id	sid	tid	token	lemma	pos
ejpam-7051	1	1	european	european	PROPN
ejpam-7051	1	2	journal	journal	PROPN
ejpam-7051	1	3	of	of	ADP
ejpam-7051	1	4	pure	pure	ADJ
ejpam-7051	1	5	and	and	CCONJ
ejpam-7051	1	6	applied	applied	ADJ
ejpam-7051	1	7	mathematics	mathematic	NOUN
ejpam-7051	1	8	2025	2025	NUM
ejpam-7051	1	9	,	,	PUNCT
ejpam-7051	1	10	vol	vol	NOUN
ejpam-7051	1	11	.	.	PROPN
ejpam-7051	1	12	18	18	NUM
ejpam-7051	1	13	,	,	PUNCT
ejpam-7051	1	14	issue	issue	NOUN
ejpam-7051	1	15	4	4	NUM
ejpam-7051	1	16	,	,	PUNCT
ejpam-7051	1	17	article	article	NOUN
ejpam-7051	1	18	number	number	NOUN
ejpam-7051	1	19	7051	7051	NUM
ejpam-7051	1	20	issn	issn	PROPN
ejpam-7051	1	21	1307	1307	NUM
ejpam-7051	1	22	-	-	SYM
ejpam-7051	1	23	5543	5543	NUM
ejpam-7051	1	24	–	–	PUNCT
ejpam-7051	1	25	ejpam.com	ejpam.com	X
ejpam-7051	1	26	published	publish	VERB
ejpam-7051	1	27	by	by	ADP
ejpam-7051	1	28	new	new	PROPN
ejpam-7051	1	29	york	york	PROPN
ejpam-7051	1	30	business	business	PROPN
ejpam-7051	1	31	global	global	PROPN
ejpam-7051	1	32	on	on	ADP
ejpam-7051	1	33	upper	upper	ADJ
ejpam-7051	1	34	and	and	CCONJ
ejpam-7051	1	35	lower	low	ADJ
ejpam-7051	1	36	µ(σ1	µ(σ1	NOUN
ejpam-7051	1	37	,	,	PUNCT
ejpam-7051	1	38	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	1	39	multifunctions	multifunction	NOUN
ejpam-7051	1	40	nipaporn	nipaporn	ADJ
ejpam-7051	1	41	chutiman1	chutiman1	NOUN
ejpam-7051	1	42	,	,	PUNCT
ejpam-7051	1	43	areeyuth	areeyuth	NOUN
ejpam-7051	1	44	sama	sama	NOUN
ejpam-7051	1	45	-	-	PUNCT
ejpam-7051	1	46	ae2	ae2	PROPN
ejpam-7051	1	47	,	,	PUNCT
ejpam-7051	1	48	chawalit	chawalit	VERB
ejpam-7051	1	49	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7051	1	50	1	1	NUM
ejpam-7051	1	51	mathematics	mathematic	NOUN
ejpam-7051	1	52	and	and	CCONJ
ejpam-7051	1	53	applied	apply	VERB
ejpam-7051	1	54	mathematics	mathematics	PROPN
ejpam-7051	1	55	research	research	NOUN
ejpam-7051	1	56	unit	unit	NOUN
ejpam-7051	1	57	,	,	PUNCT
ejpam-7051	1	58	department	department	NOUN
ejpam-7051	1	59	of	of	ADP
ejpam-7051	1	60	mathematics	mathematic	NOUN
ejpam-7051	1	61	,	,	PUNCT
ejpam-7051	1	62	faculty	faculty	NOUN
ejpam-7051	1	63	of	of	ADP
ejpam-7051	1	64	science	science	NOUN
ejpam-7051	1	65	,	,	PUNCT
ejpam-7051	1	66	mahasarakham	mahasarakham	PROPN
ejpam-7051	1	67	university	university	PROPN
ejpam-7051	1	68	,	,	PUNCT
ejpam-7051	1	69	maha	maha	PROPN
ejpam-7051	1	70	sarakham	sarakham	PROPN
ejpam-7051	1	71	,	,	PUNCT
ejpam-7051	1	72	44150	44150	NUM
ejpam-7051	1	73	,	,	PUNCT
ejpam-7051	1	74	thailand	thailand	PROPN
ejpam-7051	1	75	2	2	NUM
ejpam-7051	1	76	department	department	NOUN
ejpam-7051	1	77	of	of	ADP
ejpam-7051	1	78	mathematics	mathematic	NOUN
ejpam-7051	1	79	and	and	CCONJ
ejpam-7051	1	80	computer	computer	NOUN
ejpam-7051	1	81	science	science	NOUN
ejpam-7051	1	82	,	,	PUNCT
ejpam-7051	1	83	faculty	faculty	NOUN
ejpam-7051	1	84	of	of	ADP
ejpam-7051	1	85	science	science	NOUN
ejpam-7051	1	86	and	and	CCONJ
ejpam-7051	1	87	technology	technology	NOUN
ejpam-7051	1	88	,	,	PUNCT
ejpam-7051	1	89	prince	prince	NOUN
ejpam-7051	1	90	of	of	ADP
ejpam-7051	1	91	songkla	songkla	PROPN
ejpam-7051	1	92	university	university	PROPN
ejpam-7051	1	93	,	,	PUNCT
ejpam-7051	1	94	pattani	pattani	NOUN
ejpam-7051	1	95	campus	campus	NOUN
ejpam-7051	1	96	,	,	PUNCT
ejpam-7051	1	97	pattani	pattani	NOUN
ejpam-7051	1	98	,	,	PUNCT
ejpam-7051	1	99	94000	94000	NUM
ejpam-7051	1	100	,	,	PUNCT
ejpam-7051	1	101	thailand	thailand	PROPN
ejpam-7051	1	102	abstract	abstract	PROPN
ejpam-7051	1	103	.	.	PUNCT
ejpam-7051	2	1	this	this	DET
ejpam-7051	2	2	paper	paper	NOUN
ejpam-7051	2	3	presents	present	VERB
ejpam-7051	2	4	new	new	ADJ
ejpam-7051	2	5	concepts	concept	NOUN
ejpam-7051	2	6	of	of	ADP
ejpam-7051	2	7	continuous	continuous	ADJ
ejpam-7051	2	8	multifunctions	multifunction	NOUN
ejpam-7051	2	9	defined	define	VERB
ejpam-7051	2	10	from	from	ADP
ejpam-7051	2	11	a	a	DET
ejpam-7051	2	12	generalized	generalized	ADJ
ejpam-7051	2	13	topological	topological	ADJ
ejpam-7051	2	14	space	space	NOUN
ejpam-7051	2	15	into	into	ADP
ejpam-7051	2	16	a	a	DET
ejpam-7051	2	17	bitopological	bitopological	ADJ
ejpam-7051	2	18	space	space	NOUN
ejpam-7051	2	19	,	,	PUNCT
ejpam-7051	2	20	called	call	VERB
ejpam-7051	2	21	upper	upper	ADJ
ejpam-7051	2	22	µ(σ1	µ(σ1	NOUN
ejpam-7051	2	23	,	,	PUNCT
ejpam-7051	2	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	2	25	multifunctions	multifunction	NOUN
ejpam-7051	2	26	and	and	CCONJ
ejpam-7051	2	27	lower	low	ADJ
ejpam-7051	2	28	µ(σ1	µ(σ1	NOUN
ejpam-7051	2	29	,	,	PUNCT
ejpam-7051	2	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	2	31	multifunctions	multifunction	NOUN
ejpam-7051	2	32	.	.	PUNCT
ejpam-7051	3	1	furthermore	furthermore	ADV
ejpam-7051	3	2	,	,	PUNCT
ejpam-7051	3	3	several	several	ADJ
ejpam-7051	3	4	characterizations	characterization	NOUN
ejpam-7051	3	5	and	and	CCONJ
ejpam-7051	3	6	some	some	DET
ejpam-7051	3	7	properties	property	NOUN
ejpam-7051	3	8	concerning	concern	VERB
ejpam-7051	3	9	upper	upper	ADJ
ejpam-7051	3	10	µ(σ1	µ(σ1	NOUN
ejpam-7051	3	11	,	,	PUNCT
ejpam-7051	3	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	3	13	multifunctions	multifunction	NOUN
ejpam-7051	3	14	and	and	CCONJ
ejpam-7051	3	15	lower	low	ADJ
ejpam-7051	3	16	µ(σ1	µ(σ1	NOUN
ejpam-7051	3	17	,	,	PUNCT
ejpam-7051	3	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	3	19	multifunctions	multifunction	NOUN
ejpam-7051	3	20	are	be	AUX
ejpam-7051	3	21	discussed	discuss	VERB
ejpam-7051	3	22	.	.	PUNCT
ejpam-7051	4	1	2020	2020	NUM
ejpam-7051	4	2	mathematics	mathematic	NOUN
ejpam-7051	4	3	subject	subject	NOUN
ejpam-7051	4	4	classifications	classification	NOUN
ejpam-7051	4	5	:	:	PUNCT
ejpam-7051	4	6	54c08	54c08	NUM
ejpam-7051	4	7	,	,	PUNCT
ejpam-7051	4	8	54c60	54c60	NUM
ejpam-7051	4	9	key	key	ADJ
ejpam-7051	4	10	words	word	NOUN
ejpam-7051	4	11	and	and	CCONJ
ejpam-7051	4	12	phrases	phrase	NOUN
ejpam-7051	4	13	:	:	PUNCT
ejpam-7051	4	14	upper	upper	ADJ
ejpam-7051	4	15	µ(σ1	µ(σ1	NOUN
ejpam-7051	4	16	,	,	PUNCT
ejpam-7051	4	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	4	18	multifunction	multifunction	NOUN
ejpam-7051	4	19	,	,	PUNCT
ejpam-7051	4	20	lower	low	ADJ
ejpam-7051	4	21	µ(σ1	µ(σ1	NOUN
ejpam-7051	4	22	,	,	PUNCT
ejpam-7051	4	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	4	24	multifunction	multifunction	NOUN
ejpam-7051	4	25	1	1	NUM
ejpam-7051	4	26	.	.	PUNCT
ejpam-7051	4	27	introduction	introduction	NOUN
ejpam-7051	4	28	in	in	ADP
ejpam-7051	4	29	2002	2002	NUM
ejpam-7051	4	30	,	,	PUNCT
ejpam-7051	4	31	császár	császár	PROPN
ejpam-7051	5	1	[	[	X
ejpam-7051	5	2	1	1	NUM
ejpam-7051	5	3	]	]	PUNCT
ejpam-7051	5	4	introduced	introduce	VERB
ejpam-7051	5	5	the	the	DET
ejpam-7051	5	6	concepts	concept	NOUN
ejpam-7051	5	7	of	of	ADP
ejpam-7051	5	8	generalized	generalized	ADJ
ejpam-7051	5	9	topological	topological	ADJ
ejpam-7051	5	10	spaces	space	NOUN
ejpam-7051	5	11	and	and	CCONJ
ejpam-7051	5	12	generalized	generalized	ADJ
ejpam-7051	5	13	neighborhood	neighborhood	NOUN
ejpam-7051	5	14	systems	system	NOUN
ejpam-7051	5	15	.	.	PUNCT
ejpam-7051	6	1	the	the	DET
ejpam-7051	6	2	classes	class	NOUN
ejpam-7051	6	3	of	of	ADP
ejpam-7051	6	4	topological	topological	ADJ
ejpam-7051	6	5	spaces	space	NOUN
ejpam-7051	6	6	and	and	CCONJ
ejpam-7051	6	7	neighborhood	neighborhood	NOUN
ejpam-7051	6	8	systems	system	NOUN
ejpam-7051	6	9	are	be	AUX
ejpam-7051	6	10	contained	contain	VERB
ejpam-7051	6	11	in	in	ADP
ejpam-7051	6	12	the	the	DET
ejpam-7051	6	13	classes	class	NOUN
ejpam-7051	6	14	of	of	ADP
ejpam-7051	6	15	generalized	generalized	ADJ
ejpam-7051	6	16	topological	topological	ADJ
ejpam-7051	6	17	spaces	space	NOUN
ejpam-7051	6	18	and	and	CCONJ
ejpam-7051	6	19	generalized	generalized	ADJ
ejpam-7051	6	20	neighborhood	neighborhood	NOUN
ejpam-7051	6	21	systems	system	NOUN
ejpam-7051	6	22	,	,	PUNCT
ejpam-7051	6	23	respectively	respectively	ADV
ejpam-7051	6	24	.	.	PUNCT
ejpam-7051	7	1	moreover	moreover	ADV
ejpam-7051	7	2	,	,	PUNCT
ejpam-7051	7	3	császár	császár	PROPN
ejpam-7051	7	4	[	[	X
ejpam-7051	7	5	1	1	NUM
ejpam-7051	7	6	]	]	PUNCT
ejpam-7051	7	7	introduced	introduce	VERB
ejpam-7051	7	8	two	two	NUM
ejpam-7051	7	9	kinds	kind	NOUN
ejpam-7051	7	10	of	of	ADP
ejpam-7051	7	11	generalized	generalized	ADJ
ejpam-7051	7	12	continuous	continuous	ADJ
ejpam-7051	7	13	functions	function	NOUN
ejpam-7051	7	14	by	by	ADP
ejpam-7051	7	15	utilizing	utilize	VERB
ejpam-7051	7	16	the	the	DET
ejpam-7051	7	17	notions	notion	NOUN
ejpam-7051	7	18	of	of	ADP
ejpam-7051	7	19	generalized	generalized	ADJ
ejpam-7051	7	20	topological	topological	ADJ
ejpam-7051	7	21	spaces	space	NOUN
ejpam-7051	7	22	and	and	CCONJ
ejpam-7051	7	23	generalized	generalized	ADJ
ejpam-7051	7	24	neighborhood	neighborhood	NOUN
ejpam-7051	7	25	systems	system	NOUN
ejpam-7051	7	26	.	.	PUNCT
ejpam-7051	8	1	in	in	ADP
ejpam-7051	8	2	2009	2009	NUM
ejpam-7051	8	3	,	,	PUNCT
ejpam-7051	8	4	kanibir	kanibir	NOUN
ejpam-7051	8	5	and	and	CCONJ
ejpam-7051	8	6	reilly	reilly	ADV
ejpam-7051	9	1	[	[	X
ejpam-7051	9	2	2	2	NUM
ejpam-7051	9	3	]	]	PUNCT
ejpam-7051	9	4	extended	extend	VERB
ejpam-7051	9	5	the	the	DET
ejpam-7051	9	6	concept	concept	NOUN
ejpam-7051	9	7	of	of	ADP
ejpam-7051	9	8	generalized	generalized	ADJ
ejpam-7051	9	9	continuous	continuous	ADJ
ejpam-7051	9	10	functions	function	NOUN
ejpam-7051	9	11	to	to	ADP
ejpam-7051	9	12	multifunctions	multifunction	NOUN
ejpam-7051	9	13	and	and	CCONJ
ejpam-7051	9	14	defined	define	VERB
ejpam-7051	9	15	upper	upper	ADJ
ejpam-7051	9	16	semi	semi	ADJ
ejpam-7051	9	17	generalized	generalized	ADJ
ejpam-7051	9	18	continuous	continuous	ADJ
ejpam-7051	9	19	multifunctions	multifunction	NOUN
ejpam-7051	9	20	and	and	CCONJ
ejpam-7051	9	21	lower	low	ADJ
ejpam-7051	9	22	semi	semi	ADV
ejpam-7051	9	23	generalized	generalized	ADJ
ejpam-7051	9	24	continuous	continuous	ADJ
ejpam-7051	9	25	multifunctions	multifunction	NOUN
ejpam-7051	9	26	.	.	PUNCT
ejpam-7051	10	1	on	on	ADP
ejpam-7051	10	2	the	the	DET
ejpam-7051	10	3	other	other	ADJ
ejpam-7051	10	4	hand	hand	NOUN
ejpam-7051	10	5	,	,	PUNCT
ejpam-7051	10	6	the	the	DET
ejpam-7051	10	7	present	present	ADJ
ejpam-7051	10	8	authors	author	NOUN
ejpam-7051	10	9	introduced	introduce	VERB
ejpam-7051	10	10	and	and	CCONJ
ejpam-7051	10	11	investigated	investigate	VERB
ejpam-7051	10	12	four	four	NUM
ejpam-7051	10	13	classes	class	NOUN
ejpam-7051	10	14	of	of	ADP
ejpam-7051	10	15	multifunctions	multifunction	NOUN
ejpam-7051	10	16	defined	define	VERB
ejpam-7051	10	17	from	from	ADP
ejpam-7051	10	18	a	a	DET
ejpam-7051	10	19	generalized	generalized	ADJ
ejpam-7051	10	20	topological	topological	ADJ
ejpam-7051	10	21	space	space	NOUN
ejpam-7051	10	22	into	into	ADP
ejpam-7051	10	23	a	a	DET
ejpam-7051	10	24	generalized	generalized	ADJ
ejpam-7051	10	25	topological	topological	ADJ
ejpam-7051	10	26	space	space	NOUN
ejpam-7051	10	27	,	,	PUNCT
ejpam-7051	10	28	namely	namely	ADV
ejpam-7051	10	29	upper	upper	ADJ
ejpam-7051	10	30	β(µx	β(µx	PROPN
ejpam-7051	10	31	,	,	PUNCT
ejpam-7051	10	32	µy	µy	CCONJ
ejpam-7051	10	33	)	)	PUNCT
ejpam-7051	10	34	-continuous	-continuous	ADJ
ejpam-7051	10	35	multifunctions	multifunction	NOUN
ejpam-7051	11	1	[	[	X
ejpam-7051	11	2	3	3	NUM
ejpam-7051	11	3	]	]	PUNCT
ejpam-7051	11	4	,	,	PUNCT
ejpam-7051	11	5	lower	low	ADJ
ejpam-7051	11	6	β(µx	β(µx	PROPN
ejpam-7051	11	7	,	,	PUNCT
ejpam-7051	11	8	µy	µy	CCONJ
ejpam-7051	11	9	)	)	PUNCT
ejpam-7051	11	10	-continuous	-continuous	ADJ
ejpam-7051	11	11	multifunctions	multifunction	NOUN
ejpam-7051	12	1	[	[	X
ejpam-7051	12	2	3	3	NUM
ejpam-7051	12	3	]	]	PUNCT
ejpam-7051	12	4	,	,	PUNCT
ejpam-7051	12	5	upper	upper	ADJ
ejpam-7051	12	6	α(µx	α(µx	PROPN
ejpam-7051	12	7	,	,	PUNCT
ejpam-7051	12	8	µy	µy	CCONJ
ejpam-7051	12	9	)	)	PUNCT
ejpam-7051	12	10	-continuous	-continuous	ADJ
ejpam-7051	12	11	multifunctions	multifunction	NOUN
ejpam-7051	13	1	[	[	X
ejpam-7051	13	2	4	4	NUM
ejpam-7051	13	3	]	]	PUNCT
ejpam-7051	13	4	and	and	CCONJ
ejpam-7051	13	5	lower	low	ADJ
ejpam-7051	13	6	α(µx	α(µx	NUM
ejpam-7051	13	7	,	,	PUNCT
ejpam-7051	13	8	µy	µy	CCONJ
ejpam-7051	13	9	)	)	PUNCT
ejpam-7051	13	10	continuous	continuous	ADJ
ejpam-7051	13	11	multifunctions	multifunction	NOUN
ejpam-7051	14	1	[	[	X
ejpam-7051	14	2	4	4	NUM
ejpam-7051	14	3	]	]	PUNCT
ejpam-7051	14	4	.	.	PUNCT
ejpam-7051	15	1	pue	pue	NOUN
ejpam-7051	15	2	-	-	PUNCT
ejpam-7051	15	3	on	on	NOUN
ejpam-7051	15	4	et	et	PROPN
ejpam-7051	15	5	al	al	PROPN
ejpam-7051	15	6	.	.	PUNCT
ejpam-7051	16	1	[	[	X
ejpam-7051	16	2	5	5	NUM
ejpam-7051	16	3	]	]	PUNCT
ejpam-7051	16	4	introduced	introduce	VERB
ejpam-7051	16	5	and	and	CCONJ
ejpam-7051	16	6	studied	study	VERB
ejpam-7051	16	7	the	the	DET
ejpam-7051	16	8	concepts	concept	NOUN
ejpam-7051	16	9	of	of	ADP
ejpam-7051	16	10	∗corresponding	∗corresponde	VERB
ejpam-7051	16	11	author	author	NOUN
ejpam-7051	16	12	.	.	PUNCT
ejpam-7051	17	1	doi	doi	NOUN
ejpam-7051	17	2	:	:	PUNCT
ejpam-7051	17	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7051	https://doi.org/10.29020/nybg.ejpam.v18i4.7051	ADJ
ejpam-7051	17	4	email	email	NOUN
ejpam-7051	17	5	addresses	address	VERB
ejpam-7051	17	6	:	:	PUNCT
ejpam-7051	17	7	nipaporn.c@msu.ac.th	nipaporn.c@msu.ac.th	PROPN
ejpam-7051	17	8	(	(	PUNCT
ejpam-7051	17	9	n.	n.	NOUN
ejpam-7051	17	10	chutiman	chutiman	NOUN
ejpam-7051	17	11	)	)	PUNCT
ejpam-7051	17	12	,	,	PUNCT
ejpam-7051	17	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7051	17	14	(	(	PUNCT
ejpam-7051	17	15	a.	a.	PROPN
ejpam-7051	17	16	sama	sama	PROPN
ejpam-7051	17	17	-	-	PUNCT
ejpam-7051	17	18	ae	ae	PROPN
ejpam-7051	17	19	)	)	PUNCT
ejpam-7051	17	20	,	,	PUNCT
ejpam-7051	17	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7051	17	22	(	(	PUNCT
ejpam-7051	17	23	c.	c.	PROPN
ejpam-7051	17	24	boonpok	boonpok	PROPN
ejpam-7051	17	25	)	)	PUNCT
ejpam-7051	17	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7051	18	1	1	1	NUM
ejpam-7051	18	2	copyright	copyright	NOUN
ejpam-7051	18	3	:	:	PUNCT
ejpam-7051	18	4	©	©	PROPN
ejpam-7051	18	5	2025	2025	NUM
ejpam-7051	18	6	the	the	DET
ejpam-7051	18	7	author(s	author(s	NOUN
ejpam-7051	18	8	)	)	PUNCT
ejpam-7051	18	9	.	.	PUNCT
ejpam-7051	19	1	(	(	PUNCT
ejpam-7051	19	2	cc	cc	NOUN
ejpam-7051	19	3	by	by	ADP
ejpam-7051	19	4	-	-	PUNCT
ejpam-7051	19	5	nc	nc	PROPN
ejpam-7051	19	6	4.0	4.0	NUM
ejpam-7051	19	7	)	)	PUNCT
ejpam-7051	19	8	n.	n.	NOUN
ejpam-7051	19	9	chutiman	chutiman	NOUN
ejpam-7051	19	10	,	,	PUNCT
ejpam-7051	19	11	a.	a.	PROPN
ejpam-7051	19	12	sama	sama	PROPN
ejpam-7051	19	13	-	-	PUNCT
ejpam-7051	19	14	ae	ae	PROPN
ejpam-7051	19	15	,	,	PUNCT
ejpam-7051	19	16	c.	c.	PROPN
ejpam-7051	19	17	boonpok	boonpok	PROPN
ejpam-7051	19	18	/	/	SYM
ejpam-7051	19	19	eur	eur	PROPN
ejpam-7051	19	20	.	.	PUNCT
ejpam-7051	20	1	j.	j.	PROPN
ejpam-7051	20	2	pure	pure	PROPN
ejpam-7051	20	3	appl	appl	PROPN
ejpam-7051	20	4	.	.	PROPN
ejpam-7051	20	5	math	math	PROPN
ejpam-7051	20	6	,	,	PUNCT
ejpam-7051	20	7	18	18	NUM
ejpam-7051	20	8	(	(	PUNCT
ejpam-7051	20	9	4	4	NUM
ejpam-7051	20	10	)	)	PUNCT
ejpam-7051	20	11	(	(	PUNCT
ejpam-7051	20	12	2025	2025	NUM
ejpam-7051	20	13	)	)	PUNCT
ejpam-7051	20	14	,	,	PUNCT
ejpam-7051	20	15	7051	7051	NUM
ejpam-7051	20	16	2	2	NUM
ejpam-7051	20	17	of	of	ADP
ejpam-7051	20	18	8	8	NUM
ejpam-7051	20	19	upper	upper	ADJ
ejpam-7051	20	20	(	(	PUNCT
ejpam-7051	20	21	τ1	τ1	NOUN
ejpam-7051	20	22	,	,	PUNCT
ejpam-7051	20	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	20	24	multifunctions	multifunction	NOUN
ejpam-7051	20	25	and	and	CCONJ
ejpam-7051	20	26	lower	low	ADJ
ejpam-7051	20	27	(	(	PUNCT
ejpam-7051	20	28	τ1	τ1	NOUN
ejpam-7051	20	29	,	,	PUNCT
ejpam-7051	20	30	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	20	31	multifunctions	multifunction	NOUN
ejpam-7051	20	32	.	.	PUNCT
ejpam-7051	21	1	klanarong	klanarong	NOUN
ejpam-7051	21	2	et	et	PROPN
ejpam-7051	21	3	al	al	PROPN
ejpam-7051	21	4	.	.	PUNCT
ejpam-7051	22	1	[	[	X
ejpam-7051	22	2	6	6	NUM
ejpam-7051	22	3	]	]	PUNCT
ejpam-7051	22	4	investigated	investigate	VERB
ejpam-7051	22	5	several	several	ADJ
ejpam-7051	22	6	characterizations	characterization	NOUN
ejpam-7051	22	7	of	of	ADP
ejpam-7051	22	8	upper	upper	ADJ
ejpam-7051	22	9	(	(	PUNCT
ejpam-7051	22	10	τ1	τ1	NOUN
ejpam-7051	22	11	,	,	PUNCT
ejpam-7051	22	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	22	13	multifunctions	multifunction	NOUN
ejpam-7051	22	14	and	and	CCONJ
ejpam-7051	22	15	lower	low	ADJ
ejpam-7051	22	16	(	(	PUNCT
ejpam-7051	22	17	τ1	τ1	NOUN
ejpam-7051	22	18	,	,	PUNCT
ejpam-7051	22	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	22	20	multifunctions	multifunction	NOUN
ejpam-7051	22	21	by	by	ADP
ejpam-7051	22	22	utilizing	utilize	VERB
ejpam-7051	22	23	the	the	DET
ejpam-7051	22	24	notions	notion	NOUN
ejpam-7051	22	25	of	of	ADP
ejpam-7051	22	26	(	(	PUNCT
ejpam-7051	22	27	τ1	τ1	NOUN
ejpam-7051	22	28	,	,	PUNCT
ejpam-7051	22	29	τ2)θclosed	τ2)θclose	VERB
ejpam-7051	22	30	sets	set	NOUN
ejpam-7051	22	31	and	and	CCONJ
ejpam-7051	22	32	(	(	PUNCT
ejpam-7051	22	33	τ1	τ1	NOUN
ejpam-7051	22	34	,	,	PUNCT
ejpam-7051	22	35	τ2)θ	τ2)θ	ADJ
ejpam-7051	22	36	-	-	PUNCT
ejpam-7051	22	37	open	open	ADJ
ejpam-7051	22	38	sets	set	NOUN
ejpam-7051	22	39	.	.	PUNCT
ejpam-7051	23	1	thongmoon	thongmoon	NOUN
ejpam-7051	23	2	et	et	PROPN
ejpam-7051	23	3	al	al	PROPN
ejpam-7051	23	4	.	.	PUNCT
ejpam-7051	24	1	[	[	X
ejpam-7051	24	2	7	7	X
ejpam-7051	24	3	]	]	PUNCT
ejpam-7051	24	4	studied	study	VERB
ejpam-7051	24	5	some	some	DET
ejpam-7051	24	6	characterizations	characterization	NOUN
ejpam-7051	24	7	of	of	ADP
ejpam-7051	24	8	upper	upper	ADJ
ejpam-7051	24	9	(	(	PUNCT
ejpam-7051	24	10	τ1	τ1	NOUN
ejpam-7051	24	11	,	,	PUNCT
ejpam-7051	24	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	24	13	multifunctions	multifunction	NOUN
ejpam-7051	24	14	and	and	CCONJ
ejpam-7051	24	15	lower	low	ADJ
ejpam-7051	24	16	(	(	PUNCT
ejpam-7051	24	17	τ1	τ1	NOUN
ejpam-7051	24	18	,	,	PUNCT
ejpam-7051	24	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	24	20	multifunctions	multifunction	NOUN
ejpam-7051	24	21	by	by	ADP
ejpam-7051	24	22	using	use	VERB
ejpam-7051	24	23	τ1τ2	τ1τ2	ADJ
ejpam-7051	24	24	-	-	ADJ
ejpam-7051	24	25	δ	δ	ADJ
ejpam-7051	24	26	-	-	ADJ
ejpam-7051	24	27	open	open	ADJ
ejpam-7051	24	28	sets	set	NOUN
ejpam-7051	24	29	and	and	CCONJ
ejpam-7051	24	30	τ1τ2	τ1τ2	NOUN
ejpam-7051	24	31	-	-	ADJ
ejpam-7051	24	32	δ	δ	NOUN
ejpam-7051	24	33	-	-	PUNCT
ejpam-7051	24	34	closed	close	VERB
ejpam-7051	24	35	sets	set	NOUN
ejpam-7051	24	36	.	.	PUNCT
ejpam-7051	25	1	quite	quite	ADV
ejpam-7051	25	2	recently	recently	ADV
ejpam-7051	25	3	,	,	PUNCT
ejpam-7051	25	4	khampakdee	khampakdee	PROPN
ejpam-7051	25	5	et	et	NOUN
ejpam-7051	25	6	al	al	PROPN
ejpam-7051	25	7	.	.	PUNCT
ejpam-7051	26	1	[	[	X
ejpam-7051	26	2	8	8	NUM
ejpam-7051	26	3	]	]	PUNCT
ejpam-7051	26	4	presented	present	VERB
ejpam-7051	26	5	new	new	ADJ
ejpam-7051	26	6	classes	class	NOUN
ejpam-7051	26	7	of	of	ADP
ejpam-7051	26	8	continuous	continuous	ADJ
ejpam-7051	26	9	multifunctions	multifunction	NOUN
ejpam-7051	26	10	defined	define	VERB
ejpam-7051	26	11	from	from	ADP
ejpam-7051	26	12	an	an	DET
ejpam-7051	26	13	ideal	ideal	ADJ
ejpam-7051	26	14	topological	topological	ADJ
ejpam-7051	26	15	space	space	NOUN
ejpam-7051	26	16	into	into	ADP
ejpam-7051	26	17	a	a	DET
ejpam-7051	26	18	bitopological	bitopological	ADJ
ejpam-7051	26	19	space	space	NOUN
ejpam-7051	26	20	,	,	PUNCT
ejpam-7051	26	21	namely	namely	ADV
ejpam-7051	26	22	upper	upper	ADJ
ejpam-7051	26	23	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-7051	26	24	,	,	PUNCT
ejpam-7051	26	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	26	26	multifunctions	multifunction	NOUN
ejpam-7051	26	27	and	and	CCONJ
ejpam-7051	26	28	lower	low	ADJ
ejpam-7051	26	29	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7051	26	30	,	,	PUNCT
ejpam-7051	26	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	26	32	multifunctions	multifunction	NOUN
ejpam-7051	26	33	.	.	PUNCT
ejpam-7051	27	1	furthermore	furthermore	ADV
ejpam-7051	27	2	,	,	PUNCT
ejpam-7051	27	3	several	several	ADJ
ejpam-7051	27	4	characterizations	characterization	NOUN
ejpam-7051	27	5	and	and	CCONJ
ejpam-7051	27	6	some	some	DET
ejpam-7051	27	7	properties	property	NOUN
ejpam-7051	27	8	of	of	ADP
ejpam-7051	27	9	upper	upper	ADJ
ejpam-7051	27	10	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-7051	27	11	,	,	PUNCT
ejpam-7051	27	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	27	13	multifunctions	multifunction	NOUN
ejpam-7051	27	14	and	and	CCONJ
ejpam-7051	27	15	lower	low	ADJ
ejpam-7051	27	16	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7051	27	17	,	,	PUNCT
ejpam-7051	27	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	27	19	multifunctions	multifunction	NOUN
ejpam-7051	27	20	were	be	AUX
ejpam-7051	27	21	established	establish	VERB
ejpam-7051	27	22	in	in	ADP
ejpam-7051	27	23	[	[	X
ejpam-7051	27	24	8	8	NUM
ejpam-7051	27	25	]	]	PUNCT
ejpam-7051	27	26	.	.	PUNCT
ejpam-7051	28	1	in	in	ADP
ejpam-7051	28	2	this	this	DET
ejpam-7051	28	3	paper	paper	NOUN
ejpam-7051	28	4	,	,	PUNCT
ejpam-7051	28	5	we	we	PRON
ejpam-7051	28	6	introduce	introduce	VERB
ejpam-7051	28	7	the	the	DET
ejpam-7051	28	8	concepts	concept	NOUN
ejpam-7051	28	9	of	of	ADP
ejpam-7051	28	10	upper	upper	ADJ
ejpam-7051	28	11	µ(σ1	µ(σ1	NOUN
ejpam-7051	28	12	,	,	PUNCT
ejpam-7051	28	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	28	14	multifunctions	multifunction	NOUN
ejpam-7051	28	15	and	and	CCONJ
ejpam-7051	28	16	lower	low	ADJ
ejpam-7051	28	17	µ(σ1	µ(σ1	NOUN
ejpam-7051	28	18	,	,	PUNCT
ejpam-7051	28	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	28	20	multifunctions	multifunction	NOUN
ejpam-7051	28	21	.	.	PUNCT
ejpam-7051	29	1	we	we	PRON
ejpam-7051	29	2	also	also	ADV
ejpam-7051	29	3	investigate	investigate	VERB
ejpam-7051	29	4	several	several	ADJ
ejpam-7051	29	5	characterizations	characterization	NOUN
ejpam-7051	29	6	of	of	ADP
ejpam-7051	29	7	upper	upper	ADJ
ejpam-7051	29	8	µ(σ1	µ(σ1	NOUN
ejpam-7051	29	9	,	,	PUNCT
ejpam-7051	29	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	29	11	multifunctions	multifunction	NOUN
ejpam-7051	29	12	and	and	CCONJ
ejpam-7051	29	13	lower	low	ADJ
ejpam-7051	29	14	µ(σ1	µ(σ1	NOUN
ejpam-7051	29	15	,	,	PUNCT
ejpam-7051	29	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	29	17	multifunctions	multifunction	NOUN
ejpam-7051	29	18	.	.	PUNCT
ejpam-7051	30	1	2	2	X
ejpam-7051	30	2	.	.	X
ejpam-7051	30	3	preliminaries	preliminary	NOUN
ejpam-7051	30	4	throughout	throughout	ADP
ejpam-7051	30	5	the	the	DET
ejpam-7051	30	6	present	present	ADJ
ejpam-7051	30	7	paper	paper	NOUN
ejpam-7051	30	8	,	,	PUNCT
ejpam-7051	30	9	spaces	space	NOUN
ejpam-7051	30	10	(	(	PUNCT
ejpam-7051	30	11	x	x	NOUN
ejpam-7051	30	12	,	,	PUNCT
ejpam-7051	30	13	τ1	τ1	NOUN
ejpam-7051	30	14	,	,	PUNCT
ejpam-7051	30	15	τ2	τ2	NOUN
ejpam-7051	30	16	)	)	PUNCT
ejpam-7051	30	17	and	and	CCONJ
ejpam-7051	30	18	(	(	PUNCT
ejpam-7051	30	19	y	y	PROPN
ejpam-7051	30	20	,	,	PUNCT
ejpam-7051	30	21	σ1	σ1	PROPN
ejpam-7051	30	22	,	,	PUNCT
ejpam-7051	30	23	σ2	σ2	NOUN
ejpam-7051	30	24	)	)	PUNCT
ejpam-7051	30	25	(	(	PUNCT
ejpam-7051	30	26	or	or	CCONJ
ejpam-7051	30	27	simply	simply	ADV
ejpam-7051	30	28	x	x	X
ejpam-7051	30	29	and	and	CCONJ
ejpam-7051	30	30	y	y	PROPN
ejpam-7051	30	31	)	)	PUNCT
ejpam-7051	30	32	always	always	ADV
ejpam-7051	30	33	mean	mean	VERB
ejpam-7051	30	34	bitopological	bitopological	ADJ
ejpam-7051	30	35	spaces	space	NOUN
ejpam-7051	30	36	on	on	ADP
ejpam-7051	30	37	which	which	PRON
ejpam-7051	30	38	no	no	DET
ejpam-7051	30	39	separation	separation	NOUN
ejpam-7051	30	40	axioms	axiom	NOUN
ejpam-7051	30	41	are	be	AUX
ejpam-7051	30	42	assumed	assume	VERB
ejpam-7051	30	43	unless	unless	SCONJ
ejpam-7051	30	44	explicitly	explicitly	ADV
ejpam-7051	30	45	stated	state	VERB
ejpam-7051	30	46	.	.	PUNCT
ejpam-7051	31	1	let	let	VERB
ejpam-7051	31	2	a	a	DET
ejpam-7051	31	3	be	be	AUX
ejpam-7051	31	4	a	a	DET
ejpam-7051	31	5	subset	subset	NOUN
ejpam-7051	31	6	of	of	ADP
ejpam-7051	31	7	a	a	DET
ejpam-7051	31	8	bitopological	bitopological	ADJ
ejpam-7051	31	9	space	space	NOUN
ejpam-7051	31	10	(	(	PUNCT
ejpam-7051	31	11	x	x	NOUN
ejpam-7051	31	12	,	,	PUNCT
ejpam-7051	31	13	τ1	τ1	NOUN
ejpam-7051	31	14	,	,	PUNCT
ejpam-7051	31	15	τ2	τ2	NOUN
ejpam-7051	31	16	)	)	PUNCT
ejpam-7051	31	17	.	.	PUNCT
ejpam-7051	32	1	the	the	DET
ejpam-7051	32	2	closure	closure	NOUN
ejpam-7051	32	3	of	of	ADP
ejpam-7051	32	4	a	a	PRON
ejpam-7051	32	5	and	and	CCONJ
ejpam-7051	32	6	the	the	DET
ejpam-7051	32	7	interior	interior	NOUN
ejpam-7051	32	8	of	of	ADP
ejpam-7051	32	9	a	a	PRON
ejpam-7051	32	10	with	with	ADP
ejpam-7051	32	11	respect	respect	NOUN
ejpam-7051	32	12	to	to	ADP
ejpam-7051	32	13	τi	τi	PROPN
ejpam-7051	32	14	are	be	AUX
ejpam-7051	32	15	denoted	denote	VERB
ejpam-7051	32	16	by	by	ADP
ejpam-7051	32	17	τi	τi	NOUN
ejpam-7051	32	18	-	-	PUNCT
ejpam-7051	32	19	cl(a	cl(a	NUM
ejpam-7051	32	20	)	)	PUNCT
ejpam-7051	32	21	and	and	CCONJ
ejpam-7051	32	22	τi	τi	NOUN
ejpam-7051	32	23	-	-	PUNCT
ejpam-7051	32	24	int(a	int(a	NOUN
ejpam-7051	32	25	)	)	PUNCT
ejpam-7051	32	26	,	,	PUNCT
ejpam-7051	32	27	respectively	respectively	ADV
ejpam-7051	32	28	,	,	PUNCT
ejpam-7051	32	29	for	for	ADP
ejpam-7051	32	30	i	i	PROPN
ejpam-7051	32	31	=	=	SYM
ejpam-7051	32	32	1	1	NUM
ejpam-7051	32	33	,	,	PUNCT
ejpam-7051	32	34	2	2	NUM
ejpam-7051	32	35	.	.	X
ejpam-7051	32	36	a	a	DET
ejpam-7051	32	37	subset	subset	NOUN
ejpam-7051	32	38	a	a	PRON
ejpam-7051	32	39	of	of	ADP
ejpam-7051	32	40	a	a	DET
ejpam-7051	32	41	bitopological	bitopological	ADJ
ejpam-7051	32	42	space	space	NOUN
ejpam-7051	32	43	(	(	PUNCT
ejpam-7051	32	44	x	x	NOUN
ejpam-7051	32	45	,	,	PUNCT
ejpam-7051	32	46	τ1	τ1	NOUN
ejpam-7051	32	47	,	,	PUNCT
ejpam-7051	32	48	τ2	τ2	NOUN
ejpam-7051	32	49	)	)	PUNCT
ejpam-7051	32	50	is	be	AUX
ejpam-7051	32	51	called	call	VERB
ejpam-7051	32	52	τ1τ2closed	τ1τ2close	VERB
ejpam-7051	32	53	[	[	X
ejpam-7051	32	54	9	9	X
ejpam-7051	32	55	]	]	X
ejpam-7051	32	56	if	if	SCONJ
ejpam-7051	32	57	a	a	DET
ejpam-7051	32	58	=	=	NOUN
ejpam-7051	32	59	τ1	τ1	NOUN
ejpam-7051	32	60	-	-	PUNCT
ejpam-7051	32	61	cl(τ2	cl(τ2	NOUN
ejpam-7051	32	62	-	-	PUNCT
ejpam-7051	32	63	cl(a	cl(a	NUM
ejpam-7051	32	64	)	)	PUNCT
ejpam-7051	32	65	)	)	PUNCT
ejpam-7051	32	66	.	.	PUNCT
ejpam-7051	33	1	the	the	DET
ejpam-7051	33	2	complement	complement	NOUN
ejpam-7051	33	3	of	of	ADP
ejpam-7051	33	4	a	a	DET
ejpam-7051	33	5	τ1τ2	τ1τ2	ADJ
ejpam-7051	33	6	-	-	ADJ
ejpam-7051	33	7	closed	closed	ADJ
ejpam-7051	33	8	set	set	NOUN
ejpam-7051	33	9	is	be	AUX
ejpam-7051	33	10	called	call	VERB
ejpam-7051	33	11	τ1τ2	τ1τ2	NOUN
ejpam-7051	33	12	-	-	ADJ
ejpam-7051	33	13	open	open	ADJ
ejpam-7051	33	14	.	.	PUNCT
ejpam-7051	34	1	the	the	DET
ejpam-7051	34	2	intersection	intersection	NOUN
ejpam-7051	34	3	of	of	ADP
ejpam-7051	34	4	all	all	DET
ejpam-7051	34	5	τ1τ2	τ1τ2	ADJ
ejpam-7051	34	6	-	-	ADJ
ejpam-7051	34	7	closed	closed	ADJ
ejpam-7051	34	8	sets	set	NOUN
ejpam-7051	34	9	of	of	ADP
ejpam-7051	34	10	x	x	PUNCT
ejpam-7051	34	11	containing	contain	VERB
ejpam-7051	34	12	a	a	PRON
ejpam-7051	34	13	is	be	AUX
ejpam-7051	34	14	called	call	VERB
ejpam-7051	34	15	the	the	DET
ejpam-7051	34	16	τ1τ2	τ1τ2	NOUN
ejpam-7051	34	17	-	-	NOUN
ejpam-7051	34	18	closure	closure	NOUN
ejpam-7051	34	19	[	[	X
ejpam-7051	34	20	9	9	NUM
ejpam-7051	34	21	]	]	PUNCT
ejpam-7051	34	22	of	of	ADP
ejpam-7051	34	23	a	a	PRON
ejpam-7051	34	24	and	and	CCONJ
ejpam-7051	34	25	is	be	AUX
ejpam-7051	34	26	denoted	denote	VERB
ejpam-7051	34	27	by	by	ADP
ejpam-7051	34	28	τ1τ2	τ1τ2	NOUN
ejpam-7051	34	29	-	-	NUM
ejpam-7051	34	30	cl(a	cl(a	NUM
ejpam-7051	34	31	)	)	PUNCT
ejpam-7051	34	32	.	.	PUNCT
ejpam-7051	35	1	the	the	DET
ejpam-7051	35	2	union	union	NOUN
ejpam-7051	35	3	of	of	ADP
ejpam-7051	35	4	all	all	DET
ejpam-7051	35	5	τ1τ2	τ1τ2	ADJ
ejpam-7051	35	6	-	-	ADJ
ejpam-7051	35	7	open	open	ADJ
ejpam-7051	35	8	sets	set	NOUN
ejpam-7051	35	9	of	of	ADP
ejpam-7051	35	10	x	x	PUNCT
ejpam-7051	35	11	contained	contain	VERB
ejpam-7051	35	12	in	in	ADP
ejpam-7051	35	13	a	a	PRON
ejpam-7051	35	14	is	be	AUX
ejpam-7051	35	15	called	call	VERB
ejpam-7051	35	16	the	the	DET
ejpam-7051	35	17	τ1τ2	τ1τ2	NOUN
ejpam-7051	35	18	-	-	ADJ
ejpam-7051	35	19	interior	interior	ADJ
ejpam-7051	35	20	[	[	X
ejpam-7051	35	21	9	9	NUM
ejpam-7051	35	22	]	]	PUNCT
ejpam-7051	35	23	of	of	ADP
ejpam-7051	35	24	a	a	PRON
ejpam-7051	35	25	and	and	CCONJ
ejpam-7051	35	26	is	be	AUX
ejpam-7051	35	27	denoted	denote	VERB
ejpam-7051	35	28	by	by	ADP
ejpam-7051	35	29	τ1τ2	τ1τ2	NOUN
ejpam-7051	35	30	-	-	ADJ
ejpam-7051	35	31	int(a	int(a	NOUN
ejpam-7051	35	32	)	)	PUNCT
ejpam-7051	35	33	.	.	PUNCT
ejpam-7051	36	1	lemma	lemma	PROPN
ejpam-7051	36	2	1	1	NUM
ejpam-7051	36	3	.	.	PUNCT
ejpam-7051	37	1	[	[	X
ejpam-7051	37	2	9	9	NUM
ejpam-7051	37	3	]	]	PUNCT
ejpam-7051	37	4	let	let	VERB
ejpam-7051	37	5	a	a	PRON
ejpam-7051	37	6	and	and	CCONJ
ejpam-7051	37	7	b	b	NOUN
ejpam-7051	37	8	be	be	AUX
ejpam-7051	37	9	subsets	subset	NOUN
ejpam-7051	37	10	of	of	ADP
ejpam-7051	37	11	a	a	DET
ejpam-7051	37	12	bitopological	bitopological	ADJ
ejpam-7051	37	13	space	space	NOUN
ejpam-7051	37	14	(	(	PUNCT
ejpam-7051	37	15	x	x	NOUN
ejpam-7051	37	16	,	,	PUNCT
ejpam-7051	37	17	τ1	τ1	NOUN
ejpam-7051	37	18	,	,	PUNCT
ejpam-7051	37	19	τ2	τ2	NOUN
ejpam-7051	37	20	)	)	PUNCT
ejpam-7051	37	21	.	.	PUNCT
ejpam-7051	38	1	for	for	ADP
ejpam-7051	38	2	the	the	DET
ejpam-7051	38	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-7051	38	4	,	,	PUNCT
ejpam-7051	38	5	the	the	DET
ejpam-7051	38	6	following	follow	VERB
ejpam-7051	38	7	properties	property	NOUN
ejpam-7051	38	8	hold	hold	VERB
ejpam-7051	38	9	:	:	PUNCT
ejpam-7051	38	10	(	(	PUNCT
ejpam-7051	38	11	1	1	X
ejpam-7051	38	12	)	)	PUNCT
ejpam-7051	38	13	a	a	DET
ejpam-7051	38	14	⊆	⊆	NUM
ejpam-7051	38	15	τ1τ2	τ1τ2	NOUN
ejpam-7051	38	16	-	-	NUM
ejpam-7051	38	17	cl(a	cl(a	NUM
ejpam-7051	38	18	)	)	PUNCT
ejpam-7051	38	19	and	and	CCONJ
ejpam-7051	38	20	τ1τ2	τ1τ2	NOUN
ejpam-7051	38	21	-	-	ADJ
ejpam-7051	38	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7051	38	23	-	-	PUNCT
ejpam-7051	38	24	cl(a	cl(a	NUM
ejpam-7051	38	25	)	)	PUNCT
ejpam-7051	38	26	)	)	PUNCT
ejpam-7051	39	1	=	=	PUNCT
ejpam-7051	39	2	τ1τ2	τ1τ2	NOUN
ejpam-7051	39	3	-	-	NUM
ejpam-7051	39	4	cl(a	cl(a	NUM
ejpam-7051	39	5	)	)	PUNCT
ejpam-7051	39	6	.	.	PUNCT
ejpam-7051	40	1	(	(	PUNCT
ejpam-7051	40	2	2	2	X
ejpam-7051	40	3	)	)	PUNCT
ejpam-7051	40	4	if	if	SCONJ
ejpam-7051	40	5	a	a	DET
ejpam-7051	40	6	⊆	⊆	NUM
ejpam-7051	40	7	b	b	NOUN
ejpam-7051	40	8	,	,	PUNCT
ejpam-7051	40	9	then	then	ADV
ejpam-7051	40	10	τ1τ2	τ1τ2	NOUN
ejpam-7051	40	11	-	-	NUM
ejpam-7051	40	12	cl(a	cl(a	NUM
ejpam-7051	40	13	)	)	PUNCT
ejpam-7051	40	14	⊆	⊆	NUM
ejpam-7051	40	15	τ1τ2	τ1τ2	NOUN
ejpam-7051	40	16	-	-	NOUN
ejpam-7051	40	17	cl(b	cl(b	NOUN
ejpam-7051	40	18	)	)	PUNCT
ejpam-7051	40	19	.	.	PUNCT
ejpam-7051	41	1	(	(	PUNCT
ejpam-7051	41	2	3	3	X
ejpam-7051	41	3	)	)	PUNCT
ejpam-7051	41	4	τ1τ2	τ1τ2	NOUN
ejpam-7051	41	5	-	-	NUM
ejpam-7051	41	6	cl(a	cl(a	NUM
ejpam-7051	41	7	)	)	PUNCT
ejpam-7051	41	8	is	be	AUX
ejpam-7051	41	9	τ1τ2	τ1τ2	NOUN
ejpam-7051	41	10	-	-	ADJ
ejpam-7051	41	11	closed	closed	ADJ
ejpam-7051	41	12	.	.	PUNCT
ejpam-7051	42	1	(	(	PUNCT
ejpam-7051	42	2	4	4	X
ejpam-7051	42	3	)	)	PUNCT
ejpam-7051	42	4	a	a	PRON
ejpam-7051	42	5	is	be	AUX
ejpam-7051	42	6	τ1τ2	τ1τ2	NOUN
ejpam-7051	42	7	-	-	ADJ
ejpam-7051	42	8	closed	closed	ADJ
ejpam-7051	42	9	if	if	SCONJ
ejpam-7051	42	10	and	and	CCONJ
ejpam-7051	42	11	only	only	ADV
ejpam-7051	42	12	if	if	SCONJ
ejpam-7051	42	13	a	a	DET
ejpam-7051	42	14	=	=	PUNCT
ejpam-7051	42	15	τ1τ2	τ1τ2	NOUN
ejpam-7051	42	16	-	-	NUM
ejpam-7051	42	17	cl(a	cl(a	NUM
ejpam-7051	42	18	)	)	PUNCT
ejpam-7051	42	19	.	.	PUNCT
ejpam-7051	43	1	(	(	PUNCT
ejpam-7051	43	2	5	5	X
ejpam-7051	43	3	)	)	PUNCT
ejpam-7051	43	4	τ1τ2	τ1τ2	NOUN
ejpam-7051	43	5	-	-	NOUN
ejpam-7051	43	6	cl(x	cl(x	X
ejpam-7051	43	7	−a	−a	NOUN
ejpam-7051	43	8	)	)	PUNCT
ejpam-7051	44	1	=	=	PUNCT
ejpam-7051	44	2	x	x	X
ejpam-7051	45	1	−	−	ADP
ejpam-7051	45	2	τ1τ2	τ1τ2	NOUN
ejpam-7051	45	3	-	-	PUNCT
ejpam-7051	45	4	int(a	int(a	NOUN
ejpam-7051	45	5	)	)	PUNCT
ejpam-7051	45	6	.	.	PUNCT
ejpam-7051	46	1	a	a	DET
ejpam-7051	46	2	subset	subset	NOUN
ejpam-7051	46	3	a	a	PRON
ejpam-7051	46	4	of	of	ADP
ejpam-7051	46	5	a	a	DET
ejpam-7051	46	6	bitopological	bitopological	ADJ
ejpam-7051	46	7	space	space	NOUN
ejpam-7051	46	8	(	(	PUNCT
ejpam-7051	46	9	x	x	NOUN
ejpam-7051	46	10	,	,	PUNCT
ejpam-7051	46	11	τ1	τ1	NOUN
ejpam-7051	46	12	,	,	PUNCT
ejpam-7051	46	13	τ2	τ2	NOUN
ejpam-7051	46	14	)	)	PUNCT
ejpam-7051	46	15	is	be	AUX
ejpam-7051	46	16	said	say	VERB
ejpam-7051	46	17	to	to	PART
ejpam-7051	46	18	be	be	AUX
ejpam-7051	46	19	(	(	PUNCT
ejpam-7051	46	20	τ1	τ1	NOUN
ejpam-7051	46	21	,	,	PUNCT
ejpam-7051	46	22	τ2)r	τ2)r	NOUN
ejpam-7051	46	23	-	-	PUNCT
ejpam-7051	46	24	open	open	NOUN
ejpam-7051	47	1	[	[	X
ejpam-7051	47	2	10	10	NUM
ejpam-7051	47	3	]	]	X
ejpam-7051	47	4	(	(	PUNCT
ejpam-7051	47	5	resp	resp	NOUN
ejpam-7051	47	6	.	.	PUNCT
ejpam-7051	48	1	(	(	PUNCT
ejpam-7051	48	2	τ1	τ1	NOUN
ejpam-7051	48	3	,	,	PUNCT
ejpam-7051	48	4	τ2)s	τ2)s	NOUN
ejpam-7051	48	5	-	-	PUNCT
ejpam-7051	48	6	open	open	ADJ
ejpam-7051	48	7	[	[	X
ejpam-7051	48	8	11	11	NUM
ejpam-7051	48	9	]	]	NUM
ejpam-7051	48	10	,	,	PUNCT
ejpam-7051	48	11	(	(	PUNCT
ejpam-7051	48	12	τ1	τ1	NOUN
ejpam-7051	48	13	,	,	PUNCT
ejpam-7051	48	14	τ2)p	τ2)p	NOUN
ejpam-7051	48	15	-	-	ADJ
ejpam-7051	48	16	open	open	ADJ
ejpam-7051	48	17	[	[	X
ejpam-7051	48	18	11	11	NUM
ejpam-7051	48	19	]	]	NUM
ejpam-7051	48	20	,	,	PUNCT
ejpam-7051	48	21	(	(	PUNCT
ejpam-7051	48	22	τ1	τ1	NOUN
ejpam-7051	48	23	,	,	PUNCT
ejpam-7051	48	24	τ2)β	τ2)β	ADJ
ejpam-7051	48	25	-	-	PUNCT
ejpam-7051	48	26	open	open	NOUN
ejpam-7051	49	1	[	[	X
ejpam-7051	49	2	11	11	NUM
ejpam-7051	49	3	]	]	SYM
ejpam-7051	49	4	)	)	PUNCT
ejpam-7051	49	5	if	if	SCONJ
ejpam-7051	49	6	a	a	DET
ejpam-7051	49	7	=	=	PUNCT
ejpam-7051	49	8	τ1τ2	τ1τ2	NOUN
ejpam-7051	49	9	-	-	NOUN
ejpam-7051	49	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7051	49	11	-	-	PUNCT
ejpam-7051	49	12	cl(a	cl(a	NUM
ejpam-7051	49	13	)	)	PUNCT
ejpam-7051	49	14	)	)	PUNCT
ejpam-7051	49	15	(	(	PUNCT
ejpam-7051	49	16	resp	resp	NOUN
ejpam-7051	49	17	.	.	PUNCT
ejpam-7051	50	1	a	a	DET
ejpam-7051	50	2	⊆	⊆	NUM
ejpam-7051	50	3	τ1τ2	τ1τ2	NOUN
ejpam-7051	50	4	-	-	ADJ
ejpam-7051	50	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7051	50	6	-	-	PUNCT
ejpam-7051	50	7	int(a	int(a	NOUN
ejpam-7051	50	8	)	)	PUNCT
ejpam-7051	50	9	)	)	PUNCT
ejpam-7051	50	10	,	,	PUNCT
ejpam-7051	50	11	a	a	DET
ejpam-7051	50	12	⊆	⊆	NUM
ejpam-7051	50	13	τ1τ2	τ1τ2	NOUN
ejpam-7051	50	14	-	-	NOUN
ejpam-7051	50	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7051	50	16	-	-	PUNCT
ejpam-7051	50	17	cl(a	cl(a	NUM
ejpam-7051	50	18	)	)	PUNCT
ejpam-7051	50	19	)	)	PUNCT
ejpam-7051	50	20	,	,	PUNCT
ejpam-7051	50	21	a	a	DET
ejpam-7051	50	22	⊆	⊆	NUM
ejpam-7051	50	23	τ1τ2	τ1τ2	NOUN
ejpam-7051	50	24	-	-	PUNCT
ejpam-7051	50	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7051	50	26	-	-	PUNCT
ejpam-7051	50	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7051	50	28	-	-	PUNCT
ejpam-7051	50	29	cl(a	cl(a	NUM
ejpam-7051	50	30	)	)	PUNCT
ejpam-7051	50	31	)	)	PUNCT
ejpam-7051	50	32	)	)	PUNCT
ejpam-7051	50	33	)	)	PUNCT
ejpam-7051	50	34	.	.	PUNCT
ejpam-7051	51	1	the	the	DET
ejpam-7051	51	2	complement	complement	NOUN
ejpam-7051	51	3	of	of	ADP
ejpam-7051	51	4	a	a	DET
ejpam-7051	51	5	(	(	PUNCT
ejpam-7051	51	6	τ1	τ1	NOUN
ejpam-7051	51	7	,	,	PUNCT
ejpam-7051	51	8	τ2)r	τ2)r	NOUN
ejpam-7051	51	9	-	-	PUNCT
ejpam-7051	51	10	open	open	ADJ
ejpam-7051	51	11	(	(	PUNCT
ejpam-7051	51	12	resp	resp	NOUN
ejpam-7051	51	13	.	.	PUNCT
ejpam-7051	52	1	(	(	PUNCT
ejpam-7051	52	2	τ1	τ1	NOUN
ejpam-7051	52	3	,	,	PUNCT
ejpam-7051	52	4	τ2)s	τ2)s	NOUN
ejpam-7051	52	5	-	-	PUNCT
ejpam-7051	52	6	open	open	ADJ
ejpam-7051	52	7	,	,	PUNCT
ejpam-7051	52	8	(	(	PUNCT
ejpam-7051	52	9	τ1	τ1	NOUN
ejpam-7051	52	10	,	,	PUNCT
ejpam-7051	52	11	τ2)p	τ2)p	NOUN
ejpam-7051	52	12	-	-	ADJ
ejpam-7051	52	13	open	open	ADJ
ejpam-7051	52	14	,	,	PUNCT
ejpam-7051	52	15	(	(	PUNCT
ejpam-7051	52	16	τ1	τ1	NOUN
ejpam-7051	52	17	,	,	PUNCT
ejpam-7051	52	18	τ2)β	τ2)β	ADJ
ejpam-7051	52	19	-	-	PUNCT
ejpam-7051	52	20	open	open	ADJ
ejpam-7051	52	21	)	)	PUNCT
ejpam-7051	52	22	set	set	NOUN
ejpam-7051	52	23	is	be	AUX
ejpam-7051	52	24	said	say	VERB
ejpam-7051	52	25	to	to	PART
ejpam-7051	52	26	be	be	AUX
ejpam-7051	52	27	(	(	PUNCT
ejpam-7051	52	28	τ1	τ1	NOUN
ejpam-7051	52	29	,	,	PUNCT
ejpam-7051	52	30	τ2)r	τ2)r	NOUN
ejpam-7051	52	31	-	-	PUNCT
ejpam-7051	52	32	closed	closed	ADJ
ejpam-7051	52	33	(	(	PUNCT
ejpam-7051	52	34	resp	resp	NOUN
ejpam-7051	52	35	.	.	PUNCT
ejpam-7051	53	1	(	(	PUNCT
ejpam-7051	53	2	τ1	τ1	NOUN
ejpam-7051	53	3	,	,	PUNCT
ejpam-7051	53	4	τ2)s	τ2)s	NOUN
ejpam-7051	53	5	-	-	PUNCT
ejpam-7051	53	6	closed	closed	ADJ
ejpam-7051	53	7	,	,	PUNCT
ejpam-7051	53	8	(	(	PUNCT
ejpam-7051	53	9	τ1	τ1	NOUN
ejpam-7051	53	10	,	,	PUNCT
ejpam-7051	53	11	τ2)p	τ2)p	NOUN
ejpam-7051	53	12	-	-	PUNCT
ejpam-7051	53	13	closed	closed	ADJ
ejpam-7051	53	14	,	,	PUNCT
ejpam-7051	53	15	(	(	PUNCT
ejpam-7051	53	16	τ1	τ1	NOUN
ejpam-7051	53	17	,	,	PUNCT
ejpam-7051	53	18	τ2)p	τ2)p	NOUN
ejpam-7051	53	19	-	-	PUNCT
ejpam-7051	53	20	closed	closed	ADJ
ejpam-7051	53	21	)	)	PUNCT
ejpam-7051	53	22	.	.	PUNCT
ejpam-7051	54	1	a	a	DET
ejpam-7051	54	2	subset	subset	NOUN
ejpam-7051	54	3	a	a	PRON
ejpam-7051	54	4	of	of	ADP
ejpam-7051	54	5	a	a	DET
ejpam-7051	54	6	bitopological	bitopological	ADJ
ejpam-7051	54	7	space	space	NOUN
ejpam-7051	54	8	(	(	PUNCT
ejpam-7051	54	9	x	x	NOUN
ejpam-7051	54	10	,	,	PUNCT
ejpam-7051	54	11	τ1	τ1	NOUN
ejpam-7051	54	12	,	,	PUNCT
ejpam-7051	54	13	τ2	τ2	NOUN
ejpam-7051	54	14	)	)	PUNCT
ejpam-7051	54	15	is	be	AUX
ejpam-7051	54	16	said	say	VERB
ejpam-7051	54	17	to	to	PART
ejpam-7051	54	18	be	be	AUX
ejpam-7051	54	19	τ1τ2	τ1τ2	NOUN
ejpam-7051	54	20	-	-	ADJ
ejpam-7051	54	21	δ	δ	NOUN
ejpam-7051	54	22	-	-	NOUN
ejpam-7051	54	23	open	open	ADJ
ejpam-7051	54	24	[	[	X
ejpam-7051	54	25	12	12	NUM
ejpam-7051	54	26	]	]	X
ejpam-7051	54	27	if	if	SCONJ
ejpam-7051	54	28	a	a	PRON
ejpam-7051	54	29	is	be	AUX
ejpam-7051	54	30	the	the	DET
ejpam-7051	54	31	union	union	PROPN
ejpam-7051	54	32	n.	n.	PROPN
ejpam-7051	54	33	chutiman	chutiman	PROPN
ejpam-7051	54	34	,	,	PUNCT
ejpam-7051	54	35	a.	a.	PROPN
ejpam-7051	54	36	sama	sama	PROPN
ejpam-7051	54	37	-	-	PUNCT
ejpam-7051	54	38	ae	ae	PROPN
ejpam-7051	54	39	,	,	PUNCT
ejpam-7051	54	40	c.	c.	PROPN
ejpam-7051	54	41	boonpok	boonpok	PROPN
ejpam-7051	54	42	/	/	SYM
ejpam-7051	54	43	eur	eur	PROPN
ejpam-7051	54	44	.	.	PUNCT
ejpam-7051	55	1	j.	j.	PROPN
ejpam-7051	55	2	pure	pure	PROPN
ejpam-7051	55	3	appl	appl	PROPN
ejpam-7051	55	4	.	.	PROPN
ejpam-7051	55	5	math	math	PROPN
ejpam-7051	55	6	,	,	PUNCT
ejpam-7051	55	7	18	18	NUM
ejpam-7051	55	8	(	(	PUNCT
ejpam-7051	55	9	4	4	NUM
ejpam-7051	55	10	)	)	PUNCT
ejpam-7051	55	11	(	(	PUNCT
ejpam-7051	55	12	2025	2025	NUM
ejpam-7051	55	13	)	)	PUNCT
ejpam-7051	55	14	,	,	PUNCT
ejpam-7051	55	15	7051	7051	NUM
ejpam-7051	55	16	3	3	NUM
ejpam-7051	55	17	of	of	ADP
ejpam-7051	55	18	8	8	NUM
ejpam-7051	55	19	of	of	ADP
ejpam-7051	55	20	(	(	PUNCT
ejpam-7051	55	21	τ1	τ1	NOUN
ejpam-7051	55	22	,	,	PUNCT
ejpam-7051	55	23	τ2)r	τ2)r	ADJ
ejpam-7051	55	24	-	-	PUNCT
ejpam-7051	55	25	open	open	ADJ
ejpam-7051	55	26	sets	set	NOUN
ejpam-7051	55	27	of	of	ADP
ejpam-7051	55	28	x.	x.	NOUN
ejpam-7051	55	29	the	the	DET
ejpam-7051	55	30	complement	complement	NOUN
ejpam-7051	55	31	of	of	ADP
ejpam-7051	55	32	a	a	DET
ejpam-7051	55	33	τ1τ2	τ1τ2	ADJ
ejpam-7051	55	34	-	-	ADJ
ejpam-7051	55	35	δ	δ	NOUN
ejpam-7051	55	36	-	-	ADJ
ejpam-7051	55	37	open	open	ADJ
ejpam-7051	55	38	set	set	NOUN
ejpam-7051	55	39	is	be	AUX
ejpam-7051	55	40	called	call	VERB
ejpam-7051	55	41	τ1τ2	τ1τ2	NOUN
ejpam-7051	55	42	-	-	ADJ
ejpam-7051	55	43	δ	δ	NOUN
ejpam-7051	55	44	-	-	PUNCT
ejpam-7051	55	45	closed	closed	ADJ
ejpam-7051	55	46	[	[	X
ejpam-7051	55	47	12	12	NUM
ejpam-7051	55	48	]	]	PUNCT
ejpam-7051	55	49	.	.	PUNCT
ejpam-7051	56	1	the	the	DET
ejpam-7051	56	2	union	union	NOUN
ejpam-7051	56	3	of	of	ADP
ejpam-7051	56	4	all	all	DET
ejpam-7051	56	5	τ1τ2	τ1τ2	NOUN
ejpam-7051	56	6	-	-	ADJ
ejpam-7051	56	7	δ	δ	NOUN
ejpam-7051	56	8	-	-	ADJ
ejpam-7051	56	9	open	open	ADJ
ejpam-7051	56	10	sets	set	NOUN
ejpam-7051	56	11	of	of	ADP
ejpam-7051	56	12	x	x	PUNCT
ejpam-7051	56	13	contained	contain	VERB
ejpam-7051	56	14	in	in	ADP
ejpam-7051	56	15	a	a	PRON
ejpam-7051	56	16	is	be	AUX
ejpam-7051	56	17	called	call	VERB
ejpam-7051	56	18	the	the	DET
ejpam-7051	56	19	τ1τ2	τ1τ2	ADJ
ejpam-7051	56	20	-	-	ADJ
ejpam-7051	56	21	δ	δ	NOUN
ejpam-7051	56	22	-	-	NOUN
ejpam-7051	56	23	interior	interior	NOUN
ejpam-7051	56	24	[	[	X
ejpam-7051	56	25	12	12	NUM
ejpam-7051	56	26	]	]	PUNCT
ejpam-7051	56	27	of	of	ADP
ejpam-7051	56	28	a	a	PRON
ejpam-7051	56	29	and	and	CCONJ
ejpam-7051	56	30	is	be	AUX
ejpam-7051	56	31	denoted	denote	VERB
ejpam-7051	56	32	by	by	ADP
ejpam-7051	56	33	τ1τ2	τ1τ2	ADJ
ejpam-7051	56	34	-	-	ADJ
ejpam-7051	56	35	δ	δ	NOUN
ejpam-7051	56	36	-	-	PUNCT
ejpam-7051	56	37	int(a	int(a	PROPN
ejpam-7051	56	38	)	)	PUNCT
ejpam-7051	56	39	.	.	PUNCT
ejpam-7051	57	1	the	the	DET
ejpam-7051	57	2	intersection	intersection	NOUN
ejpam-7051	57	3	of	of	ADP
ejpam-7051	57	4	all	all	DET
ejpam-7051	57	5	τ1τ2	τ1τ2	NOUN
ejpam-7051	57	6	-	-	ADJ
ejpam-7051	57	7	δ	δ	NOUN
ejpam-7051	57	8	-	-	PUNCT
ejpam-7051	57	9	closed	close	VERB
ejpam-7051	57	10	sets	set	NOUN
ejpam-7051	57	11	of	of	ADP
ejpam-7051	57	12	x	x	PUNCT
ejpam-7051	57	13	containing	contain	VERB
ejpam-7051	57	14	a	a	PRON
ejpam-7051	57	15	is	be	AUX
ejpam-7051	57	16	called	call	VERB
ejpam-7051	57	17	the	the	DET
ejpam-7051	57	18	τ1τ2	τ1τ2	ADJ
ejpam-7051	57	19	-	-	ADJ
ejpam-7051	57	20	δ	δ	NOUN
ejpam-7051	57	21	-	-	NOUN
ejpam-7051	57	22	closure	closure	NOUN
ejpam-7051	57	23	[	[	X
ejpam-7051	57	24	12	12	NUM
ejpam-7051	57	25	]	]	PUNCT
ejpam-7051	57	26	of	of	ADP
ejpam-7051	57	27	a	a	PRON
ejpam-7051	57	28	and	and	CCONJ
ejpam-7051	57	29	is	be	AUX
ejpam-7051	57	30	denoted	denote	VERB
ejpam-7051	57	31	by	by	ADP
ejpam-7051	57	32	τ1τ2	τ1τ2	ADJ
ejpam-7051	57	33	-	-	ADJ
ejpam-7051	57	34	δ	δ	NOUN
ejpam-7051	57	35	-	-	PUNCT
ejpam-7051	57	36	cl(a	cl(a	NUM
ejpam-7051	57	37	)	)	PUNCT
ejpam-7051	57	38	.	.	PUNCT
ejpam-7051	58	1	let	let	VERB
ejpam-7051	58	2	a	a	DET
ejpam-7051	58	3	be	be	AUX
ejpam-7051	58	4	a	a	DET
ejpam-7051	58	5	subset	subset	NOUN
ejpam-7051	58	6	of	of	ADP
ejpam-7051	58	7	a	a	DET
ejpam-7051	58	8	bitopological	bitopological	ADJ
ejpam-7051	58	9	space	space	NOUN
ejpam-7051	58	10	(	(	PUNCT
ejpam-7051	58	11	x	x	NOUN
ejpam-7051	58	12	,	,	PUNCT
ejpam-7051	58	13	τ1	τ1	NOUN
ejpam-7051	58	14	,	,	PUNCT
ejpam-7051	58	15	τ2	τ2	NOUN
ejpam-7051	58	16	)	)	PUNCT
ejpam-7051	58	17	.	.	PUNCT
ejpam-7051	59	1	a	a	DET
ejpam-7051	59	2	point	point	NOUN
ejpam-7051	59	3	x	x	X
ejpam-7051	59	4	∈	∈	NOUN
ejpam-7051	59	5	x	x	PUNCT
ejpam-7051	59	6	is	be	AUX
ejpam-7051	59	7	called	call	VERB
ejpam-7051	59	8	a	a	DET
ejpam-7051	59	9	(	(	PUNCT
ejpam-7051	59	10	τ1	τ1	NOUN
ejpam-7051	59	11	,	,	PUNCT
ejpam-7051	59	12	τ2)θ	τ2)θ	ADJ
ejpam-7051	59	13	-	-	PUNCT
ejpam-7051	59	14	cluster	cluster	NOUN
ejpam-7051	59	15	point	point	NOUN
ejpam-7051	59	16	[	[	X
ejpam-7051	59	17	10	10	NUM
ejpam-7051	59	18	]	]	PUNCT
ejpam-7051	59	19	of	of	ADP
ejpam-7051	59	20	a	a	DET
ejpam-7051	59	21	if	if	SCONJ
ejpam-7051	59	22	τ1τ2	τ1τ2	NOUN
ejpam-7051	59	23	-	-	NOUN
ejpam-7051	59	24	cl(u	cl(u	NOUN
ejpam-7051	59	25	)	)	PUNCT
ejpam-7051	59	26	∩	∩	NOUN
ejpam-7051	59	27	a	a	DET
ejpam-7051	59	28	̸=	̸=	PROPN
ejpam-7051	59	29	∅	∅	NOUN
ejpam-7051	59	30	for	for	ADP
ejpam-7051	59	31	every	every	DET
ejpam-7051	59	32	τ1τ2	τ1τ2	ADJ
ejpam-7051	59	33	-	-	ADJ
ejpam-7051	59	34	open	open	ADJ
ejpam-7051	59	35	set	set	NOUN
ejpam-7051	59	36	u	u	NOUN
ejpam-7051	59	37	containing	contain	VERB
ejpam-7051	59	38	x.	x.	NOUN
ejpam-7051	59	39	the	the	DET
ejpam-7051	59	40	set	set	NOUN
ejpam-7051	59	41	of	of	ADP
ejpam-7051	59	42	all	all	DET
ejpam-7051	59	43	(	(	PUNCT
ejpam-7051	59	44	τ1	τ1	NOUN
ejpam-7051	59	45	,	,	PUNCT
ejpam-7051	59	46	τ2)θ	τ2)θ	ADJ
ejpam-7051	59	47	-	-	PUNCT
ejpam-7051	59	48	cluster	cluster	NOUN
ejpam-7051	59	49	points	point	NOUN
ejpam-7051	59	50	of	of	ADP
ejpam-7051	59	51	a	a	PRON
ejpam-7051	59	52	is	be	AUX
ejpam-7051	59	53	called	call	VERB
ejpam-7051	59	54	the	the	DET
ejpam-7051	59	55	(	(	PUNCT
ejpam-7051	59	56	τ1	τ1	NOUN
ejpam-7051	59	57	,	,	PUNCT
ejpam-7051	59	58	τ2)θ	τ2)θ	ADJ
ejpam-7051	59	59	-	-	PUNCT
ejpam-7051	59	60	closure	closure	NOUN
ejpam-7051	59	61	[	[	X
ejpam-7051	59	62	10	10	NUM
ejpam-7051	59	63	]	]	PUNCT
ejpam-7051	59	64	of	of	ADP
ejpam-7051	59	65	a	a	PRON
ejpam-7051	59	66	and	and	CCONJ
ejpam-7051	59	67	is	be	AUX
ejpam-7051	59	68	denoted	denote	VERB
ejpam-7051	59	69	by	by	ADP
ejpam-7051	59	70	(	(	PUNCT
ejpam-7051	59	71	τ1	τ1	NOUN
ejpam-7051	59	72	,	,	PUNCT
ejpam-7051	59	73	τ2)θ	τ2)θ	NOUN
ejpam-7051	59	74	-	-	PUNCT
ejpam-7051	59	75	cl(a	cl(a	NUM
ejpam-7051	59	76	)	)	PUNCT
ejpam-7051	59	77	.	.	PUNCT
ejpam-7051	60	1	a	a	DET
ejpam-7051	60	2	subset	subset	NOUN
ejpam-7051	60	3	a	a	PRON
ejpam-7051	60	4	of	of	ADP
ejpam-7051	60	5	a	a	DET
ejpam-7051	60	6	bitopological	bitopological	ADJ
ejpam-7051	60	7	space	space	NOUN
ejpam-7051	60	8	(	(	PUNCT
ejpam-7051	60	9	x	x	NOUN
ejpam-7051	60	10	,	,	PUNCT
ejpam-7051	60	11	τ1	τ1	NOUN
ejpam-7051	60	12	,	,	PUNCT
ejpam-7051	60	13	τ2	τ2	NOUN
ejpam-7051	60	14	)	)	PUNCT
ejpam-7051	60	15	is	be	AUX
ejpam-7051	60	16	said	say	VERB
ejpam-7051	60	17	to	to	PART
ejpam-7051	60	18	be	be	AUX
ejpam-7051	60	19	(	(	PUNCT
ejpam-7051	60	20	τ1	τ1	NOUN
ejpam-7051	60	21	,	,	PUNCT
ejpam-7051	60	22	τ2)θ	τ2)θ	NOUN
ejpam-7051	60	23	-	-	PUNCT
ejpam-7051	60	24	closed	closed	ADJ
ejpam-7051	60	25	[	[	X
ejpam-7051	60	26	10	10	NUM
ejpam-7051	60	27	]	]	X
ejpam-7051	60	28	if	if	SCONJ
ejpam-7051	60	29	(	(	PUNCT
ejpam-7051	60	30	τ1	τ1	NOUN
ejpam-7051	60	31	,	,	PUNCT
ejpam-7051	60	32	τ2)θ	τ2)θ	NOUN
ejpam-7051	60	33	-	-	PUNCT
ejpam-7051	60	34	cl(a	cl(a	NUM
ejpam-7051	60	35	)	)	PUNCT
ejpam-7051	61	1	=	=	PUNCT
ejpam-7051	61	2	a.	a.	NOUN
ejpam-7051	61	3	the	the	DET
ejpam-7051	61	4	complement	complement	NOUN
ejpam-7051	61	5	of	of	ADP
ejpam-7051	61	6	a	a	DET
ejpam-7051	61	7	(	(	PUNCT
ejpam-7051	61	8	τ1	τ1	NOUN
ejpam-7051	61	9	,	,	PUNCT
ejpam-7051	61	10	τ2)θ	τ2)θ	ADJ
ejpam-7051	61	11	-	-	PUNCT
ejpam-7051	61	12	closed	close	VERB
ejpam-7051	61	13	set	set	NOUN
ejpam-7051	61	14	is	be	AUX
ejpam-7051	61	15	said	say	VERB
ejpam-7051	61	16	to	to	PART
ejpam-7051	61	17	be	be	AUX
ejpam-7051	61	18	(	(	PUNCT
ejpam-7051	61	19	τ1	τ1	NOUN
ejpam-7051	61	20	,	,	PUNCT
ejpam-7051	61	21	τ2)θopen	τ2)θopen	PROPN
ejpam-7051	61	22	.	.	PUNCT
ejpam-7051	62	1	the	the	DET
ejpam-7051	62	2	union	union	NOUN
ejpam-7051	62	3	of	of	ADP
ejpam-7051	62	4	all	all	DET
ejpam-7051	62	5	(	(	PUNCT
ejpam-7051	62	6	τ1	τ1	NOUN
ejpam-7051	62	7	,	,	PUNCT
ejpam-7051	62	8	τ2)θ	τ2)θ	ADJ
ejpam-7051	62	9	-	-	PUNCT
ejpam-7051	62	10	open	open	ADJ
ejpam-7051	62	11	sets	set	NOUN
ejpam-7051	62	12	of	of	ADP
ejpam-7051	62	13	x	x	PUNCT
ejpam-7051	62	14	contained	contain	VERB
ejpam-7051	62	15	in	in	ADP
ejpam-7051	62	16	a	a	PRON
ejpam-7051	62	17	is	be	AUX
ejpam-7051	62	18	called	call	VERB
ejpam-7051	62	19	the	the	DET
ejpam-7051	62	20	(	(	PUNCT
ejpam-7051	62	21	τ1	τ1	NOUN
ejpam-7051	62	22	,	,	PUNCT
ejpam-7051	62	23	τ2)θ	τ2)θ	ADJ
ejpam-7051	62	24	-	-	PUNCT
ejpam-7051	62	25	interior	interior	NOUN
ejpam-7051	62	26	[	[	X
ejpam-7051	62	27	10	10	NUM
ejpam-7051	62	28	]	]	PUNCT
ejpam-7051	62	29	of	of	ADP
ejpam-7051	62	30	a	a	PRON
ejpam-7051	62	31	and	and	CCONJ
ejpam-7051	62	32	is	be	AUX
ejpam-7051	62	33	denoted	denote	VERB
ejpam-7051	62	34	by	by	ADP
ejpam-7051	62	35	(	(	PUNCT
ejpam-7051	62	36	τ1	τ1	NOUN
ejpam-7051	62	37	,	,	PUNCT
ejpam-7051	62	38	τ2)θ	τ2)θ	NOUN
ejpam-7051	62	39	-	-	PUNCT
ejpam-7051	62	40	int(a	int(a	NOUN
ejpam-7051	62	41	)	)	PUNCT
ejpam-7051	62	42	.	.	PUNCT
ejpam-7051	63	1	lemma	lemma	PROPN
ejpam-7051	63	2	2	2	NUM
ejpam-7051	63	3	.	.	PUNCT
ejpam-7051	64	1	[	[	X
ejpam-7051	64	2	10	10	NUM
ejpam-7051	64	3	]	]	PUNCT
ejpam-7051	64	4	for	for	ADP
ejpam-7051	64	5	a	a	DET
ejpam-7051	64	6	subset	subset	NOUN
ejpam-7051	64	7	a	a	PRON
ejpam-7051	64	8	of	of	ADP
ejpam-7051	64	9	a	a	DET
ejpam-7051	64	10	bitopological	bitopological	ADJ
ejpam-7051	64	11	space	space	NOUN
ejpam-7051	64	12	(	(	PUNCT
ejpam-7051	64	13	x	x	NOUN
ejpam-7051	64	14	,	,	PUNCT
ejpam-7051	64	15	τ1	τ1	NOUN
ejpam-7051	64	16	,	,	PUNCT
ejpam-7051	64	17	τ2	τ2	NOUN
ejpam-7051	64	18	)	)	PUNCT
ejpam-7051	64	19	,	,	PUNCT
ejpam-7051	64	20	the	the	DET
ejpam-7051	64	21	following	follow	VERB
ejpam-7051	64	22	properties	property	NOUN
ejpam-7051	64	23	hold	hold	VERB
ejpam-7051	64	24	:	:	PUNCT
ejpam-7051	64	25	(	(	PUNCT
ejpam-7051	64	26	1	1	X
ejpam-7051	64	27	)	)	PUNCT
ejpam-7051	64	28	if	if	SCONJ
ejpam-7051	64	29	a	a	PRON
ejpam-7051	64	30	is	be	AUX
ejpam-7051	64	31	τ1τ2	τ1τ2	NOUN
ejpam-7051	64	32	-	-	ADJ
ejpam-7051	64	33	open	open	ADJ
ejpam-7051	64	34	in	in	ADP
ejpam-7051	64	35	x	x	NOUN
ejpam-7051	64	36	,	,	PUNCT
ejpam-7051	64	37	then	then	ADV
ejpam-7051	64	38	τ1τ2	τ1τ2	NOUN
ejpam-7051	64	39	-	-	NUM
ejpam-7051	64	40	cl(a	cl(a	NUM
ejpam-7051	64	41	)	)	PUNCT
ejpam-7051	64	42	=	=	PUNCT
ejpam-7051	64	43	(	(	PUNCT
ejpam-7051	64	44	τ1	τ1	NOUN
ejpam-7051	64	45	,	,	PUNCT
ejpam-7051	64	46	τ2)θ	τ2)θ	NOUN
ejpam-7051	64	47	-	-	PUNCT
ejpam-7051	64	48	cl(a	cl(a	NUM
ejpam-7051	64	49	)	)	PUNCT
ejpam-7051	64	50	.	.	PUNCT
ejpam-7051	65	1	(	(	PUNCT
ejpam-7051	65	2	2	2	X
ejpam-7051	65	3	)	)	PUNCT
ejpam-7051	65	4	(	(	PUNCT
ejpam-7051	65	5	τ1	τ1	NOUN
ejpam-7051	65	6	,	,	PUNCT
ejpam-7051	65	7	τ2)θ	τ2)θ	NOUN
ejpam-7051	65	8	-	-	PUNCT
ejpam-7051	65	9	cl(a	cl(a	NUM
ejpam-7051	65	10	)	)	PUNCT
ejpam-7051	65	11	is	be	AUX
ejpam-7051	65	12	τ1τ2	τ1τ2	NOUN
ejpam-7051	65	13	-	-	ADJ
ejpam-7051	65	14	closed	closed	ADJ
ejpam-7051	65	15	in	in	ADP
ejpam-7051	65	16	x.	x.	NOUN
ejpam-7051	65	17	let	let	VERB
ejpam-7051	65	18	x	x	PRON
ejpam-7051	65	19	be	be	AUX
ejpam-7051	65	20	a	a	DET
ejpam-7051	65	21	nonempty	nonempty	ADJ
ejpam-7051	65	22	set	set	NOUN
ejpam-7051	65	23	,	,	PUNCT
ejpam-7051	65	24	and	and	CCONJ
ejpam-7051	65	25	denote	denote	VERB
ejpam-7051	65	26	p(x	p(x	PROPN
ejpam-7051	65	27	)	)	PUNCT
ejpam-7051	65	28	the	the	DET
ejpam-7051	65	29	power	power	NOUN
ejpam-7051	65	30	set	set	NOUN
ejpam-7051	65	31	of	of	ADP
ejpam-7051	65	32	x.	x.	NOUN
ejpam-7051	65	33	we	we	PRON
ejpam-7051	65	34	call	call	VERB
ejpam-7051	65	35	a	a	DET
ejpam-7051	65	36	class	class	NOUN
ejpam-7051	65	37	µ	µ	PRON
ejpam-7051	65	38	⊆	⊆	NUM
ejpam-7051	65	39	p(x	p(x	NOUN
ejpam-7051	65	40	)	)	PUNCT
ejpam-7051	65	41	a	a	DET
ejpam-7051	65	42	generalized	generalized	ADJ
ejpam-7051	65	43	topology	topology	NOUN
ejpam-7051	65	44	(	(	PUNCT
ejpam-7051	65	45	briefly	briefly	ADV
ejpam-7051	65	46	,	,	PUNCT
ejpam-7051	65	47	gt	gt	PROPN
ejpam-7051	65	48	)	)	PUNCT
ejpam-7051	65	49	if	if	SCONJ
ejpam-7051	65	50	∅	∅	NUM
ejpam-7051	65	51	∈	∈	PROPN
ejpam-7051	65	52	µ	µ	NOUN
ejpam-7051	65	53	,	,	PUNCT
ejpam-7051	65	54	and	and	CCONJ
ejpam-7051	65	55	an	an	DET
ejpam-7051	65	56	arbitrary	arbitrary	ADJ
ejpam-7051	65	57	union	union	NOUN
ejpam-7051	65	58	of	of	ADP
ejpam-7051	65	59	elements	element	NOUN
ejpam-7051	65	60	of	of	ADP
ejpam-7051	65	61	µ	µ	X
ejpam-7051	65	62	belongs	belong	VERB
ejpam-7051	65	63	to	to	ADP
ejpam-7051	65	64	µ	µ	PROPN
ejpam-7051	65	65	[	[	X
ejpam-7051	65	66	1	1	NUM
ejpam-7051	65	67	]	]	PUNCT
ejpam-7051	65	68	.	.	PUNCT
ejpam-7051	66	1	a	a	DET
ejpam-7051	66	2	set	set	NOUN
ejpam-7051	66	3	x	x	PUNCT
ejpam-7051	66	4	with	with	ADP
ejpam-7051	66	5	a	a	DET
ejpam-7051	66	6	gt	gt	PROPN
ejpam-7051	66	7	µ	µ	NOUN
ejpam-7051	66	8	on	on	ADP
ejpam-7051	66	9	it	it	PRON
ejpam-7051	66	10	is	be	AUX
ejpam-7051	66	11	said	say	VERB
ejpam-7051	66	12	to	to	PART
ejpam-7051	66	13	be	be	AUX
ejpam-7051	66	14	a	a	DET
ejpam-7051	66	15	generalized	generalized	ADJ
ejpam-7051	66	16	topological	topological	ADJ
ejpam-7051	66	17	space	space	NOUN
ejpam-7051	66	18	(	(	PUNCT
ejpam-7051	66	19	briefly	briefly	ADV
ejpam-7051	66	20	,	,	PUNCT
ejpam-7051	66	21	gts	gts	NOUN
ejpam-7051	66	22	)	)	PUNCT
ejpam-7051	66	23	and	and	CCONJ
ejpam-7051	66	24	is	be	AUX
ejpam-7051	66	25	denoted	denote	VERB
ejpam-7051	66	26	by	by	ADP
ejpam-7051	66	27	(	(	PUNCT
ejpam-7051	66	28	x,µ	x,µ	NOUN
ejpam-7051	66	29	)	)	PUNCT
ejpam-7051	66	30	.	.	PUNCT
ejpam-7051	67	1	for	for	ADP
ejpam-7051	67	2	a	a	DET
ejpam-7051	67	3	gts	gts	NOUN
ejpam-7051	67	4	(	(	PUNCT
ejpam-7051	67	5	x,µ	x,µ	NOUN
ejpam-7051	67	6	)	)	PUNCT
ejpam-7051	67	7	,	,	PUNCT
ejpam-7051	67	8	the	the	DET
ejpam-7051	67	9	elements	element	NOUN
ejpam-7051	67	10	of	of	ADP
ejpam-7051	67	11	µ	µ	NOUN
ejpam-7051	67	12	are	be	AUX
ejpam-7051	67	13	called	call	VERB
ejpam-7051	67	14	µ-open	µ-open	NOUN
ejpam-7051	67	15	sets	set	NOUN
ejpam-7051	67	16	and	and	CCONJ
ejpam-7051	67	17	the	the	DET
ejpam-7051	67	18	complements	complement	NOUN
ejpam-7051	67	19	of	of	ADP
ejpam-7051	67	20	µ-open	µ-open	NOUN
ejpam-7051	67	21	sets	set	NOUN
ejpam-7051	67	22	are	be	AUX
ejpam-7051	67	23	called	call	VERB
ejpam-7051	67	24	µ-closed	µ-close	VERB
ejpam-7051	67	25	sets	set	NOUN
ejpam-7051	67	26	.	.	PUNCT
ejpam-7051	68	1	for	for	ADP
ejpam-7051	68	2	a	a	DET
ejpam-7051	68	3	⊆	⊆	NUM
ejpam-7051	68	4	x	x	SYM
ejpam-7051	68	5	,	,	PUNCT
ejpam-7051	68	6	we	we	PRON
ejpam-7051	68	7	denote	denote	VERB
ejpam-7051	68	8	by	by	ADP
ejpam-7051	68	9	cµ(a	cµ(a	PROPN
ejpam-7051	68	10	)	)	PUNCT
ejpam-7051	68	11	the	the	DET
ejpam-7051	68	12	intersection	intersection	NOUN
ejpam-7051	68	13	of	of	ADP
ejpam-7051	68	14	all	all	DET
ejpam-7051	68	15	µ-closed	µ-close	VERB
ejpam-7051	68	16	sets	set	NOUN
ejpam-7051	68	17	containing	contain	VERB
ejpam-7051	68	18	a	a	PRON
ejpam-7051	68	19	and	and	CCONJ
ejpam-7051	68	20	by	by	ADP
ejpam-7051	68	21	iµ(a	iµ(a	PROPN
ejpam-7051	68	22	)	)	PUNCT
ejpam-7051	68	23	the	the	DET
ejpam-7051	68	24	union	union	NOUN
ejpam-7051	68	25	of	of	ADP
ejpam-7051	68	26	all	all	DET
ejpam-7051	68	27	µ-open	µ-open	NOUN
ejpam-7051	68	28	sets	set	NOUN
ejpam-7051	68	29	contained	contain	VERB
ejpam-7051	68	30	in	in	ADP
ejpam-7051	68	31	a.	a.	NOUN
ejpam-7051	68	32	then	then	ADV
ejpam-7051	68	33	,	,	PUNCT
ejpam-7051	68	34	we	we	PRON
ejpam-7051	68	35	have	have	VERB
ejpam-7051	68	36	iµ(iµ(a	iµ(iµ(a	ADJ
ejpam-7051	68	37	)	)	PUNCT
ejpam-7051	68	38	)	)	PUNCT
ejpam-7051	69	1	=	=	SYM
ejpam-7051	69	2	iµ(a	iµ(a	ADJ
ejpam-7051	69	3	)	)	PUNCT
ejpam-7051	69	4	,	,	PUNCT
ejpam-7051	69	5	cµ(cµ(a	cµ(cµ(a	PROPN
ejpam-7051	69	6	)	)	PUNCT
ejpam-7051	69	7	)	)	PUNCT
ejpam-7051	70	1	=	=	SYM
ejpam-7051	70	2	cµ(a	cµ(a	ADJ
ejpam-7051	70	3	)	)	PUNCT
ejpam-7051	70	4	,	,	PUNCT
ejpam-7051	70	5	and	and	CCONJ
ejpam-7051	70	6	iµ(a	iµ(a	ADJ
ejpam-7051	70	7	)	)	PUNCT
ejpam-7051	70	8	=	=	SYM
ejpam-7051	71	1	x	x	SYM
ejpam-7051	71	2	−	−	PROPN
ejpam-7051	71	3	cµ(x	cµ(x	SYM
ejpam-7051	71	4	−a	−a	NOUN
ejpam-7051	71	5	)	)	PUNCT
ejpam-7051	71	6	.	.	PUNCT
ejpam-7051	72	1	according	accord	VERB
ejpam-7051	72	2	to	to	ADP
ejpam-7051	72	3	[	[	X
ejpam-7051	72	4	13	13	NUM
ejpam-7051	72	5	]	]	PUNCT
ejpam-7051	72	6	,	,	PUNCT
ejpam-7051	72	7	for	for	ADP
ejpam-7051	72	8	a	a	DET
ejpam-7051	72	9	⊆	⊆	NUM
ejpam-7051	72	10	x	x	SYM
ejpam-7051	72	11	and	and	CCONJ
ejpam-7051	72	12	x	x	SYM
ejpam-7051	72	13	∈	∈	NOUN
ejpam-7051	72	14	x	x	X
ejpam-7051	72	15	,	,	PUNCT
ejpam-7051	72	16	we	we	PRON
ejpam-7051	72	17	have	have	VERB
ejpam-7051	72	18	x	x	X
ejpam-7051	72	19	∈	∈	PROPN
ejpam-7051	72	20	cµ(a	cµ(a	NOUN
ejpam-7051	72	21	)	)	PUNCT
ejpam-7051	73	1	if	if	SCONJ
ejpam-7051	73	2	and	and	CCONJ
ejpam-7051	73	3	only	only	ADV
ejpam-7051	73	4	if	if	SCONJ
ejpam-7051	73	5	x	x	PROPN
ejpam-7051	73	6	∈	∈	PROPN
ejpam-7051	73	7	m	m	PROPN
ejpam-7051	73	8	∈	∈	NOUN
ejpam-7051	73	9	µ	µ	NOUN
ejpam-7051	73	10	implies	imply	VERB
ejpam-7051	73	11	m	m	VERB
ejpam-7051	73	12	∩a	∩a	PROPN
ejpam-7051	73	13	̸=	̸=	PROPN
ejpam-7051	73	14	∅.	∅.	ADV
ejpam-7051	73	15	by	by	ADP
ejpam-7051	73	16	a	a	DET
ejpam-7051	73	17	multifunction	multifunction	NOUN
ejpam-7051	73	18	f	f	NOUN
ejpam-7051	73	19	:	:	PUNCT
ejpam-7051	73	20	x	x	X
ejpam-7051	73	21	→	→	SYM
ejpam-7051	73	22	y	y	PROPN
ejpam-7051	73	23	,	,	PUNCT
ejpam-7051	73	24	we	we	PRON
ejpam-7051	73	25	mean	mean	VERB
ejpam-7051	73	26	a	a	DET
ejpam-7051	73	27	point	point	NOUN
ejpam-7051	73	28	-	-	PUNCT
ejpam-7051	73	29	to	to	ADP
ejpam-7051	73	30	-	-	PUNCT
ejpam-7051	73	31	set	set	VERB
ejpam-7051	73	32	correspondence	correspondence	NOUN
ejpam-7051	73	33	from	from	ADP
ejpam-7051	73	34	x	x	PUNCT
ejpam-7051	73	35	into	into	ADP
ejpam-7051	73	36	y	y	PROPN
ejpam-7051	73	37	,	,	PUNCT
ejpam-7051	73	38	and	and	CCONJ
ejpam-7051	73	39	we	we	PRON
ejpam-7051	73	40	always	always	ADV
ejpam-7051	73	41	assume	assume	VERB
ejpam-7051	73	42	that	that	SCONJ
ejpam-7051	73	43	f	f	PROPN
ejpam-7051	73	44	(	(	PUNCT
ejpam-7051	73	45	x	x	X
ejpam-7051	73	46	)	)	PUNCT
ejpam-7051	73	47	̸=	̸=	NOUN
ejpam-7051	73	48	∅	∅	NOUN
ejpam-7051	73	49	for	for	ADP
ejpam-7051	73	50	all	all	PRON
ejpam-7051	73	51	x	x	SYM
ejpam-7051	73	52	∈	∈	ADJ
ejpam-7051	73	53	x.	x.	NOUN
ejpam-7051	73	54	for	for	ADP
ejpam-7051	73	55	a	a	DET
ejpam-7051	73	56	multifunction	multifunction	NOUN
ejpam-7051	74	1	f	f	NOUN
ejpam-7051	74	2	:	:	PUNCT
ejpam-7051	74	3	x	x	X
ejpam-7051	74	4	→	→	SYM
ejpam-7051	74	5	y	y	PROPN
ejpam-7051	74	6	,	,	PUNCT
ejpam-7051	74	7	we	we	PRON
ejpam-7051	74	8	shall	shall	AUX
ejpam-7051	74	9	denote	denote	VERB
ejpam-7051	74	10	the	the	DET
ejpam-7051	74	11	upper	upper	ADJ
ejpam-7051	74	12	and	and	CCONJ
ejpam-7051	74	13	lower	low	ADJ
ejpam-7051	74	14	inverse	inverse	NOUN
ejpam-7051	74	15	of	of	ADP
ejpam-7051	74	16	a	a	DET
ejpam-7051	74	17	set	set	NOUN
ejpam-7051	74	18	b	b	PROPN
ejpam-7051	74	19	of	of	ADP
ejpam-7051	74	20	y	y	PROPN
ejpam-7051	74	21	by	by	ADP
ejpam-7051	74	22	f+(b	f+(b	NOUN
ejpam-7051	74	23	)	)	PUNCT
ejpam-7051	74	24	and	and	CCONJ
ejpam-7051	74	25	f−(b	f−(b	NOUN
ejpam-7051	74	26	)	)	PUNCT
ejpam-7051	74	27	,	,	PUNCT
ejpam-7051	74	28	respectively	respectively	ADV
ejpam-7051	74	29	,	,	PUNCT
ejpam-7051	74	30	that	that	ADV
ejpam-7051	74	31	is	is	ADV
ejpam-7051	74	32	,	,	PUNCT
ejpam-7051	74	33	f+(b	f+(b	NOUN
ejpam-7051	74	34	)	)	PUNCT
ejpam-7051	74	35	=	=	PRON
ejpam-7051	75	1	{	{	PUNCT
ejpam-7051	75	2	x	x	PUNCT
ejpam-7051	75	3	∈	∈	PROPN
ejpam-7051	75	4	x	x	INTJ
ejpam-7051	76	1	|	|	NOUN
ejpam-7051	76	2	f	f	X
ejpam-7051	76	3	(	(	PUNCT
ejpam-7051	76	4	x	x	NOUN
ejpam-7051	76	5	)	)	PUNCT
ejpam-7051	76	6	⊆	⊆	NUM
ejpam-7051	76	7	b	b	NOUN
ejpam-7051	76	8	}	}	PUNCT
ejpam-7051	76	9	and	and	CCONJ
ejpam-7051	76	10	f−(b	f−(b	PROPN
ejpam-7051	76	11	)	)	PUNCT
ejpam-7051	76	12	=	=	PRON
ejpam-7051	77	1	{	{	PUNCT
ejpam-7051	77	2	x	x	PUNCT
ejpam-7051	77	3	∈	∈	PROPN
ejpam-7051	77	4	x	x	INTJ
ejpam-7051	78	1	|	|	NOUN
ejpam-7051	78	2	f	f	X
ejpam-7051	78	3	(	(	PUNCT
ejpam-7051	78	4	x	x	NOUN
ejpam-7051	78	5	)	)	PUNCT
ejpam-7051	78	6	∩	∩	NOUN
ejpam-7051	78	7	b	b	PROPN
ejpam-7051	78	8	̸=	̸=	PROPN
ejpam-7051	78	9	∅	∅	NOUN
ejpam-7051	78	10	}	}	PUNCT
ejpam-7051	78	11	.	.	PUNCT
ejpam-7051	79	1	in	in	ADP
ejpam-7051	79	2	particular	particular	ADJ
ejpam-7051	79	3	,	,	PUNCT
ejpam-7051	79	4	f−(y	f−(y	NOUN
ejpam-7051	79	5	)	)	PUNCT
ejpam-7051	79	6	=	=	SYM
ejpam-7051	80	1	{	{	PUNCT
ejpam-7051	80	2	x	x	PUNCT
ejpam-7051	80	3	∈	∈	PROPN
ejpam-7051	80	4	x	x	INTJ
ejpam-7051	81	1	|	|	ADV
ejpam-7051	81	2	y	y	PROPN
ejpam-7051	81	3	∈	∈	PROPN
ejpam-7051	81	4	f	f	X
ejpam-7051	81	5	(	(	PUNCT
ejpam-7051	81	6	x	x	NOUN
ejpam-7051	81	7	)	)	PUNCT
ejpam-7051	81	8	}	}	PUNCT
ejpam-7051	81	9	for	for	ADP
ejpam-7051	81	10	each	each	DET
ejpam-7051	81	11	point	point	NOUN
ejpam-7051	81	12	y	y	PROPN
ejpam-7051	81	13	∈	∈	PROPN
ejpam-7051	81	14	y	y	PROPN
ejpam-7051	81	15	.	.	PUNCT
ejpam-7051	82	1	for	for	ADP
ejpam-7051	82	2	each	each	DET
ejpam-7051	82	3	a	a	DET
ejpam-7051	82	4	⊆	⊆	NUM
ejpam-7051	82	5	x	x	SYM
ejpam-7051	82	6	,	,	PUNCT
ejpam-7051	82	7	f	f	PROPN
ejpam-7051	82	8	(	(	PUNCT
ejpam-7051	82	9	a	a	NOUN
ejpam-7051	82	10	)	)	PUNCT
ejpam-7051	82	11	=	=	SYM
ejpam-7051	82	12	∪x∈af	∪x∈af	NOUN
ejpam-7051	82	13	(	(	PUNCT
ejpam-7051	82	14	x	x	NOUN
ejpam-7051	82	15	)	)	PUNCT
ejpam-7051	82	16	.	.	PUNCT
ejpam-7051	83	1	3	3	X
ejpam-7051	83	2	.	.	X
ejpam-7051	83	3	upper	upper	ADJ
ejpam-7051	83	4	and	and	CCONJ
ejpam-7051	83	5	lower	low	ADJ
ejpam-7051	83	6	µ(σ1	µ(σ1	NOUN
ejpam-7051	83	7	,	,	PUNCT
ejpam-7051	83	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	83	9	multifunctions	multifunction	NOUN
ejpam-7051	83	10	in	in	ADP
ejpam-7051	83	11	this	this	DET
ejpam-7051	83	12	section	section	NOUN
ejpam-7051	83	13	,	,	PUNCT
ejpam-7051	83	14	we	we	PRON
ejpam-7051	83	15	introduce	introduce	VERB
ejpam-7051	83	16	the	the	DET
ejpam-7051	83	17	concepts	concept	NOUN
ejpam-7051	83	18	of	of	ADP
ejpam-7051	83	19	upper	upper	ADJ
ejpam-7051	83	20	µ(σ1	µ(σ1	NOUN
ejpam-7051	83	21	,	,	PUNCT
ejpam-7051	83	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	83	23	multifunctions	multifunction	NOUN
ejpam-7051	83	24	and	and	CCONJ
ejpam-7051	83	25	lower	low	ADJ
ejpam-7051	83	26	µ(σ1	µ(σ1	NOUN
ejpam-7051	83	27	,	,	PUNCT
ejpam-7051	83	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	83	29	multifunctions	multifunction	NOUN
ejpam-7051	83	30	.	.	PUNCT
ejpam-7051	84	1	furthermore	furthermore	ADV
ejpam-7051	84	2	,	,	PUNCT
ejpam-7051	84	3	several	several	ADJ
ejpam-7051	84	4	characterizations	characterization	NOUN
ejpam-7051	84	5	of	of	ADP
ejpam-7051	84	6	upper	upper	ADJ
ejpam-7051	84	7	µ(σ1	µ(σ1	NOUN
ejpam-7051	84	8	,	,	PUNCT
ejpam-7051	84	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	84	10	multifunctions	multifunction	NOUN
ejpam-7051	84	11	and	and	CCONJ
ejpam-7051	84	12	lower	low	ADJ
ejpam-7051	84	13	µ(σ1	µ(σ1	NOUN
ejpam-7051	84	14	,	,	PUNCT
ejpam-7051	84	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	84	16	multifunctions	multifunction	NOUN
ejpam-7051	84	17	are	be	AUX
ejpam-7051	84	18	discussed	discuss	VERB
ejpam-7051	84	19	.	.	PUNCT
ejpam-7051	85	1	definition	definition	NOUN
ejpam-7051	85	2	1	1	NUM
ejpam-7051	85	3	.	.	PUNCT
ejpam-7051	86	1	a	a	DET
ejpam-7051	86	2	multifunction	multifunction	NOUN
ejpam-7051	86	3	f	f	NOUN
ejpam-7051	86	4	:	:	PUNCT
ejpam-7051	86	5	(	(	PUNCT
ejpam-7051	86	6	x,µ	x,µ	NOUN
ejpam-7051	86	7	)	)	PUNCT
ejpam-7051	86	8	→	→	SYM
ejpam-7051	86	9	(	(	PUNCT
ejpam-7051	86	10	y	y	PROPN
ejpam-7051	86	11	,	,	PUNCT
ejpam-7051	86	12	σ1	σ1	PROPN
ejpam-7051	86	13	,	,	PUNCT
ejpam-7051	86	14	σ2	σ2	PROPN
ejpam-7051	86	15	)	)	PUNCT
ejpam-7051	86	16	is	be	AUX
ejpam-7051	86	17	called	call	VERB
ejpam-7051	86	18	upper	upper	ADJ
ejpam-7051	86	19	µ(σ1	µ(σ1	NOUN
ejpam-7051	86	20	,	,	PUNCT
ejpam-7051	86	21	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	86	22	at	at	ADP
ejpam-7051	86	23	a	a	DET
ejpam-7051	86	24	point	point	NOUN
ejpam-7051	86	25	x	x	SYM
ejpam-7051	86	26	∈	∈	NOUN
ejpam-7051	86	27	x	x	PUNCT
ejpam-7051	86	28	if	if	SCONJ
ejpam-7051	86	29	for	for	ADP
ejpam-7051	86	30	each	each	DET
ejpam-7051	86	31	σ1σ2	σ1σ2	VERB
ejpam-7051	86	32	-	-	ADJ
ejpam-7051	86	33	open	open	ADJ
ejpam-7051	86	34	set	set	NOUN
ejpam-7051	86	35	v	v	NOUN
ejpam-7051	86	36	of	of	ADP
ejpam-7051	86	37	y	y	PRON
ejpam-7051	86	38	such	such	ADJ
ejpam-7051	86	39	that	that	SCONJ
ejpam-7051	86	40	f	f	PROPN
ejpam-7051	86	41	(	(	PUNCT
ejpam-7051	86	42	x	x	X
ejpam-7051	86	43	)	)	PUNCT
ejpam-7051	86	44	⊆	⊆	NUM
ejpam-7051	86	45	v	v	NOUN
ejpam-7051	86	46	,	,	PUNCT
ejpam-7051	86	47	there	there	PRON
ejpam-7051	86	48	exists	exist	VERB
ejpam-7051	86	49	a	a	DET
ejpam-7051	86	50	µ-open	µ-open	NOUN
ejpam-7051	86	51	set	set	VERB
ejpam-7051	86	52	u	u	NOUN
ejpam-7051	86	53	of	of	ADP
ejpam-7051	86	54	x	x	PUNCT
ejpam-7051	86	55	containing	contain	VERB
ejpam-7051	86	56	x	x	PUNCT
ejpam-7051	86	57	such	such	ADJ
ejpam-7051	86	58	that	that	SCONJ
ejpam-7051	86	59	f	f	PROPN
ejpam-7051	86	60	(	(	PUNCT
ejpam-7051	86	61	u	u	NOUN
ejpam-7051	86	62	)	)	PUNCT
ejpam-7051	86	63	⊆	⊆	NUM
ejpam-7051	86	64	v	v	NOUN
ejpam-7051	86	65	.	.	PUNCT
ejpam-7051	87	1	a	a	DET
ejpam-7051	87	2	multifunction	multifunction	NOUN
ejpam-7051	87	3	f	f	NOUN
ejpam-7051	87	4	:	:	PUNCT
ejpam-7051	87	5	(	(	PUNCT
ejpam-7051	87	6	x,µ	x,µ	NOUN
ejpam-7051	87	7	)	)	PUNCT
ejpam-7051	87	8	→	→	SYM
ejpam-7051	87	9	(	(	PUNCT
ejpam-7051	87	10	y	y	PROPN
ejpam-7051	87	11	,	,	PUNCT
ejpam-7051	87	12	σ1	σ1	PROPN
ejpam-7051	87	13	,	,	PUNCT
ejpam-7051	87	14	σ2	σ2	PROPN
ejpam-7051	87	15	)	)	PUNCT
ejpam-7051	87	16	is	be	AUX
ejpam-7051	87	17	called	call	VERB
ejpam-7051	87	18	upper	upper	ADJ
ejpam-7051	87	19	µ(σ1	µ(σ1	NOUN
ejpam-7051	87	20	,	,	PUNCT
ejpam-7051	87	21	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	87	22	if	if	SCONJ
ejpam-7051	87	23	f	f	PROPN
ejpam-7051	87	24	is	be	AUX
ejpam-7051	87	25	upper	upper	ADJ
ejpam-7051	87	26	µ(σ1	µ(σ1	NOUN
ejpam-7051	87	27	,	,	PUNCT
ejpam-7051	87	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	87	29	at	at	ADP
ejpam-7051	87	30	each	each	DET
ejpam-7051	87	31	point	point	NOUN
ejpam-7051	87	32	x	x	PUNCT
ejpam-7051	87	33	of	of	ADP
ejpam-7051	87	34	x.	x.	PROPN
ejpam-7051	87	35	n.	n.	PROPN
ejpam-7051	87	36	chutiman	chutiman	PROPN
ejpam-7051	87	37	,	,	PUNCT
ejpam-7051	87	38	a.	a.	PROPN
ejpam-7051	87	39	sama	sama	PROPN
ejpam-7051	87	40	-	-	PUNCT
ejpam-7051	87	41	ae	ae	PROPN
ejpam-7051	87	42	,	,	PUNCT
ejpam-7051	87	43	c.	c.	PROPN
ejpam-7051	87	44	boonpok	boonpok	PROPN
ejpam-7051	87	45	/	/	SYM
ejpam-7051	87	46	eur	eur	PROPN
ejpam-7051	87	47	.	.	PUNCT
ejpam-7051	88	1	j.	j.	PROPN
ejpam-7051	88	2	pure	pure	PROPN
ejpam-7051	88	3	appl	appl	PROPN
ejpam-7051	88	4	.	.	PROPN
ejpam-7051	88	5	math	math	PROPN
ejpam-7051	88	6	,	,	PUNCT
ejpam-7051	88	7	18	18	NUM
ejpam-7051	88	8	(	(	PUNCT
ejpam-7051	88	9	4	4	NUM
ejpam-7051	88	10	)	)	PUNCT
ejpam-7051	88	11	(	(	PUNCT
ejpam-7051	88	12	2025	2025	NUM
ejpam-7051	88	13	)	)	PUNCT
ejpam-7051	88	14	,	,	PUNCT
ejpam-7051	88	15	7051	7051	NUM
ejpam-7051	88	16	4	4	NUM
ejpam-7051	88	17	of	of	ADP
ejpam-7051	88	18	8	8	NUM
ejpam-7051	88	19	theorem	theorem	NOUN
ejpam-7051	88	20	1	1	NUM
ejpam-7051	88	21	.	.	PUNCT
ejpam-7051	88	22	for	for	ADP
ejpam-7051	88	23	a	a	DET
ejpam-7051	88	24	multifunction	multifunction	NOUN
ejpam-7051	89	1	f	f	NOUN
ejpam-7051	89	2	:	:	PUNCT
ejpam-7051	89	3	(	(	PUNCT
ejpam-7051	89	4	x,µ	x,µ	NOUN
ejpam-7051	89	5	)	)	PUNCT
ejpam-7051	89	6	→	→	SYM
ejpam-7051	89	7	(	(	PUNCT
ejpam-7051	89	8	y	y	PROPN
ejpam-7051	89	9	,	,	PUNCT
ejpam-7051	89	10	σ1	σ1	PROPN
ejpam-7051	89	11	,	,	PUNCT
ejpam-7051	89	12	σ2	σ2	NOUN
ejpam-7051	89	13	)	)	PUNCT
ejpam-7051	89	14	,	,	PUNCT
ejpam-7051	89	15	the	the	DET
ejpam-7051	89	16	following	follow	VERB
ejpam-7051	89	17	properties	property	NOUN
ejpam-7051	89	18	are	be	AUX
ejpam-7051	89	19	equivalent	equivalent	ADJ
ejpam-7051	89	20	:	:	PUNCT
ejpam-7051	89	21	(	(	PUNCT
ejpam-7051	89	22	1	1	X
ejpam-7051	89	23	)	)	PUNCT
ejpam-7051	89	24	f	f	PROPN
ejpam-7051	89	25	is	be	AUX
ejpam-7051	89	26	upper	upper	ADJ
ejpam-7051	89	27	µ(σ1	µ(σ1	NOUN
ejpam-7051	89	28	,	,	PUNCT
ejpam-7051	89	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	89	30	;	;	PUNCT
ejpam-7051	89	31	(	(	PUNCT
ejpam-7051	89	32	2	2	NUM
ejpam-7051	89	33	)	)	PUNCT
ejpam-7051	89	34	f+(v	f+(v	NOUN
ejpam-7051	89	35	)	)	PUNCT
ejpam-7051	89	36	is	be	AUX
ejpam-7051	89	37	µ-open	µ-open	NOUN
ejpam-7051	89	38	in	in	ADP
ejpam-7051	89	39	x	x	PUNCT
ejpam-7051	89	40	for	for	ADP
ejpam-7051	89	41	every	every	DET
ejpam-7051	89	42	σ1σ2	σ1σ2	NOUN
ejpam-7051	89	43	-	-	ADJ
ejpam-7051	89	44	open	open	ADJ
ejpam-7051	89	45	set	set	NOUN
ejpam-7051	89	46	v	v	NOUN
ejpam-7051	89	47	of	of	ADP
ejpam-7051	89	48	y	y	PROPN
ejpam-7051	89	49	;	;	PUNCT
ejpam-7051	89	50	(	(	PUNCT
ejpam-7051	89	51	3	3	X
ejpam-7051	89	52	)	)	PUNCT
ejpam-7051	89	53	f−(k	f−(k	PROPN
ejpam-7051	89	54	)	)	PUNCT
ejpam-7051	89	55	is	be	AUX
ejpam-7051	89	56	µ-closed	µ-close	VERB
ejpam-7051	89	57	in	in	ADP
ejpam-7051	89	58	x	x	PUNCT
ejpam-7051	89	59	for	for	ADP
ejpam-7051	89	60	every	every	DET
ejpam-7051	89	61	σ1σ2	σ1σ2	NUM
ejpam-7051	89	62	-	-	PUNCT
ejpam-7051	89	63	closed	closed	ADJ
ejpam-7051	89	64	set	set	NOUN
ejpam-7051	89	65	k	k	PROPN
ejpam-7051	89	66	of	of	ADP
ejpam-7051	89	67	y	y	PROPN
ejpam-7051	89	68	;	;	PUNCT
ejpam-7051	89	69	(	(	PUNCT
ejpam-7051	89	70	4	4	NUM
ejpam-7051	89	71	)	)	PUNCT
ejpam-7051	89	72	cµ(f	cµ(f	NOUN
ejpam-7051	90	1	−(b	−(b	PROPN
ejpam-7051	90	2	)	)	PUNCT
ejpam-7051	90	3	)	)	PUNCT
ejpam-7051	91	1	⊆	⊆	X
ejpam-7051	91	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7051	91	3	-	-	PUNCT
ejpam-7051	91	4	cl(b	cl(b	NOUN
ejpam-7051	91	5	)	)	PUNCT
ejpam-7051	91	6	)	)	PUNCT
ejpam-7051	92	1	for	for	ADP
ejpam-7051	92	2	every	every	DET
ejpam-7051	92	3	subset	subset	NOUN
ejpam-7051	92	4	b	b	PROPN
ejpam-7051	92	5	of	of	ADP
ejpam-7051	92	6	y	y	PROPN
ejpam-7051	92	7	;	;	PUNCT
ejpam-7051	92	8	(	(	PUNCT
ejpam-7051	92	9	5	5	X
ejpam-7051	92	10	)	)	PUNCT
ejpam-7051	92	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7051	92	12	-	-	PUNCT
ejpam-7051	92	13	int(b	int(b	NOUN
ejpam-7051	92	14	)	)	PUNCT
ejpam-7051	92	15	)	)	PUNCT
ejpam-7051	93	1	⊆	⊆	NUM
ejpam-7051	93	2	iµ(f	iµ(f	NUM
ejpam-7051	93	3	+	+	NOUN
ejpam-7051	93	4	(	(	PUNCT
ejpam-7051	93	5	b	b	NOUN
ejpam-7051	93	6	)	)	PUNCT
ejpam-7051	93	7	)	)	PUNCT
ejpam-7051	93	8	for	for	ADP
ejpam-7051	93	9	every	every	DET
ejpam-7051	93	10	subset	subset	NOUN
ejpam-7051	93	11	b	b	PROPN
ejpam-7051	93	12	of	of	ADP
ejpam-7051	93	13	y	y	PROPN
ejpam-7051	93	14	.	.	PUNCT
ejpam-7051	94	1	proof	proof	NOUN
ejpam-7051	94	2	.	.	PUNCT
ejpam-7051	95	1	(	(	PUNCT
ejpam-7051	95	2	1	1	X
ejpam-7051	95	3	)	)	PUNCT
ejpam-7051	95	4	⇒	⇒	NOUN
ejpam-7051	95	5	(	(	PUNCT
ejpam-7051	95	6	2	2	NUM
ejpam-7051	95	7	):	):	PUNCT
ejpam-7051	95	8	let	let	VERB
ejpam-7051	95	9	v	v	PART
ejpam-7051	95	10	be	be	AUX
ejpam-7051	95	11	any	any	DET
ejpam-7051	95	12	σ1σ2	σ1σ2	NOUN
ejpam-7051	95	13	-	-	ADJ
ejpam-7051	95	14	open	open	ADJ
ejpam-7051	95	15	set	set	NOUN
ejpam-7051	95	16	of	of	ADP
ejpam-7051	95	17	y	y	PROPN
ejpam-7051	95	18	and	and	CCONJ
ejpam-7051	95	19	x	x	PROPN
ejpam-7051	95	20	∈	∈	PROPN
ejpam-7051	95	21	f+(v	f+(v	NOUN
ejpam-7051	95	22	)	)	PUNCT
ejpam-7051	95	23	.	.	PUNCT
ejpam-7051	96	1	then	then	ADV
ejpam-7051	96	2	,	,	PUNCT
ejpam-7051	96	3	f	f	PROPN
ejpam-7051	96	4	(	(	PUNCT
ejpam-7051	96	5	x	x	X
ejpam-7051	96	6	)	)	PUNCT
ejpam-7051	96	7	⊆	⊆	NUM
ejpam-7051	96	8	v	v	NOUN
ejpam-7051	96	9	and	and	CCONJ
ejpam-7051	96	10	by	by	ADP
ejpam-7051	96	11	(	(	PUNCT
ejpam-7051	96	12	1	1	NUM
ejpam-7051	96	13	)	)	PUNCT
ejpam-7051	96	14	,	,	PUNCT
ejpam-7051	96	15	there	there	PRON
ejpam-7051	96	16	exists	exist	VERB
ejpam-7051	96	17	a	a	DET
ejpam-7051	96	18	µ-open	µ-open	NOUN
ejpam-7051	96	19	set	set	VERB
ejpam-7051	96	20	u	u	NOUN
ejpam-7051	96	21	of	of	ADP
ejpam-7051	96	22	x	x	PUNCT
ejpam-7051	96	23	containing	contain	VERB
ejpam-7051	96	24	x	x	PUNCT
ejpam-7051	96	25	such	such	ADJ
ejpam-7051	96	26	that	that	SCONJ
ejpam-7051	96	27	f	f	PROPN
ejpam-7051	96	28	(	(	PUNCT
ejpam-7051	96	29	u	u	NOUN
ejpam-7051	96	30	)	)	PUNCT
ejpam-7051	96	31	⊆	⊆	NUM
ejpam-7051	96	32	v	v	NOUN
ejpam-7051	96	33	.	.	PUNCT
ejpam-7051	97	1	thus	thus	ADV
ejpam-7051	97	2	,	,	PUNCT
ejpam-7051	97	3	x	x	PUNCT
ejpam-7051	97	4	∈	∈	PROPN
ejpam-7051	97	5	u	u	NOUN
ejpam-7051	97	6	⊆	⊆	NUM
ejpam-7051	97	7	f+(v	f+(v	NOUN
ejpam-7051	97	8	)	)	PUNCT
ejpam-7051	97	9	and	and	CCONJ
ejpam-7051	97	10	hence	hence	ADV
ejpam-7051	97	11	x	x	X
ejpam-7051	97	12	∈	∈	NOUN
ejpam-7051	97	13	iµ(f	iµ(f	NUM
ejpam-7051	97	14	+	+	NOUN
ejpam-7051	97	15	(	(	PUNCT
ejpam-7051	97	16	v	v	NOUN
ejpam-7051	97	17	)	)	PUNCT
ejpam-7051	97	18	)	)	PUNCT
ejpam-7051	97	19	.	.	PUNCT
ejpam-7051	98	1	therefore	therefore	ADV
ejpam-7051	98	2	,	,	PUNCT
ejpam-7051	98	3	f+(v	f+(v	PROPN
ejpam-7051	98	4	)	)	PUNCT
ejpam-7051	98	5	⊆	⊆	NUM
ejpam-7051	98	6	iµ(f	iµ(f	NUM
ejpam-7051	98	7	+	+	NOUN
ejpam-7051	98	8	(	(	PUNCT
ejpam-7051	98	9	v	v	NOUN
ejpam-7051	98	10	)	)	PUNCT
ejpam-7051	98	11	)	)	PUNCT
ejpam-7051	98	12	.	.	PUNCT
ejpam-7051	99	1	this	this	PRON
ejpam-7051	99	2	shows	show	VERB
ejpam-7051	99	3	that	that	SCONJ
ejpam-7051	99	4	f+(v	f+(v	PROPN
ejpam-7051	99	5	)	)	PUNCT
ejpam-7051	99	6	is	be	AUX
ejpam-7051	99	7	µ-open	µ-open	NOUN
ejpam-7051	99	8	in	in	ADP
ejpam-7051	99	9	x.	x.	NOUN
ejpam-7051	99	10	(	(	PUNCT
ejpam-7051	99	11	2	2	NUM
ejpam-7051	99	12	)	)	PUNCT
ejpam-7051	99	13	⇒	⇒	NOUN
ejpam-7051	99	14	(	(	PUNCT
ejpam-7051	99	15	3	3	NUM
ejpam-7051	99	16	):	):	PUNCT
ejpam-7051	99	17	this	this	PRON
ejpam-7051	99	18	follows	follow	VERB
ejpam-7051	99	19	from	from	ADP
ejpam-7051	99	20	the	the	DET
ejpam-7051	99	21	fact	fact	NOUN
ejpam-7051	99	22	that	that	SCONJ
ejpam-7051	99	23	f+(y	f+(y	PROPN
ejpam-7051	99	24	−b	−b	ADV
ejpam-7051	99	25	)	)	PUNCT
ejpam-7051	99	26	=	=	PUNCT
ejpam-7051	100	1	x	x	X
ejpam-7051	100	2	−	−	PROPN
ejpam-7051	100	3	f−(b	f−(b	PROPN
ejpam-7051	100	4	)	)	PUNCT
ejpam-7051	100	5	for	for	ADP
ejpam-7051	100	6	every	every	DET
ejpam-7051	100	7	subset	subset	NOUN
ejpam-7051	100	8	b	b	PROPN
ejpam-7051	100	9	of	of	ADP
ejpam-7051	100	10	y	y	PROPN
ejpam-7051	100	11	.	.	PUNCT
ejpam-7051	101	1	(	(	PUNCT
ejpam-7051	101	2	3	3	X
ejpam-7051	101	3	)	)	PUNCT
ejpam-7051	101	4	⇒	⇒	NOUN
ejpam-7051	101	5	(	(	PUNCT
ejpam-7051	101	6	4	4	NUM
ejpam-7051	101	7	):	):	PUNCT
ejpam-7051	101	8	let	let	VERB
ejpam-7051	101	9	b	b	X
ejpam-7051	101	10	be	be	AUX
ejpam-7051	101	11	any	any	DET
ejpam-7051	101	12	subset	subset	NOUN
ejpam-7051	101	13	of	of	ADP
ejpam-7051	101	14	y	y	PROPN
ejpam-7051	101	15	.	.	PUNCT
ejpam-7051	102	1	then	then	ADV
ejpam-7051	102	2	,	,	PUNCT
ejpam-7051	102	3	σ1σ2	σ1σ2	NOUN
ejpam-7051	102	4	-	-	NOUN
ejpam-7051	102	5	cl(b	cl(b	NOUN
ejpam-7051	102	6	)	)	PUNCT
ejpam-7051	102	7	is	be	AUX
ejpam-7051	102	8	σ1σ2	σ1σ2	NOUN
ejpam-7051	102	9	-	-	ADJ
ejpam-7051	102	10	closed	closed	ADJ
ejpam-7051	102	11	in	in	ADP
ejpam-7051	102	12	y	y	PROPN
ejpam-7051	102	13	and	and	CCONJ
ejpam-7051	102	14	by	by	ADP
ejpam-7051	102	15	(	(	PUNCT
ejpam-7051	102	16	3	3	NUM
ejpam-7051	102	17	)	)	PUNCT
ejpam-7051	102	18	,	,	PUNCT
ejpam-7051	102	19	cµ(f−(b	cµ(f−(b	NOUN
ejpam-7051	102	20	)	)	PUNCT
ejpam-7051	102	21	)	)	PUNCT
ejpam-7051	103	1	⊆	⊆	NUM
ejpam-7051	103	2	cµ(f	cµ(f	NUM
ejpam-7051	103	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7051	103	4	-	-	NOUN
ejpam-7051	103	5	cl(b	cl(b	NOUN
ejpam-7051	103	6	)	)	PUNCT
ejpam-7051	103	7	)	)	PUNCT
ejpam-7051	103	8	)	)	PUNCT
ejpam-7051	104	1	=	=	PUNCT
ejpam-7051	104	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7051	104	3	-	-	PUNCT
ejpam-7051	104	4	cl(b	cl(b	NOUN
ejpam-7051	104	5	)	)	PUNCT
ejpam-7051	104	6	)	)	PUNCT
ejpam-7051	104	7	.	.	PUNCT
ejpam-7051	105	1	(	(	PUNCT
ejpam-7051	105	2	4	4	X
ejpam-7051	105	3	)	)	PUNCT
ejpam-7051	105	4	⇒	⇒	NOUN
ejpam-7051	105	5	(	(	PUNCT
ejpam-7051	105	6	5	5	NUM
ejpam-7051	105	7	):	):	PUNCT
ejpam-7051	105	8	let	let	VERB
ejpam-7051	105	9	b	b	X
ejpam-7051	105	10	be	be	AUX
ejpam-7051	105	11	any	any	DET
ejpam-7051	105	12	subset	subset	NOUN
ejpam-7051	105	13	of	of	ADP
ejpam-7051	105	14	y	y	PROPN
ejpam-7051	105	15	.	.	PUNCT
ejpam-7051	106	1	by	by	ADP
ejpam-7051	106	2	(	(	PUNCT
ejpam-7051	106	3	4	4	NUM
ejpam-7051	106	4	)	)	PUNCT
ejpam-7051	106	5	,	,	PUNCT
ejpam-7051	106	6	we	we	PRON
ejpam-7051	106	7	have	have	VERB
ejpam-7051	106	8	x	x	PART
ejpam-7051	106	9	−	−	NOUN
ejpam-7051	106	10	iµ(f	iµ(f	X
ejpam-7051	106	11	+	+	NOUN
ejpam-7051	106	12	(	(	PUNCT
ejpam-7051	106	13	b	b	NOUN
ejpam-7051	106	14	)	)	PUNCT
ejpam-7051	106	15	)	)	PUNCT
ejpam-7051	107	1	=	=	SYM
ejpam-7051	107	2	cµ(x	cµ(x	PUNCT
ejpam-7051	107	3	−	−	PROPN
ejpam-7051	107	4	f+(b	f+(b	NOUN
ejpam-7051	107	5	)	)	PUNCT
ejpam-7051	107	6	)	)	PUNCT
ejpam-7051	108	1	=	=	SYM
ejpam-7051	108	2	cµ(f	cµ(f	NOUN
ejpam-7051	109	1	−(y	−(y	NOUN
ejpam-7051	109	2	−b	−b	NOUN
ejpam-7051	109	3	)	)	PUNCT
ejpam-7051	109	4	)	)	PUNCT
ejpam-7051	110	1	⊆	⊆	X
ejpam-7051	110	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7051	110	3	-	-	PUNCT
ejpam-7051	110	4	cl(y	cl(y	NOUN
ejpam-7051	110	5	−b	−b	NOUN
ejpam-7051	110	6	)	)	PUNCT
ejpam-7051	110	7	)	)	PUNCT
ejpam-7051	111	1	=	=	PUNCT
ejpam-7051	111	2	f−(y	f−(y	NOUN
ejpam-7051	111	3	−	−	ADP
ejpam-7051	111	4	σ1σ2	σ1σ2	NOUN
ejpam-7051	111	5	-	-	PUNCT
ejpam-7051	111	6	int(b	int(b	NOUN
ejpam-7051	111	7	)	)	PUNCT
ejpam-7051	111	8	)	)	PUNCT
ejpam-7051	112	1	=	=	PUNCT
ejpam-7051	112	2	x	x	X
ejpam-7051	113	1	−	−	ADP
ejpam-7051	113	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7051	113	3	-	-	PUNCT
ejpam-7051	113	4	int(b	int(b	NOUN
ejpam-7051	113	5	)	)	PUNCT
ejpam-7051	113	6	)	)	PUNCT
ejpam-7051	113	7	and	and	CCONJ
ejpam-7051	113	8	so	so	ADV
ejpam-7051	113	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-7051	113	10	-	-	PUNCT
ejpam-7051	113	11	int(b	int(b	NOUN
ejpam-7051	113	12	)	)	PUNCT
ejpam-7051	113	13	)	)	PUNCT
ejpam-7051	113	14	⊆	⊆	NUM
ejpam-7051	113	15	iµ(f	iµ(f	NUM
ejpam-7051	113	16	+	+	NOUN
ejpam-7051	113	17	(	(	PUNCT
ejpam-7051	113	18	b	b	NOUN
ejpam-7051	113	19	)	)	PUNCT
ejpam-7051	113	20	)	)	PUNCT
ejpam-7051	113	21	.	.	PUNCT
ejpam-7051	114	1	(	(	PUNCT
ejpam-7051	114	2	5	5	X
ejpam-7051	114	3	)	)	PUNCT
ejpam-7051	114	4	⇒	⇒	NOUN
ejpam-7051	114	5	(	(	PUNCT
ejpam-7051	114	6	1	1	NUM
ejpam-7051	114	7	):	):	PUNCT
ejpam-7051	114	8	let	let	VERB
ejpam-7051	114	9	x	x	PUNCT
ejpam-7051	114	10	∈	∈	PROPN
ejpam-7051	114	11	x	x	X
ejpam-7051	114	12	and	and	CCONJ
ejpam-7051	114	13	v	v	X
ejpam-7051	114	14	be	be	AUX
ejpam-7051	114	15	any	any	DET
ejpam-7051	114	16	σ1σ2	σ1σ2	NOUN
ejpam-7051	114	17	-	-	ADJ
ejpam-7051	114	18	open	open	ADJ
ejpam-7051	114	19	set	set	NOUN
ejpam-7051	114	20	of	of	ADP
ejpam-7051	114	21	y	y	PRON
ejpam-7051	114	22	such	such	ADJ
ejpam-7051	114	23	that	that	SCONJ
ejpam-7051	114	24	f	f	PROPN
ejpam-7051	114	25	(	(	PUNCT
ejpam-7051	114	26	x	x	X
ejpam-7051	114	27	)	)	PUNCT
ejpam-7051	114	28	⊆	⊆	NUM
ejpam-7051	114	29	v	v	NOUN
ejpam-7051	114	30	.	.	PUNCT
ejpam-7051	115	1	then	then	ADV
ejpam-7051	115	2	,	,	PUNCT
ejpam-7051	115	3	x	x	X
ejpam-7051	115	4	∈	∈	NOUN
ejpam-7051	115	5	f+(v	f+(v	NOUN
ejpam-7051	115	6	)	)	PUNCT
ejpam-7051	116	1	=	=	PRON
ejpam-7051	116	2	iµ(f	iµ(f	NUM
ejpam-7051	116	3	+	+	ADJ
ejpam-7051	116	4	(	(	PUNCT
ejpam-7051	116	5	v	v	NOUN
ejpam-7051	116	6	)	)	PUNCT
ejpam-7051	116	7	)	)	PUNCT
ejpam-7051	116	8	.	.	PUNCT
ejpam-7051	117	1	there	there	PRON
ejpam-7051	117	2	exists	exist	VERB
ejpam-7051	117	3	a	a	DET
ejpam-7051	117	4	µ-open	µ-open	NOUN
ejpam-7051	117	5	set	set	VERB
ejpam-7051	117	6	u	u	NOUN
ejpam-7051	117	7	of	of	ADP
ejpam-7051	117	8	x	x	PUNCT
ejpam-7051	117	9	containing	contain	VERB
ejpam-7051	117	10	x	x	PUNCT
ejpam-7051	117	11	such	such	ADJ
ejpam-7051	117	12	that	that	SCONJ
ejpam-7051	117	13	u	u	NOUN
ejpam-7051	117	14	⊆	⊆	NUM
ejpam-7051	117	15	f+(v	f+(v	NOUN
ejpam-7051	117	16	)	)	PUNCT
ejpam-7051	117	17	;	;	PUNCT
ejpam-7051	117	18	hence	hence	ADV
ejpam-7051	117	19	f	f	PROPN
ejpam-7051	117	20	(	(	PUNCT
ejpam-7051	117	21	u	u	NOUN
ejpam-7051	117	22	)	)	PUNCT
ejpam-7051	117	23	⊆	⊆	NUM
ejpam-7051	117	24	v	v	NOUN
ejpam-7051	117	25	.	.	PUNCT
ejpam-7051	118	1	this	this	PRON
ejpam-7051	118	2	shows	show	VERB
ejpam-7051	118	3	that	that	SCONJ
ejpam-7051	118	4	f	f	PROPN
ejpam-7051	118	5	is	be	AUX
ejpam-7051	118	6	upper	upper	ADJ
ejpam-7051	118	7	µ(σ1	µ(σ1	NOUN
ejpam-7051	118	8	,	,	PUNCT
ejpam-7051	118	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7051	118	10	.	.	NOUN
ejpam-7051	118	11	definition	definition	NOUN
ejpam-7051	118	12	2	2	NUM
ejpam-7051	118	13	.	.	PUNCT
ejpam-7051	118	14	a	a	DET
ejpam-7051	118	15	multifunction	multifunction	NOUN
ejpam-7051	119	1	f	f	NOUN
ejpam-7051	119	2	:	:	PUNCT
ejpam-7051	119	3	(	(	PUNCT
ejpam-7051	119	4	x,µ	x,µ	NOUN
ejpam-7051	119	5	)	)	PUNCT
ejpam-7051	119	6	→	→	SYM
ejpam-7051	119	7	(	(	PUNCT
ejpam-7051	119	8	y	y	PROPN
ejpam-7051	119	9	,	,	PUNCT
ejpam-7051	119	10	σ1	σ1	PROPN
ejpam-7051	119	11	,	,	PUNCT
ejpam-7051	119	12	σ2	σ2	PROPN
ejpam-7051	119	13	)	)	PUNCT
ejpam-7051	119	14	is	be	AUX
ejpam-7051	119	15	called	call	VERB
ejpam-7051	119	16	lower	low	ADJ
ejpam-7051	119	17	µ(σ1	µ(σ1	NOUN
ejpam-7051	119	18	,	,	PUNCT
ejpam-7051	119	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	119	20	at	at	ADP
ejpam-7051	119	21	a	a	DET
ejpam-7051	119	22	point	point	NOUN
ejpam-7051	119	23	x	x	SYM
ejpam-7051	119	24	∈	∈	NOUN
ejpam-7051	119	25	x	x	PUNCT
ejpam-7051	119	26	if	if	SCONJ
ejpam-7051	119	27	for	for	ADP
ejpam-7051	119	28	each	each	DET
ejpam-7051	119	29	σ1σ2	σ1σ2	VERB
ejpam-7051	119	30	-	-	ADJ
ejpam-7051	119	31	open	open	ADJ
ejpam-7051	119	32	set	set	NOUN
ejpam-7051	119	33	v	v	NOUN
ejpam-7051	119	34	of	of	ADP
ejpam-7051	119	35	y	y	PRON
ejpam-7051	119	36	such	such	ADJ
ejpam-7051	119	37	that	that	SCONJ
ejpam-7051	119	38	f	f	PROPN
ejpam-7051	119	39	(	(	PUNCT
ejpam-7051	119	40	x	x	NOUN
ejpam-7051	119	41	)	)	PUNCT
ejpam-7051	119	42	∩	∩	NOUN
ejpam-7051	119	43	v	v	ADP
ejpam-7051	119	44	̸=	̸=	PROPN
ejpam-7051	119	45	∅	∅	NOUN
ejpam-7051	119	46	,	,	PUNCT
ejpam-7051	119	47	there	there	PRON
ejpam-7051	119	48	exists	exist	VERB
ejpam-7051	119	49	a	a	DET
ejpam-7051	119	50	µ-open	µ-open	NOUN
ejpam-7051	119	51	set	set	VERB
ejpam-7051	119	52	u	u	NOUN
ejpam-7051	119	53	of	of	ADP
ejpam-7051	119	54	x	x	PUNCT
ejpam-7051	119	55	containing	contain	VERB
ejpam-7051	119	56	x	x	PUNCT
ejpam-7051	119	57	such	such	ADJ
ejpam-7051	119	58	that	that	SCONJ
ejpam-7051	119	59	f	f	PROPN
ejpam-7051	119	60	(	(	PUNCT
ejpam-7051	119	61	z	z	NOUN
ejpam-7051	119	62	)	)	PUNCT
ejpam-7051	119	63	∩	∩	NOUN
ejpam-7051	119	64	v	v	ADP
ejpam-7051	119	65	̸=	̸=	PROPN
ejpam-7051	119	66	∅	∅	NOUN
ejpam-7051	119	67	for	for	ADP
ejpam-7051	119	68	every	every	DET
ejpam-7051	119	69	z	z	NOUN
ejpam-7051	119	70	∈	∈	PROPN
ejpam-7051	119	71	u	u	NOUN
ejpam-7051	119	72	.	.	PUNCT
ejpam-7051	120	1	a	a	DET
ejpam-7051	120	2	multifunction	multifunction	NOUN
ejpam-7051	120	3	f	f	NOUN
ejpam-7051	120	4	:	:	PUNCT
ejpam-7051	120	5	(	(	PUNCT
ejpam-7051	120	6	x,µ	x,µ	NOUN
ejpam-7051	120	7	)	)	PUNCT
ejpam-7051	120	8	→	→	SYM
ejpam-7051	120	9	(	(	PUNCT
ejpam-7051	120	10	y	y	PROPN
ejpam-7051	120	11	,	,	PUNCT
ejpam-7051	120	12	σ1	σ1	PROPN
ejpam-7051	120	13	,	,	PUNCT
ejpam-7051	120	14	σ2	σ2	PROPN
ejpam-7051	120	15	)	)	PUNCT
ejpam-7051	120	16	is	be	AUX
ejpam-7051	120	17	called	call	VERB
ejpam-7051	120	18	lower	low	ADJ
ejpam-7051	120	19	µ(σ1	µ(σ1	NOUN
ejpam-7051	120	20	,	,	PUNCT
ejpam-7051	120	21	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	120	22	if	if	SCONJ
ejpam-7051	120	23	f	f	PROPN
ejpam-7051	120	24	is	be	AUX
ejpam-7051	120	25	lower	low	ADJ
ejpam-7051	120	26	µ(σ1	µ(σ1	NOUN
ejpam-7051	120	27	,	,	PUNCT
ejpam-7051	120	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	120	29	at	at	ADP
ejpam-7051	120	30	each	each	DET
ejpam-7051	120	31	point	point	NOUN
ejpam-7051	120	32	x	x	PUNCT
ejpam-7051	120	33	of	of	ADP
ejpam-7051	120	34	x.	x.	PROPN
ejpam-7051	120	35	theorem	theorem	VERB
ejpam-7051	120	36	2	2	NUM
ejpam-7051	120	37	.	.	X
ejpam-7051	120	38	for	for	ADP
ejpam-7051	120	39	a	a	DET
ejpam-7051	120	40	multifunction	multifunction	NOUN
ejpam-7051	120	41	f	f	NOUN
ejpam-7051	120	42	:	:	PUNCT
ejpam-7051	120	43	(	(	PUNCT
ejpam-7051	120	44	x,µ	x,µ	NOUN
ejpam-7051	120	45	)	)	PUNCT
ejpam-7051	120	46	→	→	SYM
ejpam-7051	120	47	(	(	PUNCT
ejpam-7051	120	48	y	y	PROPN
ejpam-7051	120	49	,	,	PUNCT
ejpam-7051	120	50	σ1	σ1	PROPN
ejpam-7051	120	51	,	,	PUNCT
ejpam-7051	120	52	σ2	σ2	NOUN
ejpam-7051	120	53	)	)	PUNCT
ejpam-7051	120	54	,	,	PUNCT
ejpam-7051	120	55	the	the	DET
ejpam-7051	120	56	following	follow	VERB
ejpam-7051	120	57	properties	property	NOUN
ejpam-7051	120	58	are	be	AUX
ejpam-7051	120	59	equivalent	equivalent	ADJ
ejpam-7051	120	60	:	:	PUNCT
ejpam-7051	120	61	(	(	PUNCT
ejpam-7051	120	62	1	1	X
ejpam-7051	120	63	)	)	PUNCT
ejpam-7051	120	64	f	f	PROPN
ejpam-7051	120	65	is	be	AUX
ejpam-7051	120	66	lower	low	ADJ
ejpam-7051	120	67	µ(σ1	µ(σ1	NOUN
ejpam-7051	120	68	,	,	PUNCT
ejpam-7051	120	69	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	120	70	;	;	PUNCT
ejpam-7051	120	71	(	(	PUNCT
ejpam-7051	120	72	2	2	X
ejpam-7051	120	73	)	)	PUNCT
ejpam-7051	120	74	f−(v	f−(v	NOUN
ejpam-7051	120	75	)	)	PUNCT
ejpam-7051	120	76	is	be	AUX
ejpam-7051	120	77	µ-open	µ-open	NOUN
ejpam-7051	120	78	in	in	ADP
ejpam-7051	120	79	x	x	PUNCT
ejpam-7051	120	80	for	for	ADP
ejpam-7051	120	81	every	every	DET
ejpam-7051	120	82	σ1σ2	σ1σ2	NOUN
ejpam-7051	120	83	-	-	ADJ
ejpam-7051	120	84	open	open	ADJ
ejpam-7051	120	85	set	set	NOUN
ejpam-7051	120	86	v	v	NOUN
ejpam-7051	120	87	of	of	ADP
ejpam-7051	120	88	y	y	PROPN
ejpam-7051	120	89	;	;	PUNCT
ejpam-7051	120	90	(	(	PUNCT
ejpam-7051	120	91	3	3	X
ejpam-7051	120	92	)	)	PUNCT
ejpam-7051	120	93	f+(k	f+(k	NUM
ejpam-7051	120	94	)	)	PUNCT
ejpam-7051	120	95	is	be	AUX
ejpam-7051	120	96	µ-closed	µ-close	VERB
ejpam-7051	120	97	in	in	ADP
ejpam-7051	120	98	x	x	PUNCT
ejpam-7051	120	99	for	for	ADP
ejpam-7051	120	100	every	every	DET
ejpam-7051	120	101	σ1σ2	σ1σ2	NUM
ejpam-7051	120	102	-	-	PUNCT
ejpam-7051	120	103	closed	closed	ADJ
ejpam-7051	120	104	set	set	NOUN
ejpam-7051	120	105	k	k	PROPN
ejpam-7051	120	106	of	of	ADP
ejpam-7051	120	107	y	y	PROPN
ejpam-7051	120	108	;	;	PUNCT
ejpam-7051	120	109	n.	n.	PROPN
ejpam-7051	120	110	chutiman	chutiman	NOUN
ejpam-7051	120	111	,	,	PUNCT
ejpam-7051	120	112	a.	a.	PROPN
ejpam-7051	120	113	sama	sama	PROPN
ejpam-7051	120	114	-	-	PUNCT
ejpam-7051	120	115	ae	ae	PROPN
ejpam-7051	120	116	,	,	PUNCT
ejpam-7051	120	117	c.	c.	PROPN
ejpam-7051	120	118	boonpok	boonpok	PROPN
ejpam-7051	120	119	/	/	SYM
ejpam-7051	120	120	eur	eur	PROPN
ejpam-7051	120	121	.	.	PUNCT
ejpam-7051	121	1	j.	j.	PROPN
ejpam-7051	121	2	pure	pure	PROPN
ejpam-7051	121	3	appl	appl	PROPN
ejpam-7051	121	4	.	.	PROPN
ejpam-7051	121	5	math	math	PROPN
ejpam-7051	121	6	,	,	PUNCT
ejpam-7051	121	7	18	18	NUM
ejpam-7051	121	8	(	(	PUNCT
ejpam-7051	121	9	4	4	NUM
ejpam-7051	121	10	)	)	PUNCT
ejpam-7051	121	11	(	(	PUNCT
ejpam-7051	121	12	2025	2025	NUM
ejpam-7051	121	13	)	)	PUNCT
ejpam-7051	121	14	,	,	PUNCT
ejpam-7051	121	15	7051	7051	NUM
ejpam-7051	121	16	5	5	NUM
ejpam-7051	121	17	of	of	ADP
ejpam-7051	121	18	8	8	NUM
ejpam-7051	121	19	(	(	PUNCT
ejpam-7051	121	20	4	4	NUM
ejpam-7051	121	21	)	)	PUNCT
ejpam-7051	121	22	cµ(f	cµ(f	PUNCT
ejpam-7051	122	1	+	+	PROPN
ejpam-7051	122	2	(	(	PUNCT
ejpam-7051	122	3	b	b	NOUN
ejpam-7051	122	4	)	)	PUNCT
ejpam-7051	122	5	)	)	PUNCT
ejpam-7051	123	1	⊆	⊆	NUM
ejpam-7051	123	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7051	123	3	-	-	PUNCT
ejpam-7051	123	4	cl(b	cl(b	NOUN
ejpam-7051	123	5	)	)	PUNCT
ejpam-7051	123	6	)	)	PUNCT
ejpam-7051	124	1	for	for	ADP
ejpam-7051	124	2	every	every	DET
ejpam-7051	124	3	subset	subset	NOUN
ejpam-7051	124	4	b	b	PROPN
ejpam-7051	124	5	of	of	ADP
ejpam-7051	124	6	y	y	PROPN
ejpam-7051	124	7	;	;	PUNCT
ejpam-7051	124	8	(	(	PUNCT
ejpam-7051	124	9	5	5	X
ejpam-7051	124	10	)	)	PUNCT
ejpam-7051	124	11	f	f	NOUN
ejpam-7051	124	12	(	(	PUNCT
ejpam-7051	124	13	cµ(a	cµ(a	PROPN
ejpam-7051	124	14	)	)	PUNCT
ejpam-7051	124	15	)	)	PUNCT
ejpam-7051	125	1	⊆	⊆	X
ejpam-7051	125	2	σ1σ2	σ1σ2	X
ejpam-7051	125	3	-	-	NUM
ejpam-7051	125	4	cl(f	cl(f	NOUN
ejpam-7051	125	5	(	(	PUNCT
ejpam-7051	125	6	a	a	NOUN
ejpam-7051	125	7	)	)	PUNCT
ejpam-7051	125	8	)	)	PUNCT
ejpam-7051	125	9	for	for	ADP
ejpam-7051	125	10	every	every	DET
ejpam-7051	125	11	subset	subset	NOUN
ejpam-7051	125	12	a	a	PRON
ejpam-7051	125	13	of	of	ADP
ejpam-7051	125	14	x	x	PRON
ejpam-7051	125	15	;	;	PUNCT
ejpam-7051	125	16	(	(	PUNCT
ejpam-7051	125	17	6	6	X
ejpam-7051	125	18	)	)	PUNCT
ejpam-7051	125	19	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7051	125	20	-	-	PUNCT
ejpam-7051	125	21	int(b	int(b	NOUN
ejpam-7051	125	22	)	)	PUNCT
ejpam-7051	125	23	)	)	PUNCT
ejpam-7051	126	1	⊆	⊆	NUM
ejpam-7051	126	2	iµ(f	iµ(f	NOUN
ejpam-7051	126	3	−(b	−(b	NOUN
ejpam-7051	126	4	)	)	PUNCT
ejpam-7051	126	5	)	)	PUNCT
ejpam-7051	126	6	for	for	ADP
ejpam-7051	126	7	every	every	DET
ejpam-7051	126	8	subset	subset	NOUN
ejpam-7051	126	9	b	b	PROPN
ejpam-7051	126	10	of	of	ADP
ejpam-7051	126	11	y	y	PROPN
ejpam-7051	126	12	.	.	PUNCT
ejpam-7051	127	1	proof	proof	NOUN
ejpam-7051	127	2	.	.	PUNCT
ejpam-7051	128	1	we	we	PRON
ejpam-7051	128	2	prove	prove	VERB
ejpam-7051	128	3	only	only	ADV
ejpam-7051	128	4	the	the	DET
ejpam-7051	128	5	implications	implication	NOUN
ejpam-7051	128	6	(	(	PUNCT
ejpam-7051	128	7	4	4	X
ejpam-7051	128	8	)	)	PUNCT
ejpam-7051	128	9	⇒	⇒	NOUN
ejpam-7051	128	10	(	(	PUNCT
ejpam-7051	128	11	5	5	NUM
ejpam-7051	128	12	)	)	PUNCT
ejpam-7051	128	13	and	and	CCONJ
ejpam-7051	128	14	(	(	PUNCT
ejpam-7051	128	15	5	5	X
ejpam-7051	128	16	)	)	PUNCT
ejpam-7051	128	17	⇒	⇒	NOUN
ejpam-7051	128	18	(	(	PUNCT
ejpam-7051	128	19	6	6	X
ejpam-7051	128	20	)	)	PUNCT
ejpam-7051	128	21	being	be	AUX
ejpam-7051	128	22	the	the	DET
ejpam-7051	128	23	proofs	proof	NOUN
ejpam-7051	128	24	of	of	ADP
ejpam-7051	128	25	the	the	DET
ejpam-7051	128	26	other	other	ADJ
ejpam-7051	128	27	similar	similar	ADJ
ejpam-7051	128	28	to	to	ADP
ejpam-7051	128	29	those	those	PRON
ejpam-7051	128	30	of	of	ADP
ejpam-7051	128	31	theorem	theorem	NOUN
ejpam-7051	128	32	1	1	NUM
ejpam-7051	128	33	.	.	PUNCT
ejpam-7051	128	34	(	(	PUNCT
ejpam-7051	128	35	4	4	X
ejpam-7051	128	36	)	)	PUNCT
ejpam-7051	128	37	⇒	⇒	NOUN
ejpam-7051	128	38	(	(	PUNCT
ejpam-7051	128	39	5	5	NUM
ejpam-7051	128	40	):	):	PUNCT
ejpam-7051	128	41	let	let	VERB
ejpam-7051	128	42	a	a	PRON
ejpam-7051	128	43	be	be	AUX
ejpam-7051	128	44	any	any	DET
ejpam-7051	128	45	subset	subset	NOUN
ejpam-7051	128	46	of	of	ADP
ejpam-7051	128	47	x.	x.	NOUN
ejpam-7051	128	48	by	by	ADP
ejpam-7051	128	49	(	(	PUNCT
ejpam-7051	128	50	4	4	NUM
ejpam-7051	128	51	)	)	PUNCT
ejpam-7051	128	52	,	,	PUNCT
ejpam-7051	128	53	we	we	PRON
ejpam-7051	128	54	have	have	VERB
ejpam-7051	128	55	cµ(a	cµ(a	PROPN
ejpam-7051	128	56	)	)	PUNCT
ejpam-7051	128	57	⊆	⊆	NUM
ejpam-7051	128	58	cµ(f	cµ(f	X
ejpam-7051	129	1	+	+	PROPN
ejpam-7051	129	2	(	(	PUNCT
ejpam-7051	129	3	f	f	X
ejpam-7051	129	4	(	(	PUNCT
ejpam-7051	129	5	a	a	NOUN
ejpam-7051	129	6	)	)	PUNCT
ejpam-7051	129	7	)	)	PUNCT
ejpam-7051	129	8	)	)	PUNCT
ejpam-7051	129	9	⊆	⊆	X
ejpam-7051	129	10	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7051	129	11	-	-	SYM
ejpam-7051	129	12	cl(f	cl(f	NOUN
ejpam-7051	129	13	(	(	PUNCT
ejpam-7051	129	14	a	a	NOUN
ejpam-7051	129	15	)	)	PUNCT
ejpam-7051	129	16	)	)	PUNCT
ejpam-7051	129	17	)	)	PUNCT
ejpam-7051	130	1	and	and	CCONJ
ejpam-7051	130	2	hence	hence	ADV
ejpam-7051	130	3	f	f	PROPN
ejpam-7051	130	4	(	(	PUNCT
ejpam-7051	130	5	cµ(a	cµ(a	PROPN
ejpam-7051	130	6	)	)	PUNCT
ejpam-7051	130	7	)	)	PUNCT
ejpam-7051	131	1	⊆	⊆	X
ejpam-7051	131	2	σ1σ2	σ1σ2	X
ejpam-7051	131	3	-	-	NUM
ejpam-7051	131	4	cl(f	cl(f	NOUN
ejpam-7051	131	5	(	(	PUNCT
ejpam-7051	131	6	a	a	NOUN
ejpam-7051	131	7	)	)	PUNCT
ejpam-7051	131	8	)	)	PUNCT
ejpam-7051	131	9	.	.	PUNCT
ejpam-7051	132	1	(	(	PUNCT
ejpam-7051	132	2	5	5	X
ejpam-7051	132	3	)	)	PUNCT
ejpam-7051	132	4	⇒	⇒	NOUN
ejpam-7051	132	5	(	(	PUNCT
ejpam-7051	132	6	6	6	NUM
ejpam-7051	132	7	):	):	PUNCT
ejpam-7051	132	8	let	let	VERB
ejpam-7051	132	9	b	b	X
ejpam-7051	132	10	be	be	AUX
ejpam-7051	132	11	any	any	DET
ejpam-7051	132	12	subset	subset	NOUN
ejpam-7051	132	13	of	of	ADP
ejpam-7051	132	14	y	y	PROPN
ejpam-7051	132	15	.	.	PUNCT
ejpam-7051	133	1	by	by	ADP
ejpam-7051	133	2	(	(	PUNCT
ejpam-7051	133	3	5	5	NUM
ejpam-7051	133	4	)	)	PUNCT
ejpam-7051	133	5	,	,	PUNCT
ejpam-7051	133	6	f	f	PROPN
ejpam-7051	133	7	(	(	PUNCT
ejpam-7051	133	8	cµ(f	cµ(f	X
ejpam-7051	134	1	+	+	PROPN
ejpam-7051	134	2	(	(	PUNCT
ejpam-7051	134	3	y	y	NOUN
ejpam-7051	134	4	−b	−b	ADJ
ejpam-7051	134	5	)	)	PUNCT
ejpam-7051	134	6	)	)	PUNCT
ejpam-7051	134	7	)	)	PUNCT
ejpam-7051	135	1	⊆	⊆	X
ejpam-7051	135	2	σ1σ2	σ1σ2	X
ejpam-7051	135	3	-	-	NUM
ejpam-7051	135	4	cl(f	cl(f	NOUN
ejpam-7051	135	5	(	(	PUNCT
ejpam-7051	135	6	f+(y	f+(y	PROPN
ejpam-7051	135	7	−b	−b	PROPN
ejpam-7051	135	8	)	)	PUNCT
ejpam-7051	135	9	)	)	PUNCT
ejpam-7051	135	10	)	)	PUNCT
ejpam-7051	136	1	⊆	⊆	X
ejpam-7051	136	2	σ1σ2	σ1σ2	NUM
ejpam-7051	136	3	-	-	PUNCT
ejpam-7051	136	4	cl(y	cl(y	NOUN
ejpam-7051	136	5	−b	−b	NOUN
ejpam-7051	136	6	)	)	PUNCT
ejpam-7051	137	1	=	=	SYM
ejpam-7051	137	2	y	y	PROPN
ejpam-7051	137	3	−	−	ADP
ejpam-7051	137	4	σ1σ2	σ1σ2	X
ejpam-7051	137	5	-	-	PUNCT
ejpam-7051	137	6	int(b	int(b	NOUN
ejpam-7051	137	7	)	)	PUNCT
ejpam-7051	137	8	.	.	PUNCT
ejpam-7051	138	1	since	since	SCONJ
ejpam-7051	138	2	f	f	PROPN
ejpam-7051	138	3	(	(	PUNCT
ejpam-7051	138	4	cµ(f	cµ(f	X
ejpam-7051	139	1	+	+	PROPN
ejpam-7051	139	2	(	(	PUNCT
ejpam-7051	139	3	y	y	NOUN
ejpam-7051	139	4	−b	−b	ADJ
ejpam-7051	139	5	)	)	PUNCT
ejpam-7051	139	6	)	)	PUNCT
ejpam-7051	139	7	)	)	PUNCT
ejpam-7051	140	1	=	=	SYM
ejpam-7051	140	2	f	f	PROPN
ejpam-7051	140	3	(	(	PUNCT
ejpam-7051	140	4	cµ(x	cµ(x	PUNCT
ejpam-7051	140	5	−	−	PROPN
ejpam-7051	140	6	f−(b	f−(b	PROPN
ejpam-7051	140	7	)	)	PUNCT
ejpam-7051	140	8	)	)	PUNCT
ejpam-7051	140	9	)	)	PUNCT
ejpam-7051	141	1	=	=	SYM
ejpam-7051	141	2	f	f	X
ejpam-7051	141	3	(	(	PUNCT
ejpam-7051	141	4	x	x	SYM
ejpam-7051	141	5	−	−	PROPN
ejpam-7051	141	6	iµ(f	iµ(f	NOUN
ejpam-7051	141	7	−(b	−(b	INTJ
ejpam-7051	141	8	)	)	PUNCT
ejpam-7051	141	9	)	)	PUNCT
ejpam-7051	141	10	)	)	PUNCT
ejpam-7051	141	11	,	,	PUNCT
ejpam-7051	141	12	we	we	PRON
ejpam-7051	141	13	have	have	VERB
ejpam-7051	141	14	x	x	PART
ejpam-7051	141	15	−	−	PROPN
ejpam-7051	141	16	iµ(f	iµ(f	NOUN
ejpam-7051	141	17	−(b	−(b	INTJ
ejpam-7051	141	18	)	)	PUNCT
ejpam-7051	141	19	)	)	PUNCT
ejpam-7051	142	1	⊆	⊆	NUM
ejpam-7051	142	2	f+(y	f+(y	ADP
ejpam-7051	142	3	−	−	NUM
ejpam-7051	142	4	σ1σ2	σ1σ2	SYM
ejpam-7051	142	5	-	-	PUNCT
ejpam-7051	142	6	int(b	int(b	NOUN
ejpam-7051	142	7	)	)	PUNCT
ejpam-7051	142	8	)	)	PUNCT
ejpam-7051	143	1	=	=	PUNCT
ejpam-7051	143	2	x	x	X
ejpam-7051	143	3	−	−	NOUN
ejpam-7051	143	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7051	143	5	-	-	PUNCT
ejpam-7051	143	6	int(b	int(b	NOUN
ejpam-7051	143	7	)	)	PUNCT
ejpam-7051	143	8	)	)	PUNCT
ejpam-7051	143	9	and	and	CCONJ
ejpam-7051	143	10	hence	hence	ADV
ejpam-7051	143	11	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7051	143	12	-	-	PUNCT
ejpam-7051	143	13	int(b	int(b	NOUN
ejpam-7051	143	14	)	)	PUNCT
ejpam-7051	143	15	)	)	PUNCT
ejpam-7051	144	1	⊆	⊆	NUM
ejpam-7051	144	2	iµ(f	iµ(f	NOUN
ejpam-7051	144	3	−(b	−(b	NOUN
ejpam-7051	144	4	)	)	PUNCT
ejpam-7051	144	5	)	)	PUNCT
ejpam-7051	144	6	.	.	PUNCT
ejpam-7051	145	1	recall	recall	VERB
ejpam-7051	145	2	that	that	SCONJ
ejpam-7051	145	3	a	a	DET
ejpam-7051	145	4	bitopological	bitopological	ADJ
ejpam-7051	145	5	space	space	NOUN
ejpam-7051	145	6	(	(	PUNCT
ejpam-7051	145	7	x	x	NOUN
ejpam-7051	145	8	,	,	PUNCT
ejpam-7051	145	9	τ1	τ1	NOUN
ejpam-7051	145	10	,	,	PUNCT
ejpam-7051	145	11	τ2	τ2	NOUN
ejpam-7051	145	12	)	)	PUNCT
ejpam-7051	145	13	is	be	AUX
ejpam-7051	145	14	said	say	VERB
ejpam-7051	145	15	to	to	PART
ejpam-7051	145	16	be	be	AUX
ejpam-7051	145	17	(	(	PUNCT
ejpam-7051	145	18	τ1	τ1	NOUN
ejpam-7051	145	19	,	,	PUNCT
ejpam-7051	145	20	τ2)s	τ2)s	NOUN
ejpam-7051	145	21	-	-	PUNCT
ejpam-7051	145	22	regular	regular	ADJ
ejpam-7051	145	23	[	[	X
ejpam-7051	145	24	7	7	NUM
ejpam-7051	145	25	]	]	X
ejpam-7051	145	26	if	if	SCONJ
ejpam-7051	145	27	for	for	ADP
ejpam-7051	145	28	each	each	DET
ejpam-7051	145	29	(	(	PUNCT
ejpam-7051	145	30	τ1	τ1	NOUN
ejpam-7051	145	31	,	,	PUNCT
ejpam-7051	145	32	τ2)s	τ2)s	NOUN
ejpam-7051	145	33	-	-	PUNCT
ejpam-7051	145	34	closed	close	VERB
ejpam-7051	145	35	set	set	VERB
ejpam-7051	145	36	f	f	NOUN
ejpam-7051	145	37	and	and	CCONJ
ejpam-7051	145	38	each	each	DET
ejpam-7051	145	39	x	x	PROPN
ejpam-7051	145	40	̸∈	̸∈	PROPN
ejpam-7051	145	41	f	f	PROPN
ejpam-7051	145	42	,	,	PUNCT
ejpam-7051	145	43	there	there	PRON
ejpam-7051	145	44	exist	exist	VERB
ejpam-7051	145	45	disjoint	disjoint	NOUN
ejpam-7051	145	46	(	(	PUNCT
ejpam-7051	145	47	τ1	τ1	NOUN
ejpam-7051	145	48	,	,	PUNCT
ejpam-7051	145	49	τ2)s	τ2)s	NOUN
ejpam-7051	145	50	-	-	PUNCT
ejpam-7051	145	51	open	open	ADJ
ejpam-7051	145	52	sets	set	VERB
ejpam-7051	145	53	u	u	NOUN
ejpam-7051	145	54	and	and	CCONJ
ejpam-7051	145	55	v	v	ADP
ejpam-7051	145	56	such	such	ADJ
ejpam-7051	145	57	that	that	SCONJ
ejpam-7051	145	58	x	x	SYM
ejpam-7051	145	59	∈	∈	PROPN
ejpam-7051	145	60	u	u	NOUN
ejpam-7051	145	61	and	and	CCONJ
ejpam-7051	145	62	f	f	PROPN
ejpam-7051	145	63	⊆	⊆	NUM
ejpam-7051	145	64	v	v	NOUN
ejpam-7051	145	65	.	.	PUNCT
ejpam-7051	146	1	lemma	lemma	PROPN
ejpam-7051	146	2	3	3	X
ejpam-7051	146	3	.	.	PUNCT
ejpam-7051	147	1	[	[	X
ejpam-7051	147	2	7	7	X
ejpam-7051	147	3	]	]	PUNCT
ejpam-7051	147	4	a	a	DET
ejpam-7051	147	5	bitopological	bitopological	ADJ
ejpam-7051	147	6	space	space	NOUN
ejpam-7051	147	7	(	(	PUNCT
ejpam-7051	147	8	x	x	NOUN
ejpam-7051	147	9	,	,	PUNCT
ejpam-7051	147	10	τ1	τ1	NOUN
ejpam-7051	147	11	,	,	PUNCT
ejpam-7051	147	12	τ2	τ2	NOUN
ejpam-7051	147	13	)	)	PUNCT
ejpam-7051	147	14	is	be	AUX
ejpam-7051	147	15	(	(	PUNCT
ejpam-7051	147	16	τ1	τ1	NOUN
ejpam-7051	147	17	,	,	PUNCT
ejpam-7051	147	18	τ2)s	τ2)s	NOUN
ejpam-7051	147	19	-	-	PUNCT
ejpam-7051	147	20	regular	regular	ADJ
ejpam-7051	147	21	if	if	SCONJ
ejpam-7051	147	22	and	and	CCONJ
ejpam-7051	147	23	only	only	ADV
ejpam-7051	147	24	if	if	SCONJ
ejpam-7051	147	25	for	for	ADP
ejpam-7051	147	26	each	each	DET
ejpam-7051	147	27	x	x	SYM
ejpam-7051	147	28	∈	∈	PROPN
ejpam-7051	147	29	x	x	X
ejpam-7051	147	30	and	and	CCONJ
ejpam-7051	147	31	each	each	DET
ejpam-7051	147	32	(	(	PUNCT
ejpam-7051	147	33	τ1	τ1	NOUN
ejpam-7051	147	34	,	,	PUNCT
ejpam-7051	147	35	τ2)s	τ2)s	NOUN
ejpam-7051	147	36	-	-	PUNCT
ejpam-7051	147	37	open	open	ADJ
ejpam-7051	147	38	set	set	NOUN
ejpam-7051	147	39	u	u	NOUN
ejpam-7051	147	40	containing	contain	VERB
ejpam-7051	147	41	x	x	PRON
ejpam-7051	147	42	,	,	PUNCT
ejpam-7051	147	43	there	there	PRON
ejpam-7051	147	44	exists	exist	VERB
ejpam-7051	147	45	a	a	DET
ejpam-7051	147	46	(	(	PUNCT
ejpam-7051	147	47	τ1	τ1	NOUN
ejpam-7051	147	48	,	,	PUNCT
ejpam-7051	147	49	τ2)s	τ2)s	NOUN
ejpam-7051	147	50	-	-	PUNCT
ejpam-7051	147	51	open	open	NOUN
ejpam-7051	147	52	set	set	VERB
ejpam-7051	147	53	v	v	ADP
ejpam-7051	148	1	such	such	ADJ
ejpam-7051	148	2	that	that	SCONJ
ejpam-7051	148	3	x	x	SYM
ejpam-7051	148	4	∈	∈	NOUN
ejpam-7051	148	5	v	v	ADP
ejpam-7051	148	6	⊆	⊆	NUM
ejpam-7051	148	7	(	(	PUNCT
ejpam-7051	148	8	τ1	τ1	NOUN
ejpam-7051	148	9	,	,	PUNCT
ejpam-7051	148	10	τ2)-scl(v	τ2)-scl(v	NOUN
ejpam-7051	148	11	)	)	PUNCT
ejpam-7051	148	12	⊆	⊆	NUM
ejpam-7051	148	13	u	u	NOUN
ejpam-7051	148	14	.	.	PUNCT
ejpam-7051	148	15	lemma	lemma	PROPN
ejpam-7051	148	16	4	4	X
ejpam-7051	148	17	.	.	PUNCT
ejpam-7051	149	1	[	[	X
ejpam-7051	149	2	7	7	X
ejpam-7051	149	3	]	]	X
ejpam-7051	149	4	let	let	VERB
ejpam-7051	149	5	(	(	PUNCT
ejpam-7051	149	6	x	x	NOUN
ejpam-7051	149	7	,	,	PUNCT
ejpam-7051	149	8	τ1	τ1	NOUN
ejpam-7051	149	9	,	,	PUNCT
ejpam-7051	149	10	τ2	τ2	PROPN
ejpam-7051	149	11	)	)	PUNCT
ejpam-7051	149	12	be	be	VERB
ejpam-7051	149	13	a	a	DET
ejpam-7051	149	14	(	(	PUNCT
ejpam-7051	149	15	τ1	τ1	NOUN
ejpam-7051	149	16	,	,	PUNCT
ejpam-7051	149	17	τ2)s	τ2)s	NOUN
ejpam-7051	149	18	-	-	PUNCT
ejpam-7051	149	19	regular	regular	ADJ
ejpam-7051	149	20	space	space	NOUN
ejpam-7051	149	21	.	.	PUNCT
ejpam-7051	150	1	then	then	ADV
ejpam-7051	150	2	,	,	PUNCT
ejpam-7051	150	3	the	the	DET
ejpam-7051	150	4	following	follow	VERB
ejpam-7051	150	5	properties	property	NOUN
ejpam-7051	150	6	hold	hold	VERB
ejpam-7051	150	7	:	:	PUNCT
ejpam-7051	150	8	(	(	PUNCT
ejpam-7051	150	9	1	1	X
ejpam-7051	150	10	)	)	PUNCT
ejpam-7051	150	11	τ1τ2	τ1τ2	NOUN
ejpam-7051	150	12	-	-	NUM
ejpam-7051	150	13	cl(a	cl(a	NUM
ejpam-7051	150	14	)	)	PUNCT
ejpam-7051	150	15	=	=	PUNCT
ejpam-7051	151	1	τ1τ2	τ1τ2	PROPN
ejpam-7051	151	2	-	-	ADJ
ejpam-7051	151	3	δ	δ	NOUN
ejpam-7051	151	4	-	-	PUNCT
ejpam-7051	151	5	cl(a	cl(a	NUM
ejpam-7051	151	6	)	)	PUNCT
ejpam-7051	151	7	for	for	ADP
ejpam-7051	151	8	every	every	DET
ejpam-7051	151	9	subset	subset	NOUN
ejpam-7051	151	10	a	a	PRON
ejpam-7051	151	11	of	of	ADP
ejpam-7051	151	12	x.	x.	NOUN
ejpam-7051	151	13	(	(	PUNCT
ejpam-7051	151	14	2	2	NUM
ejpam-7051	151	15	)	)	PUNCT
ejpam-7051	151	16	every	every	DET
ejpam-7051	151	17	τ1τ2	τ1τ2	NOUN
ejpam-7051	151	18	-	-	ADJ
ejpam-7051	151	19	open	open	ADJ
ejpam-7051	151	20	set	set	NOUN
ejpam-7051	151	21	is	be	AUX
ejpam-7051	151	22	τ1τ2	τ1τ2	ADJ
ejpam-7051	151	23	-	-	ADJ
ejpam-7051	151	24	δ	δ	NOUN
ejpam-7051	151	25	-	-	ADJ
ejpam-7051	151	26	open	open	ADJ
ejpam-7051	151	27	.	.	PUNCT
ejpam-7051	152	1	theorem	theorem	NOUN
ejpam-7051	152	2	3	3	NUM
ejpam-7051	152	3	.	.	X
ejpam-7051	152	4	for	for	ADP
ejpam-7051	152	5	a	a	DET
ejpam-7051	152	6	multifunction	multifunction	NOUN
ejpam-7051	153	1	f	f	NOUN
ejpam-7051	153	2	:	:	PUNCT
ejpam-7051	153	3	(	(	PUNCT
ejpam-7051	153	4	x,µ	x,µ	NOUN
ejpam-7051	153	5	)	)	PUNCT
ejpam-7051	153	6	→	→	SYM
ejpam-7051	153	7	(	(	PUNCT
ejpam-7051	153	8	y	y	PROPN
ejpam-7051	153	9	,	,	PUNCT
ejpam-7051	153	10	σ1	σ1	PROPN
ejpam-7051	153	11	,	,	PUNCT
ejpam-7051	153	12	σ2	σ2	NOUN
ejpam-7051	153	13	)	)	PUNCT
ejpam-7051	153	14	,	,	PUNCT
ejpam-7051	153	15	the	the	DET
ejpam-7051	153	16	following	follow	VERB
ejpam-7051	153	17	properties	property	NOUN
ejpam-7051	153	18	are	be	AUX
ejpam-7051	153	19	equivalent	equivalent	ADJ
ejpam-7051	153	20	:	:	PUNCT
ejpam-7051	153	21	(	(	PUNCT
ejpam-7051	153	22	1	1	X
ejpam-7051	153	23	)	)	PUNCT
ejpam-7051	153	24	f	f	PROPN
ejpam-7051	153	25	is	be	AUX
ejpam-7051	153	26	upper	upper	ADJ
ejpam-7051	153	27	µ(σ1	µ(σ1	NOUN
ejpam-7051	153	28	,	,	PUNCT
ejpam-7051	153	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	153	30	;	;	PUNCT
ejpam-7051	153	31	(	(	PUNCT
ejpam-7051	153	32	2	2	X
ejpam-7051	153	33	)	)	PUNCT
ejpam-7051	153	34	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7051	153	35	-	-	PUNCT
ejpam-7051	153	36	δ	δ	NOUN
ejpam-7051	153	37	-	-	NOUN
ejpam-7051	153	38	cl(b	cl(b	NOUN
ejpam-7051	153	39	)	)	PUNCT
ejpam-7051	153	40	)	)	PUNCT
ejpam-7051	153	41	is	be	AUX
ejpam-7051	153	42	µ-closed	µ-close	VERB
ejpam-7051	153	43	in	in	ADP
ejpam-7051	153	44	x	x	PUNCT
ejpam-7051	153	45	for	for	ADP
ejpam-7051	153	46	every	every	DET
ejpam-7051	153	47	subset	subset	NOUN
ejpam-7051	153	48	b	b	PROPN
ejpam-7051	153	49	of	of	ADP
ejpam-7051	153	50	y	y	PROPN
ejpam-7051	153	51	;	;	PUNCT
ejpam-7051	153	52	(	(	PUNCT
ejpam-7051	153	53	3	3	X
ejpam-7051	153	54	)	)	PUNCT
ejpam-7051	153	55	f−(k	f−(k	PROPN
ejpam-7051	153	56	)	)	PUNCT
ejpam-7051	153	57	is	be	AUX
ejpam-7051	153	58	µ-closed	µ-close	VERB
ejpam-7051	153	59	in	in	ADP
ejpam-7051	153	60	x	x	PUNCT
ejpam-7051	153	61	for	for	ADP
ejpam-7051	153	62	every	every	DET
ejpam-7051	153	63	σ1σ2	σ1σ2	NUM
ejpam-7051	153	64	-	-	PUNCT
ejpam-7051	153	65	δ	δ	NOUN
ejpam-7051	153	66	-	-	PUNCT
ejpam-7051	153	67	closed	close	VERB
ejpam-7051	153	68	set	set	ADJ
ejpam-7051	153	69	k	k	PROPN
ejpam-7051	153	70	of	of	ADP
ejpam-7051	153	71	y	y	PROPN
ejpam-7051	153	72	;	;	PUNCT
ejpam-7051	153	73	(	(	PUNCT
ejpam-7051	153	74	4	4	X
ejpam-7051	153	75	)	)	PUNCT
ejpam-7051	153	76	f+(v	f+(v	NOUN
ejpam-7051	153	77	)	)	PUNCT
ejpam-7051	154	1	is	be	AUX
ejpam-7051	154	2	µ-open	µ-open	NOUN
ejpam-7051	154	3	in	in	ADP
ejpam-7051	154	4	x	x	PUNCT
ejpam-7051	154	5	for	for	ADP
ejpam-7051	154	6	every	every	DET
ejpam-7051	154	7	σ1σ2	σ1σ2	NUM
ejpam-7051	154	8	-	-	PUNCT
ejpam-7051	154	9	δ	δ	NOUN
ejpam-7051	154	10	-	-	ADJ
ejpam-7051	154	11	open	open	ADJ
ejpam-7051	154	12	set	set	VERB
ejpam-7051	154	13	v	v	NOUN
ejpam-7051	154	14	of	of	ADP
ejpam-7051	154	15	y	y	PROPN
ejpam-7051	154	16	.	.	PUNCT
ejpam-7051	155	1	n.	n.	PROPN
ejpam-7051	155	2	chutiman	chutiman	PROPN
ejpam-7051	155	3	,	,	PUNCT
ejpam-7051	155	4	a.	a.	PROPN
ejpam-7051	155	5	sama	sama	PROPN
ejpam-7051	155	6	-	-	PUNCT
ejpam-7051	155	7	ae	ae	PROPN
ejpam-7051	155	8	,	,	PUNCT
ejpam-7051	155	9	c.	c.	PROPN
ejpam-7051	155	10	boonpok	boonpok	PROPN
ejpam-7051	155	11	/	/	SYM
ejpam-7051	155	12	eur	eur	PROPN
ejpam-7051	155	13	.	.	PUNCT
ejpam-7051	156	1	j.	j.	PROPN
ejpam-7051	156	2	pure	pure	PROPN
ejpam-7051	156	3	appl	appl	PROPN
ejpam-7051	156	4	.	.	PROPN
ejpam-7051	156	5	math	math	PROPN
ejpam-7051	156	6	,	,	PUNCT
ejpam-7051	156	7	18	18	NUM
ejpam-7051	156	8	(	(	PUNCT
ejpam-7051	156	9	4	4	NUM
ejpam-7051	156	10	)	)	PUNCT
ejpam-7051	156	11	(	(	PUNCT
ejpam-7051	156	12	2025	2025	NUM
ejpam-7051	156	13	)	)	PUNCT
ejpam-7051	156	14	,	,	PUNCT
ejpam-7051	156	15	7051	7051	NUM
ejpam-7051	156	16	6	6	NUM
ejpam-7051	156	17	of	of	ADP
ejpam-7051	156	18	8	8	NUM
ejpam-7051	156	19	proof	proof	NOUN
ejpam-7051	156	20	.	.	PUNCT
ejpam-7051	157	1	(	(	PUNCT
ejpam-7051	157	2	1	1	X
ejpam-7051	157	3	)	)	PUNCT
ejpam-7051	157	4	⇒	⇒	NOUN
ejpam-7051	157	5	(	(	PUNCT
ejpam-7051	157	6	2	2	NUM
ejpam-7051	157	7	):	):	PUNCT
ejpam-7051	157	8	let	let	VERB
ejpam-7051	157	9	b	b	X
ejpam-7051	157	10	be	be	AUX
ejpam-7051	157	11	any	any	DET
ejpam-7051	157	12	subset	subset	NOUN
ejpam-7051	157	13	of	of	ADP
ejpam-7051	157	14	y	y	PROPN
ejpam-7051	157	15	.	.	PUNCT
ejpam-7051	158	1	by	by	ADP
ejpam-7051	158	2	lemma	lemma	PROPN
ejpam-7051	158	3	4	4	NUM
ejpam-7051	158	4	,	,	PUNCT
ejpam-7051	158	5	σ1σ2	σ1σ2	NOUN
ejpam-7051	158	6	-	-	PUNCT
ejpam-7051	158	7	δ	δ	NOUN
ejpam-7051	158	8	-	-	NOUN
ejpam-7051	158	9	cl(b	cl(b	NOUN
ejpam-7051	158	10	)	)	PUNCT
ejpam-7051	158	11	is	be	AUX
ejpam-7051	158	12	σ1σ2	σ1σ2	NOUN
ejpam-7051	158	13	-	-	ADJ
ejpam-7051	158	14	closed	closed	ADJ
ejpam-7051	158	15	in	in	ADP
ejpam-7051	158	16	y	y	PROPN
ejpam-7051	158	17	.	.	PUNCT
ejpam-7051	159	1	since	since	SCONJ
ejpam-7051	159	2	f	f	PROPN
ejpam-7051	159	3	is	be	AUX
ejpam-7051	159	4	upper	upper	ADJ
ejpam-7051	159	5	µ(σ1	µ(σ1	NOUN
ejpam-7051	159	6	,	,	PUNCT
ejpam-7051	159	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	159	8	,	,	PUNCT
ejpam-7051	159	9	by	by	ADP
ejpam-7051	159	10	theorem	theorem	NOUN
ejpam-7051	159	11	1	1	NUM
ejpam-7051	159	12	we	we	PRON
ejpam-7051	159	13	have	have	VERB
ejpam-7051	159	14	f−(σ1σ2	f−(σ1σ2	VERB
ejpam-7051	159	15	-	-	PUNCT
ejpam-7051	159	16	δ	δ	NOUN
ejpam-7051	159	17	-	-	NOUN
ejpam-7051	159	18	cl(b	cl(b	NOUN
ejpam-7051	159	19	)	)	PUNCT
ejpam-7051	159	20	)	)	PUNCT
ejpam-7051	159	21	is	be	AUX
ejpam-7051	159	22	µ-closed	µ-close	VERB
ejpam-7051	159	23	in	in	ADP
ejpam-7051	159	24	x.	x.	NOUN
ejpam-7051	159	25	(	(	PUNCT
ejpam-7051	159	26	2	2	NUM
ejpam-7051	159	27	)	)	PUNCT
ejpam-7051	159	28	⇒	⇒	NOUN
ejpam-7051	159	29	(	(	PUNCT
ejpam-7051	159	30	3	3	NUM
ejpam-7051	159	31	):	):	PUNCT
ejpam-7051	159	32	let	let	VERB
ejpam-7051	159	33	k	k	PRON
ejpam-7051	159	34	be	be	AUX
ejpam-7051	159	35	any	any	DET
ejpam-7051	159	36	σ1σ2	σ1σ2	NUM
ejpam-7051	159	37	-	-	PUNCT
ejpam-7051	159	38	δ	δ	NOUN
ejpam-7051	159	39	-	-	PUNCT
ejpam-7051	159	40	closed	closed	ADJ
ejpam-7051	159	41	set	set	NOUN
ejpam-7051	159	42	of	of	ADP
ejpam-7051	159	43	y	y	PROPN
ejpam-7051	159	44	.	.	PUNCT
ejpam-7051	160	1	then	then	ADV
ejpam-7051	160	2	,	,	PUNCT
ejpam-7051	160	3	σ1σ2	σ1σ2	PROPN
ejpam-7051	160	4	-	-	PUNCT
ejpam-7051	160	5	δ	δ	NOUN
ejpam-7051	160	6	-	-	NOUN
ejpam-7051	160	7	cl(k	cl(k	NOUN
ejpam-7051	160	8	)	)	PUNCT
ejpam-7051	161	1	=	=	SYM
ejpam-7051	162	1	k	k	PROPN
ejpam-7051	162	2	and	and	CCONJ
ejpam-7051	162	3	by	by	ADP
ejpam-7051	162	4	(	(	PUNCT
ejpam-7051	162	5	2	2	NUM
ejpam-7051	162	6	)	)	PUNCT
ejpam-7051	162	7	,	,	PUNCT
ejpam-7051	162	8	f−(k	f−(k	PROPN
ejpam-7051	162	9	)	)	PUNCT
ejpam-7051	162	10	is	be	AUX
ejpam-7051	162	11	µ-closed	µ-close	VERB
ejpam-7051	162	12	in	in	ADP
ejpam-7051	162	13	x.	x.	NOUN
ejpam-7051	162	14	(	(	PUNCT
ejpam-7051	162	15	3	3	NUM
ejpam-7051	162	16	)	)	PUNCT
ejpam-7051	162	17	⇒	⇒	NOUN
ejpam-7051	162	18	(	(	PUNCT
ejpam-7051	162	19	4	4	NUM
ejpam-7051	162	20	):	):	PUNCT
ejpam-7051	162	21	this	this	PRON
ejpam-7051	162	22	follows	follow	VERB
ejpam-7051	162	23	from	from	ADP
ejpam-7051	162	24	the	the	DET
ejpam-7051	162	25	fact	fact	NOUN
ejpam-7051	162	26	that	that	SCONJ
ejpam-7051	162	27	f+(y	f+(y	PROPN
ejpam-7051	162	28	−b	−b	ADV
ejpam-7051	162	29	)	)	PUNCT
ejpam-7051	162	30	=	=	PUNCT
ejpam-7051	163	1	x	x	SYM
ejpam-7051	163	2	−f−(b	−f−(b	PROPN
ejpam-7051	163	3	)	)	PUNCT
ejpam-7051	163	4	for	for	ADP
ejpam-7051	163	5	any	any	DET
ejpam-7051	163	6	subset	subset	NOUN
ejpam-7051	163	7	b	b	PROPN
ejpam-7051	163	8	of	of	ADP
ejpam-7051	163	9	y	y	PROPN
ejpam-7051	163	10	.	.	PUNCT
ejpam-7051	164	1	(	(	PUNCT
ejpam-7051	164	2	4	4	X
ejpam-7051	164	3	)	)	PUNCT
ejpam-7051	164	4	⇒	⇒	NOUN
ejpam-7051	164	5	(	(	PUNCT
ejpam-7051	164	6	1	1	NUM
ejpam-7051	164	7	):	):	PUNCT
ejpam-7051	164	8	let	let	VERB
ejpam-7051	164	9	v	v	PART
ejpam-7051	164	10	be	be	AUX
ejpam-7051	164	11	any	any	DET
ejpam-7051	164	12	σ1σ2	σ1σ2	NOUN
ejpam-7051	164	13	-	-	ADJ
ejpam-7051	164	14	open	open	ADJ
ejpam-7051	164	15	set	set	NOUN
ejpam-7051	164	16	of	of	ADP
ejpam-7051	164	17	y	y	PROPN
ejpam-7051	164	18	.	.	PUNCT
ejpam-7051	165	1	since	since	SCONJ
ejpam-7051	165	2	(	(	PUNCT
ejpam-7051	165	3	y	y	PROPN
ejpam-7051	165	4	,	,	PUNCT
ejpam-7051	165	5	σ1	σ1	PROPN
ejpam-7051	165	6	,	,	PUNCT
ejpam-7051	165	7	σ2	σ2	PROPN
ejpam-7051	165	8	)	)	PUNCT
ejpam-7051	165	9	is	be	AUX
ejpam-7051	165	10	(	(	PUNCT
ejpam-7051	165	11	σ1	σ1	PROPN
ejpam-7051	165	12	,	,	PUNCT
ejpam-7051	165	13	σ2)s	σ2)s	NOUN
ejpam-7051	165	14	-	-	PUNCT
ejpam-7051	165	15	regular	regular	ADJ
ejpam-7051	165	16	,	,	PUNCT
ejpam-7051	165	17	we	we	PRON
ejpam-7051	165	18	have	have	VERB
ejpam-7051	165	19	v	v	NOUN
ejpam-7051	165	20	is	be	AUX
ejpam-7051	166	1	σ1σ2	σ1σ2	NOUN
ejpam-7051	166	2	-	-	PUNCT
ejpam-7051	166	3	δ	δ	NOUN
ejpam-7051	166	4	-	-	NOUN
ejpam-7051	166	5	open	open	ADJ
ejpam-7051	166	6	in	in	ADP
ejpam-7051	166	7	y	y	PROPN
ejpam-7051	166	8	and	and	CCONJ
ejpam-7051	166	9	by	by	ADP
ejpam-7051	166	10	(	(	PUNCT
ejpam-7051	166	11	4	4	NUM
ejpam-7051	166	12	)	)	PUNCT
ejpam-7051	166	13	,	,	PUNCT
ejpam-7051	166	14	f+(v	f+(v	PROPN
ejpam-7051	166	15	)	)	PUNCT
ejpam-7051	166	16	is	be	AUX
ejpam-7051	166	17	µ-open	µ-open	NOUN
ejpam-7051	166	18	in	in	ADP
ejpam-7051	166	19	x.	x.	NOUN
ejpam-7051	166	20	thus	thus	ADV
ejpam-7051	166	21	,	,	PUNCT
ejpam-7051	166	22	f	f	PROPN
ejpam-7051	166	23	is	be	AUX
ejpam-7051	166	24	upper	upper	ADJ
ejpam-7051	166	25	µ(σ1	µ(σ1	NOUN
ejpam-7051	166	26	,	,	PUNCT
ejpam-7051	166	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	166	28	by	by	ADP
ejpam-7051	166	29	theorem	theorem	NOUN
ejpam-7051	166	30	1	1	NUM
ejpam-7051	166	31	.	.	PUNCT
ejpam-7051	166	32	theorem	theorem	NOUN
ejpam-7051	166	33	4	4	NUM
ejpam-7051	166	34	.	.	X
ejpam-7051	166	35	for	for	ADP
ejpam-7051	166	36	a	a	DET
ejpam-7051	166	37	multifunction	multifunction	NOUN
ejpam-7051	167	1	f	f	NOUN
ejpam-7051	167	2	:	:	PUNCT
ejpam-7051	167	3	(	(	PUNCT
ejpam-7051	167	4	x,µ	x,µ	NOUN
ejpam-7051	167	5	)	)	PUNCT
ejpam-7051	167	6	→	→	SYM
ejpam-7051	167	7	(	(	PUNCT
ejpam-7051	167	8	y	y	PROPN
ejpam-7051	167	9	,	,	PUNCT
ejpam-7051	167	10	σ1	σ1	PROPN
ejpam-7051	167	11	,	,	PUNCT
ejpam-7051	167	12	σ2	σ2	NOUN
ejpam-7051	167	13	)	)	PUNCT
ejpam-7051	167	14	,	,	PUNCT
ejpam-7051	167	15	the	the	DET
ejpam-7051	167	16	following	follow	VERB
ejpam-7051	167	17	properties	property	NOUN
ejpam-7051	167	18	are	be	AUX
ejpam-7051	167	19	equivalent	equivalent	ADJ
ejpam-7051	167	20	:	:	PUNCT
ejpam-7051	167	21	(	(	PUNCT
ejpam-7051	167	22	1	1	X
ejpam-7051	167	23	)	)	PUNCT
ejpam-7051	167	24	f	f	PROPN
ejpam-7051	167	25	is	be	AUX
ejpam-7051	167	26	lower	low	ADJ
ejpam-7051	167	27	µ(σ1	µ(σ1	NOUN
ejpam-7051	167	28	,	,	PUNCT
ejpam-7051	167	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	167	30	;	;	PUNCT
ejpam-7051	167	31	(	(	PUNCT
ejpam-7051	167	32	2	2	X
ejpam-7051	167	33	)	)	PUNCT
ejpam-7051	167	34	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7051	167	35	-	-	PUNCT
ejpam-7051	167	36	δ	δ	NOUN
ejpam-7051	167	37	-	-	NOUN
ejpam-7051	167	38	cl(b	cl(b	NOUN
ejpam-7051	167	39	)	)	PUNCT
ejpam-7051	167	40	)	)	PUNCT
ejpam-7051	167	41	is	be	AUX
ejpam-7051	167	42	µ-closed	µ-close	VERB
ejpam-7051	167	43	in	in	ADP
ejpam-7051	167	44	x	x	PUNCT
ejpam-7051	167	45	for	for	ADP
ejpam-7051	167	46	every	every	DET
ejpam-7051	167	47	subset	subset	NOUN
ejpam-7051	167	48	b	b	PROPN
ejpam-7051	167	49	of	of	ADP
ejpam-7051	167	50	y	y	PROPN
ejpam-7051	167	51	;	;	PUNCT
ejpam-7051	167	52	(	(	PUNCT
ejpam-7051	167	53	3	3	X
ejpam-7051	167	54	)	)	PUNCT
ejpam-7051	167	55	f+(k	f+(k	NUM
ejpam-7051	167	56	)	)	PUNCT
ejpam-7051	167	57	is	be	AUX
ejpam-7051	167	58	µ-closed	µ-close	VERB
ejpam-7051	167	59	in	in	ADP
ejpam-7051	167	60	x	x	PUNCT
ejpam-7051	167	61	for	for	ADP
ejpam-7051	167	62	every	every	DET
ejpam-7051	167	63	σ1σ2	σ1σ2	NUM
ejpam-7051	167	64	-	-	PUNCT
ejpam-7051	167	65	δ	δ	NOUN
ejpam-7051	167	66	-	-	PUNCT
ejpam-7051	167	67	closed	close	VERB
ejpam-7051	167	68	set	set	ADJ
ejpam-7051	167	69	k	k	PROPN
ejpam-7051	167	70	of	of	ADP
ejpam-7051	167	71	y	y	PROPN
ejpam-7051	167	72	;	;	PUNCT
ejpam-7051	167	73	(	(	PUNCT
ejpam-7051	167	74	4	4	X
ejpam-7051	167	75	)	)	PUNCT
ejpam-7051	167	76	f−(v	f−(v	NOUN
ejpam-7051	167	77	)	)	PUNCT
ejpam-7051	167	78	is	be	AUX
ejpam-7051	167	79	µ-open	µ-open	NOUN
ejpam-7051	167	80	in	in	ADP
ejpam-7051	167	81	x	x	PUNCT
ejpam-7051	167	82	for	for	ADP
ejpam-7051	167	83	every	every	DET
ejpam-7051	167	84	σ1σ2	σ1σ2	NUM
ejpam-7051	167	85	-	-	PUNCT
ejpam-7051	167	86	δ	δ	NOUN
ejpam-7051	167	87	-	-	ADJ
ejpam-7051	167	88	open	open	ADJ
ejpam-7051	167	89	set	set	VERB
ejpam-7051	167	90	v	v	NOUN
ejpam-7051	167	91	of	of	ADP
ejpam-7051	167	92	y	y	PROPN
ejpam-7051	167	93	.	.	PUNCT
ejpam-7051	168	1	proof	proof	NOUN
ejpam-7051	168	2	.	.	PUNCT
ejpam-7051	169	1	the	the	DET
ejpam-7051	169	2	proof	proof	NOUN
ejpam-7051	169	3	is	be	AUX
ejpam-7051	169	4	similar	similar	ADJ
ejpam-7051	169	5	to	to	ADP
ejpam-7051	169	6	that	that	PRON
ejpam-7051	169	7	of	of	ADP
ejpam-7051	169	8	theorem	theorem	ADJ
ejpam-7051	169	9	3	3	NUM
ejpam-7051	169	10	.	.	NOUN
ejpam-7051	169	11	recall	recall	VERB
ejpam-7051	169	12	that	that	SCONJ
ejpam-7051	169	13	a	a	DET
ejpam-7051	169	14	bitopological	bitopological	ADJ
ejpam-7051	169	15	space	space	NOUN
ejpam-7051	169	16	(	(	PUNCT
ejpam-7051	169	17	x	x	NOUN
ejpam-7051	169	18	,	,	PUNCT
ejpam-7051	169	19	τ1	τ1	NOUN
ejpam-7051	169	20	,	,	PUNCT
ejpam-7051	169	21	τ2	τ2	NOUN
ejpam-7051	169	22	)	)	PUNCT
ejpam-7051	169	23	is	be	AUX
ejpam-7051	169	24	said	say	VERB
ejpam-7051	169	25	to	to	PART
ejpam-7051	169	26	be	be	AUX
ejpam-7051	169	27	(	(	PUNCT
ejpam-7051	169	28	τ1	τ1	NOUN
ejpam-7051	169	29	,	,	PUNCT
ejpam-7051	169	30	τ2)-regular	τ2)-regular	ADJ
ejpam-7051	169	31	[	[	X
ejpam-7051	169	32	14	14	NUM
ejpam-7051	169	33	]	]	X
ejpam-7051	169	34	if	if	SCONJ
ejpam-7051	169	35	for	for	ADP
ejpam-7051	169	36	each	each	DET
ejpam-7051	169	37	τ1τ2	τ1τ2	ADJ
ejpam-7051	169	38	-	-	ADJ
ejpam-7051	169	39	closed	closed	ADJ
ejpam-7051	169	40	set	set	VERB
ejpam-7051	169	41	f	f	NOUN
ejpam-7051	169	42	and	and	CCONJ
ejpam-7051	169	43	each	each	DET
ejpam-7051	169	44	x	x	PROPN
ejpam-7051	169	45	̸∈	̸∈	PROPN
ejpam-7051	169	46	f	f	PROPN
ejpam-7051	169	47	,	,	PUNCT
ejpam-7051	169	48	there	there	PRON
ejpam-7051	169	49	exist	exist	VERB
ejpam-7051	169	50	disjoint	disjoint	ADJ
ejpam-7051	169	51	τ1τ2	τ1τ2	ADJ
ejpam-7051	169	52	-	-	ADJ
ejpam-7051	169	53	open	open	ADJ
ejpam-7051	169	54	sets	set	NOUN
ejpam-7051	169	55	u	u	NOUN
ejpam-7051	169	56	and	and	CCONJ
ejpam-7051	169	57	v	v	ADP
ejpam-7051	169	58	such	such	ADJ
ejpam-7051	169	59	that	that	SCONJ
ejpam-7051	169	60	x	x	SYM
ejpam-7051	169	61	∈	∈	PROPN
ejpam-7051	169	62	u	u	NOUN
ejpam-7051	169	63	and	and	CCONJ
ejpam-7051	169	64	f	f	PROPN
ejpam-7051	169	65	⊆	⊆	NUM
ejpam-7051	169	66	v	v	NOUN
ejpam-7051	169	67	.	.	PUNCT
ejpam-7051	170	1	lemma	lemma	PROPN
ejpam-7051	170	2	5	5	NUM
ejpam-7051	170	3	.	.	PUNCT
ejpam-7051	171	1	[	[	X
ejpam-7051	171	2	14	14	NUM
ejpam-7051	171	3	]	]	PUNCT
ejpam-7051	171	4	a	a	DET
ejpam-7051	171	5	bitopological	bitopological	ADJ
ejpam-7051	171	6	space	space	NOUN
ejpam-7051	171	7	(	(	PUNCT
ejpam-7051	171	8	x	x	NOUN
ejpam-7051	171	9	,	,	PUNCT
ejpam-7051	171	10	τ1	τ1	NOUN
ejpam-7051	171	11	,	,	PUNCT
ejpam-7051	171	12	τ2	τ2	NOUN
ejpam-7051	171	13	)	)	PUNCT
ejpam-7051	171	14	is	be	AUX
ejpam-7051	171	15	(	(	PUNCT
ejpam-7051	171	16	τ1	τ1	NOUN
ejpam-7051	171	17	,	,	PUNCT
ejpam-7051	171	18	τ2)-regular	τ2)-regular	ADJ
ejpam-7051	171	19	if	if	SCONJ
ejpam-7051	171	20	and	and	CCONJ
ejpam-7051	171	21	only	only	ADV
ejpam-7051	171	22	if	if	SCONJ
ejpam-7051	171	23	for	for	ADP
ejpam-7051	171	24	each	each	DET
ejpam-7051	171	25	x	x	SYM
ejpam-7051	171	26	∈	∈	PROPN
ejpam-7051	171	27	x	x	X
ejpam-7051	171	28	and	and	CCONJ
ejpam-7051	171	29	each	each	DET
ejpam-7051	171	30	τ1τ2	τ1τ2	ADJ
ejpam-7051	171	31	-	-	ADJ
ejpam-7051	171	32	open	open	ADJ
ejpam-7051	171	33	set	set	NOUN
ejpam-7051	171	34	u	u	NOUN
ejpam-7051	171	35	containing	contain	VERB
ejpam-7051	171	36	x	x	PRON
ejpam-7051	171	37	,	,	PUNCT
ejpam-7051	171	38	there	there	PRON
ejpam-7051	171	39	exists	exist	VERB
ejpam-7051	171	40	a	a	DET
ejpam-7051	171	41	τ1τ2	τ1τ2	NOUN
ejpam-7051	171	42	-	-	ADJ
ejpam-7051	171	43	open	open	ADJ
ejpam-7051	171	44	set	set	VERB
ejpam-7051	171	45	v	v	ADP
ejpam-7051	171	46	such	such	ADJ
ejpam-7051	171	47	that	that	SCONJ
ejpam-7051	171	48	x	x	SYM
ejpam-7051	171	49	∈	∈	NOUN
ejpam-7051	171	50	v	v	ADP
ejpam-7051	171	51	⊆	⊆	NUM
ejpam-7051	171	52	τ1τ2	τ1τ2	NOUN
ejpam-7051	171	53	-	-	NOUN
ejpam-7051	171	54	cl(v	cl(v	X
ejpam-7051	171	55	)	)	PUNCT
ejpam-7051	171	56	⊆	⊆	NUM
ejpam-7051	171	57	u	u	NOUN
ejpam-7051	171	58	.	.	PUNCT
ejpam-7051	172	1	lemma	lemma	PROPN
ejpam-7051	172	2	6	6	NUM
ejpam-7051	172	3	.	.	PUNCT
ejpam-7051	173	1	[	[	X
ejpam-7051	173	2	6	6	NUM
ejpam-7051	173	3	]	]	PUNCT
ejpam-7051	173	4	let	let	VERB
ejpam-7051	173	5	(	(	PUNCT
ejpam-7051	173	6	x	x	NOUN
ejpam-7051	173	7	,	,	PUNCT
ejpam-7051	173	8	τ1	τ1	NOUN
ejpam-7051	173	9	,	,	PUNCT
ejpam-7051	173	10	τ2	τ2	PROPN
ejpam-7051	173	11	)	)	PUNCT
ejpam-7051	173	12	be	be	VERB
ejpam-7051	173	13	a	a	DET
ejpam-7051	173	14	(	(	PUNCT
ejpam-7051	173	15	τ1	τ1	NOUN
ejpam-7051	173	16	,	,	PUNCT
ejpam-7051	173	17	τ2)-regular	τ2)-regular	ADJ
ejpam-7051	173	18	space	space	NOUN
ejpam-7051	173	19	.	.	PUNCT
ejpam-7051	174	1	then	then	ADV
ejpam-7051	174	2	,	,	PUNCT
ejpam-7051	174	3	the	the	DET
ejpam-7051	174	4	following	follow	VERB
ejpam-7051	174	5	properties	property	NOUN
ejpam-7051	174	6	hold	hold	VERB
ejpam-7051	174	7	:	:	PUNCT
ejpam-7051	174	8	(	(	PUNCT
ejpam-7051	174	9	1	1	X
ejpam-7051	174	10	)	)	PUNCT
ejpam-7051	174	11	τ1τ2	τ1τ2	NOUN
ejpam-7051	174	12	-	-	NUM
ejpam-7051	174	13	cl(a	cl(a	NUM
ejpam-7051	174	14	)	)	PUNCT
ejpam-7051	174	15	=	=	PUNCT
ejpam-7051	174	16	(	(	PUNCT
ejpam-7051	174	17	τ1	τ1	NOUN
ejpam-7051	174	18	,	,	PUNCT
ejpam-7051	174	19	τ2)θ	τ2)θ	NOUN
ejpam-7051	174	20	-	-	PUNCT
ejpam-7051	174	21	cl(a	cl(a	NUM
ejpam-7051	174	22	)	)	PUNCT
ejpam-7051	174	23	for	for	ADP
ejpam-7051	174	24	every	every	DET
ejpam-7051	174	25	subset	subset	NOUN
ejpam-7051	174	26	a	a	PRON
ejpam-7051	174	27	of	of	ADP
ejpam-7051	174	28	x.	x.	NOUN
ejpam-7051	174	29	(	(	PUNCT
ejpam-7051	174	30	2	2	NUM
ejpam-7051	174	31	)	)	PUNCT
ejpam-7051	174	32	every	every	DET
ejpam-7051	174	33	τ1τ2	τ1τ2	NOUN
ejpam-7051	174	34	-	-	ADJ
ejpam-7051	174	35	open	open	ADJ
ejpam-7051	174	36	set	set	NOUN
ejpam-7051	174	37	is	be	AUX
ejpam-7051	174	38	(	(	PUNCT
ejpam-7051	174	39	τ1	τ1	NOUN
ejpam-7051	174	40	,	,	PUNCT
ejpam-7051	174	41	τ2)θ	τ2)θ	ADJ
ejpam-7051	174	42	-	-	PUNCT
ejpam-7051	174	43	open	open	ADJ
ejpam-7051	174	44	.	.	PUNCT
ejpam-7051	175	1	theorem	theorem	ADJ
ejpam-7051	175	2	5	5	NUM
ejpam-7051	175	3	.	.	X
ejpam-7051	175	4	for	for	ADP
ejpam-7051	175	5	a	a	DET
ejpam-7051	175	6	multifunction	multifunction	NOUN
ejpam-7051	176	1	f	f	NOUN
ejpam-7051	176	2	:	:	PUNCT
ejpam-7051	176	3	(	(	PUNCT
ejpam-7051	176	4	x,µ	x,µ	NOUN
ejpam-7051	176	5	)	)	PUNCT
ejpam-7051	176	6	→	→	SYM
ejpam-7051	176	7	(	(	PUNCT
ejpam-7051	176	8	y	y	PROPN
ejpam-7051	176	9	,	,	PUNCT
ejpam-7051	176	10	σ1	σ1	PROPN
ejpam-7051	176	11	,	,	PUNCT
ejpam-7051	176	12	σ2	σ2	NOUN
ejpam-7051	176	13	)	)	PUNCT
ejpam-7051	176	14	,	,	PUNCT
ejpam-7051	176	15	where	where	SCONJ
ejpam-7051	176	16	(	(	PUNCT
ejpam-7051	176	17	y	y	PROPN
ejpam-7051	176	18	,	,	PUNCT
ejpam-7051	176	19	σ1	σ1	PROPN
ejpam-7051	176	20	,	,	PUNCT
ejpam-7051	176	21	σ2	σ2	PROPN
ejpam-7051	176	22	)	)	PUNCT
ejpam-7051	176	23	is	be	AUX
ejpam-7051	176	24	(	(	PUNCT
ejpam-7051	176	25	σ1	σ1	PROPN
ejpam-7051	176	26	,	,	PUNCT
ejpam-7051	176	27	σ2)regular	σ2)regular	PROPN
ejpam-7051	176	28	,	,	PUNCT
ejpam-7051	176	29	the	the	DET
ejpam-7051	176	30	following	follow	VERB
ejpam-7051	176	31	properties	property	NOUN
ejpam-7051	176	32	are	be	AUX
ejpam-7051	176	33	equivalent	equivalent	ADJ
ejpam-7051	176	34	:	:	PUNCT
ejpam-7051	176	35	(	(	PUNCT
ejpam-7051	176	36	1	1	X
ejpam-7051	176	37	)	)	PUNCT
ejpam-7051	176	38	f	f	PROPN
ejpam-7051	176	39	is	be	AUX
ejpam-7051	176	40	upper	upper	ADJ
ejpam-7051	176	41	µ(σ1	µ(σ1	NOUN
ejpam-7051	176	42	,	,	PUNCT
ejpam-7051	176	43	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	176	44	;	;	PUNCT
ejpam-7051	176	45	(	(	PUNCT
ejpam-7051	176	46	2	2	X
ejpam-7051	176	47	)	)	PUNCT
ejpam-7051	176	48	f−((σ1	f−((σ1	NOUN
ejpam-7051	176	49	,	,	PUNCT
ejpam-7051	176	50	σ2)θ	σ2)θ	ADJ
ejpam-7051	176	51	-	-	PUNCT
ejpam-7051	176	52	cl(b	cl(b	NOUN
ejpam-7051	176	53	)	)	PUNCT
ejpam-7051	176	54	)	)	PUNCT
ejpam-7051	176	55	is	be	AUX
ejpam-7051	176	56	µ-closed	µ-close	VERB
ejpam-7051	176	57	in	in	ADP
ejpam-7051	176	58	x	x	PUNCT
ejpam-7051	176	59	for	for	ADP
ejpam-7051	176	60	every	every	DET
ejpam-7051	176	61	subset	subset	NOUN
ejpam-7051	176	62	b	b	PROPN
ejpam-7051	176	63	of	of	ADP
ejpam-7051	176	64	y	y	PROPN
ejpam-7051	176	65	;	;	PUNCT
ejpam-7051	176	66	(	(	PUNCT
ejpam-7051	176	67	3	3	X
ejpam-7051	176	68	)	)	PUNCT
ejpam-7051	176	69	f−(k	f−(k	PROPN
ejpam-7051	176	70	)	)	PUNCT
ejpam-7051	176	71	is	be	AUX
ejpam-7051	176	72	µ-closed	µ-close	VERB
ejpam-7051	176	73	in	in	ADP
ejpam-7051	176	74	x	x	PUNCT
ejpam-7051	176	75	for	for	ADP
ejpam-7051	176	76	every	every	DET
ejpam-7051	176	77	(	(	PUNCT
ejpam-7051	176	78	σ1	σ1	PROPN
ejpam-7051	176	79	,	,	PUNCT
ejpam-7051	176	80	σ2)θ	σ2)θ	NOUN
ejpam-7051	176	81	-	-	PUNCT
ejpam-7051	176	82	closed	close	VERB
ejpam-7051	176	83	set	set	NOUN
ejpam-7051	176	84	k	k	PROPN
ejpam-7051	176	85	of	of	ADP
ejpam-7051	176	86	y	y	PROPN
ejpam-7051	176	87	;	;	PUNCT
ejpam-7051	176	88	(	(	PUNCT
ejpam-7051	176	89	4	4	X
ejpam-7051	176	90	)	)	PUNCT
ejpam-7051	176	91	f+(v	f+(v	NOUN
ejpam-7051	176	92	)	)	PUNCT
ejpam-7051	177	1	is	be	AUX
ejpam-7051	177	2	µ-open	µ-open	NOUN
ejpam-7051	177	3	in	in	ADP
ejpam-7051	177	4	x	x	PUNCT
ejpam-7051	177	5	for	for	ADP
ejpam-7051	177	6	every	every	DET
ejpam-7051	177	7	(	(	PUNCT
ejpam-7051	177	8	σ1	σ1	PROPN
ejpam-7051	177	9	,	,	PUNCT
ejpam-7051	177	10	σ2)θ	σ2)θ	NOUN
ejpam-7051	177	11	-	-	PUNCT
ejpam-7051	177	12	open	open	ADJ
ejpam-7051	177	13	set	set	NOUN
ejpam-7051	177	14	v	v	NOUN
ejpam-7051	177	15	of	of	ADP
ejpam-7051	177	16	y	y	PROPN
ejpam-7051	177	17	.	.	PUNCT
ejpam-7051	178	1	n.	n.	PROPN
ejpam-7051	178	2	chutiman	chutiman	PROPN
ejpam-7051	178	3	,	,	PUNCT
ejpam-7051	178	4	a.	a.	PROPN
ejpam-7051	178	5	sama	sama	PROPN
ejpam-7051	178	6	-	-	PUNCT
ejpam-7051	178	7	ae	ae	PROPN
ejpam-7051	178	8	,	,	PUNCT
ejpam-7051	178	9	c.	c.	PROPN
ejpam-7051	178	10	boonpok	boonpok	PROPN
ejpam-7051	178	11	/	/	SYM
ejpam-7051	178	12	eur	eur	PROPN
ejpam-7051	178	13	.	.	PUNCT
ejpam-7051	179	1	j.	j.	PROPN
ejpam-7051	179	2	pure	pure	PROPN
ejpam-7051	179	3	appl	appl	PROPN
ejpam-7051	179	4	.	.	PROPN
ejpam-7051	179	5	math	math	PROPN
ejpam-7051	179	6	,	,	PUNCT
ejpam-7051	179	7	18	18	NUM
ejpam-7051	179	8	(	(	PUNCT
ejpam-7051	179	9	4	4	NUM
ejpam-7051	179	10	)	)	PUNCT
ejpam-7051	179	11	(	(	PUNCT
ejpam-7051	179	12	2025	2025	NUM
ejpam-7051	179	13	)	)	PUNCT
ejpam-7051	179	14	,	,	PUNCT
ejpam-7051	179	15	7051	7051	NUM
ejpam-7051	179	16	7	7	NUM
ejpam-7051	179	17	of	of	ADP
ejpam-7051	179	18	8	8	NUM
ejpam-7051	179	19	proof	proof	NOUN
ejpam-7051	179	20	.	.	PUNCT
ejpam-7051	180	1	(	(	PUNCT
ejpam-7051	180	2	1	1	X
ejpam-7051	180	3	)	)	PUNCT
ejpam-7051	180	4	⇒	⇒	NOUN
ejpam-7051	180	5	(	(	PUNCT
ejpam-7051	180	6	2	2	NUM
ejpam-7051	180	7	):	):	PUNCT
ejpam-7051	180	8	let	let	VERB
ejpam-7051	180	9	b	b	X
ejpam-7051	180	10	be	be	AUX
ejpam-7051	180	11	any	any	DET
ejpam-7051	180	12	subset	subset	NOUN
ejpam-7051	180	13	of	of	ADP
ejpam-7051	180	14	y	y	PROPN
ejpam-7051	180	15	.	.	PUNCT
ejpam-7051	181	1	by	by	ADP
ejpam-7051	181	2	lemma	lemma	PROPN
ejpam-7051	181	3	6	6	NUM
ejpam-7051	181	4	,	,	PUNCT
ejpam-7051	181	5	(	(	PUNCT
ejpam-7051	181	6	σ1	σ1	PROPN
ejpam-7051	181	7	,	,	PUNCT
ejpam-7051	181	8	σ2)θ	σ2)θ	NOUN
ejpam-7051	181	9	-	-	PUNCT
ejpam-7051	181	10	cl(b	cl(b	NOUN
ejpam-7051	181	11	)	)	PUNCT
ejpam-7051	181	12	is	be	AUX
ejpam-7051	181	13	σ1σ2closed	σ1σ2close	VERB
ejpam-7051	181	14	in	in	ADP
ejpam-7051	181	15	y	y	PROPN
ejpam-7051	181	16	.	.	PUNCT
ejpam-7051	182	1	since	since	SCONJ
ejpam-7051	182	2	f	f	PROPN
ejpam-7051	182	3	is	be	AUX
ejpam-7051	182	4	upper	upper	ADJ
ejpam-7051	182	5	µ(σ1	µ(σ1	NOUN
ejpam-7051	182	6	,	,	PUNCT
ejpam-7051	182	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	182	8	,	,	PUNCT
ejpam-7051	182	9	by	by	ADP
ejpam-7051	182	10	theorem	theorem	NOUN
ejpam-7051	182	11	1	1	NUM
ejpam-7051	182	12	f−((σ1	f−((σ1	NOUN
ejpam-7051	182	13	,	,	PUNCT
ejpam-7051	182	14	σ2)θ	σ2)θ	ADJ
ejpam-7051	182	15	-	-	PUNCT
ejpam-7051	182	16	cl(b	cl(b	NOUN
ejpam-7051	182	17	)	)	PUNCT
ejpam-7051	182	18	)	)	PUNCT
ejpam-7051	182	19	is	be	AUX
ejpam-7051	182	20	µ-closed	µ-close	VERB
ejpam-7051	182	21	in	in	ADP
ejpam-7051	182	22	x.	x.	NOUN
ejpam-7051	182	23	(	(	PUNCT
ejpam-7051	182	24	2	2	NUM
ejpam-7051	182	25	)	)	PUNCT
ejpam-7051	182	26	⇒	⇒	NOUN
ejpam-7051	182	27	(	(	PUNCT
ejpam-7051	182	28	3	3	NUM
ejpam-7051	182	29	):	):	PUNCT
ejpam-7051	182	30	let	let	VERB
ejpam-7051	182	31	k	k	PRON
ejpam-7051	182	32	be	be	AUX
ejpam-7051	182	33	any	any	DET
ejpam-7051	182	34	(	(	PUNCT
ejpam-7051	182	35	σ1	σ1	PROPN
ejpam-7051	182	36	,	,	PUNCT
ejpam-7051	182	37	σ2)θ	σ2)θ	NOUN
ejpam-7051	182	38	-	-	PUNCT
ejpam-7051	182	39	closed	close	VERB
ejpam-7051	182	40	set	set	NOUN
ejpam-7051	182	41	of	of	ADP
ejpam-7051	182	42	y	y	PROPN
ejpam-7051	182	43	.	.	PUNCT
ejpam-7051	183	1	then	then	ADV
ejpam-7051	183	2	,	,	PUNCT
ejpam-7051	183	3	(	(	PUNCT
ejpam-7051	183	4	σ1	σ1	PROPN
ejpam-7051	183	5	,	,	PUNCT
ejpam-7051	183	6	σ2)θ	σ2)θ	NOUN
ejpam-7051	183	7	-	-	PUNCT
ejpam-7051	183	8	cl(k	cl(k	NOUN
ejpam-7051	183	9	)	)	PUNCT
ejpam-7051	184	1	=	=	SYM
ejpam-7051	185	1	k	k	PROPN
ejpam-7051	185	2	and	and	CCONJ
ejpam-7051	185	3	by	by	ADP
ejpam-7051	185	4	(	(	PUNCT
ejpam-7051	185	5	2	2	NUM
ejpam-7051	185	6	)	)	PUNCT
ejpam-7051	185	7	,	,	PUNCT
ejpam-7051	185	8	we	we	PRON
ejpam-7051	185	9	have	have	VERB
ejpam-7051	185	10	f−(k	f−(k	PROPN
ejpam-7051	185	11	)	)	PUNCT
ejpam-7051	185	12	is	be	AUX
ejpam-7051	185	13	µ-closed	µ-close	VERB
ejpam-7051	185	14	in	in	ADP
ejpam-7051	185	15	x.	x.	NOUN
ejpam-7051	185	16	(	(	PUNCT
ejpam-7051	185	17	3	3	NUM
ejpam-7051	185	18	)	)	PUNCT
ejpam-7051	185	19	⇒	⇒	NOUN
ejpam-7051	185	20	(	(	PUNCT
ejpam-7051	185	21	4	4	NUM
ejpam-7051	185	22	):	):	PUNCT
ejpam-7051	185	23	this	this	PRON
ejpam-7051	185	24	follows	follow	VERB
ejpam-7051	185	25	from	from	ADP
ejpam-7051	185	26	the	the	DET
ejpam-7051	185	27	fact	fact	NOUN
ejpam-7051	185	28	that	that	SCONJ
ejpam-7051	185	29	f+(y	f+(y	PROPN
ejpam-7051	185	30	−b	−b	ADV
ejpam-7051	185	31	)	)	PUNCT
ejpam-7051	185	32	=	=	PUNCT
ejpam-7051	185	33	x	x	SYM
ejpam-7051	185	34	−f−(b	−f−(b	PROPN
ejpam-7051	185	35	)	)	PUNCT
ejpam-7051	185	36	for	for	ADP
ejpam-7051	185	37	any	any	DET
ejpam-7051	185	38	subset	subset	NOUN
ejpam-7051	185	39	b	b	PROPN
ejpam-7051	185	40	of	of	ADP
ejpam-7051	185	41	y	y	PROPN
ejpam-7051	185	42	.	.	PUNCT
ejpam-7051	186	1	(	(	PUNCT
ejpam-7051	186	2	4	4	X
ejpam-7051	186	3	)	)	PUNCT
ejpam-7051	186	4	⇒	⇒	NOUN
ejpam-7051	186	5	(	(	PUNCT
ejpam-7051	186	6	1	1	NUM
ejpam-7051	186	7	):	):	PUNCT
ejpam-7051	186	8	let	let	VERB
ejpam-7051	186	9	v	v	PART
ejpam-7051	186	10	be	be	AUX
ejpam-7051	186	11	any	any	DET
ejpam-7051	186	12	σ1σ2	σ1σ2	NOUN
ejpam-7051	186	13	-	-	ADJ
ejpam-7051	186	14	open	open	ADJ
ejpam-7051	186	15	set	set	NOUN
ejpam-7051	186	16	of	of	ADP
ejpam-7051	186	17	y	y	PROPN
ejpam-7051	186	18	.	.	PUNCT
ejpam-7051	187	1	since	since	SCONJ
ejpam-7051	187	2	(	(	PUNCT
ejpam-7051	187	3	y	y	PROPN
ejpam-7051	187	4	,	,	PUNCT
ejpam-7051	187	5	σ1	σ1	PROPN
ejpam-7051	187	6	,	,	PUNCT
ejpam-7051	187	7	σ2	σ2	PROPN
ejpam-7051	187	8	)	)	PUNCT
ejpam-7051	187	9	is	be	AUX
ejpam-7051	187	10	(	(	PUNCT
ejpam-7051	187	11	σ1	σ1	NOUN
ejpam-7051	187	12	,	,	PUNCT
ejpam-7051	187	13	σ2)-regular	σ2)-regular	ADJ
ejpam-7051	187	14	,	,	PUNCT
ejpam-7051	187	15	we	we	PRON
ejpam-7051	187	16	have	have	VERB
ejpam-7051	187	17	v	v	NOUN
ejpam-7051	187	18	is	be	AUX
ejpam-7051	187	19	(	(	PUNCT
ejpam-7051	187	20	σ1	σ1	PROPN
ejpam-7051	187	21	,	,	PUNCT
ejpam-7051	187	22	σ2)θ	σ2)θ	NOUN
ejpam-7051	187	23	-	-	PUNCT
ejpam-7051	187	24	open	open	ADJ
ejpam-7051	187	25	in	in	ADP
ejpam-7051	187	26	y	y	PROPN
ejpam-7051	187	27	and	and	CCONJ
ejpam-7051	187	28	by	by	ADP
ejpam-7051	187	29	(	(	PUNCT
ejpam-7051	187	30	4	4	NUM
ejpam-7051	187	31	)	)	PUNCT
ejpam-7051	187	32	,	,	PUNCT
ejpam-7051	187	33	f+(v	f+(v	PROPN
ejpam-7051	187	34	)	)	PUNCT
ejpam-7051	187	35	is	be	AUX
ejpam-7051	187	36	µ-open	µ-open	NOUN
ejpam-7051	187	37	in	in	ADP
ejpam-7051	187	38	x.	x.	NOUN
ejpam-7051	187	39	thus	thus	ADV
ejpam-7051	187	40	,	,	PUNCT
ejpam-7051	187	41	f	f	PROPN
ejpam-7051	187	42	is	be	AUX
ejpam-7051	187	43	upper	upper	ADJ
ejpam-7051	187	44	µ(σ1	µ(σ1	NOUN
ejpam-7051	187	45	,	,	PUNCT
ejpam-7051	187	46	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	187	47	by	by	ADP
ejpam-7051	187	48	theorem	theorem	NOUN
ejpam-7051	187	49	1	1	NUM
ejpam-7051	187	50	.	.	PUNCT
ejpam-7051	187	51	theorem	theorem	VERB
ejpam-7051	187	52	6	6	NUM
ejpam-7051	187	53	.	.	PUNCT
ejpam-7051	187	54	for	for	ADP
ejpam-7051	187	55	a	a	DET
ejpam-7051	187	56	multifunction	multifunction	NOUN
ejpam-7051	188	1	f	f	NOUN
ejpam-7051	188	2	:	:	PUNCT
ejpam-7051	188	3	(	(	PUNCT
ejpam-7051	188	4	x,µ	x,µ	NOUN
ejpam-7051	188	5	)	)	PUNCT
ejpam-7051	188	6	→	→	SYM
ejpam-7051	188	7	(	(	PUNCT
ejpam-7051	188	8	y	y	PROPN
ejpam-7051	188	9	,	,	PUNCT
ejpam-7051	188	10	σ1	σ1	PROPN
ejpam-7051	188	11	,	,	PUNCT
ejpam-7051	188	12	σ2	σ2	NOUN
ejpam-7051	188	13	)	)	PUNCT
ejpam-7051	188	14	,	,	PUNCT
ejpam-7051	188	15	where	where	SCONJ
ejpam-7051	188	16	(	(	PUNCT
ejpam-7051	188	17	y	y	PROPN
ejpam-7051	188	18	,	,	PUNCT
ejpam-7051	188	19	σ1	σ1	PROPN
ejpam-7051	188	20	,	,	PUNCT
ejpam-7051	188	21	σ2	σ2	PROPN
ejpam-7051	188	22	)	)	PUNCT
ejpam-7051	188	23	is	be	AUX
ejpam-7051	188	24	(	(	PUNCT
ejpam-7051	188	25	σ1	σ1	PROPN
ejpam-7051	188	26	,	,	PUNCT
ejpam-7051	188	27	σ2)regular	σ2)regular	PROPN
ejpam-7051	188	28	,	,	PUNCT
ejpam-7051	188	29	the	the	DET
ejpam-7051	188	30	following	follow	VERB
ejpam-7051	188	31	properties	property	NOUN
ejpam-7051	188	32	are	be	AUX
ejpam-7051	188	33	equivalent	equivalent	ADJ
ejpam-7051	188	34	:	:	PUNCT
ejpam-7051	188	35	(	(	PUNCT
ejpam-7051	188	36	1	1	X
ejpam-7051	188	37	)	)	PUNCT
ejpam-7051	188	38	f	f	PROPN
ejpam-7051	188	39	is	be	AUX
ejpam-7051	188	40	lower	low	ADJ
ejpam-7051	188	41	µ(σ1	µ(σ1	NOUN
ejpam-7051	188	42	,	,	PUNCT
ejpam-7051	188	43	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7051	188	44	;	;	PUNCT
ejpam-7051	188	45	(	(	PUNCT
ejpam-7051	188	46	2	2	X
ejpam-7051	188	47	)	)	PUNCT
ejpam-7051	188	48	f+((σ1	f+((σ1	NOUN
ejpam-7051	188	49	,	,	PUNCT
ejpam-7051	188	50	σ2)θ	σ2)θ	ADJ
ejpam-7051	188	51	-	-	PUNCT
ejpam-7051	188	52	cl(b	cl(b	NOUN
ejpam-7051	188	53	)	)	PUNCT
ejpam-7051	188	54	)	)	PUNCT
ejpam-7051	188	55	is	be	AUX
ejpam-7051	188	56	µ-closed	µ-close	VERB
ejpam-7051	188	57	in	in	ADP
ejpam-7051	188	58	x	x	PUNCT
ejpam-7051	188	59	for	for	ADP
ejpam-7051	188	60	every	every	DET
ejpam-7051	188	61	subset	subset	NOUN
ejpam-7051	188	62	b	b	PROPN
ejpam-7051	188	63	of	of	ADP
ejpam-7051	188	64	y	y	PROPN
ejpam-7051	188	65	;	;	PUNCT
ejpam-7051	188	66	(	(	PUNCT
ejpam-7051	188	67	3	3	X
ejpam-7051	188	68	)	)	PUNCT
ejpam-7051	188	69	f+(k	f+(k	NUM
ejpam-7051	188	70	)	)	PUNCT
ejpam-7051	188	71	is	be	AUX
ejpam-7051	188	72	µ-closed	µ-close	VERB
ejpam-7051	188	73	in	in	ADP
ejpam-7051	188	74	x	x	PUNCT
ejpam-7051	188	75	for	for	SCONJ
ejpam-7051	188	76	every	every	DET
ejpam-7051	188	77	(	(	PUNCT
ejpam-7051	188	78	σ1	σ1	PROPN
ejpam-7051	188	79	,	,	PUNCT
ejpam-7051	188	80	σ2)θ	σ2)θ	NOUN
ejpam-7051	188	81	-	-	PUNCT
ejpam-7051	188	82	closed	close	VERB
ejpam-7051	188	83	set	set	NOUN
ejpam-7051	188	84	k	k	PROPN
ejpam-7051	188	85	of	of	ADP
ejpam-7051	188	86	y	y	PROPN
ejpam-7051	188	87	;	;	PUNCT
ejpam-7051	188	88	(	(	PUNCT
ejpam-7051	188	89	4	4	X
ejpam-7051	188	90	)	)	PUNCT
ejpam-7051	188	91	f−(v	f−(v	NOUN
ejpam-7051	188	92	)	)	PUNCT
ejpam-7051	188	93	is	be	AUX
ejpam-7051	188	94	µ-open	µ-open	NOUN
ejpam-7051	188	95	in	in	ADP
ejpam-7051	188	96	x	x	PUNCT
ejpam-7051	188	97	for	for	ADP
ejpam-7051	188	98	every	every	DET
ejpam-7051	188	99	(	(	PUNCT
ejpam-7051	188	100	σ1	σ1	PROPN
ejpam-7051	188	101	,	,	PUNCT
ejpam-7051	188	102	σ2)θ	σ2)θ	NOUN
ejpam-7051	188	103	-	-	PUNCT
ejpam-7051	188	104	open	open	ADJ
ejpam-7051	188	105	set	set	NOUN
ejpam-7051	188	106	v	v	NOUN
ejpam-7051	188	107	of	of	ADP
ejpam-7051	188	108	y	y	PROPN
ejpam-7051	188	109	.	.	PUNCT
ejpam-7051	189	1	proof	proof	NOUN
ejpam-7051	189	2	.	.	PUNCT
ejpam-7051	190	1	the	the	DET
ejpam-7051	190	2	proof	proof	NOUN
ejpam-7051	190	3	is	be	AUX
ejpam-7051	190	4	similar	similar	ADJ
ejpam-7051	190	5	to	to	ADP
ejpam-7051	190	6	that	that	PRON
ejpam-7051	190	7	of	of	ADP
ejpam-7051	190	8	theorem	theorem	NOUN
ejpam-7051	190	9	5	5	NUM
ejpam-7051	190	10	.	.	PUNCT
ejpam-7051	190	11	acknowledgements	acknowledgement	NOUN
ejpam-7051	190	12	this	this	DET
ejpam-7051	190	13	research	research	NOUN
ejpam-7051	190	14	project	project	NOUN
ejpam-7051	190	15	was	be	AUX
ejpam-7051	190	16	financially	financially	ADV
ejpam-7051	190	17	supported	support	VERB
ejpam-7051	190	18	by	by	ADP
ejpam-7051	190	19	mahasarakham	mahasarakham	PROPN
ejpam-7051	190	20	university	university	PROPN
ejpam-7051	190	21	.	.	PUNCT
ejpam-7051	191	1	references	reference	NOUN
ejpam-7051	191	2	[	[	X
ejpam-7051	191	3	1	1	NUM
ejpam-7051	191	4	]	]	SYM
ejpam-7051	191	5	á	á	PROPN
ejpam-7051	191	6	.	.	PUNCT
ejpam-7051	191	7	császár	császár	PROPN
ejpam-7051	191	8	.	.	PUNCT
ejpam-7051	192	1	generalized	generalize	VERB
ejpam-7051	192	2	topology	topology	NOUN
ejpam-7051	192	3	,	,	PUNCT
ejpam-7051	192	4	generalized	generalize	VERB
ejpam-7051	192	5	continuity	continuity	NOUN
ejpam-7051	192	6	.	.	PUNCT
ejpam-7051	193	1	acta	acta	PROPN
ejpam-7051	193	2	mathematica	mathematica	PROPN
ejpam-7051	193	3	hungarica	hungarica	PROPN
ejpam-7051	193	4	,	,	PUNCT
ejpam-7051	193	5	96(4):351–357	96(4):351–357	NOUN
ejpam-7051	193	6	,	,	PUNCT
ejpam-7051	193	7	2002	2002	NUM
ejpam-7051	193	8	.	.	PUNCT
ejpam-7051	194	1	[	[	X
ejpam-7051	194	2	2	2	NUM
ejpam-7051	194	3	]	]	PUNCT
ejpam-7051	194	4	a.	a.	NOUN
ejpam-7051	194	5	kanibir	kanibir	NOUN
ejpam-7051	194	6	and	and	CCONJ
ejpam-7051	194	7	i.	i.	PROPN
ejpam-7051	194	8	l.	l.	PROPN
ejpam-7051	194	9	reilly	reilly	PROPN
ejpam-7051	194	10	.	.	PUNCT
ejpam-7051	195	1	generalized	generalize	VERB
ejpam-7051	195	2	continuity	continuity	NOUN
ejpam-7051	195	3	for	for	ADP
ejpam-7051	195	4	multifunctions	multifunction	NOUN
ejpam-7051	195	5	.	.	PUNCT
ejpam-7051	196	1	acta	acta	PROPN
ejpam-7051	196	2	mathematica	mathematica	PROPN
ejpam-7051	196	3	hungarica	hungarica	PROPN
ejpam-7051	196	4	,	,	PUNCT
ejpam-7051	196	5	122(3):283–292	122(3):283–292	NUM
ejpam-7051	196	6	,	,	PUNCT
ejpam-7051	196	7	2009	2009	NUM
ejpam-7051	196	8	.	.	PUNCT
ejpam-7051	197	1	[	[	X
ejpam-7051	197	2	3	3	X
ejpam-7051	197	3	]	]	PUNCT
ejpam-7051	197	4	c.	c.	PROPN
ejpam-7051	197	5	boonpok	boonpok	PROPN
ejpam-7051	197	6	.	.	PUNCT
ejpam-7051	198	1	on	on	ADP
ejpam-7051	198	2	upper	upper	ADJ
ejpam-7051	198	3	and	and	CCONJ
ejpam-7051	198	4	lower	low	ADJ
ejpam-7051	198	5	β(µx	β(µx	PROPN
ejpam-7051	198	6	,	,	PUNCT
ejpam-7051	198	7	µy	µy	CCONJ
ejpam-7051	198	8	)	)	PUNCT
ejpam-7051	198	9	-continuous	-continuous	ADJ
ejpam-7051	198	10	multifunctions	multifunction	NOUN
ejpam-7051	198	11	.	.	PUNCT
ejpam-7051	199	1	international	international	ADJ
ejpam-7051	199	2	journal	journal	PROPN
ejpam-7051	199	3	of	of	ADP
ejpam-7051	199	4	mathematics	mathematics	PROPN
ejpam-7051	199	5	and	and	CCONJ
ejpam-7051	199	6	mathematical	mathematical	ADJ
ejpam-7051	199	7	sciences	science	NOUN
ejpam-7051	199	8	,	,	PUNCT
ejpam-7051	199	9	2012:931656	2012:931656	NUM
ejpam-7051	199	10	,	,	PUNCT
ejpam-7051	199	11	2012	2012	NUM
ejpam-7051	199	12	.	.	PUNCT
ejpam-7051	200	1	[	[	X
ejpam-7051	200	2	4	4	X
ejpam-7051	200	3	]	]	X
ejpam-7051	200	4	n.	n.	NOUN
ejpam-7051	200	5	srisarakham	srisarakham	PROPN
ejpam-7051	200	6	and	and	CCONJ
ejpam-7051	200	7	c.	c.	PROPN
ejpam-7051	200	8	boonpok	boonpok	PROPN
ejpam-7051	200	9	.	.	PUNCT
ejpam-7051	201	1	characterizations	characterization	NOUN
ejpam-7051	201	2	of	of	ADP
ejpam-7051	201	3	upper	upper	ADJ
ejpam-7051	201	4	and	and	CCONJ
ejpam-7051	201	5	lower	low	ADJ
ejpam-7051	201	6	α(µx	α(µx	NUM
ejpam-7051	201	7	,	,	PUNCT
ejpam-7051	201	8	µy	µy	CCONJ
ejpam-7051	201	9	)	)	PUNCT
ejpam-7051	201	10	continuous	continuous	ADJ
ejpam-7051	201	11	multifunctions	multifunction	NOUN
ejpam-7051	201	12	.	.	PUNCT
ejpam-7051	202	1	journal	journal	PROPN
ejpam-7051	202	2	of	of	ADP
ejpam-7051	202	3	mathematics	mathematics	PROPN
ejpam-7051	202	4	and	and	CCONJ
ejpam-7051	202	5	computer	computer	NOUN
ejpam-7051	202	6	science	science	NOUN
ejpam-7051	202	7	,	,	PUNCT
ejpam-7051	202	8	17:255	17:255	NUM
ejpam-7051	202	9	–	–	PUNCT
ejpam-7051	202	10	265	265	NUM
ejpam-7051	202	11	,	,	PUNCT
ejpam-7051	202	12	2017	2017	NUM
ejpam-7051	202	13	.	.	PUNCT
ejpam-7051	203	1	[	[	X
ejpam-7051	203	2	5	5	NUM
ejpam-7051	203	3	]	]	PUNCT
ejpam-7051	203	4	p.	p.	NOUN
ejpam-7051	203	5	pue	pue	NOUN
ejpam-7051	203	6	-	-	PUNCT
ejpam-7051	203	7	on	on	ADP
ejpam-7051	203	8	,	,	PUNCT
ejpam-7051	203	9	s.	s.	PROPN
ejpam-7051	203	10	sompong	sompong	PROPN
ejpam-7051	203	11	,	,	PUNCT
ejpam-7051	203	12	and	and	CCONJ
ejpam-7051	203	13	c.	c.	PROPN
ejpam-7051	203	14	boonpok	boonpok	PROPN
ejpam-7051	203	15	.	.	PUNCT
ejpam-7051	204	1	upper	upper	ADJ
ejpam-7051	204	2	and	and	CCONJ
ejpam-7051	204	3	lower	low	ADJ
ejpam-7051	204	4	(	(	PUNCT
ejpam-7051	204	5	τ1	τ1	NOUN
ejpam-7051	204	6	,	,	PUNCT
ejpam-7051	204	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	204	8	multifunctions	multifunction	NOUN
ejpam-7051	204	9	.	.	PUNCT
ejpam-7051	205	1	international	international	ADJ
ejpam-7051	205	2	journal	journal	PROPN
ejpam-7051	205	3	of	of	ADP
ejpam-7051	205	4	mathematics	mathematic	NOUN
ejpam-7051	205	5	and	and	CCONJ
ejpam-7051	205	6	computer	computer	NOUN
ejpam-7051	205	7	science	science	NOUN
ejpam-7051	205	8	,	,	PUNCT
ejpam-7051	205	9	19(4):1305	19(4):1305	NUM
ejpam-7051	205	10	–	–	PUNCT
ejpam-7051	205	11	1310	1310	NUM
ejpam-7051	205	12	,	,	PUNCT
ejpam-7051	205	13	2024	2024	NUM
ejpam-7051	205	14	.	.	PUNCT
ejpam-7051	206	1	[	[	X
ejpam-7051	206	2	6	6	NUM
ejpam-7051	206	3	]	]	X
ejpam-7051	206	4	c.	c.	PROPN
ejpam-7051	206	5	klanarong	klanarong	PROPN
ejpam-7051	206	6	,	,	PUNCT
ejpam-7051	206	7	s.	s.	PROPN
ejpam-7051	206	8	sompong	sompong	PROPN
ejpam-7051	206	9	,	,	PUNCT
ejpam-7051	206	10	and	and	CCONJ
ejpam-7051	206	11	c.	c.	PROPN
ejpam-7051	206	12	boonpok	boonpok	PROPN
ejpam-7051	206	13	.	.	PUNCT
ejpam-7051	207	1	(	(	PUNCT
ejpam-7051	207	2	τ1	τ1	NOUN
ejpam-7051	207	3	,	,	PUNCT
ejpam-7051	207	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7051	207	5	and	and	CCONJ
ejpam-7051	207	6	(	(	PUNCT
ejpam-7051	207	7	τ1	τ1	NOUN
ejpam-7051	207	8	,	,	PUNCT
ejpam-7051	207	9	τ2)θ	τ2)θ	ADJ
ejpam-7051	207	10	-	-	PUNCT
ejpam-7051	207	11	closed	close	VERB
ejpam-7051	207	12	sets	set	NOUN
ejpam-7051	207	13	.	.	PUNCT
ejpam-7051	208	1	international	international	ADJ
ejpam-7051	208	2	journal	journal	NOUN
ejpam-7051	208	3	of	of	ADP
ejpam-7051	208	4	mathematics	mathematic	NOUN
ejpam-7051	208	5	and	and	CCONJ
ejpam-7051	208	6	computer	computer	NOUN
ejpam-7051	208	7	science	science	NOUN
ejpam-7051	208	8	,	,	PUNCT
ejpam-7051	208	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-7051	208	10	,	,	PUNCT
ejpam-7051	208	11	2024	2024	NUM
ejpam-7051	208	12	.	.	PUNCT
ejpam-7051	209	1	n.	n.	NOUN
ejpam-7051	209	2	chutiman	chutiman	PROPN
ejpam-7051	209	3	,	,	PUNCT
ejpam-7051	209	4	a.	a.	PROPN
ejpam-7051	209	5	sama	sama	PROPN
ejpam-7051	209	6	-	-	PUNCT
ejpam-7051	209	7	ae	ae	PROPN
ejpam-7051	209	8	,	,	PUNCT
ejpam-7051	209	9	c.	c.	PROPN
ejpam-7051	209	10	boonpok	boonpok	PROPN
ejpam-7051	209	11	/	/	SYM
ejpam-7051	209	12	eur	eur	PROPN
ejpam-7051	209	13	.	.	PUNCT
ejpam-7051	210	1	j.	j.	PROPN
ejpam-7051	210	2	pure	pure	PROPN
ejpam-7051	210	3	appl	appl	PROPN
ejpam-7051	210	4	.	.	PROPN
ejpam-7051	210	5	math	math	PROPN
ejpam-7051	210	6	,	,	PUNCT
ejpam-7051	210	7	18	18	NUM
ejpam-7051	210	8	(	(	PUNCT
ejpam-7051	210	9	4	4	NUM
ejpam-7051	210	10	)	)	PUNCT
ejpam-7051	210	11	(	(	PUNCT
ejpam-7051	210	12	2025	2025	NUM
ejpam-7051	210	13	)	)	PUNCT
ejpam-7051	210	14	,	,	PUNCT
ejpam-7051	210	15	7051	7051	NUM
ejpam-7051	210	16	8	8	NUM
ejpam-7051	210	17	of	of	ADP
ejpam-7051	210	18	8	8	NUM
ejpam-7051	210	19	[	[	X
ejpam-7051	210	20	7	7	NUM
ejpam-7051	210	21	]	]	PUNCT
ejpam-7051	210	22	m.	m.	NOUN
ejpam-7051	210	23	thongmoon	thongmoon	NOUN
ejpam-7051	210	24	,	,	PUNCT
ejpam-7051	210	25	s.	s.	PROPN
ejpam-7051	210	26	sompong	sompong	PROPN
ejpam-7051	210	27	,	,	PUNCT
ejpam-7051	210	28	and	and	CCONJ
ejpam-7051	210	29	c.	c.	PROPN
ejpam-7051	210	30	boonpok	boonpok	PROPN
ejpam-7051	210	31	.	.	PUNCT
ejpam-7051	211	1	(	(	PUNCT
ejpam-7051	211	2	τ1	τ1	NOUN
ejpam-7051	211	3	,	,	PUNCT
ejpam-7051	211	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	211	5	multifunctions	multifunction	NOUN
ejpam-7051	211	6	and	and	CCONJ
ejpam-7051	211	7	τ1τ2	τ1τ2	NOUN
ejpam-7051	211	8	-	-	ADJ
ejpam-7051	211	9	δ	δ	NOUN
ejpam-7051	211	10	-	-	ADJ
ejpam-7051	211	11	open	open	ADJ
ejpam-7051	211	12	sets	set	NOUN
ejpam-7051	211	13	.	.	PUNCT
ejpam-7051	212	1	international	international	ADJ
ejpam-7051	212	2	journal	journal	NOUN
ejpam-7051	212	3	of	of	ADP
ejpam-7051	212	4	mathematics	mathematic	NOUN
ejpam-7051	212	5	and	and	CCONJ
ejpam-7051	212	6	computer	computer	NOUN
ejpam-7051	212	7	science	science	NOUN
ejpam-7051	212	8	,	,	PUNCT
ejpam-7051	212	9	19(4):1369–1375	19(4):1369–1375	NUM
ejpam-7051	212	10	,	,	PUNCT
ejpam-7051	212	11	2024	2024	NUM
ejpam-7051	212	12	.	.	PUNCT
ejpam-7051	213	1	[	[	X
ejpam-7051	213	2	8	8	X
ejpam-7051	213	3	]	]	X
ejpam-7051	213	4	j.	j.	PROPN
ejpam-7051	213	5	khampakdee	khampakdee	PROPN
ejpam-7051	213	6	,	,	PUNCT
ejpam-7051	213	7	a.	a.	PROPN
ejpam-7051	213	8	sama	sama	PROPN
ejpam-7051	213	9	-	-	PUNCT
ejpam-7051	213	10	ae	ae	PROPN
ejpam-7051	213	11	,	,	PUNCT
ejpam-7051	213	12	and	and	CCONJ
ejpam-7051	213	13	c.	c.	PROPN
ejpam-7051	213	14	boonpok	boonpok	PROPN
ejpam-7051	213	15	.	.	PUNCT
ejpam-7051	214	1	upper	upper	ADJ
ejpam-7051	214	2	and	and	CCONJ
ejpam-7051	214	3	lower	low	ADJ
ejpam-7051	214	4	continuous	continuous	ADJ
ejpam-7051	214	5	multifunctions	multifunction	NOUN
ejpam-7051	214	6	defined	define	VERB
ejpam-7051	214	7	between	between	ADP
ejpam-7051	214	8	an	an	DET
ejpam-7051	214	9	ideal	ideal	ADJ
ejpam-7051	214	10	topological	topological	ADJ
ejpam-7051	214	11	space	space	NOUN
ejpam-7051	214	12	and	and	CCONJ
ejpam-7051	214	13	a	a	DET
ejpam-7051	214	14	bitopological	bitopological	ADJ
ejpam-7051	214	15	space	space	NOUN
ejpam-7051	214	16	.	.	PUNCT
ejpam-7051	215	1	european	european	ADJ
ejpam-7051	215	2	journal	journal	PROPN
ejpam-7051	215	3	of	of	ADP
ejpam-7051	215	4	pure	pure	ADJ
ejpam-7051	215	5	and	and	CCONJ
ejpam-7051	215	6	applied	applied	ADJ
ejpam-7051	215	7	mathematics	mathematic	NOUN
ejpam-7051	215	8	,	,	PUNCT
ejpam-7051	215	9	18(3):6565	18(3):6565	NUM
ejpam-7051	215	10	,	,	PUNCT
ejpam-7051	215	11	2025	2025	NUM
ejpam-7051	215	12	.	.	PUNCT
ejpam-7051	216	1	[	[	X
ejpam-7051	216	2	9	9	NUM
ejpam-7051	216	3	]	]	PUNCT
ejpam-7051	216	4	c.	c.	PROPN
ejpam-7051	216	5	boonpok	boonpok	PROPN
ejpam-7051	216	6	,	,	PUNCT
ejpam-7051	216	7	c.	c.	PROPN
ejpam-7051	216	8	viriyapong	viriyapong	PROPN
ejpam-7051	216	9	,	,	PUNCT
ejpam-7051	216	10	and	and	CCONJ
ejpam-7051	216	11	m.	m.	NOUN
ejpam-7051	216	12	thongmoon	thongmoon	NOUN
ejpam-7051	216	13	.	.	PUNCT
ejpam-7051	217	1	on	on	ADP
ejpam-7051	217	2	upper	upper	ADJ
ejpam-7051	217	3	and	and	CCONJ
ejpam-7051	217	4	lower	low	ADJ
ejpam-7051	217	5	(	(	PUNCT
ejpam-7051	217	6	τ1	τ1	NOUN
ejpam-7051	217	7	,	,	PUNCT
ejpam-7051	217	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7051	217	9	multifunctions	multifunction	NOUN
ejpam-7051	217	10	.	.	PUNCT
ejpam-7051	218	1	journal	journal	PROPN
ejpam-7051	218	2	of	of	ADP
ejpam-7051	218	3	mathematics	mathematics	PROPN
ejpam-7051	218	4	and	and	CCONJ
ejpam-7051	218	5	computer	computer	NOUN
ejpam-7051	218	6	science	science	NOUN
ejpam-7051	218	7	,	,	PUNCT
ejpam-7051	218	8	18:282	18:282	NUM
ejpam-7051	218	9	–	–	PUNCT
ejpam-7051	218	10	293	293	NUM
ejpam-7051	218	11	,	,	PUNCT
ejpam-7051	218	12	2018	2018	NUM
ejpam-7051	218	13	.	.	PUNCT
ejpam-7051	219	1	[	[	X
ejpam-7051	219	2	10	10	NUM
ejpam-7051	219	3	]	]	X
ejpam-7051	219	4	c.	c.	PROPN
ejpam-7051	219	5	viriyapong	viriyapong	PROPN
ejpam-7051	219	6	and	and	CCONJ
ejpam-7051	219	7	c.	c.	PROPN
ejpam-7051	219	8	boonpok	boonpok	PROPN
ejpam-7051	219	9	.	.	PUNCT
ejpam-7051	220	1	(	(	PUNCT
ejpam-7051	220	2	τ1	τ1	NOUN
ejpam-7051	220	3	,	,	PUNCT
ejpam-7051	220	4	τ2)α	τ2)α	NOUN
ejpam-7051	220	5	-	-	PUNCT
ejpam-7051	220	6	continuity	continuity	NOUN
ejpam-7051	220	7	for	for	ADP
ejpam-7051	220	8	multifunctions	multifunction	NOUN
ejpam-7051	220	9	.	.	PUNCT
ejpam-7051	221	1	journal	journal	PROPN
ejpam-7051	221	2	of	of	ADP
ejpam-7051	221	3	mathematics	mathematic	NOUN
ejpam-7051	221	4	,	,	PUNCT
ejpam-7051	221	5	2020:6285763	2020:6285763	NUM
ejpam-7051	221	6	,	,	PUNCT
ejpam-7051	221	7	2020	2020	NUM
ejpam-7051	221	8	.	.	PUNCT
ejpam-7051	222	1	[	[	X
ejpam-7051	222	2	11	11	NUM
ejpam-7051	222	3	]	]	PUNCT
ejpam-7051	222	4	c.	c.	PROPN
ejpam-7051	222	5	boonpok	boonpok	PROPN
ejpam-7051	222	6	.	.	PUNCT
ejpam-7051	223	1	(	(	PUNCT
ejpam-7051	223	2	τ1	τ1	NOUN
ejpam-7051	223	3	,	,	PUNCT
ejpam-7051	223	4	τ2)δ	τ2)δ	ADJ
ejpam-7051	223	5	-	-	PUNCT
ejpam-7051	223	6	semicontinuous	semicontinuous	ADJ
ejpam-7051	223	7	multifunctions	multifunction	NOUN
ejpam-7051	223	8	.	.	PUNCT
ejpam-7051	224	1	heliyon	heliyon	NOUN
ejpam-7051	224	2	,	,	PUNCT
ejpam-7051	224	3	6	6	NUM
ejpam-7051	224	4	:	:	SYM
ejpam-7051	224	5	e05367	e05367	PROPN
ejpam-7051	224	6	,	,	PUNCT
ejpam-7051	224	7	2020	2020	NUM
ejpam-7051	224	8	.	.	PUNCT
ejpam-7051	225	1	[	[	X
ejpam-7051	225	2	12	12	NUM
ejpam-7051	225	3	]	]	X
ejpam-7051	225	4	c.	c.	PROPN
ejpam-7051	225	5	boonpok	boonpok	PROPN
ejpam-7051	225	6	and	and	CCONJ
ejpam-7051	225	7	p.	p.	NOUN
ejpam-7051	225	8	pue	pue	NOUN
ejpam-7051	225	9	-	-	PUNCT
ejpam-7051	225	10	on	on	ADP
ejpam-7051	225	11	.	.	PUNCT
ejpam-7051	226	1	characterizations	characterization	NOUN
ejpam-7051	226	2	of	of	ADP
ejpam-7051	226	3	almost	almost	ADV
ejpam-7051	226	4	(	(	PUNCT
ejpam-7051	226	5	τ1	τ1	NOUN
ejpam-7051	226	6	,	,	PUNCT
ejpam-7051	226	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7051	226	8	multifunctions	multifunction	NOUN
ejpam-7051	226	9	.	.	PUNCT
ejpam-7051	227	1	international	international	ADJ
ejpam-7051	227	2	journal	journal	NOUN
ejpam-7051	227	3	of	of	ADP
ejpam-7051	227	4	analysis	analysis	NOUN
ejpam-7051	227	5	and	and	CCONJ
ejpam-7051	227	6	applications	application	NOUN
ejpam-7051	227	7	,	,	PUNCT
ejpam-7051	227	8	22:33	22:33	NUM
ejpam-7051	227	9	,	,	PUNCT
ejpam-7051	227	10	2024	2024	NUM
ejpam-7051	227	11	.	.	PUNCT
ejpam-7051	228	1	[	[	X
ejpam-7051	228	2	13	13	NUM
ejpam-7051	228	3	]	]	SYM
ejpam-7051	228	4	á	á	PROPN
ejpam-7051	228	5	.	.	PUNCT
ejpam-7051	228	6	császár	császár	PROPN
ejpam-7051	228	7	.	.	PUNCT
ejpam-7051	229	1	δ	δ	PROPN
ejpam-7051	229	2	-	-	PUNCT
ejpam-7051	229	3	and	and	CCONJ
ejpam-7051	229	4	θ	θ	NOUN
ejpam-7051	229	5	-	-	PUNCT
ejpam-7051	229	6	modifications	modification	NOUN
ejpam-7051	229	7	of	of	ADP
ejpam-7051	229	8	generalized	generalized	ADJ
ejpam-7051	229	9	topologies	topology	NOUN
ejpam-7051	229	10	.	.	PUNCT
ejpam-7051	230	1	acta	acta	PROPN
ejpam-7051	230	2	mathematica	mathematica	PROPN
ejpam-7051	230	3	hungarica	hungarica	PROPN
ejpam-7051	230	4	,	,	PUNCT
ejpam-7051	230	5	120:274–279	120:274–279	NUM
ejpam-7051	230	6	,	,	PUNCT
ejpam-7051	230	7	2008	2008	NUM
ejpam-7051	230	8	.	.	PUNCT
ejpam-7051	231	1	[	[	X
ejpam-7051	231	2	14	14	NUM
ejpam-7051	231	3	]	]	PUNCT
ejpam-7051	231	4	m.	m.	NOUN
ejpam-7051	231	5	chiangpradit	chiangpradit	NOUN
ejpam-7051	231	6	,	,	PUNCT
ejpam-7051	231	7	s.	s.	PROPN
ejpam-7051	231	8	sompong	sompong	PROPN
ejpam-7051	231	9	,	,	PUNCT
ejpam-7051	231	10	and	and	CCONJ
ejpam-7051	231	11	c.	c.	PROPN
ejpam-7051	231	12	boonpok	boonpok	PROPN
ejpam-7051	231	13	.	.	PUNCT
ejpam-7051	232	1	on	on	ADP
ejpam-7051	232	2	characterizations	characterization	NOUN
ejpam-7051	232	3	of	of	ADP
ejpam-7051	232	4	(	(	PUNCT
ejpam-7051	232	5	τ1	τ1	NOUN
ejpam-7051	232	6	,	,	PUNCT
ejpam-7051	232	7	τ2)regular	τ2)regular	ADJ
ejpam-7051	232	8	spaces	space	NOUN
ejpam-7051	232	9	.	.	PUNCT
ejpam-7051	233	1	international	international	ADJ
ejpam-7051	233	2	journal	journal	PROPN
ejpam-7051	233	3	of	of	ADP
ejpam-7051	233	4	mathematics	mathematic	NOUN
ejpam-7051	233	5	and	and	CCONJ
ejpam-7051	233	6	computer	computer	NOUN
ejpam-7051	233	7	science	science	NOUN
ejpam-7051	233	8	,	,	PUNCT
ejpam-7051	233	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-7051	233	10	,	,	PUNCT
ejpam-7051	233	11	2024	2024	NUM
ejpam-7051	233	12	.	.	PUNCT