id	sid	tid	token	lemma	pos
ejpam-7050	1	1	european	european	PROPN
ejpam-7050	1	2	journal	journal	PROPN
ejpam-7050	1	3	of	of	ADP
ejpam-7050	1	4	pure	pure	ADJ
ejpam-7050	1	5	and	and	CCONJ
ejpam-7050	1	6	applied	applied	ADJ
ejpam-7050	1	7	mathematics	mathematic	NOUN
ejpam-7050	1	8	2025	2025	NUM
ejpam-7050	1	9	,	,	PUNCT
ejpam-7050	1	10	vol	vol	NOUN
ejpam-7050	1	11	.	.	PROPN
ejpam-7050	1	12	18	18	NUM
ejpam-7050	1	13	,	,	PUNCT
ejpam-7050	1	14	issue	issue	NOUN
ejpam-7050	1	15	4	4	NUM
ejpam-7050	1	16	,	,	PUNCT
ejpam-7050	1	17	article	article	NOUN
ejpam-7050	1	18	number	number	NOUN
ejpam-7050	1	19	7050	7050	NUM
ejpam-7050	1	20	issn	issn	PROPN
ejpam-7050	1	21	1307	1307	NUM
ejpam-7050	1	22	-	-	SYM
ejpam-7050	1	23	5543	5543	NUM
ejpam-7050	1	24	–	–	PUNCT
ejpam-7050	1	25	ejpam.com	ejpam.com	X
ejpam-7050	1	26	published	publish	VERB
ejpam-7050	1	27	by	by	ADP
ejpam-7050	1	28	new	new	PROPN
ejpam-7050	1	29	york	york	PROPN
ejpam-7050	1	30	business	business	PROPN
ejpam-7050	1	31	global	global	PROPN
ejpam-7050	1	32	on	on	ADP
ejpam-7050	1	33	upper	upper	ADJ
ejpam-7050	1	34	and	and	CCONJ
ejpam-7050	1	35	lower	low	ADJ
ejpam-7050	1	36	weakly	weakly	ADJ
ejpam-7050	1	37	τ	τ	NOUN
ejpam-7050	1	38	⋆β(σ1	⋆β(σ1	NUM
ejpam-7050	1	39	,	,	PUNCT
ejpam-7050	1	40	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	1	41	multifunctions	multifunction	NOUN
ejpam-7050	1	42	montri	montri	PROPN
ejpam-7050	1	43	thongmoon1	thongmoon1	PROPN
ejpam-7050	1	44	,	,	PUNCT
ejpam-7050	1	45	areeyuth	areeyuth	NOUN
ejpam-7050	1	46	sama	sama	NOUN
ejpam-7050	1	47	-	-	PUNCT
ejpam-7050	1	48	ae2	ae2	PROPN
ejpam-7050	1	49	,	,	PUNCT
ejpam-7050	1	50	chawalit	chawalit	VERB
ejpam-7050	1	51	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7050	1	52	1	1	NUM
ejpam-7050	1	53	mathematics	mathematic	NOUN
ejpam-7050	1	54	and	and	CCONJ
ejpam-7050	1	55	applied	apply	VERB
ejpam-7050	1	56	mathematics	mathematics	PROPN
ejpam-7050	1	57	research	research	NOUN
ejpam-7050	1	58	unit	unit	NOUN
ejpam-7050	1	59	,	,	PUNCT
ejpam-7050	1	60	department	department	NOUN
ejpam-7050	1	61	of	of	ADP
ejpam-7050	1	62	mathematics	mathematic	NOUN
ejpam-7050	1	63	,	,	PUNCT
ejpam-7050	1	64	faculty	faculty	NOUN
ejpam-7050	1	65	of	of	ADP
ejpam-7050	1	66	science	science	NOUN
ejpam-7050	1	67	,	,	PUNCT
ejpam-7050	1	68	mahasarakham	mahasarakham	PROPN
ejpam-7050	1	69	university	university	PROPN
ejpam-7050	1	70	,	,	PUNCT
ejpam-7050	1	71	maha	maha	PROPN
ejpam-7050	1	72	sarakham	sarakham	PROPN
ejpam-7050	1	73	,	,	PUNCT
ejpam-7050	1	74	44150	44150	NUM
ejpam-7050	1	75	,	,	PUNCT
ejpam-7050	1	76	thailand	thailand	PROPN
ejpam-7050	1	77	2	2	NUM
ejpam-7050	1	78	department	department	NOUN
ejpam-7050	1	79	of	of	ADP
ejpam-7050	1	80	mathematics	mathematic	NOUN
ejpam-7050	1	81	and	and	CCONJ
ejpam-7050	1	82	computer	computer	NOUN
ejpam-7050	1	83	science	science	NOUN
ejpam-7050	1	84	,	,	PUNCT
ejpam-7050	1	85	faculty	faculty	NOUN
ejpam-7050	1	86	of	of	ADP
ejpam-7050	1	87	science	science	NOUN
ejpam-7050	1	88	and	and	CCONJ
ejpam-7050	1	89	technology	technology	NOUN
ejpam-7050	1	90	,	,	PUNCT
ejpam-7050	1	91	prince	prince	NOUN
ejpam-7050	1	92	of	of	ADP
ejpam-7050	1	93	songkla	songkla	PROPN
ejpam-7050	1	94	university	university	PROPN
ejpam-7050	1	95	,	,	PUNCT
ejpam-7050	1	96	pattani	pattani	NOUN
ejpam-7050	1	97	campus	campus	NOUN
ejpam-7050	1	98	,	,	PUNCT
ejpam-7050	1	99	pattani	pattani	NOUN
ejpam-7050	1	100	,	,	PUNCT
ejpam-7050	1	101	94000	94000	NUM
ejpam-7050	1	102	,	,	PUNCT
ejpam-7050	1	103	thailand	thailand	PROPN
ejpam-7050	1	104	abstract	abstract	PROPN
ejpam-7050	1	105	.	.	PUNCT
ejpam-7050	2	1	a	a	DET
ejpam-7050	2	2	new	new	ADJ
ejpam-7050	2	3	class	class	NOUN
ejpam-7050	2	4	of	of	ADP
ejpam-7050	2	5	continuous	continuous	ADJ
ejpam-7050	2	6	multifunctions	multifunction	NOUN
ejpam-7050	2	7	between	between	ADP
ejpam-7050	2	8	an	an	DET
ejpam-7050	2	9	ideal	ideal	ADJ
ejpam-7050	2	10	topological	topological	ADJ
ejpam-7050	2	11	space	space	NOUN
ejpam-7050	2	12	and	and	CCONJ
ejpam-7050	2	13	a	a	DET
ejpam-7050	2	14	bitopological	bitopological	ADJ
ejpam-7050	2	15	space	space	NOUN
ejpam-7050	2	16	,	,	PUNCT
ejpam-7050	2	17	called	call	VERB
ejpam-7050	2	18	upper	upper	ADJ
ejpam-7050	2	19	(	(	PUNCT
ejpam-7050	2	20	lower	low	ADJ
ejpam-7050	2	21	)	)	PUNCT
ejpam-7050	2	22	weakly	weakly	ADJ
ejpam-7050	2	23	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	2	24	,	,	PUNCT
ejpam-7050	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	2	26	multifunctions	multifunction	NOUN
ejpam-7050	2	27	,	,	PUNCT
ejpam-7050	2	28	has	have	AUX
ejpam-7050	2	29	been	be	AUX
ejpam-7050	2	30	defined	define	VERB
ejpam-7050	2	31	and	and	CCONJ
ejpam-7050	2	32	studied	study	VERB
ejpam-7050	2	33	.	.	PUNCT
ejpam-7050	3	1	furthermore	furthermore	ADV
ejpam-7050	3	2	,	,	PUNCT
ejpam-7050	3	3	several	several	ADJ
ejpam-7050	3	4	characterizations	characterization	NOUN
ejpam-7050	3	5	and	and	CCONJ
ejpam-7050	3	6	some	some	DET
ejpam-7050	3	7	properties	property	NOUN
ejpam-7050	3	8	concerning	concern	VERB
ejpam-7050	3	9	upper	upper	ADJ
ejpam-7050	3	10	weakly	weakly	ADJ
ejpam-7050	3	11	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	3	12	,	,	PUNCT
ejpam-7050	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	3	14	multifunctions	multifunction	NOUN
ejpam-7050	3	15	and	and	CCONJ
ejpam-7050	3	16	lower	low	ADJ
ejpam-7050	3	17	weakly	weakly	ADJ
ejpam-7050	3	18	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	3	19	,	,	PUNCT
ejpam-7050	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	3	21	multifunctions	multifunction	NOUN
ejpam-7050	3	22	are	be	AUX
ejpam-7050	3	23	discussed	discuss	VERB
ejpam-7050	3	24	.	.	PUNCT
ejpam-7050	4	1	2020	2020	NUM
ejpam-7050	4	2	mathematics	mathematic	NOUN
ejpam-7050	4	3	subject	subject	NOUN
ejpam-7050	4	4	classifications	classification	NOUN
ejpam-7050	4	5	:	:	PUNCT
ejpam-7050	4	6	54c08	54c08	NUM
ejpam-7050	4	7	,	,	PUNCT
ejpam-7050	4	8	54c60	54c60	NUM
ejpam-7050	4	9	key	key	ADJ
ejpam-7050	4	10	words	word	NOUN
ejpam-7050	4	11	and	and	CCONJ
ejpam-7050	4	12	phrases	phrase	NOUN
ejpam-7050	4	13	:	:	PUNCT
ejpam-7050	4	14	upper	upper	ADJ
ejpam-7050	4	15	weakly	weakly	ADJ
ejpam-7050	4	16	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	4	17	,	,	PUNCT
ejpam-7050	4	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	4	19	multifunction	multifunction	NOUN
ejpam-7050	4	20	,	,	PUNCT
ejpam-7050	4	21	lower	low	ADJ
ejpam-7050	4	22	weakly	weakly	ADJ
ejpam-7050	4	23	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	4	24	,	,	PUNCT
ejpam-7050	4	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	4	26	multifunction	multifunction	NOUN
ejpam-7050	4	27	1	1	NUM
ejpam-7050	4	28	.	.	PUNCT
ejpam-7050	4	29	introduction	introduction	NOUN
ejpam-7050	4	30	in	in	ADP
ejpam-7050	4	31	1983	1983	NUM
ejpam-7050	4	32	,	,	PUNCT
ejpam-7050	4	33	abd	abd	PROPN
ejpam-7050	4	34	el	el	PROPN
ejpam-7050	4	35	-	-	PROPN
ejpam-7050	4	36	monsef	monsef	PROPN
ejpam-7050	4	37	et	et	PROPN
ejpam-7050	4	38	al	al	PROPN
ejpam-7050	4	39	.	.	PUNCT
ejpam-7050	5	1	[	[	X
ejpam-7050	5	2	1	1	X
ejpam-7050	5	3	]	]	PUNCT
ejpam-7050	5	4	introduced	introduce	VERB
ejpam-7050	5	5	and	and	CCONJ
ejpam-7050	5	6	studied	study	VERB
ejpam-7050	5	7	the	the	DET
ejpam-7050	5	8	notion	notion	NOUN
ejpam-7050	5	9	of	of	ADP
ejpam-7050	5	10	β	β	ADJ
ejpam-7050	5	11	-	-	ADJ
ejpam-7050	5	12	continuous	continuous	ADJ
ejpam-7050	5	13	functions	function	NOUN
ejpam-7050	5	14	.	.	PUNCT
ejpam-7050	6	1	nasef	nasef	NOUN
ejpam-7050	6	2	and	and	CCONJ
ejpam-7050	6	3	noiri	noiri	ADV
ejpam-7050	7	1	[	[	X
ejpam-7050	7	2	2	2	X
ejpam-7050	7	3	]	]	PUNCT
ejpam-7050	7	4	defined	define	VERB
ejpam-7050	7	5	almost	almost	ADV
ejpam-7050	7	6	β	β	ADJ
ejpam-7050	7	7	-	-	ADJ
ejpam-7050	7	8	continuous	continuous	ADJ
ejpam-7050	7	9	functions	function	NOUN
ejpam-7050	7	10	by	by	ADP
ejpam-7050	7	11	utilizing	utilize	VERB
ejpam-7050	7	12	the	the	DET
ejpam-7050	7	13	notion	notion	NOUN
ejpam-7050	7	14	of	of	ADP
ejpam-7050	7	15	β	β	ADJ
ejpam-7050	7	16	-	-	ADJ
ejpam-7050	7	17	open	open	ADJ
ejpam-7050	7	18	sets	set	NOUN
ejpam-7050	7	19	due	due	ADP
ejpam-7050	7	20	to	to	ADP
ejpam-7050	7	21	abd	abd	PROPN
ejpam-7050	7	22	el	el	PROPN
ejpam-7050	7	23	-	-	PROPN
ejpam-7050	7	24	monsef	monsef	PROPN
ejpam-7050	7	25	et	et	PROPN
ejpam-7050	7	26	al	al	PROPN
ejpam-7050	7	27	.	.	PUNCT
ejpam-7050	8	1	[	[	X
ejpam-7050	8	2	1	1	NUM
ejpam-7050	8	3	]	]	PUNCT
ejpam-7050	8	4	.	.	PUNCT
ejpam-7050	9	1	in	in	ADP
ejpam-7050	9	2	1994	1994	NUM
ejpam-7050	9	3	,	,	PUNCT
ejpam-7050	9	4	popa	popa	NOUN
ejpam-7050	9	5	and	and	CCONJ
ejpam-7050	9	6	noiri	noiri	ADV
ejpam-7050	9	7	[	[	X
ejpam-7050	9	8	3	3	X
ejpam-7050	9	9	]	]	PUNCT
ejpam-7050	9	10	introduced	introduce	VERB
ejpam-7050	9	11	the	the	DET
ejpam-7050	9	12	concept	concept	NOUN
ejpam-7050	9	13	of	of	ADP
ejpam-7050	9	14	weakly	weakly	ADJ
ejpam-7050	9	15	β	β	ADJ
ejpam-7050	9	16	-	-	ADJ
ejpam-7050	9	17	continuous	continuous	ADJ
ejpam-7050	9	18	functions	function	NOUN
ejpam-7050	9	19	and	and	CCONJ
ejpam-7050	9	20	obtained	obtain	VERB
ejpam-7050	9	21	some	some	DET
ejpam-7050	9	22	characterizations	characterization	NOUN
ejpam-7050	9	23	of	of	ADP
ejpam-7050	9	24	such	such	ADJ
ejpam-7050	9	25	functions	function	NOUN
ejpam-7050	9	26	.	.	PUNCT
ejpam-7050	10	1	the	the	DET
ejpam-7050	10	2	class	class	NOUN
ejpam-7050	10	3	of	of	ADP
ejpam-7050	10	4	almost	almost	ADV
ejpam-7050	10	5	β	β	NOUN
ejpam-7050	10	6	-	-	NOUN
ejpam-7050	10	7	continuity	continuity	NOUN
ejpam-7050	10	8	is	be	AUX
ejpam-7050	10	9	a	a	DET
ejpam-7050	10	10	generalization	generalization	NOUN
ejpam-7050	10	11	of	of	ADP
ejpam-7050	10	12	β	β	NOUN
ejpam-7050	10	13	-	-	NOUN
ejpam-7050	10	14	continuity	continuity	NOUN
ejpam-7050	10	15	and	and	CCONJ
ejpam-7050	10	16	the	the	DET
ejpam-7050	10	17	class	class	NOUN
ejpam-7050	10	18	of	of	ADP
ejpam-7050	10	19	weak	weak	ADJ
ejpam-7050	10	20	β	β	NOUN
ejpam-7050	10	21	-	-	NOUN
ejpam-7050	10	22	continuity	continuity	NOUN
ejpam-7050	10	23	is	be	AUX
ejpam-7050	10	24	a	a	DET
ejpam-7050	10	25	generalization	generalization	NOUN
ejpam-7050	10	26	of	of	ADP
ejpam-7050	10	27	almost	almost	ADV
ejpam-7050	10	28	β	β	NOUN
ejpam-7050	10	29	-	-	NOUN
ejpam-7050	10	30	continuity	continuity	NOUN
ejpam-7050	10	31	.	.	PUNCT
ejpam-7050	11	1	in	in	ADP
ejpam-7050	11	2	1999	1999	NUM
ejpam-7050	11	3	,	,	PUNCT
ejpam-7050	11	4	popa	popa	NOUN
ejpam-7050	11	5	and	and	CCONJ
ejpam-7050	11	6	noiri	noiri	ADV
ejpam-7050	11	7	[	[	X
ejpam-7050	11	8	4	4	X
ejpam-7050	11	9	]	]	PUNCT
ejpam-7050	11	10	introduced	introduce	VERB
ejpam-7050	11	11	new	new	ADJ
ejpam-7050	11	12	classes	class	NOUN
ejpam-7050	11	13	of	of	ADP
ejpam-7050	11	14	multifunctions	multifunction	NOUN
ejpam-7050	11	15	defined	define	VERB
ejpam-7050	11	16	from	from	ADP
ejpam-7050	11	17	a	a	DET
ejpam-7050	11	18	topological	topological	ADJ
ejpam-7050	11	19	space	space	NOUN
ejpam-7050	11	20	into	into	ADP
ejpam-7050	11	21	a	a	DET
ejpam-7050	11	22	topological	topological	ADJ
ejpam-7050	11	23	space	space	NOUN
ejpam-7050	11	24	,	,	PUNCT
ejpam-7050	11	25	namely	namely	ADV
ejpam-7050	11	26	upper	upper	ADJ
ejpam-7050	11	27	weakly	weakly	ADJ
ejpam-7050	11	28	β	β	ADJ
ejpam-7050	11	29	-	-	ADJ
ejpam-7050	11	30	continuous	continuous	ADJ
ejpam-7050	11	31	multifunctions	multifunction	NOUN
ejpam-7050	11	32	and	and	CCONJ
ejpam-7050	11	33	lower	low	ADJ
ejpam-7050	11	34	weakly	weakly	ADJ
ejpam-7050	11	35	β	β	ADJ
ejpam-7050	11	36	-	-	ADJ
ejpam-7050	11	37	continuous	continuous	ADJ
ejpam-7050	11	38	multifunctions	multifunction	NOUN
ejpam-7050	11	39	.	.	PUNCT
ejpam-7050	12	1	furthermore	furthermore	ADV
ejpam-7050	12	2	,	,	PUNCT
ejpam-7050	12	3	popa	popa	NOUN
ejpam-7050	12	4	and	and	CCONJ
ejpam-7050	12	5	noiri	noiri	ADV
ejpam-7050	13	1	[	[	X
ejpam-7050	13	2	5	5	NUM
ejpam-7050	13	3	]	]	PUNCT
ejpam-7050	13	4	investigated	investigate	VERB
ejpam-7050	13	5	several	several	ADJ
ejpam-7050	13	6	characterizations	characterization	NOUN
ejpam-7050	13	7	and	and	CCONJ
ejpam-7050	13	8	some	some	DET
ejpam-7050	13	9	properties	property	NOUN
ejpam-7050	13	10	of	of	ADP
ejpam-7050	13	11	upper	upper	ADJ
ejpam-7050	13	12	weakly	weakly	ADJ
ejpam-7050	13	13	β	β	X
ejpam-7050	13	14	-	-	ADJ
ejpam-7050	13	15	continuous	continuous	ADJ
ejpam-7050	13	16	multifunctions	multifunction	NOUN
ejpam-7050	13	17	and	and	CCONJ
ejpam-7050	13	18	lower	low	ADJ
ejpam-7050	13	19	weakly	weakly	ADJ
ejpam-7050	13	20	β	β	ADJ
ejpam-7050	13	21	-	-	ADJ
ejpam-7050	13	22	continuous	continuous	ADJ
ejpam-7050	13	23	multifunctions	multifunction	NOUN
ejpam-7050	13	24	.	.	PUNCT
ejpam-7050	14	1	on	on	ADP
ejpam-7050	14	2	the	the	DET
ejpam-7050	14	3	other	other	ADJ
ejpam-7050	14	4	hand	hand	NOUN
ejpam-7050	14	5	,	,	PUNCT
ejpam-7050	14	6	the	the	DET
ejpam-7050	14	7	present	present	ADJ
ejpam-7050	14	8	author	author	NOUN
ejpam-7050	14	9	introduced	introduce	VERB
ejpam-7050	14	10	and	and	CCONJ
ejpam-7050	14	11	investigated	investigate	VERB
ejpam-7050	14	12	four	four	NUM
ejpam-7050	14	13	classes	class	NOUN
ejpam-7050	14	14	of	of	ADP
ejpam-7050	14	15	multifunctions	multifunction	NOUN
ejpam-7050	14	16	defined	define	VERB
ejpam-7050	14	17	from	from	ADP
ejpam-7050	14	18	an	an	DET
ejpam-7050	14	19	ideal	ideal	ADJ
ejpam-7050	14	20	topological	topological	ADJ
ejpam-7050	14	21	space	space	NOUN
ejpam-7050	14	22	into	into	ADP
ejpam-7050	14	23	an	an	DET
ejpam-7050	14	24	ideal	ideal	ADJ
ejpam-7050	14	25	topological	topological	ADJ
ejpam-7050	14	26	space	space	NOUN
ejpam-7050	14	27	,	,	PUNCT
ejpam-7050	14	28	namely	namely	ADV
ejpam-7050	14	29	upper	upper	ADJ
ejpam-7050	14	30	weakly	weakly	ADJ
ejpam-7050	14	31	⋆-continuous	⋆-continuous	ADJ
ejpam-7050	14	32	multifunctions	multifunction	NOUN
ejpam-7050	15	1	[	[	X
ejpam-7050	15	2	6	6	NUM
ejpam-7050	15	3	]	]	PUNCT
ejpam-7050	15	4	,	,	PUNCT
ejpam-7050	15	5	∗corresponding	∗corresponde	VERB
ejpam-7050	15	6	author	author	NOUN
ejpam-7050	15	7	.	.	PUNCT
ejpam-7050	16	1	doi	doi	NOUN
ejpam-7050	16	2	:	:	PUNCT
ejpam-7050	16	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7050	https://doi.org/10.29020/nybg.ejpam.v18i4.7050	PROPN
ejpam-7050	16	4	email	email	NOUN
ejpam-7050	16	5	addresses	address	NOUN
ejpam-7050	16	6	:	:	PUNCT
ejpam-7050	16	7	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-7050	16	8	(	(	PUNCT
ejpam-7050	16	9	m.	m.	NOUN
ejpam-7050	16	10	thongmoon	thongmoon	PROPN
ejpam-7050	16	11	)	)	PUNCT
ejpam-7050	16	12	,	,	PUNCT
ejpam-7050	16	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7050	16	14	(	(	PUNCT
ejpam-7050	16	15	a.	a.	PROPN
ejpam-7050	16	16	sama	sama	PROPN
ejpam-7050	16	17	-	-	PUNCT
ejpam-7050	16	18	ae	ae	PROPN
ejpam-7050	16	19	)	)	PUNCT
ejpam-7050	16	20	,	,	PUNCT
ejpam-7050	16	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7050	16	22	(	(	PUNCT
ejpam-7050	16	23	c.	c.	PROPN
ejpam-7050	16	24	boonpok	boonpok	PROPN
ejpam-7050	16	25	)	)	PUNCT
ejpam-7050	16	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7050	17	1	1	1	NUM
ejpam-7050	17	2	copyright	copyright	NOUN
ejpam-7050	17	3	:	:	PUNCT
ejpam-7050	17	4	©	©	PROPN
ejpam-7050	17	5	2025	2025	NUM
ejpam-7050	17	6	the	the	DET
ejpam-7050	17	7	author(s	author(s	NOUN
ejpam-7050	17	8	)	)	PUNCT
ejpam-7050	17	9	.	.	PUNCT
ejpam-7050	18	1	(	(	PUNCT
ejpam-7050	18	2	cc	cc	NOUN
ejpam-7050	18	3	by	by	ADP
ejpam-7050	18	4	-	-	PUNCT
ejpam-7050	18	5	nc	nc	PROPN
ejpam-7050	18	6	4.0	4.0	NUM
ejpam-7050	18	7	)	)	PUNCT
ejpam-7050	18	8	m.	m.	NOUN
ejpam-7050	18	9	thongmoon	thongmoon	NOUN
ejpam-7050	18	10	,	,	PUNCT
ejpam-7050	18	11	a.	a.	PROPN
ejpam-7050	18	12	sama	sama	PROPN
ejpam-7050	18	13	-	-	PUNCT
ejpam-7050	18	14	ae	ae	PROPN
ejpam-7050	18	15	,	,	PUNCT
ejpam-7050	18	16	c.	c.	PROPN
ejpam-7050	18	17	boonpok	boonpok	PROPN
ejpam-7050	18	18	/	/	SYM
ejpam-7050	18	19	eur	eur	PROPN
ejpam-7050	18	20	.	.	PUNCT
ejpam-7050	19	1	j.	j.	PROPN
ejpam-7050	19	2	pure	pure	PROPN
ejpam-7050	19	3	appl	appl	PROPN
ejpam-7050	19	4	.	.	PROPN
ejpam-7050	19	5	math	math	PROPN
ejpam-7050	19	6	,	,	PUNCT
ejpam-7050	19	7	18	18	NUM
ejpam-7050	19	8	(	(	PUNCT
ejpam-7050	19	9	4	4	NUM
ejpam-7050	19	10	)	)	PUNCT
ejpam-7050	19	11	(	(	PUNCT
ejpam-7050	19	12	2025	2025	NUM
ejpam-7050	19	13	)	)	PUNCT
ejpam-7050	19	14	,	,	PUNCT
ejpam-7050	19	15	7050	7050	NUM
ejpam-7050	19	16	2	2	NUM
ejpam-7050	19	17	of	of	ADP
ejpam-7050	19	18	14	14	NUM
ejpam-7050	19	19	lower	low	ADJ
ejpam-7050	19	20	weakly	weakly	ADJ
ejpam-7050	19	21	⋆-continuous	⋆-continuous	ADJ
ejpam-7050	19	22	multifunctions	multifunction	NOUN
ejpam-7050	20	1	[	[	X
ejpam-7050	20	2	6	6	NUM
ejpam-7050	20	3	]	]	PUNCT
ejpam-7050	20	4	,	,	PUNCT
ejpam-7050	20	5	upper	upper	ADJ
ejpam-7050	20	6	weakly	weakly	ADJ
ejpam-7050	20	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7050	20	8	multifunctions	multifunction	NOUN
ejpam-7050	21	1	[	[	X
ejpam-7050	21	2	7	7	NUM
ejpam-7050	21	3	]	]	PUNCT
ejpam-7050	21	4	,	,	PUNCT
ejpam-7050	21	5	lower	low	ADJ
ejpam-7050	21	6	weakly	weakly	ADJ
ejpam-7050	21	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7050	21	8	multifunctions	multifunction	NOUN
ejpam-7050	22	1	[	[	X
ejpam-7050	22	2	7	7	NUM
ejpam-7050	22	3	]	]	PUNCT
ejpam-7050	22	4	,	,	PUNCT
ejpam-7050	22	5	upper	upper	ADJ
ejpam-7050	22	6	weakly	weakly	ADJ
ejpam-7050	22	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7050	22	8	multifunctions	multifunction	NOUN
ejpam-7050	23	1	[	[	X
ejpam-7050	23	2	8	8	NUM
ejpam-7050	23	3	]	]	PUNCT
ejpam-7050	23	4	,	,	PUNCT
ejpam-7050	23	5	lower	low	ADJ
ejpam-7050	23	6	weakly	weakly	ADJ
ejpam-7050	23	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7050	23	8	multifunctions	multifunction	NOUN
ejpam-7050	24	1	[	[	X
ejpam-7050	24	2	8	8	NUM
ejpam-7050	24	3	]	]	PUNCT
ejpam-7050	24	4	,	,	PUNCT
ejpam-7050	24	5	weakly	weakly	ADJ
ejpam-7050	24	6	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-7050	24	7	multifunctions	multifunction	NOUN
ejpam-7050	25	1	[	[	X
ejpam-7050	25	2	9	9	NUM
ejpam-7050	25	3	]	]	PUNCT
ejpam-7050	25	4	and	and	CCONJ
ejpam-7050	25	5	weakly	weakly	ADJ
ejpam-7050	25	6	pı	pı	ADJ
ejpam-7050	25	7	-	-	ADJ
ejpam-7050	25	8	continuous	continuous	ADJ
ejpam-7050	25	9	multifunctions	multifunction	NOUN
ejpam-7050	26	1	[	[	X
ejpam-7050	26	2	10	10	NUM
ejpam-7050	26	3	]	]	PUNCT
ejpam-7050	26	4	.	.	PUNCT
ejpam-7050	27	1	pue	pue	NOUN
ejpam-7050	27	2	-	-	PUNCT
ejpam-7050	27	3	on	on	NOUN
ejpam-7050	27	4	et	et	PROPN
ejpam-7050	27	5	al	al	PROPN
ejpam-7050	27	6	.	.	PUNCT
ejpam-7050	28	1	[	[	X
ejpam-7050	28	2	11	11	NUM
ejpam-7050	28	3	]	]	PUNCT
ejpam-7050	28	4	introduced	introduce	VERB
ejpam-7050	28	5	and	and	CCONJ
ejpam-7050	28	6	studied	study	VERB
ejpam-7050	28	7	two	two	NUM
ejpam-7050	28	8	classes	class	NOUN
ejpam-7050	28	9	of	of	ADP
ejpam-7050	28	10	continuous	continuous	ADJ
ejpam-7050	28	11	multifunctions	multifunction	NOUN
ejpam-7050	28	12	between	between	ADP
ejpam-7050	28	13	bitopological	bitopological	ADJ
ejpam-7050	28	14	spaces	space	NOUN
ejpam-7050	28	15	,	,	PUNCT
ejpam-7050	28	16	namely	namely	ADV
ejpam-7050	28	17	upper	upper	ADJ
ejpam-7050	28	18	(	(	PUNCT
ejpam-7050	28	19	τ1	τ1	NOUN
ejpam-7050	28	20	,	,	PUNCT
ejpam-7050	28	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7050	28	22	multifunctions	multifunction	NOUN
ejpam-7050	28	23	and	and	CCONJ
ejpam-7050	28	24	lower	low	ADJ
ejpam-7050	28	25	(	(	PUNCT
ejpam-7050	28	26	τ1	τ1	NOUN
ejpam-7050	28	27	,	,	PUNCT
ejpam-7050	28	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7050	28	29	multifunctions	multifunction	NOUN
ejpam-7050	28	30	.	.	PUNCT
ejpam-7050	29	1	thongmoon	thongmoon	NOUN
ejpam-7050	29	2	et	et	PROPN
ejpam-7050	29	3	al	al	PROPN
ejpam-7050	29	4	.	.	PUNCT
ejpam-7050	30	1	[	[	X
ejpam-7050	30	2	12	12	NUM
ejpam-7050	30	3	]	]	PUNCT
ejpam-7050	30	4	introduced	introduce	VERB
ejpam-7050	30	5	and	and	CCONJ
ejpam-7050	30	6	studied	study	VERB
ejpam-7050	30	7	the	the	DET
ejpam-7050	30	8	notions	notion	NOUN
ejpam-7050	30	9	of	of	ADP
ejpam-7050	30	10	upper	upper	ADJ
ejpam-7050	30	11	weakly	weakly	ADJ
ejpam-7050	30	12	(	(	PUNCT
ejpam-7050	30	13	τ1	τ1	NOUN
ejpam-7050	30	14	,	,	PUNCT
ejpam-7050	30	15	τ2)continuous	τ2)continuous	ADJ
ejpam-7050	30	16	multifunctions	multifunction	NOUN
ejpam-7050	30	17	and	and	CCONJ
ejpam-7050	30	18	lower	low	ADJ
ejpam-7050	30	19	weakly	weakly	ADJ
ejpam-7050	30	20	(	(	PUNCT
ejpam-7050	30	21	τ1	τ1	NOUN
ejpam-7050	30	22	,	,	PUNCT
ejpam-7050	30	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7050	30	24	multifunctions	multifunction	NOUN
ejpam-7050	30	25	.	.	PUNCT
ejpam-7050	31	1	in	in	ADP
ejpam-7050	31	2	this	this	DET
ejpam-7050	31	3	paper	paper	NOUN
ejpam-7050	31	4	,	,	PUNCT
ejpam-7050	31	5	we	we	PRON
ejpam-7050	31	6	introduce	introduce	VERB
ejpam-7050	31	7	the	the	DET
ejpam-7050	31	8	concepts	concept	NOUN
ejpam-7050	31	9	of	of	ADP
ejpam-7050	31	10	continuous	continuous	ADJ
ejpam-7050	31	11	multifunctions	multifunction	NOUN
ejpam-7050	31	12	between	between	ADP
ejpam-7050	31	13	an	an	DET
ejpam-7050	31	14	ideal	ideal	ADJ
ejpam-7050	31	15	topological	topological	ADJ
ejpam-7050	31	16	space	space	NOUN
ejpam-7050	31	17	and	and	CCONJ
ejpam-7050	31	18	a	a	DET
ejpam-7050	31	19	bitopological	bitopological	ADJ
ejpam-7050	31	20	space	space	NOUN
ejpam-7050	31	21	,	,	PUNCT
ejpam-7050	31	22	called	call	VERB
ejpam-7050	31	23	upper	upper	ADJ
ejpam-7050	31	24	weakly	weakly	ADJ
ejpam-7050	31	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	31	26	,	,	PUNCT
ejpam-7050	31	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	31	28	multifunctions	multifunction	NOUN
ejpam-7050	31	29	and	and	CCONJ
ejpam-7050	31	30	lower	low	ADJ
ejpam-7050	31	31	weakly	weakly	ADJ
ejpam-7050	31	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	31	33	,	,	PUNCT
ejpam-7050	31	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	31	35	multifunctions	multifunction	NOUN
ejpam-7050	31	36	.	.	PUNCT
ejpam-7050	32	1	we	we	PRON
ejpam-7050	32	2	also	also	ADV
ejpam-7050	32	3	investigate	investigate	VERB
ejpam-7050	32	4	several	several	ADJ
ejpam-7050	32	5	characterizations	characterization	NOUN
ejpam-7050	32	6	of	of	ADP
ejpam-7050	32	7	upper	upper	ADJ
ejpam-7050	32	8	weakly	weakly	ADJ
ejpam-7050	32	9	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	32	10	,	,	PUNCT
ejpam-7050	32	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	32	12	multifunctions	multifunction	NOUN
ejpam-7050	32	13	and	and	CCONJ
ejpam-7050	32	14	lower	low	ADJ
ejpam-7050	32	15	weakly	weakly	ADJ
ejpam-7050	32	16	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	32	17	,	,	PUNCT
ejpam-7050	32	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	32	19	multifunctions	multifunction	NOUN
ejpam-7050	32	20	.	.	PUNCT
ejpam-7050	33	1	2	2	X
ejpam-7050	33	2	.	.	X
ejpam-7050	33	3	preliminaries	preliminary	NOUN
ejpam-7050	33	4	throughout	throughout	ADP
ejpam-7050	33	5	the	the	DET
ejpam-7050	33	6	present	present	ADJ
ejpam-7050	33	7	paper	paper	NOUN
ejpam-7050	33	8	,	,	PUNCT
ejpam-7050	33	9	spaces	space	NOUN
ejpam-7050	33	10	(	(	PUNCT
ejpam-7050	33	11	x	x	NOUN
ejpam-7050	33	12	,	,	PUNCT
ejpam-7050	33	13	τ1	τ1	NOUN
ejpam-7050	33	14	,	,	PUNCT
ejpam-7050	33	15	τ2	τ2	NOUN
ejpam-7050	33	16	)	)	PUNCT
ejpam-7050	33	17	and	and	CCONJ
ejpam-7050	33	18	(	(	PUNCT
ejpam-7050	33	19	y	y	PROPN
ejpam-7050	33	20	,	,	PUNCT
ejpam-7050	33	21	σ1	σ1	PROPN
ejpam-7050	33	22	,	,	PUNCT
ejpam-7050	33	23	σ2	σ2	NOUN
ejpam-7050	33	24	)	)	PUNCT
ejpam-7050	33	25	(	(	PUNCT
ejpam-7050	33	26	or	or	CCONJ
ejpam-7050	33	27	simply	simply	ADV
ejpam-7050	33	28	x	x	X
ejpam-7050	33	29	and	and	CCONJ
ejpam-7050	33	30	y	y	PROPN
ejpam-7050	33	31	)	)	PUNCT
ejpam-7050	33	32	always	always	ADV
ejpam-7050	33	33	mean	mean	VERB
ejpam-7050	33	34	bitopological	bitopological	ADJ
ejpam-7050	33	35	spaces	space	NOUN
ejpam-7050	33	36	on	on	ADP
ejpam-7050	33	37	which	which	PRON
ejpam-7050	33	38	no	no	DET
ejpam-7050	33	39	separation	separation	NOUN
ejpam-7050	33	40	axioms	axiom	NOUN
ejpam-7050	33	41	are	be	AUX
ejpam-7050	33	42	assumed	assume	VERB
ejpam-7050	33	43	unless	unless	SCONJ
ejpam-7050	33	44	explicitly	explicitly	ADV
ejpam-7050	33	45	stated	state	VERB
ejpam-7050	33	46	.	.	PUNCT
ejpam-7050	34	1	let	let	VERB
ejpam-7050	34	2	a	a	DET
ejpam-7050	34	3	be	be	AUX
ejpam-7050	34	4	a	a	DET
ejpam-7050	34	5	subset	subset	NOUN
ejpam-7050	34	6	of	of	ADP
ejpam-7050	34	7	a	a	DET
ejpam-7050	34	8	bitopological	bitopological	ADJ
ejpam-7050	34	9	space	space	NOUN
ejpam-7050	34	10	(	(	PUNCT
ejpam-7050	34	11	x	x	NOUN
ejpam-7050	34	12	,	,	PUNCT
ejpam-7050	34	13	τ1	τ1	NOUN
ejpam-7050	34	14	,	,	PUNCT
ejpam-7050	34	15	τ2	τ2	NOUN
ejpam-7050	34	16	)	)	PUNCT
ejpam-7050	34	17	.	.	PUNCT
ejpam-7050	35	1	the	the	DET
ejpam-7050	35	2	closure	closure	NOUN
ejpam-7050	35	3	of	of	ADP
ejpam-7050	35	4	a	a	PRON
ejpam-7050	35	5	and	and	CCONJ
ejpam-7050	35	6	the	the	DET
ejpam-7050	35	7	interior	interior	NOUN
ejpam-7050	35	8	of	of	ADP
ejpam-7050	35	9	a	a	PRON
ejpam-7050	35	10	with	with	ADP
ejpam-7050	35	11	respect	respect	NOUN
ejpam-7050	35	12	to	to	ADP
ejpam-7050	35	13	τi	τi	PROPN
ejpam-7050	35	14	are	be	AUX
ejpam-7050	35	15	denoted	denote	VERB
ejpam-7050	35	16	by	by	ADP
ejpam-7050	35	17	τi	τi	NOUN
ejpam-7050	35	18	-	-	PUNCT
ejpam-7050	35	19	cl(a	cl(a	NUM
ejpam-7050	35	20	)	)	PUNCT
ejpam-7050	35	21	and	and	CCONJ
ejpam-7050	35	22	τi	τi	NOUN
ejpam-7050	35	23	-	-	PUNCT
ejpam-7050	35	24	int(a	int(a	NOUN
ejpam-7050	35	25	)	)	PUNCT
ejpam-7050	35	26	,	,	PUNCT
ejpam-7050	35	27	respectively	respectively	ADV
ejpam-7050	35	28	,	,	PUNCT
ejpam-7050	35	29	for	for	ADP
ejpam-7050	35	30	i	i	PROPN
ejpam-7050	35	31	=	=	SYM
ejpam-7050	35	32	1	1	NUM
ejpam-7050	35	33	,	,	PUNCT
ejpam-7050	35	34	2	2	NUM
ejpam-7050	35	35	.	.	X
ejpam-7050	35	36	a	a	DET
ejpam-7050	35	37	subset	subset	NOUN
ejpam-7050	35	38	a	a	PRON
ejpam-7050	35	39	of	of	ADP
ejpam-7050	35	40	a	a	DET
ejpam-7050	35	41	bitopological	bitopological	ADJ
ejpam-7050	35	42	space	space	NOUN
ejpam-7050	35	43	(	(	PUNCT
ejpam-7050	35	44	x	x	NOUN
ejpam-7050	35	45	,	,	PUNCT
ejpam-7050	35	46	τ1	τ1	NOUN
ejpam-7050	35	47	,	,	PUNCT
ejpam-7050	35	48	τ2	τ2	NOUN
ejpam-7050	35	49	)	)	PUNCT
ejpam-7050	35	50	is	be	AUX
ejpam-7050	35	51	called	call	VERB
ejpam-7050	35	52	τ1τ2	τ1τ2	VERB
ejpam-7050	35	53	-	-	ADJ
ejpam-7050	35	54	closed	closed	ADJ
ejpam-7050	35	55	[	[	X
ejpam-7050	35	56	13	13	NUM
ejpam-7050	35	57	]	]	PUNCT
ejpam-7050	35	58	if	if	SCONJ
ejpam-7050	35	59	a	a	DET
ejpam-7050	35	60	=	=	NOUN
ejpam-7050	35	61	τ1	τ1	NOUN
ejpam-7050	35	62	-	-	PUNCT
ejpam-7050	35	63	cl(τ2	cl(τ2	NOUN
ejpam-7050	35	64	-	-	PUNCT
ejpam-7050	35	65	cl(a	cl(a	NUM
ejpam-7050	35	66	)	)	PUNCT
ejpam-7050	35	67	)	)	PUNCT
ejpam-7050	35	68	.	.	PUNCT
ejpam-7050	36	1	the	the	DET
ejpam-7050	36	2	complement	complement	NOUN
ejpam-7050	36	3	of	of	ADP
ejpam-7050	36	4	a	a	DET
ejpam-7050	36	5	τ1τ2	τ1τ2	ADJ
ejpam-7050	36	6	-	-	ADJ
ejpam-7050	36	7	closed	closed	ADJ
ejpam-7050	36	8	set	set	NOUN
ejpam-7050	36	9	is	be	AUX
ejpam-7050	36	10	called	call	VERB
ejpam-7050	36	11	τ1τ2	τ1τ2	NOUN
ejpam-7050	36	12	-	-	ADJ
ejpam-7050	36	13	open	open	ADJ
ejpam-7050	36	14	.	.	PUNCT
ejpam-7050	37	1	the	the	DET
ejpam-7050	37	2	intersection	intersection	NOUN
ejpam-7050	37	3	of	of	ADP
ejpam-7050	37	4	all	all	DET
ejpam-7050	37	5	τ1τ2	τ1τ2	ADJ
ejpam-7050	37	6	-	-	ADJ
ejpam-7050	37	7	closed	closed	ADJ
ejpam-7050	37	8	sets	set	NOUN
ejpam-7050	37	9	of	of	ADP
ejpam-7050	37	10	x	x	PUNCT
ejpam-7050	37	11	containing	contain	VERB
ejpam-7050	37	12	a	a	PRON
ejpam-7050	37	13	is	be	AUX
ejpam-7050	37	14	called	call	VERB
ejpam-7050	37	15	the	the	DET
ejpam-7050	37	16	τ1τ2	τ1τ2	NOUN
ejpam-7050	37	17	-	-	NOUN
ejpam-7050	37	18	closure	closure	NOUN
ejpam-7050	37	19	[	[	X
ejpam-7050	37	20	13	13	NUM
ejpam-7050	37	21	]	]	PUNCT
ejpam-7050	37	22	of	of	ADP
ejpam-7050	37	23	a	a	PRON
ejpam-7050	37	24	and	and	CCONJ
ejpam-7050	37	25	is	be	AUX
ejpam-7050	37	26	denoted	denote	VERB
ejpam-7050	37	27	by	by	ADP
ejpam-7050	37	28	τ1τ2	τ1τ2	NOUN
ejpam-7050	37	29	-	-	NUM
ejpam-7050	37	30	cl(a	cl(a	NUM
ejpam-7050	37	31	)	)	PUNCT
ejpam-7050	37	32	.	.	PUNCT
ejpam-7050	38	1	the	the	DET
ejpam-7050	38	2	union	union	NOUN
ejpam-7050	38	3	of	of	ADP
ejpam-7050	38	4	all	all	DET
ejpam-7050	38	5	τ1τ2	τ1τ2	ADJ
ejpam-7050	38	6	-	-	ADJ
ejpam-7050	38	7	open	open	ADJ
ejpam-7050	38	8	sets	set	NOUN
ejpam-7050	38	9	of	of	ADP
ejpam-7050	38	10	x	x	PUNCT
ejpam-7050	38	11	contained	contain	VERB
ejpam-7050	38	12	in	in	ADP
ejpam-7050	38	13	a	a	PRON
ejpam-7050	38	14	is	be	AUX
ejpam-7050	38	15	called	call	VERB
ejpam-7050	38	16	the	the	DET
ejpam-7050	38	17	τ1τ2	τ1τ2	NOUN
ejpam-7050	38	18	-	-	ADJ
ejpam-7050	38	19	interior	interior	ADJ
ejpam-7050	38	20	[	[	X
ejpam-7050	38	21	13	13	NUM
ejpam-7050	38	22	]	]	PUNCT
ejpam-7050	38	23	of	of	ADP
ejpam-7050	38	24	a	a	PRON
ejpam-7050	38	25	and	and	CCONJ
ejpam-7050	38	26	is	be	AUX
ejpam-7050	38	27	denoted	denote	VERB
ejpam-7050	38	28	by	by	ADP
ejpam-7050	38	29	τ1τ2	τ1τ2	NOUN
ejpam-7050	38	30	-	-	ADJ
ejpam-7050	38	31	int(a	int(a	NOUN
ejpam-7050	38	32	)	)	PUNCT
ejpam-7050	38	33	.	.	PUNCT
ejpam-7050	39	1	lemma	lemma	PROPN
ejpam-7050	39	2	1	1	NUM
ejpam-7050	39	3	.	.	PUNCT
ejpam-7050	40	1	[	[	X
ejpam-7050	40	2	13	13	NUM
ejpam-7050	40	3	]	]	PUNCT
ejpam-7050	40	4	let	let	VERB
ejpam-7050	40	5	a	a	PRON
ejpam-7050	40	6	and	and	CCONJ
ejpam-7050	40	7	b	b	NOUN
ejpam-7050	40	8	be	be	AUX
ejpam-7050	40	9	subsets	subset	NOUN
ejpam-7050	40	10	of	of	ADP
ejpam-7050	40	11	a	a	DET
ejpam-7050	40	12	bitopological	bitopological	ADJ
ejpam-7050	40	13	space	space	NOUN
ejpam-7050	40	14	(	(	PUNCT
ejpam-7050	40	15	x	x	NOUN
ejpam-7050	40	16	,	,	PUNCT
ejpam-7050	40	17	τ1	τ1	NOUN
ejpam-7050	40	18	,	,	PUNCT
ejpam-7050	40	19	τ2	τ2	NOUN
ejpam-7050	40	20	)	)	PUNCT
ejpam-7050	40	21	.	.	PUNCT
ejpam-7050	41	1	for	for	ADP
ejpam-7050	41	2	the	the	DET
ejpam-7050	41	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-7050	41	4	,	,	PUNCT
ejpam-7050	41	5	the	the	DET
ejpam-7050	41	6	following	follow	VERB
ejpam-7050	41	7	properties	property	NOUN
ejpam-7050	41	8	hold	hold	VERB
ejpam-7050	41	9	:	:	PUNCT
ejpam-7050	41	10	(	(	PUNCT
ejpam-7050	41	11	1	1	X
ejpam-7050	41	12	)	)	PUNCT
ejpam-7050	41	13	a	a	DET
ejpam-7050	41	14	⊆	⊆	NUM
ejpam-7050	41	15	τ1τ2	τ1τ2	NOUN
ejpam-7050	41	16	-	-	NUM
ejpam-7050	41	17	cl(a	cl(a	NUM
ejpam-7050	41	18	)	)	PUNCT
ejpam-7050	41	19	and	and	CCONJ
ejpam-7050	41	20	τ1τ2	τ1τ2	NOUN
ejpam-7050	41	21	-	-	ADJ
ejpam-7050	41	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7050	41	23	-	-	PUNCT
ejpam-7050	41	24	cl(a	cl(a	NUM
ejpam-7050	41	25	)	)	PUNCT
ejpam-7050	41	26	)	)	PUNCT
ejpam-7050	42	1	=	=	PUNCT
ejpam-7050	42	2	τ1τ2	τ1τ2	NOUN
ejpam-7050	42	3	-	-	NUM
ejpam-7050	42	4	cl(a	cl(a	NUM
ejpam-7050	42	5	)	)	PUNCT
ejpam-7050	42	6	.	.	PUNCT
ejpam-7050	43	1	(	(	PUNCT
ejpam-7050	43	2	2	2	X
ejpam-7050	43	3	)	)	PUNCT
ejpam-7050	43	4	if	if	SCONJ
ejpam-7050	43	5	a	a	DET
ejpam-7050	43	6	⊆	⊆	NUM
ejpam-7050	43	7	b	b	NOUN
ejpam-7050	43	8	,	,	PUNCT
ejpam-7050	43	9	then	then	ADV
ejpam-7050	43	10	τ1τ2	τ1τ2	NOUN
ejpam-7050	43	11	-	-	NUM
ejpam-7050	43	12	cl(a	cl(a	NUM
ejpam-7050	43	13	)	)	PUNCT
ejpam-7050	43	14	⊆	⊆	NUM
ejpam-7050	43	15	τ1τ2	τ1τ2	NOUN
ejpam-7050	43	16	-	-	NOUN
ejpam-7050	43	17	cl(b	cl(b	NOUN
ejpam-7050	43	18	)	)	PUNCT
ejpam-7050	43	19	.	.	PUNCT
ejpam-7050	44	1	(	(	PUNCT
ejpam-7050	44	2	3	3	X
ejpam-7050	44	3	)	)	PUNCT
ejpam-7050	44	4	τ1τ2	τ1τ2	NOUN
ejpam-7050	44	5	-	-	NUM
ejpam-7050	44	6	cl(a	cl(a	NUM
ejpam-7050	44	7	)	)	PUNCT
ejpam-7050	44	8	is	be	AUX
ejpam-7050	44	9	τ1τ2	τ1τ2	NOUN
ejpam-7050	44	10	-	-	ADJ
ejpam-7050	44	11	closed	closed	ADJ
ejpam-7050	44	12	.	.	PUNCT
ejpam-7050	45	1	(	(	PUNCT
ejpam-7050	45	2	4	4	X
ejpam-7050	45	3	)	)	PUNCT
ejpam-7050	45	4	a	a	PRON
ejpam-7050	45	5	is	be	AUX
ejpam-7050	45	6	τ1τ2	τ1τ2	NOUN
ejpam-7050	45	7	-	-	ADJ
ejpam-7050	45	8	closed	closed	ADJ
ejpam-7050	45	9	if	if	SCONJ
ejpam-7050	45	10	and	and	CCONJ
ejpam-7050	45	11	only	only	ADV
ejpam-7050	45	12	if	if	SCONJ
ejpam-7050	45	13	a	a	DET
ejpam-7050	45	14	=	=	PUNCT
ejpam-7050	45	15	τ1τ2	τ1τ2	NOUN
ejpam-7050	45	16	-	-	NUM
ejpam-7050	45	17	cl(a	cl(a	NUM
ejpam-7050	45	18	)	)	PUNCT
ejpam-7050	45	19	.	.	PUNCT
ejpam-7050	46	1	(	(	PUNCT
ejpam-7050	46	2	5	5	X
ejpam-7050	46	3	)	)	PUNCT
ejpam-7050	46	4	τ1τ2	τ1τ2	NOUN
ejpam-7050	46	5	-	-	NOUN
ejpam-7050	46	6	cl(x	cl(x	X
ejpam-7050	46	7	−a	−a	NOUN
ejpam-7050	46	8	)	)	PUNCT
ejpam-7050	47	1	=	=	PUNCT
ejpam-7050	47	2	x	x	X
ejpam-7050	48	1	−	−	ADP
ejpam-7050	48	2	τ1τ2	τ1τ2	NOUN
ejpam-7050	48	3	-	-	PUNCT
ejpam-7050	48	4	int(a	int(a	NOUN
ejpam-7050	48	5	)	)	PUNCT
ejpam-7050	48	6	.	.	PUNCT
ejpam-7050	49	1	a	a	DET
ejpam-7050	49	2	subseta	subseta	NOUN
ejpam-7050	49	3	of	of	ADP
ejpam-7050	49	4	a	a	DET
ejpam-7050	49	5	bitopological	bitopological	ADJ
ejpam-7050	49	6	space	space	NOUN
ejpam-7050	49	7	(	(	PUNCT
ejpam-7050	49	8	x	x	NOUN
ejpam-7050	49	9	,	,	PUNCT
ejpam-7050	49	10	τ1	τ1	NOUN
ejpam-7050	49	11	,	,	PUNCT
ejpam-7050	49	12	τ2	τ2	NOUN
ejpam-7050	49	13	)	)	PUNCT
ejpam-7050	49	14	is	be	AUX
ejpam-7050	49	15	called	call	VERB
ejpam-7050	49	16	(	(	PUNCT
ejpam-7050	49	17	τ1	τ1	NOUN
ejpam-7050	49	18	,	,	PUNCT
ejpam-7050	49	19	τ2)r	τ2)r	NOUN
ejpam-7050	49	20	-	-	PUNCT
ejpam-7050	49	21	open	open	NOUN
ejpam-7050	50	1	[	[	X
ejpam-7050	50	2	14	14	NUM
ejpam-7050	50	3	]	]	X
ejpam-7050	50	4	(	(	PUNCT
ejpam-7050	50	5	resp	resp	NOUN
ejpam-7050	50	6	.	.	PUNCT
ejpam-7050	51	1	(	(	PUNCT
ejpam-7050	51	2	τ1	τ1	NOUN
ejpam-7050	51	3	,	,	PUNCT
ejpam-7050	51	4	τ2)sopen	τ2)sopen	VERB
ejpam-7050	51	5	[	[	X
ejpam-7050	51	6	15	15	NUM
ejpam-7050	51	7	]	]	PUNCT
ejpam-7050	51	8	,	,	PUNCT
ejpam-7050	51	9	(	(	PUNCT
ejpam-7050	51	10	τ1	τ1	NOUN
ejpam-7050	51	11	,	,	PUNCT
ejpam-7050	51	12	τ2)p	τ2)p	NOUN
ejpam-7050	51	13	-	-	ADJ
ejpam-7050	51	14	open	open	ADJ
ejpam-7050	51	15	[	[	X
ejpam-7050	51	16	15	15	NUM
ejpam-7050	51	17	]	]	PUNCT
ejpam-7050	51	18	,	,	PUNCT
ejpam-7050	51	19	(	(	PUNCT
ejpam-7050	51	20	τ1	τ1	NOUN
ejpam-7050	51	21	,	,	PUNCT
ejpam-7050	51	22	τ2)β	τ2)β	ADJ
ejpam-7050	51	23	-	-	PUNCT
ejpam-7050	51	24	open	open	NOUN
ejpam-7050	51	25	[	[	X
ejpam-7050	51	26	15	15	NUM
ejpam-7050	51	27	]	]	PUNCT
ejpam-7050	51	28	)	)	PUNCT
ejpam-7050	51	29	if	if	SCONJ
ejpam-7050	51	30	a	a	DET
ejpam-7050	51	31	=	=	PUNCT
ejpam-7050	51	32	τ1τ2	τ1τ2	NOUN
ejpam-7050	51	33	-	-	NOUN
ejpam-7050	51	34	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7050	51	35	-	-	PUNCT
ejpam-7050	51	36	cl(a	cl(a	NUM
ejpam-7050	51	37	)	)	PUNCT
ejpam-7050	51	38	)	)	PUNCT
ejpam-7050	51	39	(	(	PUNCT
ejpam-7050	51	40	resp	resp	NOUN
ejpam-7050	51	41	.	.	PUNCT
ejpam-7050	52	1	a	a	DET
ejpam-7050	52	2	⊆	⊆	NUM
ejpam-7050	52	3	τ1τ2	τ1τ2	NOUN
ejpam-7050	52	4	-	-	ADJ
ejpam-7050	52	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7050	52	6	-	-	PUNCT
ejpam-7050	52	7	int(a	int(a	NOUN
ejpam-7050	52	8	)	)	PUNCT
ejpam-7050	52	9	)	)	PUNCT
ejpam-7050	52	10	,	,	PUNCT
ejpam-7050	52	11	a	a	DET
ejpam-7050	52	12	⊆	⊆	NUM
ejpam-7050	52	13	τ1τ2	τ1τ2	NOUN
ejpam-7050	52	14	-	-	NOUN
ejpam-7050	52	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7050	52	16	-	-	PUNCT
ejpam-7050	52	17	cl(a	cl(a	NUM
ejpam-7050	52	18	)	)	PUNCT
ejpam-7050	52	19	)	)	PUNCT
ejpam-7050	52	20	,	,	PUNCT
ejpam-7050	52	21	a	a	DET
ejpam-7050	52	22	⊆	⊆	NUM
ejpam-7050	52	23	τ1τ2	τ1τ2	NOUN
ejpam-7050	52	24	-	-	PUNCT
ejpam-7050	52	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7050	52	26	-	-	PUNCT
ejpam-7050	52	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7050	52	28	-	-	PUNCT
ejpam-7050	52	29	cl(a	cl(a	NUM
ejpam-7050	52	30	)	)	PUNCT
ejpam-7050	52	31	)	)	PUNCT
ejpam-7050	52	32	)	)	PUNCT
ejpam-7050	52	33	)	)	PUNCT
ejpam-7050	52	34	.	.	PUNCT
ejpam-7050	53	1	the	the	DET
ejpam-7050	53	2	complement	complement	NOUN
ejpam-7050	53	3	of	of	ADP
ejpam-7050	53	4	a	a	DET
ejpam-7050	53	5	(	(	PUNCT
ejpam-7050	53	6	τ1	τ1	NOUN
ejpam-7050	53	7	,	,	PUNCT
ejpam-7050	53	8	τ2)r	τ2)r	NOUN
ejpam-7050	53	9	-	-	PUNCT
ejpam-7050	53	10	open	open	ADJ
ejpam-7050	53	11	(	(	PUNCT
ejpam-7050	53	12	resp	resp	NOUN
ejpam-7050	53	13	.	.	PUNCT
ejpam-7050	54	1	(	(	PUNCT
ejpam-7050	54	2	τ1	τ1	NOUN
ejpam-7050	54	3	,	,	PUNCT
ejpam-7050	54	4	τ2)s	τ2)s	NOUN
ejpam-7050	54	5	-	-	PUNCT
ejpam-7050	54	6	open	open	ADJ
ejpam-7050	54	7	,	,	PUNCT
ejpam-7050	54	8	(	(	PUNCT
ejpam-7050	54	9	τ1	τ1	NOUN
ejpam-7050	54	10	,	,	PUNCT
ejpam-7050	54	11	τ2)p	τ2)p	NOUN
ejpam-7050	54	12	-	-	ADJ
ejpam-7050	54	13	open	open	ADJ
ejpam-7050	54	14	,	,	PUNCT
ejpam-7050	54	15	(	(	PUNCT
ejpam-7050	54	16	τ1	τ1	NOUN
ejpam-7050	54	17	,	,	PUNCT
ejpam-7050	54	18	τ2)β	τ2)β	ADJ
ejpam-7050	54	19	-	-	PUNCT
ejpam-7050	54	20	open	open	ADJ
ejpam-7050	54	21	)	)	PUNCT
ejpam-7050	54	22	set	set	NOUN
ejpam-7050	54	23	is	be	AUX
ejpam-7050	54	24	called	call	VERB
ejpam-7050	54	25	(	(	PUNCT
ejpam-7050	54	26	τ1	τ1	NOUN
ejpam-7050	54	27	,	,	PUNCT
ejpam-7050	54	28	τ2)r	τ2)r	NOUN
ejpam-7050	54	29	-	-	PUNCT
ejpam-7050	54	30	closed	closed	ADJ
ejpam-7050	54	31	(	(	PUNCT
ejpam-7050	54	32	resp	resp	NOUN
ejpam-7050	54	33	.	.	PUNCT
ejpam-7050	55	1	(	(	PUNCT
ejpam-7050	55	2	τ1	τ1	NOUN
ejpam-7050	55	3	,	,	PUNCT
ejpam-7050	55	4	τ2)s	τ2)s	NOUN
ejpam-7050	55	5	-	-	PUNCT
ejpam-7050	55	6	closed	closed	ADJ
ejpam-7050	55	7	,	,	PUNCT
ejpam-7050	55	8	(	(	PUNCT
ejpam-7050	55	9	τ1	τ1	NOUN
ejpam-7050	55	10	,	,	PUNCT
ejpam-7050	55	11	τ2)p	τ2)p	NOUN
ejpam-7050	55	12	-	-	PUNCT
ejpam-7050	55	13	closed	closed	ADJ
ejpam-7050	55	14	,	,	PUNCT
ejpam-7050	55	15	(	(	PUNCT
ejpam-7050	55	16	τ1	τ1	NOUN
ejpam-7050	55	17	,	,	PUNCT
ejpam-7050	55	18	τ2)β	τ2)β	ADJ
ejpam-7050	55	19	-	-	PUNCT
ejpam-7050	55	20	closed	closed	ADJ
ejpam-7050	55	21	)	)	PUNCT
ejpam-7050	55	22	.	.	PUNCT
ejpam-7050	56	1	the	the	DET
ejpam-7050	56	2	intersection	intersection	NOUN
ejpam-7050	56	3	of	of	ADP
ejpam-7050	56	4	all	all	DET
ejpam-7050	56	5	(	(	PUNCT
ejpam-7050	56	6	τ1	τ1	NOUN
ejpam-7050	56	7	,	,	PUNCT
ejpam-7050	56	8	τ2)s	τ2)s	NOUN
ejpam-7050	56	9	-	-	PUNCT
ejpam-7050	56	10	closed	close	VERB
ejpam-7050	56	11	sets	set	NOUN
ejpam-7050	56	12	of	of	ADP
ejpam-7050	56	13	x	x	PUNCT
ejpam-7050	56	14	containing	contain	VERB
ejpam-7050	56	15	a	a	PRON
ejpam-7050	56	16	is	be	AUX
ejpam-7050	56	17	called	call	VERB
ejpam-7050	56	18	the	the	DET
ejpam-7050	56	19	(	(	PUNCT
ejpam-7050	56	20	τ1	τ1	NOUN
ejpam-7050	56	21	,	,	PUNCT
ejpam-7050	56	22	τ2)s	τ2)s	NOUN
ejpam-7050	56	23	-	-	PUNCT
ejpam-7050	56	24	closure	closure	NOUN
ejpam-7050	56	25	[	[	X
ejpam-7050	56	26	15	15	NUM
ejpam-7050	56	27	]	]	PUNCT
ejpam-7050	56	28	of	of	ADP
ejpam-7050	56	29	a	a	PRON
ejpam-7050	56	30	and	and	CCONJ
ejpam-7050	56	31	is	be	AUX
ejpam-7050	56	32	denoted	denote	VERB
ejpam-7050	56	33	by	by	ADP
ejpam-7050	56	34	(	(	PUNCT
ejpam-7050	56	35	τ1	τ1	NOUN
ejpam-7050	56	36	,	,	PUNCT
ejpam-7050	56	37	τ2)-scl(a	τ2)-scl(a	PROPN
ejpam-7050	56	38	)	)	PUNCT
ejpam-7050	56	39	.	.	PUNCT
ejpam-7050	57	1	the	the	DET
ejpam-7050	57	2	union	union	NOUN
ejpam-7050	57	3	of	of	ADP
ejpam-7050	57	4	all	all	DET
ejpam-7050	57	5	(	(	PUNCT
ejpam-7050	57	6	τ1	τ1	NOUN
ejpam-7050	57	7	,	,	PUNCT
ejpam-7050	57	8	τ2)s	τ2)s	NOUN
ejpam-7050	57	9	-	-	PUNCT
ejpam-7050	57	10	open	open	ADJ
ejpam-7050	57	11	sets	set	NOUN
ejpam-7050	57	12	of	of	ADP
ejpam-7050	57	13	x	x	PUNCT
ejpam-7050	57	14	contained	contain	VERB
ejpam-7050	57	15	in	in	ADP
ejpam-7050	57	16	a	a	PRON
ejpam-7050	57	17	is	be	AUX
ejpam-7050	57	18	called	call	VERB
ejpam-7050	57	19	the	the	DET
ejpam-7050	57	20	(	(	PUNCT
ejpam-7050	57	21	τ1	τ1	NOUN
ejpam-7050	57	22	,	,	PUNCT
ejpam-7050	57	23	τ2)s	τ2)s	NOUN
ejpam-7050	57	24	-	-	ADJ
ejpam-7050	57	25	interior	interior	ADJ
ejpam-7050	57	26	[	[	X
ejpam-7050	57	27	15	15	NUM
ejpam-7050	57	28	]	]	PUNCT
ejpam-7050	57	29	of	of	ADP
ejpam-7050	57	30	a	a	PRON
ejpam-7050	57	31	and	and	CCONJ
ejpam-7050	57	32	is	be	AUX
ejpam-7050	57	33	denoted	denote	VERB
ejpam-7050	57	34	by	by	ADP
ejpam-7050	57	35	(	(	PUNCT
ejpam-7050	57	36	τ1	τ1	NOUN
ejpam-7050	57	37	,	,	PUNCT
ejpam-7050	57	38	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7050	57	39	)	)	PUNCT
ejpam-7050	57	40	.	.	PUNCT
ejpam-7050	58	1	m.	m.	NOUN
ejpam-7050	58	2	thongmoon	thongmoon	PROPN
ejpam-7050	58	3	,	,	PUNCT
ejpam-7050	58	4	a.	a.	PROPN
ejpam-7050	58	5	sama	sama	PROPN
ejpam-7050	58	6	-	-	PUNCT
ejpam-7050	58	7	ae	ae	PROPN
ejpam-7050	58	8	,	,	PUNCT
ejpam-7050	58	9	c.	c.	PROPN
ejpam-7050	58	10	boonpok	boonpok	PROPN
ejpam-7050	58	11	/	/	SYM
ejpam-7050	58	12	eur	eur	PROPN
ejpam-7050	58	13	.	.	PUNCT
ejpam-7050	59	1	j.	j.	PROPN
ejpam-7050	59	2	pure	pure	PROPN
ejpam-7050	59	3	appl	appl	PROPN
ejpam-7050	59	4	.	.	PROPN
ejpam-7050	59	5	math	math	PROPN
ejpam-7050	59	6	,	,	PUNCT
ejpam-7050	59	7	18	18	NUM
ejpam-7050	59	8	(	(	PUNCT
ejpam-7050	59	9	4	4	NUM
ejpam-7050	59	10	)	)	PUNCT
ejpam-7050	59	11	(	(	PUNCT
ejpam-7050	59	12	2025	2025	NUM
ejpam-7050	59	13	)	)	PUNCT
ejpam-7050	59	14	,	,	PUNCT
ejpam-7050	59	15	7050	7050	NUM
ejpam-7050	59	16	3	3	NUM
ejpam-7050	59	17	of	of	ADP
ejpam-7050	59	18	14	14	NUM
ejpam-7050	59	19	lemma	lemma	PROPN
ejpam-7050	59	20	2	2	NUM
ejpam-7050	59	21	.	.	PUNCT
ejpam-7050	60	1	for	for	ADP
ejpam-7050	60	2	a	a	DET
ejpam-7050	60	3	subset	subset	NOUN
ejpam-7050	60	4	a	a	PRON
ejpam-7050	60	5	of	of	ADP
ejpam-7050	60	6	a	a	DET
ejpam-7050	60	7	bitopological	bitopological	ADJ
ejpam-7050	60	8	space	space	NOUN
ejpam-7050	60	9	(	(	PUNCT
ejpam-7050	60	10	x	x	NOUN
ejpam-7050	60	11	,	,	PUNCT
ejpam-7050	60	12	τ1	τ1	NOUN
ejpam-7050	60	13	,	,	PUNCT
ejpam-7050	60	14	τ2	τ2	NOUN
ejpam-7050	60	15	)	)	PUNCT
ejpam-7050	60	16	,	,	PUNCT
ejpam-7050	60	17	the	the	DET
ejpam-7050	60	18	following	follow	VERB
ejpam-7050	60	19	properties	property	NOUN
ejpam-7050	60	20	hold	hold	VERB
ejpam-7050	60	21	:	:	PUNCT
ejpam-7050	60	22	(	(	PUNCT
ejpam-7050	60	23	1	1	X
ejpam-7050	60	24	)	)	PUNCT
ejpam-7050	60	25	(	(	PUNCT
ejpam-7050	60	26	τ1	τ1	NOUN
ejpam-7050	60	27	,	,	PUNCT
ejpam-7050	60	28	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-7050	60	29	)	)	PUNCT
ejpam-7050	60	30	=	=	PUNCT
ejpam-7050	61	1	τ1τ2	τ1τ2	NOUN
ejpam-7050	61	2	-	-	NOUN
ejpam-7050	61	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7050	61	4	-	-	PUNCT
ejpam-7050	61	5	cl(a	cl(a	NUM
ejpam-7050	61	6	)	)	PUNCT
ejpam-7050	61	7	)	)	PUNCT
ejpam-7050	61	8	∪a	∪a	X
ejpam-7050	62	1	[	[	X
ejpam-7050	62	2	15	15	NUM
ejpam-7050	62	3	]	]	X
ejpam-7050	62	4	;	;	PUNCT
ejpam-7050	62	5	(	(	PUNCT
ejpam-7050	62	6	2	2	X
ejpam-7050	62	7	)	)	PUNCT
ejpam-7050	62	8	(	(	PUNCT
ejpam-7050	62	9	τ1	τ1	NOUN
ejpam-7050	62	10	,	,	PUNCT
ejpam-7050	62	11	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7050	62	12	)	)	PUNCT
ejpam-7050	63	1	=	=	PUNCT
ejpam-7050	63	2	τ1τ2	τ1τ2	NOUN
ejpam-7050	63	3	-	-	ADJ
ejpam-7050	63	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7050	63	5	-	-	PUNCT
ejpam-7050	63	6	int(a	int(a	NOUN
ejpam-7050	63	7	)	)	PUNCT
ejpam-7050	63	8	)	)	PUNCT
ejpam-7050	64	1	∩a	∩a	PROPN
ejpam-7050	65	1	[	[	X
ejpam-7050	65	2	16	16	NUM
ejpam-7050	65	3	]	]	PUNCT
ejpam-7050	65	4	.	.	PUNCT
ejpam-7050	66	1	let	let	VERB
ejpam-7050	66	2	a	a	DET
ejpam-7050	66	3	be	be	AUX
ejpam-7050	66	4	a	a	DET
ejpam-7050	66	5	subset	subset	NOUN
ejpam-7050	66	6	of	of	ADP
ejpam-7050	66	7	a	a	DET
ejpam-7050	66	8	bitopological	bitopological	ADJ
ejpam-7050	66	9	space	space	NOUN
ejpam-7050	66	10	(	(	PUNCT
ejpam-7050	66	11	x	x	NOUN
ejpam-7050	66	12	,	,	PUNCT
ejpam-7050	66	13	τ1	τ1	NOUN
ejpam-7050	66	14	,	,	PUNCT
ejpam-7050	66	15	τ2	τ2	NOUN
ejpam-7050	66	16	)	)	PUNCT
ejpam-7050	66	17	.	.	PUNCT
ejpam-7050	67	1	a	a	DET
ejpam-7050	67	2	point	point	NOUN
ejpam-7050	67	3	x	x	X
ejpam-7050	67	4	∈	∈	NOUN
ejpam-7050	67	5	x	x	PUNCT
ejpam-7050	67	6	is	be	AUX
ejpam-7050	67	7	called	call	VERB
ejpam-7050	67	8	a	a	DET
ejpam-7050	67	9	(	(	PUNCT
ejpam-7050	67	10	τ1	τ1	NOUN
ejpam-7050	67	11	,	,	PUNCT
ejpam-7050	67	12	τ2)θ	τ2)θ	ADJ
ejpam-7050	67	13	-	-	PUNCT
ejpam-7050	67	14	cluster	cluster	NOUN
ejpam-7050	67	15	point	point	NOUN
ejpam-7050	67	16	[	[	X
ejpam-7050	67	17	14	14	NUM
ejpam-7050	67	18	]	]	PUNCT
ejpam-7050	67	19	of	of	ADP
ejpam-7050	67	20	a	a	DET
ejpam-7050	67	21	if	if	SCONJ
ejpam-7050	67	22	τ1τ2	τ1τ2	ADJ
ejpam-7050	67	23	-	-	ADJ
ejpam-7050	67	24	cl(u)∩a	cl(u)∩a	ADJ
ejpam-7050	67	25	̸=	̸=	PROPN
ejpam-7050	67	26	∅	∅	NOUN
ejpam-7050	67	27	for	for	ADP
ejpam-7050	67	28	every	every	DET
ejpam-7050	67	29	τ1τ2	τ1τ2	ADJ
ejpam-7050	67	30	-	-	ADJ
ejpam-7050	67	31	open	open	ADJ
ejpam-7050	67	32	set	set	NOUN
ejpam-7050	67	33	u	u	NOUN
ejpam-7050	67	34	containing	contain	VERB
ejpam-7050	67	35	x.	x.	NOUN
ejpam-7050	67	36	the	the	DET
ejpam-7050	67	37	set	set	NOUN
ejpam-7050	67	38	of	of	ADP
ejpam-7050	67	39	all	all	DET
ejpam-7050	67	40	(	(	PUNCT
ejpam-7050	67	41	τ1	τ1	NOUN
ejpam-7050	67	42	,	,	PUNCT
ejpam-7050	67	43	τ2)θ	τ2)θ	ADJ
ejpam-7050	67	44	-	-	PUNCT
ejpam-7050	67	45	cluster	cluster	NOUN
ejpam-7050	67	46	points	point	NOUN
ejpam-7050	67	47	of	of	ADP
ejpam-7050	67	48	a	a	PRON
ejpam-7050	67	49	is	be	AUX
ejpam-7050	67	50	called	call	VERB
ejpam-7050	67	51	the	the	DET
ejpam-7050	67	52	(	(	PUNCT
ejpam-7050	67	53	τ1	τ1	NOUN
ejpam-7050	67	54	,	,	PUNCT
ejpam-7050	67	55	τ2)θ	τ2)θ	ADJ
ejpam-7050	67	56	-	-	PUNCT
ejpam-7050	67	57	closure	closure	NOUN
ejpam-7050	67	58	[	[	X
ejpam-7050	67	59	14	14	NUM
ejpam-7050	67	60	]	]	PUNCT
ejpam-7050	67	61	of	of	ADP
ejpam-7050	67	62	a	a	PRON
ejpam-7050	67	63	and	and	CCONJ
ejpam-7050	67	64	is	be	AUX
ejpam-7050	67	65	denoted	denote	VERB
ejpam-7050	67	66	by	by	ADP
ejpam-7050	67	67	(	(	PUNCT
ejpam-7050	67	68	τ1	τ1	NOUN
ejpam-7050	67	69	,	,	PUNCT
ejpam-7050	67	70	τ2)θ	τ2)θ	NOUN
ejpam-7050	67	71	-	-	PUNCT
ejpam-7050	67	72	cl(a	cl(a	NUM
ejpam-7050	67	73	)	)	PUNCT
ejpam-7050	67	74	.	.	PUNCT
ejpam-7050	68	1	a	a	DET
ejpam-7050	68	2	subset	subset	NOUN
ejpam-7050	68	3	a	a	PRON
ejpam-7050	68	4	of	of	ADP
ejpam-7050	68	5	a	a	DET
ejpam-7050	68	6	bitopological	bitopological	ADJ
ejpam-7050	68	7	space	space	NOUN
ejpam-7050	68	8	(	(	PUNCT
ejpam-7050	68	9	x	x	NOUN
ejpam-7050	68	10	,	,	PUNCT
ejpam-7050	68	11	τ1	τ1	NOUN
ejpam-7050	68	12	,	,	PUNCT
ejpam-7050	68	13	τ2	τ2	NOUN
ejpam-7050	68	14	)	)	PUNCT
ejpam-7050	68	15	is	be	AUX
ejpam-7050	68	16	said	say	VERB
ejpam-7050	68	17	to	to	PART
ejpam-7050	68	18	be	be	AUX
ejpam-7050	68	19	(	(	PUNCT
ejpam-7050	68	20	τ1	τ1	NOUN
ejpam-7050	68	21	,	,	PUNCT
ejpam-7050	68	22	τ2)θ	τ2)θ	NOUN
ejpam-7050	68	23	-	-	PUNCT
ejpam-7050	68	24	closed	closed	ADJ
ejpam-7050	68	25	[	[	X
ejpam-7050	68	26	14	14	NUM
ejpam-7050	68	27	]	]	X
ejpam-7050	68	28	if	if	SCONJ
ejpam-7050	68	29	(	(	PUNCT
ejpam-7050	68	30	τ1	τ1	NOUN
ejpam-7050	68	31	,	,	PUNCT
ejpam-7050	68	32	τ2)θ	τ2)θ	NOUN
ejpam-7050	68	33	-	-	PUNCT
ejpam-7050	68	34	cl(a	cl(a	NUM
ejpam-7050	68	35	)	)	PUNCT
ejpam-7050	69	1	=	=	PUNCT
ejpam-7050	69	2	a.	a.	NOUN
ejpam-7050	69	3	the	the	DET
ejpam-7050	69	4	complement	complement	NOUN
ejpam-7050	69	5	of	of	ADP
ejpam-7050	69	6	a	a	DET
ejpam-7050	69	7	(	(	PUNCT
ejpam-7050	69	8	τ1	τ1	NOUN
ejpam-7050	69	9	,	,	PUNCT
ejpam-7050	69	10	τ2)θ	τ2)θ	ADJ
ejpam-7050	69	11	-	-	PUNCT
ejpam-7050	69	12	closed	close	VERB
ejpam-7050	69	13	set	set	NOUN
ejpam-7050	69	14	is	be	AUX
ejpam-7050	69	15	said	say	VERB
ejpam-7050	69	16	to	to	PART
ejpam-7050	69	17	be	be	AUX
ejpam-7050	69	18	(	(	PUNCT
ejpam-7050	69	19	τ1	τ1	NOUN
ejpam-7050	69	20	,	,	PUNCT
ejpam-7050	69	21	τ2)θ	τ2)θ	NOUN
ejpam-7050	69	22	-	-	PUNCT
ejpam-7050	69	23	open	open	ADJ
ejpam-7050	69	24	.	.	PUNCT
ejpam-7050	70	1	the	the	DET
ejpam-7050	70	2	union	union	NOUN
ejpam-7050	70	3	of	of	ADP
ejpam-7050	70	4	all	all	DET
ejpam-7050	70	5	(	(	PUNCT
ejpam-7050	70	6	τ1	τ1	NOUN
ejpam-7050	70	7	,	,	PUNCT
ejpam-7050	70	8	τ2)θ	τ2)θ	ADJ
ejpam-7050	70	9	-	-	PUNCT
ejpam-7050	70	10	open	open	ADJ
ejpam-7050	70	11	sets	set	NOUN
ejpam-7050	70	12	of	of	ADP
ejpam-7050	70	13	x	x	PUNCT
ejpam-7050	70	14	contained	contain	VERB
ejpam-7050	70	15	in	in	ADP
ejpam-7050	70	16	a	a	PRON
ejpam-7050	70	17	is	be	AUX
ejpam-7050	70	18	called	call	VERB
ejpam-7050	70	19	the	the	DET
ejpam-7050	70	20	(	(	PUNCT
ejpam-7050	70	21	τ1	τ1	NOUN
ejpam-7050	70	22	,	,	PUNCT
ejpam-7050	70	23	τ2)θ	τ2)θ	ADJ
ejpam-7050	70	24	-	-	PUNCT
ejpam-7050	70	25	interior	interior	NOUN
ejpam-7050	70	26	[	[	X
ejpam-7050	70	27	14	14	NUM
ejpam-7050	70	28	]	]	PUNCT
ejpam-7050	70	29	of	of	ADP
ejpam-7050	70	30	a	a	PRON
ejpam-7050	70	31	and	and	CCONJ
ejpam-7050	70	32	is	be	AUX
ejpam-7050	70	33	denoted	denote	VERB
ejpam-7050	70	34	by	by	ADP
ejpam-7050	70	35	(	(	PUNCT
ejpam-7050	70	36	τ1	τ1	NOUN
ejpam-7050	70	37	,	,	PUNCT
ejpam-7050	70	38	τ2)θ	τ2)θ	NOUN
ejpam-7050	70	39	-	-	PUNCT
ejpam-7050	70	40	int(a	int(a	NOUN
ejpam-7050	70	41	)	)	PUNCT
ejpam-7050	70	42	.	.	PUNCT
ejpam-7050	71	1	lemma	lemma	PROPN
ejpam-7050	71	2	3	3	X
ejpam-7050	71	3	.	.	PUNCT
ejpam-7050	72	1	[	[	X
ejpam-7050	72	2	14	14	NUM
ejpam-7050	72	3	]	]	PUNCT
ejpam-7050	72	4	for	for	ADP
ejpam-7050	72	5	a	a	DET
ejpam-7050	72	6	subset	subset	NOUN
ejpam-7050	72	7	a	a	PRON
ejpam-7050	72	8	of	of	ADP
ejpam-7050	72	9	a	a	DET
ejpam-7050	72	10	bitopological	bitopological	ADJ
ejpam-7050	72	11	space	space	NOUN
ejpam-7050	72	12	(	(	PUNCT
ejpam-7050	72	13	x	x	NOUN
ejpam-7050	72	14	,	,	PUNCT
ejpam-7050	72	15	τ1	τ1	NOUN
ejpam-7050	72	16	,	,	PUNCT
ejpam-7050	72	17	τ2	τ2	NOUN
ejpam-7050	72	18	)	)	PUNCT
ejpam-7050	72	19	,	,	PUNCT
ejpam-7050	72	20	the	the	DET
ejpam-7050	72	21	following	follow	VERB
ejpam-7050	72	22	properties	property	NOUN
ejpam-7050	72	23	hold	hold	VERB
ejpam-7050	72	24	:	:	PUNCT
ejpam-7050	72	25	(	(	PUNCT
ejpam-7050	72	26	1	1	X
ejpam-7050	72	27	)	)	PUNCT
ejpam-7050	72	28	if	if	SCONJ
ejpam-7050	72	29	a	a	PRON
ejpam-7050	72	30	is	be	AUX
ejpam-7050	72	31	τ1τ2	τ1τ2	NOUN
ejpam-7050	72	32	-	-	ADJ
ejpam-7050	72	33	open	open	ADJ
ejpam-7050	72	34	in	in	ADP
ejpam-7050	72	35	x	x	NOUN
ejpam-7050	72	36	,	,	PUNCT
ejpam-7050	72	37	then	then	ADV
ejpam-7050	72	38	τ1τ2	τ1τ2	NOUN
ejpam-7050	72	39	-	-	NUM
ejpam-7050	72	40	cl(a	cl(a	NUM
ejpam-7050	72	41	)	)	PUNCT
ejpam-7050	72	42	=	=	PUNCT
ejpam-7050	72	43	(	(	PUNCT
ejpam-7050	72	44	τ1	τ1	NOUN
ejpam-7050	72	45	,	,	PUNCT
ejpam-7050	72	46	τ2)θ	τ2)θ	NOUN
ejpam-7050	72	47	-	-	PUNCT
ejpam-7050	72	48	cl(a	cl(a	NUM
ejpam-7050	72	49	)	)	PUNCT
ejpam-7050	72	50	.	.	PUNCT
ejpam-7050	73	1	(	(	PUNCT
ejpam-7050	73	2	2	2	X
ejpam-7050	73	3	)	)	PUNCT
ejpam-7050	73	4	(	(	PUNCT
ejpam-7050	73	5	τ1	τ1	NOUN
ejpam-7050	73	6	,	,	PUNCT
ejpam-7050	73	7	τ2)θ	τ2)θ	NOUN
ejpam-7050	73	8	-	-	PUNCT
ejpam-7050	73	9	cl(a	cl(a	NUM
ejpam-7050	73	10	)	)	PUNCT
ejpam-7050	73	11	is	be	AUX
ejpam-7050	73	12	τ1τ2	τ1τ2	NOUN
ejpam-7050	73	13	-	-	ADJ
ejpam-7050	73	14	closed	closed	ADJ
ejpam-7050	73	15	in	in	ADP
ejpam-7050	73	16	x.	x.	NOUN
ejpam-7050	73	17	an	an	DET
ejpam-7050	73	18	ideal	ideal	NOUN
ejpam-7050	73	19	i	i	PRON
ejpam-7050	73	20	on	on	ADP
ejpam-7050	73	21	a	a	DET
ejpam-7050	73	22	topological	topological	ADJ
ejpam-7050	73	23	space	space	NOUN
ejpam-7050	73	24	(	(	PUNCT
ejpam-7050	73	25	x	x	X
ejpam-7050	73	26	,	,	PUNCT
ejpam-7050	73	27	τ	τ	X
ejpam-7050	73	28	)	)	PUNCT
ejpam-7050	73	29	is	be	AUX
ejpam-7050	73	30	a	a	DET
ejpam-7050	73	31	nonempty	nonempty	ADJ
ejpam-7050	73	32	collection	collection	NOUN
ejpam-7050	73	33	of	of	ADP
ejpam-7050	73	34	subsets	subset	NOUN
ejpam-7050	73	35	of	of	ADP
ejpam-7050	73	36	x	x	PUNCT
ejpam-7050	73	37	satisfying	satisfy	VERB
ejpam-7050	73	38	the	the	DET
ejpam-7050	73	39	following	follow	VERB
ejpam-7050	73	40	properties	property	NOUN
ejpam-7050	73	41	:	:	PUNCT
ejpam-7050	73	42	(	(	PUNCT
ejpam-7050	73	43	1	1	X
ejpam-7050	73	44	)	)	PUNCT
ejpam-7050	73	45	a	a	DET
ejpam-7050	73	46	∈	∈	NOUN
ejpam-7050	74	1	i	i	PRON
ejpam-7050	74	2	and	and	CCONJ
ejpam-7050	74	3	b	b	X
ejpam-7050	74	4	⊆	⊆	NUM
ejpam-7050	74	5	a	a	DET
ejpam-7050	74	6	imply	imply	NOUN
ejpam-7050	74	7	b	b	X
ejpam-7050	74	8	∈	∈	PROPN
ejpam-7050	74	9	i	i	PRON
ejpam-7050	74	10	;	;	PUNCT
ejpam-7050	74	11	(	(	PUNCT
ejpam-7050	74	12	2	2	X
ejpam-7050	74	13	)	)	PUNCT
ejpam-7050	75	1	a	a	DET
ejpam-7050	75	2	∈	∈	NOUN
ejpam-7050	75	3	i	i	PRON
ejpam-7050	75	4	and	and	CCONJ
ejpam-7050	75	5	b	b	X
ejpam-7050	75	6	∈	∈	NOUN
ejpam-7050	76	1	i	i	PRON
ejpam-7050	76	2	imply	imply	VERB
ejpam-7050	76	3	a	a	DET
ejpam-7050	76	4	∪	∪	X
ejpam-7050	76	5	b	b	NOUN
ejpam-7050	76	6	∈	∈	NOUN
ejpam-7050	77	1	i	i	PRON
ejpam-7050	77	2	.	.	PUNCT
ejpam-7050	78	1	a	a	DET
ejpam-7050	78	2	topological	topological	ADJ
ejpam-7050	78	3	space	space	NOUN
ejpam-7050	78	4	(	(	PUNCT
ejpam-7050	78	5	x	x	X
ejpam-7050	78	6	,	,	PUNCT
ejpam-7050	78	7	τ	τ	X
ejpam-7050	78	8	)	)	PUNCT
ejpam-7050	78	9	with	with	ADP
ejpam-7050	78	10	an	an	DET
ejpam-7050	78	11	ideal	ideal	ADJ
ejpam-7050	78	12	i	i	PRON
ejpam-7050	78	13	on	on	ADP
ejpam-7050	78	14	x	x	SYM
ejpam-7050	78	15	is	be	AUX
ejpam-7050	78	16	called	call	VERB
ejpam-7050	78	17	an	an	DET
ejpam-7050	78	18	ideal	ideal	ADJ
ejpam-7050	78	19	topological	topological	ADJ
ejpam-7050	78	20	space	space	NOUN
ejpam-7050	78	21	and	and	CCONJ
ejpam-7050	78	22	is	be	AUX
ejpam-7050	78	23	denoted	denote	VERB
ejpam-7050	78	24	by	by	ADP
ejpam-7050	78	25	(	(	PUNCT
ejpam-7050	78	26	x	x	X
ejpam-7050	78	27	,	,	PUNCT
ejpam-7050	78	28	τ	τ	PROPN
ejpam-7050	78	29	,	,	PUNCT
ejpam-7050	78	30	i	i	NOUN
ejpam-7050	78	31	)	)	PUNCT
ejpam-7050	78	32	.	.	PUNCT
ejpam-7050	79	1	for	for	ADP
ejpam-7050	79	2	an	an	DET
ejpam-7050	79	3	ideal	ideal	ADJ
ejpam-7050	79	4	topological	topological	ADJ
ejpam-7050	79	5	space	space	NOUN
ejpam-7050	79	6	(	(	PUNCT
ejpam-7050	79	7	x	x	X
ejpam-7050	79	8	,	,	PUNCT
ejpam-7050	79	9	τ	τ	PROPN
ejpam-7050	79	10	,	,	PUNCT
ejpam-7050	79	11	i	i	PROPN
ejpam-7050	79	12	)	)	PUNCT
ejpam-7050	79	13	and	and	CCONJ
ejpam-7050	79	14	a	a	DET
ejpam-7050	79	15	subset	subset	NOUN
ejpam-7050	79	16	a	a	PRON
ejpam-7050	79	17	of	of	ADP
ejpam-7050	79	18	x	x	PRON
ejpam-7050	79	19	,	,	PUNCT
ejpam-7050	79	20	a⋆(i	a⋆(i	PROPN
ejpam-7050	79	21	)	)	PUNCT
ejpam-7050	79	22	is	be	AUX
ejpam-7050	79	23	defined	define	VERB
ejpam-7050	79	24	as	as	SCONJ
ejpam-7050	79	25	follows	follow	VERB
ejpam-7050	79	26	:	:	PUNCT
ejpam-7050	79	27	a⋆(i	a⋆(i	NOUN
ejpam-7050	79	28	)	)	PUNCT
ejpam-7050	80	1	=	=	PUNCT
ejpam-7050	80	2	{	{	PUNCT
ejpam-7050	80	3	x	x	PUNCT
ejpam-7050	80	4	∈	∈	PROPN
ejpam-7050	80	5	x	x	X
ejpam-7050	80	6	:	:	PUNCT
ejpam-7050	80	7	u	u	X
ejpam-7050	80	8	∩a	∩a	PROPN
ejpam-7050	80	9	̸∈	̸∈	PROPN
ejpam-7050	80	10	i	i	PRON
ejpam-7050	80	11	for	for	ADP
ejpam-7050	80	12	every	every	DET
ejpam-7050	80	13	open	open	ADJ
ejpam-7050	80	14	neighbourhood	neighbourhood	NOUN
ejpam-7050	80	15	u	u	NOUN
ejpam-7050	80	16	of	of	ADP
ejpam-7050	80	17	x	x	NOUN
ejpam-7050	80	18	}	}	PUNCT
ejpam-7050	80	19	.	.	PUNCT
ejpam-7050	81	1	in	in	ADP
ejpam-7050	81	2	case	case	NOUN
ejpam-7050	81	3	there	there	PRON
ejpam-7050	81	4	is	be	VERB
ejpam-7050	81	5	no	no	DET
ejpam-7050	81	6	chance	chance	NOUN
ejpam-7050	81	7	for	for	ADP
ejpam-7050	81	8	confusion	confusion	NOUN
ejpam-7050	81	9	,	,	PUNCT
ejpam-7050	81	10	a⋆(i	a⋆(i	NOUN
ejpam-7050	81	11	)	)	PUNCT
ejpam-7050	81	12	is	be	AUX
ejpam-7050	81	13	simply	simply	ADV
ejpam-7050	81	14	written	write	VERB
ejpam-7050	81	15	as	as	ADP
ejpam-7050	81	16	a⋆.	a⋆.	NOUN
ejpam-7050	81	17	in	in	ADP
ejpam-7050	81	18	[	[	X
ejpam-7050	81	19	17	17	NUM
ejpam-7050	81	20	]	]	PUNCT
ejpam-7050	81	21	,	,	PUNCT
ejpam-7050	81	22	a⋆	a⋆	ADV
ejpam-7050	81	23	is	be	AUX
ejpam-7050	81	24	called	call	VERB
ejpam-7050	81	25	the	the	DET
ejpam-7050	81	26	local	local	ADJ
ejpam-7050	81	27	function	function	NOUN
ejpam-7050	81	28	of	of	ADP
ejpam-7050	81	29	a	a	PRON
ejpam-7050	81	30	with	with	ADP
ejpam-7050	81	31	respect	respect	NOUN
ejpam-7050	81	32	to	to	ADP
ejpam-7050	81	33	i	i	PRON
ejpam-7050	81	34	and	and	CCONJ
ejpam-7050	81	35	τ	τ	PROPN
ejpam-7050	81	36	and	and	CCONJ
ejpam-7050	81	37	cl⋆(a	cl⋆(a	NUM
ejpam-7050	81	38	)	)	PUNCT
ejpam-7050	81	39	=	=	PUNCT
ejpam-7050	81	40	a⋆	a⋆	ADP
ejpam-7050	81	41	∪	∪	ADP
ejpam-7050	81	42	a	a	DET
ejpam-7050	81	43	defines	define	NOUN
ejpam-7050	81	44	a	a	DET
ejpam-7050	81	45	kuratowski	kuratowski	ADJ
ejpam-7050	81	46	closure	closure	NOUN
ejpam-7050	81	47	operator	operator	NOUN
ejpam-7050	81	48	for	for	ADP
ejpam-7050	81	49	a	a	DET
ejpam-7050	81	50	topology	topology	NOUN
ejpam-7050	81	51	τ⋆(i	τ⋆(i	NOUN
ejpam-7050	81	52	)	)	PUNCT
ejpam-7050	81	53	finer	fine	ADJ
ejpam-7050	81	54	than	than	ADP
ejpam-7050	81	55	τ	τ	PROPN
ejpam-7050	81	56	.	.	PUNCT
ejpam-7050	82	1	a	a	DET
ejpam-7050	82	2	subset	subset	NOUN
ejpam-7050	82	3	a	a	PRON
ejpam-7050	82	4	is	be	AUX
ejpam-7050	82	5	said	say	VERB
ejpam-7050	82	6	to	to	PART
ejpam-7050	82	7	be	be	AUX
ejpam-7050	82	8	⋆-closed	⋆-close	VERB
ejpam-7050	82	9	[	[	X
ejpam-7050	82	10	18	18	NUM
ejpam-7050	82	11	]	]	X
ejpam-7050	82	12	if	if	SCONJ
ejpam-7050	82	13	a⋆	a⋆	ADJ
ejpam-7050	82	14	⊆	⊆	NUM
ejpam-7050	82	15	a.	a.	NOUN
ejpam-7050	82	16	the	the	DET
ejpam-7050	82	17	interior	interior	NOUN
ejpam-7050	82	18	of	of	ADP
ejpam-7050	82	19	a	a	DET
ejpam-7050	82	20	subset	subset	NOUN
ejpam-7050	82	21	a	a	DET
ejpam-7050	82	22	in	in	ADP
ejpam-7050	82	23	(	(	PUNCT
ejpam-7050	82	24	x	x	X
ejpam-7050	82	25	,	,	PUNCT
ejpam-7050	82	26	τ⋆(i	τ⋆(i	NOUN
ejpam-7050	82	27	)	)	PUNCT
ejpam-7050	82	28	)	)	PUNCT
ejpam-7050	82	29	is	be	AUX
ejpam-7050	82	30	denoted	denote	VERB
ejpam-7050	82	31	by	by	ADP
ejpam-7050	82	32	int⋆(a	int⋆(a	NOUN
ejpam-7050	82	33	)	)	PUNCT
ejpam-7050	82	34	.	.	PUNCT
ejpam-7050	83	1	a	a	DET
ejpam-7050	83	2	subset	subset	NOUN
ejpam-7050	83	3	a	a	PRON
ejpam-7050	83	4	of	of	ADP
ejpam-7050	83	5	an	an	DET
ejpam-7050	83	6	ideal	ideal	ADJ
ejpam-7050	83	7	topological	topological	ADJ
ejpam-7050	83	8	space	space	NOUN
ejpam-7050	83	9	(	(	PUNCT
ejpam-7050	83	10	x	x	X
ejpam-7050	83	11	,	,	PUNCT
ejpam-7050	83	12	τ	τ	PROPN
ejpam-7050	83	13	,	,	PUNCT
ejpam-7050	83	14	i	i	PROPN
ejpam-7050	83	15	)	)	PUNCT
ejpam-7050	83	16	is	be	AUX
ejpam-7050	83	17	said	say	VERB
ejpam-7050	83	18	to	to	PART
ejpam-7050	83	19	be	be	AUX
ejpam-7050	83	20	r	r	NOUN
ejpam-7050	83	21	-	-	PUNCT
ejpam-7050	83	22	i	i	PRON
ejpam-7050	83	23	⋆open	⋆open	VERB
ejpam-7050	84	1	[	[	X
ejpam-7050	84	2	6	6	NUM
ejpam-7050	84	3	]	]	PUNCT
ejpam-7050	84	4	(	(	PUNCT
ejpam-7050	84	5	resp	resp	NOUN
ejpam-7050	84	6	.	.	PUNCT
ejpam-7050	85	1	i	i	PRON
ejpam-7050	85	2	⋆-preopen	⋆-preopen	VERB
ejpam-7050	86	1	[	[	X
ejpam-7050	86	2	6	6	NUM
ejpam-7050	86	3	]	]	PUNCT
ejpam-7050	86	4	,	,	PUNCT
ejpam-7050	86	5	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7050	86	6	-	-	ADJ
ejpam-7050	86	7	open	open	ADJ
ejpam-7050	86	8	[	[	X
ejpam-7050	86	9	19	19	NUM
ejpam-7050	86	10	]	]	X
ejpam-7050	86	11	(	(	PUNCT
ejpam-7050	86	12	semi	semi	NOUN
ejpam-7050	86	13	-	-	ADJ
ejpam-7050	86	14	i	i	PRON
ejpam-7050	86	15	⋆-open	⋆-open	VERB
ejpam-7050	87	1	[	[	X
ejpam-7050	87	2	20	20	NUM
ejpam-7050	87	3	]	]	NUM
ejpam-7050	87	4	)	)	PUNCT
ejpam-7050	87	5	,	,	PUNCT
ejpam-7050	87	6	τ⋆-β	τ⋆-β	PROPN
ejpam-7050	87	7	-	-	VERB
ejpam-7050	87	8	open	open	ADJ
ejpam-7050	87	9	[	[	X
ejpam-7050	87	10	19	19	NUM
ejpam-7050	87	11	]	]	X
ejpam-7050	87	12	(	(	PUNCT
ejpam-7050	87	13	semi	semi	NOUN
ejpam-7050	87	14	-	-	VERB
ejpam-7050	87	15	i	i	PRON
ejpam-7050	87	16	⋆-preopen	⋆-preopen	VERB
ejpam-7050	88	1	[	[	X
ejpam-7050	88	2	20	20	NUM
ejpam-7050	88	3	]	]	PUNCT
ejpam-7050	88	4	)	)	PUNCT
ejpam-7050	88	5	)	)	PUNCT
ejpam-7050	89	1	if	if	SCONJ
ejpam-7050	89	2	a	a	DET
ejpam-7050	89	3	=	=	PUNCT
ejpam-7050	89	4	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7050	89	5	)	)	PUNCT
ejpam-7050	89	6	)	)	PUNCT
ejpam-7050	90	1	(	(	PUNCT
ejpam-7050	90	2	resp	resp	NOUN
ejpam-7050	90	3	.	.	PUNCT
ejpam-7050	91	1	a	a	DET
ejpam-7050	91	2	⊆	⊆	NUM
ejpam-7050	91	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7050	91	4	)	)	PUNCT
ejpam-7050	91	5	)	)	PUNCT
ejpam-7050	91	6	,	,	PUNCT
ejpam-7050	91	7	a	a	DET
ejpam-7050	91	8	⊆	⊆	NUM
ejpam-7050	91	9	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7050	91	10	)	)	PUNCT
ejpam-7050	91	11	)	)	PUNCT
ejpam-7050	91	12	,	,	PUNCT
ejpam-7050	91	13	a	a	DET
ejpam-7050	91	14	⊆	⊆	NUM
ejpam-7050	91	15	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7050	91	16	)	)	PUNCT
ejpam-7050	91	17	)	)	PUNCT
ejpam-7050	91	18	)	)	PUNCT
ejpam-7050	91	19	)	)	PUNCT
ejpam-7050	91	20	.	.	PUNCT
ejpam-7050	92	1	the	the	DET
ejpam-7050	92	2	complement	complement	NOUN
ejpam-7050	92	3	of	of	ADP
ejpam-7050	92	4	a	a	DET
ejpam-7050	92	5	r	r	NOUN
ejpam-7050	92	6	-	-	PUNCT
ejpam-7050	92	7	i	i	PRON
ejpam-7050	92	8	⋆-open	⋆-open	VERB
ejpam-7050	92	9	(	(	PUNCT
ejpam-7050	92	10	resp	resp	NOUN
ejpam-7050	92	11	.	.	PUNCT
ejpam-7050	93	1	i	i	PRON
ejpam-7050	93	2	⋆-preopen	⋆-preopen	VERB
ejpam-7050	93	3	,	,	PUNCT
ejpam-7050	93	4	τ⋆-semiopen	τ⋆-semiopen	VERB
ejpam-7050	93	5	,	,	PUNCT
ejpam-7050	93	6	τ⋆-β	τ⋆-β	ADJ
ejpam-7050	93	7	-	-	ADJ
ejpam-7050	93	8	open	open	ADJ
ejpam-7050	93	9	)	)	PUNCT
ejpam-7050	93	10	set	set	NOUN
ejpam-7050	93	11	is	be	AUX
ejpam-7050	93	12	said	say	VERB
ejpam-7050	93	13	to	to	PART
ejpam-7050	93	14	be	be	AUX
ejpam-7050	93	15	r	r	NOUN
ejpam-7050	93	16	-	-	PUNCT
ejpam-7050	93	17	i	i	PRON
ejpam-7050	93	18	⋆-closed	⋆-close	VERB
ejpam-7050	93	19	(	(	PUNCT
ejpam-7050	93	20	resp	resp	NOUN
ejpam-7050	93	21	.	.	PUNCT
ejpam-7050	94	1	i	i	PRON
ejpam-7050	94	2	⋆-preclosed	⋆-preclose	VERB
ejpam-7050	94	3	,	,	PUNCT
ejpam-7050	94	4	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7050	94	5	-	-	ADJ
ejpam-7050	94	6	closed	closed	ADJ
ejpam-7050	94	7	,	,	PUNCT
ejpam-7050	94	8	τ⋆β	τ⋆β	NOUN
ejpam-7050	94	9	-	-	PUNCT
ejpam-7050	94	10	closed	closed	ADJ
ejpam-7050	94	11	)	)	PUNCT
ejpam-7050	94	12	.	.	PUNCT
ejpam-7050	95	1	for	for	ADP
ejpam-7050	95	2	a	a	DET
ejpam-7050	95	3	subset	subset	NOUN
ejpam-7050	95	4	a	a	PRON
ejpam-7050	95	5	of	of	ADP
ejpam-7050	95	6	an	an	DET
ejpam-7050	95	7	ideal	ideal	ADJ
ejpam-7050	95	8	topological	topological	ADJ
ejpam-7050	95	9	space	space	NOUN
ejpam-7050	95	10	(	(	PUNCT
ejpam-7050	95	11	x	x	X
ejpam-7050	95	12	,	,	PUNCT
ejpam-7050	95	13	τ	τ	PROPN
ejpam-7050	95	14	,	,	PUNCT
ejpam-7050	95	15	i	i	NOUN
ejpam-7050	95	16	)	)	PUNCT
ejpam-7050	95	17	,	,	PUNCT
ejpam-7050	95	18	the	the	DET
ejpam-7050	95	19	intersection	intersection	NOUN
ejpam-7050	95	20	of	of	ADP
ejpam-7050	95	21	all	all	PRON
ejpam-7050	95	22	semi	semi	NOUN
ejpam-7050	95	23	-	-	ADJ
ejpam-7050	95	24	i	i	PRON
ejpam-7050	95	25	⋆-closed	⋆-close	VERB
ejpam-7050	95	26	sets	set	NOUN
ejpam-7050	95	27	containing	contain	VERB
ejpam-7050	95	28	a	a	PRON
ejpam-7050	95	29	is	be	AUX
ejpam-7050	95	30	called	call	VERB
ejpam-7050	95	31	the	the	DET
ejpam-7050	95	32	semi	semi	NOUN
ejpam-7050	95	33	-	-	ADJ
ejpam-7050	95	34	i	i	PRON
ejpam-7050	95	35	⋆-closure	⋆-closure	NOUN
ejpam-7050	96	1	[	[	X
ejpam-7050	96	2	20	20	NUM
ejpam-7050	96	3	]	]	PUNCT
ejpam-7050	96	4	of	of	ADP
ejpam-7050	96	5	a	a	PRON
ejpam-7050	96	6	and	and	CCONJ
ejpam-7050	96	7	is	be	AUX
ejpam-7050	96	8	denoted	denote	VERB
ejpam-7050	96	9	by	by	ADP
ejpam-7050	96	10	scl⋆(a	scl⋆(a	NOUN
ejpam-7050	96	11	)	)	PUNCT
ejpam-7050	96	12	(	(	PUNCT
ejpam-7050	96	13	scli	scli	PROPN
ejpam-7050	96	14	⋆(a	⋆(a	PRON
ejpam-7050	96	15	)	)	PUNCT
ejpam-7050	97	1	[	[	X
ejpam-7050	97	2	20	20	NUM
ejpam-7050	97	3	]	]	NUM
ejpam-7050	97	4	)	)	PUNCT
ejpam-7050	97	5	.	.	PUNCT
ejpam-7050	98	1	the	the	DET
ejpam-7050	98	2	union	union	NOUN
ejpam-7050	98	3	of	of	ADP
ejpam-7050	98	4	all	all	PRON
ejpam-7050	98	5	semi	semi	ADJ
ejpam-7050	98	6	-	-	ADJ
ejpam-7050	98	7	i	i	PRON
ejpam-7050	98	8	⋆-open	⋆-open	ADJ
ejpam-7050	98	9	sets	set	NOUN
ejpam-7050	98	10	contained	contain	VERB
ejpam-7050	98	11	in	in	ADP
ejpam-7050	98	12	a	a	PRON
ejpam-7050	98	13	is	be	AUX
ejpam-7050	98	14	called	call	VERB
ejpam-7050	98	15	the	the	DET
ejpam-7050	98	16	semi	semi	NOUN
ejpam-7050	98	17	-	-	ADJ
ejpam-7050	98	18	i	i	PRON
ejpam-7050	98	19	⋆-interior	⋆-interior	PUNCT
ejpam-7050	99	1	[	[	X
ejpam-7050	99	2	20	20	NUM
ejpam-7050	99	3	]	]	PUNCT
ejpam-7050	99	4	of	of	ADP
ejpam-7050	99	5	a	a	PRON
ejpam-7050	99	6	and	and	CCONJ
ejpam-7050	99	7	is	be	AUX
ejpam-7050	99	8	denoted	denote	VERB
ejpam-7050	99	9	by	by	ADP
ejpam-7050	99	10	sint⋆(a	sint⋆(a	PROPN
ejpam-7050	99	11	)	)	PUNCT
ejpam-7050	100	1	(	(	PUNCT
ejpam-7050	100	2	sinti	sinti	PROPN
ejpam-7050	100	3	⋆(a	⋆(a	NOUN
ejpam-7050	100	4	)	)	PUNCT
ejpam-7050	101	1	[	[	X
ejpam-7050	101	2	20	20	NUM
ejpam-7050	101	3	]	]	NUM
ejpam-7050	101	4	)	)	PUNCT
ejpam-7050	101	5	.	.	PUNCT
ejpam-7050	102	1	the	the	DET
ejpam-7050	102	2	intersection	intersection	NOUN
ejpam-7050	102	3	of	of	ADP
ejpam-7050	102	4	all	all	PRON
ejpam-7050	102	5	β	β	NOUN
ejpam-7050	102	6	-	-	PUNCT
ejpam-7050	102	7	i	i	PRON
ejpam-7050	102	8	⋆-closed	⋆-close	VERB
ejpam-7050	102	9	sets	set	NOUN
ejpam-7050	102	10	containing	contain	VERB
ejpam-7050	102	11	a	a	PRON
ejpam-7050	102	12	is	be	AUX
ejpam-7050	102	13	called	call	VERB
ejpam-7050	102	14	the	the	DET
ejpam-7050	102	15	β	β	NOUN
ejpam-7050	102	16	-	-	NOUN
ejpam-7050	102	17	i	i	PRON
ejpam-7050	102	18	⋆-closure	⋆-closure	ADJ
ejpam-7050	102	19	of	of	ADP
ejpam-7050	102	20	a	a	PRON
ejpam-7050	102	21	and	and	CCONJ
ejpam-7050	102	22	is	be	AUX
ejpam-7050	102	23	denoted	denote	VERB
ejpam-7050	102	24	by	by	ADP
ejpam-7050	102	25	βcl⋆(a	βcl⋆(a	NOUN
ejpam-7050	102	26	)	)	PUNCT
ejpam-7050	102	27	.	.	PUNCT
ejpam-7050	103	1	the	the	DET
ejpam-7050	103	2	union	union	NOUN
ejpam-7050	103	3	of	of	ADP
ejpam-7050	103	4	all	all	PRON
ejpam-7050	103	5	β	β	NOUN
ejpam-7050	103	6	-	-	ADJ
ejpam-7050	103	7	i	i	PRON
ejpam-7050	103	8	⋆-open	⋆-open	ADJ
ejpam-7050	103	9	sets	set	NOUN
ejpam-7050	103	10	contained	contain	VERB
ejpam-7050	103	11	in	in	ADP
ejpam-7050	103	12	a	a	PRON
ejpam-7050	103	13	is	be	AUX
ejpam-7050	103	14	called	call	VERB
ejpam-7050	103	15	the	the	DET
ejpam-7050	103	16	β	β	NOUN
ejpam-7050	103	17	-	-	PUNCT
ejpam-7050	103	18	i	i	PRON
ejpam-7050	103	19	⋆-interior	⋆-interior	NOUN
ejpam-7050	103	20	of	of	ADP
ejpam-7050	103	21	a	a	PRON
ejpam-7050	103	22	and	and	CCONJ
ejpam-7050	103	23	is	be	AUX
ejpam-7050	103	24	denoted	denote	VERB
ejpam-7050	103	25	by	by	ADP
ejpam-7050	103	26	βint⋆(a	βint⋆(a	NOUN
ejpam-7050	103	27	)	)	PUNCT
ejpam-7050	103	28	.	.	PUNCT
ejpam-7050	104	1	m.	m.	NOUN
ejpam-7050	104	2	thongmoon	thongmoon	PROPN
ejpam-7050	104	3	,	,	PUNCT
ejpam-7050	104	4	a.	a.	PROPN
ejpam-7050	104	5	sama	sama	PROPN
ejpam-7050	104	6	-	-	PUNCT
ejpam-7050	104	7	ae	ae	PROPN
ejpam-7050	104	8	,	,	PUNCT
ejpam-7050	104	9	c.	c.	PROPN
ejpam-7050	104	10	boonpok	boonpok	PROPN
ejpam-7050	104	11	/	/	SYM
ejpam-7050	104	12	eur	eur	PROPN
ejpam-7050	104	13	.	.	PUNCT
ejpam-7050	105	1	j.	j.	PROPN
ejpam-7050	105	2	pure	pure	PROPN
ejpam-7050	105	3	appl	appl	PROPN
ejpam-7050	105	4	.	.	PROPN
ejpam-7050	105	5	math	math	PROPN
ejpam-7050	105	6	,	,	PUNCT
ejpam-7050	105	7	18	18	NUM
ejpam-7050	105	8	(	(	PUNCT
ejpam-7050	105	9	4	4	NUM
ejpam-7050	105	10	)	)	PUNCT
ejpam-7050	105	11	(	(	PUNCT
ejpam-7050	105	12	2025	2025	NUM
ejpam-7050	105	13	)	)	PUNCT
ejpam-7050	105	14	,	,	PUNCT
ejpam-7050	105	15	7050	7050	NUM
ejpam-7050	105	16	4	4	NUM
ejpam-7050	105	17	of	of	ADP
ejpam-7050	105	18	14	14	NUM
ejpam-7050	105	19	lemma	lemma	PROPN
ejpam-7050	105	20	4	4	NUM
ejpam-7050	105	21	.	.	PUNCT
ejpam-7050	106	1	for	for	ADP
ejpam-7050	106	2	a	a	DET
ejpam-7050	106	3	subset	subset	NOUN
ejpam-7050	106	4	a	a	PRON
ejpam-7050	106	5	of	of	ADP
ejpam-7050	106	6	an	an	DET
ejpam-7050	106	7	ideal	ideal	ADJ
ejpam-7050	106	8	topological	topological	ADJ
ejpam-7050	106	9	space	space	NOUN
ejpam-7050	106	10	(	(	PUNCT
ejpam-7050	106	11	x	x	X
ejpam-7050	106	12	,	,	PUNCT
ejpam-7050	106	13	τ	τ	PROPN
ejpam-7050	106	14	,	,	PUNCT
ejpam-7050	106	15	i	i	NOUN
ejpam-7050	106	16	)	)	PUNCT
ejpam-7050	106	17	,	,	PUNCT
ejpam-7050	106	18	the	the	DET
ejpam-7050	106	19	following	follow	VERB
ejpam-7050	106	20	properties	property	NOUN
ejpam-7050	106	21	hold	hold	VERB
ejpam-7050	106	22	:	:	PUNCT
ejpam-7050	106	23	(	(	PUNCT
ejpam-7050	106	24	1	1	X
ejpam-7050	106	25	)	)	PUNCT
ejpam-7050	106	26	scl⋆(a	scl⋆(a	NUM
ejpam-7050	106	27	)	)	PUNCT
ejpam-7050	106	28	=	=	PUNCT
ejpam-7050	107	1	a	a	DET
ejpam-7050	107	2	∪	∪	ADJ
ejpam-7050	107	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7050	107	4	)	)	PUNCT
ejpam-7050	107	5	)	)	PUNCT
ejpam-7050	108	1	[	[	X
ejpam-7050	108	2	20	20	NUM
ejpam-7050	108	3	]	]	PUNCT
ejpam-7050	108	4	.	.	PUNCT
ejpam-7050	109	1	(	(	PUNCT
ejpam-7050	109	2	2	2	X
ejpam-7050	109	3	)	)	PUNCT
ejpam-7050	109	4	sint⋆(a	sint⋆(a	PROPN
ejpam-7050	109	5	)	)	PUNCT
ejpam-7050	110	1	=	=	PUNCT
ejpam-7050	110	2	a	a	DET
ejpam-7050	110	3	∩	∩	ADJ
ejpam-7050	110	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7050	110	5	)	)	PUNCT
ejpam-7050	110	6	)	)	PUNCT
ejpam-7050	111	1	[	[	X
ejpam-7050	111	2	20	20	NUM
ejpam-7050	111	3	]	]	PUNCT
ejpam-7050	111	4	.	.	PUNCT
ejpam-7050	112	1	(	(	PUNCT
ejpam-7050	112	2	3	3	NUM
ejpam-7050	112	3	)	)	PUNCT
ejpam-7050	112	4	βcl⋆(a	βcl⋆(a	NUM
ejpam-7050	112	5	)	)	PUNCT
ejpam-7050	113	1	=	=	PUNCT
ejpam-7050	113	2	a	a	DET
ejpam-7050	113	3	∪	∪	ADJ
ejpam-7050	113	4	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7050	113	5	)	)	PUNCT
ejpam-7050	113	6	)	)	PUNCT
ejpam-7050	113	7	)	)	PUNCT
ejpam-7050	113	8	.	.	PUNCT
ejpam-7050	114	1	(	(	PUNCT
ejpam-7050	114	2	4	4	NUM
ejpam-7050	114	3	)	)	PUNCT
ejpam-7050	114	4	βint⋆(a	βint⋆(a	NUM
ejpam-7050	114	5	)	)	PUNCT
ejpam-7050	114	6	=	=	PUNCT
ejpam-7050	114	7	a	a	DET
ejpam-7050	114	8	∩	∩	NOUN
ejpam-7050	114	9	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7050	114	10	)	)	PUNCT
ejpam-7050	114	11	)	)	PUNCT
ejpam-7050	114	12	)	)	PUNCT
ejpam-7050	114	13	.	.	PUNCT
ejpam-7050	115	1	by	by	ADP
ejpam-7050	115	2	a	a	DET
ejpam-7050	115	3	multifunction	multifunction	NOUN
ejpam-7050	115	4	f	f	NOUN
ejpam-7050	115	5	:	:	PUNCT
ejpam-7050	115	6	x	x	X
ejpam-7050	115	7	→	→	SYM
ejpam-7050	115	8	y	y	PROPN
ejpam-7050	115	9	,	,	PUNCT
ejpam-7050	115	10	we	we	PRON
ejpam-7050	115	11	mean	mean	VERB
ejpam-7050	115	12	a	a	DET
ejpam-7050	115	13	point	point	NOUN
ejpam-7050	115	14	-	-	PUNCT
ejpam-7050	115	15	to	to	ADP
ejpam-7050	115	16	-	-	PUNCT
ejpam-7050	115	17	set	set	VERB
ejpam-7050	115	18	correspondence	correspondence	NOUN
ejpam-7050	115	19	from	from	ADP
ejpam-7050	115	20	x	x	PUNCT
ejpam-7050	115	21	into	into	ADP
ejpam-7050	115	22	y	y	PROPN
ejpam-7050	115	23	,	,	PUNCT
ejpam-7050	115	24	and	and	CCONJ
ejpam-7050	115	25	we	we	PRON
ejpam-7050	115	26	always	always	ADV
ejpam-7050	115	27	assume	assume	VERB
ejpam-7050	115	28	that	that	SCONJ
ejpam-7050	115	29	f	f	PROPN
ejpam-7050	115	30	(	(	PUNCT
ejpam-7050	115	31	x	x	X
ejpam-7050	115	32	)	)	PUNCT
ejpam-7050	115	33	̸=	̸=	NOUN
ejpam-7050	115	34	∅	∅	NOUN
ejpam-7050	115	35	for	for	ADP
ejpam-7050	115	36	all	all	PRON
ejpam-7050	115	37	x	x	SYM
ejpam-7050	115	38	∈	∈	ADJ
ejpam-7050	115	39	x.	x.	NOUN
ejpam-7050	115	40	for	for	ADP
ejpam-7050	115	41	a	a	DET
ejpam-7050	115	42	multifunction	multifunction	NOUN
ejpam-7050	115	43	f	f	NOUN
ejpam-7050	115	44	:	:	PUNCT
ejpam-7050	115	45	x	x	X
ejpam-7050	115	46	→	→	SYM
ejpam-7050	115	47	y	y	PROPN
ejpam-7050	115	48	,	,	PUNCT
ejpam-7050	115	49	we	we	PRON
ejpam-7050	115	50	shall	shall	AUX
ejpam-7050	115	51	denote	denote	VERB
ejpam-7050	115	52	the	the	DET
ejpam-7050	115	53	upper	upper	ADJ
ejpam-7050	115	54	and	and	CCONJ
ejpam-7050	115	55	lower	low	ADJ
ejpam-7050	115	56	inverse	inverse	NOUN
ejpam-7050	115	57	of	of	ADP
ejpam-7050	115	58	a	a	DET
ejpam-7050	115	59	set	set	NOUN
ejpam-7050	115	60	b	b	PROPN
ejpam-7050	115	61	of	of	ADP
ejpam-7050	115	62	y	y	PROPN
ejpam-7050	115	63	by	by	ADP
ejpam-7050	115	64	f+(b	f+(b	NOUN
ejpam-7050	115	65	)	)	PUNCT
ejpam-7050	115	66	and	and	CCONJ
ejpam-7050	115	67	f−(b	f−(b	NOUN
ejpam-7050	115	68	)	)	PUNCT
ejpam-7050	115	69	,	,	PUNCT
ejpam-7050	115	70	respectively	respectively	ADV
ejpam-7050	115	71	,	,	PUNCT
ejpam-7050	115	72	that	that	ADV
ejpam-7050	115	73	is	is	ADV
ejpam-7050	115	74	,	,	PUNCT
ejpam-7050	115	75	f+(b	f+(b	NOUN
ejpam-7050	115	76	)	)	PUNCT
ejpam-7050	115	77	=	=	PRON
ejpam-7050	116	1	{	{	PUNCT
ejpam-7050	116	2	x	x	PUNCT
ejpam-7050	116	3	∈	∈	PROPN
ejpam-7050	116	4	x	x	INTJ
ejpam-7050	117	1	|	|	NOUN
ejpam-7050	117	2	f	f	X
ejpam-7050	117	3	(	(	PUNCT
ejpam-7050	117	4	x	x	NOUN
ejpam-7050	117	5	)	)	PUNCT
ejpam-7050	117	6	⊆	⊆	NUM
ejpam-7050	117	7	b	b	NOUN
ejpam-7050	117	8	}	}	PUNCT
ejpam-7050	117	9	and	and	CCONJ
ejpam-7050	117	10	f−(b	f−(b	PROPN
ejpam-7050	117	11	)	)	PUNCT
ejpam-7050	117	12	=	=	PRON
ejpam-7050	118	1	{	{	PUNCT
ejpam-7050	118	2	x	x	PUNCT
ejpam-7050	118	3	∈	∈	PROPN
ejpam-7050	118	4	x	x	INTJ
ejpam-7050	119	1	|	|	NOUN
ejpam-7050	119	2	f	f	X
ejpam-7050	119	3	(	(	PUNCT
ejpam-7050	119	4	x	x	NOUN
ejpam-7050	119	5	)	)	PUNCT
ejpam-7050	119	6	∩	∩	NOUN
ejpam-7050	119	7	b	b	PROPN
ejpam-7050	119	8	̸=	̸=	PROPN
ejpam-7050	119	9	∅	∅	NOUN
ejpam-7050	119	10	}	}	PUNCT
ejpam-7050	119	11	.	.	PUNCT
ejpam-7050	120	1	in	in	ADP
ejpam-7050	120	2	particular	particular	ADJ
ejpam-7050	120	3	,	,	PUNCT
ejpam-7050	120	4	f−(y	f−(y	NOUN
ejpam-7050	120	5	)	)	PUNCT
ejpam-7050	120	6	=	=	SYM
ejpam-7050	121	1	{	{	PUNCT
ejpam-7050	121	2	x	x	PUNCT
ejpam-7050	121	3	∈	∈	PROPN
ejpam-7050	121	4	x	x	INTJ
ejpam-7050	122	1	|	|	ADV
ejpam-7050	122	2	y	y	PROPN
ejpam-7050	122	3	∈	∈	PROPN
ejpam-7050	122	4	f	f	X
ejpam-7050	122	5	(	(	PUNCT
ejpam-7050	122	6	x	x	NOUN
ejpam-7050	122	7	)	)	PUNCT
ejpam-7050	122	8	}	}	PUNCT
ejpam-7050	122	9	for	for	ADP
ejpam-7050	122	10	each	each	DET
ejpam-7050	122	11	point	point	NOUN
ejpam-7050	122	12	y	y	PROPN
ejpam-7050	122	13	∈	∈	PROPN
ejpam-7050	122	14	y	y	PROPN
ejpam-7050	122	15	.	.	PUNCT
ejpam-7050	123	1	for	for	ADP
ejpam-7050	123	2	each	each	DET
ejpam-7050	123	3	a	a	DET
ejpam-7050	123	4	⊆	⊆	NUM
ejpam-7050	123	5	x	x	SYM
ejpam-7050	123	6	,	,	PUNCT
ejpam-7050	123	7	f	f	PROPN
ejpam-7050	123	8	(	(	PUNCT
ejpam-7050	123	9	a	a	NOUN
ejpam-7050	123	10	)	)	PUNCT
ejpam-7050	123	11	=	=	SYM
ejpam-7050	123	12	∪x∈af	∪x∈af	NOUN
ejpam-7050	123	13	(	(	PUNCT
ejpam-7050	123	14	x	x	NOUN
ejpam-7050	123	15	)	)	PUNCT
ejpam-7050	123	16	.	.	PUNCT
ejpam-7050	124	1	3	3	X
ejpam-7050	124	2	.	.	X
ejpam-7050	124	3	upper	upper	ADJ
ejpam-7050	124	4	and	and	CCONJ
ejpam-7050	124	5	lower	low	ADJ
ejpam-7050	124	6	weakly	weakly	ADJ
ejpam-7050	124	7	τ	τ	NOUN
ejpam-7050	124	8	⋆β(σ1	⋆β(σ1	NUM
ejpam-7050	124	9	,	,	PUNCT
ejpam-7050	124	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	124	11	multifunctions	multifunction	NOUN
ejpam-7050	124	12	in	in	ADP
ejpam-7050	124	13	this	this	DET
ejpam-7050	124	14	section	section	NOUN
ejpam-7050	124	15	,	,	PUNCT
ejpam-7050	124	16	we	we	PRON
ejpam-7050	124	17	introduce	introduce	VERB
ejpam-7050	124	18	the	the	DET
ejpam-7050	124	19	notions	notion	NOUN
ejpam-7050	124	20	of	of	ADP
ejpam-7050	124	21	upper	upper	ADJ
ejpam-7050	124	22	weakly	weakly	ADJ
ejpam-7050	124	23	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	124	24	,	,	PUNCT
ejpam-7050	124	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	124	26	multifunctions	multifunction	NOUN
ejpam-7050	124	27	and	and	CCONJ
ejpam-7050	124	28	lower	low	ADJ
ejpam-7050	124	29	weakly	weakly	ADJ
ejpam-7050	124	30	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	124	31	,	,	PUNCT
ejpam-7050	124	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	124	33	multifunctions	multifunction	NOUN
ejpam-7050	124	34	.	.	PUNCT
ejpam-7050	125	1	moreover	moreover	ADV
ejpam-7050	125	2	,	,	PUNCT
ejpam-7050	125	3	several	several	ADJ
ejpam-7050	125	4	characterizations	characterization	NOUN
ejpam-7050	125	5	of	of	ADP
ejpam-7050	125	6	upper	upper	ADJ
ejpam-7050	125	7	weakly	weakly	ADJ
ejpam-7050	125	8	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	125	9	,	,	PUNCT
ejpam-7050	125	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	125	11	multifunctions	multifunction	NOUN
ejpam-7050	125	12	and	and	CCONJ
ejpam-7050	125	13	lower	low	ADJ
ejpam-7050	125	14	weakly	weakly	ADJ
ejpam-7050	125	15	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	125	16	,	,	PUNCT
ejpam-7050	125	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	125	18	multifunctions	multifunction	NOUN
ejpam-7050	125	19	discussed	discuss	VERB
ejpam-7050	125	20	.	.	PUNCT
ejpam-7050	126	1	definition	definition	NOUN
ejpam-7050	126	2	1	1	NUM
ejpam-7050	126	3	.	.	PUNCT
ejpam-7050	127	1	a	a	DET
ejpam-7050	127	2	multifunction	multifunction	NOUN
ejpam-7050	127	3	f	f	NOUN
ejpam-7050	127	4	:	:	PUNCT
ejpam-7050	127	5	(	(	PUNCT
ejpam-7050	127	6	x	x	X
ejpam-7050	127	7	,	,	PUNCT
ejpam-7050	127	8	τ	τ	PROPN
ejpam-7050	127	9	,	,	PUNCT
ejpam-7050	127	10	i	i	NOUN
ejpam-7050	127	11	)	)	PUNCT
ejpam-7050	127	12	→	→	PUNCT
ejpam-7050	127	13	(	(	PUNCT
ejpam-7050	127	14	y	y	PROPN
ejpam-7050	127	15	,	,	PUNCT
ejpam-7050	127	16	σ1	σ1	PROPN
ejpam-7050	127	17	,	,	PUNCT
ejpam-7050	127	18	σ2	σ2	PROPN
ejpam-7050	127	19	)	)	PUNCT
ejpam-7050	127	20	is	be	AUX
ejpam-7050	127	21	said	say	VERB
ejpam-7050	127	22	to	to	PART
ejpam-7050	127	23	be	be	AUX
ejpam-7050	127	24	upper	upper	ADJ
ejpam-7050	127	25	weakly	weakly	ADJ
ejpam-7050	127	26	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	127	27	,	,	PUNCT
ejpam-7050	127	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	127	29	at	at	ADP
ejpam-7050	127	30	a	a	DET
ejpam-7050	127	31	point	point	NOUN
ejpam-7050	127	32	x	x	PUNCT
ejpam-7050	127	33	of	of	ADP
ejpam-7050	127	34	x	x	PRON
ejpam-7050	127	35	if	if	SCONJ
ejpam-7050	127	36	for	for	ADP
ejpam-7050	127	37	each	each	DET
ejpam-7050	127	38	σ1σ2	σ1σ2	VERB
ejpam-7050	127	39	-	-	ADJ
ejpam-7050	127	40	open	open	ADJ
ejpam-7050	127	41	set	set	NOUN
ejpam-7050	127	42	v	v	NOUN
ejpam-7050	127	43	of	of	ADP
ejpam-7050	127	44	y	y	PRON
ejpam-7050	127	45	such	such	ADJ
ejpam-7050	127	46	that	that	SCONJ
ejpam-7050	127	47	f	f	PROPN
ejpam-7050	127	48	(	(	PUNCT
ejpam-7050	127	49	x	x	X
ejpam-7050	127	50	)	)	PUNCT
ejpam-7050	127	51	⊆	⊆	NUM
ejpam-7050	127	52	v	v	NOUN
ejpam-7050	127	53	,	,	PUNCT
ejpam-7050	127	54	there	there	PRON
ejpam-7050	127	55	exists	exist	VERB
ejpam-7050	127	56	a	a	DET
ejpam-7050	127	57	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	127	58	-	-	ADJ
ejpam-7050	127	59	open	open	ADJ
ejpam-7050	127	60	set	set	NOUN
ejpam-7050	127	61	u	u	NOUN
ejpam-7050	127	62	of	of	ADP
ejpam-7050	127	63	x	x	PUNCT
ejpam-7050	127	64	containing	contain	VERB
ejpam-7050	127	65	x	x	PUNCT
ejpam-7050	127	66	such	such	ADJ
ejpam-7050	127	67	that	that	SCONJ
ejpam-7050	127	68	f	f	PROPN
ejpam-7050	127	69	(	(	PUNCT
ejpam-7050	127	70	u	u	NOUN
ejpam-7050	127	71	)	)	PUNCT
ejpam-7050	127	72	⊆	⊆	NUM
ejpam-7050	127	73	σ1σ2	σ1σ2	NOUN
ejpam-7050	127	74	-	-	NUM
ejpam-7050	127	75	cl(v	cl(v	NOUN
ejpam-7050	127	76	)	)	PUNCT
ejpam-7050	127	77	.	.	PUNCT
ejpam-7050	128	1	a	a	DET
ejpam-7050	128	2	multifunction	multifunction	NOUN
ejpam-7050	128	3	f	f	NOUN
ejpam-7050	128	4	:	:	PUNCT
ejpam-7050	128	5	(	(	PUNCT
ejpam-7050	128	6	x	x	X
ejpam-7050	128	7	,	,	PUNCT
ejpam-7050	128	8	τ	τ	PROPN
ejpam-7050	128	9	,	,	PUNCT
ejpam-7050	128	10	i	i	NOUN
ejpam-7050	128	11	)	)	PUNCT
ejpam-7050	128	12	→	→	PUNCT
ejpam-7050	128	13	(	(	PUNCT
ejpam-7050	128	14	y	y	PROPN
ejpam-7050	128	15	,	,	PUNCT
ejpam-7050	128	16	σ1	σ1	PROPN
ejpam-7050	128	17	,	,	PUNCT
ejpam-7050	128	18	σ2	σ2	PROPN
ejpam-7050	128	19	)	)	PUNCT
ejpam-7050	128	20	is	be	AUX
ejpam-7050	128	21	said	say	VERB
ejpam-7050	128	22	to	to	PART
ejpam-7050	128	23	be	be	AUX
ejpam-7050	128	24	upper	upper	ADJ
ejpam-7050	128	25	weakly	weakly	ADJ
ejpam-7050	128	26	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	128	27	,	,	PUNCT
ejpam-7050	128	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	128	29	if	if	SCONJ
ejpam-7050	128	30	f	f	PROPN
ejpam-7050	128	31	is	be	AUX
ejpam-7050	128	32	upper	upper	ADJ
ejpam-7050	128	33	weakly	weakly	ADJ
ejpam-7050	128	34	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	128	35	,	,	PUNCT
ejpam-7050	128	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	128	37	at	at	ADP
ejpam-7050	128	38	each	each	DET
ejpam-7050	128	39	point	point	NOUN
ejpam-7050	128	40	of	of	ADP
ejpam-7050	128	41	x.	x.	NOUN
ejpam-7050	128	42	theorem	theorem	VERB
ejpam-7050	128	43	1	1	NUM
ejpam-7050	128	44	.	.	X
ejpam-7050	128	45	for	for	ADP
ejpam-7050	128	46	a	a	DET
ejpam-7050	128	47	multifunction	multifunction	NOUN
ejpam-7050	128	48	f	f	NOUN
ejpam-7050	128	49	:	:	PUNCT
ejpam-7050	128	50	(	(	PUNCT
ejpam-7050	128	51	x	x	X
ejpam-7050	128	52	,	,	PUNCT
ejpam-7050	128	53	τ	τ	PROPN
ejpam-7050	128	54	,	,	PUNCT
ejpam-7050	128	55	i	i	NOUN
ejpam-7050	128	56	)	)	PUNCT
ejpam-7050	128	57	→	→	PUNCT
ejpam-7050	128	58	(	(	PUNCT
ejpam-7050	128	59	y	y	PROPN
ejpam-7050	128	60	,	,	PUNCT
ejpam-7050	128	61	σ1	σ1	PROPN
ejpam-7050	128	62	,	,	PUNCT
ejpam-7050	128	63	σ2	σ2	NOUN
ejpam-7050	128	64	)	)	PUNCT
ejpam-7050	128	65	,	,	PUNCT
ejpam-7050	128	66	the	the	DET
ejpam-7050	128	67	following	follow	VERB
ejpam-7050	128	68	properties	property	NOUN
ejpam-7050	128	69	are	be	AUX
ejpam-7050	128	70	equivalent	equivalent	ADJ
ejpam-7050	128	71	:	:	PUNCT
ejpam-7050	128	72	(	(	PUNCT
ejpam-7050	128	73	1	1	X
ejpam-7050	128	74	)	)	PUNCT
ejpam-7050	128	75	f	f	PROPN
ejpam-7050	128	76	is	be	AUX
ejpam-7050	128	77	upper	upper	ADJ
ejpam-7050	128	78	weakly	weakly	ADJ
ejpam-7050	128	79	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	128	80	,	,	PUNCT
ejpam-7050	128	81	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	128	82	at	at	ADP
ejpam-7050	128	83	a	a	DET
ejpam-7050	128	84	point	point	NOUN
ejpam-7050	128	85	x	x	X
ejpam-7050	128	86	∈	∈	NOUN
ejpam-7050	128	87	x	x	X
ejpam-7050	128	88	;	;	PUNCT
ejpam-7050	129	1	(	(	PUNCT
ejpam-7050	129	2	2	2	X
ejpam-7050	129	3	)	)	PUNCT
ejpam-7050	129	4	x	x	SYM
ejpam-7050	129	5	∈	∈	PROPN
ejpam-7050	129	6	cl⋆(int(cl⋆(f+(σ1σ2	cl⋆(int(cl⋆(f+(σ1σ2	NOUN
ejpam-7050	129	7	-	-	PUNCT
ejpam-7050	129	8	cl(v	cl(v	NOUN
ejpam-7050	129	9	)	)	PUNCT
ejpam-7050	129	10	)	)	PUNCT
ejpam-7050	129	11	)	)	PUNCT
ejpam-7050	129	12	)	)	PUNCT
ejpam-7050	129	13	)	)	PUNCT
ejpam-7050	130	1	for	for	ADP
ejpam-7050	130	2	every	every	DET
ejpam-7050	130	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	130	4	-	-	ADJ
ejpam-7050	130	5	open	open	ADJ
ejpam-7050	130	6	set	set	NOUN
ejpam-7050	130	7	v	v	NOUN
ejpam-7050	130	8	of	of	ADP
ejpam-7050	130	9	y	y	PROPN
ejpam-7050	130	10	containing	contain	VERB
ejpam-7050	130	11	f	f	PROPN
ejpam-7050	130	12	(	(	PUNCT
ejpam-7050	130	13	x	x	NOUN
ejpam-7050	130	14	)	)	PUNCT
ejpam-7050	130	15	;	;	PUNCT
ejpam-7050	130	16	(	(	PUNCT
ejpam-7050	130	17	3	3	X
ejpam-7050	130	18	)	)	PUNCT
ejpam-7050	130	19	x	x	SYM
ejpam-7050	130	20	∈	∈	PROPN
ejpam-7050	130	21	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	130	22	-	-	NOUN
ejpam-7050	130	23	cl(v	cl(v	NOUN
ejpam-7050	130	24	)	)	PUNCT
ejpam-7050	130	25	)	)	PUNCT
ejpam-7050	130	26	)	)	PUNCT
ejpam-7050	130	27	for	for	ADP
ejpam-7050	130	28	every	every	DET
ejpam-7050	130	29	σ1σ2	σ1σ2	NOUN
ejpam-7050	130	30	-	-	ADJ
ejpam-7050	130	31	open	open	ADJ
ejpam-7050	130	32	set	set	NOUN
ejpam-7050	130	33	v	v	NOUN
ejpam-7050	130	34	of	of	ADP
ejpam-7050	130	35	y	y	PROPN
ejpam-7050	130	36	containing	contain	VERB
ejpam-7050	130	37	f	f	PROPN
ejpam-7050	130	38	(	(	PUNCT
ejpam-7050	130	39	x	x	NOUN
ejpam-7050	130	40	)	)	PUNCT
ejpam-7050	130	41	.	.	PUNCT
ejpam-7050	131	1	proof	proof	NOUN
ejpam-7050	131	2	.	.	PUNCT
ejpam-7050	132	1	(	(	PUNCT
ejpam-7050	132	2	1	1	X
ejpam-7050	132	3	)	)	PUNCT
ejpam-7050	132	4	⇒	⇒	NOUN
ejpam-7050	132	5	(	(	PUNCT
ejpam-7050	132	6	2	2	NUM
ejpam-7050	132	7	):	):	PUNCT
ejpam-7050	132	8	let	let	VERB
ejpam-7050	132	9	v	v	PART
ejpam-7050	132	10	be	be	AUX
ejpam-7050	132	11	any	any	DET
ejpam-7050	132	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	132	13	-	-	ADJ
ejpam-7050	132	14	open	open	ADJ
ejpam-7050	132	15	set	set	NOUN
ejpam-7050	132	16	of	of	ADP
ejpam-7050	132	17	y	y	PROPN
ejpam-7050	132	18	containing	contain	VERB
ejpam-7050	132	19	f	f	PROPN
ejpam-7050	132	20	(	(	PUNCT
ejpam-7050	132	21	x	x	NOUN
ejpam-7050	132	22	)	)	PUNCT
ejpam-7050	132	23	.	.	PUNCT
ejpam-7050	133	1	thus	thus	ADV
ejpam-7050	133	2	by	by	ADP
ejpam-7050	133	3	(	(	PUNCT
ejpam-7050	133	4	1	1	NUM
ejpam-7050	133	5	)	)	PUNCT
ejpam-7050	133	6	,	,	PUNCT
ejpam-7050	133	7	there	there	PRON
ejpam-7050	133	8	exists	exist	VERB
ejpam-7050	133	9	a	a	DET
ejpam-7050	133	10	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	133	11	-	-	ADJ
ejpam-7050	133	12	open	open	ADJ
ejpam-7050	133	13	set	set	NOUN
ejpam-7050	133	14	u	u	NOUN
ejpam-7050	133	15	of	of	ADP
ejpam-7050	133	16	x	x	PUNCT
ejpam-7050	133	17	containing	contain	VERB
ejpam-7050	133	18	x	x	PUNCT
ejpam-7050	133	19	such	such	ADJ
ejpam-7050	133	20	that	that	SCONJ
ejpam-7050	133	21	f	f	PROPN
ejpam-7050	133	22	(	(	PUNCT
ejpam-7050	133	23	u	u	NOUN
ejpam-7050	133	24	)	)	PUNCT
ejpam-7050	133	25	⊆	⊆	NUM
ejpam-7050	133	26	σ1σ2	σ1σ2	NOUN
ejpam-7050	133	27	-	-	NUM
ejpam-7050	133	28	cl(v	cl(v	NOUN
ejpam-7050	133	29	)	)	PUNCT
ejpam-7050	133	30	.	.	PUNCT
ejpam-7050	134	1	then	then	ADV
ejpam-7050	134	2	,	,	PUNCT
ejpam-7050	134	3	x	x	PUNCT
ejpam-7050	134	4	∈	∈	NOUN
ejpam-7050	134	5	u	u	NOUN
ejpam-7050	134	6	⊆	⊆	NUM
ejpam-7050	134	7	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	134	8	-	-	PUNCT
ejpam-7050	134	9	cl(v	cl(v	NOUN
ejpam-7050	134	10	)	)	PUNCT
ejpam-7050	134	11	)	)	PUNCT
ejpam-7050	134	12	.	.	PUNCT
ejpam-7050	135	1	since	since	SCONJ
ejpam-7050	135	2	u	u	NOUN
ejpam-7050	135	3	is	be	AUX
ejpam-7050	135	4	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	135	5	-	-	NOUN
ejpam-7050	135	6	open	open	ADJ
ejpam-7050	135	7	,	,	PUNCT
ejpam-7050	135	8	we	we	PRON
ejpam-7050	135	9	have	have	VERB
ejpam-7050	135	10	x	x	X
ejpam-7050	135	11	∈	∈	PROPN
ejpam-7050	135	12	u	u	NOUN
ejpam-7050	135	13	⊆	⊆	NUM
ejpam-7050	135	14	cl⋆(int⋆(cl⋆(u	cl⋆(int⋆(cl⋆(u	NOUN
ejpam-7050	135	15	)	)	PUNCT
ejpam-7050	135	16	)	)	PUNCT
ejpam-7050	135	17	)	)	PUNCT
ejpam-7050	136	1	⊆	⊆	NUM
ejpam-7050	136	2	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7050	136	3	-	-	PUNCT
ejpam-7050	136	4	cl(v	cl(v	NOUN
ejpam-7050	136	5	)	)	PUNCT
ejpam-7050	136	6	)	)	PUNCT
ejpam-7050	136	7	)	)	PUNCT
ejpam-7050	136	8	)	)	PUNCT
ejpam-7050	136	9	)	)	PUNCT
ejpam-7050	136	10	.	.	PUNCT
ejpam-7050	137	1	(	(	PUNCT
ejpam-7050	137	2	2	2	X
ejpam-7050	137	3	)	)	PUNCT
ejpam-7050	137	4	⇒	⇒	NOUN
ejpam-7050	137	5	(	(	PUNCT
ejpam-7050	137	6	3	3	NUM
ejpam-7050	137	7	):	):	PUNCT
ejpam-7050	137	8	let	let	VERB
ejpam-7050	137	9	v	v	PART
ejpam-7050	137	10	be	be	AUX
ejpam-7050	137	11	any	any	DET
ejpam-7050	137	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	137	13	-	-	ADJ
ejpam-7050	137	14	open	open	ADJ
ejpam-7050	137	15	set	set	NOUN
ejpam-7050	137	16	of	of	ADP
ejpam-7050	137	17	y	y	PROPN
ejpam-7050	137	18	containing	contain	VERB
ejpam-7050	137	19	f	f	PROPN
ejpam-7050	137	20	(	(	PUNCT
ejpam-7050	137	21	x	x	NOUN
ejpam-7050	137	22	)	)	PUNCT
ejpam-7050	137	23	.	.	PUNCT
ejpam-7050	138	1	then	then	ADV
ejpam-7050	138	2	by	by	ADP
ejpam-7050	138	3	(	(	PUNCT
ejpam-7050	138	4	2	2	NUM
ejpam-7050	138	5	)	)	PUNCT
ejpam-7050	138	6	,	,	PUNCT
ejpam-7050	138	7	we	we	PRON
ejpam-7050	138	8	have	have	VERB
ejpam-7050	138	9	x	x	PART
ejpam-7050	138	10	∈	∈	PROPN
ejpam-7050	138	11	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7050	138	12	-	-	PUNCT
ejpam-7050	138	13	cl(v	cl(v	NOUN
ejpam-7050	138	14	)	)	PUNCT
ejpam-7050	138	15	)	)	PUNCT
ejpam-7050	138	16	)	)	PUNCT
ejpam-7050	138	17	)	)	PUNCT
ejpam-7050	138	18	)	)	PUNCT
ejpam-7050	138	19	.	.	PUNCT
ejpam-7050	139	1	since	since	SCONJ
ejpam-7050	139	2	x	x	SYM
ejpam-7050	139	3	∈	∈	PROPN
ejpam-7050	139	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	139	5	-	-	PUNCT
ejpam-7050	139	6	cl(v	cl(v	NOUN
ejpam-7050	139	7	)	)	PUNCT
ejpam-7050	139	8	)	)	PUNCT
ejpam-7050	139	9	and	and	CCONJ
ejpam-7050	139	10	by	by	ADP
ejpam-7050	139	11	lemma	lemma	PROPN
ejpam-7050	139	12	4	4	NUM
ejpam-7050	139	13	,	,	PUNCT
ejpam-7050	139	14	x	x	SYM
ejpam-7050	139	15	∈	∈	NOUN
ejpam-7050	139	16	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	139	17	-	-	PUNCT
ejpam-7050	139	18	cl(v	cl(v	NOUN
ejpam-7050	139	19	)	)	PUNCT
ejpam-7050	139	20	)	)	PUNCT
ejpam-7050	139	21	∩	∩	ADJ
ejpam-7050	139	22	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7050	139	23	-	-	PUNCT
ejpam-7050	139	24	cl(v	cl(v	NOUN
ejpam-7050	139	25	)	)	PUNCT
ejpam-7050	139	26	)	)	PUNCT
ejpam-7050	139	27	)	)	PUNCT
ejpam-7050	139	28	)	)	PUNCT
ejpam-7050	139	29	)	)	PUNCT
ejpam-7050	140	1	=	=	SYM
ejpam-7050	140	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	140	3	-	-	NUM
ejpam-7050	140	4	cl(v	cl(v	NOUN
ejpam-7050	140	5	)	)	PUNCT
ejpam-7050	140	6	)	)	PUNCT
ejpam-7050	140	7	)	)	PUNCT
ejpam-7050	140	8	.	.	PUNCT
ejpam-7050	141	1	m.	m.	NOUN
ejpam-7050	141	2	thongmoon	thongmoon	PROPN
ejpam-7050	141	3	,	,	PUNCT
ejpam-7050	141	4	a.	a.	PROPN
ejpam-7050	141	5	sama	sama	PROPN
ejpam-7050	141	6	-	-	PUNCT
ejpam-7050	141	7	ae	ae	PROPN
ejpam-7050	141	8	,	,	PUNCT
ejpam-7050	141	9	c.	c.	PROPN
ejpam-7050	141	10	boonpok	boonpok	PROPN
ejpam-7050	141	11	/	/	SYM
ejpam-7050	141	12	eur	eur	PROPN
ejpam-7050	141	13	.	.	PUNCT
ejpam-7050	142	1	j.	j.	PROPN
ejpam-7050	142	2	pure	pure	PROPN
ejpam-7050	142	3	appl	appl	PROPN
ejpam-7050	142	4	.	.	PROPN
ejpam-7050	142	5	math	math	PROPN
ejpam-7050	142	6	,	,	PUNCT
ejpam-7050	142	7	18	18	NUM
ejpam-7050	142	8	(	(	PUNCT
ejpam-7050	142	9	4	4	NUM
ejpam-7050	142	10	)	)	PUNCT
ejpam-7050	142	11	(	(	PUNCT
ejpam-7050	142	12	2025	2025	NUM
ejpam-7050	142	13	)	)	PUNCT
ejpam-7050	142	14	,	,	PUNCT
ejpam-7050	142	15	7050	7050	NUM
ejpam-7050	142	16	5	5	NUM
ejpam-7050	142	17	of	of	ADP
ejpam-7050	142	18	14	14	NUM
ejpam-7050	142	19	(	(	PUNCT
ejpam-7050	142	20	3	3	NUM
ejpam-7050	142	21	)	)	PUNCT
ejpam-7050	142	22	⇒	⇒	NOUN
ejpam-7050	142	23	(	(	PUNCT
ejpam-7050	142	24	1	1	NUM
ejpam-7050	142	25	):	):	PUNCT
ejpam-7050	142	26	let	let	VERB
ejpam-7050	142	27	v	v	PART
ejpam-7050	142	28	be	be	AUX
ejpam-7050	142	29	any	any	DET
ejpam-7050	142	30	σ1σ2	σ1σ2	NOUN
ejpam-7050	142	31	-	-	ADJ
ejpam-7050	142	32	open	open	ADJ
ejpam-7050	142	33	set	set	NOUN
ejpam-7050	142	34	of	of	ADP
ejpam-7050	142	35	y	y	PROPN
ejpam-7050	142	36	containing	contain	VERB
ejpam-7050	142	37	f	f	PROPN
ejpam-7050	142	38	(	(	PUNCT
ejpam-7050	142	39	x	x	NOUN
ejpam-7050	142	40	)	)	PUNCT
ejpam-7050	142	41	.	.	PUNCT
ejpam-7050	143	1	by	by	ADP
ejpam-7050	143	2	(	(	PUNCT
ejpam-7050	143	3	3	3	NUM
ejpam-7050	143	4	)	)	PUNCT
ejpam-7050	143	5	,	,	PUNCT
ejpam-7050	143	6	we	we	PRON
ejpam-7050	143	7	have	have	VERB
ejpam-7050	143	8	x	x	PART
ejpam-7050	143	9	∈	∈	PROPN
ejpam-7050	143	10	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	143	11	-	-	NOUN
ejpam-7050	143	12	cl(v	cl(v	NOUN
ejpam-7050	143	13	)	)	PUNCT
ejpam-7050	143	14	)	)	PUNCT
ejpam-7050	143	15	)	)	PUNCT
ejpam-7050	144	1	and	and	CCONJ
ejpam-7050	144	2	so	so	ADV
ejpam-7050	144	3	there	there	PRON
ejpam-7050	144	4	exists	exist	VERB
ejpam-7050	144	5	a	a	DET
ejpam-7050	144	6	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	144	7	-	-	ADJ
ejpam-7050	144	8	open	open	ADJ
ejpam-7050	144	9	set	set	NOUN
ejpam-7050	144	10	u	u	NOUN
ejpam-7050	144	11	of	of	ADP
ejpam-7050	144	12	x	x	PUNCT
ejpam-7050	144	13	containing	contain	VERB
ejpam-7050	144	14	x	x	PUNCT
ejpam-7050	144	15	such	such	ADJ
ejpam-7050	144	16	that	that	SCONJ
ejpam-7050	144	17	u	u	NOUN
ejpam-7050	144	18	⊆	⊆	NUM
ejpam-7050	144	19	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	144	20	-	-	PUNCT
ejpam-7050	144	21	cl(v	cl(v	NOUN
ejpam-7050	144	22	)	)	PUNCT
ejpam-7050	144	23	)	)	PUNCT
ejpam-7050	144	24	;	;	PUNCT
ejpam-7050	144	25	hence	hence	ADV
ejpam-7050	144	26	f	f	PROPN
ejpam-7050	144	27	(	(	PUNCT
ejpam-7050	144	28	u	u	NOUN
ejpam-7050	144	29	)	)	PUNCT
ejpam-7050	144	30	⊆	⊆	NUM
ejpam-7050	144	31	σ1σ2	σ1σ2	NOUN
ejpam-7050	144	32	-	-	NUM
ejpam-7050	144	33	cl(v	cl(v	NOUN
ejpam-7050	144	34	)	)	PUNCT
ejpam-7050	144	35	.	.	PUNCT
ejpam-7050	145	1	this	this	PRON
ejpam-7050	145	2	shows	show	VERB
ejpam-7050	145	3	that	that	SCONJ
ejpam-7050	145	4	f	f	PROPN
ejpam-7050	145	5	is	be	AUX
ejpam-7050	145	6	upper	upper	ADJ
ejpam-7050	145	7	weakly	weakly	ADJ
ejpam-7050	145	8	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	145	9	,	,	PUNCT
ejpam-7050	145	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	145	11	at	at	ADP
ejpam-7050	145	12	x.	x.	NOUN
ejpam-7050	145	13	definition	definition	NOUN
ejpam-7050	145	14	2	2	NUM
ejpam-7050	145	15	.	.	PUNCT
ejpam-7050	145	16	a	a	DET
ejpam-7050	145	17	multifunction	multifunction	NOUN
ejpam-7050	146	1	f	f	NOUN
ejpam-7050	146	2	:	:	PUNCT
ejpam-7050	146	3	(	(	PUNCT
ejpam-7050	146	4	x	x	X
ejpam-7050	146	5	,	,	PUNCT
ejpam-7050	146	6	τ	τ	PROPN
ejpam-7050	146	7	,	,	PUNCT
ejpam-7050	146	8	i	i	NOUN
ejpam-7050	146	9	)	)	PUNCT
ejpam-7050	146	10	→	→	PUNCT
ejpam-7050	146	11	(	(	PUNCT
ejpam-7050	146	12	y	y	PROPN
ejpam-7050	146	13	,	,	PUNCT
ejpam-7050	146	14	σ1	σ1	PROPN
ejpam-7050	146	15	,	,	PUNCT
ejpam-7050	146	16	σ2	σ2	PROPN
ejpam-7050	146	17	)	)	PUNCT
ejpam-7050	146	18	is	be	AUX
ejpam-7050	146	19	called	call	VERB
ejpam-7050	146	20	lower	low	ADJ
ejpam-7050	146	21	weakly	weakly	ADJ
ejpam-7050	146	22	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	146	23	,	,	PUNCT
ejpam-7050	146	24	σ2)continuous	σ2)continuous	ADJ
ejpam-7050	146	25	at	at	ADP
ejpam-7050	146	26	a	a	DET
ejpam-7050	146	27	point	point	NOUN
ejpam-7050	146	28	x	x	PUNCT
ejpam-7050	146	29	of	of	ADP
ejpam-7050	146	30	x	x	PRON
ejpam-7050	146	31	if	if	SCONJ
ejpam-7050	146	32	for	for	ADP
ejpam-7050	146	33	each	each	DET
ejpam-7050	146	34	σ1σ2	σ1σ2	VERB
ejpam-7050	146	35	-	-	ADJ
ejpam-7050	146	36	open	open	ADJ
ejpam-7050	146	37	set	set	NOUN
ejpam-7050	146	38	v	v	NOUN
ejpam-7050	146	39	of	of	ADP
ejpam-7050	146	40	y	y	PRON
ejpam-7050	146	41	such	such	ADJ
ejpam-7050	146	42	that	that	SCONJ
ejpam-7050	146	43	f	f	PROPN
ejpam-7050	146	44	(	(	PUNCT
ejpam-7050	146	45	x)∩v	x)∩v	PROPN
ejpam-7050	146	46	̸=	̸=	PROPN
ejpam-7050	146	47	∅	∅	NOUN
ejpam-7050	146	48	,	,	PUNCT
ejpam-7050	146	49	there	there	PRON
ejpam-7050	146	50	exists	exist	VERB
ejpam-7050	146	51	a	a	DET
ejpam-7050	146	52	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	146	53	-	-	ADJ
ejpam-7050	146	54	open	open	ADJ
ejpam-7050	146	55	set	set	NOUN
ejpam-7050	146	56	u	u	NOUN
ejpam-7050	146	57	of	of	ADP
ejpam-7050	146	58	x	x	PUNCT
ejpam-7050	146	59	containing	contain	VERB
ejpam-7050	146	60	x	x	PUNCT
ejpam-7050	146	61	such	such	ADJ
ejpam-7050	146	62	that	that	SCONJ
ejpam-7050	146	63	f	f	PROPN
ejpam-7050	146	64	(	(	PUNCT
ejpam-7050	146	65	z	z	NOUN
ejpam-7050	146	66	)	)	PUNCT
ejpam-7050	146	67	∩	∩	NOUN
ejpam-7050	147	1	σ1σ2	σ1σ2	NOUN
ejpam-7050	147	2	-	-	NUM
ejpam-7050	147	3	cl(v	cl(v	NOUN
ejpam-7050	147	4	)	)	PUNCT
ejpam-7050	147	5	̸=	̸=	NOUN
ejpam-7050	147	6	∅	∅	NOUN
ejpam-7050	147	7	for	for	ADP
ejpam-7050	147	8	every	every	DET
ejpam-7050	147	9	z	z	NOUN
ejpam-7050	147	10	∈	∈	PROPN
ejpam-7050	147	11	u	u	NOUN
ejpam-7050	147	12	.	.	PUNCT
ejpam-7050	148	1	a	a	DET
ejpam-7050	148	2	multifunction	multifunction	NOUN
ejpam-7050	148	3	f	f	NOUN
ejpam-7050	148	4	:	:	PUNCT
ejpam-7050	148	5	(	(	PUNCT
ejpam-7050	148	6	x	x	X
ejpam-7050	148	7	,	,	PUNCT
ejpam-7050	148	8	τ	τ	PROPN
ejpam-7050	148	9	,	,	PUNCT
ejpam-7050	148	10	i	i	NOUN
ejpam-7050	148	11	)	)	PUNCT
ejpam-7050	148	12	→	→	PUNCT
ejpam-7050	148	13	(	(	PUNCT
ejpam-7050	148	14	y	y	PROPN
ejpam-7050	148	15	,	,	PUNCT
ejpam-7050	148	16	σ1	σ1	PROPN
ejpam-7050	148	17	,	,	PUNCT
ejpam-7050	148	18	σ2	σ2	PROPN
ejpam-7050	148	19	)	)	PUNCT
ejpam-7050	148	20	is	be	AUX
ejpam-7050	148	21	called	call	VERB
ejpam-7050	148	22	lower	low	ADJ
ejpam-7050	148	23	weakly	weakly	ADJ
ejpam-7050	148	24	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	148	25	,	,	PUNCT
ejpam-7050	148	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7050	148	27	if	if	SCONJ
ejpam-7050	148	28	f	f	PROPN
ejpam-7050	148	29	is	be	AUX
ejpam-7050	148	30	lower	low	ADJ
ejpam-7050	148	31	weakly	weakly	ADJ
ejpam-7050	148	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	148	33	,	,	PUNCT
ejpam-7050	148	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	148	35	at	at	ADP
ejpam-7050	148	36	each	each	DET
ejpam-7050	148	37	point	point	NOUN
ejpam-7050	148	38	of	of	ADP
ejpam-7050	148	39	x.	x.	NOUN
ejpam-7050	148	40	theorem	theorem	VERB
ejpam-7050	148	41	2	2	NUM
ejpam-7050	148	42	.	.	X
ejpam-7050	148	43	for	for	ADP
ejpam-7050	148	44	a	a	DET
ejpam-7050	148	45	multifunction	multifunction	NOUN
ejpam-7050	148	46	f	f	NOUN
ejpam-7050	148	47	:	:	PUNCT
ejpam-7050	148	48	(	(	PUNCT
ejpam-7050	148	49	x	x	X
ejpam-7050	148	50	,	,	PUNCT
ejpam-7050	148	51	τ	τ	PROPN
ejpam-7050	148	52	,	,	PUNCT
ejpam-7050	148	53	i	i	NOUN
ejpam-7050	148	54	)	)	PUNCT
ejpam-7050	148	55	→	→	PUNCT
ejpam-7050	148	56	(	(	PUNCT
ejpam-7050	148	57	y	y	PROPN
ejpam-7050	148	58	,	,	PUNCT
ejpam-7050	148	59	σ1	σ1	PROPN
ejpam-7050	148	60	,	,	PUNCT
ejpam-7050	148	61	σ2	σ2	NOUN
ejpam-7050	148	62	)	)	PUNCT
ejpam-7050	148	63	,	,	PUNCT
ejpam-7050	148	64	the	the	DET
ejpam-7050	148	65	following	follow	VERB
ejpam-7050	148	66	properties	property	NOUN
ejpam-7050	148	67	are	be	AUX
ejpam-7050	148	68	equivalent	equivalent	ADJ
ejpam-7050	148	69	:	:	PUNCT
ejpam-7050	148	70	(	(	PUNCT
ejpam-7050	148	71	1	1	X
ejpam-7050	148	72	)	)	PUNCT
ejpam-7050	148	73	f	f	PROPN
ejpam-7050	148	74	is	be	AUX
ejpam-7050	148	75	lower	low	ADJ
ejpam-7050	148	76	weakly	weakly	ADJ
ejpam-7050	148	77	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	148	78	,	,	PUNCT
ejpam-7050	148	79	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	148	80	at	at	ADP
ejpam-7050	148	81	a	a	DET
ejpam-7050	148	82	point	point	NOUN
ejpam-7050	148	83	x	x	X
ejpam-7050	148	84	∈	∈	NOUN
ejpam-7050	148	85	x	x	X
ejpam-7050	148	86	;	;	PUNCT
ejpam-7050	148	87	(	(	PUNCT
ejpam-7050	148	88	2	2	X
ejpam-7050	148	89	)	)	PUNCT
ejpam-7050	148	90	x	x	SYM
ejpam-7050	148	91	∈	∈	PROPN
ejpam-7050	148	92	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7050	148	93	-	-	PUNCT
ejpam-7050	148	94	cl(v	cl(v	NOUN
ejpam-7050	148	95	)	)	PUNCT
ejpam-7050	148	96	)	)	PUNCT
ejpam-7050	148	97	)	)	PUNCT
ejpam-7050	148	98	)	)	PUNCT
ejpam-7050	148	99	)	)	PUNCT
ejpam-7050	148	100	for	for	ADP
ejpam-7050	148	101	every	every	DET
ejpam-7050	148	102	σ1σ2	σ1σ2	NOUN
ejpam-7050	148	103	-	-	ADJ
ejpam-7050	148	104	open	open	ADJ
ejpam-7050	148	105	set	set	NOUN
ejpam-7050	148	106	v	v	NOUN
ejpam-7050	148	107	of	of	ADP
ejpam-7050	148	108	y	y	PRON
ejpam-7050	148	109	such	such	ADJ
ejpam-7050	148	110	that	that	SCONJ
ejpam-7050	148	111	f	f	PROPN
ejpam-7050	148	112	(	(	PUNCT
ejpam-7050	148	113	x	x	NOUN
ejpam-7050	148	114	)	)	PUNCT
ejpam-7050	148	115	∩	∩	NOUN
ejpam-7050	148	116	v	v	ADP
ejpam-7050	148	117	̸=	̸=	PROPN
ejpam-7050	148	118	∅	∅	NOUN
ejpam-7050	148	119	;	;	PUNCT
ejpam-7050	148	120	(	(	PUNCT
ejpam-7050	148	121	3	3	X
ejpam-7050	148	122	)	)	PUNCT
ejpam-7050	148	123	x	x	SYM
ejpam-7050	148	124	∈	∈	NOUN
ejpam-7050	148	125	βint⋆(f−(σ1σ2	βint⋆(f−(σ1σ2	NOUN
ejpam-7050	148	126	-	-	PUNCT
ejpam-7050	148	127	cl(v	cl(v	NOUN
ejpam-7050	148	128	)	)	PUNCT
ejpam-7050	148	129	)	)	PUNCT
ejpam-7050	148	130	)	)	PUNCT
ejpam-7050	148	131	for	for	ADP
ejpam-7050	148	132	every	every	DET
ejpam-7050	148	133	σ1σ2	σ1σ2	NOUN
ejpam-7050	148	134	-	-	ADJ
ejpam-7050	148	135	open	open	ADJ
ejpam-7050	148	136	set	set	NOUN
ejpam-7050	148	137	v	v	NOUN
ejpam-7050	148	138	of	of	ADP
ejpam-7050	148	139	y	y	PRON
ejpam-7050	148	140	such	such	ADJ
ejpam-7050	148	141	that	that	SCONJ
ejpam-7050	148	142	f	f	PROPN
ejpam-7050	148	143	(	(	PUNCT
ejpam-7050	148	144	x)∩v	x)∩v	PROPN
ejpam-7050	148	145	̸=	̸=	PROPN
ejpam-7050	148	146	∅.	∅.	PRON
ejpam-7050	148	147	proof	proof	NOUN
ejpam-7050	148	148	.	.	PUNCT
ejpam-7050	149	1	the	the	DET
ejpam-7050	149	2	proof	proof	NOUN
ejpam-7050	149	3	is	be	AUX
ejpam-7050	149	4	similar	similar	ADJ
ejpam-7050	149	5	to	to	ADP
ejpam-7050	149	6	that	that	PRON
ejpam-7050	149	7	of	of	ADP
ejpam-7050	149	8	theorem	theorem	NOUN
ejpam-7050	149	9	1	1	NUM
ejpam-7050	149	10	.	.	PUNCT
ejpam-7050	149	11	definition	definition	NOUN
ejpam-7050	149	12	3	3	NUM
ejpam-7050	149	13	.	.	PUNCT
ejpam-7050	150	1	a	a	DET
ejpam-7050	150	2	function	function	NOUN
ejpam-7050	150	3	f	f	NOUN
ejpam-7050	150	4	:	:	PUNCT
ejpam-7050	150	5	(	(	PUNCT
ejpam-7050	150	6	x	x	X
ejpam-7050	150	7	,	,	PUNCT
ejpam-7050	150	8	τ	τ	PROPN
ejpam-7050	150	9	,	,	PUNCT
ejpam-7050	150	10	i	i	NOUN
ejpam-7050	150	11	)	)	PUNCT
ejpam-7050	150	12	→	→	PUNCT
ejpam-7050	150	13	(	(	PUNCT
ejpam-7050	150	14	y	y	PROPN
ejpam-7050	150	15	,	,	PUNCT
ejpam-7050	150	16	σ1	σ1	PROPN
ejpam-7050	150	17	,	,	PUNCT
ejpam-7050	150	18	σ2	σ2	PROPN
ejpam-7050	150	19	)	)	PUNCT
ejpam-7050	150	20	is	be	AUX
ejpam-7050	150	21	said	say	VERB
ejpam-7050	150	22	to	to	PART
ejpam-7050	150	23	be	be	AUX
ejpam-7050	150	24	weakly	weakly	ADJ
ejpam-7050	150	25	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	150	26	,	,	PUNCT
ejpam-7050	150	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7050	150	28	at	at	ADP
ejpam-7050	150	29	a	a	DET
ejpam-7050	150	30	point	point	NOUN
ejpam-7050	150	31	x	x	SYM
ejpam-7050	150	32	∈	∈	NOUN
ejpam-7050	150	33	x	x	PUNCT
ejpam-7050	150	34	if	if	SCONJ
ejpam-7050	150	35	for	for	ADP
ejpam-7050	150	36	each	each	DET
ejpam-7050	150	37	σ1σ2	σ1σ2	VERB
ejpam-7050	150	38	-	-	ADJ
ejpam-7050	150	39	open	open	ADJ
ejpam-7050	150	40	set	set	NOUN
ejpam-7050	150	41	v	v	NOUN
ejpam-7050	150	42	of	of	ADP
ejpam-7050	150	43	y	y	NOUN
ejpam-7050	150	44	containing	contain	VERB
ejpam-7050	150	45	f(x	f(x	PROPN
ejpam-7050	150	46	)	)	PUNCT
ejpam-7050	150	47	,	,	PUNCT
ejpam-7050	150	48	there	there	PRON
ejpam-7050	150	49	exists	exist	VERB
ejpam-7050	150	50	a	a	DET
ejpam-7050	150	51	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	150	52	-	-	ADJ
ejpam-7050	150	53	open	open	ADJ
ejpam-7050	150	54	set	set	NOUN
ejpam-7050	150	55	u	u	NOUN
ejpam-7050	150	56	of	of	ADP
ejpam-7050	150	57	x	x	PUNCT
ejpam-7050	150	58	containing	contain	VERB
ejpam-7050	150	59	x	x	PUNCT
ejpam-7050	150	60	such	such	ADJ
ejpam-7050	150	61	that	that	DET
ejpam-7050	150	62	f(u	f(u	PROPN
ejpam-7050	150	63	)	)	PUNCT
ejpam-7050	151	1	⊆	⊆	NUM
ejpam-7050	151	2	σ1σ2	σ1σ2	NOUN
ejpam-7050	151	3	-	-	NUM
ejpam-7050	151	4	cl(v	cl(v	NOUN
ejpam-7050	151	5	)	)	PUNCT
ejpam-7050	151	6	.	.	PUNCT
ejpam-7050	152	1	a	a	DET
ejpam-7050	152	2	function	function	NOUN
ejpam-7050	152	3	f	f	NOUN
ejpam-7050	152	4	:	:	PUNCT
ejpam-7050	152	5	(	(	PUNCT
ejpam-7050	152	6	x	x	X
ejpam-7050	152	7	,	,	PUNCT
ejpam-7050	152	8	τ	τ	PROPN
ejpam-7050	152	9	,	,	PUNCT
ejpam-7050	152	10	i	i	NOUN
ejpam-7050	152	11	)	)	PUNCT
ejpam-7050	152	12	→	→	PUNCT
ejpam-7050	152	13	(	(	PUNCT
ejpam-7050	152	14	y	y	PROPN
ejpam-7050	152	15	,	,	PUNCT
ejpam-7050	152	16	σ1	σ1	PROPN
ejpam-7050	152	17	,	,	PUNCT
ejpam-7050	152	18	σ2	σ2	PROPN
ejpam-7050	152	19	)	)	PUNCT
ejpam-7050	152	20	is	be	AUX
ejpam-7050	152	21	said	say	VERB
ejpam-7050	152	22	to	to	PART
ejpam-7050	152	23	be	be	AUX
ejpam-7050	152	24	weakly	weakly	ADJ
ejpam-7050	152	25	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	152	26	,	,	PUNCT
ejpam-7050	152	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	152	28	if	if	SCONJ
ejpam-7050	152	29	f	f	PROPN
ejpam-7050	152	30	is	be	AUX
ejpam-7050	152	31	τ⋆β(σ1	τ⋆β(σ1	PROPN
ejpam-7050	152	32	,	,	PUNCT
ejpam-7050	152	33	σ2)continuous	σ2)continuous	ADJ
ejpam-7050	152	34	at	at	ADP
ejpam-7050	152	35	each	each	DET
ejpam-7050	152	36	point	point	NOUN
ejpam-7050	152	37	of	of	ADP
ejpam-7050	152	38	x.	x.	PROPN
ejpam-7050	152	39	corollary	corollary	NOUN
ejpam-7050	152	40	1	1	NUM
ejpam-7050	152	41	.	.	PUNCT
ejpam-7050	153	1	for	for	ADP
ejpam-7050	153	2	a	a	DET
ejpam-7050	153	3	function	function	NOUN
ejpam-7050	153	4	f	f	NOUN
ejpam-7050	153	5	:	:	PUNCT
ejpam-7050	153	6	(	(	PUNCT
ejpam-7050	153	7	x	x	X
ejpam-7050	153	8	,	,	PUNCT
ejpam-7050	153	9	τ	τ	PROPN
ejpam-7050	153	10	,	,	PUNCT
ejpam-7050	153	11	i	i	NOUN
ejpam-7050	153	12	)	)	PUNCT
ejpam-7050	153	13	→	→	PUNCT
ejpam-7050	153	14	(	(	PUNCT
ejpam-7050	153	15	y	y	PROPN
ejpam-7050	153	16	,	,	PUNCT
ejpam-7050	153	17	σ1	σ1	PROPN
ejpam-7050	153	18	,	,	PUNCT
ejpam-7050	153	19	σ2	σ2	NOUN
ejpam-7050	153	20	)	)	PUNCT
ejpam-7050	153	21	,	,	PUNCT
ejpam-7050	153	22	the	the	DET
ejpam-7050	153	23	following	follow	VERB
ejpam-7050	153	24	properties	property	NOUN
ejpam-7050	153	25	are	be	AUX
ejpam-7050	153	26	equivalent	equivalent	ADJ
ejpam-7050	153	27	:	:	PUNCT
ejpam-7050	153	28	(	(	PUNCT
ejpam-7050	153	29	1	1	X
ejpam-7050	153	30	)	)	PUNCT
ejpam-7050	153	31	f	f	PROPN
ejpam-7050	153	32	is	be	AUX
ejpam-7050	153	33	weakly	weakly	ADJ
ejpam-7050	153	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	153	35	,	,	PUNCT
ejpam-7050	153	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	153	37	at	at	ADP
ejpam-7050	153	38	a	a	DET
ejpam-7050	153	39	point	point	NOUN
ejpam-7050	153	40	x	x	X
ejpam-7050	153	41	∈	∈	NOUN
ejpam-7050	153	42	x	x	X
ejpam-7050	153	43	;	;	PUNCT
ejpam-7050	153	44	(	(	PUNCT
ejpam-7050	153	45	2	2	X
ejpam-7050	153	46	)	)	PUNCT
ejpam-7050	153	47	x	x	SYM
ejpam-7050	153	48	∈	∈	PROPN
ejpam-7050	153	49	cl⋆(int⋆(cl⋆(f−1(σ1σ2	cl⋆(int⋆(cl⋆(f−1(σ1σ2	VERB
ejpam-7050	153	50	-	-	PUNCT
ejpam-7050	153	51	cl(v	cl(v	NOUN
ejpam-7050	153	52	)	)	PUNCT
ejpam-7050	153	53	)	)	PUNCT
ejpam-7050	153	54	)	)	PUNCT
ejpam-7050	153	55	)	)	PUNCT
ejpam-7050	153	56	)	)	PUNCT
ejpam-7050	154	1	for	for	ADP
ejpam-7050	154	2	every	every	DET
ejpam-7050	154	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	154	4	-	-	ADJ
ejpam-7050	154	5	open	open	ADJ
ejpam-7050	154	6	set	set	NOUN
ejpam-7050	154	7	v	v	NOUN
ejpam-7050	154	8	of	of	ADP
ejpam-7050	154	9	y	y	NOUN
ejpam-7050	154	10	containing	contain	VERB
ejpam-7050	154	11	f(x	f(x	PROPN
ejpam-7050	154	12	)	)	PUNCT
ejpam-7050	154	13	;	;	PUNCT
ejpam-7050	154	14	(	(	PUNCT
ejpam-7050	154	15	3	3	X
ejpam-7050	154	16	)	)	PUNCT
ejpam-7050	154	17	x	x	SYM
ejpam-7050	154	18	∈	∈	PROPN
ejpam-7050	154	19	βint⋆(f−1(σ1σ2	βint⋆(f−1(σ1σ2	NOUN
ejpam-7050	154	20	-	-	X
ejpam-7050	154	21	cl(v	cl(v	NOUN
ejpam-7050	154	22	)	)	PUNCT
ejpam-7050	154	23	)	)	PUNCT
ejpam-7050	154	24	)	)	PUNCT
ejpam-7050	154	25	for	for	ADP
ejpam-7050	154	26	every	every	DET
ejpam-7050	154	27	σ1σ2	σ1σ2	NOUN
ejpam-7050	154	28	-	-	ADJ
ejpam-7050	154	29	open	open	ADJ
ejpam-7050	154	30	set	set	NOUN
ejpam-7050	154	31	v	v	NOUN
ejpam-7050	154	32	of	of	ADP
ejpam-7050	154	33	y	y	NOUN
ejpam-7050	154	34	containing	contain	VERB
ejpam-7050	154	35	f(x	f(x	PROPN
ejpam-7050	154	36	)	)	PUNCT
ejpam-7050	154	37	.	.	PUNCT
ejpam-7050	155	1	theorem	theorem	NOUN
ejpam-7050	155	2	3	3	NUM
ejpam-7050	155	3	.	.	X
ejpam-7050	155	4	for	for	ADP
ejpam-7050	155	5	a	a	DET
ejpam-7050	155	6	multifunction	multifunction	NOUN
ejpam-7050	156	1	f	f	NOUN
ejpam-7050	156	2	:	:	PUNCT
ejpam-7050	156	3	(	(	PUNCT
ejpam-7050	156	4	x	x	X
ejpam-7050	156	5	,	,	PUNCT
ejpam-7050	156	6	τ	τ	PROPN
ejpam-7050	156	7	,	,	PUNCT
ejpam-7050	156	8	i	i	NOUN
ejpam-7050	156	9	)	)	PUNCT
ejpam-7050	156	10	→	→	PUNCT
ejpam-7050	156	11	(	(	PUNCT
ejpam-7050	156	12	y	y	PROPN
ejpam-7050	156	13	,	,	PUNCT
ejpam-7050	156	14	σ1	σ1	PROPN
ejpam-7050	156	15	,	,	PUNCT
ejpam-7050	156	16	σ2	σ2	NOUN
ejpam-7050	156	17	)	)	PUNCT
ejpam-7050	156	18	,	,	PUNCT
ejpam-7050	156	19	the	the	DET
ejpam-7050	156	20	following	follow	VERB
ejpam-7050	156	21	properties	property	NOUN
ejpam-7050	156	22	are	be	AUX
ejpam-7050	156	23	equivalent	equivalent	ADJ
ejpam-7050	156	24	:	:	PUNCT
ejpam-7050	156	25	(	(	PUNCT
ejpam-7050	156	26	1	1	X
ejpam-7050	156	27	)	)	PUNCT
ejpam-7050	156	28	f	f	PROPN
ejpam-7050	156	29	is	be	AUX
ejpam-7050	156	30	upper	upper	ADJ
ejpam-7050	156	31	weakly	weakly	ADJ
ejpam-7050	156	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	156	33	,	,	PUNCT
ejpam-7050	156	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	156	35	;	;	PUNCT
ejpam-7050	156	36	m.	m.	NOUN
ejpam-7050	156	37	thongmoon	thongmoon	NOUN
ejpam-7050	156	38	,	,	PUNCT
ejpam-7050	156	39	a.	a.	PROPN
ejpam-7050	156	40	sama	sama	PROPN
ejpam-7050	156	41	-	-	PUNCT
ejpam-7050	156	42	ae	ae	PROPN
ejpam-7050	156	43	,	,	PUNCT
ejpam-7050	156	44	c.	c.	PROPN
ejpam-7050	156	45	boonpok	boonpok	PROPN
ejpam-7050	156	46	/	/	SYM
ejpam-7050	156	47	eur	eur	PROPN
ejpam-7050	156	48	.	.	PUNCT
ejpam-7050	157	1	j.	j.	PROPN
ejpam-7050	157	2	pure	pure	PROPN
ejpam-7050	157	3	appl	appl	PROPN
ejpam-7050	157	4	.	.	PROPN
ejpam-7050	157	5	math	math	PROPN
ejpam-7050	157	6	,	,	PUNCT
ejpam-7050	157	7	18	18	NUM
ejpam-7050	157	8	(	(	PUNCT
ejpam-7050	157	9	4	4	NUM
ejpam-7050	157	10	)	)	PUNCT
ejpam-7050	157	11	(	(	PUNCT
ejpam-7050	157	12	2025	2025	NUM
ejpam-7050	157	13	)	)	PUNCT
ejpam-7050	157	14	,	,	PUNCT
ejpam-7050	157	15	7050	7050	NUM
ejpam-7050	157	16	6	6	NUM
ejpam-7050	157	17	of	of	ADP
ejpam-7050	157	18	14	14	NUM
ejpam-7050	157	19	(	(	PUNCT
ejpam-7050	157	20	2	2	NUM
ejpam-7050	157	21	)	)	PUNCT
ejpam-7050	157	22	f+(v	f+(v	NOUN
ejpam-7050	157	23	)	)	PUNCT
ejpam-7050	158	1	⊆	⊆	NUM
ejpam-7050	158	2	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7050	158	3	-	-	PUNCT
ejpam-7050	158	4	cl(v	cl(v	NOUN
ejpam-7050	158	5	)	)	PUNCT
ejpam-7050	158	6	)	)	PUNCT
ejpam-7050	158	7	)	)	PUNCT
ejpam-7050	158	8	)	)	PUNCT
ejpam-7050	158	9	)	)	PUNCT
ejpam-7050	158	10	for	for	ADP
ejpam-7050	158	11	every	every	DET
ejpam-7050	158	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	158	13	-	-	ADJ
ejpam-7050	158	14	open	open	ADJ
ejpam-7050	158	15	set	set	NOUN
ejpam-7050	158	16	v	v	NOUN
ejpam-7050	158	17	of	of	ADP
ejpam-7050	158	18	y	y	PROPN
ejpam-7050	158	19	;	;	PUNCT
ejpam-7050	158	20	(	(	PUNCT
ejpam-7050	158	21	3	3	X
ejpam-7050	158	22	)	)	PUNCT
ejpam-7050	158	23	int⋆(cl⋆(int⋆(f−(v	int⋆(cl⋆(int⋆(f−(v	NOUN
ejpam-7050	158	24	)	)	PUNCT
ejpam-7050	158	25	)	)	PUNCT
ejpam-7050	158	26	)	)	PUNCT
ejpam-7050	158	27	)	)	PUNCT
ejpam-7050	159	1	⊆	⊆	X
ejpam-7050	159	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	159	3	-	-	PUNCT
ejpam-7050	159	4	cl(v	cl(v	NOUN
ejpam-7050	159	5	)	)	PUNCT
ejpam-7050	159	6	)	)	PUNCT
ejpam-7050	159	7	for	for	ADP
ejpam-7050	159	8	every	every	DET
ejpam-7050	159	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	159	10	-	-	ADJ
ejpam-7050	159	11	open	open	ADJ
ejpam-7050	159	12	set	set	NOUN
ejpam-7050	159	13	v	v	NOUN
ejpam-7050	159	14	of	of	ADP
ejpam-7050	159	15	y	y	PROPN
ejpam-7050	159	16	;	;	PUNCT
ejpam-7050	159	17	(	(	PUNCT
ejpam-7050	159	18	4	4	X
ejpam-7050	159	19	)	)	PUNCT
ejpam-7050	159	20	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-7050	159	21	-	-	PUNCT
ejpam-7050	159	22	int(k	int(k	PROPN
ejpam-7050	159	23	)	)	PUNCT
ejpam-7050	159	24	)	)	PUNCT
ejpam-7050	159	25	)	)	PUNCT
ejpam-7050	159	26	)	)	PUNCT
ejpam-7050	159	27	)	)	PUNCT
ejpam-7050	160	1	⊆	⊆	X
ejpam-7050	160	2	f−(k	f−(k	PROPN
ejpam-7050	160	3	)	)	PUNCT
ejpam-7050	160	4	for	for	ADP
ejpam-7050	160	5	every	every	DET
ejpam-7050	160	6	σ1σ2	σ1σ2	NUM
ejpam-7050	160	7	-	-	PUNCT
ejpam-7050	160	8	closed	closed	ADJ
ejpam-7050	160	9	set	set	NOUN
ejpam-7050	160	10	k	k	PROPN
ejpam-7050	160	11	of	of	ADP
ejpam-7050	160	12	y	y	PROPN
ejpam-7050	160	13	;	;	PUNCT
ejpam-7050	160	14	(	(	PUNCT
ejpam-7050	160	15	5	5	X
ejpam-7050	160	16	)	)	PUNCT
ejpam-7050	160	17	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	160	18	-	-	PUNCT
ejpam-7050	160	19	int(k	int(k	NOUN
ejpam-7050	160	20	)	)	PUNCT
ejpam-7050	160	21	)	)	PUNCT
ejpam-7050	160	22	)	)	PUNCT
ejpam-7050	161	1	⊆	⊆	X
ejpam-7050	161	2	f−(k	f−(k	PROPN
ejpam-7050	161	3	)	)	PUNCT
ejpam-7050	161	4	for	for	ADP
ejpam-7050	161	5	every	every	DET
ejpam-7050	161	6	σ1σ2	σ1σ2	NUM
ejpam-7050	161	7	-	-	PUNCT
ejpam-7050	161	8	closed	closed	ADJ
ejpam-7050	161	9	set	set	NOUN
ejpam-7050	161	10	k	k	PROPN
ejpam-7050	161	11	of	of	ADP
ejpam-7050	161	12	y	y	PROPN
ejpam-7050	161	13	;	;	PUNCT
ejpam-7050	161	14	(	(	PUNCT
ejpam-7050	161	15	6	6	X
ejpam-7050	161	16	)	)	PUNCT
ejpam-7050	161	17	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	161	18	-	-	PUNCT
ejpam-7050	161	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	161	20	-	-	PUNCT
ejpam-7050	161	21	cl(b	cl(b	NOUN
ejpam-7050	161	22	)	)	PUNCT
ejpam-7050	161	23	)	)	PUNCT
ejpam-7050	161	24	)	)	PUNCT
ejpam-7050	161	25	)	)	PUNCT
ejpam-7050	162	1	⊆	⊆	X
ejpam-7050	162	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7050	162	3	-	-	PUNCT
ejpam-7050	162	4	cl(b	cl(b	NOUN
ejpam-7050	162	5	)	)	PUNCT
ejpam-7050	162	6	)	)	PUNCT
ejpam-7050	163	1	for	for	ADP
ejpam-7050	163	2	every	every	DET
ejpam-7050	163	3	subset	subset	NOUN
ejpam-7050	163	4	b	b	PROPN
ejpam-7050	163	5	of	of	ADP
ejpam-7050	163	6	y	y	PROPN
ejpam-7050	163	7	;	;	PUNCT
ejpam-7050	163	8	(	(	PUNCT
ejpam-7050	163	9	7	7	X
ejpam-7050	163	10	)	)	PUNCT
ejpam-7050	163	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	163	12	-	-	PUNCT
ejpam-7050	163	13	int(b	int(b	NOUN
ejpam-7050	163	14	)	)	PUNCT
ejpam-7050	163	15	)	)	PUNCT
ejpam-7050	163	16	⊆	⊆	NUM
ejpam-7050	163	17	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NUM
ejpam-7050	163	18	-	-	PUNCT
ejpam-7050	163	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	163	20	-	-	PUNCT
ejpam-7050	163	21	int(b	int(b	NOUN
ejpam-7050	163	22	)	)	PUNCT
ejpam-7050	163	23	)	)	PUNCT
ejpam-7050	163	24	)	)	PUNCT
ejpam-7050	163	25	)	)	PUNCT
ejpam-7050	163	26	for	for	ADP
ejpam-7050	163	27	every	every	DET
ejpam-7050	163	28	subset	subset	NOUN
ejpam-7050	163	29	b	b	PROPN
ejpam-7050	163	30	of	of	ADP
ejpam-7050	163	31	y	y	PROPN
ejpam-7050	163	32	;	;	PUNCT
ejpam-7050	163	33	(	(	PUNCT
ejpam-7050	163	34	8)	8)	NUM
ejpam-7050	163	35	f+(v	f+(v	NOUN
ejpam-7050	163	36	)	)	PUNCT
ejpam-7050	164	1	⊆	⊆	NUM
ejpam-7050	164	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	164	3	-	-	X
ejpam-7050	164	4	cl(v	cl(v	NOUN
ejpam-7050	164	5	)	)	PUNCT
ejpam-7050	164	6	)	)	PUNCT
ejpam-7050	164	7	)	)	PUNCT
ejpam-7050	164	8	for	for	ADP
ejpam-7050	164	9	every	every	DET
ejpam-7050	164	10	σ1σ2	σ1σ2	NOUN
ejpam-7050	164	11	-	-	ADJ
ejpam-7050	164	12	open	open	ADJ
ejpam-7050	164	13	set	set	NOUN
ejpam-7050	164	14	v	v	NOUN
ejpam-7050	164	15	of	of	ADP
ejpam-7050	164	16	y	y	PROPN
ejpam-7050	164	17	;	;	PUNCT
ejpam-7050	164	18	(	(	PUNCT
ejpam-7050	164	19	9	9	NUM
ejpam-7050	164	20	)	)	PUNCT
ejpam-7050	164	21	βcl⋆(f−(v	βcl⋆(f−(v	NUM
ejpam-7050	164	22	)	)	PUNCT
ejpam-7050	164	23	)	)	PUNCT
ejpam-7050	164	24	⊆	⊆	X
ejpam-7050	164	25	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	164	26	-	-	PUNCT
ejpam-7050	164	27	cl(v	cl(v	NOUN
ejpam-7050	164	28	)	)	PUNCT
ejpam-7050	164	29	)	)	PUNCT
ejpam-7050	164	30	for	for	ADP
ejpam-7050	164	31	every	every	DET
ejpam-7050	164	32	σ1σ2	σ1σ2	NOUN
ejpam-7050	164	33	-	-	ADJ
ejpam-7050	164	34	open	open	ADJ
ejpam-7050	164	35	set	set	NOUN
ejpam-7050	164	36	v	v	NOUN
ejpam-7050	164	37	of	of	ADP
ejpam-7050	164	38	y	y	PROPN
ejpam-7050	164	39	.	.	PUNCT
ejpam-7050	165	1	proof	proof	NOUN
ejpam-7050	165	2	.	.	PUNCT
ejpam-7050	166	1	(	(	PUNCT
ejpam-7050	166	2	1	1	X
ejpam-7050	166	3	)	)	PUNCT
ejpam-7050	166	4	⇒	⇒	NOUN
ejpam-7050	166	5	(	(	PUNCT
ejpam-7050	166	6	2	2	NUM
ejpam-7050	166	7	):	):	PUNCT
ejpam-7050	166	8	let	let	VERB
ejpam-7050	166	9	v	v	PART
ejpam-7050	166	10	be	be	AUX
ejpam-7050	166	11	any	any	DET
ejpam-7050	166	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	166	13	-	-	ADJ
ejpam-7050	166	14	open	open	ADJ
ejpam-7050	166	15	set	set	NOUN
ejpam-7050	166	16	of	of	ADP
ejpam-7050	166	17	y	y	PROPN
ejpam-7050	166	18	and	and	CCONJ
ejpam-7050	166	19	x	x	PROPN
ejpam-7050	166	20	∈	∈	PROPN
ejpam-7050	166	21	f+(v	f+(v	NOUN
ejpam-7050	166	22	)	)	PUNCT
ejpam-7050	166	23	.	.	PUNCT
ejpam-7050	167	1	then	then	ADV
ejpam-7050	167	2	,	,	PUNCT
ejpam-7050	167	3	f	f	PROPN
ejpam-7050	167	4	(	(	PUNCT
ejpam-7050	167	5	x	x	X
ejpam-7050	167	6	)	)	PUNCT
ejpam-7050	167	7	⊆	⊆	NUM
ejpam-7050	167	8	v	v	NOUN
ejpam-7050	167	9	and	and	CCONJ
ejpam-7050	167	10	by	by	ADP
ejpam-7050	167	11	theorem	theorem	NOUN
ejpam-7050	167	12	1	1	NUM
ejpam-7050	167	13	,	,	PUNCT
ejpam-7050	167	14	x	x	SYM
ejpam-7050	167	15	∈	∈	NOUN
ejpam-7050	167	16	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	167	17	-	-	NOUN
ejpam-7050	167	18	cl(v	cl(v	NOUN
ejpam-7050	167	19	)	)	PUNCT
ejpam-7050	167	20	)	)	PUNCT
ejpam-7050	167	21	)	)	PUNCT
ejpam-7050	167	22	and	and	CCONJ
ejpam-7050	167	23	hence	hence	ADV
ejpam-7050	167	24	f+(v	f+(v	NOUN
ejpam-7050	167	25	)	)	PUNCT
ejpam-7050	168	1	⊆	⊆	NUM
ejpam-7050	168	2	cl⋆(int⋆(cl⋆(f+(σ1σ2	cl⋆(int⋆(cl⋆(f+(σ1σ2	NOUN
ejpam-7050	168	3	-	-	PUNCT
ejpam-7050	168	4	cl(v	cl(v	NOUN
ejpam-7050	168	5	)	)	PUNCT
ejpam-7050	168	6	)	)	PUNCT
ejpam-7050	168	7	)	)	PUNCT
ejpam-7050	168	8	)	)	PUNCT
ejpam-7050	168	9	)	)	PUNCT
ejpam-7050	168	10	by	by	ADP
ejpam-7050	168	11	lemma	lemma	PROPN
ejpam-7050	168	12	4	4	NUM
ejpam-7050	168	13	.	.	PUNCT
ejpam-7050	168	14	(	(	PUNCT
ejpam-7050	168	15	2	2	X
ejpam-7050	168	16	)	)	PUNCT
ejpam-7050	168	17	⇒	⇒	NOUN
ejpam-7050	168	18	(	(	PUNCT
ejpam-7050	168	19	3	3	NUM
ejpam-7050	168	20	):	):	PUNCT
ejpam-7050	168	21	let	let	VERB
ejpam-7050	168	22	v	v	PART
ejpam-7050	168	23	be	be	AUX
ejpam-7050	168	24	any	any	DET
ejpam-7050	168	25	σ1σ2	σ1σ2	NOUN
ejpam-7050	168	26	-	-	ADJ
ejpam-7050	168	27	open	open	ADJ
ejpam-7050	168	28	set	set	NOUN
ejpam-7050	168	29	of	of	ADP
ejpam-7050	168	30	y	y	PROPN
ejpam-7050	168	31	.	.	PUNCT
ejpam-7050	169	1	thus	thus	ADV
ejpam-7050	169	2	by	by	ADP
ejpam-7050	169	3	(	(	PUNCT
ejpam-7050	169	4	2	2	NUM
ejpam-7050	169	5	)	)	PUNCT
ejpam-7050	169	6	,	,	PUNCT
ejpam-7050	169	7	we	we	PRON
ejpam-7050	169	8	have	have	VERB
ejpam-7050	169	9	x	x	NOUN
ejpam-7050	169	10	−	−	PUNCT
ejpam-7050	169	11	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	169	12	-	-	PUNCT
ejpam-7050	169	13	cl(v	cl(v	NOUN
ejpam-7050	169	14	)	)	PUNCT
ejpam-7050	169	15	)	)	PUNCT
ejpam-7050	170	1	=	=	PUNCT
ejpam-7050	171	1	f+(y	f+(y	NOUN
ejpam-7050	171	2	−	−	NUM
ejpam-7050	171	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	171	4	-	-	NUM
ejpam-7050	171	5	cl(v	cl(v	NOUN
ejpam-7050	171	6	)	)	PUNCT
ejpam-7050	171	7	)	)	PUNCT
ejpam-7050	172	1	⊆	⊆	NUM
ejpam-7050	172	2	cl⋆(int⋆(cl⋆(f+(cl⋆(y	cl⋆(int⋆(cl⋆(f+(cl⋆(y	NUM
ejpam-7050	172	3	−	−	NUM
ejpam-7050	172	4	σ1σ2	σ1σ2	NOUN
ejpam-7050	172	5	-	-	NUM
ejpam-7050	172	6	cl(v	cl(v	NOUN
ejpam-7050	172	7	)	)	PUNCT
ejpam-7050	172	8	)	)	PUNCT
ejpam-7050	172	9	)	)	PUNCT
ejpam-7050	172	10	)	)	PUNCT
ejpam-7050	172	11	)	)	PUNCT
ejpam-7050	172	12	)	)	PUNCT
ejpam-7050	173	1	=	=	PUNCT
ejpam-7050	173	2	cl⋆(int⋆(cl⋆(f+(y	cl⋆(int⋆(cl⋆(f+(y	VERB
ejpam-7050	173	3	−	−	NOUN
ejpam-7050	173	4	σ1σ2	σ1σ2	SYM
ejpam-7050	173	5	-	-	PUNCT
ejpam-7050	173	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	173	7	-	-	PUNCT
ejpam-7050	173	8	cl(v	cl(v	NOUN
ejpam-7050	173	9	)	)	PUNCT
ejpam-7050	173	10	)	)	PUNCT
ejpam-7050	173	11	)	)	PUNCT
ejpam-7050	173	12	)	)	PUNCT
ejpam-7050	173	13	)	)	PUNCT
ejpam-7050	173	14	)	)	PUNCT
ejpam-7050	174	1	⊆	⊆	NUM
ejpam-7050	174	2	cl⋆(int⋆(cl⋆(f+(y	cl⋆(int⋆(cl⋆(f+(y	NOUN
ejpam-7050	174	3	−	−	PROPN
ejpam-7050	174	4	v	v	NOUN
ejpam-7050	174	5	)	)	PUNCT
ejpam-7050	174	6	)	)	PUNCT
ejpam-7050	174	7	)	)	PUNCT
ejpam-7050	174	8	)	)	PUNCT
ejpam-7050	175	1	=	=	PUNCT
ejpam-7050	175	2	cl⋆(int⋆(cl⋆(x	cl⋆(int⋆(cl⋆(x	NOUN
ejpam-7050	175	3	−	−	NOUN
ejpam-7050	175	4	f−(v	f−(v	NOUN
ejpam-7050	175	5	)	)	PUNCT
ejpam-7050	175	6	)	)	PUNCT
ejpam-7050	175	7	)	)	PUNCT
ejpam-7050	175	8	)	)	PUNCT
ejpam-7050	176	1	=	=	PUNCT
ejpam-7050	176	2	x	x	PUNCT
ejpam-7050	177	1	−	−	NOUN
ejpam-7050	177	2	int⋆(cl⋆(int⋆(f−(v	int⋆(cl⋆(int⋆(f−(v	NOUN
ejpam-7050	177	3	)	)	PUNCT
ejpam-7050	177	4	)	)	PUNCT
ejpam-7050	177	5	)	)	PUNCT
ejpam-7050	177	6	)	)	PUNCT
ejpam-7050	177	7	and	and	CCONJ
ejpam-7050	177	8	hence	hence	ADV
ejpam-7050	177	9	int⋆(cl⋆(int⋆(f−(v	int⋆(cl⋆(int⋆(f−(v	VERB
ejpam-7050	177	10	)	)	PUNCT
ejpam-7050	177	11	)	)	PUNCT
ejpam-7050	177	12	)	)	PUNCT
ejpam-7050	177	13	)	)	PUNCT
ejpam-7050	178	1	⊆	⊆	X
ejpam-7050	178	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	178	3	-	-	PUNCT
ejpam-7050	178	4	cl(v	cl(v	NOUN
ejpam-7050	178	5	)	)	PUNCT
ejpam-7050	178	6	)	)	PUNCT
ejpam-7050	178	7	.	.	PUNCT
ejpam-7050	179	1	(	(	PUNCT
ejpam-7050	179	2	3	3	X
ejpam-7050	179	3	)	)	PUNCT
ejpam-7050	179	4	⇒	⇒	NOUN
ejpam-7050	179	5	(	(	PUNCT
ejpam-7050	179	6	4	4	NUM
ejpam-7050	179	7	):	):	PUNCT
ejpam-7050	179	8	let	let	VERB
ejpam-7050	179	9	k	k	PRON
ejpam-7050	179	10	be	be	AUX
ejpam-7050	179	11	any	any	DET
ejpam-7050	179	12	σ1σ2	σ1σ2	NUM
ejpam-7050	179	13	-	-	PUNCT
ejpam-7050	179	14	closed	closed	ADJ
ejpam-7050	179	15	set	set	NOUN
ejpam-7050	179	16	of	of	ADP
ejpam-7050	179	17	y	y	PROPN
ejpam-7050	179	18	.	.	PUNCT
ejpam-7050	180	1	then	then	ADV
ejpam-7050	180	2	,	,	PUNCT
ejpam-7050	180	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	180	4	-	-	PUNCT
ejpam-7050	180	5	int(k	int(k	NUM
ejpam-7050	180	6	)	)	PUNCT
ejpam-7050	180	7	is	be	AUX
ejpam-7050	180	8	σ1σ2	σ1σ2	NOUN
ejpam-7050	180	9	-	-	ADJ
ejpam-7050	180	10	open	open	ADJ
ejpam-7050	180	11	in	in	ADP
ejpam-7050	180	12	y	y	PROPN
ejpam-7050	180	13	and	and	CCONJ
ejpam-7050	180	14	so	so	ADV
ejpam-7050	180	15	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	NUM
ejpam-7050	180	16	-	-	PUNCT
ejpam-7050	180	17	int(k	int(k	NOUN
ejpam-7050	180	18	)	)	PUNCT
ejpam-7050	180	19	)	)	PUNCT
ejpam-7050	180	20	)	)	PUNCT
ejpam-7050	180	21	)	)	PUNCT
ejpam-7050	180	22	)	)	PUNCT
ejpam-7050	181	1	⊆	⊆	X
ejpam-7050	181	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7050	181	3	-	-	PUNCT
ejpam-7050	181	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	181	5	-	-	PUNCT
ejpam-7050	181	6	int(k	int(k	NOUN
ejpam-7050	181	7	)	)	PUNCT
ejpam-7050	181	8	)	)	PUNCT
ejpam-7050	181	9	)	)	PUNCT
ejpam-7050	182	1	⊆	⊆	X
ejpam-7050	182	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	182	3	-	-	PUNCT
ejpam-7050	182	4	cl(k	cl(k	NOUN
ejpam-7050	182	5	)	)	PUNCT
ejpam-7050	182	6	)	)	PUNCT
ejpam-7050	183	1	=	=	SYM
ejpam-7050	183	2	f−(k	f−(k	PROPN
ejpam-7050	183	3	)	)	PUNCT
ejpam-7050	183	4	.	.	PUNCT
ejpam-7050	184	1	(	(	PUNCT
ejpam-7050	184	2	4	4	X
ejpam-7050	184	3	)	)	PUNCT
ejpam-7050	184	4	⇒	⇒	NOUN
ejpam-7050	184	5	(	(	PUNCT
ejpam-7050	184	6	5	5	NUM
ejpam-7050	184	7	):	):	PUNCT
ejpam-7050	184	8	let	let	VERB
ejpam-7050	184	9	k	k	PRON
ejpam-7050	184	10	be	be	AUX
ejpam-7050	184	11	any	any	DET
ejpam-7050	184	12	σ1σ2	σ1σ2	NUM
ejpam-7050	184	13	-	-	PUNCT
ejpam-7050	184	14	closed	closed	ADJ
ejpam-7050	184	15	set	set	NOUN
ejpam-7050	184	16	of	of	ADP
ejpam-7050	184	17	y	y	PROPN
ejpam-7050	184	18	.	.	PUNCT
ejpam-7050	185	1	then	then	ADV
ejpam-7050	185	2	,	,	PUNCT
ejpam-7050	185	3	we	we	PRON
ejpam-7050	185	4	have	have	AUX
ejpam-7050	185	5	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	VERB
ejpam-7050	185	6	-	-	PUNCT
ejpam-7050	185	7	int(k	int(k	NOUN
ejpam-7050	185	8	)	)	PUNCT
ejpam-7050	185	9	)	)	PUNCT
ejpam-7050	185	10	)	)	PUNCT
ejpam-7050	185	11	)	)	PUNCT
ejpam-7050	185	12	)	)	PUNCT
ejpam-7050	186	1	⊆	⊆	X
ejpam-7050	186	2	f−(k	f−(k	PROPN
ejpam-7050	186	3	)	)	PUNCT
ejpam-7050	186	4	and	and	CCONJ
ejpam-7050	186	5	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	186	6	-	-	PUNCT
ejpam-7050	186	7	int(k	int(k	NOUN
ejpam-7050	186	8	)	)	PUNCT
ejpam-7050	186	9	)	)	PUNCT
ejpam-7050	187	1	⊆	⊆	NUM
ejpam-7050	187	2	f−(k	f−(k	PROPN
ejpam-7050	187	3	)	)	PUNCT
ejpam-7050	187	4	.	.	PUNCT
ejpam-7050	188	1	thus	thus	ADV
ejpam-7050	188	2	by	by	ADP
ejpam-7050	188	3	lemma	lemma	PROPN
ejpam-7050	188	4	4	4	NUM
ejpam-7050	188	5	,	,	PUNCT
ejpam-7050	188	6	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	188	7	-	-	PUNCT
ejpam-7050	188	8	int(k	int(k	NOUN
ejpam-7050	188	9	)	)	PUNCT
ejpam-7050	188	10	)	)	PUNCT
ejpam-7050	188	11	)	)	PUNCT
ejpam-7050	189	1	⊆	⊆	NUM
ejpam-7050	189	2	f−(k	f−(k	PROPN
ejpam-7050	189	3	)	)	PUNCT
ejpam-7050	189	4	.	.	PUNCT
ejpam-7050	190	1	(	(	PUNCT
ejpam-7050	190	2	5	5	X
ejpam-7050	190	3	)	)	PUNCT
ejpam-7050	190	4	⇒	⇒	NOUN
ejpam-7050	190	5	(	(	PUNCT
ejpam-7050	190	6	6	6	NUM
ejpam-7050	190	7	):	):	PUNCT
ejpam-7050	190	8	let	let	VERB
ejpam-7050	190	9	b	b	X
ejpam-7050	190	10	be	be	AUX
ejpam-7050	190	11	any	any	DET
ejpam-7050	190	12	subset	subset	NOUN
ejpam-7050	190	13	of	of	ADP
ejpam-7050	190	14	y	y	PROPN
ejpam-7050	190	15	.	.	PUNCT
ejpam-7050	191	1	then	then	ADV
ejpam-7050	191	2	,	,	PUNCT
ejpam-7050	191	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	191	4	-	-	NOUN
ejpam-7050	191	5	cl(b	cl(b	NOUN
ejpam-7050	191	6	)	)	PUNCT
ejpam-7050	191	7	is	be	AUX
ejpam-7050	191	8	σ1σ2	σ1σ2	NOUN
ejpam-7050	191	9	-	-	ADJ
ejpam-7050	191	10	closed	closed	ADJ
ejpam-7050	191	11	in	in	ADP
ejpam-7050	191	12	y	y	PROPN
ejpam-7050	191	13	and	and	CCONJ
ejpam-7050	191	14	by	by	ADP
ejpam-7050	191	15	(	(	PUNCT
ejpam-7050	191	16	5	5	NUM
ejpam-7050	191	17	)	)	PUNCT
ejpam-7050	191	18	,	,	PUNCT
ejpam-7050	191	19	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	191	20	-	-	PUNCT
ejpam-7050	191	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	191	22	-	-	PUNCT
ejpam-7050	191	23	cl(b	cl(b	NOUN
ejpam-7050	191	24	)	)	PUNCT
ejpam-7050	191	25	)	)	PUNCT
ejpam-7050	191	26	)	)	PUNCT
ejpam-7050	191	27	)	)	PUNCT
ejpam-7050	192	1	⊆	⊆	X
ejpam-7050	192	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7050	192	3	-	-	PUNCT
ejpam-7050	192	4	cl(b	cl(b	NOUN
ejpam-7050	192	5	)	)	PUNCT
ejpam-7050	192	6	)	)	PUNCT
ejpam-7050	192	7	.	.	PUNCT
ejpam-7050	193	1	m.	m.	NOUN
ejpam-7050	193	2	thongmoon	thongmoon	PROPN
ejpam-7050	193	3	,	,	PUNCT
ejpam-7050	193	4	a.	a.	PROPN
ejpam-7050	193	5	sama	sama	PROPN
ejpam-7050	193	6	-	-	PUNCT
ejpam-7050	193	7	ae	ae	PROPN
ejpam-7050	193	8	,	,	PUNCT
ejpam-7050	193	9	c.	c.	PROPN
ejpam-7050	193	10	boonpok	boonpok	PROPN
ejpam-7050	193	11	/	/	SYM
ejpam-7050	193	12	eur	eur	PROPN
ejpam-7050	193	13	.	.	PUNCT
ejpam-7050	194	1	j.	j.	PROPN
ejpam-7050	194	2	pure	pure	PROPN
ejpam-7050	194	3	appl	appl	PROPN
ejpam-7050	194	4	.	.	PROPN
ejpam-7050	194	5	math	math	PROPN
ejpam-7050	194	6	,	,	PUNCT
ejpam-7050	194	7	18	18	NUM
ejpam-7050	194	8	(	(	PUNCT
ejpam-7050	194	9	4	4	NUM
ejpam-7050	194	10	)	)	PUNCT
ejpam-7050	194	11	(	(	PUNCT
ejpam-7050	194	12	2025	2025	NUM
ejpam-7050	194	13	)	)	PUNCT
ejpam-7050	194	14	,	,	PUNCT
ejpam-7050	194	15	7050	7050	NUM
ejpam-7050	194	16	7	7	NUM
ejpam-7050	194	17	of	of	ADP
ejpam-7050	194	18	14	14	NUM
ejpam-7050	194	19	(	(	PUNCT
ejpam-7050	194	20	6	6	NUM
ejpam-7050	194	21	)	)	PUNCT
ejpam-7050	194	22	⇒	⇒	NOUN
ejpam-7050	194	23	(	(	PUNCT
ejpam-7050	194	24	7	7	NUM
ejpam-7050	194	25	):	):	PUNCT
ejpam-7050	194	26	let	let	VERB
ejpam-7050	194	27	b	b	X
ejpam-7050	194	28	be	be	AUX
ejpam-7050	194	29	any	any	DET
ejpam-7050	194	30	subset	subset	NOUN
ejpam-7050	194	31	of	of	ADP
ejpam-7050	194	32	y	y	PROPN
ejpam-7050	194	33	.	.	PUNCT
ejpam-7050	195	1	by	by	ADP
ejpam-7050	195	2	(	(	PUNCT
ejpam-7050	195	3	6	6	NUM
ejpam-7050	195	4	)	)	PUNCT
ejpam-7050	195	5	,	,	PUNCT
ejpam-7050	195	6	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	195	7	-	-	PUNCT
ejpam-7050	195	8	int(b	int(b	NOUN
ejpam-7050	195	9	)	)	PUNCT
ejpam-7050	195	10	)	)	PUNCT
ejpam-7050	195	11	=	=	PUNCT
ejpam-7050	196	1	x	x	PUNCT
ejpam-7050	196	2	−	−	ADP
ejpam-7050	196	3	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7050	196	4	-	-	PUNCT
ejpam-7050	196	5	cl(y	cl(y	NOUN
ejpam-7050	196	6	−b	−b	NOUN
ejpam-7050	196	7	)	)	PUNCT
ejpam-7050	196	8	)	)	PUNCT
ejpam-7050	197	1	⊆	⊆	NUM
ejpam-7050	197	2	x	x	X
ejpam-7050	197	3	−	−	PRON
ejpam-7050	197	4	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	197	5	-	-	PUNCT
ejpam-7050	197	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	197	7	-	-	PUNCT
ejpam-7050	197	8	cl(y	cl(y	NOUN
ejpam-7050	197	9	−b	−b	NOUN
ejpam-7050	197	10	)	)	PUNCT
ejpam-7050	197	11	)	)	PUNCT
ejpam-7050	197	12	)	)	PUNCT
ejpam-7050	197	13	)	)	PUNCT
ejpam-7050	198	1	=	=	SYM
ejpam-7050	198	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	198	3	-	-	PUNCT
ejpam-7050	198	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	198	5	-	-	PUNCT
ejpam-7050	198	6	int(b	int(b	NOUN
ejpam-7050	198	7	)	)	PUNCT
ejpam-7050	198	8	)	)	PUNCT
ejpam-7050	198	9	)	)	PUNCT
ejpam-7050	198	10	)	)	PUNCT
ejpam-7050	198	11	.	.	PUNCT
ejpam-7050	199	1	(	(	PUNCT
ejpam-7050	199	2	7	7	X
ejpam-7050	199	3	)	)	PUNCT
ejpam-7050	199	4	⇒	⇒	NOUN
ejpam-7050	199	5	(	(	PUNCT
ejpam-7050	199	6	8)	8)	NUM
ejpam-7050	199	7	:	:	PUNCT
ejpam-7050	199	8	the	the	DET
ejpam-7050	199	9	proof	proof	NOUN
ejpam-7050	199	10	is	be	AUX
ejpam-7050	199	11	obvious	obvious	ADJ
ejpam-7050	199	12	.	.	PUNCT
ejpam-7050	200	1	(	(	PUNCT
ejpam-7050	200	2	8)	8)	NUM
ejpam-7050	200	3	⇒	⇒	NOUN
ejpam-7050	200	4	(	(	PUNCT
ejpam-7050	200	5	9	9	NUM
ejpam-7050	200	6	):	):	PUNCT
ejpam-7050	200	7	let	let	VERB
ejpam-7050	200	8	v	v	PART
ejpam-7050	200	9	be	be	AUX
ejpam-7050	200	10	any	any	DET
ejpam-7050	200	11	σ1σ2	σ1σ2	NOUN
ejpam-7050	200	12	-	-	ADJ
ejpam-7050	200	13	open	open	ADJ
ejpam-7050	200	14	set	set	NOUN
ejpam-7050	200	15	of	of	ADP
ejpam-7050	200	16	y	y	PROPN
ejpam-7050	200	17	.	.	PUNCT
ejpam-7050	201	1	then	then	ADV
ejpam-7050	201	2	by	by	ADP
ejpam-7050	201	3	(	(	PUNCT
ejpam-7050	201	4	8)	8)	NUM
ejpam-7050	201	5	,	,	PUNCT
ejpam-7050	201	6	we	we	PRON
ejpam-7050	201	7	have	have	VERB
ejpam-7050	201	8	βcl⋆(f−(v	βcl⋆(f−(v	NOUN
ejpam-7050	201	9	)	)	PUNCT
ejpam-7050	201	10	)	)	PUNCT
ejpam-7050	202	1	⊆	⊆	NUM
ejpam-7050	202	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	202	3	-	-	PUNCT
ejpam-7050	202	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	202	5	-	-	PUNCT
ejpam-7050	202	6	cl(v	cl(v	NOUN
ejpam-7050	202	7	)	)	PUNCT
ejpam-7050	202	8	)	)	PUNCT
ejpam-7050	202	9	)	)	PUNCT
ejpam-7050	202	10	)	)	PUNCT
ejpam-7050	203	1	=	=	SYM
ejpam-7050	203	2	βcl⋆(x	βcl⋆(x	NOUN
ejpam-7050	203	3	−	−	NOUN
ejpam-7050	204	1	f+(y	f+(y	NOUN
ejpam-7050	204	2	−	−	NOUN
ejpam-7050	204	3	σ1σ2	σ1σ2	SYM
ejpam-7050	204	4	-	-	PUNCT
ejpam-7050	204	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	204	6	-	-	PUNCT
ejpam-7050	204	7	cl(v	cl(v	NOUN
ejpam-7050	204	8	)	)	PUNCT
ejpam-7050	204	9	)	)	PUNCT
ejpam-7050	204	10	)	)	PUNCT
ejpam-7050	204	11	)	)	PUNCT
ejpam-7050	205	1	=	=	PUNCT
ejpam-7050	205	2	x	x	PUNCT
ejpam-7050	206	1	−	−	PROPN
ejpam-7050	206	2	βint⋆(f+(y	βint⋆(f+(y	NOUN
ejpam-7050	206	3	−	−	NOUN
ejpam-7050	206	4	σ1σ2	σ1σ2	SYM
ejpam-7050	206	5	-	-	PUNCT
ejpam-7050	206	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	206	7	-	-	PUNCT
ejpam-7050	206	8	cl(v	cl(v	NOUN
ejpam-7050	206	9	)	)	PUNCT
ejpam-7050	206	10	)	)	PUNCT
ejpam-7050	206	11	)	)	PUNCT
ejpam-7050	206	12	)	)	PUNCT
ejpam-7050	207	1	=	=	PUNCT
ejpam-7050	207	2	x	x	PUNCT
ejpam-7050	207	3	−	−	ADP
ejpam-7050	207	4	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	207	5	-	-	PUNCT
ejpam-7050	207	6	cl(y	cl(y	NOUN
ejpam-7050	207	7	−	−	NOUN
ejpam-7050	207	8	σ1σ2	σ1σ2	NOUN
ejpam-7050	207	9	-	-	NUM
ejpam-7050	207	10	cl(v	cl(v	NOUN
ejpam-7050	207	11	)	)	PUNCT
ejpam-7050	207	12	)	)	PUNCT
ejpam-7050	207	13	)	)	PUNCT
ejpam-7050	207	14	)	)	PUNCT
ejpam-7050	208	1	⊆	⊆	NUM
ejpam-7050	208	2	x	x	SYM
ejpam-7050	208	3	−	−	NOUN
ejpam-7050	208	4	f+(y	f+(y	NOUN
ejpam-7050	208	5	−	−	PROPN
ejpam-7050	208	6	σ1σ2	σ1σ2	NOUN
ejpam-7050	208	7	-	-	NUM
ejpam-7050	208	8	cl(v	cl(v	NOUN
ejpam-7050	208	9	)	)	PUNCT
ejpam-7050	208	10	)	)	PUNCT
ejpam-7050	209	1	=	=	PUNCT
ejpam-7050	209	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	209	3	-	-	PUNCT
ejpam-7050	209	4	cl(v	cl(v	NOUN
ejpam-7050	209	5	)	)	PUNCT
ejpam-7050	209	6	)	)	PUNCT
ejpam-7050	209	7	.	.	PUNCT
ejpam-7050	210	1	(	(	PUNCT
ejpam-7050	210	2	9	9	X
ejpam-7050	210	3	)	)	PUNCT
ejpam-7050	210	4	⇒	⇒	NOUN
ejpam-7050	210	5	(	(	PUNCT
ejpam-7050	210	6	1	1	NUM
ejpam-7050	210	7	):	):	PUNCT
ejpam-7050	210	8	let	let	VERB
ejpam-7050	210	9	x	x	PUNCT
ejpam-7050	210	10	∈	∈	PROPN
ejpam-7050	210	11	x	x	X
ejpam-7050	210	12	and	and	CCONJ
ejpam-7050	210	13	v	v	X
ejpam-7050	210	14	be	be	AUX
ejpam-7050	210	15	any	any	DET
ejpam-7050	210	16	σ1σ2	σ1σ2	NOUN
ejpam-7050	210	17	-	-	ADJ
ejpam-7050	210	18	open	open	ADJ
ejpam-7050	210	19	set	set	NOUN
ejpam-7050	210	20	of	of	ADP
ejpam-7050	210	21	y	y	PROPN
ejpam-7050	210	22	containing	contain	VERB
ejpam-7050	210	23	f	f	PROPN
ejpam-7050	210	24	(	(	PUNCT
ejpam-7050	210	25	x	x	NOUN
ejpam-7050	210	26	)	)	PUNCT
ejpam-7050	210	27	.	.	PUNCT
ejpam-7050	211	1	by	by	ADP
ejpam-7050	211	2	(	(	PUNCT
ejpam-7050	211	3	9	9	NUM
ejpam-7050	211	4	)	)	PUNCT
ejpam-7050	211	5	,	,	PUNCT
ejpam-7050	211	6	x	x	PUNCT
ejpam-7050	211	7	∈	∈	PROPN
ejpam-7050	211	8	f+(v	f+(v	NOUN
ejpam-7050	211	9	)	)	PUNCT
ejpam-7050	211	10	⊆	⊆	X
ejpam-7050	211	11	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7050	211	12	-	-	PUNCT
ejpam-7050	211	13	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	211	14	-	-	PUNCT
ejpam-7050	211	15	cl(v	cl(v	NOUN
ejpam-7050	211	16	)	)	PUNCT
ejpam-7050	211	17	)	)	PUNCT
ejpam-7050	211	18	)	)	PUNCT
ejpam-7050	212	1	=	=	PUNCT
ejpam-7050	212	2	x	x	PUNCT
ejpam-7050	212	3	−	−	ADP
ejpam-7050	212	4	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7050	212	5	-	-	PUNCT
ejpam-7050	212	6	cl(y	cl(y	NOUN
ejpam-7050	212	7	−	−	NOUN
ejpam-7050	212	8	σ1σ2	σ1σ2	NOUN
ejpam-7050	212	9	-	-	NUM
ejpam-7050	212	10	cl(v	cl(v	NOUN
ejpam-7050	212	11	)	)	PUNCT
ejpam-7050	212	12	)	)	PUNCT
ejpam-7050	212	13	)	)	PUNCT
ejpam-7050	213	1	⊆	⊆	NUM
ejpam-7050	213	2	x	x	SYM
ejpam-7050	213	3	−	−	NOUN
ejpam-7050	213	4	βcl⋆(f−(y	βcl⋆(f−(y	PUNCT
ejpam-7050	213	5	−	−	ADP
ejpam-7050	213	6	σ1σ2	σ1σ2	NOUN
ejpam-7050	213	7	-	-	NUM
ejpam-7050	213	8	cl(v	cl(v	NOUN
ejpam-7050	213	9	)	)	PUNCT
ejpam-7050	213	10	)	)	PUNCT
ejpam-7050	213	11	)	)	PUNCT
ejpam-7050	214	1	=	=	SYM
ejpam-7050	214	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	214	3	-	-	X
ejpam-7050	214	4	cl(v	cl(v	NOUN
ejpam-7050	214	5	)	)	PUNCT
ejpam-7050	214	6	)	)	PUNCT
ejpam-7050	214	7	)	)	PUNCT
ejpam-7050	215	1	and	and	CCONJ
ejpam-7050	215	2	hence	hence	ADV
ejpam-7050	215	3	f	f	PROPN
ejpam-7050	215	4	is	be	AUX
ejpam-7050	215	5	upper	upper	ADJ
ejpam-7050	215	6	weakly	weakly	ADJ
ejpam-7050	215	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	215	8	,	,	PUNCT
ejpam-7050	215	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	215	10	by	by	ADP
ejpam-7050	215	11	theorem	theorem	NOUN
ejpam-7050	215	12	1	1	NUM
ejpam-7050	215	13	.	.	PUNCT
ejpam-7050	215	14	theorem	theorem	NOUN
ejpam-7050	215	15	4	4	NUM
ejpam-7050	215	16	.	.	X
ejpam-7050	215	17	for	for	ADP
ejpam-7050	215	18	a	a	DET
ejpam-7050	215	19	multifunction	multifunction	NOUN
ejpam-7050	215	20	f	f	NOUN
ejpam-7050	215	21	:	:	PUNCT
ejpam-7050	215	22	(	(	PUNCT
ejpam-7050	215	23	x	x	X
ejpam-7050	215	24	,	,	PUNCT
ejpam-7050	215	25	τ	τ	PROPN
ejpam-7050	215	26	,	,	PUNCT
ejpam-7050	215	27	i	i	NOUN
ejpam-7050	215	28	)	)	PUNCT
ejpam-7050	215	29	→	→	PUNCT
ejpam-7050	215	30	(	(	PUNCT
ejpam-7050	215	31	y	y	PROPN
ejpam-7050	215	32	,	,	PUNCT
ejpam-7050	215	33	σ1	σ1	PROPN
ejpam-7050	215	34	,	,	PUNCT
ejpam-7050	215	35	σ2	σ2	NOUN
ejpam-7050	215	36	)	)	PUNCT
ejpam-7050	215	37	,	,	PUNCT
ejpam-7050	215	38	the	the	DET
ejpam-7050	215	39	following	follow	VERB
ejpam-7050	215	40	properties	property	NOUN
ejpam-7050	215	41	are	be	AUX
ejpam-7050	215	42	equivalent	equivalent	ADJ
ejpam-7050	215	43	:	:	PUNCT
ejpam-7050	215	44	(	(	PUNCT
ejpam-7050	215	45	1	1	X
ejpam-7050	215	46	)	)	PUNCT
ejpam-7050	215	47	f	f	PROPN
ejpam-7050	215	48	is	be	AUX
ejpam-7050	215	49	lower	low	ADJ
ejpam-7050	215	50	weakly	weakly	ADJ
ejpam-7050	215	51	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	215	52	,	,	PUNCT
ejpam-7050	215	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	215	54	;	;	PUNCT
ejpam-7050	215	55	(	(	PUNCT
ejpam-7050	215	56	2	2	X
ejpam-7050	215	57	)	)	PUNCT
ejpam-7050	215	58	f−(v	f−(v	NOUN
ejpam-7050	215	59	)	)	PUNCT
ejpam-7050	215	60	⊆	⊆	NUM
ejpam-7050	215	61	cl⋆(int⋆(cl⋆(f−(σ1σ2	cl⋆(int⋆(cl⋆(f−(σ1σ2	NOUN
ejpam-7050	215	62	-	-	PUNCT
ejpam-7050	215	63	cl(v	cl(v	NOUN
ejpam-7050	215	64	)	)	PUNCT
ejpam-7050	215	65	)	)	PUNCT
ejpam-7050	215	66	)	)	PUNCT
ejpam-7050	215	67	)	)	PUNCT
ejpam-7050	215	68	)	)	PUNCT
ejpam-7050	216	1	for	for	ADP
ejpam-7050	216	2	every	every	DET
ejpam-7050	216	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	216	4	-	-	ADJ
ejpam-7050	216	5	open	open	ADJ
ejpam-7050	216	6	set	set	NOUN
ejpam-7050	216	7	v	v	NOUN
ejpam-7050	216	8	of	of	ADP
ejpam-7050	216	9	y	y	PROPN
ejpam-7050	216	10	;	;	PUNCT
ejpam-7050	216	11	(	(	PUNCT
ejpam-7050	216	12	3	3	X
ejpam-7050	216	13	)	)	PUNCT
ejpam-7050	216	14	int⋆(cl⋆(int⋆(f+(v	int⋆(cl⋆(int⋆(f+(v	NOUN
ejpam-7050	216	15	)	)	PUNCT
ejpam-7050	216	16	)	)	PUNCT
ejpam-7050	216	17	)	)	PUNCT
ejpam-7050	216	18	)	)	PUNCT
ejpam-7050	216	19	⊆	⊆	X
ejpam-7050	216	20	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	216	21	-	-	PUNCT
ejpam-7050	216	22	cl(v	cl(v	NOUN
ejpam-7050	216	23	)	)	PUNCT
ejpam-7050	216	24	)	)	PUNCT
ejpam-7050	216	25	for	for	ADP
ejpam-7050	216	26	every	every	DET
ejpam-7050	216	27	σ1σ2	σ1σ2	NOUN
ejpam-7050	216	28	-	-	ADJ
ejpam-7050	216	29	open	open	ADJ
ejpam-7050	216	30	set	set	NOUN
ejpam-7050	216	31	v	v	NOUN
ejpam-7050	216	32	of	of	ADP
ejpam-7050	216	33	y	y	PROPN
ejpam-7050	216	34	;	;	PUNCT
ejpam-7050	216	35	(	(	PUNCT
ejpam-7050	216	36	4	4	X
ejpam-7050	216	37	)	)	PUNCT
ejpam-7050	216	38	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7050	216	39	-	-	PUNCT
ejpam-7050	216	40	int(k	int(k	NUM
ejpam-7050	216	41	)	)	PUNCT
ejpam-7050	216	42	)	)	PUNCT
ejpam-7050	216	43	)	)	PUNCT
ejpam-7050	216	44	)	)	PUNCT
ejpam-7050	216	45	)	)	PUNCT
ejpam-7050	217	1	⊆	⊆	NUM
ejpam-7050	217	2	f+(k	f+(k	NOUN
ejpam-7050	217	3	)	)	PUNCT
ejpam-7050	217	4	for	for	ADP
ejpam-7050	217	5	every	every	DET
ejpam-7050	217	6	σ1σ2	σ1σ2	NUM
ejpam-7050	217	7	-	-	PUNCT
ejpam-7050	217	8	closed	closed	ADJ
ejpam-7050	217	9	set	set	NOUN
ejpam-7050	217	10	k	k	PROPN
ejpam-7050	217	11	of	of	ADP
ejpam-7050	217	12	y	y	PROPN
ejpam-7050	217	13	;	;	PUNCT
ejpam-7050	217	14	(	(	PUNCT
ejpam-7050	217	15	5	5	X
ejpam-7050	217	16	)	)	PUNCT
ejpam-7050	217	17	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	217	18	-	-	PUNCT
ejpam-7050	217	19	int(k	int(k	NOUN
ejpam-7050	217	20	)	)	PUNCT
ejpam-7050	217	21	)	)	PUNCT
ejpam-7050	217	22	)	)	PUNCT
ejpam-7050	218	1	⊆	⊆	NUM
ejpam-7050	218	2	f+(k	f+(k	NOUN
ejpam-7050	218	3	)	)	PUNCT
ejpam-7050	218	4	for	for	ADP
ejpam-7050	218	5	every	every	DET
ejpam-7050	218	6	σ1σ2	σ1σ2	NUM
ejpam-7050	218	7	-	-	PUNCT
ejpam-7050	218	8	closed	closed	ADJ
ejpam-7050	218	9	set	set	NOUN
ejpam-7050	218	10	k	k	PROPN
ejpam-7050	218	11	of	of	ADP
ejpam-7050	218	12	y	y	PROPN
ejpam-7050	218	13	;	;	PUNCT
ejpam-7050	218	14	(	(	PUNCT
ejpam-7050	218	15	6	6	X
ejpam-7050	218	16	)	)	PUNCT
ejpam-7050	218	17	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	218	18	-	-	PUNCT
ejpam-7050	218	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	218	20	-	-	PUNCT
ejpam-7050	218	21	cl(b	cl(b	NOUN
ejpam-7050	218	22	)	)	PUNCT
ejpam-7050	218	23	)	)	PUNCT
ejpam-7050	218	24	)	)	PUNCT
ejpam-7050	218	25	)	)	PUNCT
ejpam-7050	219	1	⊆	⊆	X
ejpam-7050	219	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	219	3	-	-	PUNCT
ejpam-7050	219	4	cl(b	cl(b	NOUN
ejpam-7050	219	5	)	)	PUNCT
ejpam-7050	219	6	)	)	PUNCT
ejpam-7050	219	7	for	for	ADP
ejpam-7050	219	8	every	every	DET
ejpam-7050	219	9	subset	subset	NOUN
ejpam-7050	219	10	b	b	PROPN
ejpam-7050	219	11	of	of	ADP
ejpam-7050	219	12	y	y	PROPN
ejpam-7050	219	13	;	;	PUNCT
ejpam-7050	219	14	(	(	PUNCT
ejpam-7050	219	15	7	7	X
ejpam-7050	219	16	)	)	PUNCT
ejpam-7050	219	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	219	18	-	-	PUNCT
ejpam-7050	219	19	int(b	int(b	NOUN
ejpam-7050	219	20	)	)	PUNCT
ejpam-7050	219	21	)	)	PUNCT
ejpam-7050	219	22	⊆	⊆	X
ejpam-7050	219	23	βint⋆(f−(σ1σ2	βint⋆(f−(σ1σ2	NOUN
ejpam-7050	219	24	-	-	PUNCT
ejpam-7050	219	25	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	219	26	-	-	PUNCT
ejpam-7050	219	27	int(b	int(b	NOUN
ejpam-7050	219	28	)	)	PUNCT
ejpam-7050	219	29	)	)	PUNCT
ejpam-7050	219	30	)	)	PUNCT
ejpam-7050	219	31	)	)	PUNCT
ejpam-7050	219	32	for	for	ADP
ejpam-7050	219	33	every	every	DET
ejpam-7050	219	34	subset	subset	NOUN
ejpam-7050	219	35	b	b	PROPN
ejpam-7050	219	36	of	of	ADP
ejpam-7050	219	37	y	y	PROPN
ejpam-7050	219	38	;	;	PUNCT
ejpam-7050	219	39	(	(	PUNCT
ejpam-7050	219	40	8)	8)	NUM
ejpam-7050	219	41	f−(v	f−(v	NOUN
ejpam-7050	219	42	)	)	PUNCT
ejpam-7050	219	43	⊆	⊆	NUM
ejpam-7050	219	44	βint⋆(f−(σ1σ2	βint⋆(f−(σ1σ2	NOUN
ejpam-7050	219	45	-	-	PUNCT
ejpam-7050	219	46	cl(v	cl(v	NOUN
ejpam-7050	219	47	)	)	PUNCT
ejpam-7050	219	48	)	)	PUNCT
ejpam-7050	219	49	)	)	PUNCT
ejpam-7050	219	50	for	for	ADP
ejpam-7050	219	51	every	every	DET
ejpam-7050	219	52	σ1σ2	σ1σ2	NOUN
ejpam-7050	219	53	-	-	ADJ
ejpam-7050	219	54	open	open	ADJ
ejpam-7050	219	55	set	set	NOUN
ejpam-7050	219	56	v	v	NOUN
ejpam-7050	219	57	of	of	ADP
ejpam-7050	219	58	y	y	PROPN
ejpam-7050	219	59	;	;	PUNCT
ejpam-7050	219	60	(	(	PUNCT
ejpam-7050	219	61	9	9	X
ejpam-7050	219	62	)	)	PUNCT
ejpam-7050	219	63	βcl⋆(f+(v	βcl⋆(f+(v	NUM
ejpam-7050	219	64	)	)	PUNCT
ejpam-7050	219	65	)	)	PUNCT
ejpam-7050	220	1	⊆	⊆	X
ejpam-7050	220	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	220	3	-	-	PUNCT
ejpam-7050	220	4	cl(v	cl(v	NOUN
ejpam-7050	220	5	)	)	PUNCT
ejpam-7050	220	6	)	)	PUNCT
ejpam-7050	220	7	for	for	ADP
ejpam-7050	220	8	every	every	DET
ejpam-7050	220	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	220	10	-	-	ADJ
ejpam-7050	220	11	open	open	ADJ
ejpam-7050	220	12	set	set	NOUN
ejpam-7050	220	13	v	v	NOUN
ejpam-7050	220	14	of	of	ADP
ejpam-7050	220	15	y	y	PROPN
ejpam-7050	220	16	.	.	PUNCT
ejpam-7050	221	1	m.	m.	NOUN
ejpam-7050	221	2	thongmoon	thongmoon	PROPN
ejpam-7050	221	3	,	,	PUNCT
ejpam-7050	221	4	a.	a.	PROPN
ejpam-7050	221	5	sama	sama	PROPN
ejpam-7050	221	6	-	-	PUNCT
ejpam-7050	221	7	ae	ae	PROPN
ejpam-7050	221	8	,	,	PUNCT
ejpam-7050	221	9	c.	c.	PROPN
ejpam-7050	221	10	boonpok	boonpok	PROPN
ejpam-7050	221	11	/	/	SYM
ejpam-7050	221	12	eur	eur	PROPN
ejpam-7050	221	13	.	.	PUNCT
ejpam-7050	222	1	j.	j.	PROPN
ejpam-7050	222	2	pure	pure	PROPN
ejpam-7050	222	3	appl	appl	PROPN
ejpam-7050	222	4	.	.	PROPN
ejpam-7050	222	5	math	math	PROPN
ejpam-7050	222	6	,	,	PUNCT
ejpam-7050	222	7	18	18	NUM
ejpam-7050	222	8	(	(	PUNCT
ejpam-7050	222	9	4	4	NUM
ejpam-7050	222	10	)	)	PUNCT
ejpam-7050	222	11	(	(	PUNCT
ejpam-7050	222	12	2025	2025	NUM
ejpam-7050	222	13	)	)	PUNCT
ejpam-7050	222	14	,	,	PUNCT
ejpam-7050	222	15	7050	7050	NUM
ejpam-7050	222	16	8	8	NUM
ejpam-7050	222	17	of	of	ADP
ejpam-7050	222	18	14	14	NUM
ejpam-7050	222	19	proof	proof	NOUN
ejpam-7050	222	20	.	.	PUNCT
ejpam-7050	223	1	the	the	DET
ejpam-7050	223	2	proof	proof	NOUN
ejpam-7050	223	3	is	be	AUX
ejpam-7050	223	4	similar	similar	ADJ
ejpam-7050	223	5	to	to	ADP
ejpam-7050	223	6	that	that	PRON
ejpam-7050	223	7	of	of	ADP
ejpam-7050	223	8	theorem	theorem	ADJ
ejpam-7050	223	9	3	3	NUM
ejpam-7050	223	10	.	.	PUNCT
ejpam-7050	223	11	corollary	corollary	ADJ
ejpam-7050	223	12	2	2	NUM
ejpam-7050	223	13	.	.	PUNCT
ejpam-7050	223	14	for	for	ADP
ejpam-7050	223	15	a	a	DET
ejpam-7050	223	16	function	function	NOUN
ejpam-7050	223	17	f	f	NOUN
ejpam-7050	223	18	:	:	PUNCT
ejpam-7050	223	19	(	(	PUNCT
ejpam-7050	223	20	x	x	X
ejpam-7050	223	21	,	,	PUNCT
ejpam-7050	223	22	τ	τ	PROPN
ejpam-7050	223	23	,	,	PUNCT
ejpam-7050	223	24	i	i	NOUN
ejpam-7050	223	25	)	)	PUNCT
ejpam-7050	223	26	→	→	PUNCT
ejpam-7050	223	27	(	(	PUNCT
ejpam-7050	223	28	y	y	PROPN
ejpam-7050	223	29	,	,	PUNCT
ejpam-7050	223	30	σ1	σ1	PROPN
ejpam-7050	223	31	,	,	PUNCT
ejpam-7050	223	32	σ2	σ2	NOUN
ejpam-7050	223	33	)	)	PUNCT
ejpam-7050	223	34	,	,	PUNCT
ejpam-7050	223	35	the	the	DET
ejpam-7050	223	36	following	follow	VERB
ejpam-7050	223	37	properties	property	NOUN
ejpam-7050	223	38	are	be	AUX
ejpam-7050	223	39	equivalent	equivalent	ADJ
ejpam-7050	223	40	:	:	PUNCT
ejpam-7050	223	41	(	(	PUNCT
ejpam-7050	223	42	1	1	X
ejpam-7050	223	43	)	)	PUNCT
ejpam-7050	223	44	f	f	PROPN
ejpam-7050	223	45	is	be	AUX
ejpam-7050	223	46	weakly	weakly	ADJ
ejpam-7050	223	47	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	223	48	,	,	PUNCT
ejpam-7050	223	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	223	50	;	;	PUNCT
ejpam-7050	223	51	(	(	PUNCT
ejpam-7050	223	52	2	2	X
ejpam-7050	223	53	)	)	PUNCT
ejpam-7050	223	54	f−1(v	f−1(v	NOUN
ejpam-7050	223	55	)	)	PUNCT
ejpam-7050	224	1	⊆	⊆	NUM
ejpam-7050	224	2	cl⋆(int⋆(cl⋆(f−1(σ1σ2	cl⋆(int⋆(cl⋆(f−1(σ1σ2	NOUN
ejpam-7050	224	3	-	-	PUNCT
ejpam-7050	224	4	cl(v	cl(v	NOUN
ejpam-7050	224	5	)	)	PUNCT
ejpam-7050	224	6	)	)	PUNCT
ejpam-7050	224	7	)	)	PUNCT
ejpam-7050	224	8	)	)	PUNCT
ejpam-7050	224	9	)	)	PUNCT
ejpam-7050	225	1	for	for	ADP
ejpam-7050	225	2	every	every	DET
ejpam-7050	225	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	225	4	-	-	ADJ
ejpam-7050	225	5	open	open	ADJ
ejpam-7050	225	6	set	set	NOUN
ejpam-7050	225	7	v	v	NOUN
ejpam-7050	225	8	of	of	ADP
ejpam-7050	225	9	y	y	PROPN
ejpam-7050	225	10	;	;	PUNCT
ejpam-7050	225	11	(	(	PUNCT
ejpam-7050	225	12	3	3	X
ejpam-7050	225	13	)	)	PUNCT
ejpam-7050	225	14	int⋆(cl⋆(int⋆(f−1(v	int⋆(cl⋆(int⋆(f−1(v	NOUN
ejpam-7050	225	15	)	)	PUNCT
ejpam-7050	225	16	)	)	PUNCT
ejpam-7050	225	17	)	)	PUNCT
ejpam-7050	225	18	)	)	PUNCT
ejpam-7050	225	19	⊆	⊆	NUM
ejpam-7050	225	20	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	225	21	-	-	PUNCT
ejpam-7050	225	22	cl(v	cl(v	NOUN
ejpam-7050	225	23	)	)	PUNCT
ejpam-7050	225	24	)	)	PUNCT
ejpam-7050	225	25	for	for	ADP
ejpam-7050	225	26	every	every	DET
ejpam-7050	225	27	σ1σ2	σ1σ2	NOUN
ejpam-7050	225	28	-	-	ADJ
ejpam-7050	225	29	open	open	ADJ
ejpam-7050	225	30	set	set	NOUN
ejpam-7050	225	31	v	v	NOUN
ejpam-7050	225	32	of	of	ADP
ejpam-7050	225	33	y	y	PROPN
ejpam-7050	225	34	;	;	PUNCT
ejpam-7050	225	35	(	(	PUNCT
ejpam-7050	225	36	4	4	X
ejpam-7050	225	37	)	)	PUNCT
ejpam-7050	225	38	int⋆(cl⋆(int⋆(f−1(σ1σ2	int⋆(cl⋆(int⋆(f−1(σ1σ2	NOUN
ejpam-7050	225	39	-	-	PUNCT
ejpam-7050	225	40	int(k	int(k	PROPN
ejpam-7050	225	41	)	)	PUNCT
ejpam-7050	225	42	)	)	PUNCT
ejpam-7050	225	43	)	)	PUNCT
ejpam-7050	225	44	)	)	PUNCT
ejpam-7050	225	45	)	)	PUNCT
ejpam-7050	226	1	⊆	⊆	NUM
ejpam-7050	226	2	f−1(k	f−1(k	PROPN
ejpam-7050	226	3	)	)	PUNCT
ejpam-7050	226	4	for	for	ADP
ejpam-7050	226	5	every	every	DET
ejpam-7050	226	6	σ1σ2	σ1σ2	NUM
ejpam-7050	226	7	-	-	PUNCT
ejpam-7050	226	8	closed	closed	ADJ
ejpam-7050	226	9	set	set	NOUN
ejpam-7050	226	10	k	k	PROPN
ejpam-7050	226	11	of	of	ADP
ejpam-7050	226	12	y	y	PROPN
ejpam-7050	226	13	;	;	PUNCT
ejpam-7050	226	14	(	(	PUNCT
ejpam-7050	226	15	5	5	X
ejpam-7050	226	16	)	)	PUNCT
ejpam-7050	226	17	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	226	18	-	-	SYM
ejpam-7050	226	19	int(k	int(k	NOUN
ejpam-7050	226	20	)	)	PUNCT
ejpam-7050	226	21	)	)	PUNCT
ejpam-7050	226	22	)	)	PUNCT
ejpam-7050	227	1	⊆	⊆	NUM
ejpam-7050	227	2	f−1(k	f−1(k	PROPN
ejpam-7050	227	3	)	)	PUNCT
ejpam-7050	227	4	for	for	ADP
ejpam-7050	227	5	every	every	DET
ejpam-7050	227	6	σ1σ2	σ1σ2	NUM
ejpam-7050	227	7	-	-	PUNCT
ejpam-7050	227	8	closed	closed	ADJ
ejpam-7050	227	9	set	set	NOUN
ejpam-7050	227	10	k	k	PROPN
ejpam-7050	227	11	of	of	ADP
ejpam-7050	227	12	y	y	PROPN
ejpam-7050	227	13	;	;	PUNCT
ejpam-7050	227	14	(	(	PUNCT
ejpam-7050	227	15	6	6	X
ejpam-7050	227	16	)	)	PUNCT
ejpam-7050	227	17	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	227	18	-	-	PUNCT
ejpam-7050	227	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	227	20	-	-	PUNCT
ejpam-7050	227	21	cl(b	cl(b	NOUN
ejpam-7050	227	22	)	)	PUNCT
ejpam-7050	227	23	)	)	PUNCT
ejpam-7050	227	24	)	)	PUNCT
ejpam-7050	227	25	)	)	PUNCT
ejpam-7050	228	1	⊆	⊆	NUM
ejpam-7050	228	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	228	3	-	-	PUNCT
ejpam-7050	228	4	cl(b	cl(b	NOUN
ejpam-7050	228	5	)	)	PUNCT
ejpam-7050	228	6	)	)	PUNCT
ejpam-7050	228	7	for	for	ADP
ejpam-7050	228	8	every	every	DET
ejpam-7050	228	9	subset	subset	NOUN
ejpam-7050	228	10	b	b	PROPN
ejpam-7050	228	11	of	of	ADP
ejpam-7050	228	12	y	y	PROPN
ejpam-7050	228	13	;	;	PUNCT
ejpam-7050	228	14	(	(	PUNCT
ejpam-7050	228	15	7	7	X
ejpam-7050	228	16	)	)	PUNCT
ejpam-7050	228	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	228	18	-	-	PUNCT
ejpam-7050	228	19	int(b	int(b	NOUN
ejpam-7050	228	20	)	)	PUNCT
ejpam-7050	228	21	)	)	PUNCT
ejpam-7050	228	22	⊆	⊆	NUM
ejpam-7050	228	23	βint⋆(f−1(σ1σ2	βint⋆(f−1(σ1σ2	NOUN
ejpam-7050	228	24	-	-	PUNCT
ejpam-7050	228	25	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	228	26	-	-	PUNCT
ejpam-7050	228	27	int(b	int(b	NOUN
ejpam-7050	228	28	)	)	PUNCT
ejpam-7050	228	29	)	)	PUNCT
ejpam-7050	228	30	)	)	PUNCT
ejpam-7050	228	31	)	)	PUNCT
ejpam-7050	228	32	for	for	ADP
ejpam-7050	228	33	every	every	DET
ejpam-7050	228	34	subset	subset	NOUN
ejpam-7050	228	35	b	b	PROPN
ejpam-7050	228	36	of	of	ADP
ejpam-7050	228	37	y	y	PROPN
ejpam-7050	228	38	;	;	PUNCT
ejpam-7050	228	39	(	(	PUNCT
ejpam-7050	228	40	8)	8)	NUM
ejpam-7050	228	41	f−1(v	f−1(v	NOUN
ejpam-7050	228	42	)	)	PUNCT
ejpam-7050	229	1	⊆	⊆	NUM
ejpam-7050	229	2	βint⋆(f−1(σ1σ2	βint⋆(f−1(σ1σ2	NOUN
ejpam-7050	229	3	-	-	NOUN
ejpam-7050	229	4	cl(v	cl(v	NOUN
ejpam-7050	229	5	)	)	PUNCT
ejpam-7050	229	6	)	)	PUNCT
ejpam-7050	229	7	)	)	PUNCT
ejpam-7050	229	8	for	for	ADP
ejpam-7050	229	9	every	every	DET
ejpam-7050	229	10	σ1σ2	σ1σ2	NOUN
ejpam-7050	229	11	-	-	ADJ
ejpam-7050	229	12	open	open	ADJ
ejpam-7050	229	13	set	set	NOUN
ejpam-7050	229	14	v	v	NOUN
ejpam-7050	229	15	of	of	ADP
ejpam-7050	229	16	y	y	PROPN
ejpam-7050	229	17	;	;	PUNCT
ejpam-7050	229	18	(	(	PUNCT
ejpam-7050	229	19	9	9	X
ejpam-7050	229	20	)	)	PUNCT
ejpam-7050	229	21	βcl⋆(f−1(v	βcl⋆(f−1(v	NUM
ejpam-7050	229	22	)	)	PUNCT
ejpam-7050	229	23	)	)	PUNCT
ejpam-7050	230	1	⊆	⊆	NUM
ejpam-7050	230	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	230	3	-	-	PUNCT
ejpam-7050	230	4	cl(v	cl(v	NOUN
ejpam-7050	230	5	)	)	PUNCT
ejpam-7050	230	6	)	)	PUNCT
ejpam-7050	230	7	for	for	ADP
ejpam-7050	230	8	every	every	DET
ejpam-7050	230	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	230	10	-	-	ADJ
ejpam-7050	230	11	open	open	ADJ
ejpam-7050	230	12	set	set	NOUN
ejpam-7050	230	13	v	v	NOUN
ejpam-7050	230	14	of	of	ADP
ejpam-7050	230	15	y	y	PROPN
ejpam-7050	230	16	.	.	PUNCT
ejpam-7050	231	1	definition	definition	NOUN
ejpam-7050	231	2	4	4	NUM
ejpam-7050	231	3	.	.	PUNCT
ejpam-7050	232	1	[	[	X
ejpam-7050	232	2	21	21	NUM
ejpam-7050	232	3	]	]	PUNCT
ejpam-7050	232	4	a	a	DET
ejpam-7050	232	5	multifunction	multifunction	NOUN
ejpam-7050	232	6	f	f	NOUN
ejpam-7050	232	7	:	:	PUNCT
ejpam-7050	232	8	(	(	PUNCT
ejpam-7050	232	9	x	x	X
ejpam-7050	232	10	,	,	PUNCT
ejpam-7050	232	11	τ	τ	PROPN
ejpam-7050	232	12	,	,	PUNCT
ejpam-7050	232	13	i	i	NOUN
ejpam-7050	232	14	)	)	PUNCT
ejpam-7050	232	15	→	→	PUNCT
ejpam-7050	232	16	(	(	PUNCT
ejpam-7050	232	17	y	y	PROPN
ejpam-7050	232	18	,	,	PUNCT
ejpam-7050	232	19	σ1	σ1	PROPN
ejpam-7050	232	20	,	,	PUNCT
ejpam-7050	232	21	σ2	σ2	PROPN
ejpam-7050	232	22	)	)	PUNCT
ejpam-7050	232	23	is	be	AUX
ejpam-7050	232	24	said	say	VERB
ejpam-7050	232	25	to	to	PART
ejpam-7050	232	26	be	be	AUX
ejpam-7050	232	27	upper	upper	ADJ
ejpam-7050	232	28	almost	almost	ADV
ejpam-7050	232	29	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	232	30	,	,	PUNCT
ejpam-7050	232	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	232	32	at	at	ADP
ejpam-7050	232	33	a	a	DET
ejpam-7050	232	34	point	point	NOUN
ejpam-7050	232	35	x	x	PUNCT
ejpam-7050	232	36	of	of	ADP
ejpam-7050	232	37	x	x	PRON
ejpam-7050	232	38	if	if	SCONJ
ejpam-7050	232	39	for	for	ADP
ejpam-7050	232	40	each	each	DET
ejpam-7050	232	41	σ1σ2	σ1σ2	VERB
ejpam-7050	232	42	-	-	ADJ
ejpam-7050	232	43	open	open	ADJ
ejpam-7050	232	44	set	set	NOUN
ejpam-7050	232	45	v	v	NOUN
ejpam-7050	232	46	of	of	ADP
ejpam-7050	232	47	y	y	PRON
ejpam-7050	232	48	such	such	ADJ
ejpam-7050	232	49	that	that	SCONJ
ejpam-7050	232	50	f	f	PROPN
ejpam-7050	232	51	(	(	PUNCT
ejpam-7050	232	52	x	x	X
ejpam-7050	232	53	)	)	PUNCT
ejpam-7050	232	54	⊆	⊆	NUM
ejpam-7050	232	55	v	v	NOUN
ejpam-7050	232	56	,	,	PUNCT
ejpam-7050	232	57	there	there	PRON
ejpam-7050	232	58	exists	exist	VERB
ejpam-7050	232	59	a	a	DET
ejpam-7050	232	60	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	232	61	-	-	ADJ
ejpam-7050	232	62	open	open	ADJ
ejpam-7050	232	63	set	set	NOUN
ejpam-7050	232	64	u	u	NOUN
ejpam-7050	232	65	of	of	ADP
ejpam-7050	232	66	x	x	PUNCT
ejpam-7050	232	67	containing	contain	VERB
ejpam-7050	232	68	x	x	PUNCT
ejpam-7050	232	69	such	such	ADJ
ejpam-7050	232	70	that	that	SCONJ
ejpam-7050	232	71	f	f	PROPN
ejpam-7050	232	72	(	(	PUNCT
ejpam-7050	232	73	u	u	NOUN
ejpam-7050	232	74	)	)	PUNCT
ejpam-7050	232	75	⊆	⊆	NUM
ejpam-7050	232	76	σ1σ2	σ1σ2	X
ejpam-7050	232	77	-	-	PUNCT
ejpam-7050	232	78	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	232	79	-	-	PUNCT
ejpam-7050	232	80	cl(v	cl(v	NOUN
ejpam-7050	232	81	)	)	PUNCT
ejpam-7050	232	82	)	)	PUNCT
ejpam-7050	232	83	.	.	PUNCT
ejpam-7050	233	1	a	a	DET
ejpam-7050	233	2	multifunction	multifunction	NOUN
ejpam-7050	233	3	f	f	NOUN
ejpam-7050	233	4	:	:	PUNCT
ejpam-7050	233	5	(	(	PUNCT
ejpam-7050	233	6	x	x	X
ejpam-7050	233	7	,	,	PUNCT
ejpam-7050	233	8	τ	τ	PROPN
ejpam-7050	233	9	,	,	PUNCT
ejpam-7050	233	10	i	i	NOUN
ejpam-7050	233	11	)	)	PUNCT
ejpam-7050	233	12	→	→	PUNCT
ejpam-7050	233	13	(	(	PUNCT
ejpam-7050	233	14	y	y	PROPN
ejpam-7050	233	15	,	,	PUNCT
ejpam-7050	233	16	σ1	σ1	PROPN
ejpam-7050	233	17	,	,	PUNCT
ejpam-7050	233	18	σ2	σ2	PROPN
ejpam-7050	233	19	)	)	PUNCT
ejpam-7050	233	20	is	be	AUX
ejpam-7050	233	21	said	say	VERB
ejpam-7050	233	22	to	to	PART
ejpam-7050	233	23	be	be	AUX
ejpam-7050	233	24	upper	upper	ADJ
ejpam-7050	233	25	almost	almost	ADV
ejpam-7050	233	26	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	233	27	,	,	PUNCT
ejpam-7050	233	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	233	29	if	if	SCONJ
ejpam-7050	233	30	f	f	PROPN
ejpam-7050	233	31	is	be	AUX
ejpam-7050	233	32	upper	upper	ADJ
ejpam-7050	233	33	almost	almost	ADV
ejpam-7050	233	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	233	35	,	,	PUNCT
ejpam-7050	233	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	233	37	at	at	ADP
ejpam-7050	233	38	each	each	DET
ejpam-7050	233	39	point	point	NOUN
ejpam-7050	233	40	of	of	ADP
ejpam-7050	233	41	x.	x.	NOUN
ejpam-7050	233	42	definition	definition	NOUN
ejpam-7050	233	43	5	5	NUM
ejpam-7050	233	44	.	.	PUNCT
ejpam-7050	234	1	[	[	X
ejpam-7050	234	2	21	21	NUM
ejpam-7050	234	3	]	]	PUNCT
ejpam-7050	234	4	a	a	DET
ejpam-7050	234	5	multifunction	multifunction	NOUN
ejpam-7050	234	6	f	f	NOUN
ejpam-7050	234	7	:	:	PUNCT
ejpam-7050	234	8	(	(	PUNCT
ejpam-7050	234	9	x	x	X
ejpam-7050	234	10	,	,	PUNCT
ejpam-7050	234	11	τ	τ	PROPN
ejpam-7050	234	12	,	,	PUNCT
ejpam-7050	234	13	i	i	NOUN
ejpam-7050	234	14	)	)	PUNCT
ejpam-7050	234	15	→	→	PUNCT
ejpam-7050	234	16	(	(	PUNCT
ejpam-7050	234	17	y	y	PROPN
ejpam-7050	234	18	,	,	PUNCT
ejpam-7050	234	19	σ1	σ1	PROPN
ejpam-7050	234	20	,	,	PUNCT
ejpam-7050	234	21	σ2	σ2	PROPN
ejpam-7050	234	22	)	)	PUNCT
ejpam-7050	234	23	is	be	AUX
ejpam-7050	234	24	said	say	VERB
ejpam-7050	234	25	to	to	PART
ejpam-7050	234	26	be	be	AUX
ejpam-7050	234	27	lower	low	ADJ
ejpam-7050	234	28	almost	almost	ADV
ejpam-7050	234	29	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	234	30	,	,	PUNCT
ejpam-7050	234	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	234	32	at	at	ADP
ejpam-7050	234	33	a	a	DET
ejpam-7050	234	34	point	point	NOUN
ejpam-7050	234	35	x	x	PUNCT
ejpam-7050	234	36	of	of	ADP
ejpam-7050	234	37	x	x	PRON
ejpam-7050	234	38	if	if	SCONJ
ejpam-7050	234	39	for	for	ADP
ejpam-7050	234	40	each	each	DET
ejpam-7050	234	41	σ1σ2	σ1σ2	VERB
ejpam-7050	234	42	-	-	ADJ
ejpam-7050	234	43	open	open	ADJ
ejpam-7050	234	44	set	set	NOUN
ejpam-7050	234	45	v	v	NOUN
ejpam-7050	234	46	of	of	ADP
ejpam-7050	234	47	y	y	PRON
ejpam-7050	234	48	such	such	ADJ
ejpam-7050	234	49	that	that	SCONJ
ejpam-7050	234	50	f	f	PROPN
ejpam-7050	234	51	(	(	PUNCT
ejpam-7050	234	52	x	x	NOUN
ejpam-7050	234	53	)	)	PUNCT
ejpam-7050	234	54	∩	∩	NOUN
ejpam-7050	234	55	v	v	ADP
ejpam-7050	234	56	̸=	̸=	PROPN
ejpam-7050	234	57	∅	∅	NOUN
ejpam-7050	234	58	,	,	PUNCT
ejpam-7050	234	59	there	there	PRON
ejpam-7050	234	60	exists	exist	VERB
ejpam-7050	234	61	a	a	DET
ejpam-7050	234	62	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	234	63	-	-	ADJ
ejpam-7050	234	64	open	open	ADJ
ejpam-7050	234	65	set	set	NOUN
ejpam-7050	234	66	u	u	NOUN
ejpam-7050	234	67	of	of	ADP
ejpam-7050	234	68	x	x	PUNCT
ejpam-7050	234	69	containing	contain	VERB
ejpam-7050	234	70	x	x	PUNCT
ejpam-7050	234	71	such	such	ADJ
ejpam-7050	234	72	that	that	SCONJ
ejpam-7050	234	73	f	f	PROPN
ejpam-7050	234	74	(	(	PUNCT
ejpam-7050	234	75	z	z	NOUN
ejpam-7050	234	76	)	)	PUNCT
ejpam-7050	234	77	∩	∩	NOUN
ejpam-7050	234	78	σ1σ2	σ1σ2	X
ejpam-7050	234	79	-	-	PUNCT
ejpam-7050	234	80	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	234	81	-	-	PUNCT
ejpam-7050	234	82	cl(v	cl(v	NOUN
ejpam-7050	234	83	)	)	PUNCT
ejpam-7050	234	84	)	)	PUNCT
ejpam-7050	235	1	̸=	̸=	NOUN
ejpam-7050	235	2	∅	∅	NOUN
ejpam-7050	235	3	for	for	ADP
ejpam-7050	235	4	every	every	DET
ejpam-7050	235	5	z	z	NOUN
ejpam-7050	235	6	∈	∈	PROPN
ejpam-7050	235	7	u	u	NOUN
ejpam-7050	235	8	.	.	PUNCT
ejpam-7050	236	1	a	a	DET
ejpam-7050	236	2	multifunction	multifunction	NOUN
ejpam-7050	236	3	f	f	NOUN
ejpam-7050	236	4	:	:	PUNCT
ejpam-7050	236	5	(	(	PUNCT
ejpam-7050	236	6	x	x	X
ejpam-7050	236	7	,	,	PUNCT
ejpam-7050	236	8	τ	τ	PROPN
ejpam-7050	236	9	,	,	PUNCT
ejpam-7050	236	10	i	i	NOUN
ejpam-7050	236	11	)	)	PUNCT
ejpam-7050	236	12	→	→	PUNCT
ejpam-7050	236	13	(	(	PUNCT
ejpam-7050	236	14	y	y	PROPN
ejpam-7050	236	15	,	,	PUNCT
ejpam-7050	236	16	σ1	σ1	PROPN
ejpam-7050	236	17	,	,	PUNCT
ejpam-7050	236	18	σ2	σ2	PROPN
ejpam-7050	236	19	)	)	PUNCT
ejpam-7050	236	20	is	be	AUX
ejpam-7050	236	21	said	say	VERB
ejpam-7050	236	22	to	to	PART
ejpam-7050	236	23	be	be	AUX
ejpam-7050	236	24	lower	low	ADJ
ejpam-7050	236	25	almost	almost	ADV
ejpam-7050	236	26	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	236	27	,	,	PUNCT
ejpam-7050	236	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	236	29	if	if	SCONJ
ejpam-7050	236	30	f	f	PROPN
ejpam-7050	236	31	is	be	AUX
ejpam-7050	236	32	lower	low	ADJ
ejpam-7050	236	33	almost	almost	ADV
ejpam-7050	236	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	236	35	,	,	PUNCT
ejpam-7050	236	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	236	37	at	at	ADP
ejpam-7050	236	38	each	each	DET
ejpam-7050	236	39	point	point	NOUN
ejpam-7050	236	40	of	of	ADP
ejpam-7050	236	41	x.	x.	NOUN
ejpam-7050	236	42	remark	remark	PROPN
ejpam-7050	236	43	1	1	NUM
ejpam-7050	236	44	.	.	PUNCT
ejpam-7050	236	45	for	for	ADP
ejpam-7050	236	46	a	a	DET
ejpam-7050	236	47	multifunction	multifunction	NOUN
ejpam-7050	236	48	f	f	NOUN
ejpam-7050	236	49	:	:	PUNCT
ejpam-7050	236	50	(	(	PUNCT
ejpam-7050	236	51	x	x	X
ejpam-7050	236	52	,	,	PUNCT
ejpam-7050	236	53	τ	τ	PROPN
ejpam-7050	236	54	,	,	PUNCT
ejpam-7050	236	55	i	i	NOUN
ejpam-7050	236	56	)	)	PUNCT
ejpam-7050	236	57	→	→	PUNCT
ejpam-7050	236	58	(	(	PUNCT
ejpam-7050	236	59	y	y	PROPN
ejpam-7050	236	60	,	,	PUNCT
ejpam-7050	236	61	σ1	σ1	PROPN
ejpam-7050	236	62	,	,	PUNCT
ejpam-7050	236	63	σ2	σ2	NOUN
ejpam-7050	236	64	)	)	PUNCT
ejpam-7050	236	65	,	,	PUNCT
ejpam-7050	236	66	the	the	DET
ejpam-7050	236	67	following	follow	VERB
ejpam-7050	236	68	implication	implication	NOUN
ejpam-7050	236	69	holds	hold	VERB
ejpam-7050	236	70	:	:	PUNCT
ejpam-7050	236	71	upper	upper	ADJ
ejpam-7050	236	72	almost	almost	ADV
ejpam-7050	236	73	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	236	74	,	,	PUNCT
ejpam-7050	236	75	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7050	236	76	⇒	⇒	VERB
ejpam-7050	236	77	upper	upper	ADJ
ejpam-7050	236	78	weak	weak	ADJ
ejpam-7050	236	79	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	236	80	,	,	PUNCT
ejpam-7050	236	81	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7050	236	82	.	.	PUNCT
ejpam-7050	237	1	the	the	DET
ejpam-7050	237	2	converse	converse	NOUN
ejpam-7050	237	3	of	of	ADP
ejpam-7050	237	4	the	the	DET
ejpam-7050	237	5	implication	implication	NOUN
ejpam-7050	237	6	is	be	AUX
ejpam-7050	237	7	not	not	PART
ejpam-7050	237	8	true	true	ADJ
ejpam-7050	237	9	in	in	ADP
ejpam-7050	237	10	general	general	ADJ
ejpam-7050	237	11	.	.	PUNCT
ejpam-7050	238	1	we	we	PRON
ejpam-7050	238	2	give	give	VERB
ejpam-7050	238	3	an	an	DET
ejpam-7050	238	4	example	example	NOUN
ejpam-7050	238	5	for	for	ADP
ejpam-7050	238	6	the	the	DET
ejpam-7050	238	7	implication	implication	NOUN
ejpam-7050	238	8	as	as	SCONJ
ejpam-7050	238	9	follows	follow	VERB
ejpam-7050	238	10	.	.	PUNCT
ejpam-7050	239	1	m.	m.	NOUN
ejpam-7050	239	2	thongmoon	thongmoon	PROPN
ejpam-7050	239	3	,	,	PUNCT
ejpam-7050	239	4	a.	a.	PROPN
ejpam-7050	239	5	sama	sama	PROPN
ejpam-7050	239	6	-	-	PUNCT
ejpam-7050	239	7	ae	ae	PROPN
ejpam-7050	239	8	,	,	PUNCT
ejpam-7050	239	9	c.	c.	PROPN
ejpam-7050	239	10	boonpok	boonpok	PROPN
ejpam-7050	239	11	/	/	SYM
ejpam-7050	239	12	eur	eur	PROPN
ejpam-7050	239	13	.	.	PUNCT
ejpam-7050	240	1	j.	j.	PROPN
ejpam-7050	240	2	pure	pure	PROPN
ejpam-7050	240	3	appl	appl	PROPN
ejpam-7050	240	4	.	.	PROPN
ejpam-7050	240	5	math	math	PROPN
ejpam-7050	240	6	,	,	PUNCT
ejpam-7050	240	7	18	18	NUM
ejpam-7050	240	8	(	(	PUNCT
ejpam-7050	240	9	4	4	NUM
ejpam-7050	240	10	)	)	PUNCT
ejpam-7050	240	11	(	(	PUNCT
ejpam-7050	240	12	2025	2025	NUM
ejpam-7050	240	13	)	)	PUNCT
ejpam-7050	240	14	,	,	PUNCT
ejpam-7050	240	15	7050	7050	NUM
ejpam-7050	240	16	9	9	NUM
ejpam-7050	240	17	of	of	ADP
ejpam-7050	240	18	14	14	NUM
ejpam-7050	240	19	example	example	NOUN
ejpam-7050	241	1	1	1	NUM
ejpam-7050	241	2	.	.	PUNCT
ejpam-7050	242	1	let	let	VERB
ejpam-7050	242	2	x	x	PUNCT
ejpam-7050	242	3	=	=	PRON
ejpam-7050	242	4	{	{	PUNCT
ejpam-7050	242	5	1	1	NUM
ejpam-7050	242	6	,	,	PUNCT
ejpam-7050	242	7	2	2	NUM
ejpam-7050	242	8	,	,	PUNCT
ejpam-7050	242	9	3	3	NUM
ejpam-7050	242	10	}	}	PUNCT
ejpam-7050	242	11	with	with	ADP
ejpam-7050	242	12	a	a	DET
ejpam-7050	242	13	topology	topology	NOUN
ejpam-7050	242	14	τ	τ	X
ejpam-7050	242	15	=	=	SYM
ejpam-7050	242	16	{	{	PUNCT
ejpam-7050	242	17	∅	∅	NOUN
ejpam-7050	242	18	,	,	PUNCT
ejpam-7050	242	19	{	{	PUNCT
ejpam-7050	242	20	1	1	NUM
ejpam-7050	242	21	}	}	PUNCT
ejpam-7050	242	22	,	,	PUNCT
ejpam-7050	242	23	{	{	PUNCT
ejpam-7050	242	24	2	2	NUM
ejpam-7050	242	25	}	}	PUNCT
ejpam-7050	242	26	,	,	PUNCT
ejpam-7050	242	27	{	{	PUNCT
ejpam-7050	242	28	1	1	NUM
ejpam-7050	242	29	,	,	PUNCT
ejpam-7050	242	30	2	2	NUM
ejpam-7050	242	31	}	}	PUNCT
ejpam-7050	242	32	,	,	PUNCT
ejpam-7050	242	33	x	x	NOUN
ejpam-7050	242	34	}	}	PUNCT
ejpam-7050	242	35	and	and	CCONJ
ejpam-7050	242	36	an	an	DET
ejpam-7050	242	37	ideal	ideal	NOUN
ejpam-7050	242	38	i	i	X
ejpam-7050	242	39	=	=	SYM
ejpam-7050	242	40	{	{	PUNCT
ejpam-7050	242	41	∅	∅	NOUN
ejpam-7050	242	42	,	,	PUNCT
ejpam-7050	242	43	{	{	PUNCT
ejpam-7050	242	44	1	1	NUM
ejpam-7050	242	45	}	}	PUNCT
ejpam-7050	242	46	}	}	PUNCT
ejpam-7050	242	47	.	.	PUNCT
ejpam-7050	243	1	let	let	VERB
ejpam-7050	243	2	y	y	PROPN
ejpam-7050	243	3	=	=	PUNCT
ejpam-7050	243	4	{	{	PUNCT
ejpam-7050	243	5	a	a	PRON
ejpam-7050	243	6	,	,	PUNCT
ejpam-7050	243	7	b	b	NOUN
ejpam-7050	243	8	,	,	PUNCT
ejpam-7050	243	9	c	c	NOUN
ejpam-7050	243	10	}	}	PUNCT
ejpam-7050	243	11	with	with	ADP
ejpam-7050	243	12	topologies	topology	NOUN
ejpam-7050	243	13	σ1	σ1	NOUN
ejpam-7050	243	14	=	=	SYM
ejpam-7050	243	15	{	{	PUNCT
ejpam-7050	243	16	∅	∅	NOUN
ejpam-7050	243	17	,	,	PUNCT
ejpam-7050	243	18	{	{	PUNCT
ejpam-7050	243	19	a	a	X
ejpam-7050	243	20	}	}	PUNCT
ejpam-7050	243	21	,	,	PUNCT
ejpam-7050	243	22	{	{	PUNCT
ejpam-7050	243	23	a	a	DET
ejpam-7050	243	24	,	,	PUNCT
ejpam-7050	243	25	b	b	NOUN
ejpam-7050	243	26	}	}	PUNCT
ejpam-7050	243	27	,	,	PUNCT
ejpam-7050	243	28	y	y	PROPN
ejpam-7050	243	29	}	}	PUNCT
ejpam-7050	243	30	and	and	CCONJ
ejpam-7050	243	31	σ2	σ2	PROPN
ejpam-7050	243	32	=	=	SYM
ejpam-7050	243	33	{	{	PUNCT
ejpam-7050	243	34	∅	∅	NOUN
ejpam-7050	243	35	,	,	PUNCT
ejpam-7050	243	36	{	{	PUNCT
ejpam-7050	243	37	a	a	X
ejpam-7050	243	38	}	}	PUNCT
ejpam-7050	243	39	,	,	PUNCT
ejpam-7050	243	40	{	{	PUNCT
ejpam-7050	243	41	b	b	NOUN
ejpam-7050	243	42	}	}	PUNCT
ejpam-7050	243	43	,	,	PUNCT
ejpam-7050	243	44	{	{	PUNCT
ejpam-7050	243	45	a	a	DET
ejpam-7050	243	46	,	,	PUNCT
ejpam-7050	243	47	b	b	NOUN
ejpam-7050	243	48	}	}	PUNCT
ejpam-7050	243	49	,	,	PUNCT
ejpam-7050	243	50	y	y	PROPN
ejpam-7050	243	51	}	}	PUNCT
ejpam-7050	243	52	.	.	PUNCT
ejpam-7050	244	1	a	a	DET
ejpam-7050	244	2	multifunction	multifunction	NOUN
ejpam-7050	244	3	f	f	NOUN
ejpam-7050	244	4	:	:	PUNCT
ejpam-7050	244	5	(	(	PUNCT
ejpam-7050	244	6	x	x	X
ejpam-7050	244	7	,	,	PUNCT
ejpam-7050	244	8	τ	τ	PROPN
ejpam-7050	244	9	,	,	PUNCT
ejpam-7050	244	10	i	i	NOUN
ejpam-7050	244	11	)	)	PUNCT
ejpam-7050	244	12	→	→	PUNCT
ejpam-7050	244	13	(	(	PUNCT
ejpam-7050	244	14	y	y	PROPN
ejpam-7050	244	15	,	,	PUNCT
ejpam-7050	244	16	σ1	σ1	PROPN
ejpam-7050	244	17	,	,	PUNCT
ejpam-7050	244	18	σ2	σ2	PROPN
ejpam-7050	244	19	)	)	PUNCT
ejpam-7050	244	20	is	be	AUX
ejpam-7050	244	21	defined	define	VERB
ejpam-7050	244	22	as	as	SCONJ
ejpam-7050	244	23	follows	follow	VERB
ejpam-7050	244	24	:	:	PUNCT
ejpam-7050	244	25	f	f	X
ejpam-7050	244	26	(	(	PUNCT
ejpam-7050	244	27	1	1	X
ejpam-7050	244	28	)	)	PUNCT
ejpam-7050	244	29	=	=	PRON
ejpam-7050	245	1	{	{	PUNCT
ejpam-7050	245	2	c	c	NOUN
ejpam-7050	245	3	}	}	PUNCT
ejpam-7050	245	4	and	and	CCONJ
ejpam-7050	245	5	f	f	X
ejpam-7050	245	6	(	(	PUNCT
ejpam-7050	245	7	2	2	NUM
ejpam-7050	245	8	)	)	PUNCT
ejpam-7050	245	9	=	=	SYM
ejpam-7050	245	10	f	f	PROPN
ejpam-7050	245	11	(	(	PUNCT
ejpam-7050	245	12	3	3	NUM
ejpam-7050	245	13	)	)	PUNCT
ejpam-7050	245	14	=	=	PRON
ejpam-7050	245	15	{	{	PUNCT
ejpam-7050	245	16	a	a	PRON
ejpam-7050	245	17	,	,	PUNCT
ejpam-7050	245	18	b	b	NOUN
ejpam-7050	245	19	}	}	PUNCT
ejpam-7050	245	20	.	.	PUNCT
ejpam-7050	246	1	then	then	ADV
ejpam-7050	246	2	,	,	PUNCT
ejpam-7050	246	3	f	f	PROPN
ejpam-7050	246	4	is	be	AUX
ejpam-7050	246	5	upper	upper	ADJ
ejpam-7050	246	6	weakly	weakly	ADJ
ejpam-7050	246	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	246	8	,	,	PUNCT
ejpam-7050	246	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	246	10	but	but	CCONJ
ejpam-7050	246	11	f	f	PROPN
ejpam-7050	246	12	is	be	AUX
ejpam-7050	246	13	not	not	PART
ejpam-7050	246	14	upper	upper	ADJ
ejpam-7050	246	15	almost	almost	ADV
ejpam-7050	246	16	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	246	17	,	,	PUNCT
ejpam-7050	246	18	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7050	246	19	.	.	X
ejpam-7050	246	20	theorem	theorem	NOUN
ejpam-7050	246	21	5	5	NUM
ejpam-7050	246	22	.	.	X
ejpam-7050	246	23	for	for	ADP
ejpam-7050	246	24	a	a	DET
ejpam-7050	246	25	multifunction	multifunction	NOUN
ejpam-7050	246	26	f	f	NOUN
ejpam-7050	246	27	:	:	PUNCT
ejpam-7050	246	28	(	(	PUNCT
ejpam-7050	246	29	x	x	X
ejpam-7050	246	30	,	,	PUNCT
ejpam-7050	246	31	τ	τ	PROPN
ejpam-7050	246	32	,	,	PUNCT
ejpam-7050	246	33	i	i	NOUN
ejpam-7050	246	34	)	)	PUNCT
ejpam-7050	246	35	→	→	PUNCT
ejpam-7050	246	36	(	(	PUNCT
ejpam-7050	246	37	y	y	PROPN
ejpam-7050	246	38	,	,	PUNCT
ejpam-7050	246	39	σ1	σ1	PROPN
ejpam-7050	246	40	,	,	PUNCT
ejpam-7050	246	41	σ2	σ2	NOUN
ejpam-7050	246	42	)	)	PUNCT
ejpam-7050	246	43	,	,	PUNCT
ejpam-7050	246	44	the	the	DET
ejpam-7050	246	45	following	follow	VERB
ejpam-7050	246	46	properties	property	NOUN
ejpam-7050	246	47	are	be	AUX
ejpam-7050	246	48	equivalent	equivalent	ADJ
ejpam-7050	246	49	:	:	PUNCT
ejpam-7050	246	50	(	(	PUNCT
ejpam-7050	246	51	1	1	X
ejpam-7050	246	52	)	)	PUNCT
ejpam-7050	246	53	f	f	PROPN
ejpam-7050	246	54	is	be	AUX
ejpam-7050	246	55	upper	upper	ADJ
ejpam-7050	246	56	weakly	weakly	ADJ
ejpam-7050	246	57	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	246	58	,	,	PUNCT
ejpam-7050	246	59	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	246	60	;	;	PUNCT
ejpam-7050	246	61	(	(	PUNCT
ejpam-7050	246	62	2	2	X
ejpam-7050	246	63	)	)	PUNCT
ejpam-7050	246	64	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	246	65	-	-	PUNCT
ejpam-7050	246	66	int((σ1	int((σ1	PROPN
ejpam-7050	246	67	,	,	PUNCT
ejpam-7050	246	68	σ2)θ	σ2)θ	ADJ
ejpam-7050	246	69	-	-	PUNCT
ejpam-7050	246	70	cl(b	cl(b	NOUN
ejpam-7050	246	71	)	)	PUNCT
ejpam-7050	246	72	)	)	PUNCT
ejpam-7050	246	73	)	)	PUNCT
ejpam-7050	246	74	)	)	PUNCT
ejpam-7050	247	1	⊆	⊆	NUM
ejpam-7050	247	2	f−((σ1	f−((σ1	NOUN
ejpam-7050	247	3	,	,	PUNCT
ejpam-7050	247	4	σ2)θ	σ2)θ	ADJ
ejpam-7050	247	5	-	-	PUNCT
ejpam-7050	247	6	cl(b	cl(b	NOUN
ejpam-7050	247	7	)	)	PUNCT
ejpam-7050	247	8	)	)	PUNCT
ejpam-7050	247	9	for	for	ADP
ejpam-7050	247	10	every	every	DET
ejpam-7050	247	11	subset	subset	NOUN
ejpam-7050	247	12	b	b	PROPN
ejpam-7050	247	13	of	of	ADP
ejpam-7050	247	14	y	y	PROPN
ejpam-7050	247	15	;	;	PUNCT
ejpam-7050	247	16	(	(	PUNCT
ejpam-7050	247	17	3	3	X
ejpam-7050	247	18	)	)	PUNCT
ejpam-7050	247	19	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	247	20	-	-	PUNCT
ejpam-7050	247	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	247	22	-	-	PUNCT
ejpam-7050	247	23	cl(b	cl(b	NOUN
ejpam-7050	247	24	)	)	PUNCT
ejpam-7050	247	25	)	)	PUNCT
ejpam-7050	247	26	)	)	PUNCT
ejpam-7050	247	27	)	)	PUNCT
ejpam-7050	248	1	⊆	⊆	NUM
ejpam-7050	248	2	f−((σ1	f−((σ1	NOUN
ejpam-7050	248	3	,	,	PUNCT
ejpam-7050	248	4	σ2)θ	σ2)θ	ADJ
ejpam-7050	248	5	-	-	PUNCT
ejpam-7050	248	6	cl(b	cl(b	NOUN
ejpam-7050	248	7	)	)	PUNCT
ejpam-7050	248	8	)	)	PUNCT
ejpam-7050	248	9	for	for	ADP
ejpam-7050	248	10	every	every	DET
ejpam-7050	248	11	subset	subset	NOUN
ejpam-7050	248	12	b	b	PROPN
ejpam-7050	248	13	of	of	ADP
ejpam-7050	248	14	y	y	PROPN
ejpam-7050	248	15	;	;	PUNCT
ejpam-7050	248	16	(	(	PUNCT
ejpam-7050	248	17	4	4	X
ejpam-7050	248	18	)	)	PUNCT
ejpam-7050	248	19	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	248	20	-	-	PUNCT
ejpam-7050	248	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	248	22	-	-	PUNCT
ejpam-7050	248	23	cl(v	cl(v	NOUN
ejpam-7050	248	24	)	)	PUNCT
ejpam-7050	248	25	)	)	PUNCT
ejpam-7050	248	26	)	)	PUNCT
ejpam-7050	248	27	)	)	PUNCT
ejpam-7050	249	1	⊆	⊆	X
ejpam-7050	249	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	249	3	-	-	PUNCT
ejpam-7050	249	4	cl(v	cl(v	NOUN
ejpam-7050	249	5	)	)	PUNCT
ejpam-7050	249	6	)	)	PUNCT
ejpam-7050	249	7	for	for	ADP
ejpam-7050	249	8	every	every	DET
ejpam-7050	249	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	249	10	-	-	ADJ
ejpam-7050	249	11	open	open	ADJ
ejpam-7050	249	12	set	set	NOUN
ejpam-7050	249	13	v	v	NOUN
ejpam-7050	249	14	of	of	ADP
ejpam-7050	249	15	y	y	PROPN
ejpam-7050	249	16	;	;	PUNCT
ejpam-7050	249	17	(	(	PUNCT
ejpam-7050	249	18	5	5	X
ejpam-7050	249	19	)	)	PUNCT
ejpam-7050	249	20	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	249	21	-	-	PUNCT
ejpam-7050	249	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	249	23	-	-	PUNCT
ejpam-7050	249	24	cl(v	cl(v	NOUN
ejpam-7050	249	25	)	)	PUNCT
ejpam-7050	249	26	)	)	PUNCT
ejpam-7050	249	27	)	)	PUNCT
ejpam-7050	249	28	)	)	PUNCT
ejpam-7050	250	1	⊆	⊆	X
ejpam-7050	250	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	250	3	-	-	PUNCT
ejpam-7050	250	4	cl(v	cl(v	NOUN
ejpam-7050	250	5	)	)	PUNCT
ejpam-7050	250	6	)	)	PUNCT
ejpam-7050	250	7	for	for	ADP
ejpam-7050	250	8	every	every	DET
ejpam-7050	250	9	(	(	PUNCT
ejpam-7050	250	10	σ1	σ1	PROPN
ejpam-7050	250	11	,	,	PUNCT
ejpam-7050	250	12	σ2)p	σ2)p	NOUN
ejpam-7050	250	13	-	-	PUNCT
ejpam-7050	250	14	open	open	NOUN
ejpam-7050	250	15	set	set	NOUN
ejpam-7050	250	16	v	v	NOUN
ejpam-7050	250	17	of	of	ADP
ejpam-7050	250	18	y	y	PROPN
ejpam-7050	250	19	;	;	PUNCT
ejpam-7050	250	20	(	(	PUNCT
ejpam-7050	250	21	6	6	X
ejpam-7050	250	22	)	)	PUNCT
ejpam-7050	250	23	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	250	24	-	-	PUNCT
ejpam-7050	250	25	int(k	int(k	NOUN
ejpam-7050	250	26	)	)	PUNCT
ejpam-7050	250	27	)	)	PUNCT
ejpam-7050	250	28	)	)	PUNCT
ejpam-7050	251	1	⊆	⊆	X
ejpam-7050	251	2	f−(k	f−(k	PROPN
ejpam-7050	251	3	)	)	PUNCT
ejpam-7050	251	4	for	for	ADP
ejpam-7050	251	5	every	every	DET
ejpam-7050	251	6	(	(	PUNCT
ejpam-7050	251	7	σ1	σ1	PROPN
ejpam-7050	251	8	,	,	PUNCT
ejpam-7050	251	9	σ2)r	σ2)r	NOUN
ejpam-7050	251	10	-	-	PUNCT
ejpam-7050	251	11	closed	close	VERB
ejpam-7050	251	12	set	set	ADJ
ejpam-7050	251	13	k	k	PROPN
ejpam-7050	251	14	of	of	ADP
ejpam-7050	251	15	y	y	PROPN
ejpam-7050	251	16	;	;	PUNCT
ejpam-7050	251	17	(	(	PUNCT
ejpam-7050	251	18	7	7	X
ejpam-7050	251	19	)	)	PUNCT
ejpam-7050	251	20	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	251	21	-	-	PUNCT
ejpam-7050	251	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	251	23	-	-	PUNCT
ejpam-7050	251	24	cl(v	cl(v	NOUN
ejpam-7050	251	25	)	)	PUNCT
ejpam-7050	251	26	)	)	PUNCT
ejpam-7050	251	27	)	)	PUNCT
ejpam-7050	251	28	)	)	PUNCT
ejpam-7050	252	1	⊆	⊆	X
ejpam-7050	252	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	252	3	-	-	PUNCT
ejpam-7050	252	4	cl(v	cl(v	NOUN
ejpam-7050	252	5	)	)	PUNCT
ejpam-7050	252	6	)	)	PUNCT
ejpam-7050	252	7	for	for	ADP
ejpam-7050	252	8	every	every	DET
ejpam-7050	252	9	(	(	PUNCT
ejpam-7050	252	10	σ1	σ1	PROPN
ejpam-7050	252	11	,	,	PUNCT
ejpam-7050	252	12	σ2)β	σ2)β	NOUN
ejpam-7050	252	13	-	-	PUNCT
ejpam-7050	252	14	open	open	NOUN
ejpam-7050	252	15	set	set	NOUN
ejpam-7050	252	16	v	v	NOUN
ejpam-7050	252	17	of	of	ADP
ejpam-7050	252	18	y	y	PROPN
ejpam-7050	252	19	;	;	PUNCT
ejpam-7050	252	20	(	(	PUNCT
ejpam-7050	252	21	8)	8)	NUM
ejpam-7050	252	22	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	252	23	-	-	PUNCT
ejpam-7050	252	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	252	25	-	-	PUNCT
ejpam-7050	252	26	cl(v	cl(v	NOUN
ejpam-7050	252	27	)	)	PUNCT
ejpam-7050	252	28	)	)	PUNCT
ejpam-7050	252	29	)	)	PUNCT
ejpam-7050	252	30	)	)	PUNCT
ejpam-7050	253	1	⊆	⊆	X
ejpam-7050	253	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	253	3	-	-	PUNCT
ejpam-7050	253	4	cl(v	cl(v	NOUN
ejpam-7050	253	5	)	)	PUNCT
ejpam-7050	253	6	)	)	PUNCT
ejpam-7050	253	7	for	for	ADP
ejpam-7050	253	8	every	every	DET
ejpam-7050	253	9	(	(	PUNCT
ejpam-7050	253	10	σ1	σ1	PROPN
ejpam-7050	253	11	,	,	PUNCT
ejpam-7050	253	12	σ2)s	σ2)s	NOUN
ejpam-7050	253	13	-	-	PUNCT
ejpam-7050	253	14	open	open	NOUN
ejpam-7050	253	15	set	set	NOUN
ejpam-7050	253	16	v	v	NOUN
ejpam-7050	253	17	of	of	ADP
ejpam-7050	253	18	y	y	PROPN
ejpam-7050	253	19	.	.	PUNCT
ejpam-7050	254	1	proof	proof	NOUN
ejpam-7050	254	2	.	.	PUNCT
ejpam-7050	255	1	(	(	PUNCT
ejpam-7050	255	2	1	1	X
ejpam-7050	255	3	)	)	PUNCT
ejpam-7050	255	4	⇒	⇒	NOUN
ejpam-7050	255	5	(	(	PUNCT
ejpam-7050	255	6	2	2	NUM
ejpam-7050	255	7	):	):	PUNCT
ejpam-7050	255	8	let	let	VERB
ejpam-7050	255	9	b	b	X
ejpam-7050	255	10	be	be	AUX
ejpam-7050	255	11	any	any	DET
ejpam-7050	255	12	subset	subset	NOUN
ejpam-7050	255	13	of	of	ADP
ejpam-7050	255	14	y	y	PROPN
ejpam-7050	255	15	.	.	PUNCT
ejpam-7050	256	1	thus	thus	ADV
ejpam-7050	256	2	by	by	ADP
ejpam-7050	256	3	lemma	lemma	PROPN
ejpam-7050	256	4	3	3	NUM
ejpam-7050	256	5	,	,	PUNCT
ejpam-7050	256	6	(	(	PUNCT
ejpam-7050	256	7	σ1	σ1	PROPN
ejpam-7050	256	8	,	,	PUNCT
ejpam-7050	256	9	σ2)θ	σ2)θ	NOUN
ejpam-7050	256	10	-	-	PUNCT
ejpam-7050	256	11	cl(b	cl(b	NOUN
ejpam-7050	256	12	)	)	PUNCT
ejpam-7050	256	13	is	be	AUX
ejpam-7050	256	14	σ1σ2	σ1σ2	NOUN
ejpam-7050	256	15	-	-	ADJ
ejpam-7050	256	16	closed	closed	ADJ
ejpam-7050	256	17	in	in	ADP
ejpam-7050	256	18	y	y	PROPN
ejpam-7050	256	19	and	and	CCONJ
ejpam-7050	256	20	by	by	ADP
ejpam-7050	256	21	theorem	theorem	ADJ
ejpam-7050	256	22	3	3	NUM
ejpam-7050	256	23	,	,	PUNCT
ejpam-7050	256	24	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	256	25	-	-	PUNCT
ejpam-7050	256	26	int((σ1	int((σ1	ADJ
ejpam-7050	256	27	,	,	PUNCT
ejpam-7050	256	28	σ2)θ	σ2)θ	ADJ
ejpam-7050	256	29	-	-	PUNCT
ejpam-7050	256	30	cl(b	cl(b	NOUN
ejpam-7050	256	31	)	)	PUNCT
ejpam-7050	256	32	)	)	PUNCT
ejpam-7050	256	33	)	)	PUNCT
ejpam-7050	256	34	)	)	PUNCT
ejpam-7050	257	1	⊆	⊆	NUM
ejpam-7050	257	2	f−((σ1	f−((σ1	NOUN
ejpam-7050	257	3	,	,	PUNCT
ejpam-7050	257	4	σ2)θ	σ2)θ	ADJ
ejpam-7050	257	5	-	-	PUNCT
ejpam-7050	257	6	cl(b	cl(b	NOUN
ejpam-7050	257	7	)	)	PUNCT
ejpam-7050	257	8	)	)	PUNCT
ejpam-7050	257	9	.	.	PUNCT
ejpam-7050	258	1	(	(	PUNCT
ejpam-7050	258	2	2	2	X
ejpam-7050	258	3	)	)	PUNCT
ejpam-7050	258	4	⇒	⇒	NOUN
ejpam-7050	258	5	(	(	PUNCT
ejpam-7050	258	6	3	3	NUM
ejpam-7050	258	7	):	):	PUNCT
ejpam-7050	258	8	this	this	PRON
ejpam-7050	258	9	is	be	AUX
ejpam-7050	258	10	obvious	obvious	ADJ
ejpam-7050	258	11	since	since	SCONJ
ejpam-7050	258	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	258	13	-	-	NOUN
ejpam-7050	258	14	cl(b	cl(b	NOUN
ejpam-7050	258	15	)	)	PUNCT
ejpam-7050	258	16	⊆	⊆	NUM
ejpam-7050	258	17	(	(	PUNCT
ejpam-7050	258	18	σ1	σ1	PROPN
ejpam-7050	258	19	,	,	PUNCT
ejpam-7050	258	20	σ2)θ	σ2)θ	NOUN
ejpam-7050	258	21	-	-	PUNCT
ejpam-7050	258	22	cl(b	cl(b	NOUN
ejpam-7050	258	23	)	)	PUNCT
ejpam-7050	258	24	for	for	ADP
ejpam-7050	258	25	every	every	DET
ejpam-7050	258	26	subset	subset	NOUN
ejpam-7050	258	27	b	b	PROPN
ejpam-7050	258	28	of	of	ADP
ejpam-7050	258	29	y	y	PROPN
ejpam-7050	258	30	.	.	PUNCT
ejpam-7050	259	1	(	(	PUNCT
ejpam-7050	259	2	3	3	X
ejpam-7050	259	3	)	)	PUNCT
ejpam-7050	259	4	⇒	⇒	NOUN
ejpam-7050	259	5	(	(	PUNCT
ejpam-7050	259	6	4	4	NUM
ejpam-7050	259	7	):	):	PUNCT
ejpam-7050	259	8	this	this	PRON
ejpam-7050	259	9	is	be	AUX
ejpam-7050	259	10	obvious	obvious	ADJ
ejpam-7050	259	11	since	since	SCONJ
ejpam-7050	259	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	259	13	-	-	NOUN
ejpam-7050	259	14	cl(v	cl(v	X
ejpam-7050	259	15	)	)	PUNCT
ejpam-7050	260	1	=	=	SYM
ejpam-7050	260	2	(	(	PUNCT
ejpam-7050	260	3	σ1	σ1	PROPN
ejpam-7050	260	4	,	,	PUNCT
ejpam-7050	260	5	σ2)θ	σ2)θ	NOUN
ejpam-7050	260	6	-	-	PUNCT
ejpam-7050	260	7	cl(v	cl(v	NOUN
ejpam-7050	260	8	)	)	PUNCT
ejpam-7050	260	9	for	for	ADP
ejpam-7050	260	10	every	every	DET
ejpam-7050	260	11	σ1σ2	σ1σ2	NOUN
ejpam-7050	260	12	-	-	ADJ
ejpam-7050	260	13	open	open	ADJ
ejpam-7050	260	14	set	set	NOUN
ejpam-7050	260	15	v	v	NOUN
ejpam-7050	260	16	of	of	ADP
ejpam-7050	260	17	y	y	PROPN
ejpam-7050	260	18	.	.	PUNCT
ejpam-7050	261	1	(	(	PUNCT
ejpam-7050	261	2	4	4	X
ejpam-7050	261	3	)	)	PUNCT
ejpam-7050	261	4	⇒	⇒	NOUN
ejpam-7050	261	5	(	(	PUNCT
ejpam-7050	261	6	5	5	NUM
ejpam-7050	261	7	):	):	PUNCT
ejpam-7050	261	8	let	let	VERB
ejpam-7050	261	9	v	v	PART
ejpam-7050	261	10	be	be	AUX
ejpam-7050	261	11	any	any	DET
ejpam-7050	261	12	(	(	PUNCT
ejpam-7050	261	13	σ1	σ1	PROPN
ejpam-7050	261	14	,	,	PUNCT
ejpam-7050	261	15	σ2)p	σ2)p	NOUN
ejpam-7050	261	16	-	-	PUNCT
ejpam-7050	261	17	open	open	ADJ
ejpam-7050	261	18	set	set	NOUN
ejpam-7050	261	19	of	of	ADP
ejpam-7050	261	20	y	y	PROPN
ejpam-7050	261	21	.	.	PUNCT
ejpam-7050	262	1	then	then	ADV
ejpam-7050	262	2	,	,	PUNCT
ejpam-7050	262	3	v	v	ADP
ejpam-7050	262	4	⊆	⊆	NUM
ejpam-7050	262	5	σ1σ2	σ1σ2	NOUN
ejpam-7050	262	6	-	-	PUNCT
ejpam-7050	262	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	262	8	-	-	PUNCT
ejpam-7050	262	9	cl(v	cl(v	NOUN
ejpam-7050	262	10	)	)	PUNCT
ejpam-7050	262	11	)	)	PUNCT
ejpam-7050	262	12	and	and	CCONJ
ejpam-7050	262	13	so	so	ADV
ejpam-7050	262	14	σ1σ2	σ1σ2	NOUN
ejpam-7050	262	15	-	-	NUM
ejpam-7050	262	16	cl(v	cl(v	X
ejpam-7050	262	17	)	)	PUNCT
ejpam-7050	263	1	=	=	SYM
ejpam-7050	263	2	σ1σ2	σ1σ2	X
ejpam-7050	263	3	-	-	PUNCT
ejpam-7050	263	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	263	5	-	-	PUNCT
ejpam-7050	263	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	263	7	-	-	PUNCT
ejpam-7050	263	8	cl(v	cl(v	NOUN
ejpam-7050	263	9	)	)	PUNCT
ejpam-7050	263	10	)	)	PUNCT
ejpam-7050	263	11	)	)	PUNCT
ejpam-7050	263	12	.	.	PUNCT
ejpam-7050	264	1	now	now	ADV
ejpam-7050	264	2	,	,	PUNCT
ejpam-7050	264	3	put	put	VERB
ejpam-7050	264	4	g	g	NOUN
ejpam-7050	264	5	=	=	SYM
ejpam-7050	264	6	σ1σ2	σ1σ2	NOUN
ejpam-7050	264	7	-	-	PUNCT
ejpam-7050	264	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	264	9	-	-	PUNCT
ejpam-7050	264	10	cl(v	cl(v	NOUN
ejpam-7050	264	11	)	)	PUNCT
ejpam-7050	264	12	)	)	PUNCT
ejpam-7050	264	13	,	,	PUNCT
ejpam-7050	264	14	then	then	ADV
ejpam-7050	264	15	g	g	PROPN
ejpam-7050	264	16	is	be	AUX
ejpam-7050	264	17	σ1σ2	σ1σ2	NOUN
ejpam-7050	264	18	-	-	ADJ
ejpam-7050	264	19	open	open	ADJ
ejpam-7050	264	20	in	in	ADP
ejpam-7050	264	21	y	y	PROPN
ejpam-7050	264	22	and	and	CCONJ
ejpam-7050	264	23	σ1σ2	σ1σ2	NOUN
ejpam-7050	264	24	-	-	NUM
ejpam-7050	264	25	cl(g	cl(g	ADJ
ejpam-7050	264	26	)	)	PUNCT
ejpam-7050	264	27	=	=	PUNCT
ejpam-7050	264	28	σ1σ2	σ1σ2	NOUN
ejpam-7050	264	29	-	-	NUM
ejpam-7050	264	30	cl(v	cl(v	NOUN
ejpam-7050	264	31	)	)	PUNCT
ejpam-7050	264	32	.	.	PUNCT
ejpam-7050	265	1	thus	thus	ADV
ejpam-7050	265	2	by	by	ADP
ejpam-7050	265	3	(	(	PUNCT
ejpam-7050	265	4	4	4	NUM
ejpam-7050	265	5	)	)	PUNCT
ejpam-7050	265	6	,	,	PUNCT
ejpam-7050	265	7	we	we	PRON
ejpam-7050	265	8	have	have	VERB
ejpam-7050	265	9	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	VERB
ejpam-7050	265	10	-	-	PUNCT
ejpam-7050	265	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	265	12	-	-	PUNCT
ejpam-7050	265	13	cl(v	cl(v	NOUN
ejpam-7050	265	14	)	)	PUNCT
ejpam-7050	265	15	)	)	PUNCT
ejpam-7050	265	16	)	)	PUNCT
ejpam-7050	265	17	)	)	PUNCT
ejpam-7050	266	1	⊆	⊆	X
ejpam-7050	266	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	266	3	-	-	PUNCT
ejpam-7050	266	4	cl(v	cl(v	NOUN
ejpam-7050	266	5	)	)	PUNCT
ejpam-7050	266	6	)	)	PUNCT
ejpam-7050	266	7	.	.	PUNCT
ejpam-7050	267	1	(	(	PUNCT
ejpam-7050	267	2	5	5	X
ejpam-7050	267	3	)	)	PUNCT
ejpam-7050	267	4	⇒	⇒	NOUN
ejpam-7050	267	5	(	(	PUNCT
ejpam-7050	267	6	6	6	NUM
ejpam-7050	267	7	):	):	PUNCT
ejpam-7050	267	8	let	let	VERB
ejpam-7050	267	9	k	k	PRON
ejpam-7050	267	10	be	be	AUX
ejpam-7050	267	11	any	any	DET
ejpam-7050	267	12	(	(	PUNCT
ejpam-7050	267	13	σ1	σ1	NOUN
ejpam-7050	267	14	,	,	PUNCT
ejpam-7050	267	15	σ2)r	σ2)r	NOUN
ejpam-7050	267	16	-	-	PUNCT
ejpam-7050	267	17	closed	close	VERB
ejpam-7050	267	18	set	set	NOUN
ejpam-7050	267	19	of	of	ADP
ejpam-7050	267	20	y	y	PROPN
ejpam-7050	267	21	.	.	PUNCT
ejpam-7050	268	1	then	then	ADV
ejpam-7050	268	2	,	,	PUNCT
ejpam-7050	268	3	σ1σ2	σ1σ2	NOUN
ejpam-7050	268	4	-	-	PUNCT
ejpam-7050	268	5	int(k	int(k	NOUN
ejpam-7050	268	6	)	)	PUNCT
ejpam-7050	268	7	is	be	AUX
ejpam-7050	268	8	(	(	PUNCT
ejpam-7050	268	9	σ1	σ1	PROPN
ejpam-7050	268	10	,	,	PUNCT
ejpam-7050	268	11	σ2)p	σ2)p	NOUN
ejpam-7050	268	12	-	-	PUNCT
ejpam-7050	268	13	open	open	ADJ
ejpam-7050	268	14	in	in	ADP
ejpam-7050	268	15	y	y	PROPN
ejpam-7050	268	16	,	,	PUNCT
ejpam-7050	268	17	by	by	ADP
ejpam-7050	268	18	(	(	PUNCT
ejpam-7050	268	19	5	5	X
ejpam-7050	268	20	)	)	PUNCT
ejpam-7050	268	21	we	we	PRON
ejpam-7050	268	22	have	have	VERB
ejpam-7050	268	23	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	VERB
ejpam-7050	268	24	-	-	PUNCT
ejpam-7050	268	25	int(k	int(k	NOUN
ejpam-7050	268	26	)	)	PUNCT
ejpam-7050	268	27	)	)	PUNCT
ejpam-7050	268	28	)	)	PUNCT
ejpam-7050	269	1	=	=	SYM
ejpam-7050	269	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	VERB
ejpam-7050	269	3	-	-	PUNCT
ejpam-7050	269	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-7050	269	5	-	-	PUNCT
ejpam-7050	269	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7050	269	7	-	-	PUNCT
ejpam-7050	269	8	int(k	int(k	NOUN
ejpam-7050	269	9	)	)	PUNCT
ejpam-7050	269	10	)	)	PUNCT
ejpam-7050	269	11	)	)	PUNCT
ejpam-7050	269	12	)	)	PUNCT
ejpam-7050	269	13	)	)	PUNCT
ejpam-7050	269	14	m.	m.	NOUN
ejpam-7050	269	15	thongmoon	thongmoon	NOUN
ejpam-7050	269	16	,	,	PUNCT
ejpam-7050	269	17	a.	a.	PROPN
ejpam-7050	269	18	sama	sama	PROPN
ejpam-7050	269	19	-	-	PUNCT
ejpam-7050	269	20	ae	ae	PROPN
ejpam-7050	269	21	,	,	PUNCT
ejpam-7050	269	22	c.	c.	PROPN
ejpam-7050	269	23	boonpok	boonpok	PROPN
ejpam-7050	269	24	/	/	SYM
ejpam-7050	269	25	eur	eur	PROPN
ejpam-7050	269	26	.	.	PUNCT
ejpam-7050	270	1	j.	j.	PROPN
ejpam-7050	270	2	pure	pure	PROPN
ejpam-7050	270	3	appl	appl	PROPN
ejpam-7050	270	4	.	.	PROPN
ejpam-7050	270	5	math	math	PROPN
ejpam-7050	270	6	,	,	PUNCT
ejpam-7050	270	7	18	18	NUM
ejpam-7050	270	8	(	(	PUNCT
ejpam-7050	270	9	4	4	NUM
ejpam-7050	270	10	)	)	PUNCT
ejpam-7050	270	11	(	(	PUNCT
ejpam-7050	270	12	2025	2025	NUM
ejpam-7050	270	13	)	)	PUNCT
ejpam-7050	270	14	,	,	PUNCT
ejpam-7050	270	15	7050	7050	NUM
ejpam-7050	270	16	10	10	NUM
ejpam-7050	270	17	of	of	ADP
ejpam-7050	270	18	14	14	NUM
ejpam-7050	270	19	⊆	⊆	NUM
ejpam-7050	270	20	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	270	21	-	-	PUNCT
ejpam-7050	270	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	270	23	-	-	PUNCT
ejpam-7050	270	24	int(k	int(k	NOUN
ejpam-7050	270	25	)	)	PUNCT
ejpam-7050	270	26	)	)	PUNCT
ejpam-7050	270	27	)	)	PUNCT
ejpam-7050	271	1	=	=	SYM
ejpam-7050	271	2	f−(k	f−(k	PROPN
ejpam-7050	271	3	)	)	PUNCT
ejpam-7050	271	4	.	.	PUNCT
ejpam-7050	272	1	(	(	PUNCT
ejpam-7050	272	2	6	6	X
ejpam-7050	272	3	)	)	PUNCT
ejpam-7050	272	4	⇒	⇒	NOUN
ejpam-7050	272	5	(	(	PUNCT
ejpam-7050	272	6	7	7	NUM
ejpam-7050	272	7	):	):	PUNCT
ejpam-7050	272	8	let	let	VERB
ejpam-7050	272	9	v	v	PART
ejpam-7050	272	10	be	be	AUX
ejpam-7050	272	11	any	any	DET
ejpam-7050	272	12	(	(	PUNCT
ejpam-7050	272	13	σ1	σ1	PROPN
ejpam-7050	272	14	,	,	PUNCT
ejpam-7050	272	15	σ2)β	σ2)β	NOUN
ejpam-7050	272	16	-	-	PUNCT
ejpam-7050	272	17	open	open	ADJ
ejpam-7050	272	18	set	set	NOUN
ejpam-7050	272	19	of	of	ADP
ejpam-7050	272	20	y	y	PROPN
ejpam-7050	272	21	.	.	PUNCT
ejpam-7050	273	1	then	then	ADV
ejpam-7050	273	2	,	,	PUNCT
ejpam-7050	273	3	we	we	PRON
ejpam-7050	273	4	have	have	VERB
ejpam-7050	273	5	v	v	ADP
ejpam-7050	273	6	⊆	⊆	NUM
ejpam-7050	273	7	σ1σ2	σ1σ2	NOUN
ejpam-7050	273	8	-	-	PUNCT
ejpam-7050	273	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	273	10	-	-	PUNCT
ejpam-7050	273	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	273	12	-	-	PUNCT
ejpam-7050	273	13	cl(v	cl(v	NOUN
ejpam-7050	273	14	)	)	PUNCT
ejpam-7050	273	15	)	)	PUNCT
ejpam-7050	273	16	)	)	PUNCT
ejpam-7050	273	17	.	.	PUNCT
ejpam-7050	274	1	since	since	SCONJ
ejpam-7050	274	2	σ1σ2	σ1σ2	NOUN
ejpam-7050	274	3	-	-	NOUN
ejpam-7050	274	4	cl(v	cl(v	NOUN
ejpam-7050	274	5	)	)	PUNCT
ejpam-7050	274	6	is	be	AUX
ejpam-7050	274	7	(	(	PUNCT
ejpam-7050	274	8	σ1	σ1	NOUN
ejpam-7050	274	9	,	,	PUNCT
ejpam-7050	274	10	σ2)r	σ2)r	NOUN
ejpam-7050	274	11	-	-	PUNCT
ejpam-7050	274	12	closed	closed	ADJ
ejpam-7050	274	13	in	in	ADP
ejpam-7050	274	14	y	y	PROPN
ejpam-7050	274	15	.	.	PUNCT
ejpam-7050	275	1	thus	thus	ADV
ejpam-7050	275	2	by	by	ADP
ejpam-7050	275	3	(	(	PUNCT
ejpam-7050	275	4	6	6	NUM
ejpam-7050	275	5	)	)	PUNCT
ejpam-7050	275	6	,	,	PUNCT
ejpam-7050	275	7	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	275	8	-	-	PUNCT
ejpam-7050	275	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	275	10	-	-	PUNCT
ejpam-7050	275	11	cl(v	cl(v	NOUN
ejpam-7050	275	12	)	)	PUNCT
ejpam-7050	275	13	)	)	PUNCT
ejpam-7050	275	14	)	)	PUNCT
ejpam-7050	275	15	)	)	PUNCT
ejpam-7050	276	1	⊆	⊆	X
ejpam-7050	276	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	276	3	-	-	PUNCT
ejpam-7050	276	4	cl(v	cl(v	NOUN
ejpam-7050	276	5	)	)	PUNCT
ejpam-7050	276	6	)	)	PUNCT
ejpam-7050	276	7	.	.	PUNCT
ejpam-7050	277	1	(	(	PUNCT
ejpam-7050	277	2	7	7	X
ejpam-7050	277	3	)	)	PUNCT
ejpam-7050	277	4	⇒	⇒	NOUN
ejpam-7050	277	5	(	(	PUNCT
ejpam-7050	277	6	8)	8)	NUM
ejpam-7050	277	7	:	:	PUNCT
ejpam-7050	277	8	this	this	PRON
ejpam-7050	277	9	is	be	AUX
ejpam-7050	277	10	obvious	obvious	ADJ
ejpam-7050	277	11	since	since	SCONJ
ejpam-7050	277	12	every	every	DET
ejpam-7050	277	13	(	(	PUNCT
ejpam-7050	277	14	σ1	σ1	PROPN
ejpam-7050	277	15	,	,	PUNCT
ejpam-7050	277	16	σ2)s	σ2)s	NOUN
ejpam-7050	277	17	-	-	PUNCT
ejpam-7050	277	18	open	open	ADJ
ejpam-7050	277	19	set	set	NOUN
ejpam-7050	277	20	is	be	AUX
ejpam-7050	277	21	(	(	PUNCT
ejpam-7050	277	22	σ1	σ1	PROPN
ejpam-7050	277	23	,	,	PUNCT
ejpam-7050	277	24	σ2)β	σ2)β	NOUN
ejpam-7050	277	25	-	-	PUNCT
ejpam-7050	277	26	open	open	ADJ
ejpam-7050	277	27	.	.	PUNCT
ejpam-7050	278	1	(	(	PUNCT
ejpam-7050	278	2	8)	8)	NUM
ejpam-7050	278	3	⇒	⇒	NOUN
ejpam-7050	278	4	(	(	PUNCT
ejpam-7050	278	5	1	1	NUM
ejpam-7050	278	6	):	):	PUNCT
ejpam-7050	278	7	let	let	VERB
ejpam-7050	278	8	v	v	PART
ejpam-7050	278	9	be	be	AUX
ejpam-7050	278	10	any	any	DET
ejpam-7050	278	11	σ1σ2	σ1σ2	NOUN
ejpam-7050	278	12	-	-	ADJ
ejpam-7050	278	13	open	open	ADJ
ejpam-7050	278	14	set	set	NOUN
ejpam-7050	278	15	of	of	ADP
ejpam-7050	278	16	y	y	PROPN
ejpam-7050	278	17	.	.	PUNCT
ejpam-7050	279	1	then	then	ADV
ejpam-7050	279	2	,	,	PUNCT
ejpam-7050	279	3	since	since	SCONJ
ejpam-7050	279	4	v	v	NOUN
ejpam-7050	279	5	is	be	AUX
ejpam-7050	279	6	(	(	PUNCT
ejpam-7050	279	7	σ1	σ1	PROPN
ejpam-7050	279	8	,	,	PUNCT
ejpam-7050	279	9	σ2)s	σ2)s	NOUN
ejpam-7050	279	10	-	-	PUNCT
ejpam-7050	279	11	open	open	ADJ
ejpam-7050	279	12	set	set	NOUN
ejpam-7050	279	13	in	in	ADP
ejpam-7050	279	14	y	y	PROPN
ejpam-7050	279	15	,	,	PUNCT
ejpam-7050	279	16	by	by	ADP
ejpam-7050	279	17	(	(	PUNCT
ejpam-7050	279	18	8)	8)	NUM
ejpam-7050	279	19	we	we	PRON
ejpam-7050	279	20	have	have	AUX
ejpam-7050	279	21	βcl⋆(f−(v	βcl⋆(f−(v	NOUN
ejpam-7050	279	22	)	)	PUNCT
ejpam-7050	279	23	)	)	PUNCT
ejpam-7050	280	1	⊆	⊆	NUM
ejpam-7050	280	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	280	3	-	-	PUNCT
ejpam-7050	280	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	280	5	-	-	PUNCT
ejpam-7050	280	6	cl(v	cl(v	NOUN
ejpam-7050	280	7	)	)	PUNCT
ejpam-7050	280	8	)	)	PUNCT
ejpam-7050	280	9	)	)	PUNCT
ejpam-7050	280	10	)	)	PUNCT
ejpam-7050	280	11	⊆	⊆	X
ejpam-7050	280	12	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	280	13	-	-	PUNCT
ejpam-7050	280	14	cl(v	cl(v	NOUN
ejpam-7050	280	15	)	)	PUNCT
ejpam-7050	280	16	)	)	PUNCT
ejpam-7050	280	17	.	.	PUNCT
ejpam-7050	281	1	by	by	ADP
ejpam-7050	281	2	theorem	theorem	NOUN
ejpam-7050	281	3	3	3	NUM
ejpam-7050	281	4	,	,	PUNCT
ejpam-7050	281	5	f	f	PROPN
ejpam-7050	281	6	is	be	AUX
ejpam-7050	281	7	upper	upper	ADJ
ejpam-7050	281	8	weakly	weakly	ADJ
ejpam-7050	281	9	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	281	10	,	,	PUNCT
ejpam-7050	281	11	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7050	281	12	.	.	X
ejpam-7050	281	13	theorem	theorem	VERB
ejpam-7050	281	14	6	6	NUM
ejpam-7050	281	15	.	.	PUNCT
ejpam-7050	281	16	for	for	ADP
ejpam-7050	281	17	a	a	DET
ejpam-7050	281	18	multifunction	multifunction	NOUN
ejpam-7050	281	19	f	f	NOUN
ejpam-7050	281	20	:	:	PUNCT
ejpam-7050	281	21	(	(	PUNCT
ejpam-7050	281	22	x	x	X
ejpam-7050	281	23	,	,	PUNCT
ejpam-7050	281	24	τ	τ	PROPN
ejpam-7050	281	25	,	,	PUNCT
ejpam-7050	281	26	i	i	NOUN
ejpam-7050	281	27	)	)	PUNCT
ejpam-7050	281	28	→	→	PUNCT
ejpam-7050	281	29	(	(	PUNCT
ejpam-7050	281	30	y	y	PROPN
ejpam-7050	281	31	,	,	PUNCT
ejpam-7050	281	32	σ1	σ1	PROPN
ejpam-7050	281	33	,	,	PUNCT
ejpam-7050	281	34	σ2	σ2	NOUN
ejpam-7050	281	35	)	)	PUNCT
ejpam-7050	281	36	,	,	PUNCT
ejpam-7050	281	37	the	the	DET
ejpam-7050	281	38	following	follow	VERB
ejpam-7050	281	39	properties	property	NOUN
ejpam-7050	281	40	are	be	AUX
ejpam-7050	281	41	equivalent	equivalent	ADJ
ejpam-7050	281	42	:	:	PUNCT
ejpam-7050	281	43	(	(	PUNCT
ejpam-7050	281	44	1	1	X
ejpam-7050	281	45	)	)	PUNCT
ejpam-7050	281	46	f	f	PROPN
ejpam-7050	281	47	is	be	AUX
ejpam-7050	281	48	lower	low	ADJ
ejpam-7050	281	49	weakly	weakly	ADJ
ejpam-7050	281	50	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	281	51	,	,	PUNCT
ejpam-7050	281	52	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	281	53	;	;	PUNCT
ejpam-7050	281	54	(	(	PUNCT
ejpam-7050	281	55	2	2	X
ejpam-7050	281	56	)	)	PUNCT
ejpam-7050	281	57	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	281	58	-	-	PUNCT
ejpam-7050	281	59	int((σ1	int((σ1	ADJ
ejpam-7050	281	60	,	,	PUNCT
ejpam-7050	281	61	σ2)θ	σ2)θ	ADJ
ejpam-7050	281	62	-	-	PUNCT
ejpam-7050	281	63	cl(b	cl(b	NOUN
ejpam-7050	281	64	)	)	PUNCT
ejpam-7050	281	65	)	)	PUNCT
ejpam-7050	281	66	)	)	PUNCT
ejpam-7050	281	67	)	)	PUNCT
ejpam-7050	282	1	⊆	⊆	NUM
ejpam-7050	282	2	f+((σ1	f+((σ1	NOUN
ejpam-7050	282	3	,	,	PUNCT
ejpam-7050	282	4	σ2)θ	σ2)θ	ADJ
ejpam-7050	282	5	-	-	PUNCT
ejpam-7050	282	6	cl(b	cl(b	NOUN
ejpam-7050	282	7	)	)	PUNCT
ejpam-7050	282	8	)	)	PUNCT
ejpam-7050	282	9	for	for	ADP
ejpam-7050	282	10	every	every	DET
ejpam-7050	282	11	subset	subset	NOUN
ejpam-7050	282	12	b	b	PROPN
ejpam-7050	282	13	of	of	ADP
ejpam-7050	282	14	y	y	PROPN
ejpam-7050	282	15	;	;	PUNCT
ejpam-7050	282	16	(	(	PUNCT
ejpam-7050	282	17	3	3	X
ejpam-7050	282	18	)	)	PUNCT
ejpam-7050	282	19	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	282	20	-	-	PUNCT
ejpam-7050	282	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	282	22	-	-	PUNCT
ejpam-7050	282	23	cl(b	cl(b	NOUN
ejpam-7050	282	24	)	)	PUNCT
ejpam-7050	282	25	)	)	PUNCT
ejpam-7050	282	26	)	)	PUNCT
ejpam-7050	282	27	)	)	PUNCT
ejpam-7050	283	1	⊆	⊆	NUM
ejpam-7050	283	2	f+((σ1	f+((σ1	NOUN
ejpam-7050	283	3	,	,	PUNCT
ejpam-7050	283	4	σ2)θ	σ2)θ	ADJ
ejpam-7050	283	5	-	-	PUNCT
ejpam-7050	283	6	cl(b	cl(b	NOUN
ejpam-7050	283	7	)	)	PUNCT
ejpam-7050	283	8	)	)	PUNCT
ejpam-7050	283	9	for	for	ADP
ejpam-7050	283	10	every	every	DET
ejpam-7050	283	11	subset	subset	NOUN
ejpam-7050	283	12	b	b	PROPN
ejpam-7050	283	13	of	of	ADP
ejpam-7050	283	14	y	y	PROPN
ejpam-7050	283	15	;	;	PUNCT
ejpam-7050	283	16	(	(	PUNCT
ejpam-7050	283	17	4	4	X
ejpam-7050	283	18	)	)	PUNCT
ejpam-7050	283	19	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	283	20	-	-	PUNCT
ejpam-7050	283	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	283	22	-	-	PUNCT
ejpam-7050	283	23	cl(v	cl(v	NOUN
ejpam-7050	283	24	)	)	PUNCT
ejpam-7050	283	25	)	)	PUNCT
ejpam-7050	283	26	)	)	PUNCT
ejpam-7050	283	27	)	)	PUNCT
ejpam-7050	284	1	⊆	⊆	X
ejpam-7050	284	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	284	3	-	-	PUNCT
ejpam-7050	284	4	cl(v	cl(v	NOUN
ejpam-7050	284	5	)	)	PUNCT
ejpam-7050	284	6	)	)	PUNCT
ejpam-7050	284	7	for	for	ADP
ejpam-7050	284	8	every	every	DET
ejpam-7050	284	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	284	10	-	-	ADJ
ejpam-7050	284	11	open	open	ADJ
ejpam-7050	284	12	set	set	NOUN
ejpam-7050	284	13	v	v	NOUN
ejpam-7050	284	14	of	of	ADP
ejpam-7050	284	15	y	y	PROPN
ejpam-7050	284	16	;	;	PUNCT
ejpam-7050	284	17	(	(	PUNCT
ejpam-7050	284	18	5	5	X
ejpam-7050	284	19	)	)	PUNCT
ejpam-7050	284	20	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	284	21	-	-	PUNCT
ejpam-7050	284	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	284	23	-	-	PUNCT
ejpam-7050	284	24	cl(v	cl(v	NOUN
ejpam-7050	284	25	)	)	PUNCT
ejpam-7050	284	26	)	)	PUNCT
ejpam-7050	284	27	)	)	PUNCT
ejpam-7050	284	28	)	)	PUNCT
ejpam-7050	284	29	⊆	⊆	X
ejpam-7050	284	30	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	284	31	-	-	PUNCT
ejpam-7050	284	32	cl(v	cl(v	NOUN
ejpam-7050	284	33	)	)	PUNCT
ejpam-7050	284	34	)	)	PUNCT
ejpam-7050	284	35	for	for	ADP
ejpam-7050	284	36	every	every	DET
ejpam-7050	284	37	(	(	PUNCT
ejpam-7050	284	38	σ1	σ1	PROPN
ejpam-7050	284	39	,	,	PUNCT
ejpam-7050	284	40	σ2)p	σ2)p	NOUN
ejpam-7050	284	41	-	-	PUNCT
ejpam-7050	284	42	open	open	NOUN
ejpam-7050	284	43	set	set	NOUN
ejpam-7050	284	44	v	v	NOUN
ejpam-7050	284	45	of	of	ADP
ejpam-7050	284	46	y	y	PROPN
ejpam-7050	284	47	;	;	PUNCT
ejpam-7050	284	48	(	(	PUNCT
ejpam-7050	284	49	6	6	X
ejpam-7050	284	50	)	)	PUNCT
ejpam-7050	284	51	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	284	52	-	-	PUNCT
ejpam-7050	284	53	int(k	int(k	NOUN
ejpam-7050	284	54	)	)	PUNCT
ejpam-7050	284	55	)	)	PUNCT
ejpam-7050	284	56	)	)	PUNCT
ejpam-7050	284	57	⊆	⊆	NUM
ejpam-7050	284	58	f+(k	f+(k	NOUN
ejpam-7050	284	59	)	)	PUNCT
ejpam-7050	284	60	for	for	ADP
ejpam-7050	284	61	every	every	DET
ejpam-7050	284	62	(	(	PUNCT
ejpam-7050	284	63	σ1	σ1	PROPN
ejpam-7050	284	64	,	,	PUNCT
ejpam-7050	284	65	σ2)r	σ2)r	NOUN
ejpam-7050	284	66	-	-	PUNCT
ejpam-7050	284	67	closed	close	VERB
ejpam-7050	284	68	set	set	ADJ
ejpam-7050	284	69	k	k	PROPN
ejpam-7050	284	70	of	of	ADP
ejpam-7050	284	71	y	y	PROPN
ejpam-7050	284	72	;	;	PUNCT
ejpam-7050	284	73	(	(	PUNCT
ejpam-7050	284	74	7	7	X
ejpam-7050	284	75	)	)	PUNCT
ejpam-7050	284	76	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	284	77	-	-	PUNCT
ejpam-7050	284	78	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	284	79	-	-	PUNCT
ejpam-7050	284	80	cl(v	cl(v	NOUN
ejpam-7050	284	81	)	)	PUNCT
ejpam-7050	284	82	)	)	PUNCT
ejpam-7050	284	83	)	)	PUNCT
ejpam-7050	284	84	)	)	PUNCT
ejpam-7050	284	85	⊆	⊆	X
ejpam-7050	284	86	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	284	87	-	-	PUNCT
ejpam-7050	284	88	cl(v	cl(v	NOUN
ejpam-7050	284	89	)	)	PUNCT
ejpam-7050	284	90	)	)	PUNCT
ejpam-7050	284	91	for	for	ADP
ejpam-7050	284	92	every	every	DET
ejpam-7050	284	93	(	(	PUNCT
ejpam-7050	284	94	σ1	σ1	PROPN
ejpam-7050	284	95	,	,	PUNCT
ejpam-7050	284	96	σ2)β	σ2)β	NOUN
ejpam-7050	284	97	-	-	PUNCT
ejpam-7050	284	98	open	open	NOUN
ejpam-7050	284	99	set	set	NOUN
ejpam-7050	284	100	v	v	NOUN
ejpam-7050	284	101	of	of	ADP
ejpam-7050	284	102	y	y	PROPN
ejpam-7050	284	103	;	;	PUNCT
ejpam-7050	284	104	(	(	PUNCT
ejpam-7050	284	105	8)	8)	NUM
ejpam-7050	284	106	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7050	284	107	-	-	PUNCT
ejpam-7050	284	108	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	284	109	-	-	PUNCT
ejpam-7050	284	110	cl(v	cl(v	NOUN
ejpam-7050	284	111	)	)	PUNCT
ejpam-7050	284	112	)	)	PUNCT
ejpam-7050	284	113	)	)	PUNCT
ejpam-7050	284	114	)	)	PUNCT
ejpam-7050	284	115	⊆	⊆	X
ejpam-7050	284	116	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	284	117	-	-	PUNCT
ejpam-7050	284	118	cl(v	cl(v	NOUN
ejpam-7050	284	119	)	)	PUNCT
ejpam-7050	284	120	)	)	PUNCT
ejpam-7050	284	121	for	for	SCONJ
ejpam-7050	284	122	every	every	DET
ejpam-7050	284	123	(	(	PUNCT
ejpam-7050	284	124	σ1	σ1	PROPN
ejpam-7050	284	125	,	,	PUNCT
ejpam-7050	284	126	σ2)s	σ2)s	NOUN
ejpam-7050	284	127	-	-	PUNCT
ejpam-7050	284	128	open	open	NOUN
ejpam-7050	284	129	set	set	NOUN
ejpam-7050	284	130	v	v	NOUN
ejpam-7050	284	131	of	of	ADP
ejpam-7050	284	132	y	y	PROPN
ejpam-7050	284	133	.	.	PUNCT
ejpam-7050	285	1	proof	proof	NOUN
ejpam-7050	285	2	.	.	PUNCT
ejpam-7050	286	1	the	the	DET
ejpam-7050	286	2	proof	proof	NOUN
ejpam-7050	286	3	is	be	AUX
ejpam-7050	286	4	similar	similar	ADJ
ejpam-7050	286	5	to	to	ADP
ejpam-7050	286	6	that	that	PRON
ejpam-7050	286	7	of	of	ADP
ejpam-7050	286	8	theorem	theorem	ADJ
ejpam-7050	286	9	5	5	NUM
ejpam-7050	286	10	.	.	PUNCT
ejpam-7050	286	11	corollary	corollary	ADJ
ejpam-7050	286	12	3	3	NUM
ejpam-7050	286	13	.	.	PUNCT
ejpam-7050	287	1	for	for	ADP
ejpam-7050	287	2	a	a	DET
ejpam-7050	287	3	function	function	NOUN
ejpam-7050	287	4	f	f	NOUN
ejpam-7050	287	5	:	:	PUNCT
ejpam-7050	287	6	(	(	PUNCT
ejpam-7050	287	7	x	x	X
ejpam-7050	287	8	,	,	PUNCT
ejpam-7050	287	9	τ	τ	PROPN
ejpam-7050	287	10	,	,	PUNCT
ejpam-7050	287	11	i	i	NOUN
ejpam-7050	287	12	)	)	PUNCT
ejpam-7050	287	13	→	→	PUNCT
ejpam-7050	287	14	(	(	PUNCT
ejpam-7050	287	15	y	y	PROPN
ejpam-7050	287	16	,	,	PUNCT
ejpam-7050	287	17	σ1	σ1	PROPN
ejpam-7050	287	18	,	,	PUNCT
ejpam-7050	287	19	σ2	σ2	NOUN
ejpam-7050	287	20	)	)	PUNCT
ejpam-7050	287	21	,	,	PUNCT
ejpam-7050	287	22	the	the	DET
ejpam-7050	287	23	following	follow	VERB
ejpam-7050	287	24	properties	property	NOUN
ejpam-7050	287	25	are	be	AUX
ejpam-7050	287	26	equivalent	equivalent	ADJ
ejpam-7050	287	27	:	:	PUNCT
ejpam-7050	287	28	(	(	PUNCT
ejpam-7050	287	29	1	1	X
ejpam-7050	287	30	)	)	PUNCT
ejpam-7050	287	31	f	f	PROPN
ejpam-7050	287	32	is	be	AUX
ejpam-7050	287	33	weakly	weakly	ADJ
ejpam-7050	287	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	287	35	,	,	PUNCT
ejpam-7050	287	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	287	37	;	;	PUNCT
ejpam-7050	287	38	(	(	PUNCT
ejpam-7050	287	39	2	2	X
ejpam-7050	287	40	)	)	PUNCT
ejpam-7050	287	41	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	287	42	-	-	PUNCT
ejpam-7050	287	43	int((σ1	int((σ1	PROPN
ejpam-7050	287	44	,	,	PUNCT
ejpam-7050	287	45	σ2)θ	σ2)θ	ADJ
ejpam-7050	287	46	-	-	PUNCT
ejpam-7050	287	47	cl(b	cl(b	NOUN
ejpam-7050	287	48	)	)	PUNCT
ejpam-7050	287	49	)	)	PUNCT
ejpam-7050	287	50	)	)	PUNCT
ejpam-7050	287	51	)	)	PUNCT
ejpam-7050	288	1	⊆	⊆	NUM
ejpam-7050	288	2	f−1((σ1	f−1((σ1	NOUN
ejpam-7050	288	3	,	,	PUNCT
ejpam-7050	288	4	σ2)θ	σ2)θ	NOUN
ejpam-7050	288	5	-	-	PUNCT
ejpam-7050	288	6	cl(b	cl(b	NOUN
ejpam-7050	288	7	)	)	PUNCT
ejpam-7050	288	8	)	)	PUNCT
ejpam-7050	288	9	for	for	ADP
ejpam-7050	288	10	every	every	DET
ejpam-7050	288	11	subset	subset	NOUN
ejpam-7050	288	12	b	b	PROPN
ejpam-7050	288	13	of	of	ADP
ejpam-7050	288	14	y	y	PROPN
ejpam-7050	288	15	;	;	PUNCT
ejpam-7050	288	16	m.	m.	NOUN
ejpam-7050	288	17	thongmoon	thongmoon	NOUN
ejpam-7050	288	18	,	,	PUNCT
ejpam-7050	288	19	a.	a.	PROPN
ejpam-7050	288	20	sama	sama	PROPN
ejpam-7050	288	21	-	-	PUNCT
ejpam-7050	288	22	ae	ae	PROPN
ejpam-7050	288	23	,	,	PUNCT
ejpam-7050	288	24	c.	c.	PROPN
ejpam-7050	288	25	boonpok	boonpok	PROPN
ejpam-7050	288	26	/	/	SYM
ejpam-7050	288	27	eur	eur	PROPN
ejpam-7050	288	28	.	.	PUNCT
ejpam-7050	289	1	j.	j.	PROPN
ejpam-7050	289	2	pure	pure	PROPN
ejpam-7050	289	3	appl	appl	PROPN
ejpam-7050	289	4	.	.	PROPN
ejpam-7050	289	5	math	math	PROPN
ejpam-7050	289	6	,	,	PUNCT
ejpam-7050	289	7	18	18	NUM
ejpam-7050	289	8	(	(	PUNCT
ejpam-7050	289	9	4	4	NUM
ejpam-7050	289	10	)	)	PUNCT
ejpam-7050	289	11	(	(	PUNCT
ejpam-7050	289	12	2025	2025	NUM
ejpam-7050	289	13	)	)	PUNCT
ejpam-7050	289	14	,	,	PUNCT
ejpam-7050	289	15	7050	7050	NUM
ejpam-7050	289	16	11	11	NUM
ejpam-7050	289	17	of	of	ADP
ejpam-7050	289	18	14	14	NUM
ejpam-7050	289	19	(	(	PUNCT
ejpam-7050	289	20	3	3	NUM
ejpam-7050	289	21	)	)	PUNCT
ejpam-7050	289	22	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	289	23	-	-	PUNCT
ejpam-7050	289	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	289	25	-	-	PUNCT
ejpam-7050	289	26	cl(b	cl(b	NOUN
ejpam-7050	289	27	)	)	PUNCT
ejpam-7050	289	28	)	)	PUNCT
ejpam-7050	289	29	)	)	PUNCT
ejpam-7050	289	30	)	)	PUNCT
ejpam-7050	290	1	⊆	⊆	NUM
ejpam-7050	290	2	f−1((σ1	f−1((σ1	NOUN
ejpam-7050	290	3	,	,	PUNCT
ejpam-7050	290	4	σ2)θ	σ2)θ	NOUN
ejpam-7050	290	5	-	-	PUNCT
ejpam-7050	290	6	cl(b	cl(b	NOUN
ejpam-7050	290	7	)	)	PUNCT
ejpam-7050	290	8	)	)	PUNCT
ejpam-7050	290	9	for	for	ADP
ejpam-7050	290	10	every	every	DET
ejpam-7050	290	11	subset	subset	NOUN
ejpam-7050	290	12	b	b	PROPN
ejpam-7050	290	13	of	of	ADP
ejpam-7050	290	14	y	y	PROPN
ejpam-7050	290	15	;	;	PUNCT
ejpam-7050	290	16	(	(	PUNCT
ejpam-7050	290	17	4	4	X
ejpam-7050	290	18	)	)	PUNCT
ejpam-7050	290	19	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	290	20	-	-	PUNCT
ejpam-7050	290	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	290	22	-	-	PUNCT
ejpam-7050	290	23	cl(v	cl(v	NOUN
ejpam-7050	290	24	)	)	PUNCT
ejpam-7050	290	25	)	)	PUNCT
ejpam-7050	290	26	)	)	PUNCT
ejpam-7050	290	27	)	)	PUNCT
ejpam-7050	291	1	⊆	⊆	NUM
ejpam-7050	291	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	291	3	-	-	PUNCT
ejpam-7050	291	4	cl(v	cl(v	NOUN
ejpam-7050	291	5	)	)	PUNCT
ejpam-7050	291	6	)	)	PUNCT
ejpam-7050	291	7	for	for	ADP
ejpam-7050	291	8	every	every	DET
ejpam-7050	291	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	291	10	-	-	ADJ
ejpam-7050	291	11	open	open	ADJ
ejpam-7050	291	12	set	set	NOUN
ejpam-7050	291	13	v	v	NOUN
ejpam-7050	291	14	of	of	ADP
ejpam-7050	291	15	y	y	PROPN
ejpam-7050	291	16	;	;	PUNCT
ejpam-7050	291	17	(	(	PUNCT
ejpam-7050	291	18	5	5	X
ejpam-7050	291	19	)	)	PUNCT
ejpam-7050	291	20	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	291	21	-	-	PUNCT
ejpam-7050	291	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	291	23	-	-	PUNCT
ejpam-7050	291	24	cl(v	cl(v	NOUN
ejpam-7050	291	25	)	)	PUNCT
ejpam-7050	291	26	)	)	PUNCT
ejpam-7050	291	27	)	)	PUNCT
ejpam-7050	291	28	)	)	PUNCT
ejpam-7050	292	1	⊆	⊆	NUM
ejpam-7050	292	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	292	3	-	-	PUNCT
ejpam-7050	292	4	cl(v	cl(v	NOUN
ejpam-7050	292	5	)	)	PUNCT
ejpam-7050	292	6	)	)	PUNCT
ejpam-7050	292	7	for	for	ADP
ejpam-7050	292	8	every	every	DET
ejpam-7050	292	9	(	(	PUNCT
ejpam-7050	292	10	σ1	σ1	PROPN
ejpam-7050	292	11	,	,	PUNCT
ejpam-7050	292	12	σ2)p	σ2)p	NOUN
ejpam-7050	292	13	-	-	PUNCT
ejpam-7050	292	14	open	open	NOUN
ejpam-7050	292	15	set	set	NOUN
ejpam-7050	292	16	v	v	NOUN
ejpam-7050	292	17	of	of	ADP
ejpam-7050	292	18	y	y	PROPN
ejpam-7050	292	19	;	;	PUNCT
ejpam-7050	292	20	(	(	PUNCT
ejpam-7050	292	21	6	6	X
ejpam-7050	292	22	)	)	PUNCT
ejpam-7050	292	23	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	292	24	-	-	SYM
ejpam-7050	292	25	int(k	int(k	NOUN
ejpam-7050	292	26	)	)	PUNCT
ejpam-7050	292	27	)	)	PUNCT
ejpam-7050	292	28	)	)	PUNCT
ejpam-7050	293	1	⊆	⊆	NUM
ejpam-7050	293	2	f−1(k	f−1(k	PROPN
ejpam-7050	293	3	)	)	PUNCT
ejpam-7050	293	4	for	for	ADP
ejpam-7050	293	5	every	every	DET
ejpam-7050	293	6	(	(	PUNCT
ejpam-7050	293	7	σ1	σ1	PROPN
ejpam-7050	293	8	,	,	PUNCT
ejpam-7050	293	9	σ2)r	σ2)r	NOUN
ejpam-7050	293	10	-	-	PUNCT
ejpam-7050	293	11	closed	close	VERB
ejpam-7050	293	12	set	set	ADJ
ejpam-7050	293	13	k	k	PROPN
ejpam-7050	293	14	of	of	ADP
ejpam-7050	293	15	y	y	PROPN
ejpam-7050	293	16	;	;	PUNCT
ejpam-7050	293	17	(	(	PUNCT
ejpam-7050	293	18	7	7	X
ejpam-7050	293	19	)	)	PUNCT
ejpam-7050	293	20	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7050	293	21	-	-	PUNCT
ejpam-7050	293	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	293	23	-	-	PUNCT
ejpam-7050	293	24	cl(v	cl(v	NOUN
ejpam-7050	293	25	)	)	PUNCT
ejpam-7050	293	26	)	)	PUNCT
ejpam-7050	293	27	)	)	PUNCT
ejpam-7050	293	28	)	)	PUNCT
ejpam-7050	294	1	⊆	⊆	NUM
ejpam-7050	294	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	294	3	-	-	PUNCT
ejpam-7050	294	4	cl(v	cl(v	NOUN
ejpam-7050	294	5	)	)	PUNCT
ejpam-7050	294	6	)	)	PUNCT
ejpam-7050	294	7	for	for	ADP
ejpam-7050	294	8	every	every	DET
ejpam-7050	294	9	(	(	PUNCT
ejpam-7050	294	10	σ1	σ1	PROPN
ejpam-7050	294	11	,	,	PUNCT
ejpam-7050	294	12	σ2)β	σ2)β	NOUN
ejpam-7050	294	13	-	-	PUNCT
ejpam-7050	294	14	open	open	NOUN
ejpam-7050	294	15	set	set	NOUN
ejpam-7050	294	16	v	v	NOUN
ejpam-7050	294	17	of	of	ADP
ejpam-7050	294	18	y	y	PROPN
ejpam-7050	294	19	;	;	PUNCT
ejpam-7050	294	20	(	(	PUNCT
ejpam-7050	294	21	8)	8)	NUM
ejpam-7050	294	22	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	X
ejpam-7050	294	23	-	-	PUNCT
ejpam-7050	294	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	294	25	-	-	PUNCT
ejpam-7050	294	26	cl(v	cl(v	NOUN
ejpam-7050	294	27	)	)	PUNCT
ejpam-7050	294	28	)	)	PUNCT
ejpam-7050	294	29	)	)	PUNCT
ejpam-7050	294	30	)	)	PUNCT
ejpam-7050	295	1	⊆	⊆	NUM
ejpam-7050	295	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	295	3	-	-	PUNCT
ejpam-7050	295	4	cl(v	cl(v	NOUN
ejpam-7050	295	5	)	)	PUNCT
ejpam-7050	295	6	)	)	PUNCT
ejpam-7050	295	7	for	for	ADP
ejpam-7050	295	8	every	every	DET
ejpam-7050	295	9	(	(	PUNCT
ejpam-7050	295	10	σ1	σ1	PROPN
ejpam-7050	295	11	,	,	PUNCT
ejpam-7050	295	12	σ2)s	σ2)s	NOUN
ejpam-7050	295	13	-	-	PUNCT
ejpam-7050	295	14	open	open	NOUN
ejpam-7050	295	15	set	set	NOUN
ejpam-7050	295	16	v	v	NOUN
ejpam-7050	295	17	of	of	ADP
ejpam-7050	295	18	y	y	PROPN
ejpam-7050	295	19	.	.	PUNCT
ejpam-7050	296	1	theorem	theorem	VERB
ejpam-7050	296	2	7	7	NUM
ejpam-7050	296	3	.	.	X
ejpam-7050	296	4	for	for	ADP
ejpam-7050	296	5	a	a	DET
ejpam-7050	296	6	multifunction	multifunction	NOUN
ejpam-7050	297	1	f	f	NOUN
ejpam-7050	297	2	:	:	PUNCT
ejpam-7050	297	3	(	(	PUNCT
ejpam-7050	297	4	x	x	X
ejpam-7050	297	5	,	,	PUNCT
ejpam-7050	297	6	τ	τ	PROPN
ejpam-7050	297	7	,	,	PUNCT
ejpam-7050	297	8	i	i	NOUN
ejpam-7050	297	9	)	)	PUNCT
ejpam-7050	297	10	→	→	PUNCT
ejpam-7050	297	11	(	(	PUNCT
ejpam-7050	297	12	y	y	PROPN
ejpam-7050	297	13	,	,	PUNCT
ejpam-7050	297	14	σ1	σ1	PROPN
ejpam-7050	297	15	,	,	PUNCT
ejpam-7050	297	16	σ2	σ2	NOUN
ejpam-7050	297	17	)	)	PUNCT
ejpam-7050	297	18	,	,	PUNCT
ejpam-7050	297	19	the	the	DET
ejpam-7050	297	20	following	follow	VERB
ejpam-7050	297	21	properties	property	NOUN
ejpam-7050	297	22	are	be	AUX
ejpam-7050	297	23	equivalent	equivalent	ADJ
ejpam-7050	297	24	:	:	PUNCT
ejpam-7050	297	25	(	(	PUNCT
ejpam-7050	297	26	1	1	X
ejpam-7050	297	27	)	)	PUNCT
ejpam-7050	297	28	f	f	PROPN
ejpam-7050	297	29	is	be	AUX
ejpam-7050	297	30	upper	upper	ADJ
ejpam-7050	297	31	weakly	weakly	ADJ
ejpam-7050	297	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	297	33	,	,	PUNCT
ejpam-7050	297	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	297	35	;	;	PUNCT
ejpam-7050	297	36	(	(	PUNCT
ejpam-7050	297	37	2	2	NUM
ejpam-7050	297	38	)	)	PUNCT
ejpam-7050	297	39	βcl⋆(f−(v	βcl⋆(f−(v	NUM
ejpam-7050	297	40	)	)	PUNCT
ejpam-7050	297	41	)	)	PUNCT
ejpam-7050	298	1	⊆	⊆	X
ejpam-7050	298	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	298	3	-	-	PUNCT
ejpam-7050	298	4	cl(v	cl(v	NOUN
ejpam-7050	298	5	)	)	PUNCT
ejpam-7050	298	6	)	)	PUNCT
ejpam-7050	298	7	for	for	ADP
ejpam-7050	298	8	every	every	DET
ejpam-7050	298	9	(	(	PUNCT
ejpam-7050	298	10	σ1	σ1	PROPN
ejpam-7050	298	11	,	,	PUNCT
ejpam-7050	298	12	σ2)p	σ2)p	NOUN
ejpam-7050	298	13	-	-	PUNCT
ejpam-7050	298	14	open	open	NOUN
ejpam-7050	298	15	set	set	NOUN
ejpam-7050	298	16	v	v	NOUN
ejpam-7050	298	17	of	of	ADP
ejpam-7050	298	18	y	y	PROPN
ejpam-7050	298	19	;	;	PUNCT
ejpam-7050	298	20	(	(	PUNCT
ejpam-7050	298	21	3	3	X
ejpam-7050	298	22	)	)	PUNCT
ejpam-7050	298	23	f+(v	f+(v	NOUN
ejpam-7050	298	24	)	)	PUNCT
ejpam-7050	299	1	⊆	⊆	NUM
ejpam-7050	299	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	299	3	-	-	X
ejpam-7050	299	4	cl(v	cl(v	NOUN
ejpam-7050	299	5	)	)	PUNCT
ejpam-7050	299	6	)	)	PUNCT
ejpam-7050	299	7	)	)	PUNCT
ejpam-7050	299	8	for	for	ADP
ejpam-7050	299	9	every	every	DET
ejpam-7050	299	10	(	(	PUNCT
ejpam-7050	299	11	σ1	σ1	PROPN
ejpam-7050	299	12	,	,	PUNCT
ejpam-7050	299	13	σ2)p	σ2)p	NOUN
ejpam-7050	299	14	-	-	PUNCT
ejpam-7050	299	15	open	open	NOUN
ejpam-7050	299	16	set	set	NOUN
ejpam-7050	299	17	v	v	NOUN
ejpam-7050	299	18	of	of	ADP
ejpam-7050	299	19	y	y	PROPN
ejpam-7050	299	20	.	.	PUNCT
ejpam-7050	300	1	proof	proof	NOUN
ejpam-7050	300	2	.	.	PUNCT
ejpam-7050	301	1	(	(	PUNCT
ejpam-7050	301	2	1	1	X
ejpam-7050	301	3	)	)	PUNCT
ejpam-7050	301	4	⇒	⇒	NOUN
ejpam-7050	301	5	(	(	PUNCT
ejpam-7050	301	6	2	2	NUM
ejpam-7050	301	7	):	):	PUNCT
ejpam-7050	301	8	let	let	VERB
ejpam-7050	301	9	v	v	PART
ejpam-7050	301	10	be	be	AUX
ejpam-7050	301	11	any	any	DET
ejpam-7050	301	12	(	(	PUNCT
ejpam-7050	301	13	σ1	σ1	PROPN
ejpam-7050	301	14	,	,	PUNCT
ejpam-7050	301	15	σ2)p	σ2)p	NOUN
ejpam-7050	301	16	-	-	PUNCT
ejpam-7050	301	17	open	open	ADJ
ejpam-7050	301	18	set	set	NOUN
ejpam-7050	301	19	of	of	ADP
ejpam-7050	301	20	y	y	PROPN
ejpam-7050	301	21	.	.	PUNCT
ejpam-7050	302	1	since	since	SCONJ
ejpam-7050	302	2	f	f	PROPN
ejpam-7050	302	3	is	be	AUX
ejpam-7050	302	4	upper	upper	ADJ
ejpam-7050	302	5	weakly	weakly	ADJ
ejpam-7050	302	6	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	302	7	,	,	PUNCT
ejpam-7050	302	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	302	9	,	,	PUNCT
ejpam-7050	302	10	by	by	ADP
ejpam-7050	302	11	theorem	theorem	NOUN
ejpam-7050	302	12	3	3	NUM
ejpam-7050	302	13	we	we	PRON
ejpam-7050	302	14	have	have	VERB
ejpam-7050	302	15	βcl⋆(f−(v	βcl⋆(f−(v	NOUN
ejpam-7050	302	16	)	)	PUNCT
ejpam-7050	302	17	)	)	PUNCT
ejpam-7050	303	1	⊆	⊆	NUM
ejpam-7050	303	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	303	3	-	-	PUNCT
ejpam-7050	303	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	303	5	-	-	PUNCT
ejpam-7050	303	6	cl(v	cl(v	NOUN
ejpam-7050	303	7	)	)	PUNCT
ejpam-7050	303	8	)	)	PUNCT
ejpam-7050	303	9	)	)	PUNCT
ejpam-7050	303	10	)	)	PUNCT
ejpam-7050	303	11	⊆	⊆	X
ejpam-7050	303	12	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7050	303	13	-	-	PUNCT
ejpam-7050	303	14	cl(v	cl(v	NOUN
ejpam-7050	303	15	)	)	PUNCT
ejpam-7050	303	16	)	)	PUNCT
ejpam-7050	303	17	.	.	PUNCT
ejpam-7050	304	1	(	(	PUNCT
ejpam-7050	304	2	2	2	X
ejpam-7050	304	3	)	)	PUNCT
ejpam-7050	304	4	⇒	⇒	NOUN
ejpam-7050	304	5	(	(	PUNCT
ejpam-7050	304	6	3	3	NUM
ejpam-7050	304	7	):	):	PUNCT
ejpam-7050	304	8	let	let	VERB
ejpam-7050	304	9	v	v	PART
ejpam-7050	304	10	be	be	AUX
ejpam-7050	304	11	any	any	DET
ejpam-7050	304	12	(	(	PUNCT
ejpam-7050	304	13	σ1	σ1	PROPN
ejpam-7050	304	14	,	,	PUNCT
ejpam-7050	304	15	σ2)p	σ2)p	NOUN
ejpam-7050	304	16	-	-	PUNCT
ejpam-7050	304	17	open	open	ADJ
ejpam-7050	304	18	set	set	NOUN
ejpam-7050	304	19	of	of	ADP
ejpam-7050	304	20	y	y	PROPN
ejpam-7050	304	21	.	.	PUNCT
ejpam-7050	305	1	then	then	ADV
ejpam-7050	305	2	,	,	PUNCT
ejpam-7050	305	3	v	v	ADP
ejpam-7050	305	4	⊆	⊆	NUM
ejpam-7050	305	5	σ1σ2	σ1σ2	NOUN
ejpam-7050	305	6	-	-	PUNCT
ejpam-7050	305	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7050	305	8	-	-	PUNCT
ejpam-7050	305	9	cl(v	cl(v	NOUN
ejpam-7050	305	10	)	)	PUNCT
ejpam-7050	305	11	)	)	PUNCT
ejpam-7050	306	1	and	and	CCONJ
ejpam-7050	306	2	y	y	PROPN
ejpam-7050	306	3	−	−	PROPN
ejpam-7050	306	4	v	v	X
ejpam-7050	306	5	⊇	⊇	ADJ
ejpam-7050	306	6	σ1σ2	σ1σ2	ADV
ejpam-7050	306	7	-	-	PUNCT
ejpam-7050	306	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	306	9	-	-	PUNCT
ejpam-7050	306	10	int(y	int(y	PROPN
ejpam-7050	306	11	−	−	PROPN
ejpam-7050	306	12	v	v	NOUN
ejpam-7050	306	13	)	)	PUNCT
ejpam-7050	306	14	)	)	PUNCT
ejpam-7050	306	15	.	.	PUNCT
ejpam-7050	307	1	thus	thus	ADV
ejpam-7050	307	2	by	by	ADP
ejpam-7050	307	3	(	(	PUNCT
ejpam-7050	307	4	3	3	NUM
ejpam-7050	307	5	)	)	PUNCT
ejpam-7050	307	6	,	,	PUNCT
ejpam-7050	307	7	x	x	PUNCT
ejpam-7050	307	8	−	−	NOUN
ejpam-7050	307	9	f+(v	f+(v	NOUN
ejpam-7050	307	10	)	)	PUNCT
ejpam-7050	308	1	=	=	PUNCT
ejpam-7050	308	2	f−(y	f−(y	NOUN
ejpam-7050	308	3	−	−	ADP
ejpam-7050	308	4	v	v	NOUN
ejpam-7050	308	5	)	)	PUNCT
ejpam-7050	308	6	⊇	⊇	ADJ
ejpam-7050	308	7	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7050	308	8	-	-	PUNCT
ejpam-7050	308	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7050	308	10	-	-	PUNCT
ejpam-7050	308	11	int(y	int(y	PROPN
ejpam-7050	308	12	−	−	PROPN
ejpam-7050	308	13	v	v	NOUN
ejpam-7050	308	14	)	)	PUNCT
ejpam-7050	308	15	)	)	PUNCT
ejpam-7050	308	16	)	)	PUNCT
ejpam-7050	309	1	⊇	⊇	PROPN
ejpam-7050	309	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7050	309	3	-	-	PUNCT
ejpam-7050	309	4	int(y	int(y	PROPN
ejpam-7050	309	5	−	−	PROPN
ejpam-7050	309	6	v	v	NOUN
ejpam-7050	309	7	)	)	PUNCT
ejpam-7050	309	8	)	)	PUNCT
ejpam-7050	309	9	)	)	PUNCT
ejpam-7050	310	1	=	=	NOUN
ejpam-7050	310	2	βcl⋆(f−(y	βcl⋆(f−(y	PUNCT
ejpam-7050	310	3	−	−	ADP
ejpam-7050	310	4	σ1σ2	σ1σ2	NOUN
ejpam-7050	310	5	-	-	NUM
ejpam-7050	310	6	cl(v	cl(v	NOUN
ejpam-7050	310	7	)	)	PUNCT
ejpam-7050	310	8	)	)	PUNCT
ejpam-7050	310	9	)	)	PUNCT
ejpam-7050	311	1	=	=	PUNCT
ejpam-7050	311	2	βcl⋆(x	βcl⋆(x	NOUN
ejpam-7050	311	3	−	−	NOUN
ejpam-7050	311	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	311	5	-	-	PUNCT
ejpam-7050	311	6	cl(v	cl(v	NOUN
ejpam-7050	311	7	)	)	PUNCT
ejpam-7050	311	8	)	)	PUNCT
ejpam-7050	311	9	)	)	PUNCT
ejpam-7050	312	1	=	=	PUNCT
ejpam-7050	312	2	x	x	PUNCT
ejpam-7050	313	1	−	−	ADP
ejpam-7050	313	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	313	3	-	-	PUNCT
ejpam-7050	313	4	cl(v	cl(v	NOUN
ejpam-7050	313	5	)	)	PUNCT
ejpam-7050	313	6	)	)	PUNCT
ejpam-7050	313	7	)	)	PUNCT
ejpam-7050	313	8	and	and	CCONJ
ejpam-7050	313	9	hence	hence	ADV
ejpam-7050	313	10	f+(v	f+(v	NOUN
ejpam-7050	313	11	)	)	PUNCT
ejpam-7050	314	1	⊆	⊆	NUM
ejpam-7050	314	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	314	3	-	-	X
ejpam-7050	314	4	cl(v	cl(v	NOUN
ejpam-7050	314	5	)	)	PUNCT
ejpam-7050	314	6	)	)	PUNCT
ejpam-7050	314	7	)	)	PUNCT
ejpam-7050	314	8	.	.	PUNCT
ejpam-7050	315	1	(	(	PUNCT
ejpam-7050	315	2	3	3	X
ejpam-7050	315	3	)	)	PUNCT
ejpam-7050	315	4	⇒	⇒	NOUN
ejpam-7050	315	5	(	(	PUNCT
ejpam-7050	315	6	1	1	NUM
ejpam-7050	315	7	):	):	PUNCT
ejpam-7050	315	8	let	let	VERB
ejpam-7050	315	9	v	v	PART
ejpam-7050	315	10	be	be	AUX
ejpam-7050	315	11	any	any	DET
ejpam-7050	315	12	σ1σ2	σ1σ2	NOUN
ejpam-7050	315	13	-	-	ADJ
ejpam-7050	315	14	open	open	ADJ
ejpam-7050	315	15	set	set	NOUN
ejpam-7050	315	16	of	of	ADP
ejpam-7050	315	17	y	y	PROPN
ejpam-7050	315	18	.	.	PUNCT
ejpam-7050	316	1	then	then	ADV
ejpam-7050	316	2	,	,	PUNCT
ejpam-7050	316	3	v	v	NOUN
ejpam-7050	316	4	is	be	AUX
ejpam-7050	316	5	(	(	PUNCT
ejpam-7050	316	6	σ1	σ1	PROPN
ejpam-7050	316	7	,	,	PUNCT
ejpam-7050	316	8	σ2)p	σ2)p	NOUN
ejpam-7050	316	9	-	-	PUNCT
ejpam-7050	316	10	open	open	ADJ
ejpam-7050	316	11	in	in	ADP
ejpam-7050	316	12	y	y	PROPN
ejpam-7050	316	13	,	,	PUNCT
ejpam-7050	316	14	by	by	ADP
ejpam-7050	316	15	(	(	PUNCT
ejpam-7050	316	16	4	4	X
ejpam-7050	316	17	)	)	PUNCT
ejpam-7050	316	18	we	we	PRON
ejpam-7050	316	19	have	have	VERB
ejpam-7050	316	20	f+(v	f+(v	NOUN
ejpam-7050	316	21	)	)	PUNCT
ejpam-7050	317	1	⊆	⊆	NUM
ejpam-7050	317	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7050	317	3	-	-	X
ejpam-7050	317	4	cl(v	cl(v	NOUN
ejpam-7050	317	5	)	)	PUNCT
ejpam-7050	317	6	)	)	PUNCT
ejpam-7050	317	7	)	)	PUNCT
ejpam-7050	317	8	.	.	PUNCT
ejpam-7050	318	1	thus	thus	ADV
ejpam-7050	318	2	,	,	PUNCT
ejpam-7050	318	3	f	f	PROPN
ejpam-7050	318	4	is	be	AUX
ejpam-7050	318	5	upper	upper	ADJ
ejpam-7050	318	6	weakly	weakly	ADJ
ejpam-7050	318	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	318	8	,	,	PUNCT
ejpam-7050	318	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	318	10	by	by	ADP
ejpam-7050	318	11	theorem	theorem	NOUN
ejpam-7050	318	12	3	3	NUM
ejpam-7050	318	13	.	.	PUNCT
ejpam-7050	318	14	theorem	theorem	NOUN
ejpam-7050	318	15	8	8	NUM
ejpam-7050	318	16	.	.	PUNCT
ejpam-7050	319	1	for	for	ADP
ejpam-7050	319	2	a	a	DET
ejpam-7050	319	3	multifunction	multifunction	NOUN
ejpam-7050	319	4	f	f	NOUN
ejpam-7050	319	5	:	:	PUNCT
ejpam-7050	319	6	(	(	PUNCT
ejpam-7050	319	7	x	x	X
ejpam-7050	319	8	,	,	PUNCT
ejpam-7050	319	9	τ	τ	PROPN
ejpam-7050	319	10	,	,	PUNCT
ejpam-7050	319	11	i	i	NOUN
ejpam-7050	319	12	)	)	PUNCT
ejpam-7050	319	13	→	→	PUNCT
ejpam-7050	319	14	(	(	PUNCT
ejpam-7050	319	15	y	y	PROPN
ejpam-7050	319	16	,	,	PUNCT
ejpam-7050	319	17	σ1	σ1	PROPN
ejpam-7050	319	18	,	,	PUNCT
ejpam-7050	319	19	σ2	σ2	NOUN
ejpam-7050	319	20	)	)	PUNCT
ejpam-7050	319	21	,	,	PUNCT
ejpam-7050	319	22	the	the	DET
ejpam-7050	319	23	following	follow	VERB
ejpam-7050	319	24	properties	property	NOUN
ejpam-7050	319	25	are	be	AUX
ejpam-7050	319	26	equivalent	equivalent	ADJ
ejpam-7050	319	27	:	:	PUNCT
ejpam-7050	319	28	m.	m.	NOUN
ejpam-7050	319	29	thongmoon	thongmoon	NOUN
ejpam-7050	319	30	,	,	PUNCT
ejpam-7050	319	31	a.	a.	PROPN
ejpam-7050	319	32	sama	sama	PROPN
ejpam-7050	319	33	-	-	PUNCT
ejpam-7050	319	34	ae	ae	PROPN
ejpam-7050	319	35	,	,	PUNCT
ejpam-7050	319	36	c.	c.	PROPN
ejpam-7050	319	37	boonpok	boonpok	PROPN
ejpam-7050	319	38	/	/	SYM
ejpam-7050	319	39	eur	eur	PROPN
ejpam-7050	319	40	.	.	PUNCT
ejpam-7050	320	1	j.	j.	PROPN
ejpam-7050	320	2	pure	pure	PROPN
ejpam-7050	320	3	appl	appl	PROPN
ejpam-7050	320	4	.	.	PROPN
ejpam-7050	320	5	math	math	PROPN
ejpam-7050	320	6	,	,	PUNCT
ejpam-7050	320	7	18	18	NUM
ejpam-7050	320	8	(	(	PUNCT
ejpam-7050	320	9	4	4	NUM
ejpam-7050	320	10	)	)	PUNCT
ejpam-7050	320	11	(	(	PUNCT
ejpam-7050	320	12	2025	2025	NUM
ejpam-7050	320	13	)	)	PUNCT
ejpam-7050	320	14	,	,	PUNCT
ejpam-7050	320	15	7050	7050	NUM
ejpam-7050	320	16	12	12	NUM
ejpam-7050	320	17	of	of	ADP
ejpam-7050	320	18	14	14	NUM
ejpam-7050	320	19	(	(	PUNCT
ejpam-7050	320	20	1	1	NUM
ejpam-7050	320	21	)	)	PUNCT
ejpam-7050	320	22	f	f	PROPN
ejpam-7050	320	23	is	be	AUX
ejpam-7050	320	24	lower	low	ADJ
ejpam-7050	320	25	weakly	weakly	ADJ
ejpam-7050	320	26	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	320	27	,	,	PUNCT
ejpam-7050	320	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	320	29	;	;	PUNCT
ejpam-7050	320	30	(	(	PUNCT
ejpam-7050	320	31	2	2	X
ejpam-7050	320	32	)	)	PUNCT
ejpam-7050	320	33	βcl⋆(f+(v	βcl⋆(f+(v	NUM
ejpam-7050	320	34	)	)	PUNCT
ejpam-7050	320	35	)	)	PUNCT
ejpam-7050	321	1	⊆	⊆	X
ejpam-7050	321	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7050	321	3	-	-	PUNCT
ejpam-7050	321	4	cl(v	cl(v	NOUN
ejpam-7050	321	5	)	)	PUNCT
ejpam-7050	321	6	)	)	PUNCT
ejpam-7050	321	7	for	for	ADP
ejpam-7050	321	8	every	every	DET
ejpam-7050	321	9	(	(	PUNCT
ejpam-7050	321	10	σ1	σ1	PROPN
ejpam-7050	321	11	,	,	PUNCT
ejpam-7050	321	12	σ2)p	σ2)p	NOUN
ejpam-7050	321	13	-	-	PUNCT
ejpam-7050	321	14	open	open	NOUN
ejpam-7050	321	15	set	set	NOUN
ejpam-7050	321	16	v	v	NOUN
ejpam-7050	321	17	of	of	ADP
ejpam-7050	321	18	y	y	PROPN
ejpam-7050	321	19	;	;	PUNCT
ejpam-7050	321	20	(	(	PUNCT
ejpam-7050	321	21	3	3	X
ejpam-7050	321	22	)	)	PUNCT
ejpam-7050	321	23	f−(v	f−(v	NOUN
ejpam-7050	321	24	)	)	PUNCT
ejpam-7050	321	25	⊆	⊆	NUM
ejpam-7050	321	26	βint⋆(f−(σ1σ2	βint⋆(f−(σ1σ2	NOUN
ejpam-7050	321	27	-	-	PUNCT
ejpam-7050	321	28	cl(v	cl(v	NOUN
ejpam-7050	321	29	)	)	PUNCT
ejpam-7050	321	30	)	)	PUNCT
ejpam-7050	321	31	)	)	PUNCT
ejpam-7050	322	1	for	for	ADP
ejpam-7050	322	2	every	every	DET
ejpam-7050	322	3	(	(	PUNCT
ejpam-7050	322	4	σ1	σ1	PROPN
ejpam-7050	322	5	,	,	PUNCT
ejpam-7050	322	6	σ2)p	σ2)p	NOUN
ejpam-7050	322	7	-	-	PUNCT
ejpam-7050	322	8	open	open	NOUN
ejpam-7050	322	9	set	set	NOUN
ejpam-7050	322	10	v	v	NOUN
ejpam-7050	322	11	of	of	ADP
ejpam-7050	322	12	y	y	PROPN
ejpam-7050	322	13	.	.	PUNCT
ejpam-7050	323	1	proof	proof	NOUN
ejpam-7050	323	2	.	.	PUNCT
ejpam-7050	324	1	the	the	DET
ejpam-7050	324	2	proof	proof	NOUN
ejpam-7050	324	3	is	be	AUX
ejpam-7050	324	4	similar	similar	ADJ
ejpam-7050	324	5	to	to	ADP
ejpam-7050	324	6	that	that	PRON
ejpam-7050	324	7	of	of	ADP
ejpam-7050	324	8	theorem	theorem	ADJ
ejpam-7050	324	9	7	7	NUM
ejpam-7050	324	10	.	.	PUNCT
ejpam-7050	324	11	corollary	corollary	ADJ
ejpam-7050	324	12	4	4	NUM
ejpam-7050	324	13	.	.	PUNCT
ejpam-7050	324	14	for	for	ADP
ejpam-7050	324	15	a	a	DET
ejpam-7050	324	16	function	function	NOUN
ejpam-7050	324	17	f	f	NOUN
ejpam-7050	324	18	:	:	PUNCT
ejpam-7050	324	19	(	(	PUNCT
ejpam-7050	324	20	x	x	X
ejpam-7050	324	21	,	,	PUNCT
ejpam-7050	324	22	τ	τ	PROPN
ejpam-7050	324	23	,	,	PUNCT
ejpam-7050	324	24	i	i	NOUN
ejpam-7050	324	25	)	)	PUNCT
ejpam-7050	324	26	→	→	PUNCT
ejpam-7050	324	27	(	(	PUNCT
ejpam-7050	324	28	y	y	PROPN
ejpam-7050	324	29	,	,	PUNCT
ejpam-7050	324	30	σ1	σ1	PROPN
ejpam-7050	324	31	,	,	PUNCT
ejpam-7050	324	32	σ2	σ2	NOUN
ejpam-7050	324	33	)	)	PUNCT
ejpam-7050	324	34	,	,	PUNCT
ejpam-7050	324	35	the	the	DET
ejpam-7050	324	36	following	follow	VERB
ejpam-7050	324	37	properties	property	NOUN
ejpam-7050	324	38	are	be	AUX
ejpam-7050	324	39	equivalent	equivalent	ADJ
ejpam-7050	324	40	:	:	PUNCT
ejpam-7050	324	41	(	(	PUNCT
ejpam-7050	324	42	1	1	X
ejpam-7050	324	43	)	)	PUNCT
ejpam-7050	324	44	f	f	PROPN
ejpam-7050	324	45	is	be	AUX
ejpam-7050	324	46	weakly	weakly	ADJ
ejpam-7050	324	47	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	324	48	,	,	PUNCT
ejpam-7050	324	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	324	50	;	;	PUNCT
ejpam-7050	324	51	(	(	PUNCT
ejpam-7050	324	52	2	2	X
ejpam-7050	324	53	)	)	PUNCT
ejpam-7050	324	54	βcl⋆(f−1(v	βcl⋆(f−1(v	NUM
ejpam-7050	324	55	)	)	PUNCT
ejpam-7050	324	56	)	)	PUNCT
ejpam-7050	325	1	⊆	⊆	NUM
ejpam-7050	325	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7050	325	3	-	-	PUNCT
ejpam-7050	325	4	cl(v	cl(v	NOUN
ejpam-7050	325	5	)	)	PUNCT
ejpam-7050	325	6	)	)	PUNCT
ejpam-7050	325	7	for	for	ADP
ejpam-7050	325	8	every	every	DET
ejpam-7050	325	9	(	(	PUNCT
ejpam-7050	325	10	σ1	σ1	PROPN
ejpam-7050	325	11	,	,	PUNCT
ejpam-7050	325	12	σ2)p	σ2)p	NOUN
ejpam-7050	325	13	-	-	PUNCT
ejpam-7050	325	14	open	open	NOUN
ejpam-7050	325	15	set	set	NOUN
ejpam-7050	325	16	v	v	NOUN
ejpam-7050	325	17	of	of	ADP
ejpam-7050	325	18	y	y	PROPN
ejpam-7050	325	19	;	;	PUNCT
ejpam-7050	325	20	(	(	PUNCT
ejpam-7050	325	21	3	3	X
ejpam-7050	325	22	)	)	PUNCT
ejpam-7050	325	23	f−1(v	f−1(v	NOUN
ejpam-7050	325	24	)	)	PUNCT
ejpam-7050	326	1	⊆	⊆	NUM
ejpam-7050	326	2	βint⋆(f−1(σ1σ2	βint⋆(f−1(σ1σ2	NOUN
ejpam-7050	326	3	-	-	NOUN
ejpam-7050	326	4	cl(v	cl(v	NOUN
ejpam-7050	326	5	)	)	PUNCT
ejpam-7050	326	6	)	)	PUNCT
ejpam-7050	326	7	)	)	PUNCT
ejpam-7050	326	8	for	for	ADP
ejpam-7050	326	9	every	every	DET
ejpam-7050	326	10	(	(	PUNCT
ejpam-7050	326	11	σ1	σ1	PROPN
ejpam-7050	326	12	,	,	PUNCT
ejpam-7050	326	13	σ2)p	σ2)p	NOUN
ejpam-7050	326	14	-	-	PUNCT
ejpam-7050	326	15	open	open	NOUN
ejpam-7050	326	16	set	set	NOUN
ejpam-7050	326	17	v	v	NOUN
ejpam-7050	326	18	of	of	ADP
ejpam-7050	326	19	y	y	PROPN
ejpam-7050	326	20	.	.	PUNCT
ejpam-7050	327	1	definition	definition	NOUN
ejpam-7050	327	2	6	6	NUM
ejpam-7050	327	3	.	.	PUNCT
ejpam-7050	328	1	[	[	X
ejpam-7050	328	2	22	22	NUM
ejpam-7050	328	3	]	]	PUNCT
ejpam-7050	328	4	a	a	DET
ejpam-7050	328	5	multifunction	multifunction	NOUN
ejpam-7050	328	6	f	f	NOUN
ejpam-7050	328	7	:	:	PUNCT
ejpam-7050	328	8	(	(	PUNCT
ejpam-7050	328	9	x	x	X
ejpam-7050	328	10	,	,	PUNCT
ejpam-7050	328	11	τ	τ	PROPN
ejpam-7050	328	12	,	,	PUNCT
ejpam-7050	328	13	i	i	NOUN
ejpam-7050	328	14	)	)	PUNCT
ejpam-7050	328	15	→	→	PUNCT
ejpam-7050	328	16	(	(	PUNCT
ejpam-7050	328	17	y	y	PROPN
ejpam-7050	328	18	,	,	PUNCT
ejpam-7050	328	19	σ1	σ1	PROPN
ejpam-7050	328	20	,	,	PUNCT
ejpam-7050	328	21	σ2	σ2	PROPN
ejpam-7050	328	22	)	)	PUNCT
ejpam-7050	328	23	is	be	AUX
ejpam-7050	328	24	said	say	VERB
ejpam-7050	328	25	to	to	PART
ejpam-7050	328	26	be	be	AUX
ejpam-7050	328	27	upper	upper	ADJ
ejpam-7050	328	28	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	328	29	,	,	PUNCT
ejpam-7050	328	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	328	31	at	at	ADP
ejpam-7050	328	32	a	a	DET
ejpam-7050	328	33	point	point	NOUN
ejpam-7050	328	34	x	x	PUNCT
ejpam-7050	328	35	of	of	ADP
ejpam-7050	328	36	x	x	PRON
ejpam-7050	328	37	if	if	SCONJ
ejpam-7050	328	38	for	for	ADP
ejpam-7050	328	39	each	each	DET
ejpam-7050	328	40	σ1σ2	σ1σ2	VERB
ejpam-7050	328	41	-	-	ADJ
ejpam-7050	328	42	open	open	ADJ
ejpam-7050	328	43	set	set	NOUN
ejpam-7050	328	44	v	v	NOUN
ejpam-7050	328	45	of	of	ADP
ejpam-7050	328	46	y	y	PRON
ejpam-7050	328	47	such	such	ADJ
ejpam-7050	328	48	that	that	SCONJ
ejpam-7050	328	49	f	f	PROPN
ejpam-7050	328	50	(	(	PUNCT
ejpam-7050	328	51	x	x	X
ejpam-7050	328	52	)	)	PUNCT
ejpam-7050	328	53	⊆	⊆	NUM
ejpam-7050	328	54	v	v	NOUN
ejpam-7050	328	55	,	,	PUNCT
ejpam-7050	328	56	there	there	PRON
ejpam-7050	328	57	exists	exist	VERB
ejpam-7050	328	58	a	a	DET
ejpam-7050	328	59	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	328	60	-	-	ADJ
ejpam-7050	328	61	open	open	ADJ
ejpam-7050	328	62	set	set	NOUN
ejpam-7050	328	63	u	u	NOUN
ejpam-7050	328	64	of	of	ADP
ejpam-7050	328	65	x	x	PUNCT
ejpam-7050	328	66	containing	contain	VERB
ejpam-7050	328	67	x	x	PUNCT
ejpam-7050	328	68	such	such	ADJ
ejpam-7050	328	69	that	that	SCONJ
ejpam-7050	328	70	f	f	PROPN
ejpam-7050	328	71	(	(	PUNCT
ejpam-7050	328	72	u	u	NOUN
ejpam-7050	328	73	)	)	PUNCT
ejpam-7050	328	74	⊆	⊆	NUM
ejpam-7050	328	75	v	v	NOUN
ejpam-7050	328	76	.	.	PUNCT
ejpam-7050	329	1	a	a	DET
ejpam-7050	329	2	multifunction	multifunction	NOUN
ejpam-7050	329	3	f	f	NOUN
ejpam-7050	329	4	:	:	PUNCT
ejpam-7050	329	5	(	(	PUNCT
ejpam-7050	329	6	x	x	X
ejpam-7050	329	7	,	,	PUNCT
ejpam-7050	329	8	τ	τ	PROPN
ejpam-7050	329	9	,	,	PUNCT
ejpam-7050	329	10	i	i	NOUN
ejpam-7050	329	11	)	)	PUNCT
ejpam-7050	329	12	→	→	PUNCT
ejpam-7050	329	13	(	(	PUNCT
ejpam-7050	329	14	y	y	PROPN
ejpam-7050	329	15	,	,	PUNCT
ejpam-7050	329	16	σ1	σ1	PROPN
ejpam-7050	329	17	,	,	PUNCT
ejpam-7050	329	18	σ2	σ2	PROPN
ejpam-7050	329	19	)	)	PUNCT
ejpam-7050	329	20	is	be	AUX
ejpam-7050	329	21	said	say	VERB
ejpam-7050	329	22	to	to	PART
ejpam-7050	329	23	be	be	AUX
ejpam-7050	329	24	upper	upper	ADJ
ejpam-7050	329	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	329	26	,	,	PUNCT
ejpam-7050	329	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	329	28	if	if	SCONJ
ejpam-7050	329	29	f	f	PROPN
ejpam-7050	329	30	is	be	AUX
ejpam-7050	329	31	upper	upper	ADJ
ejpam-7050	329	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	329	33	,	,	PUNCT
ejpam-7050	329	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	329	35	at	at	ADP
ejpam-7050	329	36	each	each	DET
ejpam-7050	329	37	point	point	NOUN
ejpam-7050	329	38	of	of	ADP
ejpam-7050	329	39	x.	x.	NOUN
ejpam-7050	329	40	definition	definition	NOUN
ejpam-7050	329	41	7	7	NUM
ejpam-7050	329	42	.	.	PUNCT
ejpam-7050	330	1	[	[	X
ejpam-7050	330	2	22	22	NUM
ejpam-7050	330	3	]	]	PUNCT
ejpam-7050	330	4	a	a	DET
ejpam-7050	330	5	multifunction	multifunction	NOUN
ejpam-7050	330	6	f	f	NOUN
ejpam-7050	330	7	:	:	PUNCT
ejpam-7050	330	8	(	(	PUNCT
ejpam-7050	330	9	x	x	X
ejpam-7050	330	10	,	,	PUNCT
ejpam-7050	330	11	τ	τ	PROPN
ejpam-7050	330	12	,	,	PUNCT
ejpam-7050	330	13	i	i	NOUN
ejpam-7050	330	14	)	)	PUNCT
ejpam-7050	330	15	→	→	PUNCT
ejpam-7050	330	16	(	(	PUNCT
ejpam-7050	330	17	y	y	PROPN
ejpam-7050	330	18	,	,	PUNCT
ejpam-7050	330	19	σ1	σ1	PROPN
ejpam-7050	330	20	,	,	PUNCT
ejpam-7050	330	21	σ2	σ2	PROPN
ejpam-7050	330	22	)	)	PUNCT
ejpam-7050	330	23	is	be	AUX
ejpam-7050	330	24	said	say	VERB
ejpam-7050	330	25	to	to	PART
ejpam-7050	330	26	be	be	AUX
ejpam-7050	330	27	lower	low	ADJ
ejpam-7050	330	28	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	330	29	,	,	PUNCT
ejpam-7050	330	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	330	31	at	at	ADP
ejpam-7050	330	32	a	a	DET
ejpam-7050	330	33	point	point	NOUN
ejpam-7050	330	34	x	x	PUNCT
ejpam-7050	330	35	of	of	ADP
ejpam-7050	330	36	x	x	PRON
ejpam-7050	330	37	if	if	SCONJ
ejpam-7050	330	38	for	for	ADP
ejpam-7050	330	39	each	each	DET
ejpam-7050	330	40	σ1σ2	σ1σ2	VERB
ejpam-7050	330	41	-	-	ADJ
ejpam-7050	330	42	open	open	ADJ
ejpam-7050	330	43	set	set	NOUN
ejpam-7050	330	44	v	v	NOUN
ejpam-7050	330	45	of	of	ADP
ejpam-7050	330	46	y	y	PRON
ejpam-7050	330	47	such	such	ADJ
ejpam-7050	330	48	that	that	SCONJ
ejpam-7050	330	49	f	f	PROPN
ejpam-7050	330	50	(	(	PUNCT
ejpam-7050	330	51	x)∩v	x)∩v	PROPN
ejpam-7050	330	52	̸=	̸=	PROPN
ejpam-7050	330	53	∅	∅	NOUN
ejpam-7050	330	54	,	,	PUNCT
ejpam-7050	330	55	there	there	PRON
ejpam-7050	330	56	exists	exist	VERB
ejpam-7050	330	57	a	a	DET
ejpam-7050	330	58	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	330	59	-	-	ADJ
ejpam-7050	330	60	open	open	ADJ
ejpam-7050	330	61	set	set	NOUN
ejpam-7050	330	62	u	u	NOUN
ejpam-7050	330	63	of	of	ADP
ejpam-7050	330	64	x	x	PUNCT
ejpam-7050	330	65	containing	contain	VERB
ejpam-7050	330	66	x	x	PUNCT
ejpam-7050	330	67	such	such	ADJ
ejpam-7050	330	68	that	that	SCONJ
ejpam-7050	330	69	f	f	PROPN
ejpam-7050	330	70	(	(	PUNCT
ejpam-7050	330	71	z)∩v	z)∩v	PROPN
ejpam-7050	330	72	̸=	̸=	PROPN
ejpam-7050	330	73	∅	∅	NOUN
ejpam-7050	330	74	for	for	ADP
ejpam-7050	330	75	every	every	DET
ejpam-7050	330	76	z	z	NOUN
ejpam-7050	330	77	∈	∈	PROPN
ejpam-7050	330	78	u	u	NOUN
ejpam-7050	330	79	.	.	PUNCT
ejpam-7050	331	1	a	a	DET
ejpam-7050	331	2	multifunction	multifunction	NOUN
ejpam-7050	331	3	f	f	NOUN
ejpam-7050	331	4	:	:	PUNCT
ejpam-7050	331	5	(	(	PUNCT
ejpam-7050	331	6	x	x	X
ejpam-7050	331	7	,	,	PUNCT
ejpam-7050	331	8	τ	τ	PROPN
ejpam-7050	331	9	,	,	PUNCT
ejpam-7050	331	10	i	i	NOUN
ejpam-7050	331	11	)	)	PUNCT
ejpam-7050	331	12	→	→	PUNCT
ejpam-7050	331	13	(	(	PUNCT
ejpam-7050	331	14	y	y	PROPN
ejpam-7050	331	15	,	,	PUNCT
ejpam-7050	331	16	σ1	σ1	PROPN
ejpam-7050	331	17	,	,	PUNCT
ejpam-7050	331	18	σ2	σ2	PROPN
ejpam-7050	331	19	)	)	PUNCT
ejpam-7050	331	20	is	be	AUX
ejpam-7050	331	21	said	say	VERB
ejpam-7050	331	22	to	to	PART
ejpam-7050	331	23	be	be	AUX
ejpam-7050	331	24	lower	low	ADJ
ejpam-7050	331	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	331	26	,	,	PUNCT
ejpam-7050	331	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7050	331	28	if	if	SCONJ
ejpam-7050	331	29	f	f	PROPN
ejpam-7050	331	30	is	be	AUX
ejpam-7050	331	31	lower	low	ADJ
ejpam-7050	331	32	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	331	33	,	,	PUNCT
ejpam-7050	331	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	331	35	at	at	ADP
ejpam-7050	331	36	each	each	DET
ejpam-7050	331	37	point	point	NOUN
ejpam-7050	331	38	of	of	ADP
ejpam-7050	331	39	x.	x.	NOUN
ejpam-7050	331	40	recall	recall	VERB
ejpam-7050	331	41	that	that	SCONJ
ejpam-7050	331	42	a	a	DET
ejpam-7050	331	43	bitopological	bitopological	ADJ
ejpam-7050	331	44	space	space	NOUN
ejpam-7050	331	45	(	(	PUNCT
ejpam-7050	331	46	x	x	NOUN
ejpam-7050	331	47	,	,	PUNCT
ejpam-7050	331	48	τ1	τ1	NOUN
ejpam-7050	331	49	,	,	PUNCT
ejpam-7050	331	50	τ2	τ2	NOUN
ejpam-7050	331	51	)	)	PUNCT
ejpam-7050	331	52	is	be	AUX
ejpam-7050	331	53	said	say	VERB
ejpam-7050	331	54	to	to	PART
ejpam-7050	331	55	be	be	AUX
ejpam-7050	331	56	(	(	PUNCT
ejpam-7050	331	57	τ1	τ1	NOUN
ejpam-7050	331	58	,	,	PUNCT
ejpam-7050	331	59	τ2)-normal	τ2)-normal	ADJ
ejpam-7050	331	60	[	[	X
ejpam-7050	331	61	23	23	NUM
ejpam-7050	331	62	]	]	X
ejpam-7050	331	63	if	if	SCONJ
ejpam-7050	331	64	for	for	ADP
ejpam-7050	331	65	each	each	DET
ejpam-7050	331	66	pair	pair	NOUN
ejpam-7050	331	67	of	of	ADP
ejpam-7050	331	68	disjoint	disjoint	ADJ
ejpam-7050	331	69	τ1τ2	τ1τ2	ADJ
ejpam-7050	331	70	-	-	ADJ
ejpam-7050	331	71	closed	closed	ADJ
ejpam-7050	331	72	sets	set	NOUN
ejpam-7050	331	73	f	f	PROPN
ejpam-7050	331	74	and	and	CCONJ
ejpam-7050	331	75	f	f	PROPN
ejpam-7050	331	76	′	′	NOUN
ejpam-7050	331	77	,	,	PUNCT
ejpam-7050	331	78	there	there	PRON
ejpam-7050	331	79	exist	exist	VERB
ejpam-7050	331	80	disjoint	disjoint	ADJ
ejpam-7050	331	81	τ1τ2	τ1τ2	ADJ
ejpam-7050	331	82	-	-	ADJ
ejpam-7050	331	83	open	open	ADJ
ejpam-7050	331	84	sets	set	NOUN
ejpam-7050	331	85	u	u	NOUN
ejpam-7050	331	86	and	and	CCONJ
ejpam-7050	331	87	v	v	ADP
ejpam-7050	331	88	such	such	ADJ
ejpam-7050	331	89	that	that	SCONJ
ejpam-7050	331	90	f	f	PROPN
ejpam-7050	331	91	⊆	⊆	NUM
ejpam-7050	331	92	u	u	NOUN
ejpam-7050	331	93	and	and	CCONJ
ejpam-7050	331	94	f	f	PROPN
ejpam-7050	331	95	′	′	NUM
ejpam-7050	331	96	⊆	⊆	NUM
ejpam-7050	331	97	v	v	NOUN
ejpam-7050	331	98	.	.	PUNCT
ejpam-7050	332	1	theorem	theorem	VERB
ejpam-7050	332	2	9	9	NUM
ejpam-7050	332	3	.	.	X
ejpam-7050	332	4	for	for	ADP
ejpam-7050	332	5	a	a	DET
ejpam-7050	332	6	multifunction	multifunction	NOUN
ejpam-7050	333	1	f	f	NOUN
ejpam-7050	333	2	:	:	PUNCT
ejpam-7050	333	3	(	(	PUNCT
ejpam-7050	333	4	x	x	X
ejpam-7050	333	5	,	,	PUNCT
ejpam-7050	333	6	τ	τ	PROPN
ejpam-7050	333	7	,	,	PUNCT
ejpam-7050	333	8	i	i	NOUN
ejpam-7050	333	9	)	)	PUNCT
ejpam-7050	333	10	→	→	PUNCT
ejpam-7050	333	11	(	(	PUNCT
ejpam-7050	333	12	y	y	PROPN
ejpam-7050	333	13	,	,	PUNCT
ejpam-7050	333	14	σ1	σ1	PROPN
ejpam-7050	333	15	,	,	PUNCT
ejpam-7050	333	16	σ2	σ2	NOUN
ejpam-7050	333	17	)	)	PUNCT
ejpam-7050	333	18	such	such	ADJ
ejpam-7050	333	19	that	that	SCONJ
ejpam-7050	333	20	f	f	PROPN
ejpam-7050	333	21	(	(	PUNCT
ejpam-7050	333	22	x	x	X
ejpam-7050	333	23	)	)	PUNCT
ejpam-7050	333	24	is	be	AUX
ejpam-7050	333	25	σ1σ2	σ1σ2	NOUN
ejpam-7050	333	26	-	-	ADJ
ejpam-7050	333	27	closed	closed	ADJ
ejpam-7050	333	28	in	in	ADP
ejpam-7050	333	29	y	y	PROPN
ejpam-7050	333	30	for	for	SCONJ
ejpam-7050	333	31	each	each	DET
ejpam-7050	333	32	x	x	SYM
ejpam-7050	333	33	∈	∈	PROPN
ejpam-7050	333	34	x	x	X
ejpam-7050	333	35	and	and	CCONJ
ejpam-7050	333	36	(	(	PUNCT
ejpam-7050	333	37	y	y	PROPN
ejpam-7050	333	38	,	,	PUNCT
ejpam-7050	333	39	σ1	σ1	PROPN
ejpam-7050	333	40	,	,	PUNCT
ejpam-7050	333	41	σ2	σ2	PROPN
ejpam-7050	333	42	)	)	PUNCT
ejpam-7050	333	43	is	be	AUX
ejpam-7050	333	44	a	a	DET
ejpam-7050	333	45	(	(	PUNCT
ejpam-7050	333	46	σ1	σ1	NOUN
ejpam-7050	333	47	,	,	PUNCT
ejpam-7050	333	48	σ2)-normal	σ2)-normal	ADJ
ejpam-7050	333	49	space	space	NOUN
ejpam-7050	333	50	,	,	PUNCT
ejpam-7050	333	51	the	the	DET
ejpam-7050	333	52	following	follow	VERB
ejpam-7050	333	53	properties	property	NOUN
ejpam-7050	333	54	are	be	AUX
ejpam-7050	333	55	equivalent	equivalent	ADJ
ejpam-7050	333	56	:	:	PUNCT
ejpam-7050	333	57	(	(	PUNCT
ejpam-7050	333	58	1	1	X
ejpam-7050	333	59	)	)	PUNCT
ejpam-7050	333	60	f	f	PROPN
ejpam-7050	333	61	is	be	AUX
ejpam-7050	333	62	upper	upper	ADJ
ejpam-7050	333	63	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	333	64	,	,	PUNCT
ejpam-7050	333	65	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	333	66	;	;	PUNCT
ejpam-7050	333	67	(	(	PUNCT
ejpam-7050	333	68	2	2	X
ejpam-7050	333	69	)	)	PUNCT
ejpam-7050	333	70	f	f	PROPN
ejpam-7050	333	71	is	be	AUX
ejpam-7050	333	72	upper	upper	ADJ
ejpam-7050	333	73	almost	almost	ADV
ejpam-7050	333	74	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	333	75	,	,	PUNCT
ejpam-7050	333	76	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	333	77	;	;	PUNCT
ejpam-7050	333	78	(	(	PUNCT
ejpam-7050	333	79	3	3	X
ejpam-7050	333	80	)	)	PUNCT
ejpam-7050	333	81	f	f	PROPN
ejpam-7050	333	82	is	be	AUX
ejpam-7050	333	83	upper	upper	ADJ
ejpam-7050	333	84	weakly	weakly	ADJ
ejpam-7050	333	85	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	333	86	,	,	PUNCT
ejpam-7050	334	1	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	334	2	.	.	PUNCT
ejpam-7050	334	3	proof	proof	NOUN
ejpam-7050	334	4	.	.	PUNCT
ejpam-7050	335	1	we	we	PRON
ejpam-7050	335	2	show	show	VERB
ejpam-7050	335	3	only	only	ADV
ejpam-7050	335	4	the	the	DET
ejpam-7050	335	5	implication	implication	NOUN
ejpam-7050	335	6	(	(	PUNCT
ejpam-7050	335	7	3	3	X
ejpam-7050	335	8	)	)	PUNCT
ejpam-7050	335	9	⇒	⇒	NOUN
ejpam-7050	335	10	(	(	PUNCT
ejpam-7050	335	11	1	1	X
ejpam-7050	335	12	)	)	PUNCT
ejpam-7050	335	13	since	since	SCONJ
ejpam-7050	335	14	the	the	DET
ejpam-7050	335	15	others	other	NOUN
ejpam-7050	335	16	are	be	AUX
ejpam-7050	335	17	obvious	obvious	ADJ
ejpam-7050	335	18	.	.	PUNCT
ejpam-7050	336	1	suppose	suppose	VERB
ejpam-7050	336	2	that	that	SCONJ
ejpam-7050	336	3	f	f	PROPN
ejpam-7050	336	4	is	be	AUX
ejpam-7050	336	5	upper	upper	ADJ
ejpam-7050	336	6	weakly	weakly	ADJ
ejpam-7050	336	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	336	8	,	,	PUNCT
ejpam-7050	336	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7050	336	10	.	.	PUNCT
ejpam-7050	337	1	let	let	VERB
ejpam-7050	337	2	x	x	SYM
ejpam-7050	337	3	∈	∈	PROPN
ejpam-7050	337	4	x	x	X
ejpam-7050	337	5	and	and	CCONJ
ejpam-7050	337	6	v	v	X
ejpam-7050	337	7	be	be	AUX
ejpam-7050	337	8	any	any	DET
ejpam-7050	337	9	σ1σ2	σ1σ2	NOUN
ejpam-7050	337	10	-	-	ADJ
ejpam-7050	337	11	open	open	ADJ
ejpam-7050	337	12	set	set	NOUN
ejpam-7050	337	13	of	of	ADP
ejpam-7050	337	14	y	y	PRON
ejpam-7050	337	15	such	such	ADJ
ejpam-7050	337	16	that	that	SCONJ
ejpam-7050	337	17	f	f	PROPN
ejpam-7050	337	18	(	(	PUNCT
ejpam-7050	337	19	x	x	X
ejpam-7050	337	20	)	)	PUNCT
ejpam-7050	337	21	⊆	⊆	NUM
ejpam-7050	337	22	v	v	NOUN
ejpam-7050	337	23	.	.	PUNCT
ejpam-7050	338	1	since	since	SCONJ
ejpam-7050	338	2	f	f	PROPN
ejpam-7050	338	3	(	(	PUNCT
ejpam-7050	338	4	x	x	X
ejpam-7050	338	5	)	)	PUNCT
ejpam-7050	338	6	is	be	AUX
ejpam-7050	338	7	σ1σ2	σ1σ2	NOUN
ejpam-7050	338	8	-	-	ADJ
ejpam-7050	338	9	closed	closed	ADJ
ejpam-7050	338	10	in	in	ADP
ejpam-7050	338	11	y	y	PROPN
ejpam-7050	338	12	and	and	CCONJ
ejpam-7050	338	13	(	(	PUNCT
ejpam-7050	338	14	y	y	PROPN
ejpam-7050	338	15	,	,	PUNCT
ejpam-7050	338	16	σ1	σ1	PROPN
ejpam-7050	338	17	,	,	PUNCT
ejpam-7050	338	18	σ2	σ2	PROPN
ejpam-7050	338	19	)	)	PUNCT
ejpam-7050	338	20	is	be	AUX
ejpam-7050	338	21	(	(	PUNCT
ejpam-7050	338	22	σ1	σ1	PROPN
ejpam-7050	338	23	,	,	PUNCT
ejpam-7050	338	24	σ2)-normal	σ2)-normal	PROPN
ejpam-7050	338	25	,	,	PUNCT
ejpam-7050	338	26	there	there	PRON
ejpam-7050	338	27	exists	exist	VERB
ejpam-7050	338	28	a	a	DET
ejpam-7050	338	29	σ1σ2	σ1σ2	NUM
ejpam-7050	338	30	-	-	ADJ
ejpam-7050	338	31	open	open	ADJ
ejpam-7050	338	32	set	set	NOUN
ejpam-7050	338	33	g	g	NOUN
ejpam-7050	338	34	of	of	ADP
ejpam-7050	338	35	y	y	PRON
ejpam-7050	338	36	such	such	ADJ
ejpam-7050	338	37	that	that	SCONJ
ejpam-7050	338	38	f	f	PROPN
ejpam-7050	338	39	(	(	PUNCT
ejpam-7050	338	40	x	x	X
ejpam-7050	338	41	)	)	PUNCT
ejpam-7050	338	42	⊆	⊆	NUM
ejpam-7050	338	43	g	g	ADP
ejpam-7050	338	44	⊆	⊆	NUM
ejpam-7050	338	45	σ1σ2	σ1σ2	NOUN
ejpam-7050	338	46	-	-	PUNCT
ejpam-7050	338	47	cl(g	cl(g	ADJ
ejpam-7050	338	48	)	)	PUNCT
ejpam-7050	338	49	⊆	⊆	NUM
ejpam-7050	338	50	v	v	NOUN
ejpam-7050	338	51	.	.	PUNCT
ejpam-7050	339	1	since	since	SCONJ
ejpam-7050	339	2	f	f	PROPN
ejpam-7050	339	3	is	be	AUX
ejpam-7050	339	4	upper	upper	ADJ
ejpam-7050	339	5	weakly	weakly	ADJ
ejpam-7050	339	6	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	339	7	,	,	PUNCT
ejpam-7050	339	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	339	9	,	,	PUNCT
ejpam-7050	339	10	there	there	PRON
ejpam-7050	339	11	exists	exist	VERB
ejpam-7050	339	12	a	a	DET
ejpam-7050	339	13	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	339	14	-	-	ADJ
ejpam-7050	339	15	open	open	ADJ
ejpam-7050	339	16	set	set	NOUN
ejpam-7050	339	17	u	u	NOUN
ejpam-7050	339	18	of	of	ADP
ejpam-7050	339	19	x	x	PUNCT
ejpam-7050	339	20	containing	contain	VERB
ejpam-7050	339	21	x	x	PUNCT
ejpam-7050	339	22	such	such	ADJ
ejpam-7050	339	23	that	that	SCONJ
ejpam-7050	339	24	f	f	PROPN
ejpam-7050	339	25	(	(	PUNCT
ejpam-7050	339	26	u	u	NOUN
ejpam-7050	339	27	)	)	PUNCT
ejpam-7050	339	28	⊆	⊆	NUM
ejpam-7050	339	29	σ1σ2	σ1σ2	NOUN
ejpam-7050	339	30	-	-	PUNCT
ejpam-7050	339	31	cl(g	cl(g	NUM
ejpam-7050	339	32	)	)	PUNCT
ejpam-7050	339	33	;	;	PUNCT
ejpam-7050	339	34	hence	hence	ADV
ejpam-7050	339	35	f	f	PROPN
ejpam-7050	339	36	(	(	PUNCT
ejpam-7050	339	37	u	u	NOUN
ejpam-7050	339	38	)	)	PUNCT
ejpam-7050	339	39	⊆	⊆	NUM
ejpam-7050	339	40	v	v	NOUN
ejpam-7050	339	41	.	.	PUNCT
ejpam-7050	340	1	this	this	PRON
ejpam-7050	340	2	shows	show	VERB
ejpam-7050	340	3	that	that	SCONJ
ejpam-7050	340	4	f	f	PROPN
ejpam-7050	340	5	is	be	AUX
ejpam-7050	340	6	upper	upper	ADJ
ejpam-7050	340	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	340	8	,	,	PUNCT
ejpam-7050	340	9	σ2)continuous	σ2)continuous	ADJ
ejpam-7050	340	10	.	.	PUNCT
ejpam-7050	341	1	m.	m.	NOUN
ejpam-7050	341	2	thongmoon	thongmoon	PROPN
ejpam-7050	341	3	,	,	PUNCT
ejpam-7050	341	4	a.	a.	PROPN
ejpam-7050	341	5	sama	sama	PROPN
ejpam-7050	341	6	-	-	PUNCT
ejpam-7050	341	7	ae	ae	PROPN
ejpam-7050	341	8	,	,	PUNCT
ejpam-7050	341	9	c.	c.	PROPN
ejpam-7050	341	10	boonpok	boonpok	PROPN
ejpam-7050	341	11	/	/	SYM
ejpam-7050	341	12	eur	eur	PROPN
ejpam-7050	341	13	.	.	PUNCT
ejpam-7050	342	1	j.	j.	PROPN
ejpam-7050	342	2	pure	pure	PROPN
ejpam-7050	342	3	appl	appl	PROPN
ejpam-7050	342	4	.	.	PROPN
ejpam-7050	342	5	math	math	PROPN
ejpam-7050	342	6	,	,	PUNCT
ejpam-7050	342	7	18	18	NUM
ejpam-7050	342	8	(	(	PUNCT
ejpam-7050	342	9	4	4	NUM
ejpam-7050	342	10	)	)	PUNCT
ejpam-7050	342	11	(	(	PUNCT
ejpam-7050	342	12	2025	2025	NUM
ejpam-7050	342	13	)	)	PUNCT
ejpam-7050	342	14	,	,	PUNCT
ejpam-7050	342	15	7050	7050	NUM
ejpam-7050	342	16	13	13	NUM
ejpam-7050	342	17	of	of	ADP
ejpam-7050	342	18	14	14	NUM
ejpam-7050	342	19	theorem	theorem	ADJ
ejpam-7050	342	20	10	10	NUM
ejpam-7050	342	21	.	.	PUNCT
ejpam-7050	343	1	for	for	ADP
ejpam-7050	343	2	a	a	DET
ejpam-7050	343	3	multifunction	multifunction	NOUN
ejpam-7050	343	4	f	f	NOUN
ejpam-7050	343	5	:	:	PUNCT
ejpam-7050	343	6	(	(	PUNCT
ejpam-7050	343	7	x	x	X
ejpam-7050	343	8	,	,	PUNCT
ejpam-7050	343	9	τ	τ	PROPN
ejpam-7050	343	10	,	,	PUNCT
ejpam-7050	343	11	i	i	NOUN
ejpam-7050	343	12	)	)	PUNCT
ejpam-7050	343	13	→	→	PUNCT
ejpam-7050	343	14	(	(	PUNCT
ejpam-7050	343	15	y	y	PROPN
ejpam-7050	343	16	,	,	PUNCT
ejpam-7050	343	17	σ1	σ1	PROPN
ejpam-7050	343	18	,	,	PUNCT
ejpam-7050	343	19	σ2	σ2	NOUN
ejpam-7050	343	20	)	)	PUNCT
ejpam-7050	343	21	such	such	ADJ
ejpam-7050	343	22	that	that	SCONJ
ejpam-7050	343	23	f	f	PROPN
ejpam-7050	343	24	(	(	PUNCT
ejpam-7050	343	25	x	x	X
ejpam-7050	343	26	)	)	PUNCT
ejpam-7050	343	27	is	be	AUX
ejpam-7050	343	28	σ1σ2	σ1σ2	NOUN
ejpam-7050	343	29	-	-	ADJ
ejpam-7050	343	30	open	open	ADJ
ejpam-7050	343	31	in	in	ADP
ejpam-7050	343	32	y	y	PROPN
ejpam-7050	343	33	for	for	ADP
ejpam-7050	343	34	each	each	DET
ejpam-7050	343	35	x	x	SYM
ejpam-7050	343	36	∈	∈	PROPN
ejpam-7050	343	37	x	x	NOUN
ejpam-7050	343	38	,	,	PUNCT
ejpam-7050	343	39	the	the	DET
ejpam-7050	343	40	following	follow	VERB
ejpam-7050	343	41	properties	property	NOUN
ejpam-7050	343	42	are	be	AUX
ejpam-7050	343	43	equivalent	equivalent	ADJ
ejpam-7050	343	44	:	:	PUNCT
ejpam-7050	343	45	(	(	PUNCT
ejpam-7050	343	46	1	1	X
ejpam-7050	343	47	)	)	PUNCT
ejpam-7050	343	48	f	f	PROPN
ejpam-7050	343	49	is	be	AUX
ejpam-7050	343	50	lower	low	ADJ
ejpam-7050	343	51	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	343	52	,	,	PUNCT
ejpam-7050	343	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	343	54	;	;	PUNCT
ejpam-7050	343	55	(	(	PUNCT
ejpam-7050	343	56	2	2	X
ejpam-7050	343	57	)	)	PUNCT
ejpam-7050	343	58	f	f	PROPN
ejpam-7050	343	59	is	be	AUX
ejpam-7050	343	60	lower	low	ADJ
ejpam-7050	343	61	almost	almost	ADV
ejpam-7050	343	62	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	343	63	,	,	PUNCT
ejpam-7050	343	64	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	343	65	;	;	PUNCT
ejpam-7050	343	66	(	(	PUNCT
ejpam-7050	343	67	3	3	X
ejpam-7050	343	68	)	)	PUNCT
ejpam-7050	343	69	f	f	PROPN
ejpam-7050	343	70	is	be	AUX
ejpam-7050	343	71	lower	low	ADJ
ejpam-7050	343	72	weakly	weakly	ADJ
ejpam-7050	343	73	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	343	74	,	,	PUNCT
ejpam-7050	343	75	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7050	343	76	.	.	PUNCT
ejpam-7050	343	77	proof	proof	NOUN
ejpam-7050	343	78	.	.	PUNCT
ejpam-7050	344	1	(	(	PUNCT
ejpam-7050	344	2	1	1	X
ejpam-7050	344	3	)	)	PUNCT
ejpam-7050	344	4	⇒	⇒	NOUN
ejpam-7050	344	5	(	(	PUNCT
ejpam-7050	344	6	2	2	NUM
ejpam-7050	344	7	)	)	PUNCT
ejpam-7050	344	8	and	and	CCONJ
ejpam-7050	344	9	(	(	PUNCT
ejpam-7050	344	10	2	2	X
ejpam-7050	344	11	)	)	PUNCT
ejpam-7050	344	12	⇒	⇒	NOUN
ejpam-7050	344	13	(	(	PUNCT
ejpam-7050	344	14	3	3	NUM
ejpam-7050	344	15	):	):	PUNCT
ejpam-7050	344	16	the	the	DET
ejpam-7050	344	17	proofs	proof	NOUN
ejpam-7050	344	18	of	of	ADP
ejpam-7050	344	19	these	these	DET
ejpam-7050	344	20	implications	implication	NOUN
ejpam-7050	344	21	are	be	AUX
ejpam-7050	344	22	obvious	obvious	ADJ
ejpam-7050	344	23	.	.	PUNCT
ejpam-7050	345	1	(	(	PUNCT
ejpam-7050	345	2	3	3	X
ejpam-7050	345	3	)	)	PUNCT
ejpam-7050	345	4	⇒	⇒	NOUN
ejpam-7050	345	5	(	(	PUNCT
ejpam-7050	345	6	1	1	NUM
ejpam-7050	345	7	):	):	PUNCT
ejpam-7050	345	8	suppose	suppose	VERB
ejpam-7050	345	9	that	that	SCONJ
ejpam-7050	345	10	f	f	PROPN
ejpam-7050	345	11	is	be	AUX
ejpam-7050	345	12	lower	low	ADJ
ejpam-7050	345	13	weakly	weakly	ADJ
ejpam-7050	345	14	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	345	15	,	,	PUNCT
ejpam-7050	345	16	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7050	345	17	.	.	PUNCT
ejpam-7050	345	18	let	let	VERB
ejpam-7050	345	19	x	x	SYM
ejpam-7050	345	20	∈	∈	PROPN
ejpam-7050	345	21	x	x	X
ejpam-7050	345	22	and	and	CCONJ
ejpam-7050	345	23	v	v	X
ejpam-7050	345	24	be	be	AUX
ejpam-7050	345	25	any	any	DET
ejpam-7050	345	26	σ1σ2	σ1σ2	NOUN
ejpam-7050	345	27	-	-	ADJ
ejpam-7050	345	28	open	open	ADJ
ejpam-7050	345	29	set	set	NOUN
ejpam-7050	345	30	of	of	ADP
ejpam-7050	345	31	y	y	PRON
ejpam-7050	345	32	such	such	ADJ
ejpam-7050	345	33	that	that	SCONJ
ejpam-7050	345	34	f	f	PROPN
ejpam-7050	346	1	(	(	PUNCT
ejpam-7050	346	2	x)∩v	x)∩v	PROPN
ejpam-7050	346	3	̸=	̸=	PROPN
ejpam-7050	346	4	∅.	∅.	NOUN
ejpam-7050	346	5	then	then	ADV
ejpam-7050	346	6	,	,	PUNCT
ejpam-7050	346	7	there	there	PRON
ejpam-7050	346	8	exists	exist	VERB
ejpam-7050	346	9	a	a	DET
ejpam-7050	346	10	τ⋆-β	τ⋆-β	NOUN
ejpam-7050	346	11	-	-	ADJ
ejpam-7050	346	12	open	open	ADJ
ejpam-7050	346	13	set	set	NOUN
ejpam-7050	346	14	u	u	NOUN
ejpam-7050	346	15	of	of	ADP
ejpam-7050	346	16	x	x	PUNCT
ejpam-7050	346	17	containing	contain	VERB
ejpam-7050	346	18	x	x	PUNCT
ejpam-7050	346	19	such	such	ADJ
ejpam-7050	346	20	that	that	SCONJ
ejpam-7050	346	21	f	f	PROPN
ejpam-7050	346	22	(	(	PUNCT
ejpam-7050	346	23	z)∩σ1σ2	z)∩σ1σ2	NOUN
ejpam-7050	346	24	-	-	PUNCT
ejpam-7050	346	25	cl(v	cl(v	NOUN
ejpam-7050	346	26	)	)	PUNCT
ejpam-7050	346	27	̸=	̸=	NOUN
ejpam-7050	346	28	∅	∅	NOUN
ejpam-7050	346	29	for	for	ADP
ejpam-7050	346	30	each	each	DET
ejpam-7050	346	31	z	z	NOUN
ejpam-7050	346	32	∈	∈	PROPN
ejpam-7050	346	33	u	u	PROPN
ejpam-7050	346	34	.	.	PUNCT
ejpam-7050	347	1	since	since	SCONJ
ejpam-7050	347	2	f	f	PROPN
ejpam-7050	347	3	(	(	PUNCT
ejpam-7050	347	4	z	z	NOUN
ejpam-7050	347	5	)	)	PUNCT
ejpam-7050	347	6	is	be	AUX
ejpam-7050	347	7	σ1σ2	σ1σ2	NOUN
ejpam-7050	347	8	-	-	ADJ
ejpam-7050	347	9	open	open	ADJ
ejpam-7050	347	10	,	,	PUNCT
ejpam-7050	347	11	we	we	PRON
ejpam-7050	347	12	have	have	VERB
ejpam-7050	347	13	f	f	PROPN
ejpam-7050	347	14	(	(	PUNCT
ejpam-7050	347	15	z	z	NOUN
ejpam-7050	347	16	)	)	PUNCT
ejpam-7050	347	17	∩	∩	NOUN
ejpam-7050	347	18	v	v	ADP
ejpam-7050	347	19	̸=	̸=	PROPN
ejpam-7050	347	20	∅	∅	NOUN
ejpam-7050	347	21	for	for	ADP
ejpam-7050	347	22	each	each	DET
ejpam-7050	347	23	z	z	NOUN
ejpam-7050	347	24	∈	∈	PROPN
ejpam-7050	347	25	u	u	NOUN
ejpam-7050	347	26	and	and	CCONJ
ejpam-7050	347	27	so	so	ADV
ejpam-7050	347	28	f	f	PROPN
ejpam-7050	347	29	is	be	AUX
ejpam-7050	347	30	lower	low	ADJ
ejpam-7050	347	31	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	347	32	,	,	PUNCT
ejpam-7050	347	33	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7050	347	34	.	.	PUNCT
ejpam-7050	348	1	acknowledgements	acknowledgement	NOUN
ejpam-7050	348	2	this	this	DET
ejpam-7050	348	3	research	research	NOUN
ejpam-7050	348	4	project	project	NOUN
ejpam-7050	348	5	was	be	AUX
ejpam-7050	348	6	financially	financially	ADV
ejpam-7050	348	7	supported	support	VERB
ejpam-7050	348	8	by	by	ADP
ejpam-7050	348	9	mahasarakham	mahasarakham	PROPN
ejpam-7050	348	10	university	university	PROPN
ejpam-7050	348	11	.	.	PUNCT
ejpam-7050	349	1	references	reference	NOUN
ejpam-7050	349	2	[	[	X
ejpam-7050	349	3	1	1	NUM
ejpam-7050	349	4	]	]	PUNCT
ejpam-7050	349	5	m.	m.	NOUN
ejpam-7050	349	6	e.	e.	PROPN
ejpam-7050	349	7	abd	abd	PROPN
ejpam-7050	349	8	el	el	PROPN
ejpam-7050	349	9	-	-	PROPN
ejpam-7050	349	10	monsef	monsef	PROPN
ejpam-7050	349	11	,	,	PUNCT
ejpam-7050	349	12	s.	s.	PROPN
ejpam-7050	349	13	n.	n.	PROPN
ejpam-7050	349	14	el	el	PROPN
ejpam-7050	349	15	-	-	PROPN
ejpam-7050	349	16	deeb	deeb	PROPN
ejpam-7050	349	17	,	,	PUNCT
ejpam-7050	349	18	and	and	CCONJ
ejpam-7050	349	19	r.	r.	PROPN
ejpam-7050	349	20	a.	a.	PROPN
ejpam-7050	349	21	mahmoud	mahmoud	PROPN
ejpam-7050	349	22	.	.	PUNCT
ejpam-7050	350	1	β	β	X
ejpam-7050	350	2	-	-	ADJ
ejpam-7050	350	3	open	open	ADJ
ejpam-7050	350	4	sets	set	NOUN
ejpam-7050	350	5	and	and	CCONJ
ejpam-7050	350	6	βcontinuous	βcontinuous	ADJ
ejpam-7050	350	7	mappings	mapping	NOUN
ejpam-7050	350	8	.	.	PUNCT
ejpam-7050	351	1	bulletin	bulletin	NOUN
ejpam-7050	351	2	of	of	ADP
ejpam-7050	351	3	the	the	DET
ejpam-7050	351	4	faculty	faculty	NOUN
ejpam-7050	351	5	of	of	ADP
ejpam-7050	351	6	science	science	NOUN
ejpam-7050	351	7	,	,	PUNCT
ejpam-7050	351	8	assiut	assiut	NOUN
ejpam-7050	351	9	university	university	NOUN
ejpam-7050	351	10	,	,	PUNCT
ejpam-7050	351	11	12:77–90	12:77–90	NUM
ejpam-7050	351	12	,	,	PUNCT
ejpam-7050	351	13	1983	1983	NUM
ejpam-7050	351	14	.	.	PUNCT
ejpam-7050	352	1	[	[	X
ejpam-7050	352	2	2	2	NUM
ejpam-7050	352	3	]	]	PUNCT
ejpam-7050	352	4	a.	a.	NOUN
ejpam-7050	352	5	a.	a.	NOUN
ejpam-7050	352	6	nasef	nasef	PROPN
ejpam-7050	352	7	and	and	CCONJ
ejpam-7050	352	8	t.	t.	PROPN
ejpam-7050	352	9	noiri	noiri	PROPN
ejpam-7050	352	10	.	.	PUNCT
ejpam-7050	353	1	some	some	DET
ejpam-7050	353	2	weak	weak	ADJ
ejpam-7050	353	3	forms	form	NOUN
ejpam-7050	353	4	of	of	ADP
ejpam-7050	353	5	almost	almost	ADV
ejpam-7050	353	6	continuity	continuity	NOUN
ejpam-7050	353	7	.	.	PUNCT
ejpam-7050	354	1	acta	acta	PROPN
ejpam-7050	354	2	mathematica	mathematica	PROPN
ejpam-7050	354	3	hungarica	hungarica	PROPN
ejpam-7050	354	4	,	,	PUNCT
ejpam-7050	354	5	74(3):211–219	74(3):211–219	PROPN
ejpam-7050	354	6	,	,	PUNCT
ejpam-7050	354	7	1997	1997	NUM
ejpam-7050	354	8	.	.	PUNCT
ejpam-7050	355	1	[	[	X
ejpam-7050	355	2	3	3	X
ejpam-7050	355	3	]	]	PUNCT
ejpam-7050	355	4	v.	v.	CCONJ
ejpam-7050	355	5	popa	popa	NOUN
ejpam-7050	355	6	and	and	CCONJ
ejpam-7050	355	7	t.	t.	PROPN
ejpam-7050	355	8	noiri	noiri	PROPN
ejpam-7050	355	9	.	.	PUNCT
ejpam-7050	356	1	weakly	weakly	ADJ
ejpam-7050	356	2	β	β	X
ejpam-7050	356	3	-	-	PUNCT
ejpam-7050	356	4	continuuos	continuuos	ADJ
ejpam-7050	356	5	functions	function	NOUN
ejpam-7050	356	6	.	.	PUNCT
ejpam-7050	357	1	analele	analele	X
ejpam-7050	357	2	universitǎţii	universitǎţii	PUNCT
ejpam-7050	358	1	timişoara	timişoara	NOUN
ejpam-7050	358	2	,	,	PUNCT
ejpam-7050	358	3	seria	seria	PROPN
ejpam-7050	358	4	matematicǎ-informaticǎ	matematicǎ-informaticǎ	PROPN
ejpam-7050	358	5	,	,	PUNCT
ejpam-7050	358	6	32:83–92	32:83–92	PROPN
ejpam-7050	358	7	,	,	PUNCT
ejpam-7050	358	8	1994	1994	NUM
ejpam-7050	358	9	.	.	PUNCT
ejpam-7050	359	1	[	[	X
ejpam-7050	359	2	4	4	X
ejpam-7050	359	3	]	]	PUNCT
ejpam-7050	359	4	v.	v.	CCONJ
ejpam-7050	359	5	popa	popa	NOUN
ejpam-7050	359	6	and	and	CCONJ
ejpam-7050	359	7	t.	t.	PROPN
ejpam-7050	359	8	noiri	noiri	PROPN
ejpam-7050	359	9	.	.	PUNCT
ejpam-7050	360	1	on	on	ADP
ejpam-7050	360	2	upper	upper	ADJ
ejpam-7050	360	3	and	and	CCONJ
ejpam-7050	360	4	lower	low	ADJ
ejpam-7050	360	5	almost	almost	ADV
ejpam-7050	360	6	β	β	ADJ
ejpam-7050	360	7	-	-	ADJ
ejpam-7050	360	8	continuous	continuous	ADJ
ejpam-7050	360	9	multifunctions	multifunction	NOUN
ejpam-7050	360	10	.	.	PUNCT
ejpam-7050	361	1	acta	acta	PROPN
ejpam-7050	361	2	mathematica	mathematica	PROPN
ejpam-7050	361	3	hungarica	hungarica	PROPN
ejpam-7050	361	4	,	,	PUNCT
ejpam-7050	361	5	82(1	82(1	NOUN
ejpam-7050	361	6	-	-	PUNCT
ejpam-7050	361	7	2):57–73	2):57–73	NUM
ejpam-7050	361	8	,	,	PUNCT
ejpam-7050	361	9	1999	1999	NUM
ejpam-7050	361	10	.	.	PUNCT
ejpam-7050	362	1	[	[	X
ejpam-7050	362	2	5	5	X
ejpam-7050	362	3	]	]	PUNCT
ejpam-7050	362	4	v.	v.	CCONJ
ejpam-7050	362	5	popa	popa	NOUN
ejpam-7050	362	6	and	and	CCONJ
ejpam-7050	362	7	t.	t.	PROPN
ejpam-7050	362	8	noiri	noiri	PROPN
ejpam-7050	362	9	.	.	PUNCT
ejpam-7050	363	1	on	on	ADP
ejpam-7050	363	2	upper	upper	ADJ
ejpam-7050	363	3	and	and	CCONJ
ejpam-7050	363	4	lower	low	ADJ
ejpam-7050	363	5	weakly	weakly	ADJ
ejpam-7050	363	6	β	β	ADJ
ejpam-7050	363	7	-	-	ADJ
ejpam-7050	363	8	continuous	continuous	ADJ
ejpam-7050	363	9	multifunctions	multifunction	NOUN
ejpam-7050	363	10	.	.	PUNCT
ejpam-7050	364	1	annales	annales	PROPN
ejpam-7050	364	2	universitatis	universitatis	PROPN
ejpam-7050	364	3	scientiarum	scientiarum	PROPN
ejpam-7050	364	4	budapestinensis	budapestinensis	PROPN
ejpam-7050	364	5	de	de	PROPN
ejpam-7050	364	6	rolando	rolando	PROPN
ejpam-7050	364	7	eötvös	eötvös	PROPN
ejpam-7050	364	8	nominatae	nominatae	NOUN
ejpam-7050	364	9	,	,	PUNCT
ejpam-7050	364	10	sectio	sectio	NOUN
ejpam-7050	364	11	mathematica	mathematica	PROPN
ejpam-7050	364	12	,	,	PUNCT
ejpam-7050	364	13	43:25–48	43:25–48	PROPN
ejpam-7050	364	14	,	,	PUNCT
ejpam-7050	364	15	2000	2000	NUM
ejpam-7050	364	16	.	.	PUNCT
ejpam-7050	365	1	[	[	X
ejpam-7050	365	2	6	6	NUM
ejpam-7050	365	3	]	]	PUNCT
ejpam-7050	365	4	c.	c.	PROPN
ejpam-7050	365	5	boonpok	boonpok	PROPN
ejpam-7050	365	6	.	.	PUNCT
ejpam-7050	366	1	on	on	ADP
ejpam-7050	366	2	continuous	continuous	ADJ
ejpam-7050	366	3	multifunctions	multifunction	NOUN
ejpam-7050	366	4	in	in	ADP
ejpam-7050	366	5	ideal	ideal	ADJ
ejpam-7050	366	6	topological	topological	ADJ
ejpam-7050	366	7	spaces	space	NOUN
ejpam-7050	366	8	.	.	PUNCT
ejpam-7050	367	1	lobachevskii	lobachevskii	PROPN
ejpam-7050	367	2	journal	journal	PROPN
ejpam-7050	367	3	of	of	ADP
ejpam-7050	367	4	mathematics	mathematic	NOUN
ejpam-7050	367	5	,	,	PUNCT
ejpam-7050	367	6	40(1):24–35	40(1):24–35	NUM
ejpam-7050	367	7	,	,	PUNCT
ejpam-7050	367	8	2019	2019	NUM
ejpam-7050	367	9	.	.	PUNCT
ejpam-7050	368	1	[	[	X
ejpam-7050	368	2	7	7	X
ejpam-7050	368	3	]	]	X
ejpam-7050	368	4	c.	c.	PROPN
ejpam-7050	368	5	boonpok	boonpok	PROPN
ejpam-7050	368	6	and	and	CCONJ
ejpam-7050	368	7	j.	j.	PROPN
ejpam-7050	368	8	khampakdee	khampakdee	PROPN
ejpam-7050	368	9	.	.	PUNCT
ejpam-7050	369	1	upper	upper	ADJ
ejpam-7050	369	2	and	and	CCONJ
ejpam-7050	369	3	lower	low	ADJ
ejpam-7050	369	4	weak	weak	ADJ
ejpam-7050	369	5	sβ(⋆)-continuity	sβ(⋆)-continuity	NOUN
ejpam-7050	369	6	.	.	PUNCT
ejpam-7050	370	1	european	european	PROPN
ejpam-7050	370	2	journal	journal	PROPN
ejpam-7050	370	3	of	of	ADP
ejpam-7050	370	4	pure	pure	ADJ
ejpam-7050	370	5	and	and	CCONJ
ejpam-7050	370	6	applied	applied	ADJ
ejpam-7050	370	7	mathematics	mathematic	NOUN
ejpam-7050	370	8	,	,	PUNCT
ejpam-7050	370	9	16(4):2544–2556	16(4):2544–2556	NUM
ejpam-7050	370	10	,	,	PUNCT
ejpam-7050	370	11	2023	2023	NUM
ejpam-7050	370	12	.	.	PUNCT
ejpam-7050	371	1	[	[	X
ejpam-7050	371	2	8	8	NUM
ejpam-7050	371	3	]	]	X
ejpam-7050	371	4	c.	c.	NOUN
ejpam-7050	371	5	boonpok	boonpok	PROPN
ejpam-7050	371	6	and	and	CCONJ
ejpam-7050	371	7	p.	p.	NOUN
ejpam-7050	371	8	pue	pue	NOUN
ejpam-7050	371	9	-	-	PUNCT
ejpam-7050	371	10	on	on	ADP
ejpam-7050	371	11	.	.	PUNCT
ejpam-7050	372	1	upper	upper	ADJ
ejpam-7050	372	2	and	and	CCONJ
ejpam-7050	372	3	lower	low	ADJ
ejpam-7050	372	4	weakly	weakly	ADJ
ejpam-7050	372	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7050	372	6	multifunctions	multifunction	NOUN
ejpam-7050	372	7	.	.	PUNCT
ejpam-7050	373	1	international	international	ADJ
ejpam-7050	373	2	journal	journal	NOUN
ejpam-7050	373	3	of	of	ADP
ejpam-7050	373	4	analysis	analysis	NOUN
ejpam-7050	373	5	and	and	CCONJ
ejpam-7050	373	6	applications	application	NOUN
ejpam-7050	373	7	,	,	PUNCT
ejpam-7050	373	8	21:90	21:90	NUM
ejpam-7050	373	9	,	,	PUNCT
ejpam-7050	373	10	2023	2023	NUM
ejpam-7050	373	11	.	.	PUNCT
ejpam-7050	374	1	[	[	X
ejpam-7050	374	2	9	9	NUM
ejpam-7050	374	3	]	]	PUNCT
ejpam-7050	374	4	c.	c.	NOUN
ejpam-7050	374	5	boonpok	boonpok	PROPN
ejpam-7050	374	6	and	and	CCONJ
ejpam-7050	374	7	p.	p.	NOUN
ejpam-7050	374	8	pue	pue	NOUN
ejpam-7050	374	9	-	-	PUNCT
ejpam-7050	374	10	on	on	ADP
ejpam-7050	374	11	.	.	PUNCT
ejpam-7050	375	1	continuity	continuity	NOUN
ejpam-7050	375	2	for	for	ADP
ejpam-7050	375	3	multifunctions	multifunction	NOUN
ejpam-7050	375	4	in	in	ADP
ejpam-7050	375	5	ideal	ideal	ADJ
ejpam-7050	375	6	topological	topological	ADJ
ejpam-7050	375	7	spaces	space	NOUN
ejpam-7050	375	8	.	.	PUNCT
ejpam-7050	376	1	wseas	wseas	VERB
ejpam-7050	376	2	transactions	transaction	NOUN
ejpam-7050	376	3	on	on	ADP
ejpam-7050	376	4	mathematics	mathematic	NOUN
ejpam-7050	376	5	,	,	PUNCT
ejpam-7050	376	6	19:624–631	19:624–631	NUM
ejpam-7050	376	7	,	,	PUNCT
ejpam-7050	376	8	2020	2020	NUM
ejpam-7050	376	9	.	.	PUNCT
ejpam-7050	377	1	[	[	X
ejpam-7050	377	2	10	10	NUM
ejpam-7050	377	3	]	]	X
ejpam-7050	377	4	c.	c.	PROPN
ejpam-7050	377	5	boonpok	boonpok	PROPN
ejpam-7050	377	6	.	.	PUNCT
ejpam-7050	378	1	pı	pı	NOUN
ejpam-7050	378	2	-	-	NOUN
ejpam-7050	378	3	continuity	continuity	NOUN
ejpam-7050	378	4	and	and	CCONJ
ejpam-7050	378	5	weak	weak	ADJ
ejpam-7050	378	6	pı	pı	NOUN
ejpam-7050	378	7	-	-	NOUN
ejpam-7050	378	8	continuity	continuity	NOUN
ejpam-7050	378	9	.	.	PUNCT
ejpam-7050	379	1	carpathian	carpathian	ADJ
ejpam-7050	379	2	mathematical	mathematical	ADJ
ejpam-7050	379	3	publications	publication	NOUN
ejpam-7050	379	4	,	,	PUNCT
ejpam-7050	379	5	17(1):171–186	17(1):171–186	PROPN
ejpam-7050	379	6	,	,	PUNCT
ejpam-7050	379	7	2025	2025	NUM
ejpam-7050	379	8	.	.	PUNCT
ejpam-7050	380	1	m.	m.	NOUN
ejpam-7050	380	2	thongmoon	thongmoon	PROPN
ejpam-7050	380	3	,	,	PUNCT
ejpam-7050	380	4	a.	a.	PROPN
ejpam-7050	380	5	sama	sama	PROPN
ejpam-7050	380	6	-	-	PUNCT
ejpam-7050	380	7	ae	ae	PROPN
ejpam-7050	380	8	,	,	PUNCT
ejpam-7050	380	9	c.	c.	PROPN
ejpam-7050	380	10	boonpok	boonpok	PROPN
ejpam-7050	380	11	/	/	SYM
ejpam-7050	380	12	eur	eur	PROPN
ejpam-7050	380	13	.	.	PUNCT
ejpam-7050	381	1	j.	j.	PROPN
ejpam-7050	381	2	pure	pure	PROPN
ejpam-7050	381	3	appl	appl	PROPN
ejpam-7050	381	4	.	.	PROPN
ejpam-7050	381	5	math	math	PROPN
ejpam-7050	381	6	,	,	PUNCT
ejpam-7050	381	7	18	18	NUM
ejpam-7050	381	8	(	(	PUNCT
ejpam-7050	381	9	4	4	NUM
ejpam-7050	381	10	)	)	PUNCT
ejpam-7050	381	11	(	(	PUNCT
ejpam-7050	381	12	2025	2025	NUM
ejpam-7050	381	13	)	)	PUNCT
ejpam-7050	381	14	,	,	PUNCT
ejpam-7050	381	15	7050	7050	NUM
ejpam-7050	381	16	14	14	NUM
ejpam-7050	381	17	of	of	ADP
ejpam-7050	381	18	14	14	NUM
ejpam-7050	382	1	[	[	X
ejpam-7050	382	2	11	11	NUM
ejpam-7050	382	3	]	]	X
ejpam-7050	382	4	p.	p.	NOUN
ejpam-7050	382	5	pue	pue	NOUN
ejpam-7050	382	6	-	-	PUNCT
ejpam-7050	382	7	on	on	ADP
ejpam-7050	382	8	,	,	PUNCT
ejpam-7050	382	9	s.	s.	PROPN
ejpam-7050	382	10	sompong	sompong	PROPN
ejpam-7050	382	11	,	,	PUNCT
ejpam-7050	382	12	and	and	CCONJ
ejpam-7050	382	13	c.	c.	PROPN
ejpam-7050	382	14	boonpok	boonpok	PROPN
ejpam-7050	382	15	.	.	PUNCT
ejpam-7050	383	1	upper	upper	ADJ
ejpam-7050	383	2	and	and	CCONJ
ejpam-7050	383	3	lower	low	ADJ
ejpam-7050	383	4	(	(	PUNCT
ejpam-7050	383	5	τ1	τ1	NOUN
ejpam-7050	383	6	,	,	PUNCT
ejpam-7050	383	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7050	383	8	multifunctions	multifunction	NOUN
ejpam-7050	383	9	.	.	PUNCT
ejpam-7050	384	1	international	international	ADJ
ejpam-7050	384	2	journal	journal	PROPN
ejpam-7050	384	3	of	of	ADP
ejpam-7050	384	4	mathematics	mathematic	NOUN
ejpam-7050	384	5	and	and	CCONJ
ejpam-7050	384	6	computer	computer	NOUN
ejpam-7050	384	7	science	science	NOUN
ejpam-7050	384	8	,	,	PUNCT
ejpam-7050	384	9	19(4):1305	19(4):1305	NUM
ejpam-7050	384	10	–	–	PUNCT
ejpam-7050	384	11	1310	1310	NUM
ejpam-7050	384	12	,	,	PUNCT
ejpam-7050	384	13	2024	2024	NUM
ejpam-7050	384	14	.	.	PUNCT
ejpam-7050	385	1	[	[	X
ejpam-7050	385	2	12	12	NUM
ejpam-7050	385	3	]	]	PUNCT
ejpam-7050	385	4	m.	m.	NOUN
ejpam-7050	385	5	thongmoon	thongmoon	NOUN
ejpam-7050	385	6	,	,	PUNCT
ejpam-7050	385	7	s.	s.	PROPN
ejpam-7050	385	8	sompong	sompong	PROPN
ejpam-7050	385	9	,	,	PUNCT
ejpam-7050	385	10	and	and	CCONJ
ejpam-7050	385	11	c.	c.	PROPN
ejpam-7050	385	12	boonpok	boonpok	PROPN
ejpam-7050	385	13	.	.	PUNCT
ejpam-7050	386	1	uppr	uppr	NOUN
ejpam-7050	386	2	and	and	CCONJ
ejpam-7050	386	3	lower	low	ADJ
ejpam-7050	386	4	weak	weak	ADJ
ejpam-7050	386	5	(	(	PUNCT
ejpam-7050	386	6	τ1	τ1	NOUN
ejpam-7050	386	7	,	,	PUNCT
ejpam-7050	386	8	τ2)continuity	τ2)continuity	PROPN
ejpam-7050	386	9	.	.	PUNCT
ejpam-7050	387	1	european	european	PROPN
ejpam-7050	387	2	journal	journal	PROPN
ejpam-7050	387	3	of	of	ADP
ejpam-7050	387	4	pure	pure	ADJ
ejpam-7050	387	5	and	and	CCONJ
ejpam-7050	387	6	applied	applied	ADJ
ejpam-7050	387	7	mathematics	mathematic	NOUN
ejpam-7050	387	8	,	,	PUNCT
ejpam-7050	387	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-7050	387	10	,	,	PUNCT
ejpam-7050	387	11	2024	2024	NUM
ejpam-7050	387	12	.	.	PUNCT
ejpam-7050	388	1	[	[	X
ejpam-7050	388	2	13	13	NUM
ejpam-7050	388	3	]	]	PUNCT
ejpam-7050	388	4	c.	c.	PROPN
ejpam-7050	388	5	boonpok	boonpok	PROPN
ejpam-7050	388	6	,	,	PUNCT
ejpam-7050	388	7	c.	c.	PROPN
ejpam-7050	388	8	viriyapong	viriyapong	PROPN
ejpam-7050	388	9	,	,	PUNCT
ejpam-7050	388	10	and	and	CCONJ
ejpam-7050	388	11	m.	m.	NOUN
ejpam-7050	388	12	thongmoon	thongmoon	NOUN
ejpam-7050	388	13	.	.	PUNCT
ejpam-7050	389	1	on	on	ADP
ejpam-7050	389	2	upper	upper	ADJ
ejpam-7050	389	3	and	and	CCONJ
ejpam-7050	389	4	lower	low	ADJ
ejpam-7050	389	5	(	(	PUNCT
ejpam-7050	389	6	τ1	τ1	NOUN
ejpam-7050	389	7	,	,	PUNCT
ejpam-7050	389	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7050	389	9	multifunctions	multifunction	NOUN
ejpam-7050	389	10	.	.	PUNCT
ejpam-7050	390	1	journal	journal	PROPN
ejpam-7050	390	2	of	of	ADP
ejpam-7050	390	3	mathematics	mathematics	PROPN
ejpam-7050	390	4	and	and	CCONJ
ejpam-7050	390	5	computer	computer	NOUN
ejpam-7050	390	6	science	science	NOUN
ejpam-7050	390	7	,	,	PUNCT
ejpam-7050	390	8	18:282–293	18:282–293	NUM
ejpam-7050	390	9	,	,	PUNCT
ejpam-7050	390	10	2018	2018	NUM
ejpam-7050	390	11	.	.	PUNCT
ejpam-7050	391	1	[	[	X
ejpam-7050	391	2	14	14	NUM
ejpam-7050	391	3	]	]	X
ejpam-7050	391	4	c.	c.	PROPN
ejpam-7050	391	5	viriyapong	viriyapong	PROPN
ejpam-7050	391	6	and	and	CCONJ
ejpam-7050	391	7	c.	c.	PROPN
ejpam-7050	391	8	boonpok	boonpok	PROPN
ejpam-7050	391	9	.	.	PUNCT
ejpam-7050	392	1	(	(	PUNCT
ejpam-7050	392	2	τ1	τ1	NOUN
ejpam-7050	392	3	,	,	PUNCT
ejpam-7050	392	4	τ2)α	τ2)α	NOUN
ejpam-7050	392	5	-	-	PUNCT
ejpam-7050	392	6	continuity	continuity	NOUN
ejpam-7050	392	7	for	for	ADP
ejpam-7050	392	8	multifunctions	multifunction	NOUN
ejpam-7050	392	9	.	.	PUNCT
ejpam-7050	393	1	journal	journal	PROPN
ejpam-7050	393	2	of	of	ADP
ejpam-7050	393	3	mathematics	mathematic	NOUN
ejpam-7050	393	4	,	,	PUNCT
ejpam-7050	393	5	2020:6285763	2020:6285763	NUM
ejpam-7050	393	6	,	,	PUNCT
ejpam-7050	393	7	2020	2020	NUM
ejpam-7050	393	8	.	.	PUNCT
ejpam-7050	394	1	[	[	X
ejpam-7050	394	2	15	15	NUM
ejpam-7050	394	3	]	]	X
ejpam-7050	394	4	c.	c.	PROPN
ejpam-7050	394	5	boonpok	boonpok	PROPN
ejpam-7050	394	6	.	.	PUNCT
ejpam-7050	395	1	(	(	PUNCT
ejpam-7050	395	2	τ1	τ1	NOUN
ejpam-7050	395	3	,	,	PUNCT
ejpam-7050	395	4	τ2)δ	τ2)δ	ADJ
ejpam-7050	395	5	-	-	PUNCT
ejpam-7050	395	6	semicontinuous	semicontinuous	ADJ
ejpam-7050	395	7	multifunctions	multifunction	NOUN
ejpam-7050	395	8	.	.	PUNCT
ejpam-7050	396	1	heliyon	heliyon	NOUN
ejpam-7050	396	2	,	,	PUNCT
ejpam-7050	396	3	6	6	NUM
ejpam-7050	396	4	:	:	SYM
ejpam-7050	396	5	e05367	e05367	PROPN
ejpam-7050	396	6	,	,	PUNCT
ejpam-7050	396	7	2020	2020	NUM
ejpam-7050	396	8	.	.	PUNCT
ejpam-7050	397	1	[	[	X
ejpam-7050	397	2	16	16	NUM
ejpam-7050	397	3	]	]	X
ejpam-7050	397	4	p.	p.	NOUN
ejpam-7050	397	5	pue	pue	NOUN
ejpam-7050	397	6	-	-	PUNCT
ejpam-7050	397	7	on	on	ADP
ejpam-7050	397	8	,	,	PUNCT
ejpam-7050	397	9	s.	s.	PROPN
ejpam-7050	397	10	sompong	sompong	PROPN
ejpam-7050	397	11	,	,	PUNCT
ejpam-7050	397	12	and	and	CCONJ
ejpam-7050	397	13	c.	c.	PROPN
ejpam-7050	397	14	boonpok	boonpok	PROPN
ejpam-7050	397	15	.	.	PUNCT
ejpam-7050	398	1	almost	almost	ADV
ejpam-7050	398	2	quasi	quasi	X
ejpam-7050	398	3	(	(	PUNCT
ejpam-7050	398	4	τ1	τ1	NOUN
ejpam-7050	398	5	,	,	PUNCT
ejpam-7050	398	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7050	398	7	for	for	ADP
ejpam-7050	398	8	multifunctions	multifunction	NOUN
ejpam-7050	398	9	.	.	PUNCT
ejpam-7050	399	1	international	international	ADJ
ejpam-7050	399	2	journal	journal	NOUN
ejpam-7050	399	3	of	of	ADP
ejpam-7050	399	4	analysis	analysis	NOUN
ejpam-7050	399	5	and	and	CCONJ
ejpam-7050	399	6	applications	application	NOUN
ejpam-7050	399	7	,	,	PUNCT
ejpam-7050	399	8	22:97	22:97	NUM
ejpam-7050	399	9	,	,	PUNCT
ejpam-7050	399	10	2024	2024	NUM
ejpam-7050	399	11	.	.	PUNCT
ejpam-7050	400	1	[	[	X
ejpam-7050	400	2	17	17	NUM
ejpam-7050	400	3	]	]	PUNCT
ejpam-7050	400	4	k.	k.	PROPN
ejpam-7050	400	5	kuratowski	kuratowski	PROPN
ejpam-7050	400	6	.	.	PUNCT
ejpam-7050	401	1	topology	topology	PROPN
ejpam-7050	401	2	,	,	PUNCT
ejpam-7050	401	3	vol	vol	NOUN
ejpam-7050	401	4	.	.	PUNCT
ejpam-7050	401	5	i.	i.	PROPN
ejpam-7050	401	6	academic	academic	PROPN
ejpam-7050	401	7	press	press	PROPN
ejpam-7050	401	8	,	,	PUNCT
ejpam-7050	401	9	new	new	PROPN
ejpam-7050	401	10	york	york	PROPN
ejpam-7050	401	11	,	,	PUNCT
ejpam-7050	401	12	1966	1966	NUM
ejpam-7050	401	13	.	.	PUNCT
ejpam-7050	402	1	[	[	X
ejpam-7050	402	2	18	18	NUM
ejpam-7050	402	3	]	]	X
ejpam-7050	402	4	d.	d.	PROPN
ejpam-7050	402	5	janković	janković	PROPN
ejpam-7050	402	6	and	and	CCONJ
ejpam-7050	402	7	t.	t.	PROPN
ejpam-7050	402	8	r.	r.	PROPN
ejpam-7050	402	9	hamlett	hamlett	PROPN
ejpam-7050	402	10	.	.	PUNCT
ejpam-7050	403	1	new	new	ADJ
ejpam-7050	403	2	topologies	topology	NOUN
ejpam-7050	403	3	from	from	ADP
ejpam-7050	403	4	old	old	ADJ
ejpam-7050	403	5	via	via	ADP
ejpam-7050	403	6	ideals	ideal	NOUN
ejpam-7050	403	7	.	.	PUNCT
ejpam-7050	404	1	the	the	DET
ejpam-7050	404	2	american	american	PROPN
ejpam-7050	404	3	mathematical	mathematical	PROPN
ejpam-7050	404	4	monthly	monthly	ADV
ejpam-7050	404	5	,	,	PUNCT
ejpam-7050	404	6	97:295–310	97:295–310	PROPN
ejpam-7050	404	7	,	,	PUNCT
ejpam-7050	404	8	1990	1990	NUM
ejpam-7050	404	9	.	.	PUNCT
ejpam-7050	405	1	[	[	X
ejpam-7050	405	2	19	19	NUM
ejpam-7050	405	3	]	]	X
ejpam-7050	405	4	t.	t.	PROPN
ejpam-7050	405	5	noiri	noiri	PROPN
ejpam-7050	405	6	and	and	CCONJ
ejpam-7050	405	7	v.	v.	ADP
ejpam-7050	405	8	popa	popa	NOUN
ejpam-7050	405	9	.	.	PUNCT
ejpam-7050	406	1	on	on	ADP
ejpam-7050	406	2	(	(	PUNCT
ejpam-7050	406	3	mi	mi	ADJ
ejpam-7050	406	4	,	,	PUNCT
ejpam-7050	406	5	nj)-continuous	nj)-continuous	ADJ
ejpam-7050	406	6	multifunctions	multifunction	NOUN
ejpam-7050	406	7	.	.	PUNCT
ejpam-7050	407	1	romanian	romanian	ADJ
ejpam-7050	407	2	journal	journal	PROPN
ejpam-7050	407	3	of	of	ADP
ejpam-7050	407	4	mathematics	mathematics	PROPN
ejpam-7050	407	5	and	and	CCONJ
ejpam-7050	407	6	computer	computer	NOUN
ejpam-7050	407	7	science	science	NOUN
ejpam-7050	407	8	,	,	PUNCT
ejpam-7050	407	9	15(1):1–8	15(1):1–8	NUM
ejpam-7050	407	10	,	,	PUNCT
ejpam-7050	407	11	2025	2025	NUM
ejpam-7050	407	12	.	.	PUNCT
ejpam-7050	408	1	[	[	X
ejpam-7050	408	2	20	20	NUM
ejpam-7050	408	3	]	]	PUNCT
ejpam-7050	408	4	c.	c.	PROPN
ejpam-7050	408	5	boonpok	boonpok	PROPN
ejpam-7050	408	6	and	and	CCONJ
ejpam-7050	408	7	p.	p.	NOUN
ejpam-7050	408	8	pue	pue	NOUN
ejpam-7050	408	9	-	-	PUNCT
ejpam-7050	408	10	on	on	ADP
ejpam-7050	408	11	.	.	PUNCT
ejpam-7050	409	1	characterizations	characterization	NOUN
ejpam-7050	409	2	of	of	ADP
ejpam-7050	409	3	almost	almost	ADV
ejpam-7050	409	4	(	(	PUNCT
ejpam-7050	409	5	τ1	τ1	NOUN
ejpam-7050	409	6	,	,	PUNCT
ejpam-7050	409	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7050	409	8	multifunctions	multifunction	NOUN
ejpam-7050	409	9	.	.	PUNCT
ejpam-7050	410	1	international	international	ADJ
ejpam-7050	410	2	journal	journal	NOUN
ejpam-7050	410	3	of	of	ADP
ejpam-7050	410	4	analysis	analysis	NOUN
ejpam-7050	410	5	and	and	CCONJ
ejpam-7050	410	6	applications	application	NOUN
ejpam-7050	410	7	,	,	PUNCT
ejpam-7050	410	8	22:33	22:33	NUM
ejpam-7050	410	9	,	,	PUNCT
ejpam-7050	410	10	2024	2024	NUM
ejpam-7050	410	11	.	.	PUNCT
ejpam-7050	411	1	[	[	X
ejpam-7050	411	2	21	21	NUM
ejpam-7050	411	3	]	]	X
ejpam-7050	411	4	n.	n.	PROPN
ejpam-7050	411	5	srisarakham	srisarakham	PROPN
ejpam-7050	411	6	,	,	PUNCT
ejpam-7050	411	7	a.	a.	PROPN
ejpam-7050	411	8	sama	sama	PROPN
ejpam-7050	411	9	-	-	PUNCT
ejpam-7050	411	10	ae	ae	PROPN
ejpam-7050	411	11	,	,	PUNCT
ejpam-7050	411	12	and	and	CCONJ
ejpam-7050	411	13	c.	c.	PROPN
ejpam-7050	411	14	boonpok	boonpok	PROPN
ejpam-7050	411	15	.	.	PUNCT
ejpam-7050	412	1	almost	almost	ADV
ejpam-7050	412	2	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7050	412	3	,	,	PUNCT
ejpam-7050	412	4	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7050	412	5	for	for	ADP
ejpam-7050	412	6	multifunctions	multifunction	NOUN
ejpam-7050	412	7	.	.	PUNCT
ejpam-7050	413	1	(	(	PUNCT
ejpam-7050	413	2	submitted	submit	VERB
ejpam-7050	413	3	)	)	PUNCT
ejpam-7050	413	4	.	.	PUNCT
ejpam-7050	414	1	[	[	X
ejpam-7050	414	2	22	22	NUM
ejpam-7050	414	3	]	]	X
ejpam-7050	414	4	p.	p.	NOUN
ejpam-7050	414	5	pue	pue	NOUN
ejpam-7050	414	6	-	-	PUNCT
ejpam-7050	414	7	on	on	ADP
ejpam-7050	414	8	,	,	PUNCT
ejpam-7050	414	9	a.	a.	PROPN
ejpam-7050	414	10	sama	sama	PROPN
ejpam-7050	414	11	-	-	PUNCT
ejpam-7050	414	12	ae	ae	PROPN
ejpam-7050	414	13	,	,	PUNCT
ejpam-7050	414	14	and	and	CCONJ
ejpam-7050	414	15	c.	c.	PROPN
ejpam-7050	414	16	boonpok	boonpok	PROPN
ejpam-7050	414	17	.	.	PUNCT
ejpam-7050	415	1	upper	upper	ADJ
ejpam-7050	415	2	and	and	CCONJ
ejpam-7050	415	3	lower	low	ADJ
ejpam-7050	415	4	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7050	415	5	,	,	PUNCT
ejpam-7050	415	6	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7050	415	7	.	.	PUNCT
ejpam-7050	416	1	(	(	PUNCT
ejpam-7050	416	2	submitted	submit	VERB
ejpam-7050	416	3	)	)	PUNCT
ejpam-7050	416	4	.	.	PUNCT
ejpam-7050	417	1	[	[	X
ejpam-7050	417	2	23	23	NUM
ejpam-7050	417	3	]	]	PUNCT
ejpam-7050	417	4	m.	m.	NOUN
ejpam-7050	417	5	chiangpradit	chiangpradit	NOUN
ejpam-7050	417	6	,	,	PUNCT
ejpam-7050	417	7	s.	s.	PROPN
ejpam-7050	417	8	sompong	sompong	PROPN
ejpam-7050	417	9	,	,	PUNCT
ejpam-7050	417	10	and	and	CCONJ
ejpam-7050	417	11	c.	c.	PROPN
ejpam-7050	417	12	boonpok	boonpok	PROPN
ejpam-7050	417	13	.	.	PUNCT
ejpam-7050	418	1	on	on	ADP
ejpam-7050	418	2	characterizations	characterization	NOUN
ejpam-7050	418	3	of	of	ADP
ejpam-7050	418	4	(	(	PUNCT
ejpam-7050	418	5	τ1	τ1	NOUN
ejpam-7050	418	6	,	,	PUNCT
ejpam-7050	418	7	τ2)normal	τ2)normal	ADJ
ejpam-7050	418	8	spaces	space	NOUN
ejpam-7050	418	9	.	.	PUNCT
ejpam-7050	419	1	international	international	ADJ
ejpam-7050	419	2	journal	journal	PROPN
ejpam-7050	419	3	of	of	ADP
ejpam-7050	419	4	mathematics	mathematic	NOUN
ejpam-7050	419	5	and	and	CCONJ
ejpam-7050	419	6	computer	computer	NOUN
ejpam-7050	419	7	science	science	NOUN
ejpam-7050	419	8	,	,	PUNCT
ejpam-7050	419	9	19(4):1315–1320	19(4):1315–1320	NUM
ejpam-7050	419	10	,	,	PUNCT
ejpam-7050	419	11	2024	2024	NUM
ejpam-7050	419	12	.	.	PUNCT