id	sid	tid	token	lemma	pos
ejpam-7045	1	1	european	european	PROPN
ejpam-7045	1	2	journal	journal	PROPN
ejpam-7045	1	3	of	of	ADP
ejpam-7045	1	4	pure	pure	ADJ
ejpam-7045	1	5	and	and	CCONJ
ejpam-7045	1	6	applied	applied	ADJ
ejpam-7045	1	7	mathematics	mathematic	NOUN
ejpam-7045	1	8	2025	2025	NUM
ejpam-7045	1	9	,	,	PUNCT
ejpam-7045	1	10	vol	vol	NOUN
ejpam-7045	1	11	.	.	PROPN
ejpam-7045	1	12	18	18	NUM
ejpam-7045	1	13	,	,	PUNCT
ejpam-7045	1	14	issue	issue	NOUN
ejpam-7045	1	15	4	4	NUM
ejpam-7045	1	16	,	,	PUNCT
ejpam-7045	1	17	article	article	NOUN
ejpam-7045	1	18	number	number	NOUN
ejpam-7045	1	19	7045	7045	NUM
ejpam-7045	1	20	issn	issn	PROPN
ejpam-7045	1	21	1307	1307	NUM
ejpam-7045	1	22	-	-	SYM
ejpam-7045	1	23	5543	5543	NUM
ejpam-7045	1	24	–	–	PUNCT
ejpam-7045	1	25	ejpam.com	ejpam.com	X
ejpam-7045	1	26	published	publish	VERB
ejpam-7045	1	27	by	by	ADP
ejpam-7045	1	28	new	new	PROPN
ejpam-7045	1	29	york	york	PROPN
ejpam-7045	1	30	business	business	PROPN
ejpam-7045	1	31	global	global	PROPN
ejpam-7045	1	32	on	on	ADP
ejpam-7045	1	33	upper	upper	ADJ
ejpam-7045	1	34	and	and	CCONJ
ejpam-7045	1	35	lower	low	ADJ
ejpam-7045	1	36	almost	almost	ADV
ejpam-7045	1	37	τ	τ	NOUN
ejpam-7045	1	38	⋆α(σ1	⋆α(σ1	AUX
ejpam-7045	1	39	,	,	PUNCT
ejpam-7045	1	40	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	1	41	multifunctions	multifunction	NOUN
ejpam-7045	1	42	chokchai	chokchai	ADJ
ejpam-7045	1	43	viriyapong1	viriyapong1	NOUN
ejpam-7045	1	44	,	,	PUNCT
ejpam-7045	1	45	areeyuth	areeyuth	NOUN
ejpam-7045	1	46	sama	sama	NOUN
ejpam-7045	1	47	-	-	PUNCT
ejpam-7045	1	48	ae2	ae2	PROPN
ejpam-7045	1	49	,	,	PUNCT
ejpam-7045	1	50	chawalit	chawalit	VERB
ejpam-7045	1	51	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7045	1	52	1	1	NUM
ejpam-7045	1	53	mathematics	mathematic	NOUN
ejpam-7045	1	54	and	and	CCONJ
ejpam-7045	1	55	applied	apply	VERB
ejpam-7045	1	56	mathematics	mathematics	PROPN
ejpam-7045	1	57	research	research	NOUN
ejpam-7045	1	58	unit	unit	NOUN
ejpam-7045	1	59	,	,	PUNCT
ejpam-7045	1	60	department	department	NOUN
ejpam-7045	1	61	of	of	ADP
ejpam-7045	1	62	mathematics	mathematic	NOUN
ejpam-7045	1	63	,	,	PUNCT
ejpam-7045	1	64	faculty	faculty	NOUN
ejpam-7045	1	65	of	of	ADP
ejpam-7045	1	66	science	science	NOUN
ejpam-7045	1	67	,	,	PUNCT
ejpam-7045	1	68	mahasarakham	mahasarakham	PROPN
ejpam-7045	1	69	university	university	PROPN
ejpam-7045	1	70	,	,	PUNCT
ejpam-7045	1	71	maha	maha	PROPN
ejpam-7045	1	72	sarakham	sarakham	PROPN
ejpam-7045	1	73	,	,	PUNCT
ejpam-7045	1	74	44150	44150	NUM
ejpam-7045	1	75	,	,	PUNCT
ejpam-7045	1	76	thailand	thailand	PROPN
ejpam-7045	1	77	2	2	NUM
ejpam-7045	1	78	department	department	NOUN
ejpam-7045	1	79	of	of	ADP
ejpam-7045	1	80	mathematics	mathematic	NOUN
ejpam-7045	1	81	and	and	CCONJ
ejpam-7045	1	82	computer	computer	NOUN
ejpam-7045	1	83	science	science	NOUN
ejpam-7045	1	84	,	,	PUNCT
ejpam-7045	1	85	faculty	faculty	NOUN
ejpam-7045	1	86	of	of	ADP
ejpam-7045	1	87	science	science	NOUN
ejpam-7045	1	88	and	and	CCONJ
ejpam-7045	1	89	technology	technology	NOUN
ejpam-7045	1	90	,	,	PUNCT
ejpam-7045	1	91	prince	prince	NOUN
ejpam-7045	1	92	of	of	ADP
ejpam-7045	1	93	songkla	songkla	PROPN
ejpam-7045	1	94	university	university	PROPN
ejpam-7045	1	95	,	,	PUNCT
ejpam-7045	1	96	pattani	pattani	NOUN
ejpam-7045	1	97	campus	campus	NOUN
ejpam-7045	1	98	,	,	PUNCT
ejpam-7045	1	99	pattani	pattani	NOUN
ejpam-7045	1	100	,	,	PUNCT
ejpam-7045	1	101	94000	94000	NUM
ejpam-7045	1	102	,	,	PUNCT
ejpam-7045	1	103	thailand	thailand	PROPN
ejpam-7045	1	104	abstract	abstract	PROPN
ejpam-7045	1	105	.	.	PUNCT
ejpam-7045	2	1	this	this	DET
ejpam-7045	2	2	paper	paper	NOUN
ejpam-7045	2	3	introduces	introduce	VERB
ejpam-7045	2	4	new	new	ADJ
ejpam-7045	2	5	classes	class	NOUN
ejpam-7045	2	6	of	of	ADP
ejpam-7045	2	7	continuous	continuous	ADJ
ejpam-7045	2	8	multifunctions	multifunction	NOUN
ejpam-7045	2	9	defined	define	VERB
ejpam-7045	2	10	between	between	ADP
ejpam-7045	2	11	an	an	DET
ejpam-7045	2	12	ideal	ideal	ADJ
ejpam-7045	2	13	topological	topological	ADJ
ejpam-7045	2	14	space	space	NOUN
ejpam-7045	2	15	and	and	CCONJ
ejpam-7045	2	16	a	a	DET
ejpam-7045	2	17	bitopological	bitopological	ADJ
ejpam-7045	2	18	space	space	NOUN
ejpam-7045	2	19	,	,	PUNCT
ejpam-7045	2	20	called	call	VERB
ejpam-7045	2	21	upper	upper	ADV
ejpam-7045	2	22	almost	almost	ADV
ejpam-7045	2	23	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	2	24	,	,	PUNCT
ejpam-7045	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	2	26	multifunctions	multifunction	NOUN
ejpam-7045	2	27	and	and	CCONJ
ejpam-7045	2	28	lower	low	ADJ
ejpam-7045	2	29	almost	almost	ADV
ejpam-7045	2	30	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	2	31	,	,	PUNCT
ejpam-7045	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	2	33	multifunctions	multifunction	NOUN
ejpam-7045	2	34	.	.	PUNCT
ejpam-7045	3	1	furthermore	furthermore	ADV
ejpam-7045	3	2	,	,	PUNCT
ejpam-7045	3	3	several	several	ADJ
ejpam-7045	3	4	characterizations	characterization	NOUN
ejpam-7045	3	5	and	and	CCONJ
ejpam-7045	3	6	some	some	DET
ejpam-7045	3	7	properties	property	NOUN
ejpam-7045	3	8	concerning	concern	VERB
ejpam-7045	3	9	upper	upper	ADJ
ejpam-7045	3	10	almost	almost	ADV
ejpam-7045	3	11	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	3	12	,	,	PUNCT
ejpam-7045	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	3	14	multifunctions	multifunction	NOUN
ejpam-7045	3	15	and	and	CCONJ
ejpam-7045	3	16	lower	low	ADJ
ejpam-7045	3	17	almost	almost	ADV
ejpam-7045	3	18	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	3	19	,	,	PUNCT
ejpam-7045	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	3	21	multifunctions	multifunction	NOUN
ejpam-7045	3	22	are	be	AUX
ejpam-7045	3	23	investigated	investigate	VERB
ejpam-7045	3	24	.	.	PUNCT
ejpam-7045	4	1	2020	2020	NUM
ejpam-7045	4	2	mathematics	mathematic	NOUN
ejpam-7045	4	3	subject	subject	NOUN
ejpam-7045	4	4	classifications	classification	NOUN
ejpam-7045	4	5	:	:	PUNCT
ejpam-7045	4	6	54c08	54c08	NUM
ejpam-7045	4	7	,	,	PUNCT
ejpam-7045	4	8	54c60	54c60	NUM
ejpam-7045	4	9	key	key	ADJ
ejpam-7045	4	10	words	word	NOUN
ejpam-7045	4	11	and	and	CCONJ
ejpam-7045	4	12	phrases	phrase	NOUN
ejpam-7045	4	13	:	:	PUNCT
ejpam-7045	4	14	upper	upper	ADJ
ejpam-7045	4	15	almost	almost	ADV
ejpam-7045	4	16	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	4	17	,	,	PUNCT
ejpam-7045	4	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	4	19	multifunction	multifunction	NOUN
ejpam-7045	4	20	,	,	PUNCT
ejpam-7045	4	21	lower	low	ADJ
ejpam-7045	4	22	almost	almost	ADV
ejpam-7045	4	23	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	4	24	,	,	PUNCT
ejpam-7045	4	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	4	26	multifunction	multifunction	NOUN
ejpam-7045	4	27	1	1	NUM
ejpam-7045	4	28	.	.	PUNCT
ejpam-7045	4	29	introduction	introduction	NOUN
ejpam-7045	4	30	in	in	ADP
ejpam-7045	4	31	1988	1988	NUM
ejpam-7045	4	32	,	,	PUNCT
ejpam-7045	4	33	noiri	noiri	ADV
ejpam-7045	4	34	[	[	X
ejpam-7045	4	35	1	1	X
ejpam-7045	4	36	]	]	PUNCT
ejpam-7045	4	37	introduced	introduce	VERB
ejpam-7045	4	38	a	a	DET
ejpam-7045	4	39	class	class	NOUN
ejpam-7045	4	40	of	of	ADP
ejpam-7045	4	41	functions	function	NOUN
ejpam-7045	4	42	between	between	ADP
ejpam-7045	4	43	topological	topological	ADJ
ejpam-7045	4	44	spaces	space	NOUN
ejpam-7045	4	45	,	,	PUNCT
ejpam-7045	4	46	called	call	VERB
ejpam-7045	4	47	almost	almost	ADV
ejpam-7045	4	48	α	α	ADJ
ejpam-7045	4	49	-	-	PUNCT
ejpam-7045	4	50	continuous	continuous	ADJ
ejpam-7045	4	51	functions	function	NOUN
ejpam-7045	4	52	.	.	PUNCT
ejpam-7045	5	1	furthermore	furthermore	ADV
ejpam-7045	5	2	,	,	PUNCT
ejpam-7045	5	3	noiri	noiri	ADV
ejpam-7045	5	4	[	[	X
ejpam-7045	5	5	1	1	X
ejpam-7045	5	6	]	]	PUNCT
ejpam-7045	5	7	investigated	investigate	VERB
ejpam-7045	5	8	several	several	ADJ
ejpam-7045	5	9	characterizations	characterization	NOUN
ejpam-7045	5	10	and	and	CCONJ
ejpam-7045	5	11	some	some	DET
ejpam-7045	5	12	basic	basic	ADJ
ejpam-7045	5	13	properties	property	NOUN
ejpam-7045	5	14	of	of	ADP
ejpam-7045	5	15	almost	almost	ADV
ejpam-7045	5	16	α	α	ADJ
ejpam-7045	5	17	-	-	ADJ
ejpam-7045	5	18	continuous	continuous	ADJ
ejpam-7045	5	19	functions	function	NOUN
ejpam-7045	5	20	.	.	PUNCT
ejpam-7045	6	1	in	in	ADP
ejpam-7045	6	2	1996	1996	NUM
ejpam-7045	6	3	,	,	PUNCT
ejpam-7045	6	4	popa	popa	NOUN
ejpam-7045	6	5	and	and	CCONJ
ejpam-7045	6	6	noiri	noiri	ADV
ejpam-7045	6	7	[	[	X
ejpam-7045	6	8	2	2	X
ejpam-7045	6	9	]	]	PUNCT
ejpam-7045	6	10	extended	extend	VERB
ejpam-7045	6	11	the	the	DET
ejpam-7045	6	12	concept	concept	NOUN
ejpam-7045	6	13	of	of	ADP
ejpam-7045	6	14	almost	almost	ADV
ejpam-7045	6	15	α	α	ADJ
ejpam-7045	6	16	-	-	ADJ
ejpam-7045	6	17	continuous	continuous	ADJ
ejpam-7045	6	18	functions	function	NOUN
ejpam-7045	6	19	to	to	ADP
ejpam-7045	6	20	multifunctions	multifunction	NOUN
ejpam-7045	6	21	and	and	CCONJ
ejpam-7045	6	22	presented	present	VERB
ejpam-7045	6	23	classes	class	NOUN
ejpam-7045	6	24	of	of	ADP
ejpam-7045	6	25	multifunctions	multifunction	NOUN
ejpam-7045	6	26	defined	define	VERB
ejpam-7045	6	27	from	from	ADP
ejpam-7045	6	28	a	a	DET
ejpam-7045	6	29	topological	topological	ADJ
ejpam-7045	6	30	space	space	NOUN
ejpam-7045	6	31	into	into	ADP
ejpam-7045	6	32	a	a	DET
ejpam-7045	6	33	topological	topological	ADJ
ejpam-7045	6	34	space	space	NOUN
ejpam-7045	6	35	,	,	PUNCT
ejpam-7045	6	36	namely	namely	ADV
ejpam-7045	6	37	upper	upper	ADJ
ejpam-7045	6	38	almost	almost	ADV
ejpam-7045	6	39	α	α	ADJ
ejpam-7045	6	40	-	-	ADJ
ejpam-7045	6	41	continuous	continuous	ADJ
ejpam-7045	6	42	multifunctions	multifunction	NOUN
ejpam-7045	6	43	and	and	CCONJ
ejpam-7045	6	44	lower	low	ADJ
ejpam-7045	6	45	almost	almost	ADV
ejpam-7045	6	46	α	α	ADJ
ejpam-7045	6	47	-	-	ADJ
ejpam-7045	6	48	continuous	continuous	ADJ
ejpam-7045	6	49	multifunctions	multifunction	NOUN
ejpam-7045	6	50	.	.	PUNCT
ejpam-7045	7	1	in	in	ADP
ejpam-7045	7	2	particular	particular	ADJ
ejpam-7045	7	3	,	,	PUNCT
ejpam-7045	7	4	several	several	ADJ
ejpam-7045	7	5	characterizations	characterization	NOUN
ejpam-7045	7	6	and	and	CCONJ
ejpam-7045	7	7	some	some	DET
ejpam-7045	7	8	properties	property	NOUN
ejpam-7045	7	9	concerning	concern	VERB
ejpam-7045	7	10	upper	upper	ADJ
ejpam-7045	7	11	almost	almost	ADV
ejpam-7045	7	12	αcontinuous	αcontinuous	ADJ
ejpam-7045	7	13	multifunctions	multifunction	NOUN
ejpam-7045	7	14	and	and	CCONJ
ejpam-7045	7	15	lower	low	ADJ
ejpam-7045	7	16	almost	almost	ADV
ejpam-7045	7	17	α	α	ADJ
ejpam-7045	7	18	-	-	ADJ
ejpam-7045	7	19	continuous	continuous	ADJ
ejpam-7045	7	20	multifunctions	multifunction	NOUN
ejpam-7045	7	21	were	be	AUX
ejpam-7045	7	22	established	establish	VERB
ejpam-7045	7	23	in	in	ADP
ejpam-7045	7	24	[	[	X
ejpam-7045	7	25	2	2	NUM
ejpam-7045	7	26	]	]	PUNCT
ejpam-7045	7	27	.	.	PUNCT
ejpam-7045	8	1	on	on	ADP
ejpam-7045	8	2	the	the	DET
ejpam-7045	8	3	other	other	ADJ
ejpam-7045	8	4	hand	hand	NOUN
ejpam-7045	8	5	,	,	PUNCT
ejpam-7045	8	6	the	the	DET
ejpam-7045	8	7	present	present	ADJ
ejpam-7045	8	8	author	author	NOUN
ejpam-7045	8	9	introduced	introduce	VERB
ejpam-7045	8	10	and	and	CCONJ
ejpam-7045	8	11	studied	study	VERB
ejpam-7045	8	12	four	four	NUM
ejpam-7045	8	13	classes	class	NOUN
ejpam-7045	8	14	of	of	ADP
ejpam-7045	8	15	multifunctions	multifunction	NOUN
ejpam-7045	8	16	defined	define	VERB
ejpam-7045	8	17	from	from	ADP
ejpam-7045	8	18	an	an	DET
ejpam-7045	8	19	ideal	ideal	ADJ
ejpam-7045	8	20	topological	topological	ADJ
ejpam-7045	8	21	space	space	NOUN
ejpam-7045	8	22	into	into	ADP
ejpam-7045	8	23	an	an	DET
ejpam-7045	8	24	ideal	ideal	ADJ
ejpam-7045	8	25	topological	topological	ADJ
ejpam-7045	8	26	space	space	NOUN
ejpam-7045	8	27	,	,	PUNCT
ejpam-7045	8	28	called	call	VERB
ejpam-7045	8	29	upper	upper	ADJ
ejpam-7045	8	30	almost	almost	ADV
ejpam-7045	8	31	⋆-continuous	⋆-continuous	ADJ
ejpam-7045	8	32	multifunctions	multifunction	NOUN
ejpam-7045	9	1	[	[	X
ejpam-7045	9	2	3	3	NUM
ejpam-7045	9	3	]	]	PUNCT
ejpam-7045	9	4	,	,	PUNCT
ejpam-7045	9	5	lower	low	ADJ
ejpam-7045	9	6	almost	almost	ADV
ejpam-7045	9	7	⋆-continuous	⋆-continuous	ADJ
ejpam-7045	9	8	multifunctions	multifunction	NOUN
ejpam-7045	10	1	[	[	X
ejpam-7045	10	2	3	3	NUM
ejpam-7045	10	3	]	]	PUNCT
ejpam-7045	10	4	,	,	PUNCT
ejpam-7045	10	5	upper	upper	ADJ
ejpam-7045	10	6	almost	almost	ADV
ejpam-7045	10	7	α(⋆)-continuous	α(⋆)-continuous	ADJ
ejpam-7045	10	8	multifunctions	multifunction	NOUN
ejpam-7045	11	1	[	[	X
ejpam-7045	11	2	4	4	NUM
ejpam-7045	11	3	]	]	PUNCT
ejpam-7045	11	4	,	,	PUNCT
ejpam-7045	11	5	lower	low	ADJ
ejpam-7045	11	6	almost	almost	ADV
ejpam-7045	11	7	α(⋆)-continuous	α(⋆)-continuous	NUM
ejpam-7045	11	8	multifunctions	multifunction	NOUN
ejpam-7045	12	1	[	[	X
ejpam-7045	12	2	4	4	NUM
ejpam-7045	12	3	]	]	PUNCT
ejpam-7045	12	4	,	,	PUNCT
ejpam-7045	12	5	upper	upper	ADJ
ejpam-7045	12	6	almost	almost	ADV
ejpam-7045	12	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7045	12	8	multifunctions	multifunction	NOUN
ejpam-7045	13	1	[	[	X
ejpam-7045	13	2	5	5	NUM
ejpam-7045	13	3	]	]	PUNCT
ejpam-7045	13	4	,	,	PUNCT
ejpam-7045	13	5	lower	low	ADJ
ejpam-7045	13	6	almost	almost	ADV
ejpam-7045	13	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7045	13	8	∗corresponding	∗corresponding	NOUN
ejpam-7045	13	9	author	author	NOUN
ejpam-7045	13	10	.	.	PUNCT
ejpam-7045	14	1	doi	doi	NOUN
ejpam-7045	14	2	:	:	PUNCT
ejpam-7045	14	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7045	https://doi.org/10.29020/nybg.ejpam.v18i4.7045	ADJ
ejpam-7045	14	4	email	email	NOUN
ejpam-7045	14	5	addresses	address	NOUN
ejpam-7045	14	6	:	:	PUNCT
ejpam-7045	14	7	nongluk.h@msu.ac.th	nongluk.h@msu.ac.th	PROPN
ejpam-7045	14	8	(	(	PUNCT
ejpam-7045	14	9	c.	c.	PROPN
ejpam-7045	14	10	viriyapong	viriyapong	PROPN
ejpam-7045	14	11	)	)	PUNCT
ejpam-7045	14	12	,	,	PUNCT
ejpam-7045	14	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7045	14	14	(	(	PUNCT
ejpam-7045	14	15	a.	a.	PROPN
ejpam-7045	14	16	sama	sama	PROPN
ejpam-7045	14	17	-	-	PUNCT
ejpam-7045	14	18	ae	ae	PROPN
ejpam-7045	14	19	)	)	PUNCT
ejpam-7045	14	20	,	,	PUNCT
ejpam-7045	14	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7045	14	22	(	(	PUNCT
ejpam-7045	14	23	c.	c.	PROPN
ejpam-7045	14	24	boonpok	boonpok	PROPN
ejpam-7045	14	25	)	)	PUNCT
ejpam-7045	14	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7045	15	1	1	1	NUM
ejpam-7045	15	2	copyright	copyright	NOUN
ejpam-7045	15	3	:	:	PUNCT
ejpam-7045	15	4	©	©	PROPN
ejpam-7045	15	5	2025	2025	NUM
ejpam-7045	15	6	the	the	DET
ejpam-7045	15	7	author(s	author(s	NOUN
ejpam-7045	15	8	)	)	PUNCT
ejpam-7045	15	9	.	.	PUNCT
ejpam-7045	16	1	(	(	PUNCT
ejpam-7045	16	2	cc	cc	NOUN
ejpam-7045	16	3	by	by	ADP
ejpam-7045	16	4	-	-	PUNCT
ejpam-7045	16	5	nc	nc	PROPN
ejpam-7045	16	6	4.0	4.0	NUM
ejpam-7045	16	7	)	)	PUNCT
ejpam-7045	16	8	c.	c.	PROPN
ejpam-7045	16	9	viriyapong	viriyapong	PROPN
ejpam-7045	16	10	,	,	PUNCT
ejpam-7045	16	11	a.	a.	PROPN
ejpam-7045	16	12	sama	sama	PROPN
ejpam-7045	16	13	-	-	PUNCT
ejpam-7045	16	14	ae	ae	PROPN
ejpam-7045	16	15	,	,	PUNCT
ejpam-7045	16	16	c.	c.	PROPN
ejpam-7045	16	17	boonpok	boonpok	PROPN
ejpam-7045	16	18	/	/	SYM
ejpam-7045	16	19	eur	eur	PROPN
ejpam-7045	16	20	.	.	PUNCT
ejpam-7045	17	1	j.	j.	PROPN
ejpam-7045	17	2	pure	pure	PROPN
ejpam-7045	17	3	appl	appl	PROPN
ejpam-7045	17	4	.	.	PROPN
ejpam-7045	17	5	math	math	PROPN
ejpam-7045	17	6	,	,	PUNCT
ejpam-7045	17	7	18	18	NUM
ejpam-7045	17	8	(	(	PUNCT
ejpam-7045	17	9	4	4	NUM
ejpam-7045	17	10	)	)	PUNCT
ejpam-7045	17	11	(	(	PUNCT
ejpam-7045	17	12	2025	2025	NUM
ejpam-7045	17	13	)	)	PUNCT
ejpam-7045	17	14	,	,	PUNCT
ejpam-7045	17	15	7045	7045	NUM
ejpam-7045	17	16	2	2	NUM
ejpam-7045	17	17	of	of	ADP
ejpam-7045	17	18	12	12	NUM
ejpam-7045	17	19	multifunctions	multifunction	NOUN
ejpam-7045	17	20	[	[	X
ejpam-7045	17	21	5	5	NUM
ejpam-7045	17	22	]	]	PUNCT
ejpam-7045	17	23	and	and	CCONJ
ejpam-7045	17	24	almost	almost	ADV
ejpam-7045	17	25	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-7045	17	26	multifunctions	multifunction	NOUN
ejpam-7045	17	27	[	[	X
ejpam-7045	17	28	6	6	NUM
ejpam-7045	17	29	]	]	PUNCT
ejpam-7045	17	30	.	.	PUNCT
ejpam-7045	18	1	pue	pue	NOUN
ejpam-7045	18	2	-	-	PUNCT
ejpam-7045	18	3	on	on	NOUN
ejpam-7045	18	4	et	et	PROPN
ejpam-7045	18	5	al	al	PROPN
ejpam-7045	18	6	.	.	PUNCT
ejpam-7045	19	1	[	[	X
ejpam-7045	19	2	7	7	X
ejpam-7045	19	3	]	]	PUNCT
ejpam-7045	19	4	introduced	introduce	VERB
ejpam-7045	19	5	and	and	CCONJ
ejpam-7045	19	6	studied	study	VERB
ejpam-7045	19	7	two	two	NUM
ejpam-7045	19	8	classes	class	NOUN
ejpam-7045	19	9	of	of	ADP
ejpam-7045	19	10	multifunctions	multifunction	NOUN
ejpam-7045	19	11	between	between	ADP
ejpam-7045	19	12	bitopological	bitopological	ADJ
ejpam-7045	19	13	spaces	space	NOUN
ejpam-7045	19	14	,	,	PUNCT
ejpam-7045	19	15	namely	namely	ADV
ejpam-7045	19	16	upper	upper	ADJ
ejpam-7045	19	17	(	(	PUNCT
ejpam-7045	19	18	τ1	τ1	NOUN
ejpam-7045	19	19	,	,	PUNCT
ejpam-7045	19	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7045	19	21	multifunctions	multifunction	NOUN
ejpam-7045	19	22	and	and	CCONJ
ejpam-7045	19	23	lower	low	ADJ
ejpam-7045	19	24	(	(	PUNCT
ejpam-7045	19	25	τ1	τ1	NOUN
ejpam-7045	19	26	,	,	PUNCT
ejpam-7045	19	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7045	19	28	multifunctions	multifunction	NOUN
ejpam-7045	19	29	.	.	PUNCT
ejpam-7045	20	1	moreover	moreover	ADV
ejpam-7045	20	2	,	,	PUNCT
ejpam-7045	20	3	boonpok	boonpok	NOUN
ejpam-7045	20	4	and	and	CCONJ
ejpam-7045	20	5	pue	pue	NOUN
ejpam-7045	20	6	-	-	PUNCT
ejpam-7045	20	7	on	on	ADP
ejpam-7045	20	8	[	[	X
ejpam-7045	20	9	8	8	NUM
ejpam-7045	20	10	]	]	PUNCT
ejpam-7045	20	11	introduced	introduce	VERB
ejpam-7045	20	12	and	and	CCONJ
ejpam-7045	20	13	investigated	investigate	VERB
ejpam-7045	20	14	the	the	DET
ejpam-7045	20	15	concepts	concept	NOUN
ejpam-7045	20	16	of	of	ADP
ejpam-7045	20	17	upper	upper	ADJ
ejpam-7045	20	18	almost	almost	ADV
ejpam-7045	20	19	(	(	PUNCT
ejpam-7045	20	20	τ1	τ1	NOUN
ejpam-7045	20	21	,	,	PUNCT
ejpam-7045	20	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7045	20	23	multifunctions	multifunction	NOUN
ejpam-7045	20	24	and	and	CCONJ
ejpam-7045	20	25	lower	low	ADJ
ejpam-7045	20	26	almost	almost	ADV
ejpam-7045	20	27	(	(	PUNCT
ejpam-7045	20	28	τ1	τ1	NOUN
ejpam-7045	20	29	,	,	PUNCT
ejpam-7045	20	30	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7045	20	31	multifunctions	multifunction	NOUN
ejpam-7045	20	32	.	.	PUNCT
ejpam-7045	21	1	in	in	ADP
ejpam-7045	21	2	[	[	X
ejpam-7045	21	3	9	9	NUM
ejpam-7045	21	4	]	]	PUNCT
ejpam-7045	21	5	,	,	PUNCT
ejpam-7045	21	6	the	the	DET
ejpam-7045	21	7	present	present	ADJ
ejpam-7045	21	8	authors	author	NOUN
ejpam-7045	21	9	introduced	introduce	VERB
ejpam-7045	21	10	and	and	CCONJ
ejpam-7045	21	11	studied	study	VERB
ejpam-7045	21	12	the	the	DET
ejpam-7045	21	13	concepts	concept	NOUN
ejpam-7045	21	14	of	of	ADP
ejpam-7045	21	15	upper	upper	ADJ
ejpam-7045	21	16	almost	almost	ADV
ejpam-7045	21	17	(	(	PUNCT
ejpam-7045	21	18	τ1	τ1	NOUN
ejpam-7045	21	19	,	,	PUNCT
ejpam-7045	21	20	τ2)αcontinuous	τ2)αcontinuous	ADJ
ejpam-7045	21	21	multifunctions	multifunction	NOUN
ejpam-7045	21	22	and	and	CCONJ
ejpam-7045	21	23	lower	low	ADJ
ejpam-7045	21	24	almost	almost	ADV
ejpam-7045	21	25	(	(	PUNCT
ejpam-7045	21	26	τ1	τ1	NOUN
ejpam-7045	21	27	,	,	PUNCT
ejpam-7045	21	28	τ2)α	τ2)α	ADJ
ejpam-7045	21	29	-	-	PUNCT
ejpam-7045	21	30	continuous	continuous	ADJ
ejpam-7045	21	31	multifunctions	multifunction	NOUN
ejpam-7045	21	32	.	.	PUNCT
ejpam-7045	22	1	quite	quite	ADV
ejpam-7045	22	2	recently	recently	ADV
ejpam-7045	22	3	,	,	PUNCT
ejpam-7045	22	4	viriyapong	viriyapong	PROPN
ejpam-7045	22	5	et	et	PROPN
ejpam-7045	22	6	al	al	PROPN
ejpam-7045	22	7	.	.	PUNCT
ejpam-7045	23	1	[	[	X
ejpam-7045	23	2	10	10	NUM
ejpam-7045	23	3	]	]	PUNCT
ejpam-7045	23	4	presented	present	VERB
ejpam-7045	23	5	new	new	ADJ
ejpam-7045	23	6	classes	class	NOUN
ejpam-7045	23	7	of	of	ADP
ejpam-7045	23	8	continuous	continuous	ADJ
ejpam-7045	23	9	multifunctions	multifunction	NOUN
ejpam-7045	23	10	defined	define	VERB
ejpam-7045	23	11	from	from	ADP
ejpam-7045	23	12	an	an	DET
ejpam-7045	23	13	ideal	ideal	ADJ
ejpam-7045	23	14	topological	topological	ADJ
ejpam-7045	23	15	space	space	NOUN
ejpam-7045	23	16	into	into	ADP
ejpam-7045	23	17	a	a	DET
ejpam-7045	23	18	bitopological	bitopological	ADJ
ejpam-7045	23	19	space	space	NOUN
ejpam-7045	23	20	,	,	PUNCT
ejpam-7045	23	21	namely	namely	ADV
ejpam-7045	23	22	upper	upper	ADJ
ejpam-7045	23	23	almost	almost	ADV
ejpam-7045	23	24	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7045	23	25	,	,	PUNCT
ejpam-7045	23	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7045	23	27	multifunctions	multifunction	NOUN
ejpam-7045	23	28	and	and	CCONJ
ejpam-7045	23	29	lower	low	ADJ
ejpam-7045	23	30	almost	almost	ADV
ejpam-7045	23	31	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7045	23	32	,	,	PUNCT
ejpam-7045	23	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	23	34	multifunctions	multifunction	NOUN
ejpam-7045	23	35	.	.	PUNCT
ejpam-7045	24	1	in	in	ADP
ejpam-7045	24	2	this	this	DET
ejpam-7045	24	3	paper	paper	NOUN
ejpam-7045	24	4	,	,	PUNCT
ejpam-7045	24	5	we	we	PRON
ejpam-7045	24	6	introduce	introduce	VERB
ejpam-7045	24	7	the	the	DET
ejpam-7045	24	8	concepts	concept	NOUN
ejpam-7045	24	9	of	of	ADP
ejpam-7045	24	10	continuous	continuous	ADJ
ejpam-7045	24	11	multifunctions	multifunction	NOUN
ejpam-7045	24	12	between	between	ADP
ejpam-7045	24	13	an	an	DET
ejpam-7045	24	14	ideal	ideal	ADJ
ejpam-7045	24	15	topological	topological	ADJ
ejpam-7045	24	16	space	space	NOUN
ejpam-7045	24	17	and	and	CCONJ
ejpam-7045	24	18	a	a	DET
ejpam-7045	24	19	bitopological	bitopological	ADJ
ejpam-7045	24	20	space	space	NOUN
ejpam-7045	24	21	,	,	PUNCT
ejpam-7045	24	22	called	call	VERB
ejpam-7045	24	23	upper	upper	ADV
ejpam-7045	24	24	almost	almost	ADV
ejpam-7045	24	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	24	26	,	,	PUNCT
ejpam-7045	24	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	24	28	multifunctions	multifunction	NOUN
ejpam-7045	24	29	and	and	CCONJ
ejpam-7045	24	30	lower	low	ADJ
ejpam-7045	24	31	almost	almost	ADV
ejpam-7045	24	32	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	24	33	,	,	PUNCT
ejpam-7045	24	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	24	35	multifunctions	multifunction	NOUN
ejpam-7045	24	36	.	.	PUNCT
ejpam-7045	25	1	we	we	PRON
ejpam-7045	25	2	also	also	ADV
ejpam-7045	25	3	investigate	investigate	VERB
ejpam-7045	25	4	several	several	ADJ
ejpam-7045	25	5	characterizations	characterization	NOUN
ejpam-7045	25	6	of	of	ADP
ejpam-7045	25	7	upper	upper	ADJ
ejpam-7045	25	8	almost	almost	ADV
ejpam-7045	25	9	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	25	10	,	,	PUNCT
ejpam-7045	25	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	25	12	multifunctions	multifunction	NOUN
ejpam-7045	25	13	and	and	CCONJ
ejpam-7045	25	14	lower	low	ADJ
ejpam-7045	25	15	almost	almost	ADV
ejpam-7045	25	16	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	25	17	,	,	PUNCT
ejpam-7045	25	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	25	19	multifunctions	multifunction	NOUN
ejpam-7045	25	20	.	.	PUNCT
ejpam-7045	26	1	2	2	X
ejpam-7045	26	2	.	.	X
ejpam-7045	26	3	preliminaries	preliminary	NOUN
ejpam-7045	26	4	throughout	throughout	ADP
ejpam-7045	26	5	the	the	DET
ejpam-7045	26	6	present	present	ADJ
ejpam-7045	26	7	paper	paper	NOUN
ejpam-7045	26	8	,	,	PUNCT
ejpam-7045	26	9	spaces	space	NOUN
ejpam-7045	26	10	(	(	PUNCT
ejpam-7045	26	11	x	x	NOUN
ejpam-7045	26	12	,	,	PUNCT
ejpam-7045	26	13	τ1	τ1	NOUN
ejpam-7045	26	14	,	,	PUNCT
ejpam-7045	26	15	τ2	τ2	NOUN
ejpam-7045	26	16	)	)	PUNCT
ejpam-7045	26	17	and	and	CCONJ
ejpam-7045	26	18	(	(	PUNCT
ejpam-7045	26	19	y	y	PROPN
ejpam-7045	26	20	,	,	PUNCT
ejpam-7045	26	21	σ1	σ1	PROPN
ejpam-7045	26	22	,	,	PUNCT
ejpam-7045	26	23	σ2	σ2	NOUN
ejpam-7045	26	24	)	)	PUNCT
ejpam-7045	26	25	(	(	PUNCT
ejpam-7045	26	26	or	or	CCONJ
ejpam-7045	26	27	simply	simply	ADV
ejpam-7045	26	28	x	x	X
ejpam-7045	26	29	and	and	CCONJ
ejpam-7045	26	30	y	y	PROPN
ejpam-7045	26	31	)	)	PUNCT
ejpam-7045	26	32	always	always	ADV
ejpam-7045	26	33	mean	mean	VERB
ejpam-7045	26	34	bitopological	bitopological	ADJ
ejpam-7045	26	35	spaces	space	NOUN
ejpam-7045	26	36	on	on	ADP
ejpam-7045	26	37	which	which	PRON
ejpam-7045	26	38	no	no	DET
ejpam-7045	26	39	separation	separation	NOUN
ejpam-7045	26	40	axioms	axiom	NOUN
ejpam-7045	26	41	are	be	AUX
ejpam-7045	26	42	assumed	assume	VERB
ejpam-7045	26	43	unless	unless	SCONJ
ejpam-7045	26	44	explicitly	explicitly	ADV
ejpam-7045	26	45	stated	state	VERB
ejpam-7045	26	46	.	.	PUNCT
ejpam-7045	27	1	let	let	VERB
ejpam-7045	27	2	a	a	DET
ejpam-7045	27	3	be	be	AUX
ejpam-7045	27	4	a	a	DET
ejpam-7045	27	5	subset	subset	NOUN
ejpam-7045	27	6	of	of	ADP
ejpam-7045	27	7	a	a	DET
ejpam-7045	27	8	bitopological	bitopological	ADJ
ejpam-7045	27	9	space	space	NOUN
ejpam-7045	27	10	(	(	PUNCT
ejpam-7045	27	11	x	x	NOUN
ejpam-7045	27	12	,	,	PUNCT
ejpam-7045	27	13	τ1	τ1	NOUN
ejpam-7045	27	14	,	,	PUNCT
ejpam-7045	27	15	τ2	τ2	NOUN
ejpam-7045	27	16	)	)	PUNCT
ejpam-7045	27	17	.	.	PUNCT
ejpam-7045	28	1	the	the	DET
ejpam-7045	28	2	closure	closure	NOUN
ejpam-7045	28	3	of	of	ADP
ejpam-7045	28	4	a	a	PRON
ejpam-7045	28	5	and	and	CCONJ
ejpam-7045	28	6	the	the	DET
ejpam-7045	28	7	interior	interior	NOUN
ejpam-7045	28	8	of	of	ADP
ejpam-7045	28	9	a	a	PRON
ejpam-7045	28	10	with	with	ADP
ejpam-7045	28	11	respect	respect	NOUN
ejpam-7045	28	12	to	to	ADP
ejpam-7045	28	13	τi	τi	PROPN
ejpam-7045	28	14	are	be	AUX
ejpam-7045	28	15	denoted	denote	VERB
ejpam-7045	28	16	by	by	ADP
ejpam-7045	28	17	τi	τi	NOUN
ejpam-7045	28	18	-	-	PUNCT
ejpam-7045	28	19	cl(a	cl(a	NUM
ejpam-7045	28	20	)	)	PUNCT
ejpam-7045	28	21	and	and	CCONJ
ejpam-7045	28	22	τi	τi	NOUN
ejpam-7045	28	23	-	-	PUNCT
ejpam-7045	28	24	int(a	int(a	NOUN
ejpam-7045	28	25	)	)	PUNCT
ejpam-7045	28	26	,	,	PUNCT
ejpam-7045	28	27	respectively	respectively	ADV
ejpam-7045	28	28	,	,	PUNCT
ejpam-7045	28	29	for	for	ADP
ejpam-7045	28	30	i	i	PROPN
ejpam-7045	28	31	=	=	SYM
ejpam-7045	28	32	1	1	NUM
ejpam-7045	28	33	,	,	PUNCT
ejpam-7045	28	34	2	2	NUM
ejpam-7045	28	35	.	.	X
ejpam-7045	28	36	a	a	DET
ejpam-7045	28	37	subset	subset	NOUN
ejpam-7045	28	38	a	a	PRON
ejpam-7045	28	39	of	of	ADP
ejpam-7045	28	40	a	a	DET
ejpam-7045	28	41	bitopological	bitopological	ADJ
ejpam-7045	28	42	space	space	NOUN
ejpam-7045	28	43	(	(	PUNCT
ejpam-7045	28	44	x	x	NOUN
ejpam-7045	28	45	,	,	PUNCT
ejpam-7045	28	46	τ1	τ1	NOUN
ejpam-7045	28	47	,	,	PUNCT
ejpam-7045	28	48	τ2	τ2	NOUN
ejpam-7045	28	49	)	)	PUNCT
ejpam-7045	28	50	is	be	AUX
ejpam-7045	28	51	called	call	VERB
ejpam-7045	28	52	τ1τ2	τ1τ2	VERB
ejpam-7045	28	53	-	-	ADJ
ejpam-7045	28	54	closed	closed	ADJ
ejpam-7045	28	55	[	[	X
ejpam-7045	28	56	11	11	NUM
ejpam-7045	28	57	]	]	X
ejpam-7045	28	58	if	if	SCONJ
ejpam-7045	28	59	a	a	DET
ejpam-7045	28	60	=	=	NOUN
ejpam-7045	28	61	τ1	τ1	NOUN
ejpam-7045	28	62	-	-	PUNCT
ejpam-7045	28	63	cl(τ2	cl(τ2	NOUN
ejpam-7045	28	64	-	-	PUNCT
ejpam-7045	28	65	cl(a	cl(a	NUM
ejpam-7045	28	66	)	)	PUNCT
ejpam-7045	28	67	)	)	PUNCT
ejpam-7045	28	68	.	.	PUNCT
ejpam-7045	29	1	the	the	DET
ejpam-7045	29	2	complement	complement	NOUN
ejpam-7045	29	3	of	of	ADP
ejpam-7045	29	4	a	a	DET
ejpam-7045	29	5	τ1τ2	τ1τ2	ADJ
ejpam-7045	29	6	-	-	ADJ
ejpam-7045	29	7	closed	closed	ADJ
ejpam-7045	29	8	set	set	NOUN
ejpam-7045	29	9	is	be	AUX
ejpam-7045	29	10	called	call	VERB
ejpam-7045	29	11	τ1τ2	τ1τ2	NOUN
ejpam-7045	29	12	-	-	ADJ
ejpam-7045	29	13	open	open	ADJ
ejpam-7045	29	14	.	.	PUNCT
ejpam-7045	30	1	the	the	DET
ejpam-7045	30	2	intersection	intersection	NOUN
ejpam-7045	30	3	of	of	ADP
ejpam-7045	30	4	all	all	DET
ejpam-7045	30	5	τ1τ2	τ1τ2	ADJ
ejpam-7045	30	6	-	-	ADJ
ejpam-7045	30	7	closed	closed	ADJ
ejpam-7045	30	8	sets	set	NOUN
ejpam-7045	30	9	of	of	ADP
ejpam-7045	30	10	x	x	PUNCT
ejpam-7045	30	11	containing	contain	VERB
ejpam-7045	30	12	a	a	PRON
ejpam-7045	30	13	is	be	AUX
ejpam-7045	30	14	called	call	VERB
ejpam-7045	30	15	the	the	DET
ejpam-7045	30	16	τ1τ2	τ1τ2	NOUN
ejpam-7045	30	17	-	-	NOUN
ejpam-7045	30	18	closure	closure	NOUN
ejpam-7045	30	19	[	[	X
ejpam-7045	30	20	11	11	NUM
ejpam-7045	30	21	]	]	PUNCT
ejpam-7045	30	22	of	of	ADP
ejpam-7045	30	23	a	a	PRON
ejpam-7045	30	24	and	and	CCONJ
ejpam-7045	30	25	is	be	AUX
ejpam-7045	30	26	denoted	denote	VERB
ejpam-7045	30	27	by	by	ADP
ejpam-7045	30	28	τ1τ2	τ1τ2	NOUN
ejpam-7045	30	29	-	-	NUM
ejpam-7045	30	30	cl(a	cl(a	NUM
ejpam-7045	30	31	)	)	PUNCT
ejpam-7045	30	32	.	.	PUNCT
ejpam-7045	31	1	the	the	DET
ejpam-7045	31	2	union	union	NOUN
ejpam-7045	31	3	of	of	ADP
ejpam-7045	31	4	all	all	DET
ejpam-7045	31	5	τ1τ2	τ1τ2	ADJ
ejpam-7045	31	6	-	-	ADJ
ejpam-7045	31	7	open	open	ADJ
ejpam-7045	31	8	sets	set	NOUN
ejpam-7045	31	9	of	of	ADP
ejpam-7045	31	10	x	x	PUNCT
ejpam-7045	31	11	contained	contain	VERB
ejpam-7045	31	12	in	in	ADP
ejpam-7045	31	13	a	a	PRON
ejpam-7045	31	14	is	be	AUX
ejpam-7045	31	15	called	call	VERB
ejpam-7045	31	16	the	the	DET
ejpam-7045	31	17	τ1τ2	τ1τ2	NOUN
ejpam-7045	31	18	-	-	ADJ
ejpam-7045	31	19	interior	interior	ADJ
ejpam-7045	31	20	[	[	X
ejpam-7045	31	21	11	11	NUM
ejpam-7045	31	22	]	]	PUNCT
ejpam-7045	31	23	of	of	ADP
ejpam-7045	31	24	a	a	PRON
ejpam-7045	31	25	and	and	CCONJ
ejpam-7045	31	26	is	be	AUX
ejpam-7045	31	27	denoted	denote	VERB
ejpam-7045	31	28	by	by	ADP
ejpam-7045	31	29	τ1τ2	τ1τ2	NOUN
ejpam-7045	31	30	-	-	ADJ
ejpam-7045	31	31	int(a	int(a	NOUN
ejpam-7045	31	32	)	)	PUNCT
ejpam-7045	31	33	.	.	PUNCT
ejpam-7045	32	1	lemma	lemma	PROPN
ejpam-7045	32	2	1	1	NUM
ejpam-7045	32	3	.	.	PUNCT
ejpam-7045	33	1	[	[	X
ejpam-7045	33	2	11	11	NUM
ejpam-7045	33	3	]	]	PUNCT
ejpam-7045	33	4	let	let	VERB
ejpam-7045	33	5	a	a	PRON
ejpam-7045	33	6	and	and	CCONJ
ejpam-7045	33	7	b	b	NOUN
ejpam-7045	33	8	be	be	AUX
ejpam-7045	33	9	subsets	subset	NOUN
ejpam-7045	33	10	of	of	ADP
ejpam-7045	33	11	a	a	DET
ejpam-7045	33	12	bitopological	bitopological	ADJ
ejpam-7045	33	13	space	space	NOUN
ejpam-7045	33	14	(	(	PUNCT
ejpam-7045	33	15	x	x	NOUN
ejpam-7045	33	16	,	,	PUNCT
ejpam-7045	33	17	τ1	τ1	NOUN
ejpam-7045	33	18	,	,	PUNCT
ejpam-7045	33	19	τ2	τ2	NOUN
ejpam-7045	33	20	)	)	PUNCT
ejpam-7045	33	21	.	.	PUNCT
ejpam-7045	34	1	for	for	ADP
ejpam-7045	34	2	the	the	DET
ejpam-7045	34	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-7045	34	4	,	,	PUNCT
ejpam-7045	34	5	the	the	DET
ejpam-7045	34	6	following	follow	VERB
ejpam-7045	34	7	properties	property	NOUN
ejpam-7045	34	8	hold	hold	VERB
ejpam-7045	34	9	:	:	PUNCT
ejpam-7045	34	10	(	(	PUNCT
ejpam-7045	34	11	1	1	X
ejpam-7045	34	12	)	)	PUNCT
ejpam-7045	34	13	a	a	DET
ejpam-7045	34	14	⊆	⊆	NUM
ejpam-7045	34	15	τ1τ2	τ1τ2	NOUN
ejpam-7045	34	16	-	-	NUM
ejpam-7045	34	17	cl(a	cl(a	NUM
ejpam-7045	34	18	)	)	PUNCT
ejpam-7045	34	19	and	and	CCONJ
ejpam-7045	34	20	τ1τ2	τ1τ2	NOUN
ejpam-7045	34	21	-	-	ADJ
ejpam-7045	34	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7045	34	23	-	-	PUNCT
ejpam-7045	34	24	cl(a	cl(a	NUM
ejpam-7045	34	25	)	)	PUNCT
ejpam-7045	34	26	)	)	PUNCT
ejpam-7045	35	1	=	=	PUNCT
ejpam-7045	35	2	τ1τ2	τ1τ2	NOUN
ejpam-7045	35	3	-	-	NUM
ejpam-7045	35	4	cl(a	cl(a	NUM
ejpam-7045	35	5	)	)	PUNCT
ejpam-7045	35	6	.	.	PUNCT
ejpam-7045	36	1	(	(	PUNCT
ejpam-7045	36	2	2	2	X
ejpam-7045	36	3	)	)	PUNCT
ejpam-7045	36	4	if	if	SCONJ
ejpam-7045	36	5	a	a	DET
ejpam-7045	36	6	⊆	⊆	NUM
ejpam-7045	36	7	b	b	NOUN
ejpam-7045	36	8	,	,	PUNCT
ejpam-7045	36	9	then	then	ADV
ejpam-7045	36	10	τ1τ2	τ1τ2	NOUN
ejpam-7045	36	11	-	-	NUM
ejpam-7045	36	12	cl(a	cl(a	NUM
ejpam-7045	36	13	)	)	PUNCT
ejpam-7045	36	14	⊆	⊆	NUM
ejpam-7045	36	15	τ1τ2	τ1τ2	NOUN
ejpam-7045	36	16	-	-	NOUN
ejpam-7045	36	17	cl(b	cl(b	NOUN
ejpam-7045	36	18	)	)	PUNCT
ejpam-7045	36	19	.	.	PUNCT
ejpam-7045	37	1	(	(	PUNCT
ejpam-7045	37	2	3	3	X
ejpam-7045	37	3	)	)	PUNCT
ejpam-7045	37	4	τ1τ2	τ1τ2	NOUN
ejpam-7045	37	5	-	-	NUM
ejpam-7045	37	6	cl(a	cl(a	NUM
ejpam-7045	37	7	)	)	PUNCT
ejpam-7045	37	8	is	be	AUX
ejpam-7045	37	9	τ1τ2	τ1τ2	NOUN
ejpam-7045	37	10	-	-	ADJ
ejpam-7045	37	11	closed	closed	ADJ
ejpam-7045	37	12	.	.	PUNCT
ejpam-7045	38	1	(	(	PUNCT
ejpam-7045	38	2	4	4	X
ejpam-7045	38	3	)	)	PUNCT
ejpam-7045	38	4	a	a	PRON
ejpam-7045	38	5	is	be	AUX
ejpam-7045	38	6	τ1τ2	τ1τ2	NOUN
ejpam-7045	38	7	-	-	ADJ
ejpam-7045	38	8	closed	closed	ADJ
ejpam-7045	38	9	if	if	SCONJ
ejpam-7045	38	10	and	and	CCONJ
ejpam-7045	38	11	only	only	ADV
ejpam-7045	38	12	if	if	SCONJ
ejpam-7045	38	13	a	a	DET
ejpam-7045	38	14	=	=	PUNCT
ejpam-7045	38	15	τ1τ2	τ1τ2	NOUN
ejpam-7045	38	16	-	-	NUM
ejpam-7045	38	17	cl(a	cl(a	NUM
ejpam-7045	38	18	)	)	PUNCT
ejpam-7045	38	19	.	.	PUNCT
ejpam-7045	39	1	(	(	PUNCT
ejpam-7045	39	2	5	5	X
ejpam-7045	39	3	)	)	PUNCT
ejpam-7045	39	4	τ1τ2	τ1τ2	NOUN
ejpam-7045	39	5	-	-	NOUN
ejpam-7045	39	6	cl(x	cl(x	X
ejpam-7045	39	7	−a	−a	NOUN
ejpam-7045	39	8	)	)	PUNCT
ejpam-7045	40	1	=	=	PUNCT
ejpam-7045	40	2	x	x	X
ejpam-7045	41	1	−	−	ADP
ejpam-7045	41	2	τ1τ2	τ1τ2	NOUN
ejpam-7045	41	3	-	-	PUNCT
ejpam-7045	41	4	int(a	int(a	NOUN
ejpam-7045	41	5	)	)	PUNCT
ejpam-7045	41	6	.	.	PUNCT
ejpam-7045	42	1	a	a	DET
ejpam-7045	42	2	subset	subset	NOUN
ejpam-7045	42	3	a	a	PRON
ejpam-7045	42	4	of	of	ADP
ejpam-7045	42	5	a	a	DET
ejpam-7045	42	6	bitopological	bitopological	ADJ
ejpam-7045	42	7	space	space	NOUN
ejpam-7045	42	8	(	(	PUNCT
ejpam-7045	42	9	x	x	NOUN
ejpam-7045	42	10	,	,	PUNCT
ejpam-7045	42	11	τ1	τ1	NOUN
ejpam-7045	42	12	,	,	PUNCT
ejpam-7045	42	13	τ2	τ2	NOUN
ejpam-7045	42	14	)	)	PUNCT
ejpam-7045	42	15	is	be	AUX
ejpam-7045	42	16	said	say	VERB
ejpam-7045	42	17	to	to	PART
ejpam-7045	42	18	be	be	AUX
ejpam-7045	42	19	(	(	PUNCT
ejpam-7045	42	20	τ1	τ1	NOUN
ejpam-7045	42	21	,	,	PUNCT
ejpam-7045	42	22	τ2)r	τ2)r	NOUN
ejpam-7045	42	23	-	-	PUNCT
ejpam-7045	42	24	open	open	ADJ
ejpam-7045	43	1	[	[	X
ejpam-7045	43	2	9	9	NUM
ejpam-7045	43	3	]	]	PUNCT
ejpam-7045	43	4	(	(	PUNCT
ejpam-7045	43	5	resp	resp	NOUN
ejpam-7045	43	6	.	.	PUNCT
ejpam-7045	44	1	(	(	PUNCT
ejpam-7045	44	2	τ1	τ1	NOUN
ejpam-7045	44	3	,	,	PUNCT
ejpam-7045	44	4	τ2)s	τ2)s	NOUN
ejpam-7045	44	5	-	-	PUNCT
ejpam-7045	44	6	open	open	ADJ
ejpam-7045	44	7	[	[	X
ejpam-7045	44	8	12	12	NUM
ejpam-7045	44	9	]	]	PUNCT
ejpam-7045	44	10	,	,	PUNCT
ejpam-7045	44	11	(	(	PUNCT
ejpam-7045	44	12	τ1	τ1	NOUN
ejpam-7045	44	13	,	,	PUNCT
ejpam-7045	44	14	τ2)p	τ2)p	NOUN
ejpam-7045	44	15	-	-	ADJ
ejpam-7045	44	16	open	open	ADJ
ejpam-7045	44	17	[	[	X
ejpam-7045	44	18	12	12	NUM
ejpam-7045	44	19	]	]	PUNCT
ejpam-7045	44	20	,	,	PUNCT
ejpam-7045	44	21	(	(	PUNCT
ejpam-7045	44	22	τ1	τ1	NOUN
ejpam-7045	44	23	,	,	PUNCT
ejpam-7045	44	24	τ2)β	τ2)β	ADJ
ejpam-7045	44	25	-	-	PUNCT
ejpam-7045	44	26	open	open	NOUN
ejpam-7045	45	1	[	[	X
ejpam-7045	45	2	12	12	NUM
ejpam-7045	45	3	]	]	PUNCT
ejpam-7045	45	4	)	)	PUNCT
ejpam-7045	45	5	if	if	SCONJ
ejpam-7045	45	6	a	a	DET
ejpam-7045	45	7	=	=	PUNCT
ejpam-7045	45	8	τ1τ2	τ1τ2	NOUN
ejpam-7045	45	9	-	-	NOUN
ejpam-7045	45	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7045	45	11	-	-	PUNCT
ejpam-7045	45	12	cl(a	cl(a	NUM
ejpam-7045	45	13	)	)	PUNCT
ejpam-7045	45	14	)	)	PUNCT
ejpam-7045	45	15	(	(	PUNCT
ejpam-7045	45	16	resp	resp	NOUN
ejpam-7045	45	17	.	.	PUNCT
ejpam-7045	46	1	a	a	DET
ejpam-7045	46	2	⊆	⊆	NUM
ejpam-7045	46	3	τ1τ2	τ1τ2	NOUN
ejpam-7045	46	4	-	-	ADJ
ejpam-7045	46	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7045	46	6	-	-	PUNCT
ejpam-7045	46	7	int(a	int(a	NOUN
ejpam-7045	46	8	)	)	PUNCT
ejpam-7045	46	9	)	)	PUNCT
ejpam-7045	46	10	,	,	PUNCT
ejpam-7045	46	11	a	a	DET
ejpam-7045	46	12	⊆	⊆	NUM
ejpam-7045	46	13	τ1τ2	τ1τ2	NOUN
ejpam-7045	46	14	-	-	NOUN
ejpam-7045	46	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7045	46	16	-	-	PUNCT
ejpam-7045	46	17	cl(a	cl(a	NUM
ejpam-7045	46	18	)	)	PUNCT
ejpam-7045	46	19	)	)	PUNCT
ejpam-7045	46	20	,	,	PUNCT
ejpam-7045	46	21	a	a	DET
ejpam-7045	46	22	⊆	⊆	NUM
ejpam-7045	46	23	τ1τ2	τ1τ2	NOUN
ejpam-7045	46	24	-	-	PUNCT
ejpam-7045	46	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7045	46	26	-	-	PUNCT
ejpam-7045	46	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7045	46	28	-	-	PUNCT
ejpam-7045	46	29	cl(a	cl(a	NUM
ejpam-7045	46	30	)	)	PUNCT
ejpam-7045	46	31	)	)	PUNCT
ejpam-7045	46	32	)	)	PUNCT
ejpam-7045	46	33	)	)	PUNCT
ejpam-7045	46	34	.	.	PUNCT
ejpam-7045	47	1	the	the	DET
ejpam-7045	47	2	complement	complement	NOUN
ejpam-7045	47	3	of	of	ADP
ejpam-7045	47	4	a	a	DET
ejpam-7045	47	5	(	(	PUNCT
ejpam-7045	47	6	τ1	τ1	NOUN
ejpam-7045	47	7	,	,	PUNCT
ejpam-7045	47	8	τ2)r	τ2)r	NOUN
ejpam-7045	47	9	-	-	PUNCT
ejpam-7045	47	10	open	open	ADJ
ejpam-7045	47	11	(	(	PUNCT
ejpam-7045	47	12	resp	resp	NOUN
ejpam-7045	47	13	.	.	PUNCT
ejpam-7045	48	1	(	(	PUNCT
ejpam-7045	48	2	τ1	τ1	NOUN
ejpam-7045	48	3	,	,	PUNCT
ejpam-7045	48	4	τ2)s	τ2)s	NOUN
ejpam-7045	48	5	-	-	PUNCT
ejpam-7045	48	6	open	open	ADJ
ejpam-7045	48	7	,	,	PUNCT
ejpam-7045	48	8	(	(	PUNCT
ejpam-7045	48	9	τ1	τ1	NOUN
ejpam-7045	48	10	,	,	PUNCT
ejpam-7045	48	11	τ2)p	τ2)p	NOUN
ejpam-7045	48	12	-	-	ADJ
ejpam-7045	48	13	open	open	ADJ
ejpam-7045	48	14	,	,	PUNCT
ejpam-7045	48	15	(	(	PUNCT
ejpam-7045	48	16	τ1	τ1	NOUN
ejpam-7045	48	17	,	,	PUNCT
ejpam-7045	48	18	τ2)β	τ2)β	ADJ
ejpam-7045	48	19	-	-	PUNCT
ejpam-7045	48	20	open	open	ADJ
ejpam-7045	48	21	)	)	PUNCT
ejpam-7045	48	22	set	set	NOUN
ejpam-7045	48	23	is	be	AUX
ejpam-7045	48	24	said	say	VERB
ejpam-7045	48	25	to	to	PART
ejpam-7045	48	26	be	be	AUX
ejpam-7045	48	27	(	(	PUNCT
ejpam-7045	48	28	τ1	τ1	NOUN
ejpam-7045	48	29	,	,	PUNCT
ejpam-7045	48	30	τ2)r	τ2)r	NOUN
ejpam-7045	48	31	-	-	PUNCT
ejpam-7045	48	32	closed	closed	ADJ
ejpam-7045	48	33	(	(	PUNCT
ejpam-7045	48	34	resp	resp	NOUN
ejpam-7045	48	35	.	.	PUNCT
ejpam-7045	49	1	(	(	PUNCT
ejpam-7045	49	2	τ1	τ1	NOUN
ejpam-7045	49	3	,	,	PUNCT
ejpam-7045	49	4	τ2)s	τ2)s	NOUN
ejpam-7045	49	5	-	-	PUNCT
ejpam-7045	49	6	closed	closed	ADJ
ejpam-7045	49	7	,	,	PUNCT
ejpam-7045	49	8	(	(	PUNCT
ejpam-7045	49	9	τ1	τ1	NOUN
ejpam-7045	49	10	,	,	PUNCT
ejpam-7045	49	11	τ2)p	τ2)p	NOUN
ejpam-7045	49	12	-	-	PUNCT
ejpam-7045	49	13	closed	closed	ADJ
ejpam-7045	49	14	,	,	PUNCT
ejpam-7045	49	15	(	(	PUNCT
ejpam-7045	49	16	τ1	τ1	NOUN
ejpam-7045	49	17	,	,	PUNCT
ejpam-7045	49	18	τ2)β	τ2)β	ADJ
ejpam-7045	49	19	-	-	PUNCT
ejpam-7045	49	20	closed	closed	ADJ
ejpam-7045	49	21	)	)	PUNCT
ejpam-7045	49	22	.	.	PUNCT
ejpam-7045	50	1	the	the	DET
ejpam-7045	50	2	c.	c.	PROPN
ejpam-7045	50	3	viriyapong	viriyapong	PROPN
ejpam-7045	50	4	,	,	PUNCT
ejpam-7045	50	5	a.	a.	PROPN
ejpam-7045	50	6	sama	sama	PROPN
ejpam-7045	50	7	-	-	PUNCT
ejpam-7045	50	8	ae	ae	PROPN
ejpam-7045	50	9	,	,	PUNCT
ejpam-7045	50	10	c.	c.	PROPN
ejpam-7045	50	11	boonpok	boonpok	PROPN
ejpam-7045	50	12	/	/	SYM
ejpam-7045	50	13	eur	eur	PROPN
ejpam-7045	50	14	.	.	PUNCT
ejpam-7045	51	1	j.	j.	PROPN
ejpam-7045	51	2	pure	pure	PROPN
ejpam-7045	51	3	appl	appl	PROPN
ejpam-7045	51	4	.	.	PROPN
ejpam-7045	51	5	math	math	PROPN
ejpam-7045	51	6	,	,	PUNCT
ejpam-7045	51	7	18	18	NUM
ejpam-7045	51	8	(	(	PUNCT
ejpam-7045	51	9	4	4	NUM
ejpam-7045	51	10	)	)	PUNCT
ejpam-7045	51	11	(	(	PUNCT
ejpam-7045	51	12	2025	2025	NUM
ejpam-7045	51	13	)	)	PUNCT
ejpam-7045	51	14	,	,	PUNCT
ejpam-7045	51	15	7045	7045	NUM
ejpam-7045	51	16	3	3	NUM
ejpam-7045	51	17	of	of	ADP
ejpam-7045	51	18	12	12	NUM
ejpam-7045	51	19	intersection	intersection	NOUN
ejpam-7045	51	20	of	of	ADP
ejpam-7045	51	21	all	all	DET
ejpam-7045	51	22	(	(	PUNCT
ejpam-7045	51	23	τ1	τ1	NOUN
ejpam-7045	51	24	,	,	PUNCT
ejpam-7045	51	25	τ2)s	τ2)s	NOUN
ejpam-7045	51	26	-	-	PUNCT
ejpam-7045	51	27	closed	close	VERB
ejpam-7045	51	28	sets	set	NOUN
ejpam-7045	51	29	of	of	ADP
ejpam-7045	51	30	x	x	PUNCT
ejpam-7045	51	31	containing	contain	VERB
ejpam-7045	51	32	a	a	PRON
ejpam-7045	51	33	is	be	AUX
ejpam-7045	51	34	called	call	VERB
ejpam-7045	51	35	the	the	DET
ejpam-7045	51	36	(	(	PUNCT
ejpam-7045	51	37	τ1	τ1	NOUN
ejpam-7045	51	38	,	,	PUNCT
ejpam-7045	51	39	τ2)s	τ2)s	NOUN
ejpam-7045	51	40	-	-	PUNCT
ejpam-7045	51	41	closure	closure	NOUN
ejpam-7045	51	42	[	[	X
ejpam-7045	51	43	12	12	NUM
ejpam-7045	51	44	]	]	PUNCT
ejpam-7045	51	45	of	of	ADP
ejpam-7045	51	46	a	a	PRON
ejpam-7045	51	47	and	and	CCONJ
ejpam-7045	51	48	is	be	AUX
ejpam-7045	51	49	denoted	denote	VERB
ejpam-7045	51	50	by	by	ADP
ejpam-7045	51	51	(	(	PUNCT
ejpam-7045	51	52	τ1	τ1	NOUN
ejpam-7045	51	53	,	,	PUNCT
ejpam-7045	51	54	τ2)-scl(a	τ2)-scl(a	PROPN
ejpam-7045	51	55	)	)	PUNCT
ejpam-7045	51	56	.	.	PUNCT
ejpam-7045	52	1	the	the	DET
ejpam-7045	52	2	union	union	NOUN
ejpam-7045	52	3	of	of	ADP
ejpam-7045	52	4	all	all	DET
ejpam-7045	52	5	(	(	PUNCT
ejpam-7045	52	6	τ1	τ1	NOUN
ejpam-7045	52	7	,	,	PUNCT
ejpam-7045	52	8	τ2)s	τ2)s	NOUN
ejpam-7045	52	9	-	-	PUNCT
ejpam-7045	52	10	open	open	ADJ
ejpam-7045	52	11	sets	set	NOUN
ejpam-7045	52	12	of	of	ADP
ejpam-7045	52	13	x	x	PUNCT
ejpam-7045	52	14	contained	contain	VERB
ejpam-7045	52	15	in	in	ADP
ejpam-7045	52	16	a	a	PRON
ejpam-7045	52	17	is	be	AUX
ejpam-7045	52	18	called	call	VERB
ejpam-7045	52	19	the	the	DET
ejpam-7045	52	20	(	(	PUNCT
ejpam-7045	52	21	τ1	τ1	NOUN
ejpam-7045	52	22	,	,	PUNCT
ejpam-7045	52	23	τ2)s	τ2)s	NOUN
ejpam-7045	52	24	-	-	ADJ
ejpam-7045	52	25	interior	interior	NOUN
ejpam-7045	52	26	[	[	X
ejpam-7045	52	27	12	12	NUM
ejpam-7045	52	28	]	]	PUNCT
ejpam-7045	52	29	of	of	ADP
ejpam-7045	52	30	a	a	PRON
ejpam-7045	52	31	and	and	CCONJ
ejpam-7045	52	32	is	be	AUX
ejpam-7045	52	33	denoted	denote	VERB
ejpam-7045	52	34	by	by	ADP
ejpam-7045	52	35	(	(	PUNCT
ejpam-7045	52	36	τ1	τ1	NOUN
ejpam-7045	52	37	,	,	PUNCT
ejpam-7045	52	38	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7045	52	39	)	)	PUNCT
ejpam-7045	52	40	.	.	PUNCT
ejpam-7045	53	1	lemma	lemma	PROPN
ejpam-7045	53	2	2	2	NUM
ejpam-7045	53	3	.	.	X
ejpam-7045	54	1	for	for	ADP
ejpam-7045	54	2	a	a	DET
ejpam-7045	54	3	subset	subset	NOUN
ejpam-7045	54	4	a	a	PRON
ejpam-7045	54	5	of	of	ADP
ejpam-7045	54	6	a	a	DET
ejpam-7045	54	7	bitopological	bitopological	ADJ
ejpam-7045	54	8	space	space	NOUN
ejpam-7045	54	9	(	(	PUNCT
ejpam-7045	54	10	x	x	NOUN
ejpam-7045	54	11	,	,	PUNCT
ejpam-7045	54	12	τ1	τ1	NOUN
ejpam-7045	54	13	,	,	PUNCT
ejpam-7045	54	14	τ2	τ2	NOUN
ejpam-7045	54	15	)	)	PUNCT
ejpam-7045	54	16	,	,	PUNCT
ejpam-7045	54	17	the	the	DET
ejpam-7045	54	18	following	follow	VERB
ejpam-7045	54	19	properties	property	NOUN
ejpam-7045	54	20	hold	hold	VERB
ejpam-7045	54	21	:	:	PUNCT
ejpam-7045	54	22	(	(	PUNCT
ejpam-7045	54	23	1	1	X
ejpam-7045	54	24	)	)	PUNCT
ejpam-7045	54	25	(	(	PUNCT
ejpam-7045	54	26	τ1	τ1	NOUN
ejpam-7045	54	27	,	,	PUNCT
ejpam-7045	54	28	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-7045	54	29	)	)	PUNCT
ejpam-7045	54	30	=	=	PUNCT
ejpam-7045	55	1	τ1τ2	τ1τ2	NOUN
ejpam-7045	55	2	-	-	NOUN
ejpam-7045	55	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7045	55	4	-	-	PUNCT
ejpam-7045	55	5	cl(a	cl(a	NUM
ejpam-7045	55	6	)	)	PUNCT
ejpam-7045	55	7	)	)	PUNCT
ejpam-7045	55	8	∪a	∪a	X
ejpam-7045	56	1	[	[	X
ejpam-7045	56	2	12	12	NUM
ejpam-7045	56	3	]	]	X
ejpam-7045	56	4	;	;	PUNCT
ejpam-7045	56	5	(	(	PUNCT
ejpam-7045	56	6	2	2	X
ejpam-7045	56	7	)	)	PUNCT
ejpam-7045	56	8	(	(	PUNCT
ejpam-7045	56	9	τ1	τ1	NOUN
ejpam-7045	56	10	,	,	PUNCT
ejpam-7045	56	11	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7045	56	12	)	)	PUNCT
ejpam-7045	57	1	=	=	PUNCT
ejpam-7045	57	2	τ1τ2	τ1τ2	NOUN
ejpam-7045	57	3	-	-	ADJ
ejpam-7045	57	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7045	57	5	-	-	PUNCT
ejpam-7045	57	6	int(a	int(a	NOUN
ejpam-7045	57	7	)	)	PUNCT
ejpam-7045	57	8	)	)	PUNCT
ejpam-7045	58	1	∩a	∩a	PROPN
ejpam-7045	59	1	[	[	X
ejpam-7045	59	2	13	13	NUM
ejpam-7045	59	3	]	]	PUNCT
ejpam-7045	59	4	.	.	PUNCT
ejpam-7045	60	1	lemma	lemma	PROPN
ejpam-7045	60	2	3	3	X
ejpam-7045	60	3	.	.	PUNCT
ejpam-7045	61	1	[	[	X
ejpam-7045	61	2	14	14	NUM
ejpam-7045	61	3	]	]	PUNCT
ejpam-7045	61	4	let	let	VERB
ejpam-7045	61	5	a	a	PRON
ejpam-7045	61	6	be	be	AUX
ejpam-7045	61	7	a	a	DET
ejpam-7045	61	8	subset	subset	NOUN
ejpam-7045	61	9	of	of	ADP
ejpam-7045	61	10	a	a	DET
ejpam-7045	61	11	bitopological	bitopological	ADJ
ejpam-7045	61	12	space	space	NOUN
ejpam-7045	61	13	(	(	PUNCT
ejpam-7045	61	14	x	x	NOUN
ejpam-7045	61	15	,	,	PUNCT
ejpam-7045	61	16	τ1	τ1	NOUN
ejpam-7045	61	17	,	,	PUNCT
ejpam-7045	61	18	τ2	τ2	NOUN
ejpam-7045	61	19	)	)	PUNCT
ejpam-7045	61	20	.	.	PUNCT
ejpam-7045	62	1	if	if	SCONJ
ejpam-7045	62	2	a	a	PRON
ejpam-7045	62	3	is	be	AUX
ejpam-7045	62	4	τ1τ2	τ1τ2	NOUN
ejpam-7045	62	5	-	-	ADJ
ejpam-7045	62	6	open	open	ADJ
ejpam-7045	62	7	in	in	ADP
ejpam-7045	62	8	x	x	NOUN
ejpam-7045	62	9	,	,	PUNCT
ejpam-7045	62	10	then	then	ADV
ejpam-7045	62	11	(	(	PUNCT
ejpam-7045	62	12	τ1	τ1	NOUN
ejpam-7045	62	13	,	,	PUNCT
ejpam-7045	62	14	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-7045	62	15	)	)	PUNCT
ejpam-7045	62	16	=	=	PUNCT
ejpam-7045	63	1	τ1τ2	τ1τ2	NOUN
ejpam-7045	63	2	-	-	NOUN
ejpam-7045	63	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7045	63	4	-	-	PUNCT
ejpam-7045	63	5	cl(a	cl(a	NUM
ejpam-7045	63	6	)	)	PUNCT
ejpam-7045	63	7	)	)	PUNCT
ejpam-7045	63	8	.	.	PUNCT
ejpam-7045	64	1	a	a	DET
ejpam-7045	64	2	subset	subset	NOUN
ejpam-7045	64	3	a	a	PRON
ejpam-7045	64	4	of	of	ADP
ejpam-7045	64	5	a	a	DET
ejpam-7045	64	6	bitopological	bitopological	ADJ
ejpam-7045	64	7	space	space	NOUN
ejpam-7045	64	8	(	(	PUNCT
ejpam-7045	64	9	x	x	NOUN
ejpam-7045	64	10	,	,	PUNCT
ejpam-7045	64	11	τ1	τ1	NOUN
ejpam-7045	64	12	,	,	PUNCT
ejpam-7045	64	13	τ2	τ2	NOUN
ejpam-7045	64	14	)	)	PUNCT
ejpam-7045	64	15	is	be	AUX
ejpam-7045	64	16	said	say	VERB
ejpam-7045	64	17	to	to	PART
ejpam-7045	64	18	be	be	AUX
ejpam-7045	64	19	τ1τ2	τ1τ2	NOUN
ejpam-7045	64	20	-	-	ADJ
ejpam-7045	64	21	δ	δ	NOUN
ejpam-7045	64	22	-	-	NOUN
ejpam-7045	64	23	open	open	ADJ
ejpam-7045	64	24	[	[	X
ejpam-7045	64	25	8	8	NUM
ejpam-7045	64	26	]	]	X
ejpam-7045	64	27	if	if	SCONJ
ejpam-7045	64	28	a	a	PRON
ejpam-7045	64	29	is	be	AUX
ejpam-7045	64	30	the	the	DET
ejpam-7045	64	31	union	union	NOUN
ejpam-7045	64	32	of	of	ADP
ejpam-7045	64	33	(	(	PUNCT
ejpam-7045	64	34	τ1	τ1	NOUN
ejpam-7045	64	35	,	,	PUNCT
ejpam-7045	64	36	τ2)r	τ2)r	ADJ
ejpam-7045	64	37	-	-	PUNCT
ejpam-7045	64	38	open	open	ADJ
ejpam-7045	64	39	sets	set	NOUN
ejpam-7045	64	40	of	of	ADP
ejpam-7045	64	41	x.	x.	NOUN
ejpam-7045	64	42	the	the	DET
ejpam-7045	64	43	complement	complement	NOUN
ejpam-7045	64	44	of	of	ADP
ejpam-7045	64	45	a	a	DET
ejpam-7045	64	46	τ1τ2	τ1τ2	ADJ
ejpam-7045	64	47	-	-	ADJ
ejpam-7045	64	48	δ	δ	NOUN
ejpam-7045	64	49	-	-	ADJ
ejpam-7045	64	50	open	open	ADJ
ejpam-7045	64	51	set	set	NOUN
ejpam-7045	64	52	is	be	AUX
ejpam-7045	64	53	called	call	VERB
ejpam-7045	64	54	τ1τ2	τ1τ2	ADJ
ejpam-7045	64	55	-	-	ADJ
ejpam-7045	64	56	δclosed	δclosed	ADJ
ejpam-7045	64	57	[	[	X
ejpam-7045	64	58	8	8	NUM
ejpam-7045	64	59	]	]	PUNCT
ejpam-7045	64	60	.	.	PUNCT
ejpam-7045	65	1	the	the	DET
ejpam-7045	65	2	union	union	NOUN
ejpam-7045	65	3	of	of	ADP
ejpam-7045	65	4	all	all	DET
ejpam-7045	65	5	τ1τ2	τ1τ2	NOUN
ejpam-7045	65	6	-	-	ADJ
ejpam-7045	65	7	δ	δ	NOUN
ejpam-7045	65	8	-	-	ADJ
ejpam-7045	65	9	open	open	ADJ
ejpam-7045	65	10	sets	set	NOUN
ejpam-7045	65	11	ofx	ofx	NOUN
ejpam-7045	65	12	contained	contain	VERB
ejpam-7045	65	13	in	in	ADP
ejpam-7045	65	14	a	a	PRON
ejpam-7045	65	15	is	be	AUX
ejpam-7045	65	16	called	call	VERB
ejpam-7045	65	17	the	the	DET
ejpam-7045	65	18	τ1τ2	τ1τ2	ADJ
ejpam-7045	65	19	-	-	ADJ
ejpam-7045	65	20	δ	δ	NOUN
ejpam-7045	65	21	-	-	NOUN
ejpam-7045	65	22	interior	interior	ADJ
ejpam-7045	65	23	[	[	X
ejpam-7045	65	24	8	8	NUM
ejpam-7045	65	25	]	]	PUNCT
ejpam-7045	65	26	of	of	ADP
ejpam-7045	65	27	a	a	PRON
ejpam-7045	65	28	and	and	CCONJ
ejpam-7045	65	29	is	be	AUX
ejpam-7045	65	30	denoted	denote	VERB
ejpam-7045	65	31	by	by	ADP
ejpam-7045	65	32	τ1τ2	τ1τ2	ADJ
ejpam-7045	65	33	-	-	ADJ
ejpam-7045	65	34	δ	δ	NOUN
ejpam-7045	65	35	-	-	PUNCT
ejpam-7045	65	36	int(a	int(a	PROPN
ejpam-7045	65	37	)	)	PUNCT
ejpam-7045	65	38	.	.	PUNCT
ejpam-7045	66	1	the	the	DET
ejpam-7045	66	2	intersection	intersection	NOUN
ejpam-7045	66	3	of	of	ADP
ejpam-7045	66	4	all	all	DET
ejpam-7045	66	5	τ1τ2	τ1τ2	NOUN
ejpam-7045	66	6	-	-	ADJ
ejpam-7045	66	7	δ	δ	NOUN
ejpam-7045	66	8	-	-	PUNCT
ejpam-7045	66	9	closed	close	VERB
ejpam-7045	66	10	sets	set	NOUN
ejpam-7045	66	11	of	of	ADP
ejpam-7045	66	12	x	x	PUNCT
ejpam-7045	66	13	containing	contain	VERB
ejpam-7045	66	14	a	a	PRON
ejpam-7045	66	15	is	be	AUX
ejpam-7045	66	16	called	call	VERB
ejpam-7045	66	17	the	the	DET
ejpam-7045	66	18	τ1τ2	τ1τ2	ADJ
ejpam-7045	66	19	-	-	ADJ
ejpam-7045	66	20	δ	δ	NOUN
ejpam-7045	66	21	-	-	NOUN
ejpam-7045	66	22	closure	closure	NOUN
ejpam-7045	66	23	[	[	X
ejpam-7045	66	24	8	8	NUM
ejpam-7045	66	25	]	]	PUNCT
ejpam-7045	66	26	of	of	ADP
ejpam-7045	66	27	a	a	PRON
ejpam-7045	66	28	and	and	CCONJ
ejpam-7045	66	29	is	be	AUX
ejpam-7045	66	30	denoted	denote	VERB
ejpam-7045	66	31	by	by	ADP
ejpam-7045	66	32	τ1τ2	τ1τ2	ADJ
ejpam-7045	66	33	-	-	ADJ
ejpam-7045	66	34	δ	δ	NOUN
ejpam-7045	66	35	-	-	PUNCT
ejpam-7045	66	36	cl(a	cl(a	NUM
ejpam-7045	66	37	)	)	PUNCT
ejpam-7045	66	38	.	.	PUNCT
ejpam-7045	67	1	let	let	VERB
ejpam-7045	67	2	a	a	DET
ejpam-7045	67	3	be	be	AUX
ejpam-7045	67	4	a	a	DET
ejpam-7045	67	5	subset	subset	NOUN
ejpam-7045	67	6	of	of	ADP
ejpam-7045	67	7	a	a	DET
ejpam-7045	67	8	bitopological	bitopological	ADJ
ejpam-7045	67	9	space	space	NOUN
ejpam-7045	67	10	(	(	PUNCT
ejpam-7045	67	11	x	x	NOUN
ejpam-7045	67	12	,	,	PUNCT
ejpam-7045	67	13	τ1	τ1	NOUN
ejpam-7045	67	14	,	,	PUNCT
ejpam-7045	67	15	τ2	τ2	NOUN
ejpam-7045	67	16	)	)	PUNCT
ejpam-7045	67	17	.	.	PUNCT
ejpam-7045	68	1	a	a	DET
ejpam-7045	68	2	point	point	NOUN
ejpam-7045	68	3	x	x	X
ejpam-7045	68	4	∈	∈	NOUN
ejpam-7045	68	5	x	x	PUNCT
ejpam-7045	68	6	is	be	AUX
ejpam-7045	68	7	called	call	VERB
ejpam-7045	68	8	a	a	DET
ejpam-7045	68	9	(	(	PUNCT
ejpam-7045	68	10	τ1	τ1	NOUN
ejpam-7045	68	11	,	,	PUNCT
ejpam-7045	68	12	τ2)θ	τ2)θ	ADJ
ejpam-7045	68	13	-	-	PUNCT
ejpam-7045	68	14	cluster	cluster	NOUN
ejpam-7045	68	15	point	point	NOUN
ejpam-7045	68	16	[	[	X
ejpam-7045	68	17	9	9	NUM
ejpam-7045	68	18	]	]	PUNCT
ejpam-7045	68	19	of	of	ADP
ejpam-7045	68	20	a	a	DET
ejpam-7045	68	21	if	if	SCONJ
ejpam-7045	68	22	τ1τ2	τ1τ2	NOUN
ejpam-7045	68	23	-	-	NOUN
ejpam-7045	68	24	cl(u	cl(u	NOUN
ejpam-7045	68	25	)	)	PUNCT
ejpam-7045	68	26	∩	∩	NOUN
ejpam-7045	68	27	a	a	DET
ejpam-7045	68	28	̸=	̸=	PROPN
ejpam-7045	68	29	∅	∅	NOUN
ejpam-7045	68	30	for	for	ADP
ejpam-7045	68	31	every	every	DET
ejpam-7045	68	32	τ1τ2	τ1τ2	ADJ
ejpam-7045	68	33	-	-	ADJ
ejpam-7045	68	34	open	open	ADJ
ejpam-7045	68	35	set	set	NOUN
ejpam-7045	68	36	u	u	NOUN
ejpam-7045	68	37	containing	contain	VERB
ejpam-7045	68	38	x.	x.	NOUN
ejpam-7045	68	39	the	the	DET
ejpam-7045	68	40	set	set	NOUN
ejpam-7045	68	41	of	of	ADP
ejpam-7045	68	42	all	all	DET
ejpam-7045	68	43	(	(	PUNCT
ejpam-7045	68	44	τ1	τ1	NOUN
ejpam-7045	68	45	,	,	PUNCT
ejpam-7045	68	46	τ2)θ	τ2)θ	ADJ
ejpam-7045	68	47	-	-	PUNCT
ejpam-7045	68	48	cluster	cluster	NOUN
ejpam-7045	68	49	points	point	NOUN
ejpam-7045	68	50	of	of	ADP
ejpam-7045	68	51	a	a	PRON
ejpam-7045	68	52	is	be	AUX
ejpam-7045	68	53	called	call	VERB
ejpam-7045	68	54	the	the	DET
ejpam-7045	68	55	(	(	PUNCT
ejpam-7045	68	56	τ1	τ1	NOUN
ejpam-7045	68	57	,	,	PUNCT
ejpam-7045	68	58	τ2)θ	τ2)θ	ADJ
ejpam-7045	68	59	-	-	PUNCT
ejpam-7045	68	60	closure	closure	NOUN
ejpam-7045	68	61	[	[	X
ejpam-7045	68	62	9	9	NUM
ejpam-7045	68	63	]	]	PUNCT
ejpam-7045	68	64	of	of	ADP
ejpam-7045	68	65	a	a	PRON
ejpam-7045	68	66	and	and	CCONJ
ejpam-7045	68	67	is	be	AUX
ejpam-7045	68	68	denoted	denote	VERB
ejpam-7045	68	69	by	by	ADP
ejpam-7045	68	70	(	(	PUNCT
ejpam-7045	68	71	τ1	τ1	NOUN
ejpam-7045	68	72	,	,	PUNCT
ejpam-7045	68	73	τ2)θ	τ2)θ	NOUN
ejpam-7045	68	74	-	-	PUNCT
ejpam-7045	68	75	cl(a	cl(a	NUM
ejpam-7045	68	76	)	)	PUNCT
ejpam-7045	68	77	.	.	PUNCT
ejpam-7045	69	1	a	a	DET
ejpam-7045	69	2	subset	subset	NOUN
ejpam-7045	69	3	a	a	PRON
ejpam-7045	69	4	of	of	ADP
ejpam-7045	69	5	a	a	DET
ejpam-7045	69	6	bitopological	bitopological	ADJ
ejpam-7045	69	7	space	space	NOUN
ejpam-7045	69	8	(	(	PUNCT
ejpam-7045	69	9	x	x	NOUN
ejpam-7045	69	10	,	,	PUNCT
ejpam-7045	69	11	τ1	τ1	NOUN
ejpam-7045	69	12	,	,	PUNCT
ejpam-7045	69	13	τ2	τ2	NOUN
ejpam-7045	69	14	)	)	PUNCT
ejpam-7045	69	15	is	be	AUX
ejpam-7045	69	16	said	say	VERB
ejpam-7045	69	17	to	to	PART
ejpam-7045	69	18	be	be	AUX
ejpam-7045	69	19	(	(	PUNCT
ejpam-7045	69	20	τ1	τ1	NOUN
ejpam-7045	69	21	,	,	PUNCT
ejpam-7045	69	22	τ2)θ	τ2)θ	NOUN
ejpam-7045	69	23	-	-	PUNCT
ejpam-7045	69	24	closed	closed	ADJ
ejpam-7045	69	25	[	[	X
ejpam-7045	69	26	9	9	NUM
ejpam-7045	69	27	]	]	X
ejpam-7045	69	28	if	if	SCONJ
ejpam-7045	69	29	(	(	PUNCT
ejpam-7045	69	30	τ1	τ1	NOUN
ejpam-7045	69	31	,	,	PUNCT
ejpam-7045	69	32	τ2)θ	τ2)θ	NOUN
ejpam-7045	69	33	-	-	PUNCT
ejpam-7045	69	34	cl(a	cl(a	NUM
ejpam-7045	69	35	)	)	PUNCT
ejpam-7045	70	1	=	=	PUNCT
ejpam-7045	70	2	a.	a.	NOUN
ejpam-7045	70	3	the	the	DET
ejpam-7045	70	4	complement	complement	NOUN
ejpam-7045	70	5	of	of	ADP
ejpam-7045	70	6	a	a	DET
ejpam-7045	70	7	(	(	PUNCT
ejpam-7045	70	8	τ1	τ1	NOUN
ejpam-7045	70	9	,	,	PUNCT
ejpam-7045	70	10	τ2)θ	τ2)θ	ADJ
ejpam-7045	70	11	-	-	PUNCT
ejpam-7045	70	12	closed	close	VERB
ejpam-7045	70	13	set	set	NOUN
ejpam-7045	70	14	is	be	AUX
ejpam-7045	70	15	said	say	VERB
ejpam-7045	70	16	to	to	PART
ejpam-7045	70	17	be	be	AUX
ejpam-7045	70	18	(	(	PUNCT
ejpam-7045	70	19	τ1	τ1	NOUN
ejpam-7045	70	20	,	,	PUNCT
ejpam-7045	70	21	τ2)θopen	τ2)θopen	PROPN
ejpam-7045	70	22	.	.	PUNCT
ejpam-7045	71	1	the	the	DET
ejpam-7045	71	2	union	union	NOUN
ejpam-7045	71	3	of	of	ADP
ejpam-7045	71	4	all	all	DET
ejpam-7045	71	5	(	(	PUNCT
ejpam-7045	71	6	τ1	τ1	NOUN
ejpam-7045	71	7	,	,	PUNCT
ejpam-7045	71	8	τ2)θ	τ2)θ	ADJ
ejpam-7045	71	9	-	-	PUNCT
ejpam-7045	71	10	open	open	ADJ
ejpam-7045	71	11	sets	set	NOUN
ejpam-7045	71	12	of	of	ADP
ejpam-7045	71	13	x	x	PUNCT
ejpam-7045	71	14	contained	contain	VERB
ejpam-7045	71	15	in	in	ADP
ejpam-7045	71	16	a	a	PRON
ejpam-7045	71	17	is	be	AUX
ejpam-7045	71	18	called	call	VERB
ejpam-7045	71	19	the	the	DET
ejpam-7045	71	20	(	(	PUNCT
ejpam-7045	71	21	τ1	τ1	NOUN
ejpam-7045	71	22	,	,	PUNCT
ejpam-7045	71	23	τ2)θ	τ2)θ	ADJ
ejpam-7045	71	24	-	-	PUNCT
ejpam-7045	71	25	interior	interior	NOUN
ejpam-7045	71	26	[	[	X
ejpam-7045	71	27	9	9	NUM
ejpam-7045	71	28	]	]	PUNCT
ejpam-7045	71	29	of	of	ADP
ejpam-7045	71	30	a	a	PRON
ejpam-7045	71	31	and	and	CCONJ
ejpam-7045	71	32	is	be	AUX
ejpam-7045	71	33	denoted	denote	VERB
ejpam-7045	71	34	by	by	ADP
ejpam-7045	71	35	(	(	PUNCT
ejpam-7045	71	36	τ1	τ1	NOUN
ejpam-7045	71	37	,	,	PUNCT
ejpam-7045	71	38	τ2)θ	τ2)θ	NOUN
ejpam-7045	71	39	-	-	PUNCT
ejpam-7045	71	40	int(a	int(a	NOUN
ejpam-7045	71	41	)	)	PUNCT
ejpam-7045	71	42	.	.	PUNCT
ejpam-7045	72	1	lemma	lemma	PROPN
ejpam-7045	72	2	4	4	NUM
ejpam-7045	72	3	.	.	PUNCT
ejpam-7045	73	1	[	[	X
ejpam-7045	73	2	9	9	NUM
ejpam-7045	73	3	]	]	PUNCT
ejpam-7045	73	4	for	for	ADP
ejpam-7045	73	5	a	a	DET
ejpam-7045	73	6	subset	subset	NOUN
ejpam-7045	73	7	a	a	PRON
ejpam-7045	73	8	of	of	ADP
ejpam-7045	73	9	a	a	DET
ejpam-7045	73	10	bitopological	bitopological	ADJ
ejpam-7045	73	11	space	space	NOUN
ejpam-7045	73	12	(	(	PUNCT
ejpam-7045	73	13	x	x	NOUN
ejpam-7045	73	14	,	,	PUNCT
ejpam-7045	73	15	τ1	τ1	NOUN
ejpam-7045	73	16	,	,	PUNCT
ejpam-7045	73	17	τ2	τ2	NOUN
ejpam-7045	73	18	)	)	PUNCT
ejpam-7045	73	19	,	,	PUNCT
ejpam-7045	73	20	the	the	DET
ejpam-7045	73	21	following	follow	VERB
ejpam-7045	73	22	properties	property	NOUN
ejpam-7045	73	23	hold	hold	VERB
ejpam-7045	73	24	:	:	PUNCT
ejpam-7045	73	25	(	(	PUNCT
ejpam-7045	73	26	1	1	X
ejpam-7045	73	27	)	)	PUNCT
ejpam-7045	73	28	if	if	SCONJ
ejpam-7045	73	29	a	a	PRON
ejpam-7045	73	30	is	be	AUX
ejpam-7045	73	31	τ1τ2	τ1τ2	NOUN
ejpam-7045	73	32	-	-	ADJ
ejpam-7045	73	33	open	open	ADJ
ejpam-7045	73	34	in	in	ADP
ejpam-7045	73	35	x	x	NOUN
ejpam-7045	73	36	,	,	PUNCT
ejpam-7045	73	37	then	then	ADV
ejpam-7045	73	38	τ1τ2	τ1τ2	NOUN
ejpam-7045	73	39	-	-	NUM
ejpam-7045	73	40	cl(a	cl(a	NUM
ejpam-7045	73	41	)	)	PUNCT
ejpam-7045	73	42	=	=	PUNCT
ejpam-7045	73	43	(	(	PUNCT
ejpam-7045	73	44	τ1	τ1	NOUN
ejpam-7045	73	45	,	,	PUNCT
ejpam-7045	73	46	τ2)θ	τ2)θ	NOUN
ejpam-7045	73	47	-	-	PUNCT
ejpam-7045	73	48	cl(a	cl(a	NUM
ejpam-7045	73	49	)	)	PUNCT
ejpam-7045	73	50	.	.	PUNCT
ejpam-7045	74	1	(	(	PUNCT
ejpam-7045	74	2	2	2	X
ejpam-7045	74	3	)	)	PUNCT
ejpam-7045	74	4	(	(	PUNCT
ejpam-7045	74	5	τ1	τ1	NOUN
ejpam-7045	74	6	,	,	PUNCT
ejpam-7045	74	7	τ2)θ	τ2)θ	NOUN
ejpam-7045	74	8	-	-	PUNCT
ejpam-7045	74	9	cl(a	cl(a	NUM
ejpam-7045	74	10	)	)	PUNCT
ejpam-7045	74	11	is	be	AUX
ejpam-7045	74	12	τ1τ2	τ1τ2	NOUN
ejpam-7045	74	13	-	-	ADJ
ejpam-7045	74	14	closed	closed	ADJ
ejpam-7045	74	15	in	in	ADP
ejpam-7045	74	16	x.	x.	NOUN
ejpam-7045	74	17	an	an	DET
ejpam-7045	74	18	ideal	ideal	NOUN
ejpam-7045	74	19	i	i	PRON
ejpam-7045	74	20	on	on	ADP
ejpam-7045	74	21	a	a	DET
ejpam-7045	74	22	topological	topological	ADJ
ejpam-7045	74	23	space	space	NOUN
ejpam-7045	74	24	(	(	PUNCT
ejpam-7045	74	25	x	x	X
ejpam-7045	74	26	,	,	PUNCT
ejpam-7045	74	27	τ	τ	X
ejpam-7045	74	28	)	)	PUNCT
ejpam-7045	74	29	is	be	AUX
ejpam-7045	74	30	a	a	DET
ejpam-7045	74	31	nonempty	nonempty	ADJ
ejpam-7045	74	32	collection	collection	NOUN
ejpam-7045	74	33	of	of	ADP
ejpam-7045	74	34	subsets	subset	NOUN
ejpam-7045	74	35	of	of	ADP
ejpam-7045	74	36	x	x	PUNCT
ejpam-7045	74	37	satisfying	satisfy	VERB
ejpam-7045	74	38	the	the	DET
ejpam-7045	74	39	following	follow	VERB
ejpam-7045	74	40	properties	property	NOUN
ejpam-7045	74	41	:	:	PUNCT
ejpam-7045	74	42	(	(	PUNCT
ejpam-7045	74	43	1	1	X
ejpam-7045	74	44	)	)	PUNCT
ejpam-7045	74	45	a	a	DET
ejpam-7045	74	46	∈	∈	NOUN
ejpam-7045	75	1	i	i	PRON
ejpam-7045	75	2	and	and	CCONJ
ejpam-7045	75	3	b	b	X
ejpam-7045	75	4	⊆	⊆	NUM
ejpam-7045	75	5	a	a	DET
ejpam-7045	75	6	imply	imply	NOUN
ejpam-7045	75	7	b	b	X
ejpam-7045	75	8	∈	∈	PROPN
ejpam-7045	75	9	i	i	PRON
ejpam-7045	75	10	;	;	PUNCT
ejpam-7045	75	11	(	(	PUNCT
ejpam-7045	75	12	2	2	X
ejpam-7045	75	13	)	)	PUNCT
ejpam-7045	76	1	a	a	DET
ejpam-7045	76	2	∈	∈	NOUN
ejpam-7045	76	3	i	i	PRON
ejpam-7045	76	4	and	and	CCONJ
ejpam-7045	76	5	b	b	X
ejpam-7045	76	6	∈	∈	NOUN
ejpam-7045	77	1	i	i	PRON
ejpam-7045	77	2	imply	imply	VERB
ejpam-7045	77	3	a	a	DET
ejpam-7045	77	4	∪	∪	X
ejpam-7045	77	5	b	b	NOUN
ejpam-7045	77	6	∈	∈	NOUN
ejpam-7045	78	1	i	i	PRON
ejpam-7045	78	2	.	.	PUNCT
ejpam-7045	79	1	a	a	DET
ejpam-7045	79	2	topological	topological	ADJ
ejpam-7045	79	3	space	space	NOUN
ejpam-7045	79	4	(	(	PUNCT
ejpam-7045	79	5	x	x	X
ejpam-7045	79	6	,	,	PUNCT
ejpam-7045	79	7	τ	τ	X
ejpam-7045	79	8	)	)	PUNCT
ejpam-7045	79	9	with	with	ADP
ejpam-7045	79	10	an	an	DET
ejpam-7045	79	11	ideal	ideal	ADJ
ejpam-7045	79	12	i	i	PRON
ejpam-7045	79	13	on	on	ADP
ejpam-7045	79	14	x	x	SYM
ejpam-7045	79	15	is	be	AUX
ejpam-7045	79	16	called	call	VERB
ejpam-7045	79	17	an	an	DET
ejpam-7045	79	18	ideal	ideal	ADJ
ejpam-7045	79	19	topological	topological	ADJ
ejpam-7045	79	20	space	space	NOUN
ejpam-7045	79	21	and	and	CCONJ
ejpam-7045	79	22	is	be	AUX
ejpam-7045	79	23	denoted	denote	VERB
ejpam-7045	79	24	by	by	ADP
ejpam-7045	79	25	(	(	PUNCT
ejpam-7045	79	26	x	x	X
ejpam-7045	79	27	,	,	PUNCT
ejpam-7045	79	28	τ	τ	PROPN
ejpam-7045	79	29	,	,	PUNCT
ejpam-7045	79	30	i	i	NOUN
ejpam-7045	79	31	)	)	PUNCT
ejpam-7045	79	32	.	.	PUNCT
ejpam-7045	80	1	for	for	ADP
ejpam-7045	80	2	an	an	DET
ejpam-7045	80	3	ideal	ideal	ADJ
ejpam-7045	80	4	topological	topological	ADJ
ejpam-7045	80	5	space	space	NOUN
ejpam-7045	80	6	(	(	PUNCT
ejpam-7045	80	7	x	x	X
ejpam-7045	80	8	,	,	PUNCT
ejpam-7045	80	9	τ	τ	PROPN
ejpam-7045	80	10	,	,	PUNCT
ejpam-7045	80	11	i	i	PROPN
ejpam-7045	80	12	)	)	PUNCT
ejpam-7045	80	13	and	and	CCONJ
ejpam-7045	80	14	a	a	DET
ejpam-7045	80	15	subset	subset	NOUN
ejpam-7045	80	16	a	a	PRON
ejpam-7045	80	17	of	of	ADP
ejpam-7045	80	18	x	x	PRON
ejpam-7045	80	19	,	,	PUNCT
ejpam-7045	80	20	a⋆(i	a⋆(i	PROPN
ejpam-7045	80	21	)	)	PUNCT
ejpam-7045	80	22	is	be	AUX
ejpam-7045	80	23	defined	define	VERB
ejpam-7045	80	24	as	as	SCONJ
ejpam-7045	80	25	follows	follow	VERB
ejpam-7045	80	26	:	:	PUNCT
ejpam-7045	80	27	a⋆(i	a⋆(i	NOUN
ejpam-7045	80	28	)	)	PUNCT
ejpam-7045	81	1	=	=	PUNCT
ejpam-7045	81	2	{	{	PUNCT
ejpam-7045	81	3	x	x	PUNCT
ejpam-7045	81	4	∈	∈	PROPN
ejpam-7045	81	5	x	x	X
ejpam-7045	81	6	:	:	PUNCT
ejpam-7045	81	7	u	u	X
ejpam-7045	81	8	∩a	∩a	PROPN
ejpam-7045	81	9	̸∈	̸∈	PROPN
ejpam-7045	81	10	i	i	PRON
ejpam-7045	81	11	for	for	ADP
ejpam-7045	81	12	every	every	DET
ejpam-7045	81	13	open	open	ADJ
ejpam-7045	81	14	neighbourhood	neighbourhood	NOUN
ejpam-7045	81	15	u	u	NOUN
ejpam-7045	81	16	of	of	ADP
ejpam-7045	81	17	x	x	NOUN
ejpam-7045	81	18	}	}	PUNCT
ejpam-7045	81	19	.	.	PUNCT
ejpam-7045	82	1	in	in	ADP
ejpam-7045	82	2	case	case	NOUN
ejpam-7045	82	3	there	there	PRON
ejpam-7045	82	4	is	be	VERB
ejpam-7045	82	5	no	no	DET
ejpam-7045	82	6	chance	chance	NOUN
ejpam-7045	82	7	for	for	ADP
ejpam-7045	82	8	confusion	confusion	NOUN
ejpam-7045	82	9	,	,	PUNCT
ejpam-7045	82	10	a⋆(i	a⋆(i	NOUN
ejpam-7045	82	11	)	)	PUNCT
ejpam-7045	82	12	is	be	AUX
ejpam-7045	82	13	simply	simply	ADV
ejpam-7045	82	14	written	write	VERB
ejpam-7045	82	15	as	as	ADP
ejpam-7045	82	16	a⋆.	a⋆.	NOUN
ejpam-7045	82	17	in	in	ADP
ejpam-7045	82	18	[	[	X
ejpam-7045	82	19	15	15	NUM
ejpam-7045	82	20	]	]	PUNCT
ejpam-7045	82	21	,	,	PUNCT
ejpam-7045	82	22	a⋆	a⋆	ADV
ejpam-7045	82	23	is	be	AUX
ejpam-7045	82	24	called	call	VERB
ejpam-7045	82	25	the	the	DET
ejpam-7045	82	26	local	local	ADJ
ejpam-7045	82	27	function	function	NOUN
ejpam-7045	82	28	of	of	ADP
ejpam-7045	82	29	a	a	PRON
ejpam-7045	82	30	with	with	ADP
ejpam-7045	82	31	respect	respect	NOUN
ejpam-7045	82	32	to	to	ADP
ejpam-7045	82	33	i	i	PRON
ejpam-7045	82	34	and	and	CCONJ
ejpam-7045	82	35	τ	τ	PROPN
ejpam-7045	82	36	and	and	CCONJ
ejpam-7045	82	37	cl⋆(a	cl⋆(a	NUM
ejpam-7045	82	38	)	)	PUNCT
ejpam-7045	82	39	=	=	PUNCT
ejpam-7045	82	40	a⋆	a⋆	ADP
ejpam-7045	82	41	∪	∪	ADP
ejpam-7045	82	42	a	a	DET
ejpam-7045	82	43	defines	define	NOUN
ejpam-7045	82	44	a	a	DET
ejpam-7045	82	45	kuratowski	kuratowski	ADJ
ejpam-7045	82	46	closure	closure	NOUN
ejpam-7045	82	47	operator	operator	NOUN
ejpam-7045	82	48	for	for	ADP
ejpam-7045	82	49	a	a	DET
ejpam-7045	82	50	topology	topology	NOUN
ejpam-7045	82	51	τ⋆(i	τ⋆(i	NOUN
ejpam-7045	82	52	)	)	PUNCT
ejpam-7045	82	53	finer	fine	ADJ
ejpam-7045	82	54	than	than	ADP
ejpam-7045	82	55	τ	τ	PROPN
ejpam-7045	82	56	.	.	PUNCT
ejpam-7045	83	1	a	a	DET
ejpam-7045	83	2	subset	subset	NOUN
ejpam-7045	83	3	a	a	PRON
ejpam-7045	83	4	is	be	AUX
ejpam-7045	83	5	said	say	VERB
ejpam-7045	83	6	to	to	PART
ejpam-7045	83	7	be	be	AUX
ejpam-7045	83	8	⋆-closed	⋆-close	VERB
ejpam-7045	83	9	[	[	X
ejpam-7045	83	10	16	16	NUM
ejpam-7045	83	11	]	]	X
ejpam-7045	83	12	if	if	SCONJ
ejpam-7045	83	13	a⋆	a⋆	ADJ
ejpam-7045	83	14	⊆	⊆	NUM
ejpam-7045	83	15	a.	a.	NOUN
ejpam-7045	83	16	the	the	DET
ejpam-7045	83	17	interior	interior	NOUN
ejpam-7045	83	18	of	of	ADP
ejpam-7045	83	19	a	a	DET
ejpam-7045	83	20	subset	subset	NOUN
ejpam-7045	83	21	a	a	DET
ejpam-7045	83	22	in	in	ADP
ejpam-7045	83	23	(	(	PUNCT
ejpam-7045	83	24	x	x	X
ejpam-7045	83	25	,	,	PUNCT
ejpam-7045	83	26	τ⋆(i	τ⋆(i	NOUN
ejpam-7045	83	27	)	)	PUNCT
ejpam-7045	83	28	)	)	PUNCT
ejpam-7045	83	29	is	be	AUX
ejpam-7045	83	30	denoted	denote	VERB
ejpam-7045	83	31	by	by	ADP
ejpam-7045	83	32	int⋆(a	int⋆(a	NOUN
ejpam-7045	83	33	)	)	PUNCT
ejpam-7045	83	34	.	.	PUNCT
ejpam-7045	84	1	a	a	DET
ejpam-7045	84	2	subset	subset	NOUN
ejpam-7045	84	3	a	a	PRON
ejpam-7045	84	4	of	of	ADP
ejpam-7045	84	5	an	an	DET
ejpam-7045	84	6	ideal	ideal	ADJ
ejpam-7045	84	7	topological	topological	ADJ
ejpam-7045	84	8	space	space	NOUN
ejpam-7045	84	9	(	(	PUNCT
ejpam-7045	84	10	x	x	X
ejpam-7045	84	11	,	,	PUNCT
ejpam-7045	84	12	τ	τ	PROPN
ejpam-7045	84	13	,	,	PUNCT
ejpam-7045	84	14	i	i	PROPN
ejpam-7045	84	15	)	)	PUNCT
ejpam-7045	84	16	is	be	AUX
ejpam-7045	84	17	said	say	VERB
ejpam-7045	84	18	to	to	PART
ejpam-7045	84	19	be	be	AUX
ejpam-7045	84	20	r	r	NOUN
ejpam-7045	84	21	-	-	PUNCT
ejpam-7045	84	22	i	i	PRON
ejpam-7045	84	23	⋆-open	⋆-open	VERB
ejpam-7045	85	1	[	[	X
ejpam-7045	85	2	3	3	NUM
ejpam-7045	85	3	]	]	X
ejpam-7045	85	4	c.	c.	PROPN
ejpam-7045	85	5	viriyapong	viriyapong	PROPN
ejpam-7045	85	6	,	,	PUNCT
ejpam-7045	85	7	a.	a.	PROPN
ejpam-7045	85	8	sama	sama	PROPN
ejpam-7045	85	9	-	-	PUNCT
ejpam-7045	85	10	ae	ae	PROPN
ejpam-7045	85	11	,	,	PUNCT
ejpam-7045	85	12	c.	c.	PROPN
ejpam-7045	85	13	boonpok	boonpok	PROPN
ejpam-7045	85	14	/	/	SYM
ejpam-7045	85	15	eur	eur	PROPN
ejpam-7045	85	16	.	.	PUNCT
ejpam-7045	86	1	j.	j.	PROPN
ejpam-7045	86	2	pure	pure	PROPN
ejpam-7045	86	3	appl	appl	PROPN
ejpam-7045	86	4	.	.	PROPN
ejpam-7045	86	5	math	math	PROPN
ejpam-7045	86	6	,	,	PUNCT
ejpam-7045	86	7	18	18	NUM
ejpam-7045	86	8	(	(	PUNCT
ejpam-7045	86	9	4	4	NUM
ejpam-7045	86	10	)	)	PUNCT
ejpam-7045	86	11	(	(	PUNCT
ejpam-7045	86	12	2025	2025	NUM
ejpam-7045	86	13	)	)	PUNCT
ejpam-7045	86	14	,	,	PUNCT
ejpam-7045	86	15	7045	7045	NUM
ejpam-7045	86	16	4	4	NUM
ejpam-7045	86	17	of	of	ADP
ejpam-7045	86	18	12	12	NUM
ejpam-7045	86	19	(	(	PUNCT
ejpam-7045	86	20	resp	resp	NOUN
ejpam-7045	86	21	.	.	PUNCT
ejpam-7045	87	1	i	i	PRON
ejpam-7045	87	2	⋆-preopen	⋆-preopen	VERB
ejpam-7045	88	1	[	[	X
ejpam-7045	88	2	3	3	NUM
ejpam-7045	88	3	]	]	PUNCT
ejpam-7045	88	4	,	,	PUNCT
ejpam-7045	88	5	semi	semi	ADJ
ejpam-7045	88	6	-	-	VERB
ejpam-7045	88	7	i	i	PRON
ejpam-7045	88	8	⋆-open	⋆-open	VERB
ejpam-7045	89	1	[	[	X
ejpam-7045	89	2	17	17	NUM
ejpam-7045	89	3	]	]	PUNCT
ejpam-7045	89	4	,	,	PUNCT
ejpam-7045	89	5	semi	semi	ADJ
ejpam-7045	89	6	-	-	VERB
ejpam-7045	89	7	i	i	PRON
ejpam-7045	89	8	⋆-preopen	⋆-preopen	VERB
ejpam-7045	90	1	[	[	X
ejpam-7045	90	2	17	17	NUM
ejpam-7045	90	3	]	]	SYM
ejpam-7045	90	4	)	)	PUNCT
ejpam-7045	90	5	if	if	SCONJ
ejpam-7045	90	6	a	a	DET
ejpam-7045	90	7	=	=	PUNCT
ejpam-7045	90	8	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7045	90	9	)	)	PUNCT
ejpam-7045	90	10	)	)	PUNCT
ejpam-7045	90	11	(	(	PUNCT
ejpam-7045	90	12	resp	resp	NOUN
ejpam-7045	90	13	.	.	PUNCT
ejpam-7045	91	1	a	a	DET
ejpam-7045	91	2	⊆	⊆	NUM
ejpam-7045	91	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7045	91	4	)	)	PUNCT
ejpam-7045	91	5	)	)	PUNCT
ejpam-7045	91	6	,	,	PUNCT
ejpam-7045	91	7	a	a	DET
ejpam-7045	91	8	⊆	⊆	NUM
ejpam-7045	91	9	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7045	91	10	)	)	PUNCT
ejpam-7045	91	11	)	)	PUNCT
ejpam-7045	91	12	,	,	PUNCT
ejpam-7045	91	13	a	a	DET
ejpam-7045	91	14	⊆	⊆	NUM
ejpam-7045	91	15	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7045	91	16	)	)	PUNCT
ejpam-7045	91	17	)	)	PUNCT
ejpam-7045	91	18	)	)	PUNCT
ejpam-7045	91	19	)	)	PUNCT
ejpam-7045	91	20	.	.	PUNCT
ejpam-7045	92	1	the	the	DET
ejpam-7045	92	2	complement	complement	NOUN
ejpam-7045	92	3	of	of	ADP
ejpam-7045	92	4	a	a	DET
ejpam-7045	92	5	r	r	NOUN
ejpam-7045	92	6	-	-	PUNCT
ejpam-7045	92	7	i	i	PRON
ejpam-7045	92	8	⋆-open	⋆-open	VERB
ejpam-7045	92	9	(	(	PUNCT
ejpam-7045	92	10	resp	resp	NOUN
ejpam-7045	92	11	.	.	PUNCT
ejpam-7045	93	1	i	i	PRON
ejpam-7045	93	2	⋆-preopen	⋆-preopen	VERB
ejpam-7045	93	3	,	,	PUNCT
ejpam-7045	93	4	semi	semi	ADJ
ejpam-7045	93	5	-	-	ADJ
ejpam-7045	93	6	i	i	PRON
ejpam-7045	93	7	⋆-open	⋆-open	VERB
ejpam-7045	93	8	,	,	PUNCT
ejpam-7045	93	9	semi	semi	ADJ
ejpam-7045	93	10	-	-	VERB
ejpam-7045	93	11	i	i	PRON
ejpam-7045	93	12	⋆-preopen	⋆-preopen	ADV
ejpam-7045	93	13	)	)	PUNCT
ejpam-7045	94	1	set	set	NOUN
ejpam-7045	94	2	is	be	AUX
ejpam-7045	94	3	said	say	VERB
ejpam-7045	94	4	to	to	PART
ejpam-7045	94	5	be	be	AUX
ejpam-7045	94	6	r	r	NOUN
ejpam-7045	94	7	-	-	PUNCT
ejpam-7045	94	8	i	i	PRON
ejpam-7045	94	9	⋆-closed	⋆-close	VERB
ejpam-7045	94	10	(	(	PUNCT
ejpam-7045	94	11	resp	resp	NOUN
ejpam-7045	94	12	.	.	PUNCT
ejpam-7045	95	1	i	i	PRON
ejpam-7045	95	2	⋆-preclosed	⋆-preclose	VERB
ejpam-7045	95	3	,	,	PUNCT
ejpam-7045	95	4	semi	semi	ADV
ejpam-7045	95	5	-	-	VERB
ejpam-7045	95	6	i	i	PRON
ejpam-7045	95	7	⋆-closed	⋆-close	VERB
ejpam-7045	95	8	,	,	PUNCT
ejpam-7045	95	9	semi	semi	ADJ
ejpam-7045	95	10	-	-	VERB
ejpam-7045	95	11	i	i	PRON
ejpam-7045	95	12	⋆-preclosed	⋆-preclose	VERB
ejpam-7045	95	13	)	)	PUNCT
ejpam-7045	95	14	.	.	PUNCT
ejpam-7045	96	1	for	for	ADP
ejpam-7045	96	2	a	a	DET
ejpam-7045	96	3	subset	subset	NOUN
ejpam-7045	96	4	a	a	PRON
ejpam-7045	96	5	of	of	ADP
ejpam-7045	96	6	an	an	DET
ejpam-7045	96	7	ideal	ideal	ADJ
ejpam-7045	96	8	topological	topological	ADJ
ejpam-7045	96	9	space	space	NOUN
ejpam-7045	96	10	(	(	PUNCT
ejpam-7045	96	11	x	x	X
ejpam-7045	96	12	,	,	PUNCT
ejpam-7045	96	13	τ	τ	PROPN
ejpam-7045	96	14	,	,	PUNCT
ejpam-7045	96	15	i	i	NOUN
ejpam-7045	96	16	)	)	PUNCT
ejpam-7045	96	17	,	,	PUNCT
ejpam-7045	96	18	the	the	DET
ejpam-7045	96	19	intersection	intersection	NOUN
ejpam-7045	96	20	of	of	ADP
ejpam-7045	96	21	all	all	PRON
ejpam-7045	96	22	semi	semi	NOUN
ejpam-7045	96	23	-	-	ADJ
ejpam-7045	96	24	i	i	PRON
ejpam-7045	96	25	⋆-closed	⋆-close	VERB
ejpam-7045	96	26	sets	set	NOUN
ejpam-7045	96	27	containing	contain	VERB
ejpam-7045	96	28	a	a	PRON
ejpam-7045	96	29	is	be	AUX
ejpam-7045	96	30	called	call	VERB
ejpam-7045	96	31	the	the	DET
ejpam-7045	96	32	semi	semi	NOUN
ejpam-7045	96	33	-	-	ADJ
ejpam-7045	96	34	i	i	PRON
ejpam-7045	96	35	⋆-closure	⋆-closure	NOUN
ejpam-7045	97	1	[	[	X
ejpam-7045	97	2	17	17	NUM
ejpam-7045	97	3	]	]	PUNCT
ejpam-7045	97	4	of	of	ADP
ejpam-7045	97	5	a	a	PRON
ejpam-7045	97	6	and	and	CCONJ
ejpam-7045	97	7	is	be	AUX
ejpam-7045	97	8	denoted	denote	VERB
ejpam-7045	97	9	by	by	ADP
ejpam-7045	97	10	scl⋆(a	scl⋆(a	NOUN
ejpam-7045	97	11	)	)	PUNCT
ejpam-7045	97	12	(	(	PUNCT
ejpam-7045	97	13	scli	scli	PROPN
ejpam-7045	97	14	⋆(a	⋆(a	NUM
ejpam-7045	97	15	)	)	PUNCT
ejpam-7045	98	1	[	[	X
ejpam-7045	98	2	17	17	NUM
ejpam-7045	98	3	]	]	PUNCT
ejpam-7045	98	4	)	)	PUNCT
ejpam-7045	98	5	.	.	PUNCT
ejpam-7045	99	1	the	the	DET
ejpam-7045	99	2	union	union	NOUN
ejpam-7045	99	3	of	of	ADP
ejpam-7045	99	4	all	all	PRON
ejpam-7045	99	5	semi	semi	ADJ
ejpam-7045	99	6	-	-	ADJ
ejpam-7045	99	7	i	i	PRON
ejpam-7045	99	8	⋆-open	⋆-open	ADJ
ejpam-7045	99	9	sets	set	NOUN
ejpam-7045	99	10	contained	contain	VERB
ejpam-7045	99	11	in	in	ADP
ejpam-7045	99	12	a	a	PRON
ejpam-7045	99	13	is	be	AUX
ejpam-7045	99	14	called	call	VERB
ejpam-7045	99	15	the	the	DET
ejpam-7045	99	16	semi	semi	NOUN
ejpam-7045	99	17	-	-	ADJ
ejpam-7045	99	18	i	i	PRON
ejpam-7045	99	19	⋆-interior	⋆-interior	PUNCT
ejpam-7045	100	1	[	[	X
ejpam-7045	100	2	17	17	NUM
ejpam-7045	100	3	]	]	PUNCT
ejpam-7045	100	4	of	of	ADP
ejpam-7045	100	5	a	a	PRON
ejpam-7045	100	6	and	and	CCONJ
ejpam-7045	100	7	is	be	AUX
ejpam-7045	100	8	denoted	denote	VERB
ejpam-7045	100	9	by	by	ADP
ejpam-7045	100	10	sint⋆(a	sint⋆(a	PROPN
ejpam-7045	100	11	)	)	PUNCT
ejpam-7045	101	1	(	(	PUNCT
ejpam-7045	101	2	sinti	sinti	PROPN
ejpam-7045	101	3	⋆(a	⋆(a	NOUN
ejpam-7045	101	4	)	)	PUNCT
ejpam-7045	102	1	[	[	X
ejpam-7045	102	2	17	17	NUM
ejpam-7045	102	3	]	]	NUM
ejpam-7045	102	4	)	)	PUNCT
ejpam-7045	102	5	.	.	PUNCT
ejpam-7045	103	1	lemma	lemma	PROPN
ejpam-7045	103	2	5	5	NUM
ejpam-7045	103	3	.	.	PUNCT
ejpam-7045	104	1	[	[	X
ejpam-7045	104	2	17	17	NUM
ejpam-7045	104	3	]	]	PUNCT
ejpam-7045	104	4	for	for	ADP
ejpam-7045	104	5	a	a	DET
ejpam-7045	104	6	subset	subset	NOUN
ejpam-7045	104	7	a	a	PRON
ejpam-7045	104	8	of	of	ADP
ejpam-7045	104	9	an	an	DET
ejpam-7045	104	10	ideal	ideal	ADJ
ejpam-7045	104	11	topological	topological	ADJ
ejpam-7045	104	12	space	space	NOUN
ejpam-7045	104	13	(	(	PUNCT
ejpam-7045	104	14	x	x	X
ejpam-7045	104	15	,	,	PUNCT
ejpam-7045	104	16	τ	τ	PROPN
ejpam-7045	104	17	,	,	PUNCT
ejpam-7045	104	18	i	i	NOUN
ejpam-7045	104	19	)	)	PUNCT
ejpam-7045	104	20	,	,	PUNCT
ejpam-7045	104	21	the	the	DET
ejpam-7045	104	22	following	follow	VERB
ejpam-7045	104	23	properties	property	NOUN
ejpam-7045	104	24	hold	hold	VERB
ejpam-7045	104	25	:	:	PUNCT
ejpam-7045	104	26	(	(	PUNCT
ejpam-7045	104	27	1	1	X
ejpam-7045	104	28	)	)	PUNCT
ejpam-7045	104	29	scl⋆(a	scl⋆(a	NUM
ejpam-7045	104	30	)	)	PUNCT
ejpam-7045	104	31	=	=	PUNCT
ejpam-7045	104	32	a	a	DET
ejpam-7045	104	33	∪	∪	ADJ
ejpam-7045	104	34	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7045	104	35	)	)	PUNCT
ejpam-7045	104	36	)	)	PUNCT
ejpam-7045	104	37	.	.	PUNCT
ejpam-7045	105	1	(	(	PUNCT
ejpam-7045	105	2	2	2	X
ejpam-7045	105	3	)	)	PUNCT
ejpam-7045	105	4	sint⋆(a	sint⋆(a	PROPN
ejpam-7045	105	5	)	)	PUNCT
ejpam-7045	106	1	=	=	PUNCT
ejpam-7045	106	2	a	a	DET
ejpam-7045	106	3	∩	∩	ADJ
ejpam-7045	106	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7045	106	5	)	)	PUNCT
ejpam-7045	106	6	)	)	PUNCT
ejpam-7045	106	7	.	.	PUNCT
ejpam-7045	107	1	a	a	DET
ejpam-7045	107	2	subset	subset	NOUN
ejpam-7045	107	3	a	a	PRON
ejpam-7045	107	4	of	of	ADP
ejpam-7045	107	5	an	an	DET
ejpam-7045	107	6	ideal	ideal	ADJ
ejpam-7045	107	7	topological	topological	ADJ
ejpam-7045	107	8	space	space	NOUN
ejpam-7045	107	9	(	(	PUNCT
ejpam-7045	107	10	x	x	X
ejpam-7045	107	11	,	,	PUNCT
ejpam-7045	107	12	τ	τ	PROPN
ejpam-7045	107	13	,	,	PUNCT
ejpam-7045	107	14	i	i	PROPN
ejpam-7045	107	15	)	)	PUNCT
ejpam-7045	107	16	is	be	AUX
ejpam-7045	107	17	said	say	VERB
ejpam-7045	107	18	to	to	PART
ejpam-7045	107	19	be	be	AUX
ejpam-7045	107	20	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	107	21	-	-	PUNCT
ejpam-7045	107	22	open	open	ADJ
ejpam-7045	107	23	[	[	X
ejpam-7045	107	24	18	18	NUM
ejpam-7045	107	25	]	]	PUNCT
ejpam-7045	107	26	(	(	PUNCT
ejpam-7045	107	27	α	α	X
ejpam-7045	107	28	-	-	ADJ
ejpam-7045	107	29	i	i	PRON
ejpam-7045	107	30	⋆open	⋆open	VERB
ejpam-7045	108	1	[	[	X
ejpam-7045	108	2	19	19	NUM
ejpam-7045	108	3	]	]	PUNCT
ejpam-7045	108	4	)	)	PUNCT
ejpam-7045	108	5	if	if	SCONJ
ejpam-7045	108	6	a	a	DET
ejpam-7045	108	7	⊆	⊆	NUM
ejpam-7045	108	8	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7045	108	9	)	)	PUNCT
ejpam-7045	108	10	)	)	PUNCT
ejpam-7045	108	11	)	)	PUNCT
ejpam-7045	108	12	.	.	PUNCT
ejpam-7045	109	1	the	the	DET
ejpam-7045	109	2	complement	complement	NOUN
ejpam-7045	109	3	of	of	ADP
ejpam-7045	109	4	an	an	DET
ejpam-7045	109	5	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	109	6	-	-	PUNCT
ejpam-7045	109	7	open	open	ADJ
ejpam-7045	109	8	set	set	NOUN
ejpam-7045	109	9	is	be	AUX
ejpam-7045	109	10	said	say	VERB
ejpam-7045	109	11	to	to	PART
ejpam-7045	109	12	be	be	AUX
ejpam-7045	109	13	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	109	14	-	-	PUNCT
ejpam-7045	109	15	closed	closed	ADJ
ejpam-7045	109	16	.	.	PUNCT
ejpam-7045	110	1	lemma	lemma	PROPN
ejpam-7045	110	2	6	6	NUM
ejpam-7045	110	3	.	.	PUNCT
ejpam-7045	111	1	[	[	X
ejpam-7045	111	2	19	19	NUM
ejpam-7045	111	3	]	]	X
ejpam-7045	111	4	for	for	ADP
ejpam-7045	111	5	a	a	DET
ejpam-7045	111	6	subset	subset	NOUN
ejpam-7045	111	7	a	a	PRON
ejpam-7045	111	8	of	of	ADP
ejpam-7045	111	9	an	an	DET
ejpam-7045	111	10	ideal	ideal	ADJ
ejpam-7045	111	11	topological	topological	ADJ
ejpam-7045	111	12	space	space	NOUN
ejpam-7045	111	13	(	(	PUNCT
ejpam-7045	111	14	x	x	X
ejpam-7045	111	15	,	,	PUNCT
ejpam-7045	111	16	τ	τ	PROPN
ejpam-7045	111	17	,	,	PUNCT
ejpam-7045	111	18	i	i	NOUN
ejpam-7045	111	19	)	)	PUNCT
ejpam-7045	111	20	,	,	PUNCT
ejpam-7045	111	21	the	the	DET
ejpam-7045	111	22	following	follow	VERB
ejpam-7045	111	23	properties	property	NOUN
ejpam-7045	111	24	are	be	AUX
ejpam-7045	111	25	equivalent	equivalent	ADJ
ejpam-7045	111	26	:	:	PUNCT
ejpam-7045	111	27	(	(	PUNCT
ejpam-7045	111	28	1	1	X
ejpam-7045	111	29	)	)	PUNCT
ejpam-7045	111	30	a	a	PRON
ejpam-7045	111	31	is	be	AUX
ejpam-7045	111	32	α	α	X
ejpam-7045	111	33	-	-	ADJ
ejpam-7045	111	34	i	i	PRON
ejpam-7045	111	35	⋆-open	⋆-open	VERB
ejpam-7045	111	36	in	in	ADP
ejpam-7045	111	37	x.	x.	PROPN
ejpam-7045	111	38	(	(	PUNCT
ejpam-7045	111	39	2	2	NUM
ejpam-7045	111	40	)	)	PUNCT
ejpam-7045	111	41	g	g	ADP
ejpam-7045	111	42	⊆	⊆	NUM
ejpam-7045	111	43	a	a	DET
ejpam-7045	111	44	⊆	⊆	NUM
ejpam-7045	111	45	int⋆(cl⋆(g	int⋆(cl⋆(g	NOUN
ejpam-7045	111	46	)	)	PUNCT
ejpam-7045	111	47	)	)	PUNCT
ejpam-7045	111	48	for	for	ADP
ejpam-7045	111	49	some	some	DET
ejpam-7045	111	50	⋆-open	⋆-open	ADJ
ejpam-7045	111	51	set	set	VERB
ejpam-7045	111	52	g.	g.	PROPN
ejpam-7045	111	53	(	(	PUNCT
ejpam-7045	111	54	3	3	NUM
ejpam-7045	111	55	)	)	PUNCT
ejpam-7045	111	56	g	g	ADP
ejpam-7045	111	57	⊆	⊆	NUM
ejpam-7045	111	58	a	a	DET
ejpam-7045	111	59	⊆	⊆	NUM
ejpam-7045	111	60	scl⋆(g	scl⋆(g	NOUN
ejpam-7045	111	61	)	)	PUNCT
ejpam-7045	111	62	for	for	ADP
ejpam-7045	111	63	some	some	DET
ejpam-7045	111	64	⋆-open	⋆-open	ADJ
ejpam-7045	111	65	set	set	NOUN
ejpam-7045	111	66	g.	g.	PROPN
ejpam-7045	111	67	(	(	PUNCT
ejpam-7045	111	68	4	4	X
ejpam-7045	111	69	)	)	PUNCT
ejpam-7045	111	70	a	a	DET
ejpam-7045	111	71	⊆	⊆	NUM
ejpam-7045	111	72	scl⋆(int⋆(a	scl⋆(int⋆(a	NOUN
ejpam-7045	111	73	)	)	PUNCT
ejpam-7045	111	74	)	)	PUNCT
ejpam-7045	111	75	.	.	PUNCT
ejpam-7045	112	1	for	for	ADP
ejpam-7045	112	2	a	a	DET
ejpam-7045	112	3	subset	subset	NOUN
ejpam-7045	112	4	a	a	PRON
ejpam-7045	112	5	of	of	ADP
ejpam-7045	112	6	an	an	DET
ejpam-7045	112	7	ideal	ideal	ADJ
ejpam-7045	112	8	topological	topological	ADJ
ejpam-7045	112	9	space	space	NOUN
ejpam-7045	112	10	(	(	PUNCT
ejpam-7045	112	11	x	x	X
ejpam-7045	112	12	,	,	PUNCT
ejpam-7045	112	13	τ	τ	PROPN
ejpam-7045	112	14	,	,	PUNCT
ejpam-7045	112	15	i	i	NOUN
ejpam-7045	112	16	)	)	PUNCT
ejpam-7045	112	17	,	,	PUNCT
ejpam-7045	112	18	the	the	DET
ejpam-7045	112	19	intersection	intersection	NOUN
ejpam-7045	112	20	of	of	ADP
ejpam-7045	112	21	all	all	DET
ejpam-7045	112	22	α	α	NOUN
ejpam-7045	112	23	-	-	PUNCT
ejpam-7045	112	24	i	i	PRON
ejpam-7045	112	25	⋆closed	⋆close	VERB
ejpam-7045	112	26	sets	set	NOUN
ejpam-7045	112	27	containing	contain	VERB
ejpam-7045	112	28	a	a	PRON
ejpam-7045	112	29	is	be	AUX
ejpam-7045	112	30	called	call	VERB
ejpam-7045	112	31	the	the	DET
ejpam-7045	112	32	α	α	NOUN
ejpam-7045	112	33	-	-	NOUN
ejpam-7045	112	34	i	i	PRON
ejpam-7045	112	35	⋆-closure	⋆-closure	NOUN
ejpam-7045	113	1	[	[	X
ejpam-7045	113	2	19	19	NUM
ejpam-7045	113	3	]	]	PUNCT
ejpam-7045	113	4	of	of	ADP
ejpam-7045	113	5	a	a	PRON
ejpam-7045	113	6	and	and	CCONJ
ejpam-7045	113	7	is	be	AUX
ejpam-7045	113	8	denoted	denote	VERB
ejpam-7045	113	9	by	by	ADP
ejpam-7045	113	10	αcl⋆(a	αcl⋆(a	NUM
ejpam-7045	113	11	)	)	PUNCT
ejpam-7045	113	12	(	(	PUNCT
ejpam-7045	113	13	αcli	αcli	X
ejpam-7045	113	14	⋆(a	⋆(a	AUX
ejpam-7045	113	15	)	)	PUNCT
ejpam-7045	114	1	[	[	X
ejpam-7045	114	2	19	19	NUM
ejpam-7045	114	3	]	]	NUM
ejpam-7045	114	4	)	)	PUNCT
ejpam-7045	114	5	.	.	PUNCT
ejpam-7045	115	1	the	the	DET
ejpam-7045	115	2	α	α	PROPN
ejpam-7045	115	3	-	-	PUNCT
ejpam-7045	115	4	i	i	PRON
ejpam-7045	115	5	⋆-interior	⋆-interior	PUNCT
ejpam-7045	115	6	[	[	X
ejpam-7045	115	7	19	19	NUM
ejpam-7045	115	8	]	]	PUNCT
ejpam-7045	115	9	of	of	ADP
ejpam-7045	115	10	a	a	PRON
ejpam-7045	115	11	is	be	AUX
ejpam-7045	115	12	defined	define	VERB
ejpam-7045	115	13	by	by	ADP
ejpam-7045	115	14	the	the	DET
ejpam-7045	115	15	union	union	NOUN
ejpam-7045	115	16	of	of	ADP
ejpam-7045	115	17	all	all	DET
ejpam-7045	115	18	α	α	PROPN
ejpam-7045	115	19	-	-	ADJ
ejpam-7045	115	20	i	i	PRON
ejpam-7045	115	21	⋆-open	⋆-open	ADJ
ejpam-7045	115	22	sets	set	NOUN
ejpam-7045	115	23	contained	contain	VERB
ejpam-7045	115	24	in	in	ADP
ejpam-7045	115	25	a	a	PRON
ejpam-7045	115	26	and	and	CCONJ
ejpam-7045	115	27	is	be	AUX
ejpam-7045	115	28	denoted	denote	VERB
ejpam-7045	115	29	by	by	ADP
ejpam-7045	115	30	αint⋆(a	αint⋆(a	PROPN
ejpam-7045	115	31	)	)	PUNCT
ejpam-7045	115	32	(	(	PUNCT
ejpam-7045	115	33	αinti	αinti	X
ejpam-7045	115	34	⋆(a	⋆(a	NOUN
ejpam-7045	115	35	)	)	PUNCT
ejpam-7045	116	1	[	[	X
ejpam-7045	116	2	19	19	NUM
ejpam-7045	116	3	]	]	NUM
ejpam-7045	116	4	)	)	PUNCT
ejpam-7045	116	5	.	.	PUNCT
ejpam-7045	117	1	lemma	lemma	PROPN
ejpam-7045	117	2	7	7	NUM
ejpam-7045	117	3	.	.	PUNCT
ejpam-7045	118	1	[	[	X
ejpam-7045	118	2	19	19	NUM
ejpam-7045	118	3	]	]	X
ejpam-7045	118	4	for	for	ADP
ejpam-7045	118	5	a	a	DET
ejpam-7045	118	6	subset	subset	NOUN
ejpam-7045	118	7	a	a	PRON
ejpam-7045	118	8	of	of	ADP
ejpam-7045	118	9	an	an	DET
ejpam-7045	118	10	ideal	ideal	ADJ
ejpam-7045	118	11	topological	topological	ADJ
ejpam-7045	118	12	space	space	NOUN
ejpam-7045	118	13	(	(	PUNCT
ejpam-7045	118	14	x	x	X
ejpam-7045	118	15	,	,	PUNCT
ejpam-7045	118	16	τ	τ	PROPN
ejpam-7045	118	17	,	,	PUNCT
ejpam-7045	118	18	i	i	NOUN
ejpam-7045	118	19	)	)	PUNCT
ejpam-7045	118	20	,	,	PUNCT
ejpam-7045	118	21	the	the	DET
ejpam-7045	118	22	following	follow	VERB
ejpam-7045	118	23	properties	property	NOUN
ejpam-7045	118	24	hold	hold	VERB
ejpam-7045	118	25	:	:	PUNCT
ejpam-7045	118	26	(	(	PUNCT
ejpam-7045	118	27	1	1	X
ejpam-7045	118	28	)	)	PUNCT
ejpam-7045	118	29	a	a	PRON
ejpam-7045	118	30	is	be	AUX
ejpam-7045	118	31	α	α	X
ejpam-7045	118	32	-	-	PUNCT
ejpam-7045	118	33	i	i	PRON
ejpam-7045	118	34	⋆-closed	⋆-close	VERB
ejpam-7045	118	35	in	in	ADP
ejpam-7045	118	36	x	x	SYM
ejpam-7045	118	37	if	if	SCONJ
ejpam-7045	118	38	and	and	CCONJ
ejpam-7045	118	39	only	only	ADV
ejpam-7045	118	40	if	if	SCONJ
ejpam-7045	118	41	sint⋆(cl⋆(a	sint⋆(cl⋆(a	NOUN
ejpam-7045	118	42	)	)	PUNCT
ejpam-7045	118	43	)	)	PUNCT
ejpam-7045	119	1	⊆	⊆	NUM
ejpam-7045	119	2	a.	a.	NOUN
ejpam-7045	119	3	(	(	PUNCT
ejpam-7045	119	4	2	2	NUM
ejpam-7045	119	5	)	)	PUNCT
ejpam-7045	119	6	sint⋆(cl⋆(a	sint⋆(cl⋆(a	PROPN
ejpam-7045	119	7	)	)	PUNCT
ejpam-7045	119	8	)	)	PUNCT
ejpam-7045	119	9	=	=	PUNCT
ejpam-7045	119	10	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7045	119	11	)	)	PUNCT
ejpam-7045	119	12	)	)	PUNCT
ejpam-7045	119	13	)	)	PUNCT
ejpam-7045	119	14	.	.	PUNCT
ejpam-7045	120	1	(	(	PUNCT
ejpam-7045	120	2	3	3	X
ejpam-7045	120	3	)	)	PUNCT
ejpam-7045	120	4	αcl⋆(a	αcl⋆(a	NUM
ejpam-7045	120	5	)	)	PUNCT
ejpam-7045	121	1	=	=	PUNCT
ejpam-7045	121	2	a	a	DET
ejpam-7045	121	3	∪	∪	ADJ
ejpam-7045	121	4	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7045	121	5	)	)	PUNCT
ejpam-7045	121	6	)	)	PUNCT
ejpam-7045	121	7	)	)	PUNCT
ejpam-7045	121	8	.	.	PUNCT
ejpam-7045	122	1	(	(	PUNCT
ejpam-7045	122	2	4	4	X
ejpam-7045	122	3	)	)	PUNCT
ejpam-7045	122	4	αint⋆(a	αint⋆(a	NOUN
ejpam-7045	122	5	)	)	PUNCT
ejpam-7045	123	1	=	=	SYM
ejpam-7045	123	2	a	a	DET
ejpam-7045	123	3	∩	∩	ADJ
ejpam-7045	123	4	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7045	123	5	)	)	PUNCT
ejpam-7045	123	6	)	)	PUNCT
ejpam-7045	123	7	)	)	PUNCT
ejpam-7045	123	8	.	.	PUNCT
ejpam-7045	124	1	by	by	ADP
ejpam-7045	124	2	a	a	DET
ejpam-7045	124	3	multifunction	multifunction	NOUN
ejpam-7045	124	4	f	f	NOUN
ejpam-7045	124	5	:	:	PUNCT
ejpam-7045	124	6	x	x	X
ejpam-7045	124	7	→	→	SYM
ejpam-7045	124	8	y	y	PROPN
ejpam-7045	124	9	,	,	PUNCT
ejpam-7045	124	10	we	we	PRON
ejpam-7045	124	11	mean	mean	VERB
ejpam-7045	124	12	a	a	DET
ejpam-7045	124	13	point	point	NOUN
ejpam-7045	124	14	-	-	PUNCT
ejpam-7045	124	15	to	to	ADP
ejpam-7045	124	16	-	-	PUNCT
ejpam-7045	124	17	set	set	VERB
ejpam-7045	124	18	correspondence	correspondence	NOUN
ejpam-7045	124	19	from	from	ADP
ejpam-7045	124	20	x	x	PUNCT
ejpam-7045	124	21	into	into	ADP
ejpam-7045	124	22	y	y	PROPN
ejpam-7045	124	23	,	,	PUNCT
ejpam-7045	124	24	and	and	CCONJ
ejpam-7045	124	25	we	we	PRON
ejpam-7045	124	26	always	always	ADV
ejpam-7045	124	27	assume	assume	VERB
ejpam-7045	124	28	that	that	SCONJ
ejpam-7045	124	29	f	f	PROPN
ejpam-7045	124	30	(	(	PUNCT
ejpam-7045	124	31	x	x	X
ejpam-7045	124	32	)	)	PUNCT
ejpam-7045	124	33	̸=	̸=	NOUN
ejpam-7045	124	34	∅	∅	NOUN
ejpam-7045	124	35	for	for	ADP
ejpam-7045	124	36	all	all	PRON
ejpam-7045	124	37	x	x	SYM
ejpam-7045	124	38	∈	∈	ADJ
ejpam-7045	124	39	x.	x.	NOUN
ejpam-7045	124	40	for	for	ADP
ejpam-7045	124	41	a	a	DET
ejpam-7045	124	42	multifunction	multifunction	NOUN
ejpam-7045	124	43	f	f	NOUN
ejpam-7045	124	44	:	:	PUNCT
ejpam-7045	124	45	x	x	X
ejpam-7045	124	46	→	→	SYM
ejpam-7045	124	47	y	y	PROPN
ejpam-7045	124	48	,	,	PUNCT
ejpam-7045	124	49	we	we	PRON
ejpam-7045	124	50	shall	shall	AUX
ejpam-7045	124	51	denote	denote	VERB
ejpam-7045	124	52	the	the	DET
ejpam-7045	124	53	upper	upper	ADJ
ejpam-7045	124	54	and	and	CCONJ
ejpam-7045	124	55	lower	low	ADJ
ejpam-7045	124	56	inverse	inverse	NOUN
ejpam-7045	124	57	of	of	ADP
ejpam-7045	124	58	a	a	DET
ejpam-7045	124	59	set	set	NOUN
ejpam-7045	124	60	b	b	PROPN
ejpam-7045	124	61	of	of	ADP
ejpam-7045	124	62	y	y	PROPN
ejpam-7045	124	63	by	by	ADP
ejpam-7045	124	64	f+(b	f+(b	NOUN
ejpam-7045	124	65	)	)	PUNCT
ejpam-7045	124	66	and	and	CCONJ
ejpam-7045	124	67	f−(b	f−(b	NOUN
ejpam-7045	124	68	)	)	PUNCT
ejpam-7045	124	69	,	,	PUNCT
ejpam-7045	124	70	respectively	respectively	ADV
ejpam-7045	124	71	,	,	PUNCT
ejpam-7045	124	72	that	that	ADV
ejpam-7045	124	73	is	is	ADV
ejpam-7045	124	74	,	,	PUNCT
ejpam-7045	124	75	f+(b	f+(b	NOUN
ejpam-7045	124	76	)	)	PUNCT
ejpam-7045	124	77	=	=	PRON
ejpam-7045	125	1	{	{	PUNCT
ejpam-7045	125	2	x	x	PUNCT
ejpam-7045	125	3	∈	∈	PROPN
ejpam-7045	125	4	x	x	INTJ
ejpam-7045	126	1	|	|	NOUN
ejpam-7045	126	2	f	f	X
ejpam-7045	126	3	(	(	PUNCT
ejpam-7045	126	4	x	x	NOUN
ejpam-7045	126	5	)	)	PUNCT
ejpam-7045	126	6	⊆	⊆	NUM
ejpam-7045	126	7	b	b	NOUN
ejpam-7045	126	8	}	}	PUNCT
ejpam-7045	126	9	and	and	CCONJ
ejpam-7045	126	10	f−(b	f−(b	PROPN
ejpam-7045	126	11	)	)	PUNCT
ejpam-7045	126	12	=	=	PRON
ejpam-7045	127	1	{	{	PUNCT
ejpam-7045	127	2	x	x	PUNCT
ejpam-7045	127	3	∈	∈	PROPN
ejpam-7045	127	4	x	x	INTJ
ejpam-7045	128	1	|	|	NOUN
ejpam-7045	128	2	f	f	X
ejpam-7045	128	3	(	(	PUNCT
ejpam-7045	128	4	x	x	NOUN
ejpam-7045	128	5	)	)	PUNCT
ejpam-7045	128	6	∩	∩	NOUN
ejpam-7045	128	7	b	b	PROPN
ejpam-7045	128	8	̸=	̸=	PROPN
ejpam-7045	128	9	∅	∅	NOUN
ejpam-7045	128	10	}	}	PUNCT
ejpam-7045	128	11	.	.	PUNCT
ejpam-7045	129	1	in	in	ADP
ejpam-7045	129	2	particular	particular	ADJ
ejpam-7045	129	3	,	,	PUNCT
ejpam-7045	129	4	f−(y	f−(y	NOUN
ejpam-7045	129	5	)	)	PUNCT
ejpam-7045	129	6	=	=	SYM
ejpam-7045	130	1	{	{	PUNCT
ejpam-7045	130	2	x	x	PUNCT
ejpam-7045	130	3	∈	∈	PROPN
ejpam-7045	130	4	x	x	INTJ
ejpam-7045	131	1	|	|	ADV
ejpam-7045	131	2	y	y	PROPN
ejpam-7045	131	3	∈	∈	PROPN
ejpam-7045	131	4	f	f	X
ejpam-7045	131	5	(	(	PUNCT
ejpam-7045	131	6	x	x	NOUN
ejpam-7045	131	7	)	)	PUNCT
ejpam-7045	131	8	}	}	PUNCT
ejpam-7045	131	9	for	for	ADP
ejpam-7045	131	10	each	each	DET
ejpam-7045	131	11	point	point	NOUN
ejpam-7045	131	12	y	y	PROPN
ejpam-7045	131	13	∈	∈	PROPN
ejpam-7045	131	14	y	y	PROPN
ejpam-7045	131	15	.	.	PUNCT
ejpam-7045	132	1	for	for	ADP
ejpam-7045	132	2	each	each	PRON
ejpam-7045	132	3	a	a	DET
ejpam-7045	132	4	⊆	⊆	NUM
ejpam-7045	132	5	x	x	SYM
ejpam-7045	132	6	,	,	PUNCT
ejpam-7045	132	7	f	f	PROPN
ejpam-7045	132	8	(	(	PUNCT
ejpam-7045	132	9	a	a	NOUN
ejpam-7045	132	10	)	)	PUNCT
ejpam-7045	132	11	=	=	SYM
ejpam-7045	132	12	∪x∈af	∪x∈af	NOUN
ejpam-7045	132	13	(	(	PUNCT
ejpam-7045	132	14	x	x	NOUN
ejpam-7045	132	15	)	)	PUNCT
ejpam-7045	132	16	.	.	PUNCT
ejpam-7045	133	1	c.	c.	PROPN
ejpam-7045	133	2	viriyapong	viriyapong	PROPN
ejpam-7045	133	3	,	,	PUNCT
ejpam-7045	133	4	a.	a.	PROPN
ejpam-7045	133	5	sama	sama	PROPN
ejpam-7045	133	6	-	-	PUNCT
ejpam-7045	133	7	ae	ae	PROPN
ejpam-7045	133	8	,	,	PUNCT
ejpam-7045	133	9	c.	c.	PROPN
ejpam-7045	133	10	boonpok	boonpok	PROPN
ejpam-7045	133	11	/	/	SYM
ejpam-7045	133	12	eur	eur	PROPN
ejpam-7045	133	13	.	.	PUNCT
ejpam-7045	134	1	j.	j.	PROPN
ejpam-7045	134	2	pure	pure	PROPN
ejpam-7045	134	3	appl	appl	PROPN
ejpam-7045	134	4	.	.	PROPN
ejpam-7045	134	5	math	math	PROPN
ejpam-7045	134	6	,	,	PUNCT
ejpam-7045	134	7	18	18	NUM
ejpam-7045	134	8	(	(	PUNCT
ejpam-7045	134	9	4	4	NUM
ejpam-7045	134	10	)	)	PUNCT
ejpam-7045	134	11	(	(	PUNCT
ejpam-7045	134	12	2025	2025	NUM
ejpam-7045	134	13	)	)	PUNCT
ejpam-7045	134	14	,	,	PUNCT
ejpam-7045	134	15	7045	7045	NUM
ejpam-7045	134	16	5	5	NUM
ejpam-7045	134	17	of	of	ADP
ejpam-7045	134	18	12	12	NUM
ejpam-7045	134	19	3	3	NUM
ejpam-7045	134	20	.	.	PUNCT
ejpam-7045	134	21	upper	upper	ADJ
ejpam-7045	134	22	and	and	CCONJ
ejpam-7045	134	23	lower	low	ADJ
ejpam-7045	134	24	almost	almost	ADV
ejpam-7045	134	25	τ	τ	NOUN
ejpam-7045	134	26	⋆α(σ1	⋆α(σ1	AUX
ejpam-7045	134	27	,	,	PUNCT
ejpam-7045	134	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	134	29	multifunctions	multifunction	NOUN
ejpam-7045	134	30	in	in	ADP
ejpam-7045	134	31	this	this	DET
ejpam-7045	134	32	section	section	NOUN
ejpam-7045	134	33	,	,	PUNCT
ejpam-7045	134	34	we	we	PRON
ejpam-7045	134	35	introduce	introduce	VERB
ejpam-7045	134	36	the	the	DET
ejpam-7045	134	37	notions	notion	NOUN
ejpam-7045	134	38	of	of	ADP
ejpam-7045	134	39	upper	upper	ADJ
ejpam-7045	134	40	almost	almost	ADV
ejpam-7045	134	41	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	134	42	,	,	PUNCT
ejpam-7045	134	43	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	134	44	multifunctions	multifunction	NOUN
ejpam-7045	134	45	and	and	CCONJ
ejpam-7045	134	46	lower	low	ADJ
ejpam-7045	134	47	almost	almost	ADV
ejpam-7045	134	48	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	134	49	,	,	PUNCT
ejpam-7045	134	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	134	51	multifunctions	multifunction	NOUN
ejpam-7045	134	52	.	.	PUNCT
ejpam-7045	135	1	moreover	moreover	ADV
ejpam-7045	135	2	,	,	PUNCT
ejpam-7045	135	3	several	several	ADJ
ejpam-7045	135	4	characterizations	characterization	NOUN
ejpam-7045	135	5	of	of	ADP
ejpam-7045	135	6	upper	upper	ADJ
ejpam-7045	135	7	almost	almost	ADV
ejpam-7045	135	8	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	135	9	,	,	PUNCT
ejpam-7045	135	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	135	11	multifunctions	multifunction	NOUN
ejpam-7045	135	12	and	and	CCONJ
ejpam-7045	135	13	lower	low	ADJ
ejpam-7045	135	14	almost	almost	ADV
ejpam-7045	135	15	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	135	16	,	,	PUNCT
ejpam-7045	135	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	135	18	multifunctions	multifunction	NOUN
ejpam-7045	135	19	discussed	discuss	VERB
ejpam-7045	135	20	.	.	PUNCT
ejpam-7045	136	1	definition	definition	NOUN
ejpam-7045	136	2	1	1	NUM
ejpam-7045	136	3	.	.	PUNCT
ejpam-7045	137	1	a	a	DET
ejpam-7045	137	2	multifunction	multifunction	NOUN
ejpam-7045	137	3	f	f	NOUN
ejpam-7045	137	4	:	:	PUNCT
ejpam-7045	137	5	(	(	PUNCT
ejpam-7045	137	6	x	x	X
ejpam-7045	137	7	,	,	PUNCT
ejpam-7045	137	8	τ	τ	PROPN
ejpam-7045	137	9	,	,	PUNCT
ejpam-7045	137	10	i	i	NOUN
ejpam-7045	137	11	)	)	PUNCT
ejpam-7045	137	12	→	→	PUNCT
ejpam-7045	137	13	(	(	PUNCT
ejpam-7045	137	14	y	y	PROPN
ejpam-7045	137	15	,	,	PUNCT
ejpam-7045	137	16	σ1	σ1	PROPN
ejpam-7045	137	17	,	,	PUNCT
ejpam-7045	137	18	σ2	σ2	PROPN
ejpam-7045	137	19	)	)	PUNCT
ejpam-7045	137	20	is	be	AUX
ejpam-7045	137	21	said	say	VERB
ejpam-7045	137	22	to	to	PART
ejpam-7045	137	23	be	be	AUX
ejpam-7045	137	24	upper	upper	ADJ
ejpam-7045	137	25	almost	almost	ADV
ejpam-7045	137	26	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	137	27	,	,	PUNCT
ejpam-7045	137	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	137	29	at	at	ADP
ejpam-7045	137	30	a	a	DET
ejpam-7045	137	31	point	point	NOUN
ejpam-7045	137	32	x	x	PUNCT
ejpam-7045	137	33	of	of	ADP
ejpam-7045	137	34	x	x	PRON
ejpam-7045	137	35	if	if	SCONJ
ejpam-7045	137	36	for	for	ADP
ejpam-7045	137	37	each	each	DET
ejpam-7045	137	38	σ1σ2	σ1σ2	VERB
ejpam-7045	137	39	-	-	ADJ
ejpam-7045	137	40	open	open	ADJ
ejpam-7045	137	41	set	set	NOUN
ejpam-7045	137	42	v	v	NOUN
ejpam-7045	137	43	of	of	ADP
ejpam-7045	137	44	y	y	PRON
ejpam-7045	137	45	such	such	ADJ
ejpam-7045	137	46	that	that	SCONJ
ejpam-7045	137	47	f	f	PROPN
ejpam-7045	137	48	(	(	PUNCT
ejpam-7045	137	49	x	x	X
ejpam-7045	137	50	)	)	PUNCT
ejpam-7045	137	51	⊆	⊆	NUM
ejpam-7045	137	52	v	v	NOUN
ejpam-7045	137	53	,	,	PUNCT
ejpam-7045	137	54	there	there	PRON
ejpam-7045	137	55	exists	exist	VERB
ejpam-7045	137	56	a	a	DET
ejpam-7045	137	57	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	137	58	-	-	PUNCT
ejpam-7045	137	59	open	open	ADJ
ejpam-7045	137	60	set	set	NOUN
ejpam-7045	137	61	u	u	NOUN
ejpam-7045	137	62	of	of	ADP
ejpam-7045	137	63	x	x	PUNCT
ejpam-7045	137	64	containing	contain	VERB
ejpam-7045	137	65	x	x	PUNCT
ejpam-7045	137	66	such	such	ADJ
ejpam-7045	137	67	that	that	SCONJ
ejpam-7045	137	68	f	f	PROPN
ejpam-7045	137	69	(	(	PUNCT
ejpam-7045	137	70	u	u	NOUN
ejpam-7045	137	71	)	)	PUNCT
ejpam-7045	137	72	⊆	⊆	NUM
ejpam-7045	137	73	σ1σ2	σ1σ2	X
ejpam-7045	137	74	-	-	PUNCT
ejpam-7045	137	75	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	137	76	-	-	PUNCT
ejpam-7045	137	77	cl(v	cl(v	NOUN
ejpam-7045	137	78	)	)	PUNCT
ejpam-7045	137	79	)	)	PUNCT
ejpam-7045	137	80	.	.	PUNCT
ejpam-7045	138	1	a	a	DET
ejpam-7045	138	2	multifunction	multifunction	NOUN
ejpam-7045	138	3	f	f	NOUN
ejpam-7045	138	4	:	:	PUNCT
ejpam-7045	138	5	(	(	PUNCT
ejpam-7045	138	6	x	x	X
ejpam-7045	138	7	,	,	PUNCT
ejpam-7045	138	8	τ	τ	PROPN
ejpam-7045	138	9	,	,	PUNCT
ejpam-7045	138	10	i	i	NOUN
ejpam-7045	138	11	)	)	PUNCT
ejpam-7045	138	12	→	→	PUNCT
ejpam-7045	138	13	(	(	PUNCT
ejpam-7045	138	14	y	y	PROPN
ejpam-7045	138	15	,	,	PUNCT
ejpam-7045	138	16	σ1	σ1	PROPN
ejpam-7045	138	17	,	,	PUNCT
ejpam-7045	138	18	σ2	σ2	PROPN
ejpam-7045	138	19	)	)	PUNCT
ejpam-7045	138	20	is	be	AUX
ejpam-7045	138	21	said	say	VERB
ejpam-7045	138	22	to	to	PART
ejpam-7045	138	23	be	be	AUX
ejpam-7045	138	24	upper	upper	ADJ
ejpam-7045	138	25	almost	almost	ADV
ejpam-7045	138	26	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	138	27	,	,	PUNCT
ejpam-7045	138	28	σ2)continuous	σ2)continuous	ADJ
ejpam-7045	138	29	if	if	SCONJ
ejpam-7045	138	30	f	f	PROPN
ejpam-7045	138	31	is	be	AUX
ejpam-7045	138	32	upper	upper	ADJ
ejpam-7045	138	33	almost	almost	ADV
ejpam-7045	138	34	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	138	35	,	,	PUNCT
ejpam-7045	138	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	138	37	at	at	ADP
ejpam-7045	138	38	each	each	DET
ejpam-7045	138	39	point	point	NOUN
ejpam-7045	138	40	of	of	ADP
ejpam-7045	138	41	x.	x.	NOUN
ejpam-7045	138	42	theorem	theorem	VERB
ejpam-7045	138	43	1	1	NUM
ejpam-7045	138	44	.	.	X
ejpam-7045	138	45	for	for	ADP
ejpam-7045	138	46	a	a	DET
ejpam-7045	138	47	multifunction	multifunction	NOUN
ejpam-7045	138	48	f	f	NOUN
ejpam-7045	138	49	:	:	PUNCT
ejpam-7045	138	50	(	(	PUNCT
ejpam-7045	138	51	x	x	X
ejpam-7045	138	52	,	,	PUNCT
ejpam-7045	138	53	τ	τ	PROPN
ejpam-7045	138	54	,	,	PUNCT
ejpam-7045	138	55	i	i	NOUN
ejpam-7045	138	56	)	)	PUNCT
ejpam-7045	138	57	→	→	PUNCT
ejpam-7045	138	58	(	(	PUNCT
ejpam-7045	138	59	y	y	PROPN
ejpam-7045	138	60	,	,	PUNCT
ejpam-7045	138	61	σ1	σ1	PROPN
ejpam-7045	138	62	,	,	PUNCT
ejpam-7045	138	63	σ2	σ2	NOUN
ejpam-7045	138	64	)	)	PUNCT
ejpam-7045	138	65	,	,	PUNCT
ejpam-7045	138	66	the	the	DET
ejpam-7045	138	67	following	follow	VERB
ejpam-7045	138	68	properties	property	NOUN
ejpam-7045	138	69	are	be	AUX
ejpam-7045	138	70	equivalent	equivalent	ADJ
ejpam-7045	138	71	:	:	PUNCT
ejpam-7045	138	72	(	(	PUNCT
ejpam-7045	138	73	1	1	X
ejpam-7045	138	74	)	)	PUNCT
ejpam-7045	138	75	f	f	PROPN
ejpam-7045	138	76	is	be	AUX
ejpam-7045	138	77	upper	upper	ADJ
ejpam-7045	138	78	almost	almost	ADV
ejpam-7045	138	79	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	138	80	,	,	PUNCT
ejpam-7045	138	81	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	138	82	at	at	ADP
ejpam-7045	138	83	x	x	X
ejpam-7045	138	84	∈	∈	PROPN
ejpam-7045	138	85	x	x	X
ejpam-7045	138	86	;	;	PUNCT
ejpam-7045	138	87	(	(	PUNCT
ejpam-7045	138	88	2	2	X
ejpam-7045	138	89	)	)	PUNCT
ejpam-7045	138	90	for	for	ADP
ejpam-7045	138	91	each	each	DET
ejpam-7045	138	92	σ1σ2	σ1σ2	VERB
ejpam-7045	138	93	-	-	ADJ
ejpam-7045	138	94	open	open	ADJ
ejpam-7045	138	95	set	set	NOUN
ejpam-7045	138	96	v	v	NOUN
ejpam-7045	138	97	of	of	ADP
ejpam-7045	138	98	y	y	PROPN
ejpam-7045	138	99	containing	contain	VERB
ejpam-7045	138	100	f	f	PROPN
ejpam-7045	138	101	(	(	PUNCT
ejpam-7045	138	102	x	x	NOUN
ejpam-7045	138	103	)	)	PUNCT
ejpam-7045	138	104	,	,	PUNCT
ejpam-7045	138	105	there	there	PRON
ejpam-7045	138	106	exists	exist	VERB
ejpam-7045	138	107	a	a	DET
ejpam-7045	138	108	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	138	109	-	-	PUNCT
ejpam-7045	138	110	open	open	ADJ
ejpam-7045	138	111	set	set	NOUN
ejpam-7045	138	112	u	u	NOUN
ejpam-7045	138	113	of	of	ADP
ejpam-7045	138	114	x	x	PUNCT
ejpam-7045	138	115	containing	contain	VERB
ejpam-7045	138	116	x	x	PUNCT
ejpam-7045	138	117	such	such	ADJ
ejpam-7045	138	118	that	that	SCONJ
ejpam-7045	138	119	f	f	PROPN
ejpam-7045	138	120	(	(	PUNCT
ejpam-7045	138	121	u	u	NOUN
ejpam-7045	138	122	)	)	PUNCT
ejpam-7045	138	123	⊆	⊆	NUM
ejpam-7045	138	124	(	(	PUNCT
ejpam-7045	138	125	σ1	σ1	PROPN
ejpam-7045	138	126	,	,	PUNCT
ejpam-7045	138	127	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	138	128	)	)	PUNCT
ejpam-7045	138	129	;	;	PUNCT
ejpam-7045	138	130	(	(	PUNCT
ejpam-7045	138	131	3	3	X
ejpam-7045	138	132	)	)	PUNCT
ejpam-7045	138	133	x	x	SYM
ejpam-7045	138	134	∈	∈	NOUN
ejpam-7045	138	135	αint⋆(f+((σ1	αint⋆(f+((σ1	NOUN
ejpam-7045	138	136	,	,	PUNCT
ejpam-7045	138	137	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	138	138	)	)	PUNCT
ejpam-7045	138	139	)	)	PUNCT
ejpam-7045	138	140	)	)	PUNCT
ejpam-7045	138	141	for	for	ADP
ejpam-7045	138	142	every	every	DET
ejpam-7045	138	143	σ1σ2	σ1σ2	NOUN
ejpam-7045	138	144	-	-	ADJ
ejpam-7045	138	145	open	open	ADJ
ejpam-7045	138	146	set	set	NOUN
ejpam-7045	138	147	v	v	NOUN
ejpam-7045	138	148	of	of	ADP
ejpam-7045	138	149	y	y	PROPN
ejpam-7045	138	150	containing	contain	VERB
ejpam-7045	138	151	f	f	PROPN
ejpam-7045	138	152	(	(	PUNCT
ejpam-7045	138	153	x	x	NOUN
ejpam-7045	138	154	)	)	PUNCT
ejpam-7045	138	155	;	;	PUNCT
ejpam-7045	138	156	(	(	PUNCT
ejpam-7045	138	157	4	4	X
ejpam-7045	138	158	)	)	PUNCT
ejpam-7045	138	159	x	x	SYM
ejpam-7045	138	160	∈	∈	PROPN
ejpam-7045	138	161	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	138	162	,	,	PUNCT
ejpam-7045	138	163	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	138	164	)	)	PUNCT
ejpam-7045	138	165	)	)	PUNCT
ejpam-7045	138	166	)	)	PUNCT
ejpam-7045	138	167	)	)	PUNCT
ejpam-7045	138	168	)	)	PUNCT
ejpam-7045	138	169	for	for	ADP
ejpam-7045	138	170	every	every	DET
ejpam-7045	138	171	σ1σ2	σ1σ2	NOUN
ejpam-7045	138	172	-	-	ADJ
ejpam-7045	138	173	open	open	ADJ
ejpam-7045	138	174	set	set	NOUN
ejpam-7045	138	175	v	v	NOUN
ejpam-7045	138	176	of	of	ADP
ejpam-7045	138	177	y	y	PROPN
ejpam-7045	138	178	containing	contain	VERB
ejpam-7045	138	179	f	f	PROPN
ejpam-7045	138	180	(	(	PUNCT
ejpam-7045	138	181	x	x	NOUN
ejpam-7045	138	182	)	)	PUNCT
ejpam-7045	138	183	.	.	PUNCT
ejpam-7045	139	1	proof	proof	NOUN
ejpam-7045	139	2	.	.	PUNCT
ejpam-7045	140	1	(	(	PUNCT
ejpam-7045	140	2	1	1	X
ejpam-7045	140	3	)	)	PUNCT
ejpam-7045	140	4	⇒	⇒	NOUN
ejpam-7045	140	5	(	(	PUNCT
ejpam-7045	140	6	2	2	NUM
ejpam-7045	140	7	):	):	PUNCT
ejpam-7045	140	8	let	let	VERB
ejpam-7045	140	9	v	v	PART
ejpam-7045	140	10	be	be	AUX
ejpam-7045	140	11	any	any	DET
ejpam-7045	140	12	σ1σ2	σ1σ2	NOUN
ejpam-7045	140	13	-	-	ADJ
ejpam-7045	140	14	open	open	ADJ
ejpam-7045	140	15	set	set	NOUN
ejpam-7045	140	16	of	of	ADP
ejpam-7045	140	17	y	y	PROPN
ejpam-7045	140	18	containing	contain	VERB
ejpam-7045	140	19	f	f	PROPN
ejpam-7045	140	20	(	(	PUNCT
ejpam-7045	140	21	x	x	NOUN
ejpam-7045	140	22	)	)	PUNCT
ejpam-7045	140	23	.	.	PUNCT
ejpam-7045	141	1	then	then	ADV
ejpam-7045	141	2	,	,	PUNCT
ejpam-7045	141	3	there	there	PRON
ejpam-7045	141	4	exists	exist	VERB
ejpam-7045	141	5	a	a	DET
ejpam-7045	141	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	141	7	-	-	PUNCT
ejpam-7045	141	8	open	open	ADJ
ejpam-7045	141	9	set	set	NOUN
ejpam-7045	141	10	u	u	NOUN
ejpam-7045	141	11	containing	contain	VERB
ejpam-7045	141	12	x	x	PUNCT
ejpam-7045	141	13	such	such	ADJ
ejpam-7045	141	14	that	that	SCONJ
ejpam-7045	141	15	f	f	PROPN
ejpam-7045	141	16	(	(	PUNCT
ejpam-7045	141	17	u	u	NOUN
ejpam-7045	141	18	)	)	PUNCT
ejpam-7045	141	19	⊆	⊆	NUM
ejpam-7045	141	20	σ1σ2	σ1σ2	X
ejpam-7045	141	21	-	-	PUNCT
ejpam-7045	141	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	141	23	-	-	PUNCT
ejpam-7045	141	24	cl(v	cl(v	NOUN
ejpam-7045	141	25	)	)	PUNCT
ejpam-7045	141	26	)	)	PUNCT
ejpam-7045	141	27	and	and	CCONJ
ejpam-7045	141	28	by	by	ADP
ejpam-7045	141	29	lemma	lemma	PROPN
ejpam-7045	141	30	3	3	NUM
ejpam-7045	141	31	,	,	PUNCT
ejpam-7045	141	32	we	we	PRON
ejpam-7045	141	33	have	have	VERB
ejpam-7045	141	34	f	f	PROPN
ejpam-7045	141	35	(	(	PUNCT
ejpam-7045	141	36	u	u	NOUN
ejpam-7045	141	37	)	)	PUNCT
ejpam-7045	141	38	⊆	⊆	NUM
ejpam-7045	141	39	(	(	PUNCT
ejpam-7045	141	40	σ1	σ1	PROPN
ejpam-7045	141	41	,	,	PUNCT
ejpam-7045	141	42	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	141	43	)	)	PUNCT
ejpam-7045	141	44	.	.	PUNCT
ejpam-7045	142	1	(	(	PUNCT
ejpam-7045	142	2	2	2	X
ejpam-7045	142	3	)	)	PUNCT
ejpam-7045	142	4	⇒	⇒	NOUN
ejpam-7045	142	5	(	(	PUNCT
ejpam-7045	142	6	3	3	NUM
ejpam-7045	142	7	):	):	PUNCT
ejpam-7045	142	8	let	let	VERB
ejpam-7045	142	9	v	v	PART
ejpam-7045	142	10	be	be	AUX
ejpam-7045	142	11	any	any	DET
ejpam-7045	142	12	σ1σ2	σ1σ2	NOUN
ejpam-7045	142	13	-	-	ADJ
ejpam-7045	142	14	open	open	ADJ
ejpam-7045	142	15	set	set	NOUN
ejpam-7045	142	16	of	of	ADP
ejpam-7045	142	17	y	y	PROPN
ejpam-7045	142	18	containing	contain	VERB
ejpam-7045	142	19	f	f	PROPN
ejpam-7045	142	20	(	(	PUNCT
ejpam-7045	142	21	x	x	NOUN
ejpam-7045	142	22	)	)	PUNCT
ejpam-7045	142	23	.	.	PUNCT
ejpam-7045	143	1	by	by	ADP
ejpam-7045	143	2	(	(	PUNCT
ejpam-7045	143	3	2	2	NUM
ejpam-7045	143	4	)	)	PUNCT
ejpam-7045	143	5	,	,	PUNCT
ejpam-7045	143	6	there	there	PRON
ejpam-7045	143	7	exists	exist	VERB
ejpam-7045	143	8	a	a	DET
ejpam-7045	143	9	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	143	10	-	-	PUNCT
ejpam-7045	143	11	open	open	ADJ
ejpam-7045	143	12	set	set	NOUN
ejpam-7045	143	13	u	u	NOUN
ejpam-7045	143	14	containing	contain	VERB
ejpam-7045	143	15	x	x	PUNCT
ejpam-7045	143	16	such	such	ADJ
ejpam-7045	143	17	that	that	SCONJ
ejpam-7045	143	18	f	f	PROPN
ejpam-7045	143	19	(	(	PUNCT
ejpam-7045	143	20	u	u	NOUN
ejpam-7045	143	21	)	)	PUNCT
ejpam-7045	143	22	⊆	⊆	NUM
ejpam-7045	143	23	(	(	PUNCT
ejpam-7045	143	24	σ1	σ1	PROPN
ejpam-7045	143	25	,	,	PUNCT
ejpam-7045	143	26	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	143	27	)	)	PUNCT
ejpam-7045	143	28	and	and	CCONJ
ejpam-7045	143	29	hence	hence	ADV
ejpam-7045	143	30	u	u	X
ejpam-7045	143	31	⊆	⊆	NUM
ejpam-7045	143	32	f+((σ1	f+((σ1	NOUN
ejpam-7045	143	33	,	,	PUNCT
ejpam-7045	143	34	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	143	35	)	)	PUNCT
ejpam-7045	143	36	)	)	PUNCT
ejpam-7045	143	37	.	.	PUNCT
ejpam-7045	144	1	thus	thus	ADV
ejpam-7045	144	2	,	,	PUNCT
ejpam-7045	144	3	x	x	SYM
ejpam-7045	144	4	∈	∈	NOUN
ejpam-7045	144	5	αint⋆(f+((σ1	αint⋆(f+((σ1	NOUN
ejpam-7045	144	6	,	,	PUNCT
ejpam-7045	144	7	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	144	8	)	)	PUNCT
ejpam-7045	144	9	)	)	PUNCT
ejpam-7045	144	10	)	)	PUNCT
ejpam-7045	144	11	.	.	PUNCT
ejpam-7045	145	1	(	(	PUNCT
ejpam-7045	145	2	3	3	X
ejpam-7045	145	3	)	)	PUNCT
ejpam-7045	145	4	⇒	⇒	NOUN
ejpam-7045	145	5	(	(	PUNCT
ejpam-7045	145	6	4	4	NUM
ejpam-7045	145	7	):	):	PUNCT
ejpam-7045	145	8	let	let	VERB
ejpam-7045	145	9	v	v	PART
ejpam-7045	145	10	be	be	AUX
ejpam-7045	145	11	any	any	DET
ejpam-7045	145	12	σ1σ2	σ1σ2	NOUN
ejpam-7045	145	13	-	-	ADJ
ejpam-7045	145	14	open	open	ADJ
ejpam-7045	145	15	set	set	NOUN
ejpam-7045	145	16	of	of	ADP
ejpam-7045	145	17	y	y	PROPN
ejpam-7045	145	18	containing	contain	VERB
ejpam-7045	145	19	f	f	PROPN
ejpam-7045	145	20	(	(	PUNCT
ejpam-7045	145	21	x	x	NOUN
ejpam-7045	145	22	)	)	PUNCT
ejpam-7045	145	23	.	.	PUNCT
ejpam-7045	146	1	then	then	ADV
ejpam-7045	146	2	by	by	ADP
ejpam-7045	146	3	(	(	PUNCT
ejpam-7045	146	4	3	3	NUM
ejpam-7045	146	5	)	)	PUNCT
ejpam-7045	146	6	,	,	PUNCT
ejpam-7045	146	7	we	we	PRON
ejpam-7045	146	8	have	have	VERB
ejpam-7045	146	9	x	x	SYM
ejpam-7045	146	10	∈	∈	NOUN
ejpam-7045	146	11	αint⋆(f+((σ1	αint⋆(f+((σ1	NOUN
ejpam-7045	146	12	,	,	PUNCT
ejpam-7045	146	13	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	146	14	)	)	PUNCT
ejpam-7045	146	15	)	)	PUNCT
ejpam-7045	146	16	)	)	PUNCT
ejpam-7045	146	17	and	and	CCONJ
ejpam-7045	146	18	by	by	ADP
ejpam-7045	146	19	lemma	lemma	PROPN
ejpam-7045	146	20	7	7	NUM
ejpam-7045	146	21	,	,	PUNCT
ejpam-7045	146	22	x	x	SYM
ejpam-7045	146	23	∈	∈	NOUN
ejpam-7045	146	24	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	146	25	,	,	PUNCT
ejpam-7045	146	26	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	146	27	)	)	PUNCT
ejpam-7045	146	28	)	)	PUNCT
ejpam-7045	146	29	)	)	PUNCT
ejpam-7045	146	30	)	)	PUNCT
ejpam-7045	146	31	)	)	PUNCT
ejpam-7045	146	32	.	.	PUNCT
ejpam-7045	147	1	(	(	PUNCT
ejpam-7045	147	2	4	4	X
ejpam-7045	147	3	)	)	PUNCT
ejpam-7045	147	4	⇒	⇒	NOUN
ejpam-7045	147	5	(	(	PUNCT
ejpam-7045	147	6	1	1	NUM
ejpam-7045	147	7	):	):	PUNCT
ejpam-7045	147	8	let	let	VERB
ejpam-7045	147	9	v	v	PART
ejpam-7045	147	10	be	be	AUX
ejpam-7045	147	11	any	any	DET
ejpam-7045	147	12	σ1σ2	σ1σ2	NOUN
ejpam-7045	147	13	-	-	ADJ
ejpam-7045	147	14	open	open	ADJ
ejpam-7045	147	15	set	set	NOUN
ejpam-7045	147	16	of	of	ADP
ejpam-7045	147	17	y	y	PROPN
ejpam-7045	147	18	containing	contain	VERB
ejpam-7045	147	19	f	f	PROPN
ejpam-7045	147	20	(	(	PUNCT
ejpam-7045	147	21	x	x	NOUN
ejpam-7045	147	22	)	)	PUNCT
ejpam-7045	147	23	.	.	PUNCT
ejpam-7045	148	1	by	by	ADP
ejpam-7045	148	2	(	(	PUNCT
ejpam-7045	148	3	4	4	NUM
ejpam-7045	148	4	)	)	PUNCT
ejpam-7045	148	5	,	,	PUNCT
ejpam-7045	148	6	we	we	PRON
ejpam-7045	148	7	have	have	VERB
ejpam-7045	148	8	x	x	X
ejpam-7045	148	9	∈	∈	PROPN
ejpam-7045	148	10	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	148	11	,	,	PUNCT
ejpam-7045	148	12	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	148	13	)	)	PUNCT
ejpam-7045	148	14	)	)	PUNCT
ejpam-7045	148	15	)	)	PUNCT
ejpam-7045	148	16	)	)	PUNCT
ejpam-7045	148	17	)	)	PUNCT
ejpam-7045	148	18	and	and	CCONJ
ejpam-7045	148	19	by	by	ADP
ejpam-7045	148	20	lemma	lemma	PROPN
ejpam-7045	148	21	7	7	NUM
ejpam-7045	148	22	,	,	PUNCT
ejpam-7045	148	23	x	x	SYM
ejpam-7045	148	24	∈	∈	NOUN
ejpam-7045	148	25	αint⋆(f+((σ1	αint⋆(f+((σ1	NOUN
ejpam-7045	148	26	,	,	PUNCT
ejpam-7045	148	27	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	148	28	)	)	PUNCT
ejpam-7045	148	29	)	)	PUNCT
ejpam-7045	148	30	)	)	PUNCT
ejpam-7045	148	31	.	.	PUNCT
ejpam-7045	149	1	therefore	therefore	ADV
ejpam-7045	149	2	,	,	PUNCT
ejpam-7045	149	3	there	there	PRON
ejpam-7045	149	4	exists	exist	VERB
ejpam-7045	149	5	a	a	DET
ejpam-7045	149	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	149	7	-	-	PUNCT
ejpam-7045	149	8	open	open	ADJ
ejpam-7045	149	9	set	set	NOUN
ejpam-7045	149	10	u	u	NOUN
ejpam-7045	149	11	of	of	ADP
ejpam-7045	149	12	x	x	PUNCT
ejpam-7045	149	13	containing	contain	VERB
ejpam-7045	149	14	x	x	PUNCT
ejpam-7045	149	15	such	such	ADJ
ejpam-7045	149	16	that	that	SCONJ
ejpam-7045	149	17	u	u	NOUN
ejpam-7045	149	18	⊆	⊆	NUM
ejpam-7045	149	19	f+((σ1	f+((σ1	NOUN
ejpam-7045	149	20	,	,	PUNCT
ejpam-7045	149	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	149	22	)	)	PUNCT
ejpam-7045	149	23	)	)	PUNCT
ejpam-7045	149	24	;	;	PUNCT
ejpam-7045	149	25	hence	hence	ADV
ejpam-7045	149	26	f	f	PROPN
ejpam-7045	149	27	(	(	PUNCT
ejpam-7045	149	28	u	u	NOUN
ejpam-7045	149	29	)	)	PUNCT
ejpam-7045	149	30	⊆	⊆	NUM
ejpam-7045	149	31	(	(	PUNCT
ejpam-7045	149	32	σ1	σ1	PROPN
ejpam-7045	149	33	,	,	PUNCT
ejpam-7045	149	34	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	149	35	)	)	PUNCT
ejpam-7045	149	36	.	.	PUNCT
ejpam-7045	150	1	since	since	SCONJ
ejpam-7045	150	2	v	v	NOUN
ejpam-7045	150	3	is	be	AUX
ejpam-7045	150	4	σ1σ2	σ1σ2	NOUN
ejpam-7045	150	5	-	-	ADJ
ejpam-7045	150	6	open	open	ADJ
ejpam-7045	150	7	,	,	PUNCT
ejpam-7045	150	8	by	by	ADP
ejpam-7045	150	9	lemma	lemma	PROPN
ejpam-7045	150	10	3	3	NUM
ejpam-7045	150	11	we	we	PRON
ejpam-7045	150	12	have	have	VERB
ejpam-7045	150	13	f	f	PROPN
ejpam-7045	150	14	(	(	PUNCT
ejpam-7045	150	15	u	u	NOUN
ejpam-7045	150	16	)	)	PUNCT
ejpam-7045	150	17	⊆	⊆	NUM
ejpam-7045	150	18	σ1σ2	σ1σ2	X
ejpam-7045	150	19	-	-	PUNCT
ejpam-7045	150	20	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	150	21	-	-	PUNCT
ejpam-7045	150	22	cl(v	cl(v	NOUN
ejpam-7045	150	23	)	)	PUNCT
ejpam-7045	150	24	)	)	PUNCT
ejpam-7045	150	25	.	.	PUNCT
ejpam-7045	151	1	this	this	PRON
ejpam-7045	151	2	shows	show	VERB
ejpam-7045	151	3	that	that	SCONJ
ejpam-7045	151	4	f	f	PROPN
ejpam-7045	151	5	is	be	AUX
ejpam-7045	151	6	upper	upper	ADJ
ejpam-7045	151	7	almost	almost	ADV
ejpam-7045	151	8	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	151	9	,	,	PUNCT
ejpam-7045	151	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	151	11	at	at	ADP
ejpam-7045	151	12	x.	x.	PROPN
ejpam-7045	151	13	c.	c.	PROPN
ejpam-7045	151	14	viriyapong	viriyapong	PROPN
ejpam-7045	151	15	,	,	PUNCT
ejpam-7045	151	16	a.	a.	PROPN
ejpam-7045	151	17	sama	sama	PROPN
ejpam-7045	151	18	-	-	PUNCT
ejpam-7045	151	19	ae	ae	PROPN
ejpam-7045	151	20	,	,	PUNCT
ejpam-7045	151	21	c.	c.	PROPN
ejpam-7045	151	22	boonpok	boonpok	PROPN
ejpam-7045	151	23	/	/	SYM
ejpam-7045	151	24	eur	eur	PROPN
ejpam-7045	151	25	.	.	PUNCT
ejpam-7045	152	1	j.	j.	PROPN
ejpam-7045	152	2	pure	pure	PROPN
ejpam-7045	152	3	appl	appl	PROPN
ejpam-7045	152	4	.	.	PROPN
ejpam-7045	152	5	math	math	PROPN
ejpam-7045	152	6	,	,	PUNCT
ejpam-7045	152	7	18	18	NUM
ejpam-7045	152	8	(	(	PUNCT
ejpam-7045	152	9	4	4	NUM
ejpam-7045	152	10	)	)	PUNCT
ejpam-7045	152	11	(	(	PUNCT
ejpam-7045	152	12	2025	2025	NUM
ejpam-7045	152	13	)	)	PUNCT
ejpam-7045	152	14	,	,	PUNCT
ejpam-7045	152	15	7045	7045	NUM
ejpam-7045	152	16	6	6	NUM
ejpam-7045	152	17	of	of	ADP
ejpam-7045	152	18	12	12	NUM
ejpam-7045	152	19	definition	definition	NOUN
ejpam-7045	152	20	2	2	NUM
ejpam-7045	152	21	.	.	PUNCT
ejpam-7045	152	22	a	a	DET
ejpam-7045	152	23	multifunction	multifunction	NOUN
ejpam-7045	152	24	f	f	NOUN
ejpam-7045	152	25	:	:	PUNCT
ejpam-7045	152	26	(	(	PUNCT
ejpam-7045	152	27	x	x	X
ejpam-7045	152	28	,	,	PUNCT
ejpam-7045	152	29	τ	τ	PROPN
ejpam-7045	152	30	,	,	PUNCT
ejpam-7045	152	31	i	i	NOUN
ejpam-7045	152	32	)	)	PUNCT
ejpam-7045	152	33	→	→	PUNCT
ejpam-7045	152	34	(	(	PUNCT
ejpam-7045	152	35	y	y	PROPN
ejpam-7045	152	36	,	,	PUNCT
ejpam-7045	152	37	σ1	σ1	PROPN
ejpam-7045	152	38	,	,	PUNCT
ejpam-7045	152	39	σ2	σ2	PROPN
ejpam-7045	152	40	)	)	PUNCT
ejpam-7045	152	41	is	be	AUX
ejpam-7045	152	42	said	say	VERB
ejpam-7045	152	43	to	to	PART
ejpam-7045	152	44	be	be	AUX
ejpam-7045	152	45	lower	low	ADJ
ejpam-7045	152	46	almost	almost	ADV
ejpam-7045	152	47	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	152	48	,	,	PUNCT
ejpam-7045	152	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	152	50	at	at	ADP
ejpam-7045	152	51	a	a	DET
ejpam-7045	152	52	point	point	NOUN
ejpam-7045	152	53	x	x	SYM
ejpam-7045	152	54	∈	∈	NOUN
ejpam-7045	152	55	x	x	PUNCT
ejpam-7045	152	56	if	if	SCONJ
ejpam-7045	152	57	for	for	ADP
ejpam-7045	152	58	each	each	DET
ejpam-7045	152	59	σ1σ2	σ1σ2	VERB
ejpam-7045	152	60	-	-	ADJ
ejpam-7045	152	61	open	open	ADJ
ejpam-7045	152	62	set	set	NOUN
ejpam-7045	152	63	v	v	NOUN
ejpam-7045	152	64	of	of	ADP
ejpam-7045	152	65	y	y	PRON
ejpam-7045	152	66	such	such	ADJ
ejpam-7045	152	67	that	that	SCONJ
ejpam-7045	152	68	f	f	PROPN
ejpam-7045	152	69	(	(	PUNCT
ejpam-7045	152	70	x	x	NOUN
ejpam-7045	152	71	)	)	PUNCT
ejpam-7045	152	72	∩	∩	NOUN
ejpam-7045	152	73	v	v	ADP
ejpam-7045	152	74	̸=	̸=	PROPN
ejpam-7045	152	75	∅	∅	NOUN
ejpam-7045	152	76	,	,	PUNCT
ejpam-7045	152	77	there	there	PRON
ejpam-7045	152	78	exists	exist	VERB
ejpam-7045	152	79	a	a	DET
ejpam-7045	152	80	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	152	81	-	-	PUNCT
ejpam-7045	152	82	open	open	ADJ
ejpam-7045	152	83	set	set	NOUN
ejpam-7045	152	84	u	u	NOUN
ejpam-7045	152	85	of	of	ADP
ejpam-7045	152	86	x	x	PUNCT
ejpam-7045	152	87	containing	contain	VERB
ejpam-7045	152	88	x	x	PUNCT
ejpam-7045	152	89	such	such	ADJ
ejpam-7045	152	90	that	that	SCONJ
ejpam-7045	152	91	f	f	PROPN
ejpam-7045	152	92	(	(	PUNCT
ejpam-7045	152	93	z	z	NOUN
ejpam-7045	152	94	)	)	PUNCT
ejpam-7045	152	95	∩	∩	NOUN
ejpam-7045	152	96	σ1σ2	σ1σ2	X
ejpam-7045	152	97	-	-	PUNCT
ejpam-7045	152	98	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	152	99	-	-	PUNCT
ejpam-7045	152	100	cl(v	cl(v	NOUN
ejpam-7045	152	101	)	)	PUNCT
ejpam-7045	152	102	)	)	PUNCT
ejpam-7045	153	1	̸=	̸=	NOUN
ejpam-7045	153	2	∅	∅	NOUN
ejpam-7045	153	3	for	for	ADP
ejpam-7045	153	4	every	every	DET
ejpam-7045	153	5	z	z	NOUN
ejpam-7045	153	6	∈	∈	PROPN
ejpam-7045	153	7	u	u	NOUN
ejpam-7045	153	8	.	.	PUNCT
ejpam-7045	154	1	a	a	DET
ejpam-7045	154	2	multifunction	multifunction	NOUN
ejpam-7045	154	3	f	f	NOUN
ejpam-7045	154	4	:	:	PUNCT
ejpam-7045	154	5	(	(	PUNCT
ejpam-7045	154	6	x	x	X
ejpam-7045	154	7	,	,	PUNCT
ejpam-7045	154	8	τ	τ	PROPN
ejpam-7045	154	9	,	,	PUNCT
ejpam-7045	154	10	i	i	NOUN
ejpam-7045	154	11	)	)	PUNCT
ejpam-7045	154	12	→	→	PUNCT
ejpam-7045	154	13	(	(	PUNCT
ejpam-7045	154	14	y	y	PROPN
ejpam-7045	154	15	,	,	PUNCT
ejpam-7045	154	16	σ1	σ1	PROPN
ejpam-7045	154	17	,	,	PUNCT
ejpam-7045	154	18	σ2	σ2	PROPN
ejpam-7045	154	19	)	)	PUNCT
ejpam-7045	154	20	is	be	AUX
ejpam-7045	154	21	said	say	VERB
ejpam-7045	154	22	to	to	PART
ejpam-7045	154	23	be	be	AUX
ejpam-7045	154	24	lower	low	ADJ
ejpam-7045	154	25	almost	almost	ADV
ejpam-7045	154	26	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	154	27	,	,	PUNCT
ejpam-7045	154	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	154	29	if	if	SCONJ
ejpam-7045	154	30	f	f	PROPN
ejpam-7045	154	31	is	be	AUX
ejpam-7045	154	32	lower	low	ADJ
ejpam-7045	154	33	almost	almost	ADV
ejpam-7045	154	34	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	154	35	,	,	PUNCT
ejpam-7045	154	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	154	37	at	at	ADP
ejpam-7045	154	38	each	each	DET
ejpam-7045	154	39	point	point	NOUN
ejpam-7045	154	40	of	of	ADP
ejpam-7045	154	41	x.	x.	NOUN
ejpam-7045	154	42	theorem	theorem	VERB
ejpam-7045	154	43	2	2	NUM
ejpam-7045	154	44	.	.	X
ejpam-7045	154	45	for	for	ADP
ejpam-7045	154	46	a	a	DET
ejpam-7045	154	47	multifunction	multifunction	NOUN
ejpam-7045	154	48	f	f	NOUN
ejpam-7045	154	49	:	:	PUNCT
ejpam-7045	154	50	(	(	PUNCT
ejpam-7045	154	51	x	x	X
ejpam-7045	154	52	,	,	PUNCT
ejpam-7045	154	53	τ	τ	PROPN
ejpam-7045	154	54	,	,	PUNCT
ejpam-7045	154	55	i	i	NOUN
ejpam-7045	154	56	)	)	PUNCT
ejpam-7045	154	57	→	→	PUNCT
ejpam-7045	154	58	(	(	PUNCT
ejpam-7045	154	59	y	y	PROPN
ejpam-7045	154	60	,	,	PUNCT
ejpam-7045	154	61	σ1	σ1	PROPN
ejpam-7045	154	62	,	,	PUNCT
ejpam-7045	154	63	σ2	σ2	NOUN
ejpam-7045	154	64	)	)	PUNCT
ejpam-7045	154	65	,	,	PUNCT
ejpam-7045	154	66	the	the	DET
ejpam-7045	154	67	following	follow	VERB
ejpam-7045	154	68	properties	property	NOUN
ejpam-7045	154	69	are	be	AUX
ejpam-7045	154	70	equivalent	equivalent	ADJ
ejpam-7045	154	71	:	:	PUNCT
ejpam-7045	154	72	(	(	PUNCT
ejpam-7045	154	73	1	1	X
ejpam-7045	154	74	)	)	PUNCT
ejpam-7045	154	75	f	f	PROPN
ejpam-7045	154	76	is	be	AUX
ejpam-7045	154	77	lower	low	ADJ
ejpam-7045	154	78	almost	almost	ADV
ejpam-7045	154	79	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	154	80	,	,	PUNCT
ejpam-7045	154	81	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	154	82	at	at	ADP
ejpam-7045	154	83	x	x	X
ejpam-7045	154	84	∈	∈	PROPN
ejpam-7045	154	85	x	x	X
ejpam-7045	154	86	;	;	PUNCT
ejpam-7045	154	87	(	(	PUNCT
ejpam-7045	154	88	2	2	X
ejpam-7045	154	89	)	)	PUNCT
ejpam-7045	154	90	for	for	ADP
ejpam-7045	154	91	each	each	DET
ejpam-7045	154	92	σ1σ2	σ1σ2	VERB
ejpam-7045	154	93	-	-	ADJ
ejpam-7045	154	94	open	open	ADJ
ejpam-7045	154	95	set	set	NOUN
ejpam-7045	154	96	v	v	NOUN
ejpam-7045	154	97	of	of	ADP
ejpam-7045	154	98	y	y	PRON
ejpam-7045	154	99	such	such	ADJ
ejpam-7045	154	100	that	that	SCONJ
ejpam-7045	154	101	f	f	PROPN
ejpam-7045	154	102	(	(	PUNCT
ejpam-7045	154	103	x	x	NOUN
ejpam-7045	154	104	)	)	PUNCT
ejpam-7045	154	105	∩	∩	NOUN
ejpam-7045	154	106	v	v	ADP
ejpam-7045	154	107	̸=	̸=	PROPN
ejpam-7045	154	108	∅	∅	NOUN
ejpam-7045	154	109	,	,	PUNCT
ejpam-7045	154	110	there	there	PRON
ejpam-7045	154	111	exists	exist	VERB
ejpam-7045	154	112	a	a	DET
ejpam-7045	154	113	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	154	114	-	-	PUNCT
ejpam-7045	154	115	open	open	ADJ
ejpam-7045	154	116	set	set	NOUN
ejpam-7045	154	117	u	u	NOUN
ejpam-7045	154	118	of	of	ADP
ejpam-7045	154	119	x	x	PUNCT
ejpam-7045	154	120	containing	contain	VERB
ejpam-7045	154	121	x	x	PUNCT
ejpam-7045	154	122	such	such	ADJ
ejpam-7045	154	123	that	that	SCONJ
ejpam-7045	154	124	f	f	PROPN
ejpam-7045	154	125	(	(	PUNCT
ejpam-7045	154	126	z	z	NOUN
ejpam-7045	154	127	)	)	PUNCT
ejpam-7045	154	128	∩	∩	NOUN
ejpam-7045	154	129	(	(	PUNCT
ejpam-7045	154	130	σ1	σ1	PROPN
ejpam-7045	154	131	,	,	PUNCT
ejpam-7045	154	132	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	154	133	)	)	PUNCT
ejpam-7045	154	134	̸=	̸=	PROPN
ejpam-7045	154	135	∅	∅	NOUN
ejpam-7045	154	136	;	;	PUNCT
ejpam-7045	154	137	(	(	PUNCT
ejpam-7045	154	138	3	3	X
ejpam-7045	154	139	)	)	PUNCT
ejpam-7045	154	140	x	x	SYM
ejpam-7045	154	141	∈	∈	NOUN
ejpam-7045	154	142	αint⋆(f−((σ1	αint⋆(f−((σ1	NOUN
ejpam-7045	154	143	,	,	PUNCT
ejpam-7045	154	144	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	154	145	)	)	PUNCT
ejpam-7045	154	146	)	)	PUNCT
ejpam-7045	154	147	)	)	PUNCT
ejpam-7045	154	148	for	for	ADP
ejpam-7045	154	149	every	every	DET
ejpam-7045	154	150	σ1σ2	σ1σ2	NOUN
ejpam-7045	154	151	-	-	ADJ
ejpam-7045	154	152	open	open	ADJ
ejpam-7045	154	153	set	set	NOUN
ejpam-7045	154	154	v	v	NOUN
ejpam-7045	154	155	of	of	ADP
ejpam-7045	154	156	y	y	PRON
ejpam-7045	154	157	such	such	ADJ
ejpam-7045	154	158	that	that	SCONJ
ejpam-7045	154	159	f	f	PROPN
ejpam-7045	154	160	(	(	PUNCT
ejpam-7045	154	161	x)∩v	x)∩v	PROPN
ejpam-7045	154	162	̸=	̸=	PROPN
ejpam-7045	154	163	∅	∅	NOUN
ejpam-7045	154	164	;	;	PUNCT
ejpam-7045	154	165	(	(	PUNCT
ejpam-7045	154	166	4	4	X
ejpam-7045	154	167	)	)	PUNCT
ejpam-7045	154	168	x	x	SYM
ejpam-7045	154	169	∈	∈	PROPN
ejpam-7045	154	170	int⋆(cl⋆(int⋆(f−((σ1	int⋆(cl⋆(int⋆(f−((σ1	NOUN
ejpam-7045	154	171	,	,	PUNCT
ejpam-7045	154	172	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	154	173	)	)	PUNCT
ejpam-7045	154	174	)	)	PUNCT
ejpam-7045	154	175	)	)	PUNCT
ejpam-7045	154	176	)	)	PUNCT
ejpam-7045	154	177	)	)	PUNCT
ejpam-7045	154	178	for	for	ADP
ejpam-7045	154	179	every	every	DET
ejpam-7045	154	180	σ1σ2	σ1σ2	NOUN
ejpam-7045	154	181	-	-	ADJ
ejpam-7045	154	182	open	open	ADJ
ejpam-7045	154	183	set	set	NOUN
ejpam-7045	154	184	v	v	NOUN
ejpam-7045	154	185	of	of	ADP
ejpam-7045	154	186	y	y	PRON
ejpam-7045	154	187	such	such	ADJ
ejpam-7045	154	188	that	that	SCONJ
ejpam-7045	154	189	f	f	PROPN
ejpam-7045	154	190	(	(	PUNCT
ejpam-7045	154	191	x	x	NOUN
ejpam-7045	154	192	)	)	PUNCT
ejpam-7045	154	193	∩	∩	NOUN
ejpam-7045	154	194	v	v	ADP
ejpam-7045	154	195	̸=	̸=	PROPN
ejpam-7045	154	196	∅.	∅.	ADP
ejpam-7045	154	197	proof	proof	NOUN
ejpam-7045	154	198	.	.	PUNCT
ejpam-7045	155	1	the	the	DET
ejpam-7045	155	2	proof	proof	NOUN
ejpam-7045	155	3	is	be	AUX
ejpam-7045	155	4	similar	similar	ADJ
ejpam-7045	155	5	to	to	ADP
ejpam-7045	155	6	that	that	PRON
ejpam-7045	155	7	of	of	ADP
ejpam-7045	155	8	theorem	theorem	ADJ
ejpam-7045	155	9	1	1	NUM
ejpam-7045	155	10	.	.	PUNCT
ejpam-7045	155	11	theorem	theorem	NOUN
ejpam-7045	155	12	3	3	NUM
ejpam-7045	155	13	.	.	X
ejpam-7045	155	14	for	for	ADP
ejpam-7045	155	15	a	a	DET
ejpam-7045	155	16	multifunction	multifunction	NOUN
ejpam-7045	155	17	f	f	NOUN
ejpam-7045	155	18	:	:	PUNCT
ejpam-7045	155	19	(	(	PUNCT
ejpam-7045	155	20	x	x	X
ejpam-7045	155	21	,	,	PUNCT
ejpam-7045	155	22	τ	τ	PROPN
ejpam-7045	155	23	,	,	PUNCT
ejpam-7045	155	24	i	i	NOUN
ejpam-7045	155	25	)	)	PUNCT
ejpam-7045	155	26	→	→	PUNCT
ejpam-7045	155	27	(	(	PUNCT
ejpam-7045	155	28	y	y	PROPN
ejpam-7045	155	29	,	,	PUNCT
ejpam-7045	155	30	σ1	σ1	PROPN
ejpam-7045	155	31	,	,	PUNCT
ejpam-7045	155	32	σ2	σ2	NOUN
ejpam-7045	155	33	)	)	PUNCT
ejpam-7045	155	34	,	,	PUNCT
ejpam-7045	155	35	the	the	DET
ejpam-7045	155	36	following	follow	VERB
ejpam-7045	155	37	properties	property	NOUN
ejpam-7045	155	38	are	be	AUX
ejpam-7045	155	39	equivalent	equivalent	ADJ
ejpam-7045	155	40	:	:	PUNCT
ejpam-7045	155	41	(	(	PUNCT
ejpam-7045	155	42	1	1	X
ejpam-7045	155	43	)	)	PUNCT
ejpam-7045	155	44	f	f	PROPN
ejpam-7045	155	45	is	be	AUX
ejpam-7045	155	46	upper	upper	ADJ
ejpam-7045	155	47	almost	almost	ADV
ejpam-7045	155	48	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	155	49	,	,	PUNCT
ejpam-7045	155	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	155	51	;	;	PUNCT
ejpam-7045	155	52	(	(	PUNCT
ejpam-7045	155	53	2	2	X
ejpam-7045	155	54	)	)	PUNCT
ejpam-7045	155	55	for	for	ADP
ejpam-7045	155	56	each	each	DET
ejpam-7045	155	57	x	x	SYM
ejpam-7045	155	58	∈	∈	PROPN
ejpam-7045	155	59	x	x	X
ejpam-7045	155	60	and	and	CCONJ
ejpam-7045	155	61	each	each	DET
ejpam-7045	155	62	σ1σ2	σ1σ2	VERB
ejpam-7045	155	63	-	-	ADJ
ejpam-7045	155	64	open	open	ADJ
ejpam-7045	155	65	set	set	NOUN
ejpam-7045	155	66	v	v	NOUN
ejpam-7045	155	67	of	of	ADP
ejpam-7045	155	68	y	y	PROPN
ejpam-7045	155	69	containing	contain	VERB
ejpam-7045	155	70	f	f	PROPN
ejpam-7045	155	71	(	(	PUNCT
ejpam-7045	155	72	x	x	NOUN
ejpam-7045	155	73	)	)	PUNCT
ejpam-7045	155	74	,	,	PUNCT
ejpam-7045	155	75	there	there	PRON
ejpam-7045	155	76	exists	exist	VERB
ejpam-7045	155	77	a	a	DET
ejpam-7045	155	78	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	155	79	-	-	PUNCT
ejpam-7045	155	80	open	open	ADJ
ejpam-7045	155	81	set	set	NOUN
ejpam-7045	155	82	u	u	NOUN
ejpam-7045	155	83	of	of	ADP
ejpam-7045	155	84	x	x	PUNCT
ejpam-7045	155	85	containing	contain	VERB
ejpam-7045	155	86	x	x	PUNCT
ejpam-7045	155	87	such	such	ADJ
ejpam-7045	155	88	that	that	SCONJ
ejpam-7045	155	89	f	f	PROPN
ejpam-7045	155	90	(	(	PUNCT
ejpam-7045	155	91	u	u	NOUN
ejpam-7045	155	92	)	)	PUNCT
ejpam-7045	155	93	⊆	⊆	NUM
ejpam-7045	155	94	(	(	PUNCT
ejpam-7045	155	95	σ1	σ1	PROPN
ejpam-7045	155	96	,	,	PUNCT
ejpam-7045	155	97	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	155	98	)	)	PUNCT
ejpam-7045	155	99	;	;	PUNCT
ejpam-7045	155	100	(	(	PUNCT
ejpam-7045	155	101	3	3	X
ejpam-7045	155	102	)	)	PUNCT
ejpam-7045	155	103	for	for	ADP
ejpam-7045	155	104	each	each	DET
ejpam-7045	155	105	x	x	SYM
ejpam-7045	155	106	∈	∈	PROPN
ejpam-7045	155	107	x	x	X
ejpam-7045	155	108	and	and	CCONJ
ejpam-7045	155	109	each	each	DET
ejpam-7045	155	110	(	(	PUNCT
ejpam-7045	155	111	σ1	σ1	PROPN
ejpam-7045	155	112	,	,	PUNCT
ejpam-7045	155	113	σ2)r	σ2)r	NOUN
ejpam-7045	155	114	-	-	PUNCT
ejpam-7045	155	115	open	open	ADJ
ejpam-7045	155	116	set	set	VERB
ejpam-7045	155	117	v	v	NOUN
ejpam-7045	155	118	of	of	ADP
ejpam-7045	155	119	y	y	PROPN
ejpam-7045	155	120	containing	contain	VERB
ejpam-7045	155	121	f	f	PROPN
ejpam-7045	155	122	(	(	PUNCT
ejpam-7045	155	123	x	x	NOUN
ejpam-7045	155	124	)	)	PUNCT
ejpam-7045	155	125	,	,	PUNCT
ejpam-7045	155	126	there	there	PRON
ejpam-7045	155	127	exists	exist	VERB
ejpam-7045	155	128	a	a	DET
ejpam-7045	155	129	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	155	130	-	-	PUNCT
ejpam-7045	155	131	open	open	ADJ
ejpam-7045	155	132	set	set	NOUN
ejpam-7045	155	133	u	u	NOUN
ejpam-7045	155	134	of	of	ADP
ejpam-7045	155	135	x	x	PUNCT
ejpam-7045	155	136	containing	contain	VERB
ejpam-7045	155	137	x	x	PUNCT
ejpam-7045	155	138	such	such	ADJ
ejpam-7045	155	139	that	that	SCONJ
ejpam-7045	155	140	f	f	PROPN
ejpam-7045	155	141	(	(	PUNCT
ejpam-7045	155	142	u	u	NOUN
ejpam-7045	155	143	)	)	PUNCT
ejpam-7045	155	144	⊆	⊆	NUM
ejpam-7045	155	145	v	v	NOUN
ejpam-7045	155	146	;	;	PUNCT
ejpam-7045	155	147	(	(	PUNCT
ejpam-7045	155	148	4	4	NUM
ejpam-7045	155	149	)	)	PUNCT
ejpam-7045	155	150	f+(v	f+(v	NOUN
ejpam-7045	155	151	)	)	PUNCT
ejpam-7045	155	152	is	be	AUX
ejpam-7045	155	153	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	155	154	-	-	PUNCT
ejpam-7045	155	155	open	open	ADJ
ejpam-7045	155	156	in	in	ADP
ejpam-7045	155	157	x	x	PUNCT
ejpam-7045	155	158	for	for	ADP
ejpam-7045	155	159	every	every	DET
ejpam-7045	155	160	(	(	PUNCT
ejpam-7045	155	161	σ1	σ1	PROPN
ejpam-7045	155	162	,	,	PUNCT
ejpam-7045	155	163	σ2)r	σ2)r	NOUN
ejpam-7045	155	164	-	-	PUNCT
ejpam-7045	155	165	open	open	ADJ
ejpam-7045	155	166	set	set	VERB
ejpam-7045	155	167	v	v	NOUN
ejpam-7045	155	168	of	of	ADP
ejpam-7045	155	169	y	y	PROPN
ejpam-7045	155	170	;	;	PUNCT
ejpam-7045	155	171	(	(	PUNCT
ejpam-7045	155	172	5	5	X
ejpam-7045	155	173	)	)	PUNCT
ejpam-7045	155	174	f−(k	f−(k	PROPN
ejpam-7045	155	175	)	)	PUNCT
ejpam-7045	155	176	is	be	AUX
ejpam-7045	155	177	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	155	178	-	-	PUNCT
ejpam-7045	155	179	closed	closed	ADJ
ejpam-7045	155	180	in	in	ADP
ejpam-7045	155	181	x	x	PUNCT
ejpam-7045	155	182	for	for	ADP
ejpam-7045	155	183	every	every	DET
ejpam-7045	155	184	(	(	PUNCT
ejpam-7045	155	185	σ1	σ1	PROPN
ejpam-7045	155	186	,	,	PUNCT
ejpam-7045	156	1	σ2)r	σ2)r	NOUN
ejpam-7045	156	2	-	-	PUNCT
ejpam-7045	156	3	closed	close	VERB
ejpam-7045	156	4	set	set	ADJ
ejpam-7045	156	5	k	k	PROPN
ejpam-7045	156	6	of	of	ADP
ejpam-7045	156	7	y	y	PROPN
ejpam-7045	156	8	;	;	PUNCT
ejpam-7045	156	9	(	(	PUNCT
ejpam-7045	156	10	6	6	NUM
ejpam-7045	156	11	)	)	PUNCT
ejpam-7045	156	12	f+(v	f+(v	NOUN
ejpam-7045	156	13	)	)	PUNCT
ejpam-7045	157	1	⊆	⊆	NUM
ejpam-7045	157	2	αint⋆(f+((σ1	αint⋆(f+((σ1	NUM
ejpam-7045	157	3	,	,	PUNCT
ejpam-7045	157	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	157	5	)	)	PUNCT
ejpam-7045	157	6	)	)	PUNCT
ejpam-7045	157	7	)	)	PUNCT
ejpam-7045	158	1	for	for	ADP
ejpam-7045	158	2	every	every	DET
ejpam-7045	158	3	σ1σ2	σ1σ2	NOUN
ejpam-7045	158	4	-	-	ADJ
ejpam-7045	158	5	open	open	ADJ
ejpam-7045	158	6	set	set	NOUN
ejpam-7045	158	7	v	v	NOUN
ejpam-7045	158	8	of	of	ADP
ejpam-7045	158	9	y	y	PROPN
ejpam-7045	158	10	;	;	PUNCT
ejpam-7045	158	11	(	(	PUNCT
ejpam-7045	158	12	7	7	X
ejpam-7045	158	13	)	)	PUNCT
ejpam-7045	158	14	αcl⋆(f−((σ1	αcl⋆(f−((σ1	NOUN
ejpam-7045	158	15	,	,	PUNCT
ejpam-7045	158	16	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	158	17	)	)	PUNCT
ejpam-7045	158	18	)	)	PUNCT
ejpam-7045	158	19	)	)	PUNCT
ejpam-7045	158	20	⊆	⊆	X
ejpam-7045	158	21	f−(k	f−(k	PROPN
ejpam-7045	158	22	)	)	PUNCT
ejpam-7045	158	23	for	for	ADP
ejpam-7045	158	24	every	every	DET
ejpam-7045	158	25	σ1σ2	σ1σ2	NUM
ejpam-7045	158	26	-	-	PUNCT
ejpam-7045	158	27	closed	closed	ADJ
ejpam-7045	158	28	set	set	NOUN
ejpam-7045	158	29	k	k	PROPN
ejpam-7045	158	30	of	of	ADP
ejpam-7045	158	31	y	y	PROPN
ejpam-7045	158	32	;	;	PUNCT
ejpam-7045	158	33	(	(	PUNCT
ejpam-7045	158	34	8)	8)	NUM
ejpam-7045	158	35	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7045	158	36	-	-	PUNCT
ejpam-7045	158	37	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	158	38	-	-	PUNCT
ejpam-7045	158	39	int(k	int(k	NOUN
ejpam-7045	158	40	)	)	PUNCT
ejpam-7045	158	41	)	)	PUNCT
ejpam-7045	158	42	)	)	PUNCT
ejpam-7045	158	43	)	)	PUNCT
ejpam-7045	158	44	⊆	⊆	X
ejpam-7045	158	45	f−(k	f−(k	PROPN
ejpam-7045	158	46	)	)	PUNCT
ejpam-7045	158	47	for	for	ADP
ejpam-7045	158	48	every	every	DET
ejpam-7045	158	49	σ1σ2	σ1σ2	NUM
ejpam-7045	158	50	-	-	PUNCT
ejpam-7045	158	51	closed	closed	ADJ
ejpam-7045	158	52	set	set	NOUN
ejpam-7045	158	53	k	k	PROPN
ejpam-7045	158	54	of	of	ADP
ejpam-7045	158	55	y	y	PROPN
ejpam-7045	158	56	;	;	PUNCT
ejpam-7045	158	57	(	(	PUNCT
ejpam-7045	158	58	9	9	X
ejpam-7045	158	59	)	)	PUNCT
ejpam-7045	158	60	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7045	158	61	-	-	PUNCT
ejpam-7045	158	62	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	158	63	-	-	PUNCT
ejpam-7045	158	64	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	158	65	-	-	PUNCT
ejpam-7045	158	66	cl(b	cl(b	NOUN
ejpam-7045	158	67	)	)	PUNCT
ejpam-7045	158	68	)	)	PUNCT
ejpam-7045	158	69	)	)	PUNCT
ejpam-7045	158	70	)	)	PUNCT
ejpam-7045	158	71	)	)	PUNCT
ejpam-7045	158	72	⊆	⊆	X
ejpam-7045	158	73	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7045	158	74	-	-	PUNCT
ejpam-7045	158	75	cl(b	cl(b	NOUN
ejpam-7045	158	76	)	)	PUNCT
ejpam-7045	158	77	)	)	PUNCT
ejpam-7045	158	78	for	for	ADP
ejpam-7045	158	79	every	every	DET
ejpam-7045	158	80	subset	subset	NOUN
ejpam-7045	158	81	b	b	PROPN
ejpam-7045	158	82	of	of	ADP
ejpam-7045	158	83	y	y	PROPN
ejpam-7045	158	84	;	;	PUNCT
ejpam-7045	158	85	c.	c.	PROPN
ejpam-7045	158	86	viriyapong	viriyapong	PROPN
ejpam-7045	158	87	,	,	PUNCT
ejpam-7045	158	88	a.	a.	PROPN
ejpam-7045	158	89	sama	sama	PROPN
ejpam-7045	158	90	-	-	PUNCT
ejpam-7045	158	91	ae	ae	PROPN
ejpam-7045	158	92	,	,	PUNCT
ejpam-7045	158	93	c.	c.	PROPN
ejpam-7045	158	94	boonpok	boonpok	PROPN
ejpam-7045	158	95	/	/	SYM
ejpam-7045	158	96	eur	eur	PROPN
ejpam-7045	158	97	.	.	PUNCT
ejpam-7045	159	1	j.	j.	PROPN
ejpam-7045	159	2	pure	pure	PROPN
ejpam-7045	159	3	appl	appl	PROPN
ejpam-7045	159	4	.	.	PROPN
ejpam-7045	159	5	math	math	PROPN
ejpam-7045	159	6	,	,	PUNCT
ejpam-7045	159	7	18	18	NUM
ejpam-7045	159	8	(	(	PUNCT
ejpam-7045	159	9	4	4	NUM
ejpam-7045	159	10	)	)	PUNCT
ejpam-7045	159	11	(	(	PUNCT
ejpam-7045	159	12	2025	2025	NUM
ejpam-7045	159	13	)	)	PUNCT
ejpam-7045	159	14	,	,	PUNCT
ejpam-7045	159	15	7045	7045	NUM
ejpam-7045	159	16	7	7	NUM
ejpam-7045	159	17	of	of	ADP
ejpam-7045	159	18	12	12	NUM
ejpam-7045	159	19	(	(	PUNCT
ejpam-7045	159	20	10	10	NUM
ejpam-7045	159	21	)	)	PUNCT
ejpam-7045	159	22	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7045	159	23	-	-	PUNCT
ejpam-7045	159	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	159	25	-	-	PUNCT
ejpam-7045	159	26	int(k	int(k	NOUN
ejpam-7045	159	27	)	)	PUNCT
ejpam-7045	159	28	)	)	PUNCT
ejpam-7045	159	29	)	)	PUNCT
ejpam-7045	159	30	)	)	PUNCT
ejpam-7045	159	31	)	)	PUNCT
ejpam-7045	159	32	)	)	PUNCT
ejpam-7045	160	1	⊆	⊆	X
ejpam-7045	160	2	f−(k	f−(k	PROPN
ejpam-7045	160	3	)	)	PUNCT
ejpam-7045	160	4	for	for	ADP
ejpam-7045	160	5	every	every	DET
ejpam-7045	160	6	σ1σ2	σ1σ2	NUM
ejpam-7045	160	7	-	-	PUNCT
ejpam-7045	160	8	closed	closed	ADJ
ejpam-7045	160	9	set	set	NOUN
ejpam-7045	160	10	k	k	PROPN
ejpam-7045	160	11	of	of	ADP
ejpam-7045	160	12	y	y	PROPN
ejpam-7045	160	13	;	;	PUNCT
ejpam-7045	160	14	(	(	PUNCT
ejpam-7045	160	15	11	11	NUM
ejpam-7045	160	16	)	)	PUNCT
ejpam-7045	160	17	cl⋆(int⋆(cl⋆(f−((σ1	cl⋆(int⋆(cl⋆(f−((σ1	NOUN
ejpam-7045	160	18	,	,	PUNCT
ejpam-7045	160	19	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	160	20	)	)	PUNCT
ejpam-7045	160	21	)	)	PUNCT
ejpam-7045	160	22	)	)	PUNCT
ejpam-7045	160	23	)	)	PUNCT
ejpam-7045	160	24	)	)	PUNCT
ejpam-7045	161	1	⊆	⊆	X
ejpam-7045	161	2	f−(k	f−(k	PROPN
ejpam-7045	161	3	)	)	PUNCT
ejpam-7045	161	4	for	for	ADP
ejpam-7045	161	5	every	every	DET
ejpam-7045	161	6	σ1σ2	σ1σ2	NUM
ejpam-7045	161	7	-	-	PUNCT
ejpam-7045	161	8	closed	closed	ADJ
ejpam-7045	161	9	set	set	NOUN
ejpam-7045	161	10	k	k	PROPN
ejpam-7045	161	11	of	of	ADP
ejpam-7045	161	12	y	y	PROPN
ejpam-7045	161	13	;	;	PUNCT
ejpam-7045	161	14	(	(	PUNCT
ejpam-7045	161	15	12	12	NUM
ejpam-7045	161	16	)	)	PUNCT
ejpam-7045	161	17	f+(v	f+(v	NOUN
ejpam-7045	161	18	)	)	PUNCT
ejpam-7045	162	1	⊆	⊆	X
ejpam-7045	162	2	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	162	3	,	,	PUNCT
ejpam-7045	162	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	162	5	)	)	PUNCT
ejpam-7045	162	6	)	)	PUNCT
ejpam-7045	162	7	)	)	PUNCT
ejpam-7045	162	8	)	)	PUNCT
ejpam-7045	162	9	)	)	PUNCT
ejpam-7045	162	10	for	for	ADP
ejpam-7045	162	11	every	every	DET
ejpam-7045	162	12	σ1σ2	σ1σ2	NOUN
ejpam-7045	162	13	-	-	ADJ
ejpam-7045	162	14	open	open	ADJ
ejpam-7045	162	15	set	set	NOUN
ejpam-7045	162	16	v	v	NOUN
ejpam-7045	162	17	of	of	ADP
ejpam-7045	162	18	y	y	PROPN
ejpam-7045	162	19	.	.	PUNCT
ejpam-7045	163	1	proof	proof	NOUN
ejpam-7045	163	2	.	.	PUNCT
ejpam-7045	164	1	(	(	PUNCT
ejpam-7045	164	2	1	1	X
ejpam-7045	164	3	)	)	PUNCT
ejpam-7045	164	4	⇒	⇒	NOUN
ejpam-7045	164	5	(	(	PUNCT
ejpam-7045	164	6	2	2	NUM
ejpam-7045	164	7	):	):	PUNCT
ejpam-7045	164	8	the	the	DET
ejpam-7045	164	9	proof	proof	NOUN
ejpam-7045	164	10	follows	follow	VERB
ejpam-7045	164	11	from	from	ADP
ejpam-7045	164	12	theorem	theorem	ADJ
ejpam-7045	164	13	1	1	NUM
ejpam-7045	164	14	.	.	PUNCT
ejpam-7045	164	15	(	(	PUNCT
ejpam-7045	164	16	2	2	X
ejpam-7045	164	17	)	)	PUNCT
ejpam-7045	164	18	⇒	⇒	NOUN
ejpam-7045	164	19	(	(	PUNCT
ejpam-7045	164	20	3	3	NUM
ejpam-7045	164	21	):	):	PUNCT
ejpam-7045	164	22	the	the	DET
ejpam-7045	164	23	proof	proof	NOUN
ejpam-7045	164	24	is	be	AUX
ejpam-7045	164	25	obvious	obvious	ADJ
ejpam-7045	164	26	.	.	PUNCT
ejpam-7045	165	1	(	(	PUNCT
ejpam-7045	165	2	3	3	X
ejpam-7045	165	3	)	)	PUNCT
ejpam-7045	165	4	⇒	⇒	NOUN
ejpam-7045	165	5	(	(	PUNCT
ejpam-7045	165	6	4	4	NUM
ejpam-7045	165	7	):	):	PUNCT
ejpam-7045	165	8	let	let	VERB
ejpam-7045	165	9	v	v	PART
ejpam-7045	165	10	be	be	AUX
ejpam-7045	165	11	any	any	DET
ejpam-7045	165	12	(	(	PUNCT
ejpam-7045	165	13	σ1	σ1	NOUN
ejpam-7045	165	14	,	,	PUNCT
ejpam-7045	165	15	σ2)r	σ2)r	NOUN
ejpam-7045	165	16	-	-	PUNCT
ejpam-7045	165	17	open	open	ADJ
ejpam-7045	165	18	set	set	NOUN
ejpam-7045	165	19	of	of	ADP
ejpam-7045	165	20	y	y	PROPN
ejpam-7045	165	21	and	and	CCONJ
ejpam-7045	165	22	x	x	PROPN
ejpam-7045	165	23	∈	∈	PROPN
ejpam-7045	165	24	f+(v	f+(v	NOUN
ejpam-7045	165	25	)	)	PUNCT
ejpam-7045	165	26	.	.	PUNCT
ejpam-7045	166	1	then	then	ADV
ejpam-7045	166	2	,	,	PUNCT
ejpam-7045	166	3	f	f	PROPN
ejpam-7045	166	4	(	(	PUNCT
ejpam-7045	166	5	x	x	X
ejpam-7045	166	6	)	)	PUNCT
ejpam-7045	166	7	⊆	⊆	NUM
ejpam-7045	166	8	v	v	NOUN
ejpam-7045	166	9	and	and	CCONJ
ejpam-7045	166	10	by	by	ADP
ejpam-7045	166	11	(	(	PUNCT
ejpam-7045	166	12	3	3	NUM
ejpam-7045	166	13	)	)	PUNCT
ejpam-7045	166	14	,	,	PUNCT
ejpam-7045	166	15	there	there	PRON
ejpam-7045	166	16	exists	exist	VERB
ejpam-7045	166	17	a	a	DET
ejpam-7045	166	18	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	166	19	-	-	PUNCT
ejpam-7045	166	20	open	open	NOUN
ejpam-7045	166	21	set	set	NOUN
ejpam-7045	166	22	ux	ux	NOUN
ejpam-7045	166	23	of	of	ADP
ejpam-7045	166	24	x	x	SYM
ejpam-7045	166	25	containing	contain	VERB
ejpam-7045	166	26	x	x	PUNCT
ejpam-7045	166	27	such	such	ADJ
ejpam-7045	166	28	that	that	SCONJ
ejpam-7045	166	29	f	f	PROPN
ejpam-7045	166	30	(	(	PUNCT
ejpam-7045	166	31	ux	ux	PROPN
ejpam-7045	166	32	)	)	PUNCT
ejpam-7045	166	33	⊆	⊆	NUM
ejpam-7045	166	34	v	v	NOUN
ejpam-7045	166	35	.	.	PUNCT
ejpam-7045	167	1	thus	thus	ADV
ejpam-7045	167	2	,	,	PUNCT
ejpam-7045	167	3	x	x	PUNCT
ejpam-7045	167	4	∈	∈	PROPN
ejpam-7045	167	5	ux	ux	NOUN
ejpam-7045	167	6	⊆	⊆	NUM
ejpam-7045	167	7	f+(v	f+(v	NOUN
ejpam-7045	167	8	)	)	PUNCT
ejpam-7045	167	9	and	and	CCONJ
ejpam-7045	167	10	so	so	ADV
ejpam-7045	167	11	f+(v	f+(v	NOUN
ejpam-7045	167	12	)	)	PUNCT
ejpam-7045	168	1	=	=	PRON
ejpam-7045	168	2	∪x∈f+(v	∪x∈f+(v	X
ejpam-7045	168	3	)	)	PUNCT
ejpam-7045	168	4	ux	ux	PROPN
ejpam-7045	168	5	is	be	AUX
ejpam-7045	168	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	168	7	-	-	PUNCT
ejpam-7045	168	8	open	open	ADJ
ejpam-7045	168	9	in	in	ADP
ejpam-7045	168	10	x.	x.	NOUN
ejpam-7045	168	11	(	(	PUNCT
ejpam-7045	168	12	4	4	NUM
ejpam-7045	168	13	)	)	PUNCT
ejpam-7045	168	14	⇒	⇒	NOUN
ejpam-7045	168	15	(	(	PUNCT
ejpam-7045	168	16	5	5	NUM
ejpam-7045	168	17	):	):	PUNCT
ejpam-7045	168	18	this	this	PRON
ejpam-7045	168	19	follows	follow	VERB
ejpam-7045	168	20	from	from	ADP
ejpam-7045	168	21	the	the	DET
ejpam-7045	168	22	fact	fact	NOUN
ejpam-7045	168	23	that	that	SCONJ
ejpam-7045	168	24	f+(y	f+(y	PROPN
ejpam-7045	168	25	−b	−b	ADV
ejpam-7045	168	26	)	)	PUNCT
ejpam-7045	169	1	=	=	PUNCT
ejpam-7045	169	2	x	x	X
ejpam-7045	170	1	−	−	PROPN
ejpam-7045	170	2	f−(b	f−(b	PROPN
ejpam-7045	170	3	)	)	PUNCT
ejpam-7045	170	4	for	for	ADP
ejpam-7045	170	5	every	every	DET
ejpam-7045	170	6	subset	subset	NOUN
ejpam-7045	170	7	b	b	PROPN
ejpam-7045	170	8	of	of	ADP
ejpam-7045	170	9	y	y	PROPN
ejpam-7045	170	10	.	.	PUNCT
ejpam-7045	171	1	(	(	PUNCT
ejpam-7045	171	2	5	5	X
ejpam-7045	171	3	)	)	PUNCT
ejpam-7045	171	4	⇒	⇒	NOUN
ejpam-7045	171	5	(	(	PUNCT
ejpam-7045	171	6	6	6	NUM
ejpam-7045	171	7	):	):	PUNCT
ejpam-7045	171	8	let	let	VERB
ejpam-7045	171	9	v	v	PART
ejpam-7045	171	10	be	be	AUX
ejpam-7045	171	11	any	any	DET
ejpam-7045	171	12	σ1σ2	σ1σ2	NOUN
ejpam-7045	171	13	-	-	ADJ
ejpam-7045	171	14	open	open	ADJ
ejpam-7045	171	15	set	set	NOUN
ejpam-7045	171	16	of	of	ADP
ejpam-7045	171	17	y	y	PROPN
ejpam-7045	171	18	and	and	CCONJ
ejpam-7045	171	19	x	x	PROPN
ejpam-7045	171	20	∈	∈	PROPN
ejpam-7045	171	21	f+(v	f+(v	NOUN
ejpam-7045	171	22	)	)	PUNCT
ejpam-7045	171	23	.	.	PUNCT
ejpam-7045	172	1	then	then	ADV
ejpam-7045	172	2	,	,	PUNCT
ejpam-7045	172	3	we	we	PRON
ejpam-7045	172	4	have	have	VERB
ejpam-7045	172	5	f	f	PROPN
ejpam-7045	172	6	(	(	PUNCT
ejpam-7045	172	7	x	x	NOUN
ejpam-7045	172	8	)	)	PUNCT
ejpam-7045	172	9	⊆	⊆	NUM
ejpam-7045	172	10	v	v	ADP
ejpam-7045	172	11	⊆	⊆	NUM
ejpam-7045	172	12	(	(	PUNCT
ejpam-7045	172	13	σ1	σ1	PROPN
ejpam-7045	172	14	,	,	PUNCT
ejpam-7045	172	15	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	172	16	)	)	PUNCT
ejpam-7045	172	17	and	and	CCONJ
ejpam-7045	172	18	so	so	ADV
ejpam-7045	172	19	x	x	SYM
ejpam-7045	172	20	∈	∈	PROPN
ejpam-7045	172	21	f+((σ1	f+((σ1	NOUN
ejpam-7045	172	22	,	,	PUNCT
ejpam-7045	172	23	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	172	24	)	)	PUNCT
ejpam-7045	172	25	)	)	PUNCT
ejpam-7045	173	1	=	=	PUNCT
ejpam-7045	173	2	x−f−(y	x−f−(y	X
ejpam-7045	174	1	−(σ1	−(σ1	SYM
ejpam-7045	174	2	,	,	PUNCT
ejpam-7045	174	3	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	174	4	)	)	PUNCT
ejpam-7045	174	5	)	)	PUNCT
ejpam-7045	174	6	.	.	PUNCT
ejpam-7045	175	1	since	since	SCONJ
ejpam-7045	175	2	y	y	PROPN
ejpam-7045	175	3	−(σ1	−(σ1	AUX
ejpam-7045	175	4	,	,	PUNCT
ejpam-7045	175	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	175	6	)	)	PUNCT
ejpam-7045	175	7	is	be	AUX
ejpam-7045	175	8	(	(	PUNCT
ejpam-7045	175	9	σ1	σ1	NOUN
ejpam-7045	175	10	,	,	PUNCT
ejpam-7045	175	11	σ2)r	σ2)r	NOUN
ejpam-7045	175	12	-	-	PUNCT
ejpam-7045	175	13	closed	closed	ADJ
ejpam-7045	175	14	in	in	ADP
ejpam-7045	175	15	y	y	PROPN
ejpam-7045	175	16	and	and	CCONJ
ejpam-7045	175	17	by	by	ADP
ejpam-7045	175	18	(	(	PUNCT
ejpam-7045	175	19	5	5	NUM
ejpam-7045	175	20	)	)	PUNCT
ejpam-7045	175	21	,	,	PUNCT
ejpam-7045	175	22	f−(y	f−(y	NOUN
ejpam-7045	175	23	−(σ1	−(σ1	SYM
ejpam-7045	175	24	,	,	PUNCT
ejpam-7045	175	25	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	175	26	)	)	PUNCT
ejpam-7045	175	27	)	)	PUNCT
ejpam-7045	175	28	is	be	AUX
ejpam-7045	175	29	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	175	30	-	-	PUNCT
ejpam-7045	175	31	closed	closed	ADJ
ejpam-7045	175	32	inx	inx	NOUN
ejpam-7045	175	33	.	.	PUNCT
ejpam-7045	176	1	this	this	PRON
ejpam-7045	176	2	shows	show	VERB
ejpam-7045	176	3	that	that	SCONJ
ejpam-7045	176	4	f+((σ1	f+((σ1	NOUN
ejpam-7045	176	5	,	,	PUNCT
ejpam-7045	176	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	176	7	)	)	PUNCT
ejpam-7045	176	8	)	)	PUNCT
ejpam-7045	176	9	is	be	AUX
ejpam-7045	176	10	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	176	11	-	-	PUNCT
ejpam-7045	176	12	open	open	ADJ
ejpam-7045	176	13	in	in	ADP
ejpam-7045	176	14	x.	x.	NOUN
ejpam-7045	176	15	thus	thus	ADV
ejpam-7045	176	16	,	,	PUNCT
ejpam-7045	176	17	x	x	SYM
ejpam-7045	176	18	∈	∈	NOUN
ejpam-7045	176	19	αint⋆(f+((σ1	αint⋆(f+((σ1	NOUN
ejpam-7045	176	20	,	,	PUNCT
ejpam-7045	176	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	176	22	)	)	PUNCT
ejpam-7045	176	23	)	)	PUNCT
ejpam-7045	176	24	)	)	PUNCT
ejpam-7045	176	25	and	and	CCONJ
ejpam-7045	176	26	hence	hence	ADV
ejpam-7045	176	27	f+(v	f+(v	NOUN
ejpam-7045	176	28	)	)	PUNCT
ejpam-7045	177	1	⊆	⊆	NUM
ejpam-7045	177	2	αint⋆(f+((σ1	αint⋆(f+((σ1	NUM
ejpam-7045	177	3	,	,	PUNCT
ejpam-7045	177	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	177	5	)	)	PUNCT
ejpam-7045	177	6	)	)	PUNCT
ejpam-7045	177	7	)	)	PUNCT
ejpam-7045	177	8	.	.	PUNCT
ejpam-7045	178	1	(	(	PUNCT
ejpam-7045	178	2	6	6	X
ejpam-7045	178	3	)	)	PUNCT
ejpam-7045	178	4	⇒	⇒	NOUN
ejpam-7045	178	5	(	(	PUNCT
ejpam-7045	178	6	7	7	NUM
ejpam-7045	178	7	):	):	PUNCT
ejpam-7045	178	8	let	let	VERB
ejpam-7045	178	9	k	k	PRON
ejpam-7045	178	10	be	be	AUX
ejpam-7045	178	11	any	any	DET
ejpam-7045	178	12	σ1σ2	σ1σ2	NUM
ejpam-7045	178	13	-	-	PUNCT
ejpam-7045	178	14	closed	closed	ADJ
ejpam-7045	178	15	set	set	NOUN
ejpam-7045	178	16	of	of	ADP
ejpam-7045	178	17	y	y	PROPN
ejpam-7045	178	18	.	.	PUNCT
ejpam-7045	179	1	then	then	ADV
ejpam-7045	179	2	,	,	PUNCT
ejpam-7045	179	3	y	y	PROPN
ejpam-7045	179	4	−k	−k	PROPN
ejpam-7045	179	5	is	be	AUX
ejpam-7045	179	6	σ1σ2	σ1σ2	NOUN
ejpam-7045	179	7	-	-	ADJ
ejpam-7045	179	8	open	open	ADJ
ejpam-7045	179	9	and	and	CCONJ
ejpam-7045	179	10	by	by	ADP
ejpam-7045	179	11	(	(	PUNCT
ejpam-7045	179	12	6	6	NUM
ejpam-7045	179	13	)	)	PUNCT
ejpam-7045	179	14	,	,	PUNCT
ejpam-7045	179	15	we	we	PRON
ejpam-7045	179	16	have	have	VERB
ejpam-7045	179	17	x	x	INTJ
ejpam-7045	179	18	−	−	DET
ejpam-7045	179	19	f−(k	f−(k	PROPN
ejpam-7045	179	20	)	)	PUNCT
ejpam-7045	179	21	=	=	PUNCT
ejpam-7045	180	1	f+(y	f+(y	PROPN
ejpam-7045	180	2	−k	−k	PROPN
ejpam-7045	180	3	)	)	PUNCT
ejpam-7045	180	4	⊆	⊆	NUM
ejpam-7045	180	5	αint⋆(f+((σ1	αint⋆(f+((σ1	NUM
ejpam-7045	180	6	,	,	PUNCT
ejpam-7045	180	7	σ2)-scl(y	σ2)-scl(y	NOUN
ejpam-7045	180	8	−k	−k	NOUN
ejpam-7045	180	9	)	)	PUNCT
ejpam-7045	180	10	)	)	PUNCT
ejpam-7045	180	11	)	)	PUNCT
ejpam-7045	181	1	=	=	PUNCT
ejpam-7045	181	2	αint⋆(f+(y	αint⋆(f+(y	NOUN
ejpam-7045	181	3	−	−	PROPN
ejpam-7045	181	4	(	(	PUNCT
ejpam-7045	181	5	σ1	σ1	PROPN
ejpam-7045	181	6	,	,	PUNCT
ejpam-7045	181	7	σ2)-sint(k	σ2)-sint(k	NUM
ejpam-7045	181	8	)	)	PUNCT
ejpam-7045	181	9	)	)	PUNCT
ejpam-7045	181	10	)	)	PUNCT
ejpam-7045	182	1	=	=	PUNCT
ejpam-7045	182	2	αint⋆(x	αint⋆(x	NOUN
ejpam-7045	182	3	−	−	NOUN
ejpam-7045	182	4	f−((σ1	f−((σ1	NOUN
ejpam-7045	182	5	,	,	PUNCT
ejpam-7045	182	6	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	182	7	)	)	PUNCT
ejpam-7045	182	8	)	)	PUNCT
ejpam-7045	182	9	)	)	PUNCT
ejpam-7045	183	1	=	=	PUNCT
ejpam-7045	183	2	x	x	SYM
ejpam-7045	184	1	−	−	NOUN
ejpam-7045	184	2	αcl⋆(f−((σ1	αcl⋆(f−((σ1	NOUN
ejpam-7045	184	3	,	,	PUNCT
ejpam-7045	184	4	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	184	5	)	)	PUNCT
ejpam-7045	184	6	)	)	PUNCT
ejpam-7045	184	7	)	)	PUNCT
ejpam-7045	184	8	and	and	CCONJ
ejpam-7045	184	9	hence	hence	ADV
ejpam-7045	184	10	αcl⋆(f−((σ1	αcl⋆(f−((σ1	NUM
ejpam-7045	184	11	,	,	PUNCT
ejpam-7045	184	12	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	184	13	)	)	PUNCT
ejpam-7045	184	14	)	)	PUNCT
ejpam-7045	184	15	)	)	PUNCT
ejpam-7045	185	1	⊆	⊆	NUM
ejpam-7045	185	2	f−(k	f−(k	PROPN
ejpam-7045	185	3	)	)	PUNCT
ejpam-7045	185	4	.	.	PUNCT
ejpam-7045	186	1	(	(	PUNCT
ejpam-7045	186	2	7	7	X
ejpam-7045	186	3	)	)	PUNCT
ejpam-7045	186	4	⇒	⇒	NOUN
ejpam-7045	186	5	(	(	PUNCT
ejpam-7045	186	6	8)	8)	NUM
ejpam-7045	186	7	:	:	PUNCT
ejpam-7045	186	8	the	the	DET
ejpam-7045	186	9	proof	proof	NOUN
ejpam-7045	186	10	is	be	AUX
ejpam-7045	186	11	obvious	obvious	ADJ
ejpam-7045	186	12	since	since	SCONJ
ejpam-7045	186	13	(	(	PUNCT
ejpam-7045	186	14	σ1	σ1	PROPN
ejpam-7045	186	15	,	,	PUNCT
ejpam-7045	186	16	σ2)-sint(k	σ2)-sint(k	NUM
ejpam-7045	186	17	)	)	PUNCT
ejpam-7045	186	18	=	=	SYM
ejpam-7045	186	19	σ1σ2	σ1σ2	X
ejpam-7045	186	20	-	-	PUNCT
ejpam-7045	186	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	186	22	-	-	PUNCT
ejpam-7045	186	23	int(k	int(k	NOUN
ejpam-7045	186	24	)	)	PUNCT
ejpam-7045	186	25	)	)	PUNCT
ejpam-7045	186	26	for	for	ADP
ejpam-7045	186	27	every	every	DET
ejpam-7045	186	28	σ1σ2	σ1σ2	NUM
ejpam-7045	186	29	-	-	PUNCT
ejpam-7045	186	30	closed	closed	ADJ
ejpam-7045	186	31	set	set	NOUN
ejpam-7045	186	32	k	k	PROPN
ejpam-7045	186	33	of	of	ADP
ejpam-7045	186	34	y	y	PROPN
ejpam-7045	186	35	.	.	PUNCT
ejpam-7045	187	1	(	(	PUNCT
ejpam-7045	187	2	8)	8)	NUM
ejpam-7045	187	3	⇒	⇒	NOUN
ejpam-7045	187	4	(	(	PUNCT
ejpam-7045	187	5	9	9	NUM
ejpam-7045	187	6	):	):	PUNCT
ejpam-7045	187	7	the	the	DET
ejpam-7045	187	8	proof	proof	NOUN
ejpam-7045	187	9	is	be	AUX
ejpam-7045	187	10	obvious	obvious	ADJ
ejpam-7045	187	11	.	.	PUNCT
ejpam-7045	188	1	(	(	PUNCT
ejpam-7045	188	2	9	9	X
ejpam-7045	188	3	)	)	PUNCT
ejpam-7045	188	4	⇒	⇒	NOUN
ejpam-7045	188	5	(	(	PUNCT
ejpam-7045	188	6	10	10	NUM
ejpam-7045	188	7	):	):	PUNCT
ejpam-7045	188	8	it	it	PRON
ejpam-7045	188	9	follows	follow	VERB
ejpam-7045	188	10	from	from	ADP
ejpam-7045	188	11	lemma	lemma	PROPN
ejpam-7045	188	12	7	7	NUM
ejpam-7045	188	13	that	that	SCONJ
ejpam-7045	188	14	cl⋆(int⋆(cl⋆(b	cl⋆(int⋆(cl⋆(b	NOUN
ejpam-7045	188	15	)	)	PUNCT
ejpam-7045	188	16	)	)	PUNCT
ejpam-7045	188	17	)	)	PUNCT
ejpam-7045	189	1	⊆	⊆	NUM
ejpam-7045	189	2	αcl⋆(b	αcl⋆(b	NOUN
ejpam-7045	189	3	)	)	PUNCT
ejpam-7045	189	4	for	for	ADP
ejpam-7045	189	5	every	every	DET
ejpam-7045	189	6	subset	subset	NOUN
ejpam-7045	189	7	b	b	PROPN
ejpam-7045	189	8	of	of	ADP
ejpam-7045	189	9	y	y	PROPN
ejpam-7045	189	10	.	.	PUNCT
ejpam-7045	190	1	thus	thus	ADV
ejpam-7045	190	2	,	,	PUNCT
ejpam-7045	190	3	for	for	ADP
ejpam-7045	190	4	every	every	DET
ejpam-7045	190	5	σ1σ2	σ1σ2	NUM
ejpam-7045	190	6	-	-	PUNCT
ejpam-7045	190	7	closed	closed	ADJ
ejpam-7045	190	8	set	set	NOUN
ejpam-7045	190	9	k	k	PROPN
ejpam-7045	190	10	of	of	ADP
ejpam-7045	190	11	y	y	PROPN
ejpam-7045	190	12	,	,	PUNCT
ejpam-7045	190	13	we	we	PRON
ejpam-7045	190	14	have	have	VERB
ejpam-7045	190	15	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7045	190	16	-	-	PUNCT
ejpam-7045	190	17	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	190	18	-	-	PUNCT
ejpam-7045	190	19	int(k	int(k	NOUN
ejpam-7045	190	20	)	)	PUNCT
ejpam-7045	190	21	)	)	PUNCT
ejpam-7045	190	22	)	)	PUNCT
ejpam-7045	190	23	)	)	PUNCT
ejpam-7045	190	24	)	)	PUNCT
ejpam-7045	190	25	)	)	PUNCT
ejpam-7045	191	1	⊆	⊆	NUM
ejpam-7045	191	2	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	VERB
ejpam-7045	191	3	-	-	PUNCT
ejpam-7045	191	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	191	5	-	-	PUNCT
ejpam-7045	191	6	int(k	int(k	NOUN
ejpam-7045	191	7	)	)	PUNCT
ejpam-7045	191	8	)	)	PUNCT
ejpam-7045	191	9	)	)	PUNCT
ejpam-7045	191	10	)	)	PUNCT
ejpam-7045	192	1	=	=	SYM
ejpam-7045	192	2	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	VERB
ejpam-7045	192	3	-	-	PUNCT
ejpam-7045	192	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	192	5	-	-	PUNCT
ejpam-7045	192	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	192	7	-	-	PUNCT
ejpam-7045	192	8	cl(k	cl(k	NUM
ejpam-7045	192	9	)	)	PUNCT
ejpam-7045	192	10	)	)	PUNCT
ejpam-7045	192	11	)	)	PUNCT
ejpam-7045	192	12	)	)	PUNCT
ejpam-7045	192	13	)	)	PUNCT
ejpam-7045	193	1	⊆	⊆	X
ejpam-7045	193	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7045	193	3	-	-	PUNCT
ejpam-7045	193	4	cl(k	cl(k	NOUN
ejpam-7045	193	5	)	)	PUNCT
ejpam-7045	193	6	)	)	PUNCT
ejpam-7045	194	1	=	=	SYM
ejpam-7045	194	2	f−(k	f−(k	PROPN
ejpam-7045	194	3	)	)	PUNCT
ejpam-7045	194	4	.	.	PUNCT
ejpam-7045	195	1	(	(	PUNCT
ejpam-7045	195	2	10	10	NUM
ejpam-7045	195	3	)	)	PUNCT
ejpam-7045	195	4	⇒	⇒	NOUN
ejpam-7045	195	5	(	(	PUNCT
ejpam-7045	195	6	11	11	NUM
ejpam-7045	195	7	):	):	PUNCT
ejpam-7045	195	8	the	the	DET
ejpam-7045	195	9	proof	proof	NOUN
ejpam-7045	195	10	is	be	AUX
ejpam-7045	195	11	obvious	obvious	ADJ
ejpam-7045	195	12	since	since	SCONJ
ejpam-7045	195	13	(	(	PUNCT
ejpam-7045	195	14	σ1	σ1	PROPN
ejpam-7045	195	15	,	,	PUNCT
ejpam-7045	195	16	σ2)-sint(k	σ2)-sint(k	NUM
ejpam-7045	195	17	)	)	PUNCT
ejpam-7045	195	18	=	=	SYM
ejpam-7045	195	19	σ1σ2	σ1σ2	X
ejpam-7045	195	20	-	-	PUNCT
ejpam-7045	195	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	195	22	-	-	PUNCT
ejpam-7045	195	23	int(k	int(k	NOUN
ejpam-7045	195	24	)	)	PUNCT
ejpam-7045	195	25	)	)	PUNCT
ejpam-7045	195	26	for	for	ADP
ejpam-7045	195	27	every	every	DET
ejpam-7045	195	28	σ1σ2	σ1σ2	NUM
ejpam-7045	195	29	-	-	PUNCT
ejpam-7045	195	30	closed	closed	ADJ
ejpam-7045	195	31	set	set	NOUN
ejpam-7045	195	32	k	k	PROPN
ejpam-7045	195	33	of	of	ADP
ejpam-7045	195	34	y	y	PROPN
ejpam-7045	195	35	.	.	PUNCT
ejpam-7045	196	1	c.	c.	PROPN
ejpam-7045	196	2	viriyapong	viriyapong	PROPN
ejpam-7045	196	3	,	,	PUNCT
ejpam-7045	196	4	a.	a.	PROPN
ejpam-7045	196	5	sama	sama	PROPN
ejpam-7045	196	6	-	-	PUNCT
ejpam-7045	196	7	ae	ae	PROPN
ejpam-7045	196	8	,	,	PUNCT
ejpam-7045	196	9	c.	c.	PROPN
ejpam-7045	196	10	boonpok	boonpok	PROPN
ejpam-7045	196	11	/	/	SYM
ejpam-7045	196	12	eur	eur	PROPN
ejpam-7045	196	13	.	.	PUNCT
ejpam-7045	197	1	j.	j.	PROPN
ejpam-7045	197	2	pure	pure	PROPN
ejpam-7045	197	3	appl	appl	PROPN
ejpam-7045	197	4	.	.	PROPN
ejpam-7045	197	5	math	math	PROPN
ejpam-7045	197	6	,	,	PUNCT
ejpam-7045	197	7	18	18	NUM
ejpam-7045	197	8	(	(	PUNCT
ejpam-7045	197	9	4	4	NUM
ejpam-7045	197	10	)	)	PUNCT
ejpam-7045	197	11	(	(	PUNCT
ejpam-7045	197	12	2025	2025	NUM
ejpam-7045	197	13	)	)	PUNCT
ejpam-7045	197	14	,	,	PUNCT
ejpam-7045	197	15	7045	7045	NUM
ejpam-7045	197	16	8	8	NUM
ejpam-7045	197	17	of	of	ADP
ejpam-7045	197	18	12	12	NUM
ejpam-7045	197	19	(	(	PUNCT
ejpam-7045	197	20	11	11	NUM
ejpam-7045	197	21	)	)	PUNCT
ejpam-7045	197	22	⇒	⇒	NOUN
ejpam-7045	197	23	(	(	PUNCT
ejpam-7045	197	24	12	12	NUM
ejpam-7045	197	25	):	):	PUNCT
ejpam-7045	197	26	let	let	VERB
ejpam-7045	197	27	v	v	PART
ejpam-7045	197	28	be	be	AUX
ejpam-7045	197	29	any	any	DET
ejpam-7045	197	30	σ1σ2	σ1σ2	NOUN
ejpam-7045	197	31	-	-	ADJ
ejpam-7045	197	32	open	open	ADJ
ejpam-7045	197	33	set	set	NOUN
ejpam-7045	197	34	of	of	ADP
ejpam-7045	197	35	y	y	PROPN
ejpam-7045	197	36	.	.	PUNCT
ejpam-7045	198	1	then	then	ADV
ejpam-7045	198	2	,	,	PUNCT
ejpam-7045	198	3	y	y	PROPN
ejpam-7045	198	4	−	−	PROPN
ejpam-7045	198	5	v	v	NOUN
ejpam-7045	198	6	is	be	AUX
ejpam-7045	198	7	σ1σ2	σ1σ2	NOUN
ejpam-7045	198	8	-	-	ADJ
ejpam-7045	198	9	closed	closed	ADJ
ejpam-7045	198	10	in	in	ADP
ejpam-7045	198	11	y	y	PROPN
ejpam-7045	198	12	and	and	CCONJ
ejpam-7045	198	13	by	by	ADP
ejpam-7045	198	14	(	(	PUNCT
ejpam-7045	198	15	11	11	NUM
ejpam-7045	198	16	)	)	PUNCT
ejpam-7045	198	17	,	,	PUNCT
ejpam-7045	198	18	cl⋆(int⋆(cl⋆(f−((σ1	cl⋆(int⋆(cl⋆(f−((σ1	NOUN
ejpam-7045	198	19	,	,	PUNCT
ejpam-7045	198	20	σ2)sint(y	σ2)sint(y	X
ejpam-7045	198	21	−v	−v	NOUN
ejpam-7045	198	22	)	)	PUNCT
ejpam-7045	198	23	)	)	PUNCT
ejpam-7045	198	24	)	)	PUNCT
ejpam-7045	198	25	)	)	PUNCT
ejpam-7045	198	26	)	)	PUNCT
ejpam-7045	199	1	⊆	⊆	NUM
ejpam-7045	199	2	f−(y	f−(y	NOUN
ejpam-7045	199	3	−v	−v	NOUN
ejpam-7045	199	4	)	)	PUNCT
ejpam-7045	199	5	=	=	PUNCT
ejpam-7045	200	1	x	x	PUNCT
ejpam-7045	200	2	−f+(v	−f+(v	NOUN
ejpam-7045	200	3	)	)	PUNCT
ejpam-7045	200	4	.	.	PUNCT
ejpam-7045	201	1	moreover	moreover	ADV
ejpam-7045	201	2	,	,	PUNCT
ejpam-7045	201	3	we	we	PRON
ejpam-7045	201	4	have	have	VERB
ejpam-7045	201	5	cl⋆(int⋆(cl⋆(f−((σ1	cl⋆(int⋆(cl⋆(f−((σ1	NOUN
ejpam-7045	201	6	,	,	PUNCT
ejpam-7045	201	7	σ2)-sint(y	σ2)-sint(y	VERB
ejpam-7045	201	8	−	−	PROPN
ejpam-7045	201	9	v	v	NOUN
ejpam-7045	201	10	)	)	PUNCT
ejpam-7045	201	11	)	)	PUNCT
ejpam-7045	201	12	)	)	PUNCT
ejpam-7045	201	13	)	)	PUNCT
ejpam-7045	201	14	)	)	PUNCT
ejpam-7045	202	1	=	=	PUNCT
ejpam-7045	202	2	cl⋆(int⋆(cl⋆(f−(y	cl⋆(int⋆(cl⋆(f−(y	NUM
ejpam-7045	203	1	−	−	PROPN
ejpam-7045	203	2	(	(	PUNCT
ejpam-7045	203	3	σ1	σ1	PROPN
ejpam-7045	203	4	,	,	PUNCT
ejpam-7045	203	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	203	6	)	)	PUNCT
ejpam-7045	203	7	)	)	PUNCT
ejpam-7045	203	8	)	)	PUNCT
ejpam-7045	203	9	)	)	PUNCT
ejpam-7045	203	10	)	)	PUNCT
ejpam-7045	204	1	=	=	PUNCT
ejpam-7045	204	2	cl⋆(int⋆(cl⋆(x	cl⋆(int⋆(cl⋆(x	NOUN
ejpam-7045	204	3	−	−	NOUN
ejpam-7045	204	4	f+((σ1	f+((σ1	NOUN
ejpam-7045	204	5	,	,	PUNCT
ejpam-7045	204	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	204	7	)	)	PUNCT
ejpam-7045	204	8	)	)	PUNCT
ejpam-7045	204	9	)	)	PUNCT
ejpam-7045	204	10	)	)	PUNCT
ejpam-7045	204	11	)	)	PUNCT
ejpam-7045	205	1	=	=	PUNCT
ejpam-7045	205	2	x	x	X
ejpam-7045	206	1	−	−	NOUN
ejpam-7045	206	2	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	206	3	,	,	PUNCT
ejpam-7045	206	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	206	5	)	)	PUNCT
ejpam-7045	206	6	)	)	PUNCT
ejpam-7045	206	7	)	)	PUNCT
ejpam-7045	206	8	)	)	PUNCT
ejpam-7045	206	9	)	)	PUNCT
ejpam-7045	206	10	.	.	PUNCT
ejpam-7045	207	1	thus	thus	ADV
ejpam-7045	207	2	,	,	PUNCT
ejpam-7045	207	3	f+(v	f+(v	PROPN
ejpam-7045	207	4	)	)	PUNCT
ejpam-7045	207	5	⊆	⊆	NUM
ejpam-7045	207	6	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	207	7	,	,	PUNCT
ejpam-7045	207	8	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	207	9	)	)	PUNCT
ejpam-7045	207	10	)	)	PUNCT
ejpam-7045	207	11	)	)	PUNCT
ejpam-7045	207	12	)	)	PUNCT
ejpam-7045	207	13	)	)	PUNCT
ejpam-7045	207	14	.	.	PUNCT
ejpam-7045	208	1	(	(	PUNCT
ejpam-7045	208	2	12	12	NUM
ejpam-7045	208	3	)	)	PUNCT
ejpam-7045	208	4	⇒	⇒	NOUN
ejpam-7045	208	5	(	(	PUNCT
ejpam-7045	208	6	1	1	NUM
ejpam-7045	208	7	):	):	PUNCT
ejpam-7045	208	8	let	let	VERB
ejpam-7045	208	9	x	x	PUNCT
ejpam-7045	208	10	∈	∈	PROPN
ejpam-7045	208	11	x	x	X
ejpam-7045	208	12	and	and	CCONJ
ejpam-7045	208	13	v	v	X
ejpam-7045	208	14	be	be	AUX
ejpam-7045	208	15	any	any	DET
ejpam-7045	208	16	σ1σ2	σ1σ2	NOUN
ejpam-7045	208	17	-	-	ADJ
ejpam-7045	208	18	open	open	ADJ
ejpam-7045	208	19	set	set	NOUN
ejpam-7045	208	20	of	of	ADP
ejpam-7045	208	21	y	y	PROPN
ejpam-7045	208	22	containing	contain	VERB
ejpam-7045	208	23	f	f	PROPN
ejpam-7045	208	24	(	(	PUNCT
ejpam-7045	208	25	x	x	NOUN
ejpam-7045	208	26	)	)	PUNCT
ejpam-7045	208	27	.	.	PUNCT
ejpam-7045	209	1	by	by	ADP
ejpam-7045	209	2	(	(	PUNCT
ejpam-7045	209	3	12	12	NUM
ejpam-7045	209	4	)	)	PUNCT
ejpam-7045	209	5	,	,	PUNCT
ejpam-7045	209	6	we	we	PRON
ejpam-7045	209	7	have	have	VERB
ejpam-7045	209	8	x	x	X
ejpam-7045	209	9	∈	∈	NOUN
ejpam-7045	209	10	f+(v	f+(v	NOUN
ejpam-7045	209	11	)	)	PUNCT
ejpam-7045	210	1	⊆	⊆	X
ejpam-7045	210	2	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7045	210	3	,	,	PUNCT
ejpam-7045	210	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	210	5	)	)	PUNCT
ejpam-7045	210	6	)	)	PUNCT
ejpam-7045	210	7	)	)	PUNCT
ejpam-7045	210	8	)	)	PUNCT
ejpam-7045	210	9	)	)	PUNCT
ejpam-7045	210	10	and	and	CCONJ
ejpam-7045	210	11	hence	hence	ADV
ejpam-7045	210	12	f	f	PROPN
ejpam-7045	210	13	is	be	AUX
ejpam-7045	210	14	upper	upper	ADJ
ejpam-7045	210	15	almost	almost	ADV
ejpam-7045	210	16	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	210	17	,	,	PUNCT
ejpam-7045	210	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	210	19	at	at	ADP
ejpam-7045	210	20	x	x	PUNCT
ejpam-7045	210	21	by	by	ADP
ejpam-7045	210	22	theorem	theorem	NOUN
ejpam-7045	210	23	1	1	NUM
ejpam-7045	210	24	.	.	PUNCT
ejpam-7045	211	1	this	this	PRON
ejpam-7045	211	2	shows	show	VERB
ejpam-7045	211	3	that	that	SCONJ
ejpam-7045	211	4	f	f	PROPN
ejpam-7045	211	5	is	be	AUX
ejpam-7045	211	6	upper	upper	ADJ
ejpam-7045	211	7	almost	almost	ADV
ejpam-7045	211	8	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	211	9	,	,	PUNCT
ejpam-7045	211	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	211	11	.	.	NOUN
ejpam-7045	211	12	definition	definition	NOUN
ejpam-7045	211	13	3	3	NUM
ejpam-7045	211	14	.	.	PUNCT
ejpam-7045	212	1	[	[	X
ejpam-7045	212	2	20	20	NUM
ejpam-7045	212	3	]	]	PUNCT
ejpam-7045	212	4	a	a	DET
ejpam-7045	212	5	multifunction	multifunction	NOUN
ejpam-7045	212	6	f	f	NOUN
ejpam-7045	212	7	:	:	PUNCT
ejpam-7045	212	8	(	(	PUNCT
ejpam-7045	212	9	x	x	X
ejpam-7045	212	10	,	,	PUNCT
ejpam-7045	212	11	τ	τ	PROPN
ejpam-7045	212	12	,	,	PUNCT
ejpam-7045	212	13	i	i	NOUN
ejpam-7045	212	14	)	)	PUNCT
ejpam-7045	212	15	→	→	PUNCT
ejpam-7045	212	16	(	(	PUNCT
ejpam-7045	212	17	y	y	PROPN
ejpam-7045	212	18	,	,	PUNCT
ejpam-7045	212	19	σ1	σ1	PROPN
ejpam-7045	212	20	,	,	PUNCT
ejpam-7045	212	21	σ2	σ2	PROPN
ejpam-7045	212	22	)	)	PUNCT
ejpam-7045	212	23	is	be	AUX
ejpam-7045	212	24	said	say	VERB
ejpam-7045	212	25	to	to	PART
ejpam-7045	212	26	be	be	AUX
ejpam-7045	212	27	upper	upper	ADJ
ejpam-7045	212	28	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	212	29	,	,	PUNCT
ejpam-7045	212	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	212	31	at	at	ADP
ejpam-7045	212	32	a	a	DET
ejpam-7045	212	33	point	point	NOUN
ejpam-7045	212	34	x	x	PUNCT
ejpam-7045	212	35	of	of	ADP
ejpam-7045	212	36	x	x	PRON
ejpam-7045	212	37	if	if	SCONJ
ejpam-7045	212	38	for	for	ADP
ejpam-7045	212	39	each	each	DET
ejpam-7045	212	40	σ1σ2	σ1σ2	VERB
ejpam-7045	212	41	-	-	ADJ
ejpam-7045	212	42	open	open	ADJ
ejpam-7045	212	43	set	set	NOUN
ejpam-7045	212	44	v	v	NOUN
ejpam-7045	212	45	of	of	ADP
ejpam-7045	212	46	y	y	PRON
ejpam-7045	212	47	such	such	ADJ
ejpam-7045	212	48	that	that	SCONJ
ejpam-7045	212	49	f	f	PROPN
ejpam-7045	212	50	(	(	PUNCT
ejpam-7045	212	51	x	x	X
ejpam-7045	212	52	)	)	PUNCT
ejpam-7045	212	53	⊆	⊆	NUM
ejpam-7045	212	54	v	v	NOUN
ejpam-7045	212	55	,	,	PUNCT
ejpam-7045	212	56	there	there	PRON
ejpam-7045	212	57	exists	exist	VERB
ejpam-7045	212	58	a	a	DET
ejpam-7045	212	59	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	212	60	-	-	PUNCT
ejpam-7045	212	61	open	open	ADJ
ejpam-7045	212	62	set	set	NOUN
ejpam-7045	212	63	u	u	NOUN
ejpam-7045	212	64	of	of	ADP
ejpam-7045	212	65	x	x	PUNCT
ejpam-7045	212	66	containing	contain	VERB
ejpam-7045	212	67	x	x	PUNCT
ejpam-7045	212	68	such	such	ADJ
ejpam-7045	212	69	that	that	SCONJ
ejpam-7045	212	70	f	f	PROPN
ejpam-7045	212	71	(	(	PUNCT
ejpam-7045	212	72	u	u	NOUN
ejpam-7045	212	73	)	)	PUNCT
ejpam-7045	212	74	⊆	⊆	NUM
ejpam-7045	212	75	v	v	NOUN
ejpam-7045	212	76	.	.	PUNCT
ejpam-7045	213	1	a	a	DET
ejpam-7045	213	2	multifunction	multifunction	NOUN
ejpam-7045	213	3	f	f	NOUN
ejpam-7045	213	4	:	:	PUNCT
ejpam-7045	213	5	(	(	PUNCT
ejpam-7045	213	6	x	x	X
ejpam-7045	213	7	,	,	PUNCT
ejpam-7045	213	8	τ	τ	PROPN
ejpam-7045	213	9	,	,	PUNCT
ejpam-7045	213	10	i	i	NOUN
ejpam-7045	213	11	)	)	PUNCT
ejpam-7045	213	12	→	→	PUNCT
ejpam-7045	213	13	(	(	PUNCT
ejpam-7045	213	14	y	y	PROPN
ejpam-7045	213	15	,	,	PUNCT
ejpam-7045	213	16	σ1	σ1	PROPN
ejpam-7045	213	17	,	,	PUNCT
ejpam-7045	213	18	σ2	σ2	PROPN
ejpam-7045	213	19	)	)	PUNCT
ejpam-7045	213	20	is	be	AUX
ejpam-7045	213	21	said	say	VERB
ejpam-7045	213	22	to	to	PART
ejpam-7045	213	23	be	be	AUX
ejpam-7045	213	24	upper	upper	ADJ
ejpam-7045	213	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	213	26	,	,	PUNCT
ejpam-7045	213	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	213	28	if	if	SCONJ
ejpam-7045	213	29	f	f	PROPN
ejpam-7045	213	30	is	be	AUX
ejpam-7045	213	31	upper	upper	ADJ
ejpam-7045	213	32	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	213	33	,	,	PUNCT
ejpam-7045	213	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	213	35	at	at	ADP
ejpam-7045	213	36	each	each	DET
ejpam-7045	213	37	point	point	NOUN
ejpam-7045	213	38	of	of	ADP
ejpam-7045	213	39	x.	x.	NOUN
ejpam-7045	213	40	definition	definition	NOUN
ejpam-7045	213	41	4	4	NUM
ejpam-7045	213	42	.	.	PUNCT
ejpam-7045	214	1	[	[	X
ejpam-7045	214	2	20	20	NUM
ejpam-7045	214	3	]	]	PUNCT
ejpam-7045	214	4	a	a	DET
ejpam-7045	214	5	multifunction	multifunction	NOUN
ejpam-7045	214	6	f	f	NOUN
ejpam-7045	214	7	:	:	PUNCT
ejpam-7045	214	8	(	(	PUNCT
ejpam-7045	214	9	x	x	X
ejpam-7045	214	10	,	,	PUNCT
ejpam-7045	214	11	τ	τ	PROPN
ejpam-7045	214	12	,	,	PUNCT
ejpam-7045	214	13	i	i	NOUN
ejpam-7045	214	14	)	)	PUNCT
ejpam-7045	214	15	→	→	PUNCT
ejpam-7045	214	16	(	(	PUNCT
ejpam-7045	214	17	y	y	PROPN
ejpam-7045	214	18	,	,	PUNCT
ejpam-7045	214	19	σ1	σ1	PROPN
ejpam-7045	214	20	,	,	PUNCT
ejpam-7045	214	21	σ2	σ2	PROPN
ejpam-7045	214	22	)	)	PUNCT
ejpam-7045	214	23	is	be	AUX
ejpam-7045	214	24	said	say	VERB
ejpam-7045	214	25	to	to	PART
ejpam-7045	214	26	be	be	AUX
ejpam-7045	214	27	lower	low	ADJ
ejpam-7045	214	28	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	214	29	,	,	PUNCT
ejpam-7045	214	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	214	31	at	at	ADP
ejpam-7045	214	32	a	a	DET
ejpam-7045	214	33	point	point	NOUN
ejpam-7045	214	34	x	x	PUNCT
ejpam-7045	214	35	of	of	ADP
ejpam-7045	214	36	x	x	PRON
ejpam-7045	214	37	if	if	SCONJ
ejpam-7045	214	38	for	for	ADP
ejpam-7045	214	39	each	each	DET
ejpam-7045	214	40	σ1σ2	σ1σ2	VERB
ejpam-7045	214	41	-	-	ADJ
ejpam-7045	214	42	open	open	ADJ
ejpam-7045	214	43	set	set	NOUN
ejpam-7045	214	44	v	v	NOUN
ejpam-7045	214	45	of	of	ADP
ejpam-7045	214	46	y	y	PRON
ejpam-7045	214	47	such	such	ADJ
ejpam-7045	214	48	that	that	SCONJ
ejpam-7045	214	49	f	f	PROPN
ejpam-7045	214	50	(	(	PUNCT
ejpam-7045	214	51	x)∩v	x)∩v	PROPN
ejpam-7045	214	52	̸=	̸=	PROPN
ejpam-7045	214	53	∅	∅	NOUN
ejpam-7045	214	54	,	,	PUNCT
ejpam-7045	214	55	there	there	PRON
ejpam-7045	214	56	exists	exist	VERB
ejpam-7045	214	57	a	a	DET
ejpam-7045	214	58	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	214	59	-	-	PUNCT
ejpam-7045	214	60	open	open	ADJ
ejpam-7045	214	61	set	set	NOUN
ejpam-7045	214	62	u	u	NOUN
ejpam-7045	214	63	of	of	ADP
ejpam-7045	214	64	x	x	PUNCT
ejpam-7045	214	65	containing	contain	VERB
ejpam-7045	214	66	x	x	PUNCT
ejpam-7045	214	67	such	such	ADJ
ejpam-7045	214	68	that	that	SCONJ
ejpam-7045	214	69	f	f	PROPN
ejpam-7045	214	70	(	(	PUNCT
ejpam-7045	214	71	z)∩v	z)∩v	PROPN
ejpam-7045	214	72	̸=	̸=	PROPN
ejpam-7045	214	73	∅	∅	NOUN
ejpam-7045	214	74	for	for	ADP
ejpam-7045	214	75	every	every	DET
ejpam-7045	214	76	z	z	NOUN
ejpam-7045	214	77	∈	∈	PROPN
ejpam-7045	214	78	u	u	NOUN
ejpam-7045	214	79	.	.	PUNCT
ejpam-7045	215	1	a	a	DET
ejpam-7045	215	2	multifunction	multifunction	NOUN
ejpam-7045	215	3	f	f	NOUN
ejpam-7045	215	4	:	:	PUNCT
ejpam-7045	215	5	(	(	PUNCT
ejpam-7045	215	6	x	x	X
ejpam-7045	215	7	,	,	PUNCT
ejpam-7045	215	8	τ	τ	PROPN
ejpam-7045	215	9	,	,	PUNCT
ejpam-7045	215	10	i	i	NOUN
ejpam-7045	215	11	)	)	PUNCT
ejpam-7045	215	12	→	→	PUNCT
ejpam-7045	215	13	(	(	PUNCT
ejpam-7045	215	14	y	y	PROPN
ejpam-7045	215	15	,	,	PUNCT
ejpam-7045	215	16	σ1	σ1	PROPN
ejpam-7045	215	17	,	,	PUNCT
ejpam-7045	215	18	σ2	σ2	PROPN
ejpam-7045	215	19	)	)	PUNCT
ejpam-7045	215	20	is	be	AUX
ejpam-7045	215	21	said	say	VERB
ejpam-7045	215	22	to	to	PART
ejpam-7045	215	23	be	be	AUX
ejpam-7045	215	24	lower	low	ADJ
ejpam-7045	215	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	215	26	,	,	PUNCT
ejpam-7045	215	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7045	215	28	if	if	SCONJ
ejpam-7045	215	29	f	f	PROPN
ejpam-7045	215	30	is	be	AUX
ejpam-7045	215	31	lower	low	ADJ
ejpam-7045	215	32	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	215	33	,	,	PUNCT
ejpam-7045	215	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	215	35	at	at	ADP
ejpam-7045	215	36	each	each	DET
ejpam-7045	215	37	point	point	NOUN
ejpam-7045	215	38	of	of	ADP
ejpam-7045	215	39	x.	x.	NOUN
ejpam-7045	215	40	remark	remark	PROPN
ejpam-7045	215	41	1	1	NUM
ejpam-7045	215	42	.	.	PUNCT
ejpam-7045	215	43	for	for	ADP
ejpam-7045	215	44	a	a	DET
ejpam-7045	215	45	multifunction	multifunction	NOUN
ejpam-7045	215	46	f	f	NOUN
ejpam-7045	215	47	:	:	PUNCT
ejpam-7045	215	48	(	(	PUNCT
ejpam-7045	215	49	x	x	X
ejpam-7045	215	50	,	,	PUNCT
ejpam-7045	215	51	τ	τ	PROPN
ejpam-7045	215	52	,	,	PUNCT
ejpam-7045	215	53	i	i	NOUN
ejpam-7045	215	54	)	)	PUNCT
ejpam-7045	215	55	→	→	PUNCT
ejpam-7045	215	56	(	(	PUNCT
ejpam-7045	215	57	y	y	PROPN
ejpam-7045	215	58	,	,	PUNCT
ejpam-7045	215	59	σ1	σ1	PROPN
ejpam-7045	215	60	,	,	PUNCT
ejpam-7045	215	61	σ2	σ2	NOUN
ejpam-7045	215	62	)	)	PUNCT
ejpam-7045	215	63	,	,	PUNCT
ejpam-7045	215	64	the	the	DET
ejpam-7045	215	65	following	follow	VERB
ejpam-7045	215	66	implication	implication	NOUN
ejpam-7045	215	67	holds	hold	VERB
ejpam-7045	215	68	:	:	PUNCT
ejpam-7045	215	69	upper	upper	ADJ
ejpam-7045	215	70	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	215	71	,	,	PUNCT
ejpam-7045	215	72	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7045	215	73	⇒	⇒	NOUN
ejpam-7045	215	74	upper	upper	ADV
ejpam-7045	215	75	almost	almost	ADV
ejpam-7045	215	76	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	215	77	,	,	PUNCT
ejpam-7045	215	78	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7045	215	79	.	.	PUNCT
ejpam-7045	216	1	the	the	DET
ejpam-7045	216	2	converse	converse	NOUN
ejpam-7045	216	3	of	of	ADP
ejpam-7045	216	4	the	the	DET
ejpam-7045	216	5	implication	implication	NOUN
ejpam-7045	216	6	is	be	AUX
ejpam-7045	216	7	not	not	PART
ejpam-7045	216	8	true	true	ADJ
ejpam-7045	216	9	in	in	ADP
ejpam-7045	216	10	general	general	ADJ
ejpam-7045	216	11	.	.	PUNCT
ejpam-7045	217	1	we	we	PRON
ejpam-7045	217	2	give	give	VERB
ejpam-7045	217	3	an	an	DET
ejpam-7045	217	4	example	example	NOUN
ejpam-7045	217	5	for	for	ADP
ejpam-7045	217	6	the	the	DET
ejpam-7045	217	7	implication	implication	NOUN
ejpam-7045	217	8	as	as	SCONJ
ejpam-7045	217	9	follows	follow	VERB
ejpam-7045	217	10	.	.	PUNCT
ejpam-7045	217	11	example	example	NOUN
ejpam-7045	218	1	1	1	NUM
ejpam-7045	218	2	.	.	PUNCT
ejpam-7045	218	3	let	let	VERB
ejpam-7045	218	4	x	x	PUNCT
ejpam-7045	218	5	=	=	PRON
ejpam-7045	218	6	{	{	PUNCT
ejpam-7045	218	7	1	1	NUM
ejpam-7045	218	8	,	,	PUNCT
ejpam-7045	218	9	2	2	NUM
ejpam-7045	218	10	,	,	PUNCT
ejpam-7045	218	11	3	3	NUM
ejpam-7045	218	12	}	}	PUNCT
ejpam-7045	218	13	with	with	ADP
ejpam-7045	218	14	a	a	DET
ejpam-7045	218	15	topology	topology	NOUN
ejpam-7045	218	16	τ	τ	X
ejpam-7045	218	17	=	=	SYM
ejpam-7045	218	18	{	{	PUNCT
ejpam-7045	218	19	∅	∅	NOUN
ejpam-7045	218	20	,	,	PUNCT
ejpam-7045	218	21	{	{	PUNCT
ejpam-7045	218	22	1	1	NUM
ejpam-7045	218	23	,	,	PUNCT
ejpam-7045	218	24	2	2	NUM
ejpam-7045	218	25	}	}	PUNCT
ejpam-7045	218	26	,	,	PUNCT
ejpam-7045	218	27	x	x	NOUN
ejpam-7045	218	28	}	}	PUNCT
ejpam-7045	218	29	and	and	CCONJ
ejpam-7045	218	30	an	an	DET
ejpam-7045	218	31	ideal	ideal	NOUN
ejpam-7045	218	32	i	i	X
ejpam-7045	218	33	=	=	SYM
ejpam-7045	218	34	{	{	PUNCT
ejpam-7045	218	35	∅	∅	NOUN
ejpam-7045	218	36	,	,	PUNCT
ejpam-7045	218	37	{	{	PUNCT
ejpam-7045	218	38	3	3	NUM
ejpam-7045	218	39	}	}	PUNCT
ejpam-7045	218	40	}	}	PUNCT
ejpam-7045	218	41	.	.	PUNCT
ejpam-7045	219	1	let	let	VERB
ejpam-7045	219	2	y	y	PROPN
ejpam-7045	219	3	=	=	PUNCT
ejpam-7045	219	4	{	{	PUNCT
ejpam-7045	219	5	a	a	PRON
ejpam-7045	219	6	,	,	PUNCT
ejpam-7045	219	7	b	b	NOUN
ejpam-7045	219	8	,	,	PUNCT
ejpam-7045	219	9	c	c	NOUN
ejpam-7045	219	10	}	}	PUNCT
ejpam-7045	219	11	with	with	ADP
ejpam-7045	219	12	topologies	topology	NOUN
ejpam-7045	219	13	σ1	σ1	NOUN
ejpam-7045	219	14	=	=	SYM
ejpam-7045	219	15	{	{	PUNCT
ejpam-7045	219	16	∅	∅	NOUN
ejpam-7045	219	17	,	,	PUNCT
ejpam-7045	219	18	{	{	PUNCT
ejpam-7045	219	19	a	a	DET
ejpam-7045	219	20	,	,	PUNCT
ejpam-7045	219	21	b	b	NOUN
ejpam-7045	219	22	}	}	PUNCT
ejpam-7045	219	23	,	,	PUNCT
ejpam-7045	219	24	y	y	PROPN
ejpam-7045	219	25	}	}	PUNCT
ejpam-7045	219	26	and	and	CCONJ
ejpam-7045	219	27	σ2	σ2	PROPN
ejpam-7045	219	28	=	=	SYM
ejpam-7045	219	29	{	{	PUNCT
ejpam-7045	219	30	∅	∅	NOUN
ejpam-7045	219	31	,	,	PUNCT
ejpam-7045	219	32	{	{	PUNCT
ejpam-7045	219	33	c	c	NOUN
ejpam-7045	219	34	}	}	PUNCT
ejpam-7045	219	35	,	,	PUNCT
ejpam-7045	219	36	{	{	PUNCT
ejpam-7045	219	37	a	a	DET
ejpam-7045	219	38	,	,	PUNCT
ejpam-7045	219	39	b	b	NOUN
ejpam-7045	219	40	}	}	PUNCT
ejpam-7045	219	41	,	,	PUNCT
ejpam-7045	219	42	y	y	PROPN
ejpam-7045	219	43	}	}	PUNCT
ejpam-7045	219	44	.	.	PUNCT
ejpam-7045	220	1	a	a	DET
ejpam-7045	220	2	multifunction	multifunction	NOUN
ejpam-7045	220	3	f	f	NOUN
ejpam-7045	220	4	:	:	PUNCT
ejpam-7045	220	5	(	(	PUNCT
ejpam-7045	220	6	x	x	X
ejpam-7045	220	7	,	,	PUNCT
ejpam-7045	220	8	τ	τ	PROPN
ejpam-7045	220	9	,	,	PUNCT
ejpam-7045	220	10	i	i	NOUN
ejpam-7045	220	11	)	)	PUNCT
ejpam-7045	220	12	→	→	PUNCT
ejpam-7045	220	13	(	(	PUNCT
ejpam-7045	220	14	y	y	PROPN
ejpam-7045	220	15	,	,	PUNCT
ejpam-7045	220	16	σ1	σ1	PROPN
ejpam-7045	220	17	,	,	PUNCT
ejpam-7045	220	18	σ2	σ2	PROPN
ejpam-7045	220	19	)	)	PUNCT
ejpam-7045	220	20	is	be	AUX
ejpam-7045	220	21	defined	define	VERB
ejpam-7045	220	22	as	as	SCONJ
ejpam-7045	220	23	follows	follow	VERB
ejpam-7045	220	24	:	:	PUNCT
ejpam-7045	220	25	f	f	X
ejpam-7045	220	26	(	(	PUNCT
ejpam-7045	220	27	1	1	X
ejpam-7045	220	28	)	)	PUNCT
ejpam-7045	220	29	=	=	PRON
ejpam-7045	221	1	{	{	PUNCT
ejpam-7045	221	2	c	c	NOUN
ejpam-7045	221	3	}	}	PUNCT
ejpam-7045	221	4	and	and	CCONJ
ejpam-7045	221	5	f	f	X
ejpam-7045	221	6	(	(	PUNCT
ejpam-7045	221	7	2	2	NUM
ejpam-7045	221	8	)	)	PUNCT
ejpam-7045	221	9	=	=	PRON
ejpam-7045	221	10	{	{	PUNCT
ejpam-7045	221	11	a	a	NOUN
ejpam-7045	221	12	}	}	PUNCT
ejpam-7045	221	13	and	and	CCONJ
ejpam-7045	221	14	f	f	X
ejpam-7045	221	15	(	(	PUNCT
ejpam-7045	221	16	3	3	X
ejpam-7045	221	17	)	)	PUNCT
ejpam-7045	221	18	=	=	PRON
ejpam-7045	221	19	{	{	PUNCT
ejpam-7045	221	20	a	a	PRON
ejpam-7045	221	21	,	,	PUNCT
ejpam-7045	221	22	b	b	NOUN
ejpam-7045	221	23	}	}	PUNCT
ejpam-7045	221	24	.	.	PUNCT
ejpam-7045	222	1	then	then	ADV
ejpam-7045	222	2	f	f	PROPN
ejpam-7045	222	3	is	be	AUX
ejpam-7045	222	4	upper	upper	ADJ
ejpam-7045	222	5	almost	almost	ADV
ejpam-7045	222	6	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	222	7	,	,	PUNCT
ejpam-7045	222	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	222	9	but	but	CCONJ
ejpam-7045	222	10	f	f	PROPN
ejpam-7045	222	11	is	be	AUX
ejpam-7045	222	12	not	not	PART
ejpam-7045	222	13	upper	upper	ADJ
ejpam-7045	222	14	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	222	15	,	,	PUNCT
ejpam-7045	222	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	222	17	.	.	X
ejpam-7045	222	18	theorem	theorem	NOUN
ejpam-7045	222	19	4	4	NUM
ejpam-7045	222	20	.	.	X
ejpam-7045	222	21	for	for	ADP
ejpam-7045	222	22	a	a	DET
ejpam-7045	222	23	multifunction	multifunction	NOUN
ejpam-7045	222	24	f	f	NOUN
ejpam-7045	222	25	:	:	PUNCT
ejpam-7045	222	26	(	(	PUNCT
ejpam-7045	222	27	x	x	X
ejpam-7045	222	28	,	,	PUNCT
ejpam-7045	222	29	τ	τ	PROPN
ejpam-7045	222	30	,	,	PUNCT
ejpam-7045	222	31	i	i	NOUN
ejpam-7045	222	32	)	)	PUNCT
ejpam-7045	222	33	→	→	PUNCT
ejpam-7045	222	34	(	(	PUNCT
ejpam-7045	222	35	y	y	PROPN
ejpam-7045	222	36	,	,	PUNCT
ejpam-7045	222	37	σ1	σ1	PROPN
ejpam-7045	222	38	,	,	PUNCT
ejpam-7045	222	39	σ2	σ2	NOUN
ejpam-7045	222	40	)	)	PUNCT
ejpam-7045	222	41	,	,	PUNCT
ejpam-7045	222	42	the	the	DET
ejpam-7045	222	43	following	follow	VERB
ejpam-7045	222	44	properties	property	NOUN
ejpam-7045	222	45	are	be	AUX
ejpam-7045	222	46	equivalent	equivalent	ADJ
ejpam-7045	222	47	:	:	PUNCT
ejpam-7045	222	48	(	(	PUNCT
ejpam-7045	222	49	1	1	X
ejpam-7045	222	50	)	)	PUNCT
ejpam-7045	222	51	f	f	PROPN
ejpam-7045	222	52	is	be	AUX
ejpam-7045	222	53	lower	low	ADJ
ejpam-7045	222	54	almost	almost	ADV
ejpam-7045	222	55	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	222	56	,	,	PUNCT
ejpam-7045	222	57	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	222	58	;	;	PUNCT
ejpam-7045	222	59	(	(	PUNCT
ejpam-7045	222	60	2	2	X
ejpam-7045	222	61	)	)	PUNCT
ejpam-7045	222	62	for	for	ADP
ejpam-7045	222	63	each	each	DET
ejpam-7045	222	64	x	x	SYM
ejpam-7045	222	65	∈	∈	PROPN
ejpam-7045	222	66	x	x	X
ejpam-7045	222	67	and	and	CCONJ
ejpam-7045	222	68	each	each	DET
ejpam-7045	222	69	σ1σ2	σ1σ2	VERB
ejpam-7045	222	70	-	-	ADJ
ejpam-7045	222	71	open	open	ADJ
ejpam-7045	222	72	set	set	NOUN
ejpam-7045	222	73	v	v	NOUN
ejpam-7045	222	74	of	of	ADP
ejpam-7045	222	75	y	y	PRON
ejpam-7045	222	76	such	such	ADJ
ejpam-7045	222	77	that	that	SCONJ
ejpam-7045	222	78	f	f	PROPN
ejpam-7045	222	79	(	(	PUNCT
ejpam-7045	222	80	x	x	NOUN
ejpam-7045	222	81	)	)	PUNCT
ejpam-7045	222	82	∩	∩	NOUN
ejpam-7045	222	83	v	v	ADP
ejpam-7045	222	84	̸=	̸=	PROPN
ejpam-7045	222	85	∅	∅	NOUN
ejpam-7045	222	86	,	,	PUNCT
ejpam-7045	222	87	there	there	PRON
ejpam-7045	222	88	exists	exist	VERB
ejpam-7045	222	89	a	a	DET
ejpam-7045	222	90	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	222	91	-	-	PUNCT
ejpam-7045	222	92	open	open	ADJ
ejpam-7045	222	93	set	set	NOUN
ejpam-7045	222	94	u	u	NOUN
ejpam-7045	222	95	of	of	ADP
ejpam-7045	222	96	x	x	PUNCT
ejpam-7045	222	97	containing	contain	VERB
ejpam-7045	222	98	x	x	PUNCT
ejpam-7045	222	99	such	such	ADJ
ejpam-7045	222	100	that	that	SCONJ
ejpam-7045	222	101	u	u	NOUN
ejpam-7045	222	102	⊆	⊆	NUM
ejpam-7045	222	103	f−((σ1	f−((σ1	NOUN
ejpam-7045	222	104	,	,	PUNCT
ejpam-7045	222	105	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	222	106	)	)	PUNCT
ejpam-7045	222	107	)	)	PUNCT
ejpam-7045	222	108	;	;	PUNCT
ejpam-7045	223	1	c.	c.	PROPN
ejpam-7045	223	2	viriyapong	viriyapong	PROPN
ejpam-7045	223	3	,	,	PUNCT
ejpam-7045	223	4	a.	a.	PROPN
ejpam-7045	223	5	sama	sama	PROPN
ejpam-7045	223	6	-	-	PUNCT
ejpam-7045	223	7	ae	ae	PROPN
ejpam-7045	223	8	,	,	PUNCT
ejpam-7045	223	9	c.	c.	PROPN
ejpam-7045	223	10	boonpok	boonpok	PROPN
ejpam-7045	223	11	/	/	SYM
ejpam-7045	223	12	eur	eur	PROPN
ejpam-7045	223	13	.	.	PUNCT
ejpam-7045	224	1	j.	j.	PROPN
ejpam-7045	224	2	pure	pure	PROPN
ejpam-7045	224	3	appl	appl	PROPN
ejpam-7045	224	4	.	.	PROPN
ejpam-7045	224	5	math	math	PROPN
ejpam-7045	224	6	,	,	PUNCT
ejpam-7045	224	7	18	18	NUM
ejpam-7045	224	8	(	(	PUNCT
ejpam-7045	224	9	4	4	NUM
ejpam-7045	224	10	)	)	PUNCT
ejpam-7045	224	11	(	(	PUNCT
ejpam-7045	224	12	2025	2025	NUM
ejpam-7045	224	13	)	)	PUNCT
ejpam-7045	224	14	,	,	PUNCT
ejpam-7045	224	15	7045	7045	NUM
ejpam-7045	224	16	9	9	NUM
ejpam-7045	224	17	of	of	ADP
ejpam-7045	224	18	12	12	NUM
ejpam-7045	224	19	(	(	PUNCT
ejpam-7045	224	20	3	3	NUM
ejpam-7045	224	21	)	)	PUNCT
ejpam-7045	224	22	for	for	ADP
ejpam-7045	224	23	each	each	DET
ejpam-7045	224	24	x	x	SYM
ejpam-7045	224	25	∈	∈	PROPN
ejpam-7045	224	26	x	x	X
ejpam-7045	224	27	and	and	CCONJ
ejpam-7045	224	28	each	each	DET
ejpam-7045	224	29	(	(	PUNCT
ejpam-7045	224	30	σ1	σ1	PROPN
ejpam-7045	224	31	,	,	PUNCT
ejpam-7045	224	32	σ2)r	σ2)r	NOUN
ejpam-7045	224	33	-	-	PUNCT
ejpam-7045	224	34	open	open	ADJ
ejpam-7045	225	1	set	set	VERB
ejpam-7045	226	1	v	v	NOUN
ejpam-7045	226	2	of	of	ADP
ejpam-7045	226	3	y	y	PRON
ejpam-7045	226	4	such	such	ADJ
ejpam-7045	226	5	that	that	SCONJ
ejpam-7045	226	6	f	f	PROPN
ejpam-7045	226	7	(	(	PUNCT
ejpam-7045	226	8	x	x	NOUN
ejpam-7045	226	9	)	)	PUNCT
ejpam-7045	226	10	∩	∩	NOUN
ejpam-7045	226	11	v	v	ADP
ejpam-7045	226	12	̸=	̸=	PROPN
ejpam-7045	226	13	∅	∅	NOUN
ejpam-7045	227	1	,	,	PUNCT
ejpam-7045	227	2	there	there	PRON
ejpam-7045	227	3	exists	exist	VERB
ejpam-7045	227	4	a	a	DET
ejpam-7045	227	5	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	227	6	-	-	PUNCT
ejpam-7045	227	7	open	open	ADJ
ejpam-7045	227	8	set	set	NOUN
ejpam-7045	227	9	u	u	NOUN
ejpam-7045	227	10	of	of	ADP
ejpam-7045	227	11	x	x	PUNCT
ejpam-7045	227	12	containing	contain	VERB
ejpam-7045	227	13	x	x	PUNCT
ejpam-7045	227	14	such	such	ADJ
ejpam-7045	227	15	that	that	SCONJ
ejpam-7045	227	16	u	u	NOUN
ejpam-7045	227	17	⊆	⊆	NUM
ejpam-7045	227	18	f−(v	f−(v	NOUN
ejpam-7045	227	19	)	)	PUNCT
ejpam-7045	227	20	;	;	PUNCT
ejpam-7045	227	21	(	(	PUNCT
ejpam-7045	227	22	4	4	X
ejpam-7045	227	23	)	)	PUNCT
ejpam-7045	227	24	f−(v	f−(v	NOUN
ejpam-7045	227	25	)	)	PUNCT
ejpam-7045	227	26	is	be	AUX
ejpam-7045	227	27	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	227	28	-	-	PUNCT
ejpam-7045	227	29	open	open	ADJ
ejpam-7045	227	30	in	in	ADP
ejpam-7045	227	31	x	x	PUNCT
ejpam-7045	227	32	for	for	ADP
ejpam-7045	227	33	every	every	DET
ejpam-7045	227	34	(	(	PUNCT
ejpam-7045	227	35	σ1	σ1	PROPN
ejpam-7045	227	36	,	,	PUNCT
ejpam-7045	227	37	σ2)r	σ2)r	NOUN
ejpam-7045	227	38	-	-	PUNCT
ejpam-7045	227	39	open	open	ADJ
ejpam-7045	227	40	set	set	VERB
ejpam-7045	227	41	v	v	NOUN
ejpam-7045	227	42	of	of	ADP
ejpam-7045	227	43	y	y	PROPN
ejpam-7045	227	44	;	;	PUNCT
ejpam-7045	227	45	(	(	PUNCT
ejpam-7045	227	46	5	5	X
ejpam-7045	227	47	)	)	PUNCT
ejpam-7045	227	48	f+(k	f+(k	PROPN
ejpam-7045	227	49	)	)	PUNCT
ejpam-7045	227	50	is	be	AUX
ejpam-7045	227	51	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	227	52	-	-	PUNCT
ejpam-7045	227	53	closed	closed	ADJ
ejpam-7045	227	54	in	in	ADP
ejpam-7045	227	55	x	x	PUNCT
ejpam-7045	227	56	for	for	ADP
ejpam-7045	227	57	every	every	DET
ejpam-7045	227	58	(	(	PUNCT
ejpam-7045	227	59	σ1	σ1	PROPN
ejpam-7045	227	60	,	,	PUNCT
ejpam-7045	228	1	σ2)r	σ2)r	NOUN
ejpam-7045	228	2	-	-	PUNCT
ejpam-7045	228	3	closed	close	VERB
ejpam-7045	228	4	set	set	ADJ
ejpam-7045	228	5	k	k	PROPN
ejpam-7045	228	6	of	of	ADP
ejpam-7045	228	7	y	y	PROPN
ejpam-7045	228	8	;	;	PUNCT
ejpam-7045	228	9	(	(	PUNCT
ejpam-7045	228	10	6	6	X
ejpam-7045	228	11	)	)	PUNCT
ejpam-7045	228	12	f−(v	f−(v	NOUN
ejpam-7045	228	13	)	)	PUNCT
ejpam-7045	228	14	⊆	⊆	NUM
ejpam-7045	228	15	αint⋆(f−((σ1	αint⋆(f−((σ1	NOUN
ejpam-7045	228	16	,	,	PUNCT
ejpam-7045	228	17	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	228	18	)	)	PUNCT
ejpam-7045	228	19	)	)	PUNCT
ejpam-7045	228	20	)	)	PUNCT
ejpam-7045	229	1	for	for	ADP
ejpam-7045	229	2	every	every	DET
ejpam-7045	229	3	σ1σ2	σ1σ2	NOUN
ejpam-7045	229	4	-	-	ADJ
ejpam-7045	229	5	open	open	ADJ
ejpam-7045	229	6	set	set	NOUN
ejpam-7045	229	7	v	v	NOUN
ejpam-7045	229	8	of	of	ADP
ejpam-7045	229	9	y	y	PROPN
ejpam-7045	229	10	;	;	PUNCT
ejpam-7045	229	11	(	(	PUNCT
ejpam-7045	229	12	7	7	X
ejpam-7045	229	13	)	)	PUNCT
ejpam-7045	229	14	αcl⋆(f+((σ1	αcl⋆(f+((σ1	NOUN
ejpam-7045	229	15	,	,	PUNCT
ejpam-7045	229	16	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	229	17	)	)	PUNCT
ejpam-7045	229	18	)	)	PUNCT
ejpam-7045	229	19	)	)	PUNCT
ejpam-7045	230	1	⊆	⊆	NUM
ejpam-7045	230	2	f+(k	f+(k	NOUN
ejpam-7045	230	3	)	)	PUNCT
ejpam-7045	230	4	for	for	ADP
ejpam-7045	230	5	every	every	DET
ejpam-7045	230	6	σ1σ2	σ1σ2	NUM
ejpam-7045	230	7	-	-	PUNCT
ejpam-7045	230	8	closed	closed	ADJ
ejpam-7045	230	9	set	set	NOUN
ejpam-7045	230	10	k	k	PROPN
ejpam-7045	230	11	of	of	ADP
ejpam-7045	230	12	y	y	PROPN
ejpam-7045	230	13	;	;	PUNCT
ejpam-7045	230	14	(	(	PUNCT
ejpam-7045	230	15	8)	8)	NUM
ejpam-7045	230	16	αcl⋆(f+(σ1σ2	αcl⋆(f+(σ1σ2	NOUN
ejpam-7045	230	17	-	-	PUNCT
ejpam-7045	230	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	230	19	-	-	PUNCT
ejpam-7045	230	20	int(k	int(k	NOUN
ejpam-7045	230	21	)	)	PUNCT
ejpam-7045	230	22	)	)	PUNCT
ejpam-7045	230	23	)	)	PUNCT
ejpam-7045	230	24	)	)	PUNCT
ejpam-7045	230	25	⊆	⊆	NUM
ejpam-7045	230	26	f+(k	f+(k	NOUN
ejpam-7045	230	27	)	)	PUNCT
ejpam-7045	230	28	for	for	ADP
ejpam-7045	230	29	every	every	DET
ejpam-7045	230	30	σ1σ2	σ1σ2	NUM
ejpam-7045	230	31	-	-	PUNCT
ejpam-7045	230	32	closed	closed	ADJ
ejpam-7045	230	33	set	set	NOUN
ejpam-7045	230	34	k	k	PROPN
ejpam-7045	230	35	of	of	ADP
ejpam-7045	230	36	y	y	PROPN
ejpam-7045	230	37	;	;	PUNCT
ejpam-7045	230	38	(	(	PUNCT
ejpam-7045	230	39	9	9	X
ejpam-7045	230	40	)	)	PUNCT
ejpam-7045	230	41	αcl⋆(f+(σ1σ2	αcl⋆(f+(σ1σ2	NOUN
ejpam-7045	230	42	-	-	PUNCT
ejpam-7045	230	43	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	230	44	-	-	PUNCT
ejpam-7045	230	45	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	230	46	-	-	PUNCT
ejpam-7045	230	47	cl(b	cl(b	NOUN
ejpam-7045	230	48	)	)	PUNCT
ejpam-7045	230	49	)	)	PUNCT
ejpam-7045	230	50	)	)	PUNCT
ejpam-7045	230	51	)	)	PUNCT
ejpam-7045	230	52	)	)	PUNCT
ejpam-7045	231	1	⊆	⊆	X
ejpam-7045	231	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7045	231	3	-	-	PUNCT
ejpam-7045	231	4	cl(b	cl(b	NOUN
ejpam-7045	231	5	)	)	PUNCT
ejpam-7045	231	6	)	)	PUNCT
ejpam-7045	231	7	for	for	ADP
ejpam-7045	231	8	every	every	DET
ejpam-7045	231	9	subset	subset	NOUN
ejpam-7045	231	10	b	b	PROPN
ejpam-7045	231	11	of	of	ADP
ejpam-7045	231	12	y	y	PROPN
ejpam-7045	231	13	;	;	PUNCT
ejpam-7045	231	14	(	(	PUNCT
ejpam-7045	231	15	10	10	X
ejpam-7045	231	16	)	)	PUNCT
ejpam-7045	231	17	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7045	231	18	-	-	PUNCT
ejpam-7045	231	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	231	20	-	-	PUNCT
ejpam-7045	231	21	int(k	int(k	NOUN
ejpam-7045	231	22	)	)	PUNCT
ejpam-7045	231	23	)	)	PUNCT
ejpam-7045	231	24	)	)	PUNCT
ejpam-7045	231	25	)	)	PUNCT
ejpam-7045	231	26	)	)	PUNCT
ejpam-7045	231	27	)	)	PUNCT
ejpam-7045	231	28	⊆	⊆	NUM
ejpam-7045	231	29	f+(k	f+(k	NOUN
ejpam-7045	231	30	)	)	PUNCT
ejpam-7045	231	31	for	for	ADP
ejpam-7045	231	32	every	every	DET
ejpam-7045	231	33	σ1σ2	σ1σ2	NUM
ejpam-7045	231	34	-	-	PUNCT
ejpam-7045	231	35	closed	closed	ADJ
ejpam-7045	231	36	set	set	NOUN
ejpam-7045	231	37	k	k	PROPN
ejpam-7045	231	38	of	of	ADP
ejpam-7045	231	39	y	y	PROPN
ejpam-7045	231	40	;	;	PUNCT
ejpam-7045	231	41	(	(	PUNCT
ejpam-7045	231	42	11	11	X
ejpam-7045	231	43	)	)	PUNCT
ejpam-7045	231	44	cl⋆(int⋆(cl⋆(f+((σ1	cl⋆(int⋆(cl⋆(f+((σ1	NOUN
ejpam-7045	231	45	,	,	PUNCT
ejpam-7045	231	46	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	231	47	)	)	PUNCT
ejpam-7045	231	48	)	)	PUNCT
ejpam-7045	231	49	)	)	PUNCT
ejpam-7045	231	50	)	)	PUNCT
ejpam-7045	231	51	)	)	PUNCT
ejpam-7045	231	52	⊆	⊆	NUM
ejpam-7045	231	53	f+(k	f+(k	NOUN
ejpam-7045	231	54	)	)	PUNCT
ejpam-7045	231	55	for	for	ADP
ejpam-7045	231	56	every	every	DET
ejpam-7045	231	57	σ1σ2	σ1σ2	NUM
ejpam-7045	231	58	-	-	PUNCT
ejpam-7045	231	59	closed	closed	ADJ
ejpam-7045	231	60	set	set	NOUN
ejpam-7045	231	61	k	k	PROPN
ejpam-7045	231	62	of	of	ADP
ejpam-7045	231	63	y	y	PROPN
ejpam-7045	231	64	;	;	PUNCT
ejpam-7045	231	65	(	(	PUNCT
ejpam-7045	231	66	12	12	X
ejpam-7045	231	67	)	)	PUNCT
ejpam-7045	231	68	f−(v	f−(v	NOUN
ejpam-7045	231	69	)	)	PUNCT
ejpam-7045	231	70	⊆	⊆	NUM
ejpam-7045	231	71	int⋆(cl⋆(int⋆(f−((σ1	int⋆(cl⋆(int⋆(f−((σ1	NOUN
ejpam-7045	231	72	,	,	PUNCT
ejpam-7045	231	73	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	231	74	)	)	PUNCT
ejpam-7045	231	75	)	)	PUNCT
ejpam-7045	231	76	)	)	PUNCT
ejpam-7045	231	77	)	)	PUNCT
ejpam-7045	231	78	)	)	PUNCT
ejpam-7045	231	79	for	for	ADP
ejpam-7045	231	80	every	every	DET
ejpam-7045	231	81	σ1σ2	σ1σ2	NOUN
ejpam-7045	231	82	-	-	ADJ
ejpam-7045	231	83	open	open	ADJ
ejpam-7045	231	84	set	set	NOUN
ejpam-7045	231	85	v	v	NOUN
ejpam-7045	231	86	of	of	ADP
ejpam-7045	231	87	y	y	PROPN
ejpam-7045	231	88	.	.	PUNCT
ejpam-7045	232	1	proof	proof	NOUN
ejpam-7045	232	2	.	.	PUNCT
ejpam-7045	233	1	the	the	DET
ejpam-7045	233	2	proof	proof	NOUN
ejpam-7045	233	3	is	be	AUX
ejpam-7045	233	4	similar	similar	ADJ
ejpam-7045	233	5	to	to	ADP
ejpam-7045	233	6	that	that	PRON
ejpam-7045	233	7	of	of	ADP
ejpam-7045	233	8	theorem	theorem	ADJ
ejpam-7045	233	9	3	3	NUM
ejpam-7045	233	10	.	.	PUNCT
ejpam-7045	233	11	theorem	theorem	NOUN
ejpam-7045	233	12	5	5	NUM
ejpam-7045	233	13	.	.	X
ejpam-7045	233	14	for	for	ADP
ejpam-7045	233	15	a	a	DET
ejpam-7045	233	16	multifunction	multifunction	NOUN
ejpam-7045	233	17	f	f	NOUN
ejpam-7045	233	18	:	:	PUNCT
ejpam-7045	233	19	(	(	PUNCT
ejpam-7045	233	20	x	x	X
ejpam-7045	233	21	,	,	PUNCT
ejpam-7045	233	22	τ	τ	PROPN
ejpam-7045	233	23	,	,	PUNCT
ejpam-7045	233	24	i	i	NOUN
ejpam-7045	233	25	)	)	PUNCT
ejpam-7045	233	26	→	→	PUNCT
ejpam-7045	233	27	(	(	PUNCT
ejpam-7045	233	28	y	y	PROPN
ejpam-7045	233	29	,	,	PUNCT
ejpam-7045	233	30	σ1	σ1	PROPN
ejpam-7045	233	31	,	,	PUNCT
ejpam-7045	233	32	σ2	σ2	NOUN
ejpam-7045	233	33	)	)	PUNCT
ejpam-7045	233	34	,	,	PUNCT
ejpam-7045	233	35	the	the	DET
ejpam-7045	233	36	following	follow	VERB
ejpam-7045	233	37	properties	property	NOUN
ejpam-7045	233	38	are	be	AUX
ejpam-7045	233	39	equivalent	equivalent	ADJ
ejpam-7045	233	40	:	:	PUNCT
ejpam-7045	233	41	(	(	PUNCT
ejpam-7045	233	42	1	1	X
ejpam-7045	233	43	)	)	PUNCT
ejpam-7045	233	44	f	f	PROPN
ejpam-7045	233	45	is	be	AUX
ejpam-7045	233	46	upper	upper	ADJ
ejpam-7045	233	47	almost	almost	ADV
ejpam-7045	233	48	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	233	49	,	,	PUNCT
ejpam-7045	233	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	233	51	;	;	PUNCT
ejpam-7045	233	52	(	(	PUNCT
ejpam-7045	233	53	2	2	X
ejpam-7045	233	54	)	)	PUNCT
ejpam-7045	233	55	αcl⋆(f−(v	αcl⋆(f−(v	NUM
ejpam-7045	233	56	)	)	PUNCT
ejpam-7045	233	57	)	)	PUNCT
ejpam-7045	234	1	⊆	⊆	X
ejpam-7045	234	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7045	234	3	-	-	PUNCT
ejpam-7045	234	4	cl(v	cl(v	NOUN
ejpam-7045	234	5	)	)	PUNCT
ejpam-7045	234	6	)	)	PUNCT
ejpam-7045	234	7	for	for	ADP
ejpam-7045	234	8	every	every	DET
ejpam-7045	234	9	(	(	PUNCT
ejpam-7045	234	10	σ1	σ1	PROPN
ejpam-7045	234	11	,	,	PUNCT
ejpam-7045	234	12	σ2)β	σ2)β	NOUN
ejpam-7045	234	13	-	-	PUNCT
ejpam-7045	234	14	open	open	NOUN
ejpam-7045	234	15	set	set	NOUN
ejpam-7045	234	16	v	v	NOUN
ejpam-7045	234	17	of	of	ADP
ejpam-7045	234	18	y	y	PROPN
ejpam-7045	234	19	;	;	PUNCT
ejpam-7045	234	20	(	(	PUNCT
ejpam-7045	234	21	3	3	X
ejpam-7045	234	22	)	)	PUNCT
ejpam-7045	234	23	αcl⋆(f−(v	αcl⋆(f−(v	NUM
ejpam-7045	234	24	)	)	PUNCT
ejpam-7045	234	25	)	)	PUNCT
ejpam-7045	235	1	⊆	⊆	X
ejpam-7045	235	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7045	235	3	-	-	PUNCT
ejpam-7045	235	4	cl(v	cl(v	NOUN
ejpam-7045	235	5	)	)	PUNCT
ejpam-7045	235	6	)	)	PUNCT
ejpam-7045	235	7	for	for	ADP
ejpam-7045	235	8	every	every	DET
ejpam-7045	235	9	(	(	PUNCT
ejpam-7045	235	10	σ1	σ1	PROPN
ejpam-7045	235	11	,	,	PUNCT
ejpam-7045	235	12	σ2)s	σ2)s	NOUN
ejpam-7045	235	13	-	-	PUNCT
ejpam-7045	235	14	open	open	NOUN
ejpam-7045	235	15	set	set	NOUN
ejpam-7045	235	16	v	v	NOUN
ejpam-7045	235	17	of	of	ADP
ejpam-7045	235	18	y	y	PROPN
ejpam-7045	235	19	;	;	PUNCT
ejpam-7045	235	20	(	(	PUNCT
ejpam-7045	235	21	4	4	X
ejpam-7045	235	22	)	)	PUNCT
ejpam-7045	235	23	f+(v	f+(v	NOUN
ejpam-7045	235	24	)	)	PUNCT
ejpam-7045	236	1	⊆	⊆	X
ejpam-7045	236	2	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7045	236	3	-	-	PUNCT
ejpam-7045	236	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	236	5	-	-	PUNCT
ejpam-7045	236	6	cl(v	cl(v	NOUN
ejpam-7045	236	7	)	)	PUNCT
ejpam-7045	236	8	)	)	PUNCT
ejpam-7045	236	9	)	)	PUNCT
ejpam-7045	236	10	)	)	PUNCT
ejpam-7045	236	11	for	for	ADP
ejpam-7045	236	12	every	every	DET
ejpam-7045	236	13	(	(	PUNCT
ejpam-7045	236	14	σ1	σ1	PROPN
ejpam-7045	236	15	,	,	PUNCT
ejpam-7045	236	16	σ2)p	σ2)p	NOUN
ejpam-7045	236	17	-	-	PUNCT
ejpam-7045	236	18	open	open	NOUN
ejpam-7045	236	19	set	set	NOUN
ejpam-7045	236	20	v	v	NOUN
ejpam-7045	236	21	of	of	ADP
ejpam-7045	236	22	y	y	PROPN
ejpam-7045	236	23	.	.	PUNCT
ejpam-7045	237	1	proof	proof	NOUN
ejpam-7045	237	2	.	.	PUNCT
ejpam-7045	238	1	(	(	PUNCT
ejpam-7045	238	2	1	1	X
ejpam-7045	238	3	)	)	PUNCT
ejpam-7045	238	4	⇒	⇒	NOUN
ejpam-7045	238	5	(	(	PUNCT
ejpam-7045	238	6	2	2	NUM
ejpam-7045	238	7	):	):	PUNCT
ejpam-7045	238	8	let	let	VERB
ejpam-7045	238	9	v	v	PART
ejpam-7045	238	10	be	be	AUX
ejpam-7045	238	11	any	any	DET
ejpam-7045	238	12	(	(	PUNCT
ejpam-7045	238	13	σ1	σ1	PROPN
ejpam-7045	238	14	,	,	PUNCT
ejpam-7045	238	15	σ2)β	σ2)β	NOUN
ejpam-7045	238	16	-	-	PUNCT
ejpam-7045	238	17	open	open	ADJ
ejpam-7045	238	18	set	set	NOUN
ejpam-7045	238	19	of	of	ADP
ejpam-7045	238	20	y	y	PROPN
ejpam-7045	238	21	.	.	PUNCT
ejpam-7045	239	1	then	then	ADV
ejpam-7045	239	2	,	,	PUNCT
ejpam-7045	239	3	σ1σ2	σ1σ2	NOUN
ejpam-7045	239	4	-	-	NUM
ejpam-7045	239	5	cl(v	cl(v	NOUN
ejpam-7045	239	6	)	)	PUNCT
ejpam-7045	239	7	is	be	AUX
ejpam-7045	239	8	(	(	PUNCT
ejpam-7045	239	9	σ1	σ1	PROPN
ejpam-7045	239	10	,	,	PUNCT
ejpam-7045	239	11	σ2)rclosed	σ2)rclose	VERB
ejpam-7045	239	12	in	in	ADP
ejpam-7045	239	13	y	y	PROPN
ejpam-7045	239	14	.	.	PUNCT
ejpam-7045	240	1	since	since	SCONJ
ejpam-7045	240	2	f	f	PROPN
ejpam-7045	240	3	is	be	AUX
ejpam-7045	240	4	upper	upper	ADJ
ejpam-7045	240	5	almost	almost	ADV
ejpam-7045	240	6	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	240	7	,	,	PUNCT
ejpam-7045	240	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	240	9	,	,	PUNCT
ejpam-7045	240	10	by	by	ADP
ejpam-7045	240	11	theorem	theorem	NOUN
ejpam-7045	240	12	3	3	NUM
ejpam-7045	240	13	we	we	PRON
ejpam-7045	240	14	have	have	VERB
ejpam-7045	240	15	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7045	240	16	-	-	PUNCT
ejpam-7045	240	17	cl(v	cl(v	NOUN
ejpam-7045	240	18	)	)	PUNCT
ejpam-7045	240	19	)	)	PUNCT
ejpam-7045	240	20	is	be	AUX
ejpam-7045	240	21	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	240	22	-	-	PUNCT
ejpam-7045	240	23	closed	close	VERB
ejpam-7045	240	24	in	in	ADP
ejpam-7045	240	25	x	x	X
ejpam-7045	240	26	and	and	CCONJ
ejpam-7045	240	27	hence	hence	ADV
ejpam-7045	240	28	αcl⋆(f−(v	αcl⋆(f−(v	NUM
ejpam-7045	240	29	)	)	PUNCT
ejpam-7045	240	30	)	)	PUNCT
ejpam-7045	241	1	⊆	⊆	NUM
ejpam-7045	241	2	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7045	241	3	-	-	NOUN
ejpam-7045	241	4	cl(v	cl(v	NOUN
ejpam-7045	241	5	)	)	PUNCT
ejpam-7045	241	6	)	)	PUNCT
ejpam-7045	241	7	)	)	PUNCT
ejpam-7045	242	1	=	=	PUNCT
ejpam-7045	242	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7045	242	3	-	-	PUNCT
ejpam-7045	242	4	cl(v	cl(v	NOUN
ejpam-7045	242	5	)	)	PUNCT
ejpam-7045	242	6	)	)	PUNCT
ejpam-7045	242	7	.	.	PUNCT
ejpam-7045	243	1	(	(	PUNCT
ejpam-7045	243	2	2	2	X
ejpam-7045	243	3	)	)	PUNCT
ejpam-7045	243	4	⇒	⇒	NOUN
ejpam-7045	243	5	(	(	PUNCT
ejpam-7045	243	6	3	3	NUM
ejpam-7045	243	7	):	):	PUNCT
ejpam-7045	243	8	this	this	PRON
ejpam-7045	243	9	is	be	AUX
ejpam-7045	243	10	obvious	obvious	ADJ
ejpam-7045	243	11	since	since	SCONJ
ejpam-7045	243	12	every	every	DET
ejpam-7045	243	13	(	(	PUNCT
ejpam-7045	243	14	σ1	σ1	PROPN
ejpam-7045	243	15	,	,	PUNCT
ejpam-7045	243	16	σ2)s	σ2)s	NOUN
ejpam-7045	243	17	-	-	PUNCT
ejpam-7045	243	18	open	open	ADJ
ejpam-7045	243	19	set	set	NOUN
ejpam-7045	243	20	is	be	AUX
ejpam-7045	243	21	(	(	PUNCT
ejpam-7045	243	22	σ1	σ1	PROPN
ejpam-7045	243	23	,	,	PUNCT
ejpam-7045	243	24	σ2)β	σ2)β	NOUN
ejpam-7045	243	25	-	-	PUNCT
ejpam-7045	243	26	open	open	ADJ
ejpam-7045	243	27	.	.	PUNCT
ejpam-7045	244	1	(	(	PUNCT
ejpam-7045	244	2	3	3	X
ejpam-7045	244	3	)	)	PUNCT
ejpam-7045	244	4	⇒	⇒	NOUN
ejpam-7045	244	5	(	(	PUNCT
ejpam-7045	244	6	1	1	NUM
ejpam-7045	244	7	):	):	PUNCT
ejpam-7045	244	8	let	let	VERB
ejpam-7045	244	9	k	k	PRON
ejpam-7045	244	10	be	be	AUX
ejpam-7045	244	11	any	any	DET
ejpam-7045	244	12	(	(	PUNCT
ejpam-7045	244	13	σ1	σ1	NOUN
ejpam-7045	244	14	,	,	PUNCT
ejpam-7045	244	15	σ2)r	σ2)r	NOUN
ejpam-7045	244	16	-	-	PUNCT
ejpam-7045	244	17	closed	close	VERB
ejpam-7045	244	18	set	set	NOUN
ejpam-7045	244	19	of	of	ADP
ejpam-7045	244	20	y	y	PROPN
ejpam-7045	244	21	.	.	PUNCT
ejpam-7045	245	1	then	then	ADV
ejpam-7045	245	2	,	,	PUNCT
ejpam-7045	245	3	k	k	X
ejpam-7045	245	4	is	be	AUX
ejpam-7045	245	5	(	(	PUNCT
ejpam-7045	245	6	σ1	σ1	PROPN
ejpam-7045	245	7	,	,	PUNCT
ejpam-7045	245	8	σ2)s	σ2)s	NOUN
ejpam-7045	245	9	-	-	PUNCT
ejpam-7045	245	10	open	open	ADJ
ejpam-7045	245	11	in	in	ADP
ejpam-7045	245	12	y	y	PROPN
ejpam-7045	245	13	and	and	CCONJ
ejpam-7045	245	14	by	by	ADP
ejpam-7045	245	15	(	(	PUNCT
ejpam-7045	245	16	3	3	NUM
ejpam-7045	245	17	)	)	PUNCT
ejpam-7045	245	18	,	,	PUNCT
ejpam-7045	245	19	we	we	PRON
ejpam-7045	245	20	have	have	VERB
ejpam-7045	245	21	αcl⋆(f−(k	αcl⋆(f−(k	NOUN
ejpam-7045	245	22	)	)	PUNCT
ejpam-7045	245	23	)	)	PUNCT
ejpam-7045	246	1	⊆	⊆	X
ejpam-7045	246	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7045	246	3	-	-	PUNCT
ejpam-7045	246	4	cl(k	cl(k	NOUN
ejpam-7045	246	5	)	)	PUNCT
ejpam-7045	246	6	)	)	PUNCT
ejpam-7045	247	1	=	=	SYM
ejpam-7045	247	2	f−(k	f−(k	PROPN
ejpam-7045	247	3	)	)	PUNCT
ejpam-7045	247	4	.	.	PUNCT
ejpam-7045	248	1	thus	thus	ADV
ejpam-7045	248	2	,	,	PUNCT
ejpam-7045	248	3	f−(k	f−(k	PROPN
ejpam-7045	248	4	)	)	PUNCT
ejpam-7045	248	5	is	be	AUX
ejpam-7045	248	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	248	7	-	-	PUNCT
ejpam-7045	248	8	closed	close	VERB
ejpam-7045	248	9	in	in	ADP
ejpam-7045	248	10	x	x	PUNCT
ejpam-7045	248	11	and	and	CCONJ
ejpam-7045	248	12	hence	hence	ADV
ejpam-7045	248	13	f	f	PROPN
ejpam-7045	248	14	is	be	AUX
ejpam-7045	248	15	upper	upper	ADJ
ejpam-7045	248	16	almost	almost	ADV
ejpam-7045	248	17	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	248	18	,	,	PUNCT
ejpam-7045	248	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	248	20	by	by	ADP
ejpam-7045	248	21	theorem	theorem	NOUN
ejpam-7045	248	22	3	3	NUM
ejpam-7045	248	23	.	.	PUNCT
ejpam-7045	248	24	(	(	PUNCT
ejpam-7045	248	25	1	1	X
ejpam-7045	248	26	)	)	PUNCT
ejpam-7045	248	27	⇒	⇒	NOUN
ejpam-7045	248	28	(	(	PUNCT
ejpam-7045	248	29	4	4	NUM
ejpam-7045	248	30	):	):	PUNCT
ejpam-7045	248	31	let	let	VERB
ejpam-7045	248	32	v	v	PART
ejpam-7045	248	33	be	be	AUX
ejpam-7045	248	34	any	any	DET
ejpam-7045	248	35	(	(	PUNCT
ejpam-7045	248	36	σ1	σ1	PROPN
ejpam-7045	248	37	,	,	PUNCT
ejpam-7045	248	38	σ2)p	σ2)p	NOUN
ejpam-7045	248	39	-	-	PUNCT
ejpam-7045	248	40	open	open	ADJ
ejpam-7045	248	41	set	set	NOUN
ejpam-7045	248	42	of	of	ADP
ejpam-7045	248	43	y	y	PROPN
ejpam-7045	248	44	.	.	PUNCT
ejpam-7045	249	1	then	then	ADV
ejpam-7045	249	2	,	,	PUNCT
ejpam-7045	249	3	we	we	PRON
ejpam-7045	249	4	have	have	VERB
ejpam-7045	249	5	σ1σ2	σ1σ2	NOUN
ejpam-7045	249	6	-	-	PUNCT
ejpam-7045	249	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	249	8	-	-	PUNCT
ejpam-7045	249	9	cl(v	cl(v	NOUN
ejpam-7045	249	10	)	)	PUNCT
ejpam-7045	249	11	)	)	PUNCT
ejpam-7045	250	1	is	be	AUX
ejpam-7045	250	2	(	(	PUNCT
ejpam-7045	250	3	σ1	σ1	NOUN
ejpam-7045	250	4	,	,	PUNCT
ejpam-7045	250	5	σ2)r	σ2)r	NOUN
ejpam-7045	250	6	-	-	PUNCT
ejpam-7045	250	7	open	open	ADJ
ejpam-7045	250	8	in	in	ADP
ejpam-7045	250	9	y	y	PROPN
ejpam-7045	250	10	.	.	PUNCT
ejpam-7045	251	1	since	since	SCONJ
ejpam-7045	251	2	f	f	PROPN
ejpam-7045	251	3	is	be	AUX
ejpam-7045	251	4	upper	upper	ADJ
ejpam-7045	251	5	almost	almost	ADV
ejpam-7045	251	6	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	251	7	,	,	PUNCT
ejpam-7045	251	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	251	9	,	,	PUNCT
ejpam-7045	251	10	by	by	ADP
ejpam-7045	251	11	theorem	theorem	NOUN
ejpam-7045	251	12	3	3	NUM
ejpam-7045	251	13	we	we	PRON
ejpam-7045	251	14	have	have	AUX
ejpam-7045	251	15	f+(σ1σ2	f+(σ1σ2	VERB
ejpam-7045	251	16	-	-	PUNCT
ejpam-7045	251	17	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	251	18	-	-	PUNCT
ejpam-7045	251	19	cl(v	cl(v	NOUN
ejpam-7045	251	20	)	)	PUNCT
ejpam-7045	251	21	)	)	PUNCT
ejpam-7045	251	22	)	)	PUNCT
ejpam-7045	251	23	is	be	AUX
ejpam-7045	251	24	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	251	25	-	-	PUNCT
ejpam-7045	251	26	open	open	ADJ
ejpam-7045	251	27	in	in	ADP
ejpam-7045	251	28	x.	x.	PROPN
ejpam-7045	251	29	thus	thus	ADV
ejpam-7045	251	30	,	,	PUNCT
ejpam-7045	251	31	f+(v	f+(v	PROPN
ejpam-7045	251	32	)	)	PUNCT
ejpam-7045	252	1	⊆	⊆	X
ejpam-7045	252	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7045	252	3	-	-	PUNCT
ejpam-7045	252	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	252	5	-	-	PUNCT
ejpam-7045	252	6	cl(v	cl(v	NOUN
ejpam-7045	252	7	)	)	PUNCT
ejpam-7045	252	8	)	)	PUNCT
ejpam-7045	252	9	)	)	PUNCT
ejpam-7045	253	1	=	=	PUNCT
ejpam-7045	253	2	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7045	253	3	-	-	PUNCT
ejpam-7045	253	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	253	5	-	-	PUNCT
ejpam-7045	253	6	cl(v	cl(v	NOUN
ejpam-7045	253	7	)	)	PUNCT
ejpam-7045	253	8	)	)	PUNCT
ejpam-7045	253	9	)	)	PUNCT
ejpam-7045	253	10	)	)	PUNCT
ejpam-7045	253	11	.	.	PUNCT
ejpam-7045	254	1	c.	c.	PROPN
ejpam-7045	254	2	viriyapong	viriyapong	PROPN
ejpam-7045	254	3	,	,	PUNCT
ejpam-7045	254	4	a.	a.	PROPN
ejpam-7045	254	5	sama	sama	PROPN
ejpam-7045	254	6	-	-	PUNCT
ejpam-7045	254	7	ae	ae	PROPN
ejpam-7045	254	8	,	,	PUNCT
ejpam-7045	254	9	c.	c.	PROPN
ejpam-7045	254	10	boonpok	boonpok	PROPN
ejpam-7045	254	11	/	/	SYM
ejpam-7045	254	12	eur	eur	PROPN
ejpam-7045	254	13	.	.	PUNCT
ejpam-7045	255	1	j.	j.	PROPN
ejpam-7045	255	2	pure	pure	PROPN
ejpam-7045	255	3	appl	appl	PROPN
ejpam-7045	255	4	.	.	PROPN
ejpam-7045	255	5	math	math	PROPN
ejpam-7045	255	6	,	,	PUNCT
ejpam-7045	255	7	18	18	NUM
ejpam-7045	255	8	(	(	PUNCT
ejpam-7045	255	9	4	4	NUM
ejpam-7045	255	10	)	)	PUNCT
ejpam-7045	255	11	(	(	PUNCT
ejpam-7045	255	12	2025	2025	NUM
ejpam-7045	255	13	)	)	PUNCT
ejpam-7045	255	14	,	,	PUNCT
ejpam-7045	255	15	7045	7045	NUM
ejpam-7045	255	16	10	10	NUM
ejpam-7045	255	17	of	of	ADP
ejpam-7045	255	18	12	12	NUM
ejpam-7045	255	19	(	(	PUNCT
ejpam-7045	255	20	4	4	NUM
ejpam-7045	255	21	)	)	PUNCT
ejpam-7045	255	22	⇒	⇒	NOUN
ejpam-7045	255	23	(	(	PUNCT
ejpam-7045	255	24	1	1	NUM
ejpam-7045	255	25	):	):	PUNCT
ejpam-7045	255	26	let	let	VERB
ejpam-7045	255	27	v	v	PART
ejpam-7045	255	28	be	be	AUX
ejpam-7045	255	29	any	any	DET
ejpam-7045	255	30	(	(	PUNCT
ejpam-7045	255	31	σ1	σ1	NOUN
ejpam-7045	255	32	,	,	PUNCT
ejpam-7045	255	33	σ2)r	σ2)r	NOUN
ejpam-7045	255	34	-	-	PUNCT
ejpam-7045	255	35	open	open	ADJ
ejpam-7045	255	36	set	set	NOUN
ejpam-7045	255	37	of	of	ADP
ejpam-7045	255	38	y	y	PROPN
ejpam-7045	255	39	.	.	PUNCT
ejpam-7045	256	1	then	then	ADV
ejpam-7045	256	2	,	,	PUNCT
ejpam-7045	256	3	v	v	NOUN
ejpam-7045	256	4	is	be	AUX
ejpam-7045	256	5	(	(	PUNCT
ejpam-7045	256	6	σ1	σ1	PROPN
ejpam-7045	256	7	,	,	PUNCT
ejpam-7045	256	8	σ2)p	σ2)p	NOUN
ejpam-7045	256	9	-	-	PUNCT
ejpam-7045	256	10	open	open	ADJ
ejpam-7045	256	11	in	in	ADP
ejpam-7045	256	12	y	y	PROPN
ejpam-7045	256	13	and	and	CCONJ
ejpam-7045	256	14	by	by	ADP
ejpam-7045	256	15	(	(	PUNCT
ejpam-7045	256	16	4	4	NUM
ejpam-7045	256	17	)	)	PUNCT
ejpam-7045	256	18	,	,	PUNCT
ejpam-7045	256	19	f+(v	f+(v	PROPN
ejpam-7045	256	20	)	)	PUNCT
ejpam-7045	257	1	⊆	⊆	X
ejpam-7045	257	2	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7045	257	3	-	-	PUNCT
ejpam-7045	257	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	257	5	-	-	PUNCT
ejpam-7045	257	6	cl(v	cl(v	NOUN
ejpam-7045	257	7	)	)	PUNCT
ejpam-7045	257	8	)	)	PUNCT
ejpam-7045	257	9	)	)	PUNCT
ejpam-7045	257	10	)	)	PUNCT
ejpam-7045	257	11	=	=	PUNCT
ejpam-7045	257	12	αint⋆(f+(v	αint⋆(f+(v	PROPN
ejpam-7045	257	13	)	)	PUNCT
ejpam-7045	257	14	)	)	PUNCT
ejpam-7045	257	15	.	.	PUNCT
ejpam-7045	258	1	this	this	PRON
ejpam-7045	258	2	shows	show	VERB
ejpam-7045	258	3	that	that	SCONJ
ejpam-7045	258	4	f+(v	f+(v	PROPN
ejpam-7045	258	5	)	)	PUNCT
ejpam-7045	258	6	is	be	AUX
ejpam-7045	258	7	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	258	8	-	-	PUNCT
ejpam-7045	258	9	open	open	ADJ
ejpam-7045	258	10	in	in	ADP
ejpam-7045	258	11	x.	x.	NOUN
ejpam-7045	258	12	it	it	PRON
ejpam-7045	258	13	follows	follow	VERB
ejpam-7045	258	14	from	from	ADP
ejpam-7045	258	15	theorem	theorem	ADJ
ejpam-7045	258	16	3	3	NUM
ejpam-7045	258	17	that	that	SCONJ
ejpam-7045	258	18	f	f	PROPN
ejpam-7045	258	19	is	be	AUX
ejpam-7045	258	20	upper	upper	ADJ
ejpam-7045	258	21	almost	almost	ADV
ejpam-7045	258	22	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	258	23	,	,	PUNCT
ejpam-7045	258	24	σ2)continuous	σ2)continuous	ADJ
ejpam-7045	258	25	.	.	PUNCT
ejpam-7045	259	1	theorem	theorem	VERB
ejpam-7045	259	2	6	6	NUM
ejpam-7045	259	3	.	.	PUNCT
ejpam-7045	259	4	for	for	ADP
ejpam-7045	259	5	a	a	DET
ejpam-7045	259	6	multifunction	multifunction	NOUN
ejpam-7045	260	1	f	f	NOUN
ejpam-7045	260	2	:	:	PUNCT
ejpam-7045	260	3	(	(	PUNCT
ejpam-7045	260	4	x	x	X
ejpam-7045	260	5	,	,	PUNCT
ejpam-7045	260	6	τ	τ	PROPN
ejpam-7045	260	7	,	,	PUNCT
ejpam-7045	260	8	i	i	NOUN
ejpam-7045	260	9	)	)	PUNCT
ejpam-7045	260	10	→	→	PUNCT
ejpam-7045	260	11	(	(	PUNCT
ejpam-7045	260	12	y	y	PROPN
ejpam-7045	260	13	,	,	PUNCT
ejpam-7045	260	14	σ1	σ1	PROPN
ejpam-7045	260	15	,	,	PUNCT
ejpam-7045	260	16	σ2	σ2	NOUN
ejpam-7045	260	17	)	)	PUNCT
ejpam-7045	260	18	,	,	PUNCT
ejpam-7045	260	19	the	the	DET
ejpam-7045	260	20	following	follow	VERB
ejpam-7045	260	21	properties	property	NOUN
ejpam-7045	260	22	are	be	AUX
ejpam-7045	260	23	equivalent	equivalent	ADJ
ejpam-7045	260	24	:	:	PUNCT
ejpam-7045	260	25	(	(	PUNCT
ejpam-7045	260	26	1	1	X
ejpam-7045	260	27	)	)	PUNCT
ejpam-7045	260	28	f	f	PROPN
ejpam-7045	260	29	is	be	AUX
ejpam-7045	260	30	lower	low	ADJ
ejpam-7045	260	31	almost	almost	ADV
ejpam-7045	260	32	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	260	33	,	,	PUNCT
ejpam-7045	260	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	260	35	;	;	PUNCT
ejpam-7045	260	36	(	(	PUNCT
ejpam-7045	260	37	2	2	X
ejpam-7045	260	38	)	)	PUNCT
ejpam-7045	260	39	αcl⋆(f+(v	αcl⋆(f+(v	PROPN
ejpam-7045	260	40	)	)	PUNCT
ejpam-7045	260	41	)	)	PUNCT
ejpam-7045	261	1	⊆	⊆	X
ejpam-7045	261	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7045	261	3	-	-	PUNCT
ejpam-7045	261	4	cl(v	cl(v	NOUN
ejpam-7045	261	5	)	)	PUNCT
ejpam-7045	261	6	)	)	PUNCT
ejpam-7045	261	7	for	for	ADP
ejpam-7045	261	8	every	every	DET
ejpam-7045	261	9	(	(	PUNCT
ejpam-7045	261	10	σ1	σ1	PROPN
ejpam-7045	261	11	,	,	PUNCT
ejpam-7045	261	12	σ2)β	σ2)β	NOUN
ejpam-7045	261	13	-	-	PUNCT
ejpam-7045	261	14	open	open	NOUN
ejpam-7045	261	15	set	set	NOUN
ejpam-7045	261	16	v	v	NOUN
ejpam-7045	261	17	of	of	ADP
ejpam-7045	261	18	y	y	PROPN
ejpam-7045	261	19	;	;	PUNCT
ejpam-7045	261	20	(	(	PUNCT
ejpam-7045	261	21	3	3	X
ejpam-7045	261	22	)	)	PUNCT
ejpam-7045	261	23	αcl⋆(f+(v	αcl⋆(f+(v	PROPN
ejpam-7045	261	24	)	)	PUNCT
ejpam-7045	261	25	)	)	PUNCT
ejpam-7045	262	1	⊆	⊆	X
ejpam-7045	262	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7045	262	3	-	-	PUNCT
ejpam-7045	262	4	cl(v	cl(v	NOUN
ejpam-7045	262	5	)	)	PUNCT
ejpam-7045	262	6	)	)	PUNCT
ejpam-7045	262	7	for	for	SCONJ
ejpam-7045	262	8	every	every	DET
ejpam-7045	262	9	(	(	PUNCT
ejpam-7045	262	10	σ1	σ1	PROPN
ejpam-7045	262	11	,	,	PUNCT
ejpam-7045	262	12	σ2)s	σ2)s	NOUN
ejpam-7045	262	13	-	-	PUNCT
ejpam-7045	262	14	open	open	NOUN
ejpam-7045	262	15	set	set	NOUN
ejpam-7045	262	16	v	v	NOUN
ejpam-7045	262	17	of	of	ADP
ejpam-7045	262	18	y	y	PROPN
ejpam-7045	262	19	;	;	PUNCT
ejpam-7045	262	20	(	(	PUNCT
ejpam-7045	262	21	4	4	X
ejpam-7045	262	22	)	)	PUNCT
ejpam-7045	262	23	f−(v	f−(v	NOUN
ejpam-7045	262	24	)	)	PUNCT
ejpam-7045	262	25	⊆	⊆	NUM
ejpam-7045	262	26	αint⋆(f−(σ1σ2	αint⋆(f−(σ1σ2	NUM
ejpam-7045	262	27	-	-	PUNCT
ejpam-7045	262	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	262	29	-	-	PUNCT
ejpam-7045	262	30	cl(v	cl(v	NOUN
ejpam-7045	262	31	)	)	PUNCT
ejpam-7045	262	32	)	)	PUNCT
ejpam-7045	262	33	)	)	PUNCT
ejpam-7045	262	34	)	)	PUNCT
ejpam-7045	262	35	for	for	ADP
ejpam-7045	262	36	every	every	DET
ejpam-7045	262	37	(	(	PUNCT
ejpam-7045	262	38	σ1	σ1	PROPN
ejpam-7045	262	39	,	,	PUNCT
ejpam-7045	262	40	σ2)p	σ2)p	NOUN
ejpam-7045	262	41	-	-	PUNCT
ejpam-7045	262	42	open	open	NOUN
ejpam-7045	262	43	set	set	NOUN
ejpam-7045	262	44	v	v	NOUN
ejpam-7045	262	45	of	of	ADP
ejpam-7045	262	46	y	y	PROPN
ejpam-7045	262	47	.	.	PUNCT
ejpam-7045	263	1	proof	proof	NOUN
ejpam-7045	263	2	.	.	PUNCT
ejpam-7045	264	1	the	the	DET
ejpam-7045	264	2	proof	proof	NOUN
ejpam-7045	264	3	is	be	AUX
ejpam-7045	264	4	similar	similar	ADJ
ejpam-7045	264	5	to	to	ADP
ejpam-7045	264	6	that	that	PRON
ejpam-7045	264	7	of	of	ADP
ejpam-7045	264	8	theorem	theorem	NOUN
ejpam-7045	264	9	5	5	NUM
ejpam-7045	264	10	.	.	PUNCT
ejpam-7045	264	11	definition	definition	NOUN
ejpam-7045	264	12	5	5	NUM
ejpam-7045	264	13	.	.	PUNCT
ejpam-7045	265	1	a	a	DET
ejpam-7045	265	2	function	function	NOUN
ejpam-7045	265	3	f	f	NOUN
ejpam-7045	265	4	:	:	PUNCT
ejpam-7045	265	5	(	(	PUNCT
ejpam-7045	265	6	x	x	X
ejpam-7045	265	7	,	,	PUNCT
ejpam-7045	265	8	τ	τ	PROPN
ejpam-7045	265	9	,	,	PUNCT
ejpam-7045	265	10	i	i	NOUN
ejpam-7045	265	11	)	)	PUNCT
ejpam-7045	265	12	→	→	PUNCT
ejpam-7045	265	13	(	(	PUNCT
ejpam-7045	265	14	y	y	PROPN
ejpam-7045	265	15	,	,	PUNCT
ejpam-7045	265	16	σ1	σ1	PROPN
ejpam-7045	265	17	,	,	PUNCT
ejpam-7045	265	18	σ2	σ2	PROPN
ejpam-7045	265	19	)	)	PUNCT
ejpam-7045	265	20	is	be	AUX
ejpam-7045	265	21	said	say	VERB
ejpam-7045	265	22	to	to	PART
ejpam-7045	265	23	be	be	AUX
ejpam-7045	265	24	almost	almost	ADV
ejpam-7045	265	25	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7045	265	26	,	,	PUNCT
ejpam-7045	265	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7045	265	28	if	if	SCONJ
ejpam-7045	265	29	f−1(v	f−1(v	PROPN
ejpam-7045	265	30	)	)	PUNCT
ejpam-7045	265	31	is	be	AUX
ejpam-7045	265	32	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	265	33	-	-	PUNCT
ejpam-7045	265	34	open	open	ADJ
ejpam-7045	265	35	in	in	ADP
ejpam-7045	265	36	x	x	PUNCT
ejpam-7045	265	37	for	for	ADP
ejpam-7045	265	38	every	every	DET
ejpam-7045	265	39	(	(	PUNCT
ejpam-7045	265	40	σ1	σ1	PROPN
ejpam-7045	265	41	,	,	PUNCT
ejpam-7045	265	42	σ2)r	σ2)r	NOUN
ejpam-7045	265	43	-	-	PUNCT
ejpam-7045	265	44	open	open	ADJ
ejpam-7045	265	45	set	set	VERB
ejpam-7045	265	46	v	v	NOUN
ejpam-7045	265	47	of	of	ADP
ejpam-7045	265	48	y	y	PROPN
ejpam-7045	265	49	.	.	PUNCT
ejpam-7045	266	1	corollary	corollary	ADJ
ejpam-7045	266	2	1	1	NUM
ejpam-7045	266	3	.	.	PUNCT
ejpam-7045	267	1	for	for	ADP
ejpam-7045	267	2	a	a	DET
ejpam-7045	267	3	function	function	NOUN
ejpam-7045	267	4	f	f	NOUN
ejpam-7045	267	5	:	:	PUNCT
ejpam-7045	267	6	(	(	PUNCT
ejpam-7045	267	7	x	x	X
ejpam-7045	267	8	,	,	PUNCT
ejpam-7045	267	9	τ	τ	PROPN
ejpam-7045	267	10	,	,	PUNCT
ejpam-7045	267	11	i	i	NOUN
ejpam-7045	267	12	)	)	PUNCT
ejpam-7045	267	13	→	→	PUNCT
ejpam-7045	267	14	(	(	PUNCT
ejpam-7045	267	15	y	y	PROPN
ejpam-7045	267	16	,	,	PUNCT
ejpam-7045	267	17	σ1	σ1	PROPN
ejpam-7045	267	18	,	,	PUNCT
ejpam-7045	267	19	σ2	σ2	NOUN
ejpam-7045	267	20	)	)	PUNCT
ejpam-7045	267	21	,	,	PUNCT
ejpam-7045	267	22	the	the	DET
ejpam-7045	267	23	following	follow	VERB
ejpam-7045	267	24	properties	property	NOUN
ejpam-7045	267	25	are	be	AUX
ejpam-7045	267	26	equivalent	equivalent	ADJ
ejpam-7045	267	27	:	:	PUNCT
ejpam-7045	267	28	(	(	PUNCT
ejpam-7045	267	29	1	1	X
ejpam-7045	267	30	)	)	PUNCT
ejpam-7045	267	31	f	f	NOUN
ejpam-7045	267	32	is	be	AUX
ejpam-7045	267	33	almost	almost	ADV
ejpam-7045	267	34	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	267	35	,	,	PUNCT
ejpam-7045	267	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	267	37	;	;	PUNCT
ejpam-7045	267	38	(	(	PUNCT
ejpam-7045	267	39	2	2	X
ejpam-7045	267	40	)	)	PUNCT
ejpam-7045	267	41	for	for	ADP
ejpam-7045	267	42	each	each	DET
ejpam-7045	267	43	x	x	SYM
ejpam-7045	267	44	∈	∈	PROPN
ejpam-7045	267	45	x	x	X
ejpam-7045	267	46	and	and	CCONJ
ejpam-7045	267	47	each	each	DET
ejpam-7045	267	48	σ1σ2	σ1σ2	VERB
ejpam-7045	267	49	-	-	ADJ
ejpam-7045	267	50	open	open	ADJ
ejpam-7045	267	51	set	set	NOUN
ejpam-7045	267	52	v	v	NOUN
ejpam-7045	267	53	of	of	ADP
ejpam-7045	267	54	y	y	NOUN
ejpam-7045	267	55	containing	contain	VERB
ejpam-7045	267	56	f(x	f(x	PROPN
ejpam-7045	267	57	)	)	PUNCT
ejpam-7045	267	58	,	,	PUNCT
ejpam-7045	267	59	there	there	PRON
ejpam-7045	267	60	exists	exist	VERB
ejpam-7045	267	61	a	a	DET
ejpam-7045	267	62	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	267	63	-	-	PUNCT
ejpam-7045	267	64	open	open	ADJ
ejpam-7045	267	65	set	set	NOUN
ejpam-7045	267	66	u	u	NOUN
ejpam-7045	267	67	of	of	ADP
ejpam-7045	267	68	x	x	PUNCT
ejpam-7045	267	69	containing	contain	VERB
ejpam-7045	267	70	x	x	PUNCT
ejpam-7045	267	71	such	such	ADJ
ejpam-7045	267	72	that	that	DET
ejpam-7045	267	73	f(u	f(u	PROPN
ejpam-7045	267	74	)	)	PUNCT
ejpam-7045	267	75	⊆	⊆	NUM
ejpam-7045	267	76	(	(	PUNCT
ejpam-7045	267	77	σ1	σ1	PROPN
ejpam-7045	267	78	,	,	PUNCT
ejpam-7045	267	79	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	267	80	)	)	PUNCT
ejpam-7045	267	81	;	;	PUNCT
ejpam-7045	267	82	(	(	PUNCT
ejpam-7045	267	83	3	3	X
ejpam-7045	267	84	)	)	PUNCT
ejpam-7045	267	85	for	for	ADP
ejpam-7045	267	86	each	each	DET
ejpam-7045	267	87	x	x	SYM
ejpam-7045	267	88	∈	∈	PROPN
ejpam-7045	267	89	x	x	X
ejpam-7045	267	90	and	and	CCONJ
ejpam-7045	267	91	each	each	DET
ejpam-7045	267	92	(	(	PUNCT
ejpam-7045	267	93	σ1	σ1	PROPN
ejpam-7045	267	94	,	,	PUNCT
ejpam-7045	267	95	σ2)r	σ2)r	NOUN
ejpam-7045	267	96	-	-	PUNCT
ejpam-7045	267	97	open	open	ADJ
ejpam-7045	267	98	set	set	VERB
ejpam-7045	267	99	v	v	NOUN
ejpam-7045	267	100	of	of	ADP
ejpam-7045	267	101	y	y	NOUN
ejpam-7045	267	102	containing	contain	VERB
ejpam-7045	267	103	f(x	f(x	PROPN
ejpam-7045	267	104	)	)	PUNCT
ejpam-7045	267	105	,	,	PUNCT
ejpam-7045	267	106	there	there	PRON
ejpam-7045	267	107	exists	exist	VERB
ejpam-7045	267	108	an	an	DET
ejpam-7045	267	109	α	α	NOUN
ejpam-7045	267	110	-	-	PUNCT
ejpam-7045	267	111	i	i	PRON
ejpam-7045	267	112	⋆-open	⋆-open	VERB
ejpam-7045	267	113	set	set	VERB
ejpam-7045	267	114	u	u	NOUN
ejpam-7045	267	115	of	of	ADP
ejpam-7045	267	116	x	x	PUNCT
ejpam-7045	267	117	containing	contain	VERB
ejpam-7045	267	118	x	x	PUNCT
ejpam-7045	267	119	such	such	ADJ
ejpam-7045	267	120	that	that	DET
ejpam-7045	267	121	f(u	f(u	PROPN
ejpam-7045	267	122	)	)	PUNCT
ejpam-7045	267	123	⊆	⊆	NUM
ejpam-7045	267	124	v	v	NOUN
ejpam-7045	267	125	;	;	PUNCT
ejpam-7045	267	126	(	(	PUNCT
ejpam-7045	267	127	4	4	X
ejpam-7045	267	128	)	)	PUNCT
ejpam-7045	267	129	for	for	ADP
ejpam-7045	267	130	each	each	DET
ejpam-7045	267	131	x	x	SYM
ejpam-7045	267	132	∈	∈	PROPN
ejpam-7045	267	133	x	x	X
ejpam-7045	267	134	and	and	CCONJ
ejpam-7045	267	135	each	each	DET
ejpam-7045	267	136	σ1σ2	σ1σ2	VERB
ejpam-7045	267	137	-	-	ADJ
ejpam-7045	267	138	open	open	ADJ
ejpam-7045	267	139	set	set	NOUN
ejpam-7045	267	140	v	v	NOUN
ejpam-7045	267	141	of	of	ADP
ejpam-7045	267	142	y	y	NOUN
ejpam-7045	267	143	containing	contain	VERB
ejpam-7045	267	144	f(x	f(x	PROPN
ejpam-7045	267	145	)	)	PUNCT
ejpam-7045	267	146	,	,	PUNCT
ejpam-7045	267	147	there	there	PRON
ejpam-7045	267	148	exists	exist	VERB
ejpam-7045	267	149	a	a	DET
ejpam-7045	267	150	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	267	151	-	-	PUNCT
ejpam-7045	267	152	open	open	ADJ
ejpam-7045	267	153	set	set	NOUN
ejpam-7045	267	154	u	u	NOUN
ejpam-7045	267	155	of	of	ADP
ejpam-7045	267	156	x	x	PUNCT
ejpam-7045	267	157	containing	contain	VERB
ejpam-7045	267	158	x	x	PUNCT
ejpam-7045	267	159	such	such	ADJ
ejpam-7045	267	160	that	that	DET
ejpam-7045	267	161	f(u	f(u	PROPN
ejpam-7045	267	162	)	)	PUNCT
ejpam-7045	268	1	⊆	⊆	NUM
ejpam-7045	268	2	σ1σ2	σ1σ2	X
ejpam-7045	268	3	-	-	PUNCT
ejpam-7045	268	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	268	5	-	-	PUNCT
ejpam-7045	268	6	cl(v	cl(v	NOUN
ejpam-7045	268	7	)	)	PUNCT
ejpam-7045	268	8	)	)	PUNCT
ejpam-7045	268	9	;	;	PUNCT
ejpam-7045	268	10	(	(	PUNCT
ejpam-7045	268	11	5	5	X
ejpam-7045	268	12	)	)	PUNCT
ejpam-7045	268	13	f−1(k	f−1(k	PROPN
ejpam-7045	268	14	)	)	PUNCT
ejpam-7045	268	15	is	be	AUX
ejpam-7045	268	16	τ⋆-α	τ⋆-α	NOUN
ejpam-7045	268	17	-	-	PUNCT
ejpam-7045	268	18	closed	closed	ADJ
ejpam-7045	268	19	in	in	ADP
ejpam-7045	268	20	x	x	PUNCT
ejpam-7045	268	21	for	for	ADP
ejpam-7045	268	22	every	every	DET
ejpam-7045	268	23	(	(	PUNCT
ejpam-7045	268	24	σ1	σ1	PROPN
ejpam-7045	268	25	,	,	PUNCT
ejpam-7045	268	26	σ2)r	σ2)r	NOUN
ejpam-7045	268	27	-	-	PUNCT
ejpam-7045	268	28	closed	close	VERB
ejpam-7045	268	29	set	set	ADJ
ejpam-7045	268	30	k	k	PROPN
ejpam-7045	268	31	of	of	ADP
ejpam-7045	268	32	y	y	PROPN
ejpam-7045	268	33	;	;	PUNCT
ejpam-7045	268	34	(	(	PUNCT
ejpam-7045	268	35	6	6	X
ejpam-7045	268	36	)	)	PUNCT
ejpam-7045	268	37	f−1(v	f−1(v	NOUN
ejpam-7045	268	38	)	)	PUNCT
ejpam-7045	268	39	⊆	⊆	NUM
ejpam-7045	268	40	αint⋆(f−1((σ1	αint⋆(f−1((σ1	NUM
ejpam-7045	268	41	,	,	PUNCT
ejpam-7045	268	42	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	268	43	)	)	PUNCT
ejpam-7045	268	44	)	)	PUNCT
ejpam-7045	268	45	)	)	PUNCT
ejpam-7045	269	1	for	for	ADP
ejpam-7045	269	2	every	every	DET
ejpam-7045	269	3	σ1σ2	σ1σ2	NOUN
ejpam-7045	269	4	-	-	ADJ
ejpam-7045	269	5	open	open	ADJ
ejpam-7045	269	6	set	set	NOUN
ejpam-7045	269	7	v	v	NOUN
ejpam-7045	269	8	of	of	ADP
ejpam-7045	269	9	y	y	PROPN
ejpam-7045	269	10	;	;	PUNCT
ejpam-7045	269	11	(	(	PUNCT
ejpam-7045	269	12	7	7	X
ejpam-7045	269	13	)	)	PUNCT
ejpam-7045	269	14	αcl⋆(f−1((σ1	αcl⋆(f−1((σ1	ADJ
ejpam-7045	269	15	,	,	PUNCT
ejpam-7045	269	16	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	269	17	)	)	PUNCT
ejpam-7045	269	18	)	)	PUNCT
ejpam-7045	269	19	)	)	PUNCT
ejpam-7045	270	1	⊆	⊆	NUM
ejpam-7045	270	2	f−1(k	f−1(k	PROPN
ejpam-7045	270	3	)	)	PUNCT
ejpam-7045	270	4	for	for	ADP
ejpam-7045	270	5	every	every	DET
ejpam-7045	270	6	σ1σ2	σ1σ2	NUM
ejpam-7045	270	7	-	-	PUNCT
ejpam-7045	270	8	closed	closed	ADJ
ejpam-7045	270	9	set	set	NOUN
ejpam-7045	270	10	k	k	PROPN
ejpam-7045	270	11	of	of	ADP
ejpam-7045	270	12	y	y	PROPN
ejpam-7045	270	13	;	;	PUNCT
ejpam-7045	270	14	(	(	PUNCT
ejpam-7045	270	15	8)	8)	NUM
ejpam-7045	270	16	αcl⋆(f−1(σ1σ2	αcl⋆(f−1(σ1σ2	ADV
ejpam-7045	270	17	-	-	PUNCT
ejpam-7045	270	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	270	19	-	-	PUNCT
ejpam-7045	270	20	int(k	int(k	NOUN
ejpam-7045	270	21	)	)	PUNCT
ejpam-7045	270	22	)	)	PUNCT
ejpam-7045	270	23	)	)	PUNCT
ejpam-7045	270	24	)	)	PUNCT
ejpam-7045	271	1	⊆	⊆	NUM
ejpam-7045	271	2	f−1(k	f−1(k	PROPN
ejpam-7045	271	3	)	)	PUNCT
ejpam-7045	271	4	for	for	ADP
ejpam-7045	271	5	every	every	DET
ejpam-7045	271	6	σ1σ2	σ1σ2	NUM
ejpam-7045	271	7	-	-	PUNCT
ejpam-7045	271	8	closed	closed	ADJ
ejpam-7045	271	9	set	set	NOUN
ejpam-7045	271	10	k	k	PROPN
ejpam-7045	271	11	of	of	ADP
ejpam-7045	271	12	y	y	PROPN
ejpam-7045	271	13	;	;	PUNCT
ejpam-7045	271	14	(	(	PUNCT
ejpam-7045	271	15	9	9	X
ejpam-7045	271	16	)	)	PUNCT
ejpam-7045	271	17	αcl⋆(f−1(σ1σ2	αcl⋆(f−1(σ1σ2	NOUN
ejpam-7045	271	18	-	-	PUNCT
ejpam-7045	271	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	271	20	-	-	PUNCT
ejpam-7045	271	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	271	22	-	-	PUNCT
ejpam-7045	271	23	cl(b	cl(b	NOUN
ejpam-7045	271	24	)	)	PUNCT
ejpam-7045	271	25	)	)	PUNCT
ejpam-7045	271	26	)	)	PUNCT
ejpam-7045	271	27	)	)	PUNCT
ejpam-7045	271	28	)	)	PUNCT
ejpam-7045	272	1	⊆	⊆	NUM
ejpam-7045	272	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7045	272	3	-	-	PUNCT
ejpam-7045	272	4	cl(b	cl(b	NOUN
ejpam-7045	272	5	)	)	PUNCT
ejpam-7045	272	6	)	)	PUNCT
ejpam-7045	272	7	for	for	ADP
ejpam-7045	272	8	every	every	DET
ejpam-7045	272	9	subset	subset	NOUN
ejpam-7045	272	10	b	b	PROPN
ejpam-7045	272	11	of	of	ADP
ejpam-7045	272	12	y	y	PROPN
ejpam-7045	272	13	;	;	PUNCT
ejpam-7045	272	14	(	(	PUNCT
ejpam-7045	272	15	10	10	X
ejpam-7045	272	16	)	)	PUNCT
ejpam-7045	272	17	cl⋆(int⋆(cl⋆(f−1(σ1σ2	cl⋆(int⋆(cl⋆(f−1(σ1σ2	NOUN
ejpam-7045	272	18	-	-	PUNCT
ejpam-7045	272	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7045	272	20	-	-	PUNCT
ejpam-7045	272	21	int(k	int(k	NOUN
ejpam-7045	272	22	)	)	PUNCT
ejpam-7045	272	23	)	)	PUNCT
ejpam-7045	272	24	)	)	PUNCT
ejpam-7045	272	25	)	)	PUNCT
ejpam-7045	272	26	)	)	PUNCT
ejpam-7045	272	27	)	)	PUNCT
ejpam-7045	272	28	⊆	⊆	NUM
ejpam-7045	272	29	f−1(k	f−1(k	PROPN
ejpam-7045	272	30	)	)	PUNCT
ejpam-7045	272	31	for	for	ADP
ejpam-7045	272	32	every	every	DET
ejpam-7045	272	33	σ1σ2	σ1σ2	NUM
ejpam-7045	272	34	-	-	PUNCT
ejpam-7045	272	35	closed	closed	ADJ
ejpam-7045	272	36	set	set	NOUN
ejpam-7045	272	37	k	k	PROPN
ejpam-7045	272	38	of	of	ADP
ejpam-7045	272	39	y	y	PROPN
ejpam-7045	272	40	;	;	PUNCT
ejpam-7045	272	41	c.	c.	PROPN
ejpam-7045	272	42	viriyapong	viriyapong	PROPN
ejpam-7045	272	43	,	,	PUNCT
ejpam-7045	272	44	a.	a.	PROPN
ejpam-7045	272	45	sama	sama	PROPN
ejpam-7045	272	46	-	-	PUNCT
ejpam-7045	272	47	ae	ae	PROPN
ejpam-7045	272	48	,	,	PUNCT
ejpam-7045	272	49	c.	c.	PROPN
ejpam-7045	272	50	boonpok	boonpok	PROPN
ejpam-7045	272	51	/	/	SYM
ejpam-7045	272	52	eur	eur	PROPN
ejpam-7045	272	53	.	.	PUNCT
ejpam-7045	273	1	j.	j.	PROPN
ejpam-7045	273	2	pure	pure	PROPN
ejpam-7045	273	3	appl	appl	PROPN
ejpam-7045	273	4	.	.	PROPN
ejpam-7045	273	5	math	math	PROPN
ejpam-7045	273	6	,	,	PUNCT
ejpam-7045	273	7	18	18	NUM
ejpam-7045	273	8	(	(	PUNCT
ejpam-7045	273	9	4	4	NUM
ejpam-7045	273	10	)	)	PUNCT
ejpam-7045	273	11	(	(	PUNCT
ejpam-7045	273	12	2025	2025	NUM
ejpam-7045	273	13	)	)	PUNCT
ejpam-7045	273	14	,	,	PUNCT
ejpam-7045	273	15	7045	7045	NUM
ejpam-7045	273	16	11	11	NUM
ejpam-7045	273	17	of	of	ADP
ejpam-7045	273	18	12	12	NUM
ejpam-7045	273	19	(	(	PUNCT
ejpam-7045	273	20	11	11	NUM
ejpam-7045	273	21	)	)	PUNCT
ejpam-7045	273	22	cl⋆(int⋆(cl⋆(f−1((σ1	cl⋆(int⋆(cl⋆(f−1((σ1	NOUN
ejpam-7045	273	23	,	,	PUNCT
ejpam-7045	273	24	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7045	273	25	)	)	PUNCT
ejpam-7045	273	26	)	)	PUNCT
ejpam-7045	273	27	)	)	PUNCT
ejpam-7045	273	28	)	)	PUNCT
ejpam-7045	273	29	)	)	PUNCT
ejpam-7045	274	1	⊆	⊆	NUM
ejpam-7045	274	2	f−1(k	f−1(k	PROPN
ejpam-7045	274	3	)	)	PUNCT
ejpam-7045	274	4	for	for	ADP
ejpam-7045	274	5	every	every	DET
ejpam-7045	274	6	σ1σ2	σ1σ2	NUM
ejpam-7045	274	7	-	-	PUNCT
ejpam-7045	274	8	closed	closed	ADJ
ejpam-7045	274	9	set	set	NOUN
ejpam-7045	274	10	k	k	PROPN
ejpam-7045	274	11	of	of	ADP
ejpam-7045	274	12	y	y	PROPN
ejpam-7045	274	13	;	;	PUNCT
ejpam-7045	274	14	(	(	PUNCT
ejpam-7045	274	15	12	12	X
ejpam-7045	274	16	)	)	PUNCT
ejpam-7045	274	17	f−1(v	f−1(v	NOUN
ejpam-7045	274	18	)	)	PUNCT
ejpam-7045	274	19	⊆	⊆	NUM
ejpam-7045	274	20	int⋆(cl⋆(int⋆(f−1((σ1	int⋆(cl⋆(int⋆(f−1((σ1	NOUN
ejpam-7045	274	21	,	,	PUNCT
ejpam-7045	274	22	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7045	274	23	)	)	PUNCT
ejpam-7045	274	24	)	)	PUNCT
ejpam-7045	274	25	)	)	PUNCT
ejpam-7045	274	26	)	)	PUNCT
ejpam-7045	274	27	)	)	PUNCT
ejpam-7045	274	28	for	for	ADP
ejpam-7045	274	29	every	every	DET
ejpam-7045	274	30	σ1σ2	σ1σ2	NOUN
ejpam-7045	274	31	-	-	ADJ
ejpam-7045	274	32	open	open	ADJ
ejpam-7045	274	33	set	set	NOUN
ejpam-7045	274	34	v	v	NOUN
ejpam-7045	274	35	of	of	ADP
ejpam-7045	274	36	y	y	PROPN
ejpam-7045	274	37	.	.	PUNCT
ejpam-7045	275	1	corollary	corollary	ADJ
ejpam-7045	275	2	2	2	NUM
ejpam-7045	275	3	.	.	PUNCT
ejpam-7045	276	1	for	for	ADP
ejpam-7045	276	2	a	a	DET
ejpam-7045	276	3	function	function	NOUN
ejpam-7045	276	4	f	f	NOUN
ejpam-7045	276	5	:	:	PUNCT
ejpam-7045	276	6	(	(	PUNCT
ejpam-7045	276	7	x	x	X
ejpam-7045	276	8	,	,	PUNCT
ejpam-7045	276	9	τ	τ	PROPN
ejpam-7045	276	10	,	,	PUNCT
ejpam-7045	276	11	i	i	NOUN
ejpam-7045	276	12	)	)	PUNCT
ejpam-7045	276	13	→	→	PUNCT
ejpam-7045	276	14	(	(	PUNCT
ejpam-7045	276	15	y	y	PROPN
ejpam-7045	276	16	,	,	PUNCT
ejpam-7045	276	17	σ1	σ1	PROPN
ejpam-7045	276	18	,	,	PUNCT
ejpam-7045	276	19	σ2	σ2	NOUN
ejpam-7045	276	20	)	)	PUNCT
ejpam-7045	276	21	,	,	PUNCT
ejpam-7045	276	22	the	the	DET
ejpam-7045	276	23	following	follow	VERB
ejpam-7045	276	24	properties	property	NOUN
ejpam-7045	276	25	are	be	AUX
ejpam-7045	276	26	equivalent	equivalent	ADJ
ejpam-7045	276	27	:	:	PUNCT
ejpam-7045	276	28	(	(	PUNCT
ejpam-7045	276	29	1	1	X
ejpam-7045	276	30	)	)	PUNCT
ejpam-7045	276	31	f	f	NOUN
ejpam-7045	276	32	is	be	AUX
ejpam-7045	276	33	almost	almost	ADV
ejpam-7045	276	34	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	276	35	,	,	PUNCT
ejpam-7045	276	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	276	37	;	;	PUNCT
ejpam-7045	276	38	(	(	PUNCT
ejpam-7045	276	39	2	2	X
ejpam-7045	276	40	)	)	PUNCT
ejpam-7045	276	41	αcl⋆(f−1(v	αcl⋆(f−1(v	NUM
ejpam-7045	276	42	)	)	PUNCT
ejpam-7045	276	43	)	)	PUNCT
ejpam-7045	277	1	⊆	⊆	NUM
ejpam-7045	277	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7045	277	3	-	-	PUNCT
ejpam-7045	277	4	cl(v	cl(v	NOUN
ejpam-7045	277	5	)	)	PUNCT
ejpam-7045	277	6	)	)	PUNCT
ejpam-7045	277	7	for	for	ADP
ejpam-7045	277	8	every	every	DET
ejpam-7045	277	9	(	(	PUNCT
ejpam-7045	277	10	σ1	σ1	PROPN
ejpam-7045	277	11	,	,	PUNCT
ejpam-7045	277	12	σ2)β	σ2)β	NOUN
ejpam-7045	277	13	-	-	PUNCT
ejpam-7045	277	14	open	open	NOUN
ejpam-7045	277	15	set	set	NOUN
ejpam-7045	277	16	v	v	NOUN
ejpam-7045	277	17	of	of	ADP
ejpam-7045	277	18	y	y	PROPN
ejpam-7045	277	19	;	;	PUNCT
ejpam-7045	277	20	(	(	PUNCT
ejpam-7045	277	21	3	3	X
ejpam-7045	277	22	)	)	PUNCT
ejpam-7045	277	23	αcl⋆(f−1(v	αcl⋆(f−1(v	NUM
ejpam-7045	277	24	)	)	PUNCT
ejpam-7045	277	25	)	)	PUNCT
ejpam-7045	278	1	⊆	⊆	NUM
ejpam-7045	278	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7045	278	3	-	-	PUNCT
ejpam-7045	278	4	cl(v	cl(v	NOUN
ejpam-7045	278	5	)	)	PUNCT
ejpam-7045	278	6	)	)	PUNCT
ejpam-7045	278	7	for	for	ADP
ejpam-7045	278	8	every	every	DET
ejpam-7045	278	9	(	(	PUNCT
ejpam-7045	278	10	σ1	σ1	PROPN
ejpam-7045	278	11	,	,	PUNCT
ejpam-7045	278	12	σ2)s	σ2)s	NOUN
ejpam-7045	278	13	-	-	PUNCT
ejpam-7045	278	14	open	open	NOUN
ejpam-7045	278	15	set	set	NOUN
ejpam-7045	278	16	v	v	NOUN
ejpam-7045	278	17	of	of	ADP
ejpam-7045	278	18	y	y	PROPN
ejpam-7045	278	19	;	;	PUNCT
ejpam-7045	278	20	(	(	PUNCT
ejpam-7045	278	21	4	4	X
ejpam-7045	278	22	)	)	PUNCT
ejpam-7045	278	23	f−1(v	f−1(v	NOUN
ejpam-7045	278	24	)	)	PUNCT
ejpam-7045	278	25	⊆	⊆	NUM
ejpam-7045	278	26	αint⋆(f−1(σ1σ2	αint⋆(f−1(σ1σ2	VERB
ejpam-7045	278	27	-	-	PUNCT
ejpam-7045	278	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7045	278	29	-	-	PUNCT
ejpam-7045	278	30	cl(v	cl(v	NOUN
ejpam-7045	278	31	)	)	PUNCT
ejpam-7045	278	32	)	)	PUNCT
ejpam-7045	278	33	)	)	PUNCT
ejpam-7045	278	34	)	)	PUNCT
ejpam-7045	278	35	for	for	ADP
ejpam-7045	278	36	every	every	DET
ejpam-7045	278	37	(	(	PUNCT
ejpam-7045	278	38	σ1	σ1	PROPN
ejpam-7045	278	39	,	,	PUNCT
ejpam-7045	278	40	σ2)p	σ2)p	NOUN
ejpam-7045	278	41	-	-	PUNCT
ejpam-7045	278	42	open	open	NOUN
ejpam-7045	278	43	set	set	NOUN
ejpam-7045	278	44	v	v	NOUN
ejpam-7045	278	45	of	of	ADP
ejpam-7045	278	46	y	y	PROPN
ejpam-7045	278	47	.	.	PUNCT
ejpam-7045	279	1	4	4	X
ejpam-7045	279	2	.	.	X
ejpam-7045	279	3	conclusion	conclusion	NOUN
ejpam-7045	279	4	in	in	ADP
ejpam-7045	279	5	this	this	DET
ejpam-7045	279	6	paper	paper	NOUN
ejpam-7045	279	7	,	,	PUNCT
ejpam-7045	279	8	we	we	PRON
ejpam-7045	279	9	have	have	AUX
ejpam-7045	279	10	introduced	introduce	VERB
ejpam-7045	279	11	new	new	ADJ
ejpam-7045	279	12	classes	class	NOUN
ejpam-7045	279	13	of	of	ADP
ejpam-7045	279	14	continuous	continuous	ADJ
ejpam-7045	279	15	multifunctions	multifunction	NOUN
ejpam-7045	279	16	defined	define	VERB
ejpam-7045	279	17	from	from	ADP
ejpam-7045	279	18	an	an	DET
ejpam-7045	279	19	ideal	ideal	ADJ
ejpam-7045	279	20	topological	topological	ADJ
ejpam-7045	279	21	space	space	NOUN
ejpam-7045	279	22	into	into	ADP
ejpam-7045	279	23	a	a	DET
ejpam-7045	279	24	bitopological	bitopological	ADJ
ejpam-7045	279	25	space	space	NOUN
ejpam-7045	279	26	,	,	PUNCT
ejpam-7045	279	27	namely	namely	ADV
ejpam-7045	279	28	upper	upper	ADJ
ejpam-7045	279	29	almost	almost	ADV
ejpam-7045	279	30	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	279	31	,	,	PUNCT
ejpam-7045	279	32	σ2)continuous	σ2)continuous	ADJ
ejpam-7045	279	33	multifunctions	multifunction	NOUN
ejpam-7045	279	34	and	and	CCONJ
ejpam-7045	279	35	lower	low	ADJ
ejpam-7045	279	36	almost	almost	ADV
ejpam-7045	279	37	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	279	38	,	,	PUNCT
ejpam-7045	279	39	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	279	40	multifunctions	multifunction	NOUN
ejpam-7045	279	41	.	.	PUNCT
ejpam-7045	280	1	also	also	ADV
ejpam-7045	280	2	,	,	PUNCT
ejpam-7045	280	3	we	we	PRON
ejpam-7045	280	4	have	have	AUX
ejpam-7045	280	5	discussed	discuss	VERB
ejpam-7045	280	6	the	the	DET
ejpam-7045	280	7	relationships	relationship	NOUN
ejpam-7045	280	8	between	between	ADP
ejpam-7045	280	9	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7045	280	10	,	,	PUNCT
ejpam-7045	280	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	280	12	multifunctions	multifunction	NOUN
ejpam-7045	280	13	and	and	CCONJ
ejpam-7045	280	14	almost	almost	ADV
ejpam-7045	280	15	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7045	280	16	,	,	PUNCT
ejpam-7045	280	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	280	18	multifunctions	multifunction	NOUN
ejpam-7045	280	19	.	.	PUNCT
ejpam-7045	281	1	moreover	moreover	ADV
ejpam-7045	281	2	,	,	PUNCT
ejpam-7045	281	3	several	several	ADJ
ejpam-7045	281	4	characterizations	characterization	NOUN
ejpam-7045	281	5	and	and	CCONJ
ejpam-7045	281	6	some	some	DET
ejpam-7045	281	7	properties	property	NOUN
ejpam-7045	281	8	concerning	concern	VERB
ejpam-7045	281	9	upper	upper	ADJ
ejpam-7045	281	10	almost	almost	ADV
ejpam-7045	281	11	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	281	12	,	,	PUNCT
ejpam-7045	281	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	281	14	multifunctions	multifunction	NOUN
ejpam-7045	281	15	and	and	CCONJ
ejpam-7045	281	16	lower	low	ADJ
ejpam-7045	281	17	almost	almost	ADV
ejpam-7045	281	18	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	281	19	,	,	PUNCT
ejpam-7045	281	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7045	281	21	multifunctions	multifunction	NOUN
ejpam-7045	281	22	are	be	AUX
ejpam-7045	281	23	obtained	obtain	VERB
ejpam-7045	281	24	.	.	PUNCT
ejpam-7045	282	1	the	the	DET
ejpam-7045	282	2	ideas	idea	NOUN
ejpam-7045	282	3	and	and	CCONJ
ejpam-7045	282	4	results	result	NOUN
ejpam-7045	282	5	of	of	ADP
ejpam-7045	282	6	this	this	DET
ejpam-7045	282	7	paper	paper	NOUN
ejpam-7045	282	8	may	may	AUX
ejpam-7045	282	9	motivate	motivate	VERB
ejpam-7045	282	10	further	further	ADJ
ejpam-7045	282	11	research	research	NOUN
ejpam-7045	282	12	.	.	PUNCT
ejpam-7045	283	1	acknowledgements	acknowledgement	NOUN
ejpam-7045	283	2	this	this	DET
ejpam-7045	283	3	research	research	NOUN
ejpam-7045	283	4	project	project	NOUN
ejpam-7045	283	5	was	be	AUX
ejpam-7045	283	6	financially	financially	ADV
ejpam-7045	283	7	supported	support	VERB
ejpam-7045	283	8	by	by	ADP
ejpam-7045	283	9	mahasarakham	mahasarakham	PROPN
ejpam-7045	283	10	university	university	PROPN
ejpam-7045	283	11	.	.	PUNCT
ejpam-7045	284	1	references	reference	NOUN
ejpam-7045	284	2	[	[	X
ejpam-7045	284	3	1	1	X
ejpam-7045	284	4	]	]	PUNCT
ejpam-7045	284	5	t.	t.	PROPN
ejpam-7045	284	6	noiri	noiri	PROPN
ejpam-7045	284	7	.	.	PUNCT
ejpam-7045	285	1	almost	almost	ADV
ejpam-7045	285	2	α	α	NUM
ejpam-7045	285	3	-	-	ADJ
ejpam-7045	285	4	continuous	continuous	ADJ
ejpam-7045	285	5	functions	function	NOUN
ejpam-7045	285	6	.	.	PUNCT
ejpam-7045	286	1	kyungpook	kyungpook	PROPN
ejpam-7045	286	2	mathematical	mathematical	PROPN
ejpam-7045	286	3	journal	journal	NOUN
ejpam-7045	286	4	,	,	PUNCT
ejpam-7045	286	5	28:71	28:71	NUM
ejpam-7045	286	6	–	–	PUNCT
ejpam-7045	286	7	77	77	NUM
ejpam-7045	286	8	,	,	PUNCT
ejpam-7045	286	9	1988	1988	NUM
ejpam-7045	286	10	.	.	PUNCT
ejpam-7045	287	1	[	[	X
ejpam-7045	287	2	2	2	X
ejpam-7045	287	3	]	]	PUNCT
ejpam-7045	287	4	v.	v.	CCONJ
ejpam-7045	287	5	popa	popa	NOUN
ejpam-7045	287	6	and	and	CCONJ
ejpam-7045	287	7	t.	t.	PROPN
ejpam-7045	287	8	noiri	noiri	PROPN
ejpam-7045	287	9	.	.	PUNCT
ejpam-7045	288	1	on	on	ADP
ejpam-7045	288	2	upper	upper	ADJ
ejpam-7045	288	3	and	and	CCONJ
ejpam-7045	288	4	lower	low	ADJ
ejpam-7045	288	5	almost	almost	ADV
ejpam-7045	288	6	α	α	ADJ
ejpam-7045	288	7	-	-	ADJ
ejpam-7045	288	8	continuous	continuous	ADJ
ejpam-7045	288	9	multifunctions	multifunction	NOUN
ejpam-7045	288	10	.	.	PUNCT
ejpam-7045	289	1	demonstratio	demonstratio	PROPN
ejpam-7045	289	2	mathematica	mathematica	PROPN
ejpam-7045	289	3	,	,	PUNCT
ejpam-7045	289	4	24(2):381–396	24(2):381–396	NUM
ejpam-7045	289	5	,	,	PUNCT
ejpam-7045	289	6	1996	1996	NUM
ejpam-7045	289	7	.	.	PUNCT
ejpam-7045	290	1	[	[	X
ejpam-7045	290	2	3	3	X
ejpam-7045	290	3	]	]	PUNCT
ejpam-7045	290	4	c.	c.	PROPN
ejpam-7045	290	5	boonpok	boonpok	PROPN
ejpam-7045	290	6	.	.	PUNCT
ejpam-7045	291	1	on	on	ADP
ejpam-7045	291	2	continuous	continuous	ADJ
ejpam-7045	291	3	multifunctions	multifunction	NOUN
ejpam-7045	291	4	in	in	ADP
ejpam-7045	291	5	ideal	ideal	ADJ
ejpam-7045	291	6	topological	topological	ADJ
ejpam-7045	291	7	spaces	space	NOUN
ejpam-7045	291	8	.	.	PUNCT
ejpam-7045	292	1	lobachevskii	lobachevskii	PROPN
ejpam-7045	292	2	journal	journal	PROPN
ejpam-7045	292	3	of	of	ADP
ejpam-7045	292	4	mathematics	mathematic	NOUN
ejpam-7045	292	5	,	,	PUNCT
ejpam-7045	292	6	40(1):24–35	40(1):24–35	NUM
ejpam-7045	292	7	,	,	PUNCT
ejpam-7045	292	8	2019	2019	NUM
ejpam-7045	292	9	.	.	PUNCT
ejpam-7045	293	1	[	[	X
ejpam-7045	293	2	4	4	NUM
ejpam-7045	293	3	]	]	PUNCT
ejpam-7045	293	4	c.	c.	PROPN
ejpam-7045	293	5	boonpok	boonpok	PROPN
ejpam-7045	293	6	.	.	PUNCT
ejpam-7045	294	1	on	on	ADP
ejpam-7045	294	2	some	some	DET
ejpam-7045	294	3	types	type	NOUN
ejpam-7045	294	4	of	of	ADP
ejpam-7045	294	5	continuity	continuity	NOUN
ejpam-7045	294	6	for	for	ADP
ejpam-7045	294	7	multifunctions	multifunction	NOUN
ejpam-7045	294	8	in	in	ADP
ejpam-7045	294	9	ideal	ideal	ADJ
ejpam-7045	294	10	topological	topological	ADJ
ejpam-7045	294	11	spaces	space	NOUN
ejpam-7045	294	12	.	.	PUNCT
ejpam-7045	295	1	advances	advance	NOUN
ejpam-7045	295	2	in	in	ADP
ejpam-7045	295	3	mathematics	mathematic	NOUN
ejpam-7045	295	4	:	:	PUNCT
ejpam-7045	295	5	scientific	scientific	ADJ
ejpam-7045	295	6	journal	journal	NOUN
ejpam-7045	295	7	,	,	PUNCT
ejpam-7045	295	8	9(3):859–886	9(3):859–886	NUM
ejpam-7045	295	9	,	,	PUNCT
ejpam-7045	295	10	2020	2020	NUM
ejpam-7045	295	11	.	.	PUNCT
ejpam-7045	296	1	[	[	X
ejpam-7045	296	2	5	5	X
ejpam-7045	296	3	]	]	PUNCT
ejpam-7045	296	4	c.	c.	PROPN
ejpam-7045	296	5	boonpok	boonpok	PROPN
ejpam-7045	296	6	and	and	CCONJ
ejpam-7045	296	7	n.	n.	PROPN
ejpam-7045	296	8	srisarakham	srisarakham	PROPN
ejpam-7045	296	9	.	.	PUNCT
ejpam-7045	297	1	almost	almost	ADV
ejpam-7045	297	2	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-7045	297	3	for	for	ADP
ejpam-7045	297	4	multifunctions	multifunction	NOUN
ejpam-7045	297	5	.	.	PUNCT
ejpam-7045	298	1	international	international	ADJ
ejpam-7045	298	2	journal	journal	NOUN
ejpam-7045	298	3	of	of	ADP
ejpam-7045	298	4	analysis	analysis	NOUN
ejpam-7045	298	5	and	and	CCONJ
ejpam-7045	298	6	applications	application	NOUN
ejpam-7045	298	7	,	,	PUNCT
ejpam-7045	298	8	21:107	21:107	NUM
ejpam-7045	298	9	,	,	PUNCT
ejpam-7045	298	10	2023	2023	NUM
ejpam-7045	298	11	.	.	PUNCT
ejpam-7045	299	1	[	[	X
ejpam-7045	299	2	6	6	NUM
ejpam-7045	299	3	]	]	PUNCT
ejpam-7045	299	4	c.	c.	PROPN
ejpam-7045	299	5	boonpok	boonpok	PROPN
ejpam-7045	299	6	and	and	CCONJ
ejpam-7045	299	7	p.	p.	NOUN
ejpam-7045	299	8	pue	pue	NOUN
ejpam-7045	299	9	-	-	PUNCT
ejpam-7045	299	10	on	on	ADP
ejpam-7045	299	11	.	.	PUNCT
ejpam-7045	300	1	continuity	continuity	NOUN
ejpam-7045	300	2	for	for	ADP
ejpam-7045	300	3	multifunctions	multifunction	NOUN
ejpam-7045	300	4	in	in	ADP
ejpam-7045	300	5	ideal	ideal	ADJ
ejpam-7045	300	6	topological	topological	ADJ
ejpam-7045	300	7	spaces	space	NOUN
ejpam-7045	300	8	.	.	PUNCT
ejpam-7045	301	1	wseas	wseas	VERB
ejpam-7045	301	2	transactions	transaction	NOUN
ejpam-7045	301	3	on	on	ADP
ejpam-7045	301	4	mathematics	mathematic	NOUN
ejpam-7045	301	5	,	,	PUNCT
ejpam-7045	301	6	19:624–631	19:624–631	NUM
ejpam-7045	301	7	,	,	PUNCT
ejpam-7045	301	8	2020	2020	NUM
ejpam-7045	301	9	.	.	PUNCT
ejpam-7045	302	1	c.	c.	PROPN
ejpam-7045	302	2	viriyapong	viriyapong	PROPN
ejpam-7045	302	3	,	,	PUNCT
ejpam-7045	302	4	a.	a.	PROPN
ejpam-7045	302	5	sama	sama	PROPN
ejpam-7045	302	6	-	-	PUNCT
ejpam-7045	302	7	ae	ae	PROPN
ejpam-7045	302	8	,	,	PUNCT
ejpam-7045	302	9	c.	c.	PROPN
ejpam-7045	302	10	boonpok	boonpok	PROPN
ejpam-7045	302	11	/	/	SYM
ejpam-7045	302	12	eur	eur	PROPN
ejpam-7045	302	13	.	.	PUNCT
ejpam-7045	303	1	j.	j.	PROPN
ejpam-7045	303	2	pure	pure	PROPN
ejpam-7045	303	3	appl	appl	PROPN
ejpam-7045	303	4	.	.	PROPN
ejpam-7045	303	5	math	math	PROPN
ejpam-7045	303	6	,	,	PUNCT
ejpam-7045	303	7	18	18	NUM
ejpam-7045	303	8	(	(	PUNCT
ejpam-7045	303	9	4	4	NUM
ejpam-7045	303	10	)	)	PUNCT
ejpam-7045	303	11	(	(	PUNCT
ejpam-7045	303	12	2025	2025	NUM
ejpam-7045	303	13	)	)	PUNCT
ejpam-7045	303	14	,	,	PUNCT
ejpam-7045	303	15	7045	7045	NUM
ejpam-7045	303	16	12	12	NUM
ejpam-7045	303	17	of	of	ADP
ejpam-7045	303	18	12	12	NUM
ejpam-7045	303	19	[	[	X
ejpam-7045	303	20	7	7	NUM
ejpam-7045	303	21	]	]	X
ejpam-7045	303	22	p.	p.	NOUN
ejpam-7045	303	23	pue	pue	NOUN
ejpam-7045	303	24	-	-	PUNCT
ejpam-7045	303	25	on	on	ADP
ejpam-7045	303	26	,	,	PUNCT
ejpam-7045	303	27	s.	s.	PROPN
ejpam-7045	303	28	sompong	sompong	PROPN
ejpam-7045	303	29	,	,	PUNCT
ejpam-7045	303	30	and	and	CCONJ
ejpam-7045	303	31	c.	c.	PROPN
ejpam-7045	303	32	boonpok	boonpok	PROPN
ejpam-7045	303	33	.	.	PUNCT
ejpam-7045	304	1	upper	upper	ADJ
ejpam-7045	304	2	and	and	CCONJ
ejpam-7045	304	3	lower	low	ADJ
ejpam-7045	304	4	(	(	PUNCT
ejpam-7045	304	5	τ1	τ1	NOUN
ejpam-7045	304	6	,	,	PUNCT
ejpam-7045	304	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7045	304	8	multifunctions	multifunction	NOUN
ejpam-7045	304	9	.	.	PUNCT
ejpam-7045	305	1	international	international	ADJ
ejpam-7045	305	2	journal	journal	PROPN
ejpam-7045	305	3	of	of	ADP
ejpam-7045	305	4	mathematics	mathematic	NOUN
ejpam-7045	305	5	and	and	CCONJ
ejpam-7045	305	6	computer	computer	NOUN
ejpam-7045	305	7	science	science	NOUN
ejpam-7045	305	8	,	,	PUNCT
ejpam-7045	305	9	19(4):1305	19(4):1305	NUM
ejpam-7045	305	10	–	–	PUNCT
ejpam-7045	305	11	1310	1310	NUM
ejpam-7045	305	12	,	,	PUNCT
ejpam-7045	305	13	2024	2024	NUM
ejpam-7045	305	14	.	.	PUNCT
ejpam-7045	306	1	[	[	X
ejpam-7045	306	2	8	8	NUM
ejpam-7045	306	3	]	]	X
ejpam-7045	306	4	c.	c.	NOUN
ejpam-7045	306	5	boonpok	boonpok	PROPN
ejpam-7045	306	6	and	and	CCONJ
ejpam-7045	306	7	p.	p.	NOUN
ejpam-7045	306	8	pue	pue	NOUN
ejpam-7045	306	9	-	-	PUNCT
ejpam-7045	306	10	on	on	ADP
ejpam-7045	306	11	.	.	PUNCT
ejpam-7045	307	1	characterizations	characterization	NOUN
ejpam-7045	307	2	of	of	ADP
ejpam-7045	307	3	almost	almost	ADV
ejpam-7045	307	4	(	(	PUNCT
ejpam-7045	307	5	τ1	τ1	NOUN
ejpam-7045	307	6	,	,	PUNCT
ejpam-7045	307	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7045	307	8	multifunctions	multifunction	NOUN
ejpam-7045	307	9	.	.	PUNCT
ejpam-7045	308	1	international	international	ADJ
ejpam-7045	308	2	journal	journal	NOUN
ejpam-7045	308	3	of	of	ADP
ejpam-7045	308	4	analysis	analysis	NOUN
ejpam-7045	308	5	and	and	CCONJ
ejpam-7045	308	6	applications	application	NOUN
ejpam-7045	308	7	,	,	PUNCT
ejpam-7045	308	8	22:33	22:33	NUM
ejpam-7045	308	9	,	,	PUNCT
ejpam-7045	308	10	2024	2024	NUM
ejpam-7045	308	11	.	.	PUNCT
ejpam-7045	309	1	[	[	X
ejpam-7045	309	2	9	9	NUM
ejpam-7045	309	3	]	]	X
ejpam-7045	309	4	c.	c.	PROPN
ejpam-7045	309	5	viriyapong	viriyapong	PROPN
ejpam-7045	309	6	and	and	CCONJ
ejpam-7045	309	7	c.	c.	PROPN
ejpam-7045	309	8	boonpok	boonpok	PROPN
ejpam-7045	309	9	.	.	PUNCT
ejpam-7045	310	1	(	(	PUNCT
ejpam-7045	310	2	τ1	τ1	NOUN
ejpam-7045	310	3	,	,	PUNCT
ejpam-7045	310	4	τ2)α	τ2)α	NOUN
ejpam-7045	310	5	-	-	PUNCT
ejpam-7045	310	6	continuity	continuity	NOUN
ejpam-7045	310	7	for	for	ADP
ejpam-7045	310	8	multifunctions	multifunction	NOUN
ejpam-7045	310	9	.	.	PUNCT
ejpam-7045	311	1	journal	journal	PROPN
ejpam-7045	311	2	of	of	ADP
ejpam-7045	311	3	mathematics	mathematic	NOUN
ejpam-7045	311	4	,	,	PUNCT
ejpam-7045	311	5	2020:6285763	2020:6285763	NUM
ejpam-7045	311	6	,	,	PUNCT
ejpam-7045	311	7	2020	2020	NUM
ejpam-7045	311	8	.	.	PUNCT
ejpam-7045	312	1	[	[	X
ejpam-7045	312	2	10	10	NUM
ejpam-7045	312	3	]	]	X
ejpam-7045	312	4	c.	c.	PROPN
ejpam-7045	312	5	viriyapong	viriyapong	PROPN
ejpam-7045	312	6	,	,	PUNCT
ejpam-7045	312	7	a.	a.	PROPN
ejpam-7045	312	8	sama	sama	PROPN
ejpam-7045	312	9	-	-	PUNCT
ejpam-7045	312	10	ae	ae	PROPN
ejpam-7045	312	11	,	,	PUNCT
ejpam-7045	312	12	and	and	CCONJ
ejpam-7045	312	13	c.	c.	PROPN
ejpam-7045	312	14	boonpok	boonpok	PROPN
ejpam-7045	312	15	.	.	PUNCT
ejpam-7045	313	1	almost	almost	ADV
ejpam-7045	313	2	continuity	continuity	NOUN
ejpam-7045	313	3	for	for	ADP
ejpam-7045	313	4	multifunctions	multifunction	NOUN
ejpam-7045	313	5	defined	define	VERB
ejpam-7045	313	6	from	from	ADP
ejpam-7045	313	7	an	an	DET
ejpam-7045	313	8	ideal	ideal	ADJ
ejpam-7045	313	9	topological	topological	ADJ
ejpam-7045	313	10	space	space	NOUN
ejpam-7045	313	11	into	into	ADP
ejpam-7045	313	12	a	a	DET
ejpam-7045	313	13	bitopological	bitopological	ADJ
ejpam-7045	313	14	space	space	NOUN
ejpam-7045	313	15	.	.	PUNCT
ejpam-7045	314	1	european	european	ADJ
ejpam-7045	314	2	journal	journal	PROPN
ejpam-7045	314	3	of	of	ADP
ejpam-7045	314	4	pure	pure	ADJ
ejpam-7045	314	5	and	and	CCONJ
ejpam-7045	314	6	applied	applied	ADJ
ejpam-7045	314	7	mathematics	mathematic	NOUN
ejpam-7045	314	8	,	,	PUNCT
ejpam-7045	314	9	18(3):6566	18(3):6566	NUM
ejpam-7045	314	10	,	,	PUNCT
ejpam-7045	314	11	2025	2025	NUM
ejpam-7045	314	12	.	.	PUNCT
ejpam-7045	315	1	[	[	X
ejpam-7045	315	2	11	11	NUM
ejpam-7045	315	3	]	]	PUNCT
ejpam-7045	315	4	c.	c.	PROPN
ejpam-7045	315	5	boonpok	boonpok	PROPN
ejpam-7045	315	6	,	,	PUNCT
ejpam-7045	315	7	c.	c.	PROPN
ejpam-7045	315	8	viriyapong	viriyapong	PROPN
ejpam-7045	315	9	,	,	PUNCT
ejpam-7045	315	10	and	and	CCONJ
ejpam-7045	315	11	m.	m.	NOUN
ejpam-7045	315	12	thongmoon	thongmoon	NOUN
ejpam-7045	315	13	.	.	PUNCT
ejpam-7045	316	1	on	on	ADP
ejpam-7045	316	2	upper	upper	ADJ
ejpam-7045	316	3	and	and	CCONJ
ejpam-7045	316	4	lower	low	ADJ
ejpam-7045	316	5	(	(	PUNCT
ejpam-7045	316	6	τ1	τ1	NOUN
ejpam-7045	316	7	,	,	PUNCT
ejpam-7045	316	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7045	316	9	multifunctions	multifunction	NOUN
ejpam-7045	316	10	.	.	PUNCT
ejpam-7045	317	1	journal	journal	PROPN
ejpam-7045	317	2	of	of	ADP
ejpam-7045	317	3	mathematics	mathematics	PROPN
ejpam-7045	317	4	and	and	CCONJ
ejpam-7045	317	5	computer	computer	NOUN
ejpam-7045	317	6	science	science	NOUN
ejpam-7045	317	7	,	,	PUNCT
ejpam-7045	317	8	18:282–293	18:282–293	NUM
ejpam-7045	317	9	,	,	PUNCT
ejpam-7045	317	10	2018	2018	NUM
ejpam-7045	317	11	.	.	PUNCT
ejpam-7045	318	1	[	[	X
ejpam-7045	318	2	12	12	NUM
ejpam-7045	318	3	]	]	PUNCT
ejpam-7045	318	4	c.	c.	PROPN
ejpam-7045	318	5	boonpok	boonpok	PROPN
ejpam-7045	318	6	.	.	PUNCT
ejpam-7045	319	1	(	(	PUNCT
ejpam-7045	319	2	τ1	τ1	NOUN
ejpam-7045	319	3	,	,	PUNCT
ejpam-7045	319	4	τ2)δ	τ2)δ	ADJ
ejpam-7045	319	5	-	-	PUNCT
ejpam-7045	319	6	semicontinuous	semicontinuous	ADJ
ejpam-7045	319	7	multifunctions	multifunction	NOUN
ejpam-7045	319	8	.	.	PUNCT
ejpam-7045	320	1	heliyon	heliyon	NOUN
ejpam-7045	320	2	,	,	PUNCT
ejpam-7045	320	3	6	6	NUM
ejpam-7045	320	4	:	:	SYM
ejpam-7045	320	5	e05367	e05367	PROPN
ejpam-7045	320	6	,	,	PUNCT
ejpam-7045	320	7	2020	2020	NUM
ejpam-7045	320	8	.	.	PUNCT
ejpam-7045	321	1	[	[	X
ejpam-7045	321	2	13	13	NUM
ejpam-7045	321	3	]	]	X
ejpam-7045	321	4	p.	p.	NOUN
ejpam-7045	321	5	pue	pue	NOUN
ejpam-7045	321	6	-	-	PUNCT
ejpam-7045	321	7	on	on	ADP
ejpam-7045	321	8	,	,	PUNCT
ejpam-7045	321	9	s.	s.	PROPN
ejpam-7045	321	10	sompong	sompong	PROPN
ejpam-7045	321	11	,	,	PUNCT
ejpam-7045	321	12	and	and	CCONJ
ejpam-7045	321	13	c.	c.	PROPN
ejpam-7045	321	14	boonpok	boonpok	PROPN
ejpam-7045	321	15	.	.	PUNCT
ejpam-7045	322	1	almost	almost	ADV
ejpam-7045	322	2	quasi	quasi	X
ejpam-7045	322	3	(	(	PUNCT
ejpam-7045	322	4	τ1	τ1	NOUN
ejpam-7045	322	5	,	,	PUNCT
ejpam-7045	322	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7045	322	7	for	for	ADP
ejpam-7045	322	8	multifunctions	multifunction	NOUN
ejpam-7045	322	9	.	.	PUNCT
ejpam-7045	323	1	international	international	ADJ
ejpam-7045	323	2	journal	journal	NOUN
ejpam-7045	323	3	of	of	ADP
ejpam-7045	323	4	analysis	analysis	NOUN
ejpam-7045	323	5	and	and	CCONJ
ejpam-7045	323	6	applications	application	NOUN
ejpam-7045	323	7	,	,	PUNCT
ejpam-7045	323	8	22:97	22:97	NUM
ejpam-7045	323	9	,	,	PUNCT
ejpam-7045	323	10	2024	2024	NUM
ejpam-7045	323	11	.	.	PUNCT
ejpam-7045	324	1	[	[	X
ejpam-7045	324	2	14	14	NUM
ejpam-7045	324	3	]	]	X
ejpam-7045	324	4	c.	c.	PROPN
ejpam-7045	324	5	klanarong	klanarong	PROPN
ejpam-7045	324	6	,	,	PUNCT
ejpam-7045	324	7	a.	a.	PROPN
ejpam-7045	324	8	sama	sama	PROPN
ejpam-7045	324	9	-	-	PUNCT
ejpam-7045	324	10	ae	ae	PROPN
ejpam-7045	324	11	,	,	PUNCT
ejpam-7045	324	12	and	and	CCONJ
ejpam-7045	324	13	c.	c.	PROPN
ejpam-7045	324	14	boonpok	boonpok	PROPN
ejpam-7045	324	15	.	.	PUNCT
ejpam-7045	325	1	upper	upper	ADJ
ejpam-7045	325	2	and	and	CCONJ
ejpam-7045	325	3	lower	low	ADJ
ejpam-7045	325	4	almost	almost	ADV
ejpam-7045	325	5	(	(	PUNCT
ejpam-7045	325	6	τ1	τ1	NOUN
ejpam-7045	325	7	,	,	PUNCT
ejpam-7045	325	8	τ2)continuous	τ2)continuous	ADJ
ejpam-7045	325	9	multifunctions	multifunction	NOUN
ejpam-7045	325	10	.	.	PUNCT
ejpam-7045	326	1	european	european	ADJ
ejpam-7045	326	2	journal	journal	PROPN
ejpam-7045	326	3	of	of	ADP
ejpam-7045	326	4	pure	pure	ADJ
ejpam-7045	326	5	and	and	CCONJ
ejpam-7045	326	6	applied	applied	ADJ
ejpam-7045	326	7	mathematics	mathematic	NOUN
ejpam-7045	326	8	,	,	PUNCT
ejpam-7045	326	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-7045	326	10	,	,	PUNCT
ejpam-7045	326	11	2024	2024	NUM
ejpam-7045	326	12	.	.	PUNCT
ejpam-7045	327	1	[	[	X
ejpam-7045	327	2	15	15	NUM
ejpam-7045	327	3	]	]	PUNCT
ejpam-7045	327	4	k.	k.	PROPN
ejpam-7045	327	5	kuratowski	kuratowski	PROPN
ejpam-7045	327	6	.	.	PUNCT
ejpam-7045	328	1	topology	topology	PROPN
ejpam-7045	328	2	,	,	PUNCT
ejpam-7045	328	3	vol	vol	NOUN
ejpam-7045	328	4	.	.	PUNCT
ejpam-7045	328	5	i.	i.	PROPN
ejpam-7045	328	6	academic	academic	PROPN
ejpam-7045	328	7	press	press	PROPN
ejpam-7045	328	8	,	,	PUNCT
ejpam-7045	328	9	new	new	PROPN
ejpam-7045	328	10	york	york	PROPN
ejpam-7045	328	11	,	,	PUNCT
ejpam-7045	328	12	1966	1966	NUM
ejpam-7045	328	13	.	.	PUNCT
ejpam-7045	329	1	[	[	X
ejpam-7045	329	2	16	16	NUM
ejpam-7045	329	3	]	]	X
ejpam-7045	329	4	d.	d.	PROPN
ejpam-7045	329	5	janković	janković	PROPN
ejpam-7045	329	6	and	and	CCONJ
ejpam-7045	329	7	t.	t.	PROPN
ejpam-7045	329	8	r.	r.	PROPN
ejpam-7045	329	9	hamlett	hamlett	PROPN
ejpam-7045	329	10	.	.	PUNCT
ejpam-7045	330	1	new	new	ADJ
ejpam-7045	330	2	topologies	topology	NOUN
ejpam-7045	330	3	from	from	ADP
ejpam-7045	330	4	old	old	ADJ
ejpam-7045	330	5	via	via	ADP
ejpam-7045	330	6	ideals	ideal	NOUN
ejpam-7045	330	7	.	.	PUNCT
ejpam-7045	331	1	the	the	DET
ejpam-7045	331	2	american	american	PROPN
ejpam-7045	331	3	mathematical	mathematical	PROPN
ejpam-7045	331	4	monthly	monthly	ADV
ejpam-7045	331	5	,	,	PUNCT
ejpam-7045	331	6	97:295–310	97:295–310	PROPN
ejpam-7045	331	7	,	,	PUNCT
ejpam-7045	331	8	1990	1990	NUM
ejpam-7045	331	9	.	.	PUNCT
ejpam-7045	332	1	[	[	X
ejpam-7045	332	2	17	17	NUM
ejpam-7045	332	3	]	]	PUNCT
ejpam-7045	332	4	c.	c.	PROPN
ejpam-7045	332	5	boonpok	boonpok	PROPN
ejpam-7045	332	6	.	.	PUNCT
ejpam-7045	333	1	weak	weak	ADJ
ejpam-7045	333	2	quasi	quasi	ADJ
ejpam-7045	333	3	continuity	continuity	NOUN
ejpam-7045	333	4	for	for	ADP
ejpam-7045	333	5	multifunctions	multifunction	NOUN
ejpam-7045	333	6	in	in	ADP
ejpam-7045	333	7	ideal	ideal	ADJ
ejpam-7045	333	8	topological	topological	ADJ
ejpam-7045	333	9	spaces	space	NOUN
ejpam-7045	333	10	.	.	PUNCT
ejpam-7045	334	1	advances	advance	NOUN
ejpam-7045	334	2	in	in	ADP
ejpam-7045	334	3	mathematics	mathematic	NOUN
ejpam-7045	334	4	:	:	PUNCT
ejpam-7045	334	5	scientific	scientific	ADJ
ejpam-7045	334	6	journal	journal	NOUN
ejpam-7045	334	7	,	,	PUNCT
ejpam-7045	334	8	9(3):339–355	9(3):339–355	NUM
ejpam-7045	334	9	,	,	PUNCT
ejpam-7045	334	10	2020	2020	NUM
ejpam-7045	334	11	.	.	PUNCT
ejpam-7045	335	1	[	[	X
ejpam-7045	335	2	18	18	NUM
ejpam-7045	335	3	]	]	PUNCT
ejpam-7045	335	4	t.	t.	PROPN
ejpam-7045	335	5	noiri	noiri	PROPN
ejpam-7045	335	6	and	and	CCONJ
ejpam-7045	335	7	v.	v.	ADP
ejpam-7045	335	8	popa	popa	NOUN
ejpam-7045	335	9	.	.	PUNCT
ejpam-7045	336	1	on	on	ADP
ejpam-7045	336	2	(	(	PUNCT
ejpam-7045	336	3	mi	mi	ADJ
ejpam-7045	336	4	,	,	PUNCT
ejpam-7045	336	5	nj)-continuous	nj)-continuous	ADJ
ejpam-7045	336	6	multifunctions	multifunction	NOUN
ejpam-7045	336	7	.	.	PUNCT
ejpam-7045	337	1	romanian	romanian	ADJ
ejpam-7045	337	2	journal	journal	PROPN
ejpam-7045	337	3	of	of	ADP
ejpam-7045	337	4	mathematics	mathematics	PROPN
ejpam-7045	337	5	and	and	CCONJ
ejpam-7045	337	6	computer	computer	NOUN
ejpam-7045	337	7	science	science	NOUN
ejpam-7045	337	8	,	,	PUNCT
ejpam-7045	337	9	15(1):1–8	15(1):1–8	NUM
ejpam-7045	337	10	,	,	PUNCT
ejpam-7045	337	11	2025	2025	NUM
ejpam-7045	337	12	.	.	PUNCT
ejpam-7045	338	1	[	[	X
ejpam-7045	338	2	19	19	NUM
ejpam-7045	338	3	]	]	PUNCT
ejpam-7045	338	4	c.	c.	PROPN
ejpam-7045	338	5	boonpok	boonpok	PROPN
ejpam-7045	338	6	.	.	PUNCT
ejpam-7045	339	1	a	a	DET
ejpam-7045	339	2	study	study	NOUN
ejpam-7045	339	3	of	of	ADP
ejpam-7045	339	4	some	some	DET
ejpam-7045	339	5	forms	form	NOUN
ejpam-7045	339	6	of	of	ADP
ejpam-7045	339	7	continuity	continuity	NOUN
ejpam-7045	339	8	for	for	ADP
ejpam-7045	339	9	multifunctions	multifunction	NOUN
ejpam-7045	339	10	in	in	ADP
ejpam-7045	339	11	ideal	ideal	ADJ
ejpam-7045	339	12	topological	topological	ADJ
ejpam-7045	339	13	spaces	space	NOUN
ejpam-7045	339	14	.	.	PUNCT
ejpam-7045	340	1	mathematica	mathematica	PROPN
ejpam-7045	340	2	,	,	PUNCT
ejpam-7045	340	3	63(2):186–198	63(2):186–198	PROPN
ejpam-7045	340	4	,	,	PUNCT
ejpam-7045	340	5	2021	2021	NUM
ejpam-7045	340	6	.	.	PUNCT
ejpam-7045	341	1	[	[	X
ejpam-7045	341	2	20	20	NUM
ejpam-7045	341	3	]	]	X
ejpam-7045	341	4	j.	j.	PROPN
ejpam-7045	341	5	khampakdee	khampakdee	PROPN
ejpam-7045	341	6	,	,	PUNCT
ejpam-7045	341	7	a.	a.	PROPN
ejpam-7045	341	8	sama	sama	PROPN
ejpam-7045	341	9	-	-	PUNCT
ejpam-7045	341	10	ae	ae	PROPN
ejpam-7045	341	11	,	,	PUNCT
ejpam-7045	341	12	and	and	CCONJ
ejpam-7045	341	13	c.	c.	PROPN
ejpam-7045	341	14	boonpok	boonpok	PROPN
ejpam-7045	341	15	.	.	PUNCT
ejpam-7045	342	1	upper	upper	ADJ
ejpam-7045	342	2	and	and	CCONJ
ejpam-7045	342	3	lower	low	ADJ
ejpam-7045	342	4	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7045	342	5	,	,	PUNCT
ejpam-7045	342	6	σ2)continuity	σ2)continuity	NOUN
ejpam-7045	342	7	.	.	PUNCT
ejpam-7045	343	1	(	(	PUNCT
ejpam-7045	343	2	submitted	submit	VERB
ejpam-7045	343	3	)	)	PUNCT
ejpam-7045	343	4	.	.	PUNCT