id	sid	tid	token	lemma	pos
ejpam-4858	1	1	european	european	PROPN
ejpam-4858	1	2	journal	journal	PROPN
ejpam-4858	1	3	of	of	ADP
ejpam-4858	1	4	pure	pure	ADJ
ejpam-4858	1	5	and	and	CCONJ
ejpam-4858	1	6	applied	apply	VERB
ejpam-4858	1	7	mathematics	mathematic	NOUN
ejpam-4858	1	8	vol	vol	NOUN
ejpam-4858	1	9	.	.	PROPN
ejpam-4858	2	1	17	17	NUM
ejpam-4858	2	2	,	,	PUNCT
ejpam-4858	2	3	no	no	INTJ
ejpam-4858	2	4	.	.	NOUN
ejpam-4858	2	5	1	1	NUM
ejpam-4858	2	6	,	,	PUNCT
ejpam-4858	2	7	2024	2024	NUM
ejpam-4858	2	8	,	,	PUNCT
ejpam-4858	2	9	201	201	NUM
ejpam-4858	2	10	-	-	SYM
ejpam-4858	2	11	211	211	NUM
ejpam-4858	2	12	issn	issn	PROPN
ejpam-4858	2	13	1307	1307	NUM
ejpam-4858	2	14	-	-	SYM
ejpam-4858	2	15	5543	5543	NUM
ejpam-4858	2	16	–	–	PUNCT
ejpam-4858	3	1	ejpam.com	ejpam.com	X
ejpam-4858	3	2	published	publish	VERB
ejpam-4858	3	3	by	by	ADP
ejpam-4858	3	4	new	new	PROPN
ejpam-4858	3	5	york	york	PROPN
ejpam-4858	3	6	business	business	PROPN
ejpam-4858	3	7	global	global	PROPN
ejpam-4858	3	8	upper	upper	ADJ
ejpam-4858	3	9	and	and	CCONJ
ejpam-4858	3	10	lower	low	ADJ
ejpam-4858	3	11	α-⋆-continuity	α-⋆-continuity	PROPN
ejpam-4858	3	12	chawalit	chawalit	VERB
ejpam-4858	3	13	boonpok1	boonpok1	NOUN
ejpam-4858	3	14	,	,	PUNCT
ejpam-4858	3	15	jeeranunt	jeeranunt	PROPN
ejpam-4858	3	16	khampakdee1,∗	khampakdee1,∗	PROPN
ejpam-4858	3	17	1	1	NUM
ejpam-4858	3	18	mathematics	mathematic	NOUN
ejpam-4858	3	19	and	and	CCONJ
ejpam-4858	3	20	applied	apply	VERB
ejpam-4858	3	21	mathematics	mathematics	PROPN
ejpam-4858	3	22	research	research	NOUN
ejpam-4858	3	23	unit	unit	NOUN
ejpam-4858	3	24	,	,	PUNCT
ejpam-4858	3	25	department	department	NOUN
ejpam-4858	3	26	of	of	ADP
ejpam-4858	3	27	mathematics	mathematic	NOUN
ejpam-4858	3	28	,	,	PUNCT
ejpam-4858	3	29	faculty	faculty	NOUN
ejpam-4858	3	30	of	of	ADP
ejpam-4858	3	31	science	science	NOUN
ejpam-4858	3	32	,	,	PUNCT
ejpam-4858	3	33	mahasarakham	mahasarakham	PROPN
ejpam-4858	3	34	university	university	PROPN
ejpam-4858	3	35	,	,	PUNCT
ejpam-4858	3	36	maha	maha	PROPN
ejpam-4858	3	37	sarakham	sarakham	PROPN
ejpam-4858	3	38	,	,	PUNCT
ejpam-4858	3	39	44150	44150	NUM
ejpam-4858	3	40	,	,	PUNCT
ejpam-4858	3	41	thailand	thailand	PROPN
ejpam-4858	3	42	abstract	abstract	PROPN
ejpam-4858	3	43	.	.	PUNCT
ejpam-4858	4	1	our	our	PRON
ejpam-4858	4	2	main	main	ADJ
ejpam-4858	4	3	purpose	purpose	NOUN
ejpam-4858	4	4	is	be	AUX
ejpam-4858	4	5	to	to	PART
ejpam-4858	4	6	introduce	introduce	VERB
ejpam-4858	4	7	the	the	DET
ejpam-4858	4	8	concepts	concept	NOUN
ejpam-4858	4	9	of	of	ADP
ejpam-4858	4	10	upper	upper	ADJ
ejpam-4858	4	11	and	and	CCONJ
ejpam-4858	4	12	lower	lower	VERB
ejpam-4858	4	13	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	4	14	multifunctions	multifunction	NOUN
ejpam-4858	4	15	.	.	PUNCT
ejpam-4858	5	1	in	in	ADP
ejpam-4858	5	2	particular	particular	ADJ
ejpam-4858	5	3	,	,	PUNCT
ejpam-4858	5	4	some	some	DET
ejpam-4858	5	5	characterizations	characterization	NOUN
ejpam-4858	5	6	of	of	ADP
ejpam-4858	5	7	upper	upper	ADJ
ejpam-4858	5	8	and	and	CCONJ
ejpam-4858	5	9	lower	low	ADJ
ejpam-4858	5	10	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	5	11	multifunctions	multifunction	NOUN
ejpam-4858	5	12	are	be	AUX
ejpam-4858	5	13	investigated	investigate	VERB
ejpam-4858	5	14	.	.	PUNCT
ejpam-4858	6	1	2020	2020	NUM
ejpam-4858	6	2	mathematics	mathematic	NOUN
ejpam-4858	6	3	subject	subject	NOUN
ejpam-4858	6	4	classifications	classification	NOUN
ejpam-4858	6	5	:	:	PUNCT
ejpam-4858	6	6	54c08	54c08	NUM
ejpam-4858	6	7	,	,	PUNCT
ejpam-4858	6	8	54c60	54c60	NUM
ejpam-4858	6	9	key	key	ADJ
ejpam-4858	6	10	words	word	NOUN
ejpam-4858	6	11	and	and	CCONJ
ejpam-4858	6	12	phrases	phrase	NOUN
ejpam-4858	6	13	:	:	PUNCT
ejpam-4858	6	14	upper	upper	ADJ
ejpam-4858	6	15	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	6	16	multifunction	multifunction	NOUN
ejpam-4858	6	17	,	,	PUNCT
ejpam-4858	6	18	lower	low	ADJ
ejpam-4858	6	19	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	6	20	multifunction	multifunction	NOUN
ejpam-4858	6	21	1	1	NUM
ejpam-4858	6	22	.	.	PUNCT
ejpam-4858	7	1	introduction	introduction	NOUN
ejpam-4858	7	2	the	the	DET
ejpam-4858	7	3	field	field	NOUN
ejpam-4858	7	4	of	of	ADP
ejpam-4858	7	5	mathematical	mathematical	ADJ
ejpam-4858	7	6	science	science	NOUN
ejpam-4858	7	7	called	call	VERB
ejpam-4858	7	8	topology	topology	NOUN
ejpam-4858	7	9	is	be	AUX
ejpam-4858	7	10	concerned	concern	VERB
ejpam-4858	7	11	with	with	ADP
ejpam-4858	7	12	all	all	DET
ejpam-4858	7	13	questions	question	NOUN
ejpam-4858	7	14	directly	directly	ADV
ejpam-4858	7	15	or	or	CCONJ
ejpam-4858	7	16	indirectly	indirectly	ADV
ejpam-4858	7	17	related	relate	VERB
ejpam-4858	7	18	to	to	ADP
ejpam-4858	7	19	continuity	continuity	NOUN
ejpam-4858	7	20	.	.	PUNCT
ejpam-4858	8	1	continuity	continuity	NOUN
ejpam-4858	8	2	is	be	AUX
ejpam-4858	8	3	an	an	DET
ejpam-4858	8	4	important	important	ADJ
ejpam-4858	8	5	concept	concept	NOUN
ejpam-4858	8	6	for	for	ADP
ejpam-4858	8	7	the	the	DET
ejpam-4858	8	8	study	study	NOUN
ejpam-4858	8	9	and	and	CCONJ
ejpam-4858	8	10	investigation	investigation	NOUN
ejpam-4858	8	11	in	in	ADP
ejpam-4858	8	12	topological	topological	ADJ
ejpam-4858	8	13	spaces	space	NOUN
ejpam-4858	8	14	.	.	PUNCT
ejpam-4858	9	1	this	this	DET
ejpam-4858	9	2	concept	concept	NOUN
ejpam-4858	9	3	has	have	AUX
ejpam-4858	9	4	been	be	AUX
ejpam-4858	9	5	extended	extend	VERB
ejpam-4858	9	6	to	to	ADP
ejpam-4858	9	7	the	the	DET
ejpam-4858	9	8	setting	set	VERB
ejpam-4858	9	9	multifunctions	multifunction	NOUN
ejpam-4858	9	10	and	and	CCONJ
ejpam-4858	9	11	has	have	AUX
ejpam-4858	9	12	been	be	AUX
ejpam-4858	9	13	generalized	generalize	VERB
ejpam-4858	9	14	by	by	ADP
ejpam-4858	9	15	weaker	weak	ADJ
ejpam-4858	9	16	forms	form	NOUN
ejpam-4858	9	17	of	of	ADP
ejpam-4858	9	18	open	open	ADJ
ejpam-4858	9	19	sets	set	NOUN
ejpam-4858	9	20	.	.	PUNCT
ejpam-4858	10	1	in	in	ADP
ejpam-4858	10	2	1965	1965	NUM
ejpam-4858	10	3	,	,	PUNCT
ejpam-4858	10	4	nj̊astad	nj̊astad	NOUN
ejpam-4858	11	1	[	[	X
ejpam-4858	11	2	21	21	NUM
ejpam-4858	11	3	]	]	PUNCT
ejpam-4858	11	4	introduced	introduce	VERB
ejpam-4858	11	5	a	a	DET
ejpam-4858	11	6	weak	weak	ADJ
ejpam-4858	11	7	form	form	NOUN
ejpam-4858	11	8	of	of	ADP
ejpam-4858	11	9	open	open	ADJ
ejpam-4858	11	10	sets	set	NOUN
ejpam-4858	11	11	called	call	VERB
ejpam-4858	11	12	α	α	PRON
ejpam-4858	11	13	-	-	PUNCT
ejpam-4858	11	14	sets	set	NOUN
ejpam-4858	11	15	.	.	PUNCT
ejpam-4858	12	1	mashhour	mashhour	INTJ
ejpam-4858	12	2	et	et	PROPN
ejpam-4858	12	3	al	al	PROPN
ejpam-4858	12	4	.	.	PUNCT
ejpam-4858	13	1	[	[	X
ejpam-4858	13	2	19	19	NUM
ejpam-4858	13	3	]	]	PUNCT
ejpam-4858	13	4	defined	define	VERB
ejpam-4858	13	5	a	a	DET
ejpam-4858	13	6	function	function	NOUN
ejpam-4858	13	7	to	to	PART
ejpam-4858	13	8	be	be	AUX
ejpam-4858	13	9	α	α	X
ejpam-4858	13	10	-	-	ADJ
ejpam-4858	13	11	continuous	continuous	ADJ
ejpam-4858	13	12	if	if	SCONJ
ejpam-4858	13	13	the	the	DET
ejpam-4858	13	14	inverse	inverse	ADJ
ejpam-4858	13	15	image	image	NOUN
ejpam-4858	13	16	of	of	ADP
ejpam-4858	13	17	each	each	DET
ejpam-4858	13	18	open	open	ADJ
ejpam-4858	13	19	set	set	NOUN
ejpam-4858	13	20	is	be	AUX
ejpam-4858	13	21	an	an	DET
ejpam-4858	13	22	α	α	NOUN
ejpam-4858	13	23	-	-	PUNCT
ejpam-4858	13	24	set	set	VERB
ejpam-4858	13	25	and	and	CCONJ
ejpam-4858	13	26	obtained	obtain	VERB
ejpam-4858	13	27	several	several	ADJ
ejpam-4858	13	28	characterizations	characterization	NOUN
ejpam-4858	13	29	of	of	ADP
ejpam-4858	13	30	such	such	ADJ
ejpam-4858	13	31	functions	function	NOUN
ejpam-4858	13	32	.	.	PUNCT
ejpam-4858	14	1	noiri	noiri	PROPN
ejpam-4858	15	1	[	[	X
ejpam-4858	15	2	22	22	NUM
ejpam-4858	15	3	]	]	PUNCT
ejpam-4858	15	4	investigated	investigate	VERB
ejpam-4858	15	5	the	the	DET
ejpam-4858	15	6	relationships	relationship	NOUN
ejpam-4858	15	7	between	between	ADP
ejpam-4858	15	8	α	α	NOUN
ejpam-4858	15	9	-	-	ADJ
ejpam-4858	15	10	continuous	continuous	ADJ
ejpam-4858	15	11	functions	function	NOUN
ejpam-4858	15	12	and	and	CCONJ
ejpam-4858	15	13	several	several	ADJ
ejpam-4858	15	14	known	know	VERB
ejpam-4858	15	15	functions	function	NOUN
ejpam-4858	15	16	,	,	PUNCT
ejpam-4858	15	17	for	for	ADP
ejpam-4858	15	18	example	example	NOUN
ejpam-4858	15	19	,	,	PUNCT
ejpam-4858	15	20	almost	almost	ADV
ejpam-4858	15	21	continuous	continuous	ADJ
ejpam-4858	15	22	functions	function	NOUN
ejpam-4858	15	23	,	,	PUNCT
ejpam-4858	15	24	η	η	ADJ
ejpam-4858	15	25	-	-	ADJ
ejpam-4858	15	26	continuous	continuous	ADJ
ejpam-4858	15	27	functions	function	NOUN
ejpam-4858	15	28	,	,	PUNCT
ejpam-4858	15	29	δ	δ	NOUN
ejpam-4858	15	30	-	-	PUNCT
ejpam-4858	15	31	continuous	continuous	ADJ
ejpam-4858	15	32	functions	function	NOUN
ejpam-4858	15	33	or	or	CCONJ
ejpam-4858	15	34	irresolute	irresolute	ADJ
ejpam-4858	15	35	functions	function	NOUN
ejpam-4858	15	36	.	.	PUNCT
ejpam-4858	16	1	in	in	ADP
ejpam-4858	16	2	[	[	X
ejpam-4858	16	3	23	23	NUM
ejpam-4858	16	4	]	]	PUNCT
ejpam-4858	16	5	,	,	PUNCT
ejpam-4858	16	6	the	the	DET
ejpam-4858	16	7	present	present	ADJ
ejpam-4858	16	8	author	author	NOUN
ejpam-4858	16	9	introduced	introduce	VERB
ejpam-4858	16	10	the	the	DET
ejpam-4858	16	11	concept	concept	NOUN
ejpam-4858	16	12	of	of	ADP
ejpam-4858	16	13	almost	almost	ADV
ejpam-4858	16	14	α	α	NOUN
ejpam-4858	16	15	-	-	NOUN
ejpam-4858	16	16	continuity	continuity	NOUN
ejpam-4858	16	17	in	in	ADP
ejpam-4858	16	18	topological	topological	ADJ
ejpam-4858	16	19	spaces	space	NOUN
ejpam-4858	16	20	as	as	ADP
ejpam-4858	16	21	a	a	DET
ejpam-4858	16	22	generalization	generalization	NOUN
ejpam-4858	16	23	of	of	ADP
ejpam-4858	16	24	α	α	NOUN
ejpam-4858	16	25	-	-	PUNCT
ejpam-4858	16	26	continuity	continuity	NOUN
ejpam-4858	16	27	and	and	CCONJ
ejpam-4858	16	28	almost	almost	ADV
ejpam-4858	16	29	continuity	continuity	NOUN
ejpam-4858	16	30	.	.	PUNCT
ejpam-4858	17	1	neubrunn	neubrunn	NOUN
ejpam-4858	18	1	[	[	X
ejpam-4858	18	2	20	20	NUM
ejpam-4858	18	3	]	]	PUNCT
ejpam-4858	18	4	introduced	introduce	VERB
ejpam-4858	18	5	the	the	DET
ejpam-4858	18	6	notion	notion	NOUN
ejpam-4858	18	7	of	of	ADP
ejpam-4858	18	8	upper	upper	ADJ
ejpam-4858	18	9	(	(	PUNCT
ejpam-4858	18	10	resp	resp	NOUN
ejpam-4858	18	11	.	.	PUNCT
ejpam-4858	19	1	lower	low	ADJ
ejpam-4858	19	2	)	)	PUNCT
ejpam-4858	19	3	α	α	NUM
ejpam-4858	19	4	-	-	ADJ
ejpam-4858	19	5	continuous	continuous	ADJ
ejpam-4858	19	6	multifunctions	multifunction	NOUN
ejpam-4858	19	7	.	.	PUNCT
ejpam-4858	20	1	these	these	DET
ejpam-4858	20	2	multifunctions	multifunction	NOUN
ejpam-4858	20	3	are	be	AUX
ejpam-4858	20	4	further	far	ADV
ejpam-4858	20	5	investigated	investigate	VERB
ejpam-4858	20	6	by	by	ADP
ejpam-4858	20	7	the	the	DET
ejpam-4858	20	8	present	present	ADJ
ejpam-4858	20	9	authors	author	NOUN
ejpam-4858	20	10	[	[	X
ejpam-4858	20	11	24	24	NUM
ejpam-4858	20	12	]	]	PUNCT
ejpam-4858	20	13	.	.	PUNCT
ejpam-4858	21	1	boonpok	boonpok	PROPN
ejpam-4858	21	2	et	et	PROPN
ejpam-4858	21	3	al	al	PROPN
ejpam-4858	21	4	.	.	PUNCT
ejpam-4858	22	1	[	[	X
ejpam-4858	22	2	11	11	NUM
ejpam-4858	22	3	]	]	PUNCT
ejpam-4858	22	4	introduced	introduce	VERB
ejpam-4858	22	5	and	and	CCONJ
ejpam-4858	22	6	studied	study	VERB
ejpam-4858	22	7	the	the	DET
ejpam-4858	22	8	notions	notion	NOUN
ejpam-4858	22	9	of	of	ADP
ejpam-4858	22	10	upper	upper	ADJ
ejpam-4858	22	11	and	and	CCONJ
ejpam-4858	22	12	lower	low	ADJ
ejpam-4858	22	13	(	(	PUNCT
ejpam-4858	22	14	τ1	τ1	NOUN
ejpam-4858	22	15	,	,	PUNCT
ejpam-4858	22	16	τ2)-precontinuous	τ2)-precontinuous	ADJ
ejpam-4858	22	17	multifunctions	multifunction	NOUN
ejpam-4858	22	18	.	.	PUNCT
ejpam-4858	23	1	viriyapong	viriyapong	PROPN
ejpam-4858	23	2	and	and	CCONJ
ejpam-4858	23	3	boonpok	boonpok	VERB
ejpam-4858	24	1	[	[	X
ejpam-4858	24	2	26	26	NUM
ejpam-4858	24	3	]	]	PUNCT
ejpam-4858	24	4	introduced	introduce	VERB
ejpam-4858	24	5	and	and	CCONJ
ejpam-4858	24	6	investigated	investigate	VERB
ejpam-4858	24	7	the	the	DET
ejpam-4858	24	8	concepts	concept	NOUN
ejpam-4858	24	9	of	of	ADP
ejpam-4858	24	10	upper	upper	ADJ
ejpam-4858	24	11	and	and	CCONJ
ejpam-4858	24	12	lower	low	ADJ
ejpam-4858	24	13	(	(	PUNCT
ejpam-4858	24	14	τ1	τ1	NOUN
ejpam-4858	24	15	,	,	PUNCT
ejpam-4858	24	16	τ2)α	τ2)α	ADJ
ejpam-4858	24	17	-	-	PUNCT
ejpam-4858	24	18	continuous	continuous	ADJ
ejpam-4858	24	19	multifunctions	multifunction	NOUN
ejpam-4858	24	20	.	.	PUNCT
ejpam-4858	25	1	moreover	moreover	ADV
ejpam-4858	25	2	,	,	PUNCT
ejpam-4858	25	3	several	several	ADJ
ejpam-4858	25	4	characterizations	characterization	NOUN
ejpam-4858	25	5	of	of	ADP
ejpam-4858	25	6	upper	upper	ADJ
ejpam-4858	25	7	and	and	CCONJ
ejpam-4858	25	8	lower	low	ADJ
ejpam-4858	25	9	(	(	PUNCT
ejpam-4858	25	10	τ1	τ1	NOUN
ejpam-4858	25	11	,	,	PUNCT
ejpam-4858	25	12	τ2)δ	τ2)δ	ADJ
ejpam-4858	25	13	-	-	PUNCT
ejpam-4858	25	14	semicontinuous	semicontinuous	ADJ
ejpam-4858	25	15	multifunctions	multifunction	NOUN
ejpam-4858	25	16	were	be	AUX
ejpam-4858	25	17	established	establish	VERB
ejpam-4858	25	18	in	in	ADP
ejpam-4858	25	19	[	[	X
ejpam-4858	25	20	6	6	NUM
ejpam-4858	25	21	]	]	PUNCT
ejpam-4858	25	22	.	.	PUNCT
ejpam-4858	26	1	in	in	ADP
ejpam-4858	26	2	[	[	X
ejpam-4858	26	3	10	10	NUM
ejpam-4858	26	4	]	]	PUNCT
ejpam-4858	26	5	,	,	PUNCT
ejpam-4858	26	6	the	the	DET
ejpam-4858	26	7	authors	author	NOUN
ejpam-4858	26	8	investigated	investigate	VERB
ejpam-4858	26	9	some	some	DET
ejpam-4858	26	10	characterizations	characterization	NOUN
ejpam-4858	26	11	of	of	ADP
ejpam-4858	26	12	upper	upper	ADJ
ejpam-4858	26	13	and	and	CCONJ
ejpam-4858	26	14	lower	low	ADJ
ejpam-4858	26	15	almost	almost	ADV
ejpam-4858	26	16	weakly	weakly	ADJ
ejpam-4858	26	17	(	(	PUNCT
ejpam-4858	26	18	τ1	τ1	NOUN
ejpam-4858	26	19	,	,	PUNCT
ejpam-4858	26	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4858	26	21	multifunctions	multifunction	NOUN
ejpam-4858	26	22	.	.	PUNCT
ejpam-4858	27	1	laprom	laprom	ADP
ejpam-4858	27	2	∗corresponding	∗corresponde	VERB
ejpam-4858	27	3	author	author	NOUN
ejpam-4858	27	4	.	.	PUNCT
ejpam-4858	28	1	doi	doi	NOUN
ejpam-4858	28	2	:	:	PUNCT
ejpam-4858	28	3	https://doi.org/10.29020/nybg.ejpam.v17i1.4858	https://doi.org/10.29020/nybg.ejpam.v17i1.4858	NUM
ejpam-4858	28	4	email	email	NOUN
ejpam-4858	28	5	addresses	address	NOUN
ejpam-4858	28	6	:	:	PUNCT
ejpam-4858	29	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4858	29	2	(	(	PUNCT
ejpam-4858	29	3	c.	c.	PROPN
ejpam-4858	29	4	boonpok	boonpok	PROPN
ejpam-4858	29	5	)	)	PUNCT
ejpam-4858	29	6	,	,	PUNCT
ejpam-4858	29	7	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-4858	29	8	(	(	PUNCT
ejpam-4858	29	9	j.	j.	PROPN
ejpam-4858	29	10	khampakdee	khampakdee	PROPN
ejpam-4858	29	11	)	)	PUNCT
ejpam-4858	29	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4858	30	1	201	201	NUM
ejpam-4858	30	2	©	©	ADP
ejpam-4858	30	3	2024	2024	NUM
ejpam-4858	30	4	ejpam	ejpam	NOUN
ejpam-4858	30	5	all	all	DET
ejpam-4858	30	6	rights	right	NOUN
ejpam-4858	30	7	reserved	reserve	VERB
ejpam-4858	30	8	.	.	PUNCT
ejpam-4858	31	1	c.	c.	PROPN
ejpam-4858	31	2	boonpok	boonpok	PROPN
ejpam-4858	31	3	,	,	PUNCT
ejpam-4858	31	4	j.	j.	PROPN
ejpam-4858	31	5	khampakdee	khampakdee	PROPN
ejpam-4858	31	6	/	/	PUNCT
ejpam-4858	31	7	eur	eur	PROPN
ejpam-4858	31	8	.	.	PUNCT
ejpam-4858	32	1	j.	j.	PROPN
ejpam-4858	32	2	pure	pure	PROPN
ejpam-4858	32	3	appl	appl	PROPN
ejpam-4858	32	4	.	.	PROPN
ejpam-4858	32	5	math	math	PROPN
ejpam-4858	32	6	,	,	PUNCT
ejpam-4858	32	7	17	17	NUM
ejpam-4858	32	8	(	(	PUNCT
ejpam-4858	32	9	1	1	NUM
ejpam-4858	32	10	)	)	PUNCT
ejpam-4858	32	11	(	(	PUNCT
ejpam-4858	32	12	2024	2024	NUM
ejpam-4858	32	13	)	)	PUNCT
ejpam-4858	32	14	,	,	PUNCT
ejpam-4858	32	15	201	201	NUM
ejpam-4858	32	16	-	-	SYM
ejpam-4858	32	17	211	211	NUM
ejpam-4858	32	18	202	202	NUM
ejpam-4858	32	19	et	et	NOUN
ejpam-4858	32	20	al	al	PROPN
ejpam-4858	32	21	.	.	PUNCT
ejpam-4858	33	1	[	[	X
ejpam-4858	33	2	18	18	NUM
ejpam-4858	33	3	]	]	PUNCT
ejpam-4858	33	4	introduced	introduce	VERB
ejpam-4858	33	5	and	and	CCONJ
ejpam-4858	33	6	studied	study	VERB
ejpam-4858	33	7	the	the	DET
ejpam-4858	33	8	notions	notion	NOUN
ejpam-4858	33	9	of	of	ADP
ejpam-4858	33	10	upper	upper	ADJ
ejpam-4858	33	11	and	and	CCONJ
ejpam-4858	33	12	lower	low	ADJ
ejpam-4858	33	13	β(τ1	β(τ1	NOUN
ejpam-4858	33	14	,	,	PUNCT
ejpam-4858	33	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4858	33	16	multifunctions	multifunction	NOUN
ejpam-4858	33	17	.	.	PUNCT
ejpam-4858	34	1	the	the	DET
ejpam-4858	34	2	concept	concept	NOUN
ejpam-4858	34	3	of	of	ADP
ejpam-4858	34	4	ideal	ideal	ADJ
ejpam-4858	34	5	topological	topological	ADJ
ejpam-4858	34	6	spaces	space	NOUN
ejpam-4858	34	7	was	be	AUX
ejpam-4858	34	8	introduced	introduce	VERB
ejpam-4858	34	9	and	and	CCONJ
ejpam-4858	34	10	studied	study	VERB
ejpam-4858	34	11	by	by	ADP
ejpam-4858	34	12	kuratowski	kuratowski	PROPN
ejpam-4858	34	13	[	[	X
ejpam-4858	34	14	17	17	NUM
ejpam-4858	34	15	]	]	PUNCT
ejpam-4858	34	16	and	and	CCONJ
ejpam-4858	34	17	vaidyanathswamy	vaidyanathswamy	NOUN
ejpam-4858	34	18	[	[	X
ejpam-4858	34	19	25	25	NUM
ejpam-4858	34	20	]	]	PUNCT
ejpam-4858	34	21	.	.	PUNCT
ejpam-4858	35	1	every	every	DET
ejpam-4858	35	2	topological	topological	ADJ
ejpam-4858	35	3	space	space	NOUN
ejpam-4858	35	4	is	be	AUX
ejpam-4858	35	5	an	an	DET
ejpam-4858	35	6	ideal	ideal	ADJ
ejpam-4858	35	7	topological	topological	ADJ
ejpam-4858	35	8	space	space	NOUN
ejpam-4858	35	9	and	and	CCONJ
ejpam-4858	35	10	all	all	DET
ejpam-4858	35	11	the	the	DET
ejpam-4858	35	12	results	result	NOUN
ejpam-4858	35	13	of	of	ADP
ejpam-4858	35	14	ideal	ideal	ADJ
ejpam-4858	35	15	topological	topological	ADJ
ejpam-4858	35	16	spaces	space	NOUN
ejpam-4858	35	17	are	be	AUX
ejpam-4858	35	18	generalizations	generalization	NOUN
ejpam-4858	35	19	of	of	ADP
ejpam-4858	35	20	the	the	DET
ejpam-4858	35	21	results	result	NOUN
ejpam-4858	35	22	established	establish	VERB
ejpam-4858	35	23	in	in	ADP
ejpam-4858	35	24	topological	topological	ADJ
ejpam-4858	35	25	spaces	space	NOUN
ejpam-4858	35	26	.	.	PUNCT
ejpam-4858	36	1	in	in	ADP
ejpam-4858	36	2	1990	1990	NUM
ejpam-4858	36	3	,	,	PUNCT
ejpam-4858	36	4	janković	janković	ADJ
ejpam-4858	36	5	and	and	CCONJ
ejpam-4858	36	6	hamlett	hamlett	PROPN
ejpam-4858	37	1	[	[	X
ejpam-4858	37	2	16	16	NUM
ejpam-4858	37	3	]	]	PUNCT
ejpam-4858	37	4	introduced	introduce	VERB
ejpam-4858	37	5	the	the	DET
ejpam-4858	37	6	concept	concept	NOUN
ejpam-4858	37	7	of	of	ADP
ejpam-4858	37	8	i	i	PRON
ejpam-4858	37	9	open	open	VERB
ejpam-4858	37	10	sets	set	NOUN
ejpam-4858	37	11	in	in	ADP
ejpam-4858	37	12	ideal	ideal	ADJ
ejpam-4858	37	13	topological	topological	ADJ
ejpam-4858	37	14	spaces	space	NOUN
ejpam-4858	37	15	.	.	PUNCT
ejpam-4858	38	1	abd	abd	PROPN
ejpam-4858	38	2	el	el	PROPN
ejpam-4858	38	3	-	-	PROPN
ejpam-4858	38	4	monsef	monsef	PROPN
ejpam-4858	38	5	et	et	PROPN
ejpam-4858	38	6	al	al	PROPN
ejpam-4858	38	7	.	.	PUNCT
ejpam-4858	39	1	[	[	X
ejpam-4858	39	2	14	14	NUM
ejpam-4858	39	3	]	]	PUNCT
ejpam-4858	39	4	further	far	ADV
ejpam-4858	39	5	investigated	investigate	VERB
ejpam-4858	39	6	i	i	PRON
ejpam-4858	39	7	-open	-open	PROPN
ejpam-4858	39	8	sets	set	NOUN
ejpam-4858	39	9	and	and	CCONJ
ejpam-4858	39	10	i	i	PRON
ejpam-4858	39	11	-continuous	-continuous	ADJ
ejpam-4858	39	12	functions	function	NOUN
ejpam-4858	39	13	.	.	PUNCT
ejpam-4858	40	1	later	later	ADV
ejpam-4858	40	2	,	,	PUNCT
ejpam-4858	40	3	several	several	ADJ
ejpam-4858	40	4	authors	author	NOUN
ejpam-4858	40	5	studied	study	VERB
ejpam-4858	40	6	ideal	ideal	ADJ
ejpam-4858	40	7	topological	topological	ADJ
ejpam-4858	40	8	spaces	space	NOUN
ejpam-4858	40	9	giving	give	VERB
ejpam-4858	40	10	several	several	ADJ
ejpam-4858	40	11	convenient	convenient	ADJ
ejpam-4858	40	12	definitions	definition	NOUN
ejpam-4858	40	13	.	.	PUNCT
ejpam-4858	41	1	some	some	DET
ejpam-4858	41	2	authors	author	NOUN
ejpam-4858	41	3	obtained	obtain	VERB
ejpam-4858	41	4	decompositions	decomposition	NOUN
ejpam-4858	41	5	of	of	ADP
ejpam-4858	41	6	continuity	continuity	NOUN
ejpam-4858	41	7	.	.	PUNCT
ejpam-4858	42	1	for	for	ADP
ejpam-4858	42	2	instance	instance	NOUN
ejpam-4858	42	3	,	,	PUNCT
ejpam-4858	42	4	açikgöz	açikgöz	PROPN
ejpam-4858	42	5	et	et	NOUN
ejpam-4858	42	6	al	al	PROPN
ejpam-4858	42	7	.	.	PUNCT
ejpam-4858	43	1	[	[	X
ejpam-4858	43	2	1	1	X
ejpam-4858	43	3	]	]	PUNCT
ejpam-4858	43	4	studied	study	VERB
ejpam-4858	43	5	the	the	DET
ejpam-4858	43	6	concepts	concept	NOUN
ejpam-4858	43	7	of	of	ADP
ejpam-4858	43	8	α	α	PROPN
ejpam-4858	43	9	-	-	PUNCT
ejpam-4858	43	10	i	i	PRON
ejpam-4858	43	11	-continuity	-continuity	ADJ
ejpam-4858	43	12	and	and	CCONJ
ejpam-4858	43	13	αi	αi	VERB
ejpam-4858	43	14	-openness	-openness	PROPN
ejpam-4858	43	15	in	in	ADP
ejpam-4858	43	16	ideal	ideal	ADJ
ejpam-4858	43	17	topological	topological	ADJ
ejpam-4858	43	18	spaces	space	NOUN
ejpam-4858	43	19	and	and	CCONJ
ejpam-4858	43	20	investigated	investigate	VERB
ejpam-4858	43	21	several	several	ADJ
ejpam-4858	43	22	characterizations	characterization	NOUN
ejpam-4858	43	23	of	of	ADP
ejpam-4858	43	24	these	these	DET
ejpam-4858	43	25	functions	function	NOUN
ejpam-4858	43	26	.	.	PUNCT
ejpam-4858	44	1	hatir	hatir	PROPN
ejpam-4858	44	2	and	and	CCONJ
ejpam-4858	44	3	noiri	noiri	ADV
ejpam-4858	45	1	[	[	X
ejpam-4858	45	2	15	15	NUM
ejpam-4858	45	3	]	]	PUNCT
ejpam-4858	45	4	introduced	introduce	VERB
ejpam-4858	45	5	the	the	DET
ejpam-4858	45	6	notions	notion	NOUN
ejpam-4858	45	7	of	of	ADP
ejpam-4858	45	8	semi	semi	ADJ
ejpam-4858	45	9	-	-	ADJ
ejpam-4858	45	10	i	i	ADJ
ejpam-4858	45	11	-open	-open	NOUN
ejpam-4858	45	12	sets	set	NOUN
ejpam-4858	45	13	,	,	PUNCT
ejpam-4858	45	14	α	α	X
ejpam-4858	45	15	-	-	PUNCT
ejpam-4858	45	16	i	i	PRON
ejpam-4858	45	17	-open	-open	NOUN
ejpam-4858	45	18	sets	set	NOUN
ejpam-4858	45	19	and	and	CCONJ
ejpam-4858	45	20	β	β	X
ejpam-4858	45	21	-	-	ADJ
ejpam-4858	45	22	i	i	PRON
ejpam-4858	45	23	-open	-open	NOUN
ejpam-4858	45	24	sets	set	NOUN
ejpam-4858	45	25	via	via	ADP
ejpam-4858	45	26	idealization	idealization	NOUN
ejpam-4858	45	27	and	and	CCONJ
ejpam-4858	45	28	using	use	VERB
ejpam-4858	45	29	these	these	DET
ejpam-4858	45	30	sets	set	NOUN
ejpam-4858	45	31	obtained	obtain	VERB
ejpam-4858	45	32	new	new	ADJ
ejpam-4858	45	33	decompositions	decomposition	NOUN
ejpam-4858	45	34	of	of	ADP
ejpam-4858	45	35	continuity	continuity	NOUN
ejpam-4858	45	36	.	.	PUNCT
ejpam-4858	46	1	in	in	ADP
ejpam-4858	46	2	[	[	X
ejpam-4858	46	3	4	4	NUM
ejpam-4858	46	4	]	]	PUNCT
ejpam-4858	46	5	,	,	PUNCT
ejpam-4858	46	6	the	the	DET
ejpam-4858	46	7	author	author	NOUN
ejpam-4858	46	8	introduced	introduce	VERB
ejpam-4858	46	9	and	and	CCONJ
ejpam-4858	46	10	studied	study	VERB
ejpam-4858	46	11	the	the	DET
ejpam-4858	46	12	notions	notion	NOUN
ejpam-4858	46	13	of	of	ADP
ejpam-4858	46	14	upper	upper	ADJ
ejpam-4858	46	15	and	and	CCONJ
ejpam-4858	46	16	lower	low	ADJ
ejpam-4858	46	17	⋆continuous	⋆continuous	ADJ
ejpam-4858	46	18	multifunctions	multifunction	NOUN
ejpam-4858	46	19	.	.	PUNCT
ejpam-4858	47	1	boonpok	boonpok	PROPN
ejpam-4858	48	1	[	[	X
ejpam-4858	48	2	7	7	NUM
ejpam-4858	48	3	]	]	PUNCT
ejpam-4858	48	4	investigated	investigate	VERB
ejpam-4858	48	5	some	some	DET
ejpam-4858	48	6	characterizations	characterization	NOUN
ejpam-4858	48	7	of	of	ADP
ejpam-4858	48	8	upper	upper	ADJ
ejpam-4858	48	9	and	and	CCONJ
ejpam-4858	48	10	lower	low	ADJ
ejpam-4858	48	11	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-4858	48	12	multifunctions	multifunction	NOUN
ejpam-4858	48	13	.	.	PUNCT
ejpam-4858	49	1	furthermore	furthermore	ADV
ejpam-4858	49	2	,	,	PUNCT
ejpam-4858	49	3	several	several	ADJ
ejpam-4858	49	4	characterizations	characterization	NOUN
ejpam-4858	49	5	of	of	ADP
ejpam-4858	49	6	almost	almost	ADV
ejpam-4858	49	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	49	8	multifunctions	multifunction	NOUN
ejpam-4858	49	9	and	and	CCONJ
ejpam-4858	49	10	weakly	weakly	ADJ
ejpam-4858	49	11	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	49	12	multifunctions	multifunction	NOUN
ejpam-4858	49	13	were	be	AUX
ejpam-4858	49	14	established	establish	VERB
ejpam-4858	49	15	in	in	ADP
ejpam-4858	49	16	[	[	X
ejpam-4858	49	17	9	9	NUM
ejpam-4858	49	18	]	]	PUNCT
ejpam-4858	49	19	and	and	CCONJ
ejpam-4858	49	20	[	[	X
ejpam-4858	49	21	8	8	NUM
ejpam-4858	49	22	]	]	PUNCT
ejpam-4858	49	23	,	,	PUNCT
ejpam-4858	49	24	respectively	respectively	ADV
ejpam-4858	49	25	.	.	PUNCT
ejpam-4858	50	1	in	in	ADP
ejpam-4858	50	2	this	this	DET
ejpam-4858	50	3	paper	paper	NOUN
ejpam-4858	50	4	,	,	PUNCT
ejpam-4858	50	5	we	we	PRON
ejpam-4858	50	6	introduce	introduce	VERB
ejpam-4858	50	7	the	the	DET
ejpam-4858	50	8	notions	notion	NOUN
ejpam-4858	50	9	of	of	ADP
ejpam-4858	50	10	upper	upper	ADJ
ejpam-4858	50	11	and	and	CCONJ
ejpam-4858	50	12	lower	low	ADJ
ejpam-4858	50	13	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	50	14	multifunctions	multifunction	NOUN
ejpam-4858	50	15	.	.	PUNCT
ejpam-4858	51	1	moreover	moreover	ADV
ejpam-4858	51	2	,	,	PUNCT
ejpam-4858	51	3	some	some	DET
ejpam-4858	51	4	characterizations	characterization	NOUN
ejpam-4858	51	5	of	of	ADP
ejpam-4858	51	6	upper	upper	ADJ
ejpam-4858	51	7	and	and	CCONJ
ejpam-4858	51	8	lower	low	ADJ
ejpam-4858	51	9	α-⋆continuous	α-⋆continuous	ADJ
ejpam-4858	51	10	multifunctions	multifunction	NOUN
ejpam-4858	51	11	are	be	AUX
ejpam-4858	51	12	discussed	discuss	VERB
ejpam-4858	51	13	.	.	PUNCT
ejpam-4858	52	1	2	2	X
ejpam-4858	52	2	.	.	X
ejpam-4858	52	3	preliminaries	preliminary	NOUN
ejpam-4858	52	4	throughout	throughout	ADP
ejpam-4858	52	5	the	the	DET
ejpam-4858	52	6	present	present	ADJ
ejpam-4858	52	7	paper	paper	NOUN
ejpam-4858	52	8	,	,	PUNCT
ejpam-4858	52	9	spaces	space	NOUN
ejpam-4858	52	10	(	(	PUNCT
ejpam-4858	52	11	x	x	X
ejpam-4858	52	12	,	,	PUNCT
ejpam-4858	52	13	τ	τ	X
ejpam-4858	52	14	)	)	PUNCT
ejpam-4858	52	15	and	and	CCONJ
ejpam-4858	52	16	(	(	PUNCT
ejpam-4858	52	17	y	y	PROPN
ejpam-4858	52	18	,	,	PUNCT
ejpam-4858	52	19	σ	σ	PROPN
ejpam-4858	52	20	)	)	PUNCT
ejpam-4858	52	21	(	(	PUNCT
ejpam-4858	52	22	or	or	CCONJ
ejpam-4858	52	23	simply	simply	ADV
ejpam-4858	52	24	x	x	X
ejpam-4858	52	25	and	and	CCONJ
ejpam-4858	52	26	y	y	PROPN
ejpam-4858	52	27	)	)	PUNCT
ejpam-4858	52	28	always	always	ADV
ejpam-4858	52	29	mean	mean	VERB
ejpam-4858	52	30	topological	topological	ADJ
ejpam-4858	52	31	spaces	space	NOUN
ejpam-4858	52	32	on	on	ADP
ejpam-4858	52	33	which	which	PRON
ejpam-4858	52	34	no	no	DET
ejpam-4858	52	35	separation	separation	NOUN
ejpam-4858	52	36	axioms	axiom	NOUN
ejpam-4858	52	37	are	be	AUX
ejpam-4858	52	38	assumed	assume	VERB
ejpam-4858	52	39	unless	unless	SCONJ
ejpam-4858	52	40	explicitly	explicitly	ADV
ejpam-4858	52	41	stated	state	VERB
ejpam-4858	52	42	.	.	PUNCT
ejpam-4858	53	1	let	let	VERB
ejpam-4858	53	2	a	a	DET
ejpam-4858	53	3	be	be	AUX
ejpam-4858	53	4	a	a	DET
ejpam-4858	53	5	subset	subset	NOUN
ejpam-4858	53	6	of	of	ADP
ejpam-4858	53	7	a	a	DET
ejpam-4858	53	8	topological	topological	ADJ
ejpam-4858	53	9	space	space	NOUN
ejpam-4858	53	10	(	(	PUNCT
ejpam-4858	53	11	x	x	X
ejpam-4858	53	12	,	,	PUNCT
ejpam-4858	53	13	τ	τ	PROPN
ejpam-4858	53	14	)	)	PUNCT
ejpam-4858	53	15	.	.	PUNCT
ejpam-4858	54	1	the	the	DET
ejpam-4858	54	2	closure	closure	NOUN
ejpam-4858	54	3	of	of	ADP
ejpam-4858	54	4	a	a	PRON
ejpam-4858	54	5	and	and	CCONJ
ejpam-4858	54	6	the	the	DET
ejpam-4858	54	7	interior	interior	NOUN
ejpam-4858	54	8	of	of	ADP
ejpam-4858	54	9	a	a	PRON
ejpam-4858	54	10	are	be	AUX
ejpam-4858	54	11	denoted	denote	VERB
ejpam-4858	54	12	by	by	ADP
ejpam-4858	54	13	cl(a	cl(a	NOUN
ejpam-4858	54	14	)	)	PUNCT
ejpam-4858	54	15	and	and	CCONJ
ejpam-4858	54	16	int(a	int(a	PROPN
ejpam-4858	54	17	)	)	PUNCT
ejpam-4858	54	18	,	,	PUNCT
ejpam-4858	54	19	respectively	respectively	ADV
ejpam-4858	54	20	.	.	PUNCT
ejpam-4858	55	1	an	an	DET
ejpam-4858	55	2	ideal	ideal	NOUN
ejpam-4858	55	3	i	i	PRON
ejpam-4858	55	4	on	on	ADP
ejpam-4858	55	5	a	a	DET
ejpam-4858	55	6	topological	topological	ADJ
ejpam-4858	55	7	space	space	NOUN
ejpam-4858	55	8	(	(	PUNCT
ejpam-4858	55	9	x	x	X
ejpam-4858	55	10	,	,	PUNCT
ejpam-4858	55	11	τ	τ	X
ejpam-4858	55	12	)	)	PUNCT
ejpam-4858	55	13	is	be	AUX
ejpam-4858	55	14	a	a	DET
ejpam-4858	55	15	nonempty	nonempty	ADJ
ejpam-4858	55	16	collection	collection	NOUN
ejpam-4858	55	17	of	of	ADP
ejpam-4858	55	18	subsets	subset	NOUN
ejpam-4858	55	19	of	of	ADP
ejpam-4858	55	20	x	x	PUNCT
ejpam-4858	55	21	satisfying	satisfy	VERB
ejpam-4858	55	22	the	the	DET
ejpam-4858	55	23	following	follow	VERB
ejpam-4858	55	24	properties	property	NOUN
ejpam-4858	55	25	:	:	PUNCT
ejpam-4858	55	26	(	(	PUNCT
ejpam-4858	55	27	1	1	X
ejpam-4858	55	28	)	)	PUNCT
ejpam-4858	55	29	a	a	DET
ejpam-4858	55	30	∈	∈	NOUN
ejpam-4858	55	31	i	i	PRON
ejpam-4858	55	32	and	and	CCONJ
ejpam-4858	55	33	b	b	X
ejpam-4858	55	34	⊆	⊆	NUM
ejpam-4858	55	35	a	a	DET
ejpam-4858	55	36	imply	imply	NOUN
ejpam-4858	55	37	b	b	X
ejpam-4858	55	38	∈	∈	PROPN
ejpam-4858	55	39	i	i	PRON
ejpam-4858	55	40	;	;	PUNCT
ejpam-4858	55	41	(	(	PUNCT
ejpam-4858	55	42	2	2	X
ejpam-4858	55	43	)	)	PUNCT
ejpam-4858	56	1	a	a	PRON
ejpam-4858	56	2	∈	∈	NOUN
ejpam-4858	57	1	i	i	PRON
ejpam-4858	57	2	and	and	CCONJ
ejpam-4858	57	3	b	b	X
ejpam-4858	57	4	∈	∈	NOUN
ejpam-4858	58	1	i	i	PRON
ejpam-4858	58	2	imply	imply	VERB
ejpam-4858	58	3	a∪b	a∪b	ADJ
ejpam-4858	59	1	∈	∈	INTJ
ejpam-4858	60	1	i	i	PRON
ejpam-4858	60	2	.	.	PUNCT
ejpam-4858	61	1	a	a	DET
ejpam-4858	61	2	topological	topological	ADJ
ejpam-4858	61	3	space	space	NOUN
ejpam-4858	61	4	(	(	PUNCT
ejpam-4858	61	5	x	x	X
ejpam-4858	61	6	,	,	PUNCT
ejpam-4858	61	7	τ	τ	X
ejpam-4858	61	8	)	)	PUNCT
ejpam-4858	61	9	with	with	ADP
ejpam-4858	61	10	an	an	DET
ejpam-4858	61	11	ideal	ideal	ADJ
ejpam-4858	61	12	i	i	PRON
ejpam-4858	61	13	on	on	ADP
ejpam-4858	61	14	x	x	SYM
ejpam-4858	61	15	is	be	AUX
ejpam-4858	61	16	called	call	VERB
ejpam-4858	61	17	an	an	DET
ejpam-4858	61	18	ideal	ideal	ADJ
ejpam-4858	61	19	topological	topological	ADJ
ejpam-4858	61	20	space	space	NOUN
ejpam-4858	61	21	and	and	CCONJ
ejpam-4858	61	22	is	be	AUX
ejpam-4858	61	23	denoted	denote	VERB
ejpam-4858	61	24	by	by	ADP
ejpam-4858	61	25	(	(	PUNCT
ejpam-4858	61	26	x	x	X
ejpam-4858	61	27	,	,	PUNCT
ejpam-4858	61	28	τ	τ	PROPN
ejpam-4858	61	29	,	,	PUNCT
ejpam-4858	61	30	i	i	NOUN
ejpam-4858	61	31	)	)	PUNCT
ejpam-4858	61	32	.	.	PUNCT
ejpam-4858	62	1	for	for	ADP
ejpam-4858	62	2	an	an	DET
ejpam-4858	62	3	ideal	ideal	ADJ
ejpam-4858	62	4	topological	topological	ADJ
ejpam-4858	62	5	space	space	NOUN
ejpam-4858	62	6	(	(	PUNCT
ejpam-4858	62	7	x	x	X
ejpam-4858	62	8	,	,	PUNCT
ejpam-4858	62	9	τ	τ	PROPN
ejpam-4858	62	10	,	,	PUNCT
ejpam-4858	62	11	i	i	PROPN
ejpam-4858	62	12	)	)	PUNCT
ejpam-4858	62	13	and	and	CCONJ
ejpam-4858	62	14	a	a	DET
ejpam-4858	62	15	subset	subset	NOUN
ejpam-4858	62	16	a	a	PRON
ejpam-4858	62	17	of	of	ADP
ejpam-4858	62	18	x	x	PRON
ejpam-4858	62	19	,	,	PUNCT
ejpam-4858	62	20	a⋆(i	a⋆(i	PROPN
ejpam-4858	62	21	)	)	PUNCT
ejpam-4858	62	22	is	be	AUX
ejpam-4858	62	23	defined	define	VERB
ejpam-4858	62	24	as	as	SCONJ
ejpam-4858	62	25	follows	follow	VERB
ejpam-4858	62	26	:	:	PUNCT
ejpam-4858	62	27	a⋆(i	a⋆(i	NOUN
ejpam-4858	62	28	)	)	PUNCT
ejpam-4858	63	1	=	=	PUNCT
ejpam-4858	63	2	{	{	PUNCT
ejpam-4858	63	3	x	x	PUNCT
ejpam-4858	63	4	∈	∈	PROPN
ejpam-4858	63	5	x	x	X
ejpam-4858	63	6	:	:	PUNCT
ejpam-4858	63	7	u	u	X
ejpam-4858	63	8	∩a	∩a	PROPN
ejpam-4858	63	9	̸∈	̸∈	PROPN
ejpam-4858	63	10	i	i	PRON
ejpam-4858	63	11	for	for	ADP
ejpam-4858	63	12	every	every	DET
ejpam-4858	63	13	open	open	ADJ
ejpam-4858	63	14	neighbourhood	neighbourhood	NOUN
ejpam-4858	63	15	u	u	NOUN
ejpam-4858	63	16	of	of	ADP
ejpam-4858	63	17	x	x	NOUN
ejpam-4858	63	18	}	}	PUNCT
ejpam-4858	63	19	.	.	PUNCT
ejpam-4858	64	1	in	in	ADP
ejpam-4858	64	2	case	case	NOUN
ejpam-4858	64	3	there	there	PRON
ejpam-4858	64	4	is	be	VERB
ejpam-4858	64	5	no	no	DET
ejpam-4858	64	6	chance	chance	NOUN
ejpam-4858	64	7	for	for	ADP
ejpam-4858	64	8	confusion	confusion	NOUN
ejpam-4858	64	9	,	,	PUNCT
ejpam-4858	64	10	a⋆(i	a⋆(i	NOUN
ejpam-4858	64	11	)	)	PUNCT
ejpam-4858	64	12	is	be	AUX
ejpam-4858	64	13	simply	simply	ADV
ejpam-4858	64	14	written	write	VERB
ejpam-4858	64	15	as	as	ADP
ejpam-4858	64	16	a⋆.	a⋆.	NOUN
ejpam-4858	64	17	in	in	ADP
ejpam-4858	64	18	[	[	X
ejpam-4858	64	19	17	17	NUM
ejpam-4858	64	20	]	]	PUNCT
ejpam-4858	64	21	,	,	PUNCT
ejpam-4858	64	22	a⋆	a⋆	ADV
ejpam-4858	64	23	is	be	AUX
ejpam-4858	64	24	called	call	VERB
ejpam-4858	64	25	the	the	DET
ejpam-4858	64	26	local	local	ADJ
ejpam-4858	64	27	function	function	NOUN
ejpam-4858	64	28	of	of	ADP
ejpam-4858	64	29	a	a	PRON
ejpam-4858	64	30	with	with	ADP
ejpam-4858	64	31	respect	respect	NOUN
ejpam-4858	64	32	to	to	ADP
ejpam-4858	64	33	i	i	PRON
ejpam-4858	64	34	and	and	CCONJ
ejpam-4858	64	35	τ	τ	PROPN
ejpam-4858	64	36	and	and	CCONJ
ejpam-4858	64	37	cl⋆(a	cl⋆(a	NUM
ejpam-4858	64	38	)	)	PUNCT
ejpam-4858	64	39	=	=	PUNCT
ejpam-4858	64	40	a⋆	a⋆	ADP
ejpam-4858	64	41	∪	∪	ADP
ejpam-4858	64	42	a	a	DET
ejpam-4858	64	43	defines	define	NOUN
ejpam-4858	64	44	a	a	DET
ejpam-4858	64	45	kuratowski	kuratowski	ADJ
ejpam-4858	64	46	closure	closure	NOUN
ejpam-4858	64	47	operator	operator	NOUN
ejpam-4858	64	48	for	for	ADP
ejpam-4858	64	49	a	a	DET
ejpam-4858	64	50	topology	topology	NOUN
ejpam-4858	64	51	τ⋆(i	τ⋆(i	NOUN
ejpam-4858	64	52	)	)	PUNCT
ejpam-4858	64	53	finer	fine	ADJ
ejpam-4858	64	54	than	than	ADP
ejpam-4858	64	55	τ	τ	PROPN
ejpam-4858	64	56	.	.	PUNCT
ejpam-4858	65	1	a	a	DET
ejpam-4858	65	2	subset	subset	NOUN
ejpam-4858	65	3	a	a	PRON
ejpam-4858	65	4	is	be	AUX
ejpam-4858	65	5	said	say	VERB
ejpam-4858	65	6	to	to	PART
ejpam-4858	65	7	be	be	AUX
ejpam-4858	65	8	⋆-closed	⋆-close	VERB
ejpam-4858	65	9	[	[	X
ejpam-4858	65	10	16	16	NUM
ejpam-4858	65	11	]	]	X
ejpam-4858	65	12	if	if	SCONJ
ejpam-4858	65	13	a⋆	a⋆	ADJ
ejpam-4858	65	14	⊆	⊆	NUM
ejpam-4858	65	15	a.	a.	NOUN
ejpam-4858	65	16	the	the	DET
ejpam-4858	65	17	interior	interior	NOUN
ejpam-4858	65	18	of	of	ADP
ejpam-4858	65	19	a	a	DET
ejpam-4858	65	20	subset	subset	NOUN
ejpam-4858	65	21	a	a	DET
ejpam-4858	65	22	in	in	ADP
ejpam-4858	65	23	(	(	PUNCT
ejpam-4858	65	24	x	x	X
ejpam-4858	65	25	,	,	PUNCT
ejpam-4858	65	26	τ⋆(i	τ⋆(i	NOUN
ejpam-4858	65	27	)	)	PUNCT
ejpam-4858	65	28	)	)	PUNCT
ejpam-4858	65	29	is	be	AUX
ejpam-4858	65	30	denoted	denote	VERB
ejpam-4858	65	31	by	by	ADP
ejpam-4858	65	32	int⋆(a	int⋆(a	NOUN
ejpam-4858	65	33	)	)	PUNCT
ejpam-4858	65	34	.	.	PUNCT
ejpam-4858	66	1	a	a	DET
ejpam-4858	66	2	subset	subset	NOUN
ejpam-4858	66	3	a	a	PRON
ejpam-4858	66	4	of	of	ADP
ejpam-4858	66	5	an	an	DET
ejpam-4858	66	6	ideal	ideal	ADJ
ejpam-4858	66	7	topological	topological	ADJ
ejpam-4858	66	8	space	space	NOUN
ejpam-4858	66	9	(	(	PUNCT
ejpam-4858	66	10	x	x	X
ejpam-4858	66	11	,	,	PUNCT
ejpam-4858	66	12	τ	τ	PROPN
ejpam-4858	66	13	,	,	PUNCT
ejpam-4858	66	14	i	i	PROPN
ejpam-4858	66	15	)	)	PUNCT
ejpam-4858	66	16	is	be	AUX
ejpam-4858	66	17	said	say	VERB
ejpam-4858	66	18	to	to	PART
ejpam-4858	66	19	be	be	AUX
ejpam-4858	66	20	semi⋆-i	semi⋆-i	X
ejpam-4858	66	21	-open	-open	VERB
ejpam-4858	66	22	[	[	X
ejpam-4858	66	23	12	12	NUM
ejpam-4858	66	24	]	]	PUNCT
ejpam-4858	66	25	(	(	PUNCT
ejpam-4858	66	26	resp	resp	NOUN
ejpam-4858	66	27	.	.	PUNCT
ejpam-4858	67	1	semi	semi	ADJ
ejpam-4858	67	2	-	-	VERB
ejpam-4858	67	3	i	i	PRON
ejpam-4858	67	4	-open	-open	NOUN
ejpam-4858	68	1	[	[	X
ejpam-4858	68	2	15	15	NUM
ejpam-4858	68	3	]	]	PUNCT
ejpam-4858	68	4	)	)	PUNCT
ejpam-4858	68	5	if	if	SCONJ
ejpam-4858	68	6	a	a	DET
ejpam-4858	68	7	⊆	⊆	NUM
ejpam-4858	68	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-4858	68	9	)	)	PUNCT
ejpam-4858	68	10	)	)	PUNCT
ejpam-4858	68	11	(	(	PUNCT
ejpam-4858	68	12	resp	resp	NOUN
ejpam-4858	68	13	.	.	PUNCT
ejpam-4858	69	1	a	a	DET
ejpam-4858	69	2	⊆	⊆	NUM
ejpam-4858	69	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-4858	69	4	)	)	PUNCT
ejpam-4858	69	5	)	)	PUNCT
ejpam-4858	69	6	)	)	PUNCT
ejpam-4858	69	7	.	.	PUNCT
ejpam-4858	70	1	the	the	DET
ejpam-4858	70	2	complement	complement	NOUN
ejpam-4858	70	3	of	of	ADP
ejpam-4858	70	4	a	a	DET
ejpam-4858	70	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-4858	70	6	-open	-open	ADJ
ejpam-4858	70	7	(	(	PUNCT
ejpam-4858	70	8	resp	resp	NOUN
ejpam-4858	70	9	.	.	PUNCT
ejpam-4858	71	1	semi	semi	ADJ
ejpam-4858	71	2	-	-	VERB
ejpam-4858	71	3	i	i	PRON
ejpam-4858	71	4	-open	-open	NOUN
ejpam-4858	71	5	)	)	PUNCT
ejpam-4858	72	1	set	set	NOUN
ejpam-4858	72	2	is	be	AUX
ejpam-4858	72	3	said	say	VERB
ejpam-4858	72	4	to	to	PART
ejpam-4858	72	5	be	be	AUX
ejpam-4858	72	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-4858	72	7	-closed	-close	VERB
ejpam-4858	72	8	[	[	X
ejpam-4858	72	9	12	12	NUM
ejpam-4858	72	10	]	]	PUNCT
ejpam-4858	72	11	(	(	PUNCT
ejpam-4858	72	12	resp	resp	NOUN
ejpam-4858	72	13	.	.	PUNCT
ejpam-4858	73	1	semi	semi	ADJ
ejpam-4858	73	2	-	-	PROPN
ejpam-4858	73	3	i	i	PRON
ejpam-4858	73	4	closed	close	VERB
ejpam-4858	73	5	[	[	X
ejpam-4858	73	6	15	15	NUM
ejpam-4858	73	7	]	]	NUM
ejpam-4858	73	8	)	)	PUNCT
ejpam-4858	73	9	.	.	PUNCT
ejpam-4858	74	1	for	for	ADP
ejpam-4858	74	2	a	a	DET
ejpam-4858	74	3	subset	subset	NOUN
ejpam-4858	74	4	a	a	PRON
ejpam-4858	74	5	of	of	ADP
ejpam-4858	74	6	an	an	DET
ejpam-4858	74	7	ideal	ideal	ADJ
ejpam-4858	74	8	topological	topological	ADJ
ejpam-4858	74	9	space	space	NOUN
ejpam-4858	74	10	(	(	PUNCT
ejpam-4858	74	11	x	x	X
ejpam-4858	74	12	,	,	PUNCT
ejpam-4858	74	13	τ	τ	PROPN
ejpam-4858	74	14	,	,	PUNCT
ejpam-4858	74	15	i	i	NOUN
ejpam-4858	74	16	)	)	PUNCT
ejpam-4858	74	17	,	,	PUNCT
ejpam-4858	74	18	the	the	DET
ejpam-4858	74	19	intersection	intersection	NOUN
ejpam-4858	74	20	of	of	ADP
ejpam-4858	74	21	all	all	PRON
ejpam-4858	74	22	semi	semi	ADJ
ejpam-4858	74	23	-	-	VERB
ejpam-4858	74	24	i	i	PRON
ejpam-4858	74	25	-closed	-close	VERB
ejpam-4858	74	26	(	(	PUNCT
ejpam-4858	74	27	resp	resp	NOUN
ejpam-4858	74	28	.	.	PUNCT
ejpam-4858	75	1	semi⋆-i	semi⋆-i	PUNCT
ejpam-4858	75	2	-closed	-close	VERB
ejpam-4858	75	3	)	)	PUNCT
ejpam-4858	75	4	sets	set	NOUN
ejpam-4858	75	5	containing	contain	VERB
ejpam-4858	75	6	a	a	PRON
ejpam-4858	75	7	is	be	AUX
ejpam-4858	75	8	called	call	VERB
ejpam-4858	75	9	the	the	DET
ejpam-4858	75	10	semi	semi	ADJ
ejpam-4858	75	11	-	-	ADJ
ejpam-4858	75	12	i	i	ADJ
ejpam-4858	75	13	-closure	-closure	NOUN
ejpam-4858	75	14	[	[	X
ejpam-4858	75	15	13	13	NUM
ejpam-4858	75	16	]	]	PUNCT
ejpam-4858	75	17	(	(	PUNCT
ejpam-4858	75	18	resp	resp	NOUN
ejpam-4858	75	19	.	.	PUNCT
ejpam-4858	76	1	semi⋆-i	semi⋆-i	AUX
ejpam-4858	76	2	-closure	-closure	NOUN
ejpam-4858	76	3	[	[	X
ejpam-4858	76	4	13	13	NUM
ejpam-4858	76	5	]	]	PUNCT
ejpam-4858	76	6	)	)	PUNCT
ejpam-4858	76	7	of	of	ADP
ejpam-4858	76	8	a	a	PRON
ejpam-4858	76	9	and	and	CCONJ
ejpam-4858	76	10	is	be	AUX
ejpam-4858	76	11	denoted	denote	VERB
ejpam-4858	76	12	by	by	ADP
ejpam-4858	76	13	scli	scli	NOUN
ejpam-4858	76	14	(	(	PUNCT
ejpam-4858	76	15	a	a	NOUN
ejpam-4858	76	16	)	)	PUNCT
ejpam-4858	76	17	(	(	PUNCT
ejpam-4858	76	18	resp	resp	NOUN
ejpam-4858	76	19	.	.	PUNCT
ejpam-4858	77	1	s⋆cli	s⋆cli	ADJ
ejpam-4858	77	2	(	(	PUNCT
ejpam-4858	77	3	a	a	NOUN
ejpam-4858	77	4	)	)	PUNCT
ejpam-4858	77	5	)	)	PUNCT
ejpam-4858	77	6	.	.	PUNCT
ejpam-4858	78	1	the	the	DET
ejpam-4858	78	2	union	union	NOUN
ejpam-4858	78	3	of	of	ADP
ejpam-4858	78	4	all	all	PRON
ejpam-4858	78	5	semi	semi	ADJ
ejpam-4858	78	6	-	-	ADJ
ejpam-4858	78	7	i	i	PRON
ejpam-4858	78	8	-open	-open	ADJ
ejpam-4858	78	9	(	(	PUNCT
ejpam-4858	78	10	resp	resp	NOUN
ejpam-4858	78	11	.	.	PUNCT
ejpam-4858	78	12	semi⋆-i	semi⋆-i	X
ejpam-4858	78	13	-open	-open	VERB
ejpam-4858	78	14	)	)	PUNCT
ejpam-4858	78	15	sets	set	NOUN
ejpam-4858	78	16	contained	contain	VERB
ejpam-4858	78	17	in	in	ADP
ejpam-4858	78	18	a	a	PRON
ejpam-4858	78	19	is	be	AUX
ejpam-4858	78	20	called	call	VERB
ejpam-4858	78	21	the	the	DET
ejpam-4858	78	22	semi	semi	ADJ
ejpam-4858	78	23	-	-	ADJ
ejpam-4858	78	24	i	i	PRON
ejpam-4858	78	25	-interior	-interior	ADJ
ejpam-4858	78	26	c.	c.	PROPN
ejpam-4858	78	27	boonpok	boonpok	PROPN
ejpam-4858	78	28	,	,	PUNCT
ejpam-4858	78	29	j.	j.	PROPN
ejpam-4858	78	30	khampakdee	khampakdee	PROPN
ejpam-4858	78	31	/	/	PUNCT
ejpam-4858	78	32	eur	eur	PROPN
ejpam-4858	78	33	.	.	PUNCT
ejpam-4858	79	1	j.	j.	PROPN
ejpam-4858	79	2	pure	pure	PROPN
ejpam-4858	79	3	appl	appl	PROPN
ejpam-4858	79	4	.	.	PROPN
ejpam-4858	79	5	math	math	PROPN
ejpam-4858	79	6	,	,	PUNCT
ejpam-4858	79	7	17	17	NUM
ejpam-4858	79	8	(	(	PUNCT
ejpam-4858	79	9	1	1	NUM
ejpam-4858	79	10	)	)	PUNCT
ejpam-4858	79	11	(	(	PUNCT
ejpam-4858	79	12	2024	2024	NUM
ejpam-4858	79	13	)	)	PUNCT
ejpam-4858	79	14	,	,	PUNCT
ejpam-4858	79	15	201	201	NUM
ejpam-4858	79	16	-	-	SYM
ejpam-4858	79	17	211	211	NUM
ejpam-4858	79	18	203	203	NUM
ejpam-4858	79	19	(	(	PUNCT
ejpam-4858	79	20	resp	resp	NOUN
ejpam-4858	79	21	.	.	PUNCT
ejpam-4858	80	1	semi⋆-i	semi⋆-i	X
ejpam-4858	80	2	-interior	-interior	NOUN
ejpam-4858	80	3	)	)	PUNCT
ejpam-4858	80	4	of	of	ADP
ejpam-4858	80	5	a	a	PRON
ejpam-4858	80	6	and	and	CCONJ
ejpam-4858	80	7	is	be	AUX
ejpam-4858	80	8	denoted	denote	VERB
ejpam-4858	80	9	by	by	ADP
ejpam-4858	80	10	sinti	sinti	PROPN
ejpam-4858	80	11	(	(	PUNCT
ejpam-4858	80	12	a	a	NOUN
ejpam-4858	80	13	)	)	PUNCT
ejpam-4858	80	14	(	(	PUNCT
ejpam-4858	80	15	resp	resp	NOUN
ejpam-4858	80	16	.	.	PUNCT
ejpam-4858	81	1	s⋆inti	s⋆inti	VERB
ejpam-4858	81	2	(	(	PUNCT
ejpam-4858	81	3	a	a	NOUN
ejpam-4858	81	4	)	)	PUNCT
ejpam-4858	81	5	)	)	PUNCT
ejpam-4858	81	6	.	.	PUNCT
ejpam-4858	82	1	lemma	lemma	PROPN
ejpam-4858	82	2	1	1	NUM
ejpam-4858	82	3	.	.	PUNCT
ejpam-4858	83	1	for	for	ADP
ejpam-4858	83	2	a	a	DET
ejpam-4858	83	3	subset	subset	NOUN
ejpam-4858	83	4	a	a	PRON
ejpam-4858	83	5	of	of	ADP
ejpam-4858	83	6	an	an	DET
ejpam-4858	83	7	ideal	ideal	ADJ
ejpam-4858	83	8	topological	topological	ADJ
ejpam-4858	83	9	space	space	NOUN
ejpam-4858	83	10	(	(	PUNCT
ejpam-4858	83	11	x	x	X
ejpam-4858	83	12	,	,	PUNCT
ejpam-4858	83	13	τ	τ	PROPN
ejpam-4858	83	14	,	,	PUNCT
ejpam-4858	83	15	i	i	NOUN
ejpam-4858	83	16	)	)	PUNCT
ejpam-4858	83	17	,	,	PUNCT
ejpam-4858	83	18	the	the	DET
ejpam-4858	83	19	following	follow	VERB
ejpam-4858	83	20	properties	property	NOUN
ejpam-4858	83	21	hold	hold	VERB
ejpam-4858	83	22	:	:	PUNCT
ejpam-4858	83	23	(	(	PUNCT
ejpam-4858	83	24	1	1	X
ejpam-4858	83	25	)	)	PUNCT
ejpam-4858	83	26	if	if	SCONJ
ejpam-4858	83	27	a	a	PRON
ejpam-4858	83	28	is	be	AUX
ejpam-4858	83	29	an	an	DET
ejpam-4858	83	30	open	open	ADJ
ejpam-4858	83	31	set	set	NOUN
ejpam-4858	83	32	,	,	PUNCT
ejpam-4858	83	33	then	then	ADV
ejpam-4858	83	34	s⋆cli	s⋆cli	PROPN
ejpam-4858	83	35	(	(	PUNCT
ejpam-4858	83	36	a	a	NOUN
ejpam-4858	83	37	)	)	PUNCT
ejpam-4858	83	38	=	=	SYM
ejpam-4858	83	39	int(cl⋆(a	int(cl⋆(a	NOUN
ejpam-4858	83	40	)	)	PUNCT
ejpam-4858	83	41	)	)	PUNCT
ejpam-4858	83	42	.	.	PUNCT
ejpam-4858	84	1	(	(	PUNCT
ejpam-4858	84	2	2	2	X
ejpam-4858	84	3	)	)	PUNCT
ejpam-4858	84	4	if	if	SCONJ
ejpam-4858	84	5	a	a	PRON
ejpam-4858	84	6	is	be	AUX
ejpam-4858	84	7	a	a	DET
ejpam-4858	84	8	⋆-open	⋆-open	ADJ
ejpam-4858	84	9	set	set	NOUN
ejpam-4858	84	10	,	,	PUNCT
ejpam-4858	84	11	then	then	ADV
ejpam-4858	84	12	scli	scli	PROPN
ejpam-4858	84	13	(	(	PUNCT
ejpam-4858	84	14	a	a	X
ejpam-4858	84	15	)	)	PUNCT
ejpam-4858	84	16	=	=	SYM
ejpam-4858	84	17	int⋆(cl(a	int⋆(cl(a	PROPN
ejpam-4858	84	18	)	)	PUNCT
ejpam-4858	84	19	)	)	PUNCT
ejpam-4858	84	20	.	.	PUNCT
ejpam-4858	85	1	proof	proof	NOUN
ejpam-4858	85	2	.	.	PUNCT
ejpam-4858	86	1	(	(	PUNCT
ejpam-4858	86	2	1	1	X
ejpam-4858	86	3	)	)	PUNCT
ejpam-4858	86	4	suppose	suppose	VERB
ejpam-4858	86	5	that	that	SCONJ
ejpam-4858	86	6	a	a	PRON
ejpam-4858	86	7	is	be	AUX
ejpam-4858	86	8	an	an	DET
ejpam-4858	86	9	open	open	ADJ
ejpam-4858	86	10	set	set	NOUN
ejpam-4858	86	11	.	.	PUNCT
ejpam-4858	87	1	then	then	ADV
ejpam-4858	87	2	,	,	PUNCT
ejpam-4858	87	3	a	a	DET
ejpam-4858	87	4	⊆	⊆	NUM
ejpam-4858	87	5	int(cl⋆(a	int(cl⋆(a	NUM
ejpam-4858	87	6	)	)	PUNCT
ejpam-4858	87	7	)	)	PUNCT
ejpam-4858	87	8	and	and	CCONJ
ejpam-4858	87	9	by	by	ADP
ejpam-4858	87	10	lemma	lemma	PROPN
ejpam-4858	87	11	13(1	13(1	PROPN
ejpam-4858	87	12	)	)	PUNCT
ejpam-4858	87	13	of	of	ADP
ejpam-4858	87	14	[	[	X
ejpam-4858	87	15	13	13	NUM
ejpam-4858	87	16	]	]	PUNCT
ejpam-4858	87	17	,	,	PUNCT
ejpam-4858	87	18	we	we	PRON
ejpam-4858	87	19	have	have	VERB
ejpam-4858	87	20	s⋆cli	s⋆cli	ADJ
ejpam-4858	87	21	(	(	PUNCT
ejpam-4858	87	22	a	a	NOUN
ejpam-4858	87	23	)	)	PUNCT
ejpam-4858	87	24	=	=	NOUN
ejpam-4858	87	25	a	a	PRON
ejpam-4858	87	26	∪	∪	ADJ
ejpam-4858	87	27	int(cl⋆(a	int(cl⋆(a	NOUN
ejpam-4858	87	28	)	)	PUNCT
ejpam-4858	87	29	)	)	PUNCT
ejpam-4858	88	1	=	=	SYM
ejpam-4858	88	2	int(cl⋆(a	int(cl⋆(a	NOUN
ejpam-4858	88	3	)	)	PUNCT
ejpam-4858	88	4	)	)	PUNCT
ejpam-4858	88	5	.	.	PUNCT
ejpam-4858	89	1	(	(	PUNCT
ejpam-4858	89	2	2	2	X
ejpam-4858	89	3	)	)	PUNCT
ejpam-4858	89	4	suppose	suppose	VERB
ejpam-4858	89	5	that	that	SCONJ
ejpam-4858	89	6	a	a	PRON
ejpam-4858	89	7	is	be	AUX
ejpam-4858	89	8	a	a	DET
ejpam-4858	89	9	⋆-open	⋆-open	ADJ
ejpam-4858	89	10	set	set	NOUN
ejpam-4858	89	11	.	.	PUNCT
ejpam-4858	90	1	then	then	ADV
ejpam-4858	90	2	,	,	PUNCT
ejpam-4858	90	3	we	we	PRON
ejpam-4858	90	4	have	have	VERB
ejpam-4858	90	5	a	a	DET
ejpam-4858	90	6	⊆	⊆	NUM
ejpam-4858	90	7	int⋆(cl(a	int⋆(cl(a	NOUN
ejpam-4858	90	8	)	)	PUNCT
ejpam-4858	90	9	)	)	PUNCT
ejpam-4858	90	10	and	and	CCONJ
ejpam-4858	90	11	by	by	ADP
ejpam-4858	90	12	lemma	lemma	PROPN
ejpam-4858	90	13	13(2	13(2	PROPN
ejpam-4858	90	14	)	)	PUNCT
ejpam-4858	90	15	of	of	ADP
ejpam-4858	90	16	[	[	X
ejpam-4858	90	17	13	13	NUM
ejpam-4858	90	18	]	]	PUNCT
ejpam-4858	90	19	,	,	PUNCT
ejpam-4858	90	20	scli	scli	PROPN
ejpam-4858	90	21	(	(	PUNCT
ejpam-4858	90	22	a	a	X
ejpam-4858	90	23	)	)	PUNCT
ejpam-4858	90	24	=	=	NOUN
ejpam-4858	90	25	a	a	DET
ejpam-4858	90	26	∪	∪	ADJ
ejpam-4858	90	27	int⋆(cl(a	int⋆(cl(a	NOUN
ejpam-4858	90	28	)	)	PUNCT
ejpam-4858	90	29	)	)	PUNCT
ejpam-4858	91	1	=	=	SYM
ejpam-4858	91	2	int⋆(cl(a	int⋆(cl(a	PROPN
ejpam-4858	91	3	)	)	PUNCT
ejpam-4858	91	4	)	)	PUNCT
ejpam-4858	91	5	.	.	PUNCT
ejpam-4858	92	1	recall	recall	VERB
ejpam-4858	92	2	that	that	SCONJ
ejpam-4858	92	3	a	a	DET
ejpam-4858	92	4	subset	subset	NOUN
ejpam-4858	92	5	a	a	PRON
ejpam-4858	92	6	of	of	ADP
ejpam-4858	92	7	an	an	DET
ejpam-4858	92	8	ideal	ideal	ADJ
ejpam-4858	92	9	topological	topological	ADJ
ejpam-4858	92	10	space	space	NOUN
ejpam-4858	92	11	(	(	PUNCT
ejpam-4858	92	12	x	x	X
ejpam-4858	92	13	,	,	PUNCT
ejpam-4858	92	14	τ	τ	PROPN
ejpam-4858	92	15	,	,	PUNCT
ejpam-4858	92	16	i	i	PROPN
ejpam-4858	92	17	)	)	PUNCT
ejpam-4858	92	18	is	be	AUX
ejpam-4858	92	19	said	say	VERB
ejpam-4858	92	20	to	to	PART
ejpam-4858	92	21	be	be	AUX
ejpam-4858	92	22	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	92	23	[	[	X
ejpam-4858	92	24	2	2	NUM
ejpam-4858	92	25	]	]	PUNCT
ejpam-4858	92	26	if	if	SCONJ
ejpam-4858	92	27	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	92	28	)	)	PUNCT
ejpam-4858	92	29	)	)	PUNCT
ejpam-4858	92	30	)	)	PUNCT
ejpam-4858	93	1	⊆	⊆	NUM
ejpam-4858	93	2	a.	a.	NOUN
ejpam-4858	93	3	the	the	DET
ejpam-4858	93	4	complement	complement	NOUN
ejpam-4858	93	5	of	of	ADP
ejpam-4858	93	6	an	an	DET
ejpam-4858	93	7	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	93	8	set	set	NOUN
ejpam-4858	93	9	is	be	AUX
ejpam-4858	93	10	said	say	VERB
ejpam-4858	93	11	to	to	PART
ejpam-4858	93	12	be	be	AUX
ejpam-4858	93	13	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	93	14	.	.	PUNCT
ejpam-4858	93	15	proposition	proposition	NOUN
ejpam-4858	93	16	1	1	NUM
ejpam-4858	93	17	.	.	PUNCT
ejpam-4858	94	1	let	let	AUX
ejpam-4858	94	2	(	(	PUNCT
ejpam-4858	94	3	x	x	X
ejpam-4858	94	4	,	,	PUNCT
ejpam-4858	94	5	τ	τ	PROPN
ejpam-4858	94	6	,	,	PUNCT
ejpam-4858	94	7	i	i	PRON
ejpam-4858	94	8	)	)	PUNCT
ejpam-4858	94	9	be	be	AUX
ejpam-4858	94	10	an	an	DET
ejpam-4858	94	11	ideal	ideal	ADJ
ejpam-4858	94	12	topological	topological	ADJ
ejpam-4858	94	13	space	space	NOUN
ejpam-4858	94	14	and	and	CCONJ
ejpam-4858	94	15	{	{	PUNCT
ejpam-4858	94	16	aγ	aγ	INTJ
ejpam-4858	94	17	|	|	ADV
ejpam-4858	94	18	γ	γ	PROPN
ejpam-4858	94	19	∈	∈	PROPN
ejpam-4858	94	20	γ	γ	AUX
ejpam-4858	94	21	}	}	PUNCT
ejpam-4858	94	22	be	be	AUX
ejpam-4858	94	23	a	a	DET
ejpam-4858	94	24	family	family	NOUN
ejpam-4858	94	25	of	of	ADP
ejpam-4858	94	26	subsets	subset	NOUN
ejpam-4858	94	27	of	of	ADP
ejpam-4858	94	28	x.	x.	NOUN
ejpam-4858	94	29	if	if	SCONJ
ejpam-4858	94	30	aγ	aγ	PRON
ejpam-4858	94	31	is	be	AUX
ejpam-4858	94	32	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	94	33	for	for	ADP
ejpam-4858	94	34	each	each	DET
ejpam-4858	94	35	γ	γ	PROPN
ejpam-4858	94	36	∈	∈	PROPN
ejpam-4858	94	37	γ	γ	X
ejpam-4858	94	38	,	,	PUNCT
ejpam-4858	94	39	then	then	ADV
ejpam-4858	94	40	∩	∩	NOUN
ejpam-4858	94	41	γ∈γ	γ∈γ	ADJ
ejpam-4858	94	42	aγ	aγ	PRON
ejpam-4858	94	43	is	be	AUX
ejpam-4858	94	44	α-⋆-closed	α-⋆-closed	X
ejpam-4858	94	45	.	.	PUNCT
ejpam-4858	95	1	proof	proof	NOUN
ejpam-4858	95	2	.	.	PUNCT
ejpam-4858	96	1	suppose	suppose	VERB
ejpam-4858	96	2	that	that	SCONJ
ejpam-4858	96	3	aγ	aγ	PRON
ejpam-4858	96	4	is	be	AUX
ejpam-4858	96	5	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	96	6	for	for	ADP
ejpam-4858	96	7	each	each	DET
ejpam-4858	96	8	γ	γ	PROPN
ejpam-4858	96	9	∈	∈	PROPN
ejpam-4858	96	10	γ	γ	X
ejpam-4858	96	11	.	.	PUNCT
ejpam-4858	97	1	then	then	ADV
ejpam-4858	97	2	,	,	PUNCT
ejpam-4858	97	3	we	we	PRON
ejpam-4858	97	4	have	have	VERB
ejpam-4858	97	5	x	x	INTJ
ejpam-4858	97	6	−	−	NOUN
ejpam-4858	97	7	aγ	aγ	PRON
ejpam-4858	97	8	is	be	AUX
ejpam-4858	97	9	α-⋆open	α-⋆open	ADJ
ejpam-4858	97	10	for	for	ADP
ejpam-4858	97	11	each	each	DET
ejpam-4858	97	12	γ	γ	PROPN
ejpam-4858	97	13	∈	∈	PROPN
ejpam-4858	97	14	γ	γ	NOUN
ejpam-4858	97	15	.	.	PUNCT
ejpam-4858	98	1	thus	thus	ADV
ejpam-4858	98	2	,	,	PUNCT
ejpam-4858	98	3	∪	∪	X
ejpam-4858	98	4	γ∈γ	γ∈γ	ADJ
ejpam-4858	98	5	(	(	PUNCT
ejpam-4858	98	6	x	x	NOUN
ejpam-4858	98	7	−	−	VERB
ejpam-4858	98	8	aγ	aγ	NOUN
ejpam-4858	98	9	)	)	PUNCT
ejpam-4858	98	10	=	=	SYM
ejpam-4858	98	11	x	x	SYM
ejpam-4858	98	12	−	−	PROPN
ejpam-4858	98	13	∩	∩	NOUN
ejpam-4858	98	14	γ∈γ	γ∈γ	ADJ
ejpam-4858	98	15	aγ	aγ	PRON
ejpam-4858	98	16	is	be	AUX
ejpam-4858	98	17	α-⋆-open	α-⋆-open	NUM
ejpam-4858	98	18	and	and	CCONJ
ejpam-4858	98	19	hence	hence	ADV
ejpam-4858	98	20	∩	∩	ADV
ejpam-4858	98	21	γ∈γ	γ∈γ	ADJ
ejpam-4858	98	22	aγ	aγ	PRON
ejpam-4858	98	23	is	be	AUX
ejpam-4858	98	24	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	98	25	.	.	PUNCT
ejpam-4858	99	1	for	for	ADP
ejpam-4858	99	2	a	a	DET
ejpam-4858	99	3	subset	subset	NOUN
ejpam-4858	99	4	a	a	PRON
ejpam-4858	99	5	of	of	ADP
ejpam-4858	99	6	an	an	DET
ejpam-4858	99	7	ideal	ideal	ADJ
ejpam-4858	99	8	topological	topological	ADJ
ejpam-4858	99	9	space	space	NOUN
ejpam-4858	99	10	(	(	PUNCT
ejpam-4858	99	11	x	x	X
ejpam-4858	99	12	,	,	PUNCT
ejpam-4858	99	13	τ	τ	PROPN
ejpam-4858	99	14	,	,	PUNCT
ejpam-4858	99	15	i	i	NOUN
ejpam-4858	99	16	)	)	PUNCT
ejpam-4858	99	17	,	,	PUNCT
ejpam-4858	99	18	the	the	DET
ejpam-4858	99	19	intersection	intersection	NOUN
ejpam-4858	99	20	of	of	ADP
ejpam-4858	99	21	all	all	PRON
ejpam-4858	99	22	α-⋆-closed	α-⋆-closed	NUM
ejpam-4858	99	23	sets	set	NOUN
ejpam-4858	99	24	containing	contain	VERB
ejpam-4858	99	25	a	a	PRON
ejpam-4858	99	26	is	be	AUX
ejpam-4858	99	27	called	call	VERB
ejpam-4858	99	28	the	the	DET
ejpam-4858	99	29	α-⋆-closure	α-⋆-closure	NUM
ejpam-4858	99	30	of	of	ADP
ejpam-4858	99	31	a	a	PRON
ejpam-4858	99	32	and	and	CCONJ
ejpam-4858	99	33	is	be	AUX
ejpam-4858	99	34	denoted	denote	VERB
ejpam-4858	99	35	by	by	ADP
ejpam-4858	99	36	⋆αcl(a	⋆αcl(a	PROPN
ejpam-4858	99	37	)	)	PUNCT
ejpam-4858	99	38	.	.	PUNCT
ejpam-4858	100	1	the	the	DET
ejpam-4858	100	2	α-⋆interior	α-⋆interior	NOUN
ejpam-4858	100	3	of	of	ADP
ejpam-4858	100	4	a	a	PRON
ejpam-4858	100	5	is	be	AUX
ejpam-4858	100	6	defined	define	VERB
ejpam-4858	100	7	by	by	ADP
ejpam-4858	100	8	the	the	DET
ejpam-4858	100	9	union	union	NOUN
ejpam-4858	100	10	of	of	ADP
ejpam-4858	100	11	all	all	PRON
ejpam-4858	100	12	α-⋆-open	α-⋆-open	NUM
ejpam-4858	100	13	sets	set	NOUN
ejpam-4858	100	14	contained	contain	VERB
ejpam-4858	100	15	in	in	ADP
ejpam-4858	100	16	a	a	PRON
ejpam-4858	100	17	and	and	CCONJ
ejpam-4858	100	18	is	be	AUX
ejpam-4858	100	19	denoted	denote	VERB
ejpam-4858	100	20	by	by	ADP
ejpam-4858	100	21	⋆αint(a	⋆αint(a	PROPN
ejpam-4858	100	22	)	)	PUNCT
ejpam-4858	100	23	.	.	PUNCT
ejpam-4858	101	1	proposition	proposition	NOUN
ejpam-4858	101	2	2	2	NUM
ejpam-4858	101	3	.	.	X
ejpam-4858	101	4	for	for	ADP
ejpam-4858	101	5	a	a	DET
ejpam-4858	101	6	subset	subset	NOUN
ejpam-4858	101	7	a	a	PRON
ejpam-4858	101	8	of	of	ADP
ejpam-4858	101	9	an	an	DET
ejpam-4858	101	10	ideal	ideal	ADJ
ejpam-4858	101	11	topological	topological	ADJ
ejpam-4858	101	12	space	space	NOUN
ejpam-4858	101	13	(	(	PUNCT
ejpam-4858	101	14	x	x	X
ejpam-4858	101	15	,	,	PUNCT
ejpam-4858	101	16	τ	τ	PROPN
ejpam-4858	101	17	,	,	PUNCT
ejpam-4858	101	18	i	i	NOUN
ejpam-4858	101	19	)	)	PUNCT
ejpam-4858	101	20	,	,	PUNCT
ejpam-4858	101	21	the	the	DET
ejpam-4858	101	22	following	follow	VERB
ejpam-4858	101	23	properties	property	NOUN
ejpam-4858	101	24	hold	hold	VERB
ejpam-4858	101	25	:	:	PUNCT
ejpam-4858	101	26	(	(	PUNCT
ejpam-4858	101	27	1	1	X
ejpam-4858	101	28	)	)	PUNCT
ejpam-4858	101	29	⋆αcl(a	⋆αcl(a	NOUN
ejpam-4858	101	30	)	)	PUNCT
ejpam-4858	101	31	is	be	AUX
ejpam-4858	101	32	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	101	33	.	.	PUNCT
ejpam-4858	102	1	(	(	PUNCT
ejpam-4858	102	2	2	2	X
ejpam-4858	102	3	)	)	PUNCT
ejpam-4858	102	4	a	a	PRON
ejpam-4858	102	5	is	is	AUX
ejpam-4858	102	6	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	102	7	if	if	SCONJ
ejpam-4858	102	8	and	and	CCONJ
ejpam-4858	102	9	only	only	ADV
ejpam-4858	102	10	if	if	SCONJ
ejpam-4858	102	11	a	a	DET
ejpam-4858	102	12	=	=	PUNCT
ejpam-4858	102	13	⋆αcl(a	⋆αcl(a	NOUN
ejpam-4858	102	14	)	)	PUNCT
ejpam-4858	102	15	.	.	PUNCT
ejpam-4858	103	1	proof	proof	NOUN
ejpam-4858	103	2	.	.	PUNCT
ejpam-4858	104	1	(	(	PUNCT
ejpam-4858	104	2	1	1	X
ejpam-4858	104	3	)	)	PUNCT
ejpam-4858	104	4	follows	follow	VERB
ejpam-4858	104	5	from	from	ADP
ejpam-4858	104	6	proposition	proposition	NOUN
ejpam-4858	104	7	1	1	NUM
ejpam-4858	104	8	.	.	PUNCT
ejpam-4858	105	1	(	(	PUNCT
ejpam-4858	105	2	2	2	X
ejpam-4858	105	3	)	)	PUNCT
ejpam-4858	105	4	follows	follow	VERB
ejpam-4858	105	5	from	from	ADP
ejpam-4858	105	6	(	(	PUNCT
ejpam-4858	105	7	1	1	NUM
ejpam-4858	105	8	)	)	PUNCT
ejpam-4858	105	9	.	.	PUNCT
ejpam-4858	106	1	lemma	lemma	PROPN
ejpam-4858	106	2	2	2	NUM
ejpam-4858	106	3	.	.	X
ejpam-4858	107	1	for	for	ADP
ejpam-4858	107	2	a	a	DET
ejpam-4858	107	3	subset	subset	NOUN
ejpam-4858	107	4	a	a	PRON
ejpam-4858	107	5	of	of	ADP
ejpam-4858	107	6	an	an	DET
ejpam-4858	107	7	ideal	ideal	ADJ
ejpam-4858	107	8	topological	topological	ADJ
ejpam-4858	107	9	space	space	NOUN
ejpam-4858	107	10	(	(	PUNCT
ejpam-4858	107	11	x	x	X
ejpam-4858	107	12	,	,	PUNCT
ejpam-4858	107	13	τ	τ	PROPN
ejpam-4858	107	14	,	,	PUNCT
ejpam-4858	107	15	i	i	NOUN
ejpam-4858	107	16	)	)	PUNCT
ejpam-4858	107	17	,	,	PUNCT
ejpam-4858	107	18	the	the	DET
ejpam-4858	107	19	following	follow	VERB
ejpam-4858	107	20	properties	property	NOUN
ejpam-4858	107	21	are	be	AUX
ejpam-4858	107	22	equivalent	equivalent	ADJ
ejpam-4858	107	23	:	:	PUNCT
ejpam-4858	107	24	(	(	PUNCT
ejpam-4858	107	25	1	1	X
ejpam-4858	107	26	)	)	PUNCT
ejpam-4858	107	27	a	a	PRON
ejpam-4858	107	28	is	be	AUX
ejpam-4858	107	29	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	107	30	in	in	ADP
ejpam-4858	107	31	x	x	PRON
ejpam-4858	107	32	;	;	PUNCT
ejpam-4858	107	33	(	(	PUNCT
ejpam-4858	107	34	2	2	X
ejpam-4858	107	35	)	)	PUNCT
ejpam-4858	107	36	g	g	ADP
ejpam-4858	107	37	⊆	⊆	NUM
ejpam-4858	107	38	a	a	DET
ejpam-4858	107	39	⊆	⊆	NUM
ejpam-4858	107	40	int⋆(cl(g	int⋆(cl(g	NOUN
ejpam-4858	107	41	)	)	PUNCT
ejpam-4858	107	42	)	)	PUNCT
ejpam-4858	107	43	for	for	SCONJ
ejpam-4858	107	44	some	some	DET
ejpam-4858	107	45	⋆-open	⋆-open	NOUN
ejpam-4858	107	46	set	set	NOUN
ejpam-4858	107	47	g	g	NOUN
ejpam-4858	107	48	;	;	PUNCT
ejpam-4858	107	49	(	(	PUNCT
ejpam-4858	107	50	3	3	X
ejpam-4858	107	51	)	)	PUNCT
ejpam-4858	107	52	g	g	ADP
ejpam-4858	107	53	⊆	⊆	NUM
ejpam-4858	107	54	a	a	DET
ejpam-4858	107	55	⊆	⊆	NUM
ejpam-4858	107	56	scli	scli	NOUN
ejpam-4858	107	57	(	(	PUNCT
ejpam-4858	107	58	g	g	NOUN
ejpam-4858	107	59	)	)	PUNCT
ejpam-4858	107	60	for	for	SCONJ
ejpam-4858	107	61	some	some	DET
ejpam-4858	107	62	⋆-open	⋆-open	NOUN
ejpam-4858	107	63	set	set	NOUN
ejpam-4858	107	64	g	g	NOUN
ejpam-4858	107	65	;	;	PUNCT
ejpam-4858	107	66	(	(	PUNCT
ejpam-4858	107	67	4	4	X
ejpam-4858	107	68	)	)	PUNCT
ejpam-4858	107	69	a	a	DET
ejpam-4858	107	70	⊆	⊆	NUM
ejpam-4858	107	71	scli	scli	NOUN
ejpam-4858	107	72	(	(	PUNCT
ejpam-4858	107	73	int⋆(a	int⋆(a	NOUN
ejpam-4858	107	74	)	)	PUNCT
ejpam-4858	107	75	)	)	PUNCT
ejpam-4858	107	76	.	.	PUNCT
ejpam-4858	108	1	c.	c.	PROPN
ejpam-4858	108	2	boonpok	boonpok	PROPN
ejpam-4858	108	3	,	,	PUNCT
ejpam-4858	108	4	j.	j.	PROPN
ejpam-4858	108	5	khampakdee	khampakdee	PROPN
ejpam-4858	108	6	/	/	PUNCT
ejpam-4858	108	7	eur	eur	PROPN
ejpam-4858	108	8	.	.	PUNCT
ejpam-4858	109	1	j.	j.	PROPN
ejpam-4858	109	2	pure	pure	PROPN
ejpam-4858	109	3	appl	appl	PROPN
ejpam-4858	109	4	.	.	PROPN
ejpam-4858	109	5	math	math	PROPN
ejpam-4858	109	6	,	,	PUNCT
ejpam-4858	109	7	17	17	NUM
ejpam-4858	109	8	(	(	PUNCT
ejpam-4858	109	9	1	1	NUM
ejpam-4858	109	10	)	)	PUNCT
ejpam-4858	109	11	(	(	PUNCT
ejpam-4858	109	12	2024	2024	NUM
ejpam-4858	109	13	)	)	PUNCT
ejpam-4858	109	14	,	,	PUNCT
ejpam-4858	109	15	201	201	NUM
ejpam-4858	109	16	-	-	SYM
ejpam-4858	109	17	211	211	NUM
ejpam-4858	109	18	204	204	NUM
ejpam-4858	109	19	proof	proof	NOUN
ejpam-4858	109	20	.	.	PUNCT
ejpam-4858	110	1	(	(	PUNCT
ejpam-4858	110	2	1	1	X
ejpam-4858	110	3	)	)	PUNCT
ejpam-4858	110	4	⇒	⇒	NOUN
ejpam-4858	110	5	(	(	PUNCT
ejpam-4858	110	6	2	2	NUM
ejpam-4858	110	7	):	):	PUNCT
ejpam-4858	110	8	suppose	suppose	VERB
ejpam-4858	110	9	that	that	SCONJ
ejpam-4858	110	10	a	a	PRON
ejpam-4858	110	11	is	be	AUX
ejpam-4858	110	12	an	an	DET
ejpam-4858	110	13	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	110	14	set	set	NOUN
ejpam-4858	110	15	.	.	PUNCT
ejpam-4858	111	1	then	then	ADV
ejpam-4858	111	2	,	,	PUNCT
ejpam-4858	111	3	a	a	DET
ejpam-4858	111	4	⊆	⊆	NUM
ejpam-4858	111	5	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NUM
ejpam-4858	111	6	)	)	PUNCT
ejpam-4858	111	7	)	)	PUNCT
ejpam-4858	111	8	)	)	PUNCT
ejpam-4858	111	9	.	.	PUNCT
ejpam-4858	112	1	put	put	VERB
ejpam-4858	112	2	g	g	NOUN
ejpam-4858	112	3	=	=	PUNCT
ejpam-4858	112	4	int⋆(a	int⋆(a	NOUN
ejpam-4858	112	5	)	)	PUNCT
ejpam-4858	112	6	,	,	PUNCT
ejpam-4858	112	7	then	then	ADV
ejpam-4858	112	8	g	g	PROPN
ejpam-4858	112	9	is	be	AUX
ejpam-4858	112	10	a	a	DET
ejpam-4858	112	11	⋆-open	⋆-open	ADJ
ejpam-4858	112	12	set	set	NOUN
ejpam-4858	112	13	such	such	ADJ
ejpam-4858	112	14	that	that	SCONJ
ejpam-4858	112	15	g	g	PROPN
ejpam-4858	112	16	⊆	⊆	NUM
ejpam-4858	112	17	a	a	DET
ejpam-4858	112	18	⊆	⊆	NUM
ejpam-4858	112	19	int⋆(cl(g	int⋆(cl(g	NOUN
ejpam-4858	112	20	)	)	PUNCT
ejpam-4858	112	21	)	)	PUNCT
ejpam-4858	112	22	.	.	PUNCT
ejpam-4858	113	1	(	(	PUNCT
ejpam-4858	113	2	2	2	X
ejpam-4858	113	3	)	)	PUNCT
ejpam-4858	113	4	⇒	⇒	NOUN
ejpam-4858	113	5	(	(	PUNCT
ejpam-4858	113	6	3	3	NUM
ejpam-4858	113	7	):	):	PUNCT
ejpam-4858	113	8	this	this	PRON
ejpam-4858	113	9	follows	follow	VERB
ejpam-4858	113	10	from	from	ADP
ejpam-4858	113	11	lemma	lemma	PROPN
ejpam-4858	113	12	1(2	1(2	NUM
ejpam-4858	113	13	)	)	PUNCT
ejpam-4858	113	14	.	.	PUNCT
ejpam-4858	114	1	(	(	PUNCT
ejpam-4858	114	2	3	3	X
ejpam-4858	114	3	)	)	PUNCT
ejpam-4858	114	4	⇒	⇒	NOUN
ejpam-4858	114	5	(	(	PUNCT
ejpam-4858	114	6	4	4	NUM
ejpam-4858	114	7	):	):	PUNCT
ejpam-4858	114	8	suppose	suppose	VERB
ejpam-4858	114	9	that	that	SCONJ
ejpam-4858	114	10	g	g	PROPN
ejpam-4858	114	11	⊆	⊆	NUM
ejpam-4858	114	12	a	a	DET
ejpam-4858	114	13	⊆	⊆	NUM
ejpam-4858	114	14	scli	scli	NOUN
ejpam-4858	114	15	(	(	PUNCT
ejpam-4858	114	16	g	g	NOUN
ejpam-4858	114	17	)	)	PUNCT
ejpam-4858	114	18	for	for	ADP
ejpam-4858	114	19	some	some	DET
ejpam-4858	114	20	⋆-open	⋆-open	ADJ
ejpam-4858	114	21	set	set	VERB
ejpam-4858	114	22	g.	g.	PROPN
ejpam-4858	114	23	then	then	ADV
ejpam-4858	114	24	,	,	PUNCT
ejpam-4858	114	25	we	we	PRON
ejpam-4858	114	26	have	have	VERB
ejpam-4858	114	27	g	g	NOUN
ejpam-4858	114	28	⊆	⊆	NUM
ejpam-4858	114	29	int⋆(a	int⋆(a	NOUN
ejpam-4858	114	30	)	)	PUNCT
ejpam-4858	114	31	and	and	CCONJ
ejpam-4858	114	32	hence	hence	ADV
ejpam-4858	114	33	a	a	DET
ejpam-4858	114	34	⊆	⊆	NUM
ejpam-4858	114	35	scli	scli	NOUN
ejpam-4858	114	36	(	(	PUNCT
ejpam-4858	114	37	int⋆(a	int⋆(a	NOUN
ejpam-4858	114	38	)	)	PUNCT
ejpam-4858	114	39	)	)	PUNCT
ejpam-4858	114	40	.	.	PUNCT
ejpam-4858	115	1	(	(	PUNCT
ejpam-4858	115	2	4	4	X
ejpam-4858	115	3	)	)	PUNCT
ejpam-4858	115	4	⇒	⇒	NOUN
ejpam-4858	115	5	(	(	PUNCT
ejpam-4858	115	6	1	1	NUM
ejpam-4858	115	7	):	):	PUNCT
ejpam-4858	115	8	suppose	suppose	VERB
ejpam-4858	115	9	that	that	SCONJ
ejpam-4858	115	10	a	a	DET
ejpam-4858	115	11	⊆	⊆	NUM
ejpam-4858	115	12	scli	scli	NOUN
ejpam-4858	115	13	(	(	PUNCT
ejpam-4858	115	14	int⋆(a	int⋆(a	NOUN
ejpam-4858	115	15	)	)	PUNCT
ejpam-4858	115	16	)	)	PUNCT
ejpam-4858	115	17	.	.	PUNCT
ejpam-4858	116	1	since	since	SCONJ
ejpam-4858	116	2	int⋆(a	int⋆(a	NOUN
ejpam-4858	116	3	)	)	PUNCT
ejpam-4858	116	4	is	be	AUX
ejpam-4858	116	5	⋆-open	⋆-open	ADJ
ejpam-4858	116	6	in	in	ADP
ejpam-4858	116	7	x	x	PUNCT
ejpam-4858	116	8	and	and	CCONJ
ejpam-4858	116	9	by	by	ADP
ejpam-4858	116	10	lemma	lemma	PROPN
ejpam-4858	116	11	1(2	1(2	NUM
ejpam-4858	116	12	)	)	PUNCT
ejpam-4858	116	13	,	,	PUNCT
ejpam-4858	116	14	a	a	DET
ejpam-4858	116	15	⊆	⊆	NUM
ejpam-4858	116	16	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NUM
ejpam-4858	116	17	)	)	PUNCT
ejpam-4858	116	18	)	)	PUNCT
ejpam-4858	116	19	)	)	PUNCT
ejpam-4858	116	20	.	.	PUNCT
ejpam-4858	117	1	thus	thus	ADV
ejpam-4858	117	2	,	,	PUNCT
ejpam-4858	117	3	a	a	PRON
ejpam-4858	117	4	is	be	AUX
ejpam-4858	117	5	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	117	6	in	in	ADP
ejpam-4858	117	7	x.	x.	PROPN
ejpam-4858	117	8	lemma	lemma	PROPN
ejpam-4858	117	9	3	3	X
ejpam-4858	117	10	.	.	X
ejpam-4858	117	11	for	for	ADP
ejpam-4858	117	12	a	a	DET
ejpam-4858	117	13	subset	subset	NOUN
ejpam-4858	117	14	a	a	PRON
ejpam-4858	117	15	of	of	ADP
ejpam-4858	117	16	an	an	DET
ejpam-4858	117	17	ideal	ideal	ADJ
ejpam-4858	117	18	topological	topological	ADJ
ejpam-4858	117	19	space	space	NOUN
ejpam-4858	117	20	(	(	PUNCT
ejpam-4858	117	21	x	x	X
ejpam-4858	117	22	,	,	PUNCT
ejpam-4858	117	23	τ	τ	PROPN
ejpam-4858	117	24	,	,	PUNCT
ejpam-4858	117	25	i	i	NOUN
ejpam-4858	117	26	)	)	PUNCT
ejpam-4858	117	27	,	,	PUNCT
ejpam-4858	117	28	the	the	DET
ejpam-4858	117	29	following	follow	VERB
ejpam-4858	117	30	properties	property	NOUN
ejpam-4858	117	31	hold	hold	VERB
ejpam-4858	117	32	:	:	PUNCT
ejpam-4858	117	33	(	(	PUNCT
ejpam-4858	117	34	1	1	X
ejpam-4858	117	35	)	)	PUNCT
ejpam-4858	117	36	a	a	PRON
ejpam-4858	117	37	is	is	AUX
ejpam-4858	117	38	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	117	39	in	in	ADP
ejpam-4858	117	40	x	x	SYM
ejpam-4858	117	41	if	if	SCONJ
ejpam-4858	118	1	and	and	CCONJ
ejpam-4858	118	2	only	only	ADV
ejpam-4858	118	3	if	if	SCONJ
ejpam-4858	118	4	sinti	sinti	PROPN
ejpam-4858	118	5	(	(	PUNCT
ejpam-4858	118	6	cl⋆(a	cl⋆(a	PROPN
ejpam-4858	118	7	)	)	PUNCT
ejpam-4858	118	8	)	)	PUNCT
ejpam-4858	118	9	⊆	⊆	NUM
ejpam-4858	118	10	a.	a.	NOUN
ejpam-4858	118	11	(	(	PUNCT
ejpam-4858	118	12	2	2	X
ejpam-4858	118	13	)	)	PUNCT
ejpam-4858	118	14	sinti	sinti	PROPN
ejpam-4858	118	15	(	(	PUNCT
ejpam-4858	118	16	cl⋆(a	cl⋆(a	PROPN
ejpam-4858	118	17	)	)	PUNCT
ejpam-4858	118	18	)	)	PUNCT
ejpam-4858	118	19	=	=	PUNCT
ejpam-4858	118	20	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	118	21	)	)	PUNCT
ejpam-4858	118	22	)	)	PUNCT
ejpam-4858	118	23	)	)	PUNCT
ejpam-4858	118	24	.	.	PUNCT
ejpam-4858	119	1	(	(	PUNCT
ejpam-4858	119	2	3	3	X
ejpam-4858	119	3	)	)	PUNCT
ejpam-4858	119	4	⋆αcl(a	⋆αcl(a	NOUN
ejpam-4858	119	5	)	)	PUNCT
ejpam-4858	119	6	=	=	PUNCT
ejpam-4858	119	7	a	a	DET
ejpam-4858	119	8	∪	∪	X
ejpam-4858	119	9	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	119	10	)	)	PUNCT
ejpam-4858	119	11	)	)	PUNCT
ejpam-4858	119	12	)	)	PUNCT
ejpam-4858	119	13	.	.	PUNCT
ejpam-4858	120	1	(	(	PUNCT
ejpam-4858	120	2	4	4	X
ejpam-4858	120	3	)	)	PUNCT
ejpam-4858	120	4	⋆αint(a	⋆αint(a	NOUN
ejpam-4858	120	5	)	)	PUNCT
ejpam-4858	120	6	=	=	PUNCT
ejpam-4858	121	1	a	a	DET
ejpam-4858	121	2	∩	∩	ADJ
ejpam-4858	121	3	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	121	4	)	)	PUNCT
ejpam-4858	121	5	)	)	PUNCT
ejpam-4858	121	6	)	)	PUNCT
ejpam-4858	121	7	.	.	PUNCT
ejpam-4858	122	1	proof	proof	NOUN
ejpam-4858	122	2	.	.	PUNCT
ejpam-4858	123	1	(	(	PUNCT
ejpam-4858	123	2	1	1	X
ejpam-4858	123	3	)	)	PUNCT
ejpam-4858	123	4	follows	follow	VERB
ejpam-4858	123	5	from	from	ADP
ejpam-4858	123	6	lemma	lemma	PROPN
ejpam-4858	123	7	2	2	NUM
ejpam-4858	123	8	.	.	PUNCT
ejpam-4858	124	1	(	(	PUNCT
ejpam-4858	124	2	2	2	X
ejpam-4858	124	3	)	)	PUNCT
ejpam-4858	124	4	follows	follow	VERB
ejpam-4858	124	5	from	from	ADP
ejpam-4858	124	6	lemma	lemma	PROPN
ejpam-4858	124	7	13(1	13(1	NUM
ejpam-4858	124	8	)	)	PUNCT
ejpam-4858	124	9	of	of	ADP
ejpam-4858	124	10	[	[	X
ejpam-4858	124	11	13	13	NUM
ejpam-4858	124	12	]	]	PUNCT
ejpam-4858	124	13	.	.	PUNCT
ejpam-4858	125	1	(	(	PUNCT
ejpam-4858	125	2	3	3	X
ejpam-4858	125	3	)	)	PUNCT
ejpam-4858	125	4	we	we	PRON
ejpam-4858	125	5	observe	observe	VERB
ejpam-4858	125	6	that	that	SCONJ
ejpam-4858	125	7	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	125	8	∪	∪	ADP
ejpam-4858	125	9	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	125	10	)	)	PUNCT
ejpam-4858	125	11	)	)	PUNCT
ejpam-4858	125	12	)	)	PUNCT
ejpam-4858	125	13	)	)	PUNCT
ejpam-4858	125	14	)	)	PUNCT
ejpam-4858	125	15	)	)	PUNCT
ejpam-4858	126	1	⊆	⊆	X
ejpam-4858	126	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	126	3	∪	∪	ADV
ejpam-4858	126	4	(	(	PUNCT
ejpam-4858	126	5	cl⋆(a	cl⋆(a	NUM
ejpam-4858	126	6	)	)	PUNCT
ejpam-4858	126	7	)	)	PUNCT
ejpam-4858	126	8	)	)	PUNCT
ejpam-4858	126	9	)	)	PUNCT
ejpam-4858	126	10	)	)	PUNCT
ejpam-4858	127	1	⊆	⊆	NUM
ejpam-4858	127	2	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	127	3	)	)	PUNCT
ejpam-4858	127	4	)	)	PUNCT
ejpam-4858	127	5	)	)	PUNCT
ejpam-4858	128	1	⊆	⊆	ADP
ejpam-4858	128	2	a	a	DET
ejpam-4858	128	3	∪	∪	X
ejpam-4858	128	4	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	128	5	)	)	PUNCT
ejpam-4858	128	6	)	)	PUNCT
ejpam-4858	128	7	)	)	PUNCT
ejpam-4858	128	8	.	.	PUNCT
ejpam-4858	129	1	thus	thus	ADV
ejpam-4858	129	2	,	,	PUNCT
ejpam-4858	129	3	a	a	DET
ejpam-4858	129	4	∪	∪	ADJ
ejpam-4858	129	5	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	129	6	)	)	PUNCT
ejpam-4858	129	7	)	)	PUNCT
ejpam-4858	129	8	)	)	PUNCT
ejpam-4858	130	1	is	be	AUX
ejpam-4858	130	2	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	130	3	and	and	CCONJ
ejpam-4858	130	4	hence	hence	ADV
ejpam-4858	130	5	⋆αcl(a	⋆αcl(a	VERB
ejpam-4858	130	6	)	)	PUNCT
ejpam-4858	130	7	⊆	⊆	NUM
ejpam-4858	130	8	a	a	DET
ejpam-4858	130	9	∪	∪	X
ejpam-4858	130	10	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	130	11	)	)	PUNCT
ejpam-4858	130	12	)	)	PUNCT
ejpam-4858	130	13	)	)	PUNCT
ejpam-4858	130	14	.	.	PUNCT
ejpam-4858	131	1	on	on	ADP
ejpam-4858	131	2	the	the	DET
ejpam-4858	131	3	other	other	ADJ
ejpam-4858	131	4	hand	hand	NOUN
ejpam-4858	131	5	,	,	PUNCT
ejpam-4858	131	6	since	since	SCONJ
ejpam-4858	131	7	⋆αcl(a	⋆αcl(a	PROPN
ejpam-4858	131	8	)	)	PUNCT
ejpam-4858	131	9	is	be	AUX
ejpam-4858	131	10	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	131	11	,	,	PUNCT
ejpam-4858	131	12	we	we	PRON
ejpam-4858	131	13	have	have	VERB
ejpam-4858	131	14	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	131	15	)	)	PUNCT
ejpam-4858	131	16	)	)	PUNCT
ejpam-4858	131	17	)	)	PUNCT
ejpam-4858	132	1	⊆	⊆	X
ejpam-4858	132	2	cl⋆(int(cl⋆(⋆αcl(a	cl⋆(int(cl⋆(⋆αcl(a	NOUN
ejpam-4858	132	3	)	)	PUNCT
ejpam-4858	132	4	)	)	PUNCT
ejpam-4858	132	5	)	)	PUNCT
ejpam-4858	132	6	)	)	PUNCT
ejpam-4858	133	1	⊆	⊆	X
ejpam-4858	133	2	⋆αcl(a	⋆αcl(a	X
ejpam-4858	133	3	)	)	PUNCT
ejpam-4858	133	4	and	and	CCONJ
ejpam-4858	133	5	hence	hence	ADV
ejpam-4858	133	6	a	a	DET
ejpam-4858	133	7	∪	∪	ADJ
ejpam-4858	133	8	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	133	9	)	)	PUNCT
ejpam-4858	133	10	)	)	PUNCT
ejpam-4858	133	11	)	)	PUNCT
ejpam-4858	134	1	⊆	⊆	X
ejpam-4858	134	2	⋆αcl(a	⋆αcl(a	NOUN
ejpam-4858	134	3	)	)	PUNCT
ejpam-4858	134	4	.	.	PUNCT
ejpam-4858	135	1	thus	thus	ADV
ejpam-4858	135	2	,	,	PUNCT
ejpam-4858	135	3	⋆αcl(a	⋆αcl(a	PROPN
ejpam-4858	135	4	)	)	PUNCT
ejpam-4858	135	5	=	=	PUNCT
ejpam-4858	135	6	a	a	DET
ejpam-4858	135	7	∪	∪	X
ejpam-4858	135	8	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	135	9	)	)	PUNCT
ejpam-4858	135	10	)	)	PUNCT
ejpam-4858	135	11	)	)	PUNCT
ejpam-4858	135	12	.	.	PUNCT
ejpam-4858	136	1	(	(	PUNCT
ejpam-4858	136	2	4	4	X
ejpam-4858	136	3	)	)	PUNCT
ejpam-4858	136	4	since	since	SCONJ
ejpam-4858	136	5	⋆αint(a	⋆αint(a	PROPN
ejpam-4858	136	6	)	)	PUNCT
ejpam-4858	136	7	is	be	AUX
ejpam-4858	136	8	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	136	9	,	,	PUNCT
ejpam-4858	136	10	we	we	PRON
ejpam-4858	136	11	have	have	VERB
ejpam-4858	136	12	⋆αint(a	⋆αint(a	NOUN
ejpam-4858	136	13	)	)	PUNCT
ejpam-4858	136	14	⊆	⊆	NUM
ejpam-4858	136	15	int⋆(cl(int⋆(⋆αint(a	int⋆(cl(int⋆(⋆αint(a	NUM
ejpam-4858	136	16	)	)	PUNCT
ejpam-4858	136	17	)	)	PUNCT
ejpam-4858	136	18	)	)	PUNCT
ejpam-4858	136	19	)	)	PUNCT
ejpam-4858	137	1	⊆	⊆	NUM
ejpam-4858	137	2	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	137	3	)	)	PUNCT
ejpam-4858	137	4	)	)	PUNCT
ejpam-4858	137	5	)	)	PUNCT
ejpam-4858	137	6	and	and	CCONJ
ejpam-4858	137	7	hence	hence	ADV
ejpam-4858	137	8	⋆αint(a	⋆αint(a	PROPN
ejpam-4858	137	9	)	)	PUNCT
ejpam-4858	137	10	⊆	⊆	NUM
ejpam-4858	137	11	a	a	DET
ejpam-4858	137	12	∩	∩	ADJ
ejpam-4858	137	13	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	137	14	)	)	PUNCT
ejpam-4858	137	15	)	)	PUNCT
ejpam-4858	137	16	)	)	PUNCT
ejpam-4858	137	17	.	.	PUNCT
ejpam-4858	138	1	on	on	ADP
ejpam-4858	138	2	the	the	DET
ejpam-4858	138	3	other	other	ADJ
ejpam-4858	138	4	hand	hand	NOUN
ejpam-4858	138	5	,	,	PUNCT
ejpam-4858	138	6	we	we	PRON
ejpam-4858	138	7	have	have	VERB
ejpam-4858	138	8	a	a	DET
ejpam-4858	138	9	∩	∩	ADJ
ejpam-4858	138	10	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	138	11	)	)	PUNCT
ejpam-4858	138	12	)	)	PUNCT
ejpam-4858	138	13	)	)	PUNCT
ejpam-4858	139	1	⊆	⊆	NUM
ejpam-4858	139	2	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	139	3	)	)	PUNCT
ejpam-4858	139	4	)	)	PUNCT
ejpam-4858	139	5	)	)	PUNCT
ejpam-4858	140	1	=	=	PUNCT
ejpam-4858	140	2	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	ADJ
ejpam-4858	140	3	)	)	PUNCT
ejpam-4858	140	4	∩	∩	NOUN
ejpam-4858	140	5	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	140	6	)	)	PUNCT
ejpam-4858	140	7	)	)	PUNCT
ejpam-4858	140	8	)	)	PUNCT
ejpam-4858	140	9	)	)	PUNCT
ejpam-4858	140	10	)	)	PUNCT
ejpam-4858	141	1	=	=	PUNCT
ejpam-4858	141	2	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	ADP
ejpam-4858	141	3	∩	∩	ADJ
ejpam-4858	141	4	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	141	5	)	)	PUNCT
ejpam-4858	141	6	)	)	PUNCT
ejpam-4858	141	7	)	)	PUNCT
ejpam-4858	141	8	)	)	PUNCT
ejpam-4858	141	9	)	)	PUNCT
ejpam-4858	141	10	)	)	PUNCT
ejpam-4858	141	11	.	.	PUNCT
ejpam-4858	142	1	thus	thus	ADV
ejpam-4858	142	2	,	,	PUNCT
ejpam-4858	142	3	a	a	DET
ejpam-4858	142	4	∩	∩	ADJ
ejpam-4858	142	5	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	142	6	)	)	PUNCT
ejpam-4858	142	7	)	)	PUNCT
ejpam-4858	142	8	)	)	PUNCT
ejpam-4858	142	9	is	be	AUX
ejpam-4858	142	10	α-⋆-open	α-⋆-open	NUM
ejpam-4858	142	11	and	and	CCONJ
ejpam-4858	142	12	so	so	ADV
ejpam-4858	142	13	a	a	DET
ejpam-4858	142	14	∩	∩	ADJ
ejpam-4858	142	15	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	142	16	)	)	PUNCT
ejpam-4858	142	17	)	)	PUNCT
ejpam-4858	142	18	)	)	PUNCT
ejpam-4858	143	1	⊆	⊆	NUM
ejpam-4858	143	2	⋆αint(a	⋆αint(a	NOUN
ejpam-4858	143	3	)	)	PUNCT
ejpam-4858	143	4	.	.	PUNCT
ejpam-4858	144	1	this	this	PRON
ejpam-4858	144	2	shows	show	VERB
ejpam-4858	144	3	that	that	SCONJ
ejpam-4858	144	4	⋆αint(a	⋆αint(a	NOUN
ejpam-4858	144	5	)	)	PUNCT
ejpam-4858	144	6	=	=	PUNCT
ejpam-4858	144	7	a	a	DET
ejpam-4858	144	8	∩	∩	ADJ
ejpam-4858	144	9	int⋆(cl(int⋆(a	int⋆(cl(int⋆(a	NOUN
ejpam-4858	144	10	)	)	PUNCT
ejpam-4858	144	11	)	)	PUNCT
ejpam-4858	144	12	)	)	PUNCT
ejpam-4858	144	13	.	.	PUNCT
ejpam-4858	145	1	c.	c.	PROPN
ejpam-4858	145	2	boonpok	boonpok	PROPN
ejpam-4858	145	3	,	,	PUNCT
ejpam-4858	145	4	j.	j.	PROPN
ejpam-4858	145	5	khampakdee	khampakdee	PROPN
ejpam-4858	145	6	/	/	PUNCT
ejpam-4858	145	7	eur	eur	PROPN
ejpam-4858	145	8	.	.	PUNCT
ejpam-4858	146	1	j.	j.	PROPN
ejpam-4858	146	2	pure	pure	PROPN
ejpam-4858	146	3	appl	appl	PROPN
ejpam-4858	146	4	.	.	PROPN
ejpam-4858	146	5	math	math	PROPN
ejpam-4858	146	6	,	,	PUNCT
ejpam-4858	146	7	17	17	NUM
ejpam-4858	146	8	(	(	PUNCT
ejpam-4858	146	9	1	1	NUM
ejpam-4858	146	10	)	)	PUNCT
ejpam-4858	146	11	(	(	PUNCT
ejpam-4858	146	12	2024	2024	NUM
ejpam-4858	146	13	)	)	PUNCT
ejpam-4858	146	14	,	,	PUNCT
ejpam-4858	146	15	201	201	NUM
ejpam-4858	146	16	-	-	SYM
ejpam-4858	146	17	211	211	NUM
ejpam-4858	146	18	205	205	NUM
ejpam-4858	146	19	by	by	ADP
ejpam-4858	146	20	a	a	DET
ejpam-4858	146	21	multifunction	multifunction	NOUN
ejpam-4858	147	1	f	f	NOUN
ejpam-4858	147	2	:	:	PUNCT
ejpam-4858	147	3	x	x	X
ejpam-4858	147	4	→	→	SYM
ejpam-4858	147	5	y	y	PROPN
ejpam-4858	147	6	,	,	PUNCT
ejpam-4858	147	7	we	we	PRON
ejpam-4858	147	8	mean	mean	VERB
ejpam-4858	147	9	a	a	DET
ejpam-4858	147	10	point	point	NOUN
ejpam-4858	147	11	-	-	PUNCT
ejpam-4858	147	12	to	to	ADP
ejpam-4858	147	13	-	-	PUNCT
ejpam-4858	147	14	set	set	VERB
ejpam-4858	147	15	correspondence	correspondence	NOUN
ejpam-4858	147	16	from	from	ADP
ejpam-4858	147	17	x	x	PUNCT
ejpam-4858	147	18	into	into	ADP
ejpam-4858	147	19	y	y	PROPN
ejpam-4858	147	20	,	,	PUNCT
ejpam-4858	147	21	and	and	CCONJ
ejpam-4858	147	22	we	we	PRON
ejpam-4858	147	23	always	always	ADV
ejpam-4858	147	24	assume	assume	VERB
ejpam-4858	147	25	that	that	SCONJ
ejpam-4858	147	26	f	f	PROPN
ejpam-4858	147	27	(	(	PUNCT
ejpam-4858	147	28	x	x	X
ejpam-4858	147	29	)	)	PUNCT
ejpam-4858	147	30	̸=	̸=	NOUN
ejpam-4858	147	31	∅	∅	NOUN
ejpam-4858	147	32	for	for	ADP
ejpam-4858	147	33	all	all	PRON
ejpam-4858	147	34	x	x	SYM
ejpam-4858	147	35	∈	∈	ADJ
ejpam-4858	147	36	x.	x.	NOUN
ejpam-4858	147	37	for	for	ADP
ejpam-4858	147	38	a	a	DET
ejpam-4858	147	39	multifunction	multifunction	NOUN
ejpam-4858	147	40	f	f	NOUN
ejpam-4858	148	1	:	:	PUNCT
ejpam-4858	148	2	x	x	X
ejpam-4858	148	3	→	→	SYM
ejpam-4858	148	4	y	y	PROPN
ejpam-4858	148	5	,	,	PUNCT
ejpam-4858	148	6	following	follow	VERB
ejpam-4858	148	7	[	[	X
ejpam-4858	148	8	3	3	X
ejpam-4858	148	9	]	]	PUNCT
ejpam-4858	148	10	we	we	PRON
ejpam-4858	148	11	shall	shall	AUX
ejpam-4858	148	12	denote	denote	VERB
ejpam-4858	148	13	the	the	DET
ejpam-4858	148	14	upper	upper	ADJ
ejpam-4858	148	15	and	and	CCONJ
ejpam-4858	148	16	lower	low	ADJ
ejpam-4858	148	17	inverse	inverse	NOUN
ejpam-4858	148	18	of	of	ADP
ejpam-4858	148	19	a	a	DET
ejpam-4858	148	20	set	set	NOUN
ejpam-4858	148	21	b	b	PROPN
ejpam-4858	148	22	of	of	ADP
ejpam-4858	148	23	y	y	PROPN
ejpam-4858	148	24	by	by	ADP
ejpam-4858	148	25	f+(b	f+(b	NOUN
ejpam-4858	148	26	)	)	PUNCT
ejpam-4858	148	27	and	and	CCONJ
ejpam-4858	148	28	f−(b	f−(b	NOUN
ejpam-4858	148	29	)	)	PUNCT
ejpam-4858	148	30	,	,	PUNCT
ejpam-4858	148	31	respectively	respectively	ADV
ejpam-4858	148	32	,	,	PUNCT
ejpam-4858	148	33	that	that	ADV
ejpam-4858	148	34	is	is	ADV
ejpam-4858	148	35	,	,	PUNCT
ejpam-4858	148	36	f+(b	f+(b	NOUN
ejpam-4858	148	37	)	)	PUNCT
ejpam-4858	148	38	=	=	PRON
ejpam-4858	149	1	{	{	PUNCT
ejpam-4858	149	2	x	x	PUNCT
ejpam-4858	149	3	∈	∈	PROPN
ejpam-4858	149	4	x	x	INTJ
ejpam-4858	150	1	|	|	NOUN
ejpam-4858	150	2	f	f	X
ejpam-4858	150	3	(	(	PUNCT
ejpam-4858	150	4	x	x	NOUN
ejpam-4858	150	5	)	)	PUNCT
ejpam-4858	150	6	⊆	⊆	NUM
ejpam-4858	150	7	b	b	NOUN
ejpam-4858	150	8	}	}	PUNCT
ejpam-4858	150	9	and	and	CCONJ
ejpam-4858	150	10	f−(b	f−(b	PROPN
ejpam-4858	150	11	)	)	PUNCT
ejpam-4858	150	12	=	=	PRON
ejpam-4858	151	1	{	{	PUNCT
ejpam-4858	151	2	x	x	PUNCT
ejpam-4858	151	3	∈	∈	PROPN
ejpam-4858	151	4	x	x	INTJ
ejpam-4858	152	1	|	|	NOUN
ejpam-4858	152	2	f	f	X
ejpam-4858	152	3	(	(	PUNCT
ejpam-4858	152	4	x	x	NOUN
ejpam-4858	152	5	)	)	PUNCT
ejpam-4858	152	6	∩b	∩b	NOUN
ejpam-4858	152	7	̸=	̸=	PROPN
ejpam-4858	152	8	∅	∅	NOUN
ejpam-4858	152	9	}	}	PUNCT
ejpam-4858	152	10	.	.	PUNCT
ejpam-4858	153	1	in	in	ADP
ejpam-4858	153	2	particular	particular	ADJ
ejpam-4858	153	3	,	,	PUNCT
ejpam-4858	153	4	f−(y	f−(y	NOUN
ejpam-4858	153	5	)	)	PUNCT
ejpam-4858	153	6	=	=	SYM
ejpam-4858	154	1	{	{	PUNCT
ejpam-4858	154	2	x	x	PUNCT
ejpam-4858	154	3	∈	∈	PROPN
ejpam-4858	154	4	x	x	INTJ
ejpam-4858	155	1	|	|	ADV
ejpam-4858	155	2	y	y	PROPN
ejpam-4858	155	3	∈	∈	PROPN
ejpam-4858	155	4	f	f	X
ejpam-4858	155	5	(	(	PUNCT
ejpam-4858	155	6	x	x	NOUN
ejpam-4858	155	7	)	)	PUNCT
ejpam-4858	155	8	}	}	PUNCT
ejpam-4858	155	9	for	for	ADP
ejpam-4858	155	10	each	each	DET
ejpam-4858	155	11	point	point	NOUN
ejpam-4858	155	12	y	y	PROPN
ejpam-4858	155	13	∈	∈	PROPN
ejpam-4858	155	14	y	y	PROPN
ejpam-4858	155	15	.	.	PUNCT
ejpam-4858	156	1	for	for	ADP
ejpam-4858	156	2	each	each	DET
ejpam-4858	156	3	a	a	DET
ejpam-4858	156	4	⊆	⊆	NUM
ejpam-4858	156	5	x	x	SYM
ejpam-4858	156	6	,	,	PUNCT
ejpam-4858	156	7	f	f	PROPN
ejpam-4858	156	8	(	(	PUNCT
ejpam-4858	156	9	a	a	NOUN
ejpam-4858	156	10	)	)	PUNCT
ejpam-4858	156	11	=	=	SYM
ejpam-4858	156	12	∪x∈af	∪x∈af	NOUN
ejpam-4858	156	13	(	(	PUNCT
ejpam-4858	156	14	x	x	NOUN
ejpam-4858	156	15	)	)	PUNCT
ejpam-4858	156	16	.	.	PUNCT
ejpam-4858	157	1	3	3	X
ejpam-4858	157	2	.	.	X
ejpam-4858	157	3	upper	upper	ADJ
ejpam-4858	157	4	and	and	CCONJ
ejpam-4858	157	5	lower	lower	VERB
ejpam-4858	157	6	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	157	7	multifunctions	multifunction	NOUN
ejpam-4858	157	8	in	in	ADP
ejpam-4858	157	9	this	this	DET
ejpam-4858	157	10	section	section	NOUN
ejpam-4858	157	11	,	,	PUNCT
ejpam-4858	157	12	we	we	PRON
ejpam-4858	157	13	introduce	introduce	VERB
ejpam-4858	157	14	the	the	DET
ejpam-4858	157	15	notions	notion	NOUN
ejpam-4858	157	16	of	of	ADP
ejpam-4858	157	17	upper	upper	ADJ
ejpam-4858	157	18	and	and	CCONJ
ejpam-4858	157	19	lower	low	ADJ
ejpam-4858	157	20	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	157	21	multifunctions	multifunction	NOUN
ejpam-4858	157	22	.	.	PUNCT
ejpam-4858	158	1	moreover	moreover	ADV
ejpam-4858	158	2	,	,	PUNCT
ejpam-4858	158	3	several	several	ADJ
ejpam-4858	158	4	characterizations	characterization	NOUN
ejpam-4858	158	5	of	of	ADP
ejpam-4858	158	6	upper	upper	ADJ
ejpam-4858	158	7	and	and	CCONJ
ejpam-4858	158	8	lower	low	ADJ
ejpam-4858	158	9	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	158	10	multifunctions	multifunction	NOUN
ejpam-4858	158	11	are	be	AUX
ejpam-4858	158	12	discussed	discuss	VERB
ejpam-4858	158	13	.	.	PUNCT
ejpam-4858	159	1	definition	definition	NOUN
ejpam-4858	159	2	1	1	NUM
ejpam-4858	159	3	.	.	PUNCT
ejpam-4858	160	1	a	a	DET
ejpam-4858	160	2	multifunction	multifunction	NOUN
ejpam-4858	160	3	f	f	NOUN
ejpam-4858	160	4	:	:	PUNCT
ejpam-4858	160	5	(	(	PUNCT
ejpam-4858	160	6	x	x	X
ejpam-4858	160	7	,	,	PUNCT
ejpam-4858	160	8	τ	τ	PROPN
ejpam-4858	160	9	,	,	PUNCT
ejpam-4858	160	10	i	i	NOUN
ejpam-4858	160	11	)	)	PUNCT
ejpam-4858	160	12	→	→	PUNCT
ejpam-4858	160	13	(	(	PUNCT
ejpam-4858	160	14	y	y	PROPN
ejpam-4858	160	15	,	,	PUNCT
ejpam-4858	160	16	σ	σ	PROPN
ejpam-4858	160	17	,	,	PUNCT
ejpam-4858	160	18	j	j	PROPN
ejpam-4858	160	19	)	)	PUNCT
ejpam-4858	160	20	is	be	AUX
ejpam-4858	160	21	said	say	VERB
ejpam-4858	160	22	to	to	PART
ejpam-4858	160	23	be	be	AUX
ejpam-4858	160	24	:	:	PUNCT
ejpam-4858	160	25	(	(	PUNCT
ejpam-4858	160	26	1	1	X
ejpam-4858	160	27	)	)	PUNCT
ejpam-4858	160	28	upper	upper	ADJ
ejpam-4858	160	29	α-⋆-continuous	α-⋆-continuous	X
ejpam-4858	160	30	at	at	ADP
ejpam-4858	160	31	a	a	DET
ejpam-4858	160	32	point	point	NOUN
ejpam-4858	160	33	x	x	PUNCT
ejpam-4858	160	34	of	of	ADP
ejpam-4858	160	35	x	x	PRON
ejpam-4858	160	36	if	if	SCONJ
ejpam-4858	160	37	,	,	PUNCT
ejpam-4858	160	38	for	for	SCONJ
ejpam-4858	160	39	each	each	DET
ejpam-4858	160	40	⋆-open	⋆-open	ADV
ejpam-4858	160	41	set	set	VERB
ejpam-4858	160	42	v	v	ADP
ejpam-4858	160	43	such	such	ADJ
ejpam-4858	160	44	that	that	SCONJ
ejpam-4858	160	45	f	f	PROPN
ejpam-4858	160	46	(	(	PUNCT
ejpam-4858	160	47	x	x	X
ejpam-4858	160	48	)	)	PUNCT
ejpam-4858	160	49	⊆	⊆	NUM
ejpam-4858	160	50	v	v	NOUN
ejpam-4858	160	51	,	,	PUNCT
ejpam-4858	160	52	there	there	PRON
ejpam-4858	160	53	exists	exist	VERB
ejpam-4858	160	54	an	an	DET
ejpam-4858	160	55	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	160	56	set	set	VERB
ejpam-4858	160	57	u	u	NOUN
ejpam-4858	160	58	of	of	ADP
ejpam-4858	160	59	x	x	PUNCT
ejpam-4858	160	60	containing	contain	VERB
ejpam-4858	160	61	x	x	PUNCT
ejpam-4858	160	62	such	such	ADJ
ejpam-4858	160	63	that	that	SCONJ
ejpam-4858	160	64	f	f	PROPN
ejpam-4858	160	65	(	(	PUNCT
ejpam-4858	160	66	u	u	NOUN
ejpam-4858	160	67	)	)	PUNCT
ejpam-4858	160	68	⊆	⊆	NUM
ejpam-4858	160	69	v	v	NOUN
ejpam-4858	160	70	;	;	PUNCT
ejpam-4858	160	71	(	(	PUNCT
ejpam-4858	160	72	2	2	X
ejpam-4858	160	73	)	)	PUNCT
ejpam-4858	160	74	lower	low	ADJ
ejpam-4858	160	75	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	160	76	at	at	ADP
ejpam-4858	160	77	a	a	DET
ejpam-4858	160	78	point	point	NOUN
ejpam-4858	160	79	x	x	PUNCT
ejpam-4858	160	80	of	of	ADP
ejpam-4858	160	81	x	x	PRON
ejpam-4858	160	82	if	if	SCONJ
ejpam-4858	160	83	,	,	PUNCT
ejpam-4858	160	84	for	for	SCONJ
ejpam-4858	160	85	each	each	DET
ejpam-4858	160	86	⋆-open	⋆-open	ADV
ejpam-4858	160	87	set	set	VERB
ejpam-4858	160	88	v	v	ADP
ejpam-4858	160	89	such	such	ADJ
ejpam-4858	160	90	that	that	SCONJ
ejpam-4858	160	91	f	f	PROPN
ejpam-4858	160	92	(	(	PUNCT
ejpam-4858	160	93	x	x	NOUN
ejpam-4858	160	94	)	)	PUNCT
ejpam-4858	160	95	∩	∩	NOUN
ejpam-4858	160	96	v	v	ADP
ejpam-4858	160	97	̸=	̸=	PROPN
ejpam-4858	160	98	∅	∅	NOUN
ejpam-4858	160	99	,	,	PUNCT
ejpam-4858	160	100	there	there	PRON
ejpam-4858	160	101	exists	exist	VERB
ejpam-4858	160	102	an	an	DET
ejpam-4858	160	103	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	160	104	set	set	VERB
ejpam-4858	160	105	u	u	NOUN
ejpam-4858	160	106	of	of	ADP
ejpam-4858	160	107	x	x	PUNCT
ejpam-4858	160	108	containing	contain	VERB
ejpam-4858	160	109	x	x	PUNCT
ejpam-4858	160	110	such	such	ADJ
ejpam-4858	160	111	that	that	SCONJ
ejpam-4858	160	112	f	f	PROPN
ejpam-4858	160	113	(	(	PUNCT
ejpam-4858	160	114	z	z	NOUN
ejpam-4858	160	115	)	)	PUNCT
ejpam-4858	160	116	∩	∩	NOUN
ejpam-4858	160	117	v	v	ADP
ejpam-4858	160	118	̸=	̸=	PROPN
ejpam-4858	160	119	∅	∅	NOUN
ejpam-4858	160	120	for	for	ADP
ejpam-4858	160	121	each	each	DET
ejpam-4858	160	122	z	z	NOUN
ejpam-4858	160	123	∈	∈	PROPN
ejpam-4858	160	124	u	u	NOUN
ejpam-4858	160	125	;	;	PUNCT
ejpam-4858	160	126	(	(	PUNCT
ejpam-4858	160	127	3	3	X
ejpam-4858	160	128	)	)	PUNCT
ejpam-4858	160	129	upper	upper	ADJ
ejpam-4858	160	130	(	(	PUNCT
ejpam-4858	160	131	resp	resp	NOUN
ejpam-4858	160	132	.	.	PUNCT
ejpam-4858	161	1	lower	low	ADJ
ejpam-4858	161	2	)	)	PUNCT
ejpam-4858	161	3	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	161	4	if	if	SCONJ
ejpam-4858	161	5	f	f	PROPN
ejpam-4858	161	6	is	be	AUX
ejpam-4858	161	7	upper	upper	ADJ
ejpam-4858	161	8	(	(	PUNCT
ejpam-4858	161	9	resp	resp	NOUN
ejpam-4858	161	10	.	.	PUNCT
ejpam-4858	162	1	lower	low	ADJ
ejpam-4858	162	2	)	)	PUNCT
ejpam-4858	163	1	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	163	2	at	at	ADP
ejpam-4858	163	3	each	each	DET
ejpam-4858	163	4	point	point	NOUN
ejpam-4858	163	5	of	of	ADP
ejpam-4858	163	6	x.	x.	NOUN
ejpam-4858	163	7	theorem	theorem	VERB
ejpam-4858	163	8	1	1	NUM
ejpam-4858	163	9	.	.	X
ejpam-4858	164	1	for	for	ADP
ejpam-4858	164	2	a	a	DET
ejpam-4858	164	3	multifunction	multifunction	NOUN
ejpam-4858	164	4	f	f	NOUN
ejpam-4858	164	5	:	:	PUNCT
ejpam-4858	164	6	(	(	PUNCT
ejpam-4858	164	7	x	x	X
ejpam-4858	164	8	,	,	PUNCT
ejpam-4858	164	9	τ	τ	PROPN
ejpam-4858	164	10	,	,	PUNCT
ejpam-4858	164	11	i	i	NOUN
ejpam-4858	164	12	)	)	PUNCT
ejpam-4858	164	13	→	→	PUNCT
ejpam-4858	164	14	(	(	PUNCT
ejpam-4858	164	15	y	y	PROPN
ejpam-4858	164	16	,	,	PUNCT
ejpam-4858	164	17	σ	σ	PROPN
ejpam-4858	164	18	,	,	PUNCT
ejpam-4858	164	19	j	j	PROPN
ejpam-4858	164	20	)	)	PUNCT
ejpam-4858	164	21	,	,	PUNCT
ejpam-4858	164	22	the	the	DET
ejpam-4858	164	23	following	follow	VERB
ejpam-4858	164	24	properties	property	NOUN
ejpam-4858	164	25	are	be	AUX
ejpam-4858	164	26	equivalent	equivalent	ADJ
ejpam-4858	164	27	:	:	PUNCT
ejpam-4858	164	28	(	(	PUNCT
ejpam-4858	164	29	1	1	X
ejpam-4858	164	30	)	)	PUNCT
ejpam-4858	164	31	f	f	PROPN
ejpam-4858	164	32	is	be	AUX
ejpam-4858	164	33	upper	upper	ADJ
ejpam-4858	164	34	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	164	35	at	at	ADP
ejpam-4858	164	36	x	x	X
ejpam-4858	164	37	∈	∈	PROPN
ejpam-4858	164	38	x	x	X
ejpam-4858	164	39	;	;	PUNCT
ejpam-4858	164	40	(	(	PUNCT
ejpam-4858	164	41	2	2	X
ejpam-4858	164	42	)	)	PUNCT
ejpam-4858	164	43	x	x	SYM
ejpam-4858	164	44	∈	∈	PROPN
ejpam-4858	164	45	scli	scli	NOUN
ejpam-4858	164	46	(	(	PUNCT
ejpam-4858	164	47	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-4858	164	48	)	)	PUNCT
ejpam-4858	164	49	)	)	PUNCT
ejpam-4858	164	50	)	)	PUNCT
ejpam-4858	164	51	for	for	SCONJ
ejpam-4858	164	52	every	every	DET
ejpam-4858	164	53	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	164	54	set	set	VERB
ejpam-4858	164	55	v	v	NOUN
ejpam-4858	164	56	of	of	ADP
ejpam-4858	164	57	y	y	PROPN
ejpam-4858	164	58	containing	contain	VERB
ejpam-4858	164	59	f	f	PROPN
ejpam-4858	164	60	(	(	PUNCT
ejpam-4858	164	61	x	x	NOUN
ejpam-4858	164	62	)	)	PUNCT
ejpam-4858	164	63	;	;	PUNCT
ejpam-4858	164	64	(	(	PUNCT
ejpam-4858	164	65	3	3	X
ejpam-4858	164	66	)	)	PUNCT
ejpam-4858	164	67	x	x	SYM
ejpam-4858	164	68	∈	∈	PROPN
ejpam-4858	164	69	⋆αint(f+(v	⋆αint(f+(v	NOUN
ejpam-4858	164	70	)	)	PUNCT
ejpam-4858	164	71	)	)	PUNCT
ejpam-4858	164	72	for	for	SCONJ
ejpam-4858	164	73	every	every	DET
ejpam-4858	164	74	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	164	75	set	set	VERB
ejpam-4858	164	76	v	v	NOUN
ejpam-4858	164	77	of	of	ADP
ejpam-4858	164	78	y	y	PROPN
ejpam-4858	164	79	containing	contain	VERB
ejpam-4858	164	80	f	f	PROPN
ejpam-4858	164	81	(	(	PUNCT
ejpam-4858	164	82	x	x	NOUN
ejpam-4858	164	83	)	)	PUNCT
ejpam-4858	164	84	.	.	PUNCT
ejpam-4858	165	1	proof	proof	NOUN
ejpam-4858	165	2	.	.	PUNCT
ejpam-4858	166	1	(	(	PUNCT
ejpam-4858	166	2	1	1	X
ejpam-4858	166	3	)	)	PUNCT
ejpam-4858	166	4	⇒	⇒	NOUN
ejpam-4858	166	5	(	(	PUNCT
ejpam-4858	166	6	2	2	NUM
ejpam-4858	166	7	):	):	PUNCT
ejpam-4858	166	8	let	let	VERB
ejpam-4858	166	9	v	v	PART
ejpam-4858	166	10	be	be	AUX
ejpam-4858	166	11	any	any	DET
ejpam-4858	166	12	⋆-open	⋆-open	ADJ
ejpam-4858	166	13	set	set	NOUN
ejpam-4858	166	14	of	of	ADP
ejpam-4858	166	15	y	y	PROPN
ejpam-4858	166	16	containing	contain	VERB
ejpam-4858	166	17	f	f	PROPN
ejpam-4858	166	18	(	(	PUNCT
ejpam-4858	166	19	x	x	NOUN
ejpam-4858	166	20	)	)	PUNCT
ejpam-4858	166	21	.	.	PUNCT
ejpam-4858	167	1	then	then	ADV
ejpam-4858	167	2	,	,	PUNCT
ejpam-4858	167	3	there	there	PRON
ejpam-4858	167	4	exists	exist	VERB
ejpam-4858	167	5	an	an	DET
ejpam-4858	167	6	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	167	7	set	set	VERB
ejpam-4858	167	8	u	u	NOUN
ejpam-4858	167	9	of	of	ADP
ejpam-4858	167	10	x	x	PUNCT
ejpam-4858	167	11	containing	contain	VERB
ejpam-4858	167	12	x	x	PUNCT
ejpam-4858	167	13	such	such	ADJ
ejpam-4858	167	14	that	that	SCONJ
ejpam-4858	167	15	f	f	PROPN
ejpam-4858	167	16	(	(	PUNCT
ejpam-4858	167	17	u	u	NOUN
ejpam-4858	167	18	)	)	PUNCT
ejpam-4858	167	19	⊆	⊆	NUM
ejpam-4858	167	20	v	v	NOUN
ejpam-4858	167	21	;	;	PUNCT
ejpam-4858	167	22	hence	hence	ADV
ejpam-4858	167	23	x	x	PART
ejpam-4858	167	24	∈	∈	PROPN
ejpam-4858	167	25	u	u	NOUN
ejpam-4858	167	26	⊆	⊆	NUM
ejpam-4858	167	27	f+(v	f+(v	NOUN
ejpam-4858	167	28	)	)	PUNCT
ejpam-4858	167	29	.	.	PUNCT
ejpam-4858	168	1	since	since	SCONJ
ejpam-4858	168	2	u	u	NOUN
ejpam-4858	168	3	is	be	AUX
ejpam-4858	168	4	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	168	5	,	,	PUNCT
ejpam-4858	168	6	by	by	ADP
ejpam-4858	168	7	lemma	lemma	PROPN
ejpam-4858	168	8	2	2	NUM
ejpam-4858	168	9	,	,	PUNCT
ejpam-4858	168	10	we	we	PRON
ejpam-4858	168	11	have	have	VERB
ejpam-4858	168	12	x	x	X
ejpam-4858	168	13	∈	∈	PROPN
ejpam-4858	168	14	u	u	NOUN
ejpam-4858	168	15	⊆	⊆	NUM
ejpam-4858	168	16	scli	scli	NOUN
ejpam-4858	168	17	(	(	PUNCT
ejpam-4858	168	18	int⋆(u	int⋆(u	NOUN
ejpam-4858	168	19	)	)	PUNCT
ejpam-4858	168	20	)	)	PUNCT
ejpam-4858	169	1	⊆	⊆	NUM
ejpam-4858	169	2	scli	scli	NOUN
ejpam-4858	169	3	(	(	PUNCT
ejpam-4858	169	4	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-4858	169	5	)	)	PUNCT
ejpam-4858	169	6	)	)	PUNCT
ejpam-4858	169	7	)	)	PUNCT
ejpam-4858	169	8	.	.	PUNCT
ejpam-4858	170	1	(	(	PUNCT
ejpam-4858	170	2	2	2	X
ejpam-4858	170	3	)	)	PUNCT
ejpam-4858	170	4	⇒	⇒	NOUN
ejpam-4858	170	5	(	(	PUNCT
ejpam-4858	170	6	3	3	NUM
ejpam-4858	170	7	):	):	PUNCT
ejpam-4858	170	8	let	let	VERB
ejpam-4858	170	9	v	v	PART
ejpam-4858	170	10	be	be	AUX
ejpam-4858	170	11	any	any	DET
ejpam-4858	170	12	⋆-open	⋆-open	ADJ
ejpam-4858	170	13	set	set	NOUN
ejpam-4858	170	14	of	of	ADP
ejpam-4858	170	15	y	y	PROPN
ejpam-4858	170	16	containing	contain	VERB
ejpam-4858	170	17	f	f	PROPN
ejpam-4858	170	18	(	(	PUNCT
ejpam-4858	170	19	x	x	NOUN
ejpam-4858	170	20	)	)	PUNCT
ejpam-4858	170	21	.	.	PUNCT
ejpam-4858	171	1	then	then	ADV
ejpam-4858	171	2	by	by	ADP
ejpam-4858	171	3	(	(	PUNCT
ejpam-4858	171	4	2	2	NUM
ejpam-4858	171	5	)	)	PUNCT
ejpam-4858	171	6	,	,	PUNCT
ejpam-4858	171	7	we	we	PRON
ejpam-4858	171	8	have	have	VERB
ejpam-4858	171	9	x	x	PART
ejpam-4858	171	10	∈	∈	PROPN
ejpam-4858	171	11	scli	scli	NOUN
ejpam-4858	171	12	(	(	PUNCT
ejpam-4858	171	13	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-4858	171	14	)	)	PUNCT
ejpam-4858	171	15	)	)	PUNCT
ejpam-4858	171	16	)	)	PUNCT
ejpam-4858	171	17	and	and	CCONJ
ejpam-4858	171	18	by	by	ADP
ejpam-4858	171	19	lemma	lemma	PROPN
ejpam-4858	171	20	1(2	1(2	NUM
ejpam-4858	171	21	)	)	PUNCT
ejpam-4858	171	22	,	,	PUNCT
ejpam-4858	171	23	x	x	PUNCT
ejpam-4858	171	24	∈	∈	NOUN
ejpam-4858	171	25	int⋆(cl(int⋆(f+(v	int⋆(cl(int⋆(f+(v	NOUN
ejpam-4858	171	26	)	)	PUNCT
ejpam-4858	171	27	)	)	PUNCT
ejpam-4858	171	28	)	)	PUNCT
ejpam-4858	171	29	)	)	PUNCT
ejpam-4858	171	30	.	.	PUNCT
ejpam-4858	172	1	thus	thus	ADV
ejpam-4858	172	2	,	,	PUNCT
ejpam-4858	172	3	by	by	ADP
ejpam-4858	172	4	lemma	lemma	PROPN
ejpam-4858	172	5	3(4	3(4	NUM
ejpam-4858	172	6	)	)	PUNCT
ejpam-4858	172	7	,	,	PUNCT
ejpam-4858	172	8	x	x	PUNCT
ejpam-4858	172	9	∈	∈	PROPN
ejpam-4858	172	10	⋆αint(f+(v	⋆αint(f+(v	NOUN
ejpam-4858	172	11	)	)	PUNCT
ejpam-4858	172	12	)	)	PUNCT
ejpam-4858	172	13	.	.	PUNCT
ejpam-4858	173	1	(	(	PUNCT
ejpam-4858	173	2	3	3	X
ejpam-4858	173	3	)	)	PUNCT
ejpam-4858	173	4	⇒	⇒	NOUN
ejpam-4858	173	5	(	(	PUNCT
ejpam-4858	173	6	1	1	NUM
ejpam-4858	173	7	):	):	PUNCT
ejpam-4858	173	8	let	let	VERB
ejpam-4858	173	9	v	v	PART
ejpam-4858	173	10	be	be	AUX
ejpam-4858	173	11	any	any	DET
ejpam-4858	173	12	⋆-open	⋆-open	ADJ
ejpam-4858	173	13	set	set	NOUN
ejpam-4858	173	14	of	of	ADP
ejpam-4858	173	15	y	y	PROPN
ejpam-4858	173	16	containing	contain	VERB
ejpam-4858	173	17	f	f	PROPN
ejpam-4858	173	18	(	(	PUNCT
ejpam-4858	173	19	x	x	NOUN
ejpam-4858	173	20	)	)	PUNCT
ejpam-4858	173	21	.	.	PUNCT
ejpam-4858	174	1	by	by	ADP
ejpam-4858	174	2	(	(	PUNCT
ejpam-4858	174	3	3	3	NUM
ejpam-4858	174	4	)	)	PUNCT
ejpam-4858	174	5	,	,	PUNCT
ejpam-4858	174	6	x	x	PUNCT
ejpam-4858	174	7	∈	∈	PROPN
ejpam-4858	174	8	⋆αint(f+(v	⋆αint(f+(v	NOUN
ejpam-4858	174	9	)	)	PUNCT
ejpam-4858	174	10	)	)	PUNCT
ejpam-4858	175	1	and	and	CCONJ
ejpam-4858	175	2	so	so	ADV
ejpam-4858	175	3	there	there	PRON
ejpam-4858	175	4	exists	exist	VERB
ejpam-4858	175	5	an	an	DET
ejpam-4858	175	6	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	175	7	set	set	VERB
ejpam-4858	175	8	u	u	NOUN
ejpam-4858	175	9	of	of	ADP
ejpam-4858	175	10	x	x	PUNCT
ejpam-4858	175	11	containing	contain	VERB
ejpam-4858	175	12	x	x	PUNCT
ejpam-4858	175	13	such	such	ADJ
ejpam-4858	175	14	that	that	SCONJ
ejpam-4858	175	15	u	u	NOUN
ejpam-4858	175	16	⊆	⊆	NUM
ejpam-4858	175	17	f+(v	f+(v	NOUN
ejpam-4858	175	18	)	)	PUNCT
ejpam-4858	175	19	;	;	PUNCT
ejpam-4858	175	20	hence	hence	ADV
ejpam-4858	175	21	f	f	PROPN
ejpam-4858	175	22	(	(	PUNCT
ejpam-4858	175	23	u	u	NOUN
ejpam-4858	175	24	)	)	PUNCT
ejpam-4858	175	25	⊆	⊆	NUM
ejpam-4858	175	26	v	v	NOUN
ejpam-4858	175	27	.	.	PUNCT
ejpam-4858	176	1	this	this	PRON
ejpam-4858	176	2	shows	show	VERB
ejpam-4858	176	3	that	that	SCONJ
ejpam-4858	176	4	f	f	PROPN
ejpam-4858	176	5	is	be	AUX
ejpam-4858	176	6	upper	upper	ADJ
ejpam-4858	176	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	176	8	at	at	ADP
ejpam-4858	176	9	x.	x.	PROPN
ejpam-4858	176	10	c.	c.	PROPN
ejpam-4858	176	11	boonpok	boonpok	PROPN
ejpam-4858	176	12	,	,	PUNCT
ejpam-4858	176	13	j.	j.	PROPN
ejpam-4858	176	14	khampakdee	khampakdee	PROPN
ejpam-4858	176	15	/	/	PUNCT
ejpam-4858	176	16	eur	eur	PROPN
ejpam-4858	176	17	.	.	PUNCT
ejpam-4858	177	1	j.	j.	PROPN
ejpam-4858	177	2	pure	pure	PROPN
ejpam-4858	177	3	appl	appl	PROPN
ejpam-4858	177	4	.	.	PROPN
ejpam-4858	177	5	math	math	PROPN
ejpam-4858	177	6	,	,	PUNCT
ejpam-4858	177	7	17	17	NUM
ejpam-4858	177	8	(	(	PUNCT
ejpam-4858	177	9	1	1	NUM
ejpam-4858	177	10	)	)	PUNCT
ejpam-4858	177	11	(	(	PUNCT
ejpam-4858	177	12	2024	2024	NUM
ejpam-4858	177	13	)	)	PUNCT
ejpam-4858	177	14	,	,	PUNCT
ejpam-4858	177	15	201	201	NUM
ejpam-4858	177	16	-	-	SYM
ejpam-4858	177	17	211	211	NUM
ejpam-4858	177	18	206	206	NUM
ejpam-4858	177	19	theorem	theorem	NOUN
ejpam-4858	177	20	2	2	NUM
ejpam-4858	177	21	.	.	X
ejpam-4858	177	22	for	for	ADP
ejpam-4858	177	23	a	a	DET
ejpam-4858	177	24	multifunction	multifunction	NOUN
ejpam-4858	177	25	f	f	NOUN
ejpam-4858	177	26	:	:	PUNCT
ejpam-4858	177	27	(	(	PUNCT
ejpam-4858	177	28	x	x	X
ejpam-4858	177	29	,	,	PUNCT
ejpam-4858	177	30	τ	τ	PROPN
ejpam-4858	177	31	,	,	PUNCT
ejpam-4858	177	32	i	i	NOUN
ejpam-4858	177	33	)	)	PUNCT
ejpam-4858	177	34	→	→	PUNCT
ejpam-4858	177	35	(	(	PUNCT
ejpam-4858	177	36	y	y	PROPN
ejpam-4858	177	37	,	,	PUNCT
ejpam-4858	177	38	σ	σ	PROPN
ejpam-4858	177	39	,	,	PUNCT
ejpam-4858	177	40	j	j	PROPN
ejpam-4858	177	41	)	)	PUNCT
ejpam-4858	177	42	,	,	PUNCT
ejpam-4858	177	43	the	the	DET
ejpam-4858	177	44	following	follow	VERB
ejpam-4858	177	45	properties	property	NOUN
ejpam-4858	177	46	are	be	AUX
ejpam-4858	177	47	equivalent	equivalent	ADJ
ejpam-4858	177	48	:	:	PUNCT
ejpam-4858	177	49	(	(	PUNCT
ejpam-4858	177	50	1	1	X
ejpam-4858	177	51	)	)	PUNCT
ejpam-4858	177	52	f	f	PROPN
ejpam-4858	177	53	is	be	AUX
ejpam-4858	177	54	lower	low	ADJ
ejpam-4858	177	55	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	177	56	at	at	ADP
ejpam-4858	177	57	x	x	X
ejpam-4858	177	58	∈	∈	PROPN
ejpam-4858	177	59	x	x	X
ejpam-4858	177	60	;	;	PUNCT
ejpam-4858	177	61	(	(	PUNCT
ejpam-4858	177	62	2	2	X
ejpam-4858	177	63	)	)	PUNCT
ejpam-4858	177	64	x	x	SYM
ejpam-4858	177	65	∈	∈	PROPN
ejpam-4858	177	66	scli	scli	NOUN
ejpam-4858	177	67	(	(	PUNCT
ejpam-4858	177	68	int⋆(f−(v	int⋆(f−(v	PROPN
ejpam-4858	177	69	)	)	PUNCT
ejpam-4858	177	70	)	)	PUNCT
ejpam-4858	177	71	)	)	PUNCT
ejpam-4858	177	72	for	for	SCONJ
ejpam-4858	177	73	every	every	DET
ejpam-4858	177	74	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	177	75	set	set	VERB
ejpam-4858	177	76	v	v	NOUN
ejpam-4858	177	77	of	of	ADP
ejpam-4858	177	78	y	y	PRON
ejpam-4858	177	79	such	such	ADJ
ejpam-4858	177	80	that	that	SCONJ
ejpam-4858	177	81	f	f	PROPN
ejpam-4858	177	82	(	(	PUNCT
ejpam-4858	177	83	x	x	NOUN
ejpam-4858	177	84	)	)	PUNCT
ejpam-4858	177	85	∩	∩	NOUN
ejpam-4858	177	86	v	v	ADP
ejpam-4858	177	87	̸=	̸=	PROPN
ejpam-4858	177	88	∅	∅	NOUN
ejpam-4858	177	89	;	;	PUNCT
ejpam-4858	177	90	(	(	PUNCT
ejpam-4858	177	91	3	3	X
ejpam-4858	177	92	)	)	PUNCT
ejpam-4858	177	93	x	x	SYM
ejpam-4858	177	94	∈	∈	PROPN
ejpam-4858	177	95	⋆αint(f−(v	⋆αint(f−(v	PROPN
ejpam-4858	177	96	)	)	PUNCT
ejpam-4858	177	97	)	)	PUNCT
ejpam-4858	177	98	for	for	SCONJ
ejpam-4858	177	99	every	every	DET
ejpam-4858	177	100	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	177	101	set	set	VERB
ejpam-4858	177	102	v	v	NOUN
ejpam-4858	177	103	of	of	ADP
ejpam-4858	177	104	y	y	PRON
ejpam-4858	177	105	such	such	ADJ
ejpam-4858	177	106	that	that	SCONJ
ejpam-4858	177	107	f	f	PROPN
ejpam-4858	177	108	(	(	PUNCT
ejpam-4858	177	109	x	x	NOUN
ejpam-4858	177	110	)	)	PUNCT
ejpam-4858	177	111	∩	∩	NOUN
ejpam-4858	177	112	v	v	ADP
ejpam-4858	177	113	̸=	̸=	PROPN
ejpam-4858	177	114	∅.	∅.	ADP
ejpam-4858	177	115	proof	proof	NOUN
ejpam-4858	177	116	.	.	PUNCT
ejpam-4858	178	1	the	the	DET
ejpam-4858	178	2	proof	proof	NOUN
ejpam-4858	178	3	is	be	AUX
ejpam-4858	178	4	similar	similar	ADJ
ejpam-4858	178	5	to	to	ADP
ejpam-4858	178	6	that	that	PRON
ejpam-4858	178	7	of	of	ADP
ejpam-4858	178	8	theorem	theorem	NOUN
ejpam-4858	178	9	1	1	NUM
ejpam-4858	178	10	.	.	PUNCT
ejpam-4858	178	11	definition	definition	NOUN
ejpam-4858	178	12	2	2	NUM
ejpam-4858	178	13	.	.	PUNCT
ejpam-4858	178	14	a	a	DET
ejpam-4858	178	15	subset	subset	NOUN
ejpam-4858	178	16	n	n	NOUN
ejpam-4858	178	17	of	of	ADP
ejpam-4858	178	18	an	an	DET
ejpam-4858	178	19	ideal	ideal	ADJ
ejpam-4858	178	20	topological	topological	ADJ
ejpam-4858	178	21	space	space	NOUN
ejpam-4858	178	22	(	(	PUNCT
ejpam-4858	178	23	x	x	X
ejpam-4858	178	24	,	,	PUNCT
ejpam-4858	178	25	τ	τ	PROPN
ejpam-4858	178	26	,	,	PUNCT
ejpam-4858	178	27	i	i	PROPN
ejpam-4858	178	28	)	)	PUNCT
ejpam-4858	178	29	is	be	AUX
ejpam-4858	178	30	said	say	VERB
ejpam-4858	178	31	to	to	PART
ejpam-4858	178	32	be	be	AUX
ejpam-4858	178	33	a	a	DET
ejpam-4858	178	34	⋆neighbourhood	⋆neighbourhood	PROPN
ejpam-4858	178	35	(	(	PUNCT
ejpam-4858	178	36	resp	resp	NOUN
ejpam-4858	178	37	.	.	PUNCT
ejpam-4858	179	1	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	179	2	)	)	PUNCT
ejpam-4858	180	1	of	of	ADP
ejpam-4858	180	2	x	x	SYM
ejpam-4858	180	3	∈	∈	PROPN
ejpam-4858	180	4	x	x	INTJ
ejpam-4858	180	5	if	if	SCONJ
ejpam-4858	180	6	there	there	PRON
ejpam-4858	180	7	exists	exist	VERB
ejpam-4858	180	8	a	a	DET
ejpam-4858	180	9	⋆-open	⋆-open	ADJ
ejpam-4858	180	10	(	(	PUNCT
ejpam-4858	180	11	resp	resp	NOUN
ejpam-4858	180	12	.	.	PUNCT
ejpam-4858	180	13	α-⋆-open	α-⋆-open	VERB
ejpam-4858	180	14	)	)	PUNCT
ejpam-4858	180	15	set	set	VERB
ejpam-4858	180	16	v	v	NOUN
ejpam-4858	180	17	of	of	ADP
ejpam-4858	180	18	x	x	PUNCT
ejpam-4858	181	1	such	such	ADJ
ejpam-4858	181	2	that	that	SCONJ
ejpam-4858	181	3	x	x	SYM
ejpam-4858	181	4	∈	∈	NOUN
ejpam-4858	181	5	v	v	ADP
ejpam-4858	181	6	⊆	⊆	NUM
ejpam-4858	181	7	n	n	NOUN
ejpam-4858	181	8	.	.	PUNCT
ejpam-4858	182	1	theorem	theorem	NOUN
ejpam-4858	182	2	3	3	NUM
ejpam-4858	182	3	.	.	X
ejpam-4858	182	4	for	for	ADP
ejpam-4858	182	5	a	a	DET
ejpam-4858	182	6	multifunction	multifunction	NOUN
ejpam-4858	183	1	f	f	NOUN
ejpam-4858	183	2	:	:	PUNCT
ejpam-4858	183	3	(	(	PUNCT
ejpam-4858	183	4	x	x	X
ejpam-4858	183	5	,	,	PUNCT
ejpam-4858	183	6	τ	τ	PROPN
ejpam-4858	183	7	,	,	PUNCT
ejpam-4858	183	8	i	i	NOUN
ejpam-4858	183	9	)	)	PUNCT
ejpam-4858	183	10	→	→	PUNCT
ejpam-4858	183	11	(	(	PUNCT
ejpam-4858	183	12	y	y	PROPN
ejpam-4858	183	13	,	,	PUNCT
ejpam-4858	183	14	σ	σ	PROPN
ejpam-4858	183	15	,	,	PUNCT
ejpam-4858	183	16	j	j	PROPN
ejpam-4858	183	17	)	)	PUNCT
ejpam-4858	183	18	,	,	PUNCT
ejpam-4858	183	19	the	the	DET
ejpam-4858	183	20	following	follow	VERB
ejpam-4858	183	21	properties	property	NOUN
ejpam-4858	183	22	are	be	AUX
ejpam-4858	183	23	equivalent	equivalent	ADJ
ejpam-4858	183	24	:	:	PUNCT
ejpam-4858	183	25	(	(	PUNCT
ejpam-4858	183	26	1	1	X
ejpam-4858	183	27	)	)	PUNCT
ejpam-4858	183	28	f	f	PROPN
ejpam-4858	183	29	is	be	AUX
ejpam-4858	183	30	upper	upper	ADJ
ejpam-4858	183	31	α-⋆-continuous	α-⋆-continuous	X
ejpam-4858	183	32	;	;	PUNCT
ejpam-4858	183	33	(	(	PUNCT
ejpam-4858	183	34	2	2	NUM
ejpam-4858	183	35	)	)	PUNCT
ejpam-4858	183	36	f+(v	f+(v	NOUN
ejpam-4858	183	37	)	)	PUNCT
ejpam-4858	184	1	is	be	AUX
ejpam-4858	184	2	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	184	3	in	in	ADP
ejpam-4858	184	4	x	x	PUNCT
ejpam-4858	184	5	for	for	SCONJ
ejpam-4858	184	6	every	every	DET
ejpam-4858	184	7	⋆-open	⋆-open	NOUN
ejpam-4858	184	8	set	set	VERB
ejpam-4858	184	9	v	v	NOUN
ejpam-4858	184	10	of	of	ADP
ejpam-4858	184	11	y	y	PROPN
ejpam-4858	184	12	;	;	PUNCT
ejpam-4858	184	13	(	(	PUNCT
ejpam-4858	184	14	3	3	X
ejpam-4858	184	15	)	)	PUNCT
ejpam-4858	184	16	f−(k	f−(k	PROPN
ejpam-4858	184	17	)	)	PUNCT
ejpam-4858	184	18	is	be	AUX
ejpam-4858	184	19	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	184	20	in	in	ADP
ejpam-4858	184	21	x	x	PUNCT
ejpam-4858	184	22	for	for	ADP
ejpam-4858	184	23	every	every	DET
ejpam-4858	184	24	⋆-closed	⋆-close	VERB
ejpam-4858	184	25	set	set	NOUN
ejpam-4858	184	26	k	k	PROPN
ejpam-4858	184	27	of	of	ADP
ejpam-4858	184	28	y	y	PROPN
ejpam-4858	184	29	;	;	PUNCT
ejpam-4858	184	30	(	(	PUNCT
ejpam-4858	184	31	4	4	X
ejpam-4858	184	32	)	)	PUNCT
ejpam-4858	184	33	sinti	sinti	PROPN
ejpam-4858	184	34	(	(	PUNCT
ejpam-4858	184	35	cl⋆(f−(b	cl⋆(f−(b	NUM
ejpam-4858	184	36	)	)	PUNCT
ejpam-4858	184	37	)	)	PUNCT
ejpam-4858	184	38	)	)	PUNCT
ejpam-4858	185	1	⊆	⊆	NUM
ejpam-4858	185	2	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4858	185	3	)	)	PUNCT
ejpam-4858	185	4	)	)	PUNCT
ejpam-4858	185	5	for	for	ADP
ejpam-4858	185	6	every	every	DET
ejpam-4858	185	7	subset	subset	NOUN
ejpam-4858	185	8	b	b	PROPN
ejpam-4858	185	9	of	of	ADP
ejpam-4858	185	10	y	y	PROPN
ejpam-4858	185	11	;	;	PUNCT
ejpam-4858	185	12	(	(	PUNCT
ejpam-4858	185	13	5	5	X
ejpam-4858	185	14	)	)	PUNCT
ejpam-4858	185	15	⋆αcl(f−(b	⋆αcl(f−(b	NUM
ejpam-4858	185	16	)	)	PUNCT
ejpam-4858	185	17	)	)	PUNCT
ejpam-4858	185	18	⊆	⊆	NUM
ejpam-4858	185	19	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4858	185	20	)	)	PUNCT
ejpam-4858	185	21	)	)	PUNCT
ejpam-4858	185	22	for	for	ADP
ejpam-4858	185	23	every	every	DET
ejpam-4858	185	24	subset	subset	NOUN
ejpam-4858	185	25	b	b	PROPN
ejpam-4858	185	26	of	of	ADP
ejpam-4858	185	27	y	y	PROPN
ejpam-4858	185	28	;	;	PUNCT
ejpam-4858	185	29	(	(	PUNCT
ejpam-4858	185	30	6	6	NUM
ejpam-4858	185	31	)	)	PUNCT
ejpam-4858	185	32	for	for	ADP
ejpam-4858	185	33	each	each	DET
ejpam-4858	185	34	x	x	SYM
ejpam-4858	185	35	∈	∈	PROPN
ejpam-4858	185	36	x	x	X
ejpam-4858	185	37	and	and	CCONJ
ejpam-4858	185	38	each	each	DET
ejpam-4858	185	39	⋆-neighbourhood	⋆-neighbourhood	PUNCT
ejpam-4858	185	40	v	v	NOUN
ejpam-4858	185	41	of	of	ADP
ejpam-4858	185	42	f	f	PROPN
ejpam-4858	185	43	(	(	PUNCT
ejpam-4858	185	44	x	x	NOUN
ejpam-4858	185	45	)	)	PUNCT
ejpam-4858	185	46	,	,	PUNCT
ejpam-4858	185	47	f+(v	f+(v	PROPN
ejpam-4858	185	48	)	)	PUNCT
ejpam-4858	185	49	is	be	AUX
ejpam-4858	185	50	an	an	DET
ejpam-4858	185	51	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	185	52	of	of	ADP
ejpam-4858	185	53	x	x	PRON
ejpam-4858	185	54	;	;	PUNCT
ejpam-4858	185	55	(	(	PUNCT
ejpam-4858	185	56	7	7	X
ejpam-4858	185	57	)	)	PUNCT
ejpam-4858	185	58	for	for	ADP
ejpam-4858	185	59	each	each	DET
ejpam-4858	185	60	x	x	SYM
ejpam-4858	185	61	∈	∈	PROPN
ejpam-4858	185	62	x	x	X
ejpam-4858	185	63	and	and	CCONJ
ejpam-4858	185	64	each	each	DET
ejpam-4858	185	65	⋆-neighbourhood	⋆-neighbourhood	PUNCT
ejpam-4858	185	66	v	v	NOUN
ejpam-4858	185	67	of	of	ADP
ejpam-4858	185	68	f	f	PROPN
ejpam-4858	185	69	(	(	PUNCT
ejpam-4858	185	70	x	x	NOUN
ejpam-4858	185	71	)	)	PUNCT
ejpam-4858	185	72	,	,	PUNCT
ejpam-4858	185	73	there	there	PRON
ejpam-4858	185	74	exists	exist	VERB
ejpam-4858	185	75	an	an	DET
ejpam-4858	185	76	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	185	77	u	u	NOUN
ejpam-4858	185	78	of	of	ADP
ejpam-4858	185	79	x	x	SYM
ejpam-4858	185	80	such	such	ADJ
ejpam-4858	185	81	that	that	SCONJ
ejpam-4858	185	82	f	f	PROPN
ejpam-4858	185	83	(	(	PUNCT
ejpam-4858	185	84	u	u	NOUN
ejpam-4858	185	85	)	)	PUNCT
ejpam-4858	185	86	⊆	⊆	NUM
ejpam-4858	185	87	v	v	NOUN
ejpam-4858	185	88	.	.	PUNCT
ejpam-4858	186	1	proof	proof	NOUN
ejpam-4858	186	2	.	.	PUNCT
ejpam-4858	187	1	(	(	PUNCT
ejpam-4858	187	2	1	1	X
ejpam-4858	187	3	)	)	PUNCT
ejpam-4858	187	4	⇒	⇒	NOUN
ejpam-4858	187	5	(	(	PUNCT
ejpam-4858	187	6	2	2	NUM
ejpam-4858	187	7	):	):	PUNCT
ejpam-4858	187	8	let	let	VERB
ejpam-4858	187	9	v	v	PART
ejpam-4858	187	10	be	be	AUX
ejpam-4858	187	11	any	any	DET
ejpam-4858	187	12	⋆-open	⋆-open	ADJ
ejpam-4858	187	13	set	set	NOUN
ejpam-4858	187	14	of	of	ADP
ejpam-4858	187	15	y	y	PROPN
ejpam-4858	187	16	and	and	CCONJ
ejpam-4858	187	17	x	x	PROPN
ejpam-4858	187	18	∈	∈	PROPN
ejpam-4858	187	19	f+(v	f+(v	NOUN
ejpam-4858	187	20	)	)	PUNCT
ejpam-4858	187	21	.	.	PUNCT
ejpam-4858	188	1	then	then	ADV
ejpam-4858	188	2	,	,	PUNCT
ejpam-4858	188	3	f	f	PROPN
ejpam-4858	188	4	(	(	PUNCT
ejpam-4858	188	5	x	x	X
ejpam-4858	188	6	)	)	PUNCT
ejpam-4858	188	7	⊆	⊆	NUM
ejpam-4858	188	8	v	v	NOUN
ejpam-4858	188	9	.	.	PUNCT
ejpam-4858	189	1	since	since	SCONJ
ejpam-4858	189	2	f	f	PROPN
ejpam-4858	189	3	is	be	AUX
ejpam-4858	189	4	upper	upper	ADJ
ejpam-4858	189	5	α-⋆-continuous	α-⋆-continuous	X
ejpam-4858	189	6	at	at	ADP
ejpam-4858	189	7	x	x	X
ejpam-4858	189	8	,	,	PUNCT
ejpam-4858	189	9	there	there	PRON
ejpam-4858	189	10	exists	exist	VERB
ejpam-4858	189	11	an	an	DET
ejpam-4858	189	12	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	189	13	set	set	VERB
ejpam-4858	189	14	u	u	NOUN
ejpam-4858	189	15	of	of	ADP
ejpam-4858	189	16	x	x	PUNCT
ejpam-4858	189	17	containing	contain	VERB
ejpam-4858	189	18	x	x	PUNCT
ejpam-4858	189	19	such	such	ADJ
ejpam-4858	189	20	that	that	SCONJ
ejpam-4858	189	21	f	f	PROPN
ejpam-4858	189	22	(	(	PUNCT
ejpam-4858	189	23	u	u	NOUN
ejpam-4858	189	24	)	)	PUNCT
ejpam-4858	189	25	⊆	⊆	NUM
ejpam-4858	189	26	v	v	NOUN
ejpam-4858	189	27	;	;	PUNCT
ejpam-4858	189	28	hence	hence	ADV
ejpam-4858	189	29	x	x	PART
ejpam-4858	189	30	∈	∈	PROPN
ejpam-4858	189	31	u	u	NOUN
ejpam-4858	189	32	⊆	⊆	NUM
ejpam-4858	189	33	f+(v	f+(v	NOUN
ejpam-4858	189	34	)	)	PUNCT
ejpam-4858	189	35	.	.	PUNCT
ejpam-4858	190	1	by	by	ADP
ejpam-4858	190	2	lemma	lemma	PROPN
ejpam-4858	190	3	2	2	NUM
ejpam-4858	190	4	,	,	PUNCT
ejpam-4858	190	5	x	x	SYM
ejpam-4858	190	6	∈	∈	PROPN
ejpam-4858	190	7	u	u	NOUN
ejpam-4858	190	8	⊆	⊆	NUM
ejpam-4858	190	9	scli	scli	NOUN
ejpam-4858	190	10	(	(	PUNCT
ejpam-4858	190	11	int⋆(u	int⋆(u	NOUN
ejpam-4858	190	12	)	)	PUNCT
ejpam-4858	190	13	)	)	PUNCT
ejpam-4858	191	1	⊆	⊆	NUM
ejpam-4858	191	2	scli	scli	NOUN
ejpam-4858	191	3	(	(	PUNCT
ejpam-4858	191	4	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-4858	191	5	)	)	PUNCT
ejpam-4858	191	6	)	)	PUNCT
ejpam-4858	191	7	)	)	PUNCT
ejpam-4858	191	8	.	.	PUNCT
ejpam-4858	192	1	thus	thus	ADV
ejpam-4858	192	2	,	,	PUNCT
ejpam-4858	192	3	f+(v	f+(v	PROPN
ejpam-4858	192	4	)	)	PUNCT
ejpam-4858	192	5	⊆	⊆	NUM
ejpam-4858	192	6	scli	scli	NOUN
ejpam-4858	192	7	(	(	PUNCT
ejpam-4858	192	8	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-4858	192	9	)	)	PUNCT
ejpam-4858	192	10	)	)	PUNCT
ejpam-4858	192	11	)	)	PUNCT
ejpam-4858	192	12	.	.	PUNCT
ejpam-4858	193	1	it	it	PRON
ejpam-4858	193	2	follows	follow	VERB
ejpam-4858	193	3	from	from	ADP
ejpam-4858	193	4	lemma	lemma	PROPN
ejpam-4858	193	5	2	2	NUM
ejpam-4858	193	6	that	that	PRON
ejpam-4858	193	7	f+(v	f+(v	NOUN
ejpam-4858	193	8	)	)	PUNCT
ejpam-4858	193	9	is	be	AUX
ejpam-4858	193	10	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	193	11	in	in	ADP
ejpam-4858	193	12	x.	x.	NOUN
ejpam-4858	193	13	(	(	PUNCT
ejpam-4858	193	14	2	2	NUM
ejpam-4858	193	15	)	)	PUNCT
ejpam-4858	193	16	⇔	⇔	X
ejpam-4858	193	17	(	(	PUNCT
ejpam-4858	193	18	3	3	NUM
ejpam-4858	193	19	):	):	PUNCT
ejpam-4858	193	20	this	this	PRON
ejpam-4858	193	21	follows	follow	VERB
ejpam-4858	193	22	from	from	ADP
ejpam-4858	193	23	the	the	DET
ejpam-4858	193	24	fact	fact	NOUN
ejpam-4858	193	25	that	that	SCONJ
ejpam-4858	193	26	f+(y	f+(y	PROPN
ejpam-4858	193	27	−b	−b	ADV
ejpam-4858	193	28	)	)	PUNCT
ejpam-4858	193	29	=	=	PUNCT
ejpam-4858	194	1	x	x	SYM
ejpam-4858	194	2	−f−(b	−f−(b	PROPN
ejpam-4858	194	3	)	)	PUNCT
ejpam-4858	194	4	for	for	ADP
ejpam-4858	194	5	any	any	DET
ejpam-4858	194	6	subset	subset	NOUN
ejpam-4858	194	7	b	b	PROPN
ejpam-4858	194	8	of	of	ADP
ejpam-4858	194	9	y	y	PROPN
ejpam-4858	194	10	.	.	PUNCT
ejpam-4858	195	1	(	(	PUNCT
ejpam-4858	195	2	3	3	X
ejpam-4858	195	3	)	)	PUNCT
ejpam-4858	195	4	⇒	⇒	NOUN
ejpam-4858	195	5	(	(	PUNCT
ejpam-4858	195	6	4	4	NUM
ejpam-4858	195	7	):	):	PUNCT
ejpam-4858	195	8	let	let	VERB
ejpam-4858	195	9	b	b	X
ejpam-4858	195	10	be	be	AUX
ejpam-4858	195	11	any	any	DET
ejpam-4858	195	12	subset	subset	NOUN
ejpam-4858	195	13	of	of	ADP
ejpam-4858	195	14	y	y	PROPN
ejpam-4858	195	15	.	.	PUNCT
ejpam-4858	196	1	then	then	ADV
ejpam-4858	196	2	,	,	PUNCT
ejpam-4858	196	3	cl⋆(b	cl⋆(b	NOUN
ejpam-4858	196	4	)	)	PUNCT
ejpam-4858	196	5	is	be	AUX
ejpam-4858	196	6	⋆-closed	⋆-close	VERB
ejpam-4858	196	7	in	in	ADP
ejpam-4858	196	8	y	y	PROPN
ejpam-4858	196	9	and	and	CCONJ
ejpam-4858	196	10	by	by	ADP
ejpam-4858	196	11	(	(	PUNCT
ejpam-4858	196	12	3	3	NUM
ejpam-4858	196	13	)	)	PUNCT
ejpam-4858	196	14	,	,	PUNCT
ejpam-4858	196	15	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4858	196	16	)	)	PUNCT
ejpam-4858	196	17	)	)	PUNCT
ejpam-4858	196	18	is	be	AUX
ejpam-4858	196	19	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	196	20	in	in	ADP
ejpam-4858	196	21	x.	x.	NOUN
ejpam-4858	196	22	thus	thus	ADV
ejpam-4858	196	23	,	,	PUNCT
ejpam-4858	196	24	by	by	ADP
ejpam-4858	196	25	lemma	lemma	PROPN
ejpam-4858	196	26	3(1	3(1	NUM
ejpam-4858	196	27	)	)	PUNCT
ejpam-4858	196	28	,	,	PUNCT
ejpam-4858	196	29	sinti	sinti	PROPN
ejpam-4858	196	30	(	(	PUNCT
ejpam-4858	196	31	cl⋆(f−(b	cl⋆(f−(b	NUM
ejpam-4858	196	32	)	)	PUNCT
ejpam-4858	196	33	)	)	PUNCT
ejpam-4858	196	34	)	)	PUNCT
ejpam-4858	197	1	⊆	⊆	NUM
ejpam-4858	197	2	sinti	sinti	NOUN
ejpam-4858	197	3	(	(	PUNCT
ejpam-4858	197	4	cl⋆(f−(cl⋆(b	cl⋆(f−(cl⋆(b	PROPN
ejpam-4858	197	5	)	)	PUNCT
ejpam-4858	197	6	)	)	PUNCT
ejpam-4858	197	7	)	)	PUNCT
ejpam-4858	197	8	)	)	PUNCT
ejpam-4858	198	1	⊆	⊆	NUM
ejpam-4858	198	2	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4858	198	3	)	)	PUNCT
ejpam-4858	198	4	)	)	PUNCT
ejpam-4858	198	5	.	.	PUNCT
ejpam-4858	199	1	c.	c.	PROPN
ejpam-4858	199	2	boonpok	boonpok	PROPN
ejpam-4858	199	3	,	,	PUNCT
ejpam-4858	199	4	j.	j.	PROPN
ejpam-4858	199	5	khampakdee	khampakdee	PROPN
ejpam-4858	199	6	/	/	PUNCT
ejpam-4858	199	7	eur	eur	PROPN
ejpam-4858	199	8	.	.	PUNCT
ejpam-4858	200	1	j.	j.	PROPN
ejpam-4858	200	2	pure	pure	PROPN
ejpam-4858	200	3	appl	appl	PROPN
ejpam-4858	200	4	.	.	PROPN
ejpam-4858	200	5	math	math	PROPN
ejpam-4858	200	6	,	,	PUNCT
ejpam-4858	200	7	17	17	NUM
ejpam-4858	200	8	(	(	PUNCT
ejpam-4858	200	9	1	1	NUM
ejpam-4858	200	10	)	)	PUNCT
ejpam-4858	200	11	(	(	PUNCT
ejpam-4858	200	12	2024	2024	NUM
ejpam-4858	200	13	)	)	PUNCT
ejpam-4858	200	14	,	,	PUNCT
ejpam-4858	200	15	201	201	NUM
ejpam-4858	200	16	-	-	SYM
ejpam-4858	200	17	211	211	NUM
ejpam-4858	200	18	207	207	NUM
ejpam-4858	200	19	(	(	PUNCT
ejpam-4858	200	20	4	4	NUM
ejpam-4858	200	21	)	)	PUNCT
ejpam-4858	200	22	⇒	⇒	NOUN
ejpam-4858	200	23	(	(	PUNCT
ejpam-4858	200	24	5	5	NUM
ejpam-4858	200	25	):	):	PUNCT
ejpam-4858	200	26	let	let	VERB
ejpam-4858	200	27	b	b	X
ejpam-4858	200	28	be	be	AUX
ejpam-4858	200	29	any	any	DET
ejpam-4858	200	30	subset	subset	NOUN
ejpam-4858	200	31	of	of	ADP
ejpam-4858	200	32	y	y	PROPN
ejpam-4858	200	33	.	.	PUNCT
ejpam-4858	201	1	by	by	ADP
ejpam-4858	201	2	(	(	PUNCT
ejpam-4858	201	3	4	4	NUM
ejpam-4858	201	4	)	)	PUNCT
ejpam-4858	201	5	and	and	CCONJ
ejpam-4858	201	6	lemma	lemma	PROPN
ejpam-4858	201	7	3(3	3(3	NUM
ejpam-4858	201	8	)	)	PUNCT
ejpam-4858	201	9	,	,	PUNCT
ejpam-4858	201	10	⋆αcl(f−(b	⋆αcl(f−(b	NOUN
ejpam-4858	201	11	)	)	PUNCT
ejpam-4858	201	12	)	)	PUNCT
ejpam-4858	201	13	=	=	SYM
ejpam-4858	201	14	f−(b	f−(b	PROPN
ejpam-4858	201	15	)	)	PUNCT
ejpam-4858	201	16	∪	∪	X
ejpam-4858	201	17	sinti	sinti	PROPN
ejpam-4858	201	18	(	(	PUNCT
ejpam-4858	201	19	cl⋆(f−(b	cl⋆(f−(b	NUM
ejpam-4858	201	20	)	)	PUNCT
ejpam-4858	201	21	)	)	PUNCT
ejpam-4858	201	22	)	)	PUNCT
ejpam-4858	201	23	⊆	⊆	NUM
ejpam-4858	201	24	f−(cl⋆(b	f−(cl⋆(b	NOUN
ejpam-4858	201	25	)	)	PUNCT
ejpam-4858	201	26	)	)	PUNCT
ejpam-4858	201	27	.	.	PUNCT
ejpam-4858	202	1	(	(	PUNCT
ejpam-4858	202	2	5	5	X
ejpam-4858	202	3	)	)	PUNCT
ejpam-4858	202	4	⇒	⇒	NOUN
ejpam-4858	202	5	(	(	PUNCT
ejpam-4858	202	6	3	3	NUM
ejpam-4858	202	7	):	):	PUNCT
ejpam-4858	202	8	let	let	VERB
ejpam-4858	202	9	k	k	PRON
ejpam-4858	202	10	be	be	AUX
ejpam-4858	202	11	any	any	DET
ejpam-4858	202	12	⋆-closed	⋆-close	VERB
ejpam-4858	202	13	set	set	NOUN
ejpam-4858	202	14	of	of	ADP
ejpam-4858	202	15	y	y	PROPN
ejpam-4858	202	16	.	.	PUNCT
ejpam-4858	203	1	by	by	ADP
ejpam-4858	203	2	(	(	PUNCT
ejpam-4858	203	3	5	5	NUM
ejpam-4858	203	4	)	)	PUNCT
ejpam-4858	203	5	,	,	PUNCT
ejpam-4858	203	6	we	we	PRON
ejpam-4858	203	7	have	have	VERB
ejpam-4858	203	8	⋆αcl(f−(k	⋆αcl(f−(k	NUM
ejpam-4858	203	9	)	)	PUNCT
ejpam-4858	203	10	)	)	PUNCT
ejpam-4858	204	1	⊆	⊆	NUM
ejpam-4858	204	2	f−(cl⋆(k	f−(cl⋆(k	NOUN
ejpam-4858	204	3	)	)	PUNCT
ejpam-4858	204	4	)	)	PUNCT
ejpam-4858	205	1	=	=	SYM
ejpam-4858	205	2	f−(k	f−(k	PROPN
ejpam-4858	205	3	)	)	PUNCT
ejpam-4858	205	4	.	.	PUNCT
ejpam-4858	206	1	this	this	PRON
ejpam-4858	206	2	shows	show	VERB
ejpam-4858	206	3	that	that	SCONJ
ejpam-4858	206	4	f−(k	f−(k	PROPN
ejpam-4858	206	5	)	)	PUNCT
ejpam-4858	206	6	is	be	AUX
ejpam-4858	206	7	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	206	8	in	in	ADP
ejpam-4858	206	9	x.	x.	NOUN
ejpam-4858	206	10	(	(	PUNCT
ejpam-4858	206	11	2	2	NUM
ejpam-4858	206	12	)	)	PUNCT
ejpam-4858	206	13	⇒	⇒	NOUN
ejpam-4858	206	14	(	(	PUNCT
ejpam-4858	206	15	6	6	NUM
ejpam-4858	206	16	):	):	PUNCT
ejpam-4858	206	17	let	let	VERB
ejpam-4858	206	18	x	x	PUNCT
ejpam-4858	206	19	∈	∈	PROPN
ejpam-4858	206	20	x	x	X
ejpam-4858	206	21	and	and	CCONJ
ejpam-4858	206	22	v	v	AUX
ejpam-4858	206	23	be	be	AUX
ejpam-4858	206	24	a	a	DET
ejpam-4858	206	25	⋆-neighbourhood	⋆-neighbourhood	NOUN
ejpam-4858	206	26	of	of	ADP
ejpam-4858	206	27	f	f	PROPN
ejpam-4858	206	28	(	(	PUNCT
ejpam-4858	206	29	x	x	NOUN
ejpam-4858	206	30	)	)	PUNCT
ejpam-4858	206	31	.	.	PUNCT
ejpam-4858	207	1	then	then	ADV
ejpam-4858	207	2	,	,	PUNCT
ejpam-4858	207	3	there	there	PRON
ejpam-4858	207	4	exists	exist	VERB
ejpam-4858	207	5	a	a	DET
ejpam-4858	207	6	⋆-open	⋆-open	ADJ
ejpam-4858	207	7	set	set	VERB
ejpam-4858	207	8	g	g	NOUN
ejpam-4858	207	9	of	of	ADP
ejpam-4858	207	10	y	y	PRON
ejpam-4858	207	11	such	such	ADJ
ejpam-4858	207	12	that	that	SCONJ
ejpam-4858	207	13	f	f	PROPN
ejpam-4858	207	14	(	(	PUNCT
ejpam-4858	207	15	x	x	X
ejpam-4858	207	16	)	)	PUNCT
ejpam-4858	207	17	⊆	⊆	NUM
ejpam-4858	207	18	g	g	ADP
ejpam-4858	207	19	⊆	⊆	NUM
ejpam-4858	207	20	v	v	NOUN
ejpam-4858	207	21	.	.	PUNCT
ejpam-4858	208	1	thus	thus	ADV
ejpam-4858	208	2	,	,	PUNCT
ejpam-4858	208	3	x	x	SYM
ejpam-4858	208	4	∈	∈	NOUN
ejpam-4858	208	5	f+(g	f+(g	NOUN
ejpam-4858	208	6	)	)	PUNCT
ejpam-4858	208	7	⊆	⊆	NUM
ejpam-4858	208	8	f+(v	f+(v	NOUN
ejpam-4858	208	9	)	)	PUNCT
ejpam-4858	208	10	.	.	PUNCT
ejpam-4858	209	1	by	by	ADP
ejpam-4858	209	2	(	(	PUNCT
ejpam-4858	209	3	2	2	NUM
ejpam-4858	209	4	)	)	PUNCT
ejpam-4858	209	5	,	,	PUNCT
ejpam-4858	209	6	f+(g	f+(g	NOUN
ejpam-4858	209	7	)	)	PUNCT
ejpam-4858	209	8	is	be	AUX
ejpam-4858	209	9	α-⋆-open	α-⋆-open	NUM
ejpam-4858	209	10	and	and	CCONJ
ejpam-4858	209	11	hence	hence	ADV
ejpam-4858	209	12	f+(v	f+(v	PROPN
ejpam-4858	209	13	)	)	PUNCT
ejpam-4858	209	14	is	be	AUX
ejpam-4858	209	15	an	an	DET
ejpam-4858	209	16	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	209	17	of	of	ADP
ejpam-4858	209	18	x.	x.	NOUN
ejpam-4858	209	19	(	(	PUNCT
ejpam-4858	209	20	6	6	NUM
ejpam-4858	209	21	)	)	PUNCT
ejpam-4858	209	22	⇒	⇒	NOUN
ejpam-4858	209	23	(	(	PUNCT
ejpam-4858	209	24	7	7	NUM
ejpam-4858	209	25	):	):	PUNCT
ejpam-4858	209	26	let	let	VERB
ejpam-4858	209	27	x	x	PUNCT
ejpam-4858	209	28	∈	∈	PROPN
ejpam-4858	209	29	x	x	X
ejpam-4858	209	30	and	and	CCONJ
ejpam-4858	209	31	v	v	AUX
ejpam-4858	209	32	be	be	AUX
ejpam-4858	209	33	a	a	DET
ejpam-4858	209	34	⋆-neighbourhood	⋆-neighbourhood	NOUN
ejpam-4858	209	35	of	of	ADP
ejpam-4858	209	36	f	f	PROPN
ejpam-4858	209	37	(	(	PUNCT
ejpam-4858	209	38	x	x	NOUN
ejpam-4858	209	39	)	)	PUNCT
ejpam-4858	209	40	.	.	PUNCT
ejpam-4858	210	1	by	by	ADP
ejpam-4858	210	2	(	(	PUNCT
ejpam-4858	210	3	6	6	NUM
ejpam-4858	210	4	)	)	PUNCT
ejpam-4858	210	5	,	,	PUNCT
ejpam-4858	210	6	we	we	PRON
ejpam-4858	210	7	have	have	VERB
ejpam-4858	210	8	f+(v	f+(v	NOUN
ejpam-4858	210	9	)	)	PUNCT
ejpam-4858	211	1	is	be	AUX
ejpam-4858	211	2	an	an	DET
ejpam-4858	211	3	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	211	4	of	of	ADP
ejpam-4858	211	5	x.	x.	NOUN
ejpam-4858	211	6	put	put	VERB
ejpam-4858	211	7	u	u	NOUN
ejpam-4858	211	8	=	=	NOUN
ejpam-4858	211	9	f+(v	f+(v	PROPN
ejpam-4858	211	10	)	)	PUNCT
ejpam-4858	211	11	,	,	PUNCT
ejpam-4858	211	12	then	then	ADV
ejpam-4858	211	13	u	u	NOUN
ejpam-4858	211	14	is	be	AUX
ejpam-4858	211	15	an	an	DET
ejpam-4858	211	16	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	211	17	of	of	ADP
ejpam-4858	211	18	x	x	SYM
ejpam-4858	211	19	such	such	ADJ
ejpam-4858	211	20	that	that	SCONJ
ejpam-4858	211	21	f	f	PROPN
ejpam-4858	211	22	(	(	PUNCT
ejpam-4858	211	23	u	u	NOUN
ejpam-4858	211	24	)	)	PUNCT
ejpam-4858	211	25	⊆	⊆	NUM
ejpam-4858	211	26	v	v	NOUN
ejpam-4858	211	27	.	.	PUNCT
ejpam-4858	212	1	(	(	PUNCT
ejpam-4858	212	2	7	7	X
ejpam-4858	212	3	)	)	PUNCT
ejpam-4858	212	4	⇒	⇒	NOUN
ejpam-4858	212	5	(	(	PUNCT
ejpam-4858	212	6	1	1	NUM
ejpam-4858	212	7	):	):	PUNCT
ejpam-4858	212	8	let	let	VERB
ejpam-4858	212	9	x	x	PUNCT
ejpam-4858	212	10	∈	∈	PROPN
ejpam-4858	212	11	x	x	X
ejpam-4858	212	12	and	and	CCONJ
ejpam-4858	212	13	v	v	AUX
ejpam-4858	212	14	be	be	AUX
ejpam-4858	212	15	any	any	DET
ejpam-4858	212	16	⋆-open	⋆-open	ADJ
ejpam-4858	212	17	set	set	NOUN
ejpam-4858	212	18	of	of	ADP
ejpam-4858	212	19	y	y	PRON
ejpam-4858	212	20	such	such	ADJ
ejpam-4858	212	21	that	that	SCONJ
ejpam-4858	212	22	f	f	PROPN
ejpam-4858	212	23	(	(	PUNCT
ejpam-4858	212	24	x	x	X
ejpam-4858	212	25	)	)	PUNCT
ejpam-4858	212	26	⊆	⊆	NUM
ejpam-4858	212	27	v	v	NOUN
ejpam-4858	212	28	.	.	PUNCT
ejpam-4858	213	1	then	then	ADV
ejpam-4858	213	2	,	,	PUNCT
ejpam-4858	213	3	v	v	NOUN
ejpam-4858	213	4	is	be	AUX
ejpam-4858	213	5	a	a	DET
ejpam-4858	213	6	⋆-neighbourhood	⋆-neighbourhood	NOUN
ejpam-4858	213	7	of	of	ADP
ejpam-4858	213	8	f	f	PROPN
ejpam-4858	213	9	(	(	PUNCT
ejpam-4858	213	10	x	x	NOUN
ejpam-4858	213	11	)	)	PUNCT
ejpam-4858	213	12	and	and	CCONJ
ejpam-4858	213	13	so	so	ADV
ejpam-4858	213	14	there	there	PRON
ejpam-4858	213	15	exists	exist	VERB
ejpam-4858	213	16	an	an	DET
ejpam-4858	213	17	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	213	18	u	u	NOUN
ejpam-4858	213	19	of	of	ADP
ejpam-4858	213	20	x	x	SYM
ejpam-4858	213	21	such	such	ADJ
ejpam-4858	213	22	that	that	SCONJ
ejpam-4858	213	23	f	f	PROPN
ejpam-4858	213	24	(	(	PUNCT
ejpam-4858	213	25	u	u	NOUN
ejpam-4858	213	26	)	)	PUNCT
ejpam-4858	213	27	⊆	⊆	NUM
ejpam-4858	213	28	v	v	NOUN
ejpam-4858	213	29	.	.	PUNCT
ejpam-4858	214	1	since	since	SCONJ
ejpam-4858	214	2	u	u	NOUN
ejpam-4858	214	3	is	be	AUX
ejpam-4858	214	4	an	an	DET
ejpam-4858	214	5	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	214	6	of	of	ADP
ejpam-4858	214	7	x	x	NOUN
ejpam-4858	214	8	,	,	PUNCT
ejpam-4858	214	9	there	there	PRON
ejpam-4858	214	10	exists	exist	VERB
ejpam-4858	214	11	an	an	DET
ejpam-4858	214	12	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	214	13	set	set	VERB
ejpam-4858	214	14	g	g	NOUN
ejpam-4858	214	15	of	of	ADP
ejpam-4858	214	16	x	x	INTJ
ejpam-4858	215	1	such	such	ADJ
ejpam-4858	215	2	that	that	SCONJ
ejpam-4858	215	3	x	x	SYM
ejpam-4858	215	4	∈	∈	NOUN
ejpam-4858	215	5	g	g	NOUN
ejpam-4858	215	6	⊆	⊆	NUM
ejpam-4858	215	7	u	u	NOUN
ejpam-4858	215	8	;	;	PUNCT
ejpam-4858	215	9	hence	hence	ADV
ejpam-4858	215	10	f	f	PROPN
ejpam-4858	215	11	(	(	PUNCT
ejpam-4858	215	12	g	g	NOUN
ejpam-4858	215	13	)	)	PUNCT
ejpam-4858	215	14	⊆	⊆	NUM
ejpam-4858	215	15	v	v	NOUN
ejpam-4858	215	16	.	.	PUNCT
ejpam-4858	216	1	this	this	PRON
ejpam-4858	216	2	shows	show	VERB
ejpam-4858	216	3	that	that	SCONJ
ejpam-4858	216	4	f	f	PROPN
ejpam-4858	216	5	is	be	AUX
ejpam-4858	216	6	upper	upper	ADJ
ejpam-4858	216	7	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	216	8	.	.	PUNCT
ejpam-4858	216	9	theorem	theorem	NOUN
ejpam-4858	216	10	4	4	NUM
ejpam-4858	216	11	.	.	X
ejpam-4858	216	12	for	for	ADP
ejpam-4858	216	13	a	a	DET
ejpam-4858	216	14	multifunction	multifunction	NOUN
ejpam-4858	216	15	f	f	NOUN
ejpam-4858	216	16	:	:	PUNCT
ejpam-4858	216	17	(	(	PUNCT
ejpam-4858	216	18	x	x	X
ejpam-4858	216	19	,	,	PUNCT
ejpam-4858	216	20	τ	τ	PROPN
ejpam-4858	216	21	,	,	PUNCT
ejpam-4858	216	22	i	i	NOUN
ejpam-4858	216	23	)	)	PUNCT
ejpam-4858	216	24	→	→	PUNCT
ejpam-4858	216	25	(	(	PUNCT
ejpam-4858	216	26	y	y	PROPN
ejpam-4858	216	27	,	,	PUNCT
ejpam-4858	216	28	σ	σ	PROPN
ejpam-4858	216	29	,	,	PUNCT
ejpam-4858	216	30	j	j	PROPN
ejpam-4858	216	31	)	)	PUNCT
ejpam-4858	216	32	,	,	PUNCT
ejpam-4858	216	33	the	the	DET
ejpam-4858	216	34	following	follow	VERB
ejpam-4858	216	35	properties	property	NOUN
ejpam-4858	216	36	are	be	AUX
ejpam-4858	216	37	equivalent	equivalent	ADJ
ejpam-4858	216	38	:	:	PUNCT
ejpam-4858	216	39	(	(	PUNCT
ejpam-4858	216	40	1	1	X
ejpam-4858	216	41	)	)	PUNCT
ejpam-4858	216	42	f	f	PROPN
ejpam-4858	216	43	is	be	AUX
ejpam-4858	216	44	lower	low	ADJ
ejpam-4858	216	45	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	216	46	;	;	PUNCT
ejpam-4858	216	47	(	(	PUNCT
ejpam-4858	216	48	2	2	X
ejpam-4858	216	49	)	)	PUNCT
ejpam-4858	216	50	f−(v	f−(v	NOUN
ejpam-4858	216	51	)	)	PUNCT
ejpam-4858	216	52	is	be	AUX
ejpam-4858	216	53	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	216	54	in	in	ADP
ejpam-4858	216	55	x	x	PUNCT
ejpam-4858	216	56	for	for	SCONJ
ejpam-4858	216	57	every	every	DET
ejpam-4858	216	58	⋆-open	⋆-open	NOUN
ejpam-4858	216	59	set	set	VERB
ejpam-4858	216	60	v	v	NOUN
ejpam-4858	216	61	of	of	ADP
ejpam-4858	216	62	y	y	PROPN
ejpam-4858	216	63	;	;	PUNCT
ejpam-4858	216	64	(	(	PUNCT
ejpam-4858	216	65	3	3	X
ejpam-4858	216	66	)	)	PUNCT
ejpam-4858	216	67	f+(k	f+(k	NUM
ejpam-4858	216	68	)	)	PUNCT
ejpam-4858	216	69	is	be	AUX
ejpam-4858	216	70	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	216	71	in	in	ADP
ejpam-4858	216	72	x	x	PUNCT
ejpam-4858	216	73	for	for	ADP
ejpam-4858	216	74	every	every	DET
ejpam-4858	216	75	⋆-closed	⋆-close	VERB
ejpam-4858	216	76	set	set	NOUN
ejpam-4858	216	77	k	k	PROPN
ejpam-4858	216	78	of	of	ADP
ejpam-4858	216	79	y	y	PROPN
ejpam-4858	216	80	;	;	PUNCT
ejpam-4858	216	81	(	(	PUNCT
ejpam-4858	216	82	4	4	X
ejpam-4858	216	83	)	)	PUNCT
ejpam-4858	216	84	sinti	sinti	PROPN
ejpam-4858	216	85	(	(	PUNCT
ejpam-4858	216	86	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-4858	216	87	)	)	PUNCT
ejpam-4858	216	88	)	)	PUNCT
ejpam-4858	216	89	)	)	PUNCT
ejpam-4858	217	1	⊆	⊆	NUM
ejpam-4858	217	2	f+(cl⋆(b	f+(cl⋆(b	NOUN
ejpam-4858	217	3	)	)	PUNCT
ejpam-4858	217	4	)	)	PUNCT
ejpam-4858	217	5	for	for	ADP
ejpam-4858	217	6	every	every	DET
ejpam-4858	217	7	subset	subset	NOUN
ejpam-4858	217	8	b	b	PROPN
ejpam-4858	217	9	of	of	ADP
ejpam-4858	217	10	y	y	PROPN
ejpam-4858	217	11	;	;	PUNCT
ejpam-4858	217	12	(	(	PUNCT
ejpam-4858	217	13	5	5	X
ejpam-4858	217	14	)	)	PUNCT
ejpam-4858	217	15	⋆αcl(f+(b	⋆αcl(f+(b	NOUN
ejpam-4858	217	16	)	)	PUNCT
ejpam-4858	217	17	)	)	PUNCT
ejpam-4858	217	18	⊆	⊆	NUM
ejpam-4858	217	19	f+(cl⋆(b	f+(cl⋆(b	NOUN
ejpam-4858	217	20	)	)	PUNCT
ejpam-4858	217	21	)	)	PUNCT
ejpam-4858	217	22	for	for	ADP
ejpam-4858	217	23	every	every	DET
ejpam-4858	217	24	subset	subset	NOUN
ejpam-4858	217	25	b	b	PROPN
ejpam-4858	217	26	of	of	ADP
ejpam-4858	217	27	y	y	PROPN
ejpam-4858	217	28	;	;	PUNCT
ejpam-4858	217	29	(	(	PUNCT
ejpam-4858	217	30	6	6	X
ejpam-4858	217	31	)	)	PUNCT
ejpam-4858	217	32	f	f	NOUN
ejpam-4858	217	33	(	(	PUNCT
ejpam-4858	217	34	⋆αcl(a	⋆αcl(a	PROPN
ejpam-4858	217	35	)	)	PUNCT
ejpam-4858	217	36	)	)	PUNCT
ejpam-4858	218	1	⊆	⊆	NUM
ejpam-4858	218	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	218	3	(	(	PUNCT
ejpam-4858	218	4	a	a	NOUN
ejpam-4858	218	5	)	)	PUNCT
ejpam-4858	218	6	)	)	PUNCT
ejpam-4858	218	7	for	for	ADP
ejpam-4858	218	8	every	every	DET
ejpam-4858	218	9	subset	subset	NOUN
ejpam-4858	218	10	a	a	PRON
ejpam-4858	218	11	of	of	ADP
ejpam-4858	218	12	x	x	PRON
ejpam-4858	218	13	;	;	PUNCT
ejpam-4858	218	14	(	(	PUNCT
ejpam-4858	218	15	7	7	X
ejpam-4858	218	16	)	)	PUNCT
ejpam-4858	218	17	f	f	NOUN
ejpam-4858	218	18	(	(	PUNCT
ejpam-4858	218	19	sinti	sinti	X
ejpam-4858	218	20	(	(	PUNCT
ejpam-4858	218	21	cl⋆(a	cl⋆(a	PROPN
ejpam-4858	218	22	)	)	PUNCT
ejpam-4858	218	23	)	)	PUNCT
ejpam-4858	218	24	)	)	PUNCT
ejpam-4858	219	1	⊆	⊆	NUM
ejpam-4858	219	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	219	3	(	(	PUNCT
ejpam-4858	219	4	a	a	NOUN
ejpam-4858	219	5	)	)	PUNCT
ejpam-4858	219	6	)	)	PUNCT
ejpam-4858	219	7	for	for	ADP
ejpam-4858	219	8	every	every	DET
ejpam-4858	219	9	subset	subset	NOUN
ejpam-4858	219	10	a	a	PRON
ejpam-4858	219	11	of	of	ADP
ejpam-4858	219	12	x	x	PRON
ejpam-4858	219	13	;	;	PUNCT
ejpam-4858	219	14	(	(	PUNCT
ejpam-4858	219	15	8)	8)	NUM
ejpam-4858	219	16	f	f	X
ejpam-4858	219	17	(	(	PUNCT
ejpam-4858	219	18	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	219	19	)	)	PUNCT
ejpam-4858	219	20	)	)	PUNCT
ejpam-4858	219	21	)	)	PUNCT
ejpam-4858	219	22	)	)	PUNCT
ejpam-4858	220	1	⊆	⊆	NUM
ejpam-4858	220	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	220	3	(	(	PUNCT
ejpam-4858	220	4	a	a	NOUN
ejpam-4858	220	5	)	)	PUNCT
ejpam-4858	220	6	)	)	PUNCT
ejpam-4858	220	7	for	for	ADP
ejpam-4858	220	8	every	every	DET
ejpam-4858	220	9	subset	subset	NOUN
ejpam-4858	220	10	a	a	PRON
ejpam-4858	220	11	of	of	ADP
ejpam-4858	220	12	x.	x.	NOUN
ejpam-4858	220	13	proof	proof	NOUN
ejpam-4858	220	14	.	.	PUNCT
ejpam-4858	221	1	the	the	DET
ejpam-4858	221	2	proofs	proof	NOUN
ejpam-4858	221	3	except	except	SCONJ
ejpam-4858	221	4	for	for	ADP
ejpam-4858	221	5	the	the	DET
ejpam-4858	221	6	following	following	NOUN
ejpam-4858	221	7	are	be	AUX
ejpam-4858	221	8	similar	similar	ADJ
ejpam-4858	221	9	to	to	ADP
ejpam-4858	221	10	the	the	DET
ejpam-4858	221	11	proof	proof	NOUN
ejpam-4858	221	12	of	of	ADP
ejpam-4858	221	13	theorem	theorem	NOUN
ejpam-4858	221	14	3	3	NUM
ejpam-4858	221	15	.	.	PUNCT
ejpam-4858	221	16	(	(	PUNCT
ejpam-4858	221	17	5	5	X
ejpam-4858	221	18	)	)	PUNCT
ejpam-4858	221	19	⇒	⇒	NOUN
ejpam-4858	221	20	(	(	PUNCT
ejpam-4858	221	21	6	6	NUM
ejpam-4858	221	22	):	):	PUNCT
ejpam-4858	221	23	let	let	VERB
ejpam-4858	221	24	a	a	PRON
ejpam-4858	221	25	be	be	AUX
ejpam-4858	221	26	any	any	DET
ejpam-4858	221	27	subset	subset	NOUN
ejpam-4858	221	28	of	of	ADP
ejpam-4858	221	29	x.	x.	NOUN
ejpam-4858	221	30	since	since	SCONJ
ejpam-4858	221	31	a	a	DET
ejpam-4858	221	32	⊆	⊆	NUM
ejpam-4858	221	33	f+(f	f+(f	NOUN
ejpam-4858	221	34	(	(	PUNCT
ejpam-4858	221	35	a	a	NOUN
ejpam-4858	221	36	)	)	PUNCT
ejpam-4858	221	37	)	)	PUNCT
ejpam-4858	221	38	,	,	PUNCT
ejpam-4858	221	39	we	we	PRON
ejpam-4858	221	40	have	have	VERB
ejpam-4858	221	41	⋆αcl(a	⋆αcl(a	VERB
ejpam-4858	221	42	)	)	PUNCT
ejpam-4858	221	43	⊆	⊆	NUM
ejpam-4858	221	44	⋆αcl(f+(f	⋆αcl(f+(f	NUM
ejpam-4858	221	45	(	(	PUNCT
ejpam-4858	221	46	a	a	NOUN
ejpam-4858	221	47	)	)	PUNCT
ejpam-4858	221	48	)	)	PUNCT
ejpam-4858	221	49	)	)	PUNCT
ejpam-4858	222	1	⊆	⊆	NUM
ejpam-4858	222	2	f+(cl⋆(f	f+(cl⋆(f	X
ejpam-4858	222	3	(	(	PUNCT
ejpam-4858	222	4	a	a	NOUN
ejpam-4858	222	5	)	)	PUNCT
ejpam-4858	222	6	)	)	PUNCT
ejpam-4858	222	7	)	)	PUNCT
ejpam-4858	222	8	and	and	CCONJ
ejpam-4858	222	9	hence	hence	ADV
ejpam-4858	222	10	f	f	X
ejpam-4858	222	11	(	(	PUNCT
ejpam-4858	222	12	⋆αcl(a	⋆αcl(a	PROPN
ejpam-4858	222	13	)	)	PUNCT
ejpam-4858	222	14	)	)	PUNCT
ejpam-4858	223	1	⊆	⊆	NUM
ejpam-4858	223	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	223	3	(	(	PUNCT
ejpam-4858	223	4	a	a	NOUN
ejpam-4858	223	5	)	)	PUNCT
ejpam-4858	223	6	)	)	PUNCT
ejpam-4858	223	7	.	.	PUNCT
ejpam-4858	224	1	(	(	PUNCT
ejpam-4858	224	2	6	6	X
ejpam-4858	224	3	)	)	PUNCT
ejpam-4858	224	4	⇒	⇒	NOUN
ejpam-4858	224	5	(	(	PUNCT
ejpam-4858	224	6	7	7	NUM
ejpam-4858	224	7	):	):	PUNCT
ejpam-4858	224	8	let	let	VERB
ejpam-4858	224	9	a	a	DET
ejpam-4858	224	10	be	be	AUX
ejpam-4858	224	11	any	any	DET
ejpam-4858	224	12	subset	subset	NOUN
ejpam-4858	224	13	of	of	ADP
ejpam-4858	224	14	x.	x.	NOUN
ejpam-4858	224	15	by	by	ADP
ejpam-4858	224	16	(	(	PUNCT
ejpam-4858	224	17	6	6	NUM
ejpam-4858	224	18	)	)	PUNCT
ejpam-4858	224	19	and	and	CCONJ
ejpam-4858	224	20	lemma	lemma	PROPN
ejpam-4858	224	21	3	3	NUM
ejpam-4858	224	22	,	,	PUNCT
ejpam-4858	224	23	f	f	PROPN
ejpam-4858	224	24	(	(	PUNCT
ejpam-4858	224	25	sinti	sinti	X
ejpam-4858	224	26	(	(	PUNCT
ejpam-4858	224	27	cl⋆(a	cl⋆(a	PROPN
ejpam-4858	224	28	)	)	PUNCT
ejpam-4858	224	29	)	)	PUNCT
ejpam-4858	224	30	)	)	PUNCT
ejpam-4858	225	1	=	=	SYM
ejpam-4858	225	2	f	f	PROPN
ejpam-4858	225	3	(	(	PUNCT
ejpam-4858	225	4	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	225	5	)	)	PUNCT
ejpam-4858	225	6	)	)	PUNCT
ejpam-4858	225	7	)	)	PUNCT
ejpam-4858	225	8	)	)	PUNCT
ejpam-4858	226	1	c.	c.	PROPN
ejpam-4858	226	2	boonpok	boonpok	PROPN
ejpam-4858	226	3	,	,	PUNCT
ejpam-4858	226	4	j.	j.	PROPN
ejpam-4858	226	5	khampakdee	khampakdee	PROPN
ejpam-4858	226	6	/	/	PUNCT
ejpam-4858	226	7	eur	eur	PROPN
ejpam-4858	226	8	.	.	PUNCT
ejpam-4858	227	1	j.	j.	PROPN
ejpam-4858	227	2	pure	pure	PROPN
ejpam-4858	227	3	appl	appl	PROPN
ejpam-4858	227	4	.	.	PROPN
ejpam-4858	227	5	math	math	PROPN
ejpam-4858	227	6	,	,	PUNCT
ejpam-4858	227	7	17	17	NUM
ejpam-4858	227	8	(	(	PUNCT
ejpam-4858	227	9	1	1	NUM
ejpam-4858	227	10	)	)	PUNCT
ejpam-4858	227	11	(	(	PUNCT
ejpam-4858	227	12	2024	2024	NUM
ejpam-4858	227	13	)	)	PUNCT
ejpam-4858	227	14	,	,	PUNCT
ejpam-4858	227	15	201	201	NUM
ejpam-4858	227	16	-	-	SYM
ejpam-4858	227	17	211	211	NUM
ejpam-4858	227	18	208	208	NUM
ejpam-4858	227	19	⊆	⊆	NUM
ejpam-4858	227	20	f	f	X
ejpam-4858	227	21	(	(	PUNCT
ejpam-4858	227	22	a	a	DET
ejpam-4858	227	23	∪	∪	ADJ
ejpam-4858	227	24	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	NOUN
ejpam-4858	227	25	)	)	PUNCT
ejpam-4858	227	26	)	)	PUNCT
ejpam-4858	227	27	)	)	PUNCT
ejpam-4858	227	28	)	)	PUNCT
ejpam-4858	228	1	=	=	SYM
ejpam-4858	228	2	f	f	PROPN
ejpam-4858	228	3	(	(	PUNCT
ejpam-4858	228	4	⋆αcl(a	⋆αcl(a	PROPN
ejpam-4858	228	5	)	)	PUNCT
ejpam-4858	228	6	)	)	PUNCT
ejpam-4858	229	1	⊆	⊆	NUM
ejpam-4858	229	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	229	3	(	(	PUNCT
ejpam-4858	229	4	a	a	NOUN
ejpam-4858	229	5	)	)	PUNCT
ejpam-4858	229	6	)	)	PUNCT
ejpam-4858	229	7	.	.	PUNCT
ejpam-4858	230	1	(	(	PUNCT
ejpam-4858	230	2	7	7	X
ejpam-4858	230	3	)	)	PUNCT
ejpam-4858	230	4	⇒	⇒	NOUN
ejpam-4858	230	5	(	(	PUNCT
ejpam-4858	230	6	8)	8)	NUM
ejpam-4858	230	7	:	:	PUNCT
ejpam-4858	230	8	let	let	VERB
ejpam-4858	230	9	a	a	PRON
ejpam-4858	230	10	be	be	AUX
ejpam-4858	230	11	any	any	DET
ejpam-4858	230	12	subset	subset	NOUN
ejpam-4858	230	13	of	of	ADP
ejpam-4858	230	14	x.	x.	NOUN
ejpam-4858	230	15	by	by	ADP
ejpam-4858	230	16	(	(	PUNCT
ejpam-4858	230	17	7	7	NUM
ejpam-4858	230	18	)	)	PUNCT
ejpam-4858	230	19	and	and	CCONJ
ejpam-4858	230	20	lemma	lemma	PROPN
ejpam-4858	230	21	3(2	3(2	NUM
ejpam-4858	230	22	)	)	PUNCT
ejpam-4858	230	23	,	,	PUNCT
ejpam-4858	230	24	we	we	PRON
ejpam-4858	230	25	have	have	VERB
ejpam-4858	230	26	f	f	PROPN
ejpam-4858	230	27	(	(	PUNCT
ejpam-4858	230	28	cl⋆(int(cl⋆(a	cl⋆(int(cl⋆(a	PROPN
ejpam-4858	230	29	)	)	PUNCT
ejpam-4858	230	30	)	)	PUNCT
ejpam-4858	230	31	)	)	PUNCT
ejpam-4858	230	32	)	)	PUNCT
ejpam-4858	231	1	=	=	SYM
ejpam-4858	231	2	f	f	PROPN
ejpam-4858	231	3	(	(	PUNCT
ejpam-4858	231	4	sinti	sinti	X
ejpam-4858	231	5	(	(	PUNCT
ejpam-4858	231	6	cl⋆(a	cl⋆(a	PROPN
ejpam-4858	231	7	)	)	PUNCT
ejpam-4858	231	8	)	)	PUNCT
ejpam-4858	231	9	)	)	PUNCT
ejpam-4858	232	1	⊆	⊆	NUM
ejpam-4858	232	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	232	3	(	(	PUNCT
ejpam-4858	232	4	a	a	NOUN
ejpam-4858	232	5	)	)	PUNCT
ejpam-4858	232	6	)	)	PUNCT
ejpam-4858	232	7	.	.	PUNCT
ejpam-4858	233	1	(	(	PUNCT
ejpam-4858	233	2	8)	8)	NUM
ejpam-4858	233	3	⇒	⇒	NOUN
ejpam-4858	233	4	(	(	PUNCT
ejpam-4858	233	5	1	1	NUM
ejpam-4858	233	6	):	):	PUNCT
ejpam-4858	233	7	let	let	VERB
ejpam-4858	233	8	x	x	PUNCT
ejpam-4858	233	9	∈	∈	PROPN
ejpam-4858	233	10	x	x	X
ejpam-4858	233	11	and	and	CCONJ
ejpam-4858	233	12	v	v	AUX
ejpam-4858	233	13	be	be	AUX
ejpam-4858	233	14	any	any	DET
ejpam-4858	233	15	⋆-open	⋆-open	ADJ
ejpam-4858	233	16	set	set	NOUN
ejpam-4858	233	17	such	such	ADJ
ejpam-4858	233	18	that	that	SCONJ
ejpam-4858	233	19	f	f	PROPN
ejpam-4858	233	20	(	(	PUNCT
ejpam-4858	233	21	x	x	NOUN
ejpam-4858	233	22	)	)	PUNCT
ejpam-4858	233	23	∩	∩	NOUN
ejpam-4858	233	24	v	v	ADP
ejpam-4858	233	25	̸=	̸=	PROPN
ejpam-4858	233	26	∅.	∅.	NOUN
ejpam-4858	233	27	then	then	ADV
ejpam-4858	233	28	,	,	PUNCT
ejpam-4858	233	29	we	we	PRON
ejpam-4858	233	30	have	have	VERB
ejpam-4858	233	31	x	x	X
ejpam-4858	233	32	∈	∈	PROPN
ejpam-4858	233	33	f−(v	f−(v	NOUN
ejpam-4858	233	34	)	)	PUNCT
ejpam-4858	233	35	.	.	PUNCT
ejpam-4858	234	1	we	we	PRON
ejpam-4858	234	2	shall	shall	AUX
ejpam-4858	234	3	show	show	VERB
ejpam-4858	234	4	that	that	DET
ejpam-4858	234	5	f−(v	f−(v	NOUN
ejpam-4858	234	6	)	)	PUNCT
ejpam-4858	234	7	is	be	AUX
ejpam-4858	234	8	α-⋆-open	α-⋆-open	NUM
ejpam-4858	234	9	in	in	ADP
ejpam-4858	234	10	x.	x.	NOUN
ejpam-4858	234	11	by	by	ADP
ejpam-4858	234	12	the	the	DET
ejpam-4858	234	13	hypothesis	hypothesis	NOUN
ejpam-4858	234	14	,	,	PUNCT
ejpam-4858	234	15	f	f	PROPN
ejpam-4858	234	16	(	(	PUNCT
ejpam-4858	234	17	cl⋆(int(cl⋆(f+(y	cl⋆(int(cl⋆(f+(y	VERB
ejpam-4858	234	18	−	−	PROPN
ejpam-4858	234	19	v	v	NOUN
ejpam-4858	234	20	)	)	PUNCT
ejpam-4858	234	21	)	)	PUNCT
ejpam-4858	234	22	)	)	PUNCT
ejpam-4858	234	23	)	)	PUNCT
ejpam-4858	234	24	)	)	PUNCT
ejpam-4858	235	1	⊆	⊆	NUM
ejpam-4858	235	2	cl⋆(f	cl⋆(f	NOUN
ejpam-4858	235	3	(	(	PUNCT
ejpam-4858	235	4	f+(y	f+(y	PROPN
ejpam-4858	235	5	−	−	PROPN
ejpam-4858	235	6	v	v	NOUN
ejpam-4858	235	7	)	)	PUNCT
ejpam-4858	235	8	)	)	PUNCT
ejpam-4858	235	9	)	)	PUNCT
ejpam-4858	236	1	⊆	⊆	NUM
ejpam-4858	236	2	y	y	PROPN
ejpam-4858	236	3	−	−	PROPN
ejpam-4858	236	4	v	v	NOUN
ejpam-4858	236	5	and	and	CCONJ
ejpam-4858	236	6	hence	hence	ADV
ejpam-4858	236	7	cl⋆(int(cl⋆(f+(y	cl⋆(int(cl⋆(f+(y	VERB
ejpam-4858	236	8	−	−	PROPN
ejpam-4858	236	9	v	v	NOUN
ejpam-4858	236	10	)	)	PUNCT
ejpam-4858	236	11	)	)	PUNCT
ejpam-4858	236	12	)	)	PUNCT
ejpam-4858	236	13	)	)	PUNCT
ejpam-4858	237	1	⊆	⊆	NUM
ejpam-4858	237	2	f+(y	f+(y	ADP
ejpam-4858	237	3	−	−	PROPN
ejpam-4858	237	4	v	v	NOUN
ejpam-4858	237	5	)	)	PUNCT
ejpam-4858	237	6	=	=	PUNCT
ejpam-4858	237	7	x	x	SYM
ejpam-4858	237	8	−	−	NOUN
ejpam-4858	237	9	f−(v	f−(v	NOUN
ejpam-4858	237	10	)	)	PUNCT
ejpam-4858	237	11	.	.	PUNCT
ejpam-4858	238	1	thus	thus	ADV
ejpam-4858	238	2	,	,	PUNCT
ejpam-4858	238	3	f−(v	f−(v	ADJ
ejpam-4858	238	4	)	)	PUNCT
ejpam-4858	238	5	⊆	⊆	NUM
ejpam-4858	238	6	int⋆(cl(int⋆(f−(v	int⋆(cl(int⋆(f−(v	NOUN
ejpam-4858	238	7	)	)	PUNCT
ejpam-4858	238	8	)	)	PUNCT
ejpam-4858	238	9	)	)	PUNCT
ejpam-4858	238	10	)	)	PUNCT
ejpam-4858	239	1	and	and	CCONJ
ejpam-4858	239	2	so	so	ADV
ejpam-4858	239	3	f−(v	f−(v	ADJ
ejpam-4858	239	4	)	)	PUNCT
ejpam-4858	239	5	is	be	AUX
ejpam-4858	239	6	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	239	7	in	in	ADP
ejpam-4858	239	8	x.	x.	NOUN
ejpam-4858	239	9	put	put	VERB
ejpam-4858	239	10	u	u	NOUN
ejpam-4858	239	11	=	=	NOUN
ejpam-4858	239	12	f−(v	f−(v	PROPN
ejpam-4858	239	13	)	)	PUNCT
ejpam-4858	239	14	,	,	PUNCT
ejpam-4858	239	15	then	then	ADV
ejpam-4858	239	16	u	u	NOUN
ejpam-4858	239	17	is	be	AUX
ejpam-4858	239	18	an	an	DET
ejpam-4858	239	19	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	239	20	set	set	NOUN
ejpam-4858	239	21	of	of	ADP
ejpam-4858	239	22	x	x	PUNCT
ejpam-4858	239	23	containing	contain	VERB
ejpam-4858	239	24	x	x	PUNCT
ejpam-4858	239	25	such	such	ADJ
ejpam-4858	239	26	that	that	SCONJ
ejpam-4858	239	27	f	f	PROPN
ejpam-4858	239	28	(	(	PUNCT
ejpam-4858	239	29	z)∩v	z)∩v	PROPN
ejpam-4858	239	30	̸=	̸=	PROPN
ejpam-4858	239	31	∅	∅	NOUN
ejpam-4858	239	32	for	for	ADP
ejpam-4858	239	33	every	every	DET
ejpam-4858	239	34	z	z	NOUN
ejpam-4858	239	35	∈	∈	PROPN
ejpam-4858	239	36	u	u	NOUN
ejpam-4858	239	37	.	.	PUNCT
ejpam-4858	240	1	this	this	PRON
ejpam-4858	240	2	shows	show	VERB
ejpam-4858	240	3	that	that	SCONJ
ejpam-4858	240	4	f	f	PROPN
ejpam-4858	240	5	is	be	AUX
ejpam-4858	240	6	lower	low	ADJ
ejpam-4858	240	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	240	8	.	.	PUNCT
ejpam-4858	241	1	definition	definition	NOUN
ejpam-4858	241	2	3	3	NUM
ejpam-4858	241	3	.	.	PUNCT
ejpam-4858	242	1	a	a	DET
ejpam-4858	242	2	function	function	NOUN
ejpam-4858	242	3	f	f	NOUN
ejpam-4858	242	4	:	:	PUNCT
ejpam-4858	242	5	(	(	PUNCT
ejpam-4858	242	6	x	x	X
ejpam-4858	242	7	,	,	PUNCT
ejpam-4858	242	8	τ	τ	PROPN
ejpam-4858	242	9	,	,	PUNCT
ejpam-4858	242	10	i	i	NOUN
ejpam-4858	242	11	)	)	PUNCT
ejpam-4858	242	12	→	→	PUNCT
ejpam-4858	242	13	(	(	PUNCT
ejpam-4858	242	14	y	y	PROPN
ejpam-4858	242	15	,	,	PUNCT
ejpam-4858	242	16	σ	σ	PROPN
ejpam-4858	242	17	,	,	PUNCT
ejpam-4858	242	18	j	j	PROPN
ejpam-4858	242	19	)	)	PUNCT
ejpam-4858	242	20	is	be	AUX
ejpam-4858	242	21	called	call	VERB
ejpam-4858	242	22	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	242	23	if	if	SCONJ
ejpam-4858	242	24	f−1(v	f−1(v	PROPN
ejpam-4858	242	25	)	)	PUNCT
ejpam-4858	242	26	is	be	AUX
ejpam-4858	242	27	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	242	28	in	in	ADP
ejpam-4858	242	29	x	x	PUNCT
ejpam-4858	242	30	for	for	SCONJ
ejpam-4858	242	31	every	every	DET
ejpam-4858	242	32	⋆-open	⋆-open	NOUN
ejpam-4858	242	33	set	set	VERB
ejpam-4858	242	34	v	v	NOUN
ejpam-4858	242	35	of	of	ADP
ejpam-4858	242	36	y	y	PROPN
ejpam-4858	242	37	.	.	PUNCT
ejpam-4858	243	1	corollary	corollary	ADJ
ejpam-4858	243	2	1	1	NUM
ejpam-4858	243	3	.	.	PUNCT
ejpam-4858	244	1	for	for	ADP
ejpam-4858	244	2	a	a	DET
ejpam-4858	244	3	function	function	NOUN
ejpam-4858	244	4	f	f	NOUN
ejpam-4858	244	5	:	:	PUNCT
ejpam-4858	244	6	(	(	PUNCT
ejpam-4858	244	7	x	x	X
ejpam-4858	244	8	,	,	PUNCT
ejpam-4858	244	9	τ	τ	PROPN
ejpam-4858	244	10	,	,	PUNCT
ejpam-4858	244	11	i	i	NOUN
ejpam-4858	244	12	)	)	PUNCT
ejpam-4858	244	13	→	→	PUNCT
ejpam-4858	244	14	(	(	PUNCT
ejpam-4858	244	15	y	y	PROPN
ejpam-4858	244	16	,	,	PUNCT
ejpam-4858	244	17	σ	σ	PROPN
ejpam-4858	244	18	,	,	PUNCT
ejpam-4858	244	19	j	j	PROPN
ejpam-4858	244	20	)	)	PUNCT
ejpam-4858	244	21	,	,	PUNCT
ejpam-4858	244	22	the	the	DET
ejpam-4858	244	23	following	follow	VERB
ejpam-4858	244	24	properties	property	NOUN
ejpam-4858	244	25	are	be	AUX
ejpam-4858	244	26	equivalent	equivalent	ADJ
ejpam-4858	244	27	:	:	PUNCT
ejpam-4858	244	28	(	(	PUNCT
ejpam-4858	244	29	1	1	X
ejpam-4858	244	30	)	)	PUNCT
ejpam-4858	244	31	f	f	PROPN
ejpam-4858	244	32	is	be	AUX
ejpam-4858	244	33	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	244	34	;	;	PUNCT
ejpam-4858	244	35	(	(	PUNCT
ejpam-4858	244	36	2	2	X
ejpam-4858	244	37	)	)	PUNCT
ejpam-4858	244	38	f−1(k	f−1(k	PROPN
ejpam-4858	244	39	)	)	PUNCT
ejpam-4858	244	40	is	be	AUX
ejpam-4858	244	41	α-⋆-closed	α-⋆-close	VERB
ejpam-4858	244	42	in	in	ADP
ejpam-4858	244	43	x	x	PUNCT
ejpam-4858	244	44	for	for	ADP
ejpam-4858	244	45	every	every	DET
ejpam-4858	244	46	⋆-closed	⋆-close	VERB
ejpam-4858	244	47	set	set	NOUN
ejpam-4858	244	48	k	k	PROPN
ejpam-4858	244	49	of	of	ADP
ejpam-4858	244	50	y	y	PROPN
ejpam-4858	244	51	;	;	PUNCT
ejpam-4858	244	52	(	(	PUNCT
ejpam-4858	244	53	3	3	X
ejpam-4858	244	54	)	)	PUNCT
ejpam-4858	244	55	sinti	sinti	PROPN
ejpam-4858	244	56	(	(	PUNCT
ejpam-4858	244	57	cl⋆(f−1(b	cl⋆(f−1(b	PROPN
ejpam-4858	244	58	)	)	PUNCT
ejpam-4858	244	59	)	)	PUNCT
ejpam-4858	244	60	)	)	PUNCT
ejpam-4858	245	1	⊆	⊆	NUM
ejpam-4858	245	2	f−1(cl⋆(b	f−1(cl⋆(b	NOUN
ejpam-4858	245	3	)	)	PUNCT
ejpam-4858	245	4	)	)	PUNCT
ejpam-4858	245	5	for	for	ADP
ejpam-4858	245	6	any	any	DET
ejpam-4858	245	7	subset	subset	NOUN
ejpam-4858	245	8	b	b	PROPN
ejpam-4858	245	9	of	of	ADP
ejpam-4858	245	10	y	y	PROPN
ejpam-4858	245	11	;	;	PUNCT
ejpam-4858	245	12	(	(	PUNCT
ejpam-4858	245	13	4	4	X
ejpam-4858	245	14	)	)	PUNCT
ejpam-4858	245	15	⋆αcl(f−1(b	⋆αcl(f−1(b	PROPN
ejpam-4858	245	16	)	)	PUNCT
ejpam-4858	245	17	)	)	PUNCT
ejpam-4858	246	1	⊆	⊆	NUM
ejpam-4858	246	2	f−1(cl⋆(b	f−1(cl⋆(b	NOUN
ejpam-4858	246	3	)	)	PUNCT
ejpam-4858	246	4	)	)	PUNCT
ejpam-4858	246	5	for	for	ADP
ejpam-4858	246	6	any	any	DET
ejpam-4858	246	7	subset	subset	NOUN
ejpam-4858	246	8	b	b	PROPN
ejpam-4858	246	9	of	of	ADP
ejpam-4858	246	10	y	y	PROPN
ejpam-4858	246	11	;	;	PUNCT
ejpam-4858	246	12	(	(	PUNCT
ejpam-4858	246	13	5	5	X
ejpam-4858	246	14	)	)	PUNCT
ejpam-4858	246	15	for	for	ADP
ejpam-4858	246	16	each	each	DET
ejpam-4858	246	17	x	x	SYM
ejpam-4858	246	18	∈	∈	PROPN
ejpam-4858	246	19	x	x	X
ejpam-4858	246	20	and	and	CCONJ
ejpam-4858	246	21	each	each	DET
ejpam-4858	246	22	⋆-neighbourhood	⋆-neighbourhood	PUNCT
ejpam-4858	246	23	v	v	NOUN
ejpam-4858	246	24	of	of	ADP
ejpam-4858	246	25	f(x	f(x	PROPN
ejpam-4858	246	26	)	)	PUNCT
ejpam-4858	246	27	,	,	PUNCT
ejpam-4858	246	28	f−1(v	f−1(v	PROPN
ejpam-4858	246	29	)	)	PUNCT
ejpam-4858	246	30	is	be	AUX
ejpam-4858	246	31	an	an	DET
ejpam-4858	246	32	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	246	33	of	of	ADP
ejpam-4858	246	34	x	x	PRON
ejpam-4858	246	35	;	;	PUNCT
ejpam-4858	246	36	(	(	PUNCT
ejpam-4858	246	37	6	6	NUM
ejpam-4858	246	38	)	)	PUNCT
ejpam-4858	246	39	for	for	ADP
ejpam-4858	246	40	each	each	DET
ejpam-4858	246	41	x	x	SYM
ejpam-4858	246	42	∈	∈	PROPN
ejpam-4858	246	43	x	x	X
ejpam-4858	246	44	and	and	CCONJ
ejpam-4858	246	45	each	each	DET
ejpam-4858	246	46	⋆-neighbourhood	⋆-neighbourhood	PUNCT
ejpam-4858	246	47	v	v	NOUN
ejpam-4858	246	48	of	of	ADP
ejpam-4858	246	49	f(x	f(x	PROPN
ejpam-4858	246	50	)	)	PUNCT
ejpam-4858	246	51	,	,	PUNCT
ejpam-4858	246	52	there	there	PRON
ejpam-4858	246	53	exists	exist	VERB
ejpam-4858	246	54	an	an	DET
ejpam-4858	246	55	α-⋆-neighbourhood	α-⋆-neighbourhood	NUM
ejpam-4858	246	56	u	u	NOUN
ejpam-4858	246	57	of	of	ADP
ejpam-4858	246	58	x	x	SYM
ejpam-4858	246	59	such	such	ADJ
ejpam-4858	246	60	that	that	DET
ejpam-4858	246	61	f(u	f(u	PROPN
ejpam-4858	246	62	)	)	PUNCT
ejpam-4858	246	63	⊆	⊆	NUM
ejpam-4858	246	64	v	v	NOUN
ejpam-4858	246	65	;	;	PUNCT
ejpam-4858	246	66	(	(	PUNCT
ejpam-4858	246	67	7	7	X
ejpam-4858	246	68	)	)	PUNCT
ejpam-4858	246	69	f(⋆αcl(a	f(⋆αcl(a	NOUN
ejpam-4858	246	70	)	)	PUNCT
ejpam-4858	246	71	)	)	PUNCT
ejpam-4858	247	1	⊆	⊆	NUM
ejpam-4858	247	2	cl⋆(f(a	cl⋆(f(a	NOUN
ejpam-4858	247	3	)	)	PUNCT
ejpam-4858	247	4	)	)	PUNCT
ejpam-4858	247	5	for	for	ADP
ejpam-4858	247	6	every	every	DET
ejpam-4858	247	7	subset	subset	NOUN
ejpam-4858	247	8	a	a	PRON
ejpam-4858	247	9	of	of	ADP
ejpam-4858	247	10	x	x	PRON
ejpam-4858	247	11	;	;	PUNCT
ejpam-4858	247	12	(	(	PUNCT
ejpam-4858	247	13	8)	8)	NUM
ejpam-4858	247	14	f(sinti	f(sinti	NOUN
ejpam-4858	247	15	(	(	PUNCT
ejpam-4858	247	16	cl⋆(a	cl⋆(a	PROPN
ejpam-4858	247	17	)	)	PUNCT
ejpam-4858	247	18	)	)	PUNCT
ejpam-4858	247	19	)	)	PUNCT
ejpam-4858	248	1	⊆	⊆	NUM
ejpam-4858	248	2	cl⋆(f(a	cl⋆(f(a	NOUN
ejpam-4858	248	3	)	)	PUNCT
ejpam-4858	248	4	)	)	PUNCT
ejpam-4858	248	5	for	for	ADP
ejpam-4858	248	6	every	every	DET
ejpam-4858	248	7	subset	subset	NOUN
ejpam-4858	248	8	a	a	PRON
ejpam-4858	248	9	of	of	ADP
ejpam-4858	248	10	x	x	PRON
ejpam-4858	248	11	;	;	PUNCT
ejpam-4858	248	12	(	(	PUNCT
ejpam-4858	248	13	9	9	X
ejpam-4858	248	14	)	)	PUNCT
ejpam-4858	248	15	f(cl⋆(int(cl⋆(a	f(cl⋆(int(cl⋆(a	NOUN
ejpam-4858	248	16	)	)	PUNCT
ejpam-4858	248	17	)	)	PUNCT
ejpam-4858	248	18	)	)	PUNCT
ejpam-4858	248	19	)	)	PUNCT
ejpam-4858	249	1	⊆	⊆	NUM
ejpam-4858	249	2	cl⋆(f(a	cl⋆(f(a	NOUN
ejpam-4858	249	3	)	)	PUNCT
ejpam-4858	249	4	)	)	PUNCT
ejpam-4858	249	5	for	for	ADP
ejpam-4858	249	6	every	every	DET
ejpam-4858	249	7	subset	subset	NOUN
ejpam-4858	249	8	a	a	PRON
ejpam-4858	249	9	of	of	ADP
ejpam-4858	249	10	x.	x.	NOUN
ejpam-4858	249	11	definition	definition	NOUN
ejpam-4858	249	12	4	4	NUM
ejpam-4858	249	13	.	.	PUNCT
ejpam-4858	250	1	[	[	X
ejpam-4858	250	2	5	5	NUM
ejpam-4858	250	3	]	]	PUNCT
ejpam-4858	250	4	a	a	DET
ejpam-4858	250	5	subset	subset	NOUN
ejpam-4858	250	6	a	a	PRON
ejpam-4858	250	7	of	of	ADP
ejpam-4858	250	8	an	an	DET
ejpam-4858	250	9	ideal	ideal	ADJ
ejpam-4858	250	10	topological	topological	ADJ
ejpam-4858	250	11	space	space	NOUN
ejpam-4858	250	12	(	(	PUNCT
ejpam-4858	250	13	x	x	X
ejpam-4858	250	14	,	,	PUNCT
ejpam-4858	250	15	τ	τ	PROPN
ejpam-4858	250	16	,	,	PUNCT
ejpam-4858	250	17	i	i	PROPN
ejpam-4858	250	18	)	)	PUNCT
ejpam-4858	250	19	is	be	AUX
ejpam-4858	250	20	said	say	VERB
ejpam-4858	250	21	to	to	PART
ejpam-4858	250	22	be	be	AUX
ejpam-4858	250	23	:	:	PUNCT
ejpam-4858	250	24	(	(	PUNCT
ejpam-4858	250	25	1	1	X
ejpam-4858	250	26	)	)	PUNCT
ejpam-4858	250	27	⋆-paracompact	⋆-paracompact	NOUN
ejpam-4858	250	28	if	if	SCONJ
ejpam-4858	250	29	every	every	DET
ejpam-4858	250	30	cover	cover	NOUN
ejpam-4858	250	31	of	of	ADP
ejpam-4858	250	32	a	a	PRON
ejpam-4858	250	33	by	by	ADP
ejpam-4858	250	34	⋆-open	⋆-open	ADJ
ejpam-4858	250	35	sets	set	NOUN
ejpam-4858	250	36	of	of	ADP
ejpam-4858	250	37	x	x	VERB
ejpam-4858	250	38	is	be	AUX
ejpam-4858	250	39	refined	refine	VERB
ejpam-4858	250	40	by	by	ADP
ejpam-4858	250	41	a	a	DET
ejpam-4858	250	42	cover	cover	NOUN
ejpam-4858	250	43	of	of	ADP
ejpam-4858	250	44	a	a	PRON
ejpam-4858	250	45	which	which	PRON
ejpam-4858	250	46	consists	consist	VERB
ejpam-4858	250	47	of	of	ADP
ejpam-4858	250	48	⋆-open	⋆-open	ADJ
ejpam-4858	250	49	sets	set	NOUN
ejpam-4858	250	50	of	of	ADP
ejpam-4858	250	51	x	x	PUNCT
ejpam-4858	250	52	and	and	CCONJ
ejpam-4858	250	53	is	be	AUX
ejpam-4858	250	54	⋆-locally	⋆-locally	ADV
ejpam-4858	250	55	finite	finite	VERB
ejpam-4858	250	56	in	in	ADP
ejpam-4858	250	57	x	x	PROPN
ejpam-4858	250	58	;	;	PUNCT
ejpam-4858	250	59	c.	c.	PROPN
ejpam-4858	250	60	boonpok	boonpok	PROPN
ejpam-4858	250	61	,	,	PUNCT
ejpam-4858	250	62	j.	j.	PROPN
ejpam-4858	250	63	khampakdee	khampakdee	PROPN
ejpam-4858	250	64	/	/	PUNCT
ejpam-4858	250	65	eur	eur	PROPN
ejpam-4858	250	66	.	.	PUNCT
ejpam-4858	251	1	j.	j.	PROPN
ejpam-4858	251	2	pure	pure	PROPN
ejpam-4858	251	3	appl	appl	PROPN
ejpam-4858	251	4	.	.	PROPN
ejpam-4858	251	5	math	math	PROPN
ejpam-4858	251	6	,	,	PUNCT
ejpam-4858	251	7	17	17	NUM
ejpam-4858	251	8	(	(	PUNCT
ejpam-4858	251	9	1	1	NUM
ejpam-4858	251	10	)	)	PUNCT
ejpam-4858	251	11	(	(	PUNCT
ejpam-4858	251	12	2024	2024	NUM
ejpam-4858	251	13	)	)	PUNCT
ejpam-4858	251	14	,	,	PUNCT
ejpam-4858	251	15	201	201	NUM
ejpam-4858	251	16	-	-	SYM
ejpam-4858	251	17	211	211	NUM
ejpam-4858	251	18	209	209	NUM
ejpam-4858	251	19	(	(	PUNCT
ejpam-4858	251	20	2	2	NUM
ejpam-4858	251	21	)	)	PUNCT
ejpam-4858	251	22	⋆-regular	⋆-regular	ADJ
ejpam-4858	251	23	if	if	SCONJ
ejpam-4858	251	24	for	for	ADP
ejpam-4858	251	25	each	each	DET
ejpam-4858	251	26	x	x	SYM
ejpam-4858	251	27	∈	∈	PROPN
ejpam-4858	251	28	a	a	PRON
ejpam-4858	251	29	and	and	CCONJ
ejpam-4858	251	30	each	each	DET
ejpam-4858	251	31	⋆-open	⋆-open	ADV
ejpam-4858	251	32	set	set	VERB
ejpam-4858	251	33	u	u	NOUN
ejpam-4858	251	34	of	of	ADP
ejpam-4858	251	35	x	x	SYM
ejpam-4858	251	36	containing	contain	VERB
ejpam-4858	251	37	x	x	PRON
ejpam-4858	251	38	,	,	PUNCT
ejpam-4858	251	39	there	there	PRON
ejpam-4858	251	40	exists	exist	VERB
ejpam-4858	251	41	a	a	DET
ejpam-4858	251	42	⋆-open	⋆-open	ADJ
ejpam-4858	251	43	set	set	NOUN
ejpam-4858	251	44	v	v	NOUN
ejpam-4858	251	45	of	of	ADP
ejpam-4858	251	46	x	x	PUNCT
ejpam-4858	251	47	such	such	ADJ
ejpam-4858	251	48	that	that	SCONJ
ejpam-4858	251	49	x	x	SYM
ejpam-4858	251	50	∈	∈	NOUN
ejpam-4858	251	51	v	v	ADP
ejpam-4858	251	52	⊆	⊆	NUM
ejpam-4858	251	53	cl(v	cl(v	NOUN
ejpam-4858	251	54	)	)	PUNCT
ejpam-4858	251	55	⊆	⊆	NUM
ejpam-4858	251	56	u	u	NOUN
ejpam-4858	251	57	.	.	PUNCT
ejpam-4858	252	1	lemma	lemma	PROPN
ejpam-4858	252	2	4	4	NUM
ejpam-4858	252	3	.	.	PUNCT
ejpam-4858	253	1	[	[	X
ejpam-4858	253	2	5	5	X
ejpam-4858	253	3	]	]	PUNCT
ejpam-4858	253	4	let	let	VERB
ejpam-4858	253	5	a	a	PRON
ejpam-4858	253	6	be	be	AUX
ejpam-4858	253	7	a	a	DET
ejpam-4858	253	8	subset	subset	NOUN
ejpam-4858	253	9	of	of	ADP
ejpam-4858	253	10	an	an	DET
ejpam-4858	253	11	ideal	ideal	ADJ
ejpam-4858	253	12	topological	topological	ADJ
ejpam-4858	253	13	space	space	NOUN
ejpam-4858	253	14	(	(	PUNCT
ejpam-4858	253	15	x	x	X
ejpam-4858	253	16	,	,	PUNCT
ejpam-4858	253	17	τ	τ	PROPN
ejpam-4858	253	18	,	,	PUNCT
ejpam-4858	253	19	i	i	NOUN
ejpam-4858	253	20	)	)	PUNCT
ejpam-4858	253	21	.	.	PUNCT
ejpam-4858	254	1	if	if	SCONJ
ejpam-4858	254	2	a	a	PRON
ejpam-4858	254	3	is	be	AUX
ejpam-4858	254	4	a	a	DET
ejpam-4858	254	5	⋆-regular	⋆-regular	ADJ
ejpam-4858	254	6	⋆-paracompact	⋆-paracompact	NOUN
ejpam-4858	254	7	set	set	NOUN
ejpam-4858	254	8	of	of	ADP
ejpam-4858	254	9	x	x	PUNCT
ejpam-4858	254	10	and	and	CCONJ
ejpam-4858	254	11	each	each	DET
ejpam-4858	254	12	⋆-open	⋆-open	ADV
ejpam-4858	254	13	set	set	VERB
ejpam-4858	254	14	u	u	NOUN
ejpam-4858	254	15	containing	contain	VERB
ejpam-4858	254	16	a	a	PRON
ejpam-4858	254	17	,	,	PUNCT
ejpam-4858	254	18	then	then	ADV
ejpam-4858	254	19	there	there	PRON
ejpam-4858	254	20	exists	exist	VERB
ejpam-4858	254	21	a	a	DET
ejpam-4858	254	22	⋆-open	⋆-open	ADJ
ejpam-4858	254	23	set	set	VERB
ejpam-4858	254	24	v	v	ADP
ejpam-4858	254	25	such	such	DET
ejpam-4858	254	26	that	that	SCONJ
ejpam-4858	254	27	a	a	DET
ejpam-4858	254	28	⊆	⊆	NUM
ejpam-4858	254	29	v	v	ADP
ejpam-4858	254	30	⊆	⊆	NUM
ejpam-4858	254	31	cl(v	cl(v	NOUN
ejpam-4858	254	32	)	)	PUNCT
ejpam-4858	254	33	⊆	⊆	NUM
ejpam-4858	254	34	u	u	NOUN
ejpam-4858	254	35	.	.	PUNCT
ejpam-4858	255	1	a	a	DET
ejpam-4858	255	2	multifunction	multifunction	NOUN
ejpam-4858	255	3	f	f	NOUN
ejpam-4858	255	4	:	:	PUNCT
ejpam-4858	255	5	(	(	PUNCT
ejpam-4858	255	6	x	x	X
ejpam-4858	255	7	,	,	PUNCT
ejpam-4858	255	8	τ	τ	PROPN
ejpam-4858	255	9	,	,	PUNCT
ejpam-4858	255	10	i	i	NOUN
ejpam-4858	255	11	)	)	PUNCT
ejpam-4858	255	12	→	→	PUNCT
ejpam-4858	255	13	(	(	PUNCT
ejpam-4858	255	14	y	y	PROPN
ejpam-4858	255	15	,	,	PUNCT
ejpam-4858	255	16	σ	σ	PROPN
ejpam-4858	255	17	,	,	PUNCT
ejpam-4858	255	18	j	j	PROPN
ejpam-4858	255	19	)	)	PUNCT
ejpam-4858	255	20	is	be	AUX
ejpam-4858	255	21	called	call	VERB
ejpam-4858	255	22	punctually	punctually	ADJ
ejpam-4858	255	23	⋆-paracompact	⋆-paracompact	NOUN
ejpam-4858	255	24	(	(	PUNCT
ejpam-4858	255	25	resp	resp	NOUN
ejpam-4858	255	26	.	.	PUNCT
ejpam-4858	256	1	punctually	punctually	ADV
ejpam-4858	256	2	⋆-regular	⋆-regular	ADJ
ejpam-4858	256	3	)	)	PUNCT
ejpam-4858	256	4	if	if	SCONJ
ejpam-4858	256	5	for	for	ADP
ejpam-4858	256	6	each	each	DET
ejpam-4858	256	7	x	x	SYM
ejpam-4858	256	8	∈	∈	PROPN
ejpam-4858	256	9	x	x	X
ejpam-4858	256	10	,	,	PUNCT
ejpam-4858	256	11	f	f	PROPN
ejpam-4858	256	12	(	(	PUNCT
ejpam-4858	256	13	x	x	X
ejpam-4858	256	14	)	)	PUNCT
ejpam-4858	256	15	is	be	AUX
ejpam-4858	256	16	⋆-paracompact	⋆-paracompact	ADJ
ejpam-4858	256	17	(	(	PUNCT
ejpam-4858	256	18	resp	resp	NOUN
ejpam-4858	256	19	.	.	PUNCT
ejpam-4858	257	1	⋆-regular	⋆-regular	PROPN
ejpam-4858	257	2	)	)	PUNCT
ejpam-4858	257	3	.	.	PUNCT
ejpam-4858	258	1	by	by	ADP
ejpam-4858	258	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	258	3	)	)	PUNCT
ejpam-4858	258	4	:	:	PUNCT
ejpam-4858	258	5	(	(	PUNCT
ejpam-4858	258	6	x	x	X
ejpam-4858	258	7	,	,	PUNCT
ejpam-4858	258	8	τ	τ	PROPN
ejpam-4858	258	9	,	,	PUNCT
ejpam-4858	258	10	i	i	NOUN
ejpam-4858	258	11	)	)	PUNCT
ejpam-4858	258	12	→	→	PUNCT
ejpam-4858	258	13	(	(	PUNCT
ejpam-4858	258	14	y	y	PROPN
ejpam-4858	258	15	,	,	PUNCT
ejpam-4858	258	16	σ	σ	PROPN
ejpam-4858	258	17	,	,	PUNCT
ejpam-4858	258	18	j	j	PROPN
ejpam-4858	258	19	)	)	PUNCT
ejpam-4858	258	20	,	,	PUNCT
ejpam-4858	258	21	we	we	PRON
ejpam-4858	258	22	shall	shall	AUX
ejpam-4858	258	23	denote	denote	VERB
ejpam-4858	258	24	a	a	DET
ejpam-4858	258	25	multifunction	multifunction	NOUN
ejpam-4858	258	26	defined	define	VERB
ejpam-4858	258	27	as	as	SCONJ
ejpam-4858	258	28	follows	follow	VERB
ejpam-4858	258	29	:	:	PUNCT
ejpam-4858	259	1	[	[	X
ejpam-4858	259	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	259	3	)	)	PUNCT
ejpam-4858	259	4	]	]	PUNCT
ejpam-4858	259	5	(	(	PUNCT
ejpam-4858	259	6	x	x	X
ejpam-4858	259	7	)	)	PUNCT
ejpam-4858	259	8	=	=	SYM
ejpam-4858	259	9	⋆αclj	⋆αclj	X
ejpam-4858	259	10	(	(	PUNCT
ejpam-4858	259	11	f	f	X
ejpam-4858	259	12	(	(	PUNCT
ejpam-4858	259	13	x	x	NOUN
ejpam-4858	259	14	)	)	PUNCT
ejpam-4858	259	15	)	)	PUNCT
ejpam-4858	259	16	for	for	ADP
ejpam-4858	259	17	each	each	DET
ejpam-4858	259	18	x	x	SYM
ejpam-4858	259	19	∈	∈	PROPN
ejpam-4858	259	20	x.	x.	NOUN
ejpam-4858	259	21	lemma	lemma	PROPN
ejpam-4858	259	22	5	5	NUM
ejpam-4858	259	23	.	.	PUNCT
ejpam-4858	260	1	if	if	SCONJ
ejpam-4858	260	2	f	f	PROPN
ejpam-4858	260	3	:	:	PUNCT
ejpam-4858	260	4	(	(	PUNCT
ejpam-4858	260	5	x	x	X
ejpam-4858	260	6	,	,	PUNCT
ejpam-4858	260	7	τ	τ	PROPN
ejpam-4858	260	8	,	,	PUNCT
ejpam-4858	260	9	i	i	NOUN
ejpam-4858	260	10	)	)	PUNCT
ejpam-4858	260	11	→	→	PUNCT
ejpam-4858	260	12	(	(	PUNCT
ejpam-4858	260	13	y	y	PROPN
ejpam-4858	260	14	,	,	PUNCT
ejpam-4858	260	15	σ	σ	PROPN
ejpam-4858	260	16	,	,	PUNCT
ejpam-4858	260	17	j	j	PROPN
ejpam-4858	260	18	)	)	PUNCT
ejpam-4858	260	19	is	be	AUX
ejpam-4858	260	20	punctually	punctually	ADV
ejpam-4858	260	21	⋆-regular	⋆-regular	ADJ
ejpam-4858	260	22	and	and	CCONJ
ejpam-4858	260	23	punctually	punctually	ADJ
ejpam-4858	260	24	⋆paracompact	⋆paracompact	PROPN
ejpam-4858	260	25	,	,	PUNCT
ejpam-4858	260	26	then	then	ADV
ejpam-4858	260	27	[	[	X
ejpam-4858	260	28	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	260	29	)	)	PUNCT
ejpam-4858	260	30	]	]	PUNCT
ejpam-4858	261	1	+	+	ADJ
ejpam-4858	261	2	(	(	PUNCT
ejpam-4858	261	3	v	v	NOUN
ejpam-4858	261	4	)	)	PUNCT
ejpam-4858	261	5	=	=	PUNCT
ejpam-4858	262	1	f+(v	f+(v	NOUN
ejpam-4858	262	2	)	)	PUNCT
ejpam-4858	262	3	for	for	SCONJ
ejpam-4858	262	4	every	every	DET
ejpam-4858	262	5	⋆-open	⋆-open	NOUN
ejpam-4858	262	6	set	set	VERB
ejpam-4858	262	7	v	v	NOUN
ejpam-4858	262	8	of	of	ADP
ejpam-4858	262	9	y	y	PROPN
ejpam-4858	262	10	.	.	PUNCT
ejpam-4858	263	1	proof	proof	NOUN
ejpam-4858	263	2	.	.	PUNCT
ejpam-4858	264	1	let	let	VERB
ejpam-4858	264	2	v	v	PART
ejpam-4858	264	3	be	be	AUX
ejpam-4858	264	4	any	any	DET
ejpam-4858	264	5	⋆-open	⋆-open	ADJ
ejpam-4858	264	6	set	set	NOUN
ejpam-4858	264	7	of	of	ADP
ejpam-4858	264	8	y	y	PROPN
ejpam-4858	264	9	and	and	CCONJ
ejpam-4858	264	10	x	x	PROPN
ejpam-4858	264	11	∈	∈	PROPN
ejpam-4858	265	1	[	[	X
ejpam-4858	265	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	265	3	)	)	PUNCT
ejpam-4858	265	4	]	]	PUNCT
ejpam-4858	266	1	+	+	ADJ
ejpam-4858	266	2	(	(	PUNCT
ejpam-4858	266	3	v	v	NOUN
ejpam-4858	266	4	)	)	PUNCT
ejpam-4858	266	5	.	.	PUNCT
ejpam-4858	267	1	then	then	ADV
ejpam-4858	267	2	,	,	PUNCT
ejpam-4858	267	3	⋆αclj	⋆αclj	X
ejpam-4858	267	4	(	(	PUNCT
ejpam-4858	267	5	f	f	X
ejpam-4858	267	6	(	(	PUNCT
ejpam-4858	267	7	x	x	NOUN
ejpam-4858	267	8	)	)	PUNCT
ejpam-4858	267	9	)	)	PUNCT
ejpam-4858	267	10	⊆	⊆	NUM
ejpam-4858	267	11	v	v	NOUN
ejpam-4858	267	12	.	.	PUNCT
ejpam-4858	268	1	thus	thus	ADV
ejpam-4858	268	2	,	,	PUNCT
ejpam-4858	268	3	f	f	PROPN
ejpam-4858	268	4	(	(	PUNCT
ejpam-4858	268	5	x	x	X
ejpam-4858	268	6	)	)	PUNCT
ejpam-4858	268	7	⊆	⊆	NUM
ejpam-4858	268	8	v	v	NOUN
ejpam-4858	268	9	and	and	CCONJ
ejpam-4858	268	10	hence	hence	ADV
ejpam-4858	268	11	x	x	SYM
ejpam-4858	268	12	∈	∈	PROPN
ejpam-4858	268	13	f+(v	f+(v	NOUN
ejpam-4858	268	14	)	)	PUNCT
ejpam-4858	268	15	.	.	PUNCT
ejpam-4858	269	1	therefore	therefore	ADV
ejpam-4858	269	2	,	,	PUNCT
ejpam-4858	269	3	[	[	X
ejpam-4858	269	4	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	269	5	)	)	PUNCT
ejpam-4858	269	6	]	]	PUNCT
ejpam-4858	270	1	+	+	ADJ
ejpam-4858	270	2	(	(	PUNCT
ejpam-4858	270	3	v	v	NOUN
ejpam-4858	270	4	)	)	PUNCT
ejpam-4858	270	5	⊆	⊆	NUM
ejpam-4858	270	6	f+(v	f+(v	NOUN
ejpam-4858	270	7	)	)	PUNCT
ejpam-4858	270	8	.	.	PUNCT
ejpam-4858	271	1	on	on	ADP
ejpam-4858	271	2	the	the	DET
ejpam-4858	271	3	other	other	ADJ
ejpam-4858	271	4	hand	hand	NOUN
ejpam-4858	271	5	,	,	PUNCT
ejpam-4858	271	6	let	let	VERB
ejpam-4858	271	7	v	v	PART
ejpam-4858	271	8	be	be	AUX
ejpam-4858	271	9	any	any	DET
ejpam-4858	271	10	⋆-open	⋆-open	ADJ
ejpam-4858	271	11	set	set	NOUN
ejpam-4858	271	12	of	of	ADP
ejpam-4858	271	13	y	y	PROPN
ejpam-4858	271	14	and	and	CCONJ
ejpam-4858	271	15	x	x	PROPN
ejpam-4858	271	16	∈	∈	PROPN
ejpam-4858	271	17	f+(v	f+(v	NOUN
ejpam-4858	271	18	)	)	PUNCT
ejpam-4858	271	19	.	.	PUNCT
ejpam-4858	272	1	then	then	ADV
ejpam-4858	272	2	,	,	PUNCT
ejpam-4858	272	3	f	f	PROPN
ejpam-4858	272	4	(	(	PUNCT
ejpam-4858	272	5	x	x	X
ejpam-4858	272	6	)	)	PUNCT
ejpam-4858	272	7	⊆	⊆	NUM
ejpam-4858	272	8	v	v	NOUN
ejpam-4858	272	9	.	.	PUNCT
ejpam-4858	273	1	since	since	SCONJ
ejpam-4858	273	2	f	f	PROPN
ejpam-4858	273	3	(	(	PUNCT
ejpam-4858	273	4	x	x	X
ejpam-4858	273	5	)	)	PUNCT
ejpam-4858	273	6	is	be	AUX
ejpam-4858	273	7	punctually	punctually	ADV
ejpam-4858	273	8	⋆-regular	⋆-regular	ADJ
ejpam-4858	273	9	and	and	CCONJ
ejpam-4858	273	10	punctually	punctually	ADJ
ejpam-4858	273	11	⋆-paracompact	⋆-paracompact	NOUN
ejpam-4858	273	12	,	,	PUNCT
ejpam-4858	273	13	by	by	ADP
ejpam-4858	273	14	lemma	lemma	PROPN
ejpam-4858	273	15	4	4	NUM
ejpam-4858	273	16	,	,	PUNCT
ejpam-4858	273	17	there	there	PRON
ejpam-4858	273	18	exists	exist	VERB
ejpam-4858	273	19	a	a	DET
ejpam-4858	273	20	⋆-open	⋆-open	ADJ
ejpam-4858	273	21	set	set	NOUN
ejpam-4858	273	22	g	g	PROPN
ejpam-4858	273	23	such	such	ADJ
ejpam-4858	273	24	that	that	SCONJ
ejpam-4858	273	25	f	f	PROPN
ejpam-4858	273	26	(	(	PUNCT
ejpam-4858	273	27	x	x	X
ejpam-4858	273	28	)	)	PUNCT
ejpam-4858	273	29	⊆	⊆	NUM
ejpam-4858	273	30	g	g	ADP
ejpam-4858	273	31	⊆	⊆	NUM
ejpam-4858	273	32	cl(g	cl(g	NOUN
ejpam-4858	273	33	)	)	PUNCT
ejpam-4858	273	34	⊆	⊆	NUM
ejpam-4858	273	35	v	v	NOUN
ejpam-4858	273	36	;	;	PUNCT
ejpam-4858	273	37	hence	hence	ADV
ejpam-4858	273	38	⋆αclj	⋆αclj	NUM
ejpam-4858	273	39	(	(	PUNCT
ejpam-4858	273	40	f	f	X
ejpam-4858	273	41	(	(	PUNCT
ejpam-4858	273	42	x	x	NOUN
ejpam-4858	273	43	)	)	PUNCT
ejpam-4858	273	44	)	)	PUNCT
ejpam-4858	273	45	⊆	⊆	NUM
ejpam-4858	273	46	cl(g	cl(g	NOUN
ejpam-4858	273	47	)	)	PUNCT
ejpam-4858	273	48	⊆	⊆	NUM
ejpam-4858	273	49	v	v	NOUN
ejpam-4858	273	50	.	.	PUNCT
ejpam-4858	274	1	this	this	PRON
ejpam-4858	274	2	shows	show	VERB
ejpam-4858	274	3	that	that	SCONJ
ejpam-4858	274	4	x	x	X
ejpam-4858	274	5	∈	∈	PROPN
ejpam-4858	275	1	[	[	X
ejpam-4858	275	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	275	3	)	)	PUNCT
ejpam-4858	275	4	]	]	PUNCT
ejpam-4858	276	1	+	+	ADJ
ejpam-4858	276	2	(	(	PUNCT
ejpam-4858	276	3	v	v	NOUN
ejpam-4858	276	4	)	)	PUNCT
ejpam-4858	276	5	.	.	PUNCT
ejpam-4858	277	1	therefore	therefore	ADV
ejpam-4858	277	2	,	,	PUNCT
ejpam-4858	277	3	f+(v	f+(v	PROPN
ejpam-4858	277	4	)	)	PUNCT
ejpam-4858	278	1	⊆	⊆	NUM
ejpam-4858	278	2	[	[	X
ejpam-4858	278	3	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	278	4	)	)	PUNCT
ejpam-4858	278	5	]	]	PUNCT
ejpam-4858	279	1	+	+	ADJ
ejpam-4858	279	2	(	(	PUNCT
ejpam-4858	279	3	v	v	NOUN
ejpam-4858	279	4	)	)	PUNCT
ejpam-4858	279	5	.	.	PUNCT
ejpam-4858	280	1	consequently	consequently	ADV
ejpam-4858	280	2	,	,	PUNCT
ejpam-4858	280	3	we	we	PRON
ejpam-4858	280	4	obtain	obtain	VERB
ejpam-4858	280	5	[	[	X
ejpam-4858	280	6	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	280	7	)	)	PUNCT
ejpam-4858	280	8	]	]	PUNCT
ejpam-4858	281	1	+	+	ADJ
ejpam-4858	281	2	(	(	PUNCT
ejpam-4858	281	3	v	v	NOUN
ejpam-4858	281	4	)	)	PUNCT
ejpam-4858	281	5	=	=	PUNCT
ejpam-4858	281	6	f+(v	f+(v	NOUN
ejpam-4858	281	7	)	)	PUNCT
ejpam-4858	281	8	.	.	PUNCT
ejpam-4858	282	1	theorem	theorem	NOUN
ejpam-4858	282	2	5	5	NUM
ejpam-4858	282	3	.	.	PUNCT
ejpam-4858	283	1	let	let	VERB
ejpam-4858	283	2	f	f	NOUN
ejpam-4858	283	3	:	:	PUNCT
ejpam-4858	283	4	(	(	PUNCT
ejpam-4858	283	5	x	x	X
ejpam-4858	283	6	,	,	PUNCT
ejpam-4858	283	7	τ	τ	PROPN
ejpam-4858	283	8	,	,	PUNCT
ejpam-4858	283	9	i	i	NOUN
ejpam-4858	283	10	)	)	PUNCT
ejpam-4858	283	11	→	→	PUNCT
ejpam-4858	283	12	(	(	PUNCT
ejpam-4858	283	13	y	y	PROPN
ejpam-4858	283	14	,	,	PUNCT
ejpam-4858	283	15	σ	σ	PROPN
ejpam-4858	283	16	,	,	PUNCT
ejpam-4858	283	17	j	j	PROPN
ejpam-4858	283	18	)	)	PUNCT
ejpam-4858	283	19	be	be	AUX
ejpam-4858	283	20	punctually	punctually	ADV
ejpam-4858	283	21	⋆-regular	⋆-regular	ADJ
ejpam-4858	283	22	and	and	CCONJ
ejpam-4858	283	23	punctually	punctually	ADJ
ejpam-4858	283	24	⋆paracompact	⋆paracompact	PROPN
ejpam-4858	283	25	.	.	PUNCT
ejpam-4858	284	1	then	then	ADV
ejpam-4858	284	2	f	f	PROPN
ejpam-4858	284	3	is	be	AUX
ejpam-4858	284	4	upper	upper	ADJ
ejpam-4858	284	5	α-⋆-continuous	α-⋆-continuous	X
ejpam-4858	284	6	if	if	SCONJ
ejpam-4858	285	1	and	and	CCONJ
ejpam-4858	285	2	only	only	ADV
ejpam-4858	285	3	if	if	SCONJ
ejpam-4858	285	4	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	285	5	)	)	PUNCT
ejpam-4858	285	6	:	:	PUNCT
ejpam-4858	285	7	(	(	PUNCT
ejpam-4858	285	8	x	x	X
ejpam-4858	285	9	,	,	PUNCT
ejpam-4858	285	10	τ	τ	PROPN
ejpam-4858	285	11	,	,	PUNCT
ejpam-4858	285	12	i	i	NOUN
ejpam-4858	285	13	)	)	PUNCT
ejpam-4858	285	14	→	→	PUNCT
ejpam-4858	285	15	(	(	PUNCT
ejpam-4858	285	16	y	y	PROPN
ejpam-4858	285	17	,	,	PUNCT
ejpam-4858	285	18	σ	σ	PROPN
ejpam-4858	285	19	,	,	PUNCT
ejpam-4858	285	20	j	j	PROPN
ejpam-4858	285	21	)	)	PUNCT
ejpam-4858	285	22	is	be	AUX
ejpam-4858	285	23	upper	upper	ADJ
ejpam-4858	285	24	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	285	25	.	.	PUNCT
ejpam-4858	286	1	proof	proof	NOUN
ejpam-4858	286	2	.	.	PUNCT
ejpam-4858	287	1	suppose	suppose	VERB
ejpam-4858	287	2	that	that	SCONJ
ejpam-4858	287	3	f	f	PROPN
ejpam-4858	287	4	is	be	AUX
ejpam-4858	287	5	upper	upper	ADJ
ejpam-4858	287	6	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	287	7	.	.	PUNCT
ejpam-4858	288	1	let	let	VERB
ejpam-4858	288	2	x	x	SYM
ejpam-4858	288	3	∈	∈	PROPN
ejpam-4858	288	4	x	x	X
ejpam-4858	288	5	and	and	CCONJ
ejpam-4858	288	6	v	v	X
ejpam-4858	288	7	be	be	AUX
ejpam-4858	288	8	any	any	DET
ejpam-4858	288	9	⋆-open	⋆-open	ADJ
ejpam-4858	288	10	set	set	NOUN
ejpam-4858	288	11	of	of	ADP
ejpam-4858	288	12	y	y	PRON
ejpam-4858	288	13	such	such	ADJ
ejpam-4858	288	14	that	that	SCONJ
ejpam-4858	288	15	⋆αclj	⋆αclj	NUM
ejpam-4858	288	16	(	(	PUNCT
ejpam-4858	288	17	f	f	X
ejpam-4858	288	18	(	(	PUNCT
ejpam-4858	288	19	x	x	NOUN
ejpam-4858	288	20	)	)	PUNCT
ejpam-4858	288	21	)	)	PUNCT
ejpam-4858	288	22	⊆	⊆	NUM
ejpam-4858	288	23	v	v	NOUN
ejpam-4858	288	24	.	.	PUNCT
ejpam-4858	289	1	by	by	ADP
ejpam-4858	289	2	lemma	lemma	PROPN
ejpam-4858	289	3	5	5	NUM
ejpam-4858	289	4	,	,	PUNCT
ejpam-4858	289	5	we	we	PRON
ejpam-4858	289	6	have	have	VERB
ejpam-4858	289	7	x	x	X
ejpam-4858	289	8	∈	∈	PROPN
ejpam-4858	289	9	[	[	X
ejpam-4858	289	10	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	289	11	)	)	PUNCT
ejpam-4858	289	12	]	]	PUNCT
ejpam-4858	290	1	+	+	ADJ
ejpam-4858	290	2	(	(	PUNCT
ejpam-4858	290	3	v	v	NOUN
ejpam-4858	290	4	)	)	PUNCT
ejpam-4858	290	5	=	=	PUNCT
ejpam-4858	290	6	f+(v	f+(v	NOUN
ejpam-4858	290	7	)	)	PUNCT
ejpam-4858	290	8	.	.	PUNCT
ejpam-4858	291	1	since	since	SCONJ
ejpam-4858	291	2	f	f	PROPN
ejpam-4858	291	3	is	be	AUX
ejpam-4858	291	4	upper	upper	ADJ
ejpam-4858	291	5	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	291	6	,	,	PUNCT
ejpam-4858	291	7	there	there	PRON
ejpam-4858	291	8	exists	exist	VERB
ejpam-4858	291	9	an	an	DET
ejpam-4858	291	10	α-⋆-open	α-⋆-open	PROPN
ejpam-4858	291	11	set	set	VERB
ejpam-4858	291	12	u	u	NOUN
ejpam-4858	291	13	of	of	ADP
ejpam-4858	291	14	x	x	PUNCT
ejpam-4858	291	15	containing	contain	VERB
ejpam-4858	291	16	x	x	PUNCT
ejpam-4858	291	17	such	such	ADJ
ejpam-4858	291	18	that	that	SCONJ
ejpam-4858	291	19	f	f	PROPN
ejpam-4858	291	20	(	(	PUNCT
ejpam-4858	291	21	u	u	NOUN
ejpam-4858	291	22	)	)	PUNCT
ejpam-4858	291	23	⊆	⊆	NUM
ejpam-4858	291	24	v	v	NOUN
ejpam-4858	291	25	.	.	PUNCT
ejpam-4858	292	1	sine	sine	NOUN
ejpam-4858	292	2	f	f	PROPN
ejpam-4858	292	3	(	(	PUNCT
ejpam-4858	292	4	z	z	NOUN
ejpam-4858	292	5	)	)	PUNCT
ejpam-4858	292	6	is	be	AUX
ejpam-4858	292	7	punctually	punctually	ADV
ejpam-4858	292	8	⋆-regular	⋆-regular	ADJ
ejpam-4858	292	9	and	and	CCONJ
ejpam-4858	292	10	punctually	punctually	ADJ
ejpam-4858	292	11	⋆-paracompact	⋆-paracompact	NOUN
ejpam-4858	292	12	for	for	ADP
ejpam-4858	292	13	each	each	DET
ejpam-4858	292	14	z	z	NOUN
ejpam-4858	292	15	∈	∈	PROPN
ejpam-4858	292	16	u	u	NOUN
ejpam-4858	292	17	,	,	PUNCT
ejpam-4858	292	18	by	by	ADP
ejpam-4858	292	19	lemma	lemma	PROPN
ejpam-4858	292	20	4	4	NUM
ejpam-4858	292	21	,	,	PUNCT
ejpam-4858	292	22	there	there	PRON
ejpam-4858	292	23	exists	exist	VERB
ejpam-4858	292	24	a	a	DET
ejpam-4858	292	25	⋆-open	⋆-open	ADJ
ejpam-4858	292	26	set	set	NOUN
ejpam-4858	292	27	g	g	PROPN
ejpam-4858	292	28	such	such	ADJ
ejpam-4858	292	29	that	that	SCONJ
ejpam-4858	292	30	f	f	PROPN
ejpam-4858	292	31	(	(	PUNCT
ejpam-4858	292	32	z	z	NOUN
ejpam-4858	292	33	)	)	PUNCT
ejpam-4858	292	34	⊆	⊆	NUM
ejpam-4858	292	35	g	g	ADP
ejpam-4858	292	36	⊆	⊆	NUM
ejpam-4858	292	37	cl(g	cl(g	NOUN
ejpam-4858	292	38	)	)	PUNCT
ejpam-4858	292	39	⊆	⊆	NUM
ejpam-4858	292	40	v	v	NOUN
ejpam-4858	292	41	.	.	PUNCT
ejpam-4858	293	1	thus	thus	ADV
ejpam-4858	293	2	,	,	PUNCT
ejpam-4858	293	3	⋆αclj	⋆αclj	X
ejpam-4858	293	4	(	(	PUNCT
ejpam-4858	293	5	f	f	X
ejpam-4858	293	6	(	(	PUNCT
ejpam-4858	293	7	z	z	NOUN
ejpam-4858	293	8	)	)	PUNCT
ejpam-4858	293	9	)	)	PUNCT
ejpam-4858	293	10	⊆	⊆	NUM
ejpam-4858	293	11	cl(g	cl(g	NOUN
ejpam-4858	293	12	)	)	PUNCT
ejpam-4858	293	13	⊆	⊆	NUM
ejpam-4858	293	14	v	v	NOUN
ejpam-4858	293	15	and	and	CCONJ
ejpam-4858	293	16	hence	hence	ADV
ejpam-4858	293	17	⋆αclj	⋆αclj	NUM
ejpam-4858	293	18	(	(	PUNCT
ejpam-4858	293	19	f	f	X
ejpam-4858	293	20	(	(	PUNCT
ejpam-4858	293	21	u	u	NOUN
ejpam-4858	293	22	)	)	PUNCT
ejpam-4858	293	23	)	)	PUNCT
ejpam-4858	294	1	⊆	⊆	NUM
ejpam-4858	294	2	v	v	NOUN
ejpam-4858	294	3	.	.	PUNCT
ejpam-4858	295	1	this	this	PRON
ejpam-4858	295	2	shows	show	VERB
ejpam-4858	295	3	that	that	SCONJ
ejpam-4858	295	4	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	295	5	)	)	PUNCT
ejpam-4858	295	6	is	be	AUX
ejpam-4858	295	7	upper	upper	ADJ
ejpam-4858	295	8	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	295	9	.	.	PUNCT
ejpam-4858	296	1	conversely	conversely	ADV
ejpam-4858	296	2	,	,	PUNCT
ejpam-4858	296	3	suppose	suppose	VERB
ejpam-4858	296	4	that	that	SCONJ
ejpam-4858	296	5	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	296	6	)	)	PUNCT
ejpam-4858	296	7	is	be	AUX
ejpam-4858	296	8	upper	upper	ADJ
ejpam-4858	296	9	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	296	10	.	.	PUNCT
ejpam-4858	297	1	let	let	VERB
ejpam-4858	297	2	x	x	SYM
ejpam-4858	297	3	∈	∈	PROPN
ejpam-4858	297	4	x	x	X
ejpam-4858	297	5	and	and	CCONJ
ejpam-4858	297	6	v	v	X
ejpam-4858	297	7	be	be	AUX
ejpam-4858	297	8	any	any	DET
ejpam-4858	297	9	⋆-open	⋆-open	ADJ
ejpam-4858	297	10	set	set	NOUN
ejpam-4858	297	11	of	of	ADP
ejpam-4858	297	12	y	y	PRON
ejpam-4858	297	13	such	such	ADJ
ejpam-4858	297	14	that	that	SCONJ
ejpam-4858	297	15	f	f	PROPN
ejpam-4858	297	16	(	(	PUNCT
ejpam-4858	297	17	x	x	X
ejpam-4858	297	18	)	)	PUNCT
ejpam-4858	297	19	⊆	⊆	NUM
ejpam-4858	297	20	v	v	NOUN
ejpam-4858	297	21	.	.	PUNCT
ejpam-4858	298	1	by	by	ADP
ejpam-4858	298	2	lemma	lemma	PROPN
ejpam-4858	298	3	5	5	NUM
ejpam-4858	298	4	,	,	PUNCT
ejpam-4858	298	5	we	we	PRON
ejpam-4858	298	6	have	have	VERB
ejpam-4858	298	7	x	x	X
ejpam-4858	298	8	∈	∈	NOUN
ejpam-4858	298	9	f+(v	f+(v	NOUN
ejpam-4858	298	10	)	)	PUNCT
ejpam-4858	299	1	=	=	PUNCT
ejpam-4858	300	1	[	[	X
ejpam-4858	300	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	300	3	)	)	PUNCT
ejpam-4858	300	4	]	]	PUNCT
ejpam-4858	300	5	+	+	ADJ
ejpam-4858	300	6	(	(	PUNCT
ejpam-4858	300	7	v	v	NOUN
ejpam-4858	300	8	)	)	PUNCT
ejpam-4858	300	9	and	and	CCONJ
ejpam-4858	300	10	hence	hence	ADV
ejpam-4858	300	11	⋆αclj	⋆αclj	NUM
ejpam-4858	300	12	(	(	PUNCT
ejpam-4858	300	13	f	f	X
ejpam-4858	300	14	(	(	PUNCT
ejpam-4858	300	15	x	x	NOUN
ejpam-4858	300	16	)	)	PUNCT
ejpam-4858	300	17	)	)	PUNCT
ejpam-4858	300	18	⊆	⊆	NUM
ejpam-4858	300	19	v	v	NOUN
ejpam-4858	300	20	.	.	PUNCT
ejpam-4858	301	1	since	since	SCONJ
ejpam-4858	301	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	301	3	)	)	PUNCT
ejpam-4858	301	4	is	be	AUX
ejpam-4858	301	5	upper	upper	ADJ
ejpam-4858	301	6	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	301	7	,	,	PUNCT
ejpam-4858	301	8	there	there	PRON
ejpam-4858	301	9	exists	exist	VERB
ejpam-4858	301	10	an	an	DET
ejpam-4858	301	11	α-⋆open	α-⋆open	NOUN
ejpam-4858	301	12	set	set	NOUN
ejpam-4858	301	13	u	u	NOUN
ejpam-4858	301	14	of	of	ADP
ejpam-4858	301	15	x	x	PUNCT
ejpam-4858	301	16	containing	contain	VERB
ejpam-4858	301	17	x	x	PUNCT
ejpam-4858	301	18	such	such	ADJ
ejpam-4858	301	19	that	that	SCONJ
ejpam-4858	301	20	⋆αclj	⋆αclj	NUM
ejpam-4858	301	21	(	(	PUNCT
ejpam-4858	301	22	f	f	X
ejpam-4858	301	23	(	(	PUNCT
ejpam-4858	301	24	u	u	NOUN
ejpam-4858	301	25	)	)	PUNCT
ejpam-4858	301	26	)	)	PUNCT
ejpam-4858	301	27	⊆	⊆	NUM
ejpam-4858	301	28	v	v	NOUN
ejpam-4858	301	29	;	;	PUNCT
ejpam-4858	301	30	hence	hence	ADV
ejpam-4858	301	31	f	f	PROPN
ejpam-4858	301	32	(	(	PUNCT
ejpam-4858	301	33	u	u	NOUN
ejpam-4858	301	34	)	)	PUNCT
ejpam-4858	301	35	⊆	⊆	NUM
ejpam-4858	301	36	v	v	NOUN
ejpam-4858	301	37	.	.	PUNCT
ejpam-4858	302	1	this	this	PRON
ejpam-4858	302	2	shows	show	VERB
ejpam-4858	302	3	that	that	SCONJ
ejpam-4858	302	4	f	f	PROPN
ejpam-4858	302	5	is	be	AUX
ejpam-4858	302	6	upper	upper	ADJ
ejpam-4858	302	7	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-4858	302	8	.	.	PUNCT
ejpam-4858	303	1	lemma	lemma	PROPN
ejpam-4858	303	2	6	6	NUM
ejpam-4858	303	3	.	.	PUNCT
ejpam-4858	303	4	for	for	ADP
ejpam-4858	303	5	a	a	DET
ejpam-4858	303	6	multifunction	multifunction	NOUN
ejpam-4858	303	7	f	f	NOUN
ejpam-4858	303	8	:	:	PUNCT
ejpam-4858	303	9	(	(	PUNCT
ejpam-4858	303	10	x	x	X
ejpam-4858	303	11	,	,	PUNCT
ejpam-4858	303	12	τ	τ	PROPN
ejpam-4858	303	13	,	,	PUNCT
ejpam-4858	303	14	i	i	NOUN
ejpam-4858	303	15	)	)	PUNCT
ejpam-4858	303	16	→	→	PUNCT
ejpam-4858	303	17	(	(	PUNCT
ejpam-4858	303	18	y	y	PROPN
ejpam-4858	303	19	,	,	PUNCT
ejpam-4858	303	20	σ	σ	PROPN
ejpam-4858	303	21	,	,	PUNCT
ejpam-4858	303	22	j	j	PROPN
ejpam-4858	303	23	)	)	PUNCT
ejpam-4858	303	24	,	,	PUNCT
ejpam-4858	303	25	it	it	PRON
ejpam-4858	303	26	follows	follow	VERB
ejpam-4858	303	27	that	that	SCONJ
ejpam-4858	303	28	for	for	SCONJ
ejpam-4858	303	29	each	each	DET
ejpam-4858	303	30	⋆-open	⋆-open	ADV
ejpam-4858	303	31	set	set	VERB
ejpam-4858	303	32	v	v	NUM
ejpam-4858	303	33	of	of	ADP
ejpam-4858	303	34	y	y	PROPN
ejpam-4858	304	1	[	[	X
ejpam-4858	304	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	304	3	)	)	PUNCT
ejpam-4858	304	4	]	]	X
ejpam-4858	304	5	−(v	−(v	NOUN
ejpam-4858	304	6	)	)	PUNCT
ejpam-4858	304	7	=	=	SYM
ejpam-4858	304	8	f−(v	f−(v	ADJ
ejpam-4858	304	9	)	)	PUNCT
ejpam-4858	304	10	.	.	PUNCT
ejpam-4858	305	1	references	reference	NOUN
ejpam-4858	305	2	210	210	NUM
ejpam-4858	305	3	proof	proof	NOUN
ejpam-4858	305	4	.	.	PUNCT
ejpam-4858	306	1	suppose	suppose	VERB
ejpam-4858	306	2	that	that	SCONJ
ejpam-4858	306	3	v	v	NOUN
ejpam-4858	306	4	is	be	AUX
ejpam-4858	306	5	any	any	DET
ejpam-4858	306	6	⋆-open	⋆-open	ADJ
ejpam-4858	306	7	set	set	NOUN
ejpam-4858	306	8	of	of	ADP
ejpam-4858	306	9	y	y	PROPN
ejpam-4858	306	10	.	.	PUNCT
ejpam-4858	307	1	let	let	VERB
ejpam-4858	307	2	x	x	PUNCT
ejpam-4858	307	3	∈	∈	PROPN
ejpam-4858	307	4	[	[	X
ejpam-4858	307	5	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	307	6	)	)	PUNCT
ejpam-4858	307	7	]	]	X
ejpam-4858	307	8	−(v	−(v	NOUN
ejpam-4858	307	9	)	)	PUNCT
ejpam-4858	307	10	.	.	PUNCT
ejpam-4858	308	1	then	then	ADV
ejpam-4858	308	2	,	,	PUNCT
ejpam-4858	308	3	we	we	PRON
ejpam-4858	308	4	have	have	VERB
ejpam-4858	308	5	⋆αclj	⋆αclj	NUM
ejpam-4858	308	6	(	(	PUNCT
ejpam-4858	308	7	f	f	X
ejpam-4858	308	8	(	(	PUNCT
ejpam-4858	308	9	x	x	NOUN
ejpam-4858	308	10	)	)	PUNCT
ejpam-4858	308	11	)	)	PUNCT
ejpam-4858	308	12	∩	∩	NOUN
ejpam-4858	308	13	v	v	ADP
ejpam-4858	308	14	̸=	̸=	PROPN
ejpam-4858	308	15	∅	∅	NOUN
ejpam-4858	308	16	and	and	CCONJ
ejpam-4858	308	17	hence	hence	ADV
ejpam-4858	308	18	f	f	PROPN
ejpam-4858	308	19	(	(	PUNCT
ejpam-4858	308	20	x	x	NOUN
ejpam-4858	308	21	)	)	PUNCT
ejpam-4858	308	22	∩	∩	NOUN
ejpam-4858	308	23	v	v	ADP
ejpam-4858	308	24	̸=	̸=	PROPN
ejpam-4858	308	25	∅.	∅.	ADV
ejpam-4858	308	26	thus	thus	ADV
ejpam-4858	308	27	,	,	PUNCT
ejpam-4858	308	28	x	x	SYM
ejpam-4858	308	29	∈	∈	PROPN
ejpam-4858	308	30	f−(v	f−(v	NOUN
ejpam-4858	308	31	)	)	PUNCT
ejpam-4858	308	32	.	.	PUNCT
ejpam-4858	309	1	this	this	PRON
ejpam-4858	309	2	shows	show	VERB
ejpam-4858	309	3	that	that	SCONJ
ejpam-4858	309	4	[	[	X
ejpam-4858	309	5	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	309	6	)	)	PUNCT
ejpam-4858	309	7	]	]	X
ejpam-4858	309	8	−(v	−(v	NOUN
ejpam-4858	309	9	)	)	PUNCT
ejpam-4858	309	10	⊆	⊆	NUM
ejpam-4858	309	11	f−(v	f−(v	NOUN
ejpam-4858	309	12	)	)	PUNCT
ejpam-4858	309	13	.	.	PUNCT
ejpam-4858	310	1	on	on	ADP
ejpam-4858	310	2	the	the	DET
ejpam-4858	310	3	other	other	ADJ
ejpam-4858	310	4	hand	hand	NOUN
ejpam-4858	310	5	,	,	PUNCT
ejpam-4858	310	6	let	let	VERB
ejpam-4858	310	7	x	x	PUNCT
ejpam-4858	310	8	∈	∈	PROPN
ejpam-4858	310	9	f−(v	f−(v	NOUN
ejpam-4858	310	10	)	)	PUNCT
ejpam-4858	310	11	.	.	PUNCT
ejpam-4858	311	1	then	then	ADV
ejpam-4858	311	2	,	,	PUNCT
ejpam-4858	311	3	∅	∅	NOUN
ejpam-4858	311	4	=	=	NOUN
ejpam-4858	311	5	̸	̸	NUM
ejpam-4858	311	6	f	f	NOUN
ejpam-4858	311	7	(	(	PUNCT
ejpam-4858	311	8	x	x	NOUN
ejpam-4858	311	9	)	)	PUNCT
ejpam-4858	311	10	∩	∩	NOUN
ejpam-4858	311	11	v	v	ADP
ejpam-4858	311	12	⊆	⊆	NUM
ejpam-4858	311	13	⋆αclj	⋆αclj	NUM
ejpam-4858	311	14	(	(	PUNCT
ejpam-4858	311	15	f	f	X
ejpam-4858	311	16	(	(	PUNCT
ejpam-4858	311	17	x	x	NOUN
ejpam-4858	311	18	)	)	PUNCT
ejpam-4858	311	19	)	)	PUNCT
ejpam-4858	311	20	∩	∩	PROPN
ejpam-4858	311	21	v.	v.	CCONJ
ejpam-4858	311	22	therefore	therefore	ADV
ejpam-4858	311	23	,	,	PUNCT
ejpam-4858	311	24	x	x	PUNCT
ejpam-4858	311	25	∈	∈	PROPN
ejpam-4858	312	1	[	[	X
ejpam-4858	312	2	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	312	3	)	)	PUNCT
ejpam-4858	312	4	]	]	X
ejpam-4858	312	5	−(v	−(v	NOUN
ejpam-4858	312	6	)	)	PUNCT
ejpam-4858	312	7	.	.	PUNCT
ejpam-4858	313	1	thus	thus	ADV
ejpam-4858	313	2	,	,	PUNCT
ejpam-4858	313	3	f−(v	f−(v	ADJ
ejpam-4858	313	4	)	)	PUNCT
ejpam-4858	314	1	⊆	⊆	NUM
ejpam-4858	314	2	[	[	X
ejpam-4858	314	3	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	314	4	)	)	PUNCT
ejpam-4858	314	5	]	]	X
ejpam-4858	314	6	−(v	−(v	NOUN
ejpam-4858	314	7	)	)	PUNCT
ejpam-4858	314	8	and	and	CCONJ
ejpam-4858	314	9	hence	hence	ADV
ejpam-4858	314	10	[	[	X
ejpam-4858	314	11	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	314	12	)	)	PUNCT
ejpam-4858	314	13	]	]	X
ejpam-4858	314	14	−(v	−(v	NOUN
ejpam-4858	314	15	)	)	PUNCT
ejpam-4858	315	1	=	=	SYM
ejpam-4858	315	2	f−(v	f−(v	ADJ
ejpam-4858	315	3	)	)	PUNCT
ejpam-4858	315	4	.	.	PUNCT
ejpam-4858	316	1	theorem	theorem	VERB
ejpam-4858	316	2	6	6	NUM
ejpam-4858	316	3	.	.	PUNCT
ejpam-4858	317	1	a	a	DET
ejpam-4858	317	2	multifunction	multifunction	NOUN
ejpam-4858	317	3	f	f	NOUN
ejpam-4858	317	4	:	:	PUNCT
ejpam-4858	317	5	(	(	PUNCT
ejpam-4858	317	6	x	x	X
ejpam-4858	317	7	,	,	PUNCT
ejpam-4858	317	8	τ	τ	PROPN
ejpam-4858	317	9	,	,	PUNCT
ejpam-4858	317	10	i	i	NOUN
ejpam-4858	317	11	)	)	PUNCT
ejpam-4858	317	12	→	→	PUNCT
ejpam-4858	317	13	(	(	PUNCT
ejpam-4858	317	14	y	y	PROPN
ejpam-4858	317	15	,	,	PUNCT
ejpam-4858	317	16	σ	σ	PROPN
ejpam-4858	317	17	,	,	PUNCT
ejpam-4858	317	18	j	j	PROPN
ejpam-4858	317	19	)	)	PUNCT
ejpam-4858	317	20	is	be	AUX
ejpam-4858	317	21	lower	low	ADJ
ejpam-4858	317	22	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	317	23	if	if	SCONJ
ejpam-4858	318	1	and	and	CCONJ
ejpam-4858	318	2	only	only	ADV
ejpam-4858	318	3	if	if	SCONJ
ejpam-4858	318	4	cl⋆α(f	cl⋆α(f	NOUN
ejpam-4858	318	5	)	)	PUNCT
ejpam-4858	318	6	:	:	PUNCT
ejpam-4858	318	7	(	(	PUNCT
ejpam-4858	318	8	x	x	X
ejpam-4858	318	9	,	,	PUNCT
ejpam-4858	318	10	τ	τ	PROPN
ejpam-4858	318	11	,	,	PUNCT
ejpam-4858	318	12	i	i	NOUN
ejpam-4858	318	13	)	)	PUNCT
ejpam-4858	318	14	→	→	PUNCT
ejpam-4858	318	15	(	(	PUNCT
ejpam-4858	318	16	y	y	PROPN
ejpam-4858	318	17	,	,	PUNCT
ejpam-4858	318	18	σ	σ	PROPN
ejpam-4858	318	19	,	,	PUNCT
ejpam-4858	318	20	j	j	PROPN
ejpam-4858	318	21	)	)	PUNCT
ejpam-4858	318	22	is	be	AUX
ejpam-4858	318	23	lower	low	ADJ
ejpam-4858	318	24	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	318	25	.	.	PUNCT
ejpam-4858	319	1	proof	proof	NOUN
ejpam-4858	319	2	.	.	PUNCT
ejpam-4858	320	1	by	by	ADP
ejpam-4858	320	2	utilizing	utilize	VERB
ejpam-4858	320	3	lemma	lemma	PROPN
ejpam-4858	320	4	6	6	NUM
ejpam-4858	320	5	,	,	PUNCT
ejpam-4858	320	6	this	this	PRON
ejpam-4858	320	7	can	can	AUX
ejpam-4858	320	8	be	be	AUX
ejpam-4858	320	9	proved	prove	VERB
ejpam-4858	320	10	similarly	similarly	ADV
ejpam-4858	320	11	to	to	ADP
ejpam-4858	320	12	that	that	PRON
ejpam-4858	320	13	of	of	ADP
ejpam-4858	320	14	theorem	theorem	ADJ
ejpam-4858	320	15	5	5	NUM
ejpam-4858	320	16	.	.	PUNCT
ejpam-4858	320	17	acknowledgements	acknowledgement	NOUN
ejpam-4858	320	18	this	this	DET
ejpam-4858	320	19	research	research	NOUN
ejpam-4858	320	20	project	project	NOUN
ejpam-4858	320	21	was	be	AUX
ejpam-4858	320	22	financially	financially	ADV
ejpam-4858	320	23	supported	support	VERB
ejpam-4858	320	24	by	by	ADP
ejpam-4858	320	25	mahasarakham	mahasarakham	PROPN
ejpam-4858	320	26	university	university	PROPN
ejpam-4858	320	27	.	.	PUNCT
ejpam-4858	321	1	references	reference	NOUN
ejpam-4858	321	2	[	[	X
ejpam-4858	321	3	1	1	NUM
ejpam-4858	321	4	]	]	PUNCT
ejpam-4858	321	5	a.	a.	NOUN
ejpam-4858	321	6	açikgöz	açikgöz	PROPN
ejpam-4858	321	7	,	,	PUNCT
ejpam-4858	321	8	t.	t.	PROPN
ejpam-4858	321	9	noiri	noiri	PROPN
ejpam-4858	321	10	,	,	PUNCT
ejpam-4858	321	11	and	and	CCONJ
ejpam-4858	321	12	ş.	ş.	PROPN
ejpam-4858	321	13	yüksel	yüksel	PROPN
ejpam-4858	321	14	.	.	PUNCT
ejpam-4858	322	1	on	on	ADP
ejpam-4858	322	2	α	α	PROPN
ejpam-4858	322	3	-	-	ADJ
ejpam-4858	322	4	i	i	PRON
ejpam-4858	322	5	-continuous	-continuous	ADJ
ejpam-4858	322	6	and	and	CCONJ
ejpam-4858	322	7	α	α	NOUN
ejpam-4858	322	8	-	-	PUNCT
ejpam-4858	322	9	i	i	NOUN
ejpam-4858	322	10	-open	-open	NOUN
ejpam-4858	322	11	functions	function	NOUN
ejpam-4858	322	12	.	.	PUNCT
ejpam-4858	323	1	acta	acta	PROPN
ejpam-4858	323	2	mathematica	mathematica	PROPN
ejpam-4858	323	3	hungarica	hungarica	PROPN
ejpam-4858	323	4	,	,	PUNCT
ejpam-4858	323	5	105:27–37	105:27–37	PROPN
ejpam-4858	323	6	,	,	PUNCT
ejpam-4858	323	7	2004	2004	NUM
ejpam-4858	323	8	.	.	PUNCT
ejpam-4858	324	1	[	[	X
ejpam-4858	324	2	2	2	NUM
ejpam-4858	324	3	]	]	PUNCT
ejpam-4858	324	4	a.	a.	NOUN
ejpam-4858	324	5	açikgöz	açikgöz	PROPN
ejpam-4858	324	6	,	,	PUNCT
ejpam-4858	324	7	t.	t.	PROPN
ejpam-4858	324	8	noiri	noiri	PROPN
ejpam-4858	324	9	,	,	PUNCT
ejpam-4858	324	10	and	and	CCONJ
ejpam-4858	324	11	ş.	ş.	PROPN
ejpam-4858	324	12	yüksel	yüksel	PROPN
ejpam-4858	324	13	.	.	PUNCT
ejpam-4858	325	1	on	on	ADP
ejpam-4858	325	2	α	α	PROPN
ejpam-4858	325	3	-	-	PUNCT
ejpam-4858	325	4	operfect	operfect	ADJ
ejpam-4858	325	5	sets	set	NOUN
ejpam-4858	325	6	and	and	CCONJ
ejpam-4858	325	7	α-⋆-closed	α-⋆-closed	NUM
ejpam-4858	325	8	sets	set	NOUN
ejpam-4858	325	9	.	.	PUNCT
ejpam-4858	326	1	acta	acta	PROPN
ejpam-4858	326	2	mathematica	mathematica	PROPN
ejpam-4858	326	3	hungarica	hungarica	PROPN
ejpam-4858	326	4	,	,	PUNCT
ejpam-4858	326	5	105(1	105(1	PROPN
ejpam-4858	326	6	-	-	SYM
ejpam-4858	326	7	2):146–153	2):146–153	NUM
ejpam-4858	326	8	,	,	PUNCT
ejpam-4858	326	9	2010	2010	NUM
ejpam-4858	326	10	.	.	PUNCT
ejpam-4858	327	1	[	[	X
ejpam-4858	327	2	3	3	X
ejpam-4858	327	3	]	]	X
ejpam-4858	327	4	c.	c.	PROPN
ejpam-4858	327	5	berge	berge	PROPN
ejpam-4858	327	6	.	.	PUNCT
ejpam-4858	327	7	espaces	espace	VERB
ejpam-4858	327	8	topologiques	topologique	NOUN
ejpam-4858	327	9	fonctions	fonction	NOUN
ejpam-4858	327	10	multivoques	multivoque	NOUN
ejpam-4858	327	11	.	.	PUNCT
ejpam-4858	328	1	dunod	dunod	PROPN
ejpam-4858	328	2	,	,	PUNCT
ejpam-4858	328	3	paris	paris	PROPN
ejpam-4858	328	4	,	,	PUNCT
ejpam-4858	328	5	1959	1959	NUM
ejpam-4858	328	6	.	.	PUNCT
ejpam-4858	329	1	[	[	X
ejpam-4858	329	2	4	4	NUM
ejpam-4858	329	3	]	]	PUNCT
ejpam-4858	329	4	c.	c.	PROPN
ejpam-4858	329	5	boonpok	boonpok	PROPN
ejpam-4858	329	6	.	.	PUNCT
ejpam-4858	330	1	on	on	ADP
ejpam-4858	330	2	continuous	continuous	ADJ
ejpam-4858	330	3	multifunctions	multifunction	NOUN
ejpam-4858	330	4	in	in	ADP
ejpam-4858	330	5	ideal	ideal	ADJ
ejpam-4858	330	6	topological	topological	ADJ
ejpam-4858	330	7	spaces	space	NOUN
ejpam-4858	330	8	.	.	PUNCT
ejpam-4858	331	1	lobachevskii	lobachevskii	PROPN
ejpam-4858	331	2	journal	journal	PROPN
ejpam-4858	331	3	of	of	ADP
ejpam-4858	331	4	mathematics	mathematic	NOUN
ejpam-4858	331	5	,	,	PUNCT
ejpam-4858	331	6	40(1):24–35	40(1):24–35	NUM
ejpam-4858	331	7	,	,	PUNCT
ejpam-4858	331	8	2019	2019	NUM
ejpam-4858	331	9	.	.	PUNCT
ejpam-4858	332	1	[	[	X
ejpam-4858	332	2	5	5	X
ejpam-4858	332	3	]	]	PUNCT
ejpam-4858	332	4	c.	c.	PROPN
ejpam-4858	332	5	boonpok	boonpok	PROPN
ejpam-4858	332	6	.	.	PUNCT
ejpam-4858	333	1	on	on	ADP
ejpam-4858	333	2	some	some	DET
ejpam-4858	333	3	types	type	NOUN
ejpam-4858	333	4	of	of	ADP
ejpam-4858	333	5	continuity	continuity	NOUN
ejpam-4858	333	6	for	for	ADP
ejpam-4858	333	7	multifunctions	multifunction	NOUN
ejpam-4858	333	8	in	in	ADP
ejpam-4858	333	9	ideal	ideal	ADJ
ejpam-4858	333	10	topological	topological	ADJ
ejpam-4858	333	11	spaces	space	NOUN
ejpam-4858	333	12	.	.	PUNCT
ejpam-4858	334	1	advances	advance	NOUN
ejpam-4858	334	2	in	in	ADP
ejpam-4858	334	3	mathematics	mathematic	NOUN
ejpam-4858	334	4	:	:	PUNCT
ejpam-4858	334	5	scientific	scientific	ADJ
ejpam-4858	334	6	journal	journal	NOUN
ejpam-4858	334	7	,	,	PUNCT
ejpam-4858	334	8	9(3):859–886	9(3):859–886	NUM
ejpam-4858	334	9	,	,	PUNCT
ejpam-4858	334	10	2020	2020	NUM
ejpam-4858	334	11	.	.	PUNCT
ejpam-4858	335	1	[	[	X
ejpam-4858	335	2	6	6	NUM
ejpam-4858	335	3	]	]	PUNCT
ejpam-4858	335	4	c.	c.	PROPN
ejpam-4858	335	5	boonpok	boonpok	PROPN
ejpam-4858	335	6	.	.	PUNCT
ejpam-4858	336	1	(	(	PUNCT
ejpam-4858	336	2	τ1	τ1	NOUN
ejpam-4858	336	3	,	,	PUNCT
ejpam-4858	336	4	τ2)δ	τ2)δ	ADJ
ejpam-4858	336	5	-	-	PUNCT
ejpam-4858	336	6	continuous	continuous	ADJ
ejpam-4858	336	7	multifunctions	multifunction	NOUN
ejpam-4858	336	8	.	.	PUNCT
ejpam-4858	337	1	heliyon	heliyon	NOUN
ejpam-4858	337	2	,	,	PUNCT
ejpam-4858	337	3	2020	2020	NUM
ejpam-4858	337	4	:	:	PUNCT
ejpam-4858	337	5	e05367	e05367	PROPN
ejpam-4858	337	6	,	,	PUNCT
ejpam-4858	337	7	2020	2020	NUM
ejpam-4858	337	8	.	.	PUNCT
ejpam-4858	338	1	[	[	X
ejpam-4858	338	2	7	7	X
ejpam-4858	338	3	]	]	X
ejpam-4858	338	4	c.	c.	PROPN
ejpam-4858	338	5	boonpok	boonpok	PROPN
ejpam-4858	338	6	.	.	PUNCT
ejpam-4858	339	1	upper	upper	ADJ
ejpam-4858	339	2	and	and	CCONJ
ejpam-4858	339	3	lower	low	ADJ
ejpam-4858	339	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-4858	339	5	.	.	PUNCT
ejpam-4858	339	6	heliyon	heliyon	NOUN
ejpam-4858	339	7	,	,	PUNCT
ejpam-4858	339	8	2021	2021	NUM
ejpam-4858	339	9	:	:	PUNCT
ejpam-4858	339	10	e05986	e05986	PROPN
ejpam-4858	339	11	,	,	PUNCT
ejpam-4858	339	12	2021	2021	NUM
ejpam-4858	339	13	.	.	PUNCT
ejpam-4858	340	1	[	[	X
ejpam-4858	340	2	8	8	NUM
ejpam-4858	340	3	]	]	X
ejpam-4858	340	4	c.	c.	NOUN
ejpam-4858	340	5	boonpok	boonpok	PROPN
ejpam-4858	340	6	and	and	CCONJ
ejpam-4858	340	7	p.	p.	NOUN
ejpam-4858	340	8	pue	pue	NOUN
ejpam-4858	340	9	-	-	PUNCT
ejpam-4858	340	10	on	on	ADP
ejpam-4858	340	11	.	.	PUNCT
ejpam-4858	341	1	upper	upper	ADJ
ejpam-4858	341	2	and	and	CCONJ
ejpam-4858	341	3	lower	low	ADJ
ejpam-4858	341	4	weakly	weakly	ADJ
ejpam-4858	341	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-4858	341	6	multifunctions	multifunction	NOUN
ejpam-4858	341	7	.	.	PUNCT
ejpam-4858	342	1	international	international	ADJ
ejpam-4858	342	2	journal	journal	NOUN
ejpam-4858	342	3	of	of	ADP
ejpam-4858	342	4	analysis	analysis	NOUN
ejpam-4858	342	5	and	and	CCONJ
ejpam-4858	342	6	applications	application	NOUN
ejpam-4858	342	7	,	,	PUNCT
ejpam-4858	342	8	21:90	21:90	NUM
ejpam-4858	342	9	,	,	PUNCT
ejpam-4858	342	10	2023	2023	NUM
ejpam-4858	342	11	.	.	PUNCT
ejpam-4858	343	1	[	[	X
ejpam-4858	343	2	9	9	NUM
ejpam-4858	343	3	]	]	PUNCT
ejpam-4858	343	4	c.	c.	NOUN
ejpam-4858	343	5	boonpok	boonpok	PROPN
ejpam-4858	343	6	and	and	CCONJ
ejpam-4858	343	7	n.	n.	PROPN
ejpam-4858	343	8	srisarakham	srisarakham	PROPN
ejpam-4858	343	9	.	.	PUNCT
ejpam-4858	344	1	almost	almost	ADV
ejpam-4858	344	2	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-4858	344	3	for	for	ADP
ejpam-4858	344	4	multifunctions	multifunction	NOUN
ejpam-4858	344	5	.	.	PUNCT
ejpam-4858	345	1	international	international	ADJ
ejpam-4858	345	2	journal	journal	NOUN
ejpam-4858	345	3	of	of	ADP
ejpam-4858	345	4	analysis	analysis	NOUN
ejpam-4858	345	5	and	and	CCONJ
ejpam-4858	345	6	applications	application	NOUN
ejpam-4858	345	7	,	,	PUNCT
ejpam-4858	345	8	21:107	21:107	NUM
ejpam-4858	345	9	,	,	PUNCT
ejpam-4858	345	10	2023	2023	NUM
ejpam-4858	345	11	.	.	PUNCT
ejpam-4858	346	1	[	[	X
ejpam-4858	346	2	10	10	NUM
ejpam-4858	346	3	]	]	X
ejpam-4858	346	4	c.	c.	PROPN
ejpam-4858	346	5	boonpok	boonpok	PROPN
ejpam-4858	346	6	and	and	CCONJ
ejpam-4858	346	7	c.	c.	PROPN
ejpam-4858	346	8	viriyapong	viriyapong	PROPN
ejpam-4858	346	9	.	.	PUNCT
ejpam-4858	347	1	upper	upper	ADJ
ejpam-4858	347	2	and	and	CCONJ
ejpam-4858	347	3	lower	low	ADJ
ejpam-4858	347	4	almost	almost	ADV
ejpam-4858	347	5	weak	weak	ADJ
ejpam-4858	347	6	(	(	PUNCT
ejpam-4858	347	7	τ1	τ1	NOUN
ejpam-4858	347	8	,	,	PUNCT
ejpam-4858	347	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-4858	347	10	.	.	PUNCT
ejpam-4858	348	1	european	european	PROPN
ejpam-4858	348	2	journal	journal	PROPN
ejpam-4858	348	3	of	of	ADP
ejpam-4858	348	4	pure	pure	ADJ
ejpam-4858	348	5	and	and	CCONJ
ejpam-4858	348	6	applied	applied	ADJ
ejpam-4858	348	7	mathematics	mathematic	NOUN
ejpam-4858	348	8	,	,	PUNCT
ejpam-4858	348	9	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-4858	348	10	,	,	PUNCT
ejpam-4858	348	11	2021	2021	NUM
ejpam-4858	348	12	.	.	PUNCT
ejpam-4858	349	1	references	reference	NOUN
ejpam-4858	349	2	211	211	NUM
ejpam-4858	349	3	[	[	X
ejpam-4858	349	4	11	11	NUM
ejpam-4858	349	5	]	]	PUNCT
ejpam-4858	349	6	c.	c.	PROPN
ejpam-4858	349	7	boonpok	boonpok	PROPN
ejpam-4858	349	8	,	,	PUNCT
ejpam-4858	349	9	c.	c.	PROPN
ejpam-4858	349	10	viriyapong	viriyapong	PROPN
ejpam-4858	349	11	,	,	PUNCT
ejpam-4858	349	12	and	and	CCONJ
ejpam-4858	349	13	m.	m.	NOUN
ejpam-4858	349	14	thongmoon	thongmoon	NOUN
ejpam-4858	349	15	.	.	PUNCT
ejpam-4858	350	1	on	on	ADP
ejpam-4858	350	2	upper	upper	ADJ
ejpam-4858	350	3	and	and	CCONJ
ejpam-4858	350	4	lower	low	ADJ
ejpam-4858	350	5	(	(	PUNCT
ejpam-4858	350	6	τ1	τ1	NOUN
ejpam-4858	350	7	,	,	PUNCT
ejpam-4858	350	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-4858	350	9	multifunctions	multifunction	NOUN
ejpam-4858	350	10	.	.	PUNCT
ejpam-4858	351	1	journal	journal	PROPN
ejpam-4858	351	2	of	of	ADP
ejpam-4858	351	3	mathematics	mathematics	PROPN
ejpam-4858	351	4	and	and	CCONJ
ejpam-4858	351	5	computer	computer	NOUN
ejpam-4858	351	6	science	science	NOUN
ejpam-4858	351	7	,	,	PUNCT
ejpam-4858	351	8	18:282–293	18:282–293	NUM
ejpam-4858	351	9	,	,	PUNCT
ejpam-4858	351	10	2018	2018	NUM
ejpam-4858	351	11	.	.	PUNCT
ejpam-4858	352	1	[	[	X
ejpam-4858	352	2	12	12	NUM
ejpam-4858	352	3	]	]	X
ejpam-4858	352	4	e.	e.	PROPN
ejpam-4858	352	5	ekici	ekici	PROPN
ejpam-4858	352	6	and	and	CCONJ
ejpam-4858	352	7	t.	t.	PROPN
ejpam-4858	352	8	noiri	noiri	PROPN
ejpam-4858	352	9	.	.	PUNCT
ejpam-4858	353	1	⋆-extremally	⋆-extremally	ADV
ejpam-4858	353	2	disconnected	disconnect	VERB
ejpam-4858	353	3	ideal	ideal	ADJ
ejpam-4858	353	4	topological	topological	ADJ
ejpam-4858	353	5	spaces	space	NOUN
ejpam-4858	353	6	.	.	PUNCT
ejpam-4858	354	1	acta	acta	PROPN
ejpam-4858	354	2	mathematica	mathematica	PROPN
ejpam-4858	354	3	hungarica	hungarica	PROPN
ejpam-4858	354	4	,	,	PUNCT
ejpam-4858	354	5	122:81–90	122:81–90	NUM
ejpam-4858	354	6	,	,	PUNCT
ejpam-4858	354	7	2009	2009	NUM
ejpam-4858	354	8	.	.	PUNCT
ejpam-4858	355	1	[	[	X
ejpam-4858	355	2	13	13	NUM
ejpam-4858	355	3	]	]	X
ejpam-4858	355	4	e.	e.	PROPN
ejpam-4858	355	5	ekici	ekici	PROPN
ejpam-4858	355	6	and	and	CCONJ
ejpam-4858	355	7	t.	t.	PROPN
ejpam-4858	355	8	noiri	noiri	PROPN
ejpam-4858	355	9	.	.	PUNCT
ejpam-4858	356	1	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-4858	356	2	ideal	ideal	ADJ
ejpam-4858	356	3	topological	topological	ADJ
ejpam-4858	356	4	spaces	space	NOUN
ejpam-4858	356	5	.	.	PUNCT
ejpam-4858	357	1	annals	annal	NOUN
ejpam-4858	357	2	of	of	ADP
ejpam-4858	357	3	the	the	DET
ejpam-4858	357	4	alexandru	alexandru	PROPN
ejpam-4858	357	5	ioan	ioan	PROPN
ejpam-4858	357	6	cuza	cuza	PROPN
ejpam-4858	357	7	university	university	NOUN
ejpam-4858	357	8	-	-	PUNCT
ejpam-4858	357	9	mathematics	mathematic	NOUN
ejpam-4858	357	10	,	,	PUNCT
ejpam-4858	357	11	58:121–129	58:121–129	PROPN
ejpam-4858	357	12	,	,	PUNCT
ejpam-4858	357	13	2012	2012	NUM
ejpam-4858	357	14	.	.	PUNCT
ejpam-4858	358	1	[	[	X
ejpam-4858	358	2	14	14	NUM
ejpam-4858	358	3	]	]	PUNCT
ejpam-4858	358	4	m.	m.	PROPN
ejpam-4858	358	5	e.	e.	PROPN
ejpam-4858	358	6	abd	abd	PROPN
ejpam-4858	359	1	el	el	PROPN
ejpam-4858	359	2	-	-	PROPN
ejpam-4858	359	3	monsef	monsef	PROPN
ejpam-4858	359	4	,	,	PUNCT
ejpam-4858	359	5	e.	e.	PROPN
ejpam-4858	359	6	f.	f.	PROPN
ejpam-4858	359	7	lashien	lashien	PROPN
ejpam-4858	359	8	,	,	PUNCT
ejpam-4858	359	9	and	and	CCONJ
ejpam-4858	359	10	a.	a.	NOUN
ejpam-4858	359	11	a.	a.	NOUN
ejpam-4858	359	12	nasef	nasef	PROPN
ejpam-4858	359	13	.	.	PUNCT
ejpam-4858	360	1	on	on	ADP
ejpam-4858	360	2	i	i	PRON
ejpam-4858	360	3	-open	-open	PROPN
ejpam-4858	360	4	sets	set	NOUN
ejpam-4858	360	5	and	and	CCONJ
ejpam-4858	360	6	i	i	PRON
ejpam-4858	360	7	continuous	continuous	ADJ
ejpam-4858	360	8	functions	function	NOUN
ejpam-4858	360	9	.	.	PUNCT
ejpam-4858	361	1	kyungpook	kyungpook	PROPN
ejpam-4858	361	2	mathematical	mathematical	PROPN
ejpam-4858	361	3	journal	journal	PROPN
ejpam-4858	361	4	,	,	PUNCT
ejpam-4858	361	5	32:21–30	32:21–30	NUM
ejpam-4858	361	6	,	,	PUNCT
ejpam-4858	361	7	1992	1992	NUM
ejpam-4858	361	8	.	.	PUNCT
ejpam-4858	362	1	[	[	X
ejpam-4858	362	2	15	15	NUM
ejpam-4858	362	3	]	]	X
ejpam-4858	362	4	e.	e.	PROPN
ejpam-4858	362	5	hatir	hatir	PROPN
ejpam-4858	362	6	and	and	CCONJ
ejpam-4858	362	7	t.	t.	PROPN
ejpam-4858	362	8	noiri	noiri	PROPN
ejpam-4858	362	9	.	.	PUNCT
ejpam-4858	363	1	on	on	ADP
ejpam-4858	363	2	decompositions	decomposition	NOUN
ejpam-4858	363	3	of	of	ADP
ejpam-4858	363	4	continuity	continuity	NOUN
ejpam-4858	363	5	via	via	ADP
ejpam-4858	363	6	idealization	idealization	NOUN
ejpam-4858	363	7	.	.	PUNCT
ejpam-4858	364	1	acta	acta	PROPN
ejpam-4858	364	2	mathematica	mathematica	PROPN
ejpam-4858	364	3	hungarica	hungarica	PROPN
ejpam-4858	364	4	,	,	PUNCT
ejpam-4858	364	5	96:341–349	96:341–349	PROPN
ejpam-4858	364	6	,	,	PUNCT
ejpam-4858	364	7	2002	2002	NUM
ejpam-4858	364	8	.	.	PUNCT
ejpam-4858	365	1	[	[	X
ejpam-4858	365	2	16	16	NUM
ejpam-4858	365	3	]	]	X
ejpam-4858	365	4	d.	d.	PROPN
ejpam-4858	365	5	janković	janković	PROPN
ejpam-4858	365	6	and	and	CCONJ
ejpam-4858	365	7	t.	t.	PROPN
ejpam-4858	365	8	r.	r.	PROPN
ejpam-4858	365	9	hamlett	hamlett	PROPN
ejpam-4858	365	10	.	.	PUNCT
ejpam-4858	366	1	new	new	ADJ
ejpam-4858	366	2	topologies	topology	NOUN
ejpam-4858	366	3	from	from	ADP
ejpam-4858	366	4	old	old	ADJ
ejpam-4858	366	5	via	via	ADP
ejpam-4858	366	6	ideals	ideal	NOUN
ejpam-4858	366	7	.	.	PUNCT
ejpam-4858	367	1	the	the	DET
ejpam-4858	367	2	american	american	PROPN
ejpam-4858	367	3	mathematical	mathematical	PROPN
ejpam-4858	367	4	monthly	monthly	ADV
ejpam-4858	367	5	,	,	PUNCT
ejpam-4858	367	6	97:295–310	97:295–310	PROPN
ejpam-4858	367	7	,	,	PUNCT
ejpam-4858	367	8	1990	1990	NUM
ejpam-4858	367	9	.	.	PUNCT
ejpam-4858	368	1	[	[	X
ejpam-4858	368	2	17	17	NUM
ejpam-4858	368	3	]	]	PUNCT
ejpam-4858	368	4	k.	k.	PROPN
ejpam-4858	368	5	kuratowski	kuratowski	PROPN
ejpam-4858	368	6	.	.	PUNCT
ejpam-4858	369	1	topology	topology	PROPN
ejpam-4858	369	2	,	,	PUNCT
ejpam-4858	369	3	vol	vol	NOUN
ejpam-4858	369	4	.	.	PUNCT
ejpam-4858	369	5	i.	i.	PROPN
ejpam-4858	369	6	academic	academic	PROPN
ejpam-4858	369	7	press	press	PROPN
ejpam-4858	369	8	,	,	PUNCT
ejpam-4858	369	9	new	new	PROPN
ejpam-4858	369	10	york	york	PROPN
ejpam-4858	369	11	,	,	PUNCT
ejpam-4858	369	12	1966	1966	NUM
ejpam-4858	369	13	.	.	PUNCT
ejpam-4858	370	1	[	[	X
ejpam-4858	370	2	18	18	NUM
ejpam-4858	370	3	]	]	PUNCT
ejpam-4858	370	4	k.	k.	PROPN
ejpam-4858	370	5	laprom	laprom	PROPN
ejpam-4858	370	6	,	,	PUNCT
ejpam-4858	370	7	c.	c.	PROPN
ejpam-4858	370	8	boonpok	boonpok	PROPN
ejpam-4858	370	9	,	,	PUNCT
ejpam-4858	370	10	and	and	CCONJ
ejpam-4858	370	11	c.	c.	PROPN
ejpam-4858	370	12	viriyapong	viriyapong	PROPN
ejpam-4858	370	13	.	.	PUNCT
ejpam-4858	371	1	β(τ1	β(τ1	PROPN
ejpam-4858	371	2	,	,	PUNCT
ejpam-4858	371	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4858	371	4	multifunctions	multifunction	NOUN
ejpam-4858	371	5	on	on	ADP
ejpam-4858	371	6	bitopological	bitopological	ADJ
ejpam-4858	371	7	spaces	space	NOUN
ejpam-4858	371	8	.	.	PUNCT
ejpam-4858	372	1	journal	journal	NOUN
ejpam-4858	372	2	of	of	ADP
ejpam-4858	372	3	mathematics	mathematic	NOUN
ejpam-4858	372	4	,	,	PUNCT
ejpam-4858	372	5	2020:4020971	2020:4020971	NUM
ejpam-4858	372	6	,	,	PUNCT
ejpam-4858	372	7	2020	2020	NUM
ejpam-4858	372	8	.	.	PUNCT
ejpam-4858	373	1	[	[	X
ejpam-4858	373	2	19	19	NUM
ejpam-4858	373	3	]	]	PUNCT
ejpam-4858	373	4	a.	a.	NOUN
ejpam-4858	373	5	s.	s.	PROPN
ejpam-4858	373	6	mashhour	mashhour	PROPN
ejpam-4858	373	7	,	,	PUNCT
ejpam-4858	373	8	i.	i.	PROPN
ejpam-4858	373	9	a.	a.	PROPN
ejpam-4858	373	10	hasanein	hasanein	PROPN
ejpam-4858	373	11	,	,	PUNCT
ejpam-4858	373	12	and	and	CCONJ
ejpam-4858	373	13	s.	s.	PROPN
ejpam-4858	373	14	n.	n.	PROPN
ejpam-4858	373	15	el	el	PROPN
ejpam-4858	373	16	-	-	PROPN
ejpam-4858	373	17	deeb	deeb	PROPN
ejpam-4858	373	18	.	.	PUNCT
ejpam-4858	374	1	α	α	X
ejpam-4858	374	2	-	-	ADJ
ejpam-4858	374	3	continuous	continuous	ADJ
ejpam-4858	374	4	and	and	CCONJ
ejpam-4858	374	5	α	α	NOUN
ejpam-4858	374	6	-	-	ADJ
ejpam-4858	374	7	open	open	ADJ
ejpam-4858	374	8	mappings	mapping	NOUN
ejpam-4858	374	9	.	.	PUNCT
ejpam-4858	375	1	acta	acta	PROPN
ejpam-4858	375	2	mathematica	mathematica	PROPN
ejpam-4858	375	3	hungarica	hungarica	PROPN
ejpam-4858	375	4	,	,	PUNCT
ejpam-4858	375	5	41:213–218	41:213–218	PROPN
ejpam-4858	375	6	,	,	PUNCT
ejpam-4858	375	7	1983	1983	NUM
ejpam-4858	375	8	.	.	PUNCT
ejpam-4858	376	1	[	[	X
ejpam-4858	376	2	20	20	NUM
ejpam-4858	376	3	]	]	PUNCT
ejpam-4858	376	4	t.	t.	NOUN
ejpam-4858	376	5	neubrunn	neubrunn	PROPN
ejpam-4858	376	6	.	.	PUNCT
ejpam-4858	377	1	strongly	strongly	ADV
ejpam-4858	377	2	quasi	quasi	ADJ
ejpam-4858	377	3	-	-	ADJ
ejpam-4858	377	4	continuous	continuous	ADJ
ejpam-4858	377	5	multivalued	multivalued	ADJ
ejpam-4858	377	6	mappings	mapping	NOUN
ejpam-4858	377	7	.	.	PUNCT
ejpam-4858	378	1	general	general	ADJ
ejpam-4858	378	2	topology	topology	NOUN
ejpam-4858	378	3	and	and	CCONJ
ejpam-4858	378	4	its	its	PRON
ejpam-4858	378	5	relations	relation	NOUN
ejpam-4858	378	6	to	to	ADP
ejpam-4858	378	7	modern	modern	ADJ
ejpam-4858	378	8	analysis	analysis	NOUN
ejpam-4858	378	9	and	and	CCONJ
ejpam-4858	378	10	algebra	algebra	NOUN
ejpam-4858	378	11	vi	vi	PROPN
ejpam-4858	378	12	(	(	PUNCT
ejpam-4858	378	13	prague	prague	NOUN
ejpam-4858	378	14	1986	1986	NUM
ejpam-4858	378	15	)	)	PUNCT
ejpam-4858	378	16	heldermann	heldermann	PROPN
ejpam-4858	378	17	,	,	PUNCT
ejpam-4858	378	18	berlin	berlin	PROPN
ejpam-4858	378	19	,	,	PUNCT
ejpam-4858	378	20	pages	page	NOUN
ejpam-4858	378	21	351–359	351–359	NUM
ejpam-4858	378	22	,	,	PUNCT
ejpam-4858	378	23	1988	1988	NUM
ejpam-4858	378	24	.	.	PUNCT
ejpam-4858	379	1	[	[	X
ejpam-4858	379	2	21	21	NUM
ejpam-4858	379	3	]	]	X
ejpam-4858	379	4	o.	o.	NOUN
ejpam-4858	379	5	nj̊astad	nj̊astad	NOUN
ejpam-4858	379	6	.	.	PUNCT
ejpam-4858	380	1	on	on	ADP
ejpam-4858	380	2	some	some	DET
ejpam-4858	380	3	classes	class	NOUN
ejpam-4858	380	4	of	of	ADP
ejpam-4858	380	5	nearly	nearly	ADV
ejpam-4858	380	6	open	open	ADJ
ejpam-4858	380	7	sets	set	NOUN
ejpam-4858	380	8	.	.	PUNCT
ejpam-4858	381	1	pacific	pacific	PROPN
ejpam-4858	381	2	journal	journal	PROPN
ejpam-4858	381	3	of	of	ADP
ejpam-4858	381	4	mathematics	mathematic	NOUN
ejpam-4858	381	5	,	,	PUNCT
ejpam-4858	381	6	15:961–970	15:961–970	PROPN
ejpam-4858	381	7	,	,	PUNCT
ejpam-4858	381	8	1965	1965	NUM
ejpam-4858	381	9	.	.	PUNCT
ejpam-4858	382	1	[	[	X
ejpam-4858	382	2	22	22	NUM
ejpam-4858	382	3	]	]	PUNCT
ejpam-4858	382	4	t.	t.	PROPN
ejpam-4858	382	5	noiri	noiri	PROPN
ejpam-4858	382	6	.	.	PUNCT
ejpam-4858	383	1	on	on	ADP
ejpam-4858	383	2	α	α	NUM
ejpam-4858	383	3	-	-	ADJ
ejpam-4858	383	4	continuous	continuous	ADJ
ejpam-4858	383	5	functions	function	NOUN
ejpam-4858	383	6	.	.	PUNCT
ejpam-4858	384	1	časopis	časopis	X
ejpam-4858	384	2	pro	pro	X
ejpam-4858	384	3	pěstováńı	pěstováńı	NOUN
ejpam-4858	384	4	matematiky	matematiky	NOUN
ejpam-4858	384	5	,	,	PUNCT
ejpam-4858	384	6	109:118	109:118	NUM
ejpam-4858	384	7	–	–	PUNCT
ejpam-4858	384	8	126	126	NUM
ejpam-4858	384	9	,	,	PUNCT
ejpam-4858	384	10	1984	1984	NUM
ejpam-4858	384	11	.	.	PUNCT
ejpam-4858	385	1	[	[	X
ejpam-4858	385	2	23	23	NUM
ejpam-4858	385	3	]	]	PUNCT
ejpam-4858	385	4	t.	t.	PROPN
ejpam-4858	385	5	noiri	noiri	PROPN
ejpam-4858	385	6	.	.	PUNCT
ejpam-4858	386	1	almost	almost	ADV
ejpam-4858	386	2	α	α	NUM
ejpam-4858	386	3	-	-	ADJ
ejpam-4858	386	4	continuous	continuous	ADJ
ejpam-4858	386	5	functions	function	NOUN
ejpam-4858	386	6	.	.	PUNCT
ejpam-4858	387	1	kyungpook	kyungpook	PROPN
ejpam-4858	387	2	mathematical	mathematical	PROPN
ejpam-4858	387	3	journal	journal	PROPN
ejpam-4858	387	4	,	,	PUNCT
ejpam-4858	387	5	28:71–77	28:71–77	PROPN
ejpam-4858	387	6	,	,	PUNCT
ejpam-4858	387	7	1998	1998	NUM
ejpam-4858	387	8	.	.	PUNCT
ejpam-4858	388	1	[	[	X
ejpam-4858	388	2	24	24	NUM
ejpam-4858	388	3	]	]	PUNCT
ejpam-4858	388	4	v.	v.	CCONJ
ejpam-4858	388	5	popa	popa	NOUN
ejpam-4858	388	6	and	and	CCONJ
ejpam-4858	388	7	t.	t.	PROPN
ejpam-4858	388	8	noiri	noiri	PROPN
ejpam-4858	388	9	.	.	PUNCT
ejpam-4858	389	1	on	on	ADP
ejpam-4858	389	2	upper	upper	ADJ
ejpam-4858	389	3	and	and	CCONJ
ejpam-4858	389	4	lower	low	ADJ
ejpam-4858	389	5	α	α	ADJ
ejpam-4858	389	6	-	-	ADJ
ejpam-4858	389	7	continuous	continuous	ADJ
ejpam-4858	389	8	multifunctions	multifunction	NOUN
ejpam-4858	389	9	.	.	PUNCT
ejpam-4858	390	1	mathematica	mathematica	PROPN
ejpam-4858	390	2	slovaca	slovaca	PROPN
ejpam-4858	390	3	,	,	PUNCT
ejpam-4858	390	4	43:477–491	43:477–491	NUM
ejpam-4858	390	5	,	,	PUNCT
ejpam-4858	390	6	1993	1993	NUM
ejpam-4858	390	7	.	.	PUNCT
ejpam-4858	391	1	[	[	X
ejpam-4858	391	2	25	25	NUM
ejpam-4858	391	3	]	]	PUNCT
ejpam-4858	391	4	v.	v.	ADP
ejpam-4858	391	5	vaidyanathswamy	vaidyanathswamy	NOUN
ejpam-4858	391	6	.	.	PUNCT
ejpam-4858	392	1	the	the	DET
ejpam-4858	392	2	localization	localization	NOUN
ejpam-4858	392	3	theory	theory	NOUN
ejpam-4858	392	4	in	in	ADP
ejpam-4858	392	5	set	set	NOUN
ejpam-4858	392	6	topology	topology	NOUN
ejpam-4858	392	7	.	.	PUNCT
ejpam-4858	393	1	proceedings	proceeding	NOUN
ejpam-4858	393	2	of	of	ADP
ejpam-4858	393	3	the	the	DET
ejpam-4858	393	4	indian	indian	PROPN
ejpam-4858	393	5	academy	academy	PROPN
ejpam-4858	393	6	of	of	ADP
ejpam-4858	393	7	sciences	sciences	PROPN
ejpam-4858	393	8	,	,	PUNCT
ejpam-4858	393	9	20:51–61	20:51–61	NUM
ejpam-4858	393	10	,	,	PUNCT
ejpam-4858	393	11	1945	1945	NUM
ejpam-4858	393	12	.	.	PUNCT
ejpam-4858	394	1	[	[	X
ejpam-4858	394	2	26	26	NUM
ejpam-4858	394	3	]	]	X
ejpam-4858	394	4	c.	c.	PROPN
ejpam-4858	394	5	viriyapong	viriyapong	PROPN
ejpam-4858	394	6	and	and	CCONJ
ejpam-4858	394	7	c.	c.	PROPN
ejpam-4858	394	8	boonpok	boonpok	PROPN
ejpam-4858	394	9	.	.	PUNCT
ejpam-4858	395	1	(	(	PUNCT
ejpam-4858	395	2	τ1	τ1	NOUN
ejpam-4858	395	3	,	,	PUNCT
ejpam-4858	395	4	τ2)α	τ2)α	NOUN
ejpam-4858	395	5	-	-	PUNCT
ejpam-4858	395	6	continuity	continuity	NOUN
ejpam-4858	395	7	for	for	ADP
ejpam-4858	395	8	multifunctions	multifunction	NOUN
ejpam-4858	395	9	.	.	PUNCT
ejpam-4858	396	1	journal	journal	PROPN
ejpam-4858	396	2	of	of	ADP
ejpam-4858	396	3	mathematics	mathematic	NOUN
ejpam-4858	396	4	,	,	PUNCT
ejpam-4858	396	5	2020:6285763	2020:6285763	NUM
ejpam-4858	396	6	,	,	PUNCT
ejpam-4858	396	7	2020	2020	NUM
ejpam-4858	396	8	.	.	PUNCT