id	sid	tid	token	lemma	pos
ejpam-6570	1	1	european	european	PROPN
ejpam-6570	1	2	journal	journal	PROPN
ejpam-6570	1	3	of	of	ADP
ejpam-6570	1	4	pure	pure	ADJ
ejpam-6570	1	5	and	and	CCONJ
ejpam-6570	1	6	applied	applied	ADJ
ejpam-6570	1	7	mathematics	mathematic	NOUN
ejpam-6570	1	8	2025	2025	NUM
ejpam-6570	1	9	,	,	PUNCT
ejpam-6570	1	10	vol	vol	NOUN
ejpam-6570	1	11	.	.	PROPN
ejpam-6570	1	12	18	18	NUM
ejpam-6570	1	13	,	,	PUNCT
ejpam-6570	1	14	issue	issue	NOUN
ejpam-6570	1	15	3	3	NUM
ejpam-6570	1	16	,	,	PUNCT
ejpam-6570	1	17	article	article	NOUN
ejpam-6570	1	18	number	number	NOUN
ejpam-6570	1	19	6570	6570	NUM
ejpam-6570	1	20	issn	issn	PROPN
ejpam-6570	1	21	1307	1307	NUM
ejpam-6570	1	22	-	-	SYM
ejpam-6570	1	23	5543	5543	NUM
ejpam-6570	1	24	–	–	PUNCT
ejpam-6570	1	25	ejpam.com	ejpam.com	X
ejpam-6570	1	26	published	publish	VERB
ejpam-6570	1	27	by	by	ADP
ejpam-6570	1	28	new	new	PROPN
ejpam-6570	1	29	york	york	PROPN
ejpam-6570	1	30	business	business	PROPN
ejpam-6570	1	31	global	global	ADJ
ejpam-6570	1	32	almost	almost	ADV
ejpam-6570	1	33	weak	weak	ADJ
ejpam-6570	1	34	continuity	continuity	NOUN
ejpam-6570	1	35	for	for	ADP
ejpam-6570	1	36	multifunctions	multifunction	NOUN
ejpam-6570	1	37	defined	define	VERB
ejpam-6570	1	38	between	between	ADP
ejpam-6570	1	39	an	an	DET
ejpam-6570	1	40	ideal	ideal	ADJ
ejpam-6570	1	41	topological	topological	ADJ
ejpam-6570	1	42	space	space	NOUN
ejpam-6570	1	43	and	and	CCONJ
ejpam-6570	1	44	a	a	DET
ejpam-6570	1	45	bitopological	bitopological	ADJ
ejpam-6570	1	46	space	space	NOUN
ejpam-6570	1	47	chokchai	chokchai	ADJ
ejpam-6570	1	48	viriyapong1	viriyapong1	PROPN
ejpam-6570	1	49	,	,	PUNCT
ejpam-6570	1	50	areeyuth	areeyuth	NOUN
ejpam-6570	1	51	sama	sama	NOUN
ejpam-6570	1	52	-	-	PUNCT
ejpam-6570	1	53	ae2	ae2	PROPN
ejpam-6570	1	54	,	,	PUNCT
ejpam-6570	1	55	chawalit	chawalit	VERB
ejpam-6570	1	56	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6570	1	57	1	1	NUM
ejpam-6570	1	58	mathematics	mathematic	NOUN
ejpam-6570	1	59	and	and	CCONJ
ejpam-6570	1	60	applied	apply	VERB
ejpam-6570	1	61	mathematics	mathematics	PROPN
ejpam-6570	1	62	research	research	NOUN
ejpam-6570	1	63	unit	unit	NOUN
ejpam-6570	1	64	,	,	PUNCT
ejpam-6570	1	65	department	department	NOUN
ejpam-6570	1	66	of	of	ADP
ejpam-6570	1	67	mathematics	mathematic	NOUN
ejpam-6570	1	68	,	,	PUNCT
ejpam-6570	1	69	faculty	faculty	NOUN
ejpam-6570	1	70	of	of	ADP
ejpam-6570	1	71	science	science	NOUN
ejpam-6570	1	72	,	,	PUNCT
ejpam-6570	1	73	mahasarakham	mahasarakham	PROPN
ejpam-6570	1	74	university	university	PROPN
ejpam-6570	1	75	,	,	PUNCT
ejpam-6570	1	76	maha	maha	PROPN
ejpam-6570	1	77	sarakham	sarakham	PROPN
ejpam-6570	1	78	,	,	PUNCT
ejpam-6570	1	79	44150	44150	NUM
ejpam-6570	1	80	,	,	PUNCT
ejpam-6570	1	81	thailand	thailand	PROPN
ejpam-6570	1	82	2	2	NUM
ejpam-6570	1	83	department	department	NOUN
ejpam-6570	1	84	of	of	ADP
ejpam-6570	1	85	mathematics	mathematic	NOUN
ejpam-6570	1	86	and	and	CCONJ
ejpam-6570	1	87	computer	computer	NOUN
ejpam-6570	1	88	science	science	NOUN
ejpam-6570	1	89	,	,	PUNCT
ejpam-6570	1	90	faculty	faculty	NOUN
ejpam-6570	1	91	of	of	ADP
ejpam-6570	1	92	science	science	NOUN
ejpam-6570	1	93	and	and	CCONJ
ejpam-6570	1	94	technology	technology	NOUN
ejpam-6570	1	95	,	,	PUNCT
ejpam-6570	1	96	prince	prince	NOUN
ejpam-6570	1	97	of	of	ADP
ejpam-6570	1	98	songkla	songkla	PROPN
ejpam-6570	1	99	university	university	PROPN
ejpam-6570	1	100	,	,	PUNCT
ejpam-6570	1	101	pattani	pattani	NOUN
ejpam-6570	1	102	campus	campus	NOUN
ejpam-6570	1	103	,	,	PUNCT
ejpam-6570	1	104	pattani	pattani	NOUN
ejpam-6570	1	105	,	,	PUNCT
ejpam-6570	1	106	94000	94000	NUM
ejpam-6570	1	107	,	,	PUNCT
ejpam-6570	1	108	thailand	thailand	PROPN
ejpam-6570	1	109	abstract	abstract	PROPN
ejpam-6570	1	110	.	.	PUNCT
ejpam-6570	2	1	this	this	DET
ejpam-6570	2	2	paper	paper	NOUN
ejpam-6570	2	3	presents	present	VERB
ejpam-6570	2	4	new	new	ADJ
ejpam-6570	2	5	concepts	concept	NOUN
ejpam-6570	2	6	of	of	ADP
ejpam-6570	2	7	continuous	continuous	ADJ
ejpam-6570	2	8	multifunctions	multifunction	NOUN
ejpam-6570	2	9	,	,	PUNCT
ejpam-6570	2	10	called	call	VERB
ejpam-6570	2	11	upper	upper	ADJ
ejpam-6570	2	12	almost	almost	ADV
ejpam-6570	2	13	weakly	weakly	ADJ
ejpam-6570	2	14	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	2	15	,	,	PUNCT
ejpam-6570	2	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	2	17	multifunctions	multifunction	NOUN
ejpam-6570	2	18	and	and	CCONJ
ejpam-6570	2	19	lower	low	ADJ
ejpam-6570	2	20	almost	almost	ADV
ejpam-6570	2	21	weakly	weakly	ADJ
ejpam-6570	2	22	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	2	23	,	,	PUNCT
ejpam-6570	2	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	2	25	multifunctions	multifunction	NOUN
ejpam-6570	2	26	.	.	PUNCT
ejpam-6570	3	1	moreover	moreover	ADV
ejpam-6570	3	2	,	,	PUNCT
ejpam-6570	3	3	several	several	ADJ
ejpam-6570	3	4	characterizations	characterization	NOUN
ejpam-6570	3	5	and	and	CCONJ
ejpam-6570	3	6	some	some	DET
ejpam-6570	3	7	properties	property	NOUN
ejpam-6570	3	8	concerning	concern	VERB
ejpam-6570	3	9	upper	upper	ADJ
ejpam-6570	3	10	almost	almost	ADV
ejpam-6570	3	11	weakly	weakly	ADJ
ejpam-6570	3	12	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	3	13	,	,	PUNCT
ejpam-6570	3	14	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	3	15	multifunctions	multifunction	NOUN
ejpam-6570	3	16	and	and	CCONJ
ejpam-6570	3	17	lower	low	ADJ
ejpam-6570	3	18	almost	almost	ADV
ejpam-6570	3	19	weakly	weakly	ADJ
ejpam-6570	3	20	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	3	21	,	,	PUNCT
ejpam-6570	3	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	3	23	multifunctions	multifunction	NOUN
ejpam-6570	3	24	are	be	AUX
ejpam-6570	3	25	considered	consider	VERB
ejpam-6570	3	26	.	.	PUNCT
ejpam-6570	4	1	2020	2020	NUM
ejpam-6570	4	2	mathematics	mathematic	NOUN
ejpam-6570	4	3	subject	subject	NOUN
ejpam-6570	4	4	classifications	classification	NOUN
ejpam-6570	4	5	:	:	PUNCT
ejpam-6570	4	6	54c08	54c08	NUM
ejpam-6570	4	7	,	,	PUNCT
ejpam-6570	4	8	54c60	54c60	NUM
ejpam-6570	4	9	key	key	ADJ
ejpam-6570	4	10	words	word	NOUN
ejpam-6570	4	11	and	and	CCONJ
ejpam-6570	4	12	phrases	phrase	NOUN
ejpam-6570	4	13	:	:	PUNCT
ejpam-6570	4	14	upper	upper	ADJ
ejpam-6570	4	15	almost	almost	ADV
ejpam-6570	4	16	weakly	weakly	ADJ
ejpam-6570	4	17	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	4	18	,	,	PUNCT
ejpam-6570	4	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	4	20	multifunction	multifunction	NOUN
ejpam-6570	4	21	,	,	PUNCT
ejpam-6570	4	22	lower	low	ADJ
ejpam-6570	4	23	almost	almost	ADV
ejpam-6570	4	24	weakly	weakly	ADJ
ejpam-6570	4	25	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	4	26	,	,	PUNCT
ejpam-6570	4	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	4	28	multifunction	multifunction	NOUN
ejpam-6570	4	29	1	1	NUM
ejpam-6570	4	30	.	.	PUNCT
ejpam-6570	4	31	introduction	introduction	NOUN
ejpam-6570	4	32	topology	topology	NOUN
ejpam-6570	4	33	as	as	ADP
ejpam-6570	4	34	a	a	DET
ejpam-6570	4	35	field	field	NOUN
ejpam-6570	4	36	of	of	ADP
ejpam-6570	4	37	mathematics	mathematic	NOUN
ejpam-6570	4	38	is	be	AUX
ejpam-6570	4	39	concerned	concern	VERB
ejpam-6570	4	40	with	with	ADP
ejpam-6570	4	41	all	all	DET
ejpam-6570	4	42	questions	question	NOUN
ejpam-6570	4	43	directly	directly	ADV
ejpam-6570	4	44	or	or	CCONJ
ejpam-6570	4	45	indirectly	indirectly	ADV
ejpam-6570	4	46	related	relate	VERB
ejpam-6570	4	47	to	to	ADP
ejpam-6570	4	48	continuity	continuity	NOUN
ejpam-6570	4	49	.	.	PUNCT
ejpam-6570	5	1	singal	singal	NOUN
ejpam-6570	5	2	and	and	CCONJ
ejpam-6570	5	3	singal	singal	ADJ
ejpam-6570	5	4	[	[	X
ejpam-6570	5	5	1	1	NUM
ejpam-6570	5	6	]	]	PUNCT
ejpam-6570	5	7	introduced	introduce	VERB
ejpam-6570	5	8	the	the	DET
ejpam-6570	5	9	concept	concept	NOUN
ejpam-6570	5	10	of	of	ADP
ejpam-6570	5	11	almost	almost	ADV
ejpam-6570	5	12	continuous	continuous	ADJ
ejpam-6570	5	13	functions	function	NOUN
ejpam-6570	5	14	as	as	ADP
ejpam-6570	5	15	a	a	DET
ejpam-6570	5	16	generalization	generalization	NOUN
ejpam-6570	5	17	of	of	ADP
ejpam-6570	5	18	continuity	continuity	NOUN
ejpam-6570	5	19	.	.	PUNCT
ejpam-6570	6	1	munshi	munshi	PROPN
ejpam-6570	6	2	and	and	CCONJ
ejpam-6570	6	3	bassan	bassan	NOUN
ejpam-6570	6	4	[	[	X
ejpam-6570	6	5	2	2	NUM
ejpam-6570	6	6	]	]	PUNCT
ejpam-6570	6	7	studied	study	VERB
ejpam-6570	6	8	the	the	DET
ejpam-6570	6	9	notion	notion	NOUN
ejpam-6570	6	10	of	of	ADP
ejpam-6570	6	11	almost	almost	ADV
ejpam-6570	6	12	semi	semi	ADJ
ejpam-6570	6	13	-	-	ADJ
ejpam-6570	6	14	continuous	continuous	ADJ
ejpam-6570	6	15	functions	function	NOUN
ejpam-6570	6	16	.	.	PUNCT
ejpam-6570	7	1	noiri	noiri	ADV
ejpam-6570	8	1	[	[	X
ejpam-6570	8	2	3	3	X
ejpam-6570	8	3	]	]	PUNCT
ejpam-6570	8	4	introduced	introduce	VERB
ejpam-6570	8	5	and	and	CCONJ
ejpam-6570	8	6	investigated	investigate	VERB
ejpam-6570	8	7	the	the	DET
ejpam-6570	8	8	concept	concept	NOUN
ejpam-6570	8	9	of	of	ADP
ejpam-6570	8	10	almost	almost	ADV
ejpam-6570	8	11	α	α	NUM
ejpam-6570	8	12	-	-	ADJ
ejpam-6570	8	13	continuous	continuous	ADJ
ejpam-6570	8	14	functions	function	NOUN
ejpam-6570	8	15	.	.	PUNCT
ejpam-6570	9	1	nasef	nasef	NOUN
ejpam-6570	9	2	and	and	CCONJ
ejpam-6570	9	3	noiri	noiri	ADV
ejpam-6570	10	1	[	[	X
ejpam-6570	10	2	4	4	X
ejpam-6570	10	3	]	]	PUNCT
ejpam-6570	10	4	introduced	introduce	VERB
ejpam-6570	10	5	two	two	NUM
ejpam-6570	10	6	classes	class	NOUN
ejpam-6570	10	7	of	of	ADP
ejpam-6570	10	8	functions	function	NOUN
ejpam-6570	10	9	,	,	PUNCT
ejpam-6570	10	10	namely	namely	ADV
ejpam-6570	10	11	almost	almost	ADV
ejpam-6570	10	12	precontinuous	precontinuous	ADJ
ejpam-6570	10	13	functions	function	NOUN
ejpam-6570	10	14	and	and	CCONJ
ejpam-6570	10	15	almost	almost	ADV
ejpam-6570	10	16	β	β	ADJ
ejpam-6570	10	17	-	-	ADJ
ejpam-6570	10	18	continuous	continuous	ADJ
ejpam-6570	10	19	functions	function	NOUN
ejpam-6570	10	20	.	.	PUNCT
ejpam-6570	11	1	the	the	DET
ejpam-6570	11	2	class	class	NOUN
ejpam-6570	11	3	of	of	ADP
ejpam-6570	11	4	almost	almost	ADV
ejpam-6570	11	5	precontinuity	precontinuity	NOUN
ejpam-6570	11	6	is	be	AUX
ejpam-6570	11	7	a	a	DET
ejpam-6570	11	8	generalization	generalization	NOUN
ejpam-6570	11	9	of	of	ADP
ejpam-6570	11	10	almost	almost	ADV
ejpam-6570	11	11	α	α	NOUN
ejpam-6570	11	12	-	-	NOUN
ejpam-6570	11	13	continuity	continuity	NOUN
ejpam-6570	11	14	.	.	PUNCT
ejpam-6570	12	1	the	the	DET
ejpam-6570	12	2	class	class	NOUN
ejpam-6570	12	3	of	of	ADP
ejpam-6570	12	4	almost	almost	ADV
ejpam-6570	12	5	β	β	NOUN
ejpam-6570	12	6	-	-	NOUN
ejpam-6570	12	7	continuity	continuity	NOUN
ejpam-6570	12	8	is	be	AUX
ejpam-6570	12	9	a	a	DET
ejpam-6570	12	10	generalization	generalization	NOUN
ejpam-6570	12	11	of	of	ADP
ejpam-6570	12	12	almost	almost	ADV
ejpam-6570	12	13	semi	semi	NOUN
ejpam-6570	12	14	-	-	NOUN
ejpam-6570	12	15	continuity	continuity	NOUN
ejpam-6570	12	16	.	.	PUNCT
ejpam-6570	13	1	levine	levine	PROPN
ejpam-6570	14	1	[	[	X
ejpam-6570	14	2	5	5	NUM
ejpam-6570	14	3	]	]	PUNCT
ejpam-6570	14	4	introduced	introduce	VERB
ejpam-6570	14	5	and	and	CCONJ
ejpam-6570	14	6	investigated	investigate	VERB
ejpam-6570	14	7	the	the	DET
ejpam-6570	14	8	concept	concept	NOUN
ejpam-6570	14	9	of	of	ADP
ejpam-6570	14	10	weakly	weakly	ADJ
ejpam-6570	14	11	continuous	continuous	ADJ
ejpam-6570	14	12	functions	function	NOUN
ejpam-6570	14	13	.	.	PUNCT
ejpam-6570	15	1	husain	husain	NOUN
ejpam-6570	16	1	[	[	X
ejpam-6570	16	2	6	6	NUM
ejpam-6570	16	3	]	]	PUNCT
ejpam-6570	16	4	introduced	introduce	VERB
ejpam-6570	16	5	and	and	CCONJ
ejpam-6570	16	6	studied	study	VERB
ejpam-6570	16	7	the	the	DET
ejpam-6570	16	8	notion	notion	NOUN
ejpam-6570	16	9	of	of	ADP
ejpam-6570	16	10	almost	almost	ADV
ejpam-6570	16	11	continuous	continuous	ADJ
ejpam-6570	16	12	functions	function	NOUN
ejpam-6570	16	13	.	.	PUNCT
ejpam-6570	17	1	janković	janković	PUNCT
ejpam-6570	18	1	[	[	X
ejpam-6570	18	2	7	7	X
ejpam-6570	18	3	]	]	PUNCT
ejpam-6570	18	4	introduced	introduce	VERB
ejpam-6570	18	5	almost	almost	ADV
ejpam-6570	18	6	weak	weak	ADJ
ejpam-6570	18	7	continuity	continuity	NOUN
ejpam-6570	18	8	as	as	ADP
ejpam-6570	18	9	a	a	DET
ejpam-6570	18	10	generalization	generalization	NOUN
ejpam-6570	18	11	of	of	ADP
ejpam-6570	18	12	both	both	DET
ejpam-6570	18	13	weak	weak	ADJ
ejpam-6570	18	14	continuity	continuity	NOUN
ejpam-6570	18	15	and	and	CCONJ
ejpam-6570	18	16	almost	almost	ADV
ejpam-6570	18	17	continuity	continuity	NOUN
ejpam-6570	18	18	.	.	PUNCT
ejpam-6570	19	1	noiri	noiri	ADV
ejpam-6570	20	1	[	[	X
ejpam-6570	20	2	8	8	NUM
ejpam-6570	20	3	]	]	PUNCT
ejpam-6570	20	4	investigated	investigate	VERB
ejpam-6570	20	5	∗corresponding	∗corresponde	VERB
ejpam-6570	20	6	author	author	NOUN
ejpam-6570	20	7	.	.	PUNCT
ejpam-6570	21	1	doi	doi	NOUN
ejpam-6570	21	2	:	:	PUNCT
ejpam-6570	21	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6570	https://doi.org/10.29020/nybg.ejpam.v18i3.6570	ADJ
ejpam-6570	21	4	email	email	NOUN
ejpam-6570	21	5	addresses	address	NOUN
ejpam-6570	21	6	:	:	PUNCT
ejpam-6570	21	7	chohchai.v@msu.ac.th	chohchai.v@msu.ac.th	PROPN
ejpam-6570	21	8	(	(	PUNCT
ejpam-6570	21	9	c.	c.	PROPN
ejpam-6570	21	10	viriyapong	viriyapong	PROPN
ejpam-6570	21	11	)	)	PUNCT
ejpam-6570	21	12	,	,	PUNCT
ejpam-6570	21	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6570	21	14	(	(	PUNCT
ejpam-6570	21	15	a.	a.	PROPN
ejpam-6570	21	16	sama	sama	PROPN
ejpam-6570	21	17	-	-	PUNCT
ejpam-6570	21	18	ae	ae	PROPN
ejpam-6570	21	19	)	)	PUNCT
ejpam-6570	21	20	,	,	PUNCT
ejpam-6570	21	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6570	21	22	(	(	PUNCT
ejpam-6570	21	23	c.	c.	PROPN
ejpam-6570	21	24	boonpok	boonpok	PROPN
ejpam-6570	21	25	)	)	PUNCT
ejpam-6570	21	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6570	22	1	1	1	NUM
ejpam-6570	22	2	copyright	copyright	NOUN
ejpam-6570	22	3	:	:	PUNCT
ejpam-6570	22	4	©	©	PROPN
ejpam-6570	22	5	2025	2025	NUM
ejpam-6570	22	6	the	the	DET
ejpam-6570	22	7	author(s	author(s	NOUN
ejpam-6570	22	8	)	)	PUNCT
ejpam-6570	22	9	.	.	PUNCT
ejpam-6570	23	1	(	(	PUNCT
ejpam-6570	23	2	cc	cc	NOUN
ejpam-6570	23	3	by	by	ADP
ejpam-6570	23	4	-	-	PUNCT
ejpam-6570	23	5	nc	nc	PROPN
ejpam-6570	23	6	4.0	4.0	NUM
ejpam-6570	23	7	)	)	PUNCT
ejpam-6570	23	8	c.	c.	PROPN
ejpam-6570	23	9	viriyapong	viriyapong	PROPN
ejpam-6570	23	10	,	,	PUNCT
ejpam-6570	23	11	a.	a.	PROPN
ejpam-6570	23	12	sama	sama	PROPN
ejpam-6570	23	13	-	-	PUNCT
ejpam-6570	23	14	ae	ae	PROPN
ejpam-6570	23	15	,	,	PUNCT
ejpam-6570	23	16	c.	c.	PROPN
ejpam-6570	23	17	boonpok	boonpok	PROPN
ejpam-6570	23	18	/	/	SYM
ejpam-6570	23	19	eur	eur	PROPN
ejpam-6570	23	20	.	.	PUNCT
ejpam-6570	24	1	j.	j.	PROPN
ejpam-6570	24	2	pure	pure	PROPN
ejpam-6570	24	3	appl	appl	PROPN
ejpam-6570	24	4	.	.	PROPN
ejpam-6570	24	5	math	math	PROPN
ejpam-6570	24	6	,	,	PUNCT
ejpam-6570	24	7	18	18	NUM
ejpam-6570	24	8	(	(	PUNCT
ejpam-6570	24	9	3	3	NUM
ejpam-6570	24	10	)	)	PUNCT
ejpam-6570	24	11	(	(	PUNCT
ejpam-6570	24	12	2025	2025	NUM
ejpam-6570	24	13	)	)	PUNCT
ejpam-6570	24	14	,	,	PUNCT
ejpam-6570	24	15	6570	6570	NUM
ejpam-6570	24	16	2	2	NUM
ejpam-6570	24	17	of	of	ADP
ejpam-6570	24	18	14	14	NUM
ejpam-6570	24	19	several	several	ADJ
ejpam-6570	24	20	characterizations	characterization	NOUN
ejpam-6570	24	21	of	of	ADP
ejpam-6570	24	22	almost	almost	ADV
ejpam-6570	24	23	weakly	weakly	ADJ
ejpam-6570	24	24	continuous	continuous	ADJ
ejpam-6570	24	25	functions	function	NOUN
ejpam-6570	24	26	.	.	PUNCT
ejpam-6570	25	1	rose	rise	VERB
ejpam-6570	26	1	[	[	X
ejpam-6570	26	2	9	9	NUM
ejpam-6570	26	3	]	]	PUNCT
ejpam-6570	26	4	introduced	introduce	VERB
ejpam-6570	26	5	the	the	DET
ejpam-6570	26	6	notion	notion	NOUN
ejpam-6570	26	7	of	of	ADP
ejpam-6570	26	8	subweakly	subweakly	ADJ
ejpam-6570	26	9	continuous	continuous	ADJ
ejpam-6570	26	10	functions	function	NOUN
ejpam-6570	26	11	and	and	CCONJ
ejpam-6570	26	12	investigated	investigate	VERB
ejpam-6570	26	13	the	the	DET
ejpam-6570	26	14	relationships	relationship	NOUN
ejpam-6570	26	15	between	between	ADP
ejpam-6570	26	16	subweak	subweak	NOUN
ejpam-6570	26	17	continuity	continuity	NOUN
ejpam-6570	26	18	and	and	CCONJ
ejpam-6570	26	19	weak	weak	ADJ
ejpam-6570	26	20	continuity	continuity	NOUN
ejpam-6570	26	21	.	.	PUNCT
ejpam-6570	27	1	in	in	ADP
ejpam-6570	27	2	1993	1993	NUM
ejpam-6570	27	3	,	,	PUNCT
ejpam-6570	27	4	noiri	noiri	PRON
ejpam-6570	27	5	and	and	CCONJ
ejpam-6570	27	6	popa	popa	NOUN
ejpam-6570	27	7	[	[	X
ejpam-6570	27	8	10	10	NUM
ejpam-6570	27	9	]	]	PUNCT
ejpam-6570	27	10	extended	extend	VERB
ejpam-6570	27	11	the	the	DET
ejpam-6570	27	12	concept	concept	NOUN
ejpam-6570	27	13	of	of	ADP
ejpam-6570	27	14	almost	almost	ADV
ejpam-6570	27	15	weakly	weakly	ADJ
ejpam-6570	27	16	continuous	continuous	ADJ
ejpam-6570	27	17	functions	function	NOUN
ejpam-6570	27	18	to	to	ADP
ejpam-6570	27	19	multifunctions	multifunction	NOUN
ejpam-6570	27	20	and	and	CCONJ
ejpam-6570	27	21	defined	define	VERB
ejpam-6570	27	22	upper	upper	ADJ
ejpam-6570	27	23	almost	almost	ADV
ejpam-6570	27	24	weakly	weakly	ADJ
ejpam-6570	27	25	continuous	continuous	ADJ
ejpam-6570	27	26	multifunctions	multifunction	NOUN
ejpam-6570	27	27	and	and	CCONJ
ejpam-6570	27	28	lower	low	ADJ
ejpam-6570	27	29	almost	almost	ADV
ejpam-6570	27	30	weakly	weakly	ADJ
ejpam-6570	27	31	continuous	continuous	ADJ
ejpam-6570	27	32	multifunctions	multifunction	NOUN
ejpam-6570	27	33	.	.	PUNCT
ejpam-6570	28	1	popa	popa	NOUN
ejpam-6570	28	2	and	and	CCONJ
ejpam-6570	28	3	noiri	noiri	ADV
ejpam-6570	29	1	[	[	X
ejpam-6570	29	2	11	11	NUM
ejpam-6570	29	3	]	]	PUNCT
ejpam-6570	29	4	investigated	investigate	VERB
ejpam-6570	29	5	some	some	DET
ejpam-6570	29	6	characterizations	characterization	NOUN
ejpam-6570	29	7	and	and	CCONJ
ejpam-6570	29	8	several	several	ADJ
ejpam-6570	29	9	properties	property	NOUN
ejpam-6570	29	10	concerning	concern	VERB
ejpam-6570	29	11	upper	upper	ADJ
ejpam-6570	29	12	almost	almost	ADV
ejpam-6570	29	13	weakly	weakly	ADJ
ejpam-6570	29	14	continuous	continuous	ADJ
ejpam-6570	29	15	multifunctions	multifunction	NOUN
ejpam-6570	29	16	and	and	CCONJ
ejpam-6570	29	17	lower	low	ADJ
ejpam-6570	29	18	almost	almost	ADV
ejpam-6570	29	19	weakly	weakly	ADJ
ejpam-6570	29	20	continuous	continuous	ADJ
ejpam-6570	29	21	multifunctions	multifunction	NOUN
ejpam-6570	29	22	.	.	PUNCT
ejpam-6570	30	1	abd	abd	PROPN
ejpam-6570	30	2	el	el	PROPN
ejpam-6570	30	3	-	-	PROPN
ejpam-6570	30	4	monsef	monsef	PROPN
ejpam-6570	30	5	et	et	PROPN
ejpam-6570	30	6	al	al	PROPN
ejpam-6570	30	7	.	.	PUNCT
ejpam-6570	31	1	[	[	X
ejpam-6570	31	2	12	12	NUM
ejpam-6570	31	3	]	]	PUNCT
ejpam-6570	31	4	introduced	introduce	VERB
ejpam-6570	31	5	and	and	CCONJ
ejpam-6570	31	6	studied	study	VERB
ejpam-6570	31	7	the	the	DET
ejpam-6570	31	8	notions	notion	NOUN
ejpam-6570	31	9	of	of	ADP
ejpam-6570	31	10	i	i	PRON
ejpam-6570	31	11	-closed	-close	VERB
ejpam-6570	31	12	sets	set	NOUN
ejpam-6570	31	13	and	and	CCONJ
ejpam-6570	31	14	i	i	PRON
ejpam-6570	31	15	-continuous	-continuous	ADJ
ejpam-6570	31	16	functions	function	NOUN
ejpam-6570	31	17	.	.	PUNCT
ejpam-6570	32	1	semi	semi	ADJ
ejpam-6570	32	2	-	-	ADJ
ejpam-6570	32	3	i	i	ADJ
ejpam-6570	32	4	-open	-open	NOUN
ejpam-6570	32	5	sets	set	NOUN
ejpam-6570	32	6	,	,	PUNCT
ejpam-6570	32	7	pre	pre	ADJ
ejpam-6570	32	8	-	-	ADJ
ejpam-6570	32	9	i	i	ADJ
ejpam-6570	32	10	-open	-open	NOUN
ejpam-6570	32	11	sets	set	NOUN
ejpam-6570	32	12	,	,	PUNCT
ejpam-6570	32	13	α	α	X
ejpam-6570	32	14	-	-	PUNCT
ejpam-6570	32	15	i	i	PRON
ejpam-6570	32	16	-open	-open	NOUN
ejpam-6570	32	17	sets	set	NOUN
ejpam-6570	32	18	,	,	PUNCT
ejpam-6570	32	19	β	β	X
ejpam-6570	32	20	-	-	ADJ
ejpam-6570	32	21	i	i	PRON
ejpam-6570	32	22	-open	-open	NOUN
ejpam-6570	32	23	sets	set	NOUN
ejpam-6570	32	24	and	and	CCONJ
ejpam-6570	32	25	δ	δ	PROPN
ejpam-6570	32	26	-	-	PUNCT
ejpam-6570	32	27	i	i	PRON
ejpam-6570	32	28	-open	-open	NOUN
ejpam-6570	32	29	sets	set	NOUN
ejpam-6570	32	30	play	play	VERB
ejpam-6570	32	31	an	an	DET
ejpam-6570	32	32	important	important	ADJ
ejpam-6570	32	33	role	role	NOUN
ejpam-6570	32	34	in	in	ADP
ejpam-6570	32	35	the	the	DET
ejpam-6570	32	36	research	research	NOUN
ejpam-6570	32	37	of	of	ADP
ejpam-6570	32	38	generalizations	generalization	NOUN
ejpam-6570	32	39	of	of	ADP
ejpam-6570	32	40	continuity	continuity	NOUN
ejpam-6570	32	41	in	in	ADP
ejpam-6570	32	42	ideal	ideal	ADJ
ejpam-6570	32	43	topological	topological	ADJ
ejpam-6570	32	44	spaces	space	NOUN
ejpam-6570	32	45	.	.	PUNCT
ejpam-6570	33	1	in	in	ADP
ejpam-6570	33	2	2005	2005	NUM
ejpam-6570	33	3	,	,	PUNCT
ejpam-6570	33	4	hatir	hatir	PROPN
ejpam-6570	33	5	and	and	CCONJ
ejpam-6570	33	6	noiri	noiri	ADV
ejpam-6570	33	7	[	[	X
ejpam-6570	33	8	13	13	NUM
ejpam-6570	33	9	]	]	PUNCT
ejpam-6570	33	10	introduced	introduce	VERB
ejpam-6570	33	11	and	and	CCONJ
ejpam-6570	33	12	investigated	investigate	VERB
ejpam-6570	33	13	the	the	DET
ejpam-6570	33	14	notions	notion	NOUN
ejpam-6570	33	15	of	of	ADP
ejpam-6570	33	16	weakly	weakly	ADJ
ejpam-6570	33	17	pre	pre	ADJ
ejpam-6570	33	18	-	-	ADJ
ejpam-6570	33	19	i	i	PRON
ejpam-6570	33	20	-open	-open	NOUN
ejpam-6570	33	21	sets	set	NOUN
ejpam-6570	33	22	and	and	CCONJ
ejpam-6570	33	23	weakly	weakly	ADJ
ejpam-6570	33	24	pre	pre	ADJ
ejpam-6570	33	25	-	-	ADJ
ejpam-6570	33	26	i	i	ADJ
ejpam-6570	33	27	-continuous	-continuous	ADJ
ejpam-6570	33	28	functions	function	NOUN
ejpam-6570	33	29	.	.	PUNCT
ejpam-6570	34	1	furthermore	furthermore	ADV
ejpam-6570	34	2	,	,	PUNCT
ejpam-6570	34	3	hatir	hatir	PROPN
ejpam-6570	34	4	and	and	CCONJ
ejpam-6570	34	5	noiri	noiri	ADV
ejpam-6570	35	1	[	[	X
ejpam-6570	35	2	14	14	NUM
ejpam-6570	35	3	]	]	PUNCT
ejpam-6570	35	4	investigated	investigate	VERB
ejpam-6570	35	5	further	further	ADJ
ejpam-6570	35	6	properties	property	NOUN
ejpam-6570	35	7	of	of	ADP
ejpam-6570	35	8	semi	semi	ADJ
ejpam-6570	35	9	-	-	ADJ
ejpam-6570	35	10	i	i	PRON
ejpam-6570	35	11	-open	-open	NOUN
ejpam-6570	35	12	sets	set	NOUN
ejpam-6570	35	13	and	and	CCONJ
ejpam-6570	35	14	semii	semii	VERB
ejpam-6570	35	15	-continuous	-continuous	ADJ
ejpam-6570	35	16	functions	function	NOUN
ejpam-6570	35	17	.	.	PUNCT
ejpam-6570	36	1	on	on	ADP
ejpam-6570	36	2	the	the	DET
ejpam-6570	36	3	other	other	ADJ
ejpam-6570	36	4	hand	hand	NOUN
ejpam-6570	36	5	,	,	PUNCT
ejpam-6570	36	6	the	the	DET
ejpam-6570	36	7	present	present	ADJ
ejpam-6570	36	8	author	author	NOUN
ejpam-6570	36	9	introduced	introduce	VERB
ejpam-6570	36	10	and	and	CCONJ
ejpam-6570	36	11	studied	study	VERB
ejpam-6570	36	12	new	new	ADJ
ejpam-6570	36	13	classes	class	NOUN
ejpam-6570	36	14	of	of	ADP
ejpam-6570	36	15	multifunctions	multifunction	NOUN
ejpam-6570	36	16	between	between	ADP
ejpam-6570	36	17	ideal	ideal	ADJ
ejpam-6570	36	18	topological	topological	ADJ
ejpam-6570	36	19	spaces	space	NOUN
ejpam-6570	36	20	,	,	PUNCT
ejpam-6570	36	21	namely	namely	ADV
ejpam-6570	36	22	upper	upper	ADJ
ejpam-6570	36	23	⋆-continuous	⋆-continuous	ADJ
ejpam-6570	36	24	multifunctions	multifunction	NOUN
ejpam-6570	37	1	[	[	X
ejpam-6570	37	2	15	15	NUM
ejpam-6570	37	3	]	]	X
ejpam-6570	37	4	,	,	PUNCT
ejpam-6570	37	5	lower	low	ADJ
ejpam-6570	37	6	⋆-continuous	⋆-continuous	ADJ
ejpam-6570	37	7	multifunctions	multifunction	NOUN
ejpam-6570	38	1	[	[	X
ejpam-6570	38	2	15	15	NUM
ejpam-6570	38	3	]	]	X
ejpam-6570	38	4	,	,	PUNCT
ejpam-6570	38	5	upper	upper	ADJ
ejpam-6570	38	6	almost	almost	ADV
ejpam-6570	38	7	⋆-continuous	⋆-continuous	ADJ
ejpam-6570	38	8	multifunctions	multifunction	NOUN
ejpam-6570	39	1	[	[	X
ejpam-6570	39	2	15	15	NUM
ejpam-6570	39	3	]	]	PUNCT
ejpam-6570	39	4	,	,	PUNCT
ejpam-6570	39	5	lower	low	ADJ
ejpam-6570	39	6	almost	almost	ADV
ejpam-6570	39	7	⋆-continuous	⋆-continuous	ADJ
ejpam-6570	39	8	multifunctions	multifunction	NOUN
ejpam-6570	40	1	[	[	X
ejpam-6570	40	2	15	15	NUM
ejpam-6570	40	3	]	]	X
ejpam-6570	40	4	,	,	PUNCT
ejpam-6570	40	5	upper	upper	ADJ
ejpam-6570	40	6	weakly	weakly	ADJ
ejpam-6570	40	7	⋆-continuous	⋆-continuous	ADJ
ejpam-6570	40	8	multifunctions	multifunction	NOUN
ejpam-6570	41	1	[	[	X
ejpam-6570	41	2	15	15	NUM
ejpam-6570	41	3	]	]	PUNCT
ejpam-6570	41	4	,	,	PUNCT
ejpam-6570	41	5	lower	low	ADJ
ejpam-6570	41	6	weakly	weakly	ADJ
ejpam-6570	41	7	⋆-continuous	⋆-continuous	ADJ
ejpam-6570	41	8	multifunctions	multifunction	NOUN
ejpam-6570	42	1	[	[	X
ejpam-6570	42	2	15	15	NUM
ejpam-6570	42	3	]	]	PUNCT
ejpam-6570	42	4	,	,	PUNCT
ejpam-6570	42	5	pı	pı	ADJ
ejpam-6570	42	6	-	-	ADJ
ejpam-6570	42	7	continuous	continuous	ADJ
ejpam-6570	42	8	multifunctions	multifunction	NOUN
ejpam-6570	43	1	[	[	X
ejpam-6570	43	2	16	16	NUM
ejpam-6570	43	3	]	]	PUNCT
ejpam-6570	43	4	and	and	CCONJ
ejpam-6570	43	5	weakly	weakly	ADJ
ejpam-6570	43	6	pı	pı	ADJ
ejpam-6570	43	7	-	-	ADJ
ejpam-6570	43	8	continuous	continuous	ADJ
ejpam-6570	43	9	multifunctions	multifunction	NOUN
ejpam-6570	44	1	[	[	X
ejpam-6570	44	2	16	16	NUM
ejpam-6570	44	3	]	]	PUNCT
ejpam-6570	44	4	.	.	PUNCT
ejpam-6570	45	1	recently	recently	ADV
ejpam-6570	45	2	,	,	PUNCT
ejpam-6570	45	3	pue	pue	NOUN
ejpam-6570	45	4	-	-	PUNCT
ejpam-6570	45	5	on	on	NOUN
ejpam-6570	45	6	et	et	PROPN
ejpam-6570	45	7	al	al	PROPN
ejpam-6570	45	8	.	.	PUNCT
ejpam-6570	46	1	[	[	X
ejpam-6570	46	2	17	17	NUM
ejpam-6570	46	3	]	]	PUNCT
ejpam-6570	46	4	extended	extend	VERB
ejpam-6570	46	5	the	the	DET
ejpam-6570	46	6	idea	idea	NOUN
ejpam-6570	46	7	of	of	ADP
ejpam-6570	46	8	continuous	continuous	ADJ
ejpam-6570	46	9	multifunctions	multifunction	NOUN
ejpam-6570	46	10	to	to	ADP
ejpam-6570	46	11	bitopological	bitopological	ADJ
ejpam-6570	46	12	spaces	space	NOUN
ejpam-6570	46	13	.	.	PUNCT
ejpam-6570	47	1	klanarong	klanarong	NOUN
ejpam-6570	47	2	et	et	PROPN
ejpam-6570	47	3	al	al	PROPN
ejpam-6570	47	4	.	.	PUNCT
ejpam-6570	48	1	[	[	X
ejpam-6570	48	2	18	18	NUM
ejpam-6570	48	3	]	]	PUNCT
ejpam-6570	48	4	introduced	introduce	VERB
ejpam-6570	48	5	and	and	CCONJ
ejpam-6570	48	6	investigated	investigate	VERB
ejpam-6570	48	7	the	the	DET
ejpam-6570	48	8	concepts	concept	NOUN
ejpam-6570	48	9	of	of	ADP
ejpam-6570	48	10	upper	upper	ADJ
ejpam-6570	48	11	almost	almost	ADV
ejpam-6570	48	12	(	(	PUNCT
ejpam-6570	48	13	τ1	τ1	NOUN
ejpam-6570	48	14	,	,	PUNCT
ejpam-6570	48	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	48	16	multifunctions	multifunction	NOUN
ejpam-6570	48	17	and	and	CCONJ
ejpam-6570	48	18	lower	low	ADJ
ejpam-6570	48	19	almost	almost	ADV
ejpam-6570	48	20	(	(	PUNCT
ejpam-6570	48	21	τ1	τ1	NOUN
ejpam-6570	48	22	,	,	PUNCT
ejpam-6570	48	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	48	24	multifunctions	multifunction	NOUN
ejpam-6570	48	25	.	.	PUNCT
ejpam-6570	49	1	thongmoon	thongmoon	NOUN
ejpam-6570	49	2	et	et	PROPN
ejpam-6570	49	3	al	al	PROPN
ejpam-6570	49	4	.	.	PUNCT
ejpam-6570	50	1	[	[	X
ejpam-6570	50	2	19	19	NUM
ejpam-6570	50	3	]	]	PUNCT
ejpam-6570	50	4	introduced	introduce	VERB
ejpam-6570	50	5	and	and	CCONJ
ejpam-6570	50	6	studied	study	VERB
ejpam-6570	50	7	the	the	DET
ejpam-6570	50	8	notions	notion	NOUN
ejpam-6570	50	9	of	of	ADP
ejpam-6570	50	10	upper	upper	ADJ
ejpam-6570	50	11	weakly	weakly	ADJ
ejpam-6570	50	12	(	(	PUNCT
ejpam-6570	50	13	τ1	τ1	NOUN
ejpam-6570	50	14	,	,	PUNCT
ejpam-6570	50	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	50	16	multifunctions	multifunction	NOUN
ejpam-6570	50	17	and	and	CCONJ
ejpam-6570	50	18	lower	low	ADJ
ejpam-6570	50	19	weakly	weakly	ADJ
ejpam-6570	50	20	(	(	PUNCT
ejpam-6570	50	21	τ1	τ1	NOUN
ejpam-6570	50	22	,	,	PUNCT
ejpam-6570	50	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	50	24	multifunctions	multifunction	NOUN
ejpam-6570	50	25	.	.	PUNCT
ejpam-6570	51	1	on	on	ADP
ejpam-6570	51	2	the	the	DET
ejpam-6570	51	3	other	other	ADJ
ejpam-6570	51	4	hand	hand	NOUN
ejpam-6570	51	5	,	,	PUNCT
ejpam-6570	51	6	the	the	DET
ejpam-6570	51	7	present	present	ADJ
ejpam-6570	51	8	authors	author	NOUN
ejpam-6570	51	9	introduced	introduce	VERB
ejpam-6570	51	10	and	and	CCONJ
ejpam-6570	51	11	investigated	investigate	VERB
ejpam-6570	51	12	the	the	DET
ejpam-6570	51	13	concepts	concept	NOUN
ejpam-6570	51	14	of	of	ADP
ejpam-6570	51	15	upper	upper	ADJ
ejpam-6570	51	16	almost	almost	ADV
ejpam-6570	51	17	weakly	weakly	ADJ
ejpam-6570	51	18	(	(	PUNCT
ejpam-6570	51	19	τ1	τ1	NOUN
ejpam-6570	51	20	,	,	PUNCT
ejpam-6570	51	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	51	22	multifunctions	multifunction	NOUN
ejpam-6570	51	23	and	and	CCONJ
ejpam-6570	51	24	lower	low	ADJ
ejpam-6570	51	25	almost	almost	ADV
ejpam-6570	51	26	weakly	weakly	ADJ
ejpam-6570	51	27	(	(	PUNCT
ejpam-6570	51	28	τ1	τ1	NOUN
ejpam-6570	51	29	,	,	PUNCT
ejpam-6570	51	30	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	51	31	multifunctions	multifunction	NOUN
ejpam-6570	52	1	[	[	X
ejpam-6570	52	2	20	20	NUM
ejpam-6570	52	3	]	]	PUNCT
ejpam-6570	52	4	.	.	PUNCT
ejpam-6570	53	1	in	in	ADP
ejpam-6570	53	2	this	this	DET
ejpam-6570	53	3	paper	paper	NOUN
ejpam-6570	53	4	,	,	PUNCT
ejpam-6570	53	5	we	we	PRON
ejpam-6570	53	6	introduce	introduce	VERB
ejpam-6570	53	7	the	the	DET
ejpam-6570	53	8	concepts	concept	NOUN
ejpam-6570	53	9	of	of	ADP
ejpam-6570	53	10	upper	upper	ADJ
ejpam-6570	53	11	almost	almost	ADV
ejpam-6570	53	12	weakly	weakly	ADJ
ejpam-6570	53	13	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	53	14	,	,	PUNCT
ejpam-6570	53	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	53	16	multifunctions	multifunction	NOUN
ejpam-6570	53	17	and	and	CCONJ
ejpam-6570	53	18	lower	low	ADJ
ejpam-6570	53	19	almost	almost	ADV
ejpam-6570	53	20	weakly	weakly	ADJ
ejpam-6570	53	21	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	53	22	,	,	PUNCT
ejpam-6570	53	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	53	24	multifunctions	multifunction	NOUN
ejpam-6570	53	25	.	.	PUNCT
ejpam-6570	54	1	we	we	PRON
ejpam-6570	54	2	also	also	ADV
ejpam-6570	54	3	investigate	investigate	VERB
ejpam-6570	54	4	several	several	ADJ
ejpam-6570	54	5	characterizations	characterization	NOUN
ejpam-6570	54	6	of	of	ADP
ejpam-6570	54	7	upper	upper	ADJ
ejpam-6570	54	8	almost	almost	ADV
ejpam-6570	54	9	weakly	weakly	ADJ
ejpam-6570	54	10	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	54	11	,	,	PUNCT
ejpam-6570	54	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	54	13	multifunctions	multifunction	NOUN
ejpam-6570	54	14	and	and	CCONJ
ejpam-6570	54	15	lower	low	ADJ
ejpam-6570	54	16	almost	almost	ADV
ejpam-6570	54	17	weakly	weakly	ADJ
ejpam-6570	54	18	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	54	19	,	,	PUNCT
ejpam-6570	54	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	54	21	multifunctions	multifunction	NOUN
ejpam-6570	54	22	.	.	PUNCT
ejpam-6570	55	1	2	2	X
ejpam-6570	55	2	.	.	X
ejpam-6570	55	3	preliminaries	preliminary	NOUN
ejpam-6570	55	4	throughout	throughout	ADP
ejpam-6570	55	5	the	the	DET
ejpam-6570	55	6	present	present	ADJ
ejpam-6570	55	7	paper	paper	NOUN
ejpam-6570	55	8	,	,	PUNCT
ejpam-6570	55	9	spaces	space	NOUN
ejpam-6570	55	10	(	(	PUNCT
ejpam-6570	55	11	x	x	NOUN
ejpam-6570	55	12	,	,	PUNCT
ejpam-6570	55	13	τ1	τ1	NOUN
ejpam-6570	55	14	,	,	PUNCT
ejpam-6570	55	15	τ2	τ2	NOUN
ejpam-6570	55	16	)	)	PUNCT
ejpam-6570	55	17	and	and	CCONJ
ejpam-6570	55	18	(	(	PUNCT
ejpam-6570	55	19	y	y	PROPN
ejpam-6570	55	20	,	,	PUNCT
ejpam-6570	55	21	σ1	σ1	PROPN
ejpam-6570	55	22	,	,	PUNCT
ejpam-6570	55	23	σ2	σ2	NOUN
ejpam-6570	55	24	)	)	PUNCT
ejpam-6570	55	25	(	(	PUNCT
ejpam-6570	55	26	or	or	CCONJ
ejpam-6570	55	27	simply	simply	ADV
ejpam-6570	55	28	x	x	X
ejpam-6570	55	29	and	and	CCONJ
ejpam-6570	55	30	y	y	PROPN
ejpam-6570	55	31	)	)	PUNCT
ejpam-6570	55	32	always	always	ADV
ejpam-6570	55	33	mean	mean	VERB
ejpam-6570	55	34	bitopological	bitopological	ADJ
ejpam-6570	55	35	spaces	space	NOUN
ejpam-6570	55	36	on	on	ADP
ejpam-6570	55	37	which	which	PRON
ejpam-6570	55	38	no	no	DET
ejpam-6570	55	39	separation	separation	NOUN
ejpam-6570	55	40	axioms	axiom	NOUN
ejpam-6570	55	41	are	be	AUX
ejpam-6570	55	42	assumed	assume	VERB
ejpam-6570	55	43	unless	unless	SCONJ
ejpam-6570	55	44	explicitly	explicitly	ADV
ejpam-6570	55	45	stated	state	VERB
ejpam-6570	55	46	.	.	PUNCT
ejpam-6570	56	1	let	let	VERB
ejpam-6570	56	2	a	a	DET
ejpam-6570	56	3	be	be	AUX
ejpam-6570	56	4	a	a	DET
ejpam-6570	56	5	subset	subset	NOUN
ejpam-6570	56	6	of	of	ADP
ejpam-6570	56	7	a	a	DET
ejpam-6570	56	8	bitopological	bitopological	ADJ
ejpam-6570	56	9	space	space	NOUN
ejpam-6570	56	10	(	(	PUNCT
ejpam-6570	56	11	x	x	NOUN
ejpam-6570	56	12	,	,	PUNCT
ejpam-6570	56	13	τ1	τ1	NOUN
ejpam-6570	56	14	,	,	PUNCT
ejpam-6570	56	15	τ2	τ2	NOUN
ejpam-6570	56	16	)	)	PUNCT
ejpam-6570	56	17	.	.	PUNCT
ejpam-6570	57	1	the	the	DET
ejpam-6570	57	2	closure	closure	NOUN
ejpam-6570	57	3	of	of	ADP
ejpam-6570	57	4	a	a	PRON
ejpam-6570	57	5	and	and	CCONJ
ejpam-6570	57	6	the	the	DET
ejpam-6570	57	7	interior	interior	NOUN
ejpam-6570	57	8	of	of	ADP
ejpam-6570	57	9	a	a	PRON
ejpam-6570	57	10	with	with	ADP
ejpam-6570	57	11	respect	respect	NOUN
ejpam-6570	57	12	to	to	ADP
ejpam-6570	57	13	τi	τi	PROPN
ejpam-6570	57	14	are	be	AUX
ejpam-6570	57	15	denoted	denote	VERB
ejpam-6570	57	16	by	by	ADP
ejpam-6570	57	17	τi	τi	NOUN
ejpam-6570	57	18	-	-	PUNCT
ejpam-6570	57	19	cl(a	cl(a	NUM
ejpam-6570	57	20	)	)	PUNCT
ejpam-6570	57	21	and	and	CCONJ
ejpam-6570	57	22	τi	τi	NOUN
ejpam-6570	57	23	-	-	PUNCT
ejpam-6570	57	24	int(a	int(a	NOUN
ejpam-6570	57	25	)	)	PUNCT
ejpam-6570	57	26	,	,	PUNCT
ejpam-6570	57	27	respectively	respectively	ADV
ejpam-6570	57	28	,	,	PUNCT
ejpam-6570	57	29	for	for	ADP
ejpam-6570	57	30	i	i	PROPN
ejpam-6570	57	31	=	=	SYM
ejpam-6570	57	32	1	1	NUM
ejpam-6570	57	33	,	,	PUNCT
ejpam-6570	57	34	2	2	NUM
ejpam-6570	57	35	.	.	X
ejpam-6570	57	36	a	a	DET
ejpam-6570	57	37	subset	subset	NOUN
ejpam-6570	57	38	a	a	PRON
ejpam-6570	57	39	of	of	ADP
ejpam-6570	57	40	a	a	DET
ejpam-6570	57	41	bitopological	bitopological	ADJ
ejpam-6570	57	42	space	space	NOUN
ejpam-6570	57	43	(	(	PUNCT
ejpam-6570	57	44	x	x	NOUN
ejpam-6570	57	45	,	,	PUNCT
ejpam-6570	57	46	τ1	τ1	NOUN
ejpam-6570	57	47	,	,	PUNCT
ejpam-6570	57	48	τ2	τ2	NOUN
ejpam-6570	57	49	)	)	PUNCT
ejpam-6570	57	50	is	be	AUX
ejpam-6570	57	51	called	call	VERB
ejpam-6570	57	52	τ1τ2	τ1τ2	VERB
ejpam-6570	57	53	-	-	ADJ
ejpam-6570	57	54	closed	closed	ADJ
ejpam-6570	57	55	[	[	X
ejpam-6570	57	56	21	21	NUM
ejpam-6570	57	57	]	]	X
ejpam-6570	57	58	if	if	SCONJ
ejpam-6570	57	59	a	a	DET
ejpam-6570	57	60	=	=	NOUN
ejpam-6570	57	61	τ1	τ1	NOUN
ejpam-6570	57	62	-	-	PUNCT
ejpam-6570	57	63	cl(τ2	cl(τ2	NOUN
ejpam-6570	57	64	-	-	PUNCT
ejpam-6570	57	65	cl(a	cl(a	NUM
ejpam-6570	57	66	)	)	PUNCT
ejpam-6570	57	67	)	)	PUNCT
ejpam-6570	57	68	.	.	PUNCT
ejpam-6570	58	1	the	the	DET
ejpam-6570	58	2	complement	complement	NOUN
ejpam-6570	58	3	of	of	ADP
ejpam-6570	58	4	a	a	DET
ejpam-6570	58	5	τ1τ2	τ1τ2	ADJ
ejpam-6570	58	6	-	-	ADJ
ejpam-6570	58	7	closed	closed	ADJ
ejpam-6570	58	8	set	set	NOUN
ejpam-6570	58	9	is	be	AUX
ejpam-6570	58	10	called	call	VERB
ejpam-6570	58	11	τ1τ2	τ1τ2	NOUN
ejpam-6570	58	12	-	-	ADJ
ejpam-6570	58	13	open	open	ADJ
ejpam-6570	58	14	.	.	PUNCT
ejpam-6570	59	1	let	let	VERB
ejpam-6570	59	2	a	a	DET
ejpam-6570	59	3	be	be	AUX
ejpam-6570	59	4	a	a	DET
ejpam-6570	59	5	subset	subset	NOUN
ejpam-6570	59	6	of	of	ADP
ejpam-6570	59	7	a	a	DET
ejpam-6570	59	8	bitopological	bitopological	ADJ
ejpam-6570	59	9	space	space	NOUN
ejpam-6570	59	10	(	(	PUNCT
ejpam-6570	59	11	x	x	NOUN
ejpam-6570	59	12	,	,	PUNCT
ejpam-6570	59	13	τ1	τ1	NOUN
ejpam-6570	59	14	,	,	PUNCT
ejpam-6570	59	15	τ2	τ2	NOUN
ejpam-6570	59	16	)	)	PUNCT
ejpam-6570	59	17	.	.	PUNCT
ejpam-6570	60	1	the	the	DET
ejpam-6570	60	2	intersection	intersection	NOUN
ejpam-6570	60	3	of	of	ADP
ejpam-6570	60	4	all	all	DET
ejpam-6570	60	5	τ1τ2	τ1τ2	ADJ
ejpam-6570	60	6	-	-	ADJ
ejpam-6570	60	7	closed	closed	ADJ
ejpam-6570	60	8	sets	set	NOUN
ejpam-6570	60	9	of	of	ADP
ejpam-6570	60	10	x	x	PUNCT
ejpam-6570	60	11	containing	contain	VERB
ejpam-6570	60	12	a	a	PRON
ejpam-6570	60	13	is	be	AUX
ejpam-6570	60	14	called	call	VERB
ejpam-6570	60	15	the	the	DET
ejpam-6570	60	16	τ1τ2	τ1τ2	NOUN
ejpam-6570	60	17	-	-	NOUN
ejpam-6570	60	18	closure	closure	NOUN
ejpam-6570	60	19	[	[	X
ejpam-6570	60	20	21	21	NUM
ejpam-6570	60	21	]	]	PUNCT
ejpam-6570	60	22	of	of	ADP
ejpam-6570	60	23	a	a	PRON
ejpam-6570	60	24	and	and	CCONJ
ejpam-6570	60	25	is	be	AUX
ejpam-6570	60	26	denoted	denote	VERB
ejpam-6570	60	27	by	by	ADP
ejpam-6570	60	28	τ1τ2	τ1τ2	NOUN
ejpam-6570	60	29	-	-	NUM
ejpam-6570	60	30	cl(a	cl(a	NUM
ejpam-6570	60	31	)	)	PUNCT
ejpam-6570	60	32	.	.	PUNCT
ejpam-6570	61	1	the	the	DET
ejpam-6570	61	2	union	union	NOUN
ejpam-6570	61	3	of	of	ADP
ejpam-6570	61	4	all	all	DET
ejpam-6570	61	5	τ1τ2	τ1τ2	ADJ
ejpam-6570	61	6	-	-	ADJ
ejpam-6570	61	7	open	open	ADJ
ejpam-6570	61	8	sets	set	NOUN
ejpam-6570	61	9	of	of	ADP
ejpam-6570	61	10	x	x	PUNCT
ejpam-6570	61	11	contained	contain	VERB
ejpam-6570	61	12	in	in	ADP
ejpam-6570	61	13	a	a	PRON
ejpam-6570	61	14	is	be	AUX
ejpam-6570	61	15	called	call	VERB
ejpam-6570	61	16	the	the	DET
ejpam-6570	61	17	τ1τ2	τ1τ2	NOUN
ejpam-6570	61	18	-	-	ADJ
ejpam-6570	61	19	interior	interior	ADJ
ejpam-6570	61	20	[	[	X
ejpam-6570	61	21	21	21	NUM
ejpam-6570	61	22	]	]	PUNCT
ejpam-6570	61	23	of	of	ADP
ejpam-6570	61	24	a	a	PRON
ejpam-6570	61	25	and	and	CCONJ
ejpam-6570	61	26	is	be	AUX
ejpam-6570	61	27	denoted	denote	VERB
ejpam-6570	61	28	by	by	ADP
ejpam-6570	61	29	τ1τ2	τ1τ2	NOUN
ejpam-6570	61	30	-	-	ADJ
ejpam-6570	61	31	int(a	int(a	NOUN
ejpam-6570	61	32	)	)	PUNCT
ejpam-6570	61	33	.	.	PUNCT
ejpam-6570	62	1	a	a	DET
ejpam-6570	62	2	subset	subset	NOUN
ejpam-6570	62	3	a	a	PRON
ejpam-6570	62	4	of	of	ADP
ejpam-6570	62	5	a	a	DET
ejpam-6570	62	6	bitopological	bitopological	ADJ
ejpam-6570	62	7	space	space	NOUN
ejpam-6570	62	8	(	(	PUNCT
ejpam-6570	62	9	x	x	NOUN
ejpam-6570	62	10	,	,	PUNCT
ejpam-6570	62	11	τ1	τ1	NOUN
ejpam-6570	62	12	,	,	PUNCT
ejpam-6570	62	13	τ2	τ2	NOUN
ejpam-6570	62	14	)	)	PUNCT
ejpam-6570	62	15	is	be	AUX
ejpam-6570	62	16	said	say	VERB
ejpam-6570	62	17	to	to	PART
ejpam-6570	62	18	be	be	AUX
ejpam-6570	62	19	τ1τ2	τ1τ2	NOUN
ejpam-6570	62	20	-	-	ADJ
ejpam-6570	62	21	clopen	clopen	ADJ
ejpam-6570	63	1	[	[	X
ejpam-6570	63	2	21	21	NUM
ejpam-6570	63	3	]	]	PUNCT
ejpam-6570	63	4	if	if	SCONJ
ejpam-6570	63	5	c.	c.	PROPN
ejpam-6570	63	6	viriyapong	viriyapong	PROPN
ejpam-6570	63	7	,	,	PUNCT
ejpam-6570	63	8	a.	a.	PROPN
ejpam-6570	63	9	sama	sama	PROPN
ejpam-6570	63	10	-	-	PUNCT
ejpam-6570	63	11	ae	ae	PROPN
ejpam-6570	63	12	,	,	PUNCT
ejpam-6570	63	13	c.	c.	PROPN
ejpam-6570	63	14	boonpok	boonpok	PROPN
ejpam-6570	63	15	/	/	SYM
ejpam-6570	63	16	eur	eur	PROPN
ejpam-6570	63	17	.	.	PUNCT
ejpam-6570	64	1	j.	j.	PROPN
ejpam-6570	64	2	pure	pure	PROPN
ejpam-6570	64	3	appl	appl	PROPN
ejpam-6570	64	4	.	.	PROPN
ejpam-6570	64	5	math	math	PROPN
ejpam-6570	64	6	,	,	PUNCT
ejpam-6570	64	7	18	18	NUM
ejpam-6570	64	8	(	(	PUNCT
ejpam-6570	64	9	3	3	NUM
ejpam-6570	64	10	)	)	PUNCT
ejpam-6570	64	11	(	(	PUNCT
ejpam-6570	64	12	2025	2025	NUM
ejpam-6570	64	13	)	)	PUNCT
ejpam-6570	64	14	,	,	PUNCT
ejpam-6570	64	15	6570	6570	NUM
ejpam-6570	64	16	3	3	NUM
ejpam-6570	64	17	of	of	ADP
ejpam-6570	64	18	14	14	NUM
ejpam-6570	64	19	a	a	PRON
ejpam-6570	64	20	is	be	AUX
ejpam-6570	64	21	both	both	PRON
ejpam-6570	64	22	τ1τ2	τ1τ2	ADJ
ejpam-6570	64	23	-	-	ADJ
ejpam-6570	64	24	open	open	ADJ
ejpam-6570	64	25	and	and	CCONJ
ejpam-6570	64	26	τ1τ2	τ1τ2	NOUN
ejpam-6570	64	27	-	-	ADJ
ejpam-6570	64	28	closed	closed	ADJ
ejpam-6570	64	29	.	.	PUNCT
ejpam-6570	65	1	a	a	DET
ejpam-6570	65	2	subset	subset	NOUN
ejpam-6570	65	3	a	a	PRON
ejpam-6570	65	4	of	of	ADP
ejpam-6570	65	5	a	a	DET
ejpam-6570	65	6	bitopological	bitopological	ADJ
ejpam-6570	65	7	space	space	NOUN
ejpam-6570	65	8	(	(	PUNCT
ejpam-6570	65	9	x	x	NOUN
ejpam-6570	65	10	,	,	PUNCT
ejpam-6570	65	11	τ1	τ1	NOUN
ejpam-6570	65	12	,	,	PUNCT
ejpam-6570	65	13	τ2	τ2	NOUN
ejpam-6570	65	14	)	)	PUNCT
ejpam-6570	65	15	is	be	AUX
ejpam-6570	65	16	said	say	VERB
ejpam-6570	65	17	to	to	PART
ejpam-6570	65	18	be	be	AUX
ejpam-6570	65	19	(	(	PUNCT
ejpam-6570	65	20	τ1	τ1	NOUN
ejpam-6570	65	21	,	,	PUNCT
ejpam-6570	65	22	τ2)r	τ2)r	NOUN
ejpam-6570	65	23	-	-	PUNCT
ejpam-6570	65	24	open	open	NOUN
ejpam-6570	66	1	[	[	X
ejpam-6570	66	2	22	22	NUM
ejpam-6570	66	3	]	]	PUNCT
ejpam-6570	66	4	(	(	PUNCT
ejpam-6570	66	5	resp	resp	NOUN
ejpam-6570	66	6	.	.	PUNCT
ejpam-6570	67	1	(	(	PUNCT
ejpam-6570	67	2	τ1	τ1	NOUN
ejpam-6570	67	3	,	,	PUNCT
ejpam-6570	67	4	τ2)s	τ2)s	NOUN
ejpam-6570	67	5	-	-	PUNCT
ejpam-6570	67	6	open	open	ADJ
ejpam-6570	67	7	[	[	X
ejpam-6570	67	8	23	23	NUM
ejpam-6570	67	9	]	]	PUNCT
ejpam-6570	67	10	,	,	PUNCT
ejpam-6570	67	11	(	(	PUNCT
ejpam-6570	67	12	τ1	τ1	NOUN
ejpam-6570	67	13	,	,	PUNCT
ejpam-6570	67	14	τ2)p	τ2)p	NOUN
ejpam-6570	67	15	-	-	ADJ
ejpam-6570	67	16	open	open	ADJ
ejpam-6570	67	17	[	[	X
ejpam-6570	67	18	23	23	NUM
ejpam-6570	67	19	]	]	PUNCT
ejpam-6570	67	20	,	,	PUNCT
ejpam-6570	67	21	(	(	PUNCT
ejpam-6570	67	22	τ1	τ1	NOUN
ejpam-6570	67	23	,	,	PUNCT
ejpam-6570	67	24	τ2)β	τ2)β	ADJ
ejpam-6570	67	25	-	-	PUNCT
ejpam-6570	67	26	open	open	NOUN
ejpam-6570	68	1	[	[	X
ejpam-6570	68	2	23	23	NUM
ejpam-6570	68	3	]	]	SYM
ejpam-6570	68	4	)	)	PUNCT
ejpam-6570	68	5	if	if	SCONJ
ejpam-6570	68	6	a	a	DET
ejpam-6570	68	7	=	=	PUNCT
ejpam-6570	68	8	τ1τ2	τ1τ2	NOUN
ejpam-6570	68	9	-	-	NOUN
ejpam-6570	68	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6570	68	11	-	-	PUNCT
ejpam-6570	68	12	cl(a	cl(a	NUM
ejpam-6570	68	13	)	)	PUNCT
ejpam-6570	68	14	)	)	PUNCT
ejpam-6570	68	15	(	(	PUNCT
ejpam-6570	68	16	resp	resp	NOUN
ejpam-6570	68	17	.	.	PUNCT
ejpam-6570	69	1	a	a	DET
ejpam-6570	69	2	⊆	⊆	NUM
ejpam-6570	69	3	τ1τ2	τ1τ2	NOUN
ejpam-6570	69	4	-	-	ADJ
ejpam-6570	69	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6570	69	6	-	-	PUNCT
ejpam-6570	69	7	int(a	int(a	NOUN
ejpam-6570	69	8	)	)	PUNCT
ejpam-6570	69	9	)	)	PUNCT
ejpam-6570	69	10	,	,	PUNCT
ejpam-6570	69	11	a	a	DET
ejpam-6570	69	12	⊆	⊆	NUM
ejpam-6570	69	13	τ1τ2	τ1τ2	NOUN
ejpam-6570	69	14	-	-	NOUN
ejpam-6570	69	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6570	69	16	-	-	PUNCT
ejpam-6570	69	17	cl(a	cl(a	NUM
ejpam-6570	69	18	)	)	PUNCT
ejpam-6570	69	19	)	)	PUNCT
ejpam-6570	69	20	,	,	PUNCT
ejpam-6570	69	21	a	a	DET
ejpam-6570	69	22	⊆	⊆	NUM
ejpam-6570	69	23	τ1τ2	τ1τ2	NOUN
ejpam-6570	69	24	-	-	PUNCT
ejpam-6570	69	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6570	69	26	-	-	PUNCT
ejpam-6570	69	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6570	69	28	-	-	PUNCT
ejpam-6570	69	29	cl(a	cl(a	NUM
ejpam-6570	69	30	)	)	PUNCT
ejpam-6570	69	31	)	)	PUNCT
ejpam-6570	69	32	)	)	PUNCT
ejpam-6570	69	33	)	)	PUNCT
ejpam-6570	69	34	.	.	PUNCT
ejpam-6570	70	1	the	the	DET
ejpam-6570	70	2	complement	complement	NOUN
ejpam-6570	70	3	of	of	ADP
ejpam-6570	70	4	a	a	DET
ejpam-6570	70	5	(	(	PUNCT
ejpam-6570	70	6	τ1	τ1	NOUN
ejpam-6570	70	7	,	,	PUNCT
ejpam-6570	70	8	τ2)r	τ2)r	NOUN
ejpam-6570	70	9	-	-	PUNCT
ejpam-6570	70	10	open	open	ADJ
ejpam-6570	70	11	(	(	PUNCT
ejpam-6570	70	12	resp	resp	NOUN
ejpam-6570	70	13	.	.	PUNCT
ejpam-6570	71	1	(	(	PUNCT
ejpam-6570	71	2	τ1	τ1	NOUN
ejpam-6570	71	3	,	,	PUNCT
ejpam-6570	71	4	τ2)sopen	τ2)sopen	ADJ
ejpam-6570	71	5	,	,	PUNCT
ejpam-6570	71	6	(	(	PUNCT
ejpam-6570	71	7	τ1	τ1	NOUN
ejpam-6570	71	8	,	,	PUNCT
ejpam-6570	71	9	τ2)p	τ2)p	NOUN
ejpam-6570	71	10	-	-	ADJ
ejpam-6570	71	11	open	open	ADJ
ejpam-6570	71	12	,	,	PUNCT
ejpam-6570	71	13	(	(	PUNCT
ejpam-6570	71	14	τ1	τ1	NOUN
ejpam-6570	71	15	,	,	PUNCT
ejpam-6570	71	16	τ2)β	τ2)β	ADJ
ejpam-6570	71	17	-	-	PUNCT
ejpam-6570	71	18	open	open	ADJ
ejpam-6570	71	19	)	)	PUNCT
ejpam-6570	71	20	set	set	NOUN
ejpam-6570	71	21	is	be	AUX
ejpam-6570	71	22	called	call	VERB
ejpam-6570	71	23	(	(	PUNCT
ejpam-6570	71	24	τ1	τ1	NOUN
ejpam-6570	71	25	,	,	PUNCT
ejpam-6570	71	26	τ2)r	τ2)r	NOUN
ejpam-6570	71	27	-	-	PUNCT
ejpam-6570	71	28	closed	closed	ADJ
ejpam-6570	71	29	(	(	PUNCT
ejpam-6570	71	30	resp	resp	NOUN
ejpam-6570	71	31	.	.	PUNCT
ejpam-6570	72	1	(	(	PUNCT
ejpam-6570	72	2	τ1	τ1	NOUN
ejpam-6570	72	3	,	,	PUNCT
ejpam-6570	72	4	τ2)s	τ2)s	NOUN
ejpam-6570	72	5	-	-	PUNCT
ejpam-6570	72	6	closed	closed	ADJ
ejpam-6570	72	7	,	,	PUNCT
ejpam-6570	72	8	(	(	PUNCT
ejpam-6570	72	9	τ1	τ1	NOUN
ejpam-6570	72	10	,	,	PUNCT
ejpam-6570	72	11	τ2)p	τ2)p	NOUN
ejpam-6570	72	12	-	-	PUNCT
ejpam-6570	72	13	closed	closed	ADJ
ejpam-6570	72	14	,	,	PUNCT
ejpam-6570	72	15	(	(	PUNCT
ejpam-6570	72	16	τ1	τ1	NOUN
ejpam-6570	72	17	,	,	PUNCT
ejpam-6570	72	18	τ2)β	τ2)β	ADJ
ejpam-6570	72	19	-	-	PUNCT
ejpam-6570	72	20	closed	closed	ADJ
ejpam-6570	72	21	)	)	PUNCT
ejpam-6570	72	22	.	.	PUNCT
ejpam-6570	73	1	a	a	DET
ejpam-6570	73	2	subset	subset	NOUN
ejpam-6570	73	3	a	a	PRON
ejpam-6570	73	4	of	of	ADP
ejpam-6570	73	5	a	a	DET
ejpam-6570	73	6	bitopological	bitopological	ADJ
ejpam-6570	73	7	space	space	NOUN
ejpam-6570	73	8	(	(	PUNCT
ejpam-6570	73	9	x	x	NOUN
ejpam-6570	73	10	,	,	PUNCT
ejpam-6570	73	11	τ1	τ1	NOUN
ejpam-6570	73	12	,	,	PUNCT
ejpam-6570	73	13	τ2	τ2	NOUN
ejpam-6570	73	14	)	)	PUNCT
ejpam-6570	73	15	is	be	AUX
ejpam-6570	73	16	said	say	VERB
ejpam-6570	73	17	to	to	PART
ejpam-6570	73	18	be	be	AUX
ejpam-6570	73	19	α(τ1	α(τ1	NOUN
ejpam-6570	73	20	,	,	PUNCT
ejpam-6570	73	21	τ2)-open	τ2)-open	ADJ
ejpam-6570	73	22	[	[	X
ejpam-6570	73	23	24	24	NUM
ejpam-6570	73	24	]	]	X
ejpam-6570	73	25	if	if	SCONJ
ejpam-6570	73	26	a	a	DET
ejpam-6570	73	27	⊆	⊆	NUM
ejpam-6570	73	28	τ1τ2	τ1τ2	NOUN
ejpam-6570	73	29	-	-	PUNCT
ejpam-6570	73	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6570	73	31	-	-	PUNCT
ejpam-6570	73	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6570	73	33	-	-	PUNCT
ejpam-6570	73	34	int(a	int(a	NOUN
ejpam-6570	73	35	)	)	PUNCT
ejpam-6570	73	36	)	)	PUNCT
ejpam-6570	73	37	)	)	PUNCT
ejpam-6570	73	38	.	.	PUNCT
ejpam-6570	74	1	the	the	DET
ejpam-6570	74	2	complement	complement	NOUN
ejpam-6570	74	3	of	of	ADP
ejpam-6570	74	4	an	an	DET
ejpam-6570	74	5	α(τ1	α(τ1	NOUN
ejpam-6570	74	6	,	,	PUNCT
ejpam-6570	74	7	τ2)-open	τ2)-open	ADJ
ejpam-6570	74	8	set	set	NOUN
ejpam-6570	74	9	is	be	AUX
ejpam-6570	74	10	said	say	VERB
ejpam-6570	74	11	to	to	PART
ejpam-6570	74	12	be	be	AUX
ejpam-6570	74	13	α(τ1	α(τ1	NOUN
ejpam-6570	74	14	,	,	PUNCT
ejpam-6570	74	15	τ2)-closed	τ2)-closed	ADJ
ejpam-6570	74	16	.	.	PUNCT
ejpam-6570	75	1	for	for	ADP
ejpam-6570	75	2	a	a	DET
ejpam-6570	75	3	subset	subset	NOUN
ejpam-6570	75	4	a	a	PRON
ejpam-6570	75	5	of	of	ADP
ejpam-6570	75	6	a	a	DET
ejpam-6570	75	7	bitopological	bitopological	ADJ
ejpam-6570	75	8	space	space	NOUN
ejpam-6570	75	9	(	(	PUNCT
ejpam-6570	75	10	x	x	NOUN
ejpam-6570	75	11	,	,	PUNCT
ejpam-6570	75	12	τ1	τ1	NOUN
ejpam-6570	75	13	,	,	PUNCT
ejpam-6570	75	14	τ2	τ2	PROPN
ejpam-6570	75	15	)	)	PUNCT
ejpam-6570	75	16	,	,	PUNCT
ejpam-6570	75	17	a	a	DET
ejpam-6570	75	18	point	point	NOUN
ejpam-6570	75	19	x	x	X
ejpam-6570	75	20	∈	∈	NOUN
ejpam-6570	75	21	x	x	PUNCT
ejpam-6570	75	22	is	be	AUX
ejpam-6570	75	23	called	call	VERB
ejpam-6570	75	24	a	a	DET
ejpam-6570	75	25	(	(	PUNCT
ejpam-6570	75	26	τ1	τ1	NOUN
ejpam-6570	75	27	,	,	PUNCT
ejpam-6570	75	28	τ2)θcluster	τ2)θcluster	NOUN
ejpam-6570	75	29	point	point	NOUN
ejpam-6570	75	30	of	of	ADP
ejpam-6570	75	31	a	a	DET
ejpam-6570	75	32	if	if	SCONJ
ejpam-6570	75	33	τ1τ2	τ1τ2	NOUN
ejpam-6570	75	34	-	-	NOUN
ejpam-6570	75	35	cl(u	cl(u	NOUN
ejpam-6570	75	36	)	)	PUNCT
ejpam-6570	75	37	∩	∩	NOUN
ejpam-6570	75	38	a	a	DET
ejpam-6570	75	39	̸=	̸=	PROPN
ejpam-6570	75	40	∅	∅	NOUN
ejpam-6570	75	41	for	for	ADP
ejpam-6570	75	42	every	every	DET
ejpam-6570	75	43	τ1τ2	τ1τ2	ADJ
ejpam-6570	75	44	-	-	ADJ
ejpam-6570	75	45	open	open	ADJ
ejpam-6570	75	46	set	set	NOUN
ejpam-6570	75	47	u	u	NOUN
ejpam-6570	75	48	containing	contain	VERB
ejpam-6570	75	49	x.	x.	NOUN
ejpam-6570	75	50	the	the	DET
ejpam-6570	75	51	set	set	NOUN
ejpam-6570	75	52	of	of	ADP
ejpam-6570	75	53	all	all	DET
ejpam-6570	75	54	(	(	PUNCT
ejpam-6570	75	55	τ1	τ1	NOUN
ejpam-6570	75	56	,	,	PUNCT
ejpam-6570	75	57	τ2)θ	τ2)θ	ADJ
ejpam-6570	75	58	-	-	PUNCT
ejpam-6570	75	59	cluster	cluster	NOUN
ejpam-6570	75	60	points	point	NOUN
ejpam-6570	75	61	of	of	ADP
ejpam-6570	75	62	a	a	PRON
ejpam-6570	75	63	is	be	AUX
ejpam-6570	75	64	called	call	VERB
ejpam-6570	75	65	the	the	DET
ejpam-6570	75	66	(	(	PUNCT
ejpam-6570	75	67	τ1	τ1	NOUN
ejpam-6570	75	68	,	,	PUNCT
ejpam-6570	75	69	τ2)θ	τ2)θ	NOUN
ejpam-6570	75	70	-	-	PUNCT
ejpam-6570	75	71	closure	closure	NOUN
ejpam-6570	75	72	of	of	ADP
ejpam-6570	75	73	a	a	PRON
ejpam-6570	75	74	and	and	CCONJ
ejpam-6570	75	75	is	be	AUX
ejpam-6570	75	76	denoted	denote	VERB
ejpam-6570	75	77	by	by	ADP
ejpam-6570	75	78	(	(	PUNCT
ejpam-6570	75	79	τ1	τ1	NOUN
ejpam-6570	75	80	,	,	PUNCT
ejpam-6570	75	81	τ2)θ	τ2)θ	NOUN
ejpam-6570	75	82	-	-	PUNCT
ejpam-6570	75	83	cl(a	cl(a	NUM
ejpam-6570	75	84	)	)	PUNCT
ejpam-6570	75	85	.	.	PUNCT
ejpam-6570	76	1	a	a	DET
ejpam-6570	76	2	subset	subset	NOUN
ejpam-6570	76	3	a	a	PRON
ejpam-6570	76	4	of	of	ADP
ejpam-6570	76	5	a	a	DET
ejpam-6570	76	6	bitopological	bitopological	ADJ
ejpam-6570	76	7	space	space	NOUN
ejpam-6570	76	8	(	(	PUNCT
ejpam-6570	76	9	x	x	NOUN
ejpam-6570	76	10	,	,	PUNCT
ejpam-6570	76	11	τ1	τ1	NOUN
ejpam-6570	76	12	,	,	PUNCT
ejpam-6570	76	13	τ2	τ2	NOUN
ejpam-6570	76	14	)	)	PUNCT
ejpam-6570	76	15	is	be	AUX
ejpam-6570	76	16	said	say	VERB
ejpam-6570	76	17	to	to	PART
ejpam-6570	76	18	be	be	AUX
ejpam-6570	76	19	(	(	PUNCT
ejpam-6570	76	20	τ1	τ1	NOUN
ejpam-6570	76	21	,	,	PUNCT
ejpam-6570	76	22	τ2)θ	τ2)θ	NOUN
ejpam-6570	76	23	-	-	PUNCT
ejpam-6570	76	24	closed	closed	ADJ
ejpam-6570	76	25	if	if	SCONJ
ejpam-6570	76	26	(	(	PUNCT
ejpam-6570	76	27	τ1	τ1	NOUN
ejpam-6570	76	28	,	,	PUNCT
ejpam-6570	76	29	τ2)θ	τ2)θ	NOUN
ejpam-6570	76	30	-	-	PUNCT
ejpam-6570	76	31	cl(a	cl(a	NUM
ejpam-6570	76	32	)	)	PUNCT
ejpam-6570	76	33	=	=	PUNCT
ejpam-6570	76	34	a.	a.	NOUN
ejpam-6570	76	35	the	the	DET
ejpam-6570	76	36	complement	complement	NOUN
ejpam-6570	76	37	of	of	ADP
ejpam-6570	76	38	a	a	DET
ejpam-6570	76	39	(	(	PUNCT
ejpam-6570	76	40	τ1	τ1	NOUN
ejpam-6570	76	41	,	,	PUNCT
ejpam-6570	76	42	τ2)θ	τ2)θ	ADJ
ejpam-6570	76	43	-	-	PUNCT
ejpam-6570	76	44	closed	close	VERB
ejpam-6570	76	45	set	set	NOUN
ejpam-6570	76	46	is	be	AUX
ejpam-6570	76	47	said	say	VERB
ejpam-6570	76	48	to	to	PART
ejpam-6570	76	49	be	be	AUX
ejpam-6570	76	50	(	(	PUNCT
ejpam-6570	76	51	τ1	τ1	NOUN
ejpam-6570	76	52	,	,	PUNCT
ejpam-6570	76	53	τ2)θ	τ2)θ	NOUN
ejpam-6570	76	54	-	-	PUNCT
ejpam-6570	76	55	open	open	ADJ
ejpam-6570	76	56	.	.	PUNCT
ejpam-6570	77	1	the	the	DET
ejpam-6570	77	2	union	union	NOUN
ejpam-6570	77	3	of	of	ADP
ejpam-6570	77	4	all	all	DET
ejpam-6570	77	5	(	(	PUNCT
ejpam-6570	77	6	τ1	τ1	NOUN
ejpam-6570	77	7	,	,	PUNCT
ejpam-6570	77	8	τ2)θ	τ2)θ	ADJ
ejpam-6570	77	9	-	-	PUNCT
ejpam-6570	77	10	open	open	ADJ
ejpam-6570	77	11	sets	set	NOUN
ejpam-6570	77	12	of	of	ADP
ejpam-6570	77	13	x	x	PUNCT
ejpam-6570	77	14	contained	contain	VERB
ejpam-6570	77	15	in	in	ADP
ejpam-6570	77	16	a	a	PRON
ejpam-6570	77	17	is	be	AUX
ejpam-6570	77	18	called	call	VERB
ejpam-6570	77	19	the	the	DET
ejpam-6570	77	20	(	(	PUNCT
ejpam-6570	77	21	τ1	τ1	NOUN
ejpam-6570	77	22	,	,	PUNCT
ejpam-6570	77	23	τ2)θ	τ2)θ	ADJ
ejpam-6570	77	24	-	-	PUNCT
ejpam-6570	77	25	interior	interior	NOUN
ejpam-6570	77	26	of	of	ADP
ejpam-6570	77	27	a	a	PRON
ejpam-6570	77	28	and	and	CCONJ
ejpam-6570	77	29	is	be	AUX
ejpam-6570	77	30	denoted	denote	VERB
ejpam-6570	77	31	by	by	ADP
ejpam-6570	77	32	(	(	PUNCT
ejpam-6570	77	33	τ1	τ1	NOUN
ejpam-6570	77	34	,	,	PUNCT
ejpam-6570	77	35	τ2)θ	τ2)θ	NOUN
ejpam-6570	77	36	-	-	PUNCT
ejpam-6570	77	37	int(a	int(a	NOUN
ejpam-6570	77	38	)	)	PUNCT
ejpam-6570	78	1	[	[	X
ejpam-6570	78	2	22	22	NUM
ejpam-6570	78	3	]	]	PUNCT
ejpam-6570	78	4	.	.	PUNCT
ejpam-6570	79	1	lemma	lemma	PROPN
ejpam-6570	79	2	1	1	NUM
ejpam-6570	79	3	.	.	PUNCT
ejpam-6570	80	1	[	[	X
ejpam-6570	80	2	22	22	NUM
ejpam-6570	80	3	]	]	PUNCT
ejpam-6570	80	4	for	for	ADP
ejpam-6570	80	5	a	a	DET
ejpam-6570	80	6	subset	subset	NOUN
ejpam-6570	80	7	a	a	PRON
ejpam-6570	80	8	of	of	ADP
ejpam-6570	80	9	a	a	DET
ejpam-6570	80	10	bitopological	bitopological	ADJ
ejpam-6570	80	11	space	space	NOUN
ejpam-6570	80	12	(	(	PUNCT
ejpam-6570	80	13	x	x	NOUN
ejpam-6570	80	14	,	,	PUNCT
ejpam-6570	80	15	τ1	τ1	NOUN
ejpam-6570	80	16	,	,	PUNCT
ejpam-6570	80	17	τ2	τ2	NOUN
ejpam-6570	80	18	)	)	PUNCT
ejpam-6570	80	19	,	,	PUNCT
ejpam-6570	80	20	the	the	DET
ejpam-6570	80	21	following	follow	VERB
ejpam-6570	80	22	properties	property	NOUN
ejpam-6570	80	23	hold	hold	VERB
ejpam-6570	80	24	:	:	PUNCT
ejpam-6570	80	25	(	(	PUNCT
ejpam-6570	80	26	1	1	X
ejpam-6570	80	27	)	)	PUNCT
ejpam-6570	80	28	if	if	SCONJ
ejpam-6570	80	29	a	a	PRON
ejpam-6570	80	30	is	be	AUX
ejpam-6570	80	31	τ1τ2	τ1τ2	NOUN
ejpam-6570	80	32	-	-	ADJ
ejpam-6570	80	33	open	open	ADJ
ejpam-6570	80	34	in	in	ADP
ejpam-6570	80	35	x	x	NOUN
ejpam-6570	80	36	,	,	PUNCT
ejpam-6570	80	37	then	then	ADV
ejpam-6570	80	38	τ1τ2	τ1τ2	NOUN
ejpam-6570	80	39	-	-	NUM
ejpam-6570	80	40	cl(a	cl(a	NUM
ejpam-6570	80	41	)	)	PUNCT
ejpam-6570	80	42	=	=	PUNCT
ejpam-6570	80	43	(	(	PUNCT
ejpam-6570	80	44	τ1	τ1	NOUN
ejpam-6570	80	45	,	,	PUNCT
ejpam-6570	80	46	τ2)θ	τ2)θ	NOUN
ejpam-6570	80	47	-	-	PUNCT
ejpam-6570	80	48	cl(a	cl(a	NUM
ejpam-6570	80	49	)	)	PUNCT
ejpam-6570	80	50	.	.	PUNCT
ejpam-6570	81	1	(	(	PUNCT
ejpam-6570	81	2	2	2	X
ejpam-6570	81	3	)	)	PUNCT
ejpam-6570	81	4	(	(	PUNCT
ejpam-6570	81	5	τ1	τ1	NOUN
ejpam-6570	81	6	,	,	PUNCT
ejpam-6570	81	7	τ2)θ	τ2)θ	NOUN
ejpam-6570	81	8	-	-	PUNCT
ejpam-6570	81	9	cl(a	cl(a	NUM
ejpam-6570	81	10	)	)	PUNCT
ejpam-6570	81	11	is	be	AUX
ejpam-6570	81	12	τ1τ2	τ1τ2	NOUN
ejpam-6570	81	13	-	-	ADJ
ejpam-6570	81	14	closed	closed	ADJ
ejpam-6570	81	15	in	in	ADP
ejpam-6570	81	16	x.	x.	NOUN
ejpam-6570	81	17	an	an	DET
ejpam-6570	81	18	ideal	ideal	NOUN
ejpam-6570	81	19	i	i	PRON
ejpam-6570	81	20	on	on	ADP
ejpam-6570	81	21	a	a	DET
ejpam-6570	81	22	topological	topological	ADJ
ejpam-6570	81	23	space	space	NOUN
ejpam-6570	81	24	(	(	PUNCT
ejpam-6570	81	25	x	x	X
ejpam-6570	81	26	,	,	PUNCT
ejpam-6570	81	27	τ	τ	X
ejpam-6570	81	28	)	)	PUNCT
ejpam-6570	81	29	is	be	AUX
ejpam-6570	81	30	a	a	DET
ejpam-6570	81	31	nonempty	nonempty	ADJ
ejpam-6570	81	32	collection	collection	NOUN
ejpam-6570	81	33	of	of	ADP
ejpam-6570	81	34	subsets	subset	NOUN
ejpam-6570	81	35	of	of	ADP
ejpam-6570	81	36	x	x	PUNCT
ejpam-6570	81	37	satisfying	satisfy	VERB
ejpam-6570	81	38	the	the	DET
ejpam-6570	81	39	following	follow	VERB
ejpam-6570	81	40	properties	property	NOUN
ejpam-6570	81	41	:	:	PUNCT
ejpam-6570	81	42	(	(	PUNCT
ejpam-6570	81	43	1	1	X
ejpam-6570	81	44	)	)	PUNCT
ejpam-6570	81	45	a	a	DET
ejpam-6570	81	46	∈	∈	NOUN
ejpam-6570	82	1	i	i	PRON
ejpam-6570	82	2	and	and	CCONJ
ejpam-6570	82	3	b	b	X
ejpam-6570	82	4	⊆	⊆	NUM
ejpam-6570	82	5	a	a	DET
ejpam-6570	82	6	imply	imply	NOUN
ejpam-6570	82	7	b	b	X
ejpam-6570	82	8	∈	∈	PROPN
ejpam-6570	82	9	i	i	PRON
ejpam-6570	82	10	;	;	PUNCT
ejpam-6570	82	11	(	(	PUNCT
ejpam-6570	82	12	2	2	X
ejpam-6570	82	13	)	)	PUNCT
ejpam-6570	83	1	a	a	DET
ejpam-6570	83	2	∈	∈	NOUN
ejpam-6570	83	3	i	i	PRON
ejpam-6570	83	4	and	and	CCONJ
ejpam-6570	83	5	b	b	X
ejpam-6570	83	6	∈	∈	NOUN
ejpam-6570	84	1	i	i	PRON
ejpam-6570	84	2	imply	imply	VERB
ejpam-6570	84	3	a	a	DET
ejpam-6570	84	4	∪	∪	X
ejpam-6570	84	5	b	b	NOUN
ejpam-6570	84	6	∈	∈	NOUN
ejpam-6570	85	1	i	i	PRON
ejpam-6570	85	2	.	.	PUNCT
ejpam-6570	86	1	a	a	DET
ejpam-6570	86	2	topological	topological	ADJ
ejpam-6570	86	3	space	space	NOUN
ejpam-6570	86	4	(	(	PUNCT
ejpam-6570	86	5	x	x	X
ejpam-6570	86	6	,	,	PUNCT
ejpam-6570	86	7	τ	τ	X
ejpam-6570	86	8	)	)	PUNCT
ejpam-6570	86	9	with	with	ADP
ejpam-6570	86	10	an	an	DET
ejpam-6570	86	11	ideal	ideal	ADJ
ejpam-6570	86	12	i	i	PRON
ejpam-6570	86	13	on	on	ADP
ejpam-6570	86	14	x	x	SYM
ejpam-6570	86	15	is	be	AUX
ejpam-6570	86	16	called	call	VERB
ejpam-6570	86	17	an	an	DET
ejpam-6570	86	18	ideal	ideal	ADJ
ejpam-6570	86	19	topological	topological	ADJ
ejpam-6570	86	20	space	space	NOUN
ejpam-6570	86	21	and	and	CCONJ
ejpam-6570	86	22	is	be	AUX
ejpam-6570	86	23	denoted	denote	VERB
ejpam-6570	86	24	by	by	ADP
ejpam-6570	86	25	(	(	PUNCT
ejpam-6570	86	26	x	x	X
ejpam-6570	86	27	,	,	PUNCT
ejpam-6570	86	28	τ	τ	PROPN
ejpam-6570	86	29	,	,	PUNCT
ejpam-6570	86	30	i	i	NOUN
ejpam-6570	86	31	)	)	PUNCT
ejpam-6570	86	32	.	.	PUNCT
ejpam-6570	87	1	for	for	ADP
ejpam-6570	87	2	an	an	DET
ejpam-6570	87	3	ideal	ideal	ADJ
ejpam-6570	87	4	topological	topological	ADJ
ejpam-6570	87	5	space	space	NOUN
ejpam-6570	87	6	(	(	PUNCT
ejpam-6570	87	7	x	x	X
ejpam-6570	87	8	,	,	PUNCT
ejpam-6570	87	9	τ	τ	PROPN
ejpam-6570	87	10	,	,	PUNCT
ejpam-6570	87	11	i	i	PROPN
ejpam-6570	87	12	)	)	PUNCT
ejpam-6570	87	13	and	and	CCONJ
ejpam-6570	87	14	a	a	DET
ejpam-6570	87	15	subset	subset	NOUN
ejpam-6570	87	16	a	a	PRON
ejpam-6570	87	17	of	of	ADP
ejpam-6570	87	18	x	x	PRON
ejpam-6570	87	19	,	,	PUNCT
ejpam-6570	87	20	a⋆(i	a⋆(i	PROPN
ejpam-6570	87	21	)	)	PUNCT
ejpam-6570	87	22	is	be	AUX
ejpam-6570	87	23	defined	define	VERB
ejpam-6570	87	24	as	as	SCONJ
ejpam-6570	87	25	follows	follow	VERB
ejpam-6570	87	26	:	:	PUNCT
ejpam-6570	87	27	a⋆(i	a⋆(i	NOUN
ejpam-6570	87	28	)	)	PUNCT
ejpam-6570	88	1	=	=	PUNCT
ejpam-6570	88	2	{	{	PUNCT
ejpam-6570	88	3	x	x	PUNCT
ejpam-6570	88	4	∈	∈	PROPN
ejpam-6570	88	5	x	x	X
ejpam-6570	88	6	:	:	PUNCT
ejpam-6570	88	7	u	u	X
ejpam-6570	88	8	∩a	∩a	PROPN
ejpam-6570	88	9	̸∈	̸∈	PROPN
ejpam-6570	88	10	i	i	PRON
ejpam-6570	88	11	for	for	ADP
ejpam-6570	88	12	every	every	DET
ejpam-6570	88	13	open	open	ADJ
ejpam-6570	88	14	neighbourhood	neighbourhood	NOUN
ejpam-6570	88	15	u	u	NOUN
ejpam-6570	88	16	of	of	ADP
ejpam-6570	88	17	x	x	NOUN
ejpam-6570	88	18	}	}	PUNCT
ejpam-6570	88	19	.	.	PUNCT
ejpam-6570	89	1	in	in	ADP
ejpam-6570	89	2	case	case	NOUN
ejpam-6570	89	3	there	there	PRON
ejpam-6570	89	4	is	be	VERB
ejpam-6570	89	5	no	no	DET
ejpam-6570	89	6	chance	chance	NOUN
ejpam-6570	89	7	for	for	ADP
ejpam-6570	89	8	confusion	confusion	NOUN
ejpam-6570	89	9	,	,	PUNCT
ejpam-6570	89	10	a⋆(i	a⋆(i	NOUN
ejpam-6570	89	11	)	)	PUNCT
ejpam-6570	89	12	is	be	AUX
ejpam-6570	89	13	simply	simply	ADV
ejpam-6570	89	14	written	write	VERB
ejpam-6570	89	15	as	as	ADP
ejpam-6570	89	16	a⋆.	a⋆.	NOUN
ejpam-6570	89	17	in	in	ADP
ejpam-6570	89	18	[	[	X
ejpam-6570	89	19	25	25	NUM
ejpam-6570	89	20	]	]	PUNCT
ejpam-6570	89	21	,	,	PUNCT
ejpam-6570	89	22	a⋆	a⋆	ADV
ejpam-6570	89	23	is	be	AUX
ejpam-6570	89	24	called	call	VERB
ejpam-6570	89	25	the	the	DET
ejpam-6570	89	26	local	local	ADJ
ejpam-6570	89	27	function	function	NOUN
ejpam-6570	89	28	of	of	ADP
ejpam-6570	89	29	a	a	PRON
ejpam-6570	89	30	with	with	ADP
ejpam-6570	89	31	respect	respect	NOUN
ejpam-6570	89	32	to	to	ADP
ejpam-6570	89	33	i	i	PRON
ejpam-6570	89	34	and	and	CCONJ
ejpam-6570	89	35	τ	τ	PROPN
ejpam-6570	89	36	and	and	CCONJ
ejpam-6570	89	37	cl⋆(a	cl⋆(a	NUM
ejpam-6570	89	38	)	)	PUNCT
ejpam-6570	89	39	=	=	PUNCT
ejpam-6570	89	40	a⋆	a⋆	ADP
ejpam-6570	89	41	∪	∪	ADP
ejpam-6570	89	42	a	a	DET
ejpam-6570	89	43	defines	define	NOUN
ejpam-6570	89	44	a	a	DET
ejpam-6570	89	45	kuratowski	kuratowski	ADJ
ejpam-6570	89	46	closure	closure	NOUN
ejpam-6570	89	47	operator	operator	NOUN
ejpam-6570	89	48	for	for	ADP
ejpam-6570	89	49	a	a	DET
ejpam-6570	89	50	topology	topology	NOUN
ejpam-6570	89	51	τ⋆(i	τ⋆(i	NOUN
ejpam-6570	89	52	)	)	PUNCT
ejpam-6570	89	53	finer	fine	ADJ
ejpam-6570	89	54	than	than	ADP
ejpam-6570	89	55	τ	τ	PROPN
ejpam-6570	89	56	.	.	PUNCT
ejpam-6570	90	1	a	a	DET
ejpam-6570	90	2	subset	subset	NOUN
ejpam-6570	90	3	a	a	PRON
ejpam-6570	90	4	is	be	AUX
ejpam-6570	90	5	said	say	VERB
ejpam-6570	90	6	to	to	PART
ejpam-6570	90	7	be	be	AUX
ejpam-6570	90	8	⋆-closed	⋆-close	VERB
ejpam-6570	90	9	[	[	X
ejpam-6570	90	10	26	26	NUM
ejpam-6570	90	11	]	]	X
ejpam-6570	90	12	if	if	SCONJ
ejpam-6570	90	13	a⋆	a⋆	ADJ
ejpam-6570	90	14	⊆	⊆	NUM
ejpam-6570	90	15	a.	a.	NOUN
ejpam-6570	90	16	the	the	DET
ejpam-6570	90	17	interior	interior	NOUN
ejpam-6570	90	18	of	of	ADP
ejpam-6570	90	19	a	a	DET
ejpam-6570	90	20	subset	subset	NOUN
ejpam-6570	90	21	a	a	DET
ejpam-6570	90	22	in	in	ADP
ejpam-6570	90	23	(	(	PUNCT
ejpam-6570	90	24	x	x	X
ejpam-6570	90	25	,	,	PUNCT
ejpam-6570	90	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6570	90	27	)	)	PUNCT
ejpam-6570	90	28	)	)	PUNCT
ejpam-6570	90	29	is	be	AUX
ejpam-6570	90	30	denoted	denote	VERB
ejpam-6570	90	31	by	by	ADP
ejpam-6570	90	32	int⋆(a	int⋆(a	NOUN
ejpam-6570	90	33	)	)	PUNCT
ejpam-6570	90	34	.	.	PUNCT
ejpam-6570	91	1	a	a	DET
ejpam-6570	91	2	subset	subset	NOUN
ejpam-6570	91	3	a	a	PRON
ejpam-6570	91	4	of	of	ADP
ejpam-6570	91	5	an	an	DET
ejpam-6570	91	6	ideal	ideal	ADJ
ejpam-6570	91	7	topological	topological	ADJ
ejpam-6570	91	8	space	space	NOUN
ejpam-6570	91	9	(	(	PUNCT
ejpam-6570	91	10	x	x	X
ejpam-6570	91	11	,	,	PUNCT
ejpam-6570	91	12	τ	τ	PROPN
ejpam-6570	91	13	,	,	PUNCT
ejpam-6570	91	14	i	i	PROPN
ejpam-6570	91	15	)	)	PUNCT
ejpam-6570	91	16	is	be	AUX
ejpam-6570	91	17	said	say	VERB
ejpam-6570	91	18	to	to	PART
ejpam-6570	91	19	be	be	AUX
ejpam-6570	91	20	semi⋆-i	semi⋆-i	X
ejpam-6570	91	21	-open	-open	VERB
ejpam-6570	91	22	[	[	PUNCT
ejpam-6570	91	23	27	27	NUM
ejpam-6570	91	24	]	]	PUNCT
ejpam-6570	91	25	(	(	PUNCT
ejpam-6570	91	26	resp	resp	NOUN
ejpam-6570	91	27	.	.	PUNCT
ejpam-6570	92	1	semi	semi	ADJ
ejpam-6570	92	2	-	-	VERB
ejpam-6570	92	3	i	i	PRON
ejpam-6570	92	4	-open	-open	NOUN
ejpam-6570	93	1	[	[	X
ejpam-6570	93	2	14	14	NUM
ejpam-6570	93	3	]	]	SYM
ejpam-6570	93	4	)	)	PUNCT
ejpam-6570	93	5	if	if	SCONJ
ejpam-6570	93	6	a	a	DET
ejpam-6570	93	7	⊆	⊆	NUM
ejpam-6570	93	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6570	93	9	)	)	PUNCT
ejpam-6570	93	10	)	)	PUNCT
ejpam-6570	93	11	(	(	PUNCT
ejpam-6570	93	12	resp	resp	NOUN
ejpam-6570	93	13	.	.	PUNCT
ejpam-6570	94	1	a	a	DET
ejpam-6570	94	2	⊆	⊆	NUM
ejpam-6570	94	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6570	94	4	)	)	PUNCT
ejpam-6570	94	5	)	)	PUNCT
ejpam-6570	94	6	)	)	PUNCT
ejpam-6570	94	7	.	.	PUNCT
ejpam-6570	95	1	the	the	DET
ejpam-6570	95	2	complement	complement	NOUN
ejpam-6570	95	3	of	of	ADP
ejpam-6570	95	4	a	a	DET
ejpam-6570	95	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6570	95	6	-open	-open	ADJ
ejpam-6570	95	7	(	(	PUNCT
ejpam-6570	95	8	resp	resp	NOUN
ejpam-6570	95	9	.	.	PUNCT
ejpam-6570	96	1	semi	semi	ADJ
ejpam-6570	96	2	-	-	VERB
ejpam-6570	96	3	i	i	PRON
ejpam-6570	96	4	-open	-open	NOUN
ejpam-6570	96	5	)	)	PUNCT
ejpam-6570	97	1	set	set	NOUN
ejpam-6570	97	2	is	be	AUX
ejpam-6570	97	3	said	say	VERB
ejpam-6570	97	4	to	to	PART
ejpam-6570	97	5	be	be	AUX
ejpam-6570	97	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6570	97	7	-closed	-close	VERB
ejpam-6570	97	8	[	[	PUNCT
ejpam-6570	97	9	27	27	NUM
ejpam-6570	97	10	]	]	PUNCT
ejpam-6570	97	11	(	(	PUNCT
ejpam-6570	97	12	resp	resp	NOUN
ejpam-6570	97	13	.	.	PUNCT
ejpam-6570	98	1	semi	semi	ADJ
ejpam-6570	98	2	-	-	PROPN
ejpam-6570	98	3	i	i	PRON
ejpam-6570	98	4	closed	close	VERB
ejpam-6570	98	5	[	[	X
ejpam-6570	98	6	14	14	NUM
ejpam-6570	98	7	]	]	NUM
ejpam-6570	98	8	)	)	PUNCT
ejpam-6570	98	9	.	.	PUNCT
ejpam-6570	99	1	a	a	DET
ejpam-6570	99	2	subset	subset	NOUN
ejpam-6570	99	3	a	a	PRON
ejpam-6570	99	4	of	of	ADP
ejpam-6570	99	5	an	an	DET
ejpam-6570	99	6	ideal	ideal	ADJ
ejpam-6570	99	7	topological	topological	ADJ
ejpam-6570	99	8	space	space	NOUN
ejpam-6570	99	9	(	(	PUNCT
ejpam-6570	99	10	x	x	X
ejpam-6570	99	11	,	,	PUNCT
ejpam-6570	99	12	τ	τ	PROPN
ejpam-6570	99	13	,	,	PUNCT
ejpam-6570	99	14	i	i	PROPN
ejpam-6570	99	15	)	)	PUNCT
ejpam-6570	99	16	is	be	AUX
ejpam-6570	99	17	called	call	VERB
ejpam-6570	99	18	i	i	PRON
ejpam-6570	99	19	⋆-preopen	⋆-preopen	VERB
ejpam-6570	100	1	[	[	X
ejpam-6570	100	2	15	15	NUM
ejpam-6570	100	3	]	]	X
ejpam-6570	100	4	if	if	SCONJ
ejpam-6570	100	5	a	a	DET
ejpam-6570	100	6	⊆	⊆	NUM
ejpam-6570	100	7	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-6570	100	8	)	)	PUNCT
ejpam-6570	100	9	)	)	PUNCT
ejpam-6570	100	10	.	.	PUNCT
ejpam-6570	101	1	the	the	DET
ejpam-6570	101	2	complement	complement	NOUN
ejpam-6570	101	3	of	of	ADP
ejpam-6570	101	4	a	a	PRON
ejpam-6570	101	5	i	i	PRON
ejpam-6570	101	6	⋆-preopen	⋆-preopen	VERB
ejpam-6570	101	7	set	set	VERB
ejpam-6570	101	8	is	be	AUX
ejpam-6570	101	9	called	call	VERB
ejpam-6570	101	10	i	i	PRON
ejpam-6570	101	11	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	101	12	.	.	PUNCT
ejpam-6570	102	1	for	for	ADP
ejpam-6570	102	2	a	a	DET
ejpam-6570	102	3	subset	subset	NOUN
ejpam-6570	102	4	a	a	PRON
ejpam-6570	102	5	of	of	ADP
ejpam-6570	102	6	an	an	DET
ejpam-6570	102	7	ideal	ideal	ADJ
ejpam-6570	102	8	topological	topological	ADJ
ejpam-6570	102	9	space	space	NOUN
ejpam-6570	102	10	(	(	PUNCT
ejpam-6570	102	11	x	x	X
ejpam-6570	102	12	,	,	PUNCT
ejpam-6570	102	13	τ	τ	PROPN
ejpam-6570	102	14	,	,	PUNCT
ejpam-6570	102	15	i	i	NOUN
ejpam-6570	102	16	)	)	PUNCT
ejpam-6570	102	17	,	,	PUNCT
ejpam-6570	102	18	the	the	DET
ejpam-6570	102	19	intersection	intersection	NOUN
ejpam-6570	102	20	of	of	ADP
ejpam-6570	102	21	all	all	PRON
ejpam-6570	102	22	i	i	PRON
ejpam-6570	102	23	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	102	24	sets	set	NOUN
ejpam-6570	102	25	containing	contain	VERB
ejpam-6570	102	26	a	a	PRON
ejpam-6570	102	27	is	be	AUX
ejpam-6570	102	28	called	call	VERB
ejpam-6570	102	29	the	the	DET
ejpam-6570	102	30	⋆-preclosure	⋆-preclosure	NOUN
ejpam-6570	102	31	of	of	ADP
ejpam-6570	102	32	a	a	PRON
ejpam-6570	102	33	and	and	CCONJ
ejpam-6570	102	34	is	be	AUX
ejpam-6570	102	35	denoted	denote	VERB
ejpam-6570	102	36	by	by	ADP
ejpam-6570	102	37	pcl⋆(a	pcl⋆(a	NOUN
ejpam-6570	102	38	)	)	PUNCT
ejpam-6570	102	39	.	.	PUNCT
ejpam-6570	103	1	the	the	DET
ejpam-6570	103	2	union	union	NOUN
ejpam-6570	103	3	of	of	ADP
ejpam-6570	103	4	all	all	DET
ejpam-6570	103	5	i	i	PRON
ejpam-6570	103	6	⋆-preopen	⋆-preopen	VERB
ejpam-6570	103	7	sets	set	NOUN
ejpam-6570	103	8	contained	contain	VERB
ejpam-6570	103	9	in	in	ADP
ejpam-6570	103	10	a	a	PRON
ejpam-6570	103	11	is	be	AUX
ejpam-6570	103	12	called	call	VERB
ejpam-6570	103	13	the	the	DET
ejpam-6570	103	14	⋆-preinterior	⋆-preinterior	ADJ
ejpam-6570	103	15	of	of	ADP
ejpam-6570	103	16	a	a	PRON
ejpam-6570	103	17	and	and	CCONJ
ejpam-6570	103	18	is	be	AUX
ejpam-6570	103	19	denoted	denote	VERB
ejpam-6570	103	20	by	by	ADP
ejpam-6570	103	21	pint⋆(a	pint⋆(a	PROPN
ejpam-6570	103	22	)	)	PUNCT
ejpam-6570	103	23	.	.	PUNCT
ejpam-6570	104	1	c.	c.	PROPN
ejpam-6570	104	2	viriyapong	viriyapong	PROPN
ejpam-6570	104	3	,	,	PUNCT
ejpam-6570	104	4	a.	a.	PROPN
ejpam-6570	104	5	sama	sama	PROPN
ejpam-6570	104	6	-	-	PUNCT
ejpam-6570	104	7	ae	ae	PROPN
ejpam-6570	104	8	,	,	PUNCT
ejpam-6570	104	9	c.	c.	PROPN
ejpam-6570	104	10	boonpok	boonpok	PROPN
ejpam-6570	104	11	/	/	SYM
ejpam-6570	104	12	eur	eur	PROPN
ejpam-6570	104	13	.	.	PUNCT
ejpam-6570	105	1	j.	j.	PROPN
ejpam-6570	105	2	pure	pure	PROPN
ejpam-6570	105	3	appl	appl	PROPN
ejpam-6570	105	4	.	.	PROPN
ejpam-6570	105	5	math	math	PROPN
ejpam-6570	105	6	,	,	PUNCT
ejpam-6570	105	7	18	18	NUM
ejpam-6570	105	8	(	(	PUNCT
ejpam-6570	105	9	3	3	NUM
ejpam-6570	105	10	)	)	PUNCT
ejpam-6570	105	11	(	(	PUNCT
ejpam-6570	105	12	2025	2025	NUM
ejpam-6570	105	13	)	)	PUNCT
ejpam-6570	105	14	,	,	PUNCT
ejpam-6570	105	15	6570	6570	NUM
ejpam-6570	105	16	4	4	NUM
ejpam-6570	105	17	of	of	ADP
ejpam-6570	105	18	14	14	NUM
ejpam-6570	105	19	lemma	lemma	PROPN
ejpam-6570	105	20	2	2	NUM
ejpam-6570	105	21	.	.	PUNCT
ejpam-6570	106	1	for	for	ADP
ejpam-6570	106	2	a	a	DET
ejpam-6570	106	3	subset	subset	NOUN
ejpam-6570	106	4	a	a	PRON
ejpam-6570	106	5	of	of	ADP
ejpam-6570	106	6	an	an	DET
ejpam-6570	106	7	ideal	ideal	ADJ
ejpam-6570	106	8	topological	topological	ADJ
ejpam-6570	106	9	space	space	NOUN
ejpam-6570	106	10	(	(	PUNCT
ejpam-6570	106	11	x	x	X
ejpam-6570	106	12	,	,	PUNCT
ejpam-6570	106	13	τ	τ	PROPN
ejpam-6570	106	14	,	,	PUNCT
ejpam-6570	106	15	i	i	NOUN
ejpam-6570	106	16	)	)	PUNCT
ejpam-6570	106	17	,	,	PUNCT
ejpam-6570	106	18	the	the	DET
ejpam-6570	106	19	following	follow	VERB
ejpam-6570	106	20	properties	property	NOUN
ejpam-6570	106	21	hold	hold	VERB
ejpam-6570	106	22	:	:	PUNCT
ejpam-6570	106	23	(	(	PUNCT
ejpam-6570	106	24	1	1	X
ejpam-6570	106	25	)	)	PUNCT
ejpam-6570	106	26	pcl⋆(a	pcl⋆(a	NOUN
ejpam-6570	106	27	)	)	PUNCT
ejpam-6570	106	28	=	=	PUNCT
ejpam-6570	107	1	a	a	DET
ejpam-6570	107	2	∪	∪	ADJ
ejpam-6570	107	3	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-6570	107	4	)	)	PUNCT
ejpam-6570	107	5	)	)	PUNCT
ejpam-6570	107	6	.	.	PUNCT
ejpam-6570	108	1	(	(	PUNCT
ejpam-6570	108	2	2	2	X
ejpam-6570	108	3	)	)	PUNCT
ejpam-6570	108	4	pint⋆(a	pint⋆(a	NOUN
ejpam-6570	108	5	)	)	PUNCT
ejpam-6570	108	6	=	=	PUNCT
ejpam-6570	109	1	a	a	DET
ejpam-6570	109	2	∩	∩	NOUN
ejpam-6570	109	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-6570	109	4	)	)	PUNCT
ejpam-6570	109	5	)	)	PUNCT
ejpam-6570	109	6	.	.	PUNCT
ejpam-6570	110	1	by	by	ADP
ejpam-6570	110	2	a	a	DET
ejpam-6570	110	3	multifunction	multifunction	NOUN
ejpam-6570	110	4	f	f	NOUN
ejpam-6570	110	5	:	:	PUNCT
ejpam-6570	110	6	x	x	X
ejpam-6570	110	7	→	→	SYM
ejpam-6570	110	8	y	y	PROPN
ejpam-6570	110	9	,	,	PUNCT
ejpam-6570	110	10	we	we	PRON
ejpam-6570	110	11	mean	mean	VERB
ejpam-6570	110	12	a	a	DET
ejpam-6570	110	13	point	point	NOUN
ejpam-6570	110	14	-	-	PUNCT
ejpam-6570	110	15	to	to	ADP
ejpam-6570	110	16	-	-	PUNCT
ejpam-6570	110	17	set	set	VERB
ejpam-6570	110	18	correspondence	correspondence	NOUN
ejpam-6570	110	19	from	from	ADP
ejpam-6570	110	20	x	x	PUNCT
ejpam-6570	110	21	into	into	ADP
ejpam-6570	110	22	y	y	PROPN
ejpam-6570	110	23	,	,	PUNCT
ejpam-6570	110	24	and	and	CCONJ
ejpam-6570	110	25	always	always	ADV
ejpam-6570	110	26	assume	assume	VERB
ejpam-6570	110	27	that	that	SCONJ
ejpam-6570	111	1	f	f	PROPN
ejpam-6570	111	2	(	(	PUNCT
ejpam-6570	111	3	x	x	X
ejpam-6570	111	4	)	)	PUNCT
ejpam-6570	111	5	̸=	̸=	NOUN
ejpam-6570	111	6	∅	∅	NOUN
ejpam-6570	111	7	for	for	ADP
ejpam-6570	111	8	all	all	PRON
ejpam-6570	111	9	x	x	SYM
ejpam-6570	111	10	∈	∈	ADJ
ejpam-6570	111	11	x.	x.	NOUN
ejpam-6570	111	12	for	for	ADP
ejpam-6570	111	13	a	a	DET
ejpam-6570	111	14	multifunction	multifunction	NOUN
ejpam-6570	111	15	f	f	NOUN
ejpam-6570	111	16	:	:	PUNCT
ejpam-6570	111	17	x	x	X
ejpam-6570	111	18	→	→	SYM
ejpam-6570	111	19	y	y	PROPN
ejpam-6570	111	20	,	,	PUNCT
ejpam-6570	111	21	we	we	PRON
ejpam-6570	111	22	shall	shall	AUX
ejpam-6570	111	23	denote	denote	VERB
ejpam-6570	111	24	the	the	DET
ejpam-6570	111	25	upper	upper	ADJ
ejpam-6570	111	26	and	and	CCONJ
ejpam-6570	111	27	lower	low	ADJ
ejpam-6570	111	28	inverse	inverse	NOUN
ejpam-6570	111	29	of	of	ADP
ejpam-6570	111	30	a	a	DET
ejpam-6570	111	31	set	set	NOUN
ejpam-6570	111	32	b	b	PROPN
ejpam-6570	111	33	of	of	ADP
ejpam-6570	111	34	y	y	PROPN
ejpam-6570	111	35	by	by	ADP
ejpam-6570	111	36	f+(b	f+(b	NOUN
ejpam-6570	111	37	)	)	PUNCT
ejpam-6570	111	38	and	and	CCONJ
ejpam-6570	111	39	f−(b	f−(b	NOUN
ejpam-6570	111	40	)	)	PUNCT
ejpam-6570	111	41	,	,	PUNCT
ejpam-6570	111	42	respectively	respectively	ADV
ejpam-6570	111	43	,	,	PUNCT
ejpam-6570	111	44	that	that	ADV
ejpam-6570	111	45	is	is	ADV
ejpam-6570	111	46	,	,	PUNCT
ejpam-6570	111	47	f+(b	f+(b	NOUN
ejpam-6570	111	48	)	)	PUNCT
ejpam-6570	111	49	=	=	PRON
ejpam-6570	112	1	{	{	PUNCT
ejpam-6570	112	2	x	x	PUNCT
ejpam-6570	112	3	∈	∈	PROPN
ejpam-6570	112	4	x	x	INTJ
ejpam-6570	113	1	|	|	NOUN
ejpam-6570	113	2	f	f	X
ejpam-6570	113	3	(	(	PUNCT
ejpam-6570	113	4	x	x	NOUN
ejpam-6570	113	5	)	)	PUNCT
ejpam-6570	113	6	⊆	⊆	NUM
ejpam-6570	113	7	b	b	NOUN
ejpam-6570	113	8	}	}	PUNCT
ejpam-6570	113	9	and	and	CCONJ
ejpam-6570	113	10	f−(b	f−(b	PROPN
ejpam-6570	113	11	)	)	PUNCT
ejpam-6570	113	12	=	=	PRON
ejpam-6570	114	1	{	{	PUNCT
ejpam-6570	114	2	x	x	PUNCT
ejpam-6570	114	3	∈	∈	PROPN
ejpam-6570	114	4	x	x	INTJ
ejpam-6570	115	1	|	|	NOUN
ejpam-6570	115	2	f	f	X
ejpam-6570	115	3	(	(	PUNCT
ejpam-6570	115	4	x	x	NOUN
ejpam-6570	115	5	)	)	PUNCT
ejpam-6570	115	6	∩b	∩b	NOUN
ejpam-6570	115	7	̸=	̸=	PROPN
ejpam-6570	115	8	∅	∅	NOUN
ejpam-6570	115	9	}	}	PUNCT
ejpam-6570	115	10	.	.	PUNCT
ejpam-6570	116	1	3	3	X
ejpam-6570	116	2	.	.	X
ejpam-6570	116	3	upper	upper	ADJ
ejpam-6570	116	4	and	and	CCONJ
ejpam-6570	116	5	lower	low	ADJ
ejpam-6570	116	6	almost	almost	ADV
ejpam-6570	116	7	weakly	weakly	ADJ
ejpam-6570	116	8	τ	τ	X
ejpam-6570	116	9	⋆(σ1	⋆(σ1	NOUN
ejpam-6570	116	10	,	,	PUNCT
ejpam-6570	116	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	116	12	multifunctions	multifunction	NOUN
ejpam-6570	116	13	in	in	ADP
ejpam-6570	116	14	this	this	DET
ejpam-6570	116	15	section	section	NOUN
ejpam-6570	116	16	,	,	PUNCT
ejpam-6570	116	17	we	we	PRON
ejpam-6570	116	18	introduce	introduce	VERB
ejpam-6570	116	19	the	the	DET
ejpam-6570	116	20	notions	notion	NOUN
ejpam-6570	116	21	of	of	ADP
ejpam-6570	116	22	upper	upper	ADJ
ejpam-6570	116	23	almost	almost	ADV
ejpam-6570	116	24	weakly	weakly	ADJ
ejpam-6570	116	25	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	116	26	,	,	PUNCT
ejpam-6570	116	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	116	28	multifunctions	multifunction	NOUN
ejpam-6570	116	29	and	and	CCONJ
ejpam-6570	116	30	lower	low	ADJ
ejpam-6570	116	31	almost	almost	ADV
ejpam-6570	116	32	weakly	weakly	ADJ
ejpam-6570	116	33	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	116	34	,	,	PUNCT
ejpam-6570	116	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	116	36	multifunctions	multifunction	NOUN
ejpam-6570	116	37	.	.	PUNCT
ejpam-6570	117	1	moreover	moreover	ADV
ejpam-6570	117	2	,	,	PUNCT
ejpam-6570	117	3	several	several	ADJ
ejpam-6570	117	4	characterizations	characterization	NOUN
ejpam-6570	117	5	of	of	ADP
ejpam-6570	117	6	upper	upper	ADJ
ejpam-6570	117	7	almost	almost	ADV
ejpam-6570	117	8	weakly	weakly	ADJ
ejpam-6570	117	9	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	117	10	,	,	PUNCT
ejpam-6570	117	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	117	12	multifunctions	multifunction	NOUN
ejpam-6570	117	13	and	and	CCONJ
ejpam-6570	117	14	lower	low	ADJ
ejpam-6570	117	15	almost	almost	ADV
ejpam-6570	117	16	weakly	weakly	ADJ
ejpam-6570	117	17	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	117	18	,	,	PUNCT
ejpam-6570	117	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	117	20	multifunctions	multifunction	NOUN
ejpam-6570	117	21	are	be	AUX
ejpam-6570	117	22	discussed	discuss	VERB
ejpam-6570	117	23	.	.	PUNCT
ejpam-6570	118	1	definition	definition	NOUN
ejpam-6570	118	2	1	1	NUM
ejpam-6570	118	3	.	.	PUNCT
ejpam-6570	119	1	a	a	DET
ejpam-6570	119	2	multifunction	multifunction	NOUN
ejpam-6570	119	3	f	f	NOUN
ejpam-6570	119	4	:	:	PUNCT
ejpam-6570	119	5	(	(	PUNCT
ejpam-6570	119	6	x	x	X
ejpam-6570	119	7	,	,	PUNCT
ejpam-6570	119	8	τ	τ	PROPN
ejpam-6570	119	9	,	,	PUNCT
ejpam-6570	119	10	i	i	NOUN
ejpam-6570	119	11	)	)	PUNCT
ejpam-6570	119	12	→	→	PUNCT
ejpam-6570	119	13	(	(	PUNCT
ejpam-6570	119	14	y	y	PROPN
ejpam-6570	119	15	,	,	PUNCT
ejpam-6570	119	16	σ1	σ1	PROPN
ejpam-6570	119	17	,	,	PUNCT
ejpam-6570	119	18	σ2	σ2	PROPN
ejpam-6570	119	19	)	)	PUNCT
ejpam-6570	119	20	is	be	AUX
ejpam-6570	119	21	said	say	VERB
ejpam-6570	119	22	to	to	PART
ejpam-6570	119	23	be	be	AUX
ejpam-6570	119	24	upper	upper	ADJ
ejpam-6570	119	25	almost	almost	ADV
ejpam-6570	119	26	weakly	weakly	ADJ
ejpam-6570	119	27	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	119	28	,	,	PUNCT
ejpam-6570	119	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	119	30	if	if	SCONJ
ejpam-6570	119	31	for	for	ADP
ejpam-6570	119	32	each	each	DET
ejpam-6570	119	33	x	x	SYM
ejpam-6570	119	34	∈	∈	PROPN
ejpam-6570	119	35	x	x	X
ejpam-6570	119	36	and	and	CCONJ
ejpam-6570	119	37	each	each	DET
ejpam-6570	119	38	σ1σ2	σ1σ2	VERB
ejpam-6570	119	39	-	-	ADJ
ejpam-6570	119	40	open	open	ADJ
ejpam-6570	119	41	set	set	NOUN
ejpam-6570	119	42	v	v	NOUN
ejpam-6570	119	43	of	of	ADP
ejpam-6570	119	44	y	y	PRON
ejpam-6570	119	45	such	such	ADJ
ejpam-6570	119	46	that	that	SCONJ
ejpam-6570	119	47	f	f	PROPN
ejpam-6570	119	48	(	(	PUNCT
ejpam-6570	119	49	x	x	X
ejpam-6570	119	50	)	)	PUNCT
ejpam-6570	119	51	⊆	⊆	NUM
ejpam-6570	119	52	v	v	NOUN
ejpam-6570	119	53	,	,	PUNCT
ejpam-6570	119	54	x	x	SYM
ejpam-6570	119	55	∈	∈	NOUN
ejpam-6570	119	56	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-6570	119	57	-	-	PUNCT
ejpam-6570	119	58	cl(v	cl(v	NOUN
ejpam-6570	119	59	)	)	PUNCT
ejpam-6570	119	60	)	)	PUNCT
ejpam-6570	119	61	)	)	PUNCT
ejpam-6570	119	62	)	)	PUNCT
ejpam-6570	119	63	.	.	PUNCT
ejpam-6570	120	1	theorem	theorem	NOUN
ejpam-6570	120	2	1	1	NUM
ejpam-6570	120	3	.	.	X
ejpam-6570	120	4	for	for	ADP
ejpam-6570	120	5	a	a	DET
ejpam-6570	120	6	multifunction	multifunction	NOUN
ejpam-6570	121	1	f	f	NOUN
ejpam-6570	121	2	:	:	PUNCT
ejpam-6570	121	3	(	(	PUNCT
ejpam-6570	121	4	x	x	X
ejpam-6570	121	5	,	,	PUNCT
ejpam-6570	121	6	τ	τ	PROPN
ejpam-6570	121	7	,	,	PUNCT
ejpam-6570	121	8	i	i	NOUN
ejpam-6570	121	9	)	)	PUNCT
ejpam-6570	121	10	→	→	PUNCT
ejpam-6570	121	11	(	(	PUNCT
ejpam-6570	121	12	y	y	PROPN
ejpam-6570	121	13	,	,	PUNCT
ejpam-6570	121	14	σ1	σ1	PROPN
ejpam-6570	121	15	,	,	PUNCT
ejpam-6570	121	16	σ2	σ2	NOUN
ejpam-6570	121	17	)	)	PUNCT
ejpam-6570	121	18	,	,	PUNCT
ejpam-6570	121	19	the	the	DET
ejpam-6570	121	20	following	follow	VERB
ejpam-6570	121	21	properties	property	NOUN
ejpam-6570	121	22	are	be	AUX
ejpam-6570	121	23	equivalent	equivalent	ADJ
ejpam-6570	121	24	:	:	PUNCT
ejpam-6570	121	25	(	(	PUNCT
ejpam-6570	121	26	1	1	X
ejpam-6570	121	27	)	)	PUNCT
ejpam-6570	121	28	f	f	PROPN
ejpam-6570	121	29	is	be	AUX
ejpam-6570	121	30	upper	upper	ADJ
ejpam-6570	121	31	almost	almost	ADV
ejpam-6570	121	32	weakly	weakly	ADJ
ejpam-6570	121	33	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	121	34	,	,	PUNCT
ejpam-6570	121	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	121	36	;	;	PUNCT
ejpam-6570	121	37	(	(	PUNCT
ejpam-6570	121	38	2	2	NUM
ejpam-6570	121	39	)	)	PUNCT
ejpam-6570	121	40	f+(v	f+(v	NOUN
ejpam-6570	121	41	)	)	PUNCT
ejpam-6570	122	1	⊆	⊆	NUM
ejpam-6570	122	2	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-6570	122	3	-	-	PUNCT
ejpam-6570	122	4	cl(v	cl(v	NOUN
ejpam-6570	122	5	)	)	PUNCT
ejpam-6570	122	6	)	)	PUNCT
ejpam-6570	122	7	)	)	PUNCT
ejpam-6570	122	8	)	)	PUNCT
ejpam-6570	123	1	for	for	ADP
ejpam-6570	123	2	every	every	DET
ejpam-6570	123	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	123	4	-	-	ADJ
ejpam-6570	123	5	open	open	ADJ
ejpam-6570	123	6	set	set	NOUN
ejpam-6570	123	7	v	v	NOUN
ejpam-6570	123	8	of	of	ADP
ejpam-6570	123	9	y	y	PROPN
ejpam-6570	123	10	;	;	PUNCT
ejpam-6570	123	11	(	(	PUNCT
ejpam-6570	123	12	3	3	X
ejpam-6570	123	13	)	)	PUNCT
ejpam-6570	123	14	cl⋆(int⋆(f−(v	cl⋆(int⋆(f−(v	NOUN
ejpam-6570	123	15	)	)	PUNCT
ejpam-6570	123	16	)	)	PUNCT
ejpam-6570	123	17	)	)	PUNCT
ejpam-6570	123	18	⊆	⊆	X
ejpam-6570	123	19	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	123	20	-	-	PUNCT
ejpam-6570	123	21	cl(v	cl(v	NOUN
ejpam-6570	123	22	)	)	PUNCT
ejpam-6570	123	23	)	)	PUNCT
ejpam-6570	123	24	for	for	ADP
ejpam-6570	123	25	every	every	DET
ejpam-6570	123	26	σ1σ2	σ1σ2	NOUN
ejpam-6570	123	27	-	-	ADJ
ejpam-6570	123	28	open	open	ADJ
ejpam-6570	123	29	set	set	NOUN
ejpam-6570	123	30	v	v	NOUN
ejpam-6570	123	31	of	of	ADP
ejpam-6570	123	32	y	y	PROPN
ejpam-6570	123	33	;	;	PUNCT
ejpam-6570	123	34	(	(	PUNCT
ejpam-6570	123	35	4	4	NUM
ejpam-6570	123	36	)	)	PUNCT
ejpam-6570	123	37	pcl⋆(f−(v	pcl⋆(f−(v	NOUN
ejpam-6570	123	38	)	)	PUNCT
ejpam-6570	123	39	)	)	PUNCT
ejpam-6570	124	1	⊆	⊆	X
ejpam-6570	124	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	124	3	-	-	PUNCT
ejpam-6570	124	4	cl(v	cl(v	NOUN
ejpam-6570	124	5	)	)	PUNCT
ejpam-6570	124	6	)	)	PUNCT
ejpam-6570	124	7	for	for	ADP
ejpam-6570	124	8	every	every	DET
ejpam-6570	124	9	σ1σ2	σ1σ2	NOUN
ejpam-6570	124	10	-	-	ADJ
ejpam-6570	124	11	open	open	ADJ
ejpam-6570	124	12	set	set	NOUN
ejpam-6570	124	13	v	v	NOUN
ejpam-6570	124	14	of	of	ADP
ejpam-6570	124	15	y	y	PROPN
ejpam-6570	124	16	;	;	PUNCT
ejpam-6570	124	17	(	(	PUNCT
ejpam-6570	124	18	5	5	NUM
ejpam-6570	124	19	)	)	PUNCT
ejpam-6570	124	20	f+(v	f+(v	NOUN
ejpam-6570	124	21	)	)	PUNCT
ejpam-6570	125	1	⊆	⊆	NUM
ejpam-6570	125	2	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	125	3	-	-	NOUN
ejpam-6570	125	4	cl(v	cl(v	NOUN
ejpam-6570	125	5	)	)	PUNCT
ejpam-6570	125	6	)	)	PUNCT
ejpam-6570	125	7	)	)	PUNCT
ejpam-6570	125	8	for	for	ADP
ejpam-6570	125	9	every	every	DET
ejpam-6570	125	10	σ1σ2	σ1σ2	NOUN
ejpam-6570	125	11	-	-	ADJ
ejpam-6570	125	12	open	open	ADJ
ejpam-6570	125	13	set	set	NOUN
ejpam-6570	125	14	v	v	NOUN
ejpam-6570	125	15	of	of	ADP
ejpam-6570	125	16	y	y	PROPN
ejpam-6570	125	17	;	;	PUNCT
ejpam-6570	125	18	(	(	PUNCT
ejpam-6570	125	19	6	6	NUM
ejpam-6570	125	20	)	)	PUNCT
ejpam-6570	125	21	for	for	ADP
ejpam-6570	125	22	each	each	DET
ejpam-6570	125	23	x	x	SYM
ejpam-6570	125	24	∈	∈	PROPN
ejpam-6570	125	25	x	x	X
ejpam-6570	125	26	and	and	CCONJ
ejpam-6570	125	27	each	each	DET
ejpam-6570	125	28	σ1σ2	σ1σ2	VERB
ejpam-6570	125	29	-	-	ADJ
ejpam-6570	125	30	open	open	ADJ
ejpam-6570	125	31	set	set	NOUN
ejpam-6570	125	32	v	v	NOUN
ejpam-6570	125	33	of	of	ADP
ejpam-6570	125	34	y	y	PROPN
ejpam-6570	125	35	containing	contain	VERB
ejpam-6570	125	36	f	f	PROPN
ejpam-6570	125	37	(	(	PUNCT
ejpam-6570	125	38	x	x	NOUN
ejpam-6570	125	39	)	)	PUNCT
ejpam-6570	125	40	,	,	PUNCT
ejpam-6570	125	41	there	there	PRON
ejpam-6570	125	42	exists	exist	VERB
ejpam-6570	125	43	an	an	PRON
ejpam-6570	125	44	i	i	PRON
ejpam-6570	125	45	⋆-preopen	⋆-preopen	VERB
ejpam-6570	125	46	set	set	VERB
ejpam-6570	125	47	u	u	NOUN
ejpam-6570	125	48	of	of	ADP
ejpam-6570	125	49	x	x	PUNCT
ejpam-6570	125	50	containing	contain	VERB
ejpam-6570	125	51	x	x	PUNCT
ejpam-6570	125	52	such	such	ADJ
ejpam-6570	125	53	that	that	SCONJ
ejpam-6570	125	54	f	f	PROPN
ejpam-6570	125	55	(	(	PUNCT
ejpam-6570	125	56	u	u	NOUN
ejpam-6570	125	57	)	)	PUNCT
ejpam-6570	125	58	⊆	⊆	NUM
ejpam-6570	125	59	σ1σ2	σ1σ2	NOUN
ejpam-6570	125	60	-	-	NUM
ejpam-6570	125	61	cl(v	cl(v	NOUN
ejpam-6570	125	62	)	)	PUNCT
ejpam-6570	125	63	.	.	PUNCT
ejpam-6570	126	1	proof	proof	NOUN
ejpam-6570	126	2	.	.	PUNCT
ejpam-6570	127	1	(	(	PUNCT
ejpam-6570	127	2	1	1	X
ejpam-6570	127	3	)	)	PUNCT
ejpam-6570	127	4	⇒	⇒	NOUN
ejpam-6570	127	5	(	(	PUNCT
ejpam-6570	127	6	2	2	NUM
ejpam-6570	127	7	):	):	PUNCT
ejpam-6570	127	8	let	let	VERB
ejpam-6570	127	9	v	v	PART
ejpam-6570	127	10	be	be	AUX
ejpam-6570	127	11	any	any	DET
ejpam-6570	127	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	127	13	-	-	ADJ
ejpam-6570	127	14	open	open	ADJ
ejpam-6570	127	15	set	set	NOUN
ejpam-6570	127	16	of	of	ADP
ejpam-6570	127	17	y	y	PROPN
ejpam-6570	127	18	and	and	CCONJ
ejpam-6570	127	19	x	x	PROPN
ejpam-6570	127	20	∈	∈	PROPN
ejpam-6570	127	21	f+(v	f+(v	NOUN
ejpam-6570	127	22	)	)	PUNCT
ejpam-6570	127	23	.	.	PUNCT
ejpam-6570	128	1	then	then	ADV
ejpam-6570	128	2	,	,	PUNCT
ejpam-6570	128	3	f	f	PROPN
ejpam-6570	128	4	(	(	PUNCT
ejpam-6570	128	5	x	x	X
ejpam-6570	128	6	)	)	PUNCT
ejpam-6570	128	7	⊆	⊆	NUM
ejpam-6570	128	8	v	v	NOUN
ejpam-6570	128	9	and	and	CCONJ
ejpam-6570	128	10	by	by	ADP
ejpam-6570	128	11	(	(	PUNCT
ejpam-6570	128	12	1	1	NUM
ejpam-6570	128	13	)	)	PUNCT
ejpam-6570	128	14	,	,	PUNCT
ejpam-6570	128	15	we	we	PRON
ejpam-6570	128	16	have	have	VERB
ejpam-6570	128	17	x	x	X
ejpam-6570	128	18	∈	∈	PROPN
ejpam-6570	128	19	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-6570	128	20	-	-	PUNCT
ejpam-6570	128	21	cl(v	cl(v	NOUN
ejpam-6570	128	22	)	)	PUNCT
ejpam-6570	128	23	)	)	PUNCT
ejpam-6570	128	24	)	)	PUNCT
ejpam-6570	128	25	)	)	PUNCT
ejpam-6570	128	26	.	.	PUNCT
ejpam-6570	129	1	therefore	therefore	ADV
ejpam-6570	129	2	,	,	PUNCT
ejpam-6570	129	3	f+(v	f+(v	PROPN
ejpam-6570	129	4	)	)	PUNCT
ejpam-6570	129	5	⊆	⊆	NUM
ejpam-6570	129	6	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-6570	129	7	-	-	PUNCT
ejpam-6570	129	8	cl(v	cl(v	NOUN
ejpam-6570	129	9	)	)	PUNCT
ejpam-6570	129	10	)	)	PUNCT
ejpam-6570	129	11	)	)	PUNCT
ejpam-6570	129	12	)	)	PUNCT
ejpam-6570	129	13	.	.	PUNCT
ejpam-6570	130	1	(	(	PUNCT
ejpam-6570	130	2	2	2	X
ejpam-6570	130	3	)	)	PUNCT
ejpam-6570	130	4	⇒	⇒	NOUN
ejpam-6570	130	5	(	(	PUNCT
ejpam-6570	130	6	3	3	NUM
ejpam-6570	130	7	):	):	PUNCT
ejpam-6570	130	8	let	let	VERB
ejpam-6570	130	9	v	v	PART
ejpam-6570	130	10	be	be	AUX
ejpam-6570	130	11	any	any	DET
ejpam-6570	130	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	130	13	-	-	ADJ
ejpam-6570	130	14	open	open	ADJ
ejpam-6570	130	15	set	set	NOUN
ejpam-6570	130	16	of	of	ADP
ejpam-6570	130	17	y	y	PROPN
ejpam-6570	130	18	.	.	PUNCT
ejpam-6570	131	1	since	since	SCONJ
ejpam-6570	131	2	y	y	PROPN
ejpam-6570	131	3	−	−	PROPN
ejpam-6570	131	4	σ1σ2	σ1σ2	NOUN
ejpam-6570	131	5	-	-	NUM
ejpam-6570	131	6	cl(v	cl(v	NOUN
ejpam-6570	131	7	)	)	PUNCT
ejpam-6570	131	8	is	be	AUX
ejpam-6570	131	9	σ1σ2	σ1σ2	NOUN
ejpam-6570	131	10	-	-	ADJ
ejpam-6570	131	11	open	open	ADJ
ejpam-6570	131	12	and	and	CCONJ
ejpam-6570	131	13	by	by	ADP
ejpam-6570	131	14	(	(	PUNCT
ejpam-6570	131	15	2	2	NUM
ejpam-6570	131	16	)	)	PUNCT
ejpam-6570	131	17	,	,	PUNCT
ejpam-6570	131	18	x	x	PUNCT
ejpam-6570	131	19	−	−	ADP
ejpam-6570	131	20	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	131	21	-	-	PUNCT
ejpam-6570	131	22	cl(v	cl(v	NOUN
ejpam-6570	131	23	)	)	PUNCT
ejpam-6570	131	24	)	)	PUNCT
ejpam-6570	132	1	=	=	PUNCT
ejpam-6570	133	1	f+(y	f+(y	NOUN
ejpam-6570	133	2	−	−	NUM
ejpam-6570	133	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	133	4	-	-	NUM
ejpam-6570	133	5	cl(v	cl(v	NOUN
ejpam-6570	133	6	)	)	PUNCT
ejpam-6570	133	7	)	)	PUNCT
ejpam-6570	134	1	c.	c.	PROPN
ejpam-6570	134	2	viriyapong	viriyapong	PROPN
ejpam-6570	134	3	,	,	PUNCT
ejpam-6570	134	4	a.	a.	PROPN
ejpam-6570	134	5	sama	sama	PROPN
ejpam-6570	134	6	-	-	PUNCT
ejpam-6570	134	7	ae	ae	PROPN
ejpam-6570	134	8	,	,	PUNCT
ejpam-6570	134	9	c.	c.	PROPN
ejpam-6570	134	10	boonpok	boonpok	PROPN
ejpam-6570	134	11	/	/	SYM
ejpam-6570	134	12	eur	eur	PROPN
ejpam-6570	134	13	.	.	PUNCT
ejpam-6570	135	1	j.	j.	PROPN
ejpam-6570	135	2	pure	pure	PROPN
ejpam-6570	135	3	appl	appl	PROPN
ejpam-6570	135	4	.	.	PROPN
ejpam-6570	135	5	math	math	PROPN
ejpam-6570	135	6	,	,	PUNCT
ejpam-6570	135	7	18	18	NUM
ejpam-6570	135	8	(	(	PUNCT
ejpam-6570	135	9	3	3	NUM
ejpam-6570	135	10	)	)	PUNCT
ejpam-6570	135	11	(	(	PUNCT
ejpam-6570	135	12	2025	2025	NUM
ejpam-6570	135	13	)	)	PUNCT
ejpam-6570	135	14	,	,	PUNCT
ejpam-6570	135	15	6570	6570	NUM
ejpam-6570	135	16	5	5	NUM
ejpam-6570	135	17	of	of	ADP
ejpam-6570	135	18	14	14	NUM
ejpam-6570	135	19	⊆	⊆	NUM
ejpam-6570	135	20	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-6570	135	21	-	-	PUNCT
ejpam-6570	135	22	cl(y	cl(y	NOUN
ejpam-6570	135	23	−	−	NOUN
ejpam-6570	135	24	σ1σ2	σ1σ2	NOUN
ejpam-6570	135	25	-	-	NUM
ejpam-6570	135	26	cl(v	cl(v	NOUN
ejpam-6570	135	27	)	)	PUNCT
ejpam-6570	135	28	)	)	PUNCT
ejpam-6570	135	29	)	)	PUNCT
ejpam-6570	135	30	)	)	PUNCT
ejpam-6570	135	31	)	)	PUNCT
ejpam-6570	136	1	⊆	⊆	NUM
ejpam-6570	136	2	int⋆(cl⋆(f+(y	int⋆(cl⋆(f+(y	NOUN
ejpam-6570	136	3	−	−	PROPN
ejpam-6570	136	4	v	v	NOUN
ejpam-6570	136	5	)	)	PUNCT
ejpam-6570	136	6	)	)	PUNCT
ejpam-6570	136	7	)	)	PUNCT
ejpam-6570	137	1	=	=	PRON
ejpam-6570	137	2	int⋆(cl⋆(x	int⋆(cl⋆(x	VERB
ejpam-6570	137	3	−	−	PROPN
ejpam-6570	137	4	f−(v	f−(v	NOUN
ejpam-6570	137	5	)	)	PUNCT
ejpam-6570	137	6	)	)	PUNCT
ejpam-6570	137	7	)	)	PUNCT
ejpam-6570	138	1	=	=	PUNCT
ejpam-6570	138	2	x	x	X
ejpam-6570	139	1	−	−	NOUN
ejpam-6570	139	2	cl⋆(int⋆(f−(v	cl⋆(int⋆(f−(v	NOUN
ejpam-6570	139	3	)	)	PUNCT
ejpam-6570	139	4	)	)	PUNCT
ejpam-6570	139	5	)	)	PUNCT
ejpam-6570	139	6	.	.	PUNCT
ejpam-6570	140	1	thus	thus	ADV
ejpam-6570	140	2	,	,	PUNCT
ejpam-6570	140	3	cl⋆(int⋆(f−(v	cl⋆(int⋆(f−(v	NOUN
ejpam-6570	140	4	)	)	PUNCT
ejpam-6570	140	5	)	)	PUNCT
ejpam-6570	140	6	)	)	PUNCT
ejpam-6570	141	1	⊆	⊆	X
ejpam-6570	141	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	141	3	-	-	PUNCT
ejpam-6570	141	4	cl(v	cl(v	NOUN
ejpam-6570	141	5	)	)	PUNCT
ejpam-6570	141	6	)	)	PUNCT
ejpam-6570	141	7	.	.	PUNCT
ejpam-6570	142	1	(	(	PUNCT
ejpam-6570	142	2	3	3	X
ejpam-6570	142	3	)	)	PUNCT
ejpam-6570	142	4	⇒	⇒	NOUN
ejpam-6570	142	5	(	(	PUNCT
ejpam-6570	142	6	4	4	NUM
ejpam-6570	142	7	):	):	PUNCT
ejpam-6570	142	8	let	let	VERB
ejpam-6570	142	9	v	v	PART
ejpam-6570	142	10	be	be	AUX
ejpam-6570	142	11	any	any	DET
ejpam-6570	142	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	142	13	-	-	ADJ
ejpam-6570	142	14	open	open	ADJ
ejpam-6570	142	15	set	set	NOUN
ejpam-6570	142	16	of	of	ADP
ejpam-6570	142	17	y	y	PROPN
ejpam-6570	142	18	.	.	PUNCT
ejpam-6570	143	1	by	by	ADP
ejpam-6570	143	2	(	(	PUNCT
ejpam-6570	143	3	3	3	X
ejpam-6570	143	4	)	)	PUNCT
ejpam-6570	143	5	and	and	CCONJ
ejpam-6570	143	6	lemma	lemma	PROPN
ejpam-6570	143	7	2	2	NUM
ejpam-6570	143	8	,	,	PUNCT
ejpam-6570	143	9	pcl⋆(f−(v	pcl⋆(f−(v	NOUN
ejpam-6570	143	10	)	)	PUNCT
ejpam-6570	143	11	)	)	PUNCT
ejpam-6570	143	12	=	=	SYM
ejpam-6570	143	13	cl⋆(int⋆(f−(v	cl⋆(int⋆(f−(v	NOUN
ejpam-6570	143	14	)	)	PUNCT
ejpam-6570	143	15	)	)	PUNCT
ejpam-6570	143	16	)	)	PUNCT
ejpam-6570	143	17	∪	∪	ADP
ejpam-6570	143	18	f−(v	f−(v	NOUN
ejpam-6570	143	19	)	)	PUNCT
ejpam-6570	143	20	⊆	⊆	NUM
ejpam-6570	143	21	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	143	22	-	-	PUNCT
ejpam-6570	143	23	cl(v	cl(v	NOUN
ejpam-6570	143	24	)	)	PUNCT
ejpam-6570	143	25	)	)	PUNCT
ejpam-6570	143	26	.	.	PUNCT
ejpam-6570	144	1	(	(	PUNCT
ejpam-6570	144	2	4	4	X
ejpam-6570	144	3	)	)	PUNCT
ejpam-6570	144	4	⇒	⇒	NOUN
ejpam-6570	144	5	(	(	PUNCT
ejpam-6570	144	6	5	5	NUM
ejpam-6570	144	7	):	):	PUNCT
ejpam-6570	144	8	let	let	VERB
ejpam-6570	144	9	v	v	PART
ejpam-6570	144	10	be	be	AUX
ejpam-6570	144	11	any	any	DET
ejpam-6570	144	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	144	13	-	-	ADJ
ejpam-6570	144	14	open	open	ADJ
ejpam-6570	144	15	set	set	NOUN
ejpam-6570	144	16	of	of	ADP
ejpam-6570	144	17	y	y	PROPN
ejpam-6570	144	18	.	.	PUNCT
ejpam-6570	145	1	then	then	ADV
ejpam-6570	145	2	,	,	PUNCT
ejpam-6570	145	3	y	y	PROPN
ejpam-6570	145	4	−	−	NUM
ejpam-6570	145	5	σ1σ2	σ1σ2	NOUN
ejpam-6570	145	6	-	-	NUM
ejpam-6570	145	7	cl(v	cl(v	NOUN
ejpam-6570	145	8	)	)	PUNCT
ejpam-6570	145	9	is	be	AUX
ejpam-6570	145	10	σ1σ2	σ1σ2	NOUN
ejpam-6570	145	11	-	-	ADJ
ejpam-6570	145	12	open	open	ADJ
ejpam-6570	145	13	in	in	ADP
ejpam-6570	145	14	y	y	PROPN
ejpam-6570	145	15	.	.	PUNCT
ejpam-6570	146	1	thus	thus	ADV
ejpam-6570	146	2	by	by	ADP
ejpam-6570	146	3	(	(	PUNCT
ejpam-6570	146	4	4	4	NUM
ejpam-6570	146	5	)	)	PUNCT
ejpam-6570	146	6	,	,	PUNCT
ejpam-6570	146	7	x	x	PUNCT
ejpam-6570	146	8	−	−	VERB
ejpam-6570	146	9	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	146	10	-	-	NOUN
ejpam-6570	146	11	cl(v	cl(v	NOUN
ejpam-6570	146	12	)	)	PUNCT
ejpam-6570	146	13	)	)	PUNCT
ejpam-6570	146	14	)	)	PUNCT
ejpam-6570	147	1	=	=	SYM
ejpam-6570	147	2	pcl⋆(x	pcl⋆(x	NOUN
ejpam-6570	147	3	−	−	NOUN
ejpam-6570	147	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	147	5	-	-	PUNCT
ejpam-6570	147	6	cl(v	cl(v	NOUN
ejpam-6570	147	7	)	)	PUNCT
ejpam-6570	147	8	)	)	PUNCT
ejpam-6570	147	9	)	)	PUNCT
ejpam-6570	148	1	=	=	SYM
ejpam-6570	148	2	pcl⋆(f−(y	pcl⋆(f−(y	NOUN
ejpam-6570	148	3	−	−	PUNCT
ejpam-6570	148	4	σ1σ2	σ1σ2	NOUN
ejpam-6570	148	5	-	-	NUM
ejpam-6570	148	6	cl(v	cl(v	NOUN
ejpam-6570	148	7	)	)	PUNCT
ejpam-6570	148	8	)	)	PUNCT
ejpam-6570	148	9	)	)	PUNCT
ejpam-6570	149	1	⊆	⊆	X
ejpam-6570	149	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6570	149	3	-	-	PUNCT
ejpam-6570	149	4	cl(y	cl(y	NOUN
ejpam-6570	149	5	−	−	NOUN
ejpam-6570	149	6	σ1σ2	σ1σ2	NOUN
ejpam-6570	149	7	-	-	NUM
ejpam-6570	149	8	cl(v	cl(v	NOUN
ejpam-6570	149	9	)	)	PUNCT
ejpam-6570	149	10	)	)	PUNCT
ejpam-6570	149	11	)	)	PUNCT
ejpam-6570	150	1	⊆	⊆	NUM
ejpam-6570	150	2	f−(y	f−(y	NOUN
ejpam-6570	150	3	−	−	NOUN
ejpam-6570	150	4	v	v	NOUN
ejpam-6570	150	5	)	)	PUNCT
ejpam-6570	150	6	=	=	PUNCT
ejpam-6570	150	7	x	x	X
ejpam-6570	150	8	−	−	PROPN
ejpam-6570	150	9	f+(v	f+(v	NOUN
ejpam-6570	150	10	)	)	PUNCT
ejpam-6570	150	11	and	and	CCONJ
ejpam-6570	150	12	hence	hence	ADV
ejpam-6570	150	13	f+(v	f+(v	NOUN
ejpam-6570	150	14	)	)	PUNCT
ejpam-6570	151	1	⊆	⊆	NUM
ejpam-6570	151	2	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	151	3	-	-	NOUN
ejpam-6570	151	4	cl(v	cl(v	NOUN
ejpam-6570	151	5	)	)	PUNCT
ejpam-6570	151	6	)	)	PUNCT
ejpam-6570	151	7	)	)	PUNCT
ejpam-6570	151	8	.	.	PUNCT
ejpam-6570	152	1	(	(	PUNCT
ejpam-6570	152	2	5	5	X
ejpam-6570	152	3	)	)	PUNCT
ejpam-6570	152	4	⇒	⇒	NOUN
ejpam-6570	152	5	(	(	PUNCT
ejpam-6570	152	6	6	6	NUM
ejpam-6570	152	7	):	):	PUNCT
ejpam-6570	152	8	let	let	VERB
ejpam-6570	152	9	x	x	PUNCT
ejpam-6570	152	10	∈	∈	PROPN
ejpam-6570	152	11	x	x	X
ejpam-6570	152	12	and	and	CCONJ
ejpam-6570	152	13	v	v	X
ejpam-6570	152	14	be	be	AUX
ejpam-6570	152	15	any	any	DET
ejpam-6570	152	16	σ1σ2	σ1σ2	NOUN
ejpam-6570	152	17	-	-	ADJ
ejpam-6570	152	18	open	open	ADJ
ejpam-6570	152	19	set	set	NOUN
ejpam-6570	152	20	of	of	ADP
ejpam-6570	152	21	y	y	PROPN
ejpam-6570	152	22	containing	contain	VERB
ejpam-6570	152	23	f	f	PROPN
ejpam-6570	152	24	(	(	PUNCT
ejpam-6570	152	25	x	x	NOUN
ejpam-6570	152	26	)	)	PUNCT
ejpam-6570	152	27	.	.	PUNCT
ejpam-6570	153	1	by	by	ADP
ejpam-6570	153	2	(	(	PUNCT
ejpam-6570	153	3	5	5	NUM
ejpam-6570	153	4	)	)	PUNCT
ejpam-6570	153	5	,	,	PUNCT
ejpam-6570	153	6	x	x	PUNCT
ejpam-6570	153	7	∈	∈	PROPN
ejpam-6570	153	8	f+(v	f+(v	NOUN
ejpam-6570	153	9	)	)	PUNCT
ejpam-6570	153	10	⊆	⊆	NUM
ejpam-6570	153	11	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	153	12	-	-	NOUN
ejpam-6570	153	13	cl(v	cl(v	NOUN
ejpam-6570	153	14	)	)	PUNCT
ejpam-6570	153	15	)	)	PUNCT
ejpam-6570	153	16	)	)	PUNCT
ejpam-6570	154	1	and	and	CCONJ
ejpam-6570	154	2	there	there	PRON
ejpam-6570	154	3	exists	exist	VERB
ejpam-6570	154	4	a	a	PRON
ejpam-6570	154	5	i	i	PRON
ejpam-6570	154	6	⋆-preopen	⋆-preopen	VERB
ejpam-6570	154	7	set	set	VERB
ejpam-6570	154	8	u	u	PRON
ejpam-6570	154	9	ofx	ofx	NOUN
ejpam-6570	154	10	containing	contain	VERB
ejpam-6570	154	11	x	x	PUNCT
ejpam-6570	154	12	such	such	ADJ
ejpam-6570	154	13	that	that	SCONJ
ejpam-6570	154	14	f	f	PROPN
ejpam-6570	154	15	(	(	PUNCT
ejpam-6570	154	16	u	u	NOUN
ejpam-6570	154	17	)	)	PUNCT
ejpam-6570	154	18	⊆	⊆	NUM
ejpam-6570	154	19	σ1σ2	σ1σ2	NOUN
ejpam-6570	154	20	-	-	NUM
ejpam-6570	154	21	cl(v	cl(v	NOUN
ejpam-6570	154	22	)	)	PUNCT
ejpam-6570	154	23	.	.	PUNCT
ejpam-6570	155	1	(	(	PUNCT
ejpam-6570	155	2	6	6	X
ejpam-6570	155	3	)	)	PUNCT
ejpam-6570	155	4	⇒	⇒	NOUN
ejpam-6570	155	5	(	(	PUNCT
ejpam-6570	155	6	1	1	NUM
ejpam-6570	155	7	):	):	PUNCT
ejpam-6570	155	8	let	let	VERB
ejpam-6570	155	9	x	x	PUNCT
ejpam-6570	155	10	∈	∈	PROPN
ejpam-6570	155	11	x	x	X
ejpam-6570	155	12	and	and	CCONJ
ejpam-6570	155	13	v	v	X
ejpam-6570	155	14	be	be	AUX
ejpam-6570	155	15	any	any	DET
ejpam-6570	155	16	σ1σ2	σ1σ2	NOUN
ejpam-6570	155	17	-	-	ADJ
ejpam-6570	155	18	open	open	ADJ
ejpam-6570	155	19	set	set	NOUN
ejpam-6570	155	20	of	of	ADP
ejpam-6570	155	21	y	y	PROPN
ejpam-6570	155	22	containing	contain	VERB
ejpam-6570	155	23	f	f	PROPN
ejpam-6570	155	24	(	(	PUNCT
ejpam-6570	155	25	x	x	NOUN
ejpam-6570	155	26	)	)	PUNCT
ejpam-6570	155	27	.	.	PUNCT
ejpam-6570	156	1	by	by	ADP
ejpam-6570	156	2	(	(	PUNCT
ejpam-6570	156	3	6	6	NUM
ejpam-6570	156	4	)	)	PUNCT
ejpam-6570	156	5	,	,	PUNCT
ejpam-6570	156	6	there	there	PRON
ejpam-6570	156	7	exists	exist	VERB
ejpam-6570	156	8	an	an	PRON
ejpam-6570	156	9	i	i	PRON
ejpam-6570	156	10	⋆-preopen	⋆-preopen	VERB
ejpam-6570	156	11	set	set	VERB
ejpam-6570	156	12	u	u	NOUN
ejpam-6570	156	13	of	of	ADP
ejpam-6570	156	14	x	x	PUNCT
ejpam-6570	156	15	containing	contain	VERB
ejpam-6570	156	16	x	x	PUNCT
ejpam-6570	156	17	such	such	ADJ
ejpam-6570	156	18	that	that	SCONJ
ejpam-6570	156	19	f	f	PROPN
ejpam-6570	156	20	(	(	PUNCT
ejpam-6570	156	21	u	u	NOUN
ejpam-6570	156	22	)	)	PUNCT
ejpam-6570	156	23	⊆	⊆	NUM
ejpam-6570	156	24	σ1σ2	σ1σ2	NOUN
ejpam-6570	156	25	-	-	NUM
ejpam-6570	156	26	cl(v	cl(v	NOUN
ejpam-6570	156	27	)	)	PUNCT
ejpam-6570	156	28	;	;	PUNCT
ejpam-6570	156	29	hence	hence	ADV
ejpam-6570	156	30	u	u	NOUN
ejpam-6570	156	31	⊆	⊆	NUM
ejpam-6570	156	32	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	156	33	-	-	PUNCT
ejpam-6570	156	34	cl(v	cl(v	NOUN
ejpam-6570	156	35	)	)	PUNCT
ejpam-6570	156	36	)	)	PUNCT
ejpam-6570	156	37	.	.	PUNCT
ejpam-6570	157	1	thus	thus	ADV
ejpam-6570	157	2	,	,	PUNCT
ejpam-6570	157	3	x	x	PUNCT
ejpam-6570	157	4	∈	∈	PROPN
ejpam-6570	157	5	u	u	NOUN
ejpam-6570	157	6	⊆	⊆	NUM
ejpam-6570	157	7	int⋆(cl⋆(u	int⋆(cl⋆(u	NOUN
ejpam-6570	157	8	)	)	PUNCT
ejpam-6570	157	9	)	)	PUNCT
ejpam-6570	158	1	⊆	⊆	X
ejpam-6570	158	2	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-6570	158	3	-	-	PUNCT
ejpam-6570	158	4	cl(v	cl(v	NOUN
ejpam-6570	158	5	)	)	PUNCT
ejpam-6570	158	6	)	)	PUNCT
ejpam-6570	158	7	)	)	PUNCT
ejpam-6570	158	8	)	)	PUNCT
ejpam-6570	158	9	.	.	PUNCT
ejpam-6570	159	1	this	this	PRON
ejpam-6570	159	2	shows	show	VERB
ejpam-6570	159	3	that	that	SCONJ
ejpam-6570	159	4	f	f	PROPN
ejpam-6570	159	5	is	be	AUX
ejpam-6570	159	6	upper	upper	ADJ
ejpam-6570	159	7	almost	almost	ADV
ejpam-6570	159	8	weakly	weakly	ADJ
ejpam-6570	159	9	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	159	10	,	,	PUNCT
ejpam-6570	159	11	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6570	159	12	.	.	NOUN
ejpam-6570	159	13	definition	definition	NOUN
ejpam-6570	159	14	2	2	NUM
ejpam-6570	159	15	.	.	PUNCT
ejpam-6570	159	16	a	a	DET
ejpam-6570	159	17	multifunction	multifunction	NOUN
ejpam-6570	160	1	f	f	NOUN
ejpam-6570	160	2	:	:	PUNCT
ejpam-6570	160	3	(	(	PUNCT
ejpam-6570	160	4	x	x	X
ejpam-6570	160	5	,	,	PUNCT
ejpam-6570	160	6	τ	τ	PROPN
ejpam-6570	160	7	,	,	PUNCT
ejpam-6570	160	8	i	i	NOUN
ejpam-6570	160	9	)	)	PUNCT
ejpam-6570	160	10	→	→	PUNCT
ejpam-6570	160	11	(	(	PUNCT
ejpam-6570	160	12	y	y	PROPN
ejpam-6570	160	13	,	,	PUNCT
ejpam-6570	160	14	σ1	σ1	PROPN
ejpam-6570	160	15	,	,	PUNCT
ejpam-6570	160	16	σ2	σ2	PROPN
ejpam-6570	160	17	)	)	PUNCT
ejpam-6570	160	18	is	be	AUX
ejpam-6570	160	19	said	say	VERB
ejpam-6570	160	20	to	to	PART
ejpam-6570	160	21	be	be	AUX
ejpam-6570	160	22	lower	low	ADJ
ejpam-6570	160	23	almost	almost	ADV
ejpam-6570	160	24	weakly	weakly	ADJ
ejpam-6570	160	25	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	160	26	,	,	PUNCT
ejpam-6570	160	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	160	28	if	if	SCONJ
ejpam-6570	160	29	for	for	ADP
ejpam-6570	160	30	each	each	DET
ejpam-6570	160	31	x	x	SYM
ejpam-6570	160	32	∈	∈	PROPN
ejpam-6570	160	33	x	x	X
ejpam-6570	160	34	and	and	CCONJ
ejpam-6570	160	35	each	each	DET
ejpam-6570	160	36	σ1σ2	σ1σ2	VERB
ejpam-6570	160	37	-	-	ADJ
ejpam-6570	160	38	open	open	ADJ
ejpam-6570	160	39	set	set	NOUN
ejpam-6570	160	40	v	v	NOUN
ejpam-6570	160	41	of	of	ADP
ejpam-6570	160	42	y	y	PRON
ejpam-6570	160	43	such	such	ADJ
ejpam-6570	160	44	that	that	SCONJ
ejpam-6570	160	45	f	f	PROPN
ejpam-6570	160	46	(	(	PUNCT
ejpam-6570	160	47	x	x	NOUN
ejpam-6570	160	48	)	)	PUNCT
ejpam-6570	160	49	∩	∩	NOUN
ejpam-6570	160	50	v	v	ADP
ejpam-6570	160	51	̸=	̸=	PROPN
ejpam-6570	160	52	∅	∅	NOUN
ejpam-6570	160	53	,	,	PUNCT
ejpam-6570	160	54	x	x	SYM
ejpam-6570	160	55	∈	∈	NOUN
ejpam-6570	160	56	int⋆(cl⋆(f−(σ1σ2	int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-6570	160	57	-	-	PUNCT
ejpam-6570	160	58	cl(v	cl(v	NOUN
ejpam-6570	160	59	)	)	PUNCT
ejpam-6570	160	60	)	)	PUNCT
ejpam-6570	160	61	)	)	PUNCT
ejpam-6570	160	62	)	)	PUNCT
ejpam-6570	160	63	.	.	PUNCT
ejpam-6570	161	1	theorem	theorem	NOUN
ejpam-6570	161	2	2	2	NUM
ejpam-6570	161	3	.	.	X
ejpam-6570	161	4	for	for	ADP
ejpam-6570	161	5	a	a	DET
ejpam-6570	161	6	multifunction	multifunction	NOUN
ejpam-6570	162	1	f	f	NOUN
ejpam-6570	162	2	:	:	PUNCT
ejpam-6570	162	3	(	(	PUNCT
ejpam-6570	162	4	x	x	X
ejpam-6570	162	5	,	,	PUNCT
ejpam-6570	162	6	τ	τ	PROPN
ejpam-6570	162	7	,	,	PUNCT
ejpam-6570	162	8	i	i	NOUN
ejpam-6570	162	9	)	)	PUNCT
ejpam-6570	162	10	→	→	PUNCT
ejpam-6570	162	11	(	(	PUNCT
ejpam-6570	162	12	y	y	PROPN
ejpam-6570	162	13	,	,	PUNCT
ejpam-6570	162	14	σ1	σ1	PROPN
ejpam-6570	162	15	,	,	PUNCT
ejpam-6570	162	16	σ2	σ2	NOUN
ejpam-6570	162	17	)	)	PUNCT
ejpam-6570	162	18	,	,	PUNCT
ejpam-6570	162	19	the	the	DET
ejpam-6570	162	20	following	follow	VERB
ejpam-6570	162	21	properties	property	NOUN
ejpam-6570	162	22	are	be	AUX
ejpam-6570	162	23	equivalent	equivalent	ADJ
ejpam-6570	162	24	:	:	PUNCT
ejpam-6570	162	25	(	(	PUNCT
ejpam-6570	162	26	1	1	X
ejpam-6570	162	27	)	)	PUNCT
ejpam-6570	162	28	f	f	PROPN
ejpam-6570	162	29	is	be	AUX
ejpam-6570	162	30	lower	low	ADJ
ejpam-6570	162	31	almost	almost	ADV
ejpam-6570	162	32	weakly	weakly	ADJ
ejpam-6570	162	33	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	162	34	,	,	PUNCT
ejpam-6570	162	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	162	36	;	;	PUNCT
ejpam-6570	162	37	(	(	PUNCT
ejpam-6570	162	38	2	2	X
ejpam-6570	162	39	)	)	PUNCT
ejpam-6570	162	40	f−(v	f−(v	NOUN
ejpam-6570	162	41	)	)	PUNCT
ejpam-6570	162	42	⊆	⊆	NUM
ejpam-6570	162	43	int⋆(cl⋆(f−(σ1σ2	int⋆(cl⋆(f−(σ1σ2	NUM
ejpam-6570	162	44	-	-	PUNCT
ejpam-6570	162	45	cl(v	cl(v	NOUN
ejpam-6570	162	46	)	)	PUNCT
ejpam-6570	162	47	)	)	PUNCT
ejpam-6570	162	48	)	)	PUNCT
ejpam-6570	162	49	)	)	PUNCT
ejpam-6570	163	1	for	for	ADP
ejpam-6570	163	2	every	every	DET
ejpam-6570	163	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	163	4	-	-	ADJ
ejpam-6570	163	5	open	open	ADJ
ejpam-6570	163	6	set	set	NOUN
ejpam-6570	163	7	v	v	NOUN
ejpam-6570	163	8	of	of	ADP
ejpam-6570	163	9	y	y	PROPN
ejpam-6570	163	10	;	;	PUNCT
ejpam-6570	163	11	(	(	PUNCT
ejpam-6570	163	12	3	3	X
ejpam-6570	163	13	)	)	PUNCT
ejpam-6570	163	14	cl⋆(int⋆(f+(v	cl⋆(int⋆(f+(v	ADJ
ejpam-6570	163	15	)	)	PUNCT
ejpam-6570	163	16	)	)	PUNCT
ejpam-6570	163	17	)	)	PUNCT
ejpam-6570	164	1	⊆	⊆	X
ejpam-6570	164	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	164	3	-	-	PUNCT
ejpam-6570	164	4	cl(v	cl(v	NOUN
ejpam-6570	164	5	)	)	PUNCT
ejpam-6570	164	6	)	)	PUNCT
ejpam-6570	164	7	for	for	ADP
ejpam-6570	164	8	every	every	DET
ejpam-6570	164	9	σ1σ2	σ1σ2	NOUN
ejpam-6570	164	10	-	-	ADJ
ejpam-6570	164	11	open	open	ADJ
ejpam-6570	164	12	set	set	NOUN
ejpam-6570	164	13	v	v	NOUN
ejpam-6570	164	14	of	of	ADP
ejpam-6570	164	15	y	y	PROPN
ejpam-6570	164	16	;	;	PUNCT
ejpam-6570	164	17	(	(	PUNCT
ejpam-6570	164	18	4	4	X
ejpam-6570	164	19	)	)	PUNCT
ejpam-6570	164	20	pcl⋆(f+(v	pcl⋆(f+(v	PROPN
ejpam-6570	164	21	)	)	PUNCT
ejpam-6570	164	22	)	)	PUNCT
ejpam-6570	165	1	⊆	⊆	X
ejpam-6570	165	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	165	3	-	-	PUNCT
ejpam-6570	165	4	cl(v	cl(v	NOUN
ejpam-6570	165	5	)	)	PUNCT
ejpam-6570	165	6	)	)	PUNCT
ejpam-6570	165	7	for	for	ADP
ejpam-6570	165	8	every	every	DET
ejpam-6570	165	9	σ1σ2	σ1σ2	NOUN
ejpam-6570	165	10	-	-	ADJ
ejpam-6570	165	11	open	open	ADJ
ejpam-6570	165	12	set	set	NOUN
ejpam-6570	165	13	v	v	NOUN
ejpam-6570	165	14	of	of	ADP
ejpam-6570	165	15	y	y	PROPN
ejpam-6570	165	16	;	;	PUNCT
ejpam-6570	165	17	(	(	PUNCT
ejpam-6570	165	18	5	5	X
ejpam-6570	165	19	)	)	PUNCT
ejpam-6570	165	20	f−(v	f−(v	NOUN
ejpam-6570	165	21	)	)	PUNCT
ejpam-6570	166	1	⊆	⊆	NUM
ejpam-6570	166	2	pint⋆(f−(σ1σ2	pint⋆(f−(σ1σ2	NOUN
ejpam-6570	166	3	-	-	PUNCT
ejpam-6570	166	4	cl(v	cl(v	NOUN
ejpam-6570	166	5	)	)	PUNCT
ejpam-6570	166	6	)	)	PUNCT
ejpam-6570	166	7	)	)	PUNCT
ejpam-6570	166	8	for	for	ADP
ejpam-6570	166	9	every	every	DET
ejpam-6570	166	10	σ1σ2	σ1σ2	NOUN
ejpam-6570	166	11	-	-	ADJ
ejpam-6570	166	12	open	open	ADJ
ejpam-6570	166	13	set	set	NOUN
ejpam-6570	166	14	v	v	NOUN
ejpam-6570	166	15	of	of	ADP
ejpam-6570	166	16	y	y	PROPN
ejpam-6570	166	17	;	;	PUNCT
ejpam-6570	166	18	(	(	PUNCT
ejpam-6570	166	19	6	6	NUM
ejpam-6570	166	20	)	)	PUNCT
ejpam-6570	166	21	for	for	ADP
ejpam-6570	166	22	each	each	DET
ejpam-6570	166	23	x	x	SYM
ejpam-6570	166	24	∈	∈	PROPN
ejpam-6570	166	25	x	x	X
ejpam-6570	166	26	and	and	CCONJ
ejpam-6570	166	27	each	each	DET
ejpam-6570	166	28	σ1σ2	σ1σ2	VERB
ejpam-6570	166	29	-	-	ADJ
ejpam-6570	166	30	open	open	ADJ
ejpam-6570	166	31	set	set	NOUN
ejpam-6570	166	32	v	v	NOUN
ejpam-6570	166	33	of	of	ADP
ejpam-6570	166	34	y	y	PRON
ejpam-6570	166	35	such	such	ADJ
ejpam-6570	166	36	that	that	SCONJ
ejpam-6570	166	37	f	f	PROPN
ejpam-6570	166	38	(	(	PUNCT
ejpam-6570	166	39	x	x	NOUN
ejpam-6570	166	40	)	)	PUNCT
ejpam-6570	166	41	∩	∩	NOUN
ejpam-6570	166	42	v	v	ADP
ejpam-6570	166	43	̸=	̸=	PROPN
ejpam-6570	166	44	∅	∅	NOUN
ejpam-6570	166	45	,	,	PUNCT
ejpam-6570	166	46	there	there	PRON
ejpam-6570	166	47	exists	exist	VERB
ejpam-6570	166	48	an	an	PRON
ejpam-6570	166	49	i	i	PRON
ejpam-6570	166	50	⋆-preopen	⋆-preopen	VERB
ejpam-6570	166	51	set	set	VERB
ejpam-6570	166	52	u	u	NOUN
ejpam-6570	166	53	of	of	ADP
ejpam-6570	166	54	x	x	PUNCT
ejpam-6570	166	55	containing	contain	VERB
ejpam-6570	166	56	x	x	PUNCT
ejpam-6570	166	57	such	such	ADJ
ejpam-6570	166	58	that	that	SCONJ
ejpam-6570	166	59	f	f	PROPN
ejpam-6570	166	60	(	(	PUNCT
ejpam-6570	166	61	z	z	NOUN
ejpam-6570	166	62	)	)	PUNCT
ejpam-6570	166	63	∩	∩	NOUN
ejpam-6570	166	64	σ1σ2	σ1σ2	NOUN
ejpam-6570	166	65	-	-	NUM
ejpam-6570	166	66	cl(v	cl(v	NOUN
ejpam-6570	166	67	)	)	PUNCT
ejpam-6570	166	68	̸=	̸=	NOUN
ejpam-6570	166	69	∅	∅	NOUN
ejpam-6570	166	70	for	for	ADP
ejpam-6570	166	71	each	each	DET
ejpam-6570	166	72	z	z	NOUN
ejpam-6570	166	73	∈	∈	PROPN
ejpam-6570	166	74	u	u	PROPN
ejpam-6570	166	75	.	.	PUNCT
ejpam-6570	167	1	c.	c.	PROPN
ejpam-6570	167	2	viriyapong	viriyapong	PROPN
ejpam-6570	167	3	,	,	PUNCT
ejpam-6570	167	4	a.	a.	PROPN
ejpam-6570	167	5	sama	sama	PROPN
ejpam-6570	167	6	-	-	PUNCT
ejpam-6570	167	7	ae	ae	PROPN
ejpam-6570	167	8	,	,	PUNCT
ejpam-6570	167	9	c.	c.	PROPN
ejpam-6570	167	10	boonpok	boonpok	PROPN
ejpam-6570	167	11	/	/	SYM
ejpam-6570	167	12	eur	eur	PROPN
ejpam-6570	167	13	.	.	PUNCT
ejpam-6570	168	1	j.	j.	PROPN
ejpam-6570	168	2	pure	pure	PROPN
ejpam-6570	168	3	appl	appl	PROPN
ejpam-6570	168	4	.	.	PROPN
ejpam-6570	168	5	math	math	PROPN
ejpam-6570	168	6	,	,	PUNCT
ejpam-6570	168	7	18	18	NUM
ejpam-6570	168	8	(	(	PUNCT
ejpam-6570	168	9	3	3	NUM
ejpam-6570	168	10	)	)	PUNCT
ejpam-6570	168	11	(	(	PUNCT
ejpam-6570	168	12	2025	2025	NUM
ejpam-6570	168	13	)	)	PUNCT
ejpam-6570	168	14	,	,	PUNCT
ejpam-6570	168	15	6570	6570	NUM
ejpam-6570	168	16	6	6	NUM
ejpam-6570	168	17	of	of	ADP
ejpam-6570	168	18	14	14	NUM
ejpam-6570	168	19	proof	proof	NOUN
ejpam-6570	168	20	.	.	PUNCT
ejpam-6570	169	1	the	the	DET
ejpam-6570	169	2	proof	proof	NOUN
ejpam-6570	169	3	is	be	AUX
ejpam-6570	169	4	similar	similar	ADJ
ejpam-6570	169	5	to	to	ADP
ejpam-6570	169	6	that	that	PRON
ejpam-6570	169	7	of	of	ADP
ejpam-6570	169	8	theorem	theorem	ADJ
ejpam-6570	169	9	1	1	NUM
ejpam-6570	169	10	.	.	PUNCT
ejpam-6570	169	11	theorem	theorem	NOUN
ejpam-6570	169	12	3	3	NUM
ejpam-6570	169	13	.	.	X
ejpam-6570	169	14	for	for	ADP
ejpam-6570	169	15	a	a	DET
ejpam-6570	169	16	multifunction	multifunction	NOUN
ejpam-6570	169	17	f	f	NOUN
ejpam-6570	169	18	:	:	PUNCT
ejpam-6570	169	19	(	(	PUNCT
ejpam-6570	169	20	x	x	X
ejpam-6570	169	21	,	,	PUNCT
ejpam-6570	169	22	τ	τ	PROPN
ejpam-6570	169	23	,	,	PUNCT
ejpam-6570	169	24	i	i	NOUN
ejpam-6570	169	25	)	)	PUNCT
ejpam-6570	169	26	→	→	PUNCT
ejpam-6570	169	27	(	(	PUNCT
ejpam-6570	169	28	y	y	PROPN
ejpam-6570	169	29	,	,	PUNCT
ejpam-6570	169	30	σ1	σ1	PROPN
ejpam-6570	169	31	,	,	PUNCT
ejpam-6570	169	32	σ2	σ2	NOUN
ejpam-6570	169	33	)	)	PUNCT
ejpam-6570	169	34	,	,	PUNCT
ejpam-6570	169	35	the	the	DET
ejpam-6570	169	36	following	follow	VERB
ejpam-6570	169	37	properties	property	NOUN
ejpam-6570	169	38	are	be	AUX
ejpam-6570	169	39	equivalent	equivalent	ADJ
ejpam-6570	169	40	:	:	PUNCT
ejpam-6570	169	41	(	(	PUNCT
ejpam-6570	169	42	1	1	X
ejpam-6570	169	43	)	)	PUNCT
ejpam-6570	169	44	f	f	PROPN
ejpam-6570	169	45	is	be	AUX
ejpam-6570	169	46	upper	upper	ADJ
ejpam-6570	169	47	almost	almost	ADV
ejpam-6570	169	48	weakly	weakly	ADJ
ejpam-6570	169	49	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	169	50	,	,	PUNCT
ejpam-6570	169	51	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	169	52	;	;	PUNCT
ejpam-6570	169	53	(	(	PUNCT
ejpam-6570	169	54	2	2	X
ejpam-6570	169	55	)	)	PUNCT
ejpam-6570	169	56	cl⋆(int⋆(f−(σ1σ2	cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-6570	169	57	-	-	PUNCT
ejpam-6570	169	58	int(k	int(k	NOUN
ejpam-6570	169	59	)	)	PUNCT
ejpam-6570	169	60	)	)	PUNCT
ejpam-6570	169	61	)	)	PUNCT
ejpam-6570	169	62	)	)	PUNCT
ejpam-6570	170	1	⊆	⊆	X
ejpam-6570	170	2	f−(k	f−(k	PROPN
ejpam-6570	170	3	)	)	PUNCT
ejpam-6570	170	4	for	for	ADP
ejpam-6570	170	5	every	every	DET
ejpam-6570	170	6	σ1σ2	σ1σ2	NUM
ejpam-6570	170	7	-	-	PUNCT
ejpam-6570	170	8	closed	closed	ADJ
ejpam-6570	170	9	set	set	NOUN
ejpam-6570	170	10	k	k	PROPN
ejpam-6570	170	11	of	of	ADP
ejpam-6570	170	12	y	y	PROPN
ejpam-6570	170	13	;	;	PUNCT
ejpam-6570	170	14	(	(	PUNCT
ejpam-6570	170	15	3	3	X
ejpam-6570	170	16	)	)	PUNCT
ejpam-6570	170	17	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	170	18	-	-	PUNCT
ejpam-6570	170	19	int(k	int(k	NOUN
ejpam-6570	170	20	)	)	PUNCT
ejpam-6570	170	21	)	)	PUNCT
ejpam-6570	170	22	)	)	PUNCT
ejpam-6570	171	1	⊆	⊆	X
ejpam-6570	171	2	f−(k	f−(k	PROPN
ejpam-6570	171	3	)	)	PUNCT
ejpam-6570	171	4	for	for	ADP
ejpam-6570	171	5	every	every	DET
ejpam-6570	171	6	σ1σ2	σ1σ2	NUM
ejpam-6570	171	7	-	-	PUNCT
ejpam-6570	171	8	closed	closed	ADJ
ejpam-6570	171	9	set	set	NOUN
ejpam-6570	171	10	k	k	PROPN
ejpam-6570	171	11	of	of	ADP
ejpam-6570	171	12	y	y	PROPN
ejpam-6570	171	13	;	;	PUNCT
ejpam-6570	171	14	(	(	PUNCT
ejpam-6570	171	15	4	4	X
ejpam-6570	171	16	)	)	PUNCT
ejpam-6570	171	17	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	171	18	-	-	PUNCT
ejpam-6570	171	19	int(σ1σ2cl(b	int(σ1σ2cl(b	NOUN
ejpam-6570	171	20	)	)	PUNCT
ejpam-6570	171	21	)	)	PUNCT
ejpam-6570	171	22	)	)	PUNCT
ejpam-6570	171	23	)	)	PUNCT
ejpam-6570	172	1	⊆	⊆	X
ejpam-6570	172	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6570	172	3	-	-	PUNCT
ejpam-6570	172	4	cl(b	cl(b	NOUN
ejpam-6570	172	5	)	)	PUNCT
ejpam-6570	172	6	)	)	PUNCT
ejpam-6570	173	1	for	for	ADP
ejpam-6570	173	2	every	every	DET
ejpam-6570	173	3	subset	subset	NOUN
ejpam-6570	173	4	b	b	PROPN
ejpam-6570	173	5	of	of	ADP
ejpam-6570	173	6	y	y	PROPN
ejpam-6570	173	7	;	;	PUNCT
ejpam-6570	173	8	(	(	PUNCT
ejpam-6570	173	9	5	5	X
ejpam-6570	173	10	)	)	PUNCT
ejpam-6570	173	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	173	12	-	-	PUNCT
ejpam-6570	173	13	int(b	int(b	NOUN
ejpam-6570	173	14	)	)	PUNCT
ejpam-6570	173	15	)	)	PUNCT
ejpam-6570	173	16	⊆	⊆	NUM
ejpam-6570	173	17	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	VERB
ejpam-6570	173	18	-	-	PUNCT
ejpam-6570	173	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	173	20	-	-	PUNCT
ejpam-6570	173	21	int(b	int(b	NOUN
ejpam-6570	173	22	)	)	PUNCT
ejpam-6570	173	23	)	)	PUNCT
ejpam-6570	173	24	)	)	PUNCT
ejpam-6570	173	25	)	)	PUNCT
ejpam-6570	173	26	for	for	ADP
ejpam-6570	173	27	every	every	DET
ejpam-6570	173	28	subset	subset	NOUN
ejpam-6570	173	29	b	b	PROPN
ejpam-6570	173	30	of	of	ADP
ejpam-6570	173	31	y	y	PROPN
ejpam-6570	173	32	.	.	PUNCT
ejpam-6570	174	1	proof	proof	NOUN
ejpam-6570	174	2	.	.	PUNCT
ejpam-6570	175	1	(	(	PUNCT
ejpam-6570	175	2	1	1	X
ejpam-6570	175	3	)	)	PUNCT
ejpam-6570	175	4	⇒	⇒	NOUN
ejpam-6570	175	5	(	(	PUNCT
ejpam-6570	175	6	2	2	NUM
ejpam-6570	175	7	):	):	PUNCT
ejpam-6570	175	8	let	let	VERB
ejpam-6570	175	9	k	k	PRON
ejpam-6570	175	10	be	be	AUX
ejpam-6570	175	11	any	any	DET
ejpam-6570	175	12	σ1σ2	σ1σ2	NUM
ejpam-6570	175	13	-	-	PUNCT
ejpam-6570	175	14	closed	closed	ADJ
ejpam-6570	175	15	set	set	NOUN
ejpam-6570	175	16	of	of	ADP
ejpam-6570	175	17	y	y	PROPN
ejpam-6570	175	18	.	.	PUNCT
ejpam-6570	176	1	then	then	ADV
ejpam-6570	176	2	,	,	PUNCT
ejpam-6570	176	3	y	y	PROPN
ejpam-6570	176	4	−k	−k	PROPN
ejpam-6570	176	5	is	be	AUX
ejpam-6570	176	6	σ1σ2	σ1σ2	NOUN
ejpam-6570	176	7	-	-	ADJ
ejpam-6570	176	8	open	open	ADJ
ejpam-6570	176	9	in	in	ADP
ejpam-6570	176	10	y	y	PROPN
ejpam-6570	176	11	,	,	PUNCT
ejpam-6570	176	12	by	by	ADP
ejpam-6570	176	13	theorem	theorem	NOUN
ejpam-6570	176	14	1	1	NUM
ejpam-6570	176	15	,	,	PUNCT
ejpam-6570	176	16	we	we	PRON
ejpam-6570	176	17	have	have	VERB
ejpam-6570	176	18	x	x	INTJ
ejpam-6570	176	19	−	−	DET
ejpam-6570	176	20	f−(k	f−(k	PROPN
ejpam-6570	176	21	)	)	PUNCT
ejpam-6570	176	22	=	=	PUNCT
ejpam-6570	177	1	f+(y	f+(y	PROPN
ejpam-6570	177	2	−k	−k	PROPN
ejpam-6570	177	3	)	)	PUNCT
ejpam-6570	177	4	⊆	⊆	NUM
ejpam-6570	177	5	int⋆(cl⋆(f+(σ1σ2	int⋆(cl⋆(f+(σ1σ2	ADJ
ejpam-6570	177	6	-	-	PUNCT
ejpam-6570	177	7	cl(y	cl(y	NOUN
ejpam-6570	177	8	−k	−k	NOUN
ejpam-6570	177	9	)	)	PUNCT
ejpam-6570	177	10	)	)	PUNCT
ejpam-6570	177	11	)	)	PUNCT
ejpam-6570	177	12	)	)	PUNCT
ejpam-6570	178	1	=	=	X
ejpam-6570	178	2	int⋆(cl⋆(f+(y	int⋆(cl⋆(f+(y	NOUN
ejpam-6570	178	3	−	−	NUM
ejpam-6570	178	4	σ1σ2	σ1σ2	NUM
ejpam-6570	178	5	-	-	PUNCT
ejpam-6570	178	6	int(k	int(k	NOUN
ejpam-6570	178	7	)	)	PUNCT
ejpam-6570	178	8	)	)	PUNCT
ejpam-6570	178	9	)	)	PUNCT
ejpam-6570	178	10	)	)	PUNCT
ejpam-6570	179	1	=	=	PRON
ejpam-6570	179	2	int⋆(cl⋆(x	int⋆(cl⋆(x	VERB
ejpam-6570	179	3	−	−	NOUN
ejpam-6570	179	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	179	5	-	-	PUNCT
ejpam-6570	179	6	int(k	int(k	NOUN
ejpam-6570	179	7	)	)	PUNCT
ejpam-6570	179	8	)	)	PUNCT
ejpam-6570	179	9	)	)	PUNCT
ejpam-6570	179	10	)	)	PUNCT
ejpam-6570	180	1	=	=	PUNCT
ejpam-6570	180	2	x	x	X
ejpam-6570	180	3	−	−	ADP
ejpam-6570	180	4	cl⋆(int⋆(f−(σ1σ2	cl⋆(int⋆(f−(σ1σ2	NUM
ejpam-6570	180	5	-	-	PUNCT
ejpam-6570	180	6	int(k	int(k	NOUN
ejpam-6570	180	7	)	)	PUNCT
ejpam-6570	180	8	)	)	PUNCT
ejpam-6570	180	9	)	)	PUNCT
ejpam-6570	180	10	)	)	PUNCT
ejpam-6570	181	1	and	and	CCONJ
ejpam-6570	181	2	so	so	ADV
ejpam-6570	181	3	cl⋆(int⋆(f−(σ1σ2	cl⋆(int⋆(f−(σ1σ2	PROPN
ejpam-6570	181	4	-	-	PUNCT
ejpam-6570	181	5	int(k	int(k	NOUN
ejpam-6570	181	6	)	)	PUNCT
ejpam-6570	181	7	)	)	PUNCT
ejpam-6570	181	8	)	)	PUNCT
ejpam-6570	181	9	)	)	PUNCT
ejpam-6570	182	1	⊆	⊆	NUM
ejpam-6570	182	2	f−(k	f−(k	PROPN
ejpam-6570	182	3	)	)	PUNCT
ejpam-6570	182	4	.	.	PUNCT
ejpam-6570	183	1	(	(	PUNCT
ejpam-6570	183	2	2	2	X
ejpam-6570	183	3	)	)	PUNCT
ejpam-6570	183	4	⇒	⇒	NOUN
ejpam-6570	183	5	(	(	PUNCT
ejpam-6570	183	6	3	3	NUM
ejpam-6570	183	7	):	):	PUNCT
ejpam-6570	183	8	let	let	VERB
ejpam-6570	183	9	k	k	PRON
ejpam-6570	183	10	be	be	AUX
ejpam-6570	183	11	any	any	DET
ejpam-6570	183	12	σ1σ2	σ1σ2	NUM
ejpam-6570	183	13	-	-	PUNCT
ejpam-6570	183	14	closed	closed	ADJ
ejpam-6570	183	15	set	set	NOUN
ejpam-6570	183	16	of	of	ADP
ejpam-6570	183	17	y	y	PROPN
ejpam-6570	183	18	.	.	PUNCT
ejpam-6570	184	1	by	by	ADP
ejpam-6570	184	2	lemma	lemma	PROPN
ejpam-6570	184	3	2	2	NUM
ejpam-6570	184	4	,	,	PUNCT
ejpam-6570	184	5	we	we	PRON
ejpam-6570	184	6	have	have	AUX
ejpam-6570	184	7	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	184	8	-	-	PUNCT
ejpam-6570	184	9	int(k	int(k	NOUN
ejpam-6570	184	10	)	)	PUNCT
ejpam-6570	184	11	)	)	PUNCT
ejpam-6570	184	12	)	)	PUNCT
ejpam-6570	185	1	=	=	PUNCT
ejpam-6570	185	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	185	3	-	-	PUNCT
ejpam-6570	185	4	int(k	int(k	NOUN
ejpam-6570	185	5	)	)	PUNCT
ejpam-6570	185	6	)	)	PUNCT
ejpam-6570	185	7	∪	∪	ADP
ejpam-6570	185	8	cl⋆(int⋆(f−(σ1σ2	cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-6570	185	9	-	-	PUNCT
ejpam-6570	185	10	int(k	int(k	NOUN
ejpam-6570	185	11	)	)	PUNCT
ejpam-6570	185	12	)	)	PUNCT
ejpam-6570	185	13	)	)	PUNCT
ejpam-6570	185	14	)	)	PUNCT
ejpam-6570	186	1	⊆	⊆	NUM
ejpam-6570	186	2	f−(k	f−(k	PROPN
ejpam-6570	186	3	)	)	PUNCT
ejpam-6570	186	4	.	.	PUNCT
ejpam-6570	187	1	(	(	PUNCT
ejpam-6570	187	2	3	3	X
ejpam-6570	187	3	)	)	PUNCT
ejpam-6570	187	4	⇒	⇒	NOUN
ejpam-6570	187	5	(	(	PUNCT
ejpam-6570	187	6	4	4	NUM
ejpam-6570	187	7	):	):	PUNCT
ejpam-6570	187	8	the	the	DET
ejpam-6570	187	9	proof	proof	NOUN
ejpam-6570	187	10	is	be	AUX
ejpam-6570	187	11	obvious	obvious	ADJ
ejpam-6570	187	12	.	.	PUNCT
ejpam-6570	188	1	(	(	PUNCT
ejpam-6570	188	2	4	4	X
ejpam-6570	188	3	)	)	PUNCT
ejpam-6570	188	4	⇒	⇒	NOUN
ejpam-6570	188	5	(	(	PUNCT
ejpam-6570	188	6	5	5	NUM
ejpam-6570	188	7	):	):	PUNCT
ejpam-6570	188	8	let	let	VERB
ejpam-6570	188	9	b	b	X
ejpam-6570	188	10	be	be	AUX
ejpam-6570	188	11	any	any	DET
ejpam-6570	188	12	subset	subset	NOUN
ejpam-6570	188	13	of	of	ADP
ejpam-6570	188	14	y	y	PROPN
ejpam-6570	188	15	.	.	PUNCT
ejpam-6570	189	1	by	by	ADP
ejpam-6570	189	2	(	(	PUNCT
ejpam-6570	189	3	4	4	NUM
ejpam-6570	189	4	)	)	PUNCT
ejpam-6570	189	5	,	,	PUNCT
ejpam-6570	189	6	we	we	PRON
ejpam-6570	189	7	have	have	VERB
ejpam-6570	189	8	x	x	INTJ
ejpam-6570	189	9	−	−	PUNCT
ejpam-6570	189	10	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	ADV
ejpam-6570	189	11	-	-	PUNCT
ejpam-6570	189	12	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	189	13	-	-	PUNCT
ejpam-6570	189	14	int(b	int(b	NOUN
ejpam-6570	189	15	)	)	PUNCT
ejpam-6570	189	16	)	)	PUNCT
ejpam-6570	189	17	)	)	PUNCT
ejpam-6570	189	18	)	)	PUNCT
ejpam-6570	190	1	=	=	SYM
ejpam-6570	190	2	pcl⋆(x	pcl⋆(x	NOUN
ejpam-6570	190	3	−	−	NOUN
ejpam-6570	190	4	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-6570	190	5	-	-	PUNCT
ejpam-6570	190	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	190	7	-	-	PUNCT
ejpam-6570	190	8	int(b	int(b	NOUN
ejpam-6570	190	9	)	)	PUNCT
ejpam-6570	190	10	)	)	PUNCT
ejpam-6570	190	11	)	)	PUNCT
ejpam-6570	190	12	)	)	PUNCT
ejpam-6570	191	1	=	=	SYM
ejpam-6570	191	2	pcl⋆(f−(y	pcl⋆(f−(y	PROPN
ejpam-6570	191	3	−	−	ADP
ejpam-6570	191	4	σ1σ2	σ1σ2	X
ejpam-6570	191	5	-	-	PUNCT
ejpam-6570	191	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	191	7	-	-	PUNCT
ejpam-6570	191	8	int(b	int(b	NOUN
ejpam-6570	191	9	)	)	PUNCT
ejpam-6570	191	10	)	)	PUNCT
ejpam-6570	191	11	)	)	PUNCT
ejpam-6570	191	12	)	)	PUNCT
ejpam-6570	192	1	=	=	PRON
ejpam-6570	192	2	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	192	3	-	-	PUNCT
ejpam-6570	192	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	192	5	-	-	PUNCT
ejpam-6570	192	6	cl(y	cl(y	NOUN
ejpam-6570	192	7	−b	−b	NOUN
ejpam-6570	192	8	)	)	PUNCT
ejpam-6570	192	9	)	)	PUNCT
ejpam-6570	192	10	)	)	PUNCT
ejpam-6570	192	11	)	)	PUNCT
ejpam-6570	193	1	⊆	⊆	X
ejpam-6570	193	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6570	193	3	-	-	PUNCT
ejpam-6570	193	4	cl(y	cl(y	NOUN
ejpam-6570	193	5	−b	−b	NOUN
ejpam-6570	193	6	)	)	PUNCT
ejpam-6570	193	7	)	)	PUNCT
ejpam-6570	194	1	=	=	PUNCT
ejpam-6570	194	2	x	x	X
ejpam-6570	195	1	−	−	ADP
ejpam-6570	195	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	195	3	-	-	PUNCT
ejpam-6570	195	4	int(b	int(b	NOUN
ejpam-6570	195	5	)	)	PUNCT
ejpam-6570	195	6	)	)	PUNCT
ejpam-6570	195	7	.	.	PUNCT
ejpam-6570	196	1	thus	thus	ADV
ejpam-6570	196	2	,	,	PUNCT
ejpam-6570	196	3	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	196	4	-	-	PUNCT
ejpam-6570	196	5	int(b	int(b	NOUN
ejpam-6570	196	6	)	)	PUNCT
ejpam-6570	196	7	)	)	PUNCT
ejpam-6570	196	8	⊆	⊆	NUM
ejpam-6570	196	9	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	VERB
ejpam-6570	196	10	-	-	PUNCT
ejpam-6570	196	11	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	196	12	-	-	PUNCT
ejpam-6570	196	13	int(b	int(b	NOUN
ejpam-6570	196	14	)	)	PUNCT
ejpam-6570	196	15	)	)	PUNCT
ejpam-6570	196	16	)	)	PUNCT
ejpam-6570	196	17	)	)	PUNCT
ejpam-6570	196	18	.	.	PUNCT
ejpam-6570	197	1	(	(	PUNCT
ejpam-6570	197	2	5	5	X
ejpam-6570	197	3	)	)	PUNCT
ejpam-6570	197	4	⇒	⇒	NOUN
ejpam-6570	197	5	(	(	PUNCT
ejpam-6570	197	6	1	1	NUM
ejpam-6570	197	7	):	):	PUNCT
ejpam-6570	197	8	let	let	VERB
ejpam-6570	197	9	v	v	PART
ejpam-6570	197	10	be	be	AUX
ejpam-6570	197	11	any	any	DET
ejpam-6570	197	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	197	13	-	-	ADJ
ejpam-6570	197	14	open	open	ADJ
ejpam-6570	197	15	set	set	NOUN
ejpam-6570	197	16	of	of	ADP
ejpam-6570	197	17	y	y	PROPN
ejpam-6570	197	18	.	.	PUNCT
ejpam-6570	198	1	then	then	ADV
ejpam-6570	198	2	by	by	ADP
ejpam-6570	198	3	(	(	PUNCT
ejpam-6570	198	4	5	5	NUM
ejpam-6570	198	5	)	)	PUNCT
ejpam-6570	198	6	,	,	PUNCT
ejpam-6570	198	7	we	we	PRON
ejpam-6570	198	8	have	have	VERB
ejpam-6570	198	9	f+(v	f+(v	NOUN
ejpam-6570	198	10	)	)	PUNCT
ejpam-6570	199	1	⊆	⊆	NUM
ejpam-6570	199	2	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	199	3	-	-	NOUN
ejpam-6570	199	4	cl(v	cl(v	NOUN
ejpam-6570	199	5	)	)	PUNCT
ejpam-6570	199	6	)	)	PUNCT
ejpam-6570	199	7	)	)	PUNCT
ejpam-6570	200	1	and	and	CCONJ
ejpam-6570	200	2	hence	hence	ADV
ejpam-6570	200	3	f	f	PROPN
ejpam-6570	200	4	is	be	AUX
ejpam-6570	200	5	upper	upper	ADJ
ejpam-6570	200	6	almost	almost	ADV
ejpam-6570	200	7	weakly	weakly	ADJ
ejpam-6570	200	8	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	200	9	,	,	PUNCT
ejpam-6570	200	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	200	11	by	by	ADP
ejpam-6570	200	12	theorem	theorem	NOUN
ejpam-6570	200	13	1	1	NUM
ejpam-6570	200	14	.	.	PUNCT
ejpam-6570	200	15	theorem	theorem	NOUN
ejpam-6570	200	16	4	4	NUM
ejpam-6570	200	17	.	.	X
ejpam-6570	200	18	for	for	ADP
ejpam-6570	200	19	a	a	DET
ejpam-6570	200	20	multifunction	multifunction	NOUN
ejpam-6570	200	21	f	f	NOUN
ejpam-6570	200	22	:	:	PUNCT
ejpam-6570	200	23	(	(	PUNCT
ejpam-6570	200	24	x	x	X
ejpam-6570	200	25	,	,	PUNCT
ejpam-6570	200	26	τ	τ	PROPN
ejpam-6570	200	27	,	,	PUNCT
ejpam-6570	200	28	i	i	NOUN
ejpam-6570	200	29	)	)	PUNCT
ejpam-6570	200	30	→	→	PUNCT
ejpam-6570	200	31	(	(	PUNCT
ejpam-6570	200	32	y	y	PROPN
ejpam-6570	200	33	,	,	PUNCT
ejpam-6570	200	34	σ1	σ1	PROPN
ejpam-6570	200	35	,	,	PUNCT
ejpam-6570	200	36	σ2	σ2	NOUN
ejpam-6570	200	37	)	)	PUNCT
ejpam-6570	200	38	,	,	PUNCT
ejpam-6570	200	39	the	the	DET
ejpam-6570	200	40	following	follow	VERB
ejpam-6570	200	41	properties	property	NOUN
ejpam-6570	200	42	are	be	AUX
ejpam-6570	200	43	equivalent	equivalent	ADJ
ejpam-6570	200	44	:	:	PUNCT
ejpam-6570	200	45	c.	c.	PROPN
ejpam-6570	200	46	viriyapong	viriyapong	PROPN
ejpam-6570	200	47	,	,	PUNCT
ejpam-6570	200	48	a.	a.	PROPN
ejpam-6570	200	49	sama	sama	PROPN
ejpam-6570	200	50	-	-	PUNCT
ejpam-6570	200	51	ae	ae	PROPN
ejpam-6570	200	52	,	,	PUNCT
ejpam-6570	200	53	c.	c.	PROPN
ejpam-6570	200	54	boonpok	boonpok	PROPN
ejpam-6570	200	55	/	/	SYM
ejpam-6570	200	56	eur	eur	PROPN
ejpam-6570	200	57	.	.	PUNCT
ejpam-6570	201	1	j.	j.	PROPN
ejpam-6570	201	2	pure	pure	PROPN
ejpam-6570	201	3	appl	appl	PROPN
ejpam-6570	201	4	.	.	PROPN
ejpam-6570	201	5	math	math	PROPN
ejpam-6570	201	6	,	,	PUNCT
ejpam-6570	201	7	18	18	NUM
ejpam-6570	201	8	(	(	PUNCT
ejpam-6570	201	9	3	3	NUM
ejpam-6570	201	10	)	)	PUNCT
ejpam-6570	201	11	(	(	PUNCT
ejpam-6570	201	12	2025	2025	NUM
ejpam-6570	201	13	)	)	PUNCT
ejpam-6570	201	14	,	,	PUNCT
ejpam-6570	201	15	6570	6570	NUM
ejpam-6570	201	16	7	7	NUM
ejpam-6570	201	17	of	of	ADP
ejpam-6570	201	18	14	14	NUM
ejpam-6570	201	19	(	(	PUNCT
ejpam-6570	201	20	1	1	NUM
ejpam-6570	201	21	)	)	PUNCT
ejpam-6570	201	22	f	f	PROPN
ejpam-6570	201	23	is	be	AUX
ejpam-6570	201	24	lower	low	ADJ
ejpam-6570	201	25	almost	almost	ADV
ejpam-6570	201	26	weakly	weakly	ADJ
ejpam-6570	201	27	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	201	28	,	,	PUNCT
ejpam-6570	201	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	201	30	;	;	PUNCT
ejpam-6570	201	31	(	(	PUNCT
ejpam-6570	201	32	2	2	X
ejpam-6570	201	33	)	)	PUNCT
ejpam-6570	201	34	cl⋆(int⋆(f+(σ1σ2	cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-6570	201	35	-	-	PUNCT
ejpam-6570	201	36	int(k	int(k	NOUN
ejpam-6570	201	37	)	)	PUNCT
ejpam-6570	201	38	)	)	PUNCT
ejpam-6570	201	39	)	)	PUNCT
ejpam-6570	201	40	)	)	PUNCT
ejpam-6570	202	1	⊆	⊆	NUM
ejpam-6570	202	2	f+(k	f+(k	NOUN
ejpam-6570	202	3	)	)	PUNCT
ejpam-6570	202	4	for	for	ADP
ejpam-6570	202	5	every	every	DET
ejpam-6570	202	6	σ1σ2	σ1σ2	NUM
ejpam-6570	202	7	-	-	PUNCT
ejpam-6570	202	8	closed	closed	ADJ
ejpam-6570	202	9	set	set	NOUN
ejpam-6570	202	10	k	k	PROPN
ejpam-6570	202	11	of	of	ADP
ejpam-6570	202	12	y	y	PROPN
ejpam-6570	202	13	;	;	PUNCT
ejpam-6570	202	14	(	(	PUNCT
ejpam-6570	202	15	3	3	X
ejpam-6570	202	16	)	)	PUNCT
ejpam-6570	202	17	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	VERB
ejpam-6570	202	18	-	-	PUNCT
ejpam-6570	202	19	int(k	int(k	NOUN
ejpam-6570	202	20	)	)	PUNCT
ejpam-6570	202	21	)	)	PUNCT
ejpam-6570	202	22	)	)	PUNCT
ejpam-6570	203	1	⊆	⊆	NUM
ejpam-6570	203	2	f+(k	f+(k	NOUN
ejpam-6570	203	3	)	)	PUNCT
ejpam-6570	203	4	for	for	ADP
ejpam-6570	203	5	every	every	DET
ejpam-6570	203	6	σ1σ2	σ1σ2	NUM
ejpam-6570	203	7	-	-	PUNCT
ejpam-6570	203	8	closed	closed	ADJ
ejpam-6570	203	9	set	set	NOUN
ejpam-6570	203	10	k	k	PROPN
ejpam-6570	203	11	of	of	ADP
ejpam-6570	203	12	y	y	PROPN
ejpam-6570	203	13	;	;	PUNCT
ejpam-6570	203	14	(	(	PUNCT
ejpam-6570	203	15	4	4	X
ejpam-6570	203	16	)	)	PUNCT
ejpam-6570	203	17	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	VERB
ejpam-6570	203	18	-	-	PUNCT
ejpam-6570	203	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	203	20	-	-	PUNCT
ejpam-6570	203	21	cl(b	cl(b	NOUN
ejpam-6570	203	22	)	)	PUNCT
ejpam-6570	203	23	)	)	PUNCT
ejpam-6570	203	24	)	)	PUNCT
ejpam-6570	203	25	)	)	PUNCT
ejpam-6570	204	1	⊆	⊆	X
ejpam-6570	204	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	204	3	-	-	PUNCT
ejpam-6570	204	4	cl(b	cl(b	NOUN
ejpam-6570	204	5	)	)	PUNCT
ejpam-6570	204	6	)	)	PUNCT
ejpam-6570	204	7	for	for	ADP
ejpam-6570	204	8	every	every	DET
ejpam-6570	204	9	subset	subset	NOUN
ejpam-6570	204	10	b	b	PROPN
ejpam-6570	204	11	of	of	ADP
ejpam-6570	204	12	y	y	PROPN
ejpam-6570	204	13	;	;	PUNCT
ejpam-6570	204	14	(	(	PUNCT
ejpam-6570	204	15	5	5	X
ejpam-6570	204	16	)	)	PUNCT
ejpam-6570	204	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	204	18	-	-	PUNCT
ejpam-6570	204	19	int(b	int(b	NOUN
ejpam-6570	204	20	)	)	PUNCT
ejpam-6570	204	21	)	)	PUNCT
ejpam-6570	204	22	⊆	⊆	X
ejpam-6570	204	23	pint⋆(f−(σ1σ2	pint⋆(f−(σ1σ2	NOUN
ejpam-6570	204	24	-	-	PUNCT
ejpam-6570	204	25	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	204	26	-	-	PUNCT
ejpam-6570	204	27	int(b	int(b	NOUN
ejpam-6570	204	28	)	)	PUNCT
ejpam-6570	204	29	)	)	PUNCT
ejpam-6570	204	30	)	)	PUNCT
ejpam-6570	204	31	)	)	PUNCT
ejpam-6570	204	32	for	for	ADP
ejpam-6570	204	33	every	every	DET
ejpam-6570	204	34	subset	subset	NOUN
ejpam-6570	204	35	b	b	PROPN
ejpam-6570	204	36	of	of	ADP
ejpam-6570	204	37	y	y	PROPN
ejpam-6570	204	38	.	.	PUNCT
ejpam-6570	205	1	proof	proof	NOUN
ejpam-6570	205	2	.	.	PUNCT
ejpam-6570	206	1	the	the	DET
ejpam-6570	206	2	proof	proof	NOUN
ejpam-6570	206	3	is	be	AUX
ejpam-6570	206	4	similar	similar	ADJ
ejpam-6570	206	5	to	to	ADP
ejpam-6570	206	6	that	that	PRON
ejpam-6570	206	7	of	of	ADP
ejpam-6570	206	8	theorem	theorem	ADJ
ejpam-6570	206	9	3	3	NUM
ejpam-6570	206	10	.	.	PUNCT
ejpam-6570	206	11	theorem	theorem	NOUN
ejpam-6570	206	12	5	5	NUM
ejpam-6570	206	13	.	.	X
ejpam-6570	206	14	for	for	ADP
ejpam-6570	206	15	a	a	DET
ejpam-6570	206	16	multifunction	multifunction	NOUN
ejpam-6570	206	17	f	f	NOUN
ejpam-6570	206	18	:	:	PUNCT
ejpam-6570	206	19	(	(	PUNCT
ejpam-6570	206	20	x	x	X
ejpam-6570	206	21	,	,	PUNCT
ejpam-6570	206	22	τ	τ	PROPN
ejpam-6570	206	23	,	,	PUNCT
ejpam-6570	206	24	i	i	NOUN
ejpam-6570	206	25	)	)	PUNCT
ejpam-6570	206	26	→	→	PUNCT
ejpam-6570	206	27	(	(	PUNCT
ejpam-6570	206	28	y	y	PROPN
ejpam-6570	206	29	,	,	PUNCT
ejpam-6570	206	30	σ1	σ1	PROPN
ejpam-6570	206	31	,	,	PUNCT
ejpam-6570	206	32	σ2	σ2	NOUN
ejpam-6570	206	33	)	)	PUNCT
ejpam-6570	206	34	,	,	PUNCT
ejpam-6570	206	35	the	the	DET
ejpam-6570	206	36	following	follow	VERB
ejpam-6570	206	37	properties	property	NOUN
ejpam-6570	206	38	are	be	AUX
ejpam-6570	206	39	equivalent	equivalent	ADJ
ejpam-6570	206	40	:	:	PUNCT
ejpam-6570	206	41	(	(	PUNCT
ejpam-6570	206	42	1	1	X
ejpam-6570	206	43	)	)	PUNCT
ejpam-6570	206	44	f	f	PROPN
ejpam-6570	206	45	is	be	AUX
ejpam-6570	206	46	upper	upper	ADJ
ejpam-6570	206	47	almost	almost	ADV
ejpam-6570	206	48	weakly	weakly	ADJ
ejpam-6570	206	49	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	206	50	,	,	PUNCT
ejpam-6570	206	51	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	206	52	;	;	PUNCT
ejpam-6570	206	53	(	(	PUNCT
ejpam-6570	206	54	2	2	X
ejpam-6570	206	55	)	)	PUNCT
ejpam-6570	206	56	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	206	57	-	-	PUNCT
ejpam-6570	206	58	int((σ1	int((σ1	PROPN
ejpam-6570	206	59	,	,	PUNCT
ejpam-6570	206	60	σ2)-θcl(b	σ2)-θcl(b	NOUN
ejpam-6570	206	61	)	)	PUNCT
ejpam-6570	206	62	)	)	PUNCT
ejpam-6570	206	63	)	)	PUNCT
ejpam-6570	206	64	)	)	PUNCT
ejpam-6570	207	1	⊆	⊆	NUM
ejpam-6570	207	2	f−((σ1	f−((σ1	NOUN
ejpam-6570	207	3	,	,	PUNCT
ejpam-6570	207	4	σ2)θ	σ2)θ	ADJ
ejpam-6570	207	5	-	-	PUNCT
ejpam-6570	207	6	cl(b	cl(b	NOUN
ejpam-6570	207	7	)	)	PUNCT
ejpam-6570	207	8	)	)	PUNCT
ejpam-6570	207	9	for	for	ADP
ejpam-6570	207	10	every	every	DET
ejpam-6570	207	11	subset	subset	NOUN
ejpam-6570	207	12	b	b	PROPN
ejpam-6570	207	13	of	of	ADP
ejpam-6570	207	14	y	y	PROPN
ejpam-6570	207	15	;	;	PUNCT
ejpam-6570	207	16	(	(	PUNCT
ejpam-6570	207	17	3	3	X
ejpam-6570	207	18	)	)	PUNCT
ejpam-6570	207	19	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	207	20	-	-	PUNCT
ejpam-6570	207	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	207	22	-	-	PUNCT
ejpam-6570	207	23	cl(v	cl(v	NOUN
ejpam-6570	207	24	)	)	PUNCT
ejpam-6570	207	25	)	)	PUNCT
ejpam-6570	207	26	)	)	PUNCT
ejpam-6570	207	27	)	)	PUNCT
ejpam-6570	208	1	⊆	⊆	X
ejpam-6570	208	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	208	3	-	-	PUNCT
ejpam-6570	208	4	cl(v	cl(v	NOUN
ejpam-6570	208	5	)	)	PUNCT
ejpam-6570	208	6	)	)	PUNCT
ejpam-6570	208	7	for	for	ADP
ejpam-6570	208	8	every	every	DET
ejpam-6570	208	9	σ1σ2	σ1σ2	NOUN
ejpam-6570	208	10	-	-	ADJ
ejpam-6570	208	11	open	open	ADJ
ejpam-6570	208	12	set	set	NOUN
ejpam-6570	208	13	v	v	NOUN
ejpam-6570	208	14	of	of	ADP
ejpam-6570	208	15	y	y	PROPN
ejpam-6570	208	16	;	;	PUNCT
ejpam-6570	208	17	(	(	PUNCT
ejpam-6570	208	18	4	4	X
ejpam-6570	208	19	)	)	PUNCT
ejpam-6570	208	20	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	208	21	-	-	PUNCT
ejpam-6570	208	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	208	23	-	-	PUNCT
ejpam-6570	208	24	cl(v	cl(v	NOUN
ejpam-6570	208	25	)	)	PUNCT
ejpam-6570	208	26	)	)	PUNCT
ejpam-6570	208	27	)	)	PUNCT
ejpam-6570	208	28	)	)	PUNCT
ejpam-6570	209	1	⊆	⊆	X
ejpam-6570	209	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	209	3	-	-	PUNCT
ejpam-6570	209	4	cl(v	cl(v	NOUN
ejpam-6570	209	5	)	)	PUNCT
ejpam-6570	209	6	)	)	PUNCT
ejpam-6570	209	7	for	for	ADP
ejpam-6570	209	8	every	every	DET
ejpam-6570	209	9	(	(	PUNCT
ejpam-6570	209	10	σ1	σ1	PROPN
ejpam-6570	209	11	,	,	PUNCT
ejpam-6570	209	12	σ2)p	σ2)p	NOUN
ejpam-6570	209	13	-	-	PUNCT
ejpam-6570	209	14	open	open	NOUN
ejpam-6570	209	15	set	set	NOUN
ejpam-6570	209	16	v	v	NOUN
ejpam-6570	209	17	of	of	ADP
ejpam-6570	209	18	y	y	PROPN
ejpam-6570	209	19	;	;	PUNCT
ejpam-6570	209	20	(	(	PUNCT
ejpam-6570	209	21	5	5	X
ejpam-6570	209	22	)	)	PUNCT
ejpam-6570	209	23	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	209	24	-	-	PUNCT
ejpam-6570	209	25	int(k	int(k	NOUN
ejpam-6570	209	26	)	)	PUNCT
ejpam-6570	209	27	)	)	PUNCT
ejpam-6570	209	28	)	)	PUNCT
ejpam-6570	210	1	⊆	⊆	X
ejpam-6570	210	2	f−(k	f−(k	PROPN
ejpam-6570	210	3	)	)	PUNCT
ejpam-6570	210	4	for	for	ADP
ejpam-6570	210	5	every	every	DET
ejpam-6570	210	6	(	(	PUNCT
ejpam-6570	210	7	σ1	σ1	PROPN
ejpam-6570	210	8	,	,	PUNCT
ejpam-6570	210	9	σ2)r	σ2)r	NOUN
ejpam-6570	210	10	-	-	PUNCT
ejpam-6570	210	11	closed	close	VERB
ejpam-6570	210	12	set	set	ADJ
ejpam-6570	210	13	k	k	PROPN
ejpam-6570	210	14	of	of	ADP
ejpam-6570	210	15	y	y	PROPN
ejpam-6570	210	16	.	.	PUNCT
ejpam-6570	211	1	proof	proof	NOUN
ejpam-6570	211	2	.	.	PUNCT
ejpam-6570	212	1	(	(	PUNCT
ejpam-6570	212	2	1	1	X
ejpam-6570	212	3	)	)	PUNCT
ejpam-6570	212	4	⇒	⇒	NOUN
ejpam-6570	212	5	(	(	PUNCT
ejpam-6570	212	6	2	2	NUM
ejpam-6570	212	7	):	):	PUNCT
ejpam-6570	212	8	let	let	VERB
ejpam-6570	212	9	b	b	X
ejpam-6570	212	10	be	be	AUX
ejpam-6570	212	11	any	any	DET
ejpam-6570	212	12	subset	subset	NOUN
ejpam-6570	212	13	of	of	ADP
ejpam-6570	212	14	y	y	PROPN
ejpam-6570	212	15	.	.	PUNCT
ejpam-6570	213	1	let	let	VERB
ejpam-6570	213	2	x	x	PUNCT
ejpam-6570	213	3	∈	∈	PROPN
ejpam-6570	213	4	x	x	PUNCT
ejpam-6570	213	5	−f−((σ1	−f−((σ1	ADV
ejpam-6570	213	6	,	,	PUNCT
ejpam-6570	213	7	σ2)θ	σ2)θ	ADJ
ejpam-6570	213	8	-	-	PUNCT
ejpam-6570	213	9	cl(b	cl(b	NOUN
ejpam-6570	213	10	)	)	PUNCT
ejpam-6570	213	11	)	)	PUNCT
ejpam-6570	213	12	.	.	PUNCT
ejpam-6570	214	1	then	then	ADV
ejpam-6570	214	2	,	,	PUNCT
ejpam-6570	214	3	x	x	PUNCT
ejpam-6570	214	4	∈	∈	PROPN
ejpam-6570	214	5	f+(y	f+(y	X
ejpam-6570	214	6	−	−	PROPN
ejpam-6570	214	7	(	(	PUNCT
ejpam-6570	214	8	σ1	σ1	PROPN
ejpam-6570	214	9	,	,	PUNCT
ejpam-6570	214	10	σ2)θ	σ2)θ	NOUN
ejpam-6570	214	11	-	-	PUNCT
ejpam-6570	214	12	cl(b	cl(b	NOUN
ejpam-6570	214	13	)	)	PUNCT
ejpam-6570	214	14	)	)	PUNCT
ejpam-6570	214	15	and	and	CCONJ
ejpam-6570	214	16	(	(	PUNCT
ejpam-6570	214	17	σ1	σ1	PROPN
ejpam-6570	214	18	,	,	PUNCT
ejpam-6570	214	19	σ2)θ	σ2)θ	NOUN
ejpam-6570	214	20	-	-	PUNCT
ejpam-6570	214	21	cl(b	cl(b	NOUN
ejpam-6570	214	22	)	)	PUNCT
ejpam-6570	214	23	is	be	AUX
ejpam-6570	214	24	σ1σ2	σ1σ2	NOUN
ejpam-6570	214	25	-	-	ADJ
ejpam-6570	214	26	closed	closed	ADJ
ejpam-6570	214	27	in	in	ADP
ejpam-6570	214	28	y	y	PROPN
ejpam-6570	214	29	.	.	PUNCT
ejpam-6570	215	1	by	by	ADP
ejpam-6570	215	2	theorem	theorem	NOUN
ejpam-6570	215	3	1	1	NUM
ejpam-6570	215	4	,	,	PUNCT
ejpam-6570	215	5	there	there	PRON
ejpam-6570	215	6	exists	exist	VERB
ejpam-6570	215	7	an	an	PRON
ejpam-6570	215	8	i	i	PRON
ejpam-6570	215	9	⋆-preopen	⋆-preopen	VERB
ejpam-6570	215	10	set	set	VERB
ejpam-6570	215	11	u	u	NOUN
ejpam-6570	215	12	of	of	ADP
ejpam-6570	215	13	x	x	PUNCT
ejpam-6570	215	14	containing	contain	VERB
ejpam-6570	215	15	x	x	PUNCT
ejpam-6570	215	16	such	such	ADJ
ejpam-6570	215	17	that	that	SCONJ
ejpam-6570	215	18	u	u	NOUN
ejpam-6570	215	19	⊆	⊆	NUM
ejpam-6570	215	20	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	215	21	-	-	PUNCT
ejpam-6570	215	22	cl(y	cl(y	NOUN
ejpam-6570	215	23	−	−	PROPN
ejpam-6570	215	24	(	(	PUNCT
ejpam-6570	215	25	σ1	σ1	PROPN
ejpam-6570	215	26	,	,	PUNCT
ejpam-6570	215	27	σ2)θ	σ2)θ	NOUN
ejpam-6570	215	28	-	-	PUNCT
ejpam-6570	215	29	cl(b	cl(b	NOUN
ejpam-6570	215	30	)	)	PUNCT
ejpam-6570	215	31	)	)	PUNCT
ejpam-6570	215	32	)	)	PUNCT
ejpam-6570	216	1	=	=	PUNCT
ejpam-6570	217	1	f+(y	f+(y	NOUN
ejpam-6570	217	2	−	−	NUM
ejpam-6570	217	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	217	4	-	-	PUNCT
ejpam-6570	217	5	int((σ1	int((σ1	ADJ
ejpam-6570	217	6	,	,	PUNCT
ejpam-6570	217	7	σ2)θ	σ2)θ	ADJ
ejpam-6570	217	8	-	-	PUNCT
ejpam-6570	217	9	cl(b	cl(b	NOUN
ejpam-6570	217	10	)	)	PUNCT
ejpam-6570	217	11	)	)	PUNCT
ejpam-6570	217	12	)	)	PUNCT
ejpam-6570	218	1	=	=	PUNCT
ejpam-6570	218	2	x	x	X
ejpam-6570	218	3	−	−	NOUN
ejpam-6570	218	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	218	5	-	-	PUNCT
ejpam-6570	218	6	int((σ1	int((σ1	ADJ
ejpam-6570	218	7	,	,	PUNCT
ejpam-6570	218	8	σ2)θ	σ2)θ	ADJ
ejpam-6570	218	9	-	-	PUNCT
ejpam-6570	218	10	cl(b	cl(b	NOUN
ejpam-6570	218	11	)	)	PUNCT
ejpam-6570	218	12	)	)	PUNCT
ejpam-6570	218	13	)	)	PUNCT
ejpam-6570	218	14	.	.	PUNCT
ejpam-6570	219	1	thus	thus	ADV
ejpam-6570	219	2	,	,	PUNCT
ejpam-6570	219	3	u	u	PROPN
ejpam-6570	219	4	∩	∩	NOUN
ejpam-6570	219	5	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	219	6	-	-	PUNCT
ejpam-6570	219	7	int((σ1	int((σ1	PROPN
ejpam-6570	219	8	,	,	PUNCT
ejpam-6570	219	9	σ2)θ	σ2)θ	ADJ
ejpam-6570	219	10	-	-	PUNCT
ejpam-6570	219	11	cl(b	cl(b	NOUN
ejpam-6570	219	12	)	)	PUNCT
ejpam-6570	219	13	)	)	PUNCT
ejpam-6570	219	14	)	)	PUNCT
ejpam-6570	220	1	=	=	NOUN
ejpam-6570	220	2	∅	∅	NOUN
ejpam-6570	220	3	and	and	CCONJ
ejpam-6570	220	4	hence	hence	ADV
ejpam-6570	220	5	x	x	X
ejpam-6570	220	6	∈	∈	NOUN
ejpam-6570	220	7	x	x	X
ejpam-6570	220	8	−	−	VERB
ejpam-6570	220	9	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	220	10	-	-	PUNCT
ejpam-6570	220	11	int((σ1	int((σ1	ADJ
ejpam-6570	220	12	,	,	PUNCT
ejpam-6570	220	13	σ2)θ	σ2)θ	ADJ
ejpam-6570	220	14	-	-	PUNCT
ejpam-6570	220	15	cl(b	cl(b	NOUN
ejpam-6570	220	16	)	)	PUNCT
ejpam-6570	220	17	)	)	PUNCT
ejpam-6570	220	18	)	)	PUNCT
ejpam-6570	220	19	)	)	PUNCT
ejpam-6570	220	20	.	.	PUNCT
ejpam-6570	221	1	therefore	therefore	ADV
ejpam-6570	221	2	,	,	PUNCT
ejpam-6570	221	3	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	221	4	-	-	PUNCT
ejpam-6570	221	5	int((σ1	int((σ1	ADJ
ejpam-6570	221	6	,	,	PUNCT
ejpam-6570	221	7	σ2)θ	σ2)θ	ADJ
ejpam-6570	221	8	-	-	PUNCT
ejpam-6570	221	9	cl(b	cl(b	NOUN
ejpam-6570	221	10	)	)	PUNCT
ejpam-6570	221	11	)	)	PUNCT
ejpam-6570	221	12	)	)	PUNCT
ejpam-6570	221	13	)	)	PUNCT
ejpam-6570	222	1	⊆	⊆	NUM
ejpam-6570	222	2	f−((σ1	f−((σ1	NOUN
ejpam-6570	222	3	,	,	PUNCT
ejpam-6570	222	4	σ2)θ	σ2)θ	ADJ
ejpam-6570	222	5	-	-	PUNCT
ejpam-6570	222	6	cl(b	cl(b	NOUN
ejpam-6570	222	7	)	)	PUNCT
ejpam-6570	222	8	)	)	PUNCT
ejpam-6570	222	9	.	.	PUNCT
ejpam-6570	223	1	(	(	PUNCT
ejpam-6570	223	2	2	2	X
ejpam-6570	223	3	)	)	PUNCT
ejpam-6570	223	4	⇒	⇒	NOUN
ejpam-6570	223	5	(	(	PUNCT
ejpam-6570	223	6	3	3	NUM
ejpam-6570	223	7	):	):	PUNCT
ejpam-6570	223	8	the	the	DET
ejpam-6570	223	9	proof	proof	NOUN
ejpam-6570	223	10	is	be	AUX
ejpam-6570	223	11	obvious	obvious	ADJ
ejpam-6570	223	12	since	since	SCONJ
ejpam-6570	223	13	(	(	PUNCT
ejpam-6570	223	14	σ1	σ1	PROPN
ejpam-6570	223	15	,	,	PUNCT
ejpam-6570	223	16	σ2)θ	σ2)θ	NOUN
ejpam-6570	223	17	-	-	PUNCT
ejpam-6570	223	18	cl(v	cl(v	NOUN
ejpam-6570	223	19	)	)	PUNCT
ejpam-6570	223	20	=	=	SYM
ejpam-6570	223	21	σ1σ2	σ1σ2	NOUN
ejpam-6570	223	22	-	-	NUM
ejpam-6570	223	23	cl(v	cl(v	NOUN
ejpam-6570	223	24	)	)	PUNCT
ejpam-6570	223	25	for	for	ADP
ejpam-6570	223	26	every	every	DET
ejpam-6570	223	27	σ1σ2open	σ1σ2open	PUNCT
ejpam-6570	223	28	set	set	VERB
ejpam-6570	223	29	v	v	NOUN
ejpam-6570	223	30	of	of	ADP
ejpam-6570	223	31	y	y	PROPN
ejpam-6570	223	32	.	.	PUNCT
ejpam-6570	224	1	(	(	PUNCT
ejpam-6570	224	2	3	3	X
ejpam-6570	224	3	)	)	PUNCT
ejpam-6570	224	4	⇒	⇒	NOUN
ejpam-6570	224	5	(	(	PUNCT
ejpam-6570	224	6	4	4	NUM
ejpam-6570	224	7	):	):	PUNCT
ejpam-6570	224	8	let	let	VERB
ejpam-6570	224	9	v	v	PART
ejpam-6570	224	10	be	be	AUX
ejpam-6570	224	11	any	any	DET
ejpam-6570	224	12	(	(	PUNCT
ejpam-6570	224	13	σ1	σ1	PROPN
ejpam-6570	224	14	,	,	PUNCT
ejpam-6570	224	15	σ2)p	σ2)p	NOUN
ejpam-6570	224	16	-	-	PUNCT
ejpam-6570	224	17	open	open	ADJ
ejpam-6570	224	18	set	set	NOUN
ejpam-6570	224	19	of	of	ADP
ejpam-6570	224	20	y	y	PROPN
ejpam-6570	224	21	.	.	PUNCT
ejpam-6570	225	1	then	then	ADV
ejpam-6570	225	2	,	,	PUNCT
ejpam-6570	225	3	v	v	ADP
ejpam-6570	225	4	⊆	⊆	NUM
ejpam-6570	225	5	σ1σ2	σ1σ2	NOUN
ejpam-6570	225	6	-	-	PUNCT
ejpam-6570	225	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	225	8	-	-	PUNCT
ejpam-6570	225	9	cl(v	cl(v	NOUN
ejpam-6570	225	10	)	)	PUNCT
ejpam-6570	225	11	)	)	PUNCT
ejpam-6570	225	12	and	and	CCONJ
ejpam-6570	225	13	by	by	ADP
ejpam-6570	225	14	(	(	PUNCT
ejpam-6570	225	15	3	3	NUM
ejpam-6570	225	16	)	)	PUNCT
ejpam-6570	225	17	,	,	PUNCT
ejpam-6570	225	18	we	we	PRON
ejpam-6570	225	19	have	have	AUX
ejpam-6570	225	20	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	225	21	-	-	PUNCT
ejpam-6570	225	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	225	23	-	-	PUNCT
ejpam-6570	225	24	cl(v	cl(v	NOUN
ejpam-6570	225	25	)	)	PUNCT
ejpam-6570	225	26	)	)	PUNCT
ejpam-6570	225	27	)	)	PUNCT
ejpam-6570	225	28	)	)	PUNCT
ejpam-6570	226	1	=	=	PRON
ejpam-6570	226	2	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	226	3	-	-	PUNCT
ejpam-6570	226	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-6570	226	5	-	-	PUNCT
ejpam-6570	226	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	226	7	-	-	PUNCT
ejpam-6570	226	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	226	9	-	-	PUNCT
ejpam-6570	226	10	cl(v	cl(v	NOUN
ejpam-6570	226	11	)	)	PUNCT
ejpam-6570	226	12	)	)	PUNCT
ejpam-6570	226	13	)	)	PUNCT
ejpam-6570	226	14	)	)	PUNCT
ejpam-6570	226	15	)	)	PUNCT
ejpam-6570	226	16	)	)	PUNCT
ejpam-6570	227	1	⊆	⊆	X
ejpam-6570	227	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6570	227	3	-	-	PUNCT
ejpam-6570	227	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	227	5	-	-	PUNCT
ejpam-6570	227	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	227	7	-	-	PUNCT
ejpam-6570	227	8	int(v	int(v	NOUN
ejpam-6570	227	9	)	)	PUNCT
ejpam-6570	227	10	)	)	PUNCT
ejpam-6570	227	11	)	)	PUNCT
ejpam-6570	227	12	)	)	PUNCT
ejpam-6570	228	1	=	=	PUNCT
ejpam-6570	228	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	228	3	-	-	PUNCT
ejpam-6570	228	4	cl(v	cl(v	NOUN
ejpam-6570	228	5	)	)	PUNCT
ejpam-6570	228	6	)	)	PUNCT
ejpam-6570	228	7	.	.	PUNCT
ejpam-6570	229	1	c.	c.	PROPN
ejpam-6570	229	2	viriyapong	viriyapong	PROPN
ejpam-6570	229	3	,	,	PUNCT
ejpam-6570	229	4	a.	a.	PROPN
ejpam-6570	229	5	sama	sama	PROPN
ejpam-6570	229	6	-	-	PUNCT
ejpam-6570	229	7	ae	ae	PROPN
ejpam-6570	229	8	,	,	PUNCT
ejpam-6570	229	9	c.	c.	PROPN
ejpam-6570	229	10	boonpok	boonpok	PROPN
ejpam-6570	229	11	/	/	SYM
ejpam-6570	229	12	eur	eur	PROPN
ejpam-6570	229	13	.	.	PUNCT
ejpam-6570	230	1	j.	j.	PROPN
ejpam-6570	230	2	pure	pure	PROPN
ejpam-6570	230	3	appl	appl	PROPN
ejpam-6570	230	4	.	.	PROPN
ejpam-6570	230	5	math	math	PROPN
ejpam-6570	230	6	,	,	PUNCT
ejpam-6570	230	7	18	18	NUM
ejpam-6570	230	8	(	(	PUNCT
ejpam-6570	230	9	3	3	NUM
ejpam-6570	230	10	)	)	PUNCT
ejpam-6570	230	11	(	(	PUNCT
ejpam-6570	230	12	2025	2025	NUM
ejpam-6570	230	13	)	)	PUNCT
ejpam-6570	230	14	,	,	PUNCT
ejpam-6570	230	15	6570	6570	NUM
ejpam-6570	230	16	8	8	NUM
ejpam-6570	230	17	of	of	ADP
ejpam-6570	230	18	14	14	NUM
ejpam-6570	230	19	(	(	PUNCT
ejpam-6570	230	20	4	4	NUM
ejpam-6570	230	21	)	)	PUNCT
ejpam-6570	230	22	⇒	⇒	NOUN
ejpam-6570	230	23	(	(	PUNCT
ejpam-6570	230	24	5	5	NUM
ejpam-6570	230	25	):	):	PUNCT
ejpam-6570	230	26	let	let	VERB
ejpam-6570	230	27	k	k	PRON
ejpam-6570	230	28	be	be	AUX
ejpam-6570	230	29	any	any	DET
ejpam-6570	230	30	(	(	PUNCT
ejpam-6570	230	31	σ1	σ1	NOUN
ejpam-6570	230	32	,	,	PUNCT
ejpam-6570	230	33	σ2)r	σ2)r	NOUN
ejpam-6570	230	34	-	-	PUNCT
ejpam-6570	230	35	closed	close	VERB
ejpam-6570	230	36	set	set	NOUN
ejpam-6570	230	37	of	of	ADP
ejpam-6570	230	38	y	y	PROPN
ejpam-6570	230	39	.	.	PUNCT
ejpam-6570	231	1	then	then	ADV
ejpam-6570	231	2	,	,	PUNCT
ejpam-6570	231	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	231	4	-	-	PUNCT
ejpam-6570	231	5	int(k	int(k	NOUN
ejpam-6570	231	6	)	)	PUNCT
ejpam-6570	231	7	is	be	AUX
ejpam-6570	231	8	(	(	PUNCT
ejpam-6570	231	9	σ1	σ1	PROPN
ejpam-6570	231	10	,	,	PUNCT
ejpam-6570	231	11	σ2)p	σ2)p	NOUN
ejpam-6570	231	12	-	-	PUNCT
ejpam-6570	231	13	open	open	ADJ
ejpam-6570	231	14	in	in	ADP
ejpam-6570	231	15	y	y	PROPN
ejpam-6570	231	16	and	and	CCONJ
ejpam-6570	231	17	by	by	ADP
ejpam-6570	231	18	(	(	PUNCT
ejpam-6570	231	19	4	4	NUM
ejpam-6570	231	20	)	)	PUNCT
ejpam-6570	231	21	,	,	PUNCT
ejpam-6570	231	22	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	231	23	-	-	PUNCT
ejpam-6570	231	24	int(k	int(k	NOUN
ejpam-6570	231	25	)	)	PUNCT
ejpam-6570	231	26	)	)	PUNCT
ejpam-6570	231	27	)	)	PUNCT
ejpam-6570	232	1	=	=	PRON
ejpam-6570	232	2	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	232	3	-	-	PUNCT
ejpam-6570	232	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-6570	232	5	-	-	PUNCT
ejpam-6570	232	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6570	232	7	-	-	PUNCT
ejpam-6570	232	8	int(k	int(k	NOUN
ejpam-6570	232	9	)	)	PUNCT
ejpam-6570	232	10	)	)	PUNCT
ejpam-6570	232	11	)	)	PUNCT
ejpam-6570	232	12	)	)	PUNCT
ejpam-6570	232	13	)	)	PUNCT
ejpam-6570	233	1	⊆	⊆	X
ejpam-6570	233	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6570	233	3	-	-	PUNCT
ejpam-6570	233	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6570	233	5	-	-	PUNCT
ejpam-6570	233	6	int(k	int(k	NOUN
ejpam-6570	233	7	)	)	PUNCT
ejpam-6570	233	8	)	)	PUNCT
ejpam-6570	233	9	)	)	PUNCT
ejpam-6570	234	1	=	=	SYM
ejpam-6570	234	2	f−(k	f−(k	PROPN
ejpam-6570	234	3	)	)	PUNCT
ejpam-6570	234	4	.	.	PUNCT
ejpam-6570	235	1	(	(	PUNCT
ejpam-6570	235	2	5	5	X
ejpam-6570	235	3	)	)	PUNCT
ejpam-6570	235	4	⇒	⇒	NOUN
ejpam-6570	235	5	(	(	PUNCT
ejpam-6570	235	6	1	1	NUM
ejpam-6570	235	7	):	):	PUNCT
ejpam-6570	235	8	let	let	VERB
ejpam-6570	235	9	v	v	PART
ejpam-6570	235	10	be	be	AUX
ejpam-6570	235	11	any	any	DET
ejpam-6570	235	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	235	13	-	-	ADJ
ejpam-6570	235	14	open	open	ADJ
ejpam-6570	235	15	set	set	NOUN
ejpam-6570	235	16	of	of	ADP
ejpam-6570	235	17	y	y	PROPN
ejpam-6570	235	18	.	.	PUNCT
ejpam-6570	236	1	then	then	ADV
ejpam-6570	236	2	,	,	PUNCT
ejpam-6570	236	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	236	4	-	-	NUM
ejpam-6570	236	5	cl(v	cl(v	NOUN
ejpam-6570	236	6	)	)	PUNCT
ejpam-6570	236	7	is	be	AUX
ejpam-6570	236	8	(	(	PUNCT
ejpam-6570	236	9	σ1	σ1	NOUN
ejpam-6570	236	10	,	,	PUNCT
ejpam-6570	236	11	σ2)r	σ2)r	NOUN
ejpam-6570	236	12	-	-	PUNCT
ejpam-6570	236	13	closed	closed	ADJ
ejpam-6570	236	14	in	in	ADP
ejpam-6570	236	15	y	y	PROPN
ejpam-6570	236	16	and	and	CCONJ
ejpam-6570	236	17	by	by	ADP
ejpam-6570	236	18	(	(	PUNCT
ejpam-6570	236	19	5	5	NUM
ejpam-6570	236	20	)	)	PUNCT
ejpam-6570	236	21	,	,	PUNCT
ejpam-6570	236	22	pcl⋆(f−(v	pcl⋆(f−(v	PUNCT
ejpam-6570	236	23	)	)	PUNCT
ejpam-6570	236	24	)	)	PUNCT
ejpam-6570	237	1	⊆	⊆	X
ejpam-6570	237	2	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	237	3	-	-	PUNCT
ejpam-6570	237	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	237	5	-	-	PUNCT
ejpam-6570	237	6	cl(v	cl(v	NOUN
ejpam-6570	237	7	)	)	PUNCT
ejpam-6570	237	8	)	)	PUNCT
ejpam-6570	237	9	)	)	PUNCT
ejpam-6570	237	10	)	)	PUNCT
ejpam-6570	237	11	⊆	⊆	X
ejpam-6570	237	12	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	237	13	-	-	PUNCT
ejpam-6570	237	14	cl(v	cl(v	NOUN
ejpam-6570	237	15	)	)	PUNCT
ejpam-6570	237	16	)	)	PUNCT
ejpam-6570	237	17	.	.	PUNCT
ejpam-6570	238	1	it	it	PRON
ejpam-6570	238	2	follows	follow	VERB
ejpam-6570	238	3	from	from	ADP
ejpam-6570	238	4	theorem	theorem	ADJ
ejpam-6570	238	5	1	1	NUM
ejpam-6570	238	6	that	that	SCONJ
ejpam-6570	238	7	f	f	PROPN
ejpam-6570	238	8	is	be	AUX
ejpam-6570	238	9	upper	upper	ADJ
ejpam-6570	238	10	almost	almost	ADV
ejpam-6570	238	11	weakly	weakly	ADJ
ejpam-6570	239	1	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	239	2	,	,	PUNCT
ejpam-6570	239	3	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6570	239	4	.	.	X
ejpam-6570	239	5	theorem	theorem	VERB
ejpam-6570	239	6	6	6	NUM
ejpam-6570	239	7	.	.	PUNCT
ejpam-6570	239	8	for	for	ADP
ejpam-6570	239	9	a	a	DET
ejpam-6570	239	10	multifunction	multifunction	NOUN
ejpam-6570	240	1	f	f	NOUN
ejpam-6570	240	2	:	:	PUNCT
ejpam-6570	240	3	(	(	PUNCT
ejpam-6570	240	4	x	x	X
ejpam-6570	240	5	,	,	PUNCT
ejpam-6570	240	6	τ	τ	PROPN
ejpam-6570	240	7	,	,	PUNCT
ejpam-6570	240	8	i	i	NOUN
ejpam-6570	240	9	)	)	PUNCT
ejpam-6570	240	10	→	→	PUNCT
ejpam-6570	240	11	(	(	PUNCT
ejpam-6570	240	12	y	y	PROPN
ejpam-6570	240	13	,	,	PUNCT
ejpam-6570	240	14	σ1	σ1	PROPN
ejpam-6570	240	15	,	,	PUNCT
ejpam-6570	240	16	σ2	σ2	NOUN
ejpam-6570	240	17	)	)	PUNCT
ejpam-6570	240	18	,	,	PUNCT
ejpam-6570	240	19	the	the	DET
ejpam-6570	240	20	following	follow	VERB
ejpam-6570	240	21	properties	property	NOUN
ejpam-6570	240	22	are	be	AUX
ejpam-6570	240	23	equivalent	equivalent	ADJ
ejpam-6570	240	24	:	:	PUNCT
ejpam-6570	240	25	(	(	PUNCT
ejpam-6570	240	26	1	1	X
ejpam-6570	240	27	)	)	PUNCT
ejpam-6570	240	28	f	f	PROPN
ejpam-6570	240	29	is	be	AUX
ejpam-6570	240	30	lower	low	ADJ
ejpam-6570	240	31	almost	almost	ADV
ejpam-6570	240	32	weakly	weakly	ADJ
ejpam-6570	240	33	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	240	34	,	,	PUNCT
ejpam-6570	240	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	240	36	;	;	PUNCT
ejpam-6570	240	37	(	(	PUNCT
ejpam-6570	240	38	2	2	X
ejpam-6570	240	39	)	)	PUNCT
ejpam-6570	240	40	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	NOUN
ejpam-6570	240	41	-	-	PUNCT
ejpam-6570	240	42	int((σ1	int((σ1	NUM
ejpam-6570	240	43	,	,	PUNCT
ejpam-6570	240	44	σ2)θ	σ2)θ	ADJ
ejpam-6570	240	45	-	-	PUNCT
ejpam-6570	240	46	cl(b	cl(b	NOUN
ejpam-6570	240	47	)	)	PUNCT
ejpam-6570	240	48	)	)	PUNCT
ejpam-6570	240	49	)	)	PUNCT
ejpam-6570	240	50	)	)	PUNCT
ejpam-6570	241	1	⊆	⊆	NUM
ejpam-6570	241	2	f+((σ1	f+((σ1	NOUN
ejpam-6570	241	3	,	,	PUNCT
ejpam-6570	241	4	σ2)θ	σ2)θ	ADJ
ejpam-6570	241	5	-	-	PUNCT
ejpam-6570	241	6	cl(b	cl(b	NOUN
ejpam-6570	241	7	)	)	PUNCT
ejpam-6570	241	8	)	)	PUNCT
ejpam-6570	241	9	for	for	ADP
ejpam-6570	241	10	every	every	DET
ejpam-6570	241	11	subset	subset	NOUN
ejpam-6570	241	12	b	b	PROPN
ejpam-6570	241	13	of	of	ADP
ejpam-6570	241	14	y	y	PROPN
ejpam-6570	241	15	;	;	PUNCT
ejpam-6570	241	16	(	(	PUNCT
ejpam-6570	241	17	3	3	X
ejpam-6570	241	18	)	)	PUNCT
ejpam-6570	241	19	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	VERB
ejpam-6570	241	20	-	-	PUNCT
ejpam-6570	241	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	241	22	-	-	PUNCT
ejpam-6570	241	23	cl(v	cl(v	NOUN
ejpam-6570	241	24	)	)	PUNCT
ejpam-6570	241	25	)	)	PUNCT
ejpam-6570	241	26	)	)	PUNCT
ejpam-6570	241	27	)	)	PUNCT
ejpam-6570	242	1	⊆	⊆	X
ejpam-6570	242	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	242	3	-	-	PUNCT
ejpam-6570	242	4	cl(v	cl(v	NOUN
ejpam-6570	242	5	)	)	PUNCT
ejpam-6570	242	6	)	)	PUNCT
ejpam-6570	242	7	for	for	ADP
ejpam-6570	242	8	every	every	DET
ejpam-6570	242	9	σ1σ2	σ1σ2	NOUN
ejpam-6570	242	10	-	-	ADJ
ejpam-6570	242	11	open	open	ADJ
ejpam-6570	242	12	set	set	NOUN
ejpam-6570	242	13	v	v	NOUN
ejpam-6570	242	14	of	of	ADP
ejpam-6570	242	15	y	y	PROPN
ejpam-6570	242	16	;	;	PUNCT
ejpam-6570	242	17	(	(	PUNCT
ejpam-6570	242	18	4	4	X
ejpam-6570	242	19	)	)	PUNCT
ejpam-6570	242	20	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	VERB
ejpam-6570	242	21	-	-	PUNCT
ejpam-6570	242	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6570	242	23	-	-	PUNCT
ejpam-6570	242	24	cl(v	cl(v	NOUN
ejpam-6570	242	25	)	)	PUNCT
ejpam-6570	242	26	)	)	PUNCT
ejpam-6570	242	27	)	)	PUNCT
ejpam-6570	242	28	)	)	PUNCT
ejpam-6570	243	1	⊆	⊆	X
ejpam-6570	243	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	243	3	-	-	PUNCT
ejpam-6570	243	4	cl(v	cl(v	NOUN
ejpam-6570	243	5	)	)	PUNCT
ejpam-6570	243	6	)	)	PUNCT
ejpam-6570	243	7	for	for	ADP
ejpam-6570	243	8	every	every	DET
ejpam-6570	243	9	(	(	PUNCT
ejpam-6570	243	10	σ1	σ1	PROPN
ejpam-6570	243	11	,	,	PUNCT
ejpam-6570	243	12	σ2)p	σ2)p	NOUN
ejpam-6570	243	13	-	-	PUNCT
ejpam-6570	243	14	open	open	NOUN
ejpam-6570	243	15	set	set	NOUN
ejpam-6570	243	16	v	v	NOUN
ejpam-6570	243	17	of	of	ADP
ejpam-6570	243	18	y	y	PROPN
ejpam-6570	243	19	;	;	PUNCT
ejpam-6570	243	20	(	(	PUNCT
ejpam-6570	243	21	5	5	X
ejpam-6570	243	22	)	)	PUNCT
ejpam-6570	243	23	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	VERB
ejpam-6570	243	24	-	-	PUNCT
ejpam-6570	243	25	int(k	int(k	NOUN
ejpam-6570	243	26	)	)	PUNCT
ejpam-6570	243	27	)	)	PUNCT
ejpam-6570	243	28	)	)	PUNCT
ejpam-6570	243	29	⊆	⊆	NUM
ejpam-6570	243	30	f+(k	f+(k	NOUN
ejpam-6570	243	31	)	)	PUNCT
ejpam-6570	243	32	for	for	ADP
ejpam-6570	243	33	every	every	DET
ejpam-6570	243	34	(	(	PUNCT
ejpam-6570	243	35	σ1	σ1	PROPN
ejpam-6570	243	36	,	,	PUNCT
ejpam-6570	243	37	σ2)r	σ2)r	NOUN
ejpam-6570	243	38	-	-	PUNCT
ejpam-6570	243	39	closed	close	VERB
ejpam-6570	243	40	set	set	ADJ
ejpam-6570	243	41	k	k	PROPN
ejpam-6570	243	42	of	of	ADP
ejpam-6570	243	43	y	y	PROPN
ejpam-6570	243	44	.	.	PUNCT
ejpam-6570	244	1	proof	proof	NOUN
ejpam-6570	244	2	.	.	PUNCT
ejpam-6570	245	1	the	the	DET
ejpam-6570	245	2	proof	proof	NOUN
ejpam-6570	245	3	is	be	AUX
ejpam-6570	245	4	similar	similar	ADJ
ejpam-6570	245	5	to	to	ADP
ejpam-6570	245	6	that	that	PRON
ejpam-6570	245	7	of	of	ADP
ejpam-6570	245	8	theorem	theorem	NOUN
ejpam-6570	245	9	5	5	NUM
ejpam-6570	245	10	.	.	X
ejpam-6570	245	11	for	for	ADP
ejpam-6570	245	12	a	a	DET
ejpam-6570	245	13	multifunction	multifunction	NOUN
ejpam-6570	245	14	f	f	NOUN
ejpam-6570	245	15	:	:	PUNCT
ejpam-6570	245	16	(	(	PUNCT
ejpam-6570	245	17	x	x	X
ejpam-6570	245	18	,	,	PUNCT
ejpam-6570	245	19	τ	τ	PROPN
ejpam-6570	245	20	,	,	PUNCT
ejpam-6570	245	21	i	i	NOUN
ejpam-6570	245	22	)	)	PUNCT
ejpam-6570	245	23	→	→	PUNCT
ejpam-6570	245	24	(	(	PUNCT
ejpam-6570	245	25	y	y	PROPN
ejpam-6570	245	26	,	,	PUNCT
ejpam-6570	245	27	σ1	σ1	PROPN
ejpam-6570	245	28	,	,	PUNCT
ejpam-6570	245	29	σ2	σ2	NOUN
ejpam-6570	245	30	)	)	PUNCT
ejpam-6570	245	31	,	,	PUNCT
ejpam-6570	245	32	by	by	ADP
ejpam-6570	245	33	clfı	clfı	NOUN
ejpam-6570	245	34	:	:	PUNCT
ejpam-6570	245	35	(	(	PUNCT
ejpam-6570	245	36	x	x	X
ejpam-6570	245	37	,	,	PUNCT
ejpam-6570	245	38	τ	τ	PROPN
ejpam-6570	245	39	,	,	PUNCT
ejpam-6570	245	40	i	i	NOUN
ejpam-6570	245	41	)	)	PUNCT
ejpam-6570	245	42	→	→	PUNCT
ejpam-6570	245	43	(	(	PUNCT
ejpam-6570	245	44	y	y	PROPN
ejpam-6570	245	45	,	,	PUNCT
ejpam-6570	245	46	σ1	σ1	PROPN
ejpam-6570	245	47	,	,	PUNCT
ejpam-6570	245	48	σ2	σ2	NOUN
ejpam-6570	245	49	)	)	PUNCT
ejpam-6570	245	50	(	(	PUNCT
ejpam-6570	245	51	resp	resp	NOUN
ejpam-6570	245	52	.	.	PUNCT
ejpam-6570	246	1	pclfı	pclfı	NOUN
ejpam-6570	246	2	:	:	PUNCT
ejpam-6570	246	3	(	(	PUNCT
ejpam-6570	246	4	x	x	X
ejpam-6570	246	5	,	,	PUNCT
ejpam-6570	246	6	τ	τ	PROPN
ejpam-6570	246	7	,	,	PUNCT
ejpam-6570	246	8	i	i	NOUN
ejpam-6570	246	9	)	)	PUNCT
ejpam-6570	246	10	→	→	PUNCT
ejpam-6570	246	11	(	(	PUNCT
ejpam-6570	246	12	y	y	PROPN
ejpam-6570	246	13	,	,	PUNCT
ejpam-6570	246	14	σ1	σ1	PROPN
ejpam-6570	246	15	,	,	PUNCT
ejpam-6570	246	16	σ2	σ2	NOUN
ejpam-6570	246	17	)	)	PUNCT
ejpam-6570	246	18	)	)	PUNCT
ejpam-6570	247	1	we	we	PRON
ejpam-6570	247	2	denote	denote	VERB
ejpam-6570	247	3	a	a	DET
ejpam-6570	247	4	multifunction	multifunction	NOUN
ejpam-6570	247	5	defined	define	VERB
ejpam-6570	247	6	as	as	ADP
ejpam-6570	247	7	follows	follow	VERB
ejpam-6570	247	8	:	:	PUNCT
ejpam-6570	247	9	clfı(x	clfı(x	NUM
ejpam-6570	247	10	)	)	PUNCT
ejpam-6570	247	11	=	=	SYM
ejpam-6570	247	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	247	13	-	-	NUM
ejpam-6570	247	14	cl(f	cl(f	NOUN
ejpam-6570	247	15	(	(	PUNCT
ejpam-6570	247	16	x	x	NOUN
ejpam-6570	247	17	)	)	PUNCT
ejpam-6570	247	18	)	)	PUNCT
ejpam-6570	248	1	(	(	PUNCT
ejpam-6570	248	2	resp	resp	NOUN
ejpam-6570	248	3	.	.	PUNCT
ejpam-6570	249	1	pclfı(x	pclfı(x	NOUN
ejpam-6570	249	2	)	)	PUNCT
ejpam-6570	249	3	=	=	SYM
ejpam-6570	249	4	(	(	PUNCT
ejpam-6570	249	5	σ1	σ1	PROPN
ejpam-6570	249	6	,	,	PUNCT
ejpam-6570	249	7	σ2)-pcl(f	σ2)-pcl(f	NOUN
ejpam-6570	249	8	(	(	PUNCT
ejpam-6570	249	9	x	x	NOUN
ejpam-6570	249	10	)	)	PUNCT
ejpam-6570	249	11	)	)	PUNCT
ejpam-6570	249	12	)	)	PUNCT
ejpam-6570	249	13	for	for	ADP
ejpam-6570	249	14	each	each	DET
ejpam-6570	249	15	x	x	SYM
ejpam-6570	249	16	∈	∈	PROPN
ejpam-6570	249	17	x.	x.	NOUN
ejpam-6570	249	18	definition	definition	NOUN
ejpam-6570	249	19	3	3	NUM
ejpam-6570	249	20	.	.	PUNCT
ejpam-6570	250	1	[	[	X
ejpam-6570	250	2	21	21	NUM
ejpam-6570	250	3	]	]	X
ejpam-6570	250	4	a	a	DET
ejpam-6570	250	5	subset	subset	NOUN
ejpam-6570	250	6	a	a	PRON
ejpam-6570	250	7	of	of	ADP
ejpam-6570	250	8	a	a	DET
ejpam-6570	250	9	bitopological	bitopological	ADJ
ejpam-6570	250	10	space	space	NOUN
ejpam-6570	250	11	(	(	PUNCT
ejpam-6570	250	12	x	x	NOUN
ejpam-6570	250	13	,	,	PUNCT
ejpam-6570	250	14	τ1	τ1	NOUN
ejpam-6570	250	15	,	,	PUNCT
ejpam-6570	250	16	τ2	τ2	NOUN
ejpam-6570	250	17	)	)	PUNCT
ejpam-6570	250	18	is	be	AUX
ejpam-6570	250	19	said	say	VERB
ejpam-6570	250	20	to	to	PART
ejpam-6570	250	21	be	be	AUX
ejpam-6570	250	22	:	:	PUNCT
ejpam-6570	250	23	(	(	PUNCT
ejpam-6570	250	24	1	1	X
ejpam-6570	250	25	)	)	PUNCT
ejpam-6570	250	26	τ1τ2	τ1τ2	NOUN
ejpam-6570	250	27	-	-	NOUN
ejpam-6570	250	28	paracompact	paracompact	ADJ
ejpam-6570	250	29	if	if	SCONJ
ejpam-6570	250	30	every	every	DET
ejpam-6570	250	31	cover	cover	NOUN
ejpam-6570	250	32	of	of	ADP
ejpam-6570	250	33	a	a	PRON
ejpam-6570	250	34	by	by	ADP
ejpam-6570	250	35	τ1τ2	τ1τ2	ADJ
ejpam-6570	250	36	-	-	ADJ
ejpam-6570	250	37	open	open	ADJ
ejpam-6570	250	38	sets	set	NOUN
ejpam-6570	250	39	of	of	ADP
ejpam-6570	250	40	x	x	VERB
ejpam-6570	250	41	is	be	AUX
ejpam-6570	250	42	refined	refine	VERB
ejpam-6570	250	43	by	by	ADP
ejpam-6570	250	44	a	a	DET
ejpam-6570	250	45	cover	cover	NOUN
ejpam-6570	250	46	of	of	ADP
ejpam-6570	250	47	a	a	PRON
ejpam-6570	250	48	which	which	PRON
ejpam-6570	250	49	consists	consist	VERB
ejpam-6570	250	50	of	of	ADP
ejpam-6570	250	51	τ1τ2	τ1τ2	ADJ
ejpam-6570	250	52	-	-	ADJ
ejpam-6570	250	53	open	open	ADJ
ejpam-6570	250	54	sets	set	NOUN
ejpam-6570	250	55	of	of	ADP
ejpam-6570	250	56	x	x	PUNCT
ejpam-6570	250	57	and	and	CCONJ
ejpam-6570	250	58	is	be	AUX
ejpam-6570	250	59	τ1τ2	τ1τ2	NOUN
ejpam-6570	250	60	-	-	ADJ
ejpam-6570	250	61	locally	locally	ADV
ejpam-6570	250	62	finite	finite	NOUN
ejpam-6570	250	63	in	in	ADP
ejpam-6570	250	64	x	x	PRON
ejpam-6570	250	65	;	;	PUNCT
ejpam-6570	250	66	(	(	PUNCT
ejpam-6570	250	67	2	2	X
ejpam-6570	250	68	)	)	PUNCT
ejpam-6570	250	69	τ1τ2	τ1τ2	NOUN
ejpam-6570	250	70	-	-	NOUN
ejpam-6570	250	71	regular	regular	ADJ
ejpam-6570	250	72	if	if	SCONJ
ejpam-6570	250	73	for	for	ADP
ejpam-6570	250	74	each	each	DET
ejpam-6570	250	75	x	x	SYM
ejpam-6570	250	76	∈	∈	PROPN
ejpam-6570	250	77	a	a	PRON
ejpam-6570	250	78	and	and	CCONJ
ejpam-6570	250	79	each	each	DET
ejpam-6570	250	80	τ1τ2	τ1τ2	ADJ
ejpam-6570	250	81	-	-	ADJ
ejpam-6570	250	82	open	open	ADJ
ejpam-6570	250	83	set	set	ADJ
ejpam-6570	250	84	u	u	NOUN
ejpam-6570	250	85	of	of	ADP
ejpam-6570	250	86	x	x	PUNCT
ejpam-6570	250	87	containing	contain	VERB
ejpam-6570	250	88	x	x	PRON
ejpam-6570	250	89	,	,	PUNCT
ejpam-6570	250	90	there	there	PRON
ejpam-6570	250	91	exists	exist	VERB
ejpam-6570	250	92	a	a	DET
ejpam-6570	250	93	τ1τ2	τ1τ2	NOUN
ejpam-6570	250	94	-	-	ADJ
ejpam-6570	250	95	open	open	ADJ
ejpam-6570	250	96	set	set	NOUN
ejpam-6570	250	97	v	v	NOUN
ejpam-6570	250	98	of	of	ADP
ejpam-6570	250	99	x	x	PUNCT
ejpam-6570	250	100	such	such	ADJ
ejpam-6570	250	101	that	that	SCONJ
ejpam-6570	250	102	x	x	SYM
ejpam-6570	250	103	∈	∈	NOUN
ejpam-6570	250	104	v	v	ADP
ejpam-6570	250	105	⊆	⊆	NUM
ejpam-6570	250	106	τ1τ2	τ1τ2	NOUN
ejpam-6570	250	107	-	-	NOUN
ejpam-6570	250	108	cl(v	cl(v	X
ejpam-6570	250	109	)	)	PUNCT
ejpam-6570	250	110	⊆	⊆	NUM
ejpam-6570	250	111	u	u	NOUN
ejpam-6570	250	112	.	.	PUNCT
ejpam-6570	251	1	lemma	lemma	PROPN
ejpam-6570	251	2	3	3	X
ejpam-6570	251	3	.	.	PUNCT
ejpam-6570	252	1	[	[	X
ejpam-6570	252	2	21	21	NUM
ejpam-6570	252	3	]	]	X
ejpam-6570	252	4	if	if	SCONJ
ejpam-6570	252	5	a	a	PRON
ejpam-6570	252	6	is	be	AUX
ejpam-6570	252	7	a	a	DET
ejpam-6570	252	8	τ1τ2	τ1τ2	ADJ
ejpam-6570	252	9	-	-	ADJ
ejpam-6570	252	10	regular	regular	ADJ
ejpam-6570	252	11	τ1τ2	τ1τ2	NOUN
ejpam-6570	252	12	-	-	ADJ
ejpam-6570	252	13	paracompact	paracompact	ADJ
ejpam-6570	252	14	set	set	NOUN
ejpam-6570	252	15	of	of	ADP
ejpam-6570	252	16	a	a	DET
ejpam-6570	252	17	bitopological	bitopological	ADJ
ejpam-6570	252	18	space	space	NOUN
ejpam-6570	252	19	(	(	PUNCT
ejpam-6570	252	20	x	x	NOUN
ejpam-6570	252	21	,	,	PUNCT
ejpam-6570	252	22	τ1	τ1	NOUN
ejpam-6570	252	23	,	,	PUNCT
ejpam-6570	252	24	τ2	τ2	NOUN
ejpam-6570	252	25	)	)	PUNCT
ejpam-6570	252	26	and	and	CCONJ
ejpam-6570	252	27	u	u	NOUN
ejpam-6570	252	28	is	be	AUX
ejpam-6570	252	29	a	a	DET
ejpam-6570	252	30	τ1τ2	τ1τ2	ADJ
ejpam-6570	252	31	-	-	ADJ
ejpam-6570	252	32	open	open	ADJ
ejpam-6570	252	33	neighbourhood	neighbourhood	NOUN
ejpam-6570	252	34	of	of	ADP
ejpam-6570	252	35	a	a	PRON
ejpam-6570	252	36	,	,	PUNCT
ejpam-6570	252	37	then	then	ADV
ejpam-6570	252	38	there	there	PRON
ejpam-6570	252	39	exists	exist	VERB
ejpam-6570	252	40	a	a	DET
ejpam-6570	252	41	τ1τ2	τ1τ2	NOUN
ejpam-6570	252	42	-	-	ADJ
ejpam-6570	252	43	open	open	ADJ
ejpam-6570	252	44	set	set	NOUN
ejpam-6570	252	45	v	v	NOUN
ejpam-6570	252	46	of	of	ADP
ejpam-6570	252	47	x	x	PUNCT
ejpam-6570	252	48	such	such	ADJ
ejpam-6570	252	49	that	that	SCONJ
ejpam-6570	252	50	a	a	DET
ejpam-6570	252	51	⊆	⊆	NUM
ejpam-6570	252	52	v	v	ADP
ejpam-6570	252	53	⊆	⊆	NUM
ejpam-6570	252	54	τ1τ2	τ1τ2	NOUN
ejpam-6570	252	55	-	-	NOUN
ejpam-6570	252	56	cl(v	cl(v	X
ejpam-6570	252	57	)	)	PUNCT
ejpam-6570	252	58	⊆	⊆	NUM
ejpam-6570	252	59	u	u	NOUN
ejpam-6570	252	60	.	.	PUNCT
ejpam-6570	253	1	lemma	lemma	PROPN
ejpam-6570	253	2	4	4	X
ejpam-6570	253	3	.	.	PUNCT
ejpam-6570	254	1	if	if	SCONJ
ejpam-6570	254	2	f	f	PROPN
ejpam-6570	254	3	:	:	PUNCT
ejpam-6570	254	4	(	(	PUNCT
ejpam-6570	254	5	x	x	X
ejpam-6570	254	6	,	,	PUNCT
ejpam-6570	254	7	τ	τ	PROPN
ejpam-6570	254	8	,	,	PUNCT
ejpam-6570	254	9	i	i	NOUN
ejpam-6570	254	10	)	)	PUNCT
ejpam-6570	254	11	→	→	PUNCT
ejpam-6570	254	12	(	(	PUNCT
ejpam-6570	254	13	y	y	PROPN
ejpam-6570	254	14	,	,	PUNCT
ejpam-6570	254	15	σ1	σ1	PROPN
ejpam-6570	254	16	,	,	PUNCT
ejpam-6570	254	17	σ2	σ2	PROPN
ejpam-6570	254	18	)	)	PUNCT
ejpam-6570	254	19	is	be	AUX
ejpam-6570	254	20	a	a	DET
ejpam-6570	254	21	multifunction	multifunction	NOUN
ejpam-6570	254	22	such	such	ADJ
ejpam-6570	254	23	that	that	SCONJ
ejpam-6570	254	24	f	f	PROPN
ejpam-6570	254	25	(	(	PUNCT
ejpam-6570	254	26	x	x	X
ejpam-6570	254	27	)	)	PUNCT
ejpam-6570	254	28	is	be	AUX
ejpam-6570	254	29	σ1σ2regular	σ1σ2regular	PROPN
ejpam-6570	254	30	and	and	CCONJ
ejpam-6570	254	31	σ1σ2	σ1σ2	NOUN
ejpam-6570	254	32	-	-	ADJ
ejpam-6570	254	33	paracompact	paracompact	NOUN
ejpam-6570	254	34	for	for	ADP
ejpam-6570	254	35	each	each	DET
ejpam-6570	254	36	x	x	SYM
ejpam-6570	254	37	∈	∈	PROPN
ejpam-6570	254	38	x	x	NOUN
ejpam-6570	254	39	,	,	PUNCT
ejpam-6570	254	40	then	then	ADV
ejpam-6570	254	41	clf+	clf+	PROPN
ejpam-6570	254	42	ı	ı	PROPN
ejpam-6570	254	43	(	(	PUNCT
ejpam-6570	254	44	v	v	NOUN
ejpam-6570	254	45	)	)	PUNCT
ejpam-6570	255	1	=	=	NOUN
ejpam-6570	255	2	pclf+	pclf+	NOUN
ejpam-6570	255	3	ı	ı	PROPN
ejpam-6570	255	4	(	(	PUNCT
ejpam-6570	255	5	v	v	NOUN
ejpam-6570	255	6	)	)	PUNCT
ejpam-6570	255	7	=	=	PUNCT
ejpam-6570	255	8	f+(v	f+(v	NOUN
ejpam-6570	255	9	)	)	PUNCT
ejpam-6570	256	1	for	for	ADP
ejpam-6570	256	2	each	each	DET
ejpam-6570	256	3	σ1σ2	σ1σ2	VERB
ejpam-6570	256	4	-	-	ADJ
ejpam-6570	256	5	open	open	ADJ
ejpam-6570	256	6	set	set	NOUN
ejpam-6570	256	7	v	v	NOUN
ejpam-6570	256	8	of	of	ADP
ejpam-6570	256	9	y	y	PROPN
ejpam-6570	256	10	.	.	PUNCT
ejpam-6570	257	1	c.	c.	PROPN
ejpam-6570	257	2	viriyapong	viriyapong	PROPN
ejpam-6570	257	3	,	,	PUNCT
ejpam-6570	257	4	a.	a.	PROPN
ejpam-6570	257	5	sama	sama	PROPN
ejpam-6570	257	6	-	-	PUNCT
ejpam-6570	257	7	ae	ae	PROPN
ejpam-6570	257	8	,	,	PUNCT
ejpam-6570	257	9	c.	c.	PROPN
ejpam-6570	257	10	boonpok	boonpok	PROPN
ejpam-6570	257	11	/	/	SYM
ejpam-6570	257	12	eur	eur	PROPN
ejpam-6570	257	13	.	.	PUNCT
ejpam-6570	258	1	j.	j.	PROPN
ejpam-6570	258	2	pure	pure	PROPN
ejpam-6570	258	3	appl	appl	PROPN
ejpam-6570	258	4	.	.	PROPN
ejpam-6570	258	5	math	math	PROPN
ejpam-6570	258	6	,	,	PUNCT
ejpam-6570	258	7	18	18	NUM
ejpam-6570	258	8	(	(	PUNCT
ejpam-6570	258	9	3	3	NUM
ejpam-6570	258	10	)	)	PUNCT
ejpam-6570	258	11	(	(	PUNCT
ejpam-6570	258	12	2025	2025	NUM
ejpam-6570	258	13	)	)	PUNCT
ejpam-6570	258	14	,	,	PUNCT
ejpam-6570	258	15	6570	6570	NUM
ejpam-6570	258	16	9	9	NUM
ejpam-6570	258	17	of	of	ADP
ejpam-6570	258	18	14	14	NUM
ejpam-6570	258	19	proof	proof	NOUN
ejpam-6570	258	20	.	.	PUNCT
ejpam-6570	259	1	it	it	PRON
ejpam-6570	259	2	follows	follow	VERB
ejpam-6570	259	3	from	from	ADP
ejpam-6570	259	4	lemma	lemma	PROPN
ejpam-6570	259	5	5	5	NUM
ejpam-6570	259	6	of	of	ADP
ejpam-6570	259	7	[	[	X
ejpam-6570	259	8	28	28	NUM
ejpam-6570	259	9	]	]	PUNCT
ejpam-6570	259	10	.	.	PUNCT
ejpam-6570	260	1	theorem	theorem	ADJ
ejpam-6570	260	2	7	7	NUM
ejpam-6570	260	3	.	.	PUNCT
ejpam-6570	261	1	let	let	VERB
ejpam-6570	261	2	f	f	NOUN
ejpam-6570	261	3	:	:	PUNCT
ejpam-6570	261	4	(	(	PUNCT
ejpam-6570	261	5	x	x	X
ejpam-6570	261	6	,	,	PUNCT
ejpam-6570	261	7	τ	τ	PROPN
ejpam-6570	261	8	,	,	PUNCT
ejpam-6570	261	9	i	i	NOUN
ejpam-6570	261	10	)	)	PUNCT
ejpam-6570	261	11	→	→	PUNCT
ejpam-6570	261	12	(	(	PUNCT
ejpam-6570	261	13	y	y	PROPN
ejpam-6570	261	14	,	,	PUNCT
ejpam-6570	261	15	σ1	σ1	PROPN
ejpam-6570	261	16	,	,	PUNCT
ejpam-6570	261	17	σ2	σ2	PROPN
ejpam-6570	261	18	)	)	PUNCT
ejpam-6570	261	19	be	be	VERB
ejpam-6570	261	20	a	a	DET
ejpam-6570	261	21	multifunction	multifunction	NOUN
ejpam-6570	261	22	such	such	ADJ
ejpam-6570	261	23	that	that	SCONJ
ejpam-6570	261	24	f	f	PROPN
ejpam-6570	261	25	(	(	PUNCT
ejpam-6570	261	26	x	x	X
ejpam-6570	261	27	)	)	PUNCT
ejpam-6570	261	28	is	be	AUX
ejpam-6570	261	29	σ1σ2paracompact	σ1σ2paracompact	NUM
ejpam-6570	261	30	and	and	CCONJ
ejpam-6570	261	31	σ1σ2	σ1σ2	NOUN
ejpam-6570	261	32	-	-	ADJ
ejpam-6570	261	33	regular	regular	ADJ
ejpam-6570	261	34	for	for	ADP
ejpam-6570	261	35	each	each	DET
ejpam-6570	261	36	x	x	SYM
ejpam-6570	261	37	∈	∈	PROPN
ejpam-6570	261	38	x.	x.	NOUN
ejpam-6570	261	39	then	then	ADV
ejpam-6570	261	40	,	,	PUNCT
ejpam-6570	261	41	the	the	DET
ejpam-6570	261	42	following	follow	VERB
ejpam-6570	261	43	properties	property	NOUN
ejpam-6570	261	44	are	be	AUX
ejpam-6570	261	45	equivalent	equivalent	ADJ
ejpam-6570	261	46	:	:	PUNCT
ejpam-6570	261	47	(	(	PUNCT
ejpam-6570	261	48	1	1	X
ejpam-6570	261	49	)	)	PUNCT
ejpam-6570	261	50	f	f	PROPN
ejpam-6570	261	51	is	be	AUX
ejpam-6570	261	52	upper	upper	ADJ
ejpam-6570	261	53	almost	almost	ADV
ejpam-6570	261	54	weakly	weakly	ADJ
ejpam-6570	261	55	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	261	56	,	,	PUNCT
ejpam-6570	261	57	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	261	58	;	;	PUNCT
ejpam-6570	261	59	(	(	PUNCT
ejpam-6570	261	60	2	2	X
ejpam-6570	261	61	)	)	PUNCT
ejpam-6570	261	62	pclfı	pclfı	NOUN
ejpam-6570	261	63	is	be	AUX
ejpam-6570	261	64	upper	upper	ADJ
ejpam-6570	261	65	almost	almost	ADV
ejpam-6570	261	66	weakly	weakly	ADJ
ejpam-6570	261	67	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	261	68	,	,	PUNCT
ejpam-6570	261	69	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	261	70	;	;	PUNCT
ejpam-6570	261	71	(	(	PUNCT
ejpam-6570	261	72	3	3	X
ejpam-6570	261	73	)	)	PUNCT
ejpam-6570	261	74	clfı	clfı	NOUN
ejpam-6570	261	75	is	be	AUX
ejpam-6570	261	76	upper	upper	ADJ
ejpam-6570	261	77	almost	almost	ADV
ejpam-6570	261	78	weakly	weakly	ADJ
ejpam-6570	261	79	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	261	80	,	,	PUNCT
ejpam-6570	261	81	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	261	82	.	.	NOUN
ejpam-6570	261	83	proof	proof	NOUN
ejpam-6570	261	84	.	.	PUNCT
ejpam-6570	262	1	we	we	PRON
ejpam-6570	262	2	put	put	VERB
ejpam-6570	262	3	g	g	NOUN
ejpam-6570	262	4	=	=	PUNCT
ejpam-6570	262	5	clfı	clfı	NOUN
ejpam-6570	262	6	or	or	CCONJ
ejpam-6570	262	7	pclfı	pclfı	NOUN
ejpam-6570	262	8	in	in	ADP
ejpam-6570	262	9	the	the	DET
ejpam-6570	262	10	sequel	sequel	NOUN
ejpam-6570	262	11	.	.	PUNCT
ejpam-6570	263	1	suppose	suppose	VERB
ejpam-6570	263	2	that	that	SCONJ
ejpam-6570	263	3	f	f	PROPN
ejpam-6570	263	4	is	be	AUX
ejpam-6570	263	5	upper	upper	ADJ
ejpam-6570	263	6	almost	almost	ADV
ejpam-6570	263	7	weakly	weakly	ADJ
ejpam-6570	263	8	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	263	9	,	,	PUNCT
ejpam-6570	263	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	263	11	.	.	PUNCT
ejpam-6570	263	12	let	let	VERB
ejpam-6570	263	13	x	x	SYM
ejpam-6570	263	14	∈	∈	PROPN
ejpam-6570	263	15	x	x	X
ejpam-6570	263	16	and	and	CCONJ
ejpam-6570	263	17	v	v	X
ejpam-6570	263	18	be	be	AUX
ejpam-6570	263	19	any	any	DET
ejpam-6570	263	20	σ1σ2	σ1σ2	NOUN
ejpam-6570	263	21	-	-	ADJ
ejpam-6570	263	22	open	open	ADJ
ejpam-6570	263	23	set	set	NOUN
ejpam-6570	263	24	of	of	ADP
ejpam-6570	263	25	y	y	NOUN
ejpam-6570	263	26	containing	contain	VERB
ejpam-6570	263	27	g(x	g(x	NOUN
ejpam-6570	263	28	)	)	PUNCT
ejpam-6570	263	29	.	.	PUNCT
ejpam-6570	264	1	by	by	ADP
ejpam-6570	264	2	lemma	lemma	PROPN
ejpam-6570	264	3	4	4	NUM
ejpam-6570	264	4	,	,	PUNCT
ejpam-6570	264	5	we	we	PRON
ejpam-6570	264	6	have	have	VERB
ejpam-6570	264	7	x	x	X
ejpam-6570	264	8	∈	∈	PROPN
ejpam-6570	264	9	g+(v	g+(v	PROPN
ejpam-6570	264	10	)	)	PUNCT
ejpam-6570	264	11	=	=	PUNCT
ejpam-6570	265	1	f+(v	f+(v	NOUN
ejpam-6570	265	2	)	)	PUNCT
ejpam-6570	266	1	and	and	CCONJ
ejpam-6570	266	2	hence	hence	ADV
ejpam-6570	266	3	there	there	PRON
ejpam-6570	266	4	exists	exist	VERB
ejpam-6570	266	5	an	an	DET
ejpam-6570	266	6	i	i	PROPN
ejpam-6570	266	7	⋆preopen	⋆preopen	PROPN
ejpam-6570	266	8	set	set	VERB
ejpam-6570	266	9	u	u	NOUN
ejpam-6570	266	10	containing	contain	VERB
ejpam-6570	266	11	x	x	PUNCT
ejpam-6570	267	1	such	such	ADJ
ejpam-6570	267	2	that	that	SCONJ
ejpam-6570	267	3	f	f	PROPN
ejpam-6570	267	4	(	(	PUNCT
ejpam-6570	267	5	u	u	NOUN
ejpam-6570	267	6	)	)	PUNCT
ejpam-6570	267	7	⊆	⊆	NUM
ejpam-6570	267	8	σ1σ2	σ1σ2	NOUN
ejpam-6570	267	9	-	-	NUM
ejpam-6570	267	10	cl(v	cl(v	NOUN
ejpam-6570	267	11	)	)	PUNCT
ejpam-6570	267	12	.	.	PUNCT
ejpam-6570	268	1	since	since	SCONJ
ejpam-6570	268	2	f	f	PROPN
ejpam-6570	268	3	(	(	PUNCT
ejpam-6570	268	4	z	z	NOUN
ejpam-6570	268	5	)	)	PUNCT
ejpam-6570	268	6	is	be	AUX
ejpam-6570	268	7	σ1σ2	σ1σ2	NOUN
ejpam-6570	268	8	-	-	ADJ
ejpam-6570	268	9	paracompact	paracompact	ADJ
ejpam-6570	268	10	and	and	CCONJ
ejpam-6570	268	11	σ1σ2	σ1σ2	NOUN
ejpam-6570	268	12	-	-	ADJ
ejpam-6570	268	13	regular	regular	ADJ
ejpam-6570	268	14	for	for	ADP
ejpam-6570	268	15	each	each	DET
ejpam-6570	268	16	z	z	NOUN
ejpam-6570	268	17	∈	∈	PROPN
ejpam-6570	268	18	u	u	NOUN
ejpam-6570	268	19	,	,	PUNCT
ejpam-6570	268	20	by	by	ADP
ejpam-6570	268	21	lemma	lemma	PROPN
ejpam-6570	268	22	3	3	NUM
ejpam-6570	268	23	there	there	ADV
ejpam-6570	268	24	exists	exist	VERB
ejpam-6570	268	25	a	a	DET
ejpam-6570	268	26	σ1σ2	σ1σ2	NUM
ejpam-6570	268	27	-	-	ADJ
ejpam-6570	268	28	open	open	ADJ
ejpam-6570	268	29	set	set	NOUN
ejpam-6570	268	30	w	w	ADP
ejpam-6570	268	31	such	such	ADJ
ejpam-6570	268	32	that	that	SCONJ
ejpam-6570	268	33	f	f	PROPN
ejpam-6570	268	34	(	(	PUNCT
ejpam-6570	268	35	z	z	NOUN
ejpam-6570	268	36	)	)	PUNCT
ejpam-6570	268	37	⊆	⊆	NUM
ejpam-6570	268	38	w	w	ADP
ejpam-6570	268	39	⊆	⊆	NUM
ejpam-6570	268	40	σ1σ2	σ1σ2	NOUN
ejpam-6570	268	41	-	-	PUNCT
ejpam-6570	268	42	cl(w	cl(w	NOUN
ejpam-6570	268	43	)	)	PUNCT
ejpam-6570	268	44	⊆	⊆	NUM
ejpam-6570	268	45	v	v	NOUN
ejpam-6570	268	46	;	;	PUNCT
ejpam-6570	268	47	hence	hence	ADV
ejpam-6570	268	48	g(z	g(z	ADJ
ejpam-6570	268	49	)	)	PUNCT
ejpam-6570	268	50	⊆	⊆	NUM
ejpam-6570	268	51	σ1σ2	σ1σ2	NOUN
ejpam-6570	268	52	-	-	PUNCT
ejpam-6570	268	53	cl(w	cl(w	NOUN
ejpam-6570	268	54	)	)	PUNCT
ejpam-6570	268	55	⊆	⊆	NUM
ejpam-6570	268	56	σ1σ2	σ1σ2	NOUN
ejpam-6570	268	57	-	-	NUM
ejpam-6570	268	58	cl(v	cl(v	NOUN
ejpam-6570	268	59	)	)	PUNCT
ejpam-6570	268	60	for	for	ADP
ejpam-6570	268	61	each	each	DET
ejpam-6570	268	62	z	z	NOUN
ejpam-6570	268	63	∈	∈	PROPN
ejpam-6570	268	64	u	u	NOUN
ejpam-6570	268	65	.	.	PUNCT
ejpam-6570	269	1	thus	thus	ADV
ejpam-6570	269	2	,	,	PUNCT
ejpam-6570	269	3	g(u	g(u	PROPN
ejpam-6570	269	4	)	)	PUNCT
ejpam-6570	269	5	⊆	⊆	NUM
ejpam-6570	269	6	σ1σ2	σ1σ2	NOUN
ejpam-6570	269	7	-	-	NUM
ejpam-6570	269	8	cl(v	cl(v	NOUN
ejpam-6570	269	9	)	)	PUNCT
ejpam-6570	269	10	.	.	PUNCT
ejpam-6570	270	1	this	this	PRON
ejpam-6570	270	2	shows	show	VERB
ejpam-6570	270	3	that	that	SCONJ
ejpam-6570	270	4	g	g	PROPN
ejpam-6570	270	5	is	be	AUX
ejpam-6570	270	6	upper	upper	ADJ
ejpam-6570	270	7	almost	almost	ADV
ejpam-6570	270	8	weakly	weakly	ADJ
ejpam-6570	270	9	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	270	10	,	,	PUNCT
ejpam-6570	270	11	σ2)continuous	σ2)continuous	ADJ
ejpam-6570	270	12	.	.	PUNCT
ejpam-6570	271	1	conversely	conversely	ADV
ejpam-6570	271	2	,	,	PUNCT
ejpam-6570	271	3	suppose	suppose	VERB
ejpam-6570	271	4	that	that	SCONJ
ejpam-6570	271	5	g	g	PROPN
ejpam-6570	271	6	is	be	AUX
ejpam-6570	271	7	upper	upper	ADJ
ejpam-6570	271	8	almost	almost	ADV
ejpam-6570	271	9	weakly	weakly	ADJ
ejpam-6570	271	10	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	271	11	,	,	PUNCT
ejpam-6570	271	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	271	13	.	.	PUNCT
ejpam-6570	271	14	let	let	VERB
ejpam-6570	271	15	x	x	SYM
ejpam-6570	271	16	∈	∈	PROPN
ejpam-6570	271	17	x	x	X
ejpam-6570	271	18	and	and	CCONJ
ejpam-6570	271	19	v	v	X
ejpam-6570	271	20	be	be	AUX
ejpam-6570	271	21	any	any	DET
ejpam-6570	271	22	σ1σ2	σ1σ2	NOUN
ejpam-6570	271	23	-	-	ADJ
ejpam-6570	271	24	open	open	ADJ
ejpam-6570	271	25	set	set	NOUN
ejpam-6570	271	26	of	of	ADP
ejpam-6570	271	27	y	y	NOUN
ejpam-6570	271	28	containing	contain	VERB
ejpam-6570	271	29	g(x	g(x	NOUN
ejpam-6570	271	30	)	)	PUNCT
ejpam-6570	271	31	.	.	PUNCT
ejpam-6570	272	1	by	by	ADP
ejpam-6570	272	2	lemma	lemma	PROPN
ejpam-6570	272	3	4	4	NUM
ejpam-6570	272	4	,	,	PUNCT
ejpam-6570	272	5	we	we	PRON
ejpam-6570	272	6	have	have	VERB
ejpam-6570	272	7	x	x	X
ejpam-6570	272	8	∈	∈	NOUN
ejpam-6570	272	9	f+(v	f+(v	NOUN
ejpam-6570	272	10	)	)	PUNCT
ejpam-6570	273	1	=	=	PUNCT
ejpam-6570	273	2	g+(v	g+(v	PROPN
ejpam-6570	273	3	)	)	PUNCT
ejpam-6570	273	4	and	and	CCONJ
ejpam-6570	273	5	hence	hence	ADV
ejpam-6570	273	6	g(x	g(x	NOUN
ejpam-6570	273	7	)	)	PUNCT
ejpam-6570	273	8	⊆	⊆	NUM
ejpam-6570	273	9	v	v	NOUN
ejpam-6570	273	10	.	.	PUNCT
ejpam-6570	274	1	then	then	ADV
ejpam-6570	274	2	,	,	PUNCT
ejpam-6570	274	3	there	there	PRON
ejpam-6570	274	4	exists	exist	VERB
ejpam-6570	274	5	an	an	DET
ejpam-6570	274	6	i	i	PRON
ejpam-6570	274	7	⋆-preopen	⋆-preopen	VERB
ejpam-6570	274	8	set	set	VERB
ejpam-6570	274	9	u	u	NOUN
ejpam-6570	274	10	containing	contain	VERB
ejpam-6570	274	11	x	x	PUNCT
ejpam-6570	275	1	such	such	ADJ
ejpam-6570	275	2	that	that	SCONJ
ejpam-6570	275	3	f	f	PROPN
ejpam-6570	275	4	(	(	PUNCT
ejpam-6570	275	5	u	u	NOUN
ejpam-6570	275	6	)	)	PUNCT
ejpam-6570	275	7	⊆	⊆	NUM
ejpam-6570	275	8	σ1σ2	σ1σ2	NOUN
ejpam-6570	275	9	-	-	NUM
ejpam-6570	275	10	cl(v	cl(v	NOUN
ejpam-6570	275	11	)	)	PUNCT
ejpam-6570	275	12	.	.	PUNCT
ejpam-6570	276	1	thus	thus	ADV
ejpam-6570	276	2	,	,	PUNCT
ejpam-6570	276	3	u	u	PROPN
ejpam-6570	276	4	⊆	⊆	NUM
ejpam-6570	276	5	g+(v	g+(v	PROPN
ejpam-6570	276	6	)	)	PUNCT
ejpam-6570	276	7	=	=	PUNCT
ejpam-6570	277	1	f+(v	f+(v	NOUN
ejpam-6570	277	2	)	)	PUNCT
ejpam-6570	278	1	and	and	CCONJ
ejpam-6570	278	2	hence	hence	ADV
ejpam-6570	278	3	f	f	PROPN
ejpam-6570	278	4	(	(	PUNCT
ejpam-6570	278	5	u	u	NOUN
ejpam-6570	278	6	)	)	PUNCT
ejpam-6570	278	7	⊆	⊆	NUM
ejpam-6570	278	8	σ1σ2	σ1σ2	NOUN
ejpam-6570	278	9	-	-	NUM
ejpam-6570	278	10	cl(v	cl(v	NOUN
ejpam-6570	278	11	)	)	PUNCT
ejpam-6570	278	12	.	.	PUNCT
ejpam-6570	279	1	this	this	PRON
ejpam-6570	279	2	shows	show	VERB
ejpam-6570	279	3	that	that	SCONJ
ejpam-6570	279	4	f	f	PROPN
ejpam-6570	279	5	is	be	AUX
ejpam-6570	279	6	upper	upper	ADJ
ejpam-6570	279	7	almost	almost	ADV
ejpam-6570	279	8	weakly	weakly	ADJ
ejpam-6570	280	1	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	280	2	,	,	PUNCT
ejpam-6570	280	3	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6570	280	4	.	.	PUNCT
ejpam-6570	281	1	lemma	lemma	PROPN
ejpam-6570	281	2	5	5	NUM
ejpam-6570	281	3	.	.	PUNCT
ejpam-6570	281	4	for	for	ADP
ejpam-6570	281	5	a	a	DET
ejpam-6570	281	6	multifunction	multifunction	NOUN
ejpam-6570	281	7	f	f	NOUN
ejpam-6570	281	8	:	:	PUNCT
ejpam-6570	281	9	(	(	PUNCT
ejpam-6570	281	10	x	x	X
ejpam-6570	281	11	,	,	PUNCT
ejpam-6570	281	12	τ	τ	PROPN
ejpam-6570	281	13	,	,	PUNCT
ejpam-6570	281	14	i	i	NOUN
ejpam-6570	281	15	)	)	PUNCT
ejpam-6570	281	16	→	→	PUNCT
ejpam-6570	281	17	(	(	PUNCT
ejpam-6570	281	18	y	y	PROPN
ejpam-6570	281	19	,	,	PUNCT
ejpam-6570	281	20	σ1	σ1	PROPN
ejpam-6570	281	21	,	,	PUNCT
ejpam-6570	281	22	σ2	σ2	NOUN
ejpam-6570	281	23	)	)	PUNCT
ejpam-6570	281	24	,	,	PUNCT
ejpam-6570	281	25	clf−	clf−	PROPN
ejpam-6570	281	26	ı	ı	PROPN
ejpam-6570	281	27	(	(	PUNCT
ejpam-6570	281	28	v	v	NOUN
ejpam-6570	281	29	)	)	PUNCT
ejpam-6570	281	30	=	=	SYM
ejpam-6570	281	31	pclf−	pclf−	PROPN
ejpam-6570	281	32	ı	ı	PROPN
ejpam-6570	281	33	(	(	PUNCT
ejpam-6570	281	34	v	v	NOUN
ejpam-6570	281	35	)	)	PUNCT
ejpam-6570	281	36	=	=	SYM
ejpam-6570	281	37	f−(v	f−(v	ADJ
ejpam-6570	281	38	)	)	PUNCT
ejpam-6570	281	39	for	for	ADP
ejpam-6570	281	40	each	each	DET
ejpam-6570	281	41	σ1σ2	σ1σ2	VERB
ejpam-6570	281	42	-	-	ADJ
ejpam-6570	281	43	open	open	ADJ
ejpam-6570	281	44	set	set	NOUN
ejpam-6570	281	45	v	v	NOUN
ejpam-6570	281	46	of	of	ADP
ejpam-6570	281	47	y	y	PROPN
ejpam-6570	281	48	.	.	PUNCT
ejpam-6570	282	1	proof	proof	NOUN
ejpam-6570	282	2	.	.	PUNCT
ejpam-6570	283	1	it	it	PRON
ejpam-6570	283	2	follows	follow	VERB
ejpam-6570	283	3	from	from	ADP
ejpam-6570	283	4	lemma	lemma	PROPN
ejpam-6570	283	5	3	3	NUM
ejpam-6570	283	6	of	of	ADP
ejpam-6570	283	7	[	[	X
ejpam-6570	283	8	28	28	NUM
ejpam-6570	283	9	]	]	PUNCT
ejpam-6570	283	10	.	.	PUNCT
ejpam-6570	284	1	theorem	theorem	ADJ
ejpam-6570	284	2	8	8	NUM
ejpam-6570	284	3	.	.	PUNCT
ejpam-6570	285	1	for	for	ADP
ejpam-6570	285	2	a	a	DET
ejpam-6570	285	3	multifunction	multifunction	NOUN
ejpam-6570	285	4	f	f	NOUN
ejpam-6570	285	5	:	:	PUNCT
ejpam-6570	285	6	(	(	PUNCT
ejpam-6570	285	7	x	x	X
ejpam-6570	285	8	,	,	PUNCT
ejpam-6570	285	9	τ	τ	PROPN
ejpam-6570	285	10	,	,	PUNCT
ejpam-6570	285	11	i	i	NOUN
ejpam-6570	285	12	)	)	PUNCT
ejpam-6570	285	13	→	→	PUNCT
ejpam-6570	285	14	(	(	PUNCT
ejpam-6570	285	15	y	y	PROPN
ejpam-6570	285	16	,	,	PUNCT
ejpam-6570	285	17	σ1	σ1	PROPN
ejpam-6570	285	18	,	,	PUNCT
ejpam-6570	285	19	σ2	σ2	NOUN
ejpam-6570	285	20	)	)	PUNCT
ejpam-6570	285	21	,	,	PUNCT
ejpam-6570	285	22	the	the	DET
ejpam-6570	285	23	following	follow	VERB
ejpam-6570	285	24	properties	property	NOUN
ejpam-6570	285	25	are	be	AUX
ejpam-6570	285	26	equivalent	equivalent	ADJ
ejpam-6570	285	27	:	:	PUNCT
ejpam-6570	285	28	(	(	PUNCT
ejpam-6570	285	29	1	1	X
ejpam-6570	285	30	)	)	PUNCT
ejpam-6570	285	31	f	f	PROPN
ejpam-6570	285	32	is	be	AUX
ejpam-6570	285	33	lower	low	ADJ
ejpam-6570	285	34	almost	almost	ADV
ejpam-6570	285	35	weakly	weakly	ADJ
ejpam-6570	285	36	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	285	37	,	,	PUNCT
ejpam-6570	285	38	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	285	39	;	;	PUNCT
ejpam-6570	285	40	(	(	PUNCT
ejpam-6570	285	41	2	2	X
ejpam-6570	285	42	)	)	PUNCT
ejpam-6570	285	43	pclfı	pclfı	NOUN
ejpam-6570	285	44	is	be	AUX
ejpam-6570	285	45	lower	low	ADJ
ejpam-6570	285	46	almost	almost	ADV
ejpam-6570	285	47	weakly	weakly	ADJ
ejpam-6570	285	48	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	285	49	,	,	PUNCT
ejpam-6570	285	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	285	51	;	;	PUNCT
ejpam-6570	285	52	(	(	PUNCT
ejpam-6570	285	53	3	3	X
ejpam-6570	285	54	)	)	PUNCT
ejpam-6570	285	55	clfı	clfı	NOUN
ejpam-6570	285	56	is	be	AUX
ejpam-6570	285	57	lower	low	ADJ
ejpam-6570	285	58	almost	almost	ADV
ejpam-6570	285	59	weakly	weakly	ADJ
ejpam-6570	285	60	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	285	61	,	,	PUNCT
ejpam-6570	285	62	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	285	63	.	.	NOUN
ejpam-6570	285	64	proof	proof	NOUN
ejpam-6570	285	65	.	.	PUNCT
ejpam-6570	286	1	by	by	ADP
ejpam-6570	286	2	using	use	VERB
ejpam-6570	286	3	lemma	lemma	PROPN
ejpam-6570	286	4	5	5	NUM
ejpam-6570	286	5	this	this	PRON
ejpam-6570	286	6	can	can	AUX
ejpam-6570	286	7	be	be	AUX
ejpam-6570	286	8	shown	show	VERB
ejpam-6570	286	9	similarly	similarly	ADV
ejpam-6570	286	10	to	to	ADP
ejpam-6570	286	11	that	that	PRON
ejpam-6570	286	12	of	of	ADP
ejpam-6570	286	13	theorem	theorem	NOUN
ejpam-6570	286	14	7	7	NUM
ejpam-6570	286	15	.	.	PUNCT
ejpam-6570	286	16	the	the	DET
ejpam-6570	286	17	⋆-prefrontier	⋆-prefrontier	NOUN
ejpam-6570	286	18	of	of	ADP
ejpam-6570	286	19	a	a	DET
ejpam-6570	286	20	subset	subset	NOUN
ejpam-6570	286	21	a	a	PRON
ejpam-6570	286	22	of	of	ADP
ejpam-6570	286	23	an	an	DET
ejpam-6570	286	24	ideal	ideal	ADJ
ejpam-6570	286	25	topological	topological	ADJ
ejpam-6570	286	26	space	space	NOUN
ejpam-6570	286	27	(	(	PUNCT
ejpam-6570	286	28	x	x	X
ejpam-6570	286	29	,	,	PUNCT
ejpam-6570	286	30	τ	τ	PROPN
ejpam-6570	286	31	,	,	PUNCT
ejpam-6570	286	32	i	i	PROPN
ejpam-6570	286	33	)	)	PUNCT
ejpam-6570	286	34	,	,	PUNCT
ejpam-6570	286	35	denoted	denote	VERB
ejpam-6570	286	36	by	by	ADP
ejpam-6570	286	37	pfr⋆(a	pfr⋆(a	NOUN
ejpam-6570	286	38	)	)	PUNCT
ejpam-6570	286	39	,	,	PUNCT
ejpam-6570	286	40	is	be	AUX
ejpam-6570	286	41	defined	define	VERB
ejpam-6570	286	42	by	by	ADP
ejpam-6570	286	43	pfr⋆(a	pfr⋆(a	NOUN
ejpam-6570	286	44	)	)	PUNCT
ejpam-6570	286	45	=	=	SYM
ejpam-6570	286	46	pcl⋆(a	pcl⋆(a	NOUN
ejpam-6570	286	47	)	)	PUNCT
ejpam-6570	286	48	∩	∩	NOUN
ejpam-6570	286	49	pcl⋆(x	pcl⋆(x	NOUN
ejpam-6570	286	50	−a	−a	NOUN
ejpam-6570	286	51	)	)	PUNCT
ejpam-6570	286	52	=	=	PUNCT
ejpam-6570	287	1	pcl⋆(a)−	pcl⋆(a)−	NOUN
ejpam-6570	287	2	pint⋆(a	pint⋆(a	PROPN
ejpam-6570	287	3	)	)	PUNCT
ejpam-6570	287	4	.	.	PUNCT
ejpam-6570	288	1	c.	c.	PROPN
ejpam-6570	288	2	viriyapong	viriyapong	PROPN
ejpam-6570	288	3	,	,	PUNCT
ejpam-6570	288	4	a.	a.	PROPN
ejpam-6570	288	5	sama	sama	PROPN
ejpam-6570	288	6	-	-	PUNCT
ejpam-6570	288	7	ae	ae	PROPN
ejpam-6570	288	8	,	,	PUNCT
ejpam-6570	288	9	c.	c.	PROPN
ejpam-6570	288	10	boonpok	boonpok	PROPN
ejpam-6570	288	11	/	/	SYM
ejpam-6570	288	12	eur	eur	PROPN
ejpam-6570	288	13	.	.	PUNCT
ejpam-6570	289	1	j.	j.	PROPN
ejpam-6570	289	2	pure	pure	PROPN
ejpam-6570	289	3	appl	appl	PROPN
ejpam-6570	289	4	.	.	PROPN
ejpam-6570	289	5	math	math	PROPN
ejpam-6570	289	6	,	,	PUNCT
ejpam-6570	289	7	18	18	NUM
ejpam-6570	289	8	(	(	PUNCT
ejpam-6570	289	9	3	3	NUM
ejpam-6570	289	10	)	)	PUNCT
ejpam-6570	289	11	(	(	PUNCT
ejpam-6570	289	12	2025	2025	NUM
ejpam-6570	289	13	)	)	PUNCT
ejpam-6570	289	14	,	,	PUNCT
ejpam-6570	289	15	6570	6570	NUM
ejpam-6570	289	16	10	10	NUM
ejpam-6570	289	17	of	of	ADP
ejpam-6570	289	18	14	14	NUM
ejpam-6570	289	19	theorem	theorem	NOUN
ejpam-6570	289	20	9	9	NUM
ejpam-6570	289	21	.	.	PUNCT
ejpam-6570	290	1	the	the	DET
ejpam-6570	290	2	set	set	NOUN
ejpam-6570	290	3	of	of	ADP
ejpam-6570	290	4	all	all	DET
ejpam-6570	290	5	points	point	NOUN
ejpam-6570	290	6	x	x	PUNCT
ejpam-6570	290	7	of	of	ADP
ejpam-6570	290	8	x	x	SYM
ejpam-6570	290	9	at	at	ADP
ejpam-6570	290	10	which	which	PRON
ejpam-6570	290	11	a	a	DET
ejpam-6570	290	12	multifunction	multifunction	NOUN
ejpam-6570	291	1	f	f	NOUN
ejpam-6570	291	2	:	:	PUNCT
ejpam-6570	291	3	(	(	PUNCT
ejpam-6570	291	4	x	x	X
ejpam-6570	291	5	,	,	PUNCT
ejpam-6570	291	6	τ	τ	PROPN
ejpam-6570	291	7	,	,	PUNCT
ejpam-6570	291	8	i	i	NOUN
ejpam-6570	291	9	)	)	PUNCT
ejpam-6570	291	10	→	→	PUNCT
ejpam-6570	291	11	(	(	PUNCT
ejpam-6570	291	12	y	y	PROPN
ejpam-6570	291	13	,	,	PUNCT
ejpam-6570	291	14	σ1	σ1	PROPN
ejpam-6570	291	15	,	,	PUNCT
ejpam-6570	291	16	σ2	σ2	PROPN
ejpam-6570	291	17	)	)	PUNCT
ejpam-6570	291	18	is	be	AUX
ejpam-6570	291	19	not	not	PART
ejpam-6570	291	20	upper	upper	ADJ
ejpam-6570	291	21	almost	almost	ADV
ejpam-6570	291	22	weakly	weakly	ADJ
ejpam-6570	291	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	291	24	,	,	PUNCT
ejpam-6570	291	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	291	26	is	be	AUX
ejpam-6570	291	27	identical	identical	ADJ
ejpam-6570	291	28	with	with	ADP
ejpam-6570	291	29	the	the	DET
ejpam-6570	291	30	union	union	NOUN
ejpam-6570	291	31	of	of	ADP
ejpam-6570	291	32	the	the	DET
ejpam-6570	291	33	⋆prefrontier	⋆prefrontier	PROPN
ejpam-6570	291	34	of	of	ADP
ejpam-6570	291	35	the	the	DET
ejpam-6570	291	36	upper	upper	ADJ
ejpam-6570	291	37	inverse	inverse	NOUN
ejpam-6570	291	38	images	image	NOUN
ejpam-6570	291	39	of	of	ADP
ejpam-6570	291	40	the	the	DET
ejpam-6570	291	41	σ1σ2	σ1σ2	NOUN
ejpam-6570	291	42	-	-	NOUN
ejpam-6570	291	43	closure	closure	NOUN
ejpam-6570	291	44	of	of	ADP
ejpam-6570	291	45	σ1σ2	σ1σ2	NOUN
ejpam-6570	291	46	-	-	PUNCT
ejpam-6570	291	47	open	open	ADJ
ejpam-6570	291	48	sets	set	NOUN
ejpam-6570	291	49	containing	contain	VERB
ejpam-6570	291	50	f	f	X
ejpam-6570	291	51	(	(	PUNCT
ejpam-6570	291	52	x	x	NOUN
ejpam-6570	291	53	)	)	PUNCT
ejpam-6570	291	54	.	.	PUNCT
ejpam-6570	292	1	proof	proof	NOUN
ejpam-6570	292	2	.	.	PUNCT
ejpam-6570	293	1	let	let	VERB
ejpam-6570	293	2	x	x	PUNCT
ejpam-6570	293	3	∈	∈	PROPN
ejpam-6570	293	4	x	x	PUNCT
ejpam-6570	293	5	at	at	ADP
ejpam-6570	293	6	which	which	PRON
ejpam-6570	293	7	f	f	NOUN
ejpam-6570	293	8	is	be	AUX
ejpam-6570	293	9	not	not	PART
ejpam-6570	293	10	upper	upper	ADJ
ejpam-6570	293	11	almost	almost	ADV
ejpam-6570	293	12	weakly	weakly	ADJ
ejpam-6570	293	13	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	293	14	,	,	PUNCT
ejpam-6570	293	15	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6570	293	16	.	.	PUNCT
ejpam-6570	294	1	there	there	PRON
ejpam-6570	294	2	exists	exist	VERB
ejpam-6570	294	3	a	a	DET
ejpam-6570	294	4	σ1σ2	σ1σ2	NUM
ejpam-6570	294	5	-	-	ADJ
ejpam-6570	294	6	open	open	ADJ
ejpam-6570	294	7	set	set	NOUN
ejpam-6570	294	8	v	v	NOUN
ejpam-6570	294	9	of	of	ADP
ejpam-6570	294	10	y	y	PROPN
ejpam-6570	294	11	containing	contain	VERB
ejpam-6570	294	12	f	f	PROPN
ejpam-6570	294	13	(	(	PUNCT
ejpam-6570	294	14	x	x	X
ejpam-6570	294	15	)	)	PUNCT
ejpam-6570	294	16	such	such	ADJ
ejpam-6570	294	17	that	that	SCONJ
ejpam-6570	294	18	u	u	PROPN
ejpam-6570	294	19	∩	∩	NOUN
ejpam-6570	294	20	(	(	PUNCT
ejpam-6570	294	21	x	x	NOUN
ejpam-6570	294	22	−	−	PROPN
ejpam-6570	294	23	f+(v	f+(v	NOUN
ejpam-6570	294	24	)	)	PUNCT
ejpam-6570	294	25	)	)	PUNCT
ejpam-6570	295	1	̸=	̸=	NOUN
ejpam-6570	295	2	∅	∅	NOUN
ejpam-6570	295	3	for	for	ADP
ejpam-6570	295	4	every	every	PRON
ejpam-6570	295	5	i	i	PRON
ejpam-6570	295	6	⋆-preopen	⋆-preopen	VERB
ejpam-6570	295	7	set	set	VERB
ejpam-6570	295	8	u	u	NOUN
ejpam-6570	295	9	of	of	ADP
ejpam-6570	295	10	x	x	SYM
ejpam-6570	295	11	containing	contain	VERB
ejpam-6570	295	12	x.	x.	NOUN
ejpam-6570	295	13	therefore	therefore	ADV
ejpam-6570	295	14	,	,	PUNCT
ejpam-6570	295	15	we	we	PRON
ejpam-6570	295	16	have	have	VERB
ejpam-6570	295	17	x	x	PROPN
ejpam-6570	295	18	∈	∈	PROPN
ejpam-6570	295	19	pcl⋆(x	pcl⋆(x	NOUN
ejpam-6570	295	20	−	−	ADP
ejpam-6570	295	21	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	295	22	-	-	PUNCT
ejpam-6570	295	23	cl(v	cl(v	NOUN
ejpam-6570	295	24	)	)	PUNCT
ejpam-6570	295	25	)	)	PUNCT
ejpam-6570	295	26	)	)	PUNCT
ejpam-6570	296	1	=	=	PUNCT
ejpam-6570	296	2	x	x	PUNCT
ejpam-6570	297	1	−	−	VERB
ejpam-6570	297	2	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	297	3	-	-	NOUN
ejpam-6570	297	4	cl(v	cl(v	NOUN
ejpam-6570	297	5	)	)	PUNCT
ejpam-6570	297	6	)	)	PUNCT
ejpam-6570	297	7	)	)	PUNCT
ejpam-6570	297	8	.	.	PUNCT
ejpam-6570	298	1	since	since	SCONJ
ejpam-6570	298	2	x	x	PROPN
ejpam-6570	298	3	∈	∈	PROPN
ejpam-6570	298	4	f+(v	f+(v	NOUN
ejpam-6570	298	5	)	)	PUNCT
ejpam-6570	298	6	,	,	PUNCT
ejpam-6570	298	7	we	we	PRON
ejpam-6570	298	8	have	have	VERB
ejpam-6570	298	9	x	x	X
ejpam-6570	298	10	∈	∈	PROPN
ejpam-6570	298	11	pcl⋆(f+(σ1σ2	pcl⋆(f+(σ1σ2	NOUN
ejpam-6570	298	12	-	-	PUNCT
ejpam-6570	298	13	cl(v	cl(v	NOUN
ejpam-6570	298	14	)	)	PUNCT
ejpam-6570	298	15	)	)	PUNCT
ejpam-6570	298	16	)	)	PUNCT
ejpam-6570	299	1	and	and	CCONJ
ejpam-6570	299	2	so	so	ADV
ejpam-6570	299	3	x	x	SYM
ejpam-6570	299	4	∈	∈	PROPN
ejpam-6570	299	5	pfr⋆(f+(σ1σ2	pfr⋆(f+(σ1σ2	NOUN
ejpam-6570	299	6	-	-	PUNCT
ejpam-6570	299	7	cl(v	cl(v	NOUN
ejpam-6570	299	8	)	)	PUNCT
ejpam-6570	299	9	)	)	PUNCT
ejpam-6570	299	10	)	)	PUNCT
ejpam-6570	299	11	.	.	PUNCT
ejpam-6570	300	1	conversely	conversely	ADV
ejpam-6570	300	2	,	,	PUNCT
ejpam-6570	300	3	if	if	SCONJ
ejpam-6570	300	4	f	f	PROPN
ejpam-6570	300	5	is	be	AUX
ejpam-6570	300	6	upper	upper	ADJ
ejpam-6570	300	7	almost	almost	ADV
ejpam-6570	300	8	weakly	weakly	ADJ
ejpam-6570	300	9	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	300	10	,	,	PUNCT
ejpam-6570	300	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	300	12	,	,	PUNCT
ejpam-6570	300	13	then	then	ADV
ejpam-6570	300	14	for	for	ADP
ejpam-6570	300	15	any	any	DET
ejpam-6570	300	16	σ1σ2	σ1σ2	NOUN
ejpam-6570	300	17	-	-	ADJ
ejpam-6570	300	18	open	open	ADJ
ejpam-6570	300	19	set	set	NOUN
ejpam-6570	300	20	v	v	NOUN
ejpam-6570	300	21	of	of	ADP
ejpam-6570	300	22	y	y	PROPN
ejpam-6570	300	23	containing	contain	VERB
ejpam-6570	300	24	f	f	PROPN
ejpam-6570	300	25	(	(	PUNCT
ejpam-6570	300	26	x	x	X
ejpam-6570	300	27	)	)	PUNCT
ejpam-6570	300	28	there	there	PRON
ejpam-6570	300	29	exists	exist	VERB
ejpam-6570	300	30	an	an	PRON
ejpam-6570	300	31	i	i	PRON
ejpam-6570	300	32	⋆-preopen	⋆-preopen	VERB
ejpam-6570	300	33	set	set	VERB
ejpam-6570	300	34	u	u	NOUN
ejpam-6570	300	35	of	of	ADP
ejpam-6570	300	36	x	x	PUNCT
ejpam-6570	300	37	containing	contain	VERB
ejpam-6570	300	38	x	x	PUNCT
ejpam-6570	300	39	such	such	ADJ
ejpam-6570	300	40	that	that	SCONJ
ejpam-6570	300	41	f	f	PROPN
ejpam-6570	300	42	(	(	PUNCT
ejpam-6570	300	43	u	u	NOUN
ejpam-6570	300	44	)	)	PUNCT
ejpam-6570	300	45	⊆	⊆	NUM
ejpam-6570	300	46	σ1σ2	σ1σ2	NOUN
ejpam-6570	300	47	-	-	NUM
ejpam-6570	300	48	cl(v	cl(v	NOUN
ejpam-6570	300	49	)	)	PUNCT
ejpam-6570	300	50	;	;	PUNCT
ejpam-6570	300	51	hence	hence	ADV
ejpam-6570	300	52	u	u	NOUN
ejpam-6570	300	53	⊆	⊆	NUM
ejpam-6570	300	54	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	300	55	-	-	PUNCT
ejpam-6570	300	56	cl(v	cl(v	NOUN
ejpam-6570	300	57	)	)	PUNCT
ejpam-6570	300	58	)	)	PUNCT
ejpam-6570	300	59	.	.	PUNCT
ejpam-6570	301	1	therefore	therefore	ADV
ejpam-6570	301	2	,	,	PUNCT
ejpam-6570	301	3	x	x	X
ejpam-6570	301	4	∈	∈	NOUN
ejpam-6570	301	5	pint⋆(f+(σ1σ2	pint⋆(f+(σ1σ2	NOUN
ejpam-6570	301	6	-	-	NOUN
ejpam-6570	301	7	cl(v	cl(v	NOUN
ejpam-6570	301	8	)	)	PUNCT
ejpam-6570	301	9	)	)	PUNCT
ejpam-6570	301	10	)	)	PUNCT
ejpam-6570	301	11	.	.	PUNCT
ejpam-6570	302	1	this	this	PRON
ejpam-6570	302	2	contradicts	contradict	VERB
ejpam-6570	302	3	with	with	ADP
ejpam-6570	302	4	the	the	DET
ejpam-6570	302	5	fact	fact	NOUN
ejpam-6570	302	6	that	that	SCONJ
ejpam-6570	302	7	x	x	PUNCT
ejpam-6570	302	8	∈	∈	PROPN
ejpam-6570	302	9	pfr⋆(f+(σ1σ2	pfr⋆(f+(σ1σ2	NOUN
ejpam-6570	302	10	-	-	PUNCT
ejpam-6570	302	11	cl(v	cl(v	NOUN
ejpam-6570	302	12	)	)	PUNCT
ejpam-6570	302	13	)	)	PUNCT
ejpam-6570	302	14	)	)	PUNCT
ejpam-6570	302	15	.	.	PUNCT
ejpam-6570	303	1	thus	thus	ADV
ejpam-6570	303	2	,	,	PUNCT
ejpam-6570	303	3	f	f	PROPN
ejpam-6570	303	4	is	be	AUX
ejpam-6570	303	5	not	not	PART
ejpam-6570	303	6	upper	upper	ADJ
ejpam-6570	303	7	almost	almost	ADV
ejpam-6570	303	8	weakly	weakly	ADJ
ejpam-6570	303	9	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	303	10	,	,	PUNCT
ejpam-6570	303	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	303	12	at	at	ADP
ejpam-6570	303	13	x.	x.	NOUN
ejpam-6570	303	14	theorem	theorem	VERB
ejpam-6570	303	15	10	10	NUM
ejpam-6570	303	16	.	.	PUNCT
ejpam-6570	304	1	the	the	DET
ejpam-6570	304	2	set	set	NOUN
ejpam-6570	304	3	of	of	ADP
ejpam-6570	304	4	all	all	DET
ejpam-6570	304	5	points	point	NOUN
ejpam-6570	304	6	x	x	PUNCT
ejpam-6570	304	7	of	of	ADP
ejpam-6570	304	8	x	x	SYM
ejpam-6570	304	9	at	at	ADP
ejpam-6570	304	10	which	which	PRON
ejpam-6570	304	11	a	a	DET
ejpam-6570	304	12	multifunction	multifunction	NOUN
ejpam-6570	305	1	f	f	NOUN
ejpam-6570	305	2	:	:	PUNCT
ejpam-6570	305	3	(	(	PUNCT
ejpam-6570	305	4	x	x	X
ejpam-6570	305	5	,	,	PUNCT
ejpam-6570	305	6	τ	τ	PROPN
ejpam-6570	305	7	,	,	PUNCT
ejpam-6570	305	8	i	i	NOUN
ejpam-6570	305	9	)	)	PUNCT
ejpam-6570	305	10	→	→	PUNCT
ejpam-6570	305	11	(	(	PUNCT
ejpam-6570	305	12	y	y	PROPN
ejpam-6570	305	13	,	,	PUNCT
ejpam-6570	305	14	σ1	σ1	PROPN
ejpam-6570	305	15	,	,	PUNCT
ejpam-6570	305	16	σ2	σ2	PROPN
ejpam-6570	305	17	)	)	PUNCT
ejpam-6570	305	18	is	be	AUX
ejpam-6570	305	19	not	not	PART
ejpam-6570	305	20	lower	low	ADJ
ejpam-6570	305	21	almost	almost	ADV
ejpam-6570	305	22	weakly	weakly	ADJ
ejpam-6570	305	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	305	24	,	,	PUNCT
ejpam-6570	305	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	305	26	is	be	AUX
ejpam-6570	305	27	identical	identical	ADJ
ejpam-6570	305	28	with	with	ADP
ejpam-6570	305	29	the	the	DET
ejpam-6570	305	30	union	union	NOUN
ejpam-6570	305	31	of	of	ADP
ejpam-6570	305	32	the	the	DET
ejpam-6570	305	33	⋆prefrontier	⋆prefrontier	PROPN
ejpam-6570	305	34	of	of	ADP
ejpam-6570	305	35	the	the	DET
ejpam-6570	305	36	lower	low	ADJ
ejpam-6570	305	37	inverse	inverse	NOUN
ejpam-6570	305	38	images	image	NOUN
ejpam-6570	305	39	of	of	ADP
ejpam-6570	305	40	σ1σ2	σ1σ2	NOUN
ejpam-6570	305	41	-	-	PUNCT
ejpam-6570	305	42	closure	closure	NOUN
ejpam-6570	305	43	of	of	ADP
ejpam-6570	305	44	σ1σ2	σ1σ2	NOUN
ejpam-6570	305	45	-	-	PUNCT
ejpam-6570	305	46	open	open	ADJ
ejpam-6570	305	47	sets	set	NOUN
ejpam-6570	305	48	meeting	meet	VERB
ejpam-6570	305	49	f	f	X
ejpam-6570	305	50	(	(	PUNCT
ejpam-6570	305	51	x	x	NOUN
ejpam-6570	305	52	)	)	PUNCT
ejpam-6570	305	53	.	.	PUNCT
ejpam-6570	306	1	proof	proof	NOUN
ejpam-6570	306	2	.	.	PUNCT
ejpam-6570	307	1	the	the	DET
ejpam-6570	307	2	proof	proof	NOUN
ejpam-6570	307	3	is	be	AUX
ejpam-6570	307	4	similar	similar	ADJ
ejpam-6570	307	5	to	to	ADP
ejpam-6570	307	6	that	that	PRON
ejpam-6570	307	7	of	of	ADP
ejpam-6570	307	8	theorem	theorem	NOUN
ejpam-6570	307	9	9	9	NUM
ejpam-6570	307	10	.	.	PUNCT
ejpam-6570	307	11	definition	definition	NOUN
ejpam-6570	307	12	4	4	NUM
ejpam-6570	307	13	.	.	PUNCT
ejpam-6570	307	14	a	a	DET
ejpam-6570	307	15	multifunction	multifunction	NOUN
ejpam-6570	308	1	f	f	NOUN
ejpam-6570	308	2	:	:	PUNCT
ejpam-6570	308	3	(	(	PUNCT
ejpam-6570	308	4	x	x	X
ejpam-6570	308	5	,	,	PUNCT
ejpam-6570	308	6	τ	τ	PROPN
ejpam-6570	308	7	,	,	PUNCT
ejpam-6570	308	8	i	i	NOUN
ejpam-6570	308	9	)	)	PUNCT
ejpam-6570	308	10	→	→	PUNCT
ejpam-6570	308	11	(	(	PUNCT
ejpam-6570	308	12	y	y	PROPN
ejpam-6570	308	13	,	,	PUNCT
ejpam-6570	308	14	σ1	σ1	PROPN
ejpam-6570	308	15	,	,	PUNCT
ejpam-6570	308	16	σ2	σ2	PROPN
ejpam-6570	308	17	)	)	PUNCT
ejpam-6570	308	18	is	be	AUX
ejpam-6570	308	19	said	say	VERB
ejpam-6570	308	20	to	to	PART
ejpam-6570	308	21	be	be	AUX
ejpam-6570	308	22	upper	upper	ADJ
ejpam-6570	308	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	308	24	,	,	PUNCT
ejpam-6570	308	25	σ2)precontinuous	σ2)precontinuous	ADJ
ejpam-6570	308	26	at	at	ADP
ejpam-6570	308	27	a	a	DET
ejpam-6570	308	28	point	point	NOUN
ejpam-6570	308	29	x	x	SYM
ejpam-6570	308	30	∈	∈	NOUN
ejpam-6570	308	31	x	x	PUNCT
ejpam-6570	308	32	if	if	SCONJ
ejpam-6570	308	33	for	for	ADP
ejpam-6570	308	34	each	each	DET
ejpam-6570	308	35	σ1σ2	σ1σ2	VERB
ejpam-6570	308	36	-	-	ADJ
ejpam-6570	308	37	open	open	ADJ
ejpam-6570	308	38	set	set	NOUN
ejpam-6570	308	39	v	v	NOUN
ejpam-6570	308	40	of	of	ADP
ejpam-6570	308	41	y	y	PRON
ejpam-6570	308	42	such	such	ADJ
ejpam-6570	308	43	that	that	SCONJ
ejpam-6570	308	44	f	f	PROPN
ejpam-6570	308	45	(	(	PUNCT
ejpam-6570	308	46	x	x	X
ejpam-6570	308	47	)	)	PUNCT
ejpam-6570	308	48	⊆	⊆	NUM
ejpam-6570	308	49	v	v	NOUN
ejpam-6570	308	50	,	,	PUNCT
ejpam-6570	308	51	there	there	PRON
ejpam-6570	308	52	exists	exist	VERB
ejpam-6570	308	53	an	an	PRON
ejpam-6570	308	54	i	i	PRON
ejpam-6570	308	55	⋆-preopen	⋆-preopen	VERB
ejpam-6570	308	56	set	set	VERB
ejpam-6570	308	57	u	u	NOUN
ejpam-6570	308	58	of	of	ADP
ejpam-6570	308	59	x	x	PUNCT
ejpam-6570	308	60	containing	contain	VERB
ejpam-6570	308	61	x	x	PUNCT
ejpam-6570	308	62	such	such	ADJ
ejpam-6570	308	63	that	that	SCONJ
ejpam-6570	308	64	f	f	PROPN
ejpam-6570	308	65	(	(	PUNCT
ejpam-6570	308	66	u	u	NOUN
ejpam-6570	308	67	)	)	PUNCT
ejpam-6570	308	68	⊆	⊆	NUM
ejpam-6570	308	69	v	v	NOUN
ejpam-6570	308	70	.	.	PUNCT
ejpam-6570	309	1	a	a	DET
ejpam-6570	309	2	multifunction	multifunction	NOUN
ejpam-6570	309	3	f	f	NOUN
ejpam-6570	309	4	:	:	PUNCT
ejpam-6570	309	5	(	(	PUNCT
ejpam-6570	309	6	x	x	X
ejpam-6570	309	7	,	,	PUNCT
ejpam-6570	309	8	τ	τ	PROPN
ejpam-6570	309	9	,	,	PUNCT
ejpam-6570	309	10	i	i	NOUN
ejpam-6570	309	11	)	)	PUNCT
ejpam-6570	309	12	→	→	PUNCT
ejpam-6570	309	13	(	(	PUNCT
ejpam-6570	309	14	y	y	PROPN
ejpam-6570	309	15	,	,	PUNCT
ejpam-6570	309	16	σ1	σ1	PROPN
ejpam-6570	309	17	,	,	PUNCT
ejpam-6570	309	18	σ2	σ2	PROPN
ejpam-6570	309	19	)	)	PUNCT
ejpam-6570	309	20	is	be	AUX
ejpam-6570	309	21	said	say	VERB
ejpam-6570	309	22	to	to	PART
ejpam-6570	309	23	be	be	AUX
ejpam-6570	309	24	upper	upper	ADJ
ejpam-6570	309	25	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6570	309	26	,	,	PUNCT
ejpam-6570	309	27	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	309	28	if	if	SCONJ
ejpam-6570	309	29	f	f	PROPN
ejpam-6570	309	30	is	be	AUX
ejpam-6570	309	31	upper	upper	ADJ
ejpam-6570	309	32	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6570	309	33	,	,	PUNCT
ejpam-6570	309	34	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	309	35	at	at	ADP
ejpam-6570	309	36	each	each	DET
ejpam-6570	309	37	point	point	NOUN
ejpam-6570	309	38	x	x	PUNCT
ejpam-6570	309	39	of	of	ADP
ejpam-6570	309	40	x.	x.	PROPN
ejpam-6570	309	41	theorem	theorem	VERB
ejpam-6570	309	42	11	11	NUM
ejpam-6570	309	43	.	.	PUNCT
ejpam-6570	310	1	for	for	ADP
ejpam-6570	310	2	a	a	DET
ejpam-6570	310	3	multifunction	multifunction	NOUN
ejpam-6570	310	4	f	f	NOUN
ejpam-6570	310	5	:	:	PUNCT
ejpam-6570	310	6	(	(	PUNCT
ejpam-6570	310	7	x	x	X
ejpam-6570	310	8	,	,	PUNCT
ejpam-6570	310	9	τ	τ	PROPN
ejpam-6570	310	10	,	,	PUNCT
ejpam-6570	310	11	i	i	NOUN
ejpam-6570	310	12	)	)	PUNCT
ejpam-6570	310	13	→	→	PUNCT
ejpam-6570	310	14	(	(	PUNCT
ejpam-6570	310	15	y	y	PROPN
ejpam-6570	310	16	,	,	PUNCT
ejpam-6570	310	17	σ1	σ1	PROPN
ejpam-6570	310	18	,	,	PUNCT
ejpam-6570	310	19	σ2	σ2	NOUN
ejpam-6570	310	20	)	)	PUNCT
ejpam-6570	310	21	,	,	PUNCT
ejpam-6570	310	22	the	the	DET
ejpam-6570	310	23	following	follow	VERB
ejpam-6570	310	24	properties	property	NOUN
ejpam-6570	310	25	are	be	AUX
ejpam-6570	310	26	equivalent	equivalent	ADJ
ejpam-6570	310	27	:	:	PUNCT
ejpam-6570	310	28	(	(	PUNCT
ejpam-6570	310	29	1	1	X
ejpam-6570	310	30	)	)	PUNCT
ejpam-6570	310	31	f	f	PROPN
ejpam-6570	310	32	is	be	AUX
ejpam-6570	310	33	upper	upper	ADJ
ejpam-6570	310	34	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6570	310	35	,	,	PUNCT
ejpam-6570	310	36	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	310	37	;	;	PUNCT
ejpam-6570	310	38	(	(	PUNCT
ejpam-6570	310	39	2	2	NUM
ejpam-6570	310	40	)	)	PUNCT
ejpam-6570	310	41	f+(v	f+(v	NOUN
ejpam-6570	310	42	)	)	PUNCT
ejpam-6570	311	1	is	be	AUX
ejpam-6570	311	2	i	i	PRON
ejpam-6570	311	3	⋆-preopen	⋆-preopen	VERB
ejpam-6570	311	4	in	in	ADP
ejpam-6570	311	5	x	x	PUNCT
ejpam-6570	311	6	for	for	ADP
ejpam-6570	311	7	every	every	DET
ejpam-6570	311	8	σ1σ2	σ1σ2	NOUN
ejpam-6570	311	9	-	-	ADJ
ejpam-6570	311	10	open	open	ADJ
ejpam-6570	311	11	set	set	NOUN
ejpam-6570	311	12	v	v	NOUN
ejpam-6570	311	13	of	of	ADP
ejpam-6570	311	14	y	y	PROPN
ejpam-6570	311	15	;	;	PUNCT
ejpam-6570	311	16	(	(	PUNCT
ejpam-6570	311	17	3	3	X
ejpam-6570	311	18	)	)	PUNCT
ejpam-6570	311	19	f−(k	f−(k	PROPN
ejpam-6570	311	20	)	)	PUNCT
ejpam-6570	311	21	is	be	AUX
ejpam-6570	311	22	i	i	PRON
ejpam-6570	311	23	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	311	24	in	in	ADP
ejpam-6570	311	25	x	x	PUNCT
ejpam-6570	311	26	for	for	ADP
ejpam-6570	311	27	every	every	DET
ejpam-6570	311	28	σ1σ2	σ1σ2	NUM
ejpam-6570	311	29	-	-	PUNCT
ejpam-6570	311	30	closed	closed	ADJ
ejpam-6570	311	31	set	set	NOUN
ejpam-6570	311	32	k	k	PROPN
ejpam-6570	311	33	of	of	ADP
ejpam-6570	311	34	y	y	PROPN
ejpam-6570	311	35	;	;	PUNCT
ejpam-6570	311	36	(	(	PUNCT
ejpam-6570	311	37	4	4	X
ejpam-6570	311	38	)	)	PUNCT
ejpam-6570	311	39	pcl⋆(f−(b	pcl⋆(f−(b	NOUN
ejpam-6570	311	40	)	)	PUNCT
ejpam-6570	311	41	)	)	PUNCT
ejpam-6570	312	1	⊆	⊆	X
ejpam-6570	312	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6570	312	3	-	-	PUNCT
ejpam-6570	312	4	cl(b	cl(b	NOUN
ejpam-6570	312	5	)	)	PUNCT
ejpam-6570	312	6	)	)	PUNCT
ejpam-6570	313	1	for	for	ADP
ejpam-6570	313	2	every	every	DET
ejpam-6570	313	3	subset	subset	NOUN
ejpam-6570	313	4	b	b	PROPN
ejpam-6570	313	5	of	of	ADP
ejpam-6570	313	6	y	y	PROPN
ejpam-6570	313	7	;	;	PUNCT
ejpam-6570	313	8	c.	c.	PROPN
ejpam-6570	313	9	viriyapong	viriyapong	PROPN
ejpam-6570	313	10	,	,	PUNCT
ejpam-6570	313	11	a.	a.	PROPN
ejpam-6570	313	12	sama	sama	PROPN
ejpam-6570	313	13	-	-	PUNCT
ejpam-6570	313	14	ae	ae	PROPN
ejpam-6570	313	15	,	,	PUNCT
ejpam-6570	313	16	c.	c.	PROPN
ejpam-6570	313	17	boonpok	boonpok	PROPN
ejpam-6570	313	18	/	/	SYM
ejpam-6570	313	19	eur	eur	PROPN
ejpam-6570	313	20	.	.	PUNCT
ejpam-6570	314	1	j.	j.	PROPN
ejpam-6570	314	2	pure	pure	PROPN
ejpam-6570	314	3	appl	appl	PROPN
ejpam-6570	314	4	.	.	PROPN
ejpam-6570	314	5	math	math	PROPN
ejpam-6570	314	6	,	,	PUNCT
ejpam-6570	314	7	18	18	NUM
ejpam-6570	314	8	(	(	PUNCT
ejpam-6570	314	9	3	3	NUM
ejpam-6570	314	10	)	)	PUNCT
ejpam-6570	314	11	(	(	PUNCT
ejpam-6570	314	12	2025	2025	NUM
ejpam-6570	314	13	)	)	PUNCT
ejpam-6570	314	14	,	,	PUNCT
ejpam-6570	314	15	6570	6570	NUM
ejpam-6570	314	16	11	11	NUM
ejpam-6570	314	17	of	of	ADP
ejpam-6570	314	18	14	14	NUM
ejpam-6570	314	19	(	(	PUNCT
ejpam-6570	314	20	5	5	NUM
ejpam-6570	314	21	)	)	PUNCT
ejpam-6570	314	22	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	314	23	-	-	PUNCT
ejpam-6570	314	24	int(b	int(b	NOUN
ejpam-6570	314	25	)	)	PUNCT
ejpam-6570	314	26	)	)	PUNCT
ejpam-6570	315	1	⊆	⊆	NUM
ejpam-6570	315	2	pint⋆(f+(b	pint⋆(f+(b	PROPN
ejpam-6570	315	3	)	)	PUNCT
ejpam-6570	315	4	)	)	PUNCT
ejpam-6570	315	5	for	for	ADP
ejpam-6570	315	6	every	every	DET
ejpam-6570	315	7	subset	subset	NOUN
ejpam-6570	315	8	b	b	PROPN
ejpam-6570	315	9	of	of	ADP
ejpam-6570	315	10	y	y	PROPN
ejpam-6570	315	11	.	.	PUNCT
ejpam-6570	316	1	proof	proof	NOUN
ejpam-6570	316	2	.	.	PUNCT
ejpam-6570	317	1	(	(	PUNCT
ejpam-6570	317	2	1	1	X
ejpam-6570	317	3	)	)	PUNCT
ejpam-6570	317	4	⇒	⇒	NOUN
ejpam-6570	317	5	(	(	PUNCT
ejpam-6570	317	6	2	2	NUM
ejpam-6570	317	7	):	):	PUNCT
ejpam-6570	317	8	let	let	VERB
ejpam-6570	317	9	v	v	PART
ejpam-6570	317	10	be	be	AUX
ejpam-6570	317	11	any	any	DET
ejpam-6570	317	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	317	13	-	-	ADJ
ejpam-6570	317	14	open	open	ADJ
ejpam-6570	317	15	set	set	NOUN
ejpam-6570	317	16	of	of	ADP
ejpam-6570	317	17	y	y	PROPN
ejpam-6570	317	18	and	and	CCONJ
ejpam-6570	317	19	x	x	PROPN
ejpam-6570	317	20	∈	∈	PROPN
ejpam-6570	317	21	f+(v	f+(v	NOUN
ejpam-6570	317	22	)	)	PUNCT
ejpam-6570	317	23	.	.	PUNCT
ejpam-6570	318	1	then	then	ADV
ejpam-6570	318	2	,	,	PUNCT
ejpam-6570	318	3	f	f	PROPN
ejpam-6570	318	4	(	(	PUNCT
ejpam-6570	318	5	x	x	X
ejpam-6570	318	6	)	)	PUNCT
ejpam-6570	318	7	⊆	⊆	NUM
ejpam-6570	318	8	v	v	NOUN
ejpam-6570	318	9	and	and	CCONJ
ejpam-6570	318	10	by	by	ADP
ejpam-6570	318	11	(	(	PUNCT
ejpam-6570	318	12	1	1	NUM
ejpam-6570	318	13	)	)	PUNCT
ejpam-6570	318	14	,	,	PUNCT
ejpam-6570	318	15	there	there	PRON
ejpam-6570	318	16	exists	exist	VERB
ejpam-6570	318	17	an	an	PRON
ejpam-6570	318	18	i	i	PRON
ejpam-6570	318	19	⋆-preopen	⋆-preopen	VERB
ejpam-6570	318	20	set	set	VERB
ejpam-6570	318	21	u	u	NOUN
ejpam-6570	318	22	of	of	ADP
ejpam-6570	318	23	x	x	PUNCT
ejpam-6570	318	24	containing	contain	VERB
ejpam-6570	318	25	x	x	PUNCT
ejpam-6570	318	26	such	such	ADJ
ejpam-6570	318	27	that	that	SCONJ
ejpam-6570	318	28	f	f	PROPN
ejpam-6570	318	29	(	(	PUNCT
ejpam-6570	318	30	u	u	NOUN
ejpam-6570	318	31	)	)	PUNCT
ejpam-6570	318	32	⊆	⊆	NUM
ejpam-6570	318	33	v	v	NOUN
ejpam-6570	318	34	.	.	PUNCT
ejpam-6570	319	1	thus	thus	ADV
ejpam-6570	319	2	,	,	PUNCT
ejpam-6570	319	3	x	x	PUNCT
ejpam-6570	319	4	∈	∈	PROPN
ejpam-6570	319	5	u	u	NOUN
ejpam-6570	319	6	⊆	⊆	NUM
ejpam-6570	319	7	f+(v	f+(v	NOUN
ejpam-6570	319	8	)	)	PUNCT
ejpam-6570	319	9	and	and	CCONJ
ejpam-6570	319	10	hence	hence	ADV
ejpam-6570	319	11	x	x	PART
ejpam-6570	319	12	∈	∈	PROPN
ejpam-6570	319	13	pint⋆(f+(v	pint⋆(f+(v	PROPN
ejpam-6570	319	14	)	)	PUNCT
ejpam-6570	319	15	)	)	PUNCT
ejpam-6570	319	16	.	.	PUNCT
ejpam-6570	320	1	therefore	therefore	ADV
ejpam-6570	320	2	,	,	PUNCT
ejpam-6570	320	3	f+(v	f+(v	PROPN
ejpam-6570	320	4	)	)	PUNCT
ejpam-6570	320	5	⊆	⊆	X
ejpam-6570	320	6	pint⋆(f+(v	pint⋆(f+(v	PROPN
ejpam-6570	320	7	)	)	PUNCT
ejpam-6570	320	8	)	)	PUNCT
ejpam-6570	320	9	.	.	PUNCT
ejpam-6570	321	1	this	this	PRON
ejpam-6570	321	2	shows	show	VERB
ejpam-6570	321	3	that	that	SCONJ
ejpam-6570	321	4	f+(v	f+(v	PROPN
ejpam-6570	321	5	)	)	PUNCT
ejpam-6570	321	6	is	be	AUX
ejpam-6570	321	7	i	i	PRON
ejpam-6570	321	8	⋆-preopen	⋆-preopen	VERB
ejpam-6570	321	9	in	in	ADP
ejpam-6570	321	10	x.	x.	PROPN
ejpam-6570	321	11	(	(	PUNCT
ejpam-6570	321	12	2	2	NUM
ejpam-6570	321	13	)	)	PUNCT
ejpam-6570	321	14	⇒	⇒	NOUN
ejpam-6570	321	15	(	(	PUNCT
ejpam-6570	321	16	3	3	NUM
ejpam-6570	321	17	):	):	PUNCT
ejpam-6570	321	18	this	this	PRON
ejpam-6570	321	19	follows	follow	VERB
ejpam-6570	321	20	from	from	ADP
ejpam-6570	321	21	the	the	DET
ejpam-6570	321	22	fact	fact	NOUN
ejpam-6570	321	23	that	that	SCONJ
ejpam-6570	321	24	f+(y	f+(y	PROPN
ejpam-6570	321	25	−b	−b	ADV
ejpam-6570	321	26	)	)	PUNCT
ejpam-6570	321	27	=	=	PUNCT
ejpam-6570	322	1	x	x	X
ejpam-6570	322	2	−	−	PROPN
ejpam-6570	322	3	f−(b	f−(b	PROPN
ejpam-6570	322	4	)	)	PUNCT
ejpam-6570	322	5	for	for	ADP
ejpam-6570	322	6	every	every	DET
ejpam-6570	322	7	subset	subset	NOUN
ejpam-6570	322	8	b	b	PROPN
ejpam-6570	322	9	of	of	ADP
ejpam-6570	322	10	y	y	PROPN
ejpam-6570	322	11	.	.	PUNCT
ejpam-6570	323	1	(	(	PUNCT
ejpam-6570	323	2	3	3	X
ejpam-6570	323	3	)	)	PUNCT
ejpam-6570	323	4	⇒	⇒	NOUN
ejpam-6570	323	5	(	(	PUNCT
ejpam-6570	323	6	4	4	NUM
ejpam-6570	323	7	):	):	PUNCT
ejpam-6570	323	8	let	let	VERB
ejpam-6570	323	9	b	b	X
ejpam-6570	323	10	be	be	AUX
ejpam-6570	323	11	any	any	DET
ejpam-6570	323	12	subset	subset	NOUN
ejpam-6570	323	13	of	of	ADP
ejpam-6570	323	14	y	y	PROPN
ejpam-6570	323	15	.	.	PUNCT
ejpam-6570	324	1	then	then	ADV
ejpam-6570	324	2	,	,	PUNCT
ejpam-6570	324	3	σ1σ2	σ1σ2	NOUN
ejpam-6570	324	4	-	-	NOUN
ejpam-6570	324	5	cl(b	cl(b	NOUN
ejpam-6570	324	6	)	)	PUNCT
ejpam-6570	324	7	is	be	AUX
ejpam-6570	324	8	σ1σ2	σ1σ2	NOUN
ejpam-6570	324	9	-	-	ADJ
ejpam-6570	324	10	closed	closed	ADJ
ejpam-6570	324	11	in	in	ADP
ejpam-6570	324	12	y	y	PROPN
ejpam-6570	324	13	and	and	CCONJ
ejpam-6570	324	14	by	by	ADP
ejpam-6570	324	15	(	(	PUNCT
ejpam-6570	324	16	3	3	NUM
ejpam-6570	324	17	)	)	PUNCT
ejpam-6570	324	18	,	,	PUNCT
ejpam-6570	324	19	pcl⋆(f−(b	pcl⋆(f−(b	PROPN
ejpam-6570	324	20	)	)	PUNCT
ejpam-6570	324	21	)	)	PUNCT
ejpam-6570	325	1	⊆	⊆	X
ejpam-6570	325	2	pcl⋆(f−(σ1σ2	pcl⋆(f−(σ1σ2	VERB
ejpam-6570	325	3	-	-	PUNCT
ejpam-6570	325	4	cl(b	cl(b	NOUN
ejpam-6570	325	5	)	)	PUNCT
ejpam-6570	325	6	)	)	PUNCT
ejpam-6570	325	7	)	)	PUNCT
ejpam-6570	326	1	=	=	PUNCT
ejpam-6570	326	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6570	326	3	-	-	PUNCT
ejpam-6570	326	4	cl(b	cl(b	NOUN
ejpam-6570	326	5	)	)	PUNCT
ejpam-6570	326	6	)	)	PUNCT
ejpam-6570	326	7	.	.	PUNCT
ejpam-6570	327	1	(	(	PUNCT
ejpam-6570	327	2	4	4	X
ejpam-6570	327	3	)	)	PUNCT
ejpam-6570	327	4	⇒	⇒	NOUN
ejpam-6570	327	5	(	(	PUNCT
ejpam-6570	327	6	5	5	NUM
ejpam-6570	327	7	):	):	PUNCT
ejpam-6570	327	8	let	let	VERB
ejpam-6570	327	9	b	b	X
ejpam-6570	327	10	be	be	AUX
ejpam-6570	327	11	any	any	DET
ejpam-6570	327	12	subset	subset	NOUN
ejpam-6570	327	13	of	of	ADP
ejpam-6570	327	14	y	y	PROPN
ejpam-6570	327	15	.	.	PUNCT
ejpam-6570	328	1	by	by	ADP
ejpam-6570	328	2	(	(	PUNCT
ejpam-6570	328	3	4	4	NUM
ejpam-6570	328	4	)	)	PUNCT
ejpam-6570	328	5	,	,	PUNCT
ejpam-6570	328	6	x−pint⋆(f+(b	x−pint⋆(f+(b	NOUN
ejpam-6570	328	7	)	)	PUNCT
ejpam-6570	328	8	)	)	PUNCT
ejpam-6570	328	9	=	=	SYM
ejpam-6570	328	10	pcl⋆(x−f+(b	pcl⋆(x−f+(b	NOUN
ejpam-6570	328	11	)	)	PUNCT
ejpam-6570	328	12	)	)	PUNCT
ejpam-6570	329	1	=	=	SYM
ejpam-6570	329	2	pcl⋆(f−(y	pcl⋆(f−(y	X
ejpam-6570	329	3	−b	−b	ADJ
ejpam-6570	329	4	)	)	PUNCT
ejpam-6570	329	5	)	)	PUNCT
ejpam-6570	330	1	⊆	⊆	X
ejpam-6570	330	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6570	330	3	-	-	PUNCT
ejpam-6570	330	4	cl(y	cl(y	NOUN
ejpam-6570	330	5	−b	−b	NOUN
ejpam-6570	330	6	)	)	PUNCT
ejpam-6570	330	7	)	)	PUNCT
ejpam-6570	331	1	=	=	PUNCT
ejpam-6570	331	2	f−(y	f−(y	NOUN
ejpam-6570	331	3	−σ1σ2	−σ1σ2	NOUN
ejpam-6570	331	4	-	-	PUNCT
ejpam-6570	331	5	int(b	int(b	NOUN
ejpam-6570	331	6	)	)	PUNCT
ejpam-6570	331	7	)	)	PUNCT
ejpam-6570	332	1	=	=	PUNCT
ejpam-6570	332	2	x−f+(σ1σ2	x−f+(σ1σ2	NOUN
ejpam-6570	332	3	-	-	PUNCT
ejpam-6570	332	4	int(b	int(b	NOUN
ejpam-6570	332	5	)	)	PUNCT
ejpam-6570	332	6	)	)	PUNCT
ejpam-6570	332	7	and	and	CCONJ
ejpam-6570	332	8	hence	hence	ADV
ejpam-6570	332	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-6570	332	10	-	-	PUNCT
ejpam-6570	332	11	int(b	int(b	NOUN
ejpam-6570	332	12	)	)	PUNCT
ejpam-6570	332	13	)	)	PUNCT
ejpam-6570	333	1	⊆	⊆	NUM
ejpam-6570	333	2	pint⋆(f+(b	pint⋆(f+(b	PROPN
ejpam-6570	333	3	)	)	PUNCT
ejpam-6570	333	4	)	)	PUNCT
ejpam-6570	333	5	.	.	PUNCT
ejpam-6570	334	1	(	(	PUNCT
ejpam-6570	334	2	5	5	X
ejpam-6570	334	3	)	)	PUNCT
ejpam-6570	334	4	⇒	⇒	NOUN
ejpam-6570	334	5	(	(	PUNCT
ejpam-6570	334	6	1	1	NUM
ejpam-6570	334	7	):	):	PUNCT
ejpam-6570	334	8	let	let	VERB
ejpam-6570	334	9	x	x	PUNCT
ejpam-6570	334	10	∈	∈	PROPN
ejpam-6570	334	11	x	x	X
ejpam-6570	334	12	and	and	CCONJ
ejpam-6570	334	13	v	v	X
ejpam-6570	334	14	be	be	AUX
ejpam-6570	334	15	any	any	DET
ejpam-6570	334	16	σ1σ2	σ1σ2	NOUN
ejpam-6570	334	17	-	-	ADJ
ejpam-6570	334	18	open	open	ADJ
ejpam-6570	334	19	set	set	NOUN
ejpam-6570	334	20	of	of	ADP
ejpam-6570	334	21	y	y	PRON
ejpam-6570	334	22	such	such	ADJ
ejpam-6570	334	23	that	that	SCONJ
ejpam-6570	334	24	f	f	PROPN
ejpam-6570	334	25	(	(	PUNCT
ejpam-6570	334	26	x	x	X
ejpam-6570	334	27	)	)	PUNCT
ejpam-6570	334	28	⊆	⊆	NUM
ejpam-6570	334	29	v	v	NOUN
ejpam-6570	334	30	.	.	PUNCT
ejpam-6570	335	1	then	then	ADV
ejpam-6570	335	2	,	,	PUNCT
ejpam-6570	335	3	x	x	X
ejpam-6570	335	4	∈	∈	NOUN
ejpam-6570	335	5	f+(v	f+(v	NOUN
ejpam-6570	335	6	)	)	PUNCT
ejpam-6570	336	1	=	=	PUNCT
ejpam-6570	336	2	pint⋆(f+(v	pint⋆(f+(v	PROPN
ejpam-6570	336	3	)	)	PUNCT
ejpam-6570	336	4	)	)	PUNCT
ejpam-6570	336	5	.	.	PUNCT
ejpam-6570	337	1	there	there	PRON
ejpam-6570	337	2	exists	exist	VERB
ejpam-6570	337	3	an	an	PRON
ejpam-6570	337	4	i	i	PRON
ejpam-6570	337	5	⋆-preopen	⋆-preopen	VERB
ejpam-6570	337	6	set	set	VERB
ejpam-6570	337	7	u	u	NOUN
ejpam-6570	337	8	of	of	ADP
ejpam-6570	337	9	x	x	PUNCT
ejpam-6570	337	10	containing	contain	VERB
ejpam-6570	337	11	x	x	PUNCT
ejpam-6570	337	12	such	such	ADJ
ejpam-6570	337	13	that	that	SCONJ
ejpam-6570	337	14	u	u	NOUN
ejpam-6570	337	15	⊆	⊆	NUM
ejpam-6570	337	16	f+(v	f+(v	NOUN
ejpam-6570	337	17	)	)	PUNCT
ejpam-6570	337	18	;	;	PUNCT
ejpam-6570	337	19	hence	hence	ADV
ejpam-6570	337	20	f	f	PROPN
ejpam-6570	337	21	(	(	PUNCT
ejpam-6570	337	22	u	u	NOUN
ejpam-6570	337	23	)	)	PUNCT
ejpam-6570	337	24	⊆	⊆	NUM
ejpam-6570	337	25	v	v	NOUN
ejpam-6570	337	26	.	.	PUNCT
ejpam-6570	338	1	this	this	PRON
ejpam-6570	338	2	shows	show	VERB
ejpam-6570	338	3	that	that	SCONJ
ejpam-6570	338	4	f	f	PROPN
ejpam-6570	338	5	is	be	AUX
ejpam-6570	338	6	upper	upper	ADJ
ejpam-6570	338	7	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6570	338	8	,	,	PUNCT
ejpam-6570	338	9	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	338	10	.	.	PUNCT
ejpam-6570	338	11	definition	definition	NOUN
ejpam-6570	338	12	5	5	NUM
ejpam-6570	338	13	.	.	PUNCT
ejpam-6570	338	14	a	a	DET
ejpam-6570	338	15	multifunction	multifunction	NOUN
ejpam-6570	339	1	f	f	NOUN
ejpam-6570	339	2	:	:	PUNCT
ejpam-6570	339	3	(	(	PUNCT
ejpam-6570	339	4	x	x	X
ejpam-6570	339	5	,	,	PUNCT
ejpam-6570	339	6	τ	τ	PROPN
ejpam-6570	339	7	,	,	PUNCT
ejpam-6570	339	8	i	i	NOUN
ejpam-6570	339	9	)	)	PUNCT
ejpam-6570	339	10	→	→	PUNCT
ejpam-6570	339	11	(	(	PUNCT
ejpam-6570	339	12	y	y	PROPN
ejpam-6570	339	13	,	,	PUNCT
ejpam-6570	339	14	σ1	σ1	PROPN
ejpam-6570	339	15	,	,	PUNCT
ejpam-6570	339	16	σ2	σ2	PROPN
ejpam-6570	339	17	)	)	PUNCT
ejpam-6570	339	18	is	be	AUX
ejpam-6570	339	19	said	say	VERB
ejpam-6570	339	20	to	to	PART
ejpam-6570	339	21	be	be	AUX
ejpam-6570	339	22	lower	low	ADJ
ejpam-6570	339	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	339	24	,	,	PUNCT
ejpam-6570	339	25	σ2)precontinuous	σ2)precontinuous	ADJ
ejpam-6570	339	26	at	at	ADP
ejpam-6570	339	27	a	a	DET
ejpam-6570	339	28	point	point	NOUN
ejpam-6570	339	29	x	x	SYM
ejpam-6570	339	30	∈	∈	NOUN
ejpam-6570	339	31	x	x	PUNCT
ejpam-6570	339	32	if	if	SCONJ
ejpam-6570	339	33	for	for	ADP
ejpam-6570	339	34	each	each	DET
ejpam-6570	339	35	σ1σ2	σ1σ2	VERB
ejpam-6570	339	36	-	-	ADJ
ejpam-6570	339	37	open	open	ADJ
ejpam-6570	339	38	set	set	NOUN
ejpam-6570	339	39	v	v	NOUN
ejpam-6570	339	40	of	of	ADP
ejpam-6570	339	41	y	y	PRON
ejpam-6570	339	42	such	such	ADJ
ejpam-6570	339	43	that	that	SCONJ
ejpam-6570	339	44	f	f	PROPN
ejpam-6570	339	45	(	(	PUNCT
ejpam-6570	339	46	x	x	NOUN
ejpam-6570	339	47	)	)	PUNCT
ejpam-6570	339	48	∩	∩	NOUN
ejpam-6570	339	49	v	v	ADP
ejpam-6570	339	50	̸=	̸=	PROPN
ejpam-6570	339	51	∅	∅	NOUN
ejpam-6570	339	52	,	,	PUNCT
ejpam-6570	339	53	there	there	PRON
ejpam-6570	339	54	exists	exist	VERB
ejpam-6570	339	55	an	an	PRON
ejpam-6570	339	56	i	i	PRON
ejpam-6570	339	57	⋆-preopen	⋆-preopen	VERB
ejpam-6570	339	58	set	set	VERB
ejpam-6570	339	59	u	u	NOUN
ejpam-6570	339	60	of	of	ADP
ejpam-6570	339	61	x	x	PUNCT
ejpam-6570	339	62	containing	contain	VERB
ejpam-6570	339	63	x	x	PUNCT
ejpam-6570	339	64	such	such	ADJ
ejpam-6570	339	65	that	that	SCONJ
ejpam-6570	339	66	f	f	PROPN
ejpam-6570	339	67	(	(	PUNCT
ejpam-6570	339	68	z)∩v	z)∩v	PROPN
ejpam-6570	339	69	̸=	̸=	PROPN
ejpam-6570	339	70	∅	∅	NOUN
ejpam-6570	339	71	for	for	ADP
ejpam-6570	339	72	every	every	DET
ejpam-6570	339	73	z	z	NOUN
ejpam-6570	339	74	∈	∈	PROPN
ejpam-6570	339	75	u	u	NOUN
ejpam-6570	339	76	.	.	PUNCT
ejpam-6570	340	1	a	a	DET
ejpam-6570	340	2	multifunction	multifunction	NOUN
ejpam-6570	340	3	f	f	NOUN
ejpam-6570	340	4	:	:	PUNCT
ejpam-6570	340	5	(	(	PUNCT
ejpam-6570	340	6	x	x	X
ejpam-6570	340	7	,	,	PUNCT
ejpam-6570	340	8	τ	τ	PROPN
ejpam-6570	340	9	,	,	PUNCT
ejpam-6570	340	10	i	i	NOUN
ejpam-6570	340	11	)	)	PUNCT
ejpam-6570	340	12	→	→	PUNCT
ejpam-6570	340	13	(	(	PUNCT
ejpam-6570	340	14	y	y	PROPN
ejpam-6570	340	15	,	,	PUNCT
ejpam-6570	340	16	σ1	σ1	PROPN
ejpam-6570	340	17	,	,	PUNCT
ejpam-6570	340	18	σ2	σ2	PROPN
ejpam-6570	340	19	)	)	PUNCT
ejpam-6570	340	20	is	be	AUX
ejpam-6570	340	21	called	call	VERB
ejpam-6570	340	22	lower	low	ADJ
ejpam-6570	340	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	340	24	,	,	PUNCT
ejpam-6570	340	25	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	340	26	if	if	SCONJ
ejpam-6570	340	27	f	f	PROPN
ejpam-6570	340	28	is	be	AUX
ejpam-6570	340	29	lower	low	ADJ
ejpam-6570	340	30	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	340	31	,	,	PUNCT
ejpam-6570	340	32	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	340	33	at	at	ADP
ejpam-6570	340	34	each	each	DET
ejpam-6570	340	35	point	point	NOUN
ejpam-6570	340	36	x	x	PUNCT
ejpam-6570	340	37	of	of	ADP
ejpam-6570	340	38	x.	x.	PROPN
ejpam-6570	340	39	theorem	theorem	VERB
ejpam-6570	340	40	12	12	NUM
ejpam-6570	340	41	.	.	PUNCT
ejpam-6570	341	1	for	for	ADP
ejpam-6570	341	2	a	a	DET
ejpam-6570	341	3	multifunction	multifunction	NOUN
ejpam-6570	341	4	f	f	NOUN
ejpam-6570	341	5	:	:	PUNCT
ejpam-6570	341	6	(	(	PUNCT
ejpam-6570	341	7	x	x	X
ejpam-6570	341	8	,	,	PUNCT
ejpam-6570	341	9	τ	τ	PROPN
ejpam-6570	341	10	,	,	PUNCT
ejpam-6570	341	11	i	i	NOUN
ejpam-6570	341	12	)	)	PUNCT
ejpam-6570	341	13	→	→	PUNCT
ejpam-6570	341	14	(	(	PUNCT
ejpam-6570	341	15	y	y	PROPN
ejpam-6570	341	16	,	,	PUNCT
ejpam-6570	341	17	σ1	σ1	PROPN
ejpam-6570	341	18	,	,	PUNCT
ejpam-6570	341	19	σ2	σ2	NOUN
ejpam-6570	341	20	)	)	PUNCT
ejpam-6570	341	21	,	,	PUNCT
ejpam-6570	341	22	the	the	DET
ejpam-6570	341	23	following	follow	VERB
ejpam-6570	341	24	properties	property	NOUN
ejpam-6570	341	25	are	be	AUX
ejpam-6570	341	26	equivalent	equivalent	ADJ
ejpam-6570	341	27	:	:	PUNCT
ejpam-6570	341	28	(	(	PUNCT
ejpam-6570	341	29	1	1	X
ejpam-6570	341	30	)	)	PUNCT
ejpam-6570	341	31	f	f	PROPN
ejpam-6570	341	32	is	be	AUX
ejpam-6570	341	33	lower	low	ADJ
ejpam-6570	341	34	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	341	35	,	,	PUNCT
ejpam-6570	341	36	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	341	37	;	;	PUNCT
ejpam-6570	341	38	(	(	PUNCT
ejpam-6570	341	39	2	2	X
ejpam-6570	341	40	)	)	PUNCT
ejpam-6570	341	41	f−(v	f−(v	NOUN
ejpam-6570	341	42	)	)	PUNCT
ejpam-6570	341	43	is	be	AUX
ejpam-6570	341	44	i	i	PRON
ejpam-6570	341	45	⋆-preopen	⋆-preopen	VERB
ejpam-6570	341	46	in	in	ADP
ejpam-6570	341	47	x	x	PUNCT
ejpam-6570	341	48	for	for	ADP
ejpam-6570	341	49	every	every	DET
ejpam-6570	341	50	σ1σ2	σ1σ2	NOUN
ejpam-6570	341	51	-	-	ADJ
ejpam-6570	341	52	open	open	ADJ
ejpam-6570	341	53	set	set	NOUN
ejpam-6570	341	54	v	v	NOUN
ejpam-6570	341	55	of	of	ADP
ejpam-6570	341	56	y	y	PROPN
ejpam-6570	341	57	;	;	PUNCT
ejpam-6570	341	58	(	(	PUNCT
ejpam-6570	341	59	3	3	X
ejpam-6570	341	60	)	)	PUNCT
ejpam-6570	341	61	f+(k	f+(k	PROPN
ejpam-6570	341	62	)	)	PUNCT
ejpam-6570	341	63	is	be	AUX
ejpam-6570	341	64	i	i	PRON
ejpam-6570	341	65	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	341	66	in	in	ADP
ejpam-6570	341	67	x	x	PUNCT
ejpam-6570	341	68	for	for	ADP
ejpam-6570	341	69	every	every	DET
ejpam-6570	341	70	σ1σ2	σ1σ2	NUM
ejpam-6570	341	71	-	-	PUNCT
ejpam-6570	341	72	closed	closed	ADJ
ejpam-6570	341	73	set	set	NOUN
ejpam-6570	341	74	k	k	PROPN
ejpam-6570	341	75	of	of	ADP
ejpam-6570	341	76	y	y	PROPN
ejpam-6570	341	77	;	;	PUNCT
ejpam-6570	341	78	(	(	PUNCT
ejpam-6570	341	79	4	4	X
ejpam-6570	341	80	)	)	PUNCT
ejpam-6570	341	81	pcl⋆(f+(b	pcl⋆(f+(b	NOUN
ejpam-6570	341	82	)	)	PUNCT
ejpam-6570	341	83	)	)	PUNCT
ejpam-6570	342	1	⊆	⊆	NUM
ejpam-6570	342	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	342	3	-	-	PUNCT
ejpam-6570	342	4	cl(b	cl(b	NOUN
ejpam-6570	342	5	)	)	PUNCT
ejpam-6570	342	6	)	)	PUNCT
ejpam-6570	342	7	for	for	ADP
ejpam-6570	342	8	every	every	DET
ejpam-6570	342	9	subset	subset	NOUN
ejpam-6570	342	10	b	b	PROPN
ejpam-6570	342	11	of	of	ADP
ejpam-6570	342	12	y	y	PROPN
ejpam-6570	342	13	;	;	PUNCT
ejpam-6570	342	14	(	(	PUNCT
ejpam-6570	342	15	5	5	X
ejpam-6570	342	16	)	)	PUNCT
ejpam-6570	342	17	f	f	NOUN
ejpam-6570	342	18	(	(	PUNCT
ejpam-6570	342	19	pcl⋆(a	pcl⋆(a	NOUN
ejpam-6570	342	20	)	)	PUNCT
ejpam-6570	342	21	)	)	PUNCT
ejpam-6570	343	1	⊆	⊆	X
ejpam-6570	343	2	σ1σ2	σ1σ2	X
ejpam-6570	343	3	-	-	NUM
ejpam-6570	343	4	cl(f	cl(f	NOUN
ejpam-6570	343	5	(	(	PUNCT
ejpam-6570	343	6	a	a	NOUN
ejpam-6570	343	7	)	)	PUNCT
ejpam-6570	343	8	)	)	PUNCT
ejpam-6570	343	9	for	for	ADP
ejpam-6570	343	10	every	every	DET
ejpam-6570	343	11	subset	subset	NOUN
ejpam-6570	343	12	a	a	PRON
ejpam-6570	343	13	of	of	ADP
ejpam-6570	343	14	x	x	PRON
ejpam-6570	343	15	;	;	PUNCT
ejpam-6570	343	16	(	(	PUNCT
ejpam-6570	343	17	6	6	X
ejpam-6570	343	18	)	)	PUNCT
ejpam-6570	343	19	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	343	20	-	-	PUNCT
ejpam-6570	343	21	int(b	int(b	NOUN
ejpam-6570	343	22	)	)	PUNCT
ejpam-6570	343	23	)	)	PUNCT
ejpam-6570	343	24	⊆	⊆	NUM
ejpam-6570	343	25	pint⋆(f−(b	pint⋆(f−(b	NOUN
ejpam-6570	343	26	)	)	PUNCT
ejpam-6570	343	27	)	)	PUNCT
ejpam-6570	343	28	for	for	ADP
ejpam-6570	343	29	every	every	DET
ejpam-6570	343	30	subset	subset	NOUN
ejpam-6570	343	31	b	b	PROPN
ejpam-6570	343	32	of	of	ADP
ejpam-6570	343	33	y	y	PROPN
ejpam-6570	343	34	.	.	PUNCT
ejpam-6570	344	1	proof	proof	NOUN
ejpam-6570	344	2	.	.	PUNCT
ejpam-6570	345	1	we	we	PRON
ejpam-6570	345	2	prove	prove	VERB
ejpam-6570	345	3	only	only	ADV
ejpam-6570	345	4	the	the	DET
ejpam-6570	345	5	implications	implication	NOUN
ejpam-6570	345	6	(	(	PUNCT
ejpam-6570	345	7	4	4	X
ejpam-6570	345	8	)	)	PUNCT
ejpam-6570	345	9	⇒	⇒	NOUN
ejpam-6570	345	10	(	(	PUNCT
ejpam-6570	345	11	5	5	NUM
ejpam-6570	345	12	)	)	PUNCT
ejpam-6570	345	13	and	and	CCONJ
ejpam-6570	345	14	(	(	PUNCT
ejpam-6570	345	15	5	5	X
ejpam-6570	345	16	)	)	PUNCT
ejpam-6570	345	17	⇒	⇒	NOUN
ejpam-6570	345	18	(	(	PUNCT
ejpam-6570	345	19	6	6	X
ejpam-6570	345	20	)	)	PUNCT
ejpam-6570	345	21	being	be	AUX
ejpam-6570	345	22	the	the	DET
ejpam-6570	345	23	proofs	proof	NOUN
ejpam-6570	345	24	of	of	ADP
ejpam-6570	345	25	the	the	DET
ejpam-6570	345	26	other	other	ADJ
ejpam-6570	345	27	similar	similar	ADJ
ejpam-6570	345	28	to	to	ADP
ejpam-6570	345	29	those	those	PRON
ejpam-6570	345	30	of	of	ADP
ejpam-6570	345	31	theorem	theorem	ADJ
ejpam-6570	345	32	11	11	NUM
ejpam-6570	345	33	.	.	PUNCT
ejpam-6570	346	1	(	(	PUNCT
ejpam-6570	346	2	4	4	X
ejpam-6570	346	3	)	)	PUNCT
ejpam-6570	346	4	⇒	⇒	NOUN
ejpam-6570	346	5	(	(	PUNCT
ejpam-6570	346	6	5	5	NUM
ejpam-6570	346	7	):	):	PUNCT
ejpam-6570	346	8	let	let	VERB
ejpam-6570	346	9	a	a	PRON
ejpam-6570	346	10	be	be	AUX
ejpam-6570	346	11	any	any	DET
ejpam-6570	346	12	subset	subset	NOUN
ejpam-6570	346	13	of	of	ADP
ejpam-6570	346	14	x.	x.	NOUN
ejpam-6570	346	15	thus	thus	ADV
ejpam-6570	346	16	by	by	ADP
ejpam-6570	346	17	(	(	PUNCT
ejpam-6570	346	18	4	4	NUM
ejpam-6570	346	19	)	)	PUNCT
ejpam-6570	346	20	,	,	PUNCT
ejpam-6570	346	21	pcl⋆(a	pcl⋆(a	NOUN
ejpam-6570	346	22	)	)	PUNCT
ejpam-6570	346	23	⊆	⊆	NUM
ejpam-6570	346	24	pcl⋆(f+(f	pcl⋆(f+(f	PROPN
ejpam-6570	346	25	(	(	PUNCT
ejpam-6570	346	26	a	a	NOUN
ejpam-6570	346	27	)	)	PUNCT
ejpam-6570	346	28	)	)	PUNCT
ejpam-6570	346	29	)	)	PUNCT
ejpam-6570	346	30	⊆	⊆	X
ejpam-6570	346	31	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6570	346	32	-	-	SYM
ejpam-6570	346	33	cl(f	cl(f	NOUN
ejpam-6570	346	34	(	(	PUNCT
ejpam-6570	346	35	a	a	NOUN
ejpam-6570	346	36	)	)	PUNCT
ejpam-6570	346	37	)	)	PUNCT
ejpam-6570	346	38	)	)	PUNCT
ejpam-6570	347	1	and	and	CCONJ
ejpam-6570	347	2	so	so	ADV
ejpam-6570	347	3	f	f	X
ejpam-6570	347	4	(	(	PUNCT
ejpam-6570	347	5	pcl⋆(a	pcl⋆(a	NOUN
ejpam-6570	347	6	)	)	PUNCT
ejpam-6570	347	7	)	)	PUNCT
ejpam-6570	348	1	⊆	⊆	X
ejpam-6570	348	2	σ1σ2	σ1σ2	X
ejpam-6570	348	3	-	-	NUM
ejpam-6570	348	4	cl(f	cl(f	NOUN
ejpam-6570	348	5	(	(	PUNCT
ejpam-6570	348	6	a	a	NOUN
ejpam-6570	348	7	)	)	PUNCT
ejpam-6570	348	8	)	)	PUNCT
ejpam-6570	348	9	.	.	PUNCT
ejpam-6570	349	1	(	(	PUNCT
ejpam-6570	349	2	5	5	X
ejpam-6570	349	3	)	)	PUNCT
ejpam-6570	349	4	⇒	⇒	NOUN
ejpam-6570	349	5	(	(	PUNCT
ejpam-6570	349	6	6	6	NUM
ejpam-6570	349	7	):	):	PUNCT
ejpam-6570	349	8	let	let	VERB
ejpam-6570	349	9	b	b	X
ejpam-6570	349	10	be	be	AUX
ejpam-6570	349	11	any	any	DET
ejpam-6570	349	12	subset	subset	NOUN
ejpam-6570	349	13	of	of	ADP
ejpam-6570	349	14	y	y	PROPN
ejpam-6570	349	15	.	.	PUNCT
ejpam-6570	350	1	by	by	ADP
ejpam-6570	350	2	(	(	PUNCT
ejpam-6570	350	3	5	5	NUM
ejpam-6570	350	4	)	)	PUNCT
ejpam-6570	350	5	,	,	PUNCT
ejpam-6570	350	6	f	f	PROPN
ejpam-6570	350	7	(	(	PUNCT
ejpam-6570	350	8	pcl⋆(f+(y	pcl⋆(f+(y	PROPN
ejpam-6570	350	9	−b	−b	ADJ
ejpam-6570	350	10	)	)	PUNCT
ejpam-6570	350	11	)	)	PUNCT
ejpam-6570	350	12	)	)	PUNCT
ejpam-6570	350	13	⊆	⊆	X
ejpam-6570	350	14	σ1σ2	σ1σ2	X
ejpam-6570	350	15	-	-	NUM
ejpam-6570	350	16	cl(f	cl(f	NOUN
ejpam-6570	350	17	(	(	PUNCT
ejpam-6570	350	18	f+(y	f+(y	PROPN
ejpam-6570	350	19	−b	−b	PROPN
ejpam-6570	350	20	)	)	PUNCT
ejpam-6570	350	21	)	)	PUNCT
ejpam-6570	350	22	)	)	PUNCT
ejpam-6570	351	1	⊆	⊆	X
ejpam-6570	351	2	σ1σ2	σ1σ2	NUM
ejpam-6570	351	3	-	-	PUNCT
ejpam-6570	351	4	cl(y	cl(y	NOUN
ejpam-6570	351	5	−b	−b	NOUN
ejpam-6570	351	6	)	)	PUNCT
ejpam-6570	352	1	=	=	SYM
ejpam-6570	352	2	y	y	PROPN
ejpam-6570	352	3	−	−	ADP
ejpam-6570	352	4	σ1σ2	σ1σ2	X
ejpam-6570	352	5	-	-	PUNCT
ejpam-6570	352	6	int(b	int(b	NOUN
ejpam-6570	352	7	)	)	PUNCT
ejpam-6570	352	8	.	.	PUNCT
ejpam-6570	353	1	c.	c.	PROPN
ejpam-6570	353	2	viriyapong	viriyapong	PROPN
ejpam-6570	353	3	,	,	PUNCT
ejpam-6570	353	4	a.	a.	PROPN
ejpam-6570	353	5	sama	sama	PROPN
ejpam-6570	353	6	-	-	PUNCT
ejpam-6570	353	7	ae	ae	PROPN
ejpam-6570	353	8	,	,	PUNCT
ejpam-6570	353	9	c.	c.	PROPN
ejpam-6570	353	10	boonpok	boonpok	PROPN
ejpam-6570	353	11	/	/	SYM
ejpam-6570	353	12	eur	eur	PROPN
ejpam-6570	353	13	.	.	PUNCT
ejpam-6570	354	1	j.	j.	PROPN
ejpam-6570	354	2	pure	pure	PROPN
ejpam-6570	354	3	appl	appl	PROPN
ejpam-6570	354	4	.	.	PROPN
ejpam-6570	354	5	math	math	PROPN
ejpam-6570	354	6	,	,	PUNCT
ejpam-6570	354	7	18	18	NUM
ejpam-6570	354	8	(	(	PUNCT
ejpam-6570	354	9	3	3	NUM
ejpam-6570	354	10	)	)	PUNCT
ejpam-6570	354	11	(	(	PUNCT
ejpam-6570	354	12	2025	2025	NUM
ejpam-6570	354	13	)	)	PUNCT
ejpam-6570	354	14	,	,	PUNCT
ejpam-6570	354	15	6570	6570	NUM
ejpam-6570	354	16	12	12	NUM
ejpam-6570	354	17	of	of	ADP
ejpam-6570	354	18	14	14	NUM
ejpam-6570	354	19	since	since	SCONJ
ejpam-6570	354	20	f	f	PROPN
ejpam-6570	354	21	(	(	PUNCT
ejpam-6570	354	22	pcl⋆(f+(y	pcl⋆(f+(y	PROPN
ejpam-6570	354	23	−b	−b	ADJ
ejpam-6570	354	24	)	)	PUNCT
ejpam-6570	354	25	)	)	PUNCT
ejpam-6570	354	26	)	)	PUNCT
ejpam-6570	355	1	=	=	SYM
ejpam-6570	355	2	f	f	PROPN
ejpam-6570	355	3	(	(	PUNCT
ejpam-6570	355	4	pcl⋆(x	pcl⋆(x	NOUN
ejpam-6570	355	5	−	−	PROPN
ejpam-6570	356	1	f−(b	f−(b	PROPN
ejpam-6570	356	2	)	)	PUNCT
ejpam-6570	356	3	)	)	PUNCT
ejpam-6570	356	4	)	)	PUNCT
ejpam-6570	357	1	=	=	SYM
ejpam-6570	357	2	f	f	X
ejpam-6570	357	3	(	(	PUNCT
ejpam-6570	357	4	x	x	INTJ
ejpam-6570	357	5	−	−	PROPN
ejpam-6570	357	6	pint⋆(f−(b	pint⋆(f−(b	NOUN
ejpam-6570	357	7	)	)	PUNCT
ejpam-6570	357	8	)	)	PUNCT
ejpam-6570	357	9	)	)	PUNCT
ejpam-6570	357	10	,	,	PUNCT
ejpam-6570	357	11	we	we	PRON
ejpam-6570	357	12	have	have	VERB
ejpam-6570	357	13	x	x	X
ejpam-6570	357	14	−	−	PROPN
ejpam-6570	357	15	pint⋆(f−(b	pint⋆(f−(b	NOUN
ejpam-6570	357	16	)	)	PUNCT
ejpam-6570	357	17	)	)	PUNCT
ejpam-6570	358	1	⊆	⊆	NUM
ejpam-6570	358	2	f+(y	f+(y	ADP
ejpam-6570	358	3	−	−	NUM
ejpam-6570	358	4	σ1σ2	σ1σ2	SYM
ejpam-6570	358	5	-	-	PUNCT
ejpam-6570	358	6	int(b	int(b	NOUN
ejpam-6570	358	7	)	)	PUNCT
ejpam-6570	358	8	)	)	PUNCT
ejpam-6570	359	1	=	=	PUNCT
ejpam-6570	359	2	x	x	X
ejpam-6570	359	3	−	−	NOUN
ejpam-6570	359	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	359	5	-	-	PUNCT
ejpam-6570	359	6	int(b	int(b	NOUN
ejpam-6570	359	7	)	)	PUNCT
ejpam-6570	359	8	)	)	PUNCT
ejpam-6570	359	9	and	and	CCONJ
ejpam-6570	359	10	hence	hence	ADV
ejpam-6570	359	11	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6570	359	12	-	-	PUNCT
ejpam-6570	359	13	int(b	int(b	NOUN
ejpam-6570	359	14	)	)	PUNCT
ejpam-6570	359	15	)	)	PUNCT
ejpam-6570	360	1	⊆	⊆	NUM
ejpam-6570	360	2	pint⋆(f−(b	pint⋆(f−(b	NOUN
ejpam-6570	360	3	)	)	PUNCT
ejpam-6570	360	4	)	)	PUNCT
ejpam-6570	360	5	.	.	PUNCT
ejpam-6570	361	1	recall	recall	VERB
ejpam-6570	361	2	that	that	SCONJ
ejpam-6570	361	3	a	a	DET
ejpam-6570	361	4	bitopological	bitopological	ADJ
ejpam-6570	361	5	space	space	NOUN
ejpam-6570	361	6	(	(	PUNCT
ejpam-6570	361	7	x	x	NOUN
ejpam-6570	361	8	,	,	PUNCT
ejpam-6570	361	9	τ1	τ1	NOUN
ejpam-6570	361	10	,	,	PUNCT
ejpam-6570	361	11	τ2	τ2	NOUN
ejpam-6570	361	12	)	)	PUNCT
ejpam-6570	361	13	is	be	AUX
ejpam-6570	361	14	said	say	VERB
ejpam-6570	361	15	to	to	PART
ejpam-6570	361	16	be	be	AUX
ejpam-6570	361	17	(	(	PUNCT
ejpam-6570	361	18	τ1	τ1	NOUN
ejpam-6570	361	19	,	,	PUNCT
ejpam-6570	362	1	τ2)-regular	τ2)-regular	ADJ
ejpam-6570	362	2	[	[	X
ejpam-6570	362	3	29	29	NUM
ejpam-6570	362	4	]	]	X
ejpam-6570	362	5	if	if	SCONJ
ejpam-6570	362	6	for	for	ADP
ejpam-6570	362	7	each	each	DET
ejpam-6570	362	8	τ1τ2	τ1τ2	ADJ
ejpam-6570	362	9	-	-	ADJ
ejpam-6570	362	10	closed	closed	ADJ
ejpam-6570	362	11	set	set	VERB
ejpam-6570	362	12	f	f	NOUN
ejpam-6570	362	13	and	and	CCONJ
ejpam-6570	362	14	each	each	DET
ejpam-6570	362	15	x	x	PROPN
ejpam-6570	362	16	̸∈	̸∈	PROPN
ejpam-6570	362	17	f	f	PROPN
ejpam-6570	362	18	,	,	PUNCT
ejpam-6570	362	19	there	there	PRON
ejpam-6570	362	20	exist	exist	VERB
ejpam-6570	362	21	disjoint	disjoint	ADJ
ejpam-6570	362	22	τ1τ2	τ1τ2	ADJ
ejpam-6570	362	23	-	-	ADJ
ejpam-6570	362	24	open	open	ADJ
ejpam-6570	362	25	sets	set	NOUN
ejpam-6570	362	26	u	u	NOUN
ejpam-6570	362	27	and	and	CCONJ
ejpam-6570	362	28	v	v	ADP
ejpam-6570	362	29	such	such	ADJ
ejpam-6570	362	30	that	that	SCONJ
ejpam-6570	362	31	x	x	SYM
ejpam-6570	362	32	∈	∈	PROPN
ejpam-6570	362	33	u	u	NOUN
ejpam-6570	362	34	and	and	CCONJ
ejpam-6570	362	35	f	f	PROPN
ejpam-6570	362	36	⊆	⊆	NUM
ejpam-6570	362	37	v	v	NOUN
ejpam-6570	362	38	.	.	PUNCT
ejpam-6570	363	1	lemma	lemma	PROPN
ejpam-6570	363	2	6	6	NUM
ejpam-6570	363	3	.	.	PUNCT
ejpam-6570	364	1	[	[	X
ejpam-6570	364	2	30	30	NUM
ejpam-6570	364	3	]	]	X
ejpam-6570	364	4	let	let	VERB
ejpam-6570	364	5	(	(	PUNCT
ejpam-6570	364	6	x	x	NOUN
ejpam-6570	364	7	,	,	PUNCT
ejpam-6570	364	8	τ1	τ1	NOUN
ejpam-6570	364	9	,	,	PUNCT
ejpam-6570	364	10	τ2	τ2	PROPN
ejpam-6570	364	11	)	)	PUNCT
ejpam-6570	364	12	be	be	VERB
ejpam-6570	364	13	a	a	DET
ejpam-6570	364	14	(	(	PUNCT
ejpam-6570	364	15	τ1	τ1	NOUN
ejpam-6570	364	16	,	,	PUNCT
ejpam-6570	364	17	τ2)-regular	τ2)-regular	ADJ
ejpam-6570	364	18	space	space	NOUN
ejpam-6570	364	19	.	.	PUNCT
ejpam-6570	365	1	then	then	ADV
ejpam-6570	365	2	,	,	PUNCT
ejpam-6570	365	3	the	the	DET
ejpam-6570	365	4	following	follow	VERB
ejpam-6570	365	5	properties	property	NOUN
ejpam-6570	365	6	hold	hold	VERB
ejpam-6570	365	7	:	:	PUNCT
ejpam-6570	365	8	(	(	PUNCT
ejpam-6570	365	9	1	1	X
ejpam-6570	365	10	)	)	PUNCT
ejpam-6570	365	11	τ1τ2	τ1τ2	NOUN
ejpam-6570	365	12	-	-	NUM
ejpam-6570	365	13	cl(a	cl(a	NUM
ejpam-6570	365	14	)	)	PUNCT
ejpam-6570	365	15	=	=	PUNCT
ejpam-6570	365	16	(	(	PUNCT
ejpam-6570	365	17	τ1	τ1	NOUN
ejpam-6570	365	18	,	,	PUNCT
ejpam-6570	365	19	τ2)θ	τ2)θ	NOUN
ejpam-6570	365	20	-	-	PUNCT
ejpam-6570	365	21	cl(a	cl(a	NUM
ejpam-6570	365	22	)	)	PUNCT
ejpam-6570	365	23	for	for	ADP
ejpam-6570	365	24	every	every	DET
ejpam-6570	365	25	subset	subset	NOUN
ejpam-6570	365	26	a	a	PRON
ejpam-6570	365	27	of	of	ADP
ejpam-6570	365	28	x.	x.	NOUN
ejpam-6570	365	29	(	(	PUNCT
ejpam-6570	365	30	2	2	NUM
ejpam-6570	365	31	)	)	PUNCT
ejpam-6570	365	32	every	every	DET
ejpam-6570	365	33	τ1τ2	τ1τ2	NOUN
ejpam-6570	365	34	-	-	ADJ
ejpam-6570	365	35	open	open	ADJ
ejpam-6570	365	36	set	set	NOUN
ejpam-6570	365	37	is	be	AUX
ejpam-6570	365	38	(	(	PUNCT
ejpam-6570	365	39	τ1	τ1	NOUN
ejpam-6570	365	40	,	,	PUNCT
ejpam-6570	365	41	τ2)θ	τ2)θ	ADJ
ejpam-6570	365	42	-	-	PUNCT
ejpam-6570	365	43	open	open	ADJ
ejpam-6570	365	44	.	.	PUNCT
ejpam-6570	366	1	theorem	theorem	VERB
ejpam-6570	366	2	13	13	NUM
ejpam-6570	366	3	.	.	PUNCT
ejpam-6570	367	1	for	for	ADP
ejpam-6570	367	2	a	a	DET
ejpam-6570	367	3	multifunction	multifunction	NOUN
ejpam-6570	367	4	f	f	NOUN
ejpam-6570	367	5	:	:	PUNCT
ejpam-6570	367	6	(	(	PUNCT
ejpam-6570	367	7	x	x	X
ejpam-6570	367	8	,	,	PUNCT
ejpam-6570	367	9	τ	τ	PROPN
ejpam-6570	367	10	,	,	PUNCT
ejpam-6570	367	11	i	i	NOUN
ejpam-6570	367	12	)	)	PUNCT
ejpam-6570	367	13	→	→	PUNCT
ejpam-6570	367	14	(	(	PUNCT
ejpam-6570	367	15	y	y	PROPN
ejpam-6570	367	16	,	,	PUNCT
ejpam-6570	367	17	σ1	σ1	PROPN
ejpam-6570	367	18	,	,	PUNCT
ejpam-6570	367	19	σ2	σ2	NOUN
ejpam-6570	367	20	)	)	PUNCT
ejpam-6570	367	21	,	,	PUNCT
ejpam-6570	367	22	where	where	SCONJ
ejpam-6570	367	23	(	(	PUNCT
ejpam-6570	367	24	y	y	PROPN
ejpam-6570	367	25	,	,	PUNCT
ejpam-6570	367	26	σ1	σ1	PROPN
ejpam-6570	367	27	,	,	PUNCT
ejpam-6570	367	28	σ2	σ2	PROPN
ejpam-6570	367	29	)	)	PUNCT
ejpam-6570	367	30	is	be	AUX
ejpam-6570	367	31	(	(	PUNCT
ejpam-6570	367	32	σ1	σ1	NOUN
ejpam-6570	367	33	,	,	PUNCT
ejpam-6570	367	34	σ2)-regular	σ2)-regular	ADJ
ejpam-6570	367	35	,	,	PUNCT
ejpam-6570	367	36	the	the	DET
ejpam-6570	367	37	following	follow	VERB
ejpam-6570	367	38	properties	property	NOUN
ejpam-6570	367	39	are	be	AUX
ejpam-6570	367	40	equivalent	equivalent	ADJ
ejpam-6570	367	41	:	:	PUNCT
ejpam-6570	367	42	(	(	PUNCT
ejpam-6570	367	43	1	1	X
ejpam-6570	367	44	)	)	PUNCT
ejpam-6570	367	45	f	f	PROPN
ejpam-6570	367	46	is	be	AUX
ejpam-6570	367	47	upper	upper	ADJ
ejpam-6570	367	48	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6570	367	49	,	,	PUNCT
ejpam-6570	367	50	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	367	51	;	;	PUNCT
ejpam-6570	367	52	(	(	PUNCT
ejpam-6570	367	53	2	2	X
ejpam-6570	367	54	)	)	PUNCT
ejpam-6570	367	55	f−((σ1	f−((σ1	NOUN
ejpam-6570	367	56	,	,	PUNCT
ejpam-6570	367	57	σ2)θ	σ2)θ	ADJ
ejpam-6570	367	58	-	-	PUNCT
ejpam-6570	367	59	cl(b	cl(b	NOUN
ejpam-6570	367	60	)	)	PUNCT
ejpam-6570	367	61	)	)	PUNCT
ejpam-6570	368	1	is	be	AUX
ejpam-6570	368	2	i	i	PRON
ejpam-6570	368	3	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	368	4	in	in	ADP
ejpam-6570	368	5	x	x	PUNCT
ejpam-6570	368	6	for	for	ADP
ejpam-6570	368	7	every	every	DET
ejpam-6570	368	8	subset	subset	NOUN
ejpam-6570	368	9	b	b	PROPN
ejpam-6570	368	10	of	of	ADP
ejpam-6570	368	11	y	y	PROPN
ejpam-6570	368	12	;	;	PUNCT
ejpam-6570	368	13	(	(	PUNCT
ejpam-6570	368	14	3	3	X
ejpam-6570	368	15	)	)	PUNCT
ejpam-6570	368	16	f−(k	f−(k	PROPN
ejpam-6570	368	17	)	)	PUNCT
ejpam-6570	368	18	is	be	AUX
ejpam-6570	368	19	i	i	PRON
ejpam-6570	368	20	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	368	21	in	in	ADP
ejpam-6570	368	22	x	x	PUNCT
ejpam-6570	368	23	for	for	ADP
ejpam-6570	368	24	every	every	DET
ejpam-6570	368	25	(	(	PUNCT
ejpam-6570	368	26	σ1	σ1	PROPN
ejpam-6570	368	27	,	,	PUNCT
ejpam-6570	368	28	σ2)θ	σ2)θ	NOUN
ejpam-6570	368	29	-	-	PUNCT
ejpam-6570	368	30	closed	close	VERB
ejpam-6570	368	31	set	set	NOUN
ejpam-6570	368	32	k	k	PROPN
ejpam-6570	368	33	of	of	ADP
ejpam-6570	368	34	y	y	PROPN
ejpam-6570	368	35	;	;	PUNCT
ejpam-6570	368	36	(	(	PUNCT
ejpam-6570	368	37	4	4	X
ejpam-6570	368	38	)	)	PUNCT
ejpam-6570	369	1	f+(v	f+(v	NOUN
ejpam-6570	369	2	)	)	PUNCT
ejpam-6570	370	1	is	be	AUX
ejpam-6570	370	2	i	i	PRON
ejpam-6570	370	3	⋆-preopen	⋆-preopen	VERB
ejpam-6570	370	4	in	in	ADP
ejpam-6570	370	5	x	x	PUNCT
ejpam-6570	370	6	for	for	ADP
ejpam-6570	370	7	every	every	DET
ejpam-6570	370	8	(	(	PUNCT
ejpam-6570	370	9	σ1	σ1	PROPN
ejpam-6570	370	10	,	,	PUNCT
ejpam-6570	370	11	σ2)θ	σ2)θ	NOUN
ejpam-6570	370	12	-	-	PUNCT
ejpam-6570	370	13	open	open	ADJ
ejpam-6570	370	14	set	set	NOUN
ejpam-6570	370	15	v	v	NOUN
ejpam-6570	370	16	of	of	ADP
ejpam-6570	370	17	y	y	PROPN
ejpam-6570	370	18	.	.	PUNCT
ejpam-6570	371	1	proof	proof	NOUN
ejpam-6570	371	2	.	.	PUNCT
ejpam-6570	372	1	(	(	PUNCT
ejpam-6570	372	2	1	1	X
ejpam-6570	372	3	)	)	PUNCT
ejpam-6570	372	4	⇒	⇒	NOUN
ejpam-6570	372	5	(	(	PUNCT
ejpam-6570	372	6	2	2	NUM
ejpam-6570	372	7	):	):	PUNCT
ejpam-6570	372	8	let	let	VERB
ejpam-6570	372	9	b	b	X
ejpam-6570	372	10	be	be	AUX
ejpam-6570	372	11	any	any	DET
ejpam-6570	372	12	subset	subset	NOUN
ejpam-6570	372	13	of	of	ADP
ejpam-6570	372	14	y	y	PROPN
ejpam-6570	372	15	.	.	PUNCT
ejpam-6570	373	1	then	then	ADV
ejpam-6570	373	2	,	,	PUNCT
ejpam-6570	373	3	(	(	PUNCT
ejpam-6570	373	4	σ1	σ1	PROPN
ejpam-6570	373	5	,	,	PUNCT
ejpam-6570	373	6	σ2)θ	σ2)θ	NOUN
ejpam-6570	373	7	-	-	PUNCT
ejpam-6570	373	8	cl(b	cl(b	NOUN
ejpam-6570	373	9	)	)	PUNCT
ejpam-6570	373	10	is	be	AUX
ejpam-6570	373	11	σ1σ2	σ1σ2	NOUN
ejpam-6570	373	12	-	-	ADJ
ejpam-6570	373	13	closed	closed	ADJ
ejpam-6570	373	14	in	in	ADP
ejpam-6570	373	15	y	y	PROPN
ejpam-6570	373	16	and	and	CCONJ
ejpam-6570	373	17	by	by	ADP
ejpam-6570	373	18	theorem	theorem	NOUN
ejpam-6570	373	19	11	11	NUM
ejpam-6570	373	20	,	,	PUNCT
ejpam-6570	373	21	f−((σ1	f−((σ1	NOUN
ejpam-6570	373	22	,	,	PUNCT
ejpam-6570	373	23	σ2)θ	σ2)θ	ADJ
ejpam-6570	373	24	-	-	PUNCT
ejpam-6570	373	25	cl(b	cl(b	NOUN
ejpam-6570	373	26	)	)	PUNCT
ejpam-6570	373	27	)	)	PUNCT
ejpam-6570	374	1	is	be	AUX
ejpam-6570	374	2	i	i	PRON
ejpam-6570	374	3	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	374	4	in	in	ADP
ejpam-6570	374	5	x.	x.	PROPN
ejpam-6570	374	6	(	(	PUNCT
ejpam-6570	374	7	2	2	NUM
ejpam-6570	374	8	)	)	PUNCT
ejpam-6570	374	9	⇒	⇒	NOUN
ejpam-6570	374	10	(	(	PUNCT
ejpam-6570	374	11	3	3	NUM
ejpam-6570	374	12	):	):	PUNCT
ejpam-6570	374	13	the	the	DET
ejpam-6570	374	14	proof	proof	NOUN
ejpam-6570	374	15	is	be	AUX
ejpam-6570	374	16	obvious	obvious	ADJ
ejpam-6570	374	17	.	.	PUNCT
ejpam-6570	375	1	(	(	PUNCT
ejpam-6570	375	2	3	3	X
ejpam-6570	375	3	)	)	PUNCT
ejpam-6570	375	4	⇒	⇒	NOUN
ejpam-6570	375	5	(	(	PUNCT
ejpam-6570	375	6	4	4	NUM
ejpam-6570	375	7	):	):	PUNCT
ejpam-6570	375	8	let	let	VERB
ejpam-6570	375	9	v	v	PART
ejpam-6570	375	10	be	be	AUX
ejpam-6570	375	11	any	any	DET
ejpam-6570	375	12	(	(	PUNCT
ejpam-6570	375	13	σ1	σ1	PROPN
ejpam-6570	375	14	,	,	PUNCT
ejpam-6570	375	15	σ2)θ	σ2)θ	NOUN
ejpam-6570	375	16	-	-	PUNCT
ejpam-6570	375	17	open	open	ADJ
ejpam-6570	375	18	set	set	NOUN
ejpam-6570	375	19	of	of	ADP
ejpam-6570	375	20	y	y	PROPN
ejpam-6570	375	21	.	.	PUNCT
ejpam-6570	376	1	by	by	ADP
ejpam-6570	376	2	(	(	PUNCT
ejpam-6570	376	3	3	3	NUM
ejpam-6570	376	4	)	)	PUNCT
ejpam-6570	376	5	,	,	PUNCT
ejpam-6570	376	6	f−(y	f−(y	NOUN
ejpam-6570	376	7	−	−	NOUN
ejpam-6570	376	8	v	v	NOUN
ejpam-6570	376	9	)	)	PUNCT
ejpam-6570	376	10	is	be	AUX
ejpam-6570	376	11	i	i	PRON
ejpam-6570	376	12	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	376	13	in	in	ADP
ejpam-6570	376	14	x	x	PUNCT
ejpam-6570	376	15	and	and	CCONJ
ejpam-6570	376	16	f−(y	f−(y	NOUN
ejpam-6570	376	17	−	−	NOUN
ejpam-6570	376	18	v	v	NOUN
ejpam-6570	376	19	)	)	PUNCT
ejpam-6570	376	20	=	=	PUNCT
ejpam-6570	377	1	x	x	X
ejpam-6570	377	2	−	−	PROPN
ejpam-6570	377	3	f+(v	f+(v	NOUN
ejpam-6570	377	4	)	)	PUNCT
ejpam-6570	377	5	.	.	PUNCT
ejpam-6570	378	1	thus	thus	ADV
ejpam-6570	378	2	,	,	PUNCT
ejpam-6570	378	3	f+(v	f+(v	PROPN
ejpam-6570	378	4	)	)	PUNCT
ejpam-6570	378	5	is	be	AUX
ejpam-6570	378	6	i	i	PRON
ejpam-6570	378	7	⋆-preopen	⋆-preopen	VERB
ejpam-6570	378	8	in	in	ADP
ejpam-6570	378	9	x.	x.	PROPN
ejpam-6570	378	10	(	(	PUNCT
ejpam-6570	378	11	4	4	NUM
ejpam-6570	378	12	)	)	PUNCT
ejpam-6570	378	13	⇒	⇒	NOUN
ejpam-6570	378	14	(	(	PUNCT
ejpam-6570	378	15	1	1	NUM
ejpam-6570	378	16	):	):	PUNCT
ejpam-6570	378	17	let	let	VERB
ejpam-6570	378	18	v	v	PART
ejpam-6570	378	19	be	be	AUX
ejpam-6570	378	20	any	any	DET
ejpam-6570	378	21	σ1σ2	σ1σ2	NOUN
ejpam-6570	378	22	-	-	ADJ
ejpam-6570	378	23	open	open	ADJ
ejpam-6570	378	24	set	set	NOUN
ejpam-6570	378	25	of	of	ADP
ejpam-6570	378	26	y	y	PROPN
ejpam-6570	378	27	.	.	PUNCT
ejpam-6570	379	1	since	since	SCONJ
ejpam-6570	379	2	(	(	PUNCT
ejpam-6570	379	3	y	y	PROPN
ejpam-6570	379	4	,	,	PUNCT
ejpam-6570	379	5	σ1	σ1	PROPN
ejpam-6570	379	6	,	,	PUNCT
ejpam-6570	379	7	σ2	σ2	PROPN
ejpam-6570	379	8	)	)	PUNCT
ejpam-6570	379	9	is	be	AUX
ejpam-6570	379	10	(	(	PUNCT
ejpam-6570	379	11	σ1	σ1	NOUN
ejpam-6570	379	12	,	,	PUNCT
ejpam-6570	379	13	σ2)-regular	σ2)-regular	ADJ
ejpam-6570	379	14	,	,	PUNCT
ejpam-6570	379	15	by	by	ADP
ejpam-6570	379	16	lemma	lemma	PROPN
ejpam-6570	379	17	6	6	NUM
ejpam-6570	379	18	we	we	PRON
ejpam-6570	379	19	have	have	VERB
ejpam-6570	379	20	v	v	NOUN
ejpam-6570	379	21	is	be	AUX
ejpam-6570	379	22	(	(	PUNCT
ejpam-6570	379	23	σ1	σ1	PROPN
ejpam-6570	379	24	,	,	PUNCT
ejpam-6570	379	25	σ2)θ	σ2)θ	NOUN
ejpam-6570	379	26	-	-	PUNCT
ejpam-6570	379	27	open	open	ADJ
ejpam-6570	379	28	in	in	ADP
ejpam-6570	379	29	y	y	PROPN
ejpam-6570	379	30	and	and	CCONJ
ejpam-6570	379	31	by	by	ADP
ejpam-6570	379	32	(	(	PUNCT
ejpam-6570	379	33	4	4	NUM
ejpam-6570	379	34	)	)	PUNCT
ejpam-6570	379	35	,	,	PUNCT
ejpam-6570	380	1	f+(v	f+(v	PROPN
ejpam-6570	380	2	)	)	PUNCT
ejpam-6570	380	3	is	be	AUX
ejpam-6570	380	4	i	i	PRON
ejpam-6570	380	5	⋆-preopen	⋆-preopen	VERB
ejpam-6570	380	6	in	in	ADP
ejpam-6570	380	7	x.	x.	NOUN
ejpam-6570	380	8	thus	thus	ADV
ejpam-6570	380	9	by	by	ADP
ejpam-6570	380	10	theorem	theorem	NOUN
ejpam-6570	380	11	11	11	NUM
ejpam-6570	380	12	,	,	PUNCT
ejpam-6570	380	13	f	f	PROPN
ejpam-6570	380	14	is	be	AUX
ejpam-6570	380	15	upper	upper	ADJ
ejpam-6570	380	16	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6570	380	17	,	,	PUNCT
ejpam-6570	380	18	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	380	19	.	.	PUNCT
ejpam-6570	380	20	theorem	theorem	VERB
ejpam-6570	380	21	14	14	NUM
ejpam-6570	380	22	.	.	PUNCT
ejpam-6570	381	1	for	for	ADP
ejpam-6570	381	2	a	a	DET
ejpam-6570	381	3	multifunction	multifunction	NOUN
ejpam-6570	381	4	f	f	NOUN
ejpam-6570	381	5	:	:	PUNCT
ejpam-6570	381	6	(	(	PUNCT
ejpam-6570	381	7	x	x	X
ejpam-6570	381	8	,	,	PUNCT
ejpam-6570	381	9	τ	τ	PROPN
ejpam-6570	381	10	,	,	PUNCT
ejpam-6570	381	11	i	i	NOUN
ejpam-6570	381	12	)	)	PUNCT
ejpam-6570	381	13	→	→	PUNCT
ejpam-6570	381	14	(	(	PUNCT
ejpam-6570	381	15	y	y	PROPN
ejpam-6570	381	16	,	,	PUNCT
ejpam-6570	381	17	σ1	σ1	PROPN
ejpam-6570	381	18	,	,	PUNCT
ejpam-6570	381	19	σ2	σ2	NOUN
ejpam-6570	381	20	)	)	PUNCT
ejpam-6570	381	21	,	,	PUNCT
ejpam-6570	381	22	where	where	SCONJ
ejpam-6570	381	23	(	(	PUNCT
ejpam-6570	381	24	y	y	PROPN
ejpam-6570	381	25	,	,	PUNCT
ejpam-6570	381	26	σ1	σ1	PROPN
ejpam-6570	381	27	,	,	PUNCT
ejpam-6570	381	28	σ2	σ2	PROPN
ejpam-6570	381	29	)	)	PUNCT
ejpam-6570	381	30	is	be	AUX
ejpam-6570	381	31	(	(	PUNCT
ejpam-6570	381	32	σ1	σ1	NOUN
ejpam-6570	381	33	,	,	PUNCT
ejpam-6570	381	34	σ2)-regular	σ2)-regular	ADJ
ejpam-6570	381	35	,	,	PUNCT
ejpam-6570	381	36	the	the	DET
ejpam-6570	381	37	following	follow	VERB
ejpam-6570	381	38	properties	property	NOUN
ejpam-6570	381	39	are	be	AUX
ejpam-6570	381	40	equivalent	equivalent	ADJ
ejpam-6570	381	41	:	:	PUNCT
ejpam-6570	381	42	(	(	PUNCT
ejpam-6570	381	43	1	1	X
ejpam-6570	381	44	)	)	PUNCT
ejpam-6570	381	45	f	f	PROPN
ejpam-6570	381	46	is	be	AUX
ejpam-6570	381	47	lower	low	ADJ
ejpam-6570	381	48	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	381	49	,	,	PUNCT
ejpam-6570	381	50	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	381	51	;	;	PUNCT
ejpam-6570	381	52	(	(	PUNCT
ejpam-6570	381	53	2	2	X
ejpam-6570	381	54	)	)	PUNCT
ejpam-6570	381	55	f+((σ1	f+((σ1	NOUN
ejpam-6570	381	56	,	,	PUNCT
ejpam-6570	381	57	σ2)θ	σ2)θ	ADJ
ejpam-6570	381	58	-	-	PUNCT
ejpam-6570	381	59	cl(b	cl(b	NOUN
ejpam-6570	381	60	)	)	PUNCT
ejpam-6570	381	61	)	)	PUNCT
ejpam-6570	382	1	is	be	AUX
ejpam-6570	382	2	i	i	PRON
ejpam-6570	382	3	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	382	4	in	in	ADP
ejpam-6570	382	5	x	x	PUNCT
ejpam-6570	382	6	for	for	ADP
ejpam-6570	382	7	every	every	DET
ejpam-6570	382	8	subset	subset	NOUN
ejpam-6570	382	9	b	b	PROPN
ejpam-6570	382	10	of	of	ADP
ejpam-6570	382	11	y	y	PROPN
ejpam-6570	382	12	;	;	PUNCT
ejpam-6570	382	13	(	(	PUNCT
ejpam-6570	382	14	3	3	X
ejpam-6570	382	15	)	)	PUNCT
ejpam-6570	382	16	f+(k	f+(k	PROPN
ejpam-6570	382	17	)	)	PUNCT
ejpam-6570	382	18	is	be	AUX
ejpam-6570	382	19	i	i	PRON
ejpam-6570	382	20	⋆-preclosed	⋆-preclose	VERB
ejpam-6570	382	21	in	in	ADP
ejpam-6570	382	22	x	x	PUNCT
ejpam-6570	382	23	for	for	ADP
ejpam-6570	382	24	every	every	DET
ejpam-6570	382	25	(	(	PUNCT
ejpam-6570	382	26	σ1	σ1	PROPN
ejpam-6570	382	27	,	,	PUNCT
ejpam-6570	382	28	σ2)θ	σ2)θ	NOUN
ejpam-6570	382	29	-	-	PUNCT
ejpam-6570	382	30	closed	close	VERB
ejpam-6570	382	31	set	set	NOUN
ejpam-6570	382	32	k	k	PROPN
ejpam-6570	382	33	of	of	ADP
ejpam-6570	382	34	y	y	PROPN
ejpam-6570	382	35	;	;	PUNCT
ejpam-6570	382	36	(	(	PUNCT
ejpam-6570	382	37	4	4	X
ejpam-6570	382	38	)	)	PUNCT
ejpam-6570	382	39	f−(v	f−(v	NOUN
ejpam-6570	382	40	)	)	PUNCT
ejpam-6570	382	41	is	be	AUX
ejpam-6570	382	42	i	i	PRON
ejpam-6570	382	43	⋆-preopen	⋆-preopen	VERB
ejpam-6570	382	44	in	in	ADP
ejpam-6570	382	45	x	x	PUNCT
ejpam-6570	382	46	for	for	ADP
ejpam-6570	382	47	every	every	DET
ejpam-6570	382	48	(	(	PUNCT
ejpam-6570	382	49	σ1	σ1	PROPN
ejpam-6570	382	50	,	,	PUNCT
ejpam-6570	382	51	σ2)θ	σ2)θ	NOUN
ejpam-6570	382	52	-	-	PUNCT
ejpam-6570	382	53	open	open	ADJ
ejpam-6570	382	54	set	set	NOUN
ejpam-6570	382	55	v	v	NOUN
ejpam-6570	382	56	of	of	ADP
ejpam-6570	382	57	y	y	PROPN
ejpam-6570	382	58	;	;	PUNCT
ejpam-6570	382	59	(	(	PUNCT
ejpam-6570	382	60	5	5	X
ejpam-6570	382	61	)	)	PUNCT
ejpam-6570	382	62	f	f	PROPN
ejpam-6570	382	63	is	be	AUX
ejpam-6570	382	64	lower	low	ADJ
ejpam-6570	382	65	almost	almost	ADV
ejpam-6570	382	66	weakly	weakly	ADJ
ejpam-6570	383	1	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	383	2	,	,	PUNCT
ejpam-6570	383	3	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6570	383	4	.	.	PUNCT
ejpam-6570	384	1	c.	c.	PROPN
ejpam-6570	384	2	viriyapong	viriyapong	PROPN
ejpam-6570	384	3	,	,	PUNCT
ejpam-6570	384	4	a.	a.	PROPN
ejpam-6570	384	5	sama	sama	PROPN
ejpam-6570	384	6	-	-	PUNCT
ejpam-6570	384	7	ae	ae	PROPN
ejpam-6570	384	8	,	,	PUNCT
ejpam-6570	384	9	c.	c.	PROPN
ejpam-6570	384	10	boonpok	boonpok	PROPN
ejpam-6570	384	11	/	/	SYM
ejpam-6570	384	12	eur	eur	PROPN
ejpam-6570	384	13	.	.	PUNCT
ejpam-6570	385	1	j.	j.	PROPN
ejpam-6570	385	2	pure	pure	PROPN
ejpam-6570	385	3	appl	appl	PROPN
ejpam-6570	385	4	.	.	PROPN
ejpam-6570	385	5	math	math	PROPN
ejpam-6570	385	6	,	,	PUNCT
ejpam-6570	385	7	18	18	NUM
ejpam-6570	385	8	(	(	PUNCT
ejpam-6570	385	9	3	3	NUM
ejpam-6570	385	10	)	)	PUNCT
ejpam-6570	385	11	(	(	PUNCT
ejpam-6570	385	12	2025	2025	NUM
ejpam-6570	385	13	)	)	PUNCT
ejpam-6570	385	14	,	,	PUNCT
ejpam-6570	385	15	6570	6570	NUM
ejpam-6570	385	16	13	13	NUM
ejpam-6570	385	17	of	of	ADP
ejpam-6570	385	18	14	14	NUM
ejpam-6570	385	19	proof	proof	NOUN
ejpam-6570	385	20	.	.	PUNCT
ejpam-6570	386	1	we	we	PRON
ejpam-6570	386	2	prove	prove	VERB
ejpam-6570	386	3	only	only	ADV
ejpam-6570	386	4	the	the	DET
ejpam-6570	386	5	implication	implication	NOUN
ejpam-6570	386	6	(	(	PUNCT
ejpam-6570	386	7	5	5	NUM
ejpam-6570	386	8	)	)	PUNCT
ejpam-6570	386	9	⇒	⇒	NOUN
ejpam-6570	386	10	(	(	PUNCT
ejpam-6570	386	11	1	1	NUM
ejpam-6570	386	12	)	)	PUNCT
ejpam-6570	386	13	,	,	PUNCT
ejpam-6570	386	14	the	the	DET
ejpam-6570	386	15	proof	proof	NOUN
ejpam-6570	386	16	of	of	ADP
ejpam-6570	386	17	the	the	DET
ejpam-6570	386	18	other	other	ADJ
ejpam-6570	386	19	being	be	AUX
ejpam-6570	386	20	similar	similar	ADJ
ejpam-6570	386	21	to	to	ADP
ejpam-6570	386	22	that	that	PRON
ejpam-6570	386	23	of	of	ADP
ejpam-6570	386	24	theprem	theprem	PROPN
ejpam-6570	386	25	13	13	NUM
ejpam-6570	386	26	.	.	PUNCT
ejpam-6570	387	1	the	the	DET
ejpam-6570	387	2	proof	proof	NOUN
ejpam-6570	387	3	of	of	ADP
ejpam-6570	387	4	the	the	DET
ejpam-6570	387	5	implication	implication	NOUN
ejpam-6570	387	6	(	(	PUNCT
ejpam-6570	387	7	4	4	X
ejpam-6570	387	8	)	)	PUNCT
ejpam-6570	387	9	⇒	⇒	NOUN
ejpam-6570	387	10	(	(	PUNCT
ejpam-6570	387	11	5	5	NUM
ejpam-6570	387	12	)	)	PUNCT
ejpam-6570	387	13	is	be	AUX
ejpam-6570	387	14	obvious	obvious	ADJ
ejpam-6570	387	15	.	.	PUNCT
ejpam-6570	388	1	(	(	PUNCT
ejpam-6570	388	2	5	5	X
ejpam-6570	388	3	)	)	PUNCT
ejpam-6570	388	4	⇒	⇒	NOUN
ejpam-6570	388	5	(	(	PUNCT
ejpam-6570	388	6	1	1	NUM
ejpam-6570	388	7	):	):	PUNCT
ejpam-6570	388	8	let	let	VERB
ejpam-6570	388	9	v	v	PART
ejpam-6570	388	10	be	be	AUX
ejpam-6570	388	11	any	any	DET
ejpam-6570	388	12	σ1σ2	σ1σ2	NOUN
ejpam-6570	388	13	-	-	ADJ
ejpam-6570	388	14	open	open	ADJ
ejpam-6570	388	15	set	set	NOUN
ejpam-6570	388	16	of	of	ADP
ejpam-6570	388	17	y	y	PROPN
ejpam-6570	388	18	and	and	CCONJ
ejpam-6570	388	19	x	x	PROPN
ejpam-6570	388	20	∈	∈	PROPN
ejpam-6570	388	21	f−(v	f−(v	NOUN
ejpam-6570	388	22	)	)	PUNCT
ejpam-6570	388	23	.	.	PUNCT
ejpam-6570	389	1	then	then	ADV
ejpam-6570	389	2	,	,	PUNCT
ejpam-6570	389	3	f	f	PROPN
ejpam-6570	389	4	(	(	PUNCT
ejpam-6570	389	5	x)∩v	x)∩v	PROPN
ejpam-6570	389	6	̸=	̸=	PROPN
ejpam-6570	389	7	∅.	∅.	NOUN
ejpam-6570	389	8	since	since	SCONJ
ejpam-6570	389	9	(	(	PUNCT
ejpam-6570	389	10	y	y	PROPN
ejpam-6570	389	11	,	,	PUNCT
ejpam-6570	389	12	σ1	σ1	PROPN
ejpam-6570	389	13	,	,	PUNCT
ejpam-6570	389	14	σ2	σ2	PROPN
ejpam-6570	389	15	)	)	PUNCT
ejpam-6570	389	16	is	be	AUX
ejpam-6570	389	17	(	(	PUNCT
ejpam-6570	389	18	σ1	σ1	NOUN
ejpam-6570	389	19	,	,	PUNCT
ejpam-6570	389	20	σ2)-regular	σ2)-regular	ADJ
ejpam-6570	389	21	,	,	PUNCT
ejpam-6570	389	22	there	there	PRON
ejpam-6570	389	23	exists	exist	VERB
ejpam-6570	389	24	a	a	DET
ejpam-6570	389	25	σ1σ2	σ1σ2	NUM
ejpam-6570	389	26	-	-	ADJ
ejpam-6570	389	27	open	open	ADJ
ejpam-6570	389	28	set	set	NOUN
ejpam-6570	389	29	w	w	PROPN
ejpam-6570	389	30	of	of	ADP
ejpam-6570	389	31	y	y	PRON
ejpam-6570	389	32	such	such	ADJ
ejpam-6570	389	33	that	that	SCONJ
ejpam-6570	389	34	f	f	PROPN
ejpam-6570	389	35	(	(	PUNCT
ejpam-6570	389	36	x)∩w	x)∩w	PROPN
ejpam-6570	389	37	̸=	̸=	PROPN
ejpam-6570	389	38	∅	∅	NOUN
ejpam-6570	389	39	and	and	CCONJ
ejpam-6570	389	40	σ1σ2	σ1σ2	NOUN
ejpam-6570	389	41	-	-	PUNCT
ejpam-6570	389	42	cl(w	cl(w	NOUN
ejpam-6570	389	43	)	)	PUNCT
ejpam-6570	389	44	⊆	⊆	NUM
ejpam-6570	389	45	v	v	NOUN
ejpam-6570	389	46	.	.	PUNCT
ejpam-6570	390	1	since	since	SCONJ
ejpam-6570	390	2	f	f	PROPN
ejpam-6570	390	3	is	be	AUX
ejpam-6570	390	4	lower	low	ADJ
ejpam-6570	390	5	almost	almost	ADV
ejpam-6570	390	6	weakly	weakly	ADJ
ejpam-6570	390	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	390	8	,	,	PUNCT
ejpam-6570	390	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6570	390	10	,	,	PUNCT
ejpam-6570	390	11	by	by	ADP
ejpam-6570	390	12	theorem	theorem	NOUN
ejpam-6570	390	13	2	2	NUM
ejpam-6570	390	14	there	there	PRON
ejpam-6570	390	15	exists	exist	VERB
ejpam-6570	390	16	an	an	PRON
ejpam-6570	390	17	i	i	PRON
ejpam-6570	390	18	⋆-open	⋆-open	VERB
ejpam-6570	390	19	set	set	VERB
ejpam-6570	390	20	u	u	NOUN
ejpam-6570	390	21	of	of	ADP
ejpam-6570	390	22	x	x	PUNCT
ejpam-6570	390	23	containing	contain	VERB
ejpam-6570	390	24	x	x	PUNCT
ejpam-6570	390	25	such	such	ADJ
ejpam-6570	390	26	that	that	SCONJ
ejpam-6570	390	27	u	u	NOUN
ejpam-6570	390	28	⊆	⊆	NUM
ejpam-6570	390	29	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6570	390	30	-	-	PUNCT
ejpam-6570	390	31	cl(w	cl(w	NOUN
ejpam-6570	390	32	)	)	PUNCT
ejpam-6570	390	33	)	)	PUNCT
ejpam-6570	391	1	⊆	⊆	NUM
ejpam-6570	391	2	f−(v	f−(v	NOUN
ejpam-6570	391	3	)	)	PUNCT
ejpam-6570	391	4	.	.	PUNCT
ejpam-6570	392	1	thus	thus	ADV
ejpam-6570	392	2	,	,	PUNCT
ejpam-6570	392	3	x	x	PUNCT
ejpam-6570	392	4	∈	∈	PROPN
ejpam-6570	392	5	u	u	NOUN
ejpam-6570	392	6	⊆	⊆	NUM
ejpam-6570	392	7	int⋆(cl⋆(u	int⋆(cl⋆(u	NOUN
ejpam-6570	392	8	)	)	PUNCT
ejpam-6570	392	9	)	)	PUNCT
ejpam-6570	392	10	⊆	⊆	NUM
ejpam-6570	392	11	int⋆(cl⋆(f−(v	int⋆(cl⋆(f−(v	NOUN
ejpam-6570	392	12	)	)	PUNCT
ejpam-6570	392	13	)	)	PUNCT
ejpam-6570	392	14	)	)	PUNCT
ejpam-6570	392	15	and	and	CCONJ
ejpam-6570	392	16	hence	hence	ADV
ejpam-6570	392	17	f−(v	f−(v	ADJ
ejpam-6570	392	18	)	)	PUNCT
ejpam-6570	392	19	⊆	⊆	NUM
ejpam-6570	392	20	int⋆(cl⋆(f−(v	int⋆(cl⋆(f−(v	NOUN
ejpam-6570	392	21	)	)	PUNCT
ejpam-6570	392	22	)	)	PUNCT
ejpam-6570	392	23	)	)	PUNCT
ejpam-6570	392	24	.	.	PUNCT
ejpam-6570	393	1	therefore	therefore	ADV
ejpam-6570	393	2	,	,	PUNCT
ejpam-6570	393	3	f−(v	f−(v	ADJ
ejpam-6570	393	4	)	)	PUNCT
ejpam-6570	393	5	is	be	AUX
ejpam-6570	393	6	i	i	PRON
ejpam-6570	393	7	⋆-preopen	⋆-preopen	ADV
ejpam-6570	393	8	inx	inx	VERB
ejpam-6570	393	9	.	.	PUNCT
ejpam-6570	394	1	by	by	ADP
ejpam-6570	394	2	theorem	theorem	NOUN
ejpam-6570	394	3	12	12	NUM
ejpam-6570	394	4	,	,	PUNCT
ejpam-6570	394	5	f	f	PROPN
ejpam-6570	394	6	is	be	AUX
ejpam-6570	394	7	lower	low	ADJ
ejpam-6570	394	8	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6570	394	9	,	,	PUNCT
ejpam-6570	394	10	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-6570	394	11	.	.	PUNCT
ejpam-6570	395	1	acknowledgements	acknowledgement	NOUN
ejpam-6570	395	2	this	this	DET
ejpam-6570	395	3	research	research	NOUN
ejpam-6570	395	4	project	project	NOUN
ejpam-6570	395	5	was	be	AUX
ejpam-6570	395	6	financially	financially	ADV
ejpam-6570	395	7	supported	support	VERB
ejpam-6570	395	8	by	by	ADP
ejpam-6570	395	9	mahasarakham	mahasarakham	PROPN
ejpam-6570	395	10	university	university	PROPN
ejpam-6570	395	11	.	.	PUNCT
ejpam-6570	396	1	references	reference	NOUN
ejpam-6570	396	2	[	[	X
ejpam-6570	396	3	1	1	NUM
ejpam-6570	396	4	]	]	PUNCT
ejpam-6570	396	5	m.	m.	NOUN
ejpam-6570	396	6	k.	k.	PROPN
ejpam-6570	396	7	singal	singal	PROPN
ejpam-6570	396	8	and	and	CCONJ
ejpam-6570	396	9	a.	a.	PROPN
ejpam-6570	396	10	r.	r.	PROPN
ejpam-6570	396	11	singal	singal	PROPN
ejpam-6570	396	12	.	.	PUNCT
ejpam-6570	397	1	almost	almost	ADV
ejpam-6570	397	2	continuous	continuous	ADJ
ejpam-6570	397	3	mappings	mapping	NOUN
ejpam-6570	397	4	.	.	PUNCT
ejpam-6570	398	1	yokohama	yokohama	PROPN
ejpam-6570	398	2	mathematical	mathematical	PROPN
ejpam-6570	398	3	journal	journal	PROPN
ejpam-6570	398	4	,	,	PUNCT
ejpam-6570	398	5	16:63–73	16:63–73	PROPN
ejpam-6570	398	6	,	,	PUNCT
ejpam-6570	398	7	1968	1968	NUM
ejpam-6570	398	8	.	.	PUNCT
ejpam-6570	399	1	[	[	X
ejpam-6570	399	2	2	2	NUM
ejpam-6570	399	3	]	]	X
ejpam-6570	399	4	b.	b.	PROPN
ejpam-6570	399	5	m.	m.	PROPN
ejpam-6570	399	6	munshi	munshi	PROPN
ejpam-6570	399	7	and	and	CCONJ
ejpam-6570	399	8	d.	d.	PROPN
ejpam-6570	399	9	s.	s.	PROPN
ejpam-6570	399	10	bassan	bassan	PROPN
ejpam-6570	399	11	.	.	PUNCT
ejpam-6570	400	1	almost	almost	ADV
ejpam-6570	400	2	semi	semi	ADJ
ejpam-6570	400	3	-	-	ADJ
ejpam-6570	400	4	continuous	continuous	ADJ
ejpam-6570	400	5	mappings	mapping	NOUN
ejpam-6570	400	6	.	.	PUNCT
ejpam-6570	401	1	the	the	DET
ejpam-6570	401	2	mathematics	mathematics	PROPN
ejpam-6570	401	3	student	student	NOUN
ejpam-6570	401	4	,	,	PUNCT
ejpam-6570	401	5	49:239–248	49:239–248	PROPN
ejpam-6570	401	6	,	,	PUNCT
ejpam-6570	401	7	1981	1981	NUM
ejpam-6570	401	8	.	.	PUNCT
ejpam-6570	402	1	[	[	X
ejpam-6570	402	2	3	3	X
ejpam-6570	402	3	]	]	PUNCT
ejpam-6570	402	4	t.	t.	PROPN
ejpam-6570	402	5	noiri	noiri	PROPN
ejpam-6570	402	6	.	.	PUNCT
ejpam-6570	403	1	almost	almost	ADV
ejpam-6570	403	2	α	α	NUM
ejpam-6570	403	3	-	-	ADJ
ejpam-6570	403	4	continuous	continuous	ADJ
ejpam-6570	403	5	functions	function	NOUN
ejpam-6570	403	6	.	.	PUNCT
ejpam-6570	404	1	kyungpook	kyungpook	PROPN
ejpam-6570	404	2	mathematical	mathematical	PROPN
ejpam-6570	404	3	journal	journal	PROPN
ejpam-6570	404	4	,	,	PUNCT
ejpam-6570	404	5	28:71–77	28:71–77	PROPN
ejpam-6570	404	6	,	,	PUNCT
ejpam-6570	404	7	1988	1988	NUM
ejpam-6570	404	8	.	.	PUNCT
ejpam-6570	405	1	[	[	X
ejpam-6570	405	2	4	4	X
ejpam-6570	405	3	]	]	PUNCT
ejpam-6570	405	4	a.	a.	NOUN
ejpam-6570	405	5	a.	a.	NOUN
ejpam-6570	405	6	nasef	nasef	PROPN
ejpam-6570	405	7	and	and	CCONJ
ejpam-6570	405	8	t.	t.	PROPN
ejpam-6570	405	9	noiri	noiri	PROPN
ejpam-6570	405	10	.	.	PUNCT
ejpam-6570	406	1	some	some	DET
ejpam-6570	406	2	weak	weak	ADJ
ejpam-6570	406	3	forms	form	NOUN
ejpam-6570	406	4	of	of	ADP
ejpam-6570	406	5	almost	almost	ADV
ejpam-6570	406	6	continuity	continuity	NOUN
ejpam-6570	406	7	.	.	PUNCT
ejpam-6570	407	1	acta	acta	PROPN
ejpam-6570	407	2	mathematica	mathematica	PROPN
ejpam-6570	407	3	hungarica	hungarica	PROPN
ejpam-6570	407	4	,	,	PUNCT
ejpam-6570	407	5	74(3):211–219	74(3):211–219	PROPN
ejpam-6570	407	6	,	,	PUNCT
ejpam-6570	407	7	1997	1997	NUM
ejpam-6570	407	8	.	.	PUNCT
ejpam-6570	408	1	[	[	X
ejpam-6570	408	2	5	5	NUM
ejpam-6570	408	3	]	]	X
ejpam-6570	408	4	n.	n.	PROPN
ejpam-6570	408	5	levine	levine	PROPN
ejpam-6570	408	6	.	.	PUNCT
ejpam-6570	409	1	a	a	DET
ejpam-6570	409	2	decomposition	decomposition	NOUN
ejpam-6570	409	3	of	of	ADP
ejpam-6570	409	4	continuity	continuity	NOUN
ejpam-6570	409	5	in	in	ADP
ejpam-6570	409	6	topological	topological	ADJ
ejpam-6570	409	7	spaces	space	NOUN
ejpam-6570	409	8	.	.	PUNCT
ejpam-6570	410	1	the	the	DET
ejpam-6570	410	2	american	american	PROPN
ejpam-6570	410	3	mathematical	mathematical	PROPN
ejpam-6570	410	4	monthly	monthly	ADV
ejpam-6570	410	5	,	,	PUNCT
ejpam-6570	410	6	68:44–46	68:44–46	NUM
ejpam-6570	410	7	,	,	PUNCT
ejpam-6570	410	8	1961	1961	NUM
ejpam-6570	410	9	.	.	PUNCT
ejpam-6570	411	1	[	[	X
ejpam-6570	411	2	6	6	NUM
ejpam-6570	411	3	]	]	PUNCT
ejpam-6570	411	4	t.	t.	PROPN
ejpam-6570	411	5	husain	husain	PROPN
ejpam-6570	411	6	.	.	PUNCT
ejpam-6570	412	1	almost	almost	ADV
ejpam-6570	412	2	continuous	continuous	ADJ
ejpam-6570	412	3	mappings	mapping	NOUN
ejpam-6570	412	4	.	.	PUNCT
ejpam-6570	413	1	prace	prace	PROPN
ejpam-6570	413	2	matematyczne	matematyczne	PROPN
ejpam-6570	413	3	,	,	PUNCT
ejpam-6570	413	4	10:1–7	10:1–7	NUM
ejpam-6570	413	5	,	,	PUNCT
ejpam-6570	413	6	1966	1966	NUM
ejpam-6570	413	7	.	.	PUNCT
ejpam-6570	414	1	[	[	X
ejpam-6570	414	2	7	7	X
ejpam-6570	414	3	]	]	X
ejpam-6570	415	1	d.	d.	PROPN
ejpam-6570	415	2	s.	s.	PROPN
ejpam-6570	415	3	janković.	janković.	PROPN
ejpam-6570	415	4	θ	θ	PROPN
ejpam-6570	415	5	-	-	ADJ
ejpam-6570	415	6	regular	regular	ADJ
ejpam-6570	415	7	spaces	space	NOUN
ejpam-6570	415	8	.	.	PUNCT
ejpam-6570	416	1	international	international	ADJ
ejpam-6570	416	2	journal	journal	PROPN
ejpam-6570	416	3	of	of	ADP
ejpam-6570	416	4	mathematics	mathematics	PROPN
ejpam-6570	416	5	and	and	CCONJ
ejpam-6570	416	6	mathematical	mathematical	ADJ
ejpam-6570	416	7	sciences	science	NOUN
ejpam-6570	416	8	,	,	PUNCT
ejpam-6570	416	9	8:615–619	8:615–619	NUM
ejpam-6570	416	10	,	,	PUNCT
ejpam-6570	416	11	1985	1985	NUM
ejpam-6570	416	12	.	.	PUNCT
ejpam-6570	417	1	[	[	X
ejpam-6570	417	2	8	8	NUM
ejpam-6570	417	3	]	]	PUNCT
ejpam-6570	417	4	t.	t.	PROPN
ejpam-6570	417	5	noiri	noiri	PROPN
ejpam-6570	417	6	.	.	PUNCT
ejpam-6570	418	1	properties	property	NOUN
ejpam-6570	418	2	of	of	ADP
ejpam-6570	418	3	some	some	DET
ejpam-6570	418	4	weak	weak	ADJ
ejpam-6570	418	5	forms	form	NOUN
ejpam-6570	418	6	of	of	ADP
ejpam-6570	418	7	continuity	continuity	NOUN
ejpam-6570	418	8	.	.	PUNCT
ejpam-6570	419	1	international	international	ADJ
ejpam-6570	419	2	journal	journal	PROPN
ejpam-6570	419	3	of	of	ADP
ejpam-6570	419	4	mathematics	mathematics	PROPN
ejpam-6570	419	5	and	and	CCONJ
ejpam-6570	419	6	mathematical	mathematical	ADJ
ejpam-6570	419	7	sciences	science	NOUN
ejpam-6570	419	8	,	,	PUNCT
ejpam-6570	419	9	10(1):97–111	10(1):97–111	NUM
ejpam-6570	419	10	,	,	PUNCT
ejpam-6570	419	11	1987	1987	NUM
ejpam-6570	419	12	.	.	PUNCT
ejpam-6570	420	1	[	[	X
ejpam-6570	420	2	9	9	NUM
ejpam-6570	420	3	]	]	X
ejpam-6570	420	4	d.	d.	PROPN
ejpam-6570	420	5	a.	a.	PROPN
ejpam-6570	420	6	rose	rise	VERB
ejpam-6570	420	7	.	.	PUNCT
ejpam-6570	421	1	weak	weak	ADJ
ejpam-6570	421	2	continuity	continuity	NOUN
ejpam-6570	421	3	and	and	CCONJ
ejpam-6570	421	4	almost	almost	ADV
ejpam-6570	421	5	continuity	continuity	NOUN
ejpam-6570	421	6	.	.	PUNCT
ejpam-6570	422	1	international	international	ADJ
ejpam-6570	422	2	journal	journal	PROPN
ejpam-6570	422	3	of	of	ADP
ejpam-6570	422	4	mathematics	mathematics	PROPN
ejpam-6570	422	5	and	and	CCONJ
ejpam-6570	422	6	mathematical	mathematical	ADJ
ejpam-6570	422	7	sciences	science	NOUN
ejpam-6570	422	8	,	,	PUNCT
ejpam-6570	422	9	7:311–318	7:311–318	PROPN
ejpam-6570	422	10	,	,	PUNCT
ejpam-6570	422	11	1984	1984	NUM
ejpam-6570	422	12	.	.	PUNCT
ejpam-6570	423	1	[	[	X
ejpam-6570	423	2	10	10	NUM
ejpam-6570	423	3	]	]	PUNCT
ejpam-6570	423	4	t.	t.	PROPN
ejpam-6570	423	5	noiri	noiri	PROPN
ejpam-6570	423	6	and	and	CCONJ
ejpam-6570	423	7	v.	v.	ADP
ejpam-6570	423	8	popa	popa	NOUN
ejpam-6570	423	9	.	.	PUNCT
ejpam-6570	424	1	almost	almost	ADV
ejpam-6570	424	2	weakly	weakly	ADJ
ejpam-6570	424	3	continuous	continuous	ADJ
ejpam-6570	424	4	multifunctions	multifunction	NOUN
ejpam-6570	424	5	.	.	PUNCT
ejpam-6570	425	1	demonstratio	demonstratio	PROPN
ejpam-6570	425	2	mathematica	mathematica	PROPN
ejpam-6570	425	3	,	,	PUNCT
ejpam-6570	425	4	26(2):363–380	26(2):363–380	PROPN
ejpam-6570	425	5	,	,	PUNCT
ejpam-6570	425	6	1993	1993	NUM
ejpam-6570	425	7	.	.	PUNCT
ejpam-6570	426	1	[	[	X
ejpam-6570	426	2	11	11	NUM
ejpam-6570	426	3	]	]	PUNCT
ejpam-6570	426	4	v.	v.	CCONJ
ejpam-6570	426	5	popa	popa	NOUN
ejpam-6570	426	6	and	and	CCONJ
ejpam-6570	426	7	t.	t.	PROPN
ejpam-6570	426	8	noiri	noiri	PROPN
ejpam-6570	426	9	.	.	PUNCT
ejpam-6570	427	1	some	some	DET
ejpam-6570	427	2	properties	property	NOUN
ejpam-6570	427	3	of	of	ADP
ejpam-6570	427	4	almost	almost	ADV
ejpam-6570	427	5	weakly	weakly	ADJ
ejpam-6570	427	6	continuous	continuous	ADJ
ejpam-6570	427	7	multifunctions	multifunction	NOUN
ejpam-6570	427	8	.	.	PUNCT
ejpam-6570	428	1	demonstratio	demonstratio	PROPN
ejpam-6570	428	2	mathematica	mathematica	PROPN
ejpam-6570	428	3	,	,	PUNCT
ejpam-6570	428	4	32(3):605–614	32(3):605–614	NUM
ejpam-6570	428	5	,	,	PUNCT
ejpam-6570	428	6	1999	1999	NUM
ejpam-6570	428	7	.	.	PUNCT
ejpam-6570	429	1	[	[	X
ejpam-6570	429	2	12	12	NUM
ejpam-6570	429	3	]	]	PUNCT
ejpam-6570	429	4	m.	m.	NOUN
ejpam-6570	429	5	e.	e.	PROPN
ejpam-6570	429	6	abd	abd	PROPN
ejpam-6570	430	1	el	el	PROPN
ejpam-6570	430	2	-	-	PROPN
ejpam-6570	430	3	monsef	monsef	PROPN
ejpam-6570	430	4	,	,	PUNCT
ejpam-6570	430	5	e.	e.	PROPN
ejpam-6570	430	6	f.	f.	PROPN
ejpam-6570	430	7	lashien	lashien	PROPN
ejpam-6570	430	8	,	,	PUNCT
ejpam-6570	430	9	and	and	CCONJ
ejpam-6570	430	10	a.	a.	NOUN
ejpam-6570	430	11	a.	a.	NOUN
ejpam-6570	430	12	nasef	nasef	PROPN
ejpam-6570	430	13	.	.	PUNCT
ejpam-6570	431	1	on	on	ADP
ejpam-6570	431	2	i	i	PRON
ejpam-6570	431	3	-open	-open	PROPN
ejpam-6570	431	4	sets	set	NOUN
ejpam-6570	431	5	and	and	CCONJ
ejpam-6570	431	6	i	i	PRON
ejpam-6570	431	7	continuous	continuous	ADJ
ejpam-6570	431	8	functions	function	NOUN
ejpam-6570	431	9	.	.	PUNCT
ejpam-6570	432	1	kyungpook	kyungpook	PROPN
ejpam-6570	432	2	mathematical	mathematical	PROPN
ejpam-6570	432	3	journal	journal	PROPN
ejpam-6570	432	4	,	,	PUNCT
ejpam-6570	432	5	32:21–30	32:21–30	NUM
ejpam-6570	432	6	,	,	PUNCT
ejpam-6570	432	7	1992	1992	NUM
ejpam-6570	432	8	.	.	PUNCT
ejpam-6570	433	1	[	[	X
ejpam-6570	433	2	13	13	NUM
ejpam-6570	433	3	]	]	X
ejpam-6570	433	4	e.	e.	PROPN
ejpam-6570	433	5	hatir	hatir	PROPN
ejpam-6570	433	6	and	and	CCONJ
ejpam-6570	433	7	t.	t.	PROPN
ejpam-6570	433	8	noiri	noiri	PROPN
ejpam-6570	433	9	.	.	PUNCT
ejpam-6570	434	1	weakly	weakly	ADJ
ejpam-6570	434	2	pre	pre	ADJ
ejpam-6570	434	3	-	-	ADJ
ejpam-6570	434	4	i	i	PRON
ejpam-6570	434	5	-	-	PUNCT
ejpam-6570	434	6	open	open	ADJ
ejpam-6570	434	7	sets	set	NOUN
ejpam-6570	434	8	and	and	CCONJ
ejpam-6570	434	9	decomposition	decomposition	NOUN
ejpam-6570	434	10	of	of	ADP
ejpam-6570	434	11	continuity	continuity	NOUN
ejpam-6570	434	12	.	.	PUNCT
ejpam-6570	435	1	acta	acta	PROPN
ejpam-6570	435	2	mathematica	mathematica	PROPN
ejpam-6570	435	3	hungarica	hungarica	PROPN
ejpam-6570	435	4	,	,	PUNCT
ejpam-6570	435	5	106(3):227–238	106(3):227–238	NUM
ejpam-6570	435	6	,	,	PUNCT
ejpam-6570	435	7	2005	2005	NUM
ejpam-6570	435	8	.	.	PUNCT
ejpam-6570	436	1	c.	c.	PROPN
ejpam-6570	436	2	viriyapong	viriyapong	PROPN
ejpam-6570	436	3	,	,	PUNCT
ejpam-6570	436	4	a.	a.	PROPN
ejpam-6570	436	5	sama	sama	PROPN
ejpam-6570	436	6	-	-	PUNCT
ejpam-6570	436	7	ae	ae	PROPN
ejpam-6570	436	8	,	,	PUNCT
ejpam-6570	436	9	c.	c.	PROPN
ejpam-6570	436	10	boonpok	boonpok	PROPN
ejpam-6570	436	11	/	/	SYM
ejpam-6570	436	12	eur	eur	PROPN
ejpam-6570	436	13	.	.	PUNCT
ejpam-6570	437	1	j.	j.	PROPN
ejpam-6570	437	2	pure	pure	PROPN
ejpam-6570	437	3	appl	appl	PROPN
ejpam-6570	437	4	.	.	PROPN
ejpam-6570	437	5	math	math	PROPN
ejpam-6570	437	6	,	,	PUNCT
ejpam-6570	437	7	18	18	NUM
ejpam-6570	437	8	(	(	PUNCT
ejpam-6570	437	9	3	3	NUM
ejpam-6570	437	10	)	)	PUNCT
ejpam-6570	437	11	(	(	PUNCT
ejpam-6570	437	12	2025	2025	NUM
ejpam-6570	437	13	)	)	PUNCT
ejpam-6570	437	14	,	,	PUNCT
ejpam-6570	437	15	6570	6570	NUM
ejpam-6570	437	16	14	14	NUM
ejpam-6570	437	17	of	of	ADP
ejpam-6570	437	18	14	14	NUM
ejpam-6570	438	1	[	[	X
ejpam-6570	438	2	14	14	NUM
ejpam-6570	438	3	]	]	X
ejpam-6570	438	4	e.	e.	PROPN
ejpam-6570	438	5	hatir	hatir	PROPN
ejpam-6570	438	6	and	and	CCONJ
ejpam-6570	438	7	t.	t.	PROPN
ejpam-6570	438	8	noiri	noiri	PROPN
ejpam-6570	438	9	.	.	PUNCT
ejpam-6570	439	1	on	on	ADP
ejpam-6570	439	2	decompositions	decomposition	NOUN
ejpam-6570	439	3	of	of	ADP
ejpam-6570	439	4	continuity	continuity	NOUN
ejpam-6570	439	5	via	via	ADP
ejpam-6570	439	6	idealization	idealization	NOUN
ejpam-6570	439	7	.	.	PUNCT
ejpam-6570	440	1	acta	acta	PROPN
ejpam-6570	440	2	mathematica	mathematica	PROPN
ejpam-6570	440	3	hungarica	hungarica	PROPN
ejpam-6570	440	4	,	,	PUNCT
ejpam-6570	440	5	96:341–349	96:341–349	PROPN
ejpam-6570	440	6	,	,	PUNCT
ejpam-6570	440	7	2002	2002	NUM
ejpam-6570	440	8	.	.	PUNCT
ejpam-6570	441	1	[	[	X
ejpam-6570	441	2	15	15	NUM
ejpam-6570	441	3	]	]	X
ejpam-6570	441	4	c.	c.	PROPN
ejpam-6570	441	5	boonpok	boonpok	PROPN
ejpam-6570	441	6	.	.	PUNCT
ejpam-6570	442	1	on	on	ADP
ejpam-6570	442	2	continuous	continuous	ADJ
ejpam-6570	442	3	multifunctions	multifunction	NOUN
ejpam-6570	442	4	in	in	ADP
ejpam-6570	442	5	ideal	ideal	ADJ
ejpam-6570	442	6	topological	topological	ADJ
ejpam-6570	442	7	spaces	space	NOUN
ejpam-6570	442	8	.	.	PUNCT
ejpam-6570	443	1	lobachevskii	lobachevskii	PROPN
ejpam-6570	443	2	journal	journal	PROPN
ejpam-6570	443	3	of	of	ADP
ejpam-6570	443	4	mathematics	mathematic	NOUN
ejpam-6570	443	5	,	,	PUNCT
ejpam-6570	443	6	40(1):24–35	40(1):24–35	NUM
ejpam-6570	443	7	,	,	PUNCT
ejpam-6570	443	8	2019	2019	NUM
ejpam-6570	443	9	.	.	PUNCT
ejpam-6570	444	1	[	[	X
ejpam-6570	444	2	16	16	NUM
ejpam-6570	444	3	]	]	PUNCT
ejpam-6570	444	4	c.	c.	PROPN
ejpam-6570	444	5	boonpok	boonpok	PROPN
ejpam-6570	444	6	.	.	PUNCT
ejpam-6570	445	1	pı	pı	NOUN
ejpam-6570	445	2	-	-	NOUN
ejpam-6570	445	3	continuity	continuity	NOUN
ejpam-6570	445	4	and	and	CCONJ
ejpam-6570	445	5	weak	weak	ADJ
ejpam-6570	445	6	pı	pı	NOUN
ejpam-6570	445	7	-	-	NOUN
ejpam-6570	445	8	continuity	continuity	NOUN
ejpam-6570	445	9	.	.	PUNCT
ejpam-6570	446	1	carpathian	carpathian	ADJ
ejpam-6570	446	2	mathematical	mathematical	ADJ
ejpam-6570	446	3	publications	publication	NOUN
ejpam-6570	446	4	,	,	PUNCT
ejpam-6570	446	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6570	446	6	,	,	PUNCT
ejpam-6570	446	7	2025	2025	NUM
ejpam-6570	446	8	.	.	PUNCT
ejpam-6570	447	1	[	[	X
ejpam-6570	447	2	17	17	NUM
ejpam-6570	447	3	]	]	X
ejpam-6570	447	4	p.	p.	NOUN
ejpam-6570	447	5	pue	pue	NOUN
ejpam-6570	447	6	-	-	PUNCT
ejpam-6570	447	7	on	on	ADP
ejpam-6570	447	8	,	,	PUNCT
ejpam-6570	447	9	s.	s.	PROPN
ejpam-6570	447	10	sompong	sompong	PROPN
ejpam-6570	447	11	,	,	PUNCT
ejpam-6570	447	12	and	and	CCONJ
ejpam-6570	447	13	c.	c.	PROPN
ejpam-6570	447	14	boonpok	boonpok	PROPN
ejpam-6570	447	15	.	.	PUNCT
ejpam-6570	448	1	upper	upper	ADJ
ejpam-6570	448	2	and	and	CCONJ
ejpam-6570	448	3	lower	low	ADJ
ejpam-6570	448	4	(	(	PUNCT
ejpam-6570	448	5	τ1	τ1	NOUN
ejpam-6570	448	6	,	,	PUNCT
ejpam-6570	448	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6570	448	8	multifunctions	multifunction	NOUN
ejpam-6570	448	9	.	.	PUNCT
ejpam-6570	449	1	international	international	ADJ
ejpam-6570	449	2	journal	journal	PROPN
ejpam-6570	449	3	of	of	ADP
ejpam-6570	449	4	mathematics	mathematic	NOUN
ejpam-6570	449	5	and	and	CCONJ
ejpam-6570	449	6	computer	computer	NOUN
ejpam-6570	449	7	science	science	NOUN
ejpam-6570	449	8	,	,	PUNCT
ejpam-6570	449	9	19(4):1305	19(4):1305	NUM
ejpam-6570	449	10	–	–	PUNCT
ejpam-6570	449	11	1310	1310	NUM
ejpam-6570	449	12	,	,	PUNCT
ejpam-6570	449	13	2024	2024	NUM
ejpam-6570	449	14	.	.	PUNCT
ejpam-6570	450	1	[	[	X
ejpam-6570	450	2	18	18	NUM
ejpam-6570	450	3	]	]	X
ejpam-6570	450	4	c.	c.	PROPN
ejpam-6570	450	5	klanarong	klanarong	PROPN
ejpam-6570	450	6	,	,	PUNCT
ejpam-6570	450	7	s.	s.	PROPN
ejpam-6570	450	8	sompong	sompong	PROPN
ejpam-6570	450	9	,	,	PUNCT
ejpam-6570	450	10	and	and	CCONJ
ejpam-6570	450	11	c.	c.	PROPN
ejpam-6570	450	12	boonpok	boonpok	PROPN
ejpam-6570	450	13	.	.	PUNCT
ejpam-6570	451	1	upper	upper	ADJ
ejpam-6570	451	2	and	and	CCONJ
ejpam-6570	451	3	lower	low	ADJ
ejpam-6570	451	4	almost	almost	ADV
ejpam-6570	451	5	(	(	PUNCT
ejpam-6570	451	6	τ1	τ1	NOUN
ejpam-6570	451	7	,	,	PUNCT
ejpam-6570	451	8	τ2)continuous	τ2)continuous	ADJ
ejpam-6570	451	9	multifunctions	multifunction	NOUN
ejpam-6570	451	10	.	.	PUNCT
ejpam-6570	452	1	european	european	ADJ
ejpam-6570	452	2	journal	journal	PROPN
ejpam-6570	452	3	of	of	ADP
ejpam-6570	452	4	pure	pure	ADJ
ejpam-6570	452	5	and	and	CCONJ
ejpam-6570	452	6	applied	applied	ADJ
ejpam-6570	452	7	mathematics	mathematic	NOUN
ejpam-6570	452	8	,	,	PUNCT
ejpam-6570	452	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-6570	452	10	,	,	PUNCT
ejpam-6570	452	11	2024	2024	NUM
ejpam-6570	452	12	.	.	PUNCT
ejpam-6570	453	1	[	[	X
ejpam-6570	453	2	19	19	NUM
ejpam-6570	453	3	]	]	PUNCT
ejpam-6570	453	4	m.	m.	NOUN
ejpam-6570	453	5	thongmoon	thongmoon	NOUN
ejpam-6570	453	6	,	,	PUNCT
ejpam-6570	453	7	s.	s.	PROPN
ejpam-6570	453	8	sompong	sompong	PROPN
ejpam-6570	453	9	,	,	PUNCT
ejpam-6570	453	10	and	and	CCONJ
ejpam-6570	453	11	c.	c.	PROPN
ejpam-6570	453	12	boonpok	boonpok	PROPN
ejpam-6570	453	13	.	.	PUNCT
ejpam-6570	454	1	upper	upper	ADJ
ejpam-6570	454	2	and	and	CCONJ
ejpam-6570	454	3	lower	low	ADJ
ejpam-6570	454	4	weak	weak	ADJ
ejpam-6570	454	5	(	(	PUNCT
ejpam-6570	454	6	τ1	τ1	NOUN
ejpam-6570	454	7	,	,	PUNCT
ejpam-6570	454	8	τ2)continuity	τ2)continuity	PROPN
ejpam-6570	454	9	.	.	PUNCT
ejpam-6570	455	1	european	european	PROPN
ejpam-6570	455	2	journal	journal	PROPN
ejpam-6570	455	3	of	of	ADP
ejpam-6570	455	4	pure	pure	ADJ
ejpam-6570	455	5	and	and	CCONJ
ejpam-6570	455	6	applied	applied	ADJ
ejpam-6570	455	7	mathematics	mathematic	NOUN
ejpam-6570	455	8	,	,	PUNCT
ejpam-6570	455	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-6570	455	10	,	,	PUNCT
ejpam-6570	455	11	2024	2024	NUM
ejpam-6570	455	12	.	.	PUNCT
ejpam-6570	456	1	[	[	X
ejpam-6570	456	2	20	20	NUM
ejpam-6570	456	3	]	]	PUNCT
ejpam-6570	456	4	c.	c.	PROPN
ejpam-6570	456	5	boonpok	boonpok	PROPN
ejpam-6570	456	6	and	and	CCONJ
ejpam-6570	456	7	c.	c.	PROPN
ejpam-6570	456	8	viriyapong	viriyapong	PROPN
ejpam-6570	456	9	.	.	PUNCT
ejpam-6570	457	1	upper	upper	ADJ
ejpam-6570	457	2	and	and	CCONJ
ejpam-6570	457	3	lower	low	ADJ
ejpam-6570	457	4	almost	almost	ADV
ejpam-6570	457	5	weak	weak	ADJ
ejpam-6570	457	6	(	(	PUNCT
ejpam-6570	457	7	τ1	τ1	NOUN
ejpam-6570	457	8	,	,	PUNCT
ejpam-6570	457	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6570	457	10	.	.	PUNCT
ejpam-6570	458	1	european	european	PROPN
ejpam-6570	458	2	journal	journal	PROPN
ejpam-6570	458	3	of	of	ADP
ejpam-6570	458	4	pure	pure	ADJ
ejpam-6570	458	5	and	and	CCONJ
ejpam-6570	458	6	applied	applied	ADJ
ejpam-6570	458	7	mathematics	mathematic	NOUN
ejpam-6570	458	8	,	,	PUNCT
ejpam-6570	458	9	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-6570	458	10	,	,	PUNCT
ejpam-6570	458	11	2021	2021	NUM
ejpam-6570	458	12	.	.	PUNCT
ejpam-6570	459	1	[	[	X
ejpam-6570	459	2	21	21	NUM
ejpam-6570	459	3	]	]	X
ejpam-6570	459	4	c.	c.	PROPN
ejpam-6570	459	5	boonpok	boonpok	PROPN
ejpam-6570	459	6	,	,	PUNCT
ejpam-6570	459	7	c.	c.	PROPN
ejpam-6570	459	8	viriyapong	viriyapong	PROPN
ejpam-6570	459	9	,	,	PUNCT
ejpam-6570	459	10	and	and	CCONJ
ejpam-6570	459	11	m.	m.	NOUN
ejpam-6570	459	12	thongmoon	thongmoon	NOUN
ejpam-6570	459	13	.	.	PUNCT
ejpam-6570	460	1	on	on	ADP
ejpam-6570	460	2	upper	upper	ADJ
ejpam-6570	460	3	and	and	CCONJ
ejpam-6570	460	4	lower	low	ADJ
ejpam-6570	460	5	(	(	PUNCT
ejpam-6570	460	6	τ1	τ1	NOUN
ejpam-6570	460	7	,	,	PUNCT
ejpam-6570	460	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6570	460	9	multifunctions	multifunction	NOUN
ejpam-6570	460	10	.	.	PUNCT
ejpam-6570	461	1	journal	journal	PROPN
ejpam-6570	461	2	of	of	ADP
ejpam-6570	461	3	mathematics	mathematics	PROPN
ejpam-6570	461	4	and	and	CCONJ
ejpam-6570	461	5	computer	computer	NOUN
ejpam-6570	461	6	science	science	NOUN
ejpam-6570	461	7	,	,	PUNCT
ejpam-6570	461	8	18:282–293	18:282–293	NUM
ejpam-6570	461	9	,	,	PUNCT
ejpam-6570	461	10	2018	2018	NUM
ejpam-6570	461	11	.	.	PUNCT
ejpam-6570	462	1	[	[	X
ejpam-6570	462	2	22	22	NUM
ejpam-6570	462	3	]	]	X
ejpam-6570	462	4	c.	c.	PROPN
ejpam-6570	462	5	viriyapong	viriyapong	PROPN
ejpam-6570	462	6	and	and	CCONJ
ejpam-6570	462	7	c.	c.	PROPN
ejpam-6570	462	8	boonpok	boonpok	PROPN
ejpam-6570	462	9	.	.	PUNCT
ejpam-6570	463	1	(	(	PUNCT
ejpam-6570	463	2	τ1	τ1	NOUN
ejpam-6570	463	3	,	,	PUNCT
ejpam-6570	463	4	τ2)α	τ2)α	NOUN
ejpam-6570	463	5	-	-	PUNCT
ejpam-6570	463	6	continuity	continuity	NOUN
ejpam-6570	463	7	for	for	ADP
ejpam-6570	463	8	multifunctions	multifunction	NOUN
ejpam-6570	463	9	.	.	PUNCT
ejpam-6570	464	1	journal	journal	PROPN
ejpam-6570	464	2	of	of	ADP
ejpam-6570	464	3	mathematics	mathematic	NOUN
ejpam-6570	464	4	,	,	PUNCT
ejpam-6570	464	5	2020:6285763	2020:6285763	NUM
ejpam-6570	464	6	,	,	PUNCT
ejpam-6570	464	7	2020	2020	NUM
ejpam-6570	464	8	.	.	PUNCT
ejpam-6570	465	1	[	[	X
ejpam-6570	465	2	23	23	NUM
ejpam-6570	465	3	]	]	X
ejpam-6570	465	4	c.	c.	PROPN
ejpam-6570	465	5	boonpok	boonpok	PROPN
ejpam-6570	465	6	.	.	PUNCT
ejpam-6570	466	1	(	(	PUNCT
ejpam-6570	466	2	τ1	τ1	NOUN
ejpam-6570	466	3	,	,	PUNCT
ejpam-6570	466	4	τ2)δ	τ2)δ	ADJ
ejpam-6570	466	5	-	-	PUNCT
ejpam-6570	466	6	semicontinuous	semicontinuous	ADJ
ejpam-6570	466	7	multifunctions	multifunction	NOUN
ejpam-6570	466	8	.	.	PUNCT
ejpam-6570	467	1	heliyon	heliyon	NOUN
ejpam-6570	467	2	,	,	PUNCT
ejpam-6570	467	3	6	6	NUM
ejpam-6570	467	4	:	:	SYM
ejpam-6570	467	5	e05367	e05367	PROPN
ejpam-6570	467	6	,	,	PUNCT
ejpam-6570	467	7	2020	2020	NUM
ejpam-6570	467	8	.	.	PUNCT
ejpam-6570	468	1	[	[	X
ejpam-6570	468	2	24	24	NUM
ejpam-6570	468	3	]	]	X
ejpam-6570	468	4	n.	n.	PROPN
ejpam-6570	468	5	viriyapong	viriyapong	PROPN
ejpam-6570	468	6	,	,	PUNCT
ejpam-6570	468	7	s.	s.	PROPN
ejpam-6570	468	8	sompong	sompong	PROPN
ejpam-6570	468	9	,	,	PUNCT
ejpam-6570	468	10	and	and	CCONJ
ejpam-6570	468	11	c.	c.	PROPN
ejpam-6570	468	12	boonpok	boonpok	PROPN
ejpam-6570	468	13	.	.	PUNCT
ejpam-6570	469	1	(	(	PUNCT
ejpam-6570	469	2	τ1	τ1	NOUN
ejpam-6570	469	3	,	,	PUNCT
ejpam-6570	469	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6570	469	5	disconnectedness	disconnectedness	NOUN
ejpam-6570	469	6	in	in	ADP
ejpam-6570	469	7	bitopological	bitopological	ADJ
ejpam-6570	469	8	spaces	space	NOUN
ejpam-6570	469	9	.	.	PUNCT
ejpam-6570	470	1	international	international	ADJ
ejpam-6570	470	2	journal	journal	PROPN
ejpam-6570	470	3	of	of	ADP
ejpam-6570	470	4	mathematics	mathematic	NOUN
ejpam-6570	470	5	and	and	CCONJ
ejpam-6570	470	6	computer	computer	NOUN
ejpam-6570	470	7	science	science	NOUN
ejpam-6570	470	8	,	,	PUNCT
ejpam-6570	470	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6570	470	10	,	,	PUNCT
ejpam-6570	470	11	2024	2024	NUM
ejpam-6570	470	12	.	.	PUNCT
ejpam-6570	471	1	[	[	X
ejpam-6570	471	2	25	25	NUM
ejpam-6570	471	3	]	]	PUNCT
ejpam-6570	471	4	k.	k.	PROPN
ejpam-6570	471	5	kuratowski	kuratowski	PROPN
ejpam-6570	471	6	.	.	PUNCT
ejpam-6570	472	1	topology	topology	PROPN
ejpam-6570	472	2	,	,	PUNCT
ejpam-6570	472	3	vol	vol	NOUN
ejpam-6570	472	4	.	.	PUNCT
ejpam-6570	472	5	i.	i.	PROPN
ejpam-6570	472	6	academic	academic	PROPN
ejpam-6570	472	7	press	press	PROPN
ejpam-6570	472	8	,	,	PUNCT
ejpam-6570	472	9	new	new	PROPN
ejpam-6570	472	10	york	york	PROPN
ejpam-6570	472	11	,	,	PUNCT
ejpam-6570	472	12	1966	1966	NUM
ejpam-6570	472	13	.	.	PUNCT
ejpam-6570	473	1	[	[	X
ejpam-6570	473	2	26	26	NUM
ejpam-6570	473	3	]	]	X
ejpam-6570	473	4	d.	d.	PROPN
ejpam-6570	473	5	janković	janković	PROPN
ejpam-6570	473	6	and	and	CCONJ
ejpam-6570	473	7	t.	t.	PROPN
ejpam-6570	473	8	r.	r.	PROPN
ejpam-6570	473	9	hamlett	hamlett	PROPN
ejpam-6570	473	10	.	.	PUNCT
ejpam-6570	474	1	new	new	ADJ
ejpam-6570	474	2	topologies	topology	NOUN
ejpam-6570	474	3	from	from	ADP
ejpam-6570	474	4	old	old	ADJ
ejpam-6570	474	5	via	via	ADP
ejpam-6570	474	6	ideals	ideal	NOUN
ejpam-6570	474	7	.	.	PUNCT
ejpam-6570	475	1	the	the	DET
ejpam-6570	475	2	american	american	PROPN
ejpam-6570	475	3	mathematical	mathematical	PROPN
ejpam-6570	475	4	monthly	monthly	ADV
ejpam-6570	475	5	,	,	PUNCT
ejpam-6570	475	6	97:295–310	97:295–310	PROPN
ejpam-6570	475	7	,	,	PUNCT
ejpam-6570	475	8	1990	1990	NUM
ejpam-6570	475	9	.	.	PUNCT
ejpam-6570	476	1	[	[	X
ejpam-6570	476	2	27	27	NUM
ejpam-6570	476	3	]	]	X
ejpam-6570	476	4	e.	e.	PROPN
ejpam-6570	476	5	ekici	ekici	PROPN
ejpam-6570	476	6	and	and	CCONJ
ejpam-6570	476	7	t.	t.	PROPN
ejpam-6570	476	8	noiri	noiri	PROPN
ejpam-6570	476	9	.	.	PUNCT
ejpam-6570	477	1	⋆-extremally	⋆-extremally	ADV
ejpam-6570	477	2	disconnected	disconnect	VERB
ejpam-6570	477	3	ideal	ideal	ADJ
ejpam-6570	477	4	topological	topological	ADJ
ejpam-6570	477	5	spaces	space	NOUN
ejpam-6570	477	6	.	.	PUNCT
ejpam-6570	478	1	acta	acta	PROPN
ejpam-6570	478	2	mathematica	mathematica	PROPN
ejpam-6570	478	3	hungarica	hungarica	PROPN
ejpam-6570	478	4	,	,	PUNCT
ejpam-6570	478	5	122:81–90	122:81–90	NUM
ejpam-6570	478	6	,	,	PUNCT
ejpam-6570	478	7	2009	2009	NUM
ejpam-6570	478	8	.	.	PUNCT
ejpam-6570	479	1	[	[	X
ejpam-6570	479	2	28	28	NUM
ejpam-6570	479	3	]	]	X
ejpam-6570	479	4	p.	p.	NOUN
ejpam-6570	479	5	pue	pue	NOUN
ejpam-6570	479	6	-	-	PUNCT
ejpam-6570	479	7	on	on	ADP
ejpam-6570	479	8	,	,	PUNCT
ejpam-6570	479	9	a.	a.	PROPN
ejpam-6570	479	10	sama	sama	PROPN
ejpam-6570	479	11	-	-	PUNCT
ejpam-6570	479	12	ae	ae	PROPN
ejpam-6570	479	13	,	,	PUNCT
ejpam-6570	479	14	and	and	CCONJ
ejpam-6570	479	15	c.	c.	PROPN
ejpam-6570	479	16	boonpok	boonpok	PROPN
ejpam-6570	479	17	.	.	PUNCT
ejpam-6570	480	1	quasi	quasi	PROPN
ejpam-6570	480	2	θ(τ1	θ(τ1	PROPN
ejpam-6570	480	3	,	,	PUNCT
ejpam-6570	480	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6570	480	5	for	for	ADP
ejpam-6570	480	6	multifunctions	multifunction	NOUN
ejpam-6570	480	7	.	.	PUNCT
ejpam-6570	481	1	european	european	ADJ
ejpam-6570	481	2	journal	journal	PROPN
ejpam-6570	481	3	of	of	ADP
ejpam-6570	481	4	pure	pure	ADJ
ejpam-6570	481	5	and	and	CCONJ
ejpam-6570	481	6	applied	applied	ADJ
ejpam-6570	481	7	mathematics	mathematic	NOUN
ejpam-6570	481	8	,	,	PUNCT
ejpam-6570	481	9	18(1):5717	18(1):5717	NUM
ejpam-6570	481	10	,	,	PUNCT
ejpam-6570	481	11	2025	2025	NUM
ejpam-6570	481	12	.	.	PUNCT
ejpam-6570	482	1	[	[	X
ejpam-6570	482	2	29	29	NUM
ejpam-6570	482	3	]	]	X
ejpam-6570	482	4	m.	m.	NOUN
ejpam-6570	482	5	chiangpradit	chiangpradit	NOUN
ejpam-6570	482	6	,	,	PUNCT
ejpam-6570	482	7	s.	s.	PROPN
ejpam-6570	482	8	sompong	sompong	PROPN
ejpam-6570	482	9	,	,	PUNCT
ejpam-6570	482	10	and	and	CCONJ
ejpam-6570	482	11	c.	c.	PROPN
ejpam-6570	482	12	boonpok	boonpok	PROPN
ejpam-6570	482	13	.	.	PUNCT
ejpam-6570	483	1	on	on	ADP
ejpam-6570	483	2	characterizations	characterization	NOUN
ejpam-6570	483	3	of	of	ADP
ejpam-6570	483	4	(	(	PUNCT
ejpam-6570	483	5	τ1	τ1	NOUN
ejpam-6570	483	6	,	,	PUNCT
ejpam-6570	483	7	τ2)regular	τ2)regular	ADJ
ejpam-6570	483	8	spaces	space	NOUN
ejpam-6570	483	9	.	.	PUNCT
ejpam-6570	484	1	international	international	ADJ
ejpam-6570	484	2	of	of	ADP
ejpam-6570	484	3	journal	journal	PROPN
ejpam-6570	484	4	of	of	ADP
ejpam-6570	484	5	mathematics	mathematic	NOUN
ejpam-6570	484	6	and	and	CCONJ
ejpam-6570	484	7	computer	computer	NOUN
ejpam-6570	484	8	science	science	NOUN
ejpam-6570	484	9	,	,	PUNCT
ejpam-6570	484	10	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-6570	484	11	,	,	PUNCT
ejpam-6570	484	12	2024	2024	NUM
ejpam-6570	484	13	.	.	PUNCT
ejpam-6570	485	1	[	[	X
ejpam-6570	485	2	30	30	NUM
ejpam-6570	485	3	]	]	X
ejpam-6570	485	4	c.	c.	PROPN
ejpam-6570	485	5	klanarong	klanarong	PROPN
ejpam-6570	485	6	,	,	PUNCT
ejpam-6570	485	7	s.	s.	PROPN
ejpam-6570	485	8	sompong	sompong	PROPN
ejpam-6570	485	9	,	,	PUNCT
ejpam-6570	485	10	and	and	CCONJ
ejpam-6570	485	11	c.	c.	PROPN
ejpam-6570	485	12	boonpok	boonpok	PROPN
ejpam-6570	485	13	.	.	PUNCT
ejpam-6570	486	1	(	(	PUNCT
ejpam-6570	486	2	τ1	τ1	NOUN
ejpam-6570	486	3	,	,	PUNCT
ejpam-6570	486	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6570	486	5	and	and	CCONJ
ejpam-6570	486	6	(	(	PUNCT
ejpam-6570	486	7	τ1	τ1	NOUN
ejpam-6570	486	8	,	,	PUNCT
ejpam-6570	486	9	τ2)θ	τ2)θ	ADJ
ejpam-6570	486	10	-	-	PUNCT
ejpam-6570	486	11	closed	close	VERB
ejpam-6570	486	12	sets	set	NOUN
ejpam-6570	486	13	.	.	PUNCT
ejpam-6570	487	1	international	international	ADJ
ejpam-6570	487	2	journal	journal	NOUN
ejpam-6570	487	3	of	of	ADP
ejpam-6570	487	4	mathematics	mathematic	NOUN
ejpam-6570	487	5	and	and	CCONJ
ejpam-6570	487	6	computer	computer	NOUN
ejpam-6570	487	7	science	science	NOUN
ejpam-6570	487	8	,	,	PUNCT
ejpam-6570	487	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-6570	487	10	,	,	PUNCT
ejpam-6570	487	11	2024	2024	NUM
ejpam-6570	487	12	.	.	PUNCT