id	sid	tid	token	lemma	pos
ejpam-7049	1	1	european	european	PROPN
ejpam-7049	1	2	journal	journal	PROPN
ejpam-7049	1	3	of	of	ADP
ejpam-7049	1	4	pure	pure	ADJ
ejpam-7049	1	5	and	and	CCONJ
ejpam-7049	1	6	applied	applied	ADJ
ejpam-7049	1	7	mathematics	mathematic	NOUN
ejpam-7049	1	8	2025	2025	NUM
ejpam-7049	1	9	,	,	PUNCT
ejpam-7049	1	10	vol	vol	NOUN
ejpam-7049	1	11	.	.	PROPN
ejpam-7049	1	12	18	18	NUM
ejpam-7049	1	13	,	,	PUNCT
ejpam-7049	1	14	issue	issue	NOUN
ejpam-7049	1	15	4	4	NUM
ejpam-7049	1	16	,	,	PUNCT
ejpam-7049	1	17	article	article	NOUN
ejpam-7049	1	18	number	number	NOUN
ejpam-7049	1	19	7049	7049	NUM
ejpam-7049	1	20	issn	issn	PROPN
ejpam-7049	1	21	1307	1307	NUM
ejpam-7049	1	22	-	-	SYM
ejpam-7049	1	23	5543	5543	NUM
ejpam-7049	1	24	–	–	PUNCT
ejpam-7049	1	25	ejpam.com	ejpam.com	X
ejpam-7049	1	26	published	publish	VERB
ejpam-7049	1	27	by	by	ADP
ejpam-7049	1	28	new	new	PROPN
ejpam-7049	1	29	york	york	PROPN
ejpam-7049	1	30	business	business	PROPN
ejpam-7049	1	31	global	global	PROPN
ejpam-7049	1	32	almost	almost	ADV
ejpam-7049	1	33	τ	τ	X
ejpam-7049	1	34	⋆β(σ1	⋆β(σ1	PUNCT
ejpam-7049	1	35	,	,	PUNCT
ejpam-7049	1	36	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7049	1	37	for	for	ADP
ejpam-7049	1	38	multifunctions	multifunction	NOUN
ejpam-7049	1	39	napassanan	napassanan	PROPN
ejpam-7049	1	40	srisarakham1	srisarakham1	PROPN
ejpam-7049	1	41	,	,	PUNCT
ejpam-7049	1	42	areeyuth	areeyuth	NOUN
ejpam-7049	1	43	sama	sama	NOUN
ejpam-7049	1	44	-	-	PUNCT
ejpam-7049	1	45	ae2	ae2	PROPN
ejpam-7049	1	46	,	,	PUNCT
ejpam-7049	1	47	chawalit	chawalit	VERB
ejpam-7049	1	48	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7049	1	49	1	1	NUM
ejpam-7049	1	50	mathematics	mathematic	NOUN
ejpam-7049	1	51	and	and	CCONJ
ejpam-7049	1	52	applied	apply	VERB
ejpam-7049	1	53	mathematics	mathematics	PROPN
ejpam-7049	1	54	research	research	NOUN
ejpam-7049	1	55	unit	unit	NOUN
ejpam-7049	1	56	,	,	PUNCT
ejpam-7049	1	57	department	department	NOUN
ejpam-7049	1	58	of	of	ADP
ejpam-7049	1	59	mathematics	mathematic	NOUN
ejpam-7049	1	60	,	,	PUNCT
ejpam-7049	1	61	faculty	faculty	NOUN
ejpam-7049	1	62	of	of	ADP
ejpam-7049	1	63	science	science	NOUN
ejpam-7049	1	64	,	,	PUNCT
ejpam-7049	1	65	mahasarakham	mahasarakham	PROPN
ejpam-7049	1	66	university	university	PROPN
ejpam-7049	1	67	,	,	PUNCT
ejpam-7049	1	68	maha	maha	PROPN
ejpam-7049	1	69	sarakham	sarakham	PROPN
ejpam-7049	1	70	,	,	PUNCT
ejpam-7049	1	71	44150	44150	NUM
ejpam-7049	1	72	,	,	PUNCT
ejpam-7049	1	73	thailand	thailand	PROPN
ejpam-7049	1	74	2	2	NUM
ejpam-7049	1	75	department	department	NOUN
ejpam-7049	1	76	of	of	ADP
ejpam-7049	1	77	mathematics	mathematic	NOUN
ejpam-7049	1	78	and	and	CCONJ
ejpam-7049	1	79	computer	computer	NOUN
ejpam-7049	1	80	science	science	NOUN
ejpam-7049	1	81	,	,	PUNCT
ejpam-7049	1	82	faculty	faculty	NOUN
ejpam-7049	1	83	of	of	ADP
ejpam-7049	1	84	science	science	NOUN
ejpam-7049	1	85	and	and	CCONJ
ejpam-7049	1	86	technology	technology	NOUN
ejpam-7049	1	87	,	,	PUNCT
ejpam-7049	1	88	prince	prince	NOUN
ejpam-7049	1	89	of	of	ADP
ejpam-7049	1	90	songkla	songkla	PROPN
ejpam-7049	1	91	university	university	PROPN
ejpam-7049	1	92	,	,	PUNCT
ejpam-7049	1	93	pattani	pattani	NOUN
ejpam-7049	1	94	campus	campus	NOUN
ejpam-7049	1	95	,	,	PUNCT
ejpam-7049	1	96	pattani	pattani	NOUN
ejpam-7049	1	97	,	,	PUNCT
ejpam-7049	1	98	94000	94000	NUM
ejpam-7049	1	99	,	,	PUNCT
ejpam-7049	1	100	thailand	thailand	PROPN
ejpam-7049	1	101	abstract	abstract	PROPN
ejpam-7049	1	102	.	.	PUNCT
ejpam-7049	2	1	this	this	DET
ejpam-7049	2	2	paper	paper	NOUN
ejpam-7049	2	3	introduces	introduce	VERB
ejpam-7049	2	4	new	new	ADJ
ejpam-7049	2	5	classes	class	NOUN
ejpam-7049	2	6	of	of	ADP
ejpam-7049	2	7	continuous	continuous	ADJ
ejpam-7049	2	8	multifunctions	multifunction	NOUN
ejpam-7049	2	9	defined	define	VERB
ejpam-7049	2	10	between	between	ADP
ejpam-7049	2	11	an	an	DET
ejpam-7049	2	12	ideal	ideal	ADJ
ejpam-7049	2	13	topological	topological	ADJ
ejpam-7049	2	14	space	space	NOUN
ejpam-7049	2	15	and	and	CCONJ
ejpam-7049	2	16	a	a	DET
ejpam-7049	2	17	bitopological	bitopological	ADJ
ejpam-7049	2	18	space	space	NOUN
ejpam-7049	2	19	,	,	PUNCT
ejpam-7049	2	20	called	call	VERB
ejpam-7049	2	21	upper	upper	ADV
ejpam-7049	2	22	almost	almost	ADV
ejpam-7049	2	23	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	2	24	,	,	PUNCT
ejpam-7049	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	2	26	multifunctions	multifunction	NOUN
ejpam-7049	2	27	and	and	CCONJ
ejpam-7049	2	28	lower	low	ADJ
ejpam-7049	2	29	almost	almost	ADV
ejpam-7049	2	30	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	2	31	,	,	PUNCT
ejpam-7049	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	2	33	multifunctions	multifunction	NOUN
ejpam-7049	2	34	.	.	PUNCT
ejpam-7049	3	1	moreover	moreover	ADV
ejpam-7049	3	2	,	,	PUNCT
ejpam-7049	3	3	several	several	ADJ
ejpam-7049	3	4	characterizations	characterization	NOUN
ejpam-7049	3	5	and	and	CCONJ
ejpam-7049	3	6	some	some	DET
ejpam-7049	3	7	properties	property	NOUN
ejpam-7049	3	8	concerning	concern	VERB
ejpam-7049	3	9	upper	upper	ADJ
ejpam-7049	3	10	almost	almost	ADV
ejpam-7049	3	11	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	3	12	,	,	PUNCT
ejpam-7049	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	3	14	multifunctions	multifunction	NOUN
ejpam-7049	3	15	and	and	CCONJ
ejpam-7049	3	16	lower	low	ADJ
ejpam-7049	3	17	almost	almost	ADV
ejpam-7049	3	18	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	3	19	,	,	PUNCT
ejpam-7049	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	3	21	multifunctions	multifunction	NOUN
ejpam-7049	3	22	are	be	AUX
ejpam-7049	3	23	established	establish	VERB
ejpam-7049	3	24	.	.	PUNCT
ejpam-7049	4	1	2020	2020	NUM
ejpam-7049	4	2	mathematics	mathematics	PROPN
ejpam-7049	4	3	subject	subject	NOUN
ejpam-7049	4	4	classifications	classification	NOUN
ejpam-7049	4	5	:	:	PUNCT
ejpam-7049	4	6	54c08	54c08	NUM
ejpam-7049	4	7	,	,	PUNCT
ejpam-7049	4	8	54c60	54c60	NUM
ejpam-7049	4	9	key	key	ADJ
ejpam-7049	4	10	words	word	NOUN
ejpam-7049	4	11	and	and	CCONJ
ejpam-7049	4	12	phrases	phrase	NOUN
ejpam-7049	4	13	:	:	PUNCT
ejpam-7049	4	14	upper	upper	ADJ
ejpam-7049	4	15	almost	almost	ADV
ejpam-7049	4	16	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	4	17	,	,	PUNCT
ejpam-7049	4	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	4	19	multifunction	multifunction	NOUN
ejpam-7049	4	20	,	,	PUNCT
ejpam-7049	4	21	lower	low	ADJ
ejpam-7049	4	22	almost	almost	ADV
ejpam-7049	4	23	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	4	24	,	,	PUNCT
ejpam-7049	4	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	4	26	multifunction	multifunction	NOUN
ejpam-7049	4	27	1	1	NUM
ejpam-7049	4	28	.	.	PUNCT
ejpam-7049	4	29	introduction	introduction	NOUN
ejpam-7049	4	30	in	in	ADP
ejpam-7049	4	31	1997	1997	NUM
ejpam-7049	4	32	,	,	PUNCT
ejpam-7049	4	33	nasef	nasef	PROPN
ejpam-7049	4	34	and	and	CCONJ
ejpam-7049	4	35	noiri	noiri	ADV
ejpam-7049	5	1	[	[	X
ejpam-7049	5	2	1	1	X
ejpam-7049	5	3	]	]	PUNCT
ejpam-7049	5	4	introduced	introduce	VERB
ejpam-7049	5	5	and	and	CCONJ
ejpam-7049	5	6	investigated	investigate	VERB
ejpam-7049	5	7	two	two	NUM
ejpam-7049	5	8	classes	class	NOUN
ejpam-7049	5	9	of	of	ADP
ejpam-7049	5	10	functions	function	NOUN
ejpam-7049	5	11	defined	define	VERB
ejpam-7049	5	12	between	between	ADP
ejpam-7049	5	13	topological	topological	ADJ
ejpam-7049	5	14	spaces	space	NOUN
ejpam-7049	5	15	,	,	PUNCT
ejpam-7049	5	16	namely	namely	ADV
ejpam-7049	5	17	almost	almost	ADV
ejpam-7049	5	18	precontinuous	precontinuous	ADJ
ejpam-7049	5	19	functions	function	NOUN
ejpam-7049	5	20	and	and	CCONJ
ejpam-7049	5	21	almost	almost	ADV
ejpam-7049	5	22	βcontinuous	βcontinuous	ADJ
ejpam-7049	5	23	functions	function	NOUN
ejpam-7049	5	24	by	by	ADP
ejpam-7049	5	25	utilizing	utilize	VERB
ejpam-7049	5	26	the	the	DET
ejpam-7049	5	27	notions	notion	NOUN
ejpam-7049	5	28	of	of	ADP
ejpam-7049	5	29	preopen	preopen	ADJ
ejpam-7049	5	30	sets	set	NOUN
ejpam-7049	5	31	and	and	CCONJ
ejpam-7049	5	32	β	β	NOUN
ejpam-7049	5	33	-	-	ADJ
ejpam-7049	5	34	open	open	ADJ
ejpam-7049	5	35	sets	set	NOUN
ejpam-7049	5	36	due	due	ADP
ejpam-7049	5	37	to	to	ADP
ejpam-7049	5	38	mashhour	mashhour	PROPN
ejpam-7049	5	39	et	et	PROPN
ejpam-7049	5	40	al	al	PROPN
ejpam-7049	5	41	.	.	PUNCT
ejpam-7049	6	1	[	[	X
ejpam-7049	6	2	2	2	X
ejpam-7049	6	3	]	]	PUNCT
ejpam-7049	6	4	and	and	CCONJ
ejpam-7049	6	5	abd	abd	PROPN
ejpam-7049	6	6	el	el	PROPN
ejpam-7049	6	7	-	-	PROPN
ejpam-7049	6	8	monsef	monsef	PROPN
ejpam-7049	6	9	et	et	PROPN
ejpam-7049	6	10	al	al	PROPN
ejpam-7049	6	11	.	.	PUNCT
ejpam-7049	7	1	[	[	X
ejpam-7049	7	2	3	3	NUM
ejpam-7049	7	3	]	]	PUNCT
ejpam-7049	7	4	,	,	PUNCT
ejpam-7049	7	5	respectively	respectively	ADV
ejpam-7049	7	6	.	.	PUNCT
ejpam-7049	8	1	in	in	ADP
ejpam-7049	8	2	1998	1998	NUM
ejpam-7049	8	3	,	,	PUNCT
ejpam-7049	8	4	noiri	noiri	ADV
ejpam-7049	8	5	and	and	CCONJ
ejpam-7049	8	6	popa	popa	NOUN
ejpam-7049	8	7	[	[	X
ejpam-7049	8	8	4	4	NUM
ejpam-7049	8	9	]	]	PUNCT
ejpam-7049	8	10	investigated	investigate	VERB
ejpam-7049	8	11	several	several	ADJ
ejpam-7049	8	12	characterizations	characterization	NOUN
ejpam-7049	8	13	and	and	CCONJ
ejpam-7049	8	14	some	some	DET
ejpam-7049	8	15	properties	property	NOUN
ejpam-7049	8	16	of	of	ADP
ejpam-7049	8	17	almost	almost	ADV
ejpam-7049	8	18	β	β	ADJ
ejpam-7049	8	19	-	-	ADJ
ejpam-7049	8	20	continuous	continuous	ADJ
ejpam-7049	8	21	functions	function	NOUN
ejpam-7049	8	22	.	.	PUNCT
ejpam-7049	9	1	noiri	noiri	ADV
ejpam-7049	10	1	[	[	X
ejpam-7049	10	2	5	5	X
ejpam-7049	10	3	]	]	PUNCT
ejpam-7049	10	4	introduced	introduce	VERB
ejpam-7049	10	5	the	the	DET
ejpam-7049	10	6	concept	concept	NOUN
ejpam-7049	10	7	of	of	ADP
ejpam-7049	10	8	almost	almost	ADV
ejpam-7049	10	9	α	α	NUM
ejpam-7049	10	10	-	-	ADJ
ejpam-7049	10	11	continuous	continuous	ADJ
ejpam-7049	10	12	functions	function	NOUN
ejpam-7049	10	13	and	and	CCONJ
ejpam-7049	10	14	proved	prove	VERB
ejpam-7049	10	15	that	that	SCONJ
ejpam-7049	10	16	the	the	DET
ejpam-7049	10	17	notions	notion	NOUN
ejpam-7049	10	18	of	of	ADP
ejpam-7049	10	19	almost	almost	ADV
ejpam-7049	10	20	feeble	feeble	ADJ
ejpam-7049	10	21	continuity	continuity	NOUN
ejpam-7049	10	22	[	[	X
ejpam-7049	10	23	6	6	NUM
ejpam-7049	10	24	]	]	PUNCT
ejpam-7049	10	25	and	and	CCONJ
ejpam-7049	10	26	almost	almost	ADV
ejpam-7049	10	27	α	α	NOUN
ejpam-7049	10	28	-	-	PUNCT
ejpam-7049	10	29	continuity	continuity	NOUN
ejpam-7049	10	30	are	be	AUX
ejpam-7049	10	31	equivalent	equivalent	ADJ
ejpam-7049	10	32	.	.	PUNCT
ejpam-7049	11	1	the	the	DET
ejpam-7049	11	2	class	class	NOUN
ejpam-7049	11	3	of	of	ADP
ejpam-7049	11	4	almost	almost	ADV
ejpam-7049	11	5	precontinuity	precontinuity	NOUN
ejpam-7049	11	6	is	be	AUX
ejpam-7049	11	7	a	a	DET
ejpam-7049	11	8	generalization	generalization	NOUN
ejpam-7049	11	9	of	of	ADP
ejpam-7049	11	10	almost	almost	ADV
ejpam-7049	11	11	α	α	NOUN
ejpam-7049	11	12	-	-	PUNCT
ejpam-7049	11	13	continuity	continuity	NOUN
ejpam-7049	11	14	and	and	CCONJ
ejpam-7049	11	15	almost	almost	ADV
ejpam-7049	11	16	feeble	feeble	ADJ
ejpam-7049	11	17	continuity	continuity	NOUN
ejpam-7049	11	18	.	.	PUNCT
ejpam-7049	12	1	the	the	DET
ejpam-7049	12	2	class	class	NOUN
ejpam-7049	12	3	of	of	ADP
ejpam-7049	12	4	almost	almost	ADV
ejpam-7049	12	5	β	β	NOUN
ejpam-7049	12	6	-	-	NOUN
ejpam-7049	12	7	continuity	continuity	NOUN
ejpam-7049	12	8	is	be	AUX
ejpam-7049	12	9	a	a	DET
ejpam-7049	12	10	generalization	generalization	NOUN
ejpam-7049	12	11	of	of	ADP
ejpam-7049	12	12	almost	almost	ADV
ejpam-7049	12	13	quasi	quasi	NOUN
ejpam-7049	12	14	-	-	NOUN
ejpam-7049	12	15	continuity	continuity	NOUN
ejpam-7049	12	16	[	[	X
ejpam-7049	12	17	7	7	NUM
ejpam-7049	12	18	]	]	PUNCT
ejpam-7049	12	19	.	.	PUNCT
ejpam-7049	13	1	in	in	ADP
ejpam-7049	13	2	1999	1999	NUM
ejpam-7049	13	3	,	,	PUNCT
ejpam-7049	13	4	noiri	noiri	PRON
ejpam-7049	13	5	and	and	CCONJ
ejpam-7049	13	6	popa	popa	NOUN
ejpam-7049	13	7	[	[	X
ejpam-7049	13	8	8	8	NUM
ejpam-7049	13	9	]	]	PUNCT
ejpam-7049	13	10	extended	extend	VERB
ejpam-7049	13	11	the	the	DET
ejpam-7049	13	12	concept	concept	NOUN
ejpam-7049	13	13	of	of	ADP
ejpam-7049	13	14	almost	almost	ADV
ejpam-7049	13	15	β	β	ADJ
ejpam-7049	13	16	-	-	ADJ
ejpam-7049	13	17	continuous	continuous	ADJ
ejpam-7049	13	18	functions	function	NOUN
ejpam-7049	13	19	to	to	ADP
ejpam-7049	13	20	multifunctions	multifunction	NOUN
ejpam-7049	13	21	and	and	CCONJ
ejpam-7049	13	22	introduced	introduce	VERB
ejpam-7049	13	23	new	new	ADJ
ejpam-7049	13	24	classes	class	NOUN
ejpam-7049	13	25	of	of	ADP
ejpam-7049	13	26	multifunctions	multifunction	NOUN
ejpam-7049	13	27	defined	define	VERB
ejpam-7049	13	28	between	between	ADP
ejpam-7049	13	29	topological	topological	ADJ
ejpam-7049	13	30	spaces	space	NOUN
ejpam-7049	13	31	,	,	PUNCT
ejpam-7049	13	32	namely	namely	ADV
ejpam-7049	13	33	upper	upper	ADJ
ejpam-7049	13	34	almost	almost	ADV
ejpam-7049	13	35	β	β	ADJ
ejpam-7049	13	36	-	-	ADJ
ejpam-7049	13	37	continuous	continuous	ADJ
ejpam-7049	13	38	multifunctions	multifunction	NOUN
ejpam-7049	13	39	and	and	CCONJ
ejpam-7049	13	40	lower	low	ADJ
ejpam-7049	13	41	almost	almost	ADV
ejpam-7049	13	42	β	β	ADJ
ejpam-7049	13	43	-	-	ADJ
ejpam-7049	13	44	continuous	continuous	ADJ
ejpam-7049	13	45	multifunctions	multifunction	NOUN
ejpam-7049	13	46	.	.	PUNCT
ejpam-7049	14	1	furthermore	furthermore	ADV
ejpam-7049	14	2	,	,	PUNCT
ejpam-7049	14	3	noiri	noiri	PROPN
ejpam-7049	14	4	and	and	CCONJ
ejpam-7049	14	5	popa	popa	NOUN
ejpam-7049	14	6	[	[	X
ejpam-7049	14	7	8	8	NUM
ejpam-7049	14	8	]	]	PUNCT
ejpam-7049	14	9	investigated	investigate	VERB
ejpam-7049	14	10	several	several	ADJ
ejpam-7049	14	11	characterizations	characterization	NOUN
ejpam-7049	14	12	and	and	CCONJ
ejpam-7049	14	13	some	some	DET
ejpam-7049	14	14	properties	property	NOUN
ejpam-7049	14	15	concerning	concern	VERB
ejpam-7049	14	16	upper	upper	ADJ
ejpam-7049	14	17	almost	almost	ADV
ejpam-7049	14	18	β	β	ADJ
ejpam-7049	14	19	-	-	ADJ
ejpam-7049	14	20	continuous	continuous	ADJ
ejpam-7049	14	21	multifunctions	multifunction	NOUN
ejpam-7049	14	22	and	and	CCONJ
ejpam-7049	14	23	lower	low	ADJ
ejpam-7049	14	24	almost	almost	ADV
ejpam-7049	14	25	∗corresponding	∗corresponde	VERB
ejpam-7049	14	26	author	author	NOUN
ejpam-7049	14	27	.	.	PUNCT
ejpam-7049	15	1	doi	doi	NOUN
ejpam-7049	15	2	:	:	PUNCT
ejpam-7049	15	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7049	https://doi.org/10.29020/nybg.ejpam.v18i4.7049	NOUN
ejpam-7049	15	4	email	email	NOUN
ejpam-7049	15	5	addresses	address	VERB
ejpam-7049	15	6	:	:	PUNCT
ejpam-7049	15	7	napassanan.sri@msu.ac.th	napassanan.sri@msu.ac.th	PRON
ejpam-7049	15	8	(	(	PUNCT
ejpam-7049	15	9	n.	n.	NOUN
ejpam-7049	15	10	srisarakham	srisarakham	PROPN
ejpam-7049	15	11	)	)	PUNCT
ejpam-7049	15	12	,	,	PUNCT
ejpam-7049	15	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7049	15	14	(	(	PUNCT
ejpam-7049	15	15	a.	a.	PROPN
ejpam-7049	15	16	sama	sama	PROPN
ejpam-7049	15	17	-	-	PUNCT
ejpam-7049	15	18	ae	ae	PROPN
ejpam-7049	15	19	)	)	PUNCT
ejpam-7049	15	20	,	,	PUNCT
ejpam-7049	15	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7049	15	22	(	(	PUNCT
ejpam-7049	15	23	c.	c.	PROPN
ejpam-7049	15	24	boonpok	boonpok	PROPN
ejpam-7049	15	25	)	)	PUNCT
ejpam-7049	15	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7049	16	1	1	1	NUM
ejpam-7049	16	2	copyright	copyright	NOUN
ejpam-7049	16	3	:	:	PUNCT
ejpam-7049	16	4	©	©	PROPN
ejpam-7049	16	5	2025	2025	NUM
ejpam-7049	16	6	the	the	DET
ejpam-7049	16	7	author(s	author(s	NOUN
ejpam-7049	16	8	)	)	PUNCT
ejpam-7049	16	9	.	.	PUNCT
ejpam-7049	17	1	(	(	PUNCT
ejpam-7049	17	2	cc	cc	NOUN
ejpam-7049	17	3	by	by	ADP
ejpam-7049	17	4	-	-	PUNCT
ejpam-7049	17	5	nc	nc	PROPN
ejpam-7049	17	6	4.0	4.0	NUM
ejpam-7049	17	7	)	)	PUNCT
ejpam-7049	17	8	n.	n.	NOUN
ejpam-7049	17	9	srisarakham	srisarakham	PROPN
ejpam-7049	17	10	,	,	PUNCT
ejpam-7049	17	11	a.	a.	PROPN
ejpam-7049	17	12	sama	sama	PROPN
ejpam-7049	17	13	-	-	PUNCT
ejpam-7049	17	14	ae	ae	PROPN
ejpam-7049	17	15	,	,	PUNCT
ejpam-7049	17	16	c.	c.	PROPN
ejpam-7049	17	17	boonpok	boonpok	PROPN
ejpam-7049	17	18	/	/	SYM
ejpam-7049	17	19	eur	eur	PROPN
ejpam-7049	17	20	.	.	PUNCT
ejpam-7049	18	1	j.	j.	PROPN
ejpam-7049	18	2	pure	pure	PROPN
ejpam-7049	18	3	appl	appl	PROPN
ejpam-7049	18	4	.	.	PROPN
ejpam-7049	18	5	math	math	PROPN
ejpam-7049	18	6	,	,	PUNCT
ejpam-7049	18	7	18	18	NUM
ejpam-7049	18	8	(	(	PUNCT
ejpam-7049	18	9	4	4	NUM
ejpam-7049	18	10	)	)	PUNCT
ejpam-7049	18	11	(	(	PUNCT
ejpam-7049	18	12	2025	2025	NUM
ejpam-7049	18	13	)	)	PUNCT
ejpam-7049	18	14	,	,	PUNCT
ejpam-7049	18	15	7049	7049	NUM
ejpam-7049	18	16	2	2	NUM
ejpam-7049	18	17	of	of	ADP
ejpam-7049	18	18	11	11	NUM
ejpam-7049	18	19	β	β	NOUN
ejpam-7049	18	20	-	-	ADJ
ejpam-7049	18	21	continuous	continuous	ADJ
ejpam-7049	18	22	multifunctions	multifunction	NOUN
ejpam-7049	18	23	.	.	PUNCT
ejpam-7049	19	1	on	on	ADP
ejpam-7049	19	2	the	the	DET
ejpam-7049	19	3	other	other	ADJ
ejpam-7049	19	4	hand	hand	NOUN
ejpam-7049	19	5	,	,	PUNCT
ejpam-7049	19	6	the	the	DET
ejpam-7049	19	7	present	present	ADJ
ejpam-7049	19	8	author	author	NOUN
ejpam-7049	19	9	introduced	introduce	VERB
ejpam-7049	19	10	and	and	CCONJ
ejpam-7049	19	11	investigated	investigate	VERB
ejpam-7049	19	12	classes	class	NOUN
ejpam-7049	19	13	of	of	ADP
ejpam-7049	19	14	continuous	continuous	ADJ
ejpam-7049	19	15	multifunctions	multifunction	NOUN
ejpam-7049	19	16	defined	define	VERB
ejpam-7049	19	17	from	from	ADP
ejpam-7049	19	18	an	an	DET
ejpam-7049	19	19	ideal	ideal	ADJ
ejpam-7049	19	20	topological	topological	ADJ
ejpam-7049	19	21	space	space	NOUN
ejpam-7049	19	22	into	into	ADP
ejpam-7049	19	23	an	an	DET
ejpam-7049	19	24	ideal	ideal	ADJ
ejpam-7049	19	25	topological	topological	ADJ
ejpam-7049	19	26	space	space	NOUN
ejpam-7049	19	27	,	,	PUNCT
ejpam-7049	19	28	namely	namely	ADV
ejpam-7049	19	29	upper	upper	ADJ
ejpam-7049	19	30	almost	almost	ADV
ejpam-7049	19	31	⋆-continuous	⋆-continuous	ADJ
ejpam-7049	19	32	multifunctions	multifunction	NOUN
ejpam-7049	20	1	[	[	X
ejpam-7049	20	2	9	9	NUM
ejpam-7049	20	3	]	]	PUNCT
ejpam-7049	20	4	,	,	PUNCT
ejpam-7049	20	5	lower	low	ADJ
ejpam-7049	20	6	almost	almost	ADV
ejpam-7049	20	7	⋆-continuous	⋆-continuous	ADJ
ejpam-7049	20	8	multifunctions	multifunction	NOUN
ejpam-7049	21	1	[	[	X
ejpam-7049	21	2	9	9	NUM
ejpam-7049	21	3	]	]	PUNCT
ejpam-7049	21	4	,	,	PUNCT
ejpam-7049	21	5	upper	upper	ADJ
ejpam-7049	21	6	almost	almost	ADV
ejpam-7049	21	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7049	21	8	multifunctions	multifunction	NOUN
ejpam-7049	22	1	[	[	X
ejpam-7049	22	2	10	10	NUM
ejpam-7049	22	3	]	]	PUNCT
ejpam-7049	22	4	,	,	PUNCT
ejpam-7049	22	5	lower	low	ADJ
ejpam-7049	22	6	almost	almost	ADV
ejpam-7049	22	7	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-7049	22	8	multifunctions	multifunction	NOUN
ejpam-7049	23	1	[	[	X
ejpam-7049	23	2	10	10	NUM
ejpam-7049	23	3	]	]	PUNCT
ejpam-7049	23	4	,	,	PUNCT
ejpam-7049	23	5	upper	upper	ADJ
ejpam-7049	23	6	almost	almost	ADV
ejpam-7049	23	7	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-7049	23	8	multifunctions	multifunction	NOUN
ejpam-7049	24	1	[	[	X
ejpam-7049	24	2	11	11	NUM
ejpam-7049	24	3	]	]	PUNCT
ejpam-7049	24	4	,	,	PUNCT
ejpam-7049	24	5	lower	low	ADJ
ejpam-7049	24	6	almost	almost	ADV
ejpam-7049	24	7	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-7049	24	8	multifunctions	multifunction	NOUN
ejpam-7049	24	9	[	[	X
ejpam-7049	24	10	11	11	NUM
ejpam-7049	24	11	]	]	PUNCT
ejpam-7049	24	12	,	,	PUNCT
ejpam-7049	24	13	upper	upper	ADJ
ejpam-7049	24	14	almost	almost	ADV
ejpam-7049	24	15	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7049	24	16	multifunctions	multifunction	NOUN
ejpam-7049	25	1	[	[	X
ejpam-7049	25	2	12	12	NUM
ejpam-7049	25	3	]	]	PUNCT
ejpam-7049	25	4	,	,	PUNCT
ejpam-7049	25	5	lower	low	ADJ
ejpam-7049	25	6	almost	almost	ADV
ejpam-7049	25	7	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7049	25	8	multifunctions	multifunction	NOUN
ejpam-7049	25	9	[	[	X
ejpam-7049	25	10	12	12	NUM
ejpam-7049	25	11	]	]	PUNCT
ejpam-7049	25	12	and	and	CCONJ
ejpam-7049	26	1	almost	almost	ADV
ejpam-7049	26	2	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-7049	26	3	multifunctions	multifunction	NOUN
ejpam-7049	26	4	[	[	X
ejpam-7049	26	5	13	13	NUM
ejpam-7049	26	6	]	]	PUNCT
ejpam-7049	26	7	.	.	PUNCT
ejpam-7049	27	1	pue	pue	NOUN
ejpam-7049	27	2	-	-	PUNCT
ejpam-7049	27	3	on	on	NOUN
ejpam-7049	27	4	et	et	PROPN
ejpam-7049	27	5	al	al	PROPN
ejpam-7049	27	6	.	.	PUNCT
ejpam-7049	28	1	[	[	X
ejpam-7049	28	2	14	14	NUM
ejpam-7049	28	3	]	]	PUNCT
ejpam-7049	28	4	introduced	introduce	VERB
ejpam-7049	28	5	and	and	CCONJ
ejpam-7049	28	6	studied	study	VERB
ejpam-7049	28	7	two	two	NUM
ejpam-7049	28	8	classes	class	NOUN
ejpam-7049	28	9	of	of	ADP
ejpam-7049	28	10	multifunctions	multifunction	NOUN
ejpam-7049	28	11	between	between	ADP
ejpam-7049	28	12	bitopological	bitopological	ADJ
ejpam-7049	28	13	spaces	space	NOUN
ejpam-7049	28	14	,	,	PUNCT
ejpam-7049	28	15	called	call	VERB
ejpam-7049	28	16	upper	upper	ADJ
ejpam-7049	28	17	(	(	PUNCT
ejpam-7049	28	18	τ1	τ1	NOUN
ejpam-7049	28	19	,	,	PUNCT
ejpam-7049	28	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	28	21	multifunctions	multifunction	NOUN
ejpam-7049	28	22	and	and	CCONJ
ejpam-7049	28	23	lower	low	ADJ
ejpam-7049	28	24	(	(	PUNCT
ejpam-7049	28	25	τ1	τ1	NOUN
ejpam-7049	28	26	,	,	PUNCT
ejpam-7049	28	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	28	28	multifunctions	multifunction	NOUN
ejpam-7049	28	29	.	.	PUNCT
ejpam-7049	29	1	moreover	moreover	ADV
ejpam-7049	29	2	,	,	PUNCT
ejpam-7049	29	3	boonpok	boonpok	NOUN
ejpam-7049	29	4	and	and	CCONJ
ejpam-7049	29	5	pue	pue	NOUN
ejpam-7049	29	6	-	-	PUNCT
ejpam-7049	29	7	on	on	ADP
ejpam-7049	29	8	[	[	X
ejpam-7049	29	9	15	15	NUM
ejpam-7049	29	10	]	]	PUNCT
ejpam-7049	29	11	introduced	introduce	VERB
ejpam-7049	29	12	and	and	CCONJ
ejpam-7049	29	13	investigated	investigate	VERB
ejpam-7049	29	14	the	the	DET
ejpam-7049	29	15	concepts	concept	NOUN
ejpam-7049	29	16	of	of	ADP
ejpam-7049	29	17	upper	upper	ADJ
ejpam-7049	29	18	almost	almost	ADV
ejpam-7049	29	19	(	(	PUNCT
ejpam-7049	29	20	τ1	τ1	NOUN
ejpam-7049	29	21	,	,	PUNCT
ejpam-7049	29	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	29	23	multifunctions	multifunction	NOUN
ejpam-7049	29	24	and	and	CCONJ
ejpam-7049	29	25	lower	low	ADJ
ejpam-7049	29	26	almost	almost	ADV
ejpam-7049	29	27	(	(	PUNCT
ejpam-7049	29	28	τ1	τ1	NOUN
ejpam-7049	29	29	,	,	PUNCT
ejpam-7049	29	30	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	29	31	multifunctions	multifunction	NOUN
ejpam-7049	29	32	.	.	PUNCT
ejpam-7049	30	1	laprom	laprom	ADP
ejpam-7049	30	2	et	et	PROPN
ejpam-7049	30	3	al	al	PROPN
ejpam-7049	30	4	.	.	PUNCT
ejpam-7049	31	1	[	[	X
ejpam-7049	31	2	16	16	NUM
ejpam-7049	31	3	]	]	PUNCT
ejpam-7049	31	4	introduced	introduce	VERB
ejpam-7049	31	5	and	and	CCONJ
ejpam-7049	31	6	studied	study	VERB
ejpam-7049	31	7	the	the	DET
ejpam-7049	31	8	notions	notion	NOUN
ejpam-7049	31	9	of	of	ADP
ejpam-7049	31	10	upper	upper	ADJ
ejpam-7049	31	11	almost	almost	ADV
ejpam-7049	31	12	β(τ1	β(τ1	VERB
ejpam-7049	31	13	,	,	PUNCT
ejpam-7049	31	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	31	15	multifunctions	multifunction	NOUN
ejpam-7049	31	16	and	and	CCONJ
ejpam-7049	31	17	lower	low	ADJ
ejpam-7049	31	18	almost	almost	ADV
ejpam-7049	31	19	β(τ1	β(τ1	NOUN
ejpam-7049	31	20	,	,	PUNCT
ejpam-7049	31	21	τ2)continuous	τ2)continuous	ADJ
ejpam-7049	31	22	multifunctions	multifunction	NOUN
ejpam-7049	31	23	.	.	PUNCT
ejpam-7049	32	1	in	in	ADP
ejpam-7049	32	2	this	this	DET
ejpam-7049	32	3	paper	paper	NOUN
ejpam-7049	32	4	,	,	PUNCT
ejpam-7049	32	5	we	we	PRON
ejpam-7049	32	6	introduce	introduce	VERB
ejpam-7049	32	7	the	the	DET
ejpam-7049	32	8	concepts	concept	NOUN
ejpam-7049	32	9	of	of	ADP
ejpam-7049	32	10	continuous	continuous	ADJ
ejpam-7049	32	11	multifunctions	multifunction	NOUN
ejpam-7049	32	12	between	between	ADP
ejpam-7049	32	13	an	an	DET
ejpam-7049	32	14	ideal	ideal	ADJ
ejpam-7049	32	15	topological	topological	ADJ
ejpam-7049	32	16	space	space	NOUN
ejpam-7049	32	17	and	and	CCONJ
ejpam-7049	32	18	a	a	DET
ejpam-7049	32	19	bitopological	bitopological	ADJ
ejpam-7049	32	20	space	space	NOUN
ejpam-7049	32	21	,	,	PUNCT
ejpam-7049	32	22	called	call	VERB
ejpam-7049	32	23	upper	upper	ADV
ejpam-7049	32	24	almost	almost	ADV
ejpam-7049	32	25	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	32	26	,	,	PUNCT
ejpam-7049	32	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	32	28	multifunctions	multifunction	NOUN
ejpam-7049	32	29	and	and	CCONJ
ejpam-7049	32	30	lower	low	ADJ
ejpam-7049	32	31	almost	almost	ADV
ejpam-7049	32	32	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	32	33	,	,	PUNCT
ejpam-7049	32	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	32	35	multifunctions	multifunction	NOUN
ejpam-7049	32	36	.	.	PUNCT
ejpam-7049	33	1	we	we	PRON
ejpam-7049	33	2	also	also	ADV
ejpam-7049	33	3	investigate	investigate	VERB
ejpam-7049	33	4	several	several	ADJ
ejpam-7049	33	5	characterizations	characterization	NOUN
ejpam-7049	33	6	of	of	ADP
ejpam-7049	33	7	upper	upper	ADJ
ejpam-7049	33	8	almost	almost	ADV
ejpam-7049	33	9	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	33	10	,	,	PUNCT
ejpam-7049	33	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	33	12	multifunctions	multifunction	NOUN
ejpam-7049	33	13	and	and	CCONJ
ejpam-7049	33	14	lower	low	ADJ
ejpam-7049	33	15	almost	almost	ADV
ejpam-7049	33	16	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	33	17	,	,	PUNCT
ejpam-7049	33	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	33	19	multifunctions	multifunction	NOUN
ejpam-7049	33	20	.	.	PUNCT
ejpam-7049	34	1	2	2	X
ejpam-7049	34	2	.	.	X
ejpam-7049	34	3	preliminaries	preliminary	NOUN
ejpam-7049	34	4	throughout	throughout	ADP
ejpam-7049	34	5	the	the	DET
ejpam-7049	34	6	present	present	ADJ
ejpam-7049	34	7	paper	paper	NOUN
ejpam-7049	34	8	,	,	PUNCT
ejpam-7049	34	9	spaces	space	NOUN
ejpam-7049	34	10	(	(	PUNCT
ejpam-7049	34	11	x	x	NOUN
ejpam-7049	34	12	,	,	PUNCT
ejpam-7049	34	13	τ1	τ1	NOUN
ejpam-7049	34	14	,	,	PUNCT
ejpam-7049	34	15	τ2	τ2	NOUN
ejpam-7049	34	16	)	)	PUNCT
ejpam-7049	34	17	and	and	CCONJ
ejpam-7049	34	18	(	(	PUNCT
ejpam-7049	34	19	y	y	PROPN
ejpam-7049	34	20	,	,	PUNCT
ejpam-7049	34	21	σ1	σ1	PROPN
ejpam-7049	34	22	,	,	PUNCT
ejpam-7049	34	23	σ2	σ2	NOUN
ejpam-7049	34	24	)	)	PUNCT
ejpam-7049	34	25	(	(	PUNCT
ejpam-7049	34	26	or	or	CCONJ
ejpam-7049	34	27	simply	simply	ADV
ejpam-7049	34	28	x	x	X
ejpam-7049	34	29	and	and	CCONJ
ejpam-7049	34	30	y	y	PROPN
ejpam-7049	34	31	)	)	PUNCT
ejpam-7049	34	32	always	always	ADV
ejpam-7049	34	33	mean	mean	VERB
ejpam-7049	34	34	bitopological	bitopological	ADJ
ejpam-7049	34	35	spaces	space	NOUN
ejpam-7049	34	36	on	on	ADP
ejpam-7049	34	37	which	which	PRON
ejpam-7049	34	38	no	no	DET
ejpam-7049	34	39	separation	separation	NOUN
ejpam-7049	34	40	axioms	axiom	NOUN
ejpam-7049	34	41	are	be	AUX
ejpam-7049	34	42	assumed	assume	VERB
ejpam-7049	34	43	unless	unless	SCONJ
ejpam-7049	34	44	explicitly	explicitly	ADV
ejpam-7049	34	45	stated	state	VERB
ejpam-7049	34	46	.	.	PUNCT
ejpam-7049	35	1	let	let	VERB
ejpam-7049	35	2	a	a	DET
ejpam-7049	35	3	be	be	AUX
ejpam-7049	35	4	a	a	DET
ejpam-7049	35	5	subset	subset	NOUN
ejpam-7049	35	6	of	of	ADP
ejpam-7049	35	7	a	a	DET
ejpam-7049	35	8	bitopological	bitopological	ADJ
ejpam-7049	35	9	space	space	NOUN
ejpam-7049	35	10	(	(	PUNCT
ejpam-7049	35	11	x	x	NOUN
ejpam-7049	35	12	,	,	PUNCT
ejpam-7049	35	13	τ1	τ1	NOUN
ejpam-7049	35	14	,	,	PUNCT
ejpam-7049	35	15	τ2	τ2	NOUN
ejpam-7049	35	16	)	)	PUNCT
ejpam-7049	35	17	.	.	PUNCT
ejpam-7049	36	1	the	the	DET
ejpam-7049	36	2	closure	closure	NOUN
ejpam-7049	36	3	of	of	ADP
ejpam-7049	36	4	a	a	PRON
ejpam-7049	36	5	and	and	CCONJ
ejpam-7049	36	6	the	the	DET
ejpam-7049	36	7	interior	interior	NOUN
ejpam-7049	36	8	of	of	ADP
ejpam-7049	36	9	a	a	PRON
ejpam-7049	36	10	with	with	ADP
ejpam-7049	36	11	respect	respect	NOUN
ejpam-7049	36	12	to	to	ADP
ejpam-7049	36	13	τi	τi	PROPN
ejpam-7049	36	14	are	be	AUX
ejpam-7049	36	15	denoted	denote	VERB
ejpam-7049	36	16	by	by	ADP
ejpam-7049	36	17	τi	τi	NOUN
ejpam-7049	36	18	-	-	PUNCT
ejpam-7049	36	19	cl(a	cl(a	NUM
ejpam-7049	36	20	)	)	PUNCT
ejpam-7049	36	21	and	and	CCONJ
ejpam-7049	36	22	τi	τi	NOUN
ejpam-7049	36	23	-	-	PUNCT
ejpam-7049	36	24	int(a	int(a	NOUN
ejpam-7049	36	25	)	)	PUNCT
ejpam-7049	36	26	,	,	PUNCT
ejpam-7049	36	27	respectively	respectively	ADV
ejpam-7049	36	28	,	,	PUNCT
ejpam-7049	36	29	for	for	ADP
ejpam-7049	36	30	i	i	PROPN
ejpam-7049	36	31	=	=	SYM
ejpam-7049	36	32	1	1	NUM
ejpam-7049	36	33	,	,	PUNCT
ejpam-7049	36	34	2	2	NUM
ejpam-7049	36	35	.	.	X
ejpam-7049	36	36	a	a	DET
ejpam-7049	36	37	subset	subset	NOUN
ejpam-7049	36	38	a	a	PRON
ejpam-7049	36	39	of	of	ADP
ejpam-7049	36	40	a	a	DET
ejpam-7049	36	41	bitopological	bitopological	ADJ
ejpam-7049	36	42	space	space	NOUN
ejpam-7049	36	43	(	(	PUNCT
ejpam-7049	36	44	x	x	NOUN
ejpam-7049	36	45	,	,	PUNCT
ejpam-7049	36	46	τ1	τ1	NOUN
ejpam-7049	36	47	,	,	PUNCT
ejpam-7049	36	48	τ2	τ2	NOUN
ejpam-7049	36	49	)	)	PUNCT
ejpam-7049	36	50	is	be	AUX
ejpam-7049	36	51	called	call	VERB
ejpam-7049	36	52	τ1τ2	τ1τ2	VERB
ejpam-7049	36	53	-	-	ADJ
ejpam-7049	36	54	closed	closed	ADJ
ejpam-7049	36	55	[	[	X
ejpam-7049	36	56	17	17	NUM
ejpam-7049	36	57	]	]	PUNCT
ejpam-7049	36	58	if	if	SCONJ
ejpam-7049	36	59	a	a	DET
ejpam-7049	36	60	=	=	NOUN
ejpam-7049	36	61	τ1	τ1	NOUN
ejpam-7049	36	62	-	-	PUNCT
ejpam-7049	36	63	cl(τ2	cl(τ2	NOUN
ejpam-7049	36	64	-	-	PUNCT
ejpam-7049	36	65	cl(a	cl(a	NUM
ejpam-7049	36	66	)	)	PUNCT
ejpam-7049	36	67	)	)	PUNCT
ejpam-7049	36	68	.	.	PUNCT
ejpam-7049	37	1	the	the	DET
ejpam-7049	37	2	complement	complement	NOUN
ejpam-7049	37	3	of	of	ADP
ejpam-7049	37	4	a	a	DET
ejpam-7049	37	5	τ1τ2	τ1τ2	ADJ
ejpam-7049	37	6	-	-	ADJ
ejpam-7049	37	7	closed	closed	ADJ
ejpam-7049	37	8	set	set	NOUN
ejpam-7049	37	9	is	be	AUX
ejpam-7049	37	10	called	call	VERB
ejpam-7049	37	11	τ1τ2	τ1τ2	NOUN
ejpam-7049	37	12	-	-	ADJ
ejpam-7049	37	13	open	open	ADJ
ejpam-7049	37	14	.	.	PUNCT
ejpam-7049	38	1	the	the	DET
ejpam-7049	38	2	intersection	intersection	NOUN
ejpam-7049	38	3	of	of	ADP
ejpam-7049	38	4	all	all	DET
ejpam-7049	38	5	τ1τ2	τ1τ2	ADJ
ejpam-7049	38	6	-	-	ADJ
ejpam-7049	38	7	closed	closed	ADJ
ejpam-7049	38	8	sets	set	NOUN
ejpam-7049	38	9	of	of	ADP
ejpam-7049	38	10	x	x	PUNCT
ejpam-7049	38	11	containing	contain	VERB
ejpam-7049	38	12	a	a	PRON
ejpam-7049	38	13	is	be	AUX
ejpam-7049	38	14	called	call	VERB
ejpam-7049	38	15	the	the	DET
ejpam-7049	38	16	τ1τ2	τ1τ2	NOUN
ejpam-7049	38	17	-	-	NOUN
ejpam-7049	38	18	closure	closure	NOUN
ejpam-7049	38	19	[	[	X
ejpam-7049	38	20	17	17	NUM
ejpam-7049	38	21	]	]	PUNCT
ejpam-7049	38	22	of	of	ADP
ejpam-7049	38	23	a	a	PRON
ejpam-7049	38	24	and	and	CCONJ
ejpam-7049	38	25	is	be	AUX
ejpam-7049	38	26	denoted	denote	VERB
ejpam-7049	38	27	by	by	ADP
ejpam-7049	38	28	τ1τ2	τ1τ2	NOUN
ejpam-7049	38	29	-	-	NUM
ejpam-7049	38	30	cl(a	cl(a	NUM
ejpam-7049	38	31	)	)	PUNCT
ejpam-7049	38	32	.	.	PUNCT
ejpam-7049	39	1	the	the	DET
ejpam-7049	39	2	union	union	NOUN
ejpam-7049	39	3	of	of	ADP
ejpam-7049	39	4	all	all	DET
ejpam-7049	39	5	τ1τ2	τ1τ2	ADJ
ejpam-7049	39	6	-	-	ADJ
ejpam-7049	39	7	open	open	ADJ
ejpam-7049	39	8	sets	set	NOUN
ejpam-7049	39	9	of	of	ADP
ejpam-7049	39	10	x	x	PUNCT
ejpam-7049	39	11	contained	contain	VERB
ejpam-7049	39	12	in	in	ADP
ejpam-7049	39	13	a	a	PRON
ejpam-7049	39	14	is	be	AUX
ejpam-7049	39	15	called	call	VERB
ejpam-7049	39	16	the	the	DET
ejpam-7049	39	17	τ1τ2	τ1τ2	NOUN
ejpam-7049	39	18	-	-	ADJ
ejpam-7049	39	19	interior	interior	ADJ
ejpam-7049	39	20	[	[	X
ejpam-7049	39	21	17	17	NUM
ejpam-7049	39	22	]	]	PUNCT
ejpam-7049	39	23	of	of	ADP
ejpam-7049	39	24	a	a	PRON
ejpam-7049	39	25	and	and	CCONJ
ejpam-7049	39	26	is	be	AUX
ejpam-7049	39	27	denoted	denote	VERB
ejpam-7049	39	28	by	by	ADP
ejpam-7049	39	29	τ1τ2	τ1τ2	NOUN
ejpam-7049	39	30	-	-	ADJ
ejpam-7049	39	31	int(a	int(a	NOUN
ejpam-7049	39	32	)	)	PUNCT
ejpam-7049	39	33	.	.	PUNCT
ejpam-7049	40	1	lemma	lemma	PROPN
ejpam-7049	40	2	1	1	NUM
ejpam-7049	40	3	.	.	PUNCT
ejpam-7049	41	1	[	[	X
ejpam-7049	41	2	17	17	NUM
ejpam-7049	41	3	]	]	PUNCT
ejpam-7049	41	4	let	let	VERB
ejpam-7049	41	5	a	a	PRON
ejpam-7049	41	6	and	and	CCONJ
ejpam-7049	41	7	b	b	NOUN
ejpam-7049	41	8	be	be	AUX
ejpam-7049	41	9	subsets	subset	NOUN
ejpam-7049	41	10	of	of	ADP
ejpam-7049	41	11	a	a	DET
ejpam-7049	41	12	bitopological	bitopological	ADJ
ejpam-7049	41	13	space	space	NOUN
ejpam-7049	41	14	(	(	PUNCT
ejpam-7049	41	15	x	x	NOUN
ejpam-7049	41	16	,	,	PUNCT
ejpam-7049	41	17	τ1	τ1	NOUN
ejpam-7049	41	18	,	,	PUNCT
ejpam-7049	41	19	τ2	τ2	NOUN
ejpam-7049	41	20	)	)	PUNCT
ejpam-7049	41	21	.	.	PUNCT
ejpam-7049	42	1	for	for	ADP
ejpam-7049	42	2	the	the	DET
ejpam-7049	42	3	τ1τ2	τ1τ2	NOUN
ejpam-7049	42	4	-	-	NOUN
ejpam-7049	42	5	closure	closure	NOUN
ejpam-7049	42	6	,	,	PUNCT
ejpam-7049	42	7	the	the	DET
ejpam-7049	42	8	following	follow	VERB
ejpam-7049	42	9	properties	property	NOUN
ejpam-7049	42	10	hold	hold	VERB
ejpam-7049	42	11	:	:	PUNCT
ejpam-7049	42	12	(	(	PUNCT
ejpam-7049	42	13	1	1	X
ejpam-7049	42	14	)	)	PUNCT
ejpam-7049	42	15	a	a	DET
ejpam-7049	42	16	⊆	⊆	NUM
ejpam-7049	42	17	τ1τ2	τ1τ2	NOUN
ejpam-7049	42	18	-	-	NUM
ejpam-7049	42	19	cl(a	cl(a	NUM
ejpam-7049	42	20	)	)	PUNCT
ejpam-7049	42	21	and	and	CCONJ
ejpam-7049	42	22	τ1τ2	τ1τ2	NOUN
ejpam-7049	42	23	-	-	ADJ
ejpam-7049	42	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7049	42	25	-	-	PUNCT
ejpam-7049	42	26	cl(a	cl(a	NUM
ejpam-7049	42	27	)	)	PUNCT
ejpam-7049	42	28	)	)	PUNCT
ejpam-7049	43	1	=	=	PUNCT
ejpam-7049	43	2	τ1τ2	τ1τ2	NOUN
ejpam-7049	43	3	-	-	NUM
ejpam-7049	43	4	cl(a	cl(a	NUM
ejpam-7049	43	5	)	)	PUNCT
ejpam-7049	43	6	.	.	PUNCT
ejpam-7049	44	1	(	(	PUNCT
ejpam-7049	44	2	2	2	X
ejpam-7049	44	3	)	)	PUNCT
ejpam-7049	44	4	if	if	SCONJ
ejpam-7049	44	5	a	a	DET
ejpam-7049	44	6	⊆	⊆	NUM
ejpam-7049	44	7	b	b	NOUN
ejpam-7049	44	8	,	,	PUNCT
ejpam-7049	44	9	then	then	ADV
ejpam-7049	44	10	τ1τ2	τ1τ2	NOUN
ejpam-7049	44	11	-	-	NUM
ejpam-7049	44	12	cl(a	cl(a	NUM
ejpam-7049	44	13	)	)	PUNCT
ejpam-7049	44	14	⊆	⊆	NUM
ejpam-7049	44	15	τ1τ2	τ1τ2	NOUN
ejpam-7049	44	16	-	-	NOUN
ejpam-7049	44	17	cl(b	cl(b	NOUN
ejpam-7049	44	18	)	)	PUNCT
ejpam-7049	44	19	.	.	PUNCT
ejpam-7049	45	1	(	(	PUNCT
ejpam-7049	45	2	3	3	X
ejpam-7049	45	3	)	)	PUNCT
ejpam-7049	45	4	τ1τ2	τ1τ2	NOUN
ejpam-7049	45	5	-	-	NUM
ejpam-7049	45	6	cl(a	cl(a	NUM
ejpam-7049	45	7	)	)	PUNCT
ejpam-7049	45	8	is	be	AUX
ejpam-7049	45	9	τ1τ2	τ1τ2	NOUN
ejpam-7049	45	10	-	-	ADJ
ejpam-7049	45	11	closed	closed	ADJ
ejpam-7049	45	12	.	.	PUNCT
ejpam-7049	46	1	(	(	PUNCT
ejpam-7049	46	2	4	4	X
ejpam-7049	46	3	)	)	PUNCT
ejpam-7049	46	4	a	a	PRON
ejpam-7049	46	5	is	be	AUX
ejpam-7049	46	6	τ1τ2	τ1τ2	NOUN
ejpam-7049	46	7	-	-	ADJ
ejpam-7049	46	8	closed	closed	ADJ
ejpam-7049	46	9	if	if	SCONJ
ejpam-7049	46	10	and	and	CCONJ
ejpam-7049	46	11	only	only	ADV
ejpam-7049	46	12	if	if	SCONJ
ejpam-7049	46	13	a	a	DET
ejpam-7049	46	14	=	=	PUNCT
ejpam-7049	46	15	τ1τ2	τ1τ2	NOUN
ejpam-7049	46	16	-	-	NUM
ejpam-7049	46	17	cl(a	cl(a	NUM
ejpam-7049	46	18	)	)	PUNCT
ejpam-7049	46	19	.	.	PUNCT
ejpam-7049	47	1	(	(	PUNCT
ejpam-7049	47	2	5	5	X
ejpam-7049	47	3	)	)	PUNCT
ejpam-7049	47	4	τ1τ2	τ1τ2	NOUN
ejpam-7049	47	5	-	-	NOUN
ejpam-7049	47	6	cl(x	cl(x	X
ejpam-7049	47	7	−a	−a	NOUN
ejpam-7049	47	8	)	)	PUNCT
ejpam-7049	48	1	=	=	PUNCT
ejpam-7049	48	2	x	x	X
ejpam-7049	49	1	−	−	ADP
ejpam-7049	49	2	τ1τ2	τ1τ2	NOUN
ejpam-7049	49	3	-	-	PUNCT
ejpam-7049	49	4	int(a	int(a	NOUN
ejpam-7049	49	5	)	)	PUNCT
ejpam-7049	49	6	.	.	PUNCT
ejpam-7049	50	1	a	a	DET
ejpam-7049	50	2	subset	subset	NOUN
ejpam-7049	50	3	a	a	PRON
ejpam-7049	50	4	of	of	ADP
ejpam-7049	50	5	a	a	DET
ejpam-7049	50	6	bitopological	bitopological	ADJ
ejpam-7049	50	7	space	space	NOUN
ejpam-7049	50	8	(	(	PUNCT
ejpam-7049	50	9	x	x	NOUN
ejpam-7049	50	10	,	,	PUNCT
ejpam-7049	50	11	τ1	τ1	NOUN
ejpam-7049	50	12	,	,	PUNCT
ejpam-7049	50	13	τ2	τ2	NOUN
ejpam-7049	50	14	)	)	PUNCT
ejpam-7049	50	15	is	be	AUX
ejpam-7049	50	16	called	call	VERB
ejpam-7049	50	17	(	(	PUNCT
ejpam-7049	50	18	τ1	τ1	NOUN
ejpam-7049	50	19	,	,	PUNCT
ejpam-7049	50	20	τ2)r	τ2)r	NOUN
ejpam-7049	50	21	-	-	PUNCT
ejpam-7049	50	22	open	open	NOUN
ejpam-7049	51	1	[	[	X
ejpam-7049	51	2	18	18	NUM
ejpam-7049	51	3	]	]	PUNCT
ejpam-7049	51	4	(	(	PUNCT
ejpam-7049	51	5	resp	resp	NOUN
ejpam-7049	51	6	.	.	PUNCT
ejpam-7049	52	1	(	(	PUNCT
ejpam-7049	52	2	τ1	τ1	NOUN
ejpam-7049	52	3	,	,	PUNCT
ejpam-7049	52	4	τ2)sopen	τ2)sopen	VERB
ejpam-7049	52	5	[	[	X
ejpam-7049	52	6	19	19	NUM
ejpam-7049	52	7	]	]	PUNCT
ejpam-7049	52	8	,	,	PUNCT
ejpam-7049	52	9	(	(	PUNCT
ejpam-7049	52	10	τ1	τ1	NOUN
ejpam-7049	52	11	,	,	PUNCT
ejpam-7049	52	12	τ2)p	τ2)p	NOUN
ejpam-7049	52	13	-	-	ADJ
ejpam-7049	52	14	open	open	ADJ
ejpam-7049	52	15	[	[	X
ejpam-7049	52	16	19	19	NUM
ejpam-7049	52	17	]	]	PUNCT
ejpam-7049	52	18	,	,	PUNCT
ejpam-7049	52	19	(	(	PUNCT
ejpam-7049	52	20	τ1	τ1	NOUN
ejpam-7049	52	21	,	,	PUNCT
ejpam-7049	52	22	τ2)β	τ2)β	ADJ
ejpam-7049	52	23	-	-	PUNCT
ejpam-7049	52	24	open	open	NOUN
ejpam-7049	52	25	[	[	X
ejpam-7049	52	26	19	19	NUM
ejpam-7049	52	27	]	]	PUNCT
ejpam-7049	52	28	)	)	PUNCT
ejpam-7049	52	29	if	if	SCONJ
ejpam-7049	52	30	a	a	DET
ejpam-7049	52	31	=	=	PUNCT
ejpam-7049	52	32	τ1τ2	τ1τ2	NOUN
ejpam-7049	52	33	-	-	NOUN
ejpam-7049	52	34	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7049	52	35	-	-	PUNCT
ejpam-7049	52	36	cl(a	cl(a	NUM
ejpam-7049	52	37	)	)	PUNCT
ejpam-7049	52	38	)	)	PUNCT
ejpam-7049	52	39	(	(	PUNCT
ejpam-7049	52	40	resp	resp	NOUN
ejpam-7049	52	41	.	.	PUNCT
ejpam-7049	53	1	a	a	DET
ejpam-7049	53	2	⊆	⊆	NUM
ejpam-7049	53	3	n.	n.	NOUN
ejpam-7049	53	4	srisarakham	srisarakham	PROPN
ejpam-7049	53	5	,	,	PUNCT
ejpam-7049	53	6	a.	a.	PROPN
ejpam-7049	53	7	sama	sama	PROPN
ejpam-7049	53	8	-	-	PUNCT
ejpam-7049	53	9	ae	ae	PROPN
ejpam-7049	53	10	,	,	PUNCT
ejpam-7049	53	11	c.	c.	PROPN
ejpam-7049	53	12	boonpok	boonpok	PROPN
ejpam-7049	53	13	/	/	SYM
ejpam-7049	53	14	eur	eur	PROPN
ejpam-7049	53	15	.	.	PUNCT
ejpam-7049	54	1	j.	j.	PROPN
ejpam-7049	54	2	pure	pure	PROPN
ejpam-7049	54	3	appl	appl	PROPN
ejpam-7049	54	4	.	.	PROPN
ejpam-7049	54	5	math	math	PROPN
ejpam-7049	54	6	,	,	PUNCT
ejpam-7049	54	7	18	18	NUM
ejpam-7049	54	8	(	(	PUNCT
ejpam-7049	54	9	4	4	NUM
ejpam-7049	54	10	)	)	PUNCT
ejpam-7049	54	11	(	(	PUNCT
ejpam-7049	54	12	2025	2025	NUM
ejpam-7049	54	13	)	)	PUNCT
ejpam-7049	54	14	,	,	PUNCT
ejpam-7049	54	15	7049	7049	NUM
ejpam-7049	54	16	3	3	NUM
ejpam-7049	54	17	of	of	ADP
ejpam-7049	54	18	11	11	NUM
ejpam-7049	54	19	τ1τ2	τ1τ2	NOUN
ejpam-7049	54	20	-	-	NUM
ejpam-7049	54	21	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7049	54	22	-	-	PUNCT
ejpam-7049	54	23	int(a	int(a	NOUN
ejpam-7049	54	24	)	)	PUNCT
ejpam-7049	54	25	)	)	PUNCT
ejpam-7049	54	26	,	,	PUNCT
ejpam-7049	54	27	a	a	DET
ejpam-7049	54	28	⊆	⊆	NUM
ejpam-7049	54	29	τ1τ2	τ1τ2	NOUN
ejpam-7049	54	30	-	-	NOUN
ejpam-7049	54	31	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7049	54	32	-	-	PUNCT
ejpam-7049	54	33	cl(a	cl(a	NUM
ejpam-7049	54	34	)	)	PUNCT
ejpam-7049	54	35	)	)	PUNCT
ejpam-7049	54	36	,	,	PUNCT
ejpam-7049	54	37	a	a	DET
ejpam-7049	54	38	⊆	⊆	NUM
ejpam-7049	54	39	τ1τ2	τ1τ2	NOUN
ejpam-7049	54	40	-	-	PUNCT
ejpam-7049	54	41	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7049	54	42	-	-	PUNCT
ejpam-7049	54	43	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7049	54	44	-	-	PUNCT
ejpam-7049	54	45	cl(a	cl(a	NUM
ejpam-7049	54	46	)	)	PUNCT
ejpam-7049	54	47	)	)	PUNCT
ejpam-7049	54	48	)	)	PUNCT
ejpam-7049	54	49	)	)	PUNCT
ejpam-7049	54	50	.	.	PUNCT
ejpam-7049	55	1	the	the	DET
ejpam-7049	55	2	complement	complement	NOUN
ejpam-7049	55	3	of	of	ADP
ejpam-7049	55	4	a	a	DET
ejpam-7049	55	5	(	(	PUNCT
ejpam-7049	55	6	τ1	τ1	NOUN
ejpam-7049	55	7	,	,	PUNCT
ejpam-7049	55	8	τ2)r	τ2)r	NOUN
ejpam-7049	55	9	-	-	PUNCT
ejpam-7049	55	10	open	open	ADJ
ejpam-7049	55	11	(	(	PUNCT
ejpam-7049	55	12	resp	resp	NOUN
ejpam-7049	55	13	.	.	PUNCT
ejpam-7049	56	1	(	(	PUNCT
ejpam-7049	56	2	τ1	τ1	NOUN
ejpam-7049	56	3	,	,	PUNCT
ejpam-7049	56	4	τ2)s	τ2)s	NOUN
ejpam-7049	56	5	-	-	PUNCT
ejpam-7049	56	6	open	open	ADJ
ejpam-7049	56	7	,	,	PUNCT
ejpam-7049	56	8	(	(	PUNCT
ejpam-7049	56	9	τ1	τ1	NOUN
ejpam-7049	56	10	,	,	PUNCT
ejpam-7049	56	11	τ2)p	τ2)p	NOUN
ejpam-7049	56	12	-	-	ADJ
ejpam-7049	56	13	open	open	ADJ
ejpam-7049	56	14	,	,	PUNCT
ejpam-7049	56	15	(	(	PUNCT
ejpam-7049	56	16	τ1	τ1	NOUN
ejpam-7049	56	17	,	,	PUNCT
ejpam-7049	56	18	τ2)β	τ2)β	ADJ
ejpam-7049	56	19	-	-	PUNCT
ejpam-7049	56	20	open	open	ADJ
ejpam-7049	56	21	)	)	PUNCT
ejpam-7049	56	22	set	set	NOUN
ejpam-7049	56	23	is	be	AUX
ejpam-7049	56	24	called	call	VERB
ejpam-7049	56	25	(	(	PUNCT
ejpam-7049	56	26	τ1	τ1	NOUN
ejpam-7049	56	27	,	,	PUNCT
ejpam-7049	56	28	τ2)r	τ2)r	NOUN
ejpam-7049	56	29	-	-	PUNCT
ejpam-7049	56	30	closed	closed	ADJ
ejpam-7049	56	31	(	(	PUNCT
ejpam-7049	56	32	resp	resp	NOUN
ejpam-7049	56	33	.	.	PUNCT
ejpam-7049	57	1	(	(	PUNCT
ejpam-7049	57	2	τ1	τ1	NOUN
ejpam-7049	57	3	,	,	PUNCT
ejpam-7049	57	4	τ2)s	τ2)s	NOUN
ejpam-7049	57	5	-	-	PUNCT
ejpam-7049	57	6	closed	closed	ADJ
ejpam-7049	57	7	,	,	PUNCT
ejpam-7049	57	8	(	(	PUNCT
ejpam-7049	57	9	τ1	τ1	NOUN
ejpam-7049	57	10	,	,	PUNCT
ejpam-7049	57	11	τ2)p	τ2)p	NOUN
ejpam-7049	57	12	-	-	PUNCT
ejpam-7049	57	13	closed	closed	ADJ
ejpam-7049	57	14	,	,	PUNCT
ejpam-7049	57	15	(	(	PUNCT
ejpam-7049	57	16	τ1	τ1	NOUN
ejpam-7049	57	17	,	,	PUNCT
ejpam-7049	57	18	τ2)β	τ2)β	ADJ
ejpam-7049	57	19	-	-	PUNCT
ejpam-7049	57	20	closed	closed	ADJ
ejpam-7049	57	21	)	)	PUNCT
ejpam-7049	57	22	.	.	PUNCT
ejpam-7049	58	1	the	the	DET
ejpam-7049	58	2	intersection	intersection	NOUN
ejpam-7049	58	3	of	of	ADP
ejpam-7049	58	4	all	all	DET
ejpam-7049	58	5	(	(	PUNCT
ejpam-7049	58	6	τ1	τ1	NOUN
ejpam-7049	58	7	,	,	PUNCT
ejpam-7049	58	8	τ2)s	τ2)s	NOUN
ejpam-7049	58	9	-	-	PUNCT
ejpam-7049	58	10	closed	close	VERB
ejpam-7049	58	11	sets	set	NOUN
ejpam-7049	58	12	of	of	ADP
ejpam-7049	58	13	x	x	PUNCT
ejpam-7049	58	14	containing	contain	VERB
ejpam-7049	58	15	a	a	PRON
ejpam-7049	58	16	is	be	AUX
ejpam-7049	58	17	called	call	VERB
ejpam-7049	58	18	the	the	DET
ejpam-7049	58	19	(	(	PUNCT
ejpam-7049	58	20	τ1	τ1	NOUN
ejpam-7049	58	21	,	,	PUNCT
ejpam-7049	58	22	τ2)s	τ2)s	NOUN
ejpam-7049	58	23	-	-	PUNCT
ejpam-7049	58	24	closure	closure	NOUN
ejpam-7049	58	25	[	[	X
ejpam-7049	58	26	19	19	NUM
ejpam-7049	58	27	]	]	PUNCT
ejpam-7049	58	28	of	of	ADP
ejpam-7049	58	29	a	a	PRON
ejpam-7049	58	30	and	and	CCONJ
ejpam-7049	58	31	is	be	AUX
ejpam-7049	58	32	denoted	denote	VERB
ejpam-7049	58	33	by	by	ADP
ejpam-7049	58	34	(	(	PUNCT
ejpam-7049	58	35	τ1	τ1	NOUN
ejpam-7049	58	36	,	,	PUNCT
ejpam-7049	58	37	τ2)-scl(a	τ2)-scl(a	PROPN
ejpam-7049	58	38	)	)	PUNCT
ejpam-7049	58	39	.	.	PUNCT
ejpam-7049	59	1	the	the	DET
ejpam-7049	59	2	union	union	NOUN
ejpam-7049	59	3	of	of	ADP
ejpam-7049	59	4	all	all	DET
ejpam-7049	59	5	(	(	PUNCT
ejpam-7049	59	6	τ1	τ1	NOUN
ejpam-7049	59	7	,	,	PUNCT
ejpam-7049	59	8	τ2)s	τ2)s	NOUN
ejpam-7049	59	9	-	-	PUNCT
ejpam-7049	59	10	open	open	ADJ
ejpam-7049	59	11	sets	set	NOUN
ejpam-7049	59	12	of	of	ADP
ejpam-7049	59	13	x	x	PUNCT
ejpam-7049	59	14	contained	contain	VERB
ejpam-7049	59	15	in	in	ADP
ejpam-7049	59	16	a	a	PRON
ejpam-7049	59	17	is	be	AUX
ejpam-7049	59	18	called	call	VERB
ejpam-7049	59	19	the	the	DET
ejpam-7049	59	20	(	(	PUNCT
ejpam-7049	59	21	τ1	τ1	NOUN
ejpam-7049	59	22	,	,	PUNCT
ejpam-7049	59	23	τ2)s	τ2)s	NOUN
ejpam-7049	59	24	-	-	ADJ
ejpam-7049	59	25	interior	interior	NOUN
ejpam-7049	59	26	[	[	X
ejpam-7049	59	27	19	19	NUM
ejpam-7049	59	28	]	]	PUNCT
ejpam-7049	59	29	of	of	ADP
ejpam-7049	59	30	a	a	PRON
ejpam-7049	59	31	and	and	CCONJ
ejpam-7049	59	32	is	be	AUX
ejpam-7049	59	33	denoted	denote	VERB
ejpam-7049	59	34	by	by	ADP
ejpam-7049	59	35	(	(	PUNCT
ejpam-7049	59	36	τ1	τ1	NOUN
ejpam-7049	59	37	,	,	PUNCT
ejpam-7049	59	38	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7049	59	39	)	)	PUNCT
ejpam-7049	59	40	.	.	PUNCT
ejpam-7049	60	1	lemma	lemma	PROPN
ejpam-7049	60	2	2	2	NUM
ejpam-7049	60	3	.	.	X
ejpam-7049	61	1	for	for	ADP
ejpam-7049	61	2	a	a	DET
ejpam-7049	61	3	subset	subset	NOUN
ejpam-7049	61	4	a	a	PRON
ejpam-7049	61	5	of	of	ADP
ejpam-7049	61	6	a	a	DET
ejpam-7049	61	7	bitopological	bitopological	ADJ
ejpam-7049	61	8	space	space	NOUN
ejpam-7049	61	9	(	(	PUNCT
ejpam-7049	61	10	x	x	NOUN
ejpam-7049	61	11	,	,	PUNCT
ejpam-7049	61	12	τ1	τ1	NOUN
ejpam-7049	61	13	,	,	PUNCT
ejpam-7049	61	14	τ2	τ2	NOUN
ejpam-7049	61	15	)	)	PUNCT
ejpam-7049	61	16	,	,	PUNCT
ejpam-7049	61	17	the	the	DET
ejpam-7049	61	18	following	follow	VERB
ejpam-7049	61	19	properties	property	NOUN
ejpam-7049	61	20	hold	hold	VERB
ejpam-7049	61	21	:	:	PUNCT
ejpam-7049	61	22	(	(	PUNCT
ejpam-7049	61	23	1	1	X
ejpam-7049	61	24	)	)	PUNCT
ejpam-7049	61	25	(	(	PUNCT
ejpam-7049	61	26	τ1	τ1	NOUN
ejpam-7049	61	27	,	,	PUNCT
ejpam-7049	61	28	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-7049	61	29	)	)	PUNCT
ejpam-7049	61	30	=	=	PUNCT
ejpam-7049	62	1	τ1τ2	τ1τ2	NOUN
ejpam-7049	62	2	-	-	NOUN
ejpam-7049	62	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7049	62	4	-	-	PUNCT
ejpam-7049	62	5	cl(a	cl(a	NUM
ejpam-7049	62	6	)	)	PUNCT
ejpam-7049	62	7	)	)	PUNCT
ejpam-7049	63	1	∪a	∪a	X
ejpam-7049	64	1	[	[	X
ejpam-7049	64	2	19	19	NUM
ejpam-7049	64	3	]	]	X
ejpam-7049	64	4	;	;	PUNCT
ejpam-7049	64	5	(	(	PUNCT
ejpam-7049	64	6	2	2	X
ejpam-7049	64	7	)	)	PUNCT
ejpam-7049	64	8	(	(	PUNCT
ejpam-7049	64	9	τ1	τ1	NOUN
ejpam-7049	64	10	,	,	PUNCT
ejpam-7049	64	11	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7049	64	12	)	)	PUNCT
ejpam-7049	65	1	=	=	PUNCT
ejpam-7049	65	2	τ1τ2	τ1τ2	NOUN
ejpam-7049	65	3	-	-	ADJ
ejpam-7049	65	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7049	65	5	-	-	PUNCT
ejpam-7049	65	6	int(a	int(a	NOUN
ejpam-7049	65	7	)	)	PUNCT
ejpam-7049	65	8	)	)	PUNCT
ejpam-7049	66	1	∩a	∩a	PROPN
ejpam-7049	67	1	[	[	X
ejpam-7049	67	2	20	20	NUM
ejpam-7049	67	3	]	]	PUNCT
ejpam-7049	67	4	.	.	PUNCT
ejpam-7049	68	1	an	an	DET
ejpam-7049	68	2	ideal	ideal	NOUN
ejpam-7049	68	3	i	i	PRON
ejpam-7049	68	4	on	on	ADP
ejpam-7049	68	5	a	a	DET
ejpam-7049	68	6	topological	topological	ADJ
ejpam-7049	68	7	space	space	NOUN
ejpam-7049	68	8	(	(	PUNCT
ejpam-7049	68	9	x	x	X
ejpam-7049	68	10	,	,	PUNCT
ejpam-7049	68	11	τ	τ	X
ejpam-7049	68	12	)	)	PUNCT
ejpam-7049	68	13	is	be	AUX
ejpam-7049	68	14	a	a	DET
ejpam-7049	68	15	nonempty	nonempty	ADJ
ejpam-7049	68	16	collection	collection	NOUN
ejpam-7049	68	17	of	of	ADP
ejpam-7049	68	18	subsets	subset	NOUN
ejpam-7049	68	19	of	of	ADP
ejpam-7049	68	20	x	x	PUNCT
ejpam-7049	68	21	satisfying	satisfy	VERB
ejpam-7049	68	22	the	the	DET
ejpam-7049	68	23	following	follow	VERB
ejpam-7049	68	24	properties	property	NOUN
ejpam-7049	68	25	:	:	PUNCT
ejpam-7049	68	26	(	(	PUNCT
ejpam-7049	68	27	1	1	X
ejpam-7049	68	28	)	)	PUNCT
ejpam-7049	68	29	a	a	DET
ejpam-7049	68	30	∈	∈	NOUN
ejpam-7049	68	31	i	i	PRON
ejpam-7049	68	32	and	and	CCONJ
ejpam-7049	68	33	b	b	X
ejpam-7049	68	34	⊆	⊆	NUM
ejpam-7049	68	35	a	a	DET
ejpam-7049	68	36	imply	imply	NOUN
ejpam-7049	68	37	b	b	X
ejpam-7049	68	38	∈	∈	PROPN
ejpam-7049	68	39	i	i	PRON
ejpam-7049	68	40	;	;	PUNCT
ejpam-7049	68	41	(	(	PUNCT
ejpam-7049	68	42	2	2	X
ejpam-7049	68	43	)	)	PUNCT
ejpam-7049	69	1	a	a	PRON
ejpam-7049	69	2	∈	∈	NOUN
ejpam-7049	70	1	i	i	PRON
ejpam-7049	70	2	and	and	CCONJ
ejpam-7049	70	3	b	b	X
ejpam-7049	70	4	∈	∈	NOUN
ejpam-7049	71	1	i	i	PRON
ejpam-7049	71	2	imply	imply	VERB
ejpam-7049	71	3	a	a	DET
ejpam-7049	71	4	∪	∪	X
ejpam-7049	71	5	b	b	NOUN
ejpam-7049	71	6	∈	∈	NOUN
ejpam-7049	72	1	i	i	PRON
ejpam-7049	72	2	.	.	PUNCT
ejpam-7049	73	1	a	a	DET
ejpam-7049	73	2	topological	topological	ADJ
ejpam-7049	73	3	space	space	NOUN
ejpam-7049	73	4	(	(	PUNCT
ejpam-7049	73	5	x	x	X
ejpam-7049	73	6	,	,	PUNCT
ejpam-7049	73	7	τ	τ	X
ejpam-7049	73	8	)	)	PUNCT
ejpam-7049	73	9	with	with	ADP
ejpam-7049	73	10	an	an	DET
ejpam-7049	73	11	ideal	ideal	ADJ
ejpam-7049	73	12	i	i	PRON
ejpam-7049	73	13	on	on	ADP
ejpam-7049	73	14	x	x	SYM
ejpam-7049	73	15	is	be	AUX
ejpam-7049	73	16	called	call	VERB
ejpam-7049	73	17	an	an	DET
ejpam-7049	73	18	ideal	ideal	ADJ
ejpam-7049	73	19	topological	topological	ADJ
ejpam-7049	73	20	space	space	NOUN
ejpam-7049	73	21	and	and	CCONJ
ejpam-7049	73	22	is	be	AUX
ejpam-7049	73	23	denoted	denote	VERB
ejpam-7049	73	24	by	by	ADP
ejpam-7049	73	25	(	(	PUNCT
ejpam-7049	73	26	x	x	X
ejpam-7049	73	27	,	,	PUNCT
ejpam-7049	73	28	τ	τ	PROPN
ejpam-7049	73	29	,	,	PUNCT
ejpam-7049	73	30	i	i	NOUN
ejpam-7049	73	31	)	)	PUNCT
ejpam-7049	73	32	.	.	PUNCT
ejpam-7049	74	1	for	for	ADP
ejpam-7049	74	2	an	an	DET
ejpam-7049	74	3	ideal	ideal	ADJ
ejpam-7049	74	4	topological	topological	ADJ
ejpam-7049	74	5	space	space	NOUN
ejpam-7049	74	6	(	(	PUNCT
ejpam-7049	74	7	x	x	X
ejpam-7049	74	8	,	,	PUNCT
ejpam-7049	74	9	τ	τ	PROPN
ejpam-7049	74	10	,	,	PUNCT
ejpam-7049	74	11	i	i	PROPN
ejpam-7049	74	12	)	)	PUNCT
ejpam-7049	74	13	and	and	CCONJ
ejpam-7049	74	14	a	a	DET
ejpam-7049	74	15	subset	subset	NOUN
ejpam-7049	74	16	a	a	PRON
ejpam-7049	74	17	of	of	ADP
ejpam-7049	74	18	x	x	PRON
ejpam-7049	74	19	,	,	PUNCT
ejpam-7049	74	20	a⋆(i	a⋆(i	PROPN
ejpam-7049	74	21	)	)	PUNCT
ejpam-7049	74	22	is	be	AUX
ejpam-7049	74	23	defined	define	VERB
ejpam-7049	74	24	as	as	SCONJ
ejpam-7049	74	25	follows	follow	VERB
ejpam-7049	74	26	:	:	PUNCT
ejpam-7049	74	27	a⋆(i	a⋆(i	NOUN
ejpam-7049	74	28	)	)	PUNCT
ejpam-7049	75	1	=	=	PUNCT
ejpam-7049	75	2	{	{	PUNCT
ejpam-7049	75	3	x	x	PUNCT
ejpam-7049	75	4	∈	∈	PROPN
ejpam-7049	75	5	x	x	X
ejpam-7049	75	6	:	:	PUNCT
ejpam-7049	75	7	u	u	X
ejpam-7049	75	8	∩a	∩a	PROPN
ejpam-7049	75	9	̸∈	̸∈	PROPN
ejpam-7049	75	10	i	i	PRON
ejpam-7049	75	11	for	for	ADP
ejpam-7049	75	12	every	every	DET
ejpam-7049	75	13	open	open	ADJ
ejpam-7049	75	14	neighbourhood	neighbourhood	NOUN
ejpam-7049	75	15	u	u	NOUN
ejpam-7049	75	16	of	of	ADP
ejpam-7049	75	17	x	x	NOUN
ejpam-7049	75	18	}	}	PUNCT
ejpam-7049	75	19	.	.	PUNCT
ejpam-7049	76	1	in	in	ADP
ejpam-7049	76	2	case	case	NOUN
ejpam-7049	76	3	there	there	PRON
ejpam-7049	76	4	is	be	VERB
ejpam-7049	76	5	no	no	DET
ejpam-7049	76	6	chance	chance	NOUN
ejpam-7049	76	7	for	for	ADP
ejpam-7049	76	8	confusion	confusion	NOUN
ejpam-7049	76	9	,	,	PUNCT
ejpam-7049	76	10	a⋆(i	a⋆(i	NOUN
ejpam-7049	76	11	)	)	PUNCT
ejpam-7049	76	12	is	be	AUX
ejpam-7049	76	13	simply	simply	ADV
ejpam-7049	76	14	written	write	VERB
ejpam-7049	76	15	as	as	ADP
ejpam-7049	76	16	a⋆.	a⋆.	NOUN
ejpam-7049	76	17	in	in	ADP
ejpam-7049	76	18	[	[	X
ejpam-7049	76	19	21	21	NUM
ejpam-7049	76	20	]	]	PUNCT
ejpam-7049	76	21	,	,	PUNCT
ejpam-7049	76	22	a⋆	a⋆	ADV
ejpam-7049	76	23	is	be	AUX
ejpam-7049	76	24	called	call	VERB
ejpam-7049	76	25	the	the	DET
ejpam-7049	76	26	local	local	ADJ
ejpam-7049	76	27	function	function	NOUN
ejpam-7049	76	28	of	of	ADP
ejpam-7049	76	29	a	a	PRON
ejpam-7049	76	30	with	with	ADP
ejpam-7049	76	31	respect	respect	NOUN
ejpam-7049	76	32	to	to	ADP
ejpam-7049	76	33	i	i	PRON
ejpam-7049	76	34	and	and	CCONJ
ejpam-7049	76	35	τ	τ	PROPN
ejpam-7049	76	36	and	and	CCONJ
ejpam-7049	76	37	cl⋆(a	cl⋆(a	NUM
ejpam-7049	76	38	)	)	PUNCT
ejpam-7049	76	39	=	=	PUNCT
ejpam-7049	76	40	a⋆	a⋆	ADP
ejpam-7049	76	41	∪	∪	ADP
ejpam-7049	76	42	a	a	DET
ejpam-7049	76	43	defines	define	NOUN
ejpam-7049	76	44	a	a	DET
ejpam-7049	76	45	kuratowski	kuratowski	ADJ
ejpam-7049	76	46	closure	closure	NOUN
ejpam-7049	76	47	operator	operator	NOUN
ejpam-7049	76	48	for	for	ADP
ejpam-7049	76	49	a	a	DET
ejpam-7049	76	50	topology	topology	NOUN
ejpam-7049	76	51	τ⋆(i	τ⋆(i	NOUN
ejpam-7049	76	52	)	)	PUNCT
ejpam-7049	76	53	finer	fine	ADJ
ejpam-7049	76	54	than	than	ADP
ejpam-7049	76	55	τ	τ	PROPN
ejpam-7049	76	56	.	.	PUNCT
ejpam-7049	77	1	a	a	DET
ejpam-7049	77	2	subset	subset	NOUN
ejpam-7049	77	3	a	a	PRON
ejpam-7049	77	4	is	be	AUX
ejpam-7049	77	5	said	say	VERB
ejpam-7049	77	6	to	to	PART
ejpam-7049	77	7	be	be	AUX
ejpam-7049	77	8	⋆-closed	⋆-close	VERB
ejpam-7049	77	9	[	[	X
ejpam-7049	77	10	22	22	NUM
ejpam-7049	77	11	]	]	X
ejpam-7049	77	12	if	if	SCONJ
ejpam-7049	77	13	a⋆	a⋆	ADJ
ejpam-7049	77	14	⊆	⊆	NUM
ejpam-7049	77	15	a.	a.	NOUN
ejpam-7049	77	16	the	the	DET
ejpam-7049	77	17	interior	interior	NOUN
ejpam-7049	77	18	of	of	ADP
ejpam-7049	77	19	a	a	DET
ejpam-7049	77	20	subset	subset	NOUN
ejpam-7049	77	21	a	a	DET
ejpam-7049	77	22	in	in	ADP
ejpam-7049	77	23	(	(	PUNCT
ejpam-7049	77	24	x	x	X
ejpam-7049	77	25	,	,	PUNCT
ejpam-7049	77	26	τ⋆(i	τ⋆(i	NOUN
ejpam-7049	77	27	)	)	PUNCT
ejpam-7049	77	28	)	)	PUNCT
ejpam-7049	77	29	is	be	AUX
ejpam-7049	77	30	denoted	denote	VERB
ejpam-7049	77	31	by	by	ADP
ejpam-7049	77	32	int⋆(a	int⋆(a	NOUN
ejpam-7049	77	33	)	)	PUNCT
ejpam-7049	77	34	.	.	PUNCT
ejpam-7049	78	1	a	a	DET
ejpam-7049	78	2	subset	subset	NOUN
ejpam-7049	78	3	a	a	PRON
ejpam-7049	78	4	of	of	ADP
ejpam-7049	78	5	an	an	DET
ejpam-7049	78	6	ideal	ideal	ADJ
ejpam-7049	78	7	topological	topological	ADJ
ejpam-7049	78	8	space	space	NOUN
ejpam-7049	78	9	(	(	PUNCT
ejpam-7049	78	10	x	x	X
ejpam-7049	78	11	,	,	PUNCT
ejpam-7049	78	12	τ	τ	PROPN
ejpam-7049	78	13	,	,	PUNCT
ejpam-7049	78	14	i	i	PROPN
ejpam-7049	78	15	)	)	PUNCT
ejpam-7049	78	16	is	be	AUX
ejpam-7049	78	17	said	say	VERB
ejpam-7049	78	18	to	to	PART
ejpam-7049	78	19	be	be	AUX
ejpam-7049	78	20	r	r	NOUN
ejpam-7049	78	21	-	-	PUNCT
ejpam-7049	78	22	i	i	PRON
ejpam-7049	78	23	⋆-open	⋆-open	VERB
ejpam-7049	79	1	[	[	X
ejpam-7049	79	2	9	9	NUM
ejpam-7049	79	3	]	]	PUNCT
ejpam-7049	79	4	(	(	PUNCT
ejpam-7049	79	5	resp	resp	NOUN
ejpam-7049	79	6	.	.	PUNCT
ejpam-7049	80	1	i	i	PRON
ejpam-7049	80	2	⋆preopen	⋆preopen	VERB
ejpam-7049	81	1	[	[	X
ejpam-7049	81	2	9	9	NUM
ejpam-7049	81	3	]	]	PUNCT
ejpam-7049	81	4	,	,	PUNCT
ejpam-7049	81	5	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7049	81	6	-	-	ADJ
ejpam-7049	81	7	open	open	ADJ
ejpam-7049	81	8	[	[	X
ejpam-7049	81	9	23	23	NUM
ejpam-7049	81	10	]	]	PUNCT
ejpam-7049	81	11	(	(	PUNCT
ejpam-7049	81	12	semi	semi	NOUN
ejpam-7049	81	13	-	-	ADJ
ejpam-7049	81	14	i	i	PRON
ejpam-7049	81	15	⋆-open	⋆-open	VERB
ejpam-7049	82	1	[	[	X
ejpam-7049	82	2	15	15	NUM
ejpam-7049	82	3	]	]	NUM
ejpam-7049	82	4	)	)	PUNCT
ejpam-7049	82	5	,	,	PUNCT
ejpam-7049	82	6	τ⋆-β	τ⋆-β	PROPN
ejpam-7049	82	7	-	-	VERB
ejpam-7049	82	8	open	open	ADJ
ejpam-7049	82	9	[	[	X
ejpam-7049	82	10	23	23	NUM
ejpam-7049	82	11	]	]	PUNCT
ejpam-7049	82	12	(	(	PUNCT
ejpam-7049	82	13	semi	semi	NOUN
ejpam-7049	82	14	-	-	VERB
ejpam-7049	82	15	i	i	PRON
ejpam-7049	82	16	⋆-preopen	⋆-preopen	VERB
ejpam-7049	83	1	[	[	X
ejpam-7049	83	2	15	15	NUM
ejpam-7049	83	3	]	]	NUM
ejpam-7049	83	4	)	)	PUNCT
ejpam-7049	83	5	)	)	PUNCT
ejpam-7049	84	1	if	if	SCONJ
ejpam-7049	84	2	a	a	DET
ejpam-7049	84	3	=	=	PUNCT
ejpam-7049	84	4	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7049	84	5	)	)	PUNCT
ejpam-7049	84	6	)	)	PUNCT
ejpam-7049	85	1	(	(	PUNCT
ejpam-7049	85	2	resp	resp	NOUN
ejpam-7049	85	3	.	.	PUNCT
ejpam-7049	86	1	a	a	DET
ejpam-7049	86	2	⊆	⊆	NUM
ejpam-7049	86	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7049	86	4	)	)	PUNCT
ejpam-7049	86	5	)	)	PUNCT
ejpam-7049	86	6	,	,	PUNCT
ejpam-7049	86	7	a	a	DET
ejpam-7049	86	8	⊆	⊆	NUM
ejpam-7049	86	9	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7049	86	10	)	)	PUNCT
ejpam-7049	86	11	)	)	PUNCT
ejpam-7049	86	12	,	,	PUNCT
ejpam-7049	86	13	a	a	DET
ejpam-7049	86	14	⊆	⊆	NUM
ejpam-7049	86	15	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7049	86	16	)	)	PUNCT
ejpam-7049	86	17	)	)	PUNCT
ejpam-7049	86	18	)	)	PUNCT
ejpam-7049	86	19	)	)	PUNCT
ejpam-7049	86	20	.	.	PUNCT
ejpam-7049	87	1	the	the	DET
ejpam-7049	87	2	complement	complement	NOUN
ejpam-7049	87	3	of	of	ADP
ejpam-7049	87	4	a	a	DET
ejpam-7049	87	5	r	r	NOUN
ejpam-7049	87	6	-	-	PUNCT
ejpam-7049	87	7	i	i	PRON
ejpam-7049	87	8	⋆-open	⋆-open	VERB
ejpam-7049	87	9	(	(	PUNCT
ejpam-7049	87	10	resp	resp	NOUN
ejpam-7049	87	11	.	.	PUNCT
ejpam-7049	88	1	i	i	PRON
ejpam-7049	88	2	⋆-preopen	⋆-preopen	VERB
ejpam-7049	88	3	,	,	PUNCT
ejpam-7049	88	4	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7049	88	5	-	-	ADJ
ejpam-7049	88	6	open	open	ADJ
ejpam-7049	88	7	,	,	PUNCT
ejpam-7049	88	8	τ⋆-β	τ⋆-β	ADJ
ejpam-7049	88	9	-	-	ADJ
ejpam-7049	88	10	open	open	ADJ
ejpam-7049	88	11	)	)	PUNCT
ejpam-7049	88	12	set	set	NOUN
ejpam-7049	88	13	is	be	AUX
ejpam-7049	88	14	said	say	VERB
ejpam-7049	88	15	to	to	PART
ejpam-7049	88	16	be	be	AUX
ejpam-7049	88	17	r	r	NOUN
ejpam-7049	88	18	-	-	PUNCT
ejpam-7049	88	19	i	i	PRON
ejpam-7049	88	20	⋆-closed	⋆-close	VERB
ejpam-7049	88	21	(	(	PUNCT
ejpam-7049	88	22	resp	resp	NOUN
ejpam-7049	88	23	.	.	PUNCT
ejpam-7049	89	1	i	i	PRON
ejpam-7049	89	2	⋆-preclosed	⋆-preclose	VERB
ejpam-7049	89	3	,	,	PUNCT
ejpam-7049	89	4	τ⋆-semi	τ⋆-semi	NOUN
ejpam-7049	89	5	-	-	ADJ
ejpam-7049	89	6	closed	closed	ADJ
ejpam-7049	89	7	,	,	PUNCT
ejpam-7049	89	8	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	89	9	-	-	ADJ
ejpam-7049	89	10	closed	closed	ADJ
ejpam-7049	89	11	)	)	PUNCT
ejpam-7049	89	12	.	.	PUNCT
ejpam-7049	90	1	for	for	ADP
ejpam-7049	90	2	a	a	DET
ejpam-7049	90	3	subset	subset	NOUN
ejpam-7049	90	4	a	a	PRON
ejpam-7049	90	5	of	of	ADP
ejpam-7049	90	6	an	an	DET
ejpam-7049	90	7	ideal	ideal	ADJ
ejpam-7049	90	8	topological	topological	ADJ
ejpam-7049	90	9	space	space	NOUN
ejpam-7049	90	10	(	(	PUNCT
ejpam-7049	90	11	x	x	X
ejpam-7049	90	12	,	,	PUNCT
ejpam-7049	90	13	τ	τ	PROPN
ejpam-7049	90	14	,	,	PUNCT
ejpam-7049	90	15	i	i	NOUN
ejpam-7049	90	16	)	)	PUNCT
ejpam-7049	90	17	,	,	PUNCT
ejpam-7049	90	18	the	the	DET
ejpam-7049	90	19	intersection	intersection	NOUN
ejpam-7049	90	20	of	of	ADP
ejpam-7049	90	21	all	all	PRON
ejpam-7049	90	22	semi	semi	NOUN
ejpam-7049	90	23	-	-	ADJ
ejpam-7049	90	24	i	i	PRON
ejpam-7049	90	25	⋆-closed	⋆-close	VERB
ejpam-7049	90	26	sets	set	NOUN
ejpam-7049	90	27	containing	contain	VERB
ejpam-7049	90	28	a	a	PRON
ejpam-7049	90	29	is	be	AUX
ejpam-7049	90	30	called	call	VERB
ejpam-7049	90	31	the	the	DET
ejpam-7049	90	32	semi	semi	NOUN
ejpam-7049	90	33	-	-	ADJ
ejpam-7049	90	34	i	i	PRON
ejpam-7049	90	35	⋆-closure	⋆-closure	NOUN
ejpam-7049	91	1	[	[	X
ejpam-7049	91	2	15	15	NUM
ejpam-7049	91	3	]	]	PUNCT
ejpam-7049	91	4	of	of	ADP
ejpam-7049	91	5	a	a	PRON
ejpam-7049	91	6	and	and	CCONJ
ejpam-7049	91	7	is	be	AUX
ejpam-7049	91	8	denoted	denote	VERB
ejpam-7049	91	9	by	by	ADP
ejpam-7049	91	10	scl⋆(a	scl⋆(a	NOUN
ejpam-7049	91	11	)	)	PUNCT
ejpam-7049	91	12	(	(	PUNCT
ejpam-7049	91	13	scli	scli	PROPN
ejpam-7049	91	14	⋆(a	⋆(a	PRON
ejpam-7049	91	15	)	)	PUNCT
ejpam-7049	92	1	[	[	X
ejpam-7049	92	2	15	15	NUM
ejpam-7049	92	3	]	]	NUM
ejpam-7049	92	4	)	)	PUNCT
ejpam-7049	92	5	.	.	PUNCT
ejpam-7049	93	1	the	the	DET
ejpam-7049	93	2	union	union	NOUN
ejpam-7049	93	3	of	of	ADP
ejpam-7049	93	4	all	all	PRON
ejpam-7049	93	5	semi	semi	ADJ
ejpam-7049	93	6	-	-	ADJ
ejpam-7049	93	7	i	i	PRON
ejpam-7049	93	8	⋆-open	⋆-open	ADJ
ejpam-7049	93	9	sets	set	NOUN
ejpam-7049	93	10	contained	contain	VERB
ejpam-7049	93	11	in	in	ADP
ejpam-7049	93	12	a	a	PRON
ejpam-7049	93	13	is	be	AUX
ejpam-7049	93	14	called	call	VERB
ejpam-7049	93	15	the	the	DET
ejpam-7049	93	16	semi	semi	NOUN
ejpam-7049	93	17	-	-	ADJ
ejpam-7049	93	18	i	i	PRON
ejpam-7049	93	19	⋆-interior	⋆-interior	PUNCT
ejpam-7049	94	1	[	[	X
ejpam-7049	94	2	15	15	NUM
ejpam-7049	94	3	]	]	PUNCT
ejpam-7049	94	4	of	of	ADP
ejpam-7049	94	5	a	a	PRON
ejpam-7049	94	6	and	and	CCONJ
ejpam-7049	94	7	is	be	AUX
ejpam-7049	94	8	denoted	denote	VERB
ejpam-7049	94	9	by	by	ADP
ejpam-7049	94	10	sint⋆(a	sint⋆(a	PROPN
ejpam-7049	94	11	)	)	PUNCT
ejpam-7049	95	1	(	(	PUNCT
ejpam-7049	95	2	sinti	sinti	PROPN
ejpam-7049	95	3	⋆(a	⋆(a	NOUN
ejpam-7049	95	4	)	)	PUNCT
ejpam-7049	96	1	[	[	X
ejpam-7049	96	2	15	15	NUM
ejpam-7049	96	3	]	]	NUM
ejpam-7049	96	4	)	)	PUNCT
ejpam-7049	96	5	.	.	PUNCT
ejpam-7049	97	1	the	the	DET
ejpam-7049	97	2	intersection	intersection	NOUN
ejpam-7049	97	3	of	of	ADP
ejpam-7049	97	4	all	all	PRON
ejpam-7049	97	5	β	β	NOUN
ejpam-7049	97	6	-	-	PUNCT
ejpam-7049	97	7	i	i	PRON
ejpam-7049	97	8	⋆-closed	⋆-close	VERB
ejpam-7049	97	9	sets	set	NOUN
ejpam-7049	97	10	containing	contain	VERB
ejpam-7049	97	11	a	a	PRON
ejpam-7049	97	12	is	be	AUX
ejpam-7049	97	13	called	call	VERB
ejpam-7049	97	14	the	the	DET
ejpam-7049	97	15	β	β	NOUN
ejpam-7049	97	16	-	-	NOUN
ejpam-7049	97	17	i	i	PRON
ejpam-7049	97	18	⋆-closure	⋆-closure	ADJ
ejpam-7049	97	19	of	of	ADP
ejpam-7049	97	20	a	a	PRON
ejpam-7049	97	21	and	and	CCONJ
ejpam-7049	97	22	is	be	AUX
ejpam-7049	97	23	denoted	denote	VERB
ejpam-7049	97	24	by	by	ADP
ejpam-7049	97	25	βcl⋆(a	βcl⋆(a	NOUN
ejpam-7049	97	26	)	)	PUNCT
ejpam-7049	97	27	.	.	PUNCT
ejpam-7049	98	1	the	the	DET
ejpam-7049	98	2	union	union	NOUN
ejpam-7049	98	3	of	of	ADP
ejpam-7049	98	4	all	all	PRON
ejpam-7049	98	5	β	β	NOUN
ejpam-7049	98	6	-	-	ADJ
ejpam-7049	98	7	i	i	PRON
ejpam-7049	98	8	⋆-open	⋆-open	ADJ
ejpam-7049	98	9	sets	set	NOUN
ejpam-7049	98	10	contained	contain	VERB
ejpam-7049	98	11	in	in	ADP
ejpam-7049	98	12	a	a	PRON
ejpam-7049	98	13	is	be	AUX
ejpam-7049	98	14	called	call	VERB
ejpam-7049	98	15	the	the	DET
ejpam-7049	98	16	β	β	NOUN
ejpam-7049	98	17	-	-	PUNCT
ejpam-7049	98	18	i	i	PRON
ejpam-7049	98	19	⋆-interior	⋆-interior	NOUN
ejpam-7049	98	20	of	of	ADP
ejpam-7049	98	21	a	a	PRON
ejpam-7049	98	22	and	and	CCONJ
ejpam-7049	98	23	is	be	AUX
ejpam-7049	98	24	denoted	denote	VERB
ejpam-7049	98	25	by	by	ADP
ejpam-7049	98	26	βint⋆(a	βint⋆(a	NOUN
ejpam-7049	98	27	)	)	PUNCT
ejpam-7049	98	28	.	.	PUNCT
ejpam-7049	99	1	lemma	lemma	PROPN
ejpam-7049	99	2	3	3	X
ejpam-7049	99	3	.	.	X
ejpam-7049	100	1	for	for	ADP
ejpam-7049	100	2	a	a	DET
ejpam-7049	100	3	subset	subset	NOUN
ejpam-7049	100	4	a	a	PRON
ejpam-7049	100	5	of	of	ADP
ejpam-7049	100	6	an	an	DET
ejpam-7049	100	7	ideal	ideal	ADJ
ejpam-7049	100	8	topological	topological	ADJ
ejpam-7049	100	9	space	space	NOUN
ejpam-7049	100	10	(	(	PUNCT
ejpam-7049	100	11	x	x	X
ejpam-7049	100	12	,	,	PUNCT
ejpam-7049	100	13	τ	τ	PROPN
ejpam-7049	100	14	,	,	PUNCT
ejpam-7049	100	15	i	i	NOUN
ejpam-7049	100	16	)	)	PUNCT
ejpam-7049	100	17	,	,	PUNCT
ejpam-7049	100	18	the	the	DET
ejpam-7049	100	19	following	follow	VERB
ejpam-7049	100	20	properties	property	NOUN
ejpam-7049	100	21	hold	hold	VERB
ejpam-7049	100	22	:	:	PUNCT
ejpam-7049	100	23	(	(	PUNCT
ejpam-7049	100	24	1	1	X
ejpam-7049	100	25	)	)	PUNCT
ejpam-7049	100	26	scl⋆(a	scl⋆(a	NUM
ejpam-7049	100	27	)	)	PUNCT
ejpam-7049	100	28	=	=	PUNCT
ejpam-7049	101	1	a	a	DET
ejpam-7049	101	2	∪	∪	ADJ
ejpam-7049	101	3	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-7049	101	4	)	)	PUNCT
ejpam-7049	101	5	)	)	PUNCT
ejpam-7049	102	1	[	[	X
ejpam-7049	102	2	15	15	NUM
ejpam-7049	102	3	]	]	PUNCT
ejpam-7049	102	4	.	.	PUNCT
ejpam-7049	103	1	(	(	PUNCT
ejpam-7049	103	2	2	2	X
ejpam-7049	103	3	)	)	PUNCT
ejpam-7049	103	4	sint⋆(a	sint⋆(a	PROPN
ejpam-7049	103	5	)	)	PUNCT
ejpam-7049	104	1	=	=	PUNCT
ejpam-7049	104	2	a	a	DET
ejpam-7049	104	3	∩	∩	ADJ
ejpam-7049	104	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-7049	104	5	)	)	PUNCT
ejpam-7049	104	6	)	)	PUNCT
ejpam-7049	105	1	[	[	X
ejpam-7049	105	2	15	15	NUM
ejpam-7049	105	3	]	]	PUNCT
ejpam-7049	105	4	.	.	PUNCT
ejpam-7049	106	1	(	(	PUNCT
ejpam-7049	106	2	3	3	NUM
ejpam-7049	106	3	)	)	PUNCT
ejpam-7049	106	4	βcl⋆(a	βcl⋆(a	NUM
ejpam-7049	106	5	)	)	PUNCT
ejpam-7049	107	1	=	=	PUNCT
ejpam-7049	107	2	a	a	DET
ejpam-7049	107	3	∪	∪	ADJ
ejpam-7049	107	4	int⋆(cl⋆(int⋆(a	int⋆(cl⋆(int⋆(a	NOUN
ejpam-7049	107	5	)	)	PUNCT
ejpam-7049	107	6	)	)	PUNCT
ejpam-7049	107	7	)	)	PUNCT
ejpam-7049	107	8	.	.	PUNCT
ejpam-7049	108	1	n.	n.	PROPN
ejpam-7049	108	2	srisarakham	srisarakham	PROPN
ejpam-7049	108	3	,	,	PUNCT
ejpam-7049	108	4	a.	a.	PROPN
ejpam-7049	108	5	sama	sama	PROPN
ejpam-7049	108	6	-	-	PUNCT
ejpam-7049	108	7	ae	ae	PROPN
ejpam-7049	108	8	,	,	PUNCT
ejpam-7049	108	9	c.	c.	PROPN
ejpam-7049	108	10	boonpok	boonpok	PROPN
ejpam-7049	108	11	/	/	SYM
ejpam-7049	108	12	eur	eur	PROPN
ejpam-7049	108	13	.	.	PUNCT
ejpam-7049	109	1	j.	j.	PROPN
ejpam-7049	109	2	pure	pure	PROPN
ejpam-7049	109	3	appl	appl	PROPN
ejpam-7049	109	4	.	.	PROPN
ejpam-7049	109	5	math	math	PROPN
ejpam-7049	109	6	,	,	PUNCT
ejpam-7049	109	7	18	18	NUM
ejpam-7049	109	8	(	(	PUNCT
ejpam-7049	109	9	4	4	NUM
ejpam-7049	109	10	)	)	PUNCT
ejpam-7049	109	11	(	(	PUNCT
ejpam-7049	109	12	2025	2025	NUM
ejpam-7049	109	13	)	)	PUNCT
ejpam-7049	109	14	,	,	PUNCT
ejpam-7049	109	15	7049	7049	NUM
ejpam-7049	109	16	4	4	NUM
ejpam-7049	109	17	of	of	ADP
ejpam-7049	109	18	11	11	NUM
ejpam-7049	109	19	(	(	PUNCT
ejpam-7049	109	20	4	4	NUM
ejpam-7049	109	21	)	)	PUNCT
ejpam-7049	109	22	βint⋆(a	βint⋆(a	NUM
ejpam-7049	109	23	)	)	PUNCT
ejpam-7049	110	1	=	=	PUNCT
ejpam-7049	110	2	a	a	DET
ejpam-7049	110	3	∩	∩	NOUN
ejpam-7049	110	4	cl⋆(int⋆(cl⋆(a	cl⋆(int⋆(cl⋆(a	NOUN
ejpam-7049	110	5	)	)	PUNCT
ejpam-7049	110	6	)	)	PUNCT
ejpam-7049	110	7	)	)	PUNCT
ejpam-7049	110	8	.	.	PUNCT
ejpam-7049	111	1	by	by	ADP
ejpam-7049	111	2	a	a	DET
ejpam-7049	111	3	multifunction	multifunction	NOUN
ejpam-7049	111	4	f	f	NOUN
ejpam-7049	111	5	:	:	PUNCT
ejpam-7049	111	6	x	x	X
ejpam-7049	111	7	→	→	SYM
ejpam-7049	111	8	y	y	PROPN
ejpam-7049	111	9	,	,	PUNCT
ejpam-7049	111	10	we	we	PRON
ejpam-7049	111	11	mean	mean	VERB
ejpam-7049	111	12	a	a	DET
ejpam-7049	111	13	point	point	NOUN
ejpam-7049	111	14	-	-	PUNCT
ejpam-7049	111	15	to	to	ADP
ejpam-7049	111	16	-	-	PUNCT
ejpam-7049	111	17	set	set	VERB
ejpam-7049	111	18	correspondence	correspondence	NOUN
ejpam-7049	111	19	from	from	ADP
ejpam-7049	111	20	x	x	PUNCT
ejpam-7049	111	21	into	into	ADP
ejpam-7049	111	22	y	y	PROPN
ejpam-7049	111	23	,	,	PUNCT
ejpam-7049	111	24	and	and	CCONJ
ejpam-7049	111	25	we	we	PRON
ejpam-7049	111	26	always	always	ADV
ejpam-7049	111	27	assume	assume	VERB
ejpam-7049	111	28	that	that	SCONJ
ejpam-7049	111	29	f	f	PROPN
ejpam-7049	111	30	(	(	PUNCT
ejpam-7049	111	31	x	x	X
ejpam-7049	111	32	)	)	PUNCT
ejpam-7049	111	33	̸=	̸=	NOUN
ejpam-7049	111	34	∅	∅	NOUN
ejpam-7049	111	35	for	for	ADP
ejpam-7049	111	36	all	all	PRON
ejpam-7049	111	37	x	x	SYM
ejpam-7049	111	38	∈	∈	ADJ
ejpam-7049	111	39	x.	x.	NOUN
ejpam-7049	111	40	for	for	ADP
ejpam-7049	111	41	a	a	DET
ejpam-7049	111	42	multifunction	multifunction	NOUN
ejpam-7049	111	43	f	f	NOUN
ejpam-7049	111	44	:	:	PUNCT
ejpam-7049	111	45	x	x	X
ejpam-7049	111	46	→	→	SYM
ejpam-7049	111	47	y	y	PROPN
ejpam-7049	111	48	,	,	PUNCT
ejpam-7049	111	49	we	we	PRON
ejpam-7049	111	50	shall	shall	AUX
ejpam-7049	111	51	denote	denote	VERB
ejpam-7049	111	52	the	the	DET
ejpam-7049	111	53	upper	upper	ADJ
ejpam-7049	111	54	and	and	CCONJ
ejpam-7049	111	55	lower	low	ADJ
ejpam-7049	111	56	inverse	inverse	NOUN
ejpam-7049	111	57	of	of	ADP
ejpam-7049	111	58	a	a	DET
ejpam-7049	111	59	set	set	NOUN
ejpam-7049	111	60	b	b	PROPN
ejpam-7049	111	61	of	of	ADP
ejpam-7049	111	62	y	y	PROPN
ejpam-7049	111	63	by	by	ADP
ejpam-7049	111	64	f+(b	f+(b	NOUN
ejpam-7049	111	65	)	)	PUNCT
ejpam-7049	111	66	and	and	CCONJ
ejpam-7049	111	67	f−(b	f−(b	NOUN
ejpam-7049	111	68	)	)	PUNCT
ejpam-7049	111	69	,	,	PUNCT
ejpam-7049	111	70	respectively	respectively	ADV
ejpam-7049	111	71	,	,	PUNCT
ejpam-7049	111	72	that	that	ADV
ejpam-7049	111	73	is	is	ADV
ejpam-7049	111	74	,	,	PUNCT
ejpam-7049	111	75	f+(b	f+(b	NOUN
ejpam-7049	111	76	)	)	PUNCT
ejpam-7049	111	77	=	=	PRON
ejpam-7049	112	1	{	{	PUNCT
ejpam-7049	112	2	x	x	PUNCT
ejpam-7049	112	3	∈	∈	PROPN
ejpam-7049	112	4	x	x	INTJ
ejpam-7049	113	1	|	|	NOUN
ejpam-7049	113	2	f	f	X
ejpam-7049	113	3	(	(	PUNCT
ejpam-7049	113	4	x	x	NOUN
ejpam-7049	113	5	)	)	PUNCT
ejpam-7049	113	6	⊆	⊆	NUM
ejpam-7049	113	7	b	b	NOUN
ejpam-7049	113	8	}	}	PUNCT
ejpam-7049	113	9	and	and	CCONJ
ejpam-7049	113	10	f−(b	f−(b	PROPN
ejpam-7049	113	11	)	)	PUNCT
ejpam-7049	113	12	=	=	PRON
ejpam-7049	114	1	{	{	PUNCT
ejpam-7049	114	2	x	x	PUNCT
ejpam-7049	114	3	∈	∈	PROPN
ejpam-7049	114	4	x	x	INTJ
ejpam-7049	115	1	|	|	NOUN
ejpam-7049	115	2	f	f	X
ejpam-7049	115	3	(	(	PUNCT
ejpam-7049	115	4	x	x	NOUN
ejpam-7049	115	5	)	)	PUNCT
ejpam-7049	115	6	∩	∩	NOUN
ejpam-7049	115	7	b	b	PROPN
ejpam-7049	115	8	̸=	̸=	PROPN
ejpam-7049	115	9	∅	∅	NOUN
ejpam-7049	115	10	}	}	PUNCT
ejpam-7049	115	11	.	.	PUNCT
ejpam-7049	116	1	in	in	ADP
ejpam-7049	116	2	particular	particular	ADJ
ejpam-7049	116	3	,	,	PUNCT
ejpam-7049	116	4	f−(y	f−(y	NOUN
ejpam-7049	116	5	)	)	PUNCT
ejpam-7049	116	6	=	=	SYM
ejpam-7049	117	1	{	{	PUNCT
ejpam-7049	117	2	x	x	PUNCT
ejpam-7049	117	3	∈	∈	PROPN
ejpam-7049	117	4	x	x	INTJ
ejpam-7049	118	1	|	|	ADV
ejpam-7049	118	2	y	y	PROPN
ejpam-7049	118	3	∈	∈	PROPN
ejpam-7049	118	4	f	f	X
ejpam-7049	118	5	(	(	PUNCT
ejpam-7049	118	6	x	x	NOUN
ejpam-7049	118	7	)	)	PUNCT
ejpam-7049	118	8	}	}	PUNCT
ejpam-7049	118	9	for	for	ADP
ejpam-7049	118	10	each	each	DET
ejpam-7049	118	11	point	point	NOUN
ejpam-7049	118	12	y	y	PROPN
ejpam-7049	118	13	∈	∈	PROPN
ejpam-7049	118	14	y	y	PROPN
ejpam-7049	118	15	.	.	PUNCT
ejpam-7049	119	1	for	for	ADP
ejpam-7049	119	2	each	each	DET
ejpam-7049	119	3	a	a	DET
ejpam-7049	119	4	⊆	⊆	NUM
ejpam-7049	119	5	x	x	SYM
ejpam-7049	119	6	,	,	PUNCT
ejpam-7049	119	7	f	f	PROPN
ejpam-7049	119	8	(	(	PUNCT
ejpam-7049	119	9	a	a	NOUN
ejpam-7049	119	10	)	)	PUNCT
ejpam-7049	119	11	=	=	SYM
ejpam-7049	119	12	∪x∈af	∪x∈af	NOUN
ejpam-7049	119	13	(	(	PUNCT
ejpam-7049	119	14	x	x	NOUN
ejpam-7049	119	15	)	)	PUNCT
ejpam-7049	119	16	.	.	PUNCT
ejpam-7049	120	1	3	3	X
ejpam-7049	120	2	.	.	X
ejpam-7049	120	3	upper	upper	ADJ
ejpam-7049	120	4	and	and	CCONJ
ejpam-7049	120	5	lower	low	ADJ
ejpam-7049	120	6	almost	almost	ADV
ejpam-7049	120	7	τ	τ	NOUN
ejpam-7049	120	8	⋆β(σ1	⋆β(σ1	PUNCT
ejpam-7049	120	9	,	,	PUNCT
ejpam-7049	120	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	120	11	multifunctions	multifunction	NOUN
ejpam-7049	120	12	in	in	ADP
ejpam-7049	120	13	this	this	DET
ejpam-7049	120	14	section	section	NOUN
ejpam-7049	120	15	,	,	PUNCT
ejpam-7049	120	16	we	we	PRON
ejpam-7049	120	17	introduce	introduce	VERB
ejpam-7049	120	18	the	the	DET
ejpam-7049	120	19	notions	notion	NOUN
ejpam-7049	120	20	of	of	ADP
ejpam-7049	120	21	upper	upper	ADJ
ejpam-7049	120	22	almost	almost	ADV
ejpam-7049	120	23	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	120	24	,	,	PUNCT
ejpam-7049	120	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	120	26	multifunctions	multifunction	NOUN
ejpam-7049	120	27	and	and	CCONJ
ejpam-7049	120	28	lower	low	ADJ
ejpam-7049	120	29	almost	almost	ADV
ejpam-7049	120	30	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	120	31	,	,	PUNCT
ejpam-7049	120	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	120	33	multifunctions	multifunction	NOUN
ejpam-7049	120	34	.	.	PUNCT
ejpam-7049	121	1	moreover	moreover	ADV
ejpam-7049	121	2	,	,	PUNCT
ejpam-7049	121	3	several	several	ADJ
ejpam-7049	121	4	characterizations	characterization	NOUN
ejpam-7049	121	5	of	of	ADP
ejpam-7049	121	6	upper	upper	ADJ
ejpam-7049	121	7	almost	almost	ADV
ejpam-7049	121	8	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	121	9	,	,	PUNCT
ejpam-7049	121	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	121	11	multifunctions	multifunction	NOUN
ejpam-7049	121	12	and	and	CCONJ
ejpam-7049	121	13	lower	low	ADJ
ejpam-7049	121	14	almost	almost	ADV
ejpam-7049	121	15	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	121	16	,	,	PUNCT
ejpam-7049	121	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	121	18	multifunctions	multifunction	NOUN
ejpam-7049	121	19	discussed	discuss	VERB
ejpam-7049	121	20	.	.	PUNCT
ejpam-7049	122	1	definition	definition	NOUN
ejpam-7049	122	2	1	1	NUM
ejpam-7049	122	3	.	.	PUNCT
ejpam-7049	123	1	a	a	DET
ejpam-7049	123	2	multifunction	multifunction	NOUN
ejpam-7049	123	3	f	f	NOUN
ejpam-7049	123	4	:	:	PUNCT
ejpam-7049	123	5	(	(	PUNCT
ejpam-7049	123	6	x	x	X
ejpam-7049	123	7	,	,	PUNCT
ejpam-7049	123	8	τ	τ	PROPN
ejpam-7049	123	9	,	,	PUNCT
ejpam-7049	123	10	i	i	NOUN
ejpam-7049	123	11	)	)	PUNCT
ejpam-7049	123	12	→	→	PUNCT
ejpam-7049	123	13	(	(	PUNCT
ejpam-7049	123	14	y	y	PROPN
ejpam-7049	123	15	,	,	PUNCT
ejpam-7049	123	16	σ1	σ1	PROPN
ejpam-7049	123	17	,	,	PUNCT
ejpam-7049	123	18	σ2	σ2	PROPN
ejpam-7049	123	19	)	)	PUNCT
ejpam-7049	123	20	is	be	AUX
ejpam-7049	123	21	said	say	VERB
ejpam-7049	123	22	to	to	PART
ejpam-7049	123	23	be	be	AUX
ejpam-7049	123	24	upper	upper	ADJ
ejpam-7049	123	25	almost	almost	ADV
ejpam-7049	123	26	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	123	27	,	,	PUNCT
ejpam-7049	123	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	123	29	at	at	ADP
ejpam-7049	123	30	a	a	DET
ejpam-7049	123	31	point	point	NOUN
ejpam-7049	123	32	x	x	PUNCT
ejpam-7049	123	33	of	of	ADP
ejpam-7049	123	34	x	x	PRON
ejpam-7049	123	35	if	if	SCONJ
ejpam-7049	123	36	for	for	ADP
ejpam-7049	123	37	each	each	DET
ejpam-7049	123	38	σ1σ2	σ1σ2	VERB
ejpam-7049	123	39	-	-	ADJ
ejpam-7049	123	40	open	open	ADJ
ejpam-7049	123	41	set	set	NOUN
ejpam-7049	123	42	v	v	NOUN
ejpam-7049	123	43	of	of	ADP
ejpam-7049	123	44	y	y	PRON
ejpam-7049	123	45	such	such	ADJ
ejpam-7049	123	46	that	that	SCONJ
ejpam-7049	123	47	f	f	PROPN
ejpam-7049	123	48	(	(	PUNCT
ejpam-7049	123	49	x	x	X
ejpam-7049	123	50	)	)	PUNCT
ejpam-7049	123	51	⊆	⊆	NUM
ejpam-7049	123	52	v	v	NOUN
ejpam-7049	123	53	,	,	PUNCT
ejpam-7049	123	54	there	there	PRON
ejpam-7049	123	55	exists	exist	VERB
ejpam-7049	123	56	a	a	DET
ejpam-7049	123	57	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	123	58	-	-	ADJ
ejpam-7049	123	59	open	open	ADJ
ejpam-7049	123	60	set	set	NOUN
ejpam-7049	123	61	u	u	NOUN
ejpam-7049	123	62	of	of	ADP
ejpam-7049	123	63	x	x	PUNCT
ejpam-7049	123	64	containing	contain	VERB
ejpam-7049	123	65	x	x	PUNCT
ejpam-7049	123	66	such	such	ADJ
ejpam-7049	123	67	that	that	SCONJ
ejpam-7049	123	68	f	f	PROPN
ejpam-7049	123	69	(	(	PUNCT
ejpam-7049	123	70	u	u	NOUN
ejpam-7049	123	71	)	)	PUNCT
ejpam-7049	123	72	⊆	⊆	NUM
ejpam-7049	123	73	σ1σ2	σ1σ2	X
ejpam-7049	123	74	-	-	PUNCT
ejpam-7049	123	75	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	123	76	-	-	PUNCT
ejpam-7049	123	77	cl(v	cl(v	NOUN
ejpam-7049	123	78	)	)	PUNCT
ejpam-7049	123	79	)	)	PUNCT
ejpam-7049	123	80	.	.	PUNCT
ejpam-7049	124	1	a	a	DET
ejpam-7049	124	2	multifunction	multifunction	NOUN
ejpam-7049	124	3	f	f	NOUN
ejpam-7049	124	4	:	:	PUNCT
ejpam-7049	124	5	(	(	PUNCT
ejpam-7049	124	6	x	x	X
ejpam-7049	124	7	,	,	PUNCT
ejpam-7049	124	8	τ	τ	PROPN
ejpam-7049	124	9	,	,	PUNCT
ejpam-7049	124	10	i	i	NOUN
ejpam-7049	124	11	)	)	PUNCT
ejpam-7049	124	12	→	→	PUNCT
ejpam-7049	124	13	(	(	PUNCT
ejpam-7049	124	14	y	y	PROPN
ejpam-7049	124	15	,	,	PUNCT
ejpam-7049	124	16	σ1	σ1	PROPN
ejpam-7049	124	17	,	,	PUNCT
ejpam-7049	124	18	σ2	σ2	PROPN
ejpam-7049	124	19	)	)	PUNCT
ejpam-7049	124	20	is	be	AUX
ejpam-7049	124	21	said	say	VERB
ejpam-7049	124	22	to	to	PART
ejpam-7049	124	23	be	be	AUX
ejpam-7049	124	24	upper	upper	ADJ
ejpam-7049	124	25	almost	almost	ADV
ejpam-7049	124	26	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	124	27	,	,	PUNCT
ejpam-7049	124	28	σ2)continuous	σ2)continuous	ADJ
ejpam-7049	124	29	if	if	SCONJ
ejpam-7049	124	30	f	f	PROPN
ejpam-7049	124	31	is	be	AUX
ejpam-7049	124	32	upper	upper	ADJ
ejpam-7049	124	33	almost	almost	ADV
ejpam-7049	124	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	124	35	,	,	PUNCT
ejpam-7049	124	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	124	37	at	at	ADP
ejpam-7049	124	38	each	each	DET
ejpam-7049	124	39	point	point	NOUN
ejpam-7049	124	40	of	of	ADP
ejpam-7049	124	41	x.	x.	NOUN
ejpam-7049	124	42	theorem	theorem	VERB
ejpam-7049	124	43	1	1	NUM
ejpam-7049	124	44	.	.	PUNCT
ejpam-7049	125	1	a	a	DET
ejpam-7049	125	2	multifunction	multifunction	NOUN
ejpam-7049	125	3	f	f	NOUN
ejpam-7049	125	4	:	:	PUNCT
ejpam-7049	125	5	(	(	PUNCT
ejpam-7049	125	6	x	x	X
ejpam-7049	125	7	,	,	PUNCT
ejpam-7049	125	8	τ	τ	PROPN
ejpam-7049	125	9	,	,	PUNCT
ejpam-7049	125	10	i	i	NOUN
ejpam-7049	125	11	)	)	PUNCT
ejpam-7049	125	12	→	→	PUNCT
ejpam-7049	125	13	(	(	PUNCT
ejpam-7049	125	14	y	y	PROPN
ejpam-7049	125	15	,	,	PUNCT
ejpam-7049	125	16	σ1	σ1	PROPN
ejpam-7049	125	17	,	,	PUNCT
ejpam-7049	125	18	σ2	σ2	PROPN
ejpam-7049	125	19	)	)	PUNCT
ejpam-7049	125	20	is	be	AUX
ejpam-7049	125	21	upper	upper	ADJ
ejpam-7049	125	22	almost	almost	ADV
ejpam-7049	125	23	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	125	24	,	,	PUNCT
ejpam-7049	125	25	σ2)continuous	σ2)continuous	ADJ
ejpam-7049	125	26	at	at	ADP
ejpam-7049	125	27	x	x	X
ejpam-7049	125	28	∈	∈	PROPN
ejpam-7049	125	29	x	x	INTJ
ejpam-7049	125	30	if	if	SCONJ
ejpam-7049	125	31	and	and	CCONJ
ejpam-7049	125	32	only	only	ADV
ejpam-7049	125	33	if	if	SCONJ
ejpam-7049	125	34	x	x	SYM
ejpam-7049	125	35	∈	∈	NOUN
ejpam-7049	125	36	βint⋆(f+((σ1	βint⋆(f+((σ1	NOUN
ejpam-7049	125	37	,	,	PUNCT
ejpam-7049	125	38	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	125	39	)	)	PUNCT
ejpam-7049	125	40	)	)	PUNCT
ejpam-7049	125	41	)	)	PUNCT
ejpam-7049	125	42	for	for	ADP
ejpam-7049	125	43	every	every	DET
ejpam-7049	125	44	σ1σ2	σ1σ2	NOUN
ejpam-7049	125	45	-	-	ADJ
ejpam-7049	125	46	open	open	ADJ
ejpam-7049	125	47	set	set	NOUN
ejpam-7049	125	48	v	v	NOUN
ejpam-7049	125	49	of	of	ADP
ejpam-7049	125	50	y	y	PROPN
ejpam-7049	125	51	containing	contain	VERB
ejpam-7049	125	52	f	f	PROPN
ejpam-7049	125	53	(	(	PUNCT
ejpam-7049	125	54	x	x	NOUN
ejpam-7049	125	55	)	)	PUNCT
ejpam-7049	125	56	.	.	PUNCT
ejpam-7049	126	1	proof	proof	NOUN
ejpam-7049	126	2	.	.	PUNCT
ejpam-7049	127	1	let	let	VERB
ejpam-7049	127	2	v	v	PART
ejpam-7049	127	3	be	be	AUX
ejpam-7049	127	4	any	any	DET
ejpam-7049	127	5	σ1σ2	σ1σ2	NOUN
ejpam-7049	127	6	-	-	ADJ
ejpam-7049	127	7	open	open	ADJ
ejpam-7049	127	8	set	set	NOUN
ejpam-7049	127	9	of	of	ADP
ejpam-7049	127	10	y	y	PROPN
ejpam-7049	127	11	containing	contain	VERB
ejpam-7049	127	12	f	f	PROPN
ejpam-7049	127	13	(	(	PUNCT
ejpam-7049	127	14	x	x	NOUN
ejpam-7049	127	15	)	)	PUNCT
ejpam-7049	127	16	.	.	PUNCT
ejpam-7049	128	1	then	then	ADV
ejpam-7049	128	2	,	,	PUNCT
ejpam-7049	128	3	there	there	PRON
ejpam-7049	128	4	exists	exist	VERB
ejpam-7049	128	5	a	a	DET
ejpam-7049	128	6	τ⋆-βopen	τ⋆-βopen	ADJ
ejpam-7049	128	7	set	set	NOUN
ejpam-7049	128	8	u	u	NOUN
ejpam-7049	128	9	of	of	ADP
ejpam-7049	128	10	x	x	PUNCT
ejpam-7049	128	11	containing	contain	VERB
ejpam-7049	128	12	x	x	PUNCT
ejpam-7049	128	13	such	such	ADJ
ejpam-7049	128	14	that	that	SCONJ
ejpam-7049	128	15	f	f	PROPN
ejpam-7049	128	16	(	(	PUNCT
ejpam-7049	128	17	u	u	NOUN
ejpam-7049	128	18	)	)	PUNCT
ejpam-7049	128	19	⊆	⊆	NUM
ejpam-7049	128	20	σ1σ2	σ1σ2	X
ejpam-7049	128	21	-	-	PUNCT
ejpam-7049	128	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	128	23	-	-	PUNCT
ejpam-7049	128	24	cl(v	cl(v	NOUN
ejpam-7049	128	25	)	)	PUNCT
ejpam-7049	128	26	)	)	PUNCT
ejpam-7049	129	1	=	=	SYM
ejpam-7049	129	2	(	(	PUNCT
ejpam-7049	129	3	σ1	σ1	PROPN
ejpam-7049	129	4	,	,	PUNCT
ejpam-7049	129	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	129	6	)	)	PUNCT
ejpam-7049	129	7	;	;	PUNCT
ejpam-7049	129	8	hence	hence	ADV
ejpam-7049	129	9	u	u	NOUN
ejpam-7049	129	10	⊆	⊆	NUM
ejpam-7049	129	11	f+((σ1	f+((σ1	NOUN
ejpam-7049	129	12	,	,	PUNCT
ejpam-7049	129	13	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	129	14	)	)	PUNCT
ejpam-7049	129	15	)	)	PUNCT
ejpam-7049	129	16	.	.	PUNCT
ejpam-7049	130	1	since	since	SCONJ
ejpam-7049	130	2	u	u	NOUN
ejpam-7049	130	3	is	be	AUX
ejpam-7049	130	4	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	130	5	-	-	NOUN
ejpam-7049	130	6	open	open	ADJ
ejpam-7049	130	7	,	,	PUNCT
ejpam-7049	130	8	we	we	PRON
ejpam-7049	130	9	have	have	VERB
ejpam-7049	130	10	x	x	X
ejpam-7049	130	11	∈	∈	PROPN
ejpam-7049	130	12	u	u	NOUN
ejpam-7049	130	13	⊆	⊆	NUM
ejpam-7049	130	14	cl⋆(int⋆(cl⋆(u	cl⋆(int⋆(cl⋆(u	NOUN
ejpam-7049	130	15	)	)	PUNCT
ejpam-7049	130	16	)	)	PUNCT
ejpam-7049	130	17	)	)	PUNCT
ejpam-7049	131	1	⊆	⊆	NUM
ejpam-7049	131	2	cl⋆(int⋆(cl⋆(f+((σ1	cl⋆(int⋆(cl⋆(f+((σ1	NOUN
ejpam-7049	131	3	,	,	PUNCT
ejpam-7049	131	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	131	5	)	)	PUNCT
ejpam-7049	131	6	)	)	PUNCT
ejpam-7049	131	7	)	)	PUNCT
ejpam-7049	131	8	)	)	PUNCT
ejpam-7049	131	9	)	)	PUNCT
ejpam-7049	131	10	.	.	PUNCT
ejpam-7049	132	1	since	since	SCONJ
ejpam-7049	132	2	x	x	PROPN
ejpam-7049	132	3	∈	∈	PROPN
ejpam-7049	132	4	f+(v	f+(v	NOUN
ejpam-7049	132	5	)	)	PUNCT
ejpam-7049	132	6	⊆	⊆	NUM
ejpam-7049	132	7	f+((σ1	f+((σ1	ADJ
ejpam-7049	132	8	,	,	PUNCT
ejpam-7049	132	9	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	132	10	)	)	PUNCT
ejpam-7049	132	11	)	)	PUNCT
ejpam-7049	132	12	and	and	CCONJ
ejpam-7049	132	13	by	by	ADP
ejpam-7049	132	14	lemma	lemma	PROPN
ejpam-7049	132	15	3	3	NUM
ejpam-7049	132	16	,	,	PUNCT
ejpam-7049	132	17	x	x	SYM
ejpam-7049	132	18	∈	∈	NOUN
ejpam-7049	132	19	f+((σ1	f+((σ1	NOUN
ejpam-7049	132	20	,	,	PUNCT
ejpam-7049	132	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	132	22	)	)	PUNCT
ejpam-7049	132	23	)	)	PUNCT
ejpam-7049	132	24	∩	∩	NOUN
ejpam-7049	132	25	cl⋆(int(cl⋆((σ1	cl⋆(int(cl⋆((σ1	NOUN
ejpam-7049	132	26	,	,	PUNCT
ejpam-7049	132	27	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	132	28	)	)	PUNCT
ejpam-7049	132	29	)	)	PUNCT
ejpam-7049	132	30	)	)	PUNCT
ejpam-7049	132	31	)	)	PUNCT
ejpam-7049	132	32	=	=	PUNCT
ejpam-7049	132	33	βint⋆(f+((σ1	βint⋆(f+((σ1	NOUN
ejpam-7049	132	34	,	,	PUNCT
ejpam-7049	132	35	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	132	36	)	)	PUNCT
ejpam-7049	132	37	)	)	PUNCT
ejpam-7049	132	38	)	)	PUNCT
ejpam-7049	132	39	.	.	PUNCT
ejpam-7049	133	1	conversely	conversely	ADV
ejpam-7049	133	2	,	,	PUNCT
ejpam-7049	133	3	let	let	VERB
ejpam-7049	133	4	v	v	PART
ejpam-7049	133	5	be	be	AUX
ejpam-7049	133	6	any	any	DET
ejpam-7049	133	7	σ1σ2	σ1σ2	NOUN
ejpam-7049	133	8	-	-	ADJ
ejpam-7049	133	9	open	open	ADJ
ejpam-7049	133	10	set	set	NOUN
ejpam-7049	133	11	of	of	ADP
ejpam-7049	133	12	y	y	PROPN
ejpam-7049	133	13	containing	contain	VERB
ejpam-7049	133	14	f	f	PROPN
ejpam-7049	133	15	(	(	PUNCT
ejpam-7049	133	16	x	x	NOUN
ejpam-7049	133	17	)	)	PUNCT
ejpam-7049	133	18	.	.	PUNCT
ejpam-7049	134	1	then	then	ADV
ejpam-7049	134	2	,	,	PUNCT
ejpam-7049	134	3	we	we	PRON
ejpam-7049	134	4	have	have	VERB
ejpam-7049	134	5	x	x	NOUN
ejpam-7049	134	6	∈	∈	NOUN
ejpam-7049	134	7	βint⋆(f+((σ1	βint⋆(f+((σ1	NOUN
ejpam-7049	134	8	,	,	PUNCT
ejpam-7049	134	9	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	134	10	)	)	PUNCT
ejpam-7049	134	11	)	)	PUNCT
ejpam-7049	134	12	)	)	PUNCT
ejpam-7049	135	1	and	and	CCONJ
ejpam-7049	135	2	so	so	ADV
ejpam-7049	135	3	there	there	PRON
ejpam-7049	135	4	exists	exist	VERB
ejpam-7049	135	5	a	a	DET
ejpam-7049	135	6	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	135	7	-	-	ADJ
ejpam-7049	135	8	open	open	ADJ
ejpam-7049	135	9	set	set	NOUN
ejpam-7049	135	10	u	u	NOUN
ejpam-7049	135	11	of	of	ADP
ejpam-7049	135	12	x	x	PUNCT
ejpam-7049	135	13	containing	contain	VERB
ejpam-7049	135	14	x	x	PUNCT
ejpam-7049	135	15	such	such	ADJ
ejpam-7049	135	16	that	that	SCONJ
ejpam-7049	135	17	u	u	NOUN
ejpam-7049	135	18	⊆	⊆	NUM
ejpam-7049	135	19	f+((σ1	f+((σ1	NOUN
ejpam-7049	135	20	,	,	PUNCT
ejpam-7049	135	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	135	22	)	)	PUNCT
ejpam-7049	135	23	)	)	PUNCT
ejpam-7049	135	24	;	;	PUNCT
ejpam-7049	135	25	hence	hence	ADV
ejpam-7049	135	26	f	f	PROPN
ejpam-7049	135	27	(	(	PUNCT
ejpam-7049	135	28	u	u	NOUN
ejpam-7049	135	29	)	)	PUNCT
ejpam-7049	135	30	⊆	⊆	NUM
ejpam-7049	135	31	(	(	PUNCT
ejpam-7049	135	32	σ1	σ1	PROPN
ejpam-7049	135	33	,	,	PUNCT
ejpam-7049	135	34	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	135	35	)	)	PUNCT
ejpam-7049	135	36	=	=	SYM
ejpam-7049	135	37	σ1σ2	σ1σ2	X
ejpam-7049	135	38	-	-	PUNCT
ejpam-7049	135	39	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	135	40	-	-	PUNCT
ejpam-7049	135	41	cl(v	cl(v	NOUN
ejpam-7049	135	42	)	)	PUNCT
ejpam-7049	135	43	)	)	PUNCT
ejpam-7049	135	44	.	.	PUNCT
ejpam-7049	136	1	this	this	PRON
ejpam-7049	136	2	shows	show	VERB
ejpam-7049	136	3	that	that	SCONJ
ejpam-7049	136	4	f	f	PROPN
ejpam-7049	136	5	is	be	AUX
ejpam-7049	136	6	upper	upper	ADJ
ejpam-7049	136	7	almost	almost	ADV
ejpam-7049	136	8	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	136	9	,	,	PUNCT
ejpam-7049	136	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	136	11	at	at	ADP
ejpam-7049	136	12	x.	x.	PROPN
ejpam-7049	136	13	n.	n.	PROPN
ejpam-7049	136	14	srisarakham	srisarakham	PROPN
ejpam-7049	136	15	,	,	PUNCT
ejpam-7049	136	16	a.	a.	PROPN
ejpam-7049	136	17	sama	sama	PROPN
ejpam-7049	136	18	-	-	PUNCT
ejpam-7049	136	19	ae	ae	PROPN
ejpam-7049	136	20	,	,	PUNCT
ejpam-7049	136	21	c.	c.	PROPN
ejpam-7049	136	22	boonpok	boonpok	PROPN
ejpam-7049	136	23	/	/	SYM
ejpam-7049	136	24	eur	eur	PROPN
ejpam-7049	136	25	.	.	PUNCT
ejpam-7049	137	1	j.	j.	PROPN
ejpam-7049	137	2	pure	pure	PROPN
ejpam-7049	137	3	appl	appl	PROPN
ejpam-7049	137	4	.	.	PROPN
ejpam-7049	137	5	math	math	PROPN
ejpam-7049	137	6	,	,	PUNCT
ejpam-7049	137	7	18	18	NUM
ejpam-7049	137	8	(	(	PUNCT
ejpam-7049	137	9	4	4	NUM
ejpam-7049	137	10	)	)	PUNCT
ejpam-7049	137	11	(	(	PUNCT
ejpam-7049	137	12	2025	2025	NUM
ejpam-7049	137	13	)	)	PUNCT
ejpam-7049	137	14	,	,	PUNCT
ejpam-7049	137	15	7049	7049	NUM
ejpam-7049	137	16	5	5	NUM
ejpam-7049	137	17	of	of	ADP
ejpam-7049	137	18	11	11	NUM
ejpam-7049	137	19	definition	definition	NOUN
ejpam-7049	137	20	2	2	NUM
ejpam-7049	137	21	.	.	PUNCT
ejpam-7049	137	22	a	a	DET
ejpam-7049	137	23	multifunction	multifunction	NOUN
ejpam-7049	137	24	f	f	NOUN
ejpam-7049	137	25	:	:	PUNCT
ejpam-7049	137	26	(	(	PUNCT
ejpam-7049	137	27	x	x	X
ejpam-7049	137	28	,	,	PUNCT
ejpam-7049	137	29	τ	τ	PROPN
ejpam-7049	137	30	,	,	PUNCT
ejpam-7049	137	31	i	i	NOUN
ejpam-7049	137	32	)	)	PUNCT
ejpam-7049	137	33	→	→	PUNCT
ejpam-7049	137	34	(	(	PUNCT
ejpam-7049	137	35	y	y	PROPN
ejpam-7049	137	36	,	,	PUNCT
ejpam-7049	137	37	σ1	σ1	PROPN
ejpam-7049	137	38	,	,	PUNCT
ejpam-7049	137	39	σ2	σ2	PROPN
ejpam-7049	137	40	)	)	PUNCT
ejpam-7049	137	41	is	be	AUX
ejpam-7049	137	42	said	say	VERB
ejpam-7049	137	43	to	to	PART
ejpam-7049	137	44	be	be	AUX
ejpam-7049	137	45	lower	low	ADJ
ejpam-7049	137	46	almost	almost	ADV
ejpam-7049	137	47	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	137	48	,	,	PUNCT
ejpam-7049	137	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	137	50	at	at	ADP
ejpam-7049	137	51	a	a	DET
ejpam-7049	137	52	point	point	NOUN
ejpam-7049	137	53	x	x	PUNCT
ejpam-7049	137	54	of	of	ADP
ejpam-7049	137	55	x	x	PRON
ejpam-7049	137	56	if	if	SCONJ
ejpam-7049	137	57	for	for	ADP
ejpam-7049	137	58	each	each	DET
ejpam-7049	137	59	σ1σ2	σ1σ2	VERB
ejpam-7049	137	60	-	-	ADJ
ejpam-7049	137	61	open	open	ADJ
ejpam-7049	137	62	set	set	NOUN
ejpam-7049	137	63	v	v	NOUN
ejpam-7049	137	64	of	of	ADP
ejpam-7049	137	65	y	y	PRON
ejpam-7049	137	66	such	such	ADJ
ejpam-7049	137	67	that	that	SCONJ
ejpam-7049	137	68	f	f	PROPN
ejpam-7049	137	69	(	(	PUNCT
ejpam-7049	137	70	x	x	NOUN
ejpam-7049	137	71	)	)	PUNCT
ejpam-7049	137	72	∩	∩	NOUN
ejpam-7049	137	73	v	v	ADP
ejpam-7049	137	74	̸=	̸=	PROPN
ejpam-7049	137	75	∅	∅	NOUN
ejpam-7049	137	76	,	,	PUNCT
ejpam-7049	137	77	there	there	PRON
ejpam-7049	137	78	exists	exist	VERB
ejpam-7049	137	79	a	a	DET
ejpam-7049	137	80	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	137	81	-	-	ADJ
ejpam-7049	137	82	open	open	ADJ
ejpam-7049	137	83	set	set	NOUN
ejpam-7049	137	84	u	u	NOUN
ejpam-7049	137	85	of	of	ADP
ejpam-7049	137	86	x	x	PUNCT
ejpam-7049	137	87	containing	contain	VERB
ejpam-7049	137	88	x	x	PUNCT
ejpam-7049	137	89	such	such	ADJ
ejpam-7049	137	90	that	that	SCONJ
ejpam-7049	137	91	f	f	PROPN
ejpam-7049	137	92	(	(	PUNCT
ejpam-7049	137	93	z	z	NOUN
ejpam-7049	137	94	)	)	PUNCT
ejpam-7049	137	95	∩	∩	NOUN
ejpam-7049	137	96	σ1σ2	σ1σ2	X
ejpam-7049	137	97	-	-	PUNCT
ejpam-7049	137	98	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	137	99	-	-	PUNCT
ejpam-7049	137	100	cl(v	cl(v	NOUN
ejpam-7049	137	101	)	)	PUNCT
ejpam-7049	137	102	)	)	PUNCT
ejpam-7049	138	1	̸=	̸=	NOUN
ejpam-7049	138	2	∅	∅	NOUN
ejpam-7049	138	3	for	for	ADP
ejpam-7049	138	4	every	every	DET
ejpam-7049	138	5	z	z	NOUN
ejpam-7049	138	6	∈	∈	PROPN
ejpam-7049	138	7	u	u	NOUN
ejpam-7049	138	8	.	.	PUNCT
ejpam-7049	139	1	a	a	DET
ejpam-7049	139	2	multifunction	multifunction	NOUN
ejpam-7049	139	3	f	f	NOUN
ejpam-7049	139	4	:	:	PUNCT
ejpam-7049	139	5	(	(	PUNCT
ejpam-7049	139	6	x	x	X
ejpam-7049	139	7	,	,	PUNCT
ejpam-7049	139	8	τ	τ	PROPN
ejpam-7049	139	9	,	,	PUNCT
ejpam-7049	139	10	i	i	NOUN
ejpam-7049	139	11	)	)	PUNCT
ejpam-7049	139	12	→	→	PUNCT
ejpam-7049	139	13	(	(	PUNCT
ejpam-7049	139	14	y	y	PROPN
ejpam-7049	139	15	,	,	PUNCT
ejpam-7049	139	16	σ1	σ1	PROPN
ejpam-7049	139	17	,	,	PUNCT
ejpam-7049	139	18	σ2	σ2	PROPN
ejpam-7049	139	19	)	)	PUNCT
ejpam-7049	139	20	is	be	AUX
ejpam-7049	139	21	said	say	VERB
ejpam-7049	139	22	to	to	PART
ejpam-7049	139	23	be	be	AUX
ejpam-7049	139	24	lower	low	ADJ
ejpam-7049	139	25	almost	almost	ADV
ejpam-7049	139	26	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	139	27	,	,	PUNCT
ejpam-7049	139	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	139	29	if	if	SCONJ
ejpam-7049	139	30	f	f	PROPN
ejpam-7049	139	31	is	be	AUX
ejpam-7049	139	32	lower	low	ADJ
ejpam-7049	139	33	almost	almost	ADV
ejpam-7049	139	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	139	35	,	,	PUNCT
ejpam-7049	139	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	139	37	at	at	ADP
ejpam-7049	139	38	each	each	DET
ejpam-7049	139	39	point	point	NOUN
ejpam-7049	139	40	of	of	ADP
ejpam-7049	139	41	x.	x.	NOUN
ejpam-7049	139	42	theorem	theorem	VERB
ejpam-7049	139	43	2	2	NUM
ejpam-7049	139	44	.	.	PUNCT
ejpam-7049	139	45	a	a	DET
ejpam-7049	139	46	multifunction	multifunction	NOUN
ejpam-7049	139	47	f	f	NOUN
ejpam-7049	139	48	:	:	PUNCT
ejpam-7049	139	49	(	(	PUNCT
ejpam-7049	139	50	x	x	X
ejpam-7049	139	51	,	,	PUNCT
ejpam-7049	139	52	τ	τ	PROPN
ejpam-7049	139	53	,	,	PUNCT
ejpam-7049	139	54	i	i	NOUN
ejpam-7049	139	55	)	)	PUNCT
ejpam-7049	139	56	→	→	PUNCT
ejpam-7049	139	57	(	(	PUNCT
ejpam-7049	139	58	y	y	PROPN
ejpam-7049	139	59	,	,	PUNCT
ejpam-7049	139	60	σ1	σ1	PROPN
ejpam-7049	139	61	,	,	PUNCT
ejpam-7049	139	62	σ2	σ2	NOUN
ejpam-7049	139	63	)	)	PUNCT
ejpam-7049	139	64	is	be	AUX
ejpam-7049	139	65	lower	low	ADJ
ejpam-7049	139	66	almost	almost	ADV
ejpam-7049	139	67	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	139	68	,	,	PUNCT
ejpam-7049	139	69	σ2)continuous	σ2)continuous	ADJ
ejpam-7049	139	70	at	at	ADP
ejpam-7049	139	71	x	x	X
ejpam-7049	139	72	∈	∈	PROPN
ejpam-7049	139	73	x	x	INTJ
ejpam-7049	139	74	if	if	SCONJ
ejpam-7049	139	75	and	and	CCONJ
ejpam-7049	139	76	only	only	ADV
ejpam-7049	139	77	if	if	SCONJ
ejpam-7049	139	78	x	x	SYM
ejpam-7049	139	79	∈	∈	NOUN
ejpam-7049	139	80	βint⋆(f−((σ1	βint⋆(f−((σ1	NOUN
ejpam-7049	139	81	,	,	PUNCT
ejpam-7049	139	82	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	139	83	)	)	PUNCT
ejpam-7049	139	84	)	)	PUNCT
ejpam-7049	139	85	)	)	PUNCT
ejpam-7049	140	1	for	for	ADP
ejpam-7049	140	2	every	every	DET
ejpam-7049	140	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	140	4	-	-	ADJ
ejpam-7049	140	5	open	open	ADJ
ejpam-7049	140	6	set	set	NOUN
ejpam-7049	140	7	v	v	NOUN
ejpam-7049	140	8	of	of	ADP
ejpam-7049	140	9	y	y	PRON
ejpam-7049	140	10	such	such	ADJ
ejpam-7049	140	11	that	that	SCONJ
ejpam-7049	140	12	f	f	PROPN
ejpam-7049	140	13	(	(	PUNCT
ejpam-7049	140	14	x	x	NOUN
ejpam-7049	140	15	)	)	PUNCT
ejpam-7049	140	16	∩	∩	NOUN
ejpam-7049	140	17	v	v	ADP
ejpam-7049	140	18	̸=	̸=	PROPN
ejpam-7049	140	19	∅.	∅.	ADP
ejpam-7049	140	20	proof	proof	NOUN
ejpam-7049	140	21	.	.	PUNCT
ejpam-7049	141	1	the	the	DET
ejpam-7049	141	2	proof	proof	NOUN
ejpam-7049	141	3	is	be	AUX
ejpam-7049	141	4	similar	similar	ADJ
ejpam-7049	141	5	to	to	ADP
ejpam-7049	141	6	that	that	PRON
ejpam-7049	141	7	of	of	ADP
ejpam-7049	141	8	theorem	theorem	NOUN
ejpam-7049	141	9	1	1	NUM
ejpam-7049	141	10	.	.	PUNCT
ejpam-7049	141	11	definition	definition	NOUN
ejpam-7049	141	12	3	3	NUM
ejpam-7049	141	13	.	.	PUNCT
ejpam-7049	142	1	a	a	DET
ejpam-7049	142	2	function	function	NOUN
ejpam-7049	142	3	f	f	NOUN
ejpam-7049	142	4	:	:	PUNCT
ejpam-7049	142	5	(	(	PUNCT
ejpam-7049	142	6	x	x	X
ejpam-7049	142	7	,	,	PUNCT
ejpam-7049	142	8	τ	τ	PROPN
ejpam-7049	142	9	,	,	PUNCT
ejpam-7049	142	10	i	i	NOUN
ejpam-7049	142	11	)	)	PUNCT
ejpam-7049	142	12	→	→	PUNCT
ejpam-7049	142	13	(	(	PUNCT
ejpam-7049	142	14	y	y	PROPN
ejpam-7049	142	15	,	,	PUNCT
ejpam-7049	142	16	σ1	σ1	PROPN
ejpam-7049	142	17	,	,	PUNCT
ejpam-7049	142	18	σ2	σ2	PROPN
ejpam-7049	142	19	)	)	PUNCT
ejpam-7049	142	20	is	be	AUX
ejpam-7049	142	21	said	say	VERB
ejpam-7049	142	22	to	to	PART
ejpam-7049	142	23	be	be	AUX
ejpam-7049	142	24	almost	almost	ADV
ejpam-7049	142	25	τ⋆β(σ1	τ⋆β(σ1	ADJ
ejpam-7049	142	26	,	,	PUNCT
ejpam-7049	142	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7049	142	28	at	at	ADP
ejpam-7049	142	29	a	a	DET
ejpam-7049	142	30	point	point	NOUN
ejpam-7049	142	31	x	x	SYM
ejpam-7049	142	32	∈	∈	NOUN
ejpam-7049	142	33	x	x	PUNCT
ejpam-7049	142	34	if	if	SCONJ
ejpam-7049	142	35	for	for	ADP
ejpam-7049	142	36	each	each	DET
ejpam-7049	142	37	σ1σ2	σ1σ2	VERB
ejpam-7049	142	38	-	-	ADJ
ejpam-7049	142	39	open	open	ADJ
ejpam-7049	142	40	set	set	NOUN
ejpam-7049	142	41	v	v	NOUN
ejpam-7049	142	42	of	of	ADP
ejpam-7049	142	43	y	y	NOUN
ejpam-7049	142	44	containing	contain	VERB
ejpam-7049	142	45	f(x	f(x	PROPN
ejpam-7049	142	46	)	)	PUNCT
ejpam-7049	142	47	,	,	PUNCT
ejpam-7049	142	48	there	there	PRON
ejpam-7049	142	49	exists	exist	VERB
ejpam-7049	142	50	a	a	DET
ejpam-7049	142	51	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	142	52	-	-	ADJ
ejpam-7049	142	53	open	open	ADJ
ejpam-7049	142	54	set	set	NOUN
ejpam-7049	142	55	u	u	NOUN
ejpam-7049	142	56	of	of	ADP
ejpam-7049	142	57	x	x	PUNCT
ejpam-7049	142	58	containing	contain	VERB
ejpam-7049	142	59	x	x	PUNCT
ejpam-7049	142	60	such	such	ADJ
ejpam-7049	142	61	that	that	DET
ejpam-7049	142	62	f(u	f(u	PROPN
ejpam-7049	142	63	)	)	PUNCT
ejpam-7049	142	64	⊆	⊆	NUM
ejpam-7049	142	65	σ1σ2	σ1σ2	X
ejpam-7049	142	66	-	-	PUNCT
ejpam-7049	142	67	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	142	68	-	-	PUNCT
ejpam-7049	142	69	cl(v	cl(v	NOUN
ejpam-7049	142	70	)	)	PUNCT
ejpam-7049	142	71	)	)	PUNCT
ejpam-7049	142	72	.	.	PUNCT
ejpam-7049	143	1	a	a	DET
ejpam-7049	143	2	function	function	NOUN
ejpam-7049	143	3	f	f	NOUN
ejpam-7049	143	4	:	:	PUNCT
ejpam-7049	143	5	(	(	PUNCT
ejpam-7049	143	6	x	x	X
ejpam-7049	143	7	,	,	PUNCT
ejpam-7049	143	8	τ	τ	PROPN
ejpam-7049	143	9	,	,	PUNCT
ejpam-7049	143	10	i	i	NOUN
ejpam-7049	143	11	)	)	PUNCT
ejpam-7049	143	12	→	→	PUNCT
ejpam-7049	143	13	(	(	PUNCT
ejpam-7049	143	14	y	y	PROPN
ejpam-7049	143	15	,	,	PUNCT
ejpam-7049	143	16	σ1	σ1	PROPN
ejpam-7049	143	17	,	,	PUNCT
ejpam-7049	143	18	σ2	σ2	PROPN
ejpam-7049	143	19	)	)	PUNCT
ejpam-7049	143	20	is	be	AUX
ejpam-7049	143	21	said	say	VERB
ejpam-7049	143	22	to	to	PART
ejpam-7049	143	23	be	be	AUX
ejpam-7049	143	24	almost	almost	ADV
ejpam-7049	143	25	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	143	26	,	,	PUNCT
ejpam-7049	143	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	143	28	if	if	SCONJ
ejpam-7049	143	29	f	f	PROPN
ejpam-7049	143	30	is	be	AUX
ejpam-7049	143	31	τ⋆β(σ1	τ⋆β(σ1	PROPN
ejpam-7049	143	32	,	,	PUNCT
ejpam-7049	143	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	143	34	at	at	ADP
ejpam-7049	143	35	each	each	DET
ejpam-7049	143	36	point	point	NOUN
ejpam-7049	143	37	of	of	ADP
ejpam-7049	143	38	x.	x.	PROPN
ejpam-7049	143	39	corollary	corollary	NOUN
ejpam-7049	143	40	1	1	NUM
ejpam-7049	143	41	.	.	PUNCT
ejpam-7049	144	1	a	a	DET
ejpam-7049	144	2	function	function	NOUN
ejpam-7049	144	3	f	f	NOUN
ejpam-7049	144	4	:	:	PUNCT
ejpam-7049	144	5	(	(	PUNCT
ejpam-7049	144	6	x	x	X
ejpam-7049	144	7	,	,	PUNCT
ejpam-7049	144	8	τ	τ	PROPN
ejpam-7049	144	9	,	,	PUNCT
ejpam-7049	144	10	i	i	NOUN
ejpam-7049	144	11	)	)	PUNCT
ejpam-7049	144	12	→	→	PUNCT
ejpam-7049	144	13	(	(	PUNCT
ejpam-7049	144	14	y	y	PROPN
ejpam-7049	144	15	,	,	PUNCT
ejpam-7049	144	16	σ1	σ1	PROPN
ejpam-7049	144	17	,	,	PUNCT
ejpam-7049	144	18	σ2	σ2	PROPN
ejpam-7049	144	19	)	)	PUNCT
ejpam-7049	144	20	is	be	AUX
ejpam-7049	144	21	almost	almost	ADV
ejpam-7049	144	22	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	144	23	,	,	PUNCT
ejpam-7049	144	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	144	25	at	at	ADP
ejpam-7049	144	26	x	x	X
ejpam-7049	144	27	∈	∈	PROPN
ejpam-7049	144	28	x	x	SYM
ejpam-7049	144	29	if	if	SCONJ
ejpam-7049	145	1	and	and	CCONJ
ejpam-7049	145	2	only	only	ADV
ejpam-7049	145	3	if	if	SCONJ
ejpam-7049	145	4	x	x	SYM
ejpam-7049	145	5	∈	∈	NOUN
ejpam-7049	145	6	βint⋆(f−1((σ1	βint⋆(f−1((σ1	NUM
ejpam-7049	145	7	,	,	PUNCT
ejpam-7049	145	8	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	145	9	)	)	PUNCT
ejpam-7049	145	10	)	)	PUNCT
ejpam-7049	145	11	)	)	PUNCT
ejpam-7049	146	1	for	for	ADP
ejpam-7049	146	2	every	every	DET
ejpam-7049	146	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	146	4	-	-	ADJ
ejpam-7049	146	5	open	open	ADJ
ejpam-7049	146	6	set	set	NOUN
ejpam-7049	146	7	v	v	NOUN
ejpam-7049	146	8	of	of	ADP
ejpam-7049	146	9	y	y	NOUN
ejpam-7049	146	10	containing	contain	VERB
ejpam-7049	146	11	f(x	f(x	PROPN
ejpam-7049	146	12	)	)	PUNCT
ejpam-7049	146	13	.	.	PUNCT
ejpam-7049	147	1	theorem	theorem	NOUN
ejpam-7049	147	2	3	3	NUM
ejpam-7049	147	3	.	.	X
ejpam-7049	147	4	for	for	ADP
ejpam-7049	147	5	a	a	DET
ejpam-7049	147	6	multifunction	multifunction	NOUN
ejpam-7049	148	1	f	f	NOUN
ejpam-7049	148	2	:	:	PUNCT
ejpam-7049	148	3	(	(	PUNCT
ejpam-7049	148	4	x	x	X
ejpam-7049	148	5	,	,	PUNCT
ejpam-7049	148	6	τ	τ	PROPN
ejpam-7049	148	7	,	,	PUNCT
ejpam-7049	148	8	i	i	NOUN
ejpam-7049	148	9	)	)	PUNCT
ejpam-7049	148	10	→	→	PUNCT
ejpam-7049	148	11	(	(	PUNCT
ejpam-7049	148	12	y	y	PROPN
ejpam-7049	148	13	,	,	PUNCT
ejpam-7049	148	14	σ1	σ1	PROPN
ejpam-7049	148	15	,	,	PUNCT
ejpam-7049	148	16	σ2	σ2	NOUN
ejpam-7049	148	17	)	)	PUNCT
ejpam-7049	148	18	,	,	PUNCT
ejpam-7049	148	19	the	the	DET
ejpam-7049	148	20	following	follow	VERB
ejpam-7049	148	21	properties	property	NOUN
ejpam-7049	148	22	are	be	AUX
ejpam-7049	148	23	equivalent	equivalent	ADJ
ejpam-7049	148	24	:	:	PUNCT
ejpam-7049	148	25	(	(	PUNCT
ejpam-7049	148	26	1	1	X
ejpam-7049	148	27	)	)	PUNCT
ejpam-7049	148	28	f	f	PROPN
ejpam-7049	148	29	is	be	AUX
ejpam-7049	148	30	upper	upper	ADJ
ejpam-7049	148	31	almost	almost	ADV
ejpam-7049	148	32	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	148	33	,	,	PUNCT
ejpam-7049	148	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	148	35	;	;	PUNCT
ejpam-7049	148	36	(	(	PUNCT
ejpam-7049	148	37	2	2	X
ejpam-7049	148	38	)	)	PUNCT
ejpam-7049	148	39	for	for	ADP
ejpam-7049	148	40	each	each	DET
ejpam-7049	148	41	x	x	SYM
ejpam-7049	148	42	∈	∈	PROPN
ejpam-7049	148	43	x	x	X
ejpam-7049	148	44	and	and	CCONJ
ejpam-7049	148	45	each	each	DET
ejpam-7049	148	46	σ1σ2	σ1σ2	VERB
ejpam-7049	148	47	-	-	ADJ
ejpam-7049	148	48	open	open	ADJ
ejpam-7049	148	49	set	set	NOUN
ejpam-7049	148	50	v	v	NOUN
ejpam-7049	148	51	of	of	ADP
ejpam-7049	148	52	y	y	PROPN
ejpam-7049	148	53	containing	contain	VERB
ejpam-7049	148	54	f	f	PROPN
ejpam-7049	148	55	(	(	PUNCT
ejpam-7049	148	56	x	x	NOUN
ejpam-7049	148	57	)	)	PUNCT
ejpam-7049	148	58	,	,	PUNCT
ejpam-7049	148	59	there	there	PRON
ejpam-7049	148	60	exists	exist	VERB
ejpam-7049	148	61	a	a	DET
ejpam-7049	148	62	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	148	63	-	-	ADJ
ejpam-7049	148	64	open	open	ADJ
ejpam-7049	148	65	set	set	NOUN
ejpam-7049	148	66	u	u	NOUN
ejpam-7049	148	67	of	of	ADP
ejpam-7049	148	68	x	x	PUNCT
ejpam-7049	148	69	containing	contain	VERB
ejpam-7049	148	70	x	x	PUNCT
ejpam-7049	148	71	such	such	ADJ
ejpam-7049	148	72	that	that	SCONJ
ejpam-7049	148	73	f	f	PROPN
ejpam-7049	148	74	(	(	PUNCT
ejpam-7049	148	75	u	u	NOUN
ejpam-7049	148	76	)	)	PUNCT
ejpam-7049	148	77	⊆	⊆	NUM
ejpam-7049	148	78	(	(	PUNCT
ejpam-7049	148	79	σ1	σ1	PROPN
ejpam-7049	148	80	,	,	PUNCT
ejpam-7049	148	81	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	148	82	)	)	PUNCT
ejpam-7049	148	83	;	;	PUNCT
ejpam-7049	148	84	(	(	PUNCT
ejpam-7049	148	85	3	3	X
ejpam-7049	148	86	)	)	PUNCT
ejpam-7049	148	87	for	for	ADP
ejpam-7049	148	88	each	each	DET
ejpam-7049	148	89	x	x	SYM
ejpam-7049	148	90	∈	∈	PROPN
ejpam-7049	148	91	x	x	X
ejpam-7049	148	92	and	and	CCONJ
ejpam-7049	148	93	each	each	DET
ejpam-7049	148	94	(	(	PUNCT
ejpam-7049	148	95	σ1	σ1	PROPN
ejpam-7049	148	96	,	,	PUNCT
ejpam-7049	148	97	σ2)r	σ2)r	NOUN
ejpam-7049	148	98	-	-	PUNCT
ejpam-7049	148	99	open	open	ADJ
ejpam-7049	148	100	set	set	VERB
ejpam-7049	148	101	v	v	NOUN
ejpam-7049	148	102	of	of	ADP
ejpam-7049	148	103	y	y	PROPN
ejpam-7049	148	104	containing	contain	VERB
ejpam-7049	148	105	f	f	PROPN
ejpam-7049	148	106	(	(	PUNCT
ejpam-7049	148	107	x	x	NOUN
ejpam-7049	148	108	)	)	PUNCT
ejpam-7049	148	109	,	,	PUNCT
ejpam-7049	148	110	there	there	PRON
ejpam-7049	148	111	exists	exist	VERB
ejpam-7049	148	112	a	a	DET
ejpam-7049	148	113	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	148	114	-	-	ADJ
ejpam-7049	148	115	open	open	ADJ
ejpam-7049	148	116	set	set	NOUN
ejpam-7049	148	117	u	u	NOUN
ejpam-7049	148	118	of	of	ADP
ejpam-7049	148	119	x	x	PUNCT
ejpam-7049	148	120	containing	contain	VERB
ejpam-7049	148	121	x	x	PUNCT
ejpam-7049	148	122	such	such	ADJ
ejpam-7049	148	123	that	that	SCONJ
ejpam-7049	148	124	f	f	PROPN
ejpam-7049	148	125	(	(	PUNCT
ejpam-7049	148	126	u	u	NOUN
ejpam-7049	148	127	)	)	PUNCT
ejpam-7049	148	128	⊆	⊆	NUM
ejpam-7049	148	129	v	v	NOUN
ejpam-7049	148	130	;	;	PUNCT
ejpam-7049	148	131	(	(	PUNCT
ejpam-7049	148	132	4	4	NUM
ejpam-7049	148	133	)	)	PUNCT
ejpam-7049	148	134	f+(v	f+(v	NOUN
ejpam-7049	148	135	)	)	PUNCT
ejpam-7049	148	136	is	be	AUX
ejpam-7049	148	137	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	148	138	-	-	NOUN
ejpam-7049	148	139	open	open	ADJ
ejpam-7049	148	140	in	in	ADP
ejpam-7049	148	141	x	x	PUNCT
ejpam-7049	148	142	for	for	ADP
ejpam-7049	148	143	every	every	DET
ejpam-7049	148	144	(	(	PUNCT
ejpam-7049	148	145	σ1	σ1	PROPN
ejpam-7049	148	146	,	,	PUNCT
ejpam-7049	148	147	σ2)r	σ2)r	NOUN
ejpam-7049	148	148	-	-	PUNCT
ejpam-7049	148	149	open	open	ADJ
ejpam-7049	148	150	set	set	VERB
ejpam-7049	148	151	v	v	NOUN
ejpam-7049	148	152	of	of	ADP
ejpam-7049	148	153	y	y	PROPN
ejpam-7049	148	154	;	;	PUNCT
ejpam-7049	148	155	(	(	PUNCT
ejpam-7049	148	156	5	5	X
ejpam-7049	148	157	)	)	PUNCT
ejpam-7049	148	158	f−(k	f−(k	PROPN
ejpam-7049	148	159	)	)	PUNCT
ejpam-7049	148	160	is	be	AUX
ejpam-7049	148	161	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	148	162	-	-	VERB
ejpam-7049	148	163	closed	closed	ADJ
ejpam-7049	148	164	in	in	ADP
ejpam-7049	148	165	x	x	PUNCT
ejpam-7049	148	166	for	for	ADP
ejpam-7049	148	167	every	every	DET
ejpam-7049	148	168	(	(	PUNCT
ejpam-7049	148	169	σ1	σ1	PROPN
ejpam-7049	148	170	,	,	PUNCT
ejpam-7049	148	171	σ2)r	σ2)r	NOUN
ejpam-7049	148	172	-	-	PUNCT
ejpam-7049	148	173	closed	close	VERB
ejpam-7049	148	174	set	set	ADJ
ejpam-7049	148	175	k	k	PROPN
ejpam-7049	148	176	of	of	ADP
ejpam-7049	148	177	y	y	PROPN
ejpam-7049	148	178	;	;	PUNCT
ejpam-7049	148	179	(	(	PUNCT
ejpam-7049	148	180	6	6	NUM
ejpam-7049	148	181	)	)	PUNCT
ejpam-7049	148	182	f+(v	f+(v	NOUN
ejpam-7049	148	183	)	)	PUNCT
ejpam-7049	149	1	⊆	⊆	NUM
ejpam-7049	149	2	βint⋆(f+((σ1	βint⋆(f+((σ1	NUM
ejpam-7049	149	3	,	,	PUNCT
ejpam-7049	149	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	149	5	)	)	PUNCT
ejpam-7049	149	6	)	)	PUNCT
ejpam-7049	149	7	)	)	PUNCT
ejpam-7049	150	1	for	for	ADP
ejpam-7049	150	2	every	every	DET
ejpam-7049	150	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	150	4	-	-	ADJ
ejpam-7049	150	5	open	open	ADJ
ejpam-7049	150	6	set	set	NOUN
ejpam-7049	150	7	v	v	NOUN
ejpam-7049	150	8	of	of	ADP
ejpam-7049	150	9	y	y	PROPN
ejpam-7049	150	10	;	;	PUNCT
ejpam-7049	150	11	(	(	PUNCT
ejpam-7049	150	12	7	7	X
ejpam-7049	150	13	)	)	PUNCT
ejpam-7049	150	14	βcl⋆(f−((σ1	βcl⋆(f−((σ1	ADJ
ejpam-7049	150	15	,	,	PUNCT
ejpam-7049	150	16	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	150	17	)	)	PUNCT
ejpam-7049	150	18	)	)	PUNCT
ejpam-7049	150	19	)	)	PUNCT
ejpam-7049	151	1	⊆	⊆	X
ejpam-7049	151	2	f−(k	f−(k	PROPN
ejpam-7049	151	3	)	)	PUNCT
ejpam-7049	151	4	for	for	ADP
ejpam-7049	151	5	every	every	DET
ejpam-7049	151	6	σ1σ2	σ1σ2	NUM
ejpam-7049	151	7	-	-	PUNCT
ejpam-7049	151	8	closed	closed	ADJ
ejpam-7049	151	9	set	set	NOUN
ejpam-7049	151	10	k	k	PROPN
ejpam-7049	151	11	of	of	ADP
ejpam-7049	151	12	y	y	PROPN
ejpam-7049	151	13	;	;	PUNCT
ejpam-7049	151	14	(	(	PUNCT
ejpam-7049	151	15	8)	8)	NUM
ejpam-7049	151	16	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7049	151	17	-	-	PUNCT
ejpam-7049	151	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	151	19	-	-	PUNCT
ejpam-7049	151	20	int(k	int(k	NOUN
ejpam-7049	151	21	)	)	PUNCT
ejpam-7049	151	22	)	)	PUNCT
ejpam-7049	151	23	)	)	PUNCT
ejpam-7049	151	24	)	)	PUNCT
ejpam-7049	152	1	⊆	⊆	X
ejpam-7049	152	2	f−(k	f−(k	PROPN
ejpam-7049	152	3	)	)	PUNCT
ejpam-7049	152	4	for	for	ADP
ejpam-7049	152	5	every	every	DET
ejpam-7049	152	6	σ1σ2	σ1σ2	NUM
ejpam-7049	152	7	-	-	PUNCT
ejpam-7049	152	8	closed	closed	ADJ
ejpam-7049	152	9	set	set	NOUN
ejpam-7049	152	10	k	k	PROPN
ejpam-7049	152	11	of	of	ADP
ejpam-7049	152	12	y	y	PROPN
ejpam-7049	152	13	;	;	PUNCT
ejpam-7049	152	14	(	(	PUNCT
ejpam-7049	152	15	9	9	X
ejpam-7049	152	16	)	)	PUNCT
ejpam-7049	152	17	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7049	152	18	-	-	PUNCT
ejpam-7049	152	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	152	20	-	-	PUNCT
ejpam-7049	152	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	152	22	-	-	PUNCT
ejpam-7049	152	23	cl(b	cl(b	NOUN
ejpam-7049	152	24	)	)	PUNCT
ejpam-7049	152	25	)	)	PUNCT
ejpam-7049	152	26	)	)	PUNCT
ejpam-7049	152	27	)	)	PUNCT
ejpam-7049	152	28	)	)	PUNCT
ejpam-7049	153	1	⊆	⊆	X
ejpam-7049	153	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7049	153	3	-	-	PUNCT
ejpam-7049	153	4	cl(b	cl(b	NOUN
ejpam-7049	153	5	)	)	PUNCT
ejpam-7049	153	6	)	)	PUNCT
ejpam-7049	154	1	for	for	ADP
ejpam-7049	154	2	every	every	DET
ejpam-7049	154	3	subset	subset	NOUN
ejpam-7049	154	4	b	b	PROPN
ejpam-7049	154	5	of	of	ADP
ejpam-7049	154	6	y	y	PROPN
ejpam-7049	154	7	;	;	PUNCT
ejpam-7049	154	8	n.	n.	PROPN
ejpam-7049	154	9	srisarakham	srisarakham	PROPN
ejpam-7049	154	10	,	,	PUNCT
ejpam-7049	154	11	a.	a.	PROPN
ejpam-7049	154	12	sama	sama	PROPN
ejpam-7049	154	13	-	-	PUNCT
ejpam-7049	154	14	ae	ae	PROPN
ejpam-7049	154	15	,	,	PUNCT
ejpam-7049	154	16	c.	c.	PROPN
ejpam-7049	154	17	boonpok	boonpok	PROPN
ejpam-7049	154	18	/	/	SYM
ejpam-7049	154	19	eur	eur	PROPN
ejpam-7049	154	20	.	.	PUNCT
ejpam-7049	155	1	j.	j.	PROPN
ejpam-7049	155	2	pure	pure	PROPN
ejpam-7049	155	3	appl	appl	PROPN
ejpam-7049	155	4	.	.	PROPN
ejpam-7049	155	5	math	math	PROPN
ejpam-7049	155	6	,	,	PUNCT
ejpam-7049	155	7	18	18	NUM
ejpam-7049	155	8	(	(	PUNCT
ejpam-7049	155	9	4	4	NUM
ejpam-7049	155	10	)	)	PUNCT
ejpam-7049	155	11	(	(	PUNCT
ejpam-7049	155	12	2025	2025	NUM
ejpam-7049	155	13	)	)	PUNCT
ejpam-7049	155	14	,	,	PUNCT
ejpam-7049	155	15	7049	7049	NUM
ejpam-7049	155	16	6	6	NUM
ejpam-7049	155	17	of	of	ADP
ejpam-7049	155	18	11	11	NUM
ejpam-7049	155	19	(	(	PUNCT
ejpam-7049	155	20	10	10	NUM
ejpam-7049	155	21	)	)	PUNCT
ejpam-7049	155	22	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-7049	155	23	-	-	PUNCT
ejpam-7049	155	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	155	25	-	-	PUNCT
ejpam-7049	155	26	int(k	int(k	NOUN
ejpam-7049	155	27	)	)	PUNCT
ejpam-7049	155	28	)	)	PUNCT
ejpam-7049	155	29	)	)	PUNCT
ejpam-7049	155	30	)	)	PUNCT
ejpam-7049	155	31	)	)	PUNCT
ejpam-7049	155	32	)	)	PUNCT
ejpam-7049	156	1	⊆	⊆	X
ejpam-7049	156	2	f−(k	f−(k	PROPN
ejpam-7049	156	3	)	)	PUNCT
ejpam-7049	156	4	for	for	ADP
ejpam-7049	156	5	every	every	DET
ejpam-7049	156	6	σ1σ2	σ1σ2	NUM
ejpam-7049	156	7	-	-	PUNCT
ejpam-7049	156	8	closed	closed	ADJ
ejpam-7049	156	9	set	set	NOUN
ejpam-7049	156	10	k	k	PROPN
ejpam-7049	156	11	of	of	ADP
ejpam-7049	156	12	y	y	PROPN
ejpam-7049	156	13	;	;	PUNCT
ejpam-7049	156	14	(	(	PUNCT
ejpam-7049	156	15	11	11	X
ejpam-7049	156	16	)	)	PUNCT
ejpam-7049	156	17	int⋆(cl⋆(int⋆(f−((σ1	int⋆(cl⋆(int⋆(f−((σ1	NOUN
ejpam-7049	156	18	,	,	PUNCT
ejpam-7049	156	19	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	156	20	)	)	PUNCT
ejpam-7049	156	21	)	)	PUNCT
ejpam-7049	156	22	)	)	PUNCT
ejpam-7049	156	23	)	)	PUNCT
ejpam-7049	156	24	)	)	PUNCT
ejpam-7049	157	1	⊆	⊆	X
ejpam-7049	157	2	f−(k	f−(k	PROPN
ejpam-7049	157	3	)	)	PUNCT
ejpam-7049	157	4	for	for	ADP
ejpam-7049	157	5	every	every	DET
ejpam-7049	157	6	σ1σ2	σ1σ2	NUM
ejpam-7049	157	7	-	-	PUNCT
ejpam-7049	157	8	closed	closed	ADJ
ejpam-7049	157	9	set	set	NOUN
ejpam-7049	157	10	k	k	PROPN
ejpam-7049	157	11	of	of	ADP
ejpam-7049	157	12	y	y	PROPN
ejpam-7049	157	13	;	;	PUNCT
ejpam-7049	157	14	(	(	PUNCT
ejpam-7049	157	15	12	12	NUM
ejpam-7049	157	16	)	)	PUNCT
ejpam-7049	157	17	f+(v	f+(v	NOUN
ejpam-7049	157	18	)	)	PUNCT
ejpam-7049	158	1	⊆	⊆	NUM
ejpam-7049	158	2	cl⋆(int⋆(cl⋆(f+((σ1	cl⋆(int⋆(cl⋆(f+((σ1	NOUN
ejpam-7049	158	3	,	,	PUNCT
ejpam-7049	158	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	158	5	)	)	PUNCT
ejpam-7049	158	6	)	)	PUNCT
ejpam-7049	158	7	)	)	PUNCT
ejpam-7049	158	8	)	)	PUNCT
ejpam-7049	158	9	)	)	PUNCT
ejpam-7049	158	10	for	for	ADP
ejpam-7049	158	11	every	every	DET
ejpam-7049	158	12	σ1σ2	σ1σ2	NOUN
ejpam-7049	158	13	-	-	ADJ
ejpam-7049	158	14	open	open	ADJ
ejpam-7049	158	15	set	set	NOUN
ejpam-7049	158	16	v	v	NOUN
ejpam-7049	158	17	of	of	ADP
ejpam-7049	158	18	y	y	PROPN
ejpam-7049	158	19	.	.	PUNCT
ejpam-7049	159	1	proof	proof	NOUN
ejpam-7049	159	2	.	.	PUNCT
ejpam-7049	160	1	(	(	PUNCT
ejpam-7049	160	2	1	1	X
ejpam-7049	160	3	)	)	PUNCT
ejpam-7049	160	4	⇒	⇒	NOUN
ejpam-7049	160	5	(	(	PUNCT
ejpam-7049	160	6	2	2	NUM
ejpam-7049	160	7	)	)	PUNCT
ejpam-7049	160	8	and	and	CCONJ
ejpam-7049	160	9	(	(	PUNCT
ejpam-7049	160	10	2	2	X
ejpam-7049	160	11	)	)	PUNCT
ejpam-7049	160	12	⇒	⇒	NOUN
ejpam-7049	160	13	(	(	PUNCT
ejpam-7049	160	14	3	3	NUM
ejpam-7049	160	15	):	):	PUNCT
ejpam-7049	160	16	the	the	DET
ejpam-7049	160	17	proofs	proof	NOUN
ejpam-7049	160	18	are	be	AUX
ejpam-7049	160	19	obvious	obvious	ADJ
ejpam-7049	160	20	.	.	PUNCT
ejpam-7049	161	1	(	(	PUNCT
ejpam-7049	161	2	3	3	X
ejpam-7049	161	3	)	)	PUNCT
ejpam-7049	161	4	⇒	⇒	NOUN
ejpam-7049	161	5	(	(	PUNCT
ejpam-7049	161	6	4	4	NUM
ejpam-7049	161	7	):	):	PUNCT
ejpam-7049	161	8	let	let	VERB
ejpam-7049	161	9	v	v	PART
ejpam-7049	161	10	be	be	AUX
ejpam-7049	161	11	any	any	DET
ejpam-7049	161	12	σ1σ2	σ1σ2	NOUN
ejpam-7049	161	13	-	-	ADJ
ejpam-7049	161	14	open	open	ADJ
ejpam-7049	161	15	set	set	NOUN
ejpam-7049	161	16	of	of	ADP
ejpam-7049	161	17	y	y	PROPN
ejpam-7049	161	18	and	and	CCONJ
ejpam-7049	161	19	x	x	PROPN
ejpam-7049	161	20	∈	∈	PROPN
ejpam-7049	161	21	f+(v	f+(v	NOUN
ejpam-7049	161	22	)	)	PUNCT
ejpam-7049	161	23	.	.	PUNCT
ejpam-7049	162	1	then	then	ADV
ejpam-7049	162	2	,	,	PUNCT
ejpam-7049	162	3	f	f	PROPN
ejpam-7049	162	4	(	(	PUNCT
ejpam-7049	162	5	x	x	X
ejpam-7049	162	6	)	)	PUNCT
ejpam-7049	162	7	⊆	⊆	NUM
ejpam-7049	162	8	v	v	NOUN
ejpam-7049	162	9	and	and	CCONJ
ejpam-7049	162	10	so	so	ADV
ejpam-7049	162	11	there	there	PRON
ejpam-7049	162	12	exists	exist	VERB
ejpam-7049	162	13	a	a	DET
ejpam-7049	162	14	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	162	15	-	-	NOUN
ejpam-7049	162	16	open	open	ADJ
ejpam-7049	162	17	set	set	VERB
ejpam-7049	162	18	ux	ux	NOUN
ejpam-7049	162	19	of	of	ADP
ejpam-7049	162	20	x	x	SYM
ejpam-7049	162	21	containing	contain	VERB
ejpam-7049	162	22	x	x	PUNCT
ejpam-7049	162	23	such	such	ADJ
ejpam-7049	162	24	that	that	SCONJ
ejpam-7049	162	25	f	f	PROPN
ejpam-7049	162	26	(	(	PUNCT
ejpam-7049	162	27	ux	ux	PROPN
ejpam-7049	162	28	)	)	PUNCT
ejpam-7049	162	29	⊆	⊆	NUM
ejpam-7049	162	30	v	v	NOUN
ejpam-7049	162	31	.	.	PUNCT
ejpam-7049	163	1	thus	thus	ADV
ejpam-7049	163	2	,	,	PUNCT
ejpam-7049	163	3	x	x	PUNCT
ejpam-7049	163	4	∈	∈	PROPN
ejpam-7049	163	5	ux	ux	NOUN
ejpam-7049	163	6	⊆	⊆	NUM
ejpam-7049	163	7	f+(v	f+(v	NOUN
ejpam-7049	163	8	)	)	PUNCT
ejpam-7049	163	9	and	and	CCONJ
ejpam-7049	163	10	hence	hence	ADV
ejpam-7049	163	11	f+(v	f+(v	NOUN
ejpam-7049	163	12	)	)	PUNCT
ejpam-7049	164	1	=	=	SYM
ejpam-7049	164	2	∪x∈f+(v	∪x∈f+(v	X
ejpam-7049	164	3	)	)	PUNCT
ejpam-7049	164	4	ux	ux	PROPN
ejpam-7049	164	5	.	.	PUNCT
ejpam-7049	165	1	this	this	PRON
ejpam-7049	165	2	shows	show	VERB
ejpam-7049	165	3	that	that	SCONJ
ejpam-7049	165	4	f+(v	f+(v	PROPN
ejpam-7049	165	5	)	)	PUNCT
ejpam-7049	165	6	is	be	AUX
ejpam-7049	165	7	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	165	8	-	-	NOUN
ejpam-7049	165	9	open	open	ADJ
ejpam-7049	165	10	in	in	ADP
ejpam-7049	165	11	x.	x.	NOUN
ejpam-7049	165	12	(	(	PUNCT
ejpam-7049	165	13	4	4	NUM
ejpam-7049	165	14	)	)	PUNCT
ejpam-7049	165	15	⇒	⇒	NOUN
ejpam-7049	165	16	(	(	PUNCT
ejpam-7049	165	17	5	5	NUM
ejpam-7049	165	18	):	):	PUNCT
ejpam-7049	165	19	this	this	PRON
ejpam-7049	165	20	follows	follow	VERB
ejpam-7049	165	21	from	from	ADP
ejpam-7049	165	22	the	the	DET
ejpam-7049	165	23	fact	fact	NOUN
ejpam-7049	165	24	that	that	SCONJ
ejpam-7049	165	25	f+(y	f+(y	PROPN
ejpam-7049	165	26	−	−	PROPN
ejpam-7049	165	27	b	b	NOUN
ejpam-7049	165	28	)	)	PUNCT
ejpam-7049	165	29	=	=	SYM
ejpam-7049	165	30	y	y	PROPN
ejpam-7049	165	31	−	−	PROPN
ejpam-7049	165	32	f−(b	f−(b	PROPN
ejpam-7049	165	33	)	)	PUNCT
ejpam-7049	165	34	for	for	ADP
ejpam-7049	165	35	every	every	DET
ejpam-7049	165	36	subset	subset	NOUN
ejpam-7049	165	37	b	b	PROPN
ejpam-7049	165	38	of	of	ADP
ejpam-7049	165	39	y	y	PROPN
ejpam-7049	165	40	.	.	PUNCT
ejpam-7049	166	1	(	(	PUNCT
ejpam-7049	166	2	5	5	X
ejpam-7049	166	3	)	)	PUNCT
ejpam-7049	166	4	⇒	⇒	NOUN
ejpam-7049	166	5	(	(	PUNCT
ejpam-7049	166	6	6	6	NUM
ejpam-7049	166	7	):	):	PUNCT
ejpam-7049	166	8	let	let	VERB
ejpam-7049	166	9	v	v	PART
ejpam-7049	166	10	be	be	AUX
ejpam-7049	166	11	any	any	DET
ejpam-7049	166	12	σ1σ2	σ1σ2	NOUN
ejpam-7049	166	13	-	-	ADJ
ejpam-7049	166	14	open	open	ADJ
ejpam-7049	166	15	set	set	NOUN
ejpam-7049	166	16	of	of	ADP
ejpam-7049	166	17	y	y	PROPN
ejpam-7049	166	18	and	and	CCONJ
ejpam-7049	166	19	x	x	PROPN
ejpam-7049	166	20	∈	∈	PROPN
ejpam-7049	166	21	f+(v	f+(v	NOUN
ejpam-7049	166	22	)	)	PUNCT
ejpam-7049	166	23	.	.	PUNCT
ejpam-7049	167	1	then	then	ADV
ejpam-7049	167	2	,	,	PUNCT
ejpam-7049	167	3	f	f	PROPN
ejpam-7049	167	4	(	(	PUNCT
ejpam-7049	167	5	x	x	X
ejpam-7049	167	6	)	)	PUNCT
ejpam-7049	167	7	⊆	⊆	NUM
ejpam-7049	167	8	v	v	ADP
ejpam-7049	167	9	⊆	⊆	NUM
ejpam-7049	167	10	(	(	PUNCT
ejpam-7049	167	11	σ1	σ1	PROPN
ejpam-7049	167	12	,	,	PUNCT
ejpam-7049	167	13	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	167	14	)	)	PUNCT
ejpam-7049	167	15	and	and	CCONJ
ejpam-7049	167	16	hence	hence	ADV
ejpam-7049	167	17	x	x	PART
ejpam-7049	167	18	∈	∈	PROPN
ejpam-7049	167	19	f+((σ1	f+((σ1	NOUN
ejpam-7049	167	20	,	,	PUNCT
ejpam-7049	167	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	167	22	)	)	PUNCT
ejpam-7049	167	23	)	)	PUNCT
ejpam-7049	168	1	=	=	PUNCT
ejpam-7049	168	2	x	x	PUNCT
ejpam-7049	169	1	−	−	NOUN
ejpam-7049	169	2	f−(y	f−(y	NOUN
ejpam-7049	169	3	−	−	NOUN
ejpam-7049	169	4	(	(	PUNCT
ejpam-7049	169	5	σ1	σ1	PROPN
ejpam-7049	169	6	,	,	PUNCT
ejpam-7049	169	7	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	169	8	)	)	PUNCT
ejpam-7049	169	9	)	)	PUNCT
ejpam-7049	169	10	.	.	PUNCT
ejpam-7049	170	1	since	since	SCONJ
ejpam-7049	170	2	y	y	PROPN
ejpam-7049	170	3	−	−	PROPN
ejpam-7049	170	4	(	(	PUNCT
ejpam-7049	170	5	σ1	σ1	PROPN
ejpam-7049	170	6	,	,	PUNCT
ejpam-7049	170	7	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	170	8	)	)	PUNCT
ejpam-7049	170	9	is	be	AUX
ejpam-7049	170	10	(	(	PUNCT
ejpam-7049	170	11	σ1	σ1	NOUN
ejpam-7049	170	12	,	,	PUNCT
ejpam-7049	170	13	σ2)r	σ2)r	NOUN
ejpam-7049	170	14	-	-	PUNCT
ejpam-7049	170	15	closed	closed	ADJ
ejpam-7049	170	16	,	,	PUNCT
ejpam-7049	170	17	we	we	PRON
ejpam-7049	170	18	have	have	VERB
ejpam-7049	170	19	f−(y	f−(y	NOUN
ejpam-7049	170	20	−	−	PROPN
ejpam-7049	170	21	(	(	PUNCT
ejpam-7049	170	22	σ1	σ1	PROPN
ejpam-7049	170	23	,	,	PUNCT
ejpam-7049	170	24	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	170	25	)	)	PUNCT
ejpam-7049	170	26	)	)	PUNCT
ejpam-7049	170	27	is	be	AUX
ejpam-7049	170	28	τ⋆-βclosed	τ⋆-βclose	VERB
ejpam-7049	170	29	in	in	ADP
ejpam-7049	170	30	x.	x.	NOUN
ejpam-7049	170	31	thus	thus	ADV
ejpam-7049	170	32	,	,	PUNCT
ejpam-7049	170	33	f+((σ1	f+((σ1	NOUN
ejpam-7049	170	34	,	,	PUNCT
ejpam-7049	170	35	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	170	36	)	)	PUNCT
ejpam-7049	170	37	)	)	PUNCT
ejpam-7049	170	38	is	be	AUX
ejpam-7049	170	39	a	a	DET
ejpam-7049	170	40	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	170	41	-	-	NOUN
ejpam-7049	170	42	open	open	ADJ
ejpam-7049	170	43	set	set	NOUN
ejpam-7049	170	44	of	of	ADP
ejpam-7049	170	45	x	x	PUNCT
ejpam-7049	170	46	containing	contain	VERB
ejpam-7049	170	47	x	x	X
ejpam-7049	170	48	and	and	CCONJ
ejpam-7049	170	49	so	so	ADV
ejpam-7049	170	50	x	x	SYM
ejpam-7049	170	51	∈	∈	NOUN
ejpam-7049	170	52	βint⋆(f+((σ1	βint⋆(f+((σ1	NOUN
ejpam-7049	170	53	,	,	PUNCT
ejpam-7049	170	54	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	170	55	)	)	PUNCT
ejpam-7049	170	56	)	)	PUNCT
ejpam-7049	170	57	)	)	PUNCT
ejpam-7049	170	58	.	.	PUNCT
ejpam-7049	171	1	this	this	PRON
ejpam-7049	171	2	shows	show	VERB
ejpam-7049	171	3	that	that	SCONJ
ejpam-7049	171	4	f+(v	f+(v	PROPN
ejpam-7049	171	5	)	)	PUNCT
ejpam-7049	172	1	⊆	⊆	NUM
ejpam-7049	172	2	βint⋆(f+((σ1	βint⋆(f+((σ1	NUM
ejpam-7049	172	3	,	,	PUNCT
ejpam-7049	172	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	172	5	)	)	PUNCT
ejpam-7049	172	6	)	)	PUNCT
ejpam-7049	172	7	)	)	PUNCT
ejpam-7049	172	8	.	.	PUNCT
ejpam-7049	173	1	(	(	PUNCT
ejpam-7049	173	2	6	6	X
ejpam-7049	173	3	)	)	PUNCT
ejpam-7049	173	4	⇒	⇒	NOUN
ejpam-7049	173	5	(	(	PUNCT
ejpam-7049	173	6	7	7	NUM
ejpam-7049	173	7	):	):	PUNCT
ejpam-7049	173	8	let	let	VERB
ejpam-7049	173	9	k	k	PRON
ejpam-7049	173	10	be	be	AUX
ejpam-7049	173	11	any	any	DET
ejpam-7049	173	12	σ1σ2	σ1σ2	NUM
ejpam-7049	173	13	-	-	PUNCT
ejpam-7049	173	14	closed	closed	ADJ
ejpam-7049	173	15	set	set	NOUN
ejpam-7049	173	16	of	of	ADP
ejpam-7049	173	17	y	y	PROPN
ejpam-7049	173	18	.	.	PUNCT
ejpam-7049	174	1	then	then	ADV
ejpam-7049	174	2	,	,	PUNCT
ejpam-7049	174	3	since	since	SCONJ
ejpam-7049	174	4	y	y	PROPN
ejpam-7049	174	5	−k	−k	PROPN
ejpam-7049	174	6	is	be	AUX
ejpam-7049	174	7	σ1σ2	σ1σ2	NOUN
ejpam-7049	174	8	-	-	ADJ
ejpam-7049	174	9	open	open	ADJ
ejpam-7049	174	10	and	and	CCONJ
ejpam-7049	174	11	by	by	ADP
ejpam-7049	174	12	(	(	PUNCT
ejpam-7049	174	13	6	6	NUM
ejpam-7049	174	14	)	)	PUNCT
ejpam-7049	174	15	,	,	PUNCT
ejpam-7049	174	16	x	x	PUNCT
ejpam-7049	174	17	−	−	DET
ejpam-7049	174	18	f−(k	f−(k	PROPN
ejpam-7049	174	19	)	)	PUNCT
ejpam-7049	174	20	=	=	PUNCT
ejpam-7049	175	1	f+(y	f+(y	PROPN
ejpam-7049	175	2	−k	−k	PROPN
ejpam-7049	175	3	)	)	PUNCT
ejpam-7049	175	4	⊆	⊆	NUM
ejpam-7049	175	5	βint⋆(f+((σ1	βint⋆(f+((σ1	NUM
ejpam-7049	175	6	,	,	PUNCT
ejpam-7049	175	7	σ2)-scl(y	σ2)-scl(y	NOUN
ejpam-7049	175	8	−k	−k	PROPN
ejpam-7049	175	9	)	)	PUNCT
ejpam-7049	175	10	)	)	PUNCT
ejpam-7049	175	11	)	)	PUNCT
ejpam-7049	176	1	=	=	PUNCT
ejpam-7049	176	2	βint⋆(f+(y	βint⋆(f+(y	PROPN
ejpam-7049	176	3	−	−	PROPN
ejpam-7049	176	4	(	(	PUNCT
ejpam-7049	176	5	σ1	σ1	PROPN
ejpam-7049	176	6	,	,	PUNCT
ejpam-7049	176	7	σ2)-sint(k	σ2)-sint(k	NUM
ejpam-7049	176	8	)	)	PUNCT
ejpam-7049	176	9	)	)	PUNCT
ejpam-7049	176	10	)	)	PUNCT
ejpam-7049	177	1	=	=	PUNCT
ejpam-7049	177	2	βint⋆(x	βint⋆(x	NOUN
ejpam-7049	178	1	−	−	NOUN
ejpam-7049	178	2	f−((σ1	f−((σ1	NOUN
ejpam-7049	178	3	,	,	PUNCT
ejpam-7049	178	4	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	178	5	)	)	PUNCT
ejpam-7049	178	6	)	)	PUNCT
ejpam-7049	178	7	)	)	PUNCT
ejpam-7049	179	1	=	=	PUNCT
ejpam-7049	179	2	x	x	X
ejpam-7049	179	3	−	−	NOUN
ejpam-7049	179	4	βcl⋆(f−((σ1	βcl⋆(f−((σ1	NOUN
ejpam-7049	179	5	,	,	PUNCT
ejpam-7049	179	6	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	179	7	)	)	PUNCT
ejpam-7049	179	8	)	)	PUNCT
ejpam-7049	179	9	)	)	PUNCT
ejpam-7049	179	10	.	.	PUNCT
ejpam-7049	180	1	thus	thus	ADV
ejpam-7049	180	2	,	,	PUNCT
ejpam-7049	180	3	βcl⋆(f−((σ1	βcl⋆(f−((σ1	ADJ
ejpam-7049	180	4	,	,	PUNCT
ejpam-7049	180	5	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	180	6	)	)	PUNCT
ejpam-7049	180	7	)	)	PUNCT
ejpam-7049	180	8	)	)	PUNCT
ejpam-7049	180	9	⊆	⊆	NUM
ejpam-7049	180	10	f−(k	f−(k	PROPN
ejpam-7049	180	11	)	)	PUNCT
ejpam-7049	180	12	.	.	PUNCT
ejpam-7049	181	1	(	(	PUNCT
ejpam-7049	181	2	7	7	X
ejpam-7049	181	3	)	)	PUNCT
ejpam-7049	181	4	⇒	⇒	NOUN
ejpam-7049	181	5	(	(	PUNCT
ejpam-7049	181	6	8)	8)	NUM
ejpam-7049	181	7	:	:	PUNCT
ejpam-7049	181	8	the	the	DET
ejpam-7049	181	9	proof	proof	NOUN
ejpam-7049	181	10	is	be	AUX
ejpam-7049	181	11	obvious	obvious	ADJ
ejpam-7049	181	12	since	since	SCONJ
ejpam-7049	181	13	(	(	PUNCT
ejpam-7049	181	14	σ1	σ1	PROPN
ejpam-7049	181	15	,	,	PUNCT
ejpam-7049	181	16	σ2)-sint(k	σ2)-sint(k	NUM
ejpam-7049	181	17	)	)	PUNCT
ejpam-7049	181	18	=	=	SYM
ejpam-7049	181	19	σ1σ2	σ1σ2	X
ejpam-7049	181	20	-	-	PUNCT
ejpam-7049	181	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	181	22	-	-	PUNCT
ejpam-7049	181	23	int(k	int(k	NOUN
ejpam-7049	181	24	)	)	PUNCT
ejpam-7049	181	25	)	)	PUNCT
ejpam-7049	181	26	for	for	ADP
ejpam-7049	181	27	every	every	DET
ejpam-7049	181	28	σ1σ2	σ1σ2	NUM
ejpam-7049	181	29	-	-	PUNCT
ejpam-7049	181	30	closed	closed	ADJ
ejpam-7049	181	31	set	set	NOUN
ejpam-7049	181	32	k	k	PROPN
ejpam-7049	181	33	of	of	ADP
ejpam-7049	181	34	y	y	PROPN
ejpam-7049	181	35	.	.	PUNCT
ejpam-7049	182	1	(	(	PUNCT
ejpam-7049	182	2	8)	8)	NUM
ejpam-7049	182	3	⇒	⇒	NOUN
ejpam-7049	182	4	(	(	PUNCT
ejpam-7049	182	5	9	9	NUM
ejpam-7049	182	6	):	):	PUNCT
ejpam-7049	182	7	the	the	DET
ejpam-7049	182	8	proof	proof	NOUN
ejpam-7049	182	9	is	be	AUX
ejpam-7049	182	10	obvious	obvious	ADJ
ejpam-7049	182	11	.	.	PUNCT
ejpam-7049	183	1	(	(	PUNCT
ejpam-7049	183	2	9	9	X
ejpam-7049	183	3	)	)	PUNCT
ejpam-7049	183	4	⇒	⇒	NOUN
ejpam-7049	183	5	(	(	PUNCT
ejpam-7049	183	6	10	10	NUM
ejpam-7049	183	7	):	):	PUNCT
ejpam-7049	183	8	by	by	ADP
ejpam-7049	183	9	(	(	PUNCT
ejpam-7049	183	10	9	9	NUM
ejpam-7049	183	11	)	)	PUNCT
ejpam-7049	183	12	and	and	CCONJ
ejpam-7049	183	13	lemma	lemma	PROPN
ejpam-7049	183	14	3	3	NUM
ejpam-7049	183	15	,	,	PUNCT
ejpam-7049	183	16	int⋆(cl⋆(int⋆(f−(σ1σ2	int⋆(cl⋆(int⋆(f−(σ1σ2	NOUN
ejpam-7049	183	17	-	-	PUNCT
ejpam-7049	183	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	183	19	-	-	PUNCT
ejpam-7049	183	20	int(k	int(k	NOUN
ejpam-7049	183	21	)	)	PUNCT
ejpam-7049	183	22	)	)	PUNCT
ejpam-7049	183	23	)	)	PUNCT
ejpam-7049	183	24	)	)	PUNCT
ejpam-7049	183	25	)	)	PUNCT
ejpam-7049	183	26	)	)	PUNCT
ejpam-7049	184	1	⊆	⊆	NUM
ejpam-7049	184	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7049	184	3	-	-	PUNCT
ejpam-7049	184	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	184	5	-	-	PUNCT
ejpam-7049	184	6	int(k	int(k	NOUN
ejpam-7049	184	7	)	)	PUNCT
ejpam-7049	184	8	)	)	PUNCT
ejpam-7049	184	9	)	)	PUNCT
ejpam-7049	184	10	)	)	PUNCT
ejpam-7049	185	1	⊆	⊆	NUM
ejpam-7049	185	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7049	185	3	-	-	PUNCT
ejpam-7049	185	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	185	5	-	-	PUNCT
ejpam-7049	185	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	185	7	-	-	PUNCT
ejpam-7049	185	8	cl(k	cl(k	NUM
ejpam-7049	185	9	)	)	PUNCT
ejpam-7049	185	10	)	)	PUNCT
ejpam-7049	185	11	)	)	PUNCT
ejpam-7049	185	12	)	)	PUNCT
ejpam-7049	185	13	)	)	PUNCT
ejpam-7049	186	1	⊆	⊆	X
ejpam-7049	186	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7049	186	3	-	-	PUNCT
ejpam-7049	186	4	cl(k	cl(k	NOUN
ejpam-7049	186	5	)	)	PUNCT
ejpam-7049	186	6	)	)	PUNCT
ejpam-7049	187	1	=	=	SYM
ejpam-7049	187	2	f−(k	f−(k	PROPN
ejpam-7049	187	3	)	)	PUNCT
ejpam-7049	187	4	.	.	PUNCT
ejpam-7049	188	1	(	(	PUNCT
ejpam-7049	188	2	10	10	NUM
ejpam-7049	188	3	)	)	PUNCT
ejpam-7049	188	4	⇒	⇒	NOUN
ejpam-7049	188	5	(	(	PUNCT
ejpam-7049	188	6	11	11	NUM
ejpam-7049	188	7	):	):	PUNCT
ejpam-7049	188	8	the	the	DET
ejpam-7049	188	9	proof	proof	NOUN
ejpam-7049	188	10	is	be	AUX
ejpam-7049	188	11	obvious	obvious	ADJ
ejpam-7049	188	12	since	since	SCONJ
ejpam-7049	188	13	(	(	PUNCT
ejpam-7049	188	14	σ1	σ1	PROPN
ejpam-7049	188	15	,	,	PUNCT
ejpam-7049	188	16	σ2)-sint(k	σ2)-sint(k	NUM
ejpam-7049	188	17	)	)	PUNCT
ejpam-7049	188	18	=	=	SYM
ejpam-7049	188	19	σ1σ2	σ1σ2	X
ejpam-7049	188	20	-	-	PUNCT
ejpam-7049	188	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	188	22	-	-	PUNCT
ejpam-7049	188	23	int(k	int(k	NOUN
ejpam-7049	188	24	)	)	PUNCT
ejpam-7049	188	25	)	)	PUNCT
ejpam-7049	188	26	for	for	ADP
ejpam-7049	188	27	every	every	DET
ejpam-7049	188	28	σ1σ2	σ1σ2	NUM
ejpam-7049	188	29	-	-	PUNCT
ejpam-7049	188	30	closed	closed	ADJ
ejpam-7049	188	31	set	set	NOUN
ejpam-7049	188	32	k	k	PROPN
ejpam-7049	188	33	of	of	ADP
ejpam-7049	188	34	y	y	PROPN
ejpam-7049	188	35	.	.	PUNCT
ejpam-7049	189	1	(	(	PUNCT
ejpam-7049	189	2	11	11	NUM
ejpam-7049	189	3	)	)	PUNCT
ejpam-7049	189	4	⇒	⇒	NOUN
ejpam-7049	189	5	(	(	PUNCT
ejpam-7049	189	6	12	12	NUM
ejpam-7049	189	7	):	):	PUNCT
ejpam-7049	189	8	let	let	VERB
ejpam-7049	189	9	v	v	PART
ejpam-7049	189	10	be	be	AUX
ejpam-7049	189	11	any	any	DET
ejpam-7049	189	12	σ1σ2	σ1σ2	NOUN
ejpam-7049	189	13	-	-	ADJ
ejpam-7049	189	14	open	open	ADJ
ejpam-7049	189	15	set	set	NOUN
ejpam-7049	189	16	of	of	ADP
ejpam-7049	189	17	y	y	PROPN
ejpam-7049	189	18	.	.	PUNCT
ejpam-7049	190	1	then	then	ADV
ejpam-7049	190	2	,	,	PUNCT
ejpam-7049	190	3	y	y	PROPN
ejpam-7049	190	4	−	−	PROPN
ejpam-7049	190	5	v	v	NOUN
ejpam-7049	190	6	is	be	AUX
ejpam-7049	190	7	σ1σ2	σ1σ2	NOUN
ejpam-7049	190	8	-	-	ADJ
ejpam-7049	190	9	closed	closed	ADJ
ejpam-7049	190	10	in	in	ADP
ejpam-7049	190	11	y	y	PROPN
ejpam-7049	190	12	and	and	CCONJ
ejpam-7049	190	13	by	by	ADP
ejpam-7049	190	14	(	(	PUNCT
ejpam-7049	190	15	11	11	NUM
ejpam-7049	190	16	)	)	PUNCT
ejpam-7049	190	17	,	,	PUNCT
ejpam-7049	190	18	int⋆(cl⋆(int⋆(f−((σ1	int⋆(cl⋆(int⋆(f−((σ1	NOUN
ejpam-7049	190	19	,	,	PUNCT
ejpam-7049	190	20	σ2)-sint(y	σ2)-sint(y	VERB
ejpam-7049	190	21	−	−	PROPN
ejpam-7049	190	22	v	v	NOUN
ejpam-7049	190	23	)	)	PUNCT
ejpam-7049	190	24	)	)	PUNCT
ejpam-7049	190	25	)	)	PUNCT
ejpam-7049	190	26	)	)	PUNCT
ejpam-7049	190	27	)	)	PUNCT
ejpam-7049	191	1	⊆	⊆	NUM
ejpam-7049	191	2	f−(y	f−(y	NOUN
ejpam-7049	191	3	−	−	NOUN
ejpam-7049	191	4	v	v	NOUN
ejpam-7049	191	5	)	)	PUNCT
ejpam-7049	191	6	=	=	PUNCT
ejpam-7049	191	7	x	x	X
ejpam-7049	191	8	−	−	PROPN
ejpam-7049	191	9	f+(v	f+(v	NOUN
ejpam-7049	191	10	)	)	PUNCT
ejpam-7049	191	11	.	.	PUNCT
ejpam-7049	192	1	n.	n.	PROPN
ejpam-7049	192	2	srisarakham	srisarakham	PROPN
ejpam-7049	192	3	,	,	PUNCT
ejpam-7049	192	4	a.	a.	PROPN
ejpam-7049	192	5	sama	sama	PROPN
ejpam-7049	192	6	-	-	PUNCT
ejpam-7049	192	7	ae	ae	PROPN
ejpam-7049	192	8	,	,	PUNCT
ejpam-7049	192	9	c.	c.	PROPN
ejpam-7049	192	10	boonpok	boonpok	PROPN
ejpam-7049	192	11	/	/	SYM
ejpam-7049	192	12	eur	eur	PROPN
ejpam-7049	192	13	.	.	PUNCT
ejpam-7049	193	1	j.	j.	PROPN
ejpam-7049	193	2	pure	pure	PROPN
ejpam-7049	193	3	appl	appl	PROPN
ejpam-7049	193	4	.	.	PROPN
ejpam-7049	193	5	math	math	PROPN
ejpam-7049	193	6	,	,	PUNCT
ejpam-7049	193	7	18	18	NUM
ejpam-7049	193	8	(	(	PUNCT
ejpam-7049	193	9	4	4	NUM
ejpam-7049	193	10	)	)	PUNCT
ejpam-7049	193	11	(	(	PUNCT
ejpam-7049	193	12	2025	2025	NUM
ejpam-7049	193	13	)	)	PUNCT
ejpam-7049	193	14	,	,	PUNCT
ejpam-7049	193	15	7049	7049	NUM
ejpam-7049	193	16	7	7	NUM
ejpam-7049	193	17	of	of	ADP
ejpam-7049	193	18	11	11	NUM
ejpam-7049	193	19	moreover	moreover	ADV
ejpam-7049	193	20	,	,	PUNCT
ejpam-7049	193	21	we	we	PRON
ejpam-7049	193	22	have	have	AUX
ejpam-7049	193	23	int⋆(cl⋆(int⋆(f−((σ1	int⋆(cl⋆(int⋆(f−((σ1	NOUN
ejpam-7049	193	24	,	,	PUNCT
ejpam-7049	193	25	σ2)-sint(y	σ2)-sint(y	VERB
ejpam-7049	193	26	−	−	PROPN
ejpam-7049	193	27	v	v	NOUN
ejpam-7049	193	28	)	)	PUNCT
ejpam-7049	193	29	)	)	PUNCT
ejpam-7049	193	30	)	)	PUNCT
ejpam-7049	193	31	)	)	PUNCT
ejpam-7049	193	32	)	)	PUNCT
ejpam-7049	194	1	=	=	PUNCT
ejpam-7049	195	1	int⋆(cl⋆(int⋆(f−(y	int⋆(cl⋆(int⋆(f−(y	ADJ
ejpam-7049	195	2	−	−	PROPN
ejpam-7049	195	3	(	(	PUNCT
ejpam-7049	195	4	σ1	σ1	PROPN
ejpam-7049	195	5	,	,	PUNCT
ejpam-7049	195	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	195	7	)	)	PUNCT
ejpam-7049	195	8	)	)	PUNCT
ejpam-7049	195	9	)	)	PUNCT
ejpam-7049	195	10	)	)	PUNCT
ejpam-7049	195	11	)	)	PUNCT
ejpam-7049	196	1	=	=	PRON
ejpam-7049	196	2	int⋆(cl⋆(int⋆(x	int⋆(cl⋆(int⋆(x	VERB
ejpam-7049	196	3	−	−	NOUN
ejpam-7049	196	4	f+((σ1	f+((σ1	NOUN
ejpam-7049	196	5	,	,	PUNCT
ejpam-7049	196	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	196	7	)	)	PUNCT
ejpam-7049	196	8	)	)	PUNCT
ejpam-7049	196	9	)	)	PUNCT
ejpam-7049	196	10	)	)	PUNCT
ejpam-7049	196	11	)	)	PUNCT
ejpam-7049	197	1	=	=	PUNCT
ejpam-7049	197	2	x	x	X
ejpam-7049	197	3	−	−	NOUN
ejpam-7049	197	4	cl⋆(int⋆(cl⋆(f+((σ1	cl⋆(int⋆(cl⋆(f+((σ1	NOUN
ejpam-7049	197	5	,	,	PUNCT
ejpam-7049	197	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	197	7	)	)	PUNCT
ejpam-7049	197	8	)	)	PUNCT
ejpam-7049	197	9	)	)	PUNCT
ejpam-7049	197	10	)	)	PUNCT
ejpam-7049	197	11	)	)	PUNCT
ejpam-7049	197	12	.	.	PUNCT
ejpam-7049	198	1	thus	thus	ADV
ejpam-7049	198	2	,	,	PUNCT
ejpam-7049	198	3	f+(v	f+(v	PROPN
ejpam-7049	198	4	)	)	PUNCT
ejpam-7049	198	5	⊆	⊆	NUM
ejpam-7049	198	6	cl⋆(int⋆(cl⋆(f+((σ1	cl⋆(int⋆(cl⋆(f+((σ1	NOUN
ejpam-7049	198	7	,	,	PUNCT
ejpam-7049	198	8	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	198	9	)	)	PUNCT
ejpam-7049	198	10	)	)	PUNCT
ejpam-7049	198	11	)	)	PUNCT
ejpam-7049	198	12	)	)	PUNCT
ejpam-7049	198	13	)	)	PUNCT
ejpam-7049	198	14	.	.	PUNCT
ejpam-7049	199	1	(	(	PUNCT
ejpam-7049	199	2	12	12	NUM
ejpam-7049	199	3	)	)	PUNCT
ejpam-7049	199	4	⇒	⇒	NOUN
ejpam-7049	199	5	(	(	PUNCT
ejpam-7049	199	6	1	1	NUM
ejpam-7049	199	7	):	):	PUNCT
ejpam-7049	199	8	let	let	VERB
ejpam-7049	199	9	x	x	PRON
ejpam-7049	199	10	be	be	AUX
ejpam-7049	199	11	any	any	DET
ejpam-7049	199	12	point	point	NOUN
ejpam-7049	199	13	of	of	ADP
ejpam-7049	199	14	x	x	PUNCT
ejpam-7049	199	15	and	and	CCONJ
ejpam-7049	199	16	v	v	AUX
ejpam-7049	199	17	be	be	AUX
ejpam-7049	199	18	any	any	DET
ejpam-7049	199	19	σ1σ2	σ1σ2	NOUN
ejpam-7049	199	20	-	-	ADJ
ejpam-7049	199	21	open	open	ADJ
ejpam-7049	199	22	set	set	NOUN
ejpam-7049	199	23	of	of	ADP
ejpam-7049	199	24	y	y	PROPN
ejpam-7049	199	25	containing	contain	VERB
ejpam-7049	199	26	f	f	PROPN
ejpam-7049	199	27	(	(	PUNCT
ejpam-7049	199	28	x	x	NOUN
ejpam-7049	199	29	)	)	PUNCT
ejpam-7049	199	30	.	.	PUNCT
ejpam-7049	200	1	then	then	ADV
ejpam-7049	200	2	,	,	PUNCT
ejpam-7049	200	3	we	we	PRON
ejpam-7049	200	4	have	have	VERB
ejpam-7049	200	5	x	x	X
ejpam-7049	200	6	∈	∈	NOUN
ejpam-7049	200	7	f+(v	f+(v	NOUN
ejpam-7049	200	8	)	)	PUNCT
ejpam-7049	201	1	⊆	⊆	NUM
ejpam-7049	201	2	cl⋆(int⋆(cl⋆(f+((σ1	cl⋆(int⋆(cl⋆(f+((σ1	NOUN
ejpam-7049	201	3	,	,	PUNCT
ejpam-7049	201	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	201	5	)	)	PUNCT
ejpam-7049	201	6	)	)	PUNCT
ejpam-7049	201	7	)	)	PUNCT
ejpam-7049	201	8	)	)	PUNCT
ejpam-7049	201	9	)	)	PUNCT
ejpam-7049	201	10	and	and	CCONJ
ejpam-7049	201	11	hence	hence	ADV
ejpam-7049	201	12	x	x	X
ejpam-7049	201	13	∈	∈	NOUN
ejpam-7049	201	14	βint⋆(f+((σ1	βint⋆(f+((σ1	NOUN
ejpam-7049	201	15	,	,	PUNCT
ejpam-7049	201	16	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	201	17	)	)	PUNCT
ejpam-7049	201	18	)	)	PUNCT
ejpam-7049	201	19	)	)	PUNCT
ejpam-7049	201	20	.	.	PUNCT
ejpam-7049	202	1	thus	thus	ADV
ejpam-7049	202	2	,	,	PUNCT
ejpam-7049	202	3	f	f	PROPN
ejpam-7049	202	4	is	be	AUX
ejpam-7049	202	5	upper	upper	ADJ
ejpam-7049	202	6	almost	almost	ADV
ejpam-7049	202	7	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	202	8	,	,	PUNCT
ejpam-7049	202	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	202	10	at	at	ADP
ejpam-7049	202	11	x	x	PUNCT
ejpam-7049	202	12	by	by	ADP
ejpam-7049	202	13	theorem	theorem	NOUN
ejpam-7049	202	14	1	1	NUM
ejpam-7049	202	15	.	.	PUNCT
ejpam-7049	202	16	theorem	theorem	NOUN
ejpam-7049	202	17	4	4	NUM
ejpam-7049	202	18	.	.	X
ejpam-7049	202	19	for	for	ADP
ejpam-7049	202	20	a	a	DET
ejpam-7049	202	21	multifunction	multifunction	NOUN
ejpam-7049	202	22	f	f	NOUN
ejpam-7049	202	23	:	:	PUNCT
ejpam-7049	202	24	(	(	PUNCT
ejpam-7049	202	25	x	x	X
ejpam-7049	202	26	,	,	PUNCT
ejpam-7049	202	27	τ	τ	PROPN
ejpam-7049	202	28	,	,	PUNCT
ejpam-7049	202	29	i	i	NOUN
ejpam-7049	202	30	)	)	PUNCT
ejpam-7049	202	31	→	→	PUNCT
ejpam-7049	202	32	(	(	PUNCT
ejpam-7049	202	33	y	y	PROPN
ejpam-7049	202	34	,	,	PUNCT
ejpam-7049	202	35	σ1	σ1	PROPN
ejpam-7049	202	36	,	,	PUNCT
ejpam-7049	202	37	σ2	σ2	NOUN
ejpam-7049	202	38	)	)	PUNCT
ejpam-7049	202	39	,	,	PUNCT
ejpam-7049	202	40	the	the	DET
ejpam-7049	202	41	following	follow	VERB
ejpam-7049	202	42	properties	property	NOUN
ejpam-7049	202	43	are	be	AUX
ejpam-7049	202	44	equivalent	equivalent	ADJ
ejpam-7049	202	45	:	:	PUNCT
ejpam-7049	202	46	(	(	PUNCT
ejpam-7049	202	47	1	1	X
ejpam-7049	202	48	)	)	PUNCT
ejpam-7049	202	49	f	f	PROPN
ejpam-7049	202	50	is	be	AUX
ejpam-7049	202	51	lower	low	ADJ
ejpam-7049	202	52	almost	almost	ADV
ejpam-7049	202	53	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	202	54	,	,	PUNCT
ejpam-7049	202	55	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	202	56	;	;	PUNCT
ejpam-7049	202	57	(	(	PUNCT
ejpam-7049	202	58	2	2	X
ejpam-7049	202	59	)	)	PUNCT
ejpam-7049	202	60	for	for	ADP
ejpam-7049	202	61	each	each	DET
ejpam-7049	202	62	x	x	SYM
ejpam-7049	202	63	∈	∈	PROPN
ejpam-7049	202	64	x	x	X
ejpam-7049	202	65	and	and	CCONJ
ejpam-7049	202	66	each	each	DET
ejpam-7049	202	67	σ1σ2	σ1σ2	VERB
ejpam-7049	202	68	-	-	ADJ
ejpam-7049	202	69	open	open	ADJ
ejpam-7049	202	70	set	set	NOUN
ejpam-7049	202	71	v	v	NOUN
ejpam-7049	202	72	of	of	ADP
ejpam-7049	202	73	y	y	PRON
ejpam-7049	202	74	such	such	ADJ
ejpam-7049	202	75	that	that	SCONJ
ejpam-7049	202	76	f	f	PROPN
ejpam-7049	202	77	(	(	PUNCT
ejpam-7049	202	78	x	x	NOUN
ejpam-7049	202	79	)	)	PUNCT
ejpam-7049	202	80	∩	∩	NOUN
ejpam-7049	202	81	v	v	ADP
ejpam-7049	202	82	̸=	̸=	PROPN
ejpam-7049	202	83	∅	∅	NOUN
ejpam-7049	202	84	,	,	PUNCT
ejpam-7049	202	85	there	there	PRON
ejpam-7049	202	86	exists	exist	VERB
ejpam-7049	202	87	a	a	DET
ejpam-7049	202	88	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	202	89	-	-	ADJ
ejpam-7049	202	90	open	open	ADJ
ejpam-7049	202	91	set	set	NOUN
ejpam-7049	202	92	u	u	NOUN
ejpam-7049	202	93	of	of	ADP
ejpam-7049	202	94	x	x	PUNCT
ejpam-7049	202	95	containing	contain	VERB
ejpam-7049	202	96	x	x	PUNCT
ejpam-7049	202	97	such	such	ADJ
ejpam-7049	202	98	that	that	SCONJ
ejpam-7049	202	99	u	u	NOUN
ejpam-7049	202	100	⊆	⊆	NUM
ejpam-7049	202	101	f−((σ1	f−((σ1	NOUN
ejpam-7049	202	102	,	,	PUNCT
ejpam-7049	202	103	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	202	104	)	)	PUNCT
ejpam-7049	202	105	)	)	PUNCT
ejpam-7049	202	106	;	;	PUNCT
ejpam-7049	202	107	(	(	PUNCT
ejpam-7049	202	108	3	3	X
ejpam-7049	202	109	)	)	PUNCT
ejpam-7049	202	110	for	for	ADP
ejpam-7049	202	111	each	each	DET
ejpam-7049	202	112	x	x	SYM
ejpam-7049	202	113	∈	∈	PROPN
ejpam-7049	202	114	x	x	X
ejpam-7049	202	115	and	and	CCONJ
ejpam-7049	202	116	each	each	DET
ejpam-7049	202	117	(	(	PUNCT
ejpam-7049	202	118	σ1	σ1	PROPN
ejpam-7049	202	119	,	,	PUNCT
ejpam-7049	202	120	σ2)r	σ2)r	NOUN
ejpam-7049	202	121	-	-	PUNCT
ejpam-7049	202	122	open	open	ADJ
ejpam-7049	202	123	set	set	VERB
ejpam-7049	202	124	v	v	NOUN
ejpam-7049	202	125	of	of	ADP
ejpam-7049	202	126	y	y	PRON
ejpam-7049	202	127	such	such	ADJ
ejpam-7049	202	128	that	that	SCONJ
ejpam-7049	202	129	f	f	PROPN
ejpam-7049	202	130	(	(	PUNCT
ejpam-7049	202	131	x	x	NOUN
ejpam-7049	202	132	)	)	PUNCT
ejpam-7049	202	133	∩	∩	NOUN
ejpam-7049	202	134	v	v	ADP
ejpam-7049	202	135	̸=	̸=	PROPN
ejpam-7049	202	136	∅	∅	NOUN
ejpam-7049	202	137	,	,	PUNCT
ejpam-7049	202	138	there	there	PRON
ejpam-7049	202	139	exists	exist	VERB
ejpam-7049	202	140	a	a	DET
ejpam-7049	202	141	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	202	142	-	-	ADJ
ejpam-7049	202	143	open	open	ADJ
ejpam-7049	202	144	set	set	NOUN
ejpam-7049	202	145	u	u	NOUN
ejpam-7049	202	146	of	of	ADP
ejpam-7049	202	147	x	x	PUNCT
ejpam-7049	202	148	containing	contain	VERB
ejpam-7049	202	149	x	x	PUNCT
ejpam-7049	202	150	such	such	ADJ
ejpam-7049	202	151	that	that	SCONJ
ejpam-7049	202	152	u	u	NOUN
ejpam-7049	202	153	⊆	⊆	NUM
ejpam-7049	202	154	f−(v	f−(v	NOUN
ejpam-7049	202	155	)	)	PUNCT
ejpam-7049	202	156	;	;	PUNCT
ejpam-7049	202	157	(	(	PUNCT
ejpam-7049	202	158	4	4	X
ejpam-7049	202	159	)	)	PUNCT
ejpam-7049	202	160	f−(v	f−(v	NOUN
ejpam-7049	202	161	)	)	PUNCT
ejpam-7049	202	162	is	be	AUX
ejpam-7049	202	163	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	202	164	-	-	NOUN
ejpam-7049	202	165	open	open	ADJ
ejpam-7049	202	166	in	in	ADP
ejpam-7049	202	167	x	x	PUNCT
ejpam-7049	202	168	for	for	ADP
ejpam-7049	202	169	every	every	DET
ejpam-7049	202	170	(	(	PUNCT
ejpam-7049	202	171	σ1	σ1	PROPN
ejpam-7049	202	172	,	,	PUNCT
ejpam-7049	202	173	σ2)r	σ2)r	NOUN
ejpam-7049	202	174	-	-	PUNCT
ejpam-7049	202	175	open	open	ADJ
ejpam-7049	202	176	set	set	VERB
ejpam-7049	202	177	v	v	NOUN
ejpam-7049	202	178	of	of	ADP
ejpam-7049	202	179	y	y	PROPN
ejpam-7049	202	180	;	;	PUNCT
ejpam-7049	202	181	(	(	PUNCT
ejpam-7049	202	182	5	5	X
ejpam-7049	202	183	)	)	PUNCT
ejpam-7049	202	184	f+(k	f+(k	PROPN
ejpam-7049	202	185	)	)	PUNCT
ejpam-7049	202	186	is	be	AUX
ejpam-7049	202	187	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	202	188	-	-	VERB
ejpam-7049	202	189	closed	closed	ADJ
ejpam-7049	202	190	in	in	ADP
ejpam-7049	202	191	x	x	PUNCT
ejpam-7049	202	192	for	for	ADP
ejpam-7049	202	193	every	every	DET
ejpam-7049	202	194	(	(	PUNCT
ejpam-7049	202	195	σ1	σ1	PROPN
ejpam-7049	202	196	,	,	PUNCT
ejpam-7049	203	1	σ2)r	σ2)r	NOUN
ejpam-7049	203	2	-	-	PUNCT
ejpam-7049	203	3	closed	close	VERB
ejpam-7049	203	4	set	set	ADJ
ejpam-7049	203	5	k	k	PROPN
ejpam-7049	203	6	of	of	ADP
ejpam-7049	203	7	y	y	PROPN
ejpam-7049	203	8	;	;	PUNCT
ejpam-7049	203	9	(	(	PUNCT
ejpam-7049	203	10	6	6	X
ejpam-7049	203	11	)	)	PUNCT
ejpam-7049	203	12	f−(v	f−(v	NOUN
ejpam-7049	203	13	)	)	PUNCT
ejpam-7049	203	14	⊆	⊆	NUM
ejpam-7049	203	15	βint⋆(f−((σ1	βint⋆(f−((σ1	ADP
ejpam-7049	203	16	,	,	PUNCT
ejpam-7049	203	17	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	203	18	)	)	PUNCT
ejpam-7049	203	19	)	)	PUNCT
ejpam-7049	203	20	)	)	PUNCT
ejpam-7049	204	1	for	for	ADP
ejpam-7049	204	2	every	every	DET
ejpam-7049	204	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	204	4	-	-	ADJ
ejpam-7049	204	5	open	open	ADJ
ejpam-7049	204	6	set	set	NOUN
ejpam-7049	204	7	v	v	NOUN
ejpam-7049	204	8	of	of	ADP
ejpam-7049	204	9	y	y	PROPN
ejpam-7049	204	10	;	;	PUNCT
ejpam-7049	204	11	(	(	PUNCT
ejpam-7049	204	12	7	7	X
ejpam-7049	204	13	)	)	PUNCT
ejpam-7049	204	14	βcl⋆(f+((σ1	βcl⋆(f+((σ1	NOUN
ejpam-7049	204	15	,	,	PUNCT
ejpam-7049	204	16	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	204	17	)	)	PUNCT
ejpam-7049	204	18	)	)	PUNCT
ejpam-7049	204	19	)	)	PUNCT
ejpam-7049	205	1	⊆	⊆	NUM
ejpam-7049	205	2	f+(k	f+(k	NOUN
ejpam-7049	205	3	)	)	PUNCT
ejpam-7049	205	4	for	for	ADP
ejpam-7049	205	5	every	every	DET
ejpam-7049	205	6	σ1σ2	σ1σ2	NUM
ejpam-7049	205	7	-	-	PUNCT
ejpam-7049	205	8	closed	closed	ADJ
ejpam-7049	205	9	set	set	NOUN
ejpam-7049	205	10	k	k	PROPN
ejpam-7049	205	11	of	of	ADP
ejpam-7049	205	12	y	y	PROPN
ejpam-7049	205	13	;	;	PUNCT
ejpam-7049	205	14	(	(	PUNCT
ejpam-7049	205	15	8)	8)	NUM
ejpam-7049	205	16	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7049	205	17	-	-	PUNCT
ejpam-7049	205	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	205	19	-	-	PUNCT
ejpam-7049	205	20	int(k	int(k	NOUN
ejpam-7049	205	21	)	)	PUNCT
ejpam-7049	205	22	)	)	PUNCT
ejpam-7049	205	23	)	)	PUNCT
ejpam-7049	205	24	)	)	PUNCT
ejpam-7049	206	1	⊆	⊆	NUM
ejpam-7049	206	2	f+(k	f+(k	NOUN
ejpam-7049	206	3	)	)	PUNCT
ejpam-7049	206	4	for	for	ADP
ejpam-7049	206	5	every	every	DET
ejpam-7049	206	6	σ1σ2	σ1σ2	NUM
ejpam-7049	206	7	-	-	PUNCT
ejpam-7049	206	8	closed	closed	ADJ
ejpam-7049	206	9	set	set	NOUN
ejpam-7049	206	10	k	k	PROPN
ejpam-7049	206	11	of	of	ADP
ejpam-7049	206	12	y	y	PROPN
ejpam-7049	206	13	;	;	PUNCT
ejpam-7049	206	14	(	(	PUNCT
ejpam-7049	206	15	9	9	X
ejpam-7049	206	16	)	)	PUNCT
ejpam-7049	206	17	βcl⋆(f+(σ1σ2	βcl⋆(f+(σ1σ2	NOUN
ejpam-7049	206	18	-	-	PUNCT
ejpam-7049	206	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	206	20	-	-	PUNCT
ejpam-7049	206	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	206	22	-	-	PUNCT
ejpam-7049	206	23	cl(b	cl(b	NOUN
ejpam-7049	206	24	)	)	PUNCT
ejpam-7049	206	25	)	)	PUNCT
ejpam-7049	206	26	)	)	PUNCT
ejpam-7049	206	27	)	)	PUNCT
ejpam-7049	206	28	)	)	PUNCT
ejpam-7049	207	1	⊆	⊆	X
ejpam-7049	207	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7049	207	3	-	-	PUNCT
ejpam-7049	207	4	cl(b	cl(b	NOUN
ejpam-7049	207	5	)	)	PUNCT
ejpam-7049	207	6	)	)	PUNCT
ejpam-7049	207	7	for	for	ADP
ejpam-7049	207	8	every	every	DET
ejpam-7049	207	9	subset	subset	NOUN
ejpam-7049	207	10	b	b	PROPN
ejpam-7049	207	11	of	of	ADP
ejpam-7049	207	12	y	y	PROPN
ejpam-7049	207	13	;	;	PUNCT
ejpam-7049	207	14	(	(	PUNCT
ejpam-7049	207	15	10	10	NUM
ejpam-7049	207	16	)	)	PUNCT
ejpam-7049	207	17	int⋆(cl⋆(int⋆(f+(σ1σ2	int⋆(cl⋆(int⋆(f+(σ1σ2	NOUN
ejpam-7049	207	18	-	-	PUNCT
ejpam-7049	207	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	207	20	-	-	PUNCT
ejpam-7049	207	21	int(k	int(k	NOUN
ejpam-7049	207	22	)	)	PUNCT
ejpam-7049	207	23	)	)	PUNCT
ejpam-7049	207	24	)	)	PUNCT
ejpam-7049	207	25	)	)	PUNCT
ejpam-7049	207	26	)	)	PUNCT
ejpam-7049	207	27	)	)	PUNCT
ejpam-7049	208	1	⊆	⊆	NUM
ejpam-7049	208	2	f+(k	f+(k	NOUN
ejpam-7049	208	3	)	)	PUNCT
ejpam-7049	208	4	for	for	ADP
ejpam-7049	208	5	every	every	DET
ejpam-7049	208	6	σ1σ2	σ1σ2	NUM
ejpam-7049	208	7	-	-	PUNCT
ejpam-7049	208	8	closed	closed	ADJ
ejpam-7049	208	9	set	set	NOUN
ejpam-7049	208	10	k	k	PROPN
ejpam-7049	208	11	of	of	ADP
ejpam-7049	208	12	y	y	PROPN
ejpam-7049	208	13	;	;	PUNCT
ejpam-7049	208	14	(	(	PUNCT
ejpam-7049	208	15	11	11	X
ejpam-7049	208	16	)	)	PUNCT
ejpam-7049	208	17	int⋆(cl⋆(int⋆(f+((σ1	int⋆(cl⋆(int⋆(f+((σ1	NOUN
ejpam-7049	208	18	,	,	PUNCT
ejpam-7049	208	19	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	208	20	)	)	PUNCT
ejpam-7049	208	21	)	)	PUNCT
ejpam-7049	208	22	)	)	PUNCT
ejpam-7049	208	23	)	)	PUNCT
ejpam-7049	208	24	)	)	PUNCT
ejpam-7049	209	1	⊆	⊆	NUM
ejpam-7049	209	2	f+(k	f+(k	NOUN
ejpam-7049	209	3	)	)	PUNCT
ejpam-7049	209	4	for	for	ADP
ejpam-7049	209	5	every	every	DET
ejpam-7049	209	6	σ1σ2	σ1σ2	NUM
ejpam-7049	209	7	-	-	PUNCT
ejpam-7049	209	8	closed	closed	ADJ
ejpam-7049	209	9	set	set	NOUN
ejpam-7049	209	10	k	k	PROPN
ejpam-7049	209	11	of	of	ADP
ejpam-7049	209	12	y	y	PROPN
ejpam-7049	209	13	;	;	PUNCT
ejpam-7049	209	14	(	(	PUNCT
ejpam-7049	209	15	12	12	X
ejpam-7049	209	16	)	)	PUNCT
ejpam-7049	209	17	f−(v	f−(v	NOUN
ejpam-7049	209	18	)	)	PUNCT
ejpam-7049	209	19	⊆	⊆	NUM
ejpam-7049	209	20	cl⋆(int⋆(cl⋆(f−((σ1	cl⋆(int⋆(cl⋆(f−((σ1	NOUN
ejpam-7049	209	21	,	,	PUNCT
ejpam-7049	209	22	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	209	23	)	)	PUNCT
ejpam-7049	209	24	)	)	PUNCT
ejpam-7049	209	25	)	)	PUNCT
ejpam-7049	209	26	)	)	PUNCT
ejpam-7049	209	27	)	)	PUNCT
ejpam-7049	210	1	for	for	ADP
ejpam-7049	210	2	every	every	DET
ejpam-7049	210	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	210	4	-	-	ADJ
ejpam-7049	210	5	open	open	ADJ
ejpam-7049	210	6	set	set	NOUN
ejpam-7049	210	7	v	v	NOUN
ejpam-7049	210	8	of	of	ADP
ejpam-7049	210	9	y	y	PROPN
ejpam-7049	210	10	.	.	PUNCT
ejpam-7049	211	1	proof	proof	NOUN
ejpam-7049	211	2	.	.	PUNCT
ejpam-7049	212	1	the	the	DET
ejpam-7049	212	2	proof	proof	NOUN
ejpam-7049	212	3	is	be	AUX
ejpam-7049	212	4	similar	similar	ADJ
ejpam-7049	212	5	to	to	ADP
ejpam-7049	212	6	that	that	PRON
ejpam-7049	212	7	of	of	ADP
ejpam-7049	212	8	theorem	theorem	ADJ
ejpam-7049	212	9	3	3	NUM
ejpam-7049	212	10	.	.	PUNCT
ejpam-7049	212	11	corollary	corollary	ADJ
ejpam-7049	212	12	2	2	NUM
ejpam-7049	212	13	.	.	PUNCT
ejpam-7049	212	14	for	for	ADP
ejpam-7049	212	15	a	a	DET
ejpam-7049	212	16	function	function	NOUN
ejpam-7049	212	17	f	f	NOUN
ejpam-7049	212	18	:	:	PUNCT
ejpam-7049	212	19	(	(	PUNCT
ejpam-7049	212	20	x	x	X
ejpam-7049	212	21	,	,	PUNCT
ejpam-7049	212	22	τ	τ	PROPN
ejpam-7049	212	23	,	,	PUNCT
ejpam-7049	212	24	i	i	NOUN
ejpam-7049	212	25	)	)	PUNCT
ejpam-7049	212	26	→	→	PUNCT
ejpam-7049	212	27	(	(	PUNCT
ejpam-7049	212	28	y	y	PROPN
ejpam-7049	212	29	,	,	PUNCT
ejpam-7049	212	30	σ1	σ1	PROPN
ejpam-7049	212	31	,	,	PUNCT
ejpam-7049	212	32	σ2	σ2	NOUN
ejpam-7049	212	33	)	)	PUNCT
ejpam-7049	212	34	,	,	PUNCT
ejpam-7049	212	35	the	the	DET
ejpam-7049	212	36	following	follow	VERB
ejpam-7049	212	37	properties	property	NOUN
ejpam-7049	212	38	are	be	AUX
ejpam-7049	212	39	equivalent	equivalent	ADJ
ejpam-7049	212	40	:	:	PUNCT
ejpam-7049	212	41	n.	n.	PROPN
ejpam-7049	212	42	srisarakham	srisarakham	PROPN
ejpam-7049	212	43	,	,	PUNCT
ejpam-7049	212	44	a.	a.	PROPN
ejpam-7049	212	45	sama	sama	PROPN
ejpam-7049	212	46	-	-	PUNCT
ejpam-7049	212	47	ae	ae	PROPN
ejpam-7049	212	48	,	,	PUNCT
ejpam-7049	212	49	c.	c.	PROPN
ejpam-7049	212	50	boonpok	boonpok	PROPN
ejpam-7049	212	51	/	/	SYM
ejpam-7049	212	52	eur	eur	PROPN
ejpam-7049	212	53	.	.	PUNCT
ejpam-7049	213	1	j.	j.	PROPN
ejpam-7049	213	2	pure	pure	PROPN
ejpam-7049	213	3	appl	appl	PROPN
ejpam-7049	213	4	.	.	PROPN
ejpam-7049	213	5	math	math	PROPN
ejpam-7049	213	6	,	,	PUNCT
ejpam-7049	213	7	18	18	NUM
ejpam-7049	213	8	(	(	PUNCT
ejpam-7049	213	9	4	4	NUM
ejpam-7049	213	10	)	)	PUNCT
ejpam-7049	213	11	(	(	PUNCT
ejpam-7049	213	12	2025	2025	NUM
ejpam-7049	213	13	)	)	PUNCT
ejpam-7049	213	14	,	,	PUNCT
ejpam-7049	213	15	7049	7049	NUM
ejpam-7049	213	16	8	8	NUM
ejpam-7049	213	17	of	of	ADP
ejpam-7049	213	18	11	11	NUM
ejpam-7049	213	19	(	(	PUNCT
ejpam-7049	213	20	1	1	NUM
ejpam-7049	213	21	)	)	PUNCT
ejpam-7049	213	22	f	f	NOUN
ejpam-7049	213	23	is	be	AUX
ejpam-7049	213	24	almost	almost	ADV
ejpam-7049	213	25	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	213	26	,	,	PUNCT
ejpam-7049	213	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	213	28	;	;	PUNCT
ejpam-7049	213	29	(	(	PUNCT
ejpam-7049	213	30	2	2	X
ejpam-7049	213	31	)	)	PUNCT
ejpam-7049	213	32	for	for	ADP
ejpam-7049	213	33	each	each	DET
ejpam-7049	213	34	x	x	SYM
ejpam-7049	213	35	∈	∈	PROPN
ejpam-7049	213	36	x	x	X
ejpam-7049	213	37	and	and	CCONJ
ejpam-7049	213	38	each	each	DET
ejpam-7049	213	39	σ1σ2	σ1σ2	VERB
ejpam-7049	213	40	-	-	ADJ
ejpam-7049	213	41	open	open	ADJ
ejpam-7049	213	42	set	set	NOUN
ejpam-7049	213	43	v	v	NOUN
ejpam-7049	213	44	of	of	ADP
ejpam-7049	213	45	y	y	NOUN
ejpam-7049	213	46	containing	contain	VERB
ejpam-7049	213	47	f(x	f(x	PROPN
ejpam-7049	213	48	)	)	PUNCT
ejpam-7049	213	49	,	,	PUNCT
ejpam-7049	213	50	there	there	PRON
ejpam-7049	213	51	exists	exist	VERB
ejpam-7049	213	52	a	a	DET
ejpam-7049	213	53	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	213	54	-	-	ADJ
ejpam-7049	213	55	open	open	ADJ
ejpam-7049	213	56	set	set	NOUN
ejpam-7049	213	57	u	u	NOUN
ejpam-7049	213	58	of	of	ADP
ejpam-7049	213	59	x	x	PUNCT
ejpam-7049	213	60	containing	contain	VERB
ejpam-7049	213	61	x	x	PUNCT
ejpam-7049	213	62	such	such	ADJ
ejpam-7049	213	63	that	that	DET
ejpam-7049	213	64	f(u	f(u	PROPN
ejpam-7049	213	65	)	)	PUNCT
ejpam-7049	213	66	⊆	⊆	NUM
ejpam-7049	213	67	(	(	PUNCT
ejpam-7049	213	68	σ1	σ1	PROPN
ejpam-7049	213	69	,	,	PUNCT
ejpam-7049	213	70	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	213	71	)	)	PUNCT
ejpam-7049	213	72	;	;	PUNCT
ejpam-7049	213	73	(	(	PUNCT
ejpam-7049	213	74	3	3	X
ejpam-7049	213	75	)	)	PUNCT
ejpam-7049	213	76	for	for	ADP
ejpam-7049	213	77	each	each	DET
ejpam-7049	213	78	x	x	SYM
ejpam-7049	213	79	∈	∈	PROPN
ejpam-7049	213	80	x	x	X
ejpam-7049	213	81	and	and	CCONJ
ejpam-7049	213	82	each	each	DET
ejpam-7049	213	83	(	(	PUNCT
ejpam-7049	213	84	σ1	σ1	PROPN
ejpam-7049	213	85	,	,	PUNCT
ejpam-7049	213	86	σ2)r	σ2)r	NOUN
ejpam-7049	213	87	-	-	PUNCT
ejpam-7049	213	88	open	open	ADJ
ejpam-7049	213	89	set	set	VERB
ejpam-7049	213	90	v	v	NOUN
ejpam-7049	213	91	of	of	ADP
ejpam-7049	213	92	y	y	NOUN
ejpam-7049	213	93	containing	contain	VERB
ejpam-7049	213	94	f(x	f(x	PROPN
ejpam-7049	213	95	)	)	PUNCT
ejpam-7049	213	96	,	,	PUNCT
ejpam-7049	213	97	there	there	PRON
ejpam-7049	213	98	exists	exist	VERB
ejpam-7049	213	99	a	a	DET
ejpam-7049	213	100	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	213	101	-	-	ADJ
ejpam-7049	213	102	open	open	ADJ
ejpam-7049	213	103	set	set	NOUN
ejpam-7049	213	104	u	u	NOUN
ejpam-7049	213	105	of	of	ADP
ejpam-7049	213	106	x	x	PUNCT
ejpam-7049	213	107	containing	contain	VERB
ejpam-7049	213	108	x	x	PUNCT
ejpam-7049	213	109	such	such	ADJ
ejpam-7049	213	110	that	that	DET
ejpam-7049	213	111	f(u	f(u	PROPN
ejpam-7049	213	112	)	)	PUNCT
ejpam-7049	213	113	⊆	⊆	NUM
ejpam-7049	213	114	v	v	NOUN
ejpam-7049	213	115	;	;	PUNCT
ejpam-7049	213	116	(	(	PUNCT
ejpam-7049	213	117	4	4	X
ejpam-7049	213	118	)	)	PUNCT
ejpam-7049	213	119	f−1(v	f−1(v	NOUN
ejpam-7049	213	120	)	)	PUNCT
ejpam-7049	213	121	is	be	AUX
ejpam-7049	213	122	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	213	123	-	-	NOUN
ejpam-7049	213	124	open	open	ADJ
ejpam-7049	213	125	in	in	ADP
ejpam-7049	213	126	x	x	PUNCT
ejpam-7049	213	127	for	for	ADP
ejpam-7049	213	128	every	every	DET
ejpam-7049	213	129	(	(	PUNCT
ejpam-7049	213	130	σ1	σ1	PROPN
ejpam-7049	213	131	,	,	PUNCT
ejpam-7049	213	132	σ2)r	σ2)r	NOUN
ejpam-7049	213	133	-	-	PUNCT
ejpam-7049	213	134	open	open	ADJ
ejpam-7049	213	135	set	set	VERB
ejpam-7049	213	136	v	v	NOUN
ejpam-7049	213	137	of	of	ADP
ejpam-7049	213	138	y	y	PROPN
ejpam-7049	213	139	;	;	PUNCT
ejpam-7049	213	140	(	(	PUNCT
ejpam-7049	213	141	5	5	X
ejpam-7049	213	142	)	)	PUNCT
ejpam-7049	213	143	f−1(k	f−1(k	PROPN
ejpam-7049	213	144	)	)	PUNCT
ejpam-7049	213	145	is	be	AUX
ejpam-7049	213	146	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	213	147	-	-	VERB
ejpam-7049	213	148	closed	closed	ADJ
ejpam-7049	213	149	in	in	ADP
ejpam-7049	213	150	x	x	PUNCT
ejpam-7049	213	151	for	for	ADP
ejpam-7049	213	152	every	every	DET
ejpam-7049	213	153	(	(	PUNCT
ejpam-7049	213	154	σ1	σ1	PROPN
ejpam-7049	213	155	,	,	PUNCT
ejpam-7049	214	1	σ2)r	σ2)r	NOUN
ejpam-7049	214	2	-	-	PUNCT
ejpam-7049	214	3	closed	close	VERB
ejpam-7049	214	4	set	set	ADJ
ejpam-7049	214	5	k	k	PROPN
ejpam-7049	214	6	of	of	ADP
ejpam-7049	214	7	y	y	PROPN
ejpam-7049	214	8	;	;	PUNCT
ejpam-7049	214	9	(	(	PUNCT
ejpam-7049	214	10	6	6	X
ejpam-7049	214	11	)	)	PUNCT
ejpam-7049	214	12	f−1(v	f−1(v	NOUN
ejpam-7049	214	13	)	)	PUNCT
ejpam-7049	214	14	⊆	⊆	NUM
ejpam-7049	214	15	βint⋆(f−1((σ1	βint⋆(f−1((σ1	NUM
ejpam-7049	214	16	,	,	PUNCT
ejpam-7049	214	17	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	214	18	)	)	PUNCT
ejpam-7049	214	19	)	)	PUNCT
ejpam-7049	214	20	)	)	PUNCT
ejpam-7049	215	1	for	for	ADP
ejpam-7049	215	2	every	every	DET
ejpam-7049	215	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	215	4	-	-	ADJ
ejpam-7049	215	5	open	open	ADJ
ejpam-7049	215	6	set	set	NOUN
ejpam-7049	215	7	v	v	NOUN
ejpam-7049	215	8	of	of	ADP
ejpam-7049	215	9	y	y	PROPN
ejpam-7049	215	10	;	;	PUNCT
ejpam-7049	215	11	(	(	PUNCT
ejpam-7049	215	12	7	7	X
ejpam-7049	215	13	)	)	PUNCT
ejpam-7049	215	14	βcl⋆(f−1((σ1	βcl⋆(f−1((σ1	NOUN
ejpam-7049	215	15	,	,	PUNCT
ejpam-7049	215	16	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	215	17	)	)	PUNCT
ejpam-7049	215	18	)	)	PUNCT
ejpam-7049	215	19	)	)	PUNCT
ejpam-7049	216	1	⊆	⊆	NUM
ejpam-7049	216	2	f−1(k	f−1(k	PROPN
ejpam-7049	216	3	)	)	PUNCT
ejpam-7049	216	4	for	for	ADP
ejpam-7049	216	5	every	every	DET
ejpam-7049	216	6	σ1σ2	σ1σ2	NUM
ejpam-7049	216	7	-	-	PUNCT
ejpam-7049	216	8	closed	closed	ADJ
ejpam-7049	216	9	set	set	NOUN
ejpam-7049	216	10	k	k	PROPN
ejpam-7049	216	11	of	of	ADP
ejpam-7049	216	12	y	y	PROPN
ejpam-7049	216	13	;	;	PUNCT
ejpam-7049	216	14	(	(	PUNCT
ejpam-7049	216	15	8)	8)	NUM
ejpam-7049	216	16	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	X
ejpam-7049	216	17	-	-	PUNCT
ejpam-7049	216	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	216	19	-	-	PUNCT
ejpam-7049	216	20	int(k	int(k	NOUN
ejpam-7049	216	21	)	)	PUNCT
ejpam-7049	216	22	)	)	PUNCT
ejpam-7049	216	23	)	)	PUNCT
ejpam-7049	216	24	)	)	PUNCT
ejpam-7049	217	1	⊆	⊆	NUM
ejpam-7049	217	2	f−1(k	f−1(k	PROPN
ejpam-7049	217	3	)	)	PUNCT
ejpam-7049	217	4	for	for	ADP
ejpam-7049	217	5	every	every	DET
ejpam-7049	217	6	σ1σ2	σ1σ2	NUM
ejpam-7049	217	7	-	-	PUNCT
ejpam-7049	217	8	closed	closed	ADJ
ejpam-7049	217	9	set	set	NOUN
ejpam-7049	217	10	k	k	PROPN
ejpam-7049	217	11	of	of	ADP
ejpam-7049	217	12	y	y	PROPN
ejpam-7049	217	13	;	;	PUNCT
ejpam-7049	217	14	(	(	PUNCT
ejpam-7049	217	15	9	9	X
ejpam-7049	217	16	)	)	PUNCT
ejpam-7049	217	17	βcl⋆(f−1(σ1σ2	βcl⋆(f−1(σ1σ2	NOUN
ejpam-7049	217	18	-	-	PUNCT
ejpam-7049	217	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	217	20	-	-	PUNCT
ejpam-7049	217	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	217	22	-	-	PUNCT
ejpam-7049	217	23	cl(b	cl(b	NOUN
ejpam-7049	217	24	)	)	PUNCT
ejpam-7049	217	25	)	)	PUNCT
ejpam-7049	217	26	)	)	PUNCT
ejpam-7049	217	27	)	)	PUNCT
ejpam-7049	217	28	)	)	PUNCT
ejpam-7049	218	1	⊆	⊆	NUM
ejpam-7049	218	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7049	218	3	-	-	PUNCT
ejpam-7049	218	4	cl(b	cl(b	NOUN
ejpam-7049	218	5	)	)	PUNCT
ejpam-7049	218	6	)	)	PUNCT
ejpam-7049	218	7	for	for	ADP
ejpam-7049	218	8	every	every	DET
ejpam-7049	218	9	subset	subset	NOUN
ejpam-7049	218	10	b	b	PROPN
ejpam-7049	218	11	of	of	ADP
ejpam-7049	218	12	y	y	PROPN
ejpam-7049	218	13	;	;	PUNCT
ejpam-7049	218	14	(	(	PUNCT
ejpam-7049	218	15	10	10	X
ejpam-7049	218	16	)	)	PUNCT
ejpam-7049	218	17	int⋆(cl⋆(int⋆(f−1(σ1σ2	int⋆(cl⋆(int⋆(f−1(σ1σ2	NOUN
ejpam-7049	218	18	-	-	PUNCT
ejpam-7049	218	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	218	20	-	-	PUNCT
ejpam-7049	218	21	int(k	int(k	NOUN
ejpam-7049	218	22	)	)	PUNCT
ejpam-7049	218	23	)	)	PUNCT
ejpam-7049	218	24	)	)	PUNCT
ejpam-7049	218	25	)	)	PUNCT
ejpam-7049	218	26	)	)	PUNCT
ejpam-7049	218	27	)	)	PUNCT
ejpam-7049	218	28	⊆	⊆	NUM
ejpam-7049	218	29	f−1(k	f−1(k	PROPN
ejpam-7049	218	30	)	)	PUNCT
ejpam-7049	218	31	for	for	ADP
ejpam-7049	218	32	every	every	DET
ejpam-7049	218	33	σ1σ2	σ1σ2	NUM
ejpam-7049	218	34	-	-	PUNCT
ejpam-7049	218	35	closed	closed	ADJ
ejpam-7049	218	36	set	set	NOUN
ejpam-7049	218	37	k	k	PROPN
ejpam-7049	218	38	of	of	ADP
ejpam-7049	218	39	y	y	PROPN
ejpam-7049	218	40	;	;	PUNCT
ejpam-7049	218	41	(	(	PUNCT
ejpam-7049	218	42	11	11	X
ejpam-7049	218	43	)	)	PUNCT
ejpam-7049	218	44	int⋆(cl⋆(int⋆(f−1((σ1	int⋆(cl⋆(int⋆(f−1((σ1	NOUN
ejpam-7049	218	45	,	,	PUNCT
ejpam-7049	218	46	σ2)-sint(k	σ2)-sint(k	ADJ
ejpam-7049	218	47	)	)	PUNCT
ejpam-7049	218	48	)	)	PUNCT
ejpam-7049	218	49	)	)	PUNCT
ejpam-7049	218	50	)	)	PUNCT
ejpam-7049	218	51	)	)	PUNCT
ejpam-7049	219	1	⊆	⊆	NUM
ejpam-7049	219	2	f−1(k	f−1(k	PROPN
ejpam-7049	219	3	)	)	PUNCT
ejpam-7049	219	4	for	for	ADP
ejpam-7049	219	5	every	every	DET
ejpam-7049	219	6	σ1σ2	σ1σ2	NUM
ejpam-7049	219	7	-	-	PUNCT
ejpam-7049	219	8	closed	closed	ADJ
ejpam-7049	219	9	set	set	NOUN
ejpam-7049	219	10	k	k	PROPN
ejpam-7049	219	11	of	of	ADP
ejpam-7049	219	12	y	y	PROPN
ejpam-7049	219	13	;	;	PUNCT
ejpam-7049	219	14	(	(	PUNCT
ejpam-7049	219	15	12	12	X
ejpam-7049	219	16	)	)	PUNCT
ejpam-7049	219	17	f−1(v	f−1(v	NOUN
ejpam-7049	219	18	)	)	PUNCT
ejpam-7049	219	19	⊆	⊆	X
ejpam-7049	219	20	cl⋆(int⋆(cl⋆(f−1((σ1	cl⋆(int⋆(cl⋆(f−1((σ1	NOUN
ejpam-7049	219	21	,	,	PUNCT
ejpam-7049	219	22	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7049	219	23	)	)	PUNCT
ejpam-7049	219	24	)	)	PUNCT
ejpam-7049	219	25	)	)	PUNCT
ejpam-7049	219	26	)	)	PUNCT
ejpam-7049	219	27	)	)	PUNCT
ejpam-7049	220	1	for	for	ADP
ejpam-7049	220	2	every	every	DET
ejpam-7049	220	3	σ1σ2	σ1σ2	NOUN
ejpam-7049	220	4	-	-	ADJ
ejpam-7049	220	5	open	open	ADJ
ejpam-7049	220	6	set	set	NOUN
ejpam-7049	220	7	v	v	NOUN
ejpam-7049	220	8	of	of	ADP
ejpam-7049	220	9	y	y	PROPN
ejpam-7049	220	10	.	.	PUNCT
ejpam-7049	221	1	definition	definition	NOUN
ejpam-7049	221	2	4	4	NUM
ejpam-7049	221	3	.	.	PUNCT
ejpam-7049	222	1	[	[	X
ejpam-7049	222	2	24	24	NUM
ejpam-7049	222	3	]	]	PUNCT
ejpam-7049	222	4	a	a	DET
ejpam-7049	222	5	multifunction	multifunction	NOUN
ejpam-7049	222	6	f	f	NOUN
ejpam-7049	222	7	:	:	PUNCT
ejpam-7049	222	8	(	(	PUNCT
ejpam-7049	222	9	x	x	X
ejpam-7049	222	10	,	,	PUNCT
ejpam-7049	222	11	τ	τ	PROPN
ejpam-7049	222	12	,	,	PUNCT
ejpam-7049	222	13	i	i	NOUN
ejpam-7049	222	14	)	)	PUNCT
ejpam-7049	222	15	→	→	PUNCT
ejpam-7049	222	16	(	(	PUNCT
ejpam-7049	222	17	y	y	PROPN
ejpam-7049	222	18	,	,	PUNCT
ejpam-7049	222	19	σ1	σ1	PROPN
ejpam-7049	222	20	,	,	PUNCT
ejpam-7049	222	21	σ2	σ2	PROPN
ejpam-7049	222	22	)	)	PUNCT
ejpam-7049	222	23	is	be	AUX
ejpam-7049	222	24	said	say	VERB
ejpam-7049	222	25	to	to	PART
ejpam-7049	222	26	be	be	AUX
ejpam-7049	222	27	upper	upper	ADJ
ejpam-7049	222	28	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	222	29	,	,	PUNCT
ejpam-7049	222	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	222	31	at	at	ADP
ejpam-7049	222	32	a	a	DET
ejpam-7049	222	33	point	point	NOUN
ejpam-7049	222	34	x	x	PUNCT
ejpam-7049	222	35	of	of	ADP
ejpam-7049	222	36	x	x	PRON
ejpam-7049	222	37	if	if	SCONJ
ejpam-7049	222	38	for	for	ADP
ejpam-7049	222	39	each	each	DET
ejpam-7049	222	40	σ1σ2	σ1σ2	VERB
ejpam-7049	222	41	-	-	ADJ
ejpam-7049	222	42	open	open	ADJ
ejpam-7049	222	43	set	set	NOUN
ejpam-7049	222	44	v	v	NOUN
ejpam-7049	222	45	of	of	ADP
ejpam-7049	222	46	y	y	PRON
ejpam-7049	222	47	such	such	ADJ
ejpam-7049	222	48	that	that	SCONJ
ejpam-7049	222	49	f	f	PROPN
ejpam-7049	222	50	(	(	PUNCT
ejpam-7049	222	51	x	x	X
ejpam-7049	222	52	)	)	PUNCT
ejpam-7049	222	53	⊆	⊆	NUM
ejpam-7049	222	54	v	v	NOUN
ejpam-7049	222	55	,	,	PUNCT
ejpam-7049	222	56	there	there	PRON
ejpam-7049	222	57	exists	exist	VERB
ejpam-7049	222	58	a	a	DET
ejpam-7049	222	59	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	222	60	-	-	ADJ
ejpam-7049	222	61	open	open	ADJ
ejpam-7049	222	62	set	set	NOUN
ejpam-7049	222	63	u	u	NOUN
ejpam-7049	222	64	of	of	ADP
ejpam-7049	222	65	x	x	PUNCT
ejpam-7049	222	66	containing	contain	VERB
ejpam-7049	222	67	x	x	PUNCT
ejpam-7049	222	68	such	such	ADJ
ejpam-7049	222	69	that	that	SCONJ
ejpam-7049	222	70	f	f	PROPN
ejpam-7049	222	71	(	(	PUNCT
ejpam-7049	222	72	u	u	NOUN
ejpam-7049	222	73	)	)	PUNCT
ejpam-7049	222	74	⊆	⊆	NUM
ejpam-7049	222	75	v	v	NOUN
ejpam-7049	222	76	.	.	PUNCT
ejpam-7049	223	1	a	a	DET
ejpam-7049	223	2	multifunction	multifunction	NOUN
ejpam-7049	223	3	f	f	NOUN
ejpam-7049	223	4	:	:	PUNCT
ejpam-7049	223	5	(	(	PUNCT
ejpam-7049	223	6	x	x	X
ejpam-7049	223	7	,	,	PUNCT
ejpam-7049	223	8	τ	τ	PROPN
ejpam-7049	223	9	,	,	PUNCT
ejpam-7049	223	10	i	i	NOUN
ejpam-7049	223	11	)	)	PUNCT
ejpam-7049	223	12	→	→	PUNCT
ejpam-7049	223	13	(	(	PUNCT
ejpam-7049	223	14	y	y	PROPN
ejpam-7049	223	15	,	,	PUNCT
ejpam-7049	223	16	σ1	σ1	PROPN
ejpam-7049	223	17	,	,	PUNCT
ejpam-7049	223	18	σ2	σ2	PROPN
ejpam-7049	223	19	)	)	PUNCT
ejpam-7049	223	20	is	be	AUX
ejpam-7049	223	21	said	say	VERB
ejpam-7049	223	22	to	to	PART
ejpam-7049	223	23	be	be	AUX
ejpam-7049	223	24	upper	upper	ADJ
ejpam-7049	223	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	223	26	,	,	PUNCT
ejpam-7049	223	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	223	28	if	if	SCONJ
ejpam-7049	223	29	f	f	PROPN
ejpam-7049	223	30	is	be	AUX
ejpam-7049	223	31	upper	upper	ADJ
ejpam-7049	223	32	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	223	33	,	,	PUNCT
ejpam-7049	223	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	223	35	at	at	ADP
ejpam-7049	223	36	each	each	DET
ejpam-7049	223	37	point	point	NOUN
ejpam-7049	223	38	of	of	ADP
ejpam-7049	223	39	x.	x.	NOUN
ejpam-7049	223	40	definition	definition	NOUN
ejpam-7049	223	41	5	5	NUM
ejpam-7049	223	42	.	.	PUNCT
ejpam-7049	224	1	[	[	X
ejpam-7049	224	2	24	24	NUM
ejpam-7049	224	3	]	]	PUNCT
ejpam-7049	224	4	a	a	DET
ejpam-7049	224	5	multifunction	multifunction	NOUN
ejpam-7049	224	6	f	f	NOUN
ejpam-7049	224	7	:	:	PUNCT
ejpam-7049	224	8	(	(	PUNCT
ejpam-7049	224	9	x	x	X
ejpam-7049	224	10	,	,	PUNCT
ejpam-7049	224	11	τ	τ	PROPN
ejpam-7049	224	12	,	,	PUNCT
ejpam-7049	224	13	i	i	NOUN
ejpam-7049	224	14	)	)	PUNCT
ejpam-7049	224	15	→	→	PUNCT
ejpam-7049	224	16	(	(	PUNCT
ejpam-7049	224	17	y	y	PROPN
ejpam-7049	224	18	,	,	PUNCT
ejpam-7049	224	19	σ1	σ1	PROPN
ejpam-7049	224	20	,	,	PUNCT
ejpam-7049	224	21	σ2	σ2	PROPN
ejpam-7049	224	22	)	)	PUNCT
ejpam-7049	224	23	is	be	AUX
ejpam-7049	224	24	said	say	VERB
ejpam-7049	224	25	to	to	PART
ejpam-7049	224	26	be	be	AUX
ejpam-7049	224	27	lower	low	ADJ
ejpam-7049	224	28	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	224	29	,	,	PUNCT
ejpam-7049	224	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	224	31	at	at	ADP
ejpam-7049	224	32	a	a	DET
ejpam-7049	224	33	point	point	NOUN
ejpam-7049	224	34	x	x	PUNCT
ejpam-7049	224	35	of	of	ADP
ejpam-7049	224	36	x	x	PRON
ejpam-7049	224	37	if	if	SCONJ
ejpam-7049	224	38	for	for	ADP
ejpam-7049	224	39	each	each	DET
ejpam-7049	224	40	σ1σ2	σ1σ2	VERB
ejpam-7049	224	41	-	-	ADJ
ejpam-7049	224	42	open	open	ADJ
ejpam-7049	224	43	set	set	NOUN
ejpam-7049	224	44	v	v	NOUN
ejpam-7049	224	45	of	of	ADP
ejpam-7049	224	46	y	y	PRON
ejpam-7049	224	47	such	such	ADJ
ejpam-7049	224	48	that	that	SCONJ
ejpam-7049	224	49	f	f	PROPN
ejpam-7049	224	50	(	(	PUNCT
ejpam-7049	224	51	x)∩v	x)∩v	PROPN
ejpam-7049	224	52	̸=	̸=	PROPN
ejpam-7049	224	53	∅	∅	NOUN
ejpam-7049	224	54	,	,	PUNCT
ejpam-7049	224	55	there	there	PRON
ejpam-7049	224	56	exists	exist	VERB
ejpam-7049	224	57	a	a	DET
ejpam-7049	224	58	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	224	59	-	-	ADJ
ejpam-7049	224	60	open	open	ADJ
ejpam-7049	224	61	set	set	NOUN
ejpam-7049	224	62	u	u	NOUN
ejpam-7049	224	63	of	of	ADP
ejpam-7049	224	64	x	x	PUNCT
ejpam-7049	224	65	containing	contain	VERB
ejpam-7049	224	66	x	x	PUNCT
ejpam-7049	224	67	such	such	ADJ
ejpam-7049	224	68	that	that	SCONJ
ejpam-7049	224	69	f	f	PROPN
ejpam-7049	224	70	(	(	PUNCT
ejpam-7049	224	71	z)∩v	z)∩v	PROPN
ejpam-7049	224	72	̸=	̸=	PROPN
ejpam-7049	224	73	∅	∅	NOUN
ejpam-7049	224	74	for	for	ADP
ejpam-7049	224	75	every	every	DET
ejpam-7049	224	76	z	z	NOUN
ejpam-7049	224	77	∈	∈	PROPN
ejpam-7049	224	78	u	u	NOUN
ejpam-7049	224	79	.	.	PUNCT
ejpam-7049	225	1	a	a	DET
ejpam-7049	225	2	multifunction	multifunction	NOUN
ejpam-7049	225	3	f	f	NOUN
ejpam-7049	225	4	:	:	PUNCT
ejpam-7049	225	5	(	(	PUNCT
ejpam-7049	225	6	x	x	X
ejpam-7049	225	7	,	,	PUNCT
ejpam-7049	225	8	τ	τ	PROPN
ejpam-7049	225	9	,	,	PUNCT
ejpam-7049	225	10	i	i	NOUN
ejpam-7049	225	11	)	)	PUNCT
ejpam-7049	225	12	→	→	PUNCT
ejpam-7049	225	13	(	(	PUNCT
ejpam-7049	225	14	y	y	PROPN
ejpam-7049	225	15	,	,	PUNCT
ejpam-7049	225	16	σ1	σ1	PROPN
ejpam-7049	225	17	,	,	PUNCT
ejpam-7049	225	18	σ2	σ2	PROPN
ejpam-7049	225	19	)	)	PUNCT
ejpam-7049	225	20	is	be	AUX
ejpam-7049	225	21	said	say	VERB
ejpam-7049	225	22	to	to	PART
ejpam-7049	225	23	be	be	AUX
ejpam-7049	225	24	lower	low	ADJ
ejpam-7049	225	25	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	225	26	,	,	PUNCT
ejpam-7049	225	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7049	225	28	if	if	SCONJ
ejpam-7049	225	29	f	f	PROPN
ejpam-7049	225	30	is	be	AUX
ejpam-7049	225	31	lower	low	ADJ
ejpam-7049	225	32	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	225	33	,	,	PUNCT
ejpam-7049	225	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	225	35	at	at	ADP
ejpam-7049	225	36	each	each	DET
ejpam-7049	225	37	point	point	NOUN
ejpam-7049	225	38	of	of	ADP
ejpam-7049	225	39	x.	x.	NOUN
ejpam-7049	225	40	remark	remark	PROPN
ejpam-7049	225	41	1	1	NUM
ejpam-7049	225	42	.	.	PUNCT
ejpam-7049	225	43	for	for	ADP
ejpam-7049	225	44	a	a	DET
ejpam-7049	225	45	multifunction	multifunction	NOUN
ejpam-7049	225	46	f	f	NOUN
ejpam-7049	225	47	:	:	PUNCT
ejpam-7049	225	48	(	(	PUNCT
ejpam-7049	225	49	x	x	X
ejpam-7049	225	50	,	,	PUNCT
ejpam-7049	225	51	τ	τ	PROPN
ejpam-7049	225	52	,	,	PUNCT
ejpam-7049	225	53	i	i	NOUN
ejpam-7049	225	54	)	)	PUNCT
ejpam-7049	225	55	→	→	PUNCT
ejpam-7049	225	56	(	(	PUNCT
ejpam-7049	225	57	y	y	PROPN
ejpam-7049	225	58	,	,	PUNCT
ejpam-7049	225	59	σ1	σ1	PROPN
ejpam-7049	225	60	,	,	PUNCT
ejpam-7049	225	61	σ2	σ2	NOUN
ejpam-7049	225	62	)	)	PUNCT
ejpam-7049	225	63	,	,	PUNCT
ejpam-7049	225	64	the	the	DET
ejpam-7049	225	65	following	follow	VERB
ejpam-7049	225	66	implication	implication	NOUN
ejpam-7049	225	67	holds	hold	VERB
ejpam-7049	225	68	:	:	PUNCT
ejpam-7049	225	69	upper	upper	ADJ
ejpam-7049	225	70	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	225	71	,	,	PUNCT
ejpam-7049	225	72	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7049	225	73	⇒	⇒	NOUN
ejpam-7049	225	74	upper	upper	ADJ
ejpam-7049	225	75	almost	almost	ADV
ejpam-7049	225	76	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	225	77	,	,	PUNCT
ejpam-7049	225	78	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7049	225	79	.	.	PUNCT
ejpam-7049	226	1	the	the	DET
ejpam-7049	226	2	converse	converse	NOUN
ejpam-7049	226	3	of	of	ADP
ejpam-7049	226	4	the	the	DET
ejpam-7049	226	5	implication	implication	NOUN
ejpam-7049	226	6	is	be	AUX
ejpam-7049	226	7	not	not	PART
ejpam-7049	226	8	true	true	ADJ
ejpam-7049	226	9	in	in	ADP
ejpam-7049	226	10	general	general	ADJ
ejpam-7049	226	11	.	.	PUNCT
ejpam-7049	227	1	we	we	PRON
ejpam-7049	227	2	give	give	VERB
ejpam-7049	227	3	an	an	DET
ejpam-7049	227	4	example	example	NOUN
ejpam-7049	227	5	for	for	ADP
ejpam-7049	227	6	the	the	DET
ejpam-7049	227	7	implication	implication	NOUN
ejpam-7049	227	8	as	as	SCONJ
ejpam-7049	227	9	follows	follow	VERB
ejpam-7049	227	10	.	.	PUNCT
ejpam-7049	228	1	n.	n.	PROPN
ejpam-7049	228	2	srisarakham	srisarakham	PROPN
ejpam-7049	228	3	,	,	PUNCT
ejpam-7049	228	4	a.	a.	PROPN
ejpam-7049	228	5	sama	sama	PROPN
ejpam-7049	228	6	-	-	PUNCT
ejpam-7049	228	7	ae	ae	PROPN
ejpam-7049	228	8	,	,	PUNCT
ejpam-7049	228	9	c.	c.	PROPN
ejpam-7049	228	10	boonpok	boonpok	PROPN
ejpam-7049	228	11	/	/	SYM
ejpam-7049	228	12	eur	eur	PROPN
ejpam-7049	228	13	.	.	PUNCT
ejpam-7049	229	1	j.	j.	PROPN
ejpam-7049	229	2	pure	pure	PROPN
ejpam-7049	229	3	appl	appl	PROPN
ejpam-7049	229	4	.	.	PROPN
ejpam-7049	229	5	math	math	PROPN
ejpam-7049	229	6	,	,	PUNCT
ejpam-7049	229	7	18	18	NUM
ejpam-7049	229	8	(	(	PUNCT
ejpam-7049	229	9	4	4	NUM
ejpam-7049	229	10	)	)	PUNCT
ejpam-7049	229	11	(	(	PUNCT
ejpam-7049	229	12	2025	2025	NUM
ejpam-7049	229	13	)	)	PUNCT
ejpam-7049	229	14	,	,	PUNCT
ejpam-7049	229	15	7049	7049	NUM
ejpam-7049	229	16	9	9	NUM
ejpam-7049	229	17	of	of	ADP
ejpam-7049	229	18	11	11	NUM
ejpam-7049	229	19	example	example	NOUN
ejpam-7049	230	1	1	1	NUM
ejpam-7049	230	2	.	.	PUNCT
ejpam-7049	231	1	let	let	VERB
ejpam-7049	231	2	x	x	PUNCT
ejpam-7049	231	3	=	=	PRON
ejpam-7049	231	4	{	{	PUNCT
ejpam-7049	231	5	1	1	NUM
ejpam-7049	231	6	,	,	PUNCT
ejpam-7049	231	7	2	2	NUM
ejpam-7049	231	8	,	,	PUNCT
ejpam-7049	231	9	3	3	NUM
ejpam-7049	231	10	}	}	PUNCT
ejpam-7049	231	11	with	with	ADP
ejpam-7049	231	12	a	a	DET
ejpam-7049	231	13	topology	topology	NOUN
ejpam-7049	231	14	τ	τ	X
ejpam-7049	231	15	=	=	SYM
ejpam-7049	231	16	{	{	PUNCT
ejpam-7049	231	17	∅	∅	NOUN
ejpam-7049	231	18	,	,	PUNCT
ejpam-7049	231	19	x	x	NOUN
ejpam-7049	231	20	}	}	PUNCT
ejpam-7049	231	21	and	and	CCONJ
ejpam-7049	231	22	an	an	DET
ejpam-7049	231	23	ideal	ideal	NOUN
ejpam-7049	232	1	i	i	X
ejpam-7049	232	2	=	=	SYM
ejpam-7049	232	3	{	{	PUNCT
ejpam-7049	232	4	∅	∅	NOUN
ejpam-7049	232	5	}	}	PUNCT
ejpam-7049	232	6	.	.	PUNCT
ejpam-7049	233	1	let	let	VERB
ejpam-7049	233	2	y	y	PROPN
ejpam-7049	233	3	=	=	PUNCT
ejpam-7049	233	4	{	{	PUNCT
ejpam-7049	233	5	a	a	PRON
ejpam-7049	233	6	,	,	PUNCT
ejpam-7049	233	7	b	b	NOUN
ejpam-7049	233	8	,	,	PUNCT
ejpam-7049	233	9	c	c	NOUN
ejpam-7049	233	10	}	}	PUNCT
ejpam-7049	233	11	with	with	ADP
ejpam-7049	233	12	topologies	topology	NOUN
ejpam-7049	233	13	σ1	σ1	NOUN
ejpam-7049	233	14	=	=	SYM
ejpam-7049	233	15	{	{	PUNCT
ejpam-7049	233	16	∅	∅	NOUN
ejpam-7049	233	17	,	,	PUNCT
ejpam-7049	233	18	{	{	PUNCT
ejpam-7049	233	19	b	b	NOUN
ejpam-7049	233	20	}	}	PUNCT
ejpam-7049	233	21	,	,	PUNCT
ejpam-7049	233	22	y	y	PROPN
ejpam-7049	233	23	}	}	PUNCT
ejpam-7049	233	24	and	and	CCONJ
ejpam-7049	233	25	σ2	σ2	PROPN
ejpam-7049	233	26	=	=	SYM
ejpam-7049	233	27	{	{	PUNCT
ejpam-7049	233	28	∅	∅	NOUN
ejpam-7049	233	29	,	,	PUNCT
ejpam-7049	233	30	{	{	PUNCT
ejpam-7049	233	31	b	b	NOUN
ejpam-7049	233	32	}	}	PUNCT
ejpam-7049	233	33	,	,	PUNCT
ejpam-7049	233	34	{	{	PUNCT
ejpam-7049	233	35	a	a	DET
ejpam-7049	233	36	,	,	PUNCT
ejpam-7049	233	37	b	b	NOUN
ejpam-7049	233	38	}	}	PUNCT
ejpam-7049	233	39	,	,	PUNCT
ejpam-7049	233	40	y	y	PROPN
ejpam-7049	233	41	}	}	PUNCT
ejpam-7049	233	42	.	.	PUNCT
ejpam-7049	234	1	a	a	DET
ejpam-7049	234	2	multifunction	multifunction	NOUN
ejpam-7049	234	3	f	f	NOUN
ejpam-7049	234	4	:	:	PUNCT
ejpam-7049	234	5	(	(	PUNCT
ejpam-7049	234	6	x	x	X
ejpam-7049	234	7	,	,	PUNCT
ejpam-7049	234	8	τ	τ	PROPN
ejpam-7049	234	9	,	,	PUNCT
ejpam-7049	234	10	i	i	NOUN
ejpam-7049	234	11	)	)	PUNCT
ejpam-7049	234	12	→	→	PUNCT
ejpam-7049	234	13	(	(	PUNCT
ejpam-7049	234	14	y	y	PROPN
ejpam-7049	234	15	,	,	PUNCT
ejpam-7049	234	16	σ1	σ1	PROPN
ejpam-7049	234	17	,	,	PUNCT
ejpam-7049	234	18	σ2	σ2	PROPN
ejpam-7049	234	19	)	)	PUNCT
ejpam-7049	234	20	is	be	AUX
ejpam-7049	234	21	defined	define	VERB
ejpam-7049	234	22	as	as	SCONJ
ejpam-7049	234	23	follows	follow	VERB
ejpam-7049	234	24	:	:	PUNCT
ejpam-7049	234	25	f	f	X
ejpam-7049	234	26	(	(	PUNCT
ejpam-7049	234	27	1	1	X
ejpam-7049	234	28	)	)	PUNCT
ejpam-7049	234	29	=	=	PRON
ejpam-7049	234	30	{	{	PUNCT
ejpam-7049	234	31	b	b	NOUN
ejpam-7049	234	32	}	}	PUNCT
ejpam-7049	234	33	and	and	CCONJ
ejpam-7049	234	34	f	f	X
ejpam-7049	234	35	(	(	PUNCT
ejpam-7049	234	36	2	2	NUM
ejpam-7049	234	37	)	)	PUNCT
ejpam-7049	235	1	=	=	SYM
ejpam-7049	235	2	f	f	PROPN
ejpam-7049	235	3	(	(	PUNCT
ejpam-7049	235	4	3	3	NUM
ejpam-7049	235	5	)	)	PUNCT
ejpam-7049	235	6	=	=	PRON
ejpam-7049	235	7	{	{	PUNCT
ejpam-7049	235	8	a	a	X
ejpam-7049	235	9	,	,	PUNCT
ejpam-7049	235	10	c	c	NOUN
ejpam-7049	235	11	}	}	PUNCT
ejpam-7049	235	12	.	.	PUNCT
ejpam-7049	236	1	then	then	ADV
ejpam-7049	236	2	,	,	PUNCT
ejpam-7049	236	3	f	f	PROPN
ejpam-7049	236	4	is	be	AUX
ejpam-7049	236	5	upper	upper	ADJ
ejpam-7049	236	6	almost	almost	ADV
ejpam-7049	236	7	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	236	8	,	,	PUNCT
ejpam-7049	236	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	236	10	but	but	CCONJ
ejpam-7049	236	11	f	f	PROPN
ejpam-7049	236	12	is	be	AUX
ejpam-7049	236	13	not	not	PART
ejpam-7049	236	14	upper	upper	ADJ
ejpam-7049	236	15	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	236	16	,	,	PUNCT
ejpam-7049	236	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	236	18	,	,	PUNCT
ejpam-7049	236	19	since	since	SCONJ
ejpam-7049	236	20	{	{	PUNCT
ejpam-7049	236	21	a	a	PRON
ejpam-7049	236	22	,	,	PUNCT
ejpam-7049	236	23	c	c	NOUN
ejpam-7049	236	24	}	}	PUNCT
ejpam-7049	236	25	is	be	AUX
ejpam-7049	236	26	σ1σ2	σ1σ2	NOUN
ejpam-7049	236	27	-	-	ADJ
ejpam-7049	236	28	open	open	ADJ
ejpam-7049	236	29	in	in	ADP
ejpam-7049	236	30	y	y	PROPN
ejpam-7049	236	31	but	but	CCONJ
ejpam-7049	236	32	f+({a	f+({a	PROPN
ejpam-7049	236	33	,	,	PUNCT
ejpam-7049	236	34	c	c	NOUN
ejpam-7049	236	35	}	}	PUNCT
ejpam-7049	236	36	)	)	PUNCT
ejpam-7049	236	37	is	be	AUX
ejpam-7049	236	38	not	not	PART
ejpam-7049	236	39	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	236	40	-	-	NOUN
ejpam-7049	236	41	open	open	ADJ
ejpam-7049	236	42	in	in	ADP
ejpam-7049	236	43	x.	x.	NOUN
ejpam-7049	236	44	theorem	theorem	VERB
ejpam-7049	236	45	5	5	NUM
ejpam-7049	236	46	.	.	X
ejpam-7049	236	47	for	for	ADP
ejpam-7049	236	48	a	a	DET
ejpam-7049	236	49	multifunction	multifunction	NOUN
ejpam-7049	236	50	f	f	NOUN
ejpam-7049	236	51	:	:	PUNCT
ejpam-7049	236	52	(	(	PUNCT
ejpam-7049	236	53	x	x	X
ejpam-7049	236	54	,	,	PUNCT
ejpam-7049	236	55	τ	τ	PROPN
ejpam-7049	236	56	,	,	PUNCT
ejpam-7049	236	57	i	i	NOUN
ejpam-7049	236	58	)	)	PUNCT
ejpam-7049	236	59	→	→	PUNCT
ejpam-7049	236	60	(	(	PUNCT
ejpam-7049	236	61	y	y	PROPN
ejpam-7049	236	62	,	,	PUNCT
ejpam-7049	236	63	σ1	σ1	PROPN
ejpam-7049	236	64	,	,	PUNCT
ejpam-7049	236	65	σ2	σ2	NOUN
ejpam-7049	236	66	)	)	PUNCT
ejpam-7049	236	67	,	,	PUNCT
ejpam-7049	236	68	the	the	DET
ejpam-7049	236	69	following	follow	VERB
ejpam-7049	236	70	properties	property	NOUN
ejpam-7049	236	71	are	be	AUX
ejpam-7049	236	72	equivalent	equivalent	ADJ
ejpam-7049	236	73	:	:	PUNCT
ejpam-7049	236	74	(	(	PUNCT
ejpam-7049	236	75	1	1	X
ejpam-7049	236	76	)	)	PUNCT
ejpam-7049	236	77	f	f	PROPN
ejpam-7049	236	78	is	be	AUX
ejpam-7049	236	79	upper	upper	ADJ
ejpam-7049	236	80	almost	almost	ADV
ejpam-7049	236	81	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	236	82	,	,	PUNCT
ejpam-7049	236	83	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	236	84	;	;	PUNCT
ejpam-7049	236	85	(	(	PUNCT
ejpam-7049	236	86	2	2	NUM
ejpam-7049	236	87	)	)	PUNCT
ejpam-7049	236	88	βcl⋆(f−(v	βcl⋆(f−(v	NUM
ejpam-7049	236	89	)	)	PUNCT
ejpam-7049	236	90	)	)	PUNCT
ejpam-7049	237	1	⊆	⊆	X
ejpam-7049	237	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7049	237	3	-	-	PUNCT
ejpam-7049	237	4	cl(v	cl(v	NOUN
ejpam-7049	237	5	)	)	PUNCT
ejpam-7049	237	6	)	)	PUNCT
ejpam-7049	237	7	for	for	ADP
ejpam-7049	237	8	every	every	DET
ejpam-7049	237	9	(	(	PUNCT
ejpam-7049	237	10	σ1	σ1	PROPN
ejpam-7049	237	11	,	,	PUNCT
ejpam-7049	237	12	σ2)β	σ2)β	NOUN
ejpam-7049	237	13	-	-	PUNCT
ejpam-7049	237	14	open	open	NOUN
ejpam-7049	237	15	set	set	NOUN
ejpam-7049	237	16	v	v	NOUN
ejpam-7049	237	17	of	of	ADP
ejpam-7049	237	18	y	y	PROPN
ejpam-7049	237	19	;	;	PUNCT
ejpam-7049	237	20	(	(	PUNCT
ejpam-7049	237	21	3	3	X
ejpam-7049	237	22	)	)	PUNCT
ejpam-7049	237	23	βcl⋆(f−(v	βcl⋆(f−(v	NUM
ejpam-7049	237	24	)	)	PUNCT
ejpam-7049	237	25	)	)	PUNCT
ejpam-7049	238	1	⊆	⊆	X
ejpam-7049	238	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7049	238	3	-	-	PUNCT
ejpam-7049	238	4	cl(v	cl(v	NOUN
ejpam-7049	238	5	)	)	PUNCT
ejpam-7049	238	6	)	)	PUNCT
ejpam-7049	238	7	for	for	ADP
ejpam-7049	238	8	every	every	DET
ejpam-7049	238	9	(	(	PUNCT
ejpam-7049	238	10	σ1	σ1	PROPN
ejpam-7049	238	11	,	,	PUNCT
ejpam-7049	238	12	σ2)s	σ2)s	NOUN
ejpam-7049	238	13	-	-	PUNCT
ejpam-7049	238	14	open	open	NOUN
ejpam-7049	238	15	set	set	NOUN
ejpam-7049	238	16	v	v	NOUN
ejpam-7049	238	17	of	of	ADP
ejpam-7049	238	18	y	y	PROPN
ejpam-7049	238	19	;	;	PUNCT
ejpam-7049	238	20	(	(	PUNCT
ejpam-7049	238	21	4	4	X
ejpam-7049	238	22	)	)	PUNCT
ejpam-7049	238	23	f+(v	f+(v	NOUN
ejpam-7049	238	24	)	)	PUNCT
ejpam-7049	239	1	⊆	⊆	X
ejpam-7049	239	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7049	239	3	-	-	PUNCT
ejpam-7049	239	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	239	5	-	-	PUNCT
ejpam-7049	239	6	cl(v	cl(v	NOUN
ejpam-7049	239	7	)	)	PUNCT
ejpam-7049	239	8	)	)	PUNCT
ejpam-7049	239	9	)	)	PUNCT
ejpam-7049	239	10	)	)	PUNCT
ejpam-7049	239	11	for	for	ADP
ejpam-7049	239	12	every	every	DET
ejpam-7049	239	13	(	(	PUNCT
ejpam-7049	239	14	σ1	σ1	PROPN
ejpam-7049	239	15	,	,	PUNCT
ejpam-7049	239	16	σ2)p	σ2)p	NOUN
ejpam-7049	239	17	-	-	PUNCT
ejpam-7049	239	18	open	open	NOUN
ejpam-7049	239	19	set	set	NOUN
ejpam-7049	239	20	v	v	NOUN
ejpam-7049	239	21	of	of	ADP
ejpam-7049	239	22	y	y	PROPN
ejpam-7049	239	23	.	.	PUNCT
ejpam-7049	240	1	proof	proof	NOUN
ejpam-7049	240	2	.	.	PUNCT
ejpam-7049	241	1	(	(	PUNCT
ejpam-7049	241	2	1	1	X
ejpam-7049	241	3	)	)	PUNCT
ejpam-7049	241	4	⇒	⇒	NOUN
ejpam-7049	241	5	(	(	PUNCT
ejpam-7049	241	6	2	2	NUM
ejpam-7049	241	7	):	):	PUNCT
ejpam-7049	241	8	let	let	VERB
ejpam-7049	241	9	v	v	PART
ejpam-7049	241	10	be	be	AUX
ejpam-7049	241	11	any	any	DET
ejpam-7049	241	12	(	(	PUNCT
ejpam-7049	241	13	σ1	σ1	PROPN
ejpam-7049	241	14	,	,	PUNCT
ejpam-7049	241	15	σ2)β	σ2)β	NOUN
ejpam-7049	241	16	-	-	PUNCT
ejpam-7049	241	17	open	open	ADJ
ejpam-7049	241	18	set	set	NOUN
ejpam-7049	241	19	of	of	ADP
ejpam-7049	241	20	y	y	PROPN
ejpam-7049	241	21	.	.	PUNCT
ejpam-7049	242	1	since	since	SCONJ
ejpam-7049	242	2	σ1σ2	σ1σ2	NOUN
ejpam-7049	242	3	-	-	NOUN
ejpam-7049	242	4	cl(v	cl(v	NOUN
ejpam-7049	242	5	)	)	PUNCT
ejpam-7049	242	6	is	be	AUX
ejpam-7049	242	7	(	(	PUNCT
ejpam-7049	242	8	σ1	σ1	PROPN
ejpam-7049	242	9	,	,	PUNCT
ejpam-7049	242	10	σ2)rclosed	σ2)rclose	VERB
ejpam-7049	242	11	,	,	PUNCT
ejpam-7049	242	12	by	by	ADP
ejpam-7049	242	13	theorem	theorem	NOUN
ejpam-7049	242	14	3	3	NUM
ejpam-7049	242	15	we	we	PRON
ejpam-7049	242	16	have	have	VERB
ejpam-7049	242	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7049	242	18	-	-	PUNCT
ejpam-7049	242	19	cl(v	cl(v	NOUN
ejpam-7049	242	20	)	)	PUNCT
ejpam-7049	242	21	)	)	PUNCT
ejpam-7049	242	22	is	be	AUX
ejpam-7049	242	23	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	242	24	-	-	VERB
ejpam-7049	242	25	closed	closed	ADJ
ejpam-7049	242	26	in	in	ADP
ejpam-7049	242	27	x	x	X
ejpam-7049	242	28	and	and	CCONJ
ejpam-7049	242	29	hence	hence	ADV
ejpam-7049	242	30	βcl⋆(f−(v	βcl⋆(f−(v	NUM
ejpam-7049	242	31	)	)	PUNCT
ejpam-7049	242	32	)	)	PUNCT
ejpam-7049	243	1	⊆	⊆	X
ejpam-7049	243	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7049	243	3	-	-	PUNCT
ejpam-7049	243	4	cl(v	cl(v	NOUN
ejpam-7049	243	5	)	)	PUNCT
ejpam-7049	243	6	)	)	PUNCT
ejpam-7049	243	7	.	.	PUNCT
ejpam-7049	244	1	(	(	PUNCT
ejpam-7049	244	2	2	2	X
ejpam-7049	244	3	)	)	PUNCT
ejpam-7049	244	4	⇒	⇒	NOUN
ejpam-7049	244	5	(	(	PUNCT
ejpam-7049	244	6	3	3	NUM
ejpam-7049	244	7	):	):	PUNCT
ejpam-7049	244	8	this	this	PRON
ejpam-7049	244	9	is	be	AUX
ejpam-7049	244	10	obvious	obvious	ADJ
ejpam-7049	244	11	since	since	SCONJ
ejpam-7049	244	12	every	every	DET
ejpam-7049	244	13	(	(	PUNCT
ejpam-7049	244	14	σ1	σ1	PROPN
ejpam-7049	244	15	,	,	PUNCT
ejpam-7049	244	16	σ2)s	σ2)s	NOUN
ejpam-7049	244	17	-	-	PUNCT
ejpam-7049	244	18	open	open	ADJ
ejpam-7049	244	19	set	set	NOUN
ejpam-7049	244	20	is	be	AUX
ejpam-7049	244	21	(	(	PUNCT
ejpam-7049	244	22	σ1	σ1	PROPN
ejpam-7049	244	23	,	,	PUNCT
ejpam-7049	244	24	σ2)β	σ2)β	NOUN
ejpam-7049	244	25	-	-	PUNCT
ejpam-7049	244	26	open	open	ADJ
ejpam-7049	244	27	.	.	PUNCT
ejpam-7049	245	1	(	(	PUNCT
ejpam-7049	245	2	3	3	X
ejpam-7049	245	3	)	)	PUNCT
ejpam-7049	245	4	⇒	⇒	NOUN
ejpam-7049	245	5	(	(	PUNCT
ejpam-7049	245	6	4	4	NUM
ejpam-7049	245	7	):	):	PUNCT
ejpam-7049	245	8	let	let	VERB
ejpam-7049	245	9	v	v	PART
ejpam-7049	245	10	be	be	AUX
ejpam-7049	245	11	any	any	DET
ejpam-7049	245	12	(	(	PUNCT
ejpam-7049	245	13	σ1	σ1	PROPN
ejpam-7049	245	14	,	,	PUNCT
ejpam-7049	245	15	σ2)p	σ2)p	NOUN
ejpam-7049	245	16	-	-	PUNCT
ejpam-7049	245	17	open	open	ADJ
ejpam-7049	245	18	set	set	NOUN
ejpam-7049	245	19	of	of	ADP
ejpam-7049	245	20	y	y	PROPN
ejpam-7049	245	21	.	.	PUNCT
ejpam-7049	246	1	then	then	ADV
ejpam-7049	246	2	,	,	PUNCT
ejpam-7049	246	3	v	v	ADP
ejpam-7049	246	4	⊆	⊆	NUM
ejpam-7049	246	5	σ1σ2	σ1σ2	NOUN
ejpam-7049	246	6	-	-	PUNCT
ejpam-7049	246	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	246	8	-	-	PUNCT
ejpam-7049	246	9	cl(v	cl(v	NOUN
ejpam-7049	246	10	)	)	PUNCT
ejpam-7049	246	11	)	)	PUNCT
ejpam-7049	246	12	and	and	CCONJ
ejpam-7049	246	13	y	y	PROPN
ejpam-7049	246	14	−v	−v	NOUN
ejpam-7049	246	15	⊇	⊇	X
ejpam-7049	246	16	σ1σ2	σ1σ2	ADV
ejpam-7049	246	17	-	-	PUNCT
ejpam-7049	246	18	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	246	19	-	-	PUNCT
ejpam-7049	246	20	int(y	int(y	PROPN
ejpam-7049	246	21	−v	−v	NOUN
ejpam-7049	246	22	)	)	PUNCT
ejpam-7049	246	23	)	)	PUNCT
ejpam-7049	246	24	.	.	PUNCT
ejpam-7049	247	1	since	since	SCONJ
ejpam-7049	247	2	σ1σ2	σ1σ2	NOUN
ejpam-7049	247	3	-	-	PUNCT
ejpam-7049	247	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	247	5	-	-	PUNCT
ejpam-7049	247	6	int(y	int(y	PROPN
ejpam-7049	247	7	−v	−v	NOUN
ejpam-7049	247	8	)	)	PUNCT
ejpam-7049	247	9	)	)	PUNCT
ejpam-7049	247	10	is	be	AUX
ejpam-7049	247	11	(	(	PUNCT
ejpam-7049	247	12	σ1	σ1	PROPN
ejpam-7049	247	13	,	,	PUNCT
ejpam-7049	247	14	σ2)s	σ2)s	NOUN
ejpam-7049	247	15	-	-	PUNCT
ejpam-7049	247	16	open	open	ADJ
ejpam-7049	247	17	in	in	ADP
ejpam-7049	247	18	y	y	PROPN
ejpam-7049	247	19	and	and	CCONJ
ejpam-7049	247	20	by	by	ADP
ejpam-7049	247	21	(	(	PUNCT
ejpam-7049	247	22	3	3	NUM
ejpam-7049	247	23	)	)	PUNCT
ejpam-7049	247	24	,	,	PUNCT
ejpam-7049	247	25	x	x	PUNCT
ejpam-7049	247	26	−	−	NOUN
ejpam-7049	247	27	f+(v	f+(v	NOUN
ejpam-7049	247	28	)	)	PUNCT
ejpam-7049	248	1	=	=	PUNCT
ejpam-7049	248	2	f−(y	f−(y	NOUN
ejpam-7049	248	3	−	−	ADP
ejpam-7049	248	4	v	v	NOUN
ejpam-7049	248	5	)	)	PUNCT
ejpam-7049	248	6	⊇	⊇	ADJ
ejpam-7049	248	7	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7049	248	8	-	-	PUNCT
ejpam-7049	248	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	248	10	-	-	PUNCT
ejpam-7049	248	11	int(y	int(y	PROPN
ejpam-7049	248	12	−	−	PROPN
ejpam-7049	248	13	v	v	NOUN
ejpam-7049	248	14	)	)	PUNCT
ejpam-7049	248	15	)	)	PUNCT
ejpam-7049	248	16	)	)	PUNCT
ejpam-7049	249	1	⊇	⊇	PROPN
ejpam-7049	249	2	βcl⋆(f−(σ1σ2	βcl⋆(f−(σ1σ2	NOUN
ejpam-7049	249	3	-	-	PUNCT
ejpam-7049	249	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7049	249	5	-	-	PUNCT
ejpam-7049	249	6	int(y	int(y	PROPN
ejpam-7049	249	7	−	−	PROPN
ejpam-7049	249	8	v	v	NOUN
ejpam-7049	249	9	)	)	PUNCT
ejpam-7049	249	10	)	)	PUNCT
ejpam-7049	249	11	)	)	PUNCT
ejpam-7049	249	12	)	)	PUNCT
ejpam-7049	250	1	=	=	NOUN
ejpam-7049	250	2	βcl⋆(f−(y	βcl⋆(f−(y	PUNCT
ejpam-7049	250	3	−	−	ADP
ejpam-7049	250	4	σ1σ2	σ1σ2	X
ejpam-7049	250	5	-	-	PUNCT
ejpam-7049	250	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	250	7	-	-	PUNCT
ejpam-7049	250	8	cl(v	cl(v	NOUN
ejpam-7049	250	9	)	)	PUNCT
ejpam-7049	250	10	)	)	PUNCT
ejpam-7049	250	11	)	)	PUNCT
ejpam-7049	250	12	)	)	PUNCT
ejpam-7049	251	1	=	=	PUNCT
ejpam-7049	251	2	βcl⋆(x	βcl⋆(x	NOUN
ejpam-7049	251	3	−	−	ADP
ejpam-7049	251	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7049	251	5	-	-	PUNCT
ejpam-7049	251	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	251	7	-	-	PUNCT
ejpam-7049	251	8	cl(v	cl(v	NOUN
ejpam-7049	251	9	)	)	PUNCT
ejpam-7049	251	10	)	)	PUNCT
ejpam-7049	251	11	)	)	PUNCT
ejpam-7049	251	12	)	)	PUNCT
ejpam-7049	252	1	=	=	PUNCT
ejpam-7049	252	2	x	x	PUNCT
ejpam-7049	252	3	−	−	ADP
ejpam-7049	252	4	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7049	252	5	-	-	PUNCT
ejpam-7049	252	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	252	7	-	-	PUNCT
ejpam-7049	252	8	cl(v	cl(v	NOUN
ejpam-7049	252	9	)	)	PUNCT
ejpam-7049	252	10	)	)	PUNCT
ejpam-7049	252	11	)	)	PUNCT
ejpam-7049	252	12	)	)	PUNCT
ejpam-7049	252	13	.	.	PUNCT
ejpam-7049	253	1	thus	thus	ADV
ejpam-7049	253	2	,	,	PUNCT
ejpam-7049	253	3	f+(v	f+(v	PROPN
ejpam-7049	253	4	)	)	PUNCT
ejpam-7049	253	5	⊆	⊆	X
ejpam-7049	253	6	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7049	253	7	-	-	PUNCT
ejpam-7049	253	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	253	9	-	-	PUNCT
ejpam-7049	253	10	cl(v	cl(v	NOUN
ejpam-7049	253	11	)	)	PUNCT
ejpam-7049	253	12	)	)	PUNCT
ejpam-7049	253	13	)	)	PUNCT
ejpam-7049	253	14	)	)	PUNCT
ejpam-7049	253	15	.	.	PUNCT
ejpam-7049	254	1	(	(	PUNCT
ejpam-7049	254	2	4	4	X
ejpam-7049	254	3	)	)	PUNCT
ejpam-7049	254	4	⇒	⇒	NOUN
ejpam-7049	254	5	(	(	PUNCT
ejpam-7049	254	6	1	1	NUM
ejpam-7049	254	7	):	):	PUNCT
ejpam-7049	254	8	let	let	VERB
ejpam-7049	254	9	v	v	PART
ejpam-7049	254	10	be	be	AUX
ejpam-7049	254	11	any	any	DET
ejpam-7049	254	12	(	(	PUNCT
ejpam-7049	254	13	σ1	σ1	NOUN
ejpam-7049	254	14	,	,	PUNCT
ejpam-7049	254	15	σ2)r	σ2)r	NOUN
ejpam-7049	254	16	-	-	PUNCT
ejpam-7049	254	17	open	open	ADJ
ejpam-7049	254	18	set	set	NOUN
ejpam-7049	254	19	of	of	ADP
ejpam-7049	254	20	y	y	PROPN
ejpam-7049	254	21	.	.	PUNCT
ejpam-7049	255	1	then	then	ADV
ejpam-7049	255	2	,	,	PUNCT
ejpam-7049	255	3	v	v	NOUN
ejpam-7049	255	4	is	be	AUX
ejpam-7049	255	5	(	(	PUNCT
ejpam-7049	255	6	σ1	σ1	PROPN
ejpam-7049	255	7	,	,	PUNCT
ejpam-7049	255	8	σ2)p	σ2)p	NOUN
ejpam-7049	255	9	-	-	PUNCT
ejpam-7049	255	10	open	open	ADJ
ejpam-7049	255	11	in	in	ADP
ejpam-7049	255	12	y	y	PROPN
ejpam-7049	255	13	and	and	CCONJ
ejpam-7049	255	14	by	by	ADP
ejpam-7049	255	15	(	(	PUNCT
ejpam-7049	255	16	4	4	NUM
ejpam-7049	255	17	)	)	PUNCT
ejpam-7049	255	18	,	,	PUNCT
ejpam-7049	255	19	f+(v	f+(v	PROPN
ejpam-7049	255	20	)	)	PUNCT
ejpam-7049	256	1	⊆	⊆	X
ejpam-7049	256	2	βint⋆(f+(σ1σ2	βint⋆(f+(σ1σ2	NOUN
ejpam-7049	256	3	-	-	PUNCT
ejpam-7049	256	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	256	5	-	-	PUNCT
ejpam-7049	256	6	cl(v	cl(v	NOUN
ejpam-7049	256	7	)	)	PUNCT
ejpam-7049	256	8	)	)	PUNCT
ejpam-7049	256	9	)	)	PUNCT
ejpam-7049	256	10	)	)	PUNCT
ejpam-7049	256	11	=	=	SYM
ejpam-7049	256	12	βint⋆(f+(v	βint⋆(f+(v	PROPN
ejpam-7049	256	13	)	)	PUNCT
ejpam-7049	256	14	)	)	PUNCT
ejpam-7049	256	15	and	and	CCONJ
ejpam-7049	256	16	hence	hence	ADV
ejpam-7049	256	17	f+(v	f+(v	PROPN
ejpam-7049	256	18	)	)	PUNCT
ejpam-7049	256	19	is	be	AUX
ejpam-7049	256	20	τ⋆-β	τ⋆-β	NOUN
ejpam-7049	256	21	-	-	NOUN
ejpam-7049	256	22	open	open	ADJ
ejpam-7049	256	23	in	in	ADP
ejpam-7049	256	24	x.	x.	NOUN
ejpam-7049	256	25	it	it	PRON
ejpam-7049	256	26	follows	follow	VERB
ejpam-7049	256	27	from	from	ADP
ejpam-7049	256	28	theorem	theorem	ADJ
ejpam-7049	256	29	3	3	NUM
ejpam-7049	256	30	that	that	SCONJ
ejpam-7049	256	31	f	f	PROPN
ejpam-7049	256	32	is	be	AUX
ejpam-7049	256	33	upper	upper	ADJ
ejpam-7049	256	34	almost	almost	ADV
ejpam-7049	256	35	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	256	36	,	,	PUNCT
ejpam-7049	256	37	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7049	256	38	.	.	X
ejpam-7049	256	39	theorem	theorem	VERB
ejpam-7049	256	40	6	6	NUM
ejpam-7049	256	41	.	.	PUNCT
ejpam-7049	256	42	for	for	ADP
ejpam-7049	256	43	a	a	DET
ejpam-7049	256	44	multifunction	multifunction	NOUN
ejpam-7049	257	1	f	f	NOUN
ejpam-7049	257	2	:	:	PUNCT
ejpam-7049	257	3	(	(	PUNCT
ejpam-7049	257	4	x	x	X
ejpam-7049	257	5	,	,	PUNCT
ejpam-7049	257	6	τ	τ	PROPN
ejpam-7049	257	7	,	,	PUNCT
ejpam-7049	257	8	i	i	NOUN
ejpam-7049	257	9	)	)	PUNCT
ejpam-7049	257	10	→	→	PUNCT
ejpam-7049	257	11	(	(	PUNCT
ejpam-7049	257	12	y	y	PROPN
ejpam-7049	257	13	,	,	PUNCT
ejpam-7049	257	14	σ1	σ1	PROPN
ejpam-7049	257	15	,	,	PUNCT
ejpam-7049	257	16	σ2	σ2	NOUN
ejpam-7049	257	17	)	)	PUNCT
ejpam-7049	257	18	,	,	PUNCT
ejpam-7049	257	19	the	the	DET
ejpam-7049	257	20	following	follow	VERB
ejpam-7049	257	21	properties	property	NOUN
ejpam-7049	257	22	are	be	AUX
ejpam-7049	257	23	equivalent	equivalent	ADJ
ejpam-7049	257	24	:	:	PUNCT
ejpam-7049	257	25	(	(	PUNCT
ejpam-7049	257	26	1	1	X
ejpam-7049	257	27	)	)	PUNCT
ejpam-7049	257	28	f	f	PROPN
ejpam-7049	257	29	is	be	AUX
ejpam-7049	257	30	lower	low	ADJ
ejpam-7049	257	31	almost	almost	ADV
ejpam-7049	257	32	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	257	33	,	,	PUNCT
ejpam-7049	257	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	257	35	;	;	PUNCT
ejpam-7049	257	36	(	(	PUNCT
ejpam-7049	257	37	2	2	X
ejpam-7049	257	38	)	)	PUNCT
ejpam-7049	257	39	βcl⋆(f+(v	βcl⋆(f+(v	NUM
ejpam-7049	257	40	)	)	PUNCT
ejpam-7049	257	41	)	)	PUNCT
ejpam-7049	258	1	⊆	⊆	X
ejpam-7049	258	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7049	258	3	-	-	PUNCT
ejpam-7049	258	4	cl(v	cl(v	NOUN
ejpam-7049	258	5	)	)	PUNCT
ejpam-7049	258	6	)	)	PUNCT
ejpam-7049	258	7	for	for	ADP
ejpam-7049	258	8	every	every	DET
ejpam-7049	258	9	(	(	PUNCT
ejpam-7049	258	10	σ1	σ1	PROPN
ejpam-7049	258	11	,	,	PUNCT
ejpam-7049	258	12	σ2)β	σ2)β	NOUN
ejpam-7049	258	13	-	-	PUNCT
ejpam-7049	258	14	open	open	NOUN
ejpam-7049	258	15	set	set	NOUN
ejpam-7049	258	16	v	v	NOUN
ejpam-7049	258	17	of	of	ADP
ejpam-7049	258	18	y	y	PROPN
ejpam-7049	258	19	;	;	PUNCT
ejpam-7049	258	20	(	(	PUNCT
ejpam-7049	258	21	3	3	X
ejpam-7049	258	22	)	)	PUNCT
ejpam-7049	258	23	βcl⋆(f+(v	βcl⋆(f+(v	NUM
ejpam-7049	258	24	)	)	PUNCT
ejpam-7049	258	25	)	)	PUNCT
ejpam-7049	258	26	⊆	⊆	X
ejpam-7049	258	27	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7049	258	28	-	-	PUNCT
ejpam-7049	258	29	cl(v	cl(v	NOUN
ejpam-7049	258	30	)	)	PUNCT
ejpam-7049	258	31	)	)	PUNCT
ejpam-7049	258	32	for	for	SCONJ
ejpam-7049	258	33	every	every	DET
ejpam-7049	258	34	(	(	PUNCT
ejpam-7049	258	35	σ1	σ1	PROPN
ejpam-7049	258	36	,	,	PUNCT
ejpam-7049	258	37	σ2)s	σ2)s	NOUN
ejpam-7049	258	38	-	-	PUNCT
ejpam-7049	258	39	open	open	NOUN
ejpam-7049	258	40	set	set	NOUN
ejpam-7049	258	41	v	v	NOUN
ejpam-7049	258	42	of	of	ADP
ejpam-7049	258	43	y	y	PROPN
ejpam-7049	258	44	;	;	PUNCT
ejpam-7049	258	45	n.	n.	PROPN
ejpam-7049	258	46	srisarakham	srisarakham	PROPN
ejpam-7049	258	47	,	,	PUNCT
ejpam-7049	258	48	a.	a.	PROPN
ejpam-7049	258	49	sama	sama	PROPN
ejpam-7049	258	50	-	-	PUNCT
ejpam-7049	258	51	ae	ae	PROPN
ejpam-7049	258	52	,	,	PUNCT
ejpam-7049	258	53	c.	c.	PROPN
ejpam-7049	258	54	boonpok	boonpok	PROPN
ejpam-7049	258	55	/	/	SYM
ejpam-7049	258	56	eur	eur	PROPN
ejpam-7049	258	57	.	.	PUNCT
ejpam-7049	259	1	j.	j.	PROPN
ejpam-7049	259	2	pure	pure	PROPN
ejpam-7049	259	3	appl	appl	PROPN
ejpam-7049	259	4	.	.	PROPN
ejpam-7049	259	5	math	math	PROPN
ejpam-7049	259	6	,	,	PUNCT
ejpam-7049	259	7	18	18	NUM
ejpam-7049	259	8	(	(	PUNCT
ejpam-7049	259	9	4	4	NUM
ejpam-7049	259	10	)	)	PUNCT
ejpam-7049	259	11	(	(	PUNCT
ejpam-7049	259	12	2025	2025	NUM
ejpam-7049	259	13	)	)	PUNCT
ejpam-7049	259	14	,	,	PUNCT
ejpam-7049	259	15	7049	7049	NUM
ejpam-7049	259	16	10	10	NUM
ejpam-7049	259	17	of	of	ADP
ejpam-7049	259	18	11	11	NUM
ejpam-7049	259	19	(	(	PUNCT
ejpam-7049	259	20	4	4	NUM
ejpam-7049	259	21	)	)	PUNCT
ejpam-7049	259	22	f−(v	f−(v	NOUN
ejpam-7049	259	23	)	)	PUNCT
ejpam-7049	259	24	⊆	⊆	NUM
ejpam-7049	259	25	βint⋆(f−(σ1σ2	βint⋆(f−(σ1σ2	NOUN
ejpam-7049	259	26	-	-	PUNCT
ejpam-7049	259	27	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	259	28	-	-	PUNCT
ejpam-7049	259	29	cl(v	cl(v	NOUN
ejpam-7049	259	30	)	)	PUNCT
ejpam-7049	259	31	)	)	PUNCT
ejpam-7049	259	32	)	)	PUNCT
ejpam-7049	259	33	)	)	PUNCT
ejpam-7049	260	1	for	for	ADP
ejpam-7049	260	2	every	every	DET
ejpam-7049	260	3	(	(	PUNCT
ejpam-7049	260	4	σ1	σ1	PROPN
ejpam-7049	260	5	,	,	PUNCT
ejpam-7049	260	6	σ2)p	σ2)p	NOUN
ejpam-7049	260	7	-	-	PUNCT
ejpam-7049	260	8	open	open	NOUN
ejpam-7049	260	9	set	set	NOUN
ejpam-7049	260	10	v	v	NOUN
ejpam-7049	260	11	of	of	ADP
ejpam-7049	260	12	y	y	PROPN
ejpam-7049	260	13	.	.	PUNCT
ejpam-7049	261	1	proof	proof	NOUN
ejpam-7049	261	2	.	.	PUNCT
ejpam-7049	262	1	the	the	DET
ejpam-7049	262	2	proof	proof	NOUN
ejpam-7049	262	3	is	be	AUX
ejpam-7049	262	4	similar	similar	ADJ
ejpam-7049	262	5	to	to	ADP
ejpam-7049	262	6	that	that	PRON
ejpam-7049	262	7	of	of	ADP
ejpam-7049	262	8	theorem	theorem	ADJ
ejpam-7049	262	9	5	5	NUM
ejpam-7049	262	10	.	.	PUNCT
ejpam-7049	262	11	corollary	corollary	ADJ
ejpam-7049	262	12	3	3	NUM
ejpam-7049	262	13	.	.	PUNCT
ejpam-7049	263	1	for	for	ADP
ejpam-7049	263	2	a	a	DET
ejpam-7049	263	3	function	function	NOUN
ejpam-7049	263	4	f	f	NOUN
ejpam-7049	263	5	:	:	PUNCT
ejpam-7049	263	6	(	(	PUNCT
ejpam-7049	263	7	x	x	X
ejpam-7049	263	8	,	,	PUNCT
ejpam-7049	263	9	τ	τ	PROPN
ejpam-7049	263	10	,	,	PUNCT
ejpam-7049	263	11	i	i	NOUN
ejpam-7049	263	12	)	)	PUNCT
ejpam-7049	263	13	→	→	PUNCT
ejpam-7049	263	14	(	(	PUNCT
ejpam-7049	263	15	y	y	PROPN
ejpam-7049	263	16	,	,	PUNCT
ejpam-7049	263	17	σ1	σ1	PROPN
ejpam-7049	263	18	,	,	PUNCT
ejpam-7049	263	19	σ2	σ2	NOUN
ejpam-7049	263	20	)	)	PUNCT
ejpam-7049	263	21	,	,	PUNCT
ejpam-7049	263	22	the	the	DET
ejpam-7049	263	23	following	follow	VERB
ejpam-7049	263	24	properties	property	NOUN
ejpam-7049	263	25	are	be	AUX
ejpam-7049	263	26	equivalent	equivalent	ADJ
ejpam-7049	263	27	:	:	PUNCT
ejpam-7049	263	28	(	(	PUNCT
ejpam-7049	263	29	1	1	X
ejpam-7049	263	30	)	)	PUNCT
ejpam-7049	263	31	f	f	NOUN
ejpam-7049	263	32	is	be	AUX
ejpam-7049	263	33	almost	almost	ADV
ejpam-7049	263	34	τ⋆β(σ1	τ⋆β(σ1	ADV
ejpam-7049	263	35	,	,	PUNCT
ejpam-7049	263	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7049	263	37	;	;	PUNCT
ejpam-7049	263	38	(	(	PUNCT
ejpam-7049	263	39	2	2	X
ejpam-7049	263	40	)	)	PUNCT
ejpam-7049	263	41	βcl⋆(f−1(v	βcl⋆(f−1(v	NUM
ejpam-7049	263	42	)	)	PUNCT
ejpam-7049	263	43	)	)	PUNCT
ejpam-7049	264	1	⊆	⊆	NUM
ejpam-7049	264	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7049	264	3	-	-	PUNCT
ejpam-7049	264	4	cl(v	cl(v	NOUN
ejpam-7049	264	5	)	)	PUNCT
ejpam-7049	264	6	)	)	PUNCT
ejpam-7049	264	7	for	for	ADP
ejpam-7049	264	8	every	every	DET
ejpam-7049	264	9	(	(	PUNCT
ejpam-7049	264	10	σ1	σ1	PROPN
ejpam-7049	264	11	,	,	PUNCT
ejpam-7049	264	12	σ2)β	σ2)β	NOUN
ejpam-7049	264	13	-	-	PUNCT
ejpam-7049	264	14	open	open	NOUN
ejpam-7049	264	15	set	set	NOUN
ejpam-7049	264	16	v	v	NOUN
ejpam-7049	264	17	of	of	ADP
ejpam-7049	264	18	y	y	PROPN
ejpam-7049	264	19	;	;	PUNCT
ejpam-7049	264	20	(	(	PUNCT
ejpam-7049	264	21	3	3	X
ejpam-7049	264	22	)	)	PUNCT
ejpam-7049	264	23	βcl⋆(f−1(v	βcl⋆(f−1(v	NUM
ejpam-7049	264	24	)	)	PUNCT
ejpam-7049	264	25	)	)	PUNCT
ejpam-7049	265	1	⊆	⊆	NUM
ejpam-7049	265	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-7049	265	3	-	-	PUNCT
ejpam-7049	265	4	cl(v	cl(v	NOUN
ejpam-7049	265	5	)	)	PUNCT
ejpam-7049	265	6	)	)	PUNCT
ejpam-7049	265	7	for	for	ADP
ejpam-7049	265	8	every	every	DET
ejpam-7049	265	9	(	(	PUNCT
ejpam-7049	265	10	σ1	σ1	PROPN
ejpam-7049	265	11	,	,	PUNCT
ejpam-7049	265	12	σ2)s	σ2)s	NOUN
ejpam-7049	265	13	-	-	PUNCT
ejpam-7049	265	14	open	open	NOUN
ejpam-7049	265	15	set	set	NOUN
ejpam-7049	265	16	v	v	NOUN
ejpam-7049	265	17	of	of	ADP
ejpam-7049	265	18	y	y	PROPN
ejpam-7049	265	19	;	;	PUNCT
ejpam-7049	265	20	(	(	PUNCT
ejpam-7049	265	21	4	4	X
ejpam-7049	265	22	)	)	PUNCT
ejpam-7049	265	23	f−1(v	f−1(v	NOUN
ejpam-7049	265	24	)	)	PUNCT
ejpam-7049	265	25	⊆	⊆	NUM
ejpam-7049	265	26	βint⋆(f−1(σ1σ2	βint⋆(f−1(σ1σ2	NOUN
ejpam-7049	265	27	-	-	PUNCT
ejpam-7049	265	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7049	265	29	-	-	PUNCT
ejpam-7049	265	30	cl(v	cl(v	NOUN
ejpam-7049	265	31	)	)	PUNCT
ejpam-7049	265	32	)	)	PUNCT
ejpam-7049	265	33	)	)	PUNCT
ejpam-7049	265	34	)	)	PUNCT
ejpam-7049	266	1	for	for	ADP
ejpam-7049	266	2	every	every	DET
ejpam-7049	266	3	(	(	PUNCT
ejpam-7049	266	4	σ1	σ1	PROPN
ejpam-7049	266	5	,	,	PUNCT
ejpam-7049	266	6	σ2)p	σ2)p	NOUN
ejpam-7049	266	7	-	-	PUNCT
ejpam-7049	266	8	open	open	NOUN
ejpam-7049	266	9	set	set	NOUN
ejpam-7049	266	10	v	v	NOUN
ejpam-7049	266	11	of	of	ADP
ejpam-7049	266	12	y	y	PROPN
ejpam-7049	266	13	.	.	PUNCT
ejpam-7049	267	1	acknowledgements	acknowledgement	NOUN
ejpam-7049	267	2	this	this	DET
ejpam-7049	267	3	research	research	NOUN
ejpam-7049	267	4	project	project	NOUN
ejpam-7049	267	5	was	be	AUX
ejpam-7049	267	6	financially	financially	ADV
ejpam-7049	267	7	supported	support	VERB
ejpam-7049	267	8	by	by	ADP
ejpam-7049	267	9	mahasarakham	mahasarakham	PROPN
ejpam-7049	267	10	university	university	PROPN
ejpam-7049	267	11	.	.	PUNCT
ejpam-7049	268	1	references	reference	NOUN
ejpam-7049	268	2	[	[	X
ejpam-7049	268	3	1	1	NUM
ejpam-7049	268	4	]	]	PUNCT
ejpam-7049	268	5	a.	a.	NOUN
ejpam-7049	268	6	a.	a.	NOUN
ejpam-7049	268	7	nasef	nasef	PROPN
ejpam-7049	268	8	and	and	CCONJ
ejpam-7049	268	9	t.	t.	PROPN
ejpam-7049	268	10	noiri	noiri	PROPN
ejpam-7049	268	11	.	.	PUNCT
ejpam-7049	269	1	some	some	DET
ejpam-7049	269	2	weak	weak	ADJ
ejpam-7049	269	3	forms	form	NOUN
ejpam-7049	269	4	of	of	ADP
ejpam-7049	269	5	almost	almost	ADV
ejpam-7049	269	6	continuity	continuity	NOUN
ejpam-7049	269	7	.	.	PUNCT
ejpam-7049	270	1	acta	acta	PROPN
ejpam-7049	270	2	mathematica	mathematica	PROPN
ejpam-7049	270	3	hungarica	hungarica	PROPN
ejpam-7049	270	4	,	,	PUNCT
ejpam-7049	270	5	74(3):211–219	74(3):211–219	PROPN
ejpam-7049	270	6	,	,	PUNCT
ejpam-7049	270	7	1997	1997	NUM
ejpam-7049	270	8	.	.	PUNCT
ejpam-7049	271	1	[	[	X
ejpam-7049	271	2	2	2	NUM
ejpam-7049	271	3	]	]	PUNCT
ejpam-7049	271	4	a.	a.	NOUN
ejpam-7049	271	5	s.	s.	PROPN
ejpam-7049	271	6	mashhour	mashhour	PROPN
ejpam-7049	271	7	,	,	PUNCT
ejpam-7049	271	8	m.	m.	PROPN
ejpam-7049	271	9	e.	e.	PROPN
ejpam-7049	271	10	abd	abd	PROPN
ejpam-7049	271	11	el	el	PROPN
ejpam-7049	271	12	-	-	PROPN
ejpam-7049	271	13	monsef	monsef	ADJ
ejpam-7049	271	14	,	,	PUNCT
ejpam-7049	271	15	and	and	CCONJ
ejpam-7049	271	16	s.	s.	PROPN
ejpam-7049	271	17	n.	n.	PROPN
ejpam-7049	271	18	el	el	PROPN
ejpam-7049	271	19	-	-	PROPN
ejpam-7049	271	20	deeb	deeb	PROPN
ejpam-7049	271	21	.	.	PUNCT
ejpam-7049	272	1	on	on	ADP
ejpam-7049	272	2	precontinuous	precontinuous	ADJ
ejpam-7049	272	3	and	and	CCONJ
ejpam-7049	272	4	weak	weak	ADJ
ejpam-7049	272	5	precontinuous	precontinuous	ADJ
ejpam-7049	272	6	mappings	mapping	NOUN
ejpam-7049	272	7	.	.	PUNCT
ejpam-7049	273	1	proceedings	proceeding	NOUN
ejpam-7049	273	2	of	of	ADP
ejpam-7049	273	3	the	the	DET
ejpam-7049	273	4	mathematical	mathematical	ADJ
ejpam-7049	273	5	and	and	CCONJ
ejpam-7049	273	6	physical	physical	ADJ
ejpam-7049	273	7	society	society	NOUN
ejpam-7049	273	8	of	of	ADP
ejpam-7049	273	9	egypt	egypt	PROPN
ejpam-7049	273	10	,	,	PUNCT
ejpam-7049	273	11	53:47–53	53:47–53	NUM
ejpam-7049	273	12	,	,	PUNCT
ejpam-7049	273	13	1982	1982	NUM
ejpam-7049	273	14	.	.	PUNCT
ejpam-7049	274	1	[	[	X
ejpam-7049	274	2	3	3	X
ejpam-7049	274	3	]	]	PUNCT
ejpam-7049	274	4	m.	m.	NOUN
ejpam-7049	274	5	e.	e.	PROPN
ejpam-7049	274	6	abd	abd	PROPN
ejpam-7049	275	1	el	el	PROPN
ejpam-7049	275	2	-	-	PROPN
ejpam-7049	275	3	monsef	monsef	PROPN
ejpam-7049	275	4	,	,	PUNCT
ejpam-7049	275	5	s.	s.	PROPN
ejpam-7049	275	6	n.	n.	PROPN
ejpam-7049	275	7	el	el	PROPN
ejpam-7049	275	8	-	-	PROPN
ejpam-7049	275	9	deeb	deeb	PROPN
ejpam-7049	275	10	,	,	PUNCT
ejpam-7049	275	11	and	and	CCONJ
ejpam-7049	275	12	r.	r.	PROPN
ejpam-7049	275	13	a.	a.	PROPN
ejpam-7049	275	14	mahmoud	mahmoud	PROPN
ejpam-7049	275	15	.	.	PUNCT
ejpam-7049	276	1	β	β	X
ejpam-7049	276	2	-	-	ADJ
ejpam-7049	276	3	open	open	ADJ
ejpam-7049	276	4	sets	set	NOUN
ejpam-7049	276	5	and	and	CCONJ
ejpam-7049	276	6	βcontinuous	βcontinuous	ADJ
ejpam-7049	276	7	mappings	mapping	NOUN
ejpam-7049	276	8	.	.	PUNCT
ejpam-7049	277	1	bulletin	bulletin	NOUN
ejpam-7049	277	2	of	of	ADP
ejpam-7049	277	3	the	the	DET
ejpam-7049	277	4	faculty	faculty	NOUN
ejpam-7049	277	5	of	of	ADP
ejpam-7049	277	6	science	science	NOUN
ejpam-7049	277	7	,	,	PUNCT
ejpam-7049	277	8	assiut	assiut	NOUN
ejpam-7049	277	9	university	university	NOUN
ejpam-7049	277	10	,	,	PUNCT
ejpam-7049	277	11	12:77–90	12:77–90	NUM
ejpam-7049	277	12	,	,	PUNCT
ejpam-7049	277	13	1983	1983	NUM
ejpam-7049	277	14	.	.	PUNCT
ejpam-7049	278	1	[	[	X
ejpam-7049	278	2	4	4	X
ejpam-7049	278	3	]	]	PUNCT
ejpam-7049	278	4	t.	t.	PROPN
ejpam-7049	278	5	noiri	noiri	PROPN
ejpam-7049	278	6	and	and	CCONJ
ejpam-7049	278	7	v.	v.	ADP
ejpam-7049	278	8	popa	popa	NOUN
ejpam-7049	278	9	.	.	PUNCT
ejpam-7049	279	1	on	on	ADP
ejpam-7049	279	2	almost	almost	ADV
ejpam-7049	279	3	β	β	ADJ
ejpam-7049	279	4	-	-	ADJ
ejpam-7049	279	5	continuous	continuous	ADJ
ejpam-7049	279	6	functions	function	NOUN
ejpam-7049	279	7	.	.	PUNCT
ejpam-7049	280	1	acta	acta	PROPN
ejpam-7049	280	2	mathematica	mathematica	PROPN
ejpam-7049	280	3	hungarica	hungarica	PROPN
ejpam-7049	280	4	,	,	PUNCT
ejpam-7049	280	5	79(4):329–339	79(4):329–339	PROPN
ejpam-7049	280	6	,	,	PUNCT
ejpam-7049	280	7	1998	1998	NUM
ejpam-7049	280	8	.	.	PUNCT
ejpam-7049	281	1	[	[	X
ejpam-7049	281	2	5	5	X
ejpam-7049	281	3	]	]	PUNCT
ejpam-7049	281	4	t.	t.	PROPN
ejpam-7049	281	5	noiri	noiri	PROPN
ejpam-7049	281	6	.	.	PUNCT
ejpam-7049	282	1	almost	almost	ADV
ejpam-7049	282	2	α	α	NUM
ejpam-7049	282	3	-	-	ADJ
ejpam-7049	282	4	continuous	continuous	ADJ
ejpam-7049	282	5	functions	function	NOUN
ejpam-7049	282	6	.	.	PUNCT
ejpam-7049	283	1	kyungpook	kyungpook	PROPN
ejpam-7049	283	2	mathematical	mathematical	PROPN
ejpam-7049	283	3	journal	journal	PROPN
ejpam-7049	283	4	,	,	PUNCT
ejpam-7049	283	5	28:71–77	28:71–77	PROPN
ejpam-7049	283	6	,	,	PUNCT
ejpam-7049	283	7	1988	1988	NUM
ejpam-7049	283	8	.	.	PUNCT
ejpam-7049	284	1	[	[	X
ejpam-7049	284	2	6	6	NUM
ejpam-7049	284	3	]	]	PUNCT
ejpam-7049	284	4	s.	s.	PROPN
ejpam-7049	284	5	n.	n.	PROPN
ejpam-7049	284	6	maheshwari	maheshwari	PROPN
ejpam-7049	284	7	,	,	PUNCT
ejpam-7049	284	8	g.	g.	PROPN
ejpam-7049	284	9	i.	i.	PROPN
ejpam-7049	284	10	chae	chae	PROPN
ejpam-7049	284	11	,	,	PUNCT
ejpam-7049	284	12	and	and	CCONJ
ejpam-7049	284	13	p.	p.	PROPN
ejpam-7049	284	14	c.	c.	PROPN
ejpam-7049	284	15	jain	jain	PROPN
ejpam-7049	284	16	.	.	PUNCT
ejpam-7049	285	1	almost	almost	ADV
ejpam-7049	285	2	feebly	feebly	ADV
ejpam-7049	285	3	continuous	continuous	ADJ
ejpam-7049	285	4	functions	function	NOUN
ejpam-7049	285	5	.	.	PUNCT
ejpam-7049	286	1	ulsan	ulsan	PROPN
ejpam-7049	286	2	institute	institute	PROPN
ejpam-7049	286	3	of	of	ADP
ejpam-7049	286	4	science	science	PROPN
ejpam-7049	286	5	and	and	CCONJ
ejpam-7049	286	6	technology	technology	NOUN
ejpam-7049	286	7	report	report	NOUN
ejpam-7049	286	8	,	,	PUNCT
ejpam-7049	286	9	13:195–197	13:195–197	NUM
ejpam-7049	286	10	,	,	PUNCT
ejpam-7049	286	11	1982	1982	NUM
ejpam-7049	286	12	.	.	PUNCT
ejpam-7049	287	1	[	[	X
ejpam-7049	287	2	7	7	X
ejpam-7049	287	3	]	]	PUNCT
ejpam-7049	287	4	v.	v.	CCONJ
ejpam-7049	287	5	popa	popa	NOUN
ejpam-7049	287	6	.	.	PUNCT
ejpam-7049	288	1	on	on	ADP
ejpam-7049	288	2	the	the	DET
ejpam-7049	288	3	decomposition	decomposition	NOUN
ejpam-7049	288	4	of	of	ADP
ejpam-7049	288	5	the	the	DET
ejpam-7049	288	6	quasi	quasi	NOUN
ejpam-7049	288	7	-	-	NOUN
ejpam-7049	288	8	continuity	continuity	NOUN
ejpam-7049	288	9	in	in	ADP
ejpam-7049	288	10	topological	topological	ADJ
ejpam-7049	288	11	spaces	space	NOUN
ejpam-7049	288	12	(	(	PUNCT
ejpam-7049	288	13	roumanian	roumanian	ADJ
ejpam-7049	288	14	)	)	PUNCT
ejpam-7049	288	15	.	.	PUNCT
ejpam-7049	289	1	studii	studii	PROPN
ejpam-7049	289	2	şi	şi	PROPN
ejpam-7049	289	3	cercetǎri	cercetǎri	PROPN
ejpam-7049	289	4	de	de	ADP
ejpam-7049	289	5	matematicǎ	matematicǎ	PROPN
ejpam-7049	289	6	,	,	PUNCT
ejpam-7049	289	7	30:31–35	30:31–35	NUM
ejpam-7049	289	8	,	,	PUNCT
ejpam-7049	289	9	1978	1978	NUM
ejpam-7049	289	10	.	.	PUNCT
ejpam-7049	290	1	[	[	X
ejpam-7049	290	2	8	8	NUM
ejpam-7049	290	3	]	]	PUNCT
ejpam-7049	290	4	v.	v.	CCONJ
ejpam-7049	290	5	popa	popa	NOUN
ejpam-7049	290	6	and	and	CCONJ
ejpam-7049	290	7	t.	t.	PROPN
ejpam-7049	290	8	noiri	noiri	PROPN
ejpam-7049	290	9	.	.	PUNCT
ejpam-7049	291	1	on	on	ADP
ejpam-7049	291	2	upper	upper	ADJ
ejpam-7049	291	3	and	and	CCONJ
ejpam-7049	291	4	lower	low	ADJ
ejpam-7049	291	5	almost	almost	ADV
ejpam-7049	291	6	β	β	ADJ
ejpam-7049	291	7	-	-	ADJ
ejpam-7049	291	8	continuous	continuous	ADJ
ejpam-7049	291	9	multifunctions	multifunction	NOUN
ejpam-7049	291	10	.	.	PUNCT
ejpam-7049	292	1	acta	acta	PROPN
ejpam-7049	292	2	mathematica	mathematica	PROPN
ejpam-7049	292	3	hungarica	hungarica	PROPN
ejpam-7049	292	4	,	,	PUNCT
ejpam-7049	292	5	82(1	82(1	NOUN
ejpam-7049	292	6	-	-	PUNCT
ejpam-7049	292	7	2):57–73	2):57–73	NUM
ejpam-7049	292	8	,	,	PUNCT
ejpam-7049	292	9	1999	1999	NUM
ejpam-7049	292	10	.	.	PUNCT
ejpam-7049	293	1	[	[	X
ejpam-7049	293	2	9	9	NUM
ejpam-7049	293	3	]	]	PUNCT
ejpam-7049	293	4	c.	c.	PROPN
ejpam-7049	293	5	boonpok	boonpok	PROPN
ejpam-7049	293	6	.	.	PUNCT
ejpam-7049	294	1	on	on	ADP
ejpam-7049	294	2	continuous	continuous	ADJ
ejpam-7049	294	3	multifunctions	multifunction	NOUN
ejpam-7049	294	4	in	in	ADP
ejpam-7049	294	5	ideal	ideal	ADJ
ejpam-7049	294	6	topological	topological	ADJ
ejpam-7049	294	7	spaces	space	NOUN
ejpam-7049	294	8	.	.	PUNCT
ejpam-7049	295	1	lobachevskii	lobachevskii	PROPN
ejpam-7049	295	2	journal	journal	PROPN
ejpam-7049	295	3	of	of	ADP
ejpam-7049	295	4	mathematics	mathematic	NOUN
ejpam-7049	295	5	,	,	PUNCT
ejpam-7049	295	6	40(1):24–35	40(1):24–35	NUM
ejpam-7049	295	7	,	,	PUNCT
ejpam-7049	295	8	2019	2019	NUM
ejpam-7049	295	9	.	.	PUNCT
ejpam-7049	296	1	[	[	X
ejpam-7049	296	2	10	10	NUM
ejpam-7049	296	3	]	]	X
ejpam-7049	296	4	c.	c.	PROPN
ejpam-7049	296	5	boonpok	boonpok	PROPN
ejpam-7049	296	6	and	and	CCONJ
ejpam-7049	296	7	n.	n.	PROPN
ejpam-7049	296	8	srisarakham	srisarakham	PROPN
ejpam-7049	296	9	.	.	PUNCT
ejpam-7049	297	1	almost	almost	ADV
ejpam-7049	297	2	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-7049	297	3	for	for	ADP
ejpam-7049	297	4	multifunctions	multifunction	NOUN
ejpam-7049	297	5	.	.	PUNCT
ejpam-7049	298	1	international	international	ADJ
ejpam-7049	298	2	journal	journal	NOUN
ejpam-7049	298	3	of	of	ADP
ejpam-7049	298	4	analysis	analysis	NOUN
ejpam-7049	298	5	and	and	CCONJ
ejpam-7049	298	6	applications	application	NOUN
ejpam-7049	298	7	,	,	PUNCT
ejpam-7049	298	8	21:107	21:107	NUM
ejpam-7049	298	9	,	,	PUNCT
ejpam-7049	298	10	2023	2023	NUM
ejpam-7049	298	11	.	.	PUNCT
ejpam-7049	299	1	[	[	X
ejpam-7049	299	2	11	11	NUM
ejpam-7049	299	3	]	]	PUNCT
ejpam-7049	299	4	c.	c.	PROPN
ejpam-7049	299	5	boonpok	boonpok	PROPN
ejpam-7049	299	6	.	.	PUNCT
ejpam-7049	300	1	upper	upper	ADJ
ejpam-7049	300	2	and	and	CCONJ
ejpam-7049	300	3	lower	low	ADJ
ejpam-7049	300	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-7049	300	5	.	.	PUNCT
ejpam-7049	300	6	heliyon	heliyon	NOUN
ejpam-7049	300	7	,	,	PUNCT
ejpam-7049	300	8	7	7	NUM
ejpam-7049	300	9	:	:	PUNCT
ejpam-7049	300	10	e05986	e05986	PROPN
ejpam-7049	300	11	,	,	PUNCT
ejpam-7049	300	12	2021	2021	NUM
ejpam-7049	300	13	.	.	PUNCT
ejpam-7049	301	1	n.	n.	PROPN
ejpam-7049	301	2	srisarakham	srisarakham	PROPN
ejpam-7049	301	3	,	,	PUNCT
ejpam-7049	301	4	a.	a.	PROPN
ejpam-7049	301	5	sama	sama	PROPN
ejpam-7049	301	6	-	-	PUNCT
ejpam-7049	301	7	ae	ae	PROPN
ejpam-7049	301	8	,	,	PUNCT
ejpam-7049	301	9	c.	c.	PROPN
ejpam-7049	301	10	boonpok	boonpok	PROPN
ejpam-7049	301	11	/	/	SYM
ejpam-7049	301	12	eur	eur	PROPN
ejpam-7049	301	13	.	.	PUNCT
ejpam-7049	302	1	j.	j.	PROPN
ejpam-7049	302	2	pure	pure	PROPN
ejpam-7049	302	3	appl	appl	PROPN
ejpam-7049	302	4	.	.	PROPN
ejpam-7049	302	5	math	math	PROPN
ejpam-7049	302	6	,	,	PUNCT
ejpam-7049	302	7	18	18	NUM
ejpam-7049	302	8	(	(	PUNCT
ejpam-7049	302	9	4	4	NUM
ejpam-7049	302	10	)	)	PUNCT
ejpam-7049	302	11	(	(	PUNCT
ejpam-7049	302	12	2025	2025	NUM
ejpam-7049	302	13	)	)	PUNCT
ejpam-7049	302	14	,	,	PUNCT
ejpam-7049	302	15	7049	7049	NUM
ejpam-7049	302	16	11	11	NUM
ejpam-7049	302	17	of	of	ADP
ejpam-7049	302	18	11	11	NUM
ejpam-7049	303	1	[	[	X
ejpam-7049	303	2	12	12	NUM
ejpam-7049	303	3	]	]	PUNCT
ejpam-7049	303	4	c.	c.	PROPN
ejpam-7049	303	5	boonpok	boonpok	PROPN
ejpam-7049	303	6	and	and	CCONJ
ejpam-7049	303	7	p.	p.	NOUN
ejpam-7049	303	8	pue	pue	NOUN
ejpam-7049	303	9	-	-	PUNCT
ejpam-7049	303	10	on	on	ADP
ejpam-7049	303	11	.	.	PUNCT
ejpam-7049	304	1	upper	upper	ADJ
ejpam-7049	304	2	and	and	CCONJ
ejpam-7049	304	3	lower	low	ADJ
ejpam-7049	304	4	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-7049	304	5	multifunctions	multifunction	NOUN
ejpam-7049	304	6	.	.	PUNCT
ejpam-7049	305	1	european	european	ADJ
ejpam-7049	305	2	journal	journal	PROPN
ejpam-7049	305	3	of	of	ADP
ejpam-7049	305	4	pure	pure	ADJ
ejpam-7049	305	5	and	and	CCONJ
ejpam-7049	305	6	applied	applied	ADJ
ejpam-7049	305	7	mathematics	mathematic	NOUN
ejpam-7049	305	8	,	,	PUNCT
ejpam-7049	305	9	16(3):1634–1646	16(3):1634–1646	NUM
ejpam-7049	305	10	,	,	PUNCT
ejpam-7049	305	11	2023	2023	NUM
ejpam-7049	305	12	.	.	PUNCT
ejpam-7049	306	1	[	[	X
ejpam-7049	306	2	13	13	NUM
ejpam-7049	306	3	]	]	PUNCT
ejpam-7049	306	4	c.	c.	PROPN
ejpam-7049	306	5	boonpok	boonpok	PROPN
ejpam-7049	306	6	and	and	CCONJ
ejpam-7049	306	7	p.	p.	NOUN
ejpam-7049	306	8	pue	pue	NOUN
ejpam-7049	306	9	-	-	PUNCT
ejpam-7049	306	10	on	on	ADP
ejpam-7049	306	11	.	.	PUNCT
ejpam-7049	307	1	continuity	continuity	NOUN
ejpam-7049	307	2	for	for	ADP
ejpam-7049	307	3	multifunctions	multifunction	NOUN
ejpam-7049	307	4	in	in	ADP
ejpam-7049	307	5	ideal	ideal	ADJ
ejpam-7049	307	6	topological	topological	ADJ
ejpam-7049	307	7	spaces	space	NOUN
ejpam-7049	307	8	.	.	PUNCT
ejpam-7049	308	1	wseas	wseas	VERB
ejpam-7049	308	2	transactions	transaction	NOUN
ejpam-7049	308	3	on	on	ADP
ejpam-7049	308	4	mathematics	mathematic	NOUN
ejpam-7049	308	5	,	,	PUNCT
ejpam-7049	308	6	19:624–631	19:624–631	NUM
ejpam-7049	308	7	,	,	PUNCT
ejpam-7049	308	8	2020	2020	NUM
ejpam-7049	308	9	.	.	PUNCT
ejpam-7049	309	1	[	[	X
ejpam-7049	309	2	14	14	NUM
ejpam-7049	309	3	]	]	X
ejpam-7049	309	4	p.	p.	NOUN
ejpam-7049	309	5	pue	pue	NOUN
ejpam-7049	309	6	-	-	PUNCT
ejpam-7049	309	7	on	on	ADP
ejpam-7049	309	8	,	,	PUNCT
ejpam-7049	309	9	s.	s.	PROPN
ejpam-7049	309	10	sompong	sompong	PROPN
ejpam-7049	309	11	,	,	PUNCT
ejpam-7049	309	12	and	and	CCONJ
ejpam-7049	309	13	c.	c.	PROPN
ejpam-7049	309	14	boonpok	boonpok	PROPN
ejpam-7049	309	15	.	.	PUNCT
ejpam-7049	310	1	upper	upper	ADJ
ejpam-7049	310	2	and	and	CCONJ
ejpam-7049	310	3	lower	low	ADJ
ejpam-7049	310	4	(	(	PUNCT
ejpam-7049	310	5	τ1	τ1	NOUN
ejpam-7049	310	6	,	,	PUNCT
ejpam-7049	310	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	310	8	multifunctions	multifunction	NOUN
ejpam-7049	310	9	.	.	PUNCT
ejpam-7049	311	1	international	international	ADJ
ejpam-7049	311	2	journal	journal	PROPN
ejpam-7049	311	3	of	of	ADP
ejpam-7049	311	4	mathematics	mathematic	NOUN
ejpam-7049	311	5	and	and	CCONJ
ejpam-7049	311	6	computer	computer	NOUN
ejpam-7049	311	7	science	science	NOUN
ejpam-7049	311	8	,	,	PUNCT
ejpam-7049	311	9	19(4):1305	19(4):1305	NUM
ejpam-7049	311	10	–	–	PUNCT
ejpam-7049	311	11	1310	1310	NUM
ejpam-7049	311	12	,	,	PUNCT
ejpam-7049	311	13	2024	2024	NUM
ejpam-7049	311	14	.	.	PUNCT
ejpam-7049	312	1	[	[	X
ejpam-7049	312	2	15	15	NUM
ejpam-7049	312	3	]	]	X
ejpam-7049	312	4	c.	c.	PROPN
ejpam-7049	312	5	boonpok	boonpok	PROPN
ejpam-7049	312	6	and	and	CCONJ
ejpam-7049	312	7	p.	p.	NOUN
ejpam-7049	312	8	pue	pue	NOUN
ejpam-7049	312	9	-	-	PUNCT
ejpam-7049	312	10	on	on	ADP
ejpam-7049	312	11	.	.	PUNCT
ejpam-7049	313	1	characterizations	characterization	NOUN
ejpam-7049	313	2	of	of	ADP
ejpam-7049	313	3	almost	almost	ADV
ejpam-7049	313	4	(	(	PUNCT
ejpam-7049	313	5	τ1	τ1	NOUN
ejpam-7049	313	6	,	,	PUNCT
ejpam-7049	313	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	313	8	multifunctions	multifunction	NOUN
ejpam-7049	313	9	.	.	PUNCT
ejpam-7049	314	1	international	international	ADJ
ejpam-7049	314	2	journal	journal	NOUN
ejpam-7049	314	3	of	of	ADP
ejpam-7049	314	4	analysis	analysis	NOUN
ejpam-7049	314	5	and	and	CCONJ
ejpam-7049	314	6	applications	application	NOUN
ejpam-7049	314	7	,	,	PUNCT
ejpam-7049	314	8	22:33	22:33	NUM
ejpam-7049	314	9	,	,	PUNCT
ejpam-7049	314	10	2024	2024	NUM
ejpam-7049	314	11	.	.	PUNCT
ejpam-7049	315	1	[	[	X
ejpam-7049	315	2	16	16	NUM
ejpam-7049	315	3	]	]	PUNCT
ejpam-7049	315	4	k.	k.	PROPN
ejpam-7049	316	1	laprom	laprom	PROPN
ejpam-7049	316	2	,	,	PUNCT
ejpam-7049	316	3	c.	c.	PROPN
ejpam-7049	316	4	boonpok	boonpok	PROPN
ejpam-7049	316	5	,	,	PUNCT
ejpam-7049	316	6	and	and	CCONJ
ejpam-7049	316	7	c.	c.	PROPN
ejpam-7049	316	8	viriyapong	viriyapong	PROPN
ejpam-7049	316	9	.	.	PUNCT
ejpam-7049	317	1	β(τ1	β(τ1	PROPN
ejpam-7049	317	2	,	,	PUNCT
ejpam-7049	317	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7049	317	4	multifunctions	multifunction	NOUN
ejpam-7049	317	5	on	on	ADP
ejpam-7049	317	6	bitopological	bitopological	ADJ
ejpam-7049	317	7	spaces	space	NOUN
ejpam-7049	317	8	.	.	PUNCT
ejpam-7049	318	1	journal	journal	NOUN
ejpam-7049	318	2	of	of	ADP
ejpam-7049	318	3	mathematics	mathematic	NOUN
ejpam-7049	318	4	,	,	PUNCT
ejpam-7049	318	5	2020:4020971	2020:4020971	NUM
ejpam-7049	318	6	,	,	PUNCT
ejpam-7049	318	7	2020	2020	NUM
ejpam-7049	318	8	.	.	PUNCT
ejpam-7049	319	1	[	[	X
ejpam-7049	319	2	17	17	NUM
ejpam-7049	319	3	]	]	X
ejpam-7049	319	4	c.	c.	PROPN
ejpam-7049	319	5	boonpok	boonpok	PROPN
ejpam-7049	319	6	,	,	PUNCT
ejpam-7049	319	7	c.	c.	PROPN
ejpam-7049	319	8	viriyapong	viriyapong	PROPN
ejpam-7049	319	9	,	,	PUNCT
ejpam-7049	319	10	and	and	CCONJ
ejpam-7049	319	11	m.	m.	NOUN
ejpam-7049	319	12	thongmoon	thongmoon	NOUN
ejpam-7049	319	13	.	.	PUNCT
ejpam-7049	320	1	on	on	ADP
ejpam-7049	320	2	upper	upper	ADJ
ejpam-7049	320	3	and	and	CCONJ
ejpam-7049	320	4	lower	low	ADJ
ejpam-7049	320	5	(	(	PUNCT
ejpam-7049	320	6	τ1	τ1	NOUN
ejpam-7049	320	7	,	,	PUNCT
ejpam-7049	320	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7049	320	9	multifunctions	multifunction	NOUN
ejpam-7049	320	10	.	.	PUNCT
ejpam-7049	321	1	journal	journal	PROPN
ejpam-7049	321	2	of	of	ADP
ejpam-7049	321	3	mathematics	mathematics	PROPN
ejpam-7049	321	4	and	and	CCONJ
ejpam-7049	321	5	computer	computer	NOUN
ejpam-7049	321	6	science	science	NOUN
ejpam-7049	321	7	,	,	PUNCT
ejpam-7049	321	8	18:282	18:282	NUM
ejpam-7049	321	9	–	–	PUNCT
ejpam-7049	321	10	293	293	NUM
ejpam-7049	321	11	,	,	PUNCT
ejpam-7049	321	12	2018	2018	NUM
ejpam-7049	321	13	.	.	PUNCT
ejpam-7049	322	1	[	[	X
ejpam-7049	322	2	18	18	NUM
ejpam-7049	322	3	]	]	X
ejpam-7049	322	4	c.	c.	PROPN
ejpam-7049	322	5	viriyapong	viriyapong	PROPN
ejpam-7049	322	6	and	and	CCONJ
ejpam-7049	322	7	c.	c.	PROPN
ejpam-7049	322	8	boonpok	boonpok	PROPN
ejpam-7049	322	9	.	.	PUNCT
ejpam-7049	323	1	(	(	PUNCT
ejpam-7049	323	2	τ1	τ1	NOUN
ejpam-7049	323	3	,	,	PUNCT
ejpam-7049	323	4	τ2)α	τ2)α	NOUN
ejpam-7049	323	5	-	-	PUNCT
ejpam-7049	323	6	continuity	continuity	NOUN
ejpam-7049	323	7	for	for	ADP
ejpam-7049	323	8	multifunctions	multifunction	NOUN
ejpam-7049	323	9	.	.	PUNCT
ejpam-7049	324	1	journal	journal	PROPN
ejpam-7049	324	2	of	of	ADP
ejpam-7049	324	3	mathematics	mathematic	NOUN
ejpam-7049	324	4	,	,	PUNCT
ejpam-7049	324	5	2020:6285763	2020:6285763	NUM
ejpam-7049	324	6	,	,	PUNCT
ejpam-7049	324	7	2020	2020	NUM
ejpam-7049	324	8	.	.	PUNCT
ejpam-7049	325	1	[	[	X
ejpam-7049	325	2	19	19	NUM
ejpam-7049	325	3	]	]	PUNCT
ejpam-7049	325	4	c.	c.	PROPN
ejpam-7049	325	5	boonpok	boonpok	PROPN
ejpam-7049	325	6	.	.	PUNCT
ejpam-7049	326	1	(	(	PUNCT
ejpam-7049	326	2	τ1	τ1	NOUN
ejpam-7049	326	3	,	,	PUNCT
ejpam-7049	326	4	τ2)δ	τ2)δ	ADJ
ejpam-7049	326	5	-	-	PUNCT
ejpam-7049	326	6	semicontinuous	semicontinuous	ADJ
ejpam-7049	326	7	multifunctions	multifunction	NOUN
ejpam-7049	326	8	.	.	PUNCT
ejpam-7049	327	1	heliyon	heliyon	NOUN
ejpam-7049	327	2	,	,	PUNCT
ejpam-7049	327	3	6	6	NUM
ejpam-7049	327	4	:	:	SYM
ejpam-7049	327	5	e05367	e05367	PROPN
ejpam-7049	327	6	,	,	PUNCT
ejpam-7049	327	7	2020	2020	NUM
ejpam-7049	327	8	.	.	PUNCT
ejpam-7049	328	1	[	[	X
ejpam-7049	328	2	20	20	NUM
ejpam-7049	328	3	]	]	PUNCT
ejpam-7049	328	4	p.	p.	NOUN
ejpam-7049	328	5	pue	pue	NOUN
ejpam-7049	328	6	-	-	PUNCT
ejpam-7049	328	7	on	on	ADP
ejpam-7049	328	8	,	,	PUNCT
ejpam-7049	328	9	s.	s.	PROPN
ejpam-7049	328	10	sompong	sompong	PROPN
ejpam-7049	328	11	,	,	PUNCT
ejpam-7049	328	12	and	and	CCONJ
ejpam-7049	328	13	c.	c.	PROPN
ejpam-7049	328	14	boonpok	boonpok	PROPN
ejpam-7049	328	15	.	.	PUNCT
ejpam-7049	329	1	almost	almost	ADV
ejpam-7049	329	2	quasi	quasi	X
ejpam-7049	329	3	(	(	PUNCT
ejpam-7049	329	4	τ1	τ1	NOUN
ejpam-7049	329	5	,	,	PUNCT
ejpam-7049	329	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7049	329	7	for	for	ADP
ejpam-7049	329	8	multifunctions	multifunction	NOUN
ejpam-7049	329	9	.	.	PUNCT
ejpam-7049	330	1	international	international	ADJ
ejpam-7049	330	2	journal	journal	NOUN
ejpam-7049	330	3	of	of	ADP
ejpam-7049	330	4	analysis	analysis	NOUN
ejpam-7049	330	5	and	and	CCONJ
ejpam-7049	330	6	applications	application	NOUN
ejpam-7049	330	7	,	,	PUNCT
ejpam-7049	330	8	22:97	22:97	NUM
ejpam-7049	330	9	,	,	PUNCT
ejpam-7049	330	10	2024	2024	NUM
ejpam-7049	330	11	.	.	PUNCT
ejpam-7049	331	1	[	[	X
ejpam-7049	331	2	21	21	NUM
ejpam-7049	331	3	]	]	PUNCT
ejpam-7049	331	4	k.	k.	PROPN
ejpam-7049	331	5	kuratowski	kuratowski	PROPN
ejpam-7049	331	6	.	.	PUNCT
ejpam-7049	332	1	topology	topology	PROPN
ejpam-7049	332	2	,	,	PUNCT
ejpam-7049	332	3	vol	vol	NOUN
ejpam-7049	332	4	.	.	PUNCT
ejpam-7049	332	5	i.	i.	PROPN
ejpam-7049	332	6	academic	academic	PROPN
ejpam-7049	332	7	press	press	PROPN
ejpam-7049	332	8	,	,	PUNCT
ejpam-7049	332	9	new	new	PROPN
ejpam-7049	332	10	york	york	PROPN
ejpam-7049	332	11	,	,	PUNCT
ejpam-7049	332	12	1966	1966	NUM
ejpam-7049	332	13	.	.	PUNCT
ejpam-7049	333	1	[	[	X
ejpam-7049	333	2	22	22	NUM
ejpam-7049	333	3	]	]	X
ejpam-7049	333	4	d.	d.	PROPN
ejpam-7049	333	5	janković	janković	VERB
ejpam-7049	333	6	and	and	CCONJ
ejpam-7049	333	7	t.	t.	PROPN
ejpam-7049	333	8	r.	r.	PROPN
ejpam-7049	333	9	hamlett	hamlett	PROPN
ejpam-7049	333	10	.	.	PUNCT
ejpam-7049	334	1	new	new	ADJ
ejpam-7049	334	2	topologies	topology	NOUN
ejpam-7049	334	3	from	from	ADP
ejpam-7049	334	4	old	old	ADJ
ejpam-7049	334	5	via	via	ADP
ejpam-7049	334	6	ideals	ideal	NOUN
ejpam-7049	334	7	.	.	PUNCT
ejpam-7049	335	1	the	the	DET
ejpam-7049	335	2	american	american	PROPN
ejpam-7049	335	3	mathematical	mathematical	PROPN
ejpam-7049	335	4	monthly	monthly	ADV
ejpam-7049	335	5	,	,	PUNCT
ejpam-7049	335	6	97:295–310	97:295–310	PROPN
ejpam-7049	335	7	,	,	PUNCT
ejpam-7049	335	8	1990	1990	NUM
ejpam-7049	335	9	.	.	PUNCT
ejpam-7049	336	1	[	[	X
ejpam-7049	336	2	23	23	NUM
ejpam-7049	336	3	]	]	PUNCT
ejpam-7049	336	4	t.	t.	PROPN
ejpam-7049	336	5	noiri	noiri	PROPN
ejpam-7049	336	6	and	and	CCONJ
ejpam-7049	336	7	v.	v.	ADP
ejpam-7049	336	8	popa	popa	NOUN
ejpam-7049	336	9	.	.	PUNCT
ejpam-7049	337	1	on	on	ADP
ejpam-7049	337	2	(	(	PUNCT
ejpam-7049	337	3	mi	mi	ADJ
ejpam-7049	337	4	,	,	PUNCT
ejpam-7049	337	5	nj)-continuous	nj)-continuous	ADJ
ejpam-7049	337	6	multifunctions	multifunction	NOUN
ejpam-7049	337	7	.	.	PUNCT
ejpam-7049	338	1	romanian	romanian	ADJ
ejpam-7049	338	2	journal	journal	PROPN
ejpam-7049	338	3	of	of	ADP
ejpam-7049	338	4	mathematics	mathematics	PROPN
ejpam-7049	338	5	and	and	CCONJ
ejpam-7049	338	6	computer	computer	NOUN
ejpam-7049	338	7	science	science	NOUN
ejpam-7049	338	8	,	,	PUNCT
ejpam-7049	338	9	15(1):1–8	15(1):1–8	NUM
ejpam-7049	338	10	,	,	PUNCT
ejpam-7049	338	11	2025	2025	NUM
ejpam-7049	338	12	.	.	PUNCT
ejpam-7049	339	1	[	[	X
ejpam-7049	339	2	24	24	NUM
ejpam-7049	339	3	]	]	X
ejpam-7049	339	4	p.	p.	NOUN
ejpam-7049	339	5	pue	pue	NOUN
ejpam-7049	339	6	-	-	PUNCT
ejpam-7049	339	7	on	on	ADP
ejpam-7049	339	8	,	,	PUNCT
ejpam-7049	339	9	a.	a.	PROPN
ejpam-7049	339	10	sama	sama	PROPN
ejpam-7049	339	11	-	-	PUNCT
ejpam-7049	339	12	ae	ae	PROPN
ejpam-7049	339	13	,	,	PUNCT
ejpam-7049	339	14	and	and	CCONJ
ejpam-7049	339	15	c.	c.	PROPN
ejpam-7049	339	16	boonpok	boonpok	PROPN
ejpam-7049	339	17	.	.	PUNCT
ejpam-7049	340	1	upper	upper	ADJ
ejpam-7049	340	2	and	and	CCONJ
ejpam-7049	340	3	lower	low	ADJ
ejpam-7049	340	4	τ⋆β(σ1	τ⋆β(σ1	NOUN
ejpam-7049	340	5	,	,	PUNCT
ejpam-7049	340	6	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7049	340	7	.	.	PUNCT
ejpam-7049	341	1	(	(	PUNCT
ejpam-7049	341	2	submitted	submit	VERB
ejpam-7049	341	3	)	)	PUNCT
ejpam-7049	341	4	.	.	PUNCT