id	sid	tid	token	lemma	pos
ejpam-4732	1	1	european	european	PROPN
ejpam-4732	1	2	journal	journal	PROPN
ejpam-4732	1	3	of	of	ADP
ejpam-4732	1	4	pure	pure	ADJ
ejpam-4732	1	5	and	and	CCONJ
ejpam-4732	1	6	applied	apply	VERB
ejpam-4732	1	7	mathematics	mathematic	NOUN
ejpam-4732	1	8	vol	vol	NOUN
ejpam-4732	1	9	.	.	PUNCT
ejpam-4732	2	1	16	16	NUM
ejpam-4732	2	2	,	,	PUNCT
ejpam-4732	2	3	no	no	INTJ
ejpam-4732	2	4	.	.	NOUN
ejpam-4732	2	5	3	3	NUM
ejpam-4732	2	6	,	,	PUNCT
ejpam-4732	2	7	2023	2023	NUM
ejpam-4732	2	8	,	,	PUNCT
ejpam-4732	2	9	1634	1634	NUM
ejpam-4732	2	10	-	-	SYM
ejpam-4732	2	11	1646	1646	NUM
ejpam-4732	2	12	issn	issn	PROPN
ejpam-4732	2	13	1307	1307	NUM
ejpam-4732	2	14	-	-	SYM
ejpam-4732	2	15	5543	5543	NUM
ejpam-4732	2	16	–	–	PUNCT
ejpam-4732	2	17	ejpam.com	ejpam.com	X
ejpam-4732	2	18	published	publish	VERB
ejpam-4732	2	19	by	by	ADP
ejpam-4732	2	20	new	new	PROPN
ejpam-4732	2	21	york	york	PROPN
ejpam-4732	2	22	business	business	PROPN
ejpam-4732	2	23	global	global	PROPN
ejpam-4732	2	24	upper	upper	ADJ
ejpam-4732	2	25	and	and	CCONJ
ejpam-4732	2	26	lower	low	ADJ
ejpam-4732	2	27	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	2	28	multifunctions	multifunction	NOUN
ejpam-4732	2	29	chawalit	chawalit	VERB
ejpam-4732	2	30	boonpok1	boonpok1	PROPN
ejpam-4732	2	31	,	,	PUNCT
ejpam-4732	2	32	prapart	prapart	VERB
ejpam-4732	2	33	pue	pue	X
ejpam-4732	2	34	-	-	PUNCT
ejpam-4732	2	35	on1,∗	on1,∗	PROPN
ejpam-4732	2	36	1	1	NUM
ejpam-4732	2	37	mathematics	mathematic	NOUN
ejpam-4732	2	38	and	and	CCONJ
ejpam-4732	2	39	applied	apply	VERB
ejpam-4732	2	40	mathematics	mathematics	PROPN
ejpam-4732	2	41	research	research	NOUN
ejpam-4732	2	42	unit	unit	NOUN
ejpam-4732	2	43	,	,	PUNCT
ejpam-4732	2	44	department	department	NOUN
ejpam-4732	2	45	of	of	ADP
ejpam-4732	2	46	mathematics	mathematic	NOUN
ejpam-4732	2	47	,	,	PUNCT
ejpam-4732	2	48	faculty	faculty	NOUN
ejpam-4732	2	49	of	of	ADP
ejpam-4732	2	50	science	science	NOUN
ejpam-4732	2	51	,	,	PUNCT
ejpam-4732	2	52	mahasarakham	mahasarakham	PROPN
ejpam-4732	2	53	university	university	PROPN
ejpam-4732	2	54	,	,	PUNCT
ejpam-4732	2	55	maha	maha	PROPN
ejpam-4732	2	56	sarakham	sarakham	PROPN
ejpam-4732	2	57	,	,	PUNCT
ejpam-4732	2	58	44150	44150	NUM
ejpam-4732	2	59	,	,	PUNCT
ejpam-4732	2	60	thailand	thailand	PROPN
ejpam-4732	2	61	abstract	abstract	PROPN
ejpam-4732	2	62	.	.	PUNCT
ejpam-4732	3	1	our	our	PRON
ejpam-4732	3	2	main	main	ADJ
ejpam-4732	3	3	purpose	purpose	NOUN
ejpam-4732	3	4	is	be	AUX
ejpam-4732	3	5	to	to	PART
ejpam-4732	3	6	introduce	introduce	VERB
ejpam-4732	3	7	the	the	DET
ejpam-4732	3	8	concepts	concept	NOUN
ejpam-4732	3	9	of	of	ADP
ejpam-4732	3	10	upper	upper	ADJ
ejpam-4732	3	11	and	and	CCONJ
ejpam-4732	3	12	lower	low	ADJ
ejpam-4732	3	13	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	3	14	multifunctions	multifunction	NOUN
ejpam-4732	3	15	.	.	PUNCT
ejpam-4732	4	1	in	in	ADP
ejpam-4732	4	2	particular	particular	ADJ
ejpam-4732	4	3	,	,	PUNCT
ejpam-4732	4	4	some	some	DET
ejpam-4732	4	5	characterizations	characterization	NOUN
ejpam-4732	4	6	of	of	ADP
ejpam-4732	4	7	upper	upper	ADJ
ejpam-4732	4	8	and	and	CCONJ
ejpam-4732	4	9	lower	low	ADJ
ejpam-4732	4	10	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	4	11	multifunctions	multifunction	NOUN
ejpam-4732	4	12	are	be	AUX
ejpam-4732	4	13	investigated	investigate	VERB
ejpam-4732	4	14	.	.	PUNCT
ejpam-4732	5	1	moreover	moreover	ADV
ejpam-4732	5	2	,	,	PUNCT
ejpam-4732	5	3	the	the	DET
ejpam-4732	5	4	relationships	relationship	NOUN
ejpam-4732	5	5	between	between	ADP
ejpam-4732	5	6	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	5	7	multifunctions	multifunction	NOUN
ejpam-4732	5	8	and	and	CCONJ
ejpam-4732	5	9	almost	almost	ADV
ejpam-4732	5	10	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	5	11	multifunctions	multifunction	NOUN
ejpam-4732	5	12	are	be	AUX
ejpam-4732	5	13	discussed	discuss	VERB
ejpam-4732	5	14	.	.	PUNCT
ejpam-4732	6	1	2020	2020	NUM
ejpam-4732	6	2	mathematics	mathematic	NOUN
ejpam-4732	6	3	subject	subject	NOUN
ejpam-4732	6	4	classifications	classification	NOUN
ejpam-4732	6	5	:	:	PUNCT
ejpam-4732	6	6	54c08	54c08	NUM
ejpam-4732	6	7	,	,	PUNCT
ejpam-4732	6	8	54c60	54c60	NUM
ejpam-4732	6	9	key	key	ADJ
ejpam-4732	6	10	words	word	NOUN
ejpam-4732	6	11	and	and	CCONJ
ejpam-4732	6	12	phrases	phrase	NOUN
ejpam-4732	6	13	:	:	PUNCT
ejpam-4732	6	14	upper	upper	ADJ
ejpam-4732	6	15	sβ(⋆)-continuous	sβ(⋆)-continuous	PROPN
ejpam-4732	6	16	multifunction	multifunction	NOUN
ejpam-4732	6	17	,	,	PUNCT
ejpam-4732	6	18	lower	low	ADJ
ejpam-4732	6	19	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	6	20	multifunction	multifunction	NOUN
ejpam-4732	6	21	1	1	NUM
ejpam-4732	6	22	.	.	PUNCT
ejpam-4732	6	23	introduction	introduction	NOUN
ejpam-4732	6	24	the	the	DET
ejpam-4732	6	25	concept	concept	NOUN
ejpam-4732	6	26	of	of	ADP
ejpam-4732	6	27	semi	semi	ADJ
ejpam-4732	6	28	-	-	NOUN
ejpam-4732	6	29	continuity	continuity	NOUN
ejpam-4732	6	30	was	be	AUX
ejpam-4732	6	31	first	first	ADV
ejpam-4732	6	32	introduced	introduce	VERB
ejpam-4732	6	33	by	by	ADP
ejpam-4732	6	34	levine	levine	PROPN
ejpam-4732	7	1	[	[	X
ejpam-4732	7	2	13	13	NUM
ejpam-4732	7	3	]	]	PUNCT
ejpam-4732	7	4	.	.	PUNCT
ejpam-4732	8	1	in	in	ADP
ejpam-4732	8	2	1982	1982	NUM
ejpam-4732	8	3	,	,	PUNCT
ejpam-4732	8	4	mashhour	mashhour	PROPN
ejpam-4732	8	5	et	et	PROPN
ejpam-4732	8	6	al	al	PROPN
ejpam-4732	8	7	.	.	PUNCT
ejpam-4732	9	1	[	[	X
ejpam-4732	9	2	15	15	NUM
ejpam-4732	9	3	]	]	PUNCT
ejpam-4732	9	4	introduced	introduce	VERB
ejpam-4732	9	5	and	and	CCONJ
ejpam-4732	9	6	investigated	investigate	VERB
ejpam-4732	9	7	the	the	DET
ejpam-4732	9	8	notion	notion	NOUN
ejpam-4732	9	9	of	of	ADP
ejpam-4732	9	10	precontinuous	precontinuous	ADJ
ejpam-4732	9	11	functions	function	NOUN
ejpam-4732	9	12	.	.	PUNCT
ejpam-4732	10	1	abd	abd	PROPN
ejpam-4732	10	2	el	el	PROPN
ejpam-4732	10	3	-	-	PROPN
ejpam-4732	10	4	monsef	monsef	PROPN
ejpam-4732	10	5	et	et	PROPN
ejpam-4732	10	6	al	al	PROPN
ejpam-4732	10	7	.	.	PUNCT
ejpam-4732	11	1	[	[	X
ejpam-4732	11	2	7	7	X
ejpam-4732	11	3	]	]	PUNCT
ejpam-4732	11	4	introduced	introduce	VERB
ejpam-4732	11	5	the	the	DET
ejpam-4732	11	6	notion	notion	NOUN
ejpam-4732	11	7	of	of	ADP
ejpam-4732	11	8	β	β	ADJ
ejpam-4732	11	9	-	-	ADJ
ejpam-4732	11	10	continuous	continuous	ADJ
ejpam-4732	11	11	functions	function	NOUN
ejpam-4732	11	12	as	as	ADP
ejpam-4732	11	13	a	a	DET
ejpam-4732	11	14	generalization	generalization	NOUN
ejpam-4732	11	15	of	of	ADP
ejpam-4732	11	16	semi	semi	ADJ
ejpam-4732	11	17	-	-	ADJ
ejpam-4732	11	18	continuous	continuous	ADJ
ejpam-4732	11	19	functions	function	NOUN
ejpam-4732	11	20	[	[	X
ejpam-4732	11	21	13	13	NUM
ejpam-4732	11	22	]	]	PUNCT
ejpam-4732	11	23	and	and	CCONJ
ejpam-4732	11	24	precontinuous	precontinuous	ADJ
ejpam-4732	11	25	functions	function	NOUN
ejpam-4732	11	26	[	[	X
ejpam-4732	11	27	15	15	NUM
ejpam-4732	11	28	]	]	PUNCT
ejpam-4732	11	29	.	.	PUNCT
ejpam-4732	12	1	borśık	borśık	NOUN
ejpam-4732	12	2	and	and	CCONJ
ejpam-4732	12	3	doboš	doboš	NOUN
ejpam-4732	13	1	[	[	X
ejpam-4732	13	2	4	4	X
ejpam-4732	13	3	]	]	PUNCT
ejpam-4732	13	4	introduced	introduce	VERB
ejpam-4732	13	5	the	the	DET
ejpam-4732	13	6	notion	notion	NOUN
ejpam-4732	13	7	of	of	ADP
ejpam-4732	13	8	almost	almost	ADV
ejpam-4732	13	9	quasi	quasi	NOUN
ejpam-4732	13	10	-	-	NOUN
ejpam-4732	13	11	continuity	continuity	NOUN
ejpam-4732	13	12	which	which	PRON
ejpam-4732	13	13	is	be	AUX
ejpam-4732	13	14	weaker	weak	ADJ
ejpam-4732	13	15	than	than	ADP
ejpam-4732	13	16	that	that	PRON
ejpam-4732	13	17	of	of	ADP
ejpam-4732	13	18	quasicontinuity	quasicontinuity	NOUN
ejpam-4732	13	19	[	[	X
ejpam-4732	13	20	14	14	NUM
ejpam-4732	13	21	]	]	PUNCT
ejpam-4732	13	22	and	and	CCONJ
ejpam-4732	13	23	investigated	investigate	VERB
ejpam-4732	13	24	a	a	DET
ejpam-4732	13	25	decomposition	decomposition	NOUN
ejpam-4732	13	26	theorem	theorem	NOUN
ejpam-4732	13	27	of	of	ADP
ejpam-4732	13	28	quasi	quasi	NOUN
ejpam-4732	13	29	-	-	NOUN
ejpam-4732	13	30	continuity	continuity	NOUN
ejpam-4732	13	31	.	.	PUNCT
ejpam-4732	14	1	popa	popa	NOUN
ejpam-4732	14	2	and	and	CCONJ
ejpam-4732	14	3	noiri	noiri	ADV
ejpam-4732	15	1	[	[	X
ejpam-4732	15	2	17	17	NUM
ejpam-4732	15	3	]	]	PUNCT
ejpam-4732	15	4	investigated	investigate	VERB
ejpam-4732	15	5	some	some	DET
ejpam-4732	15	6	characterizations	characterization	NOUN
ejpam-4732	15	7	of	of	ADP
ejpam-4732	15	8	β	β	NOUN
ejpam-4732	15	9	-	-	NOUN
ejpam-4732	15	10	continuity	continuity	NOUN
ejpam-4732	15	11	and	and	CCONJ
ejpam-4732	15	12	showed	show	VERB
ejpam-4732	15	13	that	that	SCONJ
ejpam-4732	15	14	almost	almost	ADV
ejpam-4732	15	15	quasi	quasi	NOUN
ejpam-4732	15	16	-	-	NOUN
ejpam-4732	15	17	continuity	continuity	NOUN
ejpam-4732	15	18	is	be	AUX
ejpam-4732	15	19	equivalent	equivalent	ADJ
ejpam-4732	15	20	to	to	ADP
ejpam-4732	15	21	β	β	NOUN
ejpam-4732	15	22	-	-	NOUN
ejpam-4732	15	23	continuity	continuity	NOUN
ejpam-4732	15	24	.	.	PUNCT
ejpam-4732	16	1	in	in	ADP
ejpam-4732	16	2	1993	1993	NUM
ejpam-4732	16	3	,	,	PUNCT
ejpam-4732	16	4	popa	popa	NOUN
ejpam-4732	16	5	and	and	CCONJ
ejpam-4732	16	6	noiri	noiri	ADV
ejpam-4732	16	7	[	[	X
ejpam-4732	16	8	18	18	NUM
ejpam-4732	16	9	]	]	PUNCT
ejpam-4732	16	10	extended	extend	VERB
ejpam-4732	16	11	the	the	DET
ejpam-4732	16	12	concept	concept	NOUN
ejpam-4732	16	13	of	of	ADP
ejpam-4732	16	14	β	β	ADJ
ejpam-4732	16	15	-	-	ADJ
ejpam-4732	16	16	continuous	continuous	ADJ
ejpam-4732	16	17	functions	function	NOUN
ejpam-4732	16	18	to	to	ADP
ejpam-4732	16	19	multifunctions	multifunction	NOUN
ejpam-4732	16	20	and	and	CCONJ
ejpam-4732	16	21	introduced	introduce	VERB
ejpam-4732	16	22	the	the	DET
ejpam-4732	16	23	notions	notion	NOUN
ejpam-4732	16	24	of	of	ADP
ejpam-4732	16	25	upper	upper	ADJ
ejpam-4732	16	26	and	and	CCONJ
ejpam-4732	16	27	lower	low	ADJ
ejpam-4732	16	28	β	β	ADJ
ejpam-4732	16	29	-	-	ADJ
ejpam-4732	16	30	continuous	continuous	ADJ
ejpam-4732	16	31	multifunctions	multifunction	NOUN
ejpam-4732	16	32	.	.	PUNCT
ejpam-4732	17	1	moreover	moreover	ADV
ejpam-4732	17	2	,	,	PUNCT
ejpam-4732	17	3	the	the	DET
ejpam-4732	17	4	relationships	relationship	NOUN
ejpam-4732	17	5	between	between	ADP
ejpam-4732	17	6	β	β	ADJ
ejpam-4732	17	7	-	-	ADJ
ejpam-4732	17	8	continuous	continuous	ADJ
ejpam-4732	17	9	mulfunctions	mulfunction	NOUN
ejpam-4732	17	10	and	and	CCONJ
ejpam-4732	17	11	quasi	quasi	ADJ
ejpam-4732	17	12	-	-	ADJ
ejpam-4732	17	13	continuous	continuous	ADJ
ejpam-4732	17	14	multifunctions	multifunction	NOUN
ejpam-4732	17	15	were	be	AUX
ejpam-4732	17	16	established	establish	VERB
ejpam-4732	17	17	in	in	ADP
ejpam-4732	17	18	[	[	X
ejpam-4732	17	19	17	17	NUM
ejpam-4732	17	20	]	]	PUNCT
ejpam-4732	17	21	.	.	PUNCT
ejpam-4732	18	1	noiri	noiri	PROPN
ejpam-4732	18	2	and	and	CCONJ
ejpam-4732	18	3	popa	popa	NOUN
ejpam-4732	18	4	[	[	X
ejpam-4732	18	5	16	16	NUM
ejpam-4732	18	6	]	]	PUNCT
ejpam-4732	18	7	introduced	introduce	VERB
ejpam-4732	18	8	and	and	CCONJ
ejpam-4732	18	9	studied	study	VERB
ejpam-4732	18	10	the	the	DET
ejpam-4732	18	11	concepts	concept	NOUN
ejpam-4732	18	12	of	of	ADP
ejpam-4732	18	13	upper	upper	ADJ
ejpam-4732	18	14	and	and	CCONJ
ejpam-4732	18	15	lower	low	ADJ
ejpam-4732	18	16	almost	almost	ADV
ejpam-4732	18	17	β	β	ADJ
ejpam-4732	18	18	-	-	ADJ
ejpam-4732	18	19	continuous	continuous	ADJ
ejpam-4732	18	20	mulfunctions	mulfunction	NOUN
ejpam-4732	18	21	.	.	PUNCT
ejpam-4732	19	1	in	in	ADP
ejpam-4732	19	2	2003	2003	NUM
ejpam-4732	19	3	,	,	PUNCT
ejpam-4732	19	4	hatir	hatir	PROPN
ejpam-4732	19	5	et	et	PROPN
ejpam-4732	19	6	al	al	PROPN
ejpam-4732	19	7	.	.	PUNCT
ejpam-4732	20	1	[	[	X
ejpam-4732	20	2	8	8	NUM
ejpam-4732	20	3	]	]	PUNCT
ejpam-4732	20	4	introduced	introduce	VERB
ejpam-4732	20	5	and	and	CCONJ
ejpam-4732	20	6	investigated	investigate	VERB
ejpam-4732	20	7	the	the	DET
ejpam-4732	20	8	notions	notion	NOUN
ejpam-4732	20	9	of	of	ADP
ejpam-4732	20	10	strong	strong	ADJ
ejpam-4732	20	11	β	β	NOUN
ejpam-4732	20	12	-	-	ADJ
ejpam-4732	20	13	i	i	PRON
ejpam-4732	20	14	-open	-open	NOUN
ejpam-4732	20	15	sets	set	NOUN
ejpam-4732	20	16	and	and	CCONJ
ejpam-4732	20	17	strongly	strongly	ADV
ejpam-4732	20	18	β	β	X
ejpam-4732	20	19	-	-	ADJ
ejpam-4732	20	20	i	i	VERB
ejpam-4732	20	21	-continuous	-continuous	ADJ
ejpam-4732	20	22	functions	function	NOUN
ejpam-4732	20	23	in	in	ADP
ejpam-4732	20	24	ideal	ideal	ADJ
ejpam-4732	20	25	topological	topological	ADJ
ejpam-4732	20	26	spaces	space	NOUN
ejpam-4732	20	27	.	.	PUNCT
ejpam-4732	21	1	hatir	hatir	PROPN
ejpam-4732	21	2	et	et	PROPN
ejpam-4732	21	3	al	al	PROPN
ejpam-4732	21	4	.	.	PUNCT
ejpam-4732	22	1	[	[	X
ejpam-4732	22	2	9	9	NUM
ejpam-4732	22	3	]	]	PUNCT
ejpam-4732	22	4	investigated	investigate	VERB
ejpam-4732	22	5	further	further	ADJ
ejpam-4732	22	6	properties	property	NOUN
ejpam-4732	22	7	of	of	ADP
ejpam-4732	22	8	strong	strong	ADJ
ejpam-4732	22	9	β	β	NOUN
ejpam-4732	22	10	-	-	ADJ
ejpam-4732	22	11	i	i	PRON
ejpam-4732	22	12	-open	-open	NOUN
ejpam-4732	22	13	sets	set	NOUN
ejpam-4732	22	14	and	and	CCONJ
ejpam-4732	22	15	strongly	strongly	ADV
ejpam-4732	22	16	β	β	X
ejpam-4732	22	17	-	-	ADJ
ejpam-4732	22	18	i	i	VERB
ejpam-4732	22	19	continuous	continuous	ADJ
ejpam-4732	22	20	functions	function	NOUN
ejpam-4732	22	21	.	.	PUNCT
ejpam-4732	23	1	in	in	ADP
ejpam-4732	23	2	2019	2019	NUM
ejpam-4732	23	3	,	,	PUNCT
ejpam-4732	23	4	boonpok	boonpok	X
ejpam-4732	23	5	[	[	X
ejpam-4732	23	6	2	2	NUM
ejpam-4732	23	7	]	]	PUNCT
ejpam-4732	23	8	introduced	introduce	VERB
ejpam-4732	23	9	and	and	CCONJ
ejpam-4732	23	10	studied	study	VERB
ejpam-4732	23	11	the	the	DET
ejpam-4732	23	12	concepts	concept	NOUN
ejpam-4732	23	13	of	of	ADP
ejpam-4732	23	14	upper	upper	ADJ
ejpam-4732	23	15	and	and	CCONJ
ejpam-4732	23	16	lower	low	ADJ
ejpam-4732	23	17	⋆-continuous	⋆-continuous	ADJ
ejpam-4732	23	18	multifunctions	multifunction	NOUN
ejpam-4732	23	19	in	in	ADP
ejpam-4732	23	20	ideal	ideal	ADJ
ejpam-4732	23	21	topological	topological	ADJ
ejpam-4732	23	22	spaces	space	NOUN
ejpam-4732	23	23	.	.	PUNCT
ejpam-4732	24	1	in	in	ADP
ejpam-4732	24	2	[	[	X
ejpam-4732	24	3	3	3	NUM
ejpam-4732	24	4	]	]	PUNCT
ejpam-4732	24	5	,	,	PUNCT
ejpam-4732	24	6	the	the	DET
ejpam-4732	24	7	present	present	ADJ
ejpam-4732	24	8	∗corresponding	∗corresponde	VERB
ejpam-4732	24	9	author	author	NOUN
ejpam-4732	24	10	.	.	PUNCT
ejpam-4732	25	1	doi	doi	NOUN
ejpam-4732	25	2	:	:	PUNCT
ejpam-4732	25	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4732	https://doi.org/10.29020/nybg.ejpam.v16i3.4732	NUM
ejpam-4732	25	4	email	email	NOUN
ejpam-4732	25	5	addresses	address	NOUN
ejpam-4732	25	6	:	:	PUNCT
ejpam-4732	26	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4732	26	2	(	(	PUNCT
ejpam-4732	26	3	c.	c.	PROPN
ejpam-4732	26	4	boonpok	boonpok	PROPN
ejpam-4732	26	5	)	)	PUNCT
ejpam-4732	26	6	,	,	PUNCT
ejpam-4732	26	7	prapart.p@msu.ac.th	prapart.p@msu.ac.th	X
ejpam-4732	26	8	(	(	PUNCT
ejpam-4732	26	9	p.	p.	NOUN
ejpam-4732	26	10	pue	pue	NOUN
ejpam-4732	26	11	-	-	PUNCT
ejpam-4732	26	12	on	on	ADP
ejpam-4732	26	13	)	)	PUNCT
ejpam-4732	26	14	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4732	26	15	1634	1634	NUM
ejpam-4732	26	16	©	©	ADP
ejpam-4732	26	17	2023	2023	NUM
ejpam-4732	26	18	ejpam	ejpam	NOUN
ejpam-4732	26	19	all	all	DET
ejpam-4732	26	20	rights	right	NOUN
ejpam-4732	26	21	reserved	reserve	VERB
ejpam-4732	26	22	.	.	PUNCT
ejpam-4732	27	1	c.	c.	PROPN
ejpam-4732	27	2	boonpok	boonpok	PROPN
ejpam-4732	27	3	,	,	PUNCT
ejpam-4732	27	4	p.	p.	NOUN
ejpam-4732	27	5	pue	pue	NOUN
ejpam-4732	27	6	-	-	PUNCT
ejpam-4732	27	7	on	on	ADP
ejpam-4732	27	8	/	/	SYM
ejpam-4732	27	9	eur	eur	NOUN
ejpam-4732	27	10	.	.	PUNCT
ejpam-4732	28	1	j.	j.	PROPN
ejpam-4732	28	2	pure	pure	PROPN
ejpam-4732	28	3	appl	appl	PROPN
ejpam-4732	28	4	.	.	PROPN
ejpam-4732	28	5	math	math	PROPN
ejpam-4732	28	6	,	,	PUNCT
ejpam-4732	28	7	16	16	NUM
ejpam-4732	28	8	(	(	PUNCT
ejpam-4732	28	9	3	3	NUM
ejpam-4732	28	10	)	)	PUNCT
ejpam-4732	28	11	(	(	PUNCT
ejpam-4732	28	12	2023	2023	NUM
ejpam-4732	28	13	)	)	PUNCT
ejpam-4732	28	14	,	,	PUNCT
ejpam-4732	28	15	1634	1634	NUM
ejpam-4732	28	16	-	-	SYM
ejpam-4732	28	17	1646	1646	NUM
ejpam-4732	28	18	1635	1635	NUM
ejpam-4732	28	19	author	author	NOUN
ejpam-4732	28	20	introduced	introduce	VERB
ejpam-4732	28	21	and	and	CCONJ
ejpam-4732	28	22	investigated	investigate	VERB
ejpam-4732	28	23	the	the	DET
ejpam-4732	28	24	notions	notion	NOUN
ejpam-4732	28	25	of	of	ADP
ejpam-4732	28	26	upper	upper	ADJ
ejpam-4732	28	27	and	and	CCONJ
ejpam-4732	28	28	lower	low	ADJ
ejpam-4732	28	29	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-4732	28	30	multifunctions	multifunction	NOUN
ejpam-4732	28	31	.	.	PUNCT
ejpam-4732	29	1	the	the	DET
ejpam-4732	29	2	purpose	purpose	NOUN
ejpam-4732	29	3	of	of	ADP
ejpam-4732	29	4	the	the	DET
ejpam-4732	29	5	present	present	ADJ
ejpam-4732	29	6	paper	paper	NOUN
ejpam-4732	29	7	is	be	AUX
ejpam-4732	29	8	to	to	PART
ejpam-4732	29	9	introduce	introduce	VERB
ejpam-4732	29	10	the	the	DET
ejpam-4732	29	11	notions	notion	NOUN
ejpam-4732	29	12	of	of	ADP
ejpam-4732	29	13	upper	upper	ADJ
ejpam-4732	29	14	and	and	CCONJ
ejpam-4732	29	15	lower	low	ADJ
ejpam-4732	29	16	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	29	17	multifunctions	multifunction	NOUN
ejpam-4732	29	18	.	.	PUNCT
ejpam-4732	30	1	furthermore	furthermore	ADV
ejpam-4732	30	2	,	,	PUNCT
ejpam-4732	30	3	several	several	ADJ
ejpam-4732	30	4	characterizations	characterization	NOUN
ejpam-4732	30	5	of	of	ADP
ejpam-4732	30	6	upper	upper	ADJ
ejpam-4732	30	7	and	and	CCONJ
ejpam-4732	30	8	lower	low	ADJ
ejpam-4732	30	9	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	30	10	multifunctions	multifunction	NOUN
ejpam-4732	30	11	are	be	AUX
ejpam-4732	30	12	investigated	investigate	VERB
ejpam-4732	30	13	.	.	PUNCT
ejpam-4732	31	1	moreover	moreover	ADV
ejpam-4732	31	2	,	,	PUNCT
ejpam-4732	31	3	the	the	DET
ejpam-4732	31	4	relationships	relationship	NOUN
ejpam-4732	31	5	between	between	ADP
ejpam-4732	31	6	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	31	7	multifunctions	multifunction	NOUN
ejpam-4732	31	8	and	and	CCONJ
ejpam-4732	31	9	almost	almost	ADV
ejpam-4732	31	10	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	31	11	multifunctions	multifunction	NOUN
ejpam-4732	31	12	are	be	AUX
ejpam-4732	31	13	discussed	discuss	VERB
ejpam-4732	31	14	.	.	PUNCT
ejpam-4732	32	1	2	2	X
ejpam-4732	32	2	.	.	X
ejpam-4732	32	3	preliminaries	preliminary	NOUN
ejpam-4732	32	4	throughout	throughout	ADP
ejpam-4732	32	5	the	the	DET
ejpam-4732	32	6	present	present	ADJ
ejpam-4732	32	7	paper	paper	NOUN
ejpam-4732	32	8	,	,	PUNCT
ejpam-4732	32	9	spaces	space	NOUN
ejpam-4732	32	10	(	(	PUNCT
ejpam-4732	32	11	x	x	X
ejpam-4732	32	12	,	,	PUNCT
ejpam-4732	32	13	τ	τ	X
ejpam-4732	32	14	)	)	PUNCT
ejpam-4732	32	15	and	and	CCONJ
ejpam-4732	32	16	(	(	PUNCT
ejpam-4732	32	17	y	y	PROPN
ejpam-4732	32	18	,	,	PUNCT
ejpam-4732	32	19	σ	σ	PROPN
ejpam-4732	32	20	)	)	PUNCT
ejpam-4732	32	21	(	(	PUNCT
ejpam-4732	32	22	or	or	CCONJ
ejpam-4732	32	23	simply	simply	ADV
ejpam-4732	32	24	x	x	X
ejpam-4732	32	25	and	and	CCONJ
ejpam-4732	32	26	y	y	PROPN
ejpam-4732	32	27	)	)	PUNCT
ejpam-4732	32	28	always	always	ADV
ejpam-4732	32	29	mean	mean	VERB
ejpam-4732	32	30	topological	topological	ADJ
ejpam-4732	32	31	spaces	space	NOUN
ejpam-4732	32	32	on	on	ADP
ejpam-4732	32	33	which	which	PRON
ejpam-4732	32	34	no	no	DET
ejpam-4732	32	35	separation	separation	NOUN
ejpam-4732	32	36	axioms	axiom	NOUN
ejpam-4732	32	37	are	be	AUX
ejpam-4732	32	38	assumed	assume	VERB
ejpam-4732	32	39	unless	unless	SCONJ
ejpam-4732	32	40	explicitly	explicitly	ADV
ejpam-4732	32	41	stated	state	VERB
ejpam-4732	32	42	.	.	PUNCT
ejpam-4732	33	1	let	let	VERB
ejpam-4732	33	2	a	a	DET
ejpam-4732	33	3	be	be	AUX
ejpam-4732	33	4	a	a	DET
ejpam-4732	33	5	subset	subset	NOUN
ejpam-4732	33	6	of	of	ADP
ejpam-4732	33	7	a	a	DET
ejpam-4732	33	8	topological	topological	ADJ
ejpam-4732	33	9	space	space	NOUN
ejpam-4732	33	10	(	(	PUNCT
ejpam-4732	33	11	x	x	X
ejpam-4732	33	12	,	,	PUNCT
ejpam-4732	33	13	τ	τ	PROPN
ejpam-4732	33	14	)	)	PUNCT
ejpam-4732	33	15	.	.	PUNCT
ejpam-4732	34	1	the	the	DET
ejpam-4732	34	2	closure	closure	NOUN
ejpam-4732	34	3	of	of	ADP
ejpam-4732	34	4	a	a	PRON
ejpam-4732	34	5	and	and	CCONJ
ejpam-4732	34	6	the	the	DET
ejpam-4732	34	7	interior	interior	NOUN
ejpam-4732	34	8	of	of	ADP
ejpam-4732	34	9	a	a	PRON
ejpam-4732	34	10	are	be	AUX
ejpam-4732	34	11	denoted	denote	VERB
ejpam-4732	34	12	by	by	ADP
ejpam-4732	34	13	cl(a	cl(a	NOUN
ejpam-4732	34	14	)	)	PUNCT
ejpam-4732	34	15	and	and	CCONJ
ejpam-4732	34	16	int(a	int(a	PROPN
ejpam-4732	34	17	)	)	PUNCT
ejpam-4732	34	18	,	,	PUNCT
ejpam-4732	34	19	respectively	respectively	ADV
ejpam-4732	34	20	.	.	PUNCT
ejpam-4732	35	1	an	an	DET
ejpam-4732	35	2	ideal	ideal	NOUN
ejpam-4732	35	3	i	i	PRON
ejpam-4732	35	4	on	on	ADP
ejpam-4732	35	5	a	a	DET
ejpam-4732	35	6	topological	topological	ADJ
ejpam-4732	35	7	space	space	NOUN
ejpam-4732	35	8	(	(	PUNCT
ejpam-4732	35	9	x	x	X
ejpam-4732	35	10	,	,	PUNCT
ejpam-4732	35	11	τ	τ	X
ejpam-4732	35	12	)	)	PUNCT
ejpam-4732	35	13	is	be	AUX
ejpam-4732	35	14	a	a	DET
ejpam-4732	35	15	nonempty	nonempty	ADJ
ejpam-4732	35	16	collection	collection	NOUN
ejpam-4732	35	17	of	of	ADP
ejpam-4732	35	18	subsets	subset	NOUN
ejpam-4732	35	19	of	of	ADP
ejpam-4732	35	20	x	x	PUNCT
ejpam-4732	35	21	satisfying	satisfy	VERB
ejpam-4732	35	22	the	the	DET
ejpam-4732	35	23	following	follow	VERB
ejpam-4732	35	24	properties	property	NOUN
ejpam-4732	35	25	:	:	PUNCT
ejpam-4732	35	26	(	(	PUNCT
ejpam-4732	35	27	1	1	X
ejpam-4732	35	28	)	)	PUNCT
ejpam-4732	35	29	a	a	DET
ejpam-4732	35	30	∈	∈	NOUN
ejpam-4732	35	31	i	i	PRON
ejpam-4732	35	32	and	and	CCONJ
ejpam-4732	35	33	b	b	X
ejpam-4732	35	34	⊆	⊆	NUM
ejpam-4732	35	35	a	a	DET
ejpam-4732	35	36	imply	imply	NOUN
ejpam-4732	35	37	b	b	X
ejpam-4732	35	38	∈	∈	PROPN
ejpam-4732	35	39	i	i	PRON
ejpam-4732	35	40	;	;	PUNCT
ejpam-4732	35	41	(	(	PUNCT
ejpam-4732	35	42	2	2	X
ejpam-4732	35	43	)	)	PUNCT
ejpam-4732	36	1	a	a	PRON
ejpam-4732	36	2	∈	∈	NOUN
ejpam-4732	37	1	i	i	PRON
ejpam-4732	37	2	and	and	CCONJ
ejpam-4732	37	3	b	b	X
ejpam-4732	37	4	∈	∈	NOUN
ejpam-4732	38	1	i	i	PRON
ejpam-4732	38	2	imply	imply	VERB
ejpam-4732	38	3	a∪b	a∪b	ADJ
ejpam-4732	39	1	∈	∈	INTJ
ejpam-4732	40	1	i	i	PRON
ejpam-4732	40	2	.	.	PUNCT
ejpam-4732	41	1	a	a	DET
ejpam-4732	41	2	topological	topological	ADJ
ejpam-4732	41	3	space	space	NOUN
ejpam-4732	41	4	(	(	PUNCT
ejpam-4732	41	5	x	x	X
ejpam-4732	41	6	,	,	PUNCT
ejpam-4732	41	7	τ	τ	X
ejpam-4732	41	8	)	)	PUNCT
ejpam-4732	41	9	with	with	ADP
ejpam-4732	41	10	an	an	DET
ejpam-4732	41	11	ideal	ideal	ADJ
ejpam-4732	41	12	i	i	PRON
ejpam-4732	41	13	on	on	ADP
ejpam-4732	41	14	x	x	SYM
ejpam-4732	41	15	is	be	AUX
ejpam-4732	41	16	called	call	VERB
ejpam-4732	41	17	an	an	DET
ejpam-4732	41	18	ideal	ideal	ADJ
ejpam-4732	41	19	topological	topological	ADJ
ejpam-4732	41	20	space	space	NOUN
ejpam-4732	41	21	and	and	CCONJ
ejpam-4732	41	22	is	be	AUX
ejpam-4732	41	23	denoted	denote	VERB
ejpam-4732	41	24	by	by	ADP
ejpam-4732	41	25	(	(	PUNCT
ejpam-4732	41	26	x	x	X
ejpam-4732	41	27	,	,	PUNCT
ejpam-4732	41	28	τ	τ	PROPN
ejpam-4732	41	29	,	,	PUNCT
ejpam-4732	41	30	i	i	NOUN
ejpam-4732	41	31	)	)	PUNCT
ejpam-4732	41	32	.	.	PUNCT
ejpam-4732	42	1	for	for	ADP
ejpam-4732	42	2	an	an	DET
ejpam-4732	42	3	ideal	ideal	ADJ
ejpam-4732	42	4	topological	topological	ADJ
ejpam-4732	42	5	space	space	NOUN
ejpam-4732	42	6	(	(	PUNCT
ejpam-4732	42	7	x	x	X
ejpam-4732	42	8	,	,	PUNCT
ejpam-4732	42	9	τ	τ	PROPN
ejpam-4732	42	10	,	,	PUNCT
ejpam-4732	42	11	i	i	PROPN
ejpam-4732	42	12	)	)	PUNCT
ejpam-4732	42	13	and	and	CCONJ
ejpam-4732	42	14	a	a	DET
ejpam-4732	42	15	subset	subset	NOUN
ejpam-4732	42	16	a	a	PRON
ejpam-4732	42	17	of	of	ADP
ejpam-4732	42	18	x	x	PRON
ejpam-4732	42	19	,	,	PUNCT
ejpam-4732	42	20	a⋆(i	a⋆(i	PROPN
ejpam-4732	42	21	)	)	PUNCT
ejpam-4732	42	22	is	be	AUX
ejpam-4732	42	23	defined	define	VERB
ejpam-4732	42	24	as	as	SCONJ
ejpam-4732	42	25	follows	follow	VERB
ejpam-4732	42	26	:	:	PUNCT
ejpam-4732	42	27	a⋆(i	a⋆(i	NOUN
ejpam-4732	42	28	)	)	PUNCT
ejpam-4732	43	1	=	=	PUNCT
ejpam-4732	43	2	{	{	PUNCT
ejpam-4732	43	3	x	x	PUNCT
ejpam-4732	43	4	∈	∈	PROPN
ejpam-4732	43	5	x	x	X
ejpam-4732	43	6	:	:	PUNCT
ejpam-4732	43	7	u	u	X
ejpam-4732	43	8	∩a	∩a	PROPN
ejpam-4732	43	9	̸∈	̸∈	PROPN
ejpam-4732	43	10	i	i	PRON
ejpam-4732	43	11	for	for	ADP
ejpam-4732	43	12	every	every	DET
ejpam-4732	43	13	open	open	ADJ
ejpam-4732	43	14	neighbourhood	neighbourhood	NOUN
ejpam-4732	43	15	u	u	NOUN
ejpam-4732	43	16	of	of	ADP
ejpam-4732	43	17	x	x	NOUN
ejpam-4732	43	18	}	}	PUNCT
ejpam-4732	43	19	.	.	PUNCT
ejpam-4732	44	1	in	in	ADP
ejpam-4732	44	2	case	case	NOUN
ejpam-4732	44	3	there	there	PRON
ejpam-4732	44	4	is	be	VERB
ejpam-4732	44	5	no	no	DET
ejpam-4732	44	6	chance	chance	NOUN
ejpam-4732	44	7	for	for	ADP
ejpam-4732	44	8	confusion	confusion	NOUN
ejpam-4732	44	9	,	,	PUNCT
ejpam-4732	44	10	a⋆(i	a⋆(i	NOUN
ejpam-4732	44	11	)	)	PUNCT
ejpam-4732	44	12	is	be	AUX
ejpam-4732	44	13	simply	simply	ADV
ejpam-4732	44	14	written	write	VERB
ejpam-4732	44	15	as	as	ADP
ejpam-4732	44	16	a⋆.	a⋆.	NOUN
ejpam-4732	44	17	in	in	ADP
ejpam-4732	44	18	[	[	X
ejpam-4732	44	19	12	12	NUM
ejpam-4732	44	20	]	]	PUNCT
ejpam-4732	44	21	,	,	PUNCT
ejpam-4732	44	22	a⋆	a⋆	ADV
ejpam-4732	44	23	is	be	AUX
ejpam-4732	44	24	called	call	VERB
ejpam-4732	44	25	the	the	DET
ejpam-4732	44	26	local	local	ADJ
ejpam-4732	44	27	function	function	NOUN
ejpam-4732	44	28	of	of	ADP
ejpam-4732	44	29	a	a	PRON
ejpam-4732	44	30	with	with	ADP
ejpam-4732	44	31	respect	respect	NOUN
ejpam-4732	44	32	to	to	ADP
ejpam-4732	44	33	i	i	PRON
ejpam-4732	44	34	and	and	CCONJ
ejpam-4732	44	35	τ	τ	PROPN
ejpam-4732	44	36	and	and	CCONJ
ejpam-4732	44	37	cl⋆(a	cl⋆(a	NUM
ejpam-4732	44	38	)	)	PUNCT
ejpam-4732	44	39	=	=	PUNCT
ejpam-4732	44	40	a⋆	a⋆	ADP
ejpam-4732	44	41	∪	∪	ADP
ejpam-4732	44	42	a	a	DET
ejpam-4732	44	43	defines	define	NOUN
ejpam-4732	44	44	a	a	DET
ejpam-4732	44	45	kuratowski	kuratowski	ADJ
ejpam-4732	44	46	closure	closure	NOUN
ejpam-4732	44	47	operator	operator	NOUN
ejpam-4732	44	48	for	for	ADP
ejpam-4732	44	49	a	a	DET
ejpam-4732	44	50	topology	topology	NOUN
ejpam-4732	44	51	τ⋆(i	τ⋆(i	NOUN
ejpam-4732	44	52	)	)	PUNCT
ejpam-4732	44	53	finer	fine	ADJ
ejpam-4732	44	54	than	than	ADP
ejpam-4732	44	55	τ	τ	PROPN
ejpam-4732	44	56	.	.	PUNCT
ejpam-4732	45	1	a	a	DET
ejpam-4732	45	2	subset	subset	NOUN
ejpam-4732	45	3	a	a	PRON
ejpam-4732	45	4	is	be	AUX
ejpam-4732	45	5	said	say	VERB
ejpam-4732	45	6	to	to	PART
ejpam-4732	45	7	be	be	AUX
ejpam-4732	45	8	⋆-closed	⋆-close	VERB
ejpam-4732	45	9	[	[	X
ejpam-4732	45	10	11	11	NUM
ejpam-4732	45	11	]	]	X
ejpam-4732	45	12	if	if	SCONJ
ejpam-4732	45	13	a⋆	a⋆	ADJ
ejpam-4732	45	14	⊆	⊆	NUM
ejpam-4732	45	15	a.	a.	NOUN
ejpam-4732	45	16	the	the	DET
ejpam-4732	45	17	interior	interior	NOUN
ejpam-4732	45	18	of	of	ADP
ejpam-4732	45	19	a	a	DET
ejpam-4732	45	20	subset	subset	NOUN
ejpam-4732	45	21	a	a	DET
ejpam-4732	45	22	in	in	ADP
ejpam-4732	45	23	(	(	PUNCT
ejpam-4732	45	24	x	x	X
ejpam-4732	45	25	,	,	PUNCT
ejpam-4732	45	26	τ⋆(i	τ⋆(i	NOUN
ejpam-4732	45	27	)	)	PUNCT
ejpam-4732	45	28	)	)	PUNCT
ejpam-4732	45	29	is	be	AUX
ejpam-4732	45	30	denoted	denote	VERB
ejpam-4732	45	31	by	by	ADP
ejpam-4732	45	32	int⋆(a	int⋆(a	NOUN
ejpam-4732	45	33	)	)	PUNCT
ejpam-4732	45	34	.	.	PUNCT
ejpam-4732	46	1	by	by	ADP
ejpam-4732	46	2	a	a	DET
ejpam-4732	46	3	multifunction	multifunction	NOUN
ejpam-4732	46	4	f	f	NOUN
ejpam-4732	46	5	:	:	PUNCT
ejpam-4732	46	6	x	x	X
ejpam-4732	46	7	→	→	SYM
ejpam-4732	46	8	y	y	PROPN
ejpam-4732	46	9	,	,	PUNCT
ejpam-4732	46	10	we	we	PRON
ejpam-4732	46	11	mean	mean	VERB
ejpam-4732	46	12	a	a	DET
ejpam-4732	46	13	point	point	NOUN
ejpam-4732	46	14	-	-	PUNCT
ejpam-4732	46	15	to	to	ADP
ejpam-4732	46	16	-	-	PUNCT
ejpam-4732	46	17	set	set	VERB
ejpam-4732	46	18	correspondence	correspondence	NOUN
ejpam-4732	46	19	from	from	ADP
ejpam-4732	46	20	x	x	PUNCT
ejpam-4732	46	21	into	into	ADP
ejpam-4732	46	22	y	y	PROPN
ejpam-4732	46	23	,	,	PUNCT
ejpam-4732	46	24	and	and	CCONJ
ejpam-4732	46	25	we	we	PRON
ejpam-4732	46	26	always	always	ADV
ejpam-4732	46	27	assume	assume	VERB
ejpam-4732	46	28	that	that	SCONJ
ejpam-4732	46	29	f	f	PROPN
ejpam-4732	46	30	(	(	PUNCT
ejpam-4732	46	31	x	x	X
ejpam-4732	46	32	)	)	PUNCT
ejpam-4732	46	33	̸=	̸=	NOUN
ejpam-4732	46	34	∅	∅	NOUN
ejpam-4732	46	35	for	for	ADP
ejpam-4732	46	36	all	all	PRON
ejpam-4732	46	37	x	x	SYM
ejpam-4732	46	38	∈	∈	ADJ
ejpam-4732	46	39	x.	x.	NOUN
ejpam-4732	46	40	for	for	ADP
ejpam-4732	46	41	a	a	DET
ejpam-4732	46	42	multifunction	multifunction	NOUN
ejpam-4732	46	43	f	f	NOUN
ejpam-4732	47	1	:	:	PUNCT
ejpam-4732	47	2	x	x	X
ejpam-4732	47	3	→	→	SYM
ejpam-4732	47	4	y	y	PROPN
ejpam-4732	47	5	,	,	PUNCT
ejpam-4732	47	6	following	follow	VERB
ejpam-4732	47	7	[	[	X
ejpam-4732	47	8	1	1	X
ejpam-4732	47	9	]	]	PUNCT
ejpam-4732	47	10	we	we	PRON
ejpam-4732	47	11	shall	shall	AUX
ejpam-4732	47	12	denote	denote	VERB
ejpam-4732	47	13	the	the	DET
ejpam-4732	47	14	upper	upper	ADJ
ejpam-4732	47	15	and	and	CCONJ
ejpam-4732	47	16	lower	low	ADJ
ejpam-4732	47	17	inverse	inverse	NOUN
ejpam-4732	47	18	of	of	ADP
ejpam-4732	47	19	a	a	DET
ejpam-4732	47	20	set	set	NOUN
ejpam-4732	47	21	b	b	PROPN
ejpam-4732	47	22	of	of	ADP
ejpam-4732	47	23	y	y	PROPN
ejpam-4732	47	24	by	by	ADP
ejpam-4732	47	25	f+(b	f+(b	NOUN
ejpam-4732	47	26	)	)	PUNCT
ejpam-4732	47	27	and	and	CCONJ
ejpam-4732	47	28	f−(b	f−(b	NOUN
ejpam-4732	47	29	)	)	PUNCT
ejpam-4732	47	30	,	,	PUNCT
ejpam-4732	47	31	respectively	respectively	ADV
ejpam-4732	47	32	,	,	PUNCT
ejpam-4732	47	33	that	that	ADV
ejpam-4732	47	34	is	is	ADV
ejpam-4732	47	35	,	,	PUNCT
ejpam-4732	47	36	f+(b	f+(b	NOUN
ejpam-4732	47	37	)	)	PUNCT
ejpam-4732	47	38	=	=	PRON
ejpam-4732	48	1	{	{	PUNCT
ejpam-4732	48	2	x	x	PUNCT
ejpam-4732	48	3	∈	∈	PROPN
ejpam-4732	48	4	x	x	INTJ
ejpam-4732	49	1	|	|	NOUN
ejpam-4732	49	2	f	f	X
ejpam-4732	49	3	(	(	PUNCT
ejpam-4732	49	4	x	x	NOUN
ejpam-4732	49	5	)	)	PUNCT
ejpam-4732	49	6	⊆	⊆	NUM
ejpam-4732	49	7	b	b	NOUN
ejpam-4732	49	8	}	}	PUNCT
ejpam-4732	49	9	and	and	CCONJ
ejpam-4732	49	10	f−(b	f−(b	PROPN
ejpam-4732	49	11	)	)	PUNCT
ejpam-4732	49	12	=	=	PRON
ejpam-4732	50	1	{	{	PUNCT
ejpam-4732	50	2	x	x	PUNCT
ejpam-4732	50	3	∈	∈	PROPN
ejpam-4732	50	4	x	x	INTJ
ejpam-4732	51	1	|	|	NOUN
ejpam-4732	51	2	f	f	X
ejpam-4732	51	3	(	(	PUNCT
ejpam-4732	51	4	x	x	NOUN
ejpam-4732	51	5	)	)	PUNCT
ejpam-4732	51	6	∩b	∩b	NOUN
ejpam-4732	51	7	̸=	̸=	PROPN
ejpam-4732	51	8	∅	∅	NOUN
ejpam-4732	51	9	}	}	PUNCT
ejpam-4732	51	10	.	.	PUNCT
ejpam-4732	52	1	in	in	ADP
ejpam-4732	52	2	particular	particular	ADJ
ejpam-4732	52	3	,	,	PUNCT
ejpam-4732	52	4	f−(y	f−(y	NOUN
ejpam-4732	52	5	)	)	PUNCT
ejpam-4732	52	6	=	=	SYM
ejpam-4732	53	1	{	{	PUNCT
ejpam-4732	53	2	x	x	PUNCT
ejpam-4732	53	3	∈	∈	PROPN
ejpam-4732	53	4	x	x	INTJ
ejpam-4732	54	1	|	|	ADV
ejpam-4732	54	2	y	y	PROPN
ejpam-4732	54	3	∈	∈	PROPN
ejpam-4732	54	4	f	f	X
ejpam-4732	54	5	(	(	PUNCT
ejpam-4732	54	6	x	x	NOUN
ejpam-4732	54	7	)	)	PUNCT
ejpam-4732	54	8	}	}	PUNCT
ejpam-4732	54	9	for	for	ADP
ejpam-4732	54	10	each	each	DET
ejpam-4732	54	11	point	point	NOUN
ejpam-4732	54	12	y	y	PROPN
ejpam-4732	54	13	∈	∈	PROPN
ejpam-4732	54	14	y	y	PROPN
ejpam-4732	54	15	.	.	PUNCT
ejpam-4732	55	1	for	for	ADP
ejpam-4732	55	2	each	each	PRON
ejpam-4732	55	3	a	a	DET
ejpam-4732	55	4	⊆	⊆	NUM
ejpam-4732	55	5	x	x	SYM
ejpam-4732	55	6	,	,	PUNCT
ejpam-4732	55	7	f	f	PROPN
ejpam-4732	55	8	(	(	PUNCT
ejpam-4732	55	9	a	a	NOUN
ejpam-4732	55	10	)	)	PUNCT
ejpam-4732	55	11	=	=	SYM
ejpam-4732	55	12	∪x∈af	∪x∈af	NOUN
ejpam-4732	55	13	(	(	PUNCT
ejpam-4732	55	14	x	x	NOUN
ejpam-4732	55	15	)	)	PUNCT
ejpam-4732	55	16	.	.	PUNCT
ejpam-4732	56	1	lemma	lemma	PROPN
ejpam-4732	56	2	1	1	NUM
ejpam-4732	56	3	.	.	PUNCT
ejpam-4732	57	1	for	for	ADP
ejpam-4732	57	2	a	a	DET
ejpam-4732	57	3	subset	subset	NOUN
ejpam-4732	57	4	a	a	PRON
ejpam-4732	57	5	of	of	ADP
ejpam-4732	57	6	an	an	DET
ejpam-4732	57	7	ideal	ideal	ADJ
ejpam-4732	57	8	topological	topological	ADJ
ejpam-4732	57	9	space	space	NOUN
ejpam-4732	57	10	(	(	PUNCT
ejpam-4732	57	11	x	x	X
ejpam-4732	57	12	,	,	PUNCT
ejpam-4732	57	13	τ	τ	PROPN
ejpam-4732	57	14	,	,	PUNCT
ejpam-4732	57	15	i	i	NOUN
ejpam-4732	57	16	)	)	PUNCT
ejpam-4732	57	17	,	,	PUNCT
ejpam-4732	57	18	the	the	DET
ejpam-4732	57	19	following	follow	VERB
ejpam-4732	57	20	properties	property	NOUN
ejpam-4732	57	21	hold	hold	VERB
ejpam-4732	57	22	:	:	PUNCT
ejpam-4732	57	23	(	(	PUNCT
ejpam-4732	57	24	1	1	X
ejpam-4732	57	25	)	)	PUNCT
ejpam-4732	57	26	if	if	SCONJ
ejpam-4732	57	27	v	v	NOUN
ejpam-4732	57	28	∈	∈	PROPN
ejpam-4732	57	29	τ	τ	X
ejpam-4732	57	30	,	,	PUNCT
ejpam-4732	57	31	then	then	ADV
ejpam-4732	57	32	v	v	ADP
ejpam-4732	57	33	∩	∩	ADJ
ejpam-4732	57	34	cl⋆(a	cl⋆(a	NOUN
ejpam-4732	57	35	)	)	PUNCT
ejpam-4732	57	36	⊆	⊆	NUM
ejpam-4732	57	37	cl⋆(v	cl⋆(v	PROPN
ejpam-4732	57	38	∩a	∩a	PROPN
ejpam-4732	57	39	)	)	PUNCT
ejpam-4732	58	1	[	[	X
ejpam-4732	58	2	9	9	NUM
ejpam-4732	58	3	]	]	PUNCT
ejpam-4732	58	4	.	.	PUNCT
ejpam-4732	59	1	(	(	PUNCT
ejpam-4732	59	2	2	2	X
ejpam-4732	59	3	)	)	PUNCT
ejpam-4732	59	4	if	if	SCONJ
ejpam-4732	59	5	f	f	PROPN
ejpam-4732	59	6	is	be	AUX
ejpam-4732	59	7	closed	close	VERB
ejpam-4732	59	8	in	in	ADP
ejpam-4732	59	9	x	x	NOUN
ejpam-4732	59	10	,	,	PUNCT
ejpam-4732	59	11	then	then	ADV
ejpam-4732	59	12	int⋆(a	int⋆(a	PUNCT
ejpam-4732	59	13	∪	∪	PROPN
ejpam-4732	59	14	f	f	PROPN
ejpam-4732	59	15	)	)	PUNCT
ejpam-4732	59	16	⊆	⊆	NUM
ejpam-4732	59	17	int⋆(a	int⋆(a	NOUN
ejpam-4732	59	18	)	)	PUNCT
ejpam-4732	59	19	∪	∪	ADP
ejpam-4732	59	20	f	f	PROPN
ejpam-4732	59	21	.	.	PUNCT
ejpam-4732	60	1	a	a	DET
ejpam-4732	60	2	subset	subset	NOUN
ejpam-4732	60	3	a	a	PRON
ejpam-4732	60	4	of	of	ADP
ejpam-4732	60	5	an	an	DET
ejpam-4732	60	6	ideal	ideal	ADJ
ejpam-4732	60	7	topological	topological	ADJ
ejpam-4732	60	8	space	space	NOUN
ejpam-4732	60	9	(	(	PUNCT
ejpam-4732	60	10	x	x	X
ejpam-4732	60	11	,	,	PUNCT
ejpam-4732	60	12	τ	τ	PROPN
ejpam-4732	60	13	,	,	PUNCT
ejpam-4732	60	14	i	i	PROPN
ejpam-4732	60	15	)	)	PUNCT
ejpam-4732	60	16	is	be	AUX
ejpam-4732	60	17	called	call	VERB
ejpam-4732	60	18	semi	semi	ADJ
ejpam-4732	60	19	-	-	ADJ
ejpam-4732	60	20	i	i	PRON
ejpam-4732	60	21	-open	-open	NOUN
ejpam-4732	60	22	[	[	X
ejpam-4732	60	23	10	10	NUM
ejpam-4732	60	24	]	]	PUNCT
ejpam-4732	60	25	(	(	PUNCT
ejpam-4732	60	26	resp	resp	NOUN
ejpam-4732	60	27	.	.	PUNCT
ejpam-4732	61	1	pre⋆i	pre⋆i	NOUN
ejpam-4732	61	2	-open	-open	NOUN
ejpam-4732	62	1	[	[	X
ejpam-4732	62	2	5	5	NUM
ejpam-4732	62	3	]	]	PUNCT
ejpam-4732	62	4	,	,	PUNCT
ejpam-4732	62	5	strong	strong	ADJ
ejpam-4732	62	6	β	β	X
ejpam-4732	62	7	-	-	VERB
ejpam-4732	62	8	i	i	PRON
ejpam-4732	62	9	-open	-open	PROPN
ejpam-4732	63	1	[	[	X
ejpam-4732	63	2	8	8	NUM
ejpam-4732	63	3	]	]	PUNCT
ejpam-4732	63	4	)	)	PUNCT
ejpam-4732	63	5	if	if	SCONJ
ejpam-4732	63	6	a	a	DET
ejpam-4732	63	7	⊆	⊆	NUM
ejpam-4732	63	8	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-4732	63	9	)	)	PUNCT
ejpam-4732	63	10	)	)	PUNCT
ejpam-4732	63	11	(	(	PUNCT
ejpam-4732	63	12	resp	resp	NOUN
ejpam-4732	63	13	.	.	PUNCT
ejpam-4732	64	1	a	a	DET
ejpam-4732	64	2	⊆	⊆	NUM
ejpam-4732	64	3	int⋆(cl(a	int⋆(cl(a	NOUN
ejpam-4732	64	4	)	)	PUNCT
ejpam-4732	64	5	)	)	PUNCT
ejpam-4732	64	6	,	,	PUNCT
ejpam-4732	64	7	a	a	DET
ejpam-4732	64	8	⊆	⊆	NUM
ejpam-4732	64	9	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	64	10	)	)	PUNCT
ejpam-4732	64	11	)	)	PUNCT
ejpam-4732	64	12	)	)	PUNCT
ejpam-4732	64	13	)	)	PUNCT
ejpam-4732	64	14	.	.	PUNCT
ejpam-4732	65	1	the	the	DET
ejpam-4732	65	2	complement	complement	NOUN
ejpam-4732	65	3	of	of	ADP
ejpam-4732	65	4	a	a	DET
ejpam-4732	65	5	semi	semi	ADJ
ejpam-4732	65	6	-	-	ADJ
ejpam-4732	65	7	i	i	PRON
ejpam-4732	65	8	-open	-open	ADJ
ejpam-4732	65	9	(	(	PUNCT
ejpam-4732	65	10	resp	resp	NOUN
ejpam-4732	65	11	.	.	PUNCT
ejpam-4732	66	1	pre⋆i	pre⋆i	PROPN
ejpam-4732	66	2	-open	-open	PROPN
ejpam-4732	66	3	,	,	PUNCT
ejpam-4732	66	4	strong	strong	ADJ
ejpam-4732	66	5	β	β	X
ejpam-4732	66	6	-	-	PUNCT
ejpam-4732	66	7	i	i	PRON
ejpam-4732	66	8	open	open	ADJ
ejpam-4732	66	9	)	)	PUNCT
ejpam-4732	66	10	set	set	NOUN
ejpam-4732	66	11	is	be	AUX
ejpam-4732	66	12	called	call	VERB
ejpam-4732	66	13	semi	semi	ADJ
ejpam-4732	66	14	-	-	ADJ
ejpam-4732	66	15	i	i	PRON
ejpam-4732	66	16	-closed	-close	VERB
ejpam-4732	67	1	[	[	X
ejpam-4732	67	2	10	10	NUM
ejpam-4732	67	3	]	]	PUNCT
ejpam-4732	67	4	(	(	PUNCT
ejpam-4732	67	5	resp	resp	NOUN
ejpam-4732	67	6	.	.	PUNCT
ejpam-4732	68	1	pre⋆i	pre⋆i	PROPN
ejpam-4732	68	2	-closed	-close	VERB
ejpam-4732	69	1	[	[	X
ejpam-4732	69	2	5	5	NUM
ejpam-4732	69	3	]	]	PUNCT
ejpam-4732	69	4	,	,	PUNCT
ejpam-4732	69	5	strong	strong	ADJ
ejpam-4732	69	6	β	β	X
ejpam-4732	69	7	-	-	PUNCT
ejpam-4732	69	8	i	i	PRON
ejpam-4732	69	9	-closed	-close	VERB
ejpam-4732	69	10	[	[	X
ejpam-4732	69	11	8	8	NUM
ejpam-4732	69	12	]	]	NUM
ejpam-4732	69	13	)	)	PUNCT
ejpam-4732	69	14	.	.	PUNCT
ejpam-4732	70	1	the	the	DET
ejpam-4732	70	2	strong	strong	ADJ
ejpam-4732	70	3	β	β	X
ejpam-4732	70	4	-	-	ADJ
ejpam-4732	70	5	i	i	PRON
ejpam-4732	70	6	-closure	-closure	NOUN
ejpam-4732	70	7	(	(	PUNCT
ejpam-4732	70	8	resp	resp	NOUN
ejpam-4732	70	9	.	.	PUNCT
ejpam-4732	71	1	semi	semi	ADJ
ejpam-4732	71	2	-	-	ADJ
ejpam-4732	71	3	i	i	ADJ
ejpam-4732	71	4	-closure	-closure	NOUN
ejpam-4732	71	5	)	)	PUNCT
ejpam-4732	72	1	[	[	X
ejpam-4732	72	2	6	6	NUM
ejpam-4732	72	3	]	]	PUNCT
ejpam-4732	72	4	of	of	ADP
ejpam-4732	72	5	a	a	DET
ejpam-4732	72	6	subset	subset	NOUN
ejpam-4732	72	7	a	a	PRON
ejpam-4732	72	8	of	of	ADP
ejpam-4732	72	9	an	an	DET
ejpam-4732	72	10	ideal	ideal	ADJ
ejpam-4732	72	11	topological	topological	ADJ
ejpam-4732	72	12	space	space	NOUN
ejpam-4732	72	13	(	(	PUNCT
ejpam-4732	72	14	x	x	X
ejpam-4732	72	15	,	,	PUNCT
ejpam-4732	72	16	τ	τ	PROPN
ejpam-4732	72	17	,	,	PUNCT
ejpam-4732	72	18	i	i	PROPN
ejpam-4732	72	19	)	)	PUNCT
ejpam-4732	72	20	,	,	PUNCT
ejpam-4732	72	21	denoted	denote	VERB
ejpam-4732	72	22	by	by	ADP
ejpam-4732	72	23	sβcli	sβcli	NOUN
ejpam-4732	72	24	(	(	PUNCT
ejpam-4732	72	25	a	a	NOUN
ejpam-4732	72	26	)	)	PUNCT
ejpam-4732	72	27	(	(	PUNCT
ejpam-4732	72	28	resp	resp	NOUN
ejpam-4732	72	29	.	.	PUNCT
ejpam-4732	72	30	scli	scli	PROPN
ejpam-4732	72	31	(	(	PUNCT
ejpam-4732	72	32	a	a	NOUN
ejpam-4732	72	33	)	)	PUNCT
ejpam-4732	72	34	)	)	PUNCT
ejpam-4732	72	35	,	,	PUNCT
ejpam-4732	72	36	is	be	AUX
ejpam-4732	72	37	defined	define	VERB
ejpam-4732	72	38	by	by	ADP
ejpam-4732	72	39	the	the	DET
ejpam-4732	72	40	intersection	intersection	NOUN
ejpam-4732	72	41	of	of	ADP
ejpam-4732	72	42	all	all	DET
ejpam-4732	72	43	c.	c.	PROPN
ejpam-4732	72	44	boonpok	boonpok	PROPN
ejpam-4732	72	45	,	,	PUNCT
ejpam-4732	72	46	p.	p.	NOUN
ejpam-4732	72	47	pue	pue	NOUN
ejpam-4732	72	48	-	-	PUNCT
ejpam-4732	72	49	on	on	ADP
ejpam-4732	72	50	/	/	SYM
ejpam-4732	72	51	eur	eur	NOUN
ejpam-4732	72	52	.	.	PUNCT
ejpam-4732	73	1	j.	j.	PROPN
ejpam-4732	73	2	pure	pure	PROPN
ejpam-4732	73	3	appl	appl	PROPN
ejpam-4732	73	4	.	.	PROPN
ejpam-4732	73	5	math	math	PROPN
ejpam-4732	73	6	,	,	PUNCT
ejpam-4732	73	7	16	16	NUM
ejpam-4732	73	8	(	(	PUNCT
ejpam-4732	73	9	3	3	NUM
ejpam-4732	73	10	)	)	PUNCT
ejpam-4732	73	11	(	(	PUNCT
ejpam-4732	73	12	2023	2023	NUM
ejpam-4732	73	13	)	)	PUNCT
ejpam-4732	73	14	,	,	PUNCT
ejpam-4732	73	15	1634	1634	NUM
ejpam-4732	73	16	-	-	SYM
ejpam-4732	73	17	1646	1646	NUM
ejpam-4732	73	18	1636	1636	NUM
ejpam-4732	73	19	strong	strong	ADJ
ejpam-4732	73	20	β	β	X
ejpam-4732	73	21	-	-	PUNCT
ejpam-4732	73	22	i	i	PRON
ejpam-4732	73	23	-closed	-close	VERB
ejpam-4732	73	24	(	(	PUNCT
ejpam-4732	73	25	resp	resp	NOUN
ejpam-4732	73	26	.	.	PUNCT
ejpam-4732	74	1	semi	semi	ADJ
ejpam-4732	74	2	-	-	VERB
ejpam-4732	74	3	i	i	PRON
ejpam-4732	74	4	-closed	-closed	ADJ
ejpam-4732	74	5	)	)	PUNCT
ejpam-4732	74	6	sets	set	NOUN
ejpam-4732	74	7	of	of	ADP
ejpam-4732	74	8	x	x	PUNCT
ejpam-4732	74	9	containing	contain	VERB
ejpam-4732	74	10	a.	a.	NOUN
ejpam-4732	74	11	let	let	VERB
ejpam-4732	74	12	a	a	PRON
ejpam-4732	74	13	be	be	AUX
ejpam-4732	74	14	a	a	DET
ejpam-4732	74	15	subset	subset	NOUN
ejpam-4732	74	16	of	of	ADP
ejpam-4732	74	17	an	an	DET
ejpam-4732	74	18	ideal	ideal	ADJ
ejpam-4732	74	19	topological	topological	ADJ
ejpam-4732	74	20	space	space	NOUN
ejpam-4732	74	21	(	(	PUNCT
ejpam-4732	74	22	x	x	X
ejpam-4732	74	23	,	,	PUNCT
ejpam-4732	74	24	τ	τ	PROPN
ejpam-4732	74	25	,	,	PUNCT
ejpam-4732	74	26	i	i	NOUN
ejpam-4732	74	27	)	)	PUNCT
ejpam-4732	74	28	.	.	PUNCT
ejpam-4732	75	1	the	the	DET
ejpam-4732	75	2	union	union	NOUN
ejpam-4732	75	3	of	of	ADP
ejpam-4732	75	4	all	all	DET
ejpam-4732	75	5	strong	strong	ADJ
ejpam-4732	75	6	β	β	NOUN
ejpam-4732	75	7	-	-	ADJ
ejpam-4732	75	8	i	i	PRON
ejpam-4732	75	9	-open	-open	NOUN
ejpam-4732	75	10	sets	set	NOUN
ejpam-4732	75	11	of	of	ADP
ejpam-4732	75	12	x	x	PUNCT
ejpam-4732	75	13	contained	contain	VERB
ejpam-4732	75	14	in	in	ADP
ejpam-4732	75	15	a	a	PRON
ejpam-4732	75	16	is	be	AUX
ejpam-4732	75	17	called	call	VERB
ejpam-4732	75	18	the	the	DET
ejpam-4732	75	19	strong	strong	ADJ
ejpam-4732	75	20	β	β	X
ejpam-4732	75	21	-	-	ADJ
ejpam-4732	75	22	i	i	PRON
ejpam-4732	75	23	-interior	-interior	NOUN
ejpam-4732	75	24	of	of	ADP
ejpam-4732	75	25	a	a	PRON
ejpam-4732	75	26	and	and	CCONJ
ejpam-4732	75	27	is	be	AUX
ejpam-4732	75	28	denoted	denote	VERB
ejpam-4732	75	29	by	by	ADP
ejpam-4732	75	30	sβinti	sβinti	NOUN
ejpam-4732	75	31	(	(	PUNCT
ejpam-4732	75	32	a	a	NOUN
ejpam-4732	75	33	)	)	PUNCT
ejpam-4732	75	34	.	.	PUNCT
ejpam-4732	76	1	lemma	lemma	PROPN
ejpam-4732	76	2	2	2	NUM
ejpam-4732	76	3	.	.	X
ejpam-4732	77	1	for	for	ADP
ejpam-4732	77	2	a	a	DET
ejpam-4732	77	3	subset	subset	NOUN
ejpam-4732	77	4	a	a	PRON
ejpam-4732	77	5	of	of	ADP
ejpam-4732	77	6	an	an	DET
ejpam-4732	77	7	ideal	ideal	ADJ
ejpam-4732	77	8	topological	topological	ADJ
ejpam-4732	77	9	space	space	NOUN
ejpam-4732	77	10	(	(	PUNCT
ejpam-4732	77	11	x	x	X
ejpam-4732	77	12	,	,	PUNCT
ejpam-4732	77	13	τ	τ	PROPN
ejpam-4732	77	14	,	,	PUNCT
ejpam-4732	77	15	i	i	NOUN
ejpam-4732	77	16	)	)	PUNCT
ejpam-4732	77	17	,	,	PUNCT
ejpam-4732	77	18	the	the	DET
ejpam-4732	77	19	following	follow	VERB
ejpam-4732	77	20	properties	property	NOUN
ejpam-4732	77	21	hold	hold	VERB
ejpam-4732	77	22	:	:	PUNCT
ejpam-4732	77	23	(	(	PUNCT
ejpam-4732	77	24	1	1	X
ejpam-4732	77	25	)	)	PUNCT
ejpam-4732	77	26	scli	scli	NOUN
ejpam-4732	77	27	(	(	PUNCT
ejpam-4732	77	28	a	a	X
ejpam-4732	77	29	)	)	PUNCT
ejpam-4732	77	30	=	=	NOUN
ejpam-4732	77	31	a	a	DET
ejpam-4732	77	32	∪	∪	ADJ
ejpam-4732	77	33	int⋆(cl(a	int⋆(cl(a	NOUN
ejpam-4732	77	34	)	)	PUNCT
ejpam-4732	77	35	)	)	PUNCT
ejpam-4732	78	1	[	[	X
ejpam-4732	78	2	6	6	NUM
ejpam-4732	78	3	]	]	PUNCT
ejpam-4732	78	4	.	.	PUNCT
ejpam-4732	79	1	(	(	PUNCT
ejpam-4732	79	2	2	2	X
ejpam-4732	79	3	)	)	PUNCT
ejpam-4732	79	4	sβcli	sβcli	NOUN
ejpam-4732	79	5	(	(	PUNCT
ejpam-4732	79	6	a	a	NOUN
ejpam-4732	79	7	)	)	PUNCT
ejpam-4732	79	8	=	=	NOUN
ejpam-4732	79	9	a	a	DET
ejpam-4732	79	10	∪	∪	ADJ
ejpam-4732	79	11	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4732	79	12	)	)	PUNCT
ejpam-4732	79	13	)	)	PUNCT
ejpam-4732	79	14	)	)	PUNCT
ejpam-4732	80	1	[	[	X
ejpam-4732	80	2	6	6	NUM
ejpam-4732	80	3	]	]	PUNCT
ejpam-4732	80	4	.	.	PUNCT
ejpam-4732	81	1	(	(	PUNCT
ejpam-4732	81	2	3	3	X
ejpam-4732	81	3	)	)	PUNCT
ejpam-4732	81	4	sβinti	sβinti	NOUN
ejpam-4732	81	5	(	(	PUNCT
ejpam-4732	81	6	a	a	X
ejpam-4732	81	7	)	)	PUNCT
ejpam-4732	81	8	=	=	SYM
ejpam-4732	81	9	a	a	DET
ejpam-4732	81	10	∩	∩	ADJ
ejpam-4732	81	11	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	81	12	)	)	PUNCT
ejpam-4732	81	13	)	)	PUNCT
ejpam-4732	81	14	)	)	PUNCT
ejpam-4732	81	15	.	.	PUNCT
ejpam-4732	82	1	proof	proof	NOUN
ejpam-4732	82	2	.	.	PUNCT
ejpam-4732	83	1	(	(	PUNCT
ejpam-4732	83	2	3	3	X
ejpam-4732	83	3	)	)	PUNCT
ejpam-4732	83	4	we	we	PRON
ejpam-4732	83	5	observe	observe	VERB
ejpam-4732	83	6	that	that	SCONJ
ejpam-4732	83	7	a	a	DET
ejpam-4732	83	8	∩	∩	ADJ
ejpam-4732	83	9	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	83	10	)	)	PUNCT
ejpam-4732	83	11	)	)	PUNCT
ejpam-4732	83	12	)	)	PUNCT
ejpam-4732	84	1	⊆	⊆	NUM
ejpam-4732	84	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	84	3	)	)	PUNCT
ejpam-4732	84	4	)	)	PUNCT
ejpam-4732	84	5	)	)	PUNCT
ejpam-4732	85	1	=	=	PUNCT
ejpam-4732	85	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	85	3	)	)	PUNCT
ejpam-4732	85	4	∩	∩	NOUN
ejpam-4732	85	5	int(cl⋆(a	int(cl⋆(a	NOUN
ejpam-4732	85	6	)	)	PUNCT
ejpam-4732	85	7	)	)	PUNCT
ejpam-4732	85	8	)	)	PUNCT
ejpam-4732	85	9	)	)	PUNCT
ejpam-4732	86	1	⊆	⊆	X
ejpam-4732	86	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4732	86	3	∩	∩	NOUN
ejpam-4732	86	4	int(cl⋆(a	int(cl⋆(a	NOUN
ejpam-4732	86	5	)	)	PUNCT
ejpam-4732	86	6	)	)	PUNCT
ejpam-4732	86	7	)	)	PUNCT
ejpam-4732	86	8	)	)	PUNCT
ejpam-4732	86	9	)	)	PUNCT
ejpam-4732	87	1	⊆	⊆	X
ejpam-4732	87	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4732	87	3	∩	∩	ADJ
ejpam-4732	87	4	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	87	5	)	)	PUNCT
ejpam-4732	87	6	)	)	PUNCT
ejpam-4732	87	7	)	)	PUNCT
ejpam-4732	87	8	)	)	PUNCT
ejpam-4732	87	9	)	)	PUNCT
ejpam-4732	87	10	)	)	PUNCT
ejpam-4732	87	11	.	.	PUNCT
ejpam-4732	88	1	thus	thus	ADV
ejpam-4732	88	2	,	,	PUNCT
ejpam-4732	88	3	a∩cl⋆(int(cl⋆(a	a∩cl⋆(int(cl⋆(a	NOUN
ejpam-4732	88	4	)	)	PUNCT
ejpam-4732	88	5	)	)	PUNCT
ejpam-4732	88	6	)	)	PUNCT
ejpam-4732	88	7	is	be	AUX
ejpam-4732	88	8	strong	strong	ADJ
ejpam-4732	88	9	β	β	NOUN
ejpam-4732	88	10	-	-	PUNCT
ejpam-4732	88	11	i	i	PRON
ejpam-4732	88	12	-open	-open	ADJ
ejpam-4732	88	13	and	and	CCONJ
ejpam-4732	88	14	so	so	ADV
ejpam-4732	88	15	a∩cl⋆(int(cl⋆(a	a∩cl⋆(int(cl⋆(a	NOUN
ejpam-4732	88	16	)	)	PUNCT
ejpam-4732	88	17	)	)	PUNCT
ejpam-4732	88	18	)	)	PUNCT
ejpam-4732	89	1	⊆	⊆	NUM
ejpam-4732	89	2	sβinti	sβinti	NOUN
ejpam-4732	89	3	(	(	PUNCT
ejpam-4732	89	4	a	a	NOUN
ejpam-4732	89	5	)	)	PUNCT
ejpam-4732	89	6	.	.	PUNCT
ejpam-4732	90	1	on	on	ADP
ejpam-4732	90	2	the	the	DET
ejpam-4732	90	3	other	other	ADJ
ejpam-4732	90	4	hand	hand	NOUN
ejpam-4732	90	5	,	,	PUNCT
ejpam-4732	90	6	since	since	SCONJ
ejpam-4732	90	7	sβinti	sβinti	X
ejpam-4732	90	8	(	(	PUNCT
ejpam-4732	90	9	a	a	NOUN
ejpam-4732	90	10	)	)	PUNCT
ejpam-4732	90	11	is	be	AUX
ejpam-4732	90	12	strong	strong	ADJ
ejpam-4732	90	13	-	-	PUNCT
ejpam-4732	90	14	β	β	NOUN
ejpam-4732	90	15	-	-	PUNCT
ejpam-4732	90	16	i	i	PRON
ejpam-4732	90	17	-open	-open	ADJ
ejpam-4732	90	18	,	,	PUNCT
ejpam-4732	90	19	we	we	PRON
ejpam-4732	90	20	have	have	VERB
ejpam-4732	90	21	sβinti	sβinti	NOUN
ejpam-4732	90	22	(	(	PUNCT
ejpam-4732	90	23	a	a	X
ejpam-4732	90	24	)	)	PUNCT
ejpam-4732	90	25	⊆	⊆	NUM
ejpam-4732	90	26	cl⋆(int(cl⋆(sβinti	cl⋆(int(cl⋆(sβinti	X
ejpam-4732	90	27	(	(	PUNCT
ejpam-4732	90	28	a	a	NOUN
ejpam-4732	90	29	)	)	PUNCT
ejpam-4732	90	30	)	)	PUNCT
ejpam-4732	90	31	)	)	PUNCT
ejpam-4732	90	32	)	)	PUNCT
ejpam-4732	91	1	⊆	⊆	NUM
ejpam-4732	91	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	91	3	)	)	PUNCT
ejpam-4732	91	4	)	)	PUNCT
ejpam-4732	91	5	)	)	PUNCT
ejpam-4732	91	6	and	and	CCONJ
ejpam-4732	91	7	hence	hence	ADV
ejpam-4732	91	8	sβinti	sβinti	X
ejpam-4732	91	9	(	(	PUNCT
ejpam-4732	91	10	a	a	X
ejpam-4732	91	11	)	)	PUNCT
ejpam-4732	91	12	⊆	⊆	PROPN
ejpam-4732	91	13	a	a	DET
ejpam-4732	91	14	∩	∩	ADJ
ejpam-4732	91	15	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	91	16	)	)	PUNCT
ejpam-4732	91	17	)	)	PUNCT
ejpam-4732	91	18	)	)	PUNCT
ejpam-4732	91	19	.	.	PUNCT
ejpam-4732	92	1	thus	thus	ADV
ejpam-4732	92	2	,	,	PUNCT
ejpam-4732	92	3	sβinti	sβinti	NOUN
ejpam-4732	92	4	(	(	PUNCT
ejpam-4732	92	5	a	a	X
ejpam-4732	92	6	)	)	PUNCT
ejpam-4732	92	7	=	=	SYM
ejpam-4732	92	8	a	a	DET
ejpam-4732	92	9	∩	∩	ADJ
ejpam-4732	92	10	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4732	92	11	)	)	PUNCT
ejpam-4732	92	12	)	)	PUNCT
ejpam-4732	92	13	)	)	PUNCT
ejpam-4732	92	14	.	.	PUNCT
ejpam-4732	93	1	lemma	lemma	PROPN
ejpam-4732	93	2	3	3	X
ejpam-4732	93	3	.	.	PUNCT
ejpam-4732	94	1	let	let	VERB
ejpam-4732	94	2	v	v	PART
ejpam-4732	94	3	be	be	AUX
ejpam-4732	94	4	a	a	DET
ejpam-4732	94	5	subset	subset	NOUN
ejpam-4732	94	6	of	of	ADP
ejpam-4732	94	7	an	an	DET
ejpam-4732	94	8	ideal	ideal	ADJ
ejpam-4732	94	9	topological	topological	ADJ
ejpam-4732	94	10	space	space	NOUN
ejpam-4732	94	11	(	(	PUNCT
ejpam-4732	94	12	x	x	X
ejpam-4732	94	13	,	,	PUNCT
ejpam-4732	94	14	τ	τ	PROPN
ejpam-4732	94	15	,	,	PUNCT
ejpam-4732	94	16	i	i	NOUN
ejpam-4732	94	17	)	)	PUNCT
ejpam-4732	94	18	.	.	PUNCT
ejpam-4732	95	1	if	if	SCONJ
ejpam-4732	95	2	v	v	NOUN
ejpam-4732	95	3	is	be	AUX
ejpam-4732	95	4	⋆-open	⋆-open	ADJ
ejpam-4732	95	5	,	,	PUNCT
ejpam-4732	95	6	then	then	ADV
ejpam-4732	95	7	scli	scli	PROPN
ejpam-4732	95	8	(	(	PUNCT
ejpam-4732	95	9	v	v	NOUN
ejpam-4732	95	10	)	)	PUNCT
ejpam-4732	95	11	=	=	SYM
ejpam-4732	95	12	int⋆(cl(v	int⋆(cl(v	PROPN
ejpam-4732	95	13	)	)	PUNCT
ejpam-4732	95	14	)	)	PUNCT
ejpam-4732	95	15	.	.	PUNCT
ejpam-4732	96	1	proof	proof	NOUN
ejpam-4732	96	2	.	.	PUNCT
ejpam-4732	97	1	suppose	suppose	VERB
ejpam-4732	97	2	that	that	SCONJ
ejpam-4732	97	3	v	v	NOUN
ejpam-4732	97	4	is	be	AUX
ejpam-4732	97	5	⋆-open	⋆-open	ADJ
ejpam-4732	97	6	.	.	PUNCT
ejpam-4732	98	1	then	then	ADV
ejpam-4732	98	2	,	,	PUNCT
ejpam-4732	98	3	we	we	PRON
ejpam-4732	98	4	have	have	VERB
ejpam-4732	98	5	v	v	ADP
ejpam-4732	98	6	⊆	⊆	NUM
ejpam-4732	98	7	int⋆(cl(v	int⋆(cl(v	NOUN
ejpam-4732	98	8	)	)	PUNCT
ejpam-4732	98	9	)	)	PUNCT
ejpam-4732	98	10	,	,	PUNCT
ejpam-4732	98	11	by	by	ADP
ejpam-4732	98	12	lemma	lemma	PROPN
ejpam-4732	98	13	2	2	NUM
ejpam-4732	98	14	,	,	PUNCT
ejpam-4732	98	15	scli	scli	NOUN
ejpam-4732	98	16	(	(	PUNCT
ejpam-4732	98	17	v	v	NOUN
ejpam-4732	98	18	)	)	PUNCT
ejpam-4732	98	19	=	=	SYM
ejpam-4732	98	20	v	v	NOUN
ejpam-4732	98	21	∪	∪	VERB
ejpam-4732	98	22	int⋆(cl(v	int⋆(cl(v	PROPN
ejpam-4732	98	23	)	)	PUNCT
ejpam-4732	98	24	)	)	PUNCT
ejpam-4732	99	1	=	=	SYM
ejpam-4732	99	2	int⋆(cl(v	int⋆(cl(v	PROPN
ejpam-4732	99	3	)	)	PUNCT
ejpam-4732	99	4	)	)	PUNCT
ejpam-4732	99	5	.	.	PUNCT
ejpam-4732	100	1	lemma	lemma	PROPN
ejpam-4732	100	2	4	4	X
ejpam-4732	100	3	.	.	PUNCT
ejpam-4732	100	4	let	let	VERB
ejpam-4732	100	5	a	a	DET
ejpam-4732	100	6	be	be	AUX
ejpam-4732	100	7	a	a	DET
ejpam-4732	100	8	subset	subset	NOUN
ejpam-4732	100	9	of	of	ADP
ejpam-4732	100	10	an	an	DET
ejpam-4732	100	11	ideal	ideal	ADJ
ejpam-4732	100	12	topological	topological	ADJ
ejpam-4732	100	13	space	space	NOUN
ejpam-4732	100	14	(	(	PUNCT
ejpam-4732	100	15	x	x	X
ejpam-4732	100	16	,	,	PUNCT
ejpam-4732	100	17	τ	τ	PROPN
ejpam-4732	100	18	,	,	PUNCT
ejpam-4732	100	19	i	i	PROPN
ejpam-4732	100	20	)	)	PUNCT
ejpam-4732	100	21	and	and	CCONJ
ejpam-4732	100	22	x	x	PUNCT
ejpam-4732	100	23	∈	∈	PROPN
ejpam-4732	100	24	x.	x.	NOUN
ejpam-4732	100	25	then	then	ADV
ejpam-4732	100	26	,	,	PUNCT
ejpam-4732	100	27	x	x	PUNCT
ejpam-4732	100	28	∈	∈	NOUN
ejpam-4732	100	29	sβcli	sβcli	NOUN
ejpam-4732	100	30	(	(	PUNCT
ejpam-4732	100	31	a	a	X
ejpam-4732	100	32	)	)	PUNCT
ejpam-4732	100	33	if	if	SCONJ
ejpam-4732	101	1	and	and	CCONJ
ejpam-4732	101	2	only	only	ADV
ejpam-4732	101	3	if	if	SCONJ
ejpam-4732	101	4	u	u	PROPN
ejpam-4732	101	5	∩a	∩a	PROPN
ejpam-4732	101	6	̸=	̸=	PROPN
ejpam-4732	101	7	∅	∅	NOUN
ejpam-4732	101	8	for	for	ADP
ejpam-4732	101	9	every	every	DET
ejpam-4732	101	10	strong	strong	ADJ
ejpam-4732	101	11	β	β	X
ejpam-4732	101	12	-	-	ADJ
ejpam-4732	101	13	i	i	PRON
ejpam-4732	101	14	-open	-open	VERB
ejpam-4732	101	15	set	set	VERB
ejpam-4732	101	16	u	u	NOUN
ejpam-4732	101	17	containing	contain	VERB
ejpam-4732	101	18	x.	x.	NOUN
ejpam-4732	101	19	proof	proof	NOUN
ejpam-4732	101	20	.	.	PUNCT
ejpam-4732	102	1	let	let	VERB
ejpam-4732	102	2	x	x	X
ejpam-4732	102	3	∈	∈	PROPN
ejpam-4732	102	4	sβcli	sβcli	NOUN
ejpam-4732	102	5	(	(	PUNCT
ejpam-4732	102	6	a	a	NOUN
ejpam-4732	102	7	)	)	PUNCT
ejpam-4732	102	8	.	.	PUNCT
ejpam-4732	103	1	suppose	suppose	VERB
ejpam-4732	103	2	that	that	SCONJ
ejpam-4732	103	3	u	u	PRON
ejpam-4732	103	4	∩a	∩a	NOUN
ejpam-4732	103	5	=	=	NOUN
ejpam-4732	103	6	∅	∅	NOUN
ejpam-4732	103	7	for	for	ADP
ejpam-4732	103	8	some	some	DET
ejpam-4732	103	9	strong	strong	ADJ
ejpam-4732	103	10	β	β	X
ejpam-4732	103	11	-	-	ADJ
ejpam-4732	103	12	i	i	PRON
ejpam-4732	103	13	-open	-open	VERB
ejpam-4732	103	14	set	set	VERB
ejpam-4732	103	15	u	u	NOUN
ejpam-4732	103	16	of	of	ADP
ejpam-4732	103	17	x	x	SYM
ejpam-4732	103	18	containing	contain	VERB
ejpam-4732	103	19	x.	x.	NOUN
ejpam-4732	103	20	then	then	ADV
ejpam-4732	103	21	,	,	PUNCT
ejpam-4732	103	22	a	a	DET
ejpam-4732	103	23	⊆	⊆	NUM
ejpam-4732	103	24	x−u	x−u	X
ejpam-4732	103	25	and	and	CCONJ
ejpam-4732	103	26	x−u	x−u	PROPN
ejpam-4732	103	27	is	be	AUX
ejpam-4732	103	28	strong	strong	ADJ
ejpam-4732	103	29	β	β	NOUN
ejpam-4732	103	30	-	-	PUNCT
ejpam-4732	103	31	i	i	PRON
ejpam-4732	103	32	-closed	-closed	ADJ
ejpam-4732	103	33	.	.	PUNCT
ejpam-4732	104	1	since	since	SCONJ
ejpam-4732	104	2	x	x	PROPN
ejpam-4732	104	3	∈	∈	PROPN
ejpam-4732	104	4	sβcli	sβcli	NOUN
ejpam-4732	104	5	(	(	PUNCT
ejpam-4732	104	6	a	a	NOUN
ejpam-4732	104	7	)	)	PUNCT
ejpam-4732	104	8	,	,	PUNCT
ejpam-4732	104	9	we	we	PRON
ejpam-4732	104	10	have	have	VERB
ejpam-4732	104	11	x	x	PART
ejpam-4732	104	12	∈	∈	PROPN
ejpam-4732	104	13	sβcli	sβcli	NOUN
ejpam-4732	104	14	(	(	PUNCT
ejpam-4732	104	15	x	x	SYM
ejpam-4732	104	16	−	−	PROPN
ejpam-4732	104	17	u	u	NOUN
ejpam-4732	104	18	)	)	PUNCT
ejpam-4732	104	19	=	=	PUNCT
ejpam-4732	105	1	x	x	PUNCT
ejpam-4732	105	2	−	−	NOUN
ejpam-4732	105	3	u	u	NOUN
ejpam-4732	105	4	;	;	PUNCT
ejpam-4732	105	5	hence	hence	ADV
ejpam-4732	105	6	x	x	PUNCT
ejpam-4732	105	7	̸∈	̸∈	PROPN
ejpam-4732	105	8	u	u	PROPN
ejpam-4732	105	9	,	,	PUNCT
ejpam-4732	105	10	which	which	PRON
ejpam-4732	105	11	is	be	AUX
ejpam-4732	105	12	a	a	DET
ejpam-4732	105	13	contradiction	contradiction	NOUN
ejpam-4732	105	14	that	that	SCONJ
ejpam-4732	105	15	x	x	PUNCT
ejpam-4732	105	16	∈	∈	PROPN
ejpam-4732	105	17	u	u	NOUN
ejpam-4732	105	18	.	.	PUNCT
ejpam-4732	106	1	thus	thus	ADV
ejpam-4732	106	2	,	,	PUNCT
ejpam-4732	106	3	u	u	PROPN
ejpam-4732	106	4	∩a	∩a	PROPN
ejpam-4732	106	5	̸=	̸=	PROPN
ejpam-4732	106	6	∅	∅	NOUN
ejpam-4732	106	7	for	for	ADP
ejpam-4732	106	8	every	every	DET
ejpam-4732	106	9	strong	strong	ADJ
ejpam-4732	106	10	β	β	X
ejpam-4732	106	11	-	-	ADJ
ejpam-4732	106	12	i	i	PRON
ejpam-4732	106	13	-open	-open	VERB
ejpam-4732	106	14	set	set	VERB
ejpam-4732	106	15	u	u	NOUN
ejpam-4732	106	16	containing	contain	VERB
ejpam-4732	106	17	x.	x.	NOUN
ejpam-4732	106	18	conversely	conversely	ADV
ejpam-4732	106	19	,	,	PUNCT
ejpam-4732	106	20	assume	assume	VERB
ejpam-4732	106	21	that	that	SCONJ
ejpam-4732	106	22	u	u	PROPN
ejpam-4732	106	23	∩	∩	NOUN
ejpam-4732	106	24	a	a	DET
ejpam-4732	106	25	̸=	̸=	PROPN
ejpam-4732	106	26	∅	∅	NOUN
ejpam-4732	106	27	for	for	ADP
ejpam-4732	106	28	every	every	DET
ejpam-4732	106	29	strong	strong	ADJ
ejpam-4732	106	30	β	β	X
ejpam-4732	106	31	-	-	ADJ
ejpam-4732	106	32	i	i	PRON
ejpam-4732	106	33	-open	-open	VERB
ejpam-4732	106	34	set	set	VERB
ejpam-4732	106	35	u	u	NOUN
ejpam-4732	106	36	of	of	ADP
ejpam-4732	106	37	x	x	SYM
ejpam-4732	106	38	containing	contain	VERB
ejpam-4732	106	39	x.	x.	NOUN
ejpam-4732	106	40	we	we	PRON
ejpam-4732	106	41	shall	shall	AUX
ejpam-4732	106	42	show	show	VERB
ejpam-4732	106	43	that	that	SCONJ
ejpam-4732	106	44	x	x	PUNCT
ejpam-4732	106	45	∈	∈	PROPN
ejpam-4732	106	46	sβcli	sβcli	NOUN
ejpam-4732	106	47	(	(	PUNCT
ejpam-4732	106	48	a	a	NOUN
ejpam-4732	106	49	)	)	PUNCT
ejpam-4732	106	50	.	.	PUNCT
ejpam-4732	107	1	suppose	suppose	VERB
ejpam-4732	107	2	that	that	SCONJ
ejpam-4732	107	3	x	x	PROPN
ejpam-4732	107	4	̸∈	̸∈	PROPN
ejpam-4732	107	5	sβcli	sβcli	NOUN
ejpam-4732	107	6	(	(	PUNCT
ejpam-4732	107	7	a	a	NOUN
ejpam-4732	107	8	)	)	PUNCT
ejpam-4732	107	9	.	.	PUNCT
ejpam-4732	108	1	then	then	ADV
ejpam-4732	108	2	,	,	PUNCT
ejpam-4732	108	3	there	there	PRON
ejpam-4732	108	4	exists	exist	VERB
ejpam-4732	108	5	a	a	DET
ejpam-4732	108	6	strong	strong	ADJ
ejpam-4732	108	7	β	β	NOUN
ejpam-4732	108	8	-	-	PUNCT
ejpam-4732	108	9	i	i	PRON
ejpam-4732	108	10	-closed	-close	VERB
ejpam-4732	108	11	set	set	VERB
ejpam-4732	108	12	f	f	PROPN
ejpam-4732	108	13	such	such	ADJ
ejpam-4732	108	14	that	that	SCONJ
ejpam-4732	108	15	a	a	DET
ejpam-4732	108	16	⊆	⊆	NUM
ejpam-4732	108	17	f	f	PROPN
ejpam-4732	108	18	and	and	CCONJ
ejpam-4732	108	19	x	x	PROPN
ejpam-4732	108	20	̸∈	̸∈	PROPN
ejpam-4732	108	21	f	f	PROPN
ejpam-4732	108	22	.	.	PUNCT
ejpam-4732	109	1	thus	thus	ADV
ejpam-4732	109	2	,	,	PUNCT
ejpam-4732	109	3	x	x	PUNCT
ejpam-4732	109	4	−	−	PROPN
ejpam-4732	109	5	f	f	PROPN
ejpam-4732	109	6	is	be	AUX
ejpam-4732	109	7	a	a	DET
ejpam-4732	109	8	strong	strong	ADJ
ejpam-4732	109	9	β	β	NOUN
ejpam-4732	109	10	-	-	ADJ
ejpam-4732	109	11	i	i	PRON
ejpam-4732	109	12	-open	-open	NOUN
ejpam-4732	109	13	set	set	VERB
ejpam-4732	109	14	containing	contain	VERB
ejpam-4732	109	15	x	x	PUNCT
ejpam-4732	109	16	such	such	ADJ
ejpam-4732	109	17	that	that	SCONJ
ejpam-4732	109	18	(	(	PUNCT
ejpam-4732	109	19	x	x	SYM
ejpam-4732	109	20	−	−	PROPN
ejpam-4732	109	21	f	f	PROPN
ejpam-4732	109	22	)	)	PUNCT
ejpam-4732	109	23	∩	∩	NOUN
ejpam-4732	109	24	a	a	PRON
ejpam-4732	109	25	=	=	PUNCT
ejpam-4732	109	26	∅.	∅.	ADP
ejpam-4732	109	27	this	this	PRON
ejpam-4732	109	28	a	a	DET
ejpam-4732	109	29	contradiction	contradiction	NOUN
ejpam-4732	109	30	to	to	ADP
ejpam-4732	109	31	u	u	NOUN
ejpam-4732	109	32	∩	∩	NOUN
ejpam-4732	109	33	a	a	DET
ejpam-4732	109	34	̸=	̸=	PROPN
ejpam-4732	109	35	∅	∅	NOUN
ejpam-4732	109	36	;	;	PUNCT
ejpam-4732	109	37	hence	hence	ADV
ejpam-4732	109	38	x	x	PART
ejpam-4732	109	39	∈	∈	NOUN
ejpam-4732	109	40	sβcli	sβcli	NOUN
ejpam-4732	109	41	(	(	PUNCT
ejpam-4732	109	42	a	a	NOUN
ejpam-4732	109	43	)	)	PUNCT
ejpam-4732	109	44	.	.	PUNCT
ejpam-4732	110	1	c.	c.	PROPN
ejpam-4732	110	2	boonpok	boonpok	PROPN
ejpam-4732	110	3	,	,	PUNCT
ejpam-4732	110	4	p.	p.	NOUN
ejpam-4732	110	5	pue	pue	NOUN
ejpam-4732	110	6	-	-	PUNCT
ejpam-4732	110	7	on	on	ADP
ejpam-4732	110	8	/	/	SYM
ejpam-4732	110	9	eur	eur	NOUN
ejpam-4732	110	10	.	.	PUNCT
ejpam-4732	111	1	j.	j.	PROPN
ejpam-4732	111	2	pure	pure	PROPN
ejpam-4732	111	3	appl	appl	PROPN
ejpam-4732	111	4	.	.	PROPN
ejpam-4732	111	5	math	math	PROPN
ejpam-4732	111	6	,	,	PUNCT
ejpam-4732	111	7	16	16	NUM
ejpam-4732	111	8	(	(	PUNCT
ejpam-4732	111	9	3	3	NUM
ejpam-4732	111	10	)	)	PUNCT
ejpam-4732	111	11	(	(	PUNCT
ejpam-4732	111	12	2023	2023	NUM
ejpam-4732	111	13	)	)	PUNCT
ejpam-4732	111	14	,	,	PUNCT
ejpam-4732	111	15	1634	1634	NUM
ejpam-4732	111	16	-	-	SYM
ejpam-4732	111	17	1646	1646	NUM
ejpam-4732	111	18	1637	1637	NUM
ejpam-4732	111	19	lemma	lemma	PROPN
ejpam-4732	111	20	5	5	NUM
ejpam-4732	111	21	.	.	PUNCT
ejpam-4732	112	1	for	for	ADP
ejpam-4732	112	2	a	a	DET
ejpam-4732	112	3	subset	subset	NOUN
ejpam-4732	112	4	a	a	PRON
ejpam-4732	112	5	of	of	ADP
ejpam-4732	112	6	an	an	DET
ejpam-4732	112	7	ideal	ideal	ADJ
ejpam-4732	112	8	topological	topological	ADJ
ejpam-4732	112	9	space	space	NOUN
ejpam-4732	112	10	(	(	PUNCT
ejpam-4732	112	11	x	x	X
ejpam-4732	112	12	,	,	PUNCT
ejpam-4732	112	13	τ	τ	PROPN
ejpam-4732	112	14	,	,	PUNCT
ejpam-4732	112	15	i	i	NOUN
ejpam-4732	112	16	)	)	PUNCT
ejpam-4732	112	17	,	,	PUNCT
ejpam-4732	112	18	the	the	DET
ejpam-4732	112	19	following	follow	VERB
ejpam-4732	112	20	properties	property	NOUN
ejpam-4732	112	21	are	be	AUX
ejpam-4732	112	22	hold	hold	ADJ
ejpam-4732	112	23	:	:	PUNCT
ejpam-4732	112	24	(	(	PUNCT
ejpam-4732	112	25	1	1	X
ejpam-4732	112	26	)	)	PUNCT
ejpam-4732	112	27	x	x	SYM
ejpam-4732	113	1	−	−	NOUN
ejpam-4732	113	2	sβcli	sβcli	NOUN
ejpam-4732	113	3	(	(	PUNCT
ejpam-4732	113	4	a	a	X
ejpam-4732	113	5	)	)	PUNCT
ejpam-4732	113	6	=	=	NOUN
ejpam-4732	113	7	sβinti	sβinti	X
ejpam-4732	113	8	(	(	PUNCT
ejpam-4732	113	9	x	x	NOUN
ejpam-4732	113	10	−a	−a	NOUN
ejpam-4732	113	11	)	)	PUNCT
ejpam-4732	113	12	.	.	PUNCT
ejpam-4732	114	1	(	(	PUNCT
ejpam-4732	114	2	2	2	X
ejpam-4732	114	3	)	)	PUNCT
ejpam-4732	114	4	x	x	SYM
ejpam-4732	114	5	−	−	PROPN
ejpam-4732	114	6	sβinti	sβinti	NOUN
ejpam-4732	114	7	(	(	PUNCT
ejpam-4732	114	8	a	a	X
ejpam-4732	114	9	)	)	PUNCT
ejpam-4732	114	10	=	=	SYM
ejpam-4732	114	11	sβcli	sβcli	NOUN
ejpam-4732	114	12	(	(	PUNCT
ejpam-4732	114	13	x	x	NOUN
ejpam-4732	114	14	−a	−a	NOUN
ejpam-4732	114	15	)	)	PUNCT
ejpam-4732	114	16	.	.	PUNCT
ejpam-4732	115	1	proof	proof	NOUN
ejpam-4732	115	2	.	.	PUNCT
ejpam-4732	116	1	(	(	PUNCT
ejpam-4732	116	2	1	1	X
ejpam-4732	116	3	)	)	PUNCT
ejpam-4732	116	4	let	let	VERB
ejpam-4732	116	5	x	x	SYM
ejpam-4732	116	6	∈	∈	PROPN
ejpam-4732	116	7	x	x	X
ejpam-4732	116	8	−	−	X
ejpam-4732	116	9	sβcli	sβcli	NOUN
ejpam-4732	116	10	(	(	PUNCT
ejpam-4732	116	11	a	a	NOUN
ejpam-4732	116	12	)	)	PUNCT
ejpam-4732	116	13	.	.	PUNCT
ejpam-4732	117	1	then	then	ADV
ejpam-4732	117	2	,	,	PUNCT
ejpam-4732	117	3	x	x	PROPN
ejpam-4732	117	4	̸∈	̸∈	PROPN
ejpam-4732	117	5	sβcli	sβcli	NOUN
ejpam-4732	117	6	(	(	PUNCT
ejpam-4732	117	7	a	a	NOUN
ejpam-4732	117	8	)	)	PUNCT
ejpam-4732	117	9	and	and	CCONJ
ejpam-4732	117	10	there	there	PRON
ejpam-4732	117	11	exists	exist	VERB
ejpam-4732	117	12	a	a	DET
ejpam-4732	117	13	strong	strong	ADJ
ejpam-4732	117	14	β	β	NOUN
ejpam-4732	117	15	-	-	ADJ
ejpam-4732	117	16	i	i	PRON
ejpam-4732	117	17	-open	-open	NOUN
ejpam-4732	117	18	set	set	VERB
ejpam-4732	117	19	v	v	NOUN
ejpam-4732	117	20	of	of	ADP
ejpam-4732	117	21	x	x	PUNCT
ejpam-4732	117	22	containing	contain	VERB
ejpam-4732	117	23	x	x	PUNCT
ejpam-4732	118	1	such	such	ADJ
ejpam-4732	118	2	that	that	PRON
ejpam-4732	118	3	v	v	ADP
ejpam-4732	118	4	∩	∩	NOUN
ejpam-4732	118	5	a	a	DET
ejpam-4732	118	6	=	=	PUNCT
ejpam-4732	118	7	∅.	∅.	NOUN
ejpam-4732	118	8	thus	thus	ADV
ejpam-4732	118	9	,	,	PUNCT
ejpam-4732	118	10	v	v	ADP
ejpam-4732	118	11	⊆	⊆	NUM
ejpam-4732	118	12	x	x	SYM
ejpam-4732	118	13	−	−	NOUN
ejpam-4732	118	14	a	a	PRON
ejpam-4732	118	15	and	and	CCONJ
ejpam-4732	118	16	hence	hence	ADV
ejpam-4732	118	17	x	x	PART
ejpam-4732	118	18	∈	∈	PROPN
ejpam-4732	118	19	sβinti	sβinti	NOUN
ejpam-4732	118	20	(	(	PUNCT
ejpam-4732	118	21	x−a	x−a	PROPN
ejpam-4732	118	22	)	)	PUNCT
ejpam-4732	118	23	.	.	PUNCT
ejpam-4732	119	1	this	this	PRON
ejpam-4732	119	2	shows	show	VERB
ejpam-4732	119	3	that	that	SCONJ
ejpam-4732	119	4	x−sβcli	x−sβcli	PROPN
ejpam-4732	119	5	(	(	PUNCT
ejpam-4732	119	6	a	a	X
ejpam-4732	119	7	)	)	PUNCT
ejpam-4732	119	8	⊆	⊆	NUM
ejpam-4732	119	9	sβinti	sβinti	NOUN
ejpam-4732	119	10	(	(	PUNCT
ejpam-4732	119	11	x−a	x−a	PROPN
ejpam-4732	119	12	)	)	PUNCT
ejpam-4732	119	13	.	.	PUNCT
ejpam-4732	120	1	on	on	ADP
ejpam-4732	120	2	the	the	DET
ejpam-4732	120	3	other	other	ADJ
ejpam-4732	120	4	hand	hand	NOUN
ejpam-4732	120	5	,	,	PUNCT
ejpam-4732	120	6	let	let	VERB
ejpam-4732	120	7	x	x	X
ejpam-4732	120	8	∈	∈	PROPN
ejpam-4732	120	9	sβinti	sβinti	NOUN
ejpam-4732	120	10	(	(	PUNCT
ejpam-4732	120	11	x−a	x−a	PROPN
ejpam-4732	120	12	)	)	PUNCT
ejpam-4732	120	13	.	.	PUNCT
ejpam-4732	121	1	then	then	ADV
ejpam-4732	121	2	,	,	PUNCT
ejpam-4732	121	3	there	there	PRON
ejpam-4732	121	4	exists	exist	VERB
ejpam-4732	121	5	a	a	DET
ejpam-4732	121	6	strong	strong	ADJ
ejpam-4732	121	7	β	β	NOUN
ejpam-4732	121	8	-	-	ADJ
ejpam-4732	121	9	i	i	PRON
ejpam-4732	121	10	-open	-open	NOUN
ejpam-4732	121	11	set	set	VERB
ejpam-4732	121	12	v	v	NOUN
ejpam-4732	121	13	of	of	ADP
ejpam-4732	121	14	x	x	PUNCT
ejpam-4732	121	15	containing	contain	VERB
ejpam-4732	121	16	x	x	PUNCT
ejpam-4732	121	17	such	such	ADJ
ejpam-4732	121	18	that	that	PRON
ejpam-4732	121	19	v	v	ADP
ejpam-4732	121	20	⊆	⊆	NUM
ejpam-4732	121	21	x−a	x−a	NOUN
ejpam-4732	122	1	and	and	CCONJ
ejpam-4732	122	2	so	so	ADV
ejpam-4732	122	3	v	v	ADP
ejpam-4732	122	4	∩a	∩a	NOUN
ejpam-4732	122	5	=	=	PUNCT
ejpam-4732	122	6	∅.	∅.	X
ejpam-4732	122	7	by	by	ADP
ejpam-4732	122	8	lemma	lemma	PROPN
ejpam-4732	122	9	4	4	NUM
ejpam-4732	122	10	,	,	PUNCT
ejpam-4732	122	11	x	x	PROPN
ejpam-4732	122	12	̸∈	̸∈	PROPN
ejpam-4732	122	13	sβcli	sβcli	NOUN
ejpam-4732	122	14	(	(	PUNCT
ejpam-4732	122	15	a	a	NOUN
ejpam-4732	122	16	)	)	PUNCT
ejpam-4732	122	17	;	;	PUNCT
ejpam-4732	122	18	hence	hence	ADV
ejpam-4732	122	19	x	x	X
ejpam-4732	122	20	∈	∈	PROPN
ejpam-4732	122	21	x−sβcli	x−sβcli	PUNCT
ejpam-4732	122	22	(	(	PUNCT
ejpam-4732	122	23	a	a	NOUN
ejpam-4732	122	24	)	)	PUNCT
ejpam-4732	122	25	.	.	PUNCT
ejpam-4732	123	1	thus	thus	ADV
ejpam-4732	123	2	,	,	PUNCT
ejpam-4732	123	3	sβinti	sβinti	NOUN
ejpam-4732	123	4	(	(	PUNCT
ejpam-4732	123	5	x	x	SYM
ejpam-4732	123	6	−a	−a	NOUN
ejpam-4732	123	7	)	)	PUNCT
ejpam-4732	123	8	⊆	⊆	NUM
ejpam-4732	123	9	x	x	SYM
ejpam-4732	123	10	−	−	X
ejpam-4732	123	11	sβcli	sβcli	NOUN
ejpam-4732	123	12	(	(	PUNCT
ejpam-4732	123	13	a	a	NOUN
ejpam-4732	123	14	)	)	PUNCT
ejpam-4732	123	15	and	and	CCONJ
ejpam-4732	123	16	so	so	ADV
ejpam-4732	123	17	x	x	PUNCT
ejpam-4732	123	18	−	−	X
ejpam-4732	123	19	sβcli	sβcli	NOUN
ejpam-4732	123	20	(	(	PUNCT
ejpam-4732	123	21	a	a	X
ejpam-4732	123	22	)	)	PUNCT
ejpam-4732	123	23	=	=	NOUN
ejpam-4732	123	24	sβinti	sβinti	X
ejpam-4732	123	25	(	(	PUNCT
ejpam-4732	123	26	x	x	NOUN
ejpam-4732	123	27	−a	−a	NOUN
ejpam-4732	123	28	)	)	PUNCT
ejpam-4732	123	29	.	.	PUNCT
ejpam-4732	124	1	(	(	PUNCT
ejpam-4732	124	2	2	2	X
ejpam-4732	124	3	)	)	PUNCT
ejpam-4732	124	4	this	this	PRON
ejpam-4732	124	5	follows	follow	VERB
ejpam-4732	124	6	from	from	ADP
ejpam-4732	124	7	(	(	PUNCT
ejpam-4732	124	8	1	1	NUM
ejpam-4732	124	9	)	)	PUNCT
ejpam-4732	124	10	.	.	PUNCT
ejpam-4732	125	1	3	3	X
ejpam-4732	125	2	.	.	X
ejpam-4732	125	3	upper	upper	ADJ
ejpam-4732	125	4	and	and	CCONJ
ejpam-4732	125	5	lower	low	ADJ
ejpam-4732	125	6	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	125	7	multifunctions	multifunction	NOUN
ejpam-4732	125	8	in	in	ADP
ejpam-4732	125	9	this	this	DET
ejpam-4732	125	10	section	section	NOUN
ejpam-4732	125	11	,	,	PUNCT
ejpam-4732	125	12	we	we	PRON
ejpam-4732	125	13	introduce	introduce	VERB
ejpam-4732	125	14	the	the	DET
ejpam-4732	125	15	notions	notion	NOUN
ejpam-4732	125	16	of	of	ADP
ejpam-4732	125	17	upper	upper	ADJ
ejpam-4732	125	18	and	and	CCONJ
ejpam-4732	125	19	lower	low	ADJ
ejpam-4732	125	20	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	125	21	multifunctions	multifunction	NOUN
ejpam-4732	125	22	.	.	PUNCT
ejpam-4732	126	1	moreover	moreover	ADV
ejpam-4732	126	2	,	,	PUNCT
ejpam-4732	126	3	several	several	ADJ
ejpam-4732	126	4	characterizations	characterization	NOUN
ejpam-4732	126	5	of	of	ADP
ejpam-4732	126	6	upper	upper	ADJ
ejpam-4732	126	7	and	and	CCONJ
ejpam-4732	126	8	lower	low	ADJ
ejpam-4732	126	9	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	126	10	multifunctions	multifunction	NOUN
ejpam-4732	126	11	are	be	AUX
ejpam-4732	126	12	discussed	discuss	VERB
ejpam-4732	126	13	.	.	PUNCT
ejpam-4732	127	1	definition	definition	NOUN
ejpam-4732	127	2	1	1	NUM
ejpam-4732	127	3	.	.	PUNCT
ejpam-4732	128	1	a	a	DET
ejpam-4732	128	2	multifunction	multifunction	NOUN
ejpam-4732	128	3	f	f	NOUN
ejpam-4732	128	4	:	:	PUNCT
ejpam-4732	128	5	(	(	PUNCT
ejpam-4732	128	6	x	x	X
ejpam-4732	128	7	,	,	PUNCT
ejpam-4732	128	8	τ	τ	PROPN
ejpam-4732	128	9	,	,	PUNCT
ejpam-4732	128	10	i	i	NOUN
ejpam-4732	128	11	)	)	PUNCT
ejpam-4732	128	12	→	→	PUNCT
ejpam-4732	128	13	(	(	PUNCT
ejpam-4732	128	14	y	y	PROPN
ejpam-4732	128	15	,	,	PUNCT
ejpam-4732	128	16	σ	σ	PROPN
ejpam-4732	128	17	,	,	PUNCT
ejpam-4732	128	18	j	j	PROPN
ejpam-4732	128	19	)	)	PUNCT
ejpam-4732	128	20	is	be	AUX
ejpam-4732	128	21	said	say	VERB
ejpam-4732	128	22	to	to	PART
ejpam-4732	128	23	be	be	AUX
ejpam-4732	128	24	:	:	PUNCT
ejpam-4732	128	25	(	(	PUNCT
ejpam-4732	128	26	1	1	X
ejpam-4732	128	27	)	)	PUNCT
ejpam-4732	128	28	upper	upper	ADJ
ejpam-4732	128	29	sβ(⋆)-continuous	sβ(⋆)-continuous	NOUN
ejpam-4732	128	30	at	at	ADP
ejpam-4732	128	31	a	a	DET
ejpam-4732	128	32	point	point	NOUN
ejpam-4732	128	33	x	x	SYM
ejpam-4732	128	34	∈	∈	NOUN
ejpam-4732	128	35	x	x	INTJ
ejpam-4732	128	36	if	if	SCONJ
ejpam-4732	128	37	,	,	PUNCT
ejpam-4732	128	38	for	for	SCONJ
ejpam-4732	128	39	each	each	DET
ejpam-4732	128	40	⋆-open	⋆-open	ADV
ejpam-4732	128	41	set	set	VERB
ejpam-4732	128	42	v	v	NUM
ejpam-4732	128	43	of	of	ADP
ejpam-4732	128	44	y	y	PROPN
ejpam-4732	128	45	containing	contain	VERB
ejpam-4732	128	46	f	f	PROPN
ejpam-4732	128	47	(	(	PUNCT
ejpam-4732	128	48	x	x	NOUN
ejpam-4732	128	49	)	)	PUNCT
ejpam-4732	128	50	,	,	PUNCT
ejpam-4732	128	51	there	there	PRON
ejpam-4732	128	52	exists	exist	VERB
ejpam-4732	128	53	a	a	DET
ejpam-4732	128	54	strong	strong	ADJ
ejpam-4732	128	55	β	β	NOUN
ejpam-4732	128	56	-	-	ADJ
ejpam-4732	128	57	i	i	PRON
ejpam-4732	128	58	-open	-open	VERB
ejpam-4732	128	59	set	set	VERB
ejpam-4732	128	60	u	u	NOUN
ejpam-4732	128	61	of	of	ADP
ejpam-4732	128	62	x	x	PUNCT
ejpam-4732	128	63	containing	contain	VERB
ejpam-4732	128	64	x	x	PUNCT
ejpam-4732	128	65	such	such	ADJ
ejpam-4732	128	66	that	that	SCONJ
ejpam-4732	128	67	f	f	PROPN
ejpam-4732	128	68	(	(	PUNCT
ejpam-4732	128	69	u	u	NOUN
ejpam-4732	128	70	)	)	PUNCT
ejpam-4732	128	71	⊆	⊆	NUM
ejpam-4732	128	72	v	v	NOUN
ejpam-4732	128	73	;	;	PUNCT
ejpam-4732	128	74	(	(	PUNCT
ejpam-4732	128	75	2	2	X
ejpam-4732	128	76	)	)	PUNCT
ejpam-4732	128	77	lower	low	ADJ
ejpam-4732	128	78	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	128	79	at	at	ADP
ejpam-4732	128	80	a	a	DET
ejpam-4732	128	81	point	point	NOUN
ejpam-4732	128	82	x	x	SYM
ejpam-4732	128	83	∈	∈	NOUN
ejpam-4732	128	84	x	x	INTJ
ejpam-4732	128	85	if	if	SCONJ
ejpam-4732	128	86	,	,	PUNCT
ejpam-4732	128	87	for	for	SCONJ
ejpam-4732	128	88	each	each	DET
ejpam-4732	128	89	⋆-open	⋆-open	ADV
ejpam-4732	128	90	set	set	VERB
ejpam-4732	128	91	v	v	NUM
ejpam-4732	128	92	of	of	ADP
ejpam-4732	128	93	y	y	PRON
ejpam-4732	128	94	such	such	ADJ
ejpam-4732	128	95	that	that	SCONJ
ejpam-4732	128	96	f	f	PROPN
ejpam-4732	128	97	(	(	PUNCT
ejpam-4732	128	98	x	x	NOUN
ejpam-4732	128	99	)	)	PUNCT
ejpam-4732	128	100	∩	∩	NOUN
ejpam-4732	128	101	v	v	ADP
ejpam-4732	128	102	̸=	̸=	PROPN
ejpam-4732	128	103	∅	∅	NOUN
ejpam-4732	128	104	,	,	PUNCT
ejpam-4732	128	105	there	there	PRON
ejpam-4732	128	106	exists	exist	VERB
ejpam-4732	128	107	a	a	DET
ejpam-4732	128	108	strong	strong	ADJ
ejpam-4732	128	109	β	β	NOUN
ejpam-4732	128	110	-	-	ADJ
ejpam-4732	128	111	i	i	PRON
ejpam-4732	128	112	-open	-open	VERB
ejpam-4732	128	113	set	set	VERB
ejpam-4732	128	114	u	u	NOUN
ejpam-4732	128	115	of	of	ADP
ejpam-4732	128	116	x	x	PUNCT
ejpam-4732	128	117	containing	contain	VERB
ejpam-4732	128	118	x	x	PUNCT
ejpam-4732	128	119	such	such	ADJ
ejpam-4732	128	120	that	that	SCONJ
ejpam-4732	128	121	f	f	PROPN
ejpam-4732	128	122	(	(	PUNCT
ejpam-4732	128	123	z	z	NOUN
ejpam-4732	128	124	)	)	PUNCT
ejpam-4732	128	125	∩	∩	NOUN
ejpam-4732	128	126	v	v	ADP
ejpam-4732	128	127	̸=	̸=	PROPN
ejpam-4732	128	128	∅	∅	NOUN
ejpam-4732	128	129	for	for	ADP
ejpam-4732	128	130	every	every	DET
ejpam-4732	128	131	z	z	NOUN
ejpam-4732	128	132	∈	∈	PROPN
ejpam-4732	128	133	u	u	NOUN
ejpam-4732	128	134	;	;	PUNCT
ejpam-4732	128	135	(	(	PUNCT
ejpam-4732	128	136	3	3	X
ejpam-4732	128	137	)	)	PUNCT
ejpam-4732	128	138	upper	upper	ADJ
ejpam-4732	128	139	(	(	PUNCT
ejpam-4732	128	140	resp	resp	NOUN
ejpam-4732	128	141	.	.	PUNCT
ejpam-4732	129	1	lower	low	ADJ
ejpam-4732	129	2	)	)	PUNCT
ejpam-4732	129	3	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	129	4	if	if	SCONJ
ejpam-4732	129	5	f	f	PROPN
ejpam-4732	129	6	has	have	VERB
ejpam-4732	129	7	this	this	DET
ejpam-4732	129	8	property	property	NOUN
ejpam-4732	129	9	at	at	ADP
ejpam-4732	129	10	each	each	DET
ejpam-4732	129	11	point	point	NOUN
ejpam-4732	129	12	of	of	ADP
ejpam-4732	129	13	x.	x.	NOUN
ejpam-4732	129	14	theorem	theorem	VERB
ejpam-4732	129	15	1	1	NUM
ejpam-4732	129	16	.	.	PUNCT
ejpam-4732	130	1	a	a	DET
ejpam-4732	130	2	multifunction	multifunction	NOUN
ejpam-4732	130	3	f	f	NOUN
ejpam-4732	130	4	:	:	PUNCT
ejpam-4732	130	5	(	(	PUNCT
ejpam-4732	130	6	x	x	X
ejpam-4732	130	7	,	,	PUNCT
ejpam-4732	130	8	τ	τ	PROPN
ejpam-4732	130	9	,	,	PUNCT
ejpam-4732	130	10	i	i	NOUN
ejpam-4732	130	11	)	)	PUNCT
ejpam-4732	130	12	→	→	PUNCT
ejpam-4732	130	13	(	(	PUNCT
ejpam-4732	130	14	y	y	PROPN
ejpam-4732	130	15	,	,	PUNCT
ejpam-4732	130	16	σ	σ	PROPN
ejpam-4732	130	17	,	,	PUNCT
ejpam-4732	130	18	j	j	PROPN
ejpam-4732	130	19	)	)	PUNCT
ejpam-4732	130	20	is	be	AUX
ejpam-4732	130	21	upper	upper	ADJ
ejpam-4732	130	22	sβ(⋆)-continuous	sβ(⋆)-continuous	NOUN
ejpam-4732	130	23	at	at	ADP
ejpam-4732	130	24	x	x	X
ejpam-4732	130	25	∈	∈	PROPN
ejpam-4732	130	26	x	x	SYM
ejpam-4732	130	27	if	if	SCONJ
ejpam-4732	131	1	and	and	CCONJ
ejpam-4732	131	2	only	only	ADV
ejpam-4732	131	3	if	if	SCONJ
ejpam-4732	131	4	x	x	SYM
ejpam-4732	131	5	∈	∈	NOUN
ejpam-4732	131	6	sβinti	sβinti	NOUN
ejpam-4732	131	7	(	(	PUNCT
ejpam-4732	131	8	f+(v	f+(v	PROPN
ejpam-4732	131	9	)	)	PUNCT
ejpam-4732	131	10	)	)	PUNCT
ejpam-4732	131	11	for	for	ADP
ejpam-4732	131	12	every	every	DET
ejpam-4732	131	13	⋆-open	⋆-open	NOUN
ejpam-4732	131	14	set	set	VERB
ejpam-4732	131	15	v	v	NOUN
ejpam-4732	131	16	of	of	ADP
ejpam-4732	131	17	y	y	PROPN
ejpam-4732	131	18	containing	contain	VERB
ejpam-4732	131	19	f	f	PROPN
ejpam-4732	131	20	(	(	PUNCT
ejpam-4732	131	21	x	x	NOUN
ejpam-4732	131	22	)	)	PUNCT
ejpam-4732	131	23	.	.	PUNCT
ejpam-4732	132	1	proof	proof	NOUN
ejpam-4732	132	2	.	.	PUNCT
ejpam-4732	133	1	let	let	VERB
ejpam-4732	133	2	v	v	PART
ejpam-4732	133	3	be	be	AUX
ejpam-4732	133	4	any	any	DET
ejpam-4732	133	5	⋆-open	⋆-open	ADJ
ejpam-4732	133	6	set	set	NOUN
ejpam-4732	133	7	of	of	ADP
ejpam-4732	133	8	y	y	PROPN
ejpam-4732	133	9	containing	contain	VERB
ejpam-4732	133	10	f	f	PROPN
ejpam-4732	133	11	(	(	PUNCT
ejpam-4732	133	12	x	x	NOUN
ejpam-4732	133	13	)	)	PUNCT
ejpam-4732	133	14	.	.	PUNCT
ejpam-4732	134	1	then	then	ADV
ejpam-4732	134	2	,	,	PUNCT
ejpam-4732	134	3	there	there	PRON
ejpam-4732	134	4	exists	exist	VERB
ejpam-4732	134	5	a	a	DET
ejpam-4732	134	6	strong	strong	ADJ
ejpam-4732	134	7	β	β	NOUN
ejpam-4732	134	8	-	-	ADJ
ejpam-4732	134	9	i	i	PRON
ejpam-4732	134	10	-open	-open	VERB
ejpam-4732	134	11	set	set	VERB
ejpam-4732	134	12	u	u	NOUN
ejpam-4732	134	13	of	of	ADP
ejpam-4732	134	14	x	x	PUNCT
ejpam-4732	134	15	containing	contain	VERB
ejpam-4732	134	16	x	x	PUNCT
ejpam-4732	134	17	such	such	ADJ
ejpam-4732	134	18	that	that	SCONJ
ejpam-4732	134	19	f	f	PROPN
ejpam-4732	134	20	(	(	PUNCT
ejpam-4732	134	21	u	u	NOUN
ejpam-4732	134	22	)	)	PUNCT
ejpam-4732	134	23	⊆	⊆	NUM
ejpam-4732	134	24	v	v	NOUN
ejpam-4732	134	25	.	.	PUNCT
ejpam-4732	135	1	then	then	ADV
ejpam-4732	135	2	,	,	PUNCT
ejpam-4732	135	3	u	u	NOUN
ejpam-4732	135	4	⊆	⊆	NUM
ejpam-4732	135	5	f+(v	f+(v	NOUN
ejpam-4732	135	6	)	)	PUNCT
ejpam-4732	135	7	)	)	PUNCT
ejpam-4732	135	8	.	.	PUNCT
ejpam-4732	136	1	since	since	SCONJ
ejpam-4732	136	2	u	u	NOUN
ejpam-4732	136	3	is	be	AUX
ejpam-4732	136	4	strong	strong	ADJ
ejpam-4732	136	5	β	β	NOUN
ejpam-4732	136	6	-	-	PUNCT
ejpam-4732	136	7	i	i	PRON
ejpam-4732	136	8	-open	-open	ADJ
ejpam-4732	136	9	,	,	PUNCT
ejpam-4732	136	10	we	we	PRON
ejpam-4732	136	11	have	have	VERB
ejpam-4732	136	12	x	x	X
ejpam-4732	136	13	∈	∈	PROPN
ejpam-4732	136	14	u	u	NOUN
ejpam-4732	136	15	⊆	⊆	NUM
ejpam-4732	136	16	cl⋆(int(cl⋆(u	cl⋆(int(cl⋆(u	PROPN
ejpam-4732	136	17	)	)	PUNCT
ejpam-4732	136	18	)	)	PUNCT
ejpam-4732	136	19	)	)	PUNCT
ejpam-4732	137	1	⊆	⊆	NUM
ejpam-4732	137	2	cl⋆(int(cl⋆(f+(v	cl⋆(int(cl⋆(f+(v	NOUN
ejpam-4732	137	3	)	)	PUNCT
ejpam-4732	137	4	)	)	PUNCT
ejpam-4732	137	5	)	)	PUNCT
ejpam-4732	137	6	)	)	PUNCT
ejpam-4732	137	7	.	.	PUNCT
ejpam-4732	138	1	since	since	SCONJ
ejpam-4732	138	2	x	x	PROPN
ejpam-4732	138	3	∈	∈	PROPN
ejpam-4732	138	4	f+(v	f+(v	NOUN
ejpam-4732	138	5	)	)	PUNCT
ejpam-4732	138	6	and	and	CCONJ
ejpam-4732	138	7	by	by	ADP
ejpam-4732	138	8	lemma	lemma	PROPN
ejpam-4732	138	9	2	2	NUM
ejpam-4732	138	10	,	,	PUNCT
ejpam-4732	138	11	x	x	SYM
ejpam-4732	138	12	∈	∈	NOUN
ejpam-4732	138	13	f+(v	f+(v	NOUN
ejpam-4732	138	14	)	)	PUNCT
ejpam-4732	138	15	∩	∩	PROPN
ejpam-4732	138	16	cl⋆(int(cl⋆(f+(v	cl⋆(int(cl⋆(f+(v	PROPN
ejpam-4732	138	17	)	)	PUNCT
ejpam-4732	138	18	)	)	PUNCT
ejpam-4732	138	19	)	)	PUNCT
ejpam-4732	138	20	)	)	PUNCT
ejpam-4732	139	1	=	=	PRON
ejpam-4732	139	2	sβinti	sβinti	NOUN
ejpam-4732	139	3	(	(	PUNCT
ejpam-4732	139	4	f+(v	f+(v	PROPN
ejpam-4732	139	5	)	)	PUNCT
ejpam-4732	139	6	)	)	PUNCT
ejpam-4732	139	7	.	.	PUNCT
ejpam-4732	140	1	conversely	conversely	ADV
ejpam-4732	140	2	,	,	PUNCT
ejpam-4732	140	3	let	let	VERB
ejpam-4732	140	4	v	v	PART
ejpam-4732	140	5	be	be	AUX
ejpam-4732	140	6	any	any	DET
ejpam-4732	140	7	⋆-open	⋆-open	ADJ
ejpam-4732	140	8	set	set	NOUN
ejpam-4732	140	9	of	of	ADP
ejpam-4732	140	10	y	y	PROPN
ejpam-4732	140	11	containing	contain	VERB
ejpam-4732	140	12	f	f	PROPN
ejpam-4732	140	13	(	(	PUNCT
ejpam-4732	140	14	x	x	NOUN
ejpam-4732	140	15	)	)	PUNCT
ejpam-4732	140	16	.	.	PUNCT
ejpam-4732	141	1	by	by	ADP
ejpam-4732	141	2	(	(	PUNCT
ejpam-4732	141	3	2	2	NUM
ejpam-4732	141	4	)	)	PUNCT
ejpam-4732	141	5	,	,	PUNCT
ejpam-4732	141	6	x	x	PUNCT
ejpam-4732	141	7	∈	∈	PROPN
ejpam-4732	141	8	sβinti	sβinti	NOUN
ejpam-4732	141	9	(	(	PUNCT
ejpam-4732	141	10	f+(v	f+(v	PROPN
ejpam-4732	141	11	)	)	PUNCT
ejpam-4732	141	12	)	)	PUNCT
ejpam-4732	142	1	and	and	CCONJ
ejpam-4732	142	2	so	so	ADV
ejpam-4732	142	3	there	there	PRON
ejpam-4732	142	4	exists	exist	VERB
ejpam-4732	142	5	a	a	DET
ejpam-4732	142	6	strong	strong	ADJ
ejpam-4732	142	7	β	β	NOUN
ejpam-4732	142	8	-	-	ADJ
ejpam-4732	142	9	i	i	PRON
ejpam-4732	142	10	-open	-open	VERB
ejpam-4732	142	11	set	set	VERB
ejpam-4732	142	12	u	u	NOUN
ejpam-4732	142	13	of	of	ADP
ejpam-4732	142	14	x	x	PUNCT
ejpam-4732	142	15	containing	contain	VERB
ejpam-4732	142	16	x	x	PUNCT
ejpam-4732	142	17	such	such	ADJ
ejpam-4732	142	18	that	that	SCONJ
ejpam-4732	142	19	u	u	NOUN
ejpam-4732	142	20	⊆	⊆	NUM
ejpam-4732	142	21	f+(v	f+(v	NOUN
ejpam-4732	142	22	)	)	PUNCT
ejpam-4732	142	23	;	;	PUNCT
ejpam-4732	142	24	hence	hence	ADV
ejpam-4732	142	25	f	f	PROPN
ejpam-4732	142	26	(	(	PUNCT
ejpam-4732	142	27	u	u	NOUN
ejpam-4732	142	28	)	)	PUNCT
ejpam-4732	142	29	⊆	⊆	NUM
ejpam-4732	142	30	v	v	NOUN
ejpam-4732	142	31	.	.	PUNCT
ejpam-4732	143	1	this	this	PRON
ejpam-4732	143	2	shows	show	VERB
ejpam-4732	143	3	that	that	SCONJ
ejpam-4732	143	4	f	f	PROPN
ejpam-4732	143	5	is	be	AUX
ejpam-4732	143	6	upper	upper	ADJ
ejpam-4732	143	7	sβ(⋆)-continuous	sβ(⋆)-continuous	NOUN
ejpam-4732	143	8	at	at	ADP
ejpam-4732	143	9	x.	x.	NOUN
ejpam-4732	143	10	theorem	theorem	NOUN
ejpam-4732	143	11	2	2	NUM
ejpam-4732	143	12	.	.	PUNCT
ejpam-4732	143	13	a	a	DET
ejpam-4732	143	14	multifunction	multifunction	NOUN
ejpam-4732	143	15	f	f	NOUN
ejpam-4732	143	16	:	:	PUNCT
ejpam-4732	143	17	(	(	PUNCT
ejpam-4732	143	18	x	x	X
ejpam-4732	143	19	,	,	PUNCT
ejpam-4732	143	20	τ	τ	PROPN
ejpam-4732	143	21	,	,	PUNCT
ejpam-4732	143	22	i	i	NOUN
ejpam-4732	143	23	)	)	PUNCT
ejpam-4732	143	24	→	→	PUNCT
ejpam-4732	143	25	(	(	PUNCT
ejpam-4732	143	26	y	y	PROPN
ejpam-4732	143	27	,	,	PUNCT
ejpam-4732	143	28	σ	σ	PROPN
ejpam-4732	143	29	,	,	PUNCT
ejpam-4732	143	30	j	j	PROPN
ejpam-4732	143	31	)	)	PUNCT
ejpam-4732	143	32	is	be	AUX
ejpam-4732	143	33	lower	low	ADJ
ejpam-4732	143	34	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	143	35	at	at	ADP
ejpam-4732	143	36	x	x	X
ejpam-4732	143	37	∈	∈	PROPN
ejpam-4732	143	38	x	x	SYM
ejpam-4732	144	1	if	if	SCONJ
ejpam-4732	144	2	and	and	CCONJ
ejpam-4732	144	3	only	only	ADV
ejpam-4732	144	4	if	if	SCONJ
ejpam-4732	144	5	x	x	SYM
ejpam-4732	144	6	∈	∈	NOUN
ejpam-4732	144	7	sβinti	sβinti	NOUN
ejpam-4732	144	8	(	(	PUNCT
ejpam-4732	144	9	f−(v	f−(v	PROPN
ejpam-4732	144	10	)	)	PUNCT
ejpam-4732	144	11	)	)	PUNCT
ejpam-4732	144	12	for	for	ADP
ejpam-4732	144	13	every	every	DET
ejpam-4732	144	14	⋆-open	⋆-open	NOUN
ejpam-4732	144	15	set	set	VERB
ejpam-4732	144	16	v	v	NUM
ejpam-4732	144	17	of	of	ADP
ejpam-4732	144	18	y	y	PRON
ejpam-4732	144	19	such	such	ADJ
ejpam-4732	144	20	that	that	SCONJ
ejpam-4732	144	21	f	f	PROPN
ejpam-4732	144	22	(	(	PUNCT
ejpam-4732	144	23	x)∩v	x)∩v	PROPN
ejpam-4732	144	24	̸=	̸=	PROPN
ejpam-4732	144	25	∅.	∅.	PROPN
ejpam-4732	144	26	c.	c.	PROPN
ejpam-4732	144	27	boonpok	boonpok	PROPN
ejpam-4732	144	28	,	,	PUNCT
ejpam-4732	144	29	p.	p.	NOUN
ejpam-4732	144	30	pue	pue	NOUN
ejpam-4732	144	31	-	-	PUNCT
ejpam-4732	144	32	on	on	ADP
ejpam-4732	144	33	/	/	SYM
ejpam-4732	144	34	eur	eur	NOUN
ejpam-4732	144	35	.	.	PUNCT
ejpam-4732	145	1	j.	j.	PROPN
ejpam-4732	145	2	pure	pure	PROPN
ejpam-4732	145	3	appl	appl	PROPN
ejpam-4732	145	4	.	.	PROPN
ejpam-4732	145	5	math	math	PROPN
ejpam-4732	145	6	,	,	PUNCT
ejpam-4732	145	7	16	16	NUM
ejpam-4732	145	8	(	(	PUNCT
ejpam-4732	145	9	3	3	NUM
ejpam-4732	145	10	)	)	PUNCT
ejpam-4732	145	11	(	(	PUNCT
ejpam-4732	145	12	2023	2023	NUM
ejpam-4732	145	13	)	)	PUNCT
ejpam-4732	145	14	,	,	PUNCT
ejpam-4732	145	15	1634	1634	NUM
ejpam-4732	145	16	-	-	SYM
ejpam-4732	145	17	1646	1646	NUM
ejpam-4732	145	18	1638	1638	NUM
ejpam-4732	145	19	proof	proof	NOUN
ejpam-4732	145	20	.	.	PUNCT
ejpam-4732	146	1	the	the	DET
ejpam-4732	146	2	proof	proof	NOUN
ejpam-4732	146	3	is	be	AUX
ejpam-4732	146	4	similar	similar	ADJ
ejpam-4732	146	5	to	to	ADP
ejpam-4732	146	6	that	that	PRON
ejpam-4732	146	7	of	of	ADP
ejpam-4732	146	8	theorem	theorem	NOUN
ejpam-4732	146	9	1	1	NUM
ejpam-4732	146	10	.	.	PUNCT
ejpam-4732	146	11	definition	definition	NOUN
ejpam-4732	146	12	2	2	NUM
ejpam-4732	146	13	.	.	PUNCT
ejpam-4732	147	1	a	a	DET
ejpam-4732	147	2	function	function	NOUN
ejpam-4732	147	3	f	f	NOUN
ejpam-4732	147	4	:	:	PUNCT
ejpam-4732	147	5	(	(	PUNCT
ejpam-4732	147	6	x	x	X
ejpam-4732	147	7	,	,	PUNCT
ejpam-4732	147	8	τ	τ	PROPN
ejpam-4732	147	9	,	,	PUNCT
ejpam-4732	147	10	i	i	NOUN
ejpam-4732	147	11	)	)	PUNCT
ejpam-4732	147	12	→	→	PUNCT
ejpam-4732	147	13	(	(	PUNCT
ejpam-4732	147	14	y	y	PROPN
ejpam-4732	147	15	,	,	PUNCT
ejpam-4732	147	16	σ	σ	PROPN
ejpam-4732	147	17	,	,	PUNCT
ejpam-4732	147	18	j	j	PROPN
ejpam-4732	147	19	)	)	PUNCT
ejpam-4732	147	20	is	be	AUX
ejpam-4732	147	21	called	call	VERB
ejpam-4732	147	22	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	147	23	at	at	ADP
ejpam-4732	147	24	a	a	DET
ejpam-4732	147	25	point	point	NOUN
ejpam-4732	147	26	x	x	SYM
ejpam-4732	147	27	∈	∈	NOUN
ejpam-4732	147	28	x	x	INTJ
ejpam-4732	147	29	if	if	SCONJ
ejpam-4732	147	30	,	,	PUNCT
ejpam-4732	147	31	for	for	SCONJ
ejpam-4732	147	32	each	each	DET
ejpam-4732	147	33	⋆-open	⋆-open	ADV
ejpam-4732	147	34	set	set	VERB
ejpam-4732	147	35	v	v	NUM
ejpam-4732	147	36	of	of	ADP
ejpam-4732	147	37	y	y	NOUN
ejpam-4732	147	38	containing	contain	VERB
ejpam-4732	147	39	f(x	f(x	PROPN
ejpam-4732	147	40	)	)	PUNCT
ejpam-4732	147	41	,	,	PUNCT
ejpam-4732	147	42	there	there	PRON
ejpam-4732	147	43	exists	exist	VERB
ejpam-4732	147	44	a	a	DET
ejpam-4732	147	45	strong	strong	ADJ
ejpam-4732	147	46	β	β	NOUN
ejpam-4732	147	47	-	-	ADJ
ejpam-4732	147	48	i	i	PRON
ejpam-4732	147	49	-open	-open	VERB
ejpam-4732	147	50	set	set	VERB
ejpam-4732	147	51	u	u	NOUN
ejpam-4732	147	52	of	of	ADP
ejpam-4732	147	53	x	x	PUNCT
ejpam-4732	147	54	containing	contain	VERB
ejpam-4732	147	55	x	x	PUNCT
ejpam-4732	147	56	such	such	ADJ
ejpam-4732	147	57	that	that	DET
ejpam-4732	147	58	f(u	f(u	PROPN
ejpam-4732	147	59	)	)	PUNCT
ejpam-4732	147	60	⊆	⊆	NUM
ejpam-4732	147	61	v	v	NOUN
ejpam-4732	147	62	.	.	PUNCT
ejpam-4732	148	1	a	a	DET
ejpam-4732	148	2	function	function	NOUN
ejpam-4732	148	3	f	f	NOUN
ejpam-4732	148	4	:	:	PUNCT
ejpam-4732	148	5	(	(	PUNCT
ejpam-4732	148	6	x	x	X
ejpam-4732	148	7	,	,	PUNCT
ejpam-4732	148	8	τ	τ	PROPN
ejpam-4732	148	9	,	,	PUNCT
ejpam-4732	148	10	i	i	NOUN
ejpam-4732	148	11	)	)	PUNCT
ejpam-4732	148	12	→	→	PUNCT
ejpam-4732	148	13	(	(	PUNCT
ejpam-4732	148	14	y	y	PROPN
ejpam-4732	148	15	,	,	PUNCT
ejpam-4732	148	16	σ	σ	PROPN
ejpam-4732	148	17	,	,	PUNCT
ejpam-4732	148	18	j	j	PROPN
ejpam-4732	148	19	)	)	PUNCT
ejpam-4732	148	20	is	be	AUX
ejpam-4732	148	21	called	call	VERB
ejpam-4732	148	22	sβ(⋆)-continuous	sβ(⋆)-continuous	PROPN
ejpam-4732	148	23	if	if	SCONJ
ejpam-4732	148	24	f	f	PROPN
ejpam-4732	148	25	has	have	VERB
ejpam-4732	148	26	this	this	DET
ejpam-4732	148	27	property	property	NOUN
ejpam-4732	148	28	at	at	ADP
ejpam-4732	148	29	each	each	DET
ejpam-4732	148	30	point	point	NOUN
ejpam-4732	148	31	of	of	ADP
ejpam-4732	148	32	x.	x.	PROPN
ejpam-4732	148	33	corollary	corollary	NOUN
ejpam-4732	148	34	1	1	NUM
ejpam-4732	148	35	.	.	PUNCT
ejpam-4732	149	1	a	a	DET
ejpam-4732	149	2	function	function	NOUN
ejpam-4732	149	3	f	f	NOUN
ejpam-4732	149	4	:	:	PUNCT
ejpam-4732	149	5	(	(	PUNCT
ejpam-4732	149	6	x	x	X
ejpam-4732	149	7	,	,	PUNCT
ejpam-4732	149	8	τ	τ	PROPN
ejpam-4732	149	9	,	,	PUNCT
ejpam-4732	149	10	i	i	NOUN
ejpam-4732	149	11	)	)	PUNCT
ejpam-4732	149	12	→	→	PUNCT
ejpam-4732	149	13	(	(	PUNCT
ejpam-4732	149	14	y	y	PROPN
ejpam-4732	149	15	,	,	PUNCT
ejpam-4732	149	16	σ	σ	PROPN
ejpam-4732	149	17	,	,	PUNCT
ejpam-4732	149	18	j	j	PROPN
ejpam-4732	149	19	)	)	PUNCT
ejpam-4732	149	20	is	be	AUX
ejpam-4732	149	21	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	149	22	at	at	ADP
ejpam-4732	149	23	x	x	X
ejpam-4732	149	24	∈	∈	PROPN
ejpam-4732	149	25	x	x	SYM
ejpam-4732	150	1	if	if	SCONJ
ejpam-4732	151	1	and	and	CCONJ
ejpam-4732	151	2	only	only	ADV
ejpam-4732	151	3	if	if	SCONJ
ejpam-4732	151	4	x	x	SYM
ejpam-4732	151	5	∈	∈	NOUN
ejpam-4732	151	6	sβinti	sβinti	NOUN
ejpam-4732	151	7	(	(	PUNCT
ejpam-4732	151	8	f−1(v	f−1(v	PROPN
ejpam-4732	151	9	)	)	PUNCT
ejpam-4732	151	10	)	)	PUNCT
ejpam-4732	152	1	for	for	ADP
ejpam-4732	152	2	every	every	DET
ejpam-4732	152	3	⋆-open	⋆-open	NOUN
ejpam-4732	152	4	set	set	VERB
ejpam-4732	152	5	v	v	NOUN
ejpam-4732	152	6	of	of	ADP
ejpam-4732	152	7	y	y	NOUN
ejpam-4732	152	8	containing	contain	VERB
ejpam-4732	152	9	f(x	f(x	PROPN
ejpam-4732	152	10	)	)	PUNCT
ejpam-4732	152	11	.	.	PUNCT
ejpam-4732	153	1	theorem	theorem	NOUN
ejpam-4732	153	2	3	3	NUM
ejpam-4732	153	3	.	.	X
ejpam-4732	153	4	for	for	ADP
ejpam-4732	153	5	a	a	DET
ejpam-4732	153	6	multifunction	multifunction	NOUN
ejpam-4732	154	1	f	f	NOUN
ejpam-4732	154	2	:	:	PUNCT
ejpam-4732	154	3	(	(	PUNCT
ejpam-4732	154	4	x	x	X
ejpam-4732	154	5	,	,	PUNCT
ejpam-4732	154	6	τ	τ	PROPN
ejpam-4732	154	7	,	,	PUNCT
ejpam-4732	154	8	i	i	NOUN
ejpam-4732	154	9	)	)	PUNCT
ejpam-4732	154	10	→	→	PUNCT
ejpam-4732	154	11	(	(	PUNCT
ejpam-4732	154	12	y	y	PROPN
ejpam-4732	154	13	,	,	PUNCT
ejpam-4732	154	14	σ	σ	PROPN
ejpam-4732	154	15	,	,	PUNCT
ejpam-4732	154	16	j	j	PROPN
ejpam-4732	154	17	)	)	PUNCT
ejpam-4732	154	18	,	,	PUNCT
ejpam-4732	154	19	the	the	DET
ejpam-4732	154	20	following	follow	VERB
ejpam-4732	154	21	properties	property	NOUN
ejpam-4732	154	22	are	be	AUX
ejpam-4732	154	23	equivalent	equivalent	ADJ
ejpam-4732	154	24	:	:	PUNCT
ejpam-4732	154	25	(	(	PUNCT
ejpam-4732	154	26	1	1	X
ejpam-4732	154	27	)	)	PUNCT
ejpam-4732	154	28	f	f	PROPN
ejpam-4732	154	29	is	be	AUX
ejpam-4732	154	30	upper	upper	ADJ
ejpam-4732	154	31	sβ(⋆)-continuous	sβ(⋆)-continuous	NOUN
ejpam-4732	154	32	;	;	PUNCT
ejpam-4732	154	33	(	(	PUNCT
ejpam-4732	154	34	2	2	NUM
ejpam-4732	154	35	)	)	PUNCT
ejpam-4732	154	36	f+(v	f+(v	NOUN
ejpam-4732	154	37	)	)	PUNCT
ejpam-4732	155	1	is	be	AUX
ejpam-4732	155	2	strong	strong	ADJ
ejpam-4732	155	3	β	β	NOUN
ejpam-4732	155	4	-	-	PUNCT
ejpam-4732	155	5	i	i	PRON
ejpam-4732	155	6	-open	-open	VERB
ejpam-4732	155	7	in	in	ADP
ejpam-4732	155	8	x	x	PUNCT
ejpam-4732	155	9	for	for	SCONJ
ejpam-4732	155	10	every	every	DET
ejpam-4732	155	11	⋆-open	⋆-open	NOUN
ejpam-4732	155	12	set	set	VERB
ejpam-4732	155	13	v	v	NOUN
ejpam-4732	155	14	of	of	ADP
ejpam-4732	155	15	y	y	PROPN
ejpam-4732	155	16	;	;	PUNCT
ejpam-4732	155	17	(	(	PUNCT
ejpam-4732	155	18	3	3	X
ejpam-4732	155	19	)	)	PUNCT
ejpam-4732	155	20	f−(k	f−(k	PROPN
ejpam-4732	155	21	)	)	PUNCT
ejpam-4732	155	22	is	be	AUX
ejpam-4732	155	23	strong	strong	ADJ
ejpam-4732	155	24	β	β	NOUN
ejpam-4732	155	25	-	-	PUNCT
ejpam-4732	155	26	i	i	PRON
ejpam-4732	155	27	-closed	-close	VERB
ejpam-4732	155	28	in	in	ADP
ejpam-4732	155	29	x	x	PUNCT
ejpam-4732	155	30	for	for	ADP
ejpam-4732	155	31	every	every	DET
ejpam-4732	155	32	⋆-closed	⋆-close	VERB
ejpam-4732	155	33	set	set	NOUN
ejpam-4732	155	34	k	k	PROPN
ejpam-4732	155	35	of	of	ADP
ejpam-4732	155	36	y	y	PROPN
ejpam-4732	155	37	;	;	PUNCT
ejpam-4732	155	38	(	(	PUNCT
ejpam-4732	155	39	4	4	X
ejpam-4732	155	40	)	)	PUNCT
ejpam-4732	155	41	sβcli	sβcli	NOUN
ejpam-4732	155	42	(	(	PUNCT
ejpam-4732	155	43	f−(b	f−(b	PROPN
ejpam-4732	155	44	)	)	PUNCT
ejpam-4732	155	45	)	)	PUNCT
ejpam-4732	156	1	⊆	⊆	NUM
ejpam-4732	156	2	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4732	156	3	)	)	PUNCT
ejpam-4732	156	4	)	)	PUNCT
ejpam-4732	156	5	for	for	ADP
ejpam-4732	156	6	every	every	DET
ejpam-4732	156	7	subset	subset	NOUN
ejpam-4732	156	8	b	b	PROPN
ejpam-4732	156	9	of	of	ADP
ejpam-4732	156	10	y	y	PROPN
ejpam-4732	156	11	;	;	PUNCT
ejpam-4732	156	12	(	(	PUNCT
ejpam-4732	156	13	5	5	X
ejpam-4732	156	14	)	)	PUNCT
ejpam-4732	156	15	int⋆(cl(int⋆(f−(b	int⋆(cl(int⋆(f−(b	PROPN
ejpam-4732	156	16	)	)	PUNCT
ejpam-4732	156	17	)	)	PUNCT
ejpam-4732	156	18	)	)	PUNCT
ejpam-4732	156	19	)	)	PUNCT
ejpam-4732	157	1	⊆	⊆	NUM
ejpam-4732	157	2	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4732	157	3	)	)	PUNCT
ejpam-4732	157	4	)	)	PUNCT
ejpam-4732	157	5	for	for	ADP
ejpam-4732	157	6	every	every	DET
ejpam-4732	157	7	subset	subset	NOUN
ejpam-4732	157	8	b	b	PROPN
ejpam-4732	157	9	of	of	ADP
ejpam-4732	157	10	y	y	PROPN
ejpam-4732	157	11	.	.	PUNCT
ejpam-4732	158	1	proof	proof	NOUN
ejpam-4732	158	2	.	.	PUNCT
ejpam-4732	159	1	(	(	PUNCT
ejpam-4732	159	2	1	1	X
ejpam-4732	159	3	)	)	PUNCT
ejpam-4732	159	4	⇒	⇒	NOUN
ejpam-4732	159	5	(	(	PUNCT
ejpam-4732	159	6	2	2	NUM
ejpam-4732	159	7	):	):	PUNCT
ejpam-4732	159	8	let	let	VERB
ejpam-4732	159	9	v	v	PART
ejpam-4732	159	10	be	be	AUX
ejpam-4732	159	11	any	any	DET
ejpam-4732	159	12	⋆-open	⋆-open	ADJ
ejpam-4732	159	13	set	set	NOUN
ejpam-4732	159	14	of	of	ADP
ejpam-4732	159	15	y	y	PROPN
ejpam-4732	159	16	and	and	CCONJ
ejpam-4732	159	17	x	x	PROPN
ejpam-4732	159	18	∈	∈	PROPN
ejpam-4732	159	19	f+(v	f+(v	NOUN
ejpam-4732	159	20	)	)	PUNCT
ejpam-4732	159	21	.	.	PUNCT
ejpam-4732	160	1	there	there	PRON
ejpam-4732	160	2	exists	exist	VERB
ejpam-4732	160	3	a	a	DET
ejpam-4732	160	4	strong	strong	ADJ
ejpam-4732	160	5	β	β	NOUN
ejpam-4732	160	6	-	-	ADJ
ejpam-4732	160	7	i	i	PRON
ejpam-4732	160	8	-open	-open	VERB
ejpam-4732	160	9	set	set	VERB
ejpam-4732	160	10	u	u	NOUN
ejpam-4732	160	11	of	of	ADP
ejpam-4732	160	12	x	x	PUNCT
ejpam-4732	160	13	containing	contain	VERB
ejpam-4732	160	14	x	x	PUNCT
ejpam-4732	160	15	such	such	ADJ
ejpam-4732	160	16	that	that	SCONJ
ejpam-4732	160	17	f	f	PROPN
ejpam-4732	160	18	(	(	PUNCT
ejpam-4732	160	19	u	u	NOUN
ejpam-4732	160	20	)	)	PUNCT
ejpam-4732	160	21	⊆	⊆	NUM
ejpam-4732	160	22	v	v	NOUN
ejpam-4732	160	23	.	.	PUNCT
ejpam-4732	161	1	thus	thus	ADV
ejpam-4732	161	2	,	,	PUNCT
ejpam-4732	161	3	x	x	PUNCT
ejpam-4732	161	4	∈	∈	PROPN
ejpam-4732	161	5	u	u	NOUN
ejpam-4732	161	6	⊆	⊆	NUM
ejpam-4732	161	7	cl⋆(int(cl⋆(u	cl⋆(int(cl⋆(u	PROPN
ejpam-4732	161	8	)	)	PUNCT
ejpam-4732	161	9	)	)	PUNCT
ejpam-4732	161	10	)	)	PUNCT
ejpam-4732	162	1	⊆	⊆	NUM
ejpam-4732	162	2	cl⋆(int(cl⋆(f+(v	cl⋆(int(cl⋆(f+(v	NOUN
ejpam-4732	162	3	)	)	PUNCT
ejpam-4732	162	4	)	)	PUNCT
ejpam-4732	162	5	)	)	PUNCT
ejpam-4732	162	6	)	)	PUNCT
ejpam-4732	162	7	and	and	CCONJ
ejpam-4732	162	8	hence	hence	ADV
ejpam-4732	162	9	f+(v	f+(v	NOUN
ejpam-4732	162	10	)	)	PUNCT
ejpam-4732	163	1	⊆	⊆	NUM
ejpam-4732	163	2	cl⋆(int(cl⋆(f+(v	cl⋆(int(cl⋆(f+(v	NOUN
ejpam-4732	163	3	)	)	PUNCT
ejpam-4732	163	4	)	)	PUNCT
ejpam-4732	163	5	)	)	PUNCT
ejpam-4732	163	6	)	)	PUNCT
ejpam-4732	163	7	.	.	PUNCT
ejpam-4732	164	1	this	this	PRON
ejpam-4732	164	2	shows	show	VERB
ejpam-4732	164	3	that	that	SCONJ
ejpam-4732	164	4	f+(v	f+(v	NOUN
ejpam-4732	164	5	)	)	PUNCT
ejpam-4732	164	6	is	be	AUX
ejpam-4732	164	7	strong	strong	ADJ
ejpam-4732	164	8	β	β	NOUN
ejpam-4732	164	9	-	-	PUNCT
ejpam-4732	164	10	i	i	PRON
ejpam-4732	164	11	-open	-open	ADJ
ejpam-4732	164	12	in	in	ADP
ejpam-4732	164	13	x.	x.	NOUN
ejpam-4732	164	14	(	(	PUNCT
ejpam-4732	164	15	2	2	NUM
ejpam-4732	164	16	)	)	PUNCT
ejpam-4732	164	17	⇒	⇒	NOUN
ejpam-4732	164	18	(	(	PUNCT
ejpam-4732	164	19	3	3	NUM
ejpam-4732	164	20	):	):	PUNCT
ejpam-4732	164	21	this	this	PRON
ejpam-4732	164	22	follows	follow	VERB
ejpam-4732	164	23	from	from	ADP
ejpam-4732	164	24	the	the	DET
ejpam-4732	164	25	fact	fact	NOUN
ejpam-4732	164	26	that	that	SCONJ
ejpam-4732	164	27	f+(y	f+(y	PROPN
ejpam-4732	164	28	−b	−b	ADV
ejpam-4732	164	29	)	)	PUNCT
ejpam-4732	164	30	=	=	PUNCT
ejpam-4732	165	1	x	x	X
ejpam-4732	165	2	−	−	PROPN
ejpam-4732	165	3	f−(b	f−(b	PROPN
ejpam-4732	165	4	)	)	PUNCT
ejpam-4732	165	5	for	for	ADP
ejpam-4732	165	6	every	every	DET
ejpam-4732	165	7	subset	subset	NOUN
ejpam-4732	165	8	b	b	PROPN
ejpam-4732	165	9	of	of	ADP
ejpam-4732	165	10	y	y	PROPN
ejpam-4732	165	11	.	.	PUNCT
ejpam-4732	166	1	(	(	PUNCT
ejpam-4732	166	2	3	3	X
ejpam-4732	166	3	)	)	PUNCT
ejpam-4732	166	4	⇒	⇒	NOUN
ejpam-4732	166	5	(	(	PUNCT
ejpam-4732	166	6	4	4	NUM
ejpam-4732	166	7	):	):	PUNCT
ejpam-4732	166	8	for	for	ADP
ejpam-4732	166	9	any	any	DET
ejpam-4732	166	10	subset	subset	NOUN
ejpam-4732	166	11	b	b	PROPN
ejpam-4732	166	12	of	of	ADP
ejpam-4732	166	13	y	y	PROPN
ejpam-4732	166	14	,	,	PUNCT
ejpam-4732	166	15	cl⋆(b	cl⋆(b	PROPN
ejpam-4732	166	16	)	)	PUNCT
ejpam-4732	166	17	is	be	AUX
ejpam-4732	166	18	⋆-closed	⋆-close	VERB
ejpam-4732	166	19	in	in	ADP
ejpam-4732	166	20	y	y	PROPN
ejpam-4732	166	21	and	and	CCONJ
ejpam-4732	166	22	by	by	ADP
ejpam-4732	166	23	(	(	PUNCT
ejpam-4732	166	24	3	3	NUM
ejpam-4732	166	25	)	)	PUNCT
ejpam-4732	166	26	,	,	PUNCT
ejpam-4732	166	27	we	we	PRON
ejpam-4732	166	28	have	have	VERB
ejpam-4732	166	29	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4732	166	30	)	)	PUNCT
ejpam-4732	166	31	)	)	PUNCT
ejpam-4732	167	1	is	be	AUX
ejpam-4732	167	2	strong	strong	ADJ
ejpam-4732	167	3	β	β	NOUN
ejpam-4732	167	4	-	-	PUNCT
ejpam-4732	167	5	i	i	PRON
ejpam-4732	167	6	-closed	-close	VERB
ejpam-4732	167	7	in	in	ADP
ejpam-4732	167	8	x.	x.	NOUN
ejpam-4732	167	9	thus	thus	ADV
ejpam-4732	167	10	,	,	PUNCT
ejpam-4732	167	11	sβcli	sβcli	X
ejpam-4732	167	12	(	(	PUNCT
ejpam-4732	167	13	f−(b	f−(b	PROPN
ejpam-4732	167	14	)	)	PUNCT
ejpam-4732	167	15	)	)	PUNCT
ejpam-4732	168	1	⊆	⊆	NUM
ejpam-4732	168	2	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4732	168	3	)	)	PUNCT
ejpam-4732	168	4	)	)	PUNCT
ejpam-4732	168	5	.	.	PUNCT
ejpam-4732	169	1	(	(	PUNCT
ejpam-4732	169	2	4	4	X
ejpam-4732	169	3	)	)	PUNCT
ejpam-4732	169	4	⇒	⇒	NOUN
ejpam-4732	169	5	(	(	PUNCT
ejpam-4732	169	6	5	5	NUM
ejpam-4732	169	7	):	):	PUNCT
ejpam-4732	169	8	let	let	VERB
ejpam-4732	169	9	b	b	X
ejpam-4732	169	10	be	be	AUX
ejpam-4732	169	11	any	any	DET
ejpam-4732	169	12	subset	subset	NOUN
ejpam-4732	169	13	of	of	ADP
ejpam-4732	169	14	y	y	PROPN
ejpam-4732	169	15	.	.	PUNCT
ejpam-4732	170	1	by	by	ADP
ejpam-4732	170	2	(	(	PUNCT
ejpam-4732	170	3	4	4	NUM
ejpam-4732	170	4	)	)	PUNCT
ejpam-4732	170	5	and	and	CCONJ
ejpam-4732	170	6	lemma	lemma	PROPN
ejpam-4732	170	7	2	2	NUM
ejpam-4732	170	8	,	,	PUNCT
ejpam-4732	170	9	int⋆(cl(int⋆(f−(b	int⋆(cl(int⋆(f−(b	PROPN
ejpam-4732	170	10	)	)	PUNCT
ejpam-4732	170	11	)	)	PUNCT
ejpam-4732	170	12	)	)	PUNCT
ejpam-4732	170	13	)	)	PUNCT
ejpam-4732	171	1	⊆	⊆	NUM
ejpam-4732	171	2	sβcli	sβcli	NOUN
ejpam-4732	171	3	(	(	PUNCT
ejpam-4732	171	4	f−(b	f−(b	PROPN
ejpam-4732	171	5	)	)	PUNCT
ejpam-4732	171	6	)	)	PUNCT
ejpam-4732	171	7	⊆	⊆	NUM
ejpam-4732	171	8	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4732	171	9	)	)	PUNCT
ejpam-4732	171	10	)	)	PUNCT
ejpam-4732	171	11	.	.	PUNCT
ejpam-4732	172	1	(	(	PUNCT
ejpam-4732	172	2	5	5	X
ejpam-4732	172	3	)	)	PUNCT
ejpam-4732	172	4	⇒	⇒	NOUN
ejpam-4732	172	5	(	(	PUNCT
ejpam-4732	172	6	2	2	NUM
ejpam-4732	172	7	):	):	PUNCT
ejpam-4732	172	8	let	let	VERB
ejpam-4732	172	9	v	v	PART
ejpam-4732	172	10	be	be	AUX
ejpam-4732	172	11	any	any	DET
ejpam-4732	172	12	⋆-open	⋆-open	ADJ
ejpam-4732	172	13	set	set	NOUN
ejpam-4732	172	14	of	of	ADP
ejpam-4732	172	15	y	y	PROPN
ejpam-4732	172	16	.	.	PUNCT
ejpam-4732	173	1	then	then	ADV
ejpam-4732	173	2	,	,	PUNCT
ejpam-4732	173	3	y	y	PROPN
ejpam-4732	173	4	−	−	PROPN
ejpam-4732	173	5	v	v	NOUN
ejpam-4732	173	6	is	be	AUX
ejpam-4732	173	7	⋆-closed	⋆-close	VERB
ejpam-4732	173	8	in	in	ADP
ejpam-4732	173	9	y	y	PROPN
ejpam-4732	173	10	and	and	CCONJ
ejpam-4732	173	11	by	by	ADP
ejpam-4732	173	12	(	(	PUNCT
ejpam-4732	173	13	5	5	NUM
ejpam-4732	173	14	)	)	PUNCT
ejpam-4732	173	15	,	,	PUNCT
ejpam-4732	173	16	x	x	PUNCT
ejpam-4732	173	17	−	−	NOUN
ejpam-4732	173	18	f+(v	f+(v	NOUN
ejpam-4732	173	19	)	)	PUNCT
ejpam-4732	174	1	=	=	PUNCT
ejpam-4732	174	2	f−(y	f−(y	NOUN
ejpam-4732	174	3	−	−	ADP
ejpam-4732	174	4	v	v	NOUN
ejpam-4732	174	5	)	)	PUNCT
ejpam-4732	174	6	⊇	⊇	NOUN
ejpam-4732	174	7	int⋆(cl(int⋆(f−(y	int⋆(cl(int⋆(f−(y	NOUN
ejpam-4732	174	8	−	−	PROPN
ejpam-4732	174	9	v	v	NOUN
ejpam-4732	174	10	)	)	PUNCT
ejpam-4732	174	11	)	)	PUNCT
ejpam-4732	174	12	)	)	PUNCT
ejpam-4732	174	13	)	)	PUNCT
ejpam-4732	175	1	=	=	PRON
ejpam-4732	175	2	int⋆(cl(int⋆(x	int⋆(cl(int⋆(x	VERB
ejpam-4732	175	3	−	−	NOUN
ejpam-4732	175	4	f+(v	f+(v	NOUN
ejpam-4732	175	5	)	)	PUNCT
ejpam-4732	175	6	)	)	PUNCT
ejpam-4732	175	7	)	)	PUNCT
ejpam-4732	175	8	)	)	PUNCT
ejpam-4732	176	1	=	=	PUNCT
ejpam-4732	176	2	x	x	PUNCT
ejpam-4732	177	1	−	−	PROPN
ejpam-4732	177	2	cl⋆(int(cl⋆(f+(v	cl⋆(int(cl⋆(f+(v	PROPN
ejpam-4732	177	3	)	)	PUNCT
ejpam-4732	177	4	)	)	PUNCT
ejpam-4732	177	5	)	)	PUNCT
ejpam-4732	177	6	)	)	PUNCT
ejpam-4732	177	7	.	.	PUNCT
ejpam-4732	178	1	thus	thus	ADV
ejpam-4732	178	2	,	,	PUNCT
ejpam-4732	178	3	f+(v	f+(v	PROPN
ejpam-4732	178	4	)	)	PUNCT
ejpam-4732	178	5	⊆	⊆	NUM
ejpam-4732	178	6	cl⋆(int(cl⋆(f+(v	cl⋆(int(cl⋆(f+(v	NOUN
ejpam-4732	178	7	)	)	PUNCT
ejpam-4732	178	8	)	)	PUNCT
ejpam-4732	178	9	)	)	PUNCT
ejpam-4732	178	10	)	)	PUNCT
ejpam-4732	179	1	and	and	CCONJ
ejpam-4732	179	2	so	so	ADV
ejpam-4732	179	3	f+(v	f+(v	PROPN
ejpam-4732	179	4	)	)	PUNCT
ejpam-4732	179	5	is	be	AUX
ejpam-4732	179	6	strong	strong	ADJ
ejpam-4732	179	7	β	β	NOUN
ejpam-4732	179	8	-	-	PUNCT
ejpam-4732	179	9	i	i	PRON
ejpam-4732	179	10	-open	-open	ADJ
ejpam-4732	179	11	in	in	ADP
ejpam-4732	179	12	x.	x.	NOUN
ejpam-4732	179	13	(	(	PUNCT
ejpam-4732	179	14	2	2	NUM
ejpam-4732	179	15	)	)	PUNCT
ejpam-4732	179	16	⇒	⇒	NOUN
ejpam-4732	179	17	(	(	PUNCT
ejpam-4732	179	18	1	1	NUM
ejpam-4732	179	19	):	):	PUNCT
ejpam-4732	179	20	let	let	VERB
ejpam-4732	179	21	x	x	PUNCT
ejpam-4732	179	22	∈	∈	PROPN
ejpam-4732	179	23	x	x	X
ejpam-4732	179	24	and	and	CCONJ
ejpam-4732	179	25	v	v	AUX
ejpam-4732	179	26	be	be	AUX
ejpam-4732	179	27	any	any	DET
ejpam-4732	179	28	⋆-open	⋆-open	ADJ
ejpam-4732	179	29	set	set	NOUN
ejpam-4732	179	30	of	of	ADP
ejpam-4732	179	31	y	y	PROPN
ejpam-4732	179	32	containing	contain	VERB
ejpam-4732	179	33	f	f	PROPN
ejpam-4732	179	34	(	(	PUNCT
ejpam-4732	179	35	x	x	NOUN
ejpam-4732	179	36	)	)	PUNCT
ejpam-4732	179	37	.	.	PUNCT
ejpam-4732	180	1	by	by	ADP
ejpam-4732	180	2	(	(	PUNCT
ejpam-4732	180	3	2	2	NUM
ejpam-4732	180	4	)	)	PUNCT
ejpam-4732	180	5	,	,	PUNCT
ejpam-4732	180	6	we	we	PRON
ejpam-4732	180	7	have	have	VERB
ejpam-4732	180	8	f+(v	f+(v	NOUN
ejpam-4732	180	9	)	)	PUNCT
ejpam-4732	181	1	is	be	AUX
ejpam-4732	181	2	strong	strong	ADJ
ejpam-4732	181	3	β	β	NOUN
ejpam-4732	181	4	-	-	PUNCT
ejpam-4732	181	5	i	i	PRON
ejpam-4732	181	6	-open	-open	NOUN
ejpam-4732	181	7	in	in	ADP
ejpam-4732	181	8	x.	x.	NOUN
ejpam-4732	181	9	put	put	VERB
ejpam-4732	181	10	u	u	NOUN
ejpam-4732	181	11	=	=	NOUN
ejpam-4732	181	12	f+(v	f+(v	PROPN
ejpam-4732	181	13	)	)	PUNCT
ejpam-4732	181	14	.	.	PUNCT
ejpam-4732	182	1	then	then	ADV
ejpam-4732	182	2	,	,	PUNCT
ejpam-4732	182	3	u	u	NOUN
ejpam-4732	182	4	is	be	AUX
ejpam-4732	182	5	a	a	DET
ejpam-4732	182	6	strong	strong	ADJ
ejpam-4732	182	7	β	β	NOUN
ejpam-4732	182	8	-	-	ADJ
ejpam-4732	182	9	i	i	PRON
ejpam-4732	182	10	-open	-open	ADJ
ejpam-4732	182	11	set	set	NOUN
ejpam-4732	182	12	of	of	ADP
ejpam-4732	182	13	x	x	PUNCT
ejpam-4732	182	14	containing	contain	VERB
ejpam-4732	182	15	x	x	PUNCT
ejpam-4732	182	16	such	such	ADJ
ejpam-4732	182	17	that	that	SCONJ
ejpam-4732	182	18	f	f	PROPN
ejpam-4732	182	19	(	(	PUNCT
ejpam-4732	182	20	u	u	NOUN
ejpam-4732	182	21	)	)	PUNCT
ejpam-4732	182	22	⊆	⊆	NUM
ejpam-4732	182	23	v	v	NOUN
ejpam-4732	182	24	.	.	PUNCT
ejpam-4732	183	1	this	this	PRON
ejpam-4732	183	2	shows	show	VERB
ejpam-4732	183	3	that	that	SCONJ
ejpam-4732	183	4	f	f	PROPN
ejpam-4732	183	5	is	be	AUX
ejpam-4732	183	6	upper	upper	ADJ
ejpam-4732	183	7	sβ(⋆)-continuous	sβ(⋆)-continuous	PROPN
ejpam-4732	183	8	.	.	PUNCT
ejpam-4732	183	9	c.	c.	PROPN
ejpam-4732	183	10	boonpok	boonpok	PROPN
ejpam-4732	183	11	,	,	PUNCT
ejpam-4732	183	12	p.	p.	NOUN
ejpam-4732	183	13	pue	pue	NOUN
ejpam-4732	183	14	-	-	PUNCT
ejpam-4732	183	15	on	on	ADP
ejpam-4732	183	16	/	/	SYM
ejpam-4732	183	17	eur	eur	NOUN
ejpam-4732	183	18	.	.	PUNCT
ejpam-4732	184	1	j.	j.	PROPN
ejpam-4732	184	2	pure	pure	PROPN
ejpam-4732	184	3	appl	appl	PROPN
ejpam-4732	184	4	.	.	PROPN
ejpam-4732	184	5	math	math	PROPN
ejpam-4732	184	6	,	,	PUNCT
ejpam-4732	184	7	16	16	NUM
ejpam-4732	184	8	(	(	PUNCT
ejpam-4732	184	9	3	3	NUM
ejpam-4732	184	10	)	)	PUNCT
ejpam-4732	184	11	(	(	PUNCT
ejpam-4732	184	12	2023	2023	NUM
ejpam-4732	184	13	)	)	PUNCT
ejpam-4732	184	14	,	,	PUNCT
ejpam-4732	184	15	1634	1634	NUM
ejpam-4732	184	16	-	-	SYM
ejpam-4732	184	17	1646	1646	NUM
ejpam-4732	184	18	1639	1639	NUM
ejpam-4732	184	19	theorem	theorem	VERB
ejpam-4732	184	20	4	4	NUM
ejpam-4732	184	21	.	.	X
ejpam-4732	184	22	for	for	ADP
ejpam-4732	184	23	a	a	DET
ejpam-4732	184	24	multifunction	multifunction	NOUN
ejpam-4732	184	25	f	f	NOUN
ejpam-4732	184	26	:	:	PUNCT
ejpam-4732	184	27	(	(	PUNCT
ejpam-4732	184	28	x	x	X
ejpam-4732	184	29	,	,	PUNCT
ejpam-4732	184	30	τ	τ	PROPN
ejpam-4732	184	31	,	,	PUNCT
ejpam-4732	184	32	i	i	NOUN
ejpam-4732	184	33	)	)	PUNCT
ejpam-4732	185	1	→	→	PUNCT
ejpam-4732	185	2	(	(	PUNCT
ejpam-4732	185	3	y	y	PROPN
ejpam-4732	185	4	,	,	PUNCT
ejpam-4732	185	5	σ	σ	PROPN
ejpam-4732	185	6	,	,	PUNCT
ejpam-4732	185	7	j	j	PROPN
ejpam-4732	185	8	)	)	PUNCT
ejpam-4732	185	9	,	,	PUNCT
ejpam-4732	185	10	the	the	DET
ejpam-4732	185	11	following	follow	VERB
ejpam-4732	185	12	properties	property	NOUN
ejpam-4732	185	13	are	be	AUX
ejpam-4732	185	14	equivalent	equivalent	ADJ
ejpam-4732	185	15	:	:	PUNCT
ejpam-4732	185	16	(	(	PUNCT
ejpam-4732	185	17	1	1	X
ejpam-4732	185	18	)	)	PUNCT
ejpam-4732	185	19	f	f	PROPN
ejpam-4732	185	20	is	be	AUX
ejpam-4732	185	21	lower	low	ADJ
ejpam-4732	185	22	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	185	23	;	;	PUNCT
ejpam-4732	185	24	(	(	PUNCT
ejpam-4732	185	25	2	2	X
ejpam-4732	185	26	)	)	PUNCT
ejpam-4732	185	27	f−(v	f−(v	NOUN
ejpam-4732	185	28	)	)	PUNCT
ejpam-4732	185	29	is	be	AUX
ejpam-4732	185	30	strong	strong	ADJ
ejpam-4732	185	31	β	β	NOUN
ejpam-4732	185	32	-	-	PUNCT
ejpam-4732	185	33	i	i	PRON
ejpam-4732	185	34	-open	-open	VERB
ejpam-4732	185	35	in	in	ADP
ejpam-4732	185	36	x	x	PUNCT
ejpam-4732	185	37	for	for	SCONJ
ejpam-4732	185	38	every	every	DET
ejpam-4732	185	39	⋆-open	⋆-open	NOUN
ejpam-4732	185	40	set	set	VERB
ejpam-4732	185	41	v	v	NOUN
ejpam-4732	185	42	of	of	ADP
ejpam-4732	185	43	y	y	PROPN
ejpam-4732	185	44	;	;	PUNCT
ejpam-4732	185	45	(	(	PUNCT
ejpam-4732	185	46	3	3	X
ejpam-4732	185	47	)	)	PUNCT
ejpam-4732	185	48	f+(k	f+(k	PROPN
ejpam-4732	185	49	)	)	PUNCT
ejpam-4732	185	50	is	be	AUX
ejpam-4732	185	51	strong	strong	ADJ
ejpam-4732	185	52	β	β	NOUN
ejpam-4732	185	53	-	-	PUNCT
ejpam-4732	185	54	i	i	PRON
ejpam-4732	185	55	-closed	-close	VERB
ejpam-4732	185	56	in	in	ADP
ejpam-4732	185	57	x	x	PUNCT
ejpam-4732	185	58	for	for	ADP
ejpam-4732	185	59	every	every	DET
ejpam-4732	185	60	⋆-closed	⋆-close	VERB
ejpam-4732	185	61	set	set	NOUN
ejpam-4732	185	62	k	k	PROPN
ejpam-4732	185	63	of	of	ADP
ejpam-4732	185	64	y	y	PROPN
ejpam-4732	185	65	;	;	PUNCT
ejpam-4732	185	66	(	(	PUNCT
ejpam-4732	185	67	4	4	X
ejpam-4732	185	68	)	)	PUNCT
ejpam-4732	185	69	sβcli	sβcli	NOUN
ejpam-4732	185	70	(	(	PUNCT
ejpam-4732	185	71	f+(b	f+(b	NOUN
ejpam-4732	185	72	)	)	PUNCT
ejpam-4732	185	73	)	)	PUNCT
ejpam-4732	186	1	⊆	⊆	NUM
ejpam-4732	186	2	f+(cl⋆(b	f+(cl⋆(b	NOUN
ejpam-4732	186	3	)	)	PUNCT
ejpam-4732	186	4	)	)	PUNCT
ejpam-4732	186	5	for	for	ADP
ejpam-4732	186	6	every	every	DET
ejpam-4732	186	7	subset	subset	NOUN
ejpam-4732	186	8	b	b	PROPN
ejpam-4732	186	9	of	of	ADP
ejpam-4732	186	10	y	y	PROPN
ejpam-4732	186	11	;	;	PUNCT
ejpam-4732	186	12	(	(	PUNCT
ejpam-4732	186	13	5	5	X
ejpam-4732	186	14	)	)	PUNCT
ejpam-4732	186	15	int⋆(cl(int⋆(f+(b	int⋆(cl(int⋆(f+(b	NOUN
ejpam-4732	186	16	)	)	PUNCT
ejpam-4732	186	17	)	)	PUNCT
ejpam-4732	186	18	)	)	PUNCT
ejpam-4732	186	19	)	)	PUNCT
ejpam-4732	187	1	⊆	⊆	NUM
ejpam-4732	187	2	f+(cl⋆(b	f+(cl⋆(b	NOUN
ejpam-4732	187	3	)	)	PUNCT
ejpam-4732	187	4	)	)	PUNCT
ejpam-4732	187	5	for	for	ADP
ejpam-4732	187	6	every	every	DET
ejpam-4732	187	7	subset	subset	NOUN
ejpam-4732	187	8	b	b	PROPN
ejpam-4732	187	9	of	of	ADP
ejpam-4732	187	10	y	y	PROPN
ejpam-4732	187	11	;	;	PUNCT
ejpam-4732	187	12	(	(	PUNCT
ejpam-4732	187	13	6	6	X
ejpam-4732	187	14	)	)	PUNCT
ejpam-4732	187	15	f	f	NOUN
ejpam-4732	187	16	(	(	PUNCT
ejpam-4732	187	17	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4732	187	18	)	)	PUNCT
ejpam-4732	187	19	)	)	PUNCT
ejpam-4732	187	20	)	)	PUNCT
ejpam-4732	187	21	)	)	PUNCT
ejpam-4732	188	1	⊆	⊆	NUM
ejpam-4732	188	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4732	188	3	(	(	PUNCT
ejpam-4732	188	4	a	a	NOUN
ejpam-4732	188	5	)	)	PUNCT
ejpam-4732	188	6	)	)	PUNCT
ejpam-4732	188	7	for	for	ADP
ejpam-4732	188	8	every	every	DET
ejpam-4732	188	9	subset	subset	NOUN
ejpam-4732	188	10	a	a	PRON
ejpam-4732	188	11	of	of	ADP
ejpam-4732	188	12	x	x	PRON
ejpam-4732	188	13	;	;	PUNCT
ejpam-4732	188	14	(	(	PUNCT
ejpam-4732	188	15	7	7	X
ejpam-4732	188	16	)	)	PUNCT
ejpam-4732	188	17	f	f	NOUN
ejpam-4732	188	18	(	(	PUNCT
ejpam-4732	188	19	sβcli	sβcli	X
ejpam-4732	188	20	(	(	PUNCT
ejpam-4732	188	21	a	a	NOUN
ejpam-4732	188	22	)	)	PUNCT
ejpam-4732	188	23	)	)	PUNCT
ejpam-4732	189	1	⊆	⊆	NUM
ejpam-4732	189	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4732	189	3	(	(	PUNCT
ejpam-4732	189	4	a	a	NOUN
ejpam-4732	189	5	)	)	PUNCT
ejpam-4732	189	6	)	)	PUNCT
ejpam-4732	189	7	for	for	ADP
ejpam-4732	189	8	every	every	DET
ejpam-4732	189	9	subset	subset	NOUN
ejpam-4732	189	10	a	a	PRON
ejpam-4732	189	11	of	of	ADP
ejpam-4732	189	12	x.	x.	NOUN
ejpam-4732	189	13	proof	proof	NOUN
ejpam-4732	189	14	.	.	PUNCT
ejpam-4732	190	1	it	it	PRON
ejpam-4732	190	2	is	be	AUX
ejpam-4732	190	3	shown	show	VERB
ejpam-4732	190	4	similarly	similarly	ADV
ejpam-4732	190	5	to	to	ADP
ejpam-4732	190	6	the	the	DET
ejpam-4732	190	7	proof	proof	NOUN
ejpam-4732	190	8	of	of	ADP
ejpam-4732	190	9	theorem	theorem	NOUN
ejpam-4732	190	10	3	3	NUM
ejpam-4732	190	11	that	that	SCONJ
ejpam-4732	190	12	the	the	DET
ejpam-4732	190	13	statements	statement	NOUN
ejpam-4732	190	14	(	(	PUNCT
ejpam-4732	190	15	1	1	NUM
ejpam-4732	190	16	)	)	PUNCT
ejpam-4732	190	17	,	,	PUNCT
ejpam-4732	190	18	(	(	PUNCT
ejpam-4732	190	19	2	2	NUM
ejpam-4732	190	20	)	)	PUNCT
ejpam-4732	190	21	,	,	PUNCT
ejpam-4732	190	22	(	(	PUNCT
ejpam-4732	190	23	3	3	NUM
ejpam-4732	190	24	)	)	PUNCT
ejpam-4732	190	25	,	,	PUNCT
ejpam-4732	190	26	(	(	PUNCT
ejpam-4732	190	27	4	4	NUM
ejpam-4732	190	28	)	)	PUNCT
ejpam-4732	190	29	and	and	CCONJ
ejpam-4732	190	30	(	(	PUNCT
ejpam-4732	190	31	5	5	X
ejpam-4732	190	32	)	)	PUNCT
ejpam-4732	190	33	are	be	AUX
ejpam-4732	190	34	equivalent	equivalent	ADJ
ejpam-4732	190	35	.	.	PUNCT
ejpam-4732	191	1	we	we	PRON
ejpam-4732	191	2	shall	shall	AUX
ejpam-4732	191	3	prove	prove	VERB
ejpam-4732	191	4	only	only	ADV
ejpam-4732	191	5	the	the	DET
ejpam-4732	191	6	following	follow	VERB
ejpam-4732	191	7	implications	implication	NOUN
ejpam-4732	191	8	.	.	PUNCT
ejpam-4732	192	1	(	(	PUNCT
ejpam-4732	192	2	5	5	X
ejpam-4732	192	3	)	)	PUNCT
ejpam-4732	192	4	⇒	⇒	NOUN
ejpam-4732	192	5	(	(	PUNCT
ejpam-4732	192	6	6	6	NUM
ejpam-4732	192	7	):	):	PUNCT
ejpam-4732	192	8	let	let	VERB
ejpam-4732	192	9	a	a	DET
ejpam-4732	192	10	be	be	AUX
ejpam-4732	192	11	any	any	DET
ejpam-4732	192	12	subset	subset	NOUN
ejpam-4732	192	13	of	of	ADP
ejpam-4732	192	14	x.	x.	NOUN
ejpam-4732	192	15	by	by	ADP
ejpam-4732	192	16	(	(	PUNCT
ejpam-4732	192	17	5	5	NUM
ejpam-4732	192	18	)	)	PUNCT
ejpam-4732	192	19	,	,	PUNCT
ejpam-4732	192	20	we	we	PRON
ejpam-4732	192	21	have	have	VERB
ejpam-4732	192	22	int⋆(cl(int⋆(f+(f	int⋆(cl(int⋆(f+(f	NOUN
ejpam-4732	192	23	(	(	PUNCT
ejpam-4732	192	24	a	a	NOUN
ejpam-4732	192	25	)	)	PUNCT
ejpam-4732	192	26	)	)	PUNCT
ejpam-4732	192	27	)	)	PUNCT
ejpam-4732	192	28	)	)	PUNCT
ejpam-4732	192	29	)	)	PUNCT
ejpam-4732	193	1	⊆	⊆	NUM
ejpam-4732	193	2	f+(cl⋆(f	f+(cl⋆(f	X
ejpam-4732	193	3	(	(	PUNCT
ejpam-4732	193	4	a	a	NOUN
ejpam-4732	193	5	)	)	PUNCT
ejpam-4732	193	6	)	)	PUNCT
ejpam-4732	193	7	)	)	PUNCT
ejpam-4732	193	8	and	and	CCONJ
ejpam-4732	193	9	hence	hence	ADV
ejpam-4732	193	10	f	f	PROPN
ejpam-4732	193	11	(	(	PUNCT
ejpam-4732	193	12	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4732	193	13	)	)	PUNCT
ejpam-4732	193	14	)	)	PUNCT
ejpam-4732	193	15	)	)	PUNCT
ejpam-4732	193	16	)	)	PUNCT
ejpam-4732	194	1	⊆	⊆	NUM
ejpam-4732	194	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4732	194	3	(	(	PUNCT
ejpam-4732	194	4	a	a	NOUN
ejpam-4732	194	5	)	)	PUNCT
ejpam-4732	194	6	)	)	PUNCT
ejpam-4732	194	7	.	.	PUNCT
ejpam-4732	195	1	(	(	PUNCT
ejpam-4732	195	2	6	6	X
ejpam-4732	195	3	)	)	PUNCT
ejpam-4732	195	4	⇒	⇒	NOUN
ejpam-4732	195	5	(	(	PUNCT
ejpam-4732	195	6	7	7	NUM
ejpam-4732	195	7	):	):	PUNCT
ejpam-4732	195	8	let	let	VERB
ejpam-4732	195	9	a	a	DET
ejpam-4732	195	10	be	be	AUX
ejpam-4732	195	11	any	any	DET
ejpam-4732	195	12	subset	subset	NOUN
ejpam-4732	195	13	of	of	ADP
ejpam-4732	195	14	x.	x.	NOUN
ejpam-4732	195	15	by	by	ADP
ejpam-4732	195	16	(	(	PUNCT
ejpam-4732	195	17	6	6	NUM
ejpam-4732	195	18	)	)	PUNCT
ejpam-4732	195	19	and	and	CCONJ
ejpam-4732	195	20	lemma	lemma	PROPN
ejpam-4732	195	21	2	2	NUM
ejpam-4732	195	22	,	,	PUNCT
ejpam-4732	195	23	we	we	PRON
ejpam-4732	195	24	have	have	VERB
ejpam-4732	195	25	f	f	X
ejpam-4732	195	26	(	(	PUNCT
ejpam-4732	195	27	sβcli	sβcli	X
ejpam-4732	195	28	(	(	PUNCT
ejpam-4732	195	29	a	a	NOUN
ejpam-4732	195	30	)	)	PUNCT
ejpam-4732	195	31	)	)	PUNCT
ejpam-4732	196	1	=	=	SYM
ejpam-4732	196	2	f	f	PROPN
ejpam-4732	196	3	(	(	PUNCT
ejpam-4732	196	4	a	a	DET
ejpam-4732	196	5	∪	∪	ADJ
ejpam-4732	196	6	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4732	196	7	)	)	PUNCT
ejpam-4732	196	8	)	)	PUNCT
ejpam-4732	196	9	)	)	PUNCT
ejpam-4732	196	10	)	)	PUNCT
ejpam-4732	197	1	=	=	SYM
ejpam-4732	197	2	f	f	X
ejpam-4732	197	3	(	(	PUNCT
ejpam-4732	197	4	a	a	NOUN
ejpam-4732	197	5	)	)	PUNCT
ejpam-4732	197	6	∪	∪	PROPN
ejpam-4732	197	7	f	f	PROPN
ejpam-4732	197	8	(	(	PUNCT
ejpam-4732	197	9	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4732	197	10	)	)	PUNCT
ejpam-4732	197	11	)	)	PUNCT
ejpam-4732	197	12	)	)	PUNCT
ejpam-4732	197	13	)	)	PUNCT
ejpam-4732	198	1	⊆	⊆	NUM
ejpam-4732	198	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4732	198	3	(	(	PUNCT
ejpam-4732	198	4	a	a	NOUN
ejpam-4732	198	5	)	)	PUNCT
ejpam-4732	198	6	)	)	PUNCT
ejpam-4732	198	7	.	.	PUNCT
ejpam-4732	199	1	(	(	PUNCT
ejpam-4732	199	2	7	7	X
ejpam-4732	199	3	)	)	PUNCT
ejpam-4732	199	4	⇒	⇒	NOUN
ejpam-4732	199	5	(	(	PUNCT
ejpam-4732	199	6	3	3	NUM
ejpam-4732	199	7	):	):	PUNCT
ejpam-4732	199	8	let	let	VERB
ejpam-4732	199	9	k	k	PRON
ejpam-4732	199	10	be	be	AUX
ejpam-4732	199	11	any	any	DET
ejpam-4732	199	12	⋆-closed	⋆-close	VERB
ejpam-4732	199	13	set	set	NOUN
ejpam-4732	199	14	of	of	ADP
ejpam-4732	199	15	y	y	PROPN
ejpam-4732	199	16	.	.	PUNCT
ejpam-4732	200	1	by	by	ADP
ejpam-4732	200	2	(	(	PUNCT
ejpam-4732	200	3	7	7	NUM
ejpam-4732	200	4	)	)	PUNCT
ejpam-4732	200	5	,	,	PUNCT
ejpam-4732	200	6	f	f	PROPN
ejpam-4732	200	7	(	(	PUNCT
ejpam-4732	200	8	sβcli	sβcli	X
ejpam-4732	200	9	(	(	PUNCT
ejpam-4732	200	10	f+(k	f+(k	NOUN
ejpam-4732	200	11	)	)	PUNCT
ejpam-4732	200	12	)	)	PUNCT
ejpam-4732	200	13	)	)	PUNCT
ejpam-4732	201	1	⊆	⊆	NUM
ejpam-4732	201	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4732	201	3	(	(	PUNCT
ejpam-4732	201	4	f+(k	f+(k	NOUN
ejpam-4732	201	5	)	)	PUNCT
ejpam-4732	201	6	)	)	PUNCT
ejpam-4732	201	7	)	)	PUNCT
ejpam-4732	202	1	⊆	⊆	X
ejpam-4732	202	2	cl⋆(k	cl⋆(k	SYM
ejpam-4732	202	3	)	)	PUNCT
ejpam-4732	202	4	=	=	PUNCT
ejpam-4732	202	5	k.	k.	PROPN
ejpam-4732	203	1	thus	thus	ADV
ejpam-4732	203	2	,	,	PUNCT
ejpam-4732	203	3	sβcli	sβcli	NOUN
ejpam-4732	203	4	(	(	PUNCT
ejpam-4732	203	5	f+(k	f+(k	NOUN
ejpam-4732	203	6	)	)	PUNCT
ejpam-4732	203	7	)	)	PUNCT
ejpam-4732	204	1	⊆	⊆	NUM
ejpam-4732	204	2	f+(k	f+(k	NUM
ejpam-4732	204	3	)	)	PUNCT
ejpam-4732	204	4	and	and	CCONJ
ejpam-4732	204	5	hence	hence	ADV
ejpam-4732	204	6	f+(k	f+(k	NUM
ejpam-4732	204	7	)	)	PUNCT
ejpam-4732	204	8	is	be	AUX
ejpam-4732	204	9	strong	strong	ADJ
ejpam-4732	204	10	β	β	NOUN
ejpam-4732	204	11	-	-	PUNCT
ejpam-4732	204	12	i	i	PRON
ejpam-4732	204	13	-closed	-close	VERB
ejpam-4732	204	14	in	in	ADP
ejpam-4732	204	15	x.	x.	NOUN
ejpam-4732	204	16	corollary	corollary	NOUN
ejpam-4732	204	17	2	2	PROPN
ejpam-4732	204	18	.	.	PUNCT
ejpam-4732	204	19	for	for	ADP
ejpam-4732	204	20	a	a	DET
ejpam-4732	204	21	function	function	NOUN
ejpam-4732	204	22	f	f	NOUN
ejpam-4732	204	23	:	:	PUNCT
ejpam-4732	204	24	(	(	PUNCT
ejpam-4732	204	25	x	x	X
ejpam-4732	204	26	,	,	PUNCT
ejpam-4732	204	27	τ	τ	PROPN
ejpam-4732	204	28	,	,	PUNCT
ejpam-4732	204	29	i	i	NOUN
ejpam-4732	204	30	)	)	PUNCT
ejpam-4732	204	31	→	→	PUNCT
ejpam-4732	204	32	(	(	PUNCT
ejpam-4732	204	33	y	y	PROPN
ejpam-4732	204	34	,	,	PUNCT
ejpam-4732	204	35	σ	σ	PROPN
ejpam-4732	204	36	,	,	PUNCT
ejpam-4732	204	37	j	j	PROPN
ejpam-4732	204	38	)	)	PUNCT
ejpam-4732	204	39	,	,	PUNCT
ejpam-4732	204	40	the	the	DET
ejpam-4732	204	41	following	follow	VERB
ejpam-4732	204	42	properties	property	NOUN
ejpam-4732	204	43	are	be	AUX
ejpam-4732	204	44	equivalent	equivalent	ADJ
ejpam-4732	204	45	:	:	PUNCT
ejpam-4732	204	46	(	(	PUNCT
ejpam-4732	204	47	1	1	X
ejpam-4732	204	48	)	)	PUNCT
ejpam-4732	204	49	f	f	PROPN
ejpam-4732	204	50	is	be	AUX
ejpam-4732	204	51	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	204	52	;	;	PUNCT
ejpam-4732	204	53	(	(	PUNCT
ejpam-4732	204	54	2	2	X
ejpam-4732	204	55	)	)	PUNCT
ejpam-4732	204	56	f−1(v	f−1(v	NOUN
ejpam-4732	204	57	)	)	PUNCT
ejpam-4732	204	58	is	be	AUX
ejpam-4732	204	59	strong	strong	ADJ
ejpam-4732	204	60	β	β	NOUN
ejpam-4732	204	61	-	-	PUNCT
ejpam-4732	204	62	i	i	PRON
ejpam-4732	204	63	-open	-open	VERB
ejpam-4732	204	64	in	in	ADP
ejpam-4732	204	65	x	x	PUNCT
ejpam-4732	204	66	for	for	SCONJ
ejpam-4732	204	67	every	every	DET
ejpam-4732	204	68	⋆-open	⋆-open	NOUN
ejpam-4732	204	69	set	set	VERB
ejpam-4732	204	70	v	v	NOUN
ejpam-4732	204	71	of	of	ADP
ejpam-4732	204	72	y	y	PROPN
ejpam-4732	204	73	;	;	PUNCT
ejpam-4732	204	74	(	(	PUNCT
ejpam-4732	204	75	3	3	X
ejpam-4732	204	76	)	)	PUNCT
ejpam-4732	204	77	f−1(k	f−1(k	PROPN
ejpam-4732	204	78	)	)	PUNCT
ejpam-4732	204	79	is	be	AUX
ejpam-4732	204	80	strong	strong	ADJ
ejpam-4732	204	81	β	β	NOUN
ejpam-4732	204	82	-	-	PUNCT
ejpam-4732	204	83	i	i	PRON
ejpam-4732	204	84	-closed	-close	VERB
ejpam-4732	204	85	in	in	ADP
ejpam-4732	204	86	x	x	PUNCT
ejpam-4732	204	87	for	for	ADP
ejpam-4732	204	88	every	every	DET
ejpam-4732	204	89	⋆-closed	⋆-close	VERB
ejpam-4732	204	90	set	set	NOUN
ejpam-4732	204	91	k	k	PROPN
ejpam-4732	204	92	of	of	ADP
ejpam-4732	204	93	y	y	PROPN
ejpam-4732	204	94	;	;	PUNCT
ejpam-4732	204	95	(	(	PUNCT
ejpam-4732	204	96	4	4	X
ejpam-4732	204	97	)	)	PUNCT
ejpam-4732	204	98	sβcli	sβcli	NOUN
ejpam-4732	204	99	(	(	PUNCT
ejpam-4732	204	100	f−1(b	f−1(b	PROPN
ejpam-4732	204	101	)	)	PUNCT
ejpam-4732	204	102	)	)	PUNCT
ejpam-4732	205	1	⊆	⊆	NUM
ejpam-4732	205	2	f−1(cl⋆(b	f−1(cl⋆(b	NOUN
ejpam-4732	205	3	)	)	PUNCT
ejpam-4732	205	4	)	)	PUNCT
ejpam-4732	205	5	for	for	ADP
ejpam-4732	205	6	every	every	DET
ejpam-4732	205	7	subset	subset	NOUN
ejpam-4732	205	8	b	b	PROPN
ejpam-4732	205	9	of	of	ADP
ejpam-4732	205	10	y	y	PROPN
ejpam-4732	205	11	;	;	PUNCT
ejpam-4732	205	12	(	(	PUNCT
ejpam-4732	205	13	5	5	X
ejpam-4732	205	14	)	)	PUNCT
ejpam-4732	205	15	int⋆(cl(int⋆(f−1(b	int⋆(cl(int⋆(f−1(b	NOUN
ejpam-4732	205	16	)	)	PUNCT
ejpam-4732	205	17	)	)	PUNCT
ejpam-4732	205	18	)	)	PUNCT
ejpam-4732	205	19	)	)	PUNCT
ejpam-4732	206	1	⊆	⊆	NUM
ejpam-4732	206	2	f−1(cl⋆(b	f−1(cl⋆(b	NOUN
ejpam-4732	206	3	)	)	PUNCT
ejpam-4732	206	4	)	)	PUNCT
ejpam-4732	206	5	for	for	ADP
ejpam-4732	206	6	every	every	DET
ejpam-4732	206	7	subset	subset	NOUN
ejpam-4732	206	8	b	b	PROPN
ejpam-4732	206	9	of	of	ADP
ejpam-4732	206	10	y	y	PROPN
ejpam-4732	206	11	;	;	PUNCT
ejpam-4732	206	12	(	(	PUNCT
ejpam-4732	206	13	6	6	X
ejpam-4732	206	14	)	)	PUNCT
ejpam-4732	206	15	f(int⋆(cl(int⋆(a	f(int⋆(cl(int⋆(a	PROPN
ejpam-4732	206	16	)	)	PUNCT
ejpam-4732	206	17	)	)	PUNCT
ejpam-4732	206	18	)	)	PUNCT
ejpam-4732	206	19	)	)	PUNCT
ejpam-4732	207	1	⊆	⊆	NUM
ejpam-4732	207	2	cl⋆(f(a	cl⋆(f(a	NOUN
ejpam-4732	207	3	)	)	PUNCT
ejpam-4732	207	4	)	)	PUNCT
ejpam-4732	207	5	for	for	ADP
ejpam-4732	207	6	every	every	DET
ejpam-4732	207	7	subset	subset	NOUN
ejpam-4732	207	8	a	a	PRON
ejpam-4732	207	9	of	of	ADP
ejpam-4732	207	10	x	x	PRON
ejpam-4732	207	11	;	;	PUNCT
ejpam-4732	207	12	(	(	PUNCT
ejpam-4732	207	13	7	7	X
ejpam-4732	207	14	)	)	PUNCT
ejpam-4732	207	15	f(sβcli	f(sβcli	NOUN
ejpam-4732	207	16	(	(	PUNCT
ejpam-4732	207	17	a	a	NOUN
ejpam-4732	207	18	)	)	PUNCT
ejpam-4732	207	19	)	)	PUNCT
ejpam-4732	207	20	⊆	⊆	NUM
ejpam-4732	207	21	cl⋆(f(a	cl⋆(f(a	NOUN
ejpam-4732	207	22	)	)	PUNCT
ejpam-4732	207	23	)	)	PUNCT
ejpam-4732	207	24	for	for	ADP
ejpam-4732	207	25	every	every	DET
ejpam-4732	207	26	subset	subset	NOUN
ejpam-4732	207	27	a	a	PRON
ejpam-4732	207	28	of	of	ADP
ejpam-4732	207	29	x.	x.	PROPN
ejpam-4732	207	30	c.	c.	PROPN
ejpam-4732	207	31	boonpok	boonpok	PROPN
ejpam-4732	207	32	,	,	PUNCT
ejpam-4732	207	33	p.	p.	NOUN
ejpam-4732	207	34	pue	pue	NOUN
ejpam-4732	207	35	-	-	PUNCT
ejpam-4732	207	36	on	on	ADP
ejpam-4732	207	37	/	/	SYM
ejpam-4732	207	38	eur	eur	NOUN
ejpam-4732	207	39	.	.	PUNCT
ejpam-4732	208	1	j.	j.	PROPN
ejpam-4732	208	2	pure	pure	PROPN
ejpam-4732	208	3	appl	appl	PROPN
ejpam-4732	208	4	.	.	PROPN
ejpam-4732	208	5	math	math	PROPN
ejpam-4732	208	6	,	,	PUNCT
ejpam-4732	208	7	16	16	NUM
ejpam-4732	208	8	(	(	PUNCT
ejpam-4732	208	9	3	3	NUM
ejpam-4732	208	10	)	)	PUNCT
ejpam-4732	208	11	(	(	PUNCT
ejpam-4732	208	12	2023	2023	NUM
ejpam-4732	208	13	)	)	PUNCT
ejpam-4732	208	14	,	,	PUNCT
ejpam-4732	208	15	1634	1634	NUM
ejpam-4732	208	16	-	-	SYM
ejpam-4732	208	17	1646	1646	NUM
ejpam-4732	208	18	1640	1640	NUM
ejpam-4732	208	19	4	4	NUM
ejpam-4732	208	20	.	.	PUNCT
ejpam-4732	208	21	upper	upper	ADJ
ejpam-4732	208	22	and	and	CCONJ
ejpam-4732	208	23	lower	low	ADJ
ejpam-4732	208	24	almost	almost	ADV
ejpam-4732	208	25	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	208	26	multifunctions	multifunction	NOUN
ejpam-4732	208	27	we	we	PRON
ejpam-4732	208	28	begin	begin	VERB
ejpam-4732	208	29	this	this	DET
ejpam-4732	208	30	section	section	NOUN
ejpam-4732	208	31	by	by	ADP
ejpam-4732	208	32	introducing	introduce	VERB
ejpam-4732	208	33	the	the	DET
ejpam-4732	208	34	notions	notion	NOUN
ejpam-4732	208	35	of	of	ADP
ejpam-4732	208	36	upper	upper	ADJ
ejpam-4732	208	37	and	and	CCONJ
ejpam-4732	208	38	lower	low	ADJ
ejpam-4732	208	39	almost	almost	ADV
ejpam-4732	208	40	sβ(⋆)continuous	sβ(⋆)continuous	ADJ
ejpam-4732	208	41	multifunctions	multifunction	NOUN
ejpam-4732	208	42	.	.	PUNCT
ejpam-4732	209	1	definition	definition	NOUN
ejpam-4732	209	2	3	3	NUM
ejpam-4732	209	3	.	.	PUNCT
ejpam-4732	210	1	a	a	DET
ejpam-4732	210	2	multifunction	multifunction	NOUN
ejpam-4732	210	3	f	f	NOUN
ejpam-4732	210	4	:	:	PUNCT
ejpam-4732	210	5	(	(	PUNCT
ejpam-4732	210	6	x	x	X
ejpam-4732	210	7	,	,	PUNCT
ejpam-4732	210	8	τ	τ	PROPN
ejpam-4732	210	9	,	,	PUNCT
ejpam-4732	210	10	i	i	NOUN
ejpam-4732	210	11	)	)	PUNCT
ejpam-4732	210	12	→	→	PUNCT
ejpam-4732	210	13	(	(	PUNCT
ejpam-4732	210	14	y	y	PROPN
ejpam-4732	210	15	,	,	PUNCT
ejpam-4732	210	16	σ	σ	PROPN
ejpam-4732	210	17	,	,	PUNCT
ejpam-4732	210	18	j	j	PROPN
ejpam-4732	210	19	)	)	PUNCT
ejpam-4732	210	20	is	be	AUX
ejpam-4732	210	21	said	say	VERB
ejpam-4732	210	22	to	to	PART
ejpam-4732	210	23	be	be	AUX
ejpam-4732	210	24	:	:	PUNCT
ejpam-4732	210	25	(	(	PUNCT
ejpam-4732	210	26	1	1	X
ejpam-4732	210	27	)	)	PUNCT
ejpam-4732	210	28	upper	upper	ADJ
ejpam-4732	210	29	almost	almost	ADV
ejpam-4732	210	30	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	210	31	at	at	ADP
ejpam-4732	210	32	a	a	DET
ejpam-4732	210	33	point	point	NOUN
ejpam-4732	210	34	x	x	SYM
ejpam-4732	210	35	∈	∈	NOUN
ejpam-4732	210	36	x	x	INTJ
ejpam-4732	210	37	if	if	SCONJ
ejpam-4732	210	38	,	,	PUNCT
ejpam-4732	210	39	for	for	SCONJ
ejpam-4732	210	40	each	each	DET
ejpam-4732	210	41	⋆-open	⋆-open	ADV
ejpam-4732	210	42	set	set	VERB
ejpam-4732	210	43	v	v	NUM
ejpam-4732	210	44	of	of	ADP
ejpam-4732	210	45	y	y	PROPN
ejpam-4732	210	46	containing	contain	VERB
ejpam-4732	210	47	f	f	PROPN
ejpam-4732	210	48	(	(	PUNCT
ejpam-4732	210	49	x	x	NOUN
ejpam-4732	210	50	)	)	PUNCT
ejpam-4732	210	51	,	,	PUNCT
ejpam-4732	210	52	there	there	PRON
ejpam-4732	210	53	exists	exist	VERB
ejpam-4732	210	54	a	a	DET
ejpam-4732	210	55	strong	strong	ADJ
ejpam-4732	210	56	β	β	NOUN
ejpam-4732	210	57	-	-	ADJ
ejpam-4732	210	58	i	i	PRON
ejpam-4732	210	59	-open	-open	VERB
ejpam-4732	210	60	set	set	VERB
ejpam-4732	210	61	u	u	NOUN
ejpam-4732	210	62	of	of	ADP
ejpam-4732	210	63	x	x	PUNCT
ejpam-4732	210	64	containing	contain	VERB
ejpam-4732	210	65	x	x	PUNCT
ejpam-4732	210	66	such	such	ADJ
ejpam-4732	210	67	that	that	SCONJ
ejpam-4732	210	68	f	f	PROPN
ejpam-4732	210	69	(	(	PUNCT
ejpam-4732	210	70	u	u	NOUN
ejpam-4732	210	71	)	)	PUNCT
ejpam-4732	210	72	⊆	⊆	NUM
ejpam-4732	210	73	int⋆(cl(v	int⋆(cl(v	NOUN
ejpam-4732	210	74	)	)	PUNCT
ejpam-4732	210	75	)	)	PUNCT
ejpam-4732	210	76	;	;	PUNCT
ejpam-4732	210	77	(	(	PUNCT
ejpam-4732	210	78	2	2	X
ejpam-4732	210	79	)	)	PUNCT
ejpam-4732	210	80	lower	low	ADJ
ejpam-4732	210	81	almost	almost	ADV
ejpam-4732	210	82	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	210	83	at	at	ADP
ejpam-4732	210	84	a	a	DET
ejpam-4732	210	85	point	point	NOUN
ejpam-4732	210	86	x	x	SYM
ejpam-4732	210	87	∈	∈	NOUN
ejpam-4732	210	88	x	x	INTJ
ejpam-4732	210	89	if	if	SCONJ
ejpam-4732	210	90	,	,	PUNCT
ejpam-4732	210	91	for	for	SCONJ
ejpam-4732	210	92	each	each	DET
ejpam-4732	210	93	⋆-open	⋆-open	ADV
ejpam-4732	210	94	set	set	VERB
ejpam-4732	210	95	v	v	NUM
ejpam-4732	210	96	of	of	ADP
ejpam-4732	210	97	y	y	PRON
ejpam-4732	210	98	such	such	ADJ
ejpam-4732	210	99	that	that	SCONJ
ejpam-4732	210	100	f	f	PROPN
ejpam-4732	210	101	(	(	PUNCT
ejpam-4732	210	102	x	x	NOUN
ejpam-4732	210	103	)	)	PUNCT
ejpam-4732	210	104	∩	∩	NOUN
ejpam-4732	210	105	v	v	ADP
ejpam-4732	210	106	̸=	̸=	PROPN
ejpam-4732	210	107	∅	∅	NOUN
ejpam-4732	210	108	,	,	PUNCT
ejpam-4732	210	109	there	there	PRON
ejpam-4732	210	110	exists	exist	VERB
ejpam-4732	210	111	a	a	DET
ejpam-4732	210	112	strong	strong	ADJ
ejpam-4732	210	113	β	β	NOUN
ejpam-4732	210	114	-	-	ADJ
ejpam-4732	210	115	i	i	PRON
ejpam-4732	210	116	-open	-open	VERB
ejpam-4732	210	117	set	set	VERB
ejpam-4732	210	118	u	u	NOUN
ejpam-4732	210	119	of	of	ADP
ejpam-4732	210	120	x	x	PUNCT
ejpam-4732	210	121	containing	contain	VERB
ejpam-4732	210	122	x	x	PUNCT
ejpam-4732	210	123	such	such	ADJ
ejpam-4732	210	124	that	that	SCONJ
ejpam-4732	210	125	f	f	PROPN
ejpam-4732	210	126	(	(	PUNCT
ejpam-4732	210	127	z	z	NOUN
ejpam-4732	210	128	)	)	PUNCT
ejpam-4732	210	129	∩	∩	ADJ
ejpam-4732	210	130	int⋆(cl(v	int⋆(cl(v	NOUN
ejpam-4732	210	131	)	)	PUNCT
ejpam-4732	210	132	)	)	PUNCT
ejpam-4732	211	1	̸=	̸=	NOUN
ejpam-4732	211	2	∅	∅	NOUN
ejpam-4732	211	3	for	for	ADP
ejpam-4732	211	4	every	every	DET
ejpam-4732	211	5	z	z	NOUN
ejpam-4732	211	6	∈	∈	PROPN
ejpam-4732	211	7	u	u	NOUN
ejpam-4732	211	8	;	;	PUNCT
ejpam-4732	211	9	(	(	PUNCT
ejpam-4732	211	10	3	3	X
ejpam-4732	211	11	)	)	PUNCT
ejpam-4732	211	12	upper	upper	ADJ
ejpam-4732	211	13	(	(	PUNCT
ejpam-4732	211	14	resp	resp	NOUN
ejpam-4732	211	15	.	.	PUNCT
ejpam-4732	212	1	lower	low	ADJ
ejpam-4732	212	2	)	)	PUNCT
ejpam-4732	212	3	almost	almost	ADV
ejpam-4732	212	4	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-4732	212	5	if	if	SCONJ
ejpam-4732	212	6	f	f	PROPN
ejpam-4732	212	7	has	have	VERB
ejpam-4732	212	8	this	this	DET
ejpam-4732	212	9	property	property	NOUN
ejpam-4732	212	10	at	at	ADP
ejpam-4732	212	11	each	each	DET
ejpam-4732	212	12	point	point	NOUN
ejpam-4732	212	13	of	of	ADP
ejpam-4732	212	14	x.	x.	NOUN
ejpam-4732	212	15	remark	remark	PROPN
ejpam-4732	212	16	1	1	NUM
ejpam-4732	212	17	.	.	PUNCT
ejpam-4732	213	1	for	for	ADP
ejpam-4732	213	2	a	a	DET
ejpam-4732	213	3	multifunction	multifunction	NOUN
ejpam-4732	213	4	f	f	NOUN
ejpam-4732	213	5	:	:	PUNCT
ejpam-4732	213	6	(	(	PUNCT
ejpam-4732	213	7	x	x	X
ejpam-4732	213	8	,	,	PUNCT
ejpam-4732	213	9	τ	τ	PROPN
ejpam-4732	213	10	,	,	PUNCT
ejpam-4732	213	11	i	i	NOUN
ejpam-4732	213	12	)	)	PUNCT
ejpam-4732	213	13	→	→	PUNCT
ejpam-4732	213	14	(	(	PUNCT
ejpam-4732	213	15	y	y	PROPN
ejpam-4732	213	16	,	,	PUNCT
ejpam-4732	213	17	σ	σ	PROPN
ejpam-4732	213	18	,	,	PUNCT
ejpam-4732	213	19	j	j	PROPN
ejpam-4732	213	20	)	)	PUNCT
ejpam-4732	213	21	,	,	PUNCT
ejpam-4732	213	22	the	the	DET
ejpam-4732	213	23	following	follow	VERB
ejpam-4732	213	24	implication	implication	NOUN
ejpam-4732	213	25	holds	hold	VERB
ejpam-4732	213	26	:	:	PUNCT
ejpam-4732	213	27	upper	upper	ADJ
ejpam-4732	213	28	sβ(⋆)-continuity	sβ(⋆)-continuity	NOUN
ejpam-4732	213	29	⇒	⇒	NOUN
ejpam-4732	213	30	upper	upper	ADJ
ejpam-4732	213	31	almost	almost	ADV
ejpam-4732	213	32	sβ(⋆)-continuity	sβ(⋆)-continuity	NOUN
ejpam-4732	213	33	.	.	PUNCT
ejpam-4732	214	1	the	the	DET
ejpam-4732	214	2	converse	converse	NOUN
ejpam-4732	214	3	of	of	ADP
ejpam-4732	214	4	the	the	DET
ejpam-4732	214	5	implication	implication	NOUN
ejpam-4732	214	6	is	be	AUX
ejpam-4732	214	7	not	not	PART
ejpam-4732	214	8	true	true	ADJ
ejpam-4732	214	9	in	in	ADP
ejpam-4732	214	10	general	general	ADJ
ejpam-4732	214	11	.	.	PUNCT
ejpam-4732	215	1	we	we	PRON
ejpam-4732	215	2	give	give	VERB
ejpam-4732	215	3	an	an	DET
ejpam-4732	215	4	example	example	NOUN
ejpam-4732	215	5	for	for	ADP
ejpam-4732	215	6	the	the	DET
ejpam-4732	215	7	implication	implication	NOUN
ejpam-4732	215	8	as	as	SCONJ
ejpam-4732	215	9	follows	follow	VERB
ejpam-4732	215	10	.	.	PUNCT
ejpam-4732	215	11	example	example	NOUN
ejpam-4732	216	1	1	1	NUM
ejpam-4732	216	2	.	.	PUNCT
ejpam-4732	216	3	let	let	VERB
ejpam-4732	216	4	x	x	PUNCT
ejpam-4732	216	5	=	=	PRON
ejpam-4732	216	6	{	{	PUNCT
ejpam-4732	216	7	1	1	NUM
ejpam-4732	216	8	,	,	PUNCT
ejpam-4732	216	9	2	2	NUM
ejpam-4732	216	10	,	,	PUNCT
ejpam-4732	216	11	3	3	NUM
ejpam-4732	216	12	}	}	PUNCT
ejpam-4732	216	13	with	with	ADP
ejpam-4732	216	14	a	a	DET
ejpam-4732	216	15	topology	topology	NOUN
ejpam-4732	216	16	τ	τ	X
ejpam-4732	216	17	=	=	SYM
ejpam-4732	216	18	{	{	PUNCT
ejpam-4732	216	19	∅	∅	NOUN
ejpam-4732	216	20	,	,	PUNCT
ejpam-4732	216	21	x	x	NOUN
ejpam-4732	216	22	}	}	PUNCT
ejpam-4732	216	23	and	and	CCONJ
ejpam-4732	216	24	an	an	DET
ejpam-4732	216	25	ideal	ideal	NOUN
ejpam-4732	216	26	i	i	X
ejpam-4732	216	27	=	=	SYM
ejpam-4732	216	28	{	{	PUNCT
ejpam-4732	216	29	∅	∅	NOUN
ejpam-4732	216	30	}	}	PUNCT
ejpam-4732	216	31	.	.	PUNCT
ejpam-4732	217	1	let	let	VERB
ejpam-4732	217	2	y	y	PROPN
ejpam-4732	217	3	=	=	PUNCT
ejpam-4732	217	4	{	{	PUNCT
ejpam-4732	217	5	a	a	PRON
ejpam-4732	217	6	,	,	PUNCT
ejpam-4732	217	7	b	b	NOUN
ejpam-4732	217	8	,	,	PUNCT
ejpam-4732	217	9	c	c	NOUN
ejpam-4732	217	10	}	}	PUNCT
ejpam-4732	217	11	with	with	ADP
ejpam-4732	217	12	a	a	DET
ejpam-4732	217	13	topology	topology	NOUN
ejpam-4732	217	14	σ	σ	NOUN
ejpam-4732	217	15	=	=	SYM
ejpam-4732	217	16	{	{	PUNCT
ejpam-4732	217	17	∅	∅	NOUN
ejpam-4732	217	18	,	,	PUNCT
ejpam-4732	217	19	{	{	PUNCT
ejpam-4732	217	20	b	b	NOUN
ejpam-4732	217	21	}	}	PUNCT
ejpam-4732	217	22	,	,	PUNCT
ejpam-4732	217	23	y	y	PROPN
ejpam-4732	217	24	}	}	PUNCT
ejpam-4732	217	25	and	and	CCONJ
ejpam-4732	217	26	an	an	DET
ejpam-4732	217	27	ideal	ideal	NOUN
ejpam-4732	217	28	j	j	PROPN
ejpam-4732	217	29	=	=	PUNCT
ejpam-4732	217	30	{	{	PUNCT
ejpam-4732	217	31	∅	∅	NOUN
ejpam-4732	217	32	,	,	PUNCT
ejpam-4732	217	33	{	{	PUNCT
ejpam-4732	217	34	b	b	NOUN
ejpam-4732	217	35	}	}	PUNCT
ejpam-4732	217	36	}	}	PUNCT
ejpam-4732	217	37	.	.	PUNCT
ejpam-4732	218	1	a	a	DET
ejpam-4732	218	2	multifunction	multifunction	NOUN
ejpam-4732	218	3	f	f	NOUN
ejpam-4732	218	4	:	:	PUNCT
ejpam-4732	218	5	(	(	PUNCT
ejpam-4732	218	6	x	x	X
ejpam-4732	218	7	,	,	PUNCT
ejpam-4732	218	8	τ	τ	PROPN
ejpam-4732	218	9	,	,	PUNCT
ejpam-4732	218	10	i	i	NOUN
ejpam-4732	218	11	)	)	PUNCT
ejpam-4732	218	12	→	→	PUNCT
ejpam-4732	218	13	(	(	PUNCT
ejpam-4732	218	14	y	y	PROPN
ejpam-4732	218	15	,	,	PUNCT
ejpam-4732	218	16	σ	σ	PROPN
ejpam-4732	218	17	,	,	PUNCT
ejpam-4732	218	18	j	j	PROPN
ejpam-4732	218	19	)	)	PUNCT
ejpam-4732	218	20	is	be	AUX
ejpam-4732	218	21	defined	define	VERB
ejpam-4732	218	22	as	as	SCONJ
ejpam-4732	218	23	follows	follow	VERB
ejpam-4732	218	24	:	:	PUNCT
ejpam-4732	218	25	f	f	X
ejpam-4732	218	26	(	(	PUNCT
ejpam-4732	218	27	1	1	X
ejpam-4732	218	28	)	)	PUNCT
ejpam-4732	218	29	=	=	PRON
ejpam-4732	218	30	{	{	PUNCT
ejpam-4732	218	31	b	b	NOUN
ejpam-4732	218	32	}	}	PUNCT
ejpam-4732	218	33	and	and	CCONJ
ejpam-4732	218	34	f	f	X
ejpam-4732	218	35	(	(	PUNCT
ejpam-4732	218	36	2	2	NUM
ejpam-4732	218	37	)	)	PUNCT
ejpam-4732	219	1	=	=	SYM
ejpam-4732	219	2	f	f	PROPN
ejpam-4732	219	3	(	(	PUNCT
ejpam-4732	219	4	3	3	NUM
ejpam-4732	219	5	)	)	PUNCT
ejpam-4732	219	6	=	=	PRON
ejpam-4732	219	7	{	{	PUNCT
ejpam-4732	219	8	a	a	X
ejpam-4732	219	9	,	,	PUNCT
ejpam-4732	219	10	c	c	NOUN
ejpam-4732	219	11	}	}	PUNCT
ejpam-4732	219	12	.	.	PUNCT
ejpam-4732	220	1	then	then	ADV
ejpam-4732	220	2	,	,	PUNCT
ejpam-4732	220	3	f	f	PROPN
ejpam-4732	220	4	is	be	AUX
ejpam-4732	220	5	upper	upper	ADJ
ejpam-4732	220	6	almost	almost	ADV
ejpam-4732	220	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	221	1	but	but	CCONJ
ejpam-4732	221	2	f	f	PROPN
ejpam-4732	221	3	is	be	AUX
ejpam-4732	221	4	not	not	PART
ejpam-4732	221	5	upper	upper	ADJ
ejpam-4732	221	6	sβ(⋆)-continuous	sβ(⋆)-continuous	NOUN
ejpam-4732	221	7	,	,	PUNCT
ejpam-4732	221	8	since	since	SCONJ
ejpam-4732	221	9	{	{	PUNCT
ejpam-4732	221	10	a	a	PRON
ejpam-4732	221	11	,	,	PUNCT
ejpam-4732	221	12	c	c	NOUN
ejpam-4732	221	13	}	}	PUNCT
ejpam-4732	221	14	is	be	AUX
ejpam-4732	221	15	⋆-open	⋆-open	ADJ
ejpam-4732	221	16	in	in	ADP
ejpam-4732	221	17	y	y	PROPN
ejpam-4732	221	18	but	but	CCONJ
ejpam-4732	221	19	f+({a	f+({a	PROPN
ejpam-4732	221	20	,	,	PUNCT
ejpam-4732	221	21	c	c	NOUN
ejpam-4732	221	22	}	}	PUNCT
ejpam-4732	221	23	)	)	PUNCT
ejpam-4732	221	24	is	be	AUX
ejpam-4732	221	25	not	not	PART
ejpam-4732	221	26	strong	strong	ADJ
ejpam-4732	221	27	β	β	NOUN
ejpam-4732	221	28	-	-	PUNCT
ejpam-4732	221	29	i	i	PRON
ejpam-4732	221	30	-open	-open	NOUN
ejpam-4732	221	31	in	in	ADP
ejpam-4732	221	32	x.	x.	NOUN
ejpam-4732	221	33	theorem	theorem	VERB
ejpam-4732	221	34	5	5	NUM
ejpam-4732	221	35	.	.	PUNCT
ejpam-4732	221	36	a	a	DET
ejpam-4732	221	37	multifunction	multifunction	NOUN
ejpam-4732	221	38	f	f	NOUN
ejpam-4732	221	39	:	:	PUNCT
ejpam-4732	221	40	(	(	PUNCT
ejpam-4732	221	41	x	x	X
ejpam-4732	221	42	,	,	PUNCT
ejpam-4732	221	43	τ	τ	PROPN
ejpam-4732	221	44	,	,	PUNCT
ejpam-4732	221	45	i	i	NOUN
ejpam-4732	221	46	)	)	PUNCT
ejpam-4732	221	47	→	→	PUNCT
ejpam-4732	221	48	(	(	PUNCT
ejpam-4732	221	49	y	y	PROPN
ejpam-4732	221	50	,	,	PUNCT
ejpam-4732	221	51	σ	σ	PROPN
ejpam-4732	221	52	,	,	PUNCT
ejpam-4732	221	53	j	j	PROPN
ejpam-4732	221	54	)	)	PUNCT
ejpam-4732	221	55	is	be	AUX
ejpam-4732	221	56	upper	upper	ADJ
ejpam-4732	221	57	almost	almost	ADV
ejpam-4732	221	58	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	221	59	at	at	ADP
ejpam-4732	221	60	x	x	X
ejpam-4732	221	61	∈	∈	PROPN
ejpam-4732	221	62	x	x	SYM
ejpam-4732	221	63	if	if	SCONJ
ejpam-4732	222	1	and	and	CCONJ
ejpam-4732	222	2	only	only	ADV
ejpam-4732	222	3	if	if	SCONJ
ejpam-4732	222	4	x	x	SYM
ejpam-4732	222	5	∈	∈	NOUN
ejpam-4732	222	6	sβinti	sβinti	NOUN
ejpam-4732	222	7	(	(	PUNCT
ejpam-4732	222	8	f+(sclj	f+(sclj	X
ejpam-4732	222	9	(	(	PUNCT
ejpam-4732	222	10	v	v	NOUN
ejpam-4732	222	11	)	)	PUNCT
ejpam-4732	222	12	)	)	PUNCT
ejpam-4732	222	13	)	)	PUNCT
ejpam-4732	222	14	for	for	ADP
ejpam-4732	222	15	every	every	DET
ejpam-4732	222	16	⋆-open	⋆-open	NOUN
ejpam-4732	222	17	set	set	VERB
ejpam-4732	222	18	v	v	NOUN
ejpam-4732	222	19	of	of	ADP
ejpam-4732	222	20	y	y	PROPN
ejpam-4732	222	21	containing	contain	VERB
ejpam-4732	222	22	f	f	PROPN
ejpam-4732	222	23	(	(	PUNCT
ejpam-4732	222	24	x	x	NOUN
ejpam-4732	222	25	)	)	PUNCT
ejpam-4732	222	26	.	.	PUNCT
ejpam-4732	223	1	proof	proof	NOUN
ejpam-4732	223	2	.	.	PUNCT
ejpam-4732	224	1	let	let	VERB
ejpam-4732	224	2	v	v	PART
ejpam-4732	224	3	be	be	AUX
ejpam-4732	224	4	any	any	DET
ejpam-4732	224	5	⋆-open	⋆-open	ADJ
ejpam-4732	224	6	set	set	NOUN
ejpam-4732	224	7	of	of	ADP
ejpam-4732	224	8	y	y	PROPN
ejpam-4732	224	9	containing	contain	VERB
ejpam-4732	224	10	f	f	PROPN
ejpam-4732	224	11	(	(	PUNCT
ejpam-4732	224	12	x	x	NOUN
ejpam-4732	224	13	)	)	PUNCT
ejpam-4732	224	14	.	.	PUNCT
ejpam-4732	225	1	then	then	ADV
ejpam-4732	225	2	,	,	PUNCT
ejpam-4732	225	3	there	there	PRON
ejpam-4732	225	4	exists	exist	VERB
ejpam-4732	225	5	a	a	DET
ejpam-4732	225	6	strong	strong	ADJ
ejpam-4732	225	7	β	β	NOUN
ejpam-4732	225	8	-	-	ADJ
ejpam-4732	225	9	i	i	PRON
ejpam-4732	225	10	-open	-open	VERB
ejpam-4732	225	11	set	set	VERB
ejpam-4732	225	12	u	u	NOUN
ejpam-4732	225	13	of	of	ADP
ejpam-4732	225	14	x	x	PUNCT
ejpam-4732	225	15	containing	contain	VERB
ejpam-4732	225	16	x	x	PUNCT
ejpam-4732	225	17	such	such	ADJ
ejpam-4732	225	18	that	that	SCONJ
ejpam-4732	225	19	f	f	PROPN
ejpam-4732	225	20	(	(	PUNCT
ejpam-4732	225	21	u	u	NOUN
ejpam-4732	225	22	)	)	PUNCT
ejpam-4732	225	23	⊆	⊆	NUM
ejpam-4732	225	24	int⋆(cl(v	int⋆(cl(v	NOUN
ejpam-4732	225	25	)	)	PUNCT
ejpam-4732	225	26	)	)	PUNCT
ejpam-4732	226	1	=	=	PUNCT
ejpam-4732	226	2	sclj	sclj	NOUN
ejpam-4732	226	3	(	(	PUNCT
ejpam-4732	226	4	v	v	NOUN
ejpam-4732	226	5	)	)	PUNCT
ejpam-4732	226	6	;	;	PUNCT
ejpam-4732	226	7	hence	hence	ADV
ejpam-4732	226	8	u	u	NOUN
ejpam-4732	226	9	⊆	⊆	NUM
ejpam-4732	226	10	f+(sclj	f+(sclj	PROPN
ejpam-4732	226	11	(	(	PUNCT
ejpam-4732	226	12	v	v	NOUN
ejpam-4732	226	13	)	)	PUNCT
ejpam-4732	226	14	)	)	PUNCT
ejpam-4732	226	15	.	.	PUNCT
ejpam-4732	227	1	since	since	SCONJ
ejpam-4732	227	2	u	u	NOUN
ejpam-4732	227	3	is	be	AUX
ejpam-4732	227	4	strong	strong	ADJ
ejpam-4732	227	5	β	β	NOUN
ejpam-4732	227	6	-	-	PUNCT
ejpam-4732	227	7	i	i	PRON
ejpam-4732	227	8	-open	-open	ADJ
ejpam-4732	227	9	,	,	PUNCT
ejpam-4732	227	10	we	we	PRON
ejpam-4732	227	11	have	have	VERB
ejpam-4732	227	12	x	x	X
ejpam-4732	227	13	∈	∈	PROPN
ejpam-4732	227	14	u	u	NOUN
ejpam-4732	227	15	⊆	⊆	NUM
ejpam-4732	227	16	cl⋆(int(cl⋆(u	cl⋆(int(cl⋆(u	PROPN
ejpam-4732	227	17	)	)	PUNCT
ejpam-4732	227	18	)	)	PUNCT
ejpam-4732	227	19	)	)	PUNCT
ejpam-4732	228	1	⊆	⊆	X
ejpam-4732	228	2	cl⋆(int(cl⋆(f+(sclj	cl⋆(int(cl⋆(f+(sclj	NOUN
ejpam-4732	228	3	(	(	PUNCT
ejpam-4732	228	4	v	v	NOUN
ejpam-4732	228	5	)	)	PUNCT
ejpam-4732	228	6	)	)	PUNCT
ejpam-4732	228	7	)	)	PUNCT
ejpam-4732	228	8	)	)	PUNCT
ejpam-4732	228	9	)	)	PUNCT
ejpam-4732	228	10	.	.	PUNCT
ejpam-4732	229	1	since	since	SCONJ
ejpam-4732	229	2	x	x	PROPN
ejpam-4732	229	3	∈	∈	PROPN
ejpam-4732	229	4	f+(v	f+(v	NOUN
ejpam-4732	229	5	)	)	PUNCT
ejpam-4732	229	6	⊆	⊆	X
ejpam-4732	229	7	f+(sclj	f+(sclj	NUM
ejpam-4732	229	8	(	(	PUNCT
ejpam-4732	229	9	v	v	NOUN
ejpam-4732	229	10	)	)	PUNCT
ejpam-4732	229	11	)	)	PUNCT
ejpam-4732	229	12	and	and	CCONJ
ejpam-4732	229	13	by	by	ADP
ejpam-4732	229	14	lemma	lemma	PROPN
ejpam-4732	229	15	2	2	NUM
ejpam-4732	229	16	,	,	PUNCT
ejpam-4732	229	17	x	x	SYM
ejpam-4732	229	18	∈	∈	NOUN
ejpam-4732	229	19	f+(sclj	f+(sclj	X
ejpam-4732	229	20	(	(	PUNCT
ejpam-4732	229	21	v	v	NOUN
ejpam-4732	229	22	)	)	PUNCT
ejpam-4732	229	23	)	)	PUNCT
ejpam-4732	229	24	∩	∩	NOUN
ejpam-4732	229	25	cl⋆(int(cl⋆(sclj	cl⋆(int(cl⋆(sclj	PROPN
ejpam-4732	229	26	(	(	PUNCT
ejpam-4732	229	27	v	v	NOUN
ejpam-4732	229	28	)	)	PUNCT
ejpam-4732	229	29	)	)	PUNCT
ejpam-4732	229	30	)	)	PUNCT
ejpam-4732	229	31	)	)	PUNCT
ejpam-4732	230	1	=	=	PRON
ejpam-4732	230	2	sβinti	sβinti	X
ejpam-4732	230	3	(	(	PUNCT
ejpam-4732	230	4	f+(sclj	f+(sclj	X
ejpam-4732	230	5	(	(	PUNCT
ejpam-4732	230	6	v	v	NOUN
ejpam-4732	230	7	)	)	PUNCT
ejpam-4732	230	8	)	)	PUNCT
ejpam-4732	230	9	)	)	PUNCT
ejpam-4732	230	10	.	.	PUNCT
ejpam-4732	231	1	conversely	conversely	ADV
ejpam-4732	231	2	,	,	PUNCT
ejpam-4732	231	3	let	let	VERB
ejpam-4732	231	4	v	v	PART
ejpam-4732	231	5	be	be	AUX
ejpam-4732	231	6	any	any	DET
ejpam-4732	231	7	⋆-open	⋆-open	ADJ
ejpam-4732	231	8	set	set	NOUN
ejpam-4732	231	9	of	of	ADP
ejpam-4732	231	10	y	y	PROPN
ejpam-4732	231	11	containing	contain	VERB
ejpam-4732	231	12	f	f	PROPN
ejpam-4732	231	13	(	(	PUNCT
ejpam-4732	231	14	x	x	NOUN
ejpam-4732	231	15	)	)	PUNCT
ejpam-4732	231	16	.	.	PUNCT
ejpam-4732	232	1	then	then	ADV
ejpam-4732	232	2	,	,	PUNCT
ejpam-4732	232	3	we	we	PRON
ejpam-4732	232	4	have	have	VERB
ejpam-4732	232	5	x	x	PART
ejpam-4732	232	6	∈	∈	PROPN
ejpam-4732	232	7	sβinti	sβinti	NOUN
ejpam-4732	232	8	(	(	PUNCT
ejpam-4732	232	9	f+(sclj	f+(sclj	X
ejpam-4732	232	10	(	(	PUNCT
ejpam-4732	232	11	v	v	NOUN
ejpam-4732	232	12	)	)	PUNCT
ejpam-4732	232	13	)	)	PUNCT
ejpam-4732	232	14	)	)	PUNCT
ejpam-4732	233	1	c.	c.	PROPN
ejpam-4732	233	2	boonpok	boonpok	PROPN
ejpam-4732	233	3	,	,	PUNCT
ejpam-4732	233	4	p.	p.	NOUN
ejpam-4732	233	5	pue	pue	NOUN
ejpam-4732	233	6	-	-	PUNCT
ejpam-4732	233	7	on	on	ADP
ejpam-4732	233	8	/	/	SYM
ejpam-4732	233	9	eur	eur	NOUN
ejpam-4732	233	10	.	.	PUNCT
ejpam-4732	234	1	j.	j.	PROPN
ejpam-4732	234	2	pure	pure	PROPN
ejpam-4732	234	3	appl	appl	PROPN
ejpam-4732	234	4	.	.	PROPN
ejpam-4732	234	5	math	math	PROPN
ejpam-4732	234	6	,	,	PUNCT
ejpam-4732	234	7	16	16	NUM
ejpam-4732	234	8	(	(	PUNCT
ejpam-4732	234	9	3	3	NUM
ejpam-4732	234	10	)	)	PUNCT
ejpam-4732	234	11	(	(	PUNCT
ejpam-4732	234	12	2023	2023	NUM
ejpam-4732	234	13	)	)	PUNCT
ejpam-4732	234	14	,	,	PUNCT
ejpam-4732	234	15	1634	1634	NUM
ejpam-4732	234	16	-	-	SYM
ejpam-4732	234	17	1646	1646	NUM
ejpam-4732	234	18	1641	1641	NUM
ejpam-4732	235	1	and	and	CCONJ
ejpam-4732	235	2	so	so	ADV
ejpam-4732	235	3	there	there	PRON
ejpam-4732	235	4	exists	exist	VERB
ejpam-4732	235	5	a	a	DET
ejpam-4732	235	6	strong	strong	ADJ
ejpam-4732	235	7	β	β	NOUN
ejpam-4732	235	8	-	-	ADJ
ejpam-4732	235	9	i	i	PRON
ejpam-4732	235	10	-open	-open	NOUN
ejpam-4732	235	11	set	set	VERB
ejpam-4732	235	12	u	u	PRON
ejpam-4732	235	13	ofx	ofx	NOUN
ejpam-4732	235	14	containing	contain	VERB
ejpam-4732	235	15	x	x	PUNCT
ejpam-4732	235	16	such	such	ADJ
ejpam-4732	235	17	that	that	SCONJ
ejpam-4732	235	18	u	u	PROPN
ejpam-4732	236	1	⊆	⊆	NUM
ejpam-4732	236	2	f+(sclj	f+(sclj	X
ejpam-4732	236	3	(	(	PUNCT
ejpam-4732	236	4	v	v	NOUN
ejpam-4732	236	5	)	)	PUNCT
ejpam-4732	236	6	)	)	PUNCT
ejpam-4732	236	7	;	;	PUNCT
ejpam-4732	236	8	hence	hence	ADV
ejpam-4732	236	9	f	f	PROPN
ejpam-4732	236	10	(	(	PUNCT
ejpam-4732	236	11	u	u	NOUN
ejpam-4732	236	12	)	)	PUNCT
ejpam-4732	236	13	⊆	⊆	NUM
ejpam-4732	236	14	sclj	sclj	NOUN
ejpam-4732	236	15	(	(	PUNCT
ejpam-4732	236	16	v	v	NOUN
ejpam-4732	236	17	)	)	PUNCT
ejpam-4732	236	18	=	=	SYM
ejpam-4732	236	19	int⋆(cl(v	int⋆(cl(v	PROPN
ejpam-4732	236	20	)	)	PUNCT
ejpam-4732	236	21	)	)	PUNCT
ejpam-4732	236	22	.	.	PUNCT
ejpam-4732	237	1	this	this	PRON
ejpam-4732	237	2	shows	show	VERB
ejpam-4732	237	3	that	that	SCONJ
ejpam-4732	237	4	f	f	PROPN
ejpam-4732	237	5	is	be	AUX
ejpam-4732	237	6	upper	upper	ADJ
ejpam-4732	237	7	almost	almost	ADV
ejpam-4732	237	8	β(⋆)continuous	β(⋆)continuous	ADJ
ejpam-4732	237	9	at	at	SCONJ
ejpam-4732	237	10	x.	x.	NOUN
ejpam-4732	237	11	theorem	theorem	VERB
ejpam-4732	237	12	6	6	NUM
ejpam-4732	237	13	.	.	PUNCT
ejpam-4732	238	1	a	a	DET
ejpam-4732	238	2	multifunction	multifunction	NOUN
ejpam-4732	238	3	f	f	NOUN
ejpam-4732	238	4	:	:	PUNCT
ejpam-4732	238	5	(	(	PUNCT
ejpam-4732	238	6	x	x	X
ejpam-4732	238	7	,	,	PUNCT
ejpam-4732	238	8	τ	τ	PROPN
ejpam-4732	238	9	,	,	PUNCT
ejpam-4732	238	10	i	i	NOUN
ejpam-4732	238	11	)	)	PUNCT
ejpam-4732	238	12	→	→	PUNCT
ejpam-4732	238	13	(	(	PUNCT
ejpam-4732	238	14	y	y	PROPN
ejpam-4732	238	15	,	,	PUNCT
ejpam-4732	238	16	σ	σ	PROPN
ejpam-4732	238	17	,	,	PUNCT
ejpam-4732	238	18	j	j	PROPN
ejpam-4732	238	19	)	)	PUNCT
ejpam-4732	238	20	is	be	AUX
ejpam-4732	238	21	lower	low	ADJ
ejpam-4732	238	22	almost	almost	ADV
ejpam-4732	238	23	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	238	24	at	at	ADP
ejpam-4732	238	25	x	x	X
ejpam-4732	238	26	∈	∈	PROPN
ejpam-4732	238	27	x	x	SYM
ejpam-4732	238	28	if	if	SCONJ
ejpam-4732	239	1	and	and	CCONJ
ejpam-4732	239	2	only	only	ADV
ejpam-4732	239	3	if	if	SCONJ
ejpam-4732	239	4	x	x	SYM
ejpam-4732	239	5	∈	∈	NOUN
ejpam-4732	239	6	sβinti	sβinti	NOUN
ejpam-4732	239	7	(	(	PUNCT
ejpam-4732	239	8	f−(sclj	f−(sclj	NOUN
ejpam-4732	239	9	(	(	PUNCT
ejpam-4732	239	10	v	v	NOUN
ejpam-4732	239	11	)	)	PUNCT
ejpam-4732	239	12	)	)	PUNCT
ejpam-4732	239	13	)	)	PUNCT
ejpam-4732	239	14	for	for	ADP
ejpam-4732	239	15	every	every	PRON
ejpam-4732	239	16	⋆-open	⋆-open	NOUN
ejpam-4732	239	17	set	set	VERB
ejpam-4732	239	18	v	v	NUM
ejpam-4732	239	19	of	of	ADP
ejpam-4732	239	20	y	y	PRON
ejpam-4732	239	21	such	such	ADJ
ejpam-4732	239	22	that	that	SCONJ
ejpam-4732	239	23	f	f	PROPN
ejpam-4732	239	24	(	(	PUNCT
ejpam-4732	239	25	x	x	NOUN
ejpam-4732	239	26	)	)	PUNCT
ejpam-4732	239	27	∩	∩	NOUN
ejpam-4732	239	28	v	v	ADP
ejpam-4732	239	29	̸=	̸=	PROPN
ejpam-4732	239	30	∅.	∅.	ADP
ejpam-4732	239	31	proof	proof	NOUN
ejpam-4732	239	32	.	.	PUNCT
ejpam-4732	240	1	the	the	DET
ejpam-4732	240	2	proof	proof	NOUN
ejpam-4732	240	3	is	be	AUX
ejpam-4732	240	4	similar	similar	ADJ
ejpam-4732	240	5	to	to	ADP
ejpam-4732	240	6	that	that	PRON
ejpam-4732	240	7	of	of	ADP
ejpam-4732	240	8	theorem	theorem	NOUN
ejpam-4732	240	9	5	5	NUM
ejpam-4732	240	10	.	.	PUNCT
ejpam-4732	240	11	definition	definition	NOUN
ejpam-4732	240	12	4	4	NUM
ejpam-4732	240	13	.	.	PUNCT
ejpam-4732	241	1	a	a	DET
ejpam-4732	241	2	function	function	NOUN
ejpam-4732	241	3	f	f	NOUN
ejpam-4732	241	4	:	:	PUNCT
ejpam-4732	241	5	(	(	PUNCT
ejpam-4732	241	6	x	x	X
ejpam-4732	241	7	,	,	PUNCT
ejpam-4732	241	8	τ	τ	PROPN
ejpam-4732	241	9	,	,	PUNCT
ejpam-4732	241	10	i	i	NOUN
ejpam-4732	241	11	)	)	PUNCT
ejpam-4732	241	12	→	→	PUNCT
ejpam-4732	241	13	(	(	PUNCT
ejpam-4732	241	14	y	y	PROPN
ejpam-4732	241	15	,	,	PUNCT
ejpam-4732	241	16	σ	σ	PROPN
ejpam-4732	241	17	,	,	PUNCT
ejpam-4732	241	18	j	j	PROPN
ejpam-4732	241	19	)	)	PUNCT
ejpam-4732	241	20	is	be	AUX
ejpam-4732	241	21	called	call	VERB
ejpam-4732	241	22	almost	almost	ADV
ejpam-4732	241	23	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	241	24	at	at	ADP
ejpam-4732	241	25	a	a	DET
ejpam-4732	241	26	point	point	NOUN
ejpam-4732	241	27	x	x	SYM
ejpam-4732	241	28	∈	∈	NOUN
ejpam-4732	241	29	x	x	INTJ
ejpam-4732	241	30	if	if	SCONJ
ejpam-4732	241	31	,	,	PUNCT
ejpam-4732	241	32	for	for	SCONJ
ejpam-4732	241	33	each	each	DET
ejpam-4732	241	34	⋆-open	⋆-open	ADV
ejpam-4732	241	35	set	set	VERB
ejpam-4732	241	36	v	v	NUM
ejpam-4732	241	37	of	of	ADP
ejpam-4732	241	38	y	y	NOUN
ejpam-4732	241	39	containing	contain	VERB
ejpam-4732	241	40	f(x	f(x	PROPN
ejpam-4732	241	41	)	)	PUNCT
ejpam-4732	241	42	,	,	PUNCT
ejpam-4732	241	43	there	there	PRON
ejpam-4732	241	44	exists	exist	VERB
ejpam-4732	241	45	a	a	DET
ejpam-4732	241	46	strong	strong	ADJ
ejpam-4732	241	47	β	β	NOUN
ejpam-4732	241	48	-	-	ADJ
ejpam-4732	241	49	i	i	PRON
ejpam-4732	241	50	-open	-open	VERB
ejpam-4732	241	51	set	set	VERB
ejpam-4732	241	52	u	u	NOUN
ejpam-4732	241	53	of	of	ADP
ejpam-4732	241	54	x	x	PUNCT
ejpam-4732	241	55	containing	contain	VERB
ejpam-4732	241	56	x	x	PUNCT
ejpam-4732	241	57	such	such	ADJ
ejpam-4732	241	58	that	that	DET
ejpam-4732	241	59	f(u	f(u	PROPN
ejpam-4732	241	60	)	)	PUNCT
ejpam-4732	241	61	⊆	⊆	NUM
ejpam-4732	241	62	int⋆(cl(v	int⋆(cl(v	PROPN
ejpam-4732	241	63	)	)	PUNCT
ejpam-4732	241	64	)	)	PUNCT
ejpam-4732	241	65	.	.	PUNCT
ejpam-4732	242	1	a	a	DET
ejpam-4732	242	2	function	function	NOUN
ejpam-4732	242	3	f	f	NOUN
ejpam-4732	242	4	:	:	PUNCT
ejpam-4732	242	5	(	(	PUNCT
ejpam-4732	242	6	x	x	X
ejpam-4732	242	7	,	,	PUNCT
ejpam-4732	242	8	τ	τ	PROPN
ejpam-4732	242	9	,	,	PUNCT
ejpam-4732	242	10	i	i	NOUN
ejpam-4732	242	11	)	)	PUNCT
ejpam-4732	242	12	→	→	PUNCT
ejpam-4732	242	13	(	(	PUNCT
ejpam-4732	242	14	y	y	PROPN
ejpam-4732	242	15	,	,	PUNCT
ejpam-4732	242	16	σ	σ	PROPN
ejpam-4732	242	17	,	,	PUNCT
ejpam-4732	242	18	j	j	PROPN
ejpam-4732	242	19	)	)	PUNCT
ejpam-4732	242	20	is	be	AUX
ejpam-4732	242	21	called	call	VERB
ejpam-4732	242	22	almost	almost	ADV
ejpam-4732	242	23	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-4732	242	24	if	if	SCONJ
ejpam-4732	242	25	f	f	PROPN
ejpam-4732	242	26	has	have	VERB
ejpam-4732	242	27	this	this	DET
ejpam-4732	242	28	property	property	NOUN
ejpam-4732	242	29	at	at	ADP
ejpam-4732	242	30	each	each	DET
ejpam-4732	242	31	point	point	NOUN
ejpam-4732	242	32	of	of	ADP
ejpam-4732	242	33	x.	x.	PROPN
ejpam-4732	242	34	corollary	corollary	PROPN
ejpam-4732	242	35	3	3	X
ejpam-4732	242	36	.	.	PUNCT
ejpam-4732	243	1	a	a	DET
ejpam-4732	243	2	function	function	NOUN
ejpam-4732	243	3	f	f	NOUN
ejpam-4732	243	4	:	:	PUNCT
ejpam-4732	243	5	(	(	PUNCT
ejpam-4732	243	6	x	x	X
ejpam-4732	243	7	,	,	PUNCT
ejpam-4732	243	8	τ	τ	PROPN
ejpam-4732	243	9	,	,	PUNCT
ejpam-4732	243	10	i	i	NOUN
ejpam-4732	243	11	)	)	PUNCT
ejpam-4732	243	12	→	→	PUNCT
ejpam-4732	243	13	(	(	PUNCT
ejpam-4732	243	14	y	y	PROPN
ejpam-4732	243	15	,	,	PUNCT
ejpam-4732	243	16	σ	σ	PROPN
ejpam-4732	243	17	,	,	PUNCT
ejpam-4732	243	18	j	j	PROPN
ejpam-4732	243	19	)	)	PUNCT
ejpam-4732	243	20	is	be	AUX
ejpam-4732	243	21	almost	almost	ADV
ejpam-4732	243	22	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	243	23	at	at	ADP
ejpam-4732	243	24	x	x	X
ejpam-4732	243	25	∈	∈	PROPN
ejpam-4732	243	26	x	x	SYM
ejpam-4732	243	27	if	if	SCONJ
ejpam-4732	244	1	and	and	CCONJ
ejpam-4732	244	2	only	only	ADV
ejpam-4732	244	3	if	if	SCONJ
ejpam-4732	244	4	x	x	SYM
ejpam-4732	244	5	∈	∈	NOUN
ejpam-4732	244	6	sβinti	sβinti	NOUN
ejpam-4732	244	7	(	(	PUNCT
ejpam-4732	244	8	f−1(sclj	f−1(sclj	X
ejpam-4732	244	9	(	(	PUNCT
ejpam-4732	244	10	v	v	NOUN
ejpam-4732	244	11	)	)	PUNCT
ejpam-4732	244	12	)	)	PUNCT
ejpam-4732	244	13	)	)	PUNCT
ejpam-4732	244	14	for	for	ADP
ejpam-4732	244	15	every	every	DET
ejpam-4732	244	16	⋆-open	⋆-open	NOUN
ejpam-4732	244	17	set	set	VERB
ejpam-4732	244	18	v	v	NOUN
ejpam-4732	244	19	of	of	ADP
ejpam-4732	244	20	y	y	NOUN
ejpam-4732	244	21	containing	contain	VERB
ejpam-4732	244	22	f(x	f(x	PROPN
ejpam-4732	244	23	)	)	PUNCT
ejpam-4732	244	24	.	.	PUNCT
ejpam-4732	245	1	recall	recall	VERB
ejpam-4732	245	2	that	that	SCONJ
ejpam-4732	245	3	a	a	DET
ejpam-4732	245	4	subset	subset	NOUN
ejpam-4732	245	5	a	a	PRON
ejpam-4732	245	6	of	of	ADP
ejpam-4732	245	7	an	an	DET
ejpam-4732	245	8	ideal	ideal	ADJ
ejpam-4732	245	9	topological	topological	ADJ
ejpam-4732	245	10	space	space	NOUN
ejpam-4732	245	11	(	(	PUNCT
ejpam-4732	245	12	x	x	X
ejpam-4732	245	13	,	,	PUNCT
ejpam-4732	245	14	τ	τ	PROPN
ejpam-4732	245	15	,	,	PUNCT
ejpam-4732	245	16	i	i	PROPN
ejpam-4732	245	17	)	)	PUNCT
ejpam-4732	245	18	is	be	AUX
ejpam-4732	245	19	said	say	VERB
ejpam-4732	245	20	to	to	PART
ejpam-4732	245	21	be	be	AUX
ejpam-4732	245	22	r⋆-i	r⋆-i	NOUN
ejpam-4732	245	23	-open	-open	NOUN
ejpam-4732	245	24	[	[	X
ejpam-4732	245	25	2	2	NUM
ejpam-4732	245	26	]	]	PUNCT
ejpam-4732	245	27	if	if	SCONJ
ejpam-4732	245	28	a	a	DET
ejpam-4732	245	29	=	=	X
ejpam-4732	245	30	int⋆(cl(a	int⋆(cl(a	NOUN
ejpam-4732	245	31	)	)	PUNCT
ejpam-4732	245	32	)	)	PUNCT
ejpam-4732	245	33	.	.	PUNCT
ejpam-4732	246	1	the	the	DET
ejpam-4732	246	2	complement	complement	NOUN
ejpam-4732	246	3	of	of	ADP
ejpam-4732	246	4	a	a	DET
ejpam-4732	246	5	r⋆-i	r⋆-i	NOUN
ejpam-4732	246	6	-open	-open	NOUN
ejpam-4732	246	7	set	set	NOUN
ejpam-4732	246	8	is	be	AUX
ejpam-4732	246	9	said	say	VERB
ejpam-4732	246	10	to	to	PART
ejpam-4732	246	11	be	be	AUX
ejpam-4732	246	12	r⋆-i	r⋆-i	NOUN
ejpam-4732	246	13	-closed	-close	VERB
ejpam-4732	246	14	.	.	PUNCT
ejpam-4732	247	1	theorem	theorem	VERB
ejpam-4732	247	2	7	7	NUM
ejpam-4732	247	3	.	.	X
ejpam-4732	247	4	for	for	ADP
ejpam-4732	247	5	a	a	DET
ejpam-4732	247	6	multifunction	multifunction	NOUN
ejpam-4732	248	1	f	f	NOUN
ejpam-4732	248	2	:	:	PUNCT
ejpam-4732	248	3	(	(	PUNCT
ejpam-4732	248	4	x	x	X
ejpam-4732	248	5	,	,	PUNCT
ejpam-4732	248	6	τ	τ	PROPN
ejpam-4732	248	7	,	,	PUNCT
ejpam-4732	248	8	i	i	NOUN
ejpam-4732	248	9	)	)	PUNCT
ejpam-4732	248	10	→	→	PUNCT
ejpam-4732	248	11	(	(	PUNCT
ejpam-4732	248	12	y	y	PROPN
ejpam-4732	248	13	,	,	PUNCT
ejpam-4732	248	14	σ	σ	PROPN
ejpam-4732	248	15	,	,	PUNCT
ejpam-4732	248	16	j	j	PROPN
ejpam-4732	248	17	)	)	PUNCT
ejpam-4732	248	18	,	,	PUNCT
ejpam-4732	248	19	the	the	DET
ejpam-4732	248	20	following	follow	VERB
ejpam-4732	248	21	properties	property	NOUN
ejpam-4732	248	22	are	be	AUX
ejpam-4732	248	23	equivalent	equivalent	ADJ
ejpam-4732	248	24	:	:	PUNCT
ejpam-4732	248	25	(	(	PUNCT
ejpam-4732	248	26	1	1	X
ejpam-4732	248	27	)	)	PUNCT
ejpam-4732	248	28	f	f	PROPN
ejpam-4732	248	29	is	be	AUX
ejpam-4732	248	30	upper	upper	ADJ
ejpam-4732	248	31	almost	almost	ADV
ejpam-4732	248	32	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	248	33	;	;	PUNCT
ejpam-4732	248	34	(	(	PUNCT
ejpam-4732	248	35	2	2	X
ejpam-4732	248	36	)	)	PUNCT
ejpam-4732	248	37	for	for	SCONJ
ejpam-4732	248	38	each	each	DET
ejpam-4732	248	39	x	x	SYM
ejpam-4732	248	40	∈	∈	PROPN
ejpam-4732	248	41	x	x	X
ejpam-4732	248	42	and	and	CCONJ
ejpam-4732	248	43	each	each	DET
ejpam-4732	248	44	⋆-open	⋆-open	ADV
ejpam-4732	248	45	set	set	VERB
ejpam-4732	248	46	v	v	NUM
ejpam-4732	248	47	of	of	ADP
ejpam-4732	248	48	y	y	PROPN
ejpam-4732	248	49	containing	contain	VERB
ejpam-4732	248	50	f	f	PROPN
ejpam-4732	248	51	(	(	PUNCT
ejpam-4732	248	52	x	x	NOUN
ejpam-4732	248	53	)	)	PUNCT
ejpam-4732	248	54	,	,	PUNCT
ejpam-4732	248	55	there	there	PRON
ejpam-4732	248	56	exists	exist	VERB
ejpam-4732	248	57	a	a	DET
ejpam-4732	248	58	strong	strong	ADJ
ejpam-4732	248	59	β	β	NOUN
ejpam-4732	248	60	-	-	ADJ
ejpam-4732	248	61	i	i	PRON
ejpam-4732	248	62	-open	-open	VERB
ejpam-4732	248	63	set	set	VERB
ejpam-4732	248	64	u	u	NOUN
ejpam-4732	248	65	of	of	ADP
ejpam-4732	248	66	x	x	PUNCT
ejpam-4732	248	67	containing	contain	VERB
ejpam-4732	248	68	x	x	PUNCT
ejpam-4732	248	69	such	such	ADJ
ejpam-4732	248	70	that	that	SCONJ
ejpam-4732	248	71	f	f	PROPN
ejpam-4732	248	72	(	(	PUNCT
ejpam-4732	248	73	u	u	NOUN
ejpam-4732	248	74	)	)	PUNCT
ejpam-4732	248	75	⊆	⊆	NUM
ejpam-4732	248	76	sclj	sclj	NOUN
ejpam-4732	248	77	(	(	PUNCT
ejpam-4732	248	78	v	v	NOUN
ejpam-4732	248	79	)	)	PUNCT
ejpam-4732	248	80	;	;	PUNCT
ejpam-4732	248	81	(	(	PUNCT
ejpam-4732	248	82	3	3	X
ejpam-4732	248	83	)	)	PUNCT
ejpam-4732	248	84	for	for	ADP
ejpam-4732	248	85	each	each	DET
ejpam-4732	248	86	x	x	SYM
ejpam-4732	248	87	∈	∈	PROPN
ejpam-4732	248	88	x	x	X
ejpam-4732	248	89	and	and	CCONJ
ejpam-4732	248	90	each	each	DET
ejpam-4732	248	91	r⋆-j	r⋆-j	PROPN
ejpam-4732	248	92	-open	-open	PROPN
ejpam-4732	248	93	set	set	VERB
ejpam-4732	248	94	v	v	NOUN
ejpam-4732	248	95	of	of	ADP
ejpam-4732	248	96	y	y	PROPN
ejpam-4732	248	97	containing	contain	VERB
ejpam-4732	248	98	f	f	PROPN
ejpam-4732	248	99	(	(	PUNCT
ejpam-4732	248	100	x	x	NOUN
ejpam-4732	248	101	)	)	PUNCT
ejpam-4732	248	102	,	,	PUNCT
ejpam-4732	248	103	there	there	PRON
ejpam-4732	248	104	exists	exist	VERB
ejpam-4732	248	105	a	a	DET
ejpam-4732	248	106	strong	strong	ADJ
ejpam-4732	248	107	β	β	NOUN
ejpam-4732	248	108	-	-	ADJ
ejpam-4732	248	109	i	i	PRON
ejpam-4732	248	110	-open	-open	VERB
ejpam-4732	248	111	set	set	VERB
ejpam-4732	248	112	u	u	NOUN
ejpam-4732	248	113	of	of	ADP
ejpam-4732	248	114	x	x	PUNCT
ejpam-4732	248	115	containing	contain	VERB
ejpam-4732	248	116	x	x	PUNCT
ejpam-4732	248	117	such	such	ADJ
ejpam-4732	248	118	that	that	SCONJ
ejpam-4732	248	119	f	f	PROPN
ejpam-4732	248	120	(	(	PUNCT
ejpam-4732	248	121	u	u	NOUN
ejpam-4732	248	122	)	)	PUNCT
ejpam-4732	248	123	⊆	⊆	NUM
ejpam-4732	248	124	v	v	NOUN
ejpam-4732	248	125	;	;	PUNCT
ejpam-4732	248	126	(	(	PUNCT
ejpam-4732	248	127	4	4	NUM
ejpam-4732	248	128	)	)	PUNCT
ejpam-4732	248	129	f+(v	f+(v	NOUN
ejpam-4732	248	130	)	)	PUNCT
ejpam-4732	248	131	is	be	AUX
ejpam-4732	248	132	strong	strong	ADJ
ejpam-4732	248	133	β	β	NOUN
ejpam-4732	248	134	-	-	PUNCT
ejpam-4732	248	135	i	i	PRON
ejpam-4732	248	136	-open	-open	VERB
ejpam-4732	248	137	in	in	ADP
ejpam-4732	248	138	x	x	PUNCT
ejpam-4732	248	139	for	for	ADP
ejpam-4732	248	140	every	every	DET
ejpam-4732	248	141	r⋆-j	r⋆-j	PROPN
ejpam-4732	248	142	-open	-open	NOUN
ejpam-4732	248	143	set	set	VERB
ejpam-4732	248	144	v	v	NOUN
ejpam-4732	248	145	of	of	ADP
ejpam-4732	248	146	y	y	PROPN
ejpam-4732	248	147	;	;	PUNCT
ejpam-4732	248	148	(	(	PUNCT
ejpam-4732	248	149	5	5	X
ejpam-4732	248	150	)	)	PUNCT
ejpam-4732	248	151	f−(k	f−(k	PROPN
ejpam-4732	248	152	)	)	PUNCT
ejpam-4732	248	153	is	be	AUX
ejpam-4732	248	154	strong	strong	ADJ
ejpam-4732	248	155	β	β	NOUN
ejpam-4732	248	156	-	-	PUNCT
ejpam-4732	248	157	i	i	PRON
ejpam-4732	248	158	-closed	-close	VERB
ejpam-4732	248	159	in	in	ADP
ejpam-4732	248	160	x	x	PUNCT
ejpam-4732	248	161	for	for	ADP
ejpam-4732	248	162	every	every	DET
ejpam-4732	248	163	r⋆-j	r⋆-j	PROPN
ejpam-4732	248	164	-closed	-close	VERB
ejpam-4732	249	1	set	set	NOUN
ejpam-4732	249	2	k	k	PROPN
ejpam-4732	249	3	of	of	ADP
ejpam-4732	249	4	y	y	PROPN
ejpam-4732	249	5	;	;	PUNCT
ejpam-4732	249	6	(	(	PUNCT
ejpam-4732	249	7	6	6	NUM
ejpam-4732	249	8	)	)	PUNCT
ejpam-4732	249	9	f+(v	f+(v	NOUN
ejpam-4732	249	10	)	)	PUNCT
ejpam-4732	250	1	⊆	⊆	NUM
ejpam-4732	250	2	sβinti	sβinti	NOUN
ejpam-4732	250	3	(	(	PUNCT
ejpam-4732	250	4	f+(sclj	f+(sclj	X
ejpam-4732	250	5	(	(	PUNCT
ejpam-4732	250	6	v	v	NOUN
ejpam-4732	250	7	)	)	PUNCT
ejpam-4732	250	8	)	)	PUNCT
ejpam-4732	250	9	)	)	PUNCT
ejpam-4732	250	10	for	for	ADP
ejpam-4732	250	11	every	every	DET
ejpam-4732	250	12	⋆-open	⋆-open	NOUN
ejpam-4732	250	13	set	set	VERB
ejpam-4732	250	14	v	v	NOUN
ejpam-4732	250	15	of	of	ADP
ejpam-4732	250	16	y	y	PROPN
ejpam-4732	250	17	;	;	PUNCT
ejpam-4732	250	18	(	(	PUNCT
ejpam-4732	250	19	7	7	X
ejpam-4732	250	20	)	)	PUNCT
ejpam-4732	250	21	sβcli	sβcli	NOUN
ejpam-4732	250	22	(	(	PUNCT
ejpam-4732	250	23	f−(sintj	f−(sintj	NOUN
ejpam-4732	250	24	(	(	PUNCT
ejpam-4732	250	25	k	k	NOUN
ejpam-4732	250	26	)	)	PUNCT
ejpam-4732	250	27	)	)	PUNCT
ejpam-4732	250	28	)	)	PUNCT
ejpam-4732	251	1	⊆	⊆	X
ejpam-4732	251	2	f−(k	f−(k	PROPN
ejpam-4732	251	3	)	)	PUNCT
ejpam-4732	251	4	for	for	ADP
ejpam-4732	251	5	every	every	DET
ejpam-4732	251	6	⋆-closed	⋆-close	VERB
ejpam-4732	251	7	set	set	NOUN
ejpam-4732	251	8	k	k	PROPN
ejpam-4732	251	9	of	of	ADP
ejpam-4732	251	10	y	y	PROPN
ejpam-4732	251	11	;	;	PUNCT
ejpam-4732	251	12	(	(	PUNCT
ejpam-4732	251	13	8)	8)	NUM
ejpam-4732	251	14	sβcli	sβcli	NOUN
ejpam-4732	251	15	(	(	PUNCT
ejpam-4732	251	16	f−(cl⋆(int(k	f−(cl⋆(int(k	ADJ
ejpam-4732	251	17	)	)	PUNCT
ejpam-4732	251	18	)	)	PUNCT
ejpam-4732	251	19	)	)	PUNCT
ejpam-4732	251	20	)	)	PUNCT
ejpam-4732	252	1	⊆	⊆	X
ejpam-4732	252	2	f−(k	f−(k	PROPN
ejpam-4732	252	3	)	)	PUNCT
ejpam-4732	252	4	for	for	ADP
ejpam-4732	252	5	every	every	DET
ejpam-4732	252	6	⋆-closed	⋆-close	VERB
ejpam-4732	252	7	set	set	NOUN
ejpam-4732	252	8	k	k	PROPN
ejpam-4732	252	9	of	of	ADP
ejpam-4732	252	10	y	y	PROPN
ejpam-4732	252	11	;	;	PUNCT
ejpam-4732	252	12	(	(	PUNCT
ejpam-4732	252	13	9	9	X
ejpam-4732	252	14	)	)	PUNCT
ejpam-4732	252	15	sβcli	sβcli	NOUN
ejpam-4732	252	16	(	(	PUNCT
ejpam-4732	252	17	f−(cl⋆(int(cl⋆(b	f−(cl⋆(int(cl⋆(b	PROPN
ejpam-4732	252	18	)	)	PUNCT
ejpam-4732	252	19	)	)	PUNCT
ejpam-4732	252	20	)	)	PUNCT
ejpam-4732	252	21	)	)	PUNCT
ejpam-4732	252	22	)	)	PUNCT
ejpam-4732	252	23	⊆	⊆	NUM
ejpam-4732	252	24	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4732	252	25	)	)	PUNCT
ejpam-4732	252	26	)	)	PUNCT
ejpam-4732	252	27	for	for	ADP
ejpam-4732	252	28	every	every	DET
ejpam-4732	252	29	subset	subset	NOUN
ejpam-4732	252	30	b	b	PROPN
ejpam-4732	252	31	of	of	ADP
ejpam-4732	252	32	y	y	PROPN
ejpam-4732	252	33	;	;	PUNCT
ejpam-4732	252	34	(	(	PUNCT
ejpam-4732	252	35	10	10	X
ejpam-4732	252	36	)	)	PUNCT
ejpam-4732	252	37	int⋆(cl	int⋆(cl	NOUN
ejpam-4732	252	38	(	(	PUNCT
ejpam-4732	252	39	int⋆(f−(cl⋆(int(k	int⋆(f−(cl⋆(int(k	PROPN
ejpam-4732	252	40	)	)	PUNCT
ejpam-4732	252	41	)	)	PUNCT
ejpam-4732	252	42	)	)	PUNCT
ejpam-4732	252	43	)	)	PUNCT
ejpam-4732	252	44	)	)	PUNCT
ejpam-4732	252	45	)	)	PUNCT
ejpam-4732	253	1	⊆	⊆	X
ejpam-4732	253	2	f−(k	f−(k	PROPN
ejpam-4732	253	3	)	)	PUNCT
ejpam-4732	253	4	for	for	ADP
ejpam-4732	253	5	every	every	DET
ejpam-4732	253	6	⋆-closed	⋆-close	VERB
ejpam-4732	253	7	set	set	NOUN
ejpam-4732	253	8	k	k	PROPN
ejpam-4732	253	9	of	of	ADP
ejpam-4732	253	10	y	y	PROPN
ejpam-4732	253	11	;	;	PUNCT
ejpam-4732	253	12	c.	c.	PROPN
ejpam-4732	253	13	boonpok	boonpok	PROPN
ejpam-4732	253	14	,	,	PUNCT
ejpam-4732	253	15	p.	p.	NOUN
ejpam-4732	253	16	pue	pue	NOUN
ejpam-4732	253	17	-	-	PUNCT
ejpam-4732	253	18	on	on	ADP
ejpam-4732	253	19	/	/	SYM
ejpam-4732	253	20	eur	eur	NOUN
ejpam-4732	253	21	.	.	PUNCT
ejpam-4732	254	1	j.	j.	PROPN
ejpam-4732	254	2	pure	pure	PROPN
ejpam-4732	254	3	appl	appl	PROPN
ejpam-4732	254	4	.	.	PROPN
ejpam-4732	254	5	math	math	PROPN
ejpam-4732	254	6	,	,	PUNCT
ejpam-4732	254	7	16	16	NUM
ejpam-4732	254	8	(	(	PUNCT
ejpam-4732	254	9	3	3	NUM
ejpam-4732	254	10	)	)	PUNCT
ejpam-4732	254	11	(	(	PUNCT
ejpam-4732	254	12	2023	2023	NUM
ejpam-4732	254	13	)	)	PUNCT
ejpam-4732	254	14	,	,	PUNCT
ejpam-4732	254	15	1634	1634	NUM
ejpam-4732	254	16	-	-	SYM
ejpam-4732	254	17	1646	1646	NUM
ejpam-4732	254	18	1642	1642	NUM
ejpam-4732	254	19	(	(	PUNCT
ejpam-4732	254	20	11	11	NUM
ejpam-4732	254	21	)	)	PUNCT
ejpam-4732	254	22	int⋆(cl(int⋆(f−(sintj	int⋆(cl(int⋆(f−(sintj	NOUN
ejpam-4732	254	23	(	(	PUNCT
ejpam-4732	254	24	k	k	NOUN
ejpam-4732	254	25	)	)	PUNCT
ejpam-4732	254	26	)	)	PUNCT
ejpam-4732	254	27	)	)	PUNCT
ejpam-4732	254	28	)	)	PUNCT
ejpam-4732	254	29	)	)	PUNCT
ejpam-4732	255	1	⊆	⊆	X
ejpam-4732	255	2	f−(k	f−(k	PROPN
ejpam-4732	255	3	)	)	PUNCT
ejpam-4732	255	4	for	for	ADP
ejpam-4732	255	5	every	every	DET
ejpam-4732	255	6	⋆-closed	⋆-close	VERB
ejpam-4732	255	7	set	set	NOUN
ejpam-4732	255	8	k	k	PROPN
ejpam-4732	255	9	of	of	ADP
ejpam-4732	255	10	y	y	PROPN
ejpam-4732	255	11	;	;	PUNCT
ejpam-4732	255	12	(	(	PUNCT
ejpam-4732	255	13	12	12	NUM
ejpam-4732	255	14	)	)	PUNCT
ejpam-4732	255	15	f+(v	f+(v	NOUN
ejpam-4732	255	16	)	)	PUNCT
ejpam-4732	256	1	⊆	⊆	X
ejpam-4732	256	2	cl⋆(int(cl⋆(f+(sclj	cl⋆(int(cl⋆(f+(sclj	NOUN
ejpam-4732	256	3	(	(	PUNCT
ejpam-4732	256	4	v	v	NOUN
ejpam-4732	256	5	)	)	PUNCT
ejpam-4732	256	6	)	)	PUNCT
ejpam-4732	256	7	)	)	PUNCT
ejpam-4732	256	8	)	)	PUNCT
ejpam-4732	256	9	)	)	PUNCT
ejpam-4732	257	1	for	for	ADP
ejpam-4732	257	2	every	every	DET
ejpam-4732	257	3	⋆-open	⋆-open	NOUN
ejpam-4732	257	4	set	set	VERB
ejpam-4732	257	5	v	v	NOUN
ejpam-4732	257	6	of	of	ADP
ejpam-4732	257	7	y	y	PROPN
ejpam-4732	257	8	.	.	PUNCT
ejpam-4732	258	1	proof	proof	NOUN
ejpam-4732	258	2	.	.	PUNCT
ejpam-4732	259	1	(	(	PUNCT
ejpam-4732	259	2	1	1	X
ejpam-4732	259	3	)	)	PUNCT
ejpam-4732	259	4	⇒	⇒	NOUN
ejpam-4732	259	5	(	(	PUNCT
ejpam-4732	259	6	2	2	NUM
ejpam-4732	259	7	)	)	PUNCT
ejpam-4732	259	8	and	and	CCONJ
ejpam-4732	259	9	(	(	PUNCT
ejpam-4732	259	10	2	2	X
ejpam-4732	259	11	)	)	PUNCT
ejpam-4732	259	12	⇒	⇒	NOUN
ejpam-4732	259	13	(	(	PUNCT
ejpam-4732	259	14	3	3	NUM
ejpam-4732	259	15	):	):	PUNCT
ejpam-4732	259	16	the	the	DET
ejpam-4732	259	17	proofs	proof	NOUN
ejpam-4732	259	18	are	be	AUX
ejpam-4732	259	19	obvious	obvious	ADJ
ejpam-4732	259	20	.	.	PUNCT
ejpam-4732	260	1	(	(	PUNCT
ejpam-4732	260	2	3	3	X
ejpam-4732	260	3	)	)	PUNCT
ejpam-4732	260	4	⇒	⇒	NOUN
ejpam-4732	260	5	(	(	PUNCT
ejpam-4732	260	6	4	4	NUM
ejpam-4732	260	7	):	):	PUNCT
ejpam-4732	260	8	let	let	VERB
ejpam-4732	260	9	v	v	PART
ejpam-4732	260	10	be	be	AUX
ejpam-4732	260	11	any	any	DET
ejpam-4732	260	12	⋆-open	⋆-open	ADJ
ejpam-4732	260	13	set	set	NOUN
ejpam-4732	260	14	of	of	ADP
ejpam-4732	260	15	y	y	PROPN
ejpam-4732	260	16	and	and	CCONJ
ejpam-4732	260	17	x	x	PROPN
ejpam-4732	260	18	∈	∈	PROPN
ejpam-4732	260	19	f+(v	f+(v	NOUN
ejpam-4732	260	20	)	)	PUNCT
ejpam-4732	260	21	.	.	PUNCT
ejpam-4732	261	1	then	then	ADV
ejpam-4732	261	2	,	,	PUNCT
ejpam-4732	261	3	f	f	PROPN
ejpam-4732	261	4	(	(	PUNCT
ejpam-4732	261	5	x	x	X
ejpam-4732	261	6	)	)	PUNCT
ejpam-4732	261	7	⊆	⊆	NUM
ejpam-4732	261	8	v	v	NOUN
ejpam-4732	261	9	and	and	CCONJ
ejpam-4732	261	10	so	so	ADV
ejpam-4732	261	11	there	there	PRON
ejpam-4732	261	12	exists	exist	VERB
ejpam-4732	261	13	a	a	DET
ejpam-4732	261	14	strong	strong	ADJ
ejpam-4732	261	15	β	β	NOUN
ejpam-4732	261	16	-	-	ADJ
ejpam-4732	261	17	i	i	PRON
ejpam-4732	261	18	-open	-open	NOUN
ejpam-4732	261	19	set	set	VERB
ejpam-4732	261	20	ux	ux	ADP
ejpam-4732	261	21	of	of	ADP
ejpam-4732	261	22	x	x	PUNCT
ejpam-4732	261	23	containing	contain	VERB
ejpam-4732	261	24	x	x	PUNCT
ejpam-4732	261	25	such	such	ADJ
ejpam-4732	261	26	that	that	SCONJ
ejpam-4732	261	27	f	f	PROPN
ejpam-4732	261	28	(	(	PUNCT
ejpam-4732	261	29	ux	ux	PROPN
ejpam-4732	261	30	)	)	PUNCT
ejpam-4732	261	31	⊆	⊆	NUM
ejpam-4732	261	32	v	v	NOUN
ejpam-4732	261	33	.	.	PUNCT
ejpam-4732	262	1	thus	thus	ADV
ejpam-4732	262	2	,	,	PUNCT
ejpam-4732	262	3	x	x	PUNCT
ejpam-4732	262	4	∈	∈	PROPN
ejpam-4732	262	5	ux	ux	NOUN
ejpam-4732	262	6	⊆	⊆	NUM
ejpam-4732	262	7	f+(v	f+(v	NOUN
ejpam-4732	262	8	)	)	PUNCT
ejpam-4732	262	9	and	and	CCONJ
ejpam-4732	262	10	hence	hence	ADV
ejpam-4732	262	11	f+(v	f+(v	NOUN
ejpam-4732	262	12	)	)	PUNCT
ejpam-4732	263	1	=	=	SYM
ejpam-4732	263	2	∪x∈f+(v	∪x∈f+(v	X
ejpam-4732	263	3	)	)	PUNCT
ejpam-4732	263	4	ux	ux	PROPN
ejpam-4732	263	5	.	.	PUNCT
ejpam-4732	264	1	this	this	PRON
ejpam-4732	264	2	shows	show	VERB
ejpam-4732	264	3	that	that	SCONJ
ejpam-4732	264	4	f+(v	f+(v	NOUN
ejpam-4732	264	5	)	)	PUNCT
ejpam-4732	264	6	is	be	AUX
ejpam-4732	264	7	strong	strong	ADJ
ejpam-4732	264	8	β	β	NOUN
ejpam-4732	264	9	-	-	PUNCT
ejpam-4732	264	10	i	i	PRON
ejpam-4732	264	11	-open	-open	ADJ
ejpam-4732	264	12	in	in	ADP
ejpam-4732	264	13	x.	x.	NOUN
ejpam-4732	264	14	(	(	PUNCT
ejpam-4732	264	15	4	4	NUM
ejpam-4732	264	16	)	)	PUNCT
ejpam-4732	264	17	⇒	⇒	NOUN
ejpam-4732	264	18	(	(	PUNCT
ejpam-4732	264	19	5	5	NUM
ejpam-4732	264	20	):	):	PUNCT
ejpam-4732	264	21	this	this	PRON
ejpam-4732	264	22	follows	follow	VERB
ejpam-4732	264	23	from	from	ADP
ejpam-4732	264	24	the	the	DET
ejpam-4732	264	25	fact	fact	NOUN
ejpam-4732	264	26	that	that	SCONJ
ejpam-4732	264	27	f+(y	f+(y	PROPN
ejpam-4732	264	28	−b	−b	ADJ
ejpam-4732	264	29	)	)	PUNCT
ejpam-4732	265	1	=	=	SYM
ejpam-4732	265	2	y	y	PROPN
ejpam-4732	265	3	−	−	PROPN
ejpam-4732	265	4	f−(b	f−(b	PROPN
ejpam-4732	265	5	)	)	PUNCT
ejpam-4732	265	6	for	for	ADP
ejpam-4732	265	7	every	every	DET
ejpam-4732	265	8	subset	subset	NOUN
ejpam-4732	265	9	b	b	PROPN
ejpam-4732	265	10	of	of	ADP
ejpam-4732	265	11	y	y	PROPN
ejpam-4732	265	12	.	.	PUNCT
ejpam-4732	266	1	(	(	PUNCT
ejpam-4732	266	2	5	5	X
ejpam-4732	266	3	)	)	PUNCT
ejpam-4732	266	4	⇒	⇒	NOUN
ejpam-4732	266	5	(	(	PUNCT
ejpam-4732	266	6	6	6	NUM
ejpam-4732	266	7	):	):	PUNCT
ejpam-4732	266	8	let	let	VERB
ejpam-4732	266	9	v	v	PART
ejpam-4732	266	10	be	be	AUX
ejpam-4732	266	11	any	any	DET
ejpam-4732	266	12	⋆-open	⋆-open	ADJ
ejpam-4732	266	13	set	set	NOUN
ejpam-4732	266	14	of	of	ADP
ejpam-4732	266	15	y	y	PROPN
ejpam-4732	266	16	and	and	CCONJ
ejpam-4732	266	17	x	x	PROPN
ejpam-4732	266	18	∈	∈	PROPN
ejpam-4732	266	19	f+(v	f+(v	NOUN
ejpam-4732	266	20	)	)	PUNCT
ejpam-4732	266	21	.	.	PUNCT
ejpam-4732	267	1	then	then	ADV
ejpam-4732	267	2	,	,	PUNCT
ejpam-4732	267	3	f	f	PROPN
ejpam-4732	267	4	(	(	PUNCT
ejpam-4732	267	5	x	x	X
ejpam-4732	267	6	)	)	PUNCT
ejpam-4732	267	7	⊆	⊆	NUM
ejpam-4732	267	8	v	v	ADP
ejpam-4732	267	9	⊆	⊆	NUM
ejpam-4732	267	10	sclj	sclj	NOUN
ejpam-4732	267	11	(	(	PUNCT
ejpam-4732	267	12	v	v	NOUN
ejpam-4732	267	13	)	)	PUNCT
ejpam-4732	267	14	and	and	CCONJ
ejpam-4732	267	15	hence	hence	ADV
ejpam-4732	267	16	x	x	X
ejpam-4732	267	17	∈	∈	PROPN
ejpam-4732	267	18	f+(sclj	f+(sclj	X
ejpam-4732	267	19	(	(	PUNCT
ejpam-4732	267	20	v	v	NOUN
ejpam-4732	267	21	)	)	PUNCT
ejpam-4732	267	22	)	)	PUNCT
ejpam-4732	268	1	=	=	PUNCT
ejpam-4732	268	2	x	x	PUNCT
ejpam-4732	269	1	−	−	NOUN
ejpam-4732	269	2	f−(y	f−(y	NOUN
ejpam-4732	269	3	−	−	NOUN
ejpam-4732	269	4	sclj	sclj	NOUN
ejpam-4732	269	5	(	(	PUNCT
ejpam-4732	269	6	v	v	NOUN
ejpam-4732	269	7	)	)	PUNCT
ejpam-4732	269	8	)	)	PUNCT
ejpam-4732	269	9	.	.	PUNCT
ejpam-4732	270	1	since	since	SCONJ
ejpam-4732	270	2	y	y	PROPN
ejpam-4732	270	3	−	−	PROPN
ejpam-4732	270	4	sclj	sclj	NOUN
ejpam-4732	270	5	(	(	PUNCT
ejpam-4732	270	6	v	v	NOUN
ejpam-4732	270	7	)	)	PUNCT
ejpam-4732	270	8	is	be	AUX
ejpam-4732	270	9	r⋆-j	r⋆-j	PROPN
ejpam-4732	270	10	closed	closed	ADJ
ejpam-4732	270	11	,	,	PUNCT
ejpam-4732	270	12	we	we	PRON
ejpam-4732	270	13	have	have	VERB
ejpam-4732	270	14	f−(y	f−(y	NOUN
ejpam-4732	271	1	−	−	PRON
ejpam-4732	271	2	sclj	sclj	NOUN
ejpam-4732	271	3	(	(	PUNCT
ejpam-4732	271	4	v	v	NOUN
ejpam-4732	271	5	)	)	PUNCT
ejpam-4732	271	6	)	)	PUNCT
ejpam-4732	271	7	is	be	AUX
ejpam-4732	271	8	strong	strong	ADJ
ejpam-4732	271	9	β	β	NOUN
ejpam-4732	271	10	-	-	PUNCT
ejpam-4732	271	11	i	i	PRON
ejpam-4732	271	12	-closed	-close	VERB
ejpam-4732	271	13	in	in	ADP
ejpam-4732	271	14	x.	x.	NOUN
ejpam-4732	271	15	thus	thus	ADV
ejpam-4732	271	16	,	,	PUNCT
ejpam-4732	271	17	f+(sclj	f+(sclj	X
ejpam-4732	271	18	(	(	PUNCT
ejpam-4732	271	19	v	v	NOUN
ejpam-4732	271	20	)	)	PUNCT
ejpam-4732	271	21	)	)	PUNCT
ejpam-4732	271	22	is	be	AUX
ejpam-4732	271	23	a	a	DET
ejpam-4732	271	24	strong	strong	ADJ
ejpam-4732	271	25	β	β	NOUN
ejpam-4732	271	26	-	-	ADJ
ejpam-4732	271	27	i	i	PRON
ejpam-4732	271	28	-open	-open	ADJ
ejpam-4732	271	29	set	set	NOUN
ejpam-4732	271	30	of	of	ADP
ejpam-4732	271	31	x	x	PUNCT
ejpam-4732	271	32	containing	contain	VERB
ejpam-4732	271	33	x	x	X
ejpam-4732	271	34	and	and	CCONJ
ejpam-4732	271	35	so	so	ADV
ejpam-4732	271	36	x	x	SYM
ejpam-4732	271	37	∈	∈	PROPN
ejpam-4732	271	38	sβinti	sβinti	NOUN
ejpam-4732	271	39	(	(	PUNCT
ejpam-4732	271	40	f+(sclj	f+(sclj	X
ejpam-4732	271	41	(	(	PUNCT
ejpam-4732	271	42	v	v	NOUN
ejpam-4732	271	43	)	)	PUNCT
ejpam-4732	271	44	)	)	PUNCT
ejpam-4732	271	45	)	)	PUNCT
ejpam-4732	271	46	.	.	PUNCT
ejpam-4732	272	1	this	this	PRON
ejpam-4732	272	2	shows	show	VERB
ejpam-4732	272	3	that	that	SCONJ
ejpam-4732	272	4	f+(v	f+(v	PROPN
ejpam-4732	272	5	)	)	PUNCT
ejpam-4732	273	1	⊆	⊆	NUM
ejpam-4732	273	2	sβinti	sβinti	NOUN
ejpam-4732	273	3	(	(	PUNCT
ejpam-4732	273	4	f+(sclj	f+(sclj	X
ejpam-4732	273	5	(	(	PUNCT
ejpam-4732	273	6	v	v	NOUN
ejpam-4732	273	7	)	)	PUNCT
ejpam-4732	273	8	)	)	PUNCT
ejpam-4732	273	9	)	)	PUNCT
ejpam-4732	273	10	.	.	PUNCT
ejpam-4732	274	1	(	(	PUNCT
ejpam-4732	274	2	6	6	X
ejpam-4732	274	3	)	)	PUNCT
ejpam-4732	274	4	⇒	⇒	NOUN
ejpam-4732	274	5	(	(	PUNCT
ejpam-4732	274	6	7	7	NUM
ejpam-4732	274	7	):	):	PUNCT
ejpam-4732	274	8	let	let	VERB
ejpam-4732	274	9	k	k	PRON
ejpam-4732	274	10	be	be	AUX
ejpam-4732	274	11	any	any	DET
ejpam-4732	274	12	⋆-closed	⋆-close	VERB
ejpam-4732	274	13	set	set	NOUN
ejpam-4732	274	14	of	of	ADP
ejpam-4732	274	15	y	y	PROPN
ejpam-4732	274	16	.	.	PUNCT
ejpam-4732	275	1	then	then	ADV
ejpam-4732	275	2	,	,	PUNCT
ejpam-4732	275	3	since	since	SCONJ
ejpam-4732	275	4	y	y	PROPN
ejpam-4732	275	5	−k	−k	PROPN
ejpam-4732	275	6	is	be	AUX
ejpam-4732	275	7	⋆-open	⋆-open	ADJ
ejpam-4732	275	8	and	and	CCONJ
ejpam-4732	275	9	by	by	ADP
ejpam-4732	275	10	(	(	PUNCT
ejpam-4732	275	11	6	6	NUM
ejpam-4732	275	12	)	)	PUNCT
ejpam-4732	275	13	,	,	PUNCT
ejpam-4732	275	14	x	x	PUNCT
ejpam-4732	275	15	−	−	DET
ejpam-4732	275	16	f−(k	f−(k	PROPN
ejpam-4732	275	17	)	)	PUNCT
ejpam-4732	275	18	=	=	PUNCT
ejpam-4732	276	1	f+(y	f+(y	PROPN
ejpam-4732	276	2	−k	−k	PROPN
ejpam-4732	276	3	)	)	PUNCT
ejpam-4732	276	4	⊆	⊆	NUM
ejpam-4732	276	5	sβinti	sβinti	NOUN
ejpam-4732	276	6	(	(	PUNCT
ejpam-4732	276	7	f+(sclj	f+(sclj	X
ejpam-4732	276	8	(	(	PUNCT
ejpam-4732	276	9	y	y	PROPN
ejpam-4732	276	10	−k	−k	PROPN
ejpam-4732	276	11	)	)	PUNCT
ejpam-4732	276	12	)	)	PUNCT
ejpam-4732	276	13	)	)	PUNCT
ejpam-4732	277	1	=	=	PRON
ejpam-4732	277	2	sβinti	sβinti	NOUN
ejpam-4732	277	3	(	(	PUNCT
ejpam-4732	277	4	f+(y	f+(y	PROPN
ejpam-4732	277	5	−	−	PROPN
ejpam-4732	277	6	sintj	sintj	NOUN
ejpam-4732	277	7	(	(	PUNCT
ejpam-4732	277	8	k	k	NOUN
ejpam-4732	277	9	)	)	PUNCT
ejpam-4732	277	10	)	)	PUNCT
ejpam-4732	277	11	)	)	PUNCT
ejpam-4732	278	1	=	=	PRON
ejpam-4732	278	2	sβinti	sβinti	NOUN
ejpam-4732	278	3	(	(	PUNCT
ejpam-4732	278	4	x	x	SYM
ejpam-4732	278	5	−	−	PROPN
ejpam-4732	278	6	f−(sintj	f−(sintj	PROPN
ejpam-4732	278	7	(	(	PUNCT
ejpam-4732	278	8	k	k	NOUN
ejpam-4732	278	9	)	)	PUNCT
ejpam-4732	278	10	)	)	PUNCT
ejpam-4732	278	11	)	)	PUNCT
ejpam-4732	279	1	=	=	PUNCT
ejpam-4732	279	2	x	x	PUNCT
ejpam-4732	280	1	−	−	NOUN
ejpam-4732	280	2	sβcli	sβcli	NOUN
ejpam-4732	280	3	(	(	PUNCT
ejpam-4732	280	4	f−(sintj	f−(sintj	NOUN
ejpam-4732	280	5	(	(	PUNCT
ejpam-4732	280	6	k	k	NOUN
ejpam-4732	280	7	)	)	PUNCT
ejpam-4732	280	8	)	)	PUNCT
ejpam-4732	280	9	)	)	PUNCT
ejpam-4732	280	10	.	.	PUNCT
ejpam-4732	281	1	thus	thus	ADV
ejpam-4732	281	2	,	,	PUNCT
ejpam-4732	281	3	sβcli	sβcli	NOUN
ejpam-4732	281	4	(	(	PUNCT
ejpam-4732	281	5	f−(sintj	f−(sintj	NOUN
ejpam-4732	281	6	(	(	PUNCT
ejpam-4732	281	7	k	k	NOUN
ejpam-4732	281	8	)	)	PUNCT
ejpam-4732	281	9	)	)	PUNCT
ejpam-4732	281	10	)	)	PUNCT
ejpam-4732	282	1	⊆	⊆	NUM
ejpam-4732	282	2	f−(k	f−(k	PROPN
ejpam-4732	282	3	)	)	PUNCT
ejpam-4732	282	4	.	.	PUNCT
ejpam-4732	283	1	(	(	PUNCT
ejpam-4732	283	2	7	7	X
ejpam-4732	283	3	)	)	PUNCT
ejpam-4732	283	4	⇒	⇒	NOUN
ejpam-4732	283	5	(	(	PUNCT
ejpam-4732	283	6	8)	8)	NUM
ejpam-4732	283	7	:	:	PUNCT
ejpam-4732	283	8	the	the	DET
ejpam-4732	283	9	proof	proof	NOUN
ejpam-4732	283	10	is	be	AUX
ejpam-4732	283	11	obvious	obvious	ADJ
ejpam-4732	283	12	since	since	SCONJ
ejpam-4732	283	13	sintj	sintj	NOUN
ejpam-4732	283	14	(	(	PUNCT
ejpam-4732	283	15	k	k	NOUN
ejpam-4732	283	16	)	)	PUNCT
ejpam-4732	283	17	=	=	SYM
ejpam-4732	283	18	cl⋆(int(k	cl⋆(int(k	PROPN
ejpam-4732	283	19	)	)	PUNCT
ejpam-4732	283	20	)	)	PUNCT
ejpam-4732	283	21	for	for	ADP
ejpam-4732	283	22	every	every	DET
ejpam-4732	283	23	⋆-closed	⋆-close	VERB
ejpam-4732	283	24	set	set	NOUN
ejpam-4732	283	25	k	k	PROPN
ejpam-4732	283	26	of	of	ADP
ejpam-4732	283	27	y	y	PROPN
ejpam-4732	283	28	.	.	PUNCT
ejpam-4732	284	1	(	(	PUNCT
ejpam-4732	284	2	8)	8)	NUM
ejpam-4732	284	3	⇒	⇒	NOUN
ejpam-4732	284	4	(	(	PUNCT
ejpam-4732	284	5	9	9	NUM
ejpam-4732	284	6	):	):	PUNCT
ejpam-4732	284	7	the	the	DET
ejpam-4732	284	8	proof	proof	NOUN
ejpam-4732	284	9	is	be	AUX
ejpam-4732	284	10	obvious	obvious	ADJ
ejpam-4732	284	11	.	.	PUNCT
ejpam-4732	285	1	(	(	PUNCT
ejpam-4732	285	2	9	9	X
ejpam-4732	285	3	)	)	PUNCT
ejpam-4732	285	4	⇒	⇒	NOUN
ejpam-4732	285	5	(	(	PUNCT
ejpam-4732	285	6	10	10	NUM
ejpam-4732	285	7	):	):	PUNCT
ejpam-4732	285	8	by	by	ADP
ejpam-4732	285	9	(	(	PUNCT
ejpam-4732	285	10	9	9	NUM
ejpam-4732	285	11	)	)	PUNCT
ejpam-4732	285	12	and	and	CCONJ
ejpam-4732	285	13	lemma	lemma	PROPN
ejpam-4732	285	14	2	2	NUM
ejpam-4732	285	15	,	,	PUNCT
ejpam-4732	285	16	int⋆(cl(int⋆(f−(cl⋆(int⋆(k	int⋆(cl(int⋆(f−(cl⋆(int⋆(k	PROPN
ejpam-4732	285	17	)	)	PUNCT
ejpam-4732	285	18	)	)	PUNCT
ejpam-4732	285	19	)	)	PUNCT
ejpam-4732	285	20	)	)	PUNCT
ejpam-4732	285	21	)	)	PUNCT
ejpam-4732	285	22	)	)	PUNCT
ejpam-4732	286	1	⊆	⊆	NUM
ejpam-4732	286	2	sβcli	sβcli	NOUN
ejpam-4732	286	3	(	(	PUNCT
ejpam-4732	286	4	f−(cl⋆(int(k	f−(cl⋆(int(k	ADJ
ejpam-4732	286	5	)	)	PUNCT
ejpam-4732	286	6	)	)	PUNCT
ejpam-4732	286	7	)	)	PUNCT
ejpam-4732	286	8	)	)	PUNCT
ejpam-4732	287	1	⊆	⊆	X
ejpam-4732	287	2	sβcli	sβcli	NOUN
ejpam-4732	287	3	(	(	PUNCT
ejpam-4732	287	4	f−(cl⋆(int(cl⋆(k	f−(cl⋆(int(cl⋆(k	PROPN
ejpam-4732	287	5	)	)	PUNCT
ejpam-4732	287	6	)	)	PUNCT
ejpam-4732	287	7	)	)	PUNCT
ejpam-4732	287	8	)	)	PUNCT
ejpam-4732	287	9	)	)	PUNCT
ejpam-4732	288	1	⊆	⊆	X
ejpam-4732	288	2	f−(cl⋆(k	f−(cl⋆(k	NOUN
ejpam-4732	288	3	)	)	PUNCT
ejpam-4732	288	4	)	)	PUNCT
ejpam-4732	289	1	=	=	SYM
ejpam-4732	289	2	f−(k	f−(k	PROPN
ejpam-4732	289	3	)	)	PUNCT
ejpam-4732	289	4	.	.	PUNCT
ejpam-4732	290	1	(	(	PUNCT
ejpam-4732	290	2	10	10	NUM
ejpam-4732	290	3	)	)	PUNCT
ejpam-4732	290	4	⇒	⇒	NOUN
ejpam-4732	290	5	(	(	PUNCT
ejpam-4732	290	6	11	11	NUM
ejpam-4732	290	7	):	):	PUNCT
ejpam-4732	290	8	the	the	DET
ejpam-4732	290	9	proof	proof	NOUN
ejpam-4732	290	10	is	be	AUX
ejpam-4732	290	11	obvious	obvious	ADJ
ejpam-4732	290	12	since	since	SCONJ
ejpam-4732	290	13	sintj	sintj	NOUN
ejpam-4732	290	14	(	(	PUNCT
ejpam-4732	290	15	k	k	NOUN
ejpam-4732	290	16	)	)	PUNCT
ejpam-4732	290	17	=	=	SYM
ejpam-4732	290	18	cl⋆(int(k	cl⋆(int(k	PROPN
ejpam-4732	290	19	)	)	PUNCT
ejpam-4732	290	20	)	)	PUNCT
ejpam-4732	290	21	for	for	ADP
ejpam-4732	290	22	every	every	DET
ejpam-4732	290	23	⋆-closed	⋆-close	VERB
ejpam-4732	290	24	set	set	NOUN
ejpam-4732	290	25	k	k	PROPN
ejpam-4732	290	26	of	of	ADP
ejpam-4732	290	27	y	y	PROPN
ejpam-4732	290	28	.	.	PUNCT
ejpam-4732	291	1	(	(	PUNCT
ejpam-4732	291	2	11	11	NUM
ejpam-4732	291	3	)	)	PUNCT
ejpam-4732	291	4	⇒	⇒	NOUN
ejpam-4732	291	5	(	(	PUNCT
ejpam-4732	291	6	12	12	NUM
ejpam-4732	291	7	):	):	PUNCT
ejpam-4732	291	8	let	let	VERB
ejpam-4732	291	9	v	v	PART
ejpam-4732	291	10	be	be	AUX
ejpam-4732	291	11	any	any	DET
ejpam-4732	291	12	⋆-open	⋆-open	ADJ
ejpam-4732	291	13	set	set	NOUN
ejpam-4732	291	14	of	of	ADP
ejpam-4732	291	15	y	y	PROPN
ejpam-4732	291	16	.	.	PUNCT
ejpam-4732	292	1	then	then	ADV
ejpam-4732	292	2	,	,	PUNCT
ejpam-4732	292	3	y	y	PROPN
ejpam-4732	292	4	−v	−v	NOUN
ejpam-4732	292	5	is	be	AUX
ejpam-4732	292	6	⋆-closed	⋆-close	VERB
ejpam-4732	292	7	in	in	ADP
ejpam-4732	292	8	y	y	PROPN
ejpam-4732	292	9	and	and	CCONJ
ejpam-4732	292	10	by	by	ADP
ejpam-4732	292	11	(	(	PUNCT
ejpam-4732	292	12	11	11	NUM
ejpam-4732	292	13	)	)	PUNCT
ejpam-4732	292	14	,	,	PUNCT
ejpam-4732	292	15	int⋆(cl(int⋆(f−(sintj	int⋆(cl(int⋆(f−(sintj	NOUN
ejpam-4732	292	16	(	(	PUNCT
ejpam-4732	292	17	y	y	PROPN
ejpam-4732	292	18	−	−	PROPN
ejpam-4732	292	19	v	v	NOUN
ejpam-4732	292	20	)	)	PUNCT
ejpam-4732	292	21	)	)	PUNCT
ejpam-4732	292	22	)	)	PUNCT
ejpam-4732	292	23	)	)	PUNCT
ejpam-4732	292	24	)	)	PUNCT
ejpam-4732	293	1	⊆	⊆	NUM
ejpam-4732	293	2	f−(y	f−(y	NOUN
ejpam-4732	293	3	−	−	NOUN
ejpam-4732	293	4	v	v	NOUN
ejpam-4732	293	5	)	)	PUNCT
ejpam-4732	293	6	=	=	PUNCT
ejpam-4732	293	7	x	x	X
ejpam-4732	293	8	−	−	PROPN
ejpam-4732	293	9	f+(v	f+(v	NOUN
ejpam-4732	293	10	)	)	PUNCT
ejpam-4732	293	11	.	.	PUNCT
ejpam-4732	294	1	moreover	moreover	ADV
ejpam-4732	294	2	,	,	PUNCT
ejpam-4732	294	3	we	we	PRON
ejpam-4732	294	4	have	have	VERB
ejpam-4732	294	5	int⋆(cl(int⋆(f−(sintj	int⋆(cl(int⋆(f−(sintj	NOUN
ejpam-4732	294	6	(	(	PUNCT
ejpam-4732	294	7	y	y	PROPN
ejpam-4732	294	8	−	−	PROPN
ejpam-4732	294	9	v	v	NOUN
ejpam-4732	294	10	)	)	PUNCT
ejpam-4732	294	11	)	)	PUNCT
ejpam-4732	294	12	)	)	PUNCT
ejpam-4732	294	13	)	)	PUNCT
ejpam-4732	294	14	)	)	PUNCT
ejpam-4732	295	1	=	=	NOUN
ejpam-4732	295	2	int⋆(cl(int⋆(f−(y	int⋆(cl(int⋆(f−(y	NOUN
ejpam-4732	295	3	−	−	VERB
ejpam-4732	295	4	sclj	sclj	NOUN
ejpam-4732	295	5	(	(	PUNCT
ejpam-4732	295	6	v	v	NOUN
ejpam-4732	295	7	)	)	PUNCT
ejpam-4732	295	8	)	)	PUNCT
ejpam-4732	295	9	)	)	PUNCT
ejpam-4732	295	10	)	)	PUNCT
ejpam-4732	295	11	)	)	PUNCT
ejpam-4732	296	1	=	=	PRON
ejpam-4732	296	2	int⋆(cl(int⋆(x	int⋆(cl(int⋆(x	VERB
ejpam-4732	296	3	−	−	PROPN
ejpam-4732	296	4	f+(sclj	f+(sclj	ADJ
ejpam-4732	296	5	(	(	PUNCT
ejpam-4732	296	6	v	v	NOUN
ejpam-4732	296	7	)	)	PUNCT
ejpam-4732	296	8	)	)	PUNCT
ejpam-4732	296	9	)	)	PUNCT
ejpam-4732	296	10	)	)	PUNCT
ejpam-4732	296	11	)	)	PUNCT
ejpam-4732	297	1	c.	c.	PROPN
ejpam-4732	297	2	boonpok	boonpok	PROPN
ejpam-4732	297	3	,	,	PUNCT
ejpam-4732	297	4	p.	p.	NOUN
ejpam-4732	297	5	pue	pue	NOUN
ejpam-4732	297	6	-	-	PUNCT
ejpam-4732	297	7	on	on	ADP
ejpam-4732	297	8	/	/	SYM
ejpam-4732	297	9	eur	eur	NOUN
ejpam-4732	297	10	.	.	PUNCT
ejpam-4732	298	1	j.	j.	PROPN
ejpam-4732	298	2	pure	pure	PROPN
ejpam-4732	298	3	appl	appl	PROPN
ejpam-4732	298	4	.	.	PROPN
ejpam-4732	298	5	math	math	PROPN
ejpam-4732	298	6	,	,	PUNCT
ejpam-4732	298	7	16	16	NUM
ejpam-4732	298	8	(	(	PUNCT
ejpam-4732	298	9	3	3	NUM
ejpam-4732	298	10	)	)	PUNCT
ejpam-4732	298	11	(	(	PUNCT
ejpam-4732	298	12	2023	2023	NUM
ejpam-4732	298	13	)	)	PUNCT
ejpam-4732	298	14	,	,	PUNCT
ejpam-4732	298	15	1634	1634	NUM
ejpam-4732	298	16	-	-	SYM
ejpam-4732	298	17	1646	1646	NUM
ejpam-4732	298	18	1643	1643	NUM
ejpam-4732	298	19	=	=	PUNCT
ejpam-4732	298	20	x	x	SYM
ejpam-4732	299	1	−	−	NOUN
ejpam-4732	299	2	cl⋆(int(cl⋆(f+(sclj	cl⋆(int(cl⋆(f+(sclj	NOUN
ejpam-4732	299	3	(	(	PUNCT
ejpam-4732	299	4	v	v	NOUN
ejpam-4732	299	5	)	)	PUNCT
ejpam-4732	299	6	)	)	PUNCT
ejpam-4732	299	7	)	)	PUNCT
ejpam-4732	299	8	)	)	PUNCT
ejpam-4732	299	9	)	)	PUNCT
ejpam-4732	299	10	.	.	PUNCT
ejpam-4732	300	1	thus	thus	ADV
ejpam-4732	300	2	,	,	PUNCT
ejpam-4732	300	3	f+(v	f+(v	PROPN
ejpam-4732	300	4	)	)	PUNCT
ejpam-4732	300	5	⊆	⊆	NUM
ejpam-4732	300	6	cl⋆(int(cl⋆(f+(sclj	cl⋆(int(cl⋆(f+(sclj	NOUN
ejpam-4732	300	7	(	(	PUNCT
ejpam-4732	300	8	v	v	NOUN
ejpam-4732	300	9	)	)	PUNCT
ejpam-4732	300	10	)	)	PUNCT
ejpam-4732	300	11	)	)	PUNCT
ejpam-4732	300	12	)	)	PUNCT
ejpam-4732	300	13	)	)	PUNCT
ejpam-4732	300	14	.	.	PUNCT
ejpam-4732	301	1	(	(	PUNCT
ejpam-4732	301	2	12	12	NUM
ejpam-4732	301	3	)	)	PUNCT
ejpam-4732	301	4	⇒	⇒	NOUN
ejpam-4732	301	5	(	(	PUNCT
ejpam-4732	301	6	1	1	NUM
ejpam-4732	301	7	):	):	PUNCT
ejpam-4732	301	8	let	let	VERB
ejpam-4732	301	9	x	x	PRON
ejpam-4732	301	10	be	be	AUX
ejpam-4732	301	11	any	any	DET
ejpam-4732	301	12	point	point	NOUN
ejpam-4732	301	13	of	of	ADP
ejpam-4732	301	14	x	x	PUNCT
ejpam-4732	301	15	and	and	CCONJ
ejpam-4732	301	16	v	v	AUX
ejpam-4732	301	17	be	be	AUX
ejpam-4732	301	18	any	any	DET
ejpam-4732	301	19	⋆-open	⋆-open	ADJ
ejpam-4732	301	20	set	set	NOUN
ejpam-4732	301	21	of	of	ADP
ejpam-4732	301	22	y	y	PROPN
ejpam-4732	301	23	containing	contain	VERB
ejpam-4732	301	24	f	f	PROPN
ejpam-4732	301	25	(	(	PUNCT
ejpam-4732	301	26	x	x	NOUN
ejpam-4732	301	27	)	)	PUNCT
ejpam-4732	301	28	.	.	PUNCT
ejpam-4732	302	1	then	then	ADV
ejpam-4732	302	2	,	,	PUNCT
ejpam-4732	302	3	we	we	PRON
ejpam-4732	302	4	have	have	VERB
ejpam-4732	302	5	x	x	X
ejpam-4732	302	6	∈	∈	NOUN
ejpam-4732	302	7	f+(v	f+(v	NOUN
ejpam-4732	302	8	)	)	PUNCT
ejpam-4732	303	1	⊆	⊆	X
ejpam-4732	303	2	cl⋆(int(cl⋆(f+(sclj	cl⋆(int(cl⋆(f+(sclj	NOUN
ejpam-4732	303	3	(	(	PUNCT
ejpam-4732	303	4	v	v	NOUN
ejpam-4732	303	5	)	)	PUNCT
ejpam-4732	303	6	)	)	PUNCT
ejpam-4732	303	7	)	)	PUNCT
ejpam-4732	303	8	)	)	PUNCT
ejpam-4732	303	9	)	)	PUNCT
ejpam-4732	303	10	and	and	CCONJ
ejpam-4732	303	11	hence	hence	ADV
ejpam-4732	303	12	x	x	X
ejpam-4732	303	13	∈	∈	NOUN
ejpam-4732	303	14	sβinti	sβinti	NOUN
ejpam-4732	303	15	(	(	PUNCT
ejpam-4732	303	16	f+(sclj	f+(sclj	X
ejpam-4732	303	17	(	(	PUNCT
ejpam-4732	303	18	v	v	NOUN
ejpam-4732	303	19	)	)	PUNCT
ejpam-4732	303	20	)	)	PUNCT
ejpam-4732	303	21	)	)	PUNCT
ejpam-4732	303	22	.	.	PUNCT
ejpam-4732	304	1	thus	thus	ADV
ejpam-4732	304	2	,	,	PUNCT
ejpam-4732	304	3	f	f	PROPN
ejpam-4732	304	4	is	be	AUX
ejpam-4732	304	5	upper	upper	ADJ
ejpam-4732	304	6	almost	almost	ADV
ejpam-4732	304	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	304	8	at	at	ADP
ejpam-4732	304	9	x	x	PUNCT
ejpam-4732	304	10	by	by	ADP
ejpam-4732	304	11	theorem	theorem	NOUN
ejpam-4732	304	12	5	5	NUM
ejpam-4732	304	13	.	.	PUNCT
ejpam-4732	304	14	theorem	theorem	VERB
ejpam-4732	304	15	8	8	NUM
ejpam-4732	304	16	.	.	PUNCT
ejpam-4732	304	17	for	for	ADP
ejpam-4732	304	18	a	a	DET
ejpam-4732	304	19	multifunction	multifunction	NOUN
ejpam-4732	304	20	f	f	NOUN
ejpam-4732	304	21	:	:	PUNCT
ejpam-4732	304	22	(	(	PUNCT
ejpam-4732	304	23	x	x	X
ejpam-4732	304	24	,	,	PUNCT
ejpam-4732	304	25	τ	τ	PROPN
ejpam-4732	304	26	,	,	PUNCT
ejpam-4732	304	27	i	i	NOUN
ejpam-4732	304	28	)	)	PUNCT
ejpam-4732	304	29	→	→	PUNCT
ejpam-4732	304	30	(	(	PUNCT
ejpam-4732	304	31	y	y	PROPN
ejpam-4732	304	32	,	,	PUNCT
ejpam-4732	304	33	σ	σ	PROPN
ejpam-4732	304	34	,	,	PUNCT
ejpam-4732	304	35	j	j	PROPN
ejpam-4732	304	36	)	)	PUNCT
ejpam-4732	304	37	,	,	PUNCT
ejpam-4732	304	38	the	the	DET
ejpam-4732	304	39	following	follow	VERB
ejpam-4732	304	40	properties	property	NOUN
ejpam-4732	304	41	are	be	AUX
ejpam-4732	304	42	equivalent	equivalent	ADJ
ejpam-4732	304	43	:	:	PUNCT
ejpam-4732	304	44	(	(	PUNCT
ejpam-4732	304	45	1	1	X
ejpam-4732	304	46	)	)	PUNCT
ejpam-4732	304	47	f	f	PROPN
ejpam-4732	304	48	is	be	AUX
ejpam-4732	304	49	lower	low	ADJ
ejpam-4732	304	50	almost	almost	ADV
ejpam-4732	304	51	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	304	52	;	;	PUNCT
ejpam-4732	304	53	(	(	PUNCT
ejpam-4732	304	54	2	2	X
ejpam-4732	304	55	)	)	PUNCT
ejpam-4732	304	56	for	for	SCONJ
ejpam-4732	304	57	each	each	DET
ejpam-4732	304	58	x	x	SYM
ejpam-4732	304	59	∈	∈	PROPN
ejpam-4732	304	60	x	x	X
ejpam-4732	304	61	and	and	CCONJ
ejpam-4732	304	62	each	each	DET
ejpam-4732	304	63	⋆-open	⋆-open	ADV
ejpam-4732	304	64	set	set	VERB
ejpam-4732	304	65	v	v	NUM
ejpam-4732	304	66	of	of	ADP
ejpam-4732	304	67	y	y	PRON
ejpam-4732	304	68	such	such	ADJ
ejpam-4732	304	69	that	that	SCONJ
ejpam-4732	304	70	f	f	PROPN
ejpam-4732	304	71	(	(	PUNCT
ejpam-4732	304	72	x	x	NOUN
ejpam-4732	304	73	)	)	PUNCT
ejpam-4732	304	74	∩	∩	NOUN
ejpam-4732	304	75	v	v	ADP
ejpam-4732	304	76	̸=	̸=	PROPN
ejpam-4732	304	77	∅	∅	NOUN
ejpam-4732	304	78	,	,	PUNCT
ejpam-4732	304	79	there	there	PRON
ejpam-4732	304	80	exists	exist	VERB
ejpam-4732	304	81	a	a	DET
ejpam-4732	304	82	strong	strong	ADJ
ejpam-4732	304	83	β	β	NOUN
ejpam-4732	304	84	-	-	ADJ
ejpam-4732	304	85	i	i	PRON
ejpam-4732	304	86	-open	-open	VERB
ejpam-4732	304	87	set	set	VERB
ejpam-4732	304	88	u	u	NOUN
ejpam-4732	304	89	of	of	ADP
ejpam-4732	304	90	x	x	PUNCT
ejpam-4732	304	91	containing	contain	VERB
ejpam-4732	304	92	x	x	PUNCT
ejpam-4732	304	93	such	such	ADJ
ejpam-4732	304	94	that	that	SCONJ
ejpam-4732	304	95	u	u	NOUN
ejpam-4732	304	96	⊆	⊆	NUM
ejpam-4732	304	97	f−(sclj	f−(sclj	NOUN
ejpam-4732	304	98	(	(	PUNCT
ejpam-4732	304	99	v	v	NOUN
ejpam-4732	304	100	)	)	PUNCT
ejpam-4732	304	101	)	)	PUNCT
ejpam-4732	304	102	;	;	PUNCT
ejpam-4732	304	103	(	(	PUNCT
ejpam-4732	304	104	3	3	X
ejpam-4732	304	105	)	)	PUNCT
ejpam-4732	304	106	for	for	ADP
ejpam-4732	304	107	each	each	DET
ejpam-4732	304	108	x	x	SYM
ejpam-4732	304	109	∈	∈	PROPN
ejpam-4732	304	110	x	x	X
ejpam-4732	304	111	and	and	CCONJ
ejpam-4732	304	112	each	each	DET
ejpam-4732	304	113	r⋆-j	r⋆-j	PROPN
ejpam-4732	304	114	-open	-open	PROPN
ejpam-4732	304	115	set	set	VERB
ejpam-4732	304	116	v	v	NOUN
ejpam-4732	304	117	of	of	ADP
ejpam-4732	304	118	y	y	PRON
ejpam-4732	304	119	such	such	ADJ
ejpam-4732	304	120	that	that	SCONJ
ejpam-4732	304	121	f	f	PROPN
ejpam-4732	304	122	(	(	PUNCT
ejpam-4732	304	123	x)∩v	x)∩v	PROPN
ejpam-4732	304	124	̸=	̸=	PROPN
ejpam-4732	304	125	∅	∅	NOUN
ejpam-4732	304	126	,	,	PUNCT
ejpam-4732	304	127	there	there	PRON
ejpam-4732	304	128	exists	exist	VERB
ejpam-4732	304	129	a	a	DET
ejpam-4732	304	130	strong	strong	ADJ
ejpam-4732	304	131	β	β	NOUN
ejpam-4732	304	132	-	-	ADJ
ejpam-4732	304	133	i	i	PRON
ejpam-4732	304	134	-open	-open	VERB
ejpam-4732	304	135	set	set	VERB
ejpam-4732	304	136	u	u	NOUN
ejpam-4732	304	137	of	of	ADP
ejpam-4732	304	138	x	x	PUNCT
ejpam-4732	304	139	containing	contain	VERB
ejpam-4732	304	140	x	x	PUNCT
ejpam-4732	304	141	such	such	ADJ
ejpam-4732	304	142	that	that	SCONJ
ejpam-4732	304	143	u	u	NOUN
ejpam-4732	304	144	⊆	⊆	NUM
ejpam-4732	304	145	f−(v	f−(v	NOUN
ejpam-4732	304	146	)	)	PUNCT
ejpam-4732	304	147	;	;	PUNCT
ejpam-4732	304	148	(	(	PUNCT
ejpam-4732	304	149	4	4	X
ejpam-4732	304	150	)	)	PUNCT
ejpam-4732	304	151	f−(v	f−(v	NOUN
ejpam-4732	304	152	)	)	PUNCT
ejpam-4732	304	153	is	be	AUX
ejpam-4732	304	154	strong	strong	ADJ
ejpam-4732	304	155	β	β	NOUN
ejpam-4732	304	156	-	-	PUNCT
ejpam-4732	304	157	i	i	PRON
ejpam-4732	304	158	-open	-open	VERB
ejpam-4732	304	159	in	in	ADP
ejpam-4732	304	160	x	x	PUNCT
ejpam-4732	304	161	for	for	ADP
ejpam-4732	304	162	every	every	DET
ejpam-4732	304	163	r⋆-j	r⋆-j	PROPN
ejpam-4732	304	164	-open	-open	NOUN
ejpam-4732	304	165	set	set	VERB
ejpam-4732	304	166	v	v	NOUN
ejpam-4732	304	167	of	of	ADP
ejpam-4732	304	168	y	y	PROPN
ejpam-4732	304	169	;	;	PUNCT
ejpam-4732	304	170	(	(	PUNCT
ejpam-4732	304	171	5	5	X
ejpam-4732	304	172	)	)	PUNCT
ejpam-4732	304	173	f+(k	f+(k	PROPN
ejpam-4732	304	174	)	)	PUNCT
ejpam-4732	304	175	is	be	AUX
ejpam-4732	304	176	strong	strong	ADJ
ejpam-4732	304	177	β	β	NOUN
ejpam-4732	304	178	-	-	PUNCT
ejpam-4732	304	179	i	i	PRON
ejpam-4732	304	180	-closed	-close	VERB
ejpam-4732	304	181	in	in	ADP
ejpam-4732	304	182	x	x	PUNCT
ejpam-4732	304	183	for	for	ADP
ejpam-4732	304	184	every	every	DET
ejpam-4732	304	185	r⋆-j	r⋆-j	PROPN
ejpam-4732	304	186	-closed	-close	VERB
ejpam-4732	304	187	set	set	NOUN
ejpam-4732	304	188	k	k	PROPN
ejpam-4732	304	189	of	of	ADP
ejpam-4732	304	190	y	y	PROPN
ejpam-4732	304	191	;	;	PUNCT
ejpam-4732	304	192	(	(	PUNCT
ejpam-4732	304	193	6	6	X
ejpam-4732	304	194	)	)	PUNCT
ejpam-4732	304	195	f−(v	f−(v	NOUN
ejpam-4732	304	196	)	)	PUNCT
ejpam-4732	304	197	⊆	⊆	NUM
ejpam-4732	304	198	sβinti	sβinti	NOUN
ejpam-4732	304	199	(	(	PUNCT
ejpam-4732	304	200	f−(sclj	f−(sclj	NOUN
ejpam-4732	304	201	(	(	PUNCT
ejpam-4732	304	202	v	v	NOUN
ejpam-4732	304	203	)	)	PUNCT
ejpam-4732	304	204	)	)	PUNCT
ejpam-4732	304	205	)	)	PUNCT
ejpam-4732	304	206	for	for	ADP
ejpam-4732	304	207	every	every	DET
ejpam-4732	304	208	⋆-open	⋆-open	NOUN
ejpam-4732	304	209	set	set	VERB
ejpam-4732	304	210	v	v	NOUN
ejpam-4732	304	211	of	of	ADP
ejpam-4732	304	212	y	y	PROPN
ejpam-4732	304	213	;	;	PUNCT
ejpam-4732	304	214	(	(	PUNCT
ejpam-4732	304	215	7	7	X
ejpam-4732	304	216	)	)	PUNCT
ejpam-4732	304	217	sβcli	sβcli	NOUN
ejpam-4732	304	218	(	(	PUNCT
ejpam-4732	304	219	f+(sintj	f+(sintj	PROPN
ejpam-4732	304	220	(	(	PUNCT
ejpam-4732	304	221	k	k	NOUN
ejpam-4732	304	222	)	)	PUNCT
ejpam-4732	304	223	)	)	PUNCT
ejpam-4732	304	224	)	)	PUNCT
ejpam-4732	305	1	⊆	⊆	NUM
ejpam-4732	305	2	f+(k	f+(k	NOUN
ejpam-4732	305	3	)	)	PUNCT
ejpam-4732	305	4	for	for	ADP
ejpam-4732	305	5	every	every	DET
ejpam-4732	305	6	⋆-closed	⋆-close	VERB
ejpam-4732	305	7	set	set	NOUN
ejpam-4732	305	8	k	k	PROPN
ejpam-4732	305	9	of	of	ADP
ejpam-4732	305	10	y	y	PROPN
ejpam-4732	305	11	;	;	PUNCT
ejpam-4732	305	12	(	(	PUNCT
ejpam-4732	305	13	8)	8)	NUM
ejpam-4732	305	14	sβcli	sβcli	NOUN
ejpam-4732	305	15	(	(	PUNCT
ejpam-4732	305	16	f+(cl⋆(int(k	f+(cl⋆(int(k	NOUN
ejpam-4732	305	17	)	)	PUNCT
ejpam-4732	305	18	)	)	PUNCT
ejpam-4732	305	19	)	)	PUNCT
ejpam-4732	305	20	)	)	PUNCT
ejpam-4732	305	21	⊆	⊆	NUM
ejpam-4732	305	22	f+(k	f+(k	NOUN
ejpam-4732	305	23	)	)	PUNCT
ejpam-4732	305	24	for	for	ADP
ejpam-4732	305	25	every	every	DET
ejpam-4732	305	26	⋆-closed	⋆-close	VERB
ejpam-4732	305	27	set	set	NOUN
ejpam-4732	305	28	k	k	PROPN
ejpam-4732	305	29	of	of	ADP
ejpam-4732	305	30	y	y	PROPN
ejpam-4732	305	31	;	;	PUNCT
ejpam-4732	305	32	(	(	PUNCT
ejpam-4732	305	33	9	9	X
ejpam-4732	305	34	)	)	PUNCT
ejpam-4732	305	35	sβcli	sβcli	NOUN
ejpam-4732	305	36	(	(	PUNCT
ejpam-4732	305	37	f+(cl⋆(int(cl⋆(b	f+(cl⋆(int(cl⋆(b	NOUN
ejpam-4732	305	38	)	)	PUNCT
ejpam-4732	305	39	)	)	PUNCT
ejpam-4732	305	40	)	)	PUNCT
ejpam-4732	305	41	)	)	PUNCT
ejpam-4732	305	42	)	)	PUNCT
ejpam-4732	306	1	⊆	⊆	NUM
ejpam-4732	306	2	f+(cl⋆(b	f+(cl⋆(b	NOUN
ejpam-4732	306	3	)	)	PUNCT
ejpam-4732	306	4	)	)	PUNCT
ejpam-4732	306	5	for	for	ADP
ejpam-4732	306	6	every	every	DET
ejpam-4732	306	7	subset	subset	NOUN
ejpam-4732	306	8	b	b	PROPN
ejpam-4732	306	9	of	of	ADP
ejpam-4732	306	10	y	y	PROPN
ejpam-4732	306	11	;	;	PUNCT
ejpam-4732	306	12	(	(	PUNCT
ejpam-4732	306	13	10	10	X
ejpam-4732	306	14	)	)	PUNCT
ejpam-4732	306	15	int⋆(cl	int⋆(cl	NOUN
ejpam-4732	306	16	(	(	PUNCT
ejpam-4732	306	17	int⋆(f+(cl⋆(int(k	int⋆(f+(cl⋆(int(k	NOUN
ejpam-4732	306	18	)	)	PUNCT
ejpam-4732	306	19	)	)	PUNCT
ejpam-4732	306	20	)	)	PUNCT
ejpam-4732	306	21	)	)	PUNCT
ejpam-4732	306	22	)	)	PUNCT
ejpam-4732	306	23	)	)	PUNCT
ejpam-4732	307	1	⊆	⊆	NUM
ejpam-4732	307	2	f+(k	f+(k	NOUN
ejpam-4732	307	3	)	)	PUNCT
ejpam-4732	307	4	for	for	ADP
ejpam-4732	307	5	every	every	DET
ejpam-4732	307	6	⋆-closed	⋆-close	VERB
ejpam-4732	307	7	set	set	NOUN
ejpam-4732	307	8	k	k	PROPN
ejpam-4732	307	9	of	of	ADP
ejpam-4732	307	10	y	y	PROPN
ejpam-4732	307	11	;	;	PUNCT
ejpam-4732	307	12	(	(	PUNCT
ejpam-4732	307	13	11	11	X
ejpam-4732	307	14	)	)	PUNCT
ejpam-4732	307	15	int⋆(cl	int⋆(cl	NOUN
ejpam-4732	307	16	(	(	PUNCT
ejpam-4732	307	17	int⋆(f+(sintj	int⋆(f+(sintj	PROPN
ejpam-4732	307	18	(	(	PUNCT
ejpam-4732	307	19	k	k	NOUN
ejpam-4732	307	20	)	)	PUNCT
ejpam-4732	307	21	)	)	PUNCT
ejpam-4732	307	22	)	)	PUNCT
ejpam-4732	307	23	)	)	PUNCT
ejpam-4732	307	24	)	)	PUNCT
ejpam-4732	308	1	⊆	⊆	NUM
ejpam-4732	308	2	f+(k	f+(k	NOUN
ejpam-4732	308	3	)	)	PUNCT
ejpam-4732	308	4	for	for	ADP
ejpam-4732	308	5	every	every	DET
ejpam-4732	308	6	⋆-closed	⋆-close	VERB
ejpam-4732	308	7	set	set	NOUN
ejpam-4732	308	8	k	k	PROPN
ejpam-4732	308	9	of	of	ADP
ejpam-4732	308	10	y	y	PROPN
ejpam-4732	308	11	;	;	PUNCT
ejpam-4732	308	12	(	(	PUNCT
ejpam-4732	308	13	12	12	X
ejpam-4732	308	14	)	)	PUNCT
ejpam-4732	308	15	f−(v	f−(v	NOUN
ejpam-4732	308	16	)	)	PUNCT
ejpam-4732	308	17	⊆	⊆	X
ejpam-4732	308	18	cl⋆	cl⋆	PROPN
ejpam-4732	308	19	(	(	PUNCT
ejpam-4732	308	20	int(cl⋆(f−(sclj	int(cl⋆(f−(sclj	X
ejpam-4732	308	21	(	(	PUNCT
ejpam-4732	308	22	v	v	NOUN
ejpam-4732	308	23	)	)	PUNCT
ejpam-4732	308	24	)	)	PUNCT
ejpam-4732	308	25	)	)	PUNCT
ejpam-4732	308	26	)	)	PUNCT
ejpam-4732	308	27	)	)	PUNCT
ejpam-4732	308	28	for	for	ADP
ejpam-4732	308	29	every	every	DET
ejpam-4732	308	30	⋆-open	⋆-open	NOUN
ejpam-4732	308	31	set	set	VERB
ejpam-4732	308	32	v	v	NOUN
ejpam-4732	308	33	of	of	ADP
ejpam-4732	308	34	y	y	PROPN
ejpam-4732	308	35	.	.	PUNCT
ejpam-4732	309	1	proof	proof	NOUN
ejpam-4732	309	2	.	.	PUNCT
ejpam-4732	310	1	the	the	DET
ejpam-4732	310	2	proof	proof	NOUN
ejpam-4732	310	3	is	be	AUX
ejpam-4732	310	4	similar	similar	ADJ
ejpam-4732	310	5	to	to	ADP
ejpam-4732	310	6	that	that	PRON
ejpam-4732	310	7	of	of	ADP
ejpam-4732	310	8	theorem	theorem	ADJ
ejpam-4732	310	9	7	7	NUM
ejpam-4732	310	10	.	.	PUNCT
ejpam-4732	310	11	corollary	corollary	ADJ
ejpam-4732	310	12	4	4	NUM
ejpam-4732	310	13	.	.	PUNCT
ejpam-4732	310	14	for	for	ADP
ejpam-4732	310	15	a	a	DET
ejpam-4732	310	16	function	function	NOUN
ejpam-4732	310	17	f	f	NOUN
ejpam-4732	310	18	:	:	PUNCT
ejpam-4732	310	19	(	(	PUNCT
ejpam-4732	310	20	x	x	X
ejpam-4732	310	21	,	,	PUNCT
ejpam-4732	310	22	τ	τ	PROPN
ejpam-4732	310	23	,	,	PUNCT
ejpam-4732	310	24	i	i	NOUN
ejpam-4732	310	25	)	)	PUNCT
ejpam-4732	310	26	→	→	PUNCT
ejpam-4732	310	27	(	(	PUNCT
ejpam-4732	310	28	y	y	PROPN
ejpam-4732	310	29	,	,	PUNCT
ejpam-4732	310	30	σ	σ	PROPN
ejpam-4732	310	31	,	,	PUNCT
ejpam-4732	310	32	j	j	PROPN
ejpam-4732	310	33	)	)	PUNCT
ejpam-4732	310	34	,	,	PUNCT
ejpam-4732	310	35	the	the	DET
ejpam-4732	310	36	following	follow	VERB
ejpam-4732	310	37	properties	property	NOUN
ejpam-4732	310	38	are	be	AUX
ejpam-4732	310	39	equivalent	equivalent	ADJ
ejpam-4732	310	40	:	:	PUNCT
ejpam-4732	310	41	(	(	PUNCT
ejpam-4732	310	42	1	1	X
ejpam-4732	310	43	)	)	PUNCT
ejpam-4732	310	44	f	f	PROPN
ejpam-4732	310	45	is	be	AUX
ejpam-4732	310	46	almost	almost	ADV
ejpam-4732	310	47	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	310	48	;	;	PUNCT
ejpam-4732	310	49	(	(	PUNCT
ejpam-4732	310	50	2	2	X
ejpam-4732	310	51	)	)	PUNCT
ejpam-4732	310	52	for	for	SCONJ
ejpam-4732	310	53	each	each	DET
ejpam-4732	310	54	x	x	SYM
ejpam-4732	310	55	∈	∈	PROPN
ejpam-4732	310	56	x	x	X
ejpam-4732	310	57	and	and	CCONJ
ejpam-4732	310	58	each	each	DET
ejpam-4732	310	59	⋆-open	⋆-open	ADV
ejpam-4732	310	60	set	set	VERB
ejpam-4732	310	61	v	v	NUM
ejpam-4732	310	62	of	of	ADP
ejpam-4732	310	63	y	y	NOUN
ejpam-4732	310	64	containing	contain	VERB
ejpam-4732	310	65	f(x	f(x	PROPN
ejpam-4732	310	66	)	)	PUNCT
ejpam-4732	310	67	,	,	PUNCT
ejpam-4732	310	68	there	there	PRON
ejpam-4732	310	69	exists	exist	VERB
ejpam-4732	310	70	a	a	DET
ejpam-4732	310	71	strong	strong	ADJ
ejpam-4732	310	72	β	β	NOUN
ejpam-4732	310	73	-	-	ADJ
ejpam-4732	310	74	i	i	PRON
ejpam-4732	310	75	-open	-open	VERB
ejpam-4732	310	76	set	set	VERB
ejpam-4732	310	77	u	u	NOUN
ejpam-4732	310	78	of	of	ADP
ejpam-4732	310	79	x	x	PUNCT
ejpam-4732	310	80	containing	contain	VERB
ejpam-4732	310	81	x	x	PUNCT
ejpam-4732	310	82	such	such	ADJ
ejpam-4732	310	83	that	that	PRON
ejpam-4732	310	84	f(u	f(u	PROPN
ejpam-4732	310	85	)	)	PUNCT
ejpam-4732	310	86	⊆	⊆	NUM
ejpam-4732	310	87	sclj	sclj	NOUN
ejpam-4732	310	88	(	(	PUNCT
ejpam-4732	310	89	v	v	NOUN
ejpam-4732	310	90	)	)	PUNCT
ejpam-4732	310	91	;	;	PUNCT
ejpam-4732	310	92	c.	c.	PROPN
ejpam-4732	310	93	boonpok	boonpok	PROPN
ejpam-4732	310	94	,	,	PUNCT
ejpam-4732	310	95	p.	p.	NOUN
ejpam-4732	310	96	pue	pue	NOUN
ejpam-4732	310	97	-	-	PUNCT
ejpam-4732	310	98	on	on	ADP
ejpam-4732	310	99	/	/	SYM
ejpam-4732	310	100	eur	eur	NOUN
ejpam-4732	310	101	.	.	PUNCT
ejpam-4732	311	1	j.	j.	PROPN
ejpam-4732	311	2	pure	pure	PROPN
ejpam-4732	311	3	appl	appl	PROPN
ejpam-4732	311	4	.	.	PROPN
ejpam-4732	311	5	math	math	PROPN
ejpam-4732	311	6	,	,	PUNCT
ejpam-4732	311	7	16	16	NUM
ejpam-4732	311	8	(	(	PUNCT
ejpam-4732	311	9	3	3	NUM
ejpam-4732	311	10	)	)	PUNCT
ejpam-4732	311	11	(	(	PUNCT
ejpam-4732	311	12	2023	2023	NUM
ejpam-4732	311	13	)	)	PUNCT
ejpam-4732	311	14	,	,	PUNCT
ejpam-4732	311	15	1634	1634	NUM
ejpam-4732	311	16	-	-	SYM
ejpam-4732	311	17	1646	1646	NUM
ejpam-4732	311	18	1644	1644	NUM
ejpam-4732	311	19	(	(	PUNCT
ejpam-4732	311	20	3	3	NUM
ejpam-4732	311	21	)	)	PUNCT
ejpam-4732	311	22	for	for	ADP
ejpam-4732	311	23	each	each	DET
ejpam-4732	311	24	x	x	SYM
ejpam-4732	311	25	∈	∈	PROPN
ejpam-4732	311	26	x	x	X
ejpam-4732	311	27	and	and	CCONJ
ejpam-4732	311	28	each	each	DET
ejpam-4732	311	29	r⋆-j	r⋆-j	PROPN
ejpam-4732	311	30	-open	-open	PROPN
ejpam-4732	311	31	set	set	VERB
ejpam-4732	311	32	v	v	NOUN
ejpam-4732	311	33	of	of	ADP
ejpam-4732	311	34	y	y	NOUN
ejpam-4732	311	35	containing	contain	VERB
ejpam-4732	311	36	f(x	f(x	PROPN
ejpam-4732	311	37	)	)	PUNCT
ejpam-4732	312	1	,	,	PUNCT
ejpam-4732	312	2	there	there	PRON
ejpam-4732	312	3	exists	exist	VERB
ejpam-4732	312	4	a	a	DET
ejpam-4732	312	5	strong	strong	ADJ
ejpam-4732	312	6	β	β	NOUN
ejpam-4732	312	7	-	-	ADJ
ejpam-4732	312	8	i	i	PRON
ejpam-4732	312	9	-open	-open	VERB
ejpam-4732	312	10	set	set	VERB
ejpam-4732	312	11	u	u	NOUN
ejpam-4732	312	12	of	of	ADP
ejpam-4732	312	13	x	x	PUNCT
ejpam-4732	312	14	containing	contain	VERB
ejpam-4732	312	15	x	x	PUNCT
ejpam-4732	312	16	such	such	ADJ
ejpam-4732	312	17	that	that	DET
ejpam-4732	312	18	f(u	f(u	PROPN
ejpam-4732	312	19	)	)	PUNCT
ejpam-4732	312	20	⊆	⊆	NUM
ejpam-4732	312	21	v	v	NOUN
ejpam-4732	312	22	;	;	PUNCT
ejpam-4732	312	23	(	(	PUNCT
ejpam-4732	312	24	4	4	X
ejpam-4732	312	25	)	)	PUNCT
ejpam-4732	312	26	f−1(v	f−1(v	NOUN
ejpam-4732	312	27	)	)	PUNCT
ejpam-4732	312	28	is	be	AUX
ejpam-4732	312	29	strong	strong	ADJ
ejpam-4732	312	30	β	β	NOUN
ejpam-4732	312	31	-	-	PUNCT
ejpam-4732	312	32	i	i	PRON
ejpam-4732	312	33	-open	-open	VERB
ejpam-4732	312	34	in	in	ADP
ejpam-4732	312	35	x	x	PUNCT
ejpam-4732	312	36	for	for	ADP
ejpam-4732	312	37	every	every	DET
ejpam-4732	312	38	r⋆-j	r⋆-j	PROPN
ejpam-4732	312	39	-open	-open	NOUN
ejpam-4732	312	40	set	set	VERB
ejpam-4732	312	41	v	v	NOUN
ejpam-4732	312	42	of	of	ADP
ejpam-4732	312	43	y	y	PROPN
ejpam-4732	312	44	;	;	PUNCT
ejpam-4732	312	45	(	(	PUNCT
ejpam-4732	312	46	5	5	X
ejpam-4732	312	47	)	)	PUNCT
ejpam-4732	312	48	f−1(k	f−1(k	PROPN
ejpam-4732	312	49	)	)	PUNCT
ejpam-4732	312	50	is	be	AUX
ejpam-4732	312	51	strong	strong	ADJ
ejpam-4732	312	52	β	β	NOUN
ejpam-4732	312	53	-	-	PUNCT
ejpam-4732	312	54	i	i	PRON
ejpam-4732	312	55	-closed	-close	VERB
ejpam-4732	312	56	in	in	ADP
ejpam-4732	312	57	x	x	PUNCT
ejpam-4732	312	58	for	for	ADP
ejpam-4732	312	59	every	every	DET
ejpam-4732	312	60	r⋆-j	r⋆-j	PROPN
ejpam-4732	312	61	-closed	-close	VERB
ejpam-4732	312	62	set	set	NOUN
ejpam-4732	312	63	k	k	PROPN
ejpam-4732	312	64	of	of	ADP
ejpam-4732	312	65	y	y	PROPN
ejpam-4732	312	66	;	;	PUNCT
ejpam-4732	312	67	(	(	PUNCT
ejpam-4732	312	68	6	6	X
ejpam-4732	312	69	)	)	PUNCT
ejpam-4732	312	70	f−1(v	f−1(v	NOUN
ejpam-4732	312	71	)	)	PUNCT
ejpam-4732	313	1	⊆	⊆	NUM
ejpam-4732	313	2	sβinti	sβinti	NOUN
ejpam-4732	313	3	(	(	PUNCT
ejpam-4732	313	4	f−1(sclj	f−1(sclj	X
ejpam-4732	313	5	(	(	PUNCT
ejpam-4732	313	6	v	v	NOUN
ejpam-4732	313	7	)	)	PUNCT
ejpam-4732	313	8	)	)	PUNCT
ejpam-4732	313	9	)	)	PUNCT
ejpam-4732	313	10	for	for	ADP
ejpam-4732	313	11	every	every	DET
ejpam-4732	313	12	⋆-open	⋆-open	NOUN
ejpam-4732	313	13	set	set	VERB
ejpam-4732	313	14	v	v	NOUN
ejpam-4732	313	15	of	of	ADP
ejpam-4732	313	16	y	y	PROPN
ejpam-4732	313	17	;	;	PUNCT
ejpam-4732	313	18	(	(	PUNCT
ejpam-4732	313	19	7	7	X
ejpam-4732	313	20	)	)	PUNCT
ejpam-4732	313	21	sβcli	sβcli	NOUN
ejpam-4732	313	22	(	(	PUNCT
ejpam-4732	313	23	f−1(sintj	f−1(sintj	PROPN
ejpam-4732	313	24	(	(	PUNCT
ejpam-4732	313	25	k	k	NOUN
ejpam-4732	313	26	)	)	PUNCT
ejpam-4732	313	27	)	)	PUNCT
ejpam-4732	313	28	)	)	PUNCT
ejpam-4732	313	29	⊆	⊆	NUM
ejpam-4732	313	30	f−1(k	f−1(k	PROPN
ejpam-4732	313	31	)	)	PUNCT
ejpam-4732	313	32	for	for	ADP
ejpam-4732	313	33	every	every	DET
ejpam-4732	313	34	⋆-closed	⋆-close	VERB
ejpam-4732	313	35	set	set	NOUN
ejpam-4732	313	36	k	k	PROPN
ejpam-4732	313	37	of	of	ADP
ejpam-4732	313	38	y	y	PROPN
ejpam-4732	313	39	;	;	PUNCT
ejpam-4732	313	40	(	(	PUNCT
ejpam-4732	313	41	8)	8)	NUM
ejpam-4732	313	42	sβcli	sβcli	NOUN
ejpam-4732	313	43	(	(	PUNCT
ejpam-4732	313	44	f−1(cl⋆(int(k	f−1(cl⋆(int(k	ADJ
ejpam-4732	313	45	)	)	PUNCT
ejpam-4732	313	46	)	)	PUNCT
ejpam-4732	313	47	)	)	PUNCT
ejpam-4732	313	48	)	)	PUNCT
ejpam-4732	313	49	⊆	⊆	NUM
ejpam-4732	313	50	f−1(k	f−1(k	PROPN
ejpam-4732	313	51	)	)	PUNCT
ejpam-4732	313	52	for	for	ADP
ejpam-4732	313	53	every	every	DET
ejpam-4732	313	54	⋆-closed	⋆-close	VERB
ejpam-4732	313	55	set	set	NOUN
ejpam-4732	313	56	k	k	PROPN
ejpam-4732	313	57	of	of	ADP
ejpam-4732	313	58	y	y	PROPN
ejpam-4732	313	59	;	;	PUNCT
ejpam-4732	313	60	(	(	PUNCT
ejpam-4732	313	61	9	9	X
ejpam-4732	313	62	)	)	PUNCT
ejpam-4732	313	63	sβcli	sβcli	NOUN
ejpam-4732	313	64	(	(	PUNCT
ejpam-4732	313	65	f−1(cl⋆(int(cl⋆(b	f−1(cl⋆(int(cl⋆(b	NOUN
ejpam-4732	313	66	)	)	PUNCT
ejpam-4732	313	67	)	)	PUNCT
ejpam-4732	313	68	)	)	PUNCT
ejpam-4732	313	69	)	)	PUNCT
ejpam-4732	313	70	)	)	PUNCT
ejpam-4732	314	1	⊆	⊆	NUM
ejpam-4732	314	2	f−1(cl⋆(b	f−1(cl⋆(b	NOUN
ejpam-4732	314	3	)	)	PUNCT
ejpam-4732	314	4	)	)	PUNCT
ejpam-4732	314	5	for	for	ADP
ejpam-4732	314	6	every	every	DET
ejpam-4732	314	7	subset	subset	NOUN
ejpam-4732	314	8	b	b	PROPN
ejpam-4732	314	9	of	of	ADP
ejpam-4732	314	10	y	y	PROPN
ejpam-4732	314	11	;	;	PUNCT
ejpam-4732	314	12	(	(	PUNCT
ejpam-4732	314	13	10	10	X
ejpam-4732	314	14	)	)	PUNCT
ejpam-4732	314	15	int⋆(cl	int⋆(cl	NOUN
ejpam-4732	314	16	(	(	PUNCT
ejpam-4732	314	17	int⋆(f−1(cl⋆(int(k	int⋆(f−1(cl⋆(int(k	ADJ
ejpam-4732	314	18	)	)	PUNCT
ejpam-4732	314	19	)	)	PUNCT
ejpam-4732	314	20	)	)	PUNCT
ejpam-4732	314	21	)	)	PUNCT
ejpam-4732	314	22	)	)	PUNCT
ejpam-4732	314	23	)	)	PUNCT
ejpam-4732	315	1	⊆	⊆	NUM
ejpam-4732	315	2	f−1(k	f−1(k	PROPN
ejpam-4732	315	3	)	)	PUNCT
ejpam-4732	315	4	for	for	ADP
ejpam-4732	315	5	every	every	DET
ejpam-4732	315	6	⋆-closed	⋆-close	VERB
ejpam-4732	315	7	set	set	NOUN
ejpam-4732	315	8	k	k	PROPN
ejpam-4732	315	9	of	of	ADP
ejpam-4732	315	10	y	y	PROPN
ejpam-4732	315	11	;	;	PUNCT
ejpam-4732	315	12	(	(	PUNCT
ejpam-4732	315	13	11	11	X
ejpam-4732	315	14	)	)	PUNCT
ejpam-4732	315	15	int⋆(cl	int⋆(cl	NOUN
ejpam-4732	315	16	(	(	PUNCT
ejpam-4732	315	17	int⋆(f−1(sintj	int⋆(f−1(sintj	PROPN
ejpam-4732	315	18	(	(	PUNCT
ejpam-4732	315	19	k	k	NOUN
ejpam-4732	315	20	)	)	PUNCT
ejpam-4732	315	21	)	)	PUNCT
ejpam-4732	315	22	)	)	PUNCT
ejpam-4732	315	23	)	)	PUNCT
ejpam-4732	315	24	)	)	PUNCT
ejpam-4732	316	1	⊆	⊆	NUM
ejpam-4732	316	2	f−1(k	f−1(k	PROPN
ejpam-4732	316	3	)	)	PUNCT
ejpam-4732	316	4	for	for	ADP
ejpam-4732	316	5	every	every	DET
ejpam-4732	316	6	⋆-closed	⋆-close	VERB
ejpam-4732	316	7	set	set	NOUN
ejpam-4732	316	8	k	k	PROPN
ejpam-4732	316	9	of	of	ADP
ejpam-4732	316	10	y	y	PROPN
ejpam-4732	316	11	;	;	PUNCT
ejpam-4732	316	12	(	(	PUNCT
ejpam-4732	316	13	12	12	X
ejpam-4732	316	14	)	)	PUNCT
ejpam-4732	316	15	f−1(v	f−1(v	NOUN
ejpam-4732	316	16	)	)	PUNCT
ejpam-4732	317	1	⊆	⊆	X
ejpam-4732	317	2	cl⋆	cl⋆	PROPN
ejpam-4732	317	3	(	(	PUNCT
ejpam-4732	317	4	int(cl⋆(f−1(sclj	int(cl⋆(f−1(sclj	PROPN
ejpam-4732	317	5	(	(	PUNCT
ejpam-4732	317	6	v	v	NOUN
ejpam-4732	317	7	)	)	PUNCT
ejpam-4732	317	8	)	)	PUNCT
ejpam-4732	317	9	)	)	PUNCT
ejpam-4732	317	10	)	)	PUNCT
ejpam-4732	317	11	)	)	PUNCT
ejpam-4732	317	12	for	for	ADP
ejpam-4732	317	13	every	every	DET
ejpam-4732	317	14	⋆-open	⋆-open	NOUN
ejpam-4732	317	15	set	set	VERB
ejpam-4732	317	16	v	v	NOUN
ejpam-4732	317	17	of	of	ADP
ejpam-4732	317	18	y	y	PROPN
ejpam-4732	317	19	.	.	PUNCT
ejpam-4732	317	20	theorem	theorem	VERB
ejpam-4732	317	21	9	9	NUM
ejpam-4732	317	22	.	.	X
ejpam-4732	317	23	for	for	ADP
ejpam-4732	317	24	a	a	DET
ejpam-4732	317	25	multifunction	multifunction	NOUN
ejpam-4732	317	26	f	f	NOUN
ejpam-4732	317	27	:	:	PUNCT
ejpam-4732	317	28	(	(	PUNCT
ejpam-4732	317	29	x	x	X
ejpam-4732	317	30	,	,	PUNCT
ejpam-4732	317	31	τ	τ	PROPN
ejpam-4732	317	32	,	,	PUNCT
ejpam-4732	317	33	i	i	NOUN
ejpam-4732	317	34	)	)	PUNCT
ejpam-4732	318	1	→	→	PUNCT
ejpam-4732	318	2	(	(	PUNCT
ejpam-4732	318	3	y	y	PROPN
ejpam-4732	318	4	,	,	PUNCT
ejpam-4732	318	5	σ	σ	PROPN
ejpam-4732	318	6	,	,	PUNCT
ejpam-4732	318	7	j	j	PROPN
ejpam-4732	318	8	)	)	PUNCT
ejpam-4732	318	9	,	,	PUNCT
ejpam-4732	318	10	the	the	DET
ejpam-4732	318	11	following	follow	VERB
ejpam-4732	318	12	properties	property	NOUN
ejpam-4732	318	13	are	be	AUX
ejpam-4732	318	14	equivalent	equivalent	ADJ
ejpam-4732	318	15	:	:	PUNCT
ejpam-4732	318	16	(	(	PUNCT
ejpam-4732	318	17	1	1	X
ejpam-4732	318	18	)	)	PUNCT
ejpam-4732	318	19	f	f	PROPN
ejpam-4732	318	20	is	be	AUX
ejpam-4732	318	21	upper	upper	ADJ
ejpam-4732	318	22	almost	almost	ADV
ejpam-4732	318	23	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	318	24	;	;	PUNCT
ejpam-4732	318	25	(	(	PUNCT
ejpam-4732	318	26	2	2	X
ejpam-4732	318	27	)	)	PUNCT
ejpam-4732	318	28	sβcli	sβcli	NOUN
ejpam-4732	318	29	(	(	PUNCT
ejpam-4732	318	30	f−(v	f−(v	NOUN
ejpam-4732	318	31	)	)	PUNCT
ejpam-4732	318	32	)	)	PUNCT
ejpam-4732	319	1	⊆	⊆	NUM
ejpam-4732	319	2	f−(cl⋆(v	f−(cl⋆(v	NOUN
ejpam-4732	319	3	)	)	PUNCT
ejpam-4732	319	4	)	)	PUNCT
ejpam-4732	319	5	for	for	ADP
ejpam-4732	319	6	every	every	DET
ejpam-4732	319	7	strong	strong	ADJ
ejpam-4732	319	8	β	β	PROPN
ejpam-4732	319	9	-	-	ADJ
ejpam-4732	319	10	j	j	ADJ
ejpam-4732	319	11	-open	-open	NOUN
ejpam-4732	319	12	set	set	VERB
ejpam-4732	319	13	v	v	NOUN
ejpam-4732	319	14	of	of	ADP
ejpam-4732	319	15	y	y	PROPN
ejpam-4732	319	16	;	;	PUNCT
ejpam-4732	319	17	(	(	PUNCT
ejpam-4732	319	18	3	3	X
ejpam-4732	319	19	)	)	PUNCT
ejpam-4732	319	20	sβcli	sβcli	NOUN
ejpam-4732	319	21	(	(	PUNCT
ejpam-4732	319	22	f−(v	f−(v	NOUN
ejpam-4732	319	23	)	)	PUNCT
ejpam-4732	319	24	)	)	PUNCT
ejpam-4732	320	1	⊆	⊆	NUM
ejpam-4732	320	2	f−(cl⋆(v	f−(cl⋆(v	NOUN
ejpam-4732	320	3	)	)	PUNCT
ejpam-4732	320	4	)	)	PUNCT
ejpam-4732	320	5	for	for	ADP
ejpam-4732	320	6	every	every	DET
ejpam-4732	320	7	semi	semi	ADJ
ejpam-4732	320	8	-	-	ADJ
ejpam-4732	320	9	j	j	ADJ
ejpam-4732	320	10	-open	-open	NOUN
ejpam-4732	320	11	set	set	VERB
ejpam-4732	320	12	v	v	NOUN
ejpam-4732	320	13	of	of	ADP
ejpam-4732	320	14	y	y	PROPN
ejpam-4732	320	15	;	;	PUNCT
ejpam-4732	320	16	(	(	PUNCT
ejpam-4732	320	17	4	4	X
ejpam-4732	320	18	)	)	PUNCT
ejpam-4732	320	19	f+(v	f+(v	NOUN
ejpam-4732	320	20	)	)	PUNCT
ejpam-4732	321	1	⊆	⊆	NUM
ejpam-4732	321	2	sβinti	sβinti	NOUN
ejpam-4732	321	3	(	(	PUNCT
ejpam-4732	321	4	f+(int⋆(cl(v	f+(int⋆(cl(v	NOUN
ejpam-4732	321	5	)	)	PUNCT
ejpam-4732	321	6	)	)	PUNCT
ejpam-4732	321	7	)	)	PUNCT
ejpam-4732	321	8	)	)	PUNCT
ejpam-4732	321	9	for	for	ADP
ejpam-4732	321	10	every	every	DET
ejpam-4732	321	11	pre⋆j	pre⋆j	PROPN
ejpam-4732	321	12	-open	-open	NOUN
ejpam-4732	321	13	set	set	VERB
ejpam-4732	321	14	v	v	NOUN
ejpam-4732	321	15	of	of	ADP
ejpam-4732	321	16	y	y	PROPN
ejpam-4732	321	17	.	.	PUNCT
ejpam-4732	322	1	proof	proof	NOUN
ejpam-4732	322	2	.	.	PUNCT
ejpam-4732	323	1	(	(	PUNCT
ejpam-4732	323	2	1	1	X
ejpam-4732	323	3	)	)	PUNCT
ejpam-4732	323	4	⇒	⇒	NOUN
ejpam-4732	323	5	(	(	PUNCT
ejpam-4732	323	6	2	2	NUM
ejpam-4732	323	7	):	):	PUNCT
ejpam-4732	323	8	let	let	VERB
ejpam-4732	323	9	v	v	PART
ejpam-4732	323	10	be	be	AUX
ejpam-4732	323	11	any	any	PRON
ejpam-4732	323	12	strong	strong	ADJ
ejpam-4732	323	13	β	β	NOUN
ejpam-4732	323	14	-	-	ADJ
ejpam-4732	323	15	j	j	PROPN
ejpam-4732	323	16	-open	-open	NOUN
ejpam-4732	323	17	set	set	NOUN
ejpam-4732	323	18	of	of	ADP
ejpam-4732	323	19	y	y	PROPN
ejpam-4732	323	20	.	.	PUNCT
ejpam-4732	324	1	since	since	SCONJ
ejpam-4732	324	2	cl⋆(v	cl⋆(v	PROPN
ejpam-4732	324	3	)	)	PUNCT
ejpam-4732	324	4	is	be	AUX
ejpam-4732	324	5	r⋆-j	r⋆-j	PROPN
ejpam-4732	324	6	closed	closed	ADJ
ejpam-4732	324	7	,	,	PUNCT
ejpam-4732	324	8	by	by	ADP
ejpam-4732	324	9	theorem	theorem	NOUN
ejpam-4732	324	10	7	7	NUM
ejpam-4732	324	11	,	,	PUNCT
ejpam-4732	324	12	f−(cl⋆(v	f−(cl⋆(v	NOUN
ejpam-4732	324	13	)	)	PUNCT
ejpam-4732	324	14	)	)	PUNCT
ejpam-4732	324	15	is	be	AUX
ejpam-4732	324	16	strong	strong	ADJ
ejpam-4732	324	17	β	β	NOUN
ejpam-4732	324	18	-	-	PUNCT
ejpam-4732	324	19	i	i	PRON
ejpam-4732	324	20	-closed	-close	VERB
ejpam-4732	324	21	in	in	ADP
ejpam-4732	324	22	x	x	PUNCT
ejpam-4732	324	23	and	and	CCONJ
ejpam-4732	324	24	hence	hence	ADV
ejpam-4732	324	25	sβcli	sβcli	ADJ
ejpam-4732	324	26	(	(	PUNCT
ejpam-4732	324	27	f−(v	f−(v	NOUN
ejpam-4732	324	28	)	)	PUNCT
ejpam-4732	324	29	)	)	PUNCT
ejpam-4732	325	1	⊆	⊆	NUM
ejpam-4732	325	2	f−(cl⋆(v	f−(cl⋆(v	NOUN
ejpam-4732	325	3	)	)	PUNCT
ejpam-4732	325	4	)	)	PUNCT
ejpam-4732	325	5	.	.	PUNCT
ejpam-4732	326	1	(	(	PUNCT
ejpam-4732	326	2	2	2	X
ejpam-4732	326	3	)	)	PUNCT
ejpam-4732	326	4	⇒	⇒	NOUN
ejpam-4732	326	5	(	(	PUNCT
ejpam-4732	326	6	3	3	NUM
ejpam-4732	326	7	):	):	PUNCT
ejpam-4732	326	8	this	this	PRON
ejpam-4732	326	9	is	be	AUX
ejpam-4732	326	10	obvious	obvious	ADJ
ejpam-4732	326	11	since	since	SCONJ
ejpam-4732	326	12	every	every	DET
ejpam-4732	326	13	semi	semi	ADJ
ejpam-4732	326	14	-	-	ADJ
ejpam-4732	326	15	j	j	ADJ
ejpam-4732	326	16	-open	-open	NOUN
ejpam-4732	326	17	set	set	NOUN
ejpam-4732	326	18	is	be	AUX
ejpam-4732	326	19	strong	strong	ADJ
ejpam-4732	326	20	β	β	NOUN
ejpam-4732	326	21	-	-	PUNCT
ejpam-4732	326	22	j	j	NOUN
ejpam-4732	326	23	-open	-open	NOUN
ejpam-4732	326	24	.	.	PUNCT
ejpam-4732	327	1	(	(	PUNCT
ejpam-4732	327	2	3	3	X
ejpam-4732	327	3	)	)	PUNCT
ejpam-4732	327	4	⇒	⇒	NOUN
ejpam-4732	327	5	(	(	PUNCT
ejpam-4732	327	6	4	4	NUM
ejpam-4732	327	7	):	):	PUNCT
ejpam-4732	327	8	let	let	VERB
ejpam-4732	327	9	v	v	PART
ejpam-4732	327	10	be	be	AUX
ejpam-4732	327	11	any	any	DET
ejpam-4732	327	12	pre⋆j	pre⋆j	PROPN
ejpam-4732	327	13	-open	-open	ADJ
ejpam-4732	327	14	set	set	NOUN
ejpam-4732	327	15	of	of	ADP
ejpam-4732	327	16	y	y	PROPN
ejpam-4732	327	17	.	.	PUNCT
ejpam-4732	328	1	then	then	ADV
ejpam-4732	328	2	,	,	PUNCT
ejpam-4732	328	3	we	we	PRON
ejpam-4732	328	4	have	have	VERB
ejpam-4732	328	5	v	v	ADP
ejpam-4732	328	6	⊆	⊆	NUM
ejpam-4732	328	7	int⋆(cl(v	int⋆(cl(v	NOUN
ejpam-4732	328	8	)	)	PUNCT
ejpam-4732	328	9	)	)	PUNCT
ejpam-4732	329	1	and	and	CCONJ
ejpam-4732	329	2	y	y	PROPN
ejpam-4732	329	3	−	−	PROPN
ejpam-4732	329	4	v	v	PROPN
ejpam-4732	329	5	⊇	⊇	NOUN
ejpam-4732	329	6	cl⋆(int(y	cl⋆(int(y	PROPN
ejpam-4732	329	7	−	−	PROPN
ejpam-4732	329	8	v	v	NOUN
ejpam-4732	329	9	)	)	PUNCT
ejpam-4732	329	10	)	)	PUNCT
ejpam-4732	329	11	.	.	PUNCT
ejpam-4732	330	1	since	since	SCONJ
ejpam-4732	330	2	cl⋆(int(y	cl⋆(int(y	PROPN
ejpam-4732	330	3	−	−	PROPN
ejpam-4732	330	4	v	v	NOUN
ejpam-4732	330	5	)	)	PUNCT
ejpam-4732	330	6	)	)	PUNCT
ejpam-4732	330	7	is	be	AUX
ejpam-4732	330	8	semi	semi	ADJ
ejpam-4732	330	9	-	-	ADJ
ejpam-4732	330	10	j	j	ADJ
ejpam-4732	330	11	-open	-open	NOUN
ejpam-4732	330	12	in	in	ADP
ejpam-4732	330	13	y	y	PROPN
ejpam-4732	330	14	and	and	CCONJ
ejpam-4732	330	15	by	by	ADP
ejpam-4732	330	16	(	(	PUNCT
ejpam-4732	330	17	3	3	NUM
ejpam-4732	330	18	)	)	PUNCT
ejpam-4732	330	19	,	,	PUNCT
ejpam-4732	330	20	x	x	PUNCT
ejpam-4732	330	21	−	−	NOUN
ejpam-4732	330	22	f+(v	f+(v	NOUN
ejpam-4732	330	23	)	)	PUNCT
ejpam-4732	331	1	=	=	PUNCT
ejpam-4732	331	2	f−(y	f−(y	NOUN
ejpam-4732	331	3	−	−	ADP
ejpam-4732	331	4	v	v	NOUN
ejpam-4732	331	5	)	)	PUNCT
ejpam-4732	331	6	⊇	⊇	PROPN
ejpam-4732	331	7	f−(cl⋆(int(y	f−(cl⋆(int(y	VERB
ejpam-4732	331	8	−	−	PROPN
ejpam-4732	331	9	v	v	NOUN
ejpam-4732	331	10	)	)	PUNCT
ejpam-4732	331	11	)	)	PUNCT
ejpam-4732	331	12	)	)	PUNCT
ejpam-4732	332	1	⊇	⊇	NOUN
ejpam-4732	332	2	sβcli	sβcli	NOUN
ejpam-4732	332	3	(	(	PUNCT
ejpam-4732	332	4	f−(cl⋆(int(y	f−(cl⋆(int(y	NOUN
ejpam-4732	332	5	−	−	PROPN
ejpam-4732	332	6	v	v	NOUN
ejpam-4732	332	7	)	)	PUNCT
ejpam-4732	332	8	)	)	PUNCT
ejpam-4732	332	9	)	)	PUNCT
ejpam-4732	332	10	)	)	PUNCT
ejpam-4732	333	1	=	=	SYM
ejpam-4732	333	2	sβcli	sβcli	NOUN
ejpam-4732	333	3	(	(	PUNCT
ejpam-4732	333	4	f−(y	f−(y	NOUN
ejpam-4732	333	5	−	−	PROPN
ejpam-4732	333	6	int⋆(cl(v	int⋆(cl(v	NOUN
ejpam-4732	333	7	)	)	PUNCT
ejpam-4732	333	8	)	)	PUNCT
ejpam-4732	333	9	)	)	PUNCT
ejpam-4732	333	10	)	)	PUNCT
ejpam-4732	334	1	=	=	SYM
ejpam-4732	334	2	sβcli	sβcli	NOUN
ejpam-4732	334	3	(	(	PUNCT
ejpam-4732	334	4	x	x	SYM
ejpam-4732	334	5	−	−	PROPN
ejpam-4732	334	6	f+(int⋆(cl(v	f+(int⋆(cl(v	NOUN
ejpam-4732	334	7	)	)	PUNCT
ejpam-4732	334	8	)	)	PUNCT
ejpam-4732	334	9	)	)	PUNCT
ejpam-4732	334	10	)	)	PUNCT
ejpam-4732	335	1	=	=	PUNCT
ejpam-4732	335	2	x	x	PUNCT
ejpam-4732	336	1	−	−	PROPN
ejpam-4732	336	2	sβinti	sβinti	NOUN
ejpam-4732	336	3	(	(	PUNCT
ejpam-4732	336	4	f+(int⋆(cl(v	f+(int⋆(cl(v	NOUN
ejpam-4732	336	5	)	)	PUNCT
ejpam-4732	336	6	)	)	PUNCT
ejpam-4732	336	7	)	)	PUNCT
ejpam-4732	336	8	)	)	PUNCT
ejpam-4732	336	9	.	.	PUNCT
ejpam-4732	337	1	references	reference	NOUN
ejpam-4732	337	2	1645	1645	NUM
ejpam-4732	337	3	thus	thus	ADV
ejpam-4732	337	4	,	,	PUNCT
ejpam-4732	337	5	f+(v	f+(v	PROPN
ejpam-4732	337	6	)	)	PUNCT
ejpam-4732	338	1	⊆	⊆	NUM
ejpam-4732	338	2	sβinti	sβinti	NOUN
ejpam-4732	338	3	(	(	PUNCT
ejpam-4732	338	4	f+(int⋆(cl(v	f+(int⋆(cl(v	NOUN
ejpam-4732	338	5	)	)	PUNCT
ejpam-4732	338	6	)	)	PUNCT
ejpam-4732	338	7	)	)	PUNCT
ejpam-4732	338	8	)	)	PUNCT
ejpam-4732	338	9	.	.	PUNCT
ejpam-4732	339	1	(	(	PUNCT
ejpam-4732	339	2	4	4	X
ejpam-4732	339	3	)	)	PUNCT
ejpam-4732	339	4	⇒	⇒	NOUN
ejpam-4732	339	5	(	(	PUNCT
ejpam-4732	339	6	1	1	NUM
ejpam-4732	339	7	):	):	PUNCT
ejpam-4732	339	8	let	let	VERB
ejpam-4732	339	9	v	v	PART
ejpam-4732	339	10	be	be	AUX
ejpam-4732	339	11	any	any	DET
ejpam-4732	339	12	r⋆-j	r⋆-j	PROPN
ejpam-4732	339	13	-open	-open	ADJ
ejpam-4732	339	14	set	set	NOUN
ejpam-4732	339	15	of	of	ADP
ejpam-4732	339	16	y	y	PROPN
ejpam-4732	339	17	.	.	PUNCT
ejpam-4732	340	1	then	then	ADV
ejpam-4732	340	2	,	,	PUNCT
ejpam-4732	340	3	v	v	NOUN
ejpam-4732	340	4	is	be	AUX
ejpam-4732	340	5	pre⋆j	pre⋆j	PROPN
ejpam-4732	340	6	-open	-open	ADJ
ejpam-4732	340	7	in	in	ADP
ejpam-4732	340	8	y	y	PROPN
ejpam-4732	340	9	and	and	CCONJ
ejpam-4732	340	10	by	by	ADP
ejpam-4732	340	11	(	(	PUNCT
ejpam-4732	340	12	4	4	NUM
ejpam-4732	340	13	)	)	PUNCT
ejpam-4732	340	14	,	,	PUNCT
ejpam-4732	340	15	f+(v	f+(v	PROPN
ejpam-4732	340	16	)	)	PUNCT
ejpam-4732	341	1	⊆	⊆	NUM
ejpam-4732	341	2	sβinti	sβinti	NOUN
ejpam-4732	341	3	(	(	PUNCT
ejpam-4732	341	4	f+(int⋆(cl(v	f+(int⋆(cl(v	NOUN
ejpam-4732	341	5	)	)	PUNCT
ejpam-4732	341	6	)	)	PUNCT
ejpam-4732	341	7	)	)	PUNCT
ejpam-4732	341	8	)	)	PUNCT
ejpam-4732	342	1	=	=	PRON
ejpam-4732	342	2	sβinti	sβinti	NOUN
ejpam-4732	342	3	(	(	PUNCT
ejpam-4732	342	4	f+(v	f+(v	PROPN
ejpam-4732	342	5	)	)	PUNCT
ejpam-4732	342	6	)	)	PUNCT
ejpam-4732	342	7	and	and	CCONJ
ejpam-4732	342	8	hence	hence	ADV
ejpam-4732	342	9	f+(v	f+(v	PROPN
ejpam-4732	342	10	)	)	PUNCT
ejpam-4732	342	11	is	be	AUX
ejpam-4732	342	12	strong	strong	ADJ
ejpam-4732	342	13	β	β	NOUN
ejpam-4732	342	14	-	-	PUNCT
ejpam-4732	342	15	i	i	PRON
ejpam-4732	342	16	-open	-open	ADJ
ejpam-4732	342	17	in	in	ADP
ejpam-4732	342	18	x.	x.	NOUN
ejpam-4732	343	1	it	it	PRON
ejpam-4732	343	2	follows	follow	VERB
ejpam-4732	343	3	from	from	ADP
ejpam-4732	343	4	theorem	theorem	ADJ
ejpam-4732	343	5	7	7	NUM
ejpam-4732	343	6	that	that	SCONJ
ejpam-4732	343	7	f	f	PROPN
ejpam-4732	343	8	is	be	AUX
ejpam-4732	343	9	upper	upper	ADJ
ejpam-4732	343	10	almost	almost	ADV
ejpam-4732	343	11	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	343	12	.	.	PUNCT
ejpam-4732	343	13	theorem	theorem	PROPN
ejpam-4732	343	14	10	10	NUM
ejpam-4732	343	15	.	.	PUNCT
ejpam-4732	344	1	for	for	ADP
ejpam-4732	344	2	a	a	DET
ejpam-4732	344	3	multifunction	multifunction	NOUN
ejpam-4732	344	4	f	f	NOUN
ejpam-4732	344	5	:	:	PUNCT
ejpam-4732	344	6	(	(	PUNCT
ejpam-4732	344	7	x	x	X
ejpam-4732	344	8	,	,	PUNCT
ejpam-4732	344	9	τ	τ	PROPN
ejpam-4732	344	10	,	,	PUNCT
ejpam-4732	344	11	i	i	NOUN
ejpam-4732	344	12	)	)	PUNCT
ejpam-4732	344	13	→	→	PUNCT
ejpam-4732	344	14	(	(	PUNCT
ejpam-4732	344	15	y	y	PROPN
ejpam-4732	344	16	,	,	PUNCT
ejpam-4732	344	17	σ	σ	PROPN
ejpam-4732	344	18	,	,	PUNCT
ejpam-4732	344	19	j	j	PROPN
ejpam-4732	344	20	)	)	PUNCT
ejpam-4732	344	21	,	,	PUNCT
ejpam-4732	344	22	the	the	DET
ejpam-4732	344	23	following	follow	VERB
ejpam-4732	344	24	properties	property	NOUN
ejpam-4732	344	25	are	be	AUX
ejpam-4732	344	26	equivalent	equivalent	ADJ
ejpam-4732	344	27	:	:	PUNCT
ejpam-4732	344	28	(	(	PUNCT
ejpam-4732	344	29	1	1	X
ejpam-4732	344	30	)	)	PUNCT
ejpam-4732	344	31	f	f	PROPN
ejpam-4732	344	32	is	be	AUX
ejpam-4732	344	33	lower	low	ADJ
ejpam-4732	344	34	almost	almost	ADV
ejpam-4732	344	35	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	344	36	;	;	PUNCT
ejpam-4732	344	37	(	(	PUNCT
ejpam-4732	344	38	2	2	X
ejpam-4732	344	39	)	)	PUNCT
ejpam-4732	344	40	sβcli	sβcli	NOUN
ejpam-4732	344	41	(	(	PUNCT
ejpam-4732	344	42	f−(v	f−(v	NOUN
ejpam-4732	344	43	)	)	PUNCT
ejpam-4732	344	44	)	)	PUNCT
ejpam-4732	345	1	⊆	⊆	NUM
ejpam-4732	345	2	f−(cl⋆(v	f−(cl⋆(v	NOUN
ejpam-4732	345	3	)	)	PUNCT
ejpam-4732	345	4	)	)	PUNCT
ejpam-4732	345	5	for	for	ADP
ejpam-4732	345	6	every	every	DET
ejpam-4732	345	7	strong	strong	ADJ
ejpam-4732	345	8	β	β	PROPN
ejpam-4732	345	9	-	-	ADJ
ejpam-4732	345	10	j	j	ADJ
ejpam-4732	345	11	-open	-open	NOUN
ejpam-4732	345	12	set	set	VERB
ejpam-4732	345	13	v	v	NOUN
ejpam-4732	345	14	of	of	ADP
ejpam-4732	345	15	y	y	PROPN
ejpam-4732	345	16	;	;	PUNCT
ejpam-4732	345	17	(	(	PUNCT
ejpam-4732	345	18	3	3	X
ejpam-4732	345	19	)	)	PUNCT
ejpam-4732	345	20	sβcli	sβcli	NOUN
ejpam-4732	345	21	(	(	PUNCT
ejpam-4732	345	22	f−(v	f−(v	NOUN
ejpam-4732	345	23	)	)	PUNCT
ejpam-4732	345	24	)	)	PUNCT
ejpam-4732	346	1	⊆	⊆	NUM
ejpam-4732	346	2	f−(cl⋆(v	f−(cl⋆(v	NOUN
ejpam-4732	346	3	)	)	PUNCT
ejpam-4732	346	4	)	)	PUNCT
ejpam-4732	346	5	for	for	ADP
ejpam-4732	346	6	every	every	DET
ejpam-4732	346	7	semi	semi	ADJ
ejpam-4732	346	8	-	-	ADJ
ejpam-4732	346	9	j	j	ADJ
ejpam-4732	346	10	-open	-open	NOUN
ejpam-4732	346	11	set	set	VERB
ejpam-4732	346	12	v	v	NOUN
ejpam-4732	346	13	of	of	ADP
ejpam-4732	346	14	y	y	PROPN
ejpam-4732	346	15	;	;	PUNCT
ejpam-4732	346	16	(	(	PUNCT
ejpam-4732	346	17	4	4	X
ejpam-4732	346	18	)	)	PUNCT
ejpam-4732	346	19	f+(v	f+(v	NOUN
ejpam-4732	346	20	)	)	PUNCT
ejpam-4732	347	1	⊆	⊆	NUM
ejpam-4732	347	2	sβinti	sβinti	NOUN
ejpam-4732	347	3	(	(	PUNCT
ejpam-4732	347	4	f+(int⋆(cl(v	f+(int⋆(cl(v	NOUN
ejpam-4732	347	5	)	)	PUNCT
ejpam-4732	347	6	)	)	PUNCT
ejpam-4732	347	7	)	)	PUNCT
ejpam-4732	347	8	)	)	PUNCT
ejpam-4732	347	9	for	for	ADP
ejpam-4732	347	10	every	every	DET
ejpam-4732	347	11	pre⋆j	pre⋆j	PROPN
ejpam-4732	347	12	-open	-open	NOUN
ejpam-4732	347	13	set	set	VERB
ejpam-4732	347	14	v	v	NOUN
ejpam-4732	347	15	of	of	ADP
ejpam-4732	347	16	y	y	PROPN
ejpam-4732	347	17	.	.	PUNCT
ejpam-4732	348	1	proof	proof	NOUN
ejpam-4732	348	2	.	.	PUNCT
ejpam-4732	349	1	the	the	DET
ejpam-4732	349	2	proof	proof	NOUN
ejpam-4732	349	3	is	be	AUX
ejpam-4732	349	4	similar	similar	ADJ
ejpam-4732	349	5	to	to	ADP
ejpam-4732	349	6	that	that	PRON
ejpam-4732	349	7	of	of	ADP
ejpam-4732	349	8	theorem	theorem	ADJ
ejpam-4732	349	9	9	9	NUM
ejpam-4732	349	10	.	.	PUNCT
ejpam-4732	349	11	corollary	corollary	ADJ
ejpam-4732	349	12	5	5	NUM
ejpam-4732	349	13	.	.	PUNCT
ejpam-4732	350	1	for	for	ADP
ejpam-4732	350	2	a	a	DET
ejpam-4732	350	3	function	function	NOUN
ejpam-4732	350	4	f	f	NOUN
ejpam-4732	350	5	:	:	PUNCT
ejpam-4732	350	6	(	(	PUNCT
ejpam-4732	350	7	x	x	X
ejpam-4732	350	8	,	,	PUNCT
ejpam-4732	350	9	τ	τ	PROPN
ejpam-4732	350	10	,	,	PUNCT
ejpam-4732	350	11	i	i	NOUN
ejpam-4732	350	12	)	)	PUNCT
ejpam-4732	350	13	→	→	PUNCT
ejpam-4732	350	14	(	(	PUNCT
ejpam-4732	350	15	y	y	PROPN
ejpam-4732	350	16	,	,	PUNCT
ejpam-4732	350	17	σ	σ	PROPN
ejpam-4732	350	18	,	,	PUNCT
ejpam-4732	350	19	j	j	PROPN
ejpam-4732	350	20	)	)	PUNCT
ejpam-4732	350	21	,	,	PUNCT
ejpam-4732	350	22	the	the	DET
ejpam-4732	350	23	following	follow	VERB
ejpam-4732	350	24	properties	property	NOUN
ejpam-4732	350	25	are	be	AUX
ejpam-4732	350	26	equivalent	equivalent	ADJ
ejpam-4732	350	27	:	:	PUNCT
ejpam-4732	350	28	(	(	PUNCT
ejpam-4732	350	29	1	1	X
ejpam-4732	350	30	)	)	PUNCT
ejpam-4732	350	31	f	f	PROPN
ejpam-4732	350	32	is	be	AUX
ejpam-4732	350	33	almost	almost	ADV
ejpam-4732	350	34	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-4732	350	35	;	;	PUNCT
ejpam-4732	350	36	(	(	PUNCT
ejpam-4732	350	37	2	2	X
ejpam-4732	350	38	)	)	PUNCT
ejpam-4732	350	39	sβcli	sβcli	NOUN
ejpam-4732	350	40	(	(	PUNCT
ejpam-4732	350	41	f−1(v	f−1(v	NOUN
ejpam-4732	350	42	)	)	PUNCT
ejpam-4732	350	43	)	)	PUNCT
ejpam-4732	351	1	⊆	⊆	NUM
ejpam-4732	351	2	f−1(cl⋆(v	f−1(cl⋆(v	NOUN
ejpam-4732	351	3	)	)	PUNCT
ejpam-4732	351	4	)	)	PUNCT
ejpam-4732	351	5	for	for	ADP
ejpam-4732	351	6	every	every	DET
ejpam-4732	351	7	strong	strong	ADJ
ejpam-4732	351	8	β	β	PROPN
ejpam-4732	351	9	-	-	ADJ
ejpam-4732	351	10	j	j	ADJ
ejpam-4732	351	11	-open	-open	NOUN
ejpam-4732	351	12	set	set	VERB
ejpam-4732	351	13	v	v	NOUN
ejpam-4732	351	14	of	of	ADP
ejpam-4732	351	15	y	y	PROPN
ejpam-4732	351	16	;	;	PUNCT
ejpam-4732	351	17	(	(	PUNCT
ejpam-4732	351	18	3	3	X
ejpam-4732	351	19	)	)	PUNCT
ejpam-4732	351	20	sβcli	sβcli	NOUN
ejpam-4732	351	21	(	(	PUNCT
ejpam-4732	351	22	f−1(v	f−1(v	NOUN
ejpam-4732	351	23	)	)	PUNCT
ejpam-4732	351	24	)	)	PUNCT
ejpam-4732	352	1	⊆	⊆	NUM
ejpam-4732	352	2	f−1(cl⋆(v	f−1(cl⋆(v	NOUN
ejpam-4732	352	3	)	)	PUNCT
ejpam-4732	352	4	)	)	PUNCT
ejpam-4732	352	5	for	for	ADP
ejpam-4732	352	6	every	every	DET
ejpam-4732	352	7	semi	semi	ADJ
ejpam-4732	352	8	-	-	ADJ
ejpam-4732	352	9	j	j	ADJ
ejpam-4732	352	10	-open	-open	NOUN
ejpam-4732	352	11	set	set	VERB
ejpam-4732	352	12	v	v	NOUN
ejpam-4732	352	13	of	of	ADP
ejpam-4732	352	14	y	y	PROPN
ejpam-4732	352	15	;	;	PUNCT
ejpam-4732	352	16	(	(	PUNCT
ejpam-4732	352	17	4	4	X
ejpam-4732	352	18	)	)	PUNCT
ejpam-4732	352	19	f−1(v	f−1(v	NOUN
ejpam-4732	352	20	)	)	PUNCT
ejpam-4732	352	21	⊆	⊆	NUM
ejpam-4732	352	22	sβinti	sβinti	NOUN
ejpam-4732	352	23	(	(	PUNCT
ejpam-4732	352	24	f−1(int⋆(cl(v	f−1(int⋆(cl(v	NOUN
ejpam-4732	352	25	)	)	PUNCT
ejpam-4732	352	26	)	)	PUNCT
ejpam-4732	352	27	)	)	PUNCT
ejpam-4732	352	28	)	)	PUNCT
ejpam-4732	352	29	for	for	ADP
ejpam-4732	352	30	every	every	DET
ejpam-4732	352	31	pre⋆j	pre⋆j	PROPN
ejpam-4732	352	32	-open	-open	NOUN
ejpam-4732	352	33	set	set	VERB
ejpam-4732	352	34	v	v	NOUN
ejpam-4732	352	35	of	of	ADP
ejpam-4732	352	36	y	y	PROPN
ejpam-4732	352	37	.	.	PUNCT
ejpam-4732	353	1	acknowledgements	acknowledgement	NOUN
ejpam-4732	353	2	this	this	DET
ejpam-4732	353	3	research	research	NOUN
ejpam-4732	353	4	project	project	NOUN
ejpam-4732	353	5	was	be	AUX
ejpam-4732	353	6	financially	financially	ADV
ejpam-4732	353	7	supported	support	VERB
ejpam-4732	353	8	by	by	ADP
ejpam-4732	353	9	mahasarakham	mahasarakham	PROPN
ejpam-4732	353	10	university	university	PROPN
ejpam-4732	353	11	.	.	PUNCT
ejpam-4732	354	1	references	reference	NOUN
ejpam-4732	354	2	[	[	X
ejpam-4732	354	3	1	1	NUM
ejpam-4732	354	4	]	]	PUNCT
ejpam-4732	354	5	c.	c.	PROPN
ejpam-4732	354	6	berge	berge	PROPN
ejpam-4732	354	7	.	.	PUNCT
ejpam-4732	355	1	espaces	espace	VERB
ejpam-4732	355	2	topologiques	topologique	NOUN
ejpam-4732	355	3	fonctions	fonction	NOUN
ejpam-4732	355	4	multivoques	multivoque	NOUN
ejpam-4732	355	5	.	.	PUNCT
ejpam-4732	356	1	dunod	dunod	PROPN
ejpam-4732	356	2	,	,	PUNCT
ejpam-4732	356	3	paris	paris	PROPN
ejpam-4732	356	4	,	,	PUNCT
ejpam-4732	356	5	1959	1959	NUM
ejpam-4732	356	6	.	.	PUNCT
ejpam-4732	357	1	[	[	X
ejpam-4732	357	2	2	2	NUM
ejpam-4732	357	3	]	]	PUNCT
ejpam-4732	357	4	c.	c.	PROPN
ejpam-4732	357	5	boonpok	boonpok	PROPN
ejpam-4732	357	6	.	.	PUNCT
ejpam-4732	358	1	on	on	ADP
ejpam-4732	358	2	continuous	continuous	ADJ
ejpam-4732	358	3	multifunctions	multifunction	NOUN
ejpam-4732	358	4	in	in	ADP
ejpam-4732	358	5	ideal	ideal	ADJ
ejpam-4732	358	6	topological	topological	ADJ
ejpam-4732	358	7	spaces	space	NOUN
ejpam-4732	358	8	.	.	PUNCT
ejpam-4732	359	1	lobachevskii	lobachevskii	PROPN
ejpam-4732	359	2	journal	journal	PROPN
ejpam-4732	359	3	of	of	ADP
ejpam-4732	359	4	mathematics	mathematic	NOUN
ejpam-4732	359	5	,	,	PUNCT
ejpam-4732	359	6	40(1):24–35	40(1):24–35	NUM
ejpam-4732	359	7	,	,	PUNCT
ejpam-4732	359	8	2019	2019	NUM
ejpam-4732	359	9	.	.	PUNCT
ejpam-4732	360	1	[	[	X
ejpam-4732	360	2	3	3	X
ejpam-4732	360	3	]	]	PUNCT
ejpam-4732	360	4	c.	c.	PROPN
ejpam-4732	360	5	boonpok	boonpok	PROPN
ejpam-4732	360	6	.	.	PUNCT
ejpam-4732	361	1	upper	upper	ADJ
ejpam-4732	361	2	and	and	CCONJ
ejpam-4732	361	3	lower	low	ADJ
ejpam-4732	361	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-4732	361	5	.	.	PUNCT
ejpam-4732	361	6	heliyon	heliyon	NOUN
ejpam-4732	361	7	,	,	PUNCT
ejpam-4732	361	8	2021	2021	NUM
ejpam-4732	361	9	:	:	PUNCT
ejpam-4732	361	10	e05986	e05986	PROPN
ejpam-4732	361	11	,	,	PUNCT
ejpam-4732	361	12	2021	2021	NUM
ejpam-4732	361	13	.	.	PUNCT
ejpam-4732	362	1	[	[	X
ejpam-4732	362	2	4	4	X
ejpam-4732	362	3	]	]	PUNCT
ejpam-4732	362	4	j.	j.	PROPN
ejpam-4732	362	5	borśık	borśık	PROPN
ejpam-4732	362	6	and	and	CCONJ
ejpam-4732	362	7	j.	j.	PROPN
ejpam-4732	362	8	doboš.	doboš.	PROPN
ejpam-4732	362	9	on	on	ADP
ejpam-4732	362	10	decompositions	decomposition	NOUN
ejpam-4732	362	11	of	of	ADP
ejpam-4732	362	12	quasicontinuity	quasicontinuity	NOUN
ejpam-4732	362	13	.	.	PUNCT
ejpam-4732	363	1	real	real	ADJ
ejpam-4732	363	2	analysis	analysis	NOUN
ejpam-4732	363	3	exchange	exchange	NOUN
ejpam-4732	363	4	,	,	PUNCT
ejpam-4732	363	5	16:292–305	16:292–305	PROPN
ejpam-4732	363	6	,	,	PUNCT
ejpam-4732	363	7	1990/1991	1990/1991	NUM
ejpam-4732	363	8	.	.	PUNCT
ejpam-4732	364	1	[	[	X
ejpam-4732	364	2	5	5	X
ejpam-4732	364	3	]	]	PUNCT
ejpam-4732	364	4	e.	e.	PROPN
ejpam-4732	364	5	ekici	ekici	PROPN
ejpam-4732	364	6	.	.	PUNCT
ejpam-4732	365	1	on	on	ADP
ejpam-4732	365	2	aci	aci	PROPN
ejpam-4732	365	3	-sets	-sets	PROPN
ejpam-4732	365	4	,	,	PUNCT
ejpam-4732	365	5	bci	bci	NOUN
ejpam-4732	365	6	-sets	-set	NOUN
ejpam-4732	365	7	,	,	PUNCT
ejpam-4732	365	8	β⋆	β⋆	PUNCT
ejpam-4732	365	9	i	i	PRON
ejpam-4732	365	10	-open	-open	VERB
ejpam-4732	365	11	sets	set	NOUN
ejpam-4732	365	12	and	and	CCONJ
ejpam-4732	365	13	decompositions	decomposition	NOUN
ejpam-4732	365	14	of	of	ADP
ejpam-4732	365	15	continuity	continuity	NOUN
ejpam-4732	365	16	in	in	ADP
ejpam-4732	365	17	ideal	ideal	ADJ
ejpam-4732	365	18	topological	topological	ADJ
ejpam-4732	365	19	spaces	space	NOUN
ejpam-4732	365	20	.	.	PUNCT
ejpam-4732	366	1	creative	creative	ADJ
ejpam-4732	366	2	mathematics	mathematic	NOUN
ejpam-4732	366	3	informatics	informatic	NOUN
ejpam-4732	366	4	,	,	PUNCT
ejpam-4732	366	5	20:47–54	20:47–54	NUM
ejpam-4732	366	6	,	,	PUNCT
ejpam-4732	366	7	2011	2011	NUM
ejpam-4732	366	8	.	.	PUNCT
ejpam-4732	367	1	references	reference	NOUN
ejpam-4732	367	2	1646	1646	NUM
ejpam-4732	367	3	[	[	X
ejpam-4732	367	4	6	6	NUM
ejpam-4732	367	5	]	]	PUNCT
ejpam-4732	367	6	e.	e.	PROPN
ejpam-4732	367	7	ekici	ekici	PROPN
ejpam-4732	367	8	and	and	CCONJ
ejpam-4732	367	9	t.	t.	PROPN
ejpam-4732	367	10	noiri	noiri	PROPN
ejpam-4732	367	11	.	.	PUNCT
ejpam-4732	368	1	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-4732	368	2	ideal	ideal	ADJ
ejpam-4732	368	3	topological	topological	ADJ
ejpam-4732	368	4	spaces	space	NOUN
ejpam-4732	368	5	.	.	PUNCT
ejpam-4732	369	1	analele	analele	ADP
ejpam-4732	369	2	stiintifice	stiintifice	PROPN
ejpam-4732	369	3	ale	ale	PROPN
ejpam-4732	369	4	universitatii	universitatii	PROPN
ejpam-4732	370	1	al	al	PROPN
ejpam-4732	370	2	i	i	PRON
ejpam-4732	370	3	cuza	cuza	VERB
ejpam-4732	370	4	din	din	PROPN
ejpam-4732	370	5	lasi	lasi	PROPN
ejpam-4732	370	6	-	-	PUNCT
ejpam-4732	370	7	mathematica	mathematica	PROPN
ejpam-4732	370	8	,	,	PUNCT
ejpam-4732	370	9	58:121–129	58:121–129	PROPN
ejpam-4732	370	10	,	,	PUNCT
ejpam-4732	370	11	2012	2012	NUM
ejpam-4732	370	12	.	.	PUNCT
ejpam-4732	371	1	[	[	X
ejpam-4732	371	2	7	7	X
ejpam-4732	371	3	]	]	PUNCT
ejpam-4732	371	4	m.	m.	NOUN
ejpam-4732	371	5	e.	e.	PROPN
ejpam-4732	371	6	abd	abd	PROPN
ejpam-4732	372	1	el	el	PROPN
ejpam-4732	372	2	-	-	PROPN
ejpam-4732	372	3	monsef	monsef	PROPN
ejpam-4732	372	4	,	,	PUNCT
ejpam-4732	372	5	s.	s.	PROPN
ejpam-4732	372	6	n.	n.	PROPN
ejpam-4732	372	7	el	el	PROPN
ejpam-4732	372	8	-	-	PROPN
ejpam-4732	372	9	deeb	deeb	PROPN
ejpam-4732	372	10	,	,	PUNCT
ejpam-4732	372	11	and	and	CCONJ
ejpam-4732	372	12	r.	r.	PROPN
ejpam-4732	372	13	a.	a.	PROPN
ejpam-4732	372	14	mahmoud	mahmoud	PROPN
ejpam-4732	372	15	.	.	PUNCT
ejpam-4732	373	1	β	β	X
ejpam-4732	373	2	-	-	ADJ
ejpam-4732	373	3	open	open	ADJ
ejpam-4732	373	4	sets	set	NOUN
ejpam-4732	373	5	and	and	CCONJ
ejpam-4732	373	6	βcontinuous	βcontinuous	ADJ
ejpam-4732	373	7	mappings	mapping	NOUN
ejpam-4732	373	8	.	.	PUNCT
ejpam-4732	374	1	bulletin	bulletin	NOUN
ejpam-4732	374	2	of	of	ADP
ejpam-4732	374	3	the	the	DET
ejpam-4732	374	4	faculty	faculty	NOUN
ejpam-4732	374	5	of	of	ADP
ejpam-4732	374	6	science	science	NOUN
ejpam-4732	374	7	.	.	PUNCT
ejpam-4732	375	1	assiut	assiut	PROPN
ejpam-4732	375	2	university	university	PROPN
ejpam-4732	375	3	.	.	PUNCT
ejpam-4732	375	4	,	,	PUNCT
ejpam-4732	375	5	12:77–90	12:77–90	NUM
ejpam-4732	375	6	,	,	PUNCT
ejpam-4732	375	7	1983	1983	NUM
ejpam-4732	375	8	.	.	PUNCT
ejpam-4732	376	1	[	[	X
ejpam-4732	376	2	8	8	X
ejpam-4732	376	3	]	]	X
ejpam-4732	376	4	e.	e.	PROPN
ejpam-4732	376	5	hatir	hatir	PROPN
ejpam-4732	376	6	,	,	PUNCT
ejpam-4732	376	7	a.	a.	PROPN
ejpam-4732	376	8	keskin	keskin	PROPN
ejpam-4732	376	9	,	,	PUNCT
ejpam-4732	376	10	and	and	CCONJ
ejpam-4732	376	11	t.	t.	PROPN
ejpam-4732	376	12	noiri	noiri	PROPN
ejpam-4732	376	13	.	.	PUNCT
ejpam-4732	377	1	on	on	ADP
ejpam-4732	377	2	a	a	DET
ejpam-4732	377	3	new	new	ADJ
ejpam-4732	377	4	decomposition	decomposition	NOUN
ejpam-4732	377	5	of	of	ADP
ejpam-4732	377	6	continuity	continuity	NOUN
ejpam-4732	377	7	via	via	ADP
ejpam-4732	377	8	idealization	idealization	NOUN
ejpam-4732	377	9	.	.	PUNCT
ejpam-4732	378	1	jp	jp	PROPN
ejpam-4732	378	2	journal	journal	PROPN
ejpam-4732	378	3	of	of	ADP
ejpam-4732	378	4	geometry	geometry	NOUN
ejpam-4732	378	5	and	and	CCONJ
ejpam-4732	378	6	topology	topology	NOUN
ejpam-4732	378	7	,	,	PUNCT
ejpam-4732	378	8	3:53–64	3:53–64	NUM
ejpam-4732	378	9	,	,	PUNCT
ejpam-4732	378	10	2003	2003	NUM
ejpam-4732	378	11	.	.	PUNCT
ejpam-4732	379	1	[	[	X
ejpam-4732	379	2	9	9	X
ejpam-4732	379	3	]	]	PUNCT
ejpam-4732	379	4	e.	e.	PROPN
ejpam-4732	379	5	hatir	hatir	PROPN
ejpam-4732	379	6	,	,	PUNCT
ejpam-4732	379	7	a.	a.	PROPN
ejpam-4732	379	8	keskin	keskin	PROPN
ejpam-4732	379	9	,	,	PUNCT
ejpam-4732	379	10	and	and	CCONJ
ejpam-4732	379	11	t.	t.	PROPN
ejpam-4732	379	12	noiri	noiri	PROPN
ejpam-4732	379	13	.	.	PUNCT
ejpam-4732	380	1	a	a	DET
ejpam-4732	380	2	note	note	NOUN
ejpam-4732	380	3	on	on	ADP
ejpam-4732	380	4	strong	strong	ADJ
ejpam-4732	380	5	β	β	NOUN
ejpam-4732	380	6	-	-	ADJ
ejpam-4732	380	7	i	i	PRON
ejpam-4732	380	8	-open	-open	NOUN
ejpam-4732	380	9	sets	set	NOUN
ejpam-4732	380	10	and	and	CCONJ
ejpam-4732	380	11	strongly	strongly	ADV
ejpam-4732	380	12	β	β	X
ejpam-4732	380	13	-	-	ADJ
ejpam-4732	380	14	i	i	VERB
ejpam-4732	380	15	-continuous	-continuous	ADJ
ejpam-4732	380	16	functions	function	NOUN
ejpam-4732	380	17	.	.	PUNCT
ejpam-4732	381	1	acta	acta	PROPN
ejpam-4732	381	2	mathematica	mathematica	PROPN
ejpam-4732	381	3	hungarica	hungarica	PROPN
ejpam-4732	381	4	,	,	PUNCT
ejpam-4732	381	5	108:87–94	108:87–94	NUM
ejpam-4732	381	6	,	,	PUNCT
ejpam-4732	381	7	2005	2005	NUM
ejpam-4732	381	8	.	.	PUNCT
ejpam-4732	382	1	[	[	X
ejpam-4732	382	2	10	10	NUM
ejpam-4732	382	3	]	]	X
ejpam-4732	382	4	e.	e.	PROPN
ejpam-4732	382	5	hatir	hatir	PROPN
ejpam-4732	382	6	and	and	CCONJ
ejpam-4732	382	7	t.	t.	PROPN
ejpam-4732	382	8	noiri	noiri	PROPN
ejpam-4732	382	9	.	.	PUNCT
ejpam-4732	383	1	on	on	ADP
ejpam-4732	383	2	decompositions	decomposition	NOUN
ejpam-4732	383	3	of	of	ADP
ejpam-4732	383	4	continuity	continuity	NOUN
ejpam-4732	383	5	via	via	ADP
ejpam-4732	383	6	idealization	idealization	NOUN
ejpam-4732	383	7	.	.	PUNCT
ejpam-4732	384	1	acta	acta	PROPN
ejpam-4732	384	2	mathematica	mathematica	PROPN
ejpam-4732	384	3	hungarica	hungarica	PROPN
ejpam-4732	384	4	,	,	PUNCT
ejpam-4732	384	5	96:341–349	96:341–349	PROPN
ejpam-4732	384	6	,	,	PUNCT
ejpam-4732	384	7	2002	2002	NUM
ejpam-4732	384	8	.	.	PUNCT
ejpam-4732	385	1	[	[	X
ejpam-4732	385	2	11	11	NUM
ejpam-4732	385	3	]	]	X
ejpam-4732	385	4	d.	d.	PROPN
ejpam-4732	385	5	janković	janković	PROPN
ejpam-4732	385	6	and	and	CCONJ
ejpam-4732	385	7	t.	t.	PROPN
ejpam-4732	385	8	r.	r.	PROPN
ejpam-4732	385	9	hamlett	hamlett	PROPN
ejpam-4732	385	10	.	.	PUNCT
ejpam-4732	386	1	new	new	ADJ
ejpam-4732	386	2	topologies	topology	NOUN
ejpam-4732	386	3	from	from	ADP
ejpam-4732	386	4	old	old	ADJ
ejpam-4732	386	5	via	via	ADP
ejpam-4732	386	6	ideals	ideal	NOUN
ejpam-4732	386	7	.	.	PUNCT
ejpam-4732	387	1	the	the	DET
ejpam-4732	387	2	american	american	PROPN
ejpam-4732	387	3	mathematical	mathematical	PROPN
ejpam-4732	387	4	monthly	monthly	ADV
ejpam-4732	387	5	,	,	PUNCT
ejpam-4732	387	6	97:295–310	97:295–310	PROPN
ejpam-4732	387	7	,	,	PUNCT
ejpam-4732	387	8	1990	1990	NUM
ejpam-4732	387	9	.	.	PUNCT
ejpam-4732	388	1	[	[	X
ejpam-4732	388	2	12	12	NUM
ejpam-4732	388	3	]	]	PUNCT
ejpam-4732	388	4	k.	k.	PROPN
ejpam-4732	388	5	kuratowski	kuratowski	PROPN
ejpam-4732	388	6	.	.	PUNCT
ejpam-4732	389	1	topology	topology	PROPN
ejpam-4732	389	2	,	,	PUNCT
ejpam-4732	389	3	vol	vol	NOUN
ejpam-4732	389	4	.	.	PUNCT
ejpam-4732	389	5	i.	i.	PROPN
ejpam-4732	389	6	academic	academic	PROPN
ejpam-4732	389	7	press	press	PROPN
ejpam-4732	389	8	,	,	PUNCT
ejpam-4732	389	9	new	new	PROPN
ejpam-4732	389	10	york	york	PROPN
ejpam-4732	389	11	,	,	PUNCT
ejpam-4732	389	12	1966	1966	NUM
ejpam-4732	389	13	.	.	PUNCT
ejpam-4732	390	1	[	[	X
ejpam-4732	390	2	13	13	NUM
ejpam-4732	390	3	]	]	X
ejpam-4732	390	4	n.	n.	PROPN
ejpam-4732	390	5	levine	levine	PROPN
ejpam-4732	390	6	.	.	PUNCT
ejpam-4732	391	1	semi	semi	ADJ
ejpam-4732	391	2	-	-	ADJ
ejpam-4732	391	3	open	open	ADJ
ejpam-4732	391	4	sets	set	NOUN
ejpam-4732	391	5	and	and	CCONJ
ejpam-4732	391	6	semi	semi	ADJ
ejpam-4732	391	7	-	-	NOUN
ejpam-4732	391	8	continuity	continuity	NOUN
ejpam-4732	391	9	in	in	ADP
ejpam-4732	391	10	topological	topological	ADJ
ejpam-4732	391	11	spaces	space	NOUN
ejpam-4732	391	12	.	.	PUNCT
ejpam-4732	392	1	the	the	DET
ejpam-4732	392	2	american	american	PROPN
ejpam-4732	392	3	mathematical	mathematical	PROPN
ejpam-4732	392	4	monthly	monthly	ADV
ejpam-4732	392	5	,	,	PUNCT
ejpam-4732	392	6	70:36–41	70:36–41	NUM
ejpam-4732	392	7	,	,	PUNCT
ejpam-4732	392	8	1963	1963	NUM
ejpam-4732	392	9	.	.	PUNCT
ejpam-4732	393	1	[	[	X
ejpam-4732	393	2	14	14	NUM
ejpam-4732	393	3	]	]	X
ejpam-4732	393	4	s.	s.	PROPN
ejpam-4732	393	5	marcus	marcus	PROPN
ejpam-4732	393	6	.	.	PUNCT
ejpam-4732	394	1	sur	sur	PROPN
ejpam-4732	394	2	les	les	PROPN
ejpam-4732	394	3	fonctions	fonctions	PROPN
ejpam-4732	394	4	quasicontinues	quasicontinue	NOUN
ejpam-4732	394	5	au	au	PROPN
ejpam-4732	394	6	sens	sens	X
ejpam-4732	394	7	de	de	PROPN
ejpam-4732	394	8	s.	s.	PROPN
ejpam-4732	394	9	kempisty	kempisty	PROPN
ejpam-4732	394	10	.	.	PUNCT
ejpam-4732	395	1	colloquium	colloquium	NOUN
ejpam-4732	395	2	mathematicum	mathematicum	PROPN
ejpam-4732	395	3	,	,	PUNCT
ejpam-4732	395	4	8:47–53	8:47–53	NUM
ejpam-4732	395	5	,	,	PUNCT
ejpam-4732	395	6	1961	1961	NUM
ejpam-4732	395	7	.	.	PUNCT
ejpam-4732	396	1	[	[	X
ejpam-4732	396	2	15	15	NUM
ejpam-4732	396	3	]	]	X
ejpam-4732	396	4	a.	a.	NOUN
ejpam-4732	396	5	s.	s.	PROPN
ejpam-4732	396	6	mashhour	mashhour	PROPN
ejpam-4732	396	7	,	,	PUNCT
ejpam-4732	396	8	m.	m.	PROPN
ejpam-4732	396	9	e.	e.	PROPN
ejpam-4732	396	10	abd	abd	PROPN
ejpam-4732	396	11	el	el	PROPN
ejpam-4732	396	12	-	-	PROPN
ejpam-4732	396	13	monsef	monsef	ADJ
ejpam-4732	396	14	,	,	PUNCT
ejpam-4732	396	15	and	and	CCONJ
ejpam-4732	396	16	s.	s.	PROPN
ejpam-4732	396	17	n.	n.	PROPN
ejpam-4732	396	18	el	el	PROPN
ejpam-4732	396	19	-	-	PROPN
ejpam-4732	396	20	deeb	deeb	PROPN
ejpam-4732	396	21	.	.	PUNCT
ejpam-4732	397	1	on	on	ADP
ejpam-4732	397	2	precontinuous	precontinuous	ADJ
ejpam-4732	397	3	and	and	CCONJ
ejpam-4732	397	4	weak	weak	ADJ
ejpam-4732	397	5	precontinuous	precontinuous	ADJ
ejpam-4732	397	6	mappings	mapping	NOUN
ejpam-4732	397	7	.	.	PUNCT
ejpam-4732	398	1	proceedings	proceeding	NOUN
ejpam-4732	398	2	of	of	ADP
ejpam-4732	398	3	the	the	DET
ejpam-4732	398	4	mathematical	mathematical	ADJ
ejpam-4732	398	5	physical	physical	ADJ
ejpam-4732	398	6	society	society	NOUN
ejpam-4732	398	7	of	of	ADP
ejpam-4732	398	8	egypt	egypt	PROPN
ejpam-4732	398	9	,	,	PUNCT
ejpam-4732	398	10	53:47–53	53:47–53	NUM
ejpam-4732	398	11	,	,	PUNCT
ejpam-4732	398	12	1982	1982	NUM
ejpam-4732	398	13	.	.	PUNCT
ejpam-4732	399	1	[	[	X
ejpam-4732	399	2	16	16	NUM
ejpam-4732	399	3	]	]	PUNCT
ejpam-4732	399	4	t.	t.	PROPN
ejpam-4732	399	5	noiri	noiri	PROPN
ejpam-4732	399	6	and	and	CCONJ
ejpam-4732	399	7	v.	v.	ADP
ejpam-4732	399	8	popa	popa	NOUN
ejpam-4732	399	9	.	.	PUNCT
ejpam-4732	400	1	on	on	ADP
ejpam-4732	400	2	upper	upper	ADJ
ejpam-4732	400	3	and	and	CCONJ
ejpam-4732	400	4	lower	low	ADJ
ejpam-4732	400	5	almost	almost	ADV
ejpam-4732	400	6	β	β	ADJ
ejpam-4732	400	7	-	-	ADJ
ejpam-4732	400	8	continuous	continuous	ADJ
ejpam-4732	400	9	multifunctions	multifunction	NOUN
ejpam-4732	400	10	.	.	PUNCT
ejpam-4732	401	1	acta	acta	PROPN
ejpam-4732	401	2	mathematica	mathematica	PROPN
ejpam-4732	401	3	hungarica	hungarica	PROPN
ejpam-4732	401	4	,	,	PUNCT
ejpam-4732	401	5	82:57–73	82:57–73	PROPN
ejpam-4732	401	6	,	,	PUNCT
ejpam-4732	401	7	1999	1999	NUM
ejpam-4732	401	8	.	.	PUNCT
ejpam-4732	402	1	[	[	X
ejpam-4732	402	2	17	17	NUM
ejpam-4732	402	3	]	]	PUNCT
ejpam-4732	402	4	v.	v.	CCONJ
ejpam-4732	402	5	popa	popa	NOUN
ejpam-4732	402	6	and	and	CCONJ
ejpam-4732	402	7	t.	t.	PROPN
ejpam-4732	402	8	noiri	noiri	PROPN
ejpam-4732	402	9	.	.	PUNCT
ejpam-4732	403	1	on	on	ADP
ejpam-4732	403	2	β	β	ADJ
ejpam-4732	403	3	-	-	ADJ
ejpam-4732	403	4	continuous	continuous	ADJ
ejpam-4732	403	5	functions	function	NOUN
ejpam-4732	403	6	.	.	PUNCT
ejpam-4732	404	1	real	real	ADJ
ejpam-4732	404	2	analysis	analysis	NOUN
ejpam-4732	404	3	exchange	exchange	NOUN
ejpam-4732	404	4	,	,	PUNCT
ejpam-4732	404	5	18:544	18:544	NUM
ejpam-4732	404	6	–	–	PUNCT
ejpam-4732	404	7	548	548	NUM
ejpam-4732	404	8	,	,	PUNCT
ejpam-4732	404	9	1992/1993	1992/1993	NUM
ejpam-4732	404	10	.	.	PUNCT
ejpam-4732	405	1	[	[	X
ejpam-4732	405	2	18	18	NUM
ejpam-4732	405	3	]	]	PUNCT
ejpam-4732	405	4	v.	v.	CCONJ
ejpam-4732	405	5	popa	popa	NOUN
ejpam-4732	405	6	and	and	CCONJ
ejpam-4732	405	7	t.	t.	PROPN
ejpam-4732	405	8	noiri	noiri	PROPN
ejpam-4732	405	9	.	.	PUNCT
ejpam-4732	406	1	on	on	ADP
ejpam-4732	406	2	upper	upper	ADJ
ejpam-4732	406	3	and	and	CCONJ
ejpam-4732	406	4	lower	low	ADJ
ejpam-4732	406	5	β	β	ADJ
ejpam-4732	406	6	-	-	ADJ
ejpam-4732	406	7	continuous	continuous	ADJ
ejpam-4732	406	8	multifunctions	multifunction	NOUN
ejpam-4732	406	9	.	.	PUNCT
ejpam-4732	407	1	real	real	ADJ
ejpam-4732	407	2	analysis	analysis	NOUN
ejpam-4732	407	3	exchange	exchange	NOUN
ejpam-4732	407	4	,	,	PUNCT
ejpam-4732	407	5	22:362–376	22:362–376	PROPN
ejpam-4732	407	6	,	,	PUNCT
ejpam-4732	407	7	1996/1997	1996/1997	NUM
ejpam-4732	407	8	.	.	PUNCT