id	sid	tid	token	lemma	pos
ejpam-5634	1	1	european	european	PROPN
ejpam-5634	1	2	journal	journal	PROPN
ejpam-5634	1	3	of	of	ADP
ejpam-5634	1	4	pure	pure	ADJ
ejpam-5634	1	5	and	and	CCONJ
ejpam-5634	1	6	applied	applied	ADJ
ejpam-5634	1	7	mathematics	mathematic	NOUN
ejpam-5634	1	8	2025	2025	NUM
ejpam-5634	1	9	,	,	PUNCT
ejpam-5634	1	10	vol	vol	NOUN
ejpam-5634	1	11	.	.	PROPN
ejpam-5634	1	12	18	18	NUM
ejpam-5634	1	13	,	,	PUNCT
ejpam-5634	1	14	issue	issue	NOUN
ejpam-5634	1	15	1	1	NUM
ejpam-5634	1	16	,	,	PUNCT
ejpam-5634	1	17	article	article	NOUN
ejpam-5634	1	18	number	number	NOUN
ejpam-5634	1	19	5634	5634	NUM
ejpam-5634	1	20	issn	issn	PROPN
ejpam-5634	1	21	1307	1307	NUM
ejpam-5634	1	22	-	-	SYM
ejpam-5634	1	23	5543	5543	NUM
ejpam-5634	1	24	–	–	PUNCT
ejpam-5634	1	25	ejpam.com	ejpam.com	X
ejpam-5634	1	26	published	publish	VERB
ejpam-5634	1	27	by	by	ADP
ejpam-5634	1	28	new	new	PROPN
ejpam-5634	1	29	york	york	PROPN
ejpam-5634	1	30	business	business	PROPN
ejpam-5634	1	31	global	global	PROPN
ejpam-5634	1	32	s-(τ1	s-(τ1	PROPN
ejpam-5634	1	33	,	,	PUNCT
ejpam-5634	1	34	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5634	1	35	for	for	ADP
ejpam-5634	1	36	multifunctions	multifunction	NOUN
ejpam-5634	1	37	monchaya	monchaya	PROPN
ejpam-5634	1	38	chiangpradit1	chiangpradit1	PROPN
ejpam-5634	1	39	,	,	PUNCT
ejpam-5634	1	40	areeyuth	areeyuth	NOUN
ejpam-5634	1	41	sama	sama	NOUN
ejpam-5634	1	42	-	-	PUNCT
ejpam-5634	1	43	ae2	ae2	PROPN
ejpam-5634	1	44	,	,	PUNCT
ejpam-5634	1	45	chawalit	chawalit	VERB
ejpam-5634	1	46	boonpok1,∗	boonpok1,∗	NOUN
ejpam-5634	1	47	1	1	NUM
ejpam-5634	1	48	mathematics	mathematic	NOUN
ejpam-5634	1	49	and	and	CCONJ
ejpam-5634	1	50	applied	apply	VERB
ejpam-5634	1	51	mathematics	mathematics	PROPN
ejpam-5634	1	52	research	research	NOUN
ejpam-5634	1	53	unit	unit	NOUN
ejpam-5634	1	54	,	,	PUNCT
ejpam-5634	1	55	department	department	NOUN
ejpam-5634	1	56	of	of	ADP
ejpam-5634	1	57	mathematics	mathematic	NOUN
ejpam-5634	1	58	,	,	PUNCT
ejpam-5634	1	59	faculty	faculty	NOUN
ejpam-5634	1	60	of	of	ADP
ejpam-5634	1	61	science	science	NOUN
ejpam-5634	1	62	,	,	PUNCT
ejpam-5634	1	63	mahasarakham	mahasarakham	PROPN
ejpam-5634	1	64	university	university	PROPN
ejpam-5634	1	65	,	,	PUNCT
ejpam-5634	1	66	maha	maha	PROPN
ejpam-5634	1	67	sarakham	sarakham	PROPN
ejpam-5634	1	68	,	,	PUNCT
ejpam-5634	1	69	44150	44150	NUM
ejpam-5634	1	70	,	,	PUNCT
ejpam-5634	1	71	thailand	thailand	PROPN
ejpam-5634	1	72	2	2	NUM
ejpam-5634	1	73	department	department	NOUN
ejpam-5634	1	74	of	of	ADP
ejpam-5634	1	75	mathematics	mathematic	NOUN
ejpam-5634	1	76	and	and	CCONJ
ejpam-5634	1	77	computer	computer	NOUN
ejpam-5634	1	78	science	science	NOUN
ejpam-5634	1	79	,	,	PUNCT
ejpam-5634	1	80	faculty	faculty	NOUN
ejpam-5634	1	81	of	of	ADP
ejpam-5634	1	82	science	science	NOUN
ejpam-5634	1	83	and	and	CCONJ
ejpam-5634	1	84	technology	technology	NOUN
ejpam-5634	1	85	,	,	PUNCT
ejpam-5634	1	86	prince	prince	NOUN
ejpam-5634	1	87	of	of	ADP
ejpam-5634	1	88	songkla	songkla	PROPN
ejpam-5634	1	89	university	university	PROPN
ejpam-5634	1	90	,	,	PUNCT
ejpam-5634	1	91	pattani	pattani	NOUN
ejpam-5634	1	92	campus	campus	NOUN
ejpam-5634	1	93	,	,	PUNCT
ejpam-5634	1	94	pattani	pattani	NOUN
ejpam-5634	1	95	,	,	PUNCT
ejpam-5634	1	96	94000	94000	NUM
ejpam-5634	1	97	,	,	PUNCT
ejpam-5634	1	98	thailand	thailand	PROPN
ejpam-5634	1	99	abstract	abstract	PROPN
ejpam-5634	1	100	.	.	PUNCT
ejpam-5634	2	1	this	this	DET
ejpam-5634	2	2	paper	paper	NOUN
ejpam-5634	2	3	deals	deal	NOUN
ejpam-5634	2	4	with	with	ADP
ejpam-5634	2	5	the	the	DET
ejpam-5634	2	6	concepts	concept	NOUN
ejpam-5634	2	7	of	of	ADP
ejpam-5634	2	8	upper	upper	ADJ
ejpam-5634	2	9	s-(τ1	s-(τ1	PROPN
ejpam-5634	2	10	,	,	PUNCT
ejpam-5634	2	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	2	12	multifunctions	multifunction	NOUN
ejpam-5634	2	13	and	and	CCONJ
ejpam-5634	2	14	lower	low	ADJ
ejpam-5634	2	15	s-(τ1	s-(τ1	NOUN
ejpam-5634	2	16	,	,	PUNCT
ejpam-5634	2	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	2	18	multifunctions	multifunction	NOUN
ejpam-5634	2	19	.	.	PUNCT
ejpam-5634	3	1	moreover	moreover	ADV
ejpam-5634	3	2	,	,	PUNCT
ejpam-5634	3	3	several	several	ADJ
ejpam-5634	3	4	characterizations	characterization	NOUN
ejpam-5634	3	5	and	and	CCONJ
ejpam-5634	3	6	some	some	DET
ejpam-5634	3	7	properties	property	NOUN
ejpam-5634	3	8	concerning	concern	VERB
ejpam-5634	3	9	upper	upper	ADJ
ejpam-5634	3	10	s-(τ1	s-(τ1	PROPN
ejpam-5634	3	11	,	,	PUNCT
ejpam-5634	3	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	3	13	multifunctions	multifunction	NOUN
ejpam-5634	3	14	and	and	CCONJ
ejpam-5634	3	15	lower	low	ADJ
ejpam-5634	3	16	s-(τ1	s-(τ1	NOUN
ejpam-5634	3	17	,	,	PUNCT
ejpam-5634	3	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	3	19	multifunctions	multifunction	NOUN
ejpam-5634	3	20	are	be	AUX
ejpam-5634	3	21	investigated	investigate	VERB
ejpam-5634	3	22	.	.	PUNCT
ejpam-5634	4	1	2020	2020	NUM
ejpam-5634	4	2	mathematics	mathematic	NOUN
ejpam-5634	4	3	subject	subject	NOUN
ejpam-5634	4	4	classifications	classification	NOUN
ejpam-5634	4	5	:	:	PUNCT
ejpam-5634	4	6	54c08	54c08	NUM
ejpam-5634	4	7	,	,	PUNCT
ejpam-5634	4	8	54c60	54c60	NUM
ejpam-5634	4	9	key	key	ADJ
ejpam-5634	4	10	words	word	NOUN
ejpam-5634	4	11	and	and	CCONJ
ejpam-5634	4	12	phrases	phrase	NOUN
ejpam-5634	4	13	:	:	PUNCT
ejpam-5634	4	14	upper	upper	ADJ
ejpam-5634	4	15	s-(τ1	s-(τ1	PROPN
ejpam-5634	4	16	,	,	PUNCT
ejpam-5634	4	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	4	18	multifunction	multifunction	NOUN
ejpam-5634	4	19	,	,	PUNCT
ejpam-5634	4	20	lower	low	ADJ
ejpam-5634	4	21	s-(τ1	s-(τ1	PROPN
ejpam-5634	4	22	,	,	PUNCT
ejpam-5634	4	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	4	24	multifunction	multifunction	NOUN
ejpam-5634	4	25	1	1	NUM
ejpam-5634	4	26	.	.	PUNCT
ejpam-5634	5	1	introduction	introduction	NOUN
ejpam-5634	5	2	it	it	PRON
ejpam-5634	5	3	is	be	AUX
ejpam-5634	5	4	well	well	ADV
ejpam-5634	5	5	-	-	PUNCT
ejpam-5634	5	6	known	know	VERB
ejpam-5634	5	7	that	that	SCONJ
ejpam-5634	5	8	the	the	DET
ejpam-5634	5	9	branch	branch	NOUN
ejpam-5634	5	10	of	of	ADP
ejpam-5634	5	11	mathematics	mathematic	NOUN
ejpam-5634	5	12	called	call	VERB
ejpam-5634	5	13	topology	topology	NOUN
ejpam-5634	5	14	is	be	AUX
ejpam-5634	5	15	concerned	concern	VERB
ejpam-5634	5	16	with	with	ADP
ejpam-5634	5	17	all	all	DET
ejpam-5634	5	18	questions	question	NOUN
ejpam-5634	5	19	directly	directly	ADV
ejpam-5634	5	20	or	or	CCONJ
ejpam-5634	5	21	indirectly	indirectly	ADV
ejpam-5634	5	22	related	relate	VERB
ejpam-5634	5	23	to	to	ADP
ejpam-5634	5	24	continuity	continuity	NOUN
ejpam-5634	5	25	.	.	PUNCT
ejpam-5634	6	1	continuity	continuity	NOUN
ejpam-5634	6	2	is	be	AUX
ejpam-5634	6	3	an	an	DET
ejpam-5634	6	4	important	important	ADJ
ejpam-5634	6	5	concept	concept	NOUN
ejpam-5634	6	6	for	for	ADP
ejpam-5634	6	7	the	the	DET
ejpam-5634	6	8	study	study	NOUN
ejpam-5634	6	9	and	and	CCONJ
ejpam-5634	6	10	investigation	investigation	NOUN
ejpam-5634	6	11	in	in	ADP
ejpam-5634	6	12	the	the	DET
ejpam-5634	6	13	theory	theory	NOUN
ejpam-5634	6	14	of	of	ADP
ejpam-5634	6	15	classical	classical	ADJ
ejpam-5634	6	16	point	point	NOUN
ejpam-5634	6	17	set	set	VERB
ejpam-5634	6	18	topology	topology	NOUN
ejpam-5634	6	19	.	.	PUNCT
ejpam-5634	7	1	generalization	generalization	NOUN
ejpam-5634	7	2	of	of	ADP
ejpam-5634	7	3	this	this	DET
ejpam-5634	7	4	concept	concept	NOUN
ejpam-5634	7	5	by	by	ADP
ejpam-5634	7	6	using	use	VERB
ejpam-5634	7	7	stronger	strong	ADJ
ejpam-5634	7	8	and	and	CCONJ
ejpam-5634	7	9	weaker	weak	ADJ
ejpam-5634	7	10	forms	form	NOUN
ejpam-5634	7	11	of	of	ADP
ejpam-5634	7	12	open	open	ADJ
ejpam-5634	7	13	sets	set	NOUN
ejpam-5634	7	14	such	such	ADJ
ejpam-5634	7	15	as	as	ADP
ejpam-5634	7	16	semi	semi	ADJ
ejpam-5634	7	17	-	-	ADJ
ejpam-5634	7	18	open	open	ADJ
ejpam-5634	7	19	sets	set	NOUN
ejpam-5634	7	20	[	[	X
ejpam-5634	7	21	43	43	NUM
ejpam-5634	7	22	]	]	PUNCT
ejpam-5634	7	23	,	,	PUNCT
ejpam-5634	7	24	preopen	preopen	ADJ
ejpam-5634	7	25	sets	set	NOUN
ejpam-5634	7	26	[	[	X
ejpam-5634	7	27	45	45	NUM
ejpam-5634	7	28	]	]	PUNCT
ejpam-5634	7	29	,	,	PUNCT
ejpam-5634	7	30	α	α	X
ejpam-5634	7	31	-	-	ADJ
ejpam-5634	7	32	open	open	ADJ
ejpam-5634	7	33	sets	set	NOUN
ejpam-5634	7	34	[	[	X
ejpam-5634	7	35	46	46	NUM
ejpam-5634	7	36	]	]	PUNCT
ejpam-5634	7	37	,	,	PUNCT
ejpam-5634	7	38	β	β	X
ejpam-5634	7	39	-	-	ADJ
ejpam-5634	7	40	open	open	ADJ
ejpam-5634	7	41	sets	set	NOUN
ejpam-5634	7	42	[	[	X
ejpam-5634	7	43	34	34	NUM
ejpam-5634	7	44	]	]	PUNCT
ejpam-5634	7	45	and	and	CCONJ
ejpam-5634	7	46	θ	θ	ADJ
ejpam-5634	7	47	-	-	ADJ
ejpam-5634	7	48	open	open	ADJ
ejpam-5634	7	49	sets	set	NOUN
ejpam-5634	7	50	[	[	X
ejpam-5634	7	51	61	61	NUM
ejpam-5634	7	52	]	]	PUNCT
ejpam-5634	7	53	is	be	AUX
ejpam-5634	7	54	one	one	NUM
ejpam-5634	7	55	of	of	ADP
ejpam-5634	7	56	the	the	DET
ejpam-5634	7	57	main	main	ADJ
ejpam-5634	7	58	research	research	NOUN
ejpam-5634	7	59	topics	topic	NOUN
ejpam-5634	7	60	of	of	ADP
ejpam-5634	7	61	general	general	ADJ
ejpam-5634	7	62	topology	topology	NOUN
ejpam-5634	7	63	.	.	PUNCT
ejpam-5634	8	1	viriyapong	viriyapong	VERB
ejpam-5634	8	2	and	and	CCONJ
ejpam-5634	8	3	boonpok	boonpok	VERB
ejpam-5634	8	4	[	[	X
ejpam-5634	8	5	63	63	NUM
ejpam-5634	8	6	]	]	PUNCT
ejpam-5634	8	7	investigated	investigate	VERB
ejpam-5634	8	8	some	some	DET
ejpam-5634	8	9	characterizations	characterization	NOUN
ejpam-5634	8	10	of	of	ADP
ejpam-5634	8	11	(	(	PUNCT
ejpam-5634	8	12	λ	λ	PROPN
ejpam-5634	8	13	,	,	PUNCT
ejpam-5634	8	14	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	8	15	functions	function	NOUN
ejpam-5634	8	16	by	by	ADP
ejpam-5634	8	17	utilizing	utilize	VERB
ejpam-5634	8	18	the	the	DET
ejpam-5634	8	19	notions	notion	NOUN
ejpam-5634	8	20	of	of	ADP
ejpam-5634	8	21	(	(	PUNCT
ejpam-5634	8	22	λ	λ	PROPN
ejpam-5634	8	23	,	,	PUNCT
ejpam-5634	8	24	sp)-open	sp)-open	ADJ
ejpam-5634	8	25	sets	set	NOUN
ejpam-5634	8	26	and	and	CCONJ
ejpam-5634	8	27	(	(	PUNCT
ejpam-5634	8	28	λ	λ	PROPN
ejpam-5634	8	29	,	,	PUNCT
ejpam-5634	8	30	sp)-closed	sp)-close	VERB
ejpam-5634	8	31	sets	set	NOUN
ejpam-5634	8	32	due	due	ADP
ejpam-5634	8	33	to	to	ADP
ejpam-5634	8	34	boonpok	boonpok	NOUN
ejpam-5634	8	35	and	and	CCONJ
ejpam-5634	8	36	khampakdee	khampakdee	NOUN
ejpam-5634	9	1	[	[	X
ejpam-5634	9	2	12	12	NUM
ejpam-5634	9	3	]	]	PUNCT
ejpam-5634	9	4	.	.	PUNCT
ejpam-5634	10	1	dungthaisong	dungthaisong	NOUN
ejpam-5634	10	2	et	et	PROPN
ejpam-5634	10	3	al	al	PROPN
ejpam-5634	10	4	.	.	PUNCT
ejpam-5634	11	1	[	[	X
ejpam-5634	11	2	33	33	NUM
ejpam-5634	11	3	]	]	PUNCT
ejpam-5634	11	4	introduced	introduce	VERB
ejpam-5634	11	5	and	and	CCONJ
ejpam-5634	11	6	studied	study	VERB
ejpam-5634	11	7	the	the	DET
ejpam-5634	11	8	concept	concept	NOUN
ejpam-5634	11	9	of	of	ADP
ejpam-5634	11	10	g(m	g(m	ADJ
ejpam-5634	11	11	,	,	PUNCT
ejpam-5634	11	12	n)continuous	n)continuous	ADJ
ejpam-5634	11	13	functions	function	NOUN
ejpam-5634	11	14	.	.	PUNCT
ejpam-5634	12	1	duangphui	duangphui	NOUN
ejpam-5634	12	2	et	et	PROPN
ejpam-5634	12	3	al	al	PROPN
ejpam-5634	12	4	.	.	PUNCT
ejpam-5634	13	1	[	[	X
ejpam-5634	13	2	32	32	NUM
ejpam-5634	13	3	]	]	PUNCT
ejpam-5634	13	4	introduced	introduce	VERB
ejpam-5634	13	5	and	and	CCONJ
ejpam-5634	13	6	investigated	investigate	VERB
ejpam-5634	13	7	the	the	DET
ejpam-5634	13	8	notion	notion	NOUN
ejpam-5634	13	9	of	of	ADP
ejpam-5634	13	10	(	(	PUNCT
ejpam-5634	13	11	µ	µ	NOUN
ejpam-5634	13	12	,	,	PUNCT
ejpam-5634	13	13	µ′)(m	µ′)(m	VERB
ejpam-5634	13	14	,	,	PUNCT
ejpam-5634	13	15	n)-continuous	n)-continuous	ADJ
ejpam-5634	13	16	functions	function	NOUN
ejpam-5634	13	17	.	.	PUNCT
ejpam-5634	14	1	furthermore	furthermore	ADV
ejpam-5634	14	2	,	,	PUNCT
ejpam-5634	14	3	several	several	ADJ
ejpam-5634	14	4	characterizations	characterization	NOUN
ejpam-5634	14	5	and	and	CCONJ
ejpam-5634	14	6	some	some	DET
ejpam-5634	14	7	properties	property	NOUN
ejpam-5634	14	8	concerning	concern	VERB
ejpam-5634	14	9	almost	almost	ADV
ejpam-5634	14	10	(	(	PUNCT
ejpam-5634	14	11	λ	λ	PROPN
ejpam-5634	14	12	,	,	PUNCT
ejpam-5634	14	13	p)-continuous	p)-continuous	ADJ
ejpam-5634	14	14	functions	function	NOUN
ejpam-5634	14	15	,	,	PUNCT
ejpam-5634	14	16	strongly	strongly	ADV
ejpam-5634	14	17	θ(λ	θ(λ	PROPN
ejpam-5634	14	18	,	,	PUNCT
ejpam-5634	14	19	p)-continuous	p)-continuous	ADJ
ejpam-5634	14	20	functions	function	NOUN
ejpam-5634	14	21	,	,	PUNCT
ejpam-5634	14	22	almost	almost	ADV
ejpam-5634	14	23	strongly	strongly	ADV
ejpam-5634	14	24	θ(λ	θ(λ	VERB
ejpam-5634	14	25	,	,	PUNCT
ejpam-5634	14	26	p)-continuous	p)-continuous	ADJ
ejpam-5634	14	27	functions	function	NOUN
ejpam-5634	14	28	,	,	PUNCT
ejpam-5634	14	29	θ(λ	θ(λ	PROPN
ejpam-5634	14	30	,	,	PUNCT
ejpam-5634	14	31	p)-continuous	p)-continuous	ADJ
ejpam-5634	14	32	functions	function	NOUN
ejpam-5634	14	33	,	,	PUNCT
ejpam-5634	14	34	weakly	weakly	ADJ
ejpam-5634	14	35	(	(	PUNCT
ejpam-5634	14	36	λ	λ	PROPN
ejpam-5634	14	37	,	,	PUNCT
ejpam-5634	14	38	b)-continuous	b)-continuous	ADJ
ejpam-5634	14	39	functions	function	NOUN
ejpam-5634	14	40	,	,	PUNCT
ejpam-5634	14	41	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5634	14	42	functions	function	NOUN
ejpam-5634	14	43	,	,	PUNCT
ejpam-5634	14	44	(	(	PUNCT
ejpam-5634	14	45	λ	λ	NOUN
ejpam-5634	14	46	,	,	PUNCT
ejpam-5634	14	47	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-5634	14	48	functions	function	NOUN
ejpam-5634	14	49	,	,	PUNCT
ejpam-5634	14	50	⋆-continuous	⋆-continuous	ADJ
ejpam-5634	14	51	functions	function	NOUN
ejpam-5634	14	52	,	,	PUNCT
ejpam-5634	14	53	θ	θ	PROPN
ejpam-5634	14	54	-	-	ADJ
ejpam-5634	14	55	i	i	NOUN
ejpam-5634	14	56	-continuous	-continuous	ADJ
ejpam-5634	14	57	functions	function	NOUN
ejpam-5634	14	58	,	,	PUNCT
ejpam-5634	14	59	almost	almost	ADV
ejpam-5634	14	60	(	(	PUNCT
ejpam-5634	14	61	g	g	NOUN
ejpam-5634	14	62	,	,	PUNCT
ejpam-5634	14	63	m)-continuous	m)-continuous	ADJ
ejpam-5634	14	64	functions	function	NOUN
ejpam-5634	14	65	,	,	PUNCT
ejpam-5634	14	66	∗corresponding	∗corresponde	VERB
ejpam-5634	14	67	author	author	NOUN
ejpam-5634	14	68	.	.	PUNCT
ejpam-5634	15	1	doi	doi	NOUN
ejpam-5634	15	2	:	:	PUNCT
ejpam-5634	15	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5634	https://doi.org/10.29020/nybg.ejpam.v18i1.5634	NOUN
ejpam-5634	15	4	email	email	NOUN
ejpam-5634	15	5	addresses	address	VERB
ejpam-5634	15	6	:	:	PUNCT
ejpam-5634	16	1	monchaya.c@msu.ac.th	monchaya.c@msu.ac.th	PROPN
ejpam-5634	16	2	(	(	PUNCT
ejpam-5634	16	3	m.	m.	NOUN
ejpam-5634	16	4	chiangpradit	chiangpradit	NOUN
ejpam-5634	16	5	)	)	PUNCT
ejpam-5634	16	6	,	,	PUNCT
ejpam-5634	16	7	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-5634	16	8	(	(	PUNCT
ejpam-5634	16	9	a.	a.	PROPN
ejpam-5634	16	10	sama	sama	PROPN
ejpam-5634	16	11	-	-	PUNCT
ejpam-5634	16	12	ae	ae	PROPN
ejpam-5634	16	13	)	)	PUNCT
ejpam-5634	16	14	,	,	PUNCT
ejpam-5634	16	15	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5634	16	16	(	(	PUNCT
ejpam-5634	16	17	c.	c.	PROPN
ejpam-5634	16	18	boonpok	boonpok	PROPN
ejpam-5634	16	19	)	)	PUNCT
ejpam-5634	16	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5634	17	1	1	1	NUM
ejpam-5634	17	2	copyright	copyright	NOUN
ejpam-5634	17	3	:	:	PUNCT
ejpam-5634	17	4	©	©	PROPN
ejpam-5634	17	5	2025	2025	NUM
ejpam-5634	17	6	the	the	DET
ejpam-5634	17	7	author(s	author(s	NOUN
ejpam-5634	17	8	)	)	PUNCT
ejpam-5634	17	9	.	.	PUNCT
ejpam-5634	18	1	(	(	PUNCT
ejpam-5634	18	2	cc	cc	NOUN
ejpam-5634	18	3	by	by	ADP
ejpam-5634	18	4	-	-	PUNCT
ejpam-5634	18	5	nc	nc	PROPN
ejpam-5634	18	6	4.0	4.0	NUM
ejpam-5634	18	7	)	)	PUNCT
ejpam-5634	18	8	m.	m.	NOUN
ejpam-5634	18	9	chiangpradit	chiangpradit	NOUN
ejpam-5634	18	10	,	,	PUNCT
ejpam-5634	18	11	a.	a.	PROPN
ejpam-5634	18	12	sama	sama	PROPN
ejpam-5634	18	13	-	-	PUNCT
ejpam-5634	18	14	ae	ae	PROPN
ejpam-5634	18	15	,	,	PUNCT
ejpam-5634	18	16	c.	c.	PROPN
ejpam-5634	18	17	boonpok	boonpok	PROPN
ejpam-5634	18	18	/	/	SYM
ejpam-5634	18	19	eur	eur	PROPN
ejpam-5634	18	20	.	.	PUNCT
ejpam-5634	19	1	j.	j.	PROPN
ejpam-5634	19	2	pure	pure	PROPN
ejpam-5634	19	3	appl	appl	PROPN
ejpam-5634	19	4	.	.	PROPN
ejpam-5634	19	5	math	math	PROPN
ejpam-5634	19	6	,	,	PUNCT
ejpam-5634	19	7	18	18	NUM
ejpam-5634	19	8	(	(	PUNCT
ejpam-5634	19	9	1	1	NUM
ejpam-5634	19	10	)	)	PUNCT
ejpam-5634	19	11	(	(	PUNCT
ejpam-5634	19	12	2025	2025	NUM
ejpam-5634	19	13	)	)	PUNCT
ejpam-5634	19	14	,	,	PUNCT
ejpam-5634	19	15	5634	5634	NUM
ejpam-5634	19	16	2	2	NUM
ejpam-5634	19	17	of	of	ADP
ejpam-5634	19	18	12	12	NUM
ejpam-5634	19	19	pairwise	pairwise	NOUN
ejpam-5634	19	20	almost	almost	ADV
ejpam-5634	19	21	m	m	VERB
ejpam-5634	19	22	-continuous	-continuous	ADJ
ejpam-5634	19	23	functions	function	NOUN
ejpam-5634	19	24	,	,	PUNCT
ejpam-5634	19	25	(	(	PUNCT
ejpam-5634	19	26	τ1	τ1	NOUN
ejpam-5634	19	27	,	,	PUNCT
ejpam-5634	19	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	19	29	functions	function	NOUN
ejpam-5634	19	30	,	,	PUNCT
ejpam-5634	19	31	almost	almost	ADV
ejpam-5634	19	32	(	(	PUNCT
ejpam-5634	19	33	τ1	τ1	NOUN
ejpam-5634	19	34	,	,	PUNCT
ejpam-5634	19	35	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	19	36	functions	function	NOUN
ejpam-5634	19	37	,	,	PUNCT
ejpam-5634	19	38	weakly	weakly	ADJ
ejpam-5634	19	39	(	(	PUNCT
ejpam-5634	19	40	τ1	τ1	NOUN
ejpam-5634	19	41	,	,	PUNCT
ejpam-5634	19	42	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	19	43	functions	function	NOUN
ejpam-5634	19	44	,	,	PUNCT
ejpam-5634	19	45	faintly	faintly	ADV
ejpam-5634	19	46	(	(	PUNCT
ejpam-5634	19	47	τ1	τ1	PROPN
ejpam-5634	19	48	,	,	PUNCT
ejpam-5634	19	49	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	19	50	functions	function	NOUN
ejpam-5634	19	51	,	,	PUNCT
ejpam-5634	19	52	almost	almost	ADV
ejpam-5634	19	53	quasi	quasi	NOUN
ejpam-5634	19	54	(	(	PUNCT
ejpam-5634	19	55	τ1	τ1	NOUN
ejpam-5634	19	56	,	,	PUNCT
ejpam-5634	19	57	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	19	58	functions	function	NOUN
ejpam-5634	19	59	and	and	CCONJ
ejpam-5634	19	60	weakly	weakly	ADJ
ejpam-5634	19	61	quasi	quasi	NOUN
ejpam-5634	19	62	(	(	PUNCT
ejpam-5634	19	63	τ1	τ1	PROPN
ejpam-5634	19	64	,	,	PUNCT
ejpam-5634	19	65	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	19	66	functions	function	NOUN
ejpam-5634	19	67	were	be	AUX
ejpam-5634	19	68	presented	present	VERB
ejpam-5634	19	69	in	in	ADP
ejpam-5634	19	70	[	[	X
ejpam-5634	19	71	56	56	NUM
ejpam-5634	19	72	]	]	PUNCT
ejpam-5634	19	73	,	,	PUNCT
ejpam-5634	19	74	[	[	X
ejpam-5634	19	75	59	59	NUM
ejpam-5634	19	76	]	]	PUNCT
ejpam-5634	19	77	,	,	PUNCT
ejpam-5634	19	78	[	[	X
ejpam-5634	19	79	16	16	NUM
ejpam-5634	19	80	]	]	PUNCT
ejpam-5634	19	81	,	,	PUNCT
ejpam-5634	19	82	[	[	X
ejpam-5634	19	83	51	51	NUM
ejpam-5634	19	84	]	]	PUNCT
ejpam-5634	19	85	,	,	PUNCT
ejpam-5634	19	86	[	[	X
ejpam-5634	19	87	25	25	NUM
ejpam-5634	19	88	]	]	PUNCT
ejpam-5634	19	89	,	,	PUNCT
ejpam-5634	19	90	[	[	X
ejpam-5634	19	91	11	11	NUM
ejpam-5634	19	92	]	]	PUNCT
ejpam-5634	19	93	,	,	PUNCT
ejpam-5634	19	94	[	[	X
ejpam-5634	19	95	8	8	NUM
ejpam-5634	19	96	]	]	PUNCT
ejpam-5634	19	97	,	,	PUNCT
ejpam-5634	19	98	[	[	X
ejpam-5634	19	99	10	10	NUM
ejpam-5634	19	100	]	]	PUNCT
ejpam-5634	19	101	,	,	PUNCT
ejpam-5634	19	102	[	[	X
ejpam-5634	19	103	4	4	NUM
ejpam-5634	19	104	]	]	PUNCT
ejpam-5634	19	105	,	,	PUNCT
ejpam-5634	19	106	[	[	X
ejpam-5634	19	107	1	1	NUM
ejpam-5634	19	108	]	]	PUNCT
ejpam-5634	19	109	,	,	PUNCT
ejpam-5634	19	110	[	[	X
ejpam-5634	19	111	2	2	NUM
ejpam-5634	19	112	]	]	PUNCT
ejpam-5634	19	113	,	,	PUNCT
ejpam-5634	19	114	[	[	X
ejpam-5634	19	115	26	26	NUM
ejpam-5634	19	116	]	]	PUNCT
ejpam-5634	19	117	,	,	PUNCT
ejpam-5634	19	118	[	[	X
ejpam-5634	19	119	23	23	NUM
ejpam-5634	19	120	]	]	PUNCT
ejpam-5634	19	121	,	,	PUNCT
ejpam-5634	19	122	[	[	X
ejpam-5634	19	123	18	18	NUM
ejpam-5634	19	124	]	]	PUNCT
ejpam-5634	19	125	,	,	PUNCT
ejpam-5634	19	126	[	[	X
ejpam-5634	19	127	57	57	NUM
ejpam-5634	19	128	]	]	PUNCT
ejpam-5634	19	129	,	,	PUNCT
ejpam-5634	19	130	[	[	X
ejpam-5634	19	131	41	41	NUM
ejpam-5634	19	132	]	]	PUNCT
ejpam-5634	19	133	and	and	CCONJ
ejpam-5634	20	1	[	[	X
ejpam-5634	20	2	31	31	NUM
ejpam-5634	20	3	]	]	PUNCT
ejpam-5634	20	4	,	,	PUNCT
ejpam-5634	20	5	respectively	respectively	ADV
ejpam-5634	20	6	.	.	PUNCT
ejpam-5634	21	1	in	in	ADP
ejpam-5634	21	2	1965	1965	NUM
ejpam-5634	21	3	,	,	PUNCT
ejpam-5634	21	4	lee	lee	PROPN
ejpam-5634	22	1	[	[	X
ejpam-5634	22	2	42	42	NUM
ejpam-5634	22	3	]	]	PUNCT
ejpam-5634	22	4	studied	study	VERB
ejpam-5634	22	5	the	the	DET
ejpam-5634	22	6	notion	notion	NOUN
ejpam-5634	22	7	of	of	ADP
ejpam-5634	22	8	semiconnected	semiconnecte	VERB
ejpam-5634	22	9	functions	function	NOUN
ejpam-5634	22	10	.	.	PUNCT
ejpam-5634	23	1	kohli	kohli	NOUN
ejpam-5634	24	1	[	[	X
ejpam-5634	24	2	39	39	NUM
ejpam-5634	24	3	]	]	PUNCT
ejpam-5634	24	4	introduced	introduce	VERB
ejpam-5634	24	5	the	the	DET
ejpam-5634	24	6	notion	notion	NOUN
ejpam-5634	24	7	of	of	ADP
ejpam-5634	24	8	s	s	NOUN
ejpam-5634	24	9	-	-	ADJ
ejpam-5634	24	10	continuous	continuous	ADJ
ejpam-5634	24	11	functions	function	NOUN
ejpam-5634	24	12	and	and	CCONJ
ejpam-5634	24	13	investigated	investigate	VERB
ejpam-5634	24	14	several	several	ADJ
ejpam-5634	24	15	characterizations	characterization	NOUN
ejpam-5634	24	16	of	of	ADP
ejpam-5634	24	17	semilocally	semilocally	ADV
ejpam-5634	24	18	connected	connect	VERB
ejpam-5634	24	19	spaces	space	NOUN
ejpam-5634	24	20	in	in	ADP
ejpam-5634	24	21	terms	term	NOUN
ejpam-5634	24	22	of	of	ADP
ejpam-5634	24	23	s	s	NOUN
ejpam-5634	24	24	-	-	ADJ
ejpam-5634	24	25	continuous	continuous	ADJ
ejpam-5634	24	26	functions	function	NOUN
ejpam-5634	24	27	.	.	PUNCT
ejpam-5634	25	1	the	the	DET
ejpam-5634	25	2	concept	concept	NOUN
ejpam-5634	25	3	of	of	ADP
ejpam-5634	25	4	s	s	NOUN
ejpam-5634	25	5	-	-	NOUN
ejpam-5634	25	6	continuity	continuity	NOUN
ejpam-5634	25	7	as	as	ADP
ejpam-5634	25	8	a	a	DET
ejpam-5634	25	9	generalization	generalization	NOUN
ejpam-5634	25	10	of	of	ADP
ejpam-5634	25	11	continuity	continuity	NOUN
ejpam-5634	25	12	and	and	CCONJ
ejpam-5634	25	13	semiconnectedness	semiconnectedness	NOUN
ejpam-5634	25	14	.	.	PUNCT
ejpam-5634	26	1	moreover	moreover	ADV
ejpam-5634	26	2	,	,	PUNCT
ejpam-5634	26	3	kohli	kohli	PROPN
ejpam-5634	27	1	[	[	X
ejpam-5634	27	2	40	40	NUM
ejpam-5634	27	3	]	]	PUNCT
ejpam-5634	27	4	introduced	introduce	VERB
ejpam-5634	27	5	the	the	DET
ejpam-5634	27	6	concepts	concept	NOUN
ejpam-5634	27	7	of	of	ADP
ejpam-5634	27	8	s	s	NOUN
ejpam-5634	27	9	-	-	ADJ
ejpam-5634	27	10	regular	regular	ADJ
ejpam-5634	27	11	spaces	space	NOUN
ejpam-5634	27	12	and	and	CCONJ
ejpam-5634	27	13	completely	completely	ADV
ejpam-5634	27	14	s	s	NOUN
ejpam-5634	27	15	-	-	ADJ
ejpam-5634	27	16	regular	regular	ADJ
ejpam-5634	27	17	spaces	space	NOUN
ejpam-5634	27	18	and	and	CCONJ
ejpam-5634	27	19	proved	prove	VERB
ejpam-5634	27	20	that	that	SCONJ
ejpam-5634	27	21	s	s	NOUN
ejpam-5634	27	22	-	-	PUNCT
ejpam-5634	27	23	regularity	regularity	NOUN
ejpam-5634	27	24	and	and	CCONJ
ejpam-5634	27	25	complete	complete	ADJ
ejpam-5634	27	26	s	s	NOUN
ejpam-5634	27	27	-	-	PUNCT
ejpam-5634	27	28	regularity	regularity	NOUN
ejpam-5634	27	29	are	be	AUX
ejpam-5634	27	30	preserved	preserve	VERB
ejpam-5634	27	31	under	under	ADP
ejpam-5634	27	32	certain	certain	ADJ
ejpam-5634	27	33	s	s	NOUN
ejpam-5634	27	34	-	-	ADJ
ejpam-5634	27	35	continuous	continuous	ADJ
ejpam-5634	27	36	functions	function	NOUN
ejpam-5634	27	37	.	.	PUNCT
ejpam-5634	28	1	in	in	ADP
ejpam-5634	28	2	1989	1989	NUM
ejpam-5634	28	3	,	,	PUNCT
ejpam-5634	28	4	lipski	lipski	NOUN
ejpam-5634	28	5	[	[	X
ejpam-5634	28	6	44	44	NUM
ejpam-5634	28	7	]	]	PUNCT
ejpam-5634	28	8	extended	extend	VERB
ejpam-5634	28	9	the	the	DET
ejpam-5634	28	10	concept	concept	NOUN
ejpam-5634	28	11	of	of	ADP
ejpam-5634	28	12	s	s	NOUN
ejpam-5634	28	13	-	-	ADJ
ejpam-5634	28	14	continuous	continuous	ADJ
ejpam-5634	28	15	functions	function	NOUN
ejpam-5634	28	16	to	to	ADP
ejpam-5634	28	17	the	the	DET
ejpam-5634	28	18	setting	setting	NOUN
ejpam-5634	28	19	of	of	ADP
ejpam-5634	28	20	multifunctions	multifunction	NOUN
ejpam-5634	28	21	.	.	PUNCT
ejpam-5634	29	1	popa	popa	NOUN
ejpam-5634	30	1	[	[	X
ejpam-5634	30	2	47	47	NUM
ejpam-5634	30	3	]	]	PUNCT
ejpam-5634	30	4	introduced	introduce	VERB
ejpam-5634	30	5	the	the	DET
ejpam-5634	30	6	concept	concept	NOUN
ejpam-5634	30	7	of	of	ADP
ejpam-5634	30	8	precontinuous	precontinuous	ADJ
ejpam-5634	30	9	multifunctions	multifunction	NOUN
ejpam-5634	30	10	and	and	CCONJ
ejpam-5634	30	11	showed	show	VERB
ejpam-5634	30	12	that	that	SCONJ
ejpam-5634	30	13	h	h	NOUN
ejpam-5634	30	14	-	-	PUNCT
ejpam-5634	30	15	almost	almost	ADV
ejpam-5634	30	16	continuity	continuity	NOUN
ejpam-5634	30	17	and	and	CCONJ
ejpam-5634	30	18	precontinuity	precontinuity	NOUN
ejpam-5634	30	19	are	be	AUX
ejpam-5634	30	20	equivalent	equivalent	ADJ
ejpam-5634	30	21	for	for	ADP
ejpam-5634	30	22	multifunctions	multifunction	NOUN
ejpam-5634	30	23	.	.	PUNCT
ejpam-5634	31	1	ewert	ewert	PROPN
ejpam-5634	31	2	and	and	CCONJ
ejpam-5634	31	3	lipski	lipski	ADJ
ejpam-5634	31	4	[	[	X
ejpam-5634	31	5	35	35	NUM
ejpam-5634	31	6	]	]	PUNCT
ejpam-5634	31	7	introduced	introduce	VERB
ejpam-5634	31	8	and	and	CCONJ
ejpam-5634	31	9	investigated	investigate	VERB
ejpam-5634	31	10	the	the	DET
ejpam-5634	31	11	concept	concept	NOUN
ejpam-5634	31	12	of	of	ADP
ejpam-5634	31	13	s	s	NOUN
ejpam-5634	31	14	-	-	PUNCT
ejpam-5634	31	15	quasi	quasi	ADJ
ejpam-5634	31	16	-	-	ADJ
ejpam-5634	31	17	continuous	continuous	ADJ
ejpam-5634	31	18	multifunctions	multifunction	NOUN
ejpam-5634	31	19	.	.	PUNCT
ejpam-5634	32	1	popa	popa	NOUN
ejpam-5634	32	2	and	and	CCONJ
ejpam-5634	32	3	noiri	noiri	ADV
ejpam-5634	33	1	[	[	X
ejpam-5634	33	2	50	50	NUM
ejpam-5634	33	3	]	]	PUNCT
ejpam-5634	33	4	introduced	introduce	VERB
ejpam-5634	33	5	a	a	DET
ejpam-5634	33	6	new	new	ADJ
ejpam-5634	33	7	class	class	NOUN
ejpam-5634	33	8	of	of	ADP
ejpam-5634	33	9	multifunctions	multifunction	NOUN
ejpam-5634	33	10	called	call	VERB
ejpam-5634	33	11	sprecontinuous	sprecontinuous	ADJ
ejpam-5634	33	12	multifunctions	multifunction	NOUN
ejpam-5634	33	13	is	be	AUX
ejpam-5634	33	14	a	a	DET
ejpam-5634	33	15	generalization	generalization	NOUN
ejpam-5634	33	16	of	of	ADP
ejpam-5634	33	17	s	s	NOUN
ejpam-5634	33	18	-	-	ADJ
ejpam-5634	33	19	continuous	continuous	ADJ
ejpam-5634	33	20	multifunctions	multifunction	NOUN
ejpam-5634	33	21	and	and	CCONJ
ejpam-5634	33	22	precontinuous	precontinuous	ADJ
ejpam-5634	33	23	multifunctions	multifunction	NOUN
ejpam-5634	33	24	.	.	PUNCT
ejpam-5634	34	1	viriyapong	viriyapong	PROPN
ejpam-5634	34	2	and	and	CCONJ
ejpam-5634	34	3	boonpok	boonpok	VERB
ejpam-5634	35	1	[	[	X
ejpam-5634	35	2	64	64	NUM
ejpam-5634	35	3	]	]	PUNCT
ejpam-5634	35	4	introduced	introduce	VERB
ejpam-5634	35	5	and	and	CCONJ
ejpam-5634	35	6	studied	study	VERB
ejpam-5634	35	7	the	the	DET
ejpam-5634	35	8	concept	concept	NOUN
ejpam-5634	35	9	of	of	ADP
ejpam-5634	35	10	weakly	weakly	ADJ
ejpam-5634	35	11	quasi	quasi	NOUN
ejpam-5634	35	12	(	(	PUNCT
ejpam-5634	35	13	λ	λ	PROPN
ejpam-5634	35	14	,	,	PUNCT
ejpam-5634	35	15	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	35	16	multifunctions	multifunction	NOUN
ejpam-5634	35	17	.	.	PUNCT
ejpam-5634	36	1	moreover	moreover	ADV
ejpam-5634	36	2	,	,	PUNCT
ejpam-5634	36	3	several	several	ADJ
ejpam-5634	36	4	characterizations	characterization	NOUN
ejpam-5634	36	5	of	of	ADP
ejpam-5634	36	6	(	(	PUNCT
ejpam-5634	36	7	τ1	τ1	NOUN
ejpam-5634	36	8	,	,	PUNCT
ejpam-5634	36	9	τ2)δ	τ2)δ	ADJ
ejpam-5634	36	10	-	-	PUNCT
ejpam-5634	36	11	semicontinuous	semicontinuous	ADJ
ejpam-5634	36	12	multifunctions	multifunction	NOUN
ejpam-5634	36	13	,	,	PUNCT
ejpam-5634	36	14	almost	almost	ADV
ejpam-5634	36	15	weakly	weakly	ADJ
ejpam-5634	36	16	(	(	PUNCT
ejpam-5634	36	17	τ1	τ1	NOUN
ejpam-5634	36	18	,	,	PUNCT
ejpam-5634	36	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	20	multifunctions	multifunction	NOUN
ejpam-5634	36	21	,	,	PUNCT
ejpam-5634	36	22	⋆-continuous	⋆-continuous	ADJ
ejpam-5634	36	23	multifunctions	multifunction	NOUN
ejpam-5634	36	24	,	,	PUNCT
ejpam-5634	36	25	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-5634	36	26	multifunctions	multifunction	NOUN
ejpam-5634	36	27	,	,	PUNCT
ejpam-5634	36	28	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-5634	36	29	multifunctions	multifunction	NOUN
ejpam-5634	36	30	,	,	PUNCT
ejpam-5634	36	31	almost	almost	ADV
ejpam-5634	36	32	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5634	36	33	multifunctions	multifunction	NOUN
ejpam-5634	36	34	,	,	PUNCT
ejpam-5634	36	35	almost	almost	ADV
ejpam-5634	36	36	quasi	quasi	VERB
ejpam-5634	36	37	⋆-continuous	⋆-continuous	ADJ
ejpam-5634	36	38	multifunctions	multifunction	NOUN
ejpam-5634	36	39	,	,	PUNCT
ejpam-5634	36	40	weakly	weakly	ADJ
ejpam-5634	36	41	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5634	36	42	multifunctions	multifunction	NOUN
ejpam-5634	36	43	,	,	PUNCT
ejpam-5634	36	44	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-5634	36	45	multifunctions	multifunction	NOUN
ejpam-5634	36	46	,	,	PUNCT
ejpam-5634	36	47	weakly	weakly	ADJ
ejpam-5634	36	48	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-5634	36	49	multifunctions	multifunction	NOUN
ejpam-5634	36	50	,	,	PUNCT
ejpam-5634	36	51	θ(⋆)-quasi	θ(⋆)-quasi	NUM
ejpam-5634	36	52	continuous	continuous	ADJ
ejpam-5634	36	53	multifunctions	multifunction	NOUN
ejpam-5634	36	54	,	,	PUNCT
ejpam-5634	36	55	almost	almost	ADV
ejpam-5634	36	56	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-5634	36	57	multifunctions	multifunction	NOUN
ejpam-5634	36	58	,	,	PUNCT
ejpam-5634	36	59	weakly	weakly	ADJ
ejpam-5634	36	60	(	(	PUNCT
ejpam-5634	36	61	λ	λ	NOUN
ejpam-5634	36	62	,	,	PUNCT
ejpam-5634	36	63	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	36	64	multifunctions	multifunction	NOUN
ejpam-5634	36	65	,	,	PUNCT
ejpam-5634	36	66	α(λ	α(λ	PROPN
ejpam-5634	36	67	,	,	PUNCT
ejpam-5634	36	68	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	36	69	multifunctions	multifunction	NOUN
ejpam-5634	36	70	,	,	PUNCT
ejpam-5634	36	71	almost	almost	ADV
ejpam-5634	36	72	α(λ	α(λ	PROPN
ejpam-5634	36	73	,	,	PUNCT
ejpam-5634	36	74	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	36	75	multifunctions	multifunction	NOUN
ejpam-5634	36	76	,	,	PUNCT
ejpam-5634	36	77	weakly	weakly	ADJ
ejpam-5634	36	78	α(λ	α(λ	PROPN
ejpam-5634	36	79	,	,	PUNCT
ejpam-5634	36	80	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	36	81	multifunctions	multifunction	NOUN
ejpam-5634	36	82	,	,	PUNCT
ejpam-5634	36	83	almost	almost	ADV
ejpam-5634	36	84	β(λ	β(λ	NOUN
ejpam-5634	36	85	,	,	PUNCT
ejpam-5634	36	86	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	36	87	multifunctions	multifunction	NOUN
ejpam-5634	36	88	,	,	PUNCT
ejpam-5634	36	89	slightly	slightly	ADV
ejpam-5634	36	90	(	(	PUNCT
ejpam-5634	36	91	λ	λ	NOUN
ejpam-5634	36	92	,	,	PUNCT
ejpam-5634	36	93	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	36	94	multifunctions	multifunction	NOUN
ejpam-5634	36	95	,	,	PUNCT
ejpam-5634	36	96	(	(	PUNCT
ejpam-5634	36	97	τ1	τ1	NOUN
ejpam-5634	36	98	,	,	PUNCT
ejpam-5634	36	99	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	100	multifunctions	multifunction	NOUN
ejpam-5634	36	101	,	,	PUNCT
ejpam-5634	36	102	almost	almost	ADV
ejpam-5634	36	103	(	(	PUNCT
ejpam-5634	36	104	τ1	τ1	NOUN
ejpam-5634	36	105	,	,	PUNCT
ejpam-5634	36	106	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	107	multifunctions	multifunction	NOUN
ejpam-5634	36	108	,	,	PUNCT
ejpam-5634	36	109	weakly	weakly	ADJ
ejpam-5634	36	110	(	(	PUNCT
ejpam-5634	36	111	τ1	τ1	NOUN
ejpam-5634	36	112	,	,	PUNCT
ejpam-5634	36	113	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	114	multifunctions	multifunction	NOUN
ejpam-5634	36	115	,	,	PUNCT
ejpam-5634	36	116	weakly	weakly	ADJ
ejpam-5634	36	117	quasi	quasi	NOUN
ejpam-5634	36	118	(	(	PUNCT
ejpam-5634	36	119	τ1	τ1	PROPN
ejpam-5634	36	120	,	,	PUNCT
ejpam-5634	36	121	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	122	multifunctions	multifunction	NOUN
ejpam-5634	36	123	,	,	PUNCT
ejpam-5634	36	124	c-(τ1	c-(τ1	PROPN
ejpam-5634	36	125	,	,	PUNCT
ejpam-5634	36	126	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	36	127	multifunctions	multifunction	NOUN
ejpam-5634	36	128	,	,	PUNCT
ejpam-5634	36	129	c	c	NOUN
ejpam-5634	36	130	-	-	PUNCT
ejpam-5634	36	131	quasi	quasi	NOUN
ejpam-5634	36	132	(	(	PUNCT
ejpam-5634	36	133	τ1	τ1	PROPN
ejpam-5634	36	134	,	,	PUNCT
ejpam-5634	36	135	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	136	multifunctions	multifunction	NOUN
ejpam-5634	36	137	and	and	CCONJ
ejpam-5634	36	138	almost	almost	ADV
ejpam-5634	36	139	quasi	quasi	X
ejpam-5634	36	140	(	(	PUNCT
ejpam-5634	36	141	τ1	τ1	PROPN
ejpam-5634	36	142	,	,	PUNCT
ejpam-5634	36	143	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	36	144	multifunctions	multifunction	NOUN
ejpam-5634	36	145	were	be	AUX
ejpam-5634	36	146	established	establish	VERB
ejpam-5634	36	147	in	in	ADP
ejpam-5634	36	148	[	[	X
ejpam-5634	36	149	5	5	NUM
ejpam-5634	36	150	]	]	PUNCT
ejpam-5634	36	151	,	,	PUNCT
ejpam-5634	36	152	[	[	X
ejpam-5634	36	153	28	28	NUM
ejpam-5634	36	154	]	]	PUNCT
ejpam-5634	36	155	,	,	PUNCT
ejpam-5634	36	156	[	[	X
ejpam-5634	36	157	3	3	NUM
ejpam-5634	36	158	]	]	PUNCT
ejpam-5634	36	159	,	,	PUNCT
ejpam-5634	36	160	[	[	X
ejpam-5634	36	161	7	7	NUM
ejpam-5634	36	162	]	]	PUNCT
ejpam-5634	36	163	,	,	PUNCT
ejpam-5634	36	164	[	[	X
ejpam-5634	36	165	17	17	NUM
ejpam-5634	36	166	]	]	PUNCT
ejpam-5634	36	167	,	,	PUNCT
ejpam-5634	36	168	[	[	X
ejpam-5634	36	169	24	24	NUM
ejpam-5634	36	170	]	]	PUNCT
ejpam-5634	36	171	,	,	PUNCT
ejpam-5634	36	172	[	[	X
ejpam-5634	36	173	6	6	NUM
ejpam-5634	36	174	]	]	PUNCT
ejpam-5634	36	175	,	,	PUNCT
ejpam-5634	36	176	[	[	X
ejpam-5634	36	177	21	21	NUM
ejpam-5634	36	178	]	]	PUNCT
ejpam-5634	36	179	,	,	PUNCT
ejpam-5634	36	180	[	[	X
ejpam-5634	36	181	20	20	NUM
ejpam-5634	36	182	]	]	PUNCT
ejpam-5634	36	183	,	,	PUNCT
ejpam-5634	36	184	[	[	X
ejpam-5634	36	185	15	15	NUM
ejpam-5634	36	186	]	]	PUNCT
ejpam-5634	36	187	,	,	PUNCT
ejpam-5634	36	188	[	[	X
ejpam-5634	36	189	9	9	NUM
ejpam-5634	36	190	]	]	PUNCT
ejpam-5634	36	191	,	,	PUNCT
ejpam-5634	36	192	[	[	X
ejpam-5634	36	193	19	19	NUM
ejpam-5634	36	194	]	]	PUNCT
ejpam-5634	36	195	,	,	PUNCT
ejpam-5634	36	196	[	[	X
ejpam-5634	36	197	22	22	NUM
ejpam-5634	36	198	]	]	PUNCT
ejpam-5634	36	199	,	,	PUNCT
ejpam-5634	36	200	[	[	X
ejpam-5634	36	201	36	36	NUM
ejpam-5634	36	202	]	]	PUNCT
ejpam-5634	36	203	,	,	PUNCT
ejpam-5634	36	204	[	[	X
ejpam-5634	36	205	13	13	NUM
ejpam-5634	36	206	]	]	PUNCT
ejpam-5634	36	207	,	,	PUNCT
ejpam-5634	36	208	[	[	X
ejpam-5634	36	209	27	27	NUM
ejpam-5634	36	210	]	]	PUNCT
ejpam-5634	36	211	,	,	PUNCT
ejpam-5634	36	212	[	[	X
ejpam-5634	36	213	58	58	NUM
ejpam-5634	36	214	]	]	PUNCT
ejpam-5634	36	215	,	,	PUNCT
ejpam-5634	36	216	[	[	X
ejpam-5634	36	217	14	14	NUM
ejpam-5634	36	218	]	]	PUNCT
ejpam-5634	36	219	,	,	PUNCT
ejpam-5634	36	220	[	[	X
ejpam-5634	36	221	54	54	NUM
ejpam-5634	36	222	]	]	PUNCT
ejpam-5634	36	223	,	,	PUNCT
ejpam-5634	36	224	[	[	X
ejpam-5634	36	225	38	38	NUM
ejpam-5634	36	226	]	]	PUNCT
ejpam-5634	36	227	,	,	PUNCT
ejpam-5634	36	228	[	[	X
ejpam-5634	36	229	60	60	NUM
ejpam-5634	36	230	]	]	PUNCT
ejpam-5634	36	231	,	,	PUNCT
ejpam-5634	36	232	[	[	X
ejpam-5634	36	233	55	55	NUM
ejpam-5634	36	234	]	]	PUNCT
ejpam-5634	36	235	,	,	PUNCT
ejpam-5634	36	236	[	[	X
ejpam-5634	36	237	37	37	NUM
ejpam-5634	36	238	]	]	PUNCT
ejpam-5634	36	239	,	,	PUNCT
ejpam-5634	36	240	[	[	X
ejpam-5634	36	241	52	52	NUM
ejpam-5634	36	242	]	]	PUNCT
ejpam-5634	36	243	and	and	CCONJ
ejpam-5634	36	244	[	[	X
ejpam-5634	36	245	53	53	NUM
ejpam-5634	36	246	]	]	PUNCT
ejpam-5634	36	247	,	,	PUNCT
ejpam-5634	36	248	respectively	respectively	ADV
ejpam-5634	36	249	.	.	PUNCT
ejpam-5634	37	1	popa	popa	NOUN
ejpam-5634	37	2	and	and	CCONJ
ejpam-5634	37	3	noiri	noiri	ADV
ejpam-5634	38	1	[	[	X
ejpam-5634	38	2	49	49	NUM
ejpam-5634	38	3	]	]	PUNCT
ejpam-5634	38	4	introduced	introduce	VERB
ejpam-5634	38	5	and	and	CCONJ
ejpam-5634	38	6	studied	study	VERB
ejpam-5634	38	7	the	the	DET
ejpam-5634	38	8	notion	notion	NOUN
ejpam-5634	38	9	of	of	ADP
ejpam-5634	38	10	s	s	NOUN
ejpam-5634	38	11	-	-	PUNCT
ejpam-5634	38	12	β	β	NOUN
ejpam-5634	38	13	-	-	ADJ
ejpam-5634	38	14	continuous	continuous	ADJ
ejpam-5634	38	15	multifunctions	multifunction	NOUN
ejpam-5634	38	16	.	.	PUNCT
ejpam-5634	39	1	in	in	ADP
ejpam-5634	39	2	particular	particular	ADJ
ejpam-5634	39	3	,	,	PUNCT
ejpam-5634	39	4	popa	popa	NOUN
ejpam-5634	39	5	and	and	CCONJ
ejpam-5634	39	6	noiri	noiri	ADV
ejpam-5634	39	7	[	[	X
ejpam-5634	39	8	48	48	NUM
ejpam-5634	39	9	]	]	PUNCT
ejpam-5634	39	10	introduced	introduce	VERB
ejpam-5634	39	11	and	and	CCONJ
ejpam-5634	39	12	investigated	investigate	VERB
ejpam-5634	39	13	the	the	DET
ejpam-5634	39	14	concept	concept	NOUN
ejpam-5634	39	15	of	of	ADP
ejpam-5634	39	16	s	s	NOUN
ejpam-5634	39	17	-	-	PUNCT
ejpam-5634	39	18	m	m	NOUN
ejpam-5634	39	19	-	-	ADJ
ejpam-5634	39	20	continuous	continuous	ADJ
ejpam-5634	39	21	multifunctions	multifunction	NOUN
ejpam-5634	39	22	as	as	ADP
ejpam-5634	39	23	multifunctions	multifunction	NOUN
ejpam-5634	39	24	defined	define	VERB
ejpam-5634	39	25	on	on	ADP
ejpam-5634	39	26	a	a	DET
ejpam-5634	39	27	set	set	NOUN
ejpam-5634	39	28	satisfying	satisfy	VERB
ejpam-5634	39	29	some	some	DET
ejpam-5634	39	30	minimal	minimal	ADJ
ejpam-5634	39	31	conditions	condition	NOUN
ejpam-5634	39	32	.	.	PUNCT
ejpam-5634	40	1	quite	quite	ADV
ejpam-5634	40	2	recently	recently	ADV
ejpam-5634	40	3	,	,	PUNCT
ejpam-5634	40	4	viriyapong	viriyapong	PROPN
ejpam-5634	40	5	et	et	PROPN
ejpam-5634	40	6	al	al	PROPN
ejpam-5634	40	7	.	.	PUNCT
ejpam-5634	41	1	[	[	X
ejpam-5634	41	2	66	66	NUM
ejpam-5634	41	3	]	]	PUNCT
ejpam-5634	41	4	introduced	introduce	VERB
ejpam-5634	41	5	and	and	CCONJ
ejpam-5634	41	6	studied	study	VERB
ejpam-5634	41	7	the	the	DET
ejpam-5634	41	8	concept	concept	NOUN
ejpam-5634	41	9	of	of	ADP
ejpam-5634	41	10	s-(τ1	s-(τ1	PROPN
ejpam-5634	41	11	,	,	PUNCT
ejpam-5634	41	12	τ2)p	τ2)p	ADJ
ejpam-5634	41	13	-	-	PUNCT
ejpam-5634	41	14	continuous	continuous	ADJ
ejpam-5634	41	15	multifunctions	multifunction	NOUN
ejpam-5634	41	16	.	.	PUNCT
ejpam-5634	42	1	in	in	ADP
ejpam-5634	42	2	this	this	DET
ejpam-5634	42	3	paper	paper	NOUN
ejpam-5634	42	4	,	,	PUNCT
ejpam-5634	42	5	we	we	PRON
ejpam-5634	42	6	introduce	introduce	VERB
ejpam-5634	42	7	the	the	DET
ejpam-5634	42	8	concepts	concept	NOUN
ejpam-5634	42	9	of	of	ADP
ejpam-5634	42	10	upper	upper	ADJ
ejpam-5634	42	11	s-(τ1	s-(τ1	PROPN
ejpam-5634	42	12	,	,	PUNCT
ejpam-5634	42	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	42	14	multifunctions	multifunction	NOUN
ejpam-5634	42	15	and	and	CCONJ
ejpam-5634	42	16	lower	low	ADJ
ejpam-5634	42	17	s-(τ1	s-(τ1	NOUN
ejpam-5634	42	18	,	,	PUNCT
ejpam-5634	42	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	42	20	multifunctions	multifunction	NOUN
ejpam-5634	42	21	.	.	PUNCT
ejpam-5634	43	1	we	we	PRON
ejpam-5634	43	2	also	also	ADV
ejpam-5634	43	3	investigate	investigate	VERB
ejpam-5634	43	4	several	several	ADJ
ejpam-5634	43	5	characterizations	characterization	NOUN
ejpam-5634	43	6	of	of	ADP
ejpam-5634	43	7	upper	upper	ADJ
ejpam-5634	43	8	s-(τ1	s-(τ1	PROPN
ejpam-5634	43	9	,	,	PUNCT
ejpam-5634	43	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	43	11	multifunctions	multifunction	NOUN
ejpam-5634	43	12	and	and	CCONJ
ejpam-5634	43	13	lower	low	ADJ
ejpam-5634	43	14	s-(τ1	s-(τ1	NOUN
ejpam-5634	43	15	,	,	PUNCT
ejpam-5634	43	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	43	17	multifunctions	multifunction	NOUN
ejpam-5634	43	18	.	.	PUNCT
ejpam-5634	44	1	m.	m.	NOUN
ejpam-5634	44	2	chiangpradit	chiangpradit	PROPN
ejpam-5634	44	3	,	,	PUNCT
ejpam-5634	44	4	a.	a.	PROPN
ejpam-5634	44	5	sama	sama	PROPN
ejpam-5634	44	6	-	-	PUNCT
ejpam-5634	44	7	ae	ae	PROPN
ejpam-5634	44	8	,	,	PUNCT
ejpam-5634	44	9	c.	c.	PROPN
ejpam-5634	44	10	boonpok	boonpok	PROPN
ejpam-5634	44	11	/	/	SYM
ejpam-5634	44	12	eur	eur	PROPN
ejpam-5634	44	13	.	.	PUNCT
ejpam-5634	45	1	j.	j.	PROPN
ejpam-5634	45	2	pure	pure	PROPN
ejpam-5634	45	3	appl	appl	PROPN
ejpam-5634	45	4	.	.	PROPN
ejpam-5634	45	5	math	math	PROPN
ejpam-5634	45	6	,	,	PUNCT
ejpam-5634	45	7	18	18	NUM
ejpam-5634	45	8	(	(	PUNCT
ejpam-5634	45	9	1	1	NUM
ejpam-5634	45	10	)	)	PUNCT
ejpam-5634	45	11	(	(	PUNCT
ejpam-5634	45	12	2025	2025	NUM
ejpam-5634	45	13	)	)	PUNCT
ejpam-5634	45	14	,	,	PUNCT
ejpam-5634	45	15	5634	5634	NUM
ejpam-5634	45	16	3	3	NUM
ejpam-5634	45	17	of	of	ADP
ejpam-5634	45	18	12	12	NUM
ejpam-5634	45	19	2	2	NUM
ejpam-5634	45	20	.	.	PUNCT
ejpam-5634	45	21	preliminaries	preliminary	NOUN
ejpam-5634	45	22	throughout	throughout	ADP
ejpam-5634	45	23	the	the	DET
ejpam-5634	45	24	present	present	ADJ
ejpam-5634	45	25	paper	paper	NOUN
ejpam-5634	45	26	,	,	PUNCT
ejpam-5634	45	27	spaces	space	NOUN
ejpam-5634	45	28	(	(	PUNCT
ejpam-5634	45	29	x	x	NOUN
ejpam-5634	45	30	,	,	PUNCT
ejpam-5634	45	31	τ1	τ1	NOUN
ejpam-5634	45	32	,	,	PUNCT
ejpam-5634	45	33	τ2	τ2	NOUN
ejpam-5634	45	34	)	)	PUNCT
ejpam-5634	45	35	and	and	CCONJ
ejpam-5634	45	36	(	(	PUNCT
ejpam-5634	45	37	y	y	PROPN
ejpam-5634	45	38	,	,	PUNCT
ejpam-5634	45	39	σ1	σ1	PROPN
ejpam-5634	45	40	,	,	PUNCT
ejpam-5634	45	41	σ2	σ2	NOUN
ejpam-5634	45	42	)	)	PUNCT
ejpam-5634	45	43	(	(	PUNCT
ejpam-5634	45	44	or	or	CCONJ
ejpam-5634	45	45	simply	simply	ADV
ejpam-5634	45	46	x	x	X
ejpam-5634	45	47	and	and	CCONJ
ejpam-5634	45	48	y	y	PROPN
ejpam-5634	45	49	)	)	PUNCT
ejpam-5634	45	50	always	always	ADV
ejpam-5634	45	51	mean	mean	VERB
ejpam-5634	45	52	bitopological	bitopological	ADJ
ejpam-5634	45	53	spaces	space	NOUN
ejpam-5634	45	54	on	on	ADP
ejpam-5634	45	55	which	which	PRON
ejpam-5634	45	56	no	no	DET
ejpam-5634	45	57	separation	separation	NOUN
ejpam-5634	45	58	axioms	axiom	NOUN
ejpam-5634	45	59	are	be	AUX
ejpam-5634	45	60	assumed	assume	VERB
ejpam-5634	45	61	unless	unless	SCONJ
ejpam-5634	45	62	explicitly	explicitly	ADV
ejpam-5634	45	63	stated	state	VERB
ejpam-5634	45	64	.	.	PUNCT
ejpam-5634	46	1	let	let	VERB
ejpam-5634	46	2	a	a	DET
ejpam-5634	46	3	be	be	AUX
ejpam-5634	46	4	a	a	DET
ejpam-5634	46	5	subset	subset	NOUN
ejpam-5634	46	6	of	of	ADP
ejpam-5634	46	7	a	a	DET
ejpam-5634	46	8	bitopological	bitopological	ADJ
ejpam-5634	46	9	space	space	NOUN
ejpam-5634	46	10	(	(	PUNCT
ejpam-5634	46	11	x	x	NOUN
ejpam-5634	46	12	,	,	PUNCT
ejpam-5634	46	13	τ1	τ1	NOUN
ejpam-5634	46	14	,	,	PUNCT
ejpam-5634	46	15	τ2	τ2	NOUN
ejpam-5634	46	16	)	)	PUNCT
ejpam-5634	46	17	.	.	PUNCT
ejpam-5634	47	1	the	the	DET
ejpam-5634	47	2	closure	closure	NOUN
ejpam-5634	47	3	of	of	ADP
ejpam-5634	47	4	a	a	PRON
ejpam-5634	47	5	and	and	CCONJ
ejpam-5634	47	6	the	the	DET
ejpam-5634	47	7	interior	interior	NOUN
ejpam-5634	47	8	of	of	ADP
ejpam-5634	47	9	a	a	PRON
ejpam-5634	47	10	with	with	ADP
ejpam-5634	47	11	respect	respect	NOUN
ejpam-5634	47	12	to	to	ADP
ejpam-5634	47	13	τi	τi	PROPN
ejpam-5634	47	14	are	be	AUX
ejpam-5634	47	15	denoted	denote	VERB
ejpam-5634	47	16	by	by	ADP
ejpam-5634	47	17	τi	τi	NOUN
ejpam-5634	47	18	-	-	PUNCT
ejpam-5634	47	19	cl(a	cl(a	NUM
ejpam-5634	47	20	)	)	PUNCT
ejpam-5634	47	21	and	and	CCONJ
ejpam-5634	47	22	τi	τi	NOUN
ejpam-5634	47	23	-	-	PUNCT
ejpam-5634	47	24	int(a	int(a	NOUN
ejpam-5634	47	25	)	)	PUNCT
ejpam-5634	47	26	,	,	PUNCT
ejpam-5634	47	27	respectively	respectively	ADV
ejpam-5634	47	28	,	,	PUNCT
ejpam-5634	47	29	for	for	ADP
ejpam-5634	47	30	i	i	PROPN
ejpam-5634	47	31	=	=	SYM
ejpam-5634	47	32	1	1	NUM
ejpam-5634	47	33	,	,	PUNCT
ejpam-5634	47	34	2	2	NUM
ejpam-5634	47	35	.	.	X
ejpam-5634	47	36	a	a	DET
ejpam-5634	47	37	subset	subset	NOUN
ejpam-5634	47	38	a	a	PRON
ejpam-5634	47	39	of	of	ADP
ejpam-5634	47	40	a	a	DET
ejpam-5634	47	41	bitopological	bitopological	ADJ
ejpam-5634	47	42	space	space	NOUN
ejpam-5634	47	43	(	(	PUNCT
ejpam-5634	47	44	x	x	NOUN
ejpam-5634	47	45	,	,	PUNCT
ejpam-5634	47	46	τ1	τ1	NOUN
ejpam-5634	47	47	,	,	PUNCT
ejpam-5634	47	48	τ2	τ2	NOUN
ejpam-5634	47	49	)	)	PUNCT
ejpam-5634	47	50	is	be	AUX
ejpam-5634	47	51	called	call	VERB
ejpam-5634	47	52	τ1τ2	τ1τ2	VERB
ejpam-5634	47	53	-	-	ADJ
ejpam-5634	47	54	closed	closed	ADJ
ejpam-5634	47	55	[	[	X
ejpam-5634	47	56	29	29	NUM
ejpam-5634	47	57	]	]	X
ejpam-5634	47	58	if	if	SCONJ
ejpam-5634	47	59	a	a	DET
ejpam-5634	47	60	=	=	NOUN
ejpam-5634	47	61	τ1	τ1	NOUN
ejpam-5634	47	62	-	-	PUNCT
ejpam-5634	47	63	cl(τ2	cl(τ2	NOUN
ejpam-5634	47	64	-	-	PUNCT
ejpam-5634	47	65	cl(a	cl(a	NUM
ejpam-5634	47	66	)	)	PUNCT
ejpam-5634	47	67	)	)	PUNCT
ejpam-5634	47	68	.	.	PUNCT
ejpam-5634	48	1	the	the	DET
ejpam-5634	48	2	complement	complement	NOUN
ejpam-5634	48	3	of	of	ADP
ejpam-5634	48	4	a	a	DET
ejpam-5634	48	5	τ1τ2	τ1τ2	ADJ
ejpam-5634	48	6	-	-	ADJ
ejpam-5634	48	7	closed	closed	ADJ
ejpam-5634	48	8	set	set	NOUN
ejpam-5634	48	9	is	be	AUX
ejpam-5634	48	10	called	call	VERB
ejpam-5634	48	11	τ1τ2	τ1τ2	NOUN
ejpam-5634	48	12	-	-	ADJ
ejpam-5634	48	13	open	open	ADJ
ejpam-5634	48	14	.	.	PUNCT
ejpam-5634	49	1	the	the	DET
ejpam-5634	49	2	intersection	intersection	NOUN
ejpam-5634	49	3	of	of	ADP
ejpam-5634	49	4	all	all	DET
ejpam-5634	49	5	τ1τ2	τ1τ2	ADJ
ejpam-5634	49	6	-	-	ADJ
ejpam-5634	49	7	closed	closed	ADJ
ejpam-5634	49	8	sets	set	NOUN
ejpam-5634	49	9	of	of	ADP
ejpam-5634	49	10	x	x	PUNCT
ejpam-5634	49	11	containing	contain	VERB
ejpam-5634	49	12	a	a	PRON
ejpam-5634	49	13	is	be	AUX
ejpam-5634	49	14	called	call	VERB
ejpam-5634	49	15	the	the	DET
ejpam-5634	49	16	τ1τ2	τ1τ2	NOUN
ejpam-5634	49	17	-	-	NOUN
ejpam-5634	49	18	closure	closure	NOUN
ejpam-5634	49	19	[	[	X
ejpam-5634	49	20	29	29	NUM
ejpam-5634	49	21	]	]	PUNCT
ejpam-5634	49	22	of	of	ADP
ejpam-5634	49	23	a	a	PRON
ejpam-5634	49	24	and	and	CCONJ
ejpam-5634	49	25	is	be	AUX
ejpam-5634	49	26	denoted	denote	VERB
ejpam-5634	49	27	by	by	ADP
ejpam-5634	49	28	τ1τ2	τ1τ2	NOUN
ejpam-5634	49	29	-	-	NUM
ejpam-5634	49	30	cl(a	cl(a	NUM
ejpam-5634	49	31	)	)	PUNCT
ejpam-5634	49	32	.	.	PUNCT
ejpam-5634	50	1	the	the	DET
ejpam-5634	50	2	union	union	NOUN
ejpam-5634	50	3	of	of	ADP
ejpam-5634	50	4	all	all	DET
ejpam-5634	50	5	τ1τ2	τ1τ2	ADJ
ejpam-5634	50	6	-	-	ADJ
ejpam-5634	50	7	open	open	ADJ
ejpam-5634	50	8	sets	set	NOUN
ejpam-5634	50	9	of	of	ADP
ejpam-5634	50	10	x	x	PUNCT
ejpam-5634	50	11	contained	contain	VERB
ejpam-5634	50	12	in	in	ADP
ejpam-5634	50	13	a	a	PRON
ejpam-5634	50	14	is	be	AUX
ejpam-5634	50	15	called	call	VERB
ejpam-5634	50	16	the	the	DET
ejpam-5634	50	17	τ1τ2	τ1τ2	NOUN
ejpam-5634	50	18	-	-	ADJ
ejpam-5634	50	19	interior	interior	ADJ
ejpam-5634	50	20	[	[	X
ejpam-5634	50	21	29	29	NUM
ejpam-5634	50	22	]	]	PUNCT
ejpam-5634	50	23	of	of	ADP
ejpam-5634	50	24	a	a	PRON
ejpam-5634	50	25	and	and	CCONJ
ejpam-5634	50	26	is	be	AUX
ejpam-5634	50	27	denoted	denote	VERB
ejpam-5634	50	28	by	by	ADP
ejpam-5634	50	29	τ1τ2	τ1τ2	NOUN
ejpam-5634	50	30	-	-	ADJ
ejpam-5634	50	31	int(a	int(a	NOUN
ejpam-5634	50	32	)	)	PUNCT
ejpam-5634	50	33	.	.	PUNCT
ejpam-5634	51	1	lemma	lemma	PROPN
ejpam-5634	51	2	1	1	NUM
ejpam-5634	51	3	.	.	PUNCT
ejpam-5634	52	1	[	[	X
ejpam-5634	52	2	29	29	NUM
ejpam-5634	52	3	]	]	PUNCT
ejpam-5634	52	4	let	let	VERB
ejpam-5634	52	5	a	a	PRON
ejpam-5634	52	6	and	and	CCONJ
ejpam-5634	52	7	b	b	NOUN
ejpam-5634	52	8	be	be	AUX
ejpam-5634	52	9	subsets	subset	NOUN
ejpam-5634	52	10	of	of	ADP
ejpam-5634	52	11	a	a	DET
ejpam-5634	52	12	bitopological	bitopological	ADJ
ejpam-5634	52	13	space	space	NOUN
ejpam-5634	52	14	(	(	PUNCT
ejpam-5634	52	15	x	x	NOUN
ejpam-5634	52	16	,	,	PUNCT
ejpam-5634	52	17	τ1	τ1	NOUN
ejpam-5634	52	18	,	,	PUNCT
ejpam-5634	52	19	τ2	τ2	NOUN
ejpam-5634	52	20	)	)	PUNCT
ejpam-5634	52	21	.	.	PUNCT
ejpam-5634	53	1	for	for	ADP
ejpam-5634	53	2	the	the	DET
ejpam-5634	53	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-5634	53	4	,	,	PUNCT
ejpam-5634	53	5	the	the	DET
ejpam-5634	53	6	following	follow	VERB
ejpam-5634	53	7	properties	property	NOUN
ejpam-5634	53	8	hold	hold	VERB
ejpam-5634	53	9	:	:	PUNCT
ejpam-5634	53	10	(	(	PUNCT
ejpam-5634	53	11	1	1	X
ejpam-5634	53	12	)	)	PUNCT
ejpam-5634	53	13	a	a	DET
ejpam-5634	53	14	⊆	⊆	NUM
ejpam-5634	53	15	τ1τ2	τ1τ2	NOUN
ejpam-5634	53	16	-	-	NUM
ejpam-5634	53	17	cl(a	cl(a	NUM
ejpam-5634	53	18	)	)	PUNCT
ejpam-5634	53	19	and	and	CCONJ
ejpam-5634	53	20	τ1τ2	τ1τ2	NOUN
ejpam-5634	53	21	-	-	ADJ
ejpam-5634	53	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5634	53	23	-	-	PUNCT
ejpam-5634	53	24	cl(a	cl(a	NUM
ejpam-5634	53	25	)	)	PUNCT
ejpam-5634	53	26	)	)	PUNCT
ejpam-5634	54	1	=	=	PUNCT
ejpam-5634	54	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	54	3	-	-	NUM
ejpam-5634	54	4	cl(a	cl(a	NUM
ejpam-5634	54	5	)	)	PUNCT
ejpam-5634	54	6	.	.	PUNCT
ejpam-5634	55	1	(	(	PUNCT
ejpam-5634	55	2	2	2	X
ejpam-5634	55	3	)	)	PUNCT
ejpam-5634	55	4	if	if	SCONJ
ejpam-5634	55	5	a	a	DET
ejpam-5634	55	6	⊆	⊆	NUM
ejpam-5634	55	7	b	b	NOUN
ejpam-5634	55	8	,	,	PUNCT
ejpam-5634	55	9	then	then	ADV
ejpam-5634	55	10	τ1τ2	τ1τ2	NOUN
ejpam-5634	55	11	-	-	NUM
ejpam-5634	55	12	cl(a	cl(a	NUM
ejpam-5634	55	13	)	)	PUNCT
ejpam-5634	55	14	⊆	⊆	NUM
ejpam-5634	55	15	τ1τ2	τ1τ2	NOUN
ejpam-5634	55	16	-	-	NOUN
ejpam-5634	55	17	cl(b	cl(b	NOUN
ejpam-5634	55	18	)	)	PUNCT
ejpam-5634	55	19	.	.	PUNCT
ejpam-5634	56	1	(	(	PUNCT
ejpam-5634	56	2	3	3	X
ejpam-5634	56	3	)	)	PUNCT
ejpam-5634	56	4	τ1τ2	τ1τ2	NOUN
ejpam-5634	56	5	-	-	NUM
ejpam-5634	56	6	cl(a	cl(a	NUM
ejpam-5634	56	7	)	)	PUNCT
ejpam-5634	56	8	is	be	AUX
ejpam-5634	56	9	τ1τ2	τ1τ2	NOUN
ejpam-5634	56	10	-	-	ADJ
ejpam-5634	56	11	closed	closed	ADJ
ejpam-5634	56	12	.	.	PUNCT
ejpam-5634	57	1	(	(	PUNCT
ejpam-5634	57	2	4	4	X
ejpam-5634	57	3	)	)	PUNCT
ejpam-5634	57	4	a	a	PRON
ejpam-5634	57	5	is	be	AUX
ejpam-5634	57	6	τ1τ2	τ1τ2	NOUN
ejpam-5634	57	7	-	-	ADJ
ejpam-5634	57	8	closed	closed	ADJ
ejpam-5634	57	9	if	if	SCONJ
ejpam-5634	57	10	and	and	CCONJ
ejpam-5634	57	11	only	only	ADV
ejpam-5634	57	12	if	if	SCONJ
ejpam-5634	57	13	a	a	DET
ejpam-5634	57	14	=	=	PUNCT
ejpam-5634	57	15	τ1τ2	τ1τ2	NOUN
ejpam-5634	57	16	-	-	NUM
ejpam-5634	57	17	cl(a	cl(a	NUM
ejpam-5634	57	18	)	)	PUNCT
ejpam-5634	57	19	.	.	PUNCT
ejpam-5634	58	1	(	(	PUNCT
ejpam-5634	58	2	5	5	X
ejpam-5634	58	3	)	)	PUNCT
ejpam-5634	58	4	τ1τ2	τ1τ2	NOUN
ejpam-5634	58	5	-	-	NOUN
ejpam-5634	58	6	cl(x	cl(x	X
ejpam-5634	58	7	−a	−a	NOUN
ejpam-5634	58	8	)	)	PUNCT
ejpam-5634	59	1	=	=	PUNCT
ejpam-5634	59	2	x	x	X
ejpam-5634	60	1	−	−	ADP
ejpam-5634	60	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	60	3	-	-	PUNCT
ejpam-5634	60	4	int(a	int(a	NOUN
ejpam-5634	60	5	)	)	PUNCT
ejpam-5634	60	6	.	.	PUNCT
ejpam-5634	61	1	a	a	DET
ejpam-5634	61	2	subset	subset	NOUN
ejpam-5634	61	3	a	a	PRON
ejpam-5634	61	4	of	of	ADP
ejpam-5634	61	5	a	a	DET
ejpam-5634	61	6	bitopological	bitopological	ADJ
ejpam-5634	61	7	space	space	NOUN
ejpam-5634	61	8	(	(	PUNCT
ejpam-5634	61	9	x	x	NOUN
ejpam-5634	61	10	,	,	PUNCT
ejpam-5634	61	11	τ1	τ1	NOUN
ejpam-5634	61	12	,	,	PUNCT
ejpam-5634	61	13	τ2	τ2	NOUN
ejpam-5634	61	14	)	)	PUNCT
ejpam-5634	61	15	is	be	AUX
ejpam-5634	61	16	said	say	VERB
ejpam-5634	61	17	to	to	PART
ejpam-5634	61	18	be	be	AUX
ejpam-5634	61	19	τ1τ2	τ1τ2	NOUN
ejpam-5634	61	20	-	-	ADJ
ejpam-5634	61	21	clopen	clopen	ADJ
ejpam-5634	61	22	[	[	X
ejpam-5634	61	23	29	29	NUM
ejpam-5634	61	24	]	]	X
ejpam-5634	61	25	if	if	SCONJ
ejpam-5634	61	26	a	a	PRON
ejpam-5634	61	27	is	be	AUX
ejpam-5634	61	28	both	both	PRON
ejpam-5634	61	29	τ1τ2	τ1τ2	ADJ
ejpam-5634	61	30	-	-	ADJ
ejpam-5634	61	31	open	open	ADJ
ejpam-5634	61	32	and	and	CCONJ
ejpam-5634	61	33	τ1τ2	τ1τ2	NOUN
ejpam-5634	61	34	-	-	ADJ
ejpam-5634	61	35	closed	closed	ADJ
ejpam-5634	61	36	.	.	PUNCT
ejpam-5634	62	1	a	a	DET
ejpam-5634	62	2	subset	subset	NOUN
ejpam-5634	62	3	a	a	PRON
ejpam-5634	62	4	of	of	ADP
ejpam-5634	62	5	a	a	DET
ejpam-5634	62	6	bitopological	bitopological	ADJ
ejpam-5634	62	7	space	space	NOUN
ejpam-5634	62	8	(	(	PUNCT
ejpam-5634	62	9	x	x	NOUN
ejpam-5634	62	10	,	,	PUNCT
ejpam-5634	62	11	τ1	τ1	NOUN
ejpam-5634	62	12	,	,	PUNCT
ejpam-5634	62	13	τ2	τ2	NOUN
ejpam-5634	62	14	)	)	PUNCT
ejpam-5634	62	15	is	be	AUX
ejpam-5634	62	16	called	call	VERB
ejpam-5634	62	17	(	(	PUNCT
ejpam-5634	62	18	τ1	τ1	PROPN
ejpam-5634	62	19	,	,	PUNCT
ejpam-5634	62	20	τ2)ropen	τ2)ropen	ADJ
ejpam-5634	63	1	[	[	X
ejpam-5634	63	2	62	62	NUM
ejpam-5634	63	3	]	]	PUNCT
ejpam-5634	63	4	(	(	PUNCT
ejpam-5634	63	5	resp	resp	NOUN
ejpam-5634	63	6	.	.	PUNCT
ejpam-5634	64	1	(	(	PUNCT
ejpam-5634	64	2	τ1	τ1	NOUN
ejpam-5634	64	3	,	,	PUNCT
ejpam-5634	64	4	τ2)s	τ2)s	NOUN
ejpam-5634	64	5	-	-	PUNCT
ejpam-5634	64	6	open	open	ADJ
ejpam-5634	64	7	[	[	X
ejpam-5634	64	8	5	5	NUM
ejpam-5634	64	9	]	]	PUNCT
ejpam-5634	64	10	,	,	PUNCT
ejpam-5634	64	11	(	(	PUNCT
ejpam-5634	64	12	τ1	τ1	NOUN
ejpam-5634	64	13	,	,	PUNCT
ejpam-5634	64	14	τ2)p	τ2)p	NOUN
ejpam-5634	64	15	-	-	ADJ
ejpam-5634	64	16	open	open	ADJ
ejpam-5634	64	17	[	[	X
ejpam-5634	64	18	5	5	NUM
ejpam-5634	64	19	]	]	PUNCT
ejpam-5634	64	20	,	,	PUNCT
ejpam-5634	64	21	(	(	PUNCT
ejpam-5634	64	22	τ1	τ1	NOUN
ejpam-5634	64	23	,	,	PUNCT
ejpam-5634	64	24	τ2)β	τ2)β	ADJ
ejpam-5634	64	25	-	-	PUNCT
ejpam-5634	64	26	open	open	NOUN
ejpam-5634	65	1	[	[	X
ejpam-5634	65	2	5	5	NUM
ejpam-5634	65	3	]	]	PUNCT
ejpam-5634	65	4	,	,	PUNCT
ejpam-5634	65	5	α(τ1	α(τ1	NOUN
ejpam-5634	65	6	,	,	PUNCT
ejpam-5634	65	7	τ2)-open	τ2)-open	ADJ
ejpam-5634	65	8	)	)	PUNCT
ejpam-5634	66	1	[	[	X
ejpam-5634	66	2	65	65	NUM
ejpam-5634	66	3	]	]	SYM
ejpam-5634	66	4	)	)	PUNCT
ejpam-5634	66	5	if	if	SCONJ
ejpam-5634	66	6	a	a	DET
ejpam-5634	66	7	=	=	PUNCT
ejpam-5634	66	8	τ1τ2	τ1τ2	NOUN
ejpam-5634	66	9	-	-	NOUN
ejpam-5634	66	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5634	66	11	-	-	PUNCT
ejpam-5634	66	12	cl(a	cl(a	NUM
ejpam-5634	66	13	)	)	PUNCT
ejpam-5634	66	14	)	)	PUNCT
ejpam-5634	66	15	(	(	PUNCT
ejpam-5634	66	16	resp	resp	NOUN
ejpam-5634	66	17	.	.	PUNCT
ejpam-5634	67	1	a	a	DET
ejpam-5634	67	2	⊆	⊆	NUM
ejpam-5634	67	3	τ1τ2	τ1τ2	NOUN
ejpam-5634	67	4	-	-	ADJ
ejpam-5634	67	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5634	67	6	-	-	PUNCT
ejpam-5634	67	7	int(a	int(a	NOUN
ejpam-5634	67	8	)	)	PUNCT
ejpam-5634	67	9	)	)	PUNCT
ejpam-5634	67	10	,	,	PUNCT
ejpam-5634	67	11	a	a	DET
ejpam-5634	67	12	⊆	⊆	NUM
ejpam-5634	67	13	τ1τ2	τ1τ2	NOUN
ejpam-5634	67	14	-	-	NOUN
ejpam-5634	67	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5634	67	16	-	-	PUNCT
ejpam-5634	67	17	cl(a	cl(a	NUM
ejpam-5634	67	18	)	)	PUNCT
ejpam-5634	67	19	)	)	PUNCT
ejpam-5634	67	20	,	,	PUNCT
ejpam-5634	67	21	a	a	DET
ejpam-5634	67	22	⊆	⊆	NUM
ejpam-5634	67	23	τ1τ2	τ1τ2	NOUN
ejpam-5634	67	24	-	-	PUNCT
ejpam-5634	67	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5634	67	26	-	-	PUNCT
ejpam-5634	67	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5634	67	28	-	-	PUNCT
ejpam-5634	67	29	cl(a	cl(a	NUM
ejpam-5634	67	30	)	)	PUNCT
ejpam-5634	67	31	)	)	PUNCT
ejpam-5634	67	32	)	)	PUNCT
ejpam-5634	67	33	,	,	PUNCT
ejpam-5634	67	34	a	a	DET
ejpam-5634	67	35	⊆	⊆	NUM
ejpam-5634	67	36	τ1τ2	τ1τ2	NOUN
ejpam-5634	67	37	-	-	PUNCT
ejpam-5634	67	38	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5634	67	39	-	-	PUNCT
ejpam-5634	67	40	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5634	67	41	-	-	PUNCT
ejpam-5634	67	42	int(a	int(a	NOUN
ejpam-5634	67	43	)	)	PUNCT
ejpam-5634	67	44	)	)	PUNCT
ejpam-5634	67	45	)	)	PUNCT
ejpam-5634	67	46	)	)	PUNCT
ejpam-5634	67	47	.	.	PUNCT
ejpam-5634	68	1	the	the	DET
ejpam-5634	68	2	complement	complement	NOUN
ejpam-5634	68	3	of	of	ADP
ejpam-5634	68	4	a	a	DET
ejpam-5634	68	5	(	(	PUNCT
ejpam-5634	68	6	τ1	τ1	NOUN
ejpam-5634	68	7	,	,	PUNCT
ejpam-5634	68	8	τ2)r	τ2)r	NOUN
ejpam-5634	68	9	-	-	PUNCT
ejpam-5634	68	10	open	open	ADJ
ejpam-5634	68	11	(	(	PUNCT
ejpam-5634	68	12	resp	resp	NOUN
ejpam-5634	68	13	.	.	PUNCT
ejpam-5634	69	1	(	(	PUNCT
ejpam-5634	69	2	τ1	τ1	NOUN
ejpam-5634	69	3	,	,	PUNCT
ejpam-5634	69	4	τ2)s	τ2)s	NOUN
ejpam-5634	69	5	-	-	PUNCT
ejpam-5634	69	6	open	open	ADJ
ejpam-5634	69	7	,	,	PUNCT
ejpam-5634	69	8	(	(	PUNCT
ejpam-5634	69	9	τ1	τ1	NOUN
ejpam-5634	69	10	,	,	PUNCT
ejpam-5634	69	11	τ2)p	τ2)p	NOUN
ejpam-5634	69	12	-	-	ADJ
ejpam-5634	69	13	open	open	ADJ
ejpam-5634	69	14	,	,	PUNCT
ejpam-5634	69	15	(	(	PUNCT
ejpam-5634	69	16	τ1	τ1	NOUN
ejpam-5634	69	17	,	,	PUNCT
ejpam-5634	69	18	τ2)β	τ2)β	ADJ
ejpam-5634	69	19	-	-	PUNCT
ejpam-5634	69	20	open	open	ADJ
ejpam-5634	69	21	,	,	PUNCT
ejpam-5634	69	22	α(τ1	α(τ1	NOUN
ejpam-5634	69	23	,	,	PUNCT
ejpam-5634	69	24	τ2)-open	τ2)-open	ADJ
ejpam-5634	69	25	)	)	PUNCT
ejpam-5634	69	26	set	set	NOUN
ejpam-5634	69	27	is	be	AUX
ejpam-5634	69	28	called	call	VERB
ejpam-5634	69	29	(	(	PUNCT
ejpam-5634	69	30	τ1	τ1	NOUN
ejpam-5634	69	31	,	,	PUNCT
ejpam-5634	69	32	τ2)r	τ2)r	NOUN
ejpam-5634	69	33	-	-	PUNCT
ejpam-5634	69	34	closed	closed	ADJ
ejpam-5634	69	35	(	(	PUNCT
ejpam-5634	69	36	resp	resp	NOUN
ejpam-5634	69	37	.	.	PUNCT
ejpam-5634	70	1	(	(	PUNCT
ejpam-5634	70	2	τ1	τ1	NOUN
ejpam-5634	70	3	,	,	PUNCT
ejpam-5634	70	4	τ2)s	τ2)s	NOUN
ejpam-5634	70	5	-	-	PUNCT
ejpam-5634	70	6	closed	closed	ADJ
ejpam-5634	70	7	,	,	PUNCT
ejpam-5634	70	8	(	(	PUNCT
ejpam-5634	70	9	τ1	τ1	NOUN
ejpam-5634	70	10	,	,	PUNCT
ejpam-5634	70	11	τ2)p	τ2)p	NOUN
ejpam-5634	70	12	-	-	PUNCT
ejpam-5634	70	13	closed	closed	ADJ
ejpam-5634	70	14	,	,	PUNCT
ejpam-5634	70	15	(	(	PUNCT
ejpam-5634	70	16	τ1	τ1	NOUN
ejpam-5634	70	17	,	,	PUNCT
ejpam-5634	70	18	τ2)β	τ2)β	ADJ
ejpam-5634	70	19	-	-	PUNCT
ejpam-5634	70	20	closed	closed	ADJ
ejpam-5634	70	21	,	,	PUNCT
ejpam-5634	70	22	α(τ1	α(τ1	NOUN
ejpam-5634	70	23	,	,	PUNCT
ejpam-5634	70	24	τ2)closed	τ2)close	VERB
ejpam-5634	70	25	)	)	PUNCT
ejpam-5634	70	26	.	.	PUNCT
ejpam-5634	71	1	by	by	ADP
ejpam-5634	71	2	a	a	DET
ejpam-5634	71	3	multifunction	multifunction	NOUN
ejpam-5634	71	4	f	f	NOUN
ejpam-5634	71	5	:	:	PUNCT
ejpam-5634	71	6	x	x	X
ejpam-5634	71	7	→	→	SYM
ejpam-5634	71	8	y	y	PROPN
ejpam-5634	71	9	,	,	PUNCT
ejpam-5634	71	10	we	we	PRON
ejpam-5634	71	11	mean	mean	VERB
ejpam-5634	71	12	a	a	DET
ejpam-5634	71	13	point	point	NOUN
ejpam-5634	71	14	-	-	PUNCT
ejpam-5634	71	15	to	to	ADP
ejpam-5634	71	16	-	-	PUNCT
ejpam-5634	71	17	set	set	VERB
ejpam-5634	71	18	correspondence	correspondence	NOUN
ejpam-5634	71	19	from	from	ADP
ejpam-5634	71	20	x	x	PUNCT
ejpam-5634	71	21	into	into	ADP
ejpam-5634	71	22	y	y	PROPN
ejpam-5634	71	23	,	,	PUNCT
ejpam-5634	71	24	and	and	CCONJ
ejpam-5634	71	25	we	we	PRON
ejpam-5634	71	26	always	always	ADV
ejpam-5634	71	27	assume	assume	VERB
ejpam-5634	71	28	that	that	SCONJ
ejpam-5634	71	29	f	f	PROPN
ejpam-5634	71	30	(	(	PUNCT
ejpam-5634	71	31	x	x	X
ejpam-5634	71	32	)	)	PUNCT
ejpam-5634	71	33	̸=	̸=	NOUN
ejpam-5634	71	34	∅	∅	NOUN
ejpam-5634	71	35	for	for	ADP
ejpam-5634	71	36	all	all	PRON
ejpam-5634	71	37	x	x	SYM
ejpam-5634	71	38	∈	∈	ADJ
ejpam-5634	71	39	x.	x.	NOUN
ejpam-5634	71	40	for	for	ADP
ejpam-5634	71	41	a	a	DET
ejpam-5634	71	42	multifunction	multifunction	NOUN
ejpam-5634	71	43	f	f	NOUN
ejpam-5634	71	44	:	:	PUNCT
ejpam-5634	71	45	x	x	X
ejpam-5634	71	46	→	→	SYM
ejpam-5634	71	47	y	y	PROPN
ejpam-5634	71	48	,	,	PUNCT
ejpam-5634	71	49	we	we	PRON
ejpam-5634	71	50	shall	shall	AUX
ejpam-5634	71	51	denote	denote	VERB
ejpam-5634	71	52	the	the	DET
ejpam-5634	71	53	upper	upper	ADJ
ejpam-5634	71	54	and	and	CCONJ
ejpam-5634	71	55	lower	low	ADJ
ejpam-5634	71	56	inverse	inverse	NOUN
ejpam-5634	71	57	of	of	ADP
ejpam-5634	71	58	a	a	DET
ejpam-5634	71	59	set	set	NOUN
ejpam-5634	71	60	b	b	PROPN
ejpam-5634	71	61	of	of	ADP
ejpam-5634	71	62	y	y	PROPN
ejpam-5634	71	63	by	by	ADP
ejpam-5634	71	64	f+(b	f+(b	NOUN
ejpam-5634	71	65	)	)	PUNCT
ejpam-5634	71	66	and	and	CCONJ
ejpam-5634	71	67	f−(b	f−(b	NOUN
ejpam-5634	71	68	)	)	PUNCT
ejpam-5634	71	69	,	,	PUNCT
ejpam-5634	71	70	respectively	respectively	ADV
ejpam-5634	71	71	,	,	PUNCT
ejpam-5634	71	72	that	that	ADV
ejpam-5634	71	73	is	is	ADV
ejpam-5634	71	74	,	,	PUNCT
ejpam-5634	71	75	f+(b	f+(b	NOUN
ejpam-5634	71	76	)	)	PUNCT
ejpam-5634	71	77	=	=	PRON
ejpam-5634	72	1	{	{	PUNCT
ejpam-5634	72	2	x	x	PUNCT
ejpam-5634	72	3	∈	∈	PROPN
ejpam-5634	72	4	x	x	INTJ
ejpam-5634	73	1	|	|	NOUN
ejpam-5634	73	2	f	f	X
ejpam-5634	73	3	(	(	PUNCT
ejpam-5634	73	4	x	x	NOUN
ejpam-5634	73	5	)	)	PUNCT
ejpam-5634	73	6	⊆	⊆	NUM
ejpam-5634	73	7	b	b	NOUN
ejpam-5634	73	8	}	}	PUNCT
ejpam-5634	73	9	and	and	CCONJ
ejpam-5634	73	10	f−(b	f−(b	PROPN
ejpam-5634	73	11	)	)	PUNCT
ejpam-5634	73	12	=	=	PRON
ejpam-5634	74	1	{	{	PUNCT
ejpam-5634	74	2	x	x	PUNCT
ejpam-5634	74	3	∈	∈	PROPN
ejpam-5634	74	4	x	x	INTJ
ejpam-5634	75	1	|	|	NOUN
ejpam-5634	75	2	f	f	X
ejpam-5634	75	3	(	(	PUNCT
ejpam-5634	75	4	x	x	NOUN
ejpam-5634	75	5	)	)	PUNCT
ejpam-5634	75	6	∩	∩	NOUN
ejpam-5634	75	7	b	b	PROPN
ejpam-5634	75	8	̸=	̸=	PROPN
ejpam-5634	75	9	∅	∅	NOUN
ejpam-5634	75	10	}	}	PUNCT
ejpam-5634	75	11	.	.	PUNCT
ejpam-5634	76	1	in	in	ADP
ejpam-5634	76	2	particular	particular	ADJ
ejpam-5634	76	3	,	,	PUNCT
ejpam-5634	76	4	f−(y	f−(y	NOUN
ejpam-5634	76	5	)	)	PUNCT
ejpam-5634	76	6	=	=	SYM
ejpam-5634	77	1	{	{	PUNCT
ejpam-5634	77	2	x	x	PUNCT
ejpam-5634	77	3	∈	∈	PROPN
ejpam-5634	77	4	x	x	INTJ
ejpam-5634	78	1	|	|	ADV
ejpam-5634	78	2	y	y	PROPN
ejpam-5634	78	3	∈	∈	PROPN
ejpam-5634	78	4	f	f	X
ejpam-5634	78	5	(	(	PUNCT
ejpam-5634	78	6	x	x	NOUN
ejpam-5634	78	7	)	)	PUNCT
ejpam-5634	78	8	}	}	PUNCT
ejpam-5634	78	9	for	for	ADP
ejpam-5634	78	10	each	each	DET
ejpam-5634	78	11	point	point	NOUN
ejpam-5634	78	12	y	y	PROPN
ejpam-5634	78	13	∈	∈	PROPN
ejpam-5634	78	14	y	y	PROPN
ejpam-5634	78	15	.	.	PUNCT
ejpam-5634	79	1	for	for	ADP
ejpam-5634	79	2	each	each	DET
ejpam-5634	79	3	a	a	DET
ejpam-5634	79	4	⊆	⊆	NUM
ejpam-5634	79	5	x	x	SYM
ejpam-5634	79	6	,	,	PUNCT
ejpam-5634	79	7	f	f	PROPN
ejpam-5634	79	8	(	(	PUNCT
ejpam-5634	79	9	a	a	NOUN
ejpam-5634	79	10	)	)	PUNCT
ejpam-5634	79	11	=	=	SYM
ejpam-5634	79	12	∪x∈af	∪x∈af	NOUN
ejpam-5634	79	13	(	(	PUNCT
ejpam-5634	79	14	x	x	NOUN
ejpam-5634	79	15	)	)	PUNCT
ejpam-5634	79	16	.	.	PUNCT
ejpam-5634	80	1	3	3	X
ejpam-5634	80	2	.	.	X
ejpam-5634	80	3	upper	upper	ADJ
ejpam-5634	80	4	and	and	CCONJ
ejpam-5634	80	5	lower	low	ADJ
ejpam-5634	80	6	s-(τ1	s-(τ1	PROPN
ejpam-5634	80	7	,	,	PUNCT
ejpam-5634	80	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	80	9	multifunctions	multifunction	NOUN
ejpam-5634	80	10	in	in	ADP
ejpam-5634	80	11	this	this	DET
ejpam-5634	80	12	section	section	NOUN
ejpam-5634	80	13	,	,	PUNCT
ejpam-5634	80	14	we	we	PRON
ejpam-5634	80	15	introduce	introduce	VERB
ejpam-5634	80	16	the	the	DET
ejpam-5634	80	17	notions	notion	NOUN
ejpam-5634	80	18	of	of	ADP
ejpam-5634	80	19	upper	upper	ADJ
ejpam-5634	80	20	s-(τ1	s-(τ1	PROPN
ejpam-5634	80	21	,	,	PUNCT
ejpam-5634	80	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	80	23	multifunctions	multifunction	NOUN
ejpam-5634	80	24	and	and	CCONJ
ejpam-5634	80	25	lower	low	ADJ
ejpam-5634	80	26	s-(τ1	s-(τ1	NOUN
ejpam-5634	80	27	,	,	PUNCT
ejpam-5634	80	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	80	29	multifunctions	multifunction	NOUN
ejpam-5634	80	30	.	.	PUNCT
ejpam-5634	81	1	moreover	moreover	ADV
ejpam-5634	81	2	,	,	PUNCT
ejpam-5634	81	3	some	some	DET
ejpam-5634	81	4	characterizations	characterization	NOUN
ejpam-5634	81	5	of	of	ADP
ejpam-5634	81	6	upper	upper	ADJ
ejpam-5634	81	7	s-(τ1	s-(τ1	PROPN
ejpam-5634	81	8	,	,	PUNCT
ejpam-5634	81	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	81	10	multifunctions	multifunction	NOUN
ejpam-5634	81	11	and	and	CCONJ
ejpam-5634	81	12	lower	low	ADJ
ejpam-5634	81	13	s-(τ1	s-(τ1	NOUN
ejpam-5634	81	14	,	,	PUNCT
ejpam-5634	81	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	81	16	multifunctions	multifunction	NOUN
ejpam-5634	81	17	are	be	AUX
ejpam-5634	81	18	discussed	discuss	VERB
ejpam-5634	81	19	.	.	PUNCT
ejpam-5634	82	1	m.	m.	NOUN
ejpam-5634	82	2	chiangpradit	chiangpradit	PROPN
ejpam-5634	82	3	,	,	PUNCT
ejpam-5634	82	4	a.	a.	PROPN
ejpam-5634	82	5	sama	sama	PROPN
ejpam-5634	82	6	-	-	PUNCT
ejpam-5634	82	7	ae	ae	PROPN
ejpam-5634	82	8	,	,	PUNCT
ejpam-5634	82	9	c.	c.	PROPN
ejpam-5634	82	10	boonpok	boonpok	PROPN
ejpam-5634	82	11	/	/	SYM
ejpam-5634	82	12	eur	eur	PROPN
ejpam-5634	82	13	.	.	PUNCT
ejpam-5634	83	1	j.	j.	PROPN
ejpam-5634	83	2	pure	pure	PROPN
ejpam-5634	83	3	appl	appl	PROPN
ejpam-5634	83	4	.	.	PROPN
ejpam-5634	83	5	math	math	PROPN
ejpam-5634	83	6	,	,	PUNCT
ejpam-5634	83	7	18	18	NUM
ejpam-5634	83	8	(	(	PUNCT
ejpam-5634	83	9	1	1	NUM
ejpam-5634	83	10	)	)	PUNCT
ejpam-5634	83	11	(	(	PUNCT
ejpam-5634	83	12	2025	2025	NUM
ejpam-5634	83	13	)	)	PUNCT
ejpam-5634	83	14	,	,	PUNCT
ejpam-5634	83	15	5634	5634	NUM
ejpam-5634	83	16	4	4	NUM
ejpam-5634	83	17	of	of	ADP
ejpam-5634	83	18	12	12	NUM
ejpam-5634	83	19	definition	definition	NOUN
ejpam-5634	83	20	1	1	NUM
ejpam-5634	83	21	.	.	PUNCT
ejpam-5634	84	1	a	a	DET
ejpam-5634	84	2	multifunction	multifunction	NOUN
ejpam-5634	84	3	f	f	NOUN
ejpam-5634	84	4	:	:	PUNCT
ejpam-5634	84	5	(	(	PUNCT
ejpam-5634	84	6	x	x	NOUN
ejpam-5634	84	7	,	,	PUNCT
ejpam-5634	84	8	τ1	τ1	NOUN
ejpam-5634	84	9	,	,	PUNCT
ejpam-5634	84	10	τ2	τ2	NOUN
ejpam-5634	84	11	)	)	PUNCT
ejpam-5634	84	12	→	→	SYM
ejpam-5634	84	13	(	(	PUNCT
ejpam-5634	84	14	y	y	PROPN
ejpam-5634	84	15	,	,	PUNCT
ejpam-5634	84	16	σ1	σ1	PROPN
ejpam-5634	84	17	,	,	PUNCT
ejpam-5634	84	18	σ2	σ2	PROPN
ejpam-5634	84	19	)	)	PUNCT
ejpam-5634	84	20	is	be	AUX
ejpam-5634	84	21	said	say	VERB
ejpam-5634	84	22	to	to	PART
ejpam-5634	84	23	be	be	AUX
ejpam-5634	84	24	upper	upper	ADJ
ejpam-5634	84	25	s-(τ1	s-(τ1	NOUN
ejpam-5634	84	26	,	,	PUNCT
ejpam-5634	84	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	84	28	at	at	ADP
ejpam-5634	84	29	x	x	X
ejpam-5634	84	30	∈	∈	PROPN
ejpam-5634	84	31	x	x	PUNCT
ejpam-5634	84	32	if	if	SCONJ
ejpam-5634	84	33	for	for	ADP
ejpam-5634	84	34	each	each	DET
ejpam-5634	84	35	σ1σ2	σ1σ2	VERB
ejpam-5634	84	36	-	-	ADJ
ejpam-5634	84	37	open	open	ADJ
ejpam-5634	84	38	set	set	NOUN
ejpam-5634	84	39	v	v	NOUN
ejpam-5634	84	40	of	of	ADP
ejpam-5634	84	41	y	y	PROPN
ejpam-5634	84	42	containing	contain	VERB
ejpam-5634	84	43	f	f	PROPN
ejpam-5634	84	44	(	(	PUNCT
ejpam-5634	84	45	x	x	NOUN
ejpam-5634	84	46	)	)	PUNCT
ejpam-5634	84	47	and	and	CCONJ
ejpam-5634	84	48	having	have	VERB
ejpam-5634	84	49	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5634	84	50	complement	complement	NOUN
ejpam-5634	84	51	,	,	PUNCT
ejpam-5634	84	52	there	there	PRON
ejpam-5634	84	53	exists	exist	VERB
ejpam-5634	84	54	a	a	DET
ejpam-5634	84	55	τ1τ2	τ1τ2	NOUN
ejpam-5634	84	56	-	-	ADJ
ejpam-5634	84	57	open	open	ADJ
ejpam-5634	84	58	set	set	ADJ
ejpam-5634	84	59	u	u	NOUN
ejpam-5634	84	60	of	of	ADP
ejpam-5634	84	61	x	x	PUNCT
ejpam-5634	84	62	containing	contain	VERB
ejpam-5634	84	63	x	x	PUNCT
ejpam-5634	84	64	such	such	ADJ
ejpam-5634	84	65	that	that	SCONJ
ejpam-5634	84	66	f	f	PROPN
ejpam-5634	84	67	(	(	PUNCT
ejpam-5634	84	68	u	u	NOUN
ejpam-5634	84	69	)	)	PUNCT
ejpam-5634	84	70	⊆	⊆	NUM
ejpam-5634	84	71	v	v	NOUN
ejpam-5634	84	72	.	.	PUNCT
ejpam-5634	85	1	a	a	DET
ejpam-5634	85	2	multifunction	multifunction	NOUN
ejpam-5634	85	3	f	f	NOUN
ejpam-5634	85	4	:	:	PUNCT
ejpam-5634	85	5	(	(	PUNCT
ejpam-5634	85	6	x	x	NOUN
ejpam-5634	85	7	,	,	PUNCT
ejpam-5634	85	8	τ1	τ1	NOUN
ejpam-5634	85	9	,	,	PUNCT
ejpam-5634	85	10	τ2	τ2	NOUN
ejpam-5634	85	11	)	)	PUNCT
ejpam-5634	85	12	→	→	SYM
ejpam-5634	85	13	(	(	PUNCT
ejpam-5634	85	14	y	y	PROPN
ejpam-5634	85	15	,	,	PUNCT
ejpam-5634	85	16	σ1	σ1	PROPN
ejpam-5634	85	17	,	,	PUNCT
ejpam-5634	85	18	σ2	σ2	PROPN
ejpam-5634	85	19	)	)	PUNCT
ejpam-5634	85	20	is	be	AUX
ejpam-5634	85	21	said	say	VERB
ejpam-5634	85	22	to	to	PART
ejpam-5634	85	23	be	be	AUX
ejpam-5634	85	24	upper	upper	ADJ
ejpam-5634	85	25	s-(τ1	s-(τ1	PROPN
ejpam-5634	85	26	,	,	PUNCT
ejpam-5634	85	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	85	28	if	if	SCONJ
ejpam-5634	85	29	f	f	PROPN
ejpam-5634	85	30	is	be	AUX
ejpam-5634	85	31	upper	upper	ADJ
ejpam-5634	85	32	s-(τ1	s-(τ1	PROPN
ejpam-5634	85	33	,	,	PUNCT
ejpam-5634	85	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	85	35	at	at	ADP
ejpam-5634	85	36	each	each	DET
ejpam-5634	85	37	point	point	NOUN
ejpam-5634	85	38	x	x	PUNCT
ejpam-5634	85	39	of	of	ADP
ejpam-5634	85	40	x.	x.	PROPN
ejpam-5634	85	41	theorem	theorem	VERB
ejpam-5634	85	42	1	1	NUM
ejpam-5634	85	43	.	.	X
ejpam-5634	85	44	for	for	ADP
ejpam-5634	85	45	a	a	DET
ejpam-5634	85	46	multifunction	multifunction	NOUN
ejpam-5634	85	47	f	f	NOUN
ejpam-5634	85	48	:	:	PUNCT
ejpam-5634	85	49	(	(	PUNCT
ejpam-5634	85	50	x	x	NOUN
ejpam-5634	85	51	,	,	PUNCT
ejpam-5634	85	52	τ1	τ1	NOUN
ejpam-5634	85	53	,	,	PUNCT
ejpam-5634	85	54	τ2	τ2	NOUN
ejpam-5634	85	55	)	)	PUNCT
ejpam-5634	85	56	→	→	SYM
ejpam-5634	85	57	(	(	PUNCT
ejpam-5634	85	58	y	y	PROPN
ejpam-5634	85	59	,	,	PUNCT
ejpam-5634	85	60	σ1	σ1	PROPN
ejpam-5634	85	61	,	,	PUNCT
ejpam-5634	85	62	σ2	σ2	NOUN
ejpam-5634	85	63	)	)	PUNCT
ejpam-5634	85	64	,	,	PUNCT
ejpam-5634	85	65	the	the	DET
ejpam-5634	85	66	following	follow	VERB
ejpam-5634	85	67	properties	property	NOUN
ejpam-5634	85	68	are	be	AUX
ejpam-5634	85	69	equivalent	equivalent	ADJ
ejpam-5634	85	70	:	:	PUNCT
ejpam-5634	85	71	(	(	PUNCT
ejpam-5634	85	72	1	1	X
ejpam-5634	85	73	)	)	PUNCT
ejpam-5634	85	74	f	f	PROPN
ejpam-5634	85	75	is	be	AUX
ejpam-5634	85	76	upper	upper	ADJ
ejpam-5634	85	77	s-(τ1	s-(τ1	PROPN
ejpam-5634	85	78	,	,	PUNCT
ejpam-5634	85	79	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	85	80	;	;	PUNCT
ejpam-5634	85	81	(	(	PUNCT
ejpam-5634	85	82	2	2	NUM
ejpam-5634	85	83	)	)	PUNCT
ejpam-5634	85	84	f+(v	f+(v	NOUN
ejpam-5634	85	85	)	)	PUNCT
ejpam-5634	85	86	is	be	AUX
ejpam-5634	85	87	τ1τ2	τ1τ2	NOUN
ejpam-5634	85	88	-	-	ADJ
ejpam-5634	85	89	open	open	ADJ
ejpam-5634	85	90	in	in	ADP
ejpam-5634	85	91	x	x	PUNCT
ejpam-5634	85	92	for	for	ADP
ejpam-5634	85	93	every	every	DET
ejpam-5634	85	94	σ1σ2	σ1σ2	NOUN
ejpam-5634	85	95	-	-	ADJ
ejpam-5634	85	96	open	open	ADJ
ejpam-5634	85	97	set	set	NOUN
ejpam-5634	85	98	v	v	NOUN
ejpam-5634	85	99	of	of	ADP
ejpam-5634	85	100	y	y	PROPN
ejpam-5634	85	101	having	have	VERB
ejpam-5634	85	102	σ1σ2	σ1σ2	ADV
ejpam-5634	85	103	-	-	PUNCT
ejpam-5634	85	104	connected	connect	VERB
ejpam-5634	85	105	complement	complement	NOUN
ejpam-5634	85	106	;	;	PUNCT
ejpam-5634	85	107	(	(	PUNCT
ejpam-5634	85	108	3	3	X
ejpam-5634	85	109	)	)	PUNCT
ejpam-5634	85	110	f−(k	f−(k	PROPN
ejpam-5634	85	111	)	)	PUNCT
ejpam-5634	85	112	is	be	AUX
ejpam-5634	85	113	τ1τ2	τ1τ2	NOUN
ejpam-5634	85	114	-	-	ADJ
ejpam-5634	85	115	closed	closed	ADJ
ejpam-5634	85	116	in	in	ADP
ejpam-5634	85	117	x	x	PUNCT
ejpam-5634	85	118	for	for	ADP
ejpam-5634	85	119	every	every	DET
ejpam-5634	85	120	σ1σ2	σ1σ2	NOUN
ejpam-5634	85	121	-	-	ADJ
ejpam-5634	85	122	connected	connect	VERB
ejpam-5634	85	123	σ1σ2	σ1σ2	VERB
ejpam-5634	85	124	-	-	PUNCT
ejpam-5634	85	125	closed	closed	ADJ
ejpam-5634	85	126	set	set	NOUN
ejpam-5634	85	127	k	k	PROPN
ejpam-5634	85	128	of	of	ADP
ejpam-5634	85	129	y	y	PROPN
ejpam-5634	85	130	;	;	PUNCT
ejpam-5634	85	131	(	(	PUNCT
ejpam-5634	85	132	4	4	X
ejpam-5634	85	133	)	)	PUNCT
ejpam-5634	85	134	τ1τ2	τ1τ2	NOUN
ejpam-5634	85	135	-	-	NOUN
ejpam-5634	85	136	cl(f	cl(f	NOUN
ejpam-5634	85	137	−(b	−(b	PROPN
ejpam-5634	85	138	)	)	PUNCT
ejpam-5634	85	139	)	)	PUNCT
ejpam-5634	86	1	⊆	⊆	X
ejpam-5634	86	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5634	86	3	-	-	PUNCT
ejpam-5634	86	4	cl(b	cl(b	NOUN
ejpam-5634	86	5	)	)	PUNCT
ejpam-5634	86	6	)	)	PUNCT
ejpam-5634	87	1	for	for	ADP
ejpam-5634	87	2	every	every	DET
ejpam-5634	87	3	subset	subset	NOUN
ejpam-5634	87	4	b	b	PROPN
ejpam-5634	87	5	of	of	ADP
ejpam-5634	87	6	y	y	PROPN
ejpam-5634	87	7	having	have	VERB
ejpam-5634	87	8	the	the	DET
ejpam-5634	87	9	σ1σ2	σ1σ2	ADV
ejpam-5634	87	10	-	-	PUNCT
ejpam-5634	87	11	connected	connect	VERB
ejpam-5634	87	12	σ1σ2	σ1σ2	NOUN
ejpam-5634	87	13	-	-	NOUN
ejpam-5634	87	14	closure	closure	NOUN
ejpam-5634	87	15	;	;	PUNCT
ejpam-5634	87	16	(	(	PUNCT
ejpam-5634	87	17	5	5	X
ejpam-5634	87	18	)	)	PUNCT
ejpam-5634	87	19	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	87	20	-	-	PUNCT
ejpam-5634	87	21	int(b	int(b	NOUN
ejpam-5634	87	22	)	)	PUNCT
ejpam-5634	87	23	)	)	PUNCT
ejpam-5634	88	1	⊆	⊆	X
ejpam-5634	88	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	88	3	-	-	NUM
ejpam-5634	88	4	int(f	int(f	VERB
ejpam-5634	88	5	+	+	ADJ
ejpam-5634	88	6	(	(	PUNCT
ejpam-5634	88	7	b	b	NOUN
ejpam-5634	88	8	)	)	PUNCT
ejpam-5634	88	9	)	)	PUNCT
ejpam-5634	88	10	for	for	ADP
ejpam-5634	88	11	every	every	DET
ejpam-5634	88	12	subset	subset	NOUN
ejpam-5634	88	13	b	b	PROPN
ejpam-5634	88	14	of	of	ADP
ejpam-5634	88	15	y	y	PRON
ejpam-5634	88	16	such	such	ADJ
ejpam-5634	88	17	that	that	SCONJ
ejpam-5634	88	18	y−σ1σ2	y−σ1σ2	PROPN
ejpam-5634	88	19	-	-	PUNCT
ejpam-5634	88	20	int(b	int(b	NOUN
ejpam-5634	88	21	)	)	PUNCT
ejpam-5634	88	22	is	be	AUX
ejpam-5634	88	23	σ1σ2	σ1σ2	NOUN
ejpam-5634	88	24	-	-	PUNCT
ejpam-5634	88	25	connected	connect	VERB
ejpam-5634	88	26	.	.	PUNCT
ejpam-5634	89	1	proof	proof	NOUN
ejpam-5634	89	2	.	.	PUNCT
ejpam-5634	90	1	(	(	PUNCT
ejpam-5634	90	2	1	1	X
ejpam-5634	90	3	)	)	PUNCT
ejpam-5634	90	4	⇒	⇒	NOUN
ejpam-5634	90	5	(	(	PUNCT
ejpam-5634	90	6	2	2	NUM
ejpam-5634	90	7	):	):	PUNCT
ejpam-5634	90	8	let	let	VERB
ejpam-5634	90	9	v	v	PART
ejpam-5634	90	10	be	be	AUX
ejpam-5634	90	11	any	any	DET
ejpam-5634	90	12	σ1σ2	σ1σ2	NOUN
ejpam-5634	90	13	-	-	ADJ
ejpam-5634	90	14	open	open	ADJ
ejpam-5634	90	15	set	set	NOUN
ejpam-5634	90	16	of	of	ADP
ejpam-5634	90	17	y	y	PROPN
ejpam-5634	90	18	having	have	VERB
ejpam-5634	90	19	σ1σ2	σ1σ2	ADV
ejpam-5634	90	20	-	-	PUNCT
ejpam-5634	90	21	connected	connect	VERB
ejpam-5634	90	22	complement	complement	NOUN
ejpam-5634	90	23	and	and	CCONJ
ejpam-5634	90	24	x	x	PUNCT
ejpam-5634	90	25	∈	∈	PROPN
ejpam-5634	90	26	f+(v	f+(v	NOUN
ejpam-5634	90	27	)	)	PUNCT
ejpam-5634	90	28	.	.	PUNCT
ejpam-5634	91	1	then	then	ADV
ejpam-5634	91	2	,	,	PUNCT
ejpam-5634	91	3	there	there	PRON
ejpam-5634	91	4	exists	exist	VERB
ejpam-5634	91	5	a	a	DET
ejpam-5634	91	6	τ1τ2	τ1τ2	NOUN
ejpam-5634	91	7	-	-	ADJ
ejpam-5634	91	8	open	open	ADJ
ejpam-5634	91	9	set	set	ADJ
ejpam-5634	91	10	u	u	NOUN
ejpam-5634	91	11	of	of	ADP
ejpam-5634	91	12	x	x	PUNCT
ejpam-5634	91	13	containing	contain	VERB
ejpam-5634	91	14	x	x	PUNCT
ejpam-5634	91	15	such	such	ADJ
ejpam-5634	91	16	that	that	SCONJ
ejpam-5634	91	17	f	f	PROPN
ejpam-5634	91	18	(	(	PUNCT
ejpam-5634	91	19	u	u	NOUN
ejpam-5634	91	20	)	)	PUNCT
ejpam-5634	91	21	⊆	⊆	NUM
ejpam-5634	91	22	v	v	NOUN
ejpam-5634	91	23	.	.	PUNCT
ejpam-5634	92	1	therefore	therefore	ADV
ejpam-5634	92	2	,	,	PUNCT
ejpam-5634	92	3	we	we	PRON
ejpam-5634	92	4	have	have	VERB
ejpam-5634	92	5	x	x	X
ejpam-5634	92	6	∈	∈	PROPN
ejpam-5634	92	7	u	u	NOUN
ejpam-5634	92	8	⊆	⊆	NUM
ejpam-5634	92	9	f+(v	f+(v	NOUN
ejpam-5634	92	10	)	)	PUNCT
ejpam-5634	92	11	and	and	CCONJ
ejpam-5634	92	12	hence	hence	ADV
ejpam-5634	92	13	x	x	X
ejpam-5634	92	14	∈	∈	PRON
ejpam-5634	92	15	τ1τ2	τ1τ2	NOUN
ejpam-5634	92	16	-	-	NUM
ejpam-5634	92	17	int(f	int(f	VERB
ejpam-5634	92	18	+	+	ADJ
ejpam-5634	92	19	(	(	PUNCT
ejpam-5634	92	20	v	v	NOUN
ejpam-5634	92	21	)	)	PUNCT
ejpam-5634	92	22	)	)	PUNCT
ejpam-5634	92	23	.	.	PUNCT
ejpam-5634	93	1	thus	thus	ADV
ejpam-5634	93	2	,	,	PUNCT
ejpam-5634	93	3	f+(v	f+(v	PROPN
ejpam-5634	93	4	)	)	PUNCT
ejpam-5634	94	1	⊆	⊆	X
ejpam-5634	94	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	94	3	-	-	NUM
ejpam-5634	94	4	int(f	int(f	VERB
ejpam-5634	94	5	+	+	ADJ
ejpam-5634	94	6	(	(	PUNCT
ejpam-5634	94	7	v	v	NOUN
ejpam-5634	94	8	)	)	PUNCT
ejpam-5634	94	9	)	)	PUNCT
ejpam-5634	94	10	and	and	CCONJ
ejpam-5634	94	11	so	so	ADV
ejpam-5634	94	12	f+(v	f+(v	PROPN
ejpam-5634	94	13	)	)	PUNCT
ejpam-5634	94	14	is	be	AUX
ejpam-5634	94	15	τ1τ2	τ1τ2	NOUN
ejpam-5634	94	16	-	-	ADJ
ejpam-5634	94	17	open	open	ADJ
ejpam-5634	94	18	in	in	ADP
ejpam-5634	94	19	x.	x.	NOUN
ejpam-5634	94	20	(	(	PUNCT
ejpam-5634	94	21	2	2	NUM
ejpam-5634	94	22	)	)	PUNCT
ejpam-5634	94	23	⇒	⇒	NOUN
ejpam-5634	94	24	(	(	PUNCT
ejpam-5634	94	25	3	3	NUM
ejpam-5634	94	26	):	):	PUNCT
ejpam-5634	94	27	the	the	DET
ejpam-5634	94	28	proof	proof	NOUN
ejpam-5634	94	29	follows	follow	VERB
ejpam-5634	94	30	immediately	immediately	ADV
ejpam-5634	94	31	from	from	ADP
ejpam-5634	94	32	the	the	DET
ejpam-5634	94	33	fact	fact	NOUN
ejpam-5634	94	34	that	that	SCONJ
ejpam-5634	94	35	f+(y	f+(y	PROPN
ejpam-5634	94	36	−b	−b	ADV
ejpam-5634	94	37	)	)	PUNCT
ejpam-5634	94	38	=	=	SYM
ejpam-5634	95	1	x−f−(b	x−f−(b	PROPN
ejpam-5634	95	2	)	)	PUNCT
ejpam-5634	95	3	for	for	ADP
ejpam-5634	95	4	every	every	DET
ejpam-5634	95	5	subset	subset	NOUN
ejpam-5634	95	6	b	b	PROPN
ejpam-5634	95	7	of	of	ADP
ejpam-5634	95	8	y	y	PROPN
ejpam-5634	95	9	.	.	PUNCT
ejpam-5634	96	1	(	(	PUNCT
ejpam-5634	96	2	3	3	X
ejpam-5634	96	3	)	)	PUNCT
ejpam-5634	96	4	⇒	⇒	NOUN
ejpam-5634	96	5	(	(	PUNCT
ejpam-5634	96	6	4	4	NUM
ejpam-5634	96	7	):	):	PUNCT
ejpam-5634	96	8	let	let	VERB
ejpam-5634	96	9	b	b	X
ejpam-5634	96	10	be	be	AUX
ejpam-5634	96	11	any	any	DET
ejpam-5634	96	12	subset	subset	NOUN
ejpam-5634	96	13	of	of	ADP
ejpam-5634	96	14	y	y	PROPN
ejpam-5634	96	15	having	have	VERB
ejpam-5634	96	16	the	the	DET
ejpam-5634	96	17	σ1σ2	σ1σ2	ADV
ejpam-5634	96	18	-	-	PUNCT
ejpam-5634	96	19	connected	connect	VERB
ejpam-5634	96	20	σ1σ2	σ1σ2	NOUN
ejpam-5634	96	21	-	-	NOUN
ejpam-5634	96	22	closure	closure	NOUN
ejpam-5634	96	23	.	.	PUNCT
ejpam-5634	97	1	thus	thus	ADV
ejpam-5634	97	2	by	by	ADP
ejpam-5634	97	3	(	(	PUNCT
ejpam-5634	97	4	3	3	NUM
ejpam-5634	97	5	)	)	PUNCT
ejpam-5634	97	6	,	,	PUNCT
ejpam-5634	97	7	τ1τ2	τ1τ2	PROPN
ejpam-5634	97	8	-	-	NOUN
ejpam-5634	97	9	cl(f	cl(f	NOUN
ejpam-5634	97	10	−(b	−(b	PROPN
ejpam-5634	97	11	)	)	PUNCT
ejpam-5634	97	12	)	)	PUNCT
ejpam-5634	98	1	⊆	⊆	X
ejpam-5634	98	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	98	3	-	-	ADJ
ejpam-5634	98	4	cl(f	cl(f	NOUN
ejpam-5634	98	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5634	98	6	-	-	NOUN
ejpam-5634	98	7	cl(b	cl(b	NOUN
ejpam-5634	98	8	)	)	PUNCT
ejpam-5634	98	9	)	)	PUNCT
ejpam-5634	98	10	)	)	PUNCT
ejpam-5634	99	1	=	=	PUNCT
ejpam-5634	99	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5634	99	3	-	-	PUNCT
ejpam-5634	99	4	cl(b	cl(b	NOUN
ejpam-5634	99	5	)	)	PUNCT
ejpam-5634	99	6	)	)	PUNCT
ejpam-5634	99	7	.	.	PUNCT
ejpam-5634	100	1	(	(	PUNCT
ejpam-5634	100	2	4	4	X
ejpam-5634	100	3	)	)	PUNCT
ejpam-5634	100	4	⇒	⇒	NOUN
ejpam-5634	100	5	(	(	PUNCT
ejpam-5634	100	6	5	5	NUM
ejpam-5634	100	7	):	):	PUNCT
ejpam-5634	100	8	let	let	VERB
ejpam-5634	100	9	b	b	X
ejpam-5634	100	10	be	be	AUX
ejpam-5634	100	11	any	any	DET
ejpam-5634	100	12	subset	subset	NOUN
ejpam-5634	100	13	of	of	ADP
ejpam-5634	100	14	y	y	PRON
ejpam-5634	100	15	such	such	ADJ
ejpam-5634	100	16	that	that	SCONJ
ejpam-5634	100	17	y	y	PROPN
ejpam-5634	100	18	−σ1σ2	−σ1σ2	PROPN
ejpam-5634	100	19	-	-	PUNCT
ejpam-5634	100	20	int(b	int(b	NOUN
ejpam-5634	100	21	)	)	PUNCT
ejpam-5634	100	22	is	be	AUX
ejpam-5634	100	23	σ1σ2	σ1σ2	NOUN
ejpam-5634	100	24	-	-	PUNCT
ejpam-5634	100	25	connected	connect	VERB
ejpam-5634	100	26	.	.	PUNCT
ejpam-5634	101	1	by	by	ADP
ejpam-5634	101	2	(	(	PUNCT
ejpam-5634	101	3	4	4	NUM
ejpam-5634	101	4	)	)	PUNCT
ejpam-5634	101	5	,	,	PUNCT
ejpam-5634	101	6	we	we	PRON
ejpam-5634	101	7	have	have	VERB
ejpam-5634	101	8	x	x	INTJ
ejpam-5634	101	9	−	−	ADP
ejpam-5634	101	10	τ1τ2	τ1τ2	NOUN
ejpam-5634	101	11	-	-	NUM
ejpam-5634	101	12	int(f	int(f	VERB
ejpam-5634	101	13	+	+	ADJ
ejpam-5634	101	14	(	(	PUNCT
ejpam-5634	101	15	b	b	NOUN
ejpam-5634	101	16	)	)	PUNCT
ejpam-5634	101	17	)	)	PUNCT
ejpam-5634	102	1	=	=	PUNCT
ejpam-5634	102	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	102	3	-	-	NOUN
ejpam-5634	102	4	cl(x	cl(x	SYM
ejpam-5634	102	5	−	−	PROPN
ejpam-5634	102	6	f+(b	f+(b	NOUN
ejpam-5634	102	7	)	)	PUNCT
ejpam-5634	102	8	)	)	PUNCT
ejpam-5634	103	1	=	=	PUNCT
ejpam-5634	104	1	τ1τ2	τ1τ2	NOUN
ejpam-5634	104	2	-	-	PROPN
ejpam-5634	104	3	cl(f	cl(f	NOUN
ejpam-5634	104	4	−(y	−(y	NOUN
ejpam-5634	104	5	−b	−b	NOUN
ejpam-5634	104	6	)	)	PUNCT
ejpam-5634	104	7	)	)	PUNCT
ejpam-5634	105	1	⊆	⊆	X
ejpam-5634	105	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5634	105	3	-	-	PUNCT
ejpam-5634	105	4	cl(y	cl(y	NOUN
ejpam-5634	105	5	−b	−b	NOUN
ejpam-5634	105	6	)	)	PUNCT
ejpam-5634	105	7	)	)	PUNCT
ejpam-5634	106	1	=	=	PUNCT
ejpam-5634	106	2	f−(y	f−(y	NOUN
ejpam-5634	106	3	−	−	ADP
ejpam-5634	106	4	σ1σ2	σ1σ2	NOUN
ejpam-5634	106	5	-	-	PUNCT
ejpam-5634	106	6	int(b	int(b	NOUN
ejpam-5634	106	7	)	)	PUNCT
ejpam-5634	106	8	)	)	PUNCT
ejpam-5634	107	1	=	=	PUNCT
ejpam-5634	107	2	x	x	X
ejpam-5634	108	1	−	−	ADP
ejpam-5634	108	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	108	3	-	-	PUNCT
ejpam-5634	108	4	int(b	int(b	NOUN
ejpam-5634	108	5	)	)	PUNCT
ejpam-5634	108	6	)	)	PUNCT
ejpam-5634	108	7	and	and	CCONJ
ejpam-5634	108	8	hence	hence	ADV
ejpam-5634	108	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-5634	108	10	-	-	PUNCT
ejpam-5634	108	11	int(b	int(b	NOUN
ejpam-5634	108	12	)	)	PUNCT
ejpam-5634	108	13	)	)	PUNCT
ejpam-5634	109	1	⊆	⊆	X
ejpam-5634	109	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	109	3	-	-	NUM
ejpam-5634	109	4	int(f	int(f	VERB
ejpam-5634	109	5	+	+	ADJ
ejpam-5634	109	6	(	(	PUNCT
ejpam-5634	109	7	b	b	NOUN
ejpam-5634	109	8	)	)	PUNCT
ejpam-5634	109	9	)	)	PUNCT
ejpam-5634	109	10	.	.	PUNCT
ejpam-5634	110	1	(	(	PUNCT
ejpam-5634	110	2	5	5	X
ejpam-5634	110	3	)	)	PUNCT
ejpam-5634	110	4	⇒	⇒	NOUN
ejpam-5634	110	5	(	(	PUNCT
ejpam-5634	110	6	1	1	NUM
ejpam-5634	110	7	):	):	PUNCT
ejpam-5634	110	8	let	let	VERB
ejpam-5634	110	9	x	x	PUNCT
ejpam-5634	110	10	∈	∈	PROPN
ejpam-5634	110	11	x	x	X
ejpam-5634	110	12	and	and	CCONJ
ejpam-5634	110	13	v	v	X
ejpam-5634	110	14	be	be	AUX
ejpam-5634	110	15	any	any	DET
ejpam-5634	110	16	σ1σ2	σ1σ2	NOUN
ejpam-5634	110	17	-	-	ADJ
ejpam-5634	110	18	open	open	ADJ
ejpam-5634	110	19	set	set	NOUN
ejpam-5634	110	20	of	of	ADP
ejpam-5634	110	21	y	y	PROPN
ejpam-5634	110	22	containing	contain	VERB
ejpam-5634	110	23	f	f	PROPN
ejpam-5634	110	24	(	(	PUNCT
ejpam-5634	110	25	x	x	NOUN
ejpam-5634	110	26	)	)	PUNCT
ejpam-5634	110	27	and	and	CCONJ
ejpam-5634	110	28	having	have	VERB
ejpam-5634	110	29	σ1σ2	σ1σ2	NOUN
ejpam-5634	110	30	-	-	PUNCT
ejpam-5634	110	31	connected	connect	VERB
ejpam-5634	110	32	complement	complement	NOUN
ejpam-5634	110	33	.	.	PUNCT
ejpam-5634	111	1	by	by	ADP
ejpam-5634	111	2	(	(	PUNCT
ejpam-5634	111	3	5	5	NUM
ejpam-5634	111	4	)	)	PUNCT
ejpam-5634	111	5	,	,	PUNCT
ejpam-5634	111	6	f+(v	f+(v	PROPN
ejpam-5634	111	7	)	)	PUNCT
ejpam-5634	112	1	=	=	SYM
ejpam-5634	112	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	112	3	-	-	PUNCT
ejpam-5634	112	4	int(v	int(v	NOUN
ejpam-5634	112	5	)	)	PUNCT
ejpam-5634	112	6	)	)	PUNCT
ejpam-5634	113	1	⊆	⊆	X
ejpam-5634	113	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	113	3	-	-	NUM
ejpam-5634	113	4	int(f	int(f	VERB
ejpam-5634	113	5	+	+	ADJ
ejpam-5634	113	6	(	(	PUNCT
ejpam-5634	113	7	v	v	NOUN
ejpam-5634	113	8	)	)	PUNCT
ejpam-5634	113	9	)	)	PUNCT
ejpam-5634	113	10	.	.	PUNCT
ejpam-5634	114	1	then	then	ADV
ejpam-5634	114	2	,	,	PUNCT
ejpam-5634	114	3	there	there	PRON
ejpam-5634	114	4	exists	exist	VERB
ejpam-5634	114	5	a	a	DET
ejpam-5634	114	6	τ1τ2	τ1τ2	NOUN
ejpam-5634	114	7	-	-	ADJ
ejpam-5634	114	8	open	open	ADJ
ejpam-5634	114	9	set	set	ADJ
ejpam-5634	114	10	u	u	NOUN
ejpam-5634	114	11	of	of	ADP
ejpam-5634	114	12	x	x	PUNCT
ejpam-5634	114	13	containing	contain	VERB
ejpam-5634	114	14	x	x	PUNCT
ejpam-5634	114	15	such	such	ADJ
ejpam-5634	114	16	that	that	SCONJ
ejpam-5634	114	17	u	u	NOUN
ejpam-5634	114	18	⊆	⊆	NUM
ejpam-5634	114	19	f+(v	f+(v	NOUN
ejpam-5634	114	20	)	)	PUNCT
ejpam-5634	114	21	.	.	PUNCT
ejpam-5634	115	1	thus	thus	ADV
ejpam-5634	115	2	,	,	PUNCT
ejpam-5634	115	3	f	f	PROPN
ejpam-5634	115	4	(	(	PUNCT
ejpam-5634	115	5	u	u	NOUN
ejpam-5634	115	6	)	)	PUNCT
ejpam-5634	115	7	⊆	⊆	NUM
ejpam-5634	115	8	v	v	NOUN
ejpam-5634	115	9	.	.	PUNCT
ejpam-5634	116	1	this	this	PRON
ejpam-5634	116	2	shows	show	VERB
ejpam-5634	116	3	that	that	SCONJ
ejpam-5634	116	4	f	f	PROPN
ejpam-5634	116	5	is	be	AUX
ejpam-5634	116	6	upper	upper	ADJ
ejpam-5634	116	7	s-(τ1	s-(τ1	PROPN
ejpam-5634	116	8	,	,	PUNCT
ejpam-5634	116	9	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5634	116	10	.	.	PUNCT
ejpam-5634	116	11	m.	m.	NOUN
ejpam-5634	116	12	chiangpradit	chiangpradit	PROPN
ejpam-5634	116	13	,	,	PUNCT
ejpam-5634	116	14	a.	a.	PROPN
ejpam-5634	116	15	sama	sama	PROPN
ejpam-5634	116	16	-	-	PUNCT
ejpam-5634	116	17	ae	ae	PROPN
ejpam-5634	116	18	,	,	PUNCT
ejpam-5634	116	19	c.	c.	PROPN
ejpam-5634	116	20	boonpok	boonpok	PROPN
ejpam-5634	116	21	/	/	SYM
ejpam-5634	116	22	eur	eur	PROPN
ejpam-5634	116	23	.	.	PUNCT
ejpam-5634	117	1	j.	j.	PROPN
ejpam-5634	117	2	pure	pure	PROPN
ejpam-5634	117	3	appl	appl	PROPN
ejpam-5634	117	4	.	.	PROPN
ejpam-5634	117	5	math	math	PROPN
ejpam-5634	117	6	,	,	PUNCT
ejpam-5634	117	7	18	18	NUM
ejpam-5634	117	8	(	(	PUNCT
ejpam-5634	117	9	1	1	NUM
ejpam-5634	117	10	)	)	PUNCT
ejpam-5634	117	11	(	(	PUNCT
ejpam-5634	117	12	2025	2025	NUM
ejpam-5634	117	13	)	)	PUNCT
ejpam-5634	117	14	,	,	PUNCT
ejpam-5634	117	15	5634	5634	NUM
ejpam-5634	117	16	5	5	NUM
ejpam-5634	117	17	of	of	ADP
ejpam-5634	117	18	12	12	NUM
ejpam-5634	117	19	definition	definition	NOUN
ejpam-5634	117	20	2	2	NUM
ejpam-5634	117	21	.	.	PUNCT
ejpam-5634	117	22	a	a	DET
ejpam-5634	117	23	multifunction	multifunction	NOUN
ejpam-5634	117	24	f	f	NOUN
ejpam-5634	117	25	:	:	PUNCT
ejpam-5634	117	26	(	(	PUNCT
ejpam-5634	117	27	x	x	NOUN
ejpam-5634	117	28	,	,	PUNCT
ejpam-5634	117	29	τ1	τ1	NOUN
ejpam-5634	117	30	,	,	PUNCT
ejpam-5634	117	31	τ2	τ2	NOUN
ejpam-5634	117	32	)	)	PUNCT
ejpam-5634	117	33	→	→	SYM
ejpam-5634	117	34	(	(	PUNCT
ejpam-5634	117	35	y	y	PROPN
ejpam-5634	117	36	,	,	PUNCT
ejpam-5634	117	37	σ1	σ1	PROPN
ejpam-5634	117	38	,	,	PUNCT
ejpam-5634	117	39	σ2	σ2	PROPN
ejpam-5634	117	40	)	)	PUNCT
ejpam-5634	117	41	is	be	AUX
ejpam-5634	117	42	said	say	VERB
ejpam-5634	117	43	to	to	PART
ejpam-5634	117	44	be	be	AUX
ejpam-5634	117	45	lower	low	ADJ
ejpam-5634	117	46	s-(τ1	s-(τ1	NOUN
ejpam-5634	117	47	,	,	PUNCT
ejpam-5634	117	48	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	117	49	at	at	ADP
ejpam-5634	117	50	x	x	X
ejpam-5634	117	51	∈	∈	PROPN
ejpam-5634	117	52	x	x	PUNCT
ejpam-5634	117	53	if	if	SCONJ
ejpam-5634	117	54	for	for	ADP
ejpam-5634	117	55	each	each	DET
ejpam-5634	117	56	σ1σ2	σ1σ2	VERB
ejpam-5634	117	57	-	-	ADJ
ejpam-5634	117	58	open	open	ADJ
ejpam-5634	117	59	set	set	NOUN
ejpam-5634	117	60	v	v	NOUN
ejpam-5634	117	61	of	of	ADP
ejpam-5634	117	62	y	y	PRON
ejpam-5634	117	63	such	such	ADJ
ejpam-5634	117	64	that	that	SCONJ
ejpam-5634	117	65	f	f	PROPN
ejpam-5634	117	66	(	(	PUNCT
ejpam-5634	117	67	x)∩	x)∩	PROPN
ejpam-5634	117	68	v	v	ADP
ejpam-5634	117	69	̸=	̸=	PROPN
ejpam-5634	117	70	∅	∅	NOUN
ejpam-5634	117	71	and	and	CCONJ
ejpam-5634	117	72	having	have	VERB
ejpam-5634	117	73	σ1σ2	σ1σ2	NOUN
ejpam-5634	117	74	-	-	PUNCT
ejpam-5634	117	75	connected	connect	VERB
ejpam-5634	117	76	complement	complement	NOUN
ejpam-5634	117	77	,	,	PUNCT
ejpam-5634	117	78	there	there	PRON
ejpam-5634	117	79	exists	exist	VERB
ejpam-5634	117	80	a	a	DET
ejpam-5634	117	81	τ1τ2	τ1τ2	NOUN
ejpam-5634	117	82	-	-	ADJ
ejpam-5634	117	83	open	open	ADJ
ejpam-5634	117	84	set	set	ADJ
ejpam-5634	117	85	u	u	NOUN
ejpam-5634	117	86	of	of	ADP
ejpam-5634	117	87	x	x	PUNCT
ejpam-5634	117	88	containing	contain	VERB
ejpam-5634	117	89	x	x	PUNCT
ejpam-5634	117	90	such	such	ADJ
ejpam-5634	117	91	that	that	SCONJ
ejpam-5634	117	92	f	f	PROPN
ejpam-5634	117	93	(	(	PUNCT
ejpam-5634	117	94	z	z	NOUN
ejpam-5634	117	95	)	)	PUNCT
ejpam-5634	117	96	∩	∩	NOUN
ejpam-5634	117	97	v	v	ADP
ejpam-5634	117	98	̸=	̸=	PROPN
ejpam-5634	117	99	∅	∅	NOUN
ejpam-5634	117	100	for	for	ADP
ejpam-5634	117	101	each	each	DET
ejpam-5634	117	102	z	z	NOUN
ejpam-5634	117	103	∈	∈	PROPN
ejpam-5634	117	104	u	u	NOUN
ejpam-5634	117	105	.	.	PUNCT
ejpam-5634	118	1	a	a	DET
ejpam-5634	118	2	multifunction	multifunction	NOUN
ejpam-5634	118	3	f	f	NOUN
ejpam-5634	118	4	:	:	PUNCT
ejpam-5634	118	5	(	(	PUNCT
ejpam-5634	118	6	x	x	NOUN
ejpam-5634	118	7	,	,	PUNCT
ejpam-5634	118	8	τ1	τ1	NOUN
ejpam-5634	118	9	,	,	PUNCT
ejpam-5634	118	10	τ2	τ2	NOUN
ejpam-5634	118	11	)	)	PUNCT
ejpam-5634	118	12	→	→	SYM
ejpam-5634	118	13	(	(	PUNCT
ejpam-5634	118	14	y	y	PROPN
ejpam-5634	118	15	,	,	PUNCT
ejpam-5634	118	16	σ1	σ1	PROPN
ejpam-5634	118	17	,	,	PUNCT
ejpam-5634	118	18	σ2	σ2	PROPN
ejpam-5634	118	19	)	)	PUNCT
ejpam-5634	118	20	is	be	AUX
ejpam-5634	118	21	said	say	VERB
ejpam-5634	118	22	to	to	PART
ejpam-5634	118	23	be	be	AUX
ejpam-5634	118	24	lower	low	ADJ
ejpam-5634	118	25	s-(τ1	s-(τ1	NOUN
ejpam-5634	118	26	,	,	PUNCT
ejpam-5634	118	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	118	28	if	if	SCONJ
ejpam-5634	118	29	f	f	PROPN
ejpam-5634	118	30	is	be	AUX
ejpam-5634	118	31	lower	low	ADJ
ejpam-5634	118	32	s-(τ1	s-(τ1	NOUN
ejpam-5634	118	33	,	,	PUNCT
ejpam-5634	118	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	118	35	at	at	ADP
ejpam-5634	118	36	each	each	DET
ejpam-5634	118	37	point	point	NOUN
ejpam-5634	118	38	x	x	PUNCT
ejpam-5634	118	39	of	of	ADP
ejpam-5634	118	40	x.	x.	PROPN
ejpam-5634	118	41	theorem	theorem	VERB
ejpam-5634	118	42	2	2	NUM
ejpam-5634	118	43	.	.	X
ejpam-5634	118	44	for	for	ADP
ejpam-5634	118	45	a	a	DET
ejpam-5634	118	46	multifunction	multifunction	NOUN
ejpam-5634	118	47	f	f	NOUN
ejpam-5634	118	48	:	:	PUNCT
ejpam-5634	118	49	(	(	PUNCT
ejpam-5634	118	50	x	x	NOUN
ejpam-5634	118	51	,	,	PUNCT
ejpam-5634	118	52	τ1	τ1	NOUN
ejpam-5634	118	53	,	,	PUNCT
ejpam-5634	118	54	τ2	τ2	NOUN
ejpam-5634	118	55	)	)	PUNCT
ejpam-5634	118	56	→	→	SYM
ejpam-5634	118	57	(	(	PUNCT
ejpam-5634	118	58	y	y	PROPN
ejpam-5634	118	59	,	,	PUNCT
ejpam-5634	118	60	σ1	σ1	PROPN
ejpam-5634	118	61	,	,	PUNCT
ejpam-5634	118	62	σ2	σ2	NOUN
ejpam-5634	118	63	)	)	PUNCT
ejpam-5634	118	64	,	,	PUNCT
ejpam-5634	118	65	the	the	DET
ejpam-5634	118	66	following	follow	VERB
ejpam-5634	118	67	properties	property	NOUN
ejpam-5634	118	68	are	be	AUX
ejpam-5634	118	69	equivalent	equivalent	ADJ
ejpam-5634	118	70	:	:	PUNCT
ejpam-5634	118	71	(	(	PUNCT
ejpam-5634	118	72	1	1	X
ejpam-5634	118	73	)	)	PUNCT
ejpam-5634	118	74	f	f	PROPN
ejpam-5634	118	75	is	be	AUX
ejpam-5634	118	76	lower	low	ADJ
ejpam-5634	118	77	s-(τ1	s-(τ1	NOUN
ejpam-5634	118	78	,	,	PUNCT
ejpam-5634	118	79	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	118	80	;	;	PUNCT
ejpam-5634	118	81	(	(	PUNCT
ejpam-5634	118	82	2	2	X
ejpam-5634	118	83	)	)	PUNCT
ejpam-5634	118	84	f−(v	f−(v	NOUN
ejpam-5634	118	85	)	)	PUNCT
ejpam-5634	118	86	is	be	AUX
ejpam-5634	118	87	τ1τ2	τ1τ2	NOUN
ejpam-5634	118	88	-	-	ADJ
ejpam-5634	118	89	open	open	ADJ
ejpam-5634	118	90	in	in	ADP
ejpam-5634	118	91	x	x	PUNCT
ejpam-5634	118	92	for	for	ADP
ejpam-5634	118	93	every	every	DET
ejpam-5634	118	94	σ1σ2	σ1σ2	NOUN
ejpam-5634	118	95	-	-	ADJ
ejpam-5634	118	96	open	open	ADJ
ejpam-5634	118	97	set	set	NOUN
ejpam-5634	118	98	v	v	NOUN
ejpam-5634	118	99	of	of	ADP
ejpam-5634	118	100	y	y	PROPN
ejpam-5634	118	101	having	have	VERB
ejpam-5634	118	102	σ1σ2	σ1σ2	ADV
ejpam-5634	118	103	-	-	PUNCT
ejpam-5634	118	104	connected	connect	VERB
ejpam-5634	118	105	complement	complement	NOUN
ejpam-5634	118	106	;	;	PUNCT
ejpam-5634	118	107	(	(	PUNCT
ejpam-5634	118	108	3	3	X
ejpam-5634	118	109	)	)	PUNCT
ejpam-5634	118	110	f+(k	f+(k	NOUN
ejpam-5634	118	111	)	)	PUNCT
ejpam-5634	118	112	is	be	AUX
ejpam-5634	118	113	τ1τ2	τ1τ2	NOUN
ejpam-5634	118	114	-	-	ADJ
ejpam-5634	118	115	closed	closed	ADJ
ejpam-5634	118	116	in	in	ADP
ejpam-5634	118	117	x	x	PUNCT
ejpam-5634	118	118	for	for	ADP
ejpam-5634	118	119	every	every	DET
ejpam-5634	118	120	σ1σ2	σ1σ2	NOUN
ejpam-5634	118	121	-	-	ADJ
ejpam-5634	118	122	connected	connect	VERB
ejpam-5634	118	123	σ1σ2	σ1σ2	VERB
ejpam-5634	118	124	-	-	PUNCT
ejpam-5634	118	125	closed	closed	ADJ
ejpam-5634	118	126	set	set	NOUN
ejpam-5634	118	127	k	k	PROPN
ejpam-5634	118	128	of	of	ADP
ejpam-5634	118	129	y	y	PROPN
ejpam-5634	118	130	;	;	PUNCT
ejpam-5634	118	131	(	(	PUNCT
ejpam-5634	118	132	4	4	X
ejpam-5634	118	133	)	)	PUNCT
ejpam-5634	118	134	τ1τ2	τ1τ2	NOUN
ejpam-5634	118	135	-	-	NOUN
ejpam-5634	118	136	cl(f	cl(f	NOUN
ejpam-5634	118	137	+	+	NOUN
ejpam-5634	118	138	(	(	PUNCT
ejpam-5634	118	139	b	b	NOUN
ejpam-5634	118	140	)	)	PUNCT
ejpam-5634	118	141	)	)	PUNCT
ejpam-5634	119	1	⊆	⊆	NUM
ejpam-5634	119	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	119	3	-	-	PUNCT
ejpam-5634	119	4	cl(b	cl(b	NOUN
ejpam-5634	119	5	)	)	PUNCT
ejpam-5634	119	6	)	)	PUNCT
ejpam-5634	119	7	for	for	ADP
ejpam-5634	119	8	every	every	DET
ejpam-5634	119	9	subset	subset	NOUN
ejpam-5634	119	10	b	b	PROPN
ejpam-5634	119	11	of	of	ADP
ejpam-5634	119	12	y	y	PROPN
ejpam-5634	119	13	having	have	VERB
ejpam-5634	119	14	the	the	DET
ejpam-5634	119	15	σ1σ2	σ1σ2	ADV
ejpam-5634	119	16	-	-	PUNCT
ejpam-5634	119	17	connected	connect	VERB
ejpam-5634	119	18	σ1σ2	σ1σ2	NOUN
ejpam-5634	119	19	-	-	NOUN
ejpam-5634	119	20	closure	closure	NOUN
ejpam-5634	119	21	;	;	PUNCT
ejpam-5634	119	22	(	(	PUNCT
ejpam-5634	119	23	5	5	X
ejpam-5634	119	24	)	)	PUNCT
ejpam-5634	119	25	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5634	119	26	-	-	PUNCT
ejpam-5634	119	27	int(b	int(b	NOUN
ejpam-5634	119	28	)	)	PUNCT
ejpam-5634	119	29	)	)	PUNCT
ejpam-5634	120	1	⊆	⊆	X
ejpam-5634	120	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	120	3	-	-	NUM
ejpam-5634	120	4	int(f	int(f	VERB
ejpam-5634	120	5	−(b	−(b	NOUN
ejpam-5634	120	6	)	)	PUNCT
ejpam-5634	120	7	)	)	PUNCT
ejpam-5634	120	8	for	for	ADP
ejpam-5634	120	9	every	every	DET
ejpam-5634	120	10	subset	subset	NOUN
ejpam-5634	120	11	b	b	PROPN
ejpam-5634	120	12	of	of	ADP
ejpam-5634	120	13	y	y	PRON
ejpam-5634	120	14	such	such	ADJ
ejpam-5634	120	15	that	that	SCONJ
ejpam-5634	120	16	y−σ1σ2	y−σ1σ2	PROPN
ejpam-5634	120	17	-	-	PUNCT
ejpam-5634	120	18	int(b	int(b	NOUN
ejpam-5634	120	19	)	)	PUNCT
ejpam-5634	120	20	is	be	AUX
ejpam-5634	120	21	σ1σ2	σ1σ2	NOUN
ejpam-5634	120	22	-	-	PUNCT
ejpam-5634	120	23	connected	connected	ADJ
ejpam-5634	120	24	.	.	PUNCT
ejpam-5634	121	1	proof	proof	NOUN
ejpam-5634	121	2	.	.	PUNCT
ejpam-5634	122	1	the	the	DET
ejpam-5634	122	2	proof	proof	NOUN
ejpam-5634	122	3	is	be	AUX
ejpam-5634	122	4	similar	similar	ADJ
ejpam-5634	122	5	to	to	ADP
ejpam-5634	122	6	that	that	PRON
ejpam-5634	122	7	of	of	ADP
ejpam-5634	122	8	theorem	theorem	ADJ
ejpam-5634	122	9	1	1	NUM
ejpam-5634	122	10	.	.	PUNCT
ejpam-5634	122	11	corollary	corollary	ADJ
ejpam-5634	122	12	1	1	NUM
ejpam-5634	122	13	.	.	PUNCT
ejpam-5634	123	1	a	a	DET
ejpam-5634	123	2	multifunction	multifunction	NOUN
ejpam-5634	123	3	f	f	NOUN
ejpam-5634	123	4	:	:	PUNCT
ejpam-5634	123	5	(	(	PUNCT
ejpam-5634	123	6	x	x	NOUN
ejpam-5634	123	7	,	,	PUNCT
ejpam-5634	123	8	τ1	τ1	NOUN
ejpam-5634	123	9	,	,	PUNCT
ejpam-5634	123	10	τ2	τ2	NOUN
ejpam-5634	123	11	)	)	PUNCT
ejpam-5634	123	12	→	→	SYM
ejpam-5634	123	13	(	(	PUNCT
ejpam-5634	123	14	y	y	PROPN
ejpam-5634	123	15	,	,	PUNCT
ejpam-5634	123	16	σ1	σ1	PROPN
ejpam-5634	123	17	,	,	PUNCT
ejpam-5634	123	18	σ2	σ2	PROPN
ejpam-5634	123	19	)	)	PUNCT
ejpam-5634	123	20	is	be	AUX
ejpam-5634	123	21	upper	upper	ADJ
ejpam-5634	123	22	s-(τ1	s-(τ1	PROPN
ejpam-5634	123	23	,	,	PUNCT
ejpam-5634	123	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	123	25	if	if	SCONJ
ejpam-5634	123	26	f−(b	f−(b	NOUN
ejpam-5634	123	27	)	)	PUNCT
ejpam-5634	123	28	is	be	AUX
ejpam-5634	123	29	τ1τ2	τ1τ2	NOUN
ejpam-5634	123	30	-	-	ADJ
ejpam-5634	123	31	closed	closed	ADJ
ejpam-5634	123	32	in	in	ADP
ejpam-5634	123	33	x	x	PUNCT
ejpam-5634	123	34	for	for	ADP
ejpam-5634	123	35	every	every	DET
ejpam-5634	123	36	σ1σ2	σ1σ2	NOUN
ejpam-5634	123	37	-	-	ADJ
ejpam-5634	123	38	connected	connect	VERB
ejpam-5634	123	39	set	set	NOUN
ejpam-5634	123	40	b	b	PROPN
ejpam-5634	123	41	of	of	ADP
ejpam-5634	123	42	y	y	PROPN
ejpam-5634	123	43	.	.	PUNCT
ejpam-5634	124	1	proof	proof	NOUN
ejpam-5634	124	2	.	.	PUNCT
ejpam-5634	125	1	let	let	VERB
ejpam-5634	125	2	v	v	PART
ejpam-5634	125	3	be	be	AUX
ejpam-5634	125	4	any	any	DET
ejpam-5634	125	5	σ1σ2	σ1σ2	NOUN
ejpam-5634	125	6	-	-	ADJ
ejpam-5634	125	7	open	open	ADJ
ejpam-5634	125	8	set	set	NOUN
ejpam-5634	125	9	of	of	ADP
ejpam-5634	125	10	y	y	PROPN
ejpam-5634	125	11	having	have	VERB
ejpam-5634	125	12	σ1σ2	σ1σ2	ADV
ejpam-5634	125	13	-	-	PUNCT
ejpam-5634	125	14	connected	connect	VERB
ejpam-5634	125	15	complement	complement	NOUN
ejpam-5634	125	16	.	.	PUNCT
ejpam-5634	126	1	then	then	ADV
ejpam-5634	126	2	,	,	PUNCT
ejpam-5634	126	3	y	y	PROPN
ejpam-5634	126	4	−v	−v	NOUN
ejpam-5634	126	5	is	be	AUX
ejpam-5634	126	6	σ1σ2	σ1σ2	NOUN
ejpam-5634	126	7	-	-	PUNCT
ejpam-5634	126	8	connected	connected	ADJ
ejpam-5634	126	9	and	and	CCONJ
ejpam-5634	126	10	σ1σ2	σ1σ2	NOUN
ejpam-5634	126	11	-	-	PUNCT
ejpam-5634	126	12	closed	closed	ADJ
ejpam-5634	126	13	.	.	PUNCT
ejpam-5634	127	1	by	by	ADP
ejpam-5634	127	2	the	the	DET
ejpam-5634	127	3	hypothesis	hypothesis	NOUN
ejpam-5634	127	4	,	,	PUNCT
ejpam-5634	127	5	f−(y	f−(y	NOUN
ejpam-5634	127	6	−v	−v	NOUN
ejpam-5634	127	7	)	)	PUNCT
ejpam-5634	127	8	is	be	AUX
ejpam-5634	127	9	τ1τ2	τ1τ2	NOUN
ejpam-5634	127	10	-	-	ADJ
ejpam-5634	127	11	closed	closed	ADJ
ejpam-5634	127	12	in	in	ADP
ejpam-5634	127	13	x.	x.	NOUN
ejpam-5634	127	14	thus	thus	ADV
ejpam-5634	127	15	,	,	PUNCT
ejpam-5634	127	16	f+(v	f+(v	PROPN
ejpam-5634	127	17	)	)	PUNCT
ejpam-5634	128	1	is	be	AUX
ejpam-5634	128	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	128	3	-	-	ADJ
ejpam-5634	128	4	open	open	ADJ
ejpam-5634	128	5	in	in	ADP
ejpam-5634	128	6	x	x	X
ejpam-5634	128	7	and	and	CCONJ
ejpam-5634	128	8	by	by	ADP
ejpam-5634	128	9	theorem	theorem	NOUN
ejpam-5634	128	10	1	1	NUM
ejpam-5634	128	11	,	,	PUNCT
ejpam-5634	128	12	f	f	PROPN
ejpam-5634	128	13	is	be	AUX
ejpam-5634	128	14	upper	upper	ADJ
ejpam-5634	128	15	s-(τ1	s-(τ1	PROPN
ejpam-5634	128	16	,	,	PUNCT
ejpam-5634	128	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	128	18	.	.	PUNCT
ejpam-5634	129	1	corollary	corollary	ADJ
ejpam-5634	129	2	2	2	NUM
ejpam-5634	129	3	.	.	PUNCT
ejpam-5634	129	4	a	a	DET
ejpam-5634	129	5	multifunction	multifunction	NOUN
ejpam-5634	130	1	f	f	NOUN
ejpam-5634	130	2	:	:	PUNCT
ejpam-5634	130	3	(	(	PUNCT
ejpam-5634	130	4	x	x	NOUN
ejpam-5634	130	5	,	,	PUNCT
ejpam-5634	130	6	τ1	τ1	NOUN
ejpam-5634	130	7	,	,	PUNCT
ejpam-5634	130	8	τ2	τ2	NOUN
ejpam-5634	130	9	)	)	PUNCT
ejpam-5634	130	10	→	→	SYM
ejpam-5634	130	11	(	(	PUNCT
ejpam-5634	130	12	y	y	PROPN
ejpam-5634	130	13	,	,	PUNCT
ejpam-5634	130	14	σ1	σ1	PROPN
ejpam-5634	130	15	,	,	PUNCT
ejpam-5634	130	16	σ2	σ2	NOUN
ejpam-5634	130	17	)	)	PUNCT
ejpam-5634	130	18	is	be	AUX
ejpam-5634	130	19	lower	low	ADJ
ejpam-5634	130	20	s-(τ1	s-(τ1	NOUN
ejpam-5634	130	21	,	,	PUNCT
ejpam-5634	130	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	130	23	if	if	SCONJ
ejpam-5634	130	24	f+(b	f+(b	NOUN
ejpam-5634	130	25	)	)	PUNCT
ejpam-5634	130	26	is	be	AUX
ejpam-5634	130	27	τ1τ2	τ1τ2	NOUN
ejpam-5634	130	28	-	-	ADJ
ejpam-5634	130	29	closed	closed	ADJ
ejpam-5634	130	30	in	in	ADP
ejpam-5634	130	31	x	x	PUNCT
ejpam-5634	130	32	for	for	ADP
ejpam-5634	130	33	every	every	DET
ejpam-5634	130	34	σ1σ2	σ1σ2	NOUN
ejpam-5634	130	35	-	-	ADJ
ejpam-5634	130	36	connected	connect	VERB
ejpam-5634	130	37	set	set	NOUN
ejpam-5634	130	38	b	b	PROPN
ejpam-5634	130	39	of	of	ADP
ejpam-5634	130	40	y	y	PROPN
ejpam-5634	130	41	.	.	PUNCT
ejpam-5634	131	1	proof	proof	NOUN
ejpam-5634	131	2	.	.	PUNCT
ejpam-5634	132	1	the	the	DET
ejpam-5634	132	2	proof	proof	NOUN
ejpam-5634	132	3	is	be	AUX
ejpam-5634	132	4	similar	similar	ADJ
ejpam-5634	132	5	to	to	ADP
ejpam-5634	132	6	that	that	PRON
ejpam-5634	132	7	of	of	ADP
ejpam-5634	132	8	corollary	corollary	ADJ
ejpam-5634	132	9	1	1	NUM
ejpam-5634	132	10	.	.	PUNCT
ejpam-5634	132	11	definition	definition	NOUN
ejpam-5634	132	12	3	3	NUM
ejpam-5634	132	13	.	.	PUNCT
ejpam-5634	133	1	a	a	DET
ejpam-5634	133	2	function	function	NOUN
ejpam-5634	133	3	f	f	NOUN
ejpam-5634	133	4	:	:	PUNCT
ejpam-5634	133	5	(	(	PUNCT
ejpam-5634	133	6	x	x	NOUN
ejpam-5634	133	7	,	,	PUNCT
ejpam-5634	133	8	τ1	τ1	NOUN
ejpam-5634	133	9	,	,	PUNCT
ejpam-5634	133	10	τ2	τ2	NOUN
ejpam-5634	133	11	)	)	PUNCT
ejpam-5634	133	12	→	→	SYM
ejpam-5634	133	13	(	(	PUNCT
ejpam-5634	133	14	y	y	PROPN
ejpam-5634	133	15	,	,	PUNCT
ejpam-5634	133	16	σ1	σ1	PROPN
ejpam-5634	133	17	,	,	PUNCT
ejpam-5634	133	18	σ2	σ2	PROPN
ejpam-5634	133	19	)	)	PUNCT
ejpam-5634	133	20	is	be	AUX
ejpam-5634	133	21	said	say	VERB
ejpam-5634	133	22	to	to	PART
ejpam-5634	133	23	be	be	AUX
ejpam-5634	133	24	s-(τ1	s-(τ1	PROPN
ejpam-5634	133	25	,	,	PUNCT
ejpam-5634	133	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	133	27	if	if	SCONJ
ejpam-5634	133	28	for	for	ADP
ejpam-5634	133	29	x	x	SYM
ejpam-5634	133	30	∈	∈	PROPN
ejpam-5634	133	31	x	x	X
ejpam-5634	133	32	and	and	CCONJ
ejpam-5634	133	33	each	each	DET
ejpam-5634	133	34	σ1σ2	σ1σ2	VERB
ejpam-5634	133	35	-	-	ADJ
ejpam-5634	133	36	open	open	ADJ
ejpam-5634	133	37	set	set	NOUN
ejpam-5634	133	38	v	v	NOUN
ejpam-5634	133	39	of	of	ADP
ejpam-5634	133	40	y	y	NOUN
ejpam-5634	133	41	containing	contain	VERB
ejpam-5634	133	42	f(x	f(x	PROPN
ejpam-5634	133	43	)	)	PUNCT
ejpam-5634	133	44	and	and	CCONJ
ejpam-5634	133	45	having	have	VERB
ejpam-5634	133	46	σ1σ2	σ1σ2	NOUN
ejpam-5634	133	47	-	-	PUNCT
ejpam-5634	133	48	connected	connect	VERB
ejpam-5634	133	49	complement	complement	NOUN
ejpam-5634	133	50	,	,	PUNCT
ejpam-5634	133	51	there	there	PRON
ejpam-5634	133	52	exists	exist	VERB
ejpam-5634	133	53	a	a	DET
ejpam-5634	133	54	τ1τ2	τ1τ2	NOUN
ejpam-5634	133	55	-	-	ADJ
ejpam-5634	133	56	open	open	ADJ
ejpam-5634	133	57	set	set	ADJ
ejpam-5634	133	58	u	u	NOUN
ejpam-5634	133	59	of	of	ADP
ejpam-5634	133	60	x	x	PUNCT
ejpam-5634	133	61	containing	contain	VERB
ejpam-5634	133	62	x	x	PUNCT
ejpam-5634	133	63	such	such	ADJ
ejpam-5634	133	64	that	that	DET
ejpam-5634	133	65	f(u	f(u	PROPN
ejpam-5634	133	66	)	)	PUNCT
ejpam-5634	133	67	⊆	⊆	NUM
ejpam-5634	133	68	v	v	NOUN
ejpam-5634	133	69	.	.	PUNCT
ejpam-5634	134	1	corollary	corollary	ADJ
ejpam-5634	134	2	3	3	NUM
ejpam-5634	134	3	.	.	PUNCT
ejpam-5634	135	1	for	for	ADP
ejpam-5634	135	2	a	a	DET
ejpam-5634	135	3	function	function	NOUN
ejpam-5634	135	4	f	f	NOUN
ejpam-5634	135	5	:	:	PUNCT
ejpam-5634	135	6	(	(	PUNCT
ejpam-5634	135	7	x	x	NOUN
ejpam-5634	135	8	,	,	PUNCT
ejpam-5634	135	9	τ1	τ1	NOUN
ejpam-5634	135	10	,	,	PUNCT
ejpam-5634	135	11	τ2	τ2	NOUN
ejpam-5634	135	12	)	)	PUNCT
ejpam-5634	135	13	→	→	SYM
ejpam-5634	135	14	(	(	PUNCT
ejpam-5634	135	15	y	y	PROPN
ejpam-5634	135	16	,	,	PUNCT
ejpam-5634	135	17	σ1	σ1	PROPN
ejpam-5634	135	18	,	,	PUNCT
ejpam-5634	135	19	σ2	σ2	NOUN
ejpam-5634	135	20	)	)	PUNCT
ejpam-5634	135	21	,	,	PUNCT
ejpam-5634	135	22	the	the	DET
ejpam-5634	135	23	following	follow	VERB
ejpam-5634	135	24	properties	property	NOUN
ejpam-5634	135	25	are	be	AUX
ejpam-5634	135	26	equivalent	equivalent	ADJ
ejpam-5634	135	27	:	:	PUNCT
ejpam-5634	135	28	(	(	PUNCT
ejpam-5634	135	29	1	1	X
ejpam-5634	135	30	)	)	PUNCT
ejpam-5634	135	31	f	f	PROPN
ejpam-5634	135	32	is	be	AUX
ejpam-5634	135	33	s-(τ1	s-(τ1	PROPN
ejpam-5634	135	34	,	,	PUNCT
ejpam-5634	135	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	135	36	;	;	PUNCT
ejpam-5634	135	37	(	(	PUNCT
ejpam-5634	135	38	2	2	X
ejpam-5634	135	39	)	)	PUNCT
ejpam-5634	135	40	f−1(v	f−1(v	NOUN
ejpam-5634	135	41	)	)	PUNCT
ejpam-5634	135	42	is	be	AUX
ejpam-5634	135	43	τ1τ2	τ1τ2	NOUN
ejpam-5634	135	44	-	-	ADJ
ejpam-5634	135	45	open	open	ADJ
ejpam-5634	135	46	in	in	ADP
ejpam-5634	135	47	x	x	PUNCT
ejpam-5634	135	48	for	for	ADP
ejpam-5634	135	49	every	every	DET
ejpam-5634	135	50	σ1σ2	σ1σ2	NOUN
ejpam-5634	135	51	-	-	ADJ
ejpam-5634	135	52	open	open	ADJ
ejpam-5634	135	53	set	set	NOUN
ejpam-5634	135	54	v	v	NOUN
ejpam-5634	135	55	of	of	ADP
ejpam-5634	135	56	y	y	PROPN
ejpam-5634	135	57	having	have	VERB
ejpam-5634	135	58	σ1σ2	σ1σ2	ADV
ejpam-5634	135	59	-	-	PUNCT
ejpam-5634	135	60	connected	connect	VERB
ejpam-5634	135	61	complement	complement	NOUN
ejpam-5634	135	62	;	;	PUNCT
ejpam-5634	135	63	m.	m.	NOUN
ejpam-5634	135	64	chiangpradit	chiangpradit	NOUN
ejpam-5634	135	65	,	,	PUNCT
ejpam-5634	135	66	a.	a.	PROPN
ejpam-5634	135	67	sama	sama	PROPN
ejpam-5634	135	68	-	-	PUNCT
ejpam-5634	135	69	ae	ae	PROPN
ejpam-5634	135	70	,	,	PUNCT
ejpam-5634	135	71	c.	c.	PROPN
ejpam-5634	135	72	boonpok	boonpok	PROPN
ejpam-5634	135	73	/	/	SYM
ejpam-5634	135	74	eur	eur	PROPN
ejpam-5634	135	75	.	.	PUNCT
ejpam-5634	136	1	j.	j.	PROPN
ejpam-5634	136	2	pure	pure	PROPN
ejpam-5634	136	3	appl	appl	PROPN
ejpam-5634	136	4	.	.	PROPN
ejpam-5634	136	5	math	math	PROPN
ejpam-5634	136	6	,	,	PUNCT
ejpam-5634	136	7	18	18	NUM
ejpam-5634	136	8	(	(	PUNCT
ejpam-5634	136	9	1	1	NUM
ejpam-5634	136	10	)	)	PUNCT
ejpam-5634	136	11	(	(	PUNCT
ejpam-5634	136	12	2025	2025	NUM
ejpam-5634	136	13	)	)	PUNCT
ejpam-5634	136	14	,	,	PUNCT
ejpam-5634	136	15	5634	5634	NUM
ejpam-5634	136	16	6	6	NUM
ejpam-5634	136	17	of	of	ADP
ejpam-5634	136	18	12	12	NUM
ejpam-5634	136	19	(	(	PUNCT
ejpam-5634	136	20	3	3	NUM
ejpam-5634	136	21	)	)	PUNCT
ejpam-5634	136	22	f−1(k	f−1(k	PROPN
ejpam-5634	136	23	)	)	PUNCT
ejpam-5634	136	24	is	be	AUX
ejpam-5634	136	25	τ1τ2	τ1τ2	NOUN
ejpam-5634	136	26	-	-	ADJ
ejpam-5634	136	27	closed	closed	ADJ
ejpam-5634	136	28	in	in	ADP
ejpam-5634	136	29	x	x	PUNCT
ejpam-5634	136	30	for	for	ADP
ejpam-5634	136	31	every	every	DET
ejpam-5634	136	32	σ1σ2	σ1σ2	NOUN
ejpam-5634	136	33	-	-	ADJ
ejpam-5634	136	34	connected	connect	VERB
ejpam-5634	136	35	σ1σ2	σ1σ2	VERB
ejpam-5634	136	36	-	-	PUNCT
ejpam-5634	136	37	closed	closed	ADJ
ejpam-5634	136	38	set	set	NOUN
ejpam-5634	136	39	k	k	PROPN
ejpam-5634	136	40	of	of	ADP
ejpam-5634	136	41	y	y	PROPN
ejpam-5634	136	42	;	;	PUNCT
ejpam-5634	136	43	(	(	PUNCT
ejpam-5634	136	44	4	4	X
ejpam-5634	136	45	)	)	PUNCT
ejpam-5634	136	46	τ1τ2	τ1τ2	NOUN
ejpam-5634	136	47	-	-	NOUN
ejpam-5634	136	48	cl(f	cl(f	NOUN
ejpam-5634	136	49	−1(b	−1(b	NOUN
ejpam-5634	136	50	)	)	PUNCT
ejpam-5634	136	51	)	)	PUNCT
ejpam-5634	137	1	⊆	⊆	NUM
ejpam-5634	137	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5634	137	3	-	-	PUNCT
ejpam-5634	137	4	cl(b	cl(b	NOUN
ejpam-5634	137	5	)	)	PUNCT
ejpam-5634	137	6	)	)	PUNCT
ejpam-5634	137	7	for	for	ADP
ejpam-5634	137	8	every	every	DET
ejpam-5634	137	9	subset	subset	NOUN
ejpam-5634	137	10	b	b	PROPN
ejpam-5634	137	11	of	of	ADP
ejpam-5634	137	12	y	y	PROPN
ejpam-5634	137	13	having	have	VERB
ejpam-5634	137	14	the	the	DET
ejpam-5634	137	15	σ1σ2	σ1σ2	ADV
ejpam-5634	137	16	-	-	PUNCT
ejpam-5634	137	17	connected	connect	VERB
ejpam-5634	137	18	σ1σ2	σ1σ2	NOUN
ejpam-5634	137	19	-	-	NOUN
ejpam-5634	137	20	closure	closure	NOUN
ejpam-5634	137	21	;	;	PUNCT
ejpam-5634	137	22	(	(	PUNCT
ejpam-5634	137	23	5	5	X
ejpam-5634	137	24	)	)	PUNCT
ejpam-5634	137	25	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5634	137	26	-	-	PUNCT
ejpam-5634	137	27	int(b	int(b	NOUN
ejpam-5634	137	28	)	)	PUNCT
ejpam-5634	137	29	)	)	PUNCT
ejpam-5634	138	1	⊆	⊆	X
ejpam-5634	138	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	138	3	-	-	NUM
ejpam-5634	138	4	int(f	int(f	NOUN
ejpam-5634	138	5	−1(b	−1(b	NOUN
ejpam-5634	138	6	)	)	PUNCT
ejpam-5634	138	7	)	)	PUNCT
ejpam-5634	138	8	for	for	ADP
ejpam-5634	138	9	every	every	DET
ejpam-5634	138	10	subset	subset	NOUN
ejpam-5634	138	11	b	b	PROPN
ejpam-5634	138	12	of	of	ADP
ejpam-5634	138	13	y	y	PRON
ejpam-5634	138	14	such	such	ADJ
ejpam-5634	138	15	that	that	SCONJ
ejpam-5634	138	16	y−σ1σ2	y−σ1σ2	PROPN
ejpam-5634	138	17	-	-	PUNCT
ejpam-5634	138	18	int(b	int(b	NOUN
ejpam-5634	138	19	)	)	PUNCT
ejpam-5634	138	20	is	be	AUX
ejpam-5634	138	21	σ1σ2	σ1σ2	NOUN
ejpam-5634	138	22	-	-	PUNCT
ejpam-5634	138	23	connected	connect	VERB
ejpam-5634	138	24	.	.	PUNCT
ejpam-5634	139	1	for	for	ADP
ejpam-5634	139	2	a	a	DET
ejpam-5634	139	3	multifunction	multifunction	NOUN
ejpam-5634	139	4	f	f	NOUN
ejpam-5634	139	5	:	:	PUNCT
ejpam-5634	139	6	(	(	PUNCT
ejpam-5634	139	7	x	x	NOUN
ejpam-5634	139	8	,	,	PUNCT
ejpam-5634	139	9	τ1	τ1	NOUN
ejpam-5634	139	10	,	,	PUNCT
ejpam-5634	139	11	τ2	τ2	NOUN
ejpam-5634	139	12	)	)	PUNCT
ejpam-5634	139	13	→	→	SYM
ejpam-5634	139	14	(	(	PUNCT
ejpam-5634	139	15	y	y	PROPN
ejpam-5634	139	16	,	,	PUNCT
ejpam-5634	139	17	σ1	σ1	PROPN
ejpam-5634	139	18	,	,	PUNCT
ejpam-5634	139	19	σ2	σ2	PROPN
ejpam-5634	139	20	)	)	PUNCT
ejpam-5634	139	21	,	,	PUNCT
ejpam-5634	139	22	a	a	DET
ejpam-5634	139	23	multifunction	multifunction	NOUN
ejpam-5634	139	24	clf⊛	clf⊛	NOUN
ejpam-5634	139	25	:	:	PUNCT
ejpam-5634	139	26	(	(	PUNCT
ejpam-5634	139	27	x	x	NOUN
ejpam-5634	139	28	,	,	PUNCT
ejpam-5634	139	29	τ1	τ1	NOUN
ejpam-5634	139	30	,	,	PUNCT
ejpam-5634	139	31	τ2	τ2	NOUN
ejpam-5634	139	32	)	)	PUNCT
ejpam-5634	139	33	→	→	SYM
ejpam-5634	139	34	(	(	PUNCT
ejpam-5634	139	35	y	y	PROPN
ejpam-5634	139	36	,	,	PUNCT
ejpam-5634	139	37	σ1	σ1	PROPN
ejpam-5634	139	38	,	,	PUNCT
ejpam-5634	139	39	σ2	σ2	PROPN
ejpam-5634	139	40	)	)	PUNCT
ejpam-5634	139	41	is	be	AUX
ejpam-5634	139	42	defined	define	VERB
ejpam-5634	139	43	in	in	ADP
ejpam-5634	139	44	[	[	X
ejpam-5634	139	45	29	29	NUM
ejpam-5634	139	46	]	]	PUNCT
ejpam-5634	139	47	as	as	SCONJ
ejpam-5634	139	48	follows	follow	VERB
ejpam-5634	139	49	:	:	PUNCT
ejpam-5634	139	50	clf⊛(x	clf⊛(x	PROPN
ejpam-5634	139	51	)	)	PUNCT
ejpam-5634	139	52	=	=	PUNCT
ejpam-5634	140	1	σ1σ2	σ1σ2	X
ejpam-5634	140	2	-	-	NUM
ejpam-5634	140	3	cl(f	cl(f	NOUN
ejpam-5634	140	4	(	(	PUNCT
ejpam-5634	140	5	x	x	NOUN
ejpam-5634	140	6	)	)	PUNCT
ejpam-5634	140	7	)	)	PUNCT
ejpam-5634	140	8	for	for	ADP
ejpam-5634	140	9	each	each	DET
ejpam-5634	140	10	x	x	SYM
ejpam-5634	140	11	∈	∈	PROPN
ejpam-5634	140	12	x.	x.	NOUN
ejpam-5634	140	13	definition	definition	NOUN
ejpam-5634	140	14	4	4	NUM
ejpam-5634	140	15	.	.	PUNCT
ejpam-5634	141	1	[	[	X
ejpam-5634	141	2	29	29	NUM
ejpam-5634	141	3	]	]	PUNCT
ejpam-5634	141	4	a	a	DET
ejpam-5634	141	5	subset	subset	NOUN
ejpam-5634	141	6	a	a	PRON
ejpam-5634	141	7	of	of	ADP
ejpam-5634	141	8	a	a	DET
ejpam-5634	141	9	bitopological	bitopological	ADJ
ejpam-5634	141	10	space	space	NOUN
ejpam-5634	141	11	(	(	PUNCT
ejpam-5634	141	12	x	x	NOUN
ejpam-5634	141	13	,	,	PUNCT
ejpam-5634	141	14	τ1	τ1	NOUN
ejpam-5634	141	15	,	,	PUNCT
ejpam-5634	141	16	τ2	τ2	NOUN
ejpam-5634	141	17	)	)	PUNCT
ejpam-5634	141	18	is	be	AUX
ejpam-5634	141	19	said	say	VERB
ejpam-5634	141	20	to	to	PART
ejpam-5634	141	21	be	be	AUX
ejpam-5634	141	22	:	:	PUNCT
ejpam-5634	141	23	(	(	PUNCT
ejpam-5634	141	24	1	1	X
ejpam-5634	141	25	)	)	PUNCT
ejpam-5634	141	26	τ1τ2	τ1τ2	NOUN
ejpam-5634	141	27	-	-	NOUN
ejpam-5634	141	28	paracompact	paracompact	ADJ
ejpam-5634	141	29	if	if	SCONJ
ejpam-5634	141	30	every	every	DET
ejpam-5634	141	31	cover	cover	NOUN
ejpam-5634	141	32	of	of	ADP
ejpam-5634	141	33	a	a	PRON
ejpam-5634	141	34	by	by	ADP
ejpam-5634	141	35	τ1τ2	τ1τ2	ADJ
ejpam-5634	141	36	-	-	ADJ
ejpam-5634	141	37	open	open	ADJ
ejpam-5634	141	38	sets	set	NOUN
ejpam-5634	141	39	of	of	ADP
ejpam-5634	141	40	x	x	VERB
ejpam-5634	141	41	is	be	AUX
ejpam-5634	141	42	refined	refine	VERB
ejpam-5634	141	43	by	by	ADP
ejpam-5634	141	44	a	a	DET
ejpam-5634	141	45	cover	cover	NOUN
ejpam-5634	141	46	of	of	ADP
ejpam-5634	141	47	a	a	PRON
ejpam-5634	141	48	which	which	PRON
ejpam-5634	141	49	consists	consist	VERB
ejpam-5634	141	50	of	of	ADP
ejpam-5634	141	51	τ1τ2	τ1τ2	ADJ
ejpam-5634	141	52	-	-	ADJ
ejpam-5634	141	53	open	open	ADJ
ejpam-5634	141	54	sets	set	NOUN
ejpam-5634	141	55	of	of	ADP
ejpam-5634	141	56	x	x	PUNCT
ejpam-5634	141	57	and	and	CCONJ
ejpam-5634	141	58	is	be	AUX
ejpam-5634	141	59	τ1τ2	τ1τ2	NOUN
ejpam-5634	141	60	-	-	ADJ
ejpam-5634	141	61	locally	locally	ADV
ejpam-5634	141	62	finite	finite	NOUN
ejpam-5634	141	63	in	in	ADP
ejpam-5634	141	64	x	x	PRON
ejpam-5634	141	65	;	;	PUNCT
ejpam-5634	141	66	(	(	PUNCT
ejpam-5634	141	67	2	2	X
ejpam-5634	141	68	)	)	PUNCT
ejpam-5634	141	69	τ1τ2	τ1τ2	NOUN
ejpam-5634	141	70	-	-	NOUN
ejpam-5634	141	71	regular	regular	ADJ
ejpam-5634	141	72	if	if	SCONJ
ejpam-5634	141	73	for	for	ADP
ejpam-5634	141	74	each	each	DET
ejpam-5634	141	75	x	x	SYM
ejpam-5634	141	76	∈	∈	PROPN
ejpam-5634	141	77	a	a	PRON
ejpam-5634	141	78	and	and	CCONJ
ejpam-5634	141	79	each	each	DET
ejpam-5634	141	80	τ1τ2	τ1τ2	ADJ
ejpam-5634	141	81	-	-	ADJ
ejpam-5634	141	82	open	open	ADJ
ejpam-5634	141	83	set	set	ADJ
ejpam-5634	141	84	u	u	NOUN
ejpam-5634	141	85	of	of	ADP
ejpam-5634	141	86	x	x	PUNCT
ejpam-5634	141	87	containing	contain	VERB
ejpam-5634	141	88	x	x	PRON
ejpam-5634	141	89	,	,	PUNCT
ejpam-5634	141	90	there	there	PRON
ejpam-5634	141	91	exists	exist	VERB
ejpam-5634	141	92	a	a	DET
ejpam-5634	141	93	τ1τ2	τ1τ2	NOUN
ejpam-5634	141	94	-	-	ADJ
ejpam-5634	141	95	open	open	ADJ
ejpam-5634	141	96	set	set	NOUN
ejpam-5634	141	97	v	v	NOUN
ejpam-5634	141	98	of	of	ADP
ejpam-5634	141	99	x	x	PUNCT
ejpam-5634	141	100	such	such	ADJ
ejpam-5634	141	101	that	that	SCONJ
ejpam-5634	141	102	x	x	SYM
ejpam-5634	141	103	∈	∈	NOUN
ejpam-5634	141	104	v	v	ADP
ejpam-5634	141	105	⊆	⊆	NUM
ejpam-5634	141	106	τ1τ2	τ1τ2	NOUN
ejpam-5634	141	107	-	-	NOUN
ejpam-5634	141	108	cl(v	cl(v	X
ejpam-5634	141	109	)	)	PUNCT
ejpam-5634	141	110	⊆	⊆	NUM
ejpam-5634	141	111	u	u	NOUN
ejpam-5634	141	112	.	.	PUNCT
ejpam-5634	142	1	lemma	lemma	PROPN
ejpam-5634	142	2	2	2	NUM
ejpam-5634	142	3	.	.	PUNCT
ejpam-5634	143	1	[	[	X
ejpam-5634	143	2	29	29	NUM
ejpam-5634	143	3	]	]	X
ejpam-5634	143	4	if	if	SCONJ
ejpam-5634	143	5	a	a	PRON
ejpam-5634	143	6	is	be	AUX
ejpam-5634	143	7	a	a	DET
ejpam-5634	143	8	τ1τ2	τ1τ2	ADJ
ejpam-5634	143	9	-	-	ADJ
ejpam-5634	143	10	regular	regular	ADJ
ejpam-5634	143	11	τ1τ2	τ1τ2	NOUN
ejpam-5634	143	12	-	-	ADJ
ejpam-5634	143	13	paracompact	paracompact	ADJ
ejpam-5634	143	14	set	set	NOUN
ejpam-5634	143	15	of	of	ADP
ejpam-5634	143	16	a	a	DET
ejpam-5634	143	17	bitopological	bitopological	ADJ
ejpam-5634	143	18	space	space	NOUN
ejpam-5634	143	19	(	(	PUNCT
ejpam-5634	143	20	x	x	NOUN
ejpam-5634	143	21	,	,	PUNCT
ejpam-5634	143	22	τ1	τ1	NOUN
ejpam-5634	143	23	,	,	PUNCT
ejpam-5634	143	24	τ2	τ2	NOUN
ejpam-5634	143	25	)	)	PUNCT
ejpam-5634	143	26	and	and	CCONJ
ejpam-5634	143	27	u	u	NOUN
ejpam-5634	143	28	is	be	AUX
ejpam-5634	143	29	a	a	DET
ejpam-5634	143	30	τ1τ2	τ1τ2	ADJ
ejpam-5634	143	31	-	-	ADJ
ejpam-5634	143	32	open	open	ADJ
ejpam-5634	143	33	neighbourhood	neighbourhood	NOUN
ejpam-5634	143	34	of	of	ADP
ejpam-5634	143	35	a	a	PRON
ejpam-5634	143	36	,	,	PUNCT
ejpam-5634	143	37	then	then	ADV
ejpam-5634	143	38	there	there	PRON
ejpam-5634	143	39	exists	exist	VERB
ejpam-5634	143	40	a	a	DET
ejpam-5634	143	41	τ1τ2	τ1τ2	NOUN
ejpam-5634	143	42	-	-	ADJ
ejpam-5634	143	43	open	open	ADJ
ejpam-5634	143	44	set	set	NOUN
ejpam-5634	143	45	v	v	NOUN
ejpam-5634	143	46	of	of	ADP
ejpam-5634	143	47	x	x	PUNCT
ejpam-5634	143	48	such	such	ADJ
ejpam-5634	143	49	that	that	SCONJ
ejpam-5634	143	50	a	a	DET
ejpam-5634	143	51	⊆	⊆	NUM
ejpam-5634	143	52	v	v	ADP
ejpam-5634	143	53	⊆	⊆	NUM
ejpam-5634	143	54	τ1τ2	τ1τ2	NOUN
ejpam-5634	143	55	-	-	NOUN
ejpam-5634	143	56	cl(v	cl(v	X
ejpam-5634	143	57	)	)	PUNCT
ejpam-5634	143	58	⊆	⊆	NUM
ejpam-5634	143	59	u	u	NOUN
ejpam-5634	143	60	.	.	PUNCT
ejpam-5634	144	1	lemma	lemma	PROPN
ejpam-5634	144	2	3	3	X
ejpam-5634	144	3	.	.	PUNCT
ejpam-5634	145	1	[	[	X
ejpam-5634	145	2	29	29	NUM
ejpam-5634	145	3	]	]	X
ejpam-5634	145	4	if	if	SCONJ
ejpam-5634	145	5	f	f	PROPN
ejpam-5634	145	6	:	:	PUNCT
ejpam-5634	145	7	(	(	PUNCT
ejpam-5634	145	8	x	x	NOUN
ejpam-5634	145	9	,	,	PUNCT
ejpam-5634	145	10	τ1	τ1	NOUN
ejpam-5634	145	11	,	,	PUNCT
ejpam-5634	145	12	τ2	τ2	NOUN
ejpam-5634	145	13	)	)	PUNCT
ejpam-5634	145	14	→	→	SYM
ejpam-5634	145	15	(	(	PUNCT
ejpam-5634	145	16	y	y	PROPN
ejpam-5634	145	17	,	,	PUNCT
ejpam-5634	145	18	σ1	σ1	PROPN
ejpam-5634	145	19	,	,	PUNCT
ejpam-5634	145	20	σ2	σ2	PROPN
ejpam-5634	145	21	)	)	PUNCT
ejpam-5634	145	22	is	be	AUX
ejpam-5634	145	23	a	a	DET
ejpam-5634	145	24	multifunction	multifunction	NOUN
ejpam-5634	145	25	such	such	ADJ
ejpam-5634	145	26	that	that	SCONJ
ejpam-5634	145	27	f	f	PROPN
ejpam-5634	145	28	(	(	PUNCT
ejpam-5634	145	29	x	x	X
ejpam-5634	145	30	)	)	PUNCT
ejpam-5634	145	31	is	be	AUX
ejpam-5634	145	32	τ1τ2regular	τ1τ2regular	NUM
ejpam-5634	145	33	and	and	CCONJ
ejpam-5634	145	34	τ1τ2	τ1τ2	NOUN
ejpam-5634	145	35	-	-	ADJ
ejpam-5634	145	36	paracompact	paracompact	ADJ
ejpam-5634	145	37	for	for	ADP
ejpam-5634	145	38	each	each	DET
ejpam-5634	145	39	x	x	SYM
ejpam-5634	145	40	∈	∈	PROPN
ejpam-5634	145	41	x	x	NOUN
ejpam-5634	145	42	,	,	PUNCT
ejpam-5634	145	43	then	then	ADV
ejpam-5634	145	44	clf+	clf+	PROPN
ejpam-5634	145	45	⊛	⊛	X
ejpam-5634	145	46	(	(	PUNCT
ejpam-5634	145	47	v	v	NOUN
ejpam-5634	145	48	)	)	PUNCT
ejpam-5634	145	49	=	=	PUNCT
ejpam-5634	145	50	f+(v	f+(v	NOUN
ejpam-5634	145	51	)	)	PUNCT
ejpam-5634	145	52	for	for	ADP
ejpam-5634	145	53	each	each	DET
ejpam-5634	145	54	σ1σ2	σ1σ2	VERB
ejpam-5634	145	55	-	-	ADJ
ejpam-5634	145	56	open	open	ADJ
ejpam-5634	145	57	set	set	NOUN
ejpam-5634	145	58	v	v	NOUN
ejpam-5634	145	59	of	of	ADP
ejpam-5634	145	60	y	y	PROPN
ejpam-5634	145	61	.	.	PUNCT
ejpam-5634	146	1	theorem	theorem	NOUN
ejpam-5634	146	2	3	3	X
ejpam-5634	146	3	.	.	PUNCT
ejpam-5634	147	1	let	let	VERB
ejpam-5634	147	2	f	f	NOUN
ejpam-5634	147	3	:	:	PUNCT
ejpam-5634	147	4	(	(	PUNCT
ejpam-5634	147	5	x	x	NOUN
ejpam-5634	147	6	,	,	PUNCT
ejpam-5634	147	7	τ1	τ1	NOUN
ejpam-5634	147	8	,	,	PUNCT
ejpam-5634	147	9	τ2	τ2	NOUN
ejpam-5634	147	10	)	)	PUNCT
ejpam-5634	147	11	→	→	SYM
ejpam-5634	147	12	(	(	PUNCT
ejpam-5634	147	13	y	y	PROPN
ejpam-5634	147	14	,	,	PUNCT
ejpam-5634	147	15	σ1	σ1	PROPN
ejpam-5634	147	16	,	,	PUNCT
ejpam-5634	147	17	σ2	σ2	PROPN
ejpam-5634	147	18	)	)	PUNCT
ejpam-5634	147	19	be	be	VERB
ejpam-5634	147	20	a	a	DET
ejpam-5634	147	21	multifunction	multifunction	NOUN
ejpam-5634	147	22	such	such	ADJ
ejpam-5634	147	23	that	that	SCONJ
ejpam-5634	147	24	f	f	PROPN
ejpam-5634	147	25	(	(	PUNCT
ejpam-5634	147	26	x	x	X
ejpam-5634	147	27	)	)	PUNCT
ejpam-5634	147	28	is	be	AUX
ejpam-5634	147	29	σ1σ2paracompact	σ1σ2paracompact	NUM
ejpam-5634	147	30	and	and	CCONJ
ejpam-5634	147	31	σ1σ2	σ1σ2	NOUN
ejpam-5634	147	32	-	-	ADJ
ejpam-5634	147	33	regular	regular	ADJ
ejpam-5634	147	34	for	for	ADP
ejpam-5634	147	35	each	each	DET
ejpam-5634	147	36	x	x	SYM
ejpam-5634	147	37	∈	∈	PROPN
ejpam-5634	147	38	x.	x.	NOUN
ejpam-5634	147	39	then	then	ADV
ejpam-5634	147	40	,	,	PUNCT
ejpam-5634	147	41	f	f	PROPN
ejpam-5634	147	42	is	be	AUX
ejpam-5634	147	43	upper	upper	ADJ
ejpam-5634	147	44	s-(τ1	s-(τ1	PROPN
ejpam-5634	147	45	,	,	PUNCT
ejpam-5634	147	46	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	148	1	if	if	SCONJ
ejpam-5634	148	2	and	and	CCONJ
ejpam-5634	148	3	only	only	ADV
ejpam-5634	148	4	if	if	SCONJ
ejpam-5634	148	5	clf⊛	clf⊛	PROPN
ejpam-5634	148	6	:	:	PUNCT
ejpam-5634	148	7	(	(	PUNCT
ejpam-5634	148	8	x	x	NOUN
ejpam-5634	148	9	,	,	PUNCT
ejpam-5634	148	10	τ1	τ1	NOUN
ejpam-5634	148	11	,	,	PUNCT
ejpam-5634	148	12	τ2	τ2	NOUN
ejpam-5634	148	13	)	)	PUNCT
ejpam-5634	148	14	→	→	SYM
ejpam-5634	148	15	(	(	PUNCT
ejpam-5634	148	16	y	y	PROPN
ejpam-5634	148	17	,	,	PUNCT
ejpam-5634	148	18	σ1	σ1	PROPN
ejpam-5634	148	19	,	,	PUNCT
ejpam-5634	148	20	σ2	σ2	PROPN
ejpam-5634	148	21	)	)	PUNCT
ejpam-5634	148	22	is	be	AUX
ejpam-5634	148	23	upper	upper	ADJ
ejpam-5634	148	24	s-(τ1	s-(τ1	PROPN
ejpam-5634	148	25	,	,	PUNCT
ejpam-5634	148	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	148	27	.	.	PUNCT
ejpam-5634	149	1	proof	proof	NOUN
ejpam-5634	149	2	.	.	PUNCT
ejpam-5634	150	1	we	we	PRON
ejpam-5634	150	2	put	put	VERB
ejpam-5634	150	3	g	g	NOUN
ejpam-5634	150	4	=	=	PUNCT
ejpam-5634	150	5	clf⊛.	clf⊛.	NOUN
ejpam-5634	150	6	suppose	suppose	VERB
ejpam-5634	150	7	that	that	SCONJ
ejpam-5634	150	8	f	f	PROPN
ejpam-5634	150	9	is	be	AUX
ejpam-5634	150	10	upper	upper	ADJ
ejpam-5634	150	11	s-(τ1	s-(τ1	PROPN
ejpam-5634	150	12	,	,	PUNCT
ejpam-5634	150	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	150	14	.	.	PUNCT
ejpam-5634	151	1	let	let	VERB
ejpam-5634	151	2	x	x	PUNCT
ejpam-5634	151	3	∈	∈	PROPN
ejpam-5634	151	4	x	x	X
ejpam-5634	151	5	and	and	CCONJ
ejpam-5634	151	6	v	v	X
ejpam-5634	151	7	be	be	AUX
ejpam-5634	151	8	any	any	DET
ejpam-5634	151	9	σ1σ2	σ1σ2	NOUN
ejpam-5634	151	10	-	-	ADJ
ejpam-5634	151	11	open	open	ADJ
ejpam-5634	151	12	set	set	NOUN
ejpam-5634	151	13	of	of	ADP
ejpam-5634	151	14	y	y	NOUN
ejpam-5634	151	15	containing	contain	VERB
ejpam-5634	151	16	g(x	g(x	NOUN
ejpam-5634	151	17	)	)	PUNCT
ejpam-5634	151	18	and	and	CCONJ
ejpam-5634	151	19	having	have	VERB
ejpam-5634	151	20	σ1σ2	σ1σ2	NOUN
ejpam-5634	151	21	-	-	PUNCT
ejpam-5634	151	22	connected	connect	VERB
ejpam-5634	151	23	complement	complement	NOUN
ejpam-5634	151	24	.	.	PUNCT
ejpam-5634	152	1	by	by	ADP
ejpam-5634	152	2	lemma	lemma	PROPN
ejpam-5634	152	3	3	3	NUM
ejpam-5634	152	4	,	,	PUNCT
ejpam-5634	152	5	we	we	PRON
ejpam-5634	152	6	have	have	VERB
ejpam-5634	152	7	x	x	X
ejpam-5634	152	8	∈	∈	PROPN
ejpam-5634	152	9	g+(v	g+(v	PROPN
ejpam-5634	152	10	)	)	PUNCT
ejpam-5634	152	11	=	=	PUNCT
ejpam-5634	153	1	f+(v	f+(v	NOUN
ejpam-5634	153	2	)	)	PUNCT
ejpam-5634	154	1	and	and	CCONJ
ejpam-5634	154	2	hence	hence	ADV
ejpam-5634	154	3	there	there	PRON
ejpam-5634	154	4	exists	exist	VERB
ejpam-5634	154	5	a	a	DET
ejpam-5634	154	6	τ1τ2	τ1τ2	NOUN
ejpam-5634	154	7	-	-	ADJ
ejpam-5634	154	8	open	open	ADJ
ejpam-5634	154	9	set	set	ADJ
ejpam-5634	154	10	u	u	NOUN
ejpam-5634	154	11	of	of	ADP
ejpam-5634	154	12	x	x	PUNCT
ejpam-5634	154	13	containing	contain	VERB
ejpam-5634	154	14	x	x	PUNCT
ejpam-5634	154	15	such	such	ADJ
ejpam-5634	154	16	that	that	SCONJ
ejpam-5634	154	17	f	f	PROPN
ejpam-5634	154	18	(	(	PUNCT
ejpam-5634	154	19	u	u	NOUN
ejpam-5634	154	20	)	)	PUNCT
ejpam-5634	154	21	⊆	⊆	NUM
ejpam-5634	154	22	v	v	NOUN
ejpam-5634	154	23	.	.	PUNCT
ejpam-5634	155	1	since	since	SCONJ
ejpam-5634	155	2	f	f	PROPN
ejpam-5634	155	3	(	(	PUNCT
ejpam-5634	155	4	z	z	NOUN
ejpam-5634	155	5	)	)	PUNCT
ejpam-5634	155	6	is	be	AUX
ejpam-5634	155	7	σ1σ2	σ1σ2	NOUN
ejpam-5634	155	8	-	-	ADJ
ejpam-5634	155	9	paracompact	paracompact	NOUN
ejpam-5634	155	10	and	and	CCONJ
ejpam-5634	155	11	σ1σ2regular	σ1σ2regular	PROPN
ejpam-5634	155	12	for	for	ADP
ejpam-5634	155	13	each	each	DET
ejpam-5634	155	14	z	z	NOUN
ejpam-5634	155	15	∈	∈	PROPN
ejpam-5634	155	16	u	u	NOUN
ejpam-5634	155	17	,	,	PUNCT
ejpam-5634	155	18	by	by	ADP
ejpam-5634	155	19	lemma	lemma	PROPN
ejpam-5634	155	20	2	2	NUM
ejpam-5634	155	21	there	there	PRON
ejpam-5634	155	22	exists	exist	VERB
ejpam-5634	155	23	a	a	DET
ejpam-5634	155	24	τ1τ2	τ1τ2	NOUN
ejpam-5634	155	25	-	-	ADJ
ejpam-5634	155	26	open	open	ADJ
ejpam-5634	155	27	set	set	NOUN
ejpam-5634	155	28	w	w	PROPN
ejpam-5634	155	29	of	of	ADP
ejpam-5634	155	30	y	y	PRON
ejpam-5634	155	31	such	such	ADJ
ejpam-5634	155	32	that	that	SCONJ
ejpam-5634	155	33	f	f	PROPN
ejpam-5634	155	34	(	(	PUNCT
ejpam-5634	155	35	z	z	NOUN
ejpam-5634	155	36	)	)	PUNCT
ejpam-5634	155	37	⊆	⊆	NUM
ejpam-5634	155	38	w	w	ADP
ejpam-5634	155	39	⊆	⊆	NUM
ejpam-5634	155	40	σ1σ2	σ1σ2	NOUN
ejpam-5634	155	41	-	-	PUNCT
ejpam-5634	155	42	cl(w	cl(w	NOUN
ejpam-5634	155	43	)	)	PUNCT
ejpam-5634	155	44	⊆	⊆	NUM
ejpam-5634	155	45	v	v	NOUN
ejpam-5634	155	46	;	;	PUNCT
ejpam-5634	155	47	hence	hence	ADV
ejpam-5634	155	48	g(z	g(z	ADJ
ejpam-5634	155	49	)	)	PUNCT
ejpam-5634	155	50	⊆	⊆	NUM
ejpam-5634	155	51	σ1σ2	σ1σ2	NOUN
ejpam-5634	155	52	-	-	PUNCT
ejpam-5634	155	53	cl(w	cl(w	NOUN
ejpam-5634	155	54	)	)	PUNCT
ejpam-5634	155	55	⊆	⊆	NUM
ejpam-5634	155	56	v	v	NOUN
ejpam-5634	155	57	for	for	ADP
ejpam-5634	155	58	each	each	DET
ejpam-5634	155	59	z	z	NOUN
ejpam-5634	155	60	∈	∈	PROPN
ejpam-5634	155	61	u	u	NOUN
ejpam-5634	155	62	.	.	PUNCT
ejpam-5634	156	1	thus	thus	ADV
ejpam-5634	156	2	,	,	PUNCT
ejpam-5634	156	3	g(u	g(u	PROPN
ejpam-5634	156	4	)	)	PUNCT
ejpam-5634	156	5	⊆	⊆	NUM
ejpam-5634	156	6	v	v	NOUN
ejpam-5634	156	7	.	.	PUNCT
ejpam-5634	157	1	this	this	PRON
ejpam-5634	157	2	shows	show	VERB
ejpam-5634	157	3	that	that	SCONJ
ejpam-5634	157	4	g	g	PROPN
ejpam-5634	157	5	is	be	AUX
ejpam-5634	157	6	upper	upper	ADJ
ejpam-5634	157	7	s-(τ1	s-(τ1	PROPN
ejpam-5634	157	8	,	,	PUNCT
ejpam-5634	157	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	157	10	.	.	PUNCT
ejpam-5634	158	1	conversely	conversely	ADV
ejpam-5634	158	2	,	,	PUNCT
ejpam-5634	158	3	suppose	suppose	VERB
ejpam-5634	158	4	that	that	SCONJ
ejpam-5634	158	5	g	g	PROPN
ejpam-5634	158	6	is	be	AUX
ejpam-5634	158	7	upper	upper	ADJ
ejpam-5634	158	8	s-(τ1	s-(τ1	PROPN
ejpam-5634	158	9	,	,	PUNCT
ejpam-5634	158	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	158	11	.	.	PUNCT
ejpam-5634	159	1	let	let	VERB
ejpam-5634	159	2	x	x	PUNCT
ejpam-5634	159	3	∈	∈	PROPN
ejpam-5634	159	4	x	x	X
ejpam-5634	159	5	and	and	CCONJ
ejpam-5634	159	6	v	v	X
ejpam-5634	159	7	be	be	AUX
ejpam-5634	159	8	any	any	DET
ejpam-5634	159	9	σ1σ2	σ1σ2	NOUN
ejpam-5634	159	10	-	-	ADJ
ejpam-5634	159	11	open	open	ADJ
ejpam-5634	159	12	set	set	NOUN
ejpam-5634	159	13	of	of	ADP
ejpam-5634	159	14	y	y	PROPN
ejpam-5634	159	15	containing	contain	VERB
ejpam-5634	159	16	f	f	PROPN
ejpam-5634	159	17	(	(	PUNCT
ejpam-5634	159	18	x	x	NOUN
ejpam-5634	159	19	)	)	PUNCT
ejpam-5634	159	20	and	and	CCONJ
ejpam-5634	159	21	having	have	VERB
ejpam-5634	159	22	σ1σ2	σ1σ2	NOUN
ejpam-5634	159	23	-	-	PUNCT
ejpam-5634	159	24	connected	connect	VERB
ejpam-5634	159	25	complement	complement	NOUN
ejpam-5634	159	26	.	.	PUNCT
ejpam-5634	160	1	by	by	ADP
ejpam-5634	160	2	lemma	lemma	PROPN
ejpam-5634	160	3	3	3	NUM
ejpam-5634	160	4	,	,	PUNCT
ejpam-5634	160	5	we	we	PRON
ejpam-5634	160	6	have	have	VERB
ejpam-5634	160	7	x	x	X
ejpam-5634	160	8	∈	∈	NOUN
ejpam-5634	160	9	f+(v	f+(v	NOUN
ejpam-5634	160	10	)	)	PUNCT
ejpam-5634	161	1	=	=	PUNCT
ejpam-5634	161	2	g+(v	g+(v	PROPN
ejpam-5634	161	3	)	)	PUNCT
ejpam-5634	161	4	and	and	CCONJ
ejpam-5634	161	5	hence	hence	ADV
ejpam-5634	161	6	g(x	g(x	NOUN
ejpam-5634	161	7	)	)	PUNCT
ejpam-5634	161	8	⊆	⊆	NUM
ejpam-5634	161	9	v	v	NOUN
ejpam-5634	161	10	.	.	PUNCT
ejpam-5634	162	1	there	there	PRON
ejpam-5634	162	2	exists	exist	VERB
ejpam-5634	162	3	a	a	DET
ejpam-5634	162	4	τ1τ2	τ1τ2	NOUN
ejpam-5634	162	5	-	-	ADJ
ejpam-5634	162	6	open	open	ADJ
ejpam-5634	162	7	set	set	ADJ
ejpam-5634	162	8	u	u	NOUN
ejpam-5634	162	9	of	of	ADP
ejpam-5634	162	10	x	x	PUNCT
ejpam-5634	162	11	containing	contain	VERB
ejpam-5634	162	12	x	x	PUNCT
ejpam-5634	162	13	such	such	ADJ
ejpam-5634	162	14	that	that	SCONJ
ejpam-5634	162	15	g(u	g(u	PROPN
ejpam-5634	162	16	)	)	PUNCT
ejpam-5634	162	17	⊆	⊆	NUM
ejpam-5634	162	18	v	v	NOUN
ejpam-5634	162	19	.	.	PUNCT
ejpam-5634	163	1	thus	thus	ADV
ejpam-5634	163	2	,	,	PUNCT
ejpam-5634	163	3	u	u	PROPN
ejpam-5634	163	4	⊆	⊆	NUM
ejpam-5634	163	5	g+(v	g+(v	PROPN
ejpam-5634	163	6	)	)	PUNCT
ejpam-5634	163	7	=	=	PUNCT
ejpam-5634	164	1	f+(v	f+(v	NOUN
ejpam-5634	164	2	)	)	PUNCT
ejpam-5634	165	1	and	and	CCONJ
ejpam-5634	165	2	so	so	ADV
ejpam-5634	165	3	f	f	PROPN
ejpam-5634	165	4	(	(	PUNCT
ejpam-5634	165	5	u	u	NOUN
ejpam-5634	165	6	)	)	PUNCT
ejpam-5634	165	7	⊆	⊆	NUM
ejpam-5634	165	8	v	v	NOUN
ejpam-5634	165	9	.	.	PUNCT
ejpam-5634	166	1	this	this	PRON
ejpam-5634	166	2	shows	show	VERB
ejpam-5634	166	3	that	that	SCONJ
ejpam-5634	166	4	f	f	PROPN
ejpam-5634	166	5	is	be	AUX
ejpam-5634	166	6	upper	upper	ADJ
ejpam-5634	166	7	s-(τ1	s-(τ1	PROPN
ejpam-5634	166	8	,	,	PUNCT
ejpam-5634	166	9	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5634	166	10	.	.	PUNCT
ejpam-5634	166	11	m.	m.	NOUN
ejpam-5634	166	12	chiangpradit	chiangpradit	PROPN
ejpam-5634	166	13	,	,	PUNCT
ejpam-5634	166	14	a.	a.	PROPN
ejpam-5634	166	15	sama	sama	PROPN
ejpam-5634	166	16	-	-	PUNCT
ejpam-5634	166	17	ae	ae	PROPN
ejpam-5634	166	18	,	,	PUNCT
ejpam-5634	166	19	c.	c.	PROPN
ejpam-5634	166	20	boonpok	boonpok	PROPN
ejpam-5634	166	21	/	/	SYM
ejpam-5634	166	22	eur	eur	PROPN
ejpam-5634	166	23	.	.	PUNCT
ejpam-5634	167	1	j.	j.	PROPN
ejpam-5634	167	2	pure	pure	PROPN
ejpam-5634	167	3	appl	appl	PROPN
ejpam-5634	167	4	.	.	PROPN
ejpam-5634	167	5	math	math	PROPN
ejpam-5634	167	6	,	,	PUNCT
ejpam-5634	167	7	18	18	NUM
ejpam-5634	167	8	(	(	PUNCT
ejpam-5634	167	9	1	1	NUM
ejpam-5634	167	10	)	)	PUNCT
ejpam-5634	167	11	(	(	PUNCT
ejpam-5634	167	12	2025	2025	NUM
ejpam-5634	167	13	)	)	PUNCT
ejpam-5634	167	14	,	,	PUNCT
ejpam-5634	167	15	5634	5634	NUM
ejpam-5634	167	16	7	7	NUM
ejpam-5634	167	17	of	of	ADP
ejpam-5634	167	18	12	12	NUM
ejpam-5634	167	19	lemma	lemma	PROPN
ejpam-5634	167	20	4	4	NUM
ejpam-5634	167	21	.	.	PUNCT
ejpam-5634	168	1	[	[	X
ejpam-5634	168	2	29	29	NUM
ejpam-5634	168	3	]	]	PUNCT
ejpam-5634	168	4	for	for	ADP
ejpam-5634	168	5	a	a	DET
ejpam-5634	168	6	multifunction	multifunction	NOUN
ejpam-5634	168	7	f	f	NOUN
ejpam-5634	168	8	:	:	PUNCT
ejpam-5634	168	9	(	(	PUNCT
ejpam-5634	168	10	x	x	NOUN
ejpam-5634	168	11	,	,	PUNCT
ejpam-5634	168	12	τ1	τ1	NOUN
ejpam-5634	168	13	,	,	PUNCT
ejpam-5634	168	14	τ2	τ2	NOUN
ejpam-5634	168	15	)	)	PUNCT
ejpam-5634	168	16	→	→	SYM
ejpam-5634	168	17	(	(	PUNCT
ejpam-5634	168	18	y	y	PROPN
ejpam-5634	168	19	,	,	PUNCT
ejpam-5634	168	20	σ1	σ1	PROPN
ejpam-5634	168	21	,	,	PUNCT
ejpam-5634	168	22	σ2	σ2	NOUN
ejpam-5634	168	23	)	)	PUNCT
ejpam-5634	168	24	,	,	PUNCT
ejpam-5634	168	25	clf	clf	PROPN
ejpam-5634	168	26	−	−	PROPN
ejpam-5634	168	27	⊛	⊛	NUM
ejpam-5634	168	28	(	(	PUNCT
ejpam-5634	168	29	v	v	NOUN
ejpam-5634	168	30	)	)	PUNCT
ejpam-5634	168	31	=	=	SYM
ejpam-5634	168	32	f−(v	f−(v	ADJ
ejpam-5634	168	33	)	)	PUNCT
ejpam-5634	168	34	for	for	ADP
ejpam-5634	168	35	each	each	DET
ejpam-5634	168	36	σ1σ2	σ1σ2	VERB
ejpam-5634	168	37	-	-	ADJ
ejpam-5634	168	38	open	open	ADJ
ejpam-5634	168	39	set	set	NOUN
ejpam-5634	168	40	v	v	NOUN
ejpam-5634	168	41	of	of	ADP
ejpam-5634	168	42	y	y	PROPN
ejpam-5634	168	43	.	.	PUNCT
ejpam-5634	169	1	theorem	theorem	ADJ
ejpam-5634	169	2	4	4	NUM
ejpam-5634	169	3	.	.	PUNCT
ejpam-5634	169	4	a	a	DET
ejpam-5634	169	5	multifunction	multifunction	NOUN
ejpam-5634	169	6	f	f	NOUN
ejpam-5634	169	7	:	:	PUNCT
ejpam-5634	169	8	(	(	PUNCT
ejpam-5634	169	9	x	x	NOUN
ejpam-5634	169	10	,	,	PUNCT
ejpam-5634	169	11	τ1	τ1	NOUN
ejpam-5634	169	12	,	,	PUNCT
ejpam-5634	169	13	τ2	τ2	NOUN
ejpam-5634	169	14	)	)	PUNCT
ejpam-5634	169	15	→	→	SYM
ejpam-5634	169	16	(	(	PUNCT
ejpam-5634	169	17	y	y	PROPN
ejpam-5634	169	18	,	,	PUNCT
ejpam-5634	169	19	σ1	σ1	PROPN
ejpam-5634	169	20	,	,	PUNCT
ejpam-5634	169	21	σ2	σ2	NOUN
ejpam-5634	169	22	)	)	PUNCT
ejpam-5634	169	23	is	be	AUX
ejpam-5634	169	24	lower	low	ADJ
ejpam-5634	169	25	s-(τ1	s-(τ1	NOUN
ejpam-5634	169	26	,	,	PUNCT
ejpam-5634	169	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	169	28	if	if	SCONJ
ejpam-5634	169	29	and	and	CCONJ
ejpam-5634	169	30	only	only	ADV
ejpam-5634	169	31	if	if	SCONJ
ejpam-5634	169	32	clf⊛	clf⊛	PROPN
ejpam-5634	169	33	:	:	PUNCT
ejpam-5634	169	34	(	(	PUNCT
ejpam-5634	169	35	x	x	NOUN
ejpam-5634	169	36	,	,	PUNCT
ejpam-5634	169	37	τ1	τ1	NOUN
ejpam-5634	169	38	,	,	PUNCT
ejpam-5634	169	39	τ2	τ2	NOUN
ejpam-5634	169	40	)	)	PUNCT
ejpam-5634	169	41	→	→	SYM
ejpam-5634	169	42	(	(	PUNCT
ejpam-5634	169	43	y	y	PROPN
ejpam-5634	169	44	,	,	PUNCT
ejpam-5634	169	45	σ1	σ1	PROPN
ejpam-5634	169	46	,	,	PUNCT
ejpam-5634	169	47	σ2	σ2	NOUN
ejpam-5634	169	48	)	)	PUNCT
ejpam-5634	169	49	is	be	AUX
ejpam-5634	169	50	lower	low	ADJ
ejpam-5634	169	51	s-(τ1	s-(τ1	NOUN
ejpam-5634	169	52	,	,	PUNCT
ejpam-5634	169	53	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	169	54	.	.	PUNCT
ejpam-5634	170	1	proof	proof	NOUN
ejpam-5634	170	2	.	.	PUNCT
ejpam-5634	171	1	by	by	ADP
ejpam-5634	171	2	using	use	VERB
ejpam-5634	171	3	lemma	lemma	PROPN
ejpam-5634	171	4	4	4	NUM
ejpam-5634	171	5	this	this	PRON
ejpam-5634	171	6	is	be	AUX
ejpam-5634	171	7	shown	show	VERB
ejpam-5634	171	8	similarly	similarly	ADV
ejpam-5634	171	9	as	as	ADP
ejpam-5634	171	10	in	in	ADP
ejpam-5634	171	11	theorem	theorem	NOUN
ejpam-5634	171	12	3	3	NUM
ejpam-5634	171	13	.	.	PUNCT
ejpam-5634	172	1	the	the	DET
ejpam-5634	172	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	172	3	-	-	NOUN
ejpam-5634	172	4	frontier	frontier	NOUN
ejpam-5634	172	5	[	[	X
ejpam-5634	172	6	26	26	NUM
ejpam-5634	172	7	]	]	PUNCT
ejpam-5634	172	8	of	of	ADP
ejpam-5634	172	9	a	a	DET
ejpam-5634	172	10	subset	subset	NOUN
ejpam-5634	172	11	a	a	PRON
ejpam-5634	172	12	of	of	ADP
ejpam-5634	172	13	a	a	DET
ejpam-5634	172	14	bitopological	bitopological	ADJ
ejpam-5634	172	15	space	space	NOUN
ejpam-5634	172	16	(	(	PUNCT
ejpam-5634	172	17	x	x	NOUN
ejpam-5634	172	18	,	,	PUNCT
ejpam-5634	172	19	τ1	τ1	NOUN
ejpam-5634	172	20	,	,	PUNCT
ejpam-5634	172	21	τ2	τ2	PROPN
ejpam-5634	172	22	)	)	PUNCT
ejpam-5634	172	23	,	,	PUNCT
ejpam-5634	172	24	denoted	denote	VERB
ejpam-5634	172	25	by	by	ADP
ejpam-5634	172	26	τ1τ2	τ1τ2	NOUN
ejpam-5634	172	27	-	-	ADJ
ejpam-5634	172	28	fr(a	fr(a	NUM
ejpam-5634	172	29	)	)	PUNCT
ejpam-5634	172	30	,	,	PUNCT
ejpam-5634	172	31	is	be	AUX
ejpam-5634	172	32	defined	define	VERB
ejpam-5634	172	33	by	by	ADP
ejpam-5634	172	34	τ1τ2	τ1τ2	NOUN
ejpam-5634	172	35	-	-	ADJ
ejpam-5634	172	36	fr(a	fr(a	ADJ
ejpam-5634	172	37	)	)	PUNCT
ejpam-5634	173	1	=	=	PUNCT
ejpam-5634	173	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	173	3	-	-	NUM
ejpam-5634	173	4	cl(a	cl(a	NUM
ejpam-5634	173	5	)	)	PUNCT
ejpam-5634	173	6	∩	∩	NOUN
ejpam-5634	173	7	τ1τ2	τ1τ2	NOUN
ejpam-5634	173	8	-	-	ADJ
ejpam-5634	173	9	cl(x	cl(x	SYM
ejpam-5634	173	10	−a	−a	NOUN
ejpam-5634	173	11	)	)	PUNCT
ejpam-5634	173	12	=	=	PUNCT
ejpam-5634	174	1	τ1τ2	τ1τ2	ADJ
ejpam-5634	174	2	-	-	ADJ
ejpam-5634	174	3	cl(a)−	cl(a)−	ADJ
ejpam-5634	174	4	τ1τ2	τ1τ2	NOUN
ejpam-5634	174	5	-	-	ADJ
ejpam-5634	174	6	int(a	int(a	NOUN
ejpam-5634	174	7	)	)	PUNCT
ejpam-5634	174	8	.	.	PUNCT
ejpam-5634	175	1	theorem	theorem	NOUN
ejpam-5634	175	2	5	5	NUM
ejpam-5634	175	3	.	.	PUNCT
ejpam-5634	176	1	the	the	DET
ejpam-5634	176	2	set	set	NOUN
ejpam-5634	176	3	of	of	ADP
ejpam-5634	176	4	all	all	DET
ejpam-5634	176	5	points	point	NOUN
ejpam-5634	176	6	x	x	PUNCT
ejpam-5634	176	7	of	of	ADP
ejpam-5634	176	8	x	x	SYM
ejpam-5634	176	9	at	at	ADP
ejpam-5634	176	10	which	which	PRON
ejpam-5634	176	11	a	a	DET
ejpam-5634	176	12	multifunction	multifunction	NOUN
ejpam-5634	177	1	f	f	NOUN
ejpam-5634	177	2	:	:	PUNCT
ejpam-5634	177	3	(	(	PUNCT
ejpam-5634	177	4	x	x	NOUN
ejpam-5634	177	5	,	,	PUNCT
ejpam-5634	177	6	τ1	τ1	NOUN
ejpam-5634	177	7	,	,	PUNCT
ejpam-5634	177	8	τ2	τ2	NOUN
ejpam-5634	177	9	)	)	PUNCT
ejpam-5634	177	10	→	→	SYM
ejpam-5634	177	11	(	(	PUNCT
ejpam-5634	177	12	y	y	PROPN
ejpam-5634	177	13	,	,	PUNCT
ejpam-5634	177	14	σ1	σ1	PROPN
ejpam-5634	177	15	,	,	PUNCT
ejpam-5634	177	16	σ2	σ2	PROPN
ejpam-5634	177	17	)	)	PUNCT
ejpam-5634	177	18	is	be	AUX
ejpam-5634	177	19	not	not	PART
ejpam-5634	177	20	upper	upper	ADJ
ejpam-5634	177	21	s-(τ1	s-(τ1	NOUN
ejpam-5634	177	22	,	,	PUNCT
ejpam-5634	177	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	177	24	is	be	AUX
ejpam-5634	177	25	identical	identical	ADJ
ejpam-5634	177	26	with	with	ADP
ejpam-5634	177	27	the	the	DET
ejpam-5634	177	28	union	union	NOUN
ejpam-5634	177	29	of	of	ADP
ejpam-5634	177	30	the	the	DET
ejpam-5634	177	31	τ1τ2	τ1τ2	NOUN
ejpam-5634	177	32	-	-	NOUN
ejpam-5634	177	33	frontier	frontier	NOUN
ejpam-5634	177	34	of	of	ADP
ejpam-5634	177	35	the	the	DET
ejpam-5634	177	36	upper	upper	ADJ
ejpam-5634	177	37	inverse	inverse	NOUN
ejpam-5634	177	38	images	image	NOUN
ejpam-5634	177	39	of	of	ADP
ejpam-5634	177	40	σ1σ2	σ1σ2	NOUN
ejpam-5634	177	41	-	-	PUNCT
ejpam-5634	177	42	open	open	ADJ
ejpam-5634	177	43	sets	set	NOUN
ejpam-5634	177	44	containing	contain	VERB
ejpam-5634	177	45	f	f	X
ejpam-5634	177	46	(	(	PUNCT
ejpam-5634	177	47	x	x	NOUN
ejpam-5634	177	48	)	)	PUNCT
ejpam-5634	177	49	and	and	CCONJ
ejpam-5634	177	50	having	have	VERB
ejpam-5634	177	51	σ1σ2	σ1σ2	NOUN
ejpam-5634	177	52	-	-	PUNCT
ejpam-5634	177	53	connected	connect	VERB
ejpam-5634	177	54	complement	complement	NOUN
ejpam-5634	177	55	.	.	PUNCT
ejpam-5634	178	1	proof	proof	NOUN
ejpam-5634	178	2	.	.	PUNCT
ejpam-5634	179	1	let	let	VERB
ejpam-5634	179	2	x	x	PRON
ejpam-5634	179	3	be	be	AUX
ejpam-5634	179	4	a	a	DET
ejpam-5634	179	5	point	point	NOUN
ejpam-5634	179	6	of	of	ADP
ejpam-5634	179	7	x	x	PUNCT
ejpam-5634	179	8	at	at	ADP
ejpam-5634	179	9	which	which	PRON
ejpam-5634	179	10	f	f	NOUN
ejpam-5634	179	11	is	be	AUX
ejpam-5634	179	12	not	not	PART
ejpam-5634	179	13	upper	upper	ADJ
ejpam-5634	179	14	s-(τ1	s-(τ1	NOUN
ejpam-5634	179	15	,	,	PUNCT
ejpam-5634	179	16	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5634	179	17	.	.	PUNCT
ejpam-5634	180	1	then	then	ADV
ejpam-5634	180	2	,	,	PUNCT
ejpam-5634	180	3	there	there	PRON
ejpam-5634	180	4	exists	exist	VERB
ejpam-5634	180	5	a	a	DET
ejpam-5634	180	6	σ1σ2	σ1σ2	NUM
ejpam-5634	180	7	-	-	ADJ
ejpam-5634	180	8	open	open	ADJ
ejpam-5634	180	9	set	set	NOUN
ejpam-5634	180	10	v	v	NOUN
ejpam-5634	180	11	of	of	ADP
ejpam-5634	180	12	y	y	PROPN
ejpam-5634	180	13	containing	contain	VERB
ejpam-5634	180	14	f	f	PROPN
ejpam-5634	180	15	(	(	PUNCT
ejpam-5634	180	16	x	x	NOUN
ejpam-5634	180	17	)	)	PUNCT
ejpam-5634	180	18	and	and	CCONJ
ejpam-5634	180	19	having	have	VERB
ejpam-5634	180	20	σ1σ2	σ1σ2	NOUN
ejpam-5634	180	21	-	-	PUNCT
ejpam-5634	180	22	connected	connected	ADJ
ejpam-5634	180	23	complement	complement	NOUN
ejpam-5634	180	24	such	such	ADJ
ejpam-5634	180	25	that	that	SCONJ
ejpam-5634	180	26	u	u	PROPN
ejpam-5634	180	27	∩	∩	NOUN
ejpam-5634	180	28	(	(	PUNCT
ejpam-5634	180	29	x	x	NOUN
ejpam-5634	180	30	−	−	PROPN
ejpam-5634	180	31	f+(v	f+(v	NOUN
ejpam-5634	180	32	)	)	PUNCT
ejpam-5634	180	33	)	)	PUNCT
ejpam-5634	181	1	̸=	̸=	NOUN
ejpam-5634	181	2	∅	∅	NOUN
ejpam-5634	181	3	for	for	ADP
ejpam-5634	181	4	every	every	DET
ejpam-5634	181	5	τ1τ2	τ1τ2	ADJ
ejpam-5634	181	6	-	-	ADJ
ejpam-5634	181	7	open	open	ADJ
ejpam-5634	181	8	set	set	ADJ
ejpam-5634	181	9	u	u	NOUN
ejpam-5634	181	10	of	of	ADP
ejpam-5634	181	11	x	x	SYM
ejpam-5634	181	12	containing	contain	VERB
ejpam-5634	181	13	x.	x.	NOUN
ejpam-5634	181	14	therefore	therefore	ADV
ejpam-5634	181	15	,	,	PUNCT
ejpam-5634	181	16	we	we	PRON
ejpam-5634	181	17	have	have	VERB
ejpam-5634	181	18	x	x	PART
ejpam-5634	181	19	∈	∈	PROPN
ejpam-5634	181	20	τ1τ2	τ1τ2	NOUN
ejpam-5634	181	21	-	-	NOUN
ejpam-5634	181	22	cl(x	cl(x	NUM
ejpam-5634	181	23	−	−	NOUN
ejpam-5634	181	24	f+(v	f+(v	NOUN
ejpam-5634	181	25	)	)	PUNCT
ejpam-5634	181	26	)	)	PUNCT
ejpam-5634	182	1	and	and	CCONJ
ejpam-5634	182	2	hence	hence	ADV
ejpam-5634	182	3	x	x	X
ejpam-5634	182	4	∈	∈	PRON
ejpam-5634	182	5	τ1τ2	τ1τ2	NOUN
ejpam-5634	182	6	-	-	ADJ
ejpam-5634	182	7	fr(f	fr(f	PUNCT
ejpam-5634	182	8	+	+	ADJ
ejpam-5634	182	9	(	(	PUNCT
ejpam-5634	182	10	v	v	NOUN
ejpam-5634	182	11	)	)	PUNCT
ejpam-5634	182	12	)	)	PUNCT
ejpam-5634	182	13	.	.	PUNCT
ejpam-5634	183	1	conversely	conversely	ADV
ejpam-5634	183	2	,	,	PUNCT
ejpam-5634	183	3	suppose	suppose	VERB
ejpam-5634	183	4	that	that	SCONJ
ejpam-5634	183	5	v	v	NOUN
ejpam-5634	183	6	is	be	AUX
ejpam-5634	183	7	a	a	DET
ejpam-5634	183	8	σ1σ2	σ1σ2	NOUN
ejpam-5634	183	9	-	-	ADJ
ejpam-5634	183	10	open	open	ADJ
ejpam-5634	183	11	set	set	NOUN
ejpam-5634	183	12	of	of	ADP
ejpam-5634	183	13	y	y	PROPN
ejpam-5634	183	14	containing	contain	VERB
ejpam-5634	183	15	f	f	PROPN
ejpam-5634	183	16	(	(	PUNCT
ejpam-5634	183	17	x	x	NOUN
ejpam-5634	183	18	)	)	PUNCT
ejpam-5634	183	19	and	and	CCONJ
ejpam-5634	183	20	having	having	AUX
ejpam-5634	183	21	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5634	183	22	complement	complement	VERB
ejpam-5634	183	23	such	such	ADJ
ejpam-5634	183	24	that	that	SCONJ
ejpam-5634	183	25	x	x	PUNCT
ejpam-5634	183	26	∈	∈	PRON
ejpam-5634	183	27	τ1τ2	τ1τ2	NOUN
ejpam-5634	183	28	-	-	ADJ
ejpam-5634	183	29	fr(f	fr(f	PUNCT
ejpam-5634	183	30	+	+	ADJ
ejpam-5634	183	31	(	(	PUNCT
ejpam-5634	183	32	v	v	NOUN
ejpam-5634	183	33	)	)	PUNCT
ejpam-5634	183	34	)	)	PUNCT
ejpam-5634	183	35	.	.	PUNCT
ejpam-5634	184	1	if	if	SCONJ
ejpam-5634	184	2	f	f	PROPN
ejpam-5634	184	3	is	be	AUX
ejpam-5634	184	4	upper	upper	ADJ
ejpam-5634	184	5	s-(τ1	s-(τ1	PROPN
ejpam-5634	184	6	,	,	PUNCT
ejpam-5634	184	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	184	8	at	at	ADP
ejpam-5634	184	9	x	x	X
ejpam-5634	184	10	∈	∈	PROPN
ejpam-5634	184	11	x	x	NOUN
ejpam-5634	184	12	,	,	PUNCT
ejpam-5634	184	13	then	then	ADV
ejpam-5634	184	14	there	there	PRON
ejpam-5634	184	15	exists	exist	VERB
ejpam-5634	184	16	a	a	DET
ejpam-5634	184	17	τ1τ2	τ1τ2	NOUN
ejpam-5634	184	18	-	-	ADJ
ejpam-5634	184	19	open	open	ADJ
ejpam-5634	184	20	set	set	ADJ
ejpam-5634	184	21	u	u	NOUN
ejpam-5634	184	22	of	of	ADP
ejpam-5634	184	23	x	x	PUNCT
ejpam-5634	184	24	containing	contain	VERB
ejpam-5634	184	25	x	x	PUNCT
ejpam-5634	184	26	such	such	ADJ
ejpam-5634	184	27	that	that	SCONJ
ejpam-5634	184	28	u	u	NOUN
ejpam-5634	184	29	⊆	⊆	NUM
ejpam-5634	184	30	f+(v	f+(v	NOUN
ejpam-5634	184	31	)	)	PUNCT
ejpam-5634	184	32	;	;	PUNCT
ejpam-5634	184	33	hence	hence	ADV
ejpam-5634	184	34	x	x	X
ejpam-5634	184	35	∈	∈	PRON
ejpam-5634	184	36	τ1τ2	τ1τ2	NOUN
ejpam-5634	184	37	-	-	NUM
ejpam-5634	184	38	int(f	int(f	VERB
ejpam-5634	184	39	+	+	ADJ
ejpam-5634	184	40	(	(	PUNCT
ejpam-5634	184	41	v	v	NOUN
ejpam-5634	184	42	)	)	PUNCT
ejpam-5634	184	43	)	)	PUNCT
ejpam-5634	184	44	.	.	PUNCT
ejpam-5634	185	1	this	this	PRON
ejpam-5634	185	2	is	be	AUX
ejpam-5634	185	3	a	a	DET
ejpam-5634	185	4	contradiction	contradiction	NOUN
ejpam-5634	186	1	and	and	CCONJ
ejpam-5634	186	2	so	so	ADV
ejpam-5634	186	3	f	f	PROPN
ejpam-5634	186	4	is	be	AUX
ejpam-5634	186	5	not	not	PART
ejpam-5634	186	6	upper	upper	ADJ
ejpam-5634	186	7	s-(τ1	s-(τ1	NOUN
ejpam-5634	186	8	,	,	PUNCT
ejpam-5634	186	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	186	10	at	at	ADP
ejpam-5634	186	11	x.	x.	NOUN
ejpam-5634	186	12	theorem	theorem	VERB
ejpam-5634	186	13	6	6	NUM
ejpam-5634	186	14	.	.	PUNCT
ejpam-5634	187	1	the	the	DET
ejpam-5634	187	2	set	set	NOUN
ejpam-5634	187	3	of	of	ADP
ejpam-5634	187	4	all	all	DET
ejpam-5634	187	5	points	point	NOUN
ejpam-5634	187	6	x	x	PUNCT
ejpam-5634	187	7	of	of	ADP
ejpam-5634	187	8	x	x	SYM
ejpam-5634	187	9	at	at	ADP
ejpam-5634	187	10	which	which	PRON
ejpam-5634	187	11	a	a	DET
ejpam-5634	187	12	multifunction	multifunction	NOUN
ejpam-5634	188	1	f	f	NOUN
ejpam-5634	188	2	:	:	PUNCT
ejpam-5634	188	3	(	(	PUNCT
ejpam-5634	188	4	x	x	NOUN
ejpam-5634	188	5	,	,	PUNCT
ejpam-5634	188	6	τ1	τ1	NOUN
ejpam-5634	188	7	,	,	PUNCT
ejpam-5634	188	8	τ2	τ2	NOUN
ejpam-5634	188	9	)	)	PUNCT
ejpam-5634	188	10	→	→	SYM
ejpam-5634	188	11	(	(	PUNCT
ejpam-5634	188	12	y	y	PROPN
ejpam-5634	188	13	,	,	PUNCT
ejpam-5634	188	14	σ1	σ1	PROPN
ejpam-5634	188	15	,	,	PUNCT
ejpam-5634	188	16	σ2	σ2	PROPN
ejpam-5634	188	17	)	)	PUNCT
ejpam-5634	188	18	is	be	AUX
ejpam-5634	188	19	not	not	PART
ejpam-5634	188	20	lower	low	ADJ
ejpam-5634	188	21	s-(τ1	s-(τ1	NOUN
ejpam-5634	188	22	,	,	PUNCT
ejpam-5634	188	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	188	24	is	be	AUX
ejpam-5634	188	25	identical	identical	ADJ
ejpam-5634	188	26	with	with	ADP
ejpam-5634	188	27	the	the	DET
ejpam-5634	188	28	union	union	NOUN
ejpam-5634	188	29	of	of	ADP
ejpam-5634	188	30	the	the	DET
ejpam-5634	188	31	τ1τ2	τ1τ2	NOUN
ejpam-5634	188	32	-	-	NOUN
ejpam-5634	188	33	frontier	frontier	NOUN
ejpam-5634	188	34	of	of	ADP
ejpam-5634	188	35	the	the	DET
ejpam-5634	188	36	lower	low	ADJ
ejpam-5634	188	37	inverse	inverse	NOUN
ejpam-5634	188	38	images	image	NOUN
ejpam-5634	188	39	of	of	ADP
ejpam-5634	188	40	σ1σ2	σ1σ2	NOUN
ejpam-5634	188	41	-	-	PUNCT
ejpam-5634	188	42	open	open	ADJ
ejpam-5634	188	43	sets	set	NOUN
ejpam-5634	188	44	meeting	meet	VERB
ejpam-5634	188	45	f	f	X
ejpam-5634	188	46	(	(	PUNCT
ejpam-5634	188	47	x	x	NOUN
ejpam-5634	188	48	)	)	PUNCT
ejpam-5634	188	49	and	and	CCONJ
ejpam-5634	188	50	having	have	VERB
ejpam-5634	188	51	σ1σ2	σ1σ2	NOUN
ejpam-5634	188	52	-	-	PUNCT
ejpam-5634	188	53	connected	connect	VERB
ejpam-5634	188	54	complement	complement	NOUN
ejpam-5634	188	55	.	.	PUNCT
ejpam-5634	189	1	proof	proof	NOUN
ejpam-5634	189	2	.	.	PUNCT
ejpam-5634	190	1	the	the	DET
ejpam-5634	190	2	proof	proof	NOUN
ejpam-5634	190	3	is	be	AUX
ejpam-5634	190	4	similar	similar	ADJ
ejpam-5634	190	5	to	to	ADP
ejpam-5634	190	6	that	that	PRON
ejpam-5634	190	7	of	of	ADP
ejpam-5634	190	8	theorem	theorem	NOUN
ejpam-5634	190	9	5	5	NUM
ejpam-5634	190	10	.	.	X
ejpam-5634	190	11	for	for	ADP
ejpam-5634	190	12	a	a	DET
ejpam-5634	190	13	multifunction	multifunction	NOUN
ejpam-5634	190	14	f	f	NOUN
ejpam-5634	190	15	:	:	PUNCT
ejpam-5634	190	16	(	(	PUNCT
ejpam-5634	190	17	x	x	NOUN
ejpam-5634	190	18	,	,	PUNCT
ejpam-5634	190	19	τ1	τ1	NOUN
ejpam-5634	190	20	,	,	PUNCT
ejpam-5634	190	21	τ2	τ2	NOUN
ejpam-5634	190	22	)	)	PUNCT
ejpam-5634	190	23	→	→	SYM
ejpam-5634	190	24	(	(	PUNCT
ejpam-5634	190	25	y	y	PROPN
ejpam-5634	190	26	,	,	PUNCT
ejpam-5634	190	27	σ1	σ1	PROPN
ejpam-5634	190	28	,	,	PUNCT
ejpam-5634	190	29	σ2	σ2	NOUN
ejpam-5634	190	30	)	)	PUNCT
ejpam-5634	190	31	,	,	PUNCT
ejpam-5634	190	32	the	the	DET
ejpam-5634	190	33	graph	graph	NOUN
ejpam-5634	190	34	g(f	g(f	PROPN
ejpam-5634	190	35	)	)	PUNCT
ejpam-5634	191	1	=	=	PUNCT
ejpam-5634	191	2	{	{	PUNCT
ejpam-5634	191	3	(	(	PUNCT
ejpam-5634	191	4	x	x	X
ejpam-5634	191	5	,	,	PUNCT
ejpam-5634	191	6	f	f	PROPN
ejpam-5634	191	7	(	(	PUNCT
ejpam-5634	191	8	x	x	NOUN
ejpam-5634	191	9	)	)	PUNCT
ejpam-5634	191	10	)	)	PUNCT
ejpam-5634	192	1	|	|	ADV
ejpam-5634	192	2	x	x	SYM
ejpam-5634	192	3	∈	∈	NOUN
ejpam-5634	192	4	x	x	X
ejpam-5634	192	5	}	}	PUNCT
ejpam-5634	192	6	is	be	AUX
ejpam-5634	192	7	said	say	VERB
ejpam-5634	192	8	to	to	PART
ejpam-5634	192	9	be	be	AUX
ejpam-5634	192	10	strongly	strongly	ADV
ejpam-5634	192	11	(	(	PUNCT
ejpam-5634	192	12	τ1	τ1	NOUN
ejpam-5634	192	13	,	,	PUNCT
ejpam-5634	192	14	τ2)-closed	τ2)-close	VERB
ejpam-5634	192	15	if	if	SCONJ
ejpam-5634	192	16	for	for	ADP
ejpam-5634	192	17	each	each	DET
ejpam-5634	192	18	(	(	PUNCT
ejpam-5634	192	19	x	x	NOUN
ejpam-5634	192	20	,	,	PUNCT
ejpam-5634	192	21	y	y	NOUN
ejpam-5634	192	22	)	)	PUNCT
ejpam-5634	192	23	∈	∈	PROPN
ejpam-5634	192	24	(	(	PUNCT
ejpam-5634	192	25	x	x	SYM
ejpam-5634	192	26	×	×	PROPN
ejpam-5634	192	27	y	y	PROPN
ejpam-5634	192	28	)	)	PUNCT
ejpam-5634	193	1	−	−	PROPN
ejpam-5634	193	2	g(f	g(f	PROPN
ejpam-5634	193	3	)	)	PUNCT
ejpam-5634	193	4	,	,	PUNCT
ejpam-5634	193	5	there	there	PRON
ejpam-5634	193	6	exists	exist	VERB
ejpam-5634	193	7	a	a	DET
ejpam-5634	193	8	τ1τ2	τ1τ2	NOUN
ejpam-5634	193	9	-	-	ADJ
ejpam-5634	193	10	open	open	ADJ
ejpam-5634	193	11	set	set	ADJ
ejpam-5634	193	12	u	u	NOUN
ejpam-5634	193	13	of	of	ADP
ejpam-5634	193	14	x	x	PUNCT
ejpam-5634	193	15	containing	contain	VERB
ejpam-5634	193	16	x	x	X
ejpam-5634	193	17	and	and	CCONJ
ejpam-5634	193	18	a	a	DET
ejpam-5634	193	19	σ1σ2	σ1σ2	NUM
ejpam-5634	193	20	-	-	ADJ
ejpam-5634	193	21	open	open	ADJ
ejpam-5634	193	22	set	set	NOUN
ejpam-5634	193	23	v	v	NOUN
ejpam-5634	193	24	of	of	ADP
ejpam-5634	193	25	y	y	PROPN
ejpam-5634	193	26	containing	contain	VERB
ejpam-5634	193	27	y	y	PRON
ejpam-5634	193	28	such	such	ADJ
ejpam-5634	193	29	that	that	SCONJ
ejpam-5634	194	1	[	[	X
ejpam-5634	194	2	u	u	X
ejpam-5634	194	3	×	×	NOUN
ejpam-5634	194	4	σ1σ2	σ1σ2	NOUN
ejpam-5634	194	5	-	-	NUM
ejpam-5634	194	6	cl(v	cl(v	NOUN
ejpam-5634	194	7	)	)	PUNCT
ejpam-5634	194	8	]	]	PUNCT
ejpam-5634	194	9	∩g(f	∩g(f	PROPN
ejpam-5634	194	10	)	)	PUNCT
ejpam-5634	195	1	=	=	PUNCT
ejpam-5634	195	2	∅.	∅.	PRON
ejpam-5634	195	3	lemma	lemma	PROPN
ejpam-5634	195	4	5	5	NUM
ejpam-5634	195	5	.	.	PUNCT
ejpam-5634	195	6	a	a	DET
ejpam-5634	195	7	multifunction	multifunction	NOUN
ejpam-5634	195	8	f	f	NOUN
ejpam-5634	195	9	:	:	PUNCT
ejpam-5634	195	10	(	(	PUNCT
ejpam-5634	195	11	x	x	NOUN
ejpam-5634	195	12	,	,	PUNCT
ejpam-5634	195	13	τ1	τ1	NOUN
ejpam-5634	195	14	,	,	PUNCT
ejpam-5634	195	15	τ2	τ2	NOUN
ejpam-5634	195	16	)	)	PUNCT
ejpam-5634	195	17	→	→	SYM
ejpam-5634	195	18	(	(	PUNCT
ejpam-5634	195	19	y	y	PROPN
ejpam-5634	195	20	,	,	PUNCT
ejpam-5634	195	21	σ1	σ1	PROPN
ejpam-5634	195	22	,	,	PUNCT
ejpam-5634	195	23	σ2	σ2	NOUN
ejpam-5634	195	24	)	)	PUNCT
ejpam-5634	195	25	has	have	VERB
ejpam-5634	195	26	a	a	DET
ejpam-5634	195	27	strongly	strongly	ADV
ejpam-5634	195	28	(	(	PUNCT
ejpam-5634	195	29	τ1	τ1	NOUN
ejpam-5634	195	30	,	,	PUNCT
ejpam-5634	195	31	τ2)-closed	τ2)-closed	ADJ
ejpam-5634	195	32	graph	graph	NOUN
ejpam-5634	195	33	if	if	SCONJ
ejpam-5634	195	34	and	and	CCONJ
ejpam-5634	195	35	only	only	ADV
ejpam-5634	195	36	if	if	SCONJ
ejpam-5634	195	37	for	for	ADP
ejpam-5634	195	38	each	each	DET
ejpam-5634	195	39	(	(	PUNCT
ejpam-5634	195	40	x	x	NOUN
ejpam-5634	195	41	,	,	PUNCT
ejpam-5634	195	42	y	y	NOUN
ejpam-5634	195	43	)	)	PUNCT
ejpam-5634	195	44	∈	∈	PROPN
ejpam-5634	195	45	(	(	PUNCT
ejpam-5634	195	46	x	x	SYM
ejpam-5634	195	47	×	×	PROPN
ejpam-5634	195	48	y	y	PROPN
ejpam-5634	195	49	)	)	PUNCT
ejpam-5634	195	50	−g(f	−g(f	PROPN
ejpam-5634	195	51	)	)	PUNCT
ejpam-5634	195	52	,	,	PUNCT
ejpam-5634	195	53	there	there	PRON
ejpam-5634	195	54	exists	exist	VERB
ejpam-5634	195	55	a	a	DET
ejpam-5634	195	56	τ1τ2	τ1τ2	NOUN
ejpam-5634	195	57	-	-	ADJ
ejpam-5634	195	58	open	open	ADJ
ejpam-5634	195	59	set	set	ADJ
ejpam-5634	195	60	u	u	NOUN
ejpam-5634	195	61	of	of	ADP
ejpam-5634	195	62	x	x	PUNCT
ejpam-5634	195	63	containing	contain	VERB
ejpam-5634	195	64	x	x	X
ejpam-5634	195	65	and	and	CCONJ
ejpam-5634	195	66	a	a	DET
ejpam-5634	195	67	σ1σ2	σ1σ2	NUM
ejpam-5634	195	68	-	-	ADJ
ejpam-5634	195	69	open	open	ADJ
ejpam-5634	195	70	set	set	NOUN
ejpam-5634	195	71	v	v	NOUN
ejpam-5634	195	72	of	of	ADP
ejpam-5634	195	73	y	y	PROPN
ejpam-5634	195	74	containing	contain	VERB
ejpam-5634	195	75	y	y	PRON
ejpam-5634	195	76	such	such	ADJ
ejpam-5634	195	77	that	that	SCONJ
ejpam-5634	195	78	f	f	PROPN
ejpam-5634	195	79	(	(	PUNCT
ejpam-5634	195	80	u	u	NOUN
ejpam-5634	195	81	)	)	PUNCT
ejpam-5634	195	82	∩	∩	NOUN
ejpam-5634	196	1	σ1σ2	σ1σ2	NOUN
ejpam-5634	196	2	-	-	NUM
ejpam-5634	196	3	cl(v	cl(v	X
ejpam-5634	196	4	)	)	PUNCT
ejpam-5634	196	5	=	=	PUNCT
ejpam-5634	196	6	∅.	∅.	PROPN
ejpam-5634	196	7	m.	m.	NOUN
ejpam-5634	196	8	chiangpradit	chiangpradit	NOUN
ejpam-5634	196	9	,	,	PUNCT
ejpam-5634	196	10	a.	a.	PROPN
ejpam-5634	196	11	sama	sama	PROPN
ejpam-5634	196	12	-	-	PUNCT
ejpam-5634	196	13	ae	ae	PROPN
ejpam-5634	196	14	,	,	PUNCT
ejpam-5634	196	15	c.	c.	PROPN
ejpam-5634	196	16	boonpok	boonpok	PROPN
ejpam-5634	196	17	/	/	SYM
ejpam-5634	196	18	eur	eur	PROPN
ejpam-5634	196	19	.	.	PUNCT
ejpam-5634	197	1	j.	j.	PROPN
ejpam-5634	197	2	pure	pure	PROPN
ejpam-5634	197	3	appl	appl	PROPN
ejpam-5634	197	4	.	.	PROPN
ejpam-5634	197	5	math	math	PROPN
ejpam-5634	197	6	,	,	PUNCT
ejpam-5634	197	7	18	18	NUM
ejpam-5634	197	8	(	(	PUNCT
ejpam-5634	197	9	1	1	NUM
ejpam-5634	197	10	)	)	PUNCT
ejpam-5634	197	11	(	(	PUNCT
ejpam-5634	197	12	2025	2025	NUM
ejpam-5634	197	13	)	)	PUNCT
ejpam-5634	197	14	,	,	PUNCT
ejpam-5634	197	15	5634	5634	NUM
ejpam-5634	197	16	8	8	NUM
ejpam-5634	197	17	of	of	ADP
ejpam-5634	197	18	12	12	NUM
ejpam-5634	197	19	proof	proof	NOUN
ejpam-5634	197	20	.	.	PUNCT
ejpam-5634	198	1	this	this	DET
ejpam-5634	198	2	proof	proof	NOUN
ejpam-5634	198	3	is	be	AUX
ejpam-5634	198	4	obvious	obvious	ADJ
ejpam-5634	198	5	.	.	PUNCT
ejpam-5634	199	1	recall	recall	VERB
ejpam-5634	199	2	that	that	SCONJ
ejpam-5634	199	3	a	a	DET
ejpam-5634	199	4	bitopological	bitopological	ADJ
ejpam-5634	199	5	space	space	NOUN
ejpam-5634	199	6	(	(	PUNCT
ejpam-5634	199	7	x	x	NOUN
ejpam-5634	199	8	,	,	PUNCT
ejpam-5634	199	9	τ1	τ1	NOUN
ejpam-5634	199	10	,	,	PUNCT
ejpam-5634	199	11	τ2	τ2	NOUN
ejpam-5634	199	12	)	)	PUNCT
ejpam-5634	199	13	is	be	AUX
ejpam-5634	199	14	said	say	VERB
ejpam-5634	199	15	to	to	PART
ejpam-5634	199	16	be	be	AUX
ejpam-5634	199	17	(	(	PUNCT
ejpam-5634	199	18	τ1	τ1	NOUN
ejpam-5634	199	19	,	,	PUNCT
ejpam-5634	200	1	τ2)-regular	τ2)-regular	ADJ
ejpam-5634	200	2	[	[	X
ejpam-5634	200	3	30	30	NUM
ejpam-5634	200	4	]	]	X
ejpam-5634	200	5	if	if	SCONJ
ejpam-5634	200	6	for	for	ADP
ejpam-5634	200	7	each	each	DET
ejpam-5634	200	8	τ1τ2	τ1τ2	ADJ
ejpam-5634	200	9	-	-	ADJ
ejpam-5634	200	10	closed	closed	ADJ
ejpam-5634	200	11	set	set	VERB
ejpam-5634	200	12	f	f	NOUN
ejpam-5634	200	13	and	and	CCONJ
ejpam-5634	200	14	each	each	DET
ejpam-5634	200	15	point	point	NOUN
ejpam-5634	200	16	x	x	X
ejpam-5634	200	17	∈	∈	NOUN
ejpam-5634	200	18	x	x	X
ejpam-5634	201	1	−	−	PROPN
ejpam-5634	201	2	f	f	NOUN
ejpam-5634	201	3	,	,	PUNCT
ejpam-5634	201	4	there	there	PRON
ejpam-5634	201	5	exist	exist	VERB
ejpam-5634	201	6	disjoint	disjoint	ADJ
ejpam-5634	201	7	τ1τ2	τ1τ2	ADJ
ejpam-5634	201	8	-	-	ADJ
ejpam-5634	201	9	open	open	ADJ
ejpam-5634	201	10	sets	set	NOUN
ejpam-5634	201	11	u	u	NOUN
ejpam-5634	201	12	and	and	CCONJ
ejpam-5634	201	13	v	v	ADP
ejpam-5634	201	14	such	such	ADJ
ejpam-5634	201	15	that	that	SCONJ
ejpam-5634	201	16	x	x	SYM
ejpam-5634	201	17	∈	∈	PROPN
ejpam-5634	201	18	u	u	NOUN
ejpam-5634	201	19	and	and	CCONJ
ejpam-5634	201	20	f	f	PROPN
ejpam-5634	201	21	⊆	⊆	NUM
ejpam-5634	201	22	v	v	NOUN
ejpam-5634	201	23	.	.	PUNCT
ejpam-5634	202	1	definition	definition	NOUN
ejpam-5634	202	2	5	5	NUM
ejpam-5634	202	3	.	.	PUNCT
ejpam-5634	203	1	a	a	DET
ejpam-5634	203	2	bitopological	bitopological	ADJ
ejpam-5634	203	3	space	space	NOUN
ejpam-5634	203	4	(	(	PUNCT
ejpam-5634	203	5	x	x	NOUN
ejpam-5634	203	6	,	,	PUNCT
ejpam-5634	203	7	τ1	τ1	NOUN
ejpam-5634	203	8	,	,	PUNCT
ejpam-5634	203	9	τ2	τ2	NOUN
ejpam-5634	203	10	)	)	PUNCT
ejpam-5634	203	11	is	be	AUX
ejpam-5634	203	12	said	say	VERB
ejpam-5634	203	13	to	to	PART
ejpam-5634	203	14	be	be	AUX
ejpam-5634	203	15	locally	locally	ADV
ejpam-5634	203	16	τ1τ2	τ1τ2	VERB
ejpam-5634	203	17	-	-	ADJ
ejpam-5634	203	18	connected	connected	ADJ
ejpam-5634	203	19	if	if	SCONJ
ejpam-5634	203	20	for	for	ADP
ejpam-5634	203	21	each	each	DET
ejpam-5634	203	22	x	x	SYM
ejpam-5634	203	23	∈	∈	PROPN
ejpam-5634	203	24	x	x	X
ejpam-5634	203	25	and	and	CCONJ
ejpam-5634	203	26	each	each	DET
ejpam-5634	203	27	τ1τ2	τ1τ2	ADJ
ejpam-5634	203	28	-	-	ADJ
ejpam-5634	203	29	open	open	ADJ
ejpam-5634	203	30	set	set	NOUN
ejpam-5634	203	31	g	g	NOUN
ejpam-5634	203	32	of	of	ADP
ejpam-5634	203	33	x	x	PUNCT
ejpam-5634	203	34	containing	contain	VERB
ejpam-5634	203	35	x	x	PRON
ejpam-5634	203	36	,	,	PUNCT
ejpam-5634	203	37	there	there	PRON
ejpam-5634	203	38	exists	exist	VERB
ejpam-5634	203	39	a	a	DET
ejpam-5634	203	40	τ1τ2	τ1τ2	NOUN
ejpam-5634	203	41	-	-	ADJ
ejpam-5634	203	42	open	open	ADJ
ejpam-5634	203	43	τ1τ2connected	τ1τ2connecte	VERB
ejpam-5634	203	44	set	set	VERB
ejpam-5634	203	45	v	v	ADP
ejpam-5634	203	46	such	such	ADJ
ejpam-5634	203	47	that	that	SCONJ
ejpam-5634	203	48	x	x	SYM
ejpam-5634	203	49	∈	∈	NOUN
ejpam-5634	203	50	v	v	ADP
ejpam-5634	203	51	⊆	⊆	NUM
ejpam-5634	203	52	g.	g.	NOUN
ejpam-5634	203	53	theorem	theorem	VERB
ejpam-5634	203	54	7	7	NUM
ejpam-5634	203	55	.	.	PUNCT
ejpam-5634	204	1	let	let	AUX
ejpam-5634	204	2	(	(	PUNCT
ejpam-5634	204	3	y	y	PROPN
ejpam-5634	204	4	,	,	PUNCT
ejpam-5634	204	5	σ1	σ1	PROPN
ejpam-5634	204	6	,	,	PUNCT
ejpam-5634	204	7	σ2	σ2	PROPN
ejpam-5634	204	8	)	)	PUNCT
ejpam-5634	204	9	be	be	VERB
ejpam-5634	204	10	a	a	DET
ejpam-5634	204	11	(	(	PUNCT
ejpam-5634	204	12	σ1	σ1	NOUN
ejpam-5634	204	13	,	,	PUNCT
ejpam-5634	204	14	σ2)-regular	σ2)-regular	ADJ
ejpam-5634	204	15	and	and	CCONJ
ejpam-5634	204	16	locally	locally	ADV
ejpam-5634	204	17	σ1σ2	σ1σ2	ADV
ejpam-5634	204	18	-	-	PUNCT
ejpam-5634	204	19	connected	connected	ADJ
ejpam-5634	204	20	space	space	NOUN
ejpam-5634	204	21	.	.	PUNCT
ejpam-5634	205	1	if	if	SCONJ
ejpam-5634	205	2	f	f	PROPN
ejpam-5634	205	3	:	:	PUNCT
ejpam-5634	205	4	(	(	PUNCT
ejpam-5634	205	5	x	x	NOUN
ejpam-5634	205	6	,	,	PUNCT
ejpam-5634	205	7	τ1	τ1	NOUN
ejpam-5634	205	8	,	,	PUNCT
ejpam-5634	205	9	τ2	τ2	NOUN
ejpam-5634	205	10	)	)	PUNCT
ejpam-5634	205	11	→	→	SYM
ejpam-5634	205	12	(	(	PUNCT
ejpam-5634	205	13	y	y	PROPN
ejpam-5634	205	14	,	,	PUNCT
ejpam-5634	205	15	σ1	σ1	PROPN
ejpam-5634	205	16	,	,	PUNCT
ejpam-5634	205	17	σ2	σ2	PROPN
ejpam-5634	205	18	)	)	PUNCT
ejpam-5634	205	19	is	be	AUX
ejpam-5634	205	20	an	an	DET
ejpam-5634	205	21	upper	upper	ADJ
ejpam-5634	205	22	s-(τ1	s-(τ1	PROPN
ejpam-5634	205	23	,	,	PUNCT
ejpam-5634	205	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	205	25	multifunction	multifunction	NOUN
ejpam-5634	205	26	such	such	ADJ
ejpam-5634	205	27	that	that	SCONJ
ejpam-5634	205	28	f	f	PROPN
ejpam-5634	205	29	(	(	PUNCT
ejpam-5634	205	30	x	x	X
ejpam-5634	205	31	)	)	PUNCT
ejpam-5634	205	32	is	be	AUX
ejpam-5634	205	33	σ1σ2	σ1σ2	NOUN
ejpam-5634	205	34	-	-	ADJ
ejpam-5634	205	35	closed	closed	ADJ
ejpam-5634	205	36	for	for	ADP
ejpam-5634	205	37	each	each	DET
ejpam-5634	205	38	x	x	SYM
ejpam-5634	205	39	∈	∈	PROPN
ejpam-5634	205	40	x	x	NOUN
ejpam-5634	205	41	,	,	PUNCT
ejpam-5634	205	42	then	then	ADV
ejpam-5634	205	43	g(f	g(f	PROPN
ejpam-5634	205	44	)	)	PUNCT
ejpam-5634	205	45	is	be	AUX
ejpam-5634	205	46	strongly	strongly	ADV
ejpam-5634	205	47	(	(	PUNCT
ejpam-5634	205	48	τ1	τ1	NOUN
ejpam-5634	205	49	,	,	PUNCT
ejpam-5634	205	50	τ2)-closed	τ2)-closed	ADJ
ejpam-5634	205	51	.	.	PUNCT
ejpam-5634	206	1	proof	proof	NOUN
ejpam-5634	206	2	.	.	PUNCT
ejpam-5634	207	1	let	let	VERB
ejpam-5634	207	2	(	(	PUNCT
ejpam-5634	207	3	x	x	NOUN
ejpam-5634	207	4	,	,	PUNCT
ejpam-5634	207	5	y	y	NOUN
ejpam-5634	207	6	)	)	PUNCT
ejpam-5634	207	7	∈	∈	PROPN
ejpam-5634	207	8	(	(	PUNCT
ejpam-5634	207	9	x	x	SYM
ejpam-5634	207	10	×	×	PROPN
ejpam-5634	207	11	y	y	PROPN
ejpam-5634	207	12	)	)	PUNCT
ejpam-5634	208	1	−	−	PROPN
ejpam-5634	208	2	g(f	g(f	PROPN
ejpam-5634	208	3	)	)	PUNCT
ejpam-5634	208	4	.	.	PUNCT
ejpam-5634	209	1	then	then	ADV
ejpam-5634	209	2	,	,	PUNCT
ejpam-5634	209	3	y	y	PROPN
ejpam-5634	209	4	∈	∈	PROPN
ejpam-5634	209	5	y	y	PROPN
ejpam-5634	210	1	−	−	PROPN
ejpam-5634	210	2	f	f	PROPN
ejpam-5634	210	3	(	(	PUNCT
ejpam-5634	210	4	x	x	NOUN
ejpam-5634	210	5	)	)	PUNCT
ejpam-5634	210	6	.	.	PUNCT
ejpam-5634	211	1	since	since	SCONJ
ejpam-5634	211	2	(	(	PUNCT
ejpam-5634	211	3	y	y	PROPN
ejpam-5634	211	4	,	,	PUNCT
ejpam-5634	211	5	σ1	σ1	PROPN
ejpam-5634	211	6	,	,	PUNCT
ejpam-5634	211	7	σ2	σ2	PROPN
ejpam-5634	211	8	)	)	PUNCT
ejpam-5634	211	9	is	be	AUX
ejpam-5634	211	10	(	(	PUNCT
ejpam-5634	211	11	σ1	σ1	NOUN
ejpam-5634	211	12	,	,	PUNCT
ejpam-5634	211	13	σ2)-regular	σ2)-regular	ADJ
ejpam-5634	211	14	,	,	PUNCT
ejpam-5634	211	15	there	there	PRON
ejpam-5634	211	16	exist	exist	VERB
ejpam-5634	211	17	disjoint	disjoint	ADJ
ejpam-5634	211	18	σ1σ2	σ1σ2	ADJ
ejpam-5634	211	19	-	-	ADJ
ejpam-5634	211	20	open	open	ADJ
ejpam-5634	211	21	sets	set	NOUN
ejpam-5634	211	22	v	v	ADP
ejpam-5634	211	23	and	and	CCONJ
ejpam-5634	211	24	v	v	ADP
ejpam-5634	211	25	′	′	NUM
ejpam-5634	211	26	of	of	ADP
ejpam-5634	211	27	y	y	PRON
ejpam-5634	211	28	such	such	ADJ
ejpam-5634	211	29	that	that	SCONJ
ejpam-5634	211	30	f	f	PROPN
ejpam-5634	211	31	(	(	PUNCT
ejpam-5634	211	32	x	x	X
ejpam-5634	211	33	)	)	PUNCT
ejpam-5634	211	34	⊆	⊆	NUM
ejpam-5634	211	35	v	v	NOUN
ejpam-5634	211	36	and	and	CCONJ
ejpam-5634	211	37	y	y	PROPN
ejpam-5634	211	38	∈	∈	PROPN
ejpam-5634	211	39	v	v	ADP
ejpam-5634	211	40	′.	′.	NOUN
ejpam-5634	211	41	moreover	moreover	ADV
ejpam-5634	211	42	,	,	PUNCT
ejpam-5634	211	43	since	since	SCONJ
ejpam-5634	211	44	(	(	PUNCT
ejpam-5634	211	45	y	y	PROPN
ejpam-5634	211	46	,	,	PUNCT
ejpam-5634	211	47	σ1	σ1	PROPN
ejpam-5634	211	48	,	,	PUNCT
ejpam-5634	211	49	σ2	σ2	PROPN
ejpam-5634	211	50	)	)	PUNCT
ejpam-5634	211	51	is	be	AUX
ejpam-5634	211	52	locally	locally	ADV
ejpam-5634	211	53	σ1σ2	σ1σ2	VERB
ejpam-5634	211	54	-	-	PUNCT
ejpam-5634	211	55	connected	connected	ADJ
ejpam-5634	211	56	,	,	PUNCT
ejpam-5634	211	57	there	there	PRON
ejpam-5634	211	58	exists	exist	VERB
ejpam-5634	211	59	a	a	DET
ejpam-5634	211	60	σ1σ2	σ1σ2	NUM
ejpam-5634	211	61	-	-	ADJ
ejpam-5634	211	62	open	open	ADJ
ejpam-5634	211	63	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5634	211	64	set	set	VERB
ejpam-5634	211	65	w	w	PROPN
ejpam-5634	211	66	of	of	ADP
ejpam-5634	211	67	y	y	PRON
ejpam-5634	211	68	such	such	ADJ
ejpam-5634	211	69	that	that	SCONJ
ejpam-5634	211	70	y	y	PROPN
ejpam-5634	211	71	∈	∈	PROPN
ejpam-5634	211	72	w	w	ADP
ejpam-5634	211	73	⊆	⊆	NUM
ejpam-5634	211	74	σ1σ2	σ1σ2	NOUN
ejpam-5634	211	75	-	-	PUNCT
ejpam-5634	211	76	cl(w	cl(w	NOUN
ejpam-5634	211	77	)	)	PUNCT
ejpam-5634	212	1	⊆	⊆	NUM
ejpam-5634	212	2	v	v	ADP
ejpam-5634	212	3	′.	′.	NOUN
ejpam-5634	212	4	since	since	SCONJ
ejpam-5634	212	5	f	f	PROPN
ejpam-5634	212	6	is	be	AUX
ejpam-5634	212	7	upper	upper	ADJ
ejpam-5634	212	8	s-(τ1	s-(τ1	PROPN
ejpam-5634	212	9	,	,	PUNCT
ejpam-5634	212	10	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	212	11	and	and	CCONJ
ejpam-5634	212	12	y	y	PROPN
ejpam-5634	212	13	−	−	PROPN
ejpam-5634	212	14	σ1σ2	σ1σ2	NOUN
ejpam-5634	212	15	-	-	PUNCT
ejpam-5634	212	16	cl(w	cl(w	NOUN
ejpam-5634	212	17	)	)	PUNCT
ejpam-5634	212	18	is	be	AUX
ejpam-5634	212	19	a	a	DET
ejpam-5634	212	20	σ1σ2	σ1σ2	NUM
ejpam-5634	212	21	-	-	ADJ
ejpam-5634	212	22	open	open	ADJ
ejpam-5634	212	23	set	set	NOUN
ejpam-5634	212	24	having	have	VERB
ejpam-5634	212	25	σ1σ2	σ1σ2	NOUN
ejpam-5634	212	26	-	-	PUNCT
ejpam-5634	212	27	connected	connect	VERB
ejpam-5634	212	28	complement	complement	NOUN
ejpam-5634	212	29	,	,	PUNCT
ejpam-5634	212	30	there	there	PRON
ejpam-5634	212	31	exists	exist	VERB
ejpam-5634	212	32	a	a	DET
ejpam-5634	212	33	τ1τ2	τ1τ2	NOUN
ejpam-5634	212	34	-	-	ADJ
ejpam-5634	212	35	open	open	ADJ
ejpam-5634	212	36	set	set	ADJ
ejpam-5634	212	37	u	u	NOUN
ejpam-5634	212	38	of	of	ADP
ejpam-5634	212	39	x	x	PUNCT
ejpam-5634	212	40	containing	contain	VERB
ejpam-5634	212	41	x	x	PUNCT
ejpam-5634	212	42	such	such	ADJ
ejpam-5634	212	43	that	that	SCONJ
ejpam-5634	212	44	f	f	PROPN
ejpam-5634	212	45	(	(	PUNCT
ejpam-5634	212	46	u	u	NOUN
ejpam-5634	212	47	)	)	PUNCT
ejpam-5634	212	48	⊆	⊆	NUM
ejpam-5634	212	49	y	y	PROPN
ejpam-5634	212	50	−σ1σ2	−σ1σ2	PROPN
ejpam-5634	212	51	-	-	PUNCT
ejpam-5634	212	52	cl(w	cl(w	NOUN
ejpam-5634	212	53	)	)	PUNCT
ejpam-5634	212	54	.	.	PUNCT
ejpam-5634	213	1	thus	thus	ADV
ejpam-5634	213	2	,	,	PUNCT
ejpam-5634	213	3	f	f	PROPN
ejpam-5634	213	4	(	(	PUNCT
ejpam-5634	213	5	u	u	NOUN
ejpam-5634	213	6	)	)	PUNCT
ejpam-5634	213	7	∩	∩	NOUN
ejpam-5634	213	8	σ1σ2	σ1σ2	NOUN
ejpam-5634	213	9	-	-	NUM
ejpam-5634	213	10	cl(w	cl(w	NOUN
ejpam-5634	213	11	)	)	PUNCT
ejpam-5634	213	12	=	=	NOUN
ejpam-5634	213	13	∅	∅	NOUN
ejpam-5634	213	14	and	and	CCONJ
ejpam-5634	213	15	by	by	ADP
ejpam-5634	213	16	lemma	lemma	PROPN
ejpam-5634	213	17	5	5	NUM
ejpam-5634	213	18	,	,	PUNCT
ejpam-5634	213	19	g(f	g(f	PROPN
ejpam-5634	213	20	)	)	PUNCT
ejpam-5634	213	21	is	be	AUX
ejpam-5634	213	22	strongly	strongly	ADV
ejpam-5634	213	23	(	(	PUNCT
ejpam-5634	213	24	τ1	τ1	NOUN
ejpam-5634	213	25	,	,	PUNCT
ejpam-5634	213	26	τ2)-closed	τ2)-closed	PROPN
ejpam-5634	213	27	.	.	PUNCT
ejpam-5634	214	1	definition	definition	NOUN
ejpam-5634	214	2	6	6	NUM
ejpam-5634	214	3	.	.	PUNCT
ejpam-5634	215	1	[	[	X
ejpam-5634	215	2	54	54	NUM
ejpam-5634	215	3	]	]	PUNCT
ejpam-5634	215	4	a	a	DET
ejpam-5634	215	5	multifunction	multifunction	NOUN
ejpam-5634	215	6	f	f	NOUN
ejpam-5634	215	7	:	:	PUNCT
ejpam-5634	215	8	(	(	PUNCT
ejpam-5634	215	9	x	x	NOUN
ejpam-5634	215	10	,	,	PUNCT
ejpam-5634	215	11	τ1	τ1	NOUN
ejpam-5634	215	12	,	,	PUNCT
ejpam-5634	215	13	τ2	τ2	NOUN
ejpam-5634	215	14	)	)	PUNCT
ejpam-5634	215	15	→	→	SYM
ejpam-5634	215	16	(	(	PUNCT
ejpam-5634	215	17	y	y	PROPN
ejpam-5634	215	18	,	,	PUNCT
ejpam-5634	215	19	σ1	σ1	PROPN
ejpam-5634	215	20	,	,	PUNCT
ejpam-5634	215	21	σ2	σ2	PROPN
ejpam-5634	215	22	)	)	PUNCT
ejpam-5634	215	23	is	be	AUX
ejpam-5634	215	24	said	say	VERB
ejpam-5634	215	25	to	to	PART
ejpam-5634	215	26	be	be	AUX
ejpam-5634	215	27	lower	low	ADJ
ejpam-5634	215	28	(	(	PUNCT
ejpam-5634	215	29	τ1	τ1	NOUN
ejpam-5634	215	30	,	,	PUNCT
ejpam-5634	215	31	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	215	32	if	if	SCONJ
ejpam-5634	215	33	for	for	ADP
ejpam-5634	215	34	each	each	DET
ejpam-5634	215	35	x	x	SYM
ejpam-5634	215	36	∈	∈	PROPN
ejpam-5634	215	37	x	x	X
ejpam-5634	215	38	and	and	CCONJ
ejpam-5634	215	39	each	each	DET
ejpam-5634	215	40	σ1σ2	σ1σ2	VERB
ejpam-5634	215	41	-	-	ADJ
ejpam-5634	215	42	open	open	ADJ
ejpam-5634	215	43	set	set	NOUN
ejpam-5634	215	44	v	v	NOUN
ejpam-5634	215	45	of	of	ADP
ejpam-5634	215	46	y	y	PRON
ejpam-5634	215	47	such	such	ADJ
ejpam-5634	215	48	that	that	SCONJ
ejpam-5634	215	49	f	f	PROPN
ejpam-5634	215	50	(	(	PUNCT
ejpam-5634	215	51	x	x	NOUN
ejpam-5634	215	52	)	)	PUNCT
ejpam-5634	215	53	∩	∩	NOUN
ejpam-5634	215	54	v	v	ADP
ejpam-5634	215	55	̸=	̸=	PROPN
ejpam-5634	215	56	∅	∅	NOUN
ejpam-5634	215	57	,	,	PUNCT
ejpam-5634	215	58	there	there	PRON
ejpam-5634	215	59	exists	exist	VERB
ejpam-5634	215	60	a	a	DET
ejpam-5634	215	61	τ1τ2	τ1τ2	NOUN
ejpam-5634	215	62	-	-	ADJ
ejpam-5634	215	63	open	open	ADJ
ejpam-5634	215	64	set	set	ADJ
ejpam-5634	215	65	u	u	NOUN
ejpam-5634	215	66	of	of	ADP
ejpam-5634	215	67	x	x	PUNCT
ejpam-5634	215	68	containing	contain	VERB
ejpam-5634	215	69	x	x	PUNCT
ejpam-5634	215	70	such	such	ADJ
ejpam-5634	215	71	that	that	SCONJ
ejpam-5634	215	72	f	f	PROPN
ejpam-5634	215	73	(	(	PUNCT
ejpam-5634	215	74	z	z	NOUN
ejpam-5634	215	75	)	)	PUNCT
ejpam-5634	215	76	∩	∩	NOUN
ejpam-5634	215	77	v	v	ADP
ejpam-5634	215	78	̸=	̸=	PROPN
ejpam-5634	215	79	∅	∅	NOUN
ejpam-5634	215	80	for	for	ADP
ejpam-5634	215	81	each	each	DET
ejpam-5634	215	82	z	z	NOUN
ejpam-5634	215	83	∈	∈	PROPN
ejpam-5634	215	84	u	u	PROPN
ejpam-5634	215	85	.	.	PUNCT
ejpam-5634	216	1	lemma	lemma	PROPN
ejpam-5634	216	2	6	6	NUM
ejpam-5634	216	3	.	.	PUNCT
ejpam-5634	217	1	[	[	X
ejpam-5634	217	2	54	54	NUM
ejpam-5634	217	3	]	]	PUNCT
ejpam-5634	217	4	for	for	ADP
ejpam-5634	217	5	a	a	DET
ejpam-5634	217	6	multifunction	multifunction	NOUN
ejpam-5634	217	7	f	f	NOUN
ejpam-5634	217	8	:	:	PUNCT
ejpam-5634	217	9	(	(	PUNCT
ejpam-5634	217	10	x	x	NOUN
ejpam-5634	217	11	,	,	PUNCT
ejpam-5634	217	12	τ1	τ1	NOUN
ejpam-5634	217	13	,	,	PUNCT
ejpam-5634	217	14	τ2	τ2	NOUN
ejpam-5634	217	15	)	)	PUNCT
ejpam-5634	217	16	→	→	SYM
ejpam-5634	217	17	(	(	PUNCT
ejpam-5634	217	18	y	y	PROPN
ejpam-5634	217	19	,	,	PUNCT
ejpam-5634	217	20	σ1	σ1	PROPN
ejpam-5634	217	21	,	,	PUNCT
ejpam-5634	217	22	σ2	σ2	NOUN
ejpam-5634	217	23	)	)	PUNCT
ejpam-5634	217	24	,	,	PUNCT
ejpam-5634	217	25	the	the	DET
ejpam-5634	217	26	following	follow	VERB
ejpam-5634	217	27	properties	property	NOUN
ejpam-5634	217	28	are	be	AUX
ejpam-5634	217	29	equivalent	equivalent	ADJ
ejpam-5634	217	30	:	:	PUNCT
ejpam-5634	217	31	(	(	PUNCT
ejpam-5634	217	32	1	1	X
ejpam-5634	217	33	)	)	PUNCT
ejpam-5634	217	34	f	f	PROPN
ejpam-5634	217	35	is	be	AUX
ejpam-5634	217	36	lower	low	ADJ
ejpam-5634	217	37	(	(	PUNCT
ejpam-5634	217	38	τ1	τ1	NOUN
ejpam-5634	217	39	,	,	PUNCT
ejpam-5634	217	40	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	217	41	;	;	PUNCT
ejpam-5634	217	42	(	(	PUNCT
ejpam-5634	217	43	2	2	X
ejpam-5634	217	44	)	)	PUNCT
ejpam-5634	217	45	f−(v	f−(v	NOUN
ejpam-5634	217	46	)	)	PUNCT
ejpam-5634	217	47	is	be	AUX
ejpam-5634	217	48	τ1τ2	τ1τ2	NOUN
ejpam-5634	217	49	-	-	ADJ
ejpam-5634	217	50	open	open	ADJ
ejpam-5634	217	51	in	in	ADP
ejpam-5634	217	52	x	x	PUNCT
ejpam-5634	217	53	for	for	ADP
ejpam-5634	217	54	every	every	DET
ejpam-5634	217	55	σ1σ2	σ1σ2	NOUN
ejpam-5634	217	56	-	-	ADJ
ejpam-5634	217	57	open	open	ADJ
ejpam-5634	217	58	set	set	NOUN
ejpam-5634	217	59	v	v	NOUN
ejpam-5634	217	60	of	of	ADP
ejpam-5634	217	61	y	y	PROPN
ejpam-5634	217	62	;	;	PUNCT
ejpam-5634	217	63	(	(	PUNCT
ejpam-5634	217	64	3	3	X
ejpam-5634	217	65	)	)	PUNCT
ejpam-5634	217	66	f+(k	f+(k	NOUN
ejpam-5634	217	67	)	)	PUNCT
ejpam-5634	217	68	is	be	AUX
ejpam-5634	217	69	τ1τ2	τ1τ2	NOUN
ejpam-5634	217	70	-	-	ADJ
ejpam-5634	217	71	closed	closed	ADJ
ejpam-5634	217	72	in	in	ADP
ejpam-5634	217	73	x	x	PUNCT
ejpam-5634	217	74	for	for	ADP
ejpam-5634	217	75	every	every	DET
ejpam-5634	217	76	σ1σ2	σ1σ2	NUM
ejpam-5634	217	77	-	-	PUNCT
ejpam-5634	217	78	closed	closed	ADJ
ejpam-5634	217	79	set	set	NOUN
ejpam-5634	217	80	k	k	PROPN
ejpam-5634	217	81	of	of	ADP
ejpam-5634	217	82	y	y	PROPN
ejpam-5634	217	83	;	;	PUNCT
ejpam-5634	217	84	(	(	PUNCT
ejpam-5634	217	85	4	4	X
ejpam-5634	217	86	)	)	PUNCT
ejpam-5634	217	87	τ1τ2	τ1τ2	NOUN
ejpam-5634	217	88	-	-	NOUN
ejpam-5634	217	89	cl(f	cl(f	NOUN
ejpam-5634	217	90	+	+	NOUN
ejpam-5634	217	91	(	(	PUNCT
ejpam-5634	217	92	b	b	NOUN
ejpam-5634	217	93	)	)	PUNCT
ejpam-5634	217	94	)	)	PUNCT
ejpam-5634	217	95	⊆	⊆	NUM
ejpam-5634	217	96	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	217	97	-	-	PUNCT
ejpam-5634	217	98	cl(b	cl(b	NOUN
ejpam-5634	217	99	)	)	PUNCT
ejpam-5634	217	100	)	)	PUNCT
ejpam-5634	217	101	for	for	ADP
ejpam-5634	217	102	every	every	DET
ejpam-5634	217	103	subset	subset	NOUN
ejpam-5634	217	104	b	b	PROPN
ejpam-5634	217	105	of	of	ADP
ejpam-5634	217	106	y	y	PROPN
ejpam-5634	217	107	;	;	PUNCT
ejpam-5634	217	108	(	(	PUNCT
ejpam-5634	217	109	5	5	X
ejpam-5634	217	110	)	)	PUNCT
ejpam-5634	217	111	f	f	NOUN
ejpam-5634	217	112	(	(	PUNCT
ejpam-5634	217	113	τ1τ2	τ1τ2	NOUN
ejpam-5634	217	114	-	-	NUM
ejpam-5634	217	115	cl(a	cl(a	NUM
ejpam-5634	217	116	)	)	PUNCT
ejpam-5634	217	117	)	)	PUNCT
ejpam-5634	218	1	⊆	⊆	X
ejpam-5634	218	2	σ1σ2	σ1σ2	X
ejpam-5634	218	3	-	-	NUM
ejpam-5634	218	4	cl(f	cl(f	NOUN
ejpam-5634	218	5	(	(	PUNCT
ejpam-5634	218	6	a	a	NOUN
ejpam-5634	218	7	)	)	PUNCT
ejpam-5634	218	8	)	)	PUNCT
ejpam-5634	218	9	for	for	ADP
ejpam-5634	218	10	every	every	DET
ejpam-5634	218	11	subset	subset	NOUN
ejpam-5634	218	12	a	a	PRON
ejpam-5634	218	13	of	of	ADP
ejpam-5634	218	14	x	x	PRON
ejpam-5634	218	15	;	;	PUNCT
ejpam-5634	218	16	(	(	PUNCT
ejpam-5634	218	17	6	6	X
ejpam-5634	218	18	)	)	PUNCT
ejpam-5634	218	19	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5634	218	20	-	-	PUNCT
ejpam-5634	218	21	int(b	int(b	NOUN
ejpam-5634	218	22	)	)	PUNCT
ejpam-5634	218	23	)	)	PUNCT
ejpam-5634	218	24	⊆	⊆	X
ejpam-5634	218	25	τ1τ2	τ1τ2	NOUN
ejpam-5634	218	26	-	-	NUM
ejpam-5634	218	27	int(f	int(f	VERB
ejpam-5634	218	28	−(b	−(b	NOUN
ejpam-5634	218	29	)	)	PUNCT
ejpam-5634	218	30	)	)	PUNCT
ejpam-5634	218	31	for	for	ADP
ejpam-5634	218	32	every	every	DET
ejpam-5634	218	33	subset	subset	NOUN
ejpam-5634	218	34	b	b	PROPN
ejpam-5634	218	35	of	of	ADP
ejpam-5634	218	36	y	y	PROPN
ejpam-5634	218	37	.	.	PUNCT
ejpam-5634	219	1	theorem	theorem	VERB
ejpam-5634	219	2	8	8	NUM
ejpam-5634	219	3	.	.	PUNCT
ejpam-5634	220	1	if	if	SCONJ
ejpam-5634	220	2	f	f	PROPN
ejpam-5634	220	3	:	:	PUNCT
ejpam-5634	220	4	(	(	PUNCT
ejpam-5634	220	5	x	x	NOUN
ejpam-5634	220	6	,	,	PUNCT
ejpam-5634	220	7	τ1	τ1	NOUN
ejpam-5634	220	8	,	,	PUNCT
ejpam-5634	220	9	τ2	τ2	NOUN
ejpam-5634	220	10	)	)	PUNCT
ejpam-5634	220	11	→	→	SYM
ejpam-5634	220	12	(	(	PUNCT
ejpam-5634	220	13	y	y	PROPN
ejpam-5634	220	14	,	,	PUNCT
ejpam-5634	220	15	σ1	σ1	PROPN
ejpam-5634	220	16	,	,	PUNCT
ejpam-5634	220	17	σ2	σ2	NOUN
ejpam-5634	220	18	)	)	PUNCT
ejpam-5634	220	19	is	be	AUX
ejpam-5634	220	20	lower	low	ADJ
ejpam-5634	220	21	s-(τ1	s-(τ1	NOUN
ejpam-5634	220	22	,	,	PUNCT
ejpam-5634	220	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	220	24	and	and	CCONJ
ejpam-5634	220	25	f	f	X
ejpam-5634	220	26	(	(	PUNCT
ejpam-5634	220	27	a	a	NOUN
ejpam-5634	220	28	)	)	PUNCT
ejpam-5634	220	29	is	be	AUX
ejpam-5634	220	30	σ1σ2	σ1σ2	NOUN
ejpam-5634	220	31	-	-	ADJ
ejpam-5634	220	32	connected	connected	ADJ
ejpam-5634	220	33	for	for	ADP
ejpam-5634	220	34	every	every	DET
ejpam-5634	220	35	subset	subset	NOUN
ejpam-5634	220	36	a	a	PRON
ejpam-5634	220	37	of	of	ADP
ejpam-5634	220	38	x	x	PRON
ejpam-5634	220	39	,	,	PUNCT
ejpam-5634	220	40	then	then	ADV
ejpam-5634	220	41	f	f	PROPN
ejpam-5634	220	42	is	be	AUX
ejpam-5634	220	43	lower	low	ADJ
ejpam-5634	220	44	(	(	PUNCT
ejpam-5634	220	45	τ1	τ1	NOUN
ejpam-5634	220	46	,	,	PUNCT
ejpam-5634	220	47	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	220	48	.	.	PUNCT
ejpam-5634	221	1	proof	proof	NOUN
ejpam-5634	221	2	.	.	PUNCT
ejpam-5634	222	1	let	let	VERB
ejpam-5634	222	2	a	a	DET
ejpam-5634	222	3	be	be	AUX
ejpam-5634	222	4	any	any	DET
ejpam-5634	222	5	subset	subset	NOUN
ejpam-5634	222	6	of	of	ADP
ejpam-5634	222	7	x.	x.	NOUN
ejpam-5634	222	8	since	since	SCONJ
ejpam-5634	222	9	σ1σ2	σ1σ2	NOUN
ejpam-5634	222	10	-	-	NOUN
ejpam-5634	222	11	cl(f	cl(f	NOUN
ejpam-5634	222	12	(	(	PUNCT
ejpam-5634	222	13	a	a	NOUN
ejpam-5634	222	14	)	)	PUNCT
ejpam-5634	222	15	)	)	PUNCT
ejpam-5634	222	16	is	be	AUX
ejpam-5634	222	17	σ1σ2	σ1σ2	NOUN
ejpam-5634	222	18	-	-	ADJ
ejpam-5634	222	19	closed	closed	ADJ
ejpam-5634	222	20	and	and	CCONJ
ejpam-5634	222	21	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5634	222	22	,	,	PUNCT
ejpam-5634	222	23	by	by	ADP
ejpam-5634	222	24	theorem	theorem	NOUN
ejpam-5634	222	25	2	2	NUM
ejpam-5634	222	26	we	we	PRON
ejpam-5634	222	27	have	have	VERB
ejpam-5634	222	28	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	222	29	-	-	SYM
ejpam-5634	222	30	cl(f	cl(f	NOUN
ejpam-5634	222	31	(	(	PUNCT
ejpam-5634	222	32	a	a	NOUN
ejpam-5634	222	33	)	)	PUNCT
ejpam-5634	222	34	)	)	PUNCT
ejpam-5634	222	35	)	)	PUNCT
ejpam-5634	223	1	=	=	PUNCT
ejpam-5634	223	2	τ1τ2	τ1τ2	NOUN
ejpam-5634	223	3	-	-	NOUN
ejpam-5634	223	4	cl(f	cl(f	NOUN
ejpam-5634	223	5	+	+	NOUN
ejpam-5634	223	6	(	(	PUNCT
ejpam-5634	223	7	σ1σ2	σ1σ2	NOUN
ejpam-5634	223	8	-	-	NUM
ejpam-5634	223	9	cl(f	cl(f	NOUN
ejpam-5634	223	10	(	(	PUNCT
ejpam-5634	223	11	a	a	NOUN
ejpam-5634	223	12	)	)	PUNCT
ejpam-5634	223	13	)	)	PUNCT
ejpam-5634	223	14	)	)	PUNCT
ejpam-5634	223	15	)	)	PUNCT
ejpam-5634	223	16	and	and	CCONJ
ejpam-5634	223	17	a	a	DET
ejpam-5634	223	18	⊆	⊆	NUM
ejpam-5634	223	19	f+(f	f+(f	NOUN
ejpam-5634	223	20	(	(	PUNCT
ejpam-5634	223	21	a	a	NOUN
ejpam-5634	223	22	)	)	PUNCT
ejpam-5634	223	23	)	)	PUNCT
ejpam-5634	223	24	⊆	⊆	NUM
ejpam-5634	223	25	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5634	223	26	-	-	SYM
ejpam-5634	223	27	cl(f	cl(f	NOUN
ejpam-5634	223	28	(	(	PUNCT
ejpam-5634	223	29	a	a	NOUN
ejpam-5634	223	30	)	)	PUNCT
ejpam-5634	223	31	)	)	PUNCT
ejpam-5634	223	32	)	)	PUNCT
ejpam-5634	223	33	.	.	PUNCT
ejpam-5634	224	1	thus	thus	ADV
ejpam-5634	224	2	,	,	PUNCT
ejpam-5634	224	3	f	f	X
ejpam-5634	224	4	(	(	PUNCT
ejpam-5634	224	5	σ1σ2	σ1σ2	X
ejpam-5634	224	6	-	-	NUM
ejpam-5634	224	7	cl(a	cl(a	NUM
ejpam-5634	224	8	)	)	PUNCT
ejpam-5634	224	9	)	)	PUNCT
ejpam-5634	224	10	⊆	⊆	X
ejpam-5634	224	11	σ1σ2	σ1σ2	X
ejpam-5634	224	12	-	-	NUM
ejpam-5634	224	13	cl(f	cl(f	NOUN
ejpam-5634	224	14	(	(	PUNCT
ejpam-5634	224	15	a	a	NOUN
ejpam-5634	224	16	)	)	PUNCT
ejpam-5634	224	17	)	)	PUNCT
ejpam-5634	224	18	.	.	PUNCT
ejpam-5634	225	1	it	it	PRON
ejpam-5634	225	2	follows	follow	VERB
ejpam-5634	225	3	from	from	ADP
ejpam-5634	225	4	lemma	lemma	PROPN
ejpam-5634	225	5	6	6	NUM
ejpam-5634	225	6	that	that	SCONJ
ejpam-5634	225	7	f	f	PROPN
ejpam-5634	225	8	is	be	AUX
ejpam-5634	225	9	lower	low	ADJ
ejpam-5634	225	10	(	(	PUNCT
ejpam-5634	225	11	τ1	τ1	NOUN
ejpam-5634	225	12	,	,	PUNCT
ejpam-5634	225	13	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5634	225	14	.	.	PUNCT
ejpam-5634	226	1	recall	recall	VERB
ejpam-5634	226	2	that	that	SCONJ
ejpam-5634	226	3	a	a	DET
ejpam-5634	226	4	bitopological	bitopological	ADJ
ejpam-5634	226	5	space	space	NOUN
ejpam-5634	226	6	(	(	PUNCT
ejpam-5634	226	7	x	x	NOUN
ejpam-5634	226	8	,	,	PUNCT
ejpam-5634	226	9	τ1	τ1	NOUN
ejpam-5634	226	10	,	,	PUNCT
ejpam-5634	226	11	τ2	τ2	NOUN
ejpam-5634	226	12	)	)	PUNCT
ejpam-5634	226	13	is	be	AUX
ejpam-5634	226	14	said	say	VERB
ejpam-5634	226	15	to	to	PART
ejpam-5634	226	16	be	be	AUX
ejpam-5634	226	17	τ1τ2	τ1τ2	NOUN
ejpam-5634	226	18	-	-	ADJ
ejpam-5634	226	19	connected	connected	ADJ
ejpam-5634	227	1	[	[	X
ejpam-5634	227	2	29	29	NUM
ejpam-5634	227	3	]	]	X
ejpam-5634	227	4	if	if	SCONJ
ejpam-5634	227	5	x	x	PRON
ejpam-5634	227	6	can	can	AUX
ejpam-5634	227	7	not	not	PART
ejpam-5634	227	8	be	be	AUX
ejpam-5634	227	9	written	write	VERB
ejpam-5634	227	10	as	as	ADP
ejpam-5634	227	11	the	the	DET
ejpam-5634	227	12	union	union	NOUN
ejpam-5634	227	13	of	of	ADP
ejpam-5634	227	14	two	two	NUM
ejpam-5634	227	15	disjoint	disjoint	NOUN
ejpam-5634	227	16	nonempty	nonempty	ADJ
ejpam-5634	227	17	τ1τ2	τ1τ2	ADJ
ejpam-5634	227	18	-	-	ADJ
ejpam-5634	227	19	open	open	ADJ
ejpam-5634	227	20	sets	set	NOUN
ejpam-5634	227	21	.	.	PUNCT
ejpam-5634	228	1	m.	m.	NOUN
ejpam-5634	228	2	chiangpradit	chiangpradit	PROPN
ejpam-5634	228	3	,	,	PUNCT
ejpam-5634	228	4	a.	a.	PROPN
ejpam-5634	228	5	sama	sama	PROPN
ejpam-5634	228	6	-	-	PUNCT
ejpam-5634	228	7	ae	ae	PROPN
ejpam-5634	228	8	,	,	PUNCT
ejpam-5634	228	9	c.	c.	PROPN
ejpam-5634	228	10	boonpok	boonpok	PROPN
ejpam-5634	228	11	/	/	SYM
ejpam-5634	228	12	eur	eur	PROPN
ejpam-5634	228	13	.	.	PUNCT
ejpam-5634	229	1	j.	j.	PROPN
ejpam-5634	229	2	pure	pure	PROPN
ejpam-5634	229	3	appl	appl	PROPN
ejpam-5634	229	4	.	.	PROPN
ejpam-5634	229	5	math	math	PROPN
ejpam-5634	229	6	,	,	PUNCT
ejpam-5634	229	7	18	18	NUM
ejpam-5634	229	8	(	(	PUNCT
ejpam-5634	229	9	1	1	NUM
ejpam-5634	229	10	)	)	PUNCT
ejpam-5634	229	11	(	(	PUNCT
ejpam-5634	229	12	2025	2025	NUM
ejpam-5634	229	13	)	)	PUNCT
ejpam-5634	229	14	,	,	PUNCT
ejpam-5634	229	15	5634	5634	NUM
ejpam-5634	229	16	9	9	NUM
ejpam-5634	229	17	of	of	ADP
ejpam-5634	229	18	12	12	NUM
ejpam-5634	229	19	definition	definition	NOUN
ejpam-5634	229	20	7	7	NUM
ejpam-5634	229	21	.	.	PUNCT
ejpam-5634	230	1	a	a	DET
ejpam-5634	230	2	bitopological	bitopological	ADJ
ejpam-5634	230	3	space	space	NOUN
ejpam-5634	230	4	(	(	PUNCT
ejpam-5634	230	5	x	x	NOUN
ejpam-5634	230	6	,	,	PUNCT
ejpam-5634	230	7	τ1	τ1	NOUN
ejpam-5634	230	8	,	,	PUNCT
ejpam-5634	230	9	τ2	τ2	NOUN
ejpam-5634	230	10	)	)	PUNCT
ejpam-5634	230	11	is	be	AUX
ejpam-5634	230	12	said	say	VERB
ejpam-5634	230	13	to	to	PART
ejpam-5634	230	14	be	be	AUX
ejpam-5634	230	15	s	s	NOUN
ejpam-5634	230	16	-	-	PUNCT
ejpam-5634	230	17	τ1τ2	τ1τ2	ADJ
ejpam-5634	230	18	-	-	ADJ
ejpam-5634	230	19	connected	connected	ADJ
ejpam-5634	230	20	if	if	SCONJ
ejpam-5634	230	21	x	x	PRON
ejpam-5634	230	22	can	can	AUX
ejpam-5634	230	23	not	not	PART
ejpam-5634	230	24	be	be	AUX
ejpam-5634	230	25	written	write	VERB
ejpam-5634	230	26	as	as	ADP
ejpam-5634	230	27	the	the	DET
ejpam-5634	230	28	union	union	NOUN
ejpam-5634	230	29	of	of	ADP
ejpam-5634	230	30	two	two	NUM
ejpam-5634	230	31	disjoint	disjoint	NOUN
ejpam-5634	230	32	nonempty	nonempty	ADJ
ejpam-5634	230	33	τ1τ2	τ1τ2	ADJ
ejpam-5634	230	34	-	-	ADJ
ejpam-5634	230	35	open	open	ADJ
ejpam-5634	230	36	sets	set	NOUN
ejpam-5634	230	37	having	have	VERB
ejpam-5634	230	38	τ1τ2	τ1τ2	ADV
ejpam-5634	230	39	-	-	ADJ
ejpam-5634	230	40	connected	connect	VERB
ejpam-5634	230	41	complement	complement	NOUN
ejpam-5634	230	42	.	.	PUNCT
ejpam-5634	231	1	theorem	theorem	VERB
ejpam-5634	231	2	9	9	NUM
ejpam-5634	231	3	.	.	PUNCT
ejpam-5634	232	1	if	if	SCONJ
ejpam-5634	232	2	f	f	PROPN
ejpam-5634	232	3	:	:	PUNCT
ejpam-5634	232	4	(	(	PUNCT
ejpam-5634	232	5	x	x	NOUN
ejpam-5634	232	6	,	,	PUNCT
ejpam-5634	232	7	τ1	τ1	NOUN
ejpam-5634	232	8	,	,	PUNCT
ejpam-5634	232	9	τ2	τ2	NOUN
ejpam-5634	232	10	)	)	PUNCT
ejpam-5634	232	11	→	→	SYM
ejpam-5634	232	12	(	(	PUNCT
ejpam-5634	232	13	y	y	PROPN
ejpam-5634	232	14	,	,	PUNCT
ejpam-5634	232	15	σ1	σ1	PROPN
ejpam-5634	232	16	,	,	PUNCT
ejpam-5634	232	17	σ2	σ2	PROPN
ejpam-5634	232	18	)	)	PUNCT
ejpam-5634	232	19	is	be	AUX
ejpam-5634	232	20	an	an	DET
ejpam-5634	232	21	upper	upper	ADJ
ejpam-5634	232	22	or	or	CCONJ
ejpam-5634	232	23	lower	low	ADJ
ejpam-5634	232	24	s-(τ1	s-(τ1	NOUN
ejpam-5634	232	25	,	,	PUNCT
ejpam-5634	232	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	232	27	surjective	surjective	ADJ
ejpam-5634	232	28	multifunction	multifunction	NOUN
ejpam-5634	232	29	such	such	ADJ
ejpam-5634	232	30	that	that	SCONJ
ejpam-5634	232	31	f	f	PROPN
ejpam-5634	232	32	(	(	PUNCT
ejpam-5634	232	33	x	x	X
ejpam-5634	232	34	)	)	PUNCT
ejpam-5634	232	35	is	be	AUX
ejpam-5634	232	36	σ1σ2	σ1σ2	NOUN
ejpam-5634	232	37	-	-	PUNCT
ejpam-5634	232	38	connected	connected	ADJ
ejpam-5634	232	39	for	for	ADP
ejpam-5634	232	40	each	each	DET
ejpam-5634	232	41	x	x	SYM
ejpam-5634	232	42	∈	∈	PROPN
ejpam-5634	232	43	x	x	X
ejpam-5634	232	44	and	and	CCONJ
ejpam-5634	232	45	(	(	PUNCT
ejpam-5634	232	46	x	x	NOUN
ejpam-5634	232	47	,	,	PUNCT
ejpam-5634	232	48	τ1	τ1	NOUN
ejpam-5634	232	49	,	,	PUNCT
ejpam-5634	232	50	τ2	τ2	NOUN
ejpam-5634	232	51	)	)	PUNCT
ejpam-5634	232	52	is	be	AUX
ejpam-5634	232	53	τ1τ2	τ1τ2	NOUN
ejpam-5634	232	54	-	-	ADJ
ejpam-5634	232	55	connected	connected	ADJ
ejpam-5634	232	56	,	,	PUNCT
ejpam-5634	232	57	then	then	ADV
ejpam-5634	232	58	(	(	PUNCT
ejpam-5634	232	59	y	y	PROPN
ejpam-5634	232	60	,	,	PUNCT
ejpam-5634	232	61	σ1	σ1	PROPN
ejpam-5634	232	62	,	,	PUNCT
ejpam-5634	232	63	σ2	σ2	PROPN
ejpam-5634	232	64	)	)	PUNCT
ejpam-5634	232	65	is	be	AUX
ejpam-5634	232	66	s	s	NOUN
ejpam-5634	232	67	-	-	PUNCT
ejpam-5634	232	68	σ1σ2	σ1σ2	VERB
ejpam-5634	232	69	-	-	PUNCT
ejpam-5634	232	70	connected	connect	VERB
ejpam-5634	232	71	.	.	PUNCT
ejpam-5634	233	1	proof	proof	NOUN
ejpam-5634	233	2	.	.	PUNCT
ejpam-5634	234	1	suppose	suppose	VERB
ejpam-5634	234	2	that	that	SCONJ
ejpam-5634	234	3	(	(	PUNCT
ejpam-5634	234	4	y	y	PROPN
ejpam-5634	234	5	,	,	PUNCT
ejpam-5634	234	6	σ1	σ1	PROPN
ejpam-5634	234	7	,	,	PUNCT
ejpam-5634	234	8	σ2	σ2	PROPN
ejpam-5634	234	9	)	)	PUNCT
ejpam-5634	234	10	is	be	AUX
ejpam-5634	234	11	not	not	PART
ejpam-5634	234	12	s	s	NOUN
ejpam-5634	234	13	-	-	PUNCT
ejpam-5634	234	14	σ1σ2	σ1σ2	VERB
ejpam-5634	234	15	-	-	PUNCT
ejpam-5634	234	16	connected	connected	ADJ
ejpam-5634	234	17	.	.	PUNCT
ejpam-5634	235	1	there	there	PRON
ejpam-5634	235	2	exist	exist	VERB
ejpam-5634	235	3	nonempty	nonempty	ADJ
ejpam-5634	235	4	σ1σ2open	σ1σ2open	PUNCT
ejpam-5634	235	5	sets	set	VERB
ejpam-5634	235	6	u	u	NOUN
ejpam-5634	235	7	and	and	CCONJ
ejpam-5634	235	8	v	v	NOUN
ejpam-5634	235	9	of	of	ADP
ejpam-5634	235	10	y	y	PROPN
ejpam-5634	235	11	having	have	VERB
ejpam-5634	235	12	σ1σ2	σ1σ2	ADV
ejpam-5634	235	13	-	-	PUNCT
ejpam-5634	235	14	connected	connected	ADJ
ejpam-5634	235	15	complement	complement	NOUN
ejpam-5634	235	16	such	such	ADJ
ejpam-5634	235	17	that	that	SCONJ
ejpam-5634	235	18	u	u	PROPN
ejpam-5634	235	19	∩	∩	NOUN
ejpam-5634	235	20	v	v	NOUN
ejpam-5634	235	21	=	=	NOUN
ejpam-5634	235	22	∅	∅	NOUN
ejpam-5634	235	23	and	and	CCONJ
ejpam-5634	235	24	u	u	NOUN
ejpam-5634	235	25	∪	∪	NOUN
ejpam-5634	235	26	v	v	ADP
ejpam-5634	235	27	=	=	SYM
ejpam-5634	235	28	y	y	PROPN
ejpam-5634	235	29	.	.	PUNCT
ejpam-5634	236	1	since	since	SCONJ
ejpam-5634	236	2	f	f	PROPN
ejpam-5634	236	3	(	(	PUNCT
ejpam-5634	236	4	x	x	X
ejpam-5634	236	5	)	)	PUNCT
ejpam-5634	236	6	is	be	AUX
ejpam-5634	236	7	σ1σ2	σ1σ2	NOUN
ejpam-5634	236	8	-	-	PUNCT
ejpam-5634	236	9	connected	connected	ADJ
ejpam-5634	236	10	for	for	ADP
ejpam-5634	236	11	each	each	DET
ejpam-5634	236	12	x	x	SYM
ejpam-5634	236	13	∈	∈	PROPN
ejpam-5634	236	14	x	x	NOUN
ejpam-5634	236	15	,	,	PUNCT
ejpam-5634	236	16	either	either	CCONJ
ejpam-5634	236	17	f	f	PROPN
ejpam-5634	236	18	(	(	PUNCT
ejpam-5634	236	19	x	x	X
ejpam-5634	236	20	)	)	PUNCT
ejpam-5634	236	21	⊆	⊆	NUM
ejpam-5634	236	22	u	u	NOUN
ejpam-5634	236	23	or	or	CCONJ
ejpam-5634	236	24	f	f	PROPN
ejpam-5634	236	25	(	(	PUNCT
ejpam-5634	236	26	x	x	NOUN
ejpam-5634	236	27	)	)	PUNCT
ejpam-5634	236	28	⊆	⊆	NUM
ejpam-5634	236	29	v	v	NOUN
ejpam-5634	236	30	.	.	PUNCT
ejpam-5634	237	1	if	if	SCONJ
ejpam-5634	237	2	x	x	SYM
ejpam-5634	237	3	∈	∈	PROPN
ejpam-5634	237	4	f+(u∪v	f+(u∪v	PROPN
ejpam-5634	237	5	)	)	PUNCT
ejpam-5634	237	6	,	,	PUNCT
ejpam-5634	237	7	then	then	ADV
ejpam-5634	237	8	f	f	X
ejpam-5634	237	9	(	(	PUNCT
ejpam-5634	237	10	x	x	X
ejpam-5634	237	11	)	)	PUNCT
ejpam-5634	237	12	⊆	⊆	NUM
ejpam-5634	237	13	u∪v	u∪v	NOUN
ejpam-5634	237	14	and	and	CCONJ
ejpam-5634	237	15	hence	hence	ADV
ejpam-5634	237	16	x	x	PART
ejpam-5634	237	17	∈	∈	NOUN
ejpam-5634	237	18	f+(u)∪f+(v	f+(u)∪f+(v	NOUN
ejpam-5634	237	19	)	)	PUNCT
ejpam-5634	237	20	.	.	PUNCT
ejpam-5634	238	1	moreover	moreover	ADV
ejpam-5634	238	2	,	,	PUNCT
ejpam-5634	238	3	since	since	SCONJ
ejpam-5634	238	4	f	f	PROPN
ejpam-5634	238	5	is	be	AUX
ejpam-5634	238	6	surjective	surjective	ADJ
ejpam-5634	238	7	,	,	PUNCT
ejpam-5634	238	8	there	there	PRON
ejpam-5634	238	9	exist	exist	VERB
ejpam-5634	238	10	x	x	PUNCT
ejpam-5634	238	11	and	and	CCONJ
ejpam-5634	238	12	y	y	PROPN
ejpam-5634	238	13	in	in	ADP
ejpam-5634	238	14	x	x	PUNCT
ejpam-5634	238	15	such	such	ADJ
ejpam-5634	238	16	that	that	SCONJ
ejpam-5634	238	17	f	f	PROPN
ejpam-5634	238	18	(	(	PUNCT
ejpam-5634	238	19	x	x	X
ejpam-5634	238	20	)	)	PUNCT
ejpam-5634	238	21	⊆	⊆	NUM
ejpam-5634	238	22	u	u	NOUN
ejpam-5634	238	23	and	and	CCONJ
ejpam-5634	238	24	f	f	PROPN
ejpam-5634	238	25	(	(	PUNCT
ejpam-5634	238	26	y	y	PROPN
ejpam-5634	238	27	)	)	PUNCT
ejpam-5634	238	28	⊆	⊆	NUM
ejpam-5634	238	29	v	v	NOUN
ejpam-5634	238	30	;	;	PUNCT
ejpam-5634	238	31	hence	hence	ADV
ejpam-5634	238	32	x	x	SYM
ejpam-5634	238	33	∈	∈	PROPN
ejpam-5634	238	34	f+(u	f+(u	NUM
ejpam-5634	238	35	)	)	PUNCT
ejpam-5634	238	36	and	and	CCONJ
ejpam-5634	238	37	y	y	PROPN
ejpam-5634	238	38	∈	∈	PROPN
ejpam-5634	238	39	f+(v	f+(v	PROPN
ejpam-5634	238	40	)	)	PUNCT
ejpam-5634	238	41	.	.	PUNCT
ejpam-5634	239	1	therefore	therefore	ADV
ejpam-5634	239	2	,	,	PUNCT
ejpam-5634	239	3	we	we	PRON
ejpam-5634	239	4	obtain	obtain	VERB
ejpam-5634	239	5	the	the	DET
ejpam-5634	239	6	following	following	NOUN
ejpam-5634	239	7	:	:	PUNCT
ejpam-5634	239	8	(	(	PUNCT
ejpam-5634	239	9	1	1	X
ejpam-5634	239	10	)	)	PUNCT
ejpam-5634	239	11	f+(u	f+(u	NUM
ejpam-5634	239	12	)	)	PUNCT
ejpam-5634	239	13	∪	∪	ADP
ejpam-5634	239	14	f+(v	f+(v	NOUN
ejpam-5634	239	15	)	)	PUNCT
ejpam-5634	240	1	=	=	PUNCT
ejpam-5634	241	1	f+(u	f+(u	PUNCT
ejpam-5634	241	2	∪	∪	ADP
ejpam-5634	241	3	v	v	NOUN
ejpam-5634	241	4	)	)	PUNCT
ejpam-5634	241	5	=	=	SYM
ejpam-5634	242	1	x	x	X
ejpam-5634	242	2	;	;	PUNCT
ejpam-5634	242	3	(	(	PUNCT
ejpam-5634	242	4	2	2	X
ejpam-5634	242	5	)	)	PUNCT
ejpam-5634	242	6	f+(u	f+(u	NUM
ejpam-5634	242	7	)	)	PUNCT
ejpam-5634	242	8	∩	∩	NOUN
ejpam-5634	242	9	f+(v	f+(v	NOUN
ejpam-5634	242	10	)	)	PUNCT
ejpam-5634	242	11	=	=	SYM
ejpam-5634	243	1	f+(u	f+(u	NUM
ejpam-5634	243	2	∩	∩	NOUN
ejpam-5634	243	3	v	v	NOUN
ejpam-5634	243	4	)	)	PUNCT
ejpam-5634	243	5	=	=	NOUN
ejpam-5634	243	6	∅	∅	NOUN
ejpam-5634	243	7	;	;	PUNCT
ejpam-5634	243	8	(	(	PUNCT
ejpam-5634	243	9	3	3	X
ejpam-5634	243	10	)	)	PUNCT
ejpam-5634	243	11	f+(u	f+(u	NUM
ejpam-5634	243	12	)	)	PUNCT
ejpam-5634	243	13	̸=	̸=	PROPN
ejpam-5634	243	14	∅	∅	NOUN
ejpam-5634	243	15	and	and	CCONJ
ejpam-5634	243	16	f+(v	f+(v	NUM
ejpam-5634	243	17	)	)	PUNCT
ejpam-5634	244	1	̸=	̸=	PROPN
ejpam-5634	244	2	∅.	∅.	ADP
ejpam-5634	244	3	next	next	ADV
ejpam-5634	244	4	,	,	PUNCT
ejpam-5634	244	5	we	we	PRON
ejpam-5634	244	6	shall	shall	AUX
ejpam-5634	244	7	show	show	VERB
ejpam-5634	244	8	that	that	PRON
ejpam-5634	244	9	f+(u	f+(u	NUM
ejpam-5634	244	10	)	)	PUNCT
ejpam-5634	244	11	and	and	CCONJ
ejpam-5634	244	12	f+(v	f+(v	NUM
ejpam-5634	244	13	)	)	PUNCT
ejpam-5634	244	14	are	be	AUX
ejpam-5634	244	15	τ1τ2	τ1τ2	NOUN
ejpam-5634	244	16	-	-	ADJ
ejpam-5634	244	17	open	open	ADJ
ejpam-5634	244	18	in	in	ADP
ejpam-5634	244	19	x.	x.	PROPN
ejpam-5634	244	20	(	(	PUNCT
ejpam-5634	244	21	i	i	NOUN
ejpam-5634	244	22	)	)	PUNCT
ejpam-5634	244	23	let	let	VERB
ejpam-5634	244	24	f	f	PRON
ejpam-5634	244	25	be	be	AUX
ejpam-5634	244	26	upper	upper	ADJ
ejpam-5634	244	27	s(τ1	s(τ1	NOUN
ejpam-5634	244	28	,	,	PUNCT
ejpam-5634	244	29	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	244	30	.	.	PUNCT
ejpam-5634	245	1	by	by	ADP
ejpam-5634	245	2	theorem	theorem	NOUN
ejpam-5634	245	3	1	1	NUM
ejpam-5634	245	4	,	,	PUNCT
ejpam-5634	245	5	f+(u	f+(u	NUM
ejpam-5634	245	6	)	)	PUNCT
ejpam-5634	245	7	and	and	CCONJ
ejpam-5634	245	8	f+(v	f+(v	NUM
ejpam-5634	245	9	)	)	PUNCT
ejpam-5634	245	10	are	be	AUX
ejpam-5634	245	11	τ1τ2	τ1τ2	NOUN
ejpam-5634	245	12	-	-	ADJ
ejpam-5634	245	13	open	open	ADJ
ejpam-5634	245	14	in	in	ADP
ejpam-5634	245	15	x.	x.	PROPN
ejpam-5634	245	16	(	(	PUNCT
ejpam-5634	245	17	ii	ii	NOUN
ejpam-5634	245	18	)	)	PUNCT
ejpam-5634	245	19	let	let	VERB
ejpam-5634	245	20	f	f	PRON
ejpam-5634	245	21	be	be	AUX
ejpam-5634	245	22	lower	low	ADJ
ejpam-5634	245	23	s-(τ1	s-(τ1	NOUN
ejpam-5634	245	24	,	,	PUNCT
ejpam-5634	245	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	245	26	.	.	PUNCT
ejpam-5634	246	1	since	since	SCONJ
ejpam-5634	246	2	v	v	NOUN
ejpam-5634	246	3	is	be	AUX
ejpam-5634	246	4	a	a	DET
ejpam-5634	246	5	σ1σ2	σ1σ2	NUM
ejpam-5634	246	6	-	-	PUNCT
ejpam-5634	246	7	clopen	clopen	ADJ
ejpam-5634	246	8	set	set	NOUN
ejpam-5634	246	9	with	with	ADP
ejpam-5634	246	10	σ1σ2	σ1σ2	ADV
ejpam-5634	246	11	-	-	PUNCT
ejpam-5634	246	12	connected	connect	VERB
ejpam-5634	246	13	complement	complement	NOUN
ejpam-5634	246	14	,	,	PUNCT
ejpam-5634	246	15	by	by	ADP
ejpam-5634	246	16	theorem	theorem	NOUN
ejpam-5634	246	17	2	2	NUM
ejpam-5634	246	18	,	,	PUNCT
ejpam-5634	246	19	f+(v	f+(v	PROPN
ejpam-5634	246	20	)	)	PUNCT
ejpam-5634	246	21	is	be	AUX
ejpam-5634	246	22	τ1τ2	τ1τ2	NOUN
ejpam-5634	246	23	-	-	ADJ
ejpam-5634	246	24	closed	closed	ADJ
ejpam-5634	246	25	in	in	ADP
ejpam-5634	246	26	x.	x.	NOUN
ejpam-5634	246	27	therefore	therefore	ADV
ejpam-5634	246	28	,	,	PUNCT
ejpam-5634	246	29	f+(u	f+(u	NUM
ejpam-5634	246	30	)	)	PUNCT
ejpam-5634	246	31	is	be	AUX
ejpam-5634	246	32	τ1τ2	τ1τ2	NOUN
ejpam-5634	246	33	-	-	ADJ
ejpam-5634	246	34	open	open	ADJ
ejpam-5634	246	35	in	in	ADP
ejpam-5634	246	36	x.	x.	NOUN
ejpam-5634	246	37	similarly	similarly	ADV
ejpam-5634	246	38	,	,	PUNCT
ejpam-5634	246	39	we	we	PRON
ejpam-5634	246	40	obtain	obtain	VERB
ejpam-5634	246	41	f+(v	f+(v	NOUN
ejpam-5634	246	42	)	)	PUNCT
ejpam-5634	246	43	is	be	AUX
ejpam-5634	246	44	τ1τ2	τ1τ2	NOUN
ejpam-5634	246	45	-	-	ADJ
ejpam-5634	246	46	open	open	ADJ
ejpam-5634	246	47	in	in	ADP
ejpam-5634	246	48	x.	x.	NOUN
ejpam-5634	246	49	thus	thus	ADV
ejpam-5634	246	50	,	,	PUNCT
ejpam-5634	246	51	(	(	PUNCT
ejpam-5634	246	52	x	x	NOUN
ejpam-5634	246	53	,	,	PUNCT
ejpam-5634	246	54	τ1	τ1	NOUN
ejpam-5634	246	55	,	,	PUNCT
ejpam-5634	246	56	τ2	τ2	NOUN
ejpam-5634	246	57	)	)	PUNCT
ejpam-5634	246	58	is	be	AUX
ejpam-5634	246	59	not	not	PART
ejpam-5634	246	60	τ1τ2	τ1τ2	ADJ
ejpam-5634	246	61	-	-	VERB
ejpam-5634	246	62	connected	connected	ADJ
ejpam-5634	246	63	.	.	PUNCT
ejpam-5634	247	1	acknowledgements	acknowledgement	NOUN
ejpam-5634	247	2	this	this	DET
ejpam-5634	247	3	research	research	NOUN
ejpam-5634	247	4	project	project	NOUN
ejpam-5634	247	5	was	be	AUX
ejpam-5634	247	6	financially	financially	ADV
ejpam-5634	247	7	supported	support	VERB
ejpam-5634	247	8	by	by	ADP
ejpam-5634	247	9	mahasarakham	mahasarakham	PROPN
ejpam-5634	247	10	university	university	PROPN
ejpam-5634	247	11	.	.	PUNCT
ejpam-5634	248	1	references	reference	NOUN
ejpam-5634	248	2	[	[	X
ejpam-5634	248	3	1	1	NUM
ejpam-5634	248	4	]	]	PUNCT
ejpam-5634	248	5	c.	c.	PROPN
ejpam-5634	248	6	boonpok	boonpok	PROPN
ejpam-5634	248	7	.	.	PUNCT
ejpam-5634	249	1	almost	almost	ADV
ejpam-5634	249	2	(	(	PUNCT
ejpam-5634	249	3	g	g	NOUN
ejpam-5634	249	4	,	,	PUNCT
ejpam-5634	249	5	m)-continuous	m)-continuous	ADJ
ejpam-5634	249	6	functions	function	NOUN
ejpam-5634	249	7	.	.	PUNCT
ejpam-5634	250	1	international	international	ADJ
ejpam-5634	250	2	journal	journal	PROPN
ejpam-5634	250	3	of	of	ADP
ejpam-5634	250	4	mathematical	mathematical	ADJ
ejpam-5634	250	5	analysis	analysis	NOUN
ejpam-5634	250	6	,	,	PUNCT
ejpam-5634	250	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-5634	250	8	,	,	PUNCT
ejpam-5634	250	9	2010	2010	NUM
ejpam-5634	250	10	.	.	PUNCT
ejpam-5634	251	1	[	[	X
ejpam-5634	251	2	2	2	NUM
ejpam-5634	251	3	]	]	PUNCT
ejpam-5634	251	4	c.	c.	PROPN
ejpam-5634	251	5	boonpok	boonpok	PROPN
ejpam-5634	251	6	.	.	PUNCT
ejpam-5634	252	1	m	m	VERB
ejpam-5634	252	2	-continuous	-continuous	ADJ
ejpam-5634	252	3	functions	function	NOUN
ejpam-5634	252	4	in	in	ADP
ejpam-5634	252	5	biminimal	biminimal	NOUN
ejpam-5634	252	6	structure	structure	NOUN
ejpam-5634	252	7	spaces	space	NOUN
ejpam-5634	252	8	.	.	PUNCT
ejpam-5634	253	1	far	far	PROPN
ejpam-5634	253	2	east	east	PROPN
ejpam-5634	253	3	journal	journal	PROPN
ejpam-5634	253	4	of	of	ADP
ejpam-5634	253	5	mathematical	mathematical	ADJ
ejpam-5634	253	6	sciences	science	NOUN
ejpam-5634	253	7	,	,	PUNCT
ejpam-5634	253	8	43(1):41–58	43(1):41–58	NUM
ejpam-5634	253	9	,	,	PUNCT
ejpam-5634	253	10	2010	2010	NUM
ejpam-5634	253	11	.	.	PUNCT
ejpam-5634	254	1	[	[	X
ejpam-5634	254	2	3	3	X
ejpam-5634	254	3	]	]	PUNCT
ejpam-5634	254	4	c.	c.	PROPN
ejpam-5634	254	5	boonpok	boonpok	PROPN
ejpam-5634	254	6	.	.	PUNCT
ejpam-5634	255	1	on	on	ADP
ejpam-5634	255	2	continuous	continuous	ADJ
ejpam-5634	255	3	multifunctions	multifunction	NOUN
ejpam-5634	255	4	in	in	ADP
ejpam-5634	255	5	ideal	ideal	ADJ
ejpam-5634	255	6	topological	topological	ADJ
ejpam-5634	255	7	spaces	space	NOUN
ejpam-5634	255	8	.	.	PUNCT
ejpam-5634	256	1	lobachevskii	lobachevskii	PROPN
ejpam-5634	256	2	journal	journal	PROPN
ejpam-5634	256	3	of	of	ADP
ejpam-5634	256	4	mathematics	mathematic	NOUN
ejpam-5634	256	5	,	,	PUNCT
ejpam-5634	256	6	40(1):24–35	40(1):24–35	NUM
ejpam-5634	256	7	,	,	PUNCT
ejpam-5634	256	8	2019	2019	NUM
ejpam-5634	256	9	.	.	PUNCT
ejpam-5634	257	1	[	[	X
ejpam-5634	257	2	4	4	NUM
ejpam-5634	257	3	]	]	PUNCT
ejpam-5634	257	4	c.	c.	PROPN
ejpam-5634	257	5	boonpok	boonpok	PROPN
ejpam-5634	257	6	.	.	PUNCT
ejpam-5634	258	1	on	on	ADP
ejpam-5634	258	2	characterizations	characterization	NOUN
ejpam-5634	258	3	of	of	ADP
ejpam-5634	258	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5634	258	5	ideal	ideal	ADJ
ejpam-5634	258	6	topological	topological	ADJ
ejpam-5634	258	7	spaces	space	NOUN
ejpam-5634	258	8	.	.	PUNCT
ejpam-5634	259	1	journal	journal	NOUN
ejpam-5634	259	2	of	of	ADP
ejpam-5634	259	3	mathematics	mathematic	NOUN
ejpam-5634	259	4	,	,	PUNCT
ejpam-5634	259	5	2020:9387601	2020:9387601	NUM
ejpam-5634	259	6	,	,	PUNCT
ejpam-5634	259	7	2020	2020	NUM
ejpam-5634	259	8	.	.	PUNCT
ejpam-5634	260	1	[	[	X
ejpam-5634	260	2	5	5	X
ejpam-5634	260	3	]	]	PUNCT
ejpam-5634	260	4	c.	c.	PROPN
ejpam-5634	260	5	boonpok	boonpok	PROPN
ejpam-5634	260	6	.	.	PUNCT
ejpam-5634	261	1	(	(	PUNCT
ejpam-5634	261	2	τ1	τ1	NOUN
ejpam-5634	261	3	,	,	PUNCT
ejpam-5634	261	4	τ2)δ	τ2)δ	ADJ
ejpam-5634	261	5	-	-	PUNCT
ejpam-5634	261	6	semicontinuous	semicontinuous	ADJ
ejpam-5634	261	7	multifunctions	multifunction	NOUN
ejpam-5634	261	8	.	.	PUNCT
ejpam-5634	262	1	heliyon	heliyon	NOUN
ejpam-5634	262	2	,	,	PUNCT
ejpam-5634	262	3	6	6	NUM
ejpam-5634	262	4	:	:	SYM
ejpam-5634	262	5	e05367	e05367	PROPN
ejpam-5634	262	6	,	,	PUNCT
ejpam-5634	262	7	2020	2020	NUM
ejpam-5634	262	8	.	.	PUNCT
ejpam-5634	263	1	[	[	X
ejpam-5634	263	2	6	6	NUM
ejpam-5634	263	3	]	]	PUNCT
ejpam-5634	263	4	c.	c.	PROPN
ejpam-5634	263	5	boonpok	boonpok	PROPN
ejpam-5634	263	6	.	.	PUNCT
ejpam-5634	264	1	weak	weak	ADJ
ejpam-5634	264	2	quasi	quasi	ADJ
ejpam-5634	264	3	continuity	continuity	NOUN
ejpam-5634	264	4	for	for	ADP
ejpam-5634	264	5	multifunctions	multifunction	NOUN
ejpam-5634	264	6	in	in	ADP
ejpam-5634	264	7	ideal	ideal	ADJ
ejpam-5634	264	8	topological	topological	ADJ
ejpam-5634	264	9	spaces	space	NOUN
ejpam-5634	264	10	.	.	PUNCT
ejpam-5634	265	1	advances	advance	NOUN
ejpam-5634	265	2	in	in	ADP
ejpam-5634	265	3	mathematics	mathematic	NOUN
ejpam-5634	265	4	:	:	PUNCT
ejpam-5634	265	5	scientific	scientific	ADJ
ejpam-5634	265	6	journal	journal	NOUN
ejpam-5634	265	7	,	,	PUNCT
ejpam-5634	265	8	9(1):339–355	9(1):339–355	NUM
ejpam-5634	265	9	,	,	PUNCT
ejpam-5634	265	10	2020	2020	NUM
ejpam-5634	265	11	.	.	PUNCT
ejpam-5634	266	1	[	[	X
ejpam-5634	266	2	7	7	X
ejpam-5634	266	3	]	]	X
ejpam-5634	266	4	c.	c.	PROPN
ejpam-5634	266	5	boonpok	boonpok	PROPN
ejpam-5634	266	6	.	.	PUNCT
ejpam-5634	267	1	upper	upper	ADJ
ejpam-5634	267	2	and	and	CCONJ
ejpam-5634	267	3	lower	low	ADJ
ejpam-5634	267	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-5634	267	5	.	.	PUNCT
ejpam-5634	267	6	heliyon	heliyon	NOUN
ejpam-5634	267	7	,	,	PUNCT
ejpam-5634	267	8	7	7	NUM
ejpam-5634	267	9	:	:	PUNCT
ejpam-5634	267	10	e05986	e05986	PROPN
ejpam-5634	267	11	,	,	PUNCT
ejpam-5634	267	12	2021	2021	NUM
ejpam-5634	267	13	.	.	PUNCT
ejpam-5634	268	1	m.	m.	NOUN
ejpam-5634	268	2	chiangpradit	chiangpradit	PROPN
ejpam-5634	268	3	,	,	PUNCT
ejpam-5634	268	4	a.	a.	PROPN
ejpam-5634	268	5	sama	sama	PROPN
ejpam-5634	268	6	-	-	PUNCT
ejpam-5634	268	7	ae	ae	PROPN
ejpam-5634	268	8	,	,	PUNCT
ejpam-5634	268	9	c.	c.	PROPN
ejpam-5634	268	10	boonpok	boonpok	PROPN
ejpam-5634	268	11	/	/	SYM
ejpam-5634	268	12	eur	eur	PROPN
ejpam-5634	268	13	.	.	PUNCT
ejpam-5634	269	1	j.	j.	PROPN
ejpam-5634	269	2	pure	pure	PROPN
ejpam-5634	269	3	appl	appl	PROPN
ejpam-5634	269	4	.	.	PROPN
ejpam-5634	269	5	math	math	PROPN
ejpam-5634	269	6	,	,	PUNCT
ejpam-5634	269	7	18	18	NUM
ejpam-5634	269	8	(	(	PUNCT
ejpam-5634	269	9	1	1	NUM
ejpam-5634	269	10	)	)	PUNCT
ejpam-5634	269	11	(	(	PUNCT
ejpam-5634	269	12	2025	2025	NUM
ejpam-5634	269	13	)	)	PUNCT
ejpam-5634	269	14	,	,	PUNCT
ejpam-5634	269	15	5634	5634	NUM
ejpam-5634	269	16	10	10	NUM
ejpam-5634	269	17	of	of	ADP
ejpam-5634	269	18	12	12	NUM
ejpam-5634	269	19	[	[	SYM
ejpam-5634	269	20	8	8	NUM
ejpam-5634	269	21	]	]	PUNCT
ejpam-5634	269	22	c.	c.	PROPN
ejpam-5634	269	23	boonpok	boonpok	PROPN
ejpam-5634	269	24	.	.	PUNCT
ejpam-5634	270	1	on	on	ADP
ejpam-5634	270	2	some	some	DET
ejpam-5634	270	3	closed	closed	ADJ
ejpam-5634	270	4	sets	set	NOUN
ejpam-5634	270	5	and	and	CCONJ
ejpam-5634	270	6	low	low	ADJ
ejpam-5634	270	7	separation	separation	NOUN
ejpam-5634	270	8	axioms	axiom	NOUN
ejpam-5634	270	9	via	via	ADP
ejpam-5634	270	10	topological	topological	ADJ
ejpam-5634	270	11	ideals	ideal	NOUN
ejpam-5634	270	12	.	.	PUNCT
ejpam-5634	271	1	european	european	ADJ
ejpam-5634	271	2	journal	journal	PROPN
ejpam-5634	271	3	of	of	ADP
ejpam-5634	271	4	pure	pure	ADJ
ejpam-5634	271	5	and	and	CCONJ
ejpam-5634	271	6	applied	applied	ADJ
ejpam-5634	271	7	mathematics	mathematic	NOUN
ejpam-5634	271	8	,	,	PUNCT
ejpam-5634	271	9	15(3):1023–1046	15(3):1023–1046	NUM
ejpam-5634	271	10	,	,	PUNCT
ejpam-5634	271	11	2022	2022	NUM
ejpam-5634	271	12	.	.	PUNCT
ejpam-5634	272	1	[	[	X
ejpam-5634	272	2	9	9	NUM
ejpam-5634	272	3	]	]	PUNCT
ejpam-5634	272	4	c.	c.	PROPN
ejpam-5634	272	5	boonpok	boonpok	PROPN
ejpam-5634	272	6	.	.	PUNCT
ejpam-5634	273	1	θ(⋆)-quasi	θ(⋆)-quasi	DET
ejpam-5634	273	2	continuity	continuity	NOUN
ejpam-5634	273	3	for	for	ADP
ejpam-5634	273	4	multifunctions	multifunction	NOUN
ejpam-5634	273	5	.	.	PUNCT
ejpam-5634	274	1	wseas	wseas	PROPN
ejpam-5634	274	2	transactions	transaction	NOUN
ejpam-5634	274	3	on	on	ADP
ejpam-5634	274	4	mathematics	mathematic	NOUN
ejpam-5634	274	5	,	,	PUNCT
ejpam-5634	274	6	21:245–251	21:245–251	NUM
ejpam-5634	274	7	,	,	PUNCT
ejpam-5634	274	8	2022	2022	NUM
ejpam-5634	274	9	.	.	PUNCT
ejpam-5634	275	1	[	[	X
ejpam-5634	275	2	10	10	NUM
ejpam-5634	275	3	]	]	X
ejpam-5634	275	4	c.	c.	PROPN
ejpam-5634	275	5	boonpok	boonpok	PROPN
ejpam-5634	275	6	.	.	PUNCT
ejpam-5634	276	1	on	on	ADP
ejpam-5634	276	2	some	some	DET
ejpam-5634	276	3	spaces	space	NOUN
ejpam-5634	276	4	via	via	ADP
ejpam-5634	276	5	topological	topological	ADJ
ejpam-5634	276	6	ideals	ideal	NOUN
ejpam-5634	276	7	.	.	PUNCT
ejpam-5634	277	1	open	open	ADJ
ejpam-5634	277	2	mathematics	mathematic	NOUN
ejpam-5634	277	3	,	,	PUNCT
ejpam-5634	277	4	21:20230118	21:20230118	NUM
ejpam-5634	277	5	,	,	PUNCT
ejpam-5634	277	6	2023	2023	NUM
ejpam-5634	277	7	.	.	PUNCT
ejpam-5634	278	1	[	[	X
ejpam-5634	278	2	11	11	NUM
ejpam-5634	278	3	]	]	PUNCT
ejpam-5634	278	4	c.	c.	PROPN
ejpam-5634	278	5	boonpok	boonpok	PROPN
ejpam-5634	278	6	.	.	PUNCT
ejpam-5634	279	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5634	279	2	.	.	PUNCT
ejpam-5634	280	1	mathematica	mathematica	PROPN
ejpam-5634	280	2	,	,	PUNCT
ejpam-5634	280	3	65(1):31–42	65(1):31–42	NUM
ejpam-5634	280	4	,	,	PUNCT
ejpam-5634	280	5	2023	2023	NUM
ejpam-5634	280	6	.	.	PUNCT
ejpam-5634	281	1	[	[	X
ejpam-5634	281	2	12	12	NUM
ejpam-5634	281	3	]	]	X
ejpam-5634	281	4	c.	c.	PROPN
ejpam-5634	281	5	boonpok	boonpok	PROPN
ejpam-5634	281	6	and	and	CCONJ
ejpam-5634	281	7	j.	j.	PROPN
ejpam-5634	281	8	khampakdee	khampakdee	PROPN
ejpam-5634	281	9	.	.	PUNCT
ejpam-5634	282	1	(	(	PUNCT
ejpam-5634	282	2	λ	λ	NOUN
ejpam-5634	282	3	,	,	PUNCT
ejpam-5634	282	4	sp)-open	sp)-open	ADJ
ejpam-5634	282	5	sets	set	NOUN
ejpam-5634	282	6	in	in	ADP
ejpam-5634	282	7	topological	topological	ADJ
ejpam-5634	282	8	spaces	space	NOUN
ejpam-5634	282	9	.	.	PUNCT
ejpam-5634	283	1	european	european	ADJ
ejpam-5634	283	2	journal	journal	PROPN
ejpam-5634	283	3	of	of	ADP
ejpam-5634	283	4	pure	pure	ADJ
ejpam-5634	283	5	and	and	CCONJ
ejpam-5634	283	6	applied	applied	ADJ
ejpam-5634	283	7	mathematics	mathematic	NOUN
ejpam-5634	283	8	,	,	PUNCT
ejpam-5634	283	9	15(2):572–588	15(2):572–588	NUM
ejpam-5634	283	10	,	,	PUNCT
ejpam-5634	283	11	2022	2022	NUM
ejpam-5634	283	12	.	.	PUNCT
ejpam-5634	284	1	[	[	X
ejpam-5634	284	2	13	13	NUM
ejpam-5634	284	3	]	]	PUNCT
ejpam-5634	284	4	c.	c.	PROPN
ejpam-5634	284	5	boonpok	boonpok	PROPN
ejpam-5634	284	6	and	and	CCONJ
ejpam-5634	284	7	j.	j.	PROPN
ejpam-5634	284	8	khampakdee	khampakdee	PROPN
ejpam-5634	284	9	.	.	PUNCT
ejpam-5634	285	1	on	on	ADP
ejpam-5634	285	2	almost	almost	ADV
ejpam-5634	285	3	α(λ	α(λ	PROPN
ejpam-5634	285	4	,	,	PUNCT
ejpam-5634	285	5	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	285	6	multifunctions	multifunction	NOUN
ejpam-5634	285	7	.	.	PUNCT
ejpam-5634	286	1	european	european	PROPN
ejpam-5634	286	2	journal	journal	PROPN
ejpam-5634	286	3	of	of	ADP
ejpam-5634	286	4	pure	pure	ADJ
ejpam-5634	286	5	and	and	CCONJ
ejpam-5634	286	6	applied	applied	ADJ
ejpam-5634	286	7	mathematics	mathematic	NOUN
ejpam-5634	286	8	,	,	PUNCT
ejpam-5634	286	9	15(2):626–634	15(2):626–634	PROPN
ejpam-5634	286	10	,	,	PUNCT
ejpam-5634	286	11	2022	2022	NUM
ejpam-5634	286	12	.	.	PUNCT
ejpam-5634	287	1	[	[	X
ejpam-5634	287	2	14	14	NUM
ejpam-5634	287	3	]	]	X
ejpam-5634	287	4	c.	c.	PROPN
ejpam-5634	287	5	boonpok	boonpok	PROPN
ejpam-5634	287	6	and	and	CCONJ
ejpam-5634	287	7	j.	j.	PROPN
ejpam-5634	287	8	khampakdee	khampakdee	PROPN
ejpam-5634	287	9	.	.	PUNCT
ejpam-5634	288	1	slight	slight	PROPN
ejpam-5634	288	2	(	(	PUNCT
ejpam-5634	288	3	λ	λ	NOUN
ejpam-5634	288	4	,	,	PUNCT
ejpam-5634	288	5	sp)-continuity	sp)-continuity	NOUN
ejpam-5634	288	6	and	and	CCONJ
ejpam-5634	288	7	λsp	λsp	NOUN
ejpam-5634	288	8	-	-	PUNCT
ejpam-5634	288	9	extremally	extremally	ADV
ejpam-5634	288	10	disconnectedness	disconnectedness	NOUN
ejpam-5634	288	11	.	.	PUNCT
ejpam-5634	289	1	european	european	ADJ
ejpam-5634	289	2	journal	journal	PROPN
ejpam-5634	289	3	of	of	ADP
ejpam-5634	289	4	pure	pure	ADJ
ejpam-5634	289	5	and	and	CCONJ
ejpam-5634	289	6	applied	applied	ADJ
ejpam-5634	289	7	mathematics	mathematic	NOUN
ejpam-5634	289	8	,	,	PUNCT
ejpam-5634	289	9	15(3):1180–1188	15(3):1180–1188	NUM
ejpam-5634	289	10	,	,	PUNCT
ejpam-5634	289	11	2022	2022	NUM
ejpam-5634	289	12	.	.	PUNCT
ejpam-5634	290	1	[	[	X
ejpam-5634	290	2	15	15	NUM
ejpam-5634	290	3	]	]	X
ejpam-5634	290	4	c.	c.	PROPN
ejpam-5634	290	5	boonpok	boonpok	PROPN
ejpam-5634	290	6	and	and	CCONJ
ejpam-5634	290	7	j.	j.	PROPN
ejpam-5634	290	8	khampakdee	khampakdee	PROPN
ejpam-5634	290	9	.	.	PUNCT
ejpam-5634	291	1	upper	upper	ADJ
ejpam-5634	291	2	and	and	CCONJ
ejpam-5634	291	3	lower	low	ADJ
ejpam-5634	291	4	weak	weak	ADJ
ejpam-5634	291	5	sβ(⋆)-continuity	sβ(⋆)-continuity	NOUN
ejpam-5634	291	6	.	.	PUNCT
ejpam-5634	292	1	european	european	PROPN
ejpam-5634	292	2	journal	journal	PROPN
ejpam-5634	292	3	of	of	ADP
ejpam-5634	292	4	pure	pure	ADJ
ejpam-5634	292	5	and	and	CCONJ
ejpam-5634	292	6	applied	applied	ADJ
ejpam-5634	292	7	mathematics	mathematic	NOUN
ejpam-5634	292	8	,	,	PUNCT
ejpam-5634	292	9	16(4):2544–2556	16(4):2544–2556	NUM
ejpam-5634	292	10	,	,	PUNCT
ejpam-5634	292	11	2023	2023	NUM
ejpam-5634	292	12	.	.	PUNCT
ejpam-5634	293	1	[	[	X
ejpam-5634	293	2	16	16	NUM
ejpam-5634	293	3	]	]	X
ejpam-5634	293	4	c.	c.	PROPN
ejpam-5634	293	5	boonpok	boonpok	PROPN
ejpam-5634	293	6	and	and	CCONJ
ejpam-5634	293	7	j.	j.	PROPN
ejpam-5634	293	8	khampakdee	khampakdee	PROPN
ejpam-5634	293	9	.	.	PUNCT
ejpam-5634	294	1	almost	almost	ADV
ejpam-5634	294	2	strong	strong	ADJ
ejpam-5634	294	3	θ(λ	θ(λ	PROPN
ejpam-5634	294	4	,	,	PUNCT
ejpam-5634	294	5	p)-continuity	p)-continuity	NOUN
ejpam-5634	294	6	for	for	ADP
ejpam-5634	294	7	functions	function	NOUN
ejpam-5634	294	8	.	.	PUNCT
ejpam-5634	295	1	european	european	ADJ
ejpam-5634	295	2	journal	journal	PROPN
ejpam-5634	295	3	of	of	ADP
ejpam-5634	295	4	pure	pure	ADJ
ejpam-5634	295	5	and	and	CCONJ
ejpam-5634	295	6	applied	applied	ADJ
ejpam-5634	295	7	mathematics	mathematic	NOUN
ejpam-5634	295	8	,	,	PUNCT
ejpam-5634	295	9	17(1):300–309	17(1):300–309	PROPN
ejpam-5634	295	10	,	,	PUNCT
ejpam-5634	295	11	2024	2024	NUM
ejpam-5634	295	12	.	.	PUNCT
ejpam-5634	296	1	[	[	X
ejpam-5634	296	2	17	17	NUM
ejpam-5634	296	3	]	]	X
ejpam-5634	296	4	c.	c.	PROPN
ejpam-5634	296	5	boonpok	boonpok	PROPN
ejpam-5634	296	6	and	and	CCONJ
ejpam-5634	296	7	j.	j.	PROPN
ejpam-5634	296	8	khampakdee	khampakdee	PROPN
ejpam-5634	296	9	.	.	PUNCT
ejpam-5634	297	1	upper	upper	ADJ
ejpam-5634	297	2	and	and	CCONJ
ejpam-5634	297	3	lower	low	ADJ
ejpam-5634	297	4	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-5634	297	5	.	.	PUNCT
ejpam-5634	297	6	european	european	PROPN
ejpam-5634	297	7	journal	journal	PROPN
ejpam-5634	297	8	of	of	ADP
ejpam-5634	297	9	pure	pure	ADJ
ejpam-5634	297	10	and	and	CCONJ
ejpam-5634	297	11	applied	applied	ADJ
ejpam-5634	297	12	mathematics	mathematic	NOUN
ejpam-5634	297	13	,	,	PUNCT
ejpam-5634	297	14	17(1):201–211	17(1):201–211	NUM
ejpam-5634	297	15	,	,	PUNCT
ejpam-5634	297	16	2024	2024	NUM
ejpam-5634	297	17	.	.	PUNCT
ejpam-5634	298	1	[	[	X
ejpam-5634	298	2	18	18	NUM
ejpam-5634	298	3	]	]	PUNCT
ejpam-5634	298	4	c.	c.	PROPN
ejpam-5634	298	5	boonpok	boonpok	PROPN
ejpam-5634	298	6	and	and	CCONJ
ejpam-5634	298	7	c.	c.	PROPN
ejpam-5634	298	8	klanarong	klanarong	PROPN
ejpam-5634	298	9	.	.	PUNCT
ejpam-5634	299	1	on	on	ADP
ejpam-5634	299	2	weakly	weakly	ADJ
ejpam-5634	299	3	(	(	PUNCT
ejpam-5634	299	4	τ1	τ1	NOUN
ejpam-5634	299	5	,	,	PUNCT
ejpam-5634	299	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	299	7	functions	function	NOUN
ejpam-5634	299	8	.	.	PUNCT
ejpam-5634	300	1	european	european	ADJ
ejpam-5634	300	2	journal	journal	PROPN
ejpam-5634	300	3	of	of	ADP
ejpam-5634	300	4	pure	pure	ADJ
ejpam-5634	300	5	and	and	CCONJ
ejpam-5634	300	6	applied	applied	ADJ
ejpam-5634	300	7	mathematics	mathematic	NOUN
ejpam-5634	300	8	,	,	PUNCT
ejpam-5634	300	9	17(1):416–425	17(1):416–425	NUM
ejpam-5634	300	10	,	,	PUNCT
ejpam-5634	300	11	2024	2024	NUM
ejpam-5634	300	12	.	.	PUNCT
ejpam-5634	301	1	[	[	X
ejpam-5634	301	2	19	19	NUM
ejpam-5634	301	3	]	]	X
ejpam-5634	301	4	c.	c.	PROPN
ejpam-5634	301	5	boonpok	boonpok	PROPN
ejpam-5634	301	6	and	and	CCONJ
ejpam-5634	301	7	p.	p.	NOUN
ejpam-5634	301	8	pue	pue	NOUN
ejpam-5634	301	9	-	-	PUNCT
ejpam-5634	301	10	on	on	ADP
ejpam-5634	301	11	.	.	PUNCT
ejpam-5634	302	1	continuity	continuity	NOUN
ejpam-5634	302	2	for	for	ADP
ejpam-5634	302	3	multifunctions	multifunction	NOUN
ejpam-5634	302	4	in	in	ADP
ejpam-5634	302	5	ideal	ideal	ADJ
ejpam-5634	302	6	topological	topological	ADJ
ejpam-5634	302	7	spaces	space	NOUN
ejpam-5634	302	8	.	.	PUNCT
ejpam-5634	303	1	wseas	wseas	VERB
ejpam-5634	303	2	transactions	transaction	NOUN
ejpam-5634	303	3	on	on	ADP
ejpam-5634	303	4	mathematics	mathematic	NOUN
ejpam-5634	303	5	,	,	PUNCT
ejpam-5634	303	6	19:624–631	19:624–631	NUM
ejpam-5634	303	7	,	,	PUNCT
ejpam-5634	303	8	2020	2020	NUM
ejpam-5634	303	9	.	.	PUNCT
ejpam-5634	304	1	[	[	X
ejpam-5634	304	2	20	20	NUM
ejpam-5634	304	3	]	]	PUNCT
ejpam-5634	304	4	c.	c.	PROPN
ejpam-5634	304	5	boonpok	boonpok	PROPN
ejpam-5634	304	6	and	and	CCONJ
ejpam-5634	304	7	p.	p.	NOUN
ejpam-5634	304	8	pue	pue	NOUN
ejpam-5634	304	9	-	-	PUNCT
ejpam-5634	304	10	on	on	ADP
ejpam-5634	304	11	.	.	PUNCT
ejpam-5634	305	1	upper	upper	ADJ
ejpam-5634	305	2	and	and	CCONJ
ejpam-5634	305	3	lower	low	ADJ
ejpam-5634	305	4	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-5634	305	5	multifunctions	multifunction	NOUN
ejpam-5634	305	6	.	.	PUNCT
ejpam-5634	306	1	european	european	ADJ
ejpam-5634	306	2	journal	journal	PROPN
ejpam-5634	306	3	of	of	ADP
ejpam-5634	306	4	pure	pure	ADJ
ejpam-5634	306	5	and	and	CCONJ
ejpam-5634	306	6	applied	applied	ADJ
ejpam-5634	306	7	mathematics	mathematic	NOUN
ejpam-5634	306	8	,	,	PUNCT
ejpam-5634	306	9	16(3):1634–1646	16(3):1634–1646	NUM
ejpam-5634	306	10	,	,	PUNCT
ejpam-5634	306	11	2023	2023	NUM
ejpam-5634	306	12	.	.	PUNCT
ejpam-5634	307	1	[	[	X
ejpam-5634	307	2	21	21	NUM
ejpam-5634	307	3	]	]	X
ejpam-5634	307	4	c.	c.	PROPN
ejpam-5634	307	5	boonpok	boonpok	PROPN
ejpam-5634	307	6	and	and	CCONJ
ejpam-5634	307	7	p.	p.	NOUN
ejpam-5634	307	8	pue	pue	NOUN
ejpam-5634	307	9	-	-	PUNCT
ejpam-5634	307	10	on	on	ADP
ejpam-5634	307	11	.	.	PUNCT
ejpam-5634	308	1	upper	upper	ADJ
ejpam-5634	308	2	and	and	CCONJ
ejpam-5634	308	3	lower	low	ADJ
ejpam-5634	308	4	weakly	weakly	ADJ
ejpam-5634	308	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5634	308	6	multifunctions	multifunction	NOUN
ejpam-5634	308	7	.	.	PUNCT
ejpam-5634	309	1	international	international	ADJ
ejpam-5634	309	2	journal	journal	NOUN
ejpam-5634	309	3	of	of	ADP
ejpam-5634	309	4	analysis	analysis	NOUN
ejpam-5634	309	5	and	and	CCONJ
ejpam-5634	309	6	applications	application	NOUN
ejpam-5634	309	7	,	,	PUNCT
ejpam-5634	309	8	21:90	21:90	NUM
ejpam-5634	309	9	,	,	PUNCT
ejpam-5634	309	10	2023	2023	NUM
ejpam-5634	309	11	.	.	PUNCT
ejpam-5634	310	1	[	[	X
ejpam-5634	310	2	22	22	NUM
ejpam-5634	310	3	]	]	PUNCT
ejpam-5634	310	4	c.	c.	PROPN
ejpam-5634	310	5	boonpok	boonpok	PROPN
ejpam-5634	310	6	and	and	CCONJ
ejpam-5634	310	7	p.	p.	NOUN
ejpam-5634	310	8	pue	pue	NOUN
ejpam-5634	310	9	-	-	PUNCT
ejpam-5634	310	10	on	on	ADP
ejpam-5634	310	11	.	.	PUNCT
ejpam-5634	311	1	upper	upper	ADJ
ejpam-5634	311	2	and	and	CCONJ
ejpam-5634	311	3	lower	low	ADJ
ejpam-5634	311	4	weakly	weakly	ADJ
ejpam-5634	311	5	(	(	PUNCT
ejpam-5634	311	6	λ	λ	NOUN
ejpam-5634	311	7	,	,	PUNCT
ejpam-5634	311	8	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	311	9	multifunctions	multifunction	NOUN
ejpam-5634	311	10	.	.	PUNCT
ejpam-5634	312	1	european	european	PROPN
ejpam-5634	312	2	journal	journal	PROPN
ejpam-5634	312	3	of	of	ADP
ejpam-5634	312	4	pure	pure	ADJ
ejpam-5634	312	5	and	and	CCONJ
ejpam-5634	312	6	applied	applied	ADJ
ejpam-5634	312	7	mathematics	mathematic	NOUN
ejpam-5634	312	8	,	,	PUNCT
ejpam-5634	312	9	16(2):1047–1058	16(2):1047–1058	NUM
ejpam-5634	312	10	,	,	PUNCT
ejpam-5634	312	11	2023	2023	NUM
ejpam-5634	312	12	.	.	PUNCT
ejpam-5634	313	1	[	[	X
ejpam-5634	313	2	23	23	NUM
ejpam-5634	313	3	]	]	X
ejpam-5634	313	4	c.	c.	PROPN
ejpam-5634	313	5	boonpok	boonpok	PROPN
ejpam-5634	313	6	and	and	CCONJ
ejpam-5634	313	7	p.	p.	NOUN
ejpam-5634	313	8	pue	pue	NOUN
ejpam-5634	313	9	-	-	PUNCT
ejpam-5634	313	10	on	on	ADP
ejpam-5634	313	11	.	.	PUNCT
ejpam-5634	314	1	characterizations	characterization	NOUN
ejpam-5634	314	2	of	of	ADP
ejpam-5634	314	3	almost	almost	ADV
ejpam-5634	314	4	(	(	PUNCT
ejpam-5634	314	5	τ1	τ1	NOUN
ejpam-5634	314	6	,	,	PUNCT
ejpam-5634	314	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	314	8	functions	function	NOUN
ejpam-5634	314	9	.	.	PUNCT
ejpam-5634	315	1	international	international	ADJ
ejpam-5634	315	2	journal	journal	NOUN
ejpam-5634	315	3	of	of	ADP
ejpam-5634	315	4	analysis	analysis	NOUN
ejpam-5634	315	5	and	and	CCONJ
ejpam-5634	315	6	applications	application	NOUN
ejpam-5634	315	7	,	,	PUNCT
ejpam-5634	315	8	22:33	22:33	NUM
ejpam-5634	315	9	,	,	PUNCT
ejpam-5634	315	10	2024	2024	NUM
ejpam-5634	315	11	.	.	PUNCT
ejpam-5634	316	1	[	[	X
ejpam-5634	316	2	24	24	NUM
ejpam-5634	316	3	]	]	PUNCT
ejpam-5634	316	4	c.	c.	PROPN
ejpam-5634	316	5	boonpok	boonpok	PROPN
ejpam-5634	316	6	and	and	CCONJ
ejpam-5634	316	7	n.	n.	PROPN
ejpam-5634	316	8	srisarakham	srisarakham	PROPN
ejpam-5634	316	9	.	.	PUNCT
ejpam-5634	317	1	almost	almost	ADV
ejpam-5634	317	2	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-5634	317	3	for	for	ADP
ejpam-5634	317	4	multifunctions	multifunction	NOUN
ejpam-5634	317	5	.	.	PUNCT
ejpam-5634	318	1	international	international	ADJ
ejpam-5634	318	2	journal	journal	NOUN
ejpam-5634	318	3	of	of	ADP
ejpam-5634	318	4	analysis	analysis	NOUN
ejpam-5634	318	5	and	and	CCONJ
ejpam-5634	318	6	applications	application	NOUN
ejpam-5634	318	7	,	,	PUNCT
ejpam-5634	318	8	21:107	21:107	NUM
ejpam-5634	318	9	,	,	PUNCT
ejpam-5634	318	10	2023	2023	NUM
ejpam-5634	318	11	.	.	PUNCT
ejpam-5634	319	1	[	[	X
ejpam-5634	319	2	25	25	NUM
ejpam-5634	319	3	]	]	PUNCT
ejpam-5634	319	4	c.	c.	PROPN
ejpam-5634	319	5	boonpok	boonpok	PROPN
ejpam-5634	319	6	and	and	CCONJ
ejpam-5634	319	7	n.	n.	PROPN
ejpam-5634	319	8	srisarakham	srisarakham	PROPN
ejpam-5634	319	9	.	.	PUNCT
ejpam-5634	320	1	weak	weak	ADJ
ejpam-5634	320	2	forms	form	NOUN
ejpam-5634	320	3	of	of	ADP
ejpam-5634	320	4	(	(	PUNCT
ejpam-5634	320	5	λ	λ	PROPN
ejpam-5634	320	6	,	,	PUNCT
ejpam-5634	320	7	b)-open	b)-open	VERB
ejpam-5634	320	8	sets	set	NOUN
ejpam-5634	320	9	and	and	CCONJ
ejpam-5634	320	10	weak	weak	ADJ
ejpam-5634	320	11	(	(	PUNCT
ejpam-5634	320	12	λ	λ	NOUN
ejpam-5634	320	13	,	,	PUNCT
ejpam-5634	320	14	b)continuity	b)continuity	NOUN
ejpam-5634	320	15	.	.	PUNCT
ejpam-5634	321	1	european	european	PROPN
ejpam-5634	321	2	journal	journal	PROPN
ejpam-5634	321	3	of	of	ADP
ejpam-5634	321	4	pure	pure	ADJ
ejpam-5634	321	5	and	and	CCONJ
ejpam-5634	321	6	applied	applied	ADJ
ejpam-5634	321	7	mathematics	mathematic	NOUN
ejpam-5634	321	8	,	,	PUNCT
ejpam-5634	321	9	16(1):29–43	16(1):29–43	NUM
ejpam-5634	321	10	,	,	PUNCT
ejpam-5634	321	11	2023	2023	NUM
ejpam-5634	321	12	.	.	PUNCT
ejpam-5634	322	1	[	[	X
ejpam-5634	322	2	26	26	NUM
ejpam-5634	322	3	]	]	X
ejpam-5634	322	4	c.	c.	PROPN
ejpam-5634	322	5	boonpok	boonpok	PROPN
ejpam-5634	322	6	and	and	CCONJ
ejpam-5634	322	7	n.	n.	PROPN
ejpam-5634	322	8	srisarakham	srisarakham	PROPN
ejpam-5634	322	9	.	.	PUNCT
ejpam-5634	323	1	(	(	PUNCT
ejpam-5634	323	2	τ1	τ1	NOUN
ejpam-5634	323	3	,	,	PUNCT
ejpam-5634	323	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5634	323	5	for	for	ADP
ejpam-5634	323	6	functions	function	NOUN
ejpam-5634	323	7	.	.	PUNCT
ejpam-5634	324	1	asia	asia	PROPN
ejpam-5634	324	2	pacific	pacific	PROPN
ejpam-5634	324	3	journal	journal	PROPN
ejpam-5634	324	4	of	of	ADP
ejpam-5634	324	5	mathematics	mathematic	NOUN
ejpam-5634	324	6	,	,	PUNCT
ejpam-5634	324	7	11:21	11:21	NUM
ejpam-5634	324	8	,	,	PUNCT
ejpam-5634	324	9	2024	2024	NUM
ejpam-5634	324	10	.	.	PUNCT
ejpam-5634	325	1	[	[	X
ejpam-5634	325	2	27	27	NUM
ejpam-5634	325	3	]	]	X
ejpam-5634	325	4	c.	c.	PROPN
ejpam-5634	325	5	boonpok	boonpok	PROPN
ejpam-5634	325	6	and	and	CCONJ
ejpam-5634	325	7	m.	m.	NOUN
ejpam-5634	325	8	thongmoon	thongmoon	NOUN
ejpam-5634	325	9	.	.	PUNCT
ejpam-5634	326	1	weak	weak	ADJ
ejpam-5634	326	2	α(λ	α(λ	PROPN
ejpam-5634	326	3	,	,	PUNCT
ejpam-5634	326	4	sp)-continuity	sp)-continuity	NOUN
ejpam-5634	326	5	for	for	ADP
ejpam-5634	326	6	multifunctions	multifunction	NOUN
ejpam-5634	326	7	.	.	PUNCT
ejpam-5634	327	1	european	european	ADJ
ejpam-5634	327	2	journal	journal	PROPN
ejpam-5634	327	3	of	of	ADP
ejpam-5634	327	4	pure	pure	ADJ
ejpam-5634	327	5	and	and	CCONJ
ejpam-5634	327	6	applied	applied	ADJ
ejpam-5634	327	7	mathematics	mathematic	NOUN
ejpam-5634	327	8	,	,	PUNCT
ejpam-5634	327	9	16(1):465–478	16(1):465–478	NUM
ejpam-5634	327	10	,	,	PUNCT
ejpam-5634	327	11	2023	2023	NUM
ejpam-5634	327	12	.	.	PUNCT
ejpam-5634	328	1	[	[	X
ejpam-5634	328	2	28	28	NUM
ejpam-5634	328	3	]	]	X
ejpam-5634	328	4	c.	c.	PROPN
ejpam-5634	328	5	boonpok	boonpok	PROPN
ejpam-5634	328	6	and	and	CCONJ
ejpam-5634	328	7	c.	c.	PROPN
ejpam-5634	328	8	viriyapong	viriyapong	PROPN
ejpam-5634	328	9	.	.	PUNCT
ejpam-5634	329	1	upper	upper	ADJ
ejpam-5634	329	2	and	and	CCONJ
ejpam-5634	329	3	lower	low	ADJ
ejpam-5634	329	4	almost	almost	ADV
ejpam-5634	329	5	weak	weak	ADJ
ejpam-5634	329	6	(	(	PUNCT
ejpam-5634	329	7	τ1	τ1	NOUN
ejpam-5634	329	8	,	,	PUNCT
ejpam-5634	329	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5634	329	10	.	.	PUNCT
ejpam-5634	330	1	european	european	PROPN
ejpam-5634	330	2	journal	journal	PROPN
ejpam-5634	330	3	of	of	ADP
ejpam-5634	330	4	pure	pure	ADJ
ejpam-5634	330	5	and	and	CCONJ
ejpam-5634	330	6	applied	applied	ADJ
ejpam-5634	330	7	mathematics	mathematic	NOUN
ejpam-5634	330	8	,	,	PUNCT
ejpam-5634	330	9	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-5634	330	10	,	,	PUNCT
ejpam-5634	330	11	2021	2021	NUM
ejpam-5634	330	12	.	.	PUNCT
ejpam-5634	331	1	[	[	X
ejpam-5634	331	2	29	29	NUM
ejpam-5634	331	3	]	]	X
ejpam-5634	331	4	c.	c.	PROPN
ejpam-5634	331	5	boonpok	boonpok	PROPN
ejpam-5634	331	6	,	,	PUNCT
ejpam-5634	331	7	c.	c.	PROPN
ejpam-5634	331	8	viriyapong	viriyapong	PROPN
ejpam-5634	331	9	,	,	PUNCT
ejpam-5634	331	10	and	and	CCONJ
ejpam-5634	331	11	m.	m.	NOUN
ejpam-5634	331	12	thongmoon	thongmoon	NOUN
ejpam-5634	331	13	.	.	PUNCT
ejpam-5634	332	1	on	on	ADP
ejpam-5634	332	2	upper	upper	ADJ
ejpam-5634	332	3	and	and	CCONJ
ejpam-5634	332	4	lower	low	ADJ
ejpam-5634	332	5	(	(	PUNCT
ejpam-5634	332	6	τ1	τ1	NOUN
ejpam-5634	332	7	,	,	PUNCT
ejpam-5634	332	8	τ2)m	τ2)m	NOUN
ejpam-5634	332	9	.	.	PUNCT
ejpam-5634	333	1	chiangpradit	chiangpradit	NOUN
ejpam-5634	333	2	,	,	PUNCT
ejpam-5634	333	3	a.	a.	PROPN
ejpam-5634	333	4	sama	sama	PROPN
ejpam-5634	333	5	-	-	PUNCT
ejpam-5634	333	6	ae	ae	PROPN
ejpam-5634	333	7	,	,	PUNCT
ejpam-5634	333	8	c.	c.	PROPN
ejpam-5634	333	9	boonpok	boonpok	PROPN
ejpam-5634	333	10	/	/	SYM
ejpam-5634	333	11	eur	eur	PROPN
ejpam-5634	333	12	.	.	PUNCT
ejpam-5634	334	1	j.	j.	PROPN
ejpam-5634	334	2	pure	pure	PROPN
ejpam-5634	334	3	appl	appl	PROPN
ejpam-5634	334	4	.	.	PROPN
ejpam-5634	334	5	math	math	PROPN
ejpam-5634	334	6	,	,	PUNCT
ejpam-5634	334	7	18	18	NUM
ejpam-5634	334	8	(	(	PUNCT
ejpam-5634	334	9	1	1	NUM
ejpam-5634	334	10	)	)	PUNCT
ejpam-5634	334	11	(	(	PUNCT
ejpam-5634	334	12	2025	2025	NUM
ejpam-5634	334	13	)	)	PUNCT
ejpam-5634	334	14	,	,	PUNCT
ejpam-5634	334	15	5634	5634	NUM
ejpam-5634	334	16	11	11	NUM
ejpam-5634	334	17	of	of	ADP
ejpam-5634	334	18	12	12	NUM
ejpam-5634	334	19	precontinuous	precontinuous	ADJ
ejpam-5634	334	20	multifunctions	multifunction	NOUN
ejpam-5634	334	21	.	.	PUNCT
ejpam-5634	335	1	journal	journal	PROPN
ejpam-5634	335	2	of	of	ADP
ejpam-5634	335	3	mathematics	mathematics	PROPN
ejpam-5634	335	4	and	and	CCONJ
ejpam-5634	335	5	computer	computer	NOUN
ejpam-5634	335	6	science	science	NOUN
ejpam-5634	335	7	,	,	PUNCT
ejpam-5634	335	8	18:282–293	18:282–293	NUM
ejpam-5634	335	9	,	,	PUNCT
ejpam-5634	335	10	2018	2018	NUM
ejpam-5634	335	11	.	.	PUNCT
ejpam-5634	336	1	[	[	X
ejpam-5634	336	2	30	30	NUM
ejpam-5634	336	3	]	]	X
ejpam-5634	336	4	m.	m.	NOUN
ejpam-5634	336	5	chiangpradit	chiangpradit	NOUN
ejpam-5634	336	6	,	,	PUNCT
ejpam-5634	336	7	s.	s.	PROPN
ejpam-5634	336	8	sompong	sompong	PROPN
ejpam-5634	336	9	,	,	PUNCT
ejpam-5634	336	10	and	and	CCONJ
ejpam-5634	336	11	c.	c.	PROPN
ejpam-5634	336	12	boonpok	boonpok	PROPN
ejpam-5634	336	13	.	.	PUNCT
ejpam-5634	337	1	on	on	ADP
ejpam-5634	337	2	characterizations	characterization	NOUN
ejpam-5634	337	3	of	of	ADP
ejpam-5634	337	4	(	(	PUNCT
ejpam-5634	337	5	τ1	τ1	NOUN
ejpam-5634	337	6	,	,	PUNCT
ejpam-5634	337	7	τ2)regular	τ2)regular	ADJ
ejpam-5634	337	8	spaces	space	NOUN
ejpam-5634	337	9	.	.	PUNCT
ejpam-5634	338	1	international	international	ADJ
ejpam-5634	338	2	journal	journal	PROPN
ejpam-5634	338	3	of	of	ADP
ejpam-5634	338	4	mathematics	mathematic	NOUN
ejpam-5634	338	5	and	and	CCONJ
ejpam-5634	338	6	computer	computer	NOUN
ejpam-5634	338	7	science	science	NOUN
ejpam-5634	338	8	,	,	PUNCT
ejpam-5634	338	9	19(4):125	19(4):125	PROPN
ejpam-5634	338	10	,	,	PUNCT
ejpam-5634	338	11	2024	2024	NUM
ejpam-5634	338	12	.	.	PUNCT
ejpam-5634	339	1	[	[	X
ejpam-5634	339	2	31	31	NUM
ejpam-5634	339	3	]	]	PUNCT
ejpam-5634	339	4	m.	m.	NOUN
ejpam-5634	339	5	chiangpradit	chiangpradit	NOUN
ejpam-5634	339	6	,	,	PUNCT
ejpam-5634	339	7	s.	s.	PROPN
ejpam-5634	339	8	sompong	sompong	PROPN
ejpam-5634	339	9	,	,	PUNCT
ejpam-5634	339	10	and	and	CCONJ
ejpam-5634	339	11	c.	c.	PROPN
ejpam-5634	339	12	boonpok	boonpok	PROPN
ejpam-5634	339	13	.	.	PUNCT
ejpam-5634	340	1	weakly	weakly	ADJ
ejpam-5634	340	2	quasi	quasi	NOUN
ejpam-5634	340	3	(	(	PUNCT
ejpam-5634	340	4	τ1	τ1	PROPN
ejpam-5634	340	5	,	,	PUNCT
ejpam-5634	340	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	340	7	functions	function	NOUN
ejpam-5634	340	8	.	.	PUNCT
ejpam-5634	341	1	international	international	ADJ
ejpam-5634	341	2	journal	journal	NOUN
ejpam-5634	341	3	of	of	ADP
ejpam-5634	341	4	analysis	analysis	NOUN
ejpam-5634	341	5	and	and	CCONJ
ejpam-5634	341	6	applications	application	NOUN
ejpam-5634	341	7	,	,	PUNCT
ejpam-5634	341	8	22:125	22:125	NUM
ejpam-5634	341	9	,	,	PUNCT
ejpam-5634	341	10	2024	2024	NUM
ejpam-5634	341	11	.	.	PUNCT
ejpam-5634	342	1	[	[	X
ejpam-5634	342	2	32	32	NUM
ejpam-5634	342	3	]	]	PUNCT
ejpam-5634	342	4	t.	t.	PROPN
ejpam-5634	342	5	duangphui	duangphui	PROPN
ejpam-5634	342	6	,	,	PUNCT
ejpam-5634	342	7	c.	c.	PROPN
ejpam-5634	342	8	boonpok	boonpok	PROPN
ejpam-5634	342	9	,	,	PUNCT
ejpam-5634	342	10	and	and	CCONJ
ejpam-5634	342	11	c.	c.	PROPN
ejpam-5634	342	12	viriyapong	viriyapong	PROPN
ejpam-5634	342	13	.	.	PUNCT
ejpam-5634	343	1	continuous	continuous	ADJ
ejpam-5634	343	2	functions	function	NOUN
ejpam-5634	343	3	on	on	ADP
ejpam-5634	343	4	bigeneralized	bigeneralize	VERB
ejpam-5634	343	5	topological	topological	ADJ
ejpam-5634	343	6	spaces	space	NOUN
ejpam-5634	343	7	.	.	PUNCT
ejpam-5634	344	1	international	international	ADJ
ejpam-5634	344	2	journal	journal	PROPN
ejpam-5634	344	3	of	of	ADP
ejpam-5634	344	4	mathematical	mathematical	ADJ
ejpam-5634	344	5	analysis	analysis	NOUN
ejpam-5634	344	6	,	,	PUNCT
ejpam-5634	344	7	5(24):1165	5(24):1165	NUM
ejpam-5634	344	8	–	–	PUNCT
ejpam-5634	344	9	1174	1174	NUM
ejpam-5634	344	10	,	,	PUNCT
ejpam-5634	344	11	2011	2011	NUM
ejpam-5634	344	12	.	.	PUNCT
ejpam-5634	345	1	[	[	X
ejpam-5634	345	2	33	33	NUM
ejpam-5634	345	3	]	]	PUNCT
ejpam-5634	345	4	t.	t.	NOUN
ejpam-5634	345	5	dungthaisong	dungthaisong	PROPN
ejpam-5634	345	6	,	,	PUNCT
ejpam-5634	345	7	c.	c.	PROPN
ejpam-5634	345	8	boonpok	boonpok	PROPN
ejpam-5634	345	9	,	,	PUNCT
ejpam-5634	345	10	and	and	CCONJ
ejpam-5634	345	11	c.	c.	PROPN
ejpam-5634	345	12	viriyapong	viriyapong	PROPN
ejpam-5634	345	13	.	.	PUNCT
ejpam-5634	346	1	generalized	generalize	VERB
ejpam-5634	346	2	closed	close	VERB
ejpam-5634	346	3	sets	set	NOUN
ejpam-5634	346	4	in	in	ADP
ejpam-5634	346	5	bigeneralized	bigeneralize	VERB
ejpam-5634	346	6	topological	topological	ADJ
ejpam-5634	346	7	spaces	space	NOUN
ejpam-5634	346	8	.	.	PUNCT
ejpam-5634	347	1	international	international	ADJ
ejpam-5634	347	2	journal	journal	PROPN
ejpam-5634	347	3	of	of	ADP
ejpam-5634	347	4	mathematical	mathematical	ADJ
ejpam-5634	347	5	analysis	analysis	NOUN
ejpam-5634	347	6	,	,	PUNCT
ejpam-5634	347	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-5634	347	8	,	,	PUNCT
ejpam-5634	347	9	2011	2011	NUM
ejpam-5634	347	10	.	.	PUNCT
ejpam-5634	348	1	[	[	X
ejpam-5634	348	2	34	34	NUM
ejpam-5634	348	3	]	]	PUNCT
ejpam-5634	348	4	m.	m.	NOUN
ejpam-5634	348	5	e.	e.	PROPN
ejpam-5634	348	6	abd	abd	PROPN
ejpam-5634	349	1	el	el	PROPN
ejpam-5634	349	2	-	-	PROPN
ejpam-5634	349	3	monsef	monsef	PROPN
ejpam-5634	349	4	,	,	PUNCT
ejpam-5634	349	5	s.	s.	PROPN
ejpam-5634	349	6	n.	n.	PROPN
ejpam-5634	349	7	el	el	PROPN
ejpam-5634	349	8	-	-	PROPN
ejpam-5634	349	9	deeb	deeb	PROPN
ejpam-5634	349	10	,	,	PUNCT
ejpam-5634	349	11	and	and	CCONJ
ejpam-5634	349	12	r.	r.	PROPN
ejpam-5634	349	13	a.	a.	PROPN
ejpam-5634	349	14	mahmoud	mahmoud	PROPN
ejpam-5634	349	15	.	.	PUNCT
ejpam-5634	350	1	β	β	X
ejpam-5634	350	2	-	-	ADJ
ejpam-5634	350	3	open	open	ADJ
ejpam-5634	350	4	sets	set	NOUN
ejpam-5634	350	5	and	and	CCONJ
ejpam-5634	350	6	βcontinuous	βcontinuous	ADJ
ejpam-5634	350	7	mappings	mapping	NOUN
ejpam-5634	350	8	.	.	PUNCT
ejpam-5634	351	1	bulletin	bulletin	NOUN
ejpam-5634	351	2	of	of	ADP
ejpam-5634	351	3	the	the	DET
ejpam-5634	351	4	faculty	faculty	NOUN
ejpam-5634	351	5	of	of	ADP
ejpam-5634	351	6	science	science	NOUN
ejpam-5634	351	7	.	.	PUNCT
ejpam-5634	352	1	assiut	assiut	PROPN
ejpam-5634	352	2	university	university	PROPN
ejpam-5634	352	3	.	.	PUNCT
ejpam-5634	352	4	,	,	PUNCT
ejpam-5634	352	5	12:77–90	12:77–90	NUM
ejpam-5634	352	6	,	,	PUNCT
ejpam-5634	352	7	1983	1983	NUM
ejpam-5634	352	8	.	.	PUNCT
ejpam-5634	353	1	[	[	X
ejpam-5634	353	2	35	35	NUM
ejpam-5634	353	3	]	]	X
ejpam-5634	353	4	j.	j.	PROPN
ejpam-5634	353	5	ewert	ewert	PROPN
ejpam-5634	353	6	and	and	CCONJ
ejpam-5634	353	7	t.	t.	PROPN
ejpam-5634	353	8	lipski	lipski	PROPN
ejpam-5634	353	9	.	.	PUNCT
ejpam-5634	354	1	on	on	ADP
ejpam-5634	354	2	s	s	NOUN
ejpam-5634	354	3	-	-	PUNCT
ejpam-5634	354	4	quasi	quasi	ADJ
ejpam-5634	354	5	-	-	ADJ
ejpam-5634	354	6	continuous	continuous	ADJ
ejpam-5634	354	7	multivalued	multivalued	ADJ
ejpam-5634	354	8	maps	map	NOUN
ejpam-5634	354	9	.	.	PUNCT
ejpam-5634	355	1	review	review	NOUN
ejpam-5634	355	2	of	of	ADP
ejpam-5634	355	3	research	research	NOUN
ejpam-5634	355	4	,	,	PUNCT
ejpam-5634	355	5	faculty	faculty	NOUN
ejpam-5634	355	6	of	of	ADP
ejpam-5634	355	7	science	science	NOUN
ejpam-5634	355	8	,	,	PUNCT
ejpam-5634	355	9	mathematics	mathematics	NOUN
ejpam-5634	355	10	series	series	NOUN
ejpam-5634	355	11	,	,	PUNCT
ejpam-5634	355	12	20(1):167–183	20(1):167–183	PROPN
ejpam-5634	355	13	,	,	PUNCT
ejpam-5634	355	14	1990	1990	NUM
ejpam-5634	355	15	.	.	PUNCT
ejpam-5634	356	1	[	[	X
ejpam-5634	356	2	36	36	NUM
ejpam-5634	356	3	]	]	X
ejpam-5634	356	4	j.	j.	PROPN
ejpam-5634	356	5	khampakdee	khampakdee	PROPN
ejpam-5634	356	6	and	and	CCONJ
ejpam-5634	356	7	c.	c.	PROPN
ejpam-5634	356	8	boonpok	boonpok	PROPN
ejpam-5634	356	9	.	.	PUNCT
ejpam-5634	357	1	upper	upper	ADJ
ejpam-5634	357	2	and	and	CCONJ
ejpam-5634	357	3	lower	low	ADJ
ejpam-5634	357	4	α(λ	α(λ	PROPN
ejpam-5634	357	5	,	,	PUNCT
ejpam-5634	357	6	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	357	7	multifunctions	multifunction	NOUN
ejpam-5634	357	8	.	.	PUNCT
ejpam-5634	358	1	wseas	wseas	VERB
ejpam-5634	358	2	transactions	transaction	NOUN
ejpam-5634	358	3	on	on	ADP
ejpam-5634	358	4	mathematics	mathematic	NOUN
ejpam-5634	358	5	,	,	PUNCT
ejpam-5634	358	6	21:684–690	21:684–690	NUM
ejpam-5634	358	7	,	,	PUNCT
ejpam-5634	358	8	2022	2022	NUM
ejpam-5634	358	9	.	.	PUNCT
ejpam-5634	359	1	[	[	X
ejpam-5634	359	2	37	37	NUM
ejpam-5634	359	3	]	]	X
ejpam-5634	359	4	j.	j.	PROPN
ejpam-5634	359	5	khampakdee	khampakdee	PROPN
ejpam-5634	359	6	,	,	PUNCT
ejpam-5634	359	7	s.	s.	PROPN
ejpam-5634	359	8	sompong	sompong	PROPN
ejpam-5634	359	9	,	,	PUNCT
ejpam-5634	359	10	and	and	CCONJ
ejpam-5634	359	11	c.	c.	PROPN
ejpam-5634	359	12	boonpok	boonpok	PROPN
ejpam-5634	359	13	.	.	PUNCT
ejpam-5634	360	1	c-(τ1	c-(τ1	PROPN
ejpam-5634	360	2	,	,	PUNCT
ejpam-5634	360	3	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5634	360	4	for	for	ADP
ejpam-5634	360	5	multifunctions	multifunction	NOUN
ejpam-5634	360	6	.	.	PUNCT
ejpam-5634	361	1	european	european	ADJ
ejpam-5634	361	2	journal	journal	PROPN
ejpam-5634	361	3	of	of	ADP
ejpam-5634	361	4	pure	pure	ADJ
ejpam-5634	361	5	and	and	CCONJ
ejpam-5634	361	6	applied	applied	ADJ
ejpam-5634	361	7	mathematics	mathematic	NOUN
ejpam-5634	361	8	,	,	PUNCT
ejpam-5634	361	9	17(3):2289–2299	17(3):2289–2299	NUM
ejpam-5634	361	10	,	,	PUNCT
ejpam-5634	361	11	2024	2024	NUM
ejpam-5634	361	12	.	.	PUNCT
ejpam-5634	362	1	[	[	X
ejpam-5634	362	2	38	38	NUM
ejpam-5634	362	3	]	]	PUNCT
ejpam-5634	362	4	c.	c.	PROPN
ejpam-5634	362	5	klanarong	klanarong	PROPN
ejpam-5634	362	6	,	,	PUNCT
ejpam-5634	362	7	s.	s.	PROPN
ejpam-5634	362	8	sompong	sompong	PROPN
ejpam-5634	362	9	,	,	PUNCT
ejpam-5634	362	10	and	and	CCONJ
ejpam-5634	362	11	c.	c.	PROPN
ejpam-5634	362	12	boonpok	boonpok	PROPN
ejpam-5634	362	13	.	.	PUNCT
ejpam-5634	363	1	upper	upper	ADJ
ejpam-5634	363	2	and	and	CCONJ
ejpam-5634	363	3	lower	low	ADJ
ejpam-5634	363	4	almost	almost	ADV
ejpam-5634	363	5	(	(	PUNCT
ejpam-5634	363	6	τ1	τ1	NOUN
ejpam-5634	363	7	,	,	PUNCT
ejpam-5634	363	8	τ2)continuous	τ2)continuous	ADJ
ejpam-5634	363	9	multifunctions	multifunction	NOUN
ejpam-5634	363	10	.	.	PUNCT
ejpam-5634	364	1	european	european	ADJ
ejpam-5634	364	2	journal	journal	PROPN
ejpam-5634	364	3	of	of	ADP
ejpam-5634	364	4	pure	pure	ADJ
ejpam-5634	364	5	and	and	CCONJ
ejpam-5634	364	6	applied	applied	ADJ
ejpam-5634	364	7	mathematics	mathematic	NOUN
ejpam-5634	364	8	,	,	PUNCT
ejpam-5634	364	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-5634	364	10	,	,	PUNCT
ejpam-5634	364	11	2024	2024	NUM
ejpam-5634	364	12	.	.	PUNCT
ejpam-5634	365	1	[	[	X
ejpam-5634	365	2	39	39	NUM
ejpam-5634	365	3	]	]	PUNCT
ejpam-5634	365	4	j.	j.	PROPN
ejpam-5634	365	5	k.	k.	PROPN
ejpam-5634	365	6	kohli	kohli	PROPN
ejpam-5634	365	7	.	.	PUNCT
ejpam-5634	366	1	a	a	DET
ejpam-5634	366	2	class	class	NOUN
ejpam-5634	366	3	of	of	ADP
ejpam-5634	366	4	mappings	mapping	NOUN
ejpam-5634	366	5	containing	contain	VERB
ejpam-5634	366	6	all	all	PRON
ejpam-5634	366	7	continuous	continuous	ADJ
ejpam-5634	366	8	and	and	CCONJ
ejpam-5634	366	9	all	all	DET
ejpam-5634	366	10	semi	semi	ADJ
ejpam-5634	366	11	-	-	ADJ
ejpam-5634	366	12	connected	connected	ADJ
ejpam-5634	366	13	mappings	mapping	NOUN
ejpam-5634	366	14	.	.	PUNCT
ejpam-5634	367	1	proceedings	proceeding	NOUN
ejpam-5634	367	2	of	of	ADP
ejpam-5634	367	3	the	the	DET
ejpam-5634	367	4	american	american	PROPN
ejpam-5634	367	5	mathematical	mathematical	PROPN
ejpam-5634	367	6	society	society	NOUN
ejpam-5634	367	7	,	,	PUNCT
ejpam-5634	367	8	72:175–181	72:175–181	PROPN
ejpam-5634	367	9	,	,	PUNCT
ejpam-5634	367	10	1978	1978	NUM
ejpam-5634	367	11	.	.	PUNCT
ejpam-5634	368	1	[	[	X
ejpam-5634	368	2	40	40	NUM
ejpam-5634	368	3	]	]	PUNCT
ejpam-5634	368	4	j.	j.	PROPN
ejpam-5634	368	5	k.	k.	PROPN
ejpam-5634	368	6	kohli	kohli	PROPN
ejpam-5634	368	7	.	.	PUNCT
ejpam-5634	369	1	s	s	X
ejpam-5634	369	2	-	-	ADJ
ejpam-5634	369	3	continuous	continuous	ADJ
ejpam-5634	369	4	functions	function	NOUN
ejpam-5634	369	5	and	and	CCONJ
ejpam-5634	369	6	certain	certain	ADJ
ejpam-5634	369	7	weak	weak	ADJ
ejpam-5634	369	8	forms	form	NOUN
ejpam-5634	369	9	of	of	ADP
ejpam-5634	369	10	regularity	regularity	NOUN
ejpam-5634	369	11	and	and	CCONJ
ejpam-5634	369	12	complete	complete	ADJ
ejpam-5634	369	13	regularity	regularity	NOUN
ejpam-5634	369	14	.	.	PUNCT
ejpam-5634	370	1	mathematische	mathematische	PROPN
ejpam-5634	370	2	nachrichten	nachrichten	PROPN
ejpam-5634	370	3	,	,	PUNCT
ejpam-5634	370	4	97:189–196	97:189–196	NUM
ejpam-5634	370	5	,	,	PUNCT
ejpam-5634	370	6	1980	1980	NUM
ejpam-5634	370	7	.	.	PUNCT
ejpam-5634	371	1	[	[	X
ejpam-5634	371	2	41	41	NUM
ejpam-5634	371	3	]	]	X
ejpam-5634	371	4	b.	b.	PROPN
ejpam-5634	371	5	kong	kong	PROPN
ejpam-5634	371	6	-	-	PUNCT
ejpam-5634	371	7	ied	ied	PROPN
ejpam-5634	371	8	,	,	PUNCT
ejpam-5634	371	9	s.	s.	PROPN
ejpam-5634	371	10	sompong	sompong	PROPN
ejpam-5634	371	11	,	,	PUNCT
ejpam-5634	371	12	and	and	CCONJ
ejpam-5634	371	13	c.	c.	PROPN
ejpam-5634	371	14	boonpok	boonpok	PROPN
ejpam-5634	371	15	.	.	PUNCT
ejpam-5634	372	1	almost	almost	ADV
ejpam-5634	372	2	quasi	quasi	X
ejpam-5634	372	3	(	(	PUNCT
ejpam-5634	372	4	τ1	τ1	NOUN
ejpam-5634	372	5	,	,	PUNCT
ejpam-5634	372	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	372	7	functions	function	NOUN
ejpam-5634	372	8	.	.	PUNCT
ejpam-5634	373	1	asia	asia	PROPN
ejpam-5634	373	2	pacific	pacific	PROPN
ejpam-5634	373	3	journal	journal	PROPN
ejpam-5634	373	4	of	of	ADP
ejpam-5634	373	5	mathematics	mathematic	NOUN
ejpam-5634	373	6	,	,	PUNCT
ejpam-5634	373	7	11:64	11:64	NUM
ejpam-5634	373	8	,	,	PUNCT
ejpam-5634	373	9	2024	2024	NUM
ejpam-5634	373	10	.	.	PUNCT
ejpam-5634	374	1	[	[	X
ejpam-5634	374	2	42	42	NUM
ejpam-5634	374	3	]	]	X
ejpam-5634	374	4	y.	y.	PROPN
ejpam-5634	374	5	l.	l.	PROPN
ejpam-5634	374	6	lee	lee	PROPN
ejpam-5634	374	7	.	.	PUNCT
ejpam-5634	375	1	some	some	DET
ejpam-5634	375	2	characterizations	characterization	NOUN
ejpam-5634	375	3	of	of	ADP
ejpam-5634	375	4	semilocally	semilocally	ADV
ejpam-5634	375	5	connected	connected	ADJ
ejpam-5634	375	6	spaces	space	NOUN
ejpam-5634	375	7	.	.	PUNCT
ejpam-5634	376	1	proceedings	proceeding	NOUN
ejpam-5634	376	2	of	of	ADP
ejpam-5634	376	3	the	the	DET
ejpam-5634	376	4	american	american	PROPN
ejpam-5634	376	5	mathematical	mathematical	PROPN
ejpam-5634	376	6	society	society	NOUN
ejpam-5634	376	7	,	,	PUNCT
ejpam-5634	376	8	16:1318–1320	16:1318–1320	NUM
ejpam-5634	376	9	,	,	PUNCT
ejpam-5634	376	10	1965	1965	NUM
ejpam-5634	376	11	.	.	PUNCT
ejpam-5634	377	1	[	[	X
ejpam-5634	377	2	43	43	NUM
ejpam-5634	377	3	]	]	X
ejpam-5634	377	4	n.	n.	PROPN
ejpam-5634	377	5	levine	levine	PROPN
ejpam-5634	377	6	.	.	PUNCT
ejpam-5634	378	1	semi	semi	ADJ
ejpam-5634	378	2	-	-	ADJ
ejpam-5634	378	3	open	open	ADJ
ejpam-5634	378	4	sets	set	NOUN
ejpam-5634	378	5	and	and	CCONJ
ejpam-5634	378	6	semi	semi	ADJ
ejpam-5634	378	7	-	-	NOUN
ejpam-5634	378	8	continuity	continuity	NOUN
ejpam-5634	378	9	in	in	ADP
ejpam-5634	378	10	topological	topological	ADJ
ejpam-5634	378	11	spaces	space	NOUN
ejpam-5634	378	12	.	.	PUNCT
ejpam-5634	379	1	the	the	DET
ejpam-5634	379	2	american	american	PROPN
ejpam-5634	379	3	mathematical	mathematical	PROPN
ejpam-5634	379	4	monthly	monthly	ADV
ejpam-5634	379	5	,	,	PUNCT
ejpam-5634	379	6	70:36–41	70:36–41	NUM
ejpam-5634	379	7	,	,	PUNCT
ejpam-5634	379	8	1963	1963	NUM
ejpam-5634	379	9	.	.	PUNCT
ejpam-5634	380	1	[	[	X
ejpam-5634	380	2	44	44	NUM
ejpam-5634	380	3	]	]	PUNCT
ejpam-5634	380	4	t.	t.	NOUN
ejpam-5634	380	5	lipski	lipski	PROPN
ejpam-5634	380	6	.	.	PUNCT
ejpam-5634	381	1	s	s	X
ejpam-5634	381	2	-	-	ADJ
ejpam-5634	381	3	continuous	continuous	ADJ
ejpam-5634	381	4	multivalued	multivalued	ADJ
ejpam-5634	381	5	maps	map	NOUN
ejpam-5634	381	6	.	.	PUNCT
ejpam-5634	382	1	mathematical	mathematical	ADJ
ejpam-5634	382	2	chronicle	chronicle	PROPN
ejpam-5634	382	3	,	,	PUNCT
ejpam-5634	382	4	18:57–61	18:57–61	PROPN
ejpam-5634	382	5	,	,	PUNCT
ejpam-5634	382	6	1989	1989	NUM
ejpam-5634	382	7	.	.	PUNCT
ejpam-5634	383	1	[	[	X
ejpam-5634	383	2	45	45	NUM
ejpam-5634	383	3	]	]	PUNCT
ejpam-5634	383	4	a.	a.	NOUN
ejpam-5634	383	5	s.	s.	PROPN
ejpam-5634	383	6	mashhour	mashhour	PROPN
ejpam-5634	383	7	,	,	PUNCT
ejpam-5634	383	8	m.	m.	PROPN
ejpam-5634	383	9	e.	e.	PROPN
ejpam-5634	383	10	abd	abd	PROPN
ejpam-5634	383	11	el	el	PROPN
ejpam-5634	383	12	-	-	PROPN
ejpam-5634	383	13	monsef	monsef	ADJ
ejpam-5634	383	14	,	,	PUNCT
ejpam-5634	383	15	and	and	CCONJ
ejpam-5634	383	16	s.	s.	PROPN
ejpam-5634	383	17	n.	n.	PROPN
ejpam-5634	383	18	el	el	PROPN
ejpam-5634	383	19	-	-	PROPN
ejpam-5634	383	20	deeb	deeb	PROPN
ejpam-5634	383	21	.	.	PUNCT
ejpam-5634	384	1	on	on	ADP
ejpam-5634	384	2	precontinuous	precontinuous	ADJ
ejpam-5634	384	3	and	and	CCONJ
ejpam-5634	384	4	weak	weak	ADJ
ejpam-5634	384	5	precontinuous	precontinuous	ADJ
ejpam-5634	384	6	mappings	mapping	NOUN
ejpam-5634	384	7	.	.	PUNCT
ejpam-5634	385	1	proceedings	proceeding	NOUN
ejpam-5634	385	2	of	of	ADP
ejpam-5634	385	3	the	the	DET
ejpam-5634	385	4	mathematical	mathematical	ADJ
ejpam-5634	385	5	and	and	CCONJ
ejpam-5634	385	6	physical	physical	ADJ
ejpam-5634	385	7	society	society	NOUN
ejpam-5634	385	8	of	of	ADP
ejpam-5634	385	9	egypt	egypt	PROPN
ejpam-5634	385	10	,	,	PUNCT
ejpam-5634	385	11	53:47–53	53:47–53	NUM
ejpam-5634	385	12	,	,	PUNCT
ejpam-5634	385	13	1982	1982	NUM
ejpam-5634	385	14	.	.	PUNCT
ejpam-5634	386	1	[	[	X
ejpam-5634	386	2	46	46	NUM
ejpam-5634	386	3	]	]	X
ejpam-5634	386	4	o.	o.	NOUN
ejpam-5634	386	5	nj̊astad	nj̊astad	NOUN
ejpam-5634	386	6	.	.	PUNCT
ejpam-5634	387	1	on	on	ADP
ejpam-5634	387	2	some	some	DET
ejpam-5634	387	3	classes	class	NOUN
ejpam-5634	387	4	of	of	ADP
ejpam-5634	387	5	nearly	nearly	ADV
ejpam-5634	387	6	open	open	ADJ
ejpam-5634	387	7	sets	set	NOUN
ejpam-5634	387	8	.	.	PUNCT
ejpam-5634	388	1	pasific	pasific	PROPN
ejpam-5634	388	2	journal	journal	PROPN
ejpam-5634	388	3	of	of	ADP
ejpam-5634	388	4	mathematics	mathematic	NOUN
ejpam-5634	388	5	,	,	PUNCT
ejpam-5634	388	6	15:961–970	15:961–970	PROPN
ejpam-5634	388	7	,	,	PUNCT
ejpam-5634	388	8	1965	1965	NUM
ejpam-5634	388	9	.	.	PUNCT
ejpam-5634	389	1	[	[	X
ejpam-5634	389	2	47	47	NUM
ejpam-5634	389	3	]	]	PUNCT
ejpam-5634	389	4	v.	v.	CCONJ
ejpam-5634	389	5	popa	popa	NOUN
ejpam-5634	389	6	.	.	PUNCT
ejpam-5634	390	1	some	some	DET
ejpam-5634	390	2	properties	property	NOUN
ejpam-5634	390	3	of	of	ADP
ejpam-5634	390	4	h	h	NOUN
ejpam-5634	390	5	-	-	PUNCT
ejpam-5634	390	6	almost	almost	ADV
ejpam-5634	390	7	continuous	continuous	ADJ
ejpam-5634	390	8	multifunctions	multifunction	NOUN
ejpam-5634	390	9	.	.	PUNCT
ejpam-5634	391	1	problemy	problemy	PROPN
ejpam-5634	391	2	matematyczne	matematyczne	PROPN
ejpam-5634	391	3	,	,	PUNCT
ejpam-5634	391	4	10:9–26	10:9–26	NUM
ejpam-5634	391	5	,	,	PUNCT
ejpam-5634	391	6	1988	1988	NUM
ejpam-5634	391	7	.	.	PUNCT
ejpam-5634	392	1	m.	m.	NOUN
ejpam-5634	392	2	chiangpradit	chiangpradit	PROPN
ejpam-5634	392	3	,	,	PUNCT
ejpam-5634	392	4	a.	a.	PROPN
ejpam-5634	392	5	sama	sama	PROPN
ejpam-5634	392	6	-	-	PUNCT
ejpam-5634	392	7	ae	ae	PROPN
ejpam-5634	392	8	,	,	PUNCT
ejpam-5634	392	9	c.	c.	PROPN
ejpam-5634	392	10	boonpok	boonpok	PROPN
ejpam-5634	392	11	/	/	SYM
ejpam-5634	392	12	eur	eur	PROPN
ejpam-5634	392	13	.	.	PUNCT
ejpam-5634	393	1	j.	j.	PROPN
ejpam-5634	393	2	pure	pure	PROPN
ejpam-5634	393	3	appl	appl	PROPN
ejpam-5634	393	4	.	.	PROPN
ejpam-5634	393	5	math	math	PROPN
ejpam-5634	393	6	,	,	PUNCT
ejpam-5634	393	7	18	18	NUM
ejpam-5634	393	8	(	(	PUNCT
ejpam-5634	393	9	1	1	NUM
ejpam-5634	393	10	)	)	PUNCT
ejpam-5634	393	11	(	(	PUNCT
ejpam-5634	393	12	2025	2025	NUM
ejpam-5634	393	13	)	)	PUNCT
ejpam-5634	393	14	,	,	PUNCT
ejpam-5634	393	15	5634	5634	NUM
ejpam-5634	393	16	12	12	NUM
ejpam-5634	393	17	of	of	ADP
ejpam-5634	393	18	12	12	NUM
ejpam-5634	394	1	[	[	X
ejpam-5634	394	2	48	48	NUM
ejpam-5634	394	3	]	]	PUNCT
ejpam-5634	394	4	v.	v.	CCONJ
ejpam-5634	394	5	popa	popa	NOUN
ejpam-5634	394	6	and	and	CCONJ
ejpam-5634	394	7	t.	t.	PROPN
ejpam-5634	394	8	noiri	noiri	PROPN
ejpam-5634	394	9	.	.	PUNCT
ejpam-5634	395	1	a	a	DET
ejpam-5634	395	2	unified	unified	ADJ
ejpam-5634	395	3	theory	theory	NOUN
ejpam-5634	395	4	for	for	ADP
ejpam-5634	395	5	s	s	NOUN
ejpam-5634	395	6	-	-	NOUN
ejpam-5634	395	7	continuity	continuity	NOUN
ejpam-5634	395	8	of	of	ADP
ejpam-5634	395	9	multifunctions	multifunction	NOUN
ejpam-5634	395	10	.	.	PUNCT
ejpam-5634	396	1	i̇stanbul	i̇stanbul	PUNCT
ejpam-5634	397	1	üniversitesi	üniversitesi	PROPN
ejpam-5634	397	2	.	.	PUNCT
ejpam-5634	398	1	fen	fen	PROPN
ejpam-5634	398	2	fakültesi	fakültesi	PROPN
ejpam-5634	398	3	.	.	PROPN
ejpam-5634	398	4	matematik	matematik	PROPN
ejpam-5634	398	5	dergisi	dergisi	PROPN
ejpam-5634	398	6	,	,	PUNCT
ejpam-5634	398	7	59:1–15	59:1–15	NUM
ejpam-5634	398	8	,	,	PUNCT
ejpam-5634	398	9	2000	2000	NUM
ejpam-5634	398	10	.	.	PUNCT
ejpam-5634	399	1	[	[	X
ejpam-5634	399	2	49	49	X
ejpam-5634	399	3	]	]	PUNCT
ejpam-5634	399	4	v.	v.	CCONJ
ejpam-5634	399	5	popa	popa	NOUN
ejpam-5634	399	6	and	and	CCONJ
ejpam-5634	399	7	t.	t.	PROPN
ejpam-5634	399	8	noiri	noiri	PROPN
ejpam-5634	399	9	.	.	PUNCT
ejpam-5634	400	1	on	on	ADP
ejpam-5634	400	2	s	s	NOUN
ejpam-5634	400	3	-	-	PUNCT
ejpam-5634	400	4	β	β	NOUN
ejpam-5634	400	5	-	-	ADJ
ejpam-5634	400	6	continuous	continuous	ADJ
ejpam-5634	400	7	multifunctions	multifunction	NOUN
ejpam-5634	400	8	.	.	PUNCT
ejpam-5634	401	1	journal	journal	NOUN
ejpam-5634	401	2	of	of	ADP
ejpam-5634	401	3	the	the	DET
ejpam-5634	401	4	egyptian	egyptian	PROPN
ejpam-5634	401	5	mathematical	mathematical	PROPN
ejpam-5634	401	6	society	society	NOUN
ejpam-5634	401	7	,	,	PUNCT
ejpam-5634	401	8	8:127–137	8:127–137	NUM
ejpam-5634	401	9	,	,	PUNCT
ejpam-5634	401	10	2000	2000	NUM
ejpam-5634	401	11	.	.	PUNCT
ejpam-5634	402	1	[	[	X
ejpam-5634	402	2	50	50	NUM
ejpam-5634	402	3	]	]	PUNCT
ejpam-5634	402	4	v.	v.	CCONJ
ejpam-5634	402	5	popa	popa	NOUN
ejpam-5634	402	6	and	and	CCONJ
ejpam-5634	402	7	t.	t.	PROPN
ejpam-5634	402	8	noiri	noiri	PROPN
ejpam-5634	402	9	.	.	PUNCT
ejpam-5634	403	1	on	on	ADP
ejpam-5634	403	2	s	s	NOUN
ejpam-5634	403	3	-	-	ADJ
ejpam-5634	403	4	precontinuous	precontinuous	ADJ
ejpam-5634	403	5	multifunction	multifunction	NOUN
ejpam-5634	403	6	.	.	PUNCT
ejpam-5634	404	1	demonstratio	demonstratio	PROPN
ejpam-5634	404	2	mathematica	mathematica	PROPN
ejpam-5634	404	3	,	,	PUNCT
ejpam-5634	404	4	33(3):679–687	33(3):679–687	PROPN
ejpam-5634	404	5	,	,	PUNCT
ejpam-5634	404	6	2000	2000	NUM
ejpam-5634	404	7	.	.	PUNCT
ejpam-5634	405	1	[	[	X
ejpam-5634	405	2	51	51	NUM
ejpam-5634	405	3	]	]	X
ejpam-5634	405	4	p.	p.	NOUN
ejpam-5634	405	5	pue	pue	NOUN
ejpam-5634	405	6	-	-	PUNCT
ejpam-5634	405	7	on	on	ADP
ejpam-5634	405	8	and	and	CCONJ
ejpam-5634	405	9	c.	c.	PROPN
ejpam-5634	405	10	boonpok	boonpok	PROPN
ejpam-5634	405	11	.	.	PUNCT
ejpam-5634	406	1	θ(λ	θ(λ	PROPN
ejpam-5634	406	2	,	,	PUNCT
ejpam-5634	406	3	p)-continuity	p)-continuity	NOUN
ejpam-5634	406	4	for	for	ADP
ejpam-5634	406	5	functions	function	NOUN
ejpam-5634	406	6	.	.	PUNCT
ejpam-5634	407	1	international	international	ADJ
ejpam-5634	407	2	journal	journal	NOUN
ejpam-5634	407	3	of	of	ADP
ejpam-5634	407	4	mathematics	mathematic	NOUN
ejpam-5634	407	5	and	and	CCONJ
ejpam-5634	407	6	computer	computer	NOUN
ejpam-5634	407	7	science	science	NOUN
ejpam-5634	407	8	,	,	PUNCT
ejpam-5634	407	9	19(2):491–495	19(2):491–495	NUM
ejpam-5634	407	10	,	,	PUNCT
ejpam-5634	407	11	2024	2024	NUM
ejpam-5634	407	12	.	.	PUNCT
ejpam-5634	408	1	[	[	X
ejpam-5634	408	2	52	52	NUM
ejpam-5634	408	3	]	]	PUNCT
ejpam-5634	408	4	p.	p.	NOUN
ejpam-5634	408	5	pue	pue	NOUN
ejpam-5634	408	6	-	-	PUNCT
ejpam-5634	408	7	on	on	ADP
ejpam-5634	408	8	,	,	PUNCT
ejpam-5634	408	9	a.	a.	PROPN
ejpam-5634	408	10	sama	sama	PROPN
ejpam-5634	408	11	-	-	PUNCT
ejpam-5634	408	12	ae	ae	PROPN
ejpam-5634	408	13	,	,	PUNCT
ejpam-5634	408	14	and	and	CCONJ
ejpam-5634	408	15	c.	c.	PROPN
ejpam-5634	408	16	boonpok	boonpok	PROPN
ejpam-5634	408	17	.	.	PUNCT
ejpam-5634	409	1	c	c	X
ejpam-5634	409	2	-	-	PUNCT
ejpam-5634	409	3	quasi	quasi	X
ejpam-5634	409	4	(	(	PUNCT
ejpam-5634	409	5	τ1	τ1	PROPN
ejpam-5634	409	6	,	,	PUNCT
ejpam-5634	409	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	409	8	multifunctions	multifunction	NOUN
ejpam-5634	409	9	.	.	PUNCT
ejpam-5634	410	1	european	european	ADJ
ejpam-5634	410	2	journal	journal	PROPN
ejpam-5634	410	3	of	of	ADP
ejpam-5634	410	4	pure	pure	ADJ
ejpam-5634	410	5	and	and	CCONJ
ejpam-5634	410	6	applied	applied	ADJ
ejpam-5634	410	7	mathematics	mathematic	NOUN
ejpam-5634	410	8	,	,	PUNCT
ejpam-5634	410	9	17(4):3242–3253	17(4):3242–3253	NUM
ejpam-5634	410	10	,	,	PUNCT
ejpam-5634	410	11	2024	2024	NUM
ejpam-5634	410	12	.	.	PUNCT
ejpam-5634	411	1	[	[	X
ejpam-5634	411	2	53	53	NUM
ejpam-5634	411	3	]	]	PUNCT
ejpam-5634	411	4	p.	p.	NOUN
ejpam-5634	411	5	pue	pue	NOUN
ejpam-5634	411	6	-	-	PUNCT
ejpam-5634	411	7	on	on	ADP
ejpam-5634	411	8	,	,	PUNCT
ejpam-5634	411	9	s.	s.	PROPN
ejpam-5634	411	10	sompong	sompong	PROPN
ejpam-5634	411	11	,	,	PUNCT
ejpam-5634	411	12	and	and	CCONJ
ejpam-5634	411	13	c.	c.	PROPN
ejpam-5634	411	14	boonpok	boonpok	PROPN
ejpam-5634	411	15	.	.	PUNCT
ejpam-5634	412	1	almost	almost	ADV
ejpam-5634	412	2	quasi	quasi	X
ejpam-5634	412	3	(	(	PUNCT
ejpam-5634	412	4	τ1	τ1	NOUN
ejpam-5634	412	5	,	,	PUNCT
ejpam-5634	412	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5634	412	7	for	for	ADP
ejpam-5634	412	8	multifunctions	multifunction	NOUN
ejpam-5634	412	9	.	.	PUNCT
ejpam-5634	413	1	international	international	ADJ
ejpam-5634	413	2	journal	journal	NOUN
ejpam-5634	413	3	of	of	ADP
ejpam-5634	413	4	analysis	analysis	NOUN
ejpam-5634	413	5	and	and	CCONJ
ejpam-5634	413	6	applications	application	NOUN
ejpam-5634	413	7	,	,	PUNCT
ejpam-5634	413	8	22:97	22:97	NUM
ejpam-5634	413	9	,	,	PUNCT
ejpam-5634	413	10	2024	2024	NUM
ejpam-5634	413	11	.	.	PUNCT
ejpam-5634	414	1	[	[	X
ejpam-5634	414	2	54	54	NUM
ejpam-5634	414	3	]	]	PUNCT
ejpam-5634	414	4	p.	p.	NOUN
ejpam-5634	414	5	pue	pue	NOUN
ejpam-5634	414	6	-	-	PUNCT
ejpam-5634	414	7	on	on	ADP
ejpam-5634	414	8	,	,	PUNCT
ejpam-5634	414	9	s.	s.	PROPN
ejpam-5634	414	10	sompong	sompong	PROPN
ejpam-5634	414	11	,	,	PUNCT
ejpam-5634	414	12	and	and	CCONJ
ejpam-5634	414	13	c.	c.	PROPN
ejpam-5634	414	14	boonpok	boonpok	PROPN
ejpam-5634	414	15	.	.	PUNCT
ejpam-5634	415	1	upper	upper	ADJ
ejpam-5634	415	2	and	and	CCONJ
ejpam-5634	415	3	lower	low	ADJ
ejpam-5634	415	4	(	(	PUNCT
ejpam-5634	415	5	τ1	τ1	NOUN
ejpam-5634	415	6	,	,	PUNCT
ejpam-5634	415	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	415	8	mulfunctions	mulfunction	NOUN
ejpam-5634	415	9	.	.	PUNCT
ejpam-5634	416	1	international	international	ADJ
ejpam-5634	416	2	journal	journal	NOUN
ejpam-5634	416	3	of	of	ADP
ejpam-5634	416	4	mathematics	mathematic	NOUN
ejpam-5634	416	5	and	and	CCONJ
ejpam-5634	416	6	computer	computer	NOUN
ejpam-5634	416	7	science	science	NOUN
ejpam-5634	416	8	,	,	PUNCT
ejpam-5634	416	9	19(4):1305	19(4):1305	NUM
ejpam-5634	416	10	–	–	PUNCT
ejpam-5634	416	11	1310	1310	NUM
ejpam-5634	416	12	,	,	PUNCT
ejpam-5634	416	13	2024	2024	NUM
ejpam-5634	416	14	.	.	PUNCT
ejpam-5634	417	1	[	[	X
ejpam-5634	417	2	55	55	NUM
ejpam-5634	417	3	]	]	X
ejpam-5634	417	4	p.	p.	NOUN
ejpam-5634	417	5	pue	pue	NOUN
ejpam-5634	417	6	-	-	PUNCT
ejpam-5634	417	7	on	on	ADP
ejpam-5634	417	8	,	,	PUNCT
ejpam-5634	417	9	s.	s.	PROPN
ejpam-5634	417	10	sompong	sompong	PROPN
ejpam-5634	417	11	,	,	PUNCT
ejpam-5634	417	12	and	and	CCONJ
ejpam-5634	417	13	c.	c.	PROPN
ejpam-5634	417	14	boonpok	boonpok	PROPN
ejpam-5634	417	15	.	.	PUNCT
ejpam-5634	418	1	weakly	weakly	ADJ
ejpam-5634	418	2	quasi	quasi	NOUN
ejpam-5634	418	3	(	(	PUNCT
ejpam-5634	418	4	τ1	τ1	PROPN
ejpam-5634	418	5	,	,	PUNCT
ejpam-5634	418	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	418	7	multifunctions	multifunction	NOUN
ejpam-5634	418	8	.	.	PUNCT
ejpam-5634	419	1	european	european	ADJ
ejpam-5634	419	2	journal	journal	PROPN
ejpam-5634	419	3	of	of	ADP
ejpam-5634	419	4	pure	pure	ADJ
ejpam-5634	419	5	and	and	CCONJ
ejpam-5634	419	6	applied	applied	ADJ
ejpam-5634	419	7	mathematics	mathematic	NOUN
ejpam-5634	419	8	,	,	PUNCT
ejpam-5634	419	9	17(3):1553–1564	17(3):1553–1564	NUM
ejpam-5634	419	10	,	,	PUNCT
ejpam-5634	419	11	2024	2024	NUM
ejpam-5634	419	12	.	.	PUNCT
ejpam-5634	420	1	[	[	X
ejpam-5634	420	2	56	56	NUM
ejpam-5634	420	3	]	]	X
ejpam-5634	420	4	n.	n.	NOUN
ejpam-5634	420	5	srisarakham	srisarakham	PROPN
ejpam-5634	420	6	and	and	CCONJ
ejpam-5634	420	7	c.	c.	PROPN
ejpam-5634	420	8	boonpok	boonpok	PROPN
ejpam-5634	420	9	.	.	PUNCT
ejpam-5634	421	1	almost	almost	ADV
ejpam-5634	421	2	(	(	PUNCT
ejpam-5634	421	3	λ	λ	NOUN
ejpam-5634	421	4	,	,	PUNCT
ejpam-5634	421	5	p)-continuous	p)-continuous	ADJ
ejpam-5634	421	6	functions	function	NOUN
ejpam-5634	421	7	.	.	PUNCT
ejpam-5634	422	1	international	international	ADJ
ejpam-5634	422	2	journal	journal	PROPN
ejpam-5634	422	3	of	of	ADP
ejpam-5634	422	4	mathematics	mathematic	NOUN
ejpam-5634	422	5	and	and	CCONJ
ejpam-5634	422	6	computer	computer	NOUN
ejpam-5634	422	7	science	science	NOUN
ejpam-5634	422	8	,	,	PUNCT
ejpam-5634	422	9	18(2):255–259	18(2):255–259	NUM
ejpam-5634	422	10	,	,	PUNCT
ejpam-5634	422	11	2023	2023	NUM
ejpam-5634	422	12	.	.	PUNCT
ejpam-5634	423	1	[	[	X
ejpam-5634	423	2	57	57	NUM
ejpam-5634	423	3	]	]	X
ejpam-5634	423	4	n.	n.	NOUN
ejpam-5634	423	5	srisarakham	srisarakham	PROPN
ejpam-5634	423	6	,	,	PUNCT
ejpam-5634	423	7	a.	a.	PROPN
ejpam-5634	423	8	sama	sama	PROPN
ejpam-5634	423	9	-	-	PUNCT
ejpam-5634	423	10	ae	ae	PROPN
ejpam-5634	423	11	,	,	PUNCT
ejpam-5634	423	12	and	and	CCONJ
ejpam-5634	423	13	c.	c.	PROPN
ejpam-5634	423	14	boonpok	boonpok	PROPN
ejpam-5634	423	15	.	.	PUNCT
ejpam-5634	424	1	characterizations	characterization	NOUN
ejpam-5634	424	2	of	of	ADP
ejpam-5634	424	3	faintly	faintly	ADV
ejpam-5634	424	4	(	(	PUNCT
ejpam-5634	424	5	τ1	τ1	PROPN
ejpam-5634	424	6	,	,	PUNCT
ejpam-5634	424	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5634	424	8	functions	function	NOUN
ejpam-5634	424	9	.	.	PUNCT
ejpam-5634	425	1	european	european	ADJ
ejpam-5634	425	2	journal	journal	PROPN
ejpam-5634	425	3	of	of	ADP
ejpam-5634	425	4	pure	pure	ADJ
ejpam-5634	425	5	and	and	CCONJ
ejpam-5634	425	6	applied	applied	ADJ
ejpam-5634	425	7	mathematics	mathematic	NOUN
ejpam-5634	425	8	,	,	PUNCT
ejpam-5634	425	9	17(4):2753–2762	17(4):2753–2762	NUM
ejpam-5634	425	10	,	,	PUNCT
ejpam-5634	425	11	2024	2024	NUM
ejpam-5634	425	12	.	.	PUNCT
ejpam-5634	426	1	[	[	X
ejpam-5634	426	2	58	58	NUM
ejpam-5634	426	3	]	]	PUNCT
ejpam-5634	426	4	m.	m.	NOUN
ejpam-5634	426	5	thongmoon	thongmoon	NOUN
ejpam-5634	426	6	and	and	CCONJ
ejpam-5634	426	7	c.	c.	PROPN
ejpam-5634	426	8	boonpok	boonpok	PROPN
ejpam-5634	426	9	.	.	PUNCT
ejpam-5634	427	1	upper	upper	ADJ
ejpam-5634	427	2	and	and	CCONJ
ejpam-5634	427	3	lower	low	ADJ
ejpam-5634	427	4	almost	almost	ADV
ejpam-5634	427	5	β(λ	β(λ	NOUN
ejpam-5634	427	6	,	,	PUNCT
ejpam-5634	427	7	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	427	8	multifunctions	multifunction	NOUN
ejpam-5634	427	9	.	.	PUNCT
ejpam-5634	428	1	wseas	wseas	VERB
ejpam-5634	428	2	transactions	transaction	NOUN
ejpam-5634	428	3	on	on	ADP
ejpam-5634	428	4	mathematics	mathematic	NOUN
ejpam-5634	428	5	,	,	PUNCT
ejpam-5634	428	6	21:844–853	21:844–853	NUM
ejpam-5634	428	7	,	,	PUNCT
ejpam-5634	428	8	2022	2022	NUM
ejpam-5634	428	9	.	.	PUNCT
ejpam-5634	429	1	[	[	X
ejpam-5634	429	2	59	59	NUM
ejpam-5634	429	3	]	]	PUNCT
ejpam-5634	429	4	m.	m.	NOUN
ejpam-5634	429	5	thongmoon	thongmoon	NOUN
ejpam-5634	429	6	and	and	CCONJ
ejpam-5634	429	7	c.	c.	PROPN
ejpam-5634	429	8	boonpok	boonpok	PROPN
ejpam-5634	429	9	.	.	PUNCT
ejpam-5634	430	1	strongly	strongly	ADV
ejpam-5634	430	2	θ(λ	θ(λ	PROPN
ejpam-5634	430	3	,	,	PUNCT
ejpam-5634	430	4	p)-continuous	p)-continuous	ADJ
ejpam-5634	430	5	functions	function	NOUN
ejpam-5634	430	6	.	.	PUNCT
ejpam-5634	431	1	international	international	ADJ
ejpam-5634	431	2	journal	journal	PROPN
ejpam-5634	431	3	of	of	ADP
ejpam-5634	431	4	mathematics	mathematic	NOUN
ejpam-5634	431	5	and	and	CCONJ
ejpam-5634	431	6	computer	computer	NOUN
ejpam-5634	431	7	science	science	NOUN
ejpam-5634	431	8	,	,	PUNCT
ejpam-5634	431	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5634	431	10	,	,	PUNCT
ejpam-5634	431	11	2024	2024	NUM
ejpam-5634	431	12	.	.	PUNCT
ejpam-5634	432	1	[	[	X
ejpam-5634	432	2	60	60	NUM
ejpam-5634	432	3	]	]	PUNCT
ejpam-5634	432	4	m.	m.	NOUN
ejpam-5634	432	5	thongmoon	thongmoon	NOUN
ejpam-5634	432	6	,	,	PUNCT
ejpam-5634	432	7	s.	s.	PROPN
ejpam-5634	432	8	sompong	sompong	PROPN
ejpam-5634	432	9	,	,	PUNCT
ejpam-5634	432	10	and	and	CCONJ
ejpam-5634	432	11	c.	c.	PROPN
ejpam-5634	432	12	boonpok	boonpok	PROPN
ejpam-5634	432	13	.	.	PUNCT
ejpam-5634	433	1	upper	upper	ADJ
ejpam-5634	433	2	and	and	CCONJ
ejpam-5634	433	3	lower	low	ADJ
ejpam-5634	433	4	weak	weak	ADJ
ejpam-5634	433	5	(	(	PUNCT
ejpam-5634	433	6	τ1	τ1	NOUN
ejpam-5634	433	7	,	,	PUNCT
ejpam-5634	433	8	τ2)continuity	τ2)continuity	PROPN
ejpam-5634	433	9	.	.	PUNCT
ejpam-5634	434	1	european	european	PROPN
ejpam-5634	434	2	journal	journal	PROPN
ejpam-5634	434	3	of	of	ADP
ejpam-5634	434	4	pure	pure	ADJ
ejpam-5634	434	5	and	and	CCONJ
ejpam-5634	434	6	applied	applied	ADJ
ejpam-5634	434	7	mathematics	mathematic	NOUN
ejpam-5634	434	8	,	,	PUNCT
ejpam-5634	434	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-5634	434	10	,	,	PUNCT
ejpam-5634	434	11	2024	2024	NUM
ejpam-5634	434	12	.	.	PUNCT
ejpam-5634	435	1	[	[	X
ejpam-5634	435	2	61	61	NUM
ejpam-5634	435	3	]	]	X
ejpam-5634	435	4	n.	n.	NOUN
ejpam-5634	435	5	v.	v.	ADP
ejpam-5634	435	6	veličko	veličko	PROPN
ejpam-5634	435	7	.	.	PUNCT
ejpam-5634	436	1	h	h	NOUN
ejpam-5634	436	2	-	-	PUNCT
ejpam-5634	436	3	closed	close	VERB
ejpam-5634	436	4	topological	topological	ADJ
ejpam-5634	436	5	spaces	space	NOUN
ejpam-5634	436	6	.	.	PUNCT
ejpam-5634	437	1	american	american	PROPN
ejpam-5634	437	2	mathematical	mathematical	ADJ
ejpam-5634	437	3	society	society	NOUN
ejpam-5634	437	4	translations	translation	NOUN
ejpam-5634	437	5	,	,	PUNCT
ejpam-5634	437	6	78(2):102–118	78(2):102–118	NUM
ejpam-5634	437	7	,	,	PUNCT
ejpam-5634	437	8	1968	1968	NUM
ejpam-5634	437	9	.	.	PUNCT
ejpam-5634	438	1	[	[	X
ejpam-5634	438	2	62	62	NUM
ejpam-5634	438	3	]	]	PUNCT
ejpam-5634	438	4	c.	c.	PROPN
ejpam-5634	438	5	viriyapong	viriyapong	PROPN
ejpam-5634	438	6	and	and	CCONJ
ejpam-5634	438	7	c.	c.	PROPN
ejpam-5634	438	8	boonpok	boonpok	PROPN
ejpam-5634	438	9	.	.	PUNCT
ejpam-5634	439	1	(	(	PUNCT
ejpam-5634	439	2	τ1	τ1	NOUN
ejpam-5634	439	3	,	,	PUNCT
ejpam-5634	439	4	τ2)α	τ2)α	NOUN
ejpam-5634	439	5	-	-	PUNCT
ejpam-5634	439	6	continuity	continuity	NOUN
ejpam-5634	439	7	for	for	ADP
ejpam-5634	439	8	multifunctions	multifunction	NOUN
ejpam-5634	439	9	.	.	PUNCT
ejpam-5634	440	1	journal	journal	PROPN
ejpam-5634	440	2	of	of	ADP
ejpam-5634	440	3	mathematics	mathematic	NOUN
ejpam-5634	440	4	,	,	PUNCT
ejpam-5634	440	5	2020:6285763	2020:6285763	NUM
ejpam-5634	440	6	,	,	PUNCT
ejpam-5634	440	7	2020	2020	NUM
ejpam-5634	440	8	.	.	PUNCT
ejpam-5634	441	1	[	[	X
ejpam-5634	441	2	63	63	NUM
ejpam-5634	441	3	]	]	PUNCT
ejpam-5634	441	4	c.	c.	PROPN
ejpam-5634	441	5	viriyapong	viriyapong	PROPN
ejpam-5634	441	6	and	and	CCONJ
ejpam-5634	441	7	c.	c.	PROPN
ejpam-5634	441	8	boonpok	boonpok	PROPN
ejpam-5634	441	9	.	.	PUNCT
ejpam-5634	442	1	(	(	PUNCT
ejpam-5634	442	2	λ	λ	X
ejpam-5634	442	3	,	,	PUNCT
ejpam-5634	442	4	sp)-continuous	sp)-continuous	ADJ
ejpam-5634	442	5	functions	function	NOUN
ejpam-5634	442	6	.	.	PUNCT
ejpam-5634	443	1	wseas	wseas	VERB
ejpam-5634	443	2	transactions	transaction	NOUN
ejpam-5634	443	3	on	on	ADP
ejpam-5634	443	4	mathematics	mathematic	NOUN
ejpam-5634	443	5	,	,	PUNCT
ejpam-5634	443	6	21:380–385	21:380–385	NUM
ejpam-5634	443	7	,	,	PUNCT
ejpam-5634	443	8	2022	2022	NUM
ejpam-5634	443	9	.	.	PUNCT
ejpam-5634	444	1	[	[	X
ejpam-5634	444	2	64	64	NUM
ejpam-5634	444	3	]	]	PUNCT
ejpam-5634	444	4	c.	c.	PROPN
ejpam-5634	444	5	viriyapong	viriyapong	PROPN
ejpam-5634	444	6	and	and	CCONJ
ejpam-5634	444	7	c.	c.	PROPN
ejpam-5634	444	8	boonpok	boonpok	PROPN
ejpam-5634	444	9	.	.	PUNCT
ejpam-5634	445	1	weak	weak	ADJ
ejpam-5634	445	2	quasi	quasi	NOUN
ejpam-5634	445	3	(	(	PUNCT
ejpam-5634	445	4	λ	λ	PROPN
ejpam-5634	445	5	,	,	PUNCT
ejpam-5634	445	6	sp)-continuity	sp)-continuity	NOUN
ejpam-5634	445	7	for	for	ADP
ejpam-5634	445	8	multifunctions	multifunction	NOUN
ejpam-5634	445	9	.	.	PUNCT
ejpam-5634	446	1	international	international	ADJ
ejpam-5634	446	2	journal	journal	PROPN
ejpam-5634	446	3	of	of	ADP
ejpam-5634	446	4	mathematics	mathematic	NOUN
ejpam-5634	446	5	and	and	CCONJ
ejpam-5634	446	6	computer	computer	NOUN
ejpam-5634	446	7	science	science	NOUN
ejpam-5634	446	8	,	,	PUNCT
ejpam-5634	446	9	17(3):1201–1209	17(3):1201–1209	NUM
ejpam-5634	446	10	,	,	PUNCT
ejpam-5634	446	11	2022	2022	NUM
ejpam-5634	446	12	.	.	PUNCT
ejpam-5634	447	1	[	[	X
ejpam-5634	447	2	65	65	NUM
ejpam-5634	447	3	]	]	X
ejpam-5634	447	4	n.	n.	PROPN
ejpam-5634	447	5	viriyapong	viriyapong	PROPN
ejpam-5634	447	6	,	,	PUNCT
ejpam-5634	447	7	s.	s.	PROPN
ejpam-5634	447	8	sompong	sompong	PROPN
ejpam-5634	447	9	,	,	PUNCT
ejpam-5634	447	10	and	and	CCONJ
ejpam-5634	447	11	c.	c.	PROPN
ejpam-5634	447	12	boonpok	boonpok	PROPN
ejpam-5634	447	13	.	.	PUNCT
ejpam-5634	448	1	(	(	PUNCT
ejpam-5634	448	2	τ1	τ1	NOUN
ejpam-5634	448	3	,	,	PUNCT
ejpam-5634	448	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5634	448	5	disconnectedness	disconnectedness	NOUN
ejpam-5634	448	6	in	in	ADP
ejpam-5634	448	7	bitopological	bitopological	ADJ
ejpam-5634	448	8	spaces	space	NOUN
ejpam-5634	448	9	.	.	PUNCT
ejpam-5634	449	1	international	international	ADJ
ejpam-5634	449	2	journal	journal	PROPN
ejpam-5634	449	3	of	of	ADP
ejpam-5634	449	4	mathematics	mathematic	NOUN
ejpam-5634	449	5	and	and	CCONJ
ejpam-5634	449	6	computer	computer	NOUN
ejpam-5634	449	7	science	science	NOUN
ejpam-5634	449	8	,	,	PUNCT
ejpam-5634	449	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5634	449	10	,	,	PUNCT
ejpam-5634	449	11	2024	2024	NUM
ejpam-5634	449	12	.	.	PUNCT
ejpam-5634	450	1	[	[	X
ejpam-5634	450	2	66	66	NUM
ejpam-5634	450	3	]	]	X
ejpam-5634	450	4	n.	n.	PROPN
ejpam-5634	450	5	viriyapong	viriyapong	PROPN
ejpam-5634	450	6	,	,	PUNCT
ejpam-5634	450	7	s.	s.	PROPN
ejpam-5634	450	8	sompong	sompong	PROPN
ejpam-5634	450	9	,	,	PUNCT
ejpam-5634	450	10	and	and	CCONJ
ejpam-5634	450	11	c.	c.	PROPN
ejpam-5634	450	12	boonpok	boonpok	PROPN
ejpam-5634	450	13	.	.	PUNCT
ejpam-5634	451	1	upper	upper	ADJ
ejpam-5634	451	2	and	and	CCONJ
ejpam-5634	451	3	lower	low	ADJ
ejpam-5634	451	4	s-(τ1	s-(τ1	NOUN
ejpam-5634	451	5	,	,	PUNCT
ejpam-5634	451	6	τ2)p	τ2)p	ADJ
ejpam-5634	451	7	-	-	PUNCT
ejpam-5634	451	8	continuous	continuous	ADJ
ejpam-5634	451	9	multifunctions	multifunction	NOUN
ejpam-5634	451	10	.	.	PUNCT
ejpam-5634	452	1	european	european	ADJ
ejpam-5634	452	2	journal	journal	PROPN
ejpam-5634	452	3	of	of	ADP
ejpam-5634	452	4	pure	pure	ADJ
ejpam-5634	452	5	and	and	CCONJ
ejpam-5634	452	6	applied	applied	ADJ
ejpam-5634	452	7	mathematics	mathematic	NOUN
ejpam-5634	452	8	,	,	PUNCT
ejpam-5634	452	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-5634	452	10	,	,	PUNCT
ejpam-5634	452	11	2024	2024	NUM
ejpam-5634	452	12	.	.	PUNCT