	80	japan	japan	PROPN
ejpam-842	1	81	abstract	abstract	NOUN
ejpam-842	1	82	.	.	PUNCT
ejpam-842	2	1	by	by	ADP
ejpam-842	2	2	using	use	VERB
ejpam-842	2	3	is⋆g	is⋆g	ADJ
ejpam-842	2	4	-closed	-close	VERB
ejpam-842	2	5	sets	set	NOUN
ejpam-842	2	6	due	due	ADJ
ejpam-842	2	7	to	to	ADP
ejpam-842	2	8	khan	khan	PROPN
ejpam-842	2	9	and	and	CCONJ
ejpam-842	2	10	hamza	hamza	PROPN
ejpam-842	3	1	[	[	X
ejpam-842	3	2	5	5	NUM
ejpam-842	3	3	]	]	PUNCT
ejpam-842	3	4	,	,	PUNCT
ejpam-842	3	5	we	we	PRON
ejpam-842	3	6	introduce	introduce	VERB
ejpam-842	3	7	the	the	DET
ejpam-842	3	8	notion	notion	NOUN
ejpam-842	3	9	of	of	ADP
ejpam-842	3	10	is⋆	is⋆	PROPN
ejpam-842	3	11	g	g	PROPN
ejpam-842	3	12	continuous	continuous	ADJ
ejpam-842	3	13	functions	function	NOUN
ejpam-842	3	14	in	in	ADP
ejpam-842	3	15	ideal	ideal	ADJ
ejpam-842	3	16	topological	topological	ADJ
ejpam-842	3	17	spaces	space	NOUN
ejpam-842	3	18	.	.	PUNCT
ejpam-842	4	1	we	we	PRON
ejpam-842	4	2	obtain	obtain	VERB
ejpam-842	4	3	several	several	ADJ
ejpam-842	4	4	properties	property	NOUN
ejpam-842	4	5	of	of	ADP
ejpam-842	4	6	is⋆	is⋆	NOUN
ejpam-842	4	7	g	g	PROPN
ejpam-842	4	8	-continuity	-continuity	PROPN
ejpam-842	4	9	and	and	CCONJ
ejpam-842	4	10	the	the	DET
ejpam-842	4	11	relationship	relationship	NOUN
ejpam-842	4	12	between	between	ADP
ejpam-842	4	13	this	this	DET
ejpam-842	4	14	function	function	NOUN
ejpam-842	4	15	and	and	CCONJ
ejpam-842	4	16	other	other	ADJ
ejpam-842	4	17	related	related	ADJ
ejpam-842	4	18	functions	function	NOUN
ejpam-842	4	19	.	.	PUNCT
ejpam-842	5	1	2000	2000	NUM
ejpam-842	5	2	mathematics	mathematic	NOUN
ejpam-842	5	3	subject	subject	NOUN
ejpam-842	5	4	classifications	classification	NOUN
ejpam-842	5	5	:	:	PUNCT
ejpam-842	5	6	45a05	45a05	NUM
ejpam-842	5	7	,	,	PUNCT
ejpam-842	5	8	45a10	45a10	NUM
ejpam-842	5	9	key	key	ADJ
ejpam-842	5	10	words	word	NOUN
ejpam-842	5	11	and	and	CCONJ
ejpam-842	5	12	phrases	phrase	NOUN
ejpam-842	5	13	:	:	PUNCT
ejpam-842	5	14	local	local	ADJ
ejpam-842	5	15	-	-	PUNCT
ejpam-842	5	16	function	function	NOUN
ejpam-842	5	17	,	,	PUNCT
ejpam-842	5	18	is⋆g	is⋆g	X
ejpam-842	5	19	-closed	-close	VERB
ejpam-842	5	20	set	set	NOUN
ejpam-842	5	21	,	,	PUNCT
ejpam-842	5	22	is⋆g	is⋆g	ADJ
ejpam-842	5	23	-continuous	-continuous	ADJ
ejpam-842	5	24	,	,	PUNCT
ejpam-842	5	25	strong	strong	ADJ
ejpam-842	5	26	is⋆g	is⋆g	NOUN
ejpam-842	5	27	-continuous	-continuous	ADJ
ejpam-842	5	28	,	,	PUNCT
ejpam-842	5	29	weakly	weakly	ADV
ejpam-842	5	30	is⋆g	is⋆g	ADJ
ejpam-842	5	31	-continuous	-continuous	ADJ
ejpam-842	5	32	,	,	PUNCT
ejpam-842	5	33	t1/2	t1/2	NOUN
ejpam-842	5	34	-	-	NOUN
ejpam-842	5	35	space	space	NOUN
ejpam-842	5	36	.	.	PUNCT
ejpam-842	6	1	1	1	X
ejpam-842	6	2	.	.	X
ejpam-842	6	3	introduction	introduction	NOUN
ejpam-842	6	4	khan	khan	PROPN
ejpam-842	6	5	and	and	CCONJ
ejpam-842	6	6	hamza	hamza	PROPN
ejpam-842	7	1	[	[	X
ejpam-842	7	2	5	5	NUM
ejpam-842	7	3	]	]	PUNCT
ejpam-842	7	4	introduced	introduce	VERB
ejpam-842	7	5	and	and	CCONJ
ejpam-842	7	6	investigated	investigate	VERB
ejpam-842	7	7	the	the	DET
ejpam-842	7	8	notion	notion	NOUN
ejpam-842	7	9	of	of	ADP
ejpam-842	7	10	is⋆g	is⋆g	PRON
ejpam-842	7	11	-closed	-close	VERB
ejpam-842	7	12	sets	set	NOUN
ejpam-842	7	13	in	in	ADP
ejpam-842	7	14	ideal	ideal	ADJ
ejpam-842	7	15	topological	topological	ADJ
ejpam-842	7	16	spaces	space	NOUN
ejpam-842	7	17	as	as	ADP
ejpam-842	7	18	a	a	DET
ejpam-842	7	19	generalization	generalization	NOUN
ejpam-842	7	20	of	of	ADP
ejpam-842	7	21	ig	ig	PROPN
ejpam-842	7	22	-closed	-close	VERB
ejpam-842	7	23	sets	set	NOUN
ejpam-842	7	24	due	due	ADP
ejpam-842	7	25	to	to	ADP
ejpam-842	7	26	dontchev	dontchev	PROPN
ejpam-842	7	27	et	et	PROPN
ejpam-842	7	28	al	al	PROPN
ejpam-842	7	29	.	.	PUNCT
ejpam-842	8	1	[	[	X
ejpam-842	8	2	2	2	NUM
ejpam-842	8	3	]	]	PUNCT
ejpam-842	8	4	.	.	PUNCT
ejpam-842	9	1	in	in	ADP
ejpam-842	9	2	this	this	DET
ejpam-842	9	3	paper	paper	NOUN
ejpam-842	9	4	,	,	PUNCT
ejpam-842	9	5	by	by	ADP
ejpam-842	9	6	using	use	VERB
ejpam-842	9	7	is⋆g	is⋆g	ADJ
ejpam-842	9	8	-	-	PUNCT
ejpam-842	9	9	closed	closed	ADJ
ejpam-842	9	10	sets	set	NOUN
ejpam-842	9	11	we	we	PRON
ejpam-842	9	12	introduce	introduce	VERB
ejpam-842	9	13	is⋆g	is⋆g	NOUN
ejpam-842	9	14	-continuous	-continuous	ADJ
ejpam-842	9	15	functions	function	NOUN
ejpam-842	9	16	,	,	PUNCT
ejpam-842	9	17	strongly	strongly	ADV
ejpam-842	9	18	is⋆g	is⋆g	ADJ
ejpam-842	9	19	-continuous	-continuous	ADJ
ejpam-842	9	20	functions	function	NOUN
ejpam-842	9	21	and	and	CCONJ
ejpam-842	9	22	weakly	weakly	ADJ
ejpam-842	9	23	is⋆g	is⋆g	ADJ
ejpam-842	9	24	-continuous	-continuous	ADJ
ejpam-842	9	25	functions	function	NOUN
ejpam-842	9	26	.	.	PUNCT
ejpam-842	10	1	it	it	PRON
ejpam-842	10	2	turns	turn	VERB
ejpam-842	10	3	out	out	ADP
ejpam-842	10	4	that	that	SCONJ
ejpam-842	10	5	weak	weak	ADJ
ejpam-842	10	6	is⋆	is⋆	PROPN
ejpam-842	10	7	g	g	NOUN
ejpam-842	10	8	-	-	PUNCT
ejpam-842	10	9	continuity	continuity	NOUN
ejpam-842	10	10	is	be	AUX
ejpam-842	10	11	weaker	weak	ADJ
ejpam-842	10	12	than	than	ADP
ejpam-842	10	13	weak	weak	ADJ
ejpam-842	10	14	i	i	NOUN
ejpam-842	10	15	-	-	PUNCT
ejpam-842	10	16	continuity	continuity	NOUN
ejpam-842	10	17	defined	define	VERB
ejpam-842	10	18	by	by	ADP
ejpam-842	10	19	ackgoz	ackgoz	PROPN
ejpam-842	10	20	et	et	PROPN
ejpam-842	10	21	al	al	PROPN
ejpam-842	10	22	.	.	PUNCT
ejpam-842	11	1	[	[	X
ejpam-842	11	2	1	1	X
ejpam-842	11	3	]	]	PUNCT
ejpam-842	11	4	.	.	PUNCT
ejpam-842	12	1	we	we	PRON
ejpam-842	12	2	obtain	obtain	VERB
ejpam-842	12	3	several	several	ADJ
ejpam-842	12	4	properties	property	NOUN
ejpam-842	12	5	of	of	ADP
ejpam-842	12	6	is⋆g	is⋆g	ADJ
ejpam-842	12	7	-continuity	-continuity	ADJ
ejpam-842	12	8	and	and	CCONJ
ejpam-842	12	9	the	the	DET
ejpam-842	12	10	relationship	relationship	NOUN
ejpam-842	12	11	between	between	ADP
ejpam-842	12	12	this	this	DET
ejpam-842	12	13	function	function	NOUN
ejpam-842	12	14	and	and	CCONJ
ejpam-842	12	15	other	other	ADJ
ejpam-842	12	16	related	related	ADJ
ejpam-842	12	17	functions	function	NOUN
ejpam-842	12	18	.	.	PUNCT
ejpam-842	13	1	2	2	X
ejpam-842	13	2	.	.	X
ejpam-842	13	3	preliminaries	preliminary	NOUN
ejpam-842	13	4	let	let	VERB
ejpam-842	13	5	(	(	PUNCT
ejpam-842	13	6	x	x	X
ejpam-842	13	7	,	,	PUNCT
ejpam-842	13	8	τ	τ	X
ejpam-842	13	9	)	)	PUNCT
ejpam-842	13	10	be	be	VERB
ejpam-842	13	11	a	a	DET
ejpam-842	13	12	topological	topological	ADJ
ejpam-842	13	13	space	space	NOUN
ejpam-842	13	14	with	with	ADP
ejpam-842	13	15	no	no	DET
ejpam-842	13	16	separation	separation	NOUN
ejpam-842	13	17	properties	property	NOUN
ejpam-842	13	18	assumed	assume	VERB
ejpam-842	13	19	.	.	PUNCT
ejpam-842	14	1	for	for	ADP
ejpam-842	14	2	a	a	DET
ejpam-842	14	3	subset	subset	NOUN
ejpam-842	14	4	a	a	PRON
ejpam-842	14	5	of	of	ADP
ejpam-842	14	6	a	a	DET
ejpam-842	14	7	topological	topological	ADJ
ejpam-842	14	8	space	space	NOUN
ejpam-842	14	9	(	(	PUNCT
ejpam-842	14	10	x	x	X
ejpam-842	14	11	,	,	PUNCT
ejpam-842	14	12	τ	τ	PROPN
ejpam-842	14	13	)	)	PUNCT
ejpam-842	14	14	,	,	PUNCT
ejpam-842	14	15	cl(a	cl(a	NUM
ejpam-842	14	16	)	)	PUNCT
ejpam-842	14	17	and	and	CCONJ
ejpam-842	14	18	int(a	int(a	PROPN
ejpam-842	14	19	)	)	PUNCT
ejpam-842	14	20	denote	denote	VERB
ejpam-842	14	21	the	the	DET
ejpam-842	14	22	closure	closure	NOUN
ejpam-842	14	23	and	and	CCONJ
ejpam-842	14	24	interior	interior	NOUN
ejpam-842	14	25	of	of	ADP
ejpam-842	14	26	a	a	DET
ejpam-842	14	27	in	in	ADP
ejpam-842	14	28	(	(	PUNCT
ejpam-842	14	29	x	x	INTJ
ejpam-842	14	30	,	,	PUNCT
ejpam-842	14	31	τ	τ	PROPN
ejpam-842	14	32	)	)	PUNCT
ejpam-842	14	33	,	,	PUNCT
ejpam-842	14	34	respectively	respectively	ADV
ejpam-842	14	35	.	.	PUNCT
ejpam-842	15	1	an	an	DET
ejpam-842	15	2	ideal	ideal	NOUN
ejpam-842	15	3	i	i	PRON
ejpam-842	15	4	on	on	ADP
ejpam-842	15	5	a	a	DET
ejpam-842	15	6	set	set	NOUN
ejpam-842	15	7	x	x	PUNCT
ejpam-842	15	8	is	be	AUX
ejpam-842	15	9	a	a	DET
ejpam-842	15	10	non	non	ADJ
ejpam-842	15	11	-	-	ADJ
ejpam-842	15	12	empty	empty	ADJ
ejpam-842	15	13	collection	collection	NOUN
ejpam-842	15	14	of	of	ADP
ejpam-842	15	15	subsets	subset	NOUN
ejpam-842	15	16	of	of	ADP
ejpam-842	15	17	x	x	PUNCT
ejpam-842	15	18	which	which	PRON
ejpam-842	15	19	satisfies	satisfy	VERB
ejpam-842	15	20	the	the	DET
ejpam-842	15	21	following	follow	VERB
ejpam-842	15	22	properties	property	NOUN
ejpam-842	15	23	:	:	PUNCT
ejpam-842	15	24	(	(	PUNCT
ejpam-842	15	25	1	1	X
ejpam-842	15	26	)	)	PUNCT
ejpam-842	15	27	a∈	a∈	PROPN
ejpam-842	15	28	i	i	PROPN
ejpam-842	15	29	and	and	CCONJ
ejpam-842	15	30	b	b	PROPN
ejpam-842	16	1	⊂	⊂	PROPN
ejpam-842	16	2	a	a	PRON
ejpam-842	16	3	implies	imply	VERB
ejpam-842	16	4	b	b	X
ejpam-842	16	5	∈	∈	NOUN
ejpam-842	16	6	i	i	PRON
ejpam-842	16	7	,	,	PUNCT
ejpam-842	16	8	(	(	PUNCT
ejpam-842	16	9	2	2	X
ejpam-842	16	10	)	)	PUNCT
ejpam-842	16	11	a∈	a∈	PROPN
ejpam-842	16	12	i	i	PROPN
ejpam-842	16	13	and	and	CCONJ
ejpam-842	16	14	b	b	X
ejpam-842	16	15	∈	∈	PROPN
ejpam-842	16	16	i	i	PRON
ejpam-842	16	17	implies	imply	VERB
ejpam-842	16	18	a∪	a∪	PROPN
ejpam-842	17	1	b	b	X
ejpam-842	17	2	∈	∈	PROPN
ejpam-842	18	1	i	i	PRON
ejpam-842	18	2	.	.	PUNCT
ejpam-842	19	1	∗corresponding	∗corresponde	VERB
ejpam-842	19	2	author	author	NOUN
ejpam-842	19	3	.	.	PUNCT
ejpam-842	20	1	email	email	NOUN
ejpam-842	20	2	addresses	address	NOUN
ejpam-842	20	3	:	:	PUNCT
ejpam-842	20	4	profmoiz001	profmoiz001	PROPN
ejpam-842	20	5	�	�	PROPN
ejpam-842	20	6	yahoo	yahoo	PROPN
ejpam-842	20	7	.	.	PUNCT
ejpam-842	20	8	om	om	PROPN
ejpam-842	20	9	(	(	PUNCT
ejpam-842	20	10	m.	m.	PROPN
ejpam-842	20	11	khan	khan	PROPN
ejpam-842	20	12	)	)	PUNCT
ejpam-842	20	13	,	,	PUNCT
ejpam-842	20	14	t.noiri	t.noiri	ADV
ejpam-842	20	15	�	�	NOUN
ejpam-842	20	16	nifty	nifty	ADJ
ejpam-842	20	17	.	.	PUNCT
ejpam-842	21	1	om	om	PROPN
ejpam-842	21	2	(	(	PUNCT
ejpam-842	21	3	t.	t.	PROPN
ejpam-842	21	4	noiri	noiri	PROPN
ejpam-842	21	5	)	)	PUNCT
ejpam-842	21	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-842	22	1	237	237	NUM
ejpam-842	23	1	c	c	X
ejpam-842	23	2	©	©	NOUN
ejpam-842	23	3	2011	2011	NUM
ejpam-842	23	4	ejpam	ejpam	VERB
ejpam-842	23	5	all	all	DET
ejpam-842	23	6	rights	right	NOUN
ejpam-842	23	7	reserved	reserve	VERB
ejpam-842	23	8	.	.	PUNCT
ejpam-842	24	1	m.	m.	NOUN
ejpam-842	24	2	khan,̧	khan,̧	PROPN
ejpam-842	24	3	t.	t.	PROPN
ejpam-842	24	4	noiri	noiri	PROPN
ejpam-842	24	5	/	/	SYM
ejpam-842	24	6	eur	eur	PROPN
ejpam-842	24	7	.	.	PUNCT
ejpam-842	25	1	j.	j.	PROPN
ejpam-842	25	2	pure	pure	PROPN
ejpam-842	25	3	appl	appl	PROPN
ejpam-842	25	4	.	.	PROPN
ejpam-842	25	5	math	math	PROPN
ejpam-842	25	6	,	,	PUNCT
ejpam-842	25	7	4	4	NUM
ejpam-842	25	8	(	(	PUNCT
ejpam-842	25	9	2011	2011	NUM
ejpam-842	25	10	)	)	PUNCT
ejpam-842	25	11	,	,	PUNCT
ejpam-842	25	12	237	237	NUM
ejpam-842	25	13	-	-	SYM
ejpam-842	25	14	243	243	NUM
ejpam-842	25	15	238	238	NUM
ejpam-842	25	16	an	an	DET
ejpam-842	25	17	ideal	ideal	ADJ
ejpam-842	25	18	topological	topological	ADJ
ejpam-842	25	19	space	space	NOUN
ejpam-842	25	20	is	be	AUX
ejpam-842	25	21	a	a	DET
ejpam-842	25	22	topological	topological	ADJ
ejpam-842	25	23	space	space	NOUN
ejpam-842	25	24	(	(	PUNCT
ejpam-842	25	25	x	x	X
ejpam-842	25	26	,	,	PUNCT
ejpam-842	25	27	τ	τ	PROPN
ejpam-842	25	28	)	)	PUNCT
ejpam-842	25	29	with	with	ADP
ejpam-842	25	30	an	an	DET
ejpam-842	25	31	ideal	ideal	ADJ
ejpam-842	25	32	i	i	PRON
ejpam-842	25	33	on	on	ADP
ejpam-842	25	34	x	x	PUNCT
ejpam-842	25	35	and	and	CCONJ
ejpam-842	25	36	is	be	AUX
ejpam-842	25	37	denoted	denote	VERB
ejpam-842	25	38	by	by	ADP
ejpam-842	25	39	(	(	PUNCT
ejpam-842	25	40	x	x	INTJ
ejpam-842	25	41	,	,	PUNCT
ejpam-842	25	42	τ	τ	PROPN
ejpam-842	25	43	,	,	PUNCT
ejpam-842	25	44	i	i	PROPN
ejpam-842	25	45	)	)	PUNCT
ejpam-842	25	46	.	.	PUNCT
ejpam-842	26	1	for	for	ADP
ejpam-842	26	2	a	a	DET
ejpam-842	26	3	subset	subset	NOUN
ejpam-842	26	4	a	a	DET
ejpam-842	26	5	⊂	⊂	PROPN
ejpam-842	26	6	x	x	X
ejpam-842	26	7	,	,	PUNCT
ejpam-842	26	8	a⋆(i	a⋆(i	PROPN
ejpam-842	26	9	,	,	PUNCT
ejpam-842	26	10	τ	τ	X
ejpam-842	26	11	)	)	PUNCT
ejpam-842	26	12	=	=	PRON
ejpam-842	26	13	{	{	PUNCT
ejpam-842	26	14	x	x	PUNCT
ejpam-842	26	15	∈	∈	NOUN
ejpam-842	26	16	x	x	X
ejpam-842	26	17	:	:	PUNCT
ejpam-842	26	18	a∩	a∩	PROPN
ejpam-842	26	19	u	u	NOUN
ejpam-842	26	20	/∈	/∈	PUNCT
ejpam-842	27	1	i	i	PRON
ejpam-842	27	2	for	for	ADP
ejpam-842	27	3	every	every	DET
ejpam-842	27	4	u	u	PROPN
ejpam-842	27	5	∈	∈	PROPN
ejpam-842	27	6	τ(x	τ(x	NOUN
ejpam-842	27	7	)	)	PUNCT
ejpam-842	27	8	}	}	PUNCT
ejpam-842	27	9	,	,	PUNCT
ejpam-842	27	10	where	where	SCONJ
ejpam-842	27	11	,	,	PUNCT
ejpam-842	27	12	τ(x	τ(x	ADJ
ejpam-842	27	13	)	)	PUNCT
ejpam-842	27	14	=	=	PRON
ejpam-842	28	1	{	{	PUNCT
ejpam-842	28	2	u	u	X
ejpam-842	28	3	∈	∈	PROPN
ejpam-842	28	4	τ	τ	X
ejpam-842	28	5	:	:	PUNCT
ejpam-842	28	6	x	x	SYM
ejpam-842	28	7	∈	∈	PROPN
ejpam-842	28	8	u	u	NOUN
ejpam-842	28	9	}	}	PUNCT
ejpam-842	28	10	,	,	PUNCT
ejpam-842	28	11	is	be	AUX
ejpam-842	28	12	called	call	VERB
ejpam-842	28	13	the	the	DET
ejpam-842	28	14	local	local	ADJ
ejpam-842	28	15	function	function	NOUN
ejpam-842	28	16	of	of	ADP
ejpam-842	28	17	a	a	PRON
ejpam-842	28	18	with	with	ADP
ejpam-842	28	19	respect	respect	NOUN
ejpam-842	28	20	to	to	ADP
ejpam-842	28	21	i	i	PRON
ejpam-842	28	22	and	and	CCONJ
ejpam-842	28	23	τ	τ	X
ejpam-842	29	1	[	[	X
ejpam-842	29	2	4	4	NUM
ejpam-842	29	3	,	,	PUNCT
ejpam-842	29	4	6	6	NUM
ejpam-842	29	5	]	]	PUNCT
ejpam-842	29	6	.	.	PUNCT
ejpam-842	30	1	we	we	PRON
ejpam-842	30	2	simply	simply	ADV
ejpam-842	30	3	write	write	VERB
ejpam-842	30	4	a⋆	a⋆	ADV
ejpam-842	30	5	or	or	CCONJ
ejpam-842	30	6	a⋆x	a⋆x	PROPN
ejpam-842	30	7	instead	instead	ADV
ejpam-842	30	8	of	of	ADP
ejpam-842	30	9	a⋆(i	a⋆(i	NOUN
ejpam-842	30	10	,	,	PUNCT
ejpam-842	30	11	τ	τ	X
ejpam-842	30	12	)	)	PUNCT
ejpam-842	30	13	and	and	CCONJ
ejpam-842	30	14	b⋆a	b⋆a	VERB
ejpam-842	30	15	for	for	ADP
ejpam-842	30	16	b⋆(ia	b⋆(ia	PROPN
ejpam-842	30	17	,	,	PUNCT
ejpam-842	30	18	τa	τa	NOUN
ejpam-842	30	19	)	)	PUNCT
ejpam-842	30	20	in	in	ADP
ejpam-842	30	21	case	case	NOUN
ejpam-842	30	22	there	there	PRON
ejpam-842	30	23	is	be	VERB
ejpam-842	30	24	no	no	DET
ejpam-842	30	25	chance	chance	NOUN
ejpam-842	30	26	for	for	ADP
ejpam-842	30	27	confusion	confusion	NOUN
ejpam-842	30	28	.	.	PUNCT
ejpam-842	31	1	for	for	ADP
ejpam-842	31	2	every	every	DET
ejpam-842	31	3	ideal	ideal	ADJ
ejpam-842	31	4	topological	topological	ADJ
ejpam-842	31	5	space	space	NOUN
ejpam-842	31	6	(	(	PUNCT
ejpam-842	31	7	x	x	X
ejpam-842	31	8	,	,	PUNCT
ejpam-842	31	9	τ	τ	PROPN
ejpam-842	31	10	,	,	PUNCT
ejpam-842	31	11	i	i	PROPN
ejpam-842	31	12	)	)	PUNCT
ejpam-842	31	13	,	,	PUNCT
ejpam-842	31	14	there	there	PRON
ejpam-842	31	15	exists	exist	VERB
ejpam-842	31	16	a	a	DET
ejpam-842	31	17	topology	topology	NOUN
ejpam-842	31	18	τ⋆(i	τ⋆(i	NOUN
ejpam-842	31	19	)	)	PUNCT
ejpam-842	31	20	,	,	PUNCT
ejpam-842	31	21	finer	fine	ADJ
ejpam-842	31	22	than	than	ADP
ejpam-842	31	23	τ	τ	PROPN
ejpam-842	31	24	,	,	PUNCT
ejpam-842	31	25	generated	generate	VERB
ejpam-842	31	26	by	by	ADP
ejpam-842	31	27	the	the	DET
ejpam-842	31	28	base	base	NOUN
ejpam-842	31	29	β(i	β(i	PUNCT
ejpam-842	31	30	,	,	PUNCT
ejpam-842	31	31	τ	τ	X
ejpam-842	31	32	)	)	PUNCT
ejpam-842	31	33	=	=	PRON
ejpam-842	31	34	{	{	PUNCT
ejpam-842	31	35	u	u	NOUN
ejpam-842	31	36	−	−	PROPN
ejpam-842	31	37	j	j	NOUN
ejpam-842	31	38	:	:	PUNCT
ejpam-842	31	39	u	u	PROPN
ejpam-842	31	40	∈	∈	PROPN
ejpam-842	31	41	τ	τ	X
ejpam-842	31	42	and	and	CCONJ
ejpam-842	31	43	j	j	PROPN
ejpam-842	31	44	∈	∈	PROPN
ejpam-842	32	1	i	i	X
ejpam-842	32	2	}	}	PUNCT
ejpam-842	32	3	.	.	PUNCT
ejpam-842	33	1	it	it	PRON
ejpam-842	33	2	is	be	AUX
ejpam-842	33	3	known	know	VERB
ejpam-842	33	4	in	in	ADP
ejpam-842	33	5	[	[	X
ejpam-842	33	6	4	4	X
ejpam-842	33	7	]	]	PUNCT
ejpam-842	33	8	that	that	SCONJ
ejpam-842	33	9	β(i	β(i	PRON
ejpam-842	33	10	,	,	PUNCT
ejpam-842	33	11	τ	τ	X
ejpam-842	33	12	)	)	PUNCT
ejpam-842	33	13	is	be	AUX
ejpam-842	33	14	not	not	PART
ejpam-842	33	15	necessarily	necessarily	ADV
ejpam-842	33	16	a	a	DET
ejpam-842	33	17	topology	topology	NOUN
ejpam-842	33	18	.	.	PUNCT
ejpam-842	34	1	when	when	SCONJ
ejpam-842	34	2	there	there	PRON
ejpam-842	34	3	is	be	VERB
ejpam-842	34	4	no	no	DET
ejpam-842	34	5	ambiguity	ambiguity	NOUN
ejpam-842	34	6	,	,	PUNCT
ejpam-842	34	7	τ⋆(i	τ⋆(i	NOUN
ejpam-842	34	8	)	)	PUNCT
ejpam-842	34	9	is	be	AUX
ejpam-842	34	10	denoted	denote	VERB
ejpam-842	34	11	by	by	ADP
ejpam-842	34	12	τ⋆.	τ⋆.	PUNCT
ejpam-842	34	13	recall	recall	VERB
ejpam-842	34	14	that	that	SCONJ
ejpam-842	34	15	a	a	PRON
ejpam-842	34	16	is	be	AUX
ejpam-842	34	17	said	say	VERB
ejpam-842	34	18	to	to	PART
ejpam-842	34	19	be	be	AUX
ejpam-842	34	20	(	(	PUNCT
ejpam-842	34	21	)	)	PUNCT
ejpam-842	34	22	⋆-dense	⋆-dense	NOUN
ejpam-842	34	23	in	in	ADP
ejpam-842	34	24	itself	itself	PRON
ejpam-842	34	25	(	(	PUNCT
ejpam-842	34	26	resp	resp	PROPN
ejpam-842	34	27	.	.	PUNCT
ejpam-842	35	1	τ⋆-closed	τ⋆-close	VERB
ejpam-842	35	2	,	,	PUNCT
ejpam-842	35	3	(	(	PUNCT
ejpam-842	35	4	)	)	PUNCT
ejpam-842	35	5	⋆-perfect	⋆-perfect	ADJ
ejpam-842	35	6	)	)	PUNCT
ejpam-842	35	7	if	if	SCONJ
ejpam-842	35	8	a⊂	a⊂	PRON
ejpam-842	35	9	a⋆	a⋆	NOUN
ejpam-842	35	10	(	(	PUNCT
ejpam-842	35	11	resp	resp	NOUN
ejpam-842	35	12	.	.	PUNCT
ejpam-842	35	13	a⋆	a⋆	PUNCT
ejpam-842	36	1	⊂	⊂	PROPN
ejpam-842	36	2	a	a	PRON
ejpam-842	36	3	,	,	PUNCT
ejpam-842	36	4	a=	a=	ADJ
ejpam-842	36	5	a⋆	a⋆	ADJ
ejpam-842	36	6	)	)	PUNCT
ejpam-842	36	7	.	.	PUNCT
ejpam-842	37	1	for	for	ADP
ejpam-842	37	2	a	a	DET
ejpam-842	37	3	subset	subset	NOUN
ejpam-842	37	4	a	a	PRON
ejpam-842	37	5	of	of	ADP
ejpam-842	37	6	x	x	PRON
ejpam-842	37	7	,	,	PUNCT
ejpam-842	37	8	cl⋆(a	cl⋆(a	PROPN
ejpam-842	37	9	)	)	PUNCT
ejpam-842	37	10	and	and	CCONJ
ejpam-842	37	11	int⋆(a	int⋆(a	NOUN
ejpam-842	37	12	)	)	PUNCT
ejpam-842	37	13	will	will	AUX
ejpam-842	37	14	,	,	PUNCT
ejpam-842	37	15	respectively	respectively	ADV
ejpam-842	37	16	,	,	PUNCT
ejpam-842	37	17	denote	denote	VERB
ejpam-842	37	18	the	the	DET
ejpam-842	37	19	closure	closure	NOUN
ejpam-842	37	20	and	and	CCONJ
ejpam-842	37	21	interior	interior	NOUN
ejpam-842	37	22	of	of	ADP
ejpam-842	37	23	a	a	DET
ejpam-842	37	24	in	in	ADP
ejpam-842	37	25	(	(	PUNCT
ejpam-842	37	26	x	x	INTJ
ejpam-842	37	27	,	,	PUNCT
ejpam-842	37	28	τ⋆	τ⋆	NUM
ejpam-842	37	29	)	)	PUNCT
ejpam-842	37	30	.	.	PUNCT
ejpam-842	38	1	a	a	DET
ejpam-842	38	2	subset	subset	NOUN
ejpam-842	38	3	a	a	PRON
ejpam-842	38	4	of	of	ADP
ejpam-842	38	5	x	x	SYM
ejpam-842	38	6	is	be	AUX
ejpam-842	38	7	said	say	VERB
ejpam-842	38	8	to	to	PART
ejpam-842	38	9	be	be	AUX
ejpam-842	38	10	semi	semi	ADJ
ejpam-842	38	11	-	-	ADJ
ejpam-842	38	12	open	open	ADJ
ejpam-842	38	13	[	[	X
ejpam-842	38	14	7	7	NUM
ejpam-842	38	15	]	]	X
ejpam-842	38	16	if	if	SCONJ
ejpam-842	38	17	there	there	PRON
ejpam-842	38	18	exists	exist	VERB
ejpam-842	38	19	an	an	DET
ejpam-842	38	20	open	open	ADJ
ejpam-842	38	21	set	set	NOUN
ejpam-842	38	22	u	u	NOUN
ejpam-842	38	23	in	in	ADP
ejpam-842	38	24	x	x	SYM
ejpam-842	38	25	such	such	ADJ
ejpam-842	38	26	that	that	SCONJ
ejpam-842	38	27	u	u	PROPN
ejpam-842	38	28	⊂	⊂	PROPN
ejpam-842	38	29	a	a	DET
ejpam-842	38	30	⊂	⊂	PROPN
ejpam-842	38	31	cl(u	cl(u	PROPN
ejpam-842	38	32	)	)	PUNCT
ejpam-842	38	33	.	.	PUNCT
ejpam-842	39	1	the	the	DET
ejpam-842	39	2	complement	complement	NOUN
ejpam-842	39	3	of	of	ADP
ejpam-842	39	4	a	a	DET
ejpam-842	39	5	semi	semi	ADJ
ejpam-842	39	6	-	-	ADJ
ejpam-842	39	7	open	open	ADJ
ejpam-842	39	8	set	set	NOUN
ejpam-842	39	9	is	be	AUX
ejpam-842	39	10	said	say	VERB
ejpam-842	39	11	to	to	PART
ejpam-842	39	12	be	be	AUX
ejpam-842	39	13	semi	semi	ADJ
ejpam-842	39	14	-	-	ADJ
ejpam-842	39	15	closed	closed	ADJ
ejpam-842	39	16	.	.	PUNCT
ejpam-842	40	1	a	a	DET
ejpam-842	40	2	subset	subset	NOUN
ejpam-842	40	3	a	a	PRON
ejpam-842	40	4	is	be	AUX
ejpam-842	40	5	said	say	VERB
ejpam-842	40	6	to	to	PART
ejpam-842	40	7	be	be	AUX
ejpam-842	40	8	semi	semi	ADJ
ejpam-842	40	9	-	-	ADJ
ejpam-842	40	10	regular	regular	ADJ
ejpam-842	40	11	if	if	SCONJ
ejpam-842	40	12	a	a	PRON
ejpam-842	40	13	is	be	AUX
ejpam-842	40	14	semi	semi	ADJ
ejpam-842	40	15	-	-	ADJ
ejpam-842	40	16	open	open	ADJ
ejpam-842	40	17	and	and	CCONJ
ejpam-842	40	18	semi	semi	ADJ
ejpam-842	40	19	-	-	ADJ
ejpam-842	40	20	closed	closed	ADJ
ejpam-842	40	21	.	.	PUNCT
ejpam-842	41	1	a	a	DET
ejpam-842	41	2	subset	subset	NOUN
ejpam-842	41	3	a	a	PRON
ejpam-842	41	4	of	of	ADP
ejpam-842	41	5	x	x	SYM
ejpam-842	41	6	is	be	AUX
ejpam-842	41	7	said	say	VERB
ejpam-842	41	8	to	to	PART
ejpam-842	41	9	be	be	AUX
ejpam-842	41	10	generalized	generalize	VERB
ejpam-842	41	11	closed	close	VERB
ejpam-842	41	12	[	[	X
ejpam-842	41	13	8	8	NUM
ejpam-842	41	14	]	]	PUNCT
ejpam-842	41	15	(	(	PUNCT
ejpam-842	41	16	briefly	briefly	ADV
ejpam-842	41	17	,	,	PUNCT
ejpam-842	41	18	g	g	NOUN
ejpam-842	41	19	-	-	PUNCT
ejpam-842	41	20	closed	closed	ADJ
ejpam-842	41	21	)	)	PUNCT
ejpam-842	41	22	if	if	SCONJ
ejpam-842	41	23	cl(a	cl(a	NUM
ejpam-842	41	24	)	)	PUNCT
ejpam-842	42	1	⊂	⊂	PROPN
ejpam-842	42	2	u	u	NOUN
ejpam-842	42	3	whenever	whenever	SCONJ
ejpam-842	42	4	a	a	DET
ejpam-842	42	5	⊂	⊂	PROPN
ejpam-842	42	6	u	u	NOUN
ejpam-842	42	7	and	and	CCONJ
ejpam-842	42	8	u	u	NOUN
ejpam-842	42	9	is	be	AUX
ejpam-842	42	10	open	open	ADJ
ejpam-842	42	11	in	in	ADP
ejpam-842	42	12	x	x	X
ejpam-842	42	13	.	.	PUNCT
ejpam-842	43	1	the	the	DET
ejpam-842	43	2	complement	complement	NOUN
ejpam-842	43	3	of	of	ADP
ejpam-842	43	4	a	a	DET
ejpam-842	43	5	g	g	NOUN
ejpam-842	43	6	-	-	PUNCT
ejpam-842	43	7	closed	close	VERB
ejpam-842	43	8	set	set	NOUN
ejpam-842	43	9	is	be	AUX
ejpam-842	43	10	said	say	VERB
ejpam-842	43	11	to	to	PART
ejpam-842	43	12	be	be	AUX
ejpam-842	43	13	g	g	NOUN
ejpam-842	43	14	-	-	PUNCT
ejpam-842	43	15	open	open	ADJ
ejpam-842	43	16	.	.	PUNCT
ejpam-842	44	1	a	a	DET
ejpam-842	44	2	space	space	NOUN
ejpam-842	44	3	x	x	PUNCT
ejpam-842	44	4	is	be	AUX
ejpam-842	44	5	called	call	VERB
ejpam-842	44	6	a	a	DET
ejpam-842	44	7	t1/2	t1/2	NOUN
ejpam-842	44	8	-	-	PUNCT
ejpam-842	44	9	space	space	NOUN
ejpam-842	44	10	[	[	X
ejpam-842	44	11	3	3	X
ejpam-842	44	12	]	]	PUNCT
ejpam-842	44	13	if	if	SCONJ
ejpam-842	44	14	every	every	DET
ejpam-842	44	15	g	g	NOUN
ejpam-842	44	16	-	-	PUNCT
ejpam-842	44	17	closed	close	VERB
ejpam-842	44	18	set	set	NOUN
ejpam-842	44	19	in	in	ADP
ejpam-842	44	20	x	x	PUNCT
ejpam-842	44	21	is	be	AUX
ejpam-842	44	22	closed	close	VERB
ejpam-842	44	23	.	.	PUNCT
ejpam-842	45	1	recall	recall	VERB
ejpam-842	45	2	that	that	SCONJ
ejpam-842	45	3	if	if	SCONJ
ejpam-842	45	4	(	(	PUNCT
ejpam-842	45	5	x	x	X
ejpam-842	45	6	,	,	PUNCT
ejpam-842	45	7	τ	τ	PROPN
ejpam-842	45	8	,	,	PUNCT
ejpam-842	45	9	i	i	PROPN
ejpam-842	45	10	)	)	PUNCT
ejpam-842	45	11	is	be	AUX
ejpam-842	45	12	an	an	DET
ejpam-842	45	13	ideal	ideal	ADJ
ejpam-842	45	14	topological	topological	ADJ
ejpam-842	45	15	space	space	NOUN
ejpam-842	45	16	and	and	CCONJ
ejpam-842	45	17	a	a	PRON
ejpam-842	45	18	is	be	AUX
ejpam-842	45	19	a	a	DET
ejpam-842	45	20	subset	subset	NOUN
ejpam-842	45	21	of	of	ADP
ejpam-842	45	22	x	x	PRON
ejpam-842	45	23	,	,	PUNCT
ejpam-842	45	24	then	then	ADV
ejpam-842	45	25	(	(	PUNCT
ejpam-842	45	26	a	a	DET
ejpam-842	45	27	,	,	PUNCT
ejpam-842	45	28	τa	τa	PROPN
ejpam-842	45	29	,	,	PUNCT
ejpam-842	45	30	ia	ia	PROPN
ejpam-842	45	31	)	)	PUNCT
ejpam-842	45	32	is	be	AUX
ejpam-842	45	33	an	an	DET
ejpam-842	45	34	ideal	ideal	ADJ
ejpam-842	45	35	topological	topological	ADJ
ejpam-842	45	36	space	space	NOUN
ejpam-842	45	37	,	,	PUNCT
ejpam-842	45	38	where	where	SCONJ
ejpam-842	45	39	τa	τa	ADV
ejpam-842	45	40	is	be	AUX
ejpam-842	45	41	the	the	DET
ejpam-842	45	42	relative	relative	ADJ
ejpam-842	45	43	topology	topology	NOUN
ejpam-842	45	44	on	on	ADP
ejpam-842	45	45	a	a	PRON
ejpam-842	45	46	and	and	CCONJ
ejpam-842	45	47	ia	ia	NOUN
ejpam-842	45	48	=	=	SYM
ejpam-842	45	49	{	{	PUNCT
ejpam-842	45	50	a∩	a∩	PROPN
ejpam-842	45	51	j	j	PROPN
ejpam-842	45	52	:	:	PUNCT
ejpam-842	45	53	j	j	PROPN
ejpam-842	45	54	∈	∈	PROPN
ejpam-842	46	1	i	i	X
ejpam-842	46	2	}	}	PUNCT
ejpam-842	46	3	.	.	PUNCT
ejpam-842	47	1	3	3	X
ejpam-842	47	2	.	.	X
ejpam-842	47	3	is⋆	is⋆	PROPN
ejpam-842	47	4	g	g	NOUN
ejpam-842	47	5	-	-	PUNCT
ejpam-842	47	6	closed	close	VERB
ejpam-842	47	7	sets	set	NOUN
ejpam-842	47	8	the	the	DET
ejpam-842	47	9	notion	notion	NOUN
ejpam-842	47	10	of	of	ADP
ejpam-842	47	11	is⋆	is⋆	PROPN
ejpam-842	47	12	g	g	NOUN
ejpam-842	47	13	-	-	PUNCT
ejpam-842	47	14	closed	close	VERB
ejpam-842	47	15	sets	set	NOUN
ejpam-842	47	16	was	be	AUX
ejpam-842	47	17	defined	define	VERB
ejpam-842	47	18	by	by	ADP
ejpam-842	47	19	khan	khan	PROPN
ejpam-842	47	20	and	and	CCONJ
ejpam-842	47	21	hamza	hamza	PROPN
ejpam-842	48	1	[	[	X
ejpam-842	48	2	5	5	NUM
ejpam-842	48	3	]	]	PUNCT
ejpam-842	48	4	.	.	PUNCT
ejpam-842	49	1	in	in	ADP
ejpam-842	49	2	this	this	DET
ejpam-842	49	3	section	section	NOUN
ejpam-842	49	4	we	we	PRON
ejpam-842	49	5	will	will	AUX
ejpam-842	49	6	obtain	obtain	VERB
ejpam-842	49	7	further	further	ADJ
ejpam-842	49	8	properties	property	NOUN
ejpam-842	49	9	of	of	ADP
ejpam-842	49	10	is⋆g	is⋆g	NOUN
ejpam-842	49	11	-	-	PUNCT
ejpam-842	49	12	closed	closed	ADJ
ejpam-842	49	13	sets	set	NOUN
ejpam-842	49	14	in	in	ADP
ejpam-842	49	15	ideal	ideal	ADJ
ejpam-842	49	16	topological	topological	ADJ
ejpam-842	49	17	spaces	space	NOUN
ejpam-842	49	18	.	.	PUNCT
ejpam-842	50	1	definition	definition	NOUN
ejpam-842	50	2	1	1	NUM
ejpam-842	50	3	.	.	PUNCT
ejpam-842	51	1	a	a	DET
ejpam-842	51	2	subset	subset	NOUN
ejpam-842	51	3	a	a	PRON
ejpam-842	51	4	of	of	ADP
ejpam-842	51	5	a	a	DET
ejpam-842	51	6	space	space	NOUN
ejpam-842	51	7	(	(	PUNCT
ejpam-842	51	8	x	x	X
ejpam-842	51	9	,	,	PUNCT
ejpam-842	51	10	τ	τ	PROPN
ejpam-842	51	11	,	,	PUNCT
ejpam-842	51	12	i	i	PROPN
ejpam-842	51	13	)	)	PUNCT
ejpam-842	51	14	is	be	AUX
ejpam-842	51	15	said	say	VERB
ejpam-842	51	16	to	to	PART
ejpam-842	51	17	be	be	AUX
ejpam-842	51	18	is⋆g	is⋆g	ADV
ejpam-842	51	19	-closed	-close	VERB
ejpam-842	51	20	[	[	X
ejpam-842	51	21	5	5	NUM
ejpam-842	51	22	]	]	PUNCT
ejpam-842	51	23	if	if	SCONJ
ejpam-842	51	24	a⋆	a⋆	ADJ
ejpam-842	51	25	⊂	⊂	PROPN
ejpam-842	51	26	u	u	NOUN
ejpam-842	51	27	whenever	whenever	SCONJ
ejpam-842	51	28	a	a	DET
ejpam-842	51	29	⊂	⊂	PROPN
ejpam-842	51	30	u	u	NOUN
ejpam-842	51	31	and	and	CCONJ
ejpam-842	51	32	u	u	NOUN
ejpam-842	51	33	is	be	AUX
ejpam-842	51	34	semi	semi	ADJ
ejpam-842	51	35	-	-	ADJ
ejpam-842	51	36	open	open	ADJ
ejpam-842	51	37	in	in	ADP
ejpam-842	51	38	x	x	X
ejpam-842	51	39	.	.	PUNCT
ejpam-842	52	1	the	the	DET
ejpam-842	52	2	complement	complement	NOUN
ejpam-842	52	3	of	of	ADP
ejpam-842	52	4	an	an	DET
ejpam-842	52	5	is⋆g	is⋆g	NOUN
ejpam-842	52	6	-	-	PUNCT
ejpam-842	52	7	closed	closed	ADJ
ejpam-842	52	8	set	set	NOUN
ejpam-842	52	9	is	be	AUX
ejpam-842	52	10	said	say	VERB
ejpam-842	52	11	to	to	PART
ejpam-842	52	12	be	be	AUX
ejpam-842	52	13	is⋆g	is⋆g	PROPN
ejpam-842	52	14	-open	-open	NOUN
ejpam-842	52	15	,	,	PUNCT
ejpam-842	52	16	equivalently	equivalently	ADV
ejpam-842	52	17	if	if	SCONJ
ejpam-842	52	18	f	f	PROPN
ejpam-842	52	19	⊂	⊂	PROPN
ejpam-842	52	20	int⋆(a	int⋆(a	NOUN
ejpam-842	52	21	)	)	PUNCT
ejpam-842	52	22	whenever	whenever	SCONJ
ejpam-842	52	23	f	f	PROPN
ejpam-842	52	24	⊂	⊂	PROPN
ejpam-842	52	25	a	a	PRON
ejpam-842	52	26	for	for	ADP
ejpam-842	52	27	every	every	DET
ejpam-842	52	28	semi	semi	ADJ
ejpam-842	52	29	-	-	ADJ
ejpam-842	52	30	closed	closed	ADJ
ejpam-842	52	31	set	set	VERB
ejpam-842	52	32	f	f	PROPN
ejpam-842	52	33	in	in	ADP
ejpam-842	52	34	x	x	PROPN
ejpam-842	52	35	.	.	PUNCT
ejpam-842	53	1	lemma	lemma	PROPN
ejpam-842	53	2	1	1	NUM
ejpam-842	53	3	.	.	PUNCT
ejpam-842	54	1	every	every	DET
ejpam-842	54	2	open	open	ADJ
ejpam-842	54	3	set	set	NOUN
ejpam-842	54	4	is	be	AUX
ejpam-842	54	5	is⋆g	is⋆g	NOUN
ejpam-842	54	6	-	-	PUNCT
ejpam-842	54	7	open	open	ADJ
ejpam-842	54	8	.	.	PUNCT
ejpam-842	55	1	lemma	lemma	PROPN
ejpam-842	55	2	2	2	NUM
ejpam-842	55	3	(	(	PUNCT
ejpam-842	55	4	[	[	X
ejpam-842	55	5	lemma	lemma	PROPN
ejpam-842	55	6	2.7	2.7	NUM
ejpam-842	55	7	,	,	PUNCT
ejpam-842	55	8	2	2	NUM
ejpam-842	55	9	]	]	PUNCT
ejpam-842	55	10	)	)	PUNCT
ejpam-842	55	11	.	.	PUNCT
ejpam-842	56	1	let	let	AUX
ejpam-842	56	2	(	(	PUNCT
ejpam-842	56	3	x	x	X
ejpam-842	56	4	,	,	PUNCT
ejpam-842	56	5	τ	τ	PROPN
ejpam-842	56	6	,	,	PUNCT
ejpam-842	56	7	i	i	PRON
ejpam-842	56	8	)	)	PUNCT
ejpam-842	56	9	be	be	VERB
ejpam-842	56	10	an	an	DET
ejpam-842	56	11	ideal	ideal	ADJ
ejpam-842	56	12	topological	topological	ADJ
ejpam-842	56	13	space	space	NOUN
ejpam-842	56	14	and	and	CCONJ
ejpam-842	56	15	b	b	NOUN
ejpam-842	56	16	⊂	⊂	PROPN
ejpam-842	56	17	a	a	X
ejpam-842	56	18	⊂	⊂	X
ejpam-842	56	19	x	x	X
ejpam-842	56	20	.	.	PUNCT
ejpam-842	57	1	then	then	ADV
ejpam-842	57	2	b⋆(ia	b⋆(ia	PROPN
ejpam-842	57	3	,	,	PUNCT
ejpam-842	57	4	τa	τa	X
ejpam-842	57	5	)	)	PUNCT
ejpam-842	57	6	=	=	SYM
ejpam-842	57	7	b⋆(i	b⋆(i	NOUN
ejpam-842	57	8	,	,	PUNCT
ejpam-842	57	9	τ)∩	τ)∩	PROPN
ejpam-842	57	10	a.	a.	PROPN
ejpam-842	57	11	lemma	lemma	PROPN
ejpam-842	58	1	3	3	X
ejpam-842	58	2	.	.	PUNCT
ejpam-842	59	1	if	if	SCONJ
ejpam-842	59	2	u	u	NOUN
ejpam-842	59	3	is	be	AUX
ejpam-842	59	4	open	open	ADJ
ejpam-842	59	5	and	and	CCONJ
ejpam-842	59	6	a	a	PRON
ejpam-842	59	7	is	be	AUX
ejpam-842	59	8	is⋆g	is⋆g	NOUN
ejpam-842	59	9	-	-	PUNCT
ejpam-842	59	10	open	open	ADJ
ejpam-842	59	11	,	,	PUNCT
ejpam-842	59	12	then	then	ADV
ejpam-842	59	13	u	u	NOUN
ejpam-842	59	14	∩	∩	NOUN
ejpam-842	59	15	a	a	PRON
ejpam-842	59	16	is	be	AUX
ejpam-842	59	17	is⋆g	is⋆g	PROPN
ejpam-842	59	18	-open	-open	NOUN
ejpam-842	59	19	.	.	PUNCT
ejpam-842	60	1	proof	proof	NOUN
ejpam-842	60	2	.	.	PUNCT
ejpam-842	61	1	we	we	PRON
ejpam-842	61	2	prove	prove	VERB
ejpam-842	61	3	that	that	SCONJ
ejpam-842	61	4	x	x	X
ejpam-842	61	5	−	−	PROPN
ejpam-842	61	6	(	(	PUNCT
ejpam-842	61	7	u	u	NOUN
ejpam-842	61	8	∩	∩	NOUN
ejpam-842	61	9	a	a	X
ejpam-842	61	10	)	)	PUNCT
ejpam-842	61	11	is	be	AUX
ejpam-842	61	12	is⋆g	is⋆g	NOUN
ejpam-842	61	13	-	-	PUNCT
ejpam-842	61	14	closed	closed	ADJ
ejpam-842	61	15	.	.	PUNCT
ejpam-842	62	1	let	let	VERB
ejpam-842	62	2	x	x	PRON
ejpam-842	62	3	−	−	PROPN
ejpam-842	62	4	(	(	PUNCT
ejpam-842	62	5	u	u	NOUN
ejpam-842	62	6	∩	∩	NOUN
ejpam-842	62	7	a	a	X
ejpam-842	62	8	)	)	PUNCT
ejpam-842	62	9	⊂	⊂	PROPN
ejpam-842	62	10	g	g	ADP
ejpam-842	62	11	where	where	SCONJ
ejpam-842	62	12	g	g	PROPN
ejpam-842	62	13	is	be	AUX
ejpam-842	62	14	semiopen	semiopen	ADJ
ejpam-842	62	15	in	in	ADP
ejpam-842	62	16	x	x	X
ejpam-842	62	17	.	.	PUNCT
ejpam-842	63	1	this	this	PRON
ejpam-842	63	2	implies	imply	VERB
ejpam-842	63	3	(	(	PUNCT
ejpam-842	63	4	x	x	X
ejpam-842	63	5	−	−	PROPN
ejpam-842	63	6	u	u	NOUN
ejpam-842	63	7	)	)	PUNCT
ejpam-842	63	8	∪	∪	ADV
ejpam-842	63	9	(	(	PUNCT
ejpam-842	63	10	x	x	SYM
ejpam-842	63	11	−	−	PROPN
ejpam-842	63	12	a	a	X
ejpam-842	63	13	)	)	PUNCT
ejpam-842	63	14	⊂	⊂	PROPN
ejpam-842	63	15	g.	g.	PROPN
ejpam-842	64	1	since	since	SCONJ
ejpam-842	64	2	(	(	PUNCT
ejpam-842	64	3	x	x	X
ejpam-842	64	4	−	−	PROPN
ejpam-842	64	5	a	a	X
ejpam-842	64	6	)	)	PUNCT
ejpam-842	64	7	⊂	⊂	PROPN
ejpam-842	64	8	g	g	PROPN
ejpam-842	64	9	and	and	CCONJ
ejpam-842	64	10	(	(	PUNCT
ejpam-842	64	11	x	x	X
ejpam-842	64	12	−	−	NOUN
ejpam-842	64	13	a	a	NOUN
ejpam-842	64	14	)	)	PUNCT
ejpam-842	64	15	is	be	AUX
ejpam-842	64	16	is⋆gclosed	is⋆gclose	VERB
ejpam-842	64	17	in	in	ADP
ejpam-842	64	18	x	x	SYM
ejpam-842	64	19	,	,	PUNCT
ejpam-842	64	20	therefore	therefore	ADV
ejpam-842	64	21	(	(	PUNCT
ejpam-842	64	22	x	x	X
ejpam-842	64	23	−	−	PROPN
ejpam-842	64	24	a)⋆	a)⋆	PROPN
ejpam-842	64	25	⊂	⊂	PROPN
ejpam-842	64	26	g.	g.	PROPN
ejpam-842	65	1	moreover	moreover	ADV
ejpam-842	66	1	x	x	X
ejpam-842	66	2	−	−	PROPN
ejpam-842	66	3	u	u	NOUN
ejpam-842	66	4	is	be	AUX
ejpam-842	66	5	closed	close	VERB
ejpam-842	66	6	and	and	CCONJ
ejpam-842	66	7	contained	contain	VERB
ejpam-842	66	8	in	in	ADP
ejpam-842	66	9	g	g	NOUN
ejpam-842	66	10	,	,	PUNCT
ejpam-842	66	11	therefore	therefore	ADV
ejpam-842	66	12	,	,	PUNCT
ejpam-842	66	13	(	(	PUNCT
ejpam-842	66	14	x−u)⋆	x−u)⋆	PROPN
ejpam-842	66	15	⊂	⊂	PROPN
ejpam-842	66	16	cl(x−u)⊂	cl(x−u)⊂	NOUN
ejpam-842	66	17	g.	g.	NOUN
ejpam-842	66	18	hence	hence	ADV
ejpam-842	66	19	(	(	PUNCT
ejpam-842	66	20	x−(u∩a))⋆	x−(u∩a))⋆	SYM
ejpam-842	66	21	=	=	SYM
ejpam-842	66	22	(	(	PUNCT
ejpam-842	66	23	(	(	PUNCT
ejpam-842	66	24	x−u)∪(x−a))⋆	x−u)∪(x−a))⋆	X
ejpam-842	66	25	=	=	SYM
ejpam-842	66	26	(	(	PUNCT
ejpam-842	66	27	x−u)⋆∪(x−a)⋆	x−u)⋆∪(x−a)⋆	PUNCT
ejpam-842	66	28	⊂	⊂	PROPN
ejpam-842	66	29	g.	g.	PROPN
ejpam-842	67	1	this	this	PRON
ejpam-842	67	2	proves	prove	VERB
ejpam-842	67	3	that	that	SCONJ
ejpam-842	67	4	u	u	PROPN
ejpam-842	67	5	∩	∩	NOUN
ejpam-842	67	6	a	a	PRON
ejpam-842	67	7	is	be	AUX
ejpam-842	67	8	is⋆g	is⋆g	NOUN
ejpam-842	67	9	-	-	PUNCT
ejpam-842	67	10	open	open	ADJ
ejpam-842	67	11	.	.	PUNCT
ejpam-842	68	1	theorem	theorem	NOUN
ejpam-842	68	2	1	1	NUM
ejpam-842	68	3	.	.	PUNCT
ejpam-842	69	1	let	let	AUX
ejpam-842	69	2	(	(	PUNCT
ejpam-842	69	3	x	x	X
ejpam-842	69	4	,	,	PUNCT
ejpam-842	69	5	τ	τ	PROPN
ejpam-842	69	6	,	,	PUNCT
ejpam-842	69	7	i	i	PRON
ejpam-842	69	8	)	)	PUNCT
ejpam-842	69	9	be	be	VERB
ejpam-842	69	10	an	an	DET
ejpam-842	69	11	ideal	ideal	ADJ
ejpam-842	69	12	topological	topological	ADJ
ejpam-842	69	13	space	space	NOUN
ejpam-842	69	14	and	and	CCONJ
ejpam-842	69	15	b	b	NOUN
ejpam-842	69	16	⊂	⊂	X
ejpam-842	69	17	a⊂	a⊂	X
ejpam-842	70	1	x	x	X
ejpam-842	70	2	.	.	PUNCT
ejpam-842	71	1	if	if	SCONJ
ejpam-842	71	2	b	b	PROPN
ejpam-842	71	3	is	be	AUX
ejpam-842	71	4	an	an	DET
ejpam-842	71	5	is⋆g	is⋆g	ADV
ejpam-842	71	6	-closed	-close	VERB
ejpam-842	71	7	set	set	NOUN
ejpam-842	71	8	relative	relative	ADJ
ejpam-842	71	9	to	to	ADP
ejpam-842	71	10	a	a	PRON
ejpam-842	71	11	,	,	PUNCT
ejpam-842	71	12	where	where	SCONJ
ejpam-842	71	13	a	a	PRON
ejpam-842	71	14	is	be	AUX
ejpam-842	71	15	open	open	ADJ
ejpam-842	71	16	and	and	CCONJ
ejpam-842	71	17	is⋆g	is⋆g	NOUN
ejpam-842	71	18	-closed	-close	VERB
ejpam-842	71	19	in	in	ADP
ejpam-842	71	20	x	x	SYM
ejpam-842	71	21	,	,	PUNCT
ejpam-842	71	22	then	then	ADV
ejpam-842	71	23	b	b	NOUN
ejpam-842	71	24	is	be	AUX
ejpam-842	71	25	is⋆g	is⋆g	NOUN
ejpam-842	71	26	-	-	PUNCT
ejpam-842	71	27	closed	closed	ADJ
ejpam-842	71	28	in	in	ADP
ejpam-842	71	29	x	x	X
ejpam-842	71	30	.	.	PUNCT
ejpam-842	72	1	proof	proof	NOUN
ejpam-842	72	2	.	.	PUNCT
ejpam-842	73	1	let	let	VERB
ejpam-842	73	2	b	b	X
ejpam-842	73	3	⊂	⊂	PROPN
ejpam-842	73	4	g	g	PROPN
ejpam-842	73	5	,	,	PUNCT
ejpam-842	73	6	where	where	SCONJ
ejpam-842	73	7	g	g	PROPN
ejpam-842	73	8	is	be	AUX
ejpam-842	73	9	semi	semi	ADJ
ejpam-842	73	10	-	-	ADJ
ejpam-842	73	11	open	open	ADJ
ejpam-842	73	12	in	in	ADP
ejpam-842	73	13	x	x	X
ejpam-842	73	14	.	.	PUNCT
ejpam-842	74	1	then	then	ADV
ejpam-842	74	2	b	b	X
ejpam-842	74	3	⊂	⊂	PROPN
ejpam-842	74	4	a∩	a∩	PROPN
ejpam-842	74	5	g	g	PROPN
ejpam-842	74	6	and	and	CCONJ
ejpam-842	74	7	a∩	a∩	PROPN
ejpam-842	74	8	g	g	PROPN
ejpam-842	74	9	is	be	AUX
ejpam-842	74	10	semi	semi	ADJ
ejpam-842	74	11	-	-	ADJ
ejpam-842	74	12	open	open	ADJ
ejpam-842	74	13	in	in	ADP
ejpam-842	74	14	x	x	PUNCT
ejpam-842	74	15	and	and	CCONJ
ejpam-842	74	16	hence	hence	ADV
ejpam-842	74	17	in	in	ADP
ejpam-842	74	18	a.	a.	NOUN
ejpam-842	74	19	therefore	therefore	ADV
ejpam-842	74	20	b⋆a	b⋆a	VERB
ejpam-842	75	1	⊂	⊂	PROPN
ejpam-842	75	2	a∩	a∩	PROPN
ejpam-842	76	1	g.	g.	NOUN
ejpam-842	76	2	it	it	PRON
ejpam-842	76	3	follows	follow	VERB
ejpam-842	76	4	from	from	ADP
ejpam-842	76	5	lemma	lemma	PROPN
ejpam-842	76	6	2	2	NUM
ejpam-842	76	7	that	that	PRON
ejpam-842	76	8	a∩	a∩	PROPN
ejpam-842	76	9	b⋆x	b⋆x	NOUN
ejpam-842	77	1	⊂	⊂	PRON
ejpam-842	77	2	a∩	a∩	PROPN
ejpam-842	77	3	g	g	NOUN
ejpam-842	77	4	or	or	CCONJ
ejpam-842	77	5	m.	m.	NOUN
ejpam-842	77	6	khan,̧	khan,̧	PROPN
ejpam-842	77	7	t.	t.	PROPN
ejpam-842	77	8	noiri	noiri	PROPN
ejpam-842	77	9	/	/	SYM
ejpam-842	77	10	eur	eur	PROPN
ejpam-842	77	11	.	.	PUNCT
ejpam-842	78	1	j.	j.	PROPN
ejpam-842	78	2	pure	pure	PROPN
ejpam-842	78	3	appl	appl	PROPN
ejpam-842	78	4	.	.	PROPN
ejpam-842	78	5	math	math	PROPN
ejpam-842	78	6	,	,	PUNCT
ejpam-842	78	7	4	4	NUM
ejpam-842	78	8	(	(	PUNCT
ejpam-842	78	9	2011	2011	NUM
ejpam-842	78	10	)	)	PUNCT
ejpam-842	78	11	,	,	PUNCT
ejpam-842	78	12	237	237	NUM
ejpam-842	78	13	-	-	SYM
ejpam-842	78	14	243	243	NUM
ejpam-842	78	15	239	239	NUM
ejpam-842	78	16	a⊂	a⊂	NOUN
ejpam-842	78	17	g∪(x	g∪(x	NOUN
ejpam-842	78	18	−b⋆x	−b⋆x	NOUN
ejpam-842	78	19	)	)	PUNCT
ejpam-842	78	20	.	.	PUNCT
ejpam-842	79	1	by	by	ADP
ejpam-842	79	2	theorem	theorem	VERB
ejpam-842	79	3	2.3	2.3	NUM
ejpam-842	79	4	of	of	ADP
ejpam-842	79	5	[	[	X
ejpam-842	79	6	4	4	NUM
ejpam-842	79	7	]	]	PUNCT
ejpam-842	79	8	,	,	PUNCT
ejpam-842	79	9	b⋆x	b⋆x	NOUN
ejpam-842	79	10	is	be	AUX
ejpam-842	79	11	closed	close	VERB
ejpam-842	79	12	in	in	ADP
ejpam-842	79	13	x	x	NOUN
ejpam-842	79	14	and	and	CCONJ
ejpam-842	79	15	g∪(x	g∪(x	PROPN
ejpam-842	79	16	−b⋆x	−b⋆x	NOUN
ejpam-842	79	17	)	)	PUNCT
ejpam-842	79	18	is	be	AUX
ejpam-842	79	19	semi	semi	ADJ
ejpam-842	79	20	-	-	ADJ
ejpam-842	79	21	open	open	ADJ
ejpam-842	79	22	in	in	ADP
ejpam-842	79	23	x	x	X
ejpam-842	79	24	.	.	PUNCT
ejpam-842	80	1	since	since	SCONJ
ejpam-842	80	2	a	a	PRON
ejpam-842	80	3	is	be	AUX
ejpam-842	80	4	is⋆g	is⋆g	NOUN
ejpam-842	80	5	-closed	-closed	ADJ
ejpam-842	80	6	in	in	ADP
ejpam-842	80	7	x	x	SYM
ejpam-842	80	8	,	,	PUNCT
ejpam-842	80	9	a⋆x	a⋆x	PROPN
ejpam-842	80	10	⊂	⊂	X
ejpam-842	80	11	g∪(x−b⋆x	g∪(x−b⋆x	PROPN
ejpam-842	80	12	)	)	PUNCT
ejpam-842	80	13	and	and	CCONJ
ejpam-842	80	14	hence	hence	ADV
ejpam-842	80	15	b⋆	b⋆	CCONJ
ejpam-842	80	16	=	=	SYM
ejpam-842	80	17	b⋆∩a⋆	b⋆∩a⋆	PROPN
ejpam-842	80	18	⊂	⊂	NOUN
ejpam-842	80	19	b⋆∩[g∪(x−b⋆x	b⋆∩[g∪(x−b⋆x	PROPN
ejpam-842	80	20	)	)	PUNCT
ejpam-842	80	21	]	]	PUNCT
ejpam-842	80	22	⊂	⊂	PROPN
ejpam-842	80	23	g.	g.	PROPN
ejpam-842	80	24	therefore	therefore	ADV
ejpam-842	80	25	,	,	PUNCT
ejpam-842	80	26	we	we	PRON
ejpam-842	80	27	obtain	obtain	VERB
ejpam-842	80	28	b⋆x	b⋆x	NOUN
ejpam-842	81	1	⊂	⊂	PROPN
ejpam-842	81	2	g.	g.	PROPN
ejpam-842	82	1	this	this	PRON
ejpam-842	82	2	proves	prove	VERB
ejpam-842	82	3	that	that	SCONJ
ejpam-842	82	4	b	b	NOUN
ejpam-842	82	5	is	be	AUX
ejpam-842	82	6	is⋆g	is⋆g	ADV
ejpam-842	82	7	-closed	-closed	ADJ
ejpam-842	82	8	in	in	ADP
ejpam-842	82	9	x	x	X
ejpam-842	82	10	.	.	PUNCT
ejpam-842	83	1	theorem	theorem	NOUN
ejpam-842	83	2	2	2	X
ejpam-842	83	3	.	.	PUNCT
ejpam-842	84	1	let	let	VERB
ejpam-842	84	2	a	a	PRON
ejpam-842	84	3	be	be	AUX
ejpam-842	84	4	a	a	DET
ejpam-842	84	5	semi	semi	ADJ
ejpam-842	84	6	-	-	ADJ
ejpam-842	84	7	open	open	ADJ
ejpam-842	84	8	set	set	NOUN
ejpam-842	84	9	in	in	ADP
ejpam-842	84	10	a	a	DET
ejpam-842	84	11	space	space	NOUN
ejpam-842	84	12	(	(	PUNCT
ejpam-842	84	13	x	x	X
ejpam-842	84	14	,	,	PUNCT
ejpam-842	84	15	τ	τ	PROPN
ejpam-842	84	16	,	,	PUNCT
ejpam-842	84	17	i	i	PROPN
ejpam-842	84	18	)	)	PUNCT
ejpam-842	84	19	and	and	CCONJ
ejpam-842	85	1	b	b	X
ejpam-842	85	2	⊂	⊂	X
ejpam-842	85	3	a⊂	a⊂	X
ejpam-842	85	4	x	x	X
ejpam-842	85	5	.	.	PUNCT
ejpam-842	86	1	if	if	SCONJ
ejpam-842	86	2	b	b	PROPN
ejpam-842	86	3	is	be	AUX
ejpam-842	86	4	is⋆g	is⋆g	NOUN
ejpam-842	86	5	-	-	PUNCT
ejpam-842	86	6	closed	closed	ADJ
ejpam-842	86	7	in	in	ADP
ejpam-842	86	8	x	x	SYM
ejpam-842	86	9	,	,	PUNCT
ejpam-842	86	10	then	then	ADV
ejpam-842	86	11	b	b	NOUN
ejpam-842	86	12	is	be	AUX
ejpam-842	86	13	is⋆g	is⋆g	X
ejpam-842	86	14	-closed	-close	VERB
ejpam-842	86	15	relative	relative	ADJ
ejpam-842	86	16	to	to	ADP
ejpam-842	86	17	a.	a.	NOUN
ejpam-842	86	18	proof	proof	NOUN
ejpam-842	86	19	.	.	PUNCT
ejpam-842	87	1	let	let	VERB
ejpam-842	87	2	b	b	NOUN
ejpam-842	87	3	⊂	⊂	PROPN
ejpam-842	87	4	u	u	PROPN
ejpam-842	87	5	where	where	SCONJ
ejpam-842	87	6	u	u	NOUN
ejpam-842	87	7	is	be	AUX
ejpam-842	87	8	semi	semi	ADJ
ejpam-842	87	9	-	-	ADJ
ejpam-842	87	10	open	open	ADJ
ejpam-842	87	11	in	in	ADP
ejpam-842	87	12	a.	a.	NOUN
ejpam-842	87	13	then	then	ADV
ejpam-842	87	14	there	there	PRON
ejpam-842	87	15	exists	exist	VERB
ejpam-842	87	16	a	a	DET
ejpam-842	87	17	semi	semi	ADJ
ejpam-842	87	18	-	-	ADJ
ejpam-842	87	19	open	open	ADJ
ejpam-842	87	20	set	set	VERB
ejpam-842	87	21	v	v	NOUN
ejpam-842	87	22	in	in	ADP
ejpam-842	87	23	x	x	PUNCT
ejpam-842	87	24	such	such	ADJ
ejpam-842	87	25	that	that	DET
ejpam-842	87	26	u	u	NOUN
ejpam-842	87	27	=	=	NOUN
ejpam-842	87	28	a∩v	a∩v	NOUN
ejpam-842	87	29	.	.	PUNCT
ejpam-842	88	1	thus	thus	ADV
ejpam-842	88	2	b	b	X
ejpam-842	88	3	⊂	⊂	PROPN
ejpam-842	88	4	a∩v	a∩v	PROPN
ejpam-842	88	5	.	.	PUNCT
ejpam-842	89	1	now	now	ADV
ejpam-842	89	2	b	b	X
ejpam-842	89	3	⊂	⊂	PROPN
ejpam-842	89	4	v	v	PROPN
ejpam-842	89	5	implies	imply	VERB
ejpam-842	89	6	that	that	DET
ejpam-842	89	7	b⋆x	b⋆x	VERB
ejpam-842	89	8	⊂	⊂	PRON
ejpam-842	89	9	v	v	NOUN
ejpam-842	89	10	.	.	PUNCT
ejpam-842	90	1	it	it	PRON
ejpam-842	90	2	follows	follow	VERB
ejpam-842	90	3	that	that	DET
ejpam-842	90	4	a∩b⋆x	a∩b⋆x	PROPN
ejpam-842	90	5	⊂	⊂	PROPN
ejpam-842	90	6	a∩v	a∩v	PROPN
ejpam-842	90	7	.	.	PUNCT
ejpam-842	91	1	by	by	ADP
ejpam-842	91	2	lemma	lemma	PROPN
ejpam-842	91	3	2	2	NUM
ejpam-842	91	4	,	,	PUNCT
ejpam-842	91	5	b⋆a	b⋆a	X
ejpam-842	91	6	⊂	⊂	PROPN
ejpam-842	91	7	a∩	a∩	PROPN
ejpam-842	92	1	v	v	NOUN
ejpam-842	92	2	=	=	SYM
ejpam-842	92	3	u	u	PROPN
ejpam-842	92	4	.	.	PUNCT
ejpam-842	93	1	this	this	PRON
ejpam-842	93	2	proves	prove	VERB
ejpam-842	93	3	that	that	SCONJ
ejpam-842	93	4	b	b	NOUN
ejpam-842	93	5	is	be	AUX
ejpam-842	93	6	a	a	DET
ejpam-842	93	7	is⋆	is⋆	PROPN
ejpam-842	93	8	g	g	NOUN
ejpam-842	93	9	-	-	PUNCT
ejpam-842	93	10	closed	closed	ADJ
ejpam-842	93	11	relative	relative	ADJ
ejpam-842	93	12	to	to	ADP
ejpam-842	93	13	a.	a.	NOUN
ejpam-842	93	14	corollary	corollary	NOUN
ejpam-842	93	15	1	1	PROPN
ejpam-842	93	16	.	.	PUNCT
ejpam-842	94	1	let	let	VERB
ejpam-842	94	2	b	b	PROPN
ejpam-842	94	3	⊂	⊂	PROPN
ejpam-842	94	4	a	a	DET
ejpam-842	94	5	⊂	⊂	X
ejpam-842	94	6	x	x	X
ejpam-842	94	7	and	and	CCONJ
ejpam-842	94	8	a	a	DET
ejpam-842	94	9	be	be	AUX
ejpam-842	94	10	open	open	ADJ
ejpam-842	94	11	and	and	CCONJ
ejpam-842	94	12	is⋆g	is⋆g	NOUN
ejpam-842	94	13	-	-	PUNCT
ejpam-842	94	14	closed	closed	ADJ
ejpam-842	94	15	in	in	ADP
ejpam-842	94	16	(	(	PUNCT
ejpam-842	94	17	x	x	INTJ
ejpam-842	94	18	,	,	PUNCT
ejpam-842	94	19	τ	τ	PROPN
ejpam-842	94	20	,	,	PUNCT
ejpam-842	94	21	i	i	PROPN
ejpam-842	94	22	)	)	PUNCT
ejpam-842	94	23	.	.	PUNCT
ejpam-842	95	1	then	then	ADV
ejpam-842	95	2	b	b	PROPN
ejpam-842	95	3	is	be	AUX
ejpam-842	95	4	is⋆g	is⋆g	NOUN
ejpam-842	95	5	-	-	PUNCT
ejpam-842	95	6	closed	closed	ADJ
ejpam-842	95	7	relative	relative	ADJ
ejpam-842	95	8	to	to	ADP
ejpam-842	95	9	a	a	DET
ejpam-842	95	10	if	if	NOUN
ejpam-842	95	11	and	and	CCONJ
ejpam-842	95	12	only	only	ADV
ejpam-842	95	13	if	if	SCONJ
ejpam-842	95	14	b	b	NOUN
ejpam-842	95	15	is	be	AUX
ejpam-842	95	16	is⋆g	is⋆g	ADV
ejpam-842	95	17	-closed	-closed	ADJ
ejpam-842	95	18	in	in	ADP
ejpam-842	95	19	x	x	X
ejpam-842	95	20	.	.	PUNCT
ejpam-842	96	1	theorem	theorem	NOUN
ejpam-842	96	2	3	3	X
ejpam-842	96	3	.	.	PUNCT
ejpam-842	97	1	if	if	SCONJ
ejpam-842	97	2	b	b	PROPN
ejpam-842	97	3	is	be	AUX
ejpam-842	97	4	a	a	DET
ejpam-842	97	5	subset	subset	NOUN
ejpam-842	97	6	of	of	ADP
ejpam-842	97	7	a	a	DET
ejpam-842	97	8	space	space	NOUN
ejpam-842	97	9	(	(	PUNCT
ejpam-842	97	10	x	x	X
ejpam-842	97	11	,	,	PUNCT
ejpam-842	97	12	τ	τ	PROPN
ejpam-842	97	13	,	,	PUNCT
ejpam-842	97	14	i	i	NOUN
ejpam-842	97	15	)	)	PUNCT
ejpam-842	97	16	such	such	ADJ
ejpam-842	97	17	that	that	SCONJ
ejpam-842	97	18	a⊂	a⊂	NOUN
ejpam-842	97	19	b	b	X
ejpam-842	97	20	⊂	⊂	X
ejpam-842	97	21	a⋆	a⋆	NOUN
ejpam-842	97	22	and	and	CCONJ
ejpam-842	97	23	a	a	PRON
ejpam-842	97	24	is	be	AUX
ejpam-842	97	25	is⋆	is⋆	PROPN
ejpam-842	97	26	g	g	NOUN
ejpam-842	97	27	-	-	PUNCT
ejpam-842	97	28	closed	closed	ADJ
ejpam-842	97	29	in	in	ADP
ejpam-842	97	30	x	x	SYM
ejpam-842	97	31	,	,	PUNCT
ejpam-842	97	32	then	then	ADV
ejpam-842	97	33	b	b	NOUN
ejpam-842	97	34	is	be	AUX
ejpam-842	97	35	also	also	ADV
ejpam-842	97	36	is⋆	is⋆	PROPN
ejpam-842	97	37	g	g	NOUN
ejpam-842	97	38	-	-	PUNCT
ejpam-842	97	39	closed	closed	ADJ
ejpam-842	97	40	in	in	ADP
ejpam-842	97	41	x	x	X
ejpam-842	97	42	.	.	PUNCT
ejpam-842	98	1	proof	proof	NOUN
ejpam-842	98	2	.	.	PUNCT
ejpam-842	99	1	let	let	VERB
ejpam-842	99	2	g	g	PRON
ejpam-842	99	3	be	be	AUX
ejpam-842	99	4	a	a	DET
ejpam-842	99	5	semi	semi	ADJ
ejpam-842	99	6	-	-	ADJ
ejpam-842	99	7	open	open	ADJ
ejpam-842	99	8	set	set	NOUN
ejpam-842	99	9	in	in	ADP
ejpam-842	99	10	x	x	PUNCT
ejpam-842	99	11	containing	contain	VERB
ejpam-842	99	12	b	b	NOUN
ejpam-842	99	13	,	,	PUNCT
ejpam-842	99	14	then	then	ADV
ejpam-842	99	15	a	a	DET
ejpam-842	99	16	⊂	⊂	PROPN
ejpam-842	99	17	g.	g.	PROPN
ejpam-842	99	18	since	since	SCONJ
ejpam-842	99	19	a	a	PRON
ejpam-842	99	20	is	be	AUX
ejpam-842	99	21	is⋆g	is⋆g	X
ejpam-842	99	22	-closed	-closed	ADJ
ejpam-842	99	23	,	,	PUNCT
ejpam-842	99	24	therefore	therefore	ADV
ejpam-842	99	25	a⋆	a⋆	CCONJ
ejpam-842	99	26	⊂	⊂	PROPN
ejpam-842	99	27	g	g	NOUN
ejpam-842	99	28	and	and	CCONJ
ejpam-842	99	29	hence	hence	ADV
ejpam-842	99	30	b⋆	b⋆	CCONJ
ejpam-842	99	31	⊂	⊂	PROPN
ejpam-842	99	32	(	(	PUNCT
ejpam-842	99	33	a⋆)⋆	a⋆)⋆	PROPN
ejpam-842	99	34	⊂	⊂	PROPN
ejpam-842	99	35	a⋆	a⋆	PROPN
ejpam-842	100	1	⊂	⊂	PROPN
ejpam-842	100	2	g.	g.	PROPN
ejpam-842	100	3	this	this	PRON
ejpam-842	100	4	implies	imply	VERB
ejpam-842	100	5	that	that	SCONJ
ejpam-842	100	6	b	b	NOUN
ejpam-842	100	7	is	be	AUX
ejpam-842	100	8	is⋆g	is⋆g	NOUN
ejpam-842	100	9	-	-	PUNCT
ejpam-842	100	10	closed	closed	ADJ
ejpam-842	100	11	in	in	ADP
ejpam-842	100	12	x	x	X
ejpam-842	100	13	.	.	PUNCT
ejpam-842	101	1	theorem	theorem	ADJ
ejpam-842	101	2	4	4	NUM
ejpam-842	101	3	.	.	PUNCT
ejpam-842	102	1	let	let	VERB
ejpam-842	102	2	b	b	NOUN
ejpam-842	102	3	⊂	⊂	X
ejpam-842	102	4	a⊂	a⊂	X
ejpam-842	102	5	x	x	PUNCT
ejpam-842	102	6	and	and	CCONJ
ejpam-842	102	7	suppose	suppose	VERB
ejpam-842	102	8	that	that	SCONJ
ejpam-842	102	9	b	b	PROPN
ejpam-842	102	10	is	be	AUX
ejpam-842	102	11	is⋆	is⋆	NOUN
ejpam-842	102	12	g	g	NOUN
ejpam-842	102	13	-	-	PUNCT
ejpam-842	102	14	open	open	ADJ
ejpam-842	102	15	in	in	ADP
ejpam-842	102	16	x	x	PUNCT
ejpam-842	102	17	and	and	CCONJ
ejpam-842	102	18	a	a	PRON
ejpam-842	102	19	is	be	AUX
ejpam-842	102	20	a	a	DET
ejpam-842	102	21	semi	semi	ADJ
ejpam-842	102	22	-	-	ADJ
ejpam-842	102	23	regular	regular	ADJ
ejpam-842	102	24	set	set	NOUN
ejpam-842	102	25	in	in	ADP
ejpam-842	102	26	x	x	X
ejpam-842	102	27	.	.	PUNCT
ejpam-842	103	1	then	then	ADV
ejpam-842	103	2	b	b	PROPN
ejpam-842	103	3	is	be	AUX
ejpam-842	103	4	is⋆g	is⋆g	NOUN
ejpam-842	103	5	-	-	PUNCT
ejpam-842	103	6	open	open	ADJ
ejpam-842	103	7	relative	relative	ADJ
ejpam-842	103	8	to	to	ADP
ejpam-842	103	9	a.	a.	NOUN
ejpam-842	103	10	proof	proof	NOUN
ejpam-842	103	11	.	.	PUNCT
ejpam-842	104	1	we	we	PRON
ejpam-842	104	2	prove	prove	VERB
ejpam-842	104	3	that	that	SCONJ
ejpam-842	104	4	a−b	a−b	PROPN
ejpam-842	104	5	is	be	AUX
ejpam-842	104	6	is⋆g	is⋆g	NOUN
ejpam-842	104	7	-	-	PUNCT
ejpam-842	104	8	closed	closed	ADJ
ejpam-842	104	9	relative	relative	ADJ
ejpam-842	104	10	to	to	ADP
ejpam-842	104	11	a.	a.	NOUN
ejpam-842	104	12	let	let	VERB
ejpam-842	104	13	u	u	NOUN
ejpam-842	104	14	∈	∈	PROPN
ejpam-842	104	15	so(a	so(a	NOUN
ejpam-842	104	16	)	)	PUNCT
ejpam-842	104	17	such	such	ADJ
ejpam-842	104	18	that	that	SCONJ
ejpam-842	104	19	(	(	PUNCT
ejpam-842	104	20	a−b)⊂	a−b)⊂	PROPN
ejpam-842	104	21	u	u	PROPN
ejpam-842	104	22	.	.	PUNCT
ejpam-842	105	1	now	now	ADV
ejpam-842	105	2	(	(	PUNCT
ejpam-842	105	3	a−	a−	PROPN
ejpam-842	105	4	b	b	NOUN
ejpam-842	105	5	)	)	PUNCT
ejpam-842	105	6	⊂	⊂	PROPN
ejpam-842	105	7	(	(	PUNCT
ejpam-842	105	8	x	x	X
ejpam-842	105	9	−	−	PROPN
ejpam-842	105	10	b	b	X
ejpam-842	105	11	)	)	PUNCT
ejpam-842	105	12	⊂	⊂	PROPN
ejpam-842	105	13	u	u	NOUN
ejpam-842	105	14	∪	∪	X
ejpam-842	105	15	(	(	PUNCT
ejpam-842	105	16	x	x	SYM
ejpam-842	105	17	−	−	PROPN
ejpam-842	105	18	a	a	NOUN
ejpam-842	105	19	)	)	PUNCT
ejpam-842	105	20	,	,	PUNCT
ejpam-842	105	21	where	where	SCONJ
ejpam-842	105	22	u	u	PROPN
ejpam-842	105	23	∪	∪	VERB
ejpam-842	105	24	(	(	PUNCT
ejpam-842	105	25	x	x	NOUN
ejpam-842	105	26	−	−	PROPN
ejpam-842	105	27	a	a	X
ejpam-842	105	28	)	)	PUNCT
ejpam-842	105	29	∈	∈	PROPN
ejpam-842	105	30	so(x	so(x	NOUN
ejpam-842	105	31	)	)	PUNCT
ejpam-842	105	32	because	because	SCONJ
ejpam-842	105	33	a∈	a∈	PROPN
ejpam-842	105	34	sr(x	sr(x	NUM
ejpam-842	105	35	)	)	PUNCT
ejpam-842	105	36	.	.	PUNCT
ejpam-842	106	1	since	since	SCONJ
ejpam-842	106	2	x−b	x−b	NOUN
ejpam-842	106	3	is	be	AUX
ejpam-842	106	4	is⋆g	is⋆g	ADV
ejpam-842	106	5	-closed	-closed	ADJ
ejpam-842	106	6	in	in	ADP
ejpam-842	106	7	x	x	SYM
ejpam-842	106	8	,	,	PUNCT
ejpam-842	106	9	therefore	therefore	ADV
ejpam-842	106	10	(	(	PUNCT
ejpam-842	106	11	x−b)⋆x	x−b)⋆x	PROPN
ejpam-842	106	12	⊂	⊂	PROPN
ejpam-842	106	13	u∪(x−a	u∪(x−a	PROPN
ejpam-842	106	14	)	)	PUNCT
ejpam-842	106	15	or	or	CCONJ
ejpam-842	106	16	(	(	PUNCT
ejpam-842	106	17	x−b)⋆x∩a⊂	x−b)⋆x∩a⊂	PROPN
ejpam-842	106	18	(	(	PUNCT
ejpam-842	106	19	u∪(x−a))∩a⊂	u∪(x−a))∩a⊂	NOUN
ejpam-842	106	20	u	u	NOUN
ejpam-842	106	21	.	.	PUNCT
ejpam-842	107	1	by	by	ADP
ejpam-842	107	2	lemma	lemma	PROPN
ejpam-842	107	3	2	2	NUM
ejpam-842	107	4	,	,	PUNCT
ejpam-842	107	5	(	(	PUNCT
ejpam-842	107	6	a−b)⋆a	a−b)⋆a	NOUN
ejpam-842	107	7	=	=	SYM
ejpam-842	107	8	(	(	PUNCT
ejpam-842	107	9	a−b)⋆x	a−b)⋆x	NOUN
ejpam-842	107	10	∩a⊂	∩a⊂	X
ejpam-842	107	11	(	(	PUNCT
ejpam-842	107	12	x	x	PROPN
ejpam-842	107	13	−b)⋆x	−b)⋆x	PROPN
ejpam-842	107	14	∩a⊂	∩a⊂	VERB
ejpam-842	107	15	u	u	NOUN
ejpam-842	107	16	and	and	CCONJ
ejpam-842	107	17	hence	hence	ADV
ejpam-842	107	18	(	(	PUNCT
ejpam-842	107	19	a−b)⋆a	a−b)⋆a	PROPN
ejpam-842	107	20	⊂	⊂	PROPN
ejpam-842	107	21	u	u	PROPN
ejpam-842	107	22	.	.	PUNCT
ejpam-842	108	1	this	this	PRON
ejpam-842	108	2	proves	prove	VERB
ejpam-842	108	3	that	that	SCONJ
ejpam-842	108	4	b	b	NOUN
ejpam-842	108	5	is	be	AUX
ejpam-842	108	6	is⋆g	is⋆g	PRON
ejpam-842	108	7	-open	-open	ADJ
ejpam-842	108	8	relative	relative	ADJ
ejpam-842	108	9	to	to	ADP
ejpam-842	108	10	a.	a.	NOUN
ejpam-842	108	11	theorem	theorem	NOUN
ejpam-842	108	12	5	5	X
ejpam-842	108	13	.	.	PUNCT
ejpam-842	109	1	let	let	VERB
ejpam-842	109	2	b	b	NOUN
ejpam-842	109	3	⊂	⊂	X
ejpam-842	109	4	a⊂	a⊂	X
ejpam-842	109	5	x	x	X
ejpam-842	109	6	.	.	PUNCT
ejpam-842	110	1	b	b	PROPN
ejpam-842	110	2	is	be	AUX
ejpam-842	110	3	is⋆	is⋆	NOUN
ejpam-842	110	4	g	g	NOUN
ejpam-842	110	5	-	-	PUNCT
ejpam-842	110	6	open	open	ADJ
ejpam-842	110	7	in	in	ADP
ejpam-842	110	8	a	a	PRON
ejpam-842	110	9	and	and	CCONJ
ejpam-842	110	10	a	a	PRON
ejpam-842	110	11	is	be	AUX
ejpam-842	110	12	open	open	ADJ
ejpam-842	110	13	in	in	ADP
ejpam-842	110	14	x	x	PUNCT
ejpam-842	110	15	then	then	ADV
ejpam-842	110	16	b	b	PROPN
ejpam-842	110	17	is	be	AUX
ejpam-842	110	18	is⋆	is⋆	NOUN
ejpam-842	110	19	g	g	NOUN
ejpam-842	110	20	-	-	PUNCT
ejpam-842	110	21	open	open	ADJ
ejpam-842	110	22	in	in	ADP
ejpam-842	110	23	x	x	X
ejpam-842	110	24	.	.	PUNCT
ejpam-842	111	1	proof	proof	NOUN
ejpam-842	111	2	.	.	PUNCT
ejpam-842	112	1	let	let	VERB
ejpam-842	112	2	f	f	PRON
ejpam-842	112	3	be	be	AUX
ejpam-842	112	4	a	a	DET
ejpam-842	112	5	semi	semi	ADJ
ejpam-842	112	6	-	-	ADJ
ejpam-842	112	7	closed	closed	ADJ
ejpam-842	112	8	subset	subset	NOUN
ejpam-842	112	9	of	of	ADP
ejpam-842	112	10	b	b	PROPN
ejpam-842	112	11	in	in	ADP
ejpam-842	112	12	x	x	X
ejpam-842	112	13	.	.	PUNCT
ejpam-842	113	1	since	since	SCONJ
ejpam-842	113	2	a	a	PRON
ejpam-842	113	3	is	be	AUX
ejpam-842	113	4	open	open	ADJ
ejpam-842	113	5	,	,	PUNCT
ejpam-842	113	6	therefore	therefore	ADV
ejpam-842	113	7	f	f	PROPN
ejpam-842	113	8	∈	∈	PROPN
ejpam-842	113	9	sc(a	sc(a	NOUN
ejpam-842	113	10	)	)	PUNCT
ejpam-842	113	11	.	.	PUNCT
ejpam-842	114	1	since	since	SCONJ
ejpam-842	114	2	b	b	PROPN
ejpam-842	114	3	is	be	AUX
ejpam-842	114	4	is⋆	is⋆	NOUN
ejpam-842	114	5	g	g	NOUN
ejpam-842	114	6	-	-	PUNCT
ejpam-842	114	7	open	open	ADJ
ejpam-842	114	8	in	in	ADP
ejpam-842	114	9	a	a	DET
ejpam-842	114	10	,	,	PUNCT
ejpam-842	114	11	therefore	therefore	ADV
ejpam-842	114	12	f	f	PROPN
ejpam-842	114	13	⊂	⊂	PROPN
ejpam-842	114	14	int⋆a(b	int⋆a(b	PROPN
ejpam-842	114	15	)	)	PUNCT
ejpam-842	115	1	=	=	SYM
ejpam-842	115	2	a∩	a∩	PROPN
ejpam-842	116	1	int⋆x	int⋆x	PROPN
ejpam-842	116	2	(	(	PUNCT
ejpam-842	116	3	b	b	NOUN
ejpam-842	116	4	)	)	PUNCT
ejpam-842	116	5	⊂	⊂	PROPN
ejpam-842	117	1	int⋆x	int⋆x	PROPN
ejpam-842	117	2	(	(	PUNCT
ejpam-842	117	3	b	b	NOUN
ejpam-842	117	4	)	)	PUNCT
ejpam-842	117	5	.	.	PUNCT
ejpam-842	118	1	this	this	PRON
ejpam-842	118	2	proves	prove	VERB
ejpam-842	118	3	that	that	SCONJ
ejpam-842	118	4	b	b	NOUN
ejpam-842	118	5	is	be	AUX
ejpam-842	118	6	is⋆g	is⋆g	ADJ
ejpam-842	118	7	-open	-open	ADJ
ejpam-842	118	8	in	in	ADP
ejpam-842	118	9	x	x	X
ejpam-842	118	10	.	.	PUNCT
ejpam-842	119	1	4	4	X
ejpam-842	119	2	.	.	X
ejpam-842	119	3	is⋆	is⋆	PROPN
ejpam-842	119	4	g	g	NOUN
ejpam-842	119	5	-	-	PUNCT
ejpam-842	119	6	continuous	continuous	ADJ
ejpam-842	119	7	functions	function	NOUN
ejpam-842	119	8	definition	definition	NOUN
ejpam-842	119	9	2	2	NUM
ejpam-842	119	10	.	.	PUNCT
ejpam-842	120	1	a	a	DET
ejpam-842	120	2	function	function	NOUN
ejpam-842	120	3	f	f	NOUN
ejpam-842	120	4	:	:	PUNCT
ejpam-842	120	5	(	(	PUNCT
ejpam-842	120	6	x	x	X
ejpam-842	120	7	,	,	PUNCT
ejpam-842	120	8	τ	τ	PROPN
ejpam-842	120	9	,	,	PUNCT
ejpam-842	120	10	i	i	NOUN
ejpam-842	120	11	)	)	PUNCT
ejpam-842	120	12	→	→	SYM
ejpam-842	120	13	(	(	PUNCT
ejpam-842	120	14	y	y	PROPN
ejpam-842	120	15	,	,	PUNCT
ejpam-842	120	16	ω	ω	PROPN
ejpam-842	120	17	,	,	PUNCT
ejpam-842	120	18	j	j	NOUN
ejpam-842	120	19	)	)	PUNCT
ejpam-842	120	20	is	be	AUX
ejpam-842	120	21	said	say	VERB
ejpam-842	120	22	to	to	PART
ejpam-842	120	23	be	be	AUX
ejpam-842	120	24	weakly	weakly	ADJ
ejpam-842	120	25	i	i	PRON
ejpam-842	120	26	-	-	ADJ
ejpam-842	120	27	continuous	continuous	ADJ
ejpam-842	120	28	[	[	X
ejpam-842	120	29	1	1	NUM
ejpam-842	120	30	]	]	X
ejpam-842	120	31	if	if	SCONJ
ejpam-842	120	32	for	for	ADP
ejpam-842	120	33	each	each	DET
ejpam-842	120	34	x	x	SYM
ejpam-842	120	35	∈	∈	PROPN
ejpam-842	120	36	x	x	X
ejpam-842	120	37	and	and	CCONJ
ejpam-842	120	38	each	each	DET
ejpam-842	120	39	open	open	ADJ
ejpam-842	120	40	set	set	VERB
ejpam-842	120	41	v	v	NOUN
ejpam-842	120	42	in	in	ADP
ejpam-842	120	43	y	y	PROPN
ejpam-842	120	44	containing	contain	VERB
ejpam-842	120	45	f	f	PROPN
ejpam-842	120	46	(	(	PUNCT
ejpam-842	120	47	x	x	NOUN
ejpam-842	120	48	)	)	PUNCT
ejpam-842	120	49	,	,	PUNCT
ejpam-842	120	50	there	there	PRON
ejpam-842	120	51	exists	exist	VERB
ejpam-842	120	52	an	an	DET
ejpam-842	120	53	open	open	ADJ
ejpam-842	120	54	set	set	NOUN
ejpam-842	120	55	u	u	NOUN
ejpam-842	120	56	containing	contain	VERB
ejpam-842	120	57	x	x	PUNCT
ejpam-842	120	58	such	such	ADJ
ejpam-842	120	59	that	that	SCONJ
ejpam-842	120	60	f	f	PROPN
ejpam-842	120	61	(	(	PUNCT
ejpam-842	120	62	u)⊂	u)⊂	NOUN
ejpam-842	120	63	cl⋆(v	cl⋆(v	NOUN
ejpam-842	120	64	)	)	PUNCT
ejpam-842	120	65	.	.	PUNCT
ejpam-842	121	1	definition	definition	NOUN
ejpam-842	121	2	3	3	NUM
ejpam-842	121	3	.	.	PUNCT
ejpam-842	122	1	a	a	DET
ejpam-842	122	2	function	function	NOUN
ejpam-842	122	3	f	f	NOUN
ejpam-842	122	4	:	:	PUNCT
ejpam-842	122	5	(	(	PUNCT
ejpam-842	122	6	x	x	X
ejpam-842	122	7	,	,	PUNCT
ejpam-842	122	8	τ	τ	PROPN
ejpam-842	122	9	,	,	PUNCT
ejpam-842	122	10	i)→	i)→	ADJ
ejpam-842	122	11	(	(	PUNCT
ejpam-842	122	12	y	y	PROPN
ejpam-842	122	13	,	,	PUNCT
ejpam-842	122	14	ω	ω	NOUN
ejpam-842	122	15	)	)	PUNCT
ejpam-842	122	16	is	be	AUX
ejpam-842	122	17	said	say	VERB
ejpam-842	122	18	to	to	PART
ejpam-842	122	19	be	be	AUX
ejpam-842	122	20	is⋆g	is⋆g	ADV
ejpam-842	122	21	-continuous	-continuous	ADJ
ejpam-842	122	22	if	if	SCONJ
ejpam-842	122	23	for	for	ADP
ejpam-842	122	24	every	every	DET
ejpam-842	122	25	u	u	PROPN
ejpam-842	122	26	∈	∈	PROPN
ejpam-842	122	27	ω	ω	PROPN
ejpam-842	122	28	,	,	PUNCT
ejpam-842	122	29	f	f	PROPN
ejpam-842	122	30	−1(u	−1(u	X
ejpam-842	122	31	)	)	PUNCT
ejpam-842	122	32	is	be	AUX
ejpam-842	122	33	is⋆	is⋆	PROPN
ejpam-842	122	34	g	g	NOUN
ejpam-842	122	35	-	-	PUNCT
ejpam-842	122	36	open	open	ADJ
ejpam-842	122	37	in	in	ADP
ejpam-842	122	38	(	(	PUNCT
ejpam-842	122	39	x	x	INTJ
ejpam-842	122	40	,	,	PUNCT
ejpam-842	122	41	τx	τx	INTJ
ejpam-842	122	42	,	,	PUNCT
ejpam-842	122	43	i	i	PROPN
ejpam-842	122	44	)	)	PUNCT
ejpam-842	122	45	.	.	PUNCT
ejpam-842	123	1	m.	m.	NOUN
ejpam-842	123	2	khan,̧	khan,̧	PROPN
ejpam-842	123	3	t.	t.	PROPN
ejpam-842	123	4	noiri	noiri	PROPN
ejpam-842	123	5	/	/	SYM
ejpam-842	123	6	eur	eur	PROPN
ejpam-842	123	7	.	.	PUNCT
ejpam-842	124	1	j.	j.	PROPN
ejpam-842	124	2	pure	pure	PROPN
ejpam-842	124	3	appl	appl	PROPN
ejpam-842	124	4	.	.	PROPN
ejpam-842	124	5	math	math	PROPN
ejpam-842	124	6	,	,	PUNCT
ejpam-842	124	7	4	4	NUM
ejpam-842	124	8	(	(	PUNCT
ejpam-842	124	9	2011	2011	NUM
ejpam-842	124	10	)	)	PUNCT
ejpam-842	124	11	,	,	PUNCT
ejpam-842	124	12	237	237	NUM
ejpam-842	124	13	-	-	SYM
ejpam-842	124	14	243	243	NUM
ejpam-842	124	15	240	240	NUM
ejpam-842	124	16	remark	remark	NOUN
ejpam-842	124	17	1	1	NUM
ejpam-842	124	18	.	.	PUNCT
ejpam-842	125	1	every	every	DET
ejpam-842	125	2	continuous	continuous	ADJ
ejpam-842	125	3	function	function	NOUN
ejpam-842	125	4	is	be	AUX
ejpam-842	125	5	is⋆g	is⋆g	ADV
ejpam-842	125	6	-continuous	-continuous	ADJ
ejpam-842	125	7	and	and	CCONJ
ejpam-842	125	8	the	the	DET
ejpam-842	125	9	converse	converse	NOUN
ejpam-842	125	10	need	need	AUX
ejpam-842	125	11	not	not	PART
ejpam-842	125	12	be	be	AUX
ejpam-842	125	13	true	true	ADJ
ejpam-842	125	14	as	as	SCONJ
ejpam-842	125	15	seen	see	VERB
ejpam-842	125	16	from	from	ADP
ejpam-842	125	17	example	example	NOUN
ejpam-842	125	18	2	2	NUM
ejpam-842	125	19	(	(	PUNCT
ejpam-842	125	20	below	below	ADV
ejpam-842	125	21	)	)	PUNCT
ejpam-842	125	22	.	.	PUNCT
ejpam-842	126	1	definition	definition	NOUN
ejpam-842	126	2	4	4	NUM
ejpam-842	126	3	.	.	PUNCT
ejpam-842	127	1	a	a	DET
ejpam-842	127	2	function	function	NOUN
ejpam-842	127	3	f	f	NOUN
ejpam-842	127	4	:	:	PUNCT
ejpam-842	127	5	(	(	PUNCT
ejpam-842	127	6	x	x	X
ejpam-842	127	7	,	,	PUNCT
ejpam-842	127	8	τ)→	τ)→	PROPN
ejpam-842	127	9	(	(	PUNCT
ejpam-842	127	10	y	y	PROPN
ejpam-842	127	11	,	,	PUNCT
ejpam-842	127	12	ω	ω	PROPN
ejpam-842	127	13	,	,	PUNCT
ejpam-842	127	14	j	j	NOUN
ejpam-842	127	15	)	)	PUNCT
ejpam-842	127	16	is	be	AUX
ejpam-842	127	17	said	say	VERB
ejpam-842	127	18	to	to	PART
ejpam-842	127	19	be	be	AUX
ejpam-842	127	20	strongly	strongly	ADV
ejpam-842	127	21	is⋆g	is⋆g	ADJ
ejpam-842	127	22	-	-	PUNCT
ejpam-842	127	23	continuous	continuous	ADJ
ejpam-842	127	24	if	if	SCONJ
ejpam-842	127	25	for	for	ADP
ejpam-842	127	26	every	every	DET
ejpam-842	127	27	is⋆g	is⋆g	ADJ
ejpam-842	127	28	-open	-open	NOUN
ejpam-842	127	29	set	set	NOUN
ejpam-842	127	30	u	u	NOUN
ejpam-842	127	31	in	in	ADP
ejpam-842	127	32	y	y	PROPN
ejpam-842	127	33	,	,	PUNCT
ejpam-842	127	34	f	f	PROPN
ejpam-842	127	35	−1(u	−1(u	X
ejpam-842	127	36	)	)	PUNCT
ejpam-842	127	37	is	be	AUX
ejpam-842	127	38	open	open	ADJ
ejpam-842	127	39	in	in	ADP
ejpam-842	127	40	x	x	X
ejpam-842	127	41	.	.	PUNCT
ejpam-842	128	1	remark	remark	PROPN
ejpam-842	128	2	2	2	NUM
ejpam-842	128	3	.	.	PUNCT
ejpam-842	129	1	every	every	DET
ejpam-842	129	2	strongly	strongly	ADV
ejpam-842	129	3	is⋆g	is⋆g	ADJ
ejpam-842	129	4	-continuous	-continuous	ADJ
ejpam-842	129	5	function	function	NOUN
ejpam-842	129	6	is	be	AUX
ejpam-842	129	7	continuous	continuous	ADJ
ejpam-842	129	8	but	but	CCONJ
ejpam-842	129	9	the	the	DET
ejpam-842	129	10	converse	converse	NOUN
ejpam-842	129	11	is	be	AUX
ejpam-842	129	12	not	not	PART
ejpam-842	129	13	true	true	ADJ
ejpam-842	129	14	in	in	ADP
ejpam-842	129	15	general	general	ADJ
ejpam-842	129	16	.	.	PUNCT
ejpam-842	130	1	example	example	NOUN
ejpam-842	131	1	1	1	NUM
ejpam-842	131	2	.	.	PUNCT
ejpam-842	131	3	let	let	VERB
ejpam-842	131	4	x	x	PUNCT
ejpam-842	131	5	=	=	PRON
ejpam-842	131	6	{	{	PUNCT
ejpam-842	131	7	a	a	PRON
ejpam-842	131	8	,	,	PUNCT
ejpam-842	131	9	b	b	NOUN
ejpam-842	131	10	,	,	PUNCT
ejpam-842	131	11	c	c	NOUN
ejpam-842	131	12	,	,	PUNCT
ejpam-842	131	13	d	d	NOUN
ejpam-842	131	14	}	}	PUNCT
ejpam-842	131	15	with	with	ADP
ejpam-842	131	16	τ	τ	X
ejpam-842	131	17	=	=	SYM
ejpam-842	131	18	{	{	PUNCT
ejpam-842	131	19	φ	φ	PROPN
ejpam-842	131	20	,	,	PUNCT
ejpam-842	131	21	x	x	INTJ
ejpam-842	131	22	,	,	PUNCT
ejpam-842	131	23	{	{	PUNCT
ejpam-842	131	24	a	a	NOUN
ejpam-842	131	25	}	}	PUNCT
ejpam-842	131	26	,	,	PUNCT
ejpam-842	131	27	{	{	PUNCT
ejpam-842	131	28	b	b	NOUN
ejpam-842	131	29	}	}	PUNCT
ejpam-842	131	30	,	,	PUNCT
ejpam-842	131	31	{	{	PUNCT
ejpam-842	131	32	a	a	DET
ejpam-842	131	33	,	,	PUNCT
ejpam-842	131	34	b	b	NOUN
ejpam-842	131	35	}	}	PUNCT
ejpam-842	131	36	,	,	PUNCT
ejpam-842	131	37	{	{	PUNCT
ejpam-842	131	38	b	b	X
ejpam-842	131	39	,	,	PUNCT
ejpam-842	131	40	c	c	NOUN
ejpam-842	131	41	}	}	PUNCT
ejpam-842	131	42	,	,	PUNCT
ejpam-842	131	43	{	{	PUNCT
ejpam-842	131	44	a	a	DET
ejpam-842	131	45	,	,	PUNCT
ejpam-842	131	46	b	b	NOUN
ejpam-842	131	47	,	,	PUNCT
ejpam-842	131	48	c	c	NOUN
ejpam-842	131	49	}	}	PUNCT
ejpam-842	131	50	}	}	PUNCT
ejpam-842	131	51	.	.	PUNCT
ejpam-842	132	1	let	let	VERB
ejpam-842	132	2	y	y	PROPN
ejpam-842	132	3	=	=	PUNCT
ejpam-842	132	4	{	{	PUNCT
ejpam-842	132	5	a	a	PRON
ejpam-842	132	6	,	,	PUNCT
ejpam-842	132	7	b	b	NOUN
ejpam-842	132	8	,	,	PUNCT
ejpam-842	132	9	c	c	NOUN
ejpam-842	132	10	,	,	PUNCT
ejpam-842	132	11	d	d	NOUN
ejpam-842	132	12	}	}	PUNCT
ejpam-842	132	13	with	with	ADP
ejpam-842	132	14	ω	ω	PROPN
ejpam-842	132	15	=	=	SYM
ejpam-842	132	16	{	{	PUNCT
ejpam-842	132	17	φ	φ	PROPN
ejpam-842	132	18	,	,	PUNCT
ejpam-842	132	19	y	y	PROPN
ejpam-842	132	20	,	,	PUNCT
ejpam-842	132	21	{	{	PUNCT
ejpam-842	132	22	a	a	X
ejpam-842	132	23	}	}	PUNCT
ejpam-842	132	24	,	,	PUNCT
ejpam-842	132	25	{	{	PUNCT
ejpam-842	132	26	b	b	NOUN
ejpam-842	132	27	,	,	PUNCT
ejpam-842	132	28	c	c	NOUN
ejpam-842	132	29	}	}	PUNCT
ejpam-842	132	30	,	,	PUNCT
ejpam-842	132	31	{	{	PUNCT
ejpam-842	132	32	a	a	PRON
ejpam-842	132	33	,	,	PUNCT
ejpam-842	132	34	b	b	NOUN
ejpam-842	132	35	,	,	PUNCT
ejpam-842	132	36	c	c	NOUN
ejpam-842	132	37	}	}	PUNCT
ejpam-842	132	38	}	}	PUNCT
ejpam-842	132	39	and	and	CCONJ
ejpam-842	132	40	j	j	PROPN
ejpam-842	132	41	=	=	PUNCT
ejpam-842	132	42	{	{	PUNCT
ejpam-842	132	43	φ	φ	PROPN
ejpam-842	132	44	,	,	PUNCT
ejpam-842	132	45	{	{	PUNCT
ejpam-842	132	46	a	a	X
ejpam-842	132	47	}	}	PUNCT
ejpam-842	132	48	}	}	PUNCT
ejpam-842	132	49	.	.	PUNCT
ejpam-842	133	1	let	let	VERB
ejpam-842	133	2	f	f	NOUN
ejpam-842	133	3	:	:	PUNCT
ejpam-842	133	4	(	(	PUNCT
ejpam-842	133	5	x	x	X
ejpam-842	133	6	,	,	PUNCT
ejpam-842	133	7	τ)→	τ)→	PROPN
ejpam-842	133	8	(	(	PUNCT
ejpam-842	133	9	y	y	PROPN
ejpam-842	133	10	,	,	PUNCT
ejpam-842	133	11	ω	ω	PROPN
ejpam-842	133	12	,	,	PUNCT
ejpam-842	133	13	j	j	NOUN
ejpam-842	133	14	)	)	PUNCT
ejpam-842	133	15	be	be	AUX
ejpam-842	133	16	defined	define	VERB
ejpam-842	133	17	by	by	ADP
ejpam-842	133	18	f	f	PROPN
ejpam-842	133	19	(	(	PUNCT
ejpam-842	133	20	a	a	NOUN
ejpam-842	133	21	)	)	PUNCT
ejpam-842	133	22	=	=	SYM
ejpam-842	133	23	b	b	PROPN
ejpam-842	133	24	,	,	PUNCT
ejpam-842	133	25	f	f	PROPN
ejpam-842	133	26	(	(	PUNCT
ejpam-842	133	27	b	b	NOUN
ejpam-842	133	28	)	)	PUNCT
ejpam-842	133	29	=	=	SYM
ejpam-842	134	1	a	a	PROPN
ejpam-842	134	2	,	,	PUNCT
ejpam-842	134	3	f	f	PROPN
ejpam-842	134	4	(	(	PUNCT
ejpam-842	134	5	c	c	NOUN
ejpam-842	134	6	)	)	PUNCT
ejpam-842	134	7	=	=	SYM
ejpam-842	135	1	a	a	PROPN
ejpam-842	135	2	and	and	CCONJ
ejpam-842	135	3	f	f	PROPN
ejpam-842	135	4	(	(	PUNCT
ejpam-842	135	5	d	d	NOUN
ejpam-842	135	6	)	)	PUNCT
ejpam-842	135	7	=	=	SYM
ejpam-842	135	8	d.	d.	PROPN
ejpam-842	135	9	then	then	ADV
ejpam-842	135	10	f	f	PROPN
ejpam-842	135	11	is	be	AUX
ejpam-842	135	12	continuous	continuous	ADJ
ejpam-842	135	13	.	.	PUNCT
ejpam-842	136	1	let	let	VERB
ejpam-842	136	2	u	u	PRON
ejpam-842	136	3	=	=	X
ejpam-842	136	4	{	{	PUNCT
ejpam-842	136	5	a	a	X
ejpam-842	136	6	,	,	PUNCT
ejpam-842	136	7	c	c	NOUN
ejpam-842	136	8	}	}	PUNCT
ejpam-842	136	9	then	then	ADV
ejpam-842	136	10	u	u	NOUN
ejpam-842	136	11	is	be	AUX
ejpam-842	136	12	is⋆	is⋆	NOUN
ejpam-842	136	13	g	g	NOUN
ejpam-842	136	14	-	-	PUNCT
ejpam-842	136	15	open	open	ADJ
ejpam-842	136	16	in	in	ADP
ejpam-842	136	17	y	y	PROPN
ejpam-842	136	18	but	but	CCONJ
ejpam-842	136	19	f	f	PROPN
ejpam-842	136	20	−1(u	−1(u	PROPN
ejpam-842	136	21	)	)	PUNCT
ejpam-842	137	1	=	=	SYM
ejpam-842	137	2	{	{	PUNCT
ejpam-842	137	3	a	a	X
ejpam-842	137	4	,	,	PUNCT
ejpam-842	137	5	c	c	NOUN
ejpam-842	137	6	}	}	PUNCT
ejpam-842	137	7	is	be	AUX
ejpam-842	137	8	not	not	PART
ejpam-842	137	9	open	open	ADJ
ejpam-842	137	10	in	in	ADP
ejpam-842	137	11	x	x	X
ejpam-842	137	12	.	.	PUNCT
ejpam-842	138	1	hence	hence	ADV
ejpam-842	138	2	f	f	PROPN
ejpam-842	138	3	is	be	AUX
ejpam-842	138	4	not	not	PART
ejpam-842	138	5	strongly	strongly	ADV
ejpam-842	138	6	is⋆gcontinuous	is⋆gcontinuous	ADJ
ejpam-842	138	7	.	.	PUNCT
ejpam-842	139	1	definition	definition	NOUN
ejpam-842	139	2	5	5	NUM
ejpam-842	139	3	.	.	PUNCT
ejpam-842	140	1	a	a	DET
ejpam-842	140	2	function	function	NOUN
ejpam-842	140	3	f	f	NOUN
ejpam-842	140	4	:	:	PUNCT
ejpam-842	140	5	(	(	PUNCT
ejpam-842	140	6	x	x	X
ejpam-842	140	7	,	,	PUNCT
ejpam-842	140	8	τ	τ	PROPN
ejpam-842	140	9	,	,	PUNCT
ejpam-842	140	10	i)→	i)→	ADJ
ejpam-842	140	11	(	(	PUNCT
ejpam-842	140	12	y	y	PROPN
ejpam-842	140	13	,	,	PUNCT
ejpam-842	140	14	ω	ω	PROPN
ejpam-842	140	15	,	,	PUNCT
ejpam-842	140	16	j	j	NOUN
ejpam-842	140	17	)	)	PUNCT
ejpam-842	140	18	is	be	AUX
ejpam-842	140	19	said	say	VERB
ejpam-842	140	20	to	to	PART
ejpam-842	140	21	be	be	AUX
ejpam-842	140	22	weakly	weakly	ADV
ejpam-842	140	23	is⋆g	is⋆g	ADV
ejpam-842	140	24	-continuous	-continuous	ADJ
ejpam-842	140	25	if	if	SCONJ
ejpam-842	140	26	for	for	SCONJ
ejpam-842	140	27	each	each	DET
ejpam-842	140	28	x	x	SYM
ejpam-842	140	29	∈	∈	PROPN
ejpam-842	140	30	x	x	X
ejpam-842	140	31	and	and	CCONJ
ejpam-842	140	32	each	each	DET
ejpam-842	140	33	open	open	ADJ
ejpam-842	140	34	set	set	VERB
ejpam-842	140	35	v	v	NOUN
ejpam-842	140	36	in	in	ADP
ejpam-842	140	37	y	y	PROPN
ejpam-842	140	38	containing	contain	VERB
ejpam-842	140	39	f	f	PROPN
ejpam-842	140	40	(	(	PUNCT
ejpam-842	140	41	x	x	NOUN
ejpam-842	140	42	)	)	PUNCT
ejpam-842	140	43	,	,	PUNCT
ejpam-842	140	44	there	there	PRON
ejpam-842	140	45	exists	exist	VERB
ejpam-842	140	46	an	an	DET
ejpam-842	140	47	is⋆g	is⋆g	NOUN
ejpam-842	140	48	-	-	PUNCT
ejpam-842	140	49	open	open	ADJ
ejpam-842	140	50	set	set	NOUN
ejpam-842	140	51	u	u	NOUN
ejpam-842	140	52	containing	contain	VERB
ejpam-842	140	53	x	x	PUNCT
ejpam-842	140	54	such	such	ADJ
ejpam-842	140	55	that	that	SCONJ
ejpam-842	140	56	f	f	PROPN
ejpam-842	140	57	(	(	PUNCT
ejpam-842	140	58	u)⊂	u)⊂	NOUN
ejpam-842	140	59	cl⋆(v	cl⋆(v	NOUN
ejpam-842	140	60	)	)	PUNCT
ejpam-842	140	61	.	.	PUNCT
ejpam-842	141	1	remark	remark	PROPN
ejpam-842	141	2	3	3	NUM
ejpam-842	141	3	.	.	PUNCT
ejpam-842	142	1	(	(	PUNCT
ejpam-842	142	2	1	1	X
ejpam-842	142	3	)	)	PUNCT
ejpam-842	142	4	every	every	DET
ejpam-842	142	5	weakly	weakly	ADJ
ejpam-842	142	6	i	i	PRON
ejpam-842	142	7	-	-	PUNCT
ejpam-842	142	8	continuous	continuous	ADJ
ejpam-842	142	9	function	function	NOUN
ejpam-842	142	10	is	be	AUX
ejpam-842	142	11	weakly	weakly	ADV
ejpam-842	142	12	is⋆g	is⋆g	ADJ
ejpam-842	142	13	-	-	ADJ
ejpam-842	142	14	continuous	continuous	ADJ
ejpam-842	142	15	but	but	CCONJ
ejpam-842	142	16	the	the	DET
ejpam-842	142	17	converse	converse	NOUN
ejpam-842	142	18	is	be	AUX
ejpam-842	142	19	not	not	PART
ejpam-842	142	20	true	true	ADJ
ejpam-842	142	21	in	in	ADP
ejpam-842	142	22	general	general	ADJ
ejpam-842	142	23	.	.	PUNCT
ejpam-842	143	1	(	(	PUNCT
ejpam-842	143	2	2	2	X
ejpam-842	143	3	)	)	PUNCT
ejpam-842	143	4	every	every	DET
ejpam-842	143	5	is⋆g	is⋆g	ADJ
ejpam-842	143	6	-continuous	-continuous	ADJ
ejpam-842	143	7	function	function	NOUN
ejpam-842	143	8	is	be	AUX
ejpam-842	143	9	weakly	weakly	ADJ
ejpam-842	143	10	is⋆g	is⋆g	ADJ
ejpam-842	143	11	-	-	ADJ
ejpam-842	143	12	continuous	continuous	ADJ
ejpam-842	143	13	.	.	PUNCT
ejpam-842	144	1	by	by	ADP
ejpam-842	144	2	the	the	DET
ejpam-842	144	3	above	above	ADJ
ejpam-842	144	4	definitions	definition	NOUN
ejpam-842	144	5	,	,	PUNCT
ejpam-842	144	6	for	for	ADP
ejpam-842	144	7	a	a	DET
ejpam-842	144	8	function	function	NOUN
ejpam-842	144	9	f	f	NOUN
ejpam-842	144	10	:	:	PUNCT
ejpam-842	144	11	(	(	PUNCT
ejpam-842	144	12	x	x	X
ejpam-842	144	13	,	,	PUNCT
ejpam-842	144	14	τ	τ	PROPN
ejpam-842	144	15	,	,	PUNCT
ejpam-842	144	16	i	i	NOUN
ejpam-842	144	17	)	)	PUNCT
ejpam-842	144	18	→	→	SYM
ejpam-842	144	19	(	(	PUNCT
ejpam-842	144	20	y	y	PROPN
ejpam-842	144	21	,	,	PUNCT
ejpam-842	144	22	ω	ω	PROPN
ejpam-842	144	23	,	,	PUNCT
ejpam-842	144	24	j	j	PROPN
ejpam-842	144	25	)	)	PUNCT
ejpam-842	144	26	we	we	PRON
ejpam-842	144	27	obtain	obtain	VERB
ejpam-842	144	28	the	the	DET
ejpam-842	144	29	following	follow	VERB
ejpam-842	144	30	implications	implication	NOUN
ejpam-842	144	31	:	:	PUNCT
ejpam-842	144	32	strong	strong	ADJ
ejpam-842	144	33	is⋆g	is⋆g	NOUN
ejpam-842	144	34	-continuity	-continuity	ADJ
ejpam-842	144	35	⇒	⇒	NOUN
ejpam-842	144	36	continuity	continuity	NOUN
ejpam-842	144	37	⇒	⇒	VERB
ejpam-842	144	38	is⋆g	is⋆g	ADJ
ejpam-842	144	39	-	-	PUNCT
ejpam-842	144	40	continuity	continuity	NOUN
ejpam-842	144	41	⇓	⇓	PROPN
ejpam-842	144	42	⇓	⇓	PROPN
ejpam-842	144	43	weak	weak	ADJ
ejpam-842	144	44	i	i	NOUN
ejpam-842	144	45	-	-	PUNCT
ejpam-842	144	46	continuity	continuity	NOUN
ejpam-842	144	47	⇒	⇒	NOUN
ejpam-842	144	48	weak	weak	ADJ
ejpam-842	144	49	is⋆g	is⋆g	ADJ
ejpam-842	144	50	-	-	PUNCT
ejpam-842	144	51	continuity	continuity	NOUN
ejpam-842	144	52	remark	remark	NOUN
ejpam-842	144	53	4	4	NUM
ejpam-842	144	54	.	.	PUNCT
ejpam-842	144	55	is⋆	is⋆	PROPN
ejpam-842	144	56	g	g	NOUN
ejpam-842	144	57	-	-	PUNCT
ejpam-842	144	58	continuity	continuity	NOUN
ejpam-842	144	59	and	and	CCONJ
ejpam-842	144	60	weak	weak	ADJ
ejpam-842	144	61	i	i	NOUN
ejpam-842	144	62	-	-	PUNCT
ejpam-842	144	63	continuity	continuity	NOUN
ejpam-842	144	64	are	be	AUX
ejpam-842	144	65	independent	independent	ADJ
ejpam-842	144	66	of	of	ADP
ejpam-842	144	67	each	each	DET
ejpam-842	144	68	other	other	ADJ
ejpam-842	144	69	.	.	PUNCT
ejpam-842	144	70	example	example	NOUN
ejpam-842	145	1	2	2	NUM
ejpam-842	145	2	.	.	PUNCT
ejpam-842	145	3	let	let	VERB
ejpam-842	145	4	x	x	SYM
ejpam-842	145	5	=	=	PUNCT
ejpam-842	145	6	y	y	PROPN
ejpam-842	145	7	=	=	PUNCT
ejpam-842	145	8	{	{	PUNCT
ejpam-842	145	9	a	a	PRON
ejpam-842	145	10	,	,	PUNCT
ejpam-842	145	11	b	b	NOUN
ejpam-842	145	12	,	,	PUNCT
ejpam-842	145	13	c	c	NOUN
ejpam-842	145	14	,	,	PUNCT
ejpam-842	145	15	d	d	NOUN
ejpam-842	145	16	}	}	PUNCT
ejpam-842	145	17	and	and	CCONJ
ejpam-842	145	18	τ	τ	PROPN
ejpam-842	145	19	=	=	SYM
ejpam-842	145	20	ω	ω	PROPN
ejpam-842	145	21	=	=	SYM
ejpam-842	145	22	{	{	PUNCT
ejpam-842	145	23	φ	φ	PROPN
ejpam-842	145	24	,	,	PUNCT
ejpam-842	145	25	x	x	INTJ
ejpam-842	145	26	,	,	PUNCT
ejpam-842	145	27	{	{	PUNCT
ejpam-842	145	28	a	a	DET
ejpam-842	145	29	,	,	PUNCT
ejpam-842	145	30	b	b	NOUN
ejpam-842	145	31	}	}	PUNCT
ejpam-842	145	32	}	}	PUNCT
ejpam-842	145	33	with	with	ADP
ejpam-842	145	34	i	i	PRON
ejpam-842	145	35	=	=	SYM
ejpam-842	145	36	{	{	PUNCT
ejpam-842	145	37	φ	φ	PROPN
ejpam-842	145	38	,	,	PUNCT
ejpam-842	145	39	{	{	PUNCT
ejpam-842	145	40	a	a	X
ejpam-842	145	41	}	}	PUNCT
ejpam-842	145	42	,	,	PUNCT
ejpam-842	145	43	{	{	PUNCT
ejpam-842	145	44	b	b	NOUN
ejpam-842	145	45	}	}	PUNCT
ejpam-842	145	46	,	,	PUNCT
ejpam-842	145	47	{	{	PUNCT
ejpam-842	145	48	a	a	DET
ejpam-842	145	49	,	,	PUNCT
ejpam-842	145	50	b	b	NOUN
ejpam-842	145	51	}	}	PUNCT
ejpam-842	145	52	}	}	PUNCT
ejpam-842	145	53	.	.	PUNCT
ejpam-842	146	1	define	define	VERB
ejpam-842	146	2	f	f	NOUN
ejpam-842	146	3	:	:	PUNCT
ejpam-842	146	4	(	(	PUNCT
ejpam-842	146	5	x	x	X
ejpam-842	146	6	,	,	PUNCT
ejpam-842	146	7	τ	τ	PROPN
ejpam-842	146	8	,	,	PUNCT
ejpam-842	146	9	i	i	NOUN
ejpam-842	146	10	)	)	PUNCT
ejpam-842	146	11	→	→	SYM
ejpam-842	146	12	(	(	PUNCT
ejpam-842	146	13	y	y	PROPN
ejpam-842	146	14	,	,	PUNCT
ejpam-842	146	15	ω	ω	PROPN
ejpam-842	146	16	,	,	PUNCT
ejpam-842	146	17	i	i	NOUN
ejpam-842	146	18	)	)	PUNCT
ejpam-842	146	19	by	by	ADP
ejpam-842	146	20	f	f	PROPN
ejpam-842	146	21	(	(	PUNCT
ejpam-842	146	22	a	a	NOUN
ejpam-842	146	23	)	)	PUNCT
ejpam-842	146	24	=	=	SYM
ejpam-842	146	25	a	a	PROPN
ejpam-842	146	26	,	,	PUNCT
ejpam-842	146	27	f	f	PROPN
ejpam-842	146	28	(	(	PUNCT
ejpam-842	146	29	b	b	NOUN
ejpam-842	146	30	)	)	PUNCT
ejpam-842	147	1	=	=	SYM
ejpam-842	147	2	c	c	X
ejpam-842	147	3	,	,	PUNCT
ejpam-842	147	4	f	f	PROPN
ejpam-842	147	5	(	(	PUNCT
ejpam-842	147	6	c	c	NOUN
ejpam-842	147	7	)	)	PUNCT
ejpam-842	147	8	=	=	SYM
ejpam-842	147	9	b	b	PROPN
ejpam-842	147	10	and	and	CCONJ
ejpam-842	147	11	f	f	PROPN
ejpam-842	147	12	(	(	PUNCT
ejpam-842	147	13	d	d	NOUN
ejpam-842	147	14	)	)	PUNCT
ejpam-842	147	15	=	=	SYM
ejpam-842	148	1	b.	b.	PROPN
ejpam-842	148	2	then	then	ADV
ejpam-842	148	3	f	f	PROPN
ejpam-842	148	4	is	be	AUX
ejpam-842	148	5	is⋆g	is⋆g	ADV
ejpam-842	148	6	-continuous	-continuous	ADJ
ejpam-842	148	7	but	but	CCONJ
ejpam-842	148	8	not	not	PART
ejpam-842	148	9	weak	weak	ADJ
ejpam-842	148	10	i	i	NOUN
ejpam-842	148	11	-	-	NOUN
ejpam-842	148	12	continuous	continuous	ADJ
ejpam-842	148	13	.	.	PUNCT
ejpam-842	149	1	since	since	SCONJ
ejpam-842	149	2	for	for	ADP
ejpam-842	149	3	c	c	PROPN
ejpam-842	149	4	∈	∈	PROPN
ejpam-842	149	5	x	x	X
ejpam-842	149	6	,	,	PUNCT
ejpam-842	149	7	f	f	PROPN
ejpam-842	149	8	(	(	PUNCT
ejpam-842	149	9	c	c	NOUN
ejpam-842	149	10	)	)	PUNCT
ejpam-842	149	11	=	=	SYM
ejpam-842	149	12	b	b	PROPN
ejpam-842	149	13	and	and	CCONJ
ejpam-842	149	14	an	an	DET
ejpam-842	149	15	open	open	ADJ
ejpam-842	149	16	set	set	NOUN
ejpam-842	149	17	v	v	AUX
ejpam-842	149	18	=	=	PUNCT
ejpam-842	149	19	{	{	PUNCT
ejpam-842	149	20	a	a	PRON
ejpam-842	149	21	,	,	PUNCT
ejpam-842	149	22	b	b	NOUN
ejpam-842	149	23	}	}	PUNCT
ejpam-842	149	24	containing	contain	VERB
ejpam-842	149	25	f	f	X
ejpam-842	149	26	(	(	PUNCT
ejpam-842	149	27	c	c	NOUN
ejpam-842	149	28	)	)	PUNCT
ejpam-842	149	29	,	,	PUNCT
ejpam-842	149	30	the	the	DET
ejpam-842	149	31	only	only	ADJ
ejpam-842	149	32	open	open	ADJ
ejpam-842	149	33	set	set	NOUN
ejpam-842	149	34	containing	contain	VERB
ejpam-842	149	35	c	c	PROPN
ejpam-842	149	36	is	be	AUX
ejpam-842	149	37	u	u	NOUN
ejpam-842	149	38	=	=	PROPN
ejpam-842	149	39	x	x	X
ejpam-842	149	40	and	and	CCONJ
ejpam-842	149	41	f	f	PROPN
ejpam-842	149	42	(	(	PUNCT
ejpam-842	149	43	u	u	NOUN
ejpam-842	149	44	)	)	PUNCT
ejpam-842	149	45	*	*	PUNCT
ejpam-842	149	46	cl⋆(v	cl⋆(v	PROPN
ejpam-842	149	47	)	)	PUNCT
ejpam-842	150	1	=	=	PUNCT
ejpam-842	150	2	{	{	PUNCT
ejpam-842	150	3	a	a	PRON
ejpam-842	150	4	,	,	PUNCT
ejpam-842	150	5	b	b	NOUN
ejpam-842	150	6	}	}	PUNCT
ejpam-842	150	7	.	.	PUNCT
ejpam-842	151	1	example	example	NOUN
ejpam-842	152	1	3	3	X
ejpam-842	152	2	.	.	PUNCT
ejpam-842	152	3	let	let	VERB
ejpam-842	152	4	x	x	PUNCT
ejpam-842	152	5	=	=	PRON
ejpam-842	152	6	{	{	PUNCT
ejpam-842	152	7	a	a	PRON
ejpam-842	152	8	,	,	PUNCT
ejpam-842	152	9	b	b	NOUN
ejpam-842	152	10	,	,	PUNCT
ejpam-842	152	11	c	c	NOUN
ejpam-842	152	12	,	,	PUNCT
ejpam-842	152	13	d	d	NOUN
ejpam-842	152	14	}	}	PUNCT
ejpam-842	152	15	,	,	PUNCT
ejpam-842	152	16	τ	τ	X
ejpam-842	152	17	=	=	PUNCT
ejpam-842	152	18	{	{	PUNCT
ejpam-842	152	19	φ	φ	PROPN
ejpam-842	152	20	,	,	PUNCT
ejpam-842	152	21	x	x	INTJ
ejpam-842	152	22	,	,	PUNCT
ejpam-842	152	23	{	{	PUNCT
ejpam-842	152	24	a	a	PRON
ejpam-842	152	25	,	,	PUNCT
ejpam-842	152	26	b	b	NOUN
ejpam-842	152	27	}	}	PUNCT
ejpam-842	152	28	}	}	PUNCT
ejpam-842	152	29	and	and	CCONJ
ejpam-842	152	30	i	i	PRON
ejpam-842	152	31	=	=	PUNCT
ejpam-842	152	32	{	{	PUNCT
ejpam-842	152	33	φ	φ	PROPN
ejpam-842	152	34	,	,	PUNCT
ejpam-842	152	35	{	{	PUNCT
ejpam-842	152	36	a	a	X
ejpam-842	152	37	}	}	PUNCT
ejpam-842	152	38	,	,	PUNCT
ejpam-842	152	39	{	{	PUNCT
ejpam-842	152	40	b	b	NOUN
ejpam-842	152	41	}	}	PUNCT
ejpam-842	152	42	,	,	PUNCT
ejpam-842	152	43	{	{	PUNCT
ejpam-842	152	44	a	a	PRON
ejpam-842	152	45	,	,	PUNCT
ejpam-842	152	46	b	b	NOUN
ejpam-842	152	47	}	}	PUNCT
ejpam-842	152	48	}	}	PUNCT
ejpam-842	152	49	.	.	PUNCT
ejpam-842	153	1	let	let	VERB
ejpam-842	153	2	y	y	PROPN
ejpam-842	153	3	=	=	PUNCT
ejpam-842	153	4	{	{	PUNCT
ejpam-842	153	5	1,2,3,4	1,2,3,4	NUM
ejpam-842	153	6	}	}	PUNCT
ejpam-842	153	7	,	,	PUNCT
ejpam-842	153	8	ω	ω	PROPN
ejpam-842	153	9	=	=	SYM
ejpam-842	153	10	{	{	PUNCT
ejpam-842	153	11	φ	φ	PROPN
ejpam-842	153	12	,	,	PUNCT
ejpam-842	153	13	{	{	PUNCT
ejpam-842	153	14	1,2	1,2	NUM
ejpam-842	153	15	}	}	PUNCT
ejpam-842	153	16	,	,	PUNCT
ejpam-842	153	17	y	y	PROPN
ejpam-842	153	18	}	}	PUNCT
ejpam-842	153	19	and	and	CCONJ
ejpam-842	153	20	j	j	PROPN
ejpam-842	153	21	=	=	PUNCT
ejpam-842	153	22	{	{	PUNCT
ejpam-842	153	23	φ	φ	PROPN
ejpam-842	153	24	,	,	PUNCT
ejpam-842	153	25	{	{	PUNCT
ejpam-842	153	26	3	3	NUM
ejpam-842	153	27	}	}	PUNCT
ejpam-842	153	28	,	,	PUNCT
ejpam-842	153	29	{	{	PUNCT
ejpam-842	153	30	4	4	NUM
ejpam-842	153	31	}	}	PUNCT
ejpam-842	153	32	,	,	PUNCT
ejpam-842	153	33	{	{	PUNCT
ejpam-842	153	34	3,4	3,4	NUM
ejpam-842	153	35	}	}	PUNCT
ejpam-842	153	36	}	}	PUNCT
ejpam-842	153	37	.	.	PUNCT
ejpam-842	154	1	define	define	VERB
ejpam-842	154	2	f	f	NOUN
ejpam-842	154	3	:	:	PUNCT
ejpam-842	154	4	(	(	PUNCT
ejpam-842	154	5	x	x	X
ejpam-842	154	6	,	,	PUNCT
ejpam-842	154	7	τ	τ	PROPN
ejpam-842	154	8	,	,	PUNCT
ejpam-842	154	9	i)→	i)→	ADJ
ejpam-842	154	10	(	(	PUNCT
ejpam-842	154	11	y	y	PROPN
ejpam-842	154	12	,	,	PUNCT
ejpam-842	154	13	ω	ω	PROPN
ejpam-842	154	14	,	,	PUNCT
ejpam-842	154	15	i	i	NOUN
ejpam-842	154	16	)	)	PUNCT
ejpam-842	154	17	by	by	ADP
ejpam-842	154	18	f	f	PROPN
ejpam-842	154	19	(	(	PUNCT
ejpam-842	154	20	a	a	NOUN
ejpam-842	154	21	)	)	PUNCT
ejpam-842	154	22	=	=	SYM
ejpam-842	154	23	1	1	NUM
ejpam-842	154	24	,	,	PUNCT
ejpam-842	154	25	f	f	PROPN
ejpam-842	154	26	(	(	PUNCT
ejpam-842	154	27	b	b	NOUN
ejpam-842	154	28	)	)	PUNCT
ejpam-842	154	29	=	=	SYM
ejpam-842	154	30	3	3	NUM
ejpam-842	154	31	,	,	PUNCT
ejpam-842	154	32	f	f	X
ejpam-842	154	33	(	(	PUNCT
ejpam-842	154	34	c	c	NOUN
ejpam-842	154	35	)	)	PUNCT
ejpam-842	154	36	=	=	SYM
ejpam-842	154	37	2	2	NUM
ejpam-842	154	38	and	and	CCONJ
ejpam-842	154	39	f	f	PROPN
ejpam-842	154	40	(	(	PUNCT
ejpam-842	154	41	d	d	NOUN
ejpam-842	154	42	)	)	PUNCT
ejpam-842	154	43	=	=	SYM
ejpam-842	154	44	4	4	X
ejpam-842	154	45	.	.	X
ejpam-842	154	46	f	f	PROPN
ejpam-842	154	47	is	be	AUX
ejpam-842	154	48	weak	weak	ADJ
ejpam-842	154	49	i	i	NOUN
ejpam-842	154	50	-	-	PUNCT
ejpam-842	154	51	continuous	continuous	ADJ
ejpam-842	154	52	but	but	CCONJ
ejpam-842	154	53	not	not	PART
ejpam-842	154	54	is⋆g	is⋆g	ADJ
ejpam-842	154	55	-	-	PUNCT
ejpam-842	154	56	continuous	continuous	ADJ
ejpam-842	154	57	.	.	PUNCT
ejpam-842	155	1	since	since	SCONJ
ejpam-842	155	2	v	v	NUM
ejpam-842	155	3	=	=	SYM
ejpam-842	155	4	{	{	PUNCT
ejpam-842	155	5	1,2	1,2	NUM
ejpam-842	155	6	}	}	PUNCT
ejpam-842	155	7	is	be	AUX
ejpam-842	155	8	open	open	ADJ
ejpam-842	155	9	in	in	ADP
ejpam-842	155	10	y	y	PROPN
ejpam-842	155	11	but	but	CCONJ
ejpam-842	155	12	f	f	PROPN
ejpam-842	155	13	−1(v	−1(v	PROPN
ejpam-842	155	14	)	)	PUNCT
ejpam-842	156	1	=	=	PRON
ejpam-842	156	2	{	{	PUNCT
ejpam-842	156	3	a	a	X
ejpam-842	156	4	,	,	PUNCT
ejpam-842	156	5	c	c	NOUN
ejpam-842	156	6	}	}	PUNCT
ejpam-842	156	7	is	be	AUX
ejpam-842	156	8	not	not	PART
ejpam-842	156	9	is⋆	is⋆	NOUN
ejpam-842	156	10	g	g	NOUN
ejpam-842	156	11	-	-	PUNCT
ejpam-842	156	12	open	open	ADJ
ejpam-842	156	13	in	in	ADP
ejpam-842	156	14	x	x	X
ejpam-842	156	15	.	.	PUNCT
ejpam-842	157	1	theorem	theorem	NOUN
ejpam-842	157	2	6	6	NUM
ejpam-842	157	3	.	.	PUNCT
ejpam-842	158	1	let	let	VERB
ejpam-842	158	2	f	f	NOUN
ejpam-842	158	3	:	:	PUNCT
ejpam-842	158	4	(	(	PUNCT
ejpam-842	158	5	x	x	X
ejpam-842	158	6	,	,	PUNCT
ejpam-842	158	7	τ	τ	PROPN
ejpam-842	158	8	,	,	PUNCT
ejpam-842	158	9	i)→	i)→	ADJ
ejpam-842	158	10	(	(	PUNCT
ejpam-842	158	11	y	y	PROPN
ejpam-842	158	12	,	,	PUNCT
ejpam-842	158	13	ω	ω	NOUN
ejpam-842	158	14	)	)	PUNCT
ejpam-842	158	15	be	be	AUX
ejpam-842	158	16	a	a	DET
ejpam-842	158	17	function	function	NOUN
ejpam-842	158	18	.	.	PUNCT
ejpam-842	159	1	then	then	ADV
ejpam-842	159	2	,	,	PUNCT
ejpam-842	159	3	the	the	DET
ejpam-842	159	4	following	follow	VERB
ejpam-842	159	5	statements	statement	NOUN
ejpam-842	159	6	are	be	AUX
ejpam-842	159	7	equivalent	equivalent	ADJ
ejpam-842	159	8	:	:	PUNCT
ejpam-842	159	9	(	(	PUNCT
ejpam-842	159	10	1	1	X
ejpam-842	159	11	)	)	PUNCT
ejpam-842	159	12	f	f	PROPN
ejpam-842	159	13	is	be	AUX
ejpam-842	159	14	is⋆	is⋆	NOUN
ejpam-842	159	15	g	g	NOUN
ejpam-842	159	16	-	-	PUNCT
ejpam-842	159	17	continuous	continuous	ADJ
ejpam-842	159	18	.	.	PUNCT
ejpam-842	160	1	(	(	PUNCT
ejpam-842	160	2	2	2	X
ejpam-842	160	3	)	)	PUNCT
ejpam-842	160	4	the	the	DET
ejpam-842	160	5	inverse	inverse	ADJ
ejpam-842	160	6	image	image	NOUN
ejpam-842	160	7	of	of	ADP
ejpam-842	160	8	each	each	DET
ejpam-842	160	9	closed	close	VERB
ejpam-842	160	10	set	set	VERB
ejpam-842	160	11	in	in	ADP
ejpam-842	160	12	y	y	PROPN
ejpam-842	160	13	is	be	AUX
ejpam-842	160	14	is⋆g	is⋆g	ADV
ejpam-842	160	15	-closed	-closed	ADJ
ejpam-842	160	16	in	in	ADP
ejpam-842	160	17	x.	x.	PROPN
ejpam-842	160	18	m.	m.	NOUN
ejpam-842	160	19	khan,̧	khan,̧	PROPN
ejpam-842	160	20	t.	t.	PROPN
ejpam-842	160	21	noiri	noiri	PROPN
ejpam-842	160	22	/	/	SYM
ejpam-842	160	23	eur	eur	PROPN
ejpam-842	160	24	.	.	PUNCT
ejpam-842	161	1	j.	j.	PROPN
ejpam-842	161	2	pure	pure	PROPN
ejpam-842	161	3	appl	appl	PROPN
ejpam-842	161	4	.	.	PROPN
ejpam-842	161	5	math	math	PROPN
ejpam-842	161	6	,	,	PUNCT
ejpam-842	161	7	4	4	NUM
ejpam-842	161	8	(	(	PUNCT
ejpam-842	161	9	2011	2011	NUM
ejpam-842	161	10	)	)	PUNCT
ejpam-842	161	11	,	,	PUNCT
ejpam-842	161	12	237	237	NUM
ejpam-842	161	13	-	-	SYM
ejpam-842	161	14	243	243	NUM
ejpam-842	161	15	241	241	NUM
ejpam-842	161	16	(	(	PUNCT
ejpam-842	161	17	3	3	NUM
ejpam-842	161	18	)	)	PUNCT
ejpam-842	161	19	the	the	DET
ejpam-842	161	20	inverse	inverse	ADJ
ejpam-842	161	21	image	image	NOUN
ejpam-842	161	22	of	of	ADP
ejpam-842	161	23	each	each	DET
ejpam-842	161	24	open	open	ADJ
ejpam-842	161	25	set	set	NOUN
ejpam-842	161	26	in	in	ADP
ejpam-842	161	27	y	y	PROPN
ejpam-842	161	28	is	be	AUX
ejpam-842	161	29	is⋆	is⋆	NOUN
ejpam-842	161	30	g	g	NOUN
ejpam-842	161	31	-	-	PUNCT
ejpam-842	161	32	open	open	ADJ
ejpam-842	161	33	in	in	ADP
ejpam-842	161	34	x.	x.	NOUN
ejpam-842	161	35	definition	definition	NOUN
ejpam-842	161	36	6	6	NUM
ejpam-842	161	37	.	.	PUNCT
ejpam-842	162	1	an	an	DET
ejpam-842	162	2	ideal	ideal	ADJ
ejpam-842	162	3	topological	topological	ADJ
ejpam-842	162	4	space	space	NOUN
ejpam-842	162	5	(	(	PUNCT
ejpam-842	162	6	x	x	X
ejpam-842	162	7	,	,	PUNCT
ejpam-842	162	8	τ	τ	PROPN
ejpam-842	162	9	,	,	PUNCT
ejpam-842	162	10	i	i	PROPN
ejpam-842	162	11	)	)	PUNCT
ejpam-842	162	12	is	be	AUX
ejpam-842	162	13	said	say	VERB
ejpam-842	162	14	to	to	PART
ejpam-842	162	15	be	be	AUX
ejpam-842	162	16	t	t	NOUN
ejpam-842	162	17	-	-	PUNCT
ejpam-842	162	18	dense	dense	ADJ
ejpam-842	162	19	if	if	SCONJ
ejpam-842	162	20	every	every	DET
ejpam-842	162	21	subset	subset	NOUN
ejpam-842	162	22	of	of	ADP
ejpam-842	162	23	x	x	PUNCT
ejpam-842	162	24	is	be	AUX
ejpam-842	162	25	⋆-dense	⋆-dense	NOUN
ejpam-842	162	26	in	in	ADP
ejpam-842	162	27	itself	itself	PRON
ejpam-842	162	28	.	.	PUNCT
ejpam-842	163	1	definition	definition	NOUN
ejpam-842	163	2	7	7	NUM
ejpam-842	163	3	.	.	PUNCT
ejpam-842	164	1	let	let	VERB
ejpam-842	164	2	n	n	PRON
ejpam-842	164	3	be	be	AUX
ejpam-842	164	4	a	a	DET
ejpam-842	164	5	subset	subset	NOUN
ejpam-842	164	6	of	of	ADP
ejpam-842	164	7	a	a	DET
ejpam-842	164	8	space	space	NOUN
ejpam-842	164	9	(	(	PUNCT
ejpam-842	164	10	x	x	X
ejpam-842	164	11	,	,	PUNCT
ejpam-842	164	12	τ	τ	PROPN
ejpam-842	164	13	,	,	PUNCT
ejpam-842	164	14	i	i	NOUN
ejpam-842	164	15	)	)	PUNCT
ejpam-842	164	16	and	and	CCONJ
ejpam-842	164	17	x	x	PUNCT
ejpam-842	164	18	∈	∈	NOUN
ejpam-842	164	19	x	x	X
ejpam-842	164	20	.	.	PUNCT
ejpam-842	165	1	then	then	ADV
ejpam-842	165	2	n	n	PROPN
ejpam-842	165	3	is	be	AUX
ejpam-842	165	4	called	call	VERB
ejpam-842	165	5	an	an	DET
ejpam-842	165	6	is⋆	is⋆	NOUN
ejpam-842	165	7	g	g	NOUN
ejpam-842	165	8	-	-	PUNCT
ejpam-842	165	9	open	open	ADJ
ejpam-842	165	10	neighborhood	neighborhood	NOUN
ejpam-842	165	11	of	of	ADP
ejpam-842	165	12	x	x	PRON
ejpam-842	165	13	if	if	SCONJ
ejpam-842	165	14	there	there	PRON
ejpam-842	165	15	exists	exist	VERB
ejpam-842	165	16	an	an	DET
ejpam-842	165	17	is⋆g	is⋆g	ADJ
ejpam-842	165	18	-open	-open	NOUN
ejpam-842	165	19	set	set	NOUN
ejpam-842	165	20	u	u	NOUN
ejpam-842	165	21	containing	contain	VERB
ejpam-842	165	22	x	x	PUNCT
ejpam-842	165	23	such	such	ADJ
ejpam-842	165	24	that	that	SCONJ
ejpam-842	165	25	u	u	PROPN
ejpam-842	165	26	⊂	⊂	PROPN
ejpam-842	165	27	n.	n.	PROPN
ejpam-842	165	28	theorem	theorem	VERB
ejpam-842	165	29	7	7	NUM
ejpam-842	165	30	.	.	PUNCT
ejpam-842	166	1	let	let	VERB
ejpam-842	166	2	(	(	PUNCT
ejpam-842	166	3	x	x	X
ejpam-842	166	4	,	,	PUNCT
ejpam-842	166	5	τ	τ	PROPN
ejpam-842	166	6	,	,	PUNCT
ejpam-842	166	7	i	i	PRON
ejpam-842	166	8	)	)	PUNCT
ejpam-842	166	9	be	be	VERB
ejpam-842	166	10	t	t	NOUN
ejpam-842	166	11	-	-	PUNCT
ejpam-842	166	12	dense	dense	ADJ
ejpam-842	166	13	.	.	PUNCT
ejpam-842	167	1	then	then	ADV
ejpam-842	167	2	,	,	PUNCT
ejpam-842	167	3	for	for	ADP
ejpam-842	167	4	a	a	DET
ejpam-842	167	5	function	function	NOUN
ejpam-842	167	6	f	f	NOUN
ejpam-842	167	7	:	:	PUNCT
ejpam-842	167	8	(	(	PUNCT
ejpam-842	167	9	x	x	X
ejpam-842	167	10	,	,	PUNCT
ejpam-842	167	11	τ	τ	PROPN
ejpam-842	167	12	,	,	PUNCT
ejpam-842	167	13	i	i	NOUN
ejpam-842	167	14	)	)	PUNCT
ejpam-842	167	15	→	→	SYM
ejpam-842	167	16	(	(	PUNCT
ejpam-842	167	17	y	y	PROPN
ejpam-842	167	18	,	,	PUNCT
ejpam-842	167	19	ω	ω	NOUN
ejpam-842	167	20	)	)	PUNCT
ejpam-842	167	21	the	the	DET
ejpam-842	167	22	following	follow	VERB
ejpam-842	167	23	statements	statement	NOUN
ejpam-842	167	24	are	be	AUX
ejpam-842	167	25	equivalent	equivalent	ADJ
ejpam-842	167	26	:	:	PUNCT
ejpam-842	167	27	(	(	PUNCT
ejpam-842	167	28	1	1	X
ejpam-842	167	29	)	)	PUNCT
ejpam-842	167	30	f	f	PROPN
ejpam-842	167	31	is	be	AUX
ejpam-842	167	32	is⋆	is⋆	NOUN
ejpam-842	167	33	g	g	NOUN
ejpam-842	167	34	-	-	PUNCT
ejpam-842	167	35	continuous	continuous	ADJ
ejpam-842	167	36	.	.	PUNCT
ejpam-842	168	1	(	(	PUNCT
ejpam-842	168	2	2	2	NUM
ejpam-842	168	3	)	)	PUNCT
ejpam-842	168	4	for	for	ADP
ejpam-842	168	5	each	each	DET
ejpam-842	168	6	x	x	SYM
ejpam-842	168	7	∈	∈	PROPN
ejpam-842	168	8	x	x	X
ejpam-842	168	9	and	and	CCONJ
ejpam-842	168	10	each	each	DET
ejpam-842	168	11	open	open	ADJ
ejpam-842	168	12	set	set	VERB
ejpam-842	168	13	v	v	NOUN
ejpam-842	168	14	in	in	ADP
ejpam-842	168	15	y	y	PROPN
ejpam-842	168	16	with	with	ADP
ejpam-842	168	17	f	f	PROPN
ejpam-842	168	18	(	(	PUNCT
ejpam-842	168	19	x	x	X
ejpam-842	168	20	)	)	PUNCT
ejpam-842	168	21	∈	∈	NOUN
ejpam-842	168	22	v	v	NOUN
ejpam-842	168	23	,	,	PUNCT
ejpam-842	168	24	there	there	PRON
ejpam-842	168	25	exists	exist	VERB
ejpam-842	168	26	an	an	DET
ejpam-842	168	27	is⋆	is⋆	NOUN
ejpam-842	168	28	g	g	NOUN
ejpam-842	168	29	-	-	PUNCT
ejpam-842	168	30	open	open	ADJ
ejpam-842	168	31	set	set	NOUN
ejpam-842	168	32	u	u	NOUN
ejpam-842	168	33	containing	contain	VERB
ejpam-842	168	34	x	x	PUNCT
ejpam-842	168	35	such	such	ADJ
ejpam-842	168	36	that	that	SCONJ
ejpam-842	168	37	f	f	PROPN
ejpam-842	168	38	(	(	PUNCT
ejpam-842	168	39	u)⊂	u)⊂	NOUN
ejpam-842	168	40	v	v	NOUN
ejpam-842	168	41	.	.	PUNCT
ejpam-842	169	1	(	(	PUNCT
ejpam-842	169	2	3	3	X
ejpam-842	169	3	)	)	PUNCT
ejpam-842	169	4	for	for	ADP
ejpam-842	169	5	each	each	DET
ejpam-842	169	6	x	x	SYM
ejpam-842	169	7	∈	∈	PROPN
ejpam-842	169	8	x	x	X
ejpam-842	169	9	and	and	CCONJ
ejpam-842	169	10	each	each	DET
ejpam-842	169	11	open	open	ADJ
ejpam-842	169	12	set	set	VERB
ejpam-842	169	13	v	v	NOUN
ejpam-842	169	14	in	in	ADP
ejpam-842	169	15	y	y	PROPN
ejpam-842	169	16	with	with	ADP
ejpam-842	169	17	f	f	PROPN
ejpam-842	169	18	(	(	PUNCT
ejpam-842	169	19	x	x	X
ejpam-842	169	20	)	)	PUNCT
ejpam-842	169	21	∈	∈	NOUN
ejpam-842	169	22	v	v	NOUN
ejpam-842	169	23	,	,	PUNCT
ejpam-842	169	24	f	f	PROPN
ejpam-842	169	25	−1(v	−1(v	PROPN
ejpam-842	169	26	)	)	PUNCT
ejpam-842	169	27	is	be	AUX
ejpam-842	169	28	an	an	DET
ejpam-842	169	29	is⋆	is⋆	PROPN
ejpam-842	169	30	g	g	NOUN
ejpam-842	169	31	-	-	PUNCT
ejpam-842	169	32	open	open	ADJ
ejpam-842	169	33	neighborhood	neighborhood	NOUN
ejpam-842	169	34	of	of	ADP
ejpam-842	169	35	x.	x.	NOUN
ejpam-842	169	36	proof	proof	NOUN
ejpam-842	169	37	.	.	PUNCT
ejpam-842	170	1	(	(	PUNCT
ejpam-842	170	2	1	1	X
ejpam-842	170	3	)	)	PUNCT
ejpam-842	170	4	⇒	⇒	NOUN
ejpam-842	170	5	(	(	PUNCT
ejpam-842	170	6	2	2	X
ejpam-842	170	7	)	)	PUNCT
ejpam-842	170	8	let	let	VERB
ejpam-842	170	9	x	x	SYM
ejpam-842	170	10	∈	∈	NOUN
ejpam-842	170	11	x	x	PUNCT
ejpam-842	170	12	and	and	CCONJ
ejpam-842	170	13	let	let	VERB
ejpam-842	170	14	v	v	PART
ejpam-842	170	15	be	be	AUX
ejpam-842	170	16	an	an	DET
ejpam-842	170	17	open	open	ADJ
ejpam-842	170	18	set	set	NOUN
ejpam-842	170	19	in	in	ADP
ejpam-842	170	20	y	y	PRON
ejpam-842	170	21	such	such	ADJ
ejpam-842	170	22	that	that	SCONJ
ejpam-842	170	23	f	f	PROPN
ejpam-842	170	24	(	(	PUNCT
ejpam-842	170	25	x	x	X
ejpam-842	170	26	)	)	PUNCT
ejpam-842	170	27	∈	∈	NOUN
ejpam-842	170	28	v	v	NOUN
ejpam-842	170	29	.	.	PUNCT
ejpam-842	171	1	since	since	SCONJ
ejpam-842	171	2	f	f	PROPN
ejpam-842	171	3	is	be	AUX
ejpam-842	171	4	is⋆	is⋆	NOUN
ejpam-842	171	5	g	g	NOUN
ejpam-842	171	6	-	-	PUNCT
ejpam-842	171	7	continuous	continuous	ADJ
ejpam-842	171	8	,	,	PUNCT
ejpam-842	171	9	f	f	PROPN
ejpam-842	171	10	−1(v	−1(v	PROPN
ejpam-842	171	11	)	)	PUNCT
ejpam-842	171	12	is	be	AUX
ejpam-842	171	13	is⋆g	is⋆g	ADJ
ejpam-842	171	14	-open	-open	ADJ
ejpam-842	171	15	in	in	ADP
ejpam-842	171	16	x	x	X
ejpam-842	171	17	.	.	PUNCT
ejpam-842	172	1	by	by	ADP
ejpam-842	172	2	putting	put	VERB
ejpam-842	172	3	u	u	NOUN
ejpam-842	172	4	=	=	PUNCT
ejpam-842	172	5	f	f	PROPN
ejpam-842	172	6	−1(v	−1(v	PROPN
ejpam-842	172	7	)	)	PUNCT
ejpam-842	172	8	,	,	PUNCT
ejpam-842	172	9	we	we	PRON
ejpam-842	172	10	have	have	VERB
ejpam-842	172	11	x	x	X
ejpam-842	172	12	∈	∈	PROPN
ejpam-842	172	13	u	u	NOUN
ejpam-842	172	14	and	and	CCONJ
ejpam-842	172	15	f	f	PROPN
ejpam-842	172	16	(	(	PUNCT
ejpam-842	172	17	u)⊂	u)⊂	NOUN
ejpam-842	172	18	v	v	NOUN
ejpam-842	172	19	.	.	PUNCT
ejpam-842	173	1	(	(	PUNCT
ejpam-842	173	2	2	2	X
ejpam-842	173	3	)	)	PUNCT
ejpam-842	173	4	⇒	⇒	NOUN
ejpam-842	173	5	(	(	PUNCT
ejpam-842	173	6	3	3	X
ejpam-842	173	7	)	)	PUNCT
ejpam-842	173	8	let	let	VERB
ejpam-842	173	9	v	v	PART
ejpam-842	173	10	be	be	AUX
ejpam-842	173	11	an	an	DET
ejpam-842	173	12	open	open	ADJ
ejpam-842	173	13	set	set	NOUN
ejpam-842	173	14	in	in	ADP
ejpam-842	173	15	y	y	PROPN
ejpam-842	173	16	and	and	CCONJ
ejpam-842	173	17	let	let	VERB
ejpam-842	173	18	f	f	PROPN
ejpam-842	173	19	(	(	PUNCT
ejpam-842	173	20	x	x	X
ejpam-842	173	21	)	)	PUNCT
ejpam-842	173	22	∈	∈	NOUN
ejpam-842	173	23	v	v	NOUN
ejpam-842	173	24	.	.	PUNCT
ejpam-842	174	1	then	then	ADV
ejpam-842	174	2	by	by	ADP
ejpam-842	174	3	(	(	PUNCT
ejpam-842	174	4	2	2	NUM
ejpam-842	174	5	)	)	PUNCT
ejpam-842	174	6	,	,	PUNCT
ejpam-842	174	7	there	there	PRON
ejpam-842	174	8	exists	exist	VERB
ejpam-842	174	9	an	an	DET
ejpam-842	174	10	is⋆g	is⋆g	ADJ
ejpam-842	174	11	-open	-open	NOUN
ejpam-842	174	12	set	set	NOUN
ejpam-842	174	13	u	u	NOUN
ejpam-842	174	14	containing	contain	VERB
ejpam-842	174	15	x	x	PUNCT
ejpam-842	174	16	such	such	ADJ
ejpam-842	174	17	that	that	SCONJ
ejpam-842	174	18	f	f	PROPN
ejpam-842	174	19	(	(	PUNCT
ejpam-842	174	20	u	u	NOUN
ejpam-842	174	21	)	)	PUNCT
ejpam-842	174	22	⊂	⊂	PROPN
ejpam-842	174	23	v	v	NOUN
ejpam-842	174	24	.	.	PUNCT
ejpam-842	175	1	so	so	ADV
ejpam-842	175	2	x	x	SYM
ejpam-842	175	3	∈	∈	PROPN
ejpam-842	175	4	u	u	X
ejpam-842	175	5	⊂	⊂	PROPN
ejpam-842	175	6	f	f	PROPN
ejpam-842	175	7	−1(v	−1(v	PROPN
ejpam-842	175	8	)	)	PUNCT
ejpam-842	175	9	.	.	PUNCT
ejpam-842	176	1	hence	hence	ADV
ejpam-842	176	2	f	f	PROPN
ejpam-842	176	3	−1(v	−1(v	PROPN
ejpam-842	176	4	)	)	PUNCT
ejpam-842	176	5	is	be	AUX
ejpam-842	176	6	an	an	DET
ejpam-842	176	7	is⋆g	is⋆g	ADJ
ejpam-842	176	8	-open	-open	ADJ
ejpam-842	176	9	neighbourhood	neighbourhood	NOUN
ejpam-842	176	10	of	of	ADP
ejpam-842	176	11	x	x	X
ejpam-842	176	12	.	.	PUNCT
ejpam-842	177	1	(	(	PUNCT
ejpam-842	177	2	3	3	X
ejpam-842	177	3	)	)	PUNCT
ejpam-842	177	4	⇒	⇒	NOUN
ejpam-842	177	5	(	(	PUNCT
ejpam-842	177	6	1	1	X
ejpam-842	177	7	)	)	PUNCT
ejpam-842	177	8	let	let	VERB
ejpam-842	177	9	v	v	PART
ejpam-842	177	10	be	be	AUX
ejpam-842	177	11	an	an	DET
ejpam-842	177	12	open	open	ADJ
ejpam-842	177	13	set	set	NOUN
ejpam-842	177	14	in	in	ADP
ejpam-842	177	15	y	y	PROPN
ejpam-842	177	16	and	and	CCONJ
ejpam-842	177	17	let	let	VERB
ejpam-842	177	18	f	f	PROPN
ejpam-842	177	19	(	(	PUNCT
ejpam-842	177	20	x	x	X
ejpam-842	177	21	)	)	PUNCT
ejpam-842	177	22	∈	∈	NOUN
ejpam-842	177	23	v	v	NOUN
ejpam-842	177	24	.	.	PUNCT
ejpam-842	178	1	then	then	ADV
ejpam-842	178	2	by	by	ADP
ejpam-842	178	3	(	(	PUNCT
ejpam-842	178	4	3	3	NUM
ejpam-842	178	5	)	)	PUNCT
ejpam-842	178	6	,	,	PUNCT
ejpam-842	178	7	f	f	PROPN
ejpam-842	178	8	−1(v	−1(v	PROPN
ejpam-842	178	9	)	)	PUNCT
ejpam-842	178	10	is	be	AUX
ejpam-842	178	11	an	an	DET
ejpam-842	178	12	is⋆gneighborhood	is⋆gneighborhood	NOUN
ejpam-842	178	13	of	of	ADP
ejpam-842	178	14	x	x	PROPN
ejpam-842	178	15	.	.	PUNCT
ejpam-842	179	1	thus	thus	ADV
ejpam-842	179	2	for	for	ADP
ejpam-842	179	3	each	each	DET
ejpam-842	179	4	x	x	SYM
ejpam-842	179	5	∈	∈	PROPN
ejpam-842	179	6	f	f	PROPN
ejpam-842	179	7	−1(v	−1(v	PROPN
ejpam-842	179	8	)	)	PUNCT
ejpam-842	179	9	,	,	PUNCT
ejpam-842	179	10	there	there	PRON
ejpam-842	179	11	exists	exist	VERB
ejpam-842	179	12	an	an	DET
ejpam-842	179	13	is⋆g	is⋆g	ADJ
ejpam-842	179	14	-open	-open	NOUN
ejpam-842	179	15	set	set	VERB
ejpam-842	179	16	ux	ux	NOUN
ejpam-842	179	17	containing	contain	VERB
ejpam-842	179	18	x	x	PUNCT
ejpam-842	179	19	such	such	ADJ
ejpam-842	179	20	that	that	SCONJ
ejpam-842	179	21	x	x	SYM
ejpam-842	179	22	∈	∈	PROPN
ejpam-842	179	23	ux	ux	PROPN
ejpam-842	179	24	⊂	⊂	PROPN
ejpam-842	179	25	f	f	PROPN
ejpam-842	179	26	−1(v	−1(v	PROPN
ejpam-842	179	27	)	)	PUNCT
ejpam-842	179	28	.	.	PUNCT
ejpam-842	180	1	hence	hence	ADV
ejpam-842	180	2	f	f	PROPN
ejpam-842	180	3	−1(v	−1(v	PROPN
ejpam-842	180	4	)	)	PUNCT
ejpam-842	181	1	=	=	PUNCT
ejpam-842	181	2	∪x∈	∪x∈	PROPN
ejpam-842	181	3	f	f	PROPN
ejpam-842	181	4	−1(v	−1(v	PROPN
ejpam-842	181	5	)	)	PUNCT
ejpam-842	181	6	ux	ux	PROPN
ejpam-842	181	7	and	and	CCONJ
ejpam-842	181	8	so	so	ADV
ejpam-842	181	9	by	by	ADP
ejpam-842	181	10	theorem	theorem	NOUN
ejpam-842	181	11	2.12	2.12	NUM
ejpam-842	181	12	[	[	X
ejpam-842	181	13	5	5	NUM
ejpam-842	181	14	]	]	PUNCT
ejpam-842	181	15	,	,	PUNCT
ejpam-842	181	16	f	f	PROPN
ejpam-842	181	17	−1(v	−1(v	PROPN
ejpam-842	181	18	)	)	PUNCT
ejpam-842	181	19	is	be	AUX
ejpam-842	181	20	is⋆g	is⋆g	NOUN
ejpam-842	181	21	-	-	PUNCT
ejpam-842	181	22	open	open	ADJ
ejpam-842	181	23	in	in	ADP
ejpam-842	181	24	x	x	X
ejpam-842	181	25	.	.	PUNCT
ejpam-842	182	1	theorem	theorem	ADJ
ejpam-842	182	2	8	8	NUM
ejpam-842	182	3	.	.	PUNCT
ejpam-842	183	1	a	a	DET
ejpam-842	183	2	function	function	NOUN
ejpam-842	183	3	f	f	NOUN
ejpam-842	183	4	:	:	PUNCT
ejpam-842	183	5	(	(	PUNCT
ejpam-842	183	6	x	x	X
ejpam-842	183	7	,	,	PUNCT
ejpam-842	183	8	τ)→	τ)→	PROPN
ejpam-842	183	9	(	(	PUNCT
ejpam-842	183	10	y	y	PROPN
ejpam-842	183	11	,	,	PUNCT
ejpam-842	183	12	ω	ω	PROPN
ejpam-842	183	13	,	,	PUNCT
ejpam-842	183	14	j	j	NOUN
ejpam-842	183	15	)	)	PUNCT
ejpam-842	183	16	is	be	AUX
ejpam-842	183	17	strongly	strongly	ADV
ejpam-842	183	18	is⋆	is⋆	PROPN
ejpam-842	183	19	g	g	NOUN
ejpam-842	183	20	-	-	PUNCT
ejpam-842	183	21	continuous	continuous	ADJ
ejpam-842	183	22	if	if	SCONJ
ejpam-842	184	1	and	and	CCONJ
ejpam-842	184	2	only	only	ADV
ejpam-842	184	3	if	if	SCONJ
ejpam-842	184	4	the	the	DET
ejpam-842	184	5	inverse	inverse	ADJ
ejpam-842	184	6	image	image	NOUN
ejpam-842	184	7	of	of	ADP
ejpam-842	184	8	every	every	DET
ejpam-842	184	9	is⋆g	is⋆g	PROPN
ejpam-842	184	10	-closed	-close	VERB
ejpam-842	184	11	set	set	NOUN
ejpam-842	184	12	in	in	ADP
ejpam-842	184	13	y	y	PROPN
ejpam-842	184	14	is	be	AUX
ejpam-842	184	15	closed	close	VERB
ejpam-842	184	16	in	in	ADP
ejpam-842	184	17	x	x	X
ejpam-842	184	18	.	.	PUNCT
ejpam-842	185	1	theorem	theorem	ADJ
ejpam-842	185	2	9	9	NUM
ejpam-842	185	3	.	.	PUNCT
ejpam-842	186	1	(	(	PUNCT
ejpam-842	186	2	1	1	X
ejpam-842	186	3	)	)	PUNCT
ejpam-842	186	4	let	let	VERB
ejpam-842	186	5	f	f	NOUN
ejpam-842	186	6	:	:	PUNCT
ejpam-842	186	7	(	(	PUNCT
ejpam-842	186	8	x	x	X
ejpam-842	186	9	,	,	PUNCT
ejpam-842	186	10	τ)→	τ)→	PROPN
ejpam-842	186	11	(	(	PUNCT
ejpam-842	186	12	y	y	PROPN
ejpam-842	186	13	,	,	PUNCT
ejpam-842	186	14	ω	ω	PROPN
ejpam-842	186	15	,	,	PUNCT
ejpam-842	186	16	j	j	NOUN
ejpam-842	186	17	)	)	PUNCT
ejpam-842	186	18	be	be	VERB
ejpam-842	186	19	strongly	strongly	ADV
ejpam-842	186	20	is⋆	is⋆	PROPN
ejpam-842	186	21	g	g	NOUN
ejpam-842	186	22	-	-	PUNCT
ejpam-842	186	23	continuous	continuous	ADJ
ejpam-842	186	24	and	and	CCONJ
ejpam-842	186	25	h	h	NOUN
ejpam-842	186	26	:	:	PUNCT
ejpam-842	186	27	(	(	PUNCT
ejpam-842	186	28	y	y	PROPN
ejpam-842	186	29	,	,	PUNCT
ejpam-842	186	30	ω	ω	PROPN
ejpam-842	186	31	,	,	PUNCT
ejpam-842	186	32	j)→	j)→	PROPN
ejpam-842	186	33	(	(	PUNCT
ejpam-842	186	34	z	z	NOUN
ejpam-842	186	35	,	,	PUNCT
ejpam-842	186	36	σ	σ	PROPN
ejpam-842	186	37	)	)	PUNCT
ejpam-842	186	38	be	be	AUX
ejpam-842	186	39	is⋆g	is⋆g	NOUN
ejpam-842	186	40	-continuous	-continuous	ADJ
ejpam-842	186	41	,	,	PUNCT
ejpam-842	186	42	then	then	ADV
ejpam-842	186	43	h	h	PROPN
ejpam-842	186	44	◦	◦	NOUN
ejpam-842	186	45	f	f	PROPN
ejpam-842	186	46	is	be	AUX
ejpam-842	186	47	continuous	continuous	ADJ
ejpam-842	186	48	.	.	PUNCT
ejpam-842	187	1	(	(	PUNCT
ejpam-842	187	2	2	2	X
ejpam-842	187	3	)	)	PUNCT
ejpam-842	187	4	let	let	VERB
ejpam-842	187	5	f	f	NOUN
ejpam-842	187	6	:	:	PUNCT
ejpam-842	187	7	(	(	PUNCT
ejpam-842	187	8	x	x	X
ejpam-842	187	9	,	,	PUNCT
ejpam-842	187	10	τ	τ	PROPN
ejpam-842	187	11	,	,	PUNCT
ejpam-842	187	12	i	i	NOUN
ejpam-842	187	13	)	)	PUNCT
ejpam-842	187	14	→	→	SYM
ejpam-842	187	15	(	(	PUNCT
ejpam-842	187	16	y	y	PROPN
ejpam-842	187	17	,	,	PUNCT
ejpam-842	187	18	ω	ω	NOUN
ejpam-842	187	19	)	)	PUNCT
ejpam-842	187	20	be	be	AUX
ejpam-842	187	21	is⋆g	is⋆g	PROPN
ejpam-842	187	22	-continues	-continue	NOUN
ejpam-842	187	23	and	and	CCONJ
ejpam-842	187	24	g	g	NOUN
ejpam-842	187	25	:	:	PUNCT
ejpam-842	187	26	(	(	PUNCT
ejpam-842	187	27	y	y	PROPN
ejpam-842	187	28	,	,	PUNCT
ejpam-842	187	29	ω	ω	NOUN
ejpam-842	187	30	)	)	PUNCT
ejpam-842	187	31	→	→	SYM
ejpam-842	187	32	(	(	PUNCT
ejpam-842	187	33	z	z	NOUN
ejpam-842	187	34	,	,	PUNCT
ejpam-842	187	35	σ	σ	PROPN
ejpam-842	187	36	)	)	PUNCT
ejpam-842	187	37	be	be	AUX
ejpam-842	187	38	continuous	continuous	ADJ
ejpam-842	187	39	,	,	PUNCT
ejpam-842	188	1	then	then	ADV
ejpam-842	188	2	g	g	PROPN
ejpam-842	188	3	◦	◦	NOUN
ejpam-842	188	4	f	f	X
ejpam-842	188	5	:	:	PUNCT
ejpam-842	188	6	(	(	PUNCT
ejpam-842	188	7	x	x	X
ejpam-842	188	8	,	,	PUNCT
ejpam-842	188	9	τ	τ	PROPN
ejpam-842	188	10	,	,	PUNCT
ejpam-842	188	11	i)→	i)→	ADJ
ejpam-842	188	12	(	(	PUNCT
ejpam-842	188	13	z	z	NOUN
ejpam-842	188	14	,	,	PUNCT
ejpam-842	188	15	σ	σ	PROPN
ejpam-842	188	16	)	)	PUNCT
ejpam-842	188	17	is	be	AUX
ejpam-842	188	18	is⋆g	is⋆g	PROPN
ejpam-842	188	19	-continuous	-continuous	ADJ
ejpam-842	188	20	.	.	PUNCT
ejpam-842	189	1	theorem	theorem	NOUN
ejpam-842	189	2	10	10	NUM
ejpam-842	189	3	.	.	PUNCT
ejpam-842	190	1	let	let	VERB
ejpam-842	190	2	f	f	NOUN
ejpam-842	190	3	:	:	PUNCT
ejpam-842	190	4	(	(	PUNCT
ejpam-842	190	5	x	x	X
ejpam-842	190	6	,	,	PUNCT
ejpam-842	190	7	τ	τ	PROPN
ejpam-842	190	8	,	,	PUNCT
ejpam-842	190	9	i)→	i)→	ADJ
ejpam-842	190	10	(	(	PUNCT
ejpam-842	190	11	y	y	PROPN
ejpam-842	190	12	,	,	PUNCT
ejpam-842	190	13	ω	ω	NOUN
ejpam-842	190	14	)	)	PUNCT
ejpam-842	190	15	be	be	AUX
ejpam-842	190	16	is⋆g	is⋆g	ADV
ejpam-842	190	17	-continuous	-continuous	ADJ
ejpam-842	190	18	and	and	CCONJ
ejpam-842	190	19	u	u	NOUN
ejpam-842	190	20	∈	∈	PROPN
ejpam-842	190	21	ro(x	ro(x	PUNCT
ejpam-842	190	22	)	)	PUNCT
ejpam-842	190	23	.	.	PUNCT
ejpam-842	191	1	then	then	ADV
ejpam-842	191	2	the	the	DET
ejpam-842	191	3	restriction	restriction	NOUN
ejpam-842	191	4	f	f	PROPN
ejpam-842	191	5	|	|	ADV
ejpam-842	191	6	u	u	X
ejpam-842	191	7	:	:	PUNCT
ejpam-842	191	8	(	(	PUNCT
ejpam-842	191	9	u	u	NOUN
ejpam-842	191	10	,	,	PUNCT
ejpam-842	191	11	τu	τu	ADV
ejpam-842	191	12	,	,	PUNCT
ejpam-842	191	13	iu	iu	ADP
ejpam-842	191	14	)	)	PUNCT
ejpam-842	191	15	→	→	SYM
ejpam-842	191	16	(	(	PUNCT
ejpam-842	191	17	y	y	PROPN
ejpam-842	191	18	,	,	PUNCT
ejpam-842	191	19	ω	ω	NOUN
ejpam-842	191	20	)	)	PUNCT
ejpam-842	191	21	is	be	AUX
ejpam-842	191	22	is⋆g	is⋆g	ADJ
ejpam-842	191	23	-	-	PUNCT
ejpam-842	191	24	continuous	continuous	ADJ
ejpam-842	191	25	.	.	PUNCT
ejpam-842	192	1	proof	proof	NOUN
ejpam-842	192	2	.	.	PUNCT
ejpam-842	193	1	let	let	VERB
ejpam-842	193	2	v	v	PART
ejpam-842	193	3	be	be	AUX
ejpam-842	193	4	any	any	DET
ejpam-842	193	5	open	open	ADJ
ejpam-842	193	6	set	set	NOUN
ejpam-842	193	7	of	of	ADP
ejpam-842	193	8	(	(	PUNCT
ejpam-842	193	9	y	y	PROPN
ejpam-842	193	10	,	,	PUNCT
ejpam-842	193	11	τy	τy	NUM
ejpam-842	193	12	)	)	PUNCT
ejpam-842	193	13	.	.	PUNCT
ejpam-842	194	1	since	since	SCONJ
ejpam-842	194	2	f	f	PROPN
ejpam-842	194	3	is	be	AUX
ejpam-842	194	4	is⋆	is⋆	NOUN
ejpam-842	194	5	g	g	NOUN
ejpam-842	194	6	-	-	PUNCT
ejpam-842	194	7	continuous	continuous	ADJ
ejpam-842	194	8	,	,	PUNCT
ejpam-842	194	9	f	f	PROPN
ejpam-842	194	10	−1(v	−1(v	PROPN
ejpam-842	194	11	)	)	PUNCT
ejpam-842	194	12	is	be	AUX
ejpam-842	194	13	is⋆	is⋆	PROPN
ejpam-842	194	14	g	g	NOUN
ejpam-842	194	15	-	-	PUNCT
ejpam-842	194	16	open	open	ADJ
ejpam-842	194	17	in	in	ADP
ejpam-842	194	18	x	x	X
ejpam-842	194	19	.	.	PUNCT
ejpam-842	195	1	by	by	ADP
ejpam-842	195	2	theorem	theorem	NOUN
ejpam-842	195	3	2.14	2.14	NUM
ejpam-842	195	4	of	of	ADP
ejpam-842	195	5	[	[	X
ejpam-842	195	6	5	5	NUM
ejpam-842	195	7	]	]	PUNCT
ejpam-842	195	8	,	,	PUNCT
ejpam-842	195	9	f	f	PROPN
ejpam-842	195	10	−1(v	−1(v	PROPN
ejpam-842	195	11	)	)	PUNCT
ejpam-842	195	12	∩	∩	NOUN
ejpam-842	195	13	u	u	NOUN
ejpam-842	195	14	is	be	AUX
ejpam-842	195	15	is⋆g	is⋆g	X
ejpam-842	195	16	-open	-open	ADJ
ejpam-842	195	17	in	in	ADP
ejpam-842	195	18	x	x	X
ejpam-842	195	19	.	.	PUNCT
ejpam-842	196	1	thus	thus	ADV
ejpam-842	196	2	by	by	ADP
ejpam-842	196	3	theorem	theorem	NOUN
ejpam-842	196	4	4	4	NUM
ejpam-842	196	5	(	(	PUNCT
ejpam-842	196	6	f	f	NOUN
ejpam-842	196	7	|	|	ADV
ejpam-842	196	8	u)−1(v	u)−1(v	PRON
ejpam-842	196	9	)	)	PUNCT
ejpam-842	197	1	=	=	SYM
ejpam-842	197	2	f	f	PROPN
ejpam-842	197	3	−1(v	−1(v	NOUN
ejpam-842	197	4	)	)	PUNCT
ejpam-842	197	5	∩	∩	NOUN
ejpam-842	197	6	u	u	NOUN
ejpam-842	197	7	is	be	AUX
ejpam-842	197	8	is⋆	is⋆	NOUN
ejpam-842	197	9	g	g	NOUN
ejpam-842	197	10	-	-	PUNCT
ejpam-842	197	11	open	open	ADJ
ejpam-842	197	12	in	in	ADP
ejpam-842	197	13	u	u	NOUN
ejpam-842	197	14	because	because	SCONJ
ejpam-842	197	15	u	u	NOUN
ejpam-842	197	16	is	be	AUX
ejpam-842	197	17	regular	regular	ADV
ejpam-842	197	18	-	-	PUNCT
ejpam-842	197	19	open	open	ADJ
ejpam-842	197	20	in	in	ADP
ejpam-842	197	21	x	x	X
ejpam-842	197	22	.	.	PUNCT
ejpam-842	198	1	this	this	PRON
ejpam-842	198	2	proves	prove	VERB
ejpam-842	198	3	that	that	SCONJ
ejpam-842	198	4	f	f	PROPN
ejpam-842	198	5	|	|	ADV
ejpam-842	198	6	u	u	X
ejpam-842	198	7	:	:	PUNCT
ejpam-842	198	8	(	(	PUNCT
ejpam-842	198	9	u	u	NOUN
ejpam-842	198	10	,	,	PUNCT
ejpam-842	198	11	τ	τ	PROPN
ejpam-842	198	12	|	|	ADV
ejpam-842	198	13	u	u	NOUN
ejpam-842	198	14	,	,	PUNCT
ejpam-842	198	15	i	i	PRON
ejpam-842	198	16	|	|	ADV
ejpam-842	198	17	u)→	u)→	ADJ
ejpam-842	198	18	(	(	PUNCT
ejpam-842	198	19	y	y	PROPN
ejpam-842	198	20	,	,	PUNCT
ejpam-842	198	21	τy	τy	PART
ejpam-842	198	22	)	)	PUNCT
ejpam-842	198	23	is	be	AUX
ejpam-842	198	24	is⋆g	is⋆g	ADJ
ejpam-842	198	25	-	-	PUNCT
ejpam-842	198	26	continuous	continuous	ADJ
ejpam-842	198	27	.	.	PUNCT
ejpam-842	199	1	m.	m.	NOUN
ejpam-842	199	2	khan,̧	khan,̧	PROPN
ejpam-842	199	3	t.	t.	PROPN
ejpam-842	199	4	noiri	noiri	PROPN
ejpam-842	199	5	/	/	SYM
ejpam-842	199	6	eur	eur	PROPN
ejpam-842	199	7	.	.	PUNCT
ejpam-842	200	1	j.	j.	PROPN
ejpam-842	200	2	pure	pure	PROPN
ejpam-842	200	3	appl	appl	PROPN
ejpam-842	200	4	.	.	PROPN
ejpam-842	200	5	math	math	PROPN
ejpam-842	200	6	,	,	PUNCT
ejpam-842	200	7	4	4	NUM
ejpam-842	200	8	(	(	PUNCT
ejpam-842	200	9	2011	2011	NUM
ejpam-842	200	10	)	)	PUNCT
ejpam-842	200	11	,	,	PUNCT
ejpam-842	200	12	237	237	NUM
ejpam-842	200	13	-	-	SYM
ejpam-842	200	14	243	243	NUM
ejpam-842	200	15	242	242	NUM
ejpam-842	200	16	theorem	theorem	NOUN
ejpam-842	200	17	11	11	NUM
ejpam-842	200	18	.	.	PUNCT
ejpam-842	201	1	let	let	VERB
ejpam-842	201	2	f	f	NOUN
ejpam-842	201	3	:	:	PUNCT
ejpam-842	201	4	(	(	PUNCT
ejpam-842	201	5	x	x	X
ejpam-842	201	6	,	,	PUNCT
ejpam-842	201	7	τ	τ	PROPN
ejpam-842	201	8	,	,	PUNCT
ejpam-842	201	9	i	i	NOUN
ejpam-842	201	10	)	)	PUNCT
ejpam-842	201	11	→	→	SYM
ejpam-842	201	12	(	(	PUNCT
ejpam-842	201	13	y	y	PROPN
ejpam-842	201	14	,	,	PUNCT
ejpam-842	201	15	ω	ω	PROPN
ejpam-842	201	16	,	,	PUNCT
ejpam-842	201	17	j	j	NOUN
ejpam-842	201	18	)	)	PUNCT
ejpam-842	201	19	be	be	VERB
ejpam-842	201	20	a	a	DET
ejpam-842	201	21	function	function	NOUN
ejpam-842	201	22	and	and	CCONJ
ejpam-842	201	23	{	{	PUNCT
ejpam-842	201	24	uα	uα	X
ejpam-842	201	25	:	:	PUNCT
ejpam-842	201	26	α	α	PROPN
ejpam-842	201	27	∈	∈	PROPN
ejpam-842	201	28	∇	∇	X
ejpam-842	201	29	}	}	PUNCT
ejpam-842	201	30	be	be	AUX
ejpam-842	201	31	an	an	DET
ejpam-842	201	32	open	open	ADJ
ejpam-842	201	33	cover	cover	NOUN
ejpam-842	201	34	of	of	ADP
ejpam-842	201	35	a	a	DET
ejpam-842	201	36	t	t	NOUN
ejpam-842	201	37	-	-	PUNCT
ejpam-842	201	38	dense	dense	ADJ
ejpam-842	201	39	space	space	NOUN
ejpam-842	201	40	x	x	X
ejpam-842	201	41	.	.	PUNCT
ejpam-842	202	1	if	if	SCONJ
ejpam-842	202	2	the	the	DET
ejpam-842	202	3	restriction	restriction	NOUN
ejpam-842	202	4	f	f	PROPN
ejpam-842	202	5	|	|	ADV
ejpam-842	202	6	uα	uα	PROPN
ejpam-842	202	7	is	be	AUX
ejpam-842	202	8	is⋆g	is⋆g	NOUN
ejpam-842	202	9	-	-	ADJ
ejpam-842	202	10	continuous	continuous	ADJ
ejpam-842	202	11	for	for	ADP
ejpam-842	202	12	each	each	DET
ejpam-842	202	13	α	α	PROPN
ejpam-842	202	14	∈	∈	PROPN
ejpam-842	202	15	∇	∇	NOUN
ejpam-842	202	16	,	,	PUNCT
ejpam-842	202	17	then	then	ADV
ejpam-842	202	18	f	f	PROPN
ejpam-842	202	19	is	be	AUX
ejpam-842	202	20	is⋆g	is⋆g	PRON
ejpam-842	202	21	-continuous	-continuous	ADJ
ejpam-842	202	22	.	.	PUNCT
ejpam-842	203	1	proof	proof	NOUN
ejpam-842	203	2	.	.	PUNCT
ejpam-842	204	1	suppose	suppose	VERB
ejpam-842	204	2	f	f	PROPN
ejpam-842	204	3	is	be	AUX
ejpam-842	204	4	an	an	DET
ejpam-842	204	5	arbitrary	arbitrary	ADJ
ejpam-842	204	6	open	open	ADJ
ejpam-842	204	7	set	set	NOUN
ejpam-842	204	8	in	in	ADP
ejpam-842	204	9	(	(	PUNCT
ejpam-842	204	10	y	y	PROPN
ejpam-842	204	11	,	,	PUNCT
ejpam-842	204	12	ω	ω	PROPN
ejpam-842	204	13	,	,	PUNCT
ejpam-842	204	14	j	j	PROPN
ejpam-842	204	15	)	)	PUNCT
ejpam-842	204	16	.	.	PUNCT
ejpam-842	205	1	then	then	ADV
ejpam-842	205	2	for	for	ADP
ejpam-842	205	3	each	each	DET
ejpam-842	205	4	α	α	PROPN
ejpam-842	205	5	∈	∈	PROPN
ejpam-842	205	6	∇	∇	NOUN
ejpam-842	205	7	,	,	PUNCT
ejpam-842	205	8	we	we	PRON
ejpam-842	205	9	have	have	VERB
ejpam-842	205	10	(	(	PUNCT
ejpam-842	205	11	f	f	X
ejpam-842	205	12	|	|	ADV
ejpam-842	205	13	uα	uα	PROPN
ejpam-842	205	14	)	)	PUNCT
ejpam-842	205	15	−1(v	−1(v	NOUN
ejpam-842	205	16	)	)	PUNCT
ejpam-842	206	1	=	=	PUNCT
ejpam-842	206	2	f	f	PROPN
ejpam-842	206	3	−1(v	−1(v	NOUN
ejpam-842	206	4	)	)	PUNCT
ejpam-842	206	5	∩uα	∩uα	NOUN
ejpam-842	206	6	.	.	PUNCT
ejpam-842	207	1	because	because	SCONJ
ejpam-842	207	2	f	f	PROPN
ejpam-842	207	3	|	|	ADV
ejpam-842	207	4	uα	uα	PROPN
ejpam-842	207	5	is	be	AUX
ejpam-842	207	6	is⋆g	is⋆g	PROPN
ejpam-842	207	7	-continuous	-continuous	ADJ
ejpam-842	207	8	,	,	PUNCT
ejpam-842	207	9	therefore	therefore	ADV
ejpam-842	207	10	,	,	PUNCT
ejpam-842	207	11	f	f	PROPN
ejpam-842	207	12	−1(v	−1(v	PROPN
ejpam-842	207	13	)	)	PUNCT
ejpam-842	207	14	∩uα	∩uα	NOUN
ejpam-842	207	15	is	be	AUX
ejpam-842	207	16	is⋆gopen	is⋆gopen	ADJ
ejpam-842	207	17	in	in	ADP
ejpam-842	207	18	x	x	PUNCT
ejpam-842	207	19	for	for	ADP
ejpam-842	207	20	each	each	DET
ejpam-842	207	21	α	α	NOUN
ejpam-842	207	22	∈	∈	NOUN
ejpam-842	207	23	∇.	∇.	NOUN
ejpam-842	207	24	since	since	SCONJ
ejpam-842	207	25	for	for	ADP
ejpam-842	207	26	each	each	DET
ejpam-842	207	27	α	α	PROPN
ejpam-842	207	28	∈	∈	PROPN
ejpam-842	207	29	∇	∇	X
ejpam-842	207	30	,	,	PUNCT
ejpam-842	207	31	uα	uα	PROPN
ejpam-842	207	32	is	be	AUX
ejpam-842	207	33	open	open	ADJ
ejpam-842	207	34	in	in	ADP
ejpam-842	207	35	x	x	X
ejpam-842	207	36	,	,	PUNCT
ejpam-842	207	37	by	by	ADP
ejpam-842	207	38	theorem	theorem	NOUN
ejpam-842	207	39	5	5	NUM
ejpam-842	207	40	,	,	PUNCT
ejpam-842	207	41	f	f	PROPN
ejpam-842	207	42	−1(v	−1(v	PROPN
ejpam-842	207	43	)	)	PUNCT
ejpam-842	207	44	∩uα	∩uα	NOUN
ejpam-842	207	45	is	be	AUX
ejpam-842	207	46	is⋆g	is⋆g	ADJ
ejpam-842	207	47	-open	-open	ADJ
ejpam-842	207	48	in	in	ADP
ejpam-842	207	49	x	x	X
ejpam-842	207	50	.	.	PUNCT
ejpam-842	208	1	now	now	ADV
ejpam-842	208	2	since	since	SCONJ
ejpam-842	208	3	x	x	PROPN
ejpam-842	208	4	is	be	AUX
ejpam-842	208	5	t	t	NOUN
ejpam-842	208	6	-dense	-dense	NOUN
ejpam-842	208	7	,	,	PUNCT
ejpam-842	208	8	by	by	ADP
ejpam-842	208	9	[	[	PUNCT
ejpam-842	208	10	theorem	theorem	ADJ
ejpam-842	208	11	2.12	2.12	NUM
ejpam-842	208	12	5	5	NUM
ejpam-842	208	13	]	]	PUNCT
ejpam-842	208	14	,	,	PUNCT
ejpam-842	208	15	∪α∈∇	∪α∈∇	PROPN
ejpam-842	208	16	f	f	PROPN
ejpam-842	208	17	−1(v	−1(v	PROPN
ejpam-842	208	18	)	)	PUNCT
ejpam-842	208	19	∩	∩	PROPN
ejpam-842	208	20	uα	uα	PROPN
ejpam-842	208	21	=	=	SYM
ejpam-842	208	22	f	f	PROPN
ejpam-842	208	23	−1(v	−1(v	PROPN
ejpam-842	208	24	)	)	PUNCT
ejpam-842	208	25	is	be	AUX
ejpam-842	208	26	is⋆g	is⋆g	NOUN
ejpam-842	208	27	-	-	PUNCT
ejpam-842	208	28	open	open	ADJ
ejpam-842	208	29	in	in	ADP
ejpam-842	208	30	x	x	X
ejpam-842	208	31	.	.	PUNCT
ejpam-842	209	1	this	this	PRON
ejpam-842	209	2	implies	imply	VERB
ejpam-842	209	3	f	f	PROPN
ejpam-842	209	4	is	be	AUX
ejpam-842	209	5	is⋆g	is⋆g	PROPN
ejpam-842	209	6	-continuous	-continuous	ADJ
ejpam-842	209	7	.	.	PUNCT
ejpam-842	210	1	theorem	theorem	NOUN
ejpam-842	210	2	12	12	NUM
ejpam-842	210	3	.	.	PUNCT
ejpam-842	211	1	if	if	SCONJ
ejpam-842	211	2	(	(	PUNCT
ejpam-842	211	3	x	x	X
ejpam-842	211	4	,	,	PUNCT
ejpam-842	211	5	τ	τ	PROPN
ejpam-842	211	6	,	,	PUNCT
ejpam-842	211	7	i	i	PROPN
ejpam-842	211	8	)	)	PUNCT
ejpam-842	211	9	is	be	AUX
ejpam-842	211	10	a	a	DET
ejpam-842	211	11	t	t	NOUN
ejpam-842	211	12	-dense	-dense	NOUN
ejpam-842	211	13	space	space	NOUN
ejpam-842	211	14	and	and	CCONJ
ejpam-842	211	15	f	f	NOUN
ejpam-842	211	16	:	:	PUNCT
ejpam-842	211	17	(	(	PUNCT
ejpam-842	211	18	x	x	X
ejpam-842	211	19	,	,	PUNCT
ejpam-842	211	20	τ	τ	PROPN
ejpam-842	211	21	,	,	PUNCT
ejpam-842	211	22	i	i	NOUN
ejpam-842	211	23	)	)	PUNCT
ejpam-842	211	24	→	→	SYM
ejpam-842	211	25	(	(	PUNCT
ejpam-842	211	26	y	y	PROPN
ejpam-842	211	27	,	,	PUNCT
ejpam-842	211	28	ω	ω	NOUN
ejpam-842	211	29	)	)	PUNCT
ejpam-842	211	30	is	be	AUX
ejpam-842	211	31	is⋆g	is⋆g	NOUN
ejpam-842	211	32	-continuous	-continuous	ADJ
ejpam-842	211	33	,	,	PUNCT
ejpam-842	211	34	then	then	ADV
ejpam-842	211	35	graph	graph	NOUN
ejpam-842	211	36	function	function	NOUN
ejpam-842	211	37	g	g	NOUN
ejpam-842	211	38	:	:	PUNCT
ejpam-842	211	39	x	x	SYM
ejpam-842	211	40	→	→	SYM
ejpam-842	211	41	x	x	SYM
ejpam-842	211	42	×	×	PROPN
ejpam-842	211	43	y	y	PROPN
ejpam-842	211	44	,	,	PUNCT
ejpam-842	211	45	defined	define	VERB
ejpam-842	211	46	by	by	ADP
ejpam-842	211	47	g(x	g(x	NOUN
ejpam-842	211	48	)	)	PUNCT
ejpam-842	211	49	=	=	SYM
ejpam-842	212	1	(	(	PUNCT
ejpam-842	212	2	x	x	INTJ
ejpam-842	212	3	,	,	PUNCT
ejpam-842	212	4	f	f	PROPN
ejpam-842	212	5	(	(	PUNCT
ejpam-842	212	6	x	x	NOUN
ejpam-842	212	7	)	)	PUNCT
ejpam-842	212	8	)	)	PUNCT
ejpam-842	212	9	for	for	ADP
ejpam-842	212	10	each	each	DET
ejpam-842	212	11	x	x	SYM
ejpam-842	212	12	∈	∈	PROPN
ejpam-842	212	13	x	x	PUNCT
ejpam-842	212	14	,	,	PUNCT
ejpam-842	212	15	is	be	AUX
ejpam-842	212	16	is⋆	is⋆	NOUN
ejpam-842	212	17	g	g	NOUN
ejpam-842	212	18	-	-	PUNCT
ejpam-842	212	19	continuous	continuous	ADJ
ejpam-842	212	20	.	.	PUNCT
ejpam-842	213	1	proof	proof	NOUN
ejpam-842	213	2	.	.	PUNCT
ejpam-842	214	1	let	let	VERB
ejpam-842	214	2	x	x	PUNCT
ejpam-842	214	3	∈	∈	PROPN
ejpam-842	214	4	x	x	X
ejpam-842	214	5	and	and	CCONJ
ejpam-842	214	6	w	w	PROPN
ejpam-842	214	7	be	be	AUX
ejpam-842	214	8	any	any	DET
ejpam-842	214	9	open	open	ADJ
ejpam-842	214	10	set	set	NOUN
ejpam-842	214	11	in	in	ADP
ejpam-842	214	12	x	x	SYM
ejpam-842	214	13	×	×	PROPN
ejpam-842	214	14	y	y	NOUN
ejpam-842	214	15	containing	contain	VERB
ejpam-842	214	16	g(x	g(x	NOUN
ejpam-842	214	17	)	)	PUNCT
ejpam-842	215	1	=	=	SYM
ejpam-842	215	2	(	(	PUNCT
ejpam-842	215	3	x	x	INTJ
ejpam-842	215	4	,	,	PUNCT
ejpam-842	215	5	f	f	PROPN
ejpam-842	215	6	(	(	PUNCT
ejpam-842	215	7	x	x	NOUN
ejpam-842	215	8	)	)	PUNCT
ejpam-842	215	9	)	)	PUNCT
ejpam-842	215	10	.	.	PUNCT
ejpam-842	216	1	then	then	ADV
ejpam-842	216	2	there	there	PRON
ejpam-842	216	3	exists	exist	VERB
ejpam-842	216	4	a	a	DET
ejpam-842	216	5	basic	basic	ADJ
ejpam-842	216	6	open	open	ADJ
ejpam-842	216	7	set	set	NOUN
ejpam-842	216	8	u	u	NOUN
ejpam-842	216	9	×	×	NOUN
ejpam-842	216	10	v	v	ADP
ejpam-842	216	11	such	such	ADJ
ejpam-842	216	12	that	that	DET
ejpam-842	216	13	g(x	g(x	NOUN
ejpam-842	216	14	)	)	PUNCT
ejpam-842	217	1	⊂	⊂	PROPN
ejpam-842	217	2	u	u	X
ejpam-842	217	3	×	×	PROPN
ejpam-842	217	4	v	v	ADP
ejpam-842	217	5	⊂	⊂	PROPN
ejpam-842	217	6	w	w	PROPN
ejpam-842	217	7	.	.	PUNCT
ejpam-842	218	1	since	since	SCONJ
ejpam-842	218	2	f	f	PROPN
ejpam-842	218	3	is	be	AUX
ejpam-842	218	4	is⋆g	is⋆g	ADJ
ejpam-842	218	5	-	-	ADJ
ejpam-842	218	6	continuous	continuous	ADJ
ejpam-842	218	7	,	,	PUNCT
ejpam-842	218	8	there	there	PRON
ejpam-842	218	9	exits	exit	VERB
ejpam-842	218	10	an	an	DET
ejpam-842	218	11	is⋆g	is⋆g	ADJ
ejpam-842	218	12	-open	-open	ADJ
ejpam-842	218	13	set	set	VERB
ejpam-842	218	14	u1	u1	NOUN
ejpam-842	218	15	in	in	ADP
ejpam-842	218	16	x	x	PUNCT
ejpam-842	218	17	containing	contain	VERB
ejpam-842	218	18	x	x	PUNCT
ejpam-842	218	19	such	such	ADJ
ejpam-842	218	20	that	that	SCONJ
ejpam-842	218	21	f	f	PROPN
ejpam-842	218	22	(	(	PUNCT
ejpam-842	218	23	u1)⊂	u1)⊂	NOUN
ejpam-842	218	24	v	v	NOUN
ejpam-842	218	25	.	.	PUNCT
ejpam-842	219	1	by	by	ADP
ejpam-842	219	2	lemma	lemma	PROPN
ejpam-842	219	3	3	3	NUM
ejpam-842	219	4	,	,	PUNCT
ejpam-842	219	5	u1	u1	NOUN
ejpam-842	219	6	∩u	∩u	NOUN
ejpam-842	219	7	is	be	AUX
ejpam-842	219	8	is⋆g	is⋆g	ADJ
ejpam-842	219	9	-open	-open	ADJ
ejpam-842	219	10	in	in	ADP
ejpam-842	219	11	x	x	PUNCT
ejpam-842	219	12	and	and	CCONJ
ejpam-842	219	13	we	we	PRON
ejpam-842	219	14	have	have	VERB
ejpam-842	219	15	x	x	PART
ejpam-842	219	16	∈	∈	PROPN
ejpam-842	219	17	u1	u1	NOUN
ejpam-842	220	1	∩u	∩u	PROPN
ejpam-842	220	2	⊂	⊂	PROPN
ejpam-842	220	3	u	u	NOUN
ejpam-842	220	4	and	and	CCONJ
ejpam-842	220	5	g(u1	g(u1	NOUN
ejpam-842	220	6	∩u)⊂	∩u)⊂	NUM
ejpam-842	220	7	u	u	NOUN
ejpam-842	220	8	×	×	NOUN
ejpam-842	220	9	v	v	ADP
ejpam-842	220	10	⊂w	⊂w	PROPN
ejpam-842	220	11	.	.	PUNCT
ejpam-842	221	1	since	since	SCONJ
ejpam-842	221	2	x	x	PROPN
ejpam-842	221	3	is	be	AUX
ejpam-842	221	4	t	t	NOUN
ejpam-842	221	5	-dense	-dense	NOUN
ejpam-842	221	6	,	,	PUNCT
ejpam-842	221	7	therefore	therefore	ADV
ejpam-842	221	8	by	by	ADP
ejpam-842	221	9	theorem	theorem	NOUN
ejpam-842	221	10	7	7	NUM
ejpam-842	221	11	,	,	PUNCT
ejpam-842	221	12	g	g	PROPN
ejpam-842	221	13	is	be	AUX
ejpam-842	221	14	is⋆g	is⋆g	PRON
ejpam-842	221	15	-continuous	-continuous	ADJ
ejpam-842	221	16	.	.	PUNCT
ejpam-842	222	1	theorem	theorem	VERB
ejpam-842	222	2	13	13	NUM
ejpam-842	222	3	.	.	PUNCT
ejpam-842	223	1	a	a	DET
ejpam-842	223	2	function	function	NOUN
ejpam-842	223	3	f	f	NOUN
ejpam-842	223	4	:	:	PUNCT
ejpam-842	223	5	(	(	PUNCT
ejpam-842	223	6	x	x	X
ejpam-842	223	7	,	,	PUNCT
ejpam-842	223	8	τ	τ	PROPN
ejpam-842	223	9	,	,	PUNCT
ejpam-842	223	10	i)→	i)→	ADJ
ejpam-842	223	11	(	(	PUNCT
ejpam-842	223	12	y	y	PROPN
ejpam-842	223	13	,	,	PUNCT
ejpam-842	223	14	ω	ω	NOUN
ejpam-842	223	15	)	)	PUNCT
ejpam-842	223	16	is	be	AUX
ejpam-842	223	17	is⋆g	is⋆g	NOUN
ejpam-842	223	18	-	-	ADJ
ejpam-842	223	19	continuous	continuous	ADJ
ejpam-842	223	20	if	if	SCONJ
ejpam-842	223	21	the	the	DET
ejpam-842	223	22	graph	graph	NOUN
ejpam-842	223	23	function	function	VERB
ejpam-842	223	24	g	g	NOUN
ejpam-842	223	25	:	:	PUNCT
ejpam-842	223	26	x	x	SYM
ejpam-842	223	27	→	→	SYM
ejpam-842	223	28	x	x	SYM
ejpam-842	223	29	×	×	PROPN
ejpam-842	223	30	y	y	PROPN
ejpam-842	223	31	is	be	AUX
ejpam-842	223	32	is⋆g	is⋆g	NOUN
ejpam-842	223	33	-continuous	-continuous	ADJ
ejpam-842	223	34	.	.	PUNCT
ejpam-842	224	1	proof	proof	NOUN
ejpam-842	224	2	.	.	PUNCT
ejpam-842	225	1	let	let	VERB
ejpam-842	225	2	v	v	PART
ejpam-842	225	3	be	be	AUX
ejpam-842	225	4	an	an	DET
ejpam-842	225	5	open	open	ADJ
ejpam-842	225	6	set	set	NOUN
ejpam-842	225	7	in	in	ADP
ejpam-842	225	8	y	y	NOUN
ejpam-842	225	9	containing	contain	VERB
ejpam-842	225	10	f	f	PROPN
ejpam-842	225	11	(	(	PUNCT
ejpam-842	225	12	x	x	NOUN
ejpam-842	225	13	)	)	PUNCT
ejpam-842	225	14	.	.	PUNCT
ejpam-842	226	1	then	then	ADV
ejpam-842	226	2	x	x	X
ejpam-842	226	3	×	×	NOUN
ejpam-842	226	4	v	v	NOUN
ejpam-842	226	5	is	be	AUX
ejpam-842	226	6	an	an	DET
ejpam-842	226	7	open	open	ADJ
ejpam-842	226	8	set	set	NOUN
ejpam-842	226	9	in	in	ADP
ejpam-842	226	10	x	x	PUNCT
ejpam-842	226	11	×	×	PROPN
ejpam-842	226	12	y	y	PROPN
ejpam-842	226	13	and	and	CCONJ
ejpam-842	226	14	by	by	ADP
ejpam-842	226	15	the	the	DET
ejpam-842	226	16	is⋆	is⋆	PROPN
ejpam-842	226	17	g	g	NOUN
ejpam-842	226	18	-	-	PUNCT
ejpam-842	226	19	continuity	continuity	NOUN
ejpam-842	226	20	of	of	ADP
ejpam-842	226	21	g	g	NOUN
ejpam-842	226	22	,	,	PUNCT
ejpam-842	226	23	there	there	PRON
ejpam-842	226	24	exists	exist	VERB
ejpam-842	226	25	an	an	DET
ejpam-842	226	26	is⋆g	is⋆g	NOUN
ejpam-842	226	27	-	-	PUNCT
ejpam-842	226	28	open	open	ADJ
ejpam-842	226	29	set	set	NOUN
ejpam-842	226	30	u	u	NOUN
ejpam-842	226	31	in	in	ADP
ejpam-842	226	32	x	x	PUNCT
ejpam-842	226	33	containing	contain	VERB
ejpam-842	226	34	x	x	PUNCT
ejpam-842	226	35	such	such	ADJ
ejpam-842	226	36	that	that	DET
ejpam-842	226	37	g(u)⊂	g(u)⊂	NOUN
ejpam-842	226	38	x	x	X
ejpam-842	226	39	×	×	NOUN
ejpam-842	226	40	v	v	NOUN
ejpam-842	226	41	.	.	PUNCT
ejpam-842	227	1	therefore	therefore	ADV
ejpam-842	227	2	,	,	PUNCT
ejpam-842	227	3	we	we	PRON
ejpam-842	227	4	obtain	obtain	VERB
ejpam-842	227	5	f	f	NOUN
ejpam-842	227	6	(	(	PUNCT
ejpam-842	227	7	u)⊂	u)⊂	NOUN
ejpam-842	227	8	v	v	NOUN
ejpam-842	227	9	.	.	PUNCT
ejpam-842	228	1	this	this	PRON
ejpam-842	228	2	shows	show	VERB
ejpam-842	228	3	that	that	SCONJ
ejpam-842	228	4	f	f	PROPN
ejpam-842	228	5	is	be	AUX
ejpam-842	228	6	is⋆g	is⋆g	ADJ
ejpam-842	228	7	-	-	PUNCT
ejpam-842	228	8	continuous	continuous	ADJ
ejpam-842	228	9	.	.	PUNCT
ejpam-842	229	1	theorem	theorem	NOUN
ejpam-842	229	2	14	14	NUM
ejpam-842	229	3	.	.	PUNCT
ejpam-842	230	1	let	let	VERB
ejpam-842	230	2	{	{	PUNCT
ejpam-842	230	3	xα	xα	INTJ
ejpam-842	230	4	:	:	PUNCT
ejpam-842	230	5	α	α	PROPN
ejpam-842	230	6	∈	∈	PROPN
ejpam-842	230	7	∇	∇	X
ejpam-842	230	8	}	}	PUNCT
ejpam-842	230	9	be	be	AUX
ejpam-842	230	10	any	any	DET
ejpam-842	230	11	family	family	NOUN
ejpam-842	230	12	of	of	ADP
ejpam-842	230	13	topological	topological	ADJ
ejpam-842	230	14	spaces	space	NOUN
ejpam-842	230	15	.	.	PUNCT
ejpam-842	231	1	if	if	SCONJ
ejpam-842	231	2	f	f	PROPN
ejpam-842	231	3	:	:	PUNCT
ejpam-842	231	4	(	(	PUNCT
ejpam-842	231	5	x	x	X
ejpam-842	231	6	,	,	PUNCT
ejpam-842	231	7	τ	τ	PROPN
ejpam-842	231	8	,	,	PUNCT
ejpam-842	231	9	i)→	i)→	ADJ
ejpam-842	231	10	πα∈∇xα	πα∈∇xα	PROPN
ejpam-842	231	11	is	be	AUX
ejpam-842	231	12	an	an	DET
ejpam-842	231	13	is⋆g	is⋆g	ADJ
ejpam-842	231	14	-	-	PUNCT
ejpam-842	231	15	continuous	continuous	ADJ
ejpam-842	231	16	function	function	NOUN
ejpam-842	231	17	,	,	PUNCT
ejpam-842	231	18	then	then	ADV
ejpam-842	231	19	pα	pα	VERB
ejpam-842	231	20	◦	◦	NOUN
ejpam-842	231	21	f	f	X
ejpam-842	231	22	:	:	PUNCT
ejpam-842	231	23	x	x	X
ejpam-842	231	24	→	→	SYM
ejpam-842	231	25	xα	xα	X
ejpam-842	231	26	is	be	AUX
ejpam-842	231	27	is⋆g	is⋆g	ADJ
ejpam-842	231	28	-continuous	-continuous	ADJ
ejpam-842	231	29	for	for	ADP
ejpam-842	231	30	each	each	DET
ejpam-842	231	31	α	α	PROPN
ejpam-842	231	32	∈	∈	PROPN
ejpam-842	231	33	∇	∇	NOUN
ejpam-842	231	34	,	,	PUNCT
ejpam-842	231	35	where	where	SCONJ
ejpam-842	231	36	pα	pα	NOUN
ejpam-842	231	37	is	be	AUX
ejpam-842	231	38	the	the	DET
ejpam-842	231	39	projection	projection	NOUN
ejpam-842	231	40	of	of	ADP
ejpam-842	231	41	πxα	πxα	NOUN
ejpam-842	231	42	onto	onto	ADP
ejpam-842	231	43	xα	xα	PROPN
ejpam-842	231	44	.	.	PUNCT
ejpam-842	232	1	proof	proof	NOUN
ejpam-842	232	2	.	.	PUNCT
ejpam-842	233	1	we	we	PRON
ejpam-842	233	2	will	will	AUX
ejpam-842	233	3	consider	consider	VERB
ejpam-842	233	4	a	a	DET
ejpam-842	233	5	fixed	fix	VERB
ejpam-842	233	6	α0	α0	ADJ
ejpam-842	233	7	∈	∈	PROPN
ejpam-842	233	8	∇.	∇.	PRON
ejpam-842	233	9	let	let	VERB
ejpam-842	233	10	gα0	gα0	INTJ
ejpam-842	233	11	be	be	AUX
ejpam-842	233	12	an	an	DET
ejpam-842	233	13	open	open	ADJ
ejpam-842	233	14	set	set	NOUN
ejpam-842	233	15	of	of	ADP
ejpam-842	233	16	xα0	xα0	PROPN
ejpam-842	233	17	.	.	PUNCT
ejpam-842	234	1	then	then	ADV
ejpam-842	234	2	(	(	PUNCT
ejpam-842	234	3	pα0	pα0	PROPN
ejpam-842	234	4	)	)	PUNCT
ejpam-842	234	5	−1(gα0	−1(gα0	NOUN
ejpam-842	234	6	)	)	PUNCT
ejpam-842	234	7	is	be	AUX
ejpam-842	234	8	open	open	ADJ
ejpam-842	234	9	in	in	ADP
ejpam-842	234	10	πxα	πxα	PROPN
ejpam-842	234	11	.	.	PUNCT
ejpam-842	235	1	since	since	SCONJ
ejpam-842	235	2	f	f	PROPN
ejpam-842	235	3	is	be	AUX
ejpam-842	235	4	is⋆g	is⋆g	NOUN
ejpam-842	235	5	-continuous	-continuous	ADJ
ejpam-842	235	6	,	,	PUNCT
ejpam-842	235	7	f	f	PROPN
ejpam-842	235	8	−1((pα0	−1((pα0	PROPN
ejpam-842	235	9	)	)	PUNCT
ejpam-842	235	10	−1(gα0	−1(gα0	NOUN
ejpam-842	235	11	)	)	PUNCT
ejpam-842	235	12	)	)	PUNCT
ejpam-842	236	1	=	=	PRON
ejpam-842	236	2	(	(	PUNCT
ejpam-842	236	3	pα0	pα0	PROPN
ejpam-842	236	4	◦	◦	PROPN
ejpam-842	236	5	f	f	PROPN
ejpam-842	236	6	)	)	PUNCT
ejpam-842	236	7	−1(gα0	−1(gα0	NOUN
ejpam-842	236	8	)	)	PUNCT
ejpam-842	236	9	is	be	AUX
ejpam-842	236	10	is⋆	is⋆	PROPN
ejpam-842	236	11	g	g	NOUN
ejpam-842	236	12	-	-	PUNCT
ejpam-842	236	13	open	open	ADJ
ejpam-842	236	14	in	in	ADP
ejpam-842	236	15	x	x	X
ejpam-842	236	16	.	.	PUNCT
ejpam-842	237	1	thus	thus	ADV
ejpam-842	237	2	pα	pα	VERB
ejpam-842	237	3	◦	◦	NOUN
ejpam-842	237	4	f	f	PROPN
ejpam-842	237	5	is	be	AUX
ejpam-842	237	6	is⋆g	is⋆g	PRON
ejpam-842	237	7	-continuous	-continuous	ADJ
ejpam-842	237	8	.	.	PUNCT
ejpam-842	238	1	corollary	corollary	ADJ
ejpam-842	238	2	2	2	NUM
ejpam-842	238	3	.	.	PUNCT
ejpam-842	239	1	for	for	ADP
ejpam-842	239	2	any	any	DET
ejpam-842	239	3	bijective	bijective	ADJ
ejpam-842	239	4	function	function	NOUN
ejpam-842	239	5	f	f	NOUN
ejpam-842	239	6	:	:	PUNCT
ejpam-842	239	7	(	(	PUNCT
ejpam-842	239	8	x	x	X
ejpam-842	239	9	,	,	PUNCT
ejpam-842	239	10	τ)→	τ)→	PROPN
ejpam-842	239	11	(	(	PUNCT
ejpam-842	239	12	y	y	PROPN
ejpam-842	239	13	,	,	PUNCT
ejpam-842	239	14	ω	ω	PROPN
ejpam-842	239	15	,	,	PUNCT
ejpam-842	239	16	j	j	PROPN
ejpam-842	239	17	)	)	PUNCT
ejpam-842	239	18	,	,	PUNCT
ejpam-842	239	19	the	the	DET
ejpam-842	239	20	following	follow	VERB
ejpam-842	239	21	are	be	AUX
ejpam-842	239	22	equivalent	equivalent	ADJ
ejpam-842	239	23	:	:	PUNCT
ejpam-842	239	24	(	(	PUNCT
ejpam-842	239	25	1	1	X
ejpam-842	239	26	)	)	PUNCT
ejpam-842	239	27	f	f	NOUN
ejpam-842	239	28	−1	−1	NOUN
ejpam-842	239	29	:	:	PUNCT
ejpam-842	239	30	(	(	PUNCT
ejpam-842	239	31	y	y	PROPN
ejpam-842	239	32	,	,	PUNCT
ejpam-842	239	33	ω	ω	PROPN
ejpam-842	239	34	,	,	PUNCT
ejpam-842	239	35	j)→	j)→	PROPN
ejpam-842	239	36	(	(	PUNCT
ejpam-842	239	37	x	x	X
ejpam-842	239	38	,	,	PUNCT
ejpam-842	239	39	τ	τ	X
ejpam-842	239	40	)	)	PUNCT
ejpam-842	239	41	is	be	AUX
ejpam-842	239	42	is⋆g	is⋆g	NOUN
ejpam-842	239	43	-continuous	-continuous	ADJ
ejpam-842	239	44	.	.	PUNCT
ejpam-842	240	1	(	(	PUNCT
ejpam-842	240	2	2	2	X
ejpam-842	240	3	)	)	PUNCT
ejpam-842	240	4	f	f	NOUN
ejpam-842	240	5	(	(	PUNCT
ejpam-842	240	6	u	u	NOUN
ejpam-842	240	7	)	)	PUNCT
ejpam-842	240	8	is	be	AUX
ejpam-842	240	9	is⋆g	is⋆g	NOUN
ejpam-842	240	10	-	-	PUNCT
ejpam-842	240	11	open	open	ADJ
ejpam-842	240	12	in	in	ADP
ejpam-842	240	13	y	y	PROPN
ejpam-842	240	14	for	for	ADP
ejpam-842	240	15	every	every	DET
ejpam-842	240	16	open	open	ADJ
ejpam-842	240	17	set	set	NOUN
ejpam-842	240	18	u	u	NOUN
ejpam-842	240	19	in	in	ADP
ejpam-842	240	20	x	x	X
ejpam-842	240	21	.	.	PUNCT
ejpam-842	241	1	(	(	PUNCT
ejpam-842	241	2	3	3	X
ejpam-842	241	3	)	)	PUNCT
ejpam-842	241	4	f	f	NOUN
ejpam-842	241	5	(	(	PUNCT
ejpam-842	241	6	u	u	NOUN
ejpam-842	241	7	)	)	PUNCT
ejpam-842	241	8	is	be	AUX
ejpam-842	241	9	is⋆g	is⋆g	NOUN
ejpam-842	241	10	-	-	PUNCT
ejpam-842	241	11	closed	closed	ADJ
ejpam-842	241	12	in	in	ADP
ejpam-842	241	13	y	y	PROPN
ejpam-842	241	14	for	for	ADP
ejpam-842	241	15	every	every	DET
ejpam-842	241	16	closed	close	VERB
ejpam-842	241	17	set	set	VERB
ejpam-842	241	18	u	u	NOUN
ejpam-842	241	19	in	in	ADP
ejpam-842	241	20	x	x	X
ejpam-842	241	21	.	.	PUNCT
ejpam-842	242	1	proof	proof	NOUN
ejpam-842	242	2	.	.	PUNCT
ejpam-842	243	1	it	it	PRON
ejpam-842	243	2	is	be	AUX
ejpam-842	243	3	trivial	trivial	ADJ
ejpam-842	243	4	.	.	PUNCT
ejpam-842	244	1	definition	definition	NOUN
ejpam-842	244	2	8	8	NUM
ejpam-842	244	3	.	.	PUNCT
ejpam-842	245	1	an	an	DET
ejpam-842	245	2	ideal	ideal	ADJ
ejpam-842	245	3	topological	topological	ADJ
ejpam-842	245	4	space	space	NOUN
ejpam-842	245	5	(	(	PUNCT
ejpam-842	245	6	x	x	X
ejpam-842	245	7	,	,	PUNCT
ejpam-842	245	8	τ	τ	PROPN
ejpam-842	245	9	,	,	PUNCT
ejpam-842	245	10	i	i	PROPN
ejpam-842	245	11	)	)	PUNCT
ejpam-842	245	12	is	be	AUX
ejpam-842	245	13	an	an	DET
ejpam-842	245	14	ri	ri	NOUN
ejpam-842	245	15	-	-	PUNCT
ejpam-842	245	16	space	space	NOUN
ejpam-842	245	17	[	[	X
ejpam-842	245	18	1	1	NUM
ejpam-842	245	19	]	]	PUNCT
ejpam-842	245	20	,	,	PUNCT
ejpam-842	245	21	if	if	SCONJ
ejpam-842	245	22	for	for	ADP
ejpam-842	245	23	each	each	DET
ejpam-842	245	24	x	x	SYM
ejpam-842	245	25	∈	∈	PROPN
ejpam-842	245	26	x	x	X
ejpam-842	245	27	and	and	CCONJ
ejpam-842	245	28	each	each	DET
ejpam-842	245	29	open	open	ADJ
ejpam-842	245	30	neighbourhood	neighbourhood	NOUN
ejpam-842	245	31	v	v	NOUN
ejpam-842	245	32	of	of	ADP
ejpam-842	245	33	x	x	NOUN
ejpam-842	245	34	,	,	PUNCT
ejpam-842	245	35	there	there	PRON
ejpam-842	245	36	exists	exist	VERB
ejpam-842	245	37	an	an	DET
ejpam-842	245	38	open	open	ADJ
ejpam-842	245	39	neighbourhood	neighbourhood	NOUN
ejpam-842	245	40	u	u	NOUN
ejpam-842	245	41	of	of	ADP
ejpam-842	245	42	x	x	SYM
ejpam-842	245	43	such	such	ADJ
ejpam-842	245	44	that	that	SCONJ
ejpam-842	245	45	x	x	SYM
ejpam-842	245	46	∈	∈	PROPN
ejpam-842	245	47	u	u	X
ejpam-842	245	48	⊂	⊂	PROPN
ejpam-842	245	49	cl⋆(u)⊂	cl⋆(u)⊂	CCONJ
ejpam-842	245	50	v	v	NOUN
ejpam-842	245	51	.	.	PUNCT
ejpam-842	246	1	references	reference	NOUN
ejpam-842	246	2	243	243	NUM
ejpam-842	246	3	theorem	theorem	NOUN
ejpam-842	246	4	15	15	NUM
ejpam-842	246	5	.	.	PUNCT
ejpam-842	247	1	let	let	AUX
ejpam-842	247	2	(	(	PUNCT
ejpam-842	247	3	y	y	PROPN
ejpam-842	247	4	,	,	PUNCT
ejpam-842	247	5	ω	ω	PROPN
ejpam-842	247	6	,	,	PUNCT
ejpam-842	247	7	j	j	NOUN
ejpam-842	247	8	)	)	PUNCT
ejpam-842	247	9	be	be	VERB
ejpam-842	247	10	an	an	DET
ejpam-842	247	11	ri	ri	NOUN
ejpam-842	247	12	-	-	PUNCT
ejpam-842	247	13	space	space	NOUN
ejpam-842	247	14	and	and	CCONJ
ejpam-842	247	15	(	(	PUNCT
ejpam-842	247	16	x	x	X
ejpam-842	247	17	,	,	PUNCT
ejpam-842	247	18	τ	τ	PROPN
ejpam-842	247	19	,	,	PUNCT
ejpam-842	247	20	i	i	PRON
ejpam-842	247	21	)	)	PUNCT
ejpam-842	247	22	be	be	VERB
ejpam-842	247	23	a	a	DET
ejpam-842	247	24	t	t	NOUN
ejpam-842	247	25	-	-	PUNCT
ejpam-842	247	26	dense	dense	ADJ
ejpam-842	247	27	space	space	NOUN
ejpam-842	247	28	.	.	PUNCT
ejpam-842	248	1	then	then	ADV
ejpam-842	248	2	f	f	X
ejpam-842	248	3	:	:	PUNCT
ejpam-842	248	4	(	(	PUNCT
ejpam-842	248	5	x	x	X
ejpam-842	248	6	,	,	PUNCT
ejpam-842	248	7	τ	τ	PROPN
ejpam-842	248	8	,	,	PUNCT
ejpam-842	248	9	i)→	i)→	ADJ
ejpam-842	248	10	(	(	PUNCT
ejpam-842	248	11	y	y	PROPN
ejpam-842	248	12	,	,	PUNCT
ejpam-842	248	13	ω	ω	PROPN
ejpam-842	248	14	,	,	PUNCT
ejpam-842	248	15	j	j	NOUN
ejpam-842	248	16	)	)	PUNCT
ejpam-842	248	17	is	be	AUX
ejpam-842	248	18	weak	weak	ADJ
ejpam-842	248	19	is⋆g	is⋆g	NOUN
ejpam-842	248	20	-	-	ADJ
ejpam-842	248	21	continuous	continuous	ADJ
ejpam-842	248	22	if	if	SCONJ
ejpam-842	249	1	and	and	CCONJ
ejpam-842	249	2	only	only	ADV
ejpam-842	249	3	if	if	SCONJ
ejpam-842	249	4	f	f	PROPN
ejpam-842	249	5	is	be	AUX
ejpam-842	249	6	is⋆g	is⋆g	PRON
ejpam-842	249	7	-continuous	-continuous	ADJ
ejpam-842	249	8	.	.	PUNCT
ejpam-842	250	1	proof	proof	NOUN
ejpam-842	250	2	.	.	PUNCT
ejpam-842	251	1	the	the	DET
ejpam-842	251	2	sufficiency	sufficiency	NOUN
ejpam-842	251	3	is	be	AUX
ejpam-842	251	4	clear	clear	ADJ
ejpam-842	251	5	.	.	PUNCT
ejpam-842	252	1	necessity	necessity	NOUN
ejpam-842	252	2	.	.	PUNCT
ejpam-842	253	1	let	let	VERB
ejpam-842	253	2	x	x	PUNCT
ejpam-842	253	3	∈	∈	PROPN
ejpam-842	253	4	x	x	X
ejpam-842	253	5	and	and	CCONJ
ejpam-842	253	6	v	v	X
ejpam-842	253	7	be	be	AUX
ejpam-842	253	8	an	an	DET
ejpam-842	253	9	open	open	ADJ
ejpam-842	253	10	set	set	NOUN
ejpam-842	253	11	of	of	ADP
ejpam-842	253	12	y	y	PROPN
ejpam-842	253	13	containing	contain	VERB
ejpam-842	253	14	f	f	PROPN
ejpam-842	253	15	(	(	PUNCT
ejpam-842	253	16	x	x	NOUN
ejpam-842	253	17	)	)	PUNCT
ejpam-842	253	18	.	.	PUNCT
ejpam-842	254	1	since	since	SCONJ
ejpam-842	254	2	y	y	PROPN
ejpam-842	254	3	is	be	AUX
ejpam-842	254	4	an	an	DET
ejpam-842	254	5	ri	ri	NOUN
ejpam-842	254	6	-	-	PUNCT
ejpam-842	254	7	space	space	NOUN
ejpam-842	254	8	,	,	PUNCT
ejpam-842	254	9	there	there	PRON
ejpam-842	254	10	exists	exist	VERB
ejpam-842	254	11	an	an	DET
ejpam-842	254	12	open	open	ADJ
ejpam-842	254	13	set	set	NOUN
ejpam-842	254	14	w	w	PROPN
ejpam-842	254	15	of	of	ADP
ejpam-842	254	16	y	y	PRON
ejpam-842	254	17	such	such	ADJ
ejpam-842	254	18	that	that	SCONJ
ejpam-842	254	19	f	f	PROPN
ejpam-842	254	20	(	(	PUNCT
ejpam-842	254	21	x	x	X
ejpam-842	254	22	)	)	PUNCT
ejpam-842	254	23	∈	∈	PROPN
ejpam-842	254	24	w	w	PROPN
ejpam-842	254	25	⊂	⊂	PROPN
ejpam-842	254	26	cl⋆(w	cl⋆(w	PROPN
ejpam-842	254	27	)	)	PUNCT
ejpam-842	255	1	⊂	⊂	PROPN
ejpam-842	255	2	v	v	X
ejpam-842	255	3	.	.	PUNCT
ejpam-842	256	1	since	since	SCONJ
ejpam-842	256	2	f	f	PROPN
ejpam-842	256	3	is	be	AUX
ejpam-842	256	4	weakly	weakly	ADV
ejpam-842	256	5	is⋆gcontinuous	is⋆gcontinuous	ADJ
ejpam-842	256	6	,	,	PUNCT
ejpam-842	256	7	there	there	PRON
ejpam-842	256	8	exists	exist	VERB
ejpam-842	256	9	an	an	DET
ejpam-842	256	10	is⋆g	is⋆g	NOUN
ejpam-842	256	11	-	-	PUNCT
ejpam-842	256	12	open	open	ADJ
ejpam-842	256	13	set	set	NOUN
ejpam-842	256	14	u	u	PRON
ejpam-842	256	15	such	such	ADJ
ejpam-842	256	16	that	that	SCONJ
ejpam-842	256	17	x	x	SYM
ejpam-842	256	18	∈	∈	PROPN
ejpam-842	256	19	u	u	NOUN
ejpam-842	256	20	and	and	CCONJ
ejpam-842	256	21	f	f	PROPN
ejpam-842	256	22	(	(	PUNCT
ejpam-842	256	23	u	u	NOUN
ejpam-842	256	24	)	)	PUNCT
ejpam-842	256	25	⊂	⊂	PROPN
ejpam-842	256	26	cl⋆(w	cl⋆(w	PROPN
ejpam-842	256	27	)	)	PUNCT
ejpam-842	256	28	.	.	PUNCT
ejpam-842	257	1	hence	hence	ADV
ejpam-842	257	2	we	we	PRON
ejpam-842	257	3	obtain	obtain	VERB
ejpam-842	257	4	that	that	DET
ejpam-842	257	5	f	f	PROPN
ejpam-842	257	6	(	(	PUNCT
ejpam-842	257	7	u)⊂	u)⊂	NOUN
ejpam-842	257	8	cl⋆(w	cl⋆(w	NOUN
ejpam-842	257	9	)	)	PUNCT
ejpam-842	258	1	⊂	⊂	PROPN
ejpam-842	258	2	v	v	NOUN
ejpam-842	258	3	.	.	PUNCT
ejpam-842	259	1	by	by	ADP
ejpam-842	259	2	theorem	theorem	NOUN
ejpam-842	259	3	8	8	NUM
ejpam-842	259	4	,	,	PUNCT
ejpam-842	259	5	f	f	PROPN
ejpam-842	259	6	is	be	AUX
ejpam-842	259	7	is⋆g	is⋆g	ADJ
ejpam-842	259	8	-	-	PUNCT
ejpam-842	259	9	continuous	continuous	ADJ
ejpam-842	259	10	.	.	PUNCT
ejpam-842	260	1	references	reference	NOUN
ejpam-842	260	2	[	[	X
ejpam-842	260	3	1	1	X
ejpam-842	260	4	]	]	PUNCT
ejpam-842	260	5	ackgoz	ackgoz	NOUN
ejpam-842	260	6	.a	.a	PROPN
ejpam-842	260	7	,	,	PUNCT
ejpam-842	260	8	t.	t.	PROPN
ejpam-842	260	9	noiri	noiri	PROPN
ejpam-842	260	10	and	and	CCONJ
ejpam-842	260	11	s.	s.	PROPN
ejpam-842	260	12	yuksel	yuksel	PROPN
ejpam-842	260	13	,	,	PUNCT
ejpam-842	260	14	a	a	DET
ejpam-842	260	15	decomposition	decomposition	NOUN
ejpam-842	260	16	of	of	ADP
ejpam-842	260	17	continuity	continuity	NOUN
ejpam-842	260	18	in	in	ADP
ejpam-842	260	19	ideal	ideal	ADJ
ejpam-842	260	20	topological	topological	ADJ
ejpam-842	260	21	spaces	space	NOUN
ejpam-842	260	22	,	,	PUNCT
ejpam-842	260	23	acta	acta	PROPN
ejpam-842	260	24	math	math	PROPN
ejpam-842	260	25	.	.	PUNCT
ejpam-842	261	1	hungar	hungar	PROPN
ejpam-842	261	2	.	.	PUNCT
ejpam-842	261	3	,	,	PUNCT
ejpam-842	261	4	105	105	NUM
ejpam-842	261	5	,	,	PUNCT
ejpam-842	261	6	285	285	NUM
ejpam-842	261	7	-	-	SYM
ejpam-842	261	8	289	289	NUM
ejpam-842	261	9	.	.	PUNCT
ejpam-842	262	1	2004	2004	NUM
ejpam-842	262	2	.	.	PUNCT
ejpam-842	263	1	[	[	X
ejpam-842	263	2	2	2	NUM
ejpam-842	263	3	]	]	PUNCT
ejpam-842	263	4	dontchev	dontchev	PROPN
ejpam-842	263	5	.j	.j	PROPN
ejpam-842	263	6	,	,	PUNCT
ejpam-842	263	7	m.	m.	NOUN
ejpam-842	263	8	ganster	ganster	NOUN
ejpam-842	263	9	and	and	CCONJ
ejpam-842	263	10	t.	t.	PROPN
ejpam-842	263	11	noiri	noiri	PROPN
ejpam-842	263	12	,	,	PUNCT
ejpam-842	263	13	unified	unified	ADJ
ejpam-842	263	14	operations	operation	NOUN
ejpam-842	263	15	approach	approach	NOUN
ejpam-842	263	16	of	of	ADP
ejpam-842	263	17	generalized	generalized	ADJ
ejpam-842	263	18	closed	close	VERB
ejpam-842	263	19	sets	set	NOUN
ejpam-842	263	20	via	via	ADP
ejpam-842	263	21	topological	topological	ADJ
ejpam-842	263	22	ideals	ideal	NOUN
ejpam-842	263	23	,	,	PUNCT
ejpam-842	263	24	math	math	NOUN
ejpam-842	263	25	.	.	PUNCT
ejpam-842	264	1	japon	japon	PROPN
ejpam-842	264	2	.	.	PROPN
ejpam-842	264	3	,	,	PUNCT
ejpam-842	264	4	49(3	49(3	PROPN
ejpam-842	264	5	)	)	PUNCT
ejpam-842	264	6	,	,	PUNCT
ejpam-842	264	7	395	395	NUM
ejpam-842	264	8	-	-	SYM
ejpam-842	264	9	401	401	NUM
ejpam-842	264	10	.	.	PUNCT
ejpam-842	264	11	1999	1999	NUM
ejpam-842	264	12	.	.	PUNCT
ejpam-842	265	1	[	[	X
ejpam-842	265	2	3	3	X
ejpam-842	265	3	]	]	X
ejpam-842	265	4	dunham	dunham	PROPN
ejpam-842	265	5	.w	.w	PROPN
ejpam-842	265	6	,	,	PUNCT
ejpam-842	265	7	t1/2	t1/2	NOUN
ejpam-842	265	8	-	-	NOUN
ejpam-842	265	9	spaces	space	NOUN
ejpam-842	265	10	,	,	PUNCT
ejpam-842	265	11	kyungpook	kyungpook	NOUN
ejpam-842	265	12	math	math	NOUN
ejpam-842	265	13	.	.	PUNCT
ejpam-842	266	1	j.	j.	PROPN
ejpam-842	266	2	,	,	PUNCT
ejpam-842	266	3	17	17	NUM
ejpam-842	266	4	,	,	PUNCT
ejpam-842	266	5	161	161	NUM
ejpam-842	266	6	-	-	SYM
ejpam-842	266	7	169	169	NUM
ejpam-842	266	8	.	.	PUNCT
ejpam-842	266	9	1977	1977	NUM
ejpam-842	266	10	.	.	PUNCT
ejpam-842	267	1	[	[	X
ejpam-842	267	2	4	4	X
ejpam-842	267	3	]	]	PUNCT
ejpam-842	267	4	jankovic	jankovic	PROPN
ejpam-842	267	5	.j	.j	PROPN
ejpam-842	267	6	and	and	CCONJ
ejpam-842	267	7	t.r	t.r	PROPN
ejpam-842	267	8	.	.	PROPN
ejpam-842	267	9	hamlett	hamlett	PROPN
ejpam-842	267	10	,	,	PUNCT
ejpam-842	267	11	new	new	ADJ
ejpam-842	267	12	topologies	topology	NOUN
ejpam-842	267	13	from	from	ADP
ejpam-842	267	14	old	old	ADJ
ejpam-842	267	15	via	via	ADP
ejpam-842	267	16	ideals	ideal	NOUN
ejpam-842	267	17	,	,	PUNCT
ejpam-842	267	18	amer	amer	PROPN
ejpam-842	267	19	.	.	PROPN
ejpam-842	267	20	math	math	PROPN
ejpam-842	267	21	.	.	PUNCT
ejpam-842	268	1	monthly	monthly	ADV
ejpam-842	268	2	,	,	PUNCT
ejpam-842	268	3	97	97	NUM
ejpam-842	268	4	(	(	PUNCT
ejpam-842	268	5	4	4	NUM
ejpam-842	268	6	)	)	PUNCT
ejpam-842	268	7	,	,	PUNCT
ejpam-842	268	8	295	295	NUM
ejpam-842	268	9	-	-	SYM
ejpam-842	268	10	310	310	NUM
ejpam-842	268	11	.	.	NOUN
ejpam-842	268	12	1990	1990	NUM
ejpam-842	268	13	.	.	PUNCT
ejpam-842	269	1	[	[	X
ejpam-842	269	2	5	5	X
ejpam-842	269	3	]	]	PUNCT
ejpam-842	269	4	khan	khan	PROPN
ejpam-842	269	5	.m	.m	PROPN
ejpam-842	269	6	and	and	CCONJ
ejpam-842	269	7	m.	m.	PROPN
ejpam-842	269	8	hamza	hamza	PROPN
ejpam-842	269	9	,	,	PUNCT
ejpam-842	269	10	is⋆g	is⋆g	ADJ
ejpam-842	269	11	-	-	PUNCT
ejpam-842	269	12	closed	closed	ADJ
ejpam-842	269	13	sets	set	NOUN
ejpam-842	269	14	in	in	ADP
ejpam-842	269	15	ideal	ideal	ADJ
ejpam-842	269	16	topological	topological	ADJ
ejpam-842	269	17	spaces	space	NOUN
ejpam-842	269	18	,	,	PUNCT
ejpam-842	269	19	global	global	ADJ
ejpam-842	269	20	journal	journal	NOUN
ejpam-842	269	21	of	of	ADP
ejpam-842	269	22	pure	pure	ADJ
ejpam-842	269	23	and	and	CCONJ
ejpam-842	269	24	applied	applied	ADJ
ejpam-842	269	25	mathematics	mathematic	NOUN
ejpam-842	269	26	,	,	PUNCT
ejpam-842	269	27	7(1	7(1	NUM
ejpam-842	269	28	)	)	PUNCT
ejpam-842	269	29	,	,	PUNCT
ejpam-842	269	30	89	89	NUM
ejpam-842	269	31	-	-	SYM
ejpam-842	269	32	99	99	NUM
ejpam-842	269	33	.	.	PUNCT
ejpam-842	269	34	2011	2011	NUM
ejpam-842	270	1	[	[	X
ejpam-842	270	2	6	6	NUM
ejpam-842	270	3	]	]	PUNCT
ejpam-842	270	4	kuratowski	kuratowski	PROPN
ejpam-842	270	5	.k	.k	PROPN
ejpam-842	270	6	,	,	PUNCT
ejpam-842	270	7	topology	topology	PROPN
ejpam-842	270	8	i	i	PROPN
ejpam-842	270	9	,	,	PUNCT
ejpam-842	270	10	warszawa	warszawa	PROPN
ejpam-842	270	11	,	,	PUNCT
ejpam-842	270	12	1933	1933	NUM
ejpam-842	270	13	.	.	PUNCT
ejpam-842	271	1	[	[	X
ejpam-842	271	2	7	7	X
ejpam-842	271	3	]	]	X
ejpam-842	271	4	levine	levine	PROPN
ejpam-842	271	5	.n	.n	PROPN
ejpam-842	271	6	,	,	PUNCT
ejpam-842	271	7	semi	semi	ADJ
ejpam-842	271	8	-	-	ADJ
ejpam-842	271	9	open	open	ADJ
ejpam-842	271	10	sets	set	NOUN
ejpam-842	271	11	and	and	CCONJ
ejpam-842	271	12	semi	semi	ADJ
ejpam-842	271	13	-	-	NOUN
ejpam-842	271	14	continuity	continuity	NOUN
ejpam-842	271	15	in	in	ADP
ejpam-842	271	16	topological	topological	ADJ
ejpam-842	271	17	spaces	space	NOUN
ejpam-842	271	18	,	,	PUNCT
ejpam-842	271	19	amer	amer	PROPN
ejpam-842	271	20	.	.	PROPN
ejpam-842	271	21	math	math	PROPN
ejpam-842	271	22	.	.	PUNCT
ejpam-842	272	1	monthly	monthly	ADJ
ejpam-842	272	2	,	,	PUNCT
ejpam-842	272	3	70(1	70(1	NOUN
ejpam-842	272	4	)	)	PUNCT
ejpam-842	272	5	,	,	PUNCT
ejpam-842	272	6	36	36	NUM
ejpam-842	272	7	-	-	SYM
ejpam-842	272	8	41	41	NUM
ejpam-842	272	9	.	.	PUNCT
ejpam-842	273	1	1963	1963	NUM
ejpam-842	273	2	.	.	PUNCT
ejpam-842	274	1	[	[	X
ejpam-842	274	2	8	8	NUM
ejpam-842	274	3	]	]	X
ejpam-842	274	4	levine	levine	PROPN
ejpam-842	274	5	.n	.n	PROPN
ejpam-842	274	6	,	,	PUNCT
ejpam-842	274	7	generalized	generalize	VERB
ejpam-842	274	8	closed	closed	ADJ
ejpam-842	274	9	sets	set	NOUN
ejpam-842	274	10	in	in	ADP
ejpam-842	274	11	topology	topology	NOUN
ejpam-842	274	12	,	,	PUNCT
ejpam-842	274	13	rend	rend	VERB
ejpam-842	274	14	.	.	PUNCT
ejpam-842	275	1	circ	circ	PROPN
ejpam-842	275	2	.	.	PUNCT
ejpam-842	276	1	mat	mat	PROPN
ejpam-842	276	2	.	.	PUNCT
ejpam-842	276	3	palermo	palermo	PROPN
ejpam-842	276	4	(	(	PUNCT
ejpam-842	276	5	2	2	NUM
ejpam-842	276	6	)	)	PUNCT
ejpam-842	276	7	,	,	PUNCT
ejpam-842	276	8	19	19	NUM
ejpam-842	276	9	,	,	PUNCT
ejpam-842	276	10	89	89	NUM
ejpam-842	276	11	-	-	SYM
ejpam-842	276	12	96	96	NUM
ejpam-842	276	13	.	.	PUNCT
ejpam-842	276	14	1970	1970	NUM
ejpam-842	276	15	.	.	PUNCT