id	sid	tid	token	lemma	pos
ejpam-5322	1	1	european	european	PROPN
ejpam-5322	1	2	journal	journal	PROPN
ejpam-5322	1	3	of	of	ADP
ejpam-5322	1	4	pure	pure	ADJ
ejpam-5322	1	5	and	and	CCONJ
ejpam-5322	1	6	applied	apply	VERB
ejpam-5322	1	7	mathematics	mathematic	NOUN
ejpam-5322	1	8	vol	vol	NOUN
ejpam-5322	1	9	.	.	PROPN
ejpam-5322	2	1	17	17	NUM
ejpam-5322	2	2	,	,	PUNCT
ejpam-5322	2	3	no	no	INTJ
ejpam-5322	2	4	.	.	NOUN
ejpam-5322	2	5	3	3	NUM
ejpam-5322	2	6	,	,	PUNCT
ejpam-5322	2	7	2024	2024	NUM
ejpam-5322	2	8	,	,	PUNCT
ejpam-5322	2	9	2210	2210	NUM
ejpam-5322	2	10	-	-	SYM
ejpam-5322	2	11	2220	2220	NUM
ejpam-5322	2	12	issn	issn	PROPN
ejpam-5322	2	13	1307	1307	NUM
ejpam-5322	2	14	-	-	SYM
ejpam-5322	2	15	5543	5543	NUM
ejpam-5322	2	16	–	–	PUNCT
ejpam-5322	2	17	ejpam.com	ejpam.com	X
ejpam-5322	2	18	published	publish	VERB
ejpam-5322	2	19	by	by	ADP
ejpam-5322	2	20	new	new	PROPN
ejpam-5322	2	21	york	york	PROPN
ejpam-5322	2	22	business	business	PROPN
ejpam-5322	2	23	global	global	PROPN
ejpam-5322	2	24	upper	upper	ADJ
ejpam-5322	2	25	and	and	CCONJ
ejpam-5322	2	26	lower	low	ADJ
ejpam-5322	2	27	s-(τ1	s-(τ1	NOUN
ejpam-5322	2	28	,	,	PUNCT
ejpam-5322	2	29	τ2)p	τ2)p	ADJ
ejpam-5322	2	30	-	-	PUNCT
ejpam-5322	2	31	continuous	continuous	ADJ
ejpam-5322	2	32	multifunctions	multifunction	NOUN
ejpam-5322	2	33	nongluk	nongluk	PROPN
ejpam-5322	2	34	viriyapong1	viriyapong1	PROPN
ejpam-5322	2	35	,	,	PUNCT
ejpam-5322	2	36	supannee	supannee	PROPN
ejpam-5322	2	37	sompong2	sompong2	PROPN
ejpam-5322	2	38	,	,	PUNCT
ejpam-5322	2	39	chawalit	chawalit	VERB
ejpam-5322	2	40	boonpok1,∗	boonpok1,∗	NOUN
ejpam-5322	2	41	1	1	NUM
ejpam-5322	2	42	mathematics	mathematic	NOUN
ejpam-5322	2	43	and	and	CCONJ
ejpam-5322	2	44	applied	apply	VERB
ejpam-5322	2	45	mathematics	mathematics	PROPN
ejpam-5322	2	46	research	research	NOUN
ejpam-5322	2	47	unit	unit	NOUN
ejpam-5322	2	48	,	,	PUNCT
ejpam-5322	2	49	department	department	NOUN
ejpam-5322	2	50	of	of	ADP
ejpam-5322	2	51	mathematics	mathematic	NOUN
ejpam-5322	2	52	,	,	PUNCT
ejpam-5322	2	53	faculty	faculty	NOUN
ejpam-5322	2	54	of	of	ADP
ejpam-5322	2	55	science	science	NOUN
ejpam-5322	2	56	,	,	PUNCT
ejpam-5322	2	57	mahasarakham	mahasarakham	PROPN
ejpam-5322	2	58	university	university	PROPN
ejpam-5322	2	59	,	,	PUNCT
ejpam-5322	2	60	maha	maha	PROPN
ejpam-5322	2	61	sarakham	sarakham	PROPN
ejpam-5322	2	62	,	,	PUNCT
ejpam-5322	2	63	44150	44150	NUM
ejpam-5322	2	64	,	,	PUNCT
ejpam-5322	2	65	thailand	thailand	PROPN
ejpam-5322	2	66	2	2	NUM
ejpam-5322	2	67	department	department	NOUN
ejpam-5322	2	68	of	of	ADP
ejpam-5322	2	69	mathematics	mathematic	NOUN
ejpam-5322	2	70	and	and	CCONJ
ejpam-5322	2	71	statistics	statistic	NOUN
ejpam-5322	2	72	,	,	PUNCT
ejpam-5322	2	73	faculty	faculty	NOUN
ejpam-5322	2	74	of	of	ADP
ejpam-5322	2	75	science	science	NOUN
ejpam-5322	2	76	and	and	CCONJ
ejpam-5322	2	77	technology	technology	NOUN
ejpam-5322	2	78	,	,	PUNCT
ejpam-5322	2	79	sakon	sakon	PROPN
ejpam-5322	2	80	nakhon	nakhon	PROPN
ejpam-5322	2	81	rajbhat	rajbhat	PROPN
ejpam-5322	2	82	university	university	PROPN
ejpam-5322	2	83	,	,	PUNCT
ejpam-5322	2	84	sakon	sakon	PROPN
ejpam-5322	2	85	nakhon	nakhon	PROPN
ejpam-5322	2	86	,	,	PUNCT
ejpam-5322	2	87	47000	47000	NUM
ejpam-5322	2	88	,	,	PUNCT
ejpam-5322	2	89	thailand	thailand	PROPN
ejpam-5322	2	90	abstract	abstract	NOUN
ejpam-5322	2	91	.	.	PUNCT
ejpam-5322	3	1	our	our	PRON
ejpam-5322	3	2	main	main	ADJ
ejpam-5322	3	3	purpose	purpose	NOUN
ejpam-5322	3	4	is	be	AUX
ejpam-5322	3	5	to	to	PART
ejpam-5322	3	6	introduce	introduce	VERB
ejpam-5322	3	7	the	the	DET
ejpam-5322	3	8	concepts	concept	NOUN
ejpam-5322	3	9	of	of	ADP
ejpam-5322	3	10	upper	upper	ADJ
ejpam-5322	3	11	and	and	CCONJ
ejpam-5322	3	12	lower	low	ADJ
ejpam-5322	3	13	s-(τ1	s-(τ1	NOUN
ejpam-5322	3	14	,	,	PUNCT
ejpam-5322	3	15	τ2)p	τ2)p	ADJ
ejpam-5322	3	16	-	-	PUNCT
ejpam-5322	3	17	continuous	continuous	ADJ
ejpam-5322	3	18	multifunctions	multifunction	NOUN
ejpam-5322	3	19	.	.	PUNCT
ejpam-5322	4	1	furthermore	furthermore	ADV
ejpam-5322	4	2	,	,	PUNCT
ejpam-5322	4	3	several	several	ADJ
ejpam-5322	4	4	characterizations	characterization	NOUN
ejpam-5322	4	5	of	of	ADP
ejpam-5322	4	6	upper	upper	ADJ
ejpam-5322	4	7	and	and	CCONJ
ejpam-5322	4	8	lower	low	ADJ
ejpam-5322	4	9	s-(τ1	s-(τ1	NOUN
ejpam-5322	4	10	,	,	PUNCT
ejpam-5322	4	11	τ2)p	τ2)p	ADJ
ejpam-5322	4	12	-	-	PUNCT
ejpam-5322	4	13	continuous	continuous	ADJ
ejpam-5322	4	14	multifunctions	multifunction	NOUN
ejpam-5322	4	15	are	be	AUX
ejpam-5322	4	16	investigated	investigate	VERB
ejpam-5322	4	17	.	.	PUNCT
ejpam-5322	5	1	2020	2020	NUM
ejpam-5322	5	2	mathematics	mathematic	NOUN
ejpam-5322	5	3	subject	subject	NOUN
ejpam-5322	5	4	classifications	classification	NOUN
ejpam-5322	5	5	:	:	PUNCT
ejpam-5322	5	6	54c08	54c08	NUM
ejpam-5322	5	7	;	;	PUNCT
ejpam-5322	5	8	54c60	54c60	NUM
ejpam-5322	5	9	;	;	PUNCT
ejpam-5322	5	10	54e55	54e55	NUM
ejpam-5322	5	11	key	key	ADJ
ejpam-5322	5	12	words	word	NOUN
ejpam-5322	5	13	and	and	CCONJ
ejpam-5322	5	14	phrases	phrase	NOUN
ejpam-5322	5	15	:	:	PUNCT
ejpam-5322	5	16	(	(	PUNCT
ejpam-5322	5	17	τ1	τ1	NOUN
ejpam-5322	5	18	,	,	PUNCT
ejpam-5322	5	19	τ2)p	τ2)p	ADJ
ejpam-5322	5	20	-	-	PUNCT
ejpam-5322	5	21	open	open	ADJ
ejpam-5322	5	22	set	set	NOUN
ejpam-5322	5	23	;	;	PUNCT
ejpam-5322	5	24	upper	upper	ADJ
ejpam-5322	5	25	s-(τ1	s-(τ1	PROPN
ejpam-5322	5	26	,	,	PUNCT
ejpam-5322	5	27	τ2)p	τ2)p	ADJ
ejpam-5322	5	28	-	-	PUNCT
ejpam-5322	5	29	continuous	continuous	ADJ
ejpam-5322	5	30	multifunction	multifunction	NOUN
ejpam-5322	5	31	;	;	PUNCT
ejpam-5322	5	32	lower	low	ADJ
ejpam-5322	5	33	s-(τ1	s-(τ1	PROPN
ejpam-5322	5	34	,	,	PUNCT
ejpam-5322	5	35	τ2)p	τ2)p	ADJ
ejpam-5322	5	36	-	-	PUNCT
ejpam-5322	5	37	continuous	continuous	ADJ
ejpam-5322	5	38	multifunction	multifunction	NOUN
ejpam-5322	5	39	1	1	NUM
ejpam-5322	5	40	.	.	PUNCT
ejpam-5322	5	41	introduction	introduction	NOUN
ejpam-5322	5	42	in	in	ADP
ejpam-5322	5	43	1965	1965	NUM
ejpam-5322	5	44	,	,	PUNCT
ejpam-5322	5	45	lee	lee	PROPN
ejpam-5322	6	1	[	[	X
ejpam-5322	6	2	27	27	NUM
ejpam-5322	6	3	]	]	PUNCT
ejpam-5322	6	4	studied	study	VERB
ejpam-5322	6	5	the	the	DET
ejpam-5322	6	6	notion	notion	NOUN
ejpam-5322	6	7	of	of	ADP
ejpam-5322	6	8	semiconnected	semiconnecte	VERB
ejpam-5322	6	9	functions	function	NOUN
ejpam-5322	6	10	.	.	PUNCT
ejpam-5322	7	1	kohli	kohli	PROPN
ejpam-5322	8	1	[	[	X
ejpam-5322	8	2	24	24	NUM
ejpam-5322	8	3	]	]	PUNCT
ejpam-5322	8	4	introduced	introduce	VERB
ejpam-5322	8	5	the	the	DET
ejpam-5322	8	6	notion	notion	NOUN
ejpam-5322	8	7	of	of	ADP
ejpam-5322	8	8	s	s	NOUN
ejpam-5322	8	9	-	-	ADJ
ejpam-5322	8	10	continuous	continuous	ADJ
ejpam-5322	8	11	functions	function	NOUN
ejpam-5322	8	12	and	and	CCONJ
ejpam-5322	8	13	investigated	investigate	VERB
ejpam-5322	8	14	several	several	ADJ
ejpam-5322	8	15	characterizations	characterization	NOUN
ejpam-5322	8	16	of	of	ADP
ejpam-5322	8	17	semilocally	semilocally	ADV
ejpam-5322	8	18	connected	connect	VERB
ejpam-5322	8	19	spaces	space	NOUN
ejpam-5322	8	20	in	in	ADP
ejpam-5322	8	21	terms	term	NOUN
ejpam-5322	8	22	of	of	ADP
ejpam-5322	8	23	s	s	NOUN
ejpam-5322	8	24	-	-	ADJ
ejpam-5322	8	25	continuous	continuous	ADJ
ejpam-5322	8	26	functions	function	NOUN
ejpam-5322	8	27	.	.	PUNCT
ejpam-5322	9	1	the	the	DET
ejpam-5322	9	2	class	class	NOUN
ejpam-5322	9	3	of	of	ADP
ejpam-5322	9	4	s	s	NOUN
ejpam-5322	9	5	-	-	NOUN
ejpam-5322	9	6	continuity	continuity	NOUN
ejpam-5322	9	7	is	be	AUX
ejpam-5322	9	8	a	a	DET
ejpam-5322	9	9	generalization	generalization	NOUN
ejpam-5322	9	10	of	of	ADP
ejpam-5322	9	11	continuity	continuity	NOUN
ejpam-5322	9	12	and	and	CCONJ
ejpam-5322	9	13	semiconnectedness	semiconnectedness	NOUN
ejpam-5322	9	14	.	.	PUNCT
ejpam-5322	10	1	furthermore	furthermore	ADV
ejpam-5322	10	2	,	,	PUNCT
ejpam-5322	10	3	kohli	kohli	PROPN
ejpam-5322	10	4	[	[	X
ejpam-5322	10	5	25	25	NUM
ejpam-5322	10	6	]	]	PUNCT
ejpam-5322	10	7	introduced	introduce	VERB
ejpam-5322	10	8	the	the	DET
ejpam-5322	10	9	concepts	concept	NOUN
ejpam-5322	10	10	of	of	ADP
ejpam-5322	10	11	s	s	NOUN
ejpam-5322	10	12	-	-	ADJ
ejpam-5322	10	13	regular	regular	ADJ
ejpam-5322	10	14	spaces	space	NOUN
ejpam-5322	10	15	and	and	CCONJ
ejpam-5322	10	16	completely	completely	ADV
ejpam-5322	10	17	s	s	NOUN
ejpam-5322	10	18	-	-	ADJ
ejpam-5322	10	19	regular	regular	ADJ
ejpam-5322	10	20	spaces	space	NOUN
ejpam-5322	10	21	and	and	CCONJ
ejpam-5322	10	22	proved	prove	VERB
ejpam-5322	10	23	that	that	SCONJ
ejpam-5322	10	24	s	s	NOUN
ejpam-5322	10	25	-	-	PUNCT
ejpam-5322	10	26	regularity	regularity	NOUN
ejpam-5322	10	27	and	and	CCONJ
ejpam-5322	10	28	complete	complete	ADJ
ejpam-5322	10	29	s	s	NOUN
ejpam-5322	10	30	-	-	PUNCT
ejpam-5322	10	31	regularity	regularity	NOUN
ejpam-5322	10	32	are	be	AUX
ejpam-5322	10	33	preserved	preserve	VERB
ejpam-5322	10	34	under	under	ADP
ejpam-5322	10	35	certain	certain	ADJ
ejpam-5322	10	36	s	s	NOUN
ejpam-5322	10	37	-	-	ADJ
ejpam-5322	10	38	continuous	continuous	ADJ
ejpam-5322	10	39	functions	function	NOUN
ejpam-5322	10	40	.	.	PUNCT
ejpam-5322	11	1	duangphui	duangphui	NOUN
ejpam-5322	11	2	et	et	PROPN
ejpam-5322	11	3	al	al	PROPN
ejpam-5322	11	4	.	.	PUNCT
ejpam-5322	12	1	[	[	X
ejpam-5322	12	2	21	21	NUM
ejpam-5322	12	3	]	]	PUNCT
ejpam-5322	12	4	introduced	introduce	VERB
ejpam-5322	12	5	and	and	CCONJ
ejpam-5322	12	6	investigated	investigate	VERB
ejpam-5322	12	7	the	the	DET
ejpam-5322	12	8	notion	notion	NOUN
ejpam-5322	12	9	of	of	ADP
ejpam-5322	12	10	almost	almost	ADV
ejpam-5322	12	11	(	(	PUNCT
ejpam-5322	12	12	µ	µ	NUM
ejpam-5322	12	13	,	,	PUNCT
ejpam-5322	12	14	µ′)(m	µ′)(m	VERB
ejpam-5322	12	15	,	,	PUNCT
ejpam-5322	12	16	n)continuous	n)continuous	ADJ
ejpam-5322	12	17	functions	function	NOUN
ejpam-5322	12	18	.	.	PUNCT
ejpam-5322	13	1	thongmoon	thongmoon	NOUN
ejpam-5322	13	2	and	and	CCONJ
ejpam-5322	13	3	boonpok	boonpok	VERB
ejpam-5322	13	4	[	[	X
ejpam-5322	13	5	35	35	NUM
ejpam-5322	13	6	]	]	PUNCT
ejpam-5322	13	7	introduced	introduce	VERB
ejpam-5322	13	8	and	and	CCONJ
ejpam-5322	13	9	studied	study	VERB
ejpam-5322	13	10	the	the	DET
ejpam-5322	13	11	notion	notion	NOUN
ejpam-5322	13	12	of	of	ADP
ejpam-5322	13	13	strongly	strongly	ADV
ejpam-5322	13	14	θ(λ	θ(λ	ADJ
ejpam-5322	13	15	,	,	PUNCT
ejpam-5322	13	16	p)-continuous	p)-continuous	ADJ
ejpam-5322	13	17	functions	function	NOUN
ejpam-5322	13	18	.	.	PUNCT
ejpam-5322	14	1	moreover	moreover	ADV
ejpam-5322	14	2	,	,	PUNCT
ejpam-5322	14	3	several	several	ADJ
ejpam-5322	14	4	characterizations	characterization	NOUN
ejpam-5322	14	5	of	of	ADP
ejpam-5322	14	6	almost	almost	ADV
ejpam-5322	14	7	(	(	PUNCT
ejpam-5322	14	8	λ	λ	PROPN
ejpam-5322	14	9	,	,	PUNCT
ejpam-5322	14	10	p)-continuous	p)-continuous	ADJ
ejpam-5322	14	11	functions	function	NOUN
ejpam-5322	14	12	,	,	PUNCT
ejpam-5322	14	13	almost	almost	ADV
ejpam-5322	14	14	strongly	strongly	ADV
ejpam-5322	14	15	θ(λ	θ(λ	VERB
ejpam-5322	14	16	,	,	PUNCT
ejpam-5322	14	17	p)-continuous	p)-continuous	ADJ
ejpam-5322	14	18	functions	function	NOUN
ejpam-5322	14	19	,	,	PUNCT
ejpam-5322	14	20	θ(λ	θ(λ	PROPN
ejpam-5322	14	21	,	,	PUNCT
ejpam-5322	14	22	p)continuous	p)continuous	ADJ
ejpam-5322	14	23	functions	function	NOUN
ejpam-5322	14	24	,	,	PUNCT
ejpam-5322	14	25	weakly	weakly	ADJ
ejpam-5322	14	26	(	(	PUNCT
ejpam-5322	14	27	λ	λ	PROPN
ejpam-5322	14	28	,	,	PUNCT
ejpam-5322	14	29	b)-continuous	b)-continuous	ADJ
ejpam-5322	14	30	functions	function	NOUN
ejpam-5322	14	31	,	,	PUNCT
ejpam-5322	14	32	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5322	14	33	functions	function	NOUN
ejpam-5322	14	34	,	,	PUNCT
ejpam-5322	14	35	⋆-continuous	⋆-continuous	ADJ
ejpam-5322	14	36	functions	function	NOUN
ejpam-5322	14	37	,	,	PUNCT
ejpam-5322	14	38	θ	θ	PROPN
ejpam-5322	14	39	-	-	ADJ
ejpam-5322	14	40	i	i	NOUN
ejpam-5322	14	41	-continuous	-continuous	ADJ
ejpam-5322	14	42	functions	function	NOUN
ejpam-5322	14	43	,	,	PUNCT
ejpam-5322	14	44	almost	almost	ADV
ejpam-5322	14	45	(	(	PUNCT
ejpam-5322	14	46	g	g	NOUN
ejpam-5322	14	47	,	,	PUNCT
ejpam-5322	14	48	m)-continuous	m)-continuous	ADJ
ejpam-5322	14	49	functions	function	NOUN
ejpam-5322	14	50	,	,	PUNCT
ejpam-5322	14	51	(	(	PUNCT
ejpam-5322	14	52	λ	λ	NOUN
ejpam-5322	14	53	,	,	PUNCT
ejpam-5322	14	54	sp)-continuous	sp)-continuous	ADJ
ejpam-5322	14	55	functions	function	NOUN
ejpam-5322	14	56	,	,	PUNCT
ejpam-5322	14	57	δp(λ	δp(λ	NOUN
ejpam-5322	14	58	,	,	PUNCT
ejpam-5322	14	59	s)-continuous	s)-continuous	ADJ
ejpam-5322	14	60	functions	function	NOUN
ejpam-5322	14	61	,	,	PUNCT
ejpam-5322	14	62	(	(	PUNCT
ejpam-5322	14	63	λ	λ	NOUN
ejpam-5322	14	64	,	,	PUNCT
ejpam-5322	14	65	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-5322	14	66	functions	function	NOUN
ejpam-5322	14	67	,	,	PUNCT
ejpam-5322	14	68	pairwise	pairwise	NOUN
ejpam-5322	14	69	almost	almost	ADV
ejpam-5322	14	70	m	m	VERB
ejpam-5322	14	71	-continuous	-continuous	ADJ
ejpam-5322	14	72	functions	function	NOUN
ejpam-5322	14	73	,	,	PUNCT
ejpam-5322	14	74	(	(	PUNCT
ejpam-5322	14	75	τ1	τ1	NOUN
ejpam-5322	14	76	,	,	PUNCT
ejpam-5322	14	77	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	14	78	functions	function	NOUN
ejpam-5322	14	79	,	,	PUNCT
ejpam-5322	14	80	almost	almost	ADV
ejpam-5322	14	81	(	(	PUNCT
ejpam-5322	14	82	τ1	τ1	NOUN
ejpam-5322	14	83	,	,	PUNCT
ejpam-5322	14	84	τ2)continuous	τ2)continuous	ADJ
ejpam-5322	14	85	functions	function	NOUN
ejpam-5322	14	86	and	and	CCONJ
ejpam-5322	14	87	weakly	weakly	ADJ
ejpam-5322	14	88	(	(	PUNCT
ejpam-5322	14	89	τ1	τ1	NOUN
ejpam-5322	14	90	,	,	PUNCT
ejpam-5322	14	91	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	14	92	functions	function	NOUN
ejpam-5322	14	93	were	be	AUX
ejpam-5322	14	94	presented	present	VERB
ejpam-5322	14	95	in	in	ADP
ejpam-5322	14	96	[	[	X
ejpam-5322	14	97	33	33	NUM
ejpam-5322	14	98	]	]	PUNCT
ejpam-5322	14	99	,	,	PUNCT
ejpam-5322	14	100	[	[	X
ejpam-5322	14	101	11	11	NUM
ejpam-5322	14	102	]	]	PUNCT
ejpam-5322	14	103	,	,	PUNCT
ejpam-5322	14	104	[	[	X
ejpam-5322	14	105	31	31	NUM
ejpam-5322	14	106	]	]	PUNCT
ejpam-5322	14	107	,	,	PUNCT
ejpam-5322	14	108	[	[	X
ejpam-5322	14	109	16	16	NUM
ejpam-5322	14	110	]	]	PUNCT
ejpam-5322	14	111	,	,	PUNCT
ejpam-5322	14	112	[	[	X
ejpam-5322	14	113	10	10	NUM
ejpam-5322	14	114	]	]	PUNCT
ejpam-5322	14	115	,	,	PUNCT
ejpam-5322	15	1	[	[	X
ejpam-5322	15	2	9	9	NUM
ejpam-5322	15	3	]	]	PUNCT
ejpam-5322	15	4	,	,	PUNCT
ejpam-5322	15	5	[	[	X
ejpam-5322	15	6	5	5	NUM
ejpam-5322	15	7	]	]	PUNCT
ejpam-5322	15	8	,	,	PUNCT
ejpam-5322	15	9	[	[	X
ejpam-5322	15	10	2	2	NUM
ejpam-5322	15	11	]	]	PUNCT
ejpam-5322	15	12	,	,	PUNCT
ejpam-5322	15	13	[	[	X
ejpam-5322	15	14	37	37	NUM
ejpam-5322	15	15	]	]	PUNCT
ejpam-5322	15	16	,	,	PUNCT
ejpam-5322	15	17	[	[	X
ejpam-5322	15	18	34	34	NUM
ejpam-5322	15	19	]	]	PUNCT
ejpam-5322	15	20	,	,	PUNCT
ejpam-5322	15	21	[	[	X
ejpam-5322	15	22	8	8	NUM
ejpam-5322	15	23	]	]	PUNCT
ejpam-5322	15	24	,	,	PUNCT
ejpam-5322	15	25	[	[	X
ejpam-5322	15	26	3	3	NUM
ejpam-5322	15	27	]	]	PUNCT
ejpam-5322	15	28	,	,	PUNCT
ejpam-5322	15	29	[	[	X
ejpam-5322	15	30	17	17	NUM
ejpam-5322	15	31	]	]	PUNCT
ejpam-5322	15	32	,	,	PUNCT
ejpam-5322	15	33	[	[	X
ejpam-5322	15	34	15	15	NUM
ejpam-5322	15	35	]	]	PUNCT
ejpam-5322	15	36	and	and	CCONJ
ejpam-5322	15	37	[	[	X
ejpam-5322	15	38	12	12	NUM
ejpam-5322	15	39	]	]	PUNCT
ejpam-5322	15	40	,	,	PUNCT
ejpam-5322	15	41	respectively	respectively	ADV
ejpam-5322	15	42	.	.	PUNCT
ejpam-5322	16	1	∗corresponding	∗corresponde	VERB
ejpam-5322	16	2	author	author	NOUN
ejpam-5322	16	3	.	.	PUNCT
ejpam-5322	17	1	doi	doi	NOUN
ejpam-5322	17	2	:	:	PUNCT
ejpam-5322	17	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5322	https://doi.org/10.29020/nybg.ejpam.v17i3.5322	PROPN
ejpam-5322	17	4	email	email	NOUN
ejpam-5322	17	5	addresses	address	VERB
ejpam-5322	17	6	:	:	PUNCT
ejpam-5322	17	7	nongluk.h@msu.ac.th	nongluk.h@msu.ac.th	PROPN
ejpam-5322	17	8	(	(	PUNCT
ejpam-5322	17	9	n.	n.	NOUN
ejpam-5322	17	10	viriyapong	viriyapong	PROPN
ejpam-5322	17	11	)	)	PUNCT
ejpam-5322	17	12	,	,	PUNCT
ejpam-5322	17	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-5322	17	14	(	(	PUNCT
ejpam-5322	17	15	s.	s.	PROPN
ejpam-5322	17	16	sompong	sompong	PROPN
ejpam-5322	17	17	)	)	PUNCT
ejpam-5322	17	18	,	,	PUNCT
ejpam-5322	17	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5322	17	20	(	(	PUNCT
ejpam-5322	17	21	c.	c.	PROPN
ejpam-5322	17	22	boonpok	boonpok	PROPN
ejpam-5322	17	23	)	)	PUNCT
ejpam-5322	17	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5322	17	25	2210	2210	NUM
ejpam-5322	18	1	©	©	ADP
ejpam-5322	18	2	2024	2024	NUM
ejpam-5322	18	3	ejpam	ejpam	NOUN
ejpam-5322	18	4	all	all	DET
ejpam-5322	18	5	rights	right	NOUN
ejpam-5322	18	6	reserved	reserve	VERB
ejpam-5322	18	7	.	.	PUNCT
ejpam-5322	19	1	n.	n.	PROPN
ejpam-5322	19	2	viriyapong	viriyapong	PROPN
ejpam-5322	19	3	,	,	PUNCT
ejpam-5322	19	4	s.	s.	PROPN
ejpam-5322	19	5	sompong	sompong	PROPN
ejpam-5322	19	6	,	,	PUNCT
ejpam-5322	19	7	c.	c.	PROPN
ejpam-5322	19	8	boonpok	boonpok	PROPN
ejpam-5322	19	9	/	/	SYM
ejpam-5322	19	10	eur	eur	PROPN
ejpam-5322	19	11	.	.	PUNCT
ejpam-5322	20	1	j.	j.	PROPN
ejpam-5322	20	2	pure	pure	PROPN
ejpam-5322	20	3	appl	appl	PROPN
ejpam-5322	20	4	.	.	PROPN
ejpam-5322	20	5	math	math	PROPN
ejpam-5322	20	6	,	,	PUNCT
ejpam-5322	20	7	17	17	NUM
ejpam-5322	20	8	(	(	PUNCT
ejpam-5322	20	9	3	3	NUM
ejpam-5322	20	10	)	)	PUNCT
ejpam-5322	20	11	(	(	PUNCT
ejpam-5322	20	12	2024	2024	NUM
ejpam-5322	20	13	)	)	PUNCT
ejpam-5322	20	14	,	,	PUNCT
ejpam-5322	20	15	2210	2210	NUM
ejpam-5322	20	16	-	-	SYM
ejpam-5322	20	17	2220	2220	NUM
ejpam-5322	20	18	2211	2211	NUM
ejpam-5322	20	19	in	in	ADP
ejpam-5322	20	20	1989	1989	NUM
ejpam-5322	20	21	,	,	PUNCT
ejpam-5322	20	22	lipski	lipski	NOUN
ejpam-5322	21	1	[	[	X
ejpam-5322	21	2	28	28	NUM
ejpam-5322	21	3	]	]	PUNCT
ejpam-5322	21	4	extended	extend	VERB
ejpam-5322	21	5	the	the	DET
ejpam-5322	21	6	concept	concept	NOUN
ejpam-5322	21	7	of	of	ADP
ejpam-5322	21	8	s	s	NOUN
ejpam-5322	21	9	-	-	ADJ
ejpam-5322	21	10	continuous	continuous	ADJ
ejpam-5322	21	11	functions	function	NOUN
ejpam-5322	21	12	to	to	ADP
ejpam-5322	21	13	the	the	DET
ejpam-5322	21	14	setting	setting	NOUN
ejpam-5322	21	15	of	of	ADP
ejpam-5322	21	16	multifunctions	multifunction	NOUN
ejpam-5322	21	17	.	.	PUNCT
ejpam-5322	22	1	popa	popa	NOUN
ejpam-5322	22	2	[	[	X
ejpam-5322	22	3	29	29	NUM
ejpam-5322	22	4	]	]	PUNCT
ejpam-5322	22	5	introduced	introduce	VERB
ejpam-5322	22	6	the	the	DET
ejpam-5322	22	7	concept	concept	NOUN
ejpam-5322	22	8	of	of	ADP
ejpam-5322	22	9	precontinuous	precontinuous	ADJ
ejpam-5322	22	10	multifunctions	multifunction	NOUN
ejpam-5322	22	11	and	and	CCONJ
ejpam-5322	22	12	showed	show	VERB
ejpam-5322	22	13	that	that	SCONJ
ejpam-5322	22	14	h	h	NOUN
ejpam-5322	22	15	-	-	PUNCT
ejpam-5322	22	16	almost	almost	ADV
ejpam-5322	22	17	continuity	continuity	NOUN
ejpam-5322	22	18	and	and	CCONJ
ejpam-5322	22	19	precontinuity	precontinuity	NOUN
ejpam-5322	22	20	are	be	AUX
ejpam-5322	22	21	equivalent	equivalent	ADJ
ejpam-5322	22	22	for	for	ADP
ejpam-5322	22	23	multifunctions	multifunction	NOUN
ejpam-5322	22	24	.	.	PUNCT
ejpam-5322	23	1	ewert	ewert	PROPN
ejpam-5322	23	2	and	and	CCONJ
ejpam-5322	23	3	lipski	lipski	ADJ
ejpam-5322	23	4	[	[	X
ejpam-5322	23	5	22	22	NUM
ejpam-5322	23	6	]	]	PUNCT
ejpam-5322	23	7	introduced	introduce	VERB
ejpam-5322	23	8	and	and	CCONJ
ejpam-5322	23	9	investigated	investigate	VERB
ejpam-5322	23	10	the	the	DET
ejpam-5322	23	11	concept	concept	NOUN
ejpam-5322	23	12	of	of	ADP
ejpam-5322	23	13	s	s	NOUN
ejpam-5322	23	14	-	-	PUNCT
ejpam-5322	23	15	quasi	quasi	ADJ
ejpam-5322	23	16	-	-	ADJ
ejpam-5322	23	17	continuous	continuous	ADJ
ejpam-5322	23	18	multifunctions	multifunction	NOUN
ejpam-5322	23	19	.	.	PUNCT
ejpam-5322	24	1	popa	popa	NOUN
ejpam-5322	24	2	and	and	CCONJ
ejpam-5322	24	3	noiri	noiri	ADV
ejpam-5322	25	1	[	[	X
ejpam-5322	25	2	30	30	NUM
ejpam-5322	25	3	]	]	PUNCT
ejpam-5322	25	4	introduced	introduce	VERB
ejpam-5322	25	5	and	and	CCONJ
ejpam-5322	25	6	studied	study	VERB
ejpam-5322	25	7	the	the	DET
ejpam-5322	25	8	notion	notion	NOUN
ejpam-5322	25	9	of	of	ADP
ejpam-5322	25	10	s	s	NOUN
ejpam-5322	25	11	-	-	ADJ
ejpam-5322	25	12	precontinuous	precontinuous	ADJ
ejpam-5322	25	13	multifunctions	multifunction	NOUN
ejpam-5322	25	14	as	as	ADP
ejpam-5322	25	15	a	a	DET
ejpam-5322	25	16	generalization	generalization	NOUN
ejpam-5322	25	17	of	of	ADP
ejpam-5322	25	18	s	s	NOUN
ejpam-5322	25	19	-	-	ADJ
ejpam-5322	25	20	continuous	continuous	ADJ
ejpam-5322	25	21	multifunctions	multifunction	NOUN
ejpam-5322	25	22	and	and	CCONJ
ejpam-5322	25	23	precontinuous	precontinuous	ADJ
ejpam-5322	25	24	multifunctions	multifunction	NOUN
ejpam-5322	25	25	.	.	PUNCT
ejpam-5322	26	1	laprom	laprom	ADP
ejpam-5322	26	2	et	et	PROPN
ejpam-5322	26	3	al	al	PROPN
ejpam-5322	26	4	.	.	PUNCT
ejpam-5322	27	1	[	[	X
ejpam-5322	27	2	26	26	NUM
ejpam-5322	27	3	]	]	PUNCT
ejpam-5322	27	4	introduced	introduce	VERB
ejpam-5322	27	5	and	and	CCONJ
ejpam-5322	27	6	investigated	investigate	VERB
ejpam-5322	27	7	the	the	DET
ejpam-5322	27	8	concept	concept	NOUN
ejpam-5322	27	9	of	of	ADP
ejpam-5322	27	10	β(τ1	β(τ1	NOUN
ejpam-5322	27	11	,	,	PUNCT
ejpam-5322	27	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	27	13	multifunctions	multifunction	NOUN
ejpam-5322	27	14	.	.	PUNCT
ejpam-5322	28	1	in	in	ADP
ejpam-5322	28	2	particular	particular	ADJ
ejpam-5322	28	3	,	,	PUNCT
ejpam-5322	28	4	some	some	DET
ejpam-5322	28	5	characterizations	characterization	NOUN
ejpam-5322	28	6	of	of	ADP
ejpam-5322	28	7	(	(	PUNCT
ejpam-5322	28	8	τ1	τ1	NOUN
ejpam-5322	28	9	,	,	PUNCT
ejpam-5322	28	10	τ2)δ	τ2)δ	ADJ
ejpam-5322	28	11	-	-	PUNCT
ejpam-5322	28	12	semicontinuous	semicontinuous	ADJ
ejpam-5322	28	13	multifunctions	multifunction	NOUN
ejpam-5322	28	14	,	,	PUNCT
ejpam-5322	28	15	almost	almost	ADV
ejpam-5322	28	16	weakly	weakly	ADJ
ejpam-5322	28	17	⋆-continuous	⋆-continuous	ADJ
ejpam-5322	28	18	multifunctions	multifunction	NOUN
ejpam-5322	28	19	,	,	PUNCT
ejpam-5322	28	20	weakly	weakly	ADJ
ejpam-5322	28	21	⋆-continuous	⋆-continuous	ADJ
ejpam-5322	28	22	multifunctions	multifunction	NOUN
ejpam-5322	28	23	,	,	PUNCT
ejpam-5322	28	24	weakly	weakly	ADJ
ejpam-5322	28	25	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5322	28	26	multifunctions	multifunction	NOUN
ejpam-5322	28	27	,	,	PUNCT
ejpam-5322	28	28	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-5322	28	29	multifunctions	multifunction	NOUN
ejpam-5322	28	30	,	,	PUNCT
ejpam-5322	28	31	almost	almost	ADV
ejpam-5322	28	32	β(⋆)continuous	β(⋆)continuous	ADJ
ejpam-5322	28	33	multifunctions	multifunction	NOUN
ejpam-5322	28	34	,	,	PUNCT
ejpam-5322	28	35	almost	almost	ADV
ejpam-5322	28	36	weakly	weakly	ADJ
ejpam-5322	28	37	(	(	PUNCT
ejpam-5322	28	38	τ1	τ1	NOUN
ejpam-5322	28	39	,	,	PUNCT
ejpam-5322	28	40	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	28	41	multifunctions	multifunction	NOUN
ejpam-5322	28	42	,	,	PUNCT
ejpam-5322	28	43	almost	almost	ADV
ejpam-5322	28	44	(	(	PUNCT
ejpam-5322	28	45	τ1	τ1	NOUN
ejpam-5322	28	46	,	,	PUNCT
ejpam-5322	28	47	τ2)continuous	τ2)continuous	ADJ
ejpam-5322	28	48	multifunctions	multifunction	NOUN
ejpam-5322	28	49	and	and	CCONJ
ejpam-5322	28	50	(	(	PUNCT
ejpam-5322	28	51	τ1	τ1	NOUN
ejpam-5322	28	52	,	,	PUNCT
ejpam-5322	28	53	τ2)α	τ2)α	ADJ
ejpam-5322	28	54	-	-	PUNCT
ejpam-5322	28	55	continuous	continuous	ADJ
ejpam-5322	28	56	multifunctions	multifunction	NOUN
ejpam-5322	28	57	were	be	AUX
ejpam-5322	28	58	established	establish	VERB
ejpam-5322	28	59	in	in	ADP
ejpam-5322	28	60	[	[	X
ejpam-5322	28	61	6	6	NUM
ejpam-5322	28	62	]	]	PUNCT
ejpam-5322	28	63	,	,	PUNCT
ejpam-5322	28	64	[	[	X
ejpam-5322	28	65	18	18	NUM
ejpam-5322	28	66	]	]	PUNCT
ejpam-5322	28	67	,	,	PUNCT
ejpam-5322	28	68	[	[	X
ejpam-5322	28	69	4	4	NUM
ejpam-5322	28	70	]	]	PUNCT
ejpam-5322	28	71	,	,	PUNCT
ejpam-5322	28	72	[	[	X
ejpam-5322	28	73	14	14	NUM
ejpam-5322	28	74	]	]	PUNCT
ejpam-5322	28	75	,	,	PUNCT
ejpam-5322	28	76	[	[	X
ejpam-5322	28	77	13	13	NUM
ejpam-5322	28	78	]	]	PUNCT
ejpam-5322	28	79	,	,	PUNCT
ejpam-5322	28	80	[	[	X
ejpam-5322	28	81	7	7	NUM
ejpam-5322	28	82	]	]	PUNCT
ejpam-5322	28	83	,	,	PUNCT
ejpam-5322	28	84	[	[	X
ejpam-5322	28	85	19	19	NUM
ejpam-5322	28	86	]	]	PUNCT
ejpam-5322	28	87	,	,	PUNCT
ejpam-5322	28	88	[	[	X
ejpam-5322	28	89	23	23	NUM
ejpam-5322	28	90	]	]	PUNCT
ejpam-5322	28	91	and	and	CCONJ
ejpam-5322	28	92	[	[	X
ejpam-5322	28	93	36	36	NUM
ejpam-5322	28	94	]	]	PUNCT
ejpam-5322	28	95	,	,	PUNCT
ejpam-5322	28	96	respectively	respectively	ADV
ejpam-5322	28	97	.	.	PUNCT
ejpam-5322	29	1	pue	pue	NOUN
ejpam-5322	29	2	-	-	PUNCT
ejpam-5322	29	3	on	on	NOUN
ejpam-5322	29	4	et	et	PROPN
ejpam-5322	29	5	al	al	PROPN
ejpam-5322	29	6	.	.	PUNCT
ejpam-5322	30	1	[	[	X
ejpam-5322	30	2	32	32	NUM
ejpam-5322	30	3	]	]	SYM
ejpam-5322	30	4	introduce	introduce	NOUN
ejpam-5322	30	5	and	and	CCONJ
ejpam-5322	30	6	studied	study	VERB
ejpam-5322	30	7	the	the	DET
ejpam-5322	30	8	concepts	concept	NOUN
ejpam-5322	30	9	of	of	ADP
ejpam-5322	30	10	upper	upper	ADJ
ejpam-5322	30	11	and	and	CCONJ
ejpam-5322	30	12	lower	low	ADJ
ejpam-5322	30	13	(	(	PUNCT
ejpam-5322	30	14	τ1	τ1	NOUN
ejpam-5322	30	15	,	,	PUNCT
ejpam-5322	30	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	30	17	multifunctions	multifunction	NOUN
ejpam-5322	30	18	.	.	PUNCT
ejpam-5322	31	1	in	in	ADP
ejpam-5322	31	2	this	this	DET
ejpam-5322	31	3	paper	paper	NOUN
ejpam-5322	31	4	,	,	PUNCT
ejpam-5322	31	5	we	we	PRON
ejpam-5322	31	6	introduce	introduce	VERB
ejpam-5322	31	7	the	the	DET
ejpam-5322	31	8	notions	notion	NOUN
ejpam-5322	31	9	of	of	ADP
ejpam-5322	31	10	upper	upper	ADJ
ejpam-5322	31	11	and	and	CCONJ
ejpam-5322	31	12	lower	low	ADJ
ejpam-5322	31	13	s-(τ1	s-(τ1	NOUN
ejpam-5322	31	14	,	,	PUNCT
ejpam-5322	31	15	τ2)p	τ2)p	ADJ
ejpam-5322	31	16	-	-	PUNCT
ejpam-5322	31	17	continuous	continuous	ADJ
ejpam-5322	31	18	multifunctions	multifunction	NOUN
ejpam-5322	31	19	.	.	PUNCT
ejpam-5322	32	1	we	we	PRON
ejpam-5322	32	2	also	also	ADV
ejpam-5322	32	3	investigate	investigate	VERB
ejpam-5322	32	4	several	several	ADJ
ejpam-5322	32	5	characterizations	characterization	NOUN
ejpam-5322	32	6	of	of	ADP
ejpam-5322	32	7	upper	upper	ADJ
ejpam-5322	32	8	and	and	CCONJ
ejpam-5322	32	9	lower	low	ADJ
ejpam-5322	32	10	s-(τ1	s-(τ1	NOUN
ejpam-5322	32	11	,	,	PUNCT
ejpam-5322	32	12	τ2)p	τ2)p	ADJ
ejpam-5322	32	13	-	-	PUNCT
ejpam-5322	32	14	continuous	continuous	ADJ
ejpam-5322	32	15	multifunctions	multifunction	NOUN
ejpam-5322	32	16	.	.	PUNCT
ejpam-5322	33	1	2	2	X
ejpam-5322	33	2	.	.	X
ejpam-5322	33	3	preliminaries	preliminary	NOUN
ejpam-5322	33	4	throughout	throughout	ADP
ejpam-5322	33	5	the	the	DET
ejpam-5322	33	6	present	present	ADJ
ejpam-5322	33	7	paper	paper	NOUN
ejpam-5322	33	8	,	,	PUNCT
ejpam-5322	33	9	spaces	space	NOUN
ejpam-5322	33	10	(	(	PUNCT
ejpam-5322	33	11	x	x	NOUN
ejpam-5322	33	12	,	,	PUNCT
ejpam-5322	33	13	τ1	τ1	NOUN
ejpam-5322	33	14	,	,	PUNCT
ejpam-5322	33	15	τ2	τ2	NOUN
ejpam-5322	33	16	)	)	PUNCT
ejpam-5322	33	17	and	and	CCONJ
ejpam-5322	33	18	(	(	PUNCT
ejpam-5322	33	19	y	y	PROPN
ejpam-5322	33	20	,	,	PUNCT
ejpam-5322	33	21	σ1	σ1	PROPN
ejpam-5322	33	22	,	,	PUNCT
ejpam-5322	33	23	σ2	σ2	NOUN
ejpam-5322	33	24	)	)	PUNCT
ejpam-5322	33	25	(	(	PUNCT
ejpam-5322	33	26	or	or	CCONJ
ejpam-5322	33	27	simply	simply	ADV
ejpam-5322	33	28	x	x	X
ejpam-5322	33	29	and	and	CCONJ
ejpam-5322	33	30	y	y	PROPN
ejpam-5322	33	31	)	)	PUNCT
ejpam-5322	33	32	always	always	ADV
ejpam-5322	33	33	mean	mean	VERB
ejpam-5322	33	34	bitopological	bitopological	ADJ
ejpam-5322	33	35	spaces	space	NOUN
ejpam-5322	33	36	on	on	ADP
ejpam-5322	33	37	which	which	PRON
ejpam-5322	33	38	no	no	DET
ejpam-5322	33	39	separation	separation	NOUN
ejpam-5322	33	40	axioms	axiom	NOUN
ejpam-5322	33	41	are	be	AUX
ejpam-5322	33	42	assumed	assume	VERB
ejpam-5322	33	43	unless	unless	SCONJ
ejpam-5322	33	44	explicitly	explicitly	ADV
ejpam-5322	33	45	stated	state	VERB
ejpam-5322	33	46	.	.	PUNCT
ejpam-5322	34	1	let	let	VERB
ejpam-5322	34	2	a	a	DET
ejpam-5322	34	3	be	be	AUX
ejpam-5322	34	4	a	a	DET
ejpam-5322	34	5	subset	subset	NOUN
ejpam-5322	34	6	of	of	ADP
ejpam-5322	34	7	a	a	DET
ejpam-5322	34	8	bitopological	bitopological	ADJ
ejpam-5322	34	9	space	space	NOUN
ejpam-5322	34	10	(	(	PUNCT
ejpam-5322	34	11	x	x	NOUN
ejpam-5322	34	12	,	,	PUNCT
ejpam-5322	34	13	τ1	τ1	NOUN
ejpam-5322	34	14	,	,	PUNCT
ejpam-5322	34	15	τ2	τ2	NOUN
ejpam-5322	34	16	)	)	PUNCT
ejpam-5322	34	17	.	.	PUNCT
ejpam-5322	35	1	the	the	DET
ejpam-5322	35	2	closure	closure	NOUN
ejpam-5322	35	3	of	of	ADP
ejpam-5322	35	4	a	a	PRON
ejpam-5322	35	5	and	and	CCONJ
ejpam-5322	35	6	the	the	DET
ejpam-5322	35	7	interior	interior	NOUN
ejpam-5322	35	8	of	of	ADP
ejpam-5322	35	9	a	a	PRON
ejpam-5322	35	10	with	with	ADP
ejpam-5322	35	11	respect	respect	NOUN
ejpam-5322	35	12	to	to	ADP
ejpam-5322	35	13	τi	τi	PROPN
ejpam-5322	35	14	are	be	AUX
ejpam-5322	35	15	denoted	denote	VERB
ejpam-5322	35	16	by	by	ADP
ejpam-5322	35	17	τi	τi	NOUN
ejpam-5322	35	18	-	-	PUNCT
ejpam-5322	35	19	cl(a	cl(a	NUM
ejpam-5322	35	20	)	)	PUNCT
ejpam-5322	35	21	and	and	CCONJ
ejpam-5322	35	22	τi	τi	NOUN
ejpam-5322	35	23	-	-	PUNCT
ejpam-5322	35	24	int(a	int(a	NOUN
ejpam-5322	35	25	)	)	PUNCT
ejpam-5322	35	26	,	,	PUNCT
ejpam-5322	35	27	respectively	respectively	ADV
ejpam-5322	35	28	,	,	PUNCT
ejpam-5322	35	29	for	for	ADP
ejpam-5322	35	30	i	i	PROPN
ejpam-5322	35	31	=	=	SYM
ejpam-5322	35	32	1	1	NUM
ejpam-5322	35	33	,	,	PUNCT
ejpam-5322	35	34	2	2	NUM
ejpam-5322	35	35	.	.	X
ejpam-5322	35	36	a	a	DET
ejpam-5322	35	37	subset	subset	NOUN
ejpam-5322	35	38	a	a	PRON
ejpam-5322	35	39	of	of	ADP
ejpam-5322	35	40	a	a	DET
ejpam-5322	35	41	bitopological	bitopological	ADJ
ejpam-5322	35	42	space	space	NOUN
ejpam-5322	35	43	(	(	PUNCT
ejpam-5322	35	44	x	x	NOUN
ejpam-5322	35	45	,	,	PUNCT
ejpam-5322	35	46	τ1	τ1	NOUN
ejpam-5322	35	47	,	,	PUNCT
ejpam-5322	35	48	τ2	τ2	NOUN
ejpam-5322	35	49	)	)	PUNCT
ejpam-5322	35	50	is	be	AUX
ejpam-5322	35	51	called	call	VERB
ejpam-5322	35	52	τ1τ2	τ1τ2	VERB
ejpam-5322	35	53	-	-	ADJ
ejpam-5322	35	54	closed	closed	ADJ
ejpam-5322	35	55	[	[	X
ejpam-5322	35	56	20	20	NUM
ejpam-5322	35	57	]	]	PUNCT
ejpam-5322	35	58	if	if	SCONJ
ejpam-5322	35	59	a	a	DET
ejpam-5322	35	60	=	=	NOUN
ejpam-5322	35	61	τ1	τ1	NOUN
ejpam-5322	35	62	-	-	PUNCT
ejpam-5322	35	63	cl(τ2	cl(τ2	NOUN
ejpam-5322	35	64	-	-	PUNCT
ejpam-5322	35	65	cl(a	cl(a	NUM
ejpam-5322	35	66	)	)	PUNCT
ejpam-5322	35	67	)	)	PUNCT
ejpam-5322	35	68	.	.	PUNCT
ejpam-5322	36	1	the	the	DET
ejpam-5322	36	2	complement	complement	NOUN
ejpam-5322	36	3	of	of	ADP
ejpam-5322	36	4	a	a	DET
ejpam-5322	36	5	τ1τ2	τ1τ2	ADJ
ejpam-5322	36	6	-	-	ADJ
ejpam-5322	36	7	closed	closed	ADJ
ejpam-5322	36	8	set	set	NOUN
ejpam-5322	36	9	is	be	AUX
ejpam-5322	36	10	called	call	VERB
ejpam-5322	36	11	τ1τ2	τ1τ2	NOUN
ejpam-5322	36	12	-	-	ADJ
ejpam-5322	36	13	open	open	ADJ
ejpam-5322	36	14	.	.	PUNCT
ejpam-5322	37	1	the	the	DET
ejpam-5322	37	2	intersection	intersection	NOUN
ejpam-5322	37	3	of	of	ADP
ejpam-5322	37	4	all	all	DET
ejpam-5322	37	5	τ1τ2	τ1τ2	ADJ
ejpam-5322	37	6	-	-	ADJ
ejpam-5322	37	7	closed	closed	ADJ
ejpam-5322	37	8	sets	set	NOUN
ejpam-5322	37	9	of	of	ADP
ejpam-5322	37	10	x	x	PUNCT
ejpam-5322	37	11	containing	contain	VERB
ejpam-5322	37	12	a	a	PRON
ejpam-5322	37	13	is	be	AUX
ejpam-5322	37	14	called	call	VERB
ejpam-5322	37	15	the	the	DET
ejpam-5322	37	16	τ1τ2	τ1τ2	NOUN
ejpam-5322	37	17	-	-	NOUN
ejpam-5322	37	18	closure	closure	NOUN
ejpam-5322	37	19	[	[	X
ejpam-5322	37	20	20	20	NUM
ejpam-5322	37	21	]	]	PUNCT
ejpam-5322	37	22	of	of	ADP
ejpam-5322	37	23	a	a	PRON
ejpam-5322	37	24	and	and	CCONJ
ejpam-5322	37	25	is	be	AUX
ejpam-5322	37	26	denoted	denote	VERB
ejpam-5322	37	27	by	by	ADP
ejpam-5322	37	28	τ1τ2	τ1τ2	NOUN
ejpam-5322	37	29	-	-	NUM
ejpam-5322	37	30	cl(a	cl(a	NUM
ejpam-5322	37	31	)	)	PUNCT
ejpam-5322	37	32	.	.	PUNCT
ejpam-5322	38	1	the	the	DET
ejpam-5322	38	2	union	union	NOUN
ejpam-5322	38	3	of	of	ADP
ejpam-5322	38	4	all	all	DET
ejpam-5322	38	5	τ1τ2	τ1τ2	ADJ
ejpam-5322	38	6	-	-	ADJ
ejpam-5322	38	7	open	open	ADJ
ejpam-5322	38	8	sets	set	NOUN
ejpam-5322	38	9	of	of	ADP
ejpam-5322	38	10	x	x	PUNCT
ejpam-5322	38	11	contained	contain	VERB
ejpam-5322	38	12	in	in	ADP
ejpam-5322	38	13	a	a	PRON
ejpam-5322	38	14	is	be	AUX
ejpam-5322	38	15	called	call	VERB
ejpam-5322	38	16	the	the	DET
ejpam-5322	38	17	τ1τ2	τ1τ2	NOUN
ejpam-5322	38	18	-	-	ADJ
ejpam-5322	38	19	interior	interior	ADJ
ejpam-5322	38	20	[	[	X
ejpam-5322	38	21	20	20	NUM
ejpam-5322	38	22	]	]	PUNCT
ejpam-5322	38	23	of	of	ADP
ejpam-5322	38	24	a	a	PRON
ejpam-5322	38	25	and	and	CCONJ
ejpam-5322	38	26	is	be	AUX
ejpam-5322	38	27	denoted	denote	VERB
ejpam-5322	38	28	by	by	ADP
ejpam-5322	38	29	τ1τ2	τ1τ2	NOUN
ejpam-5322	38	30	-	-	ADJ
ejpam-5322	38	31	int(a	int(a	NOUN
ejpam-5322	38	32	)	)	PUNCT
ejpam-5322	38	33	.	.	PUNCT
ejpam-5322	39	1	lemma	lemma	PROPN
ejpam-5322	39	2	1	1	NUM
ejpam-5322	39	3	.	.	PUNCT
ejpam-5322	40	1	[	[	X
ejpam-5322	40	2	20	20	NUM
ejpam-5322	40	3	]	]	PUNCT
ejpam-5322	40	4	let	let	VERB
ejpam-5322	40	5	a	a	PRON
ejpam-5322	40	6	and	and	CCONJ
ejpam-5322	40	7	b	b	NOUN
ejpam-5322	40	8	be	be	AUX
ejpam-5322	40	9	subsets	subset	NOUN
ejpam-5322	40	10	of	of	ADP
ejpam-5322	40	11	a	a	DET
ejpam-5322	40	12	bitopological	bitopological	ADJ
ejpam-5322	40	13	space	space	NOUN
ejpam-5322	40	14	(	(	PUNCT
ejpam-5322	40	15	x	x	NOUN
ejpam-5322	40	16	,	,	PUNCT
ejpam-5322	40	17	τ1	τ1	NOUN
ejpam-5322	40	18	,	,	PUNCT
ejpam-5322	40	19	τ2	τ2	NOUN
ejpam-5322	40	20	)	)	PUNCT
ejpam-5322	40	21	.	.	PUNCT
ejpam-5322	41	1	for	for	ADP
ejpam-5322	41	2	the	the	DET
ejpam-5322	41	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-5322	41	4	,	,	PUNCT
ejpam-5322	41	5	the	the	DET
ejpam-5322	41	6	following	follow	VERB
ejpam-5322	41	7	properties	property	NOUN
ejpam-5322	41	8	hold	hold	VERB
ejpam-5322	41	9	:	:	PUNCT
ejpam-5322	41	10	(	(	PUNCT
ejpam-5322	41	11	1	1	X
ejpam-5322	41	12	)	)	PUNCT
ejpam-5322	41	13	a	a	DET
ejpam-5322	41	14	⊆	⊆	NUM
ejpam-5322	41	15	τ1τ2	τ1τ2	NOUN
ejpam-5322	41	16	-	-	NUM
ejpam-5322	41	17	cl(a	cl(a	NUM
ejpam-5322	41	18	)	)	PUNCT
ejpam-5322	41	19	and	and	CCONJ
ejpam-5322	41	20	τ1τ2	τ1τ2	NOUN
ejpam-5322	41	21	-	-	ADJ
ejpam-5322	41	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	41	23	-	-	PUNCT
ejpam-5322	41	24	cl(a	cl(a	NUM
ejpam-5322	41	25	)	)	PUNCT
ejpam-5322	41	26	)	)	PUNCT
ejpam-5322	42	1	=	=	PUNCT
ejpam-5322	42	2	τ1τ2	τ1τ2	NOUN
ejpam-5322	42	3	-	-	NUM
ejpam-5322	42	4	cl(a	cl(a	NUM
ejpam-5322	42	5	)	)	PUNCT
ejpam-5322	42	6	.	.	PUNCT
ejpam-5322	43	1	(	(	PUNCT
ejpam-5322	43	2	2	2	X
ejpam-5322	43	3	)	)	PUNCT
ejpam-5322	43	4	if	if	SCONJ
ejpam-5322	43	5	a	a	DET
ejpam-5322	43	6	⊆	⊆	NUM
ejpam-5322	43	7	b	b	NOUN
ejpam-5322	43	8	,	,	PUNCT
ejpam-5322	43	9	then	then	ADV
ejpam-5322	43	10	τ1τ2	τ1τ2	NOUN
ejpam-5322	43	11	-	-	NUM
ejpam-5322	43	12	cl(a	cl(a	NUM
ejpam-5322	43	13	)	)	PUNCT
ejpam-5322	43	14	⊆	⊆	NUM
ejpam-5322	43	15	τ1τ2	τ1τ2	NOUN
ejpam-5322	43	16	-	-	NOUN
ejpam-5322	43	17	cl(b	cl(b	NOUN
ejpam-5322	43	18	)	)	PUNCT
ejpam-5322	43	19	.	.	PUNCT
ejpam-5322	44	1	(	(	PUNCT
ejpam-5322	44	2	3	3	X
ejpam-5322	44	3	)	)	PUNCT
ejpam-5322	44	4	τ1τ2	τ1τ2	NOUN
ejpam-5322	44	5	-	-	NUM
ejpam-5322	44	6	cl(a	cl(a	NUM
ejpam-5322	44	7	)	)	PUNCT
ejpam-5322	44	8	is	be	AUX
ejpam-5322	44	9	τ1τ2	τ1τ2	NOUN
ejpam-5322	44	10	-	-	ADJ
ejpam-5322	44	11	closed	closed	ADJ
ejpam-5322	44	12	.	.	PUNCT
ejpam-5322	45	1	(	(	PUNCT
ejpam-5322	45	2	4	4	X
ejpam-5322	45	3	)	)	PUNCT
ejpam-5322	45	4	a	a	PRON
ejpam-5322	45	5	is	be	AUX
ejpam-5322	45	6	τ1τ2	τ1τ2	NOUN
ejpam-5322	45	7	-	-	ADJ
ejpam-5322	45	8	closed	closed	ADJ
ejpam-5322	45	9	if	if	SCONJ
ejpam-5322	45	10	and	and	CCONJ
ejpam-5322	45	11	only	only	ADV
ejpam-5322	45	12	if	if	SCONJ
ejpam-5322	45	13	a	a	DET
ejpam-5322	45	14	=	=	PUNCT
ejpam-5322	45	15	τ1τ2	τ1τ2	NOUN
ejpam-5322	45	16	-	-	NUM
ejpam-5322	45	17	cl(a	cl(a	NUM
ejpam-5322	45	18	)	)	PUNCT
ejpam-5322	45	19	.	.	PUNCT
ejpam-5322	46	1	(	(	PUNCT
ejpam-5322	46	2	5	5	X
ejpam-5322	46	3	)	)	PUNCT
ejpam-5322	46	4	τ1τ2	τ1τ2	NOUN
ejpam-5322	46	5	-	-	NOUN
ejpam-5322	46	6	cl(x	cl(x	X
ejpam-5322	46	7	−a	−a	NOUN
ejpam-5322	46	8	)	)	PUNCT
ejpam-5322	47	1	=	=	PUNCT
ejpam-5322	47	2	x	x	X
ejpam-5322	48	1	−	−	ADP
ejpam-5322	48	2	τ1τ2	τ1τ2	NOUN
ejpam-5322	48	3	-	-	PUNCT
ejpam-5322	48	4	int(a	int(a	NOUN
ejpam-5322	48	5	)	)	PUNCT
ejpam-5322	48	6	.	.	PUNCT
ejpam-5322	49	1	a	a	DET
ejpam-5322	49	2	bitopological	bitopological	ADJ
ejpam-5322	49	3	space	space	NOUN
ejpam-5322	49	4	(	(	PUNCT
ejpam-5322	49	5	x	x	NOUN
ejpam-5322	49	6	,	,	PUNCT
ejpam-5322	49	7	τ1	τ1	NOUN
ejpam-5322	49	8	,	,	PUNCT
ejpam-5322	49	9	τ2	τ2	NOUN
ejpam-5322	49	10	)	)	PUNCT
ejpam-5322	49	11	is	be	AUX
ejpam-5322	49	12	said	say	VERB
ejpam-5322	49	13	to	to	PART
ejpam-5322	49	14	be	be	AUX
ejpam-5322	49	15	τ1τ2	τ1τ2	NOUN
ejpam-5322	49	16	-	-	ADJ
ejpam-5322	49	17	connected	connected	ADJ
ejpam-5322	49	18	[	[	X
ejpam-5322	49	19	20	20	NUM
ejpam-5322	49	20	]	]	PUNCT
ejpam-5322	49	21	if	if	SCONJ
ejpam-5322	49	22	x	x	PRON
ejpam-5322	49	23	can	can	AUX
ejpam-5322	49	24	not	not	PART
ejpam-5322	49	25	be	be	AUX
ejpam-5322	49	26	written	write	VERB
ejpam-5322	49	27	as	as	ADP
ejpam-5322	49	28	the	the	DET
ejpam-5322	49	29	union	union	NOUN
ejpam-5322	49	30	of	of	ADP
ejpam-5322	49	31	two	two	NUM
ejpam-5322	49	32	nonempty	nonempty	ADV
ejpam-5322	49	33	disjoint	disjoint	NOUN
ejpam-5322	49	34	τ1τ2	τ1τ2	ADJ
ejpam-5322	49	35	-	-	ADJ
ejpam-5322	49	36	open	open	ADJ
ejpam-5322	49	37	sets	set	NOUN
ejpam-5322	49	38	.	.	PUNCT
ejpam-5322	50	1	a	a	DET
ejpam-5322	50	2	subset	subset	NOUN
ejpam-5322	50	3	a	a	PRON
ejpam-5322	50	4	of	of	ADP
ejpam-5322	50	5	a	a	DET
ejpam-5322	50	6	bitopological	bitopological	ADJ
ejpam-5322	50	7	space	space	NOUN
ejpam-5322	50	8	(	(	PUNCT
ejpam-5322	50	9	x	x	NOUN
ejpam-5322	50	10	,	,	PUNCT
ejpam-5322	50	11	τ1	τ1	NOUN
ejpam-5322	50	12	,	,	PUNCT
ejpam-5322	50	13	τ2	τ2	NOUN
ejpam-5322	50	14	)	)	PUNCT
ejpam-5322	50	15	is	be	AUX
ejpam-5322	50	16	called	call	VERB
ejpam-5322	50	17	(	(	PUNCT
ejpam-5322	50	18	τ1	τ1	NOUN
ejpam-5322	50	19	,	,	PUNCT
ejpam-5322	50	20	τ2)r	τ2)r	NOUN
ejpam-5322	50	21	-	-	PUNCT
ejpam-5322	50	22	open	open	NOUN
ejpam-5322	51	1	[	[	X
ejpam-5322	51	2	36	36	NUM
ejpam-5322	51	3	]	]	X
ejpam-5322	51	4	(	(	PUNCT
ejpam-5322	51	5	resp	resp	NOUN
ejpam-5322	51	6	.	.	PUNCT
ejpam-5322	52	1	(	(	PUNCT
ejpam-5322	52	2	τ1	τ1	NOUN
ejpam-5322	52	3	,	,	PUNCT
ejpam-5322	52	4	τ2)s	τ2)s	NOUN
ejpam-5322	52	5	-	-	PUNCT
ejpam-5322	52	6	open	open	ADJ
ejpam-5322	52	7	[	[	X
ejpam-5322	52	8	6	6	NUM
ejpam-5322	52	9	]	]	PUNCT
ejpam-5322	52	10	,	,	PUNCT
ejpam-5322	52	11	(	(	PUNCT
ejpam-5322	52	12	τ1	τ1	NOUN
ejpam-5322	52	13	,	,	PUNCT
ejpam-5322	52	14	τ2)popen	τ2)popen	ADJ
ejpam-5322	52	15	[	[	PUNCT
ejpam-5322	52	16	6	6	NUM
ejpam-5322	52	17	]	]	PUNCT
ejpam-5322	52	18	,	,	PUNCT
ejpam-5322	52	19	(	(	PUNCT
ejpam-5322	52	20	τ1	τ1	NOUN
ejpam-5322	52	21	,	,	PUNCT
ejpam-5322	52	22	τ2)β	τ2)β	ADJ
ejpam-5322	52	23	-	-	PUNCT
ejpam-5322	52	24	open	open	NOUN
ejpam-5322	53	1	[	[	X
ejpam-5322	53	2	6	6	NUM
ejpam-5322	53	3	]	]	PUNCT
ejpam-5322	53	4	,	,	PUNCT
ejpam-5322	53	5	α(τ1	α(τ1	NOUN
ejpam-5322	53	6	,	,	PUNCT
ejpam-5322	53	7	τ2)-open	τ2)-open	ADJ
ejpam-5322	53	8	)	)	PUNCT
ejpam-5322	54	1	[	[	X
ejpam-5322	54	2	38	38	NUM
ejpam-5322	54	3	]	]	SYM
ejpam-5322	54	4	)	)	PUNCT
ejpam-5322	54	5	if	if	SCONJ
ejpam-5322	54	6	a	a	DET
ejpam-5322	54	7	=	=	PUNCT
ejpam-5322	54	8	τ1τ2	τ1τ2	NOUN
ejpam-5322	54	9	-	-	NOUN
ejpam-5322	54	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5322	54	11	-	-	PUNCT
ejpam-5322	54	12	cl(a	cl(a	NUM
ejpam-5322	54	13	)	)	PUNCT
ejpam-5322	54	14	)	)	PUNCT
ejpam-5322	54	15	(	(	PUNCT
ejpam-5322	54	16	resp	resp	NOUN
ejpam-5322	54	17	.	.	PUNCT
ejpam-5322	55	1	a	a	DET
ejpam-5322	55	2	⊆	⊆	NUM
ejpam-5322	55	3	n.	n.	NOUN
ejpam-5322	55	4	viriyapong	viriyapong	PROPN
ejpam-5322	55	5	,	,	PUNCT
ejpam-5322	55	6	s.	s.	PROPN
ejpam-5322	55	7	sompong	sompong	PROPN
ejpam-5322	55	8	,	,	PUNCT
ejpam-5322	55	9	c.	c.	PROPN
ejpam-5322	55	10	boonpok	boonpok	PROPN
ejpam-5322	55	11	/	/	SYM
ejpam-5322	55	12	eur	eur	PROPN
ejpam-5322	55	13	.	.	PUNCT
ejpam-5322	56	1	j.	j.	PROPN
ejpam-5322	56	2	pure	pure	PROPN
ejpam-5322	56	3	appl	appl	PROPN
ejpam-5322	56	4	.	.	PROPN
ejpam-5322	56	5	math	math	PROPN
ejpam-5322	56	6	,	,	PUNCT
ejpam-5322	56	7	17	17	NUM
ejpam-5322	56	8	(	(	PUNCT
ejpam-5322	56	9	3	3	NUM
ejpam-5322	56	10	)	)	PUNCT
ejpam-5322	56	11	(	(	PUNCT
ejpam-5322	56	12	2024	2024	NUM
ejpam-5322	56	13	)	)	PUNCT
ejpam-5322	56	14	,	,	PUNCT
ejpam-5322	56	15	2210	2210	NUM
ejpam-5322	56	16	-	-	SYM
ejpam-5322	56	17	2220	2220	NUM
ejpam-5322	56	18	2212	2212	NUM
ejpam-5322	56	19	τ1τ2	τ1τ2	NOUN
ejpam-5322	56	20	-	-	PUNCT
ejpam-5322	56	21	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	56	22	-	-	PUNCT
ejpam-5322	56	23	int(a	int(a	NOUN
ejpam-5322	56	24	)	)	PUNCT
ejpam-5322	56	25	)	)	PUNCT
ejpam-5322	56	26	,	,	PUNCT
ejpam-5322	56	27	a	a	DET
ejpam-5322	56	28	⊆	⊆	NUM
ejpam-5322	56	29	τ1τ2	τ1τ2	NOUN
ejpam-5322	56	30	-	-	NOUN
ejpam-5322	56	31	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5322	56	32	-	-	PUNCT
ejpam-5322	56	33	cl(a	cl(a	NUM
ejpam-5322	56	34	)	)	PUNCT
ejpam-5322	56	35	)	)	PUNCT
ejpam-5322	56	36	,	,	PUNCT
ejpam-5322	56	37	a	a	DET
ejpam-5322	56	38	⊆	⊆	NUM
ejpam-5322	56	39	τ1τ2	τ1τ2	NOUN
ejpam-5322	56	40	-	-	PUNCT
ejpam-5322	56	41	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	56	42	-	-	PUNCT
ejpam-5322	56	43	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5322	56	44	-	-	PUNCT
ejpam-5322	56	45	cl(a	cl(a	NUM
ejpam-5322	56	46	)	)	PUNCT
ejpam-5322	56	47	)	)	PUNCT
ejpam-5322	56	48	)	)	PUNCT
ejpam-5322	56	49	,	,	PUNCT
ejpam-5322	56	50	a	a	DET
ejpam-5322	56	51	⊆	⊆	NUM
ejpam-5322	56	52	τ1τ2	τ1τ2	NOUN
ejpam-5322	56	53	-	-	PUNCT
ejpam-5322	56	54	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5322	56	55	-	-	PUNCT
ejpam-5322	56	56	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	56	57	-	-	PUNCT
ejpam-5322	56	58	int(a	int(a	NOUN
ejpam-5322	56	59	)	)	PUNCT
ejpam-5322	56	60	)	)	PUNCT
ejpam-5322	56	61	)	)	PUNCT
ejpam-5322	56	62	)	)	PUNCT
ejpam-5322	56	63	.	.	PUNCT
ejpam-5322	57	1	the	the	DET
ejpam-5322	57	2	complement	complement	NOUN
ejpam-5322	57	3	of	of	ADP
ejpam-5322	57	4	a	a	DET
ejpam-5322	57	5	(	(	PUNCT
ejpam-5322	57	6	τ1	τ1	NOUN
ejpam-5322	57	7	,	,	PUNCT
ejpam-5322	57	8	τ2)r	τ2)r	NOUN
ejpam-5322	57	9	-	-	PUNCT
ejpam-5322	57	10	open	open	ADJ
ejpam-5322	57	11	(	(	PUNCT
ejpam-5322	57	12	resp	resp	NOUN
ejpam-5322	57	13	.	.	PUNCT
ejpam-5322	58	1	(	(	PUNCT
ejpam-5322	58	2	τ1	τ1	NOUN
ejpam-5322	58	3	,	,	PUNCT
ejpam-5322	58	4	τ2)s	τ2)s	NOUN
ejpam-5322	58	5	-	-	PUNCT
ejpam-5322	58	6	open	open	ADJ
ejpam-5322	58	7	,	,	PUNCT
ejpam-5322	58	8	(	(	PUNCT
ejpam-5322	58	9	τ1	τ1	NOUN
ejpam-5322	58	10	,	,	PUNCT
ejpam-5322	58	11	τ2)p	τ2)p	NOUN
ejpam-5322	58	12	-	-	ADJ
ejpam-5322	58	13	open	open	ADJ
ejpam-5322	58	14	,	,	PUNCT
ejpam-5322	58	15	(	(	PUNCT
ejpam-5322	58	16	τ1	τ1	NOUN
ejpam-5322	58	17	,	,	PUNCT
ejpam-5322	58	18	τ2)β	τ2)β	ADJ
ejpam-5322	58	19	-	-	PUNCT
ejpam-5322	58	20	open	open	ADJ
ejpam-5322	58	21	,	,	PUNCT
ejpam-5322	58	22	α(τ1	α(τ1	NOUN
ejpam-5322	58	23	,	,	PUNCT
ejpam-5322	58	24	τ2)-open	τ2)-open	ADJ
ejpam-5322	58	25	)	)	PUNCT
ejpam-5322	58	26	set	set	NOUN
ejpam-5322	58	27	is	be	AUX
ejpam-5322	58	28	called	call	VERB
ejpam-5322	58	29	(	(	PUNCT
ejpam-5322	58	30	τ1	τ1	NOUN
ejpam-5322	58	31	,	,	PUNCT
ejpam-5322	58	32	τ2)r	τ2)r	NOUN
ejpam-5322	58	33	-	-	PUNCT
ejpam-5322	58	34	closed	closed	ADJ
ejpam-5322	58	35	(	(	PUNCT
ejpam-5322	58	36	resp	resp	NOUN
ejpam-5322	58	37	.	.	PUNCT
ejpam-5322	59	1	(	(	PUNCT
ejpam-5322	59	2	τ1	τ1	NOUN
ejpam-5322	59	3	,	,	PUNCT
ejpam-5322	59	4	τ2)sclosed	τ2)sclose	VERB
ejpam-5322	59	5	,	,	PUNCT
ejpam-5322	59	6	(	(	PUNCT
ejpam-5322	59	7	τ1	τ1	NOUN
ejpam-5322	59	8	,	,	PUNCT
ejpam-5322	59	9	τ2)p	τ2)p	NOUN
ejpam-5322	59	10	-	-	PUNCT
ejpam-5322	59	11	closed	closed	ADJ
ejpam-5322	59	12	,	,	PUNCT
ejpam-5322	59	13	(	(	PUNCT
ejpam-5322	59	14	τ1	τ1	NOUN
ejpam-5322	59	15	,	,	PUNCT
ejpam-5322	59	16	τ2)β	τ2)β	ADJ
ejpam-5322	59	17	-	-	PUNCT
ejpam-5322	59	18	closed	closed	ADJ
ejpam-5322	59	19	,	,	PUNCT
ejpam-5322	59	20	α(τ1	α(τ1	NOUN
ejpam-5322	59	21	,	,	PUNCT
ejpam-5322	59	22	τ2)-closed	τ2)-closed	ADJ
ejpam-5322	59	23	)	)	PUNCT
ejpam-5322	59	24	.	.	PUNCT
ejpam-5322	60	1	let	let	VERB
ejpam-5322	60	2	a	a	DET
ejpam-5322	60	3	be	be	AUX
ejpam-5322	60	4	a	a	DET
ejpam-5322	60	5	subset	subset	NOUN
ejpam-5322	60	6	of	of	ADP
ejpam-5322	60	7	a	a	DET
ejpam-5322	60	8	bitopological	bitopological	ADJ
ejpam-5322	60	9	space	space	NOUN
ejpam-5322	60	10	(	(	PUNCT
ejpam-5322	60	11	x	x	NOUN
ejpam-5322	60	12	,	,	PUNCT
ejpam-5322	60	13	τ1	τ1	NOUN
ejpam-5322	60	14	,	,	PUNCT
ejpam-5322	60	15	τ2	τ2	NOUN
ejpam-5322	60	16	)	)	PUNCT
ejpam-5322	60	17	.	.	PUNCT
ejpam-5322	61	1	the	the	DET
ejpam-5322	61	2	intersection	intersection	NOUN
ejpam-5322	61	3	of	of	ADP
ejpam-5322	61	4	all	all	DET
ejpam-5322	61	5	(	(	PUNCT
ejpam-5322	61	6	τ1	τ1	NOUN
ejpam-5322	61	7	,	,	PUNCT
ejpam-5322	61	8	τ2)p	τ2)p	ADJ
ejpam-5322	61	9	-	-	PUNCT
ejpam-5322	61	10	closed	closed	ADJ
ejpam-5322	61	11	sets	set	NOUN
ejpam-5322	61	12	of	of	ADP
ejpam-5322	61	13	x	x	PUNCT
ejpam-5322	61	14	containing	contain	VERB
ejpam-5322	61	15	a	a	PRON
ejpam-5322	61	16	is	be	AUX
ejpam-5322	61	17	called	call	VERB
ejpam-5322	61	18	the	the	DET
ejpam-5322	61	19	(	(	PUNCT
ejpam-5322	61	20	τ1	τ1	NOUN
ejpam-5322	61	21	,	,	PUNCT
ejpam-5322	61	22	τ2)p	τ2)p	NOUN
ejpam-5322	61	23	-	-	PUNCT
ejpam-5322	61	24	closure	closure	NOUN
ejpam-5322	61	25	of	of	ADP
ejpam-5322	61	26	a	a	PRON
ejpam-5322	61	27	and	and	CCONJ
ejpam-5322	61	28	is	be	AUX
ejpam-5322	61	29	denoted	denote	VERB
ejpam-5322	61	30	by	by	ADP
ejpam-5322	61	31	(	(	PUNCT
ejpam-5322	61	32	τ1	τ1	PROPN
ejpam-5322	61	33	,	,	PUNCT
ejpam-5322	61	34	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5322	61	35	)	)	PUNCT
ejpam-5322	61	36	.	.	PUNCT
ejpam-5322	62	1	the	the	DET
ejpam-5322	62	2	union	union	NOUN
ejpam-5322	62	3	of	of	ADP
ejpam-5322	62	4	all	all	DET
ejpam-5322	62	5	(	(	PUNCT
ejpam-5322	62	6	τ1	τ1	NOUN
ejpam-5322	62	7	,	,	PUNCT
ejpam-5322	62	8	τ2)p	τ2)p	ADJ
ejpam-5322	62	9	-	-	PUNCT
ejpam-5322	62	10	open	open	ADJ
ejpam-5322	62	11	sets	set	NOUN
ejpam-5322	62	12	of	of	ADP
ejpam-5322	62	13	x	x	PUNCT
ejpam-5322	62	14	contained	contain	VERB
ejpam-5322	62	15	in	in	ADP
ejpam-5322	62	16	a	a	PRON
ejpam-5322	62	17	is	be	AUX
ejpam-5322	62	18	called	call	VERB
ejpam-5322	62	19	the	the	DET
ejpam-5322	62	20	(	(	PUNCT
ejpam-5322	62	21	τ1	τ1	NOUN
ejpam-5322	62	22	,	,	PUNCT
ejpam-5322	62	23	τ2)p	τ2)p	ADJ
ejpam-5322	62	24	-	-	NOUN
ejpam-5322	62	25	interior	interior	NOUN
ejpam-5322	62	26	of	of	ADP
ejpam-5322	62	27	a	a	PRON
ejpam-5322	62	28	and	and	CCONJ
ejpam-5322	62	29	is	be	AUX
ejpam-5322	62	30	denoted	denote	VERB
ejpam-5322	62	31	by	by	ADP
ejpam-5322	62	32	(	(	PUNCT
ejpam-5322	62	33	τ1	τ1	NOUN
ejpam-5322	62	34	,	,	PUNCT
ejpam-5322	62	35	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-5322	62	36	)	)	PUNCT
ejpam-5322	62	37	.	.	PUNCT
ejpam-5322	63	1	lemma	lemma	PROPN
ejpam-5322	63	2	2	2	NUM
ejpam-5322	63	3	.	.	X
ejpam-5322	64	1	for	for	ADP
ejpam-5322	64	2	a	a	DET
ejpam-5322	64	3	subset	subset	NOUN
ejpam-5322	64	4	a	a	PRON
ejpam-5322	64	5	of	of	ADP
ejpam-5322	64	6	a	a	DET
ejpam-5322	64	7	bitopological	bitopological	ADJ
ejpam-5322	64	8	space	space	NOUN
ejpam-5322	64	9	(	(	PUNCT
ejpam-5322	64	10	x	x	NOUN
ejpam-5322	64	11	,	,	PUNCT
ejpam-5322	64	12	τ1	τ1	NOUN
ejpam-5322	64	13	,	,	PUNCT
ejpam-5322	64	14	τ2	τ2	NOUN
ejpam-5322	64	15	)	)	PUNCT
ejpam-5322	64	16	,	,	PUNCT
ejpam-5322	64	17	the	the	DET
ejpam-5322	64	18	following	follow	VERB
ejpam-5322	64	19	properties	property	NOUN
ejpam-5322	64	20	hold	hold	VERB
ejpam-5322	64	21	:	:	PUNCT
ejpam-5322	64	22	(	(	PUNCT
ejpam-5322	64	23	1	1	X
ejpam-5322	64	24	)	)	PUNCT
ejpam-5322	64	25	a	a	PRON
ejpam-5322	64	26	is	is	NOUN
ejpam-5322	64	27	(	(	PUNCT
ejpam-5322	64	28	τ1	τ1	NOUN
ejpam-5322	64	29	,	,	PUNCT
ejpam-5322	64	30	τ2)p	τ2)p	NOUN
ejpam-5322	64	31	-	-	PUNCT
ejpam-5322	64	32	closed	closed	ADJ
ejpam-5322	64	33	if	if	SCONJ
ejpam-5322	64	34	and	and	CCONJ
ejpam-5322	64	35	only	only	ADV
ejpam-5322	64	36	if	if	SCONJ
ejpam-5322	64	37	(	(	PUNCT
ejpam-5322	64	38	τ1	τ1	NOUN
ejpam-5322	64	39	,	,	PUNCT
ejpam-5322	64	40	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5322	64	41	)	)	PUNCT
ejpam-5322	64	42	=	=	SYM
ejpam-5322	64	43	a	a	X
ejpam-5322	64	44	;	;	PUNCT
ejpam-5322	64	45	(	(	PUNCT
ejpam-5322	64	46	2	2	NUM
ejpam-5322	64	47	)	)	PUNCT
ejpam-5322	64	48	(	(	PUNCT
ejpam-5322	64	49	τ1	τ1	NOUN
ejpam-5322	64	50	,	,	PUNCT
ejpam-5322	64	51	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-5322	64	52	)	)	PUNCT
ejpam-5322	64	53	=	=	PUNCT
ejpam-5322	65	1	τ1τ2	τ1τ2	NOUN
ejpam-5322	65	2	-	-	ADJ
ejpam-5322	65	3	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	65	4	-	-	PUNCT
ejpam-5322	65	5	int(a	int(a	NOUN
ejpam-5322	65	6	)	)	PUNCT
ejpam-5322	65	7	)	)	PUNCT
ejpam-5322	65	8	∪a	∪a	NUM
ejpam-5322	65	9	;	;	PUNCT
ejpam-5322	65	10	(	(	PUNCT
ejpam-5322	65	11	3	3	X
ejpam-5322	65	12	)	)	PUNCT
ejpam-5322	65	13	(	(	PUNCT
ejpam-5322	65	14	τ1	τ1	NOUN
ejpam-5322	65	15	,	,	PUNCT
ejpam-5322	65	16	τ2)-pcl((τ1	τ2)-pcl((τ1	PROPN
ejpam-5322	65	17	,	,	PUNCT
ejpam-5322	65	18	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5322	65	19	)	)	PUNCT
ejpam-5322	65	20	)	)	PUNCT
ejpam-5322	66	1	=	=	PRON
ejpam-5322	66	2	(	(	PUNCT
ejpam-5322	66	3	τ1	τ1	PROPN
ejpam-5322	66	4	,	,	PUNCT
ejpam-5322	66	5	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5322	66	6	)	)	PUNCT
ejpam-5322	66	7	.	.	PUNCT
ejpam-5322	67	1	by	by	ADP
ejpam-5322	67	2	a	a	DET
ejpam-5322	67	3	multifunction	multifunction	NOUN
ejpam-5322	67	4	f	f	NOUN
ejpam-5322	67	5	:	:	PUNCT
ejpam-5322	67	6	x	x	X
ejpam-5322	67	7	→	→	SYM
ejpam-5322	67	8	y	y	PROPN
ejpam-5322	67	9	,	,	PUNCT
ejpam-5322	67	10	we	we	PRON
ejpam-5322	67	11	mean	mean	VERB
ejpam-5322	67	12	a	a	DET
ejpam-5322	67	13	point	point	NOUN
ejpam-5322	67	14	-	-	PUNCT
ejpam-5322	67	15	to	to	ADP
ejpam-5322	67	16	-	-	PUNCT
ejpam-5322	67	17	set	set	VERB
ejpam-5322	67	18	correspondence	correspondence	NOUN
ejpam-5322	67	19	from	from	ADP
ejpam-5322	67	20	x	x	PUNCT
ejpam-5322	67	21	into	into	ADP
ejpam-5322	67	22	y	y	PROPN
ejpam-5322	67	23	,	,	PUNCT
ejpam-5322	67	24	and	and	CCONJ
ejpam-5322	67	25	we	we	PRON
ejpam-5322	67	26	always	always	ADV
ejpam-5322	67	27	assume	assume	VERB
ejpam-5322	67	28	that	that	SCONJ
ejpam-5322	67	29	f	f	PROPN
ejpam-5322	67	30	(	(	PUNCT
ejpam-5322	67	31	x	x	X
ejpam-5322	67	32	)	)	PUNCT
ejpam-5322	67	33	̸=	̸=	NOUN
ejpam-5322	67	34	∅	∅	NOUN
ejpam-5322	67	35	for	for	ADP
ejpam-5322	67	36	all	all	PRON
ejpam-5322	67	37	x	x	SYM
ejpam-5322	67	38	∈	∈	ADJ
ejpam-5322	67	39	x.	x.	NOUN
ejpam-5322	67	40	for	for	ADP
ejpam-5322	67	41	a	a	DET
ejpam-5322	67	42	multifunction	multifunction	NOUN
ejpam-5322	67	43	f	f	NOUN
ejpam-5322	68	1	:	:	PUNCT
ejpam-5322	68	2	x	x	X
ejpam-5322	68	3	→	→	SYM
ejpam-5322	68	4	y	y	PROPN
ejpam-5322	68	5	,	,	PUNCT
ejpam-5322	68	6	following	follow	VERB
ejpam-5322	68	7	[	[	X
ejpam-5322	68	8	1	1	X
ejpam-5322	68	9	]	]	PUNCT
ejpam-5322	68	10	we	we	PRON
ejpam-5322	68	11	shall	shall	AUX
ejpam-5322	68	12	denote	denote	VERB
ejpam-5322	68	13	the	the	DET
ejpam-5322	68	14	upper	upper	ADJ
ejpam-5322	68	15	and	and	CCONJ
ejpam-5322	68	16	lower	low	ADJ
ejpam-5322	68	17	inverse	inverse	NOUN
ejpam-5322	68	18	of	of	ADP
ejpam-5322	68	19	a	a	DET
ejpam-5322	68	20	set	set	NOUN
ejpam-5322	68	21	b	b	PROPN
ejpam-5322	68	22	of	of	ADP
ejpam-5322	68	23	y	y	PROPN
ejpam-5322	68	24	by	by	ADP
ejpam-5322	68	25	f+(b	f+(b	NOUN
ejpam-5322	68	26	)	)	PUNCT
ejpam-5322	68	27	and	and	CCONJ
ejpam-5322	68	28	f−(b	f−(b	NOUN
ejpam-5322	68	29	)	)	PUNCT
ejpam-5322	68	30	,	,	PUNCT
ejpam-5322	68	31	respectively	respectively	ADV
ejpam-5322	68	32	,	,	PUNCT
ejpam-5322	68	33	that	that	ADV
ejpam-5322	68	34	is	is	ADV
ejpam-5322	68	35	,	,	PUNCT
ejpam-5322	68	36	f+(b	f+(b	NOUN
ejpam-5322	68	37	)	)	PUNCT
ejpam-5322	68	38	=	=	PRON
ejpam-5322	69	1	{	{	PUNCT
ejpam-5322	69	2	x	x	PUNCT
ejpam-5322	69	3	∈	∈	PROPN
ejpam-5322	69	4	x	x	INTJ
ejpam-5322	70	1	|	|	NOUN
ejpam-5322	70	2	f	f	X
ejpam-5322	70	3	(	(	PUNCT
ejpam-5322	70	4	x	x	NOUN
ejpam-5322	70	5	)	)	PUNCT
ejpam-5322	70	6	⊆	⊆	NUM
ejpam-5322	70	7	b	b	NOUN
ejpam-5322	70	8	}	}	PUNCT
ejpam-5322	70	9	and	and	CCONJ
ejpam-5322	70	10	f−(b	f−(b	PROPN
ejpam-5322	70	11	)	)	PUNCT
ejpam-5322	70	12	=	=	PRON
ejpam-5322	71	1	{	{	PUNCT
ejpam-5322	71	2	x	x	PUNCT
ejpam-5322	71	3	∈	∈	PROPN
ejpam-5322	71	4	x	x	INTJ
ejpam-5322	72	1	|	|	NOUN
ejpam-5322	72	2	f	f	X
ejpam-5322	72	3	(	(	PUNCT
ejpam-5322	72	4	x	x	NOUN
ejpam-5322	72	5	)	)	PUNCT
ejpam-5322	72	6	∩b	∩b	NOUN
ejpam-5322	72	7	̸=	̸=	PROPN
ejpam-5322	72	8	∅	∅	NOUN
ejpam-5322	72	9	}	}	PUNCT
ejpam-5322	72	10	.	.	PUNCT
ejpam-5322	73	1	in	in	ADP
ejpam-5322	73	2	particular	particular	ADJ
ejpam-5322	73	3	,	,	PUNCT
ejpam-5322	73	4	f−(y	f−(y	NOUN
ejpam-5322	73	5	)	)	PUNCT
ejpam-5322	73	6	=	=	SYM
ejpam-5322	74	1	{	{	PUNCT
ejpam-5322	74	2	x	x	PUNCT
ejpam-5322	74	3	∈	∈	PROPN
ejpam-5322	74	4	x	x	INTJ
ejpam-5322	75	1	|	|	ADV
ejpam-5322	75	2	y	y	PROPN
ejpam-5322	75	3	∈	∈	PROPN
ejpam-5322	75	4	f	f	X
ejpam-5322	75	5	(	(	PUNCT
ejpam-5322	75	6	x	x	NOUN
ejpam-5322	75	7	)	)	PUNCT
ejpam-5322	75	8	}	}	PUNCT
ejpam-5322	75	9	for	for	ADP
ejpam-5322	75	10	each	each	DET
ejpam-5322	75	11	point	point	NOUN
ejpam-5322	75	12	y	y	PROPN
ejpam-5322	75	13	∈	∈	PROPN
ejpam-5322	75	14	y	y	PROPN
ejpam-5322	75	15	.	.	PUNCT
ejpam-5322	76	1	for	for	ADP
ejpam-5322	76	2	each	each	DET
ejpam-5322	76	3	a	a	DET
ejpam-5322	76	4	⊆	⊆	NUM
ejpam-5322	76	5	x	x	SYM
ejpam-5322	76	6	,	,	PUNCT
ejpam-5322	76	7	f	f	PROPN
ejpam-5322	76	8	(	(	PUNCT
ejpam-5322	76	9	a	a	NOUN
ejpam-5322	76	10	)	)	PUNCT
ejpam-5322	76	11	=	=	SYM
ejpam-5322	76	12	∪x∈af	∪x∈af	NOUN
ejpam-5322	76	13	(	(	PUNCT
ejpam-5322	76	14	x	x	NOUN
ejpam-5322	76	15	)	)	PUNCT
ejpam-5322	76	16	.	.	PUNCT
ejpam-5322	77	1	3	3	X
ejpam-5322	77	2	.	.	X
ejpam-5322	77	3	upper	upper	ADJ
ejpam-5322	77	4	and	and	CCONJ
ejpam-5322	77	5	lower	low	ADJ
ejpam-5322	77	6	s-(τ1	s-(τ1	NOUN
ejpam-5322	77	7	,	,	PUNCT
ejpam-5322	77	8	τ2)p	τ2)p	ADJ
ejpam-5322	77	9	-	-	ADJ
ejpam-5322	77	10	continuous	continuous	ADJ
ejpam-5322	77	11	multifunctions	multifunction	NOUN
ejpam-5322	77	12	in	in	ADP
ejpam-5322	77	13	this	this	DET
ejpam-5322	77	14	section	section	NOUN
ejpam-5322	77	15	,	,	PUNCT
ejpam-5322	77	16	we	we	PRON
ejpam-5322	77	17	introduce	introduce	VERB
ejpam-5322	77	18	the	the	DET
ejpam-5322	77	19	notions	notion	NOUN
ejpam-5322	77	20	of	of	ADP
ejpam-5322	77	21	upper	upper	ADJ
ejpam-5322	77	22	and	and	CCONJ
ejpam-5322	77	23	lower	low	ADJ
ejpam-5322	77	24	s-(τ1	s-(τ1	NOUN
ejpam-5322	77	25	,	,	PUNCT
ejpam-5322	77	26	τ2)p	τ2)p	ADJ
ejpam-5322	77	27	-	-	PUNCT
ejpam-5322	77	28	continuous	continuous	ADJ
ejpam-5322	77	29	multifunctions	multifunction	NOUN
ejpam-5322	77	30	.	.	PUNCT
ejpam-5322	78	1	moreover	moreover	ADV
ejpam-5322	78	2	,	,	PUNCT
ejpam-5322	78	3	some	some	DET
ejpam-5322	78	4	characterizations	characterization	NOUN
ejpam-5322	78	5	of	of	ADP
ejpam-5322	78	6	upper	upper	ADJ
ejpam-5322	78	7	and	and	CCONJ
ejpam-5322	78	8	lower	low	ADJ
ejpam-5322	78	9	s-(τ1	s-(τ1	NOUN
ejpam-5322	78	10	,	,	PUNCT
ejpam-5322	78	11	τ2)p	τ2)p	ADJ
ejpam-5322	78	12	-	-	PUNCT
ejpam-5322	78	13	continuous	continuous	ADJ
ejpam-5322	78	14	multifunctions	multifunction	NOUN
ejpam-5322	78	15	are	be	AUX
ejpam-5322	78	16	discussed	discuss	VERB
ejpam-5322	78	17	.	.	PUNCT
ejpam-5322	79	1	definition	definition	NOUN
ejpam-5322	79	2	1	1	NUM
ejpam-5322	79	3	.	.	PUNCT
ejpam-5322	80	1	a	a	DET
ejpam-5322	80	2	multifunction	multifunction	NOUN
ejpam-5322	80	3	f	f	NOUN
ejpam-5322	80	4	:	:	PUNCT
ejpam-5322	80	5	(	(	PUNCT
ejpam-5322	80	6	x	x	NOUN
ejpam-5322	80	7	,	,	PUNCT
ejpam-5322	80	8	τ1	τ1	NOUN
ejpam-5322	80	9	,	,	PUNCT
ejpam-5322	80	10	τ2	τ2	NOUN
ejpam-5322	80	11	)	)	PUNCT
ejpam-5322	80	12	→	→	SYM
ejpam-5322	80	13	(	(	PUNCT
ejpam-5322	80	14	y	y	PROPN
ejpam-5322	80	15	,	,	PUNCT
ejpam-5322	80	16	σ1	σ1	PROPN
ejpam-5322	80	17	,	,	PUNCT
ejpam-5322	80	18	σ2	σ2	PROPN
ejpam-5322	80	19	)	)	PUNCT
ejpam-5322	80	20	is	be	AUX
ejpam-5322	80	21	said	say	VERB
ejpam-5322	80	22	to	to	PART
ejpam-5322	80	23	be	be	AUX
ejpam-5322	80	24	upper	upper	ADJ
ejpam-5322	80	25	s-(τ1	s-(τ1	NOUN
ejpam-5322	80	26	,	,	PUNCT
ejpam-5322	80	27	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-5322	80	28	if	if	SCONJ
ejpam-5322	80	29	for	for	ADP
ejpam-5322	80	30	x	x	SYM
ejpam-5322	80	31	∈	∈	PROPN
ejpam-5322	80	32	x	x	X
ejpam-5322	80	33	and	and	CCONJ
ejpam-5322	80	34	each	each	DET
ejpam-5322	80	35	σ1σ2	σ1σ2	VERB
ejpam-5322	80	36	-	-	ADJ
ejpam-5322	80	37	open	open	ADJ
ejpam-5322	80	38	set	set	NOUN
ejpam-5322	80	39	v	v	NOUN
ejpam-5322	80	40	of	of	ADP
ejpam-5322	80	41	y	y	PROPN
ejpam-5322	80	42	containing	contain	VERB
ejpam-5322	80	43	f	f	PROPN
ejpam-5322	80	44	(	(	PUNCT
ejpam-5322	80	45	x	x	NOUN
ejpam-5322	80	46	)	)	PUNCT
ejpam-5322	80	47	and	and	CCONJ
ejpam-5322	80	48	having	have	VERB
ejpam-5322	80	49	σ1σ2	σ1σ2	NOUN
ejpam-5322	80	50	-	-	PUNCT
ejpam-5322	80	51	connected	connect	VERB
ejpam-5322	80	52	complement	complement	NOUN
ejpam-5322	80	53	,	,	PUNCT
ejpam-5322	80	54	there	there	PRON
ejpam-5322	80	55	exists	exist	VERB
ejpam-5322	80	56	a	a	DET
ejpam-5322	80	57	(	(	PUNCT
ejpam-5322	80	58	τ1	τ1	NOUN
ejpam-5322	80	59	,	,	PUNCT
ejpam-5322	80	60	τ2)p	τ2)p	ADJ
ejpam-5322	80	61	-	-	PUNCT
ejpam-5322	80	62	open	open	ADJ
ejpam-5322	80	63	set	set	NOUN
ejpam-5322	80	64	u	u	NOUN
ejpam-5322	80	65	of	of	ADP
ejpam-5322	80	66	x	x	PUNCT
ejpam-5322	80	67	containing	contain	VERB
ejpam-5322	80	68	x	x	PUNCT
ejpam-5322	80	69	such	such	ADJ
ejpam-5322	80	70	that	that	SCONJ
ejpam-5322	80	71	f	f	PROPN
ejpam-5322	80	72	(	(	PUNCT
ejpam-5322	80	73	u	u	NOUN
ejpam-5322	80	74	)	)	PUNCT
ejpam-5322	80	75	⊆	⊆	NUM
ejpam-5322	80	76	v	v	NOUN
ejpam-5322	80	77	.	.	PUNCT
ejpam-5322	81	1	theorem	theorem	NOUN
ejpam-5322	81	2	1	1	NUM
ejpam-5322	81	3	.	.	X
ejpam-5322	81	4	for	for	ADP
ejpam-5322	81	5	a	a	DET
ejpam-5322	81	6	multifunction	multifunction	NOUN
ejpam-5322	82	1	f	f	NOUN
ejpam-5322	82	2	:	:	PUNCT
ejpam-5322	82	3	(	(	PUNCT
ejpam-5322	82	4	x	x	NOUN
ejpam-5322	82	5	,	,	PUNCT
ejpam-5322	82	6	τ1	τ1	NOUN
ejpam-5322	82	7	,	,	PUNCT
ejpam-5322	82	8	τ2	τ2	NOUN
ejpam-5322	82	9	)	)	PUNCT
ejpam-5322	82	10	→	→	SYM
ejpam-5322	82	11	(	(	PUNCT
ejpam-5322	82	12	y	y	PROPN
ejpam-5322	82	13	,	,	PUNCT
ejpam-5322	82	14	σ1	σ1	PROPN
ejpam-5322	82	15	,	,	PUNCT
ejpam-5322	82	16	σ2	σ2	NOUN
ejpam-5322	82	17	)	)	PUNCT
ejpam-5322	82	18	,	,	PUNCT
ejpam-5322	82	19	the	the	DET
ejpam-5322	82	20	following	follow	VERB
ejpam-5322	82	21	properties	property	NOUN
ejpam-5322	82	22	are	be	AUX
ejpam-5322	82	23	equivalent	equivalent	ADJ
ejpam-5322	82	24	:	:	PUNCT
ejpam-5322	82	25	(	(	PUNCT
ejpam-5322	82	26	1	1	X
ejpam-5322	82	27	)	)	PUNCT
ejpam-5322	82	28	f	f	PROPN
ejpam-5322	82	29	is	be	AUX
ejpam-5322	82	30	upper	upper	ADJ
ejpam-5322	82	31	s-(τ1	s-(τ1	PROPN
ejpam-5322	82	32	,	,	PUNCT
ejpam-5322	82	33	τ2)p	τ2)p	ADJ
ejpam-5322	82	34	-	-	ADJ
ejpam-5322	82	35	continuous	continuous	ADJ
ejpam-5322	82	36	;	;	PUNCT
ejpam-5322	82	37	(	(	PUNCT
ejpam-5322	82	38	2	2	NUM
ejpam-5322	82	39	)	)	PUNCT
ejpam-5322	82	40	f+(v	f+(v	NOUN
ejpam-5322	82	41	)	)	PUNCT
ejpam-5322	83	1	is	be	AUX
ejpam-5322	83	2	(	(	PUNCT
ejpam-5322	83	3	τ1	τ1	NOUN
ejpam-5322	83	4	,	,	PUNCT
ejpam-5322	83	5	τ2)p	τ2)p	NOUN
ejpam-5322	83	6	-	-	PUNCT
ejpam-5322	83	7	open	open	ADJ
ejpam-5322	83	8	in	in	ADP
ejpam-5322	83	9	x	x	PUNCT
ejpam-5322	83	10	for	for	ADP
ejpam-5322	83	11	every	every	DET
ejpam-5322	83	12	σ1σ2	σ1σ2	NOUN
ejpam-5322	83	13	-	-	ADJ
ejpam-5322	83	14	open	open	ADJ
ejpam-5322	83	15	set	set	NOUN
ejpam-5322	83	16	v	v	NOUN
ejpam-5322	83	17	of	of	ADP
ejpam-5322	83	18	y	y	PROPN
ejpam-5322	83	19	having	have	VERB
ejpam-5322	83	20	σ1σ2	σ1σ2	ADV
ejpam-5322	83	21	-	-	PUNCT
ejpam-5322	83	22	connected	connect	VERB
ejpam-5322	83	23	complement	complement	NOUN
ejpam-5322	83	24	;	;	PUNCT
ejpam-5322	83	25	n.	n.	PROPN
ejpam-5322	83	26	viriyapong	viriyapong	PROPN
ejpam-5322	83	27	,	,	PUNCT
ejpam-5322	83	28	s.	s.	PROPN
ejpam-5322	83	29	sompong	sompong	PROPN
ejpam-5322	83	30	,	,	PUNCT
ejpam-5322	83	31	c.	c.	PROPN
ejpam-5322	83	32	boonpok	boonpok	PROPN
ejpam-5322	83	33	/	/	SYM
ejpam-5322	83	34	eur	eur	PROPN
ejpam-5322	83	35	.	.	PUNCT
ejpam-5322	84	1	j.	j.	PROPN
ejpam-5322	84	2	pure	pure	PROPN
ejpam-5322	84	3	appl	appl	PROPN
ejpam-5322	84	4	.	.	PROPN
ejpam-5322	84	5	math	math	PROPN
ejpam-5322	84	6	,	,	PUNCT
ejpam-5322	84	7	17	17	NUM
ejpam-5322	84	8	(	(	PUNCT
ejpam-5322	84	9	3	3	NUM
ejpam-5322	84	10	)	)	PUNCT
ejpam-5322	84	11	(	(	PUNCT
ejpam-5322	84	12	2024	2024	NUM
ejpam-5322	84	13	)	)	PUNCT
ejpam-5322	84	14	,	,	PUNCT
ejpam-5322	84	15	2210	2210	NUM
ejpam-5322	84	16	-	-	SYM
ejpam-5322	84	17	2220	2220	NUM
ejpam-5322	84	18	2213	2213	NUM
ejpam-5322	84	19	(	(	PUNCT
ejpam-5322	84	20	3	3	X
ejpam-5322	84	21	)	)	PUNCT
ejpam-5322	84	22	f−(k	f−(k	PROPN
ejpam-5322	84	23	)	)	PUNCT
ejpam-5322	84	24	is	be	AUX
ejpam-5322	84	25	(	(	PUNCT
ejpam-5322	84	26	τ1	τ1	NOUN
ejpam-5322	84	27	,	,	PUNCT
ejpam-5322	84	28	τ2)p	τ2)p	NOUN
ejpam-5322	84	29	-	-	PUNCT
ejpam-5322	84	30	closed	closed	ADJ
ejpam-5322	84	31	in	in	ADP
ejpam-5322	84	32	x	x	PUNCT
ejpam-5322	84	33	for	for	ADP
ejpam-5322	84	34	every	every	DET
ejpam-5322	84	35	σ1σ2	σ1σ2	NOUN
ejpam-5322	84	36	-	-	ADJ
ejpam-5322	84	37	connected	connect	VERB
ejpam-5322	84	38	σ1σ2	σ1σ2	VERB
ejpam-5322	84	39	-	-	PUNCT
ejpam-5322	84	40	closed	closed	ADJ
ejpam-5322	84	41	set	set	NOUN
ejpam-5322	84	42	k	k	PROPN
ejpam-5322	84	43	of	of	ADP
ejpam-5322	84	44	y	y	PROPN
ejpam-5322	84	45	;	;	PUNCT
ejpam-5322	84	46	(	(	PUNCT
ejpam-5322	84	47	4	4	X
ejpam-5322	84	48	)	)	PUNCT
ejpam-5322	84	49	τ1τ2	τ1τ2	NOUN
ejpam-5322	84	50	-	-	NOUN
ejpam-5322	84	51	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	84	52	-	-	PUNCT
ejpam-5322	84	53	int(f	int(f	PROPN
ejpam-5322	84	54	−(b	−(b	NOUN
ejpam-5322	84	55	)	)	PUNCT
ejpam-5322	84	56	)	)	PUNCT
ejpam-5322	84	57	)	)	PUNCT
ejpam-5322	85	1	⊆	⊆	X
ejpam-5322	85	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	85	3	-	-	PUNCT
ejpam-5322	85	4	cl(b	cl(b	NOUN
ejpam-5322	85	5	)	)	PUNCT
ejpam-5322	85	6	)	)	PUNCT
ejpam-5322	86	1	for	for	ADP
ejpam-5322	86	2	every	every	DET
ejpam-5322	86	3	subset	subset	NOUN
ejpam-5322	86	4	b	b	PROPN
ejpam-5322	86	5	of	of	ADP
ejpam-5322	86	6	y	y	PROPN
ejpam-5322	86	7	having	have	VERB
ejpam-5322	86	8	the	the	DET
ejpam-5322	86	9	σ1σ2	σ1σ2	ADV
ejpam-5322	86	10	-	-	PUNCT
ejpam-5322	86	11	connected	connect	VERB
ejpam-5322	86	12	σ1σ2	σ1σ2	NOUN
ejpam-5322	86	13	-	-	NOUN
ejpam-5322	86	14	closure	closure	NOUN
ejpam-5322	86	15	;	;	PUNCT
ejpam-5322	86	16	(	(	PUNCT
ejpam-5322	86	17	5	5	NUM
ejpam-5322	86	18	)	)	PUNCT
ejpam-5322	86	19	(	(	PUNCT
ejpam-5322	86	20	τ1	τ1	PROPN
ejpam-5322	86	21	,	,	PUNCT
ejpam-5322	86	22	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5322	86	23	−(b	−(b	PROPN
ejpam-5322	86	24	)	)	PUNCT
ejpam-5322	86	25	)	)	PUNCT
ejpam-5322	87	1	⊆	⊆	X
ejpam-5322	87	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	87	3	-	-	PUNCT
ejpam-5322	87	4	cl(b	cl(b	NOUN
ejpam-5322	87	5	)	)	PUNCT
ejpam-5322	87	6	)	)	PUNCT
ejpam-5322	88	1	for	for	ADP
ejpam-5322	88	2	every	every	DET
ejpam-5322	88	3	subset	subset	NOUN
ejpam-5322	88	4	b	b	PROPN
ejpam-5322	88	5	of	of	ADP
ejpam-5322	88	6	y	y	PROPN
ejpam-5322	88	7	having	have	VERB
ejpam-5322	88	8	the	the	DET
ejpam-5322	88	9	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5322	88	10	σ1σ2	σ1σ2	NOUN
ejpam-5322	88	11	-	-	NOUN
ejpam-5322	88	12	closure	closure	NOUN
ejpam-5322	88	13	;	;	PUNCT
ejpam-5322	88	14	(	(	PUNCT
ejpam-5322	88	15	6	6	X
ejpam-5322	88	16	)	)	PUNCT
ejpam-5322	88	17	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5322	88	18	-	-	PUNCT
ejpam-5322	88	19	int(b	int(b	NOUN
ejpam-5322	88	20	)	)	PUNCT
ejpam-5322	88	21	)	)	PUNCT
ejpam-5322	89	1	⊆	⊆	NUM
ejpam-5322	89	2	(	(	PUNCT
ejpam-5322	89	3	τ1	τ1	NOUN
ejpam-5322	89	4	,	,	PUNCT
ejpam-5322	89	5	τ2)-pint(f	τ2)-pint(f	PUNCT
ejpam-5322	90	1	+	+	ADJ
ejpam-5322	90	2	(	(	PUNCT
ejpam-5322	90	3	b	b	NOUN
ejpam-5322	90	4	)	)	PUNCT
ejpam-5322	90	5	)	)	PUNCT
ejpam-5322	90	6	for	for	ADP
ejpam-5322	90	7	every	every	DET
ejpam-5322	90	8	subset	subset	NOUN
ejpam-5322	90	9	b	b	PROPN
ejpam-5322	90	10	of	of	ADP
ejpam-5322	90	11	y	y	PRON
ejpam-5322	90	12	such	such	ADJ
ejpam-5322	90	13	that	that	SCONJ
ejpam-5322	90	14	y	y	PROPN
ejpam-5322	91	1	−	−	ADP
ejpam-5322	91	2	σ1σ2	σ1σ2	NUM
ejpam-5322	91	3	-	-	PUNCT
ejpam-5322	91	4	int(b	int(b	NOUN
ejpam-5322	91	5	)	)	PUNCT
ejpam-5322	91	6	is	be	AUX
ejpam-5322	91	7	σ1σ2	σ1σ2	NOUN
ejpam-5322	91	8	-	-	PUNCT
ejpam-5322	91	9	connected	connect	VERB
ejpam-5322	91	10	.	.	PUNCT
ejpam-5322	92	1	proof	proof	NOUN
ejpam-5322	92	2	.	.	PUNCT
ejpam-5322	93	1	(	(	PUNCT
ejpam-5322	93	2	1	1	X
ejpam-5322	93	3	)	)	PUNCT
ejpam-5322	93	4	⇒	⇒	NOUN
ejpam-5322	93	5	(	(	PUNCT
ejpam-5322	93	6	2	2	NUM
ejpam-5322	93	7	):	):	PUNCT
ejpam-5322	93	8	let	let	VERB
ejpam-5322	93	9	v	v	PART
ejpam-5322	93	10	be	be	AUX
ejpam-5322	93	11	any	any	DET
ejpam-5322	93	12	σ1σ2	σ1σ2	NOUN
ejpam-5322	93	13	-	-	ADJ
ejpam-5322	93	14	open	open	ADJ
ejpam-5322	93	15	set	set	NOUN
ejpam-5322	93	16	of	of	ADP
ejpam-5322	93	17	y	y	PROPN
ejpam-5322	93	18	having	have	VERB
ejpam-5322	93	19	σ1σ2	σ1σ2	ADV
ejpam-5322	93	20	-	-	PUNCT
ejpam-5322	93	21	connected	connect	VERB
ejpam-5322	93	22	complement	complement	NOUN
ejpam-5322	93	23	and	and	CCONJ
ejpam-5322	93	24	x	x	PUNCT
ejpam-5322	93	25	∈	∈	PROPN
ejpam-5322	93	26	f+(v	f+(v	NOUN
ejpam-5322	93	27	)	)	PUNCT
ejpam-5322	93	28	.	.	PUNCT
ejpam-5322	94	1	then	then	ADV
ejpam-5322	94	2	,	,	PUNCT
ejpam-5322	94	3	there	there	PRON
ejpam-5322	94	4	exists	exist	VERB
ejpam-5322	94	5	a	a	DET
ejpam-5322	94	6	(	(	PUNCT
ejpam-5322	94	7	τ1	τ1	NOUN
ejpam-5322	94	8	,	,	PUNCT
ejpam-5322	94	9	τ2)p	τ2)p	ADJ
ejpam-5322	94	10	-	-	PUNCT
ejpam-5322	94	11	open	open	ADJ
ejpam-5322	94	12	set	set	NOUN
ejpam-5322	94	13	u	u	NOUN
ejpam-5322	94	14	of	of	ADP
ejpam-5322	94	15	x	x	PUNCT
ejpam-5322	94	16	containing	contain	VERB
ejpam-5322	94	17	x	x	PUNCT
ejpam-5322	94	18	such	such	ADJ
ejpam-5322	94	19	that	that	SCONJ
ejpam-5322	94	20	f	f	PROPN
ejpam-5322	94	21	(	(	PUNCT
ejpam-5322	94	22	u	u	NOUN
ejpam-5322	94	23	)	)	PUNCT
ejpam-5322	94	24	⊆	⊆	NUM
ejpam-5322	94	25	v	v	NOUN
ejpam-5322	94	26	.	.	PUNCT
ejpam-5322	95	1	therefore	therefore	ADV
ejpam-5322	95	2	,	,	PUNCT
ejpam-5322	95	3	we	we	PRON
ejpam-5322	95	4	have	have	VERB
ejpam-5322	95	5	x	x	X
ejpam-5322	95	6	∈	∈	PROPN
ejpam-5322	95	7	u	u	NOUN
ejpam-5322	95	8	⊆	⊆	NUM
ejpam-5322	95	9	τ1τ2	τ1τ2	NOUN
ejpam-5322	95	10	-	-	NOUN
ejpam-5322	95	11	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5322	95	12	-	-	PUNCT
ejpam-5322	95	13	cl(f	cl(f	NOUN
ejpam-5322	95	14	+	+	NOUN
ejpam-5322	95	15	(	(	PUNCT
ejpam-5322	95	16	v	v	NOUN
ejpam-5322	95	17	)	)	PUNCT
ejpam-5322	95	18	)	)	PUNCT
ejpam-5322	95	19	)	)	PUNCT
ejpam-5322	95	20	.	.	PUNCT
ejpam-5322	96	1	thus	thus	ADV
ejpam-5322	96	2	,	,	PUNCT
ejpam-5322	96	3	f+(v	f+(v	PROPN
ejpam-5322	96	4	)	)	PUNCT
ejpam-5322	97	1	⊆	⊆	X
ejpam-5322	97	2	τ1τ2	τ1τ2	NOUN
ejpam-5322	97	3	-	-	NOUN
ejpam-5322	97	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5322	97	5	-	-	PUNCT
ejpam-5322	97	6	cl(f	cl(f	NOUN
ejpam-5322	97	7	+	+	NOUN
ejpam-5322	97	8	(	(	PUNCT
ejpam-5322	97	9	v	v	NOUN
ejpam-5322	97	10	)	)	PUNCT
ejpam-5322	97	11	)	)	PUNCT
ejpam-5322	97	12	)	)	PUNCT
ejpam-5322	97	13	and	and	CCONJ
ejpam-5322	97	14	hence	hence	ADV
ejpam-5322	97	15	f+(v	f+(v	NOUN
ejpam-5322	97	16	)	)	PUNCT
ejpam-5322	97	17	is	be	AUX
ejpam-5322	97	18	(	(	PUNCT
ejpam-5322	97	19	τ1	τ1	NOUN
ejpam-5322	97	20	,	,	PUNCT
ejpam-5322	97	21	τ2)p	τ2)p	NOUN
ejpam-5322	97	22	-	-	PUNCT
ejpam-5322	97	23	open	open	ADJ
ejpam-5322	97	24	in	in	ADP
ejpam-5322	97	25	x.	x.	NOUN
ejpam-5322	97	26	(	(	PUNCT
ejpam-5322	97	27	2	2	NUM
ejpam-5322	97	28	)	)	PUNCT
ejpam-5322	97	29	⇒	⇒	NOUN
ejpam-5322	97	30	(	(	PUNCT
ejpam-5322	97	31	3	3	NUM
ejpam-5322	97	32	):	):	PUNCT
ejpam-5322	97	33	the	the	DET
ejpam-5322	97	34	proof	proof	NOUN
ejpam-5322	97	35	follows	follow	VERB
ejpam-5322	97	36	immediately	immediately	ADV
ejpam-5322	97	37	from	from	ADP
ejpam-5322	97	38	the	the	DET
ejpam-5322	97	39	fact	fact	NOUN
ejpam-5322	97	40	that	that	SCONJ
ejpam-5322	97	41	f+(y	f+(y	PROPN
ejpam-5322	97	42	−b	−b	ADV
ejpam-5322	97	43	)	)	PUNCT
ejpam-5322	97	44	=	=	SYM
ejpam-5322	98	1	x−f−(b	x−f−(b	PROPN
ejpam-5322	98	2	)	)	PUNCT
ejpam-5322	98	3	for	for	ADP
ejpam-5322	98	4	every	every	DET
ejpam-5322	98	5	subset	subset	NOUN
ejpam-5322	98	6	b	b	PROPN
ejpam-5322	98	7	of	of	ADP
ejpam-5322	98	8	y	y	PROPN
ejpam-5322	98	9	.	.	PUNCT
ejpam-5322	99	1	(	(	PUNCT
ejpam-5322	99	2	3	3	X
ejpam-5322	99	3	)	)	PUNCT
ejpam-5322	99	4	⇒	⇒	NOUN
ejpam-5322	99	5	(	(	PUNCT
ejpam-5322	99	6	4	4	NUM
ejpam-5322	99	7	):	):	PUNCT
ejpam-5322	99	8	let	let	VERB
ejpam-5322	99	9	b	b	X
ejpam-5322	99	10	be	be	AUX
ejpam-5322	99	11	any	any	DET
ejpam-5322	99	12	subset	subset	NOUN
ejpam-5322	99	13	of	of	ADP
ejpam-5322	99	14	y	y	PROPN
ejpam-5322	99	15	having	have	VERB
ejpam-5322	99	16	the	the	DET
ejpam-5322	99	17	σ1σ2	σ1σ2	ADV
ejpam-5322	99	18	-	-	PUNCT
ejpam-5322	99	19	connected	connect	VERB
ejpam-5322	99	20	σ1σ2	σ1σ2	NOUN
ejpam-5322	99	21	-	-	NOUN
ejpam-5322	99	22	closure	closure	NOUN
ejpam-5322	99	23	.	.	PUNCT
ejpam-5322	100	1	then	then	ADV
ejpam-5322	100	2	,	,	PUNCT
ejpam-5322	100	3	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	100	4	-	-	PUNCT
ejpam-5322	100	5	cl(b	cl(b	NOUN
ejpam-5322	100	6	)	)	PUNCT
ejpam-5322	100	7	)	)	PUNCT
ejpam-5322	100	8	is	be	AUX
ejpam-5322	100	9	a	a	DET
ejpam-5322	100	10	(	(	PUNCT
ejpam-5322	100	11	τ1	τ1	NOUN
ejpam-5322	100	12	,	,	PUNCT
ejpam-5322	100	13	τ2)p	τ2)p	NOUN
ejpam-5322	100	14	-	-	PUNCT
ejpam-5322	100	15	closed	closed	ADJ
ejpam-5322	100	16	set	set	NOUN
ejpam-5322	100	17	of	of	ADP
ejpam-5322	100	18	x.	x.	NOUN
ejpam-5322	100	19	by	by	ADP
ejpam-5322	100	20	lemma	lemma	PROPN
ejpam-5322	100	21	2	2	NUM
ejpam-5322	100	22	,	,	PUNCT
ejpam-5322	100	23	we	we	PRON
ejpam-5322	100	24	have	have	VERB
ejpam-5322	100	25	τ1τ2	τ1τ2	NOUN
ejpam-5322	100	26	-	-	NOUN
ejpam-5322	100	27	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	100	28	-	-	PUNCT
ejpam-5322	100	29	int(f	int(f	PROPN
ejpam-5322	100	30	−(b	−(b	NOUN
ejpam-5322	100	31	)	)	PUNCT
ejpam-5322	100	32	)	)	PUNCT
ejpam-5322	100	33	)	)	PUNCT
ejpam-5322	101	1	⊆	⊆	X
ejpam-5322	101	2	τ1τ2	τ1τ2	NOUN
ejpam-5322	101	3	-	-	NUM
ejpam-5322	101	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	101	5	-	-	PUNCT
ejpam-5322	101	6	int(f	int(f	ADJ
ejpam-5322	101	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5322	101	8	-	-	NOUN
ejpam-5322	101	9	cl(b	cl(b	NOUN
ejpam-5322	101	10	)	)	PUNCT
ejpam-5322	101	11	)	)	PUNCT
ejpam-5322	101	12	)	)	PUNCT
ejpam-5322	101	13	)	)	PUNCT
ejpam-5322	102	1	⊆	⊆	X
ejpam-5322	102	2	(	(	PUNCT
ejpam-5322	102	3	τ1	τ1	NOUN
ejpam-5322	102	4	,	,	PUNCT
ejpam-5322	102	5	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5322	102	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5322	102	7	-	-	NOUN
ejpam-5322	102	8	cl(b	cl(b	NOUN
ejpam-5322	102	9	)	)	PUNCT
ejpam-5322	102	10	)	)	PUNCT
ejpam-5322	102	11	)	)	PUNCT
ejpam-5322	102	12	=	=	PUNCT
ejpam-5322	102	13	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	102	14	-	-	PUNCT
ejpam-5322	102	15	cl(b	cl(b	NOUN
ejpam-5322	102	16	)	)	PUNCT
ejpam-5322	102	17	)	)	PUNCT
ejpam-5322	102	18	.	.	PUNCT
ejpam-5322	103	1	(	(	PUNCT
ejpam-5322	103	2	4	4	X
ejpam-5322	103	3	)	)	PUNCT
ejpam-5322	103	4	⇒	⇒	NOUN
ejpam-5322	103	5	(	(	PUNCT
ejpam-5322	103	6	5	5	NUM
ejpam-5322	103	7	):	):	PUNCT
ejpam-5322	103	8	let	let	VERB
ejpam-5322	103	9	b	b	X
ejpam-5322	103	10	be	be	AUX
ejpam-5322	103	11	any	any	DET
ejpam-5322	103	12	subset	subset	NOUN
ejpam-5322	103	13	of	of	ADP
ejpam-5322	103	14	y	y	PROPN
ejpam-5322	103	15	having	have	VERB
ejpam-5322	103	16	the	the	DET
ejpam-5322	103	17	σ1σ2	σ1σ2	ADV
ejpam-5322	103	18	-	-	PUNCT
ejpam-5322	103	19	connected	connect	VERB
ejpam-5322	103	20	σ1σ2	σ1σ2	NOUN
ejpam-5322	103	21	-	-	NOUN
ejpam-5322	103	22	closure	closure	NOUN
ejpam-5322	103	23	.	.	PUNCT
ejpam-5322	104	1	it	it	PRON
ejpam-5322	104	2	follows	follow	VERB
ejpam-5322	104	3	from	from	ADP
ejpam-5322	104	4	lemma	lemma	PROPN
ejpam-5322	104	5	2	2	NUM
ejpam-5322	104	6	that	that	PRON
ejpam-5322	104	7	(	(	PUNCT
ejpam-5322	104	8	τ1	τ1	NOUN
ejpam-5322	104	9	,	,	PUNCT
ejpam-5322	104	10	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5322	104	11	−(b	−(b	PROPN
ejpam-5322	104	12	)	)	PUNCT
ejpam-5322	104	13	)	)	PUNCT
ejpam-5322	105	1	=	=	SYM
ejpam-5322	105	2	f−(b	f−(b	PROPN
ejpam-5322	105	3	)	)	PUNCT
ejpam-5322	105	4	∪	∪	ADP
ejpam-5322	105	5	τ1τ2	τ1τ2	NOUN
ejpam-5322	105	6	-	-	ADJ
ejpam-5322	105	7	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	105	8	-	-	PUNCT
ejpam-5322	105	9	int(f	int(f	PROPN
ejpam-5322	105	10	−(b	−(b	NOUN
ejpam-5322	105	11	)	)	PUNCT
ejpam-5322	105	12	)	)	PUNCT
ejpam-5322	105	13	)	)	PUNCT
ejpam-5322	106	1	⊆	⊆	X
ejpam-5322	106	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	106	3	-	-	PUNCT
ejpam-5322	106	4	cl(b	cl(b	NOUN
ejpam-5322	106	5	)	)	PUNCT
ejpam-5322	106	6	)	)	PUNCT
ejpam-5322	106	7	.	.	PUNCT
ejpam-5322	107	1	(	(	PUNCT
ejpam-5322	107	2	5	5	X
ejpam-5322	107	3	)	)	PUNCT
ejpam-5322	107	4	⇒	⇒	NOUN
ejpam-5322	107	5	(	(	PUNCT
ejpam-5322	107	6	6	6	NUM
ejpam-5322	107	7	):	):	PUNCT
ejpam-5322	107	8	let	let	VERB
ejpam-5322	107	9	b	b	X
ejpam-5322	107	10	be	be	AUX
ejpam-5322	107	11	any	any	DET
ejpam-5322	107	12	subset	subset	NOUN
ejpam-5322	107	13	of	of	ADP
ejpam-5322	107	14	y	y	PRON
ejpam-5322	107	15	such	such	ADJ
ejpam-5322	107	16	that	that	SCONJ
ejpam-5322	107	17	y	y	PROPN
ejpam-5322	107	18	−σ1σ2	−σ1σ2	PROPN
ejpam-5322	107	19	-	-	PUNCT
ejpam-5322	107	20	int(b	int(b	NOUN
ejpam-5322	107	21	)	)	PUNCT
ejpam-5322	107	22	is	be	AUX
ejpam-5322	107	23	σ1σ2	σ1σ2	NOUN
ejpam-5322	107	24	-	-	PUNCT
ejpam-5322	107	25	connected	connect	VERB
ejpam-5322	107	26	.	.	PUNCT
ejpam-5322	108	1	by	by	ADP
ejpam-5322	108	2	(	(	PUNCT
ejpam-5322	108	3	5	5	NUM
ejpam-5322	108	4	)	)	PUNCT
ejpam-5322	108	5	,	,	PUNCT
ejpam-5322	108	6	x	x	PUNCT
ejpam-5322	108	7	−	−	NOUN
ejpam-5322	108	8	(	(	PUNCT
ejpam-5322	108	9	τ1	τ1	NOUN
ejpam-5322	108	10	,	,	PUNCT
ejpam-5322	108	11	τ2)-int(f	τ2)-int(f	X
ejpam-5322	109	1	+	+	ADJ
ejpam-5322	109	2	(	(	PUNCT
ejpam-5322	109	3	b	b	NOUN
ejpam-5322	109	4	)	)	PUNCT
ejpam-5322	109	5	)	)	PUNCT
ejpam-5322	110	1	=	=	PRON
ejpam-5322	110	2	(	(	PUNCT
ejpam-5322	110	3	τ1	τ1	PROPN
ejpam-5322	110	4	,	,	PUNCT
ejpam-5322	110	5	τ2)-pcl(x	τ2)-pcl(x	NOUN
ejpam-5322	111	1	−	−	PROPN
ejpam-5322	111	2	f+(b	f+(b	PROPN
ejpam-5322	111	3	)	)	PUNCT
ejpam-5322	111	4	)	)	PUNCT
ejpam-5322	112	1	=	=	PRON
ejpam-5322	112	2	(	(	PUNCT
ejpam-5322	112	3	τ1	τ1	PROPN
ejpam-5322	112	4	,	,	PUNCT
ejpam-5322	112	5	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5322	112	6	−(y	−(y	NOUN
ejpam-5322	112	7	−b	−b	NOUN
ejpam-5322	112	8	)	)	PUNCT
ejpam-5322	112	9	)	)	PUNCT
ejpam-5322	113	1	⊆	⊆	X
ejpam-5322	113	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	113	3	-	-	PUNCT
ejpam-5322	113	4	cl(y	cl(y	NOUN
ejpam-5322	113	5	−b	−b	NOUN
ejpam-5322	113	6	)	)	PUNCT
ejpam-5322	113	7	)	)	PUNCT
ejpam-5322	114	1	=	=	PUNCT
ejpam-5322	114	2	f−(y	f−(y	NOUN
ejpam-5322	114	3	−	−	ADP
ejpam-5322	114	4	σ1σ2	σ1σ2	NOUN
ejpam-5322	114	5	-	-	PUNCT
ejpam-5322	114	6	int(b	int(b	NOUN
ejpam-5322	114	7	)	)	PUNCT
ejpam-5322	114	8	)	)	PUNCT
ejpam-5322	115	1	=	=	PUNCT
ejpam-5322	115	2	x	x	X
ejpam-5322	116	1	−	−	ADP
ejpam-5322	116	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5322	116	3	-	-	PUNCT
ejpam-5322	116	4	int(b	int(b	NOUN
ejpam-5322	116	5	)	)	PUNCT
ejpam-5322	116	6	)	)	PUNCT
ejpam-5322	116	7	.	.	PUNCT
ejpam-5322	117	1	n.	n.	PROPN
ejpam-5322	117	2	viriyapong	viriyapong	PROPN
ejpam-5322	117	3	,	,	PUNCT
ejpam-5322	117	4	s.	s.	PROPN
ejpam-5322	117	5	sompong	sompong	PROPN
ejpam-5322	117	6	,	,	PUNCT
ejpam-5322	117	7	c.	c.	PROPN
ejpam-5322	117	8	boonpok	boonpok	PROPN
ejpam-5322	117	9	/	/	SYM
ejpam-5322	117	10	eur	eur	PROPN
ejpam-5322	117	11	.	.	PUNCT
ejpam-5322	118	1	j.	j.	PROPN
ejpam-5322	118	2	pure	pure	PROPN
ejpam-5322	118	3	appl	appl	PROPN
ejpam-5322	118	4	.	.	PROPN
ejpam-5322	118	5	math	math	PROPN
ejpam-5322	118	6	,	,	PUNCT
ejpam-5322	118	7	17	17	NUM
ejpam-5322	118	8	(	(	PUNCT
ejpam-5322	118	9	3	3	NUM
ejpam-5322	118	10	)	)	PUNCT
ejpam-5322	118	11	(	(	PUNCT
ejpam-5322	118	12	2024	2024	NUM
ejpam-5322	118	13	)	)	PUNCT
ejpam-5322	118	14	,	,	PUNCT
ejpam-5322	118	15	2210	2210	NUM
ejpam-5322	118	16	-	-	SYM
ejpam-5322	118	17	2220	2220	NUM
ejpam-5322	118	18	2214	2214	NUM
ejpam-5322	118	19	thus	thus	ADV
ejpam-5322	118	20	,	,	PUNCT
ejpam-5322	118	21	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5322	118	22	-	-	PUNCT
ejpam-5322	118	23	int(b	int(b	NOUN
ejpam-5322	118	24	)	)	PUNCT
ejpam-5322	118	25	)	)	PUNCT
ejpam-5322	119	1	⊆	⊆	NUM
ejpam-5322	119	2	(	(	PUNCT
ejpam-5322	119	3	τ1	τ1	NOUN
ejpam-5322	119	4	,	,	PUNCT
ejpam-5322	119	5	τ2)-pint(f	τ2)-pint(f	PUNCT
ejpam-5322	120	1	+	+	ADJ
ejpam-5322	120	2	(	(	PUNCT
ejpam-5322	120	3	b	b	NOUN
ejpam-5322	120	4	)	)	PUNCT
ejpam-5322	120	5	)	)	PUNCT
ejpam-5322	120	6	.	.	PUNCT
ejpam-5322	121	1	(	(	PUNCT
ejpam-5322	121	2	6	6	X
ejpam-5322	121	3	)	)	PUNCT
ejpam-5322	121	4	⇒	⇒	NOUN
ejpam-5322	121	5	(	(	PUNCT
ejpam-5322	121	6	1	1	NUM
ejpam-5322	121	7	):	):	PUNCT
ejpam-5322	121	8	let	let	VERB
ejpam-5322	121	9	x	x	PUNCT
ejpam-5322	121	10	∈	∈	PROPN
ejpam-5322	121	11	x	x	X
ejpam-5322	121	12	and	and	CCONJ
ejpam-5322	121	13	v	v	X
ejpam-5322	121	14	be	be	AUX
ejpam-5322	121	15	any	any	DET
ejpam-5322	121	16	σ1σ2	σ1σ2	NOUN
ejpam-5322	121	17	-	-	ADJ
ejpam-5322	121	18	open	open	ADJ
ejpam-5322	121	19	set	set	NOUN
ejpam-5322	121	20	of	of	ADP
ejpam-5322	121	21	y	y	PROPN
ejpam-5322	121	22	containing	contain	VERB
ejpam-5322	121	23	f	f	PROPN
ejpam-5322	121	24	(	(	PUNCT
ejpam-5322	121	25	x	x	NOUN
ejpam-5322	121	26	)	)	PUNCT
ejpam-5322	121	27	and	and	CCONJ
ejpam-5322	121	28	having	have	VERB
ejpam-5322	121	29	σ1σ2	σ1σ2	NOUN
ejpam-5322	121	30	-	-	PUNCT
ejpam-5322	121	31	connected	connect	VERB
ejpam-5322	121	32	complement	complement	NOUN
ejpam-5322	121	33	.	.	PUNCT
ejpam-5322	122	1	by	by	ADP
ejpam-5322	122	2	(	(	PUNCT
ejpam-5322	122	3	6	6	NUM
ejpam-5322	122	4	)	)	PUNCT
ejpam-5322	122	5	,	,	PUNCT
ejpam-5322	122	6	we	we	PRON
ejpam-5322	122	7	have	have	VERB
ejpam-5322	122	8	f+(v	f+(v	NOUN
ejpam-5322	122	9	)	)	PUNCT
ejpam-5322	123	1	=	=	PUNCT
ejpam-5322	123	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5322	123	3	-	-	PUNCT
ejpam-5322	123	4	int(v	int(v	NOUN
ejpam-5322	123	5	)	)	PUNCT
ejpam-5322	123	6	)	)	PUNCT
ejpam-5322	124	1	⊆	⊆	NUM
ejpam-5322	124	2	(	(	PUNCT
ejpam-5322	124	3	τ1	τ1	NOUN
ejpam-5322	124	4	,	,	PUNCT
ejpam-5322	124	5	τ2)-pint(f	τ2)-pint(f	PUNCT
ejpam-5322	125	1	+	+	ADJ
ejpam-5322	125	2	(	(	PUNCT
ejpam-5322	125	3	v	v	NOUN
ejpam-5322	125	4	)	)	PUNCT
ejpam-5322	125	5	)	)	PUNCT
ejpam-5322	125	6	.	.	PUNCT
ejpam-5322	126	1	put	put	VERB
ejpam-5322	126	2	u	u	NOUN
ejpam-5322	126	3	=	=	PUNCT
ejpam-5322	126	4	(	(	PUNCT
ejpam-5322	126	5	τ1	τ1	PROPN
ejpam-5322	126	6	,	,	PUNCT
ejpam-5322	126	7	τ2)-pint(f	τ2)-pint(f	PUNCT
ejpam-5322	127	1	+	+	ADJ
ejpam-5322	127	2	(	(	PUNCT
ejpam-5322	127	3	v	v	NOUN
ejpam-5322	127	4	)	)	PUNCT
ejpam-5322	127	5	)	)	PUNCT
ejpam-5322	127	6	.	.	PUNCT
ejpam-5322	128	1	then	then	ADV
ejpam-5322	128	2	,	,	PUNCT
ejpam-5322	128	3	u	u	NOUN
ejpam-5322	128	4	is	be	AUX
ejpam-5322	128	5	a	a	DET
ejpam-5322	128	6	(	(	PUNCT
ejpam-5322	128	7	τ1	τ1	NOUN
ejpam-5322	128	8	,	,	PUNCT
ejpam-5322	128	9	τ2)p	τ2)p	ADJ
ejpam-5322	128	10	-	-	PUNCT
ejpam-5322	128	11	open	open	ADJ
ejpam-5322	128	12	set	set	NOUN
ejpam-5322	128	13	of	of	ADP
ejpam-5322	128	14	x	x	PUNCT
ejpam-5322	128	15	containing	contain	VERB
ejpam-5322	128	16	x	x	PUNCT
ejpam-5322	128	17	such	such	ADJ
ejpam-5322	128	18	that	that	SCONJ
ejpam-5322	128	19	f	f	PROPN
ejpam-5322	128	20	(	(	PUNCT
ejpam-5322	128	21	u	u	NOUN
ejpam-5322	128	22	)	)	PUNCT
ejpam-5322	128	23	⊆	⊆	NUM
ejpam-5322	128	24	v	v	NOUN
ejpam-5322	128	25	.	.	PUNCT
ejpam-5322	129	1	this	this	PRON
ejpam-5322	129	2	shows	show	VERB
ejpam-5322	129	3	that	that	SCONJ
ejpam-5322	129	4	f	f	PROPN
ejpam-5322	129	5	is	be	AUX
ejpam-5322	129	6	upper	upper	ADJ
ejpam-5322	129	7	s-(τ1	s-(τ1	PROPN
ejpam-5322	129	8	,	,	PUNCT
ejpam-5322	129	9	τ2)p	τ2)p	ADJ
ejpam-5322	129	10	-	-	ADJ
ejpam-5322	129	11	continuous	continuous	ADJ
ejpam-5322	129	12	.	.	PUNCT
ejpam-5322	130	1	definition	definition	NOUN
ejpam-5322	130	2	2	2	NUM
ejpam-5322	130	3	.	.	PUNCT
ejpam-5322	130	4	a	a	DET
ejpam-5322	130	5	multifunction	multifunction	NOUN
ejpam-5322	130	6	f	f	NOUN
ejpam-5322	130	7	:	:	PUNCT
ejpam-5322	130	8	(	(	PUNCT
ejpam-5322	130	9	x	x	NOUN
ejpam-5322	130	10	,	,	PUNCT
ejpam-5322	130	11	τ1	τ1	NOUN
ejpam-5322	130	12	,	,	PUNCT
ejpam-5322	130	13	τ2	τ2	NOUN
ejpam-5322	130	14	)	)	PUNCT
ejpam-5322	130	15	→	→	SYM
ejpam-5322	130	16	(	(	PUNCT
ejpam-5322	130	17	y	y	PROPN
ejpam-5322	130	18	,	,	PUNCT
ejpam-5322	130	19	σ1	σ1	PROPN
ejpam-5322	130	20	,	,	PUNCT
ejpam-5322	130	21	σ2	σ2	PROPN
ejpam-5322	130	22	)	)	PUNCT
ejpam-5322	130	23	is	be	AUX
ejpam-5322	130	24	said	say	VERB
ejpam-5322	130	25	to	to	PART
ejpam-5322	130	26	be	be	AUX
ejpam-5322	130	27	lower	low	ADJ
ejpam-5322	130	28	s-(τ1	s-(τ1	NOUN
ejpam-5322	130	29	,	,	PUNCT
ejpam-5322	130	30	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-5322	130	31	if	if	SCONJ
ejpam-5322	130	32	for	for	ADP
ejpam-5322	130	33	each	each	DET
ejpam-5322	130	34	x	x	SYM
ejpam-5322	130	35	∈	∈	PROPN
ejpam-5322	130	36	x	x	X
ejpam-5322	130	37	and	and	CCONJ
ejpam-5322	130	38	each	each	DET
ejpam-5322	130	39	σ1σ2	σ1σ2	VERB
ejpam-5322	130	40	-	-	ADJ
ejpam-5322	130	41	open	open	ADJ
ejpam-5322	130	42	set	set	NOUN
ejpam-5322	130	43	v	v	NOUN
ejpam-5322	130	44	of	of	ADP
ejpam-5322	130	45	y	y	PROPN
ejpam-5322	130	46	having	have	VERB
ejpam-5322	130	47	σ1σ2	σ1σ2	ADV
ejpam-5322	130	48	-	-	PUNCT
ejpam-5322	130	49	connected	connected	ADJ
ejpam-5322	130	50	complement	complement	NOUN
ejpam-5322	130	51	such	such	ADJ
ejpam-5322	130	52	that	that	SCONJ
ejpam-5322	130	53	f	f	PROPN
ejpam-5322	130	54	(	(	PUNCT
ejpam-5322	130	55	x)∩	x)∩	PROPN
ejpam-5322	130	56	v	v	ADP
ejpam-5322	130	57	̸=	̸=	PROPN
ejpam-5322	130	58	∅	∅	NOUN
ejpam-5322	130	59	,	,	PUNCT
ejpam-5322	130	60	there	there	PRON
ejpam-5322	130	61	exists	exist	VERB
ejpam-5322	130	62	a	a	DET
ejpam-5322	130	63	(	(	PUNCT
ejpam-5322	130	64	τ1	τ1	NOUN
ejpam-5322	130	65	,	,	PUNCT
ejpam-5322	130	66	τ2)p	τ2)p	ADJ
ejpam-5322	130	67	-	-	PUNCT
ejpam-5322	130	68	open	open	ADJ
ejpam-5322	130	69	set	set	NOUN
ejpam-5322	130	70	u	u	NOUN
ejpam-5322	130	71	of	of	ADP
ejpam-5322	130	72	x	x	PUNCT
ejpam-5322	130	73	containing	contain	VERB
ejpam-5322	130	74	x	x	PUNCT
ejpam-5322	130	75	such	such	ADJ
ejpam-5322	130	76	that	that	SCONJ
ejpam-5322	130	77	f	f	PROPN
ejpam-5322	130	78	(	(	PUNCT
ejpam-5322	130	79	z	z	NOUN
ejpam-5322	130	80	)	)	PUNCT
ejpam-5322	130	81	∩	∩	NOUN
ejpam-5322	130	82	v	v	ADP
ejpam-5322	130	83	̸=	̸=	PROPN
ejpam-5322	130	84	∅	∅	NOUN
ejpam-5322	130	85	for	for	ADP
ejpam-5322	130	86	each	each	DET
ejpam-5322	130	87	z	z	NOUN
ejpam-5322	130	88	∈	∈	PROPN
ejpam-5322	130	89	u	u	PROPN
ejpam-5322	130	90	.	.	PUNCT
ejpam-5322	131	1	theorem	theorem	NOUN
ejpam-5322	131	2	2	2	NUM
ejpam-5322	131	3	.	.	X
ejpam-5322	131	4	for	for	ADP
ejpam-5322	131	5	a	a	DET
ejpam-5322	131	6	multifunction	multifunction	NOUN
ejpam-5322	132	1	f	f	NOUN
ejpam-5322	132	2	:	:	PUNCT
ejpam-5322	132	3	(	(	PUNCT
ejpam-5322	132	4	x	x	NOUN
ejpam-5322	132	5	,	,	PUNCT
ejpam-5322	132	6	τ1	τ1	NOUN
ejpam-5322	132	7	,	,	PUNCT
ejpam-5322	132	8	τ2	τ2	NOUN
ejpam-5322	132	9	)	)	PUNCT
ejpam-5322	132	10	→	→	SYM
ejpam-5322	132	11	(	(	PUNCT
ejpam-5322	132	12	y	y	PROPN
ejpam-5322	132	13	,	,	PUNCT
ejpam-5322	132	14	σ1	σ1	PROPN
ejpam-5322	132	15	,	,	PUNCT
ejpam-5322	132	16	σ2	σ2	NOUN
ejpam-5322	132	17	)	)	PUNCT
ejpam-5322	132	18	,	,	PUNCT
ejpam-5322	132	19	the	the	DET
ejpam-5322	132	20	following	follow	VERB
ejpam-5322	132	21	properties	property	NOUN
ejpam-5322	132	22	are	be	AUX
ejpam-5322	132	23	equivalent	equivalent	ADJ
ejpam-5322	132	24	:	:	PUNCT
ejpam-5322	132	25	(	(	PUNCT
ejpam-5322	132	26	1	1	X
ejpam-5322	132	27	)	)	PUNCT
ejpam-5322	132	28	f	f	PROPN
ejpam-5322	132	29	is	be	AUX
ejpam-5322	132	30	lower	low	ADJ
ejpam-5322	132	31	s-(τ1	s-(τ1	NOUN
ejpam-5322	132	32	,	,	PUNCT
ejpam-5322	132	33	τ2)p	τ2)p	ADJ
ejpam-5322	132	34	-	-	ADJ
ejpam-5322	132	35	continuous	continuous	ADJ
ejpam-5322	132	36	;	;	PUNCT
ejpam-5322	132	37	(	(	PUNCT
ejpam-5322	132	38	2	2	X
ejpam-5322	132	39	)	)	PUNCT
ejpam-5322	132	40	f−(v	f−(v	NOUN
ejpam-5322	132	41	)	)	PUNCT
ejpam-5322	132	42	is	be	AUX
ejpam-5322	132	43	(	(	PUNCT
ejpam-5322	132	44	τ1	τ1	NOUN
ejpam-5322	132	45	,	,	PUNCT
ejpam-5322	132	46	τ2)p	τ2)p	NOUN
ejpam-5322	132	47	-	-	PUNCT
ejpam-5322	132	48	open	open	ADJ
ejpam-5322	132	49	in	in	ADP
ejpam-5322	132	50	x	x	PUNCT
ejpam-5322	132	51	for	for	ADP
ejpam-5322	132	52	every	every	DET
ejpam-5322	132	53	σ1σ2	σ1σ2	NOUN
ejpam-5322	132	54	-	-	ADJ
ejpam-5322	132	55	open	open	ADJ
ejpam-5322	132	56	set	set	NOUN
ejpam-5322	132	57	v	v	NOUN
ejpam-5322	132	58	of	of	ADP
ejpam-5322	132	59	y	y	PROPN
ejpam-5322	132	60	having	have	VERB
ejpam-5322	132	61	σ1σ2	σ1σ2	ADV
ejpam-5322	132	62	-	-	PUNCT
ejpam-5322	132	63	connected	connect	VERB
ejpam-5322	132	64	complement	complement	NOUN
ejpam-5322	132	65	;	;	PUNCT
ejpam-5322	132	66	(	(	PUNCT
ejpam-5322	132	67	3	3	X
ejpam-5322	132	68	)	)	PUNCT
ejpam-5322	132	69	f+(k	f+(k	NUM
ejpam-5322	132	70	)	)	PUNCT
ejpam-5322	132	71	is	be	AUX
ejpam-5322	132	72	(	(	PUNCT
ejpam-5322	132	73	τ1	τ1	NOUN
ejpam-5322	132	74	,	,	PUNCT
ejpam-5322	132	75	τ2)p	τ2)p	NOUN
ejpam-5322	132	76	-	-	PUNCT
ejpam-5322	132	77	closed	closed	ADJ
ejpam-5322	132	78	in	in	ADP
ejpam-5322	132	79	x	x	PUNCT
ejpam-5322	132	80	for	for	ADP
ejpam-5322	132	81	every	every	DET
ejpam-5322	132	82	σ1σ2	σ1σ2	NOUN
ejpam-5322	132	83	-	-	ADJ
ejpam-5322	132	84	connected	connect	VERB
ejpam-5322	132	85	σ1σ2	σ1σ2	VERB
ejpam-5322	132	86	-	-	PUNCT
ejpam-5322	132	87	closed	closed	ADJ
ejpam-5322	132	88	set	set	NOUN
ejpam-5322	132	89	k	k	PROPN
ejpam-5322	132	90	of	of	ADP
ejpam-5322	132	91	y	y	PROPN
ejpam-5322	132	92	;	;	PUNCT
ejpam-5322	132	93	(	(	PUNCT
ejpam-5322	132	94	4	4	X
ejpam-5322	132	95	)	)	PUNCT
ejpam-5322	132	96	τ1τ2	τ1τ2	NOUN
ejpam-5322	132	97	-	-	NOUN
ejpam-5322	132	98	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	132	99	-	-	PUNCT
ejpam-5322	132	100	int(f	int(f	VERB
ejpam-5322	132	101	+	+	ADJ
ejpam-5322	132	102	(	(	PUNCT
ejpam-5322	132	103	b	b	NOUN
ejpam-5322	132	104	)	)	PUNCT
ejpam-5322	132	105	)	)	PUNCT
ejpam-5322	132	106	)	)	PUNCT
ejpam-5322	133	1	⊆	⊆	X
ejpam-5322	133	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5322	133	3	-	-	PUNCT
ejpam-5322	133	4	cl(b	cl(b	NOUN
ejpam-5322	133	5	)	)	PUNCT
ejpam-5322	133	6	)	)	PUNCT
ejpam-5322	133	7	for	for	ADP
ejpam-5322	133	8	every	every	DET
ejpam-5322	133	9	subset	subset	NOUN
ejpam-5322	133	10	b	b	PROPN
ejpam-5322	133	11	of	of	ADP
ejpam-5322	133	12	y	y	PROPN
ejpam-5322	133	13	having	have	VERB
ejpam-5322	133	14	the	the	DET
ejpam-5322	133	15	σ1σ2	σ1σ2	ADV
ejpam-5322	133	16	-	-	PUNCT
ejpam-5322	133	17	connected	connect	VERB
ejpam-5322	133	18	σ1σ2	σ1σ2	NOUN
ejpam-5322	133	19	-	-	NOUN
ejpam-5322	133	20	closure	closure	NOUN
ejpam-5322	133	21	;	;	PUNCT
ejpam-5322	133	22	(	(	PUNCT
ejpam-5322	133	23	5	5	NUM
ejpam-5322	133	24	)	)	PUNCT
ejpam-5322	133	25	(	(	PUNCT
ejpam-5322	133	26	τ1	τ1	NOUN
ejpam-5322	133	27	,	,	PUNCT
ejpam-5322	133	28	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5322	133	29	+	+	PROPN
ejpam-5322	133	30	(	(	PUNCT
ejpam-5322	133	31	b	b	NOUN
ejpam-5322	133	32	)	)	PUNCT
ejpam-5322	133	33	)	)	PUNCT
ejpam-5322	134	1	⊆	⊆	NUM
ejpam-5322	134	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5322	134	3	-	-	PUNCT
ejpam-5322	134	4	cl(b	cl(b	NOUN
ejpam-5322	134	5	)	)	PUNCT
ejpam-5322	134	6	)	)	PUNCT
ejpam-5322	134	7	for	for	ADP
ejpam-5322	134	8	every	every	DET
ejpam-5322	134	9	subset	subset	NOUN
ejpam-5322	134	10	b	b	PROPN
ejpam-5322	134	11	of	of	ADP
ejpam-5322	134	12	y	y	PROPN
ejpam-5322	134	13	having	have	VERB
ejpam-5322	134	14	the	the	DET
ejpam-5322	134	15	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5322	134	16	σ1σ2	σ1σ2	NOUN
ejpam-5322	134	17	-	-	NOUN
ejpam-5322	134	18	closure	closure	NOUN
ejpam-5322	134	19	;	;	PUNCT
ejpam-5322	134	20	(	(	PUNCT
ejpam-5322	134	21	6	6	X
ejpam-5322	134	22	)	)	PUNCT
ejpam-5322	134	23	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5322	134	24	-	-	PUNCT
ejpam-5322	134	25	int(b	int(b	NOUN
ejpam-5322	134	26	)	)	PUNCT
ejpam-5322	134	27	)	)	PUNCT
ejpam-5322	135	1	⊆	⊆	NUM
ejpam-5322	135	2	(	(	PUNCT
ejpam-5322	135	3	τ1	τ1	NOUN
ejpam-5322	135	4	,	,	PUNCT
ejpam-5322	135	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5322	135	6	−(b	−(b	NOUN
ejpam-5322	135	7	)	)	PUNCT
ejpam-5322	135	8	)	)	PUNCT
ejpam-5322	135	9	for	for	ADP
ejpam-5322	135	10	every	every	DET
ejpam-5322	135	11	subset	subset	NOUN
ejpam-5322	135	12	b	b	PROPN
ejpam-5322	135	13	of	of	ADP
ejpam-5322	135	14	y	y	PRON
ejpam-5322	136	1	such	such	ADJ
ejpam-5322	136	2	that	that	SCONJ
ejpam-5322	136	3	y	y	PROPN
ejpam-5322	137	1	−	−	ADP
ejpam-5322	137	2	σ1σ2	σ1σ2	NUM
ejpam-5322	137	3	-	-	PUNCT
ejpam-5322	137	4	int(b	int(b	NOUN
ejpam-5322	137	5	)	)	PUNCT
ejpam-5322	137	6	is	be	AUX
ejpam-5322	137	7	σ1σ2	σ1σ2	NOUN
ejpam-5322	137	8	-	-	PUNCT
ejpam-5322	137	9	connected	connected	ADJ
ejpam-5322	137	10	.	.	PUNCT
ejpam-5322	138	1	proof	proof	NOUN
ejpam-5322	138	2	.	.	PUNCT
ejpam-5322	139	1	the	the	DET
ejpam-5322	139	2	proof	proof	NOUN
ejpam-5322	139	3	is	be	AUX
ejpam-5322	139	4	similar	similar	ADJ
ejpam-5322	139	5	to	to	ADP
ejpam-5322	139	6	that	that	PRON
ejpam-5322	139	7	of	of	ADP
ejpam-5322	139	8	theorem	theorem	NOUN
ejpam-5322	139	9	1	1	NUM
ejpam-5322	139	10	.	.	PUNCT
ejpam-5322	139	11	definition	definition	NOUN
ejpam-5322	139	12	3	3	NUM
ejpam-5322	139	13	.	.	PUNCT
ejpam-5322	140	1	a	a	DET
ejpam-5322	140	2	function	function	NOUN
ejpam-5322	140	3	:	:	PUNCT
ejpam-5322	140	4	(	(	PUNCT
ejpam-5322	140	5	x	x	NOUN
ejpam-5322	140	6	,	,	PUNCT
ejpam-5322	140	7	τ1	τ1	NOUN
ejpam-5322	140	8	,	,	PUNCT
ejpam-5322	140	9	τ2	τ2	NOUN
ejpam-5322	140	10	)	)	PUNCT
ejpam-5322	140	11	→	→	SYM
ejpam-5322	140	12	(	(	PUNCT
ejpam-5322	140	13	y	y	PROPN
ejpam-5322	140	14	,	,	PUNCT
ejpam-5322	140	15	σ1	σ1	PROPN
ejpam-5322	140	16	,	,	PUNCT
ejpam-5322	140	17	σ2	σ2	PROPN
ejpam-5322	140	18	)	)	PUNCT
ejpam-5322	140	19	is	be	AUX
ejpam-5322	140	20	said	say	VERB
ejpam-5322	140	21	to	to	PART
ejpam-5322	140	22	be	be	AUX
ejpam-5322	140	23	s-(τ1	s-(τ1	PROPN
ejpam-5322	140	24	,	,	PUNCT
ejpam-5322	140	25	τ2)p	τ2)p	ADJ
ejpam-5322	140	26	-	-	ADJ
ejpam-5322	140	27	continuous	continuous	ADJ
ejpam-5322	140	28	if	if	SCONJ
ejpam-5322	140	29	for	for	ADP
ejpam-5322	140	30	each	each	DET
ejpam-5322	140	31	point	point	NOUN
ejpam-5322	140	32	x	x	X
ejpam-5322	140	33	∈	∈	NOUN
ejpam-5322	140	34	x	x	X
ejpam-5322	140	35	and	and	CCONJ
ejpam-5322	140	36	each	each	DET
ejpam-5322	140	37	σ1σ2	σ1σ2	VERB
ejpam-5322	140	38	-	-	ADJ
ejpam-5322	140	39	open	open	ADJ
ejpam-5322	140	40	set	set	NOUN
ejpam-5322	140	41	v	v	NOUN
ejpam-5322	140	42	of	of	ADP
ejpam-5322	140	43	y	y	NOUN
ejpam-5322	140	44	containing	contain	VERB
ejpam-5322	140	45	f(x	f(x	PROPN
ejpam-5322	140	46	)	)	PUNCT
ejpam-5322	140	47	and	and	CCONJ
ejpam-5322	140	48	having	have	VERB
ejpam-5322	140	49	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5322	140	50	complement	complement	NOUN
ejpam-5322	140	51	,	,	PUNCT
ejpam-5322	140	52	there	there	PRON
ejpam-5322	140	53	exists	exist	VERB
ejpam-5322	140	54	a	a	DET
ejpam-5322	140	55	(	(	PUNCT
ejpam-5322	140	56	τ1	τ1	NOUN
ejpam-5322	140	57	,	,	PUNCT
ejpam-5322	140	58	τ2)p	τ2)p	ADJ
ejpam-5322	140	59	-	-	PUNCT
ejpam-5322	140	60	open	open	ADJ
ejpam-5322	140	61	set	set	NOUN
ejpam-5322	140	62	u	u	NOUN
ejpam-5322	140	63	of	of	ADP
ejpam-5322	140	64	x	x	PUNCT
ejpam-5322	140	65	containing	contain	VERB
ejpam-5322	140	66	x	x	PUNCT
ejpam-5322	140	67	such	such	ADJ
ejpam-5322	140	68	that	that	DET
ejpam-5322	140	69	f(u	f(u	PROPN
ejpam-5322	140	70	)	)	PUNCT
ejpam-5322	140	71	⊆	⊆	NUM
ejpam-5322	140	72	v	v	NOUN
ejpam-5322	140	73	.	.	PUNCT
ejpam-5322	141	1	corollary	corollary	ADJ
ejpam-5322	141	2	1	1	NUM
ejpam-5322	141	3	.	.	PUNCT
ejpam-5322	142	1	for	for	ADP
ejpam-5322	142	2	a	a	DET
ejpam-5322	142	3	function	function	NOUN
ejpam-5322	142	4	f	f	NOUN
ejpam-5322	142	5	:	:	PUNCT
ejpam-5322	142	6	(	(	PUNCT
ejpam-5322	142	7	x	x	NOUN
ejpam-5322	142	8	,	,	PUNCT
ejpam-5322	142	9	τ1	τ1	NOUN
ejpam-5322	142	10	,	,	PUNCT
ejpam-5322	142	11	τ2	τ2	NOUN
ejpam-5322	142	12	)	)	PUNCT
ejpam-5322	142	13	→	→	SYM
ejpam-5322	142	14	(	(	PUNCT
ejpam-5322	142	15	y	y	PROPN
ejpam-5322	142	16	,	,	PUNCT
ejpam-5322	142	17	σ1	σ1	PROPN
ejpam-5322	142	18	,	,	PUNCT
ejpam-5322	142	19	σ2	σ2	NOUN
ejpam-5322	142	20	)	)	PUNCT
ejpam-5322	142	21	,	,	PUNCT
ejpam-5322	142	22	the	the	DET
ejpam-5322	142	23	following	follow	VERB
ejpam-5322	142	24	properties	property	NOUN
ejpam-5322	142	25	are	be	AUX
ejpam-5322	142	26	equivalent	equivalent	ADJ
ejpam-5322	142	27	:	:	PUNCT
ejpam-5322	142	28	(	(	PUNCT
ejpam-5322	142	29	1	1	X
ejpam-5322	142	30	)	)	PUNCT
ejpam-5322	142	31	f	f	PROPN
ejpam-5322	142	32	is	be	AUX
ejpam-5322	142	33	s-(τ1	s-(τ1	PROPN
ejpam-5322	142	34	,	,	PUNCT
ejpam-5322	142	35	τ2)p	τ2)p	ADJ
ejpam-5322	142	36	-	-	ADJ
ejpam-5322	142	37	continuous	continuous	ADJ
ejpam-5322	142	38	;	;	PUNCT
ejpam-5322	142	39	n.	n.	PROPN
ejpam-5322	142	40	viriyapong	viriyapong	PROPN
ejpam-5322	142	41	,	,	PUNCT
ejpam-5322	142	42	s.	s.	PROPN
ejpam-5322	142	43	sompong	sompong	PROPN
ejpam-5322	142	44	,	,	PUNCT
ejpam-5322	142	45	c.	c.	PROPN
ejpam-5322	142	46	boonpok	boonpok	PROPN
ejpam-5322	142	47	/	/	SYM
ejpam-5322	142	48	eur	eur	PROPN
ejpam-5322	142	49	.	.	PUNCT
ejpam-5322	143	1	j.	j.	PROPN
ejpam-5322	143	2	pure	pure	PROPN
ejpam-5322	143	3	appl	appl	PROPN
ejpam-5322	143	4	.	.	PROPN
ejpam-5322	143	5	math	math	PROPN
ejpam-5322	143	6	,	,	PUNCT
ejpam-5322	143	7	17	17	NUM
ejpam-5322	143	8	(	(	PUNCT
ejpam-5322	143	9	3	3	NUM
ejpam-5322	143	10	)	)	PUNCT
ejpam-5322	143	11	(	(	PUNCT
ejpam-5322	143	12	2024	2024	NUM
ejpam-5322	143	13	)	)	PUNCT
ejpam-5322	143	14	,	,	PUNCT
ejpam-5322	143	15	2210	2210	NUM
ejpam-5322	143	16	-	-	SYM
ejpam-5322	143	17	2220	2220	NUM
ejpam-5322	143	18	2215	2215	NUM
ejpam-5322	143	19	(	(	PUNCT
ejpam-5322	143	20	2	2	NUM
ejpam-5322	143	21	)	)	PUNCT
ejpam-5322	143	22	f−1(v	f−1(v	NOUN
ejpam-5322	143	23	)	)	PUNCT
ejpam-5322	143	24	is	be	AUX
ejpam-5322	143	25	(	(	PUNCT
ejpam-5322	143	26	τ1	τ1	NOUN
ejpam-5322	143	27	,	,	PUNCT
ejpam-5322	143	28	τ2)p	τ2)p	NOUN
ejpam-5322	143	29	-	-	PUNCT
ejpam-5322	143	30	open	open	ADJ
ejpam-5322	143	31	in	in	ADP
ejpam-5322	143	32	x	x	PUNCT
ejpam-5322	143	33	for	for	ADP
ejpam-5322	143	34	every	every	DET
ejpam-5322	143	35	σ1σ2	σ1σ2	NOUN
ejpam-5322	143	36	-	-	ADJ
ejpam-5322	143	37	open	open	ADJ
ejpam-5322	143	38	set	set	NOUN
ejpam-5322	143	39	v	v	NOUN
ejpam-5322	143	40	of	of	ADP
ejpam-5322	143	41	y	y	PROPN
ejpam-5322	143	42	having	have	VERB
ejpam-5322	143	43	σ1σ2	σ1σ2	ADV
ejpam-5322	143	44	-	-	PUNCT
ejpam-5322	143	45	connected	connect	VERB
ejpam-5322	143	46	complement	complement	NOUN
ejpam-5322	143	47	;	;	PUNCT
ejpam-5322	143	48	(	(	PUNCT
ejpam-5322	143	49	3	3	X
ejpam-5322	143	50	)	)	PUNCT
ejpam-5322	143	51	f−1(k	f−1(k	PROPN
ejpam-5322	143	52	)	)	PUNCT
ejpam-5322	143	53	is	be	AUX
ejpam-5322	143	54	(	(	PUNCT
ejpam-5322	143	55	τ1	τ1	NOUN
ejpam-5322	143	56	,	,	PUNCT
ejpam-5322	143	57	τ2)p	τ2)p	NOUN
ejpam-5322	143	58	-	-	PUNCT
ejpam-5322	143	59	closed	closed	ADJ
ejpam-5322	143	60	in	in	ADP
ejpam-5322	143	61	x	x	PUNCT
ejpam-5322	143	62	for	for	ADP
ejpam-5322	143	63	every	every	DET
ejpam-5322	143	64	σ1σ2	σ1σ2	NOUN
ejpam-5322	143	65	-	-	ADJ
ejpam-5322	143	66	connected	connect	VERB
ejpam-5322	143	67	σ1σ2	σ1σ2	VERB
ejpam-5322	143	68	-	-	PUNCT
ejpam-5322	143	69	closed	closed	ADJ
ejpam-5322	143	70	set	set	NOUN
ejpam-5322	143	71	k	k	PROPN
ejpam-5322	143	72	of	of	ADP
ejpam-5322	143	73	y	y	PROPN
ejpam-5322	143	74	;	;	PUNCT
ejpam-5322	143	75	(	(	PUNCT
ejpam-5322	143	76	4	4	X
ejpam-5322	143	77	)	)	PUNCT
ejpam-5322	143	78	τ1τ2	τ1τ2	NOUN
ejpam-5322	143	79	-	-	NOUN
ejpam-5322	143	80	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5322	143	81	-	-	PUNCT
ejpam-5322	143	82	int(f	int(f	NOUN
ejpam-5322	143	83	−1(b	−1(b	NOUN
ejpam-5322	143	84	)	)	PUNCT
ejpam-5322	143	85	)	)	PUNCT
ejpam-5322	143	86	)	)	PUNCT
ejpam-5322	144	1	⊆	⊆	NUM
ejpam-5322	144	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5322	144	3	-	-	PUNCT
ejpam-5322	144	4	cl(b	cl(b	NOUN
ejpam-5322	144	5	)	)	PUNCT
ejpam-5322	144	6	)	)	PUNCT
ejpam-5322	144	7	for	for	ADP
ejpam-5322	144	8	every	every	DET
ejpam-5322	144	9	subset	subset	NOUN
ejpam-5322	144	10	b	b	PROPN
ejpam-5322	144	11	of	of	ADP
ejpam-5322	144	12	y	y	PROPN
ejpam-5322	144	13	having	have	VERB
ejpam-5322	144	14	the	the	DET
ejpam-5322	144	15	σ1σ2	σ1σ2	ADV
ejpam-5322	144	16	-	-	PUNCT
ejpam-5322	144	17	connected	connect	VERB
ejpam-5322	144	18	σ1σ2	σ1σ2	NOUN
ejpam-5322	144	19	-	-	NOUN
ejpam-5322	144	20	closure	closure	NOUN
ejpam-5322	144	21	;	;	PUNCT
ejpam-5322	144	22	(	(	PUNCT
ejpam-5322	144	23	5	5	NUM
ejpam-5322	144	24	)	)	PUNCT
ejpam-5322	144	25	(	(	PUNCT
ejpam-5322	144	26	τ1	τ1	NOUN
ejpam-5322	144	27	,	,	PUNCT
ejpam-5322	144	28	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5322	144	29	−1(b	−1(b	NOUN
ejpam-5322	144	30	)	)	PUNCT
ejpam-5322	144	31	)	)	PUNCT
ejpam-5322	144	32	⊆	⊆	NUM
ejpam-5322	144	33	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5322	144	34	-	-	PUNCT
ejpam-5322	144	35	cl(b	cl(b	NOUN
ejpam-5322	144	36	)	)	PUNCT
ejpam-5322	144	37	)	)	PUNCT
ejpam-5322	144	38	for	for	ADP
ejpam-5322	144	39	every	every	DET
ejpam-5322	144	40	subset	subset	NOUN
ejpam-5322	144	41	b	b	PROPN
ejpam-5322	144	42	of	of	ADP
ejpam-5322	144	43	y	y	PROPN
ejpam-5322	144	44	having	have	VERB
ejpam-5322	144	45	the	the	DET
ejpam-5322	144	46	σ1σ2connected	σ1σ2connecte	VERB
ejpam-5322	144	47	σ1σ2	σ1σ2	NOUN
ejpam-5322	144	48	-	-	NOUN
ejpam-5322	144	49	closure	closure	NOUN
ejpam-5322	144	50	;	;	PUNCT
ejpam-5322	144	51	(	(	PUNCT
ejpam-5322	144	52	6	6	X
ejpam-5322	144	53	)	)	PUNCT
ejpam-5322	144	54	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5322	144	55	-	-	PUNCT
ejpam-5322	144	56	int(b	int(b	NOUN
ejpam-5322	144	57	)	)	PUNCT
ejpam-5322	144	58	)	)	PUNCT
ejpam-5322	144	59	⊆	⊆	NUM
ejpam-5322	144	60	(	(	PUNCT
ejpam-5322	144	61	τ1	τ1	NOUN
ejpam-5322	144	62	,	,	PUNCT
ejpam-5322	144	63	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5322	144	64	−1(b	−1(b	NOUN
ejpam-5322	144	65	)	)	PUNCT
ejpam-5322	144	66	)	)	PUNCT
ejpam-5322	144	67	for	for	ADP
ejpam-5322	144	68	every	every	DET
ejpam-5322	144	69	subset	subset	NOUN
ejpam-5322	144	70	b	b	PROPN
ejpam-5322	144	71	of	of	ADP
ejpam-5322	144	72	y	y	PRON
ejpam-5322	144	73	such	such	ADJ
ejpam-5322	144	74	that	that	SCONJ
ejpam-5322	144	75	y	y	PROPN
ejpam-5322	145	1	−	−	ADP
ejpam-5322	145	2	σ1σ2	σ1σ2	NUM
ejpam-5322	145	3	-	-	PUNCT
ejpam-5322	145	4	int(b	int(b	NOUN
ejpam-5322	145	5	)	)	PUNCT
ejpam-5322	145	6	is	be	AUX
ejpam-5322	145	7	σ1σ2	σ1σ2	NOUN
ejpam-5322	145	8	-	-	PUNCT
ejpam-5322	145	9	connected	connect	VERB
ejpam-5322	145	10	.	.	PUNCT
ejpam-5322	146	1	corollary	corollary	ADJ
ejpam-5322	146	2	2	2	NUM
ejpam-5322	146	3	.	.	PUNCT
ejpam-5322	146	4	a	a	DET
ejpam-5322	146	5	multifunction	multifunction	NOUN
ejpam-5322	147	1	f	f	NOUN
ejpam-5322	147	2	:	:	PUNCT
ejpam-5322	147	3	(	(	PUNCT
ejpam-5322	147	4	x	x	NOUN
ejpam-5322	147	5	,	,	PUNCT
ejpam-5322	147	6	τ1	τ1	NOUN
ejpam-5322	147	7	,	,	PUNCT
ejpam-5322	147	8	τ2	τ2	NOUN
ejpam-5322	147	9	)	)	PUNCT
ejpam-5322	147	10	→	→	SYM
ejpam-5322	147	11	(	(	PUNCT
ejpam-5322	147	12	y	y	PROPN
ejpam-5322	147	13	,	,	PUNCT
ejpam-5322	147	14	σ1	σ1	PROPN
ejpam-5322	147	15	,	,	PUNCT
ejpam-5322	147	16	σ2	σ2	PROPN
ejpam-5322	147	17	)	)	PUNCT
ejpam-5322	147	18	is	be	AUX
ejpam-5322	147	19	upper	upper	ADJ
ejpam-5322	147	20	s-(τ1	s-(τ1	PROPN
ejpam-5322	147	21	,	,	PUNCT
ejpam-5322	147	22	τ2)p	τ2)p	ADJ
ejpam-5322	147	23	-	-	ADJ
ejpam-5322	147	24	continuous	continuous	ADJ
ejpam-5322	147	25	if	if	SCONJ
ejpam-5322	147	26	f−(v	f−(v	ADJ
ejpam-5322	147	27	)	)	PUNCT
ejpam-5322	147	28	is	be	AUX
ejpam-5322	147	29	(	(	PUNCT
ejpam-5322	147	30	τ1	τ1	NOUN
ejpam-5322	147	31	,	,	PUNCT
ejpam-5322	147	32	τ2)p	τ2)p	NOUN
ejpam-5322	147	33	-	-	PUNCT
ejpam-5322	147	34	closed	closed	ADJ
ejpam-5322	147	35	in	in	ADP
ejpam-5322	147	36	x	x	PUNCT
ejpam-5322	147	37	for	for	ADP
ejpam-5322	147	38	every	every	DET
ejpam-5322	147	39	σ1σ2	σ1σ2	NOUN
ejpam-5322	147	40	-	-	PUNCT
ejpam-5322	147	41	connected	connected	ADJ
ejpam-5322	147	42	set	set	VERB
ejpam-5322	147	43	v	v	NOUN
ejpam-5322	147	44	of	of	ADP
ejpam-5322	147	45	y	y	PROPN
ejpam-5322	147	46	.	.	PUNCT
ejpam-5322	148	1	proof	proof	NOUN
ejpam-5322	148	2	.	.	PUNCT
ejpam-5322	149	1	let	let	VERB
ejpam-5322	149	2	v	v	PART
ejpam-5322	149	3	be	be	AUX
ejpam-5322	149	4	any	any	DET
ejpam-5322	149	5	σ1σ2	σ1σ2	NOUN
ejpam-5322	149	6	-	-	ADJ
ejpam-5322	149	7	open	open	ADJ
ejpam-5322	149	8	set	set	NOUN
ejpam-5322	149	9	of	of	ADP
ejpam-5322	149	10	y	y	PROPN
ejpam-5322	149	11	having	have	VERB
ejpam-5322	149	12	σ1σ2	σ1σ2	ADV
ejpam-5322	149	13	-	-	PUNCT
ejpam-5322	149	14	connected	connect	VERB
ejpam-5322	149	15	complement	complement	NOUN
ejpam-5322	149	16	.	.	PUNCT
ejpam-5322	150	1	then	then	ADV
ejpam-5322	150	2	,	,	PUNCT
ejpam-5322	150	3	y	y	PROPN
ejpam-5322	150	4	−v	−v	NOUN
ejpam-5322	150	5	is	be	AUX
ejpam-5322	150	6	σ1σ2	σ1σ2	NOUN
ejpam-5322	150	7	-	-	PUNCT
ejpam-5322	150	8	connected	connected	ADJ
ejpam-5322	150	9	and	and	CCONJ
ejpam-5322	150	10	f−(y	f−(y	NOUN
ejpam-5322	150	11	−v	−v	NOUN
ejpam-5322	150	12	)	)	PUNCT
ejpam-5322	150	13	is	be	AUX
ejpam-5322	150	14	(	(	PUNCT
ejpam-5322	150	15	τ1	τ1	NOUN
ejpam-5322	150	16	,	,	PUNCT
ejpam-5322	150	17	τ2)p	τ2)p	NOUN
ejpam-5322	150	18	-	-	PUNCT
ejpam-5322	150	19	closed	closed	ADJ
ejpam-5322	150	20	in	in	ADP
ejpam-5322	150	21	x.	x.	NOUN
ejpam-5322	150	22	thus	thus	ADV
ejpam-5322	150	23	,	,	PUNCT
ejpam-5322	150	24	f+(v	f+(v	PROPN
ejpam-5322	150	25	)	)	PUNCT
ejpam-5322	151	1	is	be	AUX
ejpam-5322	151	2	(	(	PUNCT
ejpam-5322	151	3	τ1	τ1	NOUN
ejpam-5322	151	4	,	,	PUNCT
ejpam-5322	151	5	τ2)popen	τ2)popen	ADJ
ejpam-5322	151	6	in	in	ADP
ejpam-5322	151	7	x	x	PUNCT
ejpam-5322	151	8	and	and	CCONJ
ejpam-5322	151	9	by	by	ADP
ejpam-5322	151	10	theorem	theorem	NOUN
ejpam-5322	151	11	1	1	NUM
ejpam-5322	151	12	,	,	PUNCT
ejpam-5322	151	13	f	f	PROPN
ejpam-5322	151	14	is	be	AUX
ejpam-5322	151	15	upper	upper	ADJ
ejpam-5322	151	16	s-(τ1	s-(τ1	PROPN
ejpam-5322	151	17	,	,	PUNCT
ejpam-5322	151	18	τ2)p	τ2)p	ADJ
ejpam-5322	151	19	-	-	ADJ
ejpam-5322	151	20	continuous	continuous	ADJ
ejpam-5322	151	21	.	.	PUNCT
ejpam-5322	152	1	corollary	corollary	ADJ
ejpam-5322	152	2	3	3	NUM
ejpam-5322	152	3	.	.	PUNCT
ejpam-5322	153	1	a	a	DET
ejpam-5322	153	2	multifunction	multifunction	NOUN
ejpam-5322	153	3	f	f	NOUN
ejpam-5322	153	4	:	:	PUNCT
ejpam-5322	153	5	(	(	PUNCT
ejpam-5322	153	6	x	x	NOUN
ejpam-5322	153	7	,	,	PUNCT
ejpam-5322	153	8	τ1	τ1	NOUN
ejpam-5322	153	9	,	,	PUNCT
ejpam-5322	153	10	τ2	τ2	NOUN
ejpam-5322	153	11	)	)	PUNCT
ejpam-5322	153	12	→	→	SYM
ejpam-5322	153	13	(	(	PUNCT
ejpam-5322	153	14	y	y	PROPN
ejpam-5322	153	15	,	,	PUNCT
ejpam-5322	153	16	σ1	σ1	PROPN
ejpam-5322	153	17	,	,	PUNCT
ejpam-5322	153	18	σ2	σ2	NOUN
ejpam-5322	153	19	)	)	PUNCT
ejpam-5322	153	20	is	be	AUX
ejpam-5322	153	21	lower	low	ADJ
ejpam-5322	153	22	s-(τ1	s-(τ1	NOUN
ejpam-5322	153	23	,	,	PUNCT
ejpam-5322	153	24	τ2)p	τ2)p	ADJ
ejpam-5322	153	25	-	-	ADJ
ejpam-5322	153	26	continuous	continuous	ADJ
ejpam-5322	153	27	if	if	SCONJ
ejpam-5322	153	28	f+(v	f+(v	PROPN
ejpam-5322	153	29	)	)	PUNCT
ejpam-5322	154	1	is	be	AUX
ejpam-5322	154	2	(	(	PUNCT
ejpam-5322	154	3	τ1	τ1	NOUN
ejpam-5322	154	4	,	,	PUNCT
ejpam-5322	154	5	τ2)p	τ2)p	NOUN
ejpam-5322	154	6	-	-	PUNCT
ejpam-5322	154	7	closed	closed	ADJ
ejpam-5322	154	8	in	in	ADP
ejpam-5322	154	9	x	x	PUNCT
ejpam-5322	154	10	for	for	ADP
ejpam-5322	154	11	every	every	DET
ejpam-5322	154	12	σ1σ2	σ1σ2	NOUN
ejpam-5322	154	13	-	-	PUNCT
ejpam-5322	154	14	connected	connected	ADJ
ejpam-5322	154	15	set	set	VERB
ejpam-5322	154	16	v	v	NOUN
ejpam-5322	154	17	of	of	ADP
ejpam-5322	154	18	y	y	PROPN
ejpam-5322	154	19	.	.	PUNCT
ejpam-5322	155	1	proof	proof	NOUN
ejpam-5322	155	2	.	.	PUNCT
ejpam-5322	156	1	the	the	DET
ejpam-5322	156	2	proof	proof	NOUN
ejpam-5322	156	3	is	be	AUX
ejpam-5322	156	4	similar	similar	ADJ
ejpam-5322	156	5	to	to	ADP
ejpam-5322	156	6	that	that	PRON
ejpam-5322	156	7	of	of	ADP
ejpam-5322	156	8	corollary	corollary	ADJ
ejpam-5322	156	9	2	2	NUM
ejpam-5322	156	10	.	.	PUNCT
ejpam-5322	156	11	for	for	ADP
ejpam-5322	156	12	a	a	DET
ejpam-5322	156	13	multifunction	multifunction	NOUN
ejpam-5322	156	14	f	f	NOUN
ejpam-5322	156	15	:	:	PUNCT
ejpam-5322	156	16	(	(	PUNCT
ejpam-5322	156	17	x	x	NOUN
ejpam-5322	156	18	,	,	PUNCT
ejpam-5322	156	19	τ1	τ1	NOUN
ejpam-5322	156	20	,	,	PUNCT
ejpam-5322	156	21	τ2	τ2	NOUN
ejpam-5322	156	22	)	)	PUNCT
ejpam-5322	156	23	→	→	SYM
ejpam-5322	156	24	(	(	PUNCT
ejpam-5322	156	25	y	y	PROPN
ejpam-5322	156	26	,	,	PUNCT
ejpam-5322	156	27	σ1	σ1	PROPN
ejpam-5322	156	28	,	,	PUNCT
ejpam-5322	156	29	σ2	σ2	NOUN
ejpam-5322	156	30	)	)	PUNCT
ejpam-5322	156	31	,	,	PUNCT
ejpam-5322	156	32	by	by	ADP
ejpam-5322	156	33	clf⊛	clf⊛	PROPN
ejpam-5322	156	34	:	:	PUNCT
ejpam-5322	156	35	(	(	PUNCT
ejpam-5322	156	36	x	x	NOUN
ejpam-5322	156	37	,	,	PUNCT
ejpam-5322	156	38	τ1	τ1	NOUN
ejpam-5322	156	39	,	,	PUNCT
ejpam-5322	156	40	τ2	τ2	NOUN
ejpam-5322	156	41	)	)	PUNCT
ejpam-5322	156	42	→	→	SYM
ejpam-5322	156	43	(	(	PUNCT
ejpam-5322	156	44	y	y	PROPN
ejpam-5322	156	45	,	,	PUNCT
ejpam-5322	156	46	σ1	σ1	PROPN
ejpam-5322	156	47	,	,	PUNCT
ejpam-5322	156	48	σ2	σ2	NOUN
ejpam-5322	156	49	)	)	PUNCT
ejpam-5322	157	1	[	[	X
ejpam-5322	157	2	20	20	NUM
ejpam-5322	157	3	]	]	PUNCT
ejpam-5322	157	4	we	we	PRON
ejpam-5322	157	5	denote	denote	VERB
ejpam-5322	157	6	a	a	DET
ejpam-5322	157	7	multifunction	multifunction	NOUN
ejpam-5322	157	8	defined	define	VERB
ejpam-5322	157	9	as	as	SCONJ
ejpam-5322	157	10	follows	follow	VERB
ejpam-5322	157	11	:	:	PUNCT
ejpam-5322	157	12	clf⊛(x	clf⊛(x	PROPN
ejpam-5322	157	13	)	)	PUNCT
ejpam-5322	157	14	=	=	PUNCT
ejpam-5322	158	1	σ1σ2	σ1σ2	X
ejpam-5322	158	2	-	-	NUM
ejpam-5322	158	3	cl(f	cl(f	NOUN
ejpam-5322	158	4	(	(	PUNCT
ejpam-5322	158	5	x	x	NOUN
ejpam-5322	158	6	)	)	PUNCT
ejpam-5322	158	7	)	)	PUNCT
ejpam-5322	158	8	for	for	ADP
ejpam-5322	158	9	each	each	DET
ejpam-5322	158	10	x	x	SYM
ejpam-5322	158	11	∈	∈	PROPN
ejpam-5322	158	12	x.	x.	NOUN
ejpam-5322	158	13	definition	definition	NOUN
ejpam-5322	158	14	4	4	NUM
ejpam-5322	158	15	.	.	PUNCT
ejpam-5322	159	1	[	[	X
ejpam-5322	159	2	20	20	NUM
ejpam-5322	159	3	]	]	PUNCT
ejpam-5322	159	4	a	a	DET
ejpam-5322	159	5	subset	subset	NOUN
ejpam-5322	159	6	a	a	PRON
ejpam-5322	159	7	of	of	ADP
ejpam-5322	159	8	a	a	DET
ejpam-5322	159	9	bitopological	bitopological	ADJ
ejpam-5322	159	10	space	space	NOUN
ejpam-5322	159	11	(	(	PUNCT
ejpam-5322	159	12	x	x	NOUN
ejpam-5322	159	13	,	,	PUNCT
ejpam-5322	159	14	τ1	τ1	NOUN
ejpam-5322	159	15	,	,	PUNCT
ejpam-5322	159	16	τ2	τ2	NOUN
ejpam-5322	159	17	)	)	PUNCT
ejpam-5322	159	18	is	be	AUX
ejpam-5322	159	19	said	say	VERB
ejpam-5322	159	20	to	to	PART
ejpam-5322	159	21	be	be	AUX
ejpam-5322	159	22	:	:	PUNCT
ejpam-5322	159	23	(	(	PUNCT
ejpam-5322	159	24	1	1	X
ejpam-5322	159	25	)	)	PUNCT
ejpam-5322	159	26	τ1τ2	τ1τ2	NOUN
ejpam-5322	159	27	-	-	NOUN
ejpam-5322	159	28	paracompact	paracompact	ADJ
ejpam-5322	159	29	if	if	SCONJ
ejpam-5322	159	30	every	every	DET
ejpam-5322	159	31	cover	cover	NOUN
ejpam-5322	159	32	of	of	ADP
ejpam-5322	159	33	a	a	PRON
ejpam-5322	159	34	by	by	ADP
ejpam-5322	159	35	τ1τ2	τ1τ2	ADJ
ejpam-5322	159	36	-	-	ADJ
ejpam-5322	159	37	open	open	ADJ
ejpam-5322	159	38	sets	set	NOUN
ejpam-5322	159	39	of	of	ADP
ejpam-5322	159	40	x	x	VERB
ejpam-5322	159	41	is	be	AUX
ejpam-5322	159	42	refined	refine	VERB
ejpam-5322	159	43	by	by	ADP
ejpam-5322	159	44	a	a	DET
ejpam-5322	159	45	cover	cover	NOUN
ejpam-5322	159	46	of	of	ADP
ejpam-5322	159	47	a	a	PRON
ejpam-5322	159	48	which	which	PRON
ejpam-5322	159	49	consists	consist	VERB
ejpam-5322	159	50	of	of	ADP
ejpam-5322	159	51	τ1τ2	τ1τ2	ADJ
ejpam-5322	159	52	-	-	ADJ
ejpam-5322	159	53	open	open	ADJ
ejpam-5322	159	54	sets	set	NOUN
ejpam-5322	159	55	of	of	ADP
ejpam-5322	159	56	x	x	PUNCT
ejpam-5322	159	57	and	and	CCONJ
ejpam-5322	159	58	is	be	AUX
ejpam-5322	159	59	τ1τ2	τ1τ2	NOUN
ejpam-5322	159	60	-	-	ADJ
ejpam-5322	159	61	locally	locally	ADV
ejpam-5322	159	62	finite	finite	NOUN
ejpam-5322	159	63	in	in	ADP
ejpam-5322	159	64	x	x	PRON
ejpam-5322	159	65	;	;	PUNCT
ejpam-5322	159	66	(	(	PUNCT
ejpam-5322	159	67	2	2	X
ejpam-5322	159	68	)	)	PUNCT
ejpam-5322	159	69	τ1τ2	τ1τ2	NOUN
ejpam-5322	159	70	-	-	NOUN
ejpam-5322	159	71	regular	regular	ADJ
ejpam-5322	159	72	if	if	SCONJ
ejpam-5322	159	73	for	for	ADP
ejpam-5322	159	74	each	each	DET
ejpam-5322	159	75	x	x	SYM
ejpam-5322	159	76	∈	∈	PROPN
ejpam-5322	159	77	a	a	PRON
ejpam-5322	159	78	and	and	CCONJ
ejpam-5322	159	79	each	each	DET
ejpam-5322	159	80	τ1τ2	τ1τ2	ADJ
ejpam-5322	159	81	-	-	ADJ
ejpam-5322	159	82	open	open	ADJ
ejpam-5322	159	83	set	set	ADJ
ejpam-5322	159	84	u	u	NOUN
ejpam-5322	159	85	of	of	ADP
ejpam-5322	159	86	x	x	PUNCT
ejpam-5322	159	87	containing	contain	VERB
ejpam-5322	159	88	x	x	PRON
ejpam-5322	159	89	,	,	PUNCT
ejpam-5322	159	90	there	there	PRON
ejpam-5322	159	91	exists	exist	VERB
ejpam-5322	159	92	a	a	DET
ejpam-5322	159	93	τ1τ2	τ1τ2	NOUN
ejpam-5322	159	94	-	-	ADJ
ejpam-5322	159	95	open	open	ADJ
ejpam-5322	159	96	set	set	NOUN
ejpam-5322	159	97	v	v	NOUN
ejpam-5322	159	98	of	of	ADP
ejpam-5322	159	99	x	x	PUNCT
ejpam-5322	159	100	such	such	ADJ
ejpam-5322	159	101	that	that	SCONJ
ejpam-5322	159	102	x	x	SYM
ejpam-5322	159	103	∈	∈	NOUN
ejpam-5322	159	104	v	v	ADP
ejpam-5322	159	105	⊆	⊆	NUM
ejpam-5322	159	106	τ1τ2	τ1τ2	NOUN
ejpam-5322	159	107	-	-	NOUN
ejpam-5322	159	108	cl(v	cl(v	X
ejpam-5322	159	109	)	)	PUNCT
ejpam-5322	159	110	⊆	⊆	NUM
ejpam-5322	159	111	u	u	NOUN
ejpam-5322	159	112	.	.	PUNCT
ejpam-5322	160	1	lemma	lemma	PROPN
ejpam-5322	160	2	3	3	X
ejpam-5322	160	3	.	.	PUNCT
ejpam-5322	161	1	[	[	X
ejpam-5322	161	2	20	20	NUM
ejpam-5322	161	3	]	]	X
ejpam-5322	161	4	if	if	SCONJ
ejpam-5322	161	5	a	a	PRON
ejpam-5322	161	6	is	be	AUX
ejpam-5322	161	7	a	a	DET
ejpam-5322	161	8	τ1τ2	τ1τ2	ADJ
ejpam-5322	161	9	-	-	ADJ
ejpam-5322	161	10	regular	regular	ADJ
ejpam-5322	161	11	τ1τ2	τ1τ2	NOUN
ejpam-5322	161	12	-	-	ADJ
ejpam-5322	161	13	paracompact	paracompact	ADJ
ejpam-5322	161	14	set	set	NOUN
ejpam-5322	161	15	of	of	ADP
ejpam-5322	161	16	a	a	DET
ejpam-5322	161	17	bitopological	bitopological	ADJ
ejpam-5322	161	18	space	space	NOUN
ejpam-5322	161	19	(	(	PUNCT
ejpam-5322	161	20	x	x	NOUN
ejpam-5322	161	21	,	,	PUNCT
ejpam-5322	161	22	τ1	τ1	NOUN
ejpam-5322	161	23	,	,	PUNCT
ejpam-5322	161	24	τ2	τ2	NOUN
ejpam-5322	161	25	)	)	PUNCT
ejpam-5322	161	26	and	and	CCONJ
ejpam-5322	161	27	u	u	NOUN
ejpam-5322	161	28	is	be	AUX
ejpam-5322	161	29	a	a	DET
ejpam-5322	161	30	τ1τ2	τ1τ2	ADJ
ejpam-5322	161	31	-	-	ADJ
ejpam-5322	161	32	open	open	ADJ
ejpam-5322	161	33	neighbourhood	neighbourhood	NOUN
ejpam-5322	161	34	of	of	ADP
ejpam-5322	161	35	a	a	PRON
ejpam-5322	161	36	,	,	PUNCT
ejpam-5322	161	37	then	then	ADV
ejpam-5322	161	38	there	there	PRON
ejpam-5322	161	39	exists	exist	VERB
ejpam-5322	161	40	a	a	DET
ejpam-5322	161	41	τ1τ2	τ1τ2	NOUN
ejpam-5322	161	42	-	-	ADJ
ejpam-5322	161	43	open	open	ADJ
ejpam-5322	161	44	set	set	NOUN
ejpam-5322	161	45	v	v	NOUN
ejpam-5322	161	46	of	of	ADP
ejpam-5322	161	47	x	x	PUNCT
ejpam-5322	161	48	such	such	ADJ
ejpam-5322	161	49	that	that	SCONJ
ejpam-5322	161	50	a	a	DET
ejpam-5322	161	51	⊆	⊆	NUM
ejpam-5322	161	52	v	v	ADP
ejpam-5322	161	53	⊆	⊆	NUM
ejpam-5322	161	54	τ1τ2	τ1τ2	NOUN
ejpam-5322	161	55	-	-	NOUN
ejpam-5322	161	56	cl(v	cl(v	X
ejpam-5322	161	57	)	)	PUNCT
ejpam-5322	161	58	⊆	⊆	NUM
ejpam-5322	161	59	u	u	NOUN
ejpam-5322	161	60	.	.	PUNCT
ejpam-5322	162	1	lemma	lemma	PROPN
ejpam-5322	162	2	4	4	NUM
ejpam-5322	162	3	.	.	PUNCT
ejpam-5322	163	1	[	[	X
ejpam-5322	163	2	20	20	NUM
ejpam-5322	163	3	]	]	PUNCT
ejpam-5322	163	4	if	if	SCONJ
ejpam-5322	163	5	f	f	PROPN
ejpam-5322	163	6	:	:	PUNCT
ejpam-5322	163	7	(	(	PUNCT
ejpam-5322	163	8	x	x	NOUN
ejpam-5322	163	9	,	,	PUNCT
ejpam-5322	163	10	τ1	τ1	NOUN
ejpam-5322	163	11	,	,	PUNCT
ejpam-5322	163	12	τ2	τ2	NOUN
ejpam-5322	163	13	)	)	PUNCT
ejpam-5322	163	14	→	→	SYM
ejpam-5322	163	15	(	(	PUNCT
ejpam-5322	163	16	y	y	PROPN
ejpam-5322	163	17	,	,	PUNCT
ejpam-5322	163	18	σ1	σ1	PROPN
ejpam-5322	163	19	,	,	PUNCT
ejpam-5322	163	20	σ2	σ2	PROPN
ejpam-5322	163	21	)	)	PUNCT
ejpam-5322	163	22	is	be	AUX
ejpam-5322	163	23	a	a	DET
ejpam-5322	163	24	multifunction	multifunction	NOUN
ejpam-5322	163	25	such	such	ADJ
ejpam-5322	163	26	that	that	SCONJ
ejpam-5322	163	27	f	f	PROPN
ejpam-5322	163	28	(	(	PUNCT
ejpam-5322	163	29	x	x	X
ejpam-5322	163	30	)	)	PUNCT
ejpam-5322	163	31	is	be	AUX
ejpam-5322	163	32	τ1τ2regular	τ1τ2regular	NUM
ejpam-5322	163	33	and	and	CCONJ
ejpam-5322	163	34	τ1τ2	τ1τ2	NOUN
ejpam-5322	163	35	-	-	ADJ
ejpam-5322	163	36	paracompact	paracompact	ADJ
ejpam-5322	163	37	for	for	ADP
ejpam-5322	163	38	each	each	DET
ejpam-5322	163	39	x	x	SYM
ejpam-5322	163	40	∈	∈	PROPN
ejpam-5322	163	41	x	x	NOUN
ejpam-5322	163	42	,	,	PUNCT
ejpam-5322	163	43	then	then	ADV
ejpam-5322	163	44	clf+	clf+	PROPN
ejpam-5322	163	45	⊛	⊛	X
ejpam-5322	163	46	(	(	PUNCT
ejpam-5322	163	47	v	v	NOUN
ejpam-5322	163	48	)	)	PUNCT
ejpam-5322	163	49	=	=	PUNCT
ejpam-5322	163	50	f+(v	f+(v	NOUN
ejpam-5322	163	51	)	)	PUNCT
ejpam-5322	163	52	for	for	ADP
ejpam-5322	163	53	each	each	DET
ejpam-5322	163	54	σ1σ2	σ1σ2	VERB
ejpam-5322	163	55	-	-	ADJ
ejpam-5322	163	56	open	open	ADJ
ejpam-5322	163	57	set	set	NOUN
ejpam-5322	163	58	v	v	NOUN
ejpam-5322	163	59	of	of	ADP
ejpam-5322	163	60	y	y	PROPN
ejpam-5322	163	61	.	.	PUNCT
ejpam-5322	164	1	n.	n.	PROPN
ejpam-5322	164	2	viriyapong	viriyapong	PROPN
ejpam-5322	164	3	,	,	PUNCT
ejpam-5322	164	4	s.	s.	PROPN
ejpam-5322	164	5	sompong	sompong	PROPN
ejpam-5322	164	6	,	,	PUNCT
ejpam-5322	164	7	c.	c.	PROPN
ejpam-5322	164	8	boonpok	boonpok	PROPN
ejpam-5322	164	9	/	/	SYM
ejpam-5322	164	10	eur	eur	PROPN
ejpam-5322	164	11	.	.	PUNCT
ejpam-5322	165	1	j.	j.	PROPN
ejpam-5322	165	2	pure	pure	PROPN
ejpam-5322	165	3	appl	appl	PROPN
ejpam-5322	165	4	.	.	PROPN
ejpam-5322	165	5	math	math	PROPN
ejpam-5322	165	6	,	,	PUNCT
ejpam-5322	165	7	17	17	NUM
ejpam-5322	165	8	(	(	PUNCT
ejpam-5322	165	9	3	3	NUM
ejpam-5322	165	10	)	)	PUNCT
ejpam-5322	165	11	(	(	PUNCT
ejpam-5322	165	12	2024	2024	NUM
ejpam-5322	165	13	)	)	PUNCT
ejpam-5322	165	14	,	,	PUNCT
ejpam-5322	165	15	2210	2210	NUM
ejpam-5322	165	16	-	-	SYM
ejpam-5322	165	17	2220	2220	NUM
ejpam-5322	165	18	2216	2216	NUM
ejpam-5322	165	19	theorem	theorem	NOUN
ejpam-5322	165	20	3	3	X
ejpam-5322	165	21	.	.	PUNCT
ejpam-5322	166	1	let	let	VERB
ejpam-5322	166	2	f	f	NOUN
ejpam-5322	166	3	:	:	PUNCT
ejpam-5322	166	4	(	(	PUNCT
ejpam-5322	166	5	x	x	NOUN
ejpam-5322	166	6	,	,	PUNCT
ejpam-5322	166	7	τ1	τ1	NOUN
ejpam-5322	166	8	,	,	PUNCT
ejpam-5322	166	9	τ2	τ2	NOUN
ejpam-5322	166	10	)	)	PUNCT
ejpam-5322	166	11	→	→	SYM
ejpam-5322	166	12	(	(	PUNCT
ejpam-5322	166	13	y	y	PROPN
ejpam-5322	166	14	,	,	PUNCT
ejpam-5322	166	15	σ1	σ1	PROPN
ejpam-5322	166	16	,	,	PUNCT
ejpam-5322	166	17	σ2	σ2	PROPN
ejpam-5322	166	18	)	)	PUNCT
ejpam-5322	166	19	be	be	VERB
ejpam-5322	166	20	a	a	DET
ejpam-5322	166	21	multifunction	multifunction	NOUN
ejpam-5322	166	22	such	such	ADJ
ejpam-5322	166	23	that	that	SCONJ
ejpam-5322	166	24	f	f	PROPN
ejpam-5322	166	25	(	(	PUNCT
ejpam-5322	166	26	x	x	X
ejpam-5322	166	27	)	)	PUNCT
ejpam-5322	166	28	is	be	AUX
ejpam-5322	166	29	σ1σ2paracompact	σ1σ2paracompact	NUM
ejpam-5322	166	30	and	and	CCONJ
ejpam-5322	166	31	σ1σ2	σ1σ2	NOUN
ejpam-5322	166	32	-	-	ADJ
ejpam-5322	166	33	regular	regular	ADJ
ejpam-5322	166	34	for	for	ADP
ejpam-5322	166	35	each	each	DET
ejpam-5322	166	36	x	x	SYM
ejpam-5322	166	37	∈	∈	PROPN
ejpam-5322	166	38	x.	x.	NOUN
ejpam-5322	166	39	then	then	ADV
ejpam-5322	166	40	,	,	PUNCT
ejpam-5322	166	41	the	the	DET
ejpam-5322	166	42	following	follow	VERB
ejpam-5322	166	43	properties	property	NOUN
ejpam-5322	166	44	are	be	AUX
ejpam-5322	166	45	equivalent	equivalent	ADJ
ejpam-5322	166	46	:	:	PUNCT
ejpam-5322	166	47	(	(	PUNCT
ejpam-5322	166	48	1	1	X
ejpam-5322	166	49	)	)	PUNCT
ejpam-5322	166	50	f	f	PROPN
ejpam-5322	166	51	is	be	AUX
ejpam-5322	166	52	upper	upper	ADJ
ejpam-5322	166	53	s-(τ1	s-(τ1	PROPN
ejpam-5322	166	54	,	,	PUNCT
ejpam-5322	166	55	τ2)p	τ2)p	ADJ
ejpam-5322	166	56	-	-	ADJ
ejpam-5322	166	57	continuous	continuous	ADJ
ejpam-5322	166	58	;	;	PUNCT
ejpam-5322	166	59	(	(	PUNCT
ejpam-5322	166	60	2	2	X
ejpam-5322	166	61	)	)	PUNCT
ejpam-5322	166	62	clf⊛	clf⊛	PROPN
ejpam-5322	166	63	is	be	AUX
ejpam-5322	166	64	upper	upper	ADJ
ejpam-5322	166	65	s-(τ1	s-(τ1	PROPN
ejpam-5322	166	66	,	,	PUNCT
ejpam-5322	166	67	τ2)p	τ2)p	ADJ
ejpam-5322	166	68	-	-	ADJ
ejpam-5322	166	69	continuous	continuous	ADJ
ejpam-5322	166	70	.	.	PUNCT
ejpam-5322	167	1	proof	proof	NOUN
ejpam-5322	167	2	.	.	PUNCT
ejpam-5322	168	1	we	we	PRON
ejpam-5322	168	2	put	put	VERB
ejpam-5322	168	3	g	g	NOUN
ejpam-5322	168	4	=	=	PUNCT
ejpam-5322	168	5	clf⊛.	clf⊛.	NOUN
ejpam-5322	168	6	suppose	suppose	VERB
ejpam-5322	168	7	that	that	SCONJ
ejpam-5322	168	8	f	f	PROPN
ejpam-5322	168	9	is	be	AUX
ejpam-5322	168	10	upper	upper	ADJ
ejpam-5322	168	11	s-(τ1	s-(τ1	PROPN
ejpam-5322	168	12	,	,	PUNCT
ejpam-5322	168	13	τ2)p	τ2)p	ADJ
ejpam-5322	168	14	-	-	NOUN
ejpam-5322	168	15	continuous	continuous	ADJ
ejpam-5322	168	16	.	.	PUNCT
ejpam-5322	169	1	let	let	VERB
ejpam-5322	169	2	x	x	PUNCT
ejpam-5322	169	3	∈	∈	PROPN
ejpam-5322	169	4	x	x	X
ejpam-5322	169	5	and	and	CCONJ
ejpam-5322	169	6	v	v	X
ejpam-5322	169	7	be	be	AUX
ejpam-5322	169	8	any	any	DET
ejpam-5322	169	9	σ1σ2	σ1σ2	NOUN
ejpam-5322	169	10	-	-	ADJ
ejpam-5322	169	11	open	open	ADJ
ejpam-5322	169	12	set	set	NOUN
ejpam-5322	169	13	of	of	ADP
ejpam-5322	169	14	y	y	NOUN
ejpam-5322	169	15	containing	contain	VERB
ejpam-5322	169	16	g(x	g(x	NOUN
ejpam-5322	169	17	)	)	PUNCT
ejpam-5322	169	18	and	and	CCONJ
ejpam-5322	169	19	having	have	VERB
ejpam-5322	169	20	σ1σ2	σ1σ2	NOUN
ejpam-5322	169	21	-	-	PUNCT
ejpam-5322	169	22	connected	connect	VERB
ejpam-5322	169	23	complement	complement	NOUN
ejpam-5322	169	24	.	.	PUNCT
ejpam-5322	170	1	by	by	ADP
ejpam-5322	170	2	lemma	lemma	PROPN
ejpam-5322	170	3	4	4	NUM
ejpam-5322	170	4	,	,	PUNCT
ejpam-5322	170	5	we	we	PRON
ejpam-5322	170	6	have	have	VERB
ejpam-5322	170	7	x	x	X
ejpam-5322	170	8	∈	∈	PROPN
ejpam-5322	170	9	g+(v	g+(v	PROPN
ejpam-5322	170	10	)	)	PUNCT
ejpam-5322	170	11	=	=	PUNCT
ejpam-5322	171	1	f+(v	f+(v	NOUN
ejpam-5322	171	2	)	)	PUNCT
ejpam-5322	172	1	and	and	CCONJ
ejpam-5322	172	2	hence	hence	ADV
ejpam-5322	172	3	there	there	PRON
ejpam-5322	172	4	exists	exist	VERB
ejpam-5322	172	5	a	a	DET
ejpam-5322	172	6	τ1τ2open	τ1τ2open	ADJ
ejpam-5322	172	7	set	set	NOUN
ejpam-5322	172	8	u	u	NOUN
ejpam-5322	172	9	of	of	ADP
ejpam-5322	172	10	x	x	PUNCT
ejpam-5322	172	11	containing	contain	VERB
ejpam-5322	172	12	x	x	PUNCT
ejpam-5322	172	13	such	such	ADJ
ejpam-5322	172	14	that	that	SCONJ
ejpam-5322	172	15	f	f	PROPN
ejpam-5322	172	16	(	(	PUNCT
ejpam-5322	172	17	u	u	NOUN
ejpam-5322	172	18	)	)	PUNCT
ejpam-5322	172	19	⊆	⊆	NUM
ejpam-5322	172	20	v	v	NOUN
ejpam-5322	172	21	.	.	PUNCT
ejpam-5322	173	1	since	since	SCONJ
ejpam-5322	173	2	f	f	PROPN
ejpam-5322	173	3	(	(	PUNCT
ejpam-5322	173	4	z	z	NOUN
ejpam-5322	173	5	)	)	PUNCT
ejpam-5322	173	6	is	be	AUX
ejpam-5322	173	7	σ1σ2	σ1σ2	NOUN
ejpam-5322	173	8	-	-	ADJ
ejpam-5322	173	9	paracompact	paracompact	ADJ
ejpam-5322	173	10	and	and	CCONJ
ejpam-5322	173	11	σ1σ2	σ1σ2	NOUN
ejpam-5322	173	12	-	-	ADJ
ejpam-5322	173	13	regular	regular	ADJ
ejpam-5322	173	14	for	for	ADP
ejpam-5322	173	15	each	each	DET
ejpam-5322	173	16	z	z	NOUN
ejpam-5322	173	17	∈	∈	PROPN
ejpam-5322	173	18	u	u	NOUN
ejpam-5322	173	19	,	,	PUNCT
ejpam-5322	173	20	by	by	ADP
ejpam-5322	173	21	lemma	lemma	PROPN
ejpam-5322	173	22	3	3	NUM
ejpam-5322	173	23	there	there	ADV
ejpam-5322	173	24	exists	exist	VERB
ejpam-5322	173	25	a	a	DET
ejpam-5322	173	26	τ1τ2	τ1τ2	NOUN
ejpam-5322	173	27	-	-	ADJ
ejpam-5322	173	28	open	open	ADJ
ejpam-5322	173	29	set	set	NOUN
ejpam-5322	173	30	w	w	NOUN
ejpam-5322	173	31	of	of	ADP
ejpam-5322	173	32	x	x	SYM
ejpam-5322	173	33	such	such	ADJ
ejpam-5322	173	34	that	that	SCONJ
ejpam-5322	173	35	f	f	PROPN
ejpam-5322	173	36	(	(	PUNCT
ejpam-5322	173	37	z	z	NOUN
ejpam-5322	173	38	)	)	PUNCT
ejpam-5322	173	39	⊆	⊆	NUM
ejpam-5322	173	40	w	w	ADP
ejpam-5322	173	41	⊆	⊆	NUM
ejpam-5322	173	42	σ1σ2	σ1σ2	NOUN
ejpam-5322	173	43	-	-	PUNCT
ejpam-5322	173	44	cl(w	cl(w	NOUN
ejpam-5322	173	45	)	)	PUNCT
ejpam-5322	173	46	⊆	⊆	NUM
ejpam-5322	173	47	v	v	NOUN
ejpam-5322	173	48	;	;	PUNCT
ejpam-5322	173	49	hence	hence	ADV
ejpam-5322	173	50	g(z	g(z	ADJ
ejpam-5322	173	51	)	)	PUNCT
ejpam-5322	173	52	⊆	⊆	NUM
ejpam-5322	173	53	σ1σ2	σ1σ2	NOUN
ejpam-5322	173	54	-	-	PUNCT
ejpam-5322	173	55	cl(w	cl(w	NOUN
ejpam-5322	173	56	)	)	PUNCT
ejpam-5322	173	57	⊆	⊆	NUM
ejpam-5322	173	58	v	v	NOUN
ejpam-5322	173	59	for	for	ADP
ejpam-5322	173	60	each	each	DET
ejpam-5322	173	61	z	z	NOUN
ejpam-5322	173	62	∈	∈	PROPN
ejpam-5322	173	63	u	u	NOUN
ejpam-5322	173	64	.	.	PUNCT
ejpam-5322	174	1	thus	thus	ADV
ejpam-5322	174	2	,	,	PUNCT
ejpam-5322	174	3	g(u	g(u	PROPN
ejpam-5322	174	4	)	)	PUNCT
ejpam-5322	174	5	⊆	⊆	NUM
ejpam-5322	174	6	v	v	NOUN
ejpam-5322	174	7	and	and	CCONJ
ejpam-5322	174	8	hence	hence	ADV
ejpam-5322	174	9	g	g	PROPN
ejpam-5322	174	10	is	be	AUX
ejpam-5322	174	11	upper	upper	ADJ
ejpam-5322	174	12	s-(τ1	s-(τ1	PROPN
ejpam-5322	174	13	,	,	PUNCT
ejpam-5322	174	14	τ2)p	τ2)p	ADJ
ejpam-5322	174	15	-	-	ADJ
ejpam-5322	174	16	continuous	continuous	ADJ
ejpam-5322	174	17	.	.	PUNCT
ejpam-5322	175	1	conversely	conversely	ADV
ejpam-5322	175	2	,	,	PUNCT
ejpam-5322	175	3	suppose	suppose	VERB
ejpam-5322	175	4	that	that	SCONJ
ejpam-5322	175	5	g	g	PROPN
ejpam-5322	175	6	is	be	AUX
ejpam-5322	175	7	upper	upper	ADJ
ejpam-5322	175	8	s-(τ1	s-(τ1	PROPN
ejpam-5322	175	9	,	,	PUNCT
ejpam-5322	175	10	τ2)p	τ2)p	ADJ
ejpam-5322	175	11	-	-	NOUN
ejpam-5322	175	12	continuous	continuous	ADJ
ejpam-5322	175	13	.	.	PUNCT
ejpam-5322	176	1	let	let	VERB
ejpam-5322	176	2	x	x	PUNCT
ejpam-5322	176	3	∈	∈	PROPN
ejpam-5322	176	4	x	x	X
ejpam-5322	176	5	and	and	CCONJ
ejpam-5322	176	6	v	v	X
ejpam-5322	176	7	be	be	AUX
ejpam-5322	176	8	any	any	DET
ejpam-5322	176	9	σ1σ2	σ1σ2	NOUN
ejpam-5322	176	10	-	-	ADJ
ejpam-5322	176	11	open	open	ADJ
ejpam-5322	176	12	set	set	NOUN
ejpam-5322	176	13	of	of	ADP
ejpam-5322	176	14	y	y	PROPN
ejpam-5322	176	15	containing	contain	VERB
ejpam-5322	176	16	f	f	PROPN
ejpam-5322	176	17	(	(	PUNCT
ejpam-5322	176	18	x	x	NOUN
ejpam-5322	176	19	)	)	PUNCT
ejpam-5322	176	20	and	and	CCONJ
ejpam-5322	176	21	having	have	VERB
ejpam-5322	176	22	σ1σ2	σ1σ2	NOUN
ejpam-5322	176	23	-	-	PUNCT
ejpam-5322	176	24	connected	connect	VERB
ejpam-5322	176	25	complement	complement	NOUN
ejpam-5322	176	26	.	.	PUNCT
ejpam-5322	177	1	by	by	ADP
ejpam-5322	177	2	lemma	lemma	PROPN
ejpam-5322	177	3	4	4	NUM
ejpam-5322	177	4	,	,	PUNCT
ejpam-5322	177	5	we	we	PRON
ejpam-5322	177	6	have	have	VERB
ejpam-5322	177	7	x	x	X
ejpam-5322	177	8	∈	∈	NOUN
ejpam-5322	177	9	f+(v	f+(v	NOUN
ejpam-5322	177	10	)	)	PUNCT
ejpam-5322	178	1	=	=	PUNCT
ejpam-5322	178	2	g+(v	g+(v	PROPN
ejpam-5322	178	3	)	)	PUNCT
ejpam-5322	178	4	and	and	CCONJ
ejpam-5322	178	5	hence	hence	ADV
ejpam-5322	178	6	g(x	g(x	NOUN
ejpam-5322	178	7	)	)	PUNCT
ejpam-5322	178	8	⊆	⊆	NUM
ejpam-5322	178	9	v	v	NOUN
ejpam-5322	178	10	.	.	PUNCT
ejpam-5322	179	1	there	there	PRON
ejpam-5322	179	2	exists	exist	VERB
ejpam-5322	179	3	a	a	DET
ejpam-5322	179	4	τ1τ2	τ1τ2	NOUN
ejpam-5322	179	5	-	-	ADJ
ejpam-5322	179	6	open	open	ADJ
ejpam-5322	179	7	set	set	ADJ
ejpam-5322	179	8	u	u	NOUN
ejpam-5322	179	9	of	of	ADP
ejpam-5322	179	10	x	x	PUNCT
ejpam-5322	179	11	containing	contain	VERB
ejpam-5322	179	12	x	x	PUNCT
ejpam-5322	179	13	such	such	ADJ
ejpam-5322	179	14	that	that	SCONJ
ejpam-5322	179	15	g(u	g(u	PROPN
ejpam-5322	179	16	)	)	PUNCT
ejpam-5322	179	17	⊆	⊆	NUM
ejpam-5322	179	18	v	v	NOUN
ejpam-5322	179	19	.	.	PUNCT
ejpam-5322	180	1	thus	thus	ADV
ejpam-5322	180	2	,	,	PUNCT
ejpam-5322	180	3	u	u	PROPN
ejpam-5322	180	4	⊆	⊆	NUM
ejpam-5322	180	5	g+(v	g+(v	PROPN
ejpam-5322	180	6	)	)	PUNCT
ejpam-5322	180	7	=	=	PUNCT
ejpam-5322	181	1	f+(v	f+(v	NOUN
ejpam-5322	181	2	)	)	PUNCT
ejpam-5322	182	1	and	and	CCONJ
ejpam-5322	182	2	so	so	ADV
ejpam-5322	182	3	f	f	PROPN
ejpam-5322	182	4	(	(	PUNCT
ejpam-5322	182	5	u	u	NOUN
ejpam-5322	182	6	)	)	PUNCT
ejpam-5322	182	7	⊆	⊆	NUM
ejpam-5322	182	8	v	v	NOUN
ejpam-5322	182	9	.	.	PUNCT
ejpam-5322	183	1	this	this	PRON
ejpam-5322	183	2	shows	show	VERB
ejpam-5322	183	3	that	that	SCONJ
ejpam-5322	183	4	f	f	PROPN
ejpam-5322	183	5	is	be	AUX
ejpam-5322	183	6	upper	upper	ADJ
ejpam-5322	183	7	s-(τ1	s-(τ1	PROPN
ejpam-5322	183	8	,	,	PUNCT
ejpam-5322	183	9	τ2)p	τ2)p	ADJ
ejpam-5322	183	10	-	-	NOUN
ejpam-5322	183	11	continuous	continuous	ADJ
ejpam-5322	183	12	.	.	PUNCT
ejpam-5322	184	1	lemma	lemma	PROPN
ejpam-5322	184	2	5	5	NUM
ejpam-5322	184	3	.	.	PUNCT
ejpam-5322	185	1	[	[	X
ejpam-5322	185	2	20	20	NUM
ejpam-5322	185	3	]	]	PUNCT
ejpam-5322	185	4	for	for	ADP
ejpam-5322	185	5	a	a	DET
ejpam-5322	185	6	multifunction	multifunction	NOUN
ejpam-5322	185	7	f	f	NOUN
ejpam-5322	185	8	:	:	PUNCT
ejpam-5322	185	9	(	(	PUNCT
ejpam-5322	185	10	x	x	NOUN
ejpam-5322	185	11	,	,	PUNCT
ejpam-5322	185	12	τ1	τ1	NOUN
ejpam-5322	185	13	,	,	PUNCT
ejpam-5322	185	14	τ2	τ2	NOUN
ejpam-5322	185	15	)	)	PUNCT
ejpam-5322	185	16	→	→	SYM
ejpam-5322	185	17	(	(	PUNCT
ejpam-5322	185	18	y	y	PROPN
ejpam-5322	185	19	,	,	PUNCT
ejpam-5322	185	20	σ1	σ1	PROPN
ejpam-5322	185	21	,	,	PUNCT
ejpam-5322	185	22	σ2	σ2	NOUN
ejpam-5322	185	23	)	)	PUNCT
ejpam-5322	185	24	,	,	PUNCT
ejpam-5322	185	25	clf	clf	PROPN
ejpam-5322	185	26	−	−	PROPN
ejpam-5322	185	27	⊛	⊛	NUM
ejpam-5322	185	28	(	(	PUNCT
ejpam-5322	185	29	v	v	NOUN
ejpam-5322	185	30	)	)	PUNCT
ejpam-5322	185	31	=	=	SYM
ejpam-5322	185	32	f−(v	f−(v	ADJ
ejpam-5322	185	33	)	)	PUNCT
ejpam-5322	185	34	for	for	ADP
ejpam-5322	185	35	each	each	DET
ejpam-5322	185	36	σ1σ2	σ1σ2	VERB
ejpam-5322	185	37	-	-	ADJ
ejpam-5322	185	38	open	open	ADJ
ejpam-5322	185	39	set	set	NOUN
ejpam-5322	185	40	v	v	NOUN
ejpam-5322	185	41	of	of	ADP
ejpam-5322	185	42	y	y	PROPN
ejpam-5322	185	43	.	.	PUNCT
ejpam-5322	186	1	theorem	theorem	ADJ
ejpam-5322	186	2	4	4	NUM
ejpam-5322	186	3	.	.	X
ejpam-5322	186	4	for	for	ADP
ejpam-5322	186	5	a	a	DET
ejpam-5322	186	6	multifunction	multifunction	NOUN
ejpam-5322	187	1	f	f	NOUN
ejpam-5322	187	2	:	:	PUNCT
ejpam-5322	187	3	(	(	PUNCT
ejpam-5322	187	4	x	x	NOUN
ejpam-5322	187	5	,	,	PUNCT
ejpam-5322	187	6	τ1	τ1	NOUN
ejpam-5322	187	7	,	,	PUNCT
ejpam-5322	187	8	τ2	τ2	NOUN
ejpam-5322	187	9	)	)	PUNCT
ejpam-5322	187	10	→	→	SYM
ejpam-5322	187	11	(	(	PUNCT
ejpam-5322	187	12	y	y	PROPN
ejpam-5322	187	13	,	,	PUNCT
ejpam-5322	187	14	σ1	σ1	PROPN
ejpam-5322	187	15	,	,	PUNCT
ejpam-5322	187	16	σ2	σ2	NOUN
ejpam-5322	187	17	)	)	PUNCT
ejpam-5322	187	18	,	,	PUNCT
ejpam-5322	187	19	the	the	DET
ejpam-5322	187	20	following	follow	VERB
ejpam-5322	187	21	properties	property	NOUN
ejpam-5322	187	22	are	be	AUX
ejpam-5322	187	23	equivalent	equivalent	ADJ
ejpam-5322	187	24	:	:	PUNCT
ejpam-5322	187	25	(	(	PUNCT
ejpam-5322	187	26	1	1	X
ejpam-5322	187	27	)	)	PUNCT
ejpam-5322	187	28	f	f	PROPN
ejpam-5322	187	29	is	be	AUX
ejpam-5322	187	30	lower	low	ADJ
ejpam-5322	187	31	s-(τ1	s-(τ1	NOUN
ejpam-5322	187	32	,	,	PUNCT
ejpam-5322	187	33	τ2)p	τ2)p	ADJ
ejpam-5322	187	34	-	-	ADJ
ejpam-5322	187	35	continuous	continuous	ADJ
ejpam-5322	187	36	;	;	PUNCT
ejpam-5322	187	37	(	(	PUNCT
ejpam-5322	187	38	2	2	X
ejpam-5322	187	39	)	)	PUNCT
ejpam-5322	187	40	clf⊛	clf⊛	PROPN
ejpam-5322	187	41	is	be	AUX
ejpam-5322	187	42	lower	low	ADJ
ejpam-5322	187	43	s-(τ1	s-(τ1	NOUN
ejpam-5322	187	44	,	,	PUNCT
ejpam-5322	187	45	τ2)p	τ2)p	ADJ
ejpam-5322	187	46	-	-	ADJ
ejpam-5322	187	47	continuous	continuous	ADJ
ejpam-5322	187	48	.	.	PUNCT
ejpam-5322	188	1	proof	proof	NOUN
ejpam-5322	188	2	.	.	PUNCT
ejpam-5322	189	1	by	by	ADP
ejpam-5322	189	2	using	use	VERB
ejpam-5322	189	3	lemma	lemma	PROPN
ejpam-5322	189	4	5	5	NUM
ejpam-5322	189	5	this	this	PRON
ejpam-5322	189	6	can	can	AUX
ejpam-5322	189	7	be	be	AUX
ejpam-5322	189	8	shown	show	VERB
ejpam-5322	189	9	similarly	similarly	ADV
ejpam-5322	189	10	to	to	ADP
ejpam-5322	189	11	that	that	PRON
ejpam-5322	189	12	of	of	ADP
ejpam-5322	189	13	theorem	theorem	NOUN
ejpam-5322	189	14	3	3	NUM
ejpam-5322	189	15	.	.	PUNCT
ejpam-5322	190	1	the	the	DET
ejpam-5322	190	2	(	(	PUNCT
ejpam-5322	190	3	τ1	τ1	NOUN
ejpam-5322	190	4	,	,	PUNCT
ejpam-5322	190	5	τ2)p	τ2)p	ADJ
ejpam-5322	190	6	-	-	PUNCT
ejpam-5322	190	7	frontier	frontier	NOUN
ejpam-5322	190	8	of	of	ADP
ejpam-5322	190	9	a	a	DET
ejpam-5322	190	10	subset	subset	NOUN
ejpam-5322	190	11	a	a	PRON
ejpam-5322	190	12	of	of	ADP
ejpam-5322	190	13	a	a	DET
ejpam-5322	190	14	bitopological	bitopological	ADJ
ejpam-5322	190	15	space	space	NOUN
ejpam-5322	190	16	(	(	PUNCT
ejpam-5322	190	17	x	x	NOUN
ejpam-5322	190	18	,	,	PUNCT
ejpam-5322	190	19	τ1	τ1	NOUN
ejpam-5322	190	20	,	,	PUNCT
ejpam-5322	190	21	τ2	τ2	PROPN
ejpam-5322	190	22	)	)	PUNCT
ejpam-5322	190	23	,	,	PUNCT
ejpam-5322	190	24	denoted	denote	VERB
ejpam-5322	190	25	by	by	ADP
ejpam-5322	190	26	(	(	PUNCT
ejpam-5322	190	27	τ1	τ1	NOUN
ejpam-5322	190	28	,	,	PUNCT
ejpam-5322	190	29	τ2)-pfr(a	τ2)-pfr(a	NOUN
ejpam-5322	190	30	)	)	PUNCT
ejpam-5322	190	31	,	,	PUNCT
ejpam-5322	190	32	is	be	AUX
ejpam-5322	190	33	defined	define	VERB
ejpam-5322	190	34	by	by	ADP
ejpam-5322	190	35	(	(	PUNCT
ejpam-5322	190	36	τ1	τ1	NOUN
ejpam-5322	190	37	,	,	PUNCT
ejpam-5322	190	38	τ2)-pfr(a	τ2)-pfr(a	ADJ
ejpam-5322	190	39	)	)	PUNCT
ejpam-5322	190	40	=	=	SYM
ejpam-5322	191	1	(	(	PUNCT
ejpam-5322	191	2	τ1	τ1	PROPN
ejpam-5322	191	3	,	,	PUNCT
ejpam-5322	191	4	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-5322	191	5	)	)	PUNCT
ejpam-5322	191	6	∩	∩	NOUN
ejpam-5322	191	7	(	(	PUNCT
ejpam-5322	191	8	τ1	τ1	PROPN
ejpam-5322	191	9	,	,	PUNCT
ejpam-5322	191	10	τ2)-pcl(x	τ2)-pcl(x	NOUN
ejpam-5322	191	11	−a	−a	NOUN
ejpam-5322	191	12	)	)	PUNCT
ejpam-5322	191	13	=	=	PUNCT
ejpam-5322	191	14	(	(	PUNCT
ejpam-5322	191	15	τ1	τ1	NOUN
ejpam-5322	191	16	,	,	PUNCT
ejpam-5322	191	17	τ2)-pcl(a)−	τ2)-pcl(a)−	NOUN
ejpam-5322	191	18	(	(	PUNCT
ejpam-5322	191	19	τ1	τ1	NOUN
ejpam-5322	191	20	,	,	PUNCT
ejpam-5322	191	21	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-5322	191	22	)	)	PUNCT
ejpam-5322	191	23	.	.	PUNCT
ejpam-5322	192	1	theorem	theorem	NOUN
ejpam-5322	192	2	5	5	NUM
ejpam-5322	192	3	.	.	PUNCT
ejpam-5322	193	1	the	the	DET
ejpam-5322	193	2	set	set	NOUN
ejpam-5322	193	3	of	of	ADP
ejpam-5322	193	4	all	all	DET
ejpam-5322	193	5	points	point	NOUN
ejpam-5322	193	6	x	x	PUNCT
ejpam-5322	193	7	of	of	ADP
ejpam-5322	193	8	x	x	SYM
ejpam-5322	193	9	at	at	ADP
ejpam-5322	193	10	which	which	PRON
ejpam-5322	193	11	a	a	DET
ejpam-5322	193	12	multifunction	multifunction	NOUN
ejpam-5322	194	1	f	f	NOUN
ejpam-5322	194	2	:	:	PUNCT
ejpam-5322	194	3	(	(	PUNCT
ejpam-5322	194	4	x	x	NOUN
ejpam-5322	194	5	,	,	PUNCT
ejpam-5322	194	6	τ1	τ1	NOUN
ejpam-5322	194	7	,	,	PUNCT
ejpam-5322	194	8	τ2	τ2	NOUN
ejpam-5322	194	9	)	)	PUNCT
ejpam-5322	194	10	→	→	SYM
ejpam-5322	194	11	(	(	PUNCT
ejpam-5322	194	12	y	y	PROPN
ejpam-5322	194	13	,	,	PUNCT
ejpam-5322	194	14	σ1	σ1	PROPN
ejpam-5322	194	15	,	,	PUNCT
ejpam-5322	194	16	σ2	σ2	PROPN
ejpam-5322	194	17	)	)	PUNCT
ejpam-5322	194	18	is	be	AUX
ejpam-5322	194	19	not	not	PART
ejpam-5322	194	20	upper	upper	ADJ
ejpam-5322	194	21	s-(τ1	s-(τ1	NOUN
ejpam-5322	194	22	,	,	PUNCT
ejpam-5322	194	23	τ2)p	τ2)p	ADJ
ejpam-5322	194	24	-	-	ADJ
ejpam-5322	194	25	continuous	continuous	ADJ
ejpam-5322	194	26	is	be	AUX
ejpam-5322	194	27	identical	identical	ADJ
ejpam-5322	194	28	with	with	ADP
ejpam-5322	194	29	the	the	DET
ejpam-5322	194	30	union	union	NOUN
ejpam-5322	194	31	of	of	ADP
ejpam-5322	194	32	the	the	DET
ejpam-5322	194	33	(	(	PUNCT
ejpam-5322	194	34	τ1	τ1	NOUN
ejpam-5322	194	35	,	,	PUNCT
ejpam-5322	194	36	τ2)p	τ2)p	ADJ
ejpam-5322	194	37	-	-	PUNCT
ejpam-5322	194	38	frontier	frontier	NOUN
ejpam-5322	194	39	of	of	ADP
ejpam-5322	194	40	the	the	DET
ejpam-5322	194	41	upper	upper	ADJ
ejpam-5322	194	42	inverse	inverse	NOUN
ejpam-5322	194	43	images	image	NOUN
ejpam-5322	194	44	of	of	ADP
ejpam-5322	194	45	the	the	DET
ejpam-5322	194	46	σ1σ2	σ1σ2	NOUN
ejpam-5322	194	47	-	-	PUNCT
ejpam-5322	194	48	closures	closure	NOUN
ejpam-5322	194	49	of	of	ADP
ejpam-5322	194	50	σ1σ2	σ1σ2	NOUN
ejpam-5322	194	51	-	-	PUNCT
ejpam-5322	194	52	open	open	ADJ
ejpam-5322	194	53	sets	set	NOUN
ejpam-5322	194	54	containing	contain	VERB
ejpam-5322	194	55	f	f	X
ejpam-5322	194	56	(	(	PUNCT
ejpam-5322	194	57	x	x	NOUN
ejpam-5322	194	58	)	)	PUNCT
ejpam-5322	194	59	and	and	CCONJ
ejpam-5322	194	60	having	have	VERB
ejpam-5322	194	61	σ1σ2	σ1σ2	NOUN
ejpam-5322	194	62	-	-	PUNCT
ejpam-5322	194	63	connected	connect	VERB
ejpam-5322	194	64	complement	complement	NOUN
ejpam-5322	194	65	.	.	PUNCT
ejpam-5322	195	1	n.	n.	PROPN
ejpam-5322	195	2	viriyapong	viriyapong	PROPN
ejpam-5322	195	3	,	,	PUNCT
ejpam-5322	195	4	s.	s.	PROPN
ejpam-5322	195	5	sompong	sompong	PROPN
ejpam-5322	195	6	,	,	PUNCT
ejpam-5322	195	7	c.	c.	PROPN
ejpam-5322	195	8	boonpok	boonpok	PROPN
ejpam-5322	195	9	/	/	SYM
ejpam-5322	195	10	eur	eur	PROPN
ejpam-5322	195	11	.	.	PUNCT
ejpam-5322	196	1	j.	j.	PROPN
ejpam-5322	196	2	pure	pure	PROPN
ejpam-5322	196	3	appl	appl	PROPN
ejpam-5322	196	4	.	.	PROPN
ejpam-5322	196	5	math	math	PROPN
ejpam-5322	196	6	,	,	PUNCT
ejpam-5322	196	7	17	17	NUM
ejpam-5322	196	8	(	(	PUNCT
ejpam-5322	196	9	3	3	NUM
ejpam-5322	196	10	)	)	PUNCT
ejpam-5322	196	11	(	(	PUNCT
ejpam-5322	196	12	2024	2024	NUM
ejpam-5322	196	13	)	)	PUNCT
ejpam-5322	196	14	,	,	PUNCT
ejpam-5322	196	15	2210	2210	NUM
ejpam-5322	196	16	-	-	SYM
ejpam-5322	196	17	2220	2220	NUM
ejpam-5322	196	18	2217	2217	NUM
ejpam-5322	196	19	proof	proof	NOUN
ejpam-5322	196	20	.	.	PUNCT
ejpam-5322	196	21	suppose	suppose	VERB
ejpam-5322	196	22	that	that	SCONJ
ejpam-5322	196	23	f	f	PROPN
ejpam-5322	196	24	is	be	AUX
ejpam-5322	196	25	not	not	PART
ejpam-5322	196	26	upper	upper	ADJ
ejpam-5322	196	27	s-(τ1	s-(τ1	NOUN
ejpam-5322	196	28	,	,	PUNCT
ejpam-5322	196	29	τ2)p	τ2)p	ADJ
ejpam-5322	196	30	-	-	ADJ
ejpam-5322	196	31	continuous	continuous	ADJ
ejpam-5322	196	32	at	at	ADP
ejpam-5322	196	33	x	x	SYM
ejpam-5322	196	34	∈	∈	PROPN
ejpam-5322	196	35	x.	x.	NOUN
ejpam-5322	196	36	then	then	ADV
ejpam-5322	196	37	,	,	PUNCT
ejpam-5322	196	38	there	there	PRON
ejpam-5322	196	39	exists	exist	VERB
ejpam-5322	196	40	a	a	DET
ejpam-5322	196	41	σ1σ2	σ1σ2	NUM
ejpam-5322	196	42	-	-	ADJ
ejpam-5322	196	43	open	open	ADJ
ejpam-5322	196	44	set	set	NOUN
ejpam-5322	196	45	v	v	NOUN
ejpam-5322	196	46	of	of	ADP
ejpam-5322	196	47	y	y	PROPN
ejpam-5322	196	48	containing	contain	VERB
ejpam-5322	196	49	f	f	PROPN
ejpam-5322	196	50	(	(	PUNCT
ejpam-5322	196	51	x	x	NOUN
ejpam-5322	196	52	)	)	PUNCT
ejpam-5322	196	53	and	and	CCONJ
ejpam-5322	196	54	having	have	VERB
ejpam-5322	196	55	σ1σ2	σ1σ2	NOUN
ejpam-5322	196	56	-	-	PUNCT
ejpam-5322	196	57	connected	connected	ADJ
ejpam-5322	196	58	complement	complement	NOUN
ejpam-5322	196	59	such	such	ADJ
ejpam-5322	196	60	that	that	SCONJ
ejpam-5322	196	61	u	u	PROPN
ejpam-5322	196	62	∩	∩	NOUN
ejpam-5322	196	63	(	(	PUNCT
ejpam-5322	196	64	x−f+(v	x−f+(v	PROPN
ejpam-5322	196	65	)	)	PUNCT
ejpam-5322	196	66	)	)	PUNCT
ejpam-5322	197	1	̸=	̸=	NOUN
ejpam-5322	197	2	∅	∅	NOUN
ejpam-5322	197	3	for	for	ADP
ejpam-5322	197	4	every	every	DET
ejpam-5322	197	5	(	(	PUNCT
ejpam-5322	197	6	τ1	τ1	NOUN
ejpam-5322	197	7	,	,	PUNCT
ejpam-5322	197	8	τ2)p	τ2)p	ADJ
ejpam-5322	197	9	-	-	PUNCT
ejpam-5322	197	10	open	open	ADJ
ejpam-5322	197	11	set	set	NOUN
ejpam-5322	197	12	u	u	NOUN
ejpam-5322	197	13	of	of	ADP
ejpam-5322	197	14	x	x	SYM
ejpam-5322	197	15	containing	contain	VERB
ejpam-5322	197	16	x.	x.	NOUN
ejpam-5322	197	17	therefore	therefore	ADV
ejpam-5322	197	18	,	,	PUNCT
ejpam-5322	197	19	we	we	PRON
ejpam-5322	197	20	have	have	VERB
ejpam-5322	197	21	x	x	PROPN
ejpam-5322	197	22	∈	∈	PROPN
ejpam-5322	197	23	(	(	PUNCT
ejpam-5322	197	24	τ1	τ1	PROPN
ejpam-5322	197	25	,	,	PUNCT
ejpam-5322	197	26	τ2)-pcl(x	τ2)-pcl(x	NOUN
ejpam-5322	197	27	−	−	PROPN
ejpam-5322	197	28	f+(v	f+(v	NOUN
ejpam-5322	197	29	)	)	PUNCT
ejpam-5322	197	30	)	)	PUNCT
ejpam-5322	197	31	.	.	PUNCT
ejpam-5322	198	1	on	on	ADP
ejpam-5322	198	2	the	the	DET
ejpam-5322	198	3	other	other	ADJ
ejpam-5322	198	4	hand	hand	NOUN
ejpam-5322	198	5	,	,	PUNCT
ejpam-5322	198	6	we	we	PRON
ejpam-5322	198	7	have	have	VERB
ejpam-5322	198	8	x	x	X
ejpam-5322	198	9	∈	∈	NOUN
ejpam-5322	198	10	f+(v	f+(v	NOUN
ejpam-5322	198	11	)	)	PUNCT
ejpam-5322	199	1	⊆	⊆	NUM
ejpam-5322	199	2	(	(	PUNCT
ejpam-5322	199	3	τ1	τ1	NOUN
ejpam-5322	199	4	,	,	PUNCT
ejpam-5322	199	5	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5322	199	6	+	+	PROPN
ejpam-5322	199	7	(	(	PUNCT
ejpam-5322	199	8	v	v	NOUN
ejpam-5322	199	9	)	)	PUNCT
ejpam-5322	199	10	)	)	PUNCT
ejpam-5322	199	11	and	and	CCONJ
ejpam-5322	199	12	hence	hence	ADV
ejpam-5322	199	13	x	x	X
ejpam-5322	199	14	∈	∈	PROPN
ejpam-5322	199	15	(	(	PUNCT
ejpam-5322	199	16	τ1	τ1	NOUN
ejpam-5322	199	17	,	,	PUNCT
ejpam-5322	199	18	τ2)-pfr(f	τ2)-pfr(f	VERB
ejpam-5322	199	19	+	+	PROPN
ejpam-5322	199	20	(	(	PUNCT
ejpam-5322	199	21	v	v	NOUN
ejpam-5322	199	22	)	)	PUNCT
ejpam-5322	199	23	)	)	PUNCT
ejpam-5322	199	24	.	.	PUNCT
ejpam-5322	200	1	conversely	conversely	ADV
ejpam-5322	200	2	,	,	PUNCT
ejpam-5322	200	3	suppose	suppose	VERB
ejpam-5322	200	4	that	that	SCONJ
ejpam-5322	200	5	v	v	NOUN
ejpam-5322	200	6	is	be	AUX
ejpam-5322	200	7	a	a	DET
ejpam-5322	200	8	σ1σ2	σ1σ2	NOUN
ejpam-5322	200	9	-	-	ADJ
ejpam-5322	200	10	open	open	ADJ
ejpam-5322	200	11	set	set	NOUN
ejpam-5322	200	12	of	of	ADP
ejpam-5322	200	13	y	y	PROPN
ejpam-5322	200	14	containing	contain	VERB
ejpam-5322	200	15	f	f	PROPN
ejpam-5322	200	16	(	(	PUNCT
ejpam-5322	200	17	x	x	NOUN
ejpam-5322	200	18	)	)	PUNCT
ejpam-5322	200	19	and	and	CCONJ
ejpam-5322	200	20	having	have	VERB
ejpam-5322	200	21	σ1σ2	σ1σ2	NOUN
ejpam-5322	200	22	-	-	PUNCT
ejpam-5322	200	23	connected	connected	ADJ
ejpam-5322	200	24	complement	complement	NOUN
ejpam-5322	200	25	such	such	ADJ
ejpam-5322	200	26	that	that	SCONJ
ejpam-5322	200	27	x	x	SYM
ejpam-5322	200	28	∈	∈	PROPN
ejpam-5322	200	29	(	(	PUNCT
ejpam-5322	200	30	τ1	τ1	NOUN
ejpam-5322	200	31	,	,	PUNCT
ejpam-5322	200	32	τ2)-pfr(f	τ2)-pfr(f	VERB
ejpam-5322	200	33	+	+	PROPN
ejpam-5322	200	34	(	(	PUNCT
ejpam-5322	200	35	v	v	NOUN
ejpam-5322	200	36	)	)	PUNCT
ejpam-5322	200	37	)	)	PUNCT
ejpam-5322	200	38	.	.	PUNCT
ejpam-5322	201	1	if	if	SCONJ
ejpam-5322	201	2	f	f	PROPN
ejpam-5322	201	3	is	be	AUX
ejpam-5322	201	4	upper	upper	ADJ
ejpam-5322	201	5	s-(τ1	s-(τ1	PROPN
ejpam-5322	201	6	,	,	PUNCT
ejpam-5322	201	7	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-5322	201	8	at	at	ADP
ejpam-5322	201	9	x	x	X
ejpam-5322	201	10	∈	∈	PROPN
ejpam-5322	201	11	x	x	NOUN
ejpam-5322	201	12	,	,	PUNCT
ejpam-5322	201	13	there	there	PRON
ejpam-5322	201	14	exists	exist	VERB
ejpam-5322	201	15	a	a	DET
ejpam-5322	201	16	(	(	PUNCT
ejpam-5322	201	17	τ1	τ1	NOUN
ejpam-5322	201	18	,	,	PUNCT
ejpam-5322	201	19	τ2)p	τ2)p	ADJ
ejpam-5322	201	20	-	-	PUNCT
ejpam-5322	201	21	open	open	ADJ
ejpam-5322	201	22	set	set	NOUN
ejpam-5322	201	23	u	u	NOUN
ejpam-5322	201	24	of	of	ADP
ejpam-5322	201	25	x	x	PUNCT
ejpam-5322	201	26	containing	contain	VERB
ejpam-5322	201	27	x	x	PUNCT
ejpam-5322	201	28	such	such	ADJ
ejpam-5322	201	29	that	that	SCONJ
ejpam-5322	201	30	u	u	NOUN
ejpam-5322	201	31	⊆	⊆	NUM
ejpam-5322	201	32	f+(v	f+(v	NOUN
ejpam-5322	201	33	)	)	PUNCT
ejpam-5322	201	34	;	;	PUNCT
ejpam-5322	201	35	hence	hence	ADV
ejpam-5322	201	36	x	x	X
ejpam-5322	201	37	∈	∈	PROPN
ejpam-5322	201	38	(	(	PUNCT
ejpam-5322	201	39	τ1	τ1	NOUN
ejpam-5322	201	40	,	,	PUNCT
ejpam-5322	201	41	τ2)-pint(f	τ2)-pint(f	PUNCT
ejpam-5322	201	42	+	+	ADJ
ejpam-5322	201	43	(	(	PUNCT
ejpam-5322	201	44	v	v	NOUN
ejpam-5322	201	45	)	)	PUNCT
ejpam-5322	201	46	)	)	PUNCT
ejpam-5322	201	47	.	.	PUNCT
ejpam-5322	202	1	this	this	PRON
ejpam-5322	202	2	is	be	AUX
ejpam-5322	202	3	a	a	DET
ejpam-5322	202	4	contradiction	contradiction	NOUN
ejpam-5322	203	1	and	and	CCONJ
ejpam-5322	203	2	so	so	ADV
ejpam-5322	203	3	f	f	PROPN
ejpam-5322	203	4	is	be	AUX
ejpam-5322	203	5	not	not	PART
ejpam-5322	203	6	upper	upper	ADJ
ejpam-5322	203	7	s-(τ1	s-(τ1	NOUN
ejpam-5322	203	8	,	,	PUNCT
ejpam-5322	203	9	τ2)p	τ2)p	ADJ
ejpam-5322	203	10	-	-	ADJ
ejpam-5322	203	11	continuous	continuous	ADJ
ejpam-5322	203	12	at	at	ADP
ejpam-5322	203	13	x.	x.	NOUN
ejpam-5322	203	14	theorem	theorem	VERB
ejpam-5322	203	15	6	6	NUM
ejpam-5322	203	16	.	.	PUNCT
ejpam-5322	204	1	the	the	DET
ejpam-5322	204	2	set	set	NOUN
ejpam-5322	204	3	of	of	ADP
ejpam-5322	204	4	all	all	DET
ejpam-5322	204	5	points	point	NOUN
ejpam-5322	204	6	x	x	PUNCT
ejpam-5322	204	7	of	of	ADP
ejpam-5322	204	8	x	x	SYM
ejpam-5322	204	9	at	at	ADP
ejpam-5322	204	10	which	which	PRON
ejpam-5322	204	11	a	a	DET
ejpam-5322	204	12	multifunction	multifunction	NOUN
ejpam-5322	205	1	f	f	NOUN
ejpam-5322	205	2	:	:	PUNCT
ejpam-5322	205	3	(	(	PUNCT
ejpam-5322	205	4	x	x	NOUN
ejpam-5322	205	5	,	,	PUNCT
ejpam-5322	205	6	τ1	τ1	NOUN
ejpam-5322	205	7	,	,	PUNCT
ejpam-5322	205	8	τ2	τ2	NOUN
ejpam-5322	205	9	)	)	PUNCT
ejpam-5322	205	10	→	→	SYM
ejpam-5322	205	11	(	(	PUNCT
ejpam-5322	205	12	y	y	PROPN
ejpam-5322	205	13	,	,	PUNCT
ejpam-5322	205	14	σ1	σ1	PROPN
ejpam-5322	205	15	,	,	PUNCT
ejpam-5322	205	16	σ2	σ2	PROPN
ejpam-5322	205	17	)	)	PUNCT
ejpam-5322	205	18	is	be	AUX
ejpam-5322	205	19	not	not	PART
ejpam-5322	205	20	lower	low	ADJ
ejpam-5322	205	21	s-(τ1	s-(τ1	NOUN
ejpam-5322	205	22	,	,	PUNCT
ejpam-5322	205	23	τ2)p	τ2)p	ADJ
ejpam-5322	205	24	-	-	ADJ
ejpam-5322	205	25	continuous	continuous	ADJ
ejpam-5322	205	26	is	be	AUX
ejpam-5322	205	27	identical	identical	ADJ
ejpam-5322	205	28	with	with	ADP
ejpam-5322	205	29	the	the	DET
ejpam-5322	205	30	union	union	NOUN
ejpam-5322	205	31	of	of	ADP
ejpam-5322	205	32	the	the	DET
ejpam-5322	205	33	(	(	PUNCT
ejpam-5322	205	34	τ1	τ1	NOUN
ejpam-5322	205	35	,	,	PUNCT
ejpam-5322	205	36	τ2)p	τ2)p	ADJ
ejpam-5322	205	37	-	-	PUNCT
ejpam-5322	205	38	frontier	frontier	NOUN
ejpam-5322	205	39	of	of	ADP
ejpam-5322	205	40	the	the	DET
ejpam-5322	205	41	lower	low	ADJ
ejpam-5322	205	42	inverse	inverse	NOUN
ejpam-5322	205	43	images	image	NOUN
ejpam-5322	205	44	of	of	ADP
ejpam-5322	205	45	the	the	DET
ejpam-5322	205	46	σ1σ2	σ1σ2	NOUN
ejpam-5322	205	47	-	-	PUNCT
ejpam-5322	205	48	closures	closure	NOUN
ejpam-5322	205	49	of	of	ADP
ejpam-5322	205	50	σ1σ2	σ1σ2	NOUN
ejpam-5322	205	51	-	-	PUNCT
ejpam-5322	205	52	open	open	ADJ
ejpam-5322	205	53	sets	set	NOUN
ejpam-5322	205	54	meeting	meet	VERB
ejpam-5322	205	55	f	f	X
ejpam-5322	205	56	(	(	PUNCT
ejpam-5322	205	57	x	x	NOUN
ejpam-5322	205	58	)	)	PUNCT
ejpam-5322	205	59	and	and	CCONJ
ejpam-5322	205	60	having	have	VERB
ejpam-5322	205	61	σ1σ2	σ1σ2	NOUN
ejpam-5322	205	62	-	-	PUNCT
ejpam-5322	205	63	connected	connect	VERB
ejpam-5322	205	64	complement	complement	NOUN
ejpam-5322	205	65	.	.	PUNCT
ejpam-5322	206	1	proof	proof	NOUN
ejpam-5322	206	2	.	.	PUNCT
ejpam-5322	207	1	the	the	DET
ejpam-5322	207	2	proof	proof	NOUN
ejpam-5322	207	3	is	be	AUX
ejpam-5322	207	4	similar	similar	ADJ
ejpam-5322	207	5	to	to	ADP
ejpam-5322	207	6	that	that	PRON
ejpam-5322	207	7	of	of	ADP
ejpam-5322	207	8	theorem	theorem	NOUN
ejpam-5322	207	9	5	5	NUM
ejpam-5322	207	10	.	.	NOUN
ejpam-5322	207	11	4	4	NUM
ejpam-5322	207	12	.	.	X
ejpam-5322	207	13	conclusion	conclusion	VERB
ejpam-5322	207	14	this	this	DET
ejpam-5322	207	15	paper	paper	NOUN
ejpam-5322	207	16	deals	deal	NOUN
ejpam-5322	207	17	with	with	ADP
ejpam-5322	207	18	the	the	DET
ejpam-5322	207	19	notions	notion	NOUN
ejpam-5322	207	20	of	of	ADP
ejpam-5322	207	21	upper	upper	ADJ
ejpam-5322	207	22	and	and	CCONJ
ejpam-5322	207	23	lower	low	ADJ
ejpam-5322	207	24	s-(τ1	s-(τ1	NOUN
ejpam-5322	207	25	,	,	PUNCT
ejpam-5322	207	26	τ2)p	τ2)p	ADJ
ejpam-5322	207	27	-	-	PUNCT
ejpam-5322	207	28	continuous	continuous	ADJ
ejpam-5322	207	29	multifunctions	multifunction	NOUN
ejpam-5322	207	30	.	.	PUNCT
ejpam-5322	208	1	furthermore	furthermore	ADV
ejpam-5322	208	2	,	,	PUNCT
ejpam-5322	208	3	some	some	DET
ejpam-5322	208	4	characterizations	characterization	NOUN
ejpam-5322	208	5	and	and	CCONJ
ejpam-5322	208	6	several	several	ADJ
ejpam-5322	208	7	properties	property	NOUN
ejpam-5322	208	8	concerning	concern	VERB
ejpam-5322	208	9	upper	upper	ADJ
ejpam-5322	208	10	and	and	CCONJ
ejpam-5322	208	11	lower	low	ADJ
ejpam-5322	208	12	s-(τ1	s-(τ1	NOUN
ejpam-5322	208	13	,	,	PUNCT
ejpam-5322	208	14	τ2)p	τ2)p	ADJ
ejpam-5322	208	15	-	-	PUNCT
ejpam-5322	208	16	continuous	continuous	ADJ
ejpam-5322	208	17	multifunctions	multifunction	NOUN
ejpam-5322	208	18	are	be	AUX
ejpam-5322	208	19	established	establish	VERB
ejpam-5322	208	20	.	.	PUNCT
ejpam-5322	209	1	in	in	ADP
ejpam-5322	209	2	the	the	DET
ejpam-5322	209	3	upcoming	upcoming	ADJ
ejpam-5322	209	4	work	work	NOUN
ejpam-5322	209	5	,	,	PUNCT
ejpam-5322	209	6	we	we	PRON
ejpam-5322	209	7	plan	plan	VERB
ejpam-5322	209	8	to	to	PART
ejpam-5322	209	9	apply	apply	VERB
ejpam-5322	209	10	the	the	DET
ejpam-5322	209	11	concepts	concept	NOUN
ejpam-5322	209	12	initiated	initiate	VERB
ejpam-5322	209	13	in	in	ADP
ejpam-5322	209	14	this	this	DET
ejpam-5322	209	15	paper	paper	NOUN
ejpam-5322	209	16	to	to	PART
ejpam-5322	209	17	study	study	VERB
ejpam-5322	209	18	a	a	DET
ejpam-5322	209	19	new	new	ADJ
ejpam-5322	209	20	generalization	generalization	NOUN
ejpam-5322	209	21	of	of	ADP
ejpam-5322	209	22	upper	upper	ADJ
ejpam-5322	209	23	(	(	PUNCT
ejpam-5322	209	24	lower	low	ADJ
ejpam-5322	209	25	)	)	PUNCT
ejpam-5322	209	26	s-(τ1	s-(τ1	PROPN
ejpam-5322	209	27	,	,	PUNCT
ejpam-5322	209	28	τ2)p	τ2)p	ADJ
ejpam-5322	209	29	-	-	PUNCT
ejpam-5322	209	30	continuous	continuous	ADJ
ejpam-5322	209	31	multifunctions	multifunction	NOUN
ejpam-5322	209	32	,	,	PUNCT
ejpam-5322	209	33	namely	namely	ADV
ejpam-5322	209	34	upper	upper	ADJ
ejpam-5322	209	35	(	(	PUNCT
ejpam-5322	209	36	lower	low	ADJ
ejpam-5322	209	37	)	)	PUNCT
ejpam-5322	209	38	almost	almost	ADV
ejpam-5322	209	39	s-(τ1	s-(τ1	VERB
ejpam-5322	209	40	,	,	PUNCT
ejpam-5322	209	41	τ2)p	τ2)p	ADJ
ejpam-5322	209	42	-	-	PUNCT
ejpam-5322	209	43	continuous	continuous	ADJ
ejpam-5322	209	44	multifunctions	multifunction	NOUN
ejpam-5322	209	45	.	.	PUNCT
ejpam-5322	210	1	a	a	DET
ejpam-5322	210	2	multifunction	multifunction	NOUN
ejpam-5322	210	3	f	f	NOUN
ejpam-5322	210	4	:	:	PUNCT
ejpam-5322	210	5	(	(	PUNCT
ejpam-5322	210	6	x	x	NOUN
ejpam-5322	210	7	,	,	PUNCT
ejpam-5322	210	8	τ1	τ1	NOUN
ejpam-5322	210	9	,	,	PUNCT
ejpam-5322	210	10	τ2	τ2	NOUN
ejpam-5322	210	11	)	)	PUNCT
ejpam-5322	210	12	→	→	SYM
ejpam-5322	210	13	(	(	PUNCT
ejpam-5322	210	14	y	y	PROPN
ejpam-5322	210	15	,	,	PUNCT
ejpam-5322	210	16	σ1	σ1	PROPN
ejpam-5322	210	17	,	,	PUNCT
ejpam-5322	210	18	σ2	σ2	PROPN
ejpam-5322	210	19	)	)	PUNCT
ejpam-5322	210	20	is	be	AUX
ejpam-5322	210	21	called	call	VERB
ejpam-5322	210	22	upper	upper	ADJ
ejpam-5322	210	23	(	(	PUNCT
ejpam-5322	210	24	lower	low	ADJ
ejpam-5322	210	25	)	)	PUNCT
ejpam-5322	210	26	almost	almost	ADV
ejpam-5322	210	27	s-(τ1	s-(τ1	VERB
ejpam-5322	210	28	,	,	PUNCT
ejpam-5322	210	29	τ2)p	τ2)p	ADJ
ejpam-5322	210	30	-	-	ADJ
ejpam-5322	210	31	continuous	continuous	ADJ
ejpam-5322	210	32	multifunctions	multifunction	NOUN
ejpam-5322	210	33	if	if	SCONJ
ejpam-5322	210	34	for	for	ADP
ejpam-5322	210	35	each	each	DET
ejpam-5322	210	36	x	x	SYM
ejpam-5322	210	37	∈	∈	PROPN
ejpam-5322	210	38	x	x	X
ejpam-5322	210	39	and	and	CCONJ
ejpam-5322	210	40	each	each	DET
ejpam-5322	210	41	σ1σ2	σ1σ2	VERB
ejpam-5322	210	42	-	-	ADJ
ejpam-5322	210	43	open	open	ADJ
ejpam-5322	210	44	set	set	NOUN
ejpam-5322	210	45	v	v	NOUN
ejpam-5322	210	46	of	of	ADP
ejpam-5322	210	47	y	y	PROPN
ejpam-5322	210	48	having	have	VERB
ejpam-5322	210	49	σ1σ2	σ1σ2	ADV
ejpam-5322	210	50	-	-	PUNCT
ejpam-5322	210	51	connected	connected	ADJ
ejpam-5322	210	52	complement	complement	NOUN
ejpam-5322	210	53	such	such	ADJ
ejpam-5322	210	54	that	that	SCONJ
ejpam-5322	210	55	x	x	SYM
ejpam-5322	210	56	∈	∈	PROPN
ejpam-5322	210	57	f+(v	f+(v	NOUN
ejpam-5322	210	58	)	)	PUNCT
ejpam-5322	211	1	(	(	PUNCT
ejpam-5322	211	2	x	x	PUNCT
ejpam-5322	211	3	∈	∈	PROPN
ejpam-5322	211	4	f−(v	f−(v	NOUN
ejpam-5322	211	5	)	)	PUNCT
ejpam-5322	211	6	)	)	PUNCT
ejpam-5322	211	7	,	,	PUNCT
ejpam-5322	211	8	there	there	PRON
ejpam-5322	211	9	exists	exist	VERB
ejpam-5322	211	10	a	a	DET
ejpam-5322	211	11	(	(	PUNCT
ejpam-5322	211	12	τ1	τ1	NOUN
ejpam-5322	211	13	,	,	PUNCT
ejpam-5322	211	14	τ2)p	τ2)p	ADJ
ejpam-5322	211	15	-	-	PUNCT
ejpam-5322	211	16	open	open	ADJ
ejpam-5322	211	17	set	set	NOUN
ejpam-5322	211	18	u	u	NOUN
ejpam-5322	211	19	of	of	ADP
ejpam-5322	211	20	x	x	PUNCT
ejpam-5322	211	21	containing	contain	VERB
ejpam-5322	211	22	x	x	PUNCT
ejpam-5322	211	23	such	such	ADJ
ejpam-5322	211	24	that	that	SCONJ
ejpam-5322	211	25	u	u	NOUN
ejpam-5322	211	26	⊆	⊆	NUM
ejpam-5322	211	27	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-5322	211	28	-	-	PUNCT
ejpam-5322	211	29	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5322	211	30	-	-	PUNCT
ejpam-5322	211	31	cl(v	cl(v	NOUN
ejpam-5322	211	32	)	)	PUNCT
ejpam-5322	211	33	)	)	PUNCT
ejpam-5322	211	34	)	)	PUNCT
ejpam-5322	212	1	(	(	PUNCT
ejpam-5322	212	2	u	u	NOUN
ejpam-5322	212	3	⊆	⊆	NUM
ejpam-5322	212	4	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5322	212	5	-	-	PUNCT
ejpam-5322	212	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5322	212	7	-	-	PUNCT
ejpam-5322	212	8	cl(v	cl(v	NOUN
ejpam-5322	212	9	)	)	PUNCT
ejpam-5322	212	10	)	)	PUNCT
ejpam-5322	212	11	)	)	PUNCT
ejpam-5322	212	12	)	)	PUNCT
ejpam-5322	212	13	.	.	PUNCT
ejpam-5322	213	1	the	the	DET
ejpam-5322	213	2	class	class	NOUN
ejpam-5322	213	3	of	of	ADP
ejpam-5322	213	4	upper	upper	ADJ
ejpam-5322	213	5	(	(	PUNCT
ejpam-5322	213	6	lower	low	ADJ
ejpam-5322	213	7	)	)	PUNCT
ejpam-5322	213	8	s-(τ1	s-(τ1	PROPN
ejpam-5322	213	9	,	,	PUNCT
ejpam-5322	213	10	τ2)p	τ2)p	ADJ
ejpam-5322	213	11	-	-	PUNCT
ejpam-5322	213	12	continuous	continuous	ADJ
ejpam-5322	213	13	multifunctions	multifunction	NOUN
ejpam-5322	213	14	included	include	VERB
ejpam-5322	213	15	in	in	ADP
ejpam-5322	213	16	the	the	DET
ejpam-5322	213	17	class	class	NOUN
ejpam-5322	213	18	of	of	ADP
ejpam-5322	213	19	upper	upper	ADJ
ejpam-5322	213	20	(	(	PUNCT
ejpam-5322	213	21	lower	low	ADJ
ejpam-5322	213	22	)	)	PUNCT
ejpam-5322	213	23	almost	almost	ADV
ejpam-5322	213	24	s-(τ1	s-(τ1	VERB
ejpam-5322	213	25	,	,	PUNCT
ejpam-5322	213	26	τ2)p	τ2)p	ADJ
ejpam-5322	213	27	-	-	PUNCT
ejpam-5322	213	28	continuous	continuous	ADJ
ejpam-5322	213	29	multifunctions	multifunction	NOUN
ejpam-5322	213	30	.	.	PUNCT
ejpam-5322	214	1	acknowledgements	acknowledgement	NOUN
ejpam-5322	214	2	this	this	DET
ejpam-5322	214	3	research	research	NOUN
ejpam-5322	214	4	project	project	NOUN
ejpam-5322	214	5	was	be	AUX
ejpam-5322	214	6	financially	financially	ADV
ejpam-5322	214	7	supported	support	VERB
ejpam-5322	214	8	by	by	ADP
ejpam-5322	214	9	mahasarakham	mahasarakham	PROPN
ejpam-5322	214	10	university	university	PROPN
ejpam-5322	214	11	.	.	PUNCT
ejpam-5322	215	1	references	reference	NOUN
ejpam-5322	215	2	2218	2218	NUM
ejpam-5322	215	3	references	reference	NOUN
ejpam-5322	215	4	[	[	X
ejpam-5322	215	5	1	1	NUM
ejpam-5322	215	6	]	]	PUNCT
ejpam-5322	215	7	c.	c.	PROPN
ejpam-5322	215	8	berge	berge	PROPN
ejpam-5322	215	9	.	.	PUNCT
ejpam-5322	216	1	espaces	espace	VERB
ejpam-5322	216	2	topologiques	topologique	NOUN
ejpam-5322	216	3	fonctions	fonction	NOUN
ejpam-5322	216	4	multivoques	multivoque	NOUN
ejpam-5322	216	5	.	.	PUNCT
ejpam-5322	217	1	dunod	dunod	PROPN
ejpam-5322	217	2	,	,	PUNCT
ejpam-5322	217	3	paris	paris	PROPN
ejpam-5322	217	4	,	,	PUNCT
ejpam-5322	217	5	1959	1959	NUM
ejpam-5322	217	6	.	.	PUNCT
ejpam-5322	218	1	[	[	X
ejpam-5322	218	2	2	2	NUM
ejpam-5322	218	3	]	]	PUNCT
ejpam-5322	218	4	c.	c.	PROPN
ejpam-5322	218	5	boonpok	boonpok	PROPN
ejpam-5322	218	6	.	.	PUNCT
ejpam-5322	219	1	almost	almost	ADV
ejpam-5322	219	2	(	(	PUNCT
ejpam-5322	219	3	g	g	NOUN
ejpam-5322	219	4	,	,	PUNCT
ejpam-5322	219	5	m)-continuous	m)-continuous	ADJ
ejpam-5322	219	6	functions	function	NOUN
ejpam-5322	219	7	.	.	PUNCT
ejpam-5322	220	1	international	international	ADJ
ejpam-5322	220	2	journal	journal	PROPN
ejpam-5322	220	3	of	of	ADP
ejpam-5322	220	4	mathematical	mathematical	ADJ
ejpam-5322	220	5	analysis	analysis	NOUN
ejpam-5322	220	6	,	,	PUNCT
ejpam-5322	220	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-5322	220	8	,	,	PUNCT
ejpam-5322	220	9	2010	2010	NUM
ejpam-5322	220	10	.	.	PUNCT
ejpam-5322	221	1	[	[	X
ejpam-5322	221	2	3	3	X
ejpam-5322	221	3	]	]	PUNCT
ejpam-5322	221	4	c.	c.	PROPN
ejpam-5322	221	5	boonpok	boonpok	PROPN
ejpam-5322	221	6	.	.	PUNCT
ejpam-5322	222	1	m	m	VERB
ejpam-5322	222	2	-continuous	-continuous	ADJ
ejpam-5322	222	3	functions	function	NOUN
ejpam-5322	222	4	in	in	ADP
ejpam-5322	222	5	biminimal	biminimal	NOUN
ejpam-5322	222	6	structure	structure	NOUN
ejpam-5322	222	7	spaces	space	NOUN
ejpam-5322	222	8	.	.	PUNCT
ejpam-5322	223	1	far	far	PROPN
ejpam-5322	223	2	east	east	PROPN
ejpam-5322	223	3	journal	journal	PROPN
ejpam-5322	223	4	of	of	ADP
ejpam-5322	223	5	mathematical	mathematical	ADJ
ejpam-5322	223	6	sciences	science	NOUN
ejpam-5322	223	7	,	,	PUNCT
ejpam-5322	223	8	43(1):41–58	43(1):41–58	NUM
ejpam-5322	223	9	,	,	PUNCT
ejpam-5322	223	10	2010	2010	NUM
ejpam-5322	223	11	.	.	PUNCT
ejpam-5322	224	1	[	[	X
ejpam-5322	224	2	4	4	NUM
ejpam-5322	224	3	]	]	PUNCT
ejpam-5322	224	4	c.	c.	PROPN
ejpam-5322	224	5	boonpok	boonpok	PROPN
ejpam-5322	224	6	.	.	PUNCT
ejpam-5322	225	1	on	on	ADP
ejpam-5322	225	2	continuous	continuous	ADJ
ejpam-5322	225	3	multifunctions	multifunction	NOUN
ejpam-5322	225	4	in	in	ADP
ejpam-5322	225	5	ideal	ideal	ADJ
ejpam-5322	225	6	topological	topological	ADJ
ejpam-5322	225	7	spaces	space	NOUN
ejpam-5322	225	8	.	.	PUNCT
ejpam-5322	226	1	lobachevskii	lobachevskii	PROPN
ejpam-5322	226	2	journal	journal	PROPN
ejpam-5322	226	3	of	of	ADP
ejpam-5322	226	4	mathematics	mathematic	NOUN
ejpam-5322	226	5	,	,	PUNCT
ejpam-5322	226	6	40(1):24–35	40(1):24–35	NUM
ejpam-5322	226	7	,	,	PUNCT
ejpam-5322	226	8	2019	2019	NUM
ejpam-5322	226	9	.	.	PUNCT
ejpam-5322	227	1	[	[	X
ejpam-5322	227	2	5	5	X
ejpam-5322	227	3	]	]	PUNCT
ejpam-5322	227	4	c.	c.	PROPN
ejpam-5322	227	5	boonpok	boonpok	PROPN
ejpam-5322	227	6	.	.	PUNCT
ejpam-5322	228	1	on	on	ADP
ejpam-5322	228	2	characterizations	characterization	NOUN
ejpam-5322	228	3	of	of	ADP
ejpam-5322	228	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5322	228	5	ideal	ideal	ADJ
ejpam-5322	228	6	topological	topological	ADJ
ejpam-5322	228	7	spaces	space	NOUN
ejpam-5322	228	8	.	.	PUNCT
ejpam-5322	229	1	journal	journal	NOUN
ejpam-5322	229	2	of	of	ADP
ejpam-5322	229	3	mathematics	mathematic	NOUN
ejpam-5322	229	4	,	,	PUNCT
ejpam-5322	229	5	2020:9387601	2020:9387601	NUM
ejpam-5322	229	6	,	,	PUNCT
ejpam-5322	229	7	2020	2020	NUM
ejpam-5322	229	8	.	.	PUNCT
ejpam-5322	230	1	[	[	X
ejpam-5322	230	2	6	6	NUM
ejpam-5322	230	3	]	]	PUNCT
ejpam-5322	230	4	c.	c.	PROPN
ejpam-5322	230	5	boonpok	boonpok	PROPN
ejpam-5322	230	6	.	.	PUNCT
ejpam-5322	231	1	(	(	PUNCT
ejpam-5322	231	2	τ1	τ1	NOUN
ejpam-5322	231	3	,	,	PUNCT
ejpam-5322	231	4	τ2)δ	τ2)δ	ADJ
ejpam-5322	231	5	-	-	PUNCT
ejpam-5322	231	6	semicontinuous	semicontinuous	ADJ
ejpam-5322	231	7	multifunctions	multifunction	NOUN
ejpam-5322	231	8	.	.	PUNCT
ejpam-5322	232	1	heliyon	heliyon	NOUN
ejpam-5322	232	2	,	,	PUNCT
ejpam-5322	232	3	6	6	NUM
ejpam-5322	232	4	:	:	SYM
ejpam-5322	232	5	e05367	e05367	PROPN
ejpam-5322	232	6	,	,	PUNCT
ejpam-5322	232	7	2020	2020	NUM
ejpam-5322	232	8	.	.	PUNCT
ejpam-5322	233	1	[	[	X
ejpam-5322	233	2	7	7	X
ejpam-5322	233	3	]	]	X
ejpam-5322	233	4	c.	c.	PROPN
ejpam-5322	233	5	boonpok	boonpok	PROPN
ejpam-5322	233	6	.	.	PUNCT
ejpam-5322	234	1	upper	upper	ADJ
ejpam-5322	234	2	and	and	CCONJ
ejpam-5322	234	3	lower	low	ADJ
ejpam-5322	234	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-5322	234	5	.	.	PUNCT
ejpam-5322	234	6	heliyon	heliyon	NOUN
ejpam-5322	234	7	,	,	PUNCT
ejpam-5322	234	8	7	7	NUM
ejpam-5322	234	9	:	:	PUNCT
ejpam-5322	234	10	e05986	e05986	PROPN
ejpam-5322	234	11	,	,	PUNCT
ejpam-5322	234	12	2021	2021	NUM
ejpam-5322	234	13	.	.	PUNCT
ejpam-5322	235	1	[	[	X
ejpam-5322	235	2	8	8	NUM
ejpam-5322	235	3	]	]	X
ejpam-5322	235	4	c.	c.	PROPN
ejpam-5322	235	5	boonpok	boonpok	PROPN
ejpam-5322	235	6	.	.	PUNCT
ejpam-5322	236	1	on	on	ADP
ejpam-5322	236	2	some	some	DET
ejpam-5322	236	3	closed	closed	ADJ
ejpam-5322	236	4	sets	set	NOUN
ejpam-5322	236	5	and	and	CCONJ
ejpam-5322	236	6	low	low	ADJ
ejpam-5322	236	7	separation	separation	NOUN
ejpam-5322	236	8	axioms	axiom	NOUN
ejpam-5322	236	9	via	via	ADP
ejpam-5322	236	10	topological	topological	ADJ
ejpam-5322	236	11	ideals	ideal	NOUN
ejpam-5322	236	12	.	.	PUNCT
ejpam-5322	237	1	european	european	ADJ
ejpam-5322	237	2	journal	journal	PROPN
ejpam-5322	237	3	of	of	ADP
ejpam-5322	237	4	pure	pure	ADJ
ejpam-5322	237	5	and	and	CCONJ
ejpam-5322	237	6	applied	applied	ADJ
ejpam-5322	237	7	mathematics	mathematic	NOUN
ejpam-5322	237	8	,	,	PUNCT
ejpam-5322	237	9	15(3):300–309	15(3):300–309	NOUN
ejpam-5322	237	10	,	,	PUNCT
ejpam-5322	237	11	2022	2022	NUM
ejpam-5322	237	12	.	.	PUNCT
ejpam-5322	238	1	[	[	X
ejpam-5322	238	2	9	9	NUM
ejpam-5322	238	3	]	]	PUNCT
ejpam-5322	238	4	c.	c.	PROPN
ejpam-5322	238	5	boonpok	boonpok	PROPN
ejpam-5322	238	6	.	.	PUNCT
ejpam-5322	239	1	on	on	ADP
ejpam-5322	239	2	some	some	DET
ejpam-5322	239	3	spaces	space	NOUN
ejpam-5322	239	4	via	via	ADP
ejpam-5322	239	5	topological	topological	ADJ
ejpam-5322	239	6	ideals	ideal	NOUN
ejpam-5322	239	7	.	.	PUNCT
ejpam-5322	240	1	open	open	ADJ
ejpam-5322	240	2	mathematics	mathematic	NOUN
ejpam-5322	240	3	,	,	PUNCT
ejpam-5322	240	4	21:20230118	21:20230118	NUM
ejpam-5322	240	5	,	,	PUNCT
ejpam-5322	240	6	2023	2023	NUM
ejpam-5322	240	7	.	.	PUNCT
ejpam-5322	241	1	[	[	X
ejpam-5322	241	2	10	10	NUM
ejpam-5322	241	3	]	]	X
ejpam-5322	241	4	c.	c.	PROPN
ejpam-5322	241	5	boonpok	boonpok	PROPN
ejpam-5322	241	6	.	.	PUNCT
ejpam-5322	242	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5322	242	2	.	.	PUNCT
ejpam-5322	243	1	mathematica	mathematica	PROPN
ejpam-5322	243	2	,	,	PUNCT
ejpam-5322	243	3	65(1):31–42	65(1):31–42	NUM
ejpam-5322	243	4	,	,	PUNCT
ejpam-5322	243	5	2023	2023	NUM
ejpam-5322	243	6	.	.	PUNCT
ejpam-5322	244	1	[	[	X
ejpam-5322	244	2	11	11	NUM
ejpam-5322	244	3	]	]	X
ejpam-5322	244	4	c.	c.	PROPN
ejpam-5322	244	5	boonpok	boonpok	PROPN
ejpam-5322	244	6	and	and	CCONJ
ejpam-5322	244	7	j.	j.	PROPN
ejpam-5322	244	8	khampakdee	khampakdee	PROPN
ejpam-5322	244	9	.	.	PUNCT
ejpam-5322	245	1	almost	almost	ADV
ejpam-5322	245	2	strong	strong	ADJ
ejpam-5322	245	3	θ(λ	θ(λ	PROPN
ejpam-5322	245	4	,	,	PUNCT
ejpam-5322	245	5	p)-continuity	p)-continuity	NOUN
ejpam-5322	245	6	for	for	ADP
ejpam-5322	245	7	functions	function	NOUN
ejpam-5322	245	8	.	.	PUNCT
ejpam-5322	246	1	european	european	ADJ
ejpam-5322	246	2	journal	journal	PROPN
ejpam-5322	246	3	of	of	ADP
ejpam-5322	246	4	pure	pure	ADJ
ejpam-5322	246	5	and	and	CCONJ
ejpam-5322	246	6	applied	applied	ADJ
ejpam-5322	246	7	mathematics	mathematic	NOUN
ejpam-5322	246	8	,	,	PUNCT
ejpam-5322	246	9	17(1):300–309	17(1):300–309	PROPN
ejpam-5322	246	10	,	,	PUNCT
ejpam-5322	246	11	2024	2024	NUM
ejpam-5322	246	12	.	.	PUNCT
ejpam-5322	247	1	[	[	X
ejpam-5322	247	2	12	12	NUM
ejpam-5322	247	3	]	]	X
ejpam-5322	247	4	c.	c.	PROPN
ejpam-5322	247	5	boonpok	boonpok	PROPN
ejpam-5322	247	6	and	and	CCONJ
ejpam-5322	247	7	c.	c.	PROPN
ejpam-5322	247	8	klanarong	klanarong	PROPN
ejpam-5322	247	9	.	.	PUNCT
ejpam-5322	248	1	on	on	ADP
ejpam-5322	248	2	weakly	weakly	ADJ
ejpam-5322	248	3	(	(	PUNCT
ejpam-5322	248	4	τ1	τ1	NOUN
ejpam-5322	248	5	,	,	PUNCT
ejpam-5322	248	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	248	7	functions	function	NOUN
ejpam-5322	248	8	.	.	PUNCT
ejpam-5322	249	1	european	european	ADJ
ejpam-5322	249	2	journal	journal	PROPN
ejpam-5322	249	3	of	of	ADP
ejpam-5322	249	4	pure	pure	ADJ
ejpam-5322	249	5	and	and	CCONJ
ejpam-5322	249	6	applied	applied	ADJ
ejpam-5322	249	7	mathematics	mathematic	NOUN
ejpam-5322	249	8	,	,	PUNCT
ejpam-5322	249	9	17(1):416–425	17(1):416–425	NUM
ejpam-5322	249	10	,	,	PUNCT
ejpam-5322	249	11	2024	2024	NUM
ejpam-5322	249	12	.	.	PUNCT
ejpam-5322	250	1	[	[	X
ejpam-5322	250	2	13	13	NUM
ejpam-5322	250	3	]	]	PUNCT
ejpam-5322	250	4	c.	c.	PROPN
ejpam-5322	250	5	boonpok	boonpok	PROPN
ejpam-5322	250	6	and	and	CCONJ
ejpam-5322	250	7	p.	p.	NOUN
ejpam-5322	250	8	pue	pue	NOUN
ejpam-5322	250	9	-	-	PUNCT
ejpam-5322	250	10	on	on	ADP
ejpam-5322	250	11	.	.	PUNCT
ejpam-5322	251	1	continuity	continuity	NOUN
ejpam-5322	251	2	for	for	ADP
ejpam-5322	251	3	multifunctions	multifunction	NOUN
ejpam-5322	251	4	in	in	ADP
ejpam-5322	251	5	ideal	ideal	ADJ
ejpam-5322	251	6	topological	topological	ADJ
ejpam-5322	251	7	spaces	space	NOUN
ejpam-5322	251	8	.	.	PUNCT
ejpam-5322	252	1	wseas	wseas	VERB
ejpam-5322	252	2	transactions	transaction	NOUN
ejpam-5322	252	3	on	on	ADP
ejpam-5322	252	4	mathematics	mathematic	NOUN
ejpam-5322	252	5	,	,	PUNCT
ejpam-5322	252	6	19:624–631	19:624–631	NUM
ejpam-5322	252	7	,	,	PUNCT
ejpam-5322	252	8	2020	2020	NUM
ejpam-5322	252	9	.	.	PUNCT
ejpam-5322	253	1	[	[	X
ejpam-5322	253	2	14	14	NUM
ejpam-5322	253	3	]	]	X
ejpam-5322	253	4	c.	c.	PROPN
ejpam-5322	253	5	boonpok	boonpok	PROPN
ejpam-5322	253	6	and	and	CCONJ
ejpam-5322	253	7	p.	p.	NOUN
ejpam-5322	253	8	pue	pue	NOUN
ejpam-5322	253	9	-	-	PUNCT
ejpam-5322	253	10	on	on	ADP
ejpam-5322	253	11	.	.	PUNCT
ejpam-5322	254	1	upper	upper	ADJ
ejpam-5322	254	2	and	and	CCONJ
ejpam-5322	254	3	lower	low	ADJ
ejpam-5322	254	4	weakly	weakly	ADJ
ejpam-5322	254	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5322	254	6	multifunctions	multifunction	NOUN
ejpam-5322	254	7	.	.	PUNCT
ejpam-5322	255	1	international	international	ADJ
ejpam-5322	255	2	journal	journal	NOUN
ejpam-5322	255	3	of	of	ADP
ejpam-5322	255	4	analysis	analysis	NOUN
ejpam-5322	255	5	and	and	CCONJ
ejpam-5322	255	6	applications	application	NOUN
ejpam-5322	255	7	,	,	PUNCT
ejpam-5322	255	8	21:90	21:90	NUM
ejpam-5322	255	9	,	,	PUNCT
ejpam-5322	255	10	2023	2023	NUM
ejpam-5322	255	11	.	.	PUNCT
ejpam-5322	256	1	[	[	X
ejpam-5322	256	2	15	15	NUM
ejpam-5322	256	3	]	]	X
ejpam-5322	256	4	c.	c.	PROPN
ejpam-5322	256	5	boonpok	boonpok	PROPN
ejpam-5322	256	6	and	and	CCONJ
ejpam-5322	256	7	p.	p.	NOUN
ejpam-5322	256	8	pue	pue	NOUN
ejpam-5322	256	9	-	-	PUNCT
ejpam-5322	256	10	on	on	ADP
ejpam-5322	256	11	.	.	PUNCT
ejpam-5322	257	1	characterizations	characterization	NOUN
ejpam-5322	257	2	of	of	ADP
ejpam-5322	257	3	almost	almost	ADV
ejpam-5322	257	4	(	(	PUNCT
ejpam-5322	257	5	τ1	τ1	NOUN
ejpam-5322	257	6	,	,	PUNCT
ejpam-5322	257	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	257	8	functions	function	NOUN
ejpam-5322	257	9	.	.	PUNCT
ejpam-5322	258	1	international	international	ADJ
ejpam-5322	258	2	journal	journal	NOUN
ejpam-5322	258	3	of	of	ADP
ejpam-5322	258	4	analysis	analysis	NOUN
ejpam-5322	258	5	and	and	CCONJ
ejpam-5322	258	6	applications	application	NOUN
ejpam-5322	258	7	,	,	PUNCT
ejpam-5322	258	8	22:33	22:33	NUM
ejpam-5322	258	9	,	,	PUNCT
ejpam-5322	258	10	2024	2024	NUM
ejpam-5322	258	11	.	.	PUNCT
ejpam-5322	259	1	[	[	X
ejpam-5322	259	2	16	16	NUM
ejpam-5322	259	3	]	]	X
ejpam-5322	259	4	c.	c.	PROPN
ejpam-5322	259	5	boonpok	boonpok	PROPN
ejpam-5322	259	6	and	and	CCONJ
ejpam-5322	259	7	n.	n.	PROPN
ejpam-5322	259	8	srisarakham	srisarakham	PROPN
ejpam-5322	259	9	.	.	PUNCT
ejpam-5322	260	1	weak	weak	ADJ
ejpam-5322	260	2	forms	form	NOUN
ejpam-5322	260	3	of	of	ADP
ejpam-5322	260	4	(	(	PUNCT
ejpam-5322	260	5	λ	λ	PROPN
ejpam-5322	260	6	,	,	PUNCT
ejpam-5322	260	7	b)-open	b)-open	VERB
ejpam-5322	260	8	sets	set	NOUN
ejpam-5322	260	9	and	and	CCONJ
ejpam-5322	260	10	weak	weak	ADJ
ejpam-5322	260	11	(	(	PUNCT
ejpam-5322	260	12	λ	λ	NOUN
ejpam-5322	260	13	,	,	PUNCT
ejpam-5322	260	14	b)continuity	b)continuity	NOUN
ejpam-5322	260	15	.	.	PUNCT
ejpam-5322	261	1	european	european	PROPN
ejpam-5322	261	2	journal	journal	PROPN
ejpam-5322	261	3	of	of	ADP
ejpam-5322	261	4	pure	pure	ADJ
ejpam-5322	261	5	and	and	CCONJ
ejpam-5322	261	6	applied	applied	ADJ
ejpam-5322	261	7	mathematics	mathematic	NOUN
ejpam-5322	261	8	,	,	PUNCT
ejpam-5322	261	9	16(1):29–43	16(1):29–43	NUM
ejpam-5322	261	10	,	,	PUNCT
ejpam-5322	261	11	2023	2023	NUM
ejpam-5322	261	12	.	.	PUNCT
ejpam-5322	262	1	[	[	X
ejpam-5322	262	2	17	17	NUM
ejpam-5322	262	3	]	]	X
ejpam-5322	262	4	c.	c.	PROPN
ejpam-5322	262	5	boonpok	boonpok	PROPN
ejpam-5322	262	6	and	and	CCONJ
ejpam-5322	262	7	n.	n.	PROPN
ejpam-5322	262	8	srisarakham	srisarakham	PROPN
ejpam-5322	262	9	.	.	PUNCT
ejpam-5322	263	1	(	(	PUNCT
ejpam-5322	263	2	τ1	τ1	NOUN
ejpam-5322	263	3	,	,	PUNCT
ejpam-5322	263	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5322	263	5	for	for	ADP
ejpam-5322	263	6	functions	function	NOUN
ejpam-5322	263	7	.	.	PUNCT
ejpam-5322	264	1	asia	asia	PROPN
ejpam-5322	264	2	pacific	pacific	PROPN
ejpam-5322	264	3	journal	journal	PROPN
ejpam-5322	264	4	of	of	ADP
ejpam-5322	264	5	mathematics	mathematic	NOUN
ejpam-5322	264	6	,	,	PUNCT
ejpam-5322	264	7	11:21	11:21	NUM
ejpam-5322	264	8	,	,	PUNCT
ejpam-5322	264	9	2024	2024	NUM
ejpam-5322	264	10	.	.	PUNCT
ejpam-5322	265	1	references	reference	NOUN
ejpam-5322	265	2	2219	2219	NUM
ejpam-5322	265	3	[	[	X
ejpam-5322	265	4	18	18	NUM
ejpam-5322	265	5	]	]	PUNCT
ejpam-5322	265	6	c.	c.	PROPN
ejpam-5322	265	7	boonpok	boonpok	PROPN
ejpam-5322	265	8	and	and	CCONJ
ejpam-5322	265	9	c.	c.	PROPN
ejpam-5322	265	10	viriyapong	viriyapong	PROPN
ejpam-5322	265	11	.	.	PUNCT
ejpam-5322	266	1	almost	almost	ADV
ejpam-5322	266	2	weak	weak	ADJ
ejpam-5322	266	3	continuity	continuity	NOUN
ejpam-5322	266	4	for	for	ADP
ejpam-5322	266	5	multifunctions	multifunction	NOUN
ejpam-5322	266	6	in	in	ADP
ejpam-5322	266	7	ideal	ideal	ADJ
ejpam-5322	266	8	topological	topological	ADJ
ejpam-5322	266	9	spaces	space	NOUN
ejpam-5322	266	10	.	.	PUNCT
ejpam-5322	267	1	wseas	wseas	VERB
ejpam-5322	267	2	transactions	transaction	NOUN
ejpam-5322	267	3	on	on	ADP
ejpam-5322	267	4	mathematics	mathematic	NOUN
ejpam-5322	267	5	,	,	PUNCT
ejpam-5322	267	6	19:367–372	19:367–372	PROPN
ejpam-5322	267	7	,	,	PUNCT
ejpam-5322	267	8	2020	2020	NUM
ejpam-5322	267	9	.	.	PUNCT
ejpam-5322	268	1	[	[	X
ejpam-5322	268	2	19	19	NUM
ejpam-5322	268	3	]	]	X
ejpam-5322	268	4	c.	c.	PROPN
ejpam-5322	268	5	boonpok	boonpok	PROPN
ejpam-5322	268	6	and	and	CCONJ
ejpam-5322	268	7	c.	c.	PROPN
ejpam-5322	268	8	viriyapong	viriyapong	PROPN
ejpam-5322	268	9	.	.	PUNCT
ejpam-5322	269	1	upper	upper	ADJ
ejpam-5322	269	2	and	and	CCONJ
ejpam-5322	269	3	lower	low	ADJ
ejpam-5322	269	4	almost	almost	ADV
ejpam-5322	269	5	weak	weak	ADJ
ejpam-5322	269	6	(	(	PUNCT
ejpam-5322	269	7	τ1	τ1	NOUN
ejpam-5322	269	8	,	,	PUNCT
ejpam-5322	269	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5322	269	10	.	.	PUNCT
ejpam-5322	270	1	european	european	PROPN
ejpam-5322	270	2	journal	journal	PROPN
ejpam-5322	270	3	of	of	ADP
ejpam-5322	270	4	pure	pure	ADJ
ejpam-5322	270	5	and	and	CCONJ
ejpam-5322	270	6	applied	applied	ADJ
ejpam-5322	270	7	mathematics	mathematic	NOUN
ejpam-5322	270	8	,	,	PUNCT
ejpam-5322	270	9	14(1):1212–1225	14(1):1212–1225	NUM
ejpam-5322	270	10	,	,	PUNCT
ejpam-5322	270	11	2021	2021	NUM
ejpam-5322	270	12	.	.	PUNCT
ejpam-5322	271	1	[	[	X
ejpam-5322	271	2	20	20	NUM
ejpam-5322	271	3	]	]	PUNCT
ejpam-5322	271	4	c.	c.	PROPN
ejpam-5322	271	5	boonpok	boonpok	PROPN
ejpam-5322	271	6	,	,	PUNCT
ejpam-5322	271	7	c.	c.	PROPN
ejpam-5322	271	8	viriyapong	viriyapong	PROPN
ejpam-5322	271	9	,	,	PUNCT
ejpam-5322	271	10	and	and	CCONJ
ejpam-5322	271	11	m.	m.	NOUN
ejpam-5322	271	12	thongmoon	thongmoon	NOUN
ejpam-5322	271	13	.	.	PUNCT
ejpam-5322	272	1	on	on	ADP
ejpam-5322	272	2	upper	upper	ADJ
ejpam-5322	272	3	and	and	CCONJ
ejpam-5322	272	4	lower	low	ADJ
ejpam-5322	272	5	(	(	PUNCT
ejpam-5322	272	6	τ1	τ1	NOUN
ejpam-5322	272	7	,	,	PUNCT
ejpam-5322	272	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-5322	272	9	multifunctions	multifunction	NOUN
ejpam-5322	272	10	.	.	PUNCT
ejpam-5322	273	1	journal	journal	PROPN
ejpam-5322	273	2	of	of	ADP
ejpam-5322	273	3	mathematics	mathematics	PROPN
ejpam-5322	273	4	and	and	CCONJ
ejpam-5322	273	5	computer	computer	NOUN
ejpam-5322	273	6	science	science	NOUN
ejpam-5322	273	7	,	,	PUNCT
ejpam-5322	273	8	18:282–293	18:282–293	NUM
ejpam-5322	273	9	,	,	PUNCT
ejpam-5322	273	10	2018	2018	NUM
ejpam-5322	273	11	.	.	PUNCT
ejpam-5322	274	1	[	[	X
ejpam-5322	274	2	21	21	NUM
ejpam-5322	274	3	]	]	X
ejpam-5322	274	4	t.	t.	PROPN
ejpam-5322	274	5	duangphui	duangphui	PROPN
ejpam-5322	274	6	,	,	PUNCT
ejpam-5322	274	7	c.	c.	PROPN
ejpam-5322	274	8	boonpok	boonpok	PROPN
ejpam-5322	274	9	,	,	PUNCT
ejpam-5322	274	10	and	and	CCONJ
ejpam-5322	274	11	c.	c.	PROPN
ejpam-5322	274	12	viriyapong	viriyapong	PROPN
ejpam-5322	274	13	.	.	PUNCT
ejpam-5322	275	1	continuous	continuous	ADJ
ejpam-5322	275	2	functions	function	NOUN
ejpam-5322	275	3	on	on	ADP
ejpam-5322	275	4	bigeneralized	bigeneralize	VERB
ejpam-5322	275	5	topological	topological	ADJ
ejpam-5322	275	6	spaces	space	NOUN
ejpam-5322	275	7	.	.	PUNCT
ejpam-5322	276	1	international	international	ADJ
ejpam-5322	276	2	journal	journal	PROPN
ejpam-5322	276	3	of	of	ADP
ejpam-5322	276	4	mathematical	mathematical	ADJ
ejpam-5322	276	5	analysis	analysis	NOUN
ejpam-5322	276	6	,	,	PUNCT
ejpam-5322	276	7	5(24):1165	5(24):1165	NUM
ejpam-5322	276	8	–	–	PUNCT
ejpam-5322	276	9	1174	1174	NUM
ejpam-5322	276	10	,	,	PUNCT
ejpam-5322	276	11	2011	2011	NUM
ejpam-5322	276	12	.	.	PUNCT
ejpam-5322	277	1	[	[	X
ejpam-5322	277	2	22	22	NUM
ejpam-5322	277	3	]	]	PUNCT
ejpam-5322	277	4	j.	j.	PROPN
ejpam-5322	277	5	ewert	ewert	PROPN
ejpam-5322	277	6	and	and	CCONJ
ejpam-5322	277	7	t.	t.	PROPN
ejpam-5322	277	8	lipski	lipski	PROPN
ejpam-5322	277	9	.	.	PUNCT
ejpam-5322	278	1	on	on	ADP
ejpam-5322	278	2	s	s	NOUN
ejpam-5322	278	3	-	-	PUNCT
ejpam-5322	278	4	quasi	quasi	ADJ
ejpam-5322	278	5	-	-	ADJ
ejpam-5322	278	6	continuous	continuous	ADJ
ejpam-5322	278	7	multivalued	multivalued	ADJ
ejpam-5322	278	8	maps	map	NOUN
ejpam-5322	278	9	.	.	PUNCT
ejpam-5322	279	1	review	review	NOUN
ejpam-5322	279	2	of	of	ADP
ejpam-5322	279	3	research	research	NOUN
ejpam-5322	279	4	,	,	PUNCT
ejpam-5322	279	5	faculty	faculty	NOUN
ejpam-5322	279	6	of	of	ADP
ejpam-5322	279	7	science	science	NOUN
ejpam-5322	279	8	,	,	PUNCT
ejpam-5322	279	9	mathematics	mathematics	NOUN
ejpam-5322	279	10	series	series	NOUN
ejpam-5322	279	11	,	,	PUNCT
ejpam-5322	279	12	20(1):167–183	20(1):167–183	PROPN
ejpam-5322	279	13	,	,	PUNCT
ejpam-5322	279	14	1990	1990	NUM
ejpam-5322	279	15	.	.	PUNCT
ejpam-5322	280	1	[	[	X
ejpam-5322	280	2	23	23	NUM
ejpam-5322	280	3	]	]	X
ejpam-5322	280	4	c.	c.	PROPN
ejpam-5322	280	5	klanarong	klanarong	PROPN
ejpam-5322	280	6	,	,	PUNCT
ejpam-5322	280	7	s.	s.	PROPN
ejpam-5322	280	8	sompong	sompong	PROPN
ejpam-5322	280	9	,	,	PUNCT
ejpam-5322	280	10	and	and	CCONJ
ejpam-5322	280	11	c.	c.	PROPN
ejpam-5322	280	12	boonpok	boonpok	PROPN
ejpam-5322	280	13	.	.	PUNCT
ejpam-5322	281	1	upper	upper	ADJ
ejpam-5322	281	2	and	and	CCONJ
ejpam-5322	281	3	lower	low	ADJ
ejpam-5322	281	4	almost	almost	ADV
ejpam-5322	281	5	(	(	PUNCT
ejpam-5322	281	6	τ1	τ1	NOUN
ejpam-5322	281	7	,	,	PUNCT
ejpam-5322	281	8	τ2)continuous	τ2)continuous	ADJ
ejpam-5322	281	9	multifunctions	multifunction	NOUN
ejpam-5322	281	10	.	.	PUNCT
ejpam-5322	282	1	european	european	ADJ
ejpam-5322	282	2	journal	journal	PROPN
ejpam-5322	282	3	of	of	ADP
ejpam-5322	282	4	pure	pure	ADJ
ejpam-5322	282	5	and	and	CCONJ
ejpam-5322	282	6	applied	applied	ADJ
ejpam-5322	282	7	mathematics	mathematic	NOUN
ejpam-5322	282	8	,	,	PUNCT
ejpam-5322	282	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-5322	282	10	,	,	PUNCT
ejpam-5322	282	11	2024	2024	NUM
ejpam-5322	282	12	.	.	PUNCT
ejpam-5322	283	1	[	[	X
ejpam-5322	283	2	24	24	NUM
ejpam-5322	283	3	]	]	PUNCT
ejpam-5322	283	4	j.	j.	PROPN
ejpam-5322	283	5	k.	k.	PROPN
ejpam-5322	283	6	kohli	kohli	PROPN
ejpam-5322	283	7	.	.	PUNCT
ejpam-5322	284	1	a	a	DET
ejpam-5322	284	2	class	class	NOUN
ejpam-5322	284	3	of	of	ADP
ejpam-5322	284	4	mappings	mapping	NOUN
ejpam-5322	284	5	containing	contain	VERB
ejpam-5322	284	6	all	all	PRON
ejpam-5322	284	7	continuous	continuous	ADJ
ejpam-5322	284	8	and	and	CCONJ
ejpam-5322	284	9	all	all	DET
ejpam-5322	284	10	semi	semi	ADJ
ejpam-5322	284	11	-	-	ADJ
ejpam-5322	284	12	connected	connected	ADJ
ejpam-5322	284	13	mappings	mapping	NOUN
ejpam-5322	284	14	.	.	PUNCT
ejpam-5322	285	1	proceedings	proceeding	NOUN
ejpam-5322	285	2	of	of	ADP
ejpam-5322	285	3	the	the	DET
ejpam-5322	285	4	american	american	PROPN
ejpam-5322	285	5	mathematical	mathematical	PROPN
ejpam-5322	285	6	society	society	NOUN
ejpam-5322	285	7	,	,	PUNCT
ejpam-5322	285	8	72:175–181	72:175–181	PROPN
ejpam-5322	285	9	,	,	PUNCT
ejpam-5322	285	10	1978	1978	NUM
ejpam-5322	285	11	.	.	PUNCT
ejpam-5322	286	1	[	[	X
ejpam-5322	286	2	25	25	NUM
ejpam-5322	286	3	]	]	PUNCT
ejpam-5322	286	4	j.	j.	PROPN
ejpam-5322	286	5	k.	k.	PROPN
ejpam-5322	286	6	kohli	kohli	PROPN
ejpam-5322	286	7	.	.	PUNCT
ejpam-5322	287	1	s	s	X
ejpam-5322	287	2	-	-	ADJ
ejpam-5322	287	3	continuous	continuous	ADJ
ejpam-5322	287	4	functions	function	NOUN
ejpam-5322	287	5	and	and	CCONJ
ejpam-5322	287	6	certain	certain	ADJ
ejpam-5322	287	7	weak	weak	ADJ
ejpam-5322	287	8	forms	form	NOUN
ejpam-5322	287	9	of	of	ADP
ejpam-5322	287	10	regularity	regularity	NOUN
ejpam-5322	287	11	and	and	CCONJ
ejpam-5322	287	12	complete	complete	ADJ
ejpam-5322	287	13	regularity	regularity	NOUN
ejpam-5322	287	14	.	.	PUNCT
ejpam-5322	288	1	mathematics	mathematic	NOUN
ejpam-5322	288	2	nachrichten	nachrichten	VERB
ejpam-5322	288	3	,	,	PUNCT
ejpam-5322	288	4	97:189–196	97:189–196	NUM
ejpam-5322	288	5	,	,	PUNCT
ejpam-5322	288	6	1980	1980	NUM
ejpam-5322	288	7	.	.	PUNCT
ejpam-5322	289	1	[	[	X
ejpam-5322	289	2	26	26	NUM
ejpam-5322	289	3	]	]	PUNCT
ejpam-5322	289	4	k.	k.	PROPN
ejpam-5322	289	5	laprom	laprom	PROPN
ejpam-5322	289	6	,	,	PUNCT
ejpam-5322	289	7	c.	c.	PROPN
ejpam-5322	289	8	boonpok	boonpok	PROPN
ejpam-5322	289	9	,	,	PUNCT
ejpam-5322	289	10	and	and	CCONJ
ejpam-5322	289	11	c.	c.	PROPN
ejpam-5322	289	12	viriyapong	viriyapong	PROPN
ejpam-5322	289	13	.	.	PUNCT
ejpam-5322	290	1	β(τ1	β(τ1	PROPN
ejpam-5322	290	2	,	,	PUNCT
ejpam-5322	290	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	290	4	multifunctions	multifunction	NOUN
ejpam-5322	290	5	on	on	ADP
ejpam-5322	290	6	bitopological	bitopological	ADJ
ejpam-5322	290	7	spaces	space	NOUN
ejpam-5322	290	8	.	.	PUNCT
ejpam-5322	291	1	journal	journal	NOUN
ejpam-5322	291	2	of	of	ADP
ejpam-5322	291	3	mathematics	mathematic	NOUN
ejpam-5322	291	4	,	,	PUNCT
ejpam-5322	291	5	2020:4020971	2020:4020971	NUM
ejpam-5322	291	6	,	,	PUNCT
ejpam-5322	291	7	2020	2020	NUM
ejpam-5322	291	8	.	.	PUNCT
ejpam-5322	292	1	[	[	X
ejpam-5322	292	2	27	27	NUM
ejpam-5322	292	3	]	]	X
ejpam-5322	292	4	y.	y.	PROPN
ejpam-5322	292	5	l.	l.	PROPN
ejpam-5322	292	6	lee	lee	PROPN
ejpam-5322	292	7	.	.	PUNCT
ejpam-5322	293	1	some	some	DET
ejpam-5322	293	2	characterizations	characterization	NOUN
ejpam-5322	293	3	of	of	ADP
ejpam-5322	293	4	semilocally	semilocally	ADV
ejpam-5322	293	5	connected	connected	ADJ
ejpam-5322	293	6	spaces	space	NOUN
ejpam-5322	293	7	.	.	PUNCT
ejpam-5322	294	1	proceedings	proceeding	NOUN
ejpam-5322	294	2	of	of	ADP
ejpam-5322	294	3	the	the	DET
ejpam-5322	294	4	american	american	PROPN
ejpam-5322	294	5	mathematical	mathematical	PROPN
ejpam-5322	294	6	society	society	NOUN
ejpam-5322	294	7	,	,	PUNCT
ejpam-5322	294	8	16:1318–1320	16:1318–1320	NUM
ejpam-5322	294	9	,	,	PUNCT
ejpam-5322	294	10	1965	1965	NUM
ejpam-5322	294	11	.	.	PUNCT
ejpam-5322	295	1	[	[	X
ejpam-5322	295	2	28	28	NUM
ejpam-5322	295	3	]	]	X
ejpam-5322	295	4	t.	t.	NOUN
ejpam-5322	295	5	lipski	lipski	PROPN
ejpam-5322	295	6	.	.	PUNCT
ejpam-5322	296	1	s	s	X
ejpam-5322	296	2	-	-	ADJ
ejpam-5322	296	3	continuous	continuous	ADJ
ejpam-5322	296	4	multivalued	multivalued	ADJ
ejpam-5322	296	5	maps	map	NOUN
ejpam-5322	296	6	.	.	PUNCT
ejpam-5322	297	1	mathematical	mathematical	ADJ
ejpam-5322	297	2	chronicle	chronicle	PROPN
ejpam-5322	297	3	,	,	PUNCT
ejpam-5322	297	4	18:57–61	18:57–61	PROPN
ejpam-5322	297	5	,	,	PUNCT
ejpam-5322	297	6	1989	1989	NUM
ejpam-5322	297	7	.	.	PUNCT
ejpam-5322	298	1	[	[	X
ejpam-5322	298	2	29	29	NUM
ejpam-5322	298	3	]	]	PUNCT
ejpam-5322	298	4	v.	v.	CCONJ
ejpam-5322	298	5	popa	popa	NOUN
ejpam-5322	298	6	.	.	PUNCT
ejpam-5322	299	1	some	some	DET
ejpam-5322	299	2	properties	property	NOUN
ejpam-5322	299	3	of	of	ADP
ejpam-5322	299	4	h	h	NOUN
ejpam-5322	299	5	-	-	PUNCT
ejpam-5322	299	6	almost	almost	ADV
ejpam-5322	299	7	continuous	continuous	ADJ
ejpam-5322	299	8	multifunctions	multifunction	NOUN
ejpam-5322	299	9	.	.	PUNCT
ejpam-5322	300	1	problemy	problemy	PROPN
ejpam-5322	300	2	matematyczne	matematyczne	PROPN
ejpam-5322	300	3	,	,	PUNCT
ejpam-5322	300	4	10:9–26	10:9–26	NUM
ejpam-5322	300	5	,	,	PUNCT
ejpam-5322	300	6	1988	1988	NUM
ejpam-5322	300	7	.	.	PUNCT
ejpam-5322	301	1	[	[	X
ejpam-5322	301	2	30	30	NUM
ejpam-5322	301	3	]	]	X
ejpam-5322	301	4	v.	v.	CCONJ
ejpam-5322	301	5	popa	popa	NOUN
ejpam-5322	301	6	and	and	CCONJ
ejpam-5322	301	7	t.	t.	PROPN
ejpam-5322	301	8	noiri	noiri	PROPN
ejpam-5322	301	9	.	.	PUNCT
ejpam-5322	302	1	on	on	ADP
ejpam-5322	302	2	s	s	NOUN
ejpam-5322	302	3	-	-	ADJ
ejpam-5322	302	4	precontinuous	precontinuous	ADJ
ejpam-5322	302	5	multifunctions	multifunction	NOUN
ejpam-5322	302	6	.	.	PUNCT
ejpam-5322	303	1	demonstratio	demonstratio	PROPN
ejpam-5322	303	2	mathematica	mathematica	PROPN
ejpam-5322	303	3	,	,	PUNCT
ejpam-5322	303	4	33(3):679–687	33(3):679–687	PROPN
ejpam-5322	303	5	,	,	PUNCT
ejpam-5322	303	6	2000	2000	NUM
ejpam-5322	303	7	.	.	PUNCT
ejpam-5322	304	1	[	[	X
ejpam-5322	304	2	31	31	NUM
ejpam-5322	304	3	]	]	PUNCT
ejpam-5322	304	4	p.	p.	NOUN
ejpam-5322	304	5	pue	pue	NOUN
ejpam-5322	304	6	-	-	PUNCT
ejpam-5322	304	7	on	on	ADP
ejpam-5322	304	8	and	and	CCONJ
ejpam-5322	304	9	c.	c.	PROPN
ejpam-5322	304	10	boonpok	boonpok	PROPN
ejpam-5322	304	11	.	.	PUNCT
ejpam-5322	305	1	θ(λ	θ(λ	PROPN
ejpam-5322	305	2	,	,	PUNCT
ejpam-5322	305	3	p)-continuity	p)-continuity	NOUN
ejpam-5322	305	4	for	for	ADP
ejpam-5322	305	5	functions	function	NOUN
ejpam-5322	305	6	.	.	PUNCT
ejpam-5322	306	1	international	international	ADJ
ejpam-5322	306	2	journal	journal	NOUN
ejpam-5322	306	3	of	of	ADP
ejpam-5322	306	4	mathematics	mathematic	NOUN
ejpam-5322	306	5	and	and	CCONJ
ejpam-5322	306	6	computer	computer	NOUN
ejpam-5322	306	7	science	science	NOUN
ejpam-5322	306	8	,	,	PUNCT
ejpam-5322	306	9	19(2):491–495	19(2):491–495	NUM
ejpam-5322	306	10	,	,	PUNCT
ejpam-5322	306	11	2024	2024	NUM
ejpam-5322	306	12	.	.	PUNCT
ejpam-5322	307	1	[	[	X
ejpam-5322	307	2	32	32	NUM
ejpam-5322	307	3	]	]	PUNCT
ejpam-5322	307	4	p.	p.	NOUN
ejpam-5322	307	5	pue	pue	NOUN
ejpam-5322	307	6	-	-	PUNCT
ejpam-5322	307	7	on	on	ADP
ejpam-5322	307	8	,	,	PUNCT
ejpam-5322	307	9	s.	s.	PROPN
ejpam-5322	307	10	sompong	sompong	PROPN
ejpam-5322	307	11	,	,	PUNCT
ejpam-5322	307	12	and	and	CCONJ
ejpam-5322	307	13	c.	c.	PROPN
ejpam-5322	307	14	boonpok	boonpok	PROPN
ejpam-5322	307	15	.	.	PUNCT
ejpam-5322	308	1	upper	upper	ADJ
ejpam-5322	308	2	and	and	CCONJ
ejpam-5322	308	3	lower	low	ADJ
ejpam-5322	308	4	(	(	PUNCT
ejpam-5322	308	5	τ1	τ1	NOUN
ejpam-5322	308	6	,	,	PUNCT
ejpam-5322	308	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5322	308	8	multifunctions	multifunction	NOUN
ejpam-5322	308	9	.	.	PUNCT
ejpam-5322	309	1	international	international	ADJ
ejpam-5322	309	2	journal	journal	PROPN
ejpam-5322	309	3	of	of	ADP
ejpam-5322	309	4	mathematics	mathematic	NOUN
ejpam-5322	309	5	and	and	CCONJ
ejpam-5322	309	6	computer	computer	NOUN
ejpam-5322	309	7	science	science	NOUN
ejpam-5322	309	8	,	,	PUNCT
ejpam-5322	309	9	19(4):1305	19(4):1305	NUM
ejpam-5322	309	10	–	–	PUNCT
ejpam-5322	309	11	1310	1310	NUM
ejpam-5322	309	12	,	,	PUNCT
ejpam-5322	309	13	2024	2024	NUM
ejpam-5322	309	14	.	.	PUNCT
ejpam-5322	310	1	references	reference	NOUN
ejpam-5322	310	2	2220	2220	NUM
ejpam-5322	310	3	[	[	SYM
ejpam-5322	310	4	33	33	NUM
ejpam-5322	310	5	]	]	X
ejpam-5322	310	6	n.	n.	PROPN
ejpam-5322	310	7	srisarakham	srisarakham	PROPN
ejpam-5322	310	8	and	and	CCONJ
ejpam-5322	310	9	c.	c.	PROPN
ejpam-5322	310	10	boonpok	boonpok	PROPN
ejpam-5322	310	11	.	.	PUNCT
ejpam-5322	311	1	almost	almost	ADV
ejpam-5322	311	2	(	(	PUNCT
ejpam-5322	311	3	λ	λ	NOUN
ejpam-5322	311	4	,	,	PUNCT
ejpam-5322	311	5	p)-continuous	p)-continuous	ADJ
ejpam-5322	311	6	functions	function	NOUN
ejpam-5322	311	7	.	.	PUNCT
ejpam-5322	312	1	international	international	ADJ
ejpam-5322	312	2	journal	journal	PROPN
ejpam-5322	312	3	of	of	ADP
ejpam-5322	312	4	mathematics	mathematic	NOUN
ejpam-5322	312	5	and	and	CCONJ
ejpam-5322	312	6	computer	computer	NOUN
ejpam-5322	312	7	science	science	NOUN
ejpam-5322	312	8	,	,	PUNCT
ejpam-5322	312	9	18(2):255–259	18(2):255–259	NUM
ejpam-5322	312	10	,	,	PUNCT
ejpam-5322	312	11	2023	2023	NUM
ejpam-5322	312	12	.	.	PUNCT
ejpam-5322	313	1	[	[	X
ejpam-5322	313	2	34	34	NUM
ejpam-5322	313	3	]	]	X
ejpam-5322	313	4	n.	n.	NOUN
ejpam-5322	313	5	srisarakham	srisarakham	PROPN
ejpam-5322	313	6	and	and	CCONJ
ejpam-5322	313	7	c.	c.	PROPN
ejpam-5322	313	8	boonpok	boonpok	PROPN
ejpam-5322	313	9	.	.	PUNCT
ejpam-5322	314	1	on	on	ADP
ejpam-5322	314	2	characterizations	characterization	NOUN
ejpam-5322	314	3	of	of	ADP
ejpam-5322	314	4	δp(λ	δp(λ	NOUN
ejpam-5322	314	5	,	,	PUNCT
ejpam-5322	314	6	s)-d1	s)-d1	NOUN
ejpam-5322	314	7	spaces	space	NOUN
ejpam-5322	314	8	.	.	PUNCT
ejpam-5322	315	1	international	international	ADJ
ejpam-5322	315	2	journal	journal	PROPN
ejpam-5322	315	3	of	of	ADP
ejpam-5322	315	4	mathematics	mathematic	NOUN
ejpam-5322	315	5	and	and	CCONJ
ejpam-5322	315	6	computer	computer	NOUN
ejpam-5322	315	7	science	science	NOUN
ejpam-5322	315	8	,	,	PUNCT
ejpam-5322	315	9	18(4):743–747	18(4):743–747	PROPN
ejpam-5322	315	10	,	,	PUNCT
ejpam-5322	315	11	2023	2023	NUM
ejpam-5322	315	12	.	.	PUNCT
ejpam-5322	316	1	[	[	X
ejpam-5322	316	2	35	35	NUM
ejpam-5322	316	3	]	]	PUNCT
ejpam-5322	316	4	m.	m.	NOUN
ejpam-5322	316	5	thongmoon	thongmoon	NOUN
ejpam-5322	316	6	and	and	CCONJ
ejpam-5322	316	7	c.	c.	PROPN
ejpam-5322	316	8	boonpok	boonpok	PROPN
ejpam-5322	316	9	.	.	PUNCT
ejpam-5322	317	1	strongly	strongly	ADV
ejpam-5322	317	2	θ(λ	θ(λ	PROPN
ejpam-5322	317	3	,	,	PUNCT
ejpam-5322	317	4	p)-continuous	p)-continuous	ADJ
ejpam-5322	317	5	functions	function	NOUN
ejpam-5322	317	6	.	.	PUNCT
ejpam-5322	318	1	international	international	ADJ
ejpam-5322	318	2	journal	journal	PROPN
ejpam-5322	318	3	of	of	ADP
ejpam-5322	318	4	mathematics	mathematic	NOUN
ejpam-5322	318	5	and	and	CCONJ
ejpam-5322	318	6	computer	computer	NOUN
ejpam-5322	318	7	science	science	NOUN
ejpam-5322	318	8	,	,	PUNCT
ejpam-5322	318	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5322	318	10	,	,	PUNCT
ejpam-5322	318	11	2024	2024	NUM
ejpam-5322	318	12	.	.	PUNCT
ejpam-5322	319	1	[	[	X
ejpam-5322	319	2	36	36	NUM
ejpam-5322	319	3	]	]	X
ejpam-5322	319	4	c.	c.	PROPN
ejpam-5322	319	5	viriyapong	viriyapong	PROPN
ejpam-5322	319	6	and	and	CCONJ
ejpam-5322	319	7	c.	c.	PROPN
ejpam-5322	319	8	boonpok	boonpok	PROPN
ejpam-5322	319	9	.	.	PUNCT
ejpam-5322	320	1	(	(	PUNCT
ejpam-5322	320	2	τ1	τ1	NOUN
ejpam-5322	320	3	,	,	PUNCT
ejpam-5322	320	4	τ2)α	τ2)α	NOUN
ejpam-5322	320	5	-	-	PUNCT
ejpam-5322	320	6	continuity	continuity	NOUN
ejpam-5322	320	7	for	for	ADP
ejpam-5322	320	8	multifunctions	multifunction	NOUN
ejpam-5322	320	9	.	.	PUNCT
ejpam-5322	321	1	journal	journal	PROPN
ejpam-5322	321	2	of	of	ADP
ejpam-5322	321	3	mathematics	mathematic	NOUN
ejpam-5322	321	4	,	,	PUNCT
ejpam-5322	321	5	2020:6285763	2020:6285763	NUM
ejpam-5322	321	6	,	,	PUNCT
ejpam-5322	321	7	2020	2020	NUM
ejpam-5322	321	8	.	.	PUNCT
ejpam-5322	322	1	[	[	X
ejpam-5322	322	2	37	37	NUM
ejpam-5322	322	3	]	]	X
ejpam-5322	322	4	c.	c.	PROPN
ejpam-5322	322	5	viriyapong	viriyapong	PROPN
ejpam-5322	322	6	and	and	CCONJ
ejpam-5322	322	7	c.	c.	PROPN
ejpam-5322	322	8	boonpok	boonpok	PROPN
ejpam-5322	322	9	.	.	PUNCT
ejpam-5322	323	1	(	(	PUNCT
ejpam-5322	323	2	λ	λ	X
ejpam-5322	323	3	,	,	PUNCT
ejpam-5322	323	4	sp)-continuous	sp)-continuous	ADJ
ejpam-5322	323	5	functions	function	NOUN
ejpam-5322	323	6	.	.	PUNCT
ejpam-5322	324	1	wseas	wseas	VERB
ejpam-5322	324	2	transactions	transaction	NOUN
ejpam-5322	324	3	on	on	ADP
ejpam-5322	324	4	mathematics	mathematic	NOUN
ejpam-5322	324	5	,	,	PUNCT
ejpam-5322	324	6	21:380–385	21:380–385	NUM
ejpam-5322	324	7	,	,	PUNCT
ejpam-5322	324	8	2022	2022	NUM
ejpam-5322	324	9	.	.	PUNCT
ejpam-5322	325	1	[	[	X
ejpam-5322	325	2	38	38	NUM
ejpam-5322	325	3	]	]	X
ejpam-5322	325	4	n.	n.	PROPN
ejpam-5322	325	5	viriyapong	viriyapong	PROPN
ejpam-5322	325	6	,	,	PUNCT
ejpam-5322	325	7	s.	s.	PROPN
ejpam-5322	325	8	sompong	sompong	PROPN
ejpam-5322	325	9	,	,	PUNCT
ejpam-5322	325	10	and	and	CCONJ
ejpam-5322	325	11	c.	c.	PROPN
ejpam-5322	325	12	boonpok	boonpok	PROPN
ejpam-5322	325	13	.	.	PUNCT
ejpam-5322	326	1	(	(	PUNCT
ejpam-5322	326	2	τ1	τ1	NOUN
ejpam-5322	326	3	,	,	PUNCT
ejpam-5322	326	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5322	326	5	disconnectedness	disconnectedness	NOUN
ejpam-5322	326	6	in	in	ADP
ejpam-5322	326	7	bitopological	bitopological	ADJ
ejpam-5322	326	8	spaces	space	NOUN
ejpam-5322	326	9	.	.	PUNCT
ejpam-5322	327	1	international	international	ADJ
ejpam-5322	327	2	journal	journal	PROPN
ejpam-5322	327	3	of	of	ADP
ejpam-5322	327	4	mathematics	mathematic	NOUN
ejpam-5322	327	5	and	and	CCONJ
ejpam-5322	327	6	computer	computer	NOUN
ejpam-5322	327	7	science	science	NOUN
ejpam-5322	327	8	,	,	PUNCT
ejpam-5322	327	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5322	327	10	,	,	PUNCT
ejpam-5322	327	11	2024	2024	NUM
ejpam-5322	327	12	.	.	PUNCT