id	sid	tid	token	lemma	pos
ejpam-7048	1	1	european	european	PROPN
ejpam-7048	1	2	journal	journal	PROPN
ejpam-7048	1	3	of	of	ADP
ejpam-7048	1	4	pure	pure	ADJ
ejpam-7048	1	5	and	and	CCONJ
ejpam-7048	1	6	applied	applied	ADJ
ejpam-7048	1	7	mathematics	mathematic	NOUN
ejpam-7048	1	8	2025	2025	NUM
ejpam-7048	1	9	,	,	PUNCT
ejpam-7048	1	10	vol	vol	NOUN
ejpam-7048	1	11	.	.	PROPN
ejpam-7048	1	12	18	18	NUM
ejpam-7048	1	13	,	,	PUNCT
ejpam-7048	1	14	issue	issue	NOUN
ejpam-7048	1	15	4	4	NUM
ejpam-7048	1	16	,	,	PUNCT
ejpam-7048	1	17	article	article	NOUN
ejpam-7048	1	18	number	number	NOUN
ejpam-7048	1	19	7048	7048	NUM
ejpam-7048	1	20	issn	issn	PROPN
ejpam-7048	1	21	1307	1307	NUM
ejpam-7048	1	22	-	-	SYM
ejpam-7048	1	23	5543	5543	NUM
ejpam-7048	1	24	–	–	PUNCT
ejpam-7048	1	25	ejpam.com	ejpam.com	X
ejpam-7048	1	26	published	publish	VERB
ejpam-7048	1	27	by	by	ADP
ejpam-7048	1	28	new	new	PROPN
ejpam-7048	1	29	york	york	PROPN
ejpam-7048	1	30	business	business	PROPN
ejpam-7048	1	31	global	global	PROPN
ejpam-7048	1	32	upper	upper	ADJ
ejpam-7048	1	33	and	and	CCONJ
ejpam-7048	1	34	lower	low	ADJ
ejpam-7048	1	35	τ	τ	X
ejpam-7048	1	36	⋆β(σ1	⋆β(σ1	NUM
ejpam-7048	1	37	,	,	PUNCT
ejpam-7048	1	38	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7048	1	39	prapart	prapart	VERB
ejpam-7048	1	40	pue	pue	NOUN
ejpam-7048	1	41	-	-	PUNCT
ejpam-7048	1	42	on1	on1	PROPN
ejpam-7048	1	43	,	,	PUNCT
ejpam-7048	1	44	areeyuth	areeyuth	NOUN
ejpam-7048	1	45	sama	sama	NOUN
ejpam-7048	1	46	-	-	PUNCT
ejpam-7048	1	47	ae2	ae2	PROPN
ejpam-7048	1	48	,	,	PUNCT
ejpam-7048	1	49	chawalit	chawalit	VERB
ejpam-7048	1	50	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7048	1	51	1	1	NUM
ejpam-7048	1	52	mathematics	mathematic	NOUN
ejpam-7048	1	53	and	and	CCONJ
ejpam-7048	1	54	applied	apply	VERB
ejpam-7048	1	55	mathematics	mathematics	PROPN
ejpam-7048	1	56	research	research	NOUN
ejpam-7048	1	57	unit	unit	NOUN
ejpam-7048	1	58	,	,	PUNCT
ejpam-7048	1	59	department	department	NOUN
ejpam-7048	1	60	of	of	ADP
ejpam-7048	1	61	mathematics	mathematic	NOUN
ejpam-7048	1	62	,	,	PUNCT
ejpam-7048	1	63	faculty	faculty	NOUN
ejpam-7048	1	64	of	of	ADP
ejpam-7048	1	65	science	science	NOUN
ejpam-7048	1	66	,	,	PUNCT
ejpam-7048	1	67	mahasarakham	mahasarakham	PROPN
ejpam-7048	1	68	university	university	PROPN
ejpam-7048	1	69	,	,	PUNCT
ejpam-7048	1	70	maha	maha	PROPN
ejpam-7048	1	71	sarakham	sarakham	PROPN
ejpam-7048	1	72	,	,	PUNCT
ejpam-7048	1	73	44150	44150	NUM
ejpam-7048	1	74	,	,	PUNCT
ejpam-7048	1	75	thailand	thailand	PROPN
ejpam-7048	1	76	2	2	NUM
ejpam-7048	1	77	department	department	NOUN
ejpam-7048	1	78	of	of	ADP
ejpam-7048	1	79	mathematics	mathematic	NOUN
ejpam-7048	1	80	and	and	CCONJ
ejpam-7048	1	81	computer	computer	NOUN
ejpam-7048	1	82	science	science	NOUN
ejpam-7048	1	83	,	,	PUNCT
ejpam-7048	1	84	faculty	faculty	NOUN
ejpam-7048	1	85	of	of	ADP
ejpam-7048	1	86	science	science	NOUN
ejpam-7048	1	87	and	and	CCONJ
ejpam-7048	1	88	technology	technology	NOUN
ejpam-7048	1	89	,	,	PUNCT
ejpam-7048	1	90	prince	prince	NOUN
ejpam-7048	1	91	of	of	ADP
ejpam-7048	1	92	songkla	songkla	PROPN
ejpam-7048	1	93	university	university	PROPN
ejpam-7048	1	94	,	,	PUNCT
ejpam-7048	1	95	pattani	pattani	NOUN
ejpam-7048	1	96	campus	campus	NOUN
ejpam-7048	1	97	,	,	PUNCT
ejpam-7048	1	98	pattani	pattani	NOUN
ejpam-7048	1	99	,	,	PUNCT
ejpam-7048	1	100	94000	94000	NUM
ejpam-7048	1	101	,	,	PUNCT
ejpam-7048	1	102	thailand	thailand	PROPN
ejpam-7048	1	103	abstract	abstract	PROPN
ejpam-7048	1	104	.	.	PUNCT
ejpam-7048	2	1	a	a	DET
ejpam-7048	2	2	new	new	ADJ
ejpam-7048	2	3	class	class	NOUN
ejpam-7048	2	4	of	of	ADP
ejpam-7048	2	5	continuous	continuous	ADJ
ejpam-7048	2	6	multifunctions	multifunction	NOUN
ejpam-7048	2	7	between	between	ADP
ejpam-7048	2	8	an	an	DET
ejpam-7048	2	9	ideal	ideal	ADJ
ejpam-7048	2	10	topological	topological	ADJ
ejpam-7048	2	11	space	space	NOUN
ejpam-7048	2	12	and	and	CCONJ
ejpam-7048	2	13	a	a	DET
ejpam-7048	2	14	bitopological	bitopological	ADJ
ejpam-7048	2	15	space	space	NOUN
ejpam-7048	2	16	,	,	PUNCT
ejpam-7048	2	17	called	call	VERB
ejpam-7048	2	18	upper	upper	ADJ
ejpam-7048	2	19	(	(	PUNCT
ejpam-7048	2	20	lower	low	ADJ
ejpam-7048	2	21	)	)	PUNCT
ejpam-7048	2	22	τ⋆β(σ1	τ⋆β(σ1	PROPN
ejpam-7048	2	23	,	,	PUNCT
ejpam-7048	2	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	2	25	multifunctions	multifunction	NOUN
ejpam-7048	2	26	,	,	PUNCT
ejpam-7048	2	27	has	have	AUX
ejpam-7048	2	28	been	be	AUX
ejpam-7048	2	29	defined	define	VERB
ejpam-7048	2	30	and	and	CCONJ
ejpam-7048	2	31	studied	study	VERB
ejpam-7048	2	32	.	.	PUNCT
ejpam-7048	3	1	furthermore	furthermore	ADV
ejpam-7048	3	2	,	,	PUNCT
ejpam-7048	3	3	several	several	ADJ
ejpam-7048	3	4	characterizations	characterization	NOUN
ejpam-7048	3	5	and	and	CCONJ
ejpam-7048	3	6	some	some	DET
ejpam-7048	3	7	properties	property	NOUN
ejpam-7048	3	8	concerning	concern	VERB
ejpam-7048	3	9	upper	upper	ADJ
ejpam-7048	3	10	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	3	11	,	,	PUNCT
ejpam-7048	3	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	3	13	multifunctions	multifunction	NOUN
ejpam-7048	3	14	and	and	CCONJ
ejpam-7048	3	15	lower	low	ADJ
ejpam-7048	3	16	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	3	17	,	,	PUNCT
ejpam-7048	3	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	3	19	multifunctions	multifunction	NOUN
ejpam-7048	3	20	are	be	AUX
ejpam-7048	3	21	discussed	discuss	VERB
ejpam-7048	3	22	.	.	PUNCT
ejpam-7048	4	1	2020	2020	NUM
ejpam-7048	4	2	mathematics	mathematic	NOUN
ejpam-7048	4	3	subject	subject	NOUN
ejpam-7048	4	4	classifications	classification	NOUN
ejpam-7048	4	5	:	:	PUNCT
ejpam-7048	4	6	54c08	54c08	NUM
ejpam-7048	4	7	,	,	PUNCT
ejpam-7048	4	8	54c60	54c60	NUM
ejpam-7048	4	9	key	key	ADJ
ejpam-7048	4	10	words	word	NOUN
ejpam-7048	4	11	and	and	CCONJ
ejpam-7048	4	12	phrases	phrase	NOUN
ejpam-7048	4	13	:	:	PUNCT
ejpam-7048	4	14	upper	upper	ADJ
ejpam-7048	4	15	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	4	16	,	,	PUNCT
ejpam-7048	4	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	4	18	multifunction	multifunction	NOUN
ejpam-7048	4	19	,	,	PUNCT
ejpam-7048	4	20	lower	low	ADJ
ejpam-7048	4	21	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	4	22	,	,	PUNCT
ejpam-7048	4	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	4	24	multifunction	multifunction	NOUN
ejpam-7048	4	25	1	1	NUM
ejpam-7048	4	26	.	.	PUNCT
ejpam-7048	4	27	introduction	introduction	NOUN
ejpam-7048	4	28	the	the	DET
ejpam-7048	4	29	notion	notion	NOUN
ejpam-7048	4	30	of	of	ADP
ejpam-7048	4	31	β	β	ADJ
ejpam-7048	4	32	-	-	ADJ
ejpam-7048	4	33	continuous	continuous	ADJ
ejpam-7048	4	34	functions	function	NOUN
ejpam-7048	4	35	was	be	AUX
ejpam-7048	4	36	introduced	introduce	VERB
ejpam-7048	4	37	by	by	ADP
ejpam-7048	4	38	abd	abd	PROPN
ejpam-7048	4	39	el	el	PROPN
ejpam-7048	4	40	-	-	PROPN
ejpam-7048	4	41	monsef	monsef	PROPN
ejpam-7048	4	42	et	et	PROPN
ejpam-7048	4	43	al	al	PROPN
ejpam-7048	4	44	.	.	PUNCT
ejpam-7048	5	1	[	[	X
ejpam-7048	5	2	1	1	NUM
ejpam-7048	5	3	]	]	PUNCT
ejpam-7048	5	4	.	.	PUNCT
ejpam-7048	6	1	borsík	borsík	VERB
ejpam-7048	6	2	and	and	CCONJ
ejpam-7048	6	3	doboš	doboš	ADJ
ejpam-7048	7	1	[	[	X
ejpam-7048	7	2	2	2	NUM
ejpam-7048	7	3	]	]	PUNCT
ejpam-7048	7	4	introduced	introduce	VERB
ejpam-7048	7	5	the	the	DET
ejpam-7048	7	6	concept	concept	NOUN
ejpam-7048	7	7	of	of	ADP
ejpam-7048	7	8	almost	almost	ADV
ejpam-7048	7	9	quasicontinuity	quasicontinuity	NOUN
ejpam-7048	7	10	which	which	PRON
ejpam-7048	7	11	is	be	AUX
ejpam-7048	7	12	weaker	weak	ADJ
ejpam-7048	7	13	than	than	ADP
ejpam-7048	7	14	that	that	PRON
ejpam-7048	7	15	of	of	ADP
ejpam-7048	7	16	quasicontinuity	quasicontinuity	NOUN
ejpam-7048	7	17	[	[	X
ejpam-7048	7	18	3	3	NUM
ejpam-7048	7	19	]	]	PUNCT
ejpam-7048	7	20	.	.	PUNCT
ejpam-7048	8	1	popa	popa	NOUN
ejpam-7048	8	2	and	and	CCONJ
ejpam-7048	8	3	noiri	noiri	ADV
ejpam-7048	9	1	[	[	X
ejpam-7048	9	2	4	4	X
ejpam-7048	9	3	]	]	PUNCT
ejpam-7048	9	4	investigated	investigate	VERB
ejpam-7048	9	5	several	several	ADJ
ejpam-7048	9	6	characterizations	characterization	NOUN
ejpam-7048	9	7	of	of	ADP
ejpam-7048	9	8	β	β	NOUN
ejpam-7048	9	9	-	-	NOUN
ejpam-7048	9	10	continuity	continuity	NOUN
ejpam-7048	9	11	and	and	CCONJ
ejpam-7048	9	12	shown	show	VERB
ejpam-7048	9	13	that	that	SCONJ
ejpam-7048	9	14	almost	almost	ADV
ejpam-7048	9	15	quasi	quasi	NOUN
ejpam-7048	9	16	-	-	NOUN
ejpam-7048	9	17	continuity	continuity	NOUN
ejpam-7048	9	18	is	be	AUX
ejpam-7048	9	19	equivalent	equivalent	ADJ
ejpam-7048	9	20	to	to	ADP
ejpam-7048	9	21	β	β	NOUN
ejpam-7048	9	22	-	-	NOUN
ejpam-7048	9	23	continuity	continuity	NOUN
ejpam-7048	9	24	.	.	PUNCT
ejpam-7048	10	1	the	the	DET
ejpam-7048	10	2	equivalence	equivalence	NOUN
ejpam-7048	10	3	of	of	ADP
ejpam-7048	10	4	almost	almost	ADV
ejpam-7048	10	5	quasicontinuity	quasicontinuity	NOUN
ejpam-7048	10	6	and	and	CCONJ
ejpam-7048	10	7	β	β	NOUN
ejpam-7048	10	8	-	-	NOUN
ejpam-7048	10	9	continuity	continuity	NOUN
ejpam-7048	10	10	is	be	AUX
ejpam-7048	10	11	also	also	ADV
ejpam-7048	10	12	shown	show	VERB
ejpam-7048	10	13	by	by	ADP
ejpam-7048	10	14	borsík	borsík	NOUN
ejpam-7048	10	15	[	[	X
ejpam-7048	10	16	5	5	NUM
ejpam-7048	10	17	]	]	PUNCT
ejpam-7048	10	18	and	and	CCONJ
ejpam-7048	10	19	ewert	ewert	X
ejpam-7048	11	1	[	[	X
ejpam-7048	11	2	6	6	NUM
ejpam-7048	11	3	]	]	PUNCT
ejpam-7048	11	4	.	.	PUNCT
ejpam-7048	12	1	in	in	ADP
ejpam-7048	12	2	1996	1996	NUM
ejpam-7048	12	3	-	-	SYM
ejpam-7048	12	4	1997	1997	NUM
ejpam-7048	12	5	,	,	PUNCT
ejpam-7048	12	6	popa	popa	NOUN
ejpam-7048	12	7	and	and	CCONJ
ejpam-7048	12	8	noiri	noiri	ADV
ejpam-7048	12	9	[	[	X
ejpam-7048	12	10	7	7	X
ejpam-7048	12	11	]	]	PUNCT
ejpam-7048	12	12	extended	extend	VERB
ejpam-7048	12	13	the	the	DET
ejpam-7048	12	14	concept	concept	NOUN
ejpam-7048	12	15	of	of	ADP
ejpam-7048	12	16	β	β	ADJ
ejpam-7048	12	17	-	-	ADJ
ejpam-7048	12	18	continuous	continuous	ADJ
ejpam-7048	12	19	functions	function	NOUN
ejpam-7048	12	20	to	to	ADP
ejpam-7048	12	21	multifunctions	multifunction	NOUN
ejpam-7048	12	22	and	and	CCONJ
ejpam-7048	12	23	presented	present	VERB
ejpam-7048	12	24	new	new	ADJ
ejpam-7048	12	25	classes	class	NOUN
ejpam-7048	12	26	of	of	ADP
ejpam-7048	12	27	multifunctions	multifunction	NOUN
ejpam-7048	12	28	defined	define	VERB
ejpam-7048	12	29	from	from	ADP
ejpam-7048	12	30	a	a	DET
ejpam-7048	12	31	topological	topological	ADJ
ejpam-7048	12	32	space	space	NOUN
ejpam-7048	12	33	into	into	ADP
ejpam-7048	12	34	a	a	DET
ejpam-7048	12	35	topological	topological	ADJ
ejpam-7048	12	36	space	space	NOUN
ejpam-7048	12	37	,	,	PUNCT
ejpam-7048	12	38	namely	namely	ADV
ejpam-7048	12	39	upper	upper	ADJ
ejpam-7048	12	40	β	β	ADJ
ejpam-7048	12	41	-	-	ADJ
ejpam-7048	12	42	continuous	continuous	ADJ
ejpam-7048	12	43	multifunctions	multifunction	NOUN
ejpam-7048	12	44	and	and	CCONJ
ejpam-7048	12	45	lower	low	ADJ
ejpam-7048	12	46	β	β	ADJ
ejpam-7048	12	47	-	-	ADJ
ejpam-7048	12	48	continuous	continuous	ADJ
ejpam-7048	12	49	multifunctions	multifunction	NOUN
ejpam-7048	12	50	.	.	PUNCT
ejpam-7048	13	1	moreover	moreover	ADV
ejpam-7048	13	2	,	,	PUNCT
ejpam-7048	13	3	popa	popa	NOUN
ejpam-7048	13	4	and	and	CCONJ
ejpam-7048	13	5	noiri	noiri	ADV
ejpam-7048	14	1	[	[	X
ejpam-7048	14	2	7	7	X
ejpam-7048	14	3	]	]	PUNCT
ejpam-7048	14	4	investigated	investigate	VERB
ejpam-7048	14	5	several	several	ADJ
ejpam-7048	14	6	characterizations	characterization	NOUN
ejpam-7048	14	7	and	and	CCONJ
ejpam-7048	14	8	some	some	DET
ejpam-7048	14	9	properties	property	NOUN
ejpam-7048	14	10	concerning	concern	VERB
ejpam-7048	14	11	upper	upper	ADJ
ejpam-7048	14	12	β	β	ADJ
ejpam-7048	14	13	-	-	ADJ
ejpam-7048	14	14	continuous	continuous	ADJ
ejpam-7048	14	15	multifunctions	multifunction	NOUN
ejpam-7048	14	16	and	and	CCONJ
ejpam-7048	14	17	lower	low	ADJ
ejpam-7048	14	18	β	β	ADJ
ejpam-7048	14	19	-	-	ADJ
ejpam-7048	14	20	continuous	continuous	ADJ
ejpam-7048	14	21	multifunctions	multifunction	NOUN
ejpam-7048	14	22	.	.	PUNCT
ejpam-7048	15	1	on	on	ADP
ejpam-7048	15	2	the	the	DET
ejpam-7048	15	3	other	other	ADJ
ejpam-7048	15	4	hand	hand	NOUN
ejpam-7048	15	5	,	,	PUNCT
ejpam-7048	15	6	the	the	DET
ejpam-7048	15	7	present	present	ADJ
ejpam-7048	15	8	author	author	NOUN
ejpam-7048	15	9	introduced	introduce	VERB
ejpam-7048	15	10	and	and	CCONJ
ejpam-7048	15	11	investigated	investigate	VERB
ejpam-7048	15	12	four	four	NUM
ejpam-7048	15	13	classes	class	NOUN
ejpam-7048	15	14	of	of	ADP
ejpam-7048	15	15	multifunctions	multifunction	NOUN
ejpam-7048	15	16	defined	define	VERB
ejpam-7048	15	17	from	from	ADP
ejpam-7048	15	18	an	an	DET
ejpam-7048	15	19	ideal	ideal	ADJ
ejpam-7048	15	20	topological	topological	ADJ
ejpam-7048	15	21	space	space	NOUN
ejpam-7048	15	22	into	into	ADP
ejpam-7048	15	23	an	an	DET
ejpam-7048	15	24	ideal	ideal	ADJ
ejpam-7048	15	25	topological	topological	ADJ
ejpam-7048	15	26	space	space	NOUN
ejpam-7048	15	27	,	,	PUNCT
ejpam-7048	15	28	namely	namely	ADV
ejpam-7048	15	29	upper	upper	ADJ
ejpam-7048	15	30	⋆-continuous	⋆-continuous	ADJ
ejpam-7048	15	31	multifunctions	multifunction	NOUN
ejpam-7048	16	1	[	[	X
ejpam-7048	16	2	8	8	NUM
ejpam-7048	16	3	]	]	PUNCT
ejpam-7048	16	4	,	,	PUNCT
ejpam-7048	16	5	lower	low	ADJ
ejpam-7048	16	6	⋆-continuous	⋆-continuous	ADJ
ejpam-7048	16	7	∗corresponding	∗corresponde	VERB
ejpam-7048	16	8	author	author	NOUN
ejpam-7048	16	9	.	.	PUNCT
ejpam-7048	17	1	doi	doi	NOUN
ejpam-7048	17	2	:	:	PUNCT
ejpam-7048	17	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7048	https://doi.org/10.29020/nybg.ejpam.v18i4.7048	VERB
ejpam-7048	17	4	email	email	NOUN
ejpam-7048	17	5	addresses	address	NOUN
ejpam-7048	17	6	:	:	PUNCT
ejpam-7048	17	7	prapart.p@msu.ac.th	prapart.p@msu.ac.th	PROPN
ejpam-7048	17	8	(	(	PUNCT
ejpam-7048	17	9	p.	p.	NOUN
ejpam-7048	17	10	pue	pue	NOUN
ejpam-7048	17	11	-	-	PUNCT
ejpam-7048	17	12	on	on	ADP
ejpam-7048	17	13	)	)	PUNCT
ejpam-7048	17	14	,	,	PUNCT
ejpam-7048	17	15	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7048	17	16	(	(	PUNCT
ejpam-7048	17	17	a.	a.	PROPN
ejpam-7048	17	18	sama	sama	PROPN
ejpam-7048	17	19	-	-	PUNCT
ejpam-7048	17	20	ae	ae	PROPN
ejpam-7048	17	21	)	)	PUNCT
ejpam-7048	17	22	,	,	PUNCT
ejpam-7048	17	23	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7048	17	24	(	(	PUNCT
ejpam-7048	17	25	c.	c.	PROPN
ejpam-7048	17	26	boonpok	boonpok	PROPN
ejpam-7048	17	27	)	)	PUNCT
ejpam-7048	17	28	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7048	17	29	1	1	NUM
ejpam-7048	17	30	copyright	copyright	NOUN
ejpam-7048	17	31	:	:	PUNCT
ejpam-7048	17	32	©	©	PROPN
ejpam-7048	17	33	2025	2025	NUM
ejpam-7048	17	34	the	the	DET
ejpam-7048	17	35	author(s	author(s	NOUN
ejpam-7048	17	36	)	)	PUNCT
ejpam-7048	17	37	.	.	PUNCT
ejpam-7048	18	1	(	(	PUNCT
ejpam-7048	18	2	cc	cc	NOUN
ejpam-7048	18	3	by	by	ADP
ejpam-7048	18	4	-	-	PUNCT
ejpam-7048	18	5	nc	nc	PROPN
ejpam-7048	18	6	4.0	4.0	NUM
ejpam-7048	18	7	)	)	PUNCT
ejpam-7048	18	8	p.	p.	NOUN
ejpam-7048	18	9	pue	pue	NOUN
ejpam-7048	18	10	-	-	PUNCT
ejpam-7048	18	11	on	on	ADP
ejpam-7048	18	12	,	,	PUNCT
ejpam-7048	18	13	a.	a.	PROPN
ejpam-7048	18	14	sama	sama	PROPN
ejpam-7048	18	15	-	-	PUNCT
ejpam-7048	18	16	ae	ae	PROPN
ejpam-7048	18	17	,	,	PUNCT
ejpam-7048	18	18	c.	c.	PROPN
ejpam-7048	18	19	boonpok	boonpok	PROPN
ejpam-7048	18	20	/	/	SYM
ejpam-7048	18	21	eur	eur	PROPN
ejpam-7048	18	22	.	.	PUNCT
ejpam-7048	19	1	j.	j.	PROPN
ejpam-7048	19	2	pure	pure	PROPN
ejpam-7048	19	3	appl	appl	PROPN
ejpam-7048	19	4	.	.	PROPN
ejpam-7048	19	5	math	math	PROPN
ejpam-7048	19	6	,	,	PUNCT
ejpam-7048	19	7	18	18	NUM
ejpam-7048	19	8	(	(	PUNCT
ejpam-7048	19	9	4	4	NUM
ejpam-7048	19	10	)	)	PUNCT
ejpam-7048	19	11	(	(	PUNCT
ejpam-7048	19	12	2025	2025	NUM
ejpam-7048	19	13	)	)	PUNCT
ejpam-7048	19	14	,	,	PUNCT
ejpam-7048	19	15	7048	7048	NUM
ejpam-7048	19	16	2	2	NUM
ejpam-7048	19	17	of	of	ADP
ejpam-7048	19	18	9	9	NUM
ejpam-7048	19	19	multifunctions	multifunction	NOUN
ejpam-7048	19	20	[	[	X
ejpam-7048	19	21	8	8	NUM
ejpam-7048	19	22	]	]	PUNCT
ejpam-7048	19	23	,	,	PUNCT
ejpam-7048	19	24	upper	upper	ADJ
ejpam-7048	19	25	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-7048	19	26	multifunctions	multifunction	NOUN
ejpam-7048	20	1	[	[	X
ejpam-7048	20	2	9	9	NUM
ejpam-7048	20	3	]	]	PUNCT
ejpam-7048	20	4	,	,	PUNCT
ejpam-7048	20	5	lower	low	ADJ
ejpam-7048	20	6	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-7048	20	7	multifunctions	multifunction	NOUN
ejpam-7048	21	1	[	[	X
ejpam-7048	21	2	9	9	NUM
ejpam-7048	21	3	]	]	PUNCT
ejpam-7048	21	4	,	,	PUNCT
ejpam-7048	21	5	upper	upper	ADJ
ejpam-7048	21	6	sβ(⋆)-continuous	sβ(⋆)-continuous	PROPN
ejpam-7048	21	7	multifunctions	multifunction	NOUN
ejpam-7048	21	8	[	[	X
ejpam-7048	21	9	10	10	NUM
ejpam-7048	21	10	]	]	PUNCT
ejpam-7048	21	11	,	,	PUNCT
ejpam-7048	21	12	lower	low	ADJ
ejpam-7048	21	13	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7048	21	14	multifunctions	multifunction	NOUN
ejpam-7048	21	15	[	[	X
ejpam-7048	21	16	10	10	NUM
ejpam-7048	21	17	]	]	PUNCT
ejpam-7048	21	18	,	,	PUNCT
ejpam-7048	21	19	upper	upper	ADJ
ejpam-7048	21	20	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7048	21	21	multifunctions	multifunction	NOUN
ejpam-7048	22	1	[	[	X
ejpam-7048	22	2	11	11	NUM
ejpam-7048	22	3	]	]	PUNCT
ejpam-7048	22	4	,	,	PUNCT
ejpam-7048	22	5	lower	low	ADJ
ejpam-7048	22	6	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7048	22	7	multifunctions	multifunction	NOUN
ejpam-7048	23	1	[	[	X
ejpam-7048	23	2	11	11	NUM
ejpam-7048	23	3	]	]	PUNCT
ejpam-7048	23	4	,	,	PUNCT
ejpam-7048	23	5	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-7048	23	6	multifunctions	multifunction	NOUN
ejpam-7048	24	1	[	[	X
ejpam-7048	24	2	12	12	NUM
ejpam-7048	24	3	]	]	PUNCT
ejpam-7048	24	4	and	and	CCONJ
ejpam-7048	24	5	pı	pı	ADJ
ejpam-7048	24	6	-	-	ADJ
ejpam-7048	24	7	continuous	continuous	ADJ
ejpam-7048	24	8	multifunctions	multifunction	NOUN
ejpam-7048	25	1	[	[	X
ejpam-7048	25	2	13	13	NUM
ejpam-7048	25	3	]	]	PUNCT
ejpam-7048	25	4	.	.	PUNCT
ejpam-7048	26	1	pueon	pueon	NOUN
ejpam-7048	26	2	et	et	PROPN
ejpam-7048	26	3	al	al	PROPN
ejpam-7048	26	4	.	.	PUNCT
ejpam-7048	27	1	[	[	X
ejpam-7048	27	2	14	14	NUM
ejpam-7048	27	3	]	]	PUNCT
ejpam-7048	27	4	introduced	introduce	VERB
ejpam-7048	27	5	and	and	CCONJ
ejpam-7048	27	6	studied	study	VERB
ejpam-7048	27	7	two	two	NUM
ejpam-7048	27	8	classes	class	NOUN
ejpam-7048	27	9	of	of	ADP
ejpam-7048	27	10	multifunctions	multifunction	NOUN
ejpam-7048	27	11	between	between	ADP
ejpam-7048	27	12	bitopological	bitopological	ADJ
ejpam-7048	27	13	spaces	space	NOUN
ejpam-7048	27	14	,	,	PUNCT
ejpam-7048	27	15	namely	namely	ADV
ejpam-7048	27	16	upper	upper	ADJ
ejpam-7048	27	17	(	(	PUNCT
ejpam-7048	27	18	τ1	τ1	NOUN
ejpam-7048	27	19	,	,	PUNCT
ejpam-7048	27	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	27	21	multifunctions	multifunction	NOUN
ejpam-7048	27	22	and	and	CCONJ
ejpam-7048	27	23	lower	low	ADJ
ejpam-7048	27	24	(	(	PUNCT
ejpam-7048	27	25	τ1	τ1	NOUN
ejpam-7048	27	26	,	,	PUNCT
ejpam-7048	27	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	27	28	multifunctions	multifunction	NOUN
ejpam-7048	27	29	.	.	PUNCT
ejpam-7048	28	1	klanarong	klanarong	NOUN
ejpam-7048	28	2	et	et	PROPN
ejpam-7048	28	3	al	al	PROPN
ejpam-7048	28	4	.	.	PUNCT
ejpam-7048	29	1	[	[	X
ejpam-7048	29	2	15	15	NUM
ejpam-7048	29	3	]	]	PUNCT
ejpam-7048	29	4	investigated	investigate	VERB
ejpam-7048	29	5	several	several	ADJ
ejpam-7048	29	6	characterizations	characterization	NOUN
ejpam-7048	29	7	of	of	ADP
ejpam-7048	29	8	upper	upper	ADJ
ejpam-7048	29	9	(	(	PUNCT
ejpam-7048	29	10	τ1	τ1	NOUN
ejpam-7048	29	11	,	,	PUNCT
ejpam-7048	29	12	τ2)continuous	τ2)continuous	ADJ
ejpam-7048	29	13	multifunctions	multifunction	NOUN
ejpam-7048	29	14	and	and	CCONJ
ejpam-7048	29	15	lower	low	ADJ
ejpam-7048	29	16	(	(	PUNCT
ejpam-7048	29	17	τ1	τ1	NOUN
ejpam-7048	29	18	,	,	PUNCT
ejpam-7048	29	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	29	20	multifunctions	multifunction	NOUN
ejpam-7048	29	21	by	by	ADP
ejpam-7048	29	22	utilizing	utilize	VERB
ejpam-7048	29	23	the	the	DET
ejpam-7048	29	24	notions	notion	NOUN
ejpam-7048	29	25	of	of	ADP
ejpam-7048	29	26	(	(	PUNCT
ejpam-7048	29	27	τ1	τ1	NOUN
ejpam-7048	29	28	,	,	PUNCT
ejpam-7048	29	29	τ2)θ	τ2)θ	ADJ
ejpam-7048	29	30	-	-	PUNCT
ejpam-7048	29	31	closed	close	VERB
ejpam-7048	29	32	sets	set	NOUN
ejpam-7048	29	33	and	and	CCONJ
ejpam-7048	29	34	(	(	PUNCT
ejpam-7048	29	35	τ1	τ1	NOUN
ejpam-7048	29	36	,	,	PUNCT
ejpam-7048	29	37	τ2)θ	τ2)θ	ADJ
ejpam-7048	29	38	-	-	PUNCT
ejpam-7048	29	39	open	open	ADJ
ejpam-7048	29	40	sets	set	NOUN
ejpam-7048	29	41	.	.	PUNCT
ejpam-7048	30	1	thongmoon	thongmoon	NOUN
ejpam-7048	30	2	et	et	PROPN
ejpam-7048	30	3	al	al	PROPN
ejpam-7048	30	4	.	.	PUNCT
ejpam-7048	31	1	[	[	X
ejpam-7048	31	2	16	16	NUM
ejpam-7048	31	3	]	]	PUNCT
ejpam-7048	31	4	studied	study	VERB
ejpam-7048	31	5	some	some	DET
ejpam-7048	31	6	characterizations	characterization	NOUN
ejpam-7048	31	7	of	of	ADP
ejpam-7048	31	8	upper	upper	ADJ
ejpam-7048	31	9	(	(	PUNCT
ejpam-7048	31	10	τ1	τ1	NOUN
ejpam-7048	31	11	,	,	PUNCT
ejpam-7048	31	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	31	13	multifunctions	multifunction	NOUN
ejpam-7048	31	14	and	and	CCONJ
ejpam-7048	31	15	lower	low	ADJ
ejpam-7048	31	16	(	(	PUNCT
ejpam-7048	31	17	τ1	τ1	NOUN
ejpam-7048	31	18	,	,	PUNCT
ejpam-7048	31	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	31	20	multifunctions	multifunction	NOUN
ejpam-7048	31	21	by	by	ADP
ejpam-7048	31	22	using	use	VERB
ejpam-7048	31	23	τ1τ2	τ1τ2	ADJ
ejpam-7048	31	24	-	-	ADJ
ejpam-7048	31	25	δ	δ	ADJ
ejpam-7048	31	26	-	-	ADJ
ejpam-7048	31	27	open	open	ADJ
ejpam-7048	31	28	sets	set	NOUN
ejpam-7048	31	29	and	and	CCONJ
ejpam-7048	31	30	τ1τ2	τ1τ2	NOUN
ejpam-7048	31	31	-	-	ADJ
ejpam-7048	31	32	δ	δ	NOUN
ejpam-7048	31	33	-	-	PUNCT
ejpam-7048	31	34	closed	close	VERB
ejpam-7048	31	35	sets	set	NOUN
ejpam-7048	31	36	.	.	PUNCT
ejpam-7048	32	1	laprom	laprom	ADP
ejpam-7048	32	2	et	et	PROPN
ejpam-7048	32	3	al	al	PROPN
ejpam-7048	32	4	.	.	PUNCT
ejpam-7048	33	1	[	[	X
ejpam-7048	33	2	17	17	NUM
ejpam-7048	33	3	]	]	PUNCT
ejpam-7048	33	4	introduced	introduce	VERB
ejpam-7048	33	5	and	and	CCONJ
ejpam-7048	33	6	investigated	investigate	VERB
ejpam-7048	33	7	the	the	DET
ejpam-7048	33	8	notions	notion	NOUN
ejpam-7048	33	9	of	of	ADP
ejpam-7048	33	10	upper	upper	ADJ
ejpam-7048	33	11	β(τ1	β(τ1	NOUN
ejpam-7048	33	12	,	,	PUNCT
ejpam-7048	33	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	33	14	multifunctions	multifunction	NOUN
ejpam-7048	33	15	and	and	CCONJ
ejpam-7048	33	16	lower	low	ADJ
ejpam-7048	33	17	β(τ1	β(τ1	NOUN
ejpam-7048	33	18	,	,	PUNCT
ejpam-7048	33	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	33	20	multifunctions	multifunction	NOUN
ejpam-7048	33	21	.	.	PUNCT
ejpam-7048	34	1	in	in	ADP
ejpam-7048	34	2	this	this	DET
ejpam-7048	34	3	paper	paper	NOUN
ejpam-7048	34	4	,	,	PUNCT
ejpam-7048	34	5	we	we	PRON
ejpam-7048	34	6	introduce	introduce	VERB
ejpam-7048	34	7	the	the	DET
ejpam-7048	34	8	concepts	concept	NOUN
ejpam-7048	34	9	of	of	ADP
ejpam-7048	34	10	continuous	continuous	ADJ
ejpam-7048	34	11	multifunctions	multifunction	NOUN
ejpam-7048	34	12	between	between	ADP
ejpam-7048	34	13	an	an	DET
ejpam-7048	34	14	ideal	ideal	ADJ
ejpam-7048	34	15	topological	topological	ADJ
ejpam-7048	34	16	space	space	NOUN
ejpam-7048	34	17	and	and	CCONJ
ejpam-7048	34	18	a	a	DET
ejpam-7048	34	19	bitopological	bitopological	ADJ
ejpam-7048	34	20	space	space	NOUN
ejpam-7048	34	21	,	,	PUNCT
ejpam-7048	34	22	called	call	VERB
ejpam-7048	34	23	upper	upper	ADJ
ejpam-7048	34	24	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	34	25	,	,	PUNCT
ejpam-7048	34	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	34	27	multifunctions	multifunction	NOUN
ejpam-7048	34	28	and	and	CCONJ
ejpam-7048	34	29	lower	low	ADJ
ejpam-7048	34	30	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	34	31	,	,	PUNCT
ejpam-7048	34	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	34	33	multifunctions	multifunction	NOUN
ejpam-7048	34	34	.	.	PUNCT
ejpam-7048	35	1	we	we	PRON
ejpam-7048	35	2	also	also	ADV
ejpam-7048	35	3	investigate	investigate	VERB
ejpam-7048	35	4	several	several	ADJ
ejpam-7048	35	5	characterizations	characterization	NOUN
ejpam-7048	35	6	of	of	ADP
ejpam-7048	35	7	upper	upper	ADJ
ejpam-7048	35	8	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	35	9	,	,	PUNCT
ejpam-7048	35	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	35	11	multifunctions	multifunction	NOUN
ejpam-7048	35	12	and	and	CCONJ
ejpam-7048	35	13	lower	low	ADJ
ejpam-7048	35	14	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	35	15	,	,	PUNCT
ejpam-7048	35	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	35	17	multifunctions	multifunction	NOUN
ejpam-7048	35	18	.	.	PUNCT
ejpam-7048	36	1	2	2	X
ejpam-7048	36	2	.	.	X
ejpam-7048	36	3	preliminaries	preliminary	NOUN
ejpam-7048	36	4	throughout	throughout	ADP
ejpam-7048	36	5	the	the	DET
ejpam-7048	36	6	present	present	ADJ
ejpam-7048	36	7	paper	paper	NOUN
ejpam-7048	36	8	,	,	PUNCT
ejpam-7048	36	9	spaces	space	NOUN
ejpam-7048	36	10	(	(	PUNCT
ejpam-7048	36	11	x	x	NOUN
ejpam-7048	36	12	,	,	PUNCT
ejpam-7048	36	13	τ1	τ1	NOUN
ejpam-7048	36	14	,	,	PUNCT
ejpam-7048	36	15	τ2	τ2	NOUN
ejpam-7048	36	16	)	)	PUNCT
ejpam-7048	36	17	and	and	CCONJ
ejpam-7048	36	18	(	(	PUNCT
ejpam-7048	36	19	y	y	PROPN
ejpam-7048	36	20	,	,	PUNCT
ejpam-7048	36	21	σ1	σ1	PROPN
ejpam-7048	36	22	,	,	PUNCT
ejpam-7048	36	23	σ2	σ2	NOUN
ejpam-7048	36	24	)	)	PUNCT
ejpam-7048	36	25	(	(	PUNCT
ejpam-7048	36	26	or	or	CCONJ
ejpam-7048	36	27	simply	simply	ADV
ejpam-7048	36	28	x	x	X
ejpam-7048	36	29	and	and	CCONJ
ejpam-7048	36	30	y	y	PROPN
ejpam-7048	36	31	)	)	PUNCT
ejpam-7048	36	32	always	always	ADV
ejpam-7048	36	33	mean	mean	VERB
ejpam-7048	36	34	bitopological	bitopological	ADJ
ejpam-7048	36	35	spaces	space	NOUN
ejpam-7048	36	36	on	on	ADP
ejpam-7048	36	37	which	which	PRON
ejpam-7048	36	38	no	no	DET
ejpam-7048	36	39	separation	separation	NOUN
ejpam-7048	36	40	axioms	axiom	NOUN
ejpam-7048	36	41	are	be	AUX
ejpam-7048	36	42	assumed	assume	VERB
ejpam-7048	36	43	unless	unless	SCONJ
ejpam-7048	36	44	explicitly	explicitly	ADV
ejpam-7048	36	45	stated	state	VERB
ejpam-7048	36	46	.	.	PUNCT
ejpam-7048	37	1	let	let	VERB
ejpam-7048	37	2	a	a	DET
ejpam-7048	37	3	be	be	AUX
ejpam-7048	37	4	a	a	DET
ejpam-7048	37	5	subset	subset	NOUN
ejpam-7048	37	6	of	of	ADP
ejpam-7048	37	7	a	a	DET
ejpam-7048	37	8	bitopological	bitopological	ADJ
ejpam-7048	37	9	space	space	NOUN
ejpam-7048	37	10	(	(	PUNCT
ejpam-7048	37	11	x	x	NOUN
ejpam-7048	37	12	,	,	PUNCT
ejpam-7048	37	13	τ1	τ1	NOUN
ejpam-7048	37	14	,	,	PUNCT
ejpam-7048	37	15	τ2	τ2	NOUN
ejpam-7048	37	16	)	)	PUNCT
ejpam-7048	37	17	.	.	PUNCT
ejpam-7048	38	1	the	the	DET
ejpam-7048	38	2	closure	closure	NOUN
ejpam-7048	38	3	of	of	ADP
ejpam-7048	38	4	a	a	PRON
ejpam-7048	38	5	and	and	CCONJ
ejpam-7048	38	6	the	the	DET
ejpam-7048	38	7	interior	interior	NOUN
ejpam-7048	38	8	of	of	ADP
ejpam-7048	38	9	a	a	PRON
ejpam-7048	38	10	with	with	ADP
ejpam-7048	38	11	respect	respect	NOUN
ejpam-7048	38	12	to	to	ADP
ejpam-7048	38	13	τi	τi	PROPN
ejpam-7048	38	14	are	be	AUX
ejpam-7048	38	15	denoted	denote	VERB
ejpam-7048	38	16	by	by	ADP
ejpam-7048	38	17	τi	τi	NOUN
ejpam-7048	38	18	-	-	PUNCT
ejpam-7048	38	19	cl(a	cl(a	NUM
ejpam-7048	38	20	)	)	PUNCT
ejpam-7048	38	21	and	and	CCONJ
ejpam-7048	38	22	τi	τi	NOUN
ejpam-7048	38	23	-	-	PUNCT
ejpam-7048	38	24	int(a	int(a	NOUN
ejpam-7048	38	25	)	)	PUNCT
ejpam-7048	38	26	,	,	PUNCT
ejpam-7048	38	27	respectively	respectively	ADV
ejpam-7048	38	28	,	,	PUNCT
ejpam-7048	38	29	for	for	ADP
ejpam-7048	38	30	i	i	PROPN
ejpam-7048	38	31	=	=	SYM
ejpam-7048	38	32	1	1	NUM
ejpam-7048	38	33	,	,	PUNCT
ejpam-7048	38	34	2	2	NUM
ejpam-7048	38	35	.	.	X
ejpam-7048	38	36	a	a	DET
ejpam-7048	38	37	subset	subset	NOUN
ejpam-7048	38	38	a	a	PRON
ejpam-7048	38	39	of	of	ADP
ejpam-7048	38	40	a	a	DET
ejpam-7048	38	41	bitopological	bitopological	ADJ
ejpam-7048	38	42	space	space	NOUN
ejpam-7048	38	43	(	(	PUNCT
ejpam-7048	38	44	x	x	NOUN
ejpam-7048	38	45	,	,	PUNCT
ejpam-7048	38	46	τ1	τ1	NOUN
ejpam-7048	38	47	,	,	PUNCT
ejpam-7048	38	48	τ2	τ2	NOUN
ejpam-7048	38	49	)	)	PUNCT
ejpam-7048	38	50	is	be	AUX
ejpam-7048	38	51	called	call	VERB
ejpam-7048	38	52	τ1τ2	τ1τ2	VERB
ejpam-7048	38	53	-	-	ADJ
ejpam-7048	38	54	closed	closed	ADJ
ejpam-7048	38	55	[	[	X
ejpam-7048	38	56	18	18	NUM
ejpam-7048	38	57	]	]	PUNCT
ejpam-7048	38	58	if	if	SCONJ
ejpam-7048	38	59	a	a	DET
ejpam-7048	38	60	=	=	NOUN
ejpam-7048	38	61	τ1	τ1	NOUN
ejpam-7048	38	62	-	-	PUNCT
ejpam-7048	38	63	cl(τ2	cl(τ2	NOUN
ejpam-7048	38	64	-	-	PUNCT
ejpam-7048	38	65	cl(a	cl(a	NUM
ejpam-7048	38	66	)	)	PUNCT
ejpam-7048	38	67	)	)	PUNCT
ejpam-7048	38	68	.	.	PUNCT
ejpam-7048	39	1	the	the	DET
ejpam-7048	39	2	complement	complement	NOUN
ejpam-7048	39	3	of	of	ADP
ejpam-7048	39	4	a	a	DET
ejpam-7048	39	5	τ1τ2	τ1τ2	ADJ
ejpam-7048	39	6	-	-	ADJ
ejpam-7048	39	7	closed	closed	ADJ
ejpam-7048	39	8	set	set	NOUN
ejpam-7048	39	9	is	be	AUX
ejpam-7048	39	10	called	call	VERB
ejpam-7048	39	11	τ1τ2	τ1τ2	NOUN
ejpam-7048	39	12	-	-	ADJ
ejpam-7048	39	13	open	open	ADJ
ejpam-7048	39	14	.	.	PUNCT
ejpam-7048	40	1	the	the	DET
ejpam-7048	40	2	intersection	intersection	NOUN
ejpam-7048	40	3	of	of	ADP
ejpam-7048	40	4	all	all	DET
ejpam-7048	40	5	τ1τ2	τ1τ2	ADJ
ejpam-7048	40	6	-	-	ADJ
ejpam-7048	40	7	closed	closed	ADJ
ejpam-7048	40	8	sets	set	NOUN
ejpam-7048	40	9	of	of	ADP
ejpam-7048	40	10	x	x	PUNCT
ejpam-7048	40	11	containing	contain	VERB
ejpam-7048	40	12	a	a	PRON
ejpam-7048	40	13	is	be	AUX
ejpam-7048	40	14	called	call	VERB
ejpam-7048	40	15	the	the	DET
ejpam-7048	40	16	τ1τ2	τ1τ2	NOUN
ejpam-7048	40	17	-	-	NOUN
ejpam-7048	40	18	closure	closure	NOUN
ejpam-7048	40	19	[	[	X
ejpam-7048	40	20	18	18	NUM
ejpam-7048	40	21	]	]	PUNCT
ejpam-7048	40	22	of	of	ADP
ejpam-7048	40	23	a	a	PRON
ejpam-7048	40	24	and	and	CCONJ
ejpam-7048	40	25	is	be	AUX
ejpam-7048	40	26	denoted	denote	VERB
ejpam-7048	40	27	by	by	ADP
ejpam-7048	40	28	τ1τ2	τ1τ2	NOUN
ejpam-7048	40	29	-	-	NUM
ejpam-7048	40	30	cl(a	cl(a	NUM
ejpam-7048	40	31	)	)	PUNCT
ejpam-7048	40	32	.	.	PUNCT
ejpam-7048	41	1	the	the	DET
ejpam-7048	41	2	union	union	NOUN
ejpam-7048	41	3	of	of	ADP
ejpam-7048	41	4	all	all	DET
ejpam-7048	41	5	τ1τ2	τ1τ2	ADJ
ejpam-7048	41	6	-	-	ADJ
ejpam-7048	41	7	open	open	ADJ
ejpam-7048	41	8	sets	set	NOUN
ejpam-7048	41	9	of	of	ADP
ejpam-7048	41	10	x	x	PUNCT
ejpam-7048	41	11	contained	contain	VERB
ejpam-7048	41	12	in	in	ADP
ejpam-7048	41	13	a	a	PRON
ejpam-7048	41	14	is	be	AUX
ejpam-7048	41	15	called	call	VERB
ejpam-7048	41	16	the	the	DET
ejpam-7048	41	17	τ1τ2	τ1τ2	NOUN
ejpam-7048	41	18	-	-	ADJ
ejpam-7048	41	19	interior	interior	ADJ
ejpam-7048	41	20	[	[	X
ejpam-7048	41	21	18	18	NUM
ejpam-7048	41	22	]	]	PUNCT
ejpam-7048	41	23	of	of	ADP
ejpam-7048	41	24	a	a	PRON
ejpam-7048	41	25	and	and	CCONJ
ejpam-7048	41	26	is	be	AUX
ejpam-7048	41	27	denoted	denote	VERB
ejpam-7048	41	28	by	by	ADP
ejpam-7048	41	29	τ1τ2	τ1τ2	NOUN
ejpam-7048	41	30	-	-	ADJ
ejpam-7048	41	31	int(a	int(a	NOUN
ejpam-7048	41	32	)	)	PUNCT
ejpam-7048	41	33	.	.	PUNCT
ejpam-7048	42	1	lemma	lemma	PROPN
ejpam-7048	42	2	1	1	NUM
ejpam-7048	42	3	.	.	PUNCT
ejpam-7048	43	1	[	[	X
ejpam-7048	43	2	18	18	NUM
ejpam-7048	43	3	]	]	PUNCT
ejpam-7048	43	4	let	let	VERB
ejpam-7048	43	5	a	a	PRON
ejpam-7048	43	6	and	and	CCONJ
ejpam-7048	43	7	b	b	NOUN
ejpam-7048	43	8	be	be	AUX
ejpam-7048	43	9	subsets	subset	NOUN
ejpam-7048	43	10	of	of	ADP
ejpam-7048	43	11	a	a	DET
ejpam-7048	43	12	bitopological	bitopological	ADJ
ejpam-7048	43	13	space	space	NOUN
ejpam-7048	43	14	(	(	PUNCT
ejpam-7048	43	15	x	x	NOUN
ejpam-7048	43	16	,	,	PUNCT
ejpam-7048	43	17	τ1	τ1	NOUN
ejpam-7048	43	18	,	,	PUNCT
ejpam-7048	43	19	τ2	τ2	NOUN
ejpam-7048	43	20	)	)	PUNCT
ejpam-7048	43	21	.	.	PUNCT
ejpam-7048	44	1	for	for	ADP
ejpam-7048	44	2	the	the	DET
ejpam-7048	44	3	τ1τ2	τ1τ2	NOUN
ejpam-7048	44	4	-	-	NOUN
ejpam-7048	44	5	closure	closure	NOUN
ejpam-7048	44	6	,	,	PUNCT
ejpam-7048	44	7	the	the	DET
ejpam-7048	44	8	following	follow	VERB
ejpam-7048	44	9	properties	property	NOUN
ejpam-7048	44	10	hold	hold	VERB
ejpam-7048	44	11	:	:	PUNCT
ejpam-7048	44	12	(	(	PUNCT
ejpam-7048	44	13	1	1	X
ejpam-7048	44	14	)	)	PUNCT
ejpam-7048	44	15	a	a	DET
ejpam-7048	44	16	⊆	⊆	NUM
ejpam-7048	44	17	τ1τ2	τ1τ2	NOUN
ejpam-7048	44	18	-	-	NUM
ejpam-7048	44	19	cl(a	cl(a	NUM
ejpam-7048	44	20	)	)	PUNCT
ejpam-7048	44	21	and	and	CCONJ
ejpam-7048	44	22	τ1τ2	τ1τ2	NOUN
ejpam-7048	44	23	-	-	ADJ
ejpam-7048	44	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7048	44	25	-	-	PUNCT
ejpam-7048	44	26	cl(a	cl(a	NUM
ejpam-7048	44	27	)	)	PUNCT
ejpam-7048	44	28	)	)	PUNCT
ejpam-7048	45	1	=	=	PUNCT
ejpam-7048	45	2	τ1τ2	τ1τ2	NOUN
ejpam-7048	45	3	-	-	NUM
ejpam-7048	45	4	cl(a	cl(a	NUM
ejpam-7048	45	5	)	)	PUNCT
ejpam-7048	45	6	.	.	PUNCT
ejpam-7048	46	1	(	(	PUNCT
ejpam-7048	46	2	2	2	X
ejpam-7048	46	3	)	)	PUNCT
ejpam-7048	46	4	if	if	SCONJ
ejpam-7048	46	5	a	a	DET
ejpam-7048	46	6	⊆	⊆	NUM
ejpam-7048	46	7	b	b	NOUN
ejpam-7048	46	8	,	,	PUNCT
ejpam-7048	46	9	then	then	ADV
ejpam-7048	46	10	τ1τ2	τ1τ2	NOUN
ejpam-7048	46	11	-	-	NUM
ejpam-7048	46	12	cl(a	cl(a	NUM
ejpam-7048	46	13	)	)	PUNCT
ejpam-7048	46	14	⊆	⊆	NUM
ejpam-7048	46	15	τ1τ2	τ1τ2	NOUN
ejpam-7048	46	16	-	-	NOUN
ejpam-7048	46	17	cl(b	cl(b	NOUN
ejpam-7048	46	18	)	)	PUNCT
ejpam-7048	46	19	.	.	PUNCT
ejpam-7048	47	1	(	(	PUNCT
ejpam-7048	47	2	3	3	X
ejpam-7048	47	3	)	)	PUNCT
ejpam-7048	47	4	τ1τ2	τ1τ2	NOUN
ejpam-7048	47	5	-	-	NUM
ejpam-7048	47	6	cl(a	cl(a	NUM
ejpam-7048	47	7	)	)	PUNCT
ejpam-7048	47	8	is	be	AUX
ejpam-7048	47	9	τ1τ2	τ1τ2	NOUN
ejpam-7048	47	10	-	-	ADJ
ejpam-7048	47	11	closed	closed	ADJ
ejpam-7048	47	12	.	.	PUNCT
ejpam-7048	48	1	(	(	PUNCT
ejpam-7048	48	2	4	4	X
ejpam-7048	48	3	)	)	PUNCT
ejpam-7048	48	4	a	a	PRON
ejpam-7048	48	5	is	be	AUX
ejpam-7048	48	6	τ1τ2	τ1τ2	NOUN
ejpam-7048	48	7	-	-	ADJ
ejpam-7048	48	8	closed	closed	ADJ
ejpam-7048	48	9	if	if	SCONJ
ejpam-7048	48	10	and	and	CCONJ
ejpam-7048	48	11	only	only	ADV
ejpam-7048	48	12	if	if	SCONJ
ejpam-7048	48	13	a	a	DET
ejpam-7048	48	14	=	=	PUNCT
ejpam-7048	48	15	τ1τ2	τ1τ2	NOUN
ejpam-7048	48	16	-	-	NUM
ejpam-7048	48	17	cl(a	cl(a	NUM
ejpam-7048	48	18	)	)	PUNCT
ejpam-7048	48	19	.	.	PUNCT
ejpam-7048	49	1	(	(	PUNCT
ejpam-7048	49	2	5	5	X
ejpam-7048	49	3	)	)	PUNCT
ejpam-7048	49	4	τ1τ2	τ1τ2	NOUN
ejpam-7048	49	5	-	-	NOUN
ejpam-7048	49	6	cl(x	cl(x	X
ejpam-7048	49	7	−a	−a	NOUN
ejpam-7048	49	8	)	)	PUNCT
ejpam-7048	50	1	=	=	PUNCT
ejpam-7048	50	2	x	x	X
ejpam-7048	51	1	−	−	ADP
ejpam-7048	51	2	τ1τ2	τ1τ2	NOUN
ejpam-7048	51	3	-	-	PUNCT
ejpam-7048	51	4	int(a	int(a	NOUN
ejpam-7048	51	5	)	)	PUNCT
ejpam-7048	51	6	.	.	PUNCT
ejpam-7048	52	1	a	a	DET
ejpam-7048	52	2	subset	subset	NOUN
ejpam-7048	52	3	a	a	PRON
ejpam-7048	52	4	of	of	ADP
ejpam-7048	52	5	a	a	DET
ejpam-7048	52	6	bitopological	bitopological	ADJ
ejpam-7048	52	7	space	space	NOUN
ejpam-7048	52	8	(	(	PUNCT
ejpam-7048	52	9	x	x	NOUN
ejpam-7048	52	10	,	,	PUNCT
ejpam-7048	52	11	τ1	τ1	NOUN
ejpam-7048	52	12	,	,	PUNCT
ejpam-7048	52	13	τ2	τ2	NOUN
ejpam-7048	52	14	)	)	PUNCT
ejpam-7048	52	15	is	be	AUX
ejpam-7048	52	16	said	say	VERB
ejpam-7048	52	17	to	to	PART
ejpam-7048	52	18	be	be	AUX
ejpam-7048	52	19	(	(	PUNCT
ejpam-7048	52	20	τ1	τ1	NOUN
ejpam-7048	52	21	,	,	PUNCT
ejpam-7048	52	22	τ2)r	τ2)r	NOUN
ejpam-7048	52	23	-	-	PUNCT
ejpam-7048	52	24	open	open	NOUN
ejpam-7048	53	1	[	[	X
ejpam-7048	53	2	19	19	NUM
ejpam-7048	53	3	]	]	PUNCT
ejpam-7048	53	4	(	(	PUNCT
ejpam-7048	53	5	resp	resp	NOUN
ejpam-7048	53	6	.	.	PUNCT
ejpam-7048	54	1	(	(	PUNCT
ejpam-7048	54	2	τ1	τ1	NOUN
ejpam-7048	54	3	,	,	PUNCT
ejpam-7048	54	4	τ2)s	τ2)s	NOUN
ejpam-7048	54	5	-	-	PUNCT
ejpam-7048	54	6	open	open	ADJ
ejpam-7048	54	7	[	[	X
ejpam-7048	54	8	20	20	NUM
ejpam-7048	54	9	]	]	PUNCT
ejpam-7048	54	10	,	,	PUNCT
ejpam-7048	54	11	(	(	PUNCT
ejpam-7048	54	12	τ1	τ1	NOUN
ejpam-7048	54	13	,	,	PUNCT
ejpam-7048	54	14	τ2)p	τ2)p	NOUN
ejpam-7048	54	15	-	-	ADJ
ejpam-7048	54	16	open	open	ADJ
ejpam-7048	54	17	[	[	X
ejpam-7048	54	18	20	20	NUM
ejpam-7048	54	19	]	]	PUNCT
ejpam-7048	54	20	,	,	PUNCT
ejpam-7048	54	21	(	(	PUNCT
ejpam-7048	54	22	τ1	τ1	NOUN
ejpam-7048	54	23	,	,	PUNCT
ejpam-7048	54	24	τ2)β	τ2)β	ADJ
ejpam-7048	54	25	-	-	PUNCT
ejpam-7048	54	26	open	open	NOUN
ejpam-7048	54	27	[	[	X
ejpam-7048	54	28	20	20	NUM
ejpam-7048	54	29	]	]	SYM
ejpam-7048	54	30	)	)	PUNCT
ejpam-7048	54	31	if	if	SCONJ
ejpam-7048	54	32	a	a	DET
ejpam-7048	54	33	=	=	PUNCT
ejpam-7048	54	34	τ1τ2	τ1τ2	NOUN
ejpam-7048	54	35	-	-	NOUN
ejpam-7048	54	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7048	54	37	-	-	PUNCT
ejpam-7048	54	38	cl(a	cl(a	NUM
ejpam-7048	54	39	)	)	PUNCT
ejpam-7048	54	40	)	)	PUNCT
ejpam-7048	54	41	(	(	PUNCT
ejpam-7048	54	42	resp	resp	NOUN
ejpam-7048	54	43	.	.	PUNCT
ejpam-7048	55	1	a	a	DET
ejpam-7048	55	2	⊆	⊆	NUM
ejpam-7048	55	3	τ1τ2	τ1τ2	NOUN
ejpam-7048	55	4	-	-	ADJ
ejpam-7048	55	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7048	55	6	-	-	PUNCT
ejpam-7048	55	7	int(a	int(a	NOUN
ejpam-7048	55	8	)	)	PUNCT
ejpam-7048	55	9	)	)	PUNCT
ejpam-7048	55	10	,	,	PUNCT
ejpam-7048	55	11	a	a	DET
ejpam-7048	55	12	⊆	⊆	NUM
ejpam-7048	55	13	τ1τ2	τ1τ2	NOUN
ejpam-7048	55	14	-	-	NOUN
ejpam-7048	55	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7048	55	16	-	-	PUNCT
ejpam-7048	55	17	cl(a	cl(a	NUM
ejpam-7048	55	18	)	)	PUNCT
ejpam-7048	55	19	)	)	PUNCT
ejpam-7048	55	20	,	,	PUNCT
ejpam-7048	55	21	a	a	DET
ejpam-7048	55	22	⊆	⊆	NUM
ejpam-7048	55	23	τ1τ2	τ1τ2	NOUN
ejpam-7048	55	24	-	-	PUNCT
ejpam-7048	55	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7048	55	26	-	-	PUNCT
ejpam-7048	55	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7048	55	28	-	-	PUNCT
ejpam-7048	55	29	cl(a	cl(a	NUM
ejpam-7048	55	30	)	)	PUNCT
ejpam-7048	55	31	)	)	PUNCT
ejpam-7048	55	32	)	)	PUNCT
ejpam-7048	55	33	)	)	PUNCT
ejpam-7048	55	34	.	.	PUNCT
ejpam-7048	56	1	p.	p.	NOUN
ejpam-7048	56	2	pue	pue	NOUN
ejpam-7048	56	3	-	-	PUNCT
ejpam-7048	56	4	on	on	ADP
ejpam-7048	56	5	,	,	PUNCT
ejpam-7048	56	6	a.	a.	PROPN
ejpam-7048	56	7	sama	sama	PROPN
ejpam-7048	56	8	-	-	PUNCT
ejpam-7048	56	9	ae	ae	PROPN
ejpam-7048	56	10	,	,	PUNCT
ejpam-7048	56	11	c.	c.	PROPN
ejpam-7048	56	12	boonpok	boonpok	PROPN
ejpam-7048	56	13	/	/	SYM
ejpam-7048	56	14	eur	eur	PROPN
ejpam-7048	56	15	.	.	PUNCT
ejpam-7048	57	1	j.	j.	PROPN
ejpam-7048	57	2	pure	pure	PROPN
ejpam-7048	57	3	appl	appl	PROPN
ejpam-7048	57	4	.	.	PROPN
ejpam-7048	57	5	math	math	PROPN
ejpam-7048	57	6	,	,	PUNCT
ejpam-7048	57	7	18	18	NUM
ejpam-7048	57	8	(	(	PUNCT
ejpam-7048	57	9	4	4	NUM
ejpam-7048	57	10	)	)	PUNCT
ejpam-7048	57	11	(	(	PUNCT
ejpam-7048	57	12	2025	2025	NUM
ejpam-7048	57	13	)	)	PUNCT
ejpam-7048	57	14	,	,	PUNCT
ejpam-7048	57	15	7048	7048	NUM
ejpam-7048	57	16	3	3	NUM
ejpam-7048	57	17	of	of	ADP
ejpam-7048	57	18	9	9	NUM
ejpam-7048	57	19	the	the	DET
ejpam-7048	57	20	complement	complement	NOUN
ejpam-7048	57	21	of	of	ADP
ejpam-7048	57	22	a	a	DET
ejpam-7048	57	23	(	(	PUNCT
ejpam-7048	57	24	τ1	τ1	NOUN
ejpam-7048	57	25	,	,	PUNCT
ejpam-7048	57	26	τ2)r	τ2)r	NOUN
ejpam-7048	57	27	-	-	PUNCT
ejpam-7048	57	28	open	open	ADJ
ejpam-7048	57	29	(	(	PUNCT
ejpam-7048	57	30	resp	resp	NOUN
ejpam-7048	57	31	.	.	PUNCT
ejpam-7048	58	1	(	(	PUNCT
ejpam-7048	58	2	τ1	τ1	NOUN
ejpam-7048	58	3	,	,	PUNCT
ejpam-7048	58	4	τ2)s	τ2)s	NOUN
ejpam-7048	58	5	-	-	PUNCT
ejpam-7048	58	6	open	open	ADJ
ejpam-7048	58	7	,	,	PUNCT
ejpam-7048	58	8	(	(	PUNCT
ejpam-7048	58	9	τ1	τ1	NOUN
ejpam-7048	58	10	,	,	PUNCT
ejpam-7048	58	11	τ2)p	τ2)p	NOUN
ejpam-7048	58	12	-	-	ADJ
ejpam-7048	58	13	open	open	ADJ
ejpam-7048	58	14	,	,	PUNCT
ejpam-7048	58	15	(	(	PUNCT
ejpam-7048	58	16	τ1	τ1	NOUN
ejpam-7048	58	17	,	,	PUNCT
ejpam-7048	58	18	τ2)β	τ2)β	ADJ
ejpam-7048	58	19	-	-	PUNCT
ejpam-7048	58	20	open	open	ADJ
ejpam-7048	58	21	)	)	PUNCT
ejpam-7048	58	22	set	set	NOUN
ejpam-7048	58	23	is	be	AUX
ejpam-7048	58	24	said	say	VERB
ejpam-7048	58	25	to	to	PART
ejpam-7048	58	26	be	be	AUX
ejpam-7048	58	27	(	(	PUNCT
ejpam-7048	58	28	τ1	τ1	NOUN
ejpam-7048	58	29	,	,	PUNCT
ejpam-7048	58	30	τ2)r	τ2)r	NOUN
ejpam-7048	58	31	-	-	PUNCT
ejpam-7048	58	32	closed	closed	ADJ
ejpam-7048	58	33	(	(	PUNCT
ejpam-7048	58	34	resp	resp	NOUN
ejpam-7048	58	35	.	.	PUNCT
ejpam-7048	59	1	(	(	PUNCT
ejpam-7048	59	2	τ1	τ1	NOUN
ejpam-7048	59	3	,	,	PUNCT
ejpam-7048	59	4	τ2)s	τ2)s	NOUN
ejpam-7048	59	5	-	-	PUNCT
ejpam-7048	59	6	closed	closed	ADJ
ejpam-7048	59	7	,	,	PUNCT
ejpam-7048	59	8	(	(	PUNCT
ejpam-7048	59	9	τ1	τ1	NOUN
ejpam-7048	59	10	,	,	PUNCT
ejpam-7048	59	11	τ2)p	τ2)p	NOUN
ejpam-7048	59	12	-	-	PUNCT
ejpam-7048	59	13	closed	closed	ADJ
ejpam-7048	59	14	,	,	PUNCT
ejpam-7048	59	15	(	(	PUNCT
ejpam-7048	59	16	τ1	τ1	NOUN
ejpam-7048	59	17	,	,	PUNCT
ejpam-7048	59	18	τ2)β	τ2)β	ADJ
ejpam-7048	59	19	-	-	PUNCT
ejpam-7048	59	20	closed	closed	ADJ
ejpam-7048	59	21	)	)	PUNCT
ejpam-7048	59	22	.	.	PUNCT
ejpam-7048	60	1	a	a	DET
ejpam-7048	60	2	subset	subset	NOUN
ejpam-7048	60	3	a	a	PRON
ejpam-7048	60	4	of	of	ADP
ejpam-7048	60	5	a	a	DET
ejpam-7048	60	6	bitopological	bitopological	ADJ
ejpam-7048	60	7	space	space	NOUN
ejpam-7048	60	8	(	(	PUNCT
ejpam-7048	60	9	x	x	NOUN
ejpam-7048	60	10	,	,	PUNCT
ejpam-7048	60	11	τ1	τ1	NOUN
ejpam-7048	60	12	,	,	PUNCT
ejpam-7048	60	13	τ2	τ2	NOUN
ejpam-7048	60	14	)	)	PUNCT
ejpam-7048	60	15	is	be	AUX
ejpam-7048	60	16	said	say	VERB
ejpam-7048	60	17	to	to	PART
ejpam-7048	60	18	be	be	AUX
ejpam-7048	60	19	τ1τ2	τ1τ2	NOUN
ejpam-7048	60	20	-	-	ADJ
ejpam-7048	60	21	δ	δ	NOUN
ejpam-7048	60	22	-	-	NOUN
ejpam-7048	60	23	open	open	ADJ
ejpam-7048	60	24	[	[	X
ejpam-7048	60	25	21	21	NUM
ejpam-7048	60	26	]	]	X
ejpam-7048	60	27	if	if	SCONJ
ejpam-7048	60	28	a	a	PRON
ejpam-7048	60	29	is	be	AUX
ejpam-7048	60	30	the	the	DET
ejpam-7048	60	31	union	union	NOUN
ejpam-7048	60	32	of	of	ADP
ejpam-7048	60	33	(	(	PUNCT
ejpam-7048	60	34	τ1	τ1	NOUN
ejpam-7048	60	35	,	,	PUNCT
ejpam-7048	60	36	τ2)r	τ2)r	ADJ
ejpam-7048	60	37	-	-	PUNCT
ejpam-7048	60	38	open	open	ADJ
ejpam-7048	60	39	sets	set	NOUN
ejpam-7048	60	40	of	of	ADP
ejpam-7048	60	41	x.	x.	NOUN
ejpam-7048	60	42	the	the	DET
ejpam-7048	60	43	complement	complement	NOUN
ejpam-7048	60	44	of	of	ADP
ejpam-7048	60	45	a	a	DET
ejpam-7048	60	46	τ1τ2	τ1τ2	ADJ
ejpam-7048	60	47	-	-	ADJ
ejpam-7048	60	48	δ	δ	NOUN
ejpam-7048	60	49	-	-	ADJ
ejpam-7048	60	50	open	open	ADJ
ejpam-7048	60	51	set	set	NOUN
ejpam-7048	60	52	is	be	AUX
ejpam-7048	60	53	called	call	VERB
ejpam-7048	60	54	τ1τ2	τ1τ2	NOUN
ejpam-7048	60	55	-	-	ADJ
ejpam-7048	60	56	δ	δ	NOUN
ejpam-7048	60	57	-	-	PUNCT
ejpam-7048	60	58	closed	closed	ADJ
ejpam-7048	60	59	[	[	X
ejpam-7048	60	60	21	21	NUM
ejpam-7048	60	61	]	]	PUNCT
ejpam-7048	60	62	.	.	PUNCT
ejpam-7048	61	1	the	the	DET
ejpam-7048	61	2	union	union	NOUN
ejpam-7048	61	3	of	of	ADP
ejpam-7048	61	4	all	all	DET
ejpam-7048	61	5	τ1τ2	τ1τ2	NOUN
ejpam-7048	61	6	-	-	ADJ
ejpam-7048	61	7	δ	δ	NOUN
ejpam-7048	61	8	-	-	ADJ
ejpam-7048	61	9	open	open	ADJ
ejpam-7048	61	10	sets	set	NOUN
ejpam-7048	61	11	of	of	ADP
ejpam-7048	61	12	x	x	PUNCT
ejpam-7048	61	13	contained	contain	VERB
ejpam-7048	61	14	in	in	ADP
ejpam-7048	61	15	a	a	PRON
ejpam-7048	61	16	is	be	AUX
ejpam-7048	61	17	called	call	VERB
ejpam-7048	61	18	the	the	DET
ejpam-7048	61	19	τ1τ2	τ1τ2	ADJ
ejpam-7048	61	20	-	-	ADJ
ejpam-7048	61	21	δ	δ	NOUN
ejpam-7048	61	22	-	-	NOUN
ejpam-7048	61	23	interior	interior	NOUN
ejpam-7048	61	24	[	[	X
ejpam-7048	61	25	21	21	NUM
ejpam-7048	61	26	]	]	PUNCT
ejpam-7048	61	27	of	of	ADP
ejpam-7048	61	28	a	a	PRON
ejpam-7048	61	29	and	and	CCONJ
ejpam-7048	61	30	is	be	AUX
ejpam-7048	61	31	denoted	denote	VERB
ejpam-7048	61	32	by	by	ADP
ejpam-7048	61	33	τ1τ2	τ1τ2	ADJ
ejpam-7048	61	34	-	-	ADJ
ejpam-7048	61	35	δ	δ	NOUN
ejpam-7048	61	36	-	-	PUNCT
ejpam-7048	61	37	int(a	int(a	PROPN
ejpam-7048	61	38	)	)	PUNCT
ejpam-7048	61	39	.	.	PUNCT
ejpam-7048	62	1	the	the	DET
ejpam-7048	62	2	intersection	intersection	NOUN
ejpam-7048	62	3	of	of	ADP
ejpam-7048	62	4	all	all	DET
ejpam-7048	62	5	τ1τ2	τ1τ2	NOUN
ejpam-7048	62	6	-	-	ADJ
ejpam-7048	62	7	δ	δ	NOUN
ejpam-7048	62	8	-	-	PUNCT
ejpam-7048	62	9	closed	close	VERB
ejpam-7048	62	10	sets	set	NOUN
ejpam-7048	62	11	of	of	ADP
ejpam-7048	62	12	x	x	PUNCT
ejpam-7048	62	13	containing	contain	VERB
ejpam-7048	62	14	a	a	PRON
ejpam-7048	62	15	is	be	AUX
ejpam-7048	62	16	called	call	VERB
ejpam-7048	62	17	the	the	DET
ejpam-7048	62	18	τ1τ2	τ1τ2	ADJ
ejpam-7048	62	19	-	-	ADJ
ejpam-7048	62	20	δ	δ	NOUN
ejpam-7048	62	21	-	-	NOUN
ejpam-7048	62	22	closure	closure	NOUN
ejpam-7048	62	23	[	[	X
ejpam-7048	62	24	21	21	NUM
ejpam-7048	62	25	]	]	PUNCT
ejpam-7048	62	26	of	of	ADP
ejpam-7048	62	27	a	a	PRON
ejpam-7048	62	28	and	and	CCONJ
ejpam-7048	62	29	is	be	AUX
ejpam-7048	62	30	denoted	denote	VERB
ejpam-7048	62	31	by	by	ADP
ejpam-7048	62	32	τ1τ2	τ1τ2	ADJ
ejpam-7048	62	33	-	-	ADJ
ejpam-7048	62	34	δ	δ	NOUN
ejpam-7048	62	35	-	-	PUNCT
ejpam-7048	62	36	cl(a	cl(a	NUM
ejpam-7048	62	37	)	)	PUNCT
ejpam-7048	62	38	.	.	PUNCT
ejpam-7048	63	1	let	let	VERB
ejpam-7048	63	2	a	a	DET
ejpam-7048	63	3	be	be	AUX
ejpam-7048	63	4	a	a	DET
ejpam-7048	63	5	subset	subset	NOUN
ejpam-7048	63	6	of	of	ADP
ejpam-7048	63	7	a	a	DET
ejpam-7048	63	8	bitopological	bitopological	ADJ
ejpam-7048	63	9	space	space	NOUN
ejpam-7048	63	10	(	(	PUNCT
ejpam-7048	63	11	x	x	NOUN
ejpam-7048	63	12	,	,	PUNCT
ejpam-7048	63	13	τ1	τ1	NOUN
ejpam-7048	63	14	,	,	PUNCT
ejpam-7048	63	15	τ2	τ2	NOUN
ejpam-7048	63	16	)	)	PUNCT
ejpam-7048	63	17	.	.	PUNCT
ejpam-7048	64	1	a	a	DET
ejpam-7048	64	2	point	point	NOUN
ejpam-7048	64	3	x	x	X
ejpam-7048	64	4	∈	∈	NOUN
ejpam-7048	64	5	x	x	PUNCT
ejpam-7048	64	6	is	be	AUX
ejpam-7048	64	7	called	call	VERB
ejpam-7048	64	8	a	a	DET
ejpam-7048	64	9	(	(	PUNCT
ejpam-7048	64	10	τ1	τ1	NOUN
ejpam-7048	64	11	,	,	PUNCT
ejpam-7048	64	12	τ2)θ	τ2)θ	ADJ
ejpam-7048	64	13	-	-	PUNCT
ejpam-7048	64	14	cluster	cluster	NOUN
ejpam-7048	64	15	point	point	NOUN
ejpam-7048	64	16	[	[	X
ejpam-7048	64	17	19	19	NUM
ejpam-7048	64	18	]	]	PUNCT
ejpam-7048	64	19	of	of	ADP
ejpam-7048	64	20	a	a	DET
ejpam-7048	64	21	if	if	SCONJ
ejpam-7048	64	22	τ1τ2	τ1τ2	NOUN
ejpam-7048	64	23	-	-	NOUN
ejpam-7048	64	24	cl(u	cl(u	NOUN
ejpam-7048	64	25	)	)	PUNCT
ejpam-7048	64	26	∩	∩	NOUN
ejpam-7048	64	27	a	a	DET
ejpam-7048	64	28	̸=	̸=	PROPN
ejpam-7048	64	29	∅	∅	NOUN
ejpam-7048	64	30	for	for	ADP
ejpam-7048	64	31	every	every	DET
ejpam-7048	64	32	τ1τ2	τ1τ2	ADJ
ejpam-7048	64	33	-	-	ADJ
ejpam-7048	64	34	open	open	ADJ
ejpam-7048	64	35	set	set	NOUN
ejpam-7048	64	36	u	u	NOUN
ejpam-7048	64	37	containing	contain	VERB
ejpam-7048	64	38	x.	x.	NOUN
ejpam-7048	64	39	the	the	DET
ejpam-7048	64	40	set	set	NOUN
ejpam-7048	64	41	of	of	ADP
ejpam-7048	64	42	all	all	DET
ejpam-7048	64	43	(	(	PUNCT
ejpam-7048	64	44	τ1	τ1	NOUN
ejpam-7048	64	45	,	,	PUNCT
ejpam-7048	64	46	τ2)θ	τ2)θ	ADJ
ejpam-7048	64	47	-	-	PUNCT
ejpam-7048	64	48	cluster	cluster	NOUN
ejpam-7048	64	49	points	point	NOUN
ejpam-7048	64	50	of	of	ADP
ejpam-7048	64	51	a	a	PRON
ejpam-7048	64	52	is	be	AUX
ejpam-7048	64	53	called	call	VERB
ejpam-7048	64	54	the	the	DET
ejpam-7048	64	55	(	(	PUNCT
ejpam-7048	64	56	τ1	τ1	NOUN
ejpam-7048	64	57	,	,	PUNCT
ejpam-7048	64	58	τ2)θ	τ2)θ	ADJ
ejpam-7048	64	59	-	-	PUNCT
ejpam-7048	64	60	closure	closure	NOUN
ejpam-7048	64	61	[	[	X
ejpam-7048	64	62	19	19	NUM
ejpam-7048	64	63	]	]	PUNCT
ejpam-7048	64	64	of	of	ADP
ejpam-7048	64	65	a	a	PRON
ejpam-7048	64	66	and	and	CCONJ
ejpam-7048	64	67	is	be	AUX
ejpam-7048	64	68	denoted	denote	VERB
ejpam-7048	64	69	by	by	ADP
ejpam-7048	64	70	(	(	PUNCT
ejpam-7048	64	71	τ1	τ1	NOUN
ejpam-7048	64	72	,	,	PUNCT
ejpam-7048	64	73	τ2)θ	τ2)θ	NOUN
ejpam-7048	64	74	-	-	PUNCT
ejpam-7048	64	75	cl(a	cl(a	NUM
ejpam-7048	64	76	)	)	PUNCT
ejpam-7048	64	77	.	.	PUNCT
ejpam-7048	65	1	a	a	DET
ejpam-7048	65	2	subset	subset	NOUN
ejpam-7048	65	3	a	a	PRON
ejpam-7048	65	4	of	of	ADP
ejpam-7048	65	5	a	a	DET
ejpam-7048	65	6	bitopological	bitopological	ADJ
ejpam-7048	65	7	space	space	NOUN
ejpam-7048	65	8	(	(	PUNCT
ejpam-7048	65	9	x	x	NOUN
ejpam-7048	65	10	,	,	PUNCT
ejpam-7048	65	11	τ1	τ1	NOUN
ejpam-7048	65	12	,	,	PUNCT
ejpam-7048	65	13	τ2	τ2	NOUN
ejpam-7048	65	14	)	)	PUNCT
ejpam-7048	65	15	is	be	AUX
ejpam-7048	65	16	said	say	VERB
ejpam-7048	65	17	to	to	PART
ejpam-7048	65	18	be	be	AUX
ejpam-7048	65	19	(	(	PUNCT
ejpam-7048	65	20	τ1	τ1	NOUN
ejpam-7048	65	21	,	,	PUNCT
ejpam-7048	65	22	τ2)θ	τ2)θ	NOUN
ejpam-7048	65	23	-	-	PUNCT
ejpam-7048	65	24	closed	closed	ADJ
ejpam-7048	65	25	[	[	X
ejpam-7048	65	26	19	19	NUM
ejpam-7048	65	27	]	]	X
ejpam-7048	65	28	if	if	SCONJ
ejpam-7048	65	29	(	(	PUNCT
ejpam-7048	65	30	τ1	τ1	NOUN
ejpam-7048	65	31	,	,	PUNCT
ejpam-7048	65	32	τ2)θ	τ2)θ	NOUN
ejpam-7048	65	33	-	-	PUNCT
ejpam-7048	65	34	cl(a	cl(a	NUM
ejpam-7048	65	35	)	)	PUNCT
ejpam-7048	66	1	=	=	PUNCT
ejpam-7048	66	2	a.	a.	NOUN
ejpam-7048	66	3	the	the	DET
ejpam-7048	66	4	complement	complement	NOUN
ejpam-7048	66	5	of	of	ADP
ejpam-7048	66	6	a	a	DET
ejpam-7048	66	7	(	(	PUNCT
ejpam-7048	66	8	τ1	τ1	NOUN
ejpam-7048	66	9	,	,	PUNCT
ejpam-7048	66	10	τ2)θ	τ2)θ	ADJ
ejpam-7048	66	11	-	-	PUNCT
ejpam-7048	66	12	closed	close	VERB
ejpam-7048	66	13	set	set	NOUN
ejpam-7048	66	14	is	be	AUX
ejpam-7048	66	15	said	say	VERB
ejpam-7048	66	16	to	to	PART
ejpam-7048	66	17	be	be	AUX
ejpam-7048	66	18	(	(	PUNCT
ejpam-7048	66	19	τ1	τ1	NOUN
ejpam-7048	66	20	,	,	PUNCT
ejpam-7048	66	21	τ2)θopen	τ2)θopen	PROPN
ejpam-7048	66	22	.	.	PUNCT
ejpam-7048	67	1	the	the	DET
ejpam-7048	67	2	union	union	NOUN
ejpam-7048	67	3	of	of	ADP
ejpam-7048	67	4	all	all	DET
ejpam-7048	67	5	(	(	PUNCT
ejpam-7048	67	6	τ1	τ1	NOUN
ejpam-7048	67	7	,	,	PUNCT
ejpam-7048	67	8	τ2)θ	τ2)θ	ADJ
ejpam-7048	67	9	-	-	PUNCT
ejpam-7048	67	10	open	open	ADJ
ejpam-7048	67	11	sets	set	NOUN
ejpam-7048	67	12	of	of	ADP
ejpam-7048	67	13	x	x	PUNCT
ejpam-7048	67	14	contained	contain	VERB
ejpam-7048	67	15	in	in	ADP
ejpam-7048	67	16	a	a	PRON
ejpam-7048	67	17	is	be	AUX
ejpam-7048	67	18	called	call	VERB
ejpam-7048	67	19	the	the	DET
ejpam-7048	67	20	(	(	PUNCT
ejpam-7048	67	21	τ1	τ1	NOUN
ejpam-7048	67	22	,	,	PUNCT
ejpam-7048	67	23	τ2)θ	τ2)θ	ADJ
ejpam-7048	67	24	-	-	PUNCT
ejpam-7048	67	25	interior	interior	NOUN
ejpam-7048	67	26	[	[	X
ejpam-7048	67	27	19	19	NUM
ejpam-7048	67	28	]	]	PUNCT
ejpam-7048	67	29	of	of	ADP
ejpam-7048	67	30	a	a	PRON
ejpam-7048	67	31	and	and	CCONJ
ejpam-7048	67	32	is	be	AUX
ejpam-7048	67	33	denoted	denote	VERB
ejpam-7048	67	34	by	by	ADP
ejpam-7048	67	35	(	(	PUNCT
ejpam-7048	67	36	τ1	τ1	NOUN
ejpam-7048	67	37	,	,	PUNCT
ejpam-7048	67	38	τ2)θ	τ2)θ	NOUN
ejpam-7048	67	39	-	-	PUNCT
ejpam-7048	67	40	int(a	int(a	NOUN
ejpam-7048	67	41	)	)	PUNCT
ejpam-7048	67	42	.	.	PUNCT
ejpam-7048	68	1	a	a	DET
ejpam-7048	68	2	subset	subset	NOUN
ejpam-7048	68	3	a	a	PRON
ejpam-7048	68	4	of	of	ADP
ejpam-7048	68	5	a	a	DET
ejpam-7048	68	6	bitopological	bitopological	ADJ
ejpam-7048	68	7	space	space	NOUN
ejpam-7048	68	8	(	(	PUNCT
ejpam-7048	68	9	x	x	NOUN
ejpam-7048	68	10	,	,	PUNCT
ejpam-7048	68	11	τ1	τ1	NOUN
ejpam-7048	68	12	,	,	PUNCT
ejpam-7048	68	13	τ2	τ2	NOUN
ejpam-7048	68	14	)	)	PUNCT
ejpam-7048	68	15	is	be	AUX
ejpam-7048	68	16	called	call	VERB
ejpam-7048	68	17	τ1τ2	τ1τ2	NOUN
ejpam-7048	68	18	-	-	ADJ
ejpam-7048	68	19	nowhere	nowhere	ADV
ejpam-7048	68	20	dense	dense	ADJ
ejpam-7048	68	21	if	if	SCONJ
ejpam-7048	68	22	τ1τ2	τ1τ2	NOUN
ejpam-7048	68	23	-	-	NOUN
ejpam-7048	68	24	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7048	68	25	-	-	PUNCT
ejpam-7048	68	26	int(a	int(a	NOUN
ejpam-7048	68	27	)	)	PUNCT
ejpam-7048	68	28	)	)	PUNCT
ejpam-7048	69	1	=	=	PUNCT
ejpam-7048	69	2	∅.	∅.	VERB
ejpam-7048	69	3	an	an	DET
ejpam-7048	69	4	ideal	ideal	NOUN
ejpam-7048	69	5	i	i	PRON
ejpam-7048	69	6	on	on	ADP
ejpam-7048	69	7	a	a	DET
ejpam-7048	69	8	topological	topological	ADJ
ejpam-7048	69	9	space	space	NOUN
ejpam-7048	69	10	(	(	PUNCT
ejpam-7048	69	11	x	x	X
ejpam-7048	69	12	,	,	PUNCT
ejpam-7048	69	13	τ	τ	X
ejpam-7048	69	14	)	)	PUNCT
ejpam-7048	69	15	is	be	AUX
ejpam-7048	69	16	a	a	DET
ejpam-7048	69	17	nonempty	nonempty	ADJ
ejpam-7048	69	18	collection	collection	NOUN
ejpam-7048	69	19	of	of	ADP
ejpam-7048	69	20	subsets	subset	NOUN
ejpam-7048	69	21	of	of	ADP
ejpam-7048	69	22	x	x	PUNCT
ejpam-7048	69	23	satisfying	satisfy	VERB
ejpam-7048	69	24	the	the	DET
ejpam-7048	69	25	following	follow	VERB
ejpam-7048	69	26	properties	property	NOUN
ejpam-7048	69	27	:	:	PUNCT
ejpam-7048	69	28	(	(	PUNCT
ejpam-7048	69	29	1	1	X
ejpam-7048	69	30	)	)	PUNCT
ejpam-7048	69	31	a	a	DET
ejpam-7048	69	32	∈	∈	NOUN
ejpam-7048	69	33	i	i	PRON
ejpam-7048	69	34	and	and	CCONJ
ejpam-7048	69	35	b	b	X
ejpam-7048	69	36	⊆	⊆	NUM
ejpam-7048	69	37	a	a	DET
ejpam-7048	69	38	imply	imply	NOUN
ejpam-7048	69	39	b	b	X
ejpam-7048	69	40	∈	∈	PROPN
ejpam-7048	69	41	i	i	PRON
ejpam-7048	69	42	;	;	PUNCT
ejpam-7048	69	43	(	(	PUNCT
ejpam-7048	69	44	2	2	X
ejpam-7048	69	45	)	)	PUNCT
ejpam-7048	70	1	a	a	PRON
ejpam-7048	70	2	∈	∈	NOUN
ejpam-7048	71	1	i	i	PRON
ejpam-7048	71	2	and	and	CCONJ
ejpam-7048	71	3	b	b	X
ejpam-7048	71	4	∈	∈	NOUN
ejpam-7048	72	1	i	i	PRON
ejpam-7048	72	2	imply	imply	VERB
ejpam-7048	72	3	a	a	DET
ejpam-7048	72	4	∪	∪	X
ejpam-7048	72	5	b	b	NOUN
ejpam-7048	72	6	∈	∈	NOUN
ejpam-7048	73	1	i	i	PRON
ejpam-7048	73	2	.	.	PUNCT
ejpam-7048	74	1	a	a	DET
ejpam-7048	74	2	topological	topological	ADJ
ejpam-7048	74	3	space	space	NOUN
ejpam-7048	74	4	(	(	PUNCT
ejpam-7048	74	5	x	x	X
ejpam-7048	74	6	,	,	PUNCT
ejpam-7048	74	7	τ	τ	X
ejpam-7048	74	8	)	)	PUNCT
ejpam-7048	74	9	with	with	ADP
ejpam-7048	74	10	an	an	DET
ejpam-7048	74	11	ideal	ideal	ADJ
ejpam-7048	74	12	i	i	PRON
ejpam-7048	74	13	on	on	ADP
ejpam-7048	74	14	x	x	SYM
ejpam-7048	74	15	is	be	AUX
ejpam-7048	74	16	called	call	VERB
ejpam-7048	74	17	an	an	DET
ejpam-7048	74	18	ideal	ideal	ADJ
ejpam-7048	74	19	topological	topological	ADJ
ejpam-7048	74	20	space	space	NOUN
ejpam-7048	74	21	and	and	CCONJ
ejpam-7048	74	22	is	be	AUX
ejpam-7048	74	23	denoted	denote	VERB
ejpam-7048	74	24	by	by	ADP
ejpam-7048	74	25	(	(	PUNCT
ejpam-7048	74	26	x	x	X
ejpam-7048	74	27	,	,	PUNCT
ejpam-7048	74	28	τ	τ	PROPN
ejpam-7048	74	29	,	,	PUNCT
ejpam-7048	74	30	i	i	NOUN
ejpam-7048	74	31	)	)	PUNCT
ejpam-7048	74	32	.	.	PUNCT
ejpam-7048	75	1	for	for	ADP
ejpam-7048	75	2	an	an	DET
ejpam-7048	75	3	ideal	ideal	ADJ
ejpam-7048	75	4	topological	topological	ADJ
ejpam-7048	75	5	space	space	NOUN
ejpam-7048	75	6	(	(	PUNCT
ejpam-7048	75	7	x	x	X
ejpam-7048	75	8	,	,	PUNCT
ejpam-7048	75	9	τ	τ	PROPN
ejpam-7048	75	10	,	,	PUNCT
ejpam-7048	75	11	i	i	PROPN
ejpam-7048	75	12	)	)	PUNCT
ejpam-7048	75	13	and	and	CCONJ
ejpam-7048	75	14	a	a	DET
ejpam-7048	75	15	subset	subset	NOUN
ejpam-7048	75	16	a	a	PRON
ejpam-7048	75	17	of	of	ADP
ejpam-7048	75	18	x	x	PRON
ejpam-7048	75	19	,	,	PUNCT
ejpam-7048	75	20	a⋆(i	a⋆(i	PROPN
ejpam-7048	75	21	)	)	PUNCT
ejpam-7048	75	22	is	be	AUX
ejpam-7048	75	23	defined	define	VERB
ejpam-7048	75	24	as	as	SCONJ
ejpam-7048	75	25	follows	follow	VERB
ejpam-7048	75	26	:	:	PUNCT
ejpam-7048	75	27	a⋆(i	a⋆(i	NOUN
ejpam-7048	75	28	)	)	PUNCT
ejpam-7048	76	1	=	=	PUNCT
ejpam-7048	76	2	{	{	PUNCT
ejpam-7048	76	3	x	x	PUNCT
ejpam-7048	76	4	∈	∈	PROPN
ejpam-7048	76	5	x	x	X
ejpam-7048	76	6	:	:	PUNCT
ejpam-7048	76	7	u	u	X
ejpam-7048	76	8	∩a	∩a	PROPN
ejpam-7048	76	9	̸∈	̸∈	PROPN
ejpam-7048	76	10	i	i	PRON
ejpam-7048	76	11	for	for	ADP
ejpam-7048	76	12	every	every	DET
ejpam-7048	76	13	open	open	ADJ
ejpam-7048	76	14	neighbourhood	neighbourhood	NOUN
ejpam-7048	76	15	u	u	NOUN
ejpam-7048	76	16	of	of	ADP
ejpam-7048	76	17	x	x	NOUN
ejpam-7048	76	18	}	}	PUNCT
ejpam-7048	76	19	.	.	PUNCT
ejpam-7048	77	1	in	in	ADP
ejpam-7048	77	2	case	case	NOUN
ejpam-7048	77	3	there	there	PRON
ejpam-7048	77	4	is	be	VERB
ejpam-7048	77	5	no	no	DET
ejpam-7048	77	6	chance	chance	NOUN
ejpam-7048	77	7	for	for	ADP
ejpam-7048	77	8	confusion	confusion	NOUN
ejpam-7048	77	9	,	,	PUNCT
ejpam-7048	77	10	a⋆(i	a⋆(i	NOUN
ejpam-7048	77	11	)	)	PUNCT
ejpam-7048	77	12	is	be	AUX
ejpam-7048	77	13	simply	simply	ADV
ejpam-7048	77	14	written	write	VERB
ejpam-7048	77	15	as	as	ADP
ejpam-7048	77	16	a⋆.	a⋆.	NOUN
ejpam-7048	77	17	in	in	ADP
ejpam-7048	77	18	[	[	X
ejpam-7048	77	19	22	22	NUM
ejpam-7048	77	20	]	]	PUNCT
ejpam-7048	77	21	,	,	PUNCT
ejpam-7048	77	22	a⋆	a⋆	ADV
ejpam-7048	77	23	is	be	AUX
ejpam-7048	77	24	called	call	VERB
ejpam-7048	77	25	the	the	DET
ejpam-7048	77	26	local	local	ADJ
ejpam-7048	77	27	function	function	NOUN
ejpam-7048	77	28	of	of	ADP
ejpam-7048	77	29	a	a	PRON
ejpam-7048	77	30	with	with	ADP
ejpam-7048	77	31	respect	respect	NOUN
ejpam-7048	77	32	to	to	ADP
ejpam-7048	77	33	i	i	PRON
ejpam-7048	77	34	and	and	CCONJ
ejpam-7048	77	35	τ	τ	PROPN
ejpam-7048	77	36	and	and	CCONJ
ejpam-7048	77	37	cl⋆(a	cl⋆(a	NUM
ejpam-7048	77	38	)	)	PUNCT
ejpam-7048	77	39	=	=	PUNCT
ejpam-7048	77	40	a⋆	a⋆	ADP
ejpam-7048	77	41	∪	∪	ADP
ejpam-7048	77	42	a	a	DET
ejpam-7048	77	43	defines	define	NOUN
ejpam-7048	77	44	a	a	DET
ejpam-7048	77	45	kuratowski	kuratowski	ADJ
ejpam-7048	77	46	closure	closure	NOUN
ejpam-7048	77	47	operator	operator	NOUN
ejpam-7048	77	48	for	for	ADP
ejpam-7048	77	49	a	a	DET
ejpam-7048	77	50	topology	topology	NOUN
ejpam-7048	77	51	τ⋆(i	τ⋆(i	NOUN
ejpam-7048	77	52	)	)	PUNCT
ejpam-7048	77	53	finer	fine	ADJ
ejpam-7048	77	54	than	than	ADP
ejpam-7048	77	55	τ	τ	PROPN
ejpam-7048	77	56	.	.	PUNCT
ejpam-7048	78	1	a	a	DET
ejpam-7048	78	2	subset	subset	NOUN
ejpam-7048	78	3	a	a	PRON
ejpam-7048	78	4	is	be	AUX
ejpam-7048	78	5	said	say	VERB
ejpam-7048	78	6	to	to	PART
ejpam-7048	78	7	be	be	AUX
ejpam-7048	78	8	⋆-closed	⋆-close	VERB
ejpam-7048	78	9	[	[	X
ejpam-7048	78	10	23	23	NUM
ejpam-7048	78	11	]	]	X
ejpam-7048	78	12	if	if	SCONJ
ejpam-7048	78	13	a⋆	a⋆	ADJ
ejpam-7048	78	14	⊆	⊆	NUM
ejpam-7048	78	15	a.	a.	NOUN
ejpam-7048	78	16	the	the	DET
ejpam-7048	78	17	interior	interior	NOUN
ejpam-7048	78	18	of	of	ADP
ejpam-7048	78	19	a	a	DET
ejpam-7048	78	20	subset	subset	NOUN
ejpam-7048	78	21	a	a	DET
ejpam-7048	78	22	in	in	ADP
ejpam-7048	78	23	(	(	PUNCT
ejpam-7048	78	24	x	x	X
ejpam-7048	78	25	,	,	PUNCT
ejpam-7048	78	26	τ⋆(i	τ⋆(i	NOUN
ejpam-7048	78	27	)	)	PUNCT
ejpam-7048	78	28	)	)	PUNCT
ejpam-7048	78	29	is	be	AUX
ejpam-7048	78	30	denoted	denote	VERB
ejpam-7048	78	31	by	by	ADP
ejpam-7048	78	32	int⋆(a	int⋆(a	NOUN
ejpam-7048	78	33	)	)	PUNCT
ejpam-7048	78	34	.	.	PUNCT
ejpam-7048	79	1	a	a	DET
ejpam-7048	79	2	subset	subset	NOUN
ejpam-7048	79	3	a	a	PRON
ejpam-7048	79	4	of	of	ADP
ejpam-7048	79	5	an	an	DET
ejpam-7048	79	6	ideal	ideal	ADJ
ejpam-7048	79	7	topological	topological	ADJ
ejpam-7048	79	8	space	space	NOUN
ejpam-7048	79	9	(	(	PUNCT
ejpam-7048	79	10	x	x	X
ejpam-7048	79	11	,	,	PUNCT
ejpam-7048	79	12	τ	τ	PROPN
ejpam-7048	79	13	,	,	PUNCT
ejpam-7048	79	14	i	i	PROPN
ejpam-7048	79	15	)	)	PUNCT
ejpam-7048	79	16	is	be	AUX
ejpam-7048	79	17	said	say	VERB
ejpam-7048	79	18	to	to	PART
ejpam-7048	79	19	be	be	AUX
ejpam-7048	79	20	r	r	NOUN
ejpam-7048	79	21	-	-	PUNCT
ejpam-7048	79	22	i	i	PRON
ejpam-7048	79	23	⋆-open	⋆-open	VERB
ejpam-7048	80	1	[	[	X
ejpam-7048	80	2	8	8	NUM
ejpam-7048	80	3	]	]	PUNCT
ejpam-7048	80	4	(	(	PUNCT
ejpam-7048	80	5	resp	resp	NOUN
ejpam-7048	80	6	.	.	PUNCT
ejpam-7048	81	1	i	i	PRON
ejpam-7048	81	2	⋆preopen	⋆preopen	VERB
ejpam-7048	82	1	[	[	X
ejpam-7048	82	2	8	8	NUM
ejpam-7048	82	3	]	]	PUNCT
ejpam-7048	82	4	,	,	PUNCT
ejpam-7048	82	5	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7048	82	6	-	-	ADJ
ejpam-7048	82	7	open	open	ADJ
ejpam-7048	82	8	[	[	X
ejpam-7048	82	9	24	24	NUM
ejpam-7048	82	10	]	]	PUNCT
ejpam-7048	82	11	(	(	PUNCT
ejpam-7048	82	12	semi	semi	NOUN
ejpam-7048	82	13	-	-	ADJ
ejpam-7048	82	14	i	i	PRON
ejpam-7048	82	15	⋆-open	⋆-open	VERB
ejpam-7048	83	1	[	[	X
ejpam-7048	83	2	25	25	NUM
ejpam-7048	83	3	]	]	NUM
ejpam-7048	83	4	)	)	PUNCT
ejpam-7048	83	5	,	,	PUNCT
ejpam-7048	83	6	τ⋆-β	τ⋆-β	PROPN
ejpam-7048	83	7	-	-	ADJ
ejpam-7048	83	8	open	open	ADJ
ejpam-7048	83	9	[	[	X
ejpam-7048	83	10	24	24	NUM
ejpam-7048	83	11	]	]	PUNCT
ejpam-7048	83	12	(	(	PUNCT
ejpam-7048	83	13	semi	semi	NOUN
ejpam-7048	83	14	-	-	VERB
ejpam-7048	83	15	i	i	PRON
ejpam-7048	83	16	⋆-preopen	⋆-preopen	VERB
ejpam-7048	84	1	[	[	X
ejpam-7048	84	2	25	25	NUM
ejpam-7048	84	3	]	]	PUNCT
ejpam-7048	84	4	)	)	PUNCT
ejpam-7048	84	5	)	)	PUNCT
ejpam-7048	85	1	if	if	SCONJ
ejpam-7048	85	2	a	a	DET
ejpam-7048	85	3	=	=	PUNCT
ejpam-7048	85	4	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7048	85	5	)	)	PUNCT
ejpam-7048	85	6	)	)	PUNCT
ejpam-7048	86	1	(	(	PUNCT
ejpam-7048	86	2	resp	resp	NOUN
ejpam-7048	86	3	.	.	PUNCT
ejpam-7048	87	1	a	a	DET
ejpam-7048	87	2	⊆	⊆	NUM
ejpam-7048	87	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7048	87	4	)	)	PUNCT
ejpam-7048	87	5	)	)	PUNCT
ejpam-7048	87	6	,	,	PUNCT
ejpam-7048	87	7	a	a	DET
ejpam-7048	87	8	⊆	⊆	NUM
ejpam-7048	87	9	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7048	87	10	)	)	PUNCT
ejpam-7048	87	11	)	)	PUNCT
ejpam-7048	87	12	,	,	PUNCT
ejpam-7048	87	13	a	a	DET
ejpam-7048	87	14	⊆	⊆	NUM
ejpam-7048	87	15	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7048	87	16	)	)	PUNCT
ejpam-7048	87	17	)	)	PUNCT
ejpam-7048	87	18	)	)	PUNCT
ejpam-7048	87	19	)	)	PUNCT
ejpam-7048	87	20	.	.	PUNCT
ejpam-7048	88	1	the	the	DET
ejpam-7048	88	2	complement	complement	NOUN
ejpam-7048	88	3	of	of	ADP
ejpam-7048	88	4	a	a	DET
ejpam-7048	88	5	r	r	NOUN
ejpam-7048	88	6	-	-	PUNCT
ejpam-7048	88	7	i	i	PRON
ejpam-7048	88	8	⋆-open	⋆-open	VERB
ejpam-7048	88	9	(	(	PUNCT
ejpam-7048	88	10	resp	resp	NOUN
ejpam-7048	88	11	.	.	PUNCT
ejpam-7048	89	1	i	i	PRON
ejpam-7048	89	2	⋆-preopen	⋆-preopen	VERB
ejpam-7048	89	3	,	,	PUNCT
ejpam-7048	89	4	semi	semi	ADJ
ejpam-7048	89	5	-	-	ADJ
ejpam-7048	89	6	i	i	PRON
ejpam-7048	89	7	⋆-open	⋆-open	VERB
ejpam-7048	89	8	,	,	PUNCT
ejpam-7048	89	9	τ⋆-β	τ⋆-β	ADJ
ejpam-7048	89	10	-	-	ADJ
ejpam-7048	89	11	open	open	ADJ
ejpam-7048	89	12	)	)	PUNCT
ejpam-7048	89	13	set	set	NOUN
ejpam-7048	89	14	is	be	AUX
ejpam-7048	89	15	said	say	VERB
ejpam-7048	89	16	to	to	PART
ejpam-7048	89	17	be	be	AUX
ejpam-7048	89	18	r	r	NOUN
ejpam-7048	89	19	-	-	PUNCT
ejpam-7048	89	20	i	i	PRON
ejpam-7048	89	21	⋆-closed	⋆-close	VERB
ejpam-7048	89	22	(	(	PUNCT
ejpam-7048	89	23	resp	resp	NOUN
ejpam-7048	89	24	.	.	PUNCT
ejpam-7048	90	1	i	i	PRON
ejpam-7048	90	2	⋆-preclosed	⋆-preclose	VERB
ejpam-7048	90	3	,	,	PUNCT
ejpam-7048	90	4	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7048	90	5	-	-	ADJ
ejpam-7048	90	6	closed	closed	ADJ
ejpam-7048	90	7	,	,	PUNCT
ejpam-7048	90	8	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	90	9	-	-	ADJ
ejpam-7048	90	10	closed	closed	ADJ
ejpam-7048	90	11	)	)	PUNCT
ejpam-7048	90	12	.	.	PUNCT
ejpam-7048	91	1	for	for	ADP
ejpam-7048	91	2	a	a	DET
ejpam-7048	91	3	subset	subset	NOUN
ejpam-7048	91	4	a	a	PRON
ejpam-7048	91	5	of	of	ADP
ejpam-7048	91	6	an	an	DET
ejpam-7048	91	7	ideal	ideal	ADJ
ejpam-7048	91	8	topological	topological	ADJ
ejpam-7048	91	9	space	space	NOUN
ejpam-7048	91	10	(	(	PUNCT
ejpam-7048	91	11	x	x	X
ejpam-7048	91	12	,	,	PUNCT
ejpam-7048	91	13	τ	τ	PROPN
ejpam-7048	91	14	,	,	PUNCT
ejpam-7048	91	15	i	i	NOUN
ejpam-7048	91	16	)	)	PUNCT
ejpam-7048	91	17	,	,	PUNCT
ejpam-7048	91	18	the	the	DET
ejpam-7048	91	19	intersection	intersection	NOUN
ejpam-7048	91	20	of	of	ADP
ejpam-7048	91	21	all	all	PRON
ejpam-7048	91	22	semi	semi	NOUN
ejpam-7048	91	23	-	-	ADJ
ejpam-7048	91	24	i	i	PRON
ejpam-7048	91	25	⋆-closed	⋆-close	VERB
ejpam-7048	91	26	sets	set	NOUN
ejpam-7048	91	27	containing	contain	VERB
ejpam-7048	91	28	a	a	PRON
ejpam-7048	91	29	is	be	AUX
ejpam-7048	91	30	called	call	VERB
ejpam-7048	91	31	the	the	DET
ejpam-7048	91	32	semi	semi	NOUN
ejpam-7048	91	33	-	-	ADJ
ejpam-7048	91	34	i	i	PRON
ejpam-7048	91	35	⋆-closure	⋆-closure	NOUN
ejpam-7048	92	1	[	[	X
ejpam-7048	92	2	25	25	NUM
ejpam-7048	92	3	]	]	PUNCT
ejpam-7048	92	4	of	of	ADP
ejpam-7048	92	5	a	a	PRON
ejpam-7048	92	6	and	and	CCONJ
ejpam-7048	92	7	is	be	AUX
ejpam-7048	92	8	denoted	denote	VERB
ejpam-7048	92	9	by	by	ADP
ejpam-7048	92	10	scl⋆(a	scl⋆(a	NOUN
ejpam-7048	92	11	)	)	PUNCT
ejpam-7048	92	12	(	(	PUNCT
ejpam-7048	92	13	scli	scli	PROPN
ejpam-7048	92	14	⋆(a	⋆(a	NUM
ejpam-7048	92	15	)	)	PUNCT
ejpam-7048	93	1	[	[	X
ejpam-7048	93	2	25	25	NUM
ejpam-7048	93	3	]	]	PUNCT
ejpam-7048	93	4	)	)	PUNCT
ejpam-7048	93	5	.	.	PUNCT
ejpam-7048	94	1	the	the	DET
ejpam-7048	94	2	union	union	NOUN
ejpam-7048	94	3	of	of	ADP
ejpam-7048	94	4	all	all	PRON
ejpam-7048	94	5	semi	semi	ADJ
ejpam-7048	94	6	-	-	ADJ
ejpam-7048	94	7	i	i	PRON
ejpam-7048	94	8	⋆-open	⋆-open	ADJ
ejpam-7048	94	9	sets	set	NOUN
ejpam-7048	94	10	contained	contain	VERB
ejpam-7048	94	11	in	in	ADP
ejpam-7048	94	12	a	a	PRON
ejpam-7048	94	13	is	be	AUX
ejpam-7048	94	14	called	call	VERB
ejpam-7048	94	15	the	the	DET
ejpam-7048	94	16	semi	semi	NOUN
ejpam-7048	94	17	-	-	ADJ
ejpam-7048	94	18	i	i	PRON
ejpam-7048	94	19	⋆-interior	⋆-interior	PUNCT
ejpam-7048	95	1	[	[	X
ejpam-7048	95	2	25	25	NUM
ejpam-7048	95	3	]	]	PUNCT
ejpam-7048	95	4	of	of	ADP
ejpam-7048	95	5	a	a	PRON
ejpam-7048	95	6	and	and	CCONJ
ejpam-7048	95	7	is	be	AUX
ejpam-7048	95	8	denoted	denote	VERB
ejpam-7048	95	9	by	by	ADP
ejpam-7048	95	10	sint⋆(a	sint⋆(a	PROPN
ejpam-7048	95	11	)	)	PUNCT
ejpam-7048	96	1	(	(	PUNCT
ejpam-7048	96	2	sinti	sinti	PROPN
ejpam-7048	96	3	⋆(a	⋆(a	NOUN
ejpam-7048	96	4	)	)	PUNCT
ejpam-7048	97	1	[	[	X
ejpam-7048	97	2	25	25	NUM
ejpam-7048	97	3	]	]	PUNCT
ejpam-7048	97	4	)	)	PUNCT
ejpam-7048	97	5	.	.	PUNCT
ejpam-7048	98	1	the	the	DET
ejpam-7048	98	2	intersection	intersection	NOUN
ejpam-7048	98	3	of	of	ADP
ejpam-7048	98	4	all	all	PRON
ejpam-7048	98	5	β	β	NOUN
ejpam-7048	98	6	-	-	PUNCT
ejpam-7048	98	7	i	i	PRON
ejpam-7048	98	8	⋆-closed	⋆-close	VERB
ejpam-7048	98	9	sets	set	NOUN
ejpam-7048	98	10	containing	contain	VERB
ejpam-7048	98	11	a	a	PRON
ejpam-7048	98	12	is	be	AUX
ejpam-7048	98	13	called	call	VERB
ejpam-7048	98	14	the	the	DET
ejpam-7048	98	15	β	β	NOUN
ejpam-7048	98	16	-	-	NOUN
ejpam-7048	98	17	i	i	PRON
ejpam-7048	98	18	⋆-closure	⋆-closure	ADJ
ejpam-7048	98	19	of	of	ADP
ejpam-7048	98	20	a	a	PRON
ejpam-7048	98	21	and	and	CCONJ
ejpam-7048	98	22	is	be	AUX
ejpam-7048	98	23	denoted	denote	VERB
ejpam-7048	98	24	by	by	ADP
ejpam-7048	98	25	βcl⋆(a	βcl⋆(a	NOUN
ejpam-7048	98	26	)	)	PUNCT
ejpam-7048	98	27	.	.	PUNCT
ejpam-7048	99	1	the	the	DET
ejpam-7048	99	2	union	union	NOUN
ejpam-7048	99	3	of	of	ADP
ejpam-7048	99	4	all	all	PRON
ejpam-7048	99	5	β	β	NOUN
ejpam-7048	99	6	-	-	ADJ
ejpam-7048	99	7	i	i	PRON
ejpam-7048	99	8	⋆-open	⋆-open	ADJ
ejpam-7048	99	9	sets	set	NOUN
ejpam-7048	99	10	contained	contain	VERB
ejpam-7048	99	11	in	in	ADP
ejpam-7048	99	12	a	a	PRON
ejpam-7048	99	13	is	be	AUX
ejpam-7048	99	14	called	call	VERB
ejpam-7048	99	15	the	the	DET
ejpam-7048	99	16	β	β	NOUN
ejpam-7048	99	17	-	-	PUNCT
ejpam-7048	99	18	i	i	PRON
ejpam-7048	99	19	⋆-interior	⋆-interior	NOUN
ejpam-7048	99	20	of	of	ADP
ejpam-7048	99	21	a	a	PRON
ejpam-7048	99	22	and	and	CCONJ
ejpam-7048	99	23	is	be	AUX
ejpam-7048	99	24	denoted	denote	VERB
ejpam-7048	99	25	by	by	ADP
ejpam-7048	99	26	βint⋆(a	βint⋆(a	NOUN
ejpam-7048	99	27	)	)	PUNCT
ejpam-7048	99	28	.	.	PUNCT
ejpam-7048	100	1	lemma	lemma	PROPN
ejpam-7048	100	2	2	2	NUM
ejpam-7048	100	3	.	.	X
ejpam-7048	101	1	for	for	ADP
ejpam-7048	101	2	a	a	DET
ejpam-7048	101	3	subset	subset	NOUN
ejpam-7048	101	4	a	a	PRON
ejpam-7048	101	5	of	of	ADP
ejpam-7048	101	6	an	an	DET
ejpam-7048	101	7	ideal	ideal	ADJ
ejpam-7048	101	8	topological	topological	ADJ
ejpam-7048	101	9	space	space	NOUN
ejpam-7048	101	10	(	(	PUNCT
ejpam-7048	101	11	x	x	X
ejpam-7048	101	12	,	,	PUNCT
ejpam-7048	101	13	τ	τ	PROPN
ejpam-7048	101	14	,	,	PUNCT
ejpam-7048	101	15	i	i	NOUN
ejpam-7048	101	16	)	)	PUNCT
ejpam-7048	101	17	,	,	PUNCT
ejpam-7048	101	18	the	the	DET
ejpam-7048	101	19	following	follow	VERB
ejpam-7048	101	20	properties	property	NOUN
ejpam-7048	101	21	hold	hold	VERB
ejpam-7048	101	22	:	:	PUNCT
ejpam-7048	101	23	(	(	PUNCT
ejpam-7048	101	24	1	1	X
ejpam-7048	101	25	)	)	PUNCT
ejpam-7048	101	26	scl⋆(a	scl⋆(a	NUM
ejpam-7048	101	27	)	)	PUNCT
ejpam-7048	101	28	=	=	PUNCT
ejpam-7048	102	1	a	a	DET
ejpam-7048	102	2	∪	∪	ADJ
ejpam-7048	102	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7048	102	4	)	)	PUNCT
ejpam-7048	102	5	)	)	PUNCT
ejpam-7048	103	1	[	[	X
ejpam-7048	103	2	25	25	NUM
ejpam-7048	103	3	]	]	PUNCT
ejpam-7048	103	4	.	.	PUNCT
ejpam-7048	104	1	p.	p.	NOUN
ejpam-7048	104	2	pue	pue	NOUN
ejpam-7048	104	3	-	-	PUNCT
ejpam-7048	104	4	on	on	ADP
ejpam-7048	104	5	,	,	PUNCT
ejpam-7048	104	6	a.	a.	PROPN
ejpam-7048	104	7	sama	sama	PROPN
ejpam-7048	104	8	-	-	PUNCT
ejpam-7048	104	9	ae	ae	PROPN
ejpam-7048	104	10	,	,	PUNCT
ejpam-7048	104	11	c.	c.	PROPN
ejpam-7048	104	12	boonpok	boonpok	PROPN
ejpam-7048	104	13	/	/	SYM
ejpam-7048	104	14	eur	eur	PROPN
ejpam-7048	104	15	.	.	PUNCT
ejpam-7048	105	1	j.	j.	PROPN
ejpam-7048	105	2	pure	pure	PROPN
ejpam-7048	105	3	appl	appl	PROPN
ejpam-7048	105	4	.	.	PROPN
ejpam-7048	105	5	math	math	PROPN
ejpam-7048	105	6	,	,	PUNCT
ejpam-7048	105	7	18	18	NUM
ejpam-7048	105	8	(	(	PUNCT
ejpam-7048	105	9	4	4	NUM
ejpam-7048	105	10	)	)	PUNCT
ejpam-7048	105	11	(	(	PUNCT
ejpam-7048	105	12	2025	2025	NUM
ejpam-7048	105	13	)	)	PUNCT
ejpam-7048	105	14	,	,	PUNCT
ejpam-7048	105	15	7048	7048	NUM
ejpam-7048	105	16	4	4	NUM
ejpam-7048	105	17	of	of	ADP
ejpam-7048	105	18	9	9	NUM
ejpam-7048	105	19	(	(	PUNCT
ejpam-7048	105	20	2	2	NUM
ejpam-7048	105	21	)	)	PUNCT
ejpam-7048	105	22	sint⋆(a	sint⋆(a	PROPN
ejpam-7048	105	23	)	)	PUNCT
ejpam-7048	106	1	=	=	PUNCT
ejpam-7048	106	2	a	a	DET
ejpam-7048	106	3	∩	∩	ADJ
ejpam-7048	106	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7048	106	5	)	)	PUNCT
ejpam-7048	106	6	)	)	PUNCT
ejpam-7048	107	1	[	[	X
ejpam-7048	107	2	25	25	NUM
ejpam-7048	107	3	]	]	PUNCT
ejpam-7048	107	4	.	.	PUNCT
ejpam-7048	108	1	(	(	PUNCT
ejpam-7048	108	2	3	3	NUM
ejpam-7048	108	3	)	)	PUNCT
ejpam-7048	108	4	βcl⋆(a	βcl⋆(a	NUM
ejpam-7048	108	5	)	)	PUNCT
ejpam-7048	109	1	=	=	PUNCT
ejpam-7048	109	2	a	a	DET
ejpam-7048	109	3	∪	∪	ADJ
ejpam-7048	109	4	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7048	109	5	)	)	PUNCT
ejpam-7048	109	6	)	)	PUNCT
ejpam-7048	109	7	)	)	PUNCT
ejpam-7048	109	8	.	.	PUNCT
ejpam-7048	110	1	(	(	PUNCT
ejpam-7048	110	2	4	4	NUM
ejpam-7048	110	3	)	)	PUNCT
ejpam-7048	110	4	βint⋆(a	βint⋆(a	NUM
ejpam-7048	110	5	)	)	PUNCT
ejpam-7048	110	6	=	=	PUNCT
ejpam-7048	110	7	a	a	DET
ejpam-7048	110	8	∩	∩	NOUN
ejpam-7048	110	9	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7048	110	10	)	)	PUNCT
ejpam-7048	110	11	)	)	PUNCT
ejpam-7048	110	12	)	)	PUNCT
ejpam-7048	110	13	.	.	PUNCT
ejpam-7048	111	1	by	by	ADP
ejpam-7048	111	2	a	a	DET
ejpam-7048	111	3	multifunction	multifunction	NOUN
ejpam-7048	111	4	f	f	NOUN
ejpam-7048	111	5	:	:	PUNCT
ejpam-7048	111	6	x	x	X
ejpam-7048	111	7	→	→	SYM
ejpam-7048	111	8	y	y	PROPN
ejpam-7048	111	9	,	,	PUNCT
ejpam-7048	111	10	we	we	PRON
ejpam-7048	111	11	mean	mean	VERB
ejpam-7048	111	12	a	a	DET
ejpam-7048	111	13	point	point	NOUN
ejpam-7048	111	14	-	-	PUNCT
ejpam-7048	111	15	to	to	ADP
ejpam-7048	111	16	-	-	PUNCT
ejpam-7048	111	17	set	set	VERB
ejpam-7048	111	18	correspondence	correspondence	NOUN
ejpam-7048	111	19	from	from	ADP
ejpam-7048	111	20	x	x	PUNCT
ejpam-7048	111	21	into	into	ADP
ejpam-7048	111	22	y	y	PROPN
ejpam-7048	111	23	,	,	PUNCT
ejpam-7048	111	24	and	and	CCONJ
ejpam-7048	111	25	we	we	PRON
ejpam-7048	111	26	always	always	ADV
ejpam-7048	111	27	assume	assume	VERB
ejpam-7048	111	28	that	that	SCONJ
ejpam-7048	111	29	f	f	PROPN
ejpam-7048	111	30	(	(	PUNCT
ejpam-7048	111	31	x	x	X
ejpam-7048	111	32	)	)	PUNCT
ejpam-7048	111	33	̸=	̸=	NOUN
ejpam-7048	111	34	∅	∅	NOUN
ejpam-7048	111	35	for	for	ADP
ejpam-7048	111	36	all	all	PRON
ejpam-7048	111	37	x	x	SYM
ejpam-7048	111	38	∈	∈	ADJ
ejpam-7048	111	39	x.	x.	NOUN
ejpam-7048	111	40	for	for	ADP
ejpam-7048	111	41	a	a	DET
ejpam-7048	111	42	multifunction	multifunction	NOUN
ejpam-7048	111	43	f	f	NOUN
ejpam-7048	111	44	:	:	PUNCT
ejpam-7048	111	45	x	x	X
ejpam-7048	111	46	→	→	SYM
ejpam-7048	111	47	y	y	PROPN
ejpam-7048	111	48	,	,	PUNCT
ejpam-7048	111	49	we	we	PRON
ejpam-7048	111	50	shall	shall	AUX
ejpam-7048	111	51	denote	denote	VERB
ejpam-7048	111	52	the	the	DET
ejpam-7048	111	53	upper	upper	ADJ
ejpam-7048	111	54	and	and	CCONJ
ejpam-7048	111	55	lower	low	ADJ
ejpam-7048	111	56	inverse	inverse	NOUN
ejpam-7048	111	57	of	of	ADP
ejpam-7048	111	58	a	a	DET
ejpam-7048	111	59	set	set	NOUN
ejpam-7048	111	60	b	b	PROPN
ejpam-7048	111	61	of	of	ADP
ejpam-7048	111	62	y	y	PROPN
ejpam-7048	111	63	by	by	ADP
ejpam-7048	111	64	f+(b	f+(b	NOUN
ejpam-7048	111	65	)	)	PUNCT
ejpam-7048	111	66	and	and	CCONJ
ejpam-7048	111	67	f−(b	f−(b	NOUN
ejpam-7048	111	68	)	)	PUNCT
ejpam-7048	111	69	,	,	PUNCT
ejpam-7048	111	70	respectively	respectively	ADV
ejpam-7048	111	71	,	,	PUNCT
ejpam-7048	111	72	that	that	ADV
ejpam-7048	111	73	is	is	ADV
ejpam-7048	111	74	,	,	PUNCT
ejpam-7048	111	75	f+(b	f+(b	NOUN
ejpam-7048	111	76	)	)	PUNCT
ejpam-7048	111	77	=	=	PRON
ejpam-7048	112	1	{	{	PUNCT
ejpam-7048	112	2	x	x	PUNCT
ejpam-7048	112	3	∈	∈	PROPN
ejpam-7048	112	4	x	x	INTJ
ejpam-7048	113	1	|	|	NOUN
ejpam-7048	113	2	f	f	X
ejpam-7048	113	3	(	(	PUNCT
ejpam-7048	113	4	x	x	NOUN
ejpam-7048	113	5	)	)	PUNCT
ejpam-7048	113	6	⊆	⊆	NUM
ejpam-7048	113	7	b	b	NOUN
ejpam-7048	113	8	}	}	PUNCT
ejpam-7048	113	9	and	and	CCONJ
ejpam-7048	113	10	f−(b	f−(b	PROPN
ejpam-7048	113	11	)	)	PUNCT
ejpam-7048	113	12	=	=	PRON
ejpam-7048	114	1	{	{	PUNCT
ejpam-7048	114	2	x	x	PUNCT
ejpam-7048	114	3	∈	∈	PROPN
ejpam-7048	114	4	x	x	INTJ
ejpam-7048	115	1	|	|	NOUN
ejpam-7048	115	2	f	f	X
ejpam-7048	115	3	(	(	PUNCT
ejpam-7048	115	4	x	x	NOUN
ejpam-7048	115	5	)	)	PUNCT
ejpam-7048	115	6	∩	∩	NOUN
ejpam-7048	115	7	b	b	PROPN
ejpam-7048	115	8	̸=	̸=	PROPN
ejpam-7048	115	9	∅	∅	NOUN
ejpam-7048	115	10	}	}	PUNCT
ejpam-7048	115	11	.	.	PUNCT
ejpam-7048	116	1	in	in	ADP
ejpam-7048	116	2	particular	particular	ADJ
ejpam-7048	116	3	,	,	PUNCT
ejpam-7048	116	4	f−(y	f−(y	NOUN
ejpam-7048	116	5	)	)	PUNCT
ejpam-7048	116	6	=	=	SYM
ejpam-7048	117	1	{	{	PUNCT
ejpam-7048	117	2	x	x	PUNCT
ejpam-7048	117	3	∈	∈	PROPN
ejpam-7048	117	4	x	x	INTJ
ejpam-7048	118	1	|	|	ADV
ejpam-7048	118	2	y	y	PROPN
ejpam-7048	118	3	∈	∈	PROPN
ejpam-7048	118	4	f	f	X
ejpam-7048	118	5	(	(	PUNCT
ejpam-7048	118	6	x	x	NOUN
ejpam-7048	118	7	)	)	PUNCT
ejpam-7048	118	8	}	}	PUNCT
ejpam-7048	118	9	for	for	ADP
ejpam-7048	118	10	each	each	DET
ejpam-7048	118	11	point	point	NOUN
ejpam-7048	118	12	y	y	PROPN
ejpam-7048	118	13	∈	∈	PROPN
ejpam-7048	118	14	y	y	PROPN
ejpam-7048	118	15	.	.	PUNCT
ejpam-7048	119	1	for	for	ADP
ejpam-7048	119	2	each	each	DET
ejpam-7048	119	3	a	a	DET
ejpam-7048	119	4	⊆	⊆	NUM
ejpam-7048	119	5	x	x	SYM
ejpam-7048	119	6	,	,	PUNCT
ejpam-7048	119	7	f	f	PROPN
ejpam-7048	119	8	(	(	PUNCT
ejpam-7048	119	9	a	a	NOUN
ejpam-7048	119	10	)	)	PUNCT
ejpam-7048	119	11	=	=	SYM
ejpam-7048	119	12	∪x∈af	∪x∈af	NOUN
ejpam-7048	119	13	(	(	PUNCT
ejpam-7048	119	14	x	x	NOUN
ejpam-7048	119	15	)	)	PUNCT
ejpam-7048	119	16	.	.	PUNCT
ejpam-7048	120	1	3	3	X
ejpam-7048	120	2	.	.	X
ejpam-7048	120	3	upper	upper	ADJ
ejpam-7048	120	4	and	and	CCONJ
ejpam-7048	120	5	lower	low	ADJ
ejpam-7048	120	6	τ	τ	X
ejpam-7048	120	7	⋆β(σ1	⋆β(σ1	NUM
ejpam-7048	120	8	,	,	PUNCT
ejpam-7048	120	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	120	10	multifunctions	multifunction	NOUN
ejpam-7048	120	11	in	in	ADP
ejpam-7048	120	12	this	this	DET
ejpam-7048	120	13	section	section	NOUN
ejpam-7048	120	14	,	,	PUNCT
ejpam-7048	120	15	we	we	PRON
ejpam-7048	120	16	introduce	introduce	VERB
ejpam-7048	120	17	the	the	DET
ejpam-7048	120	18	notions	notion	NOUN
ejpam-7048	120	19	of	of	ADP
ejpam-7048	120	20	upper	upper	ADJ
ejpam-7048	120	21	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	120	22	,	,	PUNCT
ejpam-7048	120	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	120	24	multifunctions	multifunction	NOUN
ejpam-7048	120	25	and	and	CCONJ
ejpam-7048	120	26	lower	low	ADJ
ejpam-7048	120	27	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	120	28	,	,	PUNCT
ejpam-7048	120	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	120	30	multifunctions	multifunction	NOUN
ejpam-7048	120	31	.	.	PUNCT
ejpam-7048	121	1	moreover	moreover	ADV
ejpam-7048	121	2	,	,	PUNCT
ejpam-7048	121	3	several	several	ADJ
ejpam-7048	121	4	characterizations	characterization	NOUN
ejpam-7048	121	5	of	of	ADP
ejpam-7048	121	6	upper	upper	ADJ
ejpam-7048	121	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	121	8	,	,	PUNCT
ejpam-7048	121	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	121	10	multifunctions	multifunction	NOUN
ejpam-7048	121	11	and	and	CCONJ
ejpam-7048	121	12	lower	low	ADJ
ejpam-7048	121	13	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	121	14	,	,	PUNCT
ejpam-7048	121	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	121	16	multifunctions	multifunction	NOUN
ejpam-7048	121	17	discussed	discuss	VERB
ejpam-7048	121	18	.	.	PUNCT
ejpam-7048	122	1	definition	definition	NOUN
ejpam-7048	122	2	1	1	NUM
ejpam-7048	122	3	.	.	PUNCT
ejpam-7048	123	1	a	a	DET
ejpam-7048	123	2	multifunction	multifunction	NOUN
ejpam-7048	123	3	f	f	NOUN
ejpam-7048	123	4	:	:	PUNCT
ejpam-7048	123	5	(	(	PUNCT
ejpam-7048	123	6	x	x	X
ejpam-7048	123	7	,	,	PUNCT
ejpam-7048	123	8	τ	τ	PROPN
ejpam-7048	123	9	,	,	PUNCT
ejpam-7048	123	10	i	i	NOUN
ejpam-7048	123	11	)	)	PUNCT
ejpam-7048	123	12	→	→	PUNCT
ejpam-7048	123	13	(	(	PUNCT
ejpam-7048	123	14	y	y	PROPN
ejpam-7048	123	15	,	,	PUNCT
ejpam-7048	123	16	σ1	σ1	PROPN
ejpam-7048	123	17	,	,	PUNCT
ejpam-7048	123	18	σ2	σ2	PROPN
ejpam-7048	123	19	)	)	PUNCT
ejpam-7048	123	20	is	be	AUX
ejpam-7048	123	21	said	say	VERB
ejpam-7048	123	22	to	to	PART
ejpam-7048	123	23	be	be	AUX
ejpam-7048	123	24	upper	upper	ADJ
ejpam-7048	123	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	123	26	,	,	PUNCT
ejpam-7048	123	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7048	123	28	at	at	ADP
ejpam-7048	123	29	a	a	DET
ejpam-7048	123	30	point	point	NOUN
ejpam-7048	123	31	x	x	PUNCT
ejpam-7048	123	32	of	of	ADP
ejpam-7048	123	33	x	x	PRON
ejpam-7048	123	34	if	if	SCONJ
ejpam-7048	123	35	for	for	ADP
ejpam-7048	123	36	each	each	DET
ejpam-7048	123	37	σ1σ2	σ1σ2	VERB
ejpam-7048	123	38	-	-	ADJ
ejpam-7048	123	39	open	open	ADJ
ejpam-7048	123	40	set	set	NOUN
ejpam-7048	123	41	v	v	NOUN
ejpam-7048	123	42	of	of	ADP
ejpam-7048	123	43	y	y	PRON
ejpam-7048	123	44	such	such	ADJ
ejpam-7048	123	45	that	that	SCONJ
ejpam-7048	123	46	f	f	PROPN
ejpam-7048	123	47	(	(	PUNCT
ejpam-7048	123	48	x	x	X
ejpam-7048	123	49	)	)	PUNCT
ejpam-7048	123	50	⊆	⊆	NUM
ejpam-7048	123	51	v	v	NOUN
ejpam-7048	123	52	,	,	PUNCT
ejpam-7048	123	53	there	there	PRON
ejpam-7048	123	54	exists	exist	VERB
ejpam-7048	123	55	a	a	DET
ejpam-7048	123	56	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	123	57	-	-	ADJ
ejpam-7048	123	58	open	open	ADJ
ejpam-7048	123	59	set	set	NOUN
ejpam-7048	123	60	u	u	NOUN
ejpam-7048	123	61	of	of	ADP
ejpam-7048	123	62	x	x	PUNCT
ejpam-7048	123	63	containing	contain	VERB
ejpam-7048	123	64	x	x	PUNCT
ejpam-7048	123	65	such	such	ADJ
ejpam-7048	123	66	that	that	SCONJ
ejpam-7048	123	67	f	f	PROPN
ejpam-7048	123	68	(	(	PUNCT
ejpam-7048	123	69	u	u	NOUN
ejpam-7048	123	70	)	)	PUNCT
ejpam-7048	123	71	⊆	⊆	NUM
ejpam-7048	123	72	v	v	NOUN
ejpam-7048	123	73	.	.	PUNCT
ejpam-7048	124	1	a	a	DET
ejpam-7048	124	2	multifunction	multifunction	NOUN
ejpam-7048	124	3	f	f	NOUN
ejpam-7048	124	4	:	:	PUNCT
ejpam-7048	124	5	(	(	PUNCT
ejpam-7048	124	6	x	x	X
ejpam-7048	124	7	,	,	PUNCT
ejpam-7048	124	8	τ	τ	PROPN
ejpam-7048	124	9	,	,	PUNCT
ejpam-7048	124	10	i	i	NOUN
ejpam-7048	124	11	)	)	PUNCT
ejpam-7048	124	12	→	→	PUNCT
ejpam-7048	124	13	(	(	PUNCT
ejpam-7048	124	14	y	y	PROPN
ejpam-7048	124	15	,	,	PUNCT
ejpam-7048	124	16	σ1	σ1	PROPN
ejpam-7048	124	17	,	,	PUNCT
ejpam-7048	124	18	σ2	σ2	PROPN
ejpam-7048	124	19	)	)	PUNCT
ejpam-7048	124	20	is	be	AUX
ejpam-7048	124	21	said	say	VERB
ejpam-7048	124	22	to	to	PART
ejpam-7048	124	23	be	be	AUX
ejpam-7048	124	24	upper	upper	ADJ
ejpam-7048	124	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	124	26	,	,	PUNCT
ejpam-7048	124	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	124	28	if	if	SCONJ
ejpam-7048	124	29	f	f	PROPN
ejpam-7048	124	30	is	be	AUX
ejpam-7048	124	31	upper	upper	ADJ
ejpam-7048	124	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	124	33	,	,	PUNCT
ejpam-7048	124	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	124	35	at	at	ADP
ejpam-7048	124	36	each	each	DET
ejpam-7048	124	37	point	point	NOUN
ejpam-7048	124	38	of	of	ADP
ejpam-7048	124	39	x.	x.	NOUN
ejpam-7048	124	40	theorem	theorem	VERB
ejpam-7048	124	41	1	1	NUM
ejpam-7048	124	42	.	.	PUNCT
ejpam-7048	125	1	a	a	DET
ejpam-7048	125	2	multifunction	multifunction	NOUN
ejpam-7048	125	3	f	f	NOUN
ejpam-7048	125	4	:	:	PUNCT
ejpam-7048	125	5	(	(	PUNCT
ejpam-7048	125	6	x	x	X
ejpam-7048	125	7	,	,	PUNCT
ejpam-7048	125	8	τ	τ	PROPN
ejpam-7048	125	9	,	,	PUNCT
ejpam-7048	125	10	i	i	NOUN
ejpam-7048	125	11	)	)	PUNCT
ejpam-7048	125	12	→	→	PUNCT
ejpam-7048	125	13	(	(	PUNCT
ejpam-7048	125	14	y	y	PROPN
ejpam-7048	125	15	,	,	PUNCT
ejpam-7048	125	16	σ1	σ1	PROPN
ejpam-7048	125	17	,	,	PUNCT
ejpam-7048	125	18	σ2	σ2	PROPN
ejpam-7048	125	19	)	)	PUNCT
ejpam-7048	125	20	is	be	AUX
ejpam-7048	125	21	upper	upper	ADJ
ejpam-7048	125	22	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	125	23	,	,	PUNCT
ejpam-7048	125	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	125	25	at	at	ADP
ejpam-7048	125	26	x	x	X
ejpam-7048	125	27	∈	∈	PROPN
ejpam-7048	125	28	x	x	SYM
ejpam-7048	125	29	if	if	SCONJ
ejpam-7048	125	30	and	and	CCONJ
ejpam-7048	125	31	only	only	ADV
ejpam-7048	125	32	if	if	SCONJ
ejpam-7048	125	33	x	x	PROPN
ejpam-7048	125	34	∈	∈	PROPN
ejpam-7048	125	35	βint⋆(f+(v	βint⋆(f+(v	PROPN
ejpam-7048	125	36	)	)	PUNCT
ejpam-7048	125	37	)	)	PUNCT
ejpam-7048	125	38	for	for	ADP
ejpam-7048	125	39	every	every	DET
ejpam-7048	125	40	σ1σ2	σ1σ2	NOUN
ejpam-7048	125	41	-	-	ADJ
ejpam-7048	125	42	open	open	ADJ
ejpam-7048	125	43	set	set	NOUN
ejpam-7048	125	44	v	v	NOUN
ejpam-7048	125	45	of	of	ADP
ejpam-7048	125	46	y	y	PROPN
ejpam-7048	125	47	containing	contain	VERB
ejpam-7048	125	48	f	f	PROPN
ejpam-7048	125	49	(	(	PUNCT
ejpam-7048	125	50	x	x	NOUN
ejpam-7048	125	51	)	)	PUNCT
ejpam-7048	125	52	.	.	PUNCT
ejpam-7048	126	1	proof	proof	NOUN
ejpam-7048	126	2	.	.	PUNCT
ejpam-7048	127	1	let	let	VERB
ejpam-7048	127	2	v	v	PART
ejpam-7048	127	3	be	be	AUX
ejpam-7048	127	4	any	any	DET
ejpam-7048	127	5	σ1σ2	σ1σ2	NOUN
ejpam-7048	127	6	-	-	ADJ
ejpam-7048	127	7	open	open	ADJ
ejpam-7048	127	8	set	set	NOUN
ejpam-7048	127	9	of	of	ADP
ejpam-7048	127	10	y	y	PROPN
ejpam-7048	127	11	containing	contain	VERB
ejpam-7048	127	12	f	f	PROPN
ejpam-7048	127	13	(	(	PUNCT
ejpam-7048	127	14	x	x	NOUN
ejpam-7048	127	15	)	)	PUNCT
ejpam-7048	127	16	.	.	PUNCT
ejpam-7048	128	1	then	then	ADV
ejpam-7048	128	2	,	,	PUNCT
ejpam-7048	128	3	there	there	PRON
ejpam-7048	128	4	exists	exist	VERB
ejpam-7048	128	5	a	a	DET
ejpam-7048	128	6	τ⋆-βopen	τ⋆-βopen	ADJ
ejpam-7048	128	7	set	set	NOUN
ejpam-7048	128	8	u	u	NOUN
ejpam-7048	128	9	of	of	ADP
ejpam-7048	128	10	x	x	PUNCT
ejpam-7048	128	11	containing	contain	VERB
ejpam-7048	128	12	x	x	PUNCT
ejpam-7048	128	13	such	such	ADJ
ejpam-7048	128	14	that	that	SCONJ
ejpam-7048	128	15	f	f	PROPN
ejpam-7048	128	16	(	(	PUNCT
ejpam-7048	128	17	u	u	NOUN
ejpam-7048	128	18	)	)	PUNCT
ejpam-7048	128	19	⊆	⊆	NUM
ejpam-7048	128	20	v	v	NOUN
ejpam-7048	128	21	.	.	PUNCT
ejpam-7048	129	1	then	then	ADV
ejpam-7048	129	2	,	,	PUNCT
ejpam-7048	129	3	u	u	NOUN
ejpam-7048	129	4	⊆	⊆	NUM
ejpam-7048	129	5	f+(v	f+(v	NUM
ejpam-7048	129	6	)	)	PUNCT
ejpam-7048	129	7	.	.	PUNCT
ejpam-7048	130	1	since	since	SCONJ
ejpam-7048	130	2	u	u	NOUN
ejpam-7048	130	3	is	be	AUX
ejpam-7048	130	4	τ⋆β	τ⋆β	NOUN
ejpam-7048	130	5	-	-	PUNCT
ejpam-7048	130	6	open	open	ADJ
ejpam-7048	130	7	,	,	PUNCT
ejpam-7048	130	8	we	we	PRON
ejpam-7048	130	9	have	have	VERB
ejpam-7048	130	10	x	x	X
ejpam-7048	130	11	∈	∈	PROPN
ejpam-7048	130	12	u	u	NOUN
ejpam-7048	130	13	⊆	⊆	NUM
ejpam-7048	130	14	cl⋆(int⋆(cl⋆(u	cl⋆(int⋆(cl⋆(u	NOUN
ejpam-7048	130	15	)	)	PUNCT
ejpam-7048	130	16	)	)	PUNCT
ejpam-7048	130	17	)	)	PUNCT
ejpam-7048	131	1	⊆	⊆	NUM
ejpam-7048	131	2	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	131	3	)	)	PUNCT
ejpam-7048	131	4	)	)	PUNCT
ejpam-7048	131	5	)	)	PUNCT
ejpam-7048	131	6	)	)	PUNCT
ejpam-7048	131	7	.	.	PUNCT
ejpam-7048	132	1	since	since	SCONJ
ejpam-7048	132	2	x	x	PROPN
ejpam-7048	132	3	∈	∈	PROPN
ejpam-7048	132	4	f+(v	f+(v	NOUN
ejpam-7048	132	5	)	)	PUNCT
ejpam-7048	132	6	and	and	CCONJ
ejpam-7048	132	7	by	by	ADP
ejpam-7048	132	8	lemma	lemma	PROPN
ejpam-7048	132	9	2	2	NUM
ejpam-7048	132	10	,	,	PUNCT
ejpam-7048	132	11	x	x	SYM
ejpam-7048	132	12	∈	∈	NOUN
ejpam-7048	132	13	f+(v	f+(v	NOUN
ejpam-7048	132	14	)	)	PUNCT
ejpam-7048	132	15	∩	∩	NOUN
ejpam-7048	132	16	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	132	17	)	)	PUNCT
ejpam-7048	132	18	)	)	PUNCT
ejpam-7048	132	19	)	)	PUNCT
ejpam-7048	132	20	)	)	PUNCT
ejpam-7048	133	1	=	=	SYM
ejpam-7048	133	2	βint⋆(f+(v	βint⋆(f+(v	PROPN
ejpam-7048	133	3	)	)	PUNCT
ejpam-7048	133	4	)	)	PUNCT
ejpam-7048	133	5	.	.	PUNCT
ejpam-7048	134	1	conversely	conversely	ADV
ejpam-7048	134	2	,	,	PUNCT
ejpam-7048	134	3	let	let	VERB
ejpam-7048	134	4	v	v	PART
ejpam-7048	134	5	be	be	AUX
ejpam-7048	134	6	any	any	DET
ejpam-7048	134	7	σ1σ2	σ1σ2	NOUN
ejpam-7048	134	8	-	-	ADJ
ejpam-7048	134	9	open	open	ADJ
ejpam-7048	134	10	set	set	NOUN
ejpam-7048	134	11	of	of	ADP
ejpam-7048	134	12	y	y	PROPN
ejpam-7048	134	13	containing	contain	VERB
ejpam-7048	134	14	f	f	PROPN
ejpam-7048	134	15	(	(	PUNCT
ejpam-7048	134	16	x	x	NOUN
ejpam-7048	134	17	)	)	PUNCT
ejpam-7048	134	18	.	.	PUNCT
ejpam-7048	135	1	by	by	ADP
ejpam-7048	135	2	(	(	PUNCT
ejpam-7048	135	3	2	2	NUM
ejpam-7048	135	4	)	)	PUNCT
ejpam-7048	135	5	,	,	PUNCT
ejpam-7048	135	6	x	x	PUNCT
ejpam-7048	135	7	∈	∈	PROPN
ejpam-7048	135	8	sβint⋆(f+(v	sβint⋆(f+(v	PROPN
ejpam-7048	135	9	)	)	PUNCT
ejpam-7048	135	10	)	)	PUNCT
ejpam-7048	136	1	and	and	CCONJ
ejpam-7048	136	2	so	so	ADV
ejpam-7048	136	3	there	there	PRON
ejpam-7048	136	4	exists	exist	VERB
ejpam-7048	136	5	a	a	DET
ejpam-7048	136	6	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	136	7	-	-	ADJ
ejpam-7048	136	8	open	open	ADJ
ejpam-7048	136	9	set	set	NOUN
ejpam-7048	136	10	u	u	NOUN
ejpam-7048	136	11	of	of	ADP
ejpam-7048	136	12	x	x	PUNCT
ejpam-7048	136	13	containing	contain	VERB
ejpam-7048	136	14	x	x	PUNCT
ejpam-7048	136	15	such	such	ADJ
ejpam-7048	136	16	that	that	SCONJ
ejpam-7048	136	17	u	u	NOUN
ejpam-7048	136	18	⊆	⊆	NUM
ejpam-7048	136	19	f+(v	f+(v	NOUN
ejpam-7048	136	20	)	)	PUNCT
ejpam-7048	136	21	;	;	PUNCT
ejpam-7048	136	22	hence	hence	ADV
ejpam-7048	136	23	f	f	PROPN
ejpam-7048	136	24	(	(	PUNCT
ejpam-7048	136	25	u	u	NOUN
ejpam-7048	136	26	)	)	PUNCT
ejpam-7048	136	27	⊆	⊆	NUM
ejpam-7048	136	28	v	v	NOUN
ejpam-7048	136	29	.	.	PUNCT
ejpam-7048	137	1	this	this	PRON
ejpam-7048	137	2	shows	show	VERB
ejpam-7048	137	3	that	that	SCONJ
ejpam-7048	137	4	f	f	PROPN
ejpam-7048	137	5	is	be	AUX
ejpam-7048	137	6	upper	upper	ADJ
ejpam-7048	137	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	137	8	,	,	PUNCT
ejpam-7048	137	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	137	10	at	at	ADP
ejpam-7048	137	11	x.	x.	NOUN
ejpam-7048	137	12	definition	definition	NOUN
ejpam-7048	137	13	2	2	NUM
ejpam-7048	137	14	.	.	PUNCT
ejpam-7048	137	15	a	a	DET
ejpam-7048	137	16	multifunction	multifunction	NOUN
ejpam-7048	138	1	f	f	NOUN
ejpam-7048	138	2	:	:	PUNCT
ejpam-7048	138	3	(	(	PUNCT
ejpam-7048	138	4	x	x	X
ejpam-7048	138	5	,	,	PUNCT
ejpam-7048	138	6	τ	τ	PROPN
ejpam-7048	138	7	,	,	PUNCT
ejpam-7048	138	8	i	i	NOUN
ejpam-7048	138	9	)	)	PUNCT
ejpam-7048	138	10	→	→	PUNCT
ejpam-7048	138	11	(	(	PUNCT
ejpam-7048	138	12	y	y	PROPN
ejpam-7048	138	13	,	,	PUNCT
ejpam-7048	138	14	σ1	σ1	PROPN
ejpam-7048	138	15	,	,	PUNCT
ejpam-7048	138	16	σ2	σ2	PROPN
ejpam-7048	138	17	)	)	PUNCT
ejpam-7048	138	18	is	be	AUX
ejpam-7048	138	19	said	say	VERB
ejpam-7048	138	20	to	to	PART
ejpam-7048	138	21	be	be	AUX
ejpam-7048	138	22	lower	low	ADJ
ejpam-7048	138	23	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	138	24	,	,	PUNCT
ejpam-7048	138	25	σ2)continuous	σ2)continuous	ADJ
ejpam-7048	138	26	at	at	ADP
ejpam-7048	138	27	a	a	DET
ejpam-7048	138	28	point	point	NOUN
ejpam-7048	138	29	x	x	PUNCT
ejpam-7048	138	30	of	of	ADP
ejpam-7048	138	31	x	x	PRON
ejpam-7048	138	32	if	if	SCONJ
ejpam-7048	138	33	for	for	ADP
ejpam-7048	138	34	each	each	DET
ejpam-7048	138	35	σ1σ2	σ1σ2	VERB
ejpam-7048	138	36	-	-	ADJ
ejpam-7048	138	37	open	open	ADJ
ejpam-7048	138	38	set	set	NOUN
ejpam-7048	138	39	v	v	NOUN
ejpam-7048	138	40	of	of	ADP
ejpam-7048	138	41	y	y	PRON
ejpam-7048	138	42	such	such	ADJ
ejpam-7048	138	43	that	that	SCONJ
ejpam-7048	138	44	f	f	PROPN
ejpam-7048	138	45	(	(	PUNCT
ejpam-7048	138	46	x	x	NOUN
ejpam-7048	138	47	)	)	PUNCT
ejpam-7048	138	48	∩	∩	NOUN
ejpam-7048	138	49	v	v	ADP
ejpam-7048	138	50	̸=	̸=	PROPN
ejpam-7048	138	51	∅	∅	NOUN
ejpam-7048	138	52	,	,	PUNCT
ejpam-7048	138	53	there	there	PRON
ejpam-7048	138	54	exists	exist	VERB
ejpam-7048	138	55	a	a	DET
ejpam-7048	138	56	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	138	57	-	-	ADJ
ejpam-7048	138	58	open	open	ADJ
ejpam-7048	138	59	set	set	NOUN
ejpam-7048	138	60	u	u	NOUN
ejpam-7048	138	61	of	of	ADP
ejpam-7048	138	62	x	x	PUNCT
ejpam-7048	138	63	containing	contain	VERB
ejpam-7048	138	64	x	x	PUNCT
ejpam-7048	138	65	such	such	ADJ
ejpam-7048	138	66	that	that	SCONJ
ejpam-7048	138	67	f	f	PROPN
ejpam-7048	138	68	(	(	PUNCT
ejpam-7048	138	69	z)∩	z)∩	PROPN
ejpam-7048	138	70	v	v	ADP
ejpam-7048	138	71	̸=	̸=	PROPN
ejpam-7048	138	72	∅	∅	NOUN
ejpam-7048	138	73	for	for	ADP
ejpam-7048	138	74	every	every	DET
ejpam-7048	138	75	z	z	NOUN
ejpam-7048	138	76	∈	∈	PROPN
ejpam-7048	138	77	u	u	NOUN
ejpam-7048	138	78	.	.	PUNCT
ejpam-7048	139	1	a	a	DET
ejpam-7048	139	2	multifunction	multifunction	NOUN
ejpam-7048	139	3	f	f	NOUN
ejpam-7048	139	4	:	:	PUNCT
ejpam-7048	139	5	(	(	PUNCT
ejpam-7048	139	6	x	x	X
ejpam-7048	139	7	,	,	PUNCT
ejpam-7048	139	8	τ	τ	PROPN
ejpam-7048	139	9	,	,	PUNCT
ejpam-7048	139	10	i	i	NOUN
ejpam-7048	139	11	)	)	PUNCT
ejpam-7048	139	12	→	→	PUNCT
ejpam-7048	139	13	(	(	PUNCT
ejpam-7048	139	14	y	y	PROPN
ejpam-7048	139	15	,	,	PUNCT
ejpam-7048	139	16	σ1	σ1	PROPN
ejpam-7048	139	17	,	,	PUNCT
ejpam-7048	139	18	σ2	σ2	PROPN
ejpam-7048	139	19	)	)	PUNCT
ejpam-7048	139	20	is	be	AUX
ejpam-7048	139	21	said	say	VERB
ejpam-7048	139	22	to	to	PART
ejpam-7048	139	23	be	be	AUX
ejpam-7048	139	24	lower	low	ADJ
ejpam-7048	139	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	139	26	,	,	PUNCT
ejpam-7048	139	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	139	28	if	if	SCONJ
ejpam-7048	139	29	f	f	PROPN
ejpam-7048	139	30	is	be	AUX
ejpam-7048	139	31	lower	low	ADJ
ejpam-7048	139	32	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	139	33	,	,	PUNCT
ejpam-7048	139	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	139	35	at	at	ADP
ejpam-7048	139	36	each	each	DET
ejpam-7048	139	37	point	point	NOUN
ejpam-7048	139	38	of	of	ADP
ejpam-7048	139	39	x.	x.	PROPN
ejpam-7048	139	40	p.	p.	NOUN
ejpam-7048	139	41	pue	pue	PROPN
ejpam-7048	139	42	-	-	PUNCT
ejpam-7048	139	43	on	on	ADP
ejpam-7048	139	44	,	,	PUNCT
ejpam-7048	139	45	a.	a.	PROPN
ejpam-7048	139	46	sama	sama	PROPN
ejpam-7048	139	47	-	-	PUNCT
ejpam-7048	139	48	ae	ae	PROPN
ejpam-7048	139	49	,	,	PUNCT
ejpam-7048	139	50	c.	c.	PROPN
ejpam-7048	139	51	boonpok	boonpok	PROPN
ejpam-7048	139	52	/	/	SYM
ejpam-7048	139	53	eur	eur	PROPN
ejpam-7048	139	54	.	.	PUNCT
ejpam-7048	140	1	j.	j.	PROPN
ejpam-7048	140	2	pure	pure	PROPN
ejpam-7048	140	3	appl	appl	PROPN
ejpam-7048	140	4	.	.	PROPN
ejpam-7048	140	5	math	math	PROPN
ejpam-7048	140	6	,	,	PUNCT
ejpam-7048	140	7	18	18	NUM
ejpam-7048	140	8	(	(	PUNCT
ejpam-7048	140	9	4	4	NUM
ejpam-7048	140	10	)	)	PUNCT
ejpam-7048	140	11	(	(	PUNCT
ejpam-7048	140	12	2025	2025	NUM
ejpam-7048	140	13	)	)	PUNCT
ejpam-7048	140	14	,	,	PUNCT
ejpam-7048	140	15	7048	7048	NUM
ejpam-7048	140	16	5	5	NUM
ejpam-7048	140	17	of	of	ADP
ejpam-7048	140	18	9	9	NUM
ejpam-7048	140	19	theorem	theorem	NOUN
ejpam-7048	140	20	2	2	NUM
ejpam-7048	140	21	.	.	PUNCT
ejpam-7048	140	22	a	a	DET
ejpam-7048	140	23	multifunction	multifunction	NOUN
ejpam-7048	140	24	f	f	NOUN
ejpam-7048	140	25	:	:	PUNCT
ejpam-7048	140	26	(	(	PUNCT
ejpam-7048	140	27	x	x	X
ejpam-7048	140	28	,	,	PUNCT
ejpam-7048	140	29	τ	τ	PROPN
ejpam-7048	140	30	,	,	PUNCT
ejpam-7048	140	31	i	i	NOUN
ejpam-7048	140	32	)	)	PUNCT
ejpam-7048	141	1	→	→	PUNCT
ejpam-7048	141	2	(	(	PUNCT
ejpam-7048	141	3	y	y	PROPN
ejpam-7048	141	4	,	,	PUNCT
ejpam-7048	141	5	σ1	σ1	PROPN
ejpam-7048	141	6	,	,	PUNCT
ejpam-7048	141	7	σ2	σ2	NOUN
ejpam-7048	141	8	)	)	PUNCT
ejpam-7048	141	9	is	be	AUX
ejpam-7048	141	10	lower	low	ADJ
ejpam-7048	141	11	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	141	12	,	,	PUNCT
ejpam-7048	141	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	141	14	at	at	ADP
ejpam-7048	141	15	x	x	X
ejpam-7048	141	16	∈	∈	PROPN
ejpam-7048	141	17	x	x	SYM
ejpam-7048	141	18	if	if	SCONJ
ejpam-7048	141	19	and	and	CCONJ
ejpam-7048	141	20	only	only	ADV
ejpam-7048	141	21	if	if	SCONJ
ejpam-7048	141	22	x	x	SYM
ejpam-7048	141	23	∈	∈	NOUN
ejpam-7048	141	24	βint⋆(f−(v	βint⋆(f−(v	NOUN
ejpam-7048	141	25	)	)	PUNCT
ejpam-7048	141	26	)	)	PUNCT
ejpam-7048	141	27	for	for	ADP
ejpam-7048	141	28	every	every	DET
ejpam-7048	141	29	σ1σ2	σ1σ2	NOUN
ejpam-7048	141	30	-	-	ADJ
ejpam-7048	141	31	open	open	ADJ
ejpam-7048	141	32	set	set	NOUN
ejpam-7048	141	33	v	v	NOUN
ejpam-7048	141	34	of	of	ADP
ejpam-7048	141	35	y	y	PRON
ejpam-7048	141	36	such	such	ADJ
ejpam-7048	141	37	that	that	SCONJ
ejpam-7048	141	38	f	f	PROPN
ejpam-7048	141	39	(	(	PUNCT
ejpam-7048	141	40	x	x	NOUN
ejpam-7048	141	41	)	)	PUNCT
ejpam-7048	141	42	∩	∩	NOUN
ejpam-7048	141	43	v	v	ADP
ejpam-7048	141	44	̸=	̸=	PROPN
ejpam-7048	141	45	∅.	∅.	ADP
ejpam-7048	141	46	proof	proof	NOUN
ejpam-7048	141	47	.	.	PUNCT
ejpam-7048	142	1	the	the	DET
ejpam-7048	142	2	proof	proof	NOUN
ejpam-7048	142	3	is	be	AUX
ejpam-7048	142	4	similar	similar	ADJ
ejpam-7048	142	5	to	to	ADP
ejpam-7048	142	6	that	that	PRON
ejpam-7048	142	7	of	of	ADP
ejpam-7048	142	8	theorem	theorem	NOUN
ejpam-7048	142	9	1	1	NUM
ejpam-7048	142	10	.	.	PUNCT
ejpam-7048	142	11	definition	definition	NOUN
ejpam-7048	142	12	3	3	NUM
ejpam-7048	142	13	.	.	PUNCT
ejpam-7048	143	1	a	a	DET
ejpam-7048	143	2	function	function	NOUN
ejpam-7048	143	3	f	f	NOUN
ejpam-7048	143	4	:	:	PUNCT
ejpam-7048	143	5	(	(	PUNCT
ejpam-7048	143	6	x	x	X
ejpam-7048	143	7	,	,	PUNCT
ejpam-7048	143	8	τ	τ	PROPN
ejpam-7048	143	9	,	,	PUNCT
ejpam-7048	143	10	i	i	NOUN
ejpam-7048	143	11	)	)	PUNCT
ejpam-7048	143	12	→	→	PUNCT
ejpam-7048	143	13	(	(	PUNCT
ejpam-7048	143	14	y	y	PROPN
ejpam-7048	143	15	,	,	PUNCT
ejpam-7048	143	16	σ1	σ1	PROPN
ejpam-7048	143	17	,	,	PUNCT
ejpam-7048	143	18	σ2	σ2	PROPN
ejpam-7048	143	19	)	)	PUNCT
ejpam-7048	143	20	is	be	AUX
ejpam-7048	143	21	called	call	VERB
ejpam-7048	143	22	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	143	23	,	,	PUNCT
ejpam-7048	143	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	143	25	at	at	ADP
ejpam-7048	143	26	a	a	DET
ejpam-7048	143	27	point	point	NOUN
ejpam-7048	143	28	x	x	SYM
ejpam-7048	143	29	∈	∈	NOUN
ejpam-7048	143	30	x	x	PUNCT
ejpam-7048	143	31	if	if	SCONJ
ejpam-7048	143	32	for	for	ADP
ejpam-7048	143	33	each	each	DET
ejpam-7048	143	34	σ1σ2	σ1σ2	VERB
ejpam-7048	143	35	-	-	ADJ
ejpam-7048	143	36	open	open	ADJ
ejpam-7048	143	37	set	set	NOUN
ejpam-7048	143	38	v	v	NOUN
ejpam-7048	143	39	of	of	ADP
ejpam-7048	143	40	y	y	NOUN
ejpam-7048	143	41	containing	contain	VERB
ejpam-7048	143	42	f(x	f(x	PROPN
ejpam-7048	143	43	)	)	PUNCT
ejpam-7048	143	44	,	,	PUNCT
ejpam-7048	143	45	there	there	PRON
ejpam-7048	143	46	exists	exist	VERB
ejpam-7048	143	47	a	a	DET
ejpam-7048	143	48	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	143	49	-	-	ADJ
ejpam-7048	143	50	open	open	ADJ
ejpam-7048	143	51	set	set	NOUN
ejpam-7048	143	52	u	u	NOUN
ejpam-7048	143	53	of	of	ADP
ejpam-7048	143	54	x	x	PUNCT
ejpam-7048	143	55	containing	contain	VERB
ejpam-7048	143	56	x	x	PUNCT
ejpam-7048	143	57	such	such	ADJ
ejpam-7048	143	58	that	that	DET
ejpam-7048	143	59	f(u	f(u	PROPN
ejpam-7048	143	60	)	)	PUNCT
ejpam-7048	143	61	⊆	⊆	NUM
ejpam-7048	143	62	v	v	NOUN
ejpam-7048	143	63	.	.	PUNCT
ejpam-7048	144	1	a	a	DET
ejpam-7048	144	2	function	function	NOUN
ejpam-7048	144	3	f	f	NOUN
ejpam-7048	144	4	:	:	PUNCT
ejpam-7048	144	5	(	(	PUNCT
ejpam-7048	144	6	x	x	X
ejpam-7048	144	7	,	,	PUNCT
ejpam-7048	144	8	τ	τ	PROPN
ejpam-7048	144	9	,	,	PUNCT
ejpam-7048	144	10	i	i	NOUN
ejpam-7048	144	11	)	)	PUNCT
ejpam-7048	144	12	→	→	PUNCT
ejpam-7048	144	13	(	(	PUNCT
ejpam-7048	144	14	y	y	PROPN
ejpam-7048	144	15	,	,	PUNCT
ejpam-7048	144	16	σ1	σ1	PROPN
ejpam-7048	144	17	,	,	PUNCT
ejpam-7048	144	18	σ2	σ2	PROPN
ejpam-7048	144	19	)	)	PUNCT
ejpam-7048	144	20	is	be	AUX
ejpam-7048	144	21	called	call	VERB
ejpam-7048	144	22	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	144	23	,	,	PUNCT
ejpam-7048	144	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	144	25	if	if	SCONJ
ejpam-7048	144	26	f	f	PROPN
ejpam-7048	144	27	is	be	AUX
ejpam-7048	144	28	τ⋆β(σ1	τ⋆β(σ1	PROPN
ejpam-7048	144	29	,	,	PUNCT
ejpam-7048	144	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	144	31	at	at	ADP
ejpam-7048	144	32	each	each	DET
ejpam-7048	144	33	point	point	NOUN
ejpam-7048	144	34	of	of	ADP
ejpam-7048	144	35	x.	x.	PROPN
ejpam-7048	144	36	corollary	corollary	NOUN
ejpam-7048	144	37	1	1	NUM
ejpam-7048	144	38	.	.	PUNCT
ejpam-7048	145	1	a	a	DET
ejpam-7048	145	2	function	function	NOUN
ejpam-7048	145	3	f	f	NOUN
ejpam-7048	145	4	:	:	PUNCT
ejpam-7048	145	5	(	(	PUNCT
ejpam-7048	145	6	x	x	X
ejpam-7048	145	7	,	,	PUNCT
ejpam-7048	145	8	τ	τ	PROPN
ejpam-7048	145	9	,	,	PUNCT
ejpam-7048	145	10	i	i	NOUN
ejpam-7048	145	11	)	)	PUNCT
ejpam-7048	145	12	→	→	PUNCT
ejpam-7048	145	13	(	(	PUNCT
ejpam-7048	145	14	y	y	PROPN
ejpam-7048	145	15	,	,	PUNCT
ejpam-7048	145	16	σ1	σ1	PROPN
ejpam-7048	145	17	,	,	PUNCT
ejpam-7048	145	18	σ2	σ2	NOUN
ejpam-7048	145	19	)	)	PUNCT
ejpam-7048	145	20	is	be	AUX
ejpam-7048	145	21	τ⋆β(σ1	τ⋆β(σ1	PROPN
ejpam-7048	145	22	,	,	PUNCT
ejpam-7048	145	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	145	24	at	at	ADP
ejpam-7048	145	25	x	x	X
ejpam-7048	145	26	∈	∈	PROPN
ejpam-7048	145	27	x	x	SYM
ejpam-7048	145	28	if	if	SCONJ
ejpam-7048	146	1	and	and	CCONJ
ejpam-7048	146	2	only	only	ADV
ejpam-7048	146	3	if	if	SCONJ
ejpam-7048	146	4	x	x	SYM
ejpam-7048	146	5	∈	∈	PROPN
ejpam-7048	146	6	βint⋆(f−1(v	βint⋆(f−1(v	PUNCT
ejpam-7048	146	7	)	)	PUNCT
ejpam-7048	146	8	)	)	PUNCT
ejpam-7048	147	1	for	for	ADP
ejpam-7048	147	2	every	every	DET
ejpam-7048	147	3	σ1σ2	σ1σ2	NOUN
ejpam-7048	147	4	-	-	ADJ
ejpam-7048	147	5	open	open	ADJ
ejpam-7048	147	6	set	set	NOUN
ejpam-7048	147	7	v	v	NOUN
ejpam-7048	147	8	of	of	ADP
ejpam-7048	147	9	y	y	NOUN
ejpam-7048	147	10	containing	contain	VERB
ejpam-7048	147	11	f(x	f(x	PROPN
ejpam-7048	147	12	)	)	PUNCT
ejpam-7048	147	13	.	.	PUNCT
ejpam-7048	148	1	theorem	theorem	NOUN
ejpam-7048	148	2	3	3	NUM
ejpam-7048	148	3	.	.	X
ejpam-7048	148	4	for	for	ADP
ejpam-7048	148	5	a	a	DET
ejpam-7048	148	6	multifunction	multifunction	NOUN
ejpam-7048	149	1	f	f	NOUN
ejpam-7048	149	2	:	:	PUNCT
ejpam-7048	149	3	(	(	PUNCT
ejpam-7048	149	4	x	x	X
ejpam-7048	149	5	,	,	PUNCT
ejpam-7048	149	6	τ	τ	PROPN
ejpam-7048	149	7	,	,	PUNCT
ejpam-7048	149	8	i	i	NOUN
ejpam-7048	149	9	)	)	PUNCT
ejpam-7048	149	10	→	→	PUNCT
ejpam-7048	149	11	(	(	PUNCT
ejpam-7048	149	12	y	y	PROPN
ejpam-7048	149	13	,	,	PUNCT
ejpam-7048	149	14	σ1	σ1	PROPN
ejpam-7048	149	15	,	,	PUNCT
ejpam-7048	149	16	σ2	σ2	NOUN
ejpam-7048	149	17	)	)	PUNCT
ejpam-7048	149	18	,	,	PUNCT
ejpam-7048	149	19	the	the	DET
ejpam-7048	149	20	following	follow	VERB
ejpam-7048	149	21	properties	property	NOUN
ejpam-7048	149	22	are	be	AUX
ejpam-7048	149	23	equivalent	equivalent	ADJ
ejpam-7048	149	24	:	:	PUNCT
ejpam-7048	149	25	(	(	PUNCT
ejpam-7048	149	26	1	1	X
ejpam-7048	149	27	)	)	PUNCT
ejpam-7048	149	28	f	f	PROPN
ejpam-7048	149	29	is	be	AUX
ejpam-7048	149	30	upper	upper	ADJ
ejpam-7048	149	31	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	149	32	,	,	PUNCT
ejpam-7048	149	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	149	34	at	at	ADP
ejpam-7048	149	35	a	a	DET
ejpam-7048	149	36	point	point	NOUN
ejpam-7048	149	37	x	x	X
ejpam-7048	149	38	∈	∈	NOUN
ejpam-7048	149	39	x	x	X
ejpam-7048	149	40	;	;	PUNCT
ejpam-7048	149	41	(	(	PUNCT
ejpam-7048	149	42	2	2	X
ejpam-7048	149	43	)	)	PUNCT
ejpam-7048	149	44	for	for	ADP
ejpam-7048	149	45	each	each	DET
ejpam-7048	149	46	⋆-open	⋆-open	ADJ
ejpam-7048	149	47	neighborhood	neighborhood	NOUN
ejpam-7048	149	48	u	u	NOUN
ejpam-7048	149	49	of	of	ADP
ejpam-7048	149	50	x	x	PUNCT
ejpam-7048	149	51	and	and	CCONJ
ejpam-7048	149	52	each	each	DET
ejpam-7048	149	53	σ1σ2	σ1σ2	VERB
ejpam-7048	149	54	-	-	ADJ
ejpam-7048	149	55	open	open	ADJ
ejpam-7048	149	56	set	set	NOUN
ejpam-7048	149	57	v	v	NOUN
ejpam-7048	149	58	of	of	ADP
ejpam-7048	149	59	y	y	PROPN
ejpam-7048	149	60	with	with	ADP
ejpam-7048	149	61	x	x	PROPN
ejpam-7048	149	62	∈	∈	PROPN
ejpam-7048	149	63	f+(v	f+(v	NOUN
ejpam-7048	149	64	)	)	PUNCT
ejpam-7048	149	65	,	,	PUNCT
ejpam-7048	149	66	f+(v	f+(v	PROPN
ejpam-7048	149	67	)	)	PUNCT
ejpam-7048	149	68	∩	∩	NOUN
ejpam-7048	149	69	u	u	NOUN
ejpam-7048	149	70	is	be	AUX
ejpam-7048	149	71	not	not	PART
ejpam-7048	149	72	τ1τ2	τ1τ2	ADJ
ejpam-7048	149	73	-	-	ADJ
ejpam-7048	149	74	nowhere	nowhere	ADV
ejpam-7048	149	75	dense	dense	ADJ
ejpam-7048	149	76	;	;	PUNCT
ejpam-7048	149	77	(	(	PUNCT
ejpam-7048	149	78	3	3	X
ejpam-7048	149	79	)	)	PUNCT
ejpam-7048	149	80	for	for	ADP
ejpam-7048	149	81	each	each	DET
ejpam-7048	149	82	⋆-open	⋆-open	ADJ
ejpam-7048	149	83	neighborhood	neighborhood	NOUN
ejpam-7048	149	84	u	u	NOUN
ejpam-7048	149	85	of	of	ADP
ejpam-7048	149	86	x	x	PUNCT
ejpam-7048	149	87	and	and	CCONJ
ejpam-7048	149	88	each	each	DET
ejpam-7048	149	89	σ1σ2	σ1σ2	VERB
ejpam-7048	149	90	-	-	ADJ
ejpam-7048	149	91	open	open	ADJ
ejpam-7048	149	92	set	set	NOUN
ejpam-7048	149	93	v	v	NOUN
ejpam-7048	149	94	of	of	ADP
ejpam-7048	149	95	y	y	PROPN
ejpam-7048	149	96	with	with	ADP
ejpam-7048	149	97	x	x	PROPN
ejpam-7048	149	98	∈	∈	PROPN
ejpam-7048	149	99	f+(v	f+(v	NOUN
ejpam-7048	149	100	)	)	PUNCT
ejpam-7048	149	101	,	,	PUNCT
ejpam-7048	149	102	there	there	PRON
ejpam-7048	149	103	exists	exist	VERB
ejpam-7048	149	104	a	a	DET
ejpam-7048	149	105	⋆-open	⋆-open	ADJ
ejpam-7048	149	106	set	set	NOUN
ejpam-7048	149	107	g	g	NOUN
ejpam-7048	149	108	of	of	ADP
ejpam-7048	149	109	x	x	SYM
ejpam-7048	149	110	such	such	ADJ
ejpam-7048	149	111	that	that	DET
ejpam-7048	149	112	∅	∅	NOUN
ejpam-7048	149	113	̸=	̸=	PROPN
ejpam-7048	149	114	g	g	NOUN
ejpam-7048	149	115	⊆	⊆	NUM
ejpam-7048	149	116	u	u	NOUN
ejpam-7048	149	117	and	and	CCONJ
ejpam-7048	149	118	g	g	PROPN
ejpam-7048	149	119	⊆	⊆	NUM
ejpam-7048	149	120	cl⋆(f+(v	cl⋆(f+(v	PROPN
ejpam-7048	149	121	)	)	PUNCT
ejpam-7048	149	122	)	)	PUNCT
ejpam-7048	149	123	;	;	PUNCT
ejpam-7048	149	124	(	(	PUNCT
ejpam-7048	149	125	4	4	X
ejpam-7048	149	126	)	)	PUNCT
ejpam-7048	149	127	for	for	ADP
ejpam-7048	149	128	each	each	DET
ejpam-7048	149	129	σ1σ2	σ1σ2	VERB
ejpam-7048	149	130	-	-	ADJ
ejpam-7048	149	131	open	open	ADJ
ejpam-7048	149	132	set	set	NOUN
ejpam-7048	149	133	v	v	NOUN
ejpam-7048	149	134	of	of	ADP
ejpam-7048	149	135	y	y	PROPN
ejpam-7048	149	136	with	with	ADP
ejpam-7048	149	137	x	x	PROPN
ejpam-7048	149	138	∈	∈	PROPN
ejpam-7048	149	139	f+(v	f+(v	NOUN
ejpam-7048	149	140	)	)	PUNCT
ejpam-7048	149	141	,	,	PUNCT
ejpam-7048	149	142	there	there	PRON
ejpam-7048	149	143	exists	exist	VERB
ejpam-7048	149	144	a	a	DET
ejpam-7048	149	145	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7048	149	146	-	-	ADJ
ejpam-7048	149	147	open	open	ADJ
ejpam-7048	149	148	set	set	ADJ
ejpam-7048	149	149	u	u	NOUN
ejpam-7048	149	150	of	of	ADP
ejpam-7048	149	151	x	x	PUNCT
ejpam-7048	149	152	containing	contain	VERB
ejpam-7048	149	153	x	x	PUNCT
ejpam-7048	149	154	such	such	ADJ
ejpam-7048	149	155	that	that	SCONJ
ejpam-7048	149	156	u	u	PROPN
ejpam-7048	149	157	⊆	⊆	NUM
ejpam-7048	149	158	cl⋆(f+(v	cl⋆(f+(v	PROPN
ejpam-7048	149	159	)	)	PUNCT
ejpam-7048	149	160	)	)	PUNCT
ejpam-7048	149	161	;	;	PUNCT
ejpam-7048	149	162	(	(	PUNCT
ejpam-7048	149	163	5	5	X
ejpam-7048	149	164	)	)	PUNCT
ejpam-7048	149	165	x	x	SYM
ejpam-7048	149	166	∈	∈	PROPN
ejpam-7048	149	167	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	149	168	)	)	PUNCT
ejpam-7048	149	169	)	)	PUNCT
ejpam-7048	149	170	)	)	PUNCT
ejpam-7048	149	171	)	)	PUNCT
ejpam-7048	150	1	for	for	ADP
ejpam-7048	150	2	every	every	DET
ejpam-7048	150	3	σ1σ2	σ1σ2	NOUN
ejpam-7048	150	4	-	-	ADJ
ejpam-7048	150	5	open	open	ADJ
ejpam-7048	150	6	set	set	NOUN
ejpam-7048	150	7	v	v	NOUN
ejpam-7048	150	8	of	of	ADP
ejpam-7048	150	9	y	y	PROPN
ejpam-7048	150	10	with	with	ADP
ejpam-7048	150	11	x	x	PROPN
ejpam-7048	150	12	∈	∈	PROPN
ejpam-7048	150	13	f+(v	f+(v	NOUN
ejpam-7048	150	14	)	)	PUNCT
ejpam-7048	150	15	.	.	PUNCT
ejpam-7048	151	1	proof	proof	NOUN
ejpam-7048	151	2	.	.	PUNCT
ejpam-7048	152	1	(	(	PUNCT
ejpam-7048	152	2	1	1	X
ejpam-7048	152	3	)	)	PUNCT
ejpam-7048	152	4	⇒	⇒	NOUN
ejpam-7048	152	5	(	(	PUNCT
ejpam-7048	152	6	2	2	NUM
ejpam-7048	152	7	)	)	PUNCT
ejpam-7048	152	8	and	and	CCONJ
ejpam-7048	152	9	(	(	PUNCT
ejpam-7048	152	10	2	2	X
ejpam-7048	152	11	)	)	PUNCT
ejpam-7048	152	12	⇒	⇒	NOUN
ejpam-7048	152	13	(	(	PUNCT
ejpam-7048	152	14	3	3	NUM
ejpam-7048	152	15	):	):	PUNCT
ejpam-7048	152	16	the	the	DET
ejpam-7048	152	17	proof	proof	NOUN
ejpam-7048	152	18	are	be	AUX
ejpam-7048	152	19	obvious	obvious	ADJ
ejpam-7048	152	20	.	.	PUNCT
ejpam-7048	153	1	(	(	PUNCT
ejpam-7048	153	2	3	3	X
ejpam-7048	153	3	)	)	PUNCT
ejpam-7048	153	4	⇒	⇒	NOUN
ejpam-7048	153	5	(	(	PUNCT
ejpam-7048	153	6	4	4	NUM
ejpam-7048	153	7	):	):	PUNCT
ejpam-7048	153	8	let	let	VERB
ejpam-7048	153	9	v	v	PART
ejpam-7048	153	10	be	be	AUX
ejpam-7048	153	11	any	any	DET
ejpam-7048	153	12	σ1σ2	σ1σ2	NOUN
ejpam-7048	153	13	-	-	ADJ
ejpam-7048	153	14	open	open	ADJ
ejpam-7048	153	15	set	set	NOUN
ejpam-7048	153	16	of	of	ADP
ejpam-7048	153	17	y	y	PROPN
ejpam-7048	153	18	containing	contain	VERB
ejpam-7048	153	19	f	f	PROPN
ejpam-7048	153	20	(	(	PUNCT
ejpam-7048	153	21	x	x	NOUN
ejpam-7048	153	22	)	)	PUNCT
ejpam-7048	153	23	.	.	PUNCT
ejpam-7048	154	1	by	by	ADP
ejpam-7048	154	2	u	u	NOUN
ejpam-7048	154	3	⋆(x	⋆(x	NOUN
ejpam-7048	154	4	)	)	PUNCT
ejpam-7048	154	5	we	we	PRON
ejpam-7048	154	6	denote	denote	VERB
ejpam-7048	154	7	the	the	DET
ejpam-7048	154	8	family	family	NOUN
ejpam-7048	154	9	of	of	ADP
ejpam-7048	154	10	all	all	DET
ejpam-7048	154	11	⋆-open	⋆-open	ADJ
ejpam-7048	154	12	neighborhood	neighborhood	NOUN
ejpam-7048	154	13	of	of	ADP
ejpam-7048	154	14	x.	x.	NOUN
ejpam-7048	154	15	for	for	ADP
ejpam-7048	154	16	each	each	DET
ejpam-7048	154	17	u	u	PROPN
ejpam-7048	154	18	∈	∈	PROPN
ejpam-7048	154	19	u	u	NOUN
ejpam-7048	154	20	⋆(x	⋆(x	NOUN
ejpam-7048	154	21	)	)	PUNCT
ejpam-7048	154	22	,	,	PUNCT
ejpam-7048	154	23	there	there	PRON
ejpam-7048	154	24	exists	exist	VERB
ejpam-7048	154	25	a	a	DET
ejpam-7048	154	26	⋆-open	⋆-open	ADJ
ejpam-7048	154	27	set	set	NOUN
ejpam-7048	154	28	gu	gu	NOUN
ejpam-7048	154	29	of	of	ADP
ejpam-7048	154	30	x	x	SYM
ejpam-7048	154	31	such	such	ADJ
ejpam-7048	154	32	that	that	DET
ejpam-7048	154	33	∅	∅	NOUN
ejpam-7048	154	34	̸=	̸=	PROPN
ejpam-7048	154	35	gu	gu	NOUN
ejpam-7048	154	36	⊆	⊆	NUM
ejpam-7048	154	37	u	u	NOUN
ejpam-7048	154	38	and	and	CCONJ
ejpam-7048	154	39	gu	gu	VERB
ejpam-7048	154	40	⊆	⊆	NUM
ejpam-7048	154	41	cl⋆(f+(v	cl⋆(f+(v	PROPN
ejpam-7048	154	42	)	)	PUNCT
ejpam-7048	154	43	)	)	PUNCT
ejpam-7048	154	44	.	.	PUNCT
ejpam-7048	155	1	put	put	VERB
ejpam-7048	155	2	w	w	NOUN
ejpam-7048	155	3	=	=	SYM
ejpam-7048	156	1	∪{gu	∪{gu	PROPN
ejpam-7048	156	2	|	|	ADV
ejpam-7048	156	3	u	u	X
ejpam-7048	156	4	∈	∈	PROPN
ejpam-7048	156	5	u	u	NOUN
ejpam-7048	156	6	⋆(x	⋆(x	NOUN
ejpam-7048	156	7	)	)	PUNCT
ejpam-7048	156	8	}	}	PUNCT
ejpam-7048	156	9	.	.	PUNCT
ejpam-7048	157	1	then	then	ADV
ejpam-7048	157	2	,	,	PUNCT
ejpam-7048	157	3	w	w	PROPN
ejpam-7048	157	4	is	be	AUX
ejpam-7048	157	5	a	a	DET
ejpam-7048	157	6	⋆-open	⋆-open	ADJ
ejpam-7048	157	7	set	set	NOUN
ejpam-7048	157	8	of	of	ADP
ejpam-7048	157	9	x	x	PRON
ejpam-7048	157	10	,	,	PUNCT
ejpam-7048	157	11	x	x	PROPN
ejpam-7048	157	12	∈	∈	PROPN
ejpam-7048	157	13	cl⋆(w	cl⋆(w	PROPN
ejpam-7048	157	14	)	)	PUNCT
ejpam-7048	157	15	and	and	CCONJ
ejpam-7048	157	16	w	w	PROPN
ejpam-7048	157	17	⊆	⊆	NUM
ejpam-7048	157	18	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-7048	157	19	)	)	PUNCT
ejpam-7048	157	20	)	)	PUNCT
ejpam-7048	157	21	.	.	PUNCT
ejpam-7048	158	1	moreover	moreover	ADV
ejpam-7048	158	2	,	,	PUNCT
ejpam-7048	158	3	we	we	PRON
ejpam-7048	158	4	put	put	VERB
ejpam-7048	158	5	u0	u0	ADJ
ejpam-7048	158	6	=	=	NOUN
ejpam-7048	158	7	w	w	NOUN
ejpam-7048	158	8	∪	∪	X
ejpam-7048	158	9	{	{	PUNCT
ejpam-7048	158	10	x	x	NOUN
ejpam-7048	158	11	}	}	PUNCT
ejpam-7048	158	12	.	.	PUNCT
ejpam-7048	159	1	then	then	ADV
ejpam-7048	159	2	,	,	PUNCT
ejpam-7048	159	3	w	w	PROPN
ejpam-7048	159	4	⊆	⊆	NUM
ejpam-7048	159	5	u0	u0	ADJ
ejpam-7048	159	6	⊆	⊆	NUM
ejpam-7048	159	7	cl⋆(w	cl⋆(w	NOUN
ejpam-7048	159	8	)	)	PUNCT
ejpam-7048	159	9	and	and	CCONJ
ejpam-7048	159	10	u0	u0	PROPN
ejpam-7048	159	11	is	be	AUX
ejpam-7048	159	12	a	a	DET
ejpam-7048	159	13	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7048	159	14	-	-	ADJ
ejpam-7048	159	15	open	open	ADJ
ejpam-7048	159	16	set	set	NOUN
ejpam-7048	159	17	of	of	ADP
ejpam-7048	159	18	x	x	PUNCT
ejpam-7048	159	19	containing	contain	VERB
ejpam-7048	159	20	x	x	X
ejpam-7048	159	21	and	and	CCONJ
ejpam-7048	159	22	also	also	ADV
ejpam-7048	159	23	u0	u0	VERB
ejpam-7048	159	24	⊆	⊆	NUM
ejpam-7048	159	25	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-7048	159	26	)	)	PUNCT
ejpam-7048	159	27	)	)	PUNCT
ejpam-7048	159	28	.	.	PUNCT
ejpam-7048	160	1	(	(	PUNCT
ejpam-7048	160	2	4	4	X
ejpam-7048	160	3	)	)	PUNCT
ejpam-7048	160	4	⇒	⇒	NOUN
ejpam-7048	160	5	(	(	PUNCT
ejpam-7048	160	6	5	5	NUM
ejpam-7048	160	7	):	):	PUNCT
ejpam-7048	160	8	let	let	VERB
ejpam-7048	160	9	v	v	PART
ejpam-7048	160	10	be	be	AUX
ejpam-7048	160	11	any	any	DET
ejpam-7048	160	12	σ1σ2	σ1σ2	NOUN
ejpam-7048	160	13	-	-	ADJ
ejpam-7048	160	14	open	open	ADJ
ejpam-7048	160	15	set	set	NOUN
ejpam-7048	160	16	of	of	ADP
ejpam-7048	160	17	y	y	PROPN
ejpam-7048	160	18	containing	contain	VERB
ejpam-7048	160	19	f	f	PROPN
ejpam-7048	160	20	(	(	PUNCT
ejpam-7048	160	21	x	x	NOUN
ejpam-7048	160	22	)	)	PUNCT
ejpam-7048	160	23	.	.	PUNCT
ejpam-7048	161	1	there	there	PRON
ejpam-7048	161	2	exists	exist	VERB
ejpam-7048	161	3	a	a	DET
ejpam-7048	161	4	τ⋆-semiopen	τ⋆-semiopen	NOUN
ejpam-7048	161	5	set	set	NOUN
ejpam-7048	161	6	u	u	NOUN
ejpam-7048	161	7	of	of	ADP
ejpam-7048	161	8	x	x	PUNCT
ejpam-7048	161	9	containing	contain	VERB
ejpam-7048	161	10	x	x	PUNCT
ejpam-7048	161	11	such	such	ADJ
ejpam-7048	161	12	that	that	SCONJ
ejpam-7048	161	13	u	u	PROPN
ejpam-7048	161	14	⊆	⊆	NUM
ejpam-7048	161	15	cl⋆(f+(v	cl⋆(f+(v	PROPN
ejpam-7048	161	16	)	)	PUNCT
ejpam-7048	161	17	)	)	PUNCT
ejpam-7048	161	18	.	.	PUNCT
ejpam-7048	162	1	thus	thus	ADV
ejpam-7048	162	2	,	,	PUNCT
ejpam-7048	162	3	x	x	PUNCT
ejpam-7048	162	4	∈	∈	PROPN
ejpam-7048	162	5	u	u	NOUN
ejpam-7048	162	6	⊆	⊆	NUM
ejpam-7048	162	7	cl⋆(int⋆(u	cl⋆(int⋆(u	NOUN
ejpam-7048	162	8	)	)	PUNCT
ejpam-7048	162	9	)	)	PUNCT
ejpam-7048	163	1	⊆	⊆	NUM
ejpam-7048	163	2	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	163	3	)	)	PUNCT
ejpam-7048	163	4	)	)	PUNCT
ejpam-7048	163	5	)	)	PUNCT
ejpam-7048	163	6	)	)	PUNCT
ejpam-7048	163	7	.	.	PUNCT
ejpam-7048	164	1	(	(	PUNCT
ejpam-7048	164	2	5	5	X
ejpam-7048	164	3	)	)	PUNCT
ejpam-7048	164	4	⇒	⇒	NOUN
ejpam-7048	164	5	(	(	PUNCT
ejpam-7048	164	6	1	1	NUM
ejpam-7048	164	7	):	):	PUNCT
ejpam-7048	164	8	by	by	ADP
ejpam-7048	164	9	utilizing	utilize	VERB
ejpam-7048	164	10	lemma	lemma	PROPN
ejpam-7048	164	11	2	2	NUM
ejpam-7048	164	12	,	,	PUNCT
ejpam-7048	164	13	this	this	PRON
ejpam-7048	164	14	can	can	AUX
ejpam-7048	164	15	be	be	AUX
ejpam-7048	164	16	proved	prove	VERB
ejpam-7048	164	17	similarly	similarly	ADV
ejpam-7048	164	18	to	to	ADP
ejpam-7048	164	19	that	that	PRON
ejpam-7048	164	20	of	of	ADP
ejpam-7048	164	21	theorem	theorem	ADJ
ejpam-7048	164	22	1	1	NUM
ejpam-7048	164	23	.	.	PUNCT
ejpam-7048	164	24	theorem	theorem	NOUN
ejpam-7048	164	25	4	4	NUM
ejpam-7048	164	26	.	.	X
ejpam-7048	164	27	for	for	ADP
ejpam-7048	164	28	a	a	DET
ejpam-7048	164	29	multifunction	multifunction	NOUN
ejpam-7048	164	30	f	f	NOUN
ejpam-7048	164	31	:	:	PUNCT
ejpam-7048	164	32	(	(	PUNCT
ejpam-7048	164	33	x	x	X
ejpam-7048	164	34	,	,	PUNCT
ejpam-7048	164	35	τ	τ	PROPN
ejpam-7048	164	36	,	,	PUNCT
ejpam-7048	164	37	i	i	NOUN
ejpam-7048	164	38	)	)	PUNCT
ejpam-7048	164	39	→	→	PUNCT
ejpam-7048	164	40	(	(	PUNCT
ejpam-7048	164	41	y	y	PROPN
ejpam-7048	164	42	,	,	PUNCT
ejpam-7048	164	43	σ1	σ1	PROPN
ejpam-7048	164	44	,	,	PUNCT
ejpam-7048	164	45	σ2	σ2	NOUN
ejpam-7048	164	46	)	)	PUNCT
ejpam-7048	164	47	,	,	PUNCT
ejpam-7048	164	48	the	the	DET
ejpam-7048	164	49	following	follow	VERB
ejpam-7048	164	50	properties	property	NOUN
ejpam-7048	164	51	are	be	AUX
ejpam-7048	164	52	equivalent	equivalent	ADJ
ejpam-7048	164	53	:	:	PUNCT
ejpam-7048	164	54	p.	p.	NOUN
ejpam-7048	164	55	pue	pue	NOUN
ejpam-7048	164	56	-	-	PUNCT
ejpam-7048	164	57	on	on	ADP
ejpam-7048	164	58	,	,	PUNCT
ejpam-7048	164	59	a.	a.	PROPN
ejpam-7048	164	60	sama	sama	PROPN
ejpam-7048	164	61	-	-	PUNCT
ejpam-7048	164	62	ae	ae	PROPN
ejpam-7048	164	63	,	,	PUNCT
ejpam-7048	164	64	c.	c.	PROPN
ejpam-7048	164	65	boonpok	boonpok	PROPN
ejpam-7048	164	66	/	/	SYM
ejpam-7048	164	67	eur	eur	PROPN
ejpam-7048	164	68	.	.	PUNCT
ejpam-7048	165	1	j.	j.	PROPN
ejpam-7048	165	2	pure	pure	PROPN
ejpam-7048	165	3	appl	appl	PROPN
ejpam-7048	165	4	.	.	PROPN
ejpam-7048	165	5	math	math	PROPN
ejpam-7048	165	6	,	,	PUNCT
ejpam-7048	165	7	18	18	NUM
ejpam-7048	165	8	(	(	PUNCT
ejpam-7048	165	9	4	4	NUM
ejpam-7048	165	10	)	)	PUNCT
ejpam-7048	165	11	(	(	PUNCT
ejpam-7048	165	12	2025	2025	NUM
ejpam-7048	165	13	)	)	PUNCT
ejpam-7048	165	14	,	,	PUNCT
ejpam-7048	165	15	7048	7048	NUM
ejpam-7048	165	16	6	6	NUM
ejpam-7048	165	17	of	of	ADP
ejpam-7048	165	18	9	9	NUM
ejpam-7048	165	19	(	(	PUNCT
ejpam-7048	165	20	1	1	NUM
ejpam-7048	165	21	)	)	PUNCT
ejpam-7048	165	22	f	f	PROPN
ejpam-7048	165	23	is	be	AUX
ejpam-7048	165	24	lower	low	ADJ
ejpam-7048	165	25	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	165	26	,	,	PUNCT
ejpam-7048	165	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	165	28	at	at	ADP
ejpam-7048	165	29	a	a	DET
ejpam-7048	165	30	point	point	NOUN
ejpam-7048	165	31	x	x	X
ejpam-7048	165	32	∈	∈	NOUN
ejpam-7048	165	33	x	x	X
ejpam-7048	165	34	;	;	PUNCT
ejpam-7048	165	35	(	(	PUNCT
ejpam-7048	165	36	2	2	X
ejpam-7048	165	37	)	)	PUNCT
ejpam-7048	165	38	for	for	ADP
ejpam-7048	165	39	each	each	DET
ejpam-7048	165	40	⋆-open	⋆-open	ADJ
ejpam-7048	165	41	neighborhood	neighborhood	NOUN
ejpam-7048	165	42	u	u	NOUN
ejpam-7048	165	43	of	of	ADP
ejpam-7048	165	44	x	x	PUNCT
ejpam-7048	165	45	and	and	CCONJ
ejpam-7048	165	46	each	each	DET
ejpam-7048	165	47	σ1σ2	σ1σ2	VERB
ejpam-7048	165	48	-	-	ADJ
ejpam-7048	165	49	open	open	ADJ
ejpam-7048	165	50	set	set	NOUN
ejpam-7048	165	51	v	v	NOUN
ejpam-7048	165	52	of	of	ADP
ejpam-7048	165	53	y	y	PROPN
ejpam-7048	165	54	with	with	ADP
ejpam-7048	165	55	x	x	PROPN
ejpam-7048	165	56	∈	∈	PROPN
ejpam-7048	165	57	f−(v	f−(v	NOUN
ejpam-7048	165	58	)	)	PUNCT
ejpam-7048	165	59	,	,	PUNCT
ejpam-7048	165	60	f−(v	f−(v	ADJ
ejpam-7048	165	61	)	)	PUNCT
ejpam-7048	165	62	∩	∩	NOUN
ejpam-7048	165	63	u	u	NOUN
ejpam-7048	165	64	is	be	AUX
ejpam-7048	165	65	not	not	PART
ejpam-7048	165	66	τ1τ2	τ1τ2	ADJ
ejpam-7048	165	67	-	-	ADJ
ejpam-7048	165	68	nowhere	nowhere	ADV
ejpam-7048	165	69	dense	dense	ADJ
ejpam-7048	165	70	;	;	PUNCT
ejpam-7048	165	71	(	(	PUNCT
ejpam-7048	165	72	3	3	X
ejpam-7048	165	73	)	)	PUNCT
ejpam-7048	165	74	for	for	ADP
ejpam-7048	165	75	each	each	DET
ejpam-7048	165	76	⋆-open	⋆-open	ADJ
ejpam-7048	165	77	neighborhood	neighborhood	NOUN
ejpam-7048	165	78	u	u	NOUN
ejpam-7048	165	79	of	of	ADP
ejpam-7048	165	80	x	x	PUNCT
ejpam-7048	165	81	and	and	CCONJ
ejpam-7048	165	82	each	each	DET
ejpam-7048	165	83	σ1σ2	σ1σ2	VERB
ejpam-7048	165	84	-	-	ADJ
ejpam-7048	165	85	open	open	ADJ
ejpam-7048	165	86	set	set	NOUN
ejpam-7048	165	87	v	v	NOUN
ejpam-7048	165	88	of	of	ADP
ejpam-7048	165	89	y	y	PROPN
ejpam-7048	165	90	with	with	ADP
ejpam-7048	165	91	x	x	PROPN
ejpam-7048	165	92	∈	∈	PROPN
ejpam-7048	165	93	f−(v	f−(v	NOUN
ejpam-7048	165	94	)	)	PUNCT
ejpam-7048	165	95	,	,	PUNCT
ejpam-7048	165	96	there	there	PRON
ejpam-7048	165	97	exists	exist	VERB
ejpam-7048	165	98	a	a	DET
ejpam-7048	165	99	⋆-open	⋆-open	ADJ
ejpam-7048	165	100	set	set	NOUN
ejpam-7048	165	101	g	g	NOUN
ejpam-7048	165	102	of	of	ADP
ejpam-7048	165	103	x	x	SYM
ejpam-7048	165	104	such	such	ADJ
ejpam-7048	165	105	that	that	DET
ejpam-7048	165	106	∅	∅	NOUN
ejpam-7048	165	107	̸=	̸=	PROPN
ejpam-7048	165	108	g	g	NOUN
ejpam-7048	165	109	⊆	⊆	NUM
ejpam-7048	165	110	u	u	NOUN
ejpam-7048	165	111	and	and	CCONJ
ejpam-7048	165	112	g	g	NOUN
ejpam-7048	165	113	⊆	⊆	NUM
ejpam-7048	165	114	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-7048	165	115	)	)	PUNCT
ejpam-7048	165	116	)	)	PUNCT
ejpam-7048	165	117	;	;	PUNCT
ejpam-7048	165	118	(	(	PUNCT
ejpam-7048	165	119	4	4	X
ejpam-7048	165	120	)	)	PUNCT
ejpam-7048	165	121	for	for	ADP
ejpam-7048	165	122	each	each	DET
ejpam-7048	165	123	σ1σ2	σ1σ2	VERB
ejpam-7048	165	124	-	-	ADJ
ejpam-7048	165	125	open	open	ADJ
ejpam-7048	165	126	set	set	NOUN
ejpam-7048	165	127	v	v	NOUN
ejpam-7048	165	128	of	of	ADP
ejpam-7048	165	129	y	y	PROPN
ejpam-7048	165	130	with	with	ADP
ejpam-7048	165	131	x	x	PROPN
ejpam-7048	165	132	∈	∈	PROPN
ejpam-7048	165	133	f−(v	f−(v	NOUN
ejpam-7048	165	134	)	)	PUNCT
ejpam-7048	165	135	,	,	PUNCT
ejpam-7048	165	136	there	there	PRON
ejpam-7048	165	137	exists	exist	VERB
ejpam-7048	165	138	a	a	DET
ejpam-7048	165	139	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7048	165	140	-	-	ADJ
ejpam-7048	165	141	open	open	ADJ
ejpam-7048	165	142	set	set	ADJ
ejpam-7048	165	143	u	u	NOUN
ejpam-7048	165	144	of	of	ADP
ejpam-7048	165	145	x	x	PUNCT
ejpam-7048	165	146	containing	contain	VERB
ejpam-7048	165	147	x	x	PUNCT
ejpam-7048	165	148	such	such	ADJ
ejpam-7048	165	149	that	that	SCONJ
ejpam-7048	165	150	u	u	NOUN
ejpam-7048	165	151	⊆	⊆	NUM
ejpam-7048	165	152	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-7048	165	153	)	)	PUNCT
ejpam-7048	165	154	)	)	PUNCT
ejpam-7048	165	155	;	;	PUNCT
ejpam-7048	166	1	(	(	PUNCT
ejpam-7048	166	2	5	5	X
ejpam-7048	166	3	)	)	PUNCT
ejpam-7048	166	4	x	x	SYM
ejpam-7048	166	5	∈	∈	PROPN
ejpam-7048	166	6	cl⋆(int⋆(cl⋆(f−(v	cl⋆(int⋆(cl⋆(f−(v	NOUN
ejpam-7048	166	7	)	)	PUNCT
ejpam-7048	166	8	)	)	PUNCT
ejpam-7048	166	9	)	)	PUNCT
ejpam-7048	166	10	)	)	PUNCT
ejpam-7048	167	1	for	for	ADP
ejpam-7048	167	2	every	every	DET
ejpam-7048	167	3	σ1σ2	σ1σ2	NOUN
ejpam-7048	167	4	-	-	ADJ
ejpam-7048	167	5	open	open	ADJ
ejpam-7048	167	6	set	set	NOUN
ejpam-7048	167	7	v	v	NOUN
ejpam-7048	167	8	of	of	ADP
ejpam-7048	167	9	y	y	PROPN
ejpam-7048	167	10	with	with	ADP
ejpam-7048	167	11	x	x	PROPN
ejpam-7048	167	12	∈	∈	PROPN
ejpam-7048	167	13	f−(v	f−(v	NOUN
ejpam-7048	167	14	)	)	PUNCT
ejpam-7048	167	15	.	.	PUNCT
ejpam-7048	168	1	proof	proof	NOUN
ejpam-7048	168	2	.	.	PUNCT
ejpam-7048	169	1	the	the	DET
ejpam-7048	169	2	proof	proof	NOUN
ejpam-7048	169	3	is	be	AUX
ejpam-7048	169	4	similar	similar	ADJ
ejpam-7048	169	5	to	to	ADP
ejpam-7048	169	6	that	that	PRON
ejpam-7048	169	7	of	of	ADP
ejpam-7048	169	8	theorem	theorem	ADJ
ejpam-7048	169	9	3	3	NUM
ejpam-7048	169	10	.	.	PUNCT
ejpam-7048	169	11	corollary	corollary	ADJ
ejpam-7048	169	12	2	2	NUM
ejpam-7048	169	13	.	.	PUNCT
ejpam-7048	169	14	for	for	ADP
ejpam-7048	169	15	a	a	DET
ejpam-7048	169	16	function	function	NOUN
ejpam-7048	169	17	f	f	NOUN
ejpam-7048	169	18	:	:	PUNCT
ejpam-7048	169	19	(	(	PUNCT
ejpam-7048	169	20	x	x	X
ejpam-7048	169	21	,	,	PUNCT
ejpam-7048	169	22	τ	τ	PROPN
ejpam-7048	169	23	,	,	PUNCT
ejpam-7048	169	24	i	i	NOUN
ejpam-7048	169	25	)	)	PUNCT
ejpam-7048	169	26	→	→	PUNCT
ejpam-7048	169	27	(	(	PUNCT
ejpam-7048	169	28	y	y	PROPN
ejpam-7048	169	29	,	,	PUNCT
ejpam-7048	169	30	σ1	σ1	PROPN
ejpam-7048	169	31	,	,	PUNCT
ejpam-7048	169	32	σ2	σ2	NOUN
ejpam-7048	169	33	)	)	PUNCT
ejpam-7048	169	34	,	,	PUNCT
ejpam-7048	169	35	the	the	DET
ejpam-7048	169	36	following	follow	VERB
ejpam-7048	169	37	properties	property	NOUN
ejpam-7048	169	38	are	be	AUX
ejpam-7048	169	39	equivalent	equivalent	ADJ
ejpam-7048	169	40	:	:	PUNCT
ejpam-7048	169	41	(	(	PUNCT
ejpam-7048	169	42	1	1	X
ejpam-7048	169	43	)	)	PUNCT
ejpam-7048	169	44	f	f	PROPN
ejpam-7048	169	45	is	be	AUX
ejpam-7048	169	46	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	169	47	,	,	PUNCT
ejpam-7048	169	48	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	169	49	at	at	ADP
ejpam-7048	169	50	a	a	DET
ejpam-7048	169	51	point	point	NOUN
ejpam-7048	169	52	x	x	X
ejpam-7048	169	53	∈	∈	NOUN
ejpam-7048	169	54	x	x	X
ejpam-7048	169	55	;	;	PUNCT
ejpam-7048	169	56	(	(	PUNCT
ejpam-7048	169	57	2	2	X
ejpam-7048	169	58	)	)	PUNCT
ejpam-7048	169	59	for	for	ADP
ejpam-7048	169	60	each	each	DET
ejpam-7048	169	61	⋆-open	⋆-open	ADJ
ejpam-7048	169	62	neighborhood	neighborhood	NOUN
ejpam-7048	169	63	u	u	NOUN
ejpam-7048	169	64	of	of	ADP
ejpam-7048	169	65	x	x	PUNCT
ejpam-7048	169	66	and	and	CCONJ
ejpam-7048	169	67	each	each	DET
ejpam-7048	169	68	σ1σ2	σ1σ2	VERB
ejpam-7048	169	69	-	-	ADJ
ejpam-7048	169	70	open	open	ADJ
ejpam-7048	169	71	set	set	NOUN
ejpam-7048	169	72	v	v	NOUN
ejpam-7048	169	73	of	of	ADP
ejpam-7048	169	74	y	y	NOUN
ejpam-7048	169	75	containing	contain	VERB
ejpam-7048	169	76	f(x	f(x	PROPN
ejpam-7048	169	77	)	)	PUNCT
ejpam-7048	169	78	,	,	PUNCT
ejpam-7048	169	79	f−1(v	f−1(v	NOUN
ejpam-7048	169	80	)	)	PUNCT
ejpam-7048	170	1	∩	∩	NOUN
ejpam-7048	170	2	u	u	NOUN
ejpam-7048	170	3	is	be	AUX
ejpam-7048	170	4	not	not	PART
ejpam-7048	170	5	τ1τ2	τ1τ2	ADJ
ejpam-7048	170	6	-	-	ADJ
ejpam-7048	170	7	nowhere	nowhere	ADV
ejpam-7048	170	8	dense	dense	ADJ
ejpam-7048	170	9	;	;	PUNCT
ejpam-7048	170	10	(	(	PUNCT
ejpam-7048	170	11	3	3	X
ejpam-7048	170	12	)	)	PUNCT
ejpam-7048	170	13	for	for	ADP
ejpam-7048	170	14	each	each	DET
ejpam-7048	170	15	⋆-open	⋆-open	ADJ
ejpam-7048	170	16	neighborhood	neighborhood	NOUN
ejpam-7048	170	17	u	u	NOUN
ejpam-7048	170	18	of	of	ADP
ejpam-7048	170	19	x	x	PUNCT
ejpam-7048	170	20	and	and	CCONJ
ejpam-7048	170	21	each	each	DET
ejpam-7048	170	22	σ1σ2	σ1σ2	VERB
ejpam-7048	170	23	-	-	ADJ
ejpam-7048	170	24	open	open	ADJ
ejpam-7048	170	25	set	set	NOUN
ejpam-7048	170	26	v	v	NOUN
ejpam-7048	170	27	of	of	ADP
ejpam-7048	170	28	y	y	NOUN
ejpam-7048	170	29	containing	contain	VERB
ejpam-7048	170	30	f(x	f(x	PROPN
ejpam-7048	170	31	)	)	PUNCT
ejpam-7048	170	32	,	,	PUNCT
ejpam-7048	170	33	there	there	PRON
ejpam-7048	170	34	exists	exist	VERB
ejpam-7048	170	35	a	a	DET
ejpam-7048	170	36	⋆-open	⋆-open	ADJ
ejpam-7048	170	37	set	set	NOUN
ejpam-7048	170	38	g	g	NOUN
ejpam-7048	170	39	of	of	ADP
ejpam-7048	170	40	x	x	SYM
ejpam-7048	170	41	such	such	ADJ
ejpam-7048	170	42	that	that	DET
ejpam-7048	170	43	∅	∅	NOUN
ejpam-7048	170	44	̸=	̸=	PROPN
ejpam-7048	170	45	g	g	NOUN
ejpam-7048	170	46	⊆	⊆	NUM
ejpam-7048	170	47	u	u	NOUN
ejpam-7048	170	48	and	and	CCONJ
ejpam-7048	170	49	g	g	PROPN
ejpam-7048	170	50	⊆	⊆	NUM
ejpam-7048	170	51	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-7048	170	52	)	)	PUNCT
ejpam-7048	170	53	)	)	PUNCT
ejpam-7048	170	54	;	;	PUNCT
ejpam-7048	170	55	(	(	PUNCT
ejpam-7048	170	56	4	4	X
ejpam-7048	170	57	)	)	PUNCT
ejpam-7048	170	58	for	for	ADP
ejpam-7048	170	59	each	each	DET
ejpam-7048	170	60	σ1σ2	σ1σ2	VERB
ejpam-7048	170	61	-	-	ADJ
ejpam-7048	170	62	open	open	ADJ
ejpam-7048	170	63	set	set	NOUN
ejpam-7048	170	64	v	v	NOUN
ejpam-7048	170	65	of	of	ADP
ejpam-7048	170	66	y	y	NOUN
ejpam-7048	170	67	containing	contain	VERB
ejpam-7048	170	68	f(x	f(x	PROPN
ejpam-7048	170	69	)	)	PUNCT
ejpam-7048	170	70	,	,	PUNCT
ejpam-7048	170	71	there	there	PRON
ejpam-7048	170	72	exists	exist	VERB
ejpam-7048	170	73	a	a	DET
ejpam-7048	170	74	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7048	170	75	-	-	ADJ
ejpam-7048	170	76	open	open	ADJ
ejpam-7048	170	77	set	set	ADJ
ejpam-7048	170	78	u	u	NOUN
ejpam-7048	170	79	of	of	ADP
ejpam-7048	170	80	x	x	PUNCT
ejpam-7048	170	81	containing	contain	VERB
ejpam-7048	170	82	x	x	PUNCT
ejpam-7048	170	83	such	such	ADJ
ejpam-7048	170	84	that	that	SCONJ
ejpam-7048	170	85	u	u	PROPN
ejpam-7048	170	86	⊆	⊆	NUM
ejpam-7048	170	87	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-7048	170	88	)	)	PUNCT
ejpam-7048	170	89	)	)	PUNCT
ejpam-7048	170	90	;	;	PUNCT
ejpam-7048	170	91	(	(	PUNCT
ejpam-7048	170	92	5	5	X
ejpam-7048	170	93	)	)	PUNCT
ejpam-7048	170	94	x	x	SYM
ejpam-7048	170	95	∈	∈	PROPN
ejpam-7048	170	96	cl⋆(int⋆(cl⋆(f−1(v	cl⋆(int⋆(cl⋆(f−1(v	NOUN
ejpam-7048	170	97	)	)	PUNCT
ejpam-7048	170	98	)	)	PUNCT
ejpam-7048	170	99	)	)	PUNCT
ejpam-7048	170	100	)	)	PUNCT
ejpam-7048	170	101	for	for	ADP
ejpam-7048	170	102	every	every	DET
ejpam-7048	170	103	σ1σ2	σ1σ2	NOUN
ejpam-7048	170	104	-	-	ADJ
ejpam-7048	170	105	open	open	ADJ
ejpam-7048	170	106	set	set	NOUN
ejpam-7048	170	107	v	v	NOUN
ejpam-7048	170	108	of	of	ADP
ejpam-7048	170	109	y	y	NOUN
ejpam-7048	170	110	containing	contain	VERB
ejpam-7048	170	111	f(x	f(x	PROPN
ejpam-7048	170	112	)	)	PUNCT
ejpam-7048	170	113	.	.	PUNCT
ejpam-7048	171	1	theorem	theorem	ADJ
ejpam-7048	171	2	5	5	NUM
ejpam-7048	171	3	.	.	X
ejpam-7048	171	4	for	for	ADP
ejpam-7048	171	5	a	a	DET
ejpam-7048	171	6	multifunction	multifunction	NOUN
ejpam-7048	172	1	f	f	NOUN
ejpam-7048	172	2	:	:	PUNCT
ejpam-7048	172	3	(	(	PUNCT
ejpam-7048	172	4	x	x	X
ejpam-7048	172	5	,	,	PUNCT
ejpam-7048	172	6	τ	τ	PROPN
ejpam-7048	172	7	,	,	PUNCT
ejpam-7048	172	8	i	i	NOUN
ejpam-7048	172	9	)	)	PUNCT
ejpam-7048	172	10	→	→	PUNCT
ejpam-7048	172	11	(	(	PUNCT
ejpam-7048	172	12	y	y	PROPN
ejpam-7048	172	13	,	,	PUNCT
ejpam-7048	172	14	σ1	σ1	PROPN
ejpam-7048	172	15	,	,	PUNCT
ejpam-7048	172	16	σ2	σ2	NOUN
ejpam-7048	172	17	)	)	PUNCT
ejpam-7048	172	18	,	,	PUNCT
ejpam-7048	172	19	the	the	DET
ejpam-7048	172	20	following	follow	VERB
ejpam-7048	172	21	properties	property	NOUN
ejpam-7048	172	22	are	be	AUX
ejpam-7048	172	23	equivalent	equivalent	ADJ
ejpam-7048	172	24	:	:	PUNCT
ejpam-7048	172	25	(	(	PUNCT
ejpam-7048	172	26	1	1	X
ejpam-7048	172	27	)	)	PUNCT
ejpam-7048	172	28	f	f	PROPN
ejpam-7048	172	29	is	be	AUX
ejpam-7048	172	30	upper	upper	ADJ
ejpam-7048	172	31	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	172	32	,	,	PUNCT
ejpam-7048	172	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	172	34	;	;	PUNCT
ejpam-7048	172	35	(	(	PUNCT
ejpam-7048	172	36	2	2	NUM
ejpam-7048	172	37	)	)	PUNCT
ejpam-7048	172	38	f+(v	f+(v	NOUN
ejpam-7048	172	39	)	)	PUNCT
ejpam-7048	172	40	is	be	AUX
ejpam-7048	172	41	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	172	42	-	-	NOUN
ejpam-7048	172	43	open	open	ADJ
ejpam-7048	172	44	in	in	ADP
ejpam-7048	172	45	x	x	PUNCT
ejpam-7048	172	46	for	for	ADP
ejpam-7048	172	47	every	every	DET
ejpam-7048	172	48	σ1σ2	σ1σ2	NOUN
ejpam-7048	172	49	-	-	ADJ
ejpam-7048	172	50	open	open	ADJ
ejpam-7048	172	51	set	set	NOUN
ejpam-7048	172	52	v	v	NOUN
ejpam-7048	172	53	of	of	ADP
ejpam-7048	172	54	y	y	PROPN
ejpam-7048	172	55	;	;	PUNCT
ejpam-7048	172	56	(	(	PUNCT
ejpam-7048	172	57	3	3	X
ejpam-7048	172	58	)	)	PUNCT
ejpam-7048	172	59	f−(k	f−(k	PROPN
ejpam-7048	172	60	)	)	PUNCT
ejpam-7048	172	61	is	be	AUX
ejpam-7048	172	62	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	172	63	-	-	VERB
ejpam-7048	172	64	closed	closed	ADJ
ejpam-7048	172	65	in	in	ADP
ejpam-7048	172	66	x	x	PUNCT
ejpam-7048	172	67	for	for	ADP
ejpam-7048	172	68	every	every	DET
ejpam-7048	172	69	σ1σ2	σ1σ2	NUM
ejpam-7048	172	70	-	-	PUNCT
ejpam-7048	172	71	closed	closed	ADJ
ejpam-7048	172	72	set	set	NOUN
ejpam-7048	172	73	k	k	PROPN
ejpam-7048	172	74	of	of	ADP
ejpam-7048	172	75	y	y	PROPN
ejpam-7048	172	76	;	;	PUNCT
ejpam-7048	172	77	(	(	PUNCT
ejpam-7048	172	78	4	4	X
ejpam-7048	172	79	)	)	PUNCT
ejpam-7048	172	80	βcl⋆(f−(b	βcl⋆(f−(b	PUNCT
ejpam-7048	172	81	)	)	PUNCT
ejpam-7048	172	82	)	)	PUNCT
ejpam-7048	173	1	⊆	⊆	X
ejpam-7048	173	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7048	173	3	-	-	PUNCT
ejpam-7048	173	4	cl(b	cl(b	NOUN
ejpam-7048	173	5	)	)	PUNCT
ejpam-7048	173	6	)	)	PUNCT
ejpam-7048	174	1	for	for	ADP
ejpam-7048	174	2	every	every	DET
ejpam-7048	174	3	subset	subset	NOUN
ejpam-7048	174	4	b	b	PROPN
ejpam-7048	174	5	of	of	ADP
ejpam-7048	174	6	y	y	PROPN
ejpam-7048	174	7	;	;	PUNCT
ejpam-7048	174	8	(	(	PUNCT
ejpam-7048	174	9	5	5	X
ejpam-7048	174	10	)	)	PUNCT
ejpam-7048	174	11	int⋆(cl⋆(int⋆(f−(b	int⋆(cl⋆(int⋆(f−(b	NOUN
ejpam-7048	174	12	)	)	PUNCT
ejpam-7048	174	13	)	)	PUNCT
ejpam-7048	174	14	)	)	PUNCT
ejpam-7048	174	15	)	)	PUNCT
ejpam-7048	174	16	⊆	⊆	NUM
ejpam-7048	174	17	f−(σ1σ2cl(b	f−(σ1σ2cl(b	NOUN
ejpam-7048	174	18	)	)	PUNCT
ejpam-7048	174	19	)	)	PUNCT
ejpam-7048	174	20	for	for	ADP
ejpam-7048	174	21	every	every	DET
ejpam-7048	174	22	subset	subset	NOUN
ejpam-7048	174	23	b	b	PROPN
ejpam-7048	174	24	of	of	ADP
ejpam-7048	174	25	y	y	PROPN
ejpam-7048	174	26	.	.	PUNCT
ejpam-7048	175	1	proof	proof	NOUN
ejpam-7048	175	2	.	.	PUNCT
ejpam-7048	176	1	(	(	PUNCT
ejpam-7048	176	2	1	1	X
ejpam-7048	176	3	)	)	PUNCT
ejpam-7048	176	4	⇒	⇒	NOUN
ejpam-7048	176	5	(	(	PUNCT
ejpam-7048	176	6	2	2	NUM
ejpam-7048	176	7	):	):	PUNCT
ejpam-7048	176	8	let	let	VERB
ejpam-7048	176	9	v	v	PART
ejpam-7048	176	10	be	be	AUX
ejpam-7048	176	11	any	any	DET
ejpam-7048	176	12	σ1σ2	σ1σ2	NOUN
ejpam-7048	176	13	-	-	ADJ
ejpam-7048	176	14	open	open	ADJ
ejpam-7048	176	15	set	set	NOUN
ejpam-7048	176	16	of	of	ADP
ejpam-7048	176	17	y	y	PROPN
ejpam-7048	176	18	and	and	CCONJ
ejpam-7048	176	19	x	x	PROPN
ejpam-7048	176	20	∈	∈	PROPN
ejpam-7048	176	21	f+(v	f+(v	NOUN
ejpam-7048	176	22	)	)	PUNCT
ejpam-7048	176	23	.	.	PUNCT
ejpam-7048	177	1	there	there	PRON
ejpam-7048	177	2	exists	exist	VERB
ejpam-7048	177	3	a	a	DET
ejpam-7048	177	4	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	177	5	-	-	ADJ
ejpam-7048	177	6	open	open	ADJ
ejpam-7048	177	7	set	set	NOUN
ejpam-7048	177	8	u	u	NOUN
ejpam-7048	177	9	of	of	ADP
ejpam-7048	177	10	x	x	PUNCT
ejpam-7048	177	11	containing	contain	VERB
ejpam-7048	177	12	x	x	PUNCT
ejpam-7048	177	13	such	such	ADJ
ejpam-7048	177	14	that	that	SCONJ
ejpam-7048	177	15	f	f	PROPN
ejpam-7048	177	16	(	(	PUNCT
ejpam-7048	177	17	u	u	NOUN
ejpam-7048	177	18	)	)	PUNCT
ejpam-7048	177	19	⊆	⊆	NUM
ejpam-7048	177	20	v	v	NOUN
ejpam-7048	177	21	.	.	PUNCT
ejpam-7048	178	1	thus	thus	ADV
ejpam-7048	178	2	,	,	PUNCT
ejpam-7048	178	3	x	x	PUNCT
ejpam-7048	178	4	∈	∈	PROPN
ejpam-7048	178	5	u	u	NOUN
ejpam-7048	178	6	⊆	⊆	NUM
ejpam-7048	178	7	cl⋆(int⋆(cl⋆(u	cl⋆(int⋆(cl⋆(u	NOUN
ejpam-7048	178	8	)	)	PUNCT
ejpam-7048	178	9	)	)	PUNCT
ejpam-7048	178	10	)	)	PUNCT
ejpam-7048	179	1	⊆	⊆	NUM
ejpam-7048	179	2	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	179	3	)	)	PUNCT
ejpam-7048	179	4	)	)	PUNCT
ejpam-7048	179	5	)	)	PUNCT
ejpam-7048	179	6	)	)	PUNCT
ejpam-7048	179	7	and	and	CCONJ
ejpam-7048	179	8	hence	hence	ADV
ejpam-7048	179	9	f+(v	f+(v	NOUN
ejpam-7048	179	10	)	)	PUNCT
ejpam-7048	179	11	⊆	⊆	NUM
ejpam-7048	179	12	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	179	13	)	)	PUNCT
ejpam-7048	179	14	)	)	PUNCT
ejpam-7048	179	15	)	)	PUNCT
ejpam-7048	179	16	)	)	PUNCT
ejpam-7048	179	17	.	.	PUNCT
ejpam-7048	180	1	this	this	PRON
ejpam-7048	180	2	shows	show	VERB
ejpam-7048	180	3	that	that	SCONJ
ejpam-7048	180	4	f+(v	f+(v	PROPN
ejpam-7048	180	5	)	)	PUNCT
ejpam-7048	180	6	is	be	AUX
ejpam-7048	180	7	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	180	8	-	-	NOUN
ejpam-7048	180	9	open	open	ADJ
ejpam-7048	180	10	in	in	ADP
ejpam-7048	180	11	x.	x.	PROPN
ejpam-7048	180	12	p.	p.	NOUN
ejpam-7048	180	13	pue	pue	PROPN
ejpam-7048	180	14	-	-	PUNCT
ejpam-7048	180	15	on	on	ADP
ejpam-7048	180	16	,	,	PUNCT
ejpam-7048	180	17	a.	a.	PROPN
ejpam-7048	180	18	sama	sama	PROPN
ejpam-7048	180	19	-	-	PUNCT
ejpam-7048	180	20	ae	ae	PROPN
ejpam-7048	180	21	,	,	PUNCT
ejpam-7048	180	22	c.	c.	PROPN
ejpam-7048	180	23	boonpok	boonpok	PROPN
ejpam-7048	180	24	/	/	SYM
ejpam-7048	180	25	eur	eur	PROPN
ejpam-7048	180	26	.	.	PUNCT
ejpam-7048	181	1	j.	j.	PROPN
ejpam-7048	181	2	pure	pure	PROPN
ejpam-7048	181	3	appl	appl	PROPN
ejpam-7048	181	4	.	.	PROPN
ejpam-7048	181	5	math	math	PROPN
ejpam-7048	181	6	,	,	PUNCT
ejpam-7048	181	7	18	18	NUM
ejpam-7048	181	8	(	(	PUNCT
ejpam-7048	181	9	4	4	NUM
ejpam-7048	181	10	)	)	PUNCT
ejpam-7048	181	11	(	(	PUNCT
ejpam-7048	181	12	2025	2025	NUM
ejpam-7048	181	13	)	)	PUNCT
ejpam-7048	181	14	,	,	PUNCT
ejpam-7048	181	15	7048	7048	NUM
ejpam-7048	181	16	7	7	NUM
ejpam-7048	181	17	of	of	ADP
ejpam-7048	181	18	9	9	NUM
ejpam-7048	181	19	(	(	PUNCT
ejpam-7048	181	20	2	2	NUM
ejpam-7048	181	21	)	)	PUNCT
ejpam-7048	181	22	⇒	⇒	NOUN
ejpam-7048	181	23	(	(	PUNCT
ejpam-7048	181	24	3	3	NUM
ejpam-7048	181	25	):	):	PUNCT
ejpam-7048	181	26	this	this	PRON
ejpam-7048	181	27	follows	follow	VERB
ejpam-7048	181	28	from	from	ADP
ejpam-7048	181	29	the	the	DET
ejpam-7048	181	30	fact	fact	NOUN
ejpam-7048	181	31	that	that	SCONJ
ejpam-7048	181	32	f+(y	f+(y	PROPN
ejpam-7048	181	33	−b	−b	ADV
ejpam-7048	181	34	)	)	PUNCT
ejpam-7048	181	35	=	=	PUNCT
ejpam-7048	182	1	x	x	X
ejpam-7048	182	2	−	−	PROPN
ejpam-7048	182	3	f−(b	f−(b	PROPN
ejpam-7048	182	4	)	)	PUNCT
ejpam-7048	182	5	for	for	ADP
ejpam-7048	182	6	every	every	DET
ejpam-7048	182	7	subset	subset	NOUN
ejpam-7048	182	8	b	b	PROPN
ejpam-7048	182	9	of	of	ADP
ejpam-7048	182	10	y	y	PROPN
ejpam-7048	182	11	.	.	PUNCT
ejpam-7048	183	1	(	(	PUNCT
ejpam-7048	183	2	3	3	X
ejpam-7048	183	3	)	)	PUNCT
ejpam-7048	183	4	⇒	⇒	NOUN
ejpam-7048	183	5	(	(	PUNCT
ejpam-7048	183	6	4	4	NUM
ejpam-7048	183	7	):	):	PUNCT
ejpam-7048	183	8	for	for	ADP
ejpam-7048	183	9	any	any	DET
ejpam-7048	183	10	subset	subset	NOUN
ejpam-7048	183	11	b	b	PROPN
ejpam-7048	183	12	of	of	ADP
ejpam-7048	183	13	y	y	PROPN
ejpam-7048	183	14	,	,	PUNCT
ejpam-7048	183	15	σ1σ2	σ1σ2	NOUN
ejpam-7048	183	16	-	-	NOUN
ejpam-7048	183	17	cl(b	cl(b	NOUN
ejpam-7048	183	18	)	)	PUNCT
ejpam-7048	183	19	is	be	AUX
ejpam-7048	183	20	σ1σ2	σ1σ2	NOUN
ejpam-7048	183	21	-	-	ADJ
ejpam-7048	183	22	closed	closed	ADJ
ejpam-7048	183	23	in	in	ADP
ejpam-7048	183	24	y	y	PROPN
ejpam-7048	183	25	and	and	CCONJ
ejpam-7048	183	26	by	by	ADP
ejpam-7048	183	27	(	(	PUNCT
ejpam-7048	183	28	3	3	NUM
ejpam-7048	183	29	)	)	PUNCT
ejpam-7048	183	30	,	,	PUNCT
ejpam-7048	183	31	we	we	PRON
ejpam-7048	183	32	have	have	VERB
ejpam-7048	183	33	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7048	183	34	-	-	PUNCT
ejpam-7048	183	35	cl(b	cl(b	NOUN
ejpam-7048	183	36	)	)	PUNCT
ejpam-7048	183	37	)	)	PUNCT
ejpam-7048	183	38	is	be	AUX
ejpam-7048	183	39	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	183	40	-	-	VERB
ejpam-7048	183	41	closed	closed	ADJ
ejpam-7048	183	42	in	in	ADP
ejpam-7048	183	43	x.	x.	NOUN
ejpam-7048	183	44	thus	thus	ADV
ejpam-7048	183	45	,	,	PUNCT
ejpam-7048	183	46	βcl⋆(f−(b	βcl⋆(f−(b	PUNCT
ejpam-7048	183	47	)	)	PUNCT
ejpam-7048	183	48	)	)	PUNCT
ejpam-7048	184	1	⊆	⊆	X
ejpam-7048	184	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7048	184	3	-	-	PUNCT
ejpam-7048	184	4	cl(b	cl(b	NOUN
ejpam-7048	184	5	)	)	PUNCT
ejpam-7048	184	6	)	)	PUNCT
ejpam-7048	184	7	.	.	PUNCT
ejpam-7048	185	1	(	(	PUNCT
ejpam-7048	185	2	4	4	X
ejpam-7048	185	3	)	)	PUNCT
ejpam-7048	185	4	⇒	⇒	NOUN
ejpam-7048	185	5	(	(	PUNCT
ejpam-7048	185	6	5	5	NUM
ejpam-7048	185	7	):	):	PUNCT
ejpam-7048	185	8	let	let	VERB
ejpam-7048	185	9	b	b	X
ejpam-7048	185	10	be	be	AUX
ejpam-7048	185	11	any	any	DET
ejpam-7048	185	12	subset	subset	NOUN
ejpam-7048	185	13	of	of	ADP
ejpam-7048	185	14	y	y	PROPN
ejpam-7048	185	15	.	.	PUNCT
ejpam-7048	186	1	by	by	ADP
ejpam-7048	186	2	(	(	PUNCT
ejpam-7048	186	3	4	4	NUM
ejpam-7048	186	4	)	)	PUNCT
ejpam-7048	186	5	and	and	CCONJ
ejpam-7048	186	6	lemma	lemma	PROPN
ejpam-7048	186	7	2	2	NUM
ejpam-7048	186	8	,	,	PUNCT
ejpam-7048	186	9	int⋆(cl⋆(int⋆(f−(b	int⋆(cl⋆(int⋆(f−(b	NOUN
ejpam-7048	186	10	)	)	PUNCT
ejpam-7048	186	11	)	)	PUNCT
ejpam-7048	186	12	)	)	PUNCT
ejpam-7048	186	13	)	)	PUNCT
ejpam-7048	187	1	⊆	⊆	NUM
ejpam-7048	187	2	βcl⋆(f−(b	βcl⋆(f−(b	NOUN
ejpam-7048	187	3	)	)	PUNCT
ejpam-7048	187	4	)	)	PUNCT
ejpam-7048	188	1	⊆	⊆	X
ejpam-7048	188	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7048	188	3	-	-	PUNCT
ejpam-7048	188	4	cl(b	cl(b	NOUN
ejpam-7048	188	5	)	)	PUNCT
ejpam-7048	188	6	)	)	PUNCT
ejpam-7048	188	7	.	.	PUNCT
ejpam-7048	189	1	(	(	PUNCT
ejpam-7048	189	2	5	5	X
ejpam-7048	189	3	)	)	PUNCT
ejpam-7048	189	4	⇒	⇒	NOUN
ejpam-7048	189	5	(	(	PUNCT
ejpam-7048	189	6	2	2	NUM
ejpam-7048	189	7	):	):	PUNCT
ejpam-7048	189	8	let	let	VERB
ejpam-7048	189	9	v	v	PART
ejpam-7048	189	10	be	be	AUX
ejpam-7048	189	11	any	any	DET
ejpam-7048	189	12	σ1σ2	σ1σ2	NOUN
ejpam-7048	189	13	-	-	ADJ
ejpam-7048	189	14	open	open	ADJ
ejpam-7048	189	15	set	set	NOUN
ejpam-7048	189	16	of	of	ADP
ejpam-7048	189	17	y	y	PROPN
ejpam-7048	189	18	.	.	PUNCT
ejpam-7048	190	1	then	then	ADV
ejpam-7048	190	2	,	,	PUNCT
ejpam-7048	190	3	y	y	PROPN
ejpam-7048	190	4	−	−	PROPN
ejpam-7048	190	5	v	v	NOUN
ejpam-7048	190	6	is	be	AUX
ejpam-7048	190	7	σ1σ2	σ1σ2	NOUN
ejpam-7048	190	8	-	-	ADJ
ejpam-7048	190	9	closed	closed	ADJ
ejpam-7048	190	10	in	in	ADP
ejpam-7048	190	11	y	y	PROPN
ejpam-7048	190	12	and	and	CCONJ
ejpam-7048	190	13	by	by	ADP
ejpam-7048	190	14	(	(	PUNCT
ejpam-7048	190	15	5	5	NUM
ejpam-7048	190	16	)	)	PUNCT
ejpam-7048	190	17	,	,	PUNCT
ejpam-7048	190	18	x	x	PUNCT
ejpam-7048	190	19	−	−	NOUN
ejpam-7048	190	20	f+(v	f+(v	NOUN
ejpam-7048	190	21	)	)	PUNCT
ejpam-7048	191	1	=	=	PUNCT
ejpam-7048	191	2	f−(y	f−(y	NOUN
ejpam-7048	191	3	−	−	ADP
ejpam-7048	191	4	v	v	NOUN
ejpam-7048	191	5	)	)	PUNCT
ejpam-7048	191	6	⊇	⊇	NOUN
ejpam-7048	191	7	int⋆(cl⋆(int⋆(f−(y	int⋆(cl⋆(int⋆(f−(y	ADJ
ejpam-7048	191	8	−	−	PROPN
ejpam-7048	191	9	v	v	NOUN
ejpam-7048	191	10	)	)	PUNCT
ejpam-7048	191	11	)	)	PUNCT
ejpam-7048	191	12	)	)	PUNCT
ejpam-7048	191	13	)	)	PUNCT
ejpam-7048	192	1	=	=	PRON
ejpam-7048	192	2	int⋆(cl⋆(int⋆(x	int⋆(cl⋆(int⋆(x	VERB
ejpam-7048	192	3	−	−	NOUN
ejpam-7048	192	4	f+(v	f+(v	NOUN
ejpam-7048	192	5	)	)	PUNCT
ejpam-7048	192	6	)	)	PUNCT
ejpam-7048	192	7	)	)	PUNCT
ejpam-7048	192	8	)	)	PUNCT
ejpam-7048	193	1	=	=	PUNCT
ejpam-7048	193	2	x	x	X
ejpam-7048	193	3	−	−	NOUN
ejpam-7048	193	4	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	193	5	)	)	PUNCT
ejpam-7048	193	6	)	)	PUNCT
ejpam-7048	193	7	)	)	PUNCT
ejpam-7048	193	8	)	)	PUNCT
ejpam-7048	193	9	.	.	PUNCT
ejpam-7048	194	1	thus	thus	ADV
ejpam-7048	194	2	,	,	PUNCT
ejpam-7048	194	3	f+(v	f+(v	PROPN
ejpam-7048	194	4	)	)	PUNCT
ejpam-7048	194	5	⊆	⊆	NUM
ejpam-7048	194	6	cl⋆(int⋆(cl⋆(f+(v	cl⋆(int⋆(cl⋆(f+(v	PROPN
ejpam-7048	194	7	)	)	PUNCT
ejpam-7048	194	8	)	)	PUNCT
ejpam-7048	194	9	)	)	PUNCT
ejpam-7048	194	10	)	)	PUNCT
ejpam-7048	194	11	and	and	CCONJ
ejpam-7048	194	12	so	so	ADV
ejpam-7048	194	13	f+(v	f+(v	PROPN
ejpam-7048	194	14	)	)	PUNCT
ejpam-7048	194	15	is	be	AUX
ejpam-7048	194	16	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	194	17	-	-	NOUN
ejpam-7048	194	18	open	open	ADJ
ejpam-7048	194	19	in	in	ADP
ejpam-7048	194	20	x.	x.	NOUN
ejpam-7048	194	21	(	(	PUNCT
ejpam-7048	194	22	2	2	NUM
ejpam-7048	194	23	)	)	PUNCT
ejpam-7048	194	24	⇒	⇒	NOUN
ejpam-7048	194	25	(	(	PUNCT
ejpam-7048	194	26	1	1	NUM
ejpam-7048	194	27	):	):	PUNCT
ejpam-7048	194	28	let	let	VERB
ejpam-7048	194	29	x	x	PUNCT
ejpam-7048	194	30	∈	∈	PROPN
ejpam-7048	194	31	x	x	X
ejpam-7048	194	32	and	and	CCONJ
ejpam-7048	194	33	v	v	X
ejpam-7048	194	34	be	be	AUX
ejpam-7048	194	35	any	any	DET
ejpam-7048	194	36	σ1σ2	σ1σ2	NOUN
ejpam-7048	194	37	-	-	ADJ
ejpam-7048	194	38	open	open	ADJ
ejpam-7048	194	39	set	set	NOUN
ejpam-7048	194	40	of	of	ADP
ejpam-7048	194	41	y	y	PROPN
ejpam-7048	194	42	containing	contain	VERB
ejpam-7048	194	43	f	f	PROPN
ejpam-7048	194	44	(	(	PUNCT
ejpam-7048	194	45	x	x	NOUN
ejpam-7048	194	46	)	)	PUNCT
ejpam-7048	194	47	.	.	PUNCT
ejpam-7048	195	1	by	by	ADP
ejpam-7048	195	2	(	(	PUNCT
ejpam-7048	195	3	2	2	NUM
ejpam-7048	195	4	)	)	PUNCT
ejpam-7048	195	5	,	,	PUNCT
ejpam-7048	195	6	we	we	PRON
ejpam-7048	195	7	have	have	VERB
ejpam-7048	195	8	f+(v	f+(v	NOUN
ejpam-7048	195	9	)	)	PUNCT
ejpam-7048	195	10	is	be	AUX
ejpam-7048	195	11	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	195	12	-	-	NOUN
ejpam-7048	195	13	open	open	ADJ
ejpam-7048	195	14	in	in	ADP
ejpam-7048	195	15	x.	x.	NOUN
ejpam-7048	195	16	put	put	VERB
ejpam-7048	195	17	u	u	NOUN
ejpam-7048	195	18	=	=	NOUN
ejpam-7048	195	19	f+(v	f+(v	PROPN
ejpam-7048	195	20	)	)	PUNCT
ejpam-7048	195	21	.	.	PUNCT
ejpam-7048	196	1	then	then	ADV
ejpam-7048	196	2	,	,	PUNCT
ejpam-7048	196	3	u	u	NOUN
ejpam-7048	196	4	is	be	AUX
ejpam-7048	196	5	a	a	DET
ejpam-7048	196	6	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	196	7	-	-	ADJ
ejpam-7048	196	8	open	open	ADJ
ejpam-7048	196	9	set	set	NOUN
ejpam-7048	196	10	of	of	ADP
ejpam-7048	196	11	x	x	PUNCT
ejpam-7048	196	12	containing	contain	VERB
ejpam-7048	196	13	x	x	PUNCT
ejpam-7048	196	14	such	such	ADJ
ejpam-7048	196	15	that	that	SCONJ
ejpam-7048	196	16	f	f	PROPN
ejpam-7048	196	17	(	(	PUNCT
ejpam-7048	196	18	u	u	NOUN
ejpam-7048	196	19	)	)	PUNCT
ejpam-7048	196	20	⊆	⊆	NUM
ejpam-7048	196	21	v	v	NOUN
ejpam-7048	196	22	.	.	PUNCT
ejpam-7048	197	1	this	this	PRON
ejpam-7048	197	2	shows	show	VERB
ejpam-7048	197	3	that	that	SCONJ
ejpam-7048	197	4	f	f	PROPN
ejpam-7048	197	5	is	be	AUX
ejpam-7048	197	6	upper	upper	ADJ
ejpam-7048	197	7	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	197	8	,	,	PUNCT
ejpam-7048	197	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7048	197	10	.	.	X
ejpam-7048	197	11	theorem	theorem	VERB
ejpam-7048	197	12	6	6	NUM
ejpam-7048	197	13	.	.	PUNCT
ejpam-7048	197	14	for	for	ADP
ejpam-7048	197	15	a	a	DET
ejpam-7048	197	16	multifunction	multifunction	NOUN
ejpam-7048	197	17	f	f	NOUN
ejpam-7048	197	18	:	:	PUNCT
ejpam-7048	197	19	(	(	PUNCT
ejpam-7048	197	20	x	x	X
ejpam-7048	197	21	,	,	PUNCT
ejpam-7048	197	22	τ	τ	PROPN
ejpam-7048	197	23	,	,	PUNCT
ejpam-7048	197	24	i	i	NOUN
ejpam-7048	197	25	)	)	PUNCT
ejpam-7048	197	26	→	→	PUNCT
ejpam-7048	197	27	(	(	PUNCT
ejpam-7048	197	28	y	y	PROPN
ejpam-7048	197	29	,	,	PUNCT
ejpam-7048	197	30	σ1	σ1	PROPN
ejpam-7048	197	31	,	,	PUNCT
ejpam-7048	197	32	σ2	σ2	NOUN
ejpam-7048	197	33	)	)	PUNCT
ejpam-7048	197	34	,	,	PUNCT
ejpam-7048	197	35	the	the	DET
ejpam-7048	197	36	following	follow	VERB
ejpam-7048	197	37	properties	property	NOUN
ejpam-7048	197	38	are	be	AUX
ejpam-7048	197	39	equivalent	equivalent	ADJ
ejpam-7048	197	40	:	:	PUNCT
ejpam-7048	197	41	(	(	PUNCT
ejpam-7048	197	42	1	1	X
ejpam-7048	197	43	)	)	PUNCT
ejpam-7048	197	44	f	f	PROPN
ejpam-7048	197	45	is	be	AUX
ejpam-7048	197	46	lower	low	ADJ
ejpam-7048	197	47	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7048	197	48	,	,	PUNCT
ejpam-7048	197	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	197	50	;	;	PUNCT
ejpam-7048	197	51	(	(	PUNCT
ejpam-7048	197	52	2	2	X
ejpam-7048	197	53	)	)	PUNCT
ejpam-7048	197	54	f−(v	f−(v	NOUN
ejpam-7048	197	55	)	)	PUNCT
ejpam-7048	197	56	is	be	AUX
ejpam-7048	197	57	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	197	58	-	-	NOUN
ejpam-7048	197	59	open	open	ADJ
ejpam-7048	197	60	in	in	ADP
ejpam-7048	197	61	x	x	PUNCT
ejpam-7048	197	62	for	for	ADP
ejpam-7048	197	63	every	every	DET
ejpam-7048	197	64	σ1σ2	σ1σ2	NOUN
ejpam-7048	197	65	-	-	ADJ
ejpam-7048	197	66	open	open	ADJ
ejpam-7048	197	67	set	set	NOUN
ejpam-7048	197	68	v	v	NOUN
ejpam-7048	197	69	of	of	ADP
ejpam-7048	197	70	y	y	PROPN
ejpam-7048	197	71	;	;	PUNCT
ejpam-7048	197	72	(	(	PUNCT
ejpam-7048	197	73	3	3	X
ejpam-7048	197	74	)	)	PUNCT
ejpam-7048	197	75	f+(k	f+(k	NUM
ejpam-7048	197	76	)	)	PUNCT
ejpam-7048	197	77	is	be	AUX
ejpam-7048	197	78	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	197	79	-	-	VERB
ejpam-7048	197	80	closed	closed	ADJ
ejpam-7048	197	81	in	in	ADP
ejpam-7048	197	82	x	x	PUNCT
ejpam-7048	197	83	for	for	ADP
ejpam-7048	197	84	every	every	DET
ejpam-7048	197	85	σ1σ2	σ1σ2	NUM
ejpam-7048	197	86	-	-	PUNCT
ejpam-7048	197	87	closed	closed	ADJ
ejpam-7048	197	88	set	set	NOUN
ejpam-7048	197	89	k	k	PROPN
ejpam-7048	197	90	of	of	ADP
ejpam-7048	197	91	y	y	PROPN
ejpam-7048	197	92	;	;	PUNCT
ejpam-7048	197	93	(	(	PUNCT
ejpam-7048	197	94	4	4	X
ejpam-7048	197	95	)	)	PUNCT
ejpam-7048	197	96	βcl⋆(f+(b	βcl⋆(f+(b	NOUN
ejpam-7048	197	97	)	)	PUNCT
ejpam-7048	197	98	)	)	PUNCT
ejpam-7048	198	1	⊆	⊆	NUM
ejpam-7048	198	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7048	198	3	-	-	PUNCT
ejpam-7048	198	4	cl(b	cl(b	NOUN
ejpam-7048	198	5	)	)	PUNCT
ejpam-7048	198	6	)	)	PUNCT
ejpam-7048	198	7	for	for	ADP
ejpam-7048	198	8	every	every	DET
ejpam-7048	198	9	subset	subset	NOUN
ejpam-7048	198	10	b	b	PROPN
ejpam-7048	198	11	of	of	ADP
ejpam-7048	198	12	y	y	PROPN
ejpam-7048	198	13	;	;	PUNCT
ejpam-7048	198	14	(	(	PUNCT
ejpam-7048	198	15	5	5	X
ejpam-7048	198	16	)	)	PUNCT
ejpam-7048	198	17	int⋆(cl⋆(int⋆(f+(b	int⋆(cl⋆(int⋆(f+(b	NOUN
ejpam-7048	198	18	)	)	PUNCT
ejpam-7048	198	19	)	)	PUNCT
ejpam-7048	198	20	)	)	PUNCT
ejpam-7048	198	21	)	)	PUNCT
ejpam-7048	199	1	⊆	⊆	X
ejpam-7048	199	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7048	199	3	-	-	PUNCT
ejpam-7048	199	4	cl(b	cl(b	NOUN
ejpam-7048	199	5	)	)	PUNCT
ejpam-7048	199	6	)	)	PUNCT
ejpam-7048	199	7	for	for	ADP
ejpam-7048	199	8	every	every	DET
ejpam-7048	199	9	subset	subset	NOUN
ejpam-7048	199	10	b	b	PROPN
ejpam-7048	199	11	of	of	ADP
ejpam-7048	199	12	y	y	PROPN
ejpam-7048	199	13	;	;	PUNCT
ejpam-7048	199	14	(	(	PUNCT
ejpam-7048	199	15	6	6	X
ejpam-7048	199	16	)	)	PUNCT
ejpam-7048	199	17	f	f	NOUN
ejpam-7048	199	18	(	(	PUNCT
ejpam-7048	199	19	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7048	199	20	)	)	PUNCT
ejpam-7048	199	21	)	)	PUNCT
ejpam-7048	199	22	)	)	PUNCT
ejpam-7048	199	23	)	)	PUNCT
ejpam-7048	200	1	⊆	⊆	X
ejpam-7048	200	2	σ1σ2	σ1σ2	X
ejpam-7048	200	3	-	-	NUM
ejpam-7048	200	4	cl(f	cl(f	NOUN
ejpam-7048	200	5	(	(	PUNCT
ejpam-7048	200	6	a	a	NOUN
ejpam-7048	200	7	)	)	PUNCT
ejpam-7048	200	8	)	)	PUNCT
ejpam-7048	200	9	for	for	ADP
ejpam-7048	200	10	every	every	DET
ejpam-7048	200	11	subset	subset	NOUN
ejpam-7048	200	12	a	a	PRON
ejpam-7048	200	13	of	of	ADP
ejpam-7048	200	14	x	x	PRON
ejpam-7048	200	15	;	;	PUNCT
ejpam-7048	200	16	(	(	PUNCT
ejpam-7048	200	17	7	7	X
ejpam-7048	200	18	)	)	PUNCT
ejpam-7048	200	19	f	f	NOUN
ejpam-7048	200	20	(	(	PUNCT
ejpam-7048	200	21	βcl⋆(a	βcl⋆(a	NUM
ejpam-7048	200	22	)	)	PUNCT
ejpam-7048	200	23	)	)	PUNCT
ejpam-7048	201	1	⊆	⊆	X
ejpam-7048	201	2	σ1σ2	σ1σ2	X
ejpam-7048	201	3	-	-	NUM
ejpam-7048	201	4	cl(f	cl(f	NOUN
ejpam-7048	201	5	(	(	PUNCT
ejpam-7048	201	6	a	a	NOUN
ejpam-7048	201	7	)	)	PUNCT
ejpam-7048	201	8	)	)	PUNCT
ejpam-7048	201	9	for	for	ADP
ejpam-7048	201	10	every	every	DET
ejpam-7048	201	11	subset	subset	NOUN
ejpam-7048	201	12	a	a	PRON
ejpam-7048	201	13	of	of	ADP
ejpam-7048	201	14	x.	x.	NOUN
ejpam-7048	201	15	proof	proof	NOUN
ejpam-7048	201	16	.	.	PUNCT
ejpam-7048	202	1	it	it	PRON
ejpam-7048	202	2	is	be	AUX
ejpam-7048	202	3	shown	show	VERB
ejpam-7048	202	4	similarly	similarly	ADV
ejpam-7048	202	5	to	to	ADP
ejpam-7048	202	6	the	the	DET
ejpam-7048	202	7	proof	proof	NOUN
ejpam-7048	202	8	of	of	ADP
ejpam-7048	202	9	theorem	theorem	NOUN
ejpam-7048	202	10	5	5	NUM
ejpam-7048	202	11	that	that	SCONJ
ejpam-7048	202	12	the	the	DET
ejpam-7048	202	13	statements	statement	NOUN
ejpam-7048	202	14	(	(	PUNCT
ejpam-7048	202	15	1	1	NUM
ejpam-7048	202	16	)	)	PUNCT
ejpam-7048	202	17	,	,	PUNCT
ejpam-7048	202	18	(	(	PUNCT
ejpam-7048	202	19	2	2	NUM
ejpam-7048	202	20	)	)	PUNCT
ejpam-7048	202	21	,	,	PUNCT
ejpam-7048	202	22	(	(	PUNCT
ejpam-7048	202	23	3	3	NUM
ejpam-7048	202	24	)	)	PUNCT
ejpam-7048	202	25	,	,	PUNCT
ejpam-7048	202	26	(	(	PUNCT
ejpam-7048	202	27	4	4	NUM
ejpam-7048	202	28	)	)	PUNCT
ejpam-7048	202	29	and	and	CCONJ
ejpam-7048	202	30	(	(	PUNCT
ejpam-7048	202	31	5	5	X
ejpam-7048	202	32	)	)	PUNCT
ejpam-7048	202	33	are	be	AUX
ejpam-7048	202	34	equivalent	equivalent	ADJ
ejpam-7048	202	35	.	.	PUNCT
ejpam-7048	203	1	we	we	PRON
ejpam-7048	203	2	shall	shall	AUX
ejpam-7048	203	3	prove	prove	VERB
ejpam-7048	203	4	only	only	ADV
ejpam-7048	203	5	the	the	DET
ejpam-7048	203	6	following	follow	VERB
ejpam-7048	203	7	implications	implication	NOUN
ejpam-7048	203	8	.	.	PUNCT
ejpam-7048	204	1	(	(	PUNCT
ejpam-7048	204	2	5	5	X
ejpam-7048	204	3	)	)	PUNCT
ejpam-7048	204	4	⇒	⇒	NOUN
ejpam-7048	204	5	(	(	PUNCT
ejpam-7048	204	6	6	6	NUM
ejpam-7048	204	7	):	):	PUNCT
ejpam-7048	204	8	let	let	VERB
ejpam-7048	204	9	a	a	DET
ejpam-7048	204	10	be	be	AUX
ejpam-7048	204	11	any	any	DET
ejpam-7048	204	12	subset	subset	NOUN
ejpam-7048	204	13	of	of	ADP
ejpam-7048	204	14	x.	x.	NOUN
ejpam-7048	204	15	by	by	ADP
ejpam-7048	204	16	(	(	PUNCT
ejpam-7048	204	17	5	5	NUM
ejpam-7048	204	18	)	)	PUNCT
ejpam-7048	204	19	,	,	PUNCT
ejpam-7048	204	20	we	we	PRON
ejpam-7048	204	21	have	have	VERB
ejpam-7048	204	22	int⋆(cl⋆(int⋆(f+(f	int⋆(cl⋆(int⋆(f+(f	NOUN
ejpam-7048	204	23	(	(	PUNCT
ejpam-7048	204	24	a	a	NOUN
ejpam-7048	204	25	)	)	PUNCT
ejpam-7048	204	26	)	)	PUNCT
ejpam-7048	204	27	)	)	PUNCT
ejpam-7048	204	28	)	)	PUNCT
ejpam-7048	204	29	)	)	PUNCT
ejpam-7048	205	1	⊆	⊆	X
ejpam-7048	205	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7048	205	3	-	-	SYM
ejpam-7048	205	4	cl(f	cl(f	NOUN
ejpam-7048	205	5	(	(	PUNCT
ejpam-7048	205	6	a	a	NOUN
ejpam-7048	205	7	)	)	PUNCT
ejpam-7048	205	8	)	)	PUNCT
ejpam-7048	205	9	)	)	PUNCT
ejpam-7048	205	10	and	and	CCONJ
ejpam-7048	205	11	hence	hence	ADV
ejpam-7048	205	12	f	f	PROPN
ejpam-7048	205	13	(	(	PUNCT
ejpam-7048	205	14	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7048	205	15	)	)	PUNCT
ejpam-7048	205	16	)	)	PUNCT
ejpam-7048	205	17	)	)	PUNCT
ejpam-7048	205	18	)	)	PUNCT
ejpam-7048	206	1	⊆	⊆	X
ejpam-7048	206	2	σ1σ2	σ1σ2	X
ejpam-7048	206	3	-	-	NUM
ejpam-7048	206	4	cl(f	cl(f	NOUN
ejpam-7048	206	5	(	(	PUNCT
ejpam-7048	206	6	a	a	NOUN
ejpam-7048	206	7	)	)	PUNCT
ejpam-7048	206	8	)	)	PUNCT
ejpam-7048	206	9	.	.	PUNCT
ejpam-7048	207	1	(	(	PUNCT
ejpam-7048	207	2	6	6	X
ejpam-7048	207	3	)	)	PUNCT
ejpam-7048	207	4	⇒	⇒	NOUN
ejpam-7048	207	5	(	(	PUNCT
ejpam-7048	207	6	7	7	NUM
ejpam-7048	207	7	):	):	PUNCT
ejpam-7048	207	8	let	let	VERB
ejpam-7048	207	9	a	a	DET
ejpam-7048	207	10	be	be	AUX
ejpam-7048	207	11	any	any	DET
ejpam-7048	207	12	subset	subset	NOUN
ejpam-7048	207	13	of	of	ADP
ejpam-7048	207	14	x.	x.	NOUN
ejpam-7048	207	15	by	by	ADP
ejpam-7048	207	16	(	(	PUNCT
ejpam-7048	207	17	6	6	NUM
ejpam-7048	207	18	)	)	PUNCT
ejpam-7048	207	19	and	and	CCONJ
ejpam-7048	207	20	lemma	lemma	PROPN
ejpam-7048	207	21	2	2	NUM
ejpam-7048	207	22	,	,	PUNCT
ejpam-7048	207	23	we	we	PRON
ejpam-7048	207	24	have	have	VERB
ejpam-7048	207	25	f	f	PROPN
ejpam-7048	207	26	(	(	PUNCT
ejpam-7048	207	27	βcl⋆(a	βcl⋆(a	NUM
ejpam-7048	207	28	)	)	PUNCT
ejpam-7048	207	29	)	)	PUNCT
ejpam-7048	208	1	=	=	SYM
ejpam-7048	208	2	f	f	PROPN
ejpam-7048	208	3	(	(	PUNCT
ejpam-7048	208	4	a	a	DET
ejpam-7048	208	5	∪	∪	ADJ
ejpam-7048	208	6	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7048	208	7	)	)	PUNCT
ejpam-7048	208	8	)	)	PUNCT
ejpam-7048	208	9	)	)	PUNCT
ejpam-7048	208	10	)	)	PUNCT
ejpam-7048	209	1	p.	p.	NOUN
ejpam-7048	209	2	pue	pue	NOUN
ejpam-7048	209	3	-	-	PUNCT
ejpam-7048	209	4	on	on	ADP
ejpam-7048	209	5	,	,	PUNCT
ejpam-7048	209	6	a.	a.	PROPN
ejpam-7048	209	7	sama	sama	PROPN
ejpam-7048	209	8	-	-	PUNCT
ejpam-7048	209	9	ae	ae	PROPN
ejpam-7048	209	10	,	,	PUNCT
ejpam-7048	209	11	c.	c.	PROPN
ejpam-7048	209	12	boonpok	boonpok	PROPN
ejpam-7048	209	13	/	/	SYM
ejpam-7048	209	14	eur	eur	PROPN
ejpam-7048	209	15	.	.	PUNCT
ejpam-7048	210	1	j.	j.	PROPN
ejpam-7048	210	2	pure	pure	PROPN
ejpam-7048	210	3	appl	appl	PROPN
ejpam-7048	210	4	.	.	PROPN
ejpam-7048	210	5	math	math	PROPN
ejpam-7048	210	6	,	,	PUNCT
ejpam-7048	210	7	18	18	NUM
ejpam-7048	210	8	(	(	PUNCT
ejpam-7048	210	9	4	4	NUM
ejpam-7048	210	10	)	)	PUNCT
ejpam-7048	210	11	(	(	PUNCT
ejpam-7048	210	12	2025	2025	NUM
ejpam-7048	210	13	)	)	PUNCT
ejpam-7048	210	14	,	,	PUNCT
ejpam-7048	210	15	7048	7048	NUM
ejpam-7048	210	16	8	8	NUM
ejpam-7048	210	17	of	of	ADP
ejpam-7048	210	18	9	9	NUM
ejpam-7048	210	19	=	=	SYM
ejpam-7048	210	20	f	f	X
ejpam-7048	210	21	(	(	PUNCT
ejpam-7048	210	22	a	a	NOUN
ejpam-7048	210	23	)	)	PUNCT
ejpam-7048	210	24	∪	∪	PROPN
ejpam-7048	210	25	f	f	PROPN
ejpam-7048	210	26	(	(	PUNCT
ejpam-7048	210	27	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7048	210	28	)	)	PUNCT
ejpam-7048	210	29	)	)	PUNCT
ejpam-7048	210	30	)	)	PUNCT
ejpam-7048	210	31	)	)	PUNCT
ejpam-7048	211	1	⊆	⊆	X
ejpam-7048	211	2	σ1σ2	σ1σ2	X
ejpam-7048	211	3	-	-	NUM
ejpam-7048	211	4	cl(f	cl(f	NOUN
ejpam-7048	211	5	(	(	PUNCT
ejpam-7048	211	6	a	a	NOUN
ejpam-7048	211	7	)	)	PUNCT
ejpam-7048	211	8	)	)	PUNCT
ejpam-7048	211	9	.	.	PUNCT
ejpam-7048	212	1	(	(	PUNCT
ejpam-7048	212	2	7	7	X
ejpam-7048	212	3	)	)	PUNCT
ejpam-7048	212	4	⇒	⇒	NOUN
ejpam-7048	212	5	(	(	PUNCT
ejpam-7048	212	6	3	3	NUM
ejpam-7048	212	7	):	):	PUNCT
ejpam-7048	212	8	let	let	VERB
ejpam-7048	212	9	k	k	PRON
ejpam-7048	212	10	be	be	AUX
ejpam-7048	212	11	any	any	DET
ejpam-7048	212	12	σ1σ2	σ1σ2	NUM
ejpam-7048	212	13	-	-	PUNCT
ejpam-7048	212	14	closed	closed	ADJ
ejpam-7048	212	15	set	set	NOUN
ejpam-7048	212	16	of	of	ADP
ejpam-7048	212	17	y	y	PROPN
ejpam-7048	212	18	.	.	PUNCT
ejpam-7048	213	1	thus	thus	ADV
ejpam-7048	213	2	by	by	ADP
ejpam-7048	213	3	(	(	PUNCT
ejpam-7048	213	4	7	7	NUM
ejpam-7048	213	5	)	)	PUNCT
ejpam-7048	213	6	,	,	PUNCT
ejpam-7048	213	7	f	f	PROPN
ejpam-7048	213	8	(	(	PUNCT
ejpam-7048	213	9	βcl⋆(f+(k	βcl⋆(f+(k	NOUN
ejpam-7048	213	10	)	)	PUNCT
ejpam-7048	213	11	)	)	PUNCT
ejpam-7048	213	12	)	)	PUNCT
ejpam-7048	214	1	⊆	⊆	X
ejpam-7048	214	2	σ1σ2	σ1σ2	X
ejpam-7048	214	3	-	-	NUM
ejpam-7048	214	4	cl(f	cl(f	NOUN
ejpam-7048	214	5	(	(	PUNCT
ejpam-7048	214	6	f+(k	f+(k	NOUN
ejpam-7048	214	7	)	)	PUNCT
ejpam-7048	214	8	)	)	PUNCT
ejpam-7048	214	9	)	)	PUNCT
ejpam-7048	215	1	⊆	⊆	X
ejpam-7048	215	2	σ1σ2	σ1σ2	NOUN
ejpam-7048	215	3	-	-	PUNCT
ejpam-7048	215	4	cl(k	cl(k	NOUN
ejpam-7048	215	5	)	)	PUNCT
ejpam-7048	215	6	=	=	PUNCT
ejpam-7048	215	7	k.	k.	PROPN
ejpam-7048	216	1	thus	thus	ADV
ejpam-7048	216	2	,	,	PUNCT
ejpam-7048	216	3	βcl⋆(f+(k	βcl⋆(f+(k	NOUN
ejpam-7048	216	4	)	)	PUNCT
ejpam-7048	216	5	)	)	PUNCT
ejpam-7048	217	1	⊆	⊆	NUM
ejpam-7048	217	2	f+(k	f+(k	NUM
ejpam-7048	217	3	)	)	PUNCT
ejpam-7048	217	4	and	and	CCONJ
ejpam-7048	217	5	hence	hence	ADV
ejpam-7048	217	6	f+(k	f+(k	NUM
ejpam-7048	217	7	)	)	PUNCT
ejpam-7048	217	8	is	be	AUX
ejpam-7048	217	9	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	217	10	-	-	VERB
ejpam-7048	217	11	closed	closed	ADJ
ejpam-7048	217	12	in	in	ADP
ejpam-7048	217	13	x.	x.	NOUN
ejpam-7048	217	14	corollary	corollary	NOUN
ejpam-7048	217	15	3	3	PROPN
ejpam-7048	217	16	.	.	PUNCT
ejpam-7048	218	1	for	for	ADP
ejpam-7048	218	2	a	a	DET
ejpam-7048	218	3	function	function	NOUN
ejpam-7048	218	4	f	f	NOUN
ejpam-7048	218	5	:	:	PUNCT
ejpam-7048	218	6	(	(	PUNCT
ejpam-7048	218	7	x	x	X
ejpam-7048	218	8	,	,	PUNCT
ejpam-7048	218	9	τ	τ	PROPN
ejpam-7048	218	10	,	,	PUNCT
ejpam-7048	218	11	i	i	NOUN
ejpam-7048	218	12	)	)	PUNCT
ejpam-7048	218	13	→	→	PUNCT
ejpam-7048	218	14	(	(	PUNCT
ejpam-7048	218	15	y	y	PROPN
ejpam-7048	218	16	,	,	PUNCT
ejpam-7048	218	17	σ1	σ1	PROPN
ejpam-7048	218	18	,	,	PUNCT
ejpam-7048	218	19	σ2	σ2	NOUN
ejpam-7048	218	20	)	)	PUNCT
ejpam-7048	218	21	,	,	PUNCT
ejpam-7048	218	22	the	the	DET
ejpam-7048	218	23	following	follow	VERB
ejpam-7048	218	24	properties	property	NOUN
ejpam-7048	218	25	are	be	AUX
ejpam-7048	218	26	equivalent	equivalent	ADJ
ejpam-7048	218	27	:	:	PUNCT
ejpam-7048	218	28	(	(	PUNCT
ejpam-7048	218	29	1	1	X
ejpam-7048	218	30	)	)	PUNCT
ejpam-7048	218	31	f	f	PROPN
ejpam-7048	218	32	is	be	AUX
ejpam-7048	218	33	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7048	218	34	,	,	PUNCT
ejpam-7048	218	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7048	218	36	;	;	PUNCT
ejpam-7048	218	37	(	(	PUNCT
ejpam-7048	218	38	2	2	X
ejpam-7048	218	39	)	)	PUNCT
ejpam-7048	218	40	f−1(v	f−1(v	NOUN
ejpam-7048	218	41	)	)	PUNCT
ejpam-7048	218	42	is	be	AUX
ejpam-7048	218	43	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	218	44	-	-	NOUN
ejpam-7048	218	45	open	open	ADJ
ejpam-7048	218	46	in	in	ADP
ejpam-7048	218	47	x	x	PUNCT
ejpam-7048	218	48	for	for	ADP
ejpam-7048	218	49	every	every	DET
ejpam-7048	218	50	σ1σ2	σ1σ2	NOUN
ejpam-7048	218	51	-	-	ADJ
ejpam-7048	218	52	open	open	ADJ
ejpam-7048	218	53	set	set	NOUN
ejpam-7048	218	54	v	v	NOUN
ejpam-7048	218	55	of	of	ADP
ejpam-7048	218	56	y	y	PROPN
ejpam-7048	218	57	;	;	PUNCT
ejpam-7048	218	58	(	(	PUNCT
ejpam-7048	218	59	3	3	X
ejpam-7048	218	60	)	)	PUNCT
ejpam-7048	218	61	f−1(k	f−1(k	PROPN
ejpam-7048	218	62	)	)	PUNCT
ejpam-7048	218	63	is	be	AUX
ejpam-7048	218	64	τ⋆-β	τ⋆-β	NOUN
ejpam-7048	218	65	-	-	VERB
ejpam-7048	218	66	closed	closed	ADJ
ejpam-7048	218	67	in	in	ADP
ejpam-7048	218	68	x	x	PUNCT
ejpam-7048	218	69	for	for	ADP
ejpam-7048	218	70	every	every	DET
ejpam-7048	218	71	σ1σ2	σ1σ2	NUM
ejpam-7048	218	72	-	-	PUNCT
ejpam-7048	218	73	closed	closed	ADJ
ejpam-7048	218	74	set	set	NOUN
ejpam-7048	218	75	k	k	PROPN
ejpam-7048	218	76	of	of	ADP
ejpam-7048	218	77	y	y	PROPN
ejpam-7048	218	78	;	;	PUNCT
ejpam-7048	218	79	(	(	PUNCT
ejpam-7048	218	80	4	4	X
ejpam-7048	218	81	)	)	PUNCT
ejpam-7048	218	82	βcl⋆(f−1(b	βcl⋆(f−1(b	NUM
ejpam-7048	218	83	)	)	PUNCT
ejpam-7048	218	84	)	)	PUNCT
ejpam-7048	219	1	⊆	⊆	NUM
ejpam-7048	219	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7048	219	3	-	-	PUNCT
ejpam-7048	219	4	cl(b	cl(b	NOUN
ejpam-7048	219	5	)	)	PUNCT
ejpam-7048	219	6	)	)	PUNCT
ejpam-7048	219	7	for	for	ADP
ejpam-7048	219	8	every	every	DET
ejpam-7048	219	9	subset	subset	NOUN
ejpam-7048	219	10	b	b	PROPN
ejpam-7048	219	11	of	of	ADP
ejpam-7048	219	12	y	y	PROPN
ejpam-7048	219	13	;	;	PUNCT
ejpam-7048	219	14	(	(	PUNCT
ejpam-7048	219	15	5	5	X
ejpam-7048	219	16	)	)	PUNCT
ejpam-7048	219	17	int⋆(cl⋆(int⋆(f−1(b	int⋆(cl⋆(int⋆(f−1(b	ADJ
ejpam-7048	219	18	)	)	PUNCT
ejpam-7048	219	19	)	)	PUNCT
ejpam-7048	219	20	)	)	PUNCT
ejpam-7048	219	21	)	)	PUNCT
ejpam-7048	219	22	⊆	⊆	NUM
ejpam-7048	219	23	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7048	219	24	-	-	PUNCT
ejpam-7048	219	25	cl(b	cl(b	NOUN
ejpam-7048	219	26	)	)	PUNCT
ejpam-7048	219	27	)	)	PUNCT
ejpam-7048	220	1	for	for	ADP
ejpam-7048	220	2	every	every	DET
ejpam-7048	220	3	subset	subset	NOUN
ejpam-7048	220	4	b	b	PROPN
ejpam-7048	220	5	of	of	ADP
ejpam-7048	220	6	y	y	PROPN
ejpam-7048	220	7	;	;	PUNCT
ejpam-7048	220	8	(	(	PUNCT
ejpam-7048	220	9	6	6	NUM
ejpam-7048	220	10	)	)	PUNCT
ejpam-7048	220	11	f(int⋆(cl⋆(int⋆(a	f(int⋆(cl⋆(int⋆(a	NOUN
ejpam-7048	220	12	)	)	PUNCT
ejpam-7048	220	13	)	)	PUNCT
ejpam-7048	220	14	)	)	PUNCT
ejpam-7048	220	15	)	)	PUNCT
ejpam-7048	221	1	⊆	⊆	X
ejpam-7048	221	2	σ1σ2	σ1σ2	NUM
ejpam-7048	221	3	-	-	PUNCT
ejpam-7048	221	4	cl(f(a	cl(f(a	NOUN
ejpam-7048	221	5	)	)	PUNCT
ejpam-7048	221	6	)	)	PUNCT
ejpam-7048	221	7	for	for	ADP
ejpam-7048	221	8	every	every	DET
ejpam-7048	221	9	subset	subset	NOUN
ejpam-7048	221	10	a	a	PRON
ejpam-7048	221	11	of	of	ADP
ejpam-7048	221	12	x	x	PRON
ejpam-7048	221	13	;	;	PUNCT
ejpam-7048	221	14	(	(	PUNCT
ejpam-7048	221	15	7	7	X
ejpam-7048	221	16	)	)	PUNCT
ejpam-7048	221	17	f(βcl⋆(a	f(βcl⋆(a	NOUN
ejpam-7048	221	18	)	)	PUNCT
ejpam-7048	221	19	)	)	PUNCT
ejpam-7048	222	1	⊆	⊆	X
ejpam-7048	222	2	σ1σ2	σ1σ2	NUM
ejpam-7048	222	3	-	-	PUNCT
ejpam-7048	222	4	cl(f(a	cl(f(a	NOUN
ejpam-7048	222	5	)	)	PUNCT
ejpam-7048	222	6	)	)	PUNCT
ejpam-7048	222	7	for	for	ADP
ejpam-7048	222	8	every	every	DET
ejpam-7048	222	9	subset	subset	NOUN
ejpam-7048	222	10	a	a	PRON
ejpam-7048	222	11	of	of	ADP
ejpam-7048	222	12	x.	x.	NOUN
ejpam-7048	222	13	acknowledgements	acknowledgement	NOUN
ejpam-7048	222	14	this	this	DET
ejpam-7048	222	15	research	research	NOUN
ejpam-7048	222	16	project	project	NOUN
ejpam-7048	222	17	was	be	AUX
ejpam-7048	222	18	financially	financially	ADV
ejpam-7048	222	19	supported	support	VERB
ejpam-7048	222	20	by	by	ADP
ejpam-7048	222	21	mahasarakham	mahasarakham	PROPN
ejpam-7048	222	22	university	university	PROPN
ejpam-7048	222	23	.	.	PUNCT
ejpam-7048	223	1	references	reference	NOUN
ejpam-7048	223	2	[	[	X
ejpam-7048	223	3	1	1	NUM
ejpam-7048	223	4	]	]	PUNCT
ejpam-7048	223	5	m.	m.	NOUN
ejpam-7048	223	6	e.	e.	PROPN
ejpam-7048	223	7	abd	abd	PROPN
ejpam-7048	223	8	el	el	PROPN
ejpam-7048	223	9	-	-	PROPN
ejpam-7048	223	10	monsef	monsef	PROPN
ejpam-7048	223	11	,	,	PUNCT
ejpam-7048	223	12	s.	s.	PROPN
ejpam-7048	223	13	n.	n.	PROPN
ejpam-7048	223	14	el	el	PROPN
ejpam-7048	223	15	-	-	PROPN
ejpam-7048	223	16	deeb	deeb	PROPN
ejpam-7048	223	17	,	,	PUNCT
ejpam-7048	223	18	and	and	CCONJ
ejpam-7048	223	19	r.	r.	PROPN
ejpam-7048	223	20	a.	a.	PROPN
ejpam-7048	223	21	mahmoud	mahmoud	PROPN
ejpam-7048	223	22	.	.	PUNCT
ejpam-7048	224	1	β	β	X
ejpam-7048	224	2	-	-	ADJ
ejpam-7048	224	3	open	open	ADJ
ejpam-7048	224	4	sets	set	NOUN
ejpam-7048	224	5	and	and	CCONJ
ejpam-7048	224	6	βcontinuous	βcontinuous	ADJ
ejpam-7048	224	7	mappings	mapping	NOUN
ejpam-7048	224	8	.	.	PUNCT
ejpam-7048	225	1	bulletin	bulletin	NOUN
ejpam-7048	225	2	of	of	ADP
ejpam-7048	225	3	the	the	DET
ejpam-7048	225	4	faculty	faculty	NOUN
ejpam-7048	225	5	of	of	ADP
ejpam-7048	225	6	science	science	NOUN
ejpam-7048	225	7	,	,	PUNCT
ejpam-7048	225	8	assiut	assiut	NOUN
ejpam-7048	225	9	university	university	NOUN
ejpam-7048	225	10	,	,	PUNCT
ejpam-7048	225	11	12:77–90	12:77–90	NUM
ejpam-7048	225	12	,	,	PUNCT
ejpam-7048	225	13	1983	1983	NUM
ejpam-7048	225	14	.	.	PUNCT
ejpam-7048	226	1	[	[	X
ejpam-7048	226	2	2	2	X
ejpam-7048	226	3	]	]	PUNCT
ejpam-7048	226	4	j.	j.	PROPN
ejpam-7048	226	5	borsík	borsík	PROPN
ejpam-7048	226	6	and	and	CCONJ
ejpam-7048	226	7	j.	j.	PROPN
ejpam-7048	226	8	doboş	doboş	PROPN
ejpam-7048	226	9	.	.	PUNCT
ejpam-7048	227	1	on	on	ADP
ejpam-7048	227	2	decompositions	decomposition	NOUN
ejpam-7048	227	3	of	of	ADP
ejpam-7048	227	4	quasicontinuity	quasicontinuity	NOUN
ejpam-7048	227	5	.	.	PUNCT
ejpam-7048	228	1	real	real	ADJ
ejpam-7048	228	2	analysis	analysis	NOUN
ejpam-7048	228	3	exchange	exchange	NOUN
ejpam-7048	228	4	,	,	PUNCT
ejpam-7048	228	5	16:292–305	16:292–305	PROPN
ejpam-7048	228	6	,	,	PUNCT
ejpam-7048	228	7	1990	1990	NUM
ejpam-7048	228	8	-	-	SYM
ejpam-7048	228	9	1991	1991	NUM
ejpam-7048	228	10	.	.	PUNCT
ejpam-7048	229	1	[	[	X
ejpam-7048	229	2	3	3	X
ejpam-7048	229	3	]	]	PUNCT
ejpam-7048	229	4	s.	s.	PROPN
ejpam-7048	229	5	marcus	marcus	PROPN
ejpam-7048	229	6	.	.	PUNCT
ejpam-7048	230	1	sur	sur	PROPN
ejpam-7048	230	2	les	les	PROPN
ejpam-7048	230	3	fonctions	fonctions	PROPN
ejpam-7048	230	4	quasicontinues	quasicontinue	NOUN
ejpam-7048	230	5	au	au	PROPN
ejpam-7048	230	6	sens	sens	X
ejpam-7048	230	7	de	de	PROPN
ejpam-7048	230	8	s.	s.	PROPN
ejpam-7048	230	9	kempisty	kempisty	PROPN
ejpam-7048	230	10	.	.	PUNCT
ejpam-7048	231	1	colloquium	colloquium	NOUN
ejpam-7048	231	2	mathematicum	mathematicum	PROPN
ejpam-7048	231	3	,	,	PUNCT
ejpam-7048	231	4	8:47–53	8:47–53	NUM
ejpam-7048	231	5	,	,	PUNCT
ejpam-7048	231	6	1961	1961	NUM
ejpam-7048	231	7	.	.	PUNCT
ejpam-7048	232	1	[	[	X
ejpam-7048	232	2	4	4	X
ejpam-7048	232	3	]	]	PUNCT
ejpam-7048	232	4	v.	v.	CCONJ
ejpam-7048	232	5	popa	popa	NOUN
ejpam-7048	232	6	and	and	CCONJ
ejpam-7048	232	7	t.	t.	PROPN
ejpam-7048	232	8	noiri	noiri	PROPN
ejpam-7048	232	9	.	.	PUNCT
ejpam-7048	233	1	on	on	ADP
ejpam-7048	233	2	β	β	ADJ
ejpam-7048	233	3	-	-	ADJ
ejpam-7048	233	4	continuous	continuous	ADJ
ejpam-7048	233	5	functions	function	NOUN
ejpam-7048	233	6	.	.	PUNCT
ejpam-7048	234	1	real	real	ADJ
ejpam-7048	234	2	analysis	analysis	NOUN
ejpam-7048	234	3	exchange	exchange	NOUN
ejpam-7048	234	4	,	,	PUNCT
ejpam-7048	234	5	18:544	18:544	NUM
ejpam-7048	234	6	–	–	PUNCT
ejpam-7048	234	7	548	548	NUM
ejpam-7048	234	8	,	,	PUNCT
ejpam-7048	234	9	1992	1992	NUM
ejpam-7048	234	10	-	-	SYM
ejpam-7048	234	11	1993	1993	NUM
ejpam-7048	234	12	.	.	PUNCT
ejpam-7048	235	1	[	[	X
ejpam-7048	235	2	5	5	X
ejpam-7048	235	3	]	]	PUNCT
ejpam-7048	235	4	j.	j.	PROPN
ejpam-7048	235	5	borsík	borsík	PROPN
ejpam-7048	235	6	.	.	PUNCT
ejpam-7048	236	1	on	on	ADP
ejpam-7048	236	2	almost	almost	ADV
ejpam-7048	236	3	quasicontinuous	quasicontinuous	ADJ
ejpam-7048	236	4	functions	function	NOUN
ejpam-7048	236	5	.	.	PUNCT
ejpam-7048	237	1	mathematica	mathematica	PROPN
ejpam-7048	237	2	bohemica	bohemica	PROPN
ejpam-7048	237	3	,	,	PUNCT
ejpam-7048	237	4	118:241–248	118:241–248	NUM
ejpam-7048	237	5	,	,	PUNCT
ejpam-7048	237	6	1993	1993	NUM
ejpam-7048	237	7	.	.	PUNCT
ejpam-7048	238	1	[	[	X
ejpam-7048	238	2	6	6	NUM
ejpam-7048	238	3	]	]	PUNCT
ejpam-7048	238	4	j.	j.	PROPN
ejpam-7048	238	5	ewert	ewert	PROPN
ejpam-7048	238	6	.	.	PUNCT
ejpam-7048	239	1	on	on	ADP
ejpam-7048	239	2	almost	almost	ADV
ejpam-7048	239	3	quasicontinuity	quasicontinuity	NOUN
ejpam-7048	239	4	for	for	ADP
ejpam-7048	239	5	functions	function	NOUN
ejpam-7048	239	6	.	.	PUNCT
ejpam-7048	240	1	tatra	tatra	PROPN
ejpam-7048	240	2	mountains	mountain	NOUN
ejpam-7048	240	3	mathematical	mathematical	ADJ
ejpam-7048	240	4	publications	publication	NOUN
ejpam-7048	240	5	,	,	PUNCT
ejpam-7048	240	6	2:81–92	2:81–92	NOUN
ejpam-7048	240	7	,	,	PUNCT
ejpam-7048	240	8	1993	1993	NUM
ejpam-7048	240	9	.	.	PUNCT
ejpam-7048	241	1	[	[	X
ejpam-7048	241	2	7	7	X
ejpam-7048	241	3	]	]	PUNCT
ejpam-7048	241	4	v.	v.	CCONJ
ejpam-7048	241	5	popa	popa	NOUN
ejpam-7048	241	6	and	and	CCONJ
ejpam-7048	241	7	t.	t.	PROPN
ejpam-7048	241	8	noiri	noiri	PROPN
ejpam-7048	241	9	.	.	PUNCT
ejpam-7048	242	1	on	on	ADP
ejpam-7048	242	2	upper	upper	ADJ
ejpam-7048	242	3	and	and	CCONJ
ejpam-7048	242	4	lower	low	ADJ
ejpam-7048	242	5	β	β	ADJ
ejpam-7048	242	6	-	-	ADJ
ejpam-7048	242	7	continuous	continuous	ADJ
ejpam-7048	242	8	multifunctions	multifunction	NOUN
ejpam-7048	242	9	.	.	PUNCT
ejpam-7048	243	1	real	real	ADJ
ejpam-7048	243	2	analysis	analysis	NOUN
ejpam-7048	243	3	exchange	exchange	NOUN
ejpam-7048	243	4	,	,	PUNCT
ejpam-7048	243	5	22(1):362–376	22(1):362–376	NOUN
ejpam-7048	243	6	,	,	PUNCT
ejpam-7048	243	7	1996	1996	NUM
ejpam-7048	243	8	-	-	SYM
ejpam-7048	243	9	1997	1997	NUM
ejpam-7048	243	10	.	.	PUNCT
ejpam-7048	244	1	p.	p.	NOUN
ejpam-7048	244	2	pue	pue	NOUN
ejpam-7048	244	3	-	-	PUNCT
ejpam-7048	244	4	on	on	ADP
ejpam-7048	244	5	,	,	PUNCT
ejpam-7048	244	6	a.	a.	PROPN
ejpam-7048	244	7	sama	sama	PROPN
ejpam-7048	244	8	-	-	PUNCT
ejpam-7048	244	9	ae	ae	PROPN
ejpam-7048	244	10	,	,	PUNCT
ejpam-7048	244	11	c.	c.	PROPN
ejpam-7048	244	12	boonpok	boonpok	PROPN
ejpam-7048	244	13	/	/	SYM
ejpam-7048	244	14	eur	eur	PROPN
ejpam-7048	244	15	.	.	PUNCT
ejpam-7048	245	1	j.	j.	PROPN
ejpam-7048	245	2	pure	pure	PROPN
ejpam-7048	245	3	appl	appl	PROPN
ejpam-7048	245	4	.	.	PROPN
ejpam-7048	245	5	math	math	PROPN
ejpam-7048	245	6	,	,	PUNCT
ejpam-7048	245	7	18	18	NUM
ejpam-7048	245	8	(	(	PUNCT
ejpam-7048	245	9	4	4	NUM
ejpam-7048	245	10	)	)	PUNCT
ejpam-7048	245	11	(	(	PUNCT
ejpam-7048	245	12	2025	2025	NUM
ejpam-7048	245	13	)	)	PUNCT
ejpam-7048	245	14	,	,	PUNCT
ejpam-7048	245	15	7048	7048	NUM
ejpam-7048	245	16	9	9	NUM
ejpam-7048	245	17	of	of	ADP
ejpam-7048	245	18	9	9	NUM
ejpam-7048	245	19	[	[	SYM
ejpam-7048	245	20	8	8	NUM
ejpam-7048	245	21	]	]	PUNCT
ejpam-7048	245	22	c.	c.	PROPN
ejpam-7048	245	23	boonpok	boonpok	PROPN
ejpam-7048	245	24	.	.	PUNCT
ejpam-7048	246	1	on	on	ADP
ejpam-7048	246	2	continuous	continuous	ADJ
ejpam-7048	246	3	multifunctions	multifunction	NOUN
ejpam-7048	246	4	in	in	ADP
ejpam-7048	246	5	ideal	ideal	ADJ
ejpam-7048	246	6	topological	topological	ADJ
ejpam-7048	246	7	spaces	space	NOUN
ejpam-7048	246	8	.	.	PUNCT
ejpam-7048	247	1	lobachevskii	lobachevskii	PROPN
ejpam-7048	247	2	journal	journal	PROPN
ejpam-7048	247	3	of	of	ADP
ejpam-7048	247	4	mathematics	mathematic	NOUN
ejpam-7048	247	5	,	,	PUNCT
ejpam-7048	247	6	40(1):24–35	40(1):24–35	NUM
ejpam-7048	247	7	,	,	PUNCT
ejpam-7048	247	8	2019	2019	NUM
ejpam-7048	247	9	.	.	PUNCT
ejpam-7048	248	1	[	[	X
ejpam-7048	248	2	9	9	NUM
ejpam-7048	248	3	]	]	PUNCT
ejpam-7048	248	4	c.	c.	PROPN
ejpam-7048	248	5	boonpok	boonpok	PROPN
ejpam-7048	248	6	.	.	PUNCT
ejpam-7048	249	1	upper	upper	ADJ
ejpam-7048	249	2	and	and	CCONJ
ejpam-7048	249	3	lower	low	ADJ
ejpam-7048	249	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-7048	249	5	.	.	PUNCT
ejpam-7048	249	6	heliyon	heliyon	NOUN
ejpam-7048	249	7	,	,	PUNCT
ejpam-7048	249	8	7	7	NUM
ejpam-7048	249	9	:	:	PUNCT
ejpam-7048	249	10	e05986	e05986	PROPN
ejpam-7048	249	11	,	,	PUNCT
ejpam-7048	249	12	2021	2021	NUM
ejpam-7048	249	13	.	.	PUNCT
ejpam-7048	250	1	[	[	X
ejpam-7048	250	2	10	10	NUM
ejpam-7048	250	3	]	]	X
ejpam-7048	250	4	c.	c.	PROPN
ejpam-7048	250	5	boonpok	boonpok	PROPN
ejpam-7048	250	6	and	and	CCONJ
ejpam-7048	250	7	p.	p.	NOUN
ejpam-7048	250	8	pue	pue	NOUN
ejpam-7048	250	9	-	-	PUNCT
ejpam-7048	250	10	on	on	ADP
ejpam-7048	250	11	.	.	PUNCT
ejpam-7048	251	1	upper	upper	ADJ
ejpam-7048	251	2	and	and	CCONJ
ejpam-7048	251	3	lower	low	ADJ
ejpam-7048	251	4	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7048	251	5	multifunctions	multifunction	NOUN
ejpam-7048	251	6	.	.	PUNCT
ejpam-7048	252	1	european	european	ADJ
ejpam-7048	252	2	journal	journal	PROPN
ejpam-7048	252	3	of	of	ADP
ejpam-7048	252	4	pure	pure	ADJ
ejpam-7048	252	5	and	and	CCONJ
ejpam-7048	252	6	applied	applied	ADJ
ejpam-7048	252	7	mathematics	mathematic	NOUN
ejpam-7048	252	8	,	,	PUNCT
ejpam-7048	252	9	16(3):1634–1646	16(3):1634–1646	NUM
ejpam-7048	252	10	,	,	PUNCT
ejpam-7048	252	11	2023	2023	NUM
ejpam-7048	252	12	.	.	PUNCT
ejpam-7048	253	1	[	[	X
ejpam-7048	253	2	11	11	NUM
ejpam-7048	253	3	]	]	X
ejpam-7048	253	4	c.	c.	PROPN
ejpam-7048	253	5	boonpok	boonpok	PROPN
ejpam-7048	253	6	and	and	CCONJ
ejpam-7048	253	7	j.	j.	PROPN
ejpam-7048	253	8	khampakdee	khampakdee	PROPN
ejpam-7048	253	9	.	.	PUNCT
ejpam-7048	254	1	upper	upper	ADJ
ejpam-7048	254	2	and	and	CCONJ
ejpam-7048	254	3	lower	low	ADJ
ejpam-7048	254	4	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-7048	254	5	.	.	PUNCT
ejpam-7048	254	6	european	european	PROPN
ejpam-7048	254	7	journal	journal	PROPN
ejpam-7048	254	8	of	of	ADP
ejpam-7048	254	9	pure	pure	ADJ
ejpam-7048	254	10	and	and	CCONJ
ejpam-7048	254	11	applied	applied	ADJ
ejpam-7048	254	12	mathematics	mathematic	NOUN
ejpam-7048	254	13	,	,	PUNCT
ejpam-7048	254	14	17(1):201–211	17(1):201–211	NUM
ejpam-7048	254	15	,	,	PUNCT
ejpam-7048	254	16	2024	2024	NUM
ejpam-7048	254	17	.	.	PUNCT
ejpam-7048	255	1	[	[	X
ejpam-7048	255	2	12	12	NUM
ejpam-7048	255	3	]	]	PUNCT
ejpam-7048	255	4	c.	c.	PROPN
ejpam-7048	255	5	boonpok	boonpok	PROPN
ejpam-7048	255	6	.	.	PUNCT
ejpam-7048	256	1	pı	pı	NOUN
ejpam-7048	256	2	-	-	NOUN
ejpam-7048	256	3	continuity	continuity	NOUN
ejpam-7048	256	4	and	and	CCONJ
ejpam-7048	256	5	weak	weak	ADJ
ejpam-7048	256	6	pı	pı	NOUN
ejpam-7048	256	7	-	-	NOUN
ejpam-7048	256	8	continuity	continuity	NOUN
ejpam-7048	256	9	.	.	PUNCT
ejpam-7048	257	1	carpathian	carpathian	ADJ
ejpam-7048	257	2	mathematical	mathematical	ADJ
ejpam-7048	257	3	publications	publication	NOUN
ejpam-7048	257	4	,	,	PUNCT
ejpam-7048	257	5	17(1):171–186	17(1):171–186	PROPN
ejpam-7048	257	6	,	,	PUNCT
ejpam-7048	257	7	2025	2025	NUM
ejpam-7048	257	8	.	.	PUNCT
ejpam-7048	258	1	[	[	X
ejpam-7048	258	2	13	13	NUM
ejpam-7048	258	3	]	]	PUNCT
ejpam-7048	258	4	c.	c.	PROPN
ejpam-7048	258	5	boonpok	boonpok	PROPN
ejpam-7048	258	6	and	and	CCONJ
ejpam-7048	258	7	p.	p.	NOUN
ejpam-7048	258	8	pue	pue	NOUN
ejpam-7048	258	9	-	-	PUNCT
ejpam-7048	258	10	on	on	ADP
ejpam-7048	258	11	.	.	PUNCT
ejpam-7048	259	1	continuity	continuity	NOUN
ejpam-7048	259	2	for	for	ADP
ejpam-7048	259	3	multifunctions	multifunction	NOUN
ejpam-7048	259	4	in	in	ADP
ejpam-7048	259	5	ideal	ideal	ADJ
ejpam-7048	259	6	topological	topological	ADJ
ejpam-7048	259	7	spaces	space	NOUN
ejpam-7048	259	8	.	.	PUNCT
ejpam-7048	260	1	wseas	wseas	VERB
ejpam-7048	260	2	transactions	transaction	NOUN
ejpam-7048	260	3	on	on	ADP
ejpam-7048	260	4	mathematics	mathematic	NOUN
ejpam-7048	260	5	,	,	PUNCT
ejpam-7048	260	6	19:624–631	19:624–631	NUM
ejpam-7048	260	7	,	,	PUNCT
ejpam-7048	260	8	2020	2020	NUM
ejpam-7048	260	9	.	.	PUNCT
ejpam-7048	261	1	[	[	X
ejpam-7048	261	2	14	14	NUM
ejpam-7048	261	3	]	]	X
ejpam-7048	261	4	p.	p.	NOUN
ejpam-7048	261	5	pue	pue	NOUN
ejpam-7048	261	6	-	-	PUNCT
ejpam-7048	261	7	on	on	ADP
ejpam-7048	261	8	,	,	PUNCT
ejpam-7048	261	9	s.	s.	PROPN
ejpam-7048	261	10	sompong	sompong	PROPN
ejpam-7048	261	11	,	,	PUNCT
ejpam-7048	261	12	and	and	CCONJ
ejpam-7048	261	13	c.	c.	PROPN
ejpam-7048	261	14	boonpok	boonpok	PROPN
ejpam-7048	261	15	.	.	PUNCT
ejpam-7048	262	1	upper	upper	ADJ
ejpam-7048	262	2	and	and	CCONJ
ejpam-7048	262	3	lower	low	ADJ
ejpam-7048	262	4	(	(	PUNCT
ejpam-7048	262	5	τ1	τ1	NOUN
ejpam-7048	262	6	,	,	PUNCT
ejpam-7048	262	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	262	8	multifunctions	multifunction	NOUN
ejpam-7048	262	9	.	.	PUNCT
ejpam-7048	263	1	international	international	ADJ
ejpam-7048	263	2	journal	journal	PROPN
ejpam-7048	263	3	of	of	ADP
ejpam-7048	263	4	mathematics	mathematic	NOUN
ejpam-7048	263	5	and	and	CCONJ
ejpam-7048	263	6	computer	computer	NOUN
ejpam-7048	263	7	science	science	NOUN
ejpam-7048	263	8	,	,	PUNCT
ejpam-7048	263	9	19(4):1305	19(4):1305	NUM
ejpam-7048	263	10	–	–	PUNCT
ejpam-7048	263	11	1310	1310	NUM
ejpam-7048	263	12	,	,	PUNCT
ejpam-7048	263	13	2024	2024	NUM
ejpam-7048	263	14	.	.	PUNCT
ejpam-7048	264	1	[	[	X
ejpam-7048	264	2	15	15	NUM
ejpam-7048	264	3	]	]	X
ejpam-7048	264	4	c.	c.	PROPN
ejpam-7048	264	5	klanarong	klanarong	PROPN
ejpam-7048	264	6	,	,	PUNCT
ejpam-7048	264	7	s.	s.	PROPN
ejpam-7048	264	8	sompong	sompong	PROPN
ejpam-7048	264	9	,	,	PUNCT
ejpam-7048	264	10	and	and	CCONJ
ejpam-7048	264	11	c.	c.	PROPN
ejpam-7048	264	12	boonpok	boonpok	PROPN
ejpam-7048	264	13	.	.	PUNCT
ejpam-7048	265	1	(	(	PUNCT
ejpam-7048	265	2	τ1	τ1	NOUN
ejpam-7048	265	3	,	,	PUNCT
ejpam-7048	265	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7048	265	5	and	and	CCONJ
ejpam-7048	265	6	(	(	PUNCT
ejpam-7048	265	7	τ1	τ1	NOUN
ejpam-7048	265	8	,	,	PUNCT
ejpam-7048	265	9	τ2)θ	τ2)θ	ADJ
ejpam-7048	265	10	-	-	PUNCT
ejpam-7048	265	11	closed	close	VERB
ejpam-7048	265	12	sets	set	NOUN
ejpam-7048	265	13	.	.	PUNCT
ejpam-7048	266	1	international	international	ADJ
ejpam-7048	266	2	journal	journal	NOUN
ejpam-7048	266	3	of	of	ADP
ejpam-7048	266	4	mathematics	mathematic	NOUN
ejpam-7048	266	5	and	and	CCONJ
ejpam-7048	266	6	computer	computer	NOUN
ejpam-7048	266	7	science	science	NOUN
ejpam-7048	266	8	,	,	PUNCT
ejpam-7048	266	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-7048	266	10	,	,	PUNCT
ejpam-7048	266	11	2024	2024	NUM
ejpam-7048	266	12	.	.	PUNCT
ejpam-7048	267	1	[	[	X
ejpam-7048	267	2	16	16	NUM
ejpam-7048	267	3	]	]	PUNCT
ejpam-7048	267	4	m.	m.	NOUN
ejpam-7048	267	5	thongmoon	thongmoon	NOUN
ejpam-7048	267	6	,	,	PUNCT
ejpam-7048	267	7	s.	s.	PROPN
ejpam-7048	267	8	sompong	sompong	PROPN
ejpam-7048	267	9	,	,	PUNCT
ejpam-7048	267	10	and	and	CCONJ
ejpam-7048	267	11	c.	c.	PROPN
ejpam-7048	267	12	boonpok	boonpok	PROPN
ejpam-7048	267	13	.	.	PUNCT
ejpam-7048	268	1	(	(	PUNCT
ejpam-7048	268	2	τ1	τ1	NOUN
ejpam-7048	268	3	,	,	PUNCT
ejpam-7048	268	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	268	5	multifunctions	multifunction	NOUN
ejpam-7048	268	6	and	and	CCONJ
ejpam-7048	268	7	τ1τ2	τ1τ2	NOUN
ejpam-7048	268	8	-	-	ADJ
ejpam-7048	268	9	δ	δ	NOUN
ejpam-7048	268	10	-	-	ADJ
ejpam-7048	268	11	open	open	ADJ
ejpam-7048	268	12	sets	set	NOUN
ejpam-7048	268	13	.	.	PUNCT
ejpam-7048	269	1	international	international	ADJ
ejpam-7048	269	2	journal	journal	NOUN
ejpam-7048	269	3	of	of	ADP
ejpam-7048	269	4	mathematics	mathematic	NOUN
ejpam-7048	269	5	and	and	CCONJ
ejpam-7048	269	6	computer	computer	NOUN
ejpam-7048	269	7	science	science	NOUN
ejpam-7048	269	8	,	,	PUNCT
ejpam-7048	269	9	19(4):1369–1375	19(4):1369–1375	NUM
ejpam-7048	269	10	,	,	PUNCT
ejpam-7048	269	11	2024	2024	NUM
ejpam-7048	269	12	.	.	PUNCT
ejpam-7048	270	1	[	[	X
ejpam-7048	270	2	17	17	NUM
ejpam-7048	270	3	]	]	PUNCT
ejpam-7048	270	4	k.	k.	PROPN
ejpam-7048	270	5	laprom	laprom	PROPN
ejpam-7048	270	6	,	,	PUNCT
ejpam-7048	270	7	c.	c.	PROPN
ejpam-7048	270	8	boonpok	boonpok	PROPN
ejpam-7048	270	9	,	,	PUNCT
ejpam-7048	270	10	and	and	CCONJ
ejpam-7048	270	11	c.	c.	PROPN
ejpam-7048	270	12	viriyapong	viriyapong	PROPN
ejpam-7048	270	13	.	.	PUNCT
ejpam-7048	271	1	β(τ1	β(τ1	PROPN
ejpam-7048	271	2	,	,	PUNCT
ejpam-7048	271	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	271	4	multifunctions	multifunction	NOUN
ejpam-7048	271	5	on	on	ADP
ejpam-7048	271	6	bitopological	bitopological	ADJ
ejpam-7048	271	7	spaces	space	NOUN
ejpam-7048	271	8	.	.	PUNCT
ejpam-7048	272	1	journal	journal	NOUN
ejpam-7048	272	2	of	of	ADP
ejpam-7048	272	3	mathematics	mathematic	NOUN
ejpam-7048	272	4	,	,	PUNCT
ejpam-7048	272	5	2020:4020971	2020:4020971	NUM
ejpam-7048	272	6	,	,	PUNCT
ejpam-7048	272	7	2020	2020	NUM
ejpam-7048	272	8	.	.	PUNCT
ejpam-7048	273	1	[	[	X
ejpam-7048	273	2	18	18	NUM
ejpam-7048	273	3	]	]	PUNCT
ejpam-7048	273	4	c.	c.	PROPN
ejpam-7048	273	5	boonpok	boonpok	PROPN
ejpam-7048	273	6	,	,	PUNCT
ejpam-7048	273	7	c.	c.	PROPN
ejpam-7048	273	8	viriyapong	viriyapong	PROPN
ejpam-7048	273	9	,	,	PUNCT
ejpam-7048	273	10	and	and	CCONJ
ejpam-7048	273	11	m.	m.	NOUN
ejpam-7048	273	12	thongmoon	thongmoon	NOUN
ejpam-7048	273	13	.	.	PUNCT
ejpam-7048	274	1	on	on	ADP
ejpam-7048	274	2	upper	upper	ADJ
ejpam-7048	274	3	and	and	CCONJ
ejpam-7048	274	4	lower	low	ADJ
ejpam-7048	274	5	(	(	PUNCT
ejpam-7048	274	6	τ1	τ1	NOUN
ejpam-7048	274	7	,	,	PUNCT
ejpam-7048	274	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7048	274	9	multifunctions	multifunction	NOUN
ejpam-7048	274	10	.	.	PUNCT
ejpam-7048	275	1	journal	journal	PROPN
ejpam-7048	275	2	of	of	ADP
ejpam-7048	275	3	mathematics	mathematics	PROPN
ejpam-7048	275	4	and	and	CCONJ
ejpam-7048	275	5	computer	computer	NOUN
ejpam-7048	275	6	science	science	NOUN
ejpam-7048	275	7	,	,	PUNCT
ejpam-7048	275	8	18:282	18:282	NUM
ejpam-7048	275	9	–	–	PUNCT
ejpam-7048	275	10	293	293	NUM
ejpam-7048	275	11	,	,	PUNCT
ejpam-7048	275	12	2018	2018	NUM
ejpam-7048	275	13	.	.	PUNCT
ejpam-7048	276	1	[	[	X
ejpam-7048	276	2	19	19	NUM
ejpam-7048	276	3	]	]	X
ejpam-7048	276	4	c.	c.	PROPN
ejpam-7048	276	5	viriyapong	viriyapong	PROPN
ejpam-7048	276	6	and	and	CCONJ
ejpam-7048	276	7	c.	c.	PROPN
ejpam-7048	276	8	boonpok	boonpok	PROPN
ejpam-7048	276	9	.	.	PUNCT
ejpam-7048	277	1	(	(	PUNCT
ejpam-7048	277	2	τ1	τ1	NOUN
ejpam-7048	277	3	,	,	PUNCT
ejpam-7048	277	4	τ2)α	τ2)α	NOUN
ejpam-7048	277	5	-	-	PUNCT
ejpam-7048	277	6	continuity	continuity	NOUN
ejpam-7048	277	7	for	for	ADP
ejpam-7048	277	8	multifunctions	multifunction	NOUN
ejpam-7048	277	9	.	.	PUNCT
ejpam-7048	278	1	journal	journal	PROPN
ejpam-7048	278	2	of	of	ADP
ejpam-7048	278	3	mathematics	mathematic	NOUN
ejpam-7048	278	4	,	,	PUNCT
ejpam-7048	278	5	2020:6285763	2020:6285763	NUM
ejpam-7048	278	6	,	,	PUNCT
ejpam-7048	278	7	2020	2020	NUM
ejpam-7048	278	8	.	.	PUNCT
ejpam-7048	279	1	[	[	X
ejpam-7048	279	2	20	20	NUM
ejpam-7048	279	3	]	]	PUNCT
ejpam-7048	279	4	c.	c.	PROPN
ejpam-7048	279	5	boonpok	boonpok	PROPN
ejpam-7048	279	6	.	.	PUNCT
ejpam-7048	280	1	(	(	PUNCT
ejpam-7048	280	2	τ1	τ1	NOUN
ejpam-7048	280	3	,	,	PUNCT
ejpam-7048	280	4	τ2)δ	τ2)δ	ADJ
ejpam-7048	280	5	-	-	PUNCT
ejpam-7048	280	6	semicontinuous	semicontinuous	ADJ
ejpam-7048	280	7	multifunctions	multifunction	NOUN
ejpam-7048	280	8	.	.	PUNCT
ejpam-7048	281	1	heliyon	heliyon	NOUN
ejpam-7048	281	2	,	,	PUNCT
ejpam-7048	281	3	6	6	NUM
ejpam-7048	281	4	:	:	SYM
ejpam-7048	281	5	e05367	e05367	PROPN
ejpam-7048	281	6	,	,	PUNCT
ejpam-7048	281	7	2020	2020	NUM
ejpam-7048	281	8	.	.	PUNCT
ejpam-7048	282	1	[	[	X
ejpam-7048	282	2	21	21	NUM
ejpam-7048	282	3	]	]	X
ejpam-7048	282	4	c.	c.	PROPN
ejpam-7048	282	5	boonpok	boonpok	PROPN
ejpam-7048	282	6	and	and	CCONJ
ejpam-7048	282	7	p.	p.	NOUN
ejpam-7048	282	8	pue	pue	NOUN
ejpam-7048	282	9	-	-	PUNCT
ejpam-7048	282	10	on	on	ADP
ejpam-7048	282	11	.	.	PUNCT
ejpam-7048	283	1	characterizations	characterization	NOUN
ejpam-7048	283	2	of	of	ADP
ejpam-7048	283	3	almost	almost	ADV
ejpam-7048	283	4	(	(	PUNCT
ejpam-7048	283	5	τ1	τ1	NOUN
ejpam-7048	283	6	,	,	PUNCT
ejpam-7048	283	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7048	283	8	multifunctions	multifunction	NOUN
ejpam-7048	283	9	.	.	PUNCT
ejpam-7048	284	1	international	international	ADJ
ejpam-7048	284	2	journal	journal	NOUN
ejpam-7048	284	3	of	of	ADP
ejpam-7048	284	4	analysis	analysis	NOUN
ejpam-7048	284	5	and	and	CCONJ
ejpam-7048	284	6	applications	application	NOUN
ejpam-7048	284	7	,	,	PUNCT
ejpam-7048	284	8	22:33	22:33	NUM
ejpam-7048	284	9	,	,	PUNCT
ejpam-7048	284	10	2024	2024	NUM
ejpam-7048	284	11	.	.	PUNCT
ejpam-7048	285	1	[	[	X
ejpam-7048	285	2	22	22	NUM
ejpam-7048	285	3	]	]	PUNCT
ejpam-7048	285	4	k.	k.	PROPN
ejpam-7048	285	5	kuratowski	kuratowski	PROPN
ejpam-7048	285	6	.	.	PUNCT
ejpam-7048	286	1	topology	topology	PROPN
ejpam-7048	286	2	,	,	PUNCT
ejpam-7048	286	3	vol	vol	NOUN
ejpam-7048	286	4	.	.	PUNCT
ejpam-7048	286	5	i.	i.	PROPN
ejpam-7048	286	6	academic	academic	PROPN
ejpam-7048	286	7	press	press	PROPN
ejpam-7048	286	8	,	,	PUNCT
ejpam-7048	286	9	new	new	PROPN
ejpam-7048	286	10	york	york	PROPN
ejpam-7048	286	11	,	,	PUNCT
ejpam-7048	286	12	1966	1966	NUM
ejpam-7048	286	13	.	.	PUNCT
ejpam-7048	287	1	[	[	X
ejpam-7048	287	2	23	23	NUM
ejpam-7048	287	3	]	]	X
ejpam-7048	287	4	d.	d.	PROPN
ejpam-7048	287	5	janković	janković	VERB
ejpam-7048	287	6	and	and	CCONJ
ejpam-7048	287	7	t.	t.	PROPN
ejpam-7048	287	8	r.	r.	PROPN
ejpam-7048	287	9	hamlett	hamlett	PROPN
ejpam-7048	287	10	.	.	PUNCT
ejpam-7048	288	1	new	new	ADJ
ejpam-7048	288	2	topologies	topology	NOUN
ejpam-7048	288	3	from	from	ADP
ejpam-7048	288	4	old	old	ADJ
ejpam-7048	288	5	via	via	ADP
ejpam-7048	288	6	ideals	ideal	NOUN
ejpam-7048	288	7	.	.	PUNCT
ejpam-7048	289	1	the	the	DET
ejpam-7048	289	2	american	american	PROPN
ejpam-7048	289	3	mathematical	mathematical	PROPN
ejpam-7048	289	4	monthly	monthly	ADV
ejpam-7048	289	5	,	,	PUNCT
ejpam-7048	289	6	97:295–310	97:295–310	PROPN
ejpam-7048	289	7	,	,	PUNCT
ejpam-7048	289	8	1990	1990	NUM
ejpam-7048	289	9	.	.	PUNCT
ejpam-7048	290	1	[	[	X
ejpam-7048	290	2	24	24	NUM
ejpam-7048	290	3	]	]	PUNCT
ejpam-7048	290	4	t.	t.	PROPN
ejpam-7048	290	5	noiri	noiri	PROPN
ejpam-7048	290	6	and	and	CCONJ
ejpam-7048	290	7	v.	v.	ADP
ejpam-7048	290	8	popa	popa	NOUN
ejpam-7048	290	9	.	.	PUNCT
ejpam-7048	291	1	on	on	ADP
ejpam-7048	291	2	(	(	PUNCT
ejpam-7048	291	3	mi	mi	ADJ
ejpam-7048	291	4	,	,	PUNCT
ejpam-7048	291	5	nj)-continuous	nj)-continuous	ADJ
ejpam-7048	291	6	multifunctions	multifunction	NOUN
ejpam-7048	291	7	.	.	PUNCT
ejpam-7048	292	1	romanian	romanian	ADJ
ejpam-7048	292	2	journal	journal	PROPN
ejpam-7048	292	3	of	of	ADP
ejpam-7048	292	4	mathematics	mathematics	PROPN
ejpam-7048	292	5	and	and	CCONJ
ejpam-7048	292	6	computer	computer	NOUN
ejpam-7048	292	7	science	science	NOUN
ejpam-7048	292	8	,	,	PUNCT
ejpam-7048	292	9	15(1):1–8	15(1):1–8	NUM
ejpam-7048	292	10	,	,	PUNCT
ejpam-7048	292	11	2025	2025	NUM
ejpam-7048	292	12	.	.	PUNCT
ejpam-7048	293	1	[	[	X
ejpam-7048	293	2	25	25	NUM
ejpam-7048	293	3	]	]	PUNCT
ejpam-7048	293	4	c.	c.	PROPN
ejpam-7048	293	5	boonpok	boonpok	PROPN
ejpam-7048	293	6	.	.	PUNCT
ejpam-7048	294	1	weak	weak	ADJ
ejpam-7048	294	2	quasi	quasi	ADJ
ejpam-7048	294	3	continuity	continuity	NOUN
ejpam-7048	294	4	for	for	ADP
ejpam-7048	294	5	multifunctions	multifunction	NOUN
ejpam-7048	294	6	in	in	ADP
ejpam-7048	294	7	ideal	ideal	ADJ
ejpam-7048	294	8	topological	topological	ADJ
ejpam-7048	294	9	spaces	space	NOUN
ejpam-7048	294	10	.	.	PUNCT
ejpam-7048	295	1	advances	advance	NOUN
ejpam-7048	295	2	in	in	ADP
ejpam-7048	295	3	mathematics	mathematic	NOUN
ejpam-7048	295	4	:	:	PUNCT
ejpam-7048	295	5	scientific	scientific	ADJ
ejpam-7048	295	6	journal	journal	NOUN
ejpam-7048	295	7	,	,	PUNCT
ejpam-7048	295	8	9(3):339–355	9(3):339–355	NUM
ejpam-7048	295	9	,	,	PUNCT
ejpam-7048	295	10	2020	2020	NUM
ejpam-7048	295	11	.	.	PUNCT