id	sid	tid	token	lemma	pos
ejpam-7047	1	1	european	european	PROPN
ejpam-7047	1	2	journal	journal	PROPN
ejpam-7047	1	3	of	of	ADP
ejpam-7047	1	4	pure	pure	ADJ
ejpam-7047	1	5	and	and	CCONJ
ejpam-7047	1	6	applied	applied	ADJ
ejpam-7047	1	7	mathematics	mathematic	NOUN
ejpam-7047	1	8	2025	2025	NUM
ejpam-7047	1	9	,	,	PUNCT
ejpam-7047	1	10	vol	vol	NOUN
ejpam-7047	1	11	.	.	PROPN
ejpam-7047	1	12	18	18	NUM
ejpam-7047	1	13	,	,	PUNCT
ejpam-7047	1	14	issue	issue	NOUN
ejpam-7047	1	15	4	4	NUM
ejpam-7047	1	16	,	,	PUNCT
ejpam-7047	1	17	article	article	NOUN
ejpam-7047	1	18	number	number	NOUN
ejpam-7047	1	19	7047	7047	NUM
ejpam-7047	1	20	issn	issn	VERB
ejpam-7047	1	21	1307	1307	NUM
ejpam-7047	1	22	-	-	SYM
ejpam-7047	1	23	5543	5543	NUM
ejpam-7047	1	24	–	–	PUNCT
ejpam-7047	1	25	ejpam.com	ejpam.com	X
ejpam-7047	1	26	published	publish	VERB
ejpam-7047	1	27	by	by	ADP
ejpam-7047	1	28	new	new	PROPN
ejpam-7047	1	29	york	york	PROPN
ejpam-7047	1	30	business	business	PROPN
ejpam-7047	1	31	global	global	ADJ
ejpam-7047	1	32	weakly	weakly	ADJ
ejpam-7047	1	33	τ	τ	X
ejpam-7047	1	34	⋆α(σ1	⋆α(σ1	X
ejpam-7047	1	35	,	,	PUNCT
ejpam-7047	1	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	1	37	multifunctions	multifunction	NOUN
ejpam-7047	1	38	nongluk	nongluk	PROPN
ejpam-7047	1	39	viriyapong1	viriyapong1	PROPN
ejpam-7047	1	40	,	,	PUNCT
ejpam-7047	1	41	areeyuth	areeyuth	NOUN
ejpam-7047	1	42	sama	sama	NOUN
ejpam-7047	1	43	-	-	PUNCT
ejpam-7047	1	44	ae2	ae2	PROPN
ejpam-7047	1	45	,	,	PUNCT
ejpam-7047	1	46	chawalit	chawalit	VERB
ejpam-7047	1	47	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7047	1	48	1	1	NUM
ejpam-7047	1	49	mathematics	mathematic	NOUN
ejpam-7047	1	50	and	and	CCONJ
ejpam-7047	1	51	applied	apply	VERB
ejpam-7047	1	52	mathematics	mathematics	PROPN
ejpam-7047	1	53	research	research	NOUN
ejpam-7047	1	54	unit	unit	NOUN
ejpam-7047	1	55	,	,	PUNCT
ejpam-7047	1	56	department	department	NOUN
ejpam-7047	1	57	of	of	ADP
ejpam-7047	1	58	mathematics	mathematic	NOUN
ejpam-7047	1	59	,	,	PUNCT
ejpam-7047	1	60	faculty	faculty	NOUN
ejpam-7047	1	61	of	of	ADP
ejpam-7047	1	62	science	science	NOUN
ejpam-7047	1	63	,	,	PUNCT
ejpam-7047	1	64	mahasarakham	mahasarakham	PROPN
ejpam-7047	1	65	university	university	PROPN
ejpam-7047	1	66	,	,	PUNCT
ejpam-7047	1	67	maha	maha	PROPN
ejpam-7047	1	68	sarakham	sarakham	PROPN
ejpam-7047	1	69	,	,	PUNCT
ejpam-7047	1	70	44150	44150	NUM
ejpam-7047	1	71	,	,	PUNCT
ejpam-7047	1	72	thailand	thailand	PROPN
ejpam-7047	1	73	2	2	NUM
ejpam-7047	1	74	department	department	NOUN
ejpam-7047	1	75	of	of	ADP
ejpam-7047	1	76	mathematics	mathematic	NOUN
ejpam-7047	1	77	and	and	CCONJ
ejpam-7047	1	78	computer	computer	NOUN
ejpam-7047	1	79	science	science	NOUN
ejpam-7047	1	80	,	,	PUNCT
ejpam-7047	1	81	faculty	faculty	NOUN
ejpam-7047	1	82	of	of	ADP
ejpam-7047	1	83	science	science	NOUN
ejpam-7047	1	84	and	and	CCONJ
ejpam-7047	1	85	technology	technology	NOUN
ejpam-7047	1	86	,	,	PUNCT
ejpam-7047	1	87	prince	prince	NOUN
ejpam-7047	1	88	of	of	ADP
ejpam-7047	1	89	songkla	songkla	PROPN
ejpam-7047	1	90	university	university	PROPN
ejpam-7047	1	91	,	,	PUNCT
ejpam-7047	1	92	pattani	pattani	NOUN
ejpam-7047	1	93	campus	campus	NOUN
ejpam-7047	1	94	,	,	PUNCT
ejpam-7047	1	95	pattani	pattani	NOUN
ejpam-7047	1	96	,	,	PUNCT
ejpam-7047	1	97	94000	94000	NUM
ejpam-7047	1	98	,	,	PUNCT
ejpam-7047	1	99	thailand	thailand	PROPN
ejpam-7047	1	100	abstract	abstract	PROPN
ejpam-7047	1	101	.	.	PUNCT
ejpam-7047	2	1	this	this	DET
ejpam-7047	2	2	paper	paper	NOUN
ejpam-7047	2	3	presents	present	VERB
ejpam-7047	2	4	new	new	ADJ
ejpam-7047	2	5	classes	class	NOUN
ejpam-7047	2	6	of	of	ADP
ejpam-7047	2	7	continuous	continuous	ADJ
ejpam-7047	2	8	multifunctions	multifunction	NOUN
ejpam-7047	2	9	defined	define	VERB
ejpam-7047	2	10	between	between	ADP
ejpam-7047	2	11	an	an	DET
ejpam-7047	2	12	ideal	ideal	ADJ
ejpam-7047	2	13	topological	topological	ADJ
ejpam-7047	2	14	space	space	NOUN
ejpam-7047	2	15	and	and	CCONJ
ejpam-7047	2	16	a	a	DET
ejpam-7047	2	17	bitopological	bitopological	ADJ
ejpam-7047	2	18	space	space	NOUN
ejpam-7047	2	19	,	,	PUNCT
ejpam-7047	2	20	called	call	VERB
ejpam-7047	2	21	upper	upper	ADJ
ejpam-7047	2	22	weakly	weakly	ADJ
ejpam-7047	2	23	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	2	24	,	,	PUNCT
ejpam-7047	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	2	26	multifunctions	multifunction	NOUN
ejpam-7047	2	27	and	and	CCONJ
ejpam-7047	2	28	lower	low	ADJ
ejpam-7047	2	29	weakly	weakly	ADJ
ejpam-7047	2	30	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	2	31	,	,	PUNCT
ejpam-7047	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	2	33	multifunctions	multifunction	NOUN
ejpam-7047	2	34	.	.	PUNCT
ejpam-7047	3	1	furthermore	furthermore	ADV
ejpam-7047	3	2	,	,	PUNCT
ejpam-7047	3	3	several	several	ADJ
ejpam-7047	3	4	characterizations	characterization	NOUN
ejpam-7047	3	5	and	and	CCONJ
ejpam-7047	3	6	some	some	DET
ejpam-7047	3	7	properties	property	NOUN
ejpam-7047	3	8	concerning	concern	VERB
ejpam-7047	3	9	upper	upper	ADJ
ejpam-7047	3	10	weakly	weakly	ADJ
ejpam-7047	3	11	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	3	12	,	,	PUNCT
ejpam-7047	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	3	14	multifunctions	multifunction	NOUN
ejpam-7047	3	15	and	and	CCONJ
ejpam-7047	3	16	lower	low	ADJ
ejpam-7047	3	17	weakly	weakly	ADJ
ejpam-7047	3	18	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	3	19	,	,	PUNCT
ejpam-7047	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	3	21	multifunctions	multifunction	NOUN
ejpam-7047	3	22	are	be	AUX
ejpam-7047	3	23	considered	consider	VERB
ejpam-7047	3	24	.	.	PUNCT
ejpam-7047	4	1	2020	2020	NUM
ejpam-7047	4	2	mathematics	mathematic	NOUN
ejpam-7047	4	3	subject	subject	NOUN
ejpam-7047	4	4	classifications	classification	NOUN
ejpam-7047	4	5	:	:	PUNCT
ejpam-7047	4	6	54c08	54c08	NUM
ejpam-7047	4	7	,	,	PUNCT
ejpam-7047	4	8	54c60	54c60	NUM
ejpam-7047	4	9	key	key	ADJ
ejpam-7047	4	10	words	word	NOUN
ejpam-7047	4	11	and	and	CCONJ
ejpam-7047	4	12	phrases	phrase	NOUN
ejpam-7047	4	13	:	:	PUNCT
ejpam-7047	4	14	upper	upper	ADJ
ejpam-7047	4	15	weakly	weakly	ADJ
ejpam-7047	4	16	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	4	17	,	,	PUNCT
ejpam-7047	4	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	4	19	multifunction	multifunction	NOUN
ejpam-7047	4	20	,	,	PUNCT
ejpam-7047	4	21	lower	low	ADJ
ejpam-7047	4	22	weakly	weakly	ADJ
ejpam-7047	4	23	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	4	24	,	,	PUNCT
ejpam-7047	4	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	4	26	multifunction	multifunction	NOUN
ejpam-7047	4	27	1	1	NUM
ejpam-7047	4	28	.	.	PUNCT
ejpam-7047	4	29	introduction	introduction	NOUN
ejpam-7047	4	30	the	the	DET
ejpam-7047	4	31	notion	notion	NOUN
ejpam-7047	4	32	of	of	ADP
ejpam-7047	4	33	weakly	weakly	ADJ
ejpam-7047	4	34	α	α	ADJ
ejpam-7047	4	35	-	-	ADJ
ejpam-7047	4	36	continuous	continuous	ADJ
ejpam-7047	4	37	functions	function	NOUN
ejpam-7047	4	38	was	be	AUX
ejpam-7047	4	39	first	first	ADV
ejpam-7047	4	40	introduced	introduce	VERB
ejpam-7047	4	41	by	by	ADP
ejpam-7047	4	42	noiri	noiri	PROPN
ejpam-7047	4	43	[	[	X
ejpam-7047	4	44	1	1	NUM
ejpam-7047	4	45	]	]	PUNCT
ejpam-7047	4	46	.	.	PUNCT
ejpam-7047	5	1	sen	sen	PROPN
ejpam-7047	5	2	and	and	CCONJ
ejpam-7047	5	3	bhattacharyya	bhattacharyya	ADJ
ejpam-7047	6	1	[	[	X
ejpam-7047	6	2	2	2	NUM
ejpam-7047	6	3	]	]	PUNCT
ejpam-7047	6	4	investigated	investigate	VERB
ejpam-7047	6	5	several	several	ADJ
ejpam-7047	6	6	characterizations	characterization	NOUN
ejpam-7047	6	7	of	of	ADP
ejpam-7047	6	8	weakly	weakly	ADJ
ejpam-7047	6	9	α	α	ADJ
ejpam-7047	6	10	-	-	ADJ
ejpam-7047	6	11	continuous	continuous	ADJ
ejpam-7047	6	12	functions	function	NOUN
ejpam-7047	6	13	.	.	PUNCT
ejpam-7047	7	1	in	in	ADP
ejpam-7047	7	2	2002	2002	NUM
ejpam-7047	7	3	,	,	PUNCT
ejpam-7047	7	4	popa	popa	NOUN
ejpam-7047	7	5	and	and	CCONJ
ejpam-7047	7	6	noiri	noiri	ADV
ejpam-7047	7	7	[	[	X
ejpam-7047	7	8	3	3	NUM
ejpam-7047	7	9	]	]	PUNCT
ejpam-7047	7	10	extended	extend	VERB
ejpam-7047	7	11	the	the	DET
ejpam-7047	7	12	concept	concept	NOUN
ejpam-7047	7	13	of	of	ADP
ejpam-7047	7	14	α	α	NOUN
ejpam-7047	7	15	-	-	ADJ
ejpam-7047	7	16	continuous	continuous	ADJ
ejpam-7047	7	17	functions	function	NOUN
ejpam-7047	7	18	to	to	ADP
ejpam-7047	7	19	multifunctions	multifunction	NOUN
ejpam-7047	7	20	and	and	CCONJ
ejpam-7047	7	21	presented	present	VERB
ejpam-7047	7	22	two	two	NUM
ejpam-7047	7	23	classes	class	NOUN
ejpam-7047	7	24	of	of	ADP
ejpam-7047	7	25	multifunctions	multifunction	NOUN
ejpam-7047	7	26	defined	define	VERB
ejpam-7047	7	27	between	between	ADP
ejpam-7047	7	28	topological	topological	ADJ
ejpam-7047	7	29	spaces	space	NOUN
ejpam-7047	7	30	,	,	PUNCT
ejpam-7047	7	31	namely	namely	ADV
ejpam-7047	7	32	upper	upper	ADJ
ejpam-7047	7	33	weakly	weakly	ADJ
ejpam-7047	7	34	α	α	ADJ
ejpam-7047	7	35	-	-	ADJ
ejpam-7047	7	36	continuous	continuous	ADJ
ejpam-7047	7	37	multifunctions	multifunction	NOUN
ejpam-7047	7	38	and	and	CCONJ
ejpam-7047	7	39	lower	low	ADJ
ejpam-7047	7	40	weakly	weakly	ADJ
ejpam-7047	7	41	α	α	ADJ
ejpam-7047	7	42	-	-	ADJ
ejpam-7047	7	43	continuous	continuous	ADJ
ejpam-7047	7	44	multifunctions	multifunction	NOUN
ejpam-7047	7	45	.	.	PUNCT
ejpam-7047	8	1	furthermore	furthermore	ADV
ejpam-7047	8	2	,	,	PUNCT
ejpam-7047	8	3	popa	popa	NOUN
ejpam-7047	8	4	and	and	CCONJ
ejpam-7047	8	5	noiri	noiri	ADV
ejpam-7047	9	1	[	[	X
ejpam-7047	9	2	3	3	NUM
ejpam-7047	9	3	]	]	PUNCT
ejpam-7047	9	4	investigated	investigate	VERB
ejpam-7047	9	5	several	several	ADJ
ejpam-7047	9	6	characterizations	characterization	NOUN
ejpam-7047	9	7	and	and	CCONJ
ejpam-7047	9	8	some	some	DET
ejpam-7047	9	9	properties	property	NOUN
ejpam-7047	9	10	of	of	ADP
ejpam-7047	9	11	upper	upper	ADJ
ejpam-7047	9	12	weakly	weakly	ADJ
ejpam-7047	9	13	α	α	ADJ
ejpam-7047	9	14	-	-	ADJ
ejpam-7047	9	15	continuous	continuous	ADJ
ejpam-7047	9	16	multifunctions	multifunction	NOUN
ejpam-7047	9	17	and	and	CCONJ
ejpam-7047	9	18	lower	low	ADJ
ejpam-7047	9	19	weakly	weakly	ADJ
ejpam-7047	9	20	αcontinuous	αcontinuous	ADJ
ejpam-7047	9	21	multifunctions	multifunction	NOUN
ejpam-7047	9	22	.	.	PUNCT
ejpam-7047	10	1	on	on	ADP
ejpam-7047	10	2	the	the	DET
ejpam-7047	10	3	other	other	ADJ
ejpam-7047	10	4	hand	hand	NOUN
ejpam-7047	10	5	,	,	PUNCT
ejpam-7047	10	6	the	the	DET
ejpam-7047	10	7	present	present	ADJ
ejpam-7047	10	8	author	author	NOUN
ejpam-7047	10	9	introduced	introduce	VERB
ejpam-7047	10	10	and	and	CCONJ
ejpam-7047	10	11	studied	study	VERB
ejpam-7047	10	12	four	four	NUM
ejpam-7047	10	13	classes	class	NOUN
ejpam-7047	10	14	of	of	ADP
ejpam-7047	10	15	multifunctions	multifunction	NOUN
ejpam-7047	10	16	defined	define	VERB
ejpam-7047	10	17	from	from	ADP
ejpam-7047	10	18	an	an	DET
ejpam-7047	10	19	ideal	ideal	ADJ
ejpam-7047	10	20	topological	topological	ADJ
ejpam-7047	10	21	space	space	NOUN
ejpam-7047	10	22	into	into	ADP
ejpam-7047	10	23	an	an	DET
ejpam-7047	10	24	ideal	ideal	ADJ
ejpam-7047	10	25	topological	topological	ADJ
ejpam-7047	10	26	space	space	NOUN
ejpam-7047	10	27	,	,	PUNCT
ejpam-7047	10	28	called	call	VERB
ejpam-7047	10	29	upper	upper	ADJ
ejpam-7047	10	30	weakly	weakly	ADJ
ejpam-7047	10	31	⋆-continuous	⋆-continuous	ADJ
ejpam-7047	10	32	multifunctions	multifunction	NOUN
ejpam-7047	11	1	[	[	X
ejpam-7047	11	2	4	4	NUM
ejpam-7047	11	3	]	]	PUNCT
ejpam-7047	11	4	,	,	PUNCT
ejpam-7047	11	5	lower	low	ADJ
ejpam-7047	11	6	weakly	weakly	ADJ
ejpam-7047	11	7	⋆-continuous	⋆-continuous	ADJ
ejpam-7047	11	8	multifunctions	multifunction	NOUN
ejpam-7047	11	9	[	[	X
ejpam-7047	11	10	4	4	NUM
ejpam-7047	11	11	]	]	PUNCT
ejpam-7047	11	12	,	,	PUNCT
ejpam-7047	11	13	upper	upper	ADJ
ejpam-7047	11	14	weakly	weakly	ADJ
ejpam-7047	11	15	α(⋆)-continuous	α(⋆)-continuous	ADJ
ejpam-7047	11	16	multifunctions	multifunction	NOUN
ejpam-7047	12	1	[	[	X
ejpam-7047	12	2	5	5	NUM
ejpam-7047	12	3	]	]	PUNCT
ejpam-7047	12	4	,	,	PUNCT
ejpam-7047	12	5	lower	low	ADJ
ejpam-7047	12	6	weakly	weakly	ADJ
ejpam-7047	12	7	α(⋆)continuous	α(⋆)continuous	ADJ
ejpam-7047	12	8	multifunctions	multifunction	NOUN
ejpam-7047	13	1	[	[	X
ejpam-7047	13	2	5	5	NUM
ejpam-7047	13	3	]	]	PUNCT
ejpam-7047	13	4	,	,	PUNCT
ejpam-7047	13	5	upper	upper	ADJ
ejpam-7047	13	6	weakly	weakly	ADJ
ejpam-7047	13	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7047	13	8	multifunctions	multifunction	NOUN
ejpam-7047	14	1	[	[	X
ejpam-7047	14	2	6	6	NUM
ejpam-7047	14	3	]	]	PUNCT
ejpam-7047	14	4	,	,	PUNCT
ejpam-7047	14	5	lower	low	ADJ
ejpam-7047	14	6	weakly	weakly	ADJ
ejpam-7047	14	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7047	14	8	multifunctions	multifunction	NOUN
ejpam-7047	15	1	[	[	X
ejpam-7047	15	2	6	6	NUM
ejpam-7047	15	3	]	]	PUNCT
ejpam-7047	15	4	,	,	PUNCT
ejpam-7047	15	5	weakly	weakly	ADJ
ejpam-7047	15	6	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-7047	15	7	multifunctions	multifunction	NOUN
ejpam-7047	16	1	[	[	X
ejpam-7047	16	2	7	7	NUM
ejpam-7047	16	3	]	]	PUNCT
ejpam-7047	16	4	and	and	CCONJ
ejpam-7047	16	5	weakly	weakly	ADJ
ejpam-7047	16	6	pı	pı	ADJ
ejpam-7047	16	7	-	-	ADJ
ejpam-7047	16	8	continuous	continuous	ADJ
ejpam-7047	16	9	multifunctions	multifunction	NOUN
ejpam-7047	17	1	[	[	X
ejpam-7047	17	2	8	8	NUM
ejpam-7047	17	3	]	]	PUNCT
ejpam-7047	17	4	.	.	PUNCT
ejpam-7047	18	1	pue	pue	NOUN
ejpam-7047	18	2	-	-	PUNCT
ejpam-7047	18	3	on	on	NOUN
ejpam-7047	18	4	et	et	PROPN
ejpam-7047	18	5	al	al	PROPN
ejpam-7047	18	6	.	.	PUNCT
ejpam-7047	19	1	[	[	X
ejpam-7047	19	2	9	9	NUM
ejpam-7047	19	3	]	]	PUNCT
ejpam-7047	19	4	introduced	introduce	VERB
ejpam-7047	19	5	and	and	CCONJ
ejpam-7047	19	6	investigated	investigate	VERB
ejpam-7047	19	7	two	two	NUM
ejpam-7047	19	8	∗corresponding	∗corresponde	VERB
ejpam-7047	19	9	author	author	NOUN
ejpam-7047	19	10	.	.	PUNCT
ejpam-7047	20	1	doi	doi	NOUN
ejpam-7047	20	2	:	:	PUNCT
ejpam-7047	20	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7047	https://doi.org/10.29020/nybg.ejpam.v18i4.7047	NUM
ejpam-7047	20	4	email	email	NOUN
ejpam-7047	20	5	addresses	address	NOUN
ejpam-7047	20	6	:	:	PUNCT
ejpam-7047	20	7	nongluk.h@msu.ac.th	nongluk.h@msu.ac.th	PROPN
ejpam-7047	20	8	(	(	PUNCT
ejpam-7047	20	9	n.	n.	NOUN
ejpam-7047	20	10	viriyapong	viriyapong	PROPN
ejpam-7047	20	11	)	)	PUNCT
ejpam-7047	20	12	,	,	PUNCT
ejpam-7047	20	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7047	20	14	(	(	PUNCT
ejpam-7047	20	15	a.	a.	PROPN
ejpam-7047	20	16	sama	sama	PROPN
ejpam-7047	20	17	-	-	PUNCT
ejpam-7047	20	18	ae	ae	PROPN
ejpam-7047	20	19	)	)	PUNCT
ejpam-7047	20	20	,	,	PUNCT
ejpam-7047	20	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7047	20	22	(	(	PUNCT
ejpam-7047	20	23	c.	c.	PROPN
ejpam-7047	20	24	boonpok	boonpok	PROPN
ejpam-7047	20	25	)	)	PUNCT
ejpam-7047	20	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7047	21	1	1	1	NUM
ejpam-7047	21	2	copyright	copyright	NOUN
ejpam-7047	21	3	:	:	PUNCT
ejpam-7047	21	4	©	©	PROPN
ejpam-7047	21	5	2025	2025	NUM
ejpam-7047	21	6	the	the	DET
ejpam-7047	21	7	author(s	author(s	NOUN
ejpam-7047	21	8	)	)	PUNCT
ejpam-7047	21	9	.	.	PUNCT
ejpam-7047	22	1	(	(	PUNCT
ejpam-7047	22	2	cc	cc	NOUN
ejpam-7047	22	3	by	by	ADP
ejpam-7047	22	4	-	-	PUNCT
ejpam-7047	22	5	nc	nc	PROPN
ejpam-7047	22	6	4.0	4.0	NUM
ejpam-7047	22	7	)	)	PUNCT
ejpam-7047	22	8	n.	n.	NOUN
ejpam-7047	22	9	viriyapong	viriyapong	PROPN
ejpam-7047	22	10	,	,	PUNCT
ejpam-7047	22	11	a.	a.	PROPN
ejpam-7047	22	12	sama	sama	PROPN
ejpam-7047	22	13	-	-	PUNCT
ejpam-7047	22	14	ae	ae	PROPN
ejpam-7047	22	15	,	,	PUNCT
ejpam-7047	22	16	c.	c.	PROPN
ejpam-7047	22	17	boonpok	boonpok	PROPN
ejpam-7047	22	18	/	/	SYM
ejpam-7047	22	19	eur	eur	PROPN
ejpam-7047	22	20	.	.	PUNCT
ejpam-7047	23	1	j.	j.	PROPN
ejpam-7047	23	2	pure	pure	PROPN
ejpam-7047	23	3	appl	appl	PROPN
ejpam-7047	23	4	.	.	PROPN
ejpam-7047	23	5	math	math	PROPN
ejpam-7047	23	6	,	,	PUNCT
ejpam-7047	23	7	18	18	NUM
ejpam-7047	23	8	(	(	PUNCT
ejpam-7047	23	9	4	4	NUM
ejpam-7047	23	10	)	)	PUNCT
ejpam-7047	23	11	(	(	PUNCT
ejpam-7047	23	12	2025	2025	NUM
ejpam-7047	23	13	)	)	PUNCT
ejpam-7047	23	14	,	,	PUNCT
ejpam-7047	23	15	7047	7047	NUM
ejpam-7047	23	16	2	2	NUM
ejpam-7047	23	17	of	of	ADP
ejpam-7047	23	18	11	11	NUM
ejpam-7047	23	19	classes	class	NOUN
ejpam-7047	23	20	of	of	ADP
ejpam-7047	23	21	continuous	continuous	ADJ
ejpam-7047	23	22	multifunctions	multifunction	NOUN
ejpam-7047	23	23	between	between	ADP
ejpam-7047	23	24	bitopological	bitopological	ADJ
ejpam-7047	23	25	spaces	space	NOUN
ejpam-7047	23	26	,	,	PUNCT
ejpam-7047	23	27	namely	namely	ADV
ejpam-7047	23	28	upper	upper	ADJ
ejpam-7047	23	29	(	(	PUNCT
ejpam-7047	23	30	τ1	τ1	NOUN
ejpam-7047	23	31	,	,	PUNCT
ejpam-7047	23	32	τ2)continuous	τ2)continuous	ADJ
ejpam-7047	23	33	multifunctions	multifunction	NOUN
ejpam-7047	23	34	and	and	CCONJ
ejpam-7047	23	35	lower	low	ADJ
ejpam-7047	23	36	(	(	PUNCT
ejpam-7047	23	37	τ1	τ1	NOUN
ejpam-7047	23	38	,	,	PUNCT
ejpam-7047	23	39	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7047	23	40	multifunctions	multifunction	NOUN
ejpam-7047	23	41	.	.	PUNCT
ejpam-7047	24	1	thongmoon	thongmoon	NOUN
ejpam-7047	24	2	et	et	PROPN
ejpam-7047	24	3	al	al	PROPN
ejpam-7047	24	4	.	.	PUNCT
ejpam-7047	25	1	[	[	X
ejpam-7047	25	2	10	10	NUM
ejpam-7047	25	3	]	]	PUNCT
ejpam-7047	25	4	introduced	introduce	VERB
ejpam-7047	25	5	and	and	CCONJ
ejpam-7047	25	6	studied	study	VERB
ejpam-7047	25	7	the	the	DET
ejpam-7047	25	8	notions	notion	NOUN
ejpam-7047	25	9	of	of	ADP
ejpam-7047	25	10	upper	upper	ADJ
ejpam-7047	25	11	weakly	weakly	ADJ
ejpam-7047	25	12	(	(	PUNCT
ejpam-7047	25	13	τ1	τ1	NOUN
ejpam-7047	25	14	,	,	PUNCT
ejpam-7047	25	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7047	25	16	multifunctions	multifunction	NOUN
ejpam-7047	25	17	and	and	CCONJ
ejpam-7047	25	18	lower	low	ADJ
ejpam-7047	25	19	weakly	weakly	ADJ
ejpam-7047	25	20	(	(	PUNCT
ejpam-7047	25	21	τ1	τ1	NOUN
ejpam-7047	25	22	,	,	PUNCT
ejpam-7047	25	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7047	25	24	multifunctions	multifunction	NOUN
ejpam-7047	25	25	.	.	PUNCT
ejpam-7047	26	1	in	in	ADP
ejpam-7047	26	2	[	[	X
ejpam-7047	26	3	11	11	NUM
ejpam-7047	26	4	]	]	PUNCT
ejpam-7047	26	5	,	,	PUNCT
ejpam-7047	26	6	the	the	DET
ejpam-7047	26	7	present	present	ADJ
ejpam-7047	26	8	authors	author	NOUN
ejpam-7047	26	9	introduced	introduce	VERB
ejpam-7047	26	10	and	and	CCONJ
ejpam-7047	26	11	investigated	investigate	VERB
ejpam-7047	26	12	the	the	DET
ejpam-7047	26	13	concepts	concept	NOUN
ejpam-7047	26	14	of	of	ADP
ejpam-7047	26	15	upper	upper	ADJ
ejpam-7047	26	16	weakly	weakly	ADJ
ejpam-7047	26	17	(	(	PUNCT
ejpam-7047	26	18	τ1	τ1	NOUN
ejpam-7047	26	19	,	,	PUNCT
ejpam-7047	26	20	τ2)α	τ2)α	ADJ
ejpam-7047	26	21	-	-	PUNCT
ejpam-7047	26	22	continuous	continuous	ADJ
ejpam-7047	26	23	multifunctions	multifunction	NOUN
ejpam-7047	26	24	and	and	CCONJ
ejpam-7047	26	25	lower	low	ADJ
ejpam-7047	26	26	weakly	weakly	ADJ
ejpam-7047	26	27	(	(	PUNCT
ejpam-7047	26	28	τ1	τ1	NOUN
ejpam-7047	26	29	,	,	PUNCT
ejpam-7047	26	30	τ2)α	τ2)α	ADJ
ejpam-7047	26	31	-	-	PUNCT
ejpam-7047	26	32	continuous	continuous	ADJ
ejpam-7047	26	33	multifunctions	multifunction	NOUN
ejpam-7047	26	34	.	.	PUNCT
ejpam-7047	27	1	quite	quite	ADV
ejpam-7047	27	2	recently	recently	ADV
ejpam-7047	27	3	,	,	PUNCT
ejpam-7047	27	4	pue	pue	NOUN
ejpam-7047	27	5	-	-	PUNCT
ejpam-7047	27	6	on	on	NOUN
ejpam-7047	27	7	et	et	PROPN
ejpam-7047	27	8	al	al	PROPN
ejpam-7047	27	9	.	.	PUNCT
ejpam-7047	28	1	[	[	X
ejpam-7047	28	2	12	12	NUM
ejpam-7047	28	3	]	]	PUNCT
ejpam-7047	28	4	presented	present	VERB
ejpam-7047	28	5	new	new	ADJ
ejpam-7047	28	6	classes	class	NOUN
ejpam-7047	28	7	of	of	ADP
ejpam-7047	28	8	continuous	continuous	ADJ
ejpam-7047	28	9	multifunctions	multifunction	NOUN
ejpam-7047	28	10	defined	define	VERB
ejpam-7047	28	11	from	from	ADP
ejpam-7047	28	12	an	an	DET
ejpam-7047	28	13	ideal	ideal	ADJ
ejpam-7047	28	14	topological	topological	ADJ
ejpam-7047	28	15	space	space	NOUN
ejpam-7047	28	16	into	into	ADP
ejpam-7047	28	17	a	a	DET
ejpam-7047	28	18	bitopological	bitopological	ADJ
ejpam-7047	28	19	space	space	NOUN
ejpam-7047	28	20	,	,	PUNCT
ejpam-7047	28	21	namely	namely	ADV
ejpam-7047	28	22	upper	upper	ADJ
ejpam-7047	28	23	almost	almost	ADV
ejpam-7047	28	24	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7047	28	25	,	,	PUNCT
ejpam-7047	28	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	28	27	multifunctions	multifunction	NOUN
ejpam-7047	28	28	and	and	CCONJ
ejpam-7047	28	29	lower	low	ADJ
ejpam-7047	28	30	almost	almost	ADV
ejpam-7047	28	31	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7047	28	32	,	,	PUNCT
ejpam-7047	28	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	28	34	multifunctions	multifunction	NOUN
ejpam-7047	28	35	.	.	PUNCT
ejpam-7047	29	1	in	in	ADP
ejpam-7047	29	2	this	this	DET
ejpam-7047	29	3	paper	paper	NOUN
ejpam-7047	29	4	,	,	PUNCT
ejpam-7047	29	5	we	we	PRON
ejpam-7047	29	6	introduce	introduce	VERB
ejpam-7047	29	7	the	the	DET
ejpam-7047	29	8	concepts	concept	NOUN
ejpam-7047	29	9	of	of	ADP
ejpam-7047	29	10	continuous	continuous	ADJ
ejpam-7047	29	11	multifunctions	multifunction	NOUN
ejpam-7047	29	12	between	between	ADP
ejpam-7047	29	13	an	an	DET
ejpam-7047	29	14	ideal	ideal	ADJ
ejpam-7047	29	15	topological	topological	ADJ
ejpam-7047	29	16	space	space	NOUN
ejpam-7047	29	17	and	and	CCONJ
ejpam-7047	29	18	a	a	DET
ejpam-7047	29	19	bitopological	bitopological	ADJ
ejpam-7047	29	20	space	space	NOUN
ejpam-7047	29	21	,	,	PUNCT
ejpam-7047	29	22	called	call	VERB
ejpam-7047	29	23	upper	upper	ADJ
ejpam-7047	29	24	weakly	weakly	ADJ
ejpam-7047	29	25	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	29	26	,	,	PUNCT
ejpam-7047	29	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	29	28	multifunctions	multifunction	NOUN
ejpam-7047	29	29	and	and	CCONJ
ejpam-7047	29	30	lower	low	ADJ
ejpam-7047	29	31	weakly	weakly	ADJ
ejpam-7047	29	32	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	29	33	,	,	PUNCT
ejpam-7047	29	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	29	35	multifunctions	multifunction	NOUN
ejpam-7047	29	36	.	.	PUNCT
ejpam-7047	30	1	we	we	PRON
ejpam-7047	30	2	also	also	ADV
ejpam-7047	30	3	investigate	investigate	VERB
ejpam-7047	30	4	several	several	ADJ
ejpam-7047	30	5	characterizations	characterization	NOUN
ejpam-7047	30	6	of	of	ADP
ejpam-7047	30	7	upper	upper	ADJ
ejpam-7047	30	8	weakly	weakly	ADJ
ejpam-7047	30	9	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	30	10	,	,	PUNCT
ejpam-7047	30	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	30	12	multifunctions	multifunction	NOUN
ejpam-7047	30	13	and	and	CCONJ
ejpam-7047	30	14	lower	low	ADJ
ejpam-7047	30	15	weakly	weakly	ADJ
ejpam-7047	30	16	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	30	17	,	,	PUNCT
ejpam-7047	30	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	30	19	multifunctions	multifunction	NOUN
ejpam-7047	30	20	.	.	PUNCT
ejpam-7047	31	1	2	2	X
ejpam-7047	31	2	.	.	X
ejpam-7047	31	3	preliminaries	preliminary	NOUN
ejpam-7047	31	4	throughout	throughout	ADP
ejpam-7047	31	5	the	the	DET
ejpam-7047	31	6	present	present	ADJ
ejpam-7047	31	7	paper	paper	NOUN
ejpam-7047	31	8	,	,	PUNCT
ejpam-7047	31	9	spaces	space	NOUN
ejpam-7047	31	10	(	(	PUNCT
ejpam-7047	31	11	x	x	NOUN
ejpam-7047	31	12	,	,	PUNCT
ejpam-7047	31	13	τ1	τ1	NOUN
ejpam-7047	31	14	,	,	PUNCT
ejpam-7047	31	15	τ2	τ2	NOUN
ejpam-7047	31	16	)	)	PUNCT
ejpam-7047	31	17	and	and	CCONJ
ejpam-7047	31	18	(	(	PUNCT
ejpam-7047	31	19	y	y	PROPN
ejpam-7047	31	20	,	,	PUNCT
ejpam-7047	31	21	σ1	σ1	PROPN
ejpam-7047	31	22	,	,	PUNCT
ejpam-7047	31	23	σ2	σ2	NOUN
ejpam-7047	31	24	)	)	PUNCT
ejpam-7047	31	25	(	(	PUNCT
ejpam-7047	31	26	or	or	CCONJ
ejpam-7047	31	27	simply	simply	ADV
ejpam-7047	31	28	x	x	X
ejpam-7047	31	29	and	and	CCONJ
ejpam-7047	31	30	y	y	PROPN
ejpam-7047	31	31	)	)	PUNCT
ejpam-7047	31	32	always	always	ADV
ejpam-7047	31	33	mean	mean	VERB
ejpam-7047	31	34	bitopological	bitopological	ADJ
ejpam-7047	31	35	spaces	space	NOUN
ejpam-7047	31	36	on	on	ADP
ejpam-7047	31	37	which	which	PRON
ejpam-7047	31	38	no	no	DET
ejpam-7047	31	39	separation	separation	NOUN
ejpam-7047	31	40	axioms	axiom	NOUN
ejpam-7047	31	41	are	be	AUX
ejpam-7047	31	42	assumed	assume	VERB
ejpam-7047	31	43	unless	unless	SCONJ
ejpam-7047	31	44	explicitly	explicitly	ADV
ejpam-7047	31	45	stated	state	VERB
ejpam-7047	31	46	.	.	PUNCT
ejpam-7047	32	1	let	let	VERB
ejpam-7047	32	2	a	a	DET
ejpam-7047	32	3	be	be	AUX
ejpam-7047	32	4	a	a	DET
ejpam-7047	32	5	subset	subset	NOUN
ejpam-7047	32	6	of	of	ADP
ejpam-7047	32	7	a	a	DET
ejpam-7047	32	8	bitopological	bitopological	ADJ
ejpam-7047	32	9	space	space	NOUN
ejpam-7047	32	10	(	(	PUNCT
ejpam-7047	32	11	x	x	NOUN
ejpam-7047	32	12	,	,	PUNCT
ejpam-7047	32	13	τ1	τ1	NOUN
ejpam-7047	32	14	,	,	PUNCT
ejpam-7047	32	15	τ2	τ2	NOUN
ejpam-7047	32	16	)	)	PUNCT
ejpam-7047	32	17	.	.	PUNCT
ejpam-7047	33	1	the	the	DET
ejpam-7047	33	2	closure	closure	NOUN
ejpam-7047	33	3	of	of	ADP
ejpam-7047	33	4	a	a	PRON
ejpam-7047	33	5	and	and	CCONJ
ejpam-7047	33	6	the	the	DET
ejpam-7047	33	7	interior	interior	NOUN
ejpam-7047	33	8	of	of	ADP
ejpam-7047	33	9	a	a	PRON
ejpam-7047	33	10	with	with	ADP
ejpam-7047	33	11	respect	respect	NOUN
ejpam-7047	33	12	to	to	ADP
ejpam-7047	33	13	τi	τi	PROPN
ejpam-7047	33	14	are	be	AUX
ejpam-7047	33	15	denoted	denote	VERB
ejpam-7047	33	16	by	by	ADP
ejpam-7047	33	17	τi	τi	NOUN
ejpam-7047	33	18	-	-	PUNCT
ejpam-7047	33	19	cl(a	cl(a	NUM
ejpam-7047	33	20	)	)	PUNCT
ejpam-7047	33	21	and	and	CCONJ
ejpam-7047	33	22	τi	τi	NOUN
ejpam-7047	33	23	-	-	PUNCT
ejpam-7047	33	24	int(a	int(a	NOUN
ejpam-7047	33	25	)	)	PUNCT
ejpam-7047	33	26	,	,	PUNCT
ejpam-7047	33	27	respectively	respectively	ADV
ejpam-7047	33	28	,	,	PUNCT
ejpam-7047	33	29	for	for	ADP
ejpam-7047	33	30	i	i	PROPN
ejpam-7047	33	31	=	=	SYM
ejpam-7047	33	32	1	1	NUM
ejpam-7047	33	33	,	,	PUNCT
ejpam-7047	33	34	2	2	NUM
ejpam-7047	33	35	.	.	X
ejpam-7047	33	36	a	a	DET
ejpam-7047	33	37	subset	subset	NOUN
ejpam-7047	33	38	a	a	PRON
ejpam-7047	33	39	of	of	ADP
ejpam-7047	33	40	a	a	DET
ejpam-7047	33	41	bitopological	bitopological	ADJ
ejpam-7047	33	42	space	space	NOUN
ejpam-7047	33	43	(	(	PUNCT
ejpam-7047	33	44	x	x	NOUN
ejpam-7047	33	45	,	,	PUNCT
ejpam-7047	33	46	τ1	τ1	NOUN
ejpam-7047	33	47	,	,	PUNCT
ejpam-7047	33	48	τ2	τ2	NOUN
ejpam-7047	33	49	)	)	PUNCT
ejpam-7047	33	50	is	be	AUX
ejpam-7047	33	51	called	call	VERB
ejpam-7047	33	52	τ1τ2	τ1τ2	VERB
ejpam-7047	33	53	-	-	ADJ
ejpam-7047	33	54	closed	closed	ADJ
ejpam-7047	33	55	[	[	X
ejpam-7047	33	56	13	13	NUM
ejpam-7047	33	57	]	]	PUNCT
ejpam-7047	33	58	if	if	SCONJ
ejpam-7047	33	59	a	a	DET
ejpam-7047	33	60	=	=	NOUN
ejpam-7047	33	61	τ1	τ1	NOUN
ejpam-7047	33	62	-	-	PUNCT
ejpam-7047	33	63	cl(τ2	cl(τ2	NOUN
ejpam-7047	33	64	-	-	PUNCT
ejpam-7047	33	65	cl(a	cl(a	NUM
ejpam-7047	33	66	)	)	PUNCT
ejpam-7047	33	67	)	)	PUNCT
ejpam-7047	33	68	.	.	PUNCT
ejpam-7047	34	1	the	the	DET
ejpam-7047	34	2	complement	complement	NOUN
ejpam-7047	34	3	of	of	ADP
ejpam-7047	34	4	a	a	DET
ejpam-7047	34	5	τ1τ2	τ1τ2	ADJ
ejpam-7047	34	6	-	-	ADJ
ejpam-7047	34	7	closed	closed	ADJ
ejpam-7047	34	8	set	set	NOUN
ejpam-7047	34	9	is	be	AUX
ejpam-7047	34	10	called	call	VERB
ejpam-7047	34	11	τ1τ2	τ1τ2	NOUN
ejpam-7047	34	12	-	-	ADJ
ejpam-7047	34	13	open	open	ADJ
ejpam-7047	34	14	.	.	PUNCT
ejpam-7047	35	1	the	the	DET
ejpam-7047	35	2	intersection	intersection	NOUN
ejpam-7047	35	3	of	of	ADP
ejpam-7047	35	4	all	all	DET
ejpam-7047	35	5	τ1τ2	τ1τ2	ADJ
ejpam-7047	35	6	-	-	ADJ
ejpam-7047	35	7	closed	closed	ADJ
ejpam-7047	35	8	sets	set	NOUN
ejpam-7047	35	9	of	of	ADP
ejpam-7047	35	10	x	x	PUNCT
ejpam-7047	35	11	containing	contain	VERB
ejpam-7047	35	12	a	a	PRON
ejpam-7047	35	13	is	be	AUX
ejpam-7047	35	14	called	call	VERB
ejpam-7047	35	15	the	the	DET
ejpam-7047	35	16	τ1τ2	τ1τ2	NOUN
ejpam-7047	35	17	-	-	NOUN
ejpam-7047	35	18	closure	closure	NOUN
ejpam-7047	35	19	[	[	X
ejpam-7047	35	20	13	13	NUM
ejpam-7047	35	21	]	]	PUNCT
ejpam-7047	35	22	of	of	ADP
ejpam-7047	35	23	a	a	PRON
ejpam-7047	35	24	and	and	CCONJ
ejpam-7047	35	25	is	be	AUX
ejpam-7047	35	26	denoted	denote	VERB
ejpam-7047	35	27	by	by	ADP
ejpam-7047	35	28	τ1τ2	τ1τ2	NOUN
ejpam-7047	35	29	-	-	NUM
ejpam-7047	35	30	cl(a	cl(a	NUM
ejpam-7047	35	31	)	)	PUNCT
ejpam-7047	35	32	.	.	PUNCT
ejpam-7047	36	1	the	the	DET
ejpam-7047	36	2	union	union	NOUN
ejpam-7047	36	3	of	of	ADP
ejpam-7047	36	4	all	all	DET
ejpam-7047	36	5	τ1τ2	τ1τ2	ADJ
ejpam-7047	36	6	-	-	ADJ
ejpam-7047	36	7	open	open	ADJ
ejpam-7047	36	8	sets	set	NOUN
ejpam-7047	36	9	of	of	ADP
ejpam-7047	36	10	x	x	PUNCT
ejpam-7047	36	11	contained	contain	VERB
ejpam-7047	36	12	in	in	ADP
ejpam-7047	36	13	a	a	PRON
ejpam-7047	36	14	is	be	AUX
ejpam-7047	36	15	called	call	VERB
ejpam-7047	36	16	the	the	DET
ejpam-7047	36	17	τ1τ2	τ1τ2	NOUN
ejpam-7047	36	18	-	-	ADJ
ejpam-7047	36	19	interior	interior	ADJ
ejpam-7047	36	20	[	[	X
ejpam-7047	36	21	13	13	NUM
ejpam-7047	36	22	]	]	PUNCT
ejpam-7047	36	23	of	of	ADP
ejpam-7047	36	24	a	a	PRON
ejpam-7047	36	25	and	and	CCONJ
ejpam-7047	36	26	is	be	AUX
ejpam-7047	36	27	denoted	denote	VERB
ejpam-7047	36	28	by	by	ADP
ejpam-7047	36	29	τ1τ2	τ1τ2	NOUN
ejpam-7047	36	30	-	-	ADJ
ejpam-7047	36	31	int(a	int(a	NOUN
ejpam-7047	36	32	)	)	PUNCT
ejpam-7047	36	33	.	.	PUNCT
ejpam-7047	37	1	lemma	lemma	PROPN
ejpam-7047	37	2	1	1	NUM
ejpam-7047	37	3	.	.	PUNCT
ejpam-7047	38	1	[	[	X
ejpam-7047	38	2	13	13	NUM
ejpam-7047	38	3	]	]	PUNCT
ejpam-7047	38	4	let	let	VERB
ejpam-7047	38	5	a	a	PRON
ejpam-7047	38	6	and	and	CCONJ
ejpam-7047	38	7	b	b	NOUN
ejpam-7047	38	8	be	be	AUX
ejpam-7047	38	9	subsets	subset	NOUN
ejpam-7047	38	10	of	of	ADP
ejpam-7047	38	11	a	a	DET
ejpam-7047	38	12	bitopological	bitopological	ADJ
ejpam-7047	38	13	space	space	NOUN
ejpam-7047	38	14	(	(	PUNCT
ejpam-7047	38	15	x	x	NOUN
ejpam-7047	38	16	,	,	PUNCT
ejpam-7047	38	17	τ1	τ1	NOUN
ejpam-7047	38	18	,	,	PUNCT
ejpam-7047	38	19	τ2	τ2	NOUN
ejpam-7047	38	20	)	)	PUNCT
ejpam-7047	38	21	.	.	PUNCT
ejpam-7047	39	1	for	for	ADP
ejpam-7047	39	2	the	the	DET
ejpam-7047	39	3	τ1τ2	τ1τ2	NOUN
ejpam-7047	39	4	-	-	NOUN
ejpam-7047	39	5	closure	closure	NOUN
ejpam-7047	39	6	,	,	PUNCT
ejpam-7047	39	7	the	the	DET
ejpam-7047	39	8	following	follow	VERB
ejpam-7047	39	9	properties	property	NOUN
ejpam-7047	39	10	hold	hold	VERB
ejpam-7047	39	11	:	:	PUNCT
ejpam-7047	39	12	(	(	PUNCT
ejpam-7047	39	13	1	1	X
ejpam-7047	39	14	)	)	PUNCT
ejpam-7047	39	15	a	a	DET
ejpam-7047	39	16	⊆	⊆	NUM
ejpam-7047	39	17	τ1τ2	τ1τ2	NOUN
ejpam-7047	39	18	-	-	NUM
ejpam-7047	39	19	cl(a	cl(a	NUM
ejpam-7047	39	20	)	)	PUNCT
ejpam-7047	39	21	and	and	CCONJ
ejpam-7047	39	22	τ1τ2	τ1τ2	NOUN
ejpam-7047	39	23	-	-	ADJ
ejpam-7047	39	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7047	39	25	-	-	PUNCT
ejpam-7047	39	26	cl(a	cl(a	NUM
ejpam-7047	39	27	)	)	PUNCT
ejpam-7047	39	28	)	)	PUNCT
ejpam-7047	40	1	=	=	PUNCT
ejpam-7047	40	2	τ1τ2	τ1τ2	NOUN
ejpam-7047	40	3	-	-	NUM
ejpam-7047	40	4	cl(a	cl(a	NUM
ejpam-7047	40	5	)	)	PUNCT
ejpam-7047	40	6	.	.	PUNCT
ejpam-7047	41	1	(	(	PUNCT
ejpam-7047	41	2	2	2	X
ejpam-7047	41	3	)	)	PUNCT
ejpam-7047	41	4	if	if	SCONJ
ejpam-7047	41	5	a	a	DET
ejpam-7047	41	6	⊆	⊆	NUM
ejpam-7047	41	7	b	b	NOUN
ejpam-7047	41	8	,	,	PUNCT
ejpam-7047	41	9	then	then	ADV
ejpam-7047	41	10	τ1τ2	τ1τ2	NOUN
ejpam-7047	41	11	-	-	NUM
ejpam-7047	41	12	cl(a	cl(a	NUM
ejpam-7047	41	13	)	)	PUNCT
ejpam-7047	41	14	⊆	⊆	NUM
ejpam-7047	41	15	τ1τ2	τ1τ2	NOUN
ejpam-7047	41	16	-	-	NOUN
ejpam-7047	41	17	cl(b	cl(b	NOUN
ejpam-7047	41	18	)	)	PUNCT
ejpam-7047	41	19	.	.	PUNCT
ejpam-7047	42	1	(	(	PUNCT
ejpam-7047	42	2	3	3	X
ejpam-7047	42	3	)	)	PUNCT
ejpam-7047	42	4	τ1τ2	τ1τ2	NOUN
ejpam-7047	42	5	-	-	NUM
ejpam-7047	42	6	cl(a	cl(a	NUM
ejpam-7047	42	7	)	)	PUNCT
ejpam-7047	42	8	is	be	AUX
ejpam-7047	42	9	τ1τ2	τ1τ2	NOUN
ejpam-7047	42	10	-	-	ADJ
ejpam-7047	42	11	closed	closed	ADJ
ejpam-7047	42	12	.	.	PUNCT
ejpam-7047	43	1	(	(	PUNCT
ejpam-7047	43	2	4	4	X
ejpam-7047	43	3	)	)	PUNCT
ejpam-7047	43	4	a	a	PRON
ejpam-7047	43	5	is	be	AUX
ejpam-7047	43	6	τ1τ2	τ1τ2	NOUN
ejpam-7047	43	7	-	-	ADJ
ejpam-7047	43	8	closed	closed	ADJ
ejpam-7047	43	9	if	if	SCONJ
ejpam-7047	43	10	and	and	CCONJ
ejpam-7047	43	11	only	only	ADV
ejpam-7047	43	12	if	if	SCONJ
ejpam-7047	43	13	a	a	DET
ejpam-7047	43	14	=	=	PUNCT
ejpam-7047	43	15	τ1τ2	τ1τ2	NOUN
ejpam-7047	43	16	-	-	NUM
ejpam-7047	43	17	cl(a	cl(a	NUM
ejpam-7047	43	18	)	)	PUNCT
ejpam-7047	43	19	.	.	PUNCT
ejpam-7047	44	1	(	(	PUNCT
ejpam-7047	44	2	5	5	X
ejpam-7047	44	3	)	)	PUNCT
ejpam-7047	44	4	τ1τ2	τ1τ2	NOUN
ejpam-7047	44	5	-	-	NOUN
ejpam-7047	44	6	cl(x	cl(x	X
ejpam-7047	44	7	−a	−a	NOUN
ejpam-7047	44	8	)	)	PUNCT
ejpam-7047	45	1	=	=	PUNCT
ejpam-7047	45	2	x	x	X
ejpam-7047	46	1	−	−	ADP
ejpam-7047	46	2	τ1τ2	τ1τ2	NOUN
ejpam-7047	46	3	-	-	PUNCT
ejpam-7047	46	4	int(a	int(a	NOUN
ejpam-7047	46	5	)	)	PUNCT
ejpam-7047	46	6	.	.	PUNCT
ejpam-7047	47	1	a	a	DET
ejpam-7047	47	2	subset	subset	NOUN
ejpam-7047	47	3	a	a	PRON
ejpam-7047	47	4	of	of	ADP
ejpam-7047	47	5	a	a	DET
ejpam-7047	47	6	bitopological	bitopological	ADJ
ejpam-7047	47	7	space	space	NOUN
ejpam-7047	47	8	(	(	PUNCT
ejpam-7047	47	9	x	x	NOUN
ejpam-7047	47	10	,	,	PUNCT
ejpam-7047	47	11	τ1	τ1	NOUN
ejpam-7047	47	12	,	,	PUNCT
ejpam-7047	47	13	τ2	τ2	NOUN
ejpam-7047	47	14	)	)	PUNCT
ejpam-7047	47	15	is	be	AUX
ejpam-7047	47	16	said	say	VERB
ejpam-7047	47	17	to	to	PART
ejpam-7047	47	18	be	be	AUX
ejpam-7047	47	19	(	(	PUNCT
ejpam-7047	47	20	τ1	τ1	NOUN
ejpam-7047	47	21	,	,	PUNCT
ejpam-7047	47	22	τ2)r	τ2)r	NOUN
ejpam-7047	47	23	-	-	PUNCT
ejpam-7047	47	24	open	open	NOUN
ejpam-7047	47	25	[	[	X
ejpam-7047	47	26	11	11	NUM
ejpam-7047	47	27	]	]	PUNCT
ejpam-7047	47	28	(	(	PUNCT
ejpam-7047	47	29	resp	resp	NOUN
ejpam-7047	47	30	.	.	PUNCT
ejpam-7047	48	1	(	(	PUNCT
ejpam-7047	48	2	τ1	τ1	NOUN
ejpam-7047	48	3	,	,	PUNCT
ejpam-7047	48	4	τ2)s	τ2)s	NOUN
ejpam-7047	48	5	-	-	PUNCT
ejpam-7047	48	6	open	open	ADJ
ejpam-7047	48	7	[	[	X
ejpam-7047	48	8	14	14	NUM
ejpam-7047	48	9	]	]	NUM
ejpam-7047	48	10	,	,	PUNCT
ejpam-7047	48	11	(	(	PUNCT
ejpam-7047	48	12	τ1	τ1	NOUN
ejpam-7047	48	13	,	,	PUNCT
ejpam-7047	48	14	τ2)p	τ2)p	NOUN
ejpam-7047	48	15	-	-	ADJ
ejpam-7047	48	16	open	open	ADJ
ejpam-7047	48	17	[	[	X
ejpam-7047	48	18	14	14	NUM
ejpam-7047	48	19	]	]	NUM
ejpam-7047	48	20	,	,	PUNCT
ejpam-7047	48	21	(	(	PUNCT
ejpam-7047	48	22	τ1	τ1	NOUN
ejpam-7047	48	23	,	,	PUNCT
ejpam-7047	48	24	τ2)β	τ2)β	ADJ
ejpam-7047	48	25	-	-	PUNCT
ejpam-7047	48	26	open	open	NOUN
ejpam-7047	48	27	[	[	X
ejpam-7047	48	28	14	14	NUM
ejpam-7047	48	29	]	]	SYM
ejpam-7047	48	30	)	)	PUNCT
ejpam-7047	48	31	if	if	SCONJ
ejpam-7047	48	32	a	a	DET
ejpam-7047	48	33	=	=	PUNCT
ejpam-7047	48	34	τ1τ2	τ1τ2	NOUN
ejpam-7047	48	35	-	-	NOUN
ejpam-7047	48	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7047	48	37	-	-	PUNCT
ejpam-7047	48	38	cl(a	cl(a	NUM
ejpam-7047	48	39	)	)	PUNCT
ejpam-7047	48	40	)	)	PUNCT
ejpam-7047	48	41	(	(	PUNCT
ejpam-7047	48	42	resp	resp	NOUN
ejpam-7047	48	43	.	.	PUNCT
ejpam-7047	49	1	a	a	DET
ejpam-7047	49	2	⊆	⊆	NUM
ejpam-7047	49	3	τ1τ2	τ1τ2	NOUN
ejpam-7047	49	4	-	-	ADJ
ejpam-7047	49	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7047	49	6	-	-	PUNCT
ejpam-7047	49	7	int(a	int(a	NOUN
ejpam-7047	49	8	)	)	PUNCT
ejpam-7047	49	9	)	)	PUNCT
ejpam-7047	49	10	,	,	PUNCT
ejpam-7047	49	11	a	a	DET
ejpam-7047	49	12	⊆	⊆	NUM
ejpam-7047	49	13	τ1τ2	τ1τ2	NOUN
ejpam-7047	49	14	-	-	NOUN
ejpam-7047	49	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7047	49	16	-	-	PUNCT
ejpam-7047	49	17	cl(a	cl(a	NUM
ejpam-7047	49	18	)	)	PUNCT
ejpam-7047	49	19	)	)	PUNCT
ejpam-7047	49	20	,	,	PUNCT
ejpam-7047	49	21	a	a	DET
ejpam-7047	49	22	⊆	⊆	NUM
ejpam-7047	49	23	τ1τ2	τ1τ2	NOUN
ejpam-7047	49	24	-	-	PUNCT
ejpam-7047	49	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7047	49	26	-	-	PUNCT
ejpam-7047	49	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7047	49	28	-	-	PUNCT
ejpam-7047	49	29	cl(a	cl(a	NUM
ejpam-7047	49	30	)	)	PUNCT
ejpam-7047	49	31	)	)	PUNCT
ejpam-7047	49	32	)	)	PUNCT
ejpam-7047	49	33	)	)	PUNCT
ejpam-7047	49	34	.	.	PUNCT
ejpam-7047	50	1	the	the	DET
ejpam-7047	50	2	complement	complement	NOUN
ejpam-7047	50	3	of	of	ADP
ejpam-7047	50	4	a	a	DET
ejpam-7047	50	5	(	(	PUNCT
ejpam-7047	50	6	τ1	τ1	NOUN
ejpam-7047	50	7	,	,	PUNCT
ejpam-7047	50	8	τ2)r	τ2)r	NOUN
ejpam-7047	50	9	-	-	PUNCT
ejpam-7047	50	10	open	open	ADJ
ejpam-7047	50	11	(	(	PUNCT
ejpam-7047	50	12	resp	resp	NOUN
ejpam-7047	50	13	.	.	PUNCT
ejpam-7047	51	1	(	(	PUNCT
ejpam-7047	51	2	τ1	τ1	NOUN
ejpam-7047	51	3	,	,	PUNCT
ejpam-7047	51	4	τ2)s	τ2)s	NOUN
ejpam-7047	51	5	-	-	PUNCT
ejpam-7047	51	6	open	open	ADJ
ejpam-7047	51	7	,	,	PUNCT
ejpam-7047	51	8	(	(	PUNCT
ejpam-7047	51	9	τ1	τ1	NOUN
ejpam-7047	51	10	,	,	PUNCT
ejpam-7047	51	11	τ2)p	τ2)p	NOUN
ejpam-7047	51	12	-	-	ADJ
ejpam-7047	51	13	open	open	ADJ
ejpam-7047	51	14	,	,	PUNCT
ejpam-7047	51	15	(	(	PUNCT
ejpam-7047	51	16	τ1	τ1	NOUN
ejpam-7047	51	17	,	,	PUNCT
ejpam-7047	51	18	τ2)β	τ2)β	ADJ
ejpam-7047	51	19	-	-	PUNCT
ejpam-7047	51	20	open	open	ADJ
ejpam-7047	51	21	)	)	PUNCT
ejpam-7047	51	22	set	set	NOUN
ejpam-7047	51	23	is	be	AUX
ejpam-7047	51	24	said	say	VERB
ejpam-7047	51	25	to	to	PART
ejpam-7047	51	26	be	be	AUX
ejpam-7047	51	27	(	(	PUNCT
ejpam-7047	51	28	τ1	τ1	NOUN
ejpam-7047	51	29	,	,	PUNCT
ejpam-7047	51	30	τ2)r	τ2)r	NOUN
ejpam-7047	51	31	-	-	PUNCT
ejpam-7047	51	32	closed	closed	ADJ
ejpam-7047	51	33	(	(	PUNCT
ejpam-7047	51	34	resp	resp	NOUN
ejpam-7047	51	35	.	.	PUNCT
ejpam-7047	52	1	(	(	PUNCT
ejpam-7047	52	2	τ1	τ1	NOUN
ejpam-7047	52	3	,	,	PUNCT
ejpam-7047	52	4	τ2)s	τ2)s	NOUN
ejpam-7047	52	5	-	-	PUNCT
ejpam-7047	52	6	closed	closed	ADJ
ejpam-7047	52	7	,	,	PUNCT
ejpam-7047	52	8	(	(	PUNCT
ejpam-7047	52	9	τ1	τ1	NOUN
ejpam-7047	52	10	,	,	PUNCT
ejpam-7047	52	11	τ2)p	τ2)p	NOUN
ejpam-7047	52	12	-	-	PUNCT
ejpam-7047	52	13	closed	closed	ADJ
ejpam-7047	52	14	,	,	PUNCT
ejpam-7047	52	15	(	(	PUNCT
ejpam-7047	52	16	τ1	τ1	NOUN
ejpam-7047	52	17	,	,	PUNCT
ejpam-7047	52	18	τ2)β	τ2)β	ADJ
ejpam-7047	52	19	-	-	PUNCT
ejpam-7047	52	20	closed	closed	ADJ
ejpam-7047	52	21	)	)	PUNCT
ejpam-7047	52	22	.	.	PUNCT
ejpam-7047	53	1	a	a	DET
ejpam-7047	53	2	subset	subset	NOUN
ejpam-7047	53	3	a	a	PRON
ejpam-7047	53	4	of	of	ADP
ejpam-7047	53	5	a	a	DET
ejpam-7047	53	6	bitopological	bitopological	ADJ
ejpam-7047	53	7	space	space	NOUN
ejpam-7047	53	8	(	(	PUNCT
ejpam-7047	53	9	x	x	NOUN
ejpam-7047	53	10	,	,	PUNCT
ejpam-7047	53	11	τ1	τ1	NOUN
ejpam-7047	53	12	,	,	PUNCT
ejpam-7047	53	13	τ2	τ2	NOUN
ejpam-7047	53	14	)	)	PUNCT
ejpam-7047	53	15	is	be	AUX
ejpam-7047	53	16	said	say	VERB
ejpam-7047	53	17	to	to	PART
ejpam-7047	53	18	be	be	AUX
ejpam-7047	53	19	τ1τ2	τ1τ2	NOUN
ejpam-7047	53	20	-	-	ADJ
ejpam-7047	53	21	δ	δ	NOUN
ejpam-7047	53	22	-	-	NOUN
ejpam-7047	53	23	open	open	ADJ
ejpam-7047	53	24	[	[	X
ejpam-7047	53	25	15	15	NUM
ejpam-7047	53	26	]	]	X
ejpam-7047	53	27	if	if	SCONJ
ejpam-7047	53	28	a	a	PRON
ejpam-7047	53	29	is	be	AUX
ejpam-7047	53	30	the	the	DET
ejpam-7047	53	31	union	union	PROPN
ejpam-7047	53	32	n.	n.	PROPN
ejpam-7047	53	33	viriyapong	viriyapong	PROPN
ejpam-7047	53	34	,	,	PUNCT
ejpam-7047	53	35	a.	a.	PROPN
ejpam-7047	53	36	sama	sama	PROPN
ejpam-7047	53	37	-	-	PUNCT
ejpam-7047	53	38	ae	ae	PROPN
ejpam-7047	53	39	,	,	PUNCT
ejpam-7047	53	40	c.	c.	PROPN
ejpam-7047	53	41	boonpok	boonpok	PROPN
ejpam-7047	53	42	/	/	SYM
ejpam-7047	53	43	eur	eur	PROPN
ejpam-7047	53	44	.	.	PUNCT
ejpam-7047	54	1	j.	j.	PROPN
ejpam-7047	54	2	pure	pure	PROPN
ejpam-7047	54	3	appl	appl	PROPN
ejpam-7047	54	4	.	.	PROPN
ejpam-7047	54	5	math	math	PROPN
ejpam-7047	54	6	,	,	PUNCT
ejpam-7047	54	7	18	18	NUM
ejpam-7047	54	8	(	(	PUNCT
ejpam-7047	54	9	4	4	NUM
ejpam-7047	54	10	)	)	PUNCT
ejpam-7047	54	11	(	(	PUNCT
ejpam-7047	54	12	2025	2025	NUM
ejpam-7047	54	13	)	)	PUNCT
ejpam-7047	54	14	,	,	PUNCT
ejpam-7047	54	15	7047	7047	NUM
ejpam-7047	54	16	3	3	NUM
ejpam-7047	54	17	of	of	ADP
ejpam-7047	54	18	11	11	NUM
ejpam-7047	54	19	of	of	ADP
ejpam-7047	54	20	(	(	PUNCT
ejpam-7047	54	21	τ1	τ1	NOUN
ejpam-7047	54	22	,	,	PUNCT
ejpam-7047	54	23	τ2)r	τ2)r	ADJ
ejpam-7047	54	24	-	-	PUNCT
ejpam-7047	54	25	open	open	ADJ
ejpam-7047	54	26	sets	set	NOUN
ejpam-7047	54	27	of	of	ADP
ejpam-7047	54	28	x.	x.	NOUN
ejpam-7047	54	29	the	the	DET
ejpam-7047	54	30	complement	complement	NOUN
ejpam-7047	54	31	of	of	ADP
ejpam-7047	54	32	a	a	DET
ejpam-7047	54	33	τ1τ2	τ1τ2	ADJ
ejpam-7047	54	34	-	-	ADJ
ejpam-7047	54	35	δ	δ	NOUN
ejpam-7047	54	36	-	-	ADJ
ejpam-7047	54	37	open	open	ADJ
ejpam-7047	54	38	set	set	NOUN
ejpam-7047	54	39	is	be	AUX
ejpam-7047	54	40	called	call	VERB
ejpam-7047	54	41	τ1τ2	τ1τ2	NOUN
ejpam-7047	54	42	-	-	ADJ
ejpam-7047	54	43	δ	δ	NOUN
ejpam-7047	54	44	-	-	PUNCT
ejpam-7047	54	45	closed	closed	ADJ
ejpam-7047	54	46	[	[	X
ejpam-7047	54	47	15	15	NUM
ejpam-7047	54	48	]	]	PUNCT
ejpam-7047	54	49	.	.	PUNCT
ejpam-7047	55	1	the	the	DET
ejpam-7047	55	2	union	union	NOUN
ejpam-7047	55	3	of	of	ADP
ejpam-7047	55	4	all	all	DET
ejpam-7047	55	5	τ1τ2	τ1τ2	NOUN
ejpam-7047	55	6	-	-	ADJ
ejpam-7047	55	7	δ	δ	NOUN
ejpam-7047	55	8	-	-	ADJ
ejpam-7047	55	9	open	open	ADJ
ejpam-7047	55	10	sets	set	NOUN
ejpam-7047	55	11	of	of	ADP
ejpam-7047	55	12	x	x	PUNCT
ejpam-7047	55	13	contained	contain	VERB
ejpam-7047	55	14	in	in	ADP
ejpam-7047	55	15	a	a	PRON
ejpam-7047	55	16	is	be	AUX
ejpam-7047	55	17	called	call	VERB
ejpam-7047	55	18	the	the	DET
ejpam-7047	55	19	τ1τ2	τ1τ2	ADJ
ejpam-7047	55	20	-	-	ADJ
ejpam-7047	55	21	δ	δ	NOUN
ejpam-7047	55	22	-	-	NOUN
ejpam-7047	55	23	interior	interior	ADJ
ejpam-7047	55	24	[	[	X
ejpam-7047	55	25	15	15	NUM
ejpam-7047	55	26	]	]	PUNCT
ejpam-7047	55	27	of	of	ADP
ejpam-7047	55	28	a	a	PRON
ejpam-7047	55	29	and	and	CCONJ
ejpam-7047	55	30	is	be	AUX
ejpam-7047	55	31	denoted	denote	VERB
ejpam-7047	55	32	by	by	ADP
ejpam-7047	55	33	τ1τ2	τ1τ2	ADJ
ejpam-7047	55	34	-	-	ADJ
ejpam-7047	55	35	δ	δ	NOUN
ejpam-7047	55	36	-	-	PUNCT
ejpam-7047	55	37	int(a	int(a	PROPN
ejpam-7047	55	38	)	)	PUNCT
ejpam-7047	55	39	.	.	PUNCT
ejpam-7047	56	1	the	the	DET
ejpam-7047	56	2	intersection	intersection	NOUN
ejpam-7047	56	3	of	of	ADP
ejpam-7047	56	4	all	all	DET
ejpam-7047	56	5	τ1τ2	τ1τ2	NOUN
ejpam-7047	56	6	-	-	ADJ
ejpam-7047	56	7	δ	δ	NOUN
ejpam-7047	56	8	-	-	PUNCT
ejpam-7047	56	9	closed	close	VERB
ejpam-7047	56	10	sets	set	NOUN
ejpam-7047	56	11	of	of	ADP
ejpam-7047	56	12	x	x	PUNCT
ejpam-7047	56	13	containing	contain	VERB
ejpam-7047	56	14	a	a	PRON
ejpam-7047	56	15	is	be	AUX
ejpam-7047	56	16	called	call	VERB
ejpam-7047	56	17	the	the	DET
ejpam-7047	56	18	τ1τ2	τ1τ2	ADJ
ejpam-7047	56	19	-	-	ADJ
ejpam-7047	56	20	δ	δ	NOUN
ejpam-7047	56	21	-	-	NOUN
ejpam-7047	56	22	closure	closure	NOUN
ejpam-7047	56	23	[	[	X
ejpam-7047	56	24	15	15	NUM
ejpam-7047	56	25	]	]	PUNCT
ejpam-7047	56	26	of	of	ADP
ejpam-7047	56	27	a	a	PRON
ejpam-7047	56	28	and	and	CCONJ
ejpam-7047	56	29	is	be	AUX
ejpam-7047	56	30	denoted	denote	VERB
ejpam-7047	56	31	by	by	ADP
ejpam-7047	56	32	τ1τ2	τ1τ2	ADJ
ejpam-7047	56	33	-	-	ADJ
ejpam-7047	56	34	δ	δ	NOUN
ejpam-7047	56	35	-	-	PUNCT
ejpam-7047	56	36	cl(a	cl(a	NUM
ejpam-7047	56	37	)	)	PUNCT
ejpam-7047	56	38	.	.	PUNCT
ejpam-7047	57	1	let	let	VERB
ejpam-7047	57	2	a	a	DET
ejpam-7047	57	3	be	be	AUX
ejpam-7047	57	4	a	a	DET
ejpam-7047	57	5	subset	subset	NOUN
ejpam-7047	57	6	of	of	ADP
ejpam-7047	57	7	a	a	DET
ejpam-7047	57	8	bitopological	bitopological	ADJ
ejpam-7047	57	9	space	space	NOUN
ejpam-7047	57	10	(	(	PUNCT
ejpam-7047	57	11	x	x	NOUN
ejpam-7047	57	12	,	,	PUNCT
ejpam-7047	57	13	τ1	τ1	NOUN
ejpam-7047	57	14	,	,	PUNCT
ejpam-7047	57	15	τ2	τ2	NOUN
ejpam-7047	57	16	)	)	PUNCT
ejpam-7047	57	17	.	.	PUNCT
ejpam-7047	58	1	a	a	DET
ejpam-7047	58	2	point	point	NOUN
ejpam-7047	58	3	x	x	X
ejpam-7047	58	4	∈	∈	NOUN
ejpam-7047	58	5	x	x	PUNCT
ejpam-7047	58	6	is	be	AUX
ejpam-7047	58	7	called	call	VERB
ejpam-7047	58	8	a	a	DET
ejpam-7047	58	9	(	(	PUNCT
ejpam-7047	58	10	τ1	τ1	NOUN
ejpam-7047	58	11	,	,	PUNCT
ejpam-7047	58	12	τ2)θ	τ2)θ	ADJ
ejpam-7047	58	13	-	-	PUNCT
ejpam-7047	58	14	cluster	cluster	NOUN
ejpam-7047	58	15	point	point	NOUN
ejpam-7047	58	16	[	[	X
ejpam-7047	58	17	11	11	NUM
ejpam-7047	58	18	]	]	PUNCT
ejpam-7047	58	19	of	of	ADP
ejpam-7047	58	20	a	a	DET
ejpam-7047	58	21	if	if	SCONJ
ejpam-7047	58	22	τ1τ2	τ1τ2	NOUN
ejpam-7047	58	23	-	-	NOUN
ejpam-7047	58	24	cl(u	cl(u	NOUN
ejpam-7047	58	25	)	)	PUNCT
ejpam-7047	58	26	∩	∩	NOUN
ejpam-7047	58	27	a	a	DET
ejpam-7047	58	28	̸=	̸=	PROPN
ejpam-7047	58	29	∅	∅	NOUN
ejpam-7047	58	30	for	for	ADP
ejpam-7047	58	31	every	every	DET
ejpam-7047	58	32	τ1τ2	τ1τ2	ADJ
ejpam-7047	58	33	-	-	ADJ
ejpam-7047	58	34	open	open	ADJ
ejpam-7047	58	35	set	set	NOUN
ejpam-7047	58	36	u	u	NOUN
ejpam-7047	58	37	containing	contain	VERB
ejpam-7047	58	38	x.	x.	NOUN
ejpam-7047	58	39	the	the	DET
ejpam-7047	58	40	set	set	NOUN
ejpam-7047	58	41	of	of	ADP
ejpam-7047	58	42	all	all	DET
ejpam-7047	58	43	(	(	PUNCT
ejpam-7047	58	44	τ1	τ1	NOUN
ejpam-7047	58	45	,	,	PUNCT
ejpam-7047	58	46	τ2)θ	τ2)θ	ADJ
ejpam-7047	58	47	-	-	PUNCT
ejpam-7047	58	48	cluster	cluster	NOUN
ejpam-7047	58	49	points	point	NOUN
ejpam-7047	58	50	of	of	ADP
ejpam-7047	58	51	a	a	PRON
ejpam-7047	58	52	is	be	AUX
ejpam-7047	58	53	called	call	VERB
ejpam-7047	58	54	the	the	DET
ejpam-7047	58	55	(	(	PUNCT
ejpam-7047	58	56	τ1	τ1	NOUN
ejpam-7047	58	57	,	,	PUNCT
ejpam-7047	58	58	τ2)θ	τ2)θ	ADJ
ejpam-7047	58	59	-	-	PUNCT
ejpam-7047	58	60	closure	closure	NOUN
ejpam-7047	58	61	[	[	X
ejpam-7047	58	62	11	11	NUM
ejpam-7047	58	63	]	]	PUNCT
ejpam-7047	58	64	of	of	ADP
ejpam-7047	58	65	a	a	PRON
ejpam-7047	58	66	and	and	CCONJ
ejpam-7047	58	67	is	be	AUX
ejpam-7047	58	68	denoted	denote	VERB
ejpam-7047	58	69	by	by	ADP
ejpam-7047	58	70	(	(	PUNCT
ejpam-7047	58	71	τ1	τ1	NOUN
ejpam-7047	58	72	,	,	PUNCT
ejpam-7047	58	73	τ2)θ	τ2)θ	NOUN
ejpam-7047	58	74	-	-	PUNCT
ejpam-7047	58	75	cl(a	cl(a	NUM
ejpam-7047	58	76	)	)	PUNCT
ejpam-7047	58	77	.	.	PUNCT
ejpam-7047	59	1	a	a	DET
ejpam-7047	59	2	subset	subset	NOUN
ejpam-7047	59	3	a	a	PRON
ejpam-7047	59	4	of	of	ADP
ejpam-7047	59	5	a	a	DET
ejpam-7047	59	6	bitopological	bitopological	ADJ
ejpam-7047	59	7	space	space	NOUN
ejpam-7047	59	8	(	(	PUNCT
ejpam-7047	59	9	x	x	NOUN
ejpam-7047	59	10	,	,	PUNCT
ejpam-7047	59	11	τ1	τ1	NOUN
ejpam-7047	59	12	,	,	PUNCT
ejpam-7047	59	13	τ2	τ2	NOUN
ejpam-7047	59	14	)	)	PUNCT
ejpam-7047	59	15	is	be	AUX
ejpam-7047	59	16	said	say	VERB
ejpam-7047	59	17	to	to	PART
ejpam-7047	59	18	be	be	AUX
ejpam-7047	59	19	(	(	PUNCT
ejpam-7047	59	20	τ1	τ1	NOUN
ejpam-7047	59	21	,	,	PUNCT
ejpam-7047	59	22	τ2)θ	τ2)θ	NOUN
ejpam-7047	59	23	-	-	PUNCT
ejpam-7047	59	24	closed	closed	ADJ
ejpam-7047	59	25	[	[	X
ejpam-7047	59	26	11	11	NUM
ejpam-7047	59	27	]	]	X
ejpam-7047	59	28	if	if	SCONJ
ejpam-7047	59	29	(	(	PUNCT
ejpam-7047	59	30	τ1	τ1	NOUN
ejpam-7047	59	31	,	,	PUNCT
ejpam-7047	59	32	τ2)θ	τ2)θ	NOUN
ejpam-7047	59	33	-	-	PUNCT
ejpam-7047	59	34	cl(a	cl(a	NUM
ejpam-7047	59	35	)	)	PUNCT
ejpam-7047	60	1	=	=	PUNCT
ejpam-7047	60	2	a.	a.	NOUN
ejpam-7047	60	3	the	the	DET
ejpam-7047	60	4	complement	complement	NOUN
ejpam-7047	60	5	of	of	ADP
ejpam-7047	60	6	a	a	DET
ejpam-7047	60	7	(	(	PUNCT
ejpam-7047	60	8	τ1	τ1	NOUN
ejpam-7047	60	9	,	,	PUNCT
ejpam-7047	60	10	τ2)θ	τ2)θ	ADJ
ejpam-7047	60	11	-	-	PUNCT
ejpam-7047	60	12	closed	close	VERB
ejpam-7047	60	13	set	set	NOUN
ejpam-7047	60	14	is	be	AUX
ejpam-7047	60	15	said	say	VERB
ejpam-7047	60	16	to	to	PART
ejpam-7047	60	17	be	be	AUX
ejpam-7047	60	18	(	(	PUNCT
ejpam-7047	60	19	τ1	τ1	NOUN
ejpam-7047	60	20	,	,	PUNCT
ejpam-7047	60	21	τ2)θopen	τ2)θopen	PROPN
ejpam-7047	60	22	.	.	PUNCT
ejpam-7047	61	1	the	the	DET
ejpam-7047	61	2	union	union	NOUN
ejpam-7047	61	3	of	of	ADP
ejpam-7047	61	4	all	all	DET
ejpam-7047	61	5	(	(	PUNCT
ejpam-7047	61	6	τ1	τ1	NOUN
ejpam-7047	61	7	,	,	PUNCT
ejpam-7047	61	8	τ2)θ	τ2)θ	ADJ
ejpam-7047	61	9	-	-	PUNCT
ejpam-7047	61	10	open	open	ADJ
ejpam-7047	61	11	sets	set	NOUN
ejpam-7047	61	12	of	of	ADP
ejpam-7047	61	13	x	x	PUNCT
ejpam-7047	61	14	contained	contain	VERB
ejpam-7047	61	15	in	in	ADP
ejpam-7047	61	16	a	a	PRON
ejpam-7047	61	17	is	be	AUX
ejpam-7047	61	18	called	call	VERB
ejpam-7047	61	19	the	the	DET
ejpam-7047	61	20	(	(	PUNCT
ejpam-7047	61	21	τ1	τ1	NOUN
ejpam-7047	61	22	,	,	PUNCT
ejpam-7047	61	23	τ2)θ	τ2)θ	ADJ
ejpam-7047	61	24	-	-	PUNCT
ejpam-7047	61	25	interior	interior	NOUN
ejpam-7047	61	26	[	[	X
ejpam-7047	61	27	11	11	NUM
ejpam-7047	61	28	]	]	PUNCT
ejpam-7047	61	29	of	of	ADP
ejpam-7047	61	30	a	a	PRON
ejpam-7047	61	31	and	and	CCONJ
ejpam-7047	61	32	is	be	AUX
ejpam-7047	61	33	denoted	denote	VERB
ejpam-7047	61	34	by	by	ADP
ejpam-7047	61	35	(	(	PUNCT
ejpam-7047	61	36	τ1	τ1	NOUN
ejpam-7047	61	37	,	,	PUNCT
ejpam-7047	61	38	τ2)θ	τ2)θ	NOUN
ejpam-7047	61	39	-	-	PUNCT
ejpam-7047	61	40	int(a	int(a	NOUN
ejpam-7047	61	41	)	)	PUNCT
ejpam-7047	61	42	.	.	PUNCT
ejpam-7047	62	1	lemma	lemma	PROPN
ejpam-7047	62	2	2	2	NUM
ejpam-7047	62	3	.	.	PUNCT
ejpam-7047	63	1	[	[	X
ejpam-7047	63	2	11	11	NUM
ejpam-7047	63	3	]	]	PUNCT
ejpam-7047	63	4	for	for	ADP
ejpam-7047	63	5	a	a	DET
ejpam-7047	63	6	subset	subset	NOUN
ejpam-7047	63	7	a	a	PRON
ejpam-7047	63	8	of	of	ADP
ejpam-7047	63	9	a	a	DET
ejpam-7047	63	10	bitopological	bitopological	ADJ
ejpam-7047	63	11	space	space	NOUN
ejpam-7047	63	12	(	(	PUNCT
ejpam-7047	63	13	x	x	NOUN
ejpam-7047	63	14	,	,	PUNCT
ejpam-7047	63	15	τ1	τ1	NOUN
ejpam-7047	63	16	,	,	PUNCT
ejpam-7047	63	17	τ2	τ2	NOUN
ejpam-7047	63	18	)	)	PUNCT
ejpam-7047	63	19	,	,	PUNCT
ejpam-7047	63	20	the	the	DET
ejpam-7047	63	21	following	follow	VERB
ejpam-7047	63	22	properties	property	NOUN
ejpam-7047	63	23	hold	hold	VERB
ejpam-7047	63	24	:	:	PUNCT
ejpam-7047	63	25	(	(	PUNCT
ejpam-7047	63	26	1	1	X
ejpam-7047	63	27	)	)	PUNCT
ejpam-7047	63	28	if	if	SCONJ
ejpam-7047	63	29	a	a	PRON
ejpam-7047	63	30	is	be	AUX
ejpam-7047	63	31	τ1τ2	τ1τ2	NOUN
ejpam-7047	63	32	-	-	ADJ
ejpam-7047	63	33	open	open	ADJ
ejpam-7047	63	34	in	in	ADP
ejpam-7047	63	35	x	x	NOUN
ejpam-7047	63	36	,	,	PUNCT
ejpam-7047	63	37	then	then	ADV
ejpam-7047	63	38	τ1τ2	τ1τ2	NOUN
ejpam-7047	63	39	-	-	NUM
ejpam-7047	63	40	cl(a	cl(a	NUM
ejpam-7047	63	41	)	)	PUNCT
ejpam-7047	63	42	=	=	PUNCT
ejpam-7047	63	43	(	(	PUNCT
ejpam-7047	63	44	τ1	τ1	NOUN
ejpam-7047	63	45	,	,	PUNCT
ejpam-7047	63	46	τ2)θ	τ2)θ	NOUN
ejpam-7047	63	47	-	-	PUNCT
ejpam-7047	63	48	cl(a	cl(a	NUM
ejpam-7047	63	49	)	)	PUNCT
ejpam-7047	63	50	.	.	PUNCT
ejpam-7047	64	1	(	(	PUNCT
ejpam-7047	64	2	2	2	X
ejpam-7047	64	3	)	)	PUNCT
ejpam-7047	64	4	(	(	PUNCT
ejpam-7047	64	5	τ1	τ1	NOUN
ejpam-7047	64	6	,	,	PUNCT
ejpam-7047	64	7	τ2)θ	τ2)θ	NOUN
ejpam-7047	64	8	-	-	PUNCT
ejpam-7047	64	9	cl(a	cl(a	NUM
ejpam-7047	64	10	)	)	PUNCT
ejpam-7047	64	11	is	be	AUX
ejpam-7047	64	12	τ1τ2	τ1τ2	NOUN
ejpam-7047	64	13	-	-	ADJ
ejpam-7047	64	14	closed	closed	ADJ
ejpam-7047	64	15	in	in	ADP
ejpam-7047	64	16	x.	x.	NOUN
ejpam-7047	64	17	an	an	DET
ejpam-7047	64	18	ideal	ideal	NOUN
ejpam-7047	64	19	i	i	PRON
ejpam-7047	64	20	on	on	ADP
ejpam-7047	64	21	a	a	DET
ejpam-7047	64	22	topological	topological	ADJ
ejpam-7047	64	23	space	space	NOUN
ejpam-7047	64	24	(	(	PUNCT
ejpam-7047	64	25	x	x	X
ejpam-7047	64	26	,	,	PUNCT
ejpam-7047	64	27	τ	τ	X
ejpam-7047	64	28	)	)	PUNCT
ejpam-7047	64	29	is	be	AUX
ejpam-7047	64	30	a	a	DET
ejpam-7047	64	31	nonempty	nonempty	ADJ
ejpam-7047	64	32	collection	collection	NOUN
ejpam-7047	64	33	of	of	ADP
ejpam-7047	64	34	subsets	subset	NOUN
ejpam-7047	64	35	of	of	ADP
ejpam-7047	64	36	x	x	PUNCT
ejpam-7047	64	37	satisfying	satisfy	VERB
ejpam-7047	64	38	the	the	DET
ejpam-7047	64	39	following	follow	VERB
ejpam-7047	64	40	properties	property	NOUN
ejpam-7047	64	41	:	:	PUNCT
ejpam-7047	64	42	(	(	PUNCT
ejpam-7047	64	43	1	1	X
ejpam-7047	64	44	)	)	PUNCT
ejpam-7047	64	45	a	a	DET
ejpam-7047	64	46	∈	∈	NOUN
ejpam-7047	65	1	i	i	PRON
ejpam-7047	65	2	and	and	CCONJ
ejpam-7047	65	3	b	b	X
ejpam-7047	65	4	⊆	⊆	NUM
ejpam-7047	65	5	a	a	DET
ejpam-7047	65	6	imply	imply	NOUN
ejpam-7047	65	7	b	b	X
ejpam-7047	65	8	∈	∈	PROPN
ejpam-7047	65	9	i	i	PRON
ejpam-7047	65	10	;	;	PUNCT
ejpam-7047	65	11	(	(	PUNCT
ejpam-7047	65	12	2	2	X
ejpam-7047	65	13	)	)	PUNCT
ejpam-7047	66	1	a	a	DET
ejpam-7047	66	2	∈	∈	NOUN
ejpam-7047	66	3	i	i	PRON
ejpam-7047	66	4	and	and	CCONJ
ejpam-7047	66	5	b	b	X
ejpam-7047	66	6	∈	∈	NOUN
ejpam-7047	67	1	i	i	PRON
ejpam-7047	67	2	imply	imply	VERB
ejpam-7047	67	3	a	a	DET
ejpam-7047	67	4	∪	∪	X
ejpam-7047	67	5	b	b	NOUN
ejpam-7047	67	6	∈	∈	NOUN
ejpam-7047	68	1	i	i	PRON
ejpam-7047	68	2	.	.	PUNCT
ejpam-7047	69	1	a	a	DET
ejpam-7047	69	2	topological	topological	ADJ
ejpam-7047	69	3	space	space	NOUN
ejpam-7047	69	4	(	(	PUNCT
ejpam-7047	69	5	x	x	X
ejpam-7047	69	6	,	,	PUNCT
ejpam-7047	69	7	τ	τ	X
ejpam-7047	69	8	)	)	PUNCT
ejpam-7047	69	9	with	with	ADP
ejpam-7047	69	10	an	an	DET
ejpam-7047	69	11	ideal	ideal	ADJ
ejpam-7047	69	12	i	i	PRON
ejpam-7047	69	13	on	on	ADP
ejpam-7047	69	14	x	x	SYM
ejpam-7047	69	15	is	be	AUX
ejpam-7047	69	16	called	call	VERB
ejpam-7047	69	17	an	an	DET
ejpam-7047	69	18	ideal	ideal	ADJ
ejpam-7047	69	19	topological	topological	ADJ
ejpam-7047	69	20	space	space	NOUN
ejpam-7047	69	21	and	and	CCONJ
ejpam-7047	69	22	is	be	AUX
ejpam-7047	69	23	denoted	denote	VERB
ejpam-7047	69	24	by	by	ADP
ejpam-7047	69	25	(	(	PUNCT
ejpam-7047	69	26	x	x	X
ejpam-7047	69	27	,	,	PUNCT
ejpam-7047	69	28	τ	τ	PROPN
ejpam-7047	69	29	,	,	PUNCT
ejpam-7047	69	30	i	i	NOUN
ejpam-7047	69	31	)	)	PUNCT
ejpam-7047	69	32	.	.	PUNCT
ejpam-7047	70	1	for	for	ADP
ejpam-7047	70	2	an	an	DET
ejpam-7047	70	3	ideal	ideal	ADJ
ejpam-7047	70	4	topological	topological	ADJ
ejpam-7047	70	5	space	space	NOUN
ejpam-7047	70	6	(	(	PUNCT
ejpam-7047	70	7	x	x	X
ejpam-7047	70	8	,	,	PUNCT
ejpam-7047	70	9	τ	τ	PROPN
ejpam-7047	70	10	,	,	PUNCT
ejpam-7047	70	11	i	i	PROPN
ejpam-7047	70	12	)	)	PUNCT
ejpam-7047	70	13	and	and	CCONJ
ejpam-7047	70	14	a	a	DET
ejpam-7047	70	15	subset	subset	NOUN
ejpam-7047	70	16	a	a	PRON
ejpam-7047	70	17	of	of	ADP
ejpam-7047	70	18	x	x	PRON
ejpam-7047	70	19	,	,	PUNCT
ejpam-7047	70	20	a⋆(i	a⋆(i	PROPN
ejpam-7047	70	21	)	)	PUNCT
ejpam-7047	70	22	is	be	AUX
ejpam-7047	70	23	defined	define	VERB
ejpam-7047	70	24	as	as	SCONJ
ejpam-7047	70	25	follows	follow	VERB
ejpam-7047	70	26	:	:	PUNCT
ejpam-7047	70	27	a⋆(i	a⋆(i	NOUN
ejpam-7047	70	28	)	)	PUNCT
ejpam-7047	71	1	=	=	PUNCT
ejpam-7047	71	2	{	{	PUNCT
ejpam-7047	71	3	x	x	PUNCT
ejpam-7047	71	4	∈	∈	PROPN
ejpam-7047	71	5	x	x	X
ejpam-7047	71	6	:	:	PUNCT
ejpam-7047	71	7	u	u	X
ejpam-7047	71	8	∩a	∩a	PROPN
ejpam-7047	71	9	̸∈	̸∈	PROPN
ejpam-7047	71	10	i	i	PRON
ejpam-7047	71	11	for	for	ADP
ejpam-7047	71	12	every	every	DET
ejpam-7047	71	13	open	open	ADJ
ejpam-7047	71	14	neighbourhood	neighbourhood	NOUN
ejpam-7047	71	15	u	u	NOUN
ejpam-7047	71	16	of	of	ADP
ejpam-7047	71	17	x	x	NOUN
ejpam-7047	71	18	}	}	PUNCT
ejpam-7047	71	19	.	.	PUNCT
ejpam-7047	72	1	in	in	ADP
ejpam-7047	72	2	case	case	NOUN
ejpam-7047	72	3	there	there	PRON
ejpam-7047	72	4	is	be	VERB
ejpam-7047	72	5	no	no	DET
ejpam-7047	72	6	chance	chance	NOUN
ejpam-7047	72	7	for	for	ADP
ejpam-7047	72	8	confusion	confusion	NOUN
ejpam-7047	72	9	,	,	PUNCT
ejpam-7047	72	10	a⋆(i	a⋆(i	NOUN
ejpam-7047	72	11	)	)	PUNCT
ejpam-7047	72	12	is	be	AUX
ejpam-7047	72	13	simply	simply	ADV
ejpam-7047	72	14	written	write	VERB
ejpam-7047	72	15	as	as	ADP
ejpam-7047	72	16	a⋆.	a⋆.	NOUN
ejpam-7047	72	17	in	in	ADP
ejpam-7047	72	18	[	[	X
ejpam-7047	72	19	16	16	NUM
ejpam-7047	72	20	]	]	PUNCT
ejpam-7047	72	21	,	,	PUNCT
ejpam-7047	72	22	a⋆	a⋆	ADV
ejpam-7047	72	23	is	be	AUX
ejpam-7047	72	24	called	call	VERB
ejpam-7047	72	25	the	the	DET
ejpam-7047	72	26	local	local	ADJ
ejpam-7047	72	27	function	function	NOUN
ejpam-7047	72	28	of	of	ADP
ejpam-7047	72	29	a	a	PRON
ejpam-7047	72	30	with	with	ADP
ejpam-7047	72	31	respect	respect	NOUN
ejpam-7047	72	32	to	to	ADP
ejpam-7047	72	33	i	i	PRON
ejpam-7047	72	34	and	and	CCONJ
ejpam-7047	72	35	τ	τ	PROPN
ejpam-7047	72	36	and	and	CCONJ
ejpam-7047	72	37	cl⋆(a	cl⋆(a	NUM
ejpam-7047	72	38	)	)	PUNCT
ejpam-7047	72	39	=	=	PUNCT
ejpam-7047	72	40	a⋆	a⋆	ADP
ejpam-7047	72	41	∪	∪	ADP
ejpam-7047	72	42	a	a	DET
ejpam-7047	72	43	defines	define	NOUN
ejpam-7047	72	44	a	a	DET
ejpam-7047	72	45	kuratowski	kuratowski	ADJ
ejpam-7047	72	46	closure	closure	NOUN
ejpam-7047	72	47	operator	operator	NOUN
ejpam-7047	72	48	for	for	ADP
ejpam-7047	72	49	a	a	DET
ejpam-7047	72	50	topology	topology	NOUN
ejpam-7047	72	51	τ⋆(i	τ⋆(i	NOUN
ejpam-7047	72	52	)	)	PUNCT
ejpam-7047	72	53	finer	fine	ADJ
ejpam-7047	72	54	than	than	ADP
ejpam-7047	72	55	τ	τ	PROPN
ejpam-7047	72	56	.	.	PUNCT
ejpam-7047	73	1	a	a	DET
ejpam-7047	73	2	subset	subset	NOUN
ejpam-7047	73	3	a	a	PRON
ejpam-7047	73	4	is	be	AUX
ejpam-7047	73	5	said	say	VERB
ejpam-7047	73	6	to	to	PART
ejpam-7047	73	7	be	be	AUX
ejpam-7047	73	8	⋆-closed	⋆-close	VERB
ejpam-7047	73	9	[	[	X
ejpam-7047	73	10	17	17	NUM
ejpam-7047	73	11	]	]	X
ejpam-7047	73	12	if	if	SCONJ
ejpam-7047	73	13	a⋆	a⋆	ADJ
ejpam-7047	73	14	⊆	⊆	NUM
ejpam-7047	73	15	a.	a.	NOUN
ejpam-7047	73	16	the	the	DET
ejpam-7047	73	17	interior	interior	NOUN
ejpam-7047	73	18	of	of	ADP
ejpam-7047	73	19	a	a	DET
ejpam-7047	73	20	subset	subset	NOUN
ejpam-7047	73	21	a	a	DET
ejpam-7047	73	22	in	in	ADP
ejpam-7047	73	23	(	(	PUNCT
ejpam-7047	73	24	x	x	X
ejpam-7047	73	25	,	,	PUNCT
ejpam-7047	73	26	τ⋆(i	τ⋆(i	NOUN
ejpam-7047	73	27	)	)	PUNCT
ejpam-7047	73	28	)	)	PUNCT
ejpam-7047	73	29	is	be	AUX
ejpam-7047	73	30	denoted	denote	VERB
ejpam-7047	73	31	by	by	ADP
ejpam-7047	73	32	int⋆(a	int⋆(a	NOUN
ejpam-7047	73	33	)	)	PUNCT
ejpam-7047	73	34	.	.	PUNCT
ejpam-7047	74	1	a	a	DET
ejpam-7047	74	2	subset	subset	NOUN
ejpam-7047	74	3	a	a	PRON
ejpam-7047	74	4	of	of	ADP
ejpam-7047	74	5	an	an	DET
ejpam-7047	74	6	ideal	ideal	ADJ
ejpam-7047	74	7	topological	topological	ADJ
ejpam-7047	74	8	space	space	NOUN
ejpam-7047	74	9	(	(	PUNCT
ejpam-7047	74	10	x	x	X
ejpam-7047	74	11	,	,	PUNCT
ejpam-7047	74	12	τ	τ	PROPN
ejpam-7047	74	13	,	,	PUNCT
ejpam-7047	74	14	i	i	PROPN
ejpam-7047	74	15	)	)	PUNCT
ejpam-7047	74	16	is	be	AUX
ejpam-7047	74	17	said	say	VERB
ejpam-7047	74	18	to	to	PART
ejpam-7047	74	19	be	be	AUX
ejpam-7047	74	20	r	r	NOUN
ejpam-7047	74	21	-	-	PUNCT
ejpam-7047	74	22	i	i	PRON
ejpam-7047	74	23	⋆-open	⋆-open	VERB
ejpam-7047	75	1	[	[	X
ejpam-7047	75	2	4	4	NUM
ejpam-7047	75	3	]	]	PUNCT
ejpam-7047	75	4	(	(	PUNCT
ejpam-7047	75	5	resp	resp	NOUN
ejpam-7047	75	6	.	.	PUNCT
ejpam-7047	76	1	i	i	PRON
ejpam-7047	76	2	⋆-preopen	⋆-preopen	VERB
ejpam-7047	77	1	[	[	X
ejpam-7047	77	2	4	4	NUM
ejpam-7047	77	3	]	]	PUNCT
ejpam-7047	77	4	,	,	PUNCT
ejpam-7047	77	5	semi	semi	ADJ
ejpam-7047	77	6	-	-	VERB
ejpam-7047	77	7	i	i	PRON
ejpam-7047	77	8	⋆-open	⋆-open	VERB
ejpam-7047	78	1	[	[	X
ejpam-7047	78	2	18	18	NUM
ejpam-7047	78	3	]	]	PUNCT
ejpam-7047	78	4	,	,	PUNCT
ejpam-7047	78	5	semi	semi	ADJ
ejpam-7047	78	6	-	-	VERB
ejpam-7047	78	7	i	i	PRON
ejpam-7047	78	8	⋆-preopen	⋆-preopen	VERB
ejpam-7047	79	1	[	[	X
ejpam-7047	79	2	18	18	NUM
ejpam-7047	79	3	]	]	SYM
ejpam-7047	79	4	)	)	PUNCT
ejpam-7047	79	5	if	if	SCONJ
ejpam-7047	79	6	a	a	DET
ejpam-7047	79	7	=	=	PUNCT
ejpam-7047	79	8	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7047	79	9	)	)	PUNCT
ejpam-7047	79	10	)	)	PUNCT
ejpam-7047	79	11	(	(	PUNCT
ejpam-7047	79	12	resp	resp	NOUN
ejpam-7047	79	13	.	.	PUNCT
ejpam-7047	80	1	a	a	DET
ejpam-7047	80	2	⊆	⊆	NUM
ejpam-7047	80	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7047	80	4	)	)	PUNCT
ejpam-7047	80	5	)	)	PUNCT
ejpam-7047	80	6	,	,	PUNCT
ejpam-7047	80	7	a	a	DET
ejpam-7047	80	8	⊆	⊆	NUM
ejpam-7047	80	9	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7047	80	10	)	)	PUNCT
ejpam-7047	80	11	)	)	PUNCT
ejpam-7047	80	12	,	,	PUNCT
ejpam-7047	80	13	a	a	DET
ejpam-7047	80	14	⊆	⊆	NUM
ejpam-7047	80	15	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7047	80	16	)	)	PUNCT
ejpam-7047	80	17	)	)	PUNCT
ejpam-7047	80	18	)	)	PUNCT
ejpam-7047	80	19	)	)	PUNCT
ejpam-7047	80	20	.	.	PUNCT
ejpam-7047	81	1	the	the	DET
ejpam-7047	81	2	complement	complement	NOUN
ejpam-7047	81	3	of	of	ADP
ejpam-7047	81	4	a	a	DET
ejpam-7047	81	5	r	r	NOUN
ejpam-7047	81	6	-	-	PUNCT
ejpam-7047	81	7	i	i	PRON
ejpam-7047	81	8	⋆-open	⋆-open	VERB
ejpam-7047	81	9	(	(	PUNCT
ejpam-7047	81	10	resp	resp	NOUN
ejpam-7047	81	11	.	.	PUNCT
ejpam-7047	82	1	i	i	PRON
ejpam-7047	82	2	⋆-preopen	⋆-preopen	VERB
ejpam-7047	82	3	,	,	PUNCT
ejpam-7047	82	4	semi	semi	ADJ
ejpam-7047	82	5	-	-	ADJ
ejpam-7047	82	6	i	i	PRON
ejpam-7047	82	7	⋆-open	⋆-open	VERB
ejpam-7047	82	8	,	,	PUNCT
ejpam-7047	82	9	semi	semi	ADJ
ejpam-7047	82	10	-	-	VERB
ejpam-7047	82	11	i	i	PRON
ejpam-7047	82	12	⋆-preopen	⋆-preopen	ADV
ejpam-7047	82	13	)	)	PUNCT
ejpam-7047	83	1	set	set	NOUN
ejpam-7047	83	2	is	be	AUX
ejpam-7047	83	3	said	say	VERB
ejpam-7047	83	4	to	to	PART
ejpam-7047	83	5	be	be	AUX
ejpam-7047	83	6	r	r	NOUN
ejpam-7047	83	7	-	-	PUNCT
ejpam-7047	83	8	i	i	PRON
ejpam-7047	83	9	⋆-closed	⋆-close	VERB
ejpam-7047	83	10	(	(	PUNCT
ejpam-7047	83	11	resp	resp	NOUN
ejpam-7047	83	12	.	.	PUNCT
ejpam-7047	84	1	i	i	PRON
ejpam-7047	84	2	⋆-preclosed	⋆-preclose	VERB
ejpam-7047	84	3	,	,	PUNCT
ejpam-7047	84	4	semi	semi	ADV
ejpam-7047	84	5	-	-	VERB
ejpam-7047	84	6	i	i	PRON
ejpam-7047	84	7	⋆-closed	⋆-close	VERB
ejpam-7047	84	8	,	,	PUNCT
ejpam-7047	84	9	semi	semi	ADJ
ejpam-7047	84	10	-	-	VERB
ejpam-7047	84	11	i	i	PRON
ejpam-7047	84	12	⋆-preclosed	⋆-preclose	VERB
ejpam-7047	84	13	)	)	PUNCT
ejpam-7047	84	14	.	.	PUNCT
ejpam-7047	85	1	for	for	ADP
ejpam-7047	85	2	a	a	DET
ejpam-7047	85	3	subset	subset	NOUN
ejpam-7047	85	4	a	a	PRON
ejpam-7047	85	5	of	of	ADP
ejpam-7047	85	6	an	an	DET
ejpam-7047	85	7	ideal	ideal	ADJ
ejpam-7047	85	8	topological	topological	ADJ
ejpam-7047	85	9	space	space	NOUN
ejpam-7047	85	10	(	(	PUNCT
ejpam-7047	85	11	x	x	X
ejpam-7047	85	12	,	,	PUNCT
ejpam-7047	85	13	τ	τ	PROPN
ejpam-7047	85	14	,	,	PUNCT
ejpam-7047	85	15	i	i	NOUN
ejpam-7047	85	16	)	)	PUNCT
ejpam-7047	85	17	,	,	PUNCT
ejpam-7047	85	18	the	the	DET
ejpam-7047	85	19	intersection	intersection	NOUN
ejpam-7047	85	20	of	of	ADP
ejpam-7047	85	21	all	all	PRON
ejpam-7047	85	22	semi	semi	NOUN
ejpam-7047	85	23	-	-	ADJ
ejpam-7047	85	24	i	i	PRON
ejpam-7047	85	25	⋆-closed	⋆-close	VERB
ejpam-7047	85	26	sets	set	NOUN
ejpam-7047	85	27	containing	contain	VERB
ejpam-7047	85	28	a	a	PRON
ejpam-7047	85	29	is	be	AUX
ejpam-7047	85	30	called	call	VERB
ejpam-7047	85	31	the	the	DET
ejpam-7047	85	32	semi	semi	NOUN
ejpam-7047	85	33	-	-	ADJ
ejpam-7047	85	34	i	i	PRON
ejpam-7047	85	35	⋆-closure	⋆-closure	NOUN
ejpam-7047	86	1	[	[	X
ejpam-7047	86	2	18	18	NUM
ejpam-7047	86	3	]	]	PUNCT
ejpam-7047	86	4	of	of	ADP
ejpam-7047	86	5	a	a	PRON
ejpam-7047	86	6	and	and	CCONJ
ejpam-7047	86	7	is	be	AUX
ejpam-7047	86	8	denoted	denote	VERB
ejpam-7047	86	9	by	by	ADP
ejpam-7047	86	10	scl⋆(a	scl⋆(a	NOUN
ejpam-7047	86	11	)	)	PUNCT
ejpam-7047	86	12	(	(	PUNCT
ejpam-7047	86	13	scli	scli	PROPN
ejpam-7047	86	14	⋆(a	⋆(a	PRON
ejpam-7047	86	15	)	)	PUNCT
ejpam-7047	87	1	[	[	X
ejpam-7047	87	2	18	18	NUM
ejpam-7047	87	3	]	]	NUM
ejpam-7047	87	4	)	)	PUNCT
ejpam-7047	87	5	.	.	PUNCT
ejpam-7047	88	1	the	the	DET
ejpam-7047	88	2	union	union	NOUN
ejpam-7047	88	3	of	of	ADP
ejpam-7047	88	4	all	all	PRON
ejpam-7047	88	5	semi	semi	ADJ
ejpam-7047	88	6	-	-	ADJ
ejpam-7047	88	7	i	i	PRON
ejpam-7047	88	8	⋆-open	⋆-open	ADJ
ejpam-7047	88	9	sets	set	NOUN
ejpam-7047	88	10	contained	contain	VERB
ejpam-7047	88	11	in	in	ADP
ejpam-7047	88	12	a	a	PRON
ejpam-7047	88	13	is	be	AUX
ejpam-7047	88	14	called	call	VERB
ejpam-7047	88	15	the	the	DET
ejpam-7047	88	16	semi	semi	NOUN
ejpam-7047	88	17	-	-	ADJ
ejpam-7047	88	18	i	i	PRON
ejpam-7047	88	19	⋆-interior	⋆-interior	PUNCT
ejpam-7047	89	1	[	[	X
ejpam-7047	89	2	18	18	NUM
ejpam-7047	89	3	]	]	PUNCT
ejpam-7047	89	4	of	of	ADP
ejpam-7047	89	5	a	a	PRON
ejpam-7047	89	6	and	and	CCONJ
ejpam-7047	89	7	is	be	AUX
ejpam-7047	89	8	denoted	denote	VERB
ejpam-7047	89	9	by	by	ADP
ejpam-7047	89	10	sint⋆(a	sint⋆(a	PROPN
ejpam-7047	89	11	)	)	PUNCT
ejpam-7047	90	1	(	(	PUNCT
ejpam-7047	90	2	sinti	sinti	PROPN
ejpam-7047	90	3	⋆(a	⋆(a	NOUN
ejpam-7047	90	4	)	)	PUNCT
ejpam-7047	91	1	[	[	X
ejpam-7047	91	2	18	18	NUM
ejpam-7047	91	3	]	]	NUM
ejpam-7047	91	4	)	)	PUNCT
ejpam-7047	91	5	.	.	PUNCT
ejpam-7047	92	1	lemma	lemma	PROPN
ejpam-7047	92	2	3	3	X
ejpam-7047	92	3	.	.	PUNCT
ejpam-7047	93	1	[	[	X
ejpam-7047	93	2	18	18	NUM
ejpam-7047	93	3	]	]	PUNCT
ejpam-7047	93	4	for	for	ADP
ejpam-7047	93	5	a	a	DET
ejpam-7047	93	6	subset	subset	NOUN
ejpam-7047	93	7	a	a	PRON
ejpam-7047	93	8	of	of	ADP
ejpam-7047	93	9	an	an	DET
ejpam-7047	93	10	ideal	ideal	ADJ
ejpam-7047	93	11	topological	topological	ADJ
ejpam-7047	93	12	space	space	NOUN
ejpam-7047	93	13	(	(	PUNCT
ejpam-7047	93	14	x	x	X
ejpam-7047	93	15	,	,	PUNCT
ejpam-7047	93	16	τ	τ	PROPN
ejpam-7047	93	17	,	,	PUNCT
ejpam-7047	93	18	i	i	NOUN
ejpam-7047	93	19	)	)	PUNCT
ejpam-7047	93	20	,	,	PUNCT
ejpam-7047	93	21	the	the	DET
ejpam-7047	93	22	following	follow	VERB
ejpam-7047	93	23	properties	property	NOUN
ejpam-7047	93	24	hold	hold	VERB
ejpam-7047	93	25	:	:	PUNCT
ejpam-7047	93	26	(	(	PUNCT
ejpam-7047	93	27	1	1	X
ejpam-7047	93	28	)	)	PUNCT
ejpam-7047	93	29	scl⋆(a	scl⋆(a	NUM
ejpam-7047	93	30	)	)	PUNCT
ejpam-7047	93	31	=	=	PUNCT
ejpam-7047	93	32	a	a	DET
ejpam-7047	93	33	∪	∪	ADJ
ejpam-7047	93	34	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7047	93	35	)	)	PUNCT
ejpam-7047	93	36	)	)	PUNCT
ejpam-7047	93	37	.	.	PUNCT
ejpam-7047	94	1	(	(	PUNCT
ejpam-7047	94	2	2	2	X
ejpam-7047	94	3	)	)	PUNCT
ejpam-7047	94	4	sint⋆(a	sint⋆(a	PROPN
ejpam-7047	94	5	)	)	PUNCT
ejpam-7047	95	1	=	=	PUNCT
ejpam-7047	95	2	a	a	DET
ejpam-7047	95	3	∩	∩	ADJ
ejpam-7047	95	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7047	95	5	)	)	PUNCT
ejpam-7047	95	6	)	)	PUNCT
ejpam-7047	95	7	.	.	PUNCT
ejpam-7047	96	1	n.	n.	PROPN
ejpam-7047	96	2	viriyapong	viriyapong	PROPN
ejpam-7047	96	3	,	,	PUNCT
ejpam-7047	96	4	a.	a.	PROPN
ejpam-7047	96	5	sama	sama	PROPN
ejpam-7047	96	6	-	-	PUNCT
ejpam-7047	96	7	ae	ae	PROPN
ejpam-7047	96	8	,	,	PUNCT
ejpam-7047	96	9	c.	c.	PROPN
ejpam-7047	96	10	boonpok	boonpok	PROPN
ejpam-7047	96	11	/	/	SYM
ejpam-7047	96	12	eur	eur	PROPN
ejpam-7047	96	13	.	.	PUNCT
ejpam-7047	97	1	j.	j.	PROPN
ejpam-7047	97	2	pure	pure	PROPN
ejpam-7047	97	3	appl	appl	PROPN
ejpam-7047	97	4	.	.	PROPN
ejpam-7047	97	5	math	math	PROPN
ejpam-7047	97	6	,	,	PUNCT
ejpam-7047	97	7	18	18	NUM
ejpam-7047	97	8	(	(	PUNCT
ejpam-7047	97	9	4	4	NUM
ejpam-7047	97	10	)	)	PUNCT
ejpam-7047	97	11	(	(	PUNCT
ejpam-7047	97	12	2025	2025	NUM
ejpam-7047	97	13	)	)	PUNCT
ejpam-7047	97	14	,	,	PUNCT
ejpam-7047	97	15	7047	7047	NUM
ejpam-7047	97	16	4	4	NUM
ejpam-7047	97	17	of	of	ADP
ejpam-7047	97	18	11	11	NUM
ejpam-7047	97	19	a	a	DET
ejpam-7047	97	20	subset	subset	NOUN
ejpam-7047	97	21	a	a	PRON
ejpam-7047	97	22	of	of	ADP
ejpam-7047	97	23	an	an	DET
ejpam-7047	97	24	ideal	ideal	ADJ
ejpam-7047	97	25	topological	topological	ADJ
ejpam-7047	97	26	space	space	NOUN
ejpam-7047	97	27	(	(	PUNCT
ejpam-7047	97	28	x	x	X
ejpam-7047	97	29	,	,	PUNCT
ejpam-7047	97	30	τ	τ	PROPN
ejpam-7047	97	31	,	,	PUNCT
ejpam-7047	97	32	i	i	PROPN
ejpam-7047	97	33	)	)	PUNCT
ejpam-7047	97	34	is	be	AUX
ejpam-7047	97	35	called	call	VERB
ejpam-7047	97	36	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	97	37	-	-	PUNCT
ejpam-7047	97	38	open	open	ADJ
ejpam-7047	97	39	[	[	X
ejpam-7047	97	40	19	19	NUM
ejpam-7047	97	41	]	]	PUNCT
ejpam-7047	97	42	(	(	PUNCT
ejpam-7047	97	43	α	α	X
ejpam-7047	97	44	-	-	ADJ
ejpam-7047	97	45	i	i	PRON
ejpam-7047	97	46	⋆-open	⋆-open	VERB
ejpam-7047	98	1	[	[	X
ejpam-7047	98	2	20	20	NUM
ejpam-7047	98	3	]	]	SYM
ejpam-7047	98	4	)	)	PUNCT
ejpam-7047	98	5	if	if	SCONJ
ejpam-7047	98	6	a	a	DET
ejpam-7047	98	7	⊆	⊆	NUM
ejpam-7047	98	8	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7047	98	9	)	)	PUNCT
ejpam-7047	98	10	)	)	PUNCT
ejpam-7047	98	11	)	)	PUNCT
ejpam-7047	98	12	.	.	PUNCT
ejpam-7047	99	1	the	the	DET
ejpam-7047	99	2	complement	complement	NOUN
ejpam-7047	99	3	of	of	ADP
ejpam-7047	99	4	a	a	DET
ejpam-7047	99	5	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	99	6	-	-	PUNCT
ejpam-7047	99	7	open	open	ADJ
ejpam-7047	99	8	set	set	NOUN
ejpam-7047	99	9	is	be	AUX
ejpam-7047	99	10	called	call	VERB
ejpam-7047	99	11	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	99	12	-	-	PUNCT
ejpam-7047	99	13	closed	closed	ADJ
ejpam-7047	99	14	.	.	PUNCT
ejpam-7047	100	1	lemma	lemma	PROPN
ejpam-7047	100	2	4	4	NUM
ejpam-7047	100	3	.	.	PUNCT
ejpam-7047	101	1	[	[	X
ejpam-7047	101	2	20	20	NUM
ejpam-7047	101	3	]	]	PUNCT
ejpam-7047	101	4	for	for	ADP
ejpam-7047	101	5	a	a	DET
ejpam-7047	101	6	subset	subset	NOUN
ejpam-7047	101	7	a	a	PRON
ejpam-7047	101	8	of	of	ADP
ejpam-7047	101	9	an	an	DET
ejpam-7047	101	10	ideal	ideal	ADJ
ejpam-7047	101	11	topological	topological	ADJ
ejpam-7047	101	12	space	space	NOUN
ejpam-7047	101	13	(	(	PUNCT
ejpam-7047	101	14	x	x	X
ejpam-7047	101	15	,	,	PUNCT
ejpam-7047	101	16	τ	τ	PROPN
ejpam-7047	101	17	,	,	PUNCT
ejpam-7047	101	18	i	i	NOUN
ejpam-7047	101	19	)	)	PUNCT
ejpam-7047	101	20	,	,	PUNCT
ejpam-7047	101	21	the	the	DET
ejpam-7047	101	22	following	follow	VERB
ejpam-7047	101	23	properties	property	NOUN
ejpam-7047	101	24	are	be	AUX
ejpam-7047	101	25	equivalent	equivalent	ADJ
ejpam-7047	101	26	:	:	PUNCT
ejpam-7047	101	27	(	(	PUNCT
ejpam-7047	101	28	1	1	X
ejpam-7047	101	29	)	)	PUNCT
ejpam-7047	101	30	a	a	PRON
ejpam-7047	101	31	is	be	AUX
ejpam-7047	101	32	α	α	X
ejpam-7047	101	33	-	-	ADJ
ejpam-7047	101	34	i	i	PRON
ejpam-7047	101	35	⋆-open	⋆-open	VERB
ejpam-7047	101	36	in	in	ADP
ejpam-7047	101	37	x.	x.	PROPN
ejpam-7047	101	38	(	(	PUNCT
ejpam-7047	101	39	2	2	NUM
ejpam-7047	101	40	)	)	PUNCT
ejpam-7047	101	41	g	g	ADP
ejpam-7047	101	42	⊆	⊆	NUM
ejpam-7047	101	43	a	a	DET
ejpam-7047	101	44	⊆	⊆	NUM
ejpam-7047	101	45	int⋆(cl⋆(g	int⋆(cl⋆(g	NOUN
ejpam-7047	101	46	)	)	PUNCT
ejpam-7047	101	47	)	)	PUNCT
ejpam-7047	101	48	for	for	ADP
ejpam-7047	101	49	some	some	DET
ejpam-7047	101	50	⋆-open	⋆-open	ADJ
ejpam-7047	101	51	set	set	VERB
ejpam-7047	101	52	g.	g.	PROPN
ejpam-7047	101	53	(	(	PUNCT
ejpam-7047	101	54	3	3	NUM
ejpam-7047	101	55	)	)	PUNCT
ejpam-7047	101	56	g	g	ADP
ejpam-7047	101	57	⊆	⊆	NUM
ejpam-7047	101	58	a	a	DET
ejpam-7047	101	59	⊆	⊆	NUM
ejpam-7047	101	60	scl⋆(g	scl⋆(g	NOUN
ejpam-7047	101	61	)	)	PUNCT
ejpam-7047	101	62	for	for	ADP
ejpam-7047	101	63	some	some	DET
ejpam-7047	101	64	⋆-open	⋆-open	ADJ
ejpam-7047	101	65	set	set	NOUN
ejpam-7047	101	66	g.	g.	PROPN
ejpam-7047	101	67	(	(	PUNCT
ejpam-7047	101	68	4	4	X
ejpam-7047	101	69	)	)	PUNCT
ejpam-7047	101	70	a	a	DET
ejpam-7047	101	71	⊆	⊆	NUM
ejpam-7047	101	72	scl⋆(int⋆(a	scl⋆(int⋆(a	NOUN
ejpam-7047	101	73	)	)	PUNCT
ejpam-7047	101	74	)	)	PUNCT
ejpam-7047	101	75	.	.	PUNCT
ejpam-7047	102	1	for	for	ADP
ejpam-7047	102	2	a	a	DET
ejpam-7047	102	3	subset	subset	NOUN
ejpam-7047	102	4	a	a	PRON
ejpam-7047	102	5	of	of	ADP
ejpam-7047	102	6	an	an	DET
ejpam-7047	102	7	ideal	ideal	ADJ
ejpam-7047	102	8	topological	topological	ADJ
ejpam-7047	102	9	space	space	NOUN
ejpam-7047	102	10	(	(	PUNCT
ejpam-7047	102	11	x	x	X
ejpam-7047	102	12	,	,	PUNCT
ejpam-7047	102	13	τ	τ	PROPN
ejpam-7047	102	14	,	,	PUNCT
ejpam-7047	102	15	i	i	NOUN
ejpam-7047	102	16	)	)	PUNCT
ejpam-7047	102	17	,	,	PUNCT
ejpam-7047	102	18	the	the	DET
ejpam-7047	102	19	intersection	intersection	NOUN
ejpam-7047	102	20	of	of	ADP
ejpam-7047	102	21	all	all	DET
ejpam-7047	102	22	α	α	NOUN
ejpam-7047	102	23	-	-	PUNCT
ejpam-7047	102	24	i	i	PRON
ejpam-7047	102	25	⋆closed	⋆close	VERB
ejpam-7047	102	26	sets	set	NOUN
ejpam-7047	102	27	containing	contain	VERB
ejpam-7047	102	28	a	a	PRON
ejpam-7047	102	29	is	be	AUX
ejpam-7047	102	30	called	call	VERB
ejpam-7047	102	31	the	the	DET
ejpam-7047	102	32	α	α	NOUN
ejpam-7047	102	33	-	-	NOUN
ejpam-7047	102	34	i	i	PRON
ejpam-7047	102	35	⋆-closure	⋆-closure	NOUN
ejpam-7047	103	1	[	[	X
ejpam-7047	103	2	20	20	NUM
ejpam-7047	103	3	]	]	PUNCT
ejpam-7047	103	4	of	of	ADP
ejpam-7047	103	5	a	a	PRON
ejpam-7047	103	6	and	and	CCONJ
ejpam-7047	103	7	is	be	AUX
ejpam-7047	103	8	denoted	denote	VERB
ejpam-7047	103	9	by	by	ADP
ejpam-7047	103	10	αcl⋆(a	αcl⋆(a	NUM
ejpam-7047	103	11	)	)	PUNCT
ejpam-7047	103	12	(	(	PUNCT
ejpam-7047	103	13	αcli	αcli	X
ejpam-7047	103	14	⋆(a	⋆(a	AUX
ejpam-7047	103	15	)	)	PUNCT
ejpam-7047	104	1	[	[	X
ejpam-7047	104	2	20	20	NUM
ejpam-7047	104	3	]	]	NUM
ejpam-7047	104	4	)	)	PUNCT
ejpam-7047	104	5	.	.	PUNCT
ejpam-7047	105	1	the	the	DET
ejpam-7047	105	2	α	α	PROPN
ejpam-7047	105	3	-	-	PUNCT
ejpam-7047	105	4	i	i	PRON
ejpam-7047	105	5	⋆-interior	⋆-interior	PUNCT
ejpam-7047	105	6	[	[	X
ejpam-7047	105	7	20	20	NUM
ejpam-7047	105	8	]	]	PUNCT
ejpam-7047	105	9	of	of	ADP
ejpam-7047	105	10	a	a	PRON
ejpam-7047	105	11	is	be	AUX
ejpam-7047	105	12	defined	define	VERB
ejpam-7047	105	13	by	by	ADP
ejpam-7047	105	14	the	the	DET
ejpam-7047	105	15	union	union	NOUN
ejpam-7047	105	16	of	of	ADP
ejpam-7047	105	17	all	all	DET
ejpam-7047	105	18	α	α	PROPN
ejpam-7047	105	19	-	-	ADJ
ejpam-7047	105	20	i	i	PRON
ejpam-7047	105	21	⋆-open	⋆-open	ADJ
ejpam-7047	105	22	sets	set	NOUN
ejpam-7047	105	23	contained	contain	VERB
ejpam-7047	105	24	in	in	ADP
ejpam-7047	105	25	a	a	PRON
ejpam-7047	105	26	and	and	CCONJ
ejpam-7047	105	27	is	be	AUX
ejpam-7047	105	28	denoted	denote	VERB
ejpam-7047	105	29	by	by	ADP
ejpam-7047	105	30	αint⋆(a	αint⋆(a	PROPN
ejpam-7047	105	31	)	)	PUNCT
ejpam-7047	105	32	(	(	PUNCT
ejpam-7047	105	33	αinti	αinti	X
ejpam-7047	105	34	⋆(a	⋆(a	NOUN
ejpam-7047	105	35	)	)	PUNCT
ejpam-7047	106	1	[	[	X
ejpam-7047	106	2	20	20	NUM
ejpam-7047	106	3	]	]	NUM
ejpam-7047	106	4	)	)	PUNCT
ejpam-7047	106	5	.	.	PUNCT
ejpam-7047	107	1	lemma	lemma	PROPN
ejpam-7047	107	2	5	5	NUM
ejpam-7047	107	3	.	.	PUNCT
ejpam-7047	108	1	[	[	X
ejpam-7047	108	2	20	20	NUM
ejpam-7047	108	3	]	]	PUNCT
ejpam-7047	108	4	for	for	ADP
ejpam-7047	108	5	a	a	DET
ejpam-7047	108	6	subset	subset	NOUN
ejpam-7047	108	7	a	a	PRON
ejpam-7047	108	8	of	of	ADP
ejpam-7047	108	9	an	an	DET
ejpam-7047	108	10	ideal	ideal	ADJ
ejpam-7047	108	11	topological	topological	ADJ
ejpam-7047	108	12	space	space	NOUN
ejpam-7047	108	13	(	(	PUNCT
ejpam-7047	108	14	x	x	X
ejpam-7047	108	15	,	,	PUNCT
ejpam-7047	108	16	τ	τ	PROPN
ejpam-7047	108	17	,	,	PUNCT
ejpam-7047	108	18	i	i	NOUN
ejpam-7047	108	19	)	)	PUNCT
ejpam-7047	108	20	,	,	PUNCT
ejpam-7047	108	21	the	the	DET
ejpam-7047	108	22	following	follow	VERB
ejpam-7047	108	23	properties	property	NOUN
ejpam-7047	108	24	hold	hold	VERB
ejpam-7047	108	25	:	:	PUNCT
ejpam-7047	108	26	(	(	PUNCT
ejpam-7047	108	27	1	1	X
ejpam-7047	108	28	)	)	PUNCT
ejpam-7047	108	29	a	a	PRON
ejpam-7047	108	30	is	be	AUX
ejpam-7047	108	31	α	α	X
ejpam-7047	108	32	-	-	PUNCT
ejpam-7047	108	33	i	i	PRON
ejpam-7047	108	34	⋆-closed	⋆-close	VERB
ejpam-7047	108	35	in	in	ADP
ejpam-7047	108	36	x	x	SYM
ejpam-7047	108	37	if	if	SCONJ
ejpam-7047	108	38	and	and	CCONJ
ejpam-7047	108	39	only	only	ADV
ejpam-7047	108	40	if	if	SCONJ
ejpam-7047	108	41	sint⋆(cl⋆(a	sint⋆(cl⋆(a	NOUN
ejpam-7047	108	42	)	)	PUNCT
ejpam-7047	108	43	)	)	PUNCT
ejpam-7047	109	1	⊆	⊆	NUM
ejpam-7047	109	2	a.	a.	NOUN
ejpam-7047	109	3	(	(	PUNCT
ejpam-7047	109	4	2	2	NUM
ejpam-7047	109	5	)	)	PUNCT
ejpam-7047	109	6	sint⋆(cl⋆(a	sint⋆(cl⋆(a	PROPN
ejpam-7047	109	7	)	)	PUNCT
ejpam-7047	109	8	)	)	PUNCT
ejpam-7047	109	9	=	=	PUNCT
ejpam-7047	109	10	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7047	109	11	)	)	PUNCT
ejpam-7047	109	12	)	)	PUNCT
ejpam-7047	109	13	)	)	PUNCT
ejpam-7047	109	14	.	.	PUNCT
ejpam-7047	110	1	(	(	PUNCT
ejpam-7047	110	2	3	3	X
ejpam-7047	110	3	)	)	PUNCT
ejpam-7047	110	4	αcl⋆(a	αcl⋆(a	NUM
ejpam-7047	110	5	)	)	PUNCT
ejpam-7047	111	1	=	=	PUNCT
ejpam-7047	111	2	a	a	DET
ejpam-7047	111	3	∪	∪	ADJ
ejpam-7047	111	4	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7047	111	5	)	)	PUNCT
ejpam-7047	111	6	)	)	PUNCT
ejpam-7047	111	7	)	)	PUNCT
ejpam-7047	111	8	.	.	PUNCT
ejpam-7047	112	1	(	(	PUNCT
ejpam-7047	112	2	4	4	X
ejpam-7047	112	3	)	)	PUNCT
ejpam-7047	112	4	αint⋆(a	αint⋆(a	NOUN
ejpam-7047	112	5	)	)	PUNCT
ejpam-7047	113	1	=	=	SYM
ejpam-7047	113	2	a	a	DET
ejpam-7047	113	3	∩	∩	ADJ
ejpam-7047	113	4	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7047	113	5	)	)	PUNCT
ejpam-7047	113	6	)	)	PUNCT
ejpam-7047	113	7	)	)	PUNCT
ejpam-7047	113	8	.	.	PUNCT
ejpam-7047	114	1	by	by	ADP
ejpam-7047	114	2	a	a	DET
ejpam-7047	114	3	multifunction	multifunction	NOUN
ejpam-7047	114	4	f	f	NOUN
ejpam-7047	114	5	:	:	PUNCT
ejpam-7047	114	6	x	x	X
ejpam-7047	114	7	→	→	SYM
ejpam-7047	114	8	y	y	PROPN
ejpam-7047	114	9	,	,	PUNCT
ejpam-7047	114	10	we	we	PRON
ejpam-7047	114	11	mean	mean	VERB
ejpam-7047	114	12	a	a	DET
ejpam-7047	114	13	point	point	NOUN
ejpam-7047	114	14	-	-	PUNCT
ejpam-7047	114	15	to	to	ADP
ejpam-7047	114	16	-	-	PUNCT
ejpam-7047	114	17	set	set	VERB
ejpam-7047	114	18	correspondence	correspondence	NOUN
ejpam-7047	114	19	from	from	ADP
ejpam-7047	114	20	x	x	PUNCT
ejpam-7047	114	21	into	into	ADP
ejpam-7047	114	22	y	y	PROPN
ejpam-7047	114	23	,	,	PUNCT
ejpam-7047	114	24	and	and	CCONJ
ejpam-7047	114	25	we	we	PRON
ejpam-7047	114	26	always	always	ADV
ejpam-7047	114	27	assume	assume	VERB
ejpam-7047	114	28	that	that	SCONJ
ejpam-7047	114	29	f	f	PROPN
ejpam-7047	114	30	(	(	PUNCT
ejpam-7047	114	31	x	x	X
ejpam-7047	114	32	)	)	PUNCT
ejpam-7047	114	33	̸=	̸=	NOUN
ejpam-7047	114	34	∅	∅	NOUN
ejpam-7047	114	35	for	for	ADP
ejpam-7047	114	36	all	all	PRON
ejpam-7047	114	37	x	x	SYM
ejpam-7047	114	38	∈	∈	ADJ
ejpam-7047	114	39	x.	x.	NOUN
ejpam-7047	114	40	for	for	ADP
ejpam-7047	114	41	a	a	DET
ejpam-7047	114	42	multifunction	multifunction	NOUN
ejpam-7047	114	43	f	f	NOUN
ejpam-7047	114	44	:	:	PUNCT
ejpam-7047	114	45	x	x	X
ejpam-7047	114	46	→	→	SYM
ejpam-7047	114	47	y	y	PROPN
ejpam-7047	114	48	,	,	PUNCT
ejpam-7047	114	49	we	we	PRON
ejpam-7047	114	50	shall	shall	AUX
ejpam-7047	114	51	denote	denote	VERB
ejpam-7047	114	52	the	the	DET
ejpam-7047	114	53	upper	upper	ADJ
ejpam-7047	114	54	and	and	CCONJ
ejpam-7047	114	55	lower	low	ADJ
ejpam-7047	114	56	inverse	inverse	NOUN
ejpam-7047	114	57	of	of	ADP
ejpam-7047	114	58	a	a	DET
ejpam-7047	114	59	set	set	NOUN
ejpam-7047	114	60	b	b	PROPN
ejpam-7047	114	61	of	of	ADP
ejpam-7047	114	62	y	y	PROPN
ejpam-7047	114	63	by	by	ADP
ejpam-7047	114	64	f+(b	f+(b	NOUN
ejpam-7047	114	65	)	)	PUNCT
ejpam-7047	114	66	and	and	CCONJ
ejpam-7047	114	67	f−(b	f−(b	NOUN
ejpam-7047	114	68	)	)	PUNCT
ejpam-7047	114	69	,	,	PUNCT
ejpam-7047	114	70	respectively	respectively	ADV
ejpam-7047	114	71	,	,	PUNCT
ejpam-7047	114	72	that	that	ADV
ejpam-7047	114	73	is	is	ADV
ejpam-7047	114	74	,	,	PUNCT
ejpam-7047	114	75	f+(b	f+(b	NOUN
ejpam-7047	114	76	)	)	PUNCT
ejpam-7047	114	77	=	=	PRON
ejpam-7047	115	1	{	{	PUNCT
ejpam-7047	115	2	x	x	PUNCT
ejpam-7047	115	3	∈	∈	PROPN
ejpam-7047	115	4	x	x	INTJ
ejpam-7047	116	1	|	|	NOUN
ejpam-7047	116	2	f	f	X
ejpam-7047	116	3	(	(	PUNCT
ejpam-7047	116	4	x	x	NOUN
ejpam-7047	116	5	)	)	PUNCT
ejpam-7047	116	6	⊆	⊆	NUM
ejpam-7047	116	7	b	b	NOUN
ejpam-7047	116	8	}	}	PUNCT
ejpam-7047	116	9	and	and	CCONJ
ejpam-7047	116	10	f−(b	f−(b	PROPN
ejpam-7047	116	11	)	)	PUNCT
ejpam-7047	116	12	=	=	PRON
ejpam-7047	117	1	{	{	PUNCT
ejpam-7047	117	2	x	x	PUNCT
ejpam-7047	117	3	∈	∈	PROPN
ejpam-7047	117	4	x	x	INTJ
ejpam-7047	118	1	|	|	NOUN
ejpam-7047	118	2	f	f	X
ejpam-7047	118	3	(	(	PUNCT
ejpam-7047	118	4	x	x	NOUN
ejpam-7047	118	5	)	)	PUNCT
ejpam-7047	118	6	∩	∩	NOUN
ejpam-7047	118	7	b	b	PROPN
ejpam-7047	118	8	̸=	̸=	PROPN
ejpam-7047	118	9	∅	∅	NOUN
ejpam-7047	118	10	}	}	PUNCT
ejpam-7047	118	11	.	.	PUNCT
ejpam-7047	119	1	in	in	ADP
ejpam-7047	119	2	particular	particular	ADJ
ejpam-7047	119	3	,	,	PUNCT
ejpam-7047	119	4	f−(y	f−(y	NOUN
ejpam-7047	119	5	)	)	PUNCT
ejpam-7047	119	6	=	=	SYM
ejpam-7047	120	1	{	{	PUNCT
ejpam-7047	120	2	x	x	PUNCT
ejpam-7047	120	3	∈	∈	PROPN
ejpam-7047	120	4	x	x	INTJ
ejpam-7047	121	1	|	|	ADV
ejpam-7047	121	2	y	y	PROPN
ejpam-7047	121	3	∈	∈	PROPN
ejpam-7047	121	4	f	f	X
ejpam-7047	121	5	(	(	PUNCT
ejpam-7047	121	6	x	x	NOUN
ejpam-7047	121	7	)	)	PUNCT
ejpam-7047	121	8	}	}	PUNCT
ejpam-7047	121	9	for	for	ADP
ejpam-7047	121	10	each	each	DET
ejpam-7047	121	11	point	point	NOUN
ejpam-7047	121	12	y	y	PROPN
ejpam-7047	121	13	∈	∈	PROPN
ejpam-7047	121	14	y	y	PROPN
ejpam-7047	121	15	.	.	PUNCT
ejpam-7047	122	1	for	for	ADP
ejpam-7047	122	2	each	each	DET
ejpam-7047	122	3	a	a	DET
ejpam-7047	122	4	⊆	⊆	NUM
ejpam-7047	122	5	x	x	SYM
ejpam-7047	122	6	,	,	PUNCT
ejpam-7047	122	7	f	f	PROPN
ejpam-7047	122	8	(	(	PUNCT
ejpam-7047	122	9	a	a	NOUN
ejpam-7047	122	10	)	)	PUNCT
ejpam-7047	122	11	=	=	SYM
ejpam-7047	122	12	∪x∈af	∪x∈af	NOUN
ejpam-7047	122	13	(	(	PUNCT
ejpam-7047	122	14	x	x	NOUN
ejpam-7047	122	15	)	)	PUNCT
ejpam-7047	122	16	.	.	PUNCT
ejpam-7047	123	1	3	3	X
ejpam-7047	123	2	.	.	X
ejpam-7047	123	3	upper	upper	ADJ
ejpam-7047	123	4	and	and	CCONJ
ejpam-7047	123	5	lower	low	ADJ
ejpam-7047	123	6	weakly	weakly	ADJ
ejpam-7047	123	7	τ	τ	X
ejpam-7047	123	8	⋆α(σ1	⋆α(σ1	X
ejpam-7047	123	9	,	,	PUNCT
ejpam-7047	123	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	123	11	multifunctions	multifunction	NOUN
ejpam-7047	123	12	in	in	ADP
ejpam-7047	123	13	this	this	DET
ejpam-7047	123	14	section	section	NOUN
ejpam-7047	123	15	,	,	PUNCT
ejpam-7047	123	16	we	we	PRON
ejpam-7047	123	17	introduce	introduce	VERB
ejpam-7047	123	18	the	the	DET
ejpam-7047	123	19	notions	notion	NOUN
ejpam-7047	123	20	of	of	ADP
ejpam-7047	123	21	upper	upper	ADJ
ejpam-7047	123	22	weakly	weakly	ADJ
ejpam-7047	123	23	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	123	24	,	,	PUNCT
ejpam-7047	123	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	123	26	multifunctions	multifunction	NOUN
ejpam-7047	123	27	and	and	CCONJ
ejpam-7047	123	28	lower	low	ADJ
ejpam-7047	123	29	weakly	weakly	ADJ
ejpam-7047	123	30	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	123	31	,	,	PUNCT
ejpam-7047	123	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	123	33	multifunctions	multifunction	NOUN
ejpam-7047	123	34	.	.	PUNCT
ejpam-7047	124	1	moreover	moreover	ADV
ejpam-7047	124	2	,	,	PUNCT
ejpam-7047	124	3	several	several	ADJ
ejpam-7047	124	4	characterizations	characterization	NOUN
ejpam-7047	124	5	of	of	ADP
ejpam-7047	124	6	upper	upper	ADJ
ejpam-7047	124	7	weakly	weakly	ADJ
ejpam-7047	124	8	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	124	9	,	,	PUNCT
ejpam-7047	124	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	124	11	multifunctions	multifunction	NOUN
ejpam-7047	124	12	and	and	CCONJ
ejpam-7047	124	13	lower	low	ADJ
ejpam-7047	124	14	weakly	weakly	ADJ
ejpam-7047	124	15	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	124	16	,	,	PUNCT
ejpam-7047	124	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	124	18	multifunctions	multifunction	NOUN
ejpam-7047	124	19	discussed	discuss	VERB
ejpam-7047	124	20	.	.	PUNCT
ejpam-7047	125	1	definition	definition	NOUN
ejpam-7047	125	2	1	1	NUM
ejpam-7047	125	3	.	.	PUNCT
ejpam-7047	126	1	a	a	DET
ejpam-7047	126	2	multifunction	multifunction	NOUN
ejpam-7047	126	3	f	f	NOUN
ejpam-7047	126	4	:	:	PUNCT
ejpam-7047	126	5	(	(	PUNCT
ejpam-7047	126	6	x	x	X
ejpam-7047	126	7	,	,	PUNCT
ejpam-7047	126	8	τ	τ	PROPN
ejpam-7047	126	9	,	,	PUNCT
ejpam-7047	126	10	i	i	NOUN
ejpam-7047	126	11	)	)	PUNCT
ejpam-7047	126	12	→	→	PUNCT
ejpam-7047	126	13	(	(	PUNCT
ejpam-7047	126	14	y	y	PROPN
ejpam-7047	126	15	,	,	PUNCT
ejpam-7047	126	16	σ1	σ1	PROPN
ejpam-7047	126	17	,	,	PUNCT
ejpam-7047	126	18	σ2	σ2	PROPN
ejpam-7047	126	19	)	)	PUNCT
ejpam-7047	126	20	is	be	AUX
ejpam-7047	126	21	said	say	VERB
ejpam-7047	126	22	to	to	PART
ejpam-7047	126	23	be	be	AUX
ejpam-7047	126	24	upper	upper	ADJ
ejpam-7047	126	25	weakly	weakly	ADJ
ejpam-7047	126	26	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	126	27	,	,	PUNCT
ejpam-7047	126	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	126	29	at	at	ADP
ejpam-7047	126	30	a	a	DET
ejpam-7047	126	31	point	point	NOUN
ejpam-7047	126	32	x	x	PUNCT
ejpam-7047	126	33	of	of	ADP
ejpam-7047	126	34	x	x	PRON
ejpam-7047	126	35	if	if	SCONJ
ejpam-7047	126	36	for	for	ADP
ejpam-7047	126	37	each	each	DET
ejpam-7047	126	38	σ1σ2	σ1σ2	VERB
ejpam-7047	126	39	-	-	ADJ
ejpam-7047	126	40	open	open	ADJ
ejpam-7047	126	41	set	set	NOUN
ejpam-7047	126	42	v	v	NOUN
ejpam-7047	126	43	of	of	ADP
ejpam-7047	126	44	y	y	PRON
ejpam-7047	126	45	such	such	ADJ
ejpam-7047	126	46	that	that	SCONJ
ejpam-7047	126	47	f	f	PROPN
ejpam-7047	126	48	(	(	PUNCT
ejpam-7047	126	49	x	x	X
ejpam-7047	126	50	)	)	PUNCT
ejpam-7047	126	51	⊆	⊆	NUM
ejpam-7047	126	52	v	v	NOUN
ejpam-7047	126	53	,	,	PUNCT
ejpam-7047	126	54	there	there	PRON
ejpam-7047	126	55	exists	exist	VERB
ejpam-7047	126	56	a	a	DET
ejpam-7047	126	57	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	126	58	-	-	PUNCT
ejpam-7047	126	59	open	open	ADJ
ejpam-7047	126	60	set	set	NOUN
ejpam-7047	126	61	u	u	NOUN
ejpam-7047	126	62	of	of	ADP
ejpam-7047	126	63	x	x	PUNCT
ejpam-7047	126	64	containing	contain	VERB
ejpam-7047	126	65	x	x	PUNCT
ejpam-7047	126	66	such	such	ADJ
ejpam-7047	126	67	that	that	SCONJ
ejpam-7047	126	68	f	f	PROPN
ejpam-7047	126	69	(	(	PUNCT
ejpam-7047	126	70	u	u	NOUN
ejpam-7047	126	71	)	)	PUNCT
ejpam-7047	126	72	⊆	⊆	NUM
ejpam-7047	126	73	σ1σ2	σ1σ2	NOUN
ejpam-7047	126	74	-	-	NUM
ejpam-7047	126	75	cl(v	cl(v	NOUN
ejpam-7047	126	76	)	)	PUNCT
ejpam-7047	126	77	.	.	PUNCT
ejpam-7047	127	1	a	a	DET
ejpam-7047	127	2	multifunction	multifunction	NOUN
ejpam-7047	127	3	f	f	NOUN
ejpam-7047	127	4	:	:	PUNCT
ejpam-7047	127	5	(	(	PUNCT
ejpam-7047	127	6	x	x	X
ejpam-7047	127	7	,	,	PUNCT
ejpam-7047	127	8	τ	τ	PROPN
ejpam-7047	127	9	,	,	PUNCT
ejpam-7047	127	10	i	i	NOUN
ejpam-7047	127	11	)	)	PUNCT
ejpam-7047	127	12	→	→	PUNCT
ejpam-7047	127	13	(	(	PUNCT
ejpam-7047	127	14	y	y	PROPN
ejpam-7047	127	15	,	,	PUNCT
ejpam-7047	127	16	σ1	σ1	PROPN
ejpam-7047	127	17	,	,	PUNCT
ejpam-7047	127	18	σ2	σ2	PROPN
ejpam-7047	127	19	)	)	PUNCT
ejpam-7047	127	20	is	be	AUX
ejpam-7047	127	21	said	say	VERB
ejpam-7047	127	22	to	to	PART
ejpam-7047	127	23	be	be	AUX
ejpam-7047	127	24	upper	upper	ADJ
ejpam-7047	127	25	weakly	weakly	ADJ
ejpam-7047	127	26	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	127	27	,	,	PUNCT
ejpam-7047	127	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	127	29	if	if	SCONJ
ejpam-7047	127	30	f	f	PROPN
ejpam-7047	127	31	is	be	AUX
ejpam-7047	127	32	upper	upper	ADJ
ejpam-7047	127	33	weakly	weakly	ADJ
ejpam-7047	127	34	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	127	35	,	,	PUNCT
ejpam-7047	127	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	127	37	at	at	ADP
ejpam-7047	127	38	each	each	DET
ejpam-7047	127	39	point	point	NOUN
ejpam-7047	127	40	of	of	ADP
ejpam-7047	127	41	x.	x.	PROPN
ejpam-7047	127	42	n.	n.	PROPN
ejpam-7047	127	43	viriyapong	viriyapong	PROPN
ejpam-7047	127	44	,	,	PUNCT
ejpam-7047	127	45	a.	a.	PROPN
ejpam-7047	127	46	sama	sama	PROPN
ejpam-7047	127	47	-	-	PUNCT
ejpam-7047	127	48	ae	ae	PROPN
ejpam-7047	127	49	,	,	PUNCT
ejpam-7047	127	50	c.	c.	PROPN
ejpam-7047	127	51	boonpok	boonpok	PROPN
ejpam-7047	127	52	/	/	SYM
ejpam-7047	127	53	eur	eur	PROPN
ejpam-7047	127	54	.	.	PUNCT
ejpam-7047	128	1	j.	j.	PROPN
ejpam-7047	128	2	pure	pure	PROPN
ejpam-7047	128	3	appl	appl	PROPN
ejpam-7047	128	4	.	.	PROPN
ejpam-7047	128	5	math	math	PROPN
ejpam-7047	128	6	,	,	PUNCT
ejpam-7047	128	7	18	18	NUM
ejpam-7047	128	8	(	(	PUNCT
ejpam-7047	128	9	4	4	NUM
ejpam-7047	128	10	)	)	PUNCT
ejpam-7047	128	11	(	(	PUNCT
ejpam-7047	128	12	2025	2025	NUM
ejpam-7047	128	13	)	)	PUNCT
ejpam-7047	128	14	,	,	PUNCT
ejpam-7047	128	15	7047	7047	NUM
ejpam-7047	128	16	5	5	NUM
ejpam-7047	128	17	of	of	ADP
ejpam-7047	128	18	11	11	NUM
ejpam-7047	128	19	theorem	theorem	NOUN
ejpam-7047	128	20	1	1	NUM
ejpam-7047	128	21	.	.	X
ejpam-7047	128	22	for	for	ADP
ejpam-7047	128	23	a	a	DET
ejpam-7047	128	24	multifunction	multifunction	NOUN
ejpam-7047	128	25	f	f	NOUN
ejpam-7047	128	26	:	:	PUNCT
ejpam-7047	128	27	(	(	PUNCT
ejpam-7047	128	28	x	x	X
ejpam-7047	128	29	,	,	PUNCT
ejpam-7047	128	30	τ	τ	PROPN
ejpam-7047	128	31	,	,	PUNCT
ejpam-7047	128	32	i	i	NOUN
ejpam-7047	128	33	)	)	PUNCT
ejpam-7047	128	34	→	→	PUNCT
ejpam-7047	128	35	(	(	PUNCT
ejpam-7047	128	36	y	y	PROPN
ejpam-7047	128	37	,	,	PUNCT
ejpam-7047	128	38	σ1	σ1	PROPN
ejpam-7047	128	39	,	,	PUNCT
ejpam-7047	128	40	σ2	σ2	NOUN
ejpam-7047	128	41	)	)	PUNCT
ejpam-7047	128	42	,	,	PUNCT
ejpam-7047	128	43	the	the	DET
ejpam-7047	128	44	following	follow	VERB
ejpam-7047	128	45	properties	property	NOUN
ejpam-7047	128	46	are	be	AUX
ejpam-7047	128	47	equivalent	equivalent	ADJ
ejpam-7047	128	48	:	:	PUNCT
ejpam-7047	128	49	(	(	PUNCT
ejpam-7047	128	50	1	1	X
ejpam-7047	128	51	)	)	PUNCT
ejpam-7047	128	52	f	f	PROPN
ejpam-7047	128	53	is	be	AUX
ejpam-7047	128	54	upper	upper	ADJ
ejpam-7047	128	55	weakly	weakly	ADJ
ejpam-7047	128	56	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	128	57	,	,	PUNCT
ejpam-7047	128	58	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	128	59	at	at	ADP
ejpam-7047	128	60	x	x	X
ejpam-7047	128	61	∈	∈	PROPN
ejpam-7047	128	62	x	x	X
ejpam-7047	128	63	;	;	PUNCT
ejpam-7047	128	64	(	(	PUNCT
ejpam-7047	128	65	2	2	X
ejpam-7047	128	66	)	)	PUNCT
ejpam-7047	128	67	x	x	SYM
ejpam-7047	128	68	∈	∈	PROPN
ejpam-7047	128	69	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	128	70	-	-	PUNCT
ejpam-7047	128	71	cl(v	cl(v	NOUN
ejpam-7047	128	72	)	)	PUNCT
ejpam-7047	128	73	)	)	PUNCT
ejpam-7047	128	74	)	)	PUNCT
ejpam-7047	128	75	for	for	ADP
ejpam-7047	128	76	every	every	DET
ejpam-7047	128	77	σ1σ2	σ1σ2	NOUN
ejpam-7047	128	78	-	-	ADJ
ejpam-7047	128	79	open	open	ADJ
ejpam-7047	128	80	set	set	NOUN
ejpam-7047	128	81	v	v	NOUN
ejpam-7047	128	82	of	of	ADP
ejpam-7047	128	83	y	y	PROPN
ejpam-7047	128	84	containing	contain	VERB
ejpam-7047	128	85	f	f	PROPN
ejpam-7047	128	86	(	(	PUNCT
ejpam-7047	128	87	x	x	NOUN
ejpam-7047	128	88	)	)	PUNCT
ejpam-7047	128	89	;	;	PUNCT
ejpam-7047	128	90	(	(	PUNCT
ejpam-7047	128	91	3	3	X
ejpam-7047	128	92	)	)	PUNCT
ejpam-7047	128	93	x	x	SYM
ejpam-7047	128	94	∈	∈	PROPN
ejpam-7047	128	95	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	128	96	-	-	NOUN
ejpam-7047	128	97	cl(v	cl(v	NOUN
ejpam-7047	128	98	)	)	PUNCT
ejpam-7047	128	99	)	)	PUNCT
ejpam-7047	128	100	)	)	PUNCT
ejpam-7047	128	101	)	)	PUNCT
ejpam-7047	128	102	)	)	PUNCT
ejpam-7047	129	1	for	for	ADP
ejpam-7047	129	2	every	every	DET
ejpam-7047	129	3	σ1σ2	σ1σ2	NOUN
ejpam-7047	129	4	-	-	ADJ
ejpam-7047	129	5	open	open	ADJ
ejpam-7047	129	6	set	set	NOUN
ejpam-7047	129	7	v	v	NOUN
ejpam-7047	129	8	of	of	ADP
ejpam-7047	129	9	y	y	PROPN
ejpam-7047	129	10	containing	contain	VERB
ejpam-7047	129	11	f	f	PROPN
ejpam-7047	129	12	(	(	PUNCT
ejpam-7047	129	13	x	x	NOUN
ejpam-7047	129	14	)	)	PUNCT
ejpam-7047	129	15	.	.	PUNCT
ejpam-7047	130	1	proof	proof	NOUN
ejpam-7047	130	2	.	.	PUNCT
ejpam-7047	131	1	(	(	PUNCT
ejpam-7047	131	2	1	1	X
ejpam-7047	131	3	)	)	PUNCT
ejpam-7047	131	4	⇒	⇒	NOUN
ejpam-7047	131	5	(	(	PUNCT
ejpam-7047	131	6	2	2	NUM
ejpam-7047	131	7	):	):	PUNCT
ejpam-7047	131	8	let	let	VERB
ejpam-7047	131	9	v	v	PART
ejpam-7047	131	10	be	be	AUX
ejpam-7047	131	11	any	any	DET
ejpam-7047	131	12	σ1σ2	σ1σ2	NOUN
ejpam-7047	131	13	-	-	ADJ
ejpam-7047	131	14	open	open	ADJ
ejpam-7047	131	15	set	set	NOUN
ejpam-7047	131	16	of	of	ADP
ejpam-7047	131	17	y	y	PROPN
ejpam-7047	131	18	containing	contain	VERB
ejpam-7047	131	19	f	f	PROPN
ejpam-7047	131	20	(	(	PUNCT
ejpam-7047	131	21	x	x	NOUN
ejpam-7047	131	22	)	)	PUNCT
ejpam-7047	131	23	.	.	PUNCT
ejpam-7047	132	1	then	then	ADV
ejpam-7047	132	2	,	,	PUNCT
ejpam-7047	132	3	there	there	PRON
ejpam-7047	132	4	exists	exist	VERB
ejpam-7047	132	5	a	a	DET
ejpam-7047	132	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	132	7	-	-	PUNCT
ejpam-7047	132	8	open	open	ADJ
ejpam-7047	132	9	set	set	NOUN
ejpam-7047	132	10	u	u	NOUN
ejpam-7047	132	11	of	of	ADP
ejpam-7047	132	12	x	x	PUNCT
ejpam-7047	132	13	containing	contain	VERB
ejpam-7047	132	14	x	x	PUNCT
ejpam-7047	132	15	such	such	ADJ
ejpam-7047	132	16	that	that	SCONJ
ejpam-7047	132	17	f	f	PROPN
ejpam-7047	132	18	(	(	PUNCT
ejpam-7047	132	19	u	u	NOUN
ejpam-7047	132	20	)	)	PUNCT
ejpam-7047	132	21	⊆	⊆	NUM
ejpam-7047	132	22	σ1σ2	σ1σ2	NOUN
ejpam-7047	132	23	-	-	NUM
ejpam-7047	132	24	cl(v	cl(v	NOUN
ejpam-7047	132	25	)	)	PUNCT
ejpam-7047	132	26	;	;	PUNCT
ejpam-7047	132	27	hence	hence	ADV
ejpam-7047	132	28	u	u	NOUN
ejpam-7047	132	29	⊆	⊆	NUM
ejpam-7047	132	30	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	132	31	-	-	PUNCT
ejpam-7047	132	32	cl(v	cl(v	NOUN
ejpam-7047	132	33	)	)	PUNCT
ejpam-7047	132	34	)	)	PUNCT
ejpam-7047	132	35	.	.	PUNCT
ejpam-7047	133	1	thus	thus	ADV
ejpam-7047	133	2	,	,	PUNCT
ejpam-7047	133	3	x	x	PROPN
ejpam-7047	133	4	∈	∈	PROPN
ejpam-7047	133	5	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	133	6	-	-	PUNCT
ejpam-7047	133	7	cl(v	cl(v	NOUN
ejpam-7047	133	8	)	)	PUNCT
ejpam-7047	133	9	)	)	PUNCT
ejpam-7047	133	10	)	)	PUNCT
ejpam-7047	133	11	.	.	PUNCT
ejpam-7047	134	1	(	(	PUNCT
ejpam-7047	134	2	2	2	X
ejpam-7047	134	3	)	)	PUNCT
ejpam-7047	134	4	⇒	⇒	NOUN
ejpam-7047	134	5	(	(	PUNCT
ejpam-7047	134	6	3	3	NUM
ejpam-7047	134	7	):	):	PUNCT
ejpam-7047	134	8	let	let	VERB
ejpam-7047	134	9	v	v	PART
ejpam-7047	134	10	be	be	AUX
ejpam-7047	134	11	any	any	DET
ejpam-7047	134	12	σ1σ2	σ1σ2	NOUN
ejpam-7047	134	13	-	-	ADJ
ejpam-7047	134	14	open	open	ADJ
ejpam-7047	134	15	set	set	NOUN
ejpam-7047	134	16	of	of	ADP
ejpam-7047	134	17	y	y	PROPN
ejpam-7047	134	18	containing	contain	VERB
ejpam-7047	134	19	f	f	PROPN
ejpam-7047	134	20	(	(	PUNCT
ejpam-7047	134	21	x	x	NOUN
ejpam-7047	134	22	)	)	PUNCT
ejpam-7047	134	23	.	.	PUNCT
ejpam-7047	135	1	thus	thus	ADV
ejpam-7047	135	2	by	by	ADP
ejpam-7047	135	3	(	(	PUNCT
ejpam-7047	135	4	2	2	NUM
ejpam-7047	135	5	)	)	PUNCT
ejpam-7047	135	6	,	,	PUNCT
ejpam-7047	135	7	we	we	PRON
ejpam-7047	135	8	have	have	VERB
ejpam-7047	135	9	x	x	X
ejpam-7047	135	10	∈	∈	PROPN
ejpam-7047	135	11	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	135	12	-	-	PUNCT
ejpam-7047	135	13	cl(v	cl(v	NOUN
ejpam-7047	135	14	)	)	PUNCT
ejpam-7047	135	15	)	)	PUNCT
ejpam-7047	135	16	)	)	PUNCT
ejpam-7047	135	17	and	and	CCONJ
ejpam-7047	135	18	by	by	ADP
ejpam-7047	135	19	lemma	lemma	PROPN
ejpam-7047	135	20	5	5	NUM
ejpam-7047	135	21	,	,	PUNCT
ejpam-7047	135	22	x	x	SYM
ejpam-7047	135	23	∈	∈	NOUN
ejpam-7047	135	24	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	135	25	-	-	NOUN
ejpam-7047	135	26	cl(v	cl(v	NOUN
ejpam-7047	135	27	)	)	PUNCT
ejpam-7047	135	28	)	)	PUNCT
ejpam-7047	135	29	)	)	PUNCT
ejpam-7047	135	30	)	)	PUNCT
ejpam-7047	135	31	)	)	PUNCT
ejpam-7047	135	32	.	.	PUNCT
ejpam-7047	136	1	(	(	PUNCT
ejpam-7047	136	2	3	3	X
ejpam-7047	136	3	)	)	PUNCT
ejpam-7047	136	4	⇒	⇒	NOUN
ejpam-7047	136	5	(	(	PUNCT
ejpam-7047	136	6	1	1	NUM
ejpam-7047	136	7	):	):	PUNCT
ejpam-7047	136	8	let	let	VERB
ejpam-7047	136	9	v	v	PART
ejpam-7047	136	10	be	be	AUX
ejpam-7047	136	11	any	any	DET
ejpam-7047	136	12	σ1σ2	σ1σ2	NOUN
ejpam-7047	136	13	-	-	ADJ
ejpam-7047	136	14	open	open	ADJ
ejpam-7047	136	15	set	set	NOUN
ejpam-7047	136	16	of	of	ADP
ejpam-7047	136	17	y	y	PROPN
ejpam-7047	136	18	containing	contain	VERB
ejpam-7047	136	19	f	f	PROPN
ejpam-7047	136	20	(	(	PUNCT
ejpam-7047	136	21	x	x	NOUN
ejpam-7047	136	22	)	)	PUNCT
ejpam-7047	136	23	.	.	PUNCT
ejpam-7047	137	1	by	by	ADP
ejpam-7047	137	2	(	(	PUNCT
ejpam-7047	137	3	3	3	NUM
ejpam-7047	137	4	)	)	PUNCT
ejpam-7047	137	5	,	,	PUNCT
ejpam-7047	137	6	we	we	PRON
ejpam-7047	137	7	have	have	VERB
ejpam-7047	137	8	x	x	PART
ejpam-7047	137	9	∈	∈	PROPN
ejpam-7047	137	10	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	137	11	-	-	NOUN
ejpam-7047	137	12	cl(v	cl(v	NOUN
ejpam-7047	137	13	)	)	PUNCT
ejpam-7047	137	14	)	)	PUNCT
ejpam-7047	137	15	)	)	PUNCT
ejpam-7047	137	16	)	)	PUNCT
ejpam-7047	137	17	)	)	PUNCT
ejpam-7047	137	18	and	and	CCONJ
ejpam-7047	137	19	by	by	ADP
ejpam-7047	137	20	lemma	lemma	PROPN
ejpam-7047	137	21	5	5	NUM
ejpam-7047	137	22	,	,	PUNCT
ejpam-7047	137	23	x	x	SYM
ejpam-7047	137	24	∈	∈	PROPN
ejpam-7047	137	25	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	137	26	-	-	PUNCT
ejpam-7047	137	27	cl(v	cl(v	NOUN
ejpam-7047	137	28	)	)	PUNCT
ejpam-7047	137	29	)	)	PUNCT
ejpam-7047	137	30	)	)	PUNCT
ejpam-7047	137	31	.	.	PUNCT
ejpam-7047	138	1	therefore	therefore	ADV
ejpam-7047	138	2	,	,	PUNCT
ejpam-7047	138	3	there	there	PRON
ejpam-7047	138	4	exists	exist	VERB
ejpam-7047	138	5	a	a	DET
ejpam-7047	138	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	138	7	-	-	PUNCT
ejpam-7047	138	8	open	open	ADJ
ejpam-7047	138	9	set	set	NOUN
ejpam-7047	138	10	u	u	NOUN
ejpam-7047	138	11	of	of	ADP
ejpam-7047	138	12	x	x	PUNCT
ejpam-7047	138	13	containing	contain	VERB
ejpam-7047	138	14	x	x	PUNCT
ejpam-7047	138	15	such	such	ADJ
ejpam-7047	138	16	that	that	SCONJ
ejpam-7047	138	17	u	u	NOUN
ejpam-7047	138	18	⊆	⊆	NUM
ejpam-7047	138	19	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	138	20	-	-	PUNCT
ejpam-7047	138	21	cl(v	cl(v	NOUN
ejpam-7047	138	22	)	)	PUNCT
ejpam-7047	138	23	)	)	PUNCT
ejpam-7047	138	24	;	;	PUNCT
ejpam-7047	138	25	hence	hence	ADV
ejpam-7047	138	26	f	f	PROPN
ejpam-7047	138	27	(	(	PUNCT
ejpam-7047	138	28	u	u	NOUN
ejpam-7047	138	29	)	)	PUNCT
ejpam-7047	138	30	⊆	⊆	NUM
ejpam-7047	138	31	σ1σ2	σ1σ2	NOUN
ejpam-7047	138	32	-	-	NUM
ejpam-7047	138	33	cl(v	cl(v	NOUN
ejpam-7047	138	34	)	)	PUNCT
ejpam-7047	138	35	.	.	PUNCT
ejpam-7047	139	1	this	this	PRON
ejpam-7047	139	2	shows	show	VERB
ejpam-7047	139	3	that	that	SCONJ
ejpam-7047	139	4	f	f	PROPN
ejpam-7047	139	5	is	be	AUX
ejpam-7047	139	6	upper	upper	ADJ
ejpam-7047	139	7	weakly	weakly	ADJ
ejpam-7047	139	8	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	139	9	,	,	PUNCT
ejpam-7047	139	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	139	11	at	at	ADP
ejpam-7047	139	12	x.	x.	NOUN
ejpam-7047	139	13	definition	definition	NOUN
ejpam-7047	139	14	2	2	NUM
ejpam-7047	139	15	.	.	PUNCT
ejpam-7047	139	16	a	a	DET
ejpam-7047	139	17	multifunction	multifunction	NOUN
ejpam-7047	140	1	f	f	NOUN
ejpam-7047	140	2	:	:	PUNCT
ejpam-7047	140	3	(	(	PUNCT
ejpam-7047	140	4	x	x	X
ejpam-7047	140	5	,	,	PUNCT
ejpam-7047	140	6	τ	τ	PROPN
ejpam-7047	140	7	,	,	PUNCT
ejpam-7047	140	8	i	i	NOUN
ejpam-7047	140	9	)	)	PUNCT
ejpam-7047	140	10	→	→	PUNCT
ejpam-7047	140	11	(	(	PUNCT
ejpam-7047	140	12	y	y	PROPN
ejpam-7047	140	13	,	,	PUNCT
ejpam-7047	140	14	σ1	σ1	PROPN
ejpam-7047	140	15	,	,	PUNCT
ejpam-7047	140	16	σ2	σ2	PROPN
ejpam-7047	140	17	)	)	PUNCT
ejpam-7047	140	18	is	be	AUX
ejpam-7047	140	19	called	call	VERB
ejpam-7047	140	20	lower	low	ADJ
ejpam-7047	140	21	weakly	weakly	ADJ
ejpam-7047	140	22	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	140	23	,	,	PUNCT
ejpam-7047	140	24	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	140	25	at	at	ADP
ejpam-7047	140	26	a	a	DET
ejpam-7047	140	27	point	point	NOUN
ejpam-7047	140	28	x	x	PUNCT
ejpam-7047	140	29	of	of	ADP
ejpam-7047	140	30	x	x	PRON
ejpam-7047	140	31	if	if	SCONJ
ejpam-7047	140	32	for	for	ADP
ejpam-7047	140	33	each	each	DET
ejpam-7047	140	34	σ1σ2	σ1σ2	VERB
ejpam-7047	140	35	-	-	ADJ
ejpam-7047	140	36	open	open	ADJ
ejpam-7047	140	37	set	set	NOUN
ejpam-7047	140	38	v	v	NOUN
ejpam-7047	140	39	of	of	ADP
ejpam-7047	140	40	y	y	PRON
ejpam-7047	140	41	such	such	ADJ
ejpam-7047	140	42	that	that	SCONJ
ejpam-7047	140	43	f	f	PROPN
ejpam-7047	140	44	(	(	PUNCT
ejpam-7047	140	45	x	x	NOUN
ejpam-7047	140	46	)	)	PUNCT
ejpam-7047	140	47	∩	∩	NOUN
ejpam-7047	140	48	v	v	ADP
ejpam-7047	140	49	̸=	̸=	PROPN
ejpam-7047	140	50	∅	∅	NOUN
ejpam-7047	140	51	,	,	PUNCT
ejpam-7047	140	52	there	there	PRON
ejpam-7047	140	53	exists	exist	VERB
ejpam-7047	140	54	a	a	DET
ejpam-7047	140	55	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	140	56	-	-	PUNCT
ejpam-7047	140	57	open	open	ADJ
ejpam-7047	140	58	set	set	NOUN
ejpam-7047	140	59	u	u	NOUN
ejpam-7047	140	60	of	of	ADP
ejpam-7047	140	61	x	x	PUNCT
ejpam-7047	140	62	containing	contain	VERB
ejpam-7047	140	63	x	x	PUNCT
ejpam-7047	140	64	such	such	ADJ
ejpam-7047	140	65	that	that	SCONJ
ejpam-7047	140	66	σ1σ2	σ1σ2	NOUN
ejpam-7047	140	67	-	-	NUM
ejpam-7047	140	68	cl(v	cl(v	PUNCT
ejpam-7047	140	69	)	)	PUNCT
ejpam-7047	140	70	∩f	∩f	NOUN
ejpam-7047	140	71	(	(	PUNCT
ejpam-7047	140	72	z	z	X
ejpam-7047	140	73	)	)	PUNCT
ejpam-7047	140	74	̸=	̸=	NOUN
ejpam-7047	140	75	∅	∅	NOUN
ejpam-7047	140	76	for	for	ADP
ejpam-7047	140	77	every	every	DET
ejpam-7047	140	78	z	z	NOUN
ejpam-7047	140	79	∈	∈	PROPN
ejpam-7047	140	80	u	u	NOUN
ejpam-7047	140	81	.	.	PUNCT
ejpam-7047	140	82	a	a	DET
ejpam-7047	140	83	multifunction	multifunction	NOUN
ejpam-7047	140	84	f	f	NOUN
ejpam-7047	140	85	:	:	PUNCT
ejpam-7047	140	86	(	(	PUNCT
ejpam-7047	140	87	x	x	X
ejpam-7047	140	88	,	,	PUNCT
ejpam-7047	140	89	τ	τ	PROPN
ejpam-7047	140	90	,	,	PUNCT
ejpam-7047	140	91	i	i	NOUN
ejpam-7047	140	92	)	)	PUNCT
ejpam-7047	140	93	→	→	PUNCT
ejpam-7047	140	94	(	(	PUNCT
ejpam-7047	140	95	y	y	PROPN
ejpam-7047	140	96	,	,	PUNCT
ejpam-7047	140	97	σ1	σ1	PROPN
ejpam-7047	140	98	,	,	PUNCT
ejpam-7047	140	99	σ2	σ2	PROPN
ejpam-7047	140	100	)	)	PUNCT
ejpam-7047	140	101	is	be	AUX
ejpam-7047	140	102	called	call	VERB
ejpam-7047	140	103	lower	low	ADJ
ejpam-7047	140	104	weakly	weakly	ADJ
ejpam-7047	140	105	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	140	106	,	,	PUNCT
ejpam-7047	140	107	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	140	108	if	if	SCONJ
ejpam-7047	140	109	f	f	PROPN
ejpam-7047	140	110	is	be	AUX
ejpam-7047	140	111	lower	lower	ADV
ejpam-7047	140	112	weakly	weakly	ADJ
ejpam-7047	140	113	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	140	114	,	,	PUNCT
ejpam-7047	140	115	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	140	116	at	at	ADP
ejpam-7047	140	117	each	each	DET
ejpam-7047	140	118	point	point	NOUN
ejpam-7047	140	119	of	of	ADP
ejpam-7047	140	120	x.	x.	NOUN
ejpam-7047	140	121	theorem	theorem	VERB
ejpam-7047	140	122	2	2	NUM
ejpam-7047	140	123	.	.	X
ejpam-7047	140	124	for	for	ADP
ejpam-7047	140	125	a	a	DET
ejpam-7047	140	126	multifunction	multifunction	NOUN
ejpam-7047	140	127	f	f	NOUN
ejpam-7047	140	128	:	:	PUNCT
ejpam-7047	140	129	(	(	PUNCT
ejpam-7047	140	130	x	x	X
ejpam-7047	140	131	,	,	PUNCT
ejpam-7047	140	132	τ	τ	PROPN
ejpam-7047	140	133	,	,	PUNCT
ejpam-7047	140	134	i	i	NOUN
ejpam-7047	140	135	)	)	PUNCT
ejpam-7047	140	136	→	→	PUNCT
ejpam-7047	140	137	(	(	PUNCT
ejpam-7047	140	138	y	y	PROPN
ejpam-7047	140	139	,	,	PUNCT
ejpam-7047	140	140	σ1	σ1	PROPN
ejpam-7047	140	141	,	,	PUNCT
ejpam-7047	140	142	σ2	σ2	NOUN
ejpam-7047	140	143	)	)	PUNCT
ejpam-7047	140	144	,	,	PUNCT
ejpam-7047	140	145	the	the	DET
ejpam-7047	140	146	following	follow	VERB
ejpam-7047	140	147	properties	property	NOUN
ejpam-7047	140	148	are	be	AUX
ejpam-7047	140	149	equivalent	equivalent	ADJ
ejpam-7047	140	150	:	:	PUNCT
ejpam-7047	140	151	(	(	PUNCT
ejpam-7047	140	152	1	1	X
ejpam-7047	140	153	)	)	PUNCT
ejpam-7047	140	154	f	f	PROPN
ejpam-7047	140	155	is	be	AUX
ejpam-7047	140	156	lower	low	ADJ
ejpam-7047	140	157	weakly	weakly	ADJ
ejpam-7047	140	158	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	140	159	,	,	PUNCT
ejpam-7047	140	160	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	140	161	at	at	ADP
ejpam-7047	140	162	x	x	X
ejpam-7047	140	163	∈	∈	PROPN
ejpam-7047	140	164	x	x	X
ejpam-7047	140	165	;	;	PUNCT
ejpam-7047	140	166	(	(	PUNCT
ejpam-7047	140	167	2	2	X
ejpam-7047	140	168	)	)	PUNCT
ejpam-7047	140	169	x	x	SYM
ejpam-7047	140	170	∈	∈	PROPN
ejpam-7047	140	171	αint⋆(f−(σ1σ2	αint⋆(f−(σ1σ2	NOUN
ejpam-7047	140	172	-	-	NOUN
ejpam-7047	140	173	cl(v	cl(v	NOUN
ejpam-7047	140	174	)	)	PUNCT
ejpam-7047	140	175	)	)	PUNCT
ejpam-7047	140	176	)	)	PUNCT
ejpam-7047	140	177	for	for	ADP
ejpam-7047	140	178	every	every	DET
ejpam-7047	140	179	σ1σ2	σ1σ2	NOUN
ejpam-7047	140	180	-	-	ADJ
ejpam-7047	140	181	open	open	ADJ
ejpam-7047	140	182	set	set	NOUN
ejpam-7047	140	183	v	v	NOUN
ejpam-7047	140	184	of	of	ADP
ejpam-7047	140	185	y	y	PRON
ejpam-7047	140	186	such	such	ADJ
ejpam-7047	140	187	that	that	SCONJ
ejpam-7047	140	188	f	f	PROPN
ejpam-7047	140	189	(	(	PUNCT
ejpam-7047	140	190	x)∩v	x)∩v	PROPN
ejpam-7047	140	191	̸=	̸=	PROPN
ejpam-7047	140	192	∅	∅	NOUN
ejpam-7047	140	193	;	;	PUNCT
ejpam-7047	140	194	(	(	PUNCT
ejpam-7047	140	195	3	3	X
ejpam-7047	140	196	)	)	PUNCT
ejpam-7047	140	197	x	x	SYM
ejpam-7047	140	198	∈	∈	NOUN
ejpam-7047	140	199	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-7047	140	200	-	-	PUNCT
ejpam-7047	140	201	cl(v	cl(v	NOUN
ejpam-7047	140	202	)	)	PUNCT
ejpam-7047	140	203	)	)	PUNCT
ejpam-7047	140	204	)	)	PUNCT
ejpam-7047	140	205	)	)	PUNCT
ejpam-7047	140	206	)	)	PUNCT
ejpam-7047	140	207	for	for	ADP
ejpam-7047	140	208	every	every	DET
ejpam-7047	140	209	σ1σ2	σ1σ2	NOUN
ejpam-7047	140	210	-	-	ADJ
ejpam-7047	140	211	open	open	ADJ
ejpam-7047	140	212	set	set	NOUN
ejpam-7047	140	213	v	v	NOUN
ejpam-7047	140	214	of	of	ADP
ejpam-7047	140	215	y	y	PRON
ejpam-7047	140	216	such	such	ADJ
ejpam-7047	140	217	that	that	SCONJ
ejpam-7047	140	218	f	f	PROPN
ejpam-7047	140	219	(	(	PUNCT
ejpam-7047	140	220	x	x	NOUN
ejpam-7047	140	221	)	)	PUNCT
ejpam-7047	140	222	∩	∩	NOUN
ejpam-7047	140	223	v	v	ADP
ejpam-7047	140	224	̸=	̸=	PROPN
ejpam-7047	140	225	∅.	∅.	ADP
ejpam-7047	140	226	proof	proof	NOUN
ejpam-7047	140	227	.	.	PUNCT
ejpam-7047	141	1	the	the	DET
ejpam-7047	141	2	proof	proof	NOUN
ejpam-7047	141	3	is	be	AUX
ejpam-7047	141	4	similar	similar	ADJ
ejpam-7047	141	5	to	to	ADP
ejpam-7047	141	6	that	that	PRON
ejpam-7047	141	7	of	of	ADP
ejpam-7047	141	8	theorem	theorem	ADJ
ejpam-7047	141	9	1	1	NUM
ejpam-7047	141	10	.	.	PUNCT
ejpam-7047	141	11	theorem	theorem	NOUN
ejpam-7047	141	12	3	3	NUM
ejpam-7047	141	13	.	.	X
ejpam-7047	141	14	for	for	ADP
ejpam-7047	141	15	a	a	DET
ejpam-7047	141	16	multifunction	multifunction	NOUN
ejpam-7047	141	17	f	f	NOUN
ejpam-7047	141	18	:	:	PUNCT
ejpam-7047	141	19	(	(	PUNCT
ejpam-7047	141	20	x	x	X
ejpam-7047	141	21	,	,	PUNCT
ejpam-7047	141	22	τ	τ	PROPN
ejpam-7047	141	23	,	,	PUNCT
ejpam-7047	141	24	i	i	NOUN
ejpam-7047	141	25	)	)	PUNCT
ejpam-7047	141	26	→	→	PUNCT
ejpam-7047	141	27	(	(	PUNCT
ejpam-7047	141	28	y	y	PROPN
ejpam-7047	141	29	,	,	PUNCT
ejpam-7047	141	30	σ1	σ1	PROPN
ejpam-7047	141	31	,	,	PUNCT
ejpam-7047	141	32	σ2	σ2	NOUN
ejpam-7047	141	33	)	)	PUNCT
ejpam-7047	141	34	,	,	PUNCT
ejpam-7047	141	35	the	the	DET
ejpam-7047	141	36	following	follow	VERB
ejpam-7047	141	37	properties	property	NOUN
ejpam-7047	141	38	are	be	AUX
ejpam-7047	141	39	equivalent	equivalent	ADJ
ejpam-7047	141	40	:	:	PUNCT
ejpam-7047	141	41	(	(	PUNCT
ejpam-7047	141	42	1	1	X
ejpam-7047	141	43	)	)	PUNCT
ejpam-7047	141	44	f	f	PROPN
ejpam-7047	141	45	is	be	AUX
ejpam-7047	141	46	upper	upper	ADJ
ejpam-7047	141	47	weakly	weakly	ADJ
ejpam-7047	141	48	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	141	49	,	,	PUNCT
ejpam-7047	141	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	141	51	;	;	PUNCT
ejpam-7047	141	52	(	(	PUNCT
ejpam-7047	141	53	2	2	NUM
ejpam-7047	141	54	)	)	PUNCT
ejpam-7047	141	55	f+(v	f+(v	NOUN
ejpam-7047	141	56	)	)	PUNCT
ejpam-7047	142	1	⊆	⊆	NUM
ejpam-7047	142	2	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	142	3	-	-	NOUN
ejpam-7047	142	4	cl(v	cl(v	NOUN
ejpam-7047	142	5	)	)	PUNCT
ejpam-7047	142	6	)	)	PUNCT
ejpam-7047	142	7	)	)	PUNCT
ejpam-7047	142	8	)	)	PUNCT
ejpam-7047	142	9	)	)	PUNCT
ejpam-7047	143	1	for	for	ADP
ejpam-7047	143	2	every	every	DET
ejpam-7047	143	3	σ1σ2	σ1σ2	NOUN
ejpam-7047	143	4	-	-	ADJ
ejpam-7047	143	5	open	open	ADJ
ejpam-7047	143	6	set	set	NOUN
ejpam-7047	143	7	v	v	NOUN
ejpam-7047	143	8	of	of	ADP
ejpam-7047	143	9	y	y	PROPN
ejpam-7047	143	10	;	;	PUNCT
ejpam-7047	143	11	n.	n.	PROPN
ejpam-7047	143	12	viriyapong	viriyapong	PROPN
ejpam-7047	143	13	,	,	PUNCT
ejpam-7047	143	14	a.	a.	PROPN
ejpam-7047	143	15	sama	sama	PROPN
ejpam-7047	143	16	-	-	PUNCT
ejpam-7047	143	17	ae	ae	PROPN
ejpam-7047	143	18	,	,	PUNCT
ejpam-7047	143	19	c.	c.	PROPN
ejpam-7047	143	20	boonpok	boonpok	PROPN
ejpam-7047	143	21	/	/	SYM
ejpam-7047	143	22	eur	eur	PROPN
ejpam-7047	143	23	.	.	PUNCT
ejpam-7047	144	1	j.	j.	PROPN
ejpam-7047	144	2	pure	pure	PROPN
ejpam-7047	144	3	appl	appl	PROPN
ejpam-7047	144	4	.	.	PROPN
ejpam-7047	144	5	math	math	PROPN
ejpam-7047	144	6	,	,	PUNCT
ejpam-7047	144	7	18	18	NUM
ejpam-7047	144	8	(	(	PUNCT
ejpam-7047	144	9	4	4	NUM
ejpam-7047	144	10	)	)	PUNCT
ejpam-7047	144	11	(	(	PUNCT
ejpam-7047	144	12	2025	2025	NUM
ejpam-7047	144	13	)	)	PUNCT
ejpam-7047	144	14	,	,	PUNCT
ejpam-7047	144	15	7047	7047	NUM
ejpam-7047	144	16	6	6	NUM
ejpam-7047	144	17	of	of	ADP
ejpam-7047	144	18	11	11	NUM
ejpam-7047	144	19	(	(	PUNCT
ejpam-7047	144	20	3	3	NUM
ejpam-7047	144	21	)	)	PUNCT
ejpam-7047	144	22	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7047	144	23	-	-	PUNCT
ejpam-7047	144	24	int(k	int(k	NOUN
ejpam-7047	144	25	)	)	PUNCT
ejpam-7047	144	26	)	)	PUNCT
ejpam-7047	144	27	)	)	PUNCT
ejpam-7047	144	28	)	)	PUNCT
ejpam-7047	144	29	)	)	PUNCT
ejpam-7047	145	1	⊆	⊆	X
ejpam-7047	145	2	f−(k	f−(k	PROPN
ejpam-7047	145	3	)	)	PUNCT
ejpam-7047	145	4	for	for	ADP
ejpam-7047	145	5	every	every	DET
ejpam-7047	145	6	σ1σ2	σ1σ2	NUM
ejpam-7047	145	7	-	-	PUNCT
ejpam-7047	145	8	closed	closed	ADJ
ejpam-7047	145	9	set	set	NOUN
ejpam-7047	145	10	k	k	PROPN
ejpam-7047	145	11	of	of	ADP
ejpam-7047	145	12	y	y	PROPN
ejpam-7047	145	13	;	;	PUNCT
ejpam-7047	145	14	(	(	PUNCT
ejpam-7047	145	15	4	4	X
ejpam-7047	145	16	)	)	PUNCT
ejpam-7047	145	17	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	145	18	-	-	PUNCT
ejpam-7047	145	19	int(k	int(k	NUM
ejpam-7047	145	20	)	)	PUNCT
ejpam-7047	145	21	)	)	PUNCT
ejpam-7047	145	22	)	)	PUNCT
ejpam-7047	146	1	⊆	⊆	X
ejpam-7047	146	2	f−(k	f−(k	PROPN
ejpam-7047	146	3	)	)	PUNCT
ejpam-7047	146	4	for	for	ADP
ejpam-7047	146	5	every	every	DET
ejpam-7047	146	6	σ1σ2	σ1σ2	NUM
ejpam-7047	146	7	-	-	PUNCT
ejpam-7047	146	8	closed	closed	ADJ
ejpam-7047	146	9	set	set	NOUN
ejpam-7047	146	10	k	k	PROPN
ejpam-7047	146	11	of	of	ADP
ejpam-7047	146	12	y	y	PROPN
ejpam-7047	146	13	;	;	PUNCT
ejpam-7047	146	14	(	(	PUNCT
ejpam-7047	146	15	5	5	X
ejpam-7047	146	16	)	)	PUNCT
ejpam-7047	146	17	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	146	18	-	-	PUNCT
ejpam-7047	146	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	146	20	-	-	PUNCT
ejpam-7047	146	21	cl(b	cl(b	NOUN
ejpam-7047	146	22	)	)	PUNCT
ejpam-7047	146	23	)	)	PUNCT
ejpam-7047	146	24	)	)	PUNCT
ejpam-7047	146	25	)	)	PUNCT
ejpam-7047	147	1	⊆	⊆	X
ejpam-7047	147	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7047	147	3	-	-	PUNCT
ejpam-7047	147	4	cl(b	cl(b	NOUN
ejpam-7047	147	5	)	)	PUNCT
ejpam-7047	147	6	)	)	PUNCT
ejpam-7047	148	1	for	for	ADP
ejpam-7047	148	2	every	every	DET
ejpam-7047	148	3	subset	subset	NOUN
ejpam-7047	148	4	b	b	PROPN
ejpam-7047	148	5	of	of	ADP
ejpam-7047	148	6	y	y	PROPN
ejpam-7047	148	7	;	;	PUNCT
ejpam-7047	148	8	(	(	PUNCT
ejpam-7047	148	9	6	6	X
ejpam-7047	148	10	)	)	PUNCT
ejpam-7047	148	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	148	12	-	-	PUNCT
ejpam-7047	148	13	int(b	int(b	NOUN
ejpam-7047	148	14	)	)	PUNCT
ejpam-7047	148	15	)	)	PUNCT
ejpam-7047	148	16	⊆	⊆	X
ejpam-7047	148	17	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	148	18	-	-	PUNCT
ejpam-7047	148	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	148	20	-	-	PUNCT
ejpam-7047	148	21	int(b	int(b	NOUN
ejpam-7047	148	22	)	)	PUNCT
ejpam-7047	148	23	)	)	PUNCT
ejpam-7047	148	24	)	)	PUNCT
ejpam-7047	148	25	)	)	PUNCT
ejpam-7047	148	26	for	for	ADP
ejpam-7047	148	27	every	every	DET
ejpam-7047	148	28	subset	subset	NOUN
ejpam-7047	148	29	b	b	PROPN
ejpam-7047	148	30	of	of	ADP
ejpam-7047	148	31	y	y	PROPN
ejpam-7047	148	32	;	;	PUNCT
ejpam-7047	148	33	(	(	PUNCT
ejpam-7047	148	34	7	7	X
ejpam-7047	148	35	)	)	PUNCT
ejpam-7047	148	36	f+(v	f+(v	NOUN
ejpam-7047	148	37	)	)	PUNCT
ejpam-7047	148	38	⊆	⊆	X
ejpam-7047	148	39	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	148	40	-	-	PUNCT
ejpam-7047	148	41	cl(v	cl(v	NOUN
ejpam-7047	148	42	)	)	PUNCT
ejpam-7047	148	43	)	)	PUNCT
ejpam-7047	148	44	)	)	PUNCT
ejpam-7047	148	45	for	for	ADP
ejpam-7047	148	46	every	every	DET
ejpam-7047	148	47	σ1σ2	σ1σ2	NOUN
ejpam-7047	148	48	-	-	ADJ
ejpam-7047	148	49	open	open	ADJ
ejpam-7047	148	50	set	set	NOUN
ejpam-7047	148	51	v	v	NOUN
ejpam-7047	148	52	of	of	ADP
ejpam-7047	148	53	y	y	PROPN
ejpam-7047	148	54	;	;	PUNCT
ejpam-7047	148	55	(	(	PUNCT
ejpam-7047	148	56	8)	8)	NUM
ejpam-7047	148	57	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	148	58	-	-	PUNCT
ejpam-7047	148	59	int(k	int(k	NUM
ejpam-7047	148	60	)	)	PUNCT
ejpam-7047	148	61	)	)	PUNCT
ejpam-7047	148	62	)	)	PUNCT
ejpam-7047	149	1	⊆	⊆	X
ejpam-7047	149	2	f−(k	f−(k	PROPN
ejpam-7047	149	3	)	)	PUNCT
ejpam-7047	149	4	for	for	ADP
ejpam-7047	149	5	every	every	DET
ejpam-7047	149	6	(	(	PUNCT
ejpam-7047	149	7	σ1	σ1	PROPN
ejpam-7047	149	8	,	,	PUNCT
ejpam-7047	149	9	σ2)r	σ2)r	NOUN
ejpam-7047	149	10	-	-	PUNCT
ejpam-7047	149	11	closed	close	VERB
ejpam-7047	149	12	set	set	ADJ
ejpam-7047	149	13	k	k	PROPN
ejpam-7047	149	14	of	of	ADP
ejpam-7047	149	15	y	y	PROPN
ejpam-7047	149	16	;	;	PUNCT
ejpam-7047	149	17	(	(	PUNCT
ejpam-7047	149	18	9	9	X
ejpam-7047	149	19	)	)	PUNCT
ejpam-7047	149	20	αcl⋆(f−(v	αcl⋆(f−(v	NUM
ejpam-7047	149	21	)	)	PUNCT
ejpam-7047	149	22	)	)	PUNCT
ejpam-7047	150	1	⊆	⊆	X
ejpam-7047	150	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7047	150	3	-	-	PUNCT
ejpam-7047	150	4	cl(v	cl(v	NOUN
ejpam-7047	150	5	)	)	PUNCT
ejpam-7047	150	6	)	)	PUNCT
ejpam-7047	150	7	for	for	ADP
ejpam-7047	150	8	every	every	DET
ejpam-7047	150	9	σ1σ2	σ1σ2	NOUN
ejpam-7047	150	10	-	-	ADJ
ejpam-7047	150	11	open	open	ADJ
ejpam-7047	150	12	set	set	NOUN
ejpam-7047	150	13	v	v	NOUN
ejpam-7047	150	14	of	of	ADP
ejpam-7047	150	15	y	y	PROPN
ejpam-7047	150	16	;	;	PUNCT
ejpam-7047	150	17	(	(	PUNCT
ejpam-7047	150	18	10	10	NUM
ejpam-7047	150	19	)	)	PUNCT
ejpam-7047	150	20	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	150	21	-	-	PUNCT
ejpam-7047	150	22	int((σ1	int((σ1	ADJ
ejpam-7047	150	23	,	,	PUNCT
ejpam-7047	150	24	σ2)θ	σ2)θ	ADJ
ejpam-7047	150	25	-	-	PUNCT
ejpam-7047	150	26	cl(b	cl(b	NOUN
ejpam-7047	150	27	)	)	PUNCT
ejpam-7047	150	28	)	)	PUNCT
ejpam-7047	150	29	)	)	PUNCT
ejpam-7047	150	30	)	)	PUNCT
ejpam-7047	151	1	⊆	⊆	NUM
ejpam-7047	151	2	f−((σ1	f−((σ1	NOUN
ejpam-7047	151	3	,	,	PUNCT
ejpam-7047	151	4	σ2)θ	σ2)θ	ADJ
ejpam-7047	151	5	-	-	PUNCT
ejpam-7047	151	6	cl(b	cl(b	NOUN
ejpam-7047	151	7	)	)	PUNCT
ejpam-7047	151	8	)	)	PUNCT
ejpam-7047	151	9	for	for	ADP
ejpam-7047	151	10	every	every	DET
ejpam-7047	151	11	subset	subset	NOUN
ejpam-7047	151	12	b	b	PROPN
ejpam-7047	151	13	of	of	ADP
ejpam-7047	151	14	y	y	PROPN
ejpam-7047	151	15	.	.	PUNCT
ejpam-7047	152	1	proof	proof	NOUN
ejpam-7047	152	2	.	.	PUNCT
ejpam-7047	153	1	(	(	PUNCT
ejpam-7047	153	2	1	1	X
ejpam-7047	153	3	)	)	PUNCT
ejpam-7047	153	4	⇒	⇒	NOUN
ejpam-7047	153	5	(	(	PUNCT
ejpam-7047	153	6	2	2	NUM
ejpam-7047	153	7	):	):	PUNCT
ejpam-7047	153	8	let	let	VERB
ejpam-7047	153	9	v	v	PART
ejpam-7047	153	10	be	be	AUX
ejpam-7047	153	11	any	any	DET
ejpam-7047	153	12	σ1σ2	σ1σ2	NOUN
ejpam-7047	153	13	-	-	ADJ
ejpam-7047	153	14	open	open	ADJ
ejpam-7047	153	15	set	set	NOUN
ejpam-7047	153	16	of	of	ADP
ejpam-7047	153	17	y	y	PROPN
ejpam-7047	153	18	and	and	CCONJ
ejpam-7047	153	19	x	x	PROPN
ejpam-7047	153	20	∈	∈	PROPN
ejpam-7047	153	21	f+(v	f+(v	NOUN
ejpam-7047	153	22	)	)	PUNCT
ejpam-7047	153	23	.	.	PUNCT
ejpam-7047	154	1	then	then	ADV
ejpam-7047	154	2	,	,	PUNCT
ejpam-7047	154	3	f	f	PROPN
ejpam-7047	154	4	(	(	PUNCT
ejpam-7047	154	5	x	x	X
ejpam-7047	154	6	)	)	PUNCT
ejpam-7047	154	7	⊆	⊆	NUM
ejpam-7047	154	8	v	v	NOUN
ejpam-7047	154	9	and	and	CCONJ
ejpam-7047	154	10	there	there	PRON
ejpam-7047	154	11	exists	exist	VERB
ejpam-7047	154	12	a	a	DET
ejpam-7047	154	13	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	154	14	-	-	PUNCT
ejpam-7047	154	15	open	open	ADJ
ejpam-7047	154	16	set	set	NOUN
ejpam-7047	154	17	u	u	NOUN
ejpam-7047	154	18	of	of	ADP
ejpam-7047	154	19	x	x	PUNCT
ejpam-7047	154	20	containing	contain	VERB
ejpam-7047	154	21	x	x	PUNCT
ejpam-7047	154	22	such	such	ADJ
ejpam-7047	154	23	that	that	SCONJ
ejpam-7047	154	24	f	f	PROPN
ejpam-7047	154	25	(	(	PUNCT
ejpam-7047	154	26	u	u	NOUN
ejpam-7047	154	27	)	)	PUNCT
ejpam-7047	154	28	⊆	⊆	NUM
ejpam-7047	154	29	σ1σ2	σ1σ2	NOUN
ejpam-7047	154	30	-	-	NUM
ejpam-7047	154	31	cl(v	cl(v	NOUN
ejpam-7047	154	32	)	)	PUNCT
ejpam-7047	154	33	;	;	PUNCT
ejpam-7047	154	34	hence	hence	ADV
ejpam-7047	154	35	u	u	NOUN
ejpam-7047	154	36	⊆	⊆	NUM
ejpam-7047	154	37	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	154	38	-	-	PUNCT
ejpam-7047	154	39	cl(v	cl(v	NOUN
ejpam-7047	154	40	)	)	PUNCT
ejpam-7047	154	41	)	)	PUNCT
ejpam-7047	155	1	and	and	CCONJ
ejpam-7047	155	2	so	so	ADV
ejpam-7047	155	3	x	x	SYM
ejpam-7047	155	4	∈	∈	PROPN
ejpam-7047	155	5	u	u	NOUN
ejpam-7047	155	6	⊆	⊆	NUM
ejpam-7047	155	7	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	155	8	-	-	NOUN
ejpam-7047	155	9	cl(v	cl(v	NOUN
ejpam-7047	155	10	)	)	PUNCT
ejpam-7047	155	11	)	)	PUNCT
ejpam-7047	155	12	)	)	PUNCT
ejpam-7047	155	13	)	)	PUNCT
ejpam-7047	155	14	)	)	PUNCT
ejpam-7047	155	15	.	.	PUNCT
ejpam-7047	156	1	this	this	PRON
ejpam-7047	156	2	shows	show	VERB
ejpam-7047	156	3	that	that	SCONJ
ejpam-7047	156	4	f+(v	f+(v	PROPN
ejpam-7047	156	5	)	)	PUNCT
ejpam-7047	157	1	⊆	⊆	NUM
ejpam-7047	157	2	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	157	3	-	-	NOUN
ejpam-7047	157	4	cl(v	cl(v	NOUN
ejpam-7047	157	5	)	)	PUNCT
ejpam-7047	157	6	)	)	PUNCT
ejpam-7047	157	7	)	)	PUNCT
ejpam-7047	157	8	)	)	PUNCT
ejpam-7047	157	9	)	)	PUNCT
ejpam-7047	157	10	.	.	PUNCT
ejpam-7047	158	1	(	(	PUNCT
ejpam-7047	158	2	2	2	X
ejpam-7047	158	3	)	)	PUNCT
ejpam-7047	158	4	⇒	⇒	NOUN
ejpam-7047	158	5	(	(	PUNCT
ejpam-7047	158	6	3	3	NUM
ejpam-7047	158	7	):	):	PUNCT
ejpam-7047	158	8	let	let	VERB
ejpam-7047	158	9	k	k	PRON
ejpam-7047	158	10	be	be	AUX
ejpam-7047	158	11	any	any	DET
ejpam-7047	158	12	σ1σ2	σ1σ2	NUM
ejpam-7047	158	13	-	-	PUNCT
ejpam-7047	158	14	closed	closed	ADJ
ejpam-7047	158	15	set	set	NOUN
ejpam-7047	158	16	of	of	ADP
ejpam-7047	158	17	y	y	PROPN
ejpam-7047	158	18	.	.	PUNCT
ejpam-7047	159	1	then	then	ADV
ejpam-7047	159	2	,	,	PUNCT
ejpam-7047	159	3	y	y	PROPN
ejpam-7047	159	4	−k	−k	PROPN
ejpam-7047	159	5	is	be	AUX
ejpam-7047	159	6	σ1σ2	σ1σ2	NOUN
ejpam-7047	159	7	-	-	ADJ
ejpam-7047	159	8	open	open	ADJ
ejpam-7047	159	9	in	in	ADP
ejpam-7047	159	10	y	y	PROPN
ejpam-7047	159	11	and	and	CCONJ
ejpam-7047	159	12	by	by	ADP
ejpam-7047	159	13	(	(	PUNCT
ejpam-7047	159	14	2	2	NUM
ejpam-7047	159	15	)	)	PUNCT
ejpam-7047	159	16	,	,	PUNCT
ejpam-7047	159	17	we	we	PRON
ejpam-7047	159	18	have	have	VERB
ejpam-7047	159	19	x	x	INTJ
ejpam-7047	159	20	−	−	DET
ejpam-7047	159	21	f−(k	f−(k	PROPN
ejpam-7047	159	22	)	)	PUNCT
ejpam-7047	159	23	=	=	PUNCT
ejpam-7047	160	1	f+(y	f+(y	PROPN
ejpam-7047	160	2	−k	−k	PROPN
ejpam-7047	160	3	)	)	PUNCT
ejpam-7047	160	4	⊆	⊆	NUM
ejpam-7047	160	5	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	160	6	-	-	PUNCT
ejpam-7047	160	7	cl(y	cl(y	NOUN
ejpam-7047	160	8	−k	−k	NOUN
ejpam-7047	160	9	)	)	PUNCT
ejpam-7047	160	10	)	)	PUNCT
ejpam-7047	160	11	)	)	PUNCT
ejpam-7047	160	12	)	)	PUNCT
ejpam-7047	160	13	)	)	PUNCT
ejpam-7047	161	1	=	=	VERB
ejpam-7047	161	2	int⋆(cl⋆(int⋆(f+(y	int⋆(cl⋆(int⋆(f+(y	VERB
ejpam-7047	161	3	−	−	NUM
ejpam-7047	161	4	σ1σ2	σ1σ2	NUM
ejpam-7047	161	5	-	-	PUNCT
ejpam-7047	161	6	int(k	int(k	NOUN
ejpam-7047	161	7	)	)	PUNCT
ejpam-7047	161	8	)	)	PUNCT
ejpam-7047	161	9	)	)	PUNCT
ejpam-7047	161	10	)	)	PUNCT
ejpam-7047	161	11	)	)	PUNCT
ejpam-7047	162	1	=	=	PRON
ejpam-7047	162	2	int⋆(cl⋆(int⋆(x	int⋆(cl⋆(int⋆(x	VERB
ejpam-7047	162	3	−	−	NOUN
ejpam-7047	162	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7047	162	5	-	-	PUNCT
ejpam-7047	162	6	int(k	int(k	NOUN
ejpam-7047	162	7	)	)	PUNCT
ejpam-7047	162	8	)	)	PUNCT
ejpam-7047	162	9	)	)	PUNCT
ejpam-7047	162	10	)	)	PUNCT
ejpam-7047	162	11	)	)	PUNCT
ejpam-7047	163	1	=	=	PRON
ejpam-7047	163	2	int⋆(cl⋆(x	int⋆(cl⋆(x	VERB
ejpam-7047	163	3	−	−	PROPN
ejpam-7047	163	4	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	PROPN
ejpam-7047	163	5	-	-	PUNCT
ejpam-7047	163	6	int(k	int(k	NOUN
ejpam-7047	163	7	)	)	PUNCT
ejpam-7047	163	8	)	)	PUNCT
ejpam-7047	163	9	)	)	PUNCT
ejpam-7047	163	10	)	)	PUNCT
ejpam-7047	163	11	)	)	PUNCT
ejpam-7047	164	1	=	=	PRON
ejpam-7047	164	2	int⋆(x	int⋆(x	NOUN
ejpam-7047	164	3	−	−	NOUN
ejpam-7047	164	4	int⋆(cl⋆(f−(σ1σ2	int⋆(cl⋆(f−(σ1σ2	NUM
ejpam-7047	164	5	-	-	PUNCT
ejpam-7047	164	6	int(k	int(k	NOUN
ejpam-7047	164	7	)	)	PUNCT
ejpam-7047	164	8	)	)	PUNCT
ejpam-7047	164	9	)	)	PUNCT
ejpam-7047	164	10	)	)	PUNCT
ejpam-7047	164	11	)	)	PUNCT
ejpam-7047	165	1	=	=	PUNCT
ejpam-7047	166	1	x	x	X
ejpam-7047	166	2	−	−	ADP
ejpam-7047	166	3	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7047	166	4	-	-	PUNCT
ejpam-7047	166	5	int(k	int(k	NOUN
ejpam-7047	166	6	)	)	PUNCT
ejpam-7047	166	7	)	)	PUNCT
ejpam-7047	166	8	)	)	PUNCT
ejpam-7047	166	9	)	)	PUNCT
ejpam-7047	166	10	)	)	PUNCT
ejpam-7047	167	1	and	and	CCONJ
ejpam-7047	167	2	hence	hence	ADV
ejpam-7047	167	3	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7047	167	4	-	-	PUNCT
ejpam-7047	167	5	int(k	int(k	NOUN
ejpam-7047	167	6	)	)	PUNCT
ejpam-7047	167	7	)	)	PUNCT
ejpam-7047	167	8	)	)	PUNCT
ejpam-7047	167	9	)	)	PUNCT
ejpam-7047	167	10	)	)	PUNCT
ejpam-7047	168	1	⊆	⊆	NUM
ejpam-7047	168	2	f−(k	f−(k	PROPN
ejpam-7047	168	3	)	)	PUNCT
ejpam-7047	168	4	.	.	PUNCT
ejpam-7047	169	1	(	(	PUNCT
ejpam-7047	169	2	3	3	X
ejpam-7047	169	3	)	)	PUNCT
ejpam-7047	169	4	⇒	⇒	NOUN
ejpam-7047	169	5	(	(	PUNCT
ejpam-7047	169	6	4	4	NUM
ejpam-7047	169	7	):	):	PUNCT
ejpam-7047	169	8	let	let	VERB
ejpam-7047	169	9	k	k	PRON
ejpam-7047	169	10	be	be	AUX
ejpam-7047	169	11	any	any	DET
ejpam-7047	169	12	σ1σ2	σ1σ2	NUM
ejpam-7047	169	13	-	-	PUNCT
ejpam-7047	169	14	closed	closed	ADJ
ejpam-7047	169	15	set	set	NOUN
ejpam-7047	169	16	of	of	ADP
ejpam-7047	169	17	y	y	PROPN
ejpam-7047	169	18	.	.	PUNCT
ejpam-7047	170	1	by	by	ADP
ejpam-7047	170	2	(	(	PUNCT
ejpam-7047	170	3	3	3	NUM
ejpam-7047	170	4	)	)	PUNCT
ejpam-7047	170	5	,	,	PUNCT
ejpam-7047	170	6	we	we	PRON
ejpam-7047	170	7	have	have	VERB
ejpam-7047	170	8	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7047	170	9	-	-	PUNCT
ejpam-7047	170	10	int(k	int(k	NOUN
ejpam-7047	170	11	)	)	PUNCT
ejpam-7047	170	12	)	)	PUNCT
ejpam-7047	170	13	)	)	PUNCT
ejpam-7047	170	14	)	)	PUNCT
ejpam-7047	170	15	)	)	PUNCT
ejpam-7047	171	1	⊆	⊆	X
ejpam-7047	171	2	f−(k	f−(k	NOUN
ejpam-7047	171	3	)	)	PUNCT
ejpam-7047	171	4	and	and	CCONJ
ejpam-7047	171	5	hence	hence	ADV
ejpam-7047	171	6	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	171	7	-	-	PUNCT
ejpam-7047	171	8	int(k	int(k	NUM
ejpam-7047	171	9	)	)	PUNCT
ejpam-7047	171	10	)	)	PUNCT
ejpam-7047	171	11	)	)	PUNCT
ejpam-7047	172	1	⊆	⊆	X
ejpam-7047	172	2	f−(k	f−(k	PROPN
ejpam-7047	172	3	)	)	PUNCT
ejpam-7047	172	4	by	by	ADP
ejpam-7047	172	5	lemma	lemma	PROPN
ejpam-7047	172	6	5	5	NUM
ejpam-7047	172	7	.	.	PUNCT
ejpam-7047	172	8	(	(	PUNCT
ejpam-7047	172	9	4	4	X
ejpam-7047	172	10	)	)	PUNCT
ejpam-7047	172	11	⇒	⇒	NOUN
ejpam-7047	172	12	(	(	PUNCT
ejpam-7047	172	13	5	5	NUM
ejpam-7047	172	14	):	):	PUNCT
ejpam-7047	172	15	let	let	VERB
ejpam-7047	172	16	b	b	X
ejpam-7047	172	17	be	be	AUX
ejpam-7047	172	18	any	any	DET
ejpam-7047	172	19	subset	subset	NOUN
ejpam-7047	172	20	of	of	ADP
ejpam-7047	172	21	y	y	PROPN
ejpam-7047	172	22	.	.	PUNCT
ejpam-7047	173	1	then	then	ADV
ejpam-7047	173	2	,	,	PUNCT
ejpam-7047	173	3	σ1σ2	σ1σ2	NOUN
ejpam-7047	173	4	-	-	NOUN
ejpam-7047	173	5	cl(b	cl(b	NOUN
ejpam-7047	173	6	)	)	PUNCT
ejpam-7047	173	7	is	be	AUX
ejpam-7047	173	8	σ1σ2	σ1σ2	NOUN
ejpam-7047	173	9	-	-	ADJ
ejpam-7047	173	10	closed	closed	ADJ
ejpam-7047	173	11	in	in	ADP
ejpam-7047	173	12	y	y	PROPN
ejpam-7047	173	13	and	and	CCONJ
ejpam-7047	173	14	by	by	ADP
ejpam-7047	173	15	(	(	PUNCT
ejpam-7047	173	16	4	4	NUM
ejpam-7047	173	17	)	)	PUNCT
ejpam-7047	173	18	,	,	PUNCT
ejpam-7047	173	19	we	we	PRON
ejpam-7047	173	20	have	have	VERB
ejpam-7047	173	21	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	VERB
ejpam-7047	173	22	-	-	ADJ
ejpam-7047	173	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	173	24	-	-	PUNCT
ejpam-7047	173	25	cl(b	cl(b	NOUN
ejpam-7047	173	26	)	)	PUNCT
ejpam-7047	173	27	)	)	PUNCT
ejpam-7047	173	28	)	)	PUNCT
ejpam-7047	173	29	)	)	PUNCT
ejpam-7047	174	1	⊆	⊆	X
ejpam-7047	174	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7047	174	3	-	-	PUNCT
ejpam-7047	174	4	cl(b	cl(b	NOUN
ejpam-7047	174	5	)	)	PUNCT
ejpam-7047	174	6	)	)	PUNCT
ejpam-7047	174	7	.	.	PUNCT
ejpam-7047	175	1	(	(	PUNCT
ejpam-7047	175	2	5	5	X
ejpam-7047	175	3	)	)	PUNCT
ejpam-7047	175	4	⇒	⇒	NOUN
ejpam-7047	175	5	(	(	PUNCT
ejpam-7047	175	6	6	6	NUM
ejpam-7047	175	7	):	):	PUNCT
ejpam-7047	175	8	let	let	VERB
ejpam-7047	175	9	b	b	X
ejpam-7047	175	10	be	be	AUX
ejpam-7047	175	11	any	any	DET
ejpam-7047	175	12	subset	subset	NOUN
ejpam-7047	175	13	of	of	ADP
ejpam-7047	175	14	y	y	PROPN
ejpam-7047	175	15	.	.	PUNCT
ejpam-7047	176	1	by	by	ADP
ejpam-7047	176	2	(	(	PUNCT
ejpam-7047	176	3	5	5	NUM
ejpam-7047	176	4	)	)	PUNCT
ejpam-7047	176	5	,	,	PUNCT
ejpam-7047	176	6	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	176	7	-	-	PUNCT
ejpam-7047	176	8	int(b	int(b	NOUN
ejpam-7047	176	9	)	)	PUNCT
ejpam-7047	176	10	)	)	PUNCT
ejpam-7047	176	11	=	=	PUNCT
ejpam-7047	177	1	x	x	PUNCT
ejpam-7047	177	2	−	−	ADP
ejpam-7047	177	3	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7047	177	4	-	-	PUNCT
ejpam-7047	177	5	cl(y	cl(y	NOUN
ejpam-7047	177	6	−b	−b	NOUN
ejpam-7047	177	7	)	)	PUNCT
ejpam-7047	177	8	)	)	PUNCT
ejpam-7047	178	1	⊆	⊆	NUM
ejpam-7047	178	2	x	x	SYM
ejpam-7047	178	3	−	−	PRON
ejpam-7047	178	4	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	178	5	-	-	ADJ
ejpam-7047	178	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	178	7	-	-	PUNCT
ejpam-7047	178	8	cl(y	cl(y	NOUN
ejpam-7047	178	9	−b	−b	NOUN
ejpam-7047	178	10	)	)	PUNCT
ejpam-7047	178	11	)	)	PUNCT
ejpam-7047	178	12	)	)	PUNCT
ejpam-7047	178	13	)	)	PUNCT
ejpam-7047	179	1	=	=	PUNCT
ejpam-7047	180	1	x	x	PUNCT
ejpam-7047	180	2	−	−	PROPN
ejpam-7047	180	3	αcl⋆(f−(y	αcl⋆(f−(y	PUNCT
ejpam-7047	180	4	−	−	ADP
ejpam-7047	180	5	σ1σ2	σ1σ2	X
ejpam-7047	180	6	-	-	PUNCT
ejpam-7047	180	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	180	8	-	-	PUNCT
ejpam-7047	180	9	int(b	int(b	NOUN
ejpam-7047	180	10	)	)	PUNCT
ejpam-7047	180	11	)	)	PUNCT
ejpam-7047	180	12	)	)	PUNCT
ejpam-7047	180	13	)	)	PUNCT
ejpam-7047	181	1	=	=	PUNCT
ejpam-7047	181	2	x	x	PUNCT
ejpam-7047	181	3	−	−	NOUN
ejpam-7047	181	4	αcl⋆(x	αcl⋆(x	NOUN
ejpam-7047	181	5	−	−	ADP
ejpam-7047	181	6	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-7047	181	7	-	-	PUNCT
ejpam-7047	181	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	181	9	-	-	PUNCT
ejpam-7047	181	10	int(b	int(b	NOUN
ejpam-7047	181	11	)	)	PUNCT
ejpam-7047	181	12	)	)	PUNCT
ejpam-7047	181	13	)	)	PUNCT
ejpam-7047	181	14	)	)	PUNCT
ejpam-7047	182	1	n.	n.	PROPN
ejpam-7047	182	2	viriyapong	viriyapong	PROPN
ejpam-7047	182	3	,	,	PUNCT
ejpam-7047	182	4	a.	a.	PROPN
ejpam-7047	182	5	sama	sama	PROPN
ejpam-7047	182	6	-	-	PUNCT
ejpam-7047	182	7	ae	ae	PROPN
ejpam-7047	182	8	,	,	PUNCT
ejpam-7047	182	9	c.	c.	PROPN
ejpam-7047	182	10	boonpok	boonpok	PROPN
ejpam-7047	182	11	/	/	SYM
ejpam-7047	182	12	eur	eur	PROPN
ejpam-7047	182	13	.	.	PUNCT
ejpam-7047	183	1	j.	j.	PROPN
ejpam-7047	183	2	pure	pure	PROPN
ejpam-7047	183	3	appl	appl	PROPN
ejpam-7047	183	4	.	.	PROPN
ejpam-7047	183	5	math	math	PROPN
ejpam-7047	183	6	,	,	PUNCT
ejpam-7047	183	7	18	18	NUM
ejpam-7047	183	8	(	(	PUNCT
ejpam-7047	183	9	4	4	NUM
ejpam-7047	183	10	)	)	PUNCT
ejpam-7047	183	11	(	(	PUNCT
ejpam-7047	183	12	2025	2025	NUM
ejpam-7047	183	13	)	)	PUNCT
ejpam-7047	183	14	,	,	PUNCT
ejpam-7047	183	15	7047	7047	NUM
ejpam-7047	183	16	7	7	NUM
ejpam-7047	183	17	of	of	ADP
ejpam-7047	183	18	11	11	NUM
ejpam-7047	183	19	=	=	SYM
ejpam-7047	183	20	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	183	21	-	-	PUNCT
ejpam-7047	183	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	183	23	-	-	PUNCT
ejpam-7047	183	24	int(b	int(b	NOUN
ejpam-7047	183	25	)	)	PUNCT
ejpam-7047	183	26	)	)	PUNCT
ejpam-7047	183	27	)	)	PUNCT
ejpam-7047	183	28	)	)	PUNCT
ejpam-7047	183	29	.	.	PUNCT
ejpam-7047	184	1	(	(	PUNCT
ejpam-7047	184	2	6	6	X
ejpam-7047	184	3	)	)	PUNCT
ejpam-7047	184	4	⇒	⇒	NOUN
ejpam-7047	184	5	(	(	PUNCT
ejpam-7047	184	6	7	7	NUM
ejpam-7047	184	7	):	):	PUNCT
ejpam-7047	184	8	the	the	DET
ejpam-7047	184	9	proof	proof	NOUN
ejpam-7047	184	10	is	be	AUX
ejpam-7047	184	11	obvious	obvious	ADJ
ejpam-7047	184	12	.	.	PUNCT
ejpam-7047	185	1	(	(	PUNCT
ejpam-7047	185	2	7	7	X
ejpam-7047	185	3	)	)	PUNCT
ejpam-7047	185	4	⇒	⇒	NOUN
ejpam-7047	185	5	(	(	PUNCT
ejpam-7047	185	6	1	1	NUM
ejpam-7047	185	7	):	):	PUNCT
ejpam-7047	185	8	let	let	VERB
ejpam-7047	185	9	x	x	PUNCT
ejpam-7047	185	10	∈	∈	PROPN
ejpam-7047	185	11	x	x	X
ejpam-7047	185	12	and	and	CCONJ
ejpam-7047	185	13	v	v	X
ejpam-7047	185	14	be	be	AUX
ejpam-7047	185	15	any	any	DET
ejpam-7047	185	16	σ1σ2	σ1σ2	NOUN
ejpam-7047	185	17	-	-	ADJ
ejpam-7047	185	18	open	open	ADJ
ejpam-7047	185	19	set	set	NOUN
ejpam-7047	185	20	of	of	ADP
ejpam-7047	185	21	y	y	PROPN
ejpam-7047	185	22	containing	contain	VERB
ejpam-7047	185	23	f	f	PROPN
ejpam-7047	185	24	(	(	PUNCT
ejpam-7047	185	25	x	x	NOUN
ejpam-7047	185	26	)	)	PUNCT
ejpam-7047	185	27	.	.	PUNCT
ejpam-7047	186	1	it	it	PRON
ejpam-7047	186	2	follows	follow	VERB
ejpam-7047	186	3	from	from	ADP
ejpam-7047	186	4	lemma	lemma	PROPN
ejpam-7047	186	5	5	5	NUM
ejpam-7047	186	6	that	that	SCONJ
ejpam-7047	186	7	x	x	SYM
ejpam-7047	186	8	∈	∈	NOUN
ejpam-7047	186	9	f+(v	f+(v	NOUN
ejpam-7047	186	10	)	)	PUNCT
ejpam-7047	187	1	⊆	⊆	X
ejpam-7047	187	2	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	187	3	-	-	PUNCT
ejpam-7047	187	4	cl(v	cl(v	NOUN
ejpam-7047	187	5	)	)	PUNCT
ejpam-7047	187	6	)	)	PUNCT
ejpam-7047	187	7	)	)	PUNCT
ejpam-7047	187	8	⊆	⊆	X
ejpam-7047	187	9	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7047	187	10	-	-	NOUN
ejpam-7047	187	11	cl(v	cl(v	NOUN
ejpam-7047	187	12	)	)	PUNCT
ejpam-7047	187	13	)	)	PUNCT
ejpam-7047	187	14	)	)	PUNCT
ejpam-7047	187	15	)	)	PUNCT
ejpam-7047	187	16	)	)	PUNCT
ejpam-7047	187	17	and	and	CCONJ
ejpam-7047	187	18	hence	hence	ADV
ejpam-7047	187	19	f	f	PROPN
ejpam-7047	187	20	is	be	AUX
ejpam-7047	187	21	upper	upper	ADJ
ejpam-7047	187	22	weakly	weakly	ADJ
ejpam-7047	187	23	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	187	24	,	,	PUNCT
ejpam-7047	187	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	187	26	at	at	ADP
ejpam-7047	187	27	x	x	PUNCT
ejpam-7047	187	28	by	by	ADP
ejpam-7047	187	29	theorem	theorem	NOUN
ejpam-7047	187	30	1	1	NUM
ejpam-7047	187	31	.	.	PUNCT
ejpam-7047	188	1	this	this	PRON
ejpam-7047	188	2	shows	show	VERB
ejpam-7047	188	3	that	that	SCONJ
ejpam-7047	188	4	f	f	PROPN
ejpam-7047	188	5	is	be	AUX
ejpam-7047	188	6	upper	upper	ADJ
ejpam-7047	188	7	weakly	weakly	ADJ
ejpam-7047	188	8	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	188	9	,	,	PUNCT
ejpam-7047	188	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	188	11	.	.	PUNCT
ejpam-7047	189	1	(	(	PUNCT
ejpam-7047	189	2	4	4	X
ejpam-7047	189	3	)	)	PUNCT
ejpam-7047	189	4	⇒	⇒	NOUN
ejpam-7047	189	5	(	(	PUNCT
ejpam-7047	189	6	8)	8)	NUM
ejpam-7047	189	7	:	:	PUNCT
ejpam-7047	189	8	the	the	DET
ejpam-7047	189	9	proof	proof	NOUN
ejpam-7047	189	10	is	be	AUX
ejpam-7047	189	11	obvious	obvious	ADJ
ejpam-7047	189	12	.	.	PUNCT
ejpam-7047	190	1	(	(	PUNCT
ejpam-7047	190	2	8)	8)	NUM
ejpam-7047	190	3	⇒	⇒	NOUN
ejpam-7047	190	4	(	(	PUNCT
ejpam-7047	190	5	9	9	NUM
ejpam-7047	190	6	):	):	PUNCT
ejpam-7047	190	7	let	let	VERB
ejpam-7047	190	8	v	v	PART
ejpam-7047	190	9	be	be	AUX
ejpam-7047	190	10	any	any	DET
ejpam-7047	190	11	σ1σ2	σ1σ2	NOUN
ejpam-7047	190	12	-	-	ADJ
ejpam-7047	190	13	open	open	ADJ
ejpam-7047	190	14	set	set	NOUN
ejpam-7047	190	15	of	of	ADP
ejpam-7047	190	16	y	y	PROPN
ejpam-7047	190	17	.	.	PUNCT
ejpam-7047	191	1	then	then	ADV
ejpam-7047	191	2	,	,	PUNCT
ejpam-7047	191	3	we	we	PRON
ejpam-7047	191	4	have	have	VERB
ejpam-7047	191	5	σ1σ2	σ1σ2	NOUN
ejpam-7047	191	6	-	-	NUM
ejpam-7047	191	7	cl(v	cl(v	NOUN
ejpam-7047	191	8	)	)	PUNCT
ejpam-7047	191	9	is	be	AUX
ejpam-7047	191	10	(	(	PUNCT
ejpam-7047	191	11	σ1	σ1	PROPN
ejpam-7047	191	12	,	,	PUNCT
ejpam-7047	191	13	σ2)rclosed	σ2)rclose	VERB
ejpam-7047	191	14	in	in	ADP
ejpam-7047	191	15	y	y	PROPN
ejpam-7047	191	16	and	and	CCONJ
ejpam-7047	191	17	by	by	ADP
ejpam-7047	191	18	(	(	PUNCT
ejpam-7047	191	19	8)	8)	NUM
ejpam-7047	191	20	,	,	PUNCT
ejpam-7047	191	21	αcl⋆(f−(v	αcl⋆(f−(v	NUM
ejpam-7047	191	22	)	)	PUNCT
ejpam-7047	191	23	)	)	PUNCT
ejpam-7047	192	1	⊆	⊆	NUM
ejpam-7047	192	2	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	VERB
ejpam-7047	192	3	-	-	PUNCT
ejpam-7047	192	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	192	5	-	-	PUNCT
ejpam-7047	192	6	cl(v	cl(v	NOUN
ejpam-7047	192	7	)	)	PUNCT
ejpam-7047	192	8	)	)	PUNCT
ejpam-7047	192	9	)	)	PUNCT
ejpam-7047	192	10	)	)	PUNCT
ejpam-7047	193	1	⊆	⊆	X
ejpam-7047	193	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7047	193	3	-	-	PUNCT
ejpam-7047	193	4	cl(v	cl(v	NOUN
ejpam-7047	193	5	)	)	PUNCT
ejpam-7047	193	6	)	)	PUNCT
ejpam-7047	193	7	.	.	PUNCT
ejpam-7047	194	1	(	(	PUNCT
ejpam-7047	194	2	9	9	X
ejpam-7047	194	3	)	)	PUNCT
ejpam-7047	194	4	⇒	⇒	NOUN
ejpam-7047	194	5	(	(	PUNCT
ejpam-7047	194	6	7	7	NUM
ejpam-7047	194	7	):	):	PUNCT
ejpam-7047	194	8	let	let	VERB
ejpam-7047	194	9	v	v	PART
ejpam-7047	194	10	be	be	AUX
ejpam-7047	194	11	any	any	DET
ejpam-7047	194	12	σ1σ2	σ1σ2	NOUN
ejpam-7047	194	13	-	-	ADJ
ejpam-7047	194	14	open	open	ADJ
ejpam-7047	194	15	set	set	NOUN
ejpam-7047	194	16	of	of	ADP
ejpam-7047	194	17	y	y	PROPN
ejpam-7047	194	18	.	.	PUNCT
ejpam-7047	195	1	thus	thus	ADV
ejpam-7047	195	2	by	by	ADP
ejpam-7047	195	3	(	(	PUNCT
ejpam-7047	195	4	9	9	NUM
ejpam-7047	195	5	)	)	PUNCT
ejpam-7047	195	6	,	,	PUNCT
ejpam-7047	195	7	we	we	PRON
ejpam-7047	195	8	have	have	VERB
ejpam-7047	195	9	x	x	INTJ
ejpam-7047	195	10	−	−	PRON
ejpam-7047	195	11	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	195	12	-	-	PUNCT
ejpam-7047	195	13	cl(v	cl(v	NOUN
ejpam-7047	195	14	)	)	PUNCT
ejpam-7047	195	15	)	)	PUNCT
ejpam-7047	195	16	)	)	PUNCT
ejpam-7047	196	1	=	=	NOUN
ejpam-7047	196	2	αcl⋆(x	αcl⋆(x	NOUN
ejpam-7047	196	3	−	−	ADP
ejpam-7047	196	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	196	5	-	-	PUNCT
ejpam-7047	196	6	cl(v	cl(v	NOUN
ejpam-7047	196	7	)	)	PUNCT
ejpam-7047	196	8	)	)	PUNCT
ejpam-7047	196	9	)	)	PUNCT
ejpam-7047	197	1	=	=	SYM
ejpam-7047	197	2	αcl⋆(f−(y	αcl⋆(f−(y	PUNCT
ejpam-7047	197	3	−	−	ADP
ejpam-7047	197	4	σ1σ2	σ1σ2	NOUN
ejpam-7047	197	5	-	-	NUM
ejpam-7047	197	6	cl(v	cl(v	NOUN
ejpam-7047	197	7	)	)	PUNCT
ejpam-7047	197	8	)	)	PUNCT
ejpam-7047	197	9	)	)	PUNCT
ejpam-7047	198	1	⊆	⊆	X
ejpam-7047	198	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7047	198	3	-	-	PUNCT
ejpam-7047	198	4	cl(y	cl(y	NOUN
ejpam-7047	198	5	−	−	NOUN
ejpam-7047	198	6	σ1σ2	σ1σ2	NOUN
ejpam-7047	198	7	-	-	NUM
ejpam-7047	198	8	cl(v	cl(v	NOUN
ejpam-7047	198	9	)	)	PUNCT
ejpam-7047	198	10	)	)	PUNCT
ejpam-7047	198	11	)	)	PUNCT
ejpam-7047	199	1	=	=	PUNCT
ejpam-7047	199	2	x	x	X
ejpam-7047	199	3	−	−	ADP
ejpam-7047	199	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7047	199	5	-	-	PUNCT
ejpam-7047	199	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	199	7	-	-	PUNCT
ejpam-7047	199	8	cl(v	cl(v	NOUN
ejpam-7047	199	9	)	)	PUNCT
ejpam-7047	199	10	)	)	PUNCT
ejpam-7047	199	11	)	)	PUNCT
ejpam-7047	199	12	and	and	CCONJ
ejpam-7047	199	13	hence	hence	ADV
ejpam-7047	199	14	f+(v	f+(v	NOUN
ejpam-7047	199	15	)	)	PUNCT
ejpam-7047	200	1	⊆	⊆	X
ejpam-7047	200	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7047	200	3	-	-	PUNCT
ejpam-7047	200	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	200	5	-	-	PUNCT
ejpam-7047	200	6	cl(v	cl(v	NOUN
ejpam-7047	200	7	)	)	PUNCT
ejpam-7047	200	8	)	)	PUNCT
ejpam-7047	200	9	)	)	PUNCT
ejpam-7047	201	1	⊆	⊆	X
ejpam-7047	201	2	αint⋆(f+(σ1σ2	αint⋆(f+(σ1σ2	NOUN
ejpam-7047	201	3	-	-	PUNCT
ejpam-7047	201	4	cl(v	cl(v	NOUN
ejpam-7047	201	5	)	)	PUNCT
ejpam-7047	201	6	)	)	PUNCT
ejpam-7047	201	7	)	)	PUNCT
ejpam-7047	201	8	.	.	PUNCT
ejpam-7047	202	1	(	(	PUNCT
ejpam-7047	202	2	9	9	X
ejpam-7047	202	3	)	)	PUNCT
ejpam-7047	202	4	⇒	⇒	NOUN
ejpam-7047	202	5	(	(	PUNCT
ejpam-7047	202	6	10	10	NUM
ejpam-7047	202	7	):	):	PUNCT
ejpam-7047	202	8	let	let	VERB
ejpam-7047	202	9	b	b	X
ejpam-7047	202	10	be	be	AUX
ejpam-7047	202	11	any	any	DET
ejpam-7047	202	12	subset	subset	NOUN
ejpam-7047	202	13	of	of	ADP
ejpam-7047	202	14	y	y	PROPN
ejpam-7047	202	15	.	.	PUNCT
ejpam-7047	203	1	then	then	ADV
ejpam-7047	203	2	,	,	PUNCT
ejpam-7047	203	3	σ1σ2	σ1σ2	NOUN
ejpam-7047	203	4	-	-	PUNCT
ejpam-7047	203	5	int((σ1	int((σ1	ADJ
ejpam-7047	203	6	,	,	PUNCT
ejpam-7047	203	7	σ2)θ	σ2)θ	ADJ
ejpam-7047	203	8	-	-	PUNCT
ejpam-7047	203	9	cl(b	cl(b	NOUN
ejpam-7047	203	10	)	)	PUNCT
ejpam-7047	203	11	)	)	PUNCT
ejpam-7047	203	12	is	be	AUX
ejpam-7047	203	13	σ1σ2	σ1σ2	NOUN
ejpam-7047	203	14	-	-	ADJ
ejpam-7047	203	15	open	open	ADJ
ejpam-7047	203	16	in	in	ADP
ejpam-7047	203	17	y	y	PROPN
ejpam-7047	203	18	.	.	PUNCT
ejpam-7047	204	1	by	by	ADP
ejpam-7047	204	2	(	(	PUNCT
ejpam-7047	204	3	9	9	NUM
ejpam-7047	204	4	)	)	PUNCT
ejpam-7047	204	5	and	and	CCONJ
ejpam-7047	204	6	lemma	lemma	PROPN
ejpam-7047	204	7	2	2	NUM
ejpam-7047	204	8	,	,	PUNCT
ejpam-7047	204	9	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	NOUN
ejpam-7047	204	10	-	-	PUNCT
ejpam-7047	204	11	int((σ1	int((σ1	ADJ
ejpam-7047	204	12	,	,	PUNCT
ejpam-7047	204	13	σ2)θ	σ2)θ	ADJ
ejpam-7047	204	14	-	-	PUNCT
ejpam-7047	204	15	cl(b	cl(b	NOUN
ejpam-7047	204	16	)	)	PUNCT
ejpam-7047	204	17	)	)	PUNCT
ejpam-7047	204	18	)	)	PUNCT
ejpam-7047	204	19	)	)	PUNCT
ejpam-7047	205	1	⊆	⊆	X
ejpam-7047	205	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7047	205	3	-	-	PUNCT
ejpam-7047	205	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7047	205	5	-	-	PUNCT
ejpam-7047	205	6	int((σ1	int((σ1	ADJ
ejpam-7047	205	7	,	,	PUNCT
ejpam-7047	205	8	σ2)θ	σ2)θ	ADJ
ejpam-7047	205	9	-	-	PUNCT
ejpam-7047	205	10	cl(b	cl(b	NOUN
ejpam-7047	205	11	)	)	PUNCT
ejpam-7047	205	12	)	)	PUNCT
ejpam-7047	205	13	)	)	PUNCT
ejpam-7047	205	14	)	)	PUNCT
ejpam-7047	206	1	⊆	⊆	NUM
ejpam-7047	206	2	f−((σ1	f−((σ1	NOUN
ejpam-7047	206	3	,	,	PUNCT
ejpam-7047	206	4	σ2)θ	σ2)θ	NOUN
ejpam-7047	206	5	-	-	PUNCT
ejpam-7047	206	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	206	7	-	-	PUNCT
ejpam-7047	206	8	int((σ1	int((σ1	PROPN
ejpam-7047	206	9	,	,	PUNCT
ejpam-7047	206	10	σ2)θ	σ2)θ	ADJ
ejpam-7047	206	11	-	-	PUNCT
ejpam-7047	206	12	cl(b	cl(b	NOUN
ejpam-7047	206	13	)	)	PUNCT
ejpam-7047	206	14	)	)	PUNCT
ejpam-7047	206	15	)	)	PUNCT
ejpam-7047	206	16	)	)	PUNCT
ejpam-7047	207	1	⊆	⊆	NUM
ejpam-7047	207	2	f−((σ1	f−((σ1	NOUN
ejpam-7047	207	3	,	,	PUNCT
ejpam-7047	207	4	σ2)θ	σ2)θ	ADJ
ejpam-7047	207	5	-	-	PUNCT
ejpam-7047	207	6	cl(b	cl(b	NOUN
ejpam-7047	207	7	)	)	PUNCT
ejpam-7047	207	8	)	)	PUNCT
ejpam-7047	207	9	.	.	PUNCT
ejpam-7047	208	1	(	(	PUNCT
ejpam-7047	208	2	10	10	NUM
ejpam-7047	208	3	)	)	PUNCT
ejpam-7047	208	4	⇒	⇒	NOUN
ejpam-7047	208	5	(	(	PUNCT
ejpam-7047	208	6	8)	8)	NUM
ejpam-7047	208	7	:	:	PUNCT
ejpam-7047	208	8	let	let	VERB
ejpam-7047	208	9	k	k	X
ejpam-7047	208	10	be	be	AUX
ejpam-7047	208	11	any	any	DET
ejpam-7047	208	12	(	(	PUNCT
ejpam-7047	208	13	σ1	σ1	NOUN
ejpam-7047	208	14	,	,	PUNCT
ejpam-7047	208	15	σ2)r	σ2)r	NOUN
ejpam-7047	208	16	-	-	PUNCT
ejpam-7047	208	17	closed	close	VERB
ejpam-7047	208	18	set	set	NOUN
ejpam-7047	208	19	of	of	ADP
ejpam-7047	208	20	y	y	PROPN
ejpam-7047	208	21	.	.	PUNCT
ejpam-7047	209	1	then	then	ADV
ejpam-7047	209	2	by	by	ADP
ejpam-7047	209	3	(	(	PUNCT
ejpam-7047	209	4	10	10	NUM
ejpam-7047	209	5	)	)	PUNCT
ejpam-7047	209	6	and	and	CCONJ
ejpam-7047	209	7	lemma	lemma	PROPN
ejpam-7047	209	8	2	2	NUM
ejpam-7047	209	9	,	,	PUNCT
ejpam-7047	209	10	we	we	PRON
ejpam-7047	209	11	have	have	VERB
ejpam-7047	209	12	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	VERB
ejpam-7047	209	13	-	-	ADJ
ejpam-7047	209	14	int(k	int(k	NUM
ejpam-7047	209	15	)	)	PUNCT
ejpam-7047	209	16	)	)	PUNCT
ejpam-7047	209	17	)	)	PUNCT
ejpam-7047	210	1	=	=	NOUN
ejpam-7047	210	2	αcl⋆(f−(σ1σ2	αcl⋆(f−(σ1σ2	VERB
ejpam-7047	210	3	-	-	VERB
ejpam-7047	210	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-7047	210	5	-	-	PUNCT
ejpam-7047	210	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7047	210	7	-	-	PUNCT
ejpam-7047	210	8	int(k	int(k	NOUN
ejpam-7047	210	9	)	)	PUNCT
ejpam-7047	210	10	)	)	PUNCT
ejpam-7047	210	11	)	)	PUNCT
ejpam-7047	210	12	)	)	PUNCT
ejpam-7047	210	13	)	)	PUNCT
ejpam-7047	211	1	=	=	PUNCT
ejpam-7047	211	2	αcl⋆(σ1σ2	αcl⋆(σ1σ2	NOUN
ejpam-7047	211	3	-	-	PUNCT
ejpam-7047	211	4	int((σ1	int((σ1	NUM
ejpam-7047	211	5	,	,	PUNCT
ejpam-7047	211	6	σ2)θ	σ2)θ	NOUN
ejpam-7047	211	7	-	-	PUNCT
ejpam-7047	211	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	211	9	-	-	PUNCT
ejpam-7047	211	10	int(k	int(k	NOUN
ejpam-7047	211	11	)	)	PUNCT
ejpam-7047	211	12	)	)	PUNCT
ejpam-7047	211	13	)	)	PUNCT
ejpam-7047	211	14	)	)	PUNCT
ejpam-7047	212	1	⊆	⊆	NUM
ejpam-7047	212	2	f−((σ1	f−((σ1	NOUN
ejpam-7047	212	3	,	,	PUNCT
ejpam-7047	212	4	σ2)θ	σ2)θ	NOUN
ejpam-7047	212	5	-	-	PUNCT
ejpam-7047	212	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	212	7	-	-	PUNCT
ejpam-7047	212	8	int(k	int(k	NOUN
ejpam-7047	212	9	)	)	PUNCT
ejpam-7047	212	10	)	)	PUNCT
ejpam-7047	212	11	)	)	PUNCT
ejpam-7047	213	1	=	=	PUNCT
ejpam-7047	213	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7047	213	3	-	-	PUNCT
ejpam-7047	213	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	213	5	-	-	PUNCT
ejpam-7047	213	6	int(k	int(k	NOUN
ejpam-7047	213	7	)	)	PUNCT
ejpam-7047	213	8	)	)	PUNCT
ejpam-7047	213	9	)	)	PUNCT
ejpam-7047	213	10	=	=	SYM
ejpam-7047	213	11	f−(k	f−(k	PROPN
ejpam-7047	213	12	)	)	PUNCT
ejpam-7047	213	13	.	.	PUNCT
ejpam-7047	214	1	definition	definition	NOUN
ejpam-7047	214	2	3	3	NUM
ejpam-7047	214	3	.	.	PUNCT
ejpam-7047	215	1	[	[	X
ejpam-7047	215	2	21	21	NUM
ejpam-7047	215	3	]	]	PUNCT
ejpam-7047	215	4	a	a	DET
ejpam-7047	215	5	multifunction	multifunction	NOUN
ejpam-7047	215	6	f	f	NOUN
ejpam-7047	215	7	:	:	PUNCT
ejpam-7047	215	8	(	(	PUNCT
ejpam-7047	215	9	x	x	X
ejpam-7047	215	10	,	,	PUNCT
ejpam-7047	215	11	τ	τ	PROPN
ejpam-7047	215	12	,	,	PUNCT
ejpam-7047	215	13	i	i	NOUN
ejpam-7047	215	14	)	)	PUNCT
ejpam-7047	215	15	→	→	PUNCT
ejpam-7047	215	16	(	(	PUNCT
ejpam-7047	215	17	y	y	PROPN
ejpam-7047	215	18	,	,	PUNCT
ejpam-7047	215	19	σ1	σ1	PROPN
ejpam-7047	215	20	,	,	PUNCT
ejpam-7047	215	21	σ2	σ2	PROPN
ejpam-7047	215	22	)	)	PUNCT
ejpam-7047	215	23	is	be	AUX
ejpam-7047	215	24	said	say	VERB
ejpam-7047	215	25	to	to	PART
ejpam-7047	215	26	be	be	AUX
ejpam-7047	215	27	upper	upper	ADJ
ejpam-7047	215	28	almost	almost	ADV
ejpam-7047	215	29	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	215	30	,	,	PUNCT
ejpam-7047	215	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	215	32	at	at	ADP
ejpam-7047	215	33	a	a	DET
ejpam-7047	215	34	point	point	NOUN
ejpam-7047	215	35	x	x	PUNCT
ejpam-7047	215	36	of	of	ADP
ejpam-7047	215	37	x	x	PRON
ejpam-7047	215	38	if	if	SCONJ
ejpam-7047	215	39	for	for	ADP
ejpam-7047	215	40	each	each	DET
ejpam-7047	215	41	σ1σ2	σ1σ2	VERB
ejpam-7047	215	42	-	-	ADJ
ejpam-7047	215	43	open	open	ADJ
ejpam-7047	215	44	set	set	NOUN
ejpam-7047	215	45	v	v	NOUN
ejpam-7047	215	46	of	of	ADP
ejpam-7047	215	47	y	y	PRON
ejpam-7047	215	48	such	such	ADJ
ejpam-7047	215	49	that	that	SCONJ
ejpam-7047	215	50	f	f	PROPN
ejpam-7047	215	51	(	(	PUNCT
ejpam-7047	215	52	x	x	X
ejpam-7047	215	53	)	)	PUNCT
ejpam-7047	215	54	⊆	⊆	NUM
ejpam-7047	215	55	v	v	NOUN
ejpam-7047	215	56	,	,	PUNCT
ejpam-7047	215	57	there	there	PRON
ejpam-7047	215	58	exists	exist	VERB
ejpam-7047	215	59	a	a	DET
ejpam-7047	215	60	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	215	61	-	-	PUNCT
ejpam-7047	215	62	open	open	ADJ
ejpam-7047	215	63	set	set	NOUN
ejpam-7047	215	64	u	u	NOUN
ejpam-7047	215	65	of	of	ADP
ejpam-7047	215	66	x	x	PUNCT
ejpam-7047	215	67	containing	contain	VERB
ejpam-7047	215	68	x	x	PUNCT
ejpam-7047	215	69	such	such	ADJ
ejpam-7047	215	70	that	that	SCONJ
ejpam-7047	215	71	f	f	PROPN
ejpam-7047	215	72	(	(	PUNCT
ejpam-7047	215	73	u	u	NOUN
ejpam-7047	215	74	)	)	PUNCT
ejpam-7047	215	75	⊆	⊆	NUM
ejpam-7047	215	76	σ1σ2	σ1σ2	X
ejpam-7047	215	77	-	-	PUNCT
ejpam-7047	215	78	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	215	79	-	-	PUNCT
ejpam-7047	215	80	cl(v	cl(v	NOUN
ejpam-7047	215	81	)	)	PUNCT
ejpam-7047	215	82	)	)	PUNCT
ejpam-7047	215	83	.	.	PUNCT
ejpam-7047	216	1	a	a	DET
ejpam-7047	216	2	multifunction	multifunction	NOUN
ejpam-7047	216	3	f	f	NOUN
ejpam-7047	216	4	:	:	PUNCT
ejpam-7047	216	5	(	(	PUNCT
ejpam-7047	216	6	x	x	X
ejpam-7047	216	7	,	,	PUNCT
ejpam-7047	216	8	τ	τ	PROPN
ejpam-7047	216	9	,	,	PUNCT
ejpam-7047	216	10	i	i	NOUN
ejpam-7047	216	11	)	)	PUNCT
ejpam-7047	216	12	→	→	PUNCT
ejpam-7047	216	13	(	(	PUNCT
ejpam-7047	216	14	y	y	PROPN
ejpam-7047	216	15	,	,	PUNCT
ejpam-7047	216	16	σ1	σ1	PROPN
ejpam-7047	216	17	,	,	PUNCT
ejpam-7047	216	18	σ2	σ2	PROPN
ejpam-7047	216	19	)	)	PUNCT
ejpam-7047	216	20	is	be	AUX
ejpam-7047	216	21	said	say	VERB
ejpam-7047	216	22	to	to	PART
ejpam-7047	216	23	be	be	AUX
ejpam-7047	216	24	upper	upper	ADJ
ejpam-7047	216	25	almost	almost	ADV
ejpam-7047	216	26	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	216	27	,	,	PUNCT
ejpam-7047	216	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	216	29	if	if	SCONJ
ejpam-7047	216	30	f	f	PROPN
ejpam-7047	216	31	is	be	AUX
ejpam-7047	216	32	upper	upper	ADJ
ejpam-7047	216	33	almost	almost	ADV
ejpam-7047	216	34	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	216	35	,	,	PUNCT
ejpam-7047	216	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	216	37	at	at	ADP
ejpam-7047	216	38	each	each	DET
ejpam-7047	216	39	point	point	NOUN
ejpam-7047	216	40	of	of	ADP
ejpam-7047	216	41	x.	x.	PROPN
ejpam-7047	216	42	n.	n.	PROPN
ejpam-7047	216	43	viriyapong	viriyapong	PROPN
ejpam-7047	216	44	,	,	PUNCT
ejpam-7047	216	45	a.	a.	PROPN
ejpam-7047	216	46	sama	sama	PROPN
ejpam-7047	216	47	-	-	PUNCT
ejpam-7047	216	48	ae	ae	PROPN
ejpam-7047	216	49	,	,	PUNCT
ejpam-7047	216	50	c.	c.	PROPN
ejpam-7047	216	51	boonpok	boonpok	PROPN
ejpam-7047	216	52	/	/	SYM
ejpam-7047	216	53	eur	eur	PROPN
ejpam-7047	216	54	.	.	PUNCT
ejpam-7047	217	1	j.	j.	PROPN
ejpam-7047	217	2	pure	pure	PROPN
ejpam-7047	217	3	appl	appl	PROPN
ejpam-7047	217	4	.	.	PROPN
ejpam-7047	217	5	math	math	PROPN
ejpam-7047	217	6	,	,	PUNCT
ejpam-7047	217	7	18	18	NUM
ejpam-7047	217	8	(	(	PUNCT
ejpam-7047	217	9	4	4	NUM
ejpam-7047	217	10	)	)	PUNCT
ejpam-7047	217	11	(	(	PUNCT
ejpam-7047	217	12	2025	2025	NUM
ejpam-7047	217	13	)	)	PUNCT
ejpam-7047	217	14	,	,	PUNCT
ejpam-7047	217	15	7047	7047	NUM
ejpam-7047	217	16	8	8	NUM
ejpam-7047	217	17	of	of	ADP
ejpam-7047	217	18	11	11	NUM
ejpam-7047	217	19	definition	definition	NOUN
ejpam-7047	217	20	4	4	NUM
ejpam-7047	217	21	.	.	PUNCT
ejpam-7047	218	1	[	[	X
ejpam-7047	218	2	21	21	NUM
ejpam-7047	218	3	]	]	PUNCT
ejpam-7047	218	4	a	a	DET
ejpam-7047	218	5	multifunction	multifunction	NOUN
ejpam-7047	218	6	f	f	NOUN
ejpam-7047	218	7	:	:	PUNCT
ejpam-7047	218	8	(	(	PUNCT
ejpam-7047	218	9	x	x	X
ejpam-7047	218	10	,	,	PUNCT
ejpam-7047	218	11	τ	τ	PROPN
ejpam-7047	218	12	,	,	PUNCT
ejpam-7047	218	13	i	i	NOUN
ejpam-7047	218	14	)	)	PUNCT
ejpam-7047	218	15	→	→	PUNCT
ejpam-7047	218	16	(	(	PUNCT
ejpam-7047	218	17	y	y	PROPN
ejpam-7047	218	18	,	,	PUNCT
ejpam-7047	218	19	σ1	σ1	PROPN
ejpam-7047	218	20	,	,	PUNCT
ejpam-7047	218	21	σ2	σ2	PROPN
ejpam-7047	218	22	)	)	PUNCT
ejpam-7047	218	23	is	be	AUX
ejpam-7047	218	24	said	say	VERB
ejpam-7047	218	25	to	to	PART
ejpam-7047	218	26	be	be	AUX
ejpam-7047	218	27	lower	low	ADJ
ejpam-7047	218	28	almost	almost	ADV
ejpam-7047	218	29	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	218	30	,	,	PUNCT
ejpam-7047	218	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	218	32	at	at	ADP
ejpam-7047	218	33	a	a	DET
ejpam-7047	218	34	point	point	NOUN
ejpam-7047	218	35	x	x	SYM
ejpam-7047	218	36	∈	∈	NOUN
ejpam-7047	218	37	x	x	PUNCT
ejpam-7047	218	38	if	if	SCONJ
ejpam-7047	218	39	for	for	ADP
ejpam-7047	218	40	each	each	DET
ejpam-7047	218	41	σ1σ2	σ1σ2	VERB
ejpam-7047	218	42	-	-	ADJ
ejpam-7047	218	43	open	open	ADJ
ejpam-7047	218	44	set	set	NOUN
ejpam-7047	218	45	v	v	NOUN
ejpam-7047	218	46	of	of	ADP
ejpam-7047	218	47	y	y	PRON
ejpam-7047	218	48	such	such	ADJ
ejpam-7047	218	49	that	that	SCONJ
ejpam-7047	218	50	f	f	PROPN
ejpam-7047	218	51	(	(	PUNCT
ejpam-7047	218	52	x	x	NOUN
ejpam-7047	218	53	)	)	PUNCT
ejpam-7047	218	54	∩	∩	NOUN
ejpam-7047	218	55	v	v	ADP
ejpam-7047	218	56	̸=	̸=	PROPN
ejpam-7047	218	57	∅	∅	NOUN
ejpam-7047	218	58	,	,	PUNCT
ejpam-7047	218	59	there	there	PRON
ejpam-7047	218	60	exists	exist	VERB
ejpam-7047	218	61	a	a	DET
ejpam-7047	218	62	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	218	63	-	-	PUNCT
ejpam-7047	218	64	open	open	ADJ
ejpam-7047	218	65	set	set	NOUN
ejpam-7047	218	66	u	u	NOUN
ejpam-7047	218	67	of	of	ADP
ejpam-7047	218	68	x	x	PUNCT
ejpam-7047	218	69	containing	contain	VERB
ejpam-7047	218	70	x	x	PUNCT
ejpam-7047	218	71	such	such	ADJ
ejpam-7047	218	72	that	that	SCONJ
ejpam-7047	218	73	f	f	PROPN
ejpam-7047	218	74	(	(	PUNCT
ejpam-7047	218	75	z	z	NOUN
ejpam-7047	218	76	)	)	PUNCT
ejpam-7047	218	77	∩	∩	NOUN
ejpam-7047	218	78	σ1σ2	σ1σ2	X
ejpam-7047	218	79	-	-	PUNCT
ejpam-7047	218	80	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	218	81	-	-	PUNCT
ejpam-7047	218	82	cl(v	cl(v	NOUN
ejpam-7047	218	83	)	)	PUNCT
ejpam-7047	218	84	)	)	PUNCT
ejpam-7047	219	1	̸=	̸=	NOUN
ejpam-7047	219	2	∅	∅	NOUN
ejpam-7047	219	3	for	for	ADP
ejpam-7047	219	4	every	every	DET
ejpam-7047	219	5	z	z	NOUN
ejpam-7047	219	6	∈	∈	PROPN
ejpam-7047	219	7	u	u	NOUN
ejpam-7047	219	8	.	.	PUNCT
ejpam-7047	220	1	a	a	DET
ejpam-7047	220	2	multifunction	multifunction	NOUN
ejpam-7047	220	3	f	f	NOUN
ejpam-7047	220	4	:	:	PUNCT
ejpam-7047	220	5	(	(	PUNCT
ejpam-7047	220	6	x	x	X
ejpam-7047	220	7	,	,	PUNCT
ejpam-7047	220	8	τ	τ	PROPN
ejpam-7047	220	9	,	,	PUNCT
ejpam-7047	220	10	i	i	NOUN
ejpam-7047	220	11	)	)	PUNCT
ejpam-7047	220	12	→	→	PUNCT
ejpam-7047	220	13	(	(	PUNCT
ejpam-7047	220	14	y	y	PROPN
ejpam-7047	220	15	,	,	PUNCT
ejpam-7047	220	16	σ1	σ1	PROPN
ejpam-7047	220	17	,	,	PUNCT
ejpam-7047	220	18	σ2	σ2	PROPN
ejpam-7047	220	19	)	)	PUNCT
ejpam-7047	220	20	is	be	AUX
ejpam-7047	220	21	said	say	VERB
ejpam-7047	220	22	to	to	PART
ejpam-7047	220	23	be	be	AUX
ejpam-7047	220	24	lower	low	ADJ
ejpam-7047	220	25	almost	almost	ADV
ejpam-7047	220	26	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	220	27	,	,	PUNCT
ejpam-7047	220	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	220	29	if	if	SCONJ
ejpam-7047	220	30	f	f	PROPN
ejpam-7047	220	31	is	be	AUX
ejpam-7047	220	32	lower	low	ADJ
ejpam-7047	220	33	almost	almost	ADV
ejpam-7047	220	34	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	220	35	,	,	PUNCT
ejpam-7047	220	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	220	37	at	at	ADP
ejpam-7047	220	38	each	each	DET
ejpam-7047	220	39	point	point	NOUN
ejpam-7047	220	40	of	of	ADP
ejpam-7047	220	41	x.	x.	NOUN
ejpam-7047	220	42	remark	remark	PROPN
ejpam-7047	220	43	1	1	NUM
ejpam-7047	220	44	.	.	PUNCT
ejpam-7047	220	45	for	for	ADP
ejpam-7047	220	46	a	a	DET
ejpam-7047	220	47	multifunction	multifunction	NOUN
ejpam-7047	220	48	f	f	NOUN
ejpam-7047	220	49	:	:	PUNCT
ejpam-7047	220	50	(	(	PUNCT
ejpam-7047	220	51	x	x	X
ejpam-7047	220	52	,	,	PUNCT
ejpam-7047	220	53	τ	τ	PROPN
ejpam-7047	220	54	,	,	PUNCT
ejpam-7047	220	55	i	i	NOUN
ejpam-7047	220	56	)	)	PUNCT
ejpam-7047	220	57	→	→	PUNCT
ejpam-7047	220	58	(	(	PUNCT
ejpam-7047	220	59	y	y	PROPN
ejpam-7047	220	60	,	,	PUNCT
ejpam-7047	220	61	σ1	σ1	PROPN
ejpam-7047	220	62	,	,	PUNCT
ejpam-7047	220	63	σ2	σ2	NOUN
ejpam-7047	220	64	)	)	PUNCT
ejpam-7047	220	65	,	,	PUNCT
ejpam-7047	220	66	the	the	DET
ejpam-7047	220	67	following	follow	VERB
ejpam-7047	220	68	implication	implication	NOUN
ejpam-7047	220	69	holds	hold	VERB
ejpam-7047	220	70	:	:	PUNCT
ejpam-7047	220	71	upper	upper	ADJ
ejpam-7047	220	72	almost	almost	ADV
ejpam-7047	220	73	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7047	220	74	,	,	PUNCT
ejpam-7047	220	75	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7047	220	76	⇒	⇒	VERB
ejpam-7047	220	77	upper	upper	ADJ
ejpam-7047	220	78	weakly	weakly	ADJ
ejpam-7047	220	79	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	220	80	,	,	PUNCT
ejpam-7047	220	81	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7047	220	82	.	.	PUNCT
ejpam-7047	221	1	the	the	DET
ejpam-7047	221	2	converse	converse	NOUN
ejpam-7047	221	3	of	of	ADP
ejpam-7047	221	4	the	the	DET
ejpam-7047	221	5	implication	implication	NOUN
ejpam-7047	221	6	is	be	AUX
ejpam-7047	221	7	not	not	PART
ejpam-7047	221	8	true	true	ADJ
ejpam-7047	221	9	in	in	ADP
ejpam-7047	221	10	general	general	ADJ
ejpam-7047	221	11	.	.	PUNCT
ejpam-7047	222	1	we	we	PRON
ejpam-7047	222	2	give	give	VERB
ejpam-7047	222	3	an	an	DET
ejpam-7047	222	4	example	example	NOUN
ejpam-7047	222	5	for	for	ADP
ejpam-7047	222	6	the	the	DET
ejpam-7047	222	7	implication	implication	NOUN
ejpam-7047	222	8	as	as	SCONJ
ejpam-7047	222	9	follows	follow	VERB
ejpam-7047	222	10	.	.	PUNCT
ejpam-7047	222	11	example	example	NOUN
ejpam-7047	223	1	1	1	NUM
ejpam-7047	223	2	.	.	PUNCT
ejpam-7047	223	3	let	let	VERB
ejpam-7047	223	4	x	x	PUNCT
ejpam-7047	223	5	=	=	PRON
ejpam-7047	223	6	{	{	PUNCT
ejpam-7047	223	7	1	1	NUM
ejpam-7047	223	8	,	,	PUNCT
ejpam-7047	223	9	2	2	NUM
ejpam-7047	223	10	,	,	PUNCT
ejpam-7047	223	11	3	3	NUM
ejpam-7047	223	12	}	}	PUNCT
ejpam-7047	223	13	with	with	ADP
ejpam-7047	223	14	a	a	DET
ejpam-7047	223	15	topology	topology	NOUN
ejpam-7047	223	16	τ	τ	X
ejpam-7047	223	17	=	=	SYM
ejpam-7047	223	18	{	{	PUNCT
ejpam-7047	223	19	∅	∅	NOUN
ejpam-7047	223	20	,	,	PUNCT
ejpam-7047	223	21	x	x	NOUN
ejpam-7047	223	22	}	}	PUNCT
ejpam-7047	223	23	and	and	CCONJ
ejpam-7047	223	24	an	an	DET
ejpam-7047	223	25	ideal	ideal	NOUN
ejpam-7047	223	26	i	i	X
ejpam-7047	223	27	=	=	SYM
ejpam-7047	223	28	{	{	PUNCT
ejpam-7047	223	29	∅	∅	NOUN
ejpam-7047	223	30	}	}	PUNCT
ejpam-7047	223	31	.	.	PUNCT
ejpam-7047	224	1	let	let	VERB
ejpam-7047	224	2	y	y	PROPN
ejpam-7047	224	3	=	=	PUNCT
ejpam-7047	224	4	{	{	PUNCT
ejpam-7047	224	5	a	a	PRON
ejpam-7047	224	6	,	,	PUNCT
ejpam-7047	224	7	b	b	NOUN
ejpam-7047	224	8	,	,	PUNCT
ejpam-7047	224	9	c	c	NOUN
ejpam-7047	224	10	}	}	PUNCT
ejpam-7047	224	11	with	with	ADP
ejpam-7047	224	12	topologies	topology	NOUN
ejpam-7047	224	13	σ1	σ1	NOUN
ejpam-7047	224	14	=	=	SYM
ejpam-7047	224	15	{	{	PUNCT
ejpam-7047	224	16	∅	∅	NOUN
ejpam-7047	224	17	,	,	PUNCT
ejpam-7047	224	18	{	{	PUNCT
ejpam-7047	224	19	a	a	X
ejpam-7047	224	20	}	}	PUNCT
ejpam-7047	224	21	,	,	PUNCT
ejpam-7047	224	22	{	{	PUNCT
ejpam-7047	224	23	b	b	NOUN
ejpam-7047	224	24	}	}	PUNCT
ejpam-7047	224	25	,	,	PUNCT
ejpam-7047	224	26	{	{	PUNCT
ejpam-7047	224	27	a	a	DET
ejpam-7047	224	28	,	,	PUNCT
ejpam-7047	224	29	b	b	NOUN
ejpam-7047	224	30	}	}	PUNCT
ejpam-7047	224	31	,	,	PUNCT
ejpam-7047	224	32	y	y	PROPN
ejpam-7047	224	33	}	}	PUNCT
ejpam-7047	224	34	and	and	CCONJ
ejpam-7047	224	35	σ2	σ2	PROPN
ejpam-7047	224	36	=	=	SYM
ejpam-7047	224	37	{	{	PUNCT
ejpam-7047	224	38	∅	∅	NOUN
ejpam-7047	224	39	,	,	PUNCT
ejpam-7047	224	40	{	{	PUNCT
ejpam-7047	224	41	a	a	X
ejpam-7047	224	42	}	}	PUNCT
ejpam-7047	224	43	,	,	PUNCT
ejpam-7047	224	44	{	{	PUNCT
ejpam-7047	224	45	b	b	NOUN
ejpam-7047	224	46	}	}	PUNCT
ejpam-7047	224	47	,	,	PUNCT
ejpam-7047	224	48	{	{	PUNCT
ejpam-7047	224	49	a	a	DET
ejpam-7047	224	50	,	,	PUNCT
ejpam-7047	224	51	b	b	NOUN
ejpam-7047	224	52	}	}	PUNCT
ejpam-7047	224	53	,	,	PUNCT
ejpam-7047	224	54	{	{	PUNCT
ejpam-7047	224	55	a	a	X
ejpam-7047	224	56	,	,	PUNCT
ejpam-7047	224	57	c	c	NOUN
ejpam-7047	224	58	}	}	PUNCT
ejpam-7047	224	59	,	,	PUNCT
ejpam-7047	224	60	y	y	PROPN
ejpam-7047	224	61	}	}	PUNCT
ejpam-7047	224	62	.	.	PUNCT
ejpam-7047	225	1	define	define	VERB
ejpam-7047	225	2	a	a	DET
ejpam-7047	225	3	multifunction	multifunction	NOUN
ejpam-7047	225	4	f	f	NOUN
ejpam-7047	225	5	:	:	PUNCT
ejpam-7047	225	6	(	(	PUNCT
ejpam-7047	225	7	x	x	X
ejpam-7047	225	8	,	,	PUNCT
ejpam-7047	225	9	τ	τ	PROPN
ejpam-7047	225	10	,	,	PUNCT
ejpam-7047	225	11	i	i	NOUN
ejpam-7047	225	12	)	)	PUNCT
ejpam-7047	225	13	→	→	PUNCT
ejpam-7047	225	14	(	(	PUNCT
ejpam-7047	225	15	y	y	PROPN
ejpam-7047	225	16	,	,	PUNCT
ejpam-7047	225	17	σ1	σ1	PROPN
ejpam-7047	225	18	,	,	PUNCT
ejpam-7047	225	19	σ2	σ2	PROPN
ejpam-7047	225	20	)	)	PUNCT
ejpam-7047	225	21	as	as	SCONJ
ejpam-7047	225	22	follows	follow	VERB
ejpam-7047	225	23	:	:	PUNCT
ejpam-7047	225	24	f	f	X
ejpam-7047	225	25	(	(	PUNCT
ejpam-7047	225	26	1	1	X
ejpam-7047	225	27	)	)	PUNCT
ejpam-7047	225	28	=	=	PRON
ejpam-7047	225	29	{	{	PUNCT
ejpam-7047	225	30	a	a	NOUN
ejpam-7047	225	31	}	}	PUNCT
ejpam-7047	225	32	,	,	PUNCT
ejpam-7047	225	33	f	f	PROPN
ejpam-7047	225	34	(	(	PUNCT
ejpam-7047	225	35	2	2	NUM
ejpam-7047	225	36	)	)	PUNCT
ejpam-7047	225	37	=	=	PRON
ejpam-7047	225	38	{	{	PUNCT
ejpam-7047	225	39	b	b	NOUN
ejpam-7047	225	40	}	}	PUNCT
ejpam-7047	225	41	and	and	CCONJ
ejpam-7047	225	42	f	f	PROPN
ejpam-7047	225	43	(	(	PUNCT
ejpam-7047	225	44	3	3	X
ejpam-7047	225	45	)	)	PUNCT
ejpam-7047	225	46	=	=	PRON
ejpam-7047	225	47	{	{	PUNCT
ejpam-7047	225	48	a	a	X
ejpam-7047	225	49	,	,	PUNCT
ejpam-7047	225	50	c	c	NOUN
ejpam-7047	225	51	}	}	PUNCT
ejpam-7047	225	52	.	.	PUNCT
ejpam-7047	226	1	then	then	ADV
ejpam-7047	226	2	f	f	PROPN
ejpam-7047	226	3	is	be	AUX
ejpam-7047	226	4	upper	upper	ADJ
ejpam-7047	226	5	weakly	weakly	ADJ
ejpam-7047	226	6	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	226	7	,	,	PUNCT
ejpam-7047	226	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	226	9	but	but	CCONJ
ejpam-7047	226	10	f	f	PROPN
ejpam-7047	226	11	is	be	AUX
ejpam-7047	226	12	not	not	PART
ejpam-7047	226	13	upper	upper	ADJ
ejpam-7047	226	14	almost	almost	ADV
ejpam-7047	226	15	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	226	16	,	,	PUNCT
ejpam-7047	226	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	226	18	.	.	X
ejpam-7047	226	19	theorem	theorem	NOUN
ejpam-7047	226	20	4	4	NUM
ejpam-7047	226	21	.	.	X
ejpam-7047	226	22	for	for	ADP
ejpam-7047	226	23	a	a	DET
ejpam-7047	226	24	multifunction	multifunction	NOUN
ejpam-7047	226	25	f	f	NOUN
ejpam-7047	226	26	:	:	PUNCT
ejpam-7047	226	27	(	(	PUNCT
ejpam-7047	226	28	x	x	X
ejpam-7047	226	29	,	,	PUNCT
ejpam-7047	226	30	τ	τ	PROPN
ejpam-7047	226	31	,	,	PUNCT
ejpam-7047	226	32	i	i	NOUN
ejpam-7047	226	33	)	)	PUNCT
ejpam-7047	226	34	→	→	PUNCT
ejpam-7047	226	35	(	(	PUNCT
ejpam-7047	226	36	y	y	PROPN
ejpam-7047	226	37	,	,	PUNCT
ejpam-7047	226	38	σ1	σ1	PROPN
ejpam-7047	226	39	,	,	PUNCT
ejpam-7047	226	40	σ2	σ2	NOUN
ejpam-7047	226	41	)	)	PUNCT
ejpam-7047	226	42	,	,	PUNCT
ejpam-7047	226	43	the	the	DET
ejpam-7047	226	44	following	follow	VERB
ejpam-7047	226	45	properties	property	NOUN
ejpam-7047	226	46	are	be	AUX
ejpam-7047	226	47	equivalent	equivalent	ADJ
ejpam-7047	226	48	:	:	PUNCT
ejpam-7047	226	49	(	(	PUNCT
ejpam-7047	226	50	1	1	X
ejpam-7047	226	51	)	)	PUNCT
ejpam-7047	226	52	f	f	PROPN
ejpam-7047	226	53	is	be	AUX
ejpam-7047	226	54	lower	low	ADJ
ejpam-7047	226	55	weakly	weakly	ADJ
ejpam-7047	226	56	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	226	57	,	,	PUNCT
ejpam-7047	226	58	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	226	59	;	;	PUNCT
ejpam-7047	226	60	(	(	PUNCT
ejpam-7047	226	61	2	2	X
ejpam-7047	226	62	)	)	PUNCT
ejpam-7047	226	63	f−(v	f−(v	NOUN
ejpam-7047	226	64	)	)	PUNCT
ejpam-7047	226	65	⊆	⊆	NUM
ejpam-7047	226	66	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-7047	226	67	-	-	PUNCT
ejpam-7047	226	68	cl(v	cl(v	NOUN
ejpam-7047	226	69	)	)	PUNCT
ejpam-7047	226	70	)	)	PUNCT
ejpam-7047	226	71	)	)	PUNCT
ejpam-7047	226	72	)	)	PUNCT
ejpam-7047	226	73	)	)	PUNCT
ejpam-7047	227	1	for	for	ADP
ejpam-7047	227	2	every	every	DET
ejpam-7047	227	3	σ1σ2	σ1σ2	NOUN
ejpam-7047	227	4	-	-	ADJ
ejpam-7047	227	5	open	open	ADJ
ejpam-7047	227	6	set	set	NOUN
ejpam-7047	227	7	v	v	NOUN
ejpam-7047	227	8	of	of	ADP
ejpam-7047	227	9	y	y	PROPN
ejpam-7047	227	10	;	;	PUNCT
ejpam-7047	227	11	(	(	PUNCT
ejpam-7047	227	12	3	3	X
ejpam-7047	227	13	)	)	PUNCT
ejpam-7047	227	14	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7047	227	15	-	-	PUNCT
ejpam-7047	227	16	int(k	int(k	NOUN
ejpam-7047	227	17	)	)	PUNCT
ejpam-7047	227	18	)	)	PUNCT
ejpam-7047	227	19	)	)	PUNCT
ejpam-7047	227	20	)	)	PUNCT
ejpam-7047	227	21	)	)	PUNCT
ejpam-7047	227	22	⊆	⊆	NUM
ejpam-7047	227	23	f+(k	f+(k	NOUN
ejpam-7047	227	24	)	)	PUNCT
ejpam-7047	227	25	for	for	ADP
ejpam-7047	227	26	every	every	DET
ejpam-7047	227	27	σ1σ2	σ1σ2	NUM
ejpam-7047	227	28	-	-	PUNCT
ejpam-7047	227	29	closed	closed	ADJ
ejpam-7047	227	30	set	set	NOUN
ejpam-7047	227	31	k	k	PROPN
ejpam-7047	227	32	of	of	ADP
ejpam-7047	227	33	y	y	PROPN
ejpam-7047	227	34	;	;	PUNCT
ejpam-7047	227	35	(	(	PUNCT
ejpam-7047	227	36	4	4	X
ejpam-7047	227	37	)	)	PUNCT
ejpam-7047	227	38	αcl⋆(f+(σ1σ2	αcl⋆(f+(σ1σ2	NOUN
ejpam-7047	227	39	-	-	PUNCT
ejpam-7047	227	40	int(k	int(k	NOUN
ejpam-7047	227	41	)	)	PUNCT
ejpam-7047	227	42	)	)	PUNCT
ejpam-7047	227	43	)	)	PUNCT
ejpam-7047	227	44	⊆	⊆	NUM
ejpam-7047	227	45	f+(k	f+(k	NOUN
ejpam-7047	227	46	)	)	PUNCT
ejpam-7047	227	47	for	for	ADP
ejpam-7047	227	48	every	every	DET
ejpam-7047	227	49	σ1σ2	σ1σ2	NUM
ejpam-7047	227	50	-	-	PUNCT
ejpam-7047	227	51	closed	closed	ADJ
ejpam-7047	227	52	set	set	NOUN
ejpam-7047	227	53	k	k	PROPN
ejpam-7047	227	54	of	of	ADP
ejpam-7047	227	55	y	y	PROPN
ejpam-7047	227	56	;	;	PUNCT
ejpam-7047	227	57	(	(	PUNCT
ejpam-7047	227	58	5	5	X
ejpam-7047	227	59	)	)	PUNCT
ejpam-7047	227	60	αcl⋆(f+(σ1σ2	αcl⋆(f+(σ1σ2	NOUN
ejpam-7047	227	61	-	-	PUNCT
ejpam-7047	227	62	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7047	227	63	-	-	PUNCT
ejpam-7047	227	64	cl(b	cl(b	NOUN
ejpam-7047	227	65	)	)	PUNCT
ejpam-7047	227	66	)	)	PUNCT
ejpam-7047	227	67	)	)	PUNCT
ejpam-7047	227	68	)	)	PUNCT
ejpam-7047	227	69	⊆	⊆	X
ejpam-7047	227	70	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	227	71	-	-	PUNCT
ejpam-7047	227	72	cl(b	cl(b	NOUN
ejpam-7047	227	73	)	)	PUNCT
ejpam-7047	227	74	)	)	PUNCT
ejpam-7047	227	75	for	for	ADP
ejpam-7047	227	76	every	every	DET
ejpam-7047	227	77	subset	subset	NOUN
ejpam-7047	227	78	b	b	PROPN
ejpam-7047	227	79	of	of	ADP
ejpam-7047	227	80	y	y	PROPN
ejpam-7047	227	81	;	;	PUNCT
ejpam-7047	227	82	(	(	PUNCT
ejpam-7047	227	83	6	6	X
ejpam-7047	227	84	)	)	PUNCT
ejpam-7047	227	85	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7047	227	86	-	-	PUNCT
ejpam-7047	227	87	int(b	int(b	NOUN
ejpam-7047	227	88	)	)	PUNCT
ejpam-7047	227	89	)	)	PUNCT
ejpam-7047	227	90	⊆	⊆	NUM
ejpam-7047	227	91	αint⋆(f−(σ1σ2	αint⋆(f−(σ1σ2	NUM
ejpam-7047	227	92	-	-	PUNCT
ejpam-7047	227	93	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7047	227	94	-	-	PUNCT
ejpam-7047	227	95	int(b	int(b	NOUN
ejpam-7047	227	96	)	)	PUNCT
ejpam-7047	227	97	)	)	PUNCT
ejpam-7047	227	98	)	)	PUNCT
ejpam-7047	227	99	)	)	PUNCT
ejpam-7047	227	100	for	for	ADP
ejpam-7047	227	101	every	every	DET
ejpam-7047	227	102	subset	subset	NOUN
ejpam-7047	227	103	b	b	PROPN
ejpam-7047	227	104	of	of	ADP
ejpam-7047	227	105	y	y	PROPN
ejpam-7047	227	106	;	;	PUNCT
ejpam-7047	227	107	(	(	PUNCT
ejpam-7047	227	108	7	7	X
ejpam-7047	227	109	)	)	PUNCT
ejpam-7047	227	110	f−(v	f−(v	NOUN
ejpam-7047	227	111	)	)	PUNCT
ejpam-7047	227	112	⊆	⊆	NUM
ejpam-7047	227	113	αint⋆(f−(σ1σ2	αint⋆(f−(σ1σ2	NOUN
ejpam-7047	227	114	-	-	PUNCT
ejpam-7047	227	115	cl(v	cl(v	NOUN
ejpam-7047	227	116	)	)	PUNCT
ejpam-7047	227	117	)	)	PUNCT
ejpam-7047	227	118	)	)	PUNCT
ejpam-7047	227	119	for	for	ADP
ejpam-7047	227	120	every	every	DET
ejpam-7047	227	121	σ1σ2	σ1σ2	NOUN
ejpam-7047	227	122	-	-	ADJ
ejpam-7047	227	123	open	open	ADJ
ejpam-7047	227	124	set	set	NOUN
ejpam-7047	227	125	v	v	NOUN
ejpam-7047	227	126	of	of	ADP
ejpam-7047	227	127	y	y	PROPN
ejpam-7047	227	128	;	;	PUNCT
ejpam-7047	227	129	(	(	PUNCT
ejpam-7047	227	130	8)	8)	NUM
ejpam-7047	227	131	αcl⋆(f+(σ1σ2	αcl⋆(f+(σ1σ2	NOUN
ejpam-7047	227	132	-	-	PUNCT
ejpam-7047	227	133	int(k	int(k	NOUN
ejpam-7047	227	134	)	)	PUNCT
ejpam-7047	227	135	)	)	PUNCT
ejpam-7047	227	136	)	)	PUNCT
ejpam-7047	228	1	⊆	⊆	NUM
ejpam-7047	228	2	f+(k	f+(k	NOUN
ejpam-7047	228	3	)	)	PUNCT
ejpam-7047	228	4	for	for	ADP
ejpam-7047	228	5	every	every	DET
ejpam-7047	228	6	(	(	PUNCT
ejpam-7047	228	7	σ1	σ1	PROPN
ejpam-7047	228	8	,	,	PUNCT
ejpam-7047	228	9	σ2)r	σ2)r	NOUN
ejpam-7047	228	10	-	-	PUNCT
ejpam-7047	228	11	closed	close	VERB
ejpam-7047	228	12	set	set	ADJ
ejpam-7047	228	13	k	k	PROPN
ejpam-7047	228	14	of	of	ADP
ejpam-7047	228	15	y	y	PROPN
ejpam-7047	228	16	;	;	PUNCT
ejpam-7047	228	17	(	(	PUNCT
ejpam-7047	228	18	9	9	X
ejpam-7047	228	19	)	)	PUNCT
ejpam-7047	228	20	αcl⋆(f+(v	αcl⋆(f+(v	PROPN
ejpam-7047	228	21	)	)	PUNCT
ejpam-7047	228	22	)	)	PUNCT
ejpam-7047	229	1	⊆	⊆	X
ejpam-7047	229	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7047	229	3	-	-	PUNCT
ejpam-7047	229	4	cl(v	cl(v	NOUN
ejpam-7047	229	5	)	)	PUNCT
ejpam-7047	229	6	)	)	PUNCT
ejpam-7047	229	7	for	for	ADP
ejpam-7047	229	8	every	every	DET
ejpam-7047	229	9	σ1σ2	σ1σ2	NOUN
ejpam-7047	229	10	-	-	ADJ
ejpam-7047	229	11	open	open	ADJ
ejpam-7047	229	12	set	set	NOUN
ejpam-7047	229	13	v	v	NOUN
ejpam-7047	229	14	of	of	ADP
ejpam-7047	229	15	y	y	PROPN
ejpam-7047	229	16	;	;	PUNCT
ejpam-7047	229	17	(	(	PUNCT
ejpam-7047	229	18	10	10	NUM
ejpam-7047	229	19	)	)	PUNCT
ejpam-7047	229	20	αcl⋆(f+(σ1σ2	αcl⋆(f+(σ1σ2	NOUN
ejpam-7047	229	21	-	-	PUNCT
ejpam-7047	229	22	int((σ1	int((σ1	PROPN
ejpam-7047	229	23	,	,	PUNCT
ejpam-7047	229	24	σ2)θ	σ2)θ	ADJ
ejpam-7047	229	25	-	-	PUNCT
ejpam-7047	229	26	cl(b	cl(b	NOUN
ejpam-7047	229	27	)	)	PUNCT
ejpam-7047	229	28	)	)	PUNCT
ejpam-7047	229	29	)	)	PUNCT
ejpam-7047	229	30	)	)	PUNCT
ejpam-7047	230	1	⊆	⊆	NUM
ejpam-7047	230	2	f+((σ1	f+((σ1	NOUN
ejpam-7047	230	3	,	,	PUNCT
ejpam-7047	230	4	σ2)θ	σ2)θ	ADJ
ejpam-7047	230	5	-	-	PUNCT
ejpam-7047	230	6	cl(b	cl(b	NOUN
ejpam-7047	230	7	)	)	PUNCT
ejpam-7047	230	8	)	)	PUNCT
ejpam-7047	230	9	for	for	ADP
ejpam-7047	230	10	every	every	DET
ejpam-7047	230	11	subset	subset	NOUN
ejpam-7047	230	12	b	b	PROPN
ejpam-7047	230	13	of	of	ADP
ejpam-7047	230	14	y	y	PROPN
ejpam-7047	230	15	.	.	PUNCT
ejpam-7047	231	1	proof	proof	NOUN
ejpam-7047	231	2	.	.	PUNCT
ejpam-7047	232	1	the	the	DET
ejpam-7047	232	2	proof	proof	NOUN
ejpam-7047	232	3	is	be	AUX
ejpam-7047	232	4	similar	similar	ADJ
ejpam-7047	232	5	to	to	ADP
ejpam-7047	232	6	that	that	PRON
ejpam-7047	232	7	of	of	ADP
ejpam-7047	232	8	theorem	theorem	ADJ
ejpam-7047	232	9	3	3	X
ejpam-7047	232	10	.	.	PUNCT
ejpam-7047	232	11	n.	n.	PROPN
ejpam-7047	232	12	viriyapong	viriyapong	PROPN
ejpam-7047	232	13	,	,	PUNCT
ejpam-7047	232	14	a.	a.	PROPN
ejpam-7047	232	15	sama	sama	PROPN
ejpam-7047	232	16	-	-	PUNCT
ejpam-7047	232	17	ae	ae	PROPN
ejpam-7047	232	18	,	,	PUNCT
ejpam-7047	232	19	c.	c.	PROPN
ejpam-7047	232	20	boonpok	boonpok	PROPN
ejpam-7047	232	21	/	/	SYM
ejpam-7047	232	22	eur	eur	PROPN
ejpam-7047	232	23	.	.	PUNCT
ejpam-7047	233	1	j.	j.	PROPN
ejpam-7047	233	2	pure	pure	PROPN
ejpam-7047	233	3	appl	appl	PROPN
ejpam-7047	233	4	.	.	PROPN
ejpam-7047	233	5	math	math	PROPN
ejpam-7047	233	6	,	,	PUNCT
ejpam-7047	233	7	18	18	NUM
ejpam-7047	233	8	(	(	PUNCT
ejpam-7047	233	9	4	4	NUM
ejpam-7047	233	10	)	)	PUNCT
ejpam-7047	233	11	(	(	PUNCT
ejpam-7047	233	12	2025	2025	NUM
ejpam-7047	233	13	)	)	PUNCT
ejpam-7047	233	14	,	,	PUNCT
ejpam-7047	233	15	7047	7047	NUM
ejpam-7047	233	16	9	9	NUM
ejpam-7047	233	17	of	of	ADP
ejpam-7047	233	18	11	11	NUM
ejpam-7047	233	19	definition	definition	NOUN
ejpam-7047	233	20	5	5	NUM
ejpam-7047	233	21	.	.	PUNCT
ejpam-7047	234	1	a	a	DET
ejpam-7047	234	2	function	function	NOUN
ejpam-7047	234	3	f	f	NOUN
ejpam-7047	234	4	:	:	PUNCT
ejpam-7047	234	5	(	(	PUNCT
ejpam-7047	234	6	x	x	X
ejpam-7047	234	7	,	,	PUNCT
ejpam-7047	234	8	τ	τ	PROPN
ejpam-7047	234	9	,	,	PUNCT
ejpam-7047	234	10	i	i	NOUN
ejpam-7047	234	11	)	)	PUNCT
ejpam-7047	234	12	→	→	PUNCT
ejpam-7047	234	13	(	(	PUNCT
ejpam-7047	234	14	y	y	PROPN
ejpam-7047	234	15	,	,	PUNCT
ejpam-7047	234	16	σ1	σ1	PROPN
ejpam-7047	234	17	,	,	PUNCT
ejpam-7047	234	18	σ2	σ2	PROPN
ejpam-7047	234	19	)	)	PUNCT
ejpam-7047	234	20	is	be	AUX
ejpam-7047	234	21	said	say	VERB
ejpam-7047	234	22	to	to	PART
ejpam-7047	234	23	be	be	AUX
ejpam-7047	234	24	weakly	weakly	ADV
ejpam-7047	234	25	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	234	26	,	,	PUNCT
ejpam-7047	234	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	234	28	if	if	SCONJ
ejpam-7047	234	29	for	for	ADP
ejpam-7047	234	30	each	each	DET
ejpam-7047	234	31	x	x	SYM
ejpam-7047	234	32	∈	∈	PROPN
ejpam-7047	234	33	x	x	X
ejpam-7047	234	34	and	and	CCONJ
ejpam-7047	234	35	each	each	DET
ejpam-7047	234	36	σ1σ2	σ1σ2	VERB
ejpam-7047	234	37	-	-	ADJ
ejpam-7047	234	38	open	open	ADJ
ejpam-7047	234	39	set	set	NOUN
ejpam-7047	234	40	v	v	NOUN
ejpam-7047	234	41	of	of	ADP
ejpam-7047	234	42	y	y	NOUN
ejpam-7047	234	43	containing	contain	VERB
ejpam-7047	234	44	f(x	f(x	PROPN
ejpam-7047	234	45	)	)	PUNCT
ejpam-7047	234	46	,	,	PUNCT
ejpam-7047	234	47	there	there	PRON
ejpam-7047	234	48	exists	exist	VERB
ejpam-7047	234	49	a	a	DET
ejpam-7047	234	50	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	234	51	-	-	PUNCT
ejpam-7047	234	52	open	open	ADJ
ejpam-7047	234	53	set	set	NOUN
ejpam-7047	234	54	u	u	NOUN
ejpam-7047	234	55	of	of	ADP
ejpam-7047	234	56	x	x	PUNCT
ejpam-7047	234	57	containing	contain	VERB
ejpam-7047	234	58	x	x	PUNCT
ejpam-7047	234	59	such	such	ADJ
ejpam-7047	234	60	that	that	DET
ejpam-7047	234	61	f(u	f(u	PROPN
ejpam-7047	234	62	)	)	PUNCT
ejpam-7047	235	1	⊆	⊆	NUM
ejpam-7047	235	2	σ1σ2	σ1σ2	NOUN
ejpam-7047	235	3	-	-	NUM
ejpam-7047	235	4	cl(v	cl(v	NOUN
ejpam-7047	235	5	)	)	PUNCT
ejpam-7047	235	6	.	.	PUNCT
ejpam-7047	236	1	corollary	corollary	ADJ
ejpam-7047	236	2	1	1	NUM
ejpam-7047	236	3	.	.	PUNCT
ejpam-7047	237	1	for	for	ADP
ejpam-7047	237	2	a	a	DET
ejpam-7047	237	3	function	function	NOUN
ejpam-7047	237	4	f	f	NOUN
ejpam-7047	237	5	:	:	PUNCT
ejpam-7047	237	6	(	(	PUNCT
ejpam-7047	237	7	x	x	X
ejpam-7047	237	8	,	,	PUNCT
ejpam-7047	237	9	τ	τ	PROPN
ejpam-7047	237	10	,	,	PUNCT
ejpam-7047	237	11	i	i	NOUN
ejpam-7047	237	12	)	)	PUNCT
ejpam-7047	237	13	→	→	PUNCT
ejpam-7047	237	14	(	(	PUNCT
ejpam-7047	237	15	y	y	PROPN
ejpam-7047	237	16	,	,	PUNCT
ejpam-7047	237	17	σ1	σ1	PROPN
ejpam-7047	237	18	,	,	PUNCT
ejpam-7047	237	19	σ2	σ2	NOUN
ejpam-7047	237	20	)	)	PUNCT
ejpam-7047	237	21	,	,	PUNCT
ejpam-7047	237	22	the	the	DET
ejpam-7047	237	23	following	follow	VERB
ejpam-7047	237	24	properties	property	NOUN
ejpam-7047	237	25	are	be	AUX
ejpam-7047	237	26	equivalent	equivalent	ADJ
ejpam-7047	237	27	:	:	PUNCT
ejpam-7047	237	28	(	(	PUNCT
ejpam-7047	237	29	1	1	X
ejpam-7047	237	30	)	)	PUNCT
ejpam-7047	237	31	f	f	PROPN
ejpam-7047	237	32	is	be	AUX
ejpam-7047	237	33	weakly	weakly	ADV
ejpam-7047	237	34	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	237	35	,	,	PUNCT
ejpam-7047	237	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	237	37	;	;	PUNCT
ejpam-7047	237	38	(	(	PUNCT
ejpam-7047	237	39	2	2	X
ejpam-7047	237	40	)	)	PUNCT
ejpam-7047	237	41	f−1(v	f−1(v	NOUN
ejpam-7047	237	42	)	)	PUNCT
ejpam-7047	238	1	⊆	⊆	X
ejpam-7047	238	2	αint⋆(f−1(σ1σ2	αint⋆(f−1(σ1σ2	NOUN
ejpam-7047	238	3	-	-	PUNCT
ejpam-7047	238	4	cl(v	cl(v	NOUN
ejpam-7047	238	5	)	)	PUNCT
ejpam-7047	238	6	)	)	PUNCT
ejpam-7047	238	7	)	)	PUNCT
ejpam-7047	238	8	for	for	ADP
ejpam-7047	238	9	every	every	DET
ejpam-7047	238	10	σ1σ2	σ1σ2	NOUN
ejpam-7047	238	11	-	-	ADJ
ejpam-7047	238	12	open	open	ADJ
ejpam-7047	238	13	set	set	NOUN
ejpam-7047	238	14	v	v	NOUN
ejpam-7047	238	15	of	of	ADP
ejpam-7047	238	16	y	y	PROPN
ejpam-7047	238	17	;	;	PUNCT
ejpam-7047	238	18	(	(	PUNCT
ejpam-7047	238	19	3	3	X
ejpam-7047	238	20	)	)	PUNCT
ejpam-7047	238	21	αcl⋆(f−1(σ1σ2	αcl⋆(f−1(σ1σ2	NOUN
ejpam-7047	238	22	-	-	PUNCT
ejpam-7047	238	23	int(k	int(k	NUM
ejpam-7047	238	24	)	)	PUNCT
ejpam-7047	238	25	)	)	PUNCT
ejpam-7047	238	26	)	)	PUNCT
ejpam-7047	239	1	⊆	⊆	NUM
ejpam-7047	239	2	f−1(k	f−1(k	PROPN
ejpam-7047	239	3	)	)	PUNCT
ejpam-7047	239	4	for	for	ADP
ejpam-7047	239	5	every	every	DET
ejpam-7047	239	6	(	(	PUNCT
ejpam-7047	239	7	σ1	σ1	PROPN
ejpam-7047	239	8	,	,	PUNCT
ejpam-7047	239	9	σ2)r	σ2)r	NOUN
ejpam-7047	239	10	-	-	PUNCT
ejpam-7047	239	11	closed	close	VERB
ejpam-7047	239	12	set	set	ADJ
ejpam-7047	239	13	k	k	PROPN
ejpam-7047	239	14	of	of	ADP
ejpam-7047	239	15	y	y	PROPN
ejpam-7047	239	16	;	;	PUNCT
ejpam-7047	239	17	(	(	PUNCT
ejpam-7047	239	18	4	4	X
ejpam-7047	239	19	)	)	PUNCT
ejpam-7047	239	20	αcl⋆(f−1(v	αcl⋆(f−1(v	NUM
ejpam-7047	239	21	)	)	PUNCT
ejpam-7047	239	22	)	)	PUNCT
ejpam-7047	240	1	⊆	⊆	NUM
ejpam-7047	240	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7047	240	3	-	-	PUNCT
ejpam-7047	240	4	cl(v	cl(v	NOUN
ejpam-7047	240	5	)	)	PUNCT
ejpam-7047	240	6	)	)	PUNCT
ejpam-7047	240	7	for	for	ADP
ejpam-7047	240	8	every	every	DET
ejpam-7047	240	9	σ1σ2	σ1σ2	NOUN
ejpam-7047	240	10	-	-	ADJ
ejpam-7047	240	11	open	open	ADJ
ejpam-7047	240	12	set	set	NOUN
ejpam-7047	240	13	v	v	NOUN
ejpam-7047	240	14	of	of	ADP
ejpam-7047	240	15	y	y	PROPN
ejpam-7047	240	16	;	;	PUNCT
ejpam-7047	240	17	(	(	PUNCT
ejpam-7047	240	18	5	5	X
ejpam-7047	240	19	)	)	PUNCT
ejpam-7047	240	20	αcl⋆(f−1(σ1σ2	αcl⋆(f−1(σ1σ2	NOUN
ejpam-7047	240	21	-	-	PUNCT
ejpam-7047	240	22	int((σ1	int((σ1	NUM
ejpam-7047	240	23	,	,	PUNCT
ejpam-7047	240	24	σ2)θ	σ2)θ	ADJ
ejpam-7047	240	25	-	-	PUNCT
ejpam-7047	240	26	cl(b	cl(b	NOUN
ejpam-7047	240	27	)	)	PUNCT
ejpam-7047	240	28	)	)	PUNCT
ejpam-7047	240	29	)	)	PUNCT
ejpam-7047	240	30	)	)	PUNCT
ejpam-7047	241	1	⊆	⊆	NUM
ejpam-7047	241	2	f−1((σ1	f−1((σ1	NOUN
ejpam-7047	241	3	,	,	PUNCT
ejpam-7047	241	4	σ2)θ	σ2)θ	NOUN
ejpam-7047	241	5	-	-	PUNCT
ejpam-7047	241	6	cl(b	cl(b	NOUN
ejpam-7047	241	7	)	)	PUNCT
ejpam-7047	241	8	)	)	PUNCT
ejpam-7047	241	9	for	for	ADP
ejpam-7047	241	10	every	every	DET
ejpam-7047	241	11	subset	subset	NOUN
ejpam-7047	241	12	b	b	PROPN
ejpam-7047	241	13	of	of	ADP
ejpam-7047	241	14	y	y	PROPN
ejpam-7047	241	15	;	;	PUNCT
ejpam-7047	241	16	(	(	PUNCT
ejpam-7047	241	17	6	6	X
ejpam-7047	241	18	)	)	PUNCT
ejpam-7047	241	19	cl⋆(int⋆(cl⋆(f−1(v	cl⋆(int⋆(cl⋆(f−1(v	NOUN
ejpam-7047	241	20	)	)	PUNCT
ejpam-7047	241	21	)	)	PUNCT
ejpam-7047	241	22	)	)	PUNCT
ejpam-7047	241	23	)	)	PUNCT
ejpam-7047	242	1	⊆	⊆	NUM
ejpam-7047	242	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7047	242	3	-	-	PUNCT
ejpam-7047	242	4	cl(v	cl(v	NOUN
ejpam-7047	242	5	)	)	PUNCT
ejpam-7047	242	6	)	)	PUNCT
ejpam-7047	242	7	for	for	ADP
ejpam-7047	242	8	every	every	DET
ejpam-7047	242	9	σ1σ2	σ1σ2	NOUN
ejpam-7047	242	10	-	-	ADJ
ejpam-7047	242	11	open	open	ADJ
ejpam-7047	242	12	set	set	NOUN
ejpam-7047	242	13	v	v	NOUN
ejpam-7047	242	14	of	of	ADP
ejpam-7047	242	15	y	y	PROPN
ejpam-7047	242	16	;	;	PUNCT
ejpam-7047	242	17	(	(	PUNCT
ejpam-7047	242	18	7	7	X
ejpam-7047	242	19	)	)	PUNCT
ejpam-7047	242	20	f−1(v	f−1(v	NOUN
ejpam-7047	242	21	)	)	PUNCT
ejpam-7047	243	1	⊆	⊆	NUM
ejpam-7047	243	2	int⋆(cl⋆(int⋆(f−1(σ1σ2	int⋆(cl⋆(int⋆(f−1(σ1σ2	NOUN
ejpam-7047	243	3	-	-	PUNCT
ejpam-7047	243	4	cl(v	cl(v	NOUN
ejpam-7047	243	5	)	)	PUNCT
ejpam-7047	243	6	)	)	PUNCT
ejpam-7047	243	7	)	)	PUNCT
ejpam-7047	243	8	)	)	PUNCT
ejpam-7047	243	9	)	)	PUNCT
ejpam-7047	244	1	for	for	ADP
ejpam-7047	244	2	every	every	DET
ejpam-7047	244	3	σ1σ2	σ1σ2	NOUN
ejpam-7047	244	4	-	-	ADJ
ejpam-7047	244	5	open	open	ADJ
ejpam-7047	244	6	set	set	NOUN
ejpam-7047	244	7	v	v	NOUN
ejpam-7047	244	8	of	of	ADP
ejpam-7047	244	9	y	y	PROPN
ejpam-7047	244	10	;	;	PUNCT
ejpam-7047	244	11	(	(	PUNCT
ejpam-7047	244	12	8)	8)	NUM
ejpam-7047	244	13	f(cl⋆(int⋆(cl⋆(a	f(cl⋆(int⋆(cl⋆(a	NUM
ejpam-7047	244	14	)	)	PUNCT
ejpam-7047	244	15	)	)	PUNCT
ejpam-7047	244	16	)	)	PUNCT
ejpam-7047	244	17	)	)	PUNCT
ejpam-7047	245	1	⊆	⊆	X
ejpam-7047	245	2	(	(	PUNCT
ejpam-7047	245	3	σ1	σ1	PROPN
ejpam-7047	245	4	,	,	PUNCT
ejpam-7047	245	5	σ2)θ	σ2)θ	NOUN
ejpam-7047	245	6	-	-	PUNCT
ejpam-7047	245	7	cl(f(a	cl(f(a	NOUN
ejpam-7047	245	8	)	)	PUNCT
ejpam-7047	245	9	)	)	PUNCT
ejpam-7047	245	10	for	for	ADP
ejpam-7047	245	11	every	every	DET
ejpam-7047	245	12	subset	subset	NOUN
ejpam-7047	245	13	a	a	PRON
ejpam-7047	245	14	of	of	ADP
ejpam-7047	245	15	x	x	PRON
ejpam-7047	245	16	;	;	PUNCT
ejpam-7047	245	17	(	(	PUNCT
ejpam-7047	245	18	9	9	X
ejpam-7047	245	19	)	)	PUNCT
ejpam-7047	245	20	cl⋆(int⋆(cl⋆(f−1(b	cl⋆(int⋆(cl⋆(f−1(b	NUM
ejpam-7047	245	21	)	)	PUNCT
ejpam-7047	245	22	)	)	PUNCT
ejpam-7047	245	23	)	)	PUNCT
ejpam-7047	245	24	)	)	PUNCT
ejpam-7047	246	1	⊆	⊆	NUM
ejpam-7047	246	2	f−1((σ1	f−1((σ1	NOUN
ejpam-7047	246	3	,	,	PUNCT
ejpam-7047	246	4	σ2)θ	σ2)θ	NOUN
ejpam-7047	246	5	-	-	PUNCT
ejpam-7047	246	6	cl(b	cl(b	NOUN
ejpam-7047	246	7	)	)	PUNCT
ejpam-7047	246	8	)	)	PUNCT
ejpam-7047	246	9	for	for	ADP
ejpam-7047	246	10	every	every	DET
ejpam-7047	246	11	subset	subset	NOUN
ejpam-7047	246	12	b	b	PROPN
ejpam-7047	246	13	of	of	ADP
ejpam-7047	246	14	y	y	PROPN
ejpam-7047	246	15	.	.	PUNCT
ejpam-7047	247	1	definition	definition	NOUN
ejpam-7047	247	2	6	6	NUM
ejpam-7047	247	3	.	.	PUNCT
ejpam-7047	248	1	[	[	X
ejpam-7047	248	2	22	22	NUM
ejpam-7047	248	3	]	]	PUNCT
ejpam-7047	248	4	a	a	DET
ejpam-7047	248	5	function	function	NOUN
ejpam-7047	248	6	f	f	NOUN
ejpam-7047	248	7	:	:	PUNCT
ejpam-7047	248	8	(	(	PUNCT
ejpam-7047	248	9	x	x	X
ejpam-7047	248	10	,	,	PUNCT
ejpam-7047	248	11	τ	τ	PROPN
ejpam-7047	248	12	,	,	PUNCT
ejpam-7047	248	13	i	i	NOUN
ejpam-7047	248	14	)	)	PUNCT
ejpam-7047	248	15	→	→	PUNCT
ejpam-7047	248	16	(	(	PUNCT
ejpam-7047	248	17	y	y	PROPN
ejpam-7047	248	18	,	,	PUNCT
ejpam-7047	248	19	σ1	σ1	PROPN
ejpam-7047	248	20	,	,	PUNCT
ejpam-7047	248	21	σ2	σ2	PROPN
ejpam-7047	248	22	)	)	PUNCT
ejpam-7047	248	23	is	be	AUX
ejpam-7047	248	24	said	say	VERB
ejpam-7047	248	25	to	to	PART
ejpam-7047	248	26	be	be	AUX
ejpam-7047	248	27	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7047	248	28	,	,	PUNCT
ejpam-7047	248	29	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	248	30	if	if	SCONJ
ejpam-7047	248	31	for	for	ADP
ejpam-7047	248	32	each	each	DET
ejpam-7047	248	33	σ1σ2	σ1σ2	VERB
ejpam-7047	248	34	-	-	ADJ
ejpam-7047	248	35	open	open	ADJ
ejpam-7047	248	36	set	set	NOUN
ejpam-7047	248	37	v	v	NOUN
ejpam-7047	248	38	of	of	ADP
ejpam-7047	248	39	y	y	PROPN
ejpam-7047	248	40	,	,	PUNCT
ejpam-7047	248	41	f−1(v	f−1(v	PROPN
ejpam-7047	248	42	)	)	PUNCT
ejpam-7047	248	43	is	be	AUX
ejpam-7047	248	44	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	248	45	-	-	PUNCT
ejpam-7047	248	46	open	open	ADJ
ejpam-7047	248	47	in	in	ADP
ejpam-7047	248	48	x.	x.	NOUN
ejpam-7047	248	49	definition	definition	NOUN
ejpam-7047	248	50	7	7	NUM
ejpam-7047	248	51	.	.	PUNCT
ejpam-7047	249	1	[	[	X
ejpam-7047	249	2	21	21	NUM
ejpam-7047	249	3	]	]	X
ejpam-7047	249	4	a	a	DET
ejpam-7047	249	5	function	function	NOUN
ejpam-7047	249	6	f	f	NOUN
ejpam-7047	249	7	:	:	PUNCT
ejpam-7047	249	8	(	(	PUNCT
ejpam-7047	249	9	x	x	X
ejpam-7047	249	10	,	,	PUNCT
ejpam-7047	249	11	τ	τ	PROPN
ejpam-7047	249	12	,	,	PUNCT
ejpam-7047	249	13	i	i	NOUN
ejpam-7047	249	14	)	)	PUNCT
ejpam-7047	249	15	→	→	PUNCT
ejpam-7047	249	16	(	(	PUNCT
ejpam-7047	249	17	y	y	PROPN
ejpam-7047	249	18	,	,	PUNCT
ejpam-7047	249	19	σ1	σ1	PROPN
ejpam-7047	249	20	,	,	PUNCT
ejpam-7047	249	21	σ2	σ2	PROPN
ejpam-7047	249	22	)	)	PUNCT
ejpam-7047	249	23	is	be	AUX
ejpam-7047	249	24	said	say	VERB
ejpam-7047	249	25	to	to	PART
ejpam-7047	249	26	be	be	AUX
ejpam-7047	249	27	almost	almost	ADV
ejpam-7047	249	28	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7047	249	29	,	,	PUNCT
ejpam-7047	249	30	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	249	31	if	if	SCONJ
ejpam-7047	249	32	f−1(v	f−1(v	PROPN
ejpam-7047	249	33	)	)	PUNCT
ejpam-7047	249	34	is	be	AUX
ejpam-7047	249	35	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	249	36	-	-	PUNCT
ejpam-7047	249	37	open	open	ADJ
ejpam-7047	249	38	in	in	ADP
ejpam-7047	249	39	x	x	PUNCT
ejpam-7047	249	40	for	for	ADP
ejpam-7047	249	41	every	every	DET
ejpam-7047	249	42	(	(	PUNCT
ejpam-7047	249	43	σ1	σ1	PROPN
ejpam-7047	249	44	,	,	PUNCT
ejpam-7047	249	45	σ2)r	σ2)r	NOUN
ejpam-7047	249	46	-	-	PUNCT
ejpam-7047	249	47	open	open	ADJ
ejpam-7047	249	48	set	set	VERB
ejpam-7047	249	49	v	v	NOUN
ejpam-7047	249	50	of	of	ADP
ejpam-7047	249	51	y	y	PROPN
ejpam-7047	249	52	.	.	PUNCT
ejpam-7047	250	1	theorem	theorem	ADJ
ejpam-7047	250	2	5	5	NUM
ejpam-7047	250	3	.	.	X
ejpam-7047	250	4	for	for	ADP
ejpam-7047	250	5	a	a	DET
ejpam-7047	250	6	function	function	NOUN
ejpam-7047	250	7	f	f	NOUN
ejpam-7047	250	8	:	:	PUNCT
ejpam-7047	250	9	(	(	PUNCT
ejpam-7047	250	10	x	x	X
ejpam-7047	250	11	,	,	PUNCT
ejpam-7047	250	12	τ	τ	PROPN
ejpam-7047	250	13	,	,	PUNCT
ejpam-7047	250	14	i	i	NOUN
ejpam-7047	250	15	)	)	PUNCT
ejpam-7047	250	16	→	→	PUNCT
ejpam-7047	250	17	(	(	PUNCT
ejpam-7047	250	18	y	y	PROPN
ejpam-7047	250	19	,	,	PUNCT
ejpam-7047	250	20	σ1	σ1	PROPN
ejpam-7047	250	21	,	,	PUNCT
ejpam-7047	250	22	σ2	σ2	NOUN
ejpam-7047	250	23	)	)	PUNCT
ejpam-7047	251	1	such	such	ADJ
ejpam-7047	251	2	that	that	SCONJ
ejpam-7047	251	3	αint⋆(f−1(σ1σ2	αint⋆(f−1(σ1σ2	NOUN
ejpam-7047	251	4	-	-	PUNCT
ejpam-7047	251	5	cl(v	cl(v	NOUN
ejpam-7047	251	6	)	)	PUNCT
ejpam-7047	251	7	)	)	PUNCT
ejpam-7047	251	8	)	)	PUNCT
ejpam-7047	252	1	⊆	⊆	X
ejpam-7047	252	2	αint⋆(f−1(v	αint⋆(f−1(v	PROPN
ejpam-7047	252	3	)	)	PUNCT
ejpam-7047	252	4	)	)	PUNCT
ejpam-7047	252	5	for	for	ADP
ejpam-7047	252	6	every	every	DET
ejpam-7047	252	7	σ1σ2	σ1σ2	NOUN
ejpam-7047	252	8	-	-	ADJ
ejpam-7047	252	9	open	open	ADJ
ejpam-7047	252	10	set	set	NOUN
ejpam-7047	252	11	v	v	NOUN
ejpam-7047	252	12	of	of	ADP
ejpam-7047	252	13	y	y	PROPN
ejpam-7047	252	14	,	,	PUNCT
ejpam-7047	252	15	the	the	DET
ejpam-7047	252	16	following	follow	VERB
ejpam-7047	252	17	properties	property	NOUN
ejpam-7047	252	18	are	be	AUX
ejpam-7047	252	19	equivalent	equivalent	ADJ
ejpam-7047	252	20	:	:	PUNCT
ejpam-7047	252	21	(	(	PUNCT
ejpam-7047	252	22	1	1	X
ejpam-7047	252	23	)	)	PUNCT
ejpam-7047	252	24	f	f	PROPN
ejpam-7047	252	25	is	be	AUX
ejpam-7047	252	26	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7047	252	27	,	,	PUNCT
ejpam-7047	252	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	252	29	;	;	PUNCT
ejpam-7047	252	30	(	(	PUNCT
ejpam-7047	252	31	2	2	X
ejpam-7047	252	32	)	)	PUNCT
ejpam-7047	252	33	f	f	PROPN
ejpam-7047	252	34	is	be	AUX
ejpam-7047	252	35	almost	almost	ADV
ejpam-7047	252	36	τ⋆α(σ1	τ⋆α(σ1	NUM
ejpam-7047	252	37	,	,	PUNCT
ejpam-7047	252	38	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	252	39	;	;	PUNCT
ejpam-7047	252	40	(	(	PUNCT
ejpam-7047	252	41	3	3	X
ejpam-7047	252	42	)	)	PUNCT
ejpam-7047	252	43	f	f	PROPN
ejpam-7047	252	44	is	be	AUX
ejpam-7047	252	45	weakly	weakly	ADV
ejpam-7047	252	46	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	252	47	,	,	PUNCT
ejpam-7047	252	48	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	252	49	.	.	NOUN
ejpam-7047	252	50	proof	proof	NOUN
ejpam-7047	252	51	.	.	PUNCT
ejpam-7047	253	1	we	we	PRON
ejpam-7047	253	2	prove	prove	VERB
ejpam-7047	253	3	only	only	ADV
ejpam-7047	253	4	the	the	DET
ejpam-7047	253	5	implication	implication	NOUN
ejpam-7047	253	6	(	(	PUNCT
ejpam-7047	253	7	3	3	X
ejpam-7047	253	8	)	)	PUNCT
ejpam-7047	253	9	⇒	⇒	NOUN
ejpam-7047	253	10	(	(	PUNCT
ejpam-7047	253	11	1	1	NUM
ejpam-7047	253	12	)	)	PUNCT
ejpam-7047	253	13	.	.	PUNCT
ejpam-7047	254	1	suppose	suppose	VERB
ejpam-7047	254	2	that	that	SCONJ
ejpam-7047	254	3	f	f	PROPN
ejpam-7047	254	4	is	be	AUX
ejpam-7047	254	5	weakly	weakly	ADV
ejpam-7047	254	6	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	254	7	,	,	PUNCT
ejpam-7047	254	8	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	254	9	.	.	PUNCT
ejpam-7047	255	1	let	let	VERB
ejpam-7047	255	2	v	v	PART
ejpam-7047	255	3	be	be	AUX
ejpam-7047	255	4	any	any	DET
ejpam-7047	255	5	σ1σ2	σ1σ2	NOUN
ejpam-7047	255	6	-	-	ADJ
ejpam-7047	255	7	open	open	ADJ
ejpam-7047	255	8	set	set	NOUN
ejpam-7047	255	9	of	of	ADP
ejpam-7047	255	10	y	y	PROPN
ejpam-7047	255	11	.	.	PUNCT
ejpam-7047	256	1	since	since	SCONJ
ejpam-7047	256	2	f	f	PROPN
ejpam-7047	256	3	is	be	AUX
ejpam-7047	256	4	weakly	weakly	ADV
ejpam-7047	256	5	τ⋆α(σ1	τ⋆α(σ1	ADV
ejpam-7047	256	6	,	,	PUNCT
ejpam-7047	256	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7047	256	8	,	,	PUNCT
ejpam-7047	256	9	by	by	ADP
ejpam-7047	256	10	corollary	corollary	ADJ
ejpam-7047	256	11	1	1	NUM
ejpam-7047	256	12	,	,	PUNCT
ejpam-7047	256	13	we	we	PRON
ejpam-7047	256	14	have	have	VERB
ejpam-7047	256	15	f−1(v	f−1(v	NOUN
ejpam-7047	256	16	)	)	PUNCT
ejpam-7047	257	1	⊆	⊆	X
ejpam-7047	257	2	αint⋆(f−1(σ1σ2	αint⋆(f−1(σ1σ2	NOUN
ejpam-7047	257	3	-	-	PUNCT
ejpam-7047	257	4	cl(v	cl(v	NOUN
ejpam-7047	257	5	)	)	PUNCT
ejpam-7047	257	6	)	)	PUNCT
ejpam-7047	257	7	)	)	PUNCT
ejpam-7047	257	8	and	and	CCONJ
ejpam-7047	257	9	hence	hence	ADV
ejpam-7047	257	10	f−1(v	f−1(v	NOUN
ejpam-7047	257	11	)	)	PUNCT
ejpam-7047	257	12	⊆	⊆	X
ejpam-7047	257	13	αint⋆(f−1(σ1σ2	αint⋆(f−1(σ1σ2	NOUN
ejpam-7047	257	14	-	-	PUNCT
ejpam-7047	257	15	cl(v	cl(v	NOUN
ejpam-7047	257	16	)	)	PUNCT
ejpam-7047	257	17	)	)	PUNCT
ejpam-7047	257	18	)	)	PUNCT
ejpam-7047	258	1	⊆	⊆	NUM
ejpam-7047	258	2	αint⋆(f−1(v	αint⋆(f−1(v	PROPN
ejpam-7047	258	3	)	)	PUNCT
ejpam-7047	258	4	)	)	PUNCT
ejpam-7047	258	5	.	.	PUNCT
ejpam-7047	259	1	thus	thus	ADV
ejpam-7047	259	2	,	,	PUNCT
ejpam-7047	259	3	f−1(v	f−1(v	PROPN
ejpam-7047	259	4	)	)	PUNCT
ejpam-7047	259	5	is	be	AUX
ejpam-7047	259	6	τ⋆-α	τ⋆-α	NOUN
ejpam-7047	259	7	-	-	PUNCT
ejpam-7047	259	8	open	open	ADJ
ejpam-7047	259	9	in	in	ADP
ejpam-7047	259	10	x.	x.	NOUN
ejpam-7047	259	11	this	this	PRON
ejpam-7047	259	12	shows	show	VERB
ejpam-7047	259	13	that	that	SCONJ
ejpam-7047	259	14	f	f	PROPN
ejpam-7047	259	15	is	be	AUX
ejpam-7047	259	16	τ⋆α(σ1	τ⋆α(σ1	ADJ
ejpam-7047	259	17	,	,	PUNCT
ejpam-7047	259	18	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7047	259	19	.	.	PUNCT
ejpam-7047	260	1	n.	n.	PROPN
ejpam-7047	260	2	viriyapong	viriyapong	PROPN
ejpam-7047	260	3	,	,	PUNCT
ejpam-7047	260	4	a.	a.	PROPN
ejpam-7047	260	5	sama	sama	PROPN
ejpam-7047	260	6	-	-	PUNCT
ejpam-7047	260	7	ae	ae	PROPN
ejpam-7047	260	8	,	,	PUNCT
ejpam-7047	260	9	c.	c.	PROPN
ejpam-7047	260	10	boonpok	boonpok	PROPN
ejpam-7047	260	11	/	/	SYM
ejpam-7047	260	12	eur	eur	PROPN
ejpam-7047	260	13	.	.	PUNCT
ejpam-7047	261	1	j.	j.	PROPN
ejpam-7047	261	2	pure	pure	PROPN
ejpam-7047	261	3	appl	appl	PROPN
ejpam-7047	261	4	.	.	PROPN
ejpam-7047	261	5	math	math	PROPN
ejpam-7047	261	6	,	,	PUNCT
ejpam-7047	261	7	18	18	NUM
ejpam-7047	261	8	(	(	PUNCT
ejpam-7047	261	9	4	4	NUM
ejpam-7047	261	10	)	)	PUNCT
ejpam-7047	261	11	(	(	PUNCT
ejpam-7047	261	12	2025	2025	NUM
ejpam-7047	261	13	)	)	PUNCT
ejpam-7047	261	14	,	,	PUNCT
ejpam-7047	261	15	7047	7047	NUM
ejpam-7047	261	16	10	10	NUM
ejpam-7047	261	17	of	of	ADP
ejpam-7047	261	18	11	11	NUM
ejpam-7047	261	19	acknowledgements	acknowledgement	NOUN
ejpam-7047	261	20	this	this	DET
ejpam-7047	261	21	research	research	NOUN
ejpam-7047	261	22	project	project	NOUN
ejpam-7047	261	23	was	be	AUX
ejpam-7047	261	24	financially	financially	ADV
ejpam-7047	261	25	supported	support	VERB
ejpam-7047	261	26	by	by	ADP
ejpam-7047	261	27	mahasarakham	mahasarakham	PROPN
ejpam-7047	261	28	university	university	PROPN
ejpam-7047	261	29	.	.	PUNCT
ejpam-7047	262	1	references	reference	NOUN
ejpam-7047	262	2	[	[	X
ejpam-7047	262	3	1	1	X
ejpam-7047	262	4	]	]	PUNCT
ejpam-7047	262	5	t.	t.	PROPN
ejpam-7047	262	6	noiri	noiri	PROPN
ejpam-7047	262	7	.	.	PUNCT
ejpam-7047	263	1	weakly	weakly	ADJ
ejpam-7047	263	2	α	α	X
ejpam-7047	263	3	-	-	ADJ
ejpam-7047	263	4	continuous	continuous	ADJ
ejpam-7047	263	5	functions	function	NOUN
ejpam-7047	263	6	.	.	PUNCT
ejpam-7047	264	1	international	international	ADJ
ejpam-7047	264	2	journal	journal	PROPN
ejpam-7047	264	3	of	of	ADP
ejpam-7047	264	4	mathematics	mathematics	PROPN
ejpam-7047	264	5	and	and	CCONJ
ejpam-7047	264	6	mathematical	mathematical	ADJ
ejpam-7047	264	7	sciences	science	NOUN
ejpam-7047	264	8	,	,	PUNCT
ejpam-7047	264	9	10:483–490	10:483–490	NUM
ejpam-7047	264	10	,	,	PUNCT
ejpam-7047	264	11	1987	1987	NUM
ejpam-7047	264	12	.	.	PUNCT
ejpam-7047	265	1	[	[	X
ejpam-7047	265	2	2	2	NUM
ejpam-7047	265	3	]	]	PUNCT
ejpam-7047	265	4	a.	a.	NOUN
ejpam-7047	265	5	k.	k.	PROPN
ejpam-7047	265	6	sen	sen	PROPN
ejpam-7047	265	7	and	and	CCONJ
ejpam-7047	265	8	p.	p.	PROPN
ejpam-7047	265	9	bhattacharyya	bhattacharyya	PROPN
ejpam-7047	265	10	.	.	PUNCT
ejpam-7047	266	1	on	on	ADP
ejpam-7047	266	2	weakly	weakly	ADJ
ejpam-7047	266	3	α	α	NOUN
ejpam-7047	266	4	-	-	ADJ
ejpam-7047	266	5	continuous	continuous	ADJ
ejpam-7047	266	6	functions	function	NOUN
ejpam-7047	266	7	.	.	PUNCT
ejpam-7047	267	1	tamkang	tamkang	PROPN
ejpam-7047	267	2	journal	journal	PROPN
ejpam-7047	267	3	of	of	ADP
ejpam-7047	267	4	mathematics	mathematic	NOUN
ejpam-7047	267	5	,	,	PUNCT
ejpam-7047	267	6	24:445–460	24:445–460	NUM
ejpam-7047	267	7	,	,	PUNCT
ejpam-7047	267	8	1993	1993	NUM
ejpam-7047	267	9	.	.	PUNCT
ejpam-7047	268	1	[	[	X
ejpam-7047	268	2	3	3	X
ejpam-7047	268	3	]	]	PUNCT
ejpam-7047	268	4	v.	v.	CCONJ
ejpam-7047	268	5	popa	popa	NOUN
ejpam-7047	268	6	and	and	CCONJ
ejpam-7047	268	7	t.	t.	PROPN
ejpam-7047	268	8	noiri	noiri	PROPN
ejpam-7047	268	9	.	.	PUNCT
ejpam-7047	269	1	on	on	ADP
ejpam-7047	269	2	upper	upper	ADJ
ejpam-7047	269	3	and	and	CCONJ
ejpam-7047	269	4	lower	low	ADJ
ejpam-7047	269	5	weakly	weakly	ADJ
ejpam-7047	269	6	α	α	ADJ
ejpam-7047	269	7	-	-	ADJ
ejpam-7047	269	8	continuous	continuous	ADJ
ejpam-7047	269	9	multifunctions	multifunction	NOUN
ejpam-7047	269	10	.	.	PUNCT
ejpam-7047	270	1	novi	novi	PROPN
ejpam-7047	270	2	sad	sad	PROPN
ejpam-7047	270	3	journal	journal	PROPN
ejpam-7047	270	4	of	of	ADP
ejpam-7047	270	5	mathematics	mathematic	NOUN
ejpam-7047	270	6	,	,	PUNCT
ejpam-7047	270	7	32(1):7–24	32(1):7–24	NUM
ejpam-7047	270	8	,	,	PUNCT
ejpam-7047	270	9	2002	2002	NUM
ejpam-7047	270	10	.	.	PUNCT
ejpam-7047	271	1	[	[	X
ejpam-7047	271	2	4	4	NUM
ejpam-7047	271	3	]	]	PUNCT
ejpam-7047	271	4	c.	c.	PROPN
ejpam-7047	271	5	boonpok	boonpok	PROPN
ejpam-7047	271	6	.	.	PUNCT
ejpam-7047	272	1	on	on	ADP
ejpam-7047	272	2	continuous	continuous	ADJ
ejpam-7047	272	3	multifunctions	multifunction	NOUN
ejpam-7047	272	4	in	in	ADP
ejpam-7047	272	5	ideal	ideal	ADJ
ejpam-7047	272	6	topological	topological	ADJ
ejpam-7047	272	7	spaces	space	NOUN
ejpam-7047	272	8	.	.	PUNCT
ejpam-7047	273	1	lobachevskii	lobachevskii	PROPN
ejpam-7047	273	2	journal	journal	PROPN
ejpam-7047	273	3	of	of	ADP
ejpam-7047	273	4	mathematics	mathematic	NOUN
ejpam-7047	273	5	,	,	PUNCT
ejpam-7047	273	6	40(1):24–35	40(1):24–35	NUM
ejpam-7047	273	7	,	,	PUNCT
ejpam-7047	273	8	2019	2019	NUM
ejpam-7047	273	9	.	.	PUNCT
ejpam-7047	274	1	[	[	X
ejpam-7047	274	2	5	5	X
ejpam-7047	274	3	]	]	PUNCT
ejpam-7047	274	4	c.	c.	PROPN
ejpam-7047	274	5	boonpok	boonpok	PROPN
ejpam-7047	274	6	.	.	PUNCT
ejpam-7047	275	1	on	on	ADP
ejpam-7047	275	2	some	some	DET
ejpam-7047	275	3	types	type	NOUN
ejpam-7047	275	4	of	of	ADP
ejpam-7047	275	5	continuity	continuity	NOUN
ejpam-7047	275	6	for	for	ADP
ejpam-7047	275	7	multifunctions	multifunction	NOUN
ejpam-7047	275	8	in	in	ADP
ejpam-7047	275	9	ideal	ideal	ADJ
ejpam-7047	275	10	topological	topological	ADJ
ejpam-7047	275	11	spaces	space	NOUN
ejpam-7047	275	12	.	.	PUNCT
ejpam-7047	276	1	advances	advance	NOUN
ejpam-7047	276	2	in	in	ADP
ejpam-7047	276	3	mathematics	mathematic	NOUN
ejpam-7047	276	4	:	:	PUNCT
ejpam-7047	276	5	scientific	scientific	ADJ
ejpam-7047	276	6	journal	journal	NOUN
ejpam-7047	276	7	,	,	PUNCT
ejpam-7047	276	8	9(3):859–886	9(3):859–886	NUM
ejpam-7047	276	9	,	,	PUNCT
ejpam-7047	276	10	2020	2020	NUM
ejpam-7047	276	11	.	.	PUNCT
ejpam-7047	277	1	[	[	X
ejpam-7047	277	2	6	6	NUM
ejpam-7047	277	3	]	]	PUNCT
ejpam-7047	277	4	c.	c.	PROPN
ejpam-7047	277	5	boonpok	boonpok	PROPN
ejpam-7047	277	6	and	and	CCONJ
ejpam-7047	277	7	j.	j.	PROPN
ejpam-7047	277	8	khampakdee	khampakdee	PROPN
ejpam-7047	277	9	.	.	PUNCT
ejpam-7047	278	1	upper	upper	ADJ
ejpam-7047	278	2	and	and	CCONJ
ejpam-7047	278	3	lower	low	ADJ
ejpam-7047	278	4	weak	weak	ADJ
ejpam-7047	278	5	sβ(⋆)-continuity	sβ(⋆)-continuity	NOUN
ejpam-7047	278	6	.	.	PUNCT
ejpam-7047	279	1	european	european	PROPN
ejpam-7047	279	2	journal	journal	PROPN
ejpam-7047	279	3	of	of	ADP
ejpam-7047	279	4	pure	pure	ADJ
ejpam-7047	279	5	and	and	CCONJ
ejpam-7047	279	6	applied	applied	ADJ
ejpam-7047	279	7	mathematics	mathematic	NOUN
ejpam-7047	279	8	,	,	PUNCT
ejpam-7047	279	9	16(4):2544–2556	16(4):2544–2556	NUM
ejpam-7047	279	10	,	,	PUNCT
ejpam-7047	279	11	2023	2023	NUM
ejpam-7047	279	12	.	.	PUNCT
ejpam-7047	280	1	[	[	X
ejpam-7047	280	2	7	7	X
ejpam-7047	280	3	]	]	X
ejpam-7047	280	4	c.	c.	NOUN
ejpam-7047	280	5	boonpok	boonpok	PROPN
ejpam-7047	280	6	and	and	CCONJ
ejpam-7047	280	7	p.	p.	NOUN
ejpam-7047	280	8	pue	pue	NOUN
ejpam-7047	280	9	-	-	PUNCT
ejpam-7047	280	10	on	on	ADP
ejpam-7047	280	11	.	.	PUNCT
ejpam-7047	281	1	continuity	continuity	NOUN
ejpam-7047	281	2	for	for	ADP
ejpam-7047	281	3	multifunctions	multifunction	NOUN
ejpam-7047	281	4	in	in	ADP
ejpam-7047	281	5	ideal	ideal	ADJ
ejpam-7047	281	6	topological	topological	ADJ
ejpam-7047	281	7	spaces	space	NOUN
ejpam-7047	281	8	.	.	PUNCT
ejpam-7047	282	1	wseas	wseas	VERB
ejpam-7047	282	2	transactions	transaction	NOUN
ejpam-7047	282	3	on	on	ADP
ejpam-7047	282	4	mathematics	mathematic	NOUN
ejpam-7047	282	5	,	,	PUNCT
ejpam-7047	282	6	19:624–631	19:624–631	NUM
ejpam-7047	282	7	,	,	PUNCT
ejpam-7047	282	8	2020	2020	NUM
ejpam-7047	282	9	.	.	PUNCT
ejpam-7047	283	1	[	[	X
ejpam-7047	283	2	8	8	NUM
ejpam-7047	283	3	]	]	X
ejpam-7047	283	4	c.	c.	PROPN
ejpam-7047	283	5	boonpok	boonpok	PROPN
ejpam-7047	283	6	.	.	PUNCT
ejpam-7047	284	1	pı	pı	NOUN
ejpam-7047	284	2	-	-	NOUN
ejpam-7047	284	3	continuity	continuity	NOUN
ejpam-7047	284	4	and	and	CCONJ
ejpam-7047	284	5	weak	weak	ADJ
ejpam-7047	284	6	pı	pı	NOUN
ejpam-7047	284	7	-	-	NOUN
ejpam-7047	284	8	continuity	continuity	NOUN
ejpam-7047	284	9	.	.	PUNCT
ejpam-7047	285	1	carpathian	carpathian	ADJ
ejpam-7047	285	2	mathematical	mathematical	ADJ
ejpam-7047	285	3	publications	publication	NOUN
ejpam-7047	285	4	,	,	PUNCT
ejpam-7047	285	5	17(1):171–186	17(1):171–186	PROPN
ejpam-7047	285	6	,	,	PUNCT
ejpam-7047	285	7	2025	2025	NUM
ejpam-7047	285	8	.	.	PUNCT
ejpam-7047	286	1	[	[	X
ejpam-7047	286	2	9	9	NUM
ejpam-7047	286	3	]	]	X
ejpam-7047	286	4	p.	p.	NOUN
ejpam-7047	286	5	pue	pue	NOUN
ejpam-7047	286	6	-	-	PUNCT
ejpam-7047	286	7	on	on	ADP
ejpam-7047	286	8	,	,	PUNCT
ejpam-7047	286	9	s.	s.	PROPN
ejpam-7047	286	10	sompong	sompong	PROPN
ejpam-7047	286	11	,	,	PUNCT
ejpam-7047	286	12	and	and	CCONJ
ejpam-7047	286	13	c.	c.	PROPN
ejpam-7047	286	14	boonpok	boonpok	PROPN
ejpam-7047	286	15	.	.	PUNCT
ejpam-7047	287	1	upper	upper	ADJ
ejpam-7047	287	2	and	and	CCONJ
ejpam-7047	287	3	lower	low	ADJ
ejpam-7047	287	4	(	(	PUNCT
ejpam-7047	287	5	τ1	τ1	NOUN
ejpam-7047	287	6	,	,	PUNCT
ejpam-7047	287	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7047	287	8	multifunctions	multifunction	NOUN
ejpam-7047	287	9	.	.	PUNCT
ejpam-7047	288	1	international	international	ADJ
ejpam-7047	288	2	journal	journal	PROPN
ejpam-7047	288	3	of	of	ADP
ejpam-7047	288	4	mathematics	mathematic	NOUN
ejpam-7047	288	5	and	and	CCONJ
ejpam-7047	288	6	computer	computer	NOUN
ejpam-7047	288	7	science	science	NOUN
ejpam-7047	288	8	,	,	PUNCT
ejpam-7047	288	9	19(4):1305	19(4):1305	NUM
ejpam-7047	288	10	–	–	PUNCT
ejpam-7047	288	11	1310	1310	NUM
ejpam-7047	288	12	,	,	PUNCT
ejpam-7047	288	13	2024	2024	NUM
ejpam-7047	288	14	.	.	PUNCT
ejpam-7047	289	1	[	[	X
ejpam-7047	289	2	10	10	NUM
ejpam-7047	289	3	]	]	PUNCT
ejpam-7047	289	4	m.	m.	NOUN
ejpam-7047	289	5	thongmoon	thongmoon	NOUN
ejpam-7047	289	6	,	,	PUNCT
ejpam-7047	289	7	s.	s.	PROPN
ejpam-7047	289	8	sompong	sompong	PROPN
ejpam-7047	289	9	,	,	PUNCT
ejpam-7047	289	10	and	and	CCONJ
ejpam-7047	289	11	c.	c.	PROPN
ejpam-7047	289	12	boonpok	boonpok	PROPN
ejpam-7047	289	13	.	.	PUNCT
ejpam-7047	290	1	uppr	uppr	NOUN
ejpam-7047	290	2	and	and	CCONJ
ejpam-7047	290	3	lower	low	ADJ
ejpam-7047	290	4	weak	weak	ADJ
ejpam-7047	290	5	(	(	PUNCT
ejpam-7047	290	6	τ1	τ1	NOUN
ejpam-7047	290	7	,	,	PUNCT
ejpam-7047	290	8	τ2)continuity	τ2)continuity	PROPN
ejpam-7047	290	9	.	.	PUNCT
ejpam-7047	291	1	european	european	PROPN
ejpam-7047	291	2	journal	journal	PROPN
ejpam-7047	291	3	of	of	ADP
ejpam-7047	291	4	pure	pure	ADJ
ejpam-7047	291	5	and	and	CCONJ
ejpam-7047	291	6	applied	applied	ADJ
ejpam-7047	291	7	mathematics	mathematic	NOUN
ejpam-7047	291	8	,	,	PUNCT
ejpam-7047	291	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-7047	291	10	,	,	PUNCT
ejpam-7047	291	11	2024	2024	NUM
ejpam-7047	291	12	.	.	PUNCT
ejpam-7047	292	1	[	[	X
ejpam-7047	292	2	11	11	NUM
ejpam-7047	292	3	]	]	X
ejpam-7047	292	4	c.	c.	PROPN
ejpam-7047	292	5	viriyapong	viriyapong	PROPN
ejpam-7047	292	6	and	and	CCONJ
ejpam-7047	292	7	c.	c.	PROPN
ejpam-7047	292	8	boonpok	boonpok	PROPN
ejpam-7047	292	9	.	.	PUNCT
ejpam-7047	293	1	(	(	PUNCT
ejpam-7047	293	2	τ1	τ1	NOUN
ejpam-7047	293	3	,	,	PUNCT
ejpam-7047	293	4	τ2)α	τ2)α	NOUN
ejpam-7047	293	5	-	-	PUNCT
ejpam-7047	293	6	continuity	continuity	NOUN
ejpam-7047	293	7	for	for	ADP
ejpam-7047	293	8	multifunctions	multifunction	NOUN
ejpam-7047	293	9	.	.	PUNCT
ejpam-7047	294	1	journal	journal	PROPN
ejpam-7047	294	2	of	of	ADP
ejpam-7047	294	3	mathematics	mathematic	NOUN
ejpam-7047	294	4	,	,	PUNCT
ejpam-7047	294	5	2020:6285763	2020:6285763	NUM
ejpam-7047	294	6	,	,	PUNCT
ejpam-7047	294	7	2020	2020	NUM
ejpam-7047	294	8	.	.	PUNCT
ejpam-7047	295	1	[	[	X
ejpam-7047	295	2	12	12	NUM
ejpam-7047	295	3	]	]	X
ejpam-7047	295	4	p.	p.	NOUN
ejpam-7047	295	5	pue	pue	NOUN
ejpam-7047	295	6	-	-	PUNCT
ejpam-7047	295	7	on	on	ADP
ejpam-7047	295	8	,	,	PUNCT
ejpam-7047	295	9	a.	a.	PROPN
ejpam-7047	295	10	sama	sama	PROPN
ejpam-7047	295	11	-	-	PUNCT
ejpam-7047	295	12	ae	ae	PROPN
ejpam-7047	295	13	,	,	PUNCT
ejpam-7047	295	14	and	and	CCONJ
ejpam-7047	295	15	c.	c.	PROPN
ejpam-7047	295	16	boonpok	boonpok	PROPN
ejpam-7047	295	17	.	.	PUNCT
ejpam-7047	296	1	on	on	ADP
ejpam-7047	296	2	weak	weak	ADJ
ejpam-7047	296	3	forms	form	NOUN
ejpam-7047	296	4	of	of	ADP
ejpam-7047	296	5	upper	upper	ADJ
ejpam-7047	296	6	and	and	CCONJ
ejpam-7047	296	7	lower	low	ADJ
ejpam-7047	296	8	continuous	continuous	ADJ
ejpam-7047	296	9	multifunctions	multifunction	NOUN
ejpam-7047	296	10	between	between	ADP
ejpam-7047	296	11	an	an	DET
ejpam-7047	296	12	ideal	ideal	ADJ
ejpam-7047	296	13	topological	topological	ADJ
ejpam-7047	296	14	space	space	NOUN
ejpam-7047	296	15	and	and	CCONJ
ejpam-7047	296	16	a	a	DET
ejpam-7047	296	17	bitopological	bitopological	ADJ
ejpam-7047	296	18	space	space	NOUN
ejpam-7047	296	19	.	.	PUNCT
ejpam-7047	297	1	european	european	ADJ
ejpam-7047	297	2	journal	journal	PROPN
ejpam-7047	297	3	of	of	ADP
ejpam-7047	297	4	pure	pure	ADJ
ejpam-7047	297	5	and	and	CCONJ
ejpam-7047	297	6	applied	applied	ADJ
ejpam-7047	297	7	mathematics	mathematic	NOUN
ejpam-7047	297	8	,	,	PUNCT
ejpam-7047	297	9	18(3):6567	18(3):6567	NUM
ejpam-7047	297	10	,	,	PUNCT
ejpam-7047	297	11	2025	2025	NUM
ejpam-7047	297	12	.	.	PUNCT
ejpam-7047	298	1	[	[	X
ejpam-7047	298	2	13	13	NUM
ejpam-7047	298	3	]	]	PUNCT
ejpam-7047	298	4	c.	c.	PROPN
ejpam-7047	298	5	boonpok	boonpok	PROPN
ejpam-7047	298	6	,	,	PUNCT
ejpam-7047	298	7	c.	c.	PROPN
ejpam-7047	298	8	viriyapong	viriyapong	PROPN
ejpam-7047	298	9	,	,	PUNCT
ejpam-7047	298	10	and	and	CCONJ
ejpam-7047	298	11	m.	m.	NOUN
ejpam-7047	298	12	thongmoon	thongmoon	NOUN
ejpam-7047	298	13	.	.	PUNCT
ejpam-7047	299	1	on	on	ADP
ejpam-7047	299	2	upper	upper	ADJ
ejpam-7047	299	3	and	and	CCONJ
ejpam-7047	299	4	lower	low	ADJ
ejpam-7047	299	5	(	(	PUNCT
ejpam-7047	299	6	τ1	τ1	NOUN
ejpam-7047	299	7	,	,	PUNCT
ejpam-7047	299	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7047	299	9	multifunctions	multifunction	NOUN
ejpam-7047	299	10	.	.	PUNCT
ejpam-7047	300	1	journal	journal	PROPN
ejpam-7047	300	2	of	of	ADP
ejpam-7047	300	3	mathematics	mathematics	PROPN
ejpam-7047	300	4	and	and	CCONJ
ejpam-7047	300	5	computer	computer	NOUN
ejpam-7047	300	6	science	science	NOUN
ejpam-7047	300	7	,	,	PUNCT
ejpam-7047	300	8	18:282	18:282	NUM
ejpam-7047	300	9	–	–	PUNCT
ejpam-7047	300	10	293	293	NUM
ejpam-7047	300	11	,	,	PUNCT
ejpam-7047	300	12	2018	2018	NUM
ejpam-7047	300	13	.	.	PUNCT
ejpam-7047	301	1	[	[	X
ejpam-7047	301	2	14	14	NUM
ejpam-7047	301	3	]	]	X
ejpam-7047	301	4	c.	c.	PROPN
ejpam-7047	301	5	boonpok	boonpok	PROPN
ejpam-7047	301	6	.	.	PUNCT
ejpam-7047	302	1	(	(	PUNCT
ejpam-7047	302	2	τ1	τ1	NOUN
ejpam-7047	302	3	,	,	PUNCT
ejpam-7047	302	4	τ2)δ	τ2)δ	ADJ
ejpam-7047	302	5	-	-	PUNCT
ejpam-7047	302	6	semicontinuous	semicontinuous	ADJ
ejpam-7047	302	7	multifunctions	multifunction	NOUN
ejpam-7047	302	8	.	.	PUNCT
ejpam-7047	303	1	heliyon	heliyon	NOUN
ejpam-7047	303	2	,	,	PUNCT
ejpam-7047	303	3	6	6	NUM
ejpam-7047	303	4	:	:	SYM
ejpam-7047	303	5	e05367	e05367	PROPN
ejpam-7047	303	6	,	,	PUNCT
ejpam-7047	303	7	2020	2020	NUM
ejpam-7047	303	8	.	.	PUNCT
ejpam-7047	304	1	[	[	X
ejpam-7047	304	2	15	15	NUM
ejpam-7047	304	3	]	]	X
ejpam-7047	304	4	c.	c.	PROPN
ejpam-7047	304	5	boonpok	boonpok	PROPN
ejpam-7047	304	6	and	and	CCONJ
ejpam-7047	304	7	p.	p.	NOUN
ejpam-7047	304	8	pue	pue	NOUN
ejpam-7047	304	9	-	-	PUNCT
ejpam-7047	304	10	on	on	ADP
ejpam-7047	304	11	.	.	PUNCT
ejpam-7047	305	1	characterizations	characterization	NOUN
ejpam-7047	305	2	of	of	ADP
ejpam-7047	305	3	almost	almost	ADV
ejpam-7047	305	4	(	(	PUNCT
ejpam-7047	305	5	τ1	τ1	NOUN
ejpam-7047	305	6	,	,	PUNCT
ejpam-7047	305	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7047	305	8	multifunctions	multifunction	NOUN
ejpam-7047	305	9	.	.	PUNCT
ejpam-7047	306	1	international	international	ADJ
ejpam-7047	306	2	journal	journal	NOUN
ejpam-7047	306	3	of	of	ADP
ejpam-7047	306	4	analysis	analysis	NOUN
ejpam-7047	306	5	and	and	CCONJ
ejpam-7047	306	6	applications	application	NOUN
ejpam-7047	306	7	,	,	PUNCT
ejpam-7047	306	8	22:33	22:33	NUM
ejpam-7047	306	9	,	,	PUNCT
ejpam-7047	306	10	2024	2024	NUM
ejpam-7047	306	11	.	.	PUNCT
ejpam-7047	307	1	[	[	X
ejpam-7047	307	2	16	16	NUM
ejpam-7047	307	3	]	]	PUNCT
ejpam-7047	307	4	k.	k.	PROPN
ejpam-7047	307	5	kuratowski	kuratowski	PROPN
ejpam-7047	307	6	.	.	PUNCT
ejpam-7047	308	1	topology	topology	PROPN
ejpam-7047	308	2	,	,	PUNCT
ejpam-7047	308	3	vol	vol	NOUN
ejpam-7047	308	4	.	.	PUNCT
ejpam-7047	308	5	i.	i.	PROPN
ejpam-7047	308	6	academic	academic	PROPN
ejpam-7047	308	7	press	press	PROPN
ejpam-7047	308	8	,	,	PUNCT
ejpam-7047	308	9	new	new	PROPN
ejpam-7047	308	10	york	york	PROPN
ejpam-7047	308	11	,	,	PUNCT
ejpam-7047	308	12	1966	1966	NUM
ejpam-7047	308	13	.	.	PUNCT
ejpam-7047	309	1	[	[	X
ejpam-7047	309	2	17	17	NUM
ejpam-7047	309	3	]	]	X
ejpam-7047	309	4	d.	d.	PROPN
ejpam-7047	309	5	janković	janković	VERB
ejpam-7047	309	6	and	and	CCONJ
ejpam-7047	309	7	t.	t.	PROPN
ejpam-7047	309	8	r.	r.	PROPN
ejpam-7047	309	9	hamlett	hamlett	PROPN
ejpam-7047	309	10	.	.	PUNCT
ejpam-7047	310	1	new	new	ADJ
ejpam-7047	310	2	topologies	topology	NOUN
ejpam-7047	310	3	from	from	ADP
ejpam-7047	310	4	old	old	ADJ
ejpam-7047	310	5	via	via	ADP
ejpam-7047	310	6	ideals	ideal	NOUN
ejpam-7047	310	7	.	.	PUNCT
ejpam-7047	311	1	the	the	DET
ejpam-7047	311	2	american	american	PROPN
ejpam-7047	311	3	mathematical	mathematical	PROPN
ejpam-7047	311	4	monthly	monthly	ADV
ejpam-7047	311	5	,	,	PUNCT
ejpam-7047	311	6	97:295–310	97:295–310	PROPN
ejpam-7047	311	7	,	,	PUNCT
ejpam-7047	311	8	1990	1990	NUM
ejpam-7047	311	9	.	.	PUNCT
ejpam-7047	312	1	n.	n.	PROPN
ejpam-7047	312	2	viriyapong	viriyapong	PROPN
ejpam-7047	312	3	,	,	PUNCT
ejpam-7047	312	4	a.	a.	PROPN
ejpam-7047	312	5	sama	sama	PROPN
ejpam-7047	312	6	-	-	PUNCT
ejpam-7047	312	7	ae	ae	PROPN
ejpam-7047	312	8	,	,	PUNCT
ejpam-7047	312	9	c.	c.	PROPN
ejpam-7047	312	10	boonpok	boonpok	PROPN
ejpam-7047	312	11	/	/	SYM
ejpam-7047	312	12	eur	eur	PROPN
ejpam-7047	312	13	.	.	PUNCT
ejpam-7047	313	1	j.	j.	PROPN
ejpam-7047	313	2	pure	pure	PROPN
ejpam-7047	313	3	appl	appl	PROPN
ejpam-7047	313	4	.	.	PROPN
ejpam-7047	313	5	math	math	PROPN
ejpam-7047	313	6	,	,	PUNCT
ejpam-7047	313	7	18	18	NUM
ejpam-7047	313	8	(	(	PUNCT
ejpam-7047	313	9	4	4	NUM
ejpam-7047	313	10	)	)	PUNCT
ejpam-7047	313	11	(	(	PUNCT
ejpam-7047	313	12	2025	2025	NUM
ejpam-7047	313	13	)	)	PUNCT
ejpam-7047	313	14	,	,	PUNCT
ejpam-7047	313	15	7047	7047	NUM
ejpam-7047	313	16	11	11	NUM
ejpam-7047	313	17	of	of	ADP
ejpam-7047	313	18	11	11	NUM
ejpam-7047	313	19	[	[	SYM
ejpam-7047	313	20	18	18	NUM
ejpam-7047	313	21	]	]	PUNCT
ejpam-7047	313	22	c.	c.	PROPN
ejpam-7047	313	23	boonpok	boonpok	PROPN
ejpam-7047	313	24	.	.	PUNCT
ejpam-7047	314	1	weak	weak	ADJ
ejpam-7047	314	2	quasi	quasi	ADJ
ejpam-7047	314	3	continuity	continuity	NOUN
ejpam-7047	314	4	for	for	ADP
ejpam-7047	314	5	multifunctions	multifunction	NOUN
ejpam-7047	314	6	in	in	ADP
ejpam-7047	314	7	ideal	ideal	ADJ
ejpam-7047	314	8	topological	topological	ADJ
ejpam-7047	314	9	spaces	space	NOUN
ejpam-7047	314	10	.	.	PUNCT
ejpam-7047	315	1	advances	advance	NOUN
ejpam-7047	315	2	in	in	ADP
ejpam-7047	315	3	mathematics	mathematic	NOUN
ejpam-7047	315	4	:	:	PUNCT
ejpam-7047	315	5	scientific	scientific	ADJ
ejpam-7047	315	6	journal	journal	NOUN
ejpam-7047	315	7	,	,	PUNCT
ejpam-7047	315	8	9(3):339–355	9(3):339–355	NUM
ejpam-7047	315	9	,	,	PUNCT
ejpam-7047	315	10	2020	2020	NUM
ejpam-7047	315	11	.	.	PUNCT
ejpam-7047	316	1	[	[	X
ejpam-7047	316	2	19	19	NUM
ejpam-7047	316	3	]	]	X
ejpam-7047	316	4	t.	t.	PROPN
ejpam-7047	316	5	noiri	noiri	PROPN
ejpam-7047	316	6	and	and	CCONJ
ejpam-7047	316	7	v.	v.	ADP
ejpam-7047	316	8	popa	popa	NOUN
ejpam-7047	316	9	.	.	PUNCT
ejpam-7047	317	1	on	on	ADP
ejpam-7047	317	2	(	(	PUNCT
ejpam-7047	317	3	mi	mi	ADJ
ejpam-7047	317	4	,	,	PUNCT
ejpam-7047	317	5	nj)-continuous	nj)-continuous	ADJ
ejpam-7047	317	6	multifunctions	multifunction	NOUN
ejpam-7047	317	7	.	.	PUNCT
ejpam-7047	318	1	romanian	romanian	ADJ
ejpam-7047	318	2	journal	journal	PROPN
ejpam-7047	318	3	of	of	ADP
ejpam-7047	318	4	mathematics	mathematics	PROPN
ejpam-7047	318	5	and	and	CCONJ
ejpam-7047	318	6	computer	computer	NOUN
ejpam-7047	318	7	science	science	NOUN
ejpam-7047	318	8	,	,	PUNCT
ejpam-7047	318	9	15(1):1–8	15(1):1–8	NUM
ejpam-7047	318	10	,	,	PUNCT
ejpam-7047	318	11	2025	2025	NUM
ejpam-7047	318	12	.	.	PUNCT
ejpam-7047	319	1	[	[	X
ejpam-7047	319	2	20	20	NUM
ejpam-7047	319	3	]	]	PUNCT
ejpam-7047	319	4	c.	c.	PROPN
ejpam-7047	319	5	boonpok	boonpok	PROPN
ejpam-7047	319	6	.	.	PUNCT
ejpam-7047	320	1	a	a	DET
ejpam-7047	320	2	study	study	NOUN
ejpam-7047	320	3	of	of	ADP
ejpam-7047	320	4	some	some	DET
ejpam-7047	320	5	forms	form	NOUN
ejpam-7047	320	6	of	of	ADP
ejpam-7047	320	7	continuity	continuity	NOUN
ejpam-7047	320	8	for	for	ADP
ejpam-7047	320	9	multifunctions	multifunction	NOUN
ejpam-7047	320	10	in	in	ADP
ejpam-7047	320	11	ideal	ideal	ADJ
ejpam-7047	320	12	topological	topological	ADJ
ejpam-7047	320	13	spaces	space	NOUN
ejpam-7047	320	14	.	.	PUNCT
ejpam-7047	321	1	mathematica	mathematica	PROPN
ejpam-7047	321	2	,	,	PUNCT
ejpam-7047	321	3	63(2):186–198	63(2):186–198	PROPN
ejpam-7047	321	4	,	,	PUNCT
ejpam-7047	321	5	2021	2021	NUM
ejpam-7047	321	6	.	.	PUNCT
ejpam-7047	322	1	[	[	X
ejpam-7047	322	2	21	21	NUM
ejpam-7047	322	3	]	]	X
ejpam-7047	322	4	c.	c.	PROPN
ejpam-7047	322	5	viriyapong	viriyapong	PROPN
ejpam-7047	322	6	,	,	PUNCT
ejpam-7047	322	7	a.	a.	PROPN
ejpam-7047	322	8	sama	sama	PROPN
ejpam-7047	322	9	-	-	PUNCT
ejpam-7047	322	10	ae	ae	PROPN
ejpam-7047	322	11	,	,	PUNCT
ejpam-7047	322	12	and	and	CCONJ
ejpam-7047	322	13	c.	c.	PROPN
ejpam-7047	322	14	boonpok	boonpok	PROPN
ejpam-7047	322	15	.	.	PUNCT
ejpam-7047	323	1	on	on	ADP
ejpam-7047	323	2	upper	upper	ADJ
ejpam-7047	323	3	and	and	CCONJ
ejpam-7047	323	4	lower	low	ADJ
ejpam-7047	323	5	almost	almost	ADV
ejpam-7047	323	6	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	323	7	,	,	PUNCT
ejpam-7047	323	8	σ2)continuous	σ2)continuous	ADJ
ejpam-7047	323	9	multifunctions	multifunction	NOUN
ejpam-7047	323	10	.	.	PUNCT
ejpam-7047	324	1	(	(	PUNCT
ejpam-7047	324	2	accepted	accept	VERB
ejpam-7047	324	3	)	)	PUNCT
ejpam-7047	324	4	.	.	PUNCT
ejpam-7047	325	1	[	[	X
ejpam-7047	325	2	22	22	NUM
ejpam-7047	325	3	]	]	X
ejpam-7047	325	4	j.	j.	PROPN
ejpam-7047	325	5	khampakdee	khampakdee	PROPN
ejpam-7047	325	6	,	,	PUNCT
ejpam-7047	325	7	a.	a.	PROPN
ejpam-7047	325	8	sama	sama	PROPN
ejpam-7047	325	9	-	-	PUNCT
ejpam-7047	325	10	ae	ae	PROPN
ejpam-7047	325	11	,	,	PUNCT
ejpam-7047	325	12	and	and	CCONJ
ejpam-7047	325	13	c.	c.	PROPN
ejpam-7047	325	14	boonpok	boonpok	PROPN
ejpam-7047	325	15	.	.	PUNCT
ejpam-7047	326	1	upper	upper	ADJ
ejpam-7047	326	2	and	and	CCONJ
ejpam-7047	326	3	lower	low	ADJ
ejpam-7047	326	4	τ⋆α(σ1	τ⋆α(σ1	NOUN
ejpam-7047	326	5	,	,	PUNCT
ejpam-7047	326	6	σ2)continuity	σ2)continuity	NOUN
ejpam-7047	326	7	.	.	PUNCT
ejpam-7047	327	1	(	(	PUNCT
ejpam-7047	327	2	submitted	submit	VERB
ejpam-7047	327	3	)	)	PUNCT
ejpam-7047	327	4	.	.	PUNCT