id	sid	tid	token	lemma	pos
ejpam-7044	1	1	european	european	PROPN
ejpam-7044	1	2	journal	journal	PROPN
ejpam-7044	1	3	of	of	ADP
ejpam-7044	1	4	pure	pure	ADJ
ejpam-7044	1	5	and	and	CCONJ
ejpam-7044	1	6	applied	applied	ADJ
ejpam-7044	1	7	mathematics	mathematic	NOUN
ejpam-7044	1	8	2025	2025	NUM
ejpam-7044	1	9	,	,	PUNCT
ejpam-7044	1	10	vol	vol	NOUN
ejpam-7044	1	11	.	.	PROPN
ejpam-7044	1	12	18	18	NUM
ejpam-7044	1	13	,	,	PUNCT
ejpam-7044	1	14	issue	issue	NOUN
ejpam-7044	1	15	4	4	NUM
ejpam-7044	1	16	,	,	PUNCT
ejpam-7044	1	17	article	article	NOUN
ejpam-7044	1	18	number	number	NOUN
ejpam-7044	1	19	7044	7044	NUM
ejpam-7044	1	20	issn	issn	PROPN
ejpam-7044	1	21	1307	1307	NUM
ejpam-7044	1	22	-	-	SYM
ejpam-7044	1	23	5543	5543	NUM
ejpam-7044	1	24	–	–	PUNCT
ejpam-7044	1	25	ejpam.com	ejpam.com	X
ejpam-7044	1	26	published	publish	VERB
ejpam-7044	1	27	by	by	ADP
ejpam-7044	1	28	new	new	PROPN
ejpam-7044	1	29	york	york	PROPN
ejpam-7044	1	30	business	business	PROPN
ejpam-7044	1	31	global	global	PROPN
ejpam-7044	1	32	upper	upper	ADJ
ejpam-7044	1	33	and	and	CCONJ
ejpam-7044	1	34	lower	low	ADJ
ejpam-7044	1	35	τ	τ	X
ejpam-7044	1	36	⋆α(σ1	⋆α(σ1	X
ejpam-7044	1	37	,	,	PUNCT
ejpam-7044	1	38	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7044	1	39	jeeranunt	jeeranunt	NOUN
ejpam-7044	1	40	khampakdee1	khampakdee1	PROPN
ejpam-7044	1	41	,	,	PUNCT
ejpam-7044	1	42	areeyuth	areeyuth	NOUN
ejpam-7044	1	43	sama	sama	NOUN
ejpam-7044	1	44	-	-	PUNCT
ejpam-7044	1	45	ae2	ae2	PROPN
ejpam-7044	1	46	,	,	PUNCT
ejpam-7044	1	47	chawalit	chawalit	VERB
ejpam-7044	1	48	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7044	1	49	1	1	NUM
ejpam-7044	1	50	mathematics	mathematic	NOUN
ejpam-7044	1	51	and	and	CCONJ
ejpam-7044	1	52	applied	apply	VERB
ejpam-7044	1	53	mathematics	mathematics	PROPN
ejpam-7044	1	54	research	research	NOUN
ejpam-7044	1	55	unit	unit	NOUN
ejpam-7044	1	56	,	,	PUNCT
ejpam-7044	1	57	department	department	NOUN
ejpam-7044	1	58	of	of	ADP
ejpam-7044	1	59	mathematics	mathematic	NOUN
ejpam-7044	1	60	,	,	PUNCT
ejpam-7044	1	61	faculty	faculty	NOUN
ejpam-7044	1	62	of	of	ADP
ejpam-7044	1	63	science	science	NOUN
ejpam-7044	1	64	,	,	PUNCT
ejpam-7044	1	65	mahasarakham	mahasarakham	PROPN
ejpam-7044	1	66	university	university	PROPN
ejpam-7044	1	67	,	,	PUNCT
ejpam-7044	1	68	maha	maha	PROPN
ejpam-7044	1	69	sarakham	sarakham	PROPN
ejpam-7044	1	70	,	,	PUNCT
ejpam-7044	1	71	44150	44150	NUM
ejpam-7044	1	72	,	,	PUNCT
ejpam-7044	1	73	thailand	thailand	PROPN
ejpam-7044	1	74	2	2	NUM
ejpam-7044	1	75	department	department	NOUN
ejpam-7044	1	76	of	of	ADP
ejpam-7044	1	77	mathematics	mathematic	NOUN
ejpam-7044	1	78	and	and	CCONJ
ejpam-7044	1	79	computer	computer	NOUN
ejpam-7044	1	80	science	science	NOUN
ejpam-7044	1	81	,	,	PUNCT
ejpam-7044	1	82	faculty	faculty	NOUN
ejpam-7044	1	83	of	of	ADP
ejpam-7044	1	84	science	science	NOUN
ejpam-7044	1	85	and	and	CCONJ
ejpam-7044	1	86	technology	technology	NOUN
ejpam-7044	1	87	,	,	PUNCT
ejpam-7044	1	88	prince	prince	NOUN
ejpam-7044	1	89	of	of	ADP
ejpam-7044	1	90	songkla	songkla	PROPN
ejpam-7044	1	91	university	university	PROPN
ejpam-7044	1	92	,	,	PUNCT
ejpam-7044	1	93	pattani	pattani	PROPN
ejpam-7044	1	94	campus	campus	PROPN
ejpam-7044	1	95	,	,	PUNCT
ejpam-7044	1	96	patt	patt	PROPN
ejpam-7044	1	97	ronald	ronald	PROPN
ejpam-7044	1	98	94000	94000	NUM
ejpam-7044	1	99	,	,	PUNCT
ejpam-7044	1	100	thailand	thailand	PROPN
ejpam-7044	1	101	abstract	abstract	PROPN
ejpam-7044	1	102	.	.	PUNCT
ejpam-7044	2	1	a	a	DET
ejpam-7044	2	2	new	new	ADJ
ejpam-7044	2	3	class	class	NOUN
ejpam-7044	2	4	of	of	ADP
ejpam-7044	2	5	continuous	continuous	ADJ
ejpam-7044	2	6	multifunctions	multifunction	NOUN
ejpam-7044	2	7	between	between	ADP
ejpam-7044	2	8	an	an	DET
ejpam-7044	2	9	ideal	ideal	ADJ
ejpam-7044	2	10	topological	topological	ADJ
ejpam-7044	2	11	space	space	NOUN
ejpam-7044	2	12	and	and	CCONJ
ejpam-7044	2	13	a	a	DET
ejpam-7044	2	14	bitopological	bitopological	ADJ
ejpam-7044	2	15	space	space	NOUN
ejpam-7044	2	16	,	,	PUNCT
ejpam-7044	2	17	called	call	VERB
ejpam-7044	2	18	upper	upper	ADJ
ejpam-7044	2	19	(	(	PUNCT
ejpam-7044	2	20	lower	low	ADJ
ejpam-7044	2	21	)	)	PUNCT
ejpam-7044	2	22	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	2	23	,	,	PUNCT
ejpam-7044	2	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	2	25	multifunctions	multifunction	NOUN
ejpam-7044	2	26	,	,	PUNCT
ejpam-7044	2	27	has	have	AUX
ejpam-7044	2	28	been	be	AUX
ejpam-7044	2	29	defined	define	VERB
ejpam-7044	2	30	and	and	CCONJ
ejpam-7044	2	31	studied	study	VERB
ejpam-7044	2	32	.	.	PUNCT
ejpam-7044	3	1	moreover	moreover	ADV
ejpam-7044	3	2	,	,	PUNCT
ejpam-7044	3	3	several	several	ADJ
ejpam-7044	3	4	characterizations	characterization	NOUN
ejpam-7044	3	5	and	and	CCONJ
ejpam-7044	3	6	some	some	DET
ejpam-7044	3	7	properties	property	NOUN
ejpam-7044	3	8	concerning	concern	VERB
ejpam-7044	3	9	upper	upper	ADJ
ejpam-7044	3	10	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	3	11	,	,	PUNCT
ejpam-7044	3	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	3	13	multifunctions	multifunction	NOUN
ejpam-7044	3	14	and	and	CCONJ
ejpam-7044	3	15	lower	low	ADJ
ejpam-7044	3	16	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	3	17	,	,	PUNCT
ejpam-7044	3	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	3	19	multifunctions	multifunction	NOUN
ejpam-7044	3	20	are	be	AUX
ejpam-7044	3	21	established	establish	VERB
ejpam-7044	3	22	.	.	PUNCT
ejpam-7044	4	1	2020	2020	NUM
ejpam-7044	4	2	mathematics	mathematics	PROPN
ejpam-7044	4	3	subject	subject	NOUN
ejpam-7044	4	4	classifications	classification	NOUN
ejpam-7044	4	5	:	:	PUNCT
ejpam-7044	4	6	54c08	54c08	NUM
ejpam-7044	4	7	,	,	PUNCT
ejpam-7044	4	8	54c60	54c60	NUM
ejpam-7044	4	9	key	key	ADJ
ejpam-7044	4	10	words	word	NOUN
ejpam-7044	4	11	and	and	CCONJ
ejpam-7044	4	12	phrases	phrase	NOUN
ejpam-7044	4	13	:	:	PUNCT
ejpam-7044	4	14	upper	upper	ADJ
ejpam-7044	4	15	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	4	16	,	,	PUNCT
ejpam-7044	4	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	4	18	multifunction	multifunction	NOUN
ejpam-7044	4	19	,	,	PUNCT
ejpam-7044	4	20	lower	low	ADJ
ejpam-7044	4	21	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	4	22	,	,	PUNCT
ejpam-7044	4	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	4	24	multifunction	multifunction	NOUN
ejpam-7044	4	25	1	1	NUM
ejpam-7044	4	26	.	.	PUNCT
ejpam-7044	4	27	introduction	introduction	NOUN
ejpam-7044	4	28	in	in	ADP
ejpam-7044	4	29	1982	1982	NUM
ejpam-7044	4	30	,	,	PUNCT
ejpam-7044	4	31	noiri	noiri	ADV
ejpam-7044	4	32	[	[	X
ejpam-7044	4	33	1	1	X
ejpam-7044	4	34	]	]	PUNCT
ejpam-7044	4	35	introduced	introduce	VERB
ejpam-7044	4	36	a	a	DET
ejpam-7044	4	37	class	class	NOUN
ejpam-7044	4	38	of	of	ADP
ejpam-7044	4	39	functions	function	NOUN
ejpam-7044	4	40	defined	define	VERB
ejpam-7044	4	41	between	between	ADP
ejpam-7044	4	42	topological	topological	ADJ
ejpam-7044	4	43	spaces	space	NOUN
ejpam-7044	4	44	,	,	PUNCT
ejpam-7044	4	45	namely	namely	ADV
ejpam-7044	4	46	strongly	strongly	ADV
ejpam-7044	4	47	semi	semi	ADJ
ejpam-7044	4	48	-	-	ADJ
ejpam-7044	4	49	continuous	continuous	ADJ
ejpam-7044	4	50	functions	function	NOUN
ejpam-7044	4	51	.	.	PUNCT
ejpam-7044	5	1	mashhour	mashhour	INTJ
ejpam-7044	5	2	et	et	PROPN
ejpam-7044	5	3	al	al	PROPN
ejpam-7044	5	4	.	.	PUNCT
ejpam-7044	6	1	[	[	X
ejpam-7044	6	2	2	2	X
ejpam-7044	6	3	]	]	PUNCT
ejpam-7044	6	4	called	call	VERB
ejpam-7044	6	5	strongly	strongly	ADV
ejpam-7044	6	6	semicontinuous	semicontinuous	ADJ
ejpam-7044	6	7	functions	function	NOUN
ejpam-7044	6	8	α	α	PRON
ejpam-7044	6	9	-	-	ADJ
ejpam-7044	6	10	continuous	continuous	ADJ
ejpam-7044	6	11	functions	function	NOUN
ejpam-7044	6	12	and	and	CCONJ
ejpam-7044	6	13	investigated	investigate	VERB
ejpam-7044	6	14	some	some	DET
ejpam-7044	6	15	characterizations	characterization	NOUN
ejpam-7044	6	16	of	of	ADP
ejpam-7044	6	17	such	such	ADJ
ejpam-7044	6	18	functions	function	NOUN
ejpam-7044	6	19	.	.	PUNCT
ejpam-7044	7	1	in	in	ADP
ejpam-7044	7	2	1986	1986	NUM
ejpam-7044	7	3	,	,	PUNCT
ejpam-7044	7	4	neubrunn	neubrunn	X
ejpam-7044	8	1	[	[	X
ejpam-7044	8	2	3	3	NUM
ejpam-7044	8	3	]	]	PUNCT
ejpam-7044	8	4	extended	extend	VERB
ejpam-7044	8	5	the	the	DET
ejpam-7044	8	6	concept	concept	NOUN
ejpam-7044	8	7	of	of	ADP
ejpam-7044	8	8	α	α	NOUN
ejpam-7044	8	9	-	-	ADJ
ejpam-7044	8	10	continuous	continuous	ADJ
ejpam-7044	8	11	functions	function	NOUN
ejpam-7044	8	12	to	to	ADP
ejpam-7044	8	13	multifunctions	multifunction	NOUN
ejpam-7044	8	14	and	and	CCONJ
ejpam-7044	8	15	presented	present	VERB
ejpam-7044	8	16	two	two	NUM
ejpam-7044	8	17	classes	class	NOUN
ejpam-7044	8	18	of	of	ADP
ejpam-7044	8	19	multifunctions	multifunction	NOUN
ejpam-7044	8	20	defined	define	VERB
ejpam-7044	8	21	from	from	ADP
ejpam-7044	8	22	a	a	DET
ejpam-7044	8	23	topological	topological	ADJ
ejpam-7044	8	24	space	space	NOUN
ejpam-7044	8	25	into	into	ADP
ejpam-7044	8	26	a	a	DET
ejpam-7044	8	27	topological	topological	ADJ
ejpam-7044	8	28	space	space	NOUN
ejpam-7044	8	29	,	,	PUNCT
ejpam-7044	8	30	called	call	VERB
ejpam-7044	8	31	upper	upper	ADJ
ejpam-7044	8	32	α	α	ADJ
ejpam-7044	8	33	-	-	ADJ
ejpam-7044	8	34	continuous	continuous	ADJ
ejpam-7044	8	35	multifunctions	multifunction	NOUN
ejpam-7044	8	36	and	and	CCONJ
ejpam-7044	8	37	lower	low	ADJ
ejpam-7044	8	38	αcontinuous	αcontinuous	ADJ
ejpam-7044	8	39	multifunctions	multifunction	NOUN
ejpam-7044	8	40	.	.	PUNCT
ejpam-7044	9	1	in	in	ADP
ejpam-7044	9	2	1993	1993	NUM
ejpam-7044	9	3	,	,	PUNCT
ejpam-7044	9	4	popa	popa	NOUN
ejpam-7044	9	5	and	and	CCONJ
ejpam-7044	9	6	noiri	noiri	ADV
ejpam-7044	9	7	[	[	X
ejpam-7044	9	8	4	4	NUM
ejpam-7044	9	9	]	]	PUNCT
ejpam-7044	9	10	obtained	obtain	VERB
ejpam-7044	9	11	several	several	ADJ
ejpam-7044	9	12	characterizations	characterization	NOUN
ejpam-7044	9	13	and	and	CCONJ
ejpam-7044	9	14	some	some	DET
ejpam-7044	9	15	basic	basic	ADJ
ejpam-7044	9	16	properties	property	NOUN
ejpam-7044	9	17	of	of	ADP
ejpam-7044	9	18	upper	upper	ADJ
ejpam-7044	9	19	α	α	ADJ
ejpam-7044	9	20	-	-	ADJ
ejpam-7044	9	21	continuous	continuous	ADJ
ejpam-7044	9	22	multifunctions	multifunction	NOUN
ejpam-7044	9	23	and	and	CCONJ
ejpam-7044	9	24	lower	low	ADJ
ejpam-7044	9	25	α	α	ADJ
ejpam-7044	9	26	-	-	ADJ
ejpam-7044	9	27	continuous	continuous	ADJ
ejpam-7044	9	28	multifunctions	multifunction	NOUN
ejpam-7044	9	29	.	.	PUNCT
ejpam-7044	10	1	on	on	ADP
ejpam-7044	10	2	the	the	DET
ejpam-7044	10	3	other	other	ADJ
ejpam-7044	10	4	hand	hand	NOUN
ejpam-7044	10	5	,	,	PUNCT
ejpam-7044	10	6	the	the	DET
ejpam-7044	10	7	present	present	ADJ
ejpam-7044	10	8	author	author	NOUN
ejpam-7044	10	9	introduced	introduce	VERB
ejpam-7044	10	10	and	and	CCONJ
ejpam-7044	10	11	investigated	investigate	VERB
ejpam-7044	10	12	four	four	NUM
ejpam-7044	10	13	classes	class	NOUN
ejpam-7044	10	14	of	of	ADP
ejpam-7044	10	15	multifunctions	multifunction	NOUN
ejpam-7044	10	16	defined	define	VERB
ejpam-7044	10	17	from	from	ADP
ejpam-7044	10	18	an	an	DET
ejpam-7044	10	19	ideal	ideal	ADJ
ejpam-7044	10	20	topological	topological	ADJ
ejpam-7044	10	21	space	space	NOUN
ejpam-7044	10	22	into	into	ADP
ejpam-7044	10	23	an	an	DET
ejpam-7044	10	24	ideal	ideal	ADJ
ejpam-7044	10	25	topological	topological	ADJ
ejpam-7044	10	26	space	space	NOUN
ejpam-7044	10	27	,	,	PUNCT
ejpam-7044	10	28	namely	namely	ADV
ejpam-7044	10	29	upper	upper	ADJ
ejpam-7044	10	30	⋆-continuous	⋆-continuous	ADJ
ejpam-7044	10	31	multifunctions	multifunction	NOUN
ejpam-7044	11	1	[	[	X
ejpam-7044	11	2	5	5	NUM
ejpam-7044	11	3	]	]	PUNCT
ejpam-7044	11	4	,	,	PUNCT
ejpam-7044	11	5	lower	low	ADJ
ejpam-7044	11	6	⋆-continuous	⋆-continuous	ADJ
ejpam-7044	11	7	multifunctions	multifunction	NOUN
ejpam-7044	12	1	[	[	X
ejpam-7044	12	2	5	5	NUM
ejpam-7044	12	3	]	]	PUNCT
ejpam-7044	12	4	,	,	PUNCT
ejpam-7044	12	5	upper	upper	ADJ
ejpam-7044	12	6	α(⋆)-continuous	α(⋆)-continuous	ADJ
ejpam-7044	12	7	multifunctions	multifunction	NOUN
ejpam-7044	13	1	[	[	X
ejpam-7044	13	2	6	6	NUM
ejpam-7044	13	3	]	]	PUNCT
ejpam-7044	13	4	,	,	PUNCT
ejpam-7044	13	5	lower	low	ADJ
ejpam-7044	13	6	α(⋆)-continuous	α(⋆)-continuous	ADJ
ejpam-7044	13	7	multifunctions	multifunction	NOUN
ejpam-7044	14	1	[	[	X
ejpam-7044	14	2	6	6	NUM
ejpam-7044	14	3	]	]	PUNCT
ejpam-7044	14	4	,	,	PUNCT
ejpam-7044	14	5	upper	upper	ADJ
ejpam-7044	14	6	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-7044	14	7	multifunctions	multifunction	NOUN
ejpam-7044	15	1	[	[	X
ejpam-7044	15	2	7	7	NUM
ejpam-7044	15	3	]	]	PUNCT
ejpam-7044	15	4	,	,	PUNCT
ejpam-7044	15	5	lower	low	ADJ
ejpam-7044	15	6	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-7044	15	7	multifunctions	multifunction	NOUN
ejpam-7044	15	8	[	[	X
ejpam-7044	15	9	7	7	NUM
ejpam-7044	15	10	]	]	PUNCT
ejpam-7044	15	11	,	,	PUNCT
ejpam-7044	15	12	upper	upper	ADJ
ejpam-7044	15	13	∗corresponding	∗corresponding	NOUN
ejpam-7044	15	14	author	author	NOUN
ejpam-7044	15	15	.	.	PUNCT
ejpam-7044	16	1	doi	doi	NOUN
ejpam-7044	16	2	:	:	PUNCT
ejpam-7044	16	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7044	https://doi.org/10.29020/nybg.ejpam.v18i4.7044	DET
ejpam-7044	16	4	email	email	NOUN
ejpam-7044	16	5	addresses	address	VERB
ejpam-7044	16	6	:	:	PUNCT
ejpam-7044	16	7	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-7044	16	8	(	(	PUNCT
ejpam-7044	16	9	j.	j.	PROPN
ejpam-7044	16	10	khampakdee	khampakdee	PROPN
ejpam-7044	16	11	)	)	PUNCT
ejpam-7044	16	12	,	,	PUNCT
ejpam-7044	16	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7044	16	14	(	(	PUNCT
ejpam-7044	16	15	a.	a.	PROPN
ejpam-7044	16	16	sama	sama	PROPN
ejpam-7044	16	17	-	-	PUNCT
ejpam-7044	16	18	ae	ae	PROPN
ejpam-7044	16	19	)	)	PUNCT
ejpam-7044	16	20	,	,	PUNCT
ejpam-7044	16	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7044	16	22	(	(	PUNCT
ejpam-7044	16	23	c.	c.	PROPN
ejpam-7044	16	24	boonpok	boonpok	PROPN
ejpam-7044	16	25	)	)	PUNCT
ejpam-7044	16	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7044	17	1	1	1	NUM
ejpam-7044	17	2	copyright	copyright	NOUN
ejpam-7044	17	3	:	:	PUNCT
ejpam-7044	17	4	©	©	PROPN
ejpam-7044	17	5	2025	2025	NUM
ejpam-7044	17	6	the	the	DET
ejpam-7044	17	7	author(s	author(s	NOUN
ejpam-7044	17	8	)	)	PUNCT
ejpam-7044	17	9	.	.	PUNCT
ejpam-7044	18	1	(	(	PUNCT
ejpam-7044	18	2	cc	cc	NOUN
ejpam-7044	18	3	by	by	ADP
ejpam-7044	18	4	-	-	PUNCT
ejpam-7044	18	5	nc	nc	PROPN
ejpam-7044	18	6	4.0	4.0	NUM
ejpam-7044	18	7	)	)	PUNCT
ejpam-7044	18	8	j.	j.	PROPN
ejpam-7044	18	9	khampakdee	khampakdee	PROPN
ejpam-7044	18	10	,	,	PUNCT
ejpam-7044	18	11	a.	a.	PROPN
ejpam-7044	18	12	sama	sama	PROPN
ejpam-7044	18	13	-	-	PUNCT
ejpam-7044	18	14	ae	ae	PROPN
ejpam-7044	18	15	,	,	PUNCT
ejpam-7044	18	16	c.	c.	PROPN
ejpam-7044	18	17	boonpok	boonpok	PROPN
ejpam-7044	18	18	/	/	SYM
ejpam-7044	18	19	eur	eur	PROPN
ejpam-7044	18	20	.	.	PUNCT
ejpam-7044	19	1	j.	j.	PROPN
ejpam-7044	19	2	pure	pure	PROPN
ejpam-7044	19	3	appl	appl	PROPN
ejpam-7044	19	4	.	.	PROPN
ejpam-7044	19	5	math	math	PROPN
ejpam-7044	19	6	,	,	PUNCT
ejpam-7044	19	7	18	18	NUM
ejpam-7044	19	8	(	(	PUNCT
ejpam-7044	19	9	4	4	NUM
ejpam-7044	19	10	)	)	PUNCT
ejpam-7044	19	11	(	(	PUNCT
ejpam-7044	19	12	2025	2025	NUM
ejpam-7044	19	13	)	)	PUNCT
ejpam-7044	19	14	,	,	PUNCT
ejpam-7044	19	15	7044	7044	NUM
ejpam-7044	19	16	2	2	NUM
ejpam-7044	19	17	of	of	ADP
ejpam-7044	19	18	9	9	NUM
ejpam-7044	19	19	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7044	19	20	multifunctions	multifunction	NOUN
ejpam-7044	20	1	[	[	X
ejpam-7044	20	2	8	8	NUM
ejpam-7044	20	3	]	]	PUNCT
ejpam-7044	20	4	,	,	PUNCT
ejpam-7044	20	5	lower	low	ADJ
ejpam-7044	20	6	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7044	20	7	multifunctions	multifunction	NOUN
ejpam-7044	20	8	[	[	X
ejpam-7044	20	9	8	8	NUM
ejpam-7044	20	10	]	]	PUNCT
ejpam-7044	20	11	,	,	PUNCT
ejpam-7044	20	12	upper	upper	ADJ
ejpam-7044	20	13	α-⋆continuous	α-⋆continuous	ADJ
ejpam-7044	20	14	multifunctions	multifunction	NOUN
ejpam-7044	21	1	[	[	X
ejpam-7044	21	2	9	9	NUM
ejpam-7044	21	3	]	]	PUNCT
ejpam-7044	21	4	,	,	PUNCT
ejpam-7044	21	5	lower	low	ADJ
ejpam-7044	21	6	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7044	21	7	multifunctions	multifunction	NOUN
ejpam-7044	22	1	[	[	X
ejpam-7044	22	2	9	9	NUM
ejpam-7044	22	3	]	]	PUNCT
ejpam-7044	22	4	,	,	PUNCT
ejpam-7044	22	5	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-7044	22	6	multifunctions	multifunction	NOUN
ejpam-7044	23	1	[	[	X
ejpam-7044	23	2	10	10	NUM
ejpam-7044	23	3	]	]	PUNCT
ejpam-7044	23	4	and	and	CCONJ
ejpam-7044	23	5	pı	pı	ADJ
ejpam-7044	23	6	-	-	ADJ
ejpam-7044	23	7	continuous	continuous	ADJ
ejpam-7044	23	8	multifunctions	multifunction	NOUN
ejpam-7044	24	1	[	[	X
ejpam-7044	24	2	11	11	NUM
ejpam-7044	24	3	]	]	PUNCT
ejpam-7044	24	4	.	.	PUNCT
ejpam-7044	25	1	pue	pue	NOUN
ejpam-7044	25	2	-	-	PUNCT
ejpam-7044	25	3	on	on	NOUN
ejpam-7044	25	4	et	et	PROPN
ejpam-7044	25	5	al	al	PROPN
ejpam-7044	25	6	.	.	PUNCT
ejpam-7044	26	1	[	[	X
ejpam-7044	26	2	12	12	NUM
ejpam-7044	26	3	]	]	PUNCT
ejpam-7044	26	4	introduced	introduce	VERB
ejpam-7044	26	5	and	and	CCONJ
ejpam-7044	26	6	studied	study	VERB
ejpam-7044	26	7	two	two	NUM
ejpam-7044	26	8	classes	class	NOUN
ejpam-7044	26	9	of	of	ADP
ejpam-7044	26	10	multifunctions	multifunction	NOUN
ejpam-7044	26	11	between	between	ADP
ejpam-7044	26	12	bitopological	bitopological	ADJ
ejpam-7044	26	13	spaces	space	NOUN
ejpam-7044	26	14	,	,	PUNCT
ejpam-7044	26	15	namely	namely	ADV
ejpam-7044	26	16	upper	upper	ADJ
ejpam-7044	26	17	(	(	PUNCT
ejpam-7044	26	18	τ1	τ1	NOUN
ejpam-7044	26	19	,	,	PUNCT
ejpam-7044	26	20	τ2)continuous	τ2)continuous	ADJ
ejpam-7044	26	21	multifunctions	multifunction	NOUN
ejpam-7044	26	22	and	and	CCONJ
ejpam-7044	26	23	lower	low	ADJ
ejpam-7044	26	24	(	(	PUNCT
ejpam-7044	26	25	τ1	τ1	NOUN
ejpam-7044	26	26	,	,	PUNCT
ejpam-7044	26	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	26	28	multifunctions	multifunction	NOUN
ejpam-7044	26	29	.	.	PUNCT
ejpam-7044	27	1	klanarong	klanarong	NOUN
ejpam-7044	27	2	et	et	PROPN
ejpam-7044	27	3	al	al	PROPN
ejpam-7044	27	4	.	.	PUNCT
ejpam-7044	28	1	[	[	X
ejpam-7044	28	2	13	13	NUM
ejpam-7044	28	3	]	]	PUNCT
ejpam-7044	28	4	investigated	investigate	VERB
ejpam-7044	28	5	several	several	ADJ
ejpam-7044	28	6	characterizations	characterization	NOUN
ejpam-7044	28	7	of	of	ADP
ejpam-7044	28	8	upper	upper	ADJ
ejpam-7044	28	9	(	(	PUNCT
ejpam-7044	28	10	τ1	τ1	NOUN
ejpam-7044	28	11	,	,	PUNCT
ejpam-7044	28	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	28	13	multifunctions	multifunction	NOUN
ejpam-7044	28	14	and	and	CCONJ
ejpam-7044	28	15	lower	low	ADJ
ejpam-7044	28	16	(	(	PUNCT
ejpam-7044	28	17	τ1	τ1	NOUN
ejpam-7044	28	18	,	,	PUNCT
ejpam-7044	28	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	28	20	multifunctions	multifunction	NOUN
ejpam-7044	28	21	by	by	ADP
ejpam-7044	28	22	utilizing	utilize	VERB
ejpam-7044	28	23	the	the	DET
ejpam-7044	28	24	notions	notion	NOUN
ejpam-7044	28	25	of	of	ADP
ejpam-7044	28	26	(	(	PUNCT
ejpam-7044	28	27	τ1	τ1	NOUN
ejpam-7044	28	28	,	,	PUNCT
ejpam-7044	28	29	τ2)θ	τ2)θ	ADJ
ejpam-7044	28	30	-	-	PUNCT
ejpam-7044	28	31	closed	close	VERB
ejpam-7044	28	32	sets	set	NOUN
ejpam-7044	28	33	and	and	CCONJ
ejpam-7044	28	34	(	(	PUNCT
ejpam-7044	28	35	τ1	τ1	NOUN
ejpam-7044	28	36	,	,	PUNCT
ejpam-7044	28	37	τ2)θ	τ2)θ	ADJ
ejpam-7044	28	38	-	-	PUNCT
ejpam-7044	28	39	open	open	ADJ
ejpam-7044	28	40	sets	set	NOUN
ejpam-7044	28	41	.	.	PUNCT
ejpam-7044	29	1	thongmoon	thongmoon	NOUN
ejpam-7044	29	2	et	et	PROPN
ejpam-7044	29	3	al	al	PROPN
ejpam-7044	29	4	.	.	PUNCT
ejpam-7044	30	1	[	[	X
ejpam-7044	30	2	14	14	NUM
ejpam-7044	30	3	]	]	PUNCT
ejpam-7044	30	4	studied	study	VERB
ejpam-7044	30	5	some	some	DET
ejpam-7044	30	6	characterizations	characterization	NOUN
ejpam-7044	30	7	of	of	ADP
ejpam-7044	30	8	upper	upper	ADJ
ejpam-7044	30	9	(	(	PUNCT
ejpam-7044	30	10	τ1	τ1	NOUN
ejpam-7044	30	11	,	,	PUNCT
ejpam-7044	30	12	τ2)continuous	τ2)continuous	ADJ
ejpam-7044	30	13	multifunctions	multifunction	NOUN
ejpam-7044	30	14	and	and	CCONJ
ejpam-7044	30	15	lower	low	ADJ
ejpam-7044	30	16	(	(	PUNCT
ejpam-7044	30	17	τ1	τ1	NOUN
ejpam-7044	30	18	,	,	PUNCT
ejpam-7044	30	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	30	20	multifunctions	multifunction	NOUN
ejpam-7044	30	21	by	by	ADP
ejpam-7044	30	22	using	use	VERB
ejpam-7044	30	23	τ1τ2	τ1τ2	ADJ
ejpam-7044	30	24	-	-	ADJ
ejpam-7044	30	25	δopen	δopen	ADJ
ejpam-7044	30	26	sets	set	NOUN
ejpam-7044	30	27	and	and	CCONJ
ejpam-7044	30	28	τ1τ2	τ1τ2	NOUN
ejpam-7044	30	29	-	-	ADJ
ejpam-7044	30	30	δ	δ	NOUN
ejpam-7044	30	31	-	-	PUNCT
ejpam-7044	30	32	closed	close	VERB
ejpam-7044	30	33	sets	set	NOUN
ejpam-7044	30	34	.	.	PUNCT
ejpam-7044	31	1	in	in	ADP
ejpam-7044	31	2	[	[	X
ejpam-7044	31	3	15	15	NUM
ejpam-7044	31	4	]	]	PUNCT
ejpam-7044	31	5	,	,	PUNCT
ejpam-7044	31	6	the	the	DET
ejpam-7044	31	7	present	present	ADJ
ejpam-7044	31	8	authors	author	NOUN
ejpam-7044	31	9	introduced	introduce	VERB
ejpam-7044	31	10	and	and	CCONJ
ejpam-7044	31	11	investigated	investigate	VERB
ejpam-7044	31	12	the	the	DET
ejpam-7044	31	13	concepts	concept	NOUN
ejpam-7044	31	14	of	of	ADP
ejpam-7044	31	15	upper	upper	ADJ
ejpam-7044	31	16	(	(	PUNCT
ejpam-7044	31	17	τ1	τ1	NOUN
ejpam-7044	31	18	,	,	PUNCT
ejpam-7044	31	19	τ2)α	τ2)α	ADJ
ejpam-7044	31	20	-	-	PUNCT
ejpam-7044	31	21	continuous	continuous	ADJ
ejpam-7044	31	22	multifunctions	multifunction	NOUN
ejpam-7044	31	23	and	and	CCONJ
ejpam-7044	31	24	lower	low	ADJ
ejpam-7044	31	25	(	(	PUNCT
ejpam-7044	31	26	τ1	τ1	NOUN
ejpam-7044	31	27	,	,	PUNCT
ejpam-7044	31	28	τ2)α	τ2)α	ADJ
ejpam-7044	31	29	-	-	PUNCT
ejpam-7044	31	30	continuous	continuous	ADJ
ejpam-7044	31	31	multifunctions	multifunction	NOUN
ejpam-7044	31	32	.	.	PUNCT
ejpam-7044	32	1	quite	quite	ADV
ejpam-7044	32	2	recently	recently	ADV
ejpam-7044	32	3	,	,	PUNCT
ejpam-7044	32	4	khampakdee	khampakdee	PROPN
ejpam-7044	32	5	et	et	NOUN
ejpam-7044	32	6	al	al	PROPN
ejpam-7044	32	7	.	.	PUNCT
ejpam-7044	33	1	[	[	X
ejpam-7044	33	2	16	16	NUM
ejpam-7044	33	3	]	]	PUNCT
ejpam-7044	33	4	presented	present	VERB
ejpam-7044	33	5	new	new	ADJ
ejpam-7044	33	6	classes	class	NOUN
ejpam-7044	33	7	of	of	ADP
ejpam-7044	33	8	continuous	continuous	ADJ
ejpam-7044	33	9	multifunctions	multifunction	NOUN
ejpam-7044	33	10	defined	define	VERB
ejpam-7044	33	11	from	from	ADP
ejpam-7044	33	12	an	an	DET
ejpam-7044	33	13	ideal	ideal	ADJ
ejpam-7044	33	14	topological	topological	ADJ
ejpam-7044	33	15	space	space	NOUN
ejpam-7044	33	16	into	into	ADP
ejpam-7044	33	17	a	a	DET
ejpam-7044	33	18	bitopological	bitopological	ADJ
ejpam-7044	33	19	space	space	NOUN
ejpam-7044	33	20	,	,	PUNCT
ejpam-7044	33	21	namely	namely	ADV
ejpam-7044	33	22	upper	upper	ADJ
ejpam-7044	33	23	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-7044	33	24	,	,	PUNCT
ejpam-7044	33	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	33	26	multifunctions	multifunction	NOUN
ejpam-7044	33	27	and	and	CCONJ
ejpam-7044	33	28	lower	low	ADJ
ejpam-7044	33	29	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7044	33	30	,	,	PUNCT
ejpam-7044	33	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	33	32	multifunctions	multifunction	NOUN
ejpam-7044	33	33	.	.	PUNCT
ejpam-7044	34	1	in	in	ADP
ejpam-7044	34	2	this	this	DET
ejpam-7044	34	3	paper	paper	NOUN
ejpam-7044	34	4	,	,	PUNCT
ejpam-7044	34	5	we	we	PRON
ejpam-7044	34	6	introduce	introduce	VERB
ejpam-7044	34	7	the	the	DET
ejpam-7044	34	8	concepts	concept	NOUN
ejpam-7044	34	9	of	of	ADP
ejpam-7044	34	10	multifunctions	multifunction	NOUN
ejpam-7044	34	11	between	between	ADP
ejpam-7044	34	12	an	an	DET
ejpam-7044	34	13	ideal	ideal	ADJ
ejpam-7044	34	14	topological	topological	ADJ
ejpam-7044	34	15	space	space	NOUN
ejpam-7044	34	16	and	and	CCONJ
ejpam-7044	34	17	a	a	DET
ejpam-7044	34	18	bitopological	bitopological	ADJ
ejpam-7044	34	19	space	space	NOUN
ejpam-7044	34	20	,	,	PUNCT
ejpam-7044	34	21	called	call	VERB
ejpam-7044	34	22	upper	upper	ADJ
ejpam-7044	34	23	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	34	24	,	,	PUNCT
ejpam-7044	34	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	34	26	multifunctions	multifunction	NOUN
ejpam-7044	34	27	and	and	CCONJ
ejpam-7044	34	28	lower	low	ADJ
ejpam-7044	34	29	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	34	30	,	,	PUNCT
ejpam-7044	34	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	34	32	multifunctions	multifunction	NOUN
ejpam-7044	34	33	.	.	PUNCT
ejpam-7044	35	1	we	we	PRON
ejpam-7044	35	2	also	also	ADV
ejpam-7044	35	3	investigate	investigate	VERB
ejpam-7044	35	4	several	several	ADJ
ejpam-7044	35	5	characterizations	characterization	NOUN
ejpam-7044	35	6	of	of	ADP
ejpam-7044	35	7	upper	upper	ADJ
ejpam-7044	35	8	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	35	9	,	,	PUNCT
ejpam-7044	35	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	35	11	multifunctions	multifunction	NOUN
ejpam-7044	35	12	and	and	CCONJ
ejpam-7044	35	13	lower	low	ADJ
ejpam-7044	35	14	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	35	15	,	,	PUNCT
ejpam-7044	35	16	σ2)continuous	σ2)continuous	ADJ
ejpam-7044	35	17	multifunctions	multifunction	NOUN
ejpam-7044	35	18	.	.	PUNCT
ejpam-7044	36	1	2	2	X
ejpam-7044	36	2	.	.	X
ejpam-7044	36	3	preliminaries	preliminary	NOUN
ejpam-7044	36	4	throughout	throughout	ADP
ejpam-7044	36	5	the	the	DET
ejpam-7044	36	6	present	present	ADJ
ejpam-7044	36	7	paper	paper	NOUN
ejpam-7044	36	8	,	,	PUNCT
ejpam-7044	36	9	spaces	space	NOUN
ejpam-7044	36	10	(	(	PUNCT
ejpam-7044	36	11	x	x	NOUN
ejpam-7044	36	12	,	,	PUNCT
ejpam-7044	36	13	τ1	τ1	NOUN
ejpam-7044	36	14	,	,	PUNCT
ejpam-7044	36	15	τ2	τ2	NOUN
ejpam-7044	36	16	)	)	PUNCT
ejpam-7044	36	17	and	and	CCONJ
ejpam-7044	36	18	(	(	PUNCT
ejpam-7044	36	19	y	y	PROPN
ejpam-7044	36	20	,	,	PUNCT
ejpam-7044	36	21	σ1	σ1	PROPN
ejpam-7044	36	22	,	,	PUNCT
ejpam-7044	36	23	σ2	σ2	NOUN
ejpam-7044	36	24	)	)	PUNCT
ejpam-7044	36	25	(	(	PUNCT
ejpam-7044	36	26	or	or	CCONJ
ejpam-7044	36	27	simply	simply	ADV
ejpam-7044	36	28	x	x	X
ejpam-7044	36	29	and	and	CCONJ
ejpam-7044	36	30	y	y	PROPN
ejpam-7044	36	31	)	)	PUNCT
ejpam-7044	36	32	always	always	ADV
ejpam-7044	36	33	mean	mean	VERB
ejpam-7044	36	34	bitopological	bitopological	ADJ
ejpam-7044	36	35	spaces	space	NOUN
ejpam-7044	36	36	on	on	ADP
ejpam-7044	36	37	which	which	PRON
ejpam-7044	36	38	no	no	DET
ejpam-7044	36	39	separation	separation	NOUN
ejpam-7044	36	40	axioms	axiom	NOUN
ejpam-7044	36	41	are	be	AUX
ejpam-7044	36	42	assumed	assume	VERB
ejpam-7044	36	43	unless	unless	SCONJ
ejpam-7044	36	44	explicitly	explicitly	ADV
ejpam-7044	36	45	stated	state	VERB
ejpam-7044	36	46	.	.	PUNCT
ejpam-7044	37	1	let	let	VERB
ejpam-7044	37	2	a	a	DET
ejpam-7044	37	3	be	be	AUX
ejpam-7044	37	4	a	a	DET
ejpam-7044	37	5	subset	subset	NOUN
ejpam-7044	37	6	of	of	ADP
ejpam-7044	37	7	a	a	DET
ejpam-7044	37	8	bitopological	bitopological	ADJ
ejpam-7044	37	9	space	space	NOUN
ejpam-7044	37	10	(	(	PUNCT
ejpam-7044	37	11	x	x	NOUN
ejpam-7044	37	12	,	,	PUNCT
ejpam-7044	37	13	τ1	τ1	NOUN
ejpam-7044	37	14	,	,	PUNCT
ejpam-7044	37	15	τ2	τ2	NOUN
ejpam-7044	37	16	)	)	PUNCT
ejpam-7044	37	17	.	.	PUNCT
ejpam-7044	38	1	the	the	DET
ejpam-7044	38	2	closure	closure	NOUN
ejpam-7044	38	3	of	of	ADP
ejpam-7044	38	4	a	a	PRON
ejpam-7044	38	5	and	and	CCONJ
ejpam-7044	38	6	the	the	DET
ejpam-7044	38	7	interior	interior	NOUN
ejpam-7044	38	8	of	of	ADP
ejpam-7044	38	9	a	a	PRON
ejpam-7044	38	10	with	with	ADP
ejpam-7044	38	11	respect	respect	NOUN
ejpam-7044	38	12	to	to	ADP
ejpam-7044	38	13	τi	τi	PROPN
ejpam-7044	38	14	are	be	AUX
ejpam-7044	38	15	denoted	denote	VERB
ejpam-7044	38	16	by	by	ADP
ejpam-7044	38	17	τi	τi	NOUN
ejpam-7044	38	18	-	-	PUNCT
ejpam-7044	38	19	cl(a	cl(a	NUM
ejpam-7044	38	20	)	)	PUNCT
ejpam-7044	38	21	and	and	CCONJ
ejpam-7044	38	22	τi	τi	NOUN
ejpam-7044	38	23	-	-	PUNCT
ejpam-7044	38	24	int(a	int(a	NOUN
ejpam-7044	38	25	)	)	PUNCT
ejpam-7044	38	26	,	,	PUNCT
ejpam-7044	38	27	respectively	respectively	ADV
ejpam-7044	38	28	,	,	PUNCT
ejpam-7044	38	29	for	for	ADP
ejpam-7044	38	30	i	i	PROPN
ejpam-7044	38	31	=	=	SYM
ejpam-7044	38	32	1	1	NUM
ejpam-7044	38	33	,	,	PUNCT
ejpam-7044	38	34	2	2	NUM
ejpam-7044	38	35	.	.	X
ejpam-7044	38	36	a	a	DET
ejpam-7044	38	37	subset	subset	NOUN
ejpam-7044	38	38	a	a	PRON
ejpam-7044	38	39	of	of	ADP
ejpam-7044	38	40	a	a	DET
ejpam-7044	38	41	bitopological	bitopological	ADJ
ejpam-7044	38	42	space	space	NOUN
ejpam-7044	38	43	(	(	PUNCT
ejpam-7044	38	44	x	x	NOUN
ejpam-7044	38	45	,	,	PUNCT
ejpam-7044	38	46	τ1	τ1	NOUN
ejpam-7044	38	47	,	,	PUNCT
ejpam-7044	38	48	τ2	τ2	NOUN
ejpam-7044	38	49	)	)	PUNCT
ejpam-7044	38	50	is	be	AUX
ejpam-7044	38	51	called	call	VERB
ejpam-7044	38	52	τ1τ2	τ1τ2	VERB
ejpam-7044	38	53	-	-	ADJ
ejpam-7044	38	54	closed	closed	ADJ
ejpam-7044	38	55	[	[	X
ejpam-7044	38	56	17	17	NUM
ejpam-7044	38	57	]	]	PUNCT
ejpam-7044	38	58	if	if	SCONJ
ejpam-7044	38	59	a	a	DET
ejpam-7044	38	60	=	=	NOUN
ejpam-7044	38	61	τ1	τ1	NOUN
ejpam-7044	38	62	-	-	PUNCT
ejpam-7044	38	63	cl(τ2	cl(τ2	NOUN
ejpam-7044	38	64	-	-	PUNCT
ejpam-7044	38	65	cl(a	cl(a	NUM
ejpam-7044	38	66	)	)	PUNCT
ejpam-7044	38	67	)	)	PUNCT
ejpam-7044	38	68	.	.	PUNCT
ejpam-7044	39	1	the	the	DET
ejpam-7044	39	2	complement	complement	NOUN
ejpam-7044	39	3	of	of	ADP
ejpam-7044	39	4	a	a	DET
ejpam-7044	39	5	τ1τ2	τ1τ2	ADJ
ejpam-7044	39	6	-	-	ADJ
ejpam-7044	39	7	closed	closed	ADJ
ejpam-7044	39	8	set	set	NOUN
ejpam-7044	39	9	is	be	AUX
ejpam-7044	39	10	called	call	VERB
ejpam-7044	39	11	τ1τ2	τ1τ2	NOUN
ejpam-7044	39	12	-	-	ADJ
ejpam-7044	39	13	open	open	ADJ
ejpam-7044	39	14	.	.	PUNCT
ejpam-7044	40	1	the	the	DET
ejpam-7044	40	2	intersection	intersection	NOUN
ejpam-7044	40	3	of	of	ADP
ejpam-7044	40	4	all	all	DET
ejpam-7044	40	5	τ1τ2	τ1τ2	ADJ
ejpam-7044	40	6	-	-	ADJ
ejpam-7044	40	7	closed	closed	ADJ
ejpam-7044	40	8	sets	set	NOUN
ejpam-7044	40	9	of	of	ADP
ejpam-7044	40	10	x	x	PUNCT
ejpam-7044	40	11	containing	contain	VERB
ejpam-7044	40	12	a	a	PRON
ejpam-7044	40	13	is	be	AUX
ejpam-7044	40	14	called	call	VERB
ejpam-7044	40	15	the	the	DET
ejpam-7044	40	16	τ1τ2	τ1τ2	NOUN
ejpam-7044	40	17	-	-	NOUN
ejpam-7044	40	18	closure	closure	NOUN
ejpam-7044	40	19	[	[	X
ejpam-7044	40	20	17	17	NUM
ejpam-7044	40	21	]	]	PUNCT
ejpam-7044	40	22	of	of	ADP
ejpam-7044	40	23	a	a	PRON
ejpam-7044	40	24	and	and	CCONJ
ejpam-7044	40	25	is	be	AUX
ejpam-7044	40	26	denoted	denote	VERB
ejpam-7044	40	27	by	by	ADP
ejpam-7044	40	28	τ1τ2	τ1τ2	NOUN
ejpam-7044	40	29	-	-	NUM
ejpam-7044	40	30	cl(a	cl(a	NUM
ejpam-7044	40	31	)	)	PUNCT
ejpam-7044	40	32	.	.	PUNCT
ejpam-7044	41	1	the	the	DET
ejpam-7044	41	2	union	union	NOUN
ejpam-7044	41	3	of	of	ADP
ejpam-7044	41	4	all	all	DET
ejpam-7044	41	5	τ1τ2	τ1τ2	ADJ
ejpam-7044	41	6	-	-	ADJ
ejpam-7044	41	7	open	open	ADJ
ejpam-7044	41	8	sets	set	NOUN
ejpam-7044	41	9	of	of	ADP
ejpam-7044	41	10	x	x	PUNCT
ejpam-7044	41	11	contained	contain	VERB
ejpam-7044	41	12	in	in	ADP
ejpam-7044	41	13	a	a	PRON
ejpam-7044	41	14	is	be	AUX
ejpam-7044	41	15	called	call	VERB
ejpam-7044	41	16	the	the	DET
ejpam-7044	41	17	τ1τ2	τ1τ2	NOUN
ejpam-7044	41	18	-	-	ADJ
ejpam-7044	41	19	interior	interior	ADJ
ejpam-7044	41	20	[	[	X
ejpam-7044	41	21	17	17	NUM
ejpam-7044	41	22	]	]	PUNCT
ejpam-7044	41	23	of	of	ADP
ejpam-7044	41	24	a	a	PRON
ejpam-7044	41	25	and	and	CCONJ
ejpam-7044	41	26	is	be	AUX
ejpam-7044	41	27	denoted	denote	VERB
ejpam-7044	41	28	by	by	ADP
ejpam-7044	41	29	τ1τ2	τ1τ2	NOUN
ejpam-7044	41	30	-	-	ADJ
ejpam-7044	41	31	int(a	int(a	NOUN
ejpam-7044	41	32	)	)	PUNCT
ejpam-7044	41	33	.	.	PUNCT
ejpam-7044	42	1	lemma	lemma	PROPN
ejpam-7044	42	2	1	1	NUM
ejpam-7044	42	3	.	.	PUNCT
ejpam-7044	43	1	[	[	X
ejpam-7044	43	2	17	17	NUM
ejpam-7044	43	3	]	]	PUNCT
ejpam-7044	43	4	let	let	VERB
ejpam-7044	43	5	a	a	PRON
ejpam-7044	43	6	and	and	CCONJ
ejpam-7044	43	7	b	b	NOUN
ejpam-7044	43	8	be	be	AUX
ejpam-7044	43	9	subsets	subset	NOUN
ejpam-7044	43	10	of	of	ADP
ejpam-7044	43	11	a	a	DET
ejpam-7044	43	12	bitopological	bitopological	ADJ
ejpam-7044	43	13	space	space	NOUN
ejpam-7044	43	14	(	(	PUNCT
ejpam-7044	43	15	x	x	NOUN
ejpam-7044	43	16	,	,	PUNCT
ejpam-7044	43	17	τ1	τ1	NOUN
ejpam-7044	43	18	,	,	PUNCT
ejpam-7044	43	19	τ2	τ2	NOUN
ejpam-7044	43	20	)	)	PUNCT
ejpam-7044	43	21	.	.	PUNCT
ejpam-7044	44	1	for	for	ADP
ejpam-7044	44	2	the	the	DET
ejpam-7044	44	3	τ1τ2	τ1τ2	NOUN
ejpam-7044	44	4	-	-	NOUN
ejpam-7044	44	5	closure	closure	NOUN
ejpam-7044	44	6	,	,	PUNCT
ejpam-7044	44	7	the	the	DET
ejpam-7044	44	8	following	follow	VERB
ejpam-7044	44	9	properties	property	NOUN
ejpam-7044	44	10	hold	hold	VERB
ejpam-7044	44	11	:	:	PUNCT
ejpam-7044	44	12	(	(	PUNCT
ejpam-7044	44	13	1	1	X
ejpam-7044	44	14	)	)	PUNCT
ejpam-7044	44	15	a	a	DET
ejpam-7044	44	16	⊆	⊆	NUM
ejpam-7044	44	17	τ1τ2	τ1τ2	NOUN
ejpam-7044	44	18	-	-	NUM
ejpam-7044	44	19	cl(a	cl(a	NUM
ejpam-7044	44	20	)	)	PUNCT
ejpam-7044	44	21	and	and	CCONJ
ejpam-7044	44	22	τ1τ2	τ1τ2	NOUN
ejpam-7044	44	23	-	-	ADJ
ejpam-7044	44	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7044	44	25	-	-	PUNCT
ejpam-7044	44	26	cl(a	cl(a	NUM
ejpam-7044	44	27	)	)	PUNCT
ejpam-7044	44	28	)	)	PUNCT
ejpam-7044	45	1	=	=	PUNCT
ejpam-7044	45	2	τ1τ2	τ1τ2	NOUN
ejpam-7044	45	3	-	-	NUM
ejpam-7044	45	4	cl(a	cl(a	NUM
ejpam-7044	45	5	)	)	PUNCT
ejpam-7044	45	6	.	.	PUNCT
ejpam-7044	46	1	(	(	PUNCT
ejpam-7044	46	2	2	2	X
ejpam-7044	46	3	)	)	PUNCT
ejpam-7044	46	4	if	if	SCONJ
ejpam-7044	46	5	a	a	DET
ejpam-7044	46	6	⊆	⊆	NUM
ejpam-7044	46	7	b	b	NOUN
ejpam-7044	46	8	,	,	PUNCT
ejpam-7044	46	9	then	then	ADV
ejpam-7044	46	10	τ1τ2	τ1τ2	NOUN
ejpam-7044	46	11	-	-	NUM
ejpam-7044	46	12	cl(a	cl(a	NUM
ejpam-7044	46	13	)	)	PUNCT
ejpam-7044	46	14	⊆	⊆	NUM
ejpam-7044	46	15	τ1τ2	τ1τ2	NOUN
ejpam-7044	46	16	-	-	NOUN
ejpam-7044	46	17	cl(b	cl(b	NOUN
ejpam-7044	46	18	)	)	PUNCT
ejpam-7044	46	19	.	.	PUNCT
ejpam-7044	47	1	(	(	PUNCT
ejpam-7044	47	2	3	3	X
ejpam-7044	47	3	)	)	PUNCT
ejpam-7044	47	4	τ1τ2	τ1τ2	NOUN
ejpam-7044	47	5	-	-	NUM
ejpam-7044	47	6	cl(a	cl(a	NUM
ejpam-7044	47	7	)	)	PUNCT
ejpam-7044	47	8	is	be	AUX
ejpam-7044	47	9	τ1τ2	τ1τ2	NOUN
ejpam-7044	47	10	-	-	ADJ
ejpam-7044	47	11	closed	closed	ADJ
ejpam-7044	47	12	.	.	PUNCT
ejpam-7044	48	1	(	(	PUNCT
ejpam-7044	48	2	4	4	X
ejpam-7044	48	3	)	)	PUNCT
ejpam-7044	48	4	a	a	PRON
ejpam-7044	48	5	is	be	AUX
ejpam-7044	48	6	τ1τ2	τ1τ2	NOUN
ejpam-7044	48	7	-	-	ADJ
ejpam-7044	48	8	closed	closed	ADJ
ejpam-7044	48	9	if	if	SCONJ
ejpam-7044	48	10	and	and	CCONJ
ejpam-7044	48	11	only	only	ADV
ejpam-7044	48	12	if	if	SCONJ
ejpam-7044	48	13	a	a	DET
ejpam-7044	48	14	=	=	PUNCT
ejpam-7044	48	15	τ1τ2	τ1τ2	NOUN
ejpam-7044	48	16	-	-	NUM
ejpam-7044	48	17	cl(a	cl(a	NUM
ejpam-7044	48	18	)	)	PUNCT
ejpam-7044	48	19	.	.	PUNCT
ejpam-7044	49	1	(	(	PUNCT
ejpam-7044	49	2	5	5	X
ejpam-7044	49	3	)	)	PUNCT
ejpam-7044	49	4	τ1τ2	τ1τ2	NOUN
ejpam-7044	49	5	-	-	NOUN
ejpam-7044	49	6	cl(x	cl(x	X
ejpam-7044	49	7	−a	−a	NOUN
ejpam-7044	49	8	)	)	PUNCT
ejpam-7044	50	1	=	=	PUNCT
ejpam-7044	50	2	x	x	X
ejpam-7044	51	1	−	−	ADP
ejpam-7044	51	2	τ1τ2	τ1τ2	NOUN
ejpam-7044	51	3	-	-	ADJ
ejpam-7044	51	4	int(a	int(a	NOUN
ejpam-7044	51	5	)	)	PUNCT
ejpam-7044	51	6	.	.	PUNCT
ejpam-7044	52	1	j.	j.	PROPN
ejpam-7044	52	2	khampakdee	khampakdee	PROPN
ejpam-7044	52	3	,	,	PUNCT
ejpam-7044	52	4	a.	a.	PROPN
ejpam-7044	52	5	sama	sama	PROPN
ejpam-7044	52	6	-	-	PUNCT
ejpam-7044	52	7	ae	ae	PROPN
ejpam-7044	52	8	,	,	PUNCT
ejpam-7044	52	9	c.	c.	PROPN
ejpam-7044	52	10	boonpok	boonpok	PROPN
ejpam-7044	52	11	/	/	SYM
ejpam-7044	52	12	eur	eur	PROPN
ejpam-7044	52	13	.	.	PUNCT
ejpam-7044	53	1	j.	j.	PROPN
ejpam-7044	53	2	pure	pure	PROPN
ejpam-7044	53	3	appl	appl	PROPN
ejpam-7044	53	4	.	.	PROPN
ejpam-7044	53	5	math	math	PROPN
ejpam-7044	53	6	,	,	PUNCT
ejpam-7044	53	7	18	18	NUM
ejpam-7044	53	8	(	(	PUNCT
ejpam-7044	53	9	4	4	NUM
ejpam-7044	53	10	)	)	PUNCT
ejpam-7044	53	11	(	(	PUNCT
ejpam-7044	53	12	2025	2025	NUM
ejpam-7044	53	13	)	)	PUNCT
ejpam-7044	53	14	,	,	PUNCT
ejpam-7044	53	15	7044	7044	NUM
ejpam-7044	53	16	3	3	NUM
ejpam-7044	53	17	of	of	ADP
ejpam-7044	53	18	9	9	NUM
ejpam-7044	53	19	a	a	DET
ejpam-7044	53	20	subset	subset	NOUN
ejpam-7044	53	21	a	a	PRON
ejpam-7044	53	22	of	of	ADP
ejpam-7044	53	23	a	a	DET
ejpam-7044	53	24	bitopological	bitopological	ADJ
ejpam-7044	53	25	space	space	NOUN
ejpam-7044	53	26	(	(	PUNCT
ejpam-7044	53	27	x	x	NOUN
ejpam-7044	53	28	,	,	PUNCT
ejpam-7044	53	29	τ1	τ1	NOUN
ejpam-7044	53	30	,	,	PUNCT
ejpam-7044	53	31	τ2	τ2	NOUN
ejpam-7044	53	32	)	)	PUNCT
ejpam-7044	53	33	is	be	AUX
ejpam-7044	53	34	said	say	VERB
ejpam-7044	53	35	to	to	PART
ejpam-7044	53	36	be	be	AUX
ejpam-7044	53	37	(	(	PUNCT
ejpam-7044	53	38	τ1	τ1	NOUN
ejpam-7044	53	39	,	,	PUNCT
ejpam-7044	53	40	τ2)r	τ2)r	NOUN
ejpam-7044	53	41	-	-	PUNCT
ejpam-7044	53	42	open	open	NOUN
ejpam-7044	53	43	[	[	X
ejpam-7044	53	44	15	15	NUM
ejpam-7044	53	45	]	]	X
ejpam-7044	53	46	(	(	PUNCT
ejpam-7044	53	47	resp	resp	NOUN
ejpam-7044	53	48	.	.	PUNCT
ejpam-7044	54	1	(	(	PUNCT
ejpam-7044	54	2	τ1	τ1	NOUN
ejpam-7044	54	3	,	,	PUNCT
ejpam-7044	54	4	τ2)s	τ2)s	NOUN
ejpam-7044	54	5	-	-	PUNCT
ejpam-7044	54	6	open	open	ADJ
ejpam-7044	54	7	[	[	X
ejpam-7044	54	8	18	18	NUM
ejpam-7044	54	9	]	]	PUNCT
ejpam-7044	54	10	,	,	PUNCT
ejpam-7044	54	11	(	(	PUNCT
ejpam-7044	54	12	τ1	τ1	NOUN
ejpam-7044	54	13	,	,	PUNCT
ejpam-7044	54	14	τ2)p	τ2)p	NOUN
ejpam-7044	54	15	-	-	ADJ
ejpam-7044	54	16	open	open	ADJ
ejpam-7044	54	17	[	[	X
ejpam-7044	54	18	18	18	NUM
ejpam-7044	54	19	]	]	PUNCT
ejpam-7044	54	20	,	,	PUNCT
ejpam-7044	54	21	(	(	PUNCT
ejpam-7044	54	22	τ1	τ1	NOUN
ejpam-7044	54	23	,	,	PUNCT
ejpam-7044	54	24	τ2)β	τ2)β	ADJ
ejpam-7044	54	25	-	-	PUNCT
ejpam-7044	54	26	open	open	NOUN
ejpam-7044	54	27	[	[	X
ejpam-7044	54	28	18	18	NUM
ejpam-7044	54	29	]	]	SYM
ejpam-7044	54	30	)	)	PUNCT
ejpam-7044	54	31	if	if	SCONJ
ejpam-7044	54	32	a	a	DET
ejpam-7044	54	33	=	=	PUNCT
ejpam-7044	54	34	τ1τ2	τ1τ2	NOUN
ejpam-7044	54	35	-	-	NOUN
ejpam-7044	54	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7044	54	37	-	-	PUNCT
ejpam-7044	54	38	cl(a	cl(a	NUM
ejpam-7044	54	39	)	)	PUNCT
ejpam-7044	54	40	)	)	PUNCT
ejpam-7044	54	41	(	(	PUNCT
ejpam-7044	54	42	resp	resp	NOUN
ejpam-7044	54	43	.	.	PUNCT
ejpam-7044	55	1	a	a	DET
ejpam-7044	55	2	⊆	⊆	NUM
ejpam-7044	55	3	τ1τ2	τ1τ2	NOUN
ejpam-7044	55	4	-	-	ADJ
ejpam-7044	55	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7044	55	6	-	-	PUNCT
ejpam-7044	55	7	int(a	int(a	NOUN
ejpam-7044	55	8	)	)	PUNCT
ejpam-7044	55	9	)	)	PUNCT
ejpam-7044	55	10	,	,	PUNCT
ejpam-7044	55	11	a	a	DET
ejpam-7044	55	12	⊆	⊆	NUM
ejpam-7044	55	13	τ1τ2	τ1τ2	NOUN
ejpam-7044	55	14	-	-	NOUN
ejpam-7044	55	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7044	55	16	-	-	PUNCT
ejpam-7044	55	17	cl(a	cl(a	NUM
ejpam-7044	55	18	)	)	PUNCT
ejpam-7044	55	19	)	)	PUNCT
ejpam-7044	55	20	,	,	PUNCT
ejpam-7044	55	21	a	a	DET
ejpam-7044	55	22	⊆	⊆	NUM
ejpam-7044	55	23	τ1τ2	τ1τ2	NOUN
ejpam-7044	55	24	-	-	PUNCT
ejpam-7044	55	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7044	55	26	-	-	PUNCT
ejpam-7044	55	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7044	55	28	-	-	PUNCT
ejpam-7044	55	29	cl(a	cl(a	NUM
ejpam-7044	55	30	)	)	PUNCT
ejpam-7044	55	31	)	)	PUNCT
ejpam-7044	55	32	)	)	PUNCT
ejpam-7044	55	33	)	)	PUNCT
ejpam-7044	55	34	.	.	PUNCT
ejpam-7044	56	1	the	the	DET
ejpam-7044	56	2	complement	complement	NOUN
ejpam-7044	56	3	of	of	ADP
ejpam-7044	56	4	a	a	DET
ejpam-7044	56	5	(	(	PUNCT
ejpam-7044	56	6	τ1	τ1	NOUN
ejpam-7044	56	7	,	,	PUNCT
ejpam-7044	56	8	τ2)r	τ2)r	NOUN
ejpam-7044	56	9	-	-	PUNCT
ejpam-7044	56	10	open	open	ADJ
ejpam-7044	56	11	(	(	PUNCT
ejpam-7044	56	12	resp	resp	NOUN
ejpam-7044	56	13	.	.	PUNCT
ejpam-7044	57	1	(	(	PUNCT
ejpam-7044	57	2	τ1	τ1	NOUN
ejpam-7044	57	3	,	,	PUNCT
ejpam-7044	57	4	τ2)s	τ2)s	NOUN
ejpam-7044	57	5	-	-	PUNCT
ejpam-7044	57	6	open	open	ADJ
ejpam-7044	57	7	,	,	PUNCT
ejpam-7044	57	8	(	(	PUNCT
ejpam-7044	57	9	τ1	τ1	NOUN
ejpam-7044	57	10	,	,	PUNCT
ejpam-7044	57	11	τ2)p	τ2)p	NOUN
ejpam-7044	57	12	-	-	ADJ
ejpam-7044	57	13	open	open	ADJ
ejpam-7044	57	14	,	,	PUNCT
ejpam-7044	57	15	(	(	PUNCT
ejpam-7044	57	16	τ1	τ1	NOUN
ejpam-7044	57	17	,	,	PUNCT
ejpam-7044	57	18	τ2)β	τ2)β	ADJ
ejpam-7044	57	19	-	-	PUNCT
ejpam-7044	57	20	open	open	ADJ
ejpam-7044	57	21	)	)	PUNCT
ejpam-7044	57	22	set	set	NOUN
ejpam-7044	57	23	is	be	AUX
ejpam-7044	57	24	said	say	VERB
ejpam-7044	57	25	to	to	PART
ejpam-7044	57	26	be	be	AUX
ejpam-7044	57	27	(	(	PUNCT
ejpam-7044	57	28	τ1	τ1	NOUN
ejpam-7044	57	29	,	,	PUNCT
ejpam-7044	57	30	τ2)r	τ2)r	NOUN
ejpam-7044	57	31	-	-	PUNCT
ejpam-7044	57	32	closed	closed	ADJ
ejpam-7044	57	33	(	(	PUNCT
ejpam-7044	57	34	resp	resp	NOUN
ejpam-7044	57	35	.	.	PUNCT
ejpam-7044	58	1	(	(	PUNCT
ejpam-7044	58	2	τ1	τ1	NOUN
ejpam-7044	58	3	,	,	PUNCT
ejpam-7044	58	4	τ2)s	τ2)s	NOUN
ejpam-7044	58	5	-	-	PUNCT
ejpam-7044	58	6	closed	closed	ADJ
ejpam-7044	58	7	,	,	PUNCT
ejpam-7044	58	8	(	(	PUNCT
ejpam-7044	58	9	τ1	τ1	NOUN
ejpam-7044	58	10	,	,	PUNCT
ejpam-7044	58	11	τ2)p	τ2)p	NOUN
ejpam-7044	58	12	-	-	PUNCT
ejpam-7044	58	13	closed	closed	ADJ
ejpam-7044	58	14	,	,	PUNCT
ejpam-7044	58	15	(	(	PUNCT
ejpam-7044	58	16	τ1	τ1	NOUN
ejpam-7044	58	17	,	,	PUNCT
ejpam-7044	58	18	τ2)β	τ2)β	ADJ
ejpam-7044	58	19	-	-	PUNCT
ejpam-7044	58	20	closed	closed	ADJ
ejpam-7044	58	21	)	)	PUNCT
ejpam-7044	58	22	.	.	PUNCT
ejpam-7044	59	1	a	a	DET
ejpam-7044	59	2	subset	subset	NOUN
ejpam-7044	59	3	a	a	PRON
ejpam-7044	59	4	of	of	ADP
ejpam-7044	59	5	a	a	DET
ejpam-7044	59	6	bitopological	bitopological	ADJ
ejpam-7044	59	7	space	space	NOUN
ejpam-7044	59	8	(	(	PUNCT
ejpam-7044	59	9	x	x	NOUN
ejpam-7044	59	10	,	,	PUNCT
ejpam-7044	59	11	τ1	τ1	NOUN
ejpam-7044	59	12	,	,	PUNCT
ejpam-7044	59	13	τ2	τ2	NOUN
ejpam-7044	59	14	)	)	PUNCT
ejpam-7044	59	15	is	be	AUX
ejpam-7044	59	16	said	say	VERB
ejpam-7044	59	17	to	to	PART
ejpam-7044	59	18	be	be	AUX
ejpam-7044	59	19	τ1τ2	τ1τ2	NOUN
ejpam-7044	59	20	-	-	ADJ
ejpam-7044	59	21	δ	δ	NOUN
ejpam-7044	59	22	-	-	ADJ
ejpam-7044	59	23	open	open	ADJ
ejpam-7044	59	24	[	[	X
ejpam-7044	59	25	19	19	NUM
ejpam-7044	59	26	]	]	X
ejpam-7044	59	27	if	if	SCONJ
ejpam-7044	59	28	a	a	PRON
ejpam-7044	59	29	is	be	AUX
ejpam-7044	59	30	the	the	DET
ejpam-7044	59	31	union	union	NOUN
ejpam-7044	59	32	of	of	ADP
ejpam-7044	59	33	(	(	PUNCT
ejpam-7044	59	34	τ1	τ1	NOUN
ejpam-7044	59	35	,	,	PUNCT
ejpam-7044	59	36	τ2)r	τ2)r	ADJ
ejpam-7044	59	37	-	-	PUNCT
ejpam-7044	59	38	open	open	ADJ
ejpam-7044	59	39	sets	set	NOUN
ejpam-7044	59	40	of	of	ADP
ejpam-7044	59	41	x.	x.	NOUN
ejpam-7044	59	42	the	the	DET
ejpam-7044	59	43	complement	complement	NOUN
ejpam-7044	59	44	of	of	ADP
ejpam-7044	59	45	a	a	DET
ejpam-7044	59	46	τ1τ2	τ1τ2	ADJ
ejpam-7044	59	47	-	-	ADJ
ejpam-7044	59	48	δ	δ	NOUN
ejpam-7044	59	49	-	-	ADJ
ejpam-7044	59	50	open	open	ADJ
ejpam-7044	59	51	set	set	NOUN
ejpam-7044	59	52	is	be	AUX
ejpam-7044	59	53	called	call	VERB
ejpam-7044	59	54	τ1τ2	τ1τ2	NOUN
ejpam-7044	59	55	-	-	ADJ
ejpam-7044	59	56	δ	δ	NOUN
ejpam-7044	59	57	-	-	PUNCT
ejpam-7044	59	58	closed	closed	ADJ
ejpam-7044	59	59	[	[	X
ejpam-7044	59	60	19	19	NUM
ejpam-7044	59	61	]	]	PUNCT
ejpam-7044	59	62	.	.	PUNCT
ejpam-7044	60	1	the	the	DET
ejpam-7044	60	2	union	union	NOUN
ejpam-7044	60	3	of	of	ADP
ejpam-7044	60	4	all	all	DET
ejpam-7044	60	5	τ1τ2	τ1τ2	NOUN
ejpam-7044	60	6	-	-	ADJ
ejpam-7044	60	7	δ	δ	NOUN
ejpam-7044	60	8	-	-	ADJ
ejpam-7044	60	9	open	open	ADJ
ejpam-7044	60	10	sets	set	NOUN
ejpam-7044	60	11	of	of	ADP
ejpam-7044	60	12	x	x	PUNCT
ejpam-7044	60	13	contained	contain	VERB
ejpam-7044	60	14	in	in	ADP
ejpam-7044	60	15	a	a	PRON
ejpam-7044	60	16	is	be	AUX
ejpam-7044	60	17	called	call	VERB
ejpam-7044	60	18	the	the	DET
ejpam-7044	60	19	τ1τ2	τ1τ2	ADJ
ejpam-7044	60	20	-	-	ADJ
ejpam-7044	60	21	δ	δ	NOUN
ejpam-7044	60	22	-	-	NOUN
ejpam-7044	60	23	interior	interior	NOUN
ejpam-7044	60	24	[	[	X
ejpam-7044	60	25	19	19	NUM
ejpam-7044	60	26	]	]	PUNCT
ejpam-7044	60	27	of	of	ADP
ejpam-7044	60	28	a	a	PRON
ejpam-7044	60	29	and	and	CCONJ
ejpam-7044	60	30	is	be	AUX
ejpam-7044	60	31	denoted	denote	VERB
ejpam-7044	60	32	by	by	ADP
ejpam-7044	60	33	τ1τ2	τ1τ2	ADJ
ejpam-7044	60	34	-	-	ADJ
ejpam-7044	60	35	δ	δ	NOUN
ejpam-7044	60	36	-	-	PUNCT
ejpam-7044	60	37	int(a	int(a	PROPN
ejpam-7044	60	38	)	)	PUNCT
ejpam-7044	60	39	.	.	PUNCT
ejpam-7044	61	1	the	the	DET
ejpam-7044	61	2	intersection	intersection	NOUN
ejpam-7044	61	3	of	of	ADP
ejpam-7044	61	4	all	all	DET
ejpam-7044	61	5	τ1τ2	τ1τ2	NOUN
ejpam-7044	61	6	-	-	ADJ
ejpam-7044	61	7	δ	δ	NOUN
ejpam-7044	61	8	-	-	PUNCT
ejpam-7044	61	9	closed	close	VERB
ejpam-7044	61	10	sets	set	NOUN
ejpam-7044	61	11	of	of	ADP
ejpam-7044	61	12	x	x	PUNCT
ejpam-7044	61	13	containing	contain	VERB
ejpam-7044	61	14	a	a	PRON
ejpam-7044	61	15	is	be	AUX
ejpam-7044	61	16	called	call	VERB
ejpam-7044	61	17	the	the	DET
ejpam-7044	61	18	τ1τ2	τ1τ2	ADJ
ejpam-7044	61	19	-	-	ADJ
ejpam-7044	61	20	δ	δ	NOUN
ejpam-7044	61	21	-	-	NOUN
ejpam-7044	61	22	closure	closure	NOUN
ejpam-7044	61	23	[	[	X
ejpam-7044	61	24	19	19	NUM
ejpam-7044	61	25	]	]	PUNCT
ejpam-7044	61	26	of	of	ADP
ejpam-7044	61	27	a	a	PRON
ejpam-7044	61	28	and	and	CCONJ
ejpam-7044	61	29	is	be	AUX
ejpam-7044	61	30	denoted	denote	VERB
ejpam-7044	61	31	by	by	ADP
ejpam-7044	61	32	τ1τ2	τ1τ2	ADJ
ejpam-7044	61	33	-	-	ADJ
ejpam-7044	61	34	δ	δ	NOUN
ejpam-7044	61	35	-	-	PUNCT
ejpam-7044	61	36	cl(a	cl(a	NUM
ejpam-7044	61	37	)	)	PUNCT
ejpam-7044	61	38	.	.	PUNCT
ejpam-7044	62	1	let	let	VERB
ejpam-7044	62	2	a	a	DET
ejpam-7044	62	3	be	be	AUX
ejpam-7044	62	4	a	a	DET
ejpam-7044	62	5	subset	subset	NOUN
ejpam-7044	62	6	of	of	ADP
ejpam-7044	62	7	a	a	DET
ejpam-7044	62	8	bitopological	bitopological	ADJ
ejpam-7044	62	9	space	space	NOUN
ejpam-7044	62	10	(	(	PUNCT
ejpam-7044	62	11	x	x	NOUN
ejpam-7044	62	12	,	,	PUNCT
ejpam-7044	62	13	τ1	τ1	NOUN
ejpam-7044	62	14	,	,	PUNCT
ejpam-7044	62	15	τ2	τ2	NOUN
ejpam-7044	62	16	)	)	PUNCT
ejpam-7044	62	17	.	.	PUNCT
ejpam-7044	63	1	a	a	DET
ejpam-7044	63	2	point	point	NOUN
ejpam-7044	63	3	x	x	X
ejpam-7044	63	4	∈	∈	NOUN
ejpam-7044	63	5	x	x	PUNCT
ejpam-7044	63	6	is	be	AUX
ejpam-7044	63	7	called	call	VERB
ejpam-7044	63	8	a	a	DET
ejpam-7044	63	9	(	(	PUNCT
ejpam-7044	63	10	τ1	τ1	NOUN
ejpam-7044	63	11	,	,	PUNCT
ejpam-7044	63	12	τ2)θ	τ2)θ	ADJ
ejpam-7044	63	13	-	-	PUNCT
ejpam-7044	63	14	cluster	cluster	NOUN
ejpam-7044	63	15	point	point	NOUN
ejpam-7044	63	16	[	[	X
ejpam-7044	63	17	15	15	NUM
ejpam-7044	63	18	]	]	PUNCT
ejpam-7044	63	19	of	of	ADP
ejpam-7044	63	20	a	a	DET
ejpam-7044	63	21	if	if	SCONJ
ejpam-7044	63	22	τ1τ2	τ1τ2	NOUN
ejpam-7044	63	23	-	-	NOUN
ejpam-7044	63	24	cl(u	cl(u	NOUN
ejpam-7044	63	25	)	)	PUNCT
ejpam-7044	63	26	∩	∩	NOUN
ejpam-7044	63	27	a	a	DET
ejpam-7044	63	28	̸=	̸=	PROPN
ejpam-7044	63	29	∅	∅	NOUN
ejpam-7044	63	30	for	for	ADP
ejpam-7044	63	31	every	every	DET
ejpam-7044	63	32	τ1τ2	τ1τ2	ADJ
ejpam-7044	63	33	-	-	ADJ
ejpam-7044	63	34	open	open	ADJ
ejpam-7044	63	35	set	set	NOUN
ejpam-7044	63	36	u	u	NOUN
ejpam-7044	63	37	containing	contain	VERB
ejpam-7044	63	38	x.	x.	NOUN
ejpam-7044	63	39	the	the	DET
ejpam-7044	63	40	set	set	NOUN
ejpam-7044	63	41	of	of	ADP
ejpam-7044	63	42	all	all	DET
ejpam-7044	63	43	(	(	PUNCT
ejpam-7044	63	44	τ1	τ1	NOUN
ejpam-7044	63	45	,	,	PUNCT
ejpam-7044	63	46	τ2)θ	τ2)θ	ADJ
ejpam-7044	63	47	-	-	PUNCT
ejpam-7044	63	48	cluster	cluster	NOUN
ejpam-7044	63	49	points	point	NOUN
ejpam-7044	63	50	of	of	ADP
ejpam-7044	63	51	a	a	PRON
ejpam-7044	63	52	is	be	AUX
ejpam-7044	63	53	called	call	VERB
ejpam-7044	63	54	the	the	DET
ejpam-7044	63	55	(	(	PUNCT
ejpam-7044	63	56	τ1	τ1	NOUN
ejpam-7044	63	57	,	,	PUNCT
ejpam-7044	63	58	τ2)θ	τ2)θ	ADJ
ejpam-7044	63	59	-	-	PUNCT
ejpam-7044	63	60	closure	closure	NOUN
ejpam-7044	63	61	[	[	X
ejpam-7044	63	62	15	15	NUM
ejpam-7044	63	63	]	]	PUNCT
ejpam-7044	63	64	of	of	ADP
ejpam-7044	63	65	a	a	PRON
ejpam-7044	63	66	and	and	CCONJ
ejpam-7044	63	67	is	be	AUX
ejpam-7044	63	68	denoted	denote	VERB
ejpam-7044	63	69	by	by	ADP
ejpam-7044	63	70	(	(	PUNCT
ejpam-7044	63	71	τ1	τ1	NOUN
ejpam-7044	63	72	,	,	PUNCT
ejpam-7044	63	73	τ2)θ	τ2)θ	NOUN
ejpam-7044	63	74	-	-	PUNCT
ejpam-7044	63	75	cl(a	cl(a	NUM
ejpam-7044	63	76	)	)	PUNCT
ejpam-7044	63	77	.	.	PUNCT
ejpam-7044	64	1	a	a	DET
ejpam-7044	64	2	subset	subset	NOUN
ejpam-7044	64	3	a	a	PRON
ejpam-7044	64	4	of	of	ADP
ejpam-7044	64	5	a	a	DET
ejpam-7044	64	6	bitopological	bitopological	ADJ
ejpam-7044	64	7	space	space	NOUN
ejpam-7044	64	8	(	(	PUNCT
ejpam-7044	64	9	x	x	NOUN
ejpam-7044	64	10	,	,	PUNCT
ejpam-7044	64	11	τ1	τ1	NOUN
ejpam-7044	64	12	,	,	PUNCT
ejpam-7044	64	13	τ2	τ2	NOUN
ejpam-7044	64	14	)	)	PUNCT
ejpam-7044	64	15	is	be	AUX
ejpam-7044	64	16	said	say	VERB
ejpam-7044	64	17	to	to	PART
ejpam-7044	64	18	be	be	AUX
ejpam-7044	64	19	(	(	PUNCT
ejpam-7044	64	20	τ1	τ1	NOUN
ejpam-7044	64	21	,	,	PUNCT
ejpam-7044	64	22	τ2)θ	τ2)θ	NOUN
ejpam-7044	64	23	-	-	PUNCT
ejpam-7044	64	24	closed	closed	ADJ
ejpam-7044	64	25	[	[	X
ejpam-7044	64	26	15	15	NUM
ejpam-7044	64	27	]	]	X
ejpam-7044	64	28	if	if	SCONJ
ejpam-7044	64	29	(	(	PUNCT
ejpam-7044	64	30	τ1	τ1	NOUN
ejpam-7044	64	31	,	,	PUNCT
ejpam-7044	64	32	τ2)θ	τ2)θ	NOUN
ejpam-7044	64	33	-	-	PUNCT
ejpam-7044	64	34	cl(a	cl(a	NUM
ejpam-7044	64	35	)	)	PUNCT
ejpam-7044	65	1	=	=	PUNCT
ejpam-7044	65	2	a.	a.	NOUN
ejpam-7044	65	3	the	the	DET
ejpam-7044	65	4	complement	complement	NOUN
ejpam-7044	65	5	of	of	ADP
ejpam-7044	65	6	a	a	DET
ejpam-7044	65	7	(	(	PUNCT
ejpam-7044	65	8	τ1	τ1	NOUN
ejpam-7044	65	9	,	,	PUNCT
ejpam-7044	65	10	τ2)θ	τ2)θ	ADJ
ejpam-7044	65	11	-	-	PUNCT
ejpam-7044	65	12	closed	close	VERB
ejpam-7044	65	13	set	set	NOUN
ejpam-7044	65	14	is	be	AUX
ejpam-7044	65	15	said	say	VERB
ejpam-7044	65	16	to	to	PART
ejpam-7044	65	17	be	be	AUX
ejpam-7044	65	18	(	(	PUNCT
ejpam-7044	65	19	τ1	τ1	NOUN
ejpam-7044	65	20	,	,	PUNCT
ejpam-7044	65	21	τ2)θopen	τ2)θopen	PROPN
ejpam-7044	65	22	.	.	PUNCT
ejpam-7044	66	1	the	the	DET
ejpam-7044	66	2	union	union	NOUN
ejpam-7044	66	3	of	of	ADP
ejpam-7044	66	4	all	all	DET
ejpam-7044	66	5	(	(	PUNCT
ejpam-7044	66	6	τ1	τ1	NOUN
ejpam-7044	66	7	,	,	PUNCT
ejpam-7044	66	8	τ2)θ	τ2)θ	ADJ
ejpam-7044	66	9	-	-	PUNCT
ejpam-7044	66	10	open	open	ADJ
ejpam-7044	66	11	sets	set	NOUN
ejpam-7044	66	12	of	of	ADP
ejpam-7044	66	13	x	x	PUNCT
ejpam-7044	66	14	contained	contain	VERB
ejpam-7044	66	15	in	in	ADP
ejpam-7044	66	16	a	a	PRON
ejpam-7044	66	17	is	be	AUX
ejpam-7044	66	18	called	call	VERB
ejpam-7044	66	19	the	the	DET
ejpam-7044	66	20	(	(	PUNCT
ejpam-7044	66	21	τ1	τ1	NOUN
ejpam-7044	66	22	,	,	PUNCT
ejpam-7044	66	23	τ2)θ	τ2)θ	ADJ
ejpam-7044	66	24	-	-	PUNCT
ejpam-7044	66	25	interior	interior	NOUN
ejpam-7044	66	26	[	[	X
ejpam-7044	66	27	15	15	NUM
ejpam-7044	66	28	]	]	PUNCT
ejpam-7044	66	29	of	of	ADP
ejpam-7044	66	30	a	a	PRON
ejpam-7044	66	31	and	and	CCONJ
ejpam-7044	66	32	is	be	AUX
ejpam-7044	66	33	denoted	denote	VERB
ejpam-7044	66	34	by	by	ADP
ejpam-7044	66	35	(	(	PUNCT
ejpam-7044	66	36	τ1	τ1	NOUN
ejpam-7044	66	37	,	,	PUNCT
ejpam-7044	66	38	τ2)θ	τ2)θ	NOUN
ejpam-7044	66	39	-	-	PUNCT
ejpam-7044	66	40	int(a	int(a	NOUN
ejpam-7044	66	41	)	)	PUNCT
ejpam-7044	66	42	.	.	PUNCT
ejpam-7044	67	1	an	an	DET
ejpam-7044	67	2	ideal	ideal	NOUN
ejpam-7044	67	3	i	i	PRON
ejpam-7044	67	4	on	on	ADP
ejpam-7044	67	5	a	a	DET
ejpam-7044	67	6	topological	topological	ADJ
ejpam-7044	67	7	space	space	NOUN
ejpam-7044	67	8	(	(	PUNCT
ejpam-7044	67	9	x	x	X
ejpam-7044	67	10	,	,	PUNCT
ejpam-7044	67	11	τ	τ	X
ejpam-7044	67	12	)	)	PUNCT
ejpam-7044	67	13	is	be	AUX
ejpam-7044	67	14	a	a	DET
ejpam-7044	67	15	nonempty	nonempty	ADJ
ejpam-7044	67	16	collection	collection	NOUN
ejpam-7044	67	17	of	of	ADP
ejpam-7044	67	18	subsets	subset	NOUN
ejpam-7044	67	19	of	of	ADP
ejpam-7044	67	20	x	x	PUNCT
ejpam-7044	67	21	satisfying	satisfy	VERB
ejpam-7044	67	22	the	the	DET
ejpam-7044	67	23	following	follow	VERB
ejpam-7044	67	24	properties	property	NOUN
ejpam-7044	67	25	:	:	PUNCT
ejpam-7044	67	26	(	(	PUNCT
ejpam-7044	67	27	1	1	X
ejpam-7044	67	28	)	)	PUNCT
ejpam-7044	67	29	a	a	DET
ejpam-7044	67	30	∈	∈	NOUN
ejpam-7044	67	31	i	i	PRON
ejpam-7044	67	32	and	and	CCONJ
ejpam-7044	67	33	b	b	X
ejpam-7044	67	34	⊆	⊆	NUM
ejpam-7044	67	35	a	a	DET
ejpam-7044	67	36	imply	imply	NOUN
ejpam-7044	67	37	b	b	X
ejpam-7044	67	38	∈	∈	PROPN
ejpam-7044	67	39	i	i	PRON
ejpam-7044	67	40	;	;	PUNCT
ejpam-7044	67	41	(	(	PUNCT
ejpam-7044	67	42	2	2	X
ejpam-7044	67	43	)	)	PUNCT
ejpam-7044	68	1	a	a	PRON
ejpam-7044	68	2	∈	∈	NOUN
ejpam-7044	69	1	i	i	PRON
ejpam-7044	69	2	and	and	CCONJ
ejpam-7044	69	3	b	b	X
ejpam-7044	69	4	∈	∈	NOUN
ejpam-7044	70	1	i	i	PRON
ejpam-7044	70	2	imply	imply	VERB
ejpam-7044	70	3	a	a	DET
ejpam-7044	70	4	∪	∪	X
ejpam-7044	70	5	b	b	NOUN
ejpam-7044	70	6	∈	∈	NOUN
ejpam-7044	71	1	i	i	PRON
ejpam-7044	71	2	.	.	PUNCT
ejpam-7044	72	1	a	a	DET
ejpam-7044	72	2	topological	topological	ADJ
ejpam-7044	72	3	space	space	NOUN
ejpam-7044	72	4	(	(	PUNCT
ejpam-7044	72	5	x	x	X
ejpam-7044	72	6	,	,	PUNCT
ejpam-7044	72	7	τ	τ	X
ejpam-7044	72	8	)	)	PUNCT
ejpam-7044	72	9	with	with	ADP
ejpam-7044	72	10	an	an	DET
ejpam-7044	72	11	ideal	ideal	ADJ
ejpam-7044	72	12	i	i	PRON
ejpam-7044	72	13	on	on	ADP
ejpam-7044	72	14	x	x	SYM
ejpam-7044	72	15	is	be	AUX
ejpam-7044	72	16	called	call	VERB
ejpam-7044	72	17	an	an	DET
ejpam-7044	72	18	ideal	ideal	ADJ
ejpam-7044	72	19	topological	topological	ADJ
ejpam-7044	72	20	space	space	NOUN
ejpam-7044	72	21	and	and	CCONJ
ejpam-7044	72	22	is	be	AUX
ejpam-7044	72	23	denoted	denote	VERB
ejpam-7044	72	24	by	by	ADP
ejpam-7044	72	25	(	(	PUNCT
ejpam-7044	72	26	x	x	X
ejpam-7044	72	27	,	,	PUNCT
ejpam-7044	72	28	τ	τ	PROPN
ejpam-7044	72	29	,	,	PUNCT
ejpam-7044	72	30	i	i	NOUN
ejpam-7044	72	31	)	)	PUNCT
ejpam-7044	72	32	.	.	PUNCT
ejpam-7044	73	1	for	for	ADP
ejpam-7044	73	2	an	an	DET
ejpam-7044	73	3	ideal	ideal	ADJ
ejpam-7044	73	4	topological	topological	ADJ
ejpam-7044	73	5	space	space	NOUN
ejpam-7044	73	6	(	(	PUNCT
ejpam-7044	73	7	x	x	X
ejpam-7044	73	8	,	,	PUNCT
ejpam-7044	73	9	τ	τ	PROPN
ejpam-7044	73	10	,	,	PUNCT
ejpam-7044	73	11	i	i	PROPN
ejpam-7044	73	12	)	)	PUNCT
ejpam-7044	73	13	and	and	CCONJ
ejpam-7044	73	14	a	a	DET
ejpam-7044	73	15	subset	subset	NOUN
ejpam-7044	73	16	a	a	PRON
ejpam-7044	73	17	of	of	ADP
ejpam-7044	73	18	x	x	PRON
ejpam-7044	73	19	,	,	PUNCT
ejpam-7044	73	20	a⋆(i	a⋆(i	PROPN
ejpam-7044	73	21	)	)	PUNCT
ejpam-7044	73	22	is	be	AUX
ejpam-7044	73	23	defined	define	VERB
ejpam-7044	73	24	as	as	SCONJ
ejpam-7044	73	25	follows	follow	VERB
ejpam-7044	73	26	:	:	PUNCT
ejpam-7044	73	27	a⋆(i	a⋆(i	NOUN
ejpam-7044	73	28	)	)	PUNCT
ejpam-7044	74	1	=	=	PUNCT
ejpam-7044	74	2	{	{	PUNCT
ejpam-7044	74	3	x	x	PUNCT
ejpam-7044	74	4	∈	∈	PROPN
ejpam-7044	74	5	x	x	X
ejpam-7044	74	6	:	:	PUNCT
ejpam-7044	74	7	u	u	X
ejpam-7044	74	8	∩a	∩a	PROPN
ejpam-7044	74	9	̸∈	̸∈	PROPN
ejpam-7044	74	10	i	i	PRON
ejpam-7044	74	11	for	for	ADP
ejpam-7044	74	12	every	every	DET
ejpam-7044	74	13	open	open	ADJ
ejpam-7044	74	14	neighbourhood	neighbourhood	NOUN
ejpam-7044	74	15	u	u	NOUN
ejpam-7044	74	16	of	of	ADP
ejpam-7044	74	17	x	x	NOUN
ejpam-7044	74	18	}	}	PUNCT
ejpam-7044	74	19	.	.	PUNCT
ejpam-7044	75	1	in	in	ADP
ejpam-7044	75	2	case	case	NOUN
ejpam-7044	75	3	there	there	PRON
ejpam-7044	75	4	is	be	VERB
ejpam-7044	75	5	no	no	DET
ejpam-7044	75	6	chance	chance	NOUN
ejpam-7044	75	7	for	for	ADP
ejpam-7044	75	8	confusion	confusion	NOUN
ejpam-7044	75	9	,	,	PUNCT
ejpam-7044	75	10	a⋆(i	a⋆(i	NOUN
ejpam-7044	75	11	)	)	PUNCT
ejpam-7044	75	12	is	be	AUX
ejpam-7044	75	13	simply	simply	ADV
ejpam-7044	75	14	written	write	VERB
ejpam-7044	75	15	as	as	ADP
ejpam-7044	75	16	a⋆.	a⋆.	NOUN
ejpam-7044	75	17	in	in	ADP
ejpam-7044	75	18	[	[	X
ejpam-7044	75	19	20	20	NUM
ejpam-7044	75	20	]	]	PUNCT
ejpam-7044	75	21	,	,	PUNCT
ejpam-7044	75	22	a⋆	a⋆	ADV
ejpam-7044	75	23	is	be	AUX
ejpam-7044	75	24	called	call	VERB
ejpam-7044	75	25	the	the	DET
ejpam-7044	75	26	local	local	ADJ
ejpam-7044	75	27	function	function	NOUN
ejpam-7044	75	28	of	of	ADP
ejpam-7044	75	29	a	a	PRON
ejpam-7044	75	30	with	with	ADP
ejpam-7044	75	31	respect	respect	NOUN
ejpam-7044	75	32	to	to	ADP
ejpam-7044	75	33	i	i	PRON
ejpam-7044	75	34	and	and	CCONJ
ejpam-7044	75	35	τ	τ	PROPN
ejpam-7044	75	36	and	and	CCONJ
ejpam-7044	75	37	cl⋆(a	cl⋆(a	NUM
ejpam-7044	75	38	)	)	PUNCT
ejpam-7044	75	39	=	=	PUNCT
ejpam-7044	75	40	a⋆	a⋆	ADP
ejpam-7044	75	41	∪	∪	ADP
ejpam-7044	75	42	a	a	DET
ejpam-7044	75	43	defines	define	NOUN
ejpam-7044	75	44	a	a	DET
ejpam-7044	75	45	kuratowski	kuratowski	ADJ
ejpam-7044	75	46	closure	closure	NOUN
ejpam-7044	75	47	operator	operator	NOUN
ejpam-7044	75	48	for	for	ADP
ejpam-7044	75	49	a	a	DET
ejpam-7044	75	50	topology	topology	NOUN
ejpam-7044	75	51	τ⋆(i	τ⋆(i	NOUN
ejpam-7044	75	52	)	)	PUNCT
ejpam-7044	75	53	finer	fine	ADJ
ejpam-7044	75	54	than	than	ADP
ejpam-7044	75	55	τ	τ	PROPN
ejpam-7044	75	56	.	.	PUNCT
ejpam-7044	76	1	a	a	DET
ejpam-7044	76	2	subset	subset	NOUN
ejpam-7044	76	3	a	a	PRON
ejpam-7044	76	4	is	be	AUX
ejpam-7044	76	5	said	say	VERB
ejpam-7044	76	6	to	to	PART
ejpam-7044	76	7	be	be	AUX
ejpam-7044	76	8	⋆-closed	⋆-close	VERB
ejpam-7044	76	9	[	[	X
ejpam-7044	76	10	21	21	NUM
ejpam-7044	76	11	]	]	X
ejpam-7044	76	12	if	if	SCONJ
ejpam-7044	76	13	a⋆	a⋆	ADJ
ejpam-7044	76	14	⊆	⊆	NUM
ejpam-7044	76	15	a.	a.	NOUN
ejpam-7044	76	16	the	the	DET
ejpam-7044	76	17	interior	interior	NOUN
ejpam-7044	76	18	of	of	ADP
ejpam-7044	76	19	a	a	DET
ejpam-7044	76	20	subset	subset	NOUN
ejpam-7044	76	21	a	a	DET
ejpam-7044	76	22	in	in	ADP
ejpam-7044	76	23	(	(	PUNCT
ejpam-7044	76	24	x	x	X
ejpam-7044	76	25	,	,	PUNCT
ejpam-7044	76	26	τ⋆(i	τ⋆(i	NOUN
ejpam-7044	76	27	)	)	PUNCT
ejpam-7044	76	28	)	)	PUNCT
ejpam-7044	76	29	is	be	AUX
ejpam-7044	76	30	denoted	denote	VERB
ejpam-7044	76	31	by	by	ADP
ejpam-7044	76	32	int⋆(a	int⋆(a	NOUN
ejpam-7044	76	33	)	)	PUNCT
ejpam-7044	76	34	.	.	PUNCT
ejpam-7044	77	1	a	a	DET
ejpam-7044	77	2	subset	subset	NOUN
ejpam-7044	77	3	a	a	PRON
ejpam-7044	77	4	of	of	ADP
ejpam-7044	77	5	an	an	DET
ejpam-7044	77	6	ideal	ideal	ADJ
ejpam-7044	77	7	topological	topological	ADJ
ejpam-7044	77	8	space	space	NOUN
ejpam-7044	77	9	(	(	PUNCT
ejpam-7044	77	10	x	x	X
ejpam-7044	77	11	,	,	PUNCT
ejpam-7044	77	12	τ	τ	PROPN
ejpam-7044	77	13	,	,	PUNCT
ejpam-7044	77	14	i	i	PROPN
ejpam-7044	77	15	)	)	PUNCT
ejpam-7044	77	16	is	be	AUX
ejpam-7044	77	17	said	say	VERB
ejpam-7044	77	18	to	to	PART
ejpam-7044	77	19	be	be	AUX
ejpam-7044	77	20	r	r	NOUN
ejpam-7044	77	21	-	-	PUNCT
ejpam-7044	77	22	i	i	PRON
ejpam-7044	77	23	⋆-open	⋆-open	VERB
ejpam-7044	78	1	[	[	X
ejpam-7044	78	2	5	5	NUM
ejpam-7044	78	3	]	]	PUNCT
ejpam-7044	78	4	(	(	PUNCT
ejpam-7044	78	5	resp	resp	NOUN
ejpam-7044	78	6	.	.	PUNCT
ejpam-7044	79	1	i	i	PRON
ejpam-7044	79	2	⋆-preopen	⋆-preopen	VERB
ejpam-7044	80	1	[	[	X
ejpam-7044	80	2	5	5	NUM
ejpam-7044	80	3	]	]	PUNCT
ejpam-7044	80	4	,	,	PUNCT
ejpam-7044	80	5	semi	semi	ADJ
ejpam-7044	80	6	-	-	VERB
ejpam-7044	80	7	i	i	PRON
ejpam-7044	80	8	⋆-open	⋆-open	VERB
ejpam-7044	81	1	[	[	X
ejpam-7044	81	2	22	22	NUM
ejpam-7044	81	3	]	]	PUNCT
ejpam-7044	81	4	,	,	PUNCT
ejpam-7044	81	5	semi	semi	ADJ
ejpam-7044	81	6	-	-	VERB
ejpam-7044	81	7	i	i	PRON
ejpam-7044	81	8	⋆-preopen	⋆-preopen	VERB
ejpam-7044	82	1	[	[	X
ejpam-7044	82	2	22	22	NUM
ejpam-7044	82	3	]	]	SYM
ejpam-7044	82	4	)	)	PUNCT
ejpam-7044	82	5	if	if	SCONJ
ejpam-7044	82	6	a	a	DET
ejpam-7044	82	7	=	=	PUNCT
ejpam-7044	82	8	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7044	82	9	)	)	PUNCT
ejpam-7044	82	10	)	)	PUNCT
ejpam-7044	82	11	(	(	PUNCT
ejpam-7044	82	12	resp	resp	NOUN
ejpam-7044	82	13	.	.	PUNCT
ejpam-7044	83	1	a	a	DET
ejpam-7044	83	2	⊆	⊆	NUM
ejpam-7044	83	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7044	83	4	)	)	PUNCT
ejpam-7044	83	5	)	)	PUNCT
ejpam-7044	83	6	,	,	PUNCT
ejpam-7044	83	7	a	a	DET
ejpam-7044	83	8	⊆	⊆	NUM
ejpam-7044	83	9	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7044	83	10	)	)	PUNCT
ejpam-7044	83	11	)	)	PUNCT
ejpam-7044	83	12	,	,	PUNCT
ejpam-7044	83	13	a	a	DET
ejpam-7044	83	14	⊆	⊆	NUM
ejpam-7044	83	15	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	83	16	)	)	PUNCT
ejpam-7044	83	17	)	)	PUNCT
ejpam-7044	83	18	)	)	PUNCT
ejpam-7044	83	19	)	)	PUNCT
ejpam-7044	83	20	.	.	PUNCT
ejpam-7044	84	1	the	the	DET
ejpam-7044	84	2	complement	complement	NOUN
ejpam-7044	84	3	of	of	ADP
ejpam-7044	84	4	a	a	DET
ejpam-7044	84	5	r	r	NOUN
ejpam-7044	84	6	-	-	PUNCT
ejpam-7044	84	7	i	i	PRON
ejpam-7044	84	8	⋆-open	⋆-open	VERB
ejpam-7044	84	9	(	(	PUNCT
ejpam-7044	84	10	resp	resp	NOUN
ejpam-7044	84	11	.	.	PUNCT
ejpam-7044	85	1	i	i	PRON
ejpam-7044	85	2	⋆-preopen	⋆-preopen	VERB
ejpam-7044	85	3	,	,	PUNCT
ejpam-7044	85	4	semi	semi	ADJ
ejpam-7044	85	5	-	-	ADJ
ejpam-7044	85	6	i	i	PRON
ejpam-7044	85	7	⋆-open	⋆-open	VERB
ejpam-7044	85	8	,	,	PUNCT
ejpam-7044	85	9	semi	semi	ADJ
ejpam-7044	85	10	-	-	VERB
ejpam-7044	85	11	i	i	PRON
ejpam-7044	85	12	⋆-preopen	⋆-preopen	ADV
ejpam-7044	85	13	)	)	PUNCT
ejpam-7044	86	1	set	set	NOUN
ejpam-7044	86	2	is	be	AUX
ejpam-7044	86	3	said	say	VERB
ejpam-7044	86	4	to	to	PART
ejpam-7044	86	5	be	be	AUX
ejpam-7044	86	6	r	r	NOUN
ejpam-7044	86	7	-	-	PUNCT
ejpam-7044	86	8	i	i	PRON
ejpam-7044	86	9	⋆-closed	⋆-close	VERB
ejpam-7044	86	10	(	(	PUNCT
ejpam-7044	86	11	resp	resp	NOUN
ejpam-7044	86	12	.	.	PUNCT
ejpam-7044	87	1	i	i	PRON
ejpam-7044	87	2	⋆-preclosed	⋆-preclose	VERB
ejpam-7044	87	3	,	,	PUNCT
ejpam-7044	87	4	semi	semi	ADV
ejpam-7044	87	5	-	-	VERB
ejpam-7044	87	6	i	i	PRON
ejpam-7044	87	7	⋆-closed	⋆-close	VERB
ejpam-7044	87	8	,	,	PUNCT
ejpam-7044	87	9	semi	semi	ADJ
ejpam-7044	87	10	-	-	VERB
ejpam-7044	87	11	i	i	PRON
ejpam-7044	87	12	⋆-preclosed	⋆-preclose	VERB
ejpam-7044	87	13	)	)	PUNCT
ejpam-7044	87	14	.	.	PUNCT
ejpam-7044	88	1	for	for	ADP
ejpam-7044	88	2	a	a	DET
ejpam-7044	88	3	subset	subset	NOUN
ejpam-7044	88	4	a	a	PRON
ejpam-7044	88	5	of	of	ADP
ejpam-7044	88	6	an	an	DET
ejpam-7044	88	7	ideal	ideal	ADJ
ejpam-7044	88	8	topological	topological	ADJ
ejpam-7044	88	9	space	space	NOUN
ejpam-7044	88	10	(	(	PUNCT
ejpam-7044	88	11	x	x	X
ejpam-7044	88	12	,	,	PUNCT
ejpam-7044	88	13	τ	τ	PROPN
ejpam-7044	88	14	,	,	PUNCT
ejpam-7044	88	15	i	i	NOUN
ejpam-7044	88	16	)	)	PUNCT
ejpam-7044	88	17	,	,	PUNCT
ejpam-7044	88	18	the	the	DET
ejpam-7044	88	19	intersection	intersection	NOUN
ejpam-7044	88	20	of	of	ADP
ejpam-7044	88	21	all	all	PRON
ejpam-7044	88	22	semi	semi	NOUN
ejpam-7044	88	23	-	-	ADJ
ejpam-7044	88	24	i	i	PRON
ejpam-7044	88	25	⋆-closed	⋆-close	VERB
ejpam-7044	88	26	sets	set	NOUN
ejpam-7044	88	27	containing	contain	VERB
ejpam-7044	88	28	a	a	PRON
ejpam-7044	88	29	is	be	AUX
ejpam-7044	88	30	called	call	VERB
ejpam-7044	88	31	the	the	DET
ejpam-7044	88	32	semi	semi	NOUN
ejpam-7044	88	33	-	-	ADJ
ejpam-7044	88	34	i	i	PRON
ejpam-7044	88	35	⋆-closure	⋆-closure	NOUN
ejpam-7044	89	1	[	[	X
ejpam-7044	89	2	22	22	NUM
ejpam-7044	89	3	]	]	PUNCT
ejpam-7044	89	4	of	of	ADP
ejpam-7044	89	5	a	a	PRON
ejpam-7044	89	6	and	and	CCONJ
ejpam-7044	89	7	is	be	AUX
ejpam-7044	89	8	denoted	denote	VERB
ejpam-7044	89	9	by	by	ADP
ejpam-7044	89	10	scl⋆(a	scl⋆(a	NOUN
ejpam-7044	89	11	)	)	PUNCT
ejpam-7044	89	12	(	(	PUNCT
ejpam-7044	89	13	scli	scli	PROPN
ejpam-7044	89	14	⋆(a	⋆(a	PRON
ejpam-7044	89	15	)	)	PUNCT
ejpam-7044	90	1	[	[	X
ejpam-7044	90	2	22	22	NUM
ejpam-7044	90	3	]	]	PUNCT
ejpam-7044	90	4	)	)	PUNCT
ejpam-7044	90	5	.	.	PUNCT
ejpam-7044	91	1	the	the	DET
ejpam-7044	91	2	union	union	NOUN
ejpam-7044	91	3	of	of	ADP
ejpam-7044	91	4	all	all	PRON
ejpam-7044	91	5	semi	semi	ADJ
ejpam-7044	91	6	-	-	ADJ
ejpam-7044	91	7	i	i	PRON
ejpam-7044	91	8	⋆-open	⋆-open	ADJ
ejpam-7044	91	9	sets	set	NOUN
ejpam-7044	91	10	contained	contain	VERB
ejpam-7044	91	11	in	in	ADP
ejpam-7044	91	12	a	a	PRON
ejpam-7044	91	13	is	be	AUX
ejpam-7044	91	14	called	call	VERB
ejpam-7044	91	15	the	the	DET
ejpam-7044	91	16	semi	semi	NOUN
ejpam-7044	91	17	-	-	ADJ
ejpam-7044	91	18	i	i	PRON
ejpam-7044	91	19	⋆-interior	⋆-interior	PUNCT
ejpam-7044	92	1	[	[	X
ejpam-7044	92	2	22	22	NUM
ejpam-7044	92	3	]	]	PUNCT
ejpam-7044	92	4	of	of	ADP
ejpam-7044	92	5	a	a	PRON
ejpam-7044	92	6	and	and	CCONJ
ejpam-7044	92	7	is	be	AUX
ejpam-7044	92	8	denoted	denote	VERB
ejpam-7044	92	9	by	by	ADP
ejpam-7044	92	10	sint⋆(a	sint⋆(a	PROPN
ejpam-7044	92	11	)	)	PUNCT
ejpam-7044	93	1	(	(	PUNCT
ejpam-7044	93	2	sinti	sinti	PROPN
ejpam-7044	93	3	⋆(a	⋆(a	NOUN
ejpam-7044	93	4	)	)	PUNCT
ejpam-7044	94	1	[	[	X
ejpam-7044	94	2	22	22	NUM
ejpam-7044	94	3	]	]	PUNCT
ejpam-7044	94	4	)	)	PUNCT
ejpam-7044	94	5	.	.	PUNCT
ejpam-7044	95	1	lemma	lemma	PROPN
ejpam-7044	95	2	2	2	NUM
ejpam-7044	95	3	.	.	PUNCT
ejpam-7044	96	1	[	[	X
ejpam-7044	96	2	22	22	NUM
ejpam-7044	96	3	]	]	PUNCT
ejpam-7044	96	4	for	for	ADP
ejpam-7044	96	5	a	a	DET
ejpam-7044	96	6	subset	subset	NOUN
ejpam-7044	96	7	a	a	PRON
ejpam-7044	96	8	of	of	ADP
ejpam-7044	96	9	an	an	DET
ejpam-7044	96	10	ideal	ideal	ADJ
ejpam-7044	96	11	topological	topological	ADJ
ejpam-7044	96	12	space	space	NOUN
ejpam-7044	96	13	(	(	PUNCT
ejpam-7044	96	14	x	x	X
ejpam-7044	96	15	,	,	PUNCT
ejpam-7044	96	16	τ	τ	PROPN
ejpam-7044	96	17	,	,	PUNCT
ejpam-7044	96	18	i	i	NOUN
ejpam-7044	96	19	)	)	PUNCT
ejpam-7044	96	20	,	,	PUNCT
ejpam-7044	96	21	the	the	DET
ejpam-7044	96	22	following	follow	VERB
ejpam-7044	96	23	properties	property	NOUN
ejpam-7044	96	24	hold	hold	VERB
ejpam-7044	96	25	:	:	PUNCT
ejpam-7044	96	26	(	(	PUNCT
ejpam-7044	96	27	1	1	X
ejpam-7044	96	28	)	)	PUNCT
ejpam-7044	96	29	scl⋆(a	scl⋆(a	NUM
ejpam-7044	96	30	)	)	PUNCT
ejpam-7044	96	31	=	=	PUNCT
ejpam-7044	96	32	a	a	DET
ejpam-7044	96	33	∪	∪	ADJ
ejpam-7044	96	34	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7044	96	35	)	)	PUNCT
ejpam-7044	96	36	)	)	PUNCT
ejpam-7044	96	37	.	.	PUNCT
ejpam-7044	97	1	(	(	PUNCT
ejpam-7044	97	2	2	2	X
ejpam-7044	97	3	)	)	PUNCT
ejpam-7044	97	4	sint⋆(a	sint⋆(a	PROPN
ejpam-7044	97	5	)	)	PUNCT
ejpam-7044	98	1	=	=	PUNCT
ejpam-7044	98	2	a	a	DET
ejpam-7044	98	3	∩	∩	ADJ
ejpam-7044	98	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7044	98	5	)	)	PUNCT
ejpam-7044	98	6	)	)	PUNCT
ejpam-7044	98	7	.	.	PUNCT
ejpam-7044	99	1	j.	j.	PROPN
ejpam-7044	99	2	khampakdee	khampakdee	PROPN
ejpam-7044	99	3	,	,	PUNCT
ejpam-7044	99	4	a.	a.	PROPN
ejpam-7044	99	5	sama	sama	PROPN
ejpam-7044	99	6	-	-	PUNCT
ejpam-7044	99	7	ae	ae	PROPN
ejpam-7044	99	8	,	,	PUNCT
ejpam-7044	99	9	c.	c.	PROPN
ejpam-7044	99	10	boonpok	boonpok	PROPN
ejpam-7044	99	11	/	/	SYM
ejpam-7044	99	12	eur	eur	PROPN
ejpam-7044	99	13	.	.	PUNCT
ejpam-7044	100	1	j.	j.	PROPN
ejpam-7044	100	2	pure	pure	PROPN
ejpam-7044	100	3	appl	appl	PROPN
ejpam-7044	100	4	.	.	PROPN
ejpam-7044	100	5	math	math	PROPN
ejpam-7044	100	6	,	,	PUNCT
ejpam-7044	100	7	18	18	NUM
ejpam-7044	100	8	(	(	PUNCT
ejpam-7044	100	9	4	4	NUM
ejpam-7044	100	10	)	)	PUNCT
ejpam-7044	100	11	(	(	PUNCT
ejpam-7044	100	12	2025	2025	NUM
ejpam-7044	100	13	)	)	PUNCT
ejpam-7044	100	14	,	,	PUNCT
ejpam-7044	100	15	7044	7044	NUM
ejpam-7044	100	16	4	4	NUM
ejpam-7044	100	17	of	of	ADP
ejpam-7044	100	18	9	9	NUM
ejpam-7044	100	19	a	a	DET
ejpam-7044	100	20	subset	subset	NOUN
ejpam-7044	100	21	a	a	PRON
ejpam-7044	100	22	of	of	ADP
ejpam-7044	100	23	an	an	DET
ejpam-7044	100	24	ideal	ideal	ADJ
ejpam-7044	100	25	topological	topological	ADJ
ejpam-7044	100	26	space	space	NOUN
ejpam-7044	100	27	(	(	PUNCT
ejpam-7044	100	28	x	x	X
ejpam-7044	100	29	,	,	PUNCT
ejpam-7044	100	30	τ	τ	PROPN
ejpam-7044	100	31	,	,	PUNCT
ejpam-7044	100	32	i	i	PROPN
ejpam-7044	100	33	)	)	PUNCT
ejpam-7044	100	34	is	be	AUX
ejpam-7044	100	35	called	call	VERB
ejpam-7044	100	36	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	100	37	-	-	PUNCT
ejpam-7044	100	38	open	open	ADJ
ejpam-7044	100	39	[	[	X
ejpam-7044	100	40	23	23	NUM
ejpam-7044	100	41	]	]	PUNCT
ejpam-7044	100	42	(	(	PUNCT
ejpam-7044	100	43	α	α	X
ejpam-7044	100	44	-	-	ADJ
ejpam-7044	100	45	i	i	PRON
ejpam-7044	100	46	⋆-open	⋆-open	VERB
ejpam-7044	101	1	[	[	X
ejpam-7044	101	2	24	24	NUM
ejpam-7044	101	3	]	]	SYM
ejpam-7044	101	4	)	)	PUNCT
ejpam-7044	101	5	if	if	SCONJ
ejpam-7044	101	6	a	a	DET
ejpam-7044	101	7	⊆	⊆	NUM
ejpam-7044	101	8	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7044	101	9	)	)	PUNCT
ejpam-7044	101	10	)	)	PUNCT
ejpam-7044	101	11	)	)	PUNCT
ejpam-7044	101	12	.	.	PUNCT
ejpam-7044	102	1	the	the	DET
ejpam-7044	102	2	complement	complement	NOUN
ejpam-7044	102	3	of	of	ADP
ejpam-7044	102	4	a	a	DET
ejpam-7044	102	5	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	102	6	-	-	PUNCT
ejpam-7044	102	7	open	open	ADJ
ejpam-7044	102	8	set	set	NOUN
ejpam-7044	102	9	is	be	AUX
ejpam-7044	102	10	called	call	VERB
ejpam-7044	102	11	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	102	12	-	-	PUNCT
ejpam-7044	102	13	closed	closed	ADJ
ejpam-7044	102	14	.	.	PUNCT
ejpam-7044	103	1	lemma	lemma	PROPN
ejpam-7044	103	2	3	3	NUM
ejpam-7044	103	3	.	.	PUNCT
ejpam-7044	104	1	[	[	X
ejpam-7044	104	2	24	24	NUM
ejpam-7044	104	3	]	]	PUNCT
ejpam-7044	104	4	for	for	ADP
ejpam-7044	104	5	a	a	DET
ejpam-7044	104	6	subset	subset	NOUN
ejpam-7044	104	7	a	a	PRON
ejpam-7044	104	8	of	of	ADP
ejpam-7044	104	9	an	an	DET
ejpam-7044	104	10	ideal	ideal	ADJ
ejpam-7044	104	11	topological	topological	ADJ
ejpam-7044	104	12	space	space	NOUN
ejpam-7044	104	13	(	(	PUNCT
ejpam-7044	104	14	x	x	X
ejpam-7044	104	15	,	,	PUNCT
ejpam-7044	104	16	τ	τ	PROPN
ejpam-7044	104	17	,	,	PUNCT
ejpam-7044	104	18	i	i	NOUN
ejpam-7044	104	19	)	)	PUNCT
ejpam-7044	104	20	,	,	PUNCT
ejpam-7044	104	21	the	the	DET
ejpam-7044	104	22	following	follow	VERB
ejpam-7044	104	23	properties	property	NOUN
ejpam-7044	104	24	are	be	AUX
ejpam-7044	104	25	equivalent	equivalent	ADJ
ejpam-7044	104	26	:	:	PUNCT
ejpam-7044	104	27	(	(	PUNCT
ejpam-7044	104	28	1	1	X
ejpam-7044	104	29	)	)	PUNCT
ejpam-7044	104	30	a	a	PRON
ejpam-7044	104	31	is	be	AUX
ejpam-7044	104	32	α	α	X
ejpam-7044	104	33	-	-	ADJ
ejpam-7044	104	34	i	i	PRON
ejpam-7044	104	35	⋆-open	⋆-open	VERB
ejpam-7044	104	36	in	in	ADP
ejpam-7044	104	37	x.	x.	PROPN
ejpam-7044	104	38	(	(	PUNCT
ejpam-7044	104	39	2	2	NUM
ejpam-7044	104	40	)	)	PUNCT
ejpam-7044	104	41	g	g	ADP
ejpam-7044	104	42	⊆	⊆	NUM
ejpam-7044	104	43	a	a	DET
ejpam-7044	104	44	⊆	⊆	NUM
ejpam-7044	104	45	int⋆(cl⋆(g	int⋆(cl⋆(g	NOUN
ejpam-7044	104	46	)	)	PUNCT
ejpam-7044	104	47	)	)	PUNCT
ejpam-7044	104	48	for	for	ADP
ejpam-7044	104	49	some	some	DET
ejpam-7044	104	50	⋆-open	⋆-open	ADJ
ejpam-7044	104	51	set	set	VERB
ejpam-7044	104	52	g.	g.	PROPN
ejpam-7044	104	53	(	(	PUNCT
ejpam-7044	104	54	3	3	NUM
ejpam-7044	104	55	)	)	PUNCT
ejpam-7044	104	56	g	g	ADP
ejpam-7044	104	57	⊆	⊆	NUM
ejpam-7044	104	58	a	a	DET
ejpam-7044	104	59	⊆	⊆	NUM
ejpam-7044	104	60	scl⋆(g	scl⋆(g	NOUN
ejpam-7044	104	61	)	)	PUNCT
ejpam-7044	104	62	for	for	ADP
ejpam-7044	104	63	some	some	DET
ejpam-7044	104	64	⋆-open	⋆-open	ADJ
ejpam-7044	104	65	set	set	NOUN
ejpam-7044	104	66	g.	g.	PROPN
ejpam-7044	104	67	(	(	PUNCT
ejpam-7044	104	68	4	4	X
ejpam-7044	104	69	)	)	PUNCT
ejpam-7044	104	70	a	a	DET
ejpam-7044	104	71	⊆	⊆	NUM
ejpam-7044	104	72	scl⋆(int⋆(a	scl⋆(int⋆(a	NOUN
ejpam-7044	104	73	)	)	PUNCT
ejpam-7044	104	74	)	)	PUNCT
ejpam-7044	104	75	.	.	PUNCT
ejpam-7044	105	1	for	for	ADP
ejpam-7044	105	2	a	a	DET
ejpam-7044	105	3	subset	subset	NOUN
ejpam-7044	105	4	a	a	PRON
ejpam-7044	105	5	of	of	ADP
ejpam-7044	105	6	an	an	DET
ejpam-7044	105	7	ideal	ideal	ADJ
ejpam-7044	105	8	topological	topological	ADJ
ejpam-7044	105	9	space	space	NOUN
ejpam-7044	105	10	(	(	PUNCT
ejpam-7044	105	11	x	x	X
ejpam-7044	105	12	,	,	PUNCT
ejpam-7044	105	13	τ	τ	PROPN
ejpam-7044	105	14	,	,	PUNCT
ejpam-7044	105	15	i	i	NOUN
ejpam-7044	105	16	)	)	PUNCT
ejpam-7044	105	17	,	,	PUNCT
ejpam-7044	105	18	the	the	DET
ejpam-7044	105	19	intersection	intersection	NOUN
ejpam-7044	105	20	of	of	ADP
ejpam-7044	105	21	all	all	DET
ejpam-7044	105	22	α	α	NOUN
ejpam-7044	105	23	-	-	PUNCT
ejpam-7044	105	24	i	i	PRON
ejpam-7044	105	25	⋆closed	⋆close	VERB
ejpam-7044	105	26	sets	set	NOUN
ejpam-7044	105	27	containing	contain	VERB
ejpam-7044	105	28	a	a	PRON
ejpam-7044	105	29	is	be	AUX
ejpam-7044	105	30	called	call	VERB
ejpam-7044	105	31	the	the	DET
ejpam-7044	105	32	α	α	NOUN
ejpam-7044	105	33	-	-	NOUN
ejpam-7044	105	34	i	i	PRON
ejpam-7044	105	35	⋆-closure	⋆-closure	PUNCT
ejpam-7044	106	1	[	[	X
ejpam-7044	106	2	24	24	NUM
ejpam-7044	106	3	]	]	PUNCT
ejpam-7044	106	4	of	of	ADP
ejpam-7044	106	5	a	a	PRON
ejpam-7044	106	6	and	and	CCONJ
ejpam-7044	106	7	is	be	AUX
ejpam-7044	106	8	denoted	denote	VERB
ejpam-7044	106	9	by	by	ADP
ejpam-7044	106	10	αcl⋆(a	αcl⋆(a	NUM
ejpam-7044	106	11	)	)	PUNCT
ejpam-7044	106	12	(	(	PUNCT
ejpam-7044	106	13	αcli	αcli	X
ejpam-7044	106	14	⋆(a	⋆(a	AUX
ejpam-7044	106	15	)	)	PUNCT
ejpam-7044	107	1	[	[	X
ejpam-7044	107	2	24	24	NUM
ejpam-7044	107	3	]	]	PUNCT
ejpam-7044	107	4	)	)	PUNCT
ejpam-7044	107	5	.	.	PUNCT
ejpam-7044	108	1	the	the	DET
ejpam-7044	108	2	α	α	PROPN
ejpam-7044	108	3	-	-	PUNCT
ejpam-7044	108	4	i	i	PRON
ejpam-7044	108	5	⋆-interior	⋆-interior	PUNCT
ejpam-7044	108	6	[	[	X
ejpam-7044	108	7	24	24	NUM
ejpam-7044	108	8	]	]	PUNCT
ejpam-7044	108	9	of	of	ADP
ejpam-7044	108	10	a	a	PRON
ejpam-7044	108	11	is	be	AUX
ejpam-7044	108	12	defined	define	VERB
ejpam-7044	108	13	by	by	ADP
ejpam-7044	108	14	the	the	DET
ejpam-7044	108	15	union	union	NOUN
ejpam-7044	108	16	of	of	ADP
ejpam-7044	108	17	all	all	DET
ejpam-7044	108	18	α	α	PROPN
ejpam-7044	108	19	-	-	ADJ
ejpam-7044	108	20	i	i	PRON
ejpam-7044	108	21	⋆-open	⋆-open	ADJ
ejpam-7044	108	22	sets	set	NOUN
ejpam-7044	108	23	contained	contain	VERB
ejpam-7044	108	24	in	in	ADP
ejpam-7044	108	25	a	a	PRON
ejpam-7044	108	26	and	and	CCONJ
ejpam-7044	108	27	is	be	AUX
ejpam-7044	108	28	denoted	denote	VERB
ejpam-7044	108	29	by	by	ADP
ejpam-7044	108	30	αint⋆(a	αint⋆(a	PROPN
ejpam-7044	108	31	)	)	PUNCT
ejpam-7044	109	1	(	(	PUNCT
ejpam-7044	109	2	αinti	αinti	X
ejpam-7044	109	3	⋆(a	⋆(a	NOUN
ejpam-7044	109	4	)	)	PUNCT
ejpam-7044	110	1	[	[	X
ejpam-7044	110	2	24	24	NUM
ejpam-7044	110	3	]	]	PUNCT
ejpam-7044	110	4	)	)	PUNCT
ejpam-7044	110	5	.	.	PUNCT
ejpam-7044	111	1	lemma	lemma	PROPN
ejpam-7044	111	2	4	4	NUM
ejpam-7044	111	3	.	.	PUNCT
ejpam-7044	112	1	[	[	X
ejpam-7044	112	2	24	24	NUM
ejpam-7044	112	3	]	]	PUNCT
ejpam-7044	112	4	for	for	ADP
ejpam-7044	112	5	a	a	DET
ejpam-7044	112	6	subset	subset	NOUN
ejpam-7044	112	7	a	a	PRON
ejpam-7044	112	8	of	of	ADP
ejpam-7044	112	9	an	an	DET
ejpam-7044	112	10	ideal	ideal	ADJ
ejpam-7044	112	11	topological	topological	ADJ
ejpam-7044	112	12	space	space	NOUN
ejpam-7044	112	13	(	(	PUNCT
ejpam-7044	112	14	x	x	X
ejpam-7044	112	15	,	,	PUNCT
ejpam-7044	112	16	τ	τ	PROPN
ejpam-7044	112	17	,	,	PUNCT
ejpam-7044	112	18	i	i	NOUN
ejpam-7044	112	19	)	)	PUNCT
ejpam-7044	112	20	,	,	PUNCT
ejpam-7044	112	21	the	the	DET
ejpam-7044	112	22	following	follow	VERB
ejpam-7044	112	23	properties	property	NOUN
ejpam-7044	112	24	hold	hold	VERB
ejpam-7044	112	25	:	:	PUNCT
ejpam-7044	112	26	(	(	PUNCT
ejpam-7044	112	27	1	1	X
ejpam-7044	112	28	)	)	PUNCT
ejpam-7044	112	29	a	a	PRON
ejpam-7044	112	30	is	be	AUX
ejpam-7044	112	31	α	α	X
ejpam-7044	112	32	-	-	PUNCT
ejpam-7044	112	33	i	i	PRON
ejpam-7044	112	34	⋆-closed	⋆-close	VERB
ejpam-7044	112	35	in	in	ADP
ejpam-7044	112	36	x	x	SYM
ejpam-7044	112	37	if	if	SCONJ
ejpam-7044	112	38	and	and	CCONJ
ejpam-7044	112	39	only	only	ADV
ejpam-7044	112	40	if	if	SCONJ
ejpam-7044	112	41	sint⋆(cl⋆(a	sint⋆(cl⋆(a	NOUN
ejpam-7044	112	42	)	)	PUNCT
ejpam-7044	112	43	)	)	PUNCT
ejpam-7044	113	1	⊆	⊆	NUM
ejpam-7044	113	2	a.	a.	NOUN
ejpam-7044	113	3	(	(	PUNCT
ejpam-7044	113	4	2	2	NUM
ejpam-7044	113	5	)	)	PUNCT
ejpam-7044	113	6	sint⋆(cl⋆(a	sint⋆(cl⋆(a	PROPN
ejpam-7044	113	7	)	)	PUNCT
ejpam-7044	113	8	)	)	PUNCT
ejpam-7044	113	9	=	=	PUNCT
ejpam-7044	113	10	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	113	11	)	)	PUNCT
ejpam-7044	113	12	)	)	PUNCT
ejpam-7044	113	13	)	)	PUNCT
ejpam-7044	113	14	.	.	PUNCT
ejpam-7044	114	1	(	(	PUNCT
ejpam-7044	114	2	3	3	X
ejpam-7044	114	3	)	)	PUNCT
ejpam-7044	114	4	αcl⋆(a	αcl⋆(a	NUM
ejpam-7044	114	5	)	)	PUNCT
ejpam-7044	115	1	=	=	PUNCT
ejpam-7044	115	2	a	a	DET
ejpam-7044	115	3	∪	∪	ADJ
ejpam-7044	115	4	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	115	5	)	)	PUNCT
ejpam-7044	115	6	)	)	PUNCT
ejpam-7044	115	7	)	)	PUNCT
ejpam-7044	115	8	.	.	PUNCT
ejpam-7044	116	1	(	(	PUNCT
ejpam-7044	116	2	4	4	X
ejpam-7044	116	3	)	)	PUNCT
ejpam-7044	116	4	αint⋆(a	αint⋆(a	NOUN
ejpam-7044	116	5	)	)	PUNCT
ejpam-7044	117	1	=	=	SYM
ejpam-7044	117	2	a	a	DET
ejpam-7044	117	3	∩	∩	ADJ
ejpam-7044	117	4	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7044	117	5	)	)	PUNCT
ejpam-7044	117	6	)	)	PUNCT
ejpam-7044	117	7	)	)	PUNCT
ejpam-7044	117	8	.	.	PUNCT
ejpam-7044	118	1	by	by	ADP
ejpam-7044	118	2	a	a	DET
ejpam-7044	118	3	multifunction	multifunction	NOUN
ejpam-7044	118	4	f	f	NOUN
ejpam-7044	118	5	:	:	PUNCT
ejpam-7044	118	6	x	x	X
ejpam-7044	118	7	→	→	SYM
ejpam-7044	118	8	y	y	PROPN
ejpam-7044	118	9	,	,	PUNCT
ejpam-7044	118	10	we	we	PRON
ejpam-7044	118	11	mean	mean	VERB
ejpam-7044	118	12	a	a	DET
ejpam-7044	118	13	point	point	NOUN
ejpam-7044	118	14	-	-	PUNCT
ejpam-7044	118	15	to	to	ADP
ejpam-7044	118	16	-	-	PUNCT
ejpam-7044	118	17	set	set	VERB
ejpam-7044	118	18	correspondence	correspondence	NOUN
ejpam-7044	118	19	from	from	ADP
ejpam-7044	118	20	x	x	PUNCT
ejpam-7044	118	21	into	into	ADP
ejpam-7044	118	22	y	y	PROPN
ejpam-7044	118	23	,	,	PUNCT
ejpam-7044	118	24	and	and	CCONJ
ejpam-7044	118	25	we	we	PRON
ejpam-7044	118	26	always	always	ADV
ejpam-7044	118	27	assume	assume	VERB
ejpam-7044	118	28	that	that	SCONJ
ejpam-7044	118	29	f	f	PROPN
ejpam-7044	118	30	(	(	PUNCT
ejpam-7044	118	31	x	x	X
ejpam-7044	118	32	)	)	PUNCT
ejpam-7044	118	33	̸=	̸=	NOUN
ejpam-7044	118	34	∅	∅	NOUN
ejpam-7044	118	35	for	for	ADP
ejpam-7044	118	36	all	all	PRON
ejpam-7044	118	37	x	x	SYM
ejpam-7044	118	38	∈	∈	ADJ
ejpam-7044	118	39	x.	x.	NOUN
ejpam-7044	118	40	for	for	ADP
ejpam-7044	118	41	a	a	DET
ejpam-7044	118	42	multifunction	multifunction	NOUN
ejpam-7044	118	43	f	f	NOUN
ejpam-7044	118	44	:	:	PUNCT
ejpam-7044	118	45	x	x	X
ejpam-7044	118	46	→	→	SYM
ejpam-7044	118	47	y	y	PROPN
ejpam-7044	118	48	,	,	PUNCT
ejpam-7044	118	49	we	we	PRON
ejpam-7044	118	50	shall	shall	AUX
ejpam-7044	118	51	denote	denote	VERB
ejpam-7044	118	52	the	the	DET
ejpam-7044	118	53	upper	upper	ADJ
ejpam-7044	118	54	and	and	CCONJ
ejpam-7044	118	55	lower	low	ADJ
ejpam-7044	118	56	inverse	inverse	NOUN
ejpam-7044	118	57	of	of	ADP
ejpam-7044	118	58	a	a	DET
ejpam-7044	118	59	set	set	NOUN
ejpam-7044	118	60	b	b	PROPN
ejpam-7044	118	61	of	of	ADP
ejpam-7044	118	62	y	y	PROPN
ejpam-7044	118	63	by	by	ADP
ejpam-7044	118	64	f+(b	f+(b	NOUN
ejpam-7044	118	65	)	)	PUNCT
ejpam-7044	118	66	and	and	CCONJ
ejpam-7044	118	67	f−(b	f−(b	NOUN
ejpam-7044	118	68	)	)	PUNCT
ejpam-7044	118	69	,	,	PUNCT
ejpam-7044	118	70	respectively	respectively	ADV
ejpam-7044	118	71	,	,	PUNCT
ejpam-7044	118	72	that	that	ADV
ejpam-7044	118	73	is	is	ADV
ejpam-7044	118	74	,	,	PUNCT
ejpam-7044	118	75	f+(b	f+(b	NOUN
ejpam-7044	118	76	)	)	PUNCT
ejpam-7044	118	77	=	=	PRON
ejpam-7044	119	1	{	{	PUNCT
ejpam-7044	119	2	x	x	PUNCT
ejpam-7044	119	3	∈	∈	PROPN
ejpam-7044	119	4	x	x	INTJ
ejpam-7044	120	1	|	|	NOUN
ejpam-7044	120	2	f	f	X
ejpam-7044	120	3	(	(	PUNCT
ejpam-7044	120	4	x	x	NOUN
ejpam-7044	120	5	)	)	PUNCT
ejpam-7044	120	6	⊆	⊆	NUM
ejpam-7044	120	7	b	b	NOUN
ejpam-7044	120	8	}	}	PUNCT
ejpam-7044	120	9	and	and	CCONJ
ejpam-7044	120	10	f−(b	f−(b	PROPN
ejpam-7044	120	11	)	)	PUNCT
ejpam-7044	120	12	=	=	PRON
ejpam-7044	121	1	{	{	PUNCT
ejpam-7044	121	2	x	x	PUNCT
ejpam-7044	121	3	∈	∈	PROPN
ejpam-7044	121	4	x	x	INTJ
ejpam-7044	122	1	|	|	NOUN
ejpam-7044	122	2	f	f	X
ejpam-7044	122	3	(	(	PUNCT
ejpam-7044	122	4	x	x	NOUN
ejpam-7044	122	5	)	)	PUNCT
ejpam-7044	122	6	∩	∩	NOUN
ejpam-7044	122	7	b	b	PROPN
ejpam-7044	122	8	̸=	̸=	PROPN
ejpam-7044	122	9	∅	∅	NOUN
ejpam-7044	122	10	}	}	PUNCT
ejpam-7044	122	11	.	.	PUNCT
ejpam-7044	123	1	in	in	ADP
ejpam-7044	123	2	particular	particular	ADJ
ejpam-7044	123	3	,	,	PUNCT
ejpam-7044	123	4	f−(y	f−(y	NOUN
ejpam-7044	123	5	)	)	PUNCT
ejpam-7044	123	6	=	=	SYM
ejpam-7044	124	1	{	{	PUNCT
ejpam-7044	124	2	x	x	PUNCT
ejpam-7044	124	3	∈	∈	PROPN
ejpam-7044	124	4	x	x	INTJ
ejpam-7044	125	1	|	|	ADV
ejpam-7044	125	2	y	y	PROPN
ejpam-7044	125	3	∈	∈	PROPN
ejpam-7044	125	4	f	f	X
ejpam-7044	125	5	(	(	PUNCT
ejpam-7044	125	6	x	x	NOUN
ejpam-7044	125	7	)	)	PUNCT
ejpam-7044	125	8	}	}	PUNCT
ejpam-7044	125	9	for	for	ADP
ejpam-7044	125	10	each	each	DET
ejpam-7044	125	11	point	point	NOUN
ejpam-7044	125	12	y	y	PROPN
ejpam-7044	125	13	∈	∈	PROPN
ejpam-7044	125	14	y	y	PROPN
ejpam-7044	125	15	.	.	PUNCT
ejpam-7044	126	1	for	for	ADP
ejpam-7044	126	2	each	each	DET
ejpam-7044	126	3	a	a	DET
ejpam-7044	126	4	⊆	⊆	NUM
ejpam-7044	126	5	x	x	SYM
ejpam-7044	126	6	,	,	PUNCT
ejpam-7044	126	7	f	f	PROPN
ejpam-7044	126	8	(	(	PUNCT
ejpam-7044	126	9	a	a	NOUN
ejpam-7044	126	10	)	)	PUNCT
ejpam-7044	126	11	=	=	SYM
ejpam-7044	126	12	∪x∈af	∪x∈af	NOUN
ejpam-7044	126	13	(	(	PUNCT
ejpam-7044	126	14	x	x	NOUN
ejpam-7044	126	15	)	)	PUNCT
ejpam-7044	126	16	.	.	PUNCT
ejpam-7044	127	1	3	3	X
ejpam-7044	127	2	.	.	X
ejpam-7044	127	3	upper	upper	ADJ
ejpam-7044	127	4	and	and	CCONJ
ejpam-7044	127	5	lower	low	ADJ
ejpam-7044	127	6	τ	τ	X
ejpam-7044	127	7	⋆α(σ1	⋆α(σ1	X
ejpam-7044	127	8	,	,	PUNCT
ejpam-7044	127	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	127	10	multifunctions	multifunction	NOUN
ejpam-7044	127	11	in	in	ADP
ejpam-7044	127	12	this	this	DET
ejpam-7044	127	13	section	section	NOUN
ejpam-7044	127	14	,	,	PUNCT
ejpam-7044	127	15	we	we	PRON
ejpam-7044	127	16	introduce	introduce	VERB
ejpam-7044	127	17	the	the	DET
ejpam-7044	127	18	notions	notion	NOUN
ejpam-7044	127	19	of	of	ADP
ejpam-7044	127	20	upper	upper	ADJ
ejpam-7044	127	21	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	127	22	,	,	PUNCT
ejpam-7044	127	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	127	24	multifunctions	multifunction	NOUN
ejpam-7044	127	25	and	and	CCONJ
ejpam-7044	127	26	lower	low	ADJ
ejpam-7044	127	27	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	127	28	,	,	PUNCT
ejpam-7044	127	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	127	30	multifunctions	multifunction	NOUN
ejpam-7044	127	31	.	.	PUNCT
ejpam-7044	128	1	moreover	moreover	ADV
ejpam-7044	128	2	,	,	PUNCT
ejpam-7044	128	3	several	several	ADJ
ejpam-7044	128	4	characterizations	characterization	NOUN
ejpam-7044	128	5	of	of	ADP
ejpam-7044	128	6	upper	upper	ADJ
ejpam-7044	128	7	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	128	8	,	,	PUNCT
ejpam-7044	128	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	128	10	multifunctions	multifunction	NOUN
ejpam-7044	128	11	and	and	CCONJ
ejpam-7044	128	12	lower	low	ADJ
ejpam-7044	128	13	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	128	14	,	,	PUNCT
ejpam-7044	128	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	128	16	multifunctions	multifunction	NOUN
ejpam-7044	128	17	discussed	discuss	VERB
ejpam-7044	128	18	.	.	PUNCT
ejpam-7044	129	1	definition	definition	NOUN
ejpam-7044	129	2	1	1	NUM
ejpam-7044	129	3	.	.	PUNCT
ejpam-7044	130	1	a	a	DET
ejpam-7044	130	2	multifunction	multifunction	NOUN
ejpam-7044	130	3	f	f	NOUN
ejpam-7044	130	4	:	:	PUNCT
ejpam-7044	130	5	(	(	PUNCT
ejpam-7044	130	6	x	x	X
ejpam-7044	130	7	,	,	PUNCT
ejpam-7044	130	8	τ	τ	PROPN
ejpam-7044	130	9	,	,	PUNCT
ejpam-7044	130	10	i	i	NOUN
ejpam-7044	130	11	)	)	PUNCT
ejpam-7044	130	12	→	→	PUNCT
ejpam-7044	130	13	(	(	PUNCT
ejpam-7044	130	14	y	y	PROPN
ejpam-7044	130	15	,	,	PUNCT
ejpam-7044	130	16	σ1	σ1	PROPN
ejpam-7044	130	17	,	,	PUNCT
ejpam-7044	130	18	σ2	σ2	PROPN
ejpam-7044	130	19	)	)	PUNCT
ejpam-7044	130	20	is	be	AUX
ejpam-7044	130	21	said	say	VERB
ejpam-7044	130	22	to	to	PART
ejpam-7044	130	23	be	be	AUX
ejpam-7044	130	24	upper	upper	ADJ
ejpam-7044	130	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	130	26	,	,	PUNCT
ejpam-7044	130	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7044	130	28	at	at	ADP
ejpam-7044	130	29	a	a	DET
ejpam-7044	130	30	point	point	NOUN
ejpam-7044	130	31	x	x	PUNCT
ejpam-7044	130	32	of	of	ADP
ejpam-7044	130	33	x	x	PRON
ejpam-7044	130	34	if	if	SCONJ
ejpam-7044	130	35	for	for	ADP
ejpam-7044	130	36	each	each	DET
ejpam-7044	130	37	σ1σ2	σ1σ2	VERB
ejpam-7044	130	38	-	-	ADJ
ejpam-7044	130	39	open	open	ADJ
ejpam-7044	130	40	set	set	NOUN
ejpam-7044	130	41	v	v	ADP
ejpam-7044	130	42	such	such	ADJ
ejpam-7044	130	43	that	that	SCONJ
ejpam-7044	130	44	f	f	PROPN
ejpam-7044	130	45	(	(	PUNCT
ejpam-7044	130	46	x	x	X
ejpam-7044	130	47	)	)	PUNCT
ejpam-7044	130	48	⊆	⊆	NUM
ejpam-7044	130	49	v	v	NOUN
ejpam-7044	130	50	,	,	PUNCT
ejpam-7044	130	51	there	there	PRON
ejpam-7044	130	52	exists	exist	VERB
ejpam-7044	130	53	a	a	DET
ejpam-7044	130	54	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	130	55	-	-	PUNCT
ejpam-7044	130	56	open	open	ADJ
ejpam-7044	130	57	set	set	NOUN
ejpam-7044	130	58	u	u	NOUN
ejpam-7044	130	59	of	of	ADP
ejpam-7044	130	60	x	x	PUNCT
ejpam-7044	130	61	containing	contain	VERB
ejpam-7044	130	62	x	x	PUNCT
ejpam-7044	130	63	such	such	ADJ
ejpam-7044	130	64	that	that	SCONJ
ejpam-7044	130	65	f	f	PROPN
ejpam-7044	130	66	(	(	PUNCT
ejpam-7044	130	67	u	u	NOUN
ejpam-7044	130	68	)	)	PUNCT
ejpam-7044	130	69	⊆	⊆	NUM
ejpam-7044	130	70	v	v	NOUN
ejpam-7044	130	71	.	.	PUNCT
ejpam-7044	131	1	a	a	DET
ejpam-7044	131	2	multifunction	multifunction	NOUN
ejpam-7044	131	3	f	f	NOUN
ejpam-7044	131	4	:	:	PUNCT
ejpam-7044	131	5	(	(	PUNCT
ejpam-7044	131	6	x	x	X
ejpam-7044	131	7	,	,	PUNCT
ejpam-7044	131	8	τ	τ	PROPN
ejpam-7044	131	9	,	,	PUNCT
ejpam-7044	131	10	i	i	NOUN
ejpam-7044	131	11	)	)	PUNCT
ejpam-7044	131	12	→	→	PUNCT
ejpam-7044	131	13	(	(	PUNCT
ejpam-7044	131	14	y	y	PROPN
ejpam-7044	131	15	,	,	PUNCT
ejpam-7044	131	16	σ1	σ1	PROPN
ejpam-7044	131	17	,	,	PUNCT
ejpam-7044	131	18	σ2	σ2	PROPN
ejpam-7044	131	19	)	)	PUNCT
ejpam-7044	131	20	j.	j.	PROPN
ejpam-7044	131	21	khampakdee	khampakdee	PROPN
ejpam-7044	131	22	,	,	PUNCT
ejpam-7044	131	23	a.	a.	PROPN
ejpam-7044	131	24	sama	sama	PROPN
ejpam-7044	131	25	-	-	PUNCT
ejpam-7044	131	26	ae	ae	PROPN
ejpam-7044	131	27	,	,	PUNCT
ejpam-7044	131	28	c.	c.	PROPN
ejpam-7044	131	29	boonpok	boonpok	PROPN
ejpam-7044	131	30	/	/	SYM
ejpam-7044	131	31	eur	eur	PROPN
ejpam-7044	131	32	.	.	PUNCT
ejpam-7044	132	1	j.	j.	PROPN
ejpam-7044	132	2	pure	pure	PROPN
ejpam-7044	132	3	appl	appl	PROPN
ejpam-7044	132	4	.	.	PROPN
ejpam-7044	132	5	math	math	PROPN
ejpam-7044	132	6	,	,	PUNCT
ejpam-7044	132	7	18	18	NUM
ejpam-7044	132	8	(	(	PUNCT
ejpam-7044	132	9	4	4	NUM
ejpam-7044	132	10	)	)	PUNCT
ejpam-7044	132	11	(	(	PUNCT
ejpam-7044	132	12	2025	2025	NUM
ejpam-7044	132	13	)	)	PUNCT
ejpam-7044	132	14	,	,	PUNCT
ejpam-7044	132	15	7044	7044	NUM
ejpam-7044	132	16	5	5	NUM
ejpam-7044	132	17	of	of	ADP
ejpam-7044	132	18	9	9	NUM
ejpam-7044	132	19	is	be	AUX
ejpam-7044	132	20	said	say	VERB
ejpam-7044	132	21	to	to	PART
ejpam-7044	132	22	be	be	AUX
ejpam-7044	132	23	upper	upper	ADJ
ejpam-7044	132	24	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	132	25	,	,	PUNCT
ejpam-7044	132	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	132	27	if	if	SCONJ
ejpam-7044	132	28	f	f	PROPN
ejpam-7044	132	29	is	be	AUX
ejpam-7044	132	30	upper	upper	ADJ
ejpam-7044	132	31	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	132	32	,	,	PUNCT
ejpam-7044	132	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	132	34	at	at	ADP
ejpam-7044	132	35	each	each	DET
ejpam-7044	132	36	point	point	NOUN
ejpam-7044	132	37	of	of	ADP
ejpam-7044	132	38	x.	x.	NOUN
ejpam-7044	132	39	theorem	theorem	VERB
ejpam-7044	132	40	1	1	NUM
ejpam-7044	132	41	.	.	X
ejpam-7044	132	42	for	for	ADP
ejpam-7044	132	43	a	a	DET
ejpam-7044	132	44	multifunction	multifunction	NOUN
ejpam-7044	132	45	f	f	NOUN
ejpam-7044	132	46	:	:	PUNCT
ejpam-7044	132	47	(	(	PUNCT
ejpam-7044	132	48	x	x	X
ejpam-7044	132	49	,	,	PUNCT
ejpam-7044	132	50	τ	τ	PROPN
ejpam-7044	132	51	,	,	PUNCT
ejpam-7044	132	52	i	i	NOUN
ejpam-7044	132	53	)	)	PUNCT
ejpam-7044	132	54	→	→	PUNCT
ejpam-7044	132	55	(	(	PUNCT
ejpam-7044	132	56	y	y	PROPN
ejpam-7044	132	57	,	,	PUNCT
ejpam-7044	132	58	σ1	σ1	PROPN
ejpam-7044	132	59	,	,	PUNCT
ejpam-7044	132	60	σ2	σ2	NOUN
ejpam-7044	132	61	)	)	PUNCT
ejpam-7044	132	62	,	,	PUNCT
ejpam-7044	132	63	the	the	DET
ejpam-7044	132	64	following	follow	VERB
ejpam-7044	132	65	properties	property	NOUN
ejpam-7044	132	66	are	be	AUX
ejpam-7044	132	67	equivalent	equivalent	ADJ
ejpam-7044	132	68	:	:	PUNCT
ejpam-7044	132	69	(	(	PUNCT
ejpam-7044	132	70	1	1	X
ejpam-7044	132	71	)	)	PUNCT
ejpam-7044	132	72	f	f	PROPN
ejpam-7044	132	73	is	be	AUX
ejpam-7044	132	74	upper	upper	ADJ
ejpam-7044	132	75	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	132	76	,	,	PUNCT
ejpam-7044	132	77	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	132	78	at	at	ADP
ejpam-7044	132	79	x	x	X
ejpam-7044	132	80	∈	∈	PROPN
ejpam-7044	132	81	x	x	X
ejpam-7044	132	82	;	;	PUNCT
ejpam-7044	132	83	(	(	PUNCT
ejpam-7044	132	84	2	2	X
ejpam-7044	132	85	)	)	PUNCT
ejpam-7044	132	86	x	x	SYM
ejpam-7044	132	87	∈	∈	PROPN
ejpam-7044	132	88	scl⋆(int⋆(f+(v	scl⋆(int⋆(f+(v	PROPN
ejpam-7044	132	89	)	)	PUNCT
ejpam-7044	132	90	)	)	PUNCT
ejpam-7044	132	91	)	)	PUNCT
ejpam-7044	132	92	for	for	ADP
ejpam-7044	132	93	every	every	DET
ejpam-7044	132	94	σ1σ2	σ1σ2	NOUN
ejpam-7044	132	95	-	-	ADJ
ejpam-7044	132	96	open	open	ADJ
ejpam-7044	132	97	set	set	NOUN
ejpam-7044	132	98	v	v	NOUN
ejpam-7044	132	99	of	of	ADP
ejpam-7044	132	100	y	y	PROPN
ejpam-7044	132	101	containing	contain	VERB
ejpam-7044	132	102	f	f	PROPN
ejpam-7044	132	103	(	(	PUNCT
ejpam-7044	132	104	x	x	NOUN
ejpam-7044	132	105	)	)	PUNCT
ejpam-7044	132	106	;	;	PUNCT
ejpam-7044	132	107	(	(	PUNCT
ejpam-7044	132	108	3	3	X
ejpam-7044	132	109	)	)	PUNCT
ejpam-7044	132	110	x	x	SYM
ejpam-7044	132	111	∈	∈	PROPN
ejpam-7044	132	112	αint⋆(f+(v	αint⋆(f+(v	PROPN
ejpam-7044	132	113	)	)	PUNCT
ejpam-7044	132	114	)	)	PUNCT
ejpam-7044	132	115	for	for	ADP
ejpam-7044	132	116	every	every	DET
ejpam-7044	132	117	σ1σ2	σ1σ2	NOUN
ejpam-7044	132	118	-	-	ADJ
ejpam-7044	132	119	open	open	ADJ
ejpam-7044	132	120	set	set	NOUN
ejpam-7044	132	121	v	v	NOUN
ejpam-7044	132	122	of	of	ADP
ejpam-7044	132	123	y	y	PROPN
ejpam-7044	132	124	containing	contain	VERB
ejpam-7044	132	125	f	f	PROPN
ejpam-7044	132	126	(	(	PUNCT
ejpam-7044	132	127	x	x	NOUN
ejpam-7044	132	128	)	)	PUNCT
ejpam-7044	132	129	.	.	PUNCT
ejpam-7044	133	1	proof	proof	NOUN
ejpam-7044	133	2	.	.	PUNCT
ejpam-7044	134	1	(	(	PUNCT
ejpam-7044	134	2	1	1	X
ejpam-7044	134	3	)	)	PUNCT
ejpam-7044	134	4	⇒	⇒	NOUN
ejpam-7044	134	5	(	(	PUNCT
ejpam-7044	134	6	2	2	NUM
ejpam-7044	134	7	):	):	PUNCT
ejpam-7044	134	8	let	let	VERB
ejpam-7044	134	9	v	v	PART
ejpam-7044	134	10	be	be	AUX
ejpam-7044	134	11	any	any	DET
ejpam-7044	134	12	σ1σ2	σ1σ2	NOUN
ejpam-7044	134	13	-	-	ADJ
ejpam-7044	134	14	open	open	ADJ
ejpam-7044	134	15	set	set	NOUN
ejpam-7044	134	16	of	of	ADP
ejpam-7044	134	17	y	y	PROPN
ejpam-7044	134	18	containing	contain	VERB
ejpam-7044	134	19	f	f	PROPN
ejpam-7044	134	20	(	(	PUNCT
ejpam-7044	134	21	x	x	NOUN
ejpam-7044	134	22	)	)	PUNCT
ejpam-7044	134	23	.	.	PUNCT
ejpam-7044	135	1	then	then	ADV
ejpam-7044	135	2	,	,	PUNCT
ejpam-7044	135	3	there	there	PRON
ejpam-7044	135	4	exists	exist	VERB
ejpam-7044	135	5	a	a	DET
ejpam-7044	135	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	135	7	-	-	PUNCT
ejpam-7044	135	8	open	open	ADJ
ejpam-7044	135	9	set	set	NOUN
ejpam-7044	135	10	u	u	NOUN
ejpam-7044	135	11	of	of	ADP
ejpam-7044	135	12	x	x	PUNCT
ejpam-7044	135	13	containing	contain	VERB
ejpam-7044	135	14	x	x	PUNCT
ejpam-7044	135	15	such	such	ADJ
ejpam-7044	135	16	that	that	SCONJ
ejpam-7044	135	17	f	f	PROPN
ejpam-7044	135	18	(	(	PUNCT
ejpam-7044	135	19	u	u	NOUN
ejpam-7044	135	20	)	)	PUNCT
ejpam-7044	135	21	⊆	⊆	NUM
ejpam-7044	135	22	v	v	NOUN
ejpam-7044	135	23	;	;	PUNCT
ejpam-7044	135	24	hence	hence	ADV
ejpam-7044	135	25	x	x	PART
ejpam-7044	135	26	∈	∈	PROPN
ejpam-7044	135	27	u	u	NOUN
ejpam-7044	135	28	⊆	⊆	NUM
ejpam-7044	135	29	f+(v	f+(v	NOUN
ejpam-7044	135	30	)	)	PUNCT
ejpam-7044	135	31	.	.	PUNCT
ejpam-7044	136	1	since	since	SCONJ
ejpam-7044	136	2	u	u	NOUN
ejpam-7044	136	3	is	be	AUX
ejpam-7044	136	4	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	136	5	-	-	PUNCT
ejpam-7044	136	6	open	open	ADJ
ejpam-7044	136	7	,	,	PUNCT
ejpam-7044	136	8	by	by	ADP
ejpam-7044	136	9	lemma	lemma	PROPN
ejpam-7044	136	10	3	3	NUM
ejpam-7044	136	11	we	we	PRON
ejpam-7044	136	12	have	have	VERB
ejpam-7044	136	13	x	x	X
ejpam-7044	136	14	∈	∈	PROPN
ejpam-7044	136	15	u	u	NOUN
ejpam-7044	136	16	⊆	⊆	NUM
ejpam-7044	136	17	scl⋆(int⋆(u	scl⋆(int⋆(u	PROPN
ejpam-7044	136	18	)	)	PUNCT
ejpam-7044	136	19	)	)	PUNCT
ejpam-7044	136	20	⊆	⊆	NUM
ejpam-7044	136	21	scl⋆(int⋆(f+(v	scl⋆(int⋆(f+(v	PROPN
ejpam-7044	136	22	)	)	PUNCT
ejpam-7044	136	23	)	)	PUNCT
ejpam-7044	136	24	)	)	PUNCT
ejpam-7044	136	25	.	.	PUNCT
ejpam-7044	137	1	(	(	PUNCT
ejpam-7044	137	2	2	2	X
ejpam-7044	137	3	)	)	PUNCT
ejpam-7044	137	4	⇒	⇒	NOUN
ejpam-7044	137	5	(	(	PUNCT
ejpam-7044	137	6	3	3	NUM
ejpam-7044	137	7	):	):	PUNCT
ejpam-7044	137	8	let	let	VERB
ejpam-7044	137	9	v	v	PART
ejpam-7044	137	10	be	be	AUX
ejpam-7044	137	11	any	any	DET
ejpam-7044	137	12	σ1σ2	σ1σ2	NOUN
ejpam-7044	137	13	-	-	ADJ
ejpam-7044	137	14	open	open	ADJ
ejpam-7044	137	15	set	set	NOUN
ejpam-7044	137	16	of	of	ADP
ejpam-7044	137	17	y	y	PROPN
ejpam-7044	137	18	containing	contain	VERB
ejpam-7044	137	19	f	f	PROPN
ejpam-7044	137	20	(	(	PUNCT
ejpam-7044	137	21	x	x	NOUN
ejpam-7044	137	22	)	)	PUNCT
ejpam-7044	137	23	.	.	PUNCT
ejpam-7044	138	1	then	then	ADV
ejpam-7044	138	2	by	by	ADP
ejpam-7044	138	3	(	(	PUNCT
ejpam-7044	138	4	2	2	NUM
ejpam-7044	138	5	)	)	PUNCT
ejpam-7044	138	6	,	,	PUNCT
ejpam-7044	138	7	we	we	PRON
ejpam-7044	138	8	have	have	VERB
ejpam-7044	138	9	x	x	X
ejpam-7044	138	10	∈	∈	PROPN
ejpam-7044	138	11	scl⋆(int⋆(f+(v	scl⋆(int⋆(f+(v	PROPN
ejpam-7044	138	12	)	)	PUNCT
ejpam-7044	138	13	)	)	PUNCT
ejpam-7044	138	14	)	)	PUNCT
ejpam-7044	138	15	and	and	CCONJ
ejpam-7044	138	16	by	by	ADP
ejpam-7044	138	17	lemma	lemma	PROPN
ejpam-7044	138	18	2	2	NUM
ejpam-7044	138	19	,	,	PUNCT
ejpam-7044	138	20	x	x	SYM
ejpam-7044	138	21	∈	∈	NOUN
ejpam-7044	138	22	int⋆(cl⋆(int⋆(f+(v	int⋆(cl⋆(int⋆(f+(v	NOUN
ejpam-7044	138	23	)	)	PUNCT
ejpam-7044	138	24	)	)	PUNCT
ejpam-7044	138	25	)	)	PUNCT
ejpam-7044	138	26	)	)	PUNCT
ejpam-7044	138	27	.	.	PUNCT
ejpam-7044	139	1	therefore	therefore	ADV
ejpam-7044	139	2	,	,	PUNCT
ejpam-7044	139	3	x	x	PUNCT
ejpam-7044	139	4	∈	∈	PROPN
ejpam-7044	139	5	αint⋆(f+(v	αint⋆(f+(v	PROPN
ejpam-7044	139	6	)	)	PUNCT
ejpam-7044	139	7	)	)	PUNCT
ejpam-7044	139	8	by	by	ADP
ejpam-7044	139	9	lemma	lemma	PROPN
ejpam-7044	139	10	4	4	NUM
ejpam-7044	139	11	.	.	PUNCT
ejpam-7044	139	12	(	(	PUNCT
ejpam-7044	139	13	3	3	X
ejpam-7044	139	14	)	)	PUNCT
ejpam-7044	139	15	⇒	⇒	NOUN
ejpam-7044	139	16	(	(	PUNCT
ejpam-7044	139	17	1	1	NUM
ejpam-7044	139	18	):	):	PUNCT
ejpam-7044	139	19	let	let	VERB
ejpam-7044	139	20	v	v	PART
ejpam-7044	139	21	be	be	AUX
ejpam-7044	139	22	any	any	DET
ejpam-7044	139	23	σ1σ2	σ1σ2	NOUN
ejpam-7044	139	24	-	-	ADJ
ejpam-7044	139	25	open	open	ADJ
ejpam-7044	139	26	set	set	NOUN
ejpam-7044	139	27	of	of	ADP
ejpam-7044	139	28	y	y	PROPN
ejpam-7044	139	29	containing	contain	VERB
ejpam-7044	139	30	f	f	PROPN
ejpam-7044	139	31	(	(	PUNCT
ejpam-7044	139	32	x	x	NOUN
ejpam-7044	139	33	)	)	PUNCT
ejpam-7044	139	34	.	.	PUNCT
ejpam-7044	140	1	by	by	ADP
ejpam-7044	140	2	(	(	PUNCT
ejpam-7044	140	3	3	3	NUM
ejpam-7044	140	4	)	)	PUNCT
ejpam-7044	140	5	,	,	PUNCT
ejpam-7044	140	6	we	we	PRON
ejpam-7044	140	7	have	have	VERB
ejpam-7044	140	8	x	x	X
ejpam-7044	140	9	∈	∈	PROPN
ejpam-7044	140	10	αint⋆(f+(v	αint⋆(f+(v	PROPN
ejpam-7044	140	11	)	)	PUNCT
ejpam-7044	140	12	)	)	PUNCT
ejpam-7044	141	1	and	and	CCONJ
ejpam-7044	141	2	so	so	ADV
ejpam-7044	141	3	there	there	PRON
ejpam-7044	141	4	exists	exist	VERB
ejpam-7044	141	5	a	a	DET
ejpam-7044	141	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	141	7	-	-	PUNCT
ejpam-7044	141	8	open	open	ADJ
ejpam-7044	141	9	set	set	NOUN
ejpam-7044	141	10	u	u	NOUN
ejpam-7044	141	11	of	of	ADP
ejpam-7044	141	12	x	x	PUNCT
ejpam-7044	141	13	containing	contain	VERB
ejpam-7044	141	14	x	x	PUNCT
ejpam-7044	141	15	such	such	ADJ
ejpam-7044	141	16	that	that	SCONJ
ejpam-7044	141	17	u	u	NOUN
ejpam-7044	141	18	⊆	⊆	NUM
ejpam-7044	141	19	f+(v	f+(v	NOUN
ejpam-7044	141	20	)	)	PUNCT
ejpam-7044	141	21	;	;	PUNCT
ejpam-7044	141	22	hence	hence	ADV
ejpam-7044	141	23	f	f	PROPN
ejpam-7044	141	24	(	(	PUNCT
ejpam-7044	141	25	u	u	NOUN
ejpam-7044	141	26	)	)	PUNCT
ejpam-7044	141	27	⊆	⊆	NUM
ejpam-7044	141	28	v	v	NOUN
ejpam-7044	141	29	.	.	PUNCT
ejpam-7044	142	1	this	this	PRON
ejpam-7044	142	2	shows	show	VERB
ejpam-7044	142	3	that	that	SCONJ
ejpam-7044	142	4	f	f	PROPN
ejpam-7044	142	5	is	be	AUX
ejpam-7044	142	6	upper	upper	ADJ
ejpam-7044	142	7	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	142	8	,	,	PUNCT
ejpam-7044	142	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	142	10	at	at	ADP
ejpam-7044	142	11	x.	x.	NOUN
ejpam-7044	142	12	definition	definition	NOUN
ejpam-7044	142	13	2	2	NUM
ejpam-7044	142	14	.	.	PUNCT
ejpam-7044	142	15	a	a	DET
ejpam-7044	142	16	multifunction	multifunction	NOUN
ejpam-7044	143	1	f	f	NOUN
ejpam-7044	143	2	:	:	PUNCT
ejpam-7044	143	3	(	(	PUNCT
ejpam-7044	143	4	x	x	X
ejpam-7044	143	5	,	,	PUNCT
ejpam-7044	143	6	τ	τ	PROPN
ejpam-7044	143	7	,	,	PUNCT
ejpam-7044	143	8	i	i	NOUN
ejpam-7044	143	9	)	)	PUNCT
ejpam-7044	143	10	→	→	PUNCT
ejpam-7044	143	11	(	(	PUNCT
ejpam-7044	143	12	y	y	PROPN
ejpam-7044	143	13	,	,	PUNCT
ejpam-7044	143	14	σ1	σ1	PROPN
ejpam-7044	143	15	,	,	PUNCT
ejpam-7044	143	16	σ2	σ2	PROPN
ejpam-7044	143	17	)	)	PUNCT
ejpam-7044	143	18	is	be	AUX
ejpam-7044	143	19	said	say	VERB
ejpam-7044	143	20	to	to	PART
ejpam-7044	143	21	be	be	AUX
ejpam-7044	143	22	lower	low	ADJ
ejpam-7044	143	23	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	143	24	,	,	PUNCT
ejpam-7044	143	25	σ2)continuous	σ2)continuous	ADJ
ejpam-7044	143	26	at	at	ADP
ejpam-7044	143	27	a	a	DET
ejpam-7044	143	28	point	point	NOUN
ejpam-7044	143	29	x	x	PUNCT
ejpam-7044	143	30	of	of	ADP
ejpam-7044	143	31	x	x	PRON
ejpam-7044	143	32	if	if	SCONJ
ejpam-7044	143	33	for	for	ADP
ejpam-7044	143	34	each	each	DET
ejpam-7044	143	35	σ1σ2	σ1σ2	VERB
ejpam-7044	143	36	-	-	ADJ
ejpam-7044	143	37	open	open	ADJ
ejpam-7044	143	38	set	set	NOUN
ejpam-7044	143	39	v	v	ADP
ejpam-7044	143	40	such	such	ADJ
ejpam-7044	143	41	that	that	SCONJ
ejpam-7044	143	42	f	f	PROPN
ejpam-7044	143	43	(	(	PUNCT
ejpam-7044	143	44	x	x	NOUN
ejpam-7044	143	45	)	)	PUNCT
ejpam-7044	143	46	∩	∩	NOUN
ejpam-7044	143	47	v	v	ADP
ejpam-7044	143	48	̸=	̸=	PROPN
ejpam-7044	143	49	∅	∅	NOUN
ejpam-7044	143	50	,	,	PUNCT
ejpam-7044	143	51	there	there	PRON
ejpam-7044	143	52	exists	exist	VERB
ejpam-7044	143	53	a	a	DET
ejpam-7044	143	54	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	143	55	-	-	PUNCT
ejpam-7044	143	56	open	open	ADJ
ejpam-7044	143	57	set	set	NOUN
ejpam-7044	143	58	u	u	NOUN
ejpam-7044	143	59	containing	contain	VERB
ejpam-7044	143	60	x	x	PUNCT
ejpam-7044	143	61	such	such	ADJ
ejpam-7044	143	62	that	that	SCONJ
ejpam-7044	143	63	f	f	PROPN
ejpam-7044	143	64	(	(	PUNCT
ejpam-7044	143	65	z	z	NOUN
ejpam-7044	143	66	)	)	PUNCT
ejpam-7044	143	67	∩	∩	NOUN
ejpam-7044	143	68	v	v	ADP
ejpam-7044	143	69	̸=	̸=	PROPN
ejpam-7044	143	70	∅	∅	NOUN
ejpam-7044	143	71	for	for	ADP
ejpam-7044	143	72	every	every	DET
ejpam-7044	143	73	z	z	NOUN
ejpam-7044	143	74	∈	∈	PROPN
ejpam-7044	143	75	u	u	NOUN
ejpam-7044	143	76	.	.	PUNCT
ejpam-7044	144	1	a	a	DET
ejpam-7044	144	2	multifunction	multifunction	NOUN
ejpam-7044	144	3	f	f	NOUN
ejpam-7044	144	4	:	:	PUNCT
ejpam-7044	144	5	(	(	PUNCT
ejpam-7044	144	6	x	x	X
ejpam-7044	144	7	,	,	PUNCT
ejpam-7044	144	8	τ	τ	PROPN
ejpam-7044	144	9	,	,	PUNCT
ejpam-7044	144	10	i	i	NOUN
ejpam-7044	144	11	)	)	PUNCT
ejpam-7044	144	12	→	→	PUNCT
ejpam-7044	144	13	(	(	PUNCT
ejpam-7044	144	14	y	y	PROPN
ejpam-7044	144	15	,	,	PUNCT
ejpam-7044	144	16	σ1	σ1	PROPN
ejpam-7044	144	17	,	,	PUNCT
ejpam-7044	144	18	σ2	σ2	PROPN
ejpam-7044	144	19	)	)	PUNCT
ejpam-7044	144	20	is	be	AUX
ejpam-7044	144	21	said	say	VERB
ejpam-7044	144	22	to	to	PART
ejpam-7044	144	23	be	be	AUX
ejpam-7044	144	24	lower	low	ADJ
ejpam-7044	144	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	144	26	,	,	PUNCT
ejpam-7044	144	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	144	28	if	if	SCONJ
ejpam-7044	144	29	f	f	PROPN
ejpam-7044	144	30	is	be	AUX
ejpam-7044	144	31	lower	low	ADJ
ejpam-7044	144	32	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	144	33	,	,	PUNCT
ejpam-7044	144	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	144	35	at	at	ADP
ejpam-7044	144	36	each	each	DET
ejpam-7044	144	37	point	point	NOUN
ejpam-7044	144	38	of	of	ADP
ejpam-7044	144	39	x.	x.	NOUN
ejpam-7044	144	40	theorem	theorem	VERB
ejpam-7044	144	41	2	2	NUM
ejpam-7044	144	42	.	.	X
ejpam-7044	144	43	for	for	ADP
ejpam-7044	144	44	a	a	DET
ejpam-7044	144	45	multifunction	multifunction	NOUN
ejpam-7044	144	46	f	f	NOUN
ejpam-7044	144	47	:	:	PUNCT
ejpam-7044	144	48	(	(	PUNCT
ejpam-7044	144	49	x	x	X
ejpam-7044	144	50	,	,	PUNCT
ejpam-7044	144	51	τ	τ	PROPN
ejpam-7044	144	52	,	,	PUNCT
ejpam-7044	144	53	i	i	NOUN
ejpam-7044	144	54	)	)	PUNCT
ejpam-7044	144	55	→	→	PUNCT
ejpam-7044	144	56	(	(	PUNCT
ejpam-7044	144	57	y	y	PROPN
ejpam-7044	144	58	,	,	PUNCT
ejpam-7044	144	59	σ1	σ1	PROPN
ejpam-7044	144	60	,	,	PUNCT
ejpam-7044	144	61	σ2	σ2	NOUN
ejpam-7044	144	62	)	)	PUNCT
ejpam-7044	144	63	,	,	PUNCT
ejpam-7044	144	64	the	the	DET
ejpam-7044	144	65	following	follow	VERB
ejpam-7044	144	66	properties	property	NOUN
ejpam-7044	144	67	are	be	AUX
ejpam-7044	144	68	equivalent	equivalent	ADJ
ejpam-7044	144	69	:	:	PUNCT
ejpam-7044	144	70	(	(	PUNCT
ejpam-7044	144	71	1	1	X
ejpam-7044	144	72	)	)	PUNCT
ejpam-7044	144	73	f	f	PROPN
ejpam-7044	144	74	is	be	AUX
ejpam-7044	144	75	lower	low	ADJ
ejpam-7044	144	76	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	144	77	,	,	PUNCT
ejpam-7044	144	78	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	144	79	at	at	ADP
ejpam-7044	144	80	x	x	X
ejpam-7044	144	81	∈	∈	PROPN
ejpam-7044	144	82	x	x	X
ejpam-7044	144	83	;	;	PUNCT
ejpam-7044	144	84	(	(	PUNCT
ejpam-7044	144	85	2	2	X
ejpam-7044	144	86	)	)	PUNCT
ejpam-7044	144	87	x	x	SYM
ejpam-7044	144	88	∈	∈	PROPN
ejpam-7044	144	89	scl⋆(int⋆(f−(v	scl⋆(int⋆(f−(v	NUM
ejpam-7044	144	90	)	)	PUNCT
ejpam-7044	144	91	)	)	PUNCT
ejpam-7044	144	92	)	)	PUNCT
ejpam-7044	144	93	for	for	ADP
ejpam-7044	144	94	every	every	DET
ejpam-7044	144	95	σ1σ2	σ1σ2	NOUN
ejpam-7044	144	96	-	-	ADJ
ejpam-7044	144	97	open	open	ADJ
ejpam-7044	144	98	set	set	NOUN
ejpam-7044	144	99	v	v	NOUN
ejpam-7044	144	100	of	of	ADP
ejpam-7044	144	101	y	y	PRON
ejpam-7044	144	102	such	such	ADJ
ejpam-7044	144	103	that	that	SCONJ
ejpam-7044	144	104	f	f	PROPN
ejpam-7044	144	105	(	(	PUNCT
ejpam-7044	144	106	x	x	NOUN
ejpam-7044	144	107	)	)	PUNCT
ejpam-7044	144	108	∩	∩	NOUN
ejpam-7044	144	109	v	v	ADP
ejpam-7044	144	110	̸=	̸=	PROPN
ejpam-7044	144	111	∅	∅	NOUN
ejpam-7044	144	112	;	;	PUNCT
ejpam-7044	144	113	(	(	PUNCT
ejpam-7044	144	114	3	3	X
ejpam-7044	144	115	)	)	PUNCT
ejpam-7044	144	116	x	x	SYM
ejpam-7044	144	117	∈	∈	PROPN
ejpam-7044	144	118	αint⋆(f−(v	αint⋆(f−(v	ADP
ejpam-7044	144	119	)	)	PUNCT
ejpam-7044	144	120	)	)	PUNCT
ejpam-7044	144	121	for	for	ADP
ejpam-7044	144	122	every	every	DET
ejpam-7044	144	123	σ1σ2	σ1σ2	NOUN
ejpam-7044	144	124	-	-	ADJ
ejpam-7044	144	125	open	open	ADJ
ejpam-7044	144	126	set	set	NOUN
ejpam-7044	144	127	v	v	NOUN
ejpam-7044	144	128	of	of	ADP
ejpam-7044	144	129	y	y	PRON
ejpam-7044	144	130	such	such	ADJ
ejpam-7044	144	131	that	that	SCONJ
ejpam-7044	144	132	f	f	PROPN
ejpam-7044	144	133	(	(	PUNCT
ejpam-7044	144	134	x	x	NOUN
ejpam-7044	144	135	)	)	PUNCT
ejpam-7044	144	136	∩	∩	NOUN
ejpam-7044	144	137	v	v	ADP
ejpam-7044	144	138	̸=	̸=	PROPN
ejpam-7044	144	139	∅.	∅.	ADP
ejpam-7044	144	140	proof	proof	NOUN
ejpam-7044	144	141	.	.	PUNCT
ejpam-7044	145	1	the	the	DET
ejpam-7044	145	2	proof	proof	NOUN
ejpam-7044	145	3	is	be	AUX
ejpam-7044	145	4	similar	similar	ADJ
ejpam-7044	145	5	to	to	ADP
ejpam-7044	145	6	that	that	PRON
ejpam-7044	145	7	of	of	ADP
ejpam-7044	145	8	theorem	theorem	NOUN
ejpam-7044	145	9	1	1	NUM
ejpam-7044	145	10	.	.	PUNCT
ejpam-7044	145	11	definition	definition	NOUN
ejpam-7044	145	12	3	3	NUM
ejpam-7044	145	13	.	.	PUNCT
ejpam-7044	145	14	a	a	DET
ejpam-7044	145	15	subset	subset	NOUN
ejpam-7044	145	16	n	n	NOUN
ejpam-7044	145	17	of	of	ADP
ejpam-7044	145	18	an	an	DET
ejpam-7044	145	19	ideal	ideal	ADJ
ejpam-7044	145	20	topological	topological	ADJ
ejpam-7044	145	21	space	space	NOUN
ejpam-7044	145	22	(	(	PUNCT
ejpam-7044	145	23	x	x	X
ejpam-7044	145	24	,	,	PUNCT
ejpam-7044	145	25	τ	τ	PROPN
ejpam-7044	145	26	,	,	PUNCT
ejpam-7044	145	27	i	i	PROPN
ejpam-7044	145	28	)	)	PUNCT
ejpam-7044	145	29	is	be	AUX
ejpam-7044	145	30	said	say	VERB
ejpam-7044	145	31	to	to	PART
ejpam-7044	145	32	be	be	AUX
ejpam-7044	145	33	a	a	DET
ejpam-7044	145	34	τ⋆-αneighbourhood	τ⋆-αneighbourhood	NOUN
ejpam-7044	145	35	of	of	ADP
ejpam-7044	145	36	x	x	SYM
ejpam-7044	145	37	∈	∈	PROPN
ejpam-7044	145	38	x	x	INTJ
ejpam-7044	145	39	if	if	SCONJ
ejpam-7044	145	40	there	there	PRON
ejpam-7044	145	41	exists	exist	VERB
ejpam-7044	145	42	a	a	DET
ejpam-7044	145	43	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	145	44	-	-	PUNCT
ejpam-7044	145	45	open	open	NOUN
ejpam-7044	145	46	set	set	NOUN
ejpam-7044	145	47	v	v	NOUN
ejpam-7044	145	48	of	of	ADP
ejpam-7044	145	49	x	x	PUNCT
ejpam-7044	145	50	such	such	ADJ
ejpam-7044	145	51	that	that	SCONJ
ejpam-7044	145	52	x	x	SYM
ejpam-7044	145	53	∈	∈	NOUN
ejpam-7044	145	54	v	v	ADP
ejpam-7044	145	55	⊆	⊆	NUM
ejpam-7044	145	56	n	n	NOUN
ejpam-7044	145	57	.	.	PUNCT
ejpam-7044	146	1	theorem	theorem	NOUN
ejpam-7044	146	2	3	3	NUM
ejpam-7044	146	3	.	.	X
ejpam-7044	146	4	for	for	ADP
ejpam-7044	146	5	a	a	DET
ejpam-7044	146	6	multifunction	multifunction	NOUN
ejpam-7044	147	1	f	f	NOUN
ejpam-7044	147	2	:	:	PUNCT
ejpam-7044	147	3	(	(	PUNCT
ejpam-7044	147	4	x	x	X
ejpam-7044	147	5	,	,	PUNCT
ejpam-7044	147	6	τ	τ	PROPN
ejpam-7044	147	7	,	,	PUNCT
ejpam-7044	147	8	i	i	NOUN
ejpam-7044	147	9	)	)	PUNCT
ejpam-7044	147	10	→	→	PUNCT
ejpam-7044	147	11	(	(	PUNCT
ejpam-7044	147	12	y	y	PROPN
ejpam-7044	147	13	,	,	PUNCT
ejpam-7044	147	14	σ1	σ1	PROPN
ejpam-7044	147	15	,	,	PUNCT
ejpam-7044	147	16	σ2	σ2	NOUN
ejpam-7044	147	17	)	)	PUNCT
ejpam-7044	147	18	,	,	PUNCT
ejpam-7044	147	19	the	the	DET
ejpam-7044	147	20	following	follow	VERB
ejpam-7044	147	21	properties	property	NOUN
ejpam-7044	147	22	are	be	AUX
ejpam-7044	147	23	equivalent	equivalent	ADJ
ejpam-7044	147	24	:	:	PUNCT
ejpam-7044	147	25	(	(	PUNCT
ejpam-7044	147	26	1	1	X
ejpam-7044	147	27	)	)	PUNCT
ejpam-7044	147	28	f	f	PROPN
ejpam-7044	147	29	is	be	AUX
ejpam-7044	147	30	upper	upper	ADJ
ejpam-7044	147	31	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	147	32	,	,	PUNCT
ejpam-7044	147	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	147	34	;	;	PUNCT
ejpam-7044	147	35	(	(	PUNCT
ejpam-7044	147	36	2	2	NUM
ejpam-7044	147	37	)	)	PUNCT
ejpam-7044	147	38	f+(v	f+(v	NOUN
ejpam-7044	147	39	)	)	PUNCT
ejpam-7044	147	40	is	be	AUX
ejpam-7044	147	41	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	147	42	-	-	PUNCT
ejpam-7044	147	43	open	open	ADJ
ejpam-7044	147	44	in	in	ADP
ejpam-7044	147	45	x	x	PUNCT
ejpam-7044	147	46	for	for	ADP
ejpam-7044	147	47	every	every	DET
ejpam-7044	147	48	σ1σ2	σ1σ2	NOUN
ejpam-7044	147	49	-	-	ADJ
ejpam-7044	147	50	open	open	ADJ
ejpam-7044	147	51	set	set	NOUN
ejpam-7044	147	52	v	v	NOUN
ejpam-7044	147	53	of	of	ADP
ejpam-7044	147	54	y	y	PROPN
ejpam-7044	147	55	;	;	PUNCT
ejpam-7044	147	56	j.	j.	PROPN
ejpam-7044	147	57	khampakdee	khampakdee	PROPN
ejpam-7044	147	58	,	,	PUNCT
ejpam-7044	147	59	a.	a.	PROPN
ejpam-7044	147	60	sama	sama	PROPN
ejpam-7044	147	61	-	-	PUNCT
ejpam-7044	147	62	ae	ae	PROPN
ejpam-7044	147	63	,	,	PUNCT
ejpam-7044	147	64	c.	c.	PROPN
ejpam-7044	147	65	boonpok	boonpok	PROPN
ejpam-7044	147	66	/	/	SYM
ejpam-7044	147	67	eur	eur	PROPN
ejpam-7044	147	68	.	.	PUNCT
ejpam-7044	148	1	j.	j.	PROPN
ejpam-7044	148	2	pure	pure	PROPN
ejpam-7044	148	3	appl	appl	PROPN
ejpam-7044	148	4	.	.	PROPN
ejpam-7044	148	5	math	math	PROPN
ejpam-7044	148	6	,	,	PUNCT
ejpam-7044	148	7	18	18	NUM
ejpam-7044	148	8	(	(	PUNCT
ejpam-7044	148	9	4	4	NUM
ejpam-7044	148	10	)	)	PUNCT
ejpam-7044	148	11	(	(	PUNCT
ejpam-7044	148	12	2025	2025	NUM
ejpam-7044	148	13	)	)	PUNCT
ejpam-7044	148	14	,	,	PUNCT
ejpam-7044	148	15	7044	7044	NUM
ejpam-7044	148	16	6	6	NUM
ejpam-7044	148	17	of	of	ADP
ejpam-7044	148	18	9	9	NUM
ejpam-7044	148	19	(	(	PUNCT
ejpam-7044	148	20	3	3	NUM
ejpam-7044	148	21	)	)	PUNCT
ejpam-7044	148	22	f−(k	f−(k	PROPN
ejpam-7044	148	23	)	)	PUNCT
ejpam-7044	148	24	is	be	AUX
ejpam-7044	148	25	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	148	26	-	-	PUNCT
ejpam-7044	148	27	closed	closed	ADJ
ejpam-7044	148	28	in	in	ADP
ejpam-7044	148	29	x	x	PUNCT
ejpam-7044	148	30	for	for	ADP
ejpam-7044	148	31	every	every	DET
ejpam-7044	148	32	σ1σ2	σ1σ2	NUM
ejpam-7044	148	33	-	-	PUNCT
ejpam-7044	148	34	closed	closed	ADJ
ejpam-7044	148	35	set	set	NOUN
ejpam-7044	148	36	k	k	PROPN
ejpam-7044	148	37	of	of	ADP
ejpam-7044	148	38	y	y	PROPN
ejpam-7044	148	39	;	;	PUNCT
ejpam-7044	148	40	(	(	PUNCT
ejpam-7044	148	41	4	4	X
ejpam-7044	148	42	)	)	PUNCT
ejpam-7044	148	43	sint⋆(cl⋆(f−(b	sint⋆(cl⋆(f−(b	PROPN
ejpam-7044	148	44	)	)	PUNCT
ejpam-7044	148	45	)	)	PUNCT
ejpam-7044	148	46	)	)	PUNCT
ejpam-7044	149	1	⊆	⊆	X
ejpam-7044	149	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7044	149	3	-	-	PUNCT
ejpam-7044	149	4	cl(b	cl(b	NOUN
ejpam-7044	149	5	)	)	PUNCT
ejpam-7044	149	6	)	)	PUNCT
ejpam-7044	150	1	for	for	ADP
ejpam-7044	150	2	every	every	DET
ejpam-7044	150	3	subset	subset	NOUN
ejpam-7044	150	4	b	b	PROPN
ejpam-7044	150	5	of	of	ADP
ejpam-7044	150	6	y	y	PROPN
ejpam-7044	150	7	;	;	PUNCT
ejpam-7044	150	8	(	(	PUNCT
ejpam-7044	150	9	5	5	X
ejpam-7044	150	10	)	)	PUNCT
ejpam-7044	150	11	αcl⋆(f−((b	αcl⋆(f−((b	NOUN
ejpam-7044	150	12	)	)	PUNCT
ejpam-7044	150	13	)	)	PUNCT
ejpam-7044	150	14	⊆	⊆	X
ejpam-7044	150	15	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7044	150	16	-	-	PUNCT
ejpam-7044	150	17	cl(b	cl(b	NOUN
ejpam-7044	150	18	)	)	PUNCT
ejpam-7044	150	19	)	)	PUNCT
ejpam-7044	150	20	for	for	ADP
ejpam-7044	150	21	every	every	DET
ejpam-7044	150	22	subset	subset	NOUN
ejpam-7044	150	23	b	b	PROPN
ejpam-7044	150	24	of	of	ADP
ejpam-7044	150	25	y	y	PROPN
ejpam-7044	150	26	;	;	PUNCT
ejpam-7044	150	27	(	(	PUNCT
ejpam-7044	150	28	6	6	NUM
ejpam-7044	150	29	)	)	PUNCT
ejpam-7044	150	30	for	for	ADP
ejpam-7044	150	31	each	each	DET
ejpam-7044	150	32	x	x	SYM
ejpam-7044	150	33	∈	∈	PROPN
ejpam-7044	150	34	x	x	X
ejpam-7044	150	35	and	and	CCONJ
ejpam-7044	150	36	each	each	DET
ejpam-7044	150	37	σ1σ2	σ1σ2	NOUN
ejpam-7044	150	38	-	-	PUNCT
ejpam-7044	150	39	neighbourhood	neighbourhood	NOUN
ejpam-7044	150	40	v	v	NOUN
ejpam-7044	150	41	of	of	ADP
ejpam-7044	150	42	f	f	PROPN
ejpam-7044	150	43	(	(	PUNCT
ejpam-7044	150	44	x	x	NOUN
ejpam-7044	150	45	)	)	PUNCT
ejpam-7044	150	46	,	,	PUNCT
ejpam-7044	150	47	f+(v	f+(v	PROPN
ejpam-7044	150	48	)	)	PUNCT
ejpam-7044	150	49	is	be	AUX
ejpam-7044	150	50	a	a	DET
ejpam-7044	150	51	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	150	52	-	-	PUNCT
ejpam-7044	150	53	neighbourhood	neighbourhood	NOUN
ejpam-7044	150	54	of	of	ADP
ejpam-7044	150	55	x	x	PRON
ejpam-7044	150	56	;	;	PUNCT
ejpam-7044	150	57	(	(	PUNCT
ejpam-7044	150	58	7	7	X
ejpam-7044	150	59	)	)	PUNCT
ejpam-7044	150	60	for	for	ADP
ejpam-7044	150	61	each	each	DET
ejpam-7044	150	62	x	x	SYM
ejpam-7044	150	63	∈	∈	PROPN
ejpam-7044	150	64	x	x	X
ejpam-7044	150	65	and	and	CCONJ
ejpam-7044	150	66	each	each	DET
ejpam-7044	150	67	σ1σ2	σ1σ2	NOUN
ejpam-7044	150	68	-	-	PUNCT
ejpam-7044	150	69	neighbourhood	neighbourhood	NOUN
ejpam-7044	150	70	v	v	NOUN
ejpam-7044	150	71	of	of	ADP
ejpam-7044	150	72	f	f	PROPN
ejpam-7044	150	73	(	(	PUNCT
ejpam-7044	150	74	x	x	NOUN
ejpam-7044	150	75	)	)	PUNCT
ejpam-7044	150	76	,	,	PUNCT
ejpam-7044	150	77	there	there	PRON
ejpam-7044	150	78	exists	exist	VERB
ejpam-7044	150	79	a	a	DET
ejpam-7044	150	80	τ⋆-αneighbourhood	τ⋆-αneighbourhood	NOUN
ejpam-7044	150	81	u	u	NOUN
ejpam-7044	150	82	of	of	ADP
ejpam-7044	150	83	x	x	SYM
ejpam-7044	150	84	such	such	ADJ
ejpam-7044	150	85	that	that	SCONJ
ejpam-7044	150	86	f	f	PROPN
ejpam-7044	150	87	(	(	PUNCT
ejpam-7044	150	88	u	u	NOUN
ejpam-7044	150	89	)	)	PUNCT
ejpam-7044	150	90	⊆	⊆	NUM
ejpam-7044	150	91	v	v	NOUN
ejpam-7044	150	92	.	.	PUNCT
ejpam-7044	151	1	proof	proof	NOUN
ejpam-7044	151	2	.	.	PUNCT
ejpam-7044	152	1	(	(	PUNCT
ejpam-7044	152	2	1	1	X
ejpam-7044	152	3	)	)	PUNCT
ejpam-7044	152	4	⇒	⇒	NOUN
ejpam-7044	152	5	(	(	PUNCT
ejpam-7044	152	6	2	2	NUM
ejpam-7044	152	7	):	):	PUNCT
ejpam-7044	152	8	let	let	VERB
ejpam-7044	152	9	v	v	PART
ejpam-7044	152	10	be	be	AUX
ejpam-7044	152	11	any	any	DET
ejpam-7044	152	12	σ1σ2	σ1σ2	NOUN
ejpam-7044	152	13	-	-	ADJ
ejpam-7044	152	14	open	open	ADJ
ejpam-7044	152	15	set	set	NOUN
ejpam-7044	152	16	of	of	ADP
ejpam-7044	152	17	y	y	PROPN
ejpam-7044	152	18	and	and	CCONJ
ejpam-7044	152	19	x	x	PROPN
ejpam-7044	152	20	∈	∈	PROPN
ejpam-7044	152	21	f+(v	f+(v	NOUN
ejpam-7044	152	22	)	)	PUNCT
ejpam-7044	152	23	.	.	PUNCT
ejpam-7044	153	1	then	then	ADV
ejpam-7044	153	2	,	,	PUNCT
ejpam-7044	153	3	f	f	PROPN
ejpam-7044	153	4	(	(	PUNCT
ejpam-7044	153	5	x	x	X
ejpam-7044	153	6	)	)	PUNCT
ejpam-7044	153	7	⊆	⊆	NUM
ejpam-7044	153	8	v	v	NOUN
ejpam-7044	153	9	.	.	PUNCT
ejpam-7044	154	1	since	since	SCONJ
ejpam-7044	154	2	f	f	PROPN
ejpam-7044	154	3	is	be	AUX
ejpam-7044	154	4	upper	upper	ADJ
ejpam-7044	154	5	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	154	6	,	,	PUNCT
ejpam-7044	154	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	154	8	at	at	ADP
ejpam-7044	154	9	x	x	X
ejpam-7044	154	10	,	,	PUNCT
ejpam-7044	154	11	there	there	PRON
ejpam-7044	154	12	exists	exist	VERB
ejpam-7044	154	13	a	a	DET
ejpam-7044	154	14	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	154	15	-	-	PUNCT
ejpam-7044	154	16	open	open	ADJ
ejpam-7044	154	17	set	set	NOUN
ejpam-7044	154	18	u	u	NOUN
ejpam-7044	154	19	of	of	ADP
ejpam-7044	154	20	x	x	PUNCT
ejpam-7044	154	21	containing	contain	VERB
ejpam-7044	154	22	x	x	PUNCT
ejpam-7044	154	23	such	such	ADJ
ejpam-7044	154	24	that	that	SCONJ
ejpam-7044	154	25	f	f	PROPN
ejpam-7044	154	26	(	(	PUNCT
ejpam-7044	154	27	u	u	NOUN
ejpam-7044	154	28	)	)	PUNCT
ejpam-7044	154	29	⊆	⊆	NUM
ejpam-7044	154	30	v	v	NOUN
ejpam-7044	154	31	;	;	PUNCT
ejpam-7044	154	32	hence	hence	ADV
ejpam-7044	154	33	x	x	PART
ejpam-7044	154	34	∈	∈	PROPN
ejpam-7044	154	35	u	u	NOUN
ejpam-7044	154	36	⊆	⊆	NUM
ejpam-7044	154	37	f+(v	f+(v	NOUN
ejpam-7044	154	38	)	)	PUNCT
ejpam-7044	154	39	.	.	PUNCT
ejpam-7044	155	1	by	by	ADP
ejpam-7044	155	2	lemma	lemma	PROPN
ejpam-7044	155	3	3	3	NUM
ejpam-7044	155	4	,	,	PUNCT
ejpam-7044	155	5	we	we	PRON
ejpam-7044	155	6	have	have	VERB
ejpam-7044	155	7	x	x	X
ejpam-7044	155	8	∈	∈	PROPN
ejpam-7044	155	9	u	u	NOUN
ejpam-7044	155	10	⊆	⊆	NUM
ejpam-7044	155	11	scl⋆(int⋆(u	scl⋆(int⋆(u	PROPN
ejpam-7044	155	12	)	)	PUNCT
ejpam-7044	155	13	)	)	PUNCT
ejpam-7044	155	14	⊆	⊆	NUM
ejpam-7044	155	15	scl⋆(int⋆(f+(v	scl⋆(int⋆(f+(v	PROPN
ejpam-7044	155	16	)	)	PUNCT
ejpam-7044	155	17	)	)	PUNCT
ejpam-7044	155	18	)	)	PUNCT
ejpam-7044	155	19	.	.	PUNCT
ejpam-7044	156	1	thus	thus	ADV
ejpam-7044	156	2	,	,	PUNCT
ejpam-7044	156	3	f+(v	f+(v	PROPN
ejpam-7044	156	4	)	)	PUNCT
ejpam-7044	156	5	⊆	⊆	NUM
ejpam-7044	156	6	scl⋆(int⋆(f+(v	scl⋆(int⋆(f+(v	PROPN
ejpam-7044	156	7	)	)	PUNCT
ejpam-7044	156	8	)	)	PUNCT
ejpam-7044	156	9	)	)	PUNCT
ejpam-7044	156	10	.	.	PUNCT
ejpam-7044	157	1	it	it	PRON
ejpam-7044	157	2	follows	follow	VERB
ejpam-7044	157	3	from	from	ADP
ejpam-7044	157	4	lemma	lemma	PROPN
ejpam-7044	157	5	3	3	NUM
ejpam-7044	157	6	that	that	PRON
ejpam-7044	157	7	f+(v	f+(v	PROPN
ejpam-7044	157	8	)	)	PUNCT
ejpam-7044	157	9	is	be	AUX
ejpam-7044	157	10	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	157	11	-	-	PUNCT
ejpam-7044	157	12	open	open	ADJ
ejpam-7044	157	13	in	in	ADP
ejpam-7044	157	14	x.	x.	NOUN
ejpam-7044	157	15	(	(	PUNCT
ejpam-7044	157	16	2	2	X
ejpam-7044	157	17	)	)	PUNCT
ejpam-7044	157	18	⇔	⇔	X
ejpam-7044	157	19	(	(	PUNCT
ejpam-7044	157	20	3	3	NUM
ejpam-7044	157	21	):	):	PUNCT
ejpam-7044	157	22	this	this	PRON
ejpam-7044	157	23	follows	follow	VERB
ejpam-7044	157	24	from	from	ADP
ejpam-7044	157	25	the	the	DET
ejpam-7044	157	26	fact	fact	NOUN
ejpam-7044	157	27	that	that	SCONJ
ejpam-7044	157	28	f+(y	f+(y	PROPN
ejpam-7044	157	29	−b	−b	ADV
ejpam-7044	157	30	)	)	PUNCT
ejpam-7044	157	31	=	=	PUNCT
ejpam-7044	158	1	x	x	SYM
ejpam-7044	158	2	−f−(b	−f−(b	PROPN
ejpam-7044	158	3	)	)	PUNCT
ejpam-7044	158	4	for	for	ADP
ejpam-7044	158	5	any	any	DET
ejpam-7044	158	6	subset	subset	NOUN
ejpam-7044	158	7	b	b	PROPN
ejpam-7044	158	8	of	of	ADP
ejpam-7044	158	9	y	y	PROPN
ejpam-7044	158	10	.	.	PUNCT
ejpam-7044	159	1	(	(	PUNCT
ejpam-7044	159	2	3	3	X
ejpam-7044	159	3	)	)	PUNCT
ejpam-7044	159	4	⇒	⇒	NOUN
ejpam-7044	159	5	(	(	PUNCT
ejpam-7044	159	6	4	4	NUM
ejpam-7044	159	7	):	):	PUNCT
ejpam-7044	159	8	let	let	VERB
ejpam-7044	159	9	b	b	X
ejpam-7044	159	10	be	be	AUX
ejpam-7044	159	11	any	any	DET
ejpam-7044	159	12	subset	subset	NOUN
ejpam-7044	159	13	of	of	ADP
ejpam-7044	159	14	y	y	PROPN
ejpam-7044	159	15	.	.	PUNCT
ejpam-7044	160	1	then	then	ADV
ejpam-7044	160	2	,	,	PUNCT
ejpam-7044	160	3	σ1σ2	σ1σ2	NOUN
ejpam-7044	160	4	-	-	NOUN
ejpam-7044	160	5	cl(b	cl(b	NOUN
ejpam-7044	160	6	)	)	PUNCT
ejpam-7044	160	7	is	be	AUX
ejpam-7044	160	8	σ1σ2	σ1σ2	NOUN
ejpam-7044	160	9	-	-	ADJ
ejpam-7044	160	10	closed	closed	ADJ
ejpam-7044	160	11	in	in	ADP
ejpam-7044	160	12	y	y	PROPN
ejpam-7044	160	13	and	and	CCONJ
ejpam-7044	160	14	by	by	ADP
ejpam-7044	160	15	(	(	PUNCT
ejpam-7044	160	16	3	3	NUM
ejpam-7044	160	17	)	)	PUNCT
ejpam-7044	160	18	,	,	PUNCT
ejpam-7044	160	19	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7044	160	20	-	-	PUNCT
ejpam-7044	160	21	cl(b	cl(b	NOUN
ejpam-7044	160	22	)	)	PUNCT
ejpam-7044	160	23	)	)	PUNCT
ejpam-7044	161	1	is	be	AUX
ejpam-7044	161	2	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	161	3	-	-	PUNCT
ejpam-7044	161	4	closed	closed	ADJ
ejpam-7044	161	5	in	in	ADP
ejpam-7044	161	6	x.	x.	NOUN
ejpam-7044	161	7	by	by	ADP
ejpam-7044	161	8	lemma	lemma	PROPN
ejpam-7044	161	9	4	4	NUM
ejpam-7044	161	10	,	,	PUNCT
ejpam-7044	161	11	we	we	PRON
ejpam-7044	161	12	have	have	VERB
ejpam-7044	161	13	sint⋆(cl⋆(f−(b	sint⋆(cl⋆(f−(b	PROPN
ejpam-7044	161	14	)	)	PUNCT
ejpam-7044	161	15	)	)	PUNCT
ejpam-7044	161	16	)	)	PUNCT
ejpam-7044	162	1	⊆	⊆	NUM
ejpam-7044	162	2	sint⋆(cl⋆(f−(cl⋆(b	sint⋆(cl⋆(f−(cl⋆(b	NOUN
ejpam-7044	162	3	)	)	PUNCT
ejpam-7044	162	4	)	)	PUNCT
ejpam-7044	162	5	)	)	PUNCT
ejpam-7044	162	6	)	)	PUNCT
ejpam-7044	163	1	⊆	⊆	X
ejpam-7044	163	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7044	163	3	-	-	PUNCT
ejpam-7044	163	4	cl(b	cl(b	NOUN
ejpam-7044	163	5	)	)	PUNCT
ejpam-7044	163	6	)	)	PUNCT
ejpam-7044	163	7	.	.	PUNCT
ejpam-7044	164	1	(	(	PUNCT
ejpam-7044	164	2	4	4	X
ejpam-7044	164	3	)	)	PUNCT
ejpam-7044	164	4	⇒	⇒	NOUN
ejpam-7044	164	5	(	(	PUNCT
ejpam-7044	164	6	5	5	NUM
ejpam-7044	164	7	):	):	PUNCT
ejpam-7044	164	8	let	let	VERB
ejpam-7044	164	9	b	b	X
ejpam-7044	164	10	be	be	AUX
ejpam-7044	164	11	any	any	DET
ejpam-7044	164	12	subset	subset	NOUN
ejpam-7044	164	13	of	of	ADP
ejpam-7044	164	14	y	y	PROPN
ejpam-7044	164	15	.	.	PUNCT
ejpam-7044	165	1	by	by	ADP
ejpam-7044	165	2	(	(	PUNCT
ejpam-7044	165	3	4	4	NUM
ejpam-7044	165	4	)	)	PUNCT
ejpam-7044	165	5	and	and	CCONJ
ejpam-7044	165	6	lemma	lemma	PROPN
ejpam-7044	165	7	4	4	NUM
ejpam-7044	165	8	,	,	PUNCT
ejpam-7044	165	9	αcl⋆(f−(b	αcl⋆(f−(b	NOUN
ejpam-7044	165	10	)	)	PUNCT
ejpam-7044	165	11	)	)	PUNCT
ejpam-7044	166	1	=	=	SYM
ejpam-7044	166	2	f−(b	f−(b	PROPN
ejpam-7044	166	3	)	)	PUNCT
ejpam-7044	166	4	∪	∪	ADP
ejpam-7044	166	5	sint⋆(cl⋆(f−(b	sint⋆(cl⋆(f−(b	PROPN
ejpam-7044	166	6	)	)	PUNCT
ejpam-7044	166	7	)	)	PUNCT
ejpam-7044	166	8	)	)	PUNCT
ejpam-7044	167	1	⊆	⊆	X
ejpam-7044	167	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7044	167	3	-	-	PUNCT
ejpam-7044	167	4	cl(b	cl(b	NOUN
ejpam-7044	167	5	)	)	PUNCT
ejpam-7044	167	6	)	)	PUNCT
ejpam-7044	167	7	.	.	PUNCT
ejpam-7044	168	1	(	(	PUNCT
ejpam-7044	168	2	5	5	X
ejpam-7044	168	3	)	)	PUNCT
ejpam-7044	168	4	⇒	⇒	NOUN
ejpam-7044	168	5	(	(	PUNCT
ejpam-7044	168	6	3	3	NUM
ejpam-7044	168	7	):	):	PUNCT
ejpam-7044	168	8	let	let	VERB
ejpam-7044	168	9	k	k	PRON
ejpam-7044	168	10	be	be	AUX
ejpam-7044	168	11	any	any	DET
ejpam-7044	168	12	σ1σ2	σ1σ2	NUM
ejpam-7044	168	13	-	-	PUNCT
ejpam-7044	168	14	closed	closed	ADJ
ejpam-7044	168	15	set	set	NOUN
ejpam-7044	168	16	of	of	ADP
ejpam-7044	168	17	y	y	PROPN
ejpam-7044	168	18	.	.	PUNCT
ejpam-7044	169	1	thus	thus	ADV
ejpam-7044	169	2	by	by	ADP
ejpam-7044	169	3	(	(	PUNCT
ejpam-7044	169	4	5	5	NUM
ejpam-7044	169	5	)	)	PUNCT
ejpam-7044	169	6	,	,	PUNCT
ejpam-7044	169	7	we	we	PRON
ejpam-7044	169	8	have	have	VERB
ejpam-7044	169	9	αcl⋆(f−(k	αcl⋆(f−(k	NOUN
ejpam-7044	169	10	)	)	PUNCT
ejpam-7044	169	11	)	)	PUNCT
ejpam-7044	170	1	⊆	⊆	X
ejpam-7044	170	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7044	170	3	-	-	PUNCT
ejpam-7044	170	4	cl(k	cl(k	NOUN
ejpam-7044	170	5	)	)	PUNCT
ejpam-7044	170	6	)	)	PUNCT
ejpam-7044	171	1	=	=	SYM
ejpam-7044	171	2	f−(k	f−(k	PROPN
ejpam-7044	171	3	)	)	PUNCT
ejpam-7044	171	4	and	and	CCONJ
ejpam-7044	171	5	hence	hence	ADV
ejpam-7044	171	6	f−(k	f−(k	PROPN
ejpam-7044	171	7	)	)	PUNCT
ejpam-7044	171	8	is	be	AUX
ejpam-7044	171	9	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	171	10	-	-	PUNCT
ejpam-7044	171	11	closed	closed	ADJ
ejpam-7044	171	12	in	in	ADP
ejpam-7044	171	13	x.	x.	NOUN
ejpam-7044	171	14	(	(	PUNCT
ejpam-7044	171	15	2	2	NUM
ejpam-7044	171	16	)	)	PUNCT
ejpam-7044	171	17	⇒	⇒	NOUN
ejpam-7044	171	18	(	(	PUNCT
ejpam-7044	171	19	6	6	NUM
ejpam-7044	171	20	):	):	PUNCT
ejpam-7044	171	21	let	let	VERB
ejpam-7044	171	22	x	x	PUNCT
ejpam-7044	171	23	∈	∈	PROPN
ejpam-7044	171	24	x	x	X
ejpam-7044	171	25	and	and	CCONJ
ejpam-7044	171	26	v	v	AUX
ejpam-7044	171	27	be	be	AUX
ejpam-7044	171	28	a	a	DET
ejpam-7044	171	29	σ1σ2	σ1σ2	NOUN
ejpam-7044	171	30	-	-	PUNCT
ejpam-7044	171	31	neighbourhood	neighbourhood	NOUN
ejpam-7044	171	32	of	of	ADP
ejpam-7044	171	33	f	f	PROPN
ejpam-7044	171	34	(	(	PUNCT
ejpam-7044	171	35	x	x	NOUN
ejpam-7044	171	36	)	)	PUNCT
ejpam-7044	171	37	.	.	PUNCT
ejpam-7044	172	1	then	then	ADV
ejpam-7044	172	2	,	,	PUNCT
ejpam-7044	172	3	there	there	PRON
ejpam-7044	172	4	exists	exist	VERB
ejpam-7044	172	5	a	a	DET
ejpam-7044	172	6	σ1σ2	σ1σ2	NUM
ejpam-7044	172	7	-	-	ADJ
ejpam-7044	172	8	open	open	ADJ
ejpam-7044	172	9	set	set	NOUN
ejpam-7044	172	10	g	g	NOUN
ejpam-7044	172	11	of	of	ADP
ejpam-7044	172	12	y	y	PRON
ejpam-7044	172	13	such	such	ADJ
ejpam-7044	172	14	that	that	SCONJ
ejpam-7044	172	15	f	f	PROPN
ejpam-7044	172	16	(	(	PUNCT
ejpam-7044	172	17	x	x	X
ejpam-7044	172	18	)	)	PUNCT
ejpam-7044	172	19	⊆	⊆	NUM
ejpam-7044	172	20	g	g	ADP
ejpam-7044	172	21	⊆	⊆	NUM
ejpam-7044	172	22	v	v	NOUN
ejpam-7044	172	23	.	.	PUNCT
ejpam-7044	173	1	thus	thus	ADV
ejpam-7044	173	2	,	,	PUNCT
ejpam-7044	173	3	x	x	SYM
ejpam-7044	173	4	∈	∈	NOUN
ejpam-7044	173	5	f+(g	f+(g	NOUN
ejpam-7044	173	6	)	)	PUNCT
ejpam-7044	173	7	⊆	⊆	NUM
ejpam-7044	173	8	f+(v	f+(v	NOUN
ejpam-7044	173	9	)	)	PUNCT
ejpam-7044	173	10	.	.	PUNCT
ejpam-7044	174	1	by	by	ADP
ejpam-7044	174	2	(	(	PUNCT
ejpam-7044	174	3	2	2	NUM
ejpam-7044	174	4	)	)	PUNCT
ejpam-7044	174	5	,	,	PUNCT
ejpam-7044	174	6	f+(g	f+(g	NOUN
ejpam-7044	174	7	)	)	PUNCT
ejpam-7044	174	8	is	be	AUX
ejpam-7044	174	9	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	174	10	-	-	PUNCT
ejpam-7044	174	11	open	open	ADJ
ejpam-7044	174	12	in	in	ADP
ejpam-7044	174	13	x	x	X
ejpam-7044	174	14	and	and	CCONJ
ejpam-7044	174	15	so	so	ADV
ejpam-7044	174	16	f+(v	f+(v	PROPN
ejpam-7044	174	17	)	)	PUNCT
ejpam-7044	175	1	is	be	AUX
ejpam-7044	175	2	a	a	DET
ejpam-7044	175	3	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	175	4	-	-	PUNCT
ejpam-7044	175	5	neighbourhood	neighbourhood	NOUN
ejpam-7044	175	6	of	of	ADP
ejpam-7044	175	7	x.	x.	NOUN
ejpam-7044	175	8	(	(	PUNCT
ejpam-7044	175	9	6	6	NUM
ejpam-7044	175	10	)	)	PUNCT
ejpam-7044	175	11	⇒	⇒	NOUN
ejpam-7044	175	12	(	(	PUNCT
ejpam-7044	175	13	7	7	NUM
ejpam-7044	175	14	):	):	PUNCT
ejpam-7044	175	15	let	let	VERB
ejpam-7044	175	16	x	x	PUNCT
ejpam-7044	175	17	∈	∈	PROPN
ejpam-7044	175	18	x	x	X
ejpam-7044	175	19	and	and	CCONJ
ejpam-7044	175	20	v	v	AUX
ejpam-7044	175	21	be	be	AUX
ejpam-7044	175	22	a	a	DET
ejpam-7044	175	23	⋆-neighbourhood	⋆-neighbourhood	NOUN
ejpam-7044	175	24	of	of	ADP
ejpam-7044	175	25	f	f	PROPN
ejpam-7044	175	26	(	(	PUNCT
ejpam-7044	175	27	x	x	NOUN
ejpam-7044	175	28	)	)	PUNCT
ejpam-7044	175	29	.	.	PUNCT
ejpam-7044	176	1	by	by	ADP
ejpam-7044	176	2	(	(	PUNCT
ejpam-7044	176	3	6	6	NUM
ejpam-7044	176	4	)	)	PUNCT
ejpam-7044	176	5	,	,	PUNCT
ejpam-7044	176	6	we	we	PRON
ejpam-7044	176	7	have	have	VERB
ejpam-7044	176	8	f+(v	f+(v	NOUN
ejpam-7044	176	9	)	)	PUNCT
ejpam-7044	177	1	is	be	AUX
ejpam-7044	177	2	a	a	DET
ejpam-7044	177	3	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	177	4	-	-	PUNCT
ejpam-7044	177	5	neighbourhood	neighbourhood	NOUN
ejpam-7044	177	6	of	of	ADP
ejpam-7044	177	7	x.	x.	NOUN
ejpam-7044	177	8	put	put	VERB
ejpam-7044	177	9	u	u	NOUN
ejpam-7044	177	10	=	=	NOUN
ejpam-7044	177	11	f+(v	f+(v	PROPN
ejpam-7044	177	12	)	)	PUNCT
ejpam-7044	177	13	,	,	PUNCT
ejpam-7044	177	14	then	then	ADV
ejpam-7044	177	15	u	u	NOUN
ejpam-7044	177	16	is	be	AUX
ejpam-7044	177	17	a	a	DET
ejpam-7044	177	18	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	177	19	-	-	PUNCT
ejpam-7044	177	20	neighbourhood	neighbourhood	NOUN
ejpam-7044	177	21	of	of	ADP
ejpam-7044	177	22	x	x	SYM
ejpam-7044	177	23	such	such	ADJ
ejpam-7044	177	24	that	that	SCONJ
ejpam-7044	177	25	f	f	PROPN
ejpam-7044	177	26	(	(	PUNCT
ejpam-7044	177	27	u	u	NOUN
ejpam-7044	177	28	)	)	PUNCT
ejpam-7044	177	29	⊆	⊆	NUM
ejpam-7044	177	30	v	v	NOUN
ejpam-7044	177	31	.	.	PUNCT
ejpam-7044	178	1	(	(	PUNCT
ejpam-7044	178	2	7	7	X
ejpam-7044	178	3	)	)	PUNCT
ejpam-7044	178	4	⇒	⇒	NOUN
ejpam-7044	178	5	(	(	PUNCT
ejpam-7044	178	6	1	1	NUM
ejpam-7044	178	7	):	):	PUNCT
ejpam-7044	178	8	let	let	VERB
ejpam-7044	178	9	x	x	PUNCT
ejpam-7044	178	10	∈	∈	PROPN
ejpam-7044	178	11	x	x	X
ejpam-7044	178	12	and	and	CCONJ
ejpam-7044	178	13	v	v	X
ejpam-7044	178	14	be	be	AUX
ejpam-7044	178	15	any	any	DET
ejpam-7044	178	16	σ1σ2	σ1σ2	NOUN
ejpam-7044	178	17	-	-	ADJ
ejpam-7044	178	18	open	open	ADJ
ejpam-7044	178	19	set	set	NOUN
ejpam-7044	178	20	of	of	ADP
ejpam-7044	178	21	y	y	PRON
ejpam-7044	178	22	such	such	ADJ
ejpam-7044	178	23	that	that	SCONJ
ejpam-7044	178	24	f	f	PROPN
ejpam-7044	178	25	(	(	PUNCT
ejpam-7044	178	26	x	x	X
ejpam-7044	178	27	)	)	PUNCT
ejpam-7044	178	28	⊆	⊆	NUM
ejpam-7044	178	29	v	v	NOUN
ejpam-7044	178	30	.	.	PUNCT
ejpam-7044	179	1	then	then	ADV
ejpam-7044	179	2	,	,	PUNCT
ejpam-7044	179	3	v	v	NOUN
ejpam-7044	179	4	is	be	AUX
ejpam-7044	179	5	a	a	DET
ejpam-7044	179	6	σ1σ2	σ1σ2	NOUN
ejpam-7044	179	7	-	-	PUNCT
ejpam-7044	179	8	neighbourhood	neighbourhood	NOUN
ejpam-7044	179	9	of	of	ADP
ejpam-7044	179	10	f	f	PROPN
ejpam-7044	179	11	(	(	PUNCT
ejpam-7044	179	12	x	x	NOUN
ejpam-7044	179	13	)	)	PUNCT
ejpam-7044	179	14	and	and	CCONJ
ejpam-7044	179	15	so	so	ADV
ejpam-7044	179	16	there	there	PRON
ejpam-7044	179	17	exists	exist	VERB
ejpam-7044	179	18	a	a	DET
ejpam-7044	179	19	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	179	20	-	-	PUNCT
ejpam-7044	179	21	neighbourhood	neighbourhood	NOUN
ejpam-7044	179	22	u	u	NOUN
ejpam-7044	179	23	of	of	ADP
ejpam-7044	179	24	x	x	SYM
ejpam-7044	179	25	such	such	ADJ
ejpam-7044	179	26	that	that	SCONJ
ejpam-7044	179	27	f	f	PROPN
ejpam-7044	179	28	(	(	PUNCT
ejpam-7044	179	29	u	u	NOUN
ejpam-7044	179	30	)	)	PUNCT
ejpam-7044	179	31	⊆	⊆	NUM
ejpam-7044	179	32	v	v	NOUN
ejpam-7044	179	33	.	.	PUNCT
ejpam-7044	180	1	since	since	SCONJ
ejpam-7044	180	2	u	u	NOUN
ejpam-7044	180	3	is	be	AUX
ejpam-7044	180	4	a	a	DET
ejpam-7044	180	5	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	180	6	-	-	PUNCT
ejpam-7044	180	7	neighbourhood	neighbourhood	NOUN
ejpam-7044	180	8	of	of	ADP
ejpam-7044	180	9	x	x	NOUN
ejpam-7044	180	10	,	,	PUNCT
ejpam-7044	180	11	there	there	PRON
ejpam-7044	180	12	exists	exist	VERB
ejpam-7044	180	13	a	a	DET
ejpam-7044	180	14	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	180	15	-	-	PUNCT
ejpam-7044	180	16	open	open	NOUN
ejpam-7044	180	17	set	set	NOUN
ejpam-7044	180	18	g	g	NOUN
ejpam-7044	180	19	of	of	ADP
ejpam-7044	180	20	x	x	INTJ
ejpam-7044	181	1	such	such	ADJ
ejpam-7044	181	2	that	that	SCONJ
ejpam-7044	181	3	x	x	SYM
ejpam-7044	181	4	∈	∈	NOUN
ejpam-7044	181	5	g	g	NOUN
ejpam-7044	181	6	⊆	⊆	NUM
ejpam-7044	181	7	u	u	NOUN
ejpam-7044	181	8	;	;	PUNCT
ejpam-7044	181	9	hence	hence	ADV
ejpam-7044	181	10	f	f	PROPN
ejpam-7044	181	11	(	(	PUNCT
ejpam-7044	181	12	g	g	NOUN
ejpam-7044	181	13	)	)	PUNCT
ejpam-7044	181	14	⊆	⊆	NUM
ejpam-7044	181	15	v	v	NOUN
ejpam-7044	181	16	.	.	PUNCT
ejpam-7044	182	1	this	this	PRON
ejpam-7044	182	2	shows	show	VERB
ejpam-7044	182	3	that	that	SCONJ
ejpam-7044	182	4	f	f	PROPN
ejpam-7044	182	5	is	be	AUX
ejpam-7044	182	6	upper	upper	ADJ
ejpam-7044	182	7	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	182	8	,	,	PUNCT
ejpam-7044	182	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	182	10	.	.	X
ejpam-7044	182	11	theorem	theorem	NOUN
ejpam-7044	182	12	4	4	NUM
ejpam-7044	182	13	.	.	X
ejpam-7044	182	14	for	for	ADP
ejpam-7044	182	15	a	a	DET
ejpam-7044	182	16	multifunction	multifunction	NOUN
ejpam-7044	182	17	f	f	NOUN
ejpam-7044	182	18	:	:	PUNCT
ejpam-7044	182	19	(	(	PUNCT
ejpam-7044	182	20	x	x	X
ejpam-7044	182	21	,	,	PUNCT
ejpam-7044	182	22	τ	τ	PROPN
ejpam-7044	182	23	,	,	PUNCT
ejpam-7044	182	24	i	i	NOUN
ejpam-7044	182	25	)	)	PUNCT
ejpam-7044	182	26	→	→	PUNCT
ejpam-7044	182	27	(	(	PUNCT
ejpam-7044	182	28	y	y	PROPN
ejpam-7044	182	29	,	,	PUNCT
ejpam-7044	182	30	σ1	σ1	PROPN
ejpam-7044	182	31	,	,	PUNCT
ejpam-7044	182	32	σ2	σ2	NOUN
ejpam-7044	182	33	)	)	PUNCT
ejpam-7044	182	34	,	,	PUNCT
ejpam-7044	182	35	the	the	DET
ejpam-7044	182	36	following	follow	VERB
ejpam-7044	182	37	properties	property	NOUN
ejpam-7044	182	38	are	be	AUX
ejpam-7044	182	39	equivalent	equivalent	ADJ
ejpam-7044	182	40	:	:	PUNCT
ejpam-7044	182	41	j.	j.	PROPN
ejpam-7044	182	42	khampakdee	khampakdee	PROPN
ejpam-7044	182	43	,	,	PUNCT
ejpam-7044	182	44	a.	a.	PROPN
ejpam-7044	182	45	sama	sama	PROPN
ejpam-7044	182	46	-	-	PUNCT
ejpam-7044	182	47	ae	ae	PROPN
ejpam-7044	182	48	,	,	PUNCT
ejpam-7044	182	49	c.	c.	PROPN
ejpam-7044	182	50	boonpok	boonpok	PROPN
ejpam-7044	182	51	/	/	SYM
ejpam-7044	182	52	eur	eur	PROPN
ejpam-7044	182	53	.	.	PUNCT
ejpam-7044	183	1	j.	j.	PROPN
ejpam-7044	183	2	pure	pure	PROPN
ejpam-7044	183	3	appl	appl	PROPN
ejpam-7044	183	4	.	.	PROPN
ejpam-7044	183	5	math	math	PROPN
ejpam-7044	183	6	,	,	PUNCT
ejpam-7044	183	7	18	18	NUM
ejpam-7044	183	8	(	(	PUNCT
ejpam-7044	183	9	4	4	NUM
ejpam-7044	183	10	)	)	PUNCT
ejpam-7044	183	11	(	(	PUNCT
ejpam-7044	183	12	2025	2025	NUM
ejpam-7044	183	13	)	)	PUNCT
ejpam-7044	183	14	,	,	PUNCT
ejpam-7044	183	15	7044	7044	NUM
ejpam-7044	183	16	7	7	NUM
ejpam-7044	183	17	of	of	ADP
ejpam-7044	183	18	9	9	NUM
ejpam-7044	183	19	(	(	PUNCT
ejpam-7044	183	20	1	1	NUM
ejpam-7044	183	21	)	)	PUNCT
ejpam-7044	183	22	f	f	PROPN
ejpam-7044	183	23	is	be	AUX
ejpam-7044	183	24	lower	low	ADJ
ejpam-7044	183	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	183	26	,	,	PUNCT
ejpam-7044	183	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	183	28	;	;	PUNCT
ejpam-7044	183	29	(	(	PUNCT
ejpam-7044	183	30	2	2	X
ejpam-7044	183	31	)	)	PUNCT
ejpam-7044	183	32	f−(v	f−(v	NOUN
ejpam-7044	183	33	)	)	PUNCT
ejpam-7044	183	34	is	be	AUX
ejpam-7044	183	35	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	183	36	-	-	PUNCT
ejpam-7044	183	37	open	open	ADJ
ejpam-7044	183	38	in	in	ADP
ejpam-7044	183	39	x	x	PUNCT
ejpam-7044	183	40	for	for	ADP
ejpam-7044	183	41	every	every	DET
ejpam-7044	183	42	σ1σ2	σ1σ2	NOUN
ejpam-7044	183	43	-	-	ADJ
ejpam-7044	183	44	open	open	ADJ
ejpam-7044	183	45	set	set	NOUN
ejpam-7044	183	46	v	v	NOUN
ejpam-7044	183	47	of	of	ADP
ejpam-7044	183	48	y	y	PROPN
ejpam-7044	183	49	;	;	PUNCT
ejpam-7044	183	50	(	(	PUNCT
ejpam-7044	183	51	3	3	X
ejpam-7044	183	52	)	)	PUNCT
ejpam-7044	183	53	f+(k	f+(k	NUM
ejpam-7044	183	54	)	)	PUNCT
ejpam-7044	183	55	is	be	AUX
ejpam-7044	183	56	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	183	57	-	-	PUNCT
ejpam-7044	183	58	closed	closed	ADJ
ejpam-7044	183	59	in	in	ADP
ejpam-7044	183	60	x	x	PUNCT
ejpam-7044	183	61	for	for	ADP
ejpam-7044	183	62	every	every	DET
ejpam-7044	183	63	σ1σ2	σ1σ2	NUM
ejpam-7044	183	64	-	-	PUNCT
ejpam-7044	183	65	closed	closed	ADJ
ejpam-7044	183	66	set	set	NOUN
ejpam-7044	183	67	k	k	PROPN
ejpam-7044	183	68	of	of	ADP
ejpam-7044	183	69	y	y	PROPN
ejpam-7044	183	70	;	;	PUNCT
ejpam-7044	183	71	(	(	PUNCT
ejpam-7044	183	72	4	4	X
ejpam-7044	183	73	)	)	PUNCT
ejpam-7044	183	74	sint⋆(cl⋆(f+(b	sint⋆(cl⋆(f+(b	NOUN
ejpam-7044	183	75	)	)	PUNCT
ejpam-7044	183	76	)	)	PUNCT
ejpam-7044	183	77	)	)	PUNCT
ejpam-7044	184	1	⊆	⊆	X
ejpam-7044	184	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7044	184	3	-	-	PUNCT
ejpam-7044	184	4	cl(b	cl(b	NOUN
ejpam-7044	184	5	)	)	PUNCT
ejpam-7044	184	6	)	)	PUNCT
ejpam-7044	184	7	for	for	ADP
ejpam-7044	184	8	every	every	DET
ejpam-7044	184	9	subset	subset	NOUN
ejpam-7044	184	10	b	b	PROPN
ejpam-7044	184	11	of	of	ADP
ejpam-7044	184	12	y	y	PROPN
ejpam-7044	184	13	;	;	PUNCT
ejpam-7044	184	14	(	(	PUNCT
ejpam-7044	184	15	5	5	X
ejpam-7044	184	16	)	)	PUNCT
ejpam-7044	184	17	αcl⋆(f+(b	αcl⋆(f+(b	NOUN
ejpam-7044	184	18	)	)	PUNCT
ejpam-7044	184	19	)	)	PUNCT
ejpam-7044	184	20	⊆	⊆	NUM
ejpam-7044	184	21	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7044	184	22	-	-	PUNCT
ejpam-7044	184	23	cl(b	cl(b	NOUN
ejpam-7044	184	24	)	)	PUNCT
ejpam-7044	184	25	)	)	PUNCT
ejpam-7044	184	26	for	for	ADP
ejpam-7044	184	27	every	every	DET
ejpam-7044	184	28	subset	subset	NOUN
ejpam-7044	184	29	b	b	PROPN
ejpam-7044	184	30	of	of	ADP
ejpam-7044	184	31	y	y	PROPN
ejpam-7044	184	32	;	;	PUNCT
ejpam-7044	184	33	(	(	PUNCT
ejpam-7044	184	34	6	6	X
ejpam-7044	184	35	)	)	PUNCT
ejpam-7044	184	36	f	f	NOUN
ejpam-7044	184	37	(	(	PUNCT
ejpam-7044	184	38	αcl⋆(a	αcl⋆(a	NUM
ejpam-7044	184	39	)	)	PUNCT
ejpam-7044	184	40	)	)	PUNCT
ejpam-7044	184	41	⊆	⊆	X
ejpam-7044	184	42	σ1σ2	σ1σ2	X
ejpam-7044	184	43	-	-	NUM
ejpam-7044	184	44	cl(f	cl(f	NOUN
ejpam-7044	184	45	(	(	PUNCT
ejpam-7044	184	46	a	a	NOUN
ejpam-7044	184	47	)	)	PUNCT
ejpam-7044	184	48	)	)	PUNCT
ejpam-7044	184	49	for	for	ADP
ejpam-7044	184	50	every	every	DET
ejpam-7044	184	51	subset	subset	NOUN
ejpam-7044	184	52	a	a	PRON
ejpam-7044	184	53	of	of	ADP
ejpam-7044	184	54	x	x	PRON
ejpam-7044	184	55	;	;	PUNCT
ejpam-7044	184	56	(	(	PUNCT
ejpam-7044	184	57	7	7	X
ejpam-7044	184	58	)	)	PUNCT
ejpam-7044	184	59	f	f	NOUN
ejpam-7044	184	60	(	(	PUNCT
ejpam-7044	184	61	sint⋆(cl⋆(a	sint⋆(cl⋆(a	PROPN
ejpam-7044	184	62	)	)	PUNCT
ejpam-7044	184	63	)	)	PUNCT
ejpam-7044	184	64	)	)	PUNCT
ejpam-7044	184	65	⊆	⊆	X
ejpam-7044	184	66	σ1σ2	σ1σ2	X
ejpam-7044	184	67	-	-	NUM
ejpam-7044	184	68	cl(f	cl(f	NOUN
ejpam-7044	184	69	(	(	PUNCT
ejpam-7044	184	70	a	a	NOUN
ejpam-7044	184	71	)	)	PUNCT
ejpam-7044	184	72	)	)	PUNCT
ejpam-7044	184	73	for	for	ADP
ejpam-7044	184	74	every	every	DET
ejpam-7044	184	75	subset	subset	NOUN
ejpam-7044	184	76	a	a	PRON
ejpam-7044	184	77	of	of	ADP
ejpam-7044	184	78	x	x	PRON
ejpam-7044	184	79	;	;	PUNCT
ejpam-7044	184	80	(	(	PUNCT
ejpam-7044	184	81	8)	8)	NUM
ejpam-7044	184	82	f	f	X
ejpam-7044	184	83	(	(	PUNCT
ejpam-7044	184	84	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	184	85	)	)	PUNCT
ejpam-7044	184	86	)	)	PUNCT
ejpam-7044	184	87	)	)	PUNCT
ejpam-7044	184	88	)	)	PUNCT
ejpam-7044	184	89	⊆	⊆	X
ejpam-7044	184	90	σ1σ2	σ1σ2	X
ejpam-7044	184	91	-	-	NUM
ejpam-7044	184	92	cl(f	cl(f	NOUN
ejpam-7044	184	93	(	(	PUNCT
ejpam-7044	184	94	a	a	NOUN
ejpam-7044	184	95	)	)	PUNCT
ejpam-7044	184	96	)	)	PUNCT
ejpam-7044	184	97	for	for	ADP
ejpam-7044	184	98	every	every	DET
ejpam-7044	184	99	subset	subset	NOUN
ejpam-7044	184	100	a	a	PRON
ejpam-7044	184	101	of	of	ADP
ejpam-7044	184	102	x.	x.	NOUN
ejpam-7044	184	103	proof	proof	NOUN
ejpam-7044	184	104	.	.	PUNCT
ejpam-7044	185	1	the	the	DET
ejpam-7044	185	2	proofs	proof	NOUN
ejpam-7044	185	3	except	except	SCONJ
ejpam-7044	185	4	for	for	ADP
ejpam-7044	185	5	the	the	DET
ejpam-7044	185	6	following	following	NOUN
ejpam-7044	185	7	are	be	AUX
ejpam-7044	185	8	similar	similar	ADJ
ejpam-7044	185	9	to	to	ADP
ejpam-7044	185	10	the	the	DET
ejpam-7044	185	11	proof	proof	NOUN
ejpam-7044	185	12	of	of	ADP
ejpam-7044	185	13	theorem	theorem	NOUN
ejpam-7044	185	14	3	3	NUM
ejpam-7044	185	15	.	.	PUNCT
ejpam-7044	185	16	(	(	PUNCT
ejpam-7044	185	17	5	5	X
ejpam-7044	185	18	)	)	PUNCT
ejpam-7044	185	19	⇒	⇒	NOUN
ejpam-7044	185	20	(	(	PUNCT
ejpam-7044	185	21	6	6	NUM
ejpam-7044	185	22	):	):	PUNCT
ejpam-7044	185	23	let	let	VERB
ejpam-7044	185	24	a	a	PRON
ejpam-7044	185	25	be	be	AUX
ejpam-7044	185	26	any	any	DET
ejpam-7044	185	27	subset	subset	NOUN
ejpam-7044	185	28	of	of	ADP
ejpam-7044	185	29	x.	x.	NOUN
ejpam-7044	185	30	since	since	SCONJ
ejpam-7044	185	31	a	a	DET
ejpam-7044	185	32	⊆	⊆	NUM
ejpam-7044	185	33	f+(f	f+(f	NOUN
ejpam-7044	185	34	(	(	PUNCT
ejpam-7044	185	35	a	a	NOUN
ejpam-7044	185	36	)	)	PUNCT
ejpam-7044	185	37	)	)	PUNCT
ejpam-7044	185	38	,	,	PUNCT
ejpam-7044	185	39	we	we	PRON
ejpam-7044	185	40	have	have	VERB
ejpam-7044	185	41	αcl⋆(a	αcl⋆(a	NOUN
ejpam-7044	185	42	)	)	PUNCT
ejpam-7044	185	43	⊆	⊆	NUM
ejpam-7044	185	44	αcl⋆(f+(f	αcl⋆(f+(f	NUM
ejpam-7044	185	45	(	(	PUNCT
ejpam-7044	185	46	a	a	NOUN
ejpam-7044	185	47	)	)	PUNCT
ejpam-7044	185	48	)	)	PUNCT
ejpam-7044	185	49	)	)	PUNCT
ejpam-7044	186	1	⊆	⊆	X
ejpam-7044	186	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7044	186	3	-	-	SYM
ejpam-7044	186	4	cl(f	cl(f	NOUN
ejpam-7044	186	5	(	(	PUNCT
ejpam-7044	186	6	a	a	NOUN
ejpam-7044	186	7	)	)	PUNCT
ejpam-7044	186	8	)	)	PUNCT
ejpam-7044	186	9	)	)	PUNCT
ejpam-7044	187	1	and	and	CCONJ
ejpam-7044	187	2	so	so	ADV
ejpam-7044	187	3	f	f	PROPN
ejpam-7044	187	4	(	(	PUNCT
ejpam-7044	187	5	αcl⋆(a	αcl⋆(a	NUM
ejpam-7044	187	6	)	)	PUNCT
ejpam-7044	187	7	)	)	PUNCT
ejpam-7044	188	1	⊆	⊆	X
ejpam-7044	188	2	σ1σ2	σ1σ2	X
ejpam-7044	188	3	-	-	NUM
ejpam-7044	188	4	cl(f	cl(f	NOUN
ejpam-7044	188	5	(	(	PUNCT
ejpam-7044	188	6	a	a	NOUN
ejpam-7044	188	7	)	)	PUNCT
ejpam-7044	188	8	)	)	PUNCT
ejpam-7044	188	9	.	.	PUNCT
ejpam-7044	189	1	(	(	PUNCT
ejpam-7044	189	2	6	6	X
ejpam-7044	189	3	)	)	PUNCT
ejpam-7044	189	4	⇒	⇒	NOUN
ejpam-7044	189	5	(	(	PUNCT
ejpam-7044	189	6	7	7	NUM
ejpam-7044	189	7	):	):	PUNCT
ejpam-7044	189	8	let	let	VERB
ejpam-7044	189	9	a	a	DET
ejpam-7044	189	10	be	be	AUX
ejpam-7044	189	11	any	any	DET
ejpam-7044	189	12	subset	subset	NOUN
ejpam-7044	189	13	of	of	ADP
ejpam-7044	189	14	x.	x.	NOUN
ejpam-7044	189	15	by	by	ADP
ejpam-7044	189	16	(	(	PUNCT
ejpam-7044	189	17	6	6	NUM
ejpam-7044	189	18	)	)	PUNCT
ejpam-7044	189	19	and	and	CCONJ
ejpam-7044	189	20	lemma	lemma	PROPN
ejpam-7044	189	21	4	4	NUM
ejpam-7044	189	22	,	,	PUNCT
ejpam-7044	189	23	f	f	PROPN
ejpam-7044	189	24	(	(	PUNCT
ejpam-7044	189	25	sint⋆(cl⋆(a	sint⋆(cl⋆(a	PROPN
ejpam-7044	189	26	)	)	PUNCT
ejpam-7044	189	27	)	)	PUNCT
ejpam-7044	189	28	)	)	PUNCT
ejpam-7044	190	1	=	=	SYM
ejpam-7044	190	2	f	f	X
ejpam-7044	190	3	(	(	PUNCT
ejpam-7044	190	4	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	190	5	)	)	PUNCT
ejpam-7044	190	6	)	)	PUNCT
ejpam-7044	190	7	)	)	PUNCT
ejpam-7044	190	8	)	)	PUNCT
ejpam-7044	191	1	⊆	⊆	NUM
ejpam-7044	191	2	f	f	X
ejpam-7044	191	3	(	(	PUNCT
ejpam-7044	191	4	a	a	DET
ejpam-7044	191	5	∪	∪	ADJ
ejpam-7044	191	6	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	191	7	)	)	PUNCT
ejpam-7044	191	8	)	)	PUNCT
ejpam-7044	191	9	)	)	PUNCT
ejpam-7044	191	10	)	)	PUNCT
ejpam-7044	192	1	=	=	SYM
ejpam-7044	192	2	f	f	PROPN
ejpam-7044	192	3	(	(	PUNCT
ejpam-7044	192	4	αcl⋆(a	αcl⋆(a	NUM
ejpam-7044	192	5	)	)	PUNCT
ejpam-7044	192	6	)	)	PUNCT
ejpam-7044	193	1	⊆	⊆	X
ejpam-7044	193	2	σ1σ2	σ1σ2	X
ejpam-7044	193	3	-	-	NUM
ejpam-7044	193	4	cl(f	cl(f	NOUN
ejpam-7044	193	5	(	(	PUNCT
ejpam-7044	193	6	a	a	NOUN
ejpam-7044	193	7	)	)	PUNCT
ejpam-7044	193	8	)	)	PUNCT
ejpam-7044	193	9	.	.	PUNCT
ejpam-7044	194	1	(	(	PUNCT
ejpam-7044	194	2	7	7	X
ejpam-7044	194	3	)	)	PUNCT
ejpam-7044	194	4	⇒	⇒	NOUN
ejpam-7044	194	5	(	(	PUNCT
ejpam-7044	194	6	8)	8)	NUM
ejpam-7044	194	7	:	:	PUNCT
ejpam-7044	194	8	let	let	VERB
ejpam-7044	194	9	a	a	PRON
ejpam-7044	194	10	be	be	AUX
ejpam-7044	194	11	any	any	DET
ejpam-7044	194	12	subset	subset	NOUN
ejpam-7044	194	13	of	of	ADP
ejpam-7044	194	14	x.	x.	NOUN
ejpam-7044	194	15	by	by	ADP
ejpam-7044	194	16	(	(	PUNCT
ejpam-7044	194	17	7	7	NUM
ejpam-7044	194	18	)	)	PUNCT
ejpam-7044	194	19	and	and	CCONJ
ejpam-7044	194	20	lemma	lemma	PROPN
ejpam-7044	194	21	4	4	NUM
ejpam-7044	194	22	,	,	PUNCT
ejpam-7044	194	23	we	we	PRON
ejpam-7044	194	24	have	have	VERB
ejpam-7044	194	25	f	f	X
ejpam-7044	194	26	(	(	PUNCT
ejpam-7044	194	27	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	194	28	)	)	PUNCT
ejpam-7044	194	29	)	)	PUNCT
ejpam-7044	194	30	)	)	PUNCT
ejpam-7044	194	31	)	)	PUNCT
ejpam-7044	195	1	=	=	SYM
ejpam-7044	195	2	f	f	X
ejpam-7044	195	3	(	(	PUNCT
ejpam-7044	195	4	sint⋆(cl⋆(a	sint⋆(cl⋆(a	PROPN
ejpam-7044	195	5	)	)	PUNCT
ejpam-7044	195	6	)	)	PUNCT
ejpam-7044	195	7	)	)	PUNCT
ejpam-7044	196	1	⊆	⊆	X
ejpam-7044	196	2	σ1σ2	σ1σ2	X
ejpam-7044	196	3	-	-	NUM
ejpam-7044	196	4	cl(f	cl(f	NOUN
ejpam-7044	196	5	(	(	PUNCT
ejpam-7044	196	6	a	a	NOUN
ejpam-7044	196	7	)	)	PUNCT
ejpam-7044	196	8	)	)	PUNCT
ejpam-7044	196	9	.	.	PUNCT
ejpam-7044	197	1	(	(	PUNCT
ejpam-7044	197	2	8)	8)	NUM
ejpam-7044	197	3	⇒	⇒	NOUN
ejpam-7044	197	4	(	(	PUNCT
ejpam-7044	197	5	1	1	NUM
ejpam-7044	197	6	):	):	PUNCT
ejpam-7044	197	7	let	let	VERB
ejpam-7044	197	8	x	x	PUNCT
ejpam-7044	197	9	∈	∈	PROPN
ejpam-7044	197	10	x	x	X
ejpam-7044	197	11	and	and	CCONJ
ejpam-7044	197	12	v	v	X
ejpam-7044	197	13	be	be	AUX
ejpam-7044	197	14	any	any	DET
ejpam-7044	197	15	σ1σ2	σ1σ2	NOUN
ejpam-7044	197	16	-	-	ADJ
ejpam-7044	197	17	open	open	ADJ
ejpam-7044	197	18	set	set	NOUN
ejpam-7044	197	19	such	such	ADJ
ejpam-7044	197	20	that	that	SCONJ
ejpam-7044	197	21	f	f	PROPN
ejpam-7044	197	22	(	(	PUNCT
ejpam-7044	197	23	x	x	NOUN
ejpam-7044	197	24	)	)	PUNCT
ejpam-7044	197	25	∩	∩	NOUN
ejpam-7044	197	26	v	v	ADP
ejpam-7044	197	27	̸=	̸=	PROPN
ejpam-7044	197	28	∅.	∅.	NOUN
ejpam-7044	197	29	then	then	ADV
ejpam-7044	197	30	,	,	PUNCT
ejpam-7044	197	31	we	we	PRON
ejpam-7044	197	32	have	have	VERB
ejpam-7044	197	33	x	x	X
ejpam-7044	197	34	∈	∈	PROPN
ejpam-7044	197	35	f−(v	f−(v	NOUN
ejpam-7044	197	36	)	)	PUNCT
ejpam-7044	197	37	.	.	PUNCT
ejpam-7044	198	1	we	we	PRON
ejpam-7044	198	2	shall	shall	AUX
ejpam-7044	198	3	show	show	VERB
ejpam-7044	198	4	that	that	DET
ejpam-7044	198	5	f−(v	f−(v	NOUN
ejpam-7044	198	6	)	)	PUNCT
ejpam-7044	198	7	is	be	AUX
ejpam-7044	198	8	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	198	9	-	-	PUNCT
ejpam-7044	198	10	open	open	ADJ
ejpam-7044	198	11	in	in	ADP
ejpam-7044	198	12	x.	x.	NOUN
ejpam-7044	198	13	by	by	ADP
ejpam-7044	198	14	the	the	DET
ejpam-7044	198	15	hypothesis	hypothesis	NOUN
ejpam-7044	198	16	,	,	PUNCT
ejpam-7044	198	17	f	f	PROPN
ejpam-7044	198	18	(	(	PUNCT
ejpam-7044	198	19	cl⋆(int⋆(cl⋆(f+(y	cl⋆(int⋆(cl⋆(f+(y	VERB
ejpam-7044	198	20	−	−	PROPN
ejpam-7044	198	21	v	v	NOUN
ejpam-7044	198	22	)	)	PUNCT
ejpam-7044	198	23	)	)	PUNCT
ejpam-7044	198	24	)	)	PUNCT
ejpam-7044	198	25	)	)	PUNCT
ejpam-7044	198	26	)	)	PUNCT
ejpam-7044	199	1	⊆	⊆	X
ejpam-7044	199	2	σ1σ2	σ1σ2	X
ejpam-7044	199	3	-	-	NUM
ejpam-7044	199	4	cl(f	cl(f	NOUN
ejpam-7044	199	5	(	(	PUNCT
ejpam-7044	199	6	f+(y	f+(y	PROPN
ejpam-7044	199	7	−	−	PROPN
ejpam-7044	199	8	v	v	NOUN
ejpam-7044	199	9	)	)	PUNCT
ejpam-7044	199	10	)	)	PUNCT
ejpam-7044	199	11	)	)	PUNCT
ejpam-7044	200	1	⊆	⊆	NUM
ejpam-7044	200	2	y	y	PROPN
ejpam-7044	200	3	−	−	PROPN
ejpam-7044	200	4	v	v	NOUN
ejpam-7044	200	5	and	and	CCONJ
ejpam-7044	200	6	hence	hence	ADV
ejpam-7044	200	7	cl⋆(int⋆(cl⋆(f+(y	cl⋆(int⋆(cl⋆(f+(y	VERB
ejpam-7044	200	8	−	−	PROPN
ejpam-7044	200	9	v	v	NOUN
ejpam-7044	200	10	)	)	PUNCT
ejpam-7044	200	11	)	)	PUNCT
ejpam-7044	200	12	)	)	PUNCT
ejpam-7044	200	13	)	)	PUNCT
ejpam-7044	201	1	⊆	⊆	NUM
ejpam-7044	201	2	f+(y	f+(y	ADP
ejpam-7044	201	3	−	−	PROPN
ejpam-7044	201	4	v	v	NOUN
ejpam-7044	201	5	)	)	PUNCT
ejpam-7044	201	6	=	=	PUNCT
ejpam-7044	201	7	x	x	SYM
ejpam-7044	201	8	−	−	NOUN
ejpam-7044	201	9	f−(v	f−(v	NOUN
ejpam-7044	201	10	)	)	PUNCT
ejpam-7044	201	11	.	.	PUNCT
ejpam-7044	202	1	thus	thus	ADV
ejpam-7044	202	2	,	,	PUNCT
ejpam-7044	202	3	f−(v	f−(v	ADJ
ejpam-7044	202	4	)	)	PUNCT
ejpam-7044	202	5	⊆	⊆	NUM
ejpam-7044	202	6	int⋆(cl⋆(int⋆(f−(v	int⋆(cl⋆(int⋆(f−(v	NOUN
ejpam-7044	202	7	)	)	PUNCT
ejpam-7044	202	8	)	)	PUNCT
ejpam-7044	202	9	)	)	PUNCT
ejpam-7044	202	10	)	)	PUNCT
ejpam-7044	203	1	and	and	CCONJ
ejpam-7044	203	2	so	so	ADV
ejpam-7044	203	3	f−(v	f−(v	ADJ
ejpam-7044	203	4	)	)	PUNCT
ejpam-7044	203	5	is	be	AUX
ejpam-7044	203	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	203	7	-	-	PUNCT
ejpam-7044	203	8	open	open	ADJ
ejpam-7044	203	9	in	in	ADP
ejpam-7044	203	10	x.	x.	NOUN
ejpam-7044	203	11	put	put	VERB
ejpam-7044	203	12	u	u	NOUN
ejpam-7044	203	13	=	=	NOUN
ejpam-7044	203	14	f−(v	f−(v	PROPN
ejpam-7044	203	15	)	)	PUNCT
ejpam-7044	203	16	.	.	PUNCT
ejpam-7044	204	1	then	then	ADV
ejpam-7044	204	2	,	,	PUNCT
ejpam-7044	204	3	u	u	NOUN
ejpam-7044	204	4	is	be	AUX
ejpam-7044	204	5	a	a	DET
ejpam-7044	204	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	204	7	-	-	PUNCT
ejpam-7044	204	8	open	open	ADJ
ejpam-7044	204	9	set	set	NOUN
ejpam-7044	204	10	of	of	ADP
ejpam-7044	204	11	x	x	PUNCT
ejpam-7044	204	12	containing	contain	VERB
ejpam-7044	204	13	x	x	PUNCT
ejpam-7044	204	14	such	such	ADJ
ejpam-7044	204	15	that	that	SCONJ
ejpam-7044	204	16	f	f	PROPN
ejpam-7044	204	17	(	(	PUNCT
ejpam-7044	204	18	z	z	NOUN
ejpam-7044	204	19	)	)	PUNCT
ejpam-7044	204	20	∩	∩	NOUN
ejpam-7044	204	21	v	v	ADP
ejpam-7044	204	22	̸=	̸=	PROPN
ejpam-7044	204	23	∅	∅	NOUN
ejpam-7044	204	24	for	for	ADP
ejpam-7044	204	25	every	every	DET
ejpam-7044	204	26	z	z	NOUN
ejpam-7044	204	27	∈	∈	PROPN
ejpam-7044	204	28	u	u	NOUN
ejpam-7044	204	29	.	.	PUNCT
ejpam-7044	205	1	this	this	PRON
ejpam-7044	205	2	shows	show	VERB
ejpam-7044	205	3	that	that	SCONJ
ejpam-7044	205	4	f	f	PROPN
ejpam-7044	205	5	is	be	AUX
ejpam-7044	205	6	lower	low	ADJ
ejpam-7044	205	7	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7044	205	8	,	,	PUNCT
ejpam-7044	205	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	205	10	.	.	NOUN
ejpam-7044	205	11	definition	definition	NOUN
ejpam-7044	205	12	4	4	NUM
ejpam-7044	205	13	.	.	PUNCT
ejpam-7044	206	1	a	a	DET
ejpam-7044	206	2	function	function	NOUN
ejpam-7044	206	3	f	f	NOUN
ejpam-7044	206	4	:	:	PUNCT
ejpam-7044	206	5	(	(	PUNCT
ejpam-7044	206	6	x	x	X
ejpam-7044	206	7	,	,	PUNCT
ejpam-7044	206	8	τ	τ	PROPN
ejpam-7044	206	9	,	,	PUNCT
ejpam-7044	206	10	i	i	NOUN
ejpam-7044	206	11	)	)	PUNCT
ejpam-7044	206	12	→	→	PUNCT
ejpam-7044	206	13	(	(	PUNCT
ejpam-7044	206	14	y	y	PROPN
ejpam-7044	206	15	,	,	PUNCT
ejpam-7044	206	16	σ1	σ1	PROPN
ejpam-7044	206	17	,	,	PUNCT
ejpam-7044	206	18	σ2	σ2	PROPN
ejpam-7044	206	19	)	)	PUNCT
ejpam-7044	206	20	is	be	AUX
ejpam-7044	206	21	said	say	VERB
ejpam-7044	206	22	to	to	PART
ejpam-7044	206	23	be	be	AUX
ejpam-7044	206	24	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7044	206	25	,	,	PUNCT
ejpam-7044	206	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	206	27	if	if	SCONJ
ejpam-7044	206	28	for	for	ADP
ejpam-7044	206	29	every	every	DET
ejpam-7044	206	30	σ1σ2	σ1σ2	NUM
ejpam-7044	206	31	-	-	ADJ
ejpam-7044	206	32	open	open	ADJ
ejpam-7044	206	33	set	set	NOUN
ejpam-7044	206	34	v	v	NOUN
ejpam-7044	206	35	of	of	ADP
ejpam-7044	206	36	y	y	PROPN
ejpam-7044	206	37	,	,	PUNCT
ejpam-7044	206	38	f−1(v	f−1(v	PROPN
ejpam-7044	206	39	)	)	PUNCT
ejpam-7044	206	40	is	be	AUX
ejpam-7044	206	41	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	206	42	-	-	PUNCT
ejpam-7044	206	43	open	open	ADJ
ejpam-7044	206	44	in	in	ADP
ejpam-7044	206	45	x.	x.	PROPN
ejpam-7044	206	46	j.	j.	PROPN
ejpam-7044	206	47	khampakdee	khampakdee	PROPN
ejpam-7044	206	48	,	,	PUNCT
ejpam-7044	206	49	a.	a.	PROPN
ejpam-7044	206	50	sama	sama	PROPN
ejpam-7044	206	51	-	-	PUNCT
ejpam-7044	206	52	ae	ae	PROPN
ejpam-7044	206	53	,	,	PUNCT
ejpam-7044	206	54	c.	c.	PROPN
ejpam-7044	206	55	boonpok	boonpok	PROPN
ejpam-7044	206	56	/	/	SYM
ejpam-7044	206	57	eur	eur	PROPN
ejpam-7044	206	58	.	.	PUNCT
ejpam-7044	207	1	j.	j.	PROPN
ejpam-7044	207	2	pure	pure	PROPN
ejpam-7044	207	3	appl	appl	PROPN
ejpam-7044	207	4	.	.	PROPN
ejpam-7044	207	5	math	math	PROPN
ejpam-7044	207	6	,	,	PUNCT
ejpam-7044	207	7	18	18	NUM
ejpam-7044	207	8	(	(	PUNCT
ejpam-7044	207	9	4	4	NUM
ejpam-7044	207	10	)	)	PUNCT
ejpam-7044	207	11	(	(	PUNCT
ejpam-7044	207	12	2025	2025	NUM
ejpam-7044	207	13	)	)	PUNCT
ejpam-7044	207	14	,	,	PUNCT
ejpam-7044	207	15	7044	7044	NUM
ejpam-7044	207	16	8	8	NUM
ejpam-7044	207	17	of	of	ADP
ejpam-7044	207	18	9	9	NUM
ejpam-7044	207	19	corollary	corollary	ADJ
ejpam-7044	207	20	1	1	NUM
ejpam-7044	207	21	.	.	PUNCT
ejpam-7044	208	1	for	for	ADP
ejpam-7044	208	2	a	a	DET
ejpam-7044	208	3	function	function	NOUN
ejpam-7044	208	4	f	f	NOUN
ejpam-7044	208	5	:	:	PUNCT
ejpam-7044	208	6	(	(	PUNCT
ejpam-7044	208	7	x	x	X
ejpam-7044	208	8	,	,	PUNCT
ejpam-7044	208	9	τ	τ	PROPN
ejpam-7044	208	10	,	,	PUNCT
ejpam-7044	208	11	i	i	NOUN
ejpam-7044	208	12	)	)	PUNCT
ejpam-7044	208	13	→	→	PUNCT
ejpam-7044	208	14	(	(	PUNCT
ejpam-7044	208	15	y	y	PROPN
ejpam-7044	208	16	,	,	PUNCT
ejpam-7044	208	17	σ1	σ1	PROPN
ejpam-7044	208	18	,	,	PUNCT
ejpam-7044	208	19	σ2	σ2	NOUN
ejpam-7044	208	20	)	)	PUNCT
ejpam-7044	208	21	,	,	PUNCT
ejpam-7044	208	22	the	the	DET
ejpam-7044	208	23	following	follow	VERB
ejpam-7044	208	24	properties	property	NOUN
ejpam-7044	208	25	are	be	AUX
ejpam-7044	208	26	equivalent	equivalent	ADJ
ejpam-7044	208	27	:	:	PUNCT
ejpam-7044	208	28	(	(	PUNCT
ejpam-7044	208	29	1	1	X
ejpam-7044	208	30	)	)	PUNCT
ejpam-7044	208	31	f	f	PROPN
ejpam-7044	208	32	is	be	AUX
ejpam-7044	208	33	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7044	208	34	,	,	PUNCT
ejpam-7044	208	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7044	208	36	;	;	PUNCT
ejpam-7044	208	37	(	(	PUNCT
ejpam-7044	208	38	2	2	X
ejpam-7044	208	39	)	)	PUNCT
ejpam-7044	208	40	f−1(k	f−1(k	PROPN
ejpam-7044	208	41	)	)	PUNCT
ejpam-7044	208	42	is	be	AUX
ejpam-7044	208	43	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	208	44	-	-	PUNCT
ejpam-7044	208	45	closed	closed	ADJ
ejpam-7044	208	46	in	in	ADP
ejpam-7044	208	47	x	x	PUNCT
ejpam-7044	208	48	for	for	ADP
ejpam-7044	208	49	every	every	DET
ejpam-7044	208	50	σ1σ2	σ1σ2	NUM
ejpam-7044	208	51	-	-	PUNCT
ejpam-7044	208	52	closed	closed	ADJ
ejpam-7044	208	53	set	set	NOUN
ejpam-7044	208	54	k	k	PROPN
ejpam-7044	208	55	of	of	ADP
ejpam-7044	208	56	y	y	PROPN
ejpam-7044	208	57	;	;	PUNCT
ejpam-7044	208	58	(	(	PUNCT
ejpam-7044	208	59	3	3	X
ejpam-7044	208	60	)	)	PUNCT
ejpam-7044	208	61	sint⋆(cl⋆(f−1(b	sint⋆(cl⋆(f−1(b	ADJ
ejpam-7044	208	62	)	)	PUNCT
ejpam-7044	208	63	)	)	PUNCT
ejpam-7044	208	64	)	)	PUNCT
ejpam-7044	209	1	⊆	⊆	NUM
ejpam-7044	209	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7044	209	3	-	-	PUNCT
ejpam-7044	209	4	cl(b	cl(b	NOUN
ejpam-7044	209	5	)	)	PUNCT
ejpam-7044	209	6	)	)	PUNCT
ejpam-7044	209	7	for	for	ADP
ejpam-7044	209	8	every	every	DET
ejpam-7044	209	9	subset	subset	NOUN
ejpam-7044	209	10	b	b	PROPN
ejpam-7044	209	11	of	of	ADP
ejpam-7044	209	12	y	y	PROPN
ejpam-7044	209	13	;	;	PUNCT
ejpam-7044	209	14	(	(	PUNCT
ejpam-7044	209	15	4	4	X
ejpam-7044	209	16	)	)	PUNCT
ejpam-7044	209	17	αcl⋆(f−1(b	αcl⋆(f−1(b	NUM
ejpam-7044	209	18	)	)	PUNCT
ejpam-7044	209	19	)	)	PUNCT
ejpam-7044	209	20	⊆	⊆	NUM
ejpam-7044	209	21	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7044	209	22	-	-	PUNCT
ejpam-7044	209	23	cl(b	cl(b	NOUN
ejpam-7044	209	24	)	)	PUNCT
ejpam-7044	209	25	)	)	PUNCT
ejpam-7044	209	26	for	for	ADP
ejpam-7044	209	27	every	every	DET
ejpam-7044	209	28	subset	subset	NOUN
ejpam-7044	209	29	b	b	PROPN
ejpam-7044	209	30	of	of	ADP
ejpam-7044	209	31	y	y	PROPN
ejpam-7044	209	32	;	;	PUNCT
ejpam-7044	209	33	(	(	PUNCT
ejpam-7044	209	34	5	5	X
ejpam-7044	209	35	)	)	PUNCT
ejpam-7044	209	36	for	for	ADP
ejpam-7044	209	37	each	each	DET
ejpam-7044	209	38	x	x	SYM
ejpam-7044	209	39	∈	∈	PROPN
ejpam-7044	209	40	x	x	X
ejpam-7044	209	41	and	and	CCONJ
ejpam-7044	209	42	each	each	DET
ejpam-7044	209	43	σ1σ2	σ1σ2	NOUN
ejpam-7044	209	44	-	-	PUNCT
ejpam-7044	209	45	neighbourhood	neighbourhood	NOUN
ejpam-7044	209	46	v	v	NOUN
ejpam-7044	209	47	of	of	ADP
ejpam-7044	209	48	f(x	f(x	PROPN
ejpam-7044	209	49	)	)	PUNCT
ejpam-7044	209	50	,	,	PUNCT
ejpam-7044	209	51	f−1(v	f−1(v	PROPN
ejpam-7044	209	52	)	)	PUNCT
ejpam-7044	209	53	is	be	AUX
ejpam-7044	209	54	a	a	DET
ejpam-7044	209	55	τ⋆-α	τ⋆-α	NOUN
ejpam-7044	209	56	-	-	PUNCT
ejpam-7044	209	57	neighbourhood	neighbourhood	NOUN
ejpam-7044	209	58	of	of	ADP
ejpam-7044	209	59	x	x	PRON
ejpam-7044	209	60	;	;	PUNCT
ejpam-7044	209	61	(	(	PUNCT
ejpam-7044	209	62	6	6	NUM
ejpam-7044	209	63	)	)	PUNCT
ejpam-7044	209	64	for	for	ADP
ejpam-7044	209	65	each	each	DET
ejpam-7044	209	66	x	x	SYM
ejpam-7044	209	67	∈	∈	PROPN
ejpam-7044	209	68	x	x	X
ejpam-7044	209	69	and	and	CCONJ
ejpam-7044	209	70	each	each	DET
ejpam-7044	209	71	σ1σ2	σ1σ2	NOUN
ejpam-7044	209	72	-	-	PUNCT
ejpam-7044	209	73	neighbourhood	neighbourhood	NOUN
ejpam-7044	209	74	v	v	NOUN
ejpam-7044	209	75	of	of	ADP
ejpam-7044	209	76	f(x	f(x	PROPN
ejpam-7044	209	77	)	)	PUNCT
ejpam-7044	209	78	,	,	PUNCT
ejpam-7044	209	79	there	there	PRON
ejpam-7044	209	80	exists	exist	VERB
ejpam-7044	209	81	a	a	DET
ejpam-7044	209	82	τ⋆-αneighbourhood	τ⋆-αneighbourhood	NOUN
ejpam-7044	209	83	u	u	NOUN
ejpam-7044	209	84	of	of	ADP
ejpam-7044	209	85	x	x	SYM
ejpam-7044	209	86	such	such	ADJ
ejpam-7044	209	87	that	that	DET
ejpam-7044	209	88	f(u	f(u	PROPN
ejpam-7044	209	89	)	)	PUNCT
ejpam-7044	209	90	⊆	⊆	NUM
ejpam-7044	209	91	v	v	NOUN
ejpam-7044	209	92	;	;	PUNCT
ejpam-7044	209	93	(	(	PUNCT
ejpam-7044	209	94	7	7	X
ejpam-7044	209	95	)	)	PUNCT
ejpam-7044	209	96	f(αcl⋆(a	f(αcl⋆(a	NOUN
ejpam-7044	209	97	)	)	PUNCT
ejpam-7044	209	98	)	)	PUNCT
ejpam-7044	210	1	⊆	⊆	X
ejpam-7044	210	2	σ1σ2	σ1σ2	NUM
ejpam-7044	210	3	-	-	PUNCT
ejpam-7044	210	4	cl(f(a	cl(f(a	NOUN
ejpam-7044	210	5	)	)	PUNCT
ejpam-7044	210	6	)	)	PUNCT
ejpam-7044	210	7	for	for	ADP
ejpam-7044	210	8	every	every	DET
ejpam-7044	210	9	subset	subset	NOUN
ejpam-7044	210	10	a	a	PRON
ejpam-7044	210	11	of	of	ADP
ejpam-7044	210	12	x	x	PRON
ejpam-7044	210	13	;	;	PUNCT
ejpam-7044	210	14	(	(	PUNCT
ejpam-7044	210	15	8)	8)	NUM
ejpam-7044	210	16	f(sint⋆(cl⋆(a	f(sint⋆(cl⋆(a	NOUN
ejpam-7044	210	17	)	)	PUNCT
ejpam-7044	210	18	)	)	PUNCT
ejpam-7044	210	19	)	)	PUNCT
ejpam-7044	211	1	⊆	⊆	X
ejpam-7044	211	2	σ1σ2	σ1σ2	NUM
ejpam-7044	211	3	-	-	PUNCT
ejpam-7044	211	4	cl(f(a	cl(f(a	NOUN
ejpam-7044	211	5	)	)	PUNCT
ejpam-7044	211	6	)	)	PUNCT
ejpam-7044	211	7	for	for	ADP
ejpam-7044	211	8	every	every	DET
ejpam-7044	211	9	subset	subset	NOUN
ejpam-7044	211	10	a	a	PRON
ejpam-7044	211	11	of	of	ADP
ejpam-7044	211	12	x	x	PRON
ejpam-7044	211	13	;	;	PUNCT
ejpam-7044	211	14	(	(	PUNCT
ejpam-7044	211	15	9	9	X
ejpam-7044	211	16	)	)	PUNCT
ejpam-7044	211	17	f(cl⋆(int⋆(cl⋆(a	f(cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7044	211	18	)	)	PUNCT
ejpam-7044	211	19	)	)	PUNCT
ejpam-7044	211	20	)	)	PUNCT
ejpam-7044	211	21	)	)	PUNCT
ejpam-7044	212	1	⊆	⊆	X
ejpam-7044	212	2	σ1σ2	σ1σ2	NUM
ejpam-7044	212	3	-	-	PUNCT
ejpam-7044	212	4	cl(f(a	cl(f(a	NOUN
ejpam-7044	212	5	)	)	PUNCT
ejpam-7044	212	6	)	)	PUNCT
ejpam-7044	212	7	for	for	ADP
ejpam-7044	212	8	every	every	DET
ejpam-7044	212	9	subset	subset	NOUN
ejpam-7044	212	10	a	a	PRON
ejpam-7044	212	11	of	of	ADP
ejpam-7044	212	12	x.	x.	NOUN
ejpam-7044	212	13	acknowledgements	acknowledgement	NOUN
ejpam-7044	212	14	this	this	DET
ejpam-7044	212	15	research	research	NOUN
ejpam-7044	212	16	project	project	NOUN
ejpam-7044	212	17	was	be	AUX
ejpam-7044	212	18	financially	financially	ADV
ejpam-7044	212	19	supported	support	VERB
ejpam-7044	212	20	by	by	ADP
ejpam-7044	212	21	mahasarakham	mahasarakham	PROPN
ejpam-7044	212	22	university	university	PROPN
ejpam-7044	212	23	.	.	PUNCT
ejpam-7044	213	1	references	reference	NOUN
ejpam-7044	213	2	[	[	X
ejpam-7044	213	3	1	1	X
ejpam-7044	213	4	]	]	PUNCT
ejpam-7044	213	5	t.	t.	PROPN
ejpam-7044	213	6	noiri	noiri	PROPN
ejpam-7044	213	7	.	.	PUNCT
ejpam-7044	214	1	a	a	DET
ejpam-7044	214	2	function	function	NOUN
ejpam-7044	214	3	which	which	PRON
ejpam-7044	214	4	preserves	preserve	VERB
ejpam-7044	214	5	connected	connected	ADJ
ejpam-7044	214	6	spaces	space	NOUN
ejpam-7044	214	7	.	.	PUNCT
ejpam-7044	215	1	časopis	časopis	PROPN
ejpam-7044	215	2	pěstování	pěstování	PROPN
ejpam-7044	215	3	matematiky	matematiky	PROPN
ejpam-7044	215	4	,	,	PUNCT
ejpam-7044	215	5	107:393–396	107:393–396	NUM
ejpam-7044	215	6	,	,	PUNCT
ejpam-7044	215	7	1982	1982	NUM
ejpam-7044	215	8	.	.	PUNCT
ejpam-7044	216	1	[	[	X
ejpam-7044	216	2	2	2	NUM
ejpam-7044	216	3	]	]	PUNCT
ejpam-7044	216	4	a.	a.	NOUN
ejpam-7044	216	5	s.	s.	PROPN
ejpam-7044	216	6	mashhour	mashhour	PROPN
ejpam-7044	216	7	,	,	PUNCT
ejpam-7044	216	8	i.	i.	PROPN
ejpam-7044	216	9	a.	a.	PROPN
ejpam-7044	216	10	hasanein	hasanein	PROPN
ejpam-7044	216	11	,	,	PUNCT
ejpam-7044	216	12	and	and	CCONJ
ejpam-7044	216	13	s.	s.	PROPN
ejpam-7044	216	14	n.	n.	PROPN
ejpam-7044	216	15	el	el	PROPN
ejpam-7044	216	16	-	-	PROPN
ejpam-7044	216	17	deeb	deeb	PROPN
ejpam-7044	216	18	.	.	PUNCT
ejpam-7044	217	1	α	α	X
ejpam-7044	217	2	-	-	ADJ
ejpam-7044	217	3	continuous	continuous	ADJ
ejpam-7044	217	4	and	and	CCONJ
ejpam-7044	217	5	α	α	NOUN
ejpam-7044	217	6	-	-	ADJ
ejpam-7044	217	7	open	open	ADJ
ejpam-7044	217	8	mappings	mapping	NOUN
ejpam-7044	217	9	.	.	PUNCT
ejpam-7044	218	1	acta	acta	PROPN
ejpam-7044	218	2	mathematica	mathematica	PROPN
ejpam-7044	218	3	hungarica	hungarica	PROPN
ejpam-7044	218	4	,	,	PUNCT
ejpam-7044	218	5	41:213–218	41:213–218	PROPN
ejpam-7044	218	6	,	,	PUNCT
ejpam-7044	218	7	1983	1983	NUM
ejpam-7044	218	8	.	.	PUNCT
ejpam-7044	219	1	[	[	X
ejpam-7044	219	2	3	3	X
ejpam-7044	219	3	]	]	PUNCT
ejpam-7044	219	4	t.	t.	NOUN
ejpam-7044	219	5	neubrunn	neubrunn	PROPN
ejpam-7044	219	6	.	.	PUNCT
ejpam-7044	220	1	strongly	strongly	ADV
ejpam-7044	220	2	quasi	quasi	ADJ
ejpam-7044	220	3	-	-	ADJ
ejpam-7044	220	4	continuous	continuous	ADJ
ejpam-7044	220	5	multivalued	multivalued	ADJ
ejpam-7044	220	6	mappings	mapping	NOUN
ejpam-7044	220	7	.	.	PUNCT
ejpam-7044	221	1	general	general	ADJ
ejpam-7044	221	2	topology	topology	NOUN
ejpam-7044	221	3	and	and	CCONJ
ejpam-7044	221	4	its	its	PRON
ejpam-7044	221	5	relations	relation	NOUN
ejpam-7044	221	6	to	to	ADP
ejpam-7044	221	7	modern	modern	ADJ
ejpam-7044	221	8	analysis	analysis	NOUN
ejpam-7044	221	9	and	and	CCONJ
ejpam-7044	221	10	algebra	algebra	NOUN
ejpam-7044	221	11	vi	vi	PROPN
ejpam-7044	221	12	,	,	PUNCT
ejpam-7044	221	13	proceedings	proceeding	NOUN
ejpam-7044	221	14	of	of	ADP
ejpam-7044	221	15	the	the	DET
ejpam-7044	221	16	symposium	symposium	NOUN
ejpam-7044	221	17	,	,	PUNCT
ejpam-7044	221	18	prague	prague	NOUN
ejpam-7044	221	19	,	,	PUNCT
ejpam-7044	221	20	1968	1968	NUM
ejpam-7044	221	21	,	,	PUNCT
ejpam-7044	221	22	heldermann	heldermann	PROPN
ejpam-7044	221	23	verlag	verlag	PROPN
ejpam-7044	221	24	berlin	berlin	PROPN
ejpam-7044	221	25	,	,	PUNCT
ejpam-7044	221	26	1988	1988	NUM
ejpam-7044	221	27	,	,	PUNCT
ejpam-7044	221	28	pages	page	NOUN
ejpam-7044	221	29	351–359	351–359	NUM
ejpam-7044	221	30	.	.	PUNCT
ejpam-7044	222	1	[	[	X
ejpam-7044	222	2	4	4	X
ejpam-7044	222	3	]	]	PUNCT
ejpam-7044	222	4	v.	v.	CCONJ
ejpam-7044	222	5	popa	popa	NOUN
ejpam-7044	222	6	and	and	CCONJ
ejpam-7044	222	7	t.	t.	PROPN
ejpam-7044	222	8	noiri	noiri	PROPN
ejpam-7044	222	9	.	.	PUNCT
ejpam-7044	223	1	on	on	ADP
ejpam-7044	223	2	upper	upper	ADJ
ejpam-7044	223	3	and	and	CCONJ
ejpam-7044	223	4	lower	low	ADJ
ejpam-7044	223	5	α	α	ADJ
ejpam-7044	223	6	-	-	ADJ
ejpam-7044	223	7	continuous	continuous	ADJ
ejpam-7044	223	8	multifunctions	multifunction	NOUN
ejpam-7044	223	9	.	.	PUNCT
ejpam-7044	224	1	mathematica	mathematica	PROPN
ejpam-7044	224	2	slovaca	slovaca	PROPN
ejpam-7044	224	3	,	,	PUNCT
ejpam-7044	224	4	43(4):477–491	43(4):477–491	NOUN
ejpam-7044	224	5	,	,	PUNCT
ejpam-7044	224	6	1993	1993	NUM
ejpam-7044	224	7	.	.	PUNCT
ejpam-7044	225	1	[	[	X
ejpam-7044	225	2	5	5	X
ejpam-7044	225	3	]	]	PUNCT
ejpam-7044	225	4	c.	c.	PROPN
ejpam-7044	225	5	boonpok	boonpok	PROPN
ejpam-7044	225	6	.	.	PUNCT
ejpam-7044	226	1	on	on	ADP
ejpam-7044	226	2	continuous	continuous	ADJ
ejpam-7044	226	3	multifunctions	multifunction	NOUN
ejpam-7044	226	4	in	in	ADP
ejpam-7044	226	5	ideal	ideal	ADJ
ejpam-7044	226	6	topological	topological	ADJ
ejpam-7044	226	7	spaces	space	NOUN
ejpam-7044	226	8	.	.	PUNCT
ejpam-7044	227	1	lobachevskii	lobachevskii	PROPN
ejpam-7044	227	2	journal	journal	PROPN
ejpam-7044	227	3	of	of	ADP
ejpam-7044	227	4	mathematics	mathematic	NOUN
ejpam-7044	227	5	,	,	PUNCT
ejpam-7044	227	6	40(1):24–35	40(1):24–35	NUM
ejpam-7044	227	7	,	,	PUNCT
ejpam-7044	227	8	2019	2019	NUM
ejpam-7044	227	9	.	.	PUNCT
ejpam-7044	228	1	[	[	X
ejpam-7044	228	2	6	6	NUM
ejpam-7044	228	3	]	]	PUNCT
ejpam-7044	228	4	c.	c.	PROPN
ejpam-7044	228	5	boonpok	boonpok	PROPN
ejpam-7044	228	6	.	.	PUNCT
ejpam-7044	229	1	on	on	ADP
ejpam-7044	229	2	some	some	DET
ejpam-7044	229	3	types	type	NOUN
ejpam-7044	229	4	of	of	ADP
ejpam-7044	229	5	continuity	continuity	NOUN
ejpam-7044	229	6	for	for	ADP
ejpam-7044	229	7	multifunctions	multifunction	NOUN
ejpam-7044	229	8	in	in	ADP
ejpam-7044	229	9	ideal	ideal	ADJ
ejpam-7044	229	10	topological	topological	ADJ
ejpam-7044	229	11	spaces	space	NOUN
ejpam-7044	229	12	.	.	PUNCT
ejpam-7044	230	1	advances	advance	NOUN
ejpam-7044	230	2	in	in	ADP
ejpam-7044	230	3	mathematics	mathematic	NOUN
ejpam-7044	230	4	:	:	PUNCT
ejpam-7044	230	5	scientific	scientific	ADJ
ejpam-7044	230	6	journal	journal	NOUN
ejpam-7044	230	7	,	,	PUNCT
ejpam-7044	230	8	9(3):859–886	9(3):859–886	NUM
ejpam-7044	230	9	,	,	PUNCT
ejpam-7044	230	10	2020	2020	NUM
ejpam-7044	230	11	.	.	PUNCT
ejpam-7044	231	1	[	[	X
ejpam-7044	231	2	7	7	X
ejpam-7044	231	3	]	]	X
ejpam-7044	231	4	c.	c.	PROPN
ejpam-7044	231	5	boonpok	boonpok	PROPN
ejpam-7044	231	6	.	.	PUNCT
ejpam-7044	232	1	upper	upper	ADJ
ejpam-7044	232	2	and	and	CCONJ
ejpam-7044	232	3	lower	low	ADJ
ejpam-7044	232	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-7044	232	5	.	.	PUNCT
ejpam-7044	232	6	heliyon	heliyon	NOUN
ejpam-7044	232	7	,	,	PUNCT
ejpam-7044	232	8	7	7	NUM
ejpam-7044	232	9	:	:	PUNCT
ejpam-7044	232	10	e05986	e05986	PROPN
ejpam-7044	232	11	,	,	PUNCT
ejpam-7044	232	12	2021	2021	NUM
ejpam-7044	232	13	.	.	PUNCT
ejpam-7044	233	1	[	[	X
ejpam-7044	233	2	8	8	NUM
ejpam-7044	233	3	]	]	X
ejpam-7044	233	4	c.	c.	NOUN
ejpam-7044	233	5	boonpok	boonpok	PROPN
ejpam-7044	233	6	and	and	CCONJ
ejpam-7044	233	7	p.	p.	NOUN
ejpam-7044	233	8	pue	pue	NOUN
ejpam-7044	233	9	-	-	PUNCT
ejpam-7044	233	10	on	on	ADP
ejpam-7044	233	11	.	.	PUNCT
ejpam-7044	234	1	upper	upper	ADJ
ejpam-7044	234	2	and	and	CCONJ
ejpam-7044	234	3	lower	low	ADJ
ejpam-7044	234	4	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7044	234	5	multifunctions	multifunction	NOUN
ejpam-7044	234	6	.	.	PUNCT
ejpam-7044	235	1	european	european	ADJ
ejpam-7044	235	2	journal	journal	PROPN
ejpam-7044	235	3	of	of	ADP
ejpam-7044	235	4	pure	pure	ADJ
ejpam-7044	235	5	and	and	CCONJ
ejpam-7044	235	6	applied	applied	ADJ
ejpam-7044	235	7	mathematics	mathematic	NOUN
ejpam-7044	235	8	,	,	PUNCT
ejpam-7044	235	9	16(3):1634–1646	16(3):1634–1646	NUM
ejpam-7044	235	10	,	,	PUNCT
ejpam-7044	235	11	2023	2023	NUM
ejpam-7044	235	12	.	.	PUNCT
ejpam-7044	236	1	j.	j.	PROPN
ejpam-7044	236	2	khampakdee	khampakdee	PROPN
ejpam-7044	236	3	,	,	PUNCT
ejpam-7044	236	4	a.	a.	PROPN
ejpam-7044	236	5	sama	sama	PROPN
ejpam-7044	236	6	-	-	PUNCT
ejpam-7044	236	7	ae	ae	PROPN
ejpam-7044	236	8	,	,	PUNCT
ejpam-7044	236	9	c.	c.	PROPN
ejpam-7044	236	10	boonpok	boonpok	PROPN
ejpam-7044	236	11	/	/	SYM
ejpam-7044	236	12	eur	eur	PROPN
ejpam-7044	236	13	.	.	PUNCT
ejpam-7044	237	1	j.	j.	PROPN
ejpam-7044	237	2	pure	pure	PROPN
ejpam-7044	237	3	appl	appl	PROPN
ejpam-7044	237	4	.	.	PROPN
ejpam-7044	237	5	math	math	PROPN
ejpam-7044	237	6	,	,	PUNCT
ejpam-7044	237	7	18	18	NUM
ejpam-7044	237	8	(	(	PUNCT
ejpam-7044	237	9	4	4	NUM
ejpam-7044	237	10	)	)	PUNCT
ejpam-7044	237	11	(	(	PUNCT
ejpam-7044	237	12	2025	2025	NUM
ejpam-7044	237	13	)	)	PUNCT
ejpam-7044	237	14	,	,	PUNCT
ejpam-7044	237	15	7044	7044	NUM
ejpam-7044	237	16	9	9	NUM
ejpam-7044	237	17	of	of	ADP
ejpam-7044	237	18	9	9	NUM
ejpam-7044	237	19	[	[	SYM
ejpam-7044	237	20	9	9	NUM
ejpam-7044	237	21	]	]	PUNCT
ejpam-7044	237	22	c.	c.	NOUN
ejpam-7044	237	23	boonpok	boonpok	PROPN
ejpam-7044	237	24	and	and	CCONJ
ejpam-7044	237	25	j.	j.	PROPN
ejpam-7044	237	26	khampakdee	khampakdee	PROPN
ejpam-7044	237	27	.	.	PUNCT
ejpam-7044	238	1	upper	upper	ADJ
ejpam-7044	238	2	and	and	CCONJ
ejpam-7044	238	3	lower	low	ADJ
ejpam-7044	238	4	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-7044	238	5	.	.	PUNCT
ejpam-7044	238	6	european	european	PROPN
ejpam-7044	238	7	journal	journal	PROPN
ejpam-7044	238	8	of	of	ADP
ejpam-7044	238	9	pure	pure	ADJ
ejpam-7044	238	10	and	and	CCONJ
ejpam-7044	238	11	applied	applied	ADJ
ejpam-7044	238	12	mathematics	mathematic	NOUN
ejpam-7044	238	13	,	,	PUNCT
ejpam-7044	238	14	17(1):201–211	17(1):201–211	NUM
ejpam-7044	238	15	,	,	PUNCT
ejpam-7044	238	16	2024	2024	NUM
ejpam-7044	238	17	.	.	PUNCT
ejpam-7044	239	1	[	[	X
ejpam-7044	239	2	10	10	NUM
ejpam-7044	239	3	]	]	X
ejpam-7044	239	4	c.	c.	PROPN
ejpam-7044	239	5	boonpok	boonpok	PROPN
ejpam-7044	239	6	and	and	CCONJ
ejpam-7044	239	7	p.	p.	NOUN
ejpam-7044	239	8	pue	pue	NOUN
ejpam-7044	239	9	-	-	PUNCT
ejpam-7044	239	10	on	on	ADP
ejpam-7044	239	11	.	.	PUNCT
ejpam-7044	240	1	continuity	continuity	NOUN
ejpam-7044	240	2	for	for	ADP
ejpam-7044	240	3	multifunctions	multifunction	NOUN
ejpam-7044	240	4	in	in	ADP
ejpam-7044	240	5	ideal	ideal	ADJ
ejpam-7044	240	6	topological	topological	ADJ
ejpam-7044	240	7	spaces	space	NOUN
ejpam-7044	240	8	.	.	PUNCT
ejpam-7044	241	1	wseas	wseas	VERB
ejpam-7044	241	2	transactions	transaction	NOUN
ejpam-7044	241	3	on	on	ADP
ejpam-7044	241	4	mathematics	mathematic	NOUN
ejpam-7044	241	5	,	,	PUNCT
ejpam-7044	241	6	19:624–631	19:624–631	NUM
ejpam-7044	241	7	,	,	PUNCT
ejpam-7044	241	8	2020	2020	NUM
ejpam-7044	241	9	.	.	PUNCT
ejpam-7044	242	1	[	[	X
ejpam-7044	242	2	11	11	NUM
ejpam-7044	242	3	]	]	PUNCT
ejpam-7044	242	4	c.	c.	PROPN
ejpam-7044	242	5	boonpok	boonpok	PROPN
ejpam-7044	242	6	.	.	PUNCT
ejpam-7044	243	1	pı	pı	NOUN
ejpam-7044	243	2	-	-	NOUN
ejpam-7044	243	3	continuity	continuity	NOUN
ejpam-7044	243	4	and	and	CCONJ
ejpam-7044	243	5	weak	weak	ADJ
ejpam-7044	243	6	pı	pı	NOUN
ejpam-7044	243	7	-	-	NOUN
ejpam-7044	243	8	continuity	continuity	NOUN
ejpam-7044	243	9	.	.	PUNCT
ejpam-7044	244	1	carpathian	carpathian	ADJ
ejpam-7044	244	2	mathematical	mathematical	ADJ
ejpam-7044	244	3	publications	publication	NOUN
ejpam-7044	244	4	,	,	PUNCT
ejpam-7044	244	5	17(1):171–186	17(1):171–186	PROPN
ejpam-7044	244	6	,	,	PUNCT
ejpam-7044	244	7	2025	2025	NUM
ejpam-7044	244	8	.	.	PUNCT
ejpam-7044	245	1	[	[	X
ejpam-7044	245	2	12	12	NUM
ejpam-7044	245	3	]	]	X
ejpam-7044	245	4	p.	p.	NOUN
ejpam-7044	245	5	pue	pue	NOUN
ejpam-7044	245	6	-	-	PUNCT
ejpam-7044	245	7	on	on	ADP
ejpam-7044	245	8	,	,	PUNCT
ejpam-7044	245	9	s.	s.	PROPN
ejpam-7044	245	10	sompong	sompong	PROPN
ejpam-7044	245	11	,	,	PUNCT
ejpam-7044	245	12	and	and	CCONJ
ejpam-7044	245	13	c.	c.	PROPN
ejpam-7044	245	14	boonpok	boonpok	PROPN
ejpam-7044	245	15	.	.	PUNCT
ejpam-7044	246	1	upper	upper	ADJ
ejpam-7044	246	2	and	and	CCONJ
ejpam-7044	246	3	lower	low	ADJ
ejpam-7044	246	4	(	(	PUNCT
ejpam-7044	246	5	τ1	τ1	NOUN
ejpam-7044	246	6	,	,	PUNCT
ejpam-7044	246	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	246	8	multifunctions	multifunction	NOUN
ejpam-7044	246	9	.	.	PUNCT
ejpam-7044	247	1	international	international	ADJ
ejpam-7044	247	2	journal	journal	PROPN
ejpam-7044	247	3	of	of	ADP
ejpam-7044	247	4	mathematics	mathematic	NOUN
ejpam-7044	247	5	and	and	CCONJ
ejpam-7044	247	6	computer	computer	NOUN
ejpam-7044	247	7	science	science	NOUN
ejpam-7044	247	8	,	,	PUNCT
ejpam-7044	247	9	19(4):1305	19(4):1305	NUM
ejpam-7044	247	10	–	–	PUNCT
ejpam-7044	247	11	1310	1310	NUM
ejpam-7044	247	12	,	,	PUNCT
ejpam-7044	247	13	2024	2024	NUM
ejpam-7044	247	14	.	.	PUNCT
ejpam-7044	248	1	[	[	X
ejpam-7044	248	2	13	13	NUM
ejpam-7044	248	3	]	]	X
ejpam-7044	248	4	c.	c.	PROPN
ejpam-7044	248	5	klanarong	klanarong	PROPN
ejpam-7044	248	6	,	,	PUNCT
ejpam-7044	248	7	s.	s.	PROPN
ejpam-7044	248	8	sompong	sompong	PROPN
ejpam-7044	248	9	,	,	PUNCT
ejpam-7044	248	10	and	and	CCONJ
ejpam-7044	248	11	c.	c.	PROPN
ejpam-7044	248	12	boonpok	boonpok	PROPN
ejpam-7044	248	13	.	.	PUNCT
ejpam-7044	249	1	(	(	PUNCT
ejpam-7044	249	2	τ1	τ1	NOUN
ejpam-7044	249	3	,	,	PUNCT
ejpam-7044	249	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7044	249	5	and	and	CCONJ
ejpam-7044	249	6	(	(	PUNCT
ejpam-7044	249	7	τ1	τ1	NOUN
ejpam-7044	249	8	,	,	PUNCT
ejpam-7044	249	9	τ2)θ	τ2)θ	ADJ
ejpam-7044	249	10	-	-	PUNCT
ejpam-7044	249	11	closed	close	VERB
ejpam-7044	249	12	sets	set	NOUN
ejpam-7044	249	13	.	.	PUNCT
ejpam-7044	250	1	international	international	ADJ
ejpam-7044	250	2	journal	journal	NOUN
ejpam-7044	250	3	of	of	ADP
ejpam-7044	250	4	mathematics	mathematic	NOUN
ejpam-7044	250	5	and	and	CCONJ
ejpam-7044	250	6	computer	computer	NOUN
ejpam-7044	250	7	science	science	NOUN
ejpam-7044	250	8	,	,	PUNCT
ejpam-7044	250	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-7044	250	10	,	,	PUNCT
ejpam-7044	250	11	2024	2024	NUM
ejpam-7044	250	12	.	.	PUNCT
ejpam-7044	251	1	[	[	X
ejpam-7044	251	2	14	14	NUM
ejpam-7044	251	3	]	]	PUNCT
ejpam-7044	251	4	m.	m.	NOUN
ejpam-7044	251	5	thongmoon	thongmoon	NOUN
ejpam-7044	251	6	,	,	PUNCT
ejpam-7044	251	7	s.	s.	PROPN
ejpam-7044	251	8	sompong	sompong	PROPN
ejpam-7044	251	9	,	,	PUNCT
ejpam-7044	251	10	and	and	CCONJ
ejpam-7044	251	11	c.	c.	PROPN
ejpam-7044	251	12	boonpok	boonpok	PROPN
ejpam-7044	251	13	.	.	PUNCT
ejpam-7044	252	1	(	(	PUNCT
ejpam-7044	252	2	τ1	τ1	NOUN
ejpam-7044	252	3	,	,	PUNCT
ejpam-7044	252	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	252	5	multifunctions	multifunction	NOUN
ejpam-7044	252	6	and	and	CCONJ
ejpam-7044	252	7	τ1τ2	τ1τ2	NOUN
ejpam-7044	252	8	-	-	ADJ
ejpam-7044	252	9	δ	δ	NOUN
ejpam-7044	252	10	-	-	ADJ
ejpam-7044	252	11	open	open	ADJ
ejpam-7044	252	12	sets	set	NOUN
ejpam-7044	252	13	.	.	PUNCT
ejpam-7044	253	1	international	international	ADJ
ejpam-7044	253	2	journal	journal	NOUN
ejpam-7044	253	3	of	of	ADP
ejpam-7044	253	4	mathematics	mathematic	NOUN
ejpam-7044	253	5	and	and	CCONJ
ejpam-7044	253	6	computer	computer	NOUN
ejpam-7044	253	7	science	science	NOUN
ejpam-7044	253	8	,	,	PUNCT
ejpam-7044	253	9	19(4):1369–1375	19(4):1369–1375	NUM
ejpam-7044	253	10	,	,	PUNCT
ejpam-7044	253	11	2024	2024	NUM
ejpam-7044	253	12	.	.	PUNCT
ejpam-7044	254	1	[	[	X
ejpam-7044	254	2	15	15	NUM
ejpam-7044	254	3	]	]	X
ejpam-7044	254	4	c.	c.	PROPN
ejpam-7044	254	5	viriyapong	viriyapong	PROPN
ejpam-7044	254	6	and	and	CCONJ
ejpam-7044	254	7	c.	c.	PROPN
ejpam-7044	254	8	boonpok	boonpok	PROPN
ejpam-7044	254	9	.	.	PUNCT
ejpam-7044	255	1	(	(	PUNCT
ejpam-7044	255	2	τ1	τ1	NOUN
ejpam-7044	255	3	,	,	PUNCT
ejpam-7044	255	4	τ2)α	τ2)α	NOUN
ejpam-7044	255	5	-	-	PUNCT
ejpam-7044	255	6	continuity	continuity	NOUN
ejpam-7044	255	7	for	for	ADP
ejpam-7044	255	8	multifunctions	multifunction	NOUN
ejpam-7044	255	9	.	.	PUNCT
ejpam-7044	256	1	journal	journal	PROPN
ejpam-7044	256	2	of	of	ADP
ejpam-7044	256	3	mathematics	mathematic	NOUN
ejpam-7044	256	4	,	,	PUNCT
ejpam-7044	256	5	2020:6285763	2020:6285763	NUM
ejpam-7044	256	6	,	,	PUNCT
ejpam-7044	256	7	2020	2020	NUM
ejpam-7044	256	8	.	.	PUNCT
ejpam-7044	257	1	[	[	X
ejpam-7044	257	2	16	16	NUM
ejpam-7044	257	3	]	]	X
ejpam-7044	257	4	j.	j.	PROPN
ejpam-7044	257	5	khampakdee	khampakdee	PROPN
ejpam-7044	257	6	,	,	PUNCT
ejpam-7044	257	7	a.	a.	PROPN
ejpam-7044	257	8	sama	sama	PROPN
ejpam-7044	257	9	-	-	PUNCT
ejpam-7044	257	10	ae	ae	PROPN
ejpam-7044	257	11	,	,	PUNCT
ejpam-7044	257	12	and	and	CCONJ
ejpam-7044	257	13	c.	c.	PROPN
ejpam-7044	257	14	boonpok	boonpok	PROPN
ejpam-7044	257	15	.	.	PUNCT
ejpam-7044	258	1	upper	upper	ADJ
ejpam-7044	258	2	and	and	CCONJ
ejpam-7044	258	3	lower	low	ADJ
ejpam-7044	258	4	continuous	continuous	ADJ
ejpam-7044	258	5	multifunctions	multifunction	NOUN
ejpam-7044	258	6	defined	define	VERB
ejpam-7044	258	7	between	between	ADP
ejpam-7044	258	8	an	an	DET
ejpam-7044	258	9	ideal	ideal	ADJ
ejpam-7044	258	10	topological	topological	ADJ
ejpam-7044	258	11	space	space	NOUN
ejpam-7044	258	12	and	and	CCONJ
ejpam-7044	258	13	a	a	DET
ejpam-7044	258	14	bitopological	bitopological	ADJ
ejpam-7044	258	15	space	space	NOUN
ejpam-7044	258	16	.	.	PUNCT
ejpam-7044	259	1	european	european	ADJ
ejpam-7044	259	2	journal	journal	PROPN
ejpam-7044	259	3	of	of	ADP
ejpam-7044	259	4	pure	pure	ADJ
ejpam-7044	259	5	and	and	CCONJ
ejpam-7044	259	6	applied	applied	ADJ
ejpam-7044	259	7	mathematics	mathematic	NOUN
ejpam-7044	259	8	,	,	PUNCT
ejpam-7044	259	9	18(3):6565	18(3):6565	NUM
ejpam-7044	259	10	,	,	PUNCT
ejpam-7044	259	11	2025	2025	NUM
ejpam-7044	259	12	.	.	PUNCT
ejpam-7044	260	1	[	[	X
ejpam-7044	260	2	17	17	NUM
ejpam-7044	260	3	]	]	X
ejpam-7044	260	4	c.	c.	PROPN
ejpam-7044	260	5	boonpok	boonpok	PROPN
ejpam-7044	260	6	,	,	PUNCT
ejpam-7044	260	7	c.	c.	PROPN
ejpam-7044	260	8	viriyapong	viriyapong	PROPN
ejpam-7044	260	9	,	,	PUNCT
ejpam-7044	260	10	and	and	CCONJ
ejpam-7044	260	11	m.	m.	NOUN
ejpam-7044	260	12	thongmoon	thongmoon	NOUN
ejpam-7044	260	13	.	.	PUNCT
ejpam-7044	261	1	on	on	ADP
ejpam-7044	261	2	upper	upper	ADJ
ejpam-7044	261	3	and	and	CCONJ
ejpam-7044	261	4	lower	low	ADJ
ejpam-7044	261	5	(	(	PUNCT
ejpam-7044	261	6	τ1	τ1	NOUN
ejpam-7044	261	7	,	,	PUNCT
ejpam-7044	261	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7044	261	9	multifunctions	multifunction	NOUN
ejpam-7044	261	10	.	.	PUNCT
ejpam-7044	262	1	journal	journal	PROPN
ejpam-7044	262	2	of	of	ADP
ejpam-7044	262	3	mathematics	mathematics	PROPN
ejpam-7044	262	4	and	and	CCONJ
ejpam-7044	262	5	computer	computer	NOUN
ejpam-7044	262	6	science	science	NOUN
ejpam-7044	262	7	,	,	PUNCT
ejpam-7044	262	8	18:282	18:282	NUM
ejpam-7044	262	9	–	–	PUNCT
ejpam-7044	262	10	293	293	NUM
ejpam-7044	262	11	,	,	PUNCT
ejpam-7044	262	12	2018	2018	NUM
ejpam-7044	262	13	.	.	PUNCT
ejpam-7044	263	1	[	[	X
ejpam-7044	263	2	18	18	NUM
ejpam-7044	263	3	]	]	PUNCT
ejpam-7044	263	4	c.	c.	PROPN
ejpam-7044	263	5	boonpok	boonpok	PROPN
ejpam-7044	263	6	.	.	PUNCT
ejpam-7044	264	1	(	(	PUNCT
ejpam-7044	264	2	τ1	τ1	NOUN
ejpam-7044	264	3	,	,	PUNCT
ejpam-7044	264	4	τ2)δ	τ2)δ	ADJ
ejpam-7044	264	5	-	-	PUNCT
ejpam-7044	264	6	semicontinuous	semicontinuous	ADJ
ejpam-7044	264	7	multifunctions	multifunction	NOUN
ejpam-7044	264	8	.	.	PUNCT
ejpam-7044	265	1	heliyon	heliyon	NOUN
ejpam-7044	265	2	,	,	PUNCT
ejpam-7044	265	3	6	6	NUM
ejpam-7044	265	4	:	:	SYM
ejpam-7044	265	5	e05367	e05367	PROPN
ejpam-7044	265	6	,	,	PUNCT
ejpam-7044	265	7	2020	2020	NUM
ejpam-7044	265	8	.	.	PUNCT
ejpam-7044	266	1	[	[	X
ejpam-7044	266	2	19	19	NUM
ejpam-7044	266	3	]	]	X
ejpam-7044	266	4	c.	c.	PROPN
ejpam-7044	266	5	boonpok	boonpok	PROPN
ejpam-7044	266	6	and	and	CCONJ
ejpam-7044	266	7	p.	p.	NOUN
ejpam-7044	266	8	pue	pue	NOUN
ejpam-7044	266	9	-	-	PUNCT
ejpam-7044	266	10	on	on	ADP
ejpam-7044	266	11	.	.	PUNCT
ejpam-7044	267	1	characterizations	characterization	NOUN
ejpam-7044	267	2	of	of	ADP
ejpam-7044	267	3	almost	almost	ADV
ejpam-7044	267	4	(	(	PUNCT
ejpam-7044	267	5	τ1	τ1	NOUN
ejpam-7044	267	6	,	,	PUNCT
ejpam-7044	267	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7044	267	8	multifunctions	multifunction	NOUN
ejpam-7044	267	9	.	.	PUNCT
ejpam-7044	268	1	international	international	ADJ
ejpam-7044	268	2	journal	journal	NOUN
ejpam-7044	268	3	of	of	ADP
ejpam-7044	268	4	analysis	analysis	NOUN
ejpam-7044	268	5	and	and	CCONJ
ejpam-7044	268	6	applications	application	NOUN
ejpam-7044	268	7	,	,	PUNCT
ejpam-7044	268	8	22:33	22:33	NUM
ejpam-7044	268	9	,	,	PUNCT
ejpam-7044	268	10	2024	2024	NUM
ejpam-7044	268	11	.	.	PUNCT
ejpam-7044	269	1	[	[	X
ejpam-7044	269	2	20	20	NUM
ejpam-7044	269	3	]	]	PUNCT
ejpam-7044	269	4	k.	k.	PROPN
ejpam-7044	269	5	kuratowski	kuratowski	PROPN
ejpam-7044	269	6	.	.	PUNCT
ejpam-7044	270	1	topology	topology	PROPN
ejpam-7044	270	2	,	,	PUNCT
ejpam-7044	270	3	vol	vol	NOUN
ejpam-7044	270	4	.	.	PUNCT
ejpam-7044	270	5	i.	i.	PROPN
ejpam-7044	270	6	academic	academic	PROPN
ejpam-7044	270	7	press	press	PROPN
ejpam-7044	270	8	,	,	PUNCT
ejpam-7044	270	9	new	new	PROPN
ejpam-7044	270	10	york	york	PROPN
ejpam-7044	270	11	,	,	PUNCT
ejpam-7044	270	12	1966	1966	NUM
ejpam-7044	270	13	.	.	PUNCT
ejpam-7044	271	1	[	[	X
ejpam-7044	271	2	21	21	NUM
ejpam-7044	271	3	]	]	X
ejpam-7044	271	4	d.	d.	PROPN
ejpam-7044	271	5	janković	janković	PROPN
ejpam-7044	271	6	and	and	CCONJ
ejpam-7044	271	7	t.	t.	PROPN
ejpam-7044	271	8	r.	r.	PROPN
ejpam-7044	271	9	hamlett	hamlett	PROPN
ejpam-7044	271	10	.	.	PUNCT
ejpam-7044	272	1	new	new	ADJ
ejpam-7044	272	2	topologies	topology	NOUN
ejpam-7044	272	3	from	from	ADP
ejpam-7044	272	4	old	old	ADJ
ejpam-7044	272	5	via	via	ADP
ejpam-7044	272	6	ideals	ideal	NOUN
ejpam-7044	272	7	.	.	PUNCT
ejpam-7044	273	1	the	the	DET
ejpam-7044	273	2	american	american	PROPN
ejpam-7044	273	3	mathematical	mathematical	PROPN
ejpam-7044	273	4	monthly	monthly	ADV
ejpam-7044	273	5	,	,	PUNCT
ejpam-7044	273	6	97:295–310	97:295–310	PROPN
ejpam-7044	273	7	,	,	PUNCT
ejpam-7044	273	8	1990	1990	NUM
ejpam-7044	273	9	.	.	PUNCT
ejpam-7044	274	1	[	[	X
ejpam-7044	274	2	22	22	NUM
ejpam-7044	274	3	]	]	PUNCT
ejpam-7044	274	4	c.	c.	PROPN
ejpam-7044	274	5	boonpok	boonpok	PROPN
ejpam-7044	274	6	.	.	PUNCT
ejpam-7044	275	1	weak	weak	ADJ
ejpam-7044	275	2	quasi	quasi	ADJ
ejpam-7044	275	3	continuity	continuity	NOUN
ejpam-7044	275	4	for	for	ADP
ejpam-7044	275	5	multifunctions	multifunction	NOUN
ejpam-7044	275	6	in	in	ADP
ejpam-7044	275	7	ideal	ideal	ADJ
ejpam-7044	275	8	topological	topological	ADJ
ejpam-7044	275	9	spaces	space	NOUN
ejpam-7044	275	10	.	.	PUNCT
ejpam-7044	276	1	advances	advance	NOUN
ejpam-7044	276	2	in	in	ADP
ejpam-7044	276	3	mathematics	mathematic	NOUN
ejpam-7044	276	4	:	:	PUNCT
ejpam-7044	276	5	scientific	scientific	ADJ
ejpam-7044	276	6	journal	journal	NOUN
ejpam-7044	276	7	,	,	PUNCT
ejpam-7044	276	8	9(3):339–355	9(3):339–355	NUM
ejpam-7044	276	9	,	,	PUNCT
ejpam-7044	276	10	2020	2020	NUM
ejpam-7044	276	11	.	.	PUNCT
ejpam-7044	277	1	[	[	X
ejpam-7044	277	2	23	23	NUM
ejpam-7044	277	3	]	]	PUNCT
ejpam-7044	277	4	t.	t.	PROPN
ejpam-7044	277	5	noiri	noiri	PROPN
ejpam-7044	277	6	and	and	CCONJ
ejpam-7044	277	7	v.	v.	ADP
ejpam-7044	277	8	popa	popa	NOUN
ejpam-7044	277	9	.	.	PUNCT
ejpam-7044	278	1	on	on	ADP
ejpam-7044	278	2	(	(	PUNCT
ejpam-7044	278	3	mi	mi	ADJ
ejpam-7044	278	4	,	,	PUNCT
ejpam-7044	278	5	nj)-continuous	nj)-continuous	ADJ
ejpam-7044	278	6	multifunctions	multifunction	NOUN
ejpam-7044	278	7	.	.	PUNCT
ejpam-7044	279	1	romanian	romanian	ADJ
ejpam-7044	279	2	journal	journal	PROPN
ejpam-7044	279	3	of	of	ADP
ejpam-7044	279	4	mathematics	mathematics	PROPN
ejpam-7044	279	5	and	and	CCONJ
ejpam-7044	279	6	computer	computer	NOUN
ejpam-7044	279	7	science	science	NOUN
ejpam-7044	279	8	,	,	PUNCT
ejpam-7044	279	9	15(1):1–8	15(1):1–8	NUM
ejpam-7044	279	10	,	,	PUNCT
ejpam-7044	279	11	2025	2025	NUM
ejpam-7044	279	12	.	.	PUNCT
ejpam-7044	280	1	[	[	X
ejpam-7044	280	2	24	24	NUM
ejpam-7044	280	3	]	]	PUNCT
ejpam-7044	280	4	c.	c.	PROPN
ejpam-7044	280	5	boonpok	boonpok	PROPN
ejpam-7044	280	6	.	.	PUNCT
ejpam-7044	281	1	a	a	DET
ejpam-7044	281	2	study	study	NOUN
ejpam-7044	281	3	of	of	ADP
ejpam-7044	281	4	some	some	DET
ejpam-7044	281	5	forms	form	NOUN
ejpam-7044	281	6	of	of	ADP
ejpam-7044	281	7	continuity	continuity	NOUN
ejpam-7044	281	8	for	for	ADP
ejpam-7044	281	9	multifunctions	multifunction	NOUN
ejpam-7044	281	10	in	in	ADP
ejpam-7044	281	11	ideal	ideal	ADJ
ejpam-7044	281	12	topological	topological	ADJ
ejpam-7044	281	13	spaces	space	NOUN
ejpam-7044	281	14	.	.	PUNCT
ejpam-7044	282	1	mathematica	mathematica	PROPN
ejpam-7044	282	2	,	,	PUNCT
ejpam-7044	282	3	63(2):186–198	63(2):186–198	PROPN
ejpam-7044	282	4	,	,	PUNCT
ejpam-7044	282	5	2021	2021	NUM
ejpam-7044	282	6	.	.	PUNCT