id	sid	tid	token	lemma	pos
ejpam-1245	1	1	4_aljarrah.dvi	4_aljarrah.dvi	PROPN
ejpam-1245	1	2	european	european	PROPN
ejpam-1245	1	3	journal	journal	PROPN
ejpam-1245	1	4	of	of	ADP
ejpam-1245	1	5	pure	pure	ADJ
ejpam-1245	1	6	and	and	CCONJ
ejpam-1245	1	7	applied	apply	VERB
ejpam-1245	1	8	mathematics	mathematic	NOUN
ejpam-1245	1	9	vol	vol	NOUN
ejpam-1245	1	10	.	.	PROPN
ejpam-1245	1	11	5	5	NUM
ejpam-1245	1	12	,	,	PUNCT
ejpam-1245	1	13	no	no	INTJ
ejpam-1245	1	14	.	.	NOUN
ejpam-1245	1	15	2	2	NUM
ejpam-1245	1	16	,	,	PUNCT
ejpam-1245	1	17	2012	2012	NUM
ejpam-1245	1	18	,	,	PUNCT
ejpam-1245	1	19	129	129	NUM
ejpam-1245	1	20	-	-	SYM
ejpam-1245	1	21	140	140	NUM
ejpam-1245	1	22	issn	issn	PROPN
ejpam-1245	1	23	1307	1307	NUM
ejpam-1245	1	24	-	-	SYM
ejpam-1245	1	25	5543	5543	NUM
ejpam-1245	1	26	–	–	PUNCT
ejpam-1245	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-1245	1	28	on	on	ADP
ejpam-1245	1	29	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	1	30	functions	function	NOUN
ejpam-1245	1	31	heyam	heyam	PROPN
ejpam-1245	1	32	hussein	hussein	PROPN
ejpam-1245	1	33	aljarrah∗	aljarrah∗	NOUN
ejpam-1245	1	34	,	,	PUNCT
ejpam-1245	1	35	mohd	mohd	PROPN
ejpam-1245	1	36	salmi	salmi	PROPN
ejpam-1245	1	37	md	md	PROPN
ejpam-1245	1	38	noorani	noorani	PROPN
ejpam-1245	1	39	school	school	PROPN
ejpam-1245	1	40	of	of	ADP
ejpam-1245	1	41	mathematical	mathematical	ADJ
ejpam-1245	1	42	sciences	science	NOUN
ejpam-1245	1	43	,	,	PUNCT
ejpam-1245	1	44	faculty	faculty	NOUN
ejpam-1245	1	45	of	of	ADP
ejpam-1245	1	46	science	science	NOUN
ejpam-1245	1	47	and	and	CCONJ
ejpam-1245	1	48	technology	technology	NOUN
ejpam-1245	1	49	,	,	PUNCT
ejpam-1245	1	50	universiti	universiti	PROPN
ejpam-1245	1	51	kebangsaan	kebangsaan	PROPN
ejpam-1245	1	52	malaysia	malaysia	PROPN
ejpam-1245	1	53	,	,	PUNCT
ejpam-1245	1	54	43600	43600	NUM
ejpam-1245	1	55	ukm	ukm	PROPN
ejpam-1245	1	56	bangi	bangi	PROPN
ejpam-1245	1	57	,	,	PUNCT
ejpam-1245	1	58	selangor	selangor	PROPN
ejpam-1245	1	59	,	,	PUNCT
ejpam-1245	1	60	malaysia	malaysia	PROPN
ejpam-1245	1	61	abstract	abstract	NOUN
ejpam-1245	1	62	.	.	PUNCT
ejpam-1245	2	1	a	a	DET
ejpam-1245	2	2	subset	subset	NOUN
ejpam-1245	2	3	a	a	PRON
ejpam-1245	2	4	of	of	ADP
ejpam-1245	2	5	topological	topological	ADJ
ejpam-1245	2	6	space	space	NOUN
ejpam-1245	2	7	(	(	PUNCT
ejpam-1245	2	8	x	x	X
ejpam-1245	2	9	,	,	PUNCT
ejpam-1245	2	10	τ	τ	X
ejpam-1245	2	11	)	)	PUNCT
ejpam-1245	2	12	is	be	AUX
ejpam-1245	2	13	said	say	VERB
ejpam-1245	2	14	to	to	PART
ejpam-1245	2	15	be	be	AUX
ejpam-1245	2	16	ωβ−open	ωβ−open	PRON
ejpam-1245	2	17	[	[	X
ejpam-1245	2	18	3	3	X
ejpam-1245	2	19	]	]	X
ejpam-1245	2	20	if	if	SCONJ
ejpam-1245	2	21	for	for	ADP
ejpam-1245	2	22	every	every	DET
ejpam-1245	2	23	x	x	SYM
ejpam-1245	2	24	∈	∈	PROPN
ejpam-1245	2	25	a	a	DET
ejpam-1245	2	26	there	there	PRON
ejpam-1245	2	27	exists	exist	VERB
ejpam-1245	2	28	an	an	DET
ejpam-1245	2	29	β−open	β−open	PUNCT
ejpam-1245	2	30	set	set	VERB
ejpam-1245	2	31	u	u	NOUN
ejpam-1245	2	32	containing	contain	VERB
ejpam-1245	2	33	x	x	PUNCT
ejpam-1245	2	34	such	such	ADJ
ejpam-1245	2	35	that	that	DET
ejpam-1245	2	36	u	u	NOUN
ejpam-1245	2	37	−	−	PROPN
ejpam-1245	2	38	a	a	PRON
ejpam-1245	2	39	is	be	AUX
ejpam-1245	2	40	a	a	DET
ejpam-1245	2	41	countable	countable	ADJ
ejpam-1245	2	42	.	.	PUNCT
ejpam-1245	3	1	in	in	ADP
ejpam-1245	3	2	this	this	DET
ejpam-1245	3	3	paper	paper	NOUN
ejpam-1245	3	4	,	,	PUNCT
ejpam-1245	3	5	we	we	PRON
ejpam-1245	3	6	introduce	introduce	VERB
ejpam-1245	3	7	and	and	CCONJ
ejpam-1245	3	8	study	study	VERB
ejpam-1245	3	9	new	new	ADJ
ejpam-1245	3	10	class	class	NOUN
ejpam-1245	3	11	of	of	ADP
ejpam-1245	3	12	function	function	NOUN
ejpam-1245	3	13	which	which	PRON
ejpam-1245	3	14	is	be	AUX
ejpam-1245	3	15	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	3	16	functions	function	NOUN
ejpam-1245	3	17	by	by	ADP
ejpam-1245	3	18	using	use	VERB
ejpam-1245	3	19	the	the	DET
ejpam-1245	3	20	notion	notion	NOUN
ejpam-1245	3	21	of	of	ADP
ejpam-1245	3	22	ωβ−open	ωβ−open	PROPN
ejpam-1245	3	23	sets	set	NOUN
ejpam-1245	3	24	.	.	PUNCT
ejpam-1245	4	1	this	this	DET
ejpam-1245	4	2	new	new	ADJ
ejpam-1245	4	3	class	class	NOUN
ejpam-1245	4	4	of	of	ADP
ejpam-1245	4	5	function	function	NOUN
ejpam-1245	4	6	defines	define	NOUN
ejpam-1245	4	7	as	as	ADP
ejpam-1245	4	8	a	a	DET
ejpam-1245	4	9	function	function	NOUN
ejpam-1245	4	10	f	f	NOUN
ejpam-1245	4	11	:	:	PUNCT
ejpam-1245	4	12	(	(	PUNCT
ejpam-1245	4	13	x	x	X
ejpam-1245	4	14	,	,	PUNCT
ejpam-1245	4	15	τ)→	τ)→	PROPN
ejpam-1245	4	16	(	(	PUNCT
ejpam-1245	4	17	y	y	PROPN
ejpam-1245	4	18	,	,	PUNCT
ejpam-1245	4	19	σ	σ	PROPN
ejpam-1245	4	20	)	)	PUNCT
ejpam-1245	4	21	from	from	ADP
ejpam-1245	4	22	a	a	DET
ejpam-1245	4	23	topological	topological	ADJ
ejpam-1245	4	24	space	space	NOUN
ejpam-1245	4	25	(	(	PUNCT
ejpam-1245	4	26	x	x	X
ejpam-1245	4	27	,	,	PUNCT
ejpam-1245	4	28	τ	τ	PROPN
ejpam-1245	4	29	)	)	PUNCT
ejpam-1245	4	30	into	into	ADP
ejpam-1245	4	31	a	a	DET
ejpam-1245	4	32	topological	topological	ADJ
ejpam-1245	4	33	space	space	NOUN
ejpam-1245	4	34	(	(	PUNCT
ejpam-1245	4	35	y	y	PROPN
ejpam-1245	4	36	,	,	PUNCT
ejpam-1245	4	37	σ	σ	PROPN
ejpam-1245	4	38	)	)	PUNCT
ejpam-1245	4	39	is	be	AUX
ejpam-1245	4	40	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	4	41	function	function	NOUN
ejpam-1245	4	42	if	if	SCONJ
ejpam-1245	4	43	and	and	CCONJ
ejpam-1245	4	44	only	only	ADV
ejpam-1245	4	45	if	if	SCONJ
ejpam-1245	4	46	for	for	ADP
ejpam-1245	4	47	each	each	DET
ejpam-1245	4	48	x	x	SYM
ejpam-1245	4	49	∈	∈	PROPN
ejpam-1245	4	50	x	x	X
ejpam-1245	4	51	and	and	CCONJ
ejpam-1245	4	52	each	each	DET
ejpam-1245	4	53	open	open	ADJ
ejpam-1245	4	54	set	set	VERB
ejpam-1245	4	55	v	v	NOUN
ejpam-1245	4	56	in	in	ADP
ejpam-1245	4	57	(	(	PUNCT
ejpam-1245	4	58	y	y	PROPN
ejpam-1245	4	59	,	,	PUNCT
ejpam-1245	4	60	σ	σ	PROPN
ejpam-1245	4	61	)	)	PUNCT
ejpam-1245	4	62	containing	contain	VERB
ejpam-1245	4	63	f	f	PROPN
ejpam-1245	4	64	(	(	PUNCT
ejpam-1245	4	65	x	x	X
ejpam-1245	4	66	)	)	PUNCT
ejpam-1245	4	67	there	there	PRON
ejpam-1245	4	68	exists	exist	VERB
ejpam-1245	4	69	an	an	DET
ejpam-1245	4	70	ωβ−open	ωβ−open	PROPN
ejpam-1245	4	71	set	set	NOUN
ejpam-1245	4	72	u	u	NOUN
ejpam-1245	4	73	containing	contain	VERB
ejpam-1245	4	74	x	x	PUNCT
ejpam-1245	4	75	such	such	ADJ
ejpam-1245	4	76	that	that	SCONJ
ejpam-1245	4	77	f	f	PROPN
ejpam-1245	4	78	(	(	PUNCT
ejpam-1245	4	79	u)⊆	u)⊆	PROPN
ejpam-1245	4	80	v	v	NOUN
ejpam-1245	4	81	.	.	PUNCT
ejpam-1245	5	1	we	we	PRON
ejpam-1245	5	2	give	give	VERB
ejpam-1245	5	3	some	some	DET
ejpam-1245	5	4	characterizations	characterization	NOUN
ejpam-1245	5	5	of	of	ADP
ejpam-1245	5	6	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	5	7	functions	function	NOUN
ejpam-1245	5	8	,	,	PUNCT
ejpam-1245	5	9	define	define	VERB
ejpam-1245	5	10	ωβ−irresolute	ωβ−irresolute	ADP
ejpam-1245	5	11	and	and	CCONJ
ejpam-1245	5	12	ωβ−open	ωβ−open	PROPN
ejpam-1245	5	13	function	function	NOUN
ejpam-1245	5	14	.	.	PUNCT
ejpam-1245	6	1	finally	finally	ADV
ejpam-1245	6	2	,	,	PUNCT
ejpam-1245	6	3	we	we	PRON
ejpam-1245	6	4	find	find	VERB
ejpam-1245	6	5	relationship	relationship	NOUN
ejpam-1245	6	6	between	between	ADP
ejpam-1245	6	7	these	these	DET
ejpam-1245	6	8	type	type	NOUN
ejpam-1245	6	9	of	of	ADP
ejpam-1245	6	10	function	function	NOUN
ejpam-1245	6	11	.	.	PUNCT
ejpam-1245	7	1	2010	2010	NUM
ejpam-1245	7	2	mathematics	mathematic	NOUN
ejpam-1245	7	3	subject	subject	NOUN
ejpam-1245	7	4	classifications	classification	NOUN
ejpam-1245	7	5	:	:	PUNCT
ejpam-1245	7	6	54c05	54c05	NUM
ejpam-1245	7	7	,	,	PUNCT
ejpam-1245	7	8	54c08	54c08	NUM
ejpam-1245	7	9	,	,	PUNCT
ejpam-1245	7	10	54c10	54c10	NUM
ejpam-1245	7	11	key	key	ADJ
ejpam-1245	7	12	words	word	NOUN
ejpam-1245	7	13	and	and	CCONJ
ejpam-1245	7	14	phrases	phrase	NOUN
ejpam-1245	7	15	:	:	PUNCT
ejpam-1245	7	16	ωβ−open	ωβ−open	PROPN
ejpam-1245	7	17	set	set	PROPN
ejpam-1245	7	18	,	,	PUNCT
ejpam-1245	7	19	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	7	20	,	,	PUNCT
ejpam-1245	7	21	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	7	22	,	,	PUNCT
ejpam-1245	7	23	ωβ−open	ωβ−open	ADJ
ejpam-1245	7	24	functions	function	NOUN
ejpam-1245	7	25	,	,	PUNCT
ejpam-1245	7	26	ωβ−cloesd	ωβ−cloesd	NOUN
ejpam-1245	7	27	functions	function	NOUN
ejpam-1245	7	28	.	.	PUNCT
ejpam-1245	8	1	1	1	X
ejpam-1245	8	2	.	.	X
ejpam-1245	8	3	introduction	introduction	NOUN
ejpam-1245	8	4	throughout	throughout	ADP
ejpam-1245	8	5	the	the	DET
ejpam-1245	8	6	present	present	ADJ
ejpam-1245	8	7	paper	paper	NOUN
ejpam-1245	8	8	,	,	PUNCT
ejpam-1245	8	9	a	a	DET
ejpam-1245	8	10	space	space	NOUN
ejpam-1245	8	11	mean	mean	NOUN
ejpam-1245	8	12	topological	topological	ADJ
ejpam-1245	8	13	space	space	NOUN
ejpam-1245	8	14	on	on	ADP
ejpam-1245	8	15	which	which	PRON
ejpam-1245	8	16	no	no	DET
ejpam-1245	8	17	separation	separation	NOUN
ejpam-1245	8	18	axiom	axiom	NOUN
ejpam-1245	8	19	is	be	AUX
ejpam-1245	8	20	assumed	assume	VERB
ejpam-1245	8	21	unless	unless	SCONJ
ejpam-1245	8	22	explicitly	explicitly	ADV
ejpam-1245	8	23	stated	state	VERB
ejpam-1245	8	24	.	.	PUNCT
ejpam-1245	9	1	let	let	VERB
ejpam-1245	9	2	a	a	DET
ejpam-1245	9	3	be	be	AUX
ejpam-1245	9	4	a	a	DET
ejpam-1245	9	5	subset	subset	NOUN
ejpam-1245	9	6	of	of	ADP
ejpam-1245	9	7	a	a	DET
ejpam-1245	9	8	space	space	NOUN
ejpam-1245	9	9	(	(	PUNCT
ejpam-1245	9	10	x	x	X
ejpam-1245	9	11	,	,	PUNCT
ejpam-1245	9	12	τ	τ	PROPN
ejpam-1245	9	13	)	)	PUNCT
ejpam-1245	9	14	.	.	PUNCT
ejpam-1245	10	1	the	the	DET
ejpam-1245	10	2	closure	closure	NOUN
ejpam-1245	10	3	of	of	ADP
ejpam-1245	10	4	a	a	PRON
ejpam-1245	10	5	and	and	CCONJ
ejpam-1245	10	6	interior	interior	ADJ
ejpam-1245	10	7	of	of	ADP
ejpam-1245	10	8	a	a	PRON
ejpam-1245	10	9	in	in	ADP
ejpam-1245	10	10	(	(	PUNCT
ejpam-1245	10	11	x	x	INTJ
ejpam-1245	10	12	,	,	PUNCT
ejpam-1245	10	13	τ	τ	X
ejpam-1245	10	14	)	)	PUNCT
ejpam-1245	10	15	are	be	AUX
ejpam-1245	10	16	denoted	denote	VERB
ejpam-1245	10	17	by	by	ADP
ejpam-1245	10	18	int(a	int(a	PROPN
ejpam-1245	10	19	)	)	PUNCT
ejpam-1245	10	20	and	and	CCONJ
ejpam-1245	10	21	cl(a	cl(a	NUM
ejpam-1245	10	22	)	)	PUNCT
ejpam-1245	10	23	,	,	PUNCT
ejpam-1245	10	24	respectively	respectively	ADV
ejpam-1245	10	25	.	.	PUNCT
ejpam-1245	11	1	a	a	DET
ejpam-1245	11	2	subset	subset	NOUN
ejpam-1245	11	3	a	a	PRON
ejpam-1245	11	4	of	of	ADP
ejpam-1245	11	5	a	a	DET
ejpam-1245	11	6	space	space	NOUN
ejpam-1245	11	7	(	(	PUNCT
ejpam-1245	11	8	x	x	X
ejpam-1245	11	9	,	,	PUNCT
ejpam-1245	11	10	τ	τ	X
ejpam-1245	11	11	)	)	PUNCT
ejpam-1245	11	12	is	be	AUX
ejpam-1245	11	13	said	say	VERB
ejpam-1245	11	14	to	to	PART
ejpam-1245	11	15	be	be	AUX
ejpam-1245	11	16	b−open	b−open	ADJ
ejpam-1245	11	17	[	[	X
ejpam-1245	11	18	4	4	NUM
ejpam-1245	11	19	]	]	PUNCT
ejpam-1245	11	20	,	,	PUNCT
ejpam-1245	11	21	(	(	PUNCT
ejpam-1245	11	22	reps	rep	NOUN
ejpam-1245	11	23	.	.	PROPN
ejpam-1245	11	24	β−open	β−open	PUNCT
ejpam-1245	12	1	[	[	X
ejpam-1245	12	2	7	7	NUM
ejpam-1245	12	3	]	]	PUNCT
ejpam-1245	12	4	)	)	PUNCT
ejpam-1245	12	5	if	if	SCONJ
ejpam-1245	12	6	a	a	DET
ejpam-1245	12	7	⊆	⊆	NUM
ejpam-1245	12	8	int(cl(a	int(cl(a	PROPN
ejpam-1245	12	9	)	)	PUNCT
ejpam-1245	12	10	)	)	PUNCT
ejpam-1245	12	11	∪	∪	ADP
ejpam-1245	12	12	cl(int(a	cl(int(a	PROPN
ejpam-1245	12	13	)	)	PUNCT
ejpam-1245	12	14	)	)	PUNCT
ejpam-1245	12	15	,	,	PUNCT
ejpam-1245	12	16	(	(	PUNCT
ejpam-1245	12	17	resp	resp	NOUN
ejpam-1245	12	18	.	.	PUNCT
ejpam-1245	13	1	a⊆	a⊆	PROPN
ejpam-1245	13	2	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-1245	13	3	)	)	PUNCT
ejpam-1245	13	4	)	)	PUNCT
ejpam-1245	13	5	)	)	PUNCT
ejpam-1245	13	6	)	)	PUNCT
ejpam-1245	13	7	.	.	PUNCT
ejpam-1245	14	1	recall	recall	VERB
ejpam-1245	14	2	that	that	SCONJ
ejpam-1245	14	3	a	a	DET
ejpam-1245	14	4	subset	subset	NOUN
ejpam-1245	14	5	a	a	PRON
ejpam-1245	14	6	of	of	ADP
ejpam-1245	14	7	a	a	DET
ejpam-1245	14	8	space	space	NOUN
ejpam-1245	14	9	(	(	PUNCT
ejpam-1245	14	10	x	x	X
ejpam-1245	14	11	,	,	PUNCT
ejpam-1245	14	12	τ	τ	X
ejpam-1245	14	13	)	)	PUNCT
ejpam-1245	14	14	is	be	AUX
ejpam-1245	14	15	said	say	VERB
ejpam-1245	14	16	to	to	PART
ejpam-1245	14	17	be	be	AUX
ejpam-1245	14	18	ωβ−open	ωβ−open	PRON
ejpam-1245	15	1	[	[	X
ejpam-1245	15	2	3	3	NUM
ejpam-1245	15	3	]	]	PUNCT
ejpam-1245	15	4	(	(	PUNCT
ejpam-1245	15	5	resp	resp	NOUN
ejpam-1245	15	6	.	.	PUNCT
ejpam-1245	16	1	ωb−open	ωb−open	VERB
ejpam-1245	17	1	[	[	PUNCT
ejpam-1245	17	2	9	9	NUM
ejpam-1245	17	3	]	]	PUNCT
ejpam-1245	17	4	,	,	PUNCT
ejpam-1245	17	5	ω−open	ω−open	PROPN
ejpam-1245	18	1	[	[	X
ejpam-1245	18	2	2	2	NUM
ejpam-1245	18	3	]	]	PUNCT
ejpam-1245	18	4	)	)	PUNCT
ejpam-1245	18	5	set	set	VERB
ejpam-1245	18	6	if	if	SCONJ
ejpam-1245	18	7	for	for	ADP
ejpam-1245	18	8	every	every	DET
ejpam-1245	18	9	x	x	SYM
ejpam-1245	18	10	∈	∈	PROPN
ejpam-1245	18	11	a	a	DET
ejpam-1245	18	12	there	there	PRON
ejpam-1245	18	13	exists	exist	VERB
ejpam-1245	18	14	an	an	DET
ejpam-1245	18	15	β−open	β−open	PUNCT
ejpam-1245	18	16	(	(	PUNCT
ejpam-1245	18	17	resp	resp	NOUN
ejpam-1245	18	18	.	.	PUNCT
ejpam-1245	19	1	b−open	b−open	ADJ
ejpam-1245	19	2	,	,	PUNCT
ejpam-1245	19	3	open	open	ADJ
ejpam-1245	19	4	)	)	PUNCT
ejpam-1245	19	5	set	set	VERB
ejpam-1245	19	6	u	u	NOUN
ejpam-1245	19	7	containing	contain	VERB
ejpam-1245	19	8	x	x	PUNCT
ejpam-1245	19	9	such	such	ADJ
ejpam-1245	19	10	that	that	DET
ejpam-1245	19	11	u	u	NOUN
ejpam-1245	19	12	−	−	PROPN
ejpam-1245	19	13	a	a	PRON
ejpam-1245	19	14	is	be	AUX
ejpam-1245	19	15	a	a	DET
ejpam-1245	19	16	countable	countable	NOUN
ejpam-1245	19	17	.	.	PUNCT
ejpam-1245	20	1	we	we	PRON
ejpam-1245	20	2	write	write	VERB
ejpam-1245	20	3	ωβo(x	ωβo(x	PROPN
ejpam-1245	20	4	,	,	PUNCT
ejpam-1245	20	5	τ	τ	X
ejpam-1245	20	6	)	)	PUNCT
ejpam-1245	20	7	(	(	PUNCT
ejpam-1245	20	8	resp	resp	NOUN
ejpam-1245	20	9	.	.	PUNCT
ejpam-1245	21	1	ωbo(x	ωbo(x	PROPN
ejpam-1245	21	2	,	,	PUNCT
ejpam-1245	21	3	τ	τ	PROPN
ejpam-1245	21	4	)	)	PUNCT
ejpam-1245	21	5	,	,	PUNCT
ejpam-1245	21	6	βo(x	βo(x	PUNCT
ejpam-1245	21	7	,	,	PUNCT
ejpam-1245	21	8	τ	τ	PROPN
ejpam-1245	21	9	)	)	PUNCT
ejpam-1245	21	10	,	,	PUNCT
ejpam-1245	21	11	ωo(x	ωo(x	PROPN
ejpam-1245	21	12	,	,	PUNCT
ejpam-1245	21	13	τ	τ	PROPN
ejpam-1245	21	14	)	)	PUNCT
ejpam-1245	21	15	,	,	PUNCT
ejpam-1245	21	16	bo(x	bo(x	NUM
ejpam-1245	21	17	,	,	PUNCT
ejpam-1245	21	18	τ	τ	PROPN
ejpam-1245	21	19	)	)	PUNCT
ejpam-1245	21	20	)	)	PUNCT
ejpam-1245	21	21	to	to	PART
ejpam-1245	21	22	denote	denote	VERB
ejpam-1245	21	23	the	the	DET
ejpam-1245	21	24	family	family	NOUN
ejpam-1245	21	25	of	of	ADP
ejpam-1245	21	26	all	all	DET
ejpam-1245	21	27	ωβ−open	ωβ−open	PROPN
ejpam-1245	21	28	(	(	PUNCT
ejpam-1245	21	29	resp	resp	NOUN
ejpam-1245	21	30	.	.	PUNCT
ejpam-1245	22	1	ωb−open	ωb−open	ADJ
ejpam-1245	22	2	,	,	PUNCT
ejpam-1245	22	3	β−open	β−open	ADJ
ejpam-1245	22	4	,	,	PUNCT
ejpam-1245	22	5	ω−open	ω−open	NOUN
ejpam-1245	22	6	,	,	PUNCT
ejpam-1245	22	7	b−open	b−open	ADJ
ejpam-1245	22	8	)	)	PUNCT
ejpam-1245	22	9	subsets	subset	NOUN
ejpam-1245	22	10	of	of	ADP
ejpam-1245	22	11	(	(	PUNCT
ejpam-1245	22	12	x	x	PROPN
ejpam-1245	22	13	,	,	PUNCT
ejpam-1245	22	14	τ	τ	PROPN
ejpam-1245	22	15	)	)	PUNCT
ejpam-1245	22	16	.	.	PUNCT
ejpam-1245	23	1	∗corresponding	∗corresponde	VERB
ejpam-1245	23	2	author	author	NOUN
ejpam-1245	23	3	.	.	PUNCT
ejpam-1245	24	1	email	email	NOUN
ejpam-1245	24	2	addresses	address	NOUN
ejpam-1245	24	3	:	:	PUNCT
ejpam-1245	24	4	hiamaljarah	hiamaljarah	PROPN
ejpam-1245	24	5	�	�	PROPN
ejpam-1245	24	6	yahoo	yahoo	PROPN
ejpam-1245	24	7	.	.	PUNCT
ejpam-1245	25	1	om	om	PROPN
ejpam-1245	25	2	(	(	PUNCT
ejpam-1245	25	3	h.	h.	PROPN
ejpam-1245	25	4	aljarrah	aljarrah	PROPN
ejpam-1245	25	5	)	)	PUNCT
ejpam-1245	25	6	,	,	PUNCT
ejpam-1245	25	7	msn�ukm.my	msn�ukm.my	PROPN
ejpam-1245	25	8	(	(	PUNCT
ejpam-1245	25	9	m.	m.	NOUN
ejpam-1245	25	10	noorani	noorani	PROPN
ejpam-1245	25	11	)	)	PUNCT
ejpam-1245	25	12	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1245	26	1	129	129	NUM
ejpam-1245	26	2	c	c	X
ejpam-1245	26	3	©	©	PROPN
ejpam-1245	26	4	2012	2012	NUM
ejpam-1245	26	5	ejpam	ejpam	VERB
ejpam-1245	26	6	all	all	DET
ejpam-1245	26	7	rights	right	NOUN
ejpam-1245	26	8	reserved	reserve	VERB
ejpam-1245	26	9	.	.	PUNCT
ejpam-1245	27	1	h.	h.	PROPN
ejpam-1245	27	2	aljarrah	aljarrah	PROPN
ejpam-1245	27	3	,	,	PUNCT
ejpam-1245	27	4	m.	m.	NOUN
ejpam-1245	27	5	noorani	noorani	PROPN
ejpam-1245	27	6	/	/	SYM
ejpam-1245	27	7	eur	eur	PROPN
ejpam-1245	27	8	.	.	PUNCT
ejpam-1245	28	1	j.	j.	PROPN
ejpam-1245	28	2	pure	pure	PROPN
ejpam-1245	28	3	appl	appl	PROPN
ejpam-1245	28	4	.	.	PROPN
ejpam-1245	28	5	math	math	PROPN
ejpam-1245	28	6	,	,	PUNCT
ejpam-1245	28	7	5	5	NUM
ejpam-1245	28	8	(	(	PUNCT
ejpam-1245	28	9	2012	2012	NUM
ejpam-1245	28	10	)	)	PUNCT
ejpam-1245	28	11	,	,	PUNCT
ejpam-1245	28	12	129	129	NUM
ejpam-1245	28	13	-	-	SYM
ejpam-1245	28	14	140	140	NUM
ejpam-1245	28	15	130	130	NUM
ejpam-1245	28	16	definition	definition	NOUN
ejpam-1245	28	17	1	1	NUM
ejpam-1245	28	18	.	.	PUNCT
ejpam-1245	29	1	a	a	DET
ejpam-1245	29	2	function	function	NOUN
ejpam-1245	29	3	f	f	NOUN
ejpam-1245	29	4	:	:	PUNCT
ejpam-1245	29	5	(	(	PUNCT
ejpam-1245	29	6	x	x	X
ejpam-1245	29	7	,	,	PUNCT
ejpam-1245	29	8	τ)→	τ)→	PROPN
ejpam-1245	29	9	(	(	PUNCT
ejpam-1245	29	10	y	y	PROPN
ejpam-1245	29	11	,	,	PUNCT
ejpam-1245	29	12	σ	σ	PROPN
ejpam-1245	29	13	)	)	PUNCT
ejpam-1245	29	14	is	be	AUX
ejpam-1245	29	15	called	call	VERB
ejpam-1245	29	16	ω−continuous	ω−continuous	PUNCT
ejpam-1245	30	1	[	[	X
ejpam-1245	30	2	6	6	NUM
ejpam-1245	30	3	]	]	PUNCT
ejpam-1245	30	4	(	(	PUNCT
ejpam-1245	30	5	resp	resp	NOUN
ejpam-1245	30	6	.	.	PUNCT
ejpam-1245	31	1	ωb−continuous	ωb−continuous	ADJ
ejpam-1245	31	2	[	[	X
ejpam-1245	31	3	9	9	NUM
ejpam-1245	31	4	]	]	SYM
ejpam-1245	31	5	)	)	PUNCT
ejpam-1245	31	6	if	if	SCONJ
ejpam-1245	31	7	for	for	ADP
ejpam-1245	31	8	every	every	DET
ejpam-1245	31	9	x	x	SYM
ejpam-1245	31	10	∈	∈	PROPN
ejpam-1245	31	11	x	x	X
ejpam-1245	31	12	and	and	CCONJ
ejpam-1245	31	13	each	each	DET
ejpam-1245	31	14	open	open	ADJ
ejpam-1245	31	15	set	set	VERB
ejpam-1245	31	16	v	v	NOUN
ejpam-1245	31	17	in	in	ADP
ejpam-1245	31	18	(	(	PUNCT
ejpam-1245	31	19	y	y	PROPN
ejpam-1245	31	20	,	,	PUNCT
ejpam-1245	31	21	σ	σ	PROPN
ejpam-1245	31	22	)	)	PUNCT
ejpam-1245	31	23	containing	contain	VERB
ejpam-1245	31	24	f	f	PROPN
ejpam-1245	31	25	(	(	PUNCT
ejpam-1245	31	26	x	x	X
ejpam-1245	31	27	)	)	PUNCT
ejpam-1245	31	28	there	there	PRON
ejpam-1245	31	29	exists	exist	VERB
ejpam-1245	31	30	an	an	DET
ejpam-1245	31	31	ωo(x	ωo(x	PROPN
ejpam-1245	31	32	,	,	PUNCT
ejpam-1245	31	33	τ	τ	X
ejpam-1245	31	34	)	)	PUNCT
ejpam-1245	31	35	(	(	PUNCT
ejpam-1245	31	36	resp	resp	NOUN
ejpam-1245	31	37	.	.	PUNCT
ejpam-1245	32	1	ωbo(x	ωbo(x	PROPN
ejpam-1245	32	2	,	,	PUNCT
ejpam-1245	32	3	τ	τ	PROPN
ejpam-1245	32	4	)	)	PUNCT
ejpam-1245	32	5	)	)	PUNCT
ejpam-1245	32	6	set	set	VERB
ejpam-1245	32	7	u	u	NOUN
ejpam-1245	32	8	containing	contain	VERB
ejpam-1245	32	9	x	x	PUNCT
ejpam-1245	32	10	such	such	ADJ
ejpam-1245	32	11	that	that	SCONJ
ejpam-1245	32	12	f	f	PROPN
ejpam-1245	32	13	(	(	PUNCT
ejpam-1245	32	14	u)⊆	u)⊆	PROPN
ejpam-1245	32	15	v	v	NOUN
ejpam-1245	32	16	.	.	PUNCT
ejpam-1245	33	1	lemma	lemma	PROPN
ejpam-1245	33	2	1	1	NUM
ejpam-1245	33	3	.	.	PUNCT
ejpam-1245	34	1	[	[	X
ejpam-1245	34	2	3	3	X
ejpam-1245	34	3	]	]	X
ejpam-1245	34	4	let	let	VERB
ejpam-1245	34	5	(	(	PUNCT
ejpam-1245	34	6	x	x	X
ejpam-1245	34	7	,	,	PUNCT
ejpam-1245	34	8	τ	τ	X
ejpam-1245	34	9	)	)	PUNCT
ejpam-1245	34	10	be	be	VERB
ejpam-1245	34	11	a	a	DET
ejpam-1245	34	12	topological	topological	ADJ
ejpam-1245	34	13	space	space	NOUN
ejpam-1245	34	14	:	:	PUNCT
ejpam-1245	34	15	i.	i.	PROPN
ejpam-1245	34	16	the	the	DET
ejpam-1245	34	17	union	union	NOUN
ejpam-1245	34	18	of	of	ADP
ejpam-1245	34	19	any	any	DET
ejpam-1245	34	20	family	family	NOUN
ejpam-1245	34	21	of	of	ADP
ejpam-1245	34	22	ωβo(x	ωβo(x	PROPN
ejpam-1245	34	23	,	,	PUNCT
ejpam-1245	34	24	τ	τ	X
ejpam-1245	34	25	)	)	PUNCT
ejpam-1245	34	26	sets	set	NOUN
ejpam-1245	34	27	is	be	AUX
ejpam-1245	34	28	ωβo(x	ωβo(x	PROPN
ejpam-1245	34	29	,	,	PUNCT
ejpam-1245	34	30	τ	τ	PROPN
ejpam-1245	34	31	)	)	PUNCT
ejpam-1245	34	32	.	.	PUNCT
ejpam-1245	35	1	ii	ii	PROPN
ejpam-1245	35	2	.	.	PUNCT
ejpam-1245	36	1	the	the	DET
ejpam-1245	36	2	intersection	intersection	NOUN
ejpam-1245	36	3	of	of	ADP
ejpam-1245	36	4	an	an	DET
ejpam-1245	36	5	ωβo(x	ωβo(x	PROPN
ejpam-1245	36	6	,	,	PUNCT
ejpam-1245	36	7	τ	τ	NOUN
ejpam-1245	36	8	)	)	PUNCT
ejpam-1245	36	9	set	set	NOUN
ejpam-1245	36	10	and	and	CCONJ
ejpam-1245	36	11	open	open	ADJ
ejpam-1245	36	12	set	set	NOUN
ejpam-1245	36	13	is	be	AUX
ejpam-1245	36	14	ωβo(x	ωβo(x	PROPN
ejpam-1245	36	15	,	,	PUNCT
ejpam-1245	36	16	τ	τ	PROPN
ejpam-1245	36	17	)	)	PUNCT
ejpam-1245	36	18	.	.	PUNCT
ejpam-1245	37	1	theorem	theorem	NOUN
ejpam-1245	37	2	1	1	NUM
ejpam-1245	37	3	.	.	PUNCT
ejpam-1245	38	1	[	[	X
ejpam-1245	38	2	3	3	X
ejpam-1245	38	3	]	]	X
ejpam-1245	38	4	let	let	VERB
ejpam-1245	38	5	(	(	PUNCT
ejpam-1245	38	6	y	y	NOUN
ejpam-1245	38	7	,	,	PUNCT
ejpam-1245	38	8	τy	τy	PART
ejpam-1245	38	9	)	)	PUNCT
ejpam-1245	38	10	be	be	AUX
ejpam-1245	38	11	a	a	DET
ejpam-1245	38	12	subspace	subspace	NOUN
ejpam-1245	38	13	of	of	ADP
ejpam-1245	38	14	(	(	PUNCT
ejpam-1245	38	15	x	x	PROPN
ejpam-1245	38	16	,	,	PUNCT
ejpam-1245	38	17	τ	τ	PROPN
ejpam-1245	38	18	)	)	PUNCT
ejpam-1245	38	19	,	,	PUNCT
ejpam-1245	38	20	a⊆	a⊆	VERB
ejpam-1245	38	21	y	y	PROPN
ejpam-1245	38	22	and	and	CCONJ
ejpam-1245	38	23	y	y	PROPN
ejpam-1245	38	24	is	be	AUX
ejpam-1245	38	25	βo(x	βo(x	PUNCT
ejpam-1245	38	26	,	,	PUNCT
ejpam-1245	38	27	τ	τ	X
ejpam-1245	38	28	)	)	PUNCT
ejpam-1245	38	29	sets	set	NOUN
ejpam-1245	38	30	.	.	PUNCT
ejpam-1245	39	1	then	then	ADV
ejpam-1245	39	2	a∈ωβo(x	a∈ωβo(x	PROPN
ejpam-1245	39	3	,	,	PUNCT
ejpam-1245	39	4	τ	τ	X
ejpam-1245	39	5	)	)	PUNCT
ejpam-1245	39	6	if	if	SCONJ
ejpam-1245	39	7	and	and	CCONJ
ejpam-1245	39	8	only	only	ADV
ejpam-1245	39	9	if	if	SCONJ
ejpam-1245	39	10	a∈ωβo(y	a∈ωβo(y	VERB
ejpam-1245	39	11	,	,	PUNCT
ejpam-1245	39	12	τy	τy	PRON
ejpam-1245	39	13	)	)	PUNCT
ejpam-1245	39	14	.	.	PUNCT
ejpam-1245	40	1	theorem	theorem	NOUN
ejpam-1245	40	2	2	2	NUM
ejpam-1245	40	3	.	.	PUNCT
ejpam-1245	41	1	[	[	X
ejpam-1245	41	2	3	3	X
ejpam-1245	41	3	]	]	PUNCT
ejpam-1245	41	4	let	let	VERB
ejpam-1245	41	5	a	a	PRON
ejpam-1245	41	6	be	be	AUX
ejpam-1245	41	7	a	a	DET
ejpam-1245	41	8	subset	subset	NOUN
ejpam-1245	41	9	of	of	ADP
ejpam-1245	41	10	a	a	DET
ejpam-1245	41	11	topological	topological	ADJ
ejpam-1245	41	12	space	space	NOUN
ejpam-1245	41	13	(	(	PUNCT
ejpam-1245	41	14	x	x	X
ejpam-1245	41	15	,	,	PUNCT
ejpam-1245	41	16	τ	τ	PROPN
ejpam-1245	41	17	)	)	PUNCT
ejpam-1245	41	18	.	.	PUNCT
ejpam-1245	42	1	then	then	ADV
ejpam-1245	42	2	x	x	X
ejpam-1245	42	3	∈ωβ	∈ωβ	PROPN
ejpam-1245	42	4	cl(a	cl(a	X
ejpam-1245	42	5	)	)	PUNCT
ejpam-1245	42	6	if	if	SCONJ
ejpam-1245	42	7	and	and	CCONJ
ejpam-1245	42	8	only	only	ADV
ejpam-1245	42	9	if	if	SCONJ
ejpam-1245	42	10	for	for	ADP
ejpam-1245	42	11	every	every	DET
ejpam-1245	42	12	ωβo(x	ωβo(x	PROPN
ejpam-1245	42	13	,	,	PUNCT
ejpam-1245	42	14	τ	τ	X
ejpam-1245	42	15	)	)	PUNCT
ejpam-1245	42	16	set	set	VERB
ejpam-1245	42	17	u	u	NOUN
ejpam-1245	42	18	containing	contain	VERB
ejpam-1245	42	19	x	x	PRON
ejpam-1245	42	20	,	,	PUNCT
ejpam-1245	42	21	a∩	a∩	PROPN
ejpam-1245	42	22	u	u	PROPN
ejpam-1245	42	23	6=	6=	PROPN
ejpam-1245	42	24	φ	φ	PROPN
ejpam-1245	42	25	.	.	PUNCT
ejpam-1245	43	1	theorem	theorem	NOUN
ejpam-1245	43	2	3	3	NUM
ejpam-1245	43	3	.	.	PUNCT
ejpam-1245	44	1	[	[	X
ejpam-1245	44	2	5	5	X
ejpam-1245	44	3	]	]	PUNCT
ejpam-1245	44	4	if	if	SCONJ
ejpam-1245	44	5	f	f	PROPN
ejpam-1245	44	6	:	:	PUNCT
ejpam-1245	44	7	(	(	PUNCT
ejpam-1245	44	8	x	x	X
ejpam-1245	44	9	,	,	PUNCT
ejpam-1245	44	10	τ)→	τ)→	PROPN
ejpam-1245	44	11	(	(	PUNCT
ejpam-1245	44	12	y	y	PROPN
ejpam-1245	44	13	,	,	PUNCT
ejpam-1245	44	14	σ	σ	PROPN
ejpam-1245	44	15	)	)	PUNCT
ejpam-1245	44	16	is	be	AUX
ejpam-1245	44	17	an	an	DET
ejpam-1245	44	18	open	open	ADJ
ejpam-1245	44	19	continuous	continuous	ADJ
ejpam-1245	44	20	function	function	NOUN
ejpam-1245	44	21	,	,	PUNCT
ejpam-1245	44	22	then	then	ADV
ejpam-1245	44	23	f	f	PROPN
ejpam-1245	44	24	−1(cl(a	−1(cl(a	PROPN
ejpam-1245	44	25	)	)	PUNCT
ejpam-1245	44	26	)	)	PUNCT
ejpam-1245	45	1	=	=	X
ejpam-1245	45	2	cl	cl	NOUN
ejpam-1245	45	3	(	(	PUNCT
ejpam-1245	45	4	f	f	PROPN
ejpam-1245	45	5	−1(a	−1(a	ADP
ejpam-1245	45	6	)	)	PUNCT
ejpam-1245	45	7	)	)	PUNCT
ejpam-1245	45	8	.	.	PUNCT
ejpam-1245	46	1	2	2	X
ejpam-1245	46	2	.	.	NUM
ejpam-1245	46	3	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	46	4	functions	function	NOUN
ejpam-1245	46	5	definition	definition	NOUN
ejpam-1245	46	6	2	2	NUM
ejpam-1245	46	7	.	.	PUNCT
ejpam-1245	47	1	a	a	DET
ejpam-1245	47	2	function	function	NOUN
ejpam-1245	47	3	f	f	NOUN
ejpam-1245	47	4	:	:	PUNCT
ejpam-1245	47	5	(	(	PUNCT
ejpam-1245	47	6	x	x	X
ejpam-1245	47	7	,	,	PUNCT
ejpam-1245	47	8	τ)→	τ)→	PROPN
ejpam-1245	47	9	(	(	PUNCT
ejpam-1245	47	10	y	y	PROPN
ejpam-1245	47	11	,	,	PUNCT
ejpam-1245	47	12	σ	σ	PROPN
ejpam-1245	47	13	)	)	PUNCT
ejpam-1245	47	14	is	be	AUX
ejpam-1245	47	15	called	call	VERB
ejpam-1245	47	16	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	47	17	at	at	ADP
ejpam-1245	47	18	a	a	DET
ejpam-1245	47	19	point	point	NOUN
ejpam-1245	47	20	x	x	SYM
ejpam-1245	47	21	∈	∈	NOUN
ejpam-1245	47	22	x	x	X
ejpam-1245	47	23	,	,	PUNCT
ejpam-1245	47	24	if	if	SCONJ
ejpam-1245	47	25	for	for	ADP
ejpam-1245	47	26	every	every	DET
ejpam-1245	47	27	open	open	NOUN
ejpam-1245	47	28	set	set	VERB
ejpam-1245	47	29	v	v	NOUN
ejpam-1245	47	30	in	in	ADP
ejpam-1245	47	31	(	(	PUNCT
ejpam-1245	47	32	y	y	PROPN
ejpam-1245	47	33	,	,	PUNCT
ejpam-1245	47	34	σ	σ	PROPN
ejpam-1245	47	35	)	)	PUNCT
ejpam-1245	47	36	containing	contain	VERB
ejpam-1245	47	37	f	f	PROPN
ejpam-1245	47	38	(	(	PUNCT
ejpam-1245	47	39	x	x	X
ejpam-1245	47	40	)	)	PUNCT
ejpam-1245	47	41	there	there	PRON
ejpam-1245	47	42	exists	exist	VERB
ejpam-1245	47	43	an	an	DET
ejpam-1245	47	44	ωβo(x	ωβo(x	PROPN
ejpam-1245	47	45	,	,	PUNCT
ejpam-1245	47	46	τ	τ	X
ejpam-1245	47	47	)	)	PUNCT
ejpam-1245	47	48	set	set	VERB
ejpam-1245	47	49	u	u	NOUN
ejpam-1245	47	50	containing	contain	VERB
ejpam-1245	47	51	x	x	PUNCT
ejpam-1245	47	52	such	such	ADJ
ejpam-1245	47	53	that	that	SCONJ
ejpam-1245	47	54	f	f	PROPN
ejpam-1245	47	55	(	(	PUNCT
ejpam-1245	47	56	u	u	NOUN
ejpam-1245	47	57	)	)	PUNCT
ejpam-1245	47	58	⊆	⊆	NUM
ejpam-1245	47	59	v	v	NOUN
ejpam-1245	47	60	.	.	PUNCT
ejpam-1245	48	1	if	if	SCONJ
ejpam-1245	48	2	f	f	PROPN
ejpam-1245	48	3	is	be	AUX
ejpam-1245	48	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	48	5	at	at	ADP
ejpam-1245	48	6	each	each	DET
ejpam-1245	48	7	point	point	NOUN
ejpam-1245	48	8	of	of	ADP
ejpam-1245	48	9	x	x	PRON
ejpam-1245	48	10	then	then	ADV
ejpam-1245	48	11	f	f	PROPN
ejpam-1245	48	12	is	be	AUX
ejpam-1245	48	13	said	say	VERB
ejpam-1245	48	14	to	to	PART
ejpam-1245	48	15	be	be	AUX
ejpam-1245	48	16	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	48	17	on	on	ADP
ejpam-1245	48	18	x	x	X
ejpam-1245	48	19	.	.	PUNCT
ejpam-1245	48	20	definition	definition	NOUN
ejpam-1245	48	21	3	3	X
ejpam-1245	48	22	.	.	PUNCT
ejpam-1245	49	1	let	let	AUX
ejpam-1245	49	2	(	(	PUNCT
ejpam-1245	49	3	x	x	X
ejpam-1245	49	4	,	,	PUNCT
ejpam-1245	49	5	τ	τ	X
ejpam-1245	49	6	)	)	PUNCT
ejpam-1245	49	7	be	be	VERB
ejpam-1245	49	8	any	any	DET
ejpam-1245	49	9	space	space	NOUN
ejpam-1245	49	10	,	,	PUNCT
ejpam-1245	49	11	a	a	DET
ejpam-1245	49	12	set	set	NOUN
ejpam-1245	49	13	a⊆	a⊆	NOUN
ejpam-1245	49	14	x	x	VERB
ejpam-1245	49	15	is	be	AUX
ejpam-1245	49	16	said	say	VERB
ejpam-1245	49	17	to	to	PART
ejpam-1245	49	18	be	be	AUX
ejpam-1245	49	19	ωβ−neighborhood	ωβ−neighborhood	PROPN
ejpam-1245	49	20	of	of	ADP
ejpam-1245	49	21	a	a	DET
ejpam-1245	49	22	point	point	NOUN
ejpam-1245	49	23	x	x	PUNCT
ejpam-1245	49	24	in	in	ADP
ejpam-1245	49	25	x	x	PUNCT
ejpam-1245	49	26	if	if	SCONJ
ejpam-1245	49	27	and	and	CCONJ
ejpam-1245	49	28	only	only	ADV
ejpam-1245	49	29	if	if	SCONJ
ejpam-1245	49	30	there	there	PRON
ejpam-1245	49	31	exists	exist	VERB
ejpam-1245	49	32	a	a	DET
ejpam-1245	49	33	ωβo(x	ωβo(x	PROPN
ejpam-1245	49	34	,	,	PUNCT
ejpam-1245	49	35	τ	τ	X
ejpam-1245	49	36	)	)	PUNCT
ejpam-1245	49	37	set	set	VERB
ejpam-1245	49	38	u	u	NOUN
ejpam-1245	49	39	containing	contain	VERB
ejpam-1245	49	40	x	x	PUNCT
ejpam-1245	49	41	such	such	ADJ
ejpam-1245	49	42	that	that	SCONJ
ejpam-1245	49	43	u	u	PROPN
ejpam-1245	49	44	⊆	⊆	NUM
ejpam-1245	49	45	a.	a.	NOUN
ejpam-1245	49	46	theorem	theorem	NOUN
ejpam-1245	49	47	4	4	X
ejpam-1245	49	48	.	.	PUNCT
ejpam-1245	50	1	let	let	VERB
ejpam-1245	50	2	f	f	NOUN
ejpam-1245	50	3	:	:	PUNCT
ejpam-1245	50	4	(	(	PUNCT
ejpam-1245	50	5	x	x	X
ejpam-1245	50	6	,	,	PUNCT
ejpam-1245	50	7	τ)→	τ)→	PROPN
ejpam-1245	50	8	(	(	PUNCT
ejpam-1245	50	9	y	y	PROPN
ejpam-1245	50	10	,	,	PUNCT
ejpam-1245	50	11	σ	σ	PROPN
ejpam-1245	50	12	)	)	PUNCT
ejpam-1245	50	13	be	be	VERB
ejpam-1245	50	14	a	a	DET
ejpam-1245	50	15	function	function	NOUN
ejpam-1245	50	16	,	,	PUNCT
ejpam-1245	50	17	where	where	SCONJ
ejpam-1245	50	18	x	x	PUNCT
ejpam-1245	50	19	and	and	CCONJ
ejpam-1245	50	20	y	y	PROPN
ejpam-1245	50	21	are	be	AUX
ejpam-1245	50	22	topological	topological	ADJ
ejpam-1245	50	23	space	space	NOUN
ejpam-1245	50	24	.	.	PUNCT
ejpam-1245	51	1	then	then	ADV
ejpam-1245	51	2	the	the	DET
ejpam-1245	51	3	following	follow	VERB
ejpam-1245	51	4	are	be	AUX
ejpam-1245	51	5	equivalent	equivalent	ADJ
ejpam-1245	51	6	:	:	PUNCT
ejpam-1245	51	7	i.	i.	NOUN
ejpam-1245	51	8	the	the	DET
ejpam-1245	51	9	function	function	NOUN
ejpam-1245	51	10	f	f	PROPN
ejpam-1245	51	11	is	be	AUX
ejpam-1245	51	12	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	51	13	.	.	PUNCT
ejpam-1245	51	14	ii	ii	PROPN
ejpam-1245	51	15	.	.	PUNCT
ejpam-1245	52	1	for	for	ADP
ejpam-1245	52	2	each	each	DET
ejpam-1245	52	3	open	open	ADJ
ejpam-1245	52	4	set	set	VERB
ejpam-1245	52	5	v	v	ADP
ejpam-1245	52	6	⊂	⊂	PROPN
ejpam-1245	52	7	y	y	PROPN
ejpam-1245	52	8	,	,	PUNCT
ejpam-1245	52	9	f	f	PROPN
ejpam-1245	52	10	−1(v	−1(v	PROPN
ejpam-1245	52	11	)	)	PUNCT
ejpam-1245	52	12	is	be	AUX
ejpam-1245	52	13	ωβo(x	ωβo(x	PROPN
ejpam-1245	52	14	,	,	PUNCT
ejpam-1245	52	15	τ	τ	PROPN
ejpam-1245	52	16	)	)	PUNCT
ejpam-1245	52	17	.	.	PUNCT
ejpam-1245	53	1	iii	iii	X
ejpam-1245	53	2	.	.	PUNCT
ejpam-1245	54	1	for	for	ADP
ejpam-1245	54	2	each	each	DET
ejpam-1245	54	3	x	x	SYM
ejpam-1245	54	4	∈	∈	PROPN
ejpam-1245	54	5	x	x	X
ejpam-1245	54	6	,	,	PUNCT
ejpam-1245	54	7	the	the	DET
ejpam-1245	54	8	inverse	inverse	NOUN
ejpam-1245	54	9	of	of	ADP
ejpam-1245	54	10	every	every	DET
ejpam-1245	54	11	neighborhood	neighborhood	NOUN
ejpam-1245	54	12	of	of	ADP
ejpam-1245	54	13	f	f	PROPN
ejpam-1245	54	14	(	(	PUNCT
ejpam-1245	54	15	x	x	X
ejpam-1245	54	16	)	)	PUNCT
ejpam-1245	54	17	is	be	AUX
ejpam-1245	54	18	an	an	DET
ejpam-1245	54	19	ωβ−neighborhood	ωβ−neighborhood	NOUN
ejpam-1245	54	20	of	of	ADP
ejpam-1245	54	21	x.	x.	PROPN
ejpam-1245	54	22	iv	iv	PROPN
ejpam-1245	54	23	.	.	PUNCT
ejpam-1245	55	1	for	for	ADP
ejpam-1245	55	2	each	each	DET
ejpam-1245	55	3	x	x	SYM
ejpam-1245	55	4	∈	∈	PROPN
ejpam-1245	55	5	x	x	X
ejpam-1245	55	6	and	and	CCONJ
ejpam-1245	55	7	each	each	DET
ejpam-1245	55	8	neighborhood	neighborhood	NOUN
ejpam-1245	55	9	nx	nx	X
ejpam-1245	55	10	of	of	ADP
ejpam-1245	55	11	f	f	PROPN
ejpam-1245	55	12	(	(	PUNCT
ejpam-1245	55	13	x	x	NOUN
ejpam-1245	55	14	)	)	PUNCT
ejpam-1245	55	15	,	,	PUNCT
ejpam-1245	55	16	there	there	PRON
ejpam-1245	55	17	is	be	VERB
ejpam-1245	55	18	an	an	DET
ejpam-1245	55	19	ωβ−neighborhood	ωβ−neighborhood	NOUN
ejpam-1245	55	20	v	v	NOUN
ejpam-1245	55	21	of	of	ADP
ejpam-1245	55	22	x	x	PUNCT
ejpam-1245	55	23	such	such	ADJ
ejpam-1245	55	24	that	that	SCONJ
ejpam-1245	55	25	f	f	PROPN
ejpam-1245	55	26	(	(	PUNCT
ejpam-1245	55	27	u)⊆	u)⊆	PROPN
ejpam-1245	55	28	nx	nx	X
ejpam-1245	55	29	.	.	PUNCT
ejpam-1245	56	1	v.	v.	ADP
ejpam-1245	56	2	for	for	ADP
ejpam-1245	56	3	each	each	DET
ejpam-1245	56	4	closed	close	VERB
ejpam-1245	56	5	set	set	VERB
ejpam-1245	56	6	m	m	PROPN
ejpam-1245	56	7	⊂	⊂	PROPN
ejpam-1245	56	8	y	y	PROPN
ejpam-1245	56	9	,	,	PUNCT
ejpam-1245	56	10	f	f	PROPN
ejpam-1245	56	11	−1(m	−1(m	PROPN
ejpam-1245	56	12	)	)	PUNCT
ejpam-1245	56	13	is	be	AUX
ejpam-1245	56	14	ωβ−closed	ωβ−close	VERB
ejpam-1245	56	15	in	in	ADP
ejpam-1245	56	16	x	x	X
ejpam-1245	56	17	.	.	PUNCT
ejpam-1245	57	1	vi	vi	X
ejpam-1245	57	2	.	.	NOUN
ejpam-1245	57	3	for	for	ADP
ejpam-1245	57	4	each	each	DET
ejpam-1245	57	5	subset	subset	VERB
ejpam-1245	57	6	a⊂	a⊂	NOUN
ejpam-1245	57	7	x	x	X
ejpam-1245	57	8	,	,	PUNCT
ejpam-1245	57	9	f	f	PROPN
ejpam-1245	57	10	(	(	PUNCT
ejpam-1245	57	11	ωβ	ωβ	NOUN
ejpam-1245	57	12	cl(a	cl(a	NUM
ejpam-1245	57	13	)	)	PUNCT
ejpam-1245	57	14	)	)	PUNCT
ejpam-1245	58	1	⊂	⊂	PROPN
ejpam-1245	58	2	cl	cl	PROPN
ejpam-1245	58	3	(	(	PUNCT
ejpam-1245	58	4	f	f	PROPN
ejpam-1245	58	5	(	(	PUNCT
ejpam-1245	58	6	a	a	NOUN
ejpam-1245	58	7	)	)	PUNCT
ejpam-1245	58	8	)	)	PUNCT
ejpam-1245	58	9	.	.	PUNCT
ejpam-1245	59	1	vii	vii	PROPN
ejpam-1245	59	2	.	.	PROPN
ejpam-1245	60	1	for	for	ADP
ejpam-1245	60	2	each	each	DET
ejpam-1245	60	3	subset	subset	NOUN
ejpam-1245	60	4	b	b	PROPN
ejpam-1245	60	5	⊂	⊂	PROPN
ejpam-1245	60	6	y	y	PROPN
ejpam-1245	60	7	,	,	PUNCT
ejpam-1245	60	8	ωβ	ωβ	X
ejpam-1245	60	9	cl	cl	NOUN
ejpam-1245	60	10	(	(	PUNCT
ejpam-1245	60	11	f	f	NOUN
ejpam-1245	60	12	−1(b))⊆	−1(b))⊆	PROPN
ejpam-1245	60	13	(	(	PUNCT
ejpam-1245	60	14	f	f	PROPN
ejpam-1245	60	15	−1(cl(b	−1(cl(b	PROPN
ejpam-1245	60	16	)	)	PUNCT
ejpam-1245	60	17	)	)	PUNCT
ejpam-1245	60	18	)	)	PUNCT
ejpam-1245	60	19	.	.	PUNCT
ejpam-1245	61	1	h.	h.	PROPN
ejpam-1245	61	2	aljarrah	aljarrah	PROPN
ejpam-1245	61	3	,	,	PUNCT
ejpam-1245	61	4	m.	m.	NOUN
ejpam-1245	61	5	noorani	noorani	PROPN
ejpam-1245	61	6	/	/	SYM
ejpam-1245	61	7	eur	eur	PROPN
ejpam-1245	61	8	.	.	PUNCT
ejpam-1245	62	1	j.	j.	PROPN
ejpam-1245	62	2	pure	pure	PROPN
ejpam-1245	62	3	appl	appl	PROPN
ejpam-1245	62	4	.	.	PROPN
ejpam-1245	62	5	math	math	PROPN
ejpam-1245	62	6	,	,	PUNCT
ejpam-1245	62	7	5	5	NUM
ejpam-1245	62	8	(	(	PUNCT
ejpam-1245	62	9	2012	2012	NUM
ejpam-1245	62	10	)	)	PUNCT
ejpam-1245	62	11	,	,	PUNCT
ejpam-1245	62	12	129	129	NUM
ejpam-1245	62	13	-	-	SYM
ejpam-1245	62	14	140	140	NUM
ejpam-1245	62	15	131	131	NUM
ejpam-1245	62	16	proof	proof	NOUN
ejpam-1245	62	17	.	.	PUNCT
ejpam-1245	63	1	(	(	PUNCT
ejpam-1245	63	2	i	i	PROPN
ejpam-1245	63	3	→	→	SYM
ejpam-1245	63	4	ii	ii	PROPN
ejpam-1245	63	5	)	)	PUNCT
ejpam-1245	63	6	let	let	VERB
ejpam-1245	63	7	v	v	PART
ejpam-1245	63	8	be	be	AUX
ejpam-1245	63	9	open	open	ADJ
ejpam-1245	63	10	in	in	ADP
ejpam-1245	63	11	y	y	PROPN
ejpam-1245	63	12	and	and	CCONJ
ejpam-1245	64	1	x	x	SYM
ejpam-1245	64	2	∈	∈	PROPN
ejpam-1245	64	3	f	f	PROPN
ejpam-1245	64	4	−1(v	−1(v	PROPN
ejpam-1245	64	5	)	)	PUNCT
ejpam-1245	65	1	then	then	ADV
ejpam-1245	65	2	f	f	X
ejpam-1245	65	3	(	(	PUNCT
ejpam-1245	65	4	x	x	X
ejpam-1245	65	5	)	)	PUNCT
ejpam-1245	65	6	∈	∈	NOUN
ejpam-1245	65	7	v	v	NOUN
ejpam-1245	65	8	,	,	PUNCT
ejpam-1245	65	9	by	by	ADP
ejpam-1245	65	10	(	(	PUNCT
ejpam-1245	65	11	i	i	NOUN
ejpam-1245	65	12	)	)	PUNCT
ejpam-1245	65	13	,	,	PUNCT
ejpam-1245	65	14	there	there	PRON
ejpam-1245	65	15	exists	exist	VERB
ejpam-1245	65	16	an	an	DET
ejpam-1245	65	17	ωβo(x	ωβo(x	PROPN
ejpam-1245	65	18	,	,	PUNCT
ejpam-1245	65	19	τ	τ	PROPN
ejpam-1245	65	20	)	)	PUNCT
ejpam-1245	65	21	set	set	VERB
ejpam-1245	65	22	ux	ux	INTJ
ejpam-1245	65	23	in	in	ADP
ejpam-1245	65	24	x	x	PUNCT
ejpam-1245	65	25	containing	contain	VERB
ejpam-1245	65	26	x	x	PROPN
ejpam-1245	65	27	and	and	CCONJ
ejpam-1245	65	28	f	f	PROPN
ejpam-1245	65	29	(	(	PUNCT
ejpam-1245	65	30	ux	ux	PROPN
ejpam-1245	65	31	)	)	PUNCT
ejpam-1245	65	32	⊆	⊆	NUM
ejpam-1245	65	33	v	v	NOUN
ejpam-1245	65	34	.	.	PUNCT
ejpam-1245	66	1	then	then	ADV
ejpam-1245	66	2	x	x	SYM
ejpam-1245	66	3	∈	∈	PROPN
ejpam-1245	66	4	ux	ux	NOUN
ejpam-1245	66	5	⊆	⊆	NUM
ejpam-1245	66	6	f	f	PROPN
ejpam-1245	66	7	−1(v	−1(v	PROPN
ejpam-1245	66	8	)	)	PUNCT
ejpam-1245	66	9	and	and	CCONJ
ejpam-1245	66	10	hence	hence	ADV
ejpam-1245	66	11	f	f	PROPN
ejpam-1245	66	12	−1(v	−1(v	PROPN
ejpam-1245	66	13	)	)	PUNCT
ejpam-1245	67	1	=	=	PUNCT
ejpam-1245	67	2	∪	∪	ADP
ejpam-1245	67	3	x∈	x∈	PROPN
ejpam-1245	67	4	f	f	PROPN
ejpam-1245	67	5	−1(v	−1(v	PROPN
ejpam-1245	67	6	)	)	PUNCT
ejpam-1245	67	7	ux	ux	INTJ
ejpam-1245	67	8	.	.	PUNCT
ejpam-1245	68	1	by	by	ADP
ejpam-1245	68	2	lemma	lemma	PROPN
ejpam-1245	68	3	1(i	1(i	NUM
ejpam-1245	68	4	)	)	PUNCT
ejpam-1245	68	5	,	,	PUNCT
ejpam-1245	68	6	f	f	PROPN
ejpam-1245	68	7	−1(v	−1(v	PROPN
ejpam-1245	68	8	)	)	PUNCT
ejpam-1245	68	9	∈	∈	PROPN
ejpam-1245	68	10	ωβo(x	ωβo(x	PROPN
ejpam-1245	68	11	,	,	PUNCT
ejpam-1245	68	12	τ	τ	PROPN
ejpam-1245	68	13	)	)	PUNCT
ejpam-1245	68	14	,	,	PUNCT
ejpam-1245	68	15	which	which	PRON
ejpam-1245	68	16	implies	imply	VERB
ejpam-1245	68	17	that	that	SCONJ
ejpam-1245	68	18	f	f	PROPN
ejpam-1245	68	19	is	be	AUX
ejpam-1245	68	20	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	68	21	.	.	PUNCT
ejpam-1245	69	1	(	(	PUNCT
ejpam-1245	69	2	ii→	ii→	PROPN
ejpam-1245	69	3	iii	iii	NOUN
ejpam-1245	69	4	)	)	PUNCT
ejpam-1245	69	5	for	for	ADP
ejpam-1245	69	6	x	x	SYM
ejpam-1245	69	7	∈	∈	PROPN
ejpam-1245	69	8	x	x	PUNCT
ejpam-1245	69	9	,	,	PUNCT
ejpam-1245	69	10	let	let	VERB
ejpam-1245	69	11	v	v	PART
ejpam-1245	69	12	be	be	AUX
ejpam-1245	69	13	the	the	DET
ejpam-1245	69	14	neighborhood	neighborhood	NOUN
ejpam-1245	69	15	of	of	ADP
ejpam-1245	69	16	f	f	PROPN
ejpam-1245	69	17	(	(	PUNCT
ejpam-1245	69	18	x	x	X
ejpam-1245	69	19	)	)	PUNCT
ejpam-1245	69	20	then	then	ADV
ejpam-1245	69	21	f	f	X
ejpam-1245	69	22	(	(	PUNCT
ejpam-1245	69	23	x	x	X
ejpam-1245	69	24	)	)	PUNCT
ejpam-1245	69	25	∈w	∈w	VERB
ejpam-1245	69	26	⊆	⊆	NUM
ejpam-1245	69	27	v	v	NOUN
ejpam-1245	69	28	,	,	PUNCT
ejpam-1245	69	29	where	where	SCONJ
ejpam-1245	69	30	w	w	NOUN
ejpam-1245	69	31	is	be	AUX
ejpam-1245	69	32	open	open	ADJ
ejpam-1245	69	33	in	in	ADP
ejpam-1245	69	34	y	y	PROPN
ejpam-1245	69	35	.	.	PUNCT
ejpam-1245	70	1	by	by	ADP
ejpam-1245	70	2	(	(	PUNCT
ejpam-1245	70	3	ii	ii	NOUN
ejpam-1245	70	4	)	)	PUNCT
ejpam-1245	70	5	,	,	PUNCT
ejpam-1245	70	6	f	f	PROPN
ejpam-1245	70	7	−1(w	−1(w	ADV
ejpam-1245	70	8	)	)	PUNCT
ejpam-1245	70	9	∈ωβo(x	∈ωβo(x	PROPN
ejpam-1245	70	10	,	,	PUNCT
ejpam-1245	70	11	τ	τ	PROPN
ejpam-1245	70	12	)	)	PUNCT
ejpam-1245	70	13	,	,	PUNCT
ejpam-1245	70	14	and	and	CCONJ
ejpam-1245	70	15	x	x	X
ejpam-1245	70	16	∈	∈	PROPN
ejpam-1245	70	17	f	f	X
ejpam-1245	70	18	−1(w	−1(w	ADV
ejpam-1245	70	19	)	)	PUNCT
ejpam-1245	70	20	⊆	⊆	NUM
ejpam-1245	70	21	f	f	PROPN
ejpam-1245	70	22	−1(v	−1(v	NOUN
ejpam-1245	70	23	)	)	PUNCT
ejpam-1245	70	24	.	.	PUNCT
ejpam-1245	71	1	then	then	ADV
ejpam-1245	71	2	by	by	ADP
ejpam-1245	71	3	definition	definition	NOUN
ejpam-1245	71	4	3	3	NUM
ejpam-1245	71	5	,	,	PUNCT
ejpam-1245	71	6	f	f	PROPN
ejpam-1245	71	7	−1(v	−1(v	PROPN
ejpam-1245	71	8	)	)	PUNCT
ejpam-1245	71	9	is	be	AUX
ejpam-1245	71	10	ωβ−neighborhood	ωβ−neighborhood	X
ejpam-1245	71	11	of	of	ADP
ejpam-1245	71	12	x	x	X
ejpam-1245	71	13	.	.	PUNCT
ejpam-1245	72	1	(	(	PUNCT
ejpam-1245	72	2	iii→	iii→	NOUN
ejpam-1245	72	3	iv	iv	NUM
ejpam-1245	72	4	)	)	PUNCT
ejpam-1245	72	5	for	for	ADP
ejpam-1245	72	6	x	x	SYM
ejpam-1245	72	7	∈	∈	PROPN
ejpam-1245	72	8	x	x	X
ejpam-1245	72	9	and	and	CCONJ
ejpam-1245	72	10	nx	nx	PROPN
ejpam-1245	72	11	be	be	AUX
ejpam-1245	72	12	a	a	DET
ejpam-1245	72	13	neighborhood	neighborhood	NOUN
ejpam-1245	72	14	of	of	ADP
ejpam-1245	72	15	f	f	PROPN
ejpam-1245	72	16	(	(	PUNCT
ejpam-1245	72	17	x	x	NOUN
ejpam-1245	72	18	)	)	PUNCT
ejpam-1245	72	19	.	.	PUNCT
ejpam-1245	73	1	then	then	ADV
ejpam-1245	73	2	v	v	X
ejpam-1245	73	3	=	=	SYM
ejpam-1245	73	4	f	f	PROPN
ejpam-1245	73	5	−1(nx	−1(nx	NOUN
ejpam-1245	73	6	)	)	PUNCT
ejpam-1245	73	7	is	be	AUX
ejpam-1245	73	8	anωβ−neighborhood	anωβ−neighborhood	NOUN
ejpam-1245	73	9	of	of	ADP
ejpam-1245	73	10	x	x	X
ejpam-1245	73	11	and	and	CCONJ
ejpam-1245	73	12	f	f	PROPN
ejpam-1245	73	13	(	(	PUNCT
ejpam-1245	73	14	v	v	NOUN
ejpam-1245	73	15	)	)	PUNCT
ejpam-1245	74	1	=	=	SYM
ejpam-1245	74	2	f	f	X
ejpam-1245	74	3	(	(	PUNCT
ejpam-1245	74	4	f	f	X
ejpam-1245	74	5	−1(nx))⊂	−1(nx))⊂	PRON
ejpam-1245	74	6	nx	nx	X
ejpam-1245	74	7	.	.	PUNCT
ejpam-1245	75	1	(	(	PUNCT
ejpam-1245	75	2	iv	iv	X
ejpam-1245	75	3	→	→	SYM
ejpam-1245	75	4	v	v	NOUN
ejpam-1245	75	5	)	)	PUNCT
ejpam-1245	75	6	for	for	ADP
ejpam-1245	75	7	any	any	DET
ejpam-1245	75	8	x	x	SYM
ejpam-1245	75	9	∈	∈	PROPN
ejpam-1245	75	10	x	x	X
ejpam-1245	75	11	−	−	PROPN
ejpam-1245	75	12	f	f	PROPN
ejpam-1245	75	13	−1(m	−1(m	PROPN
ejpam-1245	75	14	)	)	PUNCT
ejpam-1245	75	15	,	,	PUNCT
ejpam-1245	75	16	f	f	PROPN
ejpam-1245	75	17	(	(	PUNCT
ejpam-1245	75	18	x	x	X
ejpam-1245	75	19	)	)	PUNCT
ejpam-1245	75	20	∈	∈	PROPN
ejpam-1245	76	1	y	y	NOUN
ejpam-1245	77	1	−	−	NOUN
ejpam-1245	77	2	m	m	VERB
ejpam-1245	77	3	.	.	PUNCT
ejpam-1245	78	1	since	since	SCONJ
ejpam-1245	78	2	m	m	PROPN
ejpam-1245	78	3	is	be	AUX
ejpam-1245	78	4	closed	close	VERB
ejpam-1245	78	5	,	,	PUNCT
ejpam-1245	78	6	the	the	DET
ejpam-1245	78	7	set	set	NOUN
ejpam-1245	78	8	y	y	PROPN
ejpam-1245	78	9	−	−	PROPN
ejpam-1245	78	10	m	m	VERB
ejpam-1245	78	11	is	be	AUX
ejpam-1245	78	12	neighborhood	neighborhood	NOUN
ejpam-1245	78	13	of	of	ADP
ejpam-1245	78	14	f	f	PROPN
ejpam-1245	78	15	(	(	PUNCT
ejpam-1245	78	16	x	x	NOUN
ejpam-1245	78	17	)	)	PUNCT
ejpam-1245	78	18	,	,	PUNCT
ejpam-1245	78	19	hence	hence	ADV
ejpam-1245	78	20	there	there	PRON
ejpam-1245	78	21	is	be	VERB
ejpam-1245	78	22	a	a	DET
ejpam-1245	78	23	ωβ−neighborhood	ωβ−neighborhood	NOUN
ejpam-1245	78	24	v	v	NOUN
ejpam-1245	78	25	of	of	ADP
ejpam-1245	78	26	x	x	PUNCT
ejpam-1245	79	1	such	such	ADJ
ejpam-1245	79	2	that	that	SCONJ
ejpam-1245	79	3	f	f	PROPN
ejpam-1245	79	4	(	(	PUNCT
ejpam-1245	79	5	v	v	NOUN
ejpam-1245	79	6	)	)	PUNCT
ejpam-1245	79	7	⊂	⊂	PROPN
ejpam-1245	80	1	y	y	PROPN
ejpam-1245	80	2	−m	−m	PROPN
ejpam-1245	80	3	,	,	PUNCT
ejpam-1245	80	4	there	there	PRON
ejpam-1245	80	5	exists	exist	VERB
ejpam-1245	80	6	an	an	DET
ejpam-1245	80	7	ωβo(x	ωβo(x	PROPN
ejpam-1245	80	8	,	,	PUNCT
ejpam-1245	80	9	τ	τ	PROPN
ejpam-1245	80	10	)	)	PUNCT
ejpam-1245	80	11	set	set	VERB
ejpam-1245	80	12	ux	ux	INTJ
ejpam-1245	80	13	in	in	ADP
ejpam-1245	80	14	x	x	PUNCT
ejpam-1245	80	15	containing	contain	VERB
ejpam-1245	80	16	x	x	X
ejpam-1245	80	17	and	and	CCONJ
ejpam-1245	80	18	ux	ux	NUM
ejpam-1245	80	19	⊆	⊆	NUM
ejpam-1245	80	20	v	v	ADP
ejpam-1245	80	21	⊆	⊆	NUM
ejpam-1245	80	22	x	x	SYM
ejpam-1245	80	23	−	−	PROPN
ejpam-1245	80	24	f	f	PROPN
ejpam-1245	80	25	−1(m	−1(m	PROPN
ejpam-1245	80	26	)	)	PUNCT
ejpam-1245	80	27	,	,	PUNCT
ejpam-1245	80	28	take	take	VERB
ejpam-1245	80	29	(	(	PUNCT
ejpam-1245	80	30	x	x	NOUN
ejpam-1245	80	31	−	−	PROPN
ejpam-1245	80	32	f	f	PROPN
ejpam-1245	80	33	−1(m	−1(m	PROPN
ejpam-1245	80	34	)	)	PUNCT
ejpam-1245	80	35	)	)	PUNCT
ejpam-1245	81	1	=	=	PUNCT
ejpam-1245	81	2	∪	∪	ADP
ejpam-1245	81	3	x∈	x∈	PROPN
ejpam-1245	81	4	f	f	PROPN
ejpam-1245	81	5	−1(y−m	−1(y−m	PROPN
ejpam-1245	81	6	)	)	PUNCT
ejpam-1245	81	7	ux	ux	INTJ
ejpam-1245	81	8	.	.	PUNCT
ejpam-1245	82	1	by	by	ADP
ejpam-1245	82	2	lemma	lemma	PROPN
ejpam-1245	82	3	1(i	1(i	NUM
ejpam-1245	82	4	)	)	PUNCT
ejpam-1245	82	5	,	,	PUNCT
ejpam-1245	82	6	the	the	DET
ejpam-1245	82	7	set	set	NOUN
ejpam-1245	82	8	(	(	PUNCT
ejpam-1245	82	9	x	x	NOUN
ejpam-1245	82	10	−	−	PROPN
ejpam-1245	82	11	f	f	PROPN
ejpam-1245	82	12	−1(m	−1(m	PROPN
ejpam-1245	82	13	)	)	PUNCT
ejpam-1245	82	14	)	)	PUNCT
ejpam-1245	83	1	∈	∈	PROPN
ejpam-1245	83	2	ωβo(x	ωβo(x	PROPN
ejpam-1245	83	3	,	,	PUNCT
ejpam-1245	83	4	τ	τ	PROPN
ejpam-1245	83	5	)	)	PUNCT
ejpam-1245	83	6	,	,	PUNCT
ejpam-1245	83	7	which	which	PRON
ejpam-1245	83	8	implies	imply	VERB
ejpam-1245	83	9	f	f	PROPN
ejpam-1245	83	10	−1(m	−1(m	PROPN
ejpam-1245	83	11	)	)	PUNCT
ejpam-1245	83	12	is	be	AUX
ejpam-1245	83	13	ωβc(x	ωβc(x	PROPN
ejpam-1245	83	14	,	,	PUNCT
ejpam-1245	83	15	τ	τ	PROPN
ejpam-1245	83	16	)	)	PUNCT
ejpam-1245	83	17	.	.	PUNCT
ejpam-1245	84	1	(	(	PUNCT
ejpam-1245	84	2	v→	v→	ADP
ejpam-1245	84	3	vi	vi	NOUN
ejpam-1245	84	4	)	)	PUNCT
ejpam-1245	84	5	let	let	VERB
ejpam-1245	84	6	a⊆	a⊆	VERB
ejpam-1245	84	7	x	x	PRON
ejpam-1245	84	8	,	,	PUNCT
ejpam-1245	84	9	since	since	SCONJ
ejpam-1245	84	10	cl	cl	NOUN
ejpam-1245	84	11	(	(	PUNCT
ejpam-1245	84	12	f	f	X
ejpam-1245	84	13	(	(	PUNCT
ejpam-1245	84	14	a	a	NOUN
ejpam-1245	84	15	)	)	PUNCT
ejpam-1245	84	16	)	)	PUNCT
ejpam-1245	84	17	is	be	AUX
ejpam-1245	84	18	a	a	DET
ejpam-1245	84	19	closed	closed	ADJ
ejpam-1245	84	20	set	set	NOUN
ejpam-1245	84	21	in	in	ADP
ejpam-1245	84	22	y	y	PROPN
ejpam-1245	84	23	by	by	ADP
ejpam-1245	84	24	(	(	PUNCT
ejpam-1245	84	25	vi	vi	NOUN
ejpam-1245	84	26	)	)	PUNCT
ejpam-1245	84	27	,	,	PUNCT
ejpam-1245	84	28	f	f	PROPN
ejpam-1245	84	29	−1(cl	−1(cl	PROPN
ejpam-1245	84	30	(	(	PUNCT
ejpam-1245	84	31	f	f	PROPN
ejpam-1245	84	32	(	(	PUNCT
ejpam-1245	84	33	a	a	NOUN
ejpam-1245	84	34	)	)	PUNCT
ejpam-1245	84	35	)	)	PUNCT
ejpam-1245	84	36	)	)	PUNCT
ejpam-1245	84	37	is	be	AUX
ejpam-1245	84	38	an	an	DET
ejpam-1245	84	39	ωβc(x	ωβc(x	PROPN
ejpam-1245	84	40	,	,	PUNCT
ejpam-1245	84	41	τ	τ	PROPN
ejpam-1245	84	42	)	)	PUNCT
ejpam-1245	84	43	set	set	NOUN
ejpam-1245	84	44	containing	contain	VERB
ejpam-1245	84	45	a	a	PRON
ejpam-1245	84	46	,	,	PUNCT
ejpam-1245	84	47	then	then	ADV
ejpam-1245	84	48	f	f	X
ejpam-1245	84	49	(	(	PUNCT
ejpam-1245	84	50	ωβ	ωβ	NOUN
ejpam-1245	84	51	cl(a	cl(a	NUM
ejpam-1245	84	52	)	)	PUNCT
ejpam-1245	84	53	)	)	PUNCT
ejpam-1245	85	1	⊂	⊂	PROPN
ejpam-1245	85	2	cl	cl	PROPN
ejpam-1245	85	3	(	(	PUNCT
ejpam-1245	85	4	f	f	PROPN
ejpam-1245	85	5	(	(	PUNCT
ejpam-1245	85	6	a	a	NOUN
ejpam-1245	85	7	)	)	PUNCT
ejpam-1245	85	8	)	)	PUNCT
ejpam-1245	85	9	.	.	PUNCT
ejpam-1245	86	1	(	(	PUNCT
ejpam-1245	86	2	vi→	vi→	ADV
ejpam-1245	86	3	vii	vii	PROPN
ejpam-1245	86	4	)	)	PUNCT
ejpam-1245	86	5	let	let	VERB
ejpam-1245	86	6	b	b	PROPN
ejpam-1245	86	7	⊂	⊂	PROPN
ejpam-1245	86	8	y	y	PROPN
ejpam-1245	86	9	.	.	PUNCT
ejpam-1245	87	1	by	by	ADP
ejpam-1245	87	2	(	(	PUNCT
ejpam-1245	87	3	vi	vi	NOUN
ejpam-1245	87	4	)	)	PUNCT
ejpam-1245	87	5	,	,	PUNCT
ejpam-1245	87	6	f	f	PROPN
ejpam-1245	87	7	(	(	PUNCT
ejpam-1245	87	8	ωβ	ωβ	INTJ
ejpam-1245	87	9	cl	cl	NOUN
ejpam-1245	87	10	(	(	PUNCT
ejpam-1245	87	11	f	f	NOUN
ejpam-1245	87	12	−1(b)))⊆	−1(b)))⊆	X
ejpam-1245	87	13	cl(b	cl(b	PROPN
ejpam-1245	87	14	)	)	PUNCT
ejpam-1245	87	15	,	,	PUNCT
ejpam-1245	87	16	so	so	ADV
ejpam-1245	87	17	ωβ	ωβ	ADV
ejpam-1245	87	18	cl	cl	NOUN
ejpam-1245	87	19	(	(	PUNCT
ejpam-1245	87	20	f	f	PROPN
ejpam-1245	87	21	−1(b	−1(b	NOUN
ejpam-1245	87	22	)	)	PUNCT
ejpam-1245	87	23	)	)	PUNCT
ejpam-1245	88	1	⊆	⊆	NUM
ejpam-1245	88	2	f	f	PROPN
ejpam-1245	88	3	−1(cl(b	−1(cl(b	PROPN
ejpam-1245	88	4	)	)	PUNCT
ejpam-1245	88	5	)	)	PUNCT
ejpam-1245	88	6	.	.	PUNCT
ejpam-1245	89	1	(	(	PUNCT
ejpam-1245	89	2	vii	vii	PROPN
ejpam-1245	89	3	→	→	SYM
ejpam-1245	89	4	i	i	PROPN
ejpam-1245	89	5	)	)	PUNCT
ejpam-1245	89	6	suppose	suppose	VERB
ejpam-1245	89	7	on	on	ADP
ejpam-1245	89	8	the	the	DET
ejpam-1245	89	9	contrary	contrary	NOUN
ejpam-1245	89	10	that	that	SCONJ
ejpam-1245	89	11	f	f	PROPN
ejpam-1245	89	12	is	be	AUX
ejpam-1245	89	13	not	not	PART
ejpam-1245	89	14	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	89	15	.	.	PUNCT
ejpam-1245	90	1	so	so	ADV
ejpam-1245	90	2	there	there	PRON
ejpam-1245	90	3	exist	exist	VERB
ejpam-1245	90	4	x	x	X
ejpam-1245	90	5	∈	∈	PROPN
ejpam-1245	90	6	x	x	X
ejpam-1245	90	7	and	and	CCONJ
ejpam-1245	90	8	v	v	ADP
ejpam-1245	90	9	∈	∈	PROPN
ejpam-1245	90	10	σ	σ	NOUN
ejpam-1245	90	11	with	with	ADP
ejpam-1245	90	12	f	f	PROPN
ejpam-1245	90	13	(	(	PUNCT
ejpam-1245	90	14	x	x	X
ejpam-1245	90	15	)	)	PUNCT
ejpam-1245	90	16	∈	∈	NOUN
ejpam-1245	90	17	v	v	ADP
ejpam-1245	90	18	such	such	ADJ
ejpam-1245	90	19	that	that	PRON
ejpam-1245	90	20	for	for	ADP
ejpam-1245	90	21	all	all	DET
ejpam-1245	90	22	ωβo(x	ωβo(x	PROPN
ejpam-1245	90	23	,	,	PUNCT
ejpam-1245	90	24	τ	τ	X
ejpam-1245	90	25	)	)	PUNCT
ejpam-1245	90	26	sets	set	VERB
ejpam-1245	90	27	u	u	NOUN
ejpam-1245	90	28	with	with	ADP
ejpam-1245	90	29	x	x	PROPN
ejpam-1245	90	30	∈	∈	PROPN
ejpam-1245	90	31	u	u	NOUN
ejpam-1245	90	32	and	and	CCONJ
ejpam-1245	90	33	f	f	PROPN
ejpam-1245	90	34	(	(	PUNCT
ejpam-1245	90	35	u	u	NOUN
ejpam-1245	90	36	)	)	PUNCT
ejpam-1245	90	37	6⊂	6⊂	NUM
ejpam-1245	90	38	(	(	PUNCT
ejpam-1245	90	39	v	v	NOUN
ejpam-1245	90	40	)	)	PUNCT
ejpam-1245	90	41	i.e.	i.e.	X
ejpam-1245	90	42	f	f	X
ejpam-1245	90	43	(	(	PUNCT
ejpam-1245	90	44	u	u	NOUN
ejpam-1245	90	45	)	)	PUNCT
ejpam-1245	90	46	∩	∩	NOUN
ejpam-1245	90	47	(	(	PUNCT
ejpam-1245	90	48	y	y	PROPN
ejpam-1245	90	49	−	−	PROPN
ejpam-1245	90	50	v	v	NOUN
ejpam-1245	90	51	)	)	PUNCT
ejpam-1245	90	52	6=	6=	ADP
ejpam-1245	91	1	φ	φ	PROPN
ejpam-1245	91	2	.	.	PUNCT
ejpam-1245	92	1	therefore	therefore	ADV
ejpam-1245	92	2	,	,	PUNCT
ejpam-1245	92	3	by	by	ADP
ejpam-1245	92	4	theorem	theorem	NOUN
ejpam-1245	92	5	2	2	NUM
ejpam-1245	92	6	,	,	PUNCT
ejpam-1245	92	7	x	x	SYM
ejpam-1245	92	8	∈	∈	ADJ
ejpam-1245	92	9	ωβ	ωβ	NOUN
ejpam-1245	92	10	cl	cl	NOUN
ejpam-1245	92	11	(	(	PUNCT
ejpam-1245	92	12	f	f	PROPN
ejpam-1245	92	13	−1(y	−1(y	PRON
ejpam-1245	92	14	−	−	PROPN
ejpam-1245	92	15	v	v	NOUN
ejpam-1245	92	16	)	)	PUNCT
ejpam-1245	92	17	)	)	PUNCT
ejpam-1245	92	18	and	and	CCONJ
ejpam-1245	92	19	so	so	ADV
ejpam-1245	92	20	by	by	ADP
ejpam-1245	92	21	(	(	PUNCT
ejpam-1245	92	22	vii	vii	PROPN
ejpam-1245	92	23	)	)	PUNCT
ejpam-1245	92	24	,	,	PUNCT
ejpam-1245	92	25	f	f	PROPN
ejpam-1245	92	26	(	(	PUNCT
ejpam-1245	92	27	x	x	X
ejpam-1245	92	28	)	)	PUNCT
ejpam-1245	92	29	∈	∈	PROPN
ejpam-1245	92	30	cl(y	cl(y	PUNCT
ejpam-1245	92	31	−	−	NOUN
ejpam-1245	92	32	v	v	NOUN
ejpam-1245	92	33	)	)	PUNCT
ejpam-1245	92	34	,	,	PUNCT
ejpam-1245	92	35	thus	thus	ADV
ejpam-1245	92	36	for	for	ADP
ejpam-1245	92	37	all	all	DET
ejpam-1245	92	38	open	open	ADJ
ejpam-1245	92	39	sets	set	NOUN
ejpam-1245	92	40	v	v	ADP
ejpam-1245	92	41	in	in	ADP
ejpam-1245	92	42	(	(	PUNCT
ejpam-1245	92	43	y	y	PROPN
ejpam-1245	92	44	,	,	PUNCT
ejpam-1245	92	45	σ	σ	PROPN
ejpam-1245	92	46	)	)	PUNCT
ejpam-1245	92	47	containing	contain	VERB
ejpam-1245	92	48	f	f	PROPN
ejpam-1245	92	49	(	(	PUNCT
ejpam-1245	92	50	x	x	X
ejpam-1245	92	51	)	)	PUNCT
ejpam-1245	92	52	,	,	PUNCT
ejpam-1245	92	53	the	the	DET
ejpam-1245	92	54	set	set	NOUN
ejpam-1245	92	55	v	v	ADP
ejpam-1245	92	56	∩	∩	NOUN
ejpam-1245	92	57	(	(	PUNCT
ejpam-1245	92	58	y	y	PROPN
ejpam-1245	92	59	−	−	PROPN
ejpam-1245	92	60	v	v	NOUN
ejpam-1245	92	61	)	)	PUNCT
ejpam-1245	92	62	6=	6=	ADP
ejpam-1245	93	1	φ	φ	PROPN
ejpam-1245	93	2	,	,	PUNCT
ejpam-1245	93	3	a	a	DET
ejpam-1245	93	4	contradiction	contradiction	NOUN
ejpam-1245	93	5	.	.	PUNCT
ejpam-1245	94	1	therefore	therefore	ADV
ejpam-1245	94	2	,	,	PUNCT
ejpam-1245	94	3	f	f	PROPN
ejpam-1245	94	4	is	be	AUX
ejpam-1245	94	5	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	94	6	.	.	PUNCT
ejpam-1245	95	1	definition	definition	NOUN
ejpam-1245	95	2	4	4	NUM
ejpam-1245	95	3	.	.	PUNCT
ejpam-1245	96	1	for	for	ADP
ejpam-1245	96	2	any	any	DET
ejpam-1245	96	3	subset	subset	NOUN
ejpam-1245	96	4	a	a	PRON
ejpam-1245	96	5	of	of	ADP
ejpam-1245	96	6	a	a	DET
ejpam-1245	96	7	topological	topological	ADJ
ejpam-1245	96	8	space	space	NOUN
ejpam-1245	96	9	(	(	PUNCT
ejpam-1245	96	10	x	x	X
ejpam-1245	96	11	,	,	PUNCT
ejpam-1245	96	12	τ	τ	PROPN
ejpam-1245	96	13	)	)	PUNCT
ejpam-1245	96	14	the	the	DET
ejpam-1245	96	15	frontier	frontier	NOUN
ejpam-1245	96	16	of	of	ADP
ejpam-1245	96	17	a	a	DET
ejpam-1245	96	18	,	,	PUNCT
ejpam-1245	96	19	denoted	denote	VERB
ejpam-1245	96	20	byωβfr(a	byωβfr(a	NOUN
ejpam-1245	96	21	)	)	PUNCT
ejpam-1245	96	22	,	,	PUNCT
ejpam-1245	96	23	is	be	AUX
ejpam-1245	96	24	define	define	VERB
ejpam-1245	96	25	as	as	ADP
ejpam-1245	96	26	ωβ	ωβ	PROPN
ejpam-1245	96	27	cl(a)∩ωβ	cl(a)∩ωβ	PROPN
ejpam-1245	96	28	cl(x	cl(x	NOUN
ejpam-1245	96	29	−	−	PROPN
ejpam-1245	96	30	a	a	X
ejpam-1245	96	31	)	)	PUNCT
ejpam-1245	96	32	.	.	PUNCT
ejpam-1245	97	1	theorem	theorem	NOUN
ejpam-1245	97	2	5	5	NUM
ejpam-1245	97	3	.	.	PUNCT
ejpam-1245	98	1	let	let	VERB
ejpam-1245	98	2	(	(	PUNCT
ejpam-1245	98	3	x	x	X
ejpam-1245	98	4	,	,	PUNCT
ejpam-1245	98	5	τ	τ	PROPN
ejpam-1245	98	6	)	)	PUNCT
ejpam-1245	98	7	,	,	PUNCT
ejpam-1245	98	8	(	(	PUNCT
ejpam-1245	98	9	y	y	PROPN
ejpam-1245	98	10	,	,	PUNCT
ejpam-1245	98	11	σ	σ	PROPN
ejpam-1245	98	12	)	)	PUNCT
ejpam-1245	98	13	be	be	VERB
ejpam-1245	98	14	a	a	DET
ejpam-1245	98	15	topological	topological	ADJ
ejpam-1245	98	16	space	space	NOUN
ejpam-1245	98	17	and	and	CCONJ
ejpam-1245	98	18	f	f	NOUN
ejpam-1245	98	19	:	:	PUNCT
ejpam-1245	98	20	(	(	PUNCT
ejpam-1245	98	21	x	x	X
ejpam-1245	98	22	,	,	PUNCT
ejpam-1245	98	23	τ	τ	PROPN
ejpam-1245	98	24	)	)	PUNCT
ejpam-1245	98	25	→	→	SYM
ejpam-1245	98	26	(	(	PUNCT
ejpam-1245	98	27	y	y	PROPN
ejpam-1245	98	28	,	,	PUNCT
ejpam-1245	98	29	σ	σ	PROPN
ejpam-1245	98	30	)	)	PUNCT
ejpam-1245	98	31	be	be	AUX
ejpam-1245	98	32	a	a	DET
ejpam-1245	98	33	function	function	NOUN
ejpam-1245	98	34	.	.	PUNCT
ejpam-1245	99	1	then	then	ADV
ejpam-1245	99	2	x	x	X
ejpam-1245	99	3	−ωβ	−ωβ	NOUN
ejpam-1245	99	4	c	c	PROPN
ejpam-1245	99	5	(	(	PUNCT
ejpam-1245	99	6	f	f	PROPN
ejpam-1245	99	7	)	)	PUNCT
ejpam-1245	99	8	=	=	SYM
ejpam-1245	99	9	∪{ωβfr	∪{ωβfr	NOUN
ejpam-1245	99	10	(	(	PUNCT
ejpam-1245	99	11	f	f	PROPN
ejpam-1245	99	12	−1(v	−1(v	PROPN
ejpam-1245	99	13	)	)	PUNCT
ejpam-1245	99	14	)	)	PUNCT
ejpam-1245	99	15	:	:	PUNCT
ejpam-1245	100	1	v	v	X
ejpam-1245	100	2	∈	∈	PROPN
ejpam-1245	100	3	σ	σ	PROPN
ejpam-1245	100	4	,	,	PUNCT
ejpam-1245	100	5	f	f	PROPN
ejpam-1245	100	6	(	(	PUNCT
ejpam-1245	100	7	x	x	X
ejpam-1245	100	8	)	)	PUNCT
ejpam-1245	100	9	∈	∈	PROPN
ejpam-1245	100	10	v	v	NOUN
ejpam-1245	100	11	,	,	PUNCT
ejpam-1245	100	12	x	x	X
ejpam-1245	100	13	∈	∈	NOUN
ejpam-1245	100	14	x	x	PUNCT
ejpam-1245	100	15	}	}	PUNCT
ejpam-1245	100	16	where	where	SCONJ
ejpam-1245	100	17	ωβ	ωβ	ADP
ejpam-1245	100	18	c	c	X
ejpam-1245	100	19	(	(	PUNCT
ejpam-1245	100	20	f	f	PROPN
ejpam-1245	100	21	)	)	PUNCT
ejpam-1245	100	22	denotes	denote	VERB
ejpam-1245	100	23	the	the	DET
ejpam-1245	100	24	set	set	NOUN
ejpam-1245	100	25	of	of	ADP
ejpam-1245	100	26	points	point	NOUN
ejpam-1245	100	27	at	at	ADP
ejpam-1245	100	28	which	which	PRON
ejpam-1245	100	29	f	f	PROPN
ejpam-1245	100	30	is	be	AUX
ejpam-1245	100	31	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	100	32	.	.	PUNCT
ejpam-1245	101	1	proof	proof	NOUN
ejpam-1245	101	2	.	.	PUNCT
ejpam-1245	102	1	let	let	VERB
ejpam-1245	102	2	x	x	SYM
ejpam-1245	102	3	∈	∈	PROPN
ejpam-1245	102	4	x	x	X
ejpam-1245	102	5	−ωβ	−ωβ	NOUN
ejpam-1245	102	6	c	c	X
ejpam-1245	102	7	(	(	PUNCT
ejpam-1245	102	8	f	f	PROPN
ejpam-1245	102	9	)	)	PUNCT
ejpam-1245	102	10	.	.	PUNCT
ejpam-1245	103	1	then	then	ADV
ejpam-1245	103	2	for	for	ADP
ejpam-1245	103	3	every	every	DET
ejpam-1245	103	4	ωβo(x	ωβo(x	PROPN
ejpam-1245	103	5	,	,	PUNCT
ejpam-1245	103	6	τ	τ	X
ejpam-1245	103	7	)	)	PUNCT
ejpam-1245	103	8	set	set	VERB
ejpam-1245	103	9	u	u	NOUN
ejpam-1245	103	10	containing	contain	VERB
ejpam-1245	103	11	x	x	PUNCT
ejpam-1245	103	12	there	there	PRON
ejpam-1245	103	13	exists	exist	VERB
ejpam-1245	103	14	open	open	ADJ
ejpam-1245	103	15	sets	set	NOUN
ejpam-1245	103	16	v	v	ADP
ejpam-1245	103	17	in	in	ADP
ejpam-1245	103	18	(	(	PUNCT
ejpam-1245	103	19	y	y	PROPN
ejpam-1245	103	20	,	,	PUNCT
ejpam-1245	103	21	σ	σ	PROPN
ejpam-1245	103	22	)	)	PUNCT
ejpam-1245	103	23	containing	contain	VERB
ejpam-1245	103	24	f	f	PROPN
ejpam-1245	103	25	(	(	PUNCT
ejpam-1245	103	26	x	x	X
ejpam-1245	103	27	)	)	PUNCT
ejpam-1245	103	28	such	such	ADJ
ejpam-1245	103	29	f	f	PROPN
ejpam-1245	103	30	(	(	PUNCT
ejpam-1245	103	31	u	u	NOUN
ejpam-1245	103	32	)	)	PUNCT
ejpam-1245	103	33	6⊂	6⊂	NUM
ejpam-1245	103	34	v	v	NOUN
ejpam-1245	103	35	,	,	PUNCT
ejpam-1245	103	36	hence	hence	ADV
ejpam-1245	103	37	u	u	NOUN
ejpam-1245	103	38	∩	∩	NOUN
ejpam-1245	103	39	(	(	PUNCT
ejpam-1245	103	40	x	x	SYM
ejpam-1245	103	41	−	−	PROPN
ejpam-1245	103	42	f	f	PROPN
ejpam-1245	103	43	−1(v	−1(v	PROPN
ejpam-1245	103	44	)	)	PUNCT
ejpam-1245	103	45	)	)	PUNCT
ejpam-1245	104	1	6=	6=	ADP
ejpam-1245	104	2	φ	φ	PROPN
ejpam-1245	104	3	for	for	ADP
ejpam-1245	104	4	every	every	DET
ejpam-1245	104	5	ωβo(x	ωβo(x	PROPN
ejpam-1245	104	6	,	,	PUNCT
ejpam-1245	104	7	τ	τ	X
ejpam-1245	104	8	)	)	PUNCT
ejpam-1245	104	9	set	set	VERB
ejpam-1245	104	10	u	u	NOUN
ejpam-1245	104	11	containing	contain	VERB
ejpam-1245	104	12	x	x	X
ejpam-1245	104	13	.	.	PUNCT
ejpam-1245	105	1	therefore	therefore	ADV
ejpam-1245	105	2	,	,	PUNCT
ejpam-1245	105	3	by	by	ADP
ejpam-1245	105	4	theorem	theorem	NOUN
ejpam-1245	105	5	2	2	NUM
ejpam-1245	105	6	x	x	SYM
ejpam-1245	105	7	∈ωβ	∈ωβ	NOUN
ejpam-1245	105	8	cl(x	cl(x	X
ejpam-1245	105	9	−	−	PROPN
ejpam-1245	105	10	f	f	PROPN
ejpam-1245	105	11	−1(v	−1(v	PROPN
ejpam-1245	105	12	)	)	PUNCT
ejpam-1245	105	13	)	)	PUNCT
ejpam-1245	105	14	.	.	PUNCT
ejpam-1245	106	1	then	then	ADV
ejpam-1245	106	2	x	x	SYM
ejpam-1245	106	3	∈	∈	PROPN
ejpam-1245	106	4	f	f	X
ejpam-1245	106	5	−1(v	−1(v	NOUN
ejpam-1245	106	6	)	)	PUNCT
ejpam-1245	106	7	∩ωβ	∩ωβ	PROPN
ejpam-1245	106	8	cl(x	cl(x	PUNCT
ejpam-1245	106	9	−	−	PROPN
ejpam-1245	106	10	f	f	PROPN
ejpam-1245	106	11	−1(v	−1(v	PROPN
ejpam-1245	106	12	)	)	PUNCT
ejpam-1245	106	13	)	)	PUNCT
ejpam-1245	107	1	⊆ωβfr	⊆ωβfr	PROPN
ejpam-1245	107	2	(	(	PUNCT
ejpam-1245	107	3	f	f	PROPN
ejpam-1245	107	4	−1(v	−1(v	PROPN
ejpam-1245	107	5	)	)	PUNCT
ejpam-1245	107	6	)	)	PUNCT
ejpam-1245	107	7	.	.	PUNCT
ejpam-1245	108	1	hence	hence	ADV
ejpam-1245	108	2	,	,	PUNCT
ejpam-1245	108	3	x	x	X
ejpam-1245	108	4	−ωβ	−ωβ	X
ejpam-1245	108	5	c	c	X
ejpam-1245	108	6	(	(	PUNCT
ejpam-1245	108	7	f	f	PROPN
ejpam-1245	108	8	)	)	PUNCT
ejpam-1245	108	9	⊆	⊆	NUM
ejpam-1245	108	10	∪{ωβfr	∪{ωβfr	SYM
ejpam-1245	108	11	(	(	PUNCT
ejpam-1245	108	12	f	f	PROPN
ejpam-1245	108	13	−1(v	−1(v	PROPN
ejpam-1245	108	14	)	)	PUNCT
ejpam-1245	108	15	)	)	PUNCT
ejpam-1245	108	16	,	,	PUNCT
ejpam-1245	108	17	v	v	X
ejpam-1245	108	18	∈	∈	PROPN
ejpam-1245	108	19	σ	σ	PROPN
ejpam-1245	108	20	,	,	PUNCT
ejpam-1245	108	21	f	f	PROPN
ejpam-1245	108	22	(	(	PUNCT
ejpam-1245	108	23	x	x	X
ejpam-1245	108	24	)	)	PUNCT
ejpam-1245	108	25	∈	∈	PROPN
ejpam-1245	108	26	v	v	NOUN
ejpam-1245	108	27	,	,	PUNCT
ejpam-1245	108	28	x	x	X
ejpam-1245	108	29	∈	∈	NOUN
ejpam-1245	108	30	x	x	PUNCT
ejpam-1245	108	31	}	}	PUNCT
ejpam-1245	108	32	.	.	PUNCT
ejpam-1245	109	1	conversely	conversely	ADV
ejpam-1245	109	2	,	,	PUNCT
ejpam-1245	109	3	let	let	VERB
ejpam-1245	109	4	x	x	PUNCT
ejpam-1245	109	5	/∈	/∈	PUNCT
ejpam-1245	110	1	x	x	X
ejpam-1245	111	1	−ωβ	−ωβ	NOUN
ejpam-1245	112	1	c	c	X
ejpam-1245	113	1	(	(	PUNCT
ejpam-1245	113	2	f	f	PROPN
ejpam-1245	113	3	)	)	PUNCT
ejpam-1245	113	4	.	.	PUNCT
ejpam-1245	114	1	then	then	ADV
ejpam-1245	114	2	for	for	ADP
ejpam-1245	114	3	each	each	DET
ejpam-1245	114	4	open	open	ADJ
ejpam-1245	114	5	sets	set	NOUN
ejpam-1245	114	6	v	v	ADP
ejpam-1245	114	7	in	in	ADP
ejpam-1245	114	8	(	(	PUNCT
ejpam-1245	114	9	y	y	PROPN
ejpam-1245	114	10	,	,	PUNCT
ejpam-1245	114	11	σ	σ	PROPN
ejpam-1245	114	12	)	)	PUNCT
ejpam-1245	114	13	containing	contain	VERB
ejpam-1245	114	14	f	f	PROPN
ejpam-1245	114	15	(	(	PUNCT
ejpam-1245	114	16	x	x	X
ejpam-1245	114	17	)	)	PUNCT
ejpam-1245	114	18	,	,	PUNCT
ejpam-1245	114	19	f	f	PROPN
ejpam-1245	114	20	−1(v	−1(v	PROPN
ejpam-1245	114	21	)	)	PUNCT
ejpam-1245	114	22	is	be	AUX
ejpam-1245	114	23	ωβo(x	ωβo(x	PROPN
ejpam-1245	114	24	,	,	PUNCT
ejpam-1245	114	25	τ	τ	X
ejpam-1245	114	26	)	)	PUNCT
ejpam-1245	114	27	containing	contain	VERB
ejpam-1245	114	28	x	x	SYM
ejpam-1245	114	29	,	,	PUNCT
ejpam-1245	114	30	thus	thus	ADV
ejpam-1245	114	31	for	for	ADP
ejpam-1245	114	32	every	every	DET
ejpam-1245	114	33	v	v	NOUN
ejpam-1245	114	34	∈	∈	PROPN
ejpam-1245	114	35	σ	σ	NOUN
ejpam-1245	114	36	containing	contain	VERB
ejpam-1245	114	37	f	f	PROPN
ejpam-1245	114	38	(	(	PUNCT
ejpam-1245	114	39	x	x	NOUN
ejpam-1245	114	40	)	)	PUNCT
ejpam-1245	114	41	,	,	PUNCT
ejpam-1245	114	42	x	x	PUNCT
ejpam-1245	114	43	∈	∈	NOUN
ejpam-1245	114	44	ωβ	ωβ	X
ejpam-1245	114	45	int	int	NOUN
ejpam-1245	114	46	(	(	PUNCT
ejpam-1245	114	47	f	f	PROPN
ejpam-1245	114	48	−1(v	−1(v	PROPN
ejpam-1245	114	49	)	)	PUNCT
ejpam-1245	114	50	)	)	PUNCT
ejpam-1245	114	51	and	and	CCONJ
ejpam-1245	115	1	hence	hence	ADV
ejpam-1245	115	2	x	x	PROPN
ejpam-1245	115	3	/∈	/∈	PUNCT
ejpam-1245	115	4	ωβfr	ωβfr	PROPN
ejpam-1245	115	5	(	(	PUNCT
ejpam-1245	115	6	f	f	PROPN
ejpam-1245	115	7	−1(v	−1(v	PROPN
ejpam-1245	115	8	)	)	PUNCT
ejpam-1245	115	9	)	)	PUNCT
ejpam-1245	115	10	.	.	PUNCT
ejpam-1245	116	1	so	so	ADV
ejpam-1245	116	2	∪{ωβfr	∪{ωβfr	VERB
ejpam-1245	116	3	(	(	PUNCT
ejpam-1245	116	4	f	f	PROPN
ejpam-1245	116	5	−1(v	−1(v	PROPN
ejpam-1245	116	6	)	)	PUNCT
ejpam-1245	116	7	)	)	PUNCT
ejpam-1245	116	8	:	:	PUNCT
ejpam-1245	116	9	v	v	X
ejpam-1245	116	10	∈	∈	PROPN
ejpam-1245	116	11	σ	σ	PROPN
ejpam-1245	116	12	,	,	PUNCT
ejpam-1245	116	13	f	f	PROPN
ejpam-1245	116	14	(	(	PUNCT
ejpam-1245	116	15	x	x	X
ejpam-1245	116	16	)	)	PUNCT
ejpam-1245	116	17	∈	∈	PROPN
ejpam-1245	116	18	v	v	NOUN
ejpam-1245	116	19	,	,	PUNCT
ejpam-1245	116	20	x	x	X
ejpam-1245	116	21	∈	∈	NOUN
ejpam-1245	116	22	x	x	SYM
ejpam-1245	116	23	}	}	PUNCT
ejpam-1245	116	24	⊆	⊆	NUM
ejpam-1245	116	25	x	x	SYM
ejpam-1245	116	26	−ωβ	−ωβ	NOUN
ejpam-1245	116	27	c	c	X
ejpam-1245	116	28	(	(	PUNCT
ejpam-1245	116	29	f	f	PROPN
ejpam-1245	116	30	)	)	PUNCT
ejpam-1245	116	31	.	.	PUNCT
ejpam-1245	117	1	corollary	corollary	ADJ
ejpam-1245	117	2	1	1	NUM
ejpam-1245	117	3	.	.	PUNCT
ejpam-1245	118	1	a	a	DET
ejpam-1245	118	2	function	function	NOUN
ejpam-1245	118	3	f	f	NOUN
ejpam-1245	118	4	:	:	PUNCT
ejpam-1245	118	5	(	(	PUNCT
ejpam-1245	118	6	x	x	X
ejpam-1245	118	7	,	,	PUNCT
ejpam-1245	118	8	τ)→	τ)→	PROPN
ejpam-1245	118	9	(	(	PUNCT
ejpam-1245	118	10	y	y	PROPN
ejpam-1245	118	11	,	,	PUNCT
ejpam-1245	118	12	σ	σ	PROPN
ejpam-1245	118	13	)	)	PUNCT
ejpam-1245	118	14	is	be	AUX
ejpam-1245	118	15	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	118	16	if	if	SCONJ
ejpam-1245	118	17	and	and	CCONJ
ejpam-1245	118	18	only	only	ADV
ejpam-1245	118	19	if	if	SCONJ
ejpam-1245	118	20	f	f	PROPN
ejpam-1245	118	21	−1(int(g))⊆ωβ	−1(int(g))⊆ωβ	PROPN
ejpam-1245	118	22	int	int	NOUN
ejpam-1245	118	23	(	(	PUNCT
ejpam-1245	118	24	f	f	NOUN
ejpam-1245	118	25	−1(g	−1(g	NOUN
ejpam-1245	118	26	)	)	PUNCT
ejpam-1245	118	27	)	)	PUNCT
ejpam-1245	118	28	,	,	PUNCT
ejpam-1245	118	29	for	for	ADP
ejpam-1245	118	30	any	any	DET
ejpam-1245	118	31	subset	subset	NOUN
ejpam-1245	118	32	g	g	PROPN
ejpam-1245	118	33	⊆	⊆	PROPN
ejpam-1245	118	34	y	y	PROPN
ejpam-1245	118	35	.	.	PUNCT
ejpam-1245	119	1	proof	proof	NOUN
ejpam-1245	119	2	.	.	PUNCT
ejpam-1245	120	1	necessity	necessity	NOUN
ejpam-1245	120	2	.	.	PUNCT
ejpam-1245	121	1	let	let	VERB
ejpam-1245	121	2	g	g	NOUN
ejpam-1245	121	3	be	be	AUX
ejpam-1245	121	4	any	any	DET
ejpam-1245	121	5	subset	subset	NOUN
ejpam-1245	121	6	of	of	ADP
ejpam-1245	121	7	y	y	PROPN
ejpam-1245	121	8	.	.	PUNCT
ejpam-1245	122	1	since	since	SCONJ
ejpam-1245	122	2	f	f	PROPN
ejpam-1245	122	3	is	be	AUX
ejpam-1245	122	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	122	5	,	,	PUNCT
ejpam-1245	122	6	f	f	PROPN
ejpam-1245	122	7	−1(int(g	−1(int(g	PROPN
ejpam-1245	122	8	)	)	PUNCT
ejpam-1245	122	9	)	)	PUNCT
ejpam-1245	122	10	is	be	AUX
ejpam-1245	122	11	ωβo(x	ωβo(x	PROPN
ejpam-1245	122	12	,	,	PUNCT
ejpam-1245	122	13	τ	τ	NOUN
ejpam-1245	122	14	)	)	PUNCT
ejpam-1245	122	15	set	set	NOUN
ejpam-1245	122	16	.	.	PUNCT
ejpam-1245	123	1	as	as	ADP
ejpam-1245	123	2	f	f	PROPN
ejpam-1245	123	3	−1(int(g))⊆	−1(int(g))⊆	PROPN
ejpam-1245	123	4	f	f	PROPN
ejpam-1245	123	5	−1(g	−1(g	NOUN
ejpam-1245	123	6	)	)	PUNCT
ejpam-1245	123	7	,	,	PUNCT
ejpam-1245	123	8	then	then	ADV
ejpam-1245	123	9	f	f	PROPN
ejpam-1245	123	10	−1(int(g))⊆ωβ	−1(int(g))⊆ωβ	PROPN
ejpam-1245	123	11	int	int	NOUN
ejpam-1245	123	12	(	(	PUNCT
ejpam-1245	123	13	f	f	NOUN
ejpam-1245	123	14	−1(g	−1(g	NOUN
ejpam-1245	123	15	)	)	PUNCT
ejpam-1245	123	16	)	)	PUNCT
ejpam-1245	123	17	.	.	PUNCT
ejpam-1245	124	1	h.	h.	PROPN
ejpam-1245	124	2	aljarrah	aljarrah	PROPN
ejpam-1245	124	3	,	,	PUNCT
ejpam-1245	124	4	m.	m.	NOUN
ejpam-1245	124	5	noorani	noorani	PROPN
ejpam-1245	124	6	/	/	SYM
ejpam-1245	124	7	eur	eur	PROPN
ejpam-1245	124	8	.	.	PUNCT
ejpam-1245	125	1	j.	j.	PROPN
ejpam-1245	125	2	pure	pure	PROPN
ejpam-1245	125	3	appl	appl	PROPN
ejpam-1245	125	4	.	.	PROPN
ejpam-1245	125	5	math	math	PROPN
ejpam-1245	125	6	,	,	PUNCT
ejpam-1245	125	7	5	5	NUM
ejpam-1245	125	8	(	(	PUNCT
ejpam-1245	125	9	2012	2012	NUM
ejpam-1245	125	10	)	)	PUNCT
ejpam-1245	125	11	,	,	PUNCT
ejpam-1245	125	12	129	129	NUM
ejpam-1245	125	13	-	-	SYM
ejpam-1245	125	14	140	140	NUM
ejpam-1245	125	15	132	132	NUM
ejpam-1245	125	16	sufficiency	sufficiency	NOUN
ejpam-1245	125	17	.	.	PUNCT
ejpam-1245	126	1	let	let	VERB
ejpam-1245	126	2	x	x	PUNCT
ejpam-1245	126	3	∈	∈	PROPN
ejpam-1245	126	4	x	x	X
ejpam-1245	126	5	and	and	CCONJ
ejpam-1245	126	6	v	v	ADP
ejpam-1245	126	7	∈	∈	PROPN
ejpam-1245	126	8	σ	σ	NOUN
ejpam-1245	126	9	with	with	ADP
ejpam-1245	126	10	f	f	PROPN
ejpam-1245	126	11	(	(	PUNCT
ejpam-1245	126	12	x	x	X
ejpam-1245	126	13	)	)	PUNCT
ejpam-1245	126	14	∈	∈	NOUN
ejpam-1245	126	15	v	v	NOUN
ejpam-1245	126	16	.	.	PUNCT
ejpam-1245	127	1	then	then	ADV
ejpam-1245	127	2	x	x	SYM
ejpam-1245	127	3	∈	∈	PROPN
ejpam-1245	127	4	f	f	X
ejpam-1245	127	5	−1(v	−1(v	PROPN
ejpam-1245	127	6	)	)	PUNCT
ejpam-1245	127	7	and	and	CCONJ
ejpam-1245	127	8	so	so	ADV
ejpam-1245	127	9	by	by	ADP
ejpam-1245	127	10	assumption	assumption	NOUN
ejpam-1245	127	11	x	x	X
ejpam-1245	127	12	∈	∈	PROPN
ejpam-1245	127	13	ωβ	ωβ	X
ejpam-1245	127	14	int	int	NOUN
ejpam-1245	127	15	(	(	PUNCT
ejpam-1245	127	16	f	f	PROPN
ejpam-1245	127	17	−1(v	−1(v	PROPN
ejpam-1245	127	18	)	)	PUNCT
ejpam-1245	127	19	)	)	PUNCT
ejpam-1245	127	20	.	.	PUNCT
ejpam-1245	128	1	there	there	PRON
ejpam-1245	128	2	exists	exist	VERB
ejpam-1245	128	3	an	an	DET
ejpam-1245	128	4	ωβo(x	ωβo(x	PROPN
ejpam-1245	128	5	,	,	PUNCT
ejpam-1245	128	6	τ	τ	PROPN
ejpam-1245	128	7	)	)	PUNCT
ejpam-1245	128	8	such	such	ADJ
ejpam-1245	128	9	that	that	SCONJ
ejpam-1245	128	10	x	x	SYM
ejpam-1245	128	11	∈	∈	NOUN
ejpam-1245	128	12	u	u	NOUN
ejpam-1245	128	13	⊆	⊆	NUM
ejpam-1245	128	14	f	f	PROPN
ejpam-1245	128	15	−1(v	−1(v	NOUN
ejpam-1245	128	16	)	)	PUNCT
ejpam-1245	128	17	.	.	PUNCT
ejpam-1245	129	1	hence	hence	ADV
ejpam-1245	129	2	f	f	PROPN
ejpam-1245	129	3	(	(	PUNCT
ejpam-1245	129	4	x	x	X
ejpam-1245	129	5	)	)	PUNCT
ejpam-1245	129	6	∈	∈	PROPN
ejpam-1245	129	7	f	f	PROPN
ejpam-1245	129	8	(	(	PUNCT
ejpam-1245	129	9	u)⊆	u)⊆	PROPN
ejpam-1245	129	10	v	v	NOUN
ejpam-1245	129	11	and	and	CCONJ
ejpam-1245	129	12	the	the	DET
ejpam-1245	129	13	result	result	NOUN
ejpam-1245	129	14	follows	follow	VERB
ejpam-1245	129	15	.	.	PUNCT
ejpam-1245	130	1	note	note	VERB
ejpam-1245	130	2	that	that	SCONJ
ejpam-1245	130	3	if	if	SCONJ
ejpam-1245	130	4	x	x	PRON
ejpam-1245	130	5	is	be	AUX
ejpam-1245	130	6	a	a	DET
ejpam-1245	130	7	countable	countable	ADJ
ejpam-1245	130	8	set	set	NOUN
ejpam-1245	130	9	then	then	ADV
ejpam-1245	130	10	every	every	DET
ejpam-1245	130	11	function	function	NOUN
ejpam-1245	130	12	f	f	NOUN
ejpam-1245	130	13	:	:	PUNCT
ejpam-1245	130	14	(	(	PUNCT
ejpam-1245	130	15	x	x	X
ejpam-1245	130	16	,	,	PUNCT
ejpam-1245	130	17	τ)→	τ)→	PROPN
ejpam-1245	130	18	(	(	PUNCT
ejpam-1245	130	19	y	y	PROPN
ejpam-1245	130	20	,	,	PUNCT
ejpam-1245	130	21	σ	σ	PROPN
ejpam-1245	130	22	)	)	PUNCT
ejpam-1245	130	23	isωβ−continuous	isωβ−continuous	ADJ
ejpam-1245	130	24	.	.	PUNCT
ejpam-1245	131	1	the	the	DET
ejpam-1245	131	2	following	follow	VERB
ejpam-1245	131	3	diagram	diagram	NOUN
ejpam-1245	131	4	follows	follow	VERB
ejpam-1245	131	5	immediately	immediately	ADV
ejpam-1245	131	6	from	from	ADP
ejpam-1245	131	7	the	the	DET
ejpam-1245	131	8	definitions	definition	NOUN
ejpam-1245	131	9	in	in	ADP
ejpam-1245	131	10	which	which	PRON
ejpam-1245	131	11	none	none	NOUN
ejpam-1245	131	12	of	of	ADP
ejpam-1245	131	13	the	the	DET
ejpam-1245	131	14	implications	implication	NOUN
ejpam-1245	131	15	is	be	AUX
ejpam-1245	131	16	reversible	reversible	ADJ
ejpam-1245	131	17	.	.	PUNCT
ejpam-1245	132	1	continuous	continuous	ADJ
ejpam-1245	132	2	→	→	NOUN
ejpam-1245	132	3	b−continuous	b−continuous	ADJ
ejpam-1245	132	4	→	→	SYM
ejpam-1245	132	5	β−continuous	β−continuous	ADJ
ejpam-1245	132	6	↓	↓	PROPN
ejpam-1245	132	7	↓	↓	PROPN
ejpam-1245	132	8	↓	↓	PROPN
ejpam-1245	132	9	ω−continuous	ω−continuous	PART
ejpam-1245	132	10	→	→	SYM
ejpam-1245	132	11	ωb−continuous	ωb−continuous	ADJ
ejpam-1245	132	12	→	→	SYM
ejpam-1245	132	13	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	132	14	example	example	NOUN
ejpam-1245	132	15	1	1	X
ejpam-1245	132	16	.	.	PUNCT
ejpam-1245	133	1	let	let	VERB
ejpam-1245	133	2	x	x	PUNCT
ejpam-1245	133	3	=	=	PUNCT
ejpam-1245	133	4	r	r	NOUN
ejpam-1245	133	5	with	with	ADP
ejpam-1245	133	6	the	the	DET
ejpam-1245	133	7	topology	topology	NOUN
ejpam-1245	133	8	τ	τ	NOUN
ejpam-1245	133	9	=	=	PUNCT
ejpam-1245	133	10	τu	τu	ADP
ejpam-1245	133	11	and	and	CCONJ
ejpam-1245	133	12	y	y	PROPN
ejpam-1245	133	13	=	=	PUNCT
ejpam-1245	133	14	{	{	PUNCT
ejpam-1245	133	15	0,1	0,1	NOUN
ejpam-1245	133	16	}	}	PUNCT
ejpam-1245	133	17	with	with	ADP
ejpam-1245	133	18	the	the	DET
ejpam-1245	133	19	topology	topology	NOUN
ejpam-1245	133	20	σ	σ	NOUN
ejpam-1245	133	21	=	=	SYM
ejpam-1245	133	22	{	{	PUNCT
ejpam-1245	133	23	φ	φ	PROPN
ejpam-1245	133	24	,	,	PUNCT
ejpam-1245	133	25	y	y	PROPN
ejpam-1245	133	26	,	,	PUNCT
ejpam-1245	133	27	{	{	PUNCT
ejpam-1245	133	28	0	0	NUM
ejpam-1245	133	29	}	}	PUNCT
ejpam-1245	133	30	}	}	PUNCT
ejpam-1245	133	31	.	.	PUNCT
ejpam-1245	134	1	let	let	VERB
ejpam-1245	134	2	f	f	NOUN
ejpam-1245	134	3	:	:	PUNCT
ejpam-1245	134	4	(	(	PUNCT
ejpam-1245	134	5	x	x	X
ejpam-1245	134	6	,	,	PUNCT
ejpam-1245	134	7	τ)→	τ)→	PROPN
ejpam-1245	134	8	(	(	PUNCT
ejpam-1245	134	9	y	y	PROPN
ejpam-1245	134	10	,	,	PUNCT
ejpam-1245	134	11	σ	σ	PROPN
ejpam-1245	134	12	)	)	PUNCT
ejpam-1245	134	13	be	be	VERB
ejpam-1245	134	14	the	the	DET
ejpam-1245	134	15	function	function	NOUN
ejpam-1245	134	16	defined	define	VERB
ejpam-1245	134	17	by	by	ADP
ejpam-1245	134	18	f	f	PROPN
ejpam-1245	134	19	(	(	PUNCT
ejpam-1245	134	20	x	x	NOUN
ejpam-1245	134	21	)	)	PUNCT
ejpam-1245	134	22	=	=	SYM
ejpam-1245	135	1	(	(	PUNCT
ejpam-1245	135	2	1	1	NUM
ejpam-1245	135	3	x	x	SYM
ejpam-1245	135	4	∈	∈	PROPN
ejpam-1245	135	5	r−q	r−q	NOUN
ejpam-1245	135	6	0	0	NUM
ejpam-1245	135	7	x	x	SYM
ejpam-1245	135	8	∈q	∈q	NOUN
ejpam-1245	135	9	then	then	ADV
ejpam-1245	135	10	f	f	PROPN
ejpam-1245	135	11	is	be	AUX
ejpam-1245	135	12	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	135	13	but	but	CCONJ
ejpam-1245	135	14	it	it	PRON
ejpam-1245	135	15	is	be	AUX
ejpam-1245	135	16	neither	neither	CCONJ
ejpam-1245	135	17	continuous	continuous	ADJ
ejpam-1245	135	18	nor	nor	CCONJ
ejpam-1245	135	19	ω−continuous	ω−continuous	ADJ
ejpam-1245	135	20	.	.	NOUN
ejpam-1245	135	21	example	example	NOUN
ejpam-1245	136	1	2	2	NUM
ejpam-1245	136	2	.	.	PUNCT
ejpam-1245	136	3	let	let	VERB
ejpam-1245	136	4	x	x	PUNCT
ejpam-1245	136	5	=	=	PUNCT
ejpam-1245	136	6	{	{	PUNCT
ejpam-1245	136	7	1,2,3	1,2,3	NOUN
ejpam-1245	136	8	}	}	PUNCT
ejpam-1245	136	9	with	with	ADP
ejpam-1245	136	10	the	the	DET
ejpam-1245	136	11	topology	topology	NOUN
ejpam-1245	136	12	τ	τ	NOUN
ejpam-1245	136	13	=	=	PUNCT
ejpam-1245	136	14	{	{	PUNCT
ejpam-1245	136	15	x	x	PROPN
ejpam-1245	136	16	,	,	PUNCT
ejpam-1245	136	17	φ	φ	PROPN
ejpam-1245	136	18	,	,	PUNCT
ejpam-1245	136	19	{	{	PUNCT
ejpam-1245	136	20	1	1	NUM
ejpam-1245	136	21	}	}	PUNCT
ejpam-1245	136	22	,	,	PUNCT
ejpam-1245	136	23	{	{	PUNCT
ejpam-1245	136	24	2	2	NUM
ejpam-1245	136	25	}	}	PUNCT
ejpam-1245	136	26	,	,	PUNCT
ejpam-1245	136	27	{	{	PUNCT
ejpam-1245	136	28	1,2	1,2	NUM
ejpam-1245	136	29	}	}	PUNCT
ejpam-1245	136	30	}	}	PUNCT
ejpam-1245	136	31	and	and	CCONJ
ejpam-1245	136	32	y	y	PROPN
ejpam-1245	136	33	=	=	PUNCT
ejpam-1245	136	34	{	{	PUNCT
ejpam-1245	136	35	a	a	DET
ejpam-1245	136	36	,	,	PUNCT
ejpam-1245	136	37	b	b	NOUN
ejpam-1245	136	38	}	}	PUNCT
ejpam-1245	136	39	with	with	ADP
ejpam-1245	136	40	the	the	DET
ejpam-1245	136	41	topology	topology	NOUN
ejpam-1245	136	42	σ	σ	NOUN
ejpam-1245	136	43	=	=	SYM
ejpam-1245	136	44	{	{	PUNCT
ejpam-1245	136	45	φ	φ	PROPN
ejpam-1245	136	46	,	,	PUNCT
ejpam-1245	136	47	y	y	PROPN
ejpam-1245	136	48	,	,	PUNCT
ejpam-1245	136	49	{	{	PUNCT
ejpam-1245	136	50	a	a	X
ejpam-1245	136	51	}	}	PUNCT
ejpam-1245	136	52	}	}	PUNCT
ejpam-1245	136	53	.	.	PUNCT
ejpam-1245	137	1	let	let	VERB
ejpam-1245	137	2	f	f	NOUN
ejpam-1245	137	3	:	:	PUNCT
ejpam-1245	137	4	(	(	PUNCT
ejpam-1245	137	5	x	x	X
ejpam-1245	137	6	,	,	PUNCT
ejpam-1245	137	7	τ)→	τ)→	PROPN
ejpam-1245	137	8	(	(	PUNCT
ejpam-1245	137	9	y	y	PROPN
ejpam-1245	137	10	,	,	PUNCT
ejpam-1245	137	11	σ	σ	PROPN
ejpam-1245	137	12	)	)	PUNCT
ejpam-1245	137	13	be	be	VERB
ejpam-1245	137	14	the	the	DET
ejpam-1245	137	15	function	function	NOUN
ejpam-1245	137	16	defined	define	VERB
ejpam-1245	137	17	by	by	ADP
ejpam-1245	137	18	f	f	PROPN
ejpam-1245	137	19	(	(	PUNCT
ejpam-1245	137	20	x	x	NOUN
ejpam-1245	137	21	)	)	PUNCT
ejpam-1245	137	22	=	=	SYM
ejpam-1245	138	1	(	(	PUNCT
ejpam-1245	138	2	b	b	X
ejpam-1245	138	3	x	x	SYM
ejpam-1245	138	4	=	=	PUNCT
ejpam-1245	138	5	{	{	PUNCT
ejpam-1245	138	6	1,2	1,2	NUM
ejpam-1245	138	7	}	}	PUNCT
ejpam-1245	138	8	a	a	DET
ejpam-1245	138	9	x	x	SYM
ejpam-1245	138	10	=	=	SYM
ejpam-1245	138	11	3	3	NUM
ejpam-1245	138	12	then	then	ADV
ejpam-1245	138	13	f	f	PROPN
ejpam-1245	138	14	is	be	AUX
ejpam-1245	138	15	not	not	PART
ejpam-1245	138	16	β−continuous	β−continuous	ADJ
ejpam-1245	138	17	,	,	PUNCT
ejpam-1245	138	18	but	but	CCONJ
ejpam-1245	138	19	it	it	PRON
ejpam-1245	138	20	can	can	AUX
ejpam-1245	138	21	be	be	AUX
ejpam-1245	138	22	easily	easily	ADV
ejpam-1245	138	23	seen	see	VERB
ejpam-1245	138	24	that	that	SCONJ
ejpam-1245	138	25	f	f	PROPN
ejpam-1245	138	26	is	be	AUX
ejpam-1245	138	27	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	138	28	.	.	PUNCT
ejpam-1245	138	29	example	example	NOUN
ejpam-1245	139	1	3	3	X
ejpam-1245	139	2	.	.	PUNCT
ejpam-1245	140	1	let	let	VERB
ejpam-1245	140	2	x	x	PUNCT
ejpam-1245	140	3	=	=	PUNCT
ejpam-1245	140	4	r	r	NOUN
ejpam-1245	140	5	with	with	ADP
ejpam-1245	140	6	the	the	DET
ejpam-1245	140	7	topology	topology	NOUN
ejpam-1245	140	8	τ	τ	NOUN
ejpam-1245	140	9	=	=	PUNCT
ejpam-1245	140	10	τu	τu	ADP
ejpam-1245	140	11	and	and	CCONJ
ejpam-1245	140	12	y	y	PROPN
ejpam-1245	141	1	=	=	PUNCT
ejpam-1245	141	2	{	{	PUNCT
ejpam-1245	141	3	a	a	DET
ejpam-1245	141	4	,	,	PUNCT
ejpam-1245	141	5	b	b	NOUN
ejpam-1245	141	6	}	}	PUNCT
ejpam-1245	141	7	with	with	ADP
ejpam-1245	141	8	the	the	DET
ejpam-1245	141	9	topology	topology	NOUN
ejpam-1245	141	10	σ	σ	NOUN
ejpam-1245	141	11	=	=	SYM
ejpam-1245	141	12	{	{	PUNCT
ejpam-1245	141	13	φ	φ	PROPN
ejpam-1245	141	14	,	,	PUNCT
ejpam-1245	141	15	y	y	PROPN
ejpam-1245	141	16	,	,	PUNCT
ejpam-1245	141	17	{	{	PUNCT
ejpam-1245	141	18	a	a	X
ejpam-1245	141	19	}	}	PUNCT
ejpam-1245	141	20	}	}	PUNCT
ejpam-1245	141	21	.	.	PUNCT
ejpam-1245	142	1	let	let	VERB
ejpam-1245	142	2	f	f	NOUN
ejpam-1245	142	3	:	:	PUNCT
ejpam-1245	142	4	(	(	PUNCT
ejpam-1245	142	5	x	x	X
ejpam-1245	142	6	,	,	PUNCT
ejpam-1245	142	7	τ)→	τ)→	PROPN
ejpam-1245	142	8	(	(	PUNCT
ejpam-1245	142	9	y	y	PROPN
ejpam-1245	142	10	,	,	PUNCT
ejpam-1245	142	11	σ	σ	PROPN
ejpam-1245	142	12	)	)	PUNCT
ejpam-1245	142	13	be	be	VERB
ejpam-1245	142	14	the	the	DET
ejpam-1245	142	15	function	function	NOUN
ejpam-1245	142	16	defined	define	VERB
ejpam-1245	142	17	by	by	ADP
ejpam-1245	142	18	f	f	PROPN
ejpam-1245	142	19	(	(	PUNCT
ejpam-1245	142	20	x	x	NOUN
ejpam-1245	142	21	)	)	PUNCT
ejpam-1245	143	1	=	=	SYM
ejpam-1245	144	1	(	(	PUNCT
ejpam-1245	144	2	a	a	DET
ejpam-1245	144	3	x	x	X
ejpam-1245	144	4	∈	∈	PROPN
ejpam-1245	145	1	[	[	X
ejpam-1245	145	2	0,2)∩r−q	0,2)∩r−q	NOUN
ejpam-1245	145	3	b	b	X
ejpam-1245	145	4	x	x	SYM
ejpam-1245	145	5	∈	∈	PROPN
ejpam-1245	145	6	[	[	X
ejpam-1245	145	7	0,2)∩q	0,2)∩q	NUM
ejpam-1245	145	8	then	then	ADV
ejpam-1245	145	9	f	f	PROPN
ejpam-1245	145	10	is	be	AUX
ejpam-1245	145	11	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	145	12	,	,	PUNCT
ejpam-1245	145	13	but	but	CCONJ
ejpam-1245	145	14	it	it	PRON
ejpam-1245	145	15	is	be	AUX
ejpam-1245	145	16	not	not	PART
ejpam-1245	145	17	ωb−continuous	ωb−continuous	ADJ
ejpam-1245	145	18	.	.	PUNCT
ejpam-1245	146	1	proposition	proposition	NOUN
ejpam-1245	146	2	1	1	NUM
ejpam-1245	146	3	.	.	PUNCT
ejpam-1245	147	1	if	if	SCONJ
ejpam-1245	147	2	f	f	PROPN
ejpam-1245	147	3	:	:	PUNCT
ejpam-1245	147	4	(	(	PUNCT
ejpam-1245	147	5	x	x	X
ejpam-1245	147	6	,	,	PUNCT
ejpam-1245	147	7	τ)→	τ)→	PROPN
ejpam-1245	147	8	(	(	PUNCT
ejpam-1245	147	9	y	y	PROPN
ejpam-1245	147	10	,	,	PUNCT
ejpam-1245	147	11	σ	σ	PROPN
ejpam-1245	147	12	)	)	PUNCT
ejpam-1245	147	13	is	be	AUX
ejpam-1245	147	14	an	an	DET
ejpam-1245	147	15	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	147	16	function	function	NOUN
ejpam-1245	147	17	and	and	CCONJ
ejpam-1245	147	18	a	a	PRON
ejpam-1245	147	19	is	be	AUX
ejpam-1245	147	20	an	an	DET
ejpam-1245	147	21	open	open	ADJ
ejpam-1245	147	22	set	set	NOUN
ejpam-1245	147	23	in	in	ADP
ejpam-1245	147	24	x	x	SYM
ejpam-1245	147	25	,	,	PUNCT
ejpam-1245	147	26	then	then	ADV
ejpam-1245	147	27	the	the	DET
ejpam-1245	147	28	restriction	restriction	NOUN
ejpam-1245	147	29	f	f	PROPN
ejpam-1245	147	30	|a	|a	VERB
ejpam-1245	147	31	:	:	PUNCT
ejpam-1245	147	32	(	(	PUNCT
ejpam-1245	147	33	a	a	X
ejpam-1245	147	34	,	,	PUNCT
ejpam-1245	147	35	τa)→	τa)→	X
ejpam-1245	147	36	(	(	PUNCT
ejpam-1245	147	37	y	y	PROPN
ejpam-1245	147	38	,	,	PUNCT
ejpam-1245	147	39	σ	σ	PROPN
ejpam-1245	147	40	)	)	PUNCT
ejpam-1245	147	41	is	be	AUX
ejpam-1245	147	42	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	147	43	.	.	PUNCT
ejpam-1245	148	1	proof	proof	NOUN
ejpam-1245	148	2	.	.	PUNCT
ejpam-1245	149	1	since	since	SCONJ
ejpam-1245	149	2	f	f	PROPN
ejpam-1245	149	3	is	be	AUX
ejpam-1245	149	4	an	an	DET
ejpam-1245	149	5	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	149	6	,	,	PUNCT
ejpam-1245	149	7	for	for	ADP
ejpam-1245	149	8	any	any	DET
ejpam-1245	149	9	open	open	ADJ
ejpam-1245	149	10	set	set	VERB
ejpam-1245	149	11	v	v	NOUN
ejpam-1245	149	12	in	in	ADP
ejpam-1245	149	13	y	y	PROPN
ejpam-1245	149	14	,	,	PUNCT
ejpam-1245	149	15	f	f	PROPN
ejpam-1245	149	16	−1(v	−1(v	PROPN
ejpam-1245	149	17	)	)	PUNCT
ejpam-1245	149	18	is	be	AUX
ejpam-1245	149	19	a	a	DET
ejpam-1245	149	20	ωβo(x	ωβo(x	PROPN
ejpam-1245	149	21	,	,	PUNCT
ejpam-1245	149	22	τ	τ	NOUN
ejpam-1245	149	23	)	)	PUNCT
ejpam-1245	149	24	set	set	NOUN
ejpam-1245	149	25	.	.	PUNCT
ejpam-1245	150	1	hence	hence	ADV
ejpam-1245	150	2	by	by	ADP
ejpam-1245	150	3	lemma	lemma	PROPN
ejpam-1245	150	4	1(ii	1(ii	NUM
ejpam-1245	150	5	)	)	PUNCT
ejpam-1245	150	6	,	,	PUNCT
ejpam-1245	150	7	f	f	PROPN
ejpam-1245	150	8	−1(v	−1(v	PROPN
ejpam-1245	150	9	)	)	PUNCT
ejpam-1245	151	1	∩a	∩a	PROPN
ejpam-1245	151	2	is	be	AUX
ejpam-1245	151	3	a	a	DET
ejpam-1245	151	4	ωβo(x	ωβo(x	PROPN
ejpam-1245	151	5	,	,	PUNCT
ejpam-1245	151	6	τ	τ	PROPN
ejpam-1245	151	7	)	)	PUNCT
ejpam-1245	151	8	since	since	SCONJ
ejpam-1245	151	9	a	a	PRON
ejpam-1245	151	10	is	be	AUX
ejpam-1245	151	11	an	an	DET
ejpam-1245	151	12	open	open	ADJ
ejpam-1245	151	13	set	set	NOUN
ejpam-1245	151	14	.	.	PUNCT
ejpam-1245	152	1	therefore	therefore	ADV
ejpam-1245	152	2	,	,	PUNCT
ejpam-1245	152	3	by	by	ADP
ejpam-1245	152	4	theorem	theorem	NOUN
ejpam-1245	152	5	1	1	NUM
ejpam-1245	152	6	,	,	PUNCT
ejpam-1245	152	7	(	(	PUNCT
ejpam-1245	152	8	f	f	X
ejpam-1245	152	9	|a)−1(v	|a)−1(v	NOUN
ejpam-1245	152	10	)	)	PUNCT
ejpam-1245	153	1	=	=	SYM
ejpam-1245	153	2	f	f	PROPN
ejpam-1245	153	3	−1(v	−1(v	NOUN
ejpam-1245	153	4	)	)	PUNCT
ejpam-1245	154	1	∩a	∩a	PROPN
ejpam-1245	154	2	is	be	AUX
ejpam-1245	154	3	ωβo(a	ωβo(a	PROPN
ejpam-1245	154	4	,	,	PUNCT
ejpam-1245	154	5	τa	τa	NOUN
ejpam-1245	154	6	)	)	PUNCT
ejpam-1245	154	7	sets	set	NOUN
ejpam-1245	154	8	,	,	PUNCT
ejpam-1245	154	9	which	which	PRON
ejpam-1245	154	10	implies	imply	VERB
ejpam-1245	154	11	that	that	SCONJ
ejpam-1245	154	12	f	f	PROPN
ejpam-1245	154	13	|a	|a	VERB
ejpam-1245	154	14	is	be	AUX
ejpam-1245	154	15	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	154	16	function	function	NOUN
ejpam-1245	154	17	.	.	PUNCT
ejpam-1245	155	1	observe	observe	VERB
ejpam-1245	155	2	that	that	SCONJ
ejpam-1245	155	3	the	the	DET
ejpam-1245	155	4	above	above	ADJ
ejpam-1245	155	5	theorem	theorem	NOUN
ejpam-1245	155	6	is	be	AUX
ejpam-1245	155	7	not	not	PART
ejpam-1245	155	8	true	true	ADJ
ejpam-1245	155	9	if	if	SCONJ
ejpam-1245	155	10	a	a	PRON
ejpam-1245	155	11	were	be	AUX
ejpam-1245	155	12	taken	take	VERB
ejpam-1245	155	13	to	to	PART
ejpam-1245	155	14	be	be	AUX
ejpam-1245	155	15	βo(x	βo(x	PUNCT
ejpam-1245	155	16	,	,	PUNCT
ejpam-1245	155	17	τ	τ	X
ejpam-1245	155	18	)	)	PUNCT
ejpam-1245	155	19	sets	set	NOUN
ejpam-1245	155	20	or	or	CCONJ
ejpam-1245	155	21	ωo(x	ωo(x	PROPN
ejpam-1245	155	22	,	,	PUNCT
ejpam-1245	155	23	τ	τ	PROPN
ejpam-1245	155	24	)	)	PUNCT
ejpam-1245	155	25	,	,	PUNCT
ejpam-1245	155	26	as	as	SCONJ
ejpam-1245	155	27	it	it	PRON
ejpam-1245	155	28	shown	show	VERB
ejpam-1245	155	29	in	in	ADP
ejpam-1245	155	30	the	the	DET
ejpam-1245	155	31	next	next	ADJ
ejpam-1245	155	32	examples	example	NOUN
ejpam-1245	155	33	.	.	PUNCT
ejpam-1245	156	1	h.	h.	PROPN
ejpam-1245	156	2	aljarrah	aljarrah	PROPN
ejpam-1245	156	3	,	,	PUNCT
ejpam-1245	156	4	m.	m.	NOUN
ejpam-1245	156	5	noorani	noorani	PROPN
ejpam-1245	156	6	/	/	SYM
ejpam-1245	156	7	eur	eur	PROPN
ejpam-1245	156	8	.	.	PUNCT
ejpam-1245	157	1	j.	j.	PROPN
ejpam-1245	157	2	pure	pure	PROPN
ejpam-1245	157	3	appl	appl	PROPN
ejpam-1245	157	4	.	.	PROPN
ejpam-1245	157	5	math	math	PROPN
ejpam-1245	157	6	,	,	PUNCT
ejpam-1245	157	7	5	5	NUM
ejpam-1245	157	8	(	(	PUNCT
ejpam-1245	157	9	2012	2012	NUM
ejpam-1245	157	10	)	)	PUNCT
ejpam-1245	157	11	,	,	PUNCT
ejpam-1245	157	12	129	129	NUM
ejpam-1245	157	13	-	-	SYM
ejpam-1245	157	14	140	140	NUM
ejpam-1245	157	15	133	133	NUM
ejpam-1245	157	16	example	example	NOUN
ejpam-1245	157	17	4	4	NUM
ejpam-1245	157	18	.	.	PUNCT
ejpam-1245	158	1	let	let	VERB
ejpam-1245	158	2	x	x	PUNCT
ejpam-1245	158	3	=	=	PUNCT
ejpam-1245	158	4	r	r	NOUN
ejpam-1245	158	5	with	with	ADP
ejpam-1245	158	6	the	the	DET
ejpam-1245	158	7	topology	topology	NOUN
ejpam-1245	158	8	τ	τ	PROPN
ejpam-1245	158	9	=	=	PUNCT
ejpam-1245	158	10	τcoc	τcoc	PROPN
ejpam-1245	158	11	and	and	CCONJ
ejpam-1245	158	12	y	y	PROPN
ejpam-1245	158	13	=	=	PUNCT
ejpam-1245	158	14	{	{	PUNCT
ejpam-1245	158	15	0,1	0,1	NOUN
ejpam-1245	158	16	}	}	PUNCT
ejpam-1245	158	17	with	with	ADP
ejpam-1245	158	18	the	the	DET
ejpam-1245	158	19	topology	topology	NOUN
ejpam-1245	158	20	σ	σ	NOUN
ejpam-1245	158	21	=	=	SYM
ejpam-1245	158	22	{	{	PUNCT
ejpam-1245	158	23	φ	φ	PROPN
ejpam-1245	158	24	,	,	PUNCT
ejpam-1245	158	25	y	y	PROPN
ejpam-1245	158	26	,	,	PUNCT
ejpam-1245	158	27	{	{	PUNCT
ejpam-1245	158	28	1	1	NUM
ejpam-1245	158	29	}	}	PUNCT
ejpam-1245	158	30	}	}	PUNCT
ejpam-1245	158	31	.	.	PUNCT
ejpam-1245	159	1	let	let	VERB
ejpam-1245	159	2	f	f	NOUN
ejpam-1245	159	3	:	:	PUNCT
ejpam-1245	159	4	(	(	PUNCT
ejpam-1245	159	5	x	x	X
ejpam-1245	159	6	,	,	PUNCT
ejpam-1245	159	7	τ)→	τ)→	PROPN
ejpam-1245	159	8	(	(	PUNCT
ejpam-1245	159	9	y	y	PROPN
ejpam-1245	159	10	,	,	PUNCT
ejpam-1245	159	11	σ	σ	PROPN
ejpam-1245	159	12	)	)	PUNCT
ejpam-1245	159	13	be	be	VERB
ejpam-1245	159	14	the	the	DET
ejpam-1245	159	15	function	function	NOUN
ejpam-1245	159	16	defined	define	VERB
ejpam-1245	159	17	by	by	ADP
ejpam-1245	159	18	f	f	PROPN
ejpam-1245	159	19	(	(	PUNCT
ejpam-1245	159	20	x	x	NOUN
ejpam-1245	159	21	)	)	PUNCT
ejpam-1245	159	22	=	=	SYM
ejpam-1245	160	1	(	(	PUNCT
ejpam-1245	160	2	1	1	NUM
ejpam-1245	160	3	x	x	SYM
ejpam-1245	160	4	∈	∈	PROPN
ejpam-1245	160	5	(	(	PUNCT
ejpam-1245	160	6	0,1	0,1	NOUN
ejpam-1245	160	7	]	]	PUNCT
ejpam-1245	160	8	0	0	NUM
ejpam-1245	160	9	x	x	SYM
ejpam-1245	160	10	/∈	/∈	INTJ
ejpam-1245	160	11	(	(	PUNCT
ejpam-1245	160	12	0,1	0,1	NUM
ejpam-1245	160	13	]	]	PUNCT
ejpam-1245	160	14	it	it	PRON
ejpam-1245	160	15	can	can	AUX
ejpam-1245	160	16	be	be	AUX
ejpam-1245	160	17	easily	easily	ADV
ejpam-1245	160	18	seen	see	VERB
ejpam-1245	160	19	that	that	SCONJ
ejpam-1245	160	20	f	f	PROPN
ejpam-1245	160	21	is	be	AUX
ejpam-1245	160	22	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	160	23	.	.	PUNCT
ejpam-1245	161	1	we	we	PRON
ejpam-1245	161	2	take	take	VERB
ejpam-1245	161	3	a=	a=	ADV
ejpam-1245	161	4	(	(	PUNCT
ejpam-1245	161	5	0,1	0,1	NUM
ejpam-1245	161	6	]	]	PUNCT
ejpam-1245	161	7	.	.	PUNCT
ejpam-1245	162	1	then	then	ADV
ejpam-1245	162	2	a	a	DET
ejpam-1245	162	3	∈	∈	PROPN
ejpam-1245	162	4	βo(x	βo(x	PUNCT
ejpam-1245	162	5	,	,	PUNCT
ejpam-1245	162	6	τ	τ	X
ejpam-1245	162	7	)	)	PUNCT
ejpam-1245	162	8	and	and	CCONJ
ejpam-1245	162	9	f	f	PROPN
ejpam-1245	162	10	|a	|a	VERB
ejpam-1245	162	11	is	be	AUX
ejpam-1245	162	12	not	not	PART
ejpam-1245	162	13	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	162	14	since	since	SCONJ
ejpam-1245	162	15	(	(	PUNCT
ejpam-1245	162	16	f	f	NOUN
ejpam-1245	162	17	|a)−1(1	|a)−1(1	PROPN
ejpam-1245	162	18	)	)	PUNCT
ejpam-1245	162	19	=	=	PRON
ejpam-1245	162	20	{	{	PUNCT
ejpam-1245	162	21	1	1	NUM
ejpam-1245	162	22	}	}	PUNCT
ejpam-1245	162	23	/∈ωβo(a	/∈ωβo(a	NUM
ejpam-1245	162	24	,	,	PUNCT
ejpam-1245	162	25	τa	τa	PROPN
ejpam-1245	162	26	)	)	PUNCT
ejpam-1245	162	27	.	.	PUNCT
ejpam-1245	163	1	example	example	NOUN
ejpam-1245	164	1	5	5	NUM
ejpam-1245	164	2	.	.	PUNCT
ejpam-1245	164	3	let	let	VERB
ejpam-1245	164	4	x	x	PUNCT
ejpam-1245	164	5	=	=	PUNCT
ejpam-1245	164	6	r	r	NOUN
ejpam-1245	164	7	with	with	ADP
ejpam-1245	164	8	the	the	DET
ejpam-1245	164	9	topology	topology	NOUN
ejpam-1245	164	10	τ	τ	NOUN
ejpam-1245	164	11	=	=	PUNCT
ejpam-1245	164	12	τu	τu	ADP
ejpam-1245	164	13	and	and	CCONJ
ejpam-1245	164	14	y	y	PROPN
ejpam-1245	164	15	=	=	PUNCT
ejpam-1245	164	16	{	{	PUNCT
ejpam-1245	164	17	0,1	0,1	NOUN
ejpam-1245	164	18	}	}	PUNCT
ejpam-1245	164	19	with	with	ADP
ejpam-1245	164	20	the	the	DET
ejpam-1245	164	21	topology	topology	NOUN
ejpam-1245	164	22	σ	σ	NOUN
ejpam-1245	164	23	=	=	SYM
ejpam-1245	164	24	{	{	PUNCT
ejpam-1245	164	25	φ	φ	PROPN
ejpam-1245	164	26	,	,	PUNCT
ejpam-1245	164	27	y	y	PROPN
ejpam-1245	164	28	,	,	PUNCT
ejpam-1245	164	29	{	{	PUNCT
ejpam-1245	164	30	1	1	NUM
ejpam-1245	164	31	}	}	PUNCT
ejpam-1245	164	32	}	}	PUNCT
ejpam-1245	164	33	.	.	PUNCT
ejpam-1245	165	1	let	let	VERB
ejpam-1245	165	2	f	f	NOUN
ejpam-1245	165	3	:	:	PUNCT
ejpam-1245	165	4	(	(	PUNCT
ejpam-1245	165	5	x	x	X
ejpam-1245	165	6	,	,	PUNCT
ejpam-1245	165	7	τ)→	τ)→	PROPN
ejpam-1245	165	8	(	(	PUNCT
ejpam-1245	165	9	y	y	PROPN
ejpam-1245	165	10	,	,	PUNCT
ejpam-1245	165	11	σ	σ	PROPN
ejpam-1245	165	12	)	)	PUNCT
ejpam-1245	165	13	be	be	VERB
ejpam-1245	165	14	the	the	DET
ejpam-1245	165	15	function	function	NOUN
ejpam-1245	165	16	defined	define	VERB
ejpam-1245	165	17	by	by	ADP
ejpam-1245	165	18	f	f	PROPN
ejpam-1245	165	19	(	(	PUNCT
ejpam-1245	165	20	x	x	NOUN
ejpam-1245	165	21	)	)	PUNCT
ejpam-1245	165	22	=	=	SYM
ejpam-1245	166	1	(	(	PUNCT
ejpam-1245	166	2	1	1	NUM
ejpam-1245	166	3	x	x	X
ejpam-1245	166	4	=	=	PUNCT
ejpam-1245	166	5	p	p	VERB
ejpam-1245	166	6	2	2	NUM
ejpam-1245	166	7	0	0	NUM
ejpam-1245	166	8	x	x	SYM
ejpam-1245	166	9	∈q	∈q	NOUN
ejpam-1245	166	10	it	it	PRON
ejpam-1245	166	11	can	can	AUX
ejpam-1245	166	12	be	be	AUX
ejpam-1245	166	13	easily	easily	ADV
ejpam-1245	166	14	seen	see	VERB
ejpam-1245	166	15	that	that	SCONJ
ejpam-1245	166	16	f	f	PROPN
ejpam-1245	166	17	is	be	AUX
ejpam-1245	166	18	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	166	19	.	.	PUNCT
ejpam-1245	167	1	we	we	PRON
ejpam-1245	167	2	take	take	VERB
ejpam-1245	167	3	a=	a=	ADV
ejpam-1245	167	4	r−q	r−q	VERB
ejpam-1245	167	5	.	.	PUNCT
ejpam-1245	168	1	then	then	ADV
ejpam-1245	168	2	a∈	a∈	PROPN
ejpam-1245	168	3	ωo(x	ωo(x	PROPN
ejpam-1245	168	4	,	,	PUNCT
ejpam-1245	168	5	τ	τ	X
ejpam-1245	168	6	)	)	PUNCT
ejpam-1245	168	7	and	and	CCONJ
ejpam-1245	168	8	f	f	PROPN
ejpam-1245	168	9	|a	|a	VERB
ejpam-1245	168	10	is	be	AUX
ejpam-1245	168	11	not	not	PART
ejpam-1245	168	12	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	168	13	since	since	SCONJ
ejpam-1245	168	14	(	(	PUNCT
ejpam-1245	168	15	f	f	X
ejpam-1245	168	16	|a)−1(y	|a)−1(y	ADV
ejpam-1245	168	17	)	)	PUNCT
ejpam-1245	169	1	=	=	PRON
ejpam-1245	169	2	{	{	PUNCT
ejpam-1245	169	3	p2	p2	PROPN
ejpam-1245	169	4	}	}	PUNCT
ejpam-1245	169	5	/∈ωβo(a	/∈ωβo(a	NUM
ejpam-1245	169	6	,	,	PUNCT
ejpam-1245	169	7	τa	τa	PROPN
ejpam-1245	169	8	)	)	PUNCT
ejpam-1245	169	9	.	.	PUNCT
ejpam-1245	170	1	definition	definition	NOUN
ejpam-1245	170	2	5	5	NUM
ejpam-1245	170	3	.	.	PUNCT
ejpam-1245	171	1	[	[	X
ejpam-1245	171	2	7	7	X
ejpam-1245	171	3	]	]	X
ejpam-1245	171	4	a	a	DET
ejpam-1245	171	5	cover	cover	NOUN
ejpam-1245	171	6	υ	υ	NOUN
ejpam-1245	171	7	=	=	PUNCT
ejpam-1245	171	8	{	{	PUNCT
ejpam-1245	171	9	uα	uα	X
ejpam-1245	171	10	:	:	PUNCT
ejpam-1245	171	11	α	α	PROPN
ejpam-1245	171	12	∈	∈	PROPN
ejpam-1245	171	13	∆	∆	PROPN
ejpam-1245	171	14	}	}	PUNCT
ejpam-1245	171	15	of	of	ADP
ejpam-1245	171	16	subset	subset	NOUN
ejpam-1245	171	17	of	of	ADP
ejpam-1245	171	18	x	x	PRON
ejpam-1245	171	19	is	be	AUX
ejpam-1245	171	20	called	call	VERB
ejpam-1245	171	21	a	a	DET
ejpam-1245	171	22	βo(x	βo(x	PUNCT
ejpam-1245	171	23	,	,	PUNCT
ejpam-1245	171	24	τ	τ	X
ejpam-1245	171	25	)	)	PUNCT
ejpam-1245	171	26	cover	cover	VERB
ejpam-1245	171	27	if	if	SCONJ
ejpam-1245	171	28	uα	uα	PROPN
ejpam-1245	171	29	is	be	AUX
ejpam-1245	171	30	βo(x	βo(x	PUNCT
ejpam-1245	171	31	,	,	PUNCT
ejpam-1245	171	32	τ	τ	X
ejpam-1245	171	33	)	)	PUNCT
ejpam-1245	171	34	for	for	ADP
ejpam-1245	171	35	each	each	DET
ejpam-1245	171	36	α	α	PRON
ejpam-1245	171	37	∈∆.	∈∆.	PROPN
ejpam-1245	171	38	now	now	ADV
ejpam-1245	171	39	we	we	PRON
ejpam-1245	171	40	prove	prove	VERB
ejpam-1245	171	41	the	the	DET
ejpam-1245	171	42	following	follow	VERB
ejpam-1245	171	43	proposition	proposition	NOUN
ejpam-1245	171	44	.	.	PUNCT
ejpam-1245	172	1	proposition	proposition	NOUN
ejpam-1245	172	2	2	2	NUM
ejpam-1245	172	3	.	.	PUNCT
ejpam-1245	173	1	let	let	VERB
ejpam-1245	173	2	f	f	NOUN
ejpam-1245	173	3	:	:	PUNCT
ejpam-1245	173	4	(	(	PUNCT
ejpam-1245	173	5	x	x	X
ejpam-1245	173	6	,	,	PUNCT
ejpam-1245	173	7	τ)→	τ)→	PROPN
ejpam-1245	173	8	(	(	PUNCT
ejpam-1245	173	9	y	y	PROPN
ejpam-1245	173	10	,	,	PUNCT
ejpam-1245	173	11	σ	σ	PROPN
ejpam-1245	173	12	)	)	PUNCT
ejpam-1245	173	13	be	be	VERB
ejpam-1245	173	14	any	any	DET
ejpam-1245	173	15	function	function	NOUN
ejpam-1245	173	16	and	and	CCONJ
ejpam-1245	173	17	a	a	DET
ejpam-1245	173	18	=	=	PUNCT
ejpam-1245	173	19	{	{	PUNCT
ejpam-1245	173	20	aα	aα	NOUN
ejpam-1245	173	21	:	:	PUNCT
ejpam-1245	173	22	α	α	PROPN
ejpam-1245	173	23	∈	∈	PROPN
ejpam-1245	173	24	∆	∆	PROPN
ejpam-1245	173	25	}	}	PUNCT
ejpam-1245	173	26	be	be	AUX
ejpam-1245	173	27	a	a	DET
ejpam-1245	173	28	βo(x	βo(x	PUNCT
ejpam-1245	173	29	,	,	PUNCT
ejpam-1245	173	30	τ	τ	X
ejpam-1245	173	31	)	)	PUNCT
ejpam-1245	173	32	cover	cover	NOUN
ejpam-1245	173	33	of	of	ADP
ejpam-1245	173	34	x	x	X
ejpam-1245	173	35	.	.	PUNCT
ejpam-1245	174	1	if	if	SCONJ
ejpam-1245	174	2	the	the	DET
ejpam-1245	174	3	restriction	restriction	NOUN
ejpam-1245	174	4	,	,	PUNCT
ejpam-1245	174	5	f	f	PROPN
ejpam-1245	174	6	|aα	|aα	PROPN
ejpam-1245	174	7	:	:	PUNCT
ejpam-1245	174	8	(	(	PUNCT
ejpam-1245	174	9	aα	aα	NOUN
ejpam-1245	174	10	,	,	PUNCT
ejpam-1245	174	11	τaα	τaα	NOUN
ejpam-1245	174	12	)	)	PUNCT
ejpam-1245	174	13	→	→	SYM
ejpam-1245	174	14	(	(	PUNCT
ejpam-1245	174	15	y	y	PROPN
ejpam-1245	174	16	,	,	PUNCT
ejpam-1245	174	17	σ	σ	PROPN
ejpam-1245	174	18	)	)	PUNCT
ejpam-1245	174	19	is	be	AUX
ejpam-1245	174	20	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	174	21	for	for	ADP
ejpam-1245	174	22	each	each	DET
ejpam-1245	174	23	α	α	NOUN
ejpam-1245	174	24	∈	∈	PROPN
ejpam-1245	174	25	∆	∆	PROPN
ejpam-1245	174	26	,	,	PUNCT
ejpam-1245	174	27	then	then	ADV
ejpam-1245	174	28	f	f	PROPN
ejpam-1245	174	29	is	be	AUX
ejpam-1245	174	30	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	174	31	.	.	PUNCT
ejpam-1245	175	1	proof	proof	NOUN
ejpam-1245	175	2	.	.	PUNCT
ejpam-1245	176	1	let	let	VERB
ejpam-1245	176	2	v	v	PART
ejpam-1245	176	3	be	be	AUX
ejpam-1245	176	4	any	any	DET
ejpam-1245	176	5	open	open	ADJ
ejpam-1245	176	6	set	set	NOUN
ejpam-1245	176	7	in	in	ADP
ejpam-1245	176	8	y	y	PROPN
ejpam-1245	176	9	.	.	PUNCT
ejpam-1245	177	1	since	since	SCONJ
ejpam-1245	177	2	f	f	PROPN
ejpam-1245	177	3	|aα	|aα	PROPN
ejpam-1245	177	4	is	be	AUX
ejpam-1245	177	5	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	177	6	,	,	PUNCT
ejpam-1245	177	7	then	then	ADV
ejpam-1245	177	8	for	for	SCONJ
ejpam-1245	177	9	each	each	DET
ejpam-1245	177	10	α	α	NOUN
ejpam-1245	177	11	∈∆	∈∆	NOUN
ejpam-1245	177	12	,	,	PUNCT
ejpam-1245	177	13	we	we	PRON
ejpam-1245	177	14	have	have	VERB
ejpam-1245	177	15	(	(	PUNCT
ejpam-1245	177	16	f	f	X
ejpam-1245	177	17	|a)−1(v	|a)−1(v	NOUN
ejpam-1245	177	18	)	)	PUNCT
ejpam-1245	177	19	=	=	SYM
ejpam-1245	178	1	f	f	PROPN
ejpam-1245	178	2	−1(v	−1(v	NOUN
ejpam-1245	178	3	)	)	PUNCT
ejpam-1245	179	1	∩aα	∩aα	NOUN
ejpam-1245	179	2	∈ωβo(aα	∈ωβo(aα	PROPN
ejpam-1245	179	3	,	,	PUNCT
ejpam-1245	179	4	τaα	τaα	NOUN
ejpam-1245	179	5	)	)	PUNCT
ejpam-1245	179	6	.	.	PUNCT
ejpam-1245	180	1	so	so	ADV
ejpam-1245	180	2	by	by	ADP
ejpam-1245	180	3	theorem	theorem	NOUN
ejpam-1245	180	4	1	1	NUM
ejpam-1245	180	5	,	,	PUNCT
ejpam-1245	180	6	f	f	PROPN
ejpam-1245	180	7	−1(v	−1(v	PROPN
ejpam-1245	180	8	)	)	PUNCT
ejpam-1245	180	9	∩aα	∩aα	PROPN
ejpam-1245	180	10	∈ωβo(x	∈ωβo(x	PROPN
ejpam-1245	180	11	,	,	PUNCT
ejpam-1245	180	12	τ	τ	PROPN
ejpam-1245	180	13	)	)	PUNCT
ejpam-1245	180	14	for	for	ADP
ejpam-1245	180	15	each	each	DET
ejpam-1245	180	16	α	α	PRON
ejpam-1245	180	17	∈∆.	∈∆.	PROPN
ejpam-1245	180	18	take	take	VERB
ejpam-1245	180	19	f	f	PROPN
ejpam-1245	180	20	−1(v	−1(v	NOUN
ejpam-1245	180	21	)	)	PUNCT
ejpam-1245	180	22	=	=	SYM
ejpam-1245	180	23	∪	∪	ADP
ejpam-1245	180	24	α∈∆	α∈∆	PROPN
ejpam-1245	180	25	(	(	PUNCT
ejpam-1245	180	26	f	f	PROPN
ejpam-1245	180	27	−1(v	−1(v	PROPN
ejpam-1245	180	28	)	)	PUNCT
ejpam-1245	180	29	∩	∩	PROPN
ejpam-1245	180	30	aα	aα	NOUN
ejpam-1245	180	31	)	)	PUNCT
ejpam-1245	180	32	.	.	PUNCT
ejpam-1245	181	1	by	by	ADP
ejpam-1245	181	2	lemma	lemma	PROPN
ejpam-1245	181	3	1(i	1(i	NUM
ejpam-1245	181	4	)	)	PUNCT
ejpam-1245	181	5	f	f	PROPN
ejpam-1245	181	6	−1(v	−1(v	PROPN
ejpam-1245	181	7	)	)	PUNCT
ejpam-1245	181	8	∈ωβo(x	∈ωβo(x	PROPN
ejpam-1245	181	9	,	,	PUNCT
ejpam-1245	181	10	τ	τ	PROPN
ejpam-1245	181	11	)	)	PUNCT
ejpam-1245	181	12	.	.	PUNCT
ejpam-1245	182	1	corollary	corollary	ADJ
ejpam-1245	182	2	2	2	NUM
ejpam-1245	182	3	.	.	PUNCT
ejpam-1245	183	1	let	let	VERB
ejpam-1245	183	2	f	f	NOUN
ejpam-1245	183	3	:	:	PUNCT
ejpam-1245	183	4	(	(	PUNCT
ejpam-1245	183	5	x	x	X
ejpam-1245	183	6	,	,	PUNCT
ejpam-1245	183	7	τ	τ	PROPN
ejpam-1245	183	8	)	)	PUNCT
ejpam-1245	183	9	→	→	SYM
ejpam-1245	183	10	(	(	PUNCT
ejpam-1245	183	11	y	y	PROPN
ejpam-1245	183	12	,	,	PUNCT
ejpam-1245	183	13	σ	σ	PROPN
ejpam-1245	183	14	)	)	PUNCT
ejpam-1245	183	15	be	be	VERB
ejpam-1245	183	16	any	any	DET
ejpam-1245	183	17	function	function	NOUN
ejpam-1245	183	18	and	and	CCONJ
ejpam-1245	183	19	a	a	DET
ejpam-1245	183	20	=	=	PUNCT
ejpam-1245	183	21	{	{	PUNCT
ejpam-1245	183	22	aα	aα	NOUN
ejpam-1245	183	23	:	:	PUNCT
ejpam-1245	183	24	α	α	PROPN
ejpam-1245	183	25	∈	∈	NOUN
ejpam-1245	183	26	∆	∆	PROPN
ejpam-1245	183	27	}	}	PUNCT
ejpam-1245	183	28	a	a	DET
ejpam-1245	183	29	open	open	ADJ
ejpam-1245	183	30	cover	cover	NOUN
ejpam-1245	183	31	of	of	ADP
ejpam-1245	183	32	x	x	X
ejpam-1245	183	33	.	.	PUNCT
ejpam-1245	184	1	if	if	SCONJ
ejpam-1245	184	2	the	the	DET
ejpam-1245	184	3	restriction	restriction	NOUN
ejpam-1245	184	4	,	,	PUNCT
ejpam-1245	184	5	f	f	PROPN
ejpam-1245	184	6	|aα	|aα	PROPN
ejpam-1245	184	7	:	:	PUNCT
ejpam-1245	184	8	(	(	PUNCT
ejpam-1245	184	9	aα	aα	NOUN
ejpam-1245	184	10	,	,	PUNCT
ejpam-1245	184	11	τaα	τaα	NOUN
ejpam-1245	184	12	)	)	PUNCT
ejpam-1245	184	13	→	→	PUNCT
ejpam-1245	184	14	(	(	PUNCT
ejpam-1245	184	15	y	y	PROPN
ejpam-1245	184	16	,	,	PUNCT
ejpam-1245	184	17	σ	σ	PROPN
ejpam-1245	184	18	)	)	PUNCT
ejpam-1245	184	19	is	be	AUX
ejpam-1245	184	20	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	184	21	for	for	ADP
ejpam-1245	184	22	each	each	DET
ejpam-1245	184	23	α	α	NOUN
ejpam-1245	184	24	∈	∈	PROPN
ejpam-1245	184	25	∆	∆	PROPN
ejpam-1245	184	26	,	,	PUNCT
ejpam-1245	184	27	then	then	ADV
ejpam-1245	184	28	f	f	PROPN
ejpam-1245	184	29	is	be	AUX
ejpam-1245	184	30	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	184	31	.	.	PUNCT
ejpam-1245	185	1	the	the	DET
ejpam-1245	185	2	composition	composition	NOUN
ejpam-1245	185	3	g	g	PROPN
ejpam-1245	185	4	◦	◦	NOUN
ejpam-1245	185	5	f	f	X
ejpam-1245	185	6	:	:	PUNCT
ejpam-1245	185	7	(	(	PUNCT
ejpam-1245	185	8	x	x	X
ejpam-1245	185	9	,	,	PUNCT
ejpam-1245	185	10	τ)→	τ)→	PROPN
ejpam-1245	185	11	(	(	PUNCT
ejpam-1245	185	12	z	z	NOUN
ejpam-1245	185	13	,	,	PUNCT
ejpam-1245	185	14	ρ	ρ	PROPN
ejpam-1245	185	15	)	)	PUNCT
ejpam-1245	185	16	of	of	ADP
ejpam-1245	185	17	a	a	DET
ejpam-1245	185	18	continuous	continuous	ADJ
ejpam-1245	185	19	function	function	NOUN
ejpam-1245	185	20	f	f	NOUN
ejpam-1245	185	21	:	:	PUNCT
ejpam-1245	185	22	(	(	PUNCT
ejpam-1245	185	23	x	x	X
ejpam-1245	185	24	,	,	PUNCT
ejpam-1245	185	25	τ)→	τ)→	PROPN
ejpam-1245	185	26	(	(	PUNCT
ejpam-1245	185	27	y	y	PROPN
ejpam-1245	185	28	,	,	PUNCT
ejpam-1245	185	29	σ	σ	PROPN
ejpam-1245	185	30	)	)	PUNCT
ejpam-1245	185	31	and	and	CCONJ
ejpam-1245	185	32	an	an	DET
ejpam-1245	185	33	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	185	34	function	function	NOUN
ejpam-1245	185	35	g	g	NOUN
ejpam-1245	185	36	:	:	PUNCT
ejpam-1245	185	37	(	(	PUNCT
ejpam-1245	185	38	y	y	PROPN
ejpam-1245	185	39	,	,	PUNCT
ejpam-1245	185	40	σ	σ	PROPN
ejpam-1245	185	41	)	)	PUNCT
ejpam-1245	185	42	→	→	SYM
ejpam-1245	185	43	(	(	PUNCT
ejpam-1245	185	44	z	z	NOUN
ejpam-1245	185	45	,	,	PUNCT
ejpam-1245	185	46	ρ	ρ	PROPN
ejpam-1245	185	47	)	)	PUNCT
ejpam-1245	185	48	is	be	AUX
ejpam-1245	185	49	not	not	PART
ejpam-1245	185	50	necessarily	necessarily	ADV
ejpam-1245	185	51	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	185	52	function	function	NOUN
ejpam-1245	185	53	as	as	ADP
ejpam-1245	185	54	the	the	DET
ejpam-1245	185	55	following	follow	VERB
ejpam-1245	185	56	example	example	NOUN
ejpam-1245	185	57	shows	show	NOUN
ejpam-1245	185	58	.	.	PUNCT
ejpam-1245	186	1	thus	thus	ADV
ejpam-1245	186	2	,	,	PUNCT
ejpam-1245	186	3	the	the	DET
ejpam-1245	186	4	composition	composition	NOUN
ejpam-1245	186	5	of	of	ADP
ejpam-1245	186	6	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	186	7	functions	function	NOUN
ejpam-1245	186	8	need	need	AUX
ejpam-1245	186	9	not	not	PART
ejpam-1245	186	10	be	be	AUX
ejpam-1245	186	11	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	186	12	.	.	PUNCT
ejpam-1245	186	13	example	example	NOUN
ejpam-1245	187	1	6	6	NUM
ejpam-1245	187	2	.	.	PUNCT
ejpam-1245	188	1	let	let	VERB
ejpam-1245	188	2	x	x	PUNCT
ejpam-1245	188	3	=	=	PUNCT
ejpam-1245	188	4	r	r	NOUN
ejpam-1245	188	5	with	with	ADP
ejpam-1245	188	6	the	the	DET
ejpam-1245	188	7	topology	topology	NOUN
ejpam-1245	188	8	τ	τ	PROPN
ejpam-1245	188	9	=	=	SYM
ejpam-1245	188	10	τcoc	τcoc	PROPN
ejpam-1245	188	11	,	,	PUNCT
ejpam-1245	188	12	y={1,2	y={1,2	ADJ
ejpam-1245	188	13	}	}	PUNCT
ejpam-1245	188	14	with	with	ADP
ejpam-1245	188	15	the	the	DET
ejpam-1245	188	16	topology	topology	NOUN
ejpam-1245	188	17	σ	σ	NOUN
ejpam-1245	188	18	=	=	SYM
ejpam-1245	188	19	{	{	PUNCT
ejpam-1245	188	20	φ	φ	PROPN
ejpam-1245	188	21	,	,	PUNCT
ejpam-1245	188	22	y	y	PROPN
ejpam-1245	188	23	,	,	PUNCT
ejpam-1245	188	24	{	{	PUNCT
ejpam-1245	188	25	1	1	NUM
ejpam-1245	188	26	}	}	PUNCT
ejpam-1245	188	27	}	}	PUNCT
ejpam-1245	188	28	and	and	CCONJ
ejpam-1245	188	29	z	z	NOUN
ejpam-1245	188	30	=	=	SYM
ejpam-1245	188	31	{	{	PUNCT
ejpam-1245	188	32	a	a	DET
ejpam-1245	188	33	,	,	PUNCT
ejpam-1245	188	34	b	b	NOUN
ejpam-1245	188	35	}	}	PUNCT
ejpam-1245	188	36	with	with	ADP
ejpam-1245	188	37	the	the	DET
ejpam-1245	188	38	topology	topology	NOUN
ejpam-1245	188	39	ρ	ρ	PROPN
ejpam-1245	188	40	=	=	SYM
ejpam-1245	188	41	�	�	PROPN
ejpam-1245	188	42	φ	φ	PROPN
ejpam-1245	188	43	,	,	PUNCT
ejpam-1245	188	44	z	z	PROPN
ejpam-1245	188	45	,	,	PUNCT
ejpam-1245	188	46	{	{	PUNCT
ejpam-1245	188	47	a	a	X
ejpam-1245	188	48	}	}	PUNCT
ejpam-1245	188	49	.	.	PUNCT
ejpam-1245	189	1	let	let	VERB
ejpam-1245	189	2	f	f	NOUN
ejpam-1245	189	3	:	:	PUNCT
ejpam-1245	189	4	(	(	PUNCT
ejpam-1245	189	5	x	x	X
ejpam-1245	189	6	,	,	PUNCT
ejpam-1245	189	7	τ	τ	PROPN
ejpam-1245	189	8	)	)	PUNCT
ejpam-1245	189	9	→	→	SYM
ejpam-1245	189	10	(	(	PUNCT
ejpam-1245	189	11	y	y	PROPN
ejpam-1245	189	12	,	,	PUNCT
ejpam-1245	189	13	σ	σ	PROPN
ejpam-1245	189	14	)	)	PUNCT
ejpam-1245	189	15	be	be	VERB
ejpam-1245	189	16	the	the	DET
ejpam-1245	189	17	function	function	NOUN
ejpam-1245	189	18	defined	define	VERB
ejpam-1245	189	19	by	by	ADP
ejpam-1245	189	20	f	f	PROPN
ejpam-1245	189	21	(	(	PUNCT
ejpam-1245	189	22	x	x	NOUN
ejpam-1245	189	23	)	)	PUNCT
ejpam-1245	189	24	=	=	SYM
ejpam-1245	190	1	(	(	PUNCT
ejpam-1245	190	2	1	1	NUM
ejpam-1245	190	3	x	x	SYM
ejpam-1245	190	4	∈	∈	PROPN
ejpam-1245	190	5	r−q	r−q	NOUN
ejpam-1245	190	6	2	2	NUM
ejpam-1245	190	7	x	x	SYM
ejpam-1245	190	8	∈q	∈q	PROPN
ejpam-1245	190	9	h.	h.	PROPN
ejpam-1245	190	10	aljarrah	aljarrah	PROPN
ejpam-1245	190	11	,	,	PUNCT
ejpam-1245	190	12	m.	m.	NOUN
ejpam-1245	190	13	noorani	noorani	PROPN
ejpam-1245	190	14	/	/	SYM
ejpam-1245	190	15	eur	eur	PROPN
ejpam-1245	190	16	.	.	PUNCT
ejpam-1245	191	1	j.	j.	PROPN
ejpam-1245	191	2	pure	pure	PROPN
ejpam-1245	191	3	appl	appl	PROPN
ejpam-1245	191	4	.	.	PROPN
ejpam-1245	191	5	math	math	PROPN
ejpam-1245	191	6	,	,	PUNCT
ejpam-1245	191	7	5	5	NUM
ejpam-1245	191	8	(	(	PUNCT
ejpam-1245	191	9	2012	2012	NUM
ejpam-1245	191	10	)	)	PUNCT
ejpam-1245	191	11	,	,	PUNCT
ejpam-1245	191	12	129	129	NUM
ejpam-1245	191	13	-	-	SYM
ejpam-1245	191	14	140	140	NUM
ejpam-1245	191	15	134	134	NUM
ejpam-1245	191	16	and	and	CCONJ
ejpam-1245	191	17	g	g	NOUN
ejpam-1245	191	18	:	:	PUNCT
ejpam-1245	191	19	(	(	PUNCT
ejpam-1245	191	20	x	x	X
ejpam-1245	191	21	,	,	PUNCT
ejpam-1245	191	22	σ)→	σ)→	PROPN
ejpam-1245	191	23	(	(	PUNCT
ejpam-1245	191	24	y	y	PROPN
ejpam-1245	191	25	,	,	PUNCT
ejpam-1245	191	26	ρ	ρ	PROPN
ejpam-1245	191	27	)	)	PUNCT
ejpam-1245	191	28	be	be	VERB
ejpam-1245	191	29	the	the	DET
ejpam-1245	191	30	function	function	NOUN
ejpam-1245	191	31	defined	define	VERB
ejpam-1245	191	32	by	by	ADP
ejpam-1245	191	33	g(x	g(x	NOUN
ejpam-1245	191	34	)	)	PUNCT
ejpam-1245	192	1	=	=	PRON
ejpam-1245	192	2	(	(	PUNCT
ejpam-1245	192	3	a	a	PRON
ejpam-1245	192	4	x	x	X
ejpam-1245	192	5	=	=	SYM
ejpam-1245	192	6	2	2	NUM
ejpam-1245	192	7	b	b	NOUN
ejpam-1245	192	8	x	x	SYM
ejpam-1245	192	9	=	=	SYM
ejpam-1245	192	10	1	1	NUM
ejpam-1245	192	11	then	then	ADV
ejpam-1245	192	12	f	f	PROPN
ejpam-1245	192	13	is	be	AUX
ejpam-1245	192	14	continuous	continuous	ADJ
ejpam-1245	192	15	(	(	PUNCT
ejpam-1245	192	16	hence	hence	ADV
ejpam-1245	192	17	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	192	18	)	)	PUNCT
ejpam-1245	192	19	and	and	CCONJ
ejpam-1245	192	20	g	g	PROPN
ejpam-1245	192	21	is	be	AUX
ejpam-1245	192	22	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	192	23	.	.	PUNCT
ejpam-1245	193	1	however	however	ADV
ejpam-1245	193	2	g	g	PROPN
ejpam-1245	193	3	◦	◦	PROPN
ejpam-1245	193	4	f	f	PROPN
ejpam-1245	193	5	is	be	AUX
ejpam-1245	193	6	not	not	PART
ejpam-1245	193	7	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	193	8	,	,	PUNCT
ejpam-1245	193	9	because	because	SCONJ
ejpam-1245	193	10	(	(	PUNCT
ejpam-1245	193	11	g	g	PROPN
ejpam-1245	193	12	◦	◦	NOUN
ejpam-1245	193	13	f	f	PROPN
ejpam-1245	193	14	)	)	PUNCT
ejpam-1245	193	15	−1({a	−1({a	PROPN
ejpam-1245	193	16	}	}	PUNCT
ejpam-1245	193	17	)	)	PUNCT
ejpam-1245	194	1	=	=	PUNCT
ejpam-1245	194	2	q	q	PROPN
ejpam-1245	194	3	/∈ωβo(x	/∈ωβo(x	PUNCT
ejpam-1245	194	4	,	,	PUNCT
ejpam-1245	194	5	τ	τ	PROPN
ejpam-1245	194	6	)	)	PUNCT
ejpam-1245	194	7	.	.	PUNCT
ejpam-1245	195	1	proposition	proposition	NOUN
ejpam-1245	195	2	3	3	NUM
ejpam-1245	195	3	.	.	PUNCT
ejpam-1245	196	1	if	if	SCONJ
ejpam-1245	196	2	f	f	PROPN
ejpam-1245	196	3	:	:	PUNCT
ejpam-1245	196	4	(	(	PUNCT
ejpam-1245	196	5	x	x	X
ejpam-1245	196	6	,	,	PUNCT
ejpam-1245	196	7	τ)→	τ)→	PROPN
ejpam-1245	196	8	(	(	PUNCT
ejpam-1245	196	9	y	y	PROPN
ejpam-1245	196	10	,	,	PUNCT
ejpam-1245	196	11	σ	σ	PROPN
ejpam-1245	196	12	)	)	PUNCT
ejpam-1245	196	13	is	be	AUX
ejpam-1245	196	14	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	196	15	and	and	CCONJ
ejpam-1245	196	16	g	g	NOUN
ejpam-1245	196	17	:	:	PUNCT
ejpam-1245	196	18	(	(	PUNCT
ejpam-1245	196	19	y	y	NOUN
ejpam-1245	196	20	,	,	PUNCT
ejpam-1245	196	21	σ)→	σ)→	PROPN
ejpam-1245	196	22	(	(	PUNCT
ejpam-1245	196	23	z	z	PROPN
ejpam-1245	196	24	,	,	PUNCT
ejpam-1245	196	25	ρ	ρ	PROPN
ejpam-1245	196	26	)	)	PUNCT
ejpam-1245	196	27	is	be	AUX
ejpam-1245	196	28	continuous	continuous	ADJ
ejpam-1245	196	29	,	,	PUNCT
ejpam-1245	196	30	then	then	ADV
ejpam-1245	196	31	g	g	PROPN
ejpam-1245	196	32	◦	◦	NOUN
ejpam-1245	196	33	f	f	X
ejpam-1245	196	34	:	:	PUNCT
ejpam-1245	196	35	(	(	PUNCT
ejpam-1245	196	36	x	x	X
ejpam-1245	196	37	,	,	PUNCT
ejpam-1245	196	38	τ)→	τ)→	PROPN
ejpam-1245	196	39	(	(	PUNCT
ejpam-1245	196	40	z	z	NOUN
ejpam-1245	196	41	,	,	PUNCT
ejpam-1245	196	42	ρ	ρ	PROPN
ejpam-1245	196	43	)	)	PUNCT
ejpam-1245	196	44	is	be	AUX
ejpam-1245	196	45	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	196	46	.	.	PUNCT
ejpam-1245	197	1	proof	proof	NOUN
ejpam-1245	197	2	.	.	PUNCT
ejpam-1245	198	1	let	let	VERB
ejpam-1245	198	2	x	x	PUNCT
ejpam-1245	198	3	∈	∈	PROPN
ejpam-1245	198	4	x	x	X
ejpam-1245	198	5	and	and	CCONJ
ejpam-1245	198	6	v	v	ADP
ejpam-1245	198	7	∈	∈	PROPN
ejpam-1245	198	8	ρ	ρ	NOUN
ejpam-1245	198	9	with	with	ADP
ejpam-1245	198	10	(	(	PUNCT
ejpam-1245	198	11	g	g	PROPN
ejpam-1245	198	12	◦	◦	NOUN
ejpam-1245	198	13	f	f	PROPN
ejpam-1245	198	14	)	)	PUNCT
ejpam-1245	198	15	(	(	PUNCT
ejpam-1245	198	16	x	x	X
ejpam-1245	198	17	)	)	PUNCT
ejpam-1245	198	18	∈	∈	PROPN
ejpam-1245	198	19	v	v	NOUN
ejpam-1245	198	20	and	and	CCONJ
ejpam-1245	198	21	f	f	PROPN
ejpam-1245	198	22	(	(	PUNCT
ejpam-1245	198	23	x	x	X
ejpam-1245	198	24	)	)	PUNCT
ejpam-1245	198	25	∈	∈	PROPN
ejpam-1245	198	26	y	y	PROPN
ejpam-1245	198	27	,	,	PUNCT
ejpam-1245	198	28	since	since	SCONJ
ejpam-1245	198	29	g	g	PROPN
ejpam-1245	198	30	is	be	AUX
ejpam-1245	198	31	continuous	continuous	ADJ
ejpam-1245	198	32	,	,	PUNCT
ejpam-1245	198	33	there	there	PRON
ejpam-1245	198	34	exists	exist	VERB
ejpam-1245	198	35	open	open	ADJ
ejpam-1245	198	36	sets	set	NOUN
ejpam-1245	198	37	w	w	VERB
ejpam-1245	198	38	in	in	ADP
ejpam-1245	198	39	(	(	PUNCT
ejpam-1245	198	40	z	z	NOUN
ejpam-1245	198	41	,	,	PUNCT
ejpam-1245	198	42	ρ	ρ	PROPN
ejpam-1245	198	43	)	)	PUNCT
ejpam-1245	198	44	with	with	ADP
ejpam-1245	198	45	f	f	PROPN
ejpam-1245	198	46	(	(	PUNCT
ejpam-1245	198	47	x	x	X
ejpam-1245	198	48	)	)	PUNCT
ejpam-1245	198	49	∈	∈	PROPN
ejpam-1245	198	50	w	w	NOUN
ejpam-1245	198	51	and	and	CCONJ
ejpam-1245	198	52	g(w	g(w	PROPN
ejpam-1245	198	53	)	)	PUNCT
ejpam-1245	199	1	⊆	⊆	NUM
ejpam-1245	199	2	v	v	NOUN
ejpam-1245	199	3	.	.	PUNCT
ejpam-1245	200	1	moreover	moreover	ADV
ejpam-1245	200	2	f	f	PROPN
ejpam-1245	200	3	is	be	AUX
ejpam-1245	200	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	200	5	there	there	PRON
ejpam-1245	200	6	existsωβo(x	existsωβo(x	NUM
ejpam-1245	200	7	,	,	PUNCT
ejpam-1245	200	8	τ	τ	X
ejpam-1245	200	9	)	)	PUNCT
ejpam-1245	200	10	say	say	VERB
ejpam-1245	200	11	u	u	NOUN
ejpam-1245	200	12	containing	contain	VERB
ejpam-1245	200	13	x	x	PUNCT
ejpam-1245	200	14	such	such	ADJ
ejpam-1245	200	15	that	that	SCONJ
ejpam-1245	200	16	f	f	PROPN
ejpam-1245	200	17	(	(	PUNCT
ejpam-1245	200	18	u)⊆w	u)⊆w	PROPN
ejpam-1245	200	19	.	.	PUNCT
ejpam-1245	201	1	now	now	ADV
ejpam-1245	201	2	(	(	PUNCT
ejpam-1245	201	3	g	g	PROPN
ejpam-1245	201	4	◦	◦	NOUN
ejpam-1245	201	5	f	f	PROPN
ejpam-1245	201	6	)	)	PUNCT
ejpam-1245	201	7	(	(	PUNCT
ejpam-1245	201	8	u)⊆	u)⊆	PROPN
ejpam-1245	201	9	g(w	g(w	PROPN
ejpam-1245	201	10	)	)	PUNCT
ejpam-1245	201	11	⊆	⊆	NUM
ejpam-1245	201	12	v	v	NOUN
ejpam-1245	201	13	.	.	PUNCT
ejpam-1245	202	1	we	we	PRON
ejpam-1245	202	2	note	note	VERB
ejpam-1245	202	3	that	that	SCONJ
ejpam-1245	202	4	proposition	proposition	NOUN
ejpam-1245	202	5	3	3	NUM
ejpam-1245	202	6	is	be	AUX
ejpam-1245	202	7	not	not	PART
ejpam-1245	202	8	true	true	ADJ
ejpam-1245	202	9	if	if	SCONJ
ejpam-1245	202	10	g	g	PROPN
ejpam-1245	202	11	is	be	AUX
ejpam-1245	202	12	assumed	assume	VERB
ejpam-1245	202	13	to	to	PART
ejpam-1245	202	14	be	be	AUX
ejpam-1245	202	15	only	only	ADV
ejpam-1245	202	16	ω−continuous	ω−continuous	ADJ
ejpam-1245	202	17	or	or	CCONJ
ejpam-1245	202	18	β−continuous	β−continuous	ADJ
ejpam-1245	202	19	as	as	SCONJ
ejpam-1245	202	20	it	it	PRON
ejpam-1245	202	21	is	be	AUX
ejpam-1245	202	22	shown	show	VERB
ejpam-1245	202	23	in	in	ADP
ejpam-1245	202	24	the	the	DET
ejpam-1245	202	25	next	next	ADJ
ejpam-1245	202	26	example	example	NOUN
ejpam-1245	202	27	.	.	PUNCT
ejpam-1245	203	1	example	example	NOUN
ejpam-1245	204	1	7	7	X
ejpam-1245	204	2	.	.	X
ejpam-1245	205	1	consider	consider	VERB
ejpam-1245	205	2	x	x	X
ejpam-1245	205	3	=	=	SYM
ejpam-1245	205	4	r	r	NOUN
ejpam-1245	205	5	with	with	ADP
ejpam-1245	205	6	the	the	DET
ejpam-1245	205	7	topology	topology	NOUN
ejpam-1245	205	8	τ	τ	PROPN
ejpam-1245	205	9	=	=	PUNCT
ejpam-1245	205	10	τcoc	τcoc	PROPN
ejpam-1245	205	11	,	,	PUNCT
ejpam-1245	205	12	y	y	PROPN
ejpam-1245	205	13	=	=	PUNCT
ejpam-1245	205	14	{	{	PUNCT
ejpam-1245	205	15	a	a	PRON
ejpam-1245	205	16	,	,	PUNCT
ejpam-1245	205	17	b	b	NOUN
ejpam-1245	205	18	,	,	PUNCT
ejpam-1245	205	19	c	c	NOUN
ejpam-1245	205	20	}	}	PUNCT
ejpam-1245	205	21	with	with	ADP
ejpam-1245	205	22	the	the	DET
ejpam-1245	205	23	topology	topology	NOUN
ejpam-1245	205	24	σ	σ	NOUN
ejpam-1245	205	25	=	=	SYM
ejpam-1245	205	26	{	{	PUNCT
ejpam-1245	205	27	φ	φ	PROPN
ejpam-1245	205	28	,	,	PUNCT
ejpam-1245	205	29	y	y	PROPN
ejpam-1245	205	30	,	,	PUNCT
ejpam-1245	205	31	{	{	PUNCT
ejpam-1245	205	32	a	a	X
ejpam-1245	205	33	}	}	PUNCT
ejpam-1245	205	34	,	,	PUNCT
ejpam-1245	205	35	{	{	PUNCT
ejpam-1245	205	36	b	b	NOUN
ejpam-1245	205	37	}	}	PUNCT
ejpam-1245	205	38	,	,	PUNCT
ejpam-1245	205	39	{	{	PUNCT
ejpam-1245	205	40	a	a	PRON
ejpam-1245	205	41	,	,	PUNCT
ejpam-1245	205	42	b	b	NOUN
ejpam-1245	205	43	}	}	PUNCT
ejpam-1245	205	44	}	}	PUNCT
ejpam-1245	205	45	and	and	CCONJ
ejpam-1245	205	46	z	z	NOUN
ejpam-1245	205	47	=	=	PRON
ejpam-1245	206	1	{	{	PUNCT
ejpam-1245	206	2	1,2,3,4}with	1,2,3,4}with	NUM
ejpam-1245	206	3	the	the	DET
ejpam-1245	206	4	topologyρ	topologyρ	NOUN
ejpam-1245	206	5	=	=	SYM
ejpam-1245	206	6	{	{	PUNCT
ejpam-1245	206	7	φ	φ	PROPN
ejpam-1245	206	8	,	,	PUNCT
ejpam-1245	206	9	z	z	NOUN
ejpam-1245	206	10	,	,	PUNCT
ejpam-1245	206	11	{	{	PUNCT
ejpam-1245	206	12	1	1	NUM
ejpam-1245	206	13	}	}	PUNCT
ejpam-1245	206	14	,	,	PUNCT
ejpam-1245	206	15	{	{	PUNCT
ejpam-1245	206	16	1,2	1,2	NUM
ejpam-1245	206	17	}	}	PUNCT
ejpam-1245	206	18	,	,	PUNCT
ejpam-1245	206	19	{	{	PUNCT
ejpam-1245	206	20	1,2,3	1,2,3	NUM
ejpam-1245	206	21	}	}	PUNCT
ejpam-1245	206	22	}	}	PUNCT
ejpam-1245	206	23	.	.	PUNCT
ejpam-1245	207	1	let	let	VERB
ejpam-1245	207	2	f	f	NOUN
ejpam-1245	207	3	:	:	PUNCT
ejpam-1245	207	4	(	(	PUNCT
ejpam-1245	207	5	x	x	X
ejpam-1245	207	6	,	,	PUNCT
ejpam-1245	207	7	τ)→	τ)→	PROPN
ejpam-1245	207	8	(	(	PUNCT
ejpam-1245	207	9	y	y	PROPN
ejpam-1245	207	10	,	,	PUNCT
ejpam-1245	207	11	σ	σ	PROPN
ejpam-1245	207	12	)	)	PUNCT
ejpam-1245	207	13	be	be	VERB
ejpam-1245	207	14	the	the	DET
ejpam-1245	207	15	function	function	NOUN
ejpam-1245	207	16	define	define	NOUN
ejpam-1245	207	17	by	by	ADP
ejpam-1245	207	18	f	f	PROPN
ejpam-1245	207	19	(	(	PUNCT
ejpam-1245	207	20	x	x	NOUN
ejpam-1245	207	21	)	)	PUNCT
ejpam-1245	208	1	=	=	SYM
ejpam-1245	208	2	(	(	PUNCT
ejpam-1245	208	3	a	a	DET
ejpam-1245	208	4	x	x	X
ejpam-1245	208	5	∈	∈	PROPN
ejpam-1245	208	6	r−q	r−q	NOUN
ejpam-1245	208	7	c	c	NOUN
ejpam-1245	208	8	x	x	PUNCT
ejpam-1245	208	9	∈q	∈q	NOUN
ejpam-1245	208	10	and	and	CCONJ
ejpam-1245	208	11	g	g	NOUN
ejpam-1245	208	12	:	:	PUNCT
ejpam-1245	208	13	(	(	PUNCT
ejpam-1245	208	14	y	y	NOUN
ejpam-1245	208	15	,	,	PUNCT
ejpam-1245	208	16	σ)→	σ)→	PROPN
ejpam-1245	208	17	(	(	PUNCT
ejpam-1245	208	18	z	z	PROPN
ejpam-1245	208	19	,	,	PUNCT
ejpam-1245	208	20	ρ	ρ	PROPN
ejpam-1245	208	21	)	)	PUNCT
ejpam-1245	208	22	be	be	VERB
ejpam-1245	208	23	the	the	DET
ejpam-1245	208	24	function	function	NOUN
ejpam-1245	208	25	define	define	NOUN
ejpam-1245	208	26	by	by	ADP
ejpam-1245	208	27	g(x	g(x	NOUN
ejpam-1245	208	28	)	)	PUNCT
ejpam-1245	209	1	=	=	PUNCT
ejpam-1245	209	2			PROPN
ejpam-1245	209	3			X
ejpam-1245	209	4			ADJ
ejpam-1245	209	5	1	1	NUM
ejpam-1245	209	6	x	x	SYM
ejpam-1245	209	7	=	=	PUNCT
ejpam-1245	209	8	a	a	DET
ejpam-1245	209	9	3	3	NUM
ejpam-1245	209	10	x	x	X
ejpam-1245	209	11	=	=	SYM
ejpam-1245	209	12	b	b	X
ejpam-1245	209	13	2	2	NUM
ejpam-1245	209	14	x	x	X
ejpam-1245	209	15	=	=	SYM
ejpam-1245	209	16	c	c	NOUN
ejpam-1245	209	17	then	then	ADV
ejpam-1245	209	18	f	f	PROPN
ejpam-1245	209	19	is	be	AUX
ejpam-1245	209	20	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	209	21	,	,	PUNCT
ejpam-1245	209	22	g	g	PROPN
ejpam-1245	209	23	is	be	AUX
ejpam-1245	209	24	ω−continuous	ω−continuous	ADJ
ejpam-1245	209	25	and	and	CCONJ
ejpam-1245	209	26	β−continuous	β−continuous	PRON
ejpam-1245	209	27	function	function	NOUN
ejpam-1245	209	28	but	but	CCONJ
ejpam-1245	209	29	g	g	PROPN
ejpam-1245	209	30	◦	◦	NOUN
ejpam-1245	209	31	f	f	PROPN
ejpam-1245	209	32	is	be	AUX
ejpam-1245	209	33	not	not	PART
ejpam-1245	209	34	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	209	35	since	since	SCONJ
ejpam-1245	209	36	(	(	PUNCT
ejpam-1245	209	37	g	g	PROPN
ejpam-1245	209	38	◦	◦	NOUN
ejpam-1245	209	39	f	f	PROPN
ejpam-1245	209	40	)	)	PUNCT
ejpam-1245	209	41	−1(2	−1(2	PROPN
ejpam-1245	209	42	)	)	PUNCT
ejpam-1245	210	1	=	=	PUNCT
ejpam-1245	211	1	q	q	PROPN
ejpam-1245	211	2	/∈ωβo(x	/∈ωβo(x	PUNCT
ejpam-1245	211	3	,	,	PUNCT
ejpam-1245	211	4	τ	τ	PROPN
ejpam-1245	211	5	)	)	PUNCT
ejpam-1245	211	6	.	.	PUNCT
ejpam-1245	212	1	corollary	corollary	ADJ
ejpam-1245	212	2	3	3	X
ejpam-1245	212	3	.	.	PUNCT
ejpam-1245	213	1	if	if	SCONJ
ejpam-1245	213	2	f	f	PROPN
ejpam-1245	213	3	:	:	PUNCT
ejpam-1245	213	4	(	(	PUNCT
ejpam-1245	213	5	x	x	X
ejpam-1245	213	6	,	,	PUNCT
ejpam-1245	213	7	τ)→	τ)→	PROPN
ejpam-1245	213	8	∏	∏	NUM
ejpam-1245	213	9	α∈∆	α∈∆	X
ejpam-1245	213	10	xα	xα	X
ejpam-1245	213	11	is	be	AUX
ejpam-1245	213	12	an	an	DET
ejpam-1245	213	13	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	213	14	function	function	NOUN
ejpam-1245	213	15	from	from	ADP
ejpam-1245	213	16	a	a	DET
ejpam-1245	213	17	space	space	NOUN
ejpam-1245	213	18	(	(	PUNCT
ejpam-1245	213	19	x	x	X
ejpam-1245	213	20	,	,	PUNCT
ejpam-1245	213	21	τ	τ	PROPN
ejpam-1245	213	22	)	)	PUNCT
ejpam-1245	213	23	into	into	ADP
ejpam-1245	213	24	a	a	DET
ejpam-1245	213	25	product	product	NOUN
ejpam-1245	213	26	space	space	NOUN
ejpam-1245	213	27	∏	∏	PROPN
ejpam-1245	213	28	α∈∆	α∈∆	X
ejpam-1245	213	29	xα	xα	PROPN
ejpam-1245	213	30	,	,	PUNCT
ejpam-1245	213	31	then	then	ADV
ejpam-1245	213	32	pα	pα	VERB
ejpam-1245	213	33	◦	◦	PROPN
ejpam-1245	213	34	f	f	PROPN
ejpam-1245	213	35	is	be	AUX
ejpam-1245	213	36	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	213	37	for	for	ADP
ejpam-1245	213	38	each	each	DET
ejpam-1245	213	39	α	α	NOUN
ejpam-1245	213	40	∈	∈	NOUN
ejpam-1245	213	41	∆	∆	PROPN
ejpam-1245	213	42	,	,	PUNCT
ejpam-1245	213	43	where	where	SCONJ
ejpam-1245	213	44	pα	pα	NOUN
ejpam-1245	213	45	is	be	AUX
ejpam-1245	213	46	the	the	DET
ejpam-1245	213	47	projection	projection	NOUN
ejpam-1245	213	48	function	function	NOUN
ejpam-1245	213	49	from	from	ADP
ejpam-1245	213	50	the	the	DET
ejpam-1245	213	51	product	product	NOUN
ejpam-1245	213	52	space	space	NOUN
ejpam-1245	213	53	∏	∏	PROPN
ejpam-1245	213	54	α∈∆	α∈∆	X
ejpam-1245	214	1	xα	xα	INTJ
ejpam-1245	214	2	onto	onto	ADP
ejpam-1245	214	3	the	the	DET
ejpam-1245	214	4	space	space	NOUN
ejpam-1245	214	5	xα	xα	INTJ
ejpam-1245	214	6	for	for	ADP
ejpam-1245	214	7	each	each	DET
ejpam-1245	214	8	α	α	PRON
ejpam-1245	214	9	∈∆.	∈∆.	PROPN
ejpam-1245	214	10	theorem	theorem	VERB
ejpam-1245	214	11	6	6	NUM
ejpam-1245	214	12	.	.	PUNCT
ejpam-1245	215	1	let	let	VERB
ejpam-1245	215	2	x	x	PRON
ejpam-1245	215	3	and	and	CCONJ
ejpam-1245	215	4	y	y	PROPN
ejpam-1245	215	5	be	be	AUX
ejpam-1245	215	6	a	a	DET
ejpam-1245	215	7	topological	topological	ADJ
ejpam-1245	215	8	spaces	space	NOUN
ejpam-1245	215	9	,	,	PUNCT
ejpam-1245	215	10	let	let	VERB
ejpam-1245	215	11	f	f	X
ejpam-1245	215	12	:	:	PUNCT
ejpam-1245	215	13	(	(	PUNCT
ejpam-1245	215	14	x	x	X
ejpam-1245	215	15	,	,	PUNCT
ejpam-1245	215	16	τ	τ	PROPN
ejpam-1245	215	17	)	)	PUNCT
ejpam-1245	215	18	→	→	SYM
ejpam-1245	215	19	(	(	PUNCT
ejpam-1245	215	20	y	y	PROPN
ejpam-1245	215	21	,	,	PUNCT
ejpam-1245	215	22	σ	σ	PROPN
ejpam-1245	215	23	)	)	PUNCT
ejpam-1245	215	24	be	be	AUX
ejpam-1245	215	25	a	a	DET
ejpam-1245	215	26	function	function	NOUN
ejpam-1245	215	27	and	and	CCONJ
ejpam-1245	215	28	g	g	NOUN
ejpam-1245	215	29	:	:	PUNCT
ejpam-1245	215	30	(	(	PUNCT
ejpam-1245	215	31	x	x	X
ejpam-1245	215	32	,	,	PUNCT
ejpam-1245	215	33	τ)→	τ)→	PROPN
ejpam-1245	215	34	(	(	PUNCT
ejpam-1245	215	35	x	x	SYM
ejpam-1245	215	36	×	×	PROPN
ejpam-1245	215	37	y	y	PROPN
ejpam-1245	215	38	,	,	PUNCT
ejpam-1245	215	39	τ×σ	τ×σ	PUNCT
ejpam-1245	215	40	)	)	PUNCT
ejpam-1245	215	41	be	be	VERB
ejpam-1245	215	42	the	the	DET
ejpam-1245	215	43	graph	graph	NOUN
ejpam-1245	215	44	function	function	NOUN
ejpam-1245	215	45	of	of	ADP
ejpam-1245	215	46	f	f	PROPN
ejpam-1245	215	47	given	give	VERB
ejpam-1245	215	48	by	by	ADP
ejpam-1245	215	49	g(x	g(x	NOUN
ejpam-1245	215	50	)	)	PUNCT
ejpam-1245	216	1	=	=	SYM
ejpam-1245	216	2	(	(	PUNCT
ejpam-1245	216	3	x	x	INTJ
ejpam-1245	216	4	,	,	PUNCT
ejpam-1245	216	5	f	f	PROPN
ejpam-1245	216	6	(	(	PUNCT
ejpam-1245	216	7	x	x	NOUN
ejpam-1245	216	8	)	)	PUNCT
ejpam-1245	216	9	)	)	PUNCT
ejpam-1245	216	10	for	for	ADP
ejpam-1245	216	11	every	every	DET
ejpam-1245	216	12	point	point	NOUN
ejpam-1245	216	13	x	x	X
ejpam-1245	216	14	∈	∈	NOUN
ejpam-1245	216	15	x	x	X
ejpam-1245	216	16	.	.	PUNCT
ejpam-1245	217	1	then	then	ADV
ejpam-1245	217	2	g	g	PROPN
ejpam-1245	217	3	is	be	AUX
ejpam-1245	217	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	217	5	if	if	SCONJ
ejpam-1245	217	6	and	and	CCONJ
ejpam-1245	217	7	only	only	ADV
ejpam-1245	217	8	if	if	SCONJ
ejpam-1245	217	9	f	f	PROPN
ejpam-1245	217	10	is	be	AUX
ejpam-1245	217	11	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	217	12	.	.	PUNCT
ejpam-1245	218	1	proof	proof	NOUN
ejpam-1245	218	2	.	.	PUNCT
ejpam-1245	219	1	assume	assume	VERB
ejpam-1245	219	2	that	that	SCONJ
ejpam-1245	219	3	g	g	PROPN
ejpam-1245	219	4	is	be	AUX
ejpam-1245	219	5	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	219	6	.	.	PUNCT
ejpam-1245	220	1	now	now	ADV
ejpam-1245	220	2	f	f	X
ejpam-1245	220	3	=	=	SYM
ejpam-1245	220	4	py	py	PROPN
ejpam-1245	220	5	◦	◦	NOUN
ejpam-1245	220	6	g	g	PRON
ejpam-1245	220	7	where	where	SCONJ
ejpam-1245	220	8	py	py	INTJ
ejpam-1245	220	9	:	:	PUNCT
ejpam-1245	220	10	x	x	X
ejpam-1245	220	11	×y	×y	X
ejpam-1245	220	12	→	→	SYM
ejpam-1245	220	13	y	y	PROPN
ejpam-1245	220	14	,	,	PUNCT
ejpam-1245	220	15	then	then	ADV
ejpam-1245	220	16	f	f	PROPN
ejpam-1245	220	17	is	be	AUX
ejpam-1245	220	18	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	220	19	by	by	ADP
ejpam-1245	220	20	corollary	corollary	ADJ
ejpam-1245	220	21	3	3	NUM
ejpam-1245	220	22	.	.	PUNCT
ejpam-1245	221	1	conversely	conversely	ADV
ejpam-1245	221	2	,	,	PUNCT
ejpam-1245	221	3	assume	assume	VERB
ejpam-1245	221	4	that	that	SCONJ
ejpam-1245	221	5	f	f	PROPN
ejpam-1245	221	6	is	be	AUX
ejpam-1245	221	7	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	221	8	.	.	PUNCT
ejpam-1245	222	1	let	let	VERB
ejpam-1245	222	2	x	x	SYM
ejpam-1245	222	3	∈	∈	PROPN
ejpam-1245	222	4	x	x	X
ejpam-1245	222	5	and	and	CCONJ
ejpam-1245	222	6	h.	h.	PROPN
ejpam-1245	222	7	aljarrah	aljarrah	PROPN
ejpam-1245	222	8	,	,	PUNCT
ejpam-1245	222	9	m.	m.	NOUN
ejpam-1245	222	10	noorani	noorani	PROPN
ejpam-1245	222	11	/	/	SYM
ejpam-1245	222	12	eur	eur	PROPN
ejpam-1245	222	13	.	.	PUNCT
ejpam-1245	223	1	j.	j.	PROPN
ejpam-1245	223	2	pure	pure	PROPN
ejpam-1245	223	3	appl	appl	PROPN
ejpam-1245	223	4	.	.	PROPN
ejpam-1245	223	5	math	math	PROPN
ejpam-1245	223	6	,	,	PUNCT
ejpam-1245	223	7	5	5	NUM
ejpam-1245	223	8	(	(	PUNCT
ejpam-1245	223	9	2012	2012	NUM
ejpam-1245	223	10	)	)	PUNCT
ejpam-1245	223	11	,	,	PUNCT
ejpam-1245	223	12	129	129	NUM
ejpam-1245	223	13	-	-	SYM
ejpam-1245	223	14	140	140	NUM
ejpam-1245	223	15	135	135	NUM
ejpam-1245	223	16	w	w	NOUN
ejpam-1245	223	17	be	be	AUX
ejpam-1245	223	18	any	any	DET
ejpam-1245	223	19	open	open	ADJ
ejpam-1245	223	20	set	set	NOUN
ejpam-1245	223	21	in	in	ADP
ejpam-1245	223	22	x	x	SYM
ejpam-1245	223	23	×	×	PROPN
ejpam-1245	223	24	y	y	NOUN
ejpam-1245	223	25	containing	contain	VERB
ejpam-1245	223	26	g(x	g(x	NOUN
ejpam-1245	223	27	)	)	PUNCT
ejpam-1245	223	28	.	.	PUNCT
ejpam-1245	224	1	then	then	ADV
ejpam-1245	224	2	there	there	PRON
ejpam-1245	224	3	exist	exist	VERB
ejpam-1245	224	4	open	open	ADJ
ejpam-1245	224	5	sets	set	NOUN
ejpam-1245	224	6	u	u	NOUN
ejpam-1245	224	7	⊆	⊆	NUM
ejpam-1245	224	8	x	x	NOUN
ejpam-1245	224	9	and	and	CCONJ
ejpam-1245	224	10	v	v	ADP
ejpam-1245	224	11	⊆	⊆	NUM
ejpam-1245	224	12	y	y	PROPN
ejpam-1245	224	13	such	such	ADJ
ejpam-1245	224	14	that	that	DET
ejpam-1245	224	15	g(x	g(x	NOUN
ejpam-1245	224	16	)	)	PUNCT
ejpam-1245	225	1	=	=	SYM
ejpam-1245	225	2	(	(	PUNCT
ejpam-1245	225	3	x	x	INTJ
ejpam-1245	225	4	,	,	PUNCT
ejpam-1245	225	5	f	f	PROPN
ejpam-1245	225	6	(	(	PUNCT
ejpam-1245	225	7	x	x	NOUN
ejpam-1245	225	8	)	)	PUNCT
ejpam-1245	225	9	)	)	PUNCT
ejpam-1245	226	1	∈	∈	PROPN
ejpam-1245	226	2	u	u	NOUN
ejpam-1245	226	3	×	×	NOUN
ejpam-1245	226	4	v	v	ADP
ejpam-1245	226	5	⊆w	⊆w	NOUN
ejpam-1245	226	6	.	.	PUNCT
ejpam-1245	227	1	since	since	SCONJ
ejpam-1245	227	2	f	f	PROPN
ejpam-1245	227	3	is	be	AUX
ejpam-1245	227	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	227	5	,	,	PUNCT
ejpam-1245	227	6	there	there	PRON
ejpam-1245	227	7	exists	exist	VERB
ejpam-1245	227	8	ωβo(x	ωβo(x	PROPN
ejpam-1245	227	9	,	,	PUNCT
ejpam-1245	227	10	τ	τ	X
ejpam-1245	227	11	)	)	PUNCT
ejpam-1245	227	12	sets	set	VERB
ejpam-1245	227	13	u1	u1	NOUN
ejpam-1245	227	14	in	in	ADP
ejpam-1245	227	15	containing	contain	VERB
ejpam-1245	227	16	x	x	PUNCT
ejpam-1245	227	17	such	such	ADJ
ejpam-1245	227	18	that	that	SCONJ
ejpam-1245	227	19	f	f	PROPN
ejpam-1245	227	20	(	(	PUNCT
ejpam-1245	227	21	u1	u1	PROPN
ejpam-1245	227	22	)	)	PUNCT
ejpam-1245	227	23	⊆	⊆	NUM
ejpam-1245	227	24	v	v	NOUN
ejpam-1245	227	25	.	.	PUNCT
ejpam-1245	228	1	put	put	VERB
ejpam-1245	228	2	h	h	NOUN
ejpam-1245	229	1	=	=	SYM
ejpam-1245	229	2	u	u	NOUN
ejpam-1245	229	3	∩	∩	NOUN
ejpam-1245	229	4	u1	u1	NOUN
ejpam-1245	229	5	.	.	PUNCT
ejpam-1245	230	1	then	then	ADV
ejpam-1245	230	2	h	h	PROPN
ejpam-1245	230	3	∈	∈	PROPN
ejpam-1245	230	4	ωβo(x	ωβo(x	PROPN
ejpam-1245	230	5	,	,	PUNCT
ejpam-1245	230	6	τ	τ	PROPN
ejpam-1245	230	7	)	)	PUNCT
ejpam-1245	230	8	,	,	PUNCT
ejpam-1245	230	9	by	by	ADP
ejpam-1245	230	10	lemma	lemma	PROPN
ejpam-1245	230	11	1(ii	1(ii	NUM
ejpam-1245	230	12	)	)	PUNCT
ejpam-1245	230	13	,	,	PUNCT
ejpam-1245	230	14	such	such	ADJ
ejpam-1245	230	15	that	that	SCONJ
ejpam-1245	230	16	x	x	SYM
ejpam-1245	230	17	∈	∈	PROPN
ejpam-1245	230	18	h	h	NOUN
ejpam-1245	230	19	and	and	CCONJ
ejpam-1245	230	20	f	f	PROPN
ejpam-1245	230	21	(	(	PUNCT
ejpam-1245	230	22	h)⊆	h)⊆	PROPN
ejpam-1245	230	23	v	v	NOUN
ejpam-1245	230	24	.	.	PUNCT
ejpam-1245	231	1	therefore	therefore	ADV
ejpam-1245	231	2	we	we	PRON
ejpam-1245	231	3	have	have	VERB
ejpam-1245	231	4	g(h)⊆	g(h)⊆	NOUN
ejpam-1245	231	5	u	u	NOUN
ejpam-1245	231	6	×	×	NOUN
ejpam-1245	231	7	v	v	ADP
ejpam-1245	231	8	⊆w	⊆w	NOUN
ejpam-1245	231	9	.	.	PUNCT
ejpam-1245	232	1	thus	thus	ADV
ejpam-1245	232	2	g	g	PROPN
ejpam-1245	232	3	is	be	AUX
ejpam-1245	232	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	232	5	.	.	PUNCT
ejpam-1245	233	1	definition	definition	NOUN
ejpam-1245	233	2	6	6	NUM
ejpam-1245	233	3	.	.	PUNCT
ejpam-1245	234	1	[	[	X
ejpam-1245	234	2	8	8	NUM
ejpam-1245	234	3	]	]	X
ejpam-1245	234	4	a	a	DET
ejpam-1245	234	5	function	function	NOUN
ejpam-1245	234	6	f	f	NOUN
ejpam-1245	234	7	:	:	PUNCT
ejpam-1245	234	8	(	(	PUNCT
ejpam-1245	234	9	x	x	X
ejpam-1245	234	10	,	,	PUNCT
ejpam-1245	234	11	τ)→	τ)→	PROPN
ejpam-1245	234	12	(	(	PUNCT
ejpam-1245	234	13	y	y	PROPN
ejpam-1245	234	14	,	,	PUNCT
ejpam-1245	234	15	σ	σ	PROPN
ejpam-1245	234	16	)	)	PUNCT
ejpam-1245	234	17	is	be	AUX
ejpam-1245	234	18	called	call	VERB
ejpam-1245	234	19	pre	pre	ADJ
ejpam-1245	234	20	-	-	ADJ
ejpam-1245	234	21	semi	semi	ADJ
ejpam-1245	234	22	-	-	ADJ
ejpam-1245	234	23	preopen	preopen	ADJ
ejpam-1245	234	24	if	if	SCONJ
ejpam-1245	234	25	the	the	DET
ejpam-1245	234	26	image	image	NOUN
ejpam-1245	234	27	of	of	ADP
ejpam-1245	234	28	each	each	DET
ejpam-1245	234	29	semi	semi	ADJ
ejpam-1245	234	30	-	-	ADJ
ejpam-1245	234	31	preopen	preopen	ADJ
ejpam-1245	234	32	set	set	NOUN
ejpam-1245	234	33	in	in	ADP
ejpam-1245	234	34	x	x	PRON
ejpam-1245	234	35	is	be	AUX
ejpam-1245	234	36	a	a	DET
ejpam-1245	234	37	semi	semi	ADJ
ejpam-1245	234	38	-	-	ADJ
ejpam-1245	234	39	preopen	preopen	ADJ
ejpam-1245	234	40	set	set	NOUN
ejpam-1245	234	41	in	in	ADP
ejpam-1245	234	42	y	y	PROPN
ejpam-1245	234	43	.	.	PUNCT
ejpam-1245	235	1	theorem	theorem	ADJ
ejpam-1245	235	2	7	7	NUM
ejpam-1245	235	3	.	.	PUNCT
ejpam-1245	236	1	let	let	VERB
ejpam-1245	236	2	f	f	NOUN
ejpam-1245	236	3	:	:	PUNCT
ejpam-1245	236	4	(	(	PUNCT
ejpam-1245	236	5	x	x	X
ejpam-1245	236	6	,	,	PUNCT
ejpam-1245	236	7	τ)→	τ)→	PROPN
ejpam-1245	236	8	(	(	PUNCT
ejpam-1245	236	9	y	y	PROPN
ejpam-1245	236	10	,	,	PUNCT
ejpam-1245	236	11	σ	σ	PROPN
ejpam-1245	236	12	)	)	PUNCT
ejpam-1245	236	13	be	be	AUX
ejpam-1245	236	14	an	an	DET
ejpam-1245	236	15	pre	pre	ADJ
ejpam-1245	236	16	-	-	ADJ
ejpam-1245	236	17	semi	semi	ADJ
ejpam-1245	236	18	-	-	ADJ
ejpam-1245	236	19	preopen	preopen	ADJ
ejpam-1245	236	20	surjection	surjection	NOUN
ejpam-1245	236	21	and	and	CCONJ
ejpam-1245	236	22	let	let	VERB
ejpam-1245	236	23	g	g	NOUN
ejpam-1245	236	24	:	:	PUNCT
ejpam-1245	236	25	(	(	PUNCT
ejpam-1245	236	26	y	y	NOUN
ejpam-1245	236	27	,	,	PUNCT
ejpam-1245	236	28	σ)→	σ)→	PROPN
ejpam-1245	236	29	(	(	PUNCT
ejpam-1245	236	30	z	z	PROPN
ejpam-1245	236	31	,	,	PUNCT
ejpam-1245	236	32	ρ	ρ	PROPN
ejpam-1245	236	33	)	)	PUNCT
ejpam-1245	236	34	such	such	ADJ
ejpam-1245	236	35	that	that	SCONJ
ejpam-1245	236	36	g	g	PROPN
ejpam-1245	236	37	◦	◦	NOUN
ejpam-1245	236	38	f	f	X
ejpam-1245	236	39	:	:	PUNCT
ejpam-1245	236	40	(	(	PUNCT
ejpam-1245	236	41	x	x	X
ejpam-1245	236	42	,	,	PUNCT
ejpam-1245	236	43	τ)→	τ)→	PROPN
ejpam-1245	236	44	(	(	PUNCT
ejpam-1245	236	45	z	z	NOUN
ejpam-1245	236	46	,	,	PUNCT
ejpam-1245	236	47	ρ	ρ	PROPN
ejpam-1245	236	48	)	)	PUNCT
ejpam-1245	236	49	isωβ−containuous	isωβ−containuous	ADJ
ejpam-1245	236	50	,	,	PUNCT
ejpam-1245	236	51	then	then	ADV
ejpam-1245	236	52	g	g	PROPN
ejpam-1245	236	53	isωβ−containuous	isωβ−containuous	ADJ
ejpam-1245	236	54	.	.	PUNCT
ejpam-1245	237	1	proof	proof	NOUN
ejpam-1245	237	2	.	.	PUNCT
ejpam-1245	238	1	at	at	ADP
ejpam-1245	238	2	first	first	ADV
ejpam-1245	238	3	we	we	PRON
ejpam-1245	238	4	show	show	VERB
ejpam-1245	238	5	if	if	SCONJ
ejpam-1245	238	6	f	f	PROPN
ejpam-1245	238	7	:	:	PUNCT
ejpam-1245	238	8	(	(	PUNCT
ejpam-1245	238	9	x	x	X
ejpam-1245	238	10	,	,	PUNCT
ejpam-1245	238	11	τ)→	τ)→	PROPN
ejpam-1245	238	12	(	(	PUNCT
ejpam-1245	238	13	y	y	PROPN
ejpam-1245	238	14	,	,	PUNCT
ejpam-1245	238	15	σ	σ	PROPN
ejpam-1245	238	16	)	)	PUNCT
ejpam-1245	238	17	be	be	AUX
ejpam-1245	238	18	an	an	DET
ejpam-1245	238	19	pre	pre	ADJ
ejpam-1245	238	20	-	-	ADJ
ejpam-1245	238	21	semi	semi	ADJ
ejpam-1245	238	22	-	-	ADJ
ejpam-1245	238	23	preopen	preopen	ADJ
ejpam-1245	238	24	function	function	NOUN
ejpam-1245	238	25	and	and	CCONJ
ejpam-1245	238	26	u	u	NOUN
ejpam-1245	238	27	∈	∈	PROPN
ejpam-1245	238	28	ωβo(x	ωβo(x	PROPN
ejpam-1245	238	29	,	,	PUNCT
ejpam-1245	238	30	τ	τ	PROPN
ejpam-1245	238	31	)	)	PUNCT
ejpam-1245	238	32	,	,	PUNCT
ejpam-1245	238	33	then	then	ADV
ejpam-1245	238	34	f	f	PROPN
ejpam-1245	238	35	(	(	PUNCT
ejpam-1245	238	36	u	u	NOUN
ejpam-1245	238	37	)	)	PUNCT
ejpam-1245	238	38	∈	∈	PROPN
ejpam-1245	238	39	ωβo(y	ωβo(y	NUM
ejpam-1245	238	40	,	,	PUNCT
ejpam-1245	238	41	σ	σ	NOUN
ejpam-1245	238	42	)	)	PUNCT
ejpam-1245	238	43	.	.	PUNCT
ejpam-1245	239	1	so	so	ADV
ejpam-1245	239	2	let	let	VERB
ejpam-1245	239	3	u	u	PRON
ejpam-1245	239	4	∈	∈	PROPN
ejpam-1245	239	5	ωβo(x	ωβo(x	PROPN
ejpam-1245	239	6	,	,	PUNCT
ejpam-1245	239	7	τ	τ	PROPN
ejpam-1245	239	8	)	)	PUNCT
ejpam-1245	239	9	then	then	ADV
ejpam-1245	239	10	for	for	ADP
ejpam-1245	239	11	all	all	DET
ejpam-1245	239	12	x	x	SYM
ejpam-1245	239	13	∈	∈	ADJ
ejpam-1245	239	14	u	u	NOUN
ejpam-1245	239	15	there	there	PRON
ejpam-1245	239	16	exists	exist	VERB
ejpam-1245	239	17	βo(x	βo(x	PUNCT
ejpam-1245	239	18	,	,	PUNCT
ejpam-1245	239	19	τ	τ	X
ejpam-1245	239	20	)	)	PUNCT
ejpam-1245	239	21	sets	set	VERB
ejpam-1245	239	22	u1	u1	NOUN
ejpam-1245	239	23	in	in	ADP
ejpam-1245	239	24	(	(	PUNCT
ejpam-1245	239	25	x	x	INTJ
ejpam-1245	239	26	,	,	PUNCT
ejpam-1245	239	27	τ	τ	X
ejpam-1245	239	28	)	)	PUNCT
ejpam-1245	239	29	containing	contain	VERB
ejpam-1245	239	30	x	x	NOUN
ejpam-1245	239	31	and	and	CCONJ
ejpam-1245	239	32	u1	u1	PROPN
ejpam-1245	239	33	−	−	PROPN
ejpam-1245	239	34	u	u	NOUN
ejpam-1245	240	1	⊆	⊆	NUM
ejpam-1245	240	2	c	c	NOUN
ejpam-1245	240	3	where	where	SCONJ
ejpam-1245	240	4	c	c	NOUN
ejpam-1245	240	5	is	be	AUX
ejpam-1245	240	6	a	a	DET
ejpam-1245	240	7	countable	countable	ADJ
ejpam-1245	240	8	set	set	NOUN
ejpam-1245	240	9	.	.	PUNCT
ejpam-1245	241	1	thus	thus	ADV
ejpam-1245	241	2	f	f	X
ejpam-1245	241	3	(	(	PUNCT
ejpam-1245	241	4	u1)−	u1)−	PROPN
ejpam-1245	241	5	f	f	X
ejpam-1245	241	6	(	(	PUNCT
ejpam-1245	241	7	u)⊆	u)⊆	PROPN
ejpam-1245	241	8	f	f	X
ejpam-1245	241	9	(	(	PUNCT
ejpam-1245	241	10	c	c	NOUN
ejpam-1245	241	11	)	)	PUNCT
ejpam-1245	241	12	where	where	SCONJ
ejpam-1245	241	13	f	f	PROPN
ejpam-1245	241	14	(	(	PUNCT
ejpam-1245	241	15	c	c	NOUN
ejpam-1245	241	16	)	)	PUNCT
ejpam-1245	241	17	is	be	AUX
ejpam-1245	241	18	a	a	DET
ejpam-1245	241	19	countable	countable	ADJ
ejpam-1245	241	20	set	set	NOUN
ejpam-1245	241	21	.	.	PUNCT
ejpam-1245	242	1	this	this	PRON
ejpam-1245	242	2	implies	imply	VERB
ejpam-1245	242	3	f	f	PROPN
ejpam-1245	242	4	(	(	PUNCT
ejpam-1245	242	5	u	u	NOUN
ejpam-1245	242	6	)	)	PUNCT
ejpam-1245	242	7	∈ωβo(y	∈ωβo(y	NOUN
ejpam-1245	242	8	,	,	PUNCT
ejpam-1245	242	9	σ	σ	NOUN
ejpam-1245	242	10	)	)	PUNCT
ejpam-1245	242	11	.	.	PUNCT
ejpam-1245	243	1	now	now	ADV
ejpam-1245	243	2	,	,	PUNCT
ejpam-1245	243	3	let	let	VERB
ejpam-1245	243	4	y	y	PROPN
ejpam-1245	243	5	∈	∈	PROPN
ejpam-1245	243	6	y	y	PROPN
ejpam-1245	243	7	and	and	CCONJ
ejpam-1245	243	8	let	let	VERB
ejpam-1245	243	9	v	v	NUM
ejpam-1245	243	10	∈	∈	PROPN
ejpam-1245	243	11	ρ	ρ	NOUN
ejpam-1245	243	12	with	with	ADP
ejpam-1245	243	13	g(y	g(y	NOUN
ejpam-1245	243	14	)	)	PUNCT
ejpam-1245	243	15	∈	∈	PROPN
ejpam-1245	243	16	v	v	NOUN
ejpam-1245	243	17	.	.	PUNCT
ejpam-1245	244	1	choose	choose	VERB
ejpam-1245	244	2	x	x	PUNCT
ejpam-1245	244	3	∈	∈	PROPN
ejpam-1245	244	4	x	x	PUNCT
ejpam-1245	245	1	such	such	ADJ
ejpam-1245	245	2	that	that	SCONJ
ejpam-1245	245	3	f	f	PROPN
ejpam-1245	245	4	(	(	PUNCT
ejpam-1245	245	5	x	x	X
ejpam-1245	245	6	)	)	PUNCT
ejpam-1245	245	7	=	=	SYM
ejpam-1245	245	8	y.	y.	NOUN
ejpam-1245	245	9	since	since	SCONJ
ejpam-1245	245	10	g	g	PROPN
ejpam-1245	245	11	◦	◦	PROPN
ejpam-1245	245	12	f	f	PROPN
ejpam-1245	245	13	is	be	AUX
ejpam-1245	245	14	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	245	15	there	there	PRON
ejpam-1245	245	16	exists	exist	VERB
ejpam-1245	245	17	u	u	PROPN
ejpam-1245	245	18	∈	∈	PROPN
ejpam-1245	245	19	ωβo(x	ωβo(x	PROPN
ejpam-1245	245	20	,	,	PUNCT
ejpam-1245	245	21	τ	τ	PROPN
ejpam-1245	245	22	)	)	PUNCT
ejpam-1245	245	23	with	with	ADP
ejpam-1245	245	24	x	x	PROPN
ejpam-1245	245	25	∈	∈	PROPN
ejpam-1245	245	26	u	u	NOUN
ejpam-1245	245	27	and	and	CCONJ
ejpam-1245	245	28	g	g	PROPN
ejpam-1245	245	29	(	(	PUNCT
ejpam-1245	245	30	f	f	PROPN
ejpam-1245	245	31	(	(	PUNCT
ejpam-1245	245	32	u	u	NOUN
ejpam-1245	245	33	)	)	PUNCT
ejpam-1245	245	34	)	)	PUNCT
ejpam-1245	246	1	⊆	⊆	NUM
ejpam-1245	246	2	v	v	NOUN
ejpam-1245	246	3	.	.	PUNCT
ejpam-1245	247	1	but	but	CCONJ
ejpam-1245	247	2	f	f	PROPN
ejpam-1245	247	3	is	be	AUX
ejpam-1245	247	4	pre	pre	ADJ
ejpam-1245	247	5	-	-	ADJ
ejpam-1245	247	6	semi	semi	ADJ
ejpam-1245	247	7	-	-	ADJ
ejpam-1245	247	8	preopen	preopen	ADJ
ejpam-1245	247	9	function	function	NOUN
ejpam-1245	247	10	therefore	therefore	ADV
ejpam-1245	247	11	,	,	PUNCT
ejpam-1245	247	12	by	by	ADP
ejpam-1245	247	13	assumption	assumption	NOUN
ejpam-1245	247	14	,	,	PUNCT
ejpam-1245	247	15	f	f	PROPN
ejpam-1245	247	16	(	(	PUNCT
ejpam-1245	247	17	u	u	NOUN
ejpam-1245	247	18	)	)	PUNCT
ejpam-1245	247	19	∈	∈	PROPN
ejpam-1245	247	20	ωβo(y	ωβo(y	NUM
ejpam-1245	247	21	,	,	PUNCT
ejpam-1245	247	22	σ	σ	NOUN
ejpam-1245	247	23	)	)	PUNCT
ejpam-1245	247	24	with	with	ADP
ejpam-1245	247	25	f	f	PROPN
ejpam-1245	247	26	(	(	PUNCT
ejpam-1245	247	27	x	x	X
ejpam-1245	247	28	)	)	PUNCT
ejpam-1245	247	29	∈	∈	PROPN
ejpam-1245	247	30	f	f	X
ejpam-1245	247	31	(	(	PUNCT
ejpam-1245	247	32	u	u	NOUN
ejpam-1245	247	33	)	)	PUNCT
ejpam-1245	247	34	.	.	PUNCT
ejpam-1245	248	1	so	so	ADV
ejpam-1245	248	2	we	we	PRON
ejpam-1245	248	3	get	get	VERB
ejpam-1245	248	4	the	the	DET
ejpam-1245	248	5	result	result	NOUN
ejpam-1245	248	6	.	.	PUNCT
ejpam-1245	249	1	corollary	corollary	ADJ
ejpam-1245	249	2	4	4	NUM
ejpam-1245	249	3	.	.	PUNCT
ejpam-1245	250	1	let	let	VERB
ejpam-1245	250	2	fα	fα	VERB
ejpam-1245	250	3	:	:	PUNCT
ejpam-1245	250	4	(	(	PUNCT
ejpam-1245	250	5	xα	xα	INTJ
ejpam-1245	250	6	,	,	PUNCT
ejpam-1245	250	7	τα)→	τα)→	NOUN
ejpam-1245	250	8	(	(	PUNCT
ejpam-1245	250	9	yα	yα	NOUN
ejpam-1245	250	10	,	,	PUNCT
ejpam-1245	250	11	τα	τα	PROPN
ejpam-1245	250	12	)	)	PUNCT
ejpam-1245	250	13	be	be	AUX
ejpam-1245	250	14	a	a	DET
ejpam-1245	250	15	function	function	NOUN
ejpam-1245	250	16	for	for	ADP
ejpam-1245	250	17	each	each	DET
ejpam-1245	250	18	α	α	NOUN
ejpam-1245	250	19	∈	∈	PROPN
ejpam-1245	251	1	∆.	∆.	NOUN
ejpam-1245	251	2	if	if	SCONJ
ejpam-1245	251	3	the	the	DET
ejpam-1245	251	4	product	product	NOUN
ejpam-1245	251	5	function	function	VERB
ejpam-1245	251	6	f	f	PROPN
ejpam-1245	251	7	=	=	SYM
ejpam-1245	251	8	∏	∏	PROPN
ejpam-1245	251	9	α∈∆	α∈∆	NOUN
ejpam-1245	251	10	fα	fα	NOUN
ejpam-1245	251	11	:	:	PUNCT
ejpam-1245	251	12	∏	∏	NUM
ejpam-1245	251	13	α∈∆	α∈∆	X
ejpam-1245	251	14	xα→	xα→	X
ejpam-1245	251	15	∏	∏	NUM
ejpam-1245	251	16	α∈∆	α∈∆	NOUN
ejpam-1245	251	17	yα	yα	NOUN
ejpam-1245	251	18	is	be	AUX
ejpam-1245	251	19	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	251	20	,	,	PUNCT
ejpam-1245	251	21	then	then	ADV
ejpam-1245	251	22	fα	fα	NOUN
ejpam-1245	251	23	is	be	AUX
ejpam-1245	251	24	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	251	25	.	.	PUNCT
ejpam-1245	252	1	proof	proof	NOUN
ejpam-1245	252	2	.	.	PUNCT
ejpam-1245	253	1	at	at	ADP
ejpam-1245	253	2	first	first	ADV
ejpam-1245	253	3	we	we	PRON
ejpam-1245	253	4	prove	prove	VERB
ejpam-1245	253	5	that	that	SCONJ
ejpam-1245	253	6	any	any	DET
ejpam-1245	253	7	projection	projection	NOUN
ejpam-1245	253	8	function	function	NOUN
ejpam-1245	253	9	is	be	AUX
ejpam-1245	253	10	pre	pre	ADJ
ejpam-1245	253	11	-	-	ADJ
ejpam-1245	253	12	semi	semi	ADJ
ejpam-1245	253	13	-	-	ADJ
ejpam-1245	253	14	preopen	preopen	ADJ
ejpam-1245	253	15	function	function	NOUN
ejpam-1245	253	16	.	.	PUNCT
ejpam-1245	254	1	let	let	VERB
ejpam-1245	254	2	u	u	PRON
ejpam-1245	254	3	∈	∈	PROPN
ejpam-1245	254	4	βo(x	βo(x	PUNCT
ejpam-1245	254	5	,	,	PUNCT
ejpam-1245	254	6	τ	τ	X
ejpam-1245	254	7	)	)	PUNCT
ejpam-1245	254	8	hence	hence	ADV
ejpam-1245	254	9	f	f	PROPN
ejpam-1245	254	10	(	(	PUNCT
ejpam-1245	254	11	u	u	NOUN
ejpam-1245	254	12	)	)	PUNCT
ejpam-1245	254	13	⊆	⊆	NUM
ejpam-1245	254	14	f	f	PROPN
ejpam-1245	254	15	(	(	PUNCT
ejpam-1245	254	16	cl(int(cl(u	cl(int(cl(u	PROPN
ejpam-1245	254	17	)	)	PUNCT
ejpam-1245	254	18	)	)	PUNCT
ejpam-1245	254	19	)	)	PUNCT
ejpam-1245	254	20	)	)	PUNCT
ejpam-1245	254	21	,	,	PUNCT
ejpam-1245	254	22	by	by	ADP
ejpam-1245	254	23	using	use	VERB
ejpam-1245	254	24	the	the	DET
ejpam-1245	254	25	assumption	assumption	NOUN
ejpam-1245	254	26	that	that	SCONJ
ejpam-1245	254	27	f	f	PROPN
ejpam-1245	254	28	is	be	AUX
ejpam-1245	254	29	open	open	ADJ
ejpam-1245	254	30	and	and	CCONJ
ejpam-1245	254	31	continuous	continuous	ADJ
ejpam-1245	254	32	surjective	surjective	NOUN
ejpam-1245	254	33	,	,	PUNCT
ejpam-1245	254	34	f	f	PROPN
ejpam-1245	254	35	(	(	PUNCT
ejpam-1245	254	36	u)⊆	u)⊆	NUM
ejpam-1245	254	37	cl(int(cl	cl(int(cl	PROPN
ejpam-1245	254	38	(	(	PUNCT
ejpam-1245	254	39	f	f	PROPN
ejpam-1245	254	40	(	(	PUNCT
ejpam-1245	254	41	u	u	NOUN
ejpam-1245	254	42	)	)	PUNCT
ejpam-1245	254	43	)	)	PUNCT
ejpam-1245	254	44	)	)	PUNCT
ejpam-1245	254	45	)	)	PUNCT
ejpam-1245	254	46	.	.	PUNCT
ejpam-1245	255	1	thus	thus	ADV
ejpam-1245	255	2	f	f	X
ejpam-1245	255	3	(	(	PUNCT
ejpam-1245	255	4	u	u	NOUN
ejpam-1245	255	5	)	)	PUNCT
ejpam-1245	255	6	∈	∈	PROPN
ejpam-1245	255	7	βo(y	βo(y	PUNCT
ejpam-1245	255	8	,	,	PUNCT
ejpam-1245	255	9	σ	σ	PROPN
ejpam-1245	255	10	)	)	PUNCT
ejpam-1245	255	11	.	.	PUNCT
ejpam-1245	256	1	now	now	ADV
ejpam-1245	256	2	for	for	SCONJ
ejpam-1245	256	3	each	each	DET
ejpam-1245	256	4	β	β	NOUN
ejpam-1245	256	5	∈∆	∈∆	NOUN
ejpam-1245	256	6	,	,	PUNCT
ejpam-1245	256	7	let	let	VERB
ejpam-1245	256	8	pβ	pβ	ADV
ejpam-1245	256	9	:	:	PUNCT
ejpam-1245	256	10	∏	∏	X
ejpam-1245	256	11	α∈∆	α∈∆	X
ejpam-1245	256	12	xα→	xα→	PUNCT
ejpam-1245	257	1	xβ	xβ	NOUN
ejpam-1245	257	2	and	and	CCONJ
ejpam-1245	257	3	qβ	qβ	X
ejpam-1245	257	4	:	:	PUNCT
ejpam-1245	257	5	∏	∏	NUM
ejpam-1245	257	6	α∈∆	α∈∆	PROPN
ejpam-1245	257	7	yα→	yα→	NOUN
ejpam-1245	257	8	yβ	yβ	NOUN
ejpam-1245	257	9	be	be	AUX
ejpam-1245	257	10	the	the	DET
ejpam-1245	257	11	projections	projection	NOUN
ejpam-1245	257	12	,	,	PUNCT
ejpam-1245	257	13	then	then	ADV
ejpam-1245	257	14	we	we	PRON
ejpam-1245	257	15	have	have	VERB
ejpam-1245	257	16	qβ	qβ	NOUN
ejpam-1245	257	17	◦	◦	NOUN
ejpam-1245	257	18	f	f	NOUN
ejpam-1245	258	1	=	=	X
ejpam-1245	258	2	fβ	fβ	ADP
ejpam-1245	258	3	◦	◦	VERB
ejpam-1245	258	4	pβ	pβ	ADV
ejpam-1245	258	5	for	for	ADP
ejpam-1245	258	6	each	each	PRON
ejpam-1245	258	7	β	β	X
ejpam-1245	258	8	∈	∈	PROPN
ejpam-1245	259	1	∆.	∆.	NOUN
ejpam-1245	259	2	since	since	SCONJ
ejpam-1245	259	3	f	f	PROPN
ejpam-1245	259	4	is	be	AUX
ejpam-1245	259	5	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	259	6	and	and	CCONJ
ejpam-1245	259	7	qβ	qβ	PROPN
ejpam-1245	259	8	is	be	AUX
ejpam-1245	259	9	continuous	continuous	ADJ
ejpam-1245	259	10	,	,	PUNCT
ejpam-1245	259	11	by	by	ADP
ejpam-1245	259	12	proposition	proposition	NOUN
ejpam-1245	259	13	3	3	NUM
ejpam-1245	259	14	qβ	qβ	NOUN
ejpam-1245	259	15	◦	◦	NOUN
ejpam-1245	259	16	f	f	PROPN
ejpam-1245	259	17	is	be	AUX
ejpam-1245	259	18	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	259	19	and	and	CCONJ
ejpam-1245	259	20	hence	hence	ADV
ejpam-1245	259	21	fβ	fβ	ADP
ejpam-1245	259	22	◦	◦	NOUN
ejpam-1245	259	23	pβ	pβ	ADV
ejpam-1245	259	24	is	be	AUX
ejpam-1245	259	25	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	259	26	function	function	NOUN
ejpam-1245	259	27	.	.	PUNCT
ejpam-1245	260	1	since	since	SCONJ
ejpam-1245	260	2	pβ	pβ	ADV
ejpam-1245	260	3	is	be	AUX
ejpam-1245	260	4	pre	pre	ADJ
ejpam-1245	260	5	-	-	ADJ
ejpam-1245	260	6	semi	semi	ADJ
ejpam-1245	260	7	-	-	ADJ
ejpam-1245	260	8	preopen	preopen	ADJ
ejpam-1245	260	9	function	function	NOUN
ejpam-1245	260	10	it	it	PRON
ejpam-1245	260	11	follows	follow	VERB
ejpam-1245	260	12	from	from	ADP
ejpam-1245	260	13	theorem	theorem	NOUN
ejpam-1245	260	14	7	7	NUM
ejpam-1245	260	15	that	that	SCONJ
ejpam-1245	260	16	fβ	fβ	NOUN
ejpam-1245	260	17	is	be	AUX
ejpam-1245	260	18	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	260	19	function	function	NOUN
ejpam-1245	260	20	.	.	PUNCT
ejpam-1245	261	1	theorem	theorem	VERB
ejpam-1245	261	2	8	8	NUM
ejpam-1245	261	3	.	.	PUNCT
ejpam-1245	262	1	[	[	X
ejpam-1245	262	2	3	3	X
ejpam-1245	262	3	]	]	PUNCT
ejpam-1245	262	4	for	for	ADP
ejpam-1245	262	5	any	any	DET
ejpam-1245	262	6	space	space	NOUN
ejpam-1245	262	7	x	x	SYM
ejpam-1245	262	8	,	,	PUNCT
ejpam-1245	262	9	the	the	DET
ejpam-1245	262	10	following	follow	VERB
ejpam-1245	262	11	properties	property	NOUN
ejpam-1245	262	12	are	be	AUX
ejpam-1245	262	13	equivalent	equivalent	ADJ
ejpam-1245	262	14	:	:	PUNCT
ejpam-1245	262	15	i.	i.	NOUN
ejpam-1245	262	16	x	x	PROPN
ejpam-1245	262	17	is	be	AUX
ejpam-1245	262	18	β−lindelőf	β−lindelőf	NOUN
ejpam-1245	262	19	.	.	PUNCT
ejpam-1245	262	20	ii	ii	PROPN
ejpam-1245	262	21	.	.	PUNCT
ejpam-1245	263	1	every	every	DET
ejpam-1245	263	2	ωβo(x	ωβo(x	PROPN
ejpam-1245	263	3	,	,	PUNCT
ejpam-1245	263	4	τ	τ	NOUN
ejpam-1245	263	5	)	)	PUNCT
ejpam-1245	263	6	cover	cover	NOUN
ejpam-1245	263	7	of	of	ADP
ejpam-1245	263	8	x	x	PUNCT
ejpam-1245	263	9	has	have	VERB
ejpam-1245	263	10	a	a	DET
ejpam-1245	263	11	countable	countable	ADJ
ejpam-1245	263	12	subcover	subcover	NOUN
ejpam-1245	263	13	.	.	PUNCT
ejpam-1245	264	1	proposition	proposition	NOUN
ejpam-1245	264	2	4	4	NUM
ejpam-1245	264	3	.	.	PUNCT
ejpam-1245	265	1	let	let	VERB
ejpam-1245	265	2	f	f	NOUN
ejpam-1245	265	3	:	:	PUNCT
ejpam-1245	265	4	(	(	PUNCT
ejpam-1245	265	5	x	x	X
ejpam-1245	265	6	,	,	PUNCT
ejpam-1245	265	7	τ	τ	PROPN
ejpam-1245	265	8	)	)	PUNCT
ejpam-1245	265	9	→	→	SYM
ejpam-1245	265	10	(	(	PUNCT
ejpam-1245	265	11	y	y	PROPN
ejpam-1245	265	12	,	,	PUNCT
ejpam-1245	265	13	σ	σ	PROPN
ejpam-1245	265	14	)	)	PUNCT
ejpam-1245	265	15	be	be	VERB
ejpam-1245	265	16	an	an	DET
ejpam-1245	265	17	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	265	18	surjective	surjective	ADJ
ejpam-1245	265	19	function	function	NOUN
ejpam-1245	265	20	.	.	PUNCT
ejpam-1245	266	1	if	if	SCONJ
ejpam-1245	266	2	x	x	PRON
ejpam-1245	266	3	is	be	AUX
ejpam-1245	266	4	β−lindelőf	β−lindelőf	NOUN
ejpam-1245	266	5	,	,	PUNCT
ejpam-1245	266	6	then	then	ADV
ejpam-1245	266	7	y	y	PROPN
ejpam-1245	266	8	is	be	AUX
ejpam-1245	266	9	lindelőf	lindelőf	ADJ
ejpam-1245	266	10	.	.	PUNCT
ejpam-1245	266	11	proof	proof	NOUN
ejpam-1245	266	12	.	.	PUNCT
ejpam-1245	267	1	let	let	VERB
ejpam-1245	267	2	{	{	PUNCT
ejpam-1245	267	3	vα	vα	X
ejpam-1245	267	4	:	:	PUNCT
ejpam-1245	267	5	α	α	PRON
ejpam-1245	267	6	∈∆	∈∆	AUX
ejpam-1245	267	7	}	}	PUNCT
ejpam-1245	267	8	be	be	AUX
ejpam-1245	267	9	an	an	DET
ejpam-1245	267	10	open	open	ADJ
ejpam-1245	267	11	cover	cover	NOUN
ejpam-1245	267	12	of	of	ADP
ejpam-1245	267	13	y	y	PROPN
ejpam-1245	267	14	.	.	PUNCT
ejpam-1245	268	1	then	then	ADV
ejpam-1245	268	2	{	{	PUNCT
ejpam-1245	268	3	f	f	PROPN
ejpam-1245	268	4	−1(vα	−1(vα	PROPN
ejpam-1245	268	5	)	)	PUNCT
ejpam-1245	268	6	:	:	PUNCT
ejpam-1245	268	7	α	α	PROPN
ejpam-1245	268	8	∈∆	∈∆	NOUN
ejpam-1245	268	9	}	}	PUNCT
ejpam-1245	268	10	isωβo(x	isωβo(x	PROPN
ejpam-1245	268	11	,	,	PUNCT
ejpam-1245	268	12	τ	τ	X
ejpam-1245	268	13	)	)	PUNCT
ejpam-1245	268	14	cover	cover	NOUN
ejpam-1245	268	15	of	of	ADP
ejpam-1245	268	16	x	x	SYM
ejpam-1245	268	17	(	(	PUNCT
ejpam-1245	268	18	since	since	SCONJ
ejpam-1245	268	19	f	f	PROPN
ejpam-1245	268	20	is	be	AUX
ejpam-1245	268	21	ωβ−	ωβ−	NUM
ejpam-1245	268	22	continuous	continuous	ADJ
ejpam-1245	268	23	)	)	PUNCT
ejpam-1245	268	24	.	.	PUNCT
ejpam-1245	269	1	since	since	SCONJ
ejpam-1245	269	2	x	x	PROPN
ejpam-1245	269	3	is	be	AUX
ejpam-1245	269	4	β−lindelőf	β−lindelőf	NOUN
ejpam-1245	269	5	,	,	PUNCT
ejpam-1245	269	6	by	by	ADP
ejpam-1245	269	7	theorem	theorem	NOUN
ejpam-1245	269	8	8	8	NUM
ejpam-1245	269	9	,	,	PUNCT
ejpam-1245	269	10	x	x	PRON
ejpam-1245	269	11	has	have	VERB
ejpam-1245	269	12	a	a	DET
ejpam-1245	269	13	countable	countable	ADJ
ejpam-1245	269	14	subcover	subcover	NOUN
ejpam-1245	269	15	,	,	PUNCT
ejpam-1245	269	16	say	say	VERB
ejpam-1245	269	17	f	f	PROPN
ejpam-1245	269	18	−1(vα1	−1(vα1	PROPN
ejpam-1245	269	19	)	)	PUNCT
ejpam-1245	269	20	,	,	PUNCT
ejpam-1245	269	21	f	f	PROPN
ejpam-1245	269	22	−1(vα2	−1(vα2	PROPN
ejpam-1245	269	23	)	)	PUNCT
ejpam-1245	269	24	,	,	PUNCT
ejpam-1245	269	25	.	.	PUNCT
ejpam-1245	269	26	.	.	PUNCT
ejpam-1245	270	1	.	.	PUNCT
ejpam-1245	271	1	,	,	PUNCT
ejpam-1245	271	2	f	f	PROPN
ejpam-1245	271	3	−1(vαn	−1(vαn	PROPN
ejpam-1245	271	4	)	)	PUNCT
ejpam-1245	271	5	,	,	PUNCT
ejpam-1245	271	6	.	.	PUNCT
ejpam-1245	271	7	.	.	PUNCT
ejpam-1245	272	1	.	.	PUNCT
ejpam-1245	273	1	,	,	PUNCT
ejpam-1245	273	2	thus	thus	ADV
ejpam-1245	273	3	vα1	vα1	NOUN
ejpam-1245	273	4	,	,	PUNCT
ejpam-1245	273	5	vα2	vα2	NOUN
ejpam-1245	273	6	,	,	PUNCT
ejpam-1245	273	7	.	.	PUNCT
ejpam-1245	273	8	.	.	PUNCT
ejpam-1245	273	9	.	.	PUNCT
ejpam-1245	274	1	,	,	PUNCT
ejpam-1245	274	2	vαn	vαn	NOUN
ejpam-1245	274	3	,	,	PUNCT
ejpam-1245	274	4	.	.	PUNCT
ejpam-1245	274	5	.	.	PUNCT
ejpam-1245	275	1	.	.	PUNCT
ejpam-1245	276	1	is	be	AUX
ejpam-1245	276	2	a	a	DET
ejpam-1245	276	3	subcover	subcover	NOUN
ejpam-1245	276	4	of	of	ADP
ejpam-1245	276	5	{	{	PUNCT
ejpam-1245	276	6	vα	vα	X
ejpam-1245	276	7	:	:	PUNCT
ejpam-1245	276	8	α	α	PROPN
ejpam-1245	276	9	∈∆	∈∆	NOUN
ejpam-1245	276	10	}	}	PUNCT
ejpam-1245	276	11	of	of	ADP
ejpam-1245	276	12	y	y	PROPN
ejpam-1245	276	13	.	.	PUNCT
ejpam-1245	277	1	this	this	PRON
ejpam-1245	277	2	shows	show	VERB
ejpam-1245	277	3	that	that	SCONJ
ejpam-1245	277	4	y	y	PROPN
ejpam-1245	277	5	is	be	AUX
ejpam-1245	277	6	lindelőf	lindelőf	ADJ
ejpam-1245	277	7	.	.	PUNCT
ejpam-1245	277	8	corollary	corollary	ADJ
ejpam-1245	277	9	5	5	NUM
ejpam-1245	277	10	.	.	PUNCT
ejpam-1245	278	1	let	let	VERB
ejpam-1245	278	2	f	f	NOUN
ejpam-1245	278	3	:	:	PUNCT
ejpam-1245	278	4	(	(	PUNCT
ejpam-1245	278	5	x	x	X
ejpam-1245	278	6	,	,	PUNCT
ejpam-1245	278	7	τ)→	τ)→	PROPN
ejpam-1245	278	8	(	(	PUNCT
ejpam-1245	278	9	y	y	PROPN
ejpam-1245	278	10	,	,	PUNCT
ejpam-1245	278	11	σ	σ	PROPN
ejpam-1245	278	12	)	)	PUNCT
ejpam-1245	278	13	be	be	AUX
ejpam-1245	278	14	a	a	DET
ejpam-1245	278	15	β−continuous	β−continuous	ADJ
ejpam-1245	278	16	(	(	PUNCT
ejpam-1245	278	17	or	or	CCONJ
ejpam-1245	278	18	ω−continuous	ω−continuous	NUM
ejpam-1245	278	19	)	)	PUNCT
ejpam-1245	278	20	surjective	surjective	ADJ
ejpam-1245	278	21	function	function	NOUN
ejpam-1245	278	22	.	.	PUNCT
ejpam-1245	279	1	if	if	SCONJ
ejpam-1245	279	2	x	x	PRON
ejpam-1245	279	3	is	be	AUX
ejpam-1245	279	4	β−lindelőf	β−lindelőf	NOUN
ejpam-1245	279	5	,	,	PUNCT
ejpam-1245	279	6	then	then	ADV
ejpam-1245	279	7	y	y	PROPN
ejpam-1245	279	8	is	be	AUX
ejpam-1245	279	9	lindelőf	lindelőf	PROPN
ejpam-1245	279	10	.	.	PUNCT
ejpam-1245	279	11	h.	h.	PROPN
ejpam-1245	279	12	aljarrah	aljarrah	PROPN
ejpam-1245	279	13	,	,	PUNCT
ejpam-1245	279	14	m.	m.	NOUN
ejpam-1245	279	15	noorani	noorani	PROPN
ejpam-1245	279	16	/	/	SYM
ejpam-1245	279	17	eur	eur	PROPN
ejpam-1245	279	18	.	.	PUNCT
ejpam-1245	280	1	j.	j.	PROPN
ejpam-1245	280	2	pure	pure	PROPN
ejpam-1245	280	3	appl	appl	PROPN
ejpam-1245	280	4	.	.	PROPN
ejpam-1245	280	5	math	math	PROPN
ejpam-1245	280	6	,	,	PUNCT
ejpam-1245	280	7	5	5	NUM
ejpam-1245	280	8	(	(	PUNCT
ejpam-1245	280	9	2012	2012	NUM
ejpam-1245	280	10	)	)	PUNCT
ejpam-1245	280	11	,	,	PUNCT
ejpam-1245	280	12	129	129	NUM
ejpam-1245	280	13	-	-	SYM
ejpam-1245	280	14	140	140	NUM
ejpam-1245	280	15	136	136	NUM
ejpam-1245	280	16	3	3	NUM
ejpam-1245	280	17	.	.	PUNCT
ejpam-1245	280	18	ωβ−irresolute	ωβ−irresolute	NOUN
ejpam-1245	280	19	functions	function	NOUN
ejpam-1245	280	20	definition	definition	NOUN
ejpam-1245	280	21	7	7	NUM
ejpam-1245	280	22	.	.	PUNCT
ejpam-1245	281	1	a	a	DET
ejpam-1245	281	2	function	function	NOUN
ejpam-1245	281	3	f	f	NOUN
ejpam-1245	281	4	:	:	PUNCT
ejpam-1245	281	5	(	(	PUNCT
ejpam-1245	281	6	x	x	X
ejpam-1245	281	7	,	,	PUNCT
ejpam-1245	281	8	τ)→	τ)→	PROPN
ejpam-1245	281	9	(	(	PUNCT
ejpam-1245	281	10	y	y	PROPN
ejpam-1245	281	11	,	,	PUNCT
ejpam-1245	281	12	σ	σ	PROPN
ejpam-1245	281	13	)	)	PUNCT
ejpam-1245	281	14	is	be	AUX
ejpam-1245	281	15	called	call	VERB
ejpam-1245	281	16	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	281	17	if	if	SCONJ
ejpam-1245	281	18	the	the	DET
ejpam-1245	281	19	inverse	inverse	ADJ
ejpam-1245	281	20	image	image	NOUN
ejpam-1245	281	21	of	of	ADP
ejpam-1245	281	22	each	each	DET
ejpam-1245	281	23	ωβo(y	ωβo(y	NUM
ejpam-1245	281	24	,	,	PUNCT
ejpam-1245	281	25	σ	σ	NOUN
ejpam-1245	281	26	)	)	PUNCT
ejpam-1245	281	27	set	set	NOUN
ejpam-1245	281	28	is	be	AUX
ejpam-1245	281	29	an	an	DET
ejpam-1245	281	30	ωβo(x	ωβo(x	PROPN
ejpam-1245	281	31	,	,	PUNCT
ejpam-1245	281	32	τ	τ	NOUN
ejpam-1245	281	33	)	)	PUNCT
ejpam-1245	281	34	set	set	NOUN
ejpam-1245	281	35	.	.	PUNCT
ejpam-1245	282	1	note	note	VERB
ejpam-1245	282	2	that	that	SCONJ
ejpam-1245	282	3	every	every	DET
ejpam-1245	282	4	ωβ−irresolute	ωβ−irresolute	ADJ
ejpam-1245	282	5	function	function	NOUN
ejpam-1245	282	6	is	be	AUX
ejpam-1245	282	7	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	282	8	but	but	CCONJ
ejpam-1245	282	9	the	the	DET
ejpam-1245	282	10	converse	converse	NOUN
ejpam-1245	282	11	is	be	AUX
ejpam-1245	282	12	not	not	PART
ejpam-1245	282	13	true	true	ADJ
ejpam-1245	282	14	,	,	PUNCT
ejpam-1245	282	15	which	which	PRON
ejpam-1245	282	16	is	be	AUX
ejpam-1245	282	17	shown	show	VERB
ejpam-1245	282	18	by	by	ADP
ejpam-1245	282	19	the	the	DET
ejpam-1245	282	20	following	follow	VERB
ejpam-1245	282	21	example	example	NOUN
ejpam-1245	282	22	.	.	PUNCT
ejpam-1245	283	1	example	example	NOUN
ejpam-1245	283	2	8	8	NUM
ejpam-1245	283	3	.	.	PUNCT
ejpam-1245	284	1	let	let	VERB
ejpam-1245	284	2	x	x	PUNCT
ejpam-1245	284	3	=	=	PUNCT
ejpam-1245	284	4	r	r	NOUN
ejpam-1245	284	5	with	with	ADP
ejpam-1245	284	6	the	the	DET
ejpam-1245	284	7	topologies	topology	NOUN
ejpam-1245	284	8	τ	τ	X
ejpam-1245	284	9	=	=	X
ejpam-1245	284	10	τcoc	τcoc	PROPN
ejpam-1245	284	11	and	and	CCONJ
ejpam-1245	284	12	let	let	VERB
ejpam-1245	284	13	y	y	PROPN
ejpam-1245	284	14	=	=	PUNCT
ejpam-1245	284	15	{	{	PUNCT
ejpam-1245	284	16	1,2	1,2	NUM
ejpam-1245	284	17	}	}	PUNCT
ejpam-1245	284	18	with	with	ADP
ejpam-1245	284	19	the	the	DET
ejpam-1245	284	20	topology	topology	NOUN
ejpam-1245	284	21	σ	σ	PROPN
ejpam-1245	284	22	=	=	PROPN
ejpam-1245	284	23	�	�	PROPN
ejpam-1245	284	24	φ	φ	PROPN
ejpam-1245	284	25	,	,	PUNCT
ejpam-1245	284	26	y	y	PROPN
ejpam-1245	284	27	,	,	PUNCT
ejpam-1245	284	28	{	{	PUNCT
ejpam-1245	284	29	2	2	NUM
ejpam-1245	284	30	}	}	PUNCT
ejpam-1245	284	31	.	.	PUNCT
ejpam-1245	285	1	let	let	VERB
ejpam-1245	285	2	f	f	NOUN
ejpam-1245	285	3	:	:	PUNCT
ejpam-1245	285	4	(	(	PUNCT
ejpam-1245	285	5	x	x	X
ejpam-1245	285	6	,	,	PUNCT
ejpam-1245	285	7	τ)→	τ)→	PROPN
ejpam-1245	285	8	(	(	PUNCT
ejpam-1245	285	9	y	y	PROPN
ejpam-1245	285	10	,	,	PUNCT
ejpam-1245	285	11	σ	σ	PROPN
ejpam-1245	285	12	)	)	PUNCT
ejpam-1245	285	13	be	be	VERB
ejpam-1245	285	14	the	the	DET
ejpam-1245	285	15	function	function	NOUN
ejpam-1245	285	16	defined	define	VERB
ejpam-1245	285	17	by	by	ADP
ejpam-1245	285	18	f	f	PROPN
ejpam-1245	285	19	(	(	PUNCT
ejpam-1245	285	20	x	x	NOUN
ejpam-1245	285	21	)	)	PUNCT
ejpam-1245	285	22	=	=	SYM
ejpam-1245	286	1	(	(	PUNCT
ejpam-1245	286	2	1	1	NUM
ejpam-1245	286	3	x	x	SYM
ejpam-1245	286	4	∈q	∈q	NOUN
ejpam-1245	286	5	2	2	NUM
ejpam-1245	286	6	x	x	SYM
ejpam-1245	286	7	∈	∈	PROPN
ejpam-1245	286	8	r−q	r−q	NOUN
ejpam-1245	286	9	then	then	ADV
ejpam-1245	286	10	f	f	PROPN
ejpam-1245	286	11	is	be	AUX
ejpam-1245	286	12	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	286	13	but	but	CCONJ
ejpam-1245	286	14	not	not	PART
ejpam-1245	286	15	ωβ−irresolute	ωβ−irresolute	NOUN
ejpam-1245	286	16	since	since	SCONJ
ejpam-1245	286	17	f	f	PROPN
ejpam-1245	286	18	−1({1	−1({1	PROPN
ejpam-1245	286	19	}	}	PUNCT
ejpam-1245	286	20	)	)	PUNCT
ejpam-1245	287	1	=	=	PUNCT
ejpam-1245	287	2	q	q	PROPN
ejpam-1245	287	3	/∈ωβo(x	/∈ωβo(x	PUNCT
ejpam-1245	287	4	,	,	PUNCT
ejpam-1245	287	5	τ	τ	PROPN
ejpam-1245	287	6	)	)	PUNCT
ejpam-1245	287	7	.	.	PUNCT
ejpam-1245	288	1	theorem	theorem	NOUN
ejpam-1245	288	2	9	9	NUM
ejpam-1245	288	3	.	.	PUNCT
ejpam-1245	289	1	let	let	VERB
ejpam-1245	289	2	f	f	NOUN
ejpam-1245	289	3	:	:	PUNCT
ejpam-1245	289	4	(	(	PUNCT
ejpam-1245	289	5	x	x	X
ejpam-1245	289	6	,	,	PUNCT
ejpam-1245	289	7	τ)→	τ)→	PROPN
ejpam-1245	289	8	(	(	PUNCT
ejpam-1245	289	9	y	y	PROPN
ejpam-1245	289	10	,	,	PUNCT
ejpam-1245	289	11	σ	σ	PROPN
ejpam-1245	289	12	)	)	PUNCT
ejpam-1245	289	13	be	be	AUX
ejpam-1245	289	14	a	a	DET
ejpam-1245	289	15	function	function	NOUN
ejpam-1245	289	16	.	.	PUNCT
ejpam-1245	290	1	then	then	ADV
ejpam-1245	290	2	the	the	DET
ejpam-1245	290	3	following	follow	VERB
ejpam-1245	290	4	conditions	condition	NOUN
ejpam-1245	290	5	are	be	AUX
ejpam-1245	290	6	equivalent	equivalent	ADJ
ejpam-1245	290	7	:	:	PUNCT
ejpam-1245	290	8	i.	i.	NOUN
ejpam-1245	290	9	the	the	DET
ejpam-1245	290	10	function	function	NOUN
ejpam-1245	290	11	f	f	PROPN
ejpam-1245	290	12	is	be	AUX
ejpam-1245	290	13	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	290	14	.	.	PUNCT
ejpam-1245	290	15	ii	ii	PROPN
ejpam-1245	290	16	.	.	PUNCT
ejpam-1245	291	1	for	for	ADP
ejpam-1245	291	2	each	each	DET
ejpam-1245	291	3	x	x	SYM
ejpam-1245	291	4	∈	∈	PROPN
ejpam-1245	291	5	x	x	X
ejpam-1245	291	6	and	and	CCONJ
ejpam-1245	291	7	v	v	ADP
ejpam-1245	291	8	∈	∈	NOUN
ejpam-1245	291	9	ωβo(y	ωβo(y	NUM
ejpam-1245	291	10	,	,	PUNCT
ejpam-1245	291	11	σ	σ	NOUN
ejpam-1245	291	12	)	)	PUNCT
ejpam-1245	291	13	containing	contain	VERB
ejpam-1245	291	14	f	f	PROPN
ejpam-1245	291	15	(	(	PUNCT
ejpam-1245	291	16	x	x	NOUN
ejpam-1245	291	17	)	)	PUNCT
ejpam-1245	291	18	,	,	PUNCT
ejpam-1245	291	19	there	there	PRON
ejpam-1245	291	20	exists	exist	VERB
ejpam-1245	291	21	u	u	PROPN
ejpam-1245	291	22	∈	∈	PROPN
ejpam-1245	291	23	ωβo(x	ωβo(x	PROPN
ejpam-1245	291	24	,	,	PUNCT
ejpam-1245	291	25	τ	τ	X
ejpam-1245	291	26	)	)	PUNCT
ejpam-1245	291	27	containing	contain	VERB
ejpam-1245	291	28	x	x	PROPN
ejpam-1245	291	29	and	and	CCONJ
ejpam-1245	291	30	f	f	PROPN
ejpam-1245	291	31	(	(	PUNCT
ejpam-1245	291	32	u)⊆	u)⊆	PROPN
ejpam-1245	291	33	v	v	NOUN
ejpam-1245	291	34	.	.	PUNCT
ejpam-1245	292	1	iii	iii	X
ejpam-1245	292	2	.	.	PUNCT
ejpam-1245	293	1	for	for	ADP
ejpam-1245	293	2	each	each	DET
ejpam-1245	293	3	x	x	SYM
ejpam-1245	293	4	∈	∈	PROPN
ejpam-1245	293	5	x	x	X
ejpam-1245	293	6	,	,	PUNCT
ejpam-1245	293	7	the	the	DET
ejpam-1245	293	8	inverse	inverse	NOUN
ejpam-1245	293	9	of	of	ADP
ejpam-1245	293	10	every	every	DET
ejpam-1245	293	11	ωβ−neighbourhood	ωβ−neighbourhood	NOUN
ejpam-1245	293	12	of	of	ADP
ejpam-1245	293	13	f	f	PROPN
ejpam-1245	293	14	(	(	PUNCT
ejpam-1245	293	15	x	x	X
ejpam-1245	293	16	)	)	PUNCT
ejpam-1245	293	17	is	be	AUX
ejpam-1245	293	18	ωβ−	ωβ−	X
ejpam-1245	293	19	neighbourhood	neighbourhood	NOUN
ejpam-1245	293	20	of	of	ADP
ejpam-1245	293	21	x.	x.	NOUN
ejpam-1245	293	22	iv	iv	PROPN
ejpam-1245	293	23	.	.	PUNCT
ejpam-1245	294	1	for	for	ADP
ejpam-1245	294	2	each	each	DET
ejpam-1245	294	3	x	x	SYM
ejpam-1245	294	4	∈	∈	PROPN
ejpam-1245	294	5	x	x	X
ejpam-1245	294	6	and	and	CCONJ
ejpam-1245	294	7	ωβ−neighbourhood	ωβ−neighbourhood	NUM
ejpam-1245	294	8	v	v	NOUN
ejpam-1245	294	9	of	of	ADP
ejpam-1245	294	10	f	f	PROPN
ejpam-1245	294	11	(	(	PUNCT
ejpam-1245	294	12	x	x	NOUN
ejpam-1245	294	13	)	)	PUNCT
ejpam-1245	294	14	,	,	PUNCT
ejpam-1245	294	15	there	there	PRON
ejpam-1245	294	16	exists	exist	VERB
ejpam-1245	294	17	ωβ−neighbourhood	ωβ−neighbourhood	PROPN
ejpam-1245	294	18	u	u	NOUN
ejpam-1245	294	19	of	of	ADP
ejpam-1245	294	20	x	x	SYM
ejpam-1245	294	21	such	such	ADJ
ejpam-1245	294	22	that	that	SCONJ
ejpam-1245	294	23	f	f	PROPN
ejpam-1245	294	24	(	(	PUNCT
ejpam-1245	294	25	u)⊆	u)⊆	PROPN
ejpam-1245	294	26	v	v	NOUN
ejpam-1245	294	27	.	.	PUNCT
ejpam-1245	295	1	proof	proof	NOUN
ejpam-1245	295	2	.	.	PUNCT
ejpam-1245	296	1	(	(	PUNCT
ejpam-1245	296	2	i→	i→	PROPN
ejpam-1245	296	3	ii	ii	NOUN
ejpam-1245	296	4	)	)	PUNCT
ejpam-1245	296	5	assume	assume	VERB
ejpam-1245	296	6	x	x	X
ejpam-1245	296	7	∈	∈	PROPN
ejpam-1245	296	8	x	x	X
ejpam-1245	296	9	and	and	CCONJ
ejpam-1245	296	10	v	v	X
ejpam-1245	296	11	isωβo(y	isωβo(y	PROPN
ejpam-1245	296	12	,	,	PUNCT
ejpam-1245	296	13	σ	σ	PROPN
ejpam-1245	296	14	)	)	PUNCT
ejpam-1245	296	15	containing	contain	VERB
ejpam-1245	296	16	f	f	PROPN
ejpam-1245	296	17	(	(	PUNCT
ejpam-1245	296	18	x	x	NOUN
ejpam-1245	296	19	)	)	PUNCT
ejpam-1245	296	20	,	,	PUNCT
ejpam-1245	296	21	since	since	SCONJ
ejpam-1245	296	22	f	f	PROPN
ejpam-1245	296	23	isωβ−irresolute	isωβ−irresolute	VERB
ejpam-1245	296	24	then	then	ADV
ejpam-1245	296	25	f	f	PROPN
ejpam-1245	296	26	−1(v	−1(v	PROPN
ejpam-1245	296	27	)	)	PUNCT
ejpam-1245	297	1	∈ωβo(x	∈ωβo(x	PROPN
ejpam-1245	297	2	,	,	PUNCT
ejpam-1245	297	3	τ	τ	PROPN
ejpam-1245	297	4	)	)	PUNCT
ejpam-1245	297	5	containing	contain	VERB
ejpam-1245	297	6	x	x	PUNCT
ejpam-1245	297	7	and	and	CCONJ
ejpam-1245	297	8	hence	hence	ADV
ejpam-1245	297	9	f	f	PROPN
ejpam-1245	298	1	(	(	PUNCT
ejpam-1245	298	2	f	f	PROPN
ejpam-1245	298	3	−1(v	−1(v	PROPN
ejpam-1245	298	4	)	)	PUNCT
ejpam-1245	298	5	)	)	PUNCT
ejpam-1245	299	1	⊆	⊆	NUM
ejpam-1245	299	2	v	v	NOUN
ejpam-1245	299	3	.	.	PUNCT
ejpam-1245	300	1	(	(	PUNCT
ejpam-1245	300	2	ii	ii	PROPN
ejpam-1245	300	3	→	→	SYM
ejpam-1245	300	4	iii	iii	X
ejpam-1245	300	5	)	)	PUNCT
ejpam-1245	300	6	assume	assume	VERB
ejpam-1245	300	7	x	x	X
ejpam-1245	300	8	∈	∈	PROPN
ejpam-1245	300	9	x	x	X
ejpam-1245	300	10	and	and	CCONJ
ejpam-1245	300	11	v	v	NOUN
ejpam-1245	300	12	is	be	AUX
ejpam-1245	300	13	ωβ−neighbourhood	ωβ−neighbourhood	NUM
ejpam-1245	300	14	of	of	ADP
ejpam-1245	300	15	f	f	PROPN
ejpam-1245	300	16	(	(	PUNCT
ejpam-1245	300	17	x	x	NOUN
ejpam-1245	300	18	)	)	PUNCT
ejpam-1245	300	19	,	,	PUNCT
ejpam-1245	300	20	by	by	ADP
ejpam-1245	300	21	definition	definition	NOUN
ejpam-1245	300	22	3	3	NUM
ejpam-1245	300	23	there	there	ADV
ejpam-1245	300	24	exists	exist	VERB
ejpam-1245	300	25	v1	v1	PROPN
ejpam-1245	300	26	∈	∈	PROPN
ejpam-1245	300	27	ωβo(y	ωβo(y	NUM
ejpam-1245	300	28	,	,	PUNCT
ejpam-1245	300	29	σ	σ	NOUN
ejpam-1245	300	30	)	)	PUNCT
ejpam-1245	300	31	such	such	ADJ
ejpam-1245	300	32	that	that	SCONJ
ejpam-1245	300	33	f	f	PROPN
ejpam-1245	300	34	(	(	PUNCT
ejpam-1245	300	35	x	x	X
ejpam-1245	300	36	)	)	PUNCT
ejpam-1245	300	37	∈	∈	NOUN
ejpam-1245	300	38	v1	v1	NOUN
ejpam-1245	300	39	⊆	⊆	NUM
ejpam-1245	300	40	v	v	NOUN
ejpam-1245	300	41	,	,	PUNCT
ejpam-1245	300	42	there	there	PRON
ejpam-1245	300	43	exists	exist	VERB
ejpam-1245	300	44	u	u	PROPN
ejpam-1245	300	45	∈	∈	PROPN
ejpam-1245	300	46	ωβo(x	ωβo(x	PROPN
ejpam-1245	300	47	,	,	PUNCT
ejpam-1245	300	48	τ	τ	X
ejpam-1245	300	49	)	)	PUNCT
ejpam-1245	300	50	containing	contain	VERB
ejpam-1245	300	51	x	x	PROPN
ejpam-1245	300	52	and	and	CCONJ
ejpam-1245	300	53	f	f	PROPN
ejpam-1245	300	54	(	(	PUNCT
ejpam-1245	300	55	u	u	NOUN
ejpam-1245	300	56	)	)	PUNCT
ejpam-1245	300	57	⊆v1	⊆v1	PROPN
ejpam-1245	300	58	,	,	PUNCT
ejpam-1245	300	59	x	x	X
ejpam-1245	300	60	∈	∈	PROPN
ejpam-1245	300	61	u	u	NOUN
ejpam-1245	301	1	⊆	⊆	NUM
ejpam-1245	301	2	f	f	PROPN
ejpam-1245	301	3	−1(v1	−1(v1	NOUN
ejpam-1245	301	4	)	)	PUNCT
ejpam-1245	301	5	⊆	⊆	NUM
ejpam-1245	301	6	f	f	PROPN
ejpam-1245	301	7	−1(v	−1(v	NOUN
ejpam-1245	301	8	)	)	PUNCT
ejpam-1245	301	9	.	.	PUNCT
ejpam-1245	302	1	hence	hence	ADV
ejpam-1245	302	2	by	by	ADP
ejpam-1245	302	3	use	use	NOUN
ejpam-1245	302	4	definition	definition	NOUN
ejpam-1245	302	5	3	3	NUM
ejpam-1245	302	6	,	,	PUNCT
ejpam-1245	302	7	f	f	PROPN
ejpam-1245	302	8	−1(v	−1(v	PROPN
ejpam-1245	302	9	)	)	PUNCT
ejpam-1245	302	10	is	be	AUX
ejpam-1245	302	11	ωβ−neighbourhood	ωβ−neighbourhood	NUM
ejpam-1245	302	12	of	of	ADP
ejpam-1245	302	13	x	x	X
ejpam-1245	302	14	.	.	PUNCT
ejpam-1245	303	1	(	(	PUNCT
ejpam-1245	303	2	iii→	iii→	NOUN
ejpam-1245	303	3	iv	iv	NUM
ejpam-1245	303	4	)	)	PUNCT
ejpam-1245	303	5	let	let	VERB
ejpam-1245	303	6	v	v	NOUN
ejpam-1245	303	7	is	be	AUX
ejpam-1245	303	8	ωβ−neighbourhood	ωβ−neighbourhood	NUM
ejpam-1245	303	9	of	of	ADP
ejpam-1245	303	10	f	f	PROPN
ejpam-1245	303	11	(	(	PUNCT
ejpam-1245	303	12	x	x	NOUN
ejpam-1245	303	13	)	)	PUNCT
ejpam-1245	303	14	,	,	PUNCT
ejpam-1245	303	15	by	by	ADP
ejpam-1245	303	16	(	(	PUNCT
ejpam-1245	303	17	iii	iii	NOUN
ejpam-1245	303	18	)	)	PUNCT
ejpam-1245	303	19	,	,	PUNCT
ejpam-1245	304	1	f	f	PROPN
ejpam-1245	304	2	−1(v	−1(v	PROPN
ejpam-1245	304	3	)	)	PUNCT
ejpam-1245	304	4	is	be	AUX
ejpam-1245	304	5	ωβ−neighbourhood	ωβ−neighbourhood	NUM
ejpam-1245	304	6	of	of	ADP
ejpam-1245	304	7	x	x	PROPN
ejpam-1245	304	8	and	and	CCONJ
ejpam-1245	304	9	f	f	PROPN
ejpam-1245	304	10	(	(	PUNCT
ejpam-1245	304	11	f	f	PROPN
ejpam-1245	304	12	−1(v	−1(v	PROPN
ejpam-1245	304	13	)	)	PUNCT
ejpam-1245	304	14	)	)	PUNCT
ejpam-1245	305	1	⊆	⊆	NUM
ejpam-1245	305	2	v	v	NOUN
ejpam-1245	305	3	.	.	PUNCT
ejpam-1245	305	4	(	(	PUNCT
ejpam-1245	305	5	iv→	iv→	X
ejpam-1245	305	6	i	i	PRON
ejpam-1245	305	7	)	)	PUNCT
ejpam-1245	305	8	for	for	ADP
ejpam-1245	305	9	each	each	DET
ejpam-1245	305	10	x	x	SYM
ejpam-1245	305	11	∈	∈	PROPN
ejpam-1245	305	12	x	x	PUNCT
ejpam-1245	305	13	,	,	PUNCT
ejpam-1245	305	14	let	let	VERB
ejpam-1245	305	15	v	v	ADP
ejpam-1245	305	16	∈ωβo(y	∈ωβo(y	NOUN
ejpam-1245	305	17	,	,	PUNCT
ejpam-1245	305	18	σ	σ	NOUN
ejpam-1245	305	19	)	)	PUNCT
ejpam-1245	305	20	containing	contain	VERB
ejpam-1245	305	21	f	f	PROPN
ejpam-1245	305	22	(	(	PUNCT
ejpam-1245	305	23	x	x	NOUN
ejpam-1245	305	24	)	)	PUNCT
ejpam-1245	305	25	.	.	PUNCT
ejpam-1245	306	1	put	put	VERB
ejpam-1245	306	2	a=	a=	PROPN
ejpam-1245	306	3	f	f	PROPN
ejpam-1245	306	4	−1(v	−1(v	PROPN
ejpam-1245	306	5	)	)	PUNCT
ejpam-1245	306	6	,	,	PUNCT
ejpam-1245	306	7	let	let	VERB
ejpam-1245	306	8	x	x	PUNCT
ejpam-1245	306	9	∈	∈	VERB
ejpam-1245	306	10	a.	a.	NOUN
ejpam-1245	307	1	then	then	ADV
ejpam-1245	307	2	f	f	PROPN
ejpam-1245	307	3	(	(	PUNCT
ejpam-1245	307	4	x	x	X
ejpam-1245	307	5	)	)	PUNCT
ejpam-1245	307	6	∈	∈	NOUN
ejpam-1245	307	7	v	v	NOUN
ejpam-1245	307	8	.	.	PUNCT
ejpam-1245	308	1	since	since	SCONJ
ejpam-1245	308	2	v	v	NUM
ejpam-1245	308	3	∈	∈	PROPN
ejpam-1245	308	4	ωβo(y	ωβo(y	NUM
ejpam-1245	308	5	,	,	PUNCT
ejpam-1245	308	6	σ	σ	PROPN
ejpam-1245	308	7	)	)	PUNCT
ejpam-1245	308	8	then	then	ADV
ejpam-1245	308	9	v	v	NOUN
ejpam-1245	308	10	is	be	AUX
ejpam-1245	308	11	a	a	DET
ejpam-1245	308	12	ωβ−neighbourhood	ωβ−neighbourhood	NOUN
ejpam-1245	308	13	of	of	ADP
ejpam-1245	308	14	f	f	PROPN
ejpam-1245	308	15	(	(	PUNCT
ejpam-1245	308	16	x	x	NOUN
ejpam-1245	308	17	)	)	PUNCT
ejpam-1245	308	18	.	.	PUNCT
ejpam-1245	309	1	so	so	ADV
ejpam-1245	309	2	by	by	ADP
ejpam-1245	309	3	hypothesis	hypothesis	NOUN
ejpam-1245	309	4	,	,	PUNCT
ejpam-1245	309	5	a=	a=	PROPN
ejpam-1245	309	6	f	f	X
ejpam-1245	309	7	−1(v	−1(v	PROPN
ejpam-1245	309	8	)	)	PUNCT
ejpam-1245	309	9	is	be	AUX
ejpam-1245	309	10	ωβ−neighbourhood	ωβ−neighbourhood	NUM
ejpam-1245	309	11	of	of	ADP
ejpam-1245	309	12	x	x	X
ejpam-1245	309	13	.	.	PUNCT
ejpam-1245	310	1	hence	hence	ADV
ejpam-1245	310	2	by	by	ADP
ejpam-1245	310	3	definition	definition	NOUN
ejpam-1245	310	4	3	3	NUM
ejpam-1245	310	5	there	there	PRON
ejpam-1245	310	6	exists	exist	VERB
ejpam-1245	310	7	ax	ax	NOUN
ejpam-1245	310	8	∈	∈	PROPN
ejpam-1245	310	9	ωβo(x	ωβo(x	PROPN
ejpam-1245	310	10	,	,	PUNCT
ejpam-1245	310	11	τ	τ	X
ejpam-1245	310	12	)	)	PUNCT
ejpam-1245	310	13	such	such	ADJ
ejpam-1245	310	14	that	that	SCONJ
ejpam-1245	310	15	x	x	SYM
ejpam-1245	310	16	∈	∈	NOUN
ejpam-1245	310	17	ax	ax	NOUN
ejpam-1245	310	18	⊆	⊆	NUM
ejpam-1245	310	19	a.	a.	NOUN
ejpam-1245	310	20	thus	thus	ADV
ejpam-1245	310	21	,	,	PUNCT
ejpam-1245	310	22	by	by	ADP
ejpam-1245	310	23	lemma	lemma	PROPN
ejpam-1245	310	24	1(i	1(i	NUM
ejpam-1245	310	25	)	)	PUNCT
ejpam-1245	310	26	a	a	DET
ejpam-1245	310	27	=	=	NOUN
ejpam-1245	310	28	∪	∪	ADJ
ejpam-1245	310	29	x∈a	x∈a	ADJ
ejpam-1245	310	30	ax	ax	NOUN
ejpam-1245	310	31	is	be	AUX
ejpam-1245	310	32	ωβo(x	ωβo(x	PROPN
ejpam-1245	310	33	,	,	PUNCT
ejpam-1245	310	34	τ	τ	NOUN
ejpam-1245	310	35	)	)	PUNCT
ejpam-1245	310	36	set	set	NOUN
ejpam-1245	310	37	.	.	PUNCT
ejpam-1245	311	1	therefore	therefore	ADV
ejpam-1245	311	2	,	,	PUNCT
ejpam-1245	311	3	f	f	PROPN
ejpam-1245	311	4	is	be	AUX
ejpam-1245	311	5	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	311	6	.	.	PUNCT
ejpam-1245	311	7	theorem	theorem	VERB
ejpam-1245	311	8	10	10	NUM
ejpam-1245	311	9	.	.	PUNCT
ejpam-1245	312	1	the	the	DET
ejpam-1245	312	2	following	follow	VERB
ejpam-1245	312	3	conditions	condition	NOUN
ejpam-1245	312	4	are	be	AUX
ejpam-1245	312	5	equivalent	equivalent	ADJ
ejpam-1245	312	6	for	for	ADP
ejpam-1245	312	7	a	a	DET
ejpam-1245	312	8	function	function	NOUN
ejpam-1245	312	9	f	f	NOUN
ejpam-1245	312	10	:	:	PUNCT
ejpam-1245	312	11	(	(	PUNCT
ejpam-1245	312	12	x	x	X
ejpam-1245	312	13	,	,	PUNCT
ejpam-1245	312	14	τ)→	τ)→	PROPN
ejpam-1245	312	15	(	(	PUNCT
ejpam-1245	312	16	y	y	PROPN
ejpam-1245	312	17	,	,	PUNCT
ejpam-1245	312	18	σ	σ	PROPN
ejpam-1245	312	19	):	):	PUNCT
ejpam-1245	312	20	i.	i.	PROPN
ejpam-1245	312	21	f	f	PROPN
ejpam-1245	312	22	is	be	AUX
ejpam-1245	312	23	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	312	24	.	.	PUNCT
ejpam-1245	313	1	h.	h.	PROPN
ejpam-1245	313	2	aljarrah	aljarrah	PROPN
ejpam-1245	313	3	,	,	PUNCT
ejpam-1245	313	4	m.	m.	NOUN
ejpam-1245	313	5	noorani	noorani	PROPN
ejpam-1245	313	6	/	/	SYM
ejpam-1245	313	7	eur	eur	PROPN
ejpam-1245	313	8	.	.	PUNCT
ejpam-1245	314	1	j.	j.	PROPN
ejpam-1245	314	2	pure	pure	PROPN
ejpam-1245	314	3	appl	appl	PROPN
ejpam-1245	314	4	.	.	PROPN
ejpam-1245	314	5	math	math	PROPN
ejpam-1245	314	6	,	,	PUNCT
ejpam-1245	314	7	5	5	NUM
ejpam-1245	314	8	(	(	PUNCT
ejpam-1245	314	9	2012	2012	NUM
ejpam-1245	314	10	)	)	PUNCT
ejpam-1245	314	11	,	,	PUNCT
ejpam-1245	314	12	129	129	NUM
ejpam-1245	314	13	-	-	SYM
ejpam-1245	314	14	140	140	NUM
ejpam-1245	314	15	137	137	NUM
ejpam-1245	314	16	ii	ii	NOUN
ejpam-1245	314	17	.	.	PUNCT
ejpam-1245	315	1	for	for	ADP
ejpam-1245	315	2	each	each	DET
ejpam-1245	315	3	ωβc(y	ωβc(y	PROPN
ejpam-1245	315	4	,	,	PUNCT
ejpam-1245	315	5	σ	σ	NOUN
ejpam-1245	315	6	)	)	PUNCT
ejpam-1245	315	7	subset	subset	NOUN
ejpam-1245	315	8	c	c	PROPN
ejpam-1245	315	9	of	of	ADP
ejpam-1245	315	10	y	y	PROPN
ejpam-1245	315	11	,	,	PUNCT
ejpam-1245	315	12	f	f	PROPN
ejpam-1245	315	13	−1(c	−1(c	PROPN
ejpam-1245	315	14	)	)	PUNCT
ejpam-1245	315	15	is	be	AUX
ejpam-1245	315	16	ωβc(x	ωβc(x	PROPN
ejpam-1245	315	17	,	,	PUNCT
ejpam-1245	315	18	τ	τ	PROPN
ejpam-1245	315	19	)	)	PUNCT
ejpam-1245	315	20	.	.	PUNCT
ejpam-1245	316	1	iii	iii	X
ejpam-1245	316	2	.	.	PROPN
ejpam-1245	317	1	for	for	ADP
ejpam-1245	317	2	each	each	DET
ejpam-1245	317	3	subset	subset	VERB
ejpam-1245	317	4	a	a	PRON
ejpam-1245	317	5	of	of	ADP
ejpam-1245	317	6	x	x	SYM
ejpam-1245	317	7	,	,	PUNCT
ejpam-1245	317	8	f	f	PROPN
ejpam-1245	317	9	(	(	PUNCT
ejpam-1245	317	10	ωβ	ωβ	INTJ
ejpam-1245	317	11	cl(a))⊆ωβ	cl(a))⊆ωβ	PROPN
ejpam-1245	317	12	cl	cl	PROPN
ejpam-1245	317	13	(	(	PUNCT
ejpam-1245	317	14	f	f	X
ejpam-1245	317	15	(	(	PUNCT
ejpam-1245	317	16	a	a	NOUN
ejpam-1245	317	17	)	)	PUNCT
ejpam-1245	317	18	)	)	PUNCT
ejpam-1245	317	19	.	.	PUNCT
ejpam-1245	318	1	proof	proof	NOUN
ejpam-1245	318	2	.	.	PUNCT
ejpam-1245	319	1	(	(	PUNCT
ejpam-1245	319	2	i	i	PROPN
ejpam-1245	319	3	→	→	SYM
ejpam-1245	319	4	ii	ii	PROPN
ejpam-1245	319	5	)	)	PUNCT
ejpam-1245	319	6	let	let	VERB
ejpam-1245	319	7	c	c	PRON
ejpam-1245	319	8	be	be	AUX
ejpam-1245	319	9	ωβc(y	ωβc(y	PROPN
ejpam-1245	319	10	,	,	PUNCT
ejpam-1245	319	11	σ	σ	NOUN
ejpam-1245	319	12	)	)	PUNCT
ejpam-1245	319	13	subset	subset	NOUN
ejpam-1245	319	14	of	of	ADP
ejpam-1245	319	15	y	y	PROPN
ejpam-1245	319	16	.	.	PUNCT
ejpam-1245	320	1	then	then	ADV
ejpam-1245	320	2	x	x	X
ejpam-1245	320	3	−	−	PROPN
ejpam-1245	320	4	f	f	PROPN
ejpam-1245	320	5	−1(c	−1(c	ADV
ejpam-1245	320	6	)	)	PUNCT
ejpam-1245	320	7	∈	∈	PROPN
ejpam-1245	320	8	ωβo(x	ωβo(x	PROPN
ejpam-1245	320	9	,	,	PUNCT
ejpam-1245	320	10	τ	τ	PROPN
ejpam-1245	320	11	)	)	PUNCT
ejpam-1245	320	12	,	,	PUNCT
ejpam-1245	320	13	which	which	PRON
ejpam-1245	320	14	implies	imply	VERB
ejpam-1245	320	15	that	that	SCONJ
ejpam-1245	320	16	f	f	PROPN
ejpam-1245	320	17	−1(c	−1(c	ADV
ejpam-1245	320	18	)	)	PUNCT
ejpam-1245	320	19	is	be	AUX
ejpam-1245	320	20	ωβc(x	ωβc(x	PROPN
ejpam-1245	320	21	,	,	PUNCT
ejpam-1245	320	22	τ	τ	PROPN
ejpam-1245	320	23	)	)	PUNCT
ejpam-1245	320	24	.	.	PUNCT
ejpam-1245	321	1	(	(	PUNCT
ejpam-1245	321	2	ii	ii	X
ejpam-1245	321	3	→	→	SYM
ejpam-1245	321	4	iii	iii	X
ejpam-1245	321	5	)	)	PUNCT
ejpam-1245	321	6	let	let	VERB
ejpam-1245	321	7	a	a	PRON
ejpam-1245	321	8	be	be	AUX
ejpam-1245	321	9	a	a	DET
ejpam-1245	321	10	subset	subset	NOUN
ejpam-1245	321	11	of	of	ADP
ejpam-1245	321	12	x	x	SYM
ejpam-1245	321	13	,	,	PUNCT
ejpam-1245	321	14	since	since	SCONJ
ejpam-1245	321	15	a	a	DET
ejpam-1245	321	16	⊂	⊂	PROPN
ejpam-1245	321	17	f	f	X
ejpam-1245	321	18	−1	−1	PROPN
ejpam-1245	321	19	(	(	PUNCT
ejpam-1245	321	20	f	f	PROPN
ejpam-1245	321	21	(	(	PUNCT
ejpam-1245	321	22	a	a	NOUN
ejpam-1245	321	23	)	)	PUNCT
ejpam-1245	321	24	)	)	PUNCT
ejpam-1245	321	25	,	,	PUNCT
ejpam-1245	321	26	we	we	PRON
ejpam-1245	321	27	have	have	VERB
ejpam-1245	321	28	a	a	DET
ejpam-1245	321	29	⊂	⊂	PROPN
ejpam-1245	321	30	f	f	X
ejpam-1245	321	31	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	321	32	cl	cl	PROPN
ejpam-1245	321	33	(	(	PUNCT
ejpam-1245	321	34	f	f	X
ejpam-1245	321	35	(	(	PUNCT
ejpam-1245	321	36	a	a	NOUN
ejpam-1245	321	37	)	)	PUNCT
ejpam-1245	321	38	)	)	PUNCT
ejpam-1245	321	39	)	)	PUNCT
ejpam-1245	321	40	.	.	PUNCT
ejpam-1245	322	1	now	now	ADV
ejpam-1245	322	2	by	by	ADP
ejpam-1245	322	3	(	(	PUNCT
ejpam-1245	322	4	ii	ii	NOUN
ejpam-1245	322	5	)	)	PUNCT
ejpam-1245	322	6	,	,	PUNCT
ejpam-1245	322	7	f	f	PROPN
ejpam-1245	322	8	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	322	9	cl	cl	PROPN
ejpam-1245	322	10	(	(	PUNCT
ejpam-1245	322	11	f	f	X
ejpam-1245	322	12	(	(	PUNCT
ejpam-1245	322	13	a	a	NOUN
ejpam-1245	322	14	)	)	PUNCT
ejpam-1245	322	15	)	)	PUNCT
ejpam-1245	322	16	)	)	PUNCT
ejpam-1245	322	17	is	be	AUX
ejpam-1245	322	18	ωβc(x	ωβc(x	PROPN
ejpam-1245	322	19	,	,	PUNCT
ejpam-1245	322	20	τ	τ	PROPN
ejpam-1245	322	21	)	)	PUNCT
ejpam-1245	322	22	set	set	NOUN
ejpam-1245	322	23	containing	contain	VERB
ejpam-1245	322	24	a	a	DET
ejpam-1245	322	25	then	then	ADV
ejpam-1245	322	26	ωβ	ωβ	NOUN
ejpam-1245	322	27	cl(a	cl(a	NUM
ejpam-1245	322	28	)	)	PUNCT
ejpam-1245	322	29	⊆	⊆	NUM
ejpam-1245	322	30	f	f	PROPN
ejpam-1245	322	31	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	322	32	cl	cl	PROPN
ejpam-1245	322	33	(	(	PUNCT
ejpam-1245	322	34	f	f	X
ejpam-1245	322	35	(	(	PUNCT
ejpam-1245	322	36	a	a	NOUN
ejpam-1245	322	37	)	)	PUNCT
ejpam-1245	322	38	)	)	PUNCT
ejpam-1245	322	39	)	)	PUNCT
ejpam-1245	322	40	,	,	PUNCT
ejpam-1245	322	41	which	which	PRON
ejpam-1245	322	42	implies	imply	VERB
ejpam-1245	322	43	f	f	PROPN
ejpam-1245	322	44	(	(	PUNCT
ejpam-1245	322	45	ωβ	ωβ	INTJ
ejpam-1245	322	46	cl(a))⊆ωβ	cl(a))⊆ωβ	PROPN
ejpam-1245	322	47	cl	cl	PROPN
ejpam-1245	322	48	(	(	PUNCT
ejpam-1245	322	49	f	f	X
ejpam-1245	322	50	(	(	PUNCT
ejpam-1245	322	51	a	a	NOUN
ejpam-1245	322	52	)	)	PUNCT
ejpam-1245	322	53	)	)	PUNCT
ejpam-1245	322	54	.	.	PUNCT
ejpam-1245	323	1	(	(	PUNCT
ejpam-1245	323	2	iii	iii	X
ejpam-1245	323	3	→	→	SYM
ejpam-1245	323	4	iv	iv	NUM
ejpam-1245	323	5	)	)	PUNCT
ejpam-1245	323	6	let	let	VERB
ejpam-1245	323	7	b	b	PROPN
ejpam-1245	323	8	⊂	⊂	PROPN
ejpam-1245	323	9	y	y	PROPN
ejpam-1245	323	10	,	,	PUNCT
ejpam-1245	323	11	by	by	ADP
ejpam-1245	323	12	(	(	PUNCT
ejpam-1245	323	13	iii	iii	X
ejpam-1245	323	14	)	)	PUNCT
ejpam-1245	323	15	f	f	NOUN
ejpam-1245	323	16	(	(	PUNCT
ejpam-1245	323	17	ωβ	ωβ	INTJ
ejpam-1245	323	18	cl	cl	NOUN
ejpam-1245	323	19	(	(	PUNCT
ejpam-1245	323	20	f	f	PROPN
ejpam-1245	323	21	−1(b	−1(b	NOUN
ejpam-1245	323	22	)	)	PUNCT
ejpam-1245	323	23	)	)	PUNCT
ejpam-1245	323	24	)	)	PUNCT
ejpam-1245	324	1	⊆	⊆	NUM
ejpam-1245	324	2	ωβ	ωβ	X
ejpam-1245	324	3	cl	cl	NOUN
ejpam-1245	324	4	(	(	PUNCT
ejpam-1245	324	5	f	f	PROPN
ejpam-1245	324	6	(	(	PUNCT
ejpam-1245	324	7	f	f	PROPN
ejpam-1245	324	8	−1(b	−1(b	NOUN
ejpam-1245	324	9	)	)	PUNCT
ejpam-1245	324	10	)	)	PUNCT
ejpam-1245	324	11	)	)	PUNCT
ejpam-1245	325	1	⊆	⊆	NUM
ejpam-1245	325	2	ωβ	ωβ	NOUN
ejpam-1245	325	3	cl(b	cl(b	NOUN
ejpam-1245	325	4	)	)	PUNCT
ejpam-1245	325	5	,	,	PUNCT
ejpam-1245	325	6	hence	hence	ADV
ejpam-1245	325	7	ωβ	ωβ	ADP
ejpam-1245	325	8	cl	cl	NOUN
ejpam-1245	325	9	(	(	PUNCT
ejpam-1245	325	10	f	f	PROPN
ejpam-1245	325	11	−1(b	−1(b	NOUN
ejpam-1245	325	12	)	)	PUNCT
ejpam-1245	325	13	)	)	PUNCT
ejpam-1245	326	1	⊆	⊆	NUM
ejpam-1245	326	2	f	f	PROPN
ejpam-1245	326	3	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	326	4	cl(b	cl(b	PROPN
ejpam-1245	326	5	)	)	PUNCT
ejpam-1245	326	6	)	)	PUNCT
ejpam-1245	326	7	.	.	PUNCT
ejpam-1245	327	1	(	(	PUNCT
ejpam-1245	327	2	iv	iv	X
ejpam-1245	327	3	→	→	SYM
ejpam-1245	327	4	i	i	NOUN
ejpam-1245	327	5	)	)	PUNCT
ejpam-1245	327	6	suppose	suppose	VERB
ejpam-1245	327	7	f	f	PROPN
ejpam-1245	327	8	is	be	AUX
ejpam-1245	327	9	not	not	PART
ejpam-1245	327	10	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	327	11	.	.	PUNCT
ejpam-1245	328	1	so	so	ADV
ejpam-1245	328	2	there	there	PRON
ejpam-1245	328	3	exist	exist	VERB
ejpam-1245	328	4	x	x	X
ejpam-1245	328	5	∈	∈	PROPN
ejpam-1245	328	6	x	x	X
ejpam-1245	328	7	and	and	CCONJ
ejpam-1245	328	8	v	v	ADP
ejpam-1245	328	9	∈	∈	NOUN
ejpam-1245	328	10	ωβo(y	ωβo(y	NUM
ejpam-1245	328	11	,	,	PUNCT
ejpam-1245	328	12	σ	σ	NOUN
ejpam-1245	328	13	)	)	PUNCT
ejpam-1245	328	14	with	with	ADP
ejpam-1245	328	15	f	f	PROPN
ejpam-1245	328	16	(	(	PUNCT
ejpam-1245	328	17	x	x	X
ejpam-1245	328	18	)	)	PUNCT
ejpam-1245	328	19	∈	∈	NOUN
ejpam-1245	328	20	v	v	ADP
ejpam-1245	328	21	such	such	ADJ
ejpam-1245	328	22	that	that	PRON
ejpam-1245	328	23	for	for	ADP
ejpam-1245	328	24	allωβo(x	allωβo(x	PROPN
ejpam-1245	328	25	,	,	PUNCT
ejpam-1245	328	26	τ	τ	PROPN
ejpam-1245	328	27	)	)	PUNCT
ejpam-1245	328	28	set	set	VERB
ejpam-1245	328	29	u	u	NOUN
ejpam-1245	328	30	with	with	ADP
ejpam-1245	328	31	x	x	PROPN
ejpam-1245	328	32	∈	∈	PROPN
ejpam-1245	328	33	u	u	NOUN
ejpam-1245	328	34	and	and	CCONJ
ejpam-1245	328	35	f	f	PROPN
ejpam-1245	328	36	(	(	PUNCT
ejpam-1245	328	37	u	u	NOUN
ejpam-1245	328	38	)	)	PUNCT
ejpam-1245	328	39	6⊂	6⊂	NUM
ejpam-1245	328	40	(	(	PUNCT
ejpam-1245	328	41	v	v	NOUN
ejpam-1245	328	42	)	)	PUNCT
ejpam-1245	328	43	i.e.	i.e.	X
ejpam-1245	328	44	f	f	X
ejpam-1245	328	45	(	(	PUNCT
ejpam-1245	328	46	u)∩(y−v	u)∩(y−v	PROPN
ejpam-1245	328	47	)	)	PUNCT
ejpam-1245	328	48	6=	6=	ADP
ejpam-1245	329	1	φ	φ	PROPN
ejpam-1245	329	2	.	.	PUNCT
ejpam-1245	330	1	therefore	therefore	ADV
ejpam-1245	330	2	,	,	PUNCT
ejpam-1245	330	3	by	by	ADP
ejpam-1245	330	4	(	(	PUNCT
ejpam-1245	330	5	vii	vii	PROPN
ejpam-1245	330	6	)	)	PUNCT
ejpam-1245	330	7	,	,	PUNCT
ejpam-1245	330	8	x	x	PUNCT
ejpam-1245	330	9	∈	∈	PROPN
ejpam-1245	330	10	f	f	PROPN
ejpam-1245	330	11	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	330	12	cl(y	cl(y	NOUN
ejpam-1245	330	13	−	−	PROPN
ejpam-1245	330	14	v	v	NOUN
ejpam-1245	330	15	)	)	PUNCT
ejpam-1245	330	16	)	)	PUNCT
ejpam-1245	330	17	.	.	PUNCT
ejpam-1245	331	1	so	so	ADV
ejpam-1245	331	2	by	by	ADP
ejpam-1245	331	3	theorem	theorem	NOUN
ejpam-1245	331	4	2	2	NUM
ejpam-1245	331	5	,	,	PUNCT
ejpam-1245	331	6	f	f	PROPN
ejpam-1245	331	7	(	(	PUNCT
ejpam-1245	331	8	x	x	NOUN
ejpam-1245	331	9	)	)	PUNCT
ejpam-1245	331	10	∈ωβ	∈ωβ	NOUN
ejpam-1245	331	11	cl(y	cl(y	NOUN
ejpam-1245	331	12	−	−	NOUN
ejpam-1245	331	13	v	v	NOUN
ejpam-1245	331	14	)	)	PUNCT
ejpam-1245	331	15	.	.	PUNCT
ejpam-1245	332	1	thus	thus	ADV
ejpam-1245	332	2	for	for	ADP
ejpam-1245	332	3	all	all	DET
ejpam-1245	332	4	ωβo(y	ωβo(y	NUM
ejpam-1245	332	5	,	,	PUNCT
ejpam-1245	332	6	σ	σ	NOUN
ejpam-1245	332	7	)	)	PUNCT
ejpam-1245	332	8	sets	set	VERB
ejpam-1245	332	9	v	v	ADP
ejpam-1245	332	10	containing	contain	VERB
ejpam-1245	332	11	f	f	X
ejpam-1245	332	12	(	(	PUNCT
ejpam-1245	332	13	x	x	NOUN
ejpam-1245	332	14	)	)	PUNCT
ejpam-1245	332	15	,	,	PUNCT
ejpam-1245	332	16	so	so	ADV
ejpam-1245	332	17	v	v	ADP
ejpam-1245	332	18	∩	∩	NOUN
ejpam-1245	332	19	(	(	PUNCT
ejpam-1245	332	20	y	y	PROPN
ejpam-1245	332	21	−	−	PROPN
ejpam-1245	332	22	v	v	NOUN
ejpam-1245	332	23	)	)	PUNCT
ejpam-1245	332	24	6=	6=	ADP
ejpam-1245	333	1	φ	φ	PROPN
ejpam-1245	333	2	,	,	PUNCT
ejpam-1245	333	3	a	a	DET
ejpam-1245	333	4	contradiction	contradiction	NOUN
ejpam-1245	333	5	.	.	PUNCT
ejpam-1245	334	1	therefore	therefore	ADV
ejpam-1245	334	2	,	,	PUNCT
ejpam-1245	334	3	f	f	PROPN
ejpam-1245	334	4	is	be	AUX
ejpam-1245	334	5	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	334	6	.	.	PUNCT
ejpam-1245	334	7	theorem	theorem	NOUN
ejpam-1245	334	8	11	11	NUM
ejpam-1245	334	9	.	.	PUNCT
ejpam-1245	335	1	let	let	VERB
ejpam-1245	335	2	f	f	NOUN
ejpam-1245	335	3	:	:	PUNCT
ejpam-1245	335	4	(	(	PUNCT
ejpam-1245	335	5	x	x	X
ejpam-1245	335	6	,	,	PUNCT
ejpam-1245	335	7	τ	τ	PROPN
ejpam-1245	335	8	)	)	PUNCT
ejpam-1245	335	9	→	→	SYM
ejpam-1245	335	10	(	(	PUNCT
ejpam-1245	335	11	y	y	PROPN
ejpam-1245	335	12	,	,	PUNCT
ejpam-1245	335	13	σ	σ	PROPN
ejpam-1245	335	14	)	)	PUNCT
ejpam-1245	335	15	be	be	AUX
ejpam-1245	335	16	a	a	DET
ejpam-1245	335	17	function	function	NOUN
ejpam-1245	335	18	.	.	PUNCT
ejpam-1245	336	1	then	then	ADV
ejpam-1245	336	2	f	f	PROPN
ejpam-1245	336	3	is	be	AUX
ejpam-1245	336	4	ωβ−irresolute	ωβ−irresolute	ADJ
ejpam-1245	336	5	if	if	SCONJ
ejpam-1245	337	1	and	and	CCONJ
ejpam-1245	337	2	only	only	ADV
ejpam-1245	337	3	if	if	SCONJ
ejpam-1245	337	4	f	f	PROPN
ejpam-1245	337	5	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	337	6	int(b	int(b	PROPN
ejpam-1245	337	7	)	)	PUNCT
ejpam-1245	337	8	)	)	PUNCT
ejpam-1245	338	1	⊆ωβ	⊆ωβ	VERB
ejpam-1245	338	2	int	int	NOUN
ejpam-1245	338	3	(	(	PUNCT
ejpam-1245	338	4	f	f	PROPN
ejpam-1245	338	5	−1(b	−1(b	NOUN
ejpam-1245	338	6	)	)	PUNCT
ejpam-1245	338	7	)	)	PUNCT
ejpam-1245	338	8	.	.	PUNCT
ejpam-1245	339	1	proof	proof	NOUN
ejpam-1245	339	2	.	.	PUNCT
ejpam-1245	340	1	necessity	necessity	NOUN
ejpam-1245	340	2	.	.	PUNCT
ejpam-1245	341	1	let	let	VERB
ejpam-1245	341	2	b	b	X
ejpam-1245	341	3	be	be	AUX
ejpam-1245	341	4	any	any	DET
ejpam-1245	341	5	subset	subset	NOUN
ejpam-1245	341	6	of	of	ADP
ejpam-1245	341	7	y	y	PROPN
ejpam-1245	341	8	.	.	PUNCT
ejpam-1245	342	1	since	since	SCONJ
ejpam-1245	342	2	f	f	PROPN
ejpam-1245	342	3	isωβ−irrrsolute	isωβ−irrrsolute	VERB
ejpam-1245	342	4	,	,	PUNCT
ejpam-1245	342	5	we	we	PRON
ejpam-1245	342	6	have	have	VERB
ejpam-1245	342	7	f	f	PROPN
ejpam-1245	342	8	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	342	9	int(b	int(b	PROPN
ejpam-1245	342	10	)	)	PUNCT
ejpam-1245	342	11	)	)	PUNCT
ejpam-1245	342	12	is	be	AUX
ejpam-1245	342	13	ωβo(x	ωβo(x	PROPN
ejpam-1245	342	14	,	,	PUNCT
ejpam-1245	342	15	τ	τ	NOUN
ejpam-1245	342	16	)	)	PUNCT
ejpam-1245	342	17	set	set	NOUN
ejpam-1245	342	18	.	.	PUNCT
ejpam-1245	343	1	as	as	ADP
ejpam-1245	343	2	f	f	PROPN
ejpam-1245	343	3	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	343	4	int(b	int(b	PROPN
ejpam-1245	343	5	)	)	PUNCT
ejpam-1245	343	6	)	)	PUNCT
ejpam-1245	344	1	⊆	⊆	NUM
ejpam-1245	344	2	f	f	PROPN
ejpam-1245	344	3	−1(b	−1(b	NOUN
ejpam-1245	344	4	)	)	PUNCT
ejpam-1245	344	5	,	,	PUNCT
ejpam-1245	344	6	then	then	ADV
ejpam-1245	344	7	f	f	PROPN
ejpam-1245	344	8	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	344	9	int(b	int(b	PROPN
ejpam-1245	344	10	)	)	PUNCT
ejpam-1245	344	11	)	)	PUNCT
ejpam-1245	345	1	⊆ωβ	⊆ωβ	VERB
ejpam-1245	345	2	int	int	NOUN
ejpam-1245	345	3	(	(	PUNCT
ejpam-1245	345	4	f	f	PROPN
ejpam-1245	345	5	−1(b	−1(b	NOUN
ejpam-1245	345	6	)	)	PUNCT
ejpam-1245	345	7	)	)	PUNCT
ejpam-1245	345	8	.	.	PUNCT
ejpam-1245	346	1	sufficiency	sufficiency	PROPN
ejpam-1245	346	2	.	.	PUNCT
ejpam-1245	347	1	let	let	VERB
ejpam-1245	347	2	x	x	PUNCT
ejpam-1245	347	3	∈	∈	PROPN
ejpam-1245	347	4	x	x	X
ejpam-1245	347	5	and	and	CCONJ
ejpam-1245	347	6	v	v	ADP
ejpam-1245	347	7	∈	∈	NOUN
ejpam-1245	347	8	ωβo(y	ωβo(y	NUM
ejpam-1245	347	9	,	,	PUNCT
ejpam-1245	347	10	σ	σ	NOUN
ejpam-1245	347	11	)	)	PUNCT
ejpam-1245	347	12	with	with	ADP
ejpam-1245	347	13	f	f	PROPN
ejpam-1245	347	14	(	(	PUNCT
ejpam-1245	347	15	x	x	X
ejpam-1245	347	16	)	)	PUNCT
ejpam-1245	347	17	∈	∈	NOUN
ejpam-1245	347	18	v	v	NOUN
ejpam-1245	347	19	.	.	PUNCT
ejpam-1245	348	1	then	then	ADV
ejpam-1245	348	2	x	x	SYM
ejpam-1245	348	3	∈	∈	PROPN
ejpam-1245	348	4	f	f	X
ejpam-1245	348	5	−1(v	−1(v	PROPN
ejpam-1245	348	6	)	)	PUNCT
ejpam-1245	348	7	and	and	CCONJ
ejpam-1245	348	8	so	so	ADV
ejpam-1245	348	9	by	by	ADP
ejpam-1245	348	10	assumption	assumption	NOUN
ejpam-1245	348	11	x	x	X
ejpam-1245	348	12	∈	∈	PROPN
ejpam-1245	348	13	ωβ	ωβ	X
ejpam-1245	348	14	int	int	NOUN
ejpam-1245	348	15	(	(	PUNCT
ejpam-1245	348	16	f	f	PROPN
ejpam-1245	348	17	−1(v	−1(v	PROPN
ejpam-1245	348	18	)	)	PUNCT
ejpam-1245	348	19	)	)	PUNCT
ejpam-1245	348	20	.	.	PUNCT
ejpam-1245	349	1	there	there	PRON
ejpam-1245	349	2	exists	exist	VERB
ejpam-1245	349	3	an	an	DET
ejpam-1245	349	4	ωβo(x	ωβo(x	PROPN
ejpam-1245	349	5	,	,	PUNCT
ejpam-1245	349	6	τ	τ	NOUN
ejpam-1245	349	7	)	)	PUNCT
ejpam-1245	349	8	sets	set	VERB
ejpam-1245	349	9	such	such	ADJ
ejpam-1245	349	10	that	that	SCONJ
ejpam-1245	349	11	x	x	SYM
ejpam-1245	349	12	∈	∈	NOUN
ejpam-1245	349	13	u	u	NOUN
ejpam-1245	349	14	⊆	⊆	NUM
ejpam-1245	349	15	f	f	PROPN
ejpam-1245	349	16	−1(v	−1(v	NOUN
ejpam-1245	349	17	)	)	PUNCT
ejpam-1245	349	18	.	.	PUNCT
ejpam-1245	350	1	hence	hence	ADV
ejpam-1245	350	2	f	f	PROPN
ejpam-1245	350	3	(	(	PUNCT
ejpam-1245	350	4	x	x	X
ejpam-1245	350	5	)	)	PUNCT
ejpam-1245	350	6	∈	∈	PROPN
ejpam-1245	350	7	f	f	PROPN
ejpam-1245	350	8	(	(	PUNCT
ejpam-1245	350	9	u)⊆	u)⊆	PROPN
ejpam-1245	350	10	v	v	NOUN
ejpam-1245	350	11	and	and	CCONJ
ejpam-1245	350	12	the	the	DET
ejpam-1245	350	13	result	result	NOUN
ejpam-1245	350	14	follows	follow	VERB
ejpam-1245	350	15	.	.	PUNCT
ejpam-1245	351	1	proposition	proposition	NOUN
ejpam-1245	351	2	5	5	NUM
ejpam-1245	351	3	.	.	PUNCT
ejpam-1245	352	1	if	if	SCONJ
ejpam-1245	352	2	f	f	PROPN
ejpam-1245	352	3	:	:	PUNCT
ejpam-1245	352	4	(	(	PUNCT
ejpam-1245	352	5	x	x	X
ejpam-1245	352	6	,	,	PUNCT
ejpam-1245	352	7	τ)→	τ)→	PROPN
ejpam-1245	352	8	(	(	PUNCT
ejpam-1245	352	9	y	y	PROPN
ejpam-1245	352	10	,	,	PUNCT
ejpam-1245	352	11	σ	σ	PROPN
ejpam-1245	352	12	)	)	PUNCT
ejpam-1245	352	13	isωβ−irresolute	isωβ−irresolute	VERB
ejpam-1245	352	14	and	and	CCONJ
ejpam-1245	352	15	g	g	NOUN
ejpam-1245	352	16	:	:	PUNCT
ejpam-1245	352	17	(	(	PUNCT
ejpam-1245	352	18	y	y	NOUN
ejpam-1245	352	19	,	,	PUNCT
ejpam-1245	352	20	σ)→	σ)→	PROPN
ejpam-1245	352	21	(	(	PUNCT
ejpam-1245	352	22	z	z	PROPN
ejpam-1245	352	23	,	,	PUNCT
ejpam-1245	352	24	ρ	ρ	PROPN
ejpam-1245	352	25	)	)	PUNCT
ejpam-1245	352	26	isωβ−continuous	isωβ−continuous	ADJ
ejpam-1245	352	27	,	,	PUNCT
ejpam-1245	352	28	then	then	ADV
ejpam-1245	352	29	g	g	PROPN
ejpam-1245	352	30	◦	◦	PROPN
ejpam-1245	352	31	f	f	PROPN
ejpam-1245	352	32	is	be	AUX
ejpam-1245	352	33	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	352	34	.	.	PUNCT
ejpam-1245	353	1	proof	proof	NOUN
ejpam-1245	353	2	.	.	PUNCT
ejpam-1245	354	1	let	let	VERB
ejpam-1245	354	2	x	x	PUNCT
ejpam-1245	354	3	∈	∈	PROPN
ejpam-1245	354	4	x	x	PUNCT
ejpam-1245	354	5	and	and	CCONJ
ejpam-1245	354	6	let	let	VERB
ejpam-1245	354	7	v	v	PART
ejpam-1245	354	8	be	be	AUX
ejpam-1245	354	9	any	any	DET
ejpam-1245	354	10	open	open	ADJ
ejpam-1245	354	11	set	set	NOUN
ejpam-1245	354	12	in	in	ADP
ejpam-1245	354	13	(	(	PUNCT
ejpam-1245	354	14	z	z	NOUN
ejpam-1245	354	15	,	,	PUNCT
ejpam-1245	354	16	ρ	ρ	PROPN
ejpam-1245	354	17	)	)	PUNCT
ejpam-1245	354	18	containing	contain	VERB
ejpam-1245	354	19	g	g	NOUN
ejpam-1245	354	20	(	(	PUNCT
ejpam-1245	354	21	f	f	PROPN
ejpam-1245	354	22	(	(	PUNCT
ejpam-1245	354	23	x	x	NOUN
ejpam-1245	354	24	)	)	PUNCT
ejpam-1245	354	25	)	)	PUNCT
ejpam-1245	354	26	.	.	PUNCT
ejpam-1245	355	1	since	since	SCONJ
ejpam-1245	355	2	g	g	PROPN
ejpam-1245	355	3	is	be	AUX
ejpam-1245	355	4	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	355	5	,	,	PUNCT
ejpam-1245	355	6	there	there	PRON
ejpam-1245	355	7	exists	exist	VERB
ejpam-1245	355	8	an	an	DET
ejpam-1245	355	9	ωβo(y	ωβo(y	PROPN
ejpam-1245	355	10	,	,	PUNCT
ejpam-1245	355	11	σ	σ	NOUN
ejpam-1245	355	12	)	)	PUNCT
ejpam-1245	355	13	set	set	VERB
ejpam-1245	355	14	w	w	NOUN
ejpam-1245	355	15	containing	contain	VERB
ejpam-1245	355	16	f	f	PROPN
ejpam-1245	355	17	(	(	PUNCT
ejpam-1245	355	18	x	x	X
ejpam-1245	355	19	)	)	PUNCT
ejpam-1245	355	20	such	such	ADJ
ejpam-1245	355	21	that	that	SCONJ
ejpam-1245	355	22	g(w	g(w	PROPN
ejpam-1245	355	23	)	)	PUNCT
ejpam-1245	355	24	⊆	⊆	NUM
ejpam-1245	355	25	v	v	NOUN
ejpam-1245	355	26	.	.	PUNCT
ejpam-1245	356	1	put	put	VERB
ejpam-1245	356	2	u	u	NOUN
ejpam-1245	356	3	=	=	PUNCT
ejpam-1245	356	4	f	f	PROPN
ejpam-1245	356	5	−1(w	−1(w	ADV
ejpam-1245	356	6	)	)	PUNCT
ejpam-1245	356	7	since	since	SCONJ
ejpam-1245	356	8	f	f	PROPN
ejpam-1245	356	9	is	be	AUX
ejpam-1245	356	10	ωβ−irresolute	ωβ−irresolute	ADJ
ejpam-1245	356	11	,	,	PUNCT
ejpam-1245	356	12	then	then	ADV
ejpam-1245	356	13	u	u	PROPN
ejpam-1245	356	14	∈ωβo(x	∈ωβo(x	PROPN
ejpam-1245	356	15	,	,	PUNCT
ejpam-1245	356	16	τ	τ	PROPN
ejpam-1245	356	17	)	)	PUNCT
ejpam-1245	356	18	such	such	ADJ
ejpam-1245	356	19	that	that	SCONJ
ejpam-1245	356	20	x	x	SYM
ejpam-1245	356	21	∈	∈	PROPN
ejpam-1245	356	22	u	u	NOUN
ejpam-1245	356	23	and	and	CCONJ
ejpam-1245	356	24	g	g	PROPN
ejpam-1245	356	25	(	(	PUNCT
ejpam-1245	356	26	f	f	X
ejpam-1245	356	27	(	(	PUNCT
ejpam-1245	356	28	u))⊆	u))⊆	X
ejpam-1245	356	29	g(w	g(w	PROPN
ejpam-1245	356	30	)	)	PUNCT
ejpam-1245	356	31	⊆	⊆	NUM
ejpam-1245	356	32	v	v	NOUN
ejpam-1245	356	33	.	.	PUNCT
ejpam-1245	357	1	hence	hence	ADV
ejpam-1245	357	2	g	g	PROPN
ejpam-1245	357	3	◦	◦	PROPN
ejpam-1245	357	4	f	f	PROPN
ejpam-1245	357	5	is	be	AUX
ejpam-1245	357	6	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	357	7	.	.	PUNCT
ejpam-1245	358	1	corollary	corollary	ADJ
ejpam-1245	358	2	6	6	NUM
ejpam-1245	358	3	.	.	PUNCT
ejpam-1245	359	1	if	if	SCONJ
ejpam-1245	359	2	f	f	PROPN
ejpam-1245	359	3	:	:	PUNCT
ejpam-1245	359	4	(	(	PUNCT
ejpam-1245	359	5	x	x	X
ejpam-1245	359	6	,	,	PUNCT
ejpam-1245	359	7	τ)→	τ)→	PROPN
ejpam-1245	359	8	(	(	PUNCT
ejpam-1245	359	9	y	y	PROPN
ejpam-1245	359	10	,	,	PUNCT
ejpam-1245	359	11	σ	σ	PROPN
ejpam-1245	359	12	)	)	PUNCT
ejpam-1245	359	13	isωβ−irresolute	isωβ−irresolute	VERB
ejpam-1245	359	14	and	and	CCONJ
ejpam-1245	359	15	g	g	NOUN
ejpam-1245	359	16	:	:	PUNCT
ejpam-1245	359	17	(	(	PUNCT
ejpam-1245	359	18	y	y	NOUN
ejpam-1245	359	19	,	,	PUNCT
ejpam-1245	359	20	σ)→	σ)→	PROPN
ejpam-1245	359	21	(	(	PUNCT
ejpam-1245	359	22	z	z	PROPN
ejpam-1245	359	23	,	,	PUNCT
ejpam-1245	359	24	ρ	ρ	PROPN
ejpam-1245	359	25	)	)	PUNCT
ejpam-1245	359	26	isωb−continuous	isωb−continuous	ADJ
ejpam-1245	359	27	,	,	PUNCT
ejpam-1245	359	28	then	then	ADV
ejpam-1245	359	29	g	g	PROPN
ejpam-1245	359	30	◦	◦	PROPN
ejpam-1245	359	31	f	f	PROPN
ejpam-1245	359	32	is	be	AUX
ejpam-1245	359	33	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	359	34	.	.	PUNCT
ejpam-1245	360	1	recall	recall	VERB
ejpam-1245	360	2	that	that	SCONJ
ejpam-1245	360	3	a	a	DET
ejpam-1245	360	4	function	function	NOUN
ejpam-1245	360	5	f	f	NOUN
ejpam-1245	360	6	:	:	PUNCT
ejpam-1245	360	7	(	(	PUNCT
ejpam-1245	360	8	x	x	X
ejpam-1245	360	9	,	,	PUNCT
ejpam-1245	360	10	τ	τ	PROPN
ejpam-1245	360	11	)	)	PUNCT
ejpam-1245	360	12	→	→	SYM
ejpam-1245	360	13	(	(	PUNCT
ejpam-1245	360	14	y	y	PROPN
ejpam-1245	360	15	,	,	PUNCT
ejpam-1245	360	16	σ	σ	PROPN
ejpam-1245	360	17	)	)	PUNCT
ejpam-1245	360	18	is	be	AUX
ejpam-1245	360	19	said	say	VERB
ejpam-1245	360	20	to	to	PART
ejpam-1245	360	21	be	be	AUX
ejpam-1245	360	22	ω−irresolute	ω−irresolute	VERB
ejpam-1245	360	23	[	[	X
ejpam-1245	360	24	1	1	NUM
ejpam-1245	360	25	]	]	X
ejpam-1245	360	26	if	if	SCONJ
ejpam-1245	360	27	the	the	DET
ejpam-1245	360	28	inverse	inverse	ADJ
ejpam-1245	360	29	image	image	NOUN
ejpam-1245	360	30	of	of	ADP
ejpam-1245	360	31	each	each	DET
ejpam-1245	360	32	ωo(y	ωo(y	NUM
ejpam-1245	360	33	,	,	PUNCT
ejpam-1245	360	34	σ	σ	X
ejpam-1245	360	35	)	)	PUNCT
ejpam-1245	360	36	set	set	NOUN
ejpam-1245	360	37	is	be	AUX
ejpam-1245	360	38	an	an	DET
ejpam-1245	360	39	ωo(x	ωo(x	PROPN
ejpam-1245	360	40	,	,	PUNCT
ejpam-1245	360	41	τ	τ	PROPN
ejpam-1245	360	42	)	)	PUNCT
ejpam-1245	360	43	.	.	PUNCT
ejpam-1245	361	1	theorem	theorem	NOUN
ejpam-1245	361	2	12	12	NUM
ejpam-1245	361	3	.	.	PUNCT
ejpam-1245	362	1	if	if	SCONJ
ejpam-1245	362	2	f	f	PROPN
ejpam-1245	362	3	:	:	PUNCT
ejpam-1245	362	4	(	(	PUNCT
ejpam-1245	362	5	x	x	X
ejpam-1245	362	6	,	,	PUNCT
ejpam-1245	362	7	τ)→	τ)→	PROPN
ejpam-1245	362	8	(	(	PUNCT
ejpam-1245	362	9	y	y	PROPN
ejpam-1245	362	10	,	,	PUNCT
ejpam-1245	362	11	σ	σ	PROPN
ejpam-1245	362	12	)	)	PUNCT
ejpam-1245	362	13	isω−irresolute	isω−irresolute	NOUN
ejpam-1245	362	14	and	and	CCONJ
ejpam-1245	362	15	every	every	DET
ejpam-1245	362	16	βo(y	βo(y	NUM
ejpam-1245	362	17	,	,	PUNCT
ejpam-1245	362	18	σ	σ	PROPN
ejpam-1245	362	19	)	)	PUNCT
ejpam-1245	362	20	set	set	NOUN
ejpam-1245	362	21	is	be	AUX
ejpam-1245	362	22	closed	close	VERB
ejpam-1245	362	23	in	in	ADP
ejpam-1245	362	24	the	the	DET
ejpam-1245	362	25	space	space	NOUN
ejpam-1245	362	26	(	(	PUNCT
ejpam-1245	362	27	y	y	PROPN
ejpam-1245	362	28	,	,	PUNCT
ejpam-1245	362	29	σ	σ	PROPN
ejpam-1245	362	30	)	)	PUNCT
ejpam-1245	362	31	then	then	ADV
ejpam-1245	362	32	f	f	PROPN
ejpam-1245	362	33	is	be	AUX
ejpam-1245	362	34	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	362	35	.	.	PUNCT
ejpam-1245	362	36	proof	proof	NOUN
ejpam-1245	362	37	.	.	PUNCT
ejpam-1245	363	1	let	let	VERB
ejpam-1245	363	2	u	u	PRON
ejpam-1245	363	3	be	be	AUX
ejpam-1245	363	4	any	any	DET
ejpam-1245	363	5	ωβo(y	ωβo(y	NUM
ejpam-1245	363	6	,	,	PUNCT
ejpam-1245	363	7	σ	σ	NOUN
ejpam-1245	363	8	)	)	PUNCT
ejpam-1245	363	9	set	set	NOUN
ejpam-1245	363	10	,	,	PUNCT
ejpam-1245	363	11	then	then	ADV
ejpam-1245	363	12	for	for	ADP
ejpam-1245	363	13	all	all	DET
ejpam-1245	363	14	y	y	PROPN
ejpam-1245	363	15	∈	∈	PROPN
ejpam-1245	363	16	y	y	PROPN
ejpam-1245	363	17	,	,	PUNCT
ejpam-1245	363	18	there	there	PRON
ejpam-1245	363	19	exists	exist	VERB
ejpam-1245	363	20	βo(y	βo(y	PUNCT
ejpam-1245	363	21	,	,	PUNCT
ejpam-1245	363	22	σ	σ	NOUN
ejpam-1245	363	23	)	)	PUNCT
ejpam-1245	363	24	sets	set	VERB
ejpam-1245	363	25	u1	u1	NOUN
ejpam-1245	363	26	containing	contain	VERB
ejpam-1245	363	27	x	x	PUNCT
ejpam-1245	363	28	such	such	ADJ
ejpam-1245	363	29	that	that	DET
ejpam-1245	363	30	u1	u1	NOUN
ejpam-1245	363	31	−	−	PROPN
ejpam-1245	363	32	u	u	NOUN
ejpam-1245	363	33	is	be	AUX
ejpam-1245	363	34	a	a	DET
ejpam-1245	363	35	countable	countable	ADJ
ejpam-1245	363	36	,	,	PUNCT
ejpam-1245	363	37	thus	thus	ADV
ejpam-1245	363	38	by	by	ADP
ejpam-1245	363	39	assumption	assumption	NOUN
ejpam-1245	363	40	u1	u1	NOUN
ejpam-1245	363	41	⊆	⊆	NUM
ejpam-1245	363	42	cl(int(cl(u1)))⊆	cl(int(cl(u1)))⊆	NOUN
ejpam-1245	363	43	int(u1	int(u1	NOUN
ejpam-1245	363	44	)	)	PUNCT
ejpam-1245	363	45	,	,	PUNCT
ejpam-1245	363	46	so	so	ADV
ejpam-1245	363	47	u1	u1	NOUN
ejpam-1245	363	48	is	be	AUX
ejpam-1245	363	49	open	open	ADJ
ejpam-1245	363	50	sets	set	NOUN
ejpam-1245	363	51	in	in	ADP
ejpam-1245	363	52	(	(	PUNCT
ejpam-1245	363	53	y	y	PROPN
ejpam-1245	363	54	,	,	PUNCT
ejpam-1245	363	55	σ	σ	PROPN
ejpam-1245	363	56	)	)	PUNCT
ejpam-1245	363	57	,	,	PUNCT
ejpam-1245	363	58	hence	hence	ADV
ejpam-1245	363	59	u	u	NOUN
ejpam-1245	363	60	∈	∈	PROPN
ejpam-1245	363	61	ωo(y	ωo(y	NUM
ejpam-1245	363	62	,	,	PUNCT
ejpam-1245	363	63	σ	σ	NOUN
ejpam-1245	363	64	)	)	PUNCT
ejpam-1245	363	65	.	.	PUNCT
ejpam-1245	364	1	since	since	SCONJ
ejpam-1245	364	2	f	f	PROPN
ejpam-1245	364	3	is	be	AUX
ejpam-1245	364	4	ω−irresolute	ω−irresolute	NOUN
ejpam-1245	364	5	,	,	PUNCT
ejpam-1245	364	6	then	then	ADV
ejpam-1245	364	7	f	f	PROPN
ejpam-1245	364	8	−1(u	−1(u	PROPN
ejpam-1245	364	9	)	)	PUNCT
ejpam-1245	364	10	∈ωo(x	∈ωo(x	NOUN
ejpam-1245	364	11	,	,	PUNCT
ejpam-1245	364	12	τ)⊆ωβo(x	τ)⊆ωβo(x	PROPN
ejpam-1245	364	13	,	,	PUNCT
ejpam-1245	364	14	τ	τ	PROPN
ejpam-1245	364	15	)	)	PUNCT
ejpam-1245	364	16	.	.	PUNCT
ejpam-1245	365	1	h.	h.	PROPN
ejpam-1245	365	2	aljarrah	aljarrah	PROPN
ejpam-1245	365	3	,	,	PUNCT
ejpam-1245	365	4	m.	m.	NOUN
ejpam-1245	365	5	noorani	noorani	PROPN
ejpam-1245	365	6	/	/	SYM
ejpam-1245	365	7	eur	eur	PROPN
ejpam-1245	365	8	.	.	PUNCT
ejpam-1245	366	1	j.	j.	PROPN
ejpam-1245	366	2	pure	pure	PROPN
ejpam-1245	366	3	appl	appl	PROPN
ejpam-1245	366	4	.	.	PROPN
ejpam-1245	366	5	math	math	PROPN
ejpam-1245	366	6	,	,	PUNCT
ejpam-1245	366	7	5	5	NUM
ejpam-1245	366	8	(	(	PUNCT
ejpam-1245	366	9	2012	2012	NUM
ejpam-1245	366	10	)	)	PUNCT
ejpam-1245	366	11	,	,	PUNCT
ejpam-1245	366	12	129	129	NUM
ejpam-1245	366	13	-	-	SYM
ejpam-1245	366	14	140	140	NUM
ejpam-1245	366	15	138	138	NUM
ejpam-1245	366	16	proposition	proposition	NOUN
ejpam-1245	366	17	6	6	NUM
ejpam-1245	366	18	.	.	PUNCT
ejpam-1245	367	1	let	let	VERB
ejpam-1245	367	2	f	f	NOUN
ejpam-1245	367	3	:	:	PUNCT
ejpam-1245	367	4	(	(	PUNCT
ejpam-1245	367	5	x	x	X
ejpam-1245	367	6	,	,	PUNCT
ejpam-1245	367	7	τ)→	τ)→	PROPN
ejpam-1245	367	8	(	(	PUNCT
ejpam-1245	367	9	y	y	PROPN
ejpam-1245	367	10	,	,	PUNCT
ejpam-1245	367	11	σ	σ	PROPN
ejpam-1245	367	12	)	)	PUNCT
ejpam-1245	367	13	be	be	AUX
ejpam-1245	367	14	an	an	DET
ejpam-1245	367	15	open	open	ADJ
ejpam-1245	367	16	continuous	continuous	ADJ
ejpam-1245	367	17	function	function	NOUN
ejpam-1245	367	18	and	and	CCONJ
ejpam-1245	367	19	every	every	DET
ejpam-1245	367	20	ωβo(y	ωβo(y	NUM
ejpam-1245	367	21	,	,	PUNCT
ejpam-1245	367	22	σ	σ	NOUN
ejpam-1245	367	23	)	)	PUNCT
ejpam-1245	367	24	is	be	AUX
ejpam-1245	367	25	closed	close	VERB
ejpam-1245	367	26	in	in	ADP
ejpam-1245	367	27	the	the	DET
ejpam-1245	367	28	space	space	NOUN
ejpam-1245	367	29	(	(	PUNCT
ejpam-1245	367	30	y	y	PROPN
ejpam-1245	367	31	,	,	PUNCT
ejpam-1245	367	32	σ	σ	PROPN
ejpam-1245	367	33	)	)	PUNCT
ejpam-1245	367	34	then	then	ADV
ejpam-1245	367	35	f	f	PROPN
ejpam-1245	367	36	is	be	AUX
ejpam-1245	367	37	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	367	38	.	.	PUNCT
ejpam-1245	367	39	proof	proof	NOUN
ejpam-1245	367	40	.	.	PUNCT
ejpam-1245	368	1	let	let	VERB
ejpam-1245	368	2	u	u	PRON
ejpam-1245	368	3	∈ωβo(y	∈ωβo(y	NOUN
ejpam-1245	368	4	,	,	PUNCT
ejpam-1245	368	5	σ	σ	NOUN
ejpam-1245	368	6	)	)	PUNCT
ejpam-1245	368	7	,	,	PUNCT
ejpam-1245	368	8	by	by	ADP
ejpam-1245	368	9	theorem	theorem	NOUN
ejpam-1245	368	10	3	3	NUM
ejpam-1245	368	11	,	,	PUNCT
ejpam-1245	368	12	ωβ	ωβ	X
ejpam-1245	368	13	cl	cl	NOUN
ejpam-1245	368	14	(	(	PUNCT
ejpam-1245	368	15	f	f	PROPN
ejpam-1245	368	16	−1(u	−1(u	NOUN
ejpam-1245	368	17	)	)	PUNCT
ejpam-1245	368	18	)	)	PUNCT
ejpam-1245	369	1	⊆	⊆	NUM
ejpam-1245	369	2	cl	cl	NOUN
ejpam-1245	369	3	(	(	PUNCT
ejpam-1245	369	4	f	f	PROPN
ejpam-1245	369	5	−1(u	−1(u	NOUN
ejpam-1245	369	6	)	)	PUNCT
ejpam-1245	369	7	)	)	PUNCT
ejpam-1245	369	8	=	=	PUNCT
ejpam-1245	370	1	f	f	PROPN
ejpam-1245	371	1	−1(cl(u))⊆	−1(cl(u))⊆	PUNCT
ejpam-1245	371	2	f	f	PROPN
ejpam-1245	371	3	−1(ωβ	−1(ωβ	PROPN
ejpam-1245	371	4	cl(u	cl(u	PROPN
ejpam-1245	371	5	)	)	PUNCT
ejpam-1245	371	6	)	)	PUNCT
ejpam-1245	372	1	,	,	PUNCT
ejpam-1245	372	2	hence	hence	ADV
ejpam-1245	372	3	f	f	PROPN
ejpam-1245	372	4	is	be	AUX
ejpam-1245	372	5	ωβ−irresolute	ωβ−irresolute	ADJ
ejpam-1245	372	6	,	,	PUNCT
ejpam-1245	372	7	by	by	ADP
ejpam-1245	372	8	theorem	theorem	NOUN
ejpam-1245	372	9	10	10	NUM
ejpam-1245	372	10	.	.	PUNCT
ejpam-1245	373	1	in	in	ADP
ejpam-1245	373	2	[	[	X
ejpam-1245	373	3	3	3	NUM
ejpam-1245	373	4	]	]	PUNCT
ejpam-1245	373	5	,	,	PUNCT
ejpam-1245	373	6	aljarrah	aljarrah	PROPN
ejpam-1245	373	7	and	and	CCONJ
ejpam-1245	373	8	noorani	noorani	ADV
ejpam-1245	373	9	define	define	VERB
ejpam-1245	373	10	the	the	DET
ejpam-1245	373	11	ωβ	ωβ	ADJ
ejpam-1245	373	12	−	−	PROPN
ejpam-1245	373	13	t2	t2	PROPN
ejpam-1245	373	14	as	as	SCONJ
ejpam-1245	373	15	if	if	SCONJ
ejpam-1245	373	16	for	for	ADP
ejpam-1245	373	17	each	each	DET
ejpam-1245	373	18	two	two	NUM
ejpam-1245	373	19	distinct	distinct	ADJ
ejpam-1245	373	20	point	point	NOUN
ejpam-1245	373	21	x	x	X
ejpam-1245	373	22	,	,	PUNCT
ejpam-1245	373	23	y	y	PROPN
ejpam-1245	373	24	∈	∈	PROPN
ejpam-1245	373	25	x	x	INTJ
ejpam-1245	373	26	,	,	PUNCT
ejpam-1245	373	27	there	there	PRON
ejpam-1245	373	28	exists	exist	VERB
ejpam-1245	373	29	u	u	NOUN
ejpam-1245	373	30	,	,	PUNCT
ejpam-1245	373	31	v	v	PROPN
ejpam-1245	373	32	∈ωβo(x	∈ωβo(x	PROPN
ejpam-1245	373	33	,	,	PUNCT
ejpam-1245	373	34	τ	τ	PROPN
ejpam-1245	373	35	)	)	PUNCT
ejpam-1245	373	36	such	such	ADJ
ejpam-1245	373	37	that	that	SCONJ
ejpam-1245	373	38	x	x	SYM
ejpam-1245	373	39	∈	∈	PROPN
ejpam-1245	373	40	u	u	NOUN
ejpam-1245	373	41	,	,	PUNCT
ejpam-1245	373	42	y	y	PROPN
ejpam-1245	373	43	∈	∈	PROPN
ejpam-1245	373	44	v	v	NOUN
ejpam-1245	373	45	and	and	CCONJ
ejpam-1245	373	46	u	u	NOUN
ejpam-1245	373	47	∩	∩	NOUN
ejpam-1245	373	48	v	v	NOUN
ejpam-1245	373	49	=	=	SYM
ejpam-1245	373	50	φ	φ	PROPN
ejpam-1245	373	51	.	.	PUNCT
ejpam-1245	373	52	theorem	theorem	VERB
ejpam-1245	373	53	13	13	NUM
ejpam-1245	373	54	.	.	PUNCT
ejpam-1245	374	1	if	if	SCONJ
ejpam-1245	374	2	f	f	PROPN
ejpam-1245	374	3	:	:	PUNCT
ejpam-1245	374	4	(	(	PUNCT
ejpam-1245	374	5	x	x	X
ejpam-1245	374	6	,	,	PUNCT
ejpam-1245	374	7	τ	τ	PROPN
ejpam-1245	374	8	)	)	PUNCT
ejpam-1245	374	9	→	→	SYM
ejpam-1245	374	10	(	(	PUNCT
ejpam-1245	374	11	y	y	PROPN
ejpam-1245	374	12	,	,	PUNCT
ejpam-1245	374	13	σ	σ	PROPN
ejpam-1245	374	14	)	)	PUNCT
ejpam-1245	374	15	is	be	AUX
ejpam-1245	374	16	an	an	DET
ejpam-1245	374	17	ωβ−irresolute	ωβ−irresolute	ADJ
ejpam-1245	374	18	injective	injective	ADJ
ejpam-1245	374	19	function	function	NOUN
ejpam-1245	374	20	and	and	CCONJ
ejpam-1245	374	21	the	the	DET
ejpam-1245	374	22	space	space	NOUN
ejpam-1245	374	23	y	y	PROPN
ejpam-1245	374	24	is	be	AUX
ejpam-1245	374	25	ωβ	ωβ	ADP
ejpam-1245	374	26	−	−	PROPN
ejpam-1245	374	27	t2	t2	NOUN
ejpam-1245	374	28	,	,	PUNCT
ejpam-1245	374	29	then	then	ADV
ejpam-1245	374	30	x	x	PUNCT
ejpam-1245	374	31	is	be	AUX
ejpam-1245	374	32	ωβ	ωβ	ADP
ejpam-1245	374	33	−	−	PROPN
ejpam-1245	374	34	t2	t2	NOUN
ejpam-1245	374	35	.	.	PUNCT
ejpam-1245	375	1	proof	proof	NOUN
ejpam-1245	375	2	.	.	PUNCT
ejpam-1245	376	1	let	let	VERB
ejpam-1245	376	2	x1	x1	PROPN
ejpam-1245	376	3	and	and	CCONJ
ejpam-1245	376	4	x2	x2	PROPN
ejpam-1245	376	5	be	be	VERB
ejpam-1245	376	6	two	two	NUM
ejpam-1245	376	7	distinct	distinct	ADJ
ejpam-1245	376	8	points	point	NOUN
ejpam-1245	376	9	of	of	ADP
ejpam-1245	376	10	x	x	X
ejpam-1245	376	11	.	.	PUNCT
ejpam-1245	377	1	since	since	SCONJ
ejpam-1245	377	2	f	f	PROPN
ejpam-1245	377	3	is	be	AUX
ejpam-1245	377	4	injective	injective	ADJ
ejpam-1245	377	5	and	and	CCONJ
ejpam-1245	377	6	y	y	PROPN
ejpam-1245	377	7	isωβ−t2	isωβ−t2	PROPN
ejpam-1245	377	8	,	,	PUNCT
ejpam-1245	377	9	there	there	PRON
ejpam-1245	377	10	exist	exist	VERB
ejpam-1245	377	11	v1	v1	NOUN
ejpam-1245	377	12	,	,	PUNCT
ejpam-1245	377	13	v2	v2	PROPN
ejpam-1245	377	14	∈ωβo(y	∈ωβo(y	NOUN
ejpam-1245	377	15	,	,	PUNCT
ejpam-1245	377	16	σ	σ	NOUN
ejpam-1245	377	17	)	)	PUNCT
ejpam-1245	377	18	such	such	ADJ
ejpam-1245	377	19	that	that	SCONJ
ejpam-1245	377	20	f	f	PROPN
ejpam-1245	377	21	(	(	PUNCT
ejpam-1245	377	22	x1	x1	PROPN
ejpam-1245	377	23	)	)	PUNCT
ejpam-1245	377	24	∈	∈	PROPN
ejpam-1245	377	25	v1	v1	NOUN
ejpam-1245	377	26	,	,	PUNCT
ejpam-1245	377	27	f	f	PROPN
ejpam-1245	377	28	(	(	PUNCT
ejpam-1245	377	29	x2	x2	ADJ
ejpam-1245	377	30	)	)	PUNCT
ejpam-1245	377	31	∈	∈	PROPN
ejpam-1245	377	32	v2	v2	NOUN
ejpam-1245	377	33	and	and	CCONJ
ejpam-1245	377	34	v1∩v2	v1∩v2	PROPN
ejpam-1245	377	35	=	=	SYM
ejpam-1245	377	36	φ	φ	PROPN
ejpam-1245	377	37	.	.	PUNCT
ejpam-1245	378	1	now	now	ADV
ejpam-1245	378	2	x1	x1	PROPN
ejpam-1245	378	3	∈	∈	PROPN
ejpam-1245	378	4	f	f	PROPN
ejpam-1245	378	5	−1(v1	−1(v1	X
ejpam-1245	378	6	)	)	PUNCT
ejpam-1245	378	7	,	,	PUNCT
ejpam-1245	378	8	x2	x2	PROPN
ejpam-1245	378	9	∈	∈	PROPN
ejpam-1245	378	10	f	f	PROPN
ejpam-1245	378	11	−1(v2	−1(v2	NOUN
ejpam-1245	378	12	)	)	PUNCT
ejpam-1245	378	13	and	and	CCONJ
ejpam-1245	378	14	f	f	PROPN
ejpam-1245	378	15	−1(v1	−1(v1	PROPN
ejpam-1245	378	16	∩	∩	NOUN
ejpam-1245	378	17	v2	v2	NOUN
ejpam-1245	378	18	)	)	PUNCT
ejpam-1245	378	19	=	=	SYM
ejpam-1245	379	1	f	f	PROPN
ejpam-1245	379	2	−1(v1	−1(v1	SYM
ejpam-1245	379	3	)	)	PUNCT
ejpam-1245	379	4	∩	∩	PROPN
ejpam-1245	379	5	f	f	PROPN
ejpam-1245	379	6	−1(v2	−1(v2	PROPN
ejpam-1245	379	7	)	)	PUNCT
ejpam-1245	379	8	=	=	SYM
ejpam-1245	380	1	φ	φ	PROPN
ejpam-1245	380	2	.	.	PUNCT
ejpam-1245	381	1	since	since	SCONJ
ejpam-1245	381	2	f	f	PROPN
ejpam-1245	381	3	is	be	AUX
ejpam-1245	381	4	ωβ−irresolute	ωβ−irresolute	PROPN
ejpam-1245	381	5	then	then	ADV
ejpam-1245	381	6	f	f	PROPN
ejpam-1245	381	7	−1(v1	−1(v1	PROPN
ejpam-1245	381	8	)	)	PUNCT
ejpam-1245	381	9	,	,	PUNCT
ejpam-1245	381	10	f	f	PROPN
ejpam-1245	381	11	−1(v2	−1(v2	NOUN
ejpam-1245	381	12	)	)	PUNCT
ejpam-1245	381	13	is	be	AUX
ejpam-1245	381	14	ωβo(x	ωβo(x	PROPN
ejpam-1245	381	15	,	,	PUNCT
ejpam-1245	381	16	τ	τ	PROPN
ejpam-1245	381	17	)	)	PUNCT
ejpam-1245	381	18	.	.	PUNCT
ejpam-1245	382	1	hence	hence	ADV
ejpam-1245	382	2	x	x	PRON
ejpam-1245	382	3	is	be	AUX
ejpam-1245	382	4	ωβ	ωβ	ADP
ejpam-1245	382	5	−	−	PROPN
ejpam-1245	382	6	t2	t2	NOUN
ejpam-1245	382	7	.	.	PUNCT
ejpam-1245	383	1	definition	definition	NOUN
ejpam-1245	383	2	8	8	NUM
ejpam-1245	383	3	.	.	PUNCT
ejpam-1245	384	1	a	a	DET
ejpam-1245	384	2	space	space	NOUN
ejpam-1245	384	3	x	x	PUNCT
ejpam-1245	384	4	is	be	AUX
ejpam-1245	384	5	said	say	VERB
ejpam-1245	384	6	to	to	PART
ejpam-1245	384	7	be	be	AUX
ejpam-1245	384	8	ωβ−connected	ωβ−connecte	VERB
ejpam-1245	384	9	if	if	SCONJ
ejpam-1245	384	10	there	there	PRON
ejpam-1245	384	11	exist	exist	VERB
ejpam-1245	384	12	disjoint	disjoint	NOUN
ejpam-1245	384	13	ωβo(x	ωβo(x	PROPN
ejpam-1245	384	14	,	,	PUNCT
ejpam-1245	384	15	τ	τ	X
ejpam-1245	384	16	)	)	PUNCT
ejpam-1245	384	17	sets	set	VERB
ejpam-1245	384	18	a	a	DET
ejpam-1245	384	19	and	and	CCONJ
ejpam-1245	384	20	b	b	NOUN
ejpam-1245	384	21	such	such	ADJ
ejpam-1245	384	22	that	that	PRON
ejpam-1245	385	1	a∪	a∪	PROPN
ejpam-1245	386	1	b	b	NOUN
ejpam-1245	386	2	=	=	NOUN
ejpam-1245	386	3	x	x	X
ejpam-1245	386	4	.	.	PUNCT
ejpam-1245	387	1	proposition	proposition	NOUN
ejpam-1245	387	2	7	7	NUM
ejpam-1245	387	3	.	.	PUNCT
ejpam-1245	388	1	if	if	SCONJ
ejpam-1245	388	2	f	f	PROPN
ejpam-1245	388	3	:	:	PUNCT
ejpam-1245	388	4	(	(	PUNCT
ejpam-1245	388	5	x	x	X
ejpam-1245	388	6	,	,	PUNCT
ejpam-1245	388	7	τ)→	τ)→	PROPN
ejpam-1245	388	8	(	(	PUNCT
ejpam-1245	388	9	y	y	PROPN
ejpam-1245	388	10	,	,	PUNCT
ejpam-1245	388	11	σ	σ	PROPN
ejpam-1245	388	12	)	)	PUNCT
ejpam-1245	388	13	is	be	AUX
ejpam-1245	388	14	anωβ−irresolute	anωβ−irresolute	DET
ejpam-1245	388	15	surjective	surjective	ADJ
ejpam-1245	388	16	function	function	NOUN
ejpam-1245	388	17	and	and	CCONJ
ejpam-1245	388	18	x	x	VERB
ejpam-1245	388	19	isωβ−connected	isωβ−connecte	VERB
ejpam-1245	388	20	,	,	PUNCT
ejpam-1245	388	21	then	then	ADV
ejpam-1245	388	22	y	y	PROPN
ejpam-1245	388	23	is	be	AUX
ejpam-1245	388	24	ωβ−connected	ωβ−connecte	VERB
ejpam-1245	388	25	.	.	PUNCT
ejpam-1245	389	1	proof	proof	NOUN
ejpam-1245	389	2	.	.	PUNCT
ejpam-1245	390	1	suppose	suppose	VERB
ejpam-1245	390	2	y	y	PRON
ejpam-1245	390	3	is	be	AUX
ejpam-1245	390	4	not	not	PART
ejpam-1245	390	5	ωβ−connected	ωβ−connecte	VERB
ejpam-1245	390	6	.	.	PUNCT
ejpam-1245	391	1	then	then	ADV
ejpam-1245	391	2	there	there	PRON
ejpam-1245	391	3	exist	exist	VERB
ejpam-1245	391	4	disjoint	disjoint	NOUN
ejpam-1245	391	5	ωβo(y	ωβo(y	NUM
ejpam-1245	391	6	,	,	PUNCT
ejpam-1245	391	7	σ	σ	NOUN
ejpam-1245	391	8	)	)	PUNCT
ejpam-1245	391	9	sets	set	VERB
ejpam-1245	391	10	a	a	PRON
ejpam-1245	391	11	and	and	CCONJ
ejpam-1245	391	12	b	b	NOUN
ejpam-1245	391	13	such	such	ADJ
ejpam-1245	391	14	that	that	SCONJ
ejpam-1245	391	15	a	a	DET
ejpam-1245	391	16	∪	∪	NOUN
ejpam-1245	391	17	b	b	NOUN
ejpam-1245	391	18	=	=	SYM
ejpam-1245	391	19	y	y	PROPN
ejpam-1245	391	20	.	.	PUNCT
ejpam-1245	392	1	since	since	SCONJ
ejpam-1245	392	2	f	f	PROPN
ejpam-1245	392	3	is	be	AUX
ejpam-1245	392	4	ωβ−irresolute	ωβ−irresolute	ADP
ejpam-1245	392	5	surjective	surjective	ADJ
ejpam-1245	392	6	,	,	PUNCT
ejpam-1245	392	7	f	f	PROPN
ejpam-1245	392	8	−1(a	−1(a	CCONJ
ejpam-1245	392	9	)	)	PUNCT
ejpam-1245	392	10	and	and	CCONJ
ejpam-1245	392	11	f	f	PROPN
ejpam-1245	392	12	−1(b	−1(b	ADV
ejpam-1245	392	13	)	)	PUNCT
ejpam-1245	392	14	are	be	AUX
ejpam-1245	392	15	nonempty	nonempty	X
ejpam-1245	392	16	ωβo(x	ωβo(x	NUM
ejpam-1245	392	17	,	,	PUNCT
ejpam-1245	392	18	τ	τ	NOUN
ejpam-1245	392	19	)	)	PUNCT
ejpam-1245	392	20	sets	set	NOUN
ejpam-1245	392	21	.	.	PUNCT
ejpam-1245	393	1	moreover	moreover	ADV
ejpam-1245	393	2	f	f	PROPN
ejpam-1245	393	3	−1(a	−1(a	PROPN
ejpam-1245	393	4	)	)	PUNCT
ejpam-1245	393	5	∪	∪	ADP
ejpam-1245	393	6	f	f	PROPN
ejpam-1245	393	7	−1(b	−1(b	NOUN
ejpam-1245	393	8	)	)	PUNCT
ejpam-1245	393	9	=	=	SYM
ejpam-1245	394	1	x	x	X
ejpam-1245	394	2	.	.	PUNCT
ejpam-1245	395	1	this	this	PRON
ejpam-1245	395	2	is	be	AUX
ejpam-1245	395	3	show	show	NOUN
ejpam-1245	395	4	that	that	SCONJ
ejpam-1245	395	5	(	(	PUNCT
ejpam-1245	395	6	x	x	X
ejpam-1245	395	7	,	,	PUNCT
ejpam-1245	395	8	τ	τ	X
ejpam-1245	395	9	)	)	PUNCT
ejpam-1245	395	10	is	be	AUX
ejpam-1245	395	11	not	not	PART
ejpam-1245	395	12	ωβ−connected	ωβ−connecte	VERB
ejpam-1245	395	13	,	,	PUNCT
ejpam-1245	395	14	which	which	PRON
ejpam-1245	395	15	is	be	AUX
ejpam-1245	395	16	a	a	DET
ejpam-1245	395	17	contradiction	contradiction	NOUN
ejpam-1245	395	18	.	.	PUNCT
ejpam-1245	396	1	hence	hence	ADV
ejpam-1245	396	2	(	(	PUNCT
ejpam-1245	396	3	y	y	PROPN
ejpam-1245	396	4	,	,	PUNCT
ejpam-1245	396	5	σ	σ	PROPN
ejpam-1245	396	6	)	)	PUNCT
ejpam-1245	396	7	is	be	AUX
ejpam-1245	396	8	ωβ−connected	ωβ−connecte	VERB
ejpam-1245	396	9	.	.	PUNCT
ejpam-1245	397	1	4	4	X
ejpam-1245	397	2	.	.	X
ejpam-1245	397	3	ωβ−open	ωβ−open	VERB
ejpam-1245	397	4	and	and	CCONJ
ejpam-1245	397	5	ωβ−closed	ωβ−close	VERB
ejpam-1245	397	6	functions	function	NOUN
ejpam-1245	397	7	definition	definition	NOUN
ejpam-1245	397	8	9	9	NUM
ejpam-1245	397	9	.	.	PUNCT
ejpam-1245	398	1	a	a	DET
ejpam-1245	398	2	function	function	NOUN
ejpam-1245	398	3	f	f	NOUN
ejpam-1245	398	4	:	:	PUNCT
ejpam-1245	398	5	(	(	PUNCT
ejpam-1245	398	6	x	x	X
ejpam-1245	398	7	,	,	PUNCT
ejpam-1245	398	8	τ)→	τ)→	PROPN
ejpam-1245	398	9	(	(	PUNCT
ejpam-1245	398	10	y	y	PROPN
ejpam-1245	398	11	,	,	PUNCT
ejpam-1245	398	12	σ	σ	PROPN
ejpam-1245	398	13	)	)	PUNCT
ejpam-1245	398	14	is	be	AUX
ejpam-1245	398	15	called	call	VERB
ejpam-1245	398	16	ωβ−open	ωβ−open	PROPN
ejpam-1245	398	17	(	(	PUNCT
ejpam-1245	398	18	resp	resp	NOUN
ejpam-1245	398	19	.	.	PUNCT
ejpam-1245	399	1	ωβ−closed	ωβ−close	VERB
ejpam-1245	399	2	)	)	PUNCT
ejpam-1245	399	3	if	if	SCONJ
ejpam-1245	399	4	the	the	DET
ejpam-1245	399	5	image	image	NOUN
ejpam-1245	399	6	of	of	ADP
ejpam-1245	399	7	each	each	PRON
ejpam-1245	399	8	open	open	ADJ
ejpam-1245	399	9	(	(	PUNCT
ejpam-1245	399	10	resp	resp	NOUN
ejpam-1245	399	11	.	.	PUNCT
ejpam-1245	400	1	closed	close	VERB
ejpam-1245	400	2	)	)	PUNCT
ejpam-1245	400	3	set	set	VERB
ejpam-1245	400	4	in	in	ADP
ejpam-1245	400	5	(	(	PUNCT
ejpam-1245	400	6	x	x	INTJ
ejpam-1245	400	7	,	,	PUNCT
ejpam-1245	400	8	τ	τ	X
ejpam-1245	400	9	)	)	PUNCT
ejpam-1245	400	10	is	be	AUX
ejpam-1245	400	11	an	an	DET
ejpam-1245	400	12	ωβo(y	ωβo(y	PROPN
ejpam-1245	400	13	,	,	PUNCT
ejpam-1245	400	14	σ	σ	NOUN
ejpam-1245	400	15	)	)	PUNCT
ejpam-1245	400	16	(	(	PUNCT
ejpam-1245	400	17	resp	resp	NOUN
ejpam-1245	400	18	.	.	PUNCT
ejpam-1245	401	1	ωβc(y	ωβc(y	PROPN
ejpam-1245	401	2	,	,	PUNCT
ejpam-1245	401	3	σ	σ	NOUN
ejpam-1245	401	4	)	)	PUNCT
ejpam-1245	401	5	)	)	PUNCT
ejpam-1245	401	6	.	.	PUNCT
ejpam-1245	402	1	note	note	VERB
ejpam-1245	402	2	that	that	SCONJ
ejpam-1245	402	3	every	every	DET
ejpam-1245	402	4	open	open	ADJ
ejpam-1245	402	5	(	(	PUNCT
ejpam-1245	402	6	closed	closed	ADJ
ejpam-1245	402	7	)	)	PUNCT
ejpam-1245	402	8	function	function	NOUN
ejpam-1245	402	9	is	be	AUX
ejpam-1245	402	10	ωβ−open	ωβ−open	PROPN
ejpam-1245	402	11	(	(	PUNCT
ejpam-1245	402	12	resp	resp	NOUN
ejpam-1245	402	13	.	.	PUNCT
ejpam-1245	403	1	ωβ−closed	ωβ−close	VERB
ejpam-1245	403	2	)	)	PUNCT
ejpam-1245	403	3	function	function	NOUN
ejpam-1245	403	4	,	,	PUNCT
ejpam-1245	403	5	but	but	CCONJ
ejpam-1245	403	6	the	the	DET
ejpam-1245	403	7	converse	converse	NOUN
ejpam-1245	403	8	is	be	AUX
ejpam-1245	403	9	not	not	PART
ejpam-1245	403	10	true	true	ADJ
ejpam-1245	403	11	,	,	PUNCT
ejpam-1245	403	12	which	which	PRON
ejpam-1245	403	13	is	be	AUX
ejpam-1245	403	14	shown	show	VERB
ejpam-1245	403	15	by	by	ADP
ejpam-1245	403	16	the	the	DET
ejpam-1245	403	17	following	follow	VERB
ejpam-1245	403	18	example	example	NOUN
ejpam-1245	403	19	.	.	PUNCT
ejpam-1245	404	1	example	example	NOUN
ejpam-1245	405	1	9	9	NUM
ejpam-1245	405	2	.	.	PUNCT
ejpam-1245	406	1	let	let	VERB
ejpam-1245	406	2	x	x	PUNCT
ejpam-1245	406	3	=	=	PRON
ejpam-1245	406	4	{	{	PUNCT
ejpam-1245	406	5	a	a	DET
ejpam-1245	406	6	,	,	PUNCT
ejpam-1245	406	7	b	b	NOUN
ejpam-1245	406	8	}	}	PUNCT
ejpam-1245	406	9	with	with	ADP
ejpam-1245	406	10	the	the	DET
ejpam-1245	406	11	topology	topology	NOUN
ejpam-1245	406	12	τ	τ	X
ejpam-1245	406	13	=	=	SYM
ejpam-1245	406	14	{	{	PUNCT
ejpam-1245	406	15	φ	φ	PROPN
ejpam-1245	406	16	,	,	PUNCT
ejpam-1245	406	17	x	x	INTJ
ejpam-1245	406	18	,	,	PUNCT
ejpam-1245	406	19	{	{	PUNCT
ejpam-1245	406	20	a	a	X
ejpam-1245	406	21	}	}	PUNCT
ejpam-1245	406	22	}	}	PUNCT
ejpam-1245	406	23	and	and	CCONJ
ejpam-1245	406	24	y	y	PROPN
ejpam-1245	406	25	=	=	PUNCT
ejpam-1245	406	26	{	{	PUNCT
ejpam-1245	406	27	1,2,3	1,2,3	NUM
ejpam-1245	406	28	}	}	PUNCT
ejpam-1245	406	29	with	with	ADP
ejpam-1245	406	30	the	the	DET
ejpam-1245	406	31	topology	topology	NOUN
ejpam-1245	406	32	σ	σ	NOUN
ejpam-1245	406	33	=	=	SYM
ejpam-1245	406	34	{	{	PUNCT
ejpam-1245	406	35	φ	φ	PROPN
ejpam-1245	406	36	,	,	PUNCT
ejpam-1245	406	37	x	x	INTJ
ejpam-1245	406	38	,	,	PUNCT
ejpam-1245	406	39	{	{	PUNCT
ejpam-1245	406	40	1	1	NUM
ejpam-1245	406	41	}	}	PUNCT
ejpam-1245	406	42	,	,	PUNCT
ejpam-1245	406	43	{	{	PUNCT
ejpam-1245	406	44	2	2	NUM
ejpam-1245	406	45	}	}	PUNCT
ejpam-1245	406	46	,	,	PUNCT
ejpam-1245	406	47	{	{	PUNCT
ejpam-1245	406	48	1,2	1,2	NUM
ejpam-1245	406	49	}	}	PUNCT
ejpam-1245	406	50	}	}	PUNCT
ejpam-1245	406	51	.	.	PUNCT
ejpam-1245	407	1	let	let	VERB
ejpam-1245	407	2	f	f	NOUN
ejpam-1245	407	3	:	:	PUNCT
ejpam-1245	407	4	(	(	PUNCT
ejpam-1245	407	5	x	x	X
ejpam-1245	407	6	,	,	PUNCT
ejpam-1245	407	7	τ)→	τ)→	PROPN
ejpam-1245	407	8	(	(	PUNCT
ejpam-1245	407	9	y	y	PROPN
ejpam-1245	407	10	,	,	PUNCT
ejpam-1245	407	11	σ	σ	PROPN
ejpam-1245	407	12	)	)	PUNCT
ejpam-1245	407	13	be	be	VERB
ejpam-1245	407	14	the	the	DET
ejpam-1245	407	15	function	function	NOUN
ejpam-1245	407	16	define	define	NOUN
ejpam-1245	407	17	by	by	ADP
ejpam-1245	407	18	f	f	PROPN
ejpam-1245	407	19	(	(	PUNCT
ejpam-1245	407	20	x	x	NOUN
ejpam-1245	407	21	)	)	PUNCT
ejpam-1245	407	22	=	=	SYM
ejpam-1245	407	23	3	3	NUM
ejpam-1245	407	24	for	for	ADP
ejpam-1245	407	25	all	all	DET
ejpam-1245	407	26	x	x	SYM
ejpam-1245	407	27	∈	∈	NOUN
ejpam-1245	407	28	x	x	X
ejpam-1245	407	29	.	.	PUNCT
ejpam-1245	408	1	then	then	ADV
ejpam-1245	408	2	f	f	PROPN
ejpam-1245	408	3	is	be	AUX
ejpam-1245	408	4	ωβ−open	ωβ−open	ADJ
ejpam-1245	408	5	and	and	CCONJ
ejpam-1245	408	6	ωβ−closed	ωβ−close	VERB
ejpam-1245	408	7	function	function	NOUN
ejpam-1245	408	8	,	,	PUNCT
ejpam-1245	408	9	but	but	CCONJ
ejpam-1245	408	10	it	it	PRON
ejpam-1245	408	11	is	be	AUX
ejpam-1245	408	12	neither	neither	CCONJ
ejpam-1245	408	13	open	open	ADJ
ejpam-1245	408	14	nor	nor	CCONJ
ejpam-1245	408	15	closed	closed	ADJ
ejpam-1245	408	16	function	function	NOUN
ejpam-1245	408	17	.	.	PUNCT
ejpam-1245	409	1	proposition	proposition	NOUN
ejpam-1245	409	2	8	8	NUM
ejpam-1245	409	3	.	.	PUNCT
ejpam-1245	410	1	a	a	DET
ejpam-1245	410	2	function	function	NOUN
ejpam-1245	410	3	f	f	NOUN
ejpam-1245	410	4	:	:	PUNCT
ejpam-1245	410	5	(	(	PUNCT
ejpam-1245	410	6	x	x	X
ejpam-1245	410	7	,	,	PUNCT
ejpam-1245	410	8	τ	τ	PROPN
ejpam-1245	410	9	)	)	PUNCT
ejpam-1245	410	10	→	→	SYM
ejpam-1245	410	11	(	(	PUNCT
ejpam-1245	410	12	y	y	PROPN
ejpam-1245	410	13	,	,	PUNCT
ejpam-1245	410	14	σ	σ	PROPN
ejpam-1245	410	15	)	)	PUNCT
ejpam-1245	410	16	is	be	AUX
ejpam-1245	410	17	ωβ−open	ωβ−open	ADJ
ejpam-1245	410	18	if	if	SCONJ
ejpam-1245	410	19	and	and	CCONJ
ejpam-1245	410	20	only	only	ADV
ejpam-1245	410	21	if	if	SCONJ
ejpam-1245	410	22	for	for	ADP
ejpam-1245	410	23	each	each	DET
ejpam-1245	410	24	x	x	SYM
ejpam-1245	410	25	∈	∈	PROPN
ejpam-1245	410	26	x	x	X
ejpam-1245	410	27	and	and	CCONJ
ejpam-1245	410	28	each	each	DET
ejpam-1245	410	29	open	open	ADJ
ejpam-1245	410	30	set	set	VERB
ejpam-1245	410	31	u	u	NOUN
ejpam-1245	410	32	of	of	ADP
ejpam-1245	410	33	x	x	PUNCT
ejpam-1245	410	34	containing	contain	VERB
ejpam-1245	410	35	x	x	PRON
ejpam-1245	410	36	,	,	PUNCT
ejpam-1245	410	37	there	there	PRON
ejpam-1245	410	38	exists	exist	VERB
ejpam-1245	410	39	an	an	DET
ejpam-1245	410	40	ωβo(y	ωβo(y	PROPN
ejpam-1245	410	41	,	,	PUNCT
ejpam-1245	410	42	σ	σ	NOUN
ejpam-1245	410	43	)	)	PUNCT
ejpam-1245	410	44	set	set	VERB
ejpam-1245	410	45	w	w	NOUN
ejpam-1245	410	46	containing	contain	VERB
ejpam-1245	410	47	f	f	PROPN
ejpam-1245	410	48	(	(	PUNCT
ejpam-1245	410	49	x	x	X
ejpam-1245	410	50	)	)	PUNCT
ejpam-1245	411	1	such	such	ADJ
ejpam-1245	411	2	that	that	PRON
ejpam-1245	411	3	w	w	PROPN
ejpam-1245	411	4	⊂	⊂	PROPN
ejpam-1245	411	5	f	f	X
ejpam-1245	411	6	(	(	PUNCT
ejpam-1245	411	7	u	u	NOUN
ejpam-1245	411	8	)	)	PUNCT
ejpam-1245	411	9	.	.	PUNCT
ejpam-1245	412	1	theorem	theorem	PROPN
ejpam-1245	412	2	14	14	NUM
ejpam-1245	412	3	.	.	PUNCT
ejpam-1245	413	1	let	let	VERB
ejpam-1245	413	2	f	f	NOUN
ejpam-1245	413	3	:	:	PUNCT
ejpam-1245	413	4	(	(	PUNCT
ejpam-1245	413	5	x	x	X
ejpam-1245	413	6	,	,	PUNCT
ejpam-1245	413	7	τ)→	τ)→	PROPN
ejpam-1245	413	8	(	(	PUNCT
ejpam-1245	413	9	y	y	PROPN
ejpam-1245	413	10	,	,	PUNCT
ejpam-1245	413	11	σ	σ	PROPN
ejpam-1245	413	12	)	)	PUNCT
ejpam-1245	413	13	be	be	VERB
ejpam-1245	413	14	a	a	DET
ejpam-1245	413	15	function	function	NOUN
ejpam-1245	413	16	from	from	ADP
ejpam-1245	413	17	space	space	NOUN
ejpam-1245	413	18	(	(	PUNCT
ejpam-1245	413	19	x	x	X
ejpam-1245	413	20	,	,	PUNCT
ejpam-1245	413	21	τ	τ	PROPN
ejpam-1245	413	22	)	)	PUNCT
ejpam-1245	413	23	into	into	ADP
ejpam-1245	413	24	a	a	DET
ejpam-1245	413	25	space	space	NOUN
ejpam-1245	413	26	(	(	PUNCT
ejpam-1245	413	27	y	y	PROPN
ejpam-1245	413	28	,	,	PUNCT
ejpam-1245	413	29	σ	σ	PROPN
ejpam-1245	413	30	)	)	PUNCT
ejpam-1245	413	31	.	.	PUNCT
ejpam-1245	414	1	then	then	ADV
ejpam-1245	414	2	f	f	PROPN
ejpam-1245	414	3	is	be	AUX
ejpam-1245	414	4	ωβ−closed	ωβ−close	VERB
ejpam-1245	414	5	if	if	SCONJ
ejpam-1245	414	6	and	and	CCONJ
ejpam-1245	414	7	only	only	ADV
ejpam-1245	414	8	if	if	SCONJ
ejpam-1245	414	9	ωβ	ωβ	NOUN
ejpam-1245	414	10	cl	cl	NOUN
ejpam-1245	414	11	(	(	PUNCT
ejpam-1245	414	12	f	f	PROPN
ejpam-1245	414	13	(	(	PUNCT
ejpam-1245	414	14	a))⊆	a))⊆	PROPN
ejpam-1245	414	15	f	f	X
ejpam-1245	414	16	(	(	PUNCT
ejpam-1245	414	17	ωβ	ωβ	NOUN
ejpam-1245	414	18	cl(a	cl(a	NUM
ejpam-1245	414	19	)	)	PUNCT
ejpam-1245	414	20	)	)	PUNCT
ejpam-1245	414	21	for	for	SCONJ
ejpam-1245	414	22	each	each	PRON
ejpam-1245	414	23	set	set	VERB
ejpam-1245	414	24	a	a	DET
ejpam-1245	414	25	subset	subset	NOUN
ejpam-1245	414	26	of	of	ADP
ejpam-1245	414	27	(	(	PUNCT
ejpam-1245	414	28	x	x	PROPN
ejpam-1245	414	29	,	,	PUNCT
ejpam-1245	414	30	τ	τ	PROPN
ejpam-1245	414	31	)	)	PUNCT
ejpam-1245	414	32	.	.	PUNCT
ejpam-1245	415	1	references	reference	NOUN
ejpam-1245	415	2	139	139	NUM
ejpam-1245	415	3	proof	proof	NOUN
ejpam-1245	415	4	.	.	PUNCT
ejpam-1245	416	1	let	let	VERB
ejpam-1245	416	2	f	f	PROPN
ejpam-1245	416	3	is	be	AUX
ejpam-1245	416	4	ωβ−closed	ωβ−close	VERB
ejpam-1245	416	5	function	function	NOUN
ejpam-1245	416	6	and	and	CCONJ
ejpam-1245	416	7	a	a	DET
ejpam-1245	416	8	any	any	DET
ejpam-1245	416	9	subset	subset	NOUN
ejpam-1245	416	10	of	of	ADP
ejpam-1245	416	11	x	x	X
ejpam-1245	416	12	.	.	PUNCT
ejpam-1245	417	1	then	then	ADV
ejpam-1245	417	2	f	f	X
ejpam-1245	417	3	(	(	PUNCT
ejpam-1245	417	4	a	a	PROPN
ejpam-1245	417	5	)	)	PUNCT
ejpam-1245	417	6	⊂	⊂	PROPN
ejpam-1245	417	7	f	f	X
ejpam-1245	417	8	(	(	PUNCT
ejpam-1245	417	9	ωβ	ωβ	NOUN
ejpam-1245	417	10	cl(a	cl(a	NUM
ejpam-1245	417	11	)	)	PUNCT
ejpam-1245	417	12	)	)	PUNCT
ejpam-1245	418	1	∈	∈	PROPN
ejpam-1245	418	2	ωβc(y	ωβc(y	PROPN
ejpam-1245	418	3	,	,	PUNCT
ejpam-1245	418	4	σ	σ	PROPN
ejpam-1245	418	5	)	)	PUNCT
ejpam-1245	418	6	,	,	PUNCT
ejpam-1245	418	7	it	it	PRON
ejpam-1245	418	8	follows	follow	VERB
ejpam-1245	418	9	that	that	SCONJ
ejpam-1245	418	10	ωβ	ωβ	NOUN
ejpam-1245	418	11	cl	cl	NOUN
ejpam-1245	418	12	(	(	PUNCT
ejpam-1245	418	13	f	f	X
ejpam-1245	418	14	(	(	PUNCT
ejpam-1245	418	15	a	a	NOUN
ejpam-1245	418	16	)	)	PUNCT
ejpam-1245	418	17	)	)	PUNCT
ejpam-1245	419	1	⊂	⊂	PROPN
ejpam-1245	419	2	f	f	X
ejpam-1245	419	3	(	(	PUNCT
ejpam-1245	419	4	ωβ	ωβ	NOUN
ejpam-1245	419	5	cl(a	cl(a	NUM
ejpam-1245	419	6	)	)	PUNCT
ejpam-1245	419	7	)	)	PUNCT
ejpam-1245	419	8	.	.	PUNCT
ejpam-1245	420	1	conversely	conversely	ADV
ejpam-1245	420	2	,	,	PUNCT
ejpam-1245	420	3	assume	assume	VERB
ejpam-1245	420	4	that	that	SCONJ
ejpam-1245	420	5	b	b	X
ejpam-1245	420	6	∈	∈	PROPN
ejpam-1245	420	7	ωβc(x	ωβc(x	PROPN
ejpam-1245	420	8	,	,	PUNCT
ejpam-1245	420	9	τ	τ	PROPN
ejpam-1245	420	10	)	)	PUNCT
ejpam-1245	420	11	.	.	PUNCT
ejpam-1245	421	1	then	then	ADV
ejpam-1245	421	2	ωβ	ωβ	PROPN
ejpam-1245	421	3	cl	cl	NOUN
ejpam-1245	421	4	(	(	PUNCT
ejpam-1245	421	5	f	f	PROPN
ejpam-1245	421	6	(	(	PUNCT
ejpam-1245	421	7	b	b	NOUN
ejpam-1245	421	8	)	)	PUNCT
ejpam-1245	421	9	)	)	PUNCT
ejpam-1245	422	1	⊂	⊂	PROPN
ejpam-1245	422	2	f	f	X
ejpam-1245	422	3	(	(	PUNCT
ejpam-1245	422	4	ωβ	ωβ	INTJ
ejpam-1245	422	5	cl(b	cl(b	NOUN
ejpam-1245	422	6	)	)	PUNCT
ejpam-1245	422	7	)	)	PUNCT
ejpam-1245	423	1	=	=	SYM
ejpam-1245	423	2	f	f	X
ejpam-1245	423	3	(	(	PUNCT
ejpam-1245	423	4	b	b	NOUN
ejpam-1245	423	5	)	)	PUNCT
ejpam-1245	423	6	.	.	PUNCT
ejpam-1245	424	1	thus	thus	ADV
ejpam-1245	424	2	we	we	PRON
ejpam-1245	424	3	obtain	obtain	VERB
ejpam-1245	424	4	that	that	PRON
ejpam-1245	424	5	ωβ	ωβ	ADJ
ejpam-1245	424	6	cl	cl	NOUN
ejpam-1245	424	7	(	(	PUNCT
ejpam-1245	424	8	f	f	PROPN
ejpam-1245	424	9	(	(	PUNCT
ejpam-1245	424	10	b	b	NOUN
ejpam-1245	424	11	)	)	PUNCT
ejpam-1245	424	12	)	)	PUNCT
ejpam-1245	425	1	=	=	SYM
ejpam-1245	425	2	f	f	X
ejpam-1245	425	3	(	(	PUNCT
ejpam-1245	425	4	b	b	NOUN
ejpam-1245	425	5	)	)	PUNCT
ejpam-1245	425	6	,	,	PUNCT
ejpam-1245	425	7	so	so	CCONJ
ejpam-1245	425	8	f	f	PROPN
ejpam-1245	425	9	is	be	AUX
ejpam-1245	425	10	ωβ−closed	ωβ−close	VERB
ejpam-1245	425	11	function	function	NOUN
ejpam-1245	425	12	.	.	PUNCT
ejpam-1245	426	1	proposition	proposition	NOUN
ejpam-1245	426	2	9	9	NUM
ejpam-1245	426	3	.	.	PUNCT
ejpam-1245	427	1	let	let	VERB
ejpam-1245	427	2	f	f	NOUN
ejpam-1245	427	3	:	:	PUNCT
ejpam-1245	427	4	(	(	PUNCT
ejpam-1245	427	5	x	x	X
ejpam-1245	427	6	,	,	PUNCT
ejpam-1245	427	7	τ)→	τ)→	PROPN
ejpam-1245	427	8	(	(	PUNCT
ejpam-1245	427	9	y	y	PROPN
ejpam-1245	427	10	,	,	PUNCT
ejpam-1245	427	11	σ	σ	PROPN
ejpam-1245	427	12	)	)	PUNCT
ejpam-1245	427	13	be	be	VERB
ejpam-1245	427	14	a	a	DET
ejpam-1245	427	15	continuous	continuous	ADJ
ejpam-1245	427	16	surjection	surjection	NOUN
ejpam-1245	427	17	function	function	NOUN
ejpam-1245	427	18	and	and	CCONJ
ejpam-1245	427	19	let	let	VERB
ejpam-1245	427	20	g	g	NOUN
ejpam-1245	427	21	:	:	PUNCT
ejpam-1245	427	22	(	(	PUNCT
ejpam-1245	427	23	y	y	PROPN
ejpam-1245	427	24	,	,	PUNCT
ejpam-1245	427	25	σ	σ	PROPN
ejpam-1245	427	26	)	)	PUNCT
ejpam-1245	427	27	→	→	SYM
ejpam-1245	427	28	(	(	PUNCT
ejpam-1245	427	29	z	z	NOUN
ejpam-1245	427	30	,	,	PUNCT
ejpam-1245	427	31	ρ	ρ	PROPN
ejpam-1245	427	32	)	)	PUNCT
ejpam-1245	427	33	be	be	VERB
ejpam-1245	427	34	such	such	ADJ
ejpam-1245	427	35	that	that	SCONJ
ejpam-1245	427	36	g	g	PROPN
ejpam-1245	427	37	◦	◦	NOUN
ejpam-1245	428	1	f	f	X
ejpam-1245	428	2	:	:	PUNCT
ejpam-1245	428	3	(	(	PUNCT
ejpam-1245	428	4	x	x	X
ejpam-1245	428	5	,	,	PUNCT
ejpam-1245	428	6	τ	τ	PROPN
ejpam-1245	428	7	)	)	PUNCT
ejpam-1245	428	8	→	→	SYM
ejpam-1245	428	9	(	(	PUNCT
ejpam-1245	428	10	z	z	NOUN
ejpam-1245	428	11	,	,	PUNCT
ejpam-1245	428	12	ρ	ρ	PROPN
ejpam-1245	428	13	)	)	PUNCT
ejpam-1245	428	14	is	be	AUX
ejpam-1245	428	15	ωβ−open	ωβ−open	PROPN
ejpam-1245	428	16	function	function	NOUN
ejpam-1245	428	17	,	,	PUNCT
ejpam-1245	428	18	then	then	ADV
ejpam-1245	428	19	g	g	PROPN
ejpam-1245	428	20	is	be	AUX
ejpam-1245	428	21	ωβ−open	ωβ−open	ADJ
ejpam-1245	428	22	.	.	PUNCT
ejpam-1245	429	1	proof	proof	NOUN
ejpam-1245	429	2	.	.	PUNCT
ejpam-1245	430	1	let	let	VERB
ejpam-1245	430	2	y	y	PROPN
ejpam-1245	430	3	∈	∈	PROPN
ejpam-1245	430	4	y	y	PROPN
ejpam-1245	430	5	and	and	CCONJ
ejpam-1245	430	6	let	let	VERB
ejpam-1245	430	7	v	v	NUM
ejpam-1245	430	8	∈	∈	PROPN
ejpam-1245	430	9	ρ	ρ	NOUN
ejpam-1245	430	10	with	with	ADP
ejpam-1245	430	11	g(y	g(y	NOUN
ejpam-1245	430	12	)	)	PUNCT
ejpam-1245	430	13	∈	∈	PROPN
ejpam-1245	430	14	v	v	NOUN
ejpam-1245	430	15	.	.	PUNCT
ejpam-1245	431	1	choose	choose	VERB
ejpam-1245	431	2	x	x	PUNCT
ejpam-1245	431	3	∈	∈	PROPN
ejpam-1245	431	4	x	x	PUNCT
ejpam-1245	432	1	such	such	ADJ
ejpam-1245	432	2	that	that	SCONJ
ejpam-1245	432	3	f	f	PROPN
ejpam-1245	432	4	(	(	PUNCT
ejpam-1245	432	5	x	x	X
ejpam-1245	432	6	)	)	PUNCT
ejpam-1245	432	7	=	=	SYM
ejpam-1245	432	8	y.	y.	NOUN
ejpam-1245	432	9	since	since	SCONJ
ejpam-1245	432	10	g	g	PROPN
ejpam-1245	432	11	◦	◦	PROPN
ejpam-1245	432	12	f	f	PROPN
ejpam-1245	432	13	is	be	AUX
ejpam-1245	432	14	ωβ−open	ωβ−open	PROPN
ejpam-1245	432	15	function	function	NOUN
ejpam-1245	432	16	,	,	PUNCT
ejpam-1245	432	17	then	then	ADV
ejpam-1245	432	18	g(v	g(v	X
ejpam-1245	432	19	)	)	PUNCT
ejpam-1245	433	1	=	=	SYM
ejpam-1245	433	2	g	g	PROPN
ejpam-1245	433	3	◦	◦	NOUN
ejpam-1245	433	4	f	f	PROPN
ejpam-1245	434	1	(	(	PUNCT
ejpam-1245	434	2	f	f	PROPN
ejpam-1245	434	3	−1(v	−1(v	PROPN
ejpam-1245	434	4	)	)	PUNCT
ejpam-1245	434	5	)	)	PUNCT
ejpam-1245	435	1	∈	∈	PROPN
ejpam-1245	435	2	ωβo(z	ωβo(z	PROPN
ejpam-1245	435	3	,	,	PUNCT
ejpam-1245	435	4	ρ	ρ	PROPN
ejpam-1245	435	5	)	)	PUNCT
ejpam-1245	435	6	.	.	PUNCT
ejpam-1245	436	1	this	this	PRON
ejpam-1245	436	2	is	be	AUX
ejpam-1245	436	3	show	show	NOUN
ejpam-1245	436	4	that	that	SCONJ
ejpam-1245	436	5	g	g	PROPN
ejpam-1245	436	6	is	be	AUX
ejpam-1245	436	7	ωβ−open	ωβ−open	PROPN
ejpam-1245	436	8	function	function	NOUN
ejpam-1245	436	9	.	.	PUNCT
ejpam-1245	437	1	the	the	DET
ejpam-1245	437	2	following	follow	VERB
ejpam-1245	437	3	examples	example	NOUN
ejpam-1245	437	4	show	show	VERB
ejpam-1245	437	5	that	that	SCONJ
ejpam-1245	437	6	theωβ−open	theωβ−open	ADJ
ejpam-1245	437	7	function	function	NOUN
ejpam-1245	437	8	is	be	AUX
ejpam-1245	437	9	independent	independent	ADJ
ejpam-1245	437	10	withωβ−irresolute	withωβ−irresolute	ADJ
ejpam-1245	437	11	and	and	CCONJ
ejpam-1245	437	12	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	437	13	function	function	NOUN
ejpam-1245	437	14	.	.	PUNCT
ejpam-1245	438	1	example	example	NOUN
ejpam-1245	439	1	10	10	NUM
ejpam-1245	439	2	.	.	PUNCT
ejpam-1245	440	1	let	let	VERB
ejpam-1245	440	2	x	x	PUNCT
ejpam-1245	440	3	=	=	PUNCT
ejpam-1245	440	4	r	r	NOUN
ejpam-1245	440	5	with	with	ADP
ejpam-1245	440	6	the	the	DET
ejpam-1245	440	7	topologies	topology	NOUN
ejpam-1245	440	8	τ	τ	X
ejpam-1245	440	9	=	=	X
ejpam-1245	440	10	τcoc	τcoc	PROPN
ejpam-1245	440	11	and	and	CCONJ
ejpam-1245	440	12	let	let	VERB
ejpam-1245	440	13	y	y	PROPN
ejpam-1245	440	14	=	=	PUNCT
ejpam-1245	440	15	{	{	PUNCT
ejpam-1245	440	16	1,2	1,2	NUM
ejpam-1245	440	17	}	}	PUNCT
ejpam-1245	440	18	with	with	ADP
ejpam-1245	440	19	the	the	DET
ejpam-1245	440	20	topology	topology	NOUN
ejpam-1245	440	21	ρ	ρ	PROPN
ejpam-1245	440	22	=	=	SYM
ejpam-1245	440	23	�	�	PROPN
ejpam-1245	440	24	φ	φ	PROPN
ejpam-1245	440	25	,	,	PUNCT
ejpam-1245	440	26	y	y	PROPN
ejpam-1245	440	27	,	,	PUNCT
ejpam-1245	440	28	{	{	PUNCT
ejpam-1245	440	29	2	2	NUM
ejpam-1245	440	30	}	}	PUNCT
ejpam-1245	440	31	.	.	PUNCT
ejpam-1245	441	1	let	let	VERB
ejpam-1245	441	2	f	f	NOUN
ejpam-1245	441	3	:	:	PUNCT
ejpam-1245	441	4	(	(	PUNCT
ejpam-1245	441	5	x	x	X
ejpam-1245	441	6	,	,	PUNCT
ejpam-1245	441	7	τ)→	τ)→	PROPN
ejpam-1245	441	8	(	(	PUNCT
ejpam-1245	441	9	y	y	PROPN
ejpam-1245	441	10	,	,	PUNCT
ejpam-1245	441	11	σ	σ	PROPN
ejpam-1245	441	12	)	)	PUNCT
ejpam-1245	441	13	be	be	VERB
ejpam-1245	441	14	the	the	DET
ejpam-1245	441	15	function	function	NOUN
ejpam-1245	441	16	defined	define	VERB
ejpam-1245	441	17	by	by	ADP
ejpam-1245	441	18	f	f	PROPN
ejpam-1245	441	19	(	(	PUNCT
ejpam-1245	441	20	x	x	NOUN
ejpam-1245	441	21	)	)	PUNCT
ejpam-1245	441	22	=	=	SYM
ejpam-1245	442	1	(	(	PUNCT
ejpam-1245	442	2	1	1	NUM
ejpam-1245	442	3	x	x	SYM
ejpam-1245	442	4	∈	∈	PROPN
ejpam-1245	442	5	r−q	r−q	NOUN
ejpam-1245	442	6	2	2	NUM
ejpam-1245	442	7	x	x	SYM
ejpam-1245	442	8	∈q	∈q	NOUN
ejpam-1245	442	9	then	then	ADV
ejpam-1245	442	10	f	f	PROPN
ejpam-1245	442	11	is	be	AUX
ejpam-1245	442	12	not	not	PART
ejpam-1245	442	13	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	442	14	,	,	PUNCT
ejpam-1245	442	15	but	but	CCONJ
ejpam-1245	442	16	it	it	PRON
ejpam-1245	442	17	can	can	AUX
ejpam-1245	442	18	easily	easily	ADV
ejpam-1245	442	19	seen	see	VERB
ejpam-1245	442	20	that	that	SCONJ
ejpam-1245	442	21	f	f	PROPN
ejpam-1245	442	22	(	(	PUNCT
ejpam-1245	442	23	x	x	X
ejpam-1245	442	24	)	)	PUNCT
ejpam-1245	442	25	is	be	AUX
ejpam-1245	442	26	ωβ−open	ωβ−open	PROPN
ejpam-1245	442	27	function	function	NOUN
ejpam-1245	442	28	.	.	PUNCT
ejpam-1245	443	1	example	example	NOUN
ejpam-1245	444	1	11	11	NUM
ejpam-1245	444	2	.	.	PUNCT
ejpam-1245	445	1	let	let	VERB
ejpam-1245	445	2	x	x	PUNCT
ejpam-1245	445	3	=	=	PUNCT
ejpam-1245	445	4	{	{	PUNCT
ejpam-1245	445	5	1,2	1,2	NUM
ejpam-1245	445	6	}	}	PUNCT
ejpam-1245	445	7	with	with	ADP
ejpam-1245	445	8	the	the	DET
ejpam-1245	445	9	topology	topology	NOUN
ejpam-1245	445	10	τ	τ	PROPN
ejpam-1245	445	11	=	=	SYM
ejpam-1245	445	12	�	�	PROPN
ejpam-1245	445	13	φ	φ	PROPN
ejpam-1245	445	14	,	,	PUNCT
ejpam-1245	445	15	x	x	INTJ
ejpam-1245	445	16	,	,	PUNCT
ejpam-1245	445	17	{	{	PUNCT
ejpam-1245	445	18	1	1	NUM
ejpam-1245	445	19	}	}	PUNCT
ejpam-1245	445	20	and	and	CCONJ
ejpam-1245	445	21	let	let	VERB
ejpam-1245	445	22	y	y	NOUN
ejpam-1245	445	23	=	=	NOUN
ejpam-1245	445	24	r	r	NOUN
ejpam-1245	445	25	with	with	ADP
ejpam-1245	445	26	the	the	DET
ejpam-1245	445	27	topologies	topology	NOUN
ejpam-1245	445	28	σ	σ	NOUN
ejpam-1245	445	29	=	=	PUNCT
ejpam-1245	445	30	τcoc	τcoc	PROPN
ejpam-1245	445	31	.	.	PUNCT
ejpam-1245	446	1	let	let	VERB
ejpam-1245	446	2	f	f	NOUN
ejpam-1245	446	3	:	:	PUNCT
ejpam-1245	446	4	(	(	PUNCT
ejpam-1245	446	5	x	x	X
ejpam-1245	446	6	,	,	PUNCT
ejpam-1245	446	7	τ)→	τ)→	PROPN
ejpam-1245	446	8	(	(	PUNCT
ejpam-1245	446	9	y	y	PROPN
ejpam-1245	446	10	,	,	PUNCT
ejpam-1245	446	11	σ	σ	PROPN
ejpam-1245	446	12	)	)	PUNCT
ejpam-1245	446	13	be	be	VERB
ejpam-1245	446	14	the	the	DET
ejpam-1245	446	15	function	function	NOUN
ejpam-1245	446	16	defined	define	VERB
ejpam-1245	446	17	by	by	ADP
ejpam-1245	446	18	f	f	PROPN
ejpam-1245	446	19	(	(	PUNCT
ejpam-1245	446	20	x	x	NOUN
ejpam-1245	446	21	)	)	PUNCT
ejpam-1245	446	22	=	=	SYM
ejpam-1245	447	1	(	(	PUNCT
ejpam-1245	447	2	r−q	r−q	NOUN
ejpam-1245	447	3	x	x	SYM
ejpam-1245	447	4	=	=	SYM
ejpam-1245	447	5	2	2	NUM
ejpam-1245	447	6	q	q	NOUN
ejpam-1245	447	7	x	x	SYM
ejpam-1245	447	8	=	=	NOUN
ejpam-1245	447	9	1	1	NUM
ejpam-1245	448	1	then	then	ADV
ejpam-1245	448	2	f	f	PROPN
ejpam-1245	448	3	is	be	AUX
ejpam-1245	448	4	not	not	PART
ejpam-1245	448	5	ωβ−open	ωβ−open	ADJ
ejpam-1245	448	6	,	,	PUNCT
ejpam-1245	448	7	but	but	CCONJ
ejpam-1245	448	8	it	it	PRON
ejpam-1245	448	9	can	can	AUX
ejpam-1245	448	10	easily	easily	ADV
ejpam-1245	448	11	seen	see	VERB
ejpam-1245	448	12	that	that	SCONJ
ejpam-1245	448	13	f	f	PROPN
ejpam-1245	448	14	is	be	AUX
ejpam-1245	448	15	ωβ−continuous	ωβ−continuous	ADJ
ejpam-1245	448	16	and	and	CCONJ
ejpam-1245	448	17	ωβ−irresolute	ωβ−irresolute	ADP
ejpam-1245	448	18	function	function	NOUN
ejpam-1245	448	19	.	.	PUNCT
ejpam-1245	449	1	example	example	NOUN
ejpam-1245	449	2	12	12	NUM
ejpam-1245	449	3	.	.	PUNCT
ejpam-1245	450	1	consider	consider	VERB
ejpam-1245	450	2	the	the	DET
ejpam-1245	450	3	function	function	NOUN
ejpam-1245	450	4	f	f	PROPN
ejpam-1245	450	5	in	in	ADP
ejpam-1245	450	6	the	the	DET
ejpam-1245	450	7	example	example	NOUN
ejpam-1245	450	8	8	8	NUM
ejpam-1245	450	9	which	which	PRON
ejpam-1245	450	10	isωβ−open	isωβ−open	ADP
ejpam-1245	450	11	,	,	PUNCT
ejpam-1245	450	12	but	but	CCONJ
ejpam-1245	450	13	notωβ−irresolute	notωβ−irresolute	X
ejpam-1245	450	14	.	.	PUNCT
ejpam-1245	451	1	acknowledgements	acknowledgement	NOUN
ejpam-1245	451	2	:	:	PUNCT
ejpam-1245	451	3	this	this	DET
ejpam-1245	451	4	work	work	NOUN
ejpam-1245	451	5	is	be	AUX
ejpam-1245	451	6	financially	financially	ADV
ejpam-1245	451	7	supported	support	VERB
ejpam-1245	451	8	by	by	ADP
ejpam-1245	451	9	the	the	DET
ejpam-1245	451	10	malaysian	malaysian	PROPN
ejpam-1245	451	11	ministry	ministry	PROPN
ejpam-1245	451	12	of	of	ADP
ejpam-1245	451	13	science	science	PROPN
ejpam-1245	451	14	,	,	PUNCT
ejpam-1245	451	15	technology	technology	NOUN
ejpam-1245	451	16	and	and	CCONJ
ejpam-1245	451	17	environment	environment	NOUN
ejpam-1245	451	18	,	,	PUNCT
ejpam-1245	451	19	science	science	NOUN
ejpam-1245	451	20	fund	fund	NOUN
ejpam-1245	451	21	grant	grant	NOUN
ejpam-1245	451	22	no	no	INTJ
ejpam-1245	451	23	.	.	PUNCT
ejpam-1245	452	1	ukm	ukm	PROPN
ejpam-1245	452	2	-	-	PUNCT
ejpam-1245	452	3	st-06	st-06	NOUN
ejpam-1245	452	4	-	-	PUNCT
ejpam-1245	452	5	frgs0146	frgs0146	NOUN
ejpam-1245	452	6	-	-	PUNCT
ejpam-1245	452	7	2010	2010	NUM
ejpam-1245	452	8	.	.	PUNCT
ejpam-1245	453	1	references	reference	NOUN
ejpam-1245	453	2	[	[	X
ejpam-1245	453	3	1	1	NUM
ejpam-1245	453	4	]	]	X
ejpam-1245	453	5	k	k	PROPN
ejpam-1245	453	6	al	al	PROPN
ejpam-1245	453	7	-	-	PROPN
ejpam-1245	453	8	zoubi	zoubi	PROPN
ejpam-1245	453	9	.	.	PUNCT
ejpam-1245	454	1	semi	semi	ADV
ejpam-1245	454	2	ω−	ω−	ADP
ejpam-1245	454	3	containuous	containuous	ADJ
ejpam-1245	454	4	functions	function	NOUN
ejpam-1245	454	5	.	.	PUNCT
ejpam-1245	455	1	abhath	abhath	PROPN
ejpam-1245	455	2	al	al	PROPN
ejpam-1245	455	3	-	-	PUNCT
ejpam-1245	455	4	yarmouk	yarmouk	PRON
ejpam-1245	455	5	,	,	PUNCT
ejpam-1245	455	6	12(1):119–131	12(1):119–131	PROPN
ejpam-1245	455	7	,	,	PUNCT
ejpam-1245	455	8	2003	2003	NUM
ejpam-1245	455	9	.	.	PUNCT
ejpam-1245	456	1	[	[	X
ejpam-1245	456	2	2	2	NUM
ejpam-1245	456	3	]	]	X
ejpam-1245	456	4	k	k	PROPN
ejpam-1245	456	5	al	al	PROPN
ejpam-1245	456	6	-	-	PROPN
ejpam-1245	456	7	zoubi	zoubi	PROPN
ejpam-1245	456	8	and	and	CCONJ
ejpam-1245	456	9	b	b	PROPN
ejpam-1245	456	10	al	al	PROPN
ejpam-1245	456	11	-	-	PUNCT
ejpam-1245	456	12	nashef	nashef	PROPN
ejpam-1245	456	13	.	.	PUNCT
ejpam-1245	457	1	the	the	DET
ejpam-1245	457	2	topology	topology	NOUN
ejpam-1245	457	3	of	of	ADP
ejpam-1245	457	4	ω−	ω−	ADP
ejpam-1245	457	5	open	open	ADJ
ejpam-1245	457	6	subsets	subset	NOUN
ejpam-1245	457	7	.	.	PUNCT
ejpam-1245	458	1	al	al	PROPN
ejpam-1245	458	2	-	-	PUNCT
ejpam-1245	458	3	manarah	manarah	PROPN
ejpam-1245	458	4	journal	journal	NOUN
ejpam-1245	458	5	,	,	PUNCT
ejpam-1245	458	6	9(2):169–179	9(2):169–179	NUM
ejpam-1245	458	7	,	,	PUNCT
ejpam-1245	458	8	2003	2003	NUM
ejpam-1245	458	9	.	.	PUNCT
ejpam-1245	459	1	[	[	X
ejpam-1245	459	2	3	3	X
ejpam-1245	459	3	]	]	X
ejpam-1245	459	4	h	h	NOUN
ejpam-1245	459	5	aljarrah	aljarrah	PROPN
ejpam-1245	459	6	and	and	CCONJ
ejpam-1245	459	7	m	m	NOUN
ejpam-1245	459	8	noorani	noorani	ADJ
ejpam-1245	459	9	.	.	PUNCT
ejpam-1245	460	1	on	on	ADP
ejpam-1245	460	2	ωβ−	ωβ−	NUM
ejpam-1245	460	3	open	open	ADJ
ejpam-1245	460	4	sets	set	NOUN
ejpam-1245	460	5	.	.	PUNCT
ejpam-1245	461	1	submitted	submit	VERB
ejpam-1245	461	2	.	.	PUNCT
ejpam-1245	462	1	references	reference	NOUN
ejpam-1245	462	2	140	140	NUM
ejpam-1245	462	3	[	[	X
ejpam-1245	462	4	4	4	NUM
ejpam-1245	462	5	]	]	X
ejpam-1245	462	6	d	d	X
ejpam-1245	462	7	andrijevic	andrijevic	VERB
ejpam-1245	462	8	.	.	PUNCT
ejpam-1245	463	1	on	on	ADP
ejpam-1245	463	2	b	b	X
ejpam-1245	463	3	-	-	PUNCT
ejpam-1245	463	4	open	open	ADJ
ejpam-1245	463	5	sets	set	NOUN
ejpam-1245	463	6	.	.	PUNCT
ejpam-1245	464	1	mat	mat	X
ejpam-1245	464	2	.	.	PROPN
ejpam-1245	464	3	vesnik	vesnik	PROPN
ejpam-1245	464	4	,	,	PUNCT
ejpam-1245	464	5	48:59–64	48:59–64	PROPN
ejpam-1245	464	6	,	,	PUNCT
ejpam-1245	464	7	1996	1996	NUM
ejpam-1245	464	8	.	.	PUNCT
ejpam-1245	465	1	[	[	X
ejpam-1245	465	2	5	5	NUM
ejpam-1245	465	3	]	]	PUNCT
ejpam-1245	465	4	s	s	VERB
ejpam-1245	465	5	crossley	crossley	NOUN
ejpam-1245	465	6	and	and	CCONJ
ejpam-1245	465	7	s	s	VERB
ejpam-1245	465	8	hildebrand	hildebrand	NOUN
ejpam-1245	465	9	.	.	PUNCT
ejpam-1245	466	1	semi	semi	ADJ
ejpam-1245	466	2	-	-	ADJ
ejpam-1245	466	3	topological	topological	ADJ
ejpam-1245	466	4	properties	property	NOUN
ejpam-1245	466	5	.	.	PUNCT
ejpam-1245	467	1	fund	fund	PROPN
ejpam-1245	467	2	.	.	PUNCT
ejpam-1245	468	1	math	math	NOUN
ejpam-1245	468	2	.	.	PUNCT
ejpam-1245	469	1	,	,	PUNCT
ejpam-1245	469	2	74(3):233–254	74(3):233–254	PROPN
ejpam-1245	469	3	,	,	PUNCT
ejpam-1245	469	4	1972	1972	NUM
ejpam-1245	469	5	.	.	PUNCT
ejpam-1245	470	1	[	[	X
ejpam-1245	470	2	6	6	NUM
ejpam-1245	470	3	]	]	PUNCT
ejpam-1245	470	4	h	h	PROPN
ejpam-1245	470	5	hdeib	hdeib	PROPN
ejpam-1245	470	6	.	.	PUNCT
ejpam-1245	471	1	ω−	ω−	ADP
ejpam-1245	471	2	continuous	continuous	ADJ
ejpam-1245	471	3	functions	function	NOUN
ejpam-1245	471	4	.	.	PUNCT
ejpam-1245	472	1	dirasat	dirasat	PROPN
ejpam-1245	472	2	,	,	PUNCT
ejpam-1245	472	3	xvi:136–142	xvi:136–142	NOUN
ejpam-1245	472	4	,	,	PUNCT
ejpam-1245	472	5	1996	1996	NUM
ejpam-1245	472	6	.	.	PUNCT
ejpam-1245	473	1	[	[	X
ejpam-1245	473	2	7	7	NUM
ejpam-1245	473	3	]	]	X
ejpam-1245	473	4	m	m	AUX
ejpam-1245	473	5	monesf	monesf	ADJ
ejpam-1245	473	6	,	,	PUNCT
ejpam-1245	473	7	s	s	PROPN
ejpam-1245	473	8	el	el	PROPN
ejpam-1245	473	9	deeb	deeb	PROPN
ejpam-1245	473	10	,	,	PUNCT
ejpam-1245	473	11	and	and	CCONJ
ejpam-1245	473	12	r	r	PROPN
ejpam-1245	473	13	mahmoud	mahmoud	PROPN
ejpam-1245	473	14	.	.	PUNCT
ejpam-1245	473	15	β−open	β−open	PUNCT
ejpam-1245	474	1	sets	set	NOUN
ejpam-1245	474	2	and	and	CCONJ
ejpam-1245	474	3	β−continuous	β−continuous	PRON
ejpam-1245	474	4	mapping	mapping	NOUN
ejpam-1245	474	5	.	.	PUNCT
ejpam-1245	475	1	bull	bull	NOUN
ejpam-1245	475	2	.	.	PUNCT
ejpam-1245	476	1	fac	fac	PROPN
ejpam-1245	476	2	.	.	PUNCT
ejpam-1245	477	1	sci	sci	PROPN
ejpam-1245	477	2	.	.	PUNCT
ejpam-1245	477	3	assiut	assiut	PROPN
ejpam-1245	477	4	univ	univ	PROPN
ejpam-1245	477	5	.	.	PROPN
ejpam-1245	477	6	,	,	PUNCT
ejpam-1245	477	7	c	c	PROPN
ejpam-1245	477	8	12:77–90	12:77–90	NUM
ejpam-1245	477	9	,	,	PUNCT
ejpam-1245	477	10	1983	1983	NUM
ejpam-1245	477	11	.	.	PUNCT
ejpam-1245	478	1	[	[	X
ejpam-1245	478	2	8	8	NUM
ejpam-1245	478	3	]	]	X
ejpam-1245	478	4	g	g	PROPN
ejpam-1245	478	5	navalagi	navalagi	NOUN
ejpam-1245	478	6	.	.	PUNCT
ejpam-1245	479	1	semi	semi	ADJ
ejpam-1245	479	2	-	-	ADJ
ejpam-1245	479	3	precontinuous	precontinuous	ADJ
ejpam-1245	479	4	functions	function	NOUN
ejpam-1245	479	5	and	and	CCONJ
ejpam-1245	479	6	properties	property	NOUN
ejpam-1245	479	7	of	of	ADP
ejpam-1245	479	8	generalized	generalized	ADJ
ejpam-1245	479	9	semi	semi	ADJ
ejpam-1245	479	10	-	-	ADJ
ejpam-1245	479	11	preclosed	preclosed	ADJ
ejpam-1245	479	12	sets	set	NOUN
ejpam-1245	479	13	in	in	ADP
ejpam-1245	479	14	topological	topological	ADJ
ejpam-1245	479	15	spaces	space	NOUN
ejpam-1245	479	16	.	.	PUNCT
ejpam-1245	480	1	internat	internat	PROPN
ejpam-1245	480	2	.	.	PUNCT
ejpam-1245	481	1	j.	j.	PROPN
ejpam-1245	481	2	math	math	PROPN
ejpam-1245	481	3	.	.	PUNCT
ejpam-1245	482	1	math	math	NOUN
ejpam-1245	482	2	.	.	PUNCT
ejpam-1245	483	1	sci	sci	PROPN
ejpam-1245	483	2	.	.	PROPN
ejpam-1245	483	3	,	,	PUNCT
ejpam-1245	483	4	29(2):58–98	29(2):58–98	NUM
ejpam-1245	483	5	,	,	PUNCT
ejpam-1245	483	6	2002	2002	NUM
ejpam-1245	483	7	.	.	PUNCT
ejpam-1245	484	1	[	[	X
ejpam-1245	484	2	9	9	NUM
ejpam-1245	484	3	]	]	PUNCT
ejpam-1245	484	4	t	t	PROPN
ejpam-1245	484	5	noiri	noiri	PROPN
ejpam-1245	484	6	,	,	PUNCT
ejpam-1245	484	7	a	a	DET
ejpam-1245	484	8	al	al	PROPN
ejpam-1245	484	9	-	-	PUNCT
ejpam-1245	484	10	omari	omari	PROPN
ejpam-1245	484	11	,	,	PUNCT
ejpam-1245	484	12	and	and	CCONJ
ejpam-1245	484	13	m	m	NOUN
ejpam-1245	484	14	noorani	noorani	ADJ
ejpam-1245	484	15	.	.	PUNCT
ejpam-1245	485	1	on	on	ADP
ejpam-1245	485	2	ωb−	ωb−	NUM
ejpam-1245	485	3	open	open	ADJ
ejpam-1245	485	4	sets	set	NOUN
ejpam-1245	485	5	and	and	CCONJ
ejpam-1245	485	6	b	b	X
ejpam-1245	485	7	-	-	PUNCT
ejpam-1245	485	8	lindelof	lindelof	NOUN
ejpam-1245	485	9	spaces	space	NOUN
ejpam-1245	485	10	.	.	PUNCT
ejpam-1245	486	1	eur	eur	PROPN
ejpam-1245	486	2	.	.	PUNCT
ejpam-1245	487	1	j.	j.	PROPN
ejpam-1245	487	2	pure	pure	PROPN
ejpam-1245	487	3	appl	appl	PROPN
ejpam-1245	487	4	.	.	PUNCT
ejpam-1245	487	5	math	math	PROPN
ejpam-1245	487	6	.	.	PUNCT
ejpam-1245	487	7	,	,	PUNCT
ejpam-1245	487	8	1:3–9	1:3–9	NUM
ejpam-1245	487	9	,	,	PUNCT
ejpam-1245	487	10	2008	2008	NUM
ejpam-1245	487	11	.	.	PUNCT