id	sid	tid	token	lemma	pos
ejpam-6569	1	1	european	european	PROPN
ejpam-6569	1	2	journal	journal	PROPN
ejpam-6569	1	3	of	of	ADP
ejpam-6569	1	4	pure	pure	ADJ
ejpam-6569	1	5	and	and	CCONJ
ejpam-6569	1	6	applied	applied	ADJ
ejpam-6569	1	7	mathematics	mathematic	NOUN
ejpam-6569	1	8	2025	2025	NUM
ejpam-6569	1	9	,	,	PUNCT
ejpam-6569	1	10	vol	vol	NOUN
ejpam-6569	1	11	.	.	PROPN
ejpam-6569	1	12	18	18	NUM
ejpam-6569	1	13	,	,	PUNCT
ejpam-6569	1	14	issue	issue	NOUN
ejpam-6569	1	15	3	3	NUM
ejpam-6569	1	16	,	,	PUNCT
ejpam-6569	1	17	article	article	NOUN
ejpam-6569	1	18	number	number	NOUN
ejpam-6569	1	19	6569	6569	NUM
ejpam-6569	1	20	issn	issn	VERB
ejpam-6569	1	21	1307	1307	NUM
ejpam-6569	1	22	-	-	SYM
ejpam-6569	1	23	5543	5543	NUM
ejpam-6569	1	24	–	–	PUNCT
ejpam-6569	1	25	ejpam.com	ejpam.com	X
ejpam-6569	1	26	published	publish	VERB
ejpam-6569	1	27	by	by	ADP
ejpam-6569	1	28	new	new	PROPN
ejpam-6569	1	29	york	york	PROPN
ejpam-6569	1	30	business	business	PROPN
ejpam-6569	1	31	global	global	PROPN
ejpam-6569	1	32	upper	upper	ADJ
ejpam-6569	1	33	and	and	CCONJ
ejpam-6569	1	34	lower	low	ADJ
ejpam-6569	1	35	quasi	quasi	NOUN
ejpam-6569	1	36	θτ	θτ	ADP
ejpam-6569	1	37	⋆(σ1	⋆(σ1	PROPN
ejpam-6569	1	38	,	,	PUNCT
ejpam-6569	1	39	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6569	1	40	napassanan	napassanan	NOUN
ejpam-6569	1	41	srisarakham1	srisarakham1	PROPN
ejpam-6569	1	42	,	,	PUNCT
ejpam-6569	1	43	areeyuth	areeyuth	NOUN
ejpam-6569	1	44	sama	sama	NOUN
ejpam-6569	1	45	-	-	PUNCT
ejpam-6569	1	46	ae2	ae2	PROPN
ejpam-6569	1	47	,	,	PUNCT
ejpam-6569	1	48	chawalit	chawalit	VERB
ejpam-6569	1	49	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6569	1	50	1	1	NUM
ejpam-6569	1	51	mathematics	mathematic	NOUN
ejpam-6569	1	52	and	and	CCONJ
ejpam-6569	1	53	applied	apply	VERB
ejpam-6569	1	54	mathematics	mathematics	PROPN
ejpam-6569	1	55	research	research	NOUN
ejpam-6569	1	56	unit	unit	NOUN
ejpam-6569	1	57	,	,	PUNCT
ejpam-6569	1	58	department	department	NOUN
ejpam-6569	1	59	of	of	ADP
ejpam-6569	1	60	mathematics	mathematic	NOUN
ejpam-6569	1	61	,	,	PUNCT
ejpam-6569	1	62	faculty	faculty	NOUN
ejpam-6569	1	63	of	of	ADP
ejpam-6569	1	64	science	science	NOUN
ejpam-6569	1	65	,	,	PUNCT
ejpam-6569	1	66	mahasarakham	mahasarakham	PROPN
ejpam-6569	1	67	university	university	PROPN
ejpam-6569	1	68	,	,	PUNCT
ejpam-6569	1	69	maha	maha	PROPN
ejpam-6569	1	70	sarakham	sarakham	PROPN
ejpam-6569	1	71	,	,	PUNCT
ejpam-6569	1	72	44150	44150	NUM
ejpam-6569	1	73	,	,	PUNCT
ejpam-6569	1	74	thailand	thailand	PROPN
ejpam-6569	1	75	2	2	NUM
ejpam-6569	1	76	department	department	NOUN
ejpam-6569	1	77	of	of	ADP
ejpam-6569	1	78	mathematics	mathematic	NOUN
ejpam-6569	1	79	and	and	CCONJ
ejpam-6569	1	80	computer	computer	NOUN
ejpam-6569	1	81	science	science	NOUN
ejpam-6569	1	82	,	,	PUNCT
ejpam-6569	1	83	faculty	faculty	NOUN
ejpam-6569	1	84	of	of	ADP
ejpam-6569	1	85	science	science	NOUN
ejpam-6569	1	86	and	and	CCONJ
ejpam-6569	1	87	technology	technology	NOUN
ejpam-6569	1	88	,	,	PUNCT
ejpam-6569	1	89	prince	prince	NOUN
ejpam-6569	1	90	of	of	ADP
ejpam-6569	1	91	songkla	songkla	PROPN
ejpam-6569	1	92	university	university	PROPN
ejpam-6569	1	93	,	,	PUNCT
ejpam-6569	1	94	pattani	pattani	NOUN
ejpam-6569	1	95	campus	campus	NOUN
ejpam-6569	1	96	,	,	PUNCT
ejpam-6569	1	97	pattani	pattani	NOUN
ejpam-6569	1	98	,	,	PUNCT
ejpam-6569	1	99	94000	94000	NUM
ejpam-6569	1	100	,	,	PUNCT
ejpam-6569	1	101	thailand	thailand	PROPN
ejpam-6569	1	102	abstract	abstract	PROPN
ejpam-6569	1	103	.	.	PUNCT
ejpam-6569	2	1	this	this	DET
ejpam-6569	2	2	paper	paper	NOUN
ejpam-6569	2	3	presents	present	VERB
ejpam-6569	2	4	new	new	ADJ
ejpam-6569	2	5	concepts	concept	NOUN
ejpam-6569	2	6	of	of	ADP
ejpam-6569	2	7	continuous	continuous	ADJ
ejpam-6569	2	8	multifunctions	multifunction	NOUN
ejpam-6569	2	9	,	,	PUNCT
ejpam-6569	2	10	called	call	VERB
ejpam-6569	2	11	upper	upper	ADJ
ejpam-6569	2	12	quasi	quasi	NOUN
ejpam-6569	2	13	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	2	14	,	,	PUNCT
ejpam-6569	2	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	2	16	multifunctions	multifunction	NOUN
ejpam-6569	2	17	and	and	CCONJ
ejpam-6569	2	18	lower	low	ADJ
ejpam-6569	2	19	quasi	quasi	NOUN
ejpam-6569	2	20	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	2	21	,	,	PUNCT
ejpam-6569	2	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	2	23	multifunctions	multifunction	NOUN
ejpam-6569	2	24	.	.	PUNCT
ejpam-6569	3	1	moreover	moreover	ADV
ejpam-6569	3	2	,	,	PUNCT
ejpam-6569	3	3	several	several	ADJ
ejpam-6569	3	4	characterizations	characterization	NOUN
ejpam-6569	3	5	and	and	CCONJ
ejpam-6569	3	6	some	some	DET
ejpam-6569	3	7	properties	property	NOUN
ejpam-6569	3	8	concerning	concern	VERB
ejpam-6569	3	9	upper	upper	ADJ
ejpam-6569	3	10	quasi	quasi	NOUN
ejpam-6569	3	11	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	3	12	,	,	PUNCT
ejpam-6569	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	3	14	multifunctions	multifunction	NOUN
ejpam-6569	3	15	and	and	CCONJ
ejpam-6569	3	16	lower	low	ADJ
ejpam-6569	3	17	quasi	quasi	NOUN
ejpam-6569	3	18	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	3	19	,	,	PUNCT
ejpam-6569	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	3	21	multifunctions	multifunction	NOUN
ejpam-6569	3	22	are	be	AUX
ejpam-6569	3	23	considered	consider	VERB
ejpam-6569	3	24	.	.	PUNCT
ejpam-6569	4	1	2020	2020	NUM
ejpam-6569	4	2	mathematics	mathematic	NOUN
ejpam-6569	4	3	subject	subject	NOUN
ejpam-6569	4	4	classifications	classification	NOUN
ejpam-6569	4	5	:	:	PUNCT
ejpam-6569	4	6	54c08	54c08	NUM
ejpam-6569	4	7	,	,	PUNCT
ejpam-6569	4	8	54c60	54c60	NUM
ejpam-6569	4	9	key	key	ADJ
ejpam-6569	4	10	words	word	NOUN
ejpam-6569	4	11	and	and	CCONJ
ejpam-6569	4	12	phrases	phrase	NOUN
ejpam-6569	4	13	:	:	PUNCT
ejpam-6569	4	14	upper	upper	ADJ
ejpam-6569	4	15	quasi	quasi	NOUN
ejpam-6569	4	16	θτ⋆(σ1	θτ⋆(σ1	PROPN
ejpam-6569	4	17	,	,	PUNCT
ejpam-6569	4	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	4	19	multifunction	multifunction	NOUN
ejpam-6569	4	20	,	,	PUNCT
ejpam-6569	4	21	lower	low	ADJ
ejpam-6569	4	22	quasi	quasi	NOUN
ejpam-6569	4	23	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	4	24	,	,	PUNCT
ejpam-6569	4	25	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	4	26	multifunction	multifunction	NOUN
ejpam-6569	4	27	1	1	NUM
ejpam-6569	4	28	.	.	PUNCT
ejpam-6569	4	29	introduction	introduction	NOUN
ejpam-6569	4	30	in	in	ADP
ejpam-6569	4	31	1963	1963	NUM
ejpam-6569	4	32	,	,	PUNCT
ejpam-6569	4	33	levine	levine	PROPN
ejpam-6569	5	1	[	[	X
ejpam-6569	5	2	1	1	NUM
ejpam-6569	5	3	]	]	PUNCT
ejpam-6569	5	4	introduced	introduce	VERB
ejpam-6569	5	5	and	and	CCONJ
ejpam-6569	5	6	studied	study	VERB
ejpam-6569	5	7	the	the	DET
ejpam-6569	5	8	notion	notion	NOUN
ejpam-6569	5	9	of	of	ADP
ejpam-6569	5	10	semi	semi	ADJ
ejpam-6569	5	11	-	-	ADJ
ejpam-6569	5	12	continuous	continuous	ADJ
ejpam-6569	5	13	functions	function	NOUN
ejpam-6569	5	14	.	.	PUNCT
ejpam-6569	6	1	arya	arya	PROPN
ejpam-6569	6	2	and	and	CCONJ
ejpam-6569	6	3	bhamini	bhamini	PROPN
ejpam-6569	7	1	[	[	X
ejpam-6569	7	2	2	2	NUM
ejpam-6569	7	3	]	]	PUNCT
ejpam-6569	7	4	introduced	introduce	VERB
ejpam-6569	7	5	the	the	DET
ejpam-6569	7	6	concept	concept	NOUN
ejpam-6569	7	7	of	of	ADP
ejpam-6569	7	8	θ	θ	NOUN
ejpam-6569	7	9	-	-	PUNCT
ejpam-6569	7	10	semi	semi	NOUN
ejpam-6569	7	11	-	-	NOUN
ejpam-6569	7	12	continuity	continuity	NOUN
ejpam-6569	7	13	as	as	ADP
ejpam-6569	7	14	a	a	DET
ejpam-6569	7	15	generalization	generalization	NOUN
ejpam-6569	7	16	of	of	ADP
ejpam-6569	7	17	semi	semi	NOUN
ejpam-6569	7	18	-	-	NOUN
ejpam-6569	7	19	continuity	continuity	NOUN
ejpam-6569	7	20	.	.	PUNCT
ejpam-6569	8	1	noiri	noiri	ADV
ejpam-6569	9	1	[	[	X
ejpam-6569	9	2	3	3	X
ejpam-6569	9	3	]	]	PUNCT
ejpam-6569	9	4	and	and	CCONJ
ejpam-6569	9	5	jafari	jafari	ADJ
ejpam-6569	9	6	and	and	CCONJ
ejpam-6569	9	7	noiri	noiri	ADV
ejpam-6569	10	1	[	[	X
ejpam-6569	10	2	4	4	X
ejpam-6569	10	3	]	]	PUNCT
ejpam-6569	10	4	have	have	AUX
ejpam-6569	10	5	further	far	ADV
ejpam-6569	10	6	investigated	investigate	VERB
ejpam-6569	10	7	some	some	DET
ejpam-6569	10	8	characterizations	characterization	NOUN
ejpam-6569	10	9	of	of	ADP
ejpam-6569	10	10	θ	θ	NOUN
ejpam-6569	10	11	-	-	PUNCT
ejpam-6569	10	12	semi	semi	ADJ
ejpam-6569	10	13	-	-	ADJ
ejpam-6569	10	14	continuous	continuous	ADJ
ejpam-6569	10	15	functions	function	NOUN
ejpam-6569	10	16	.	.	PUNCT
ejpam-6569	11	1	marcus	marcus	PROPN
ejpam-6569	12	1	[	[	X
ejpam-6569	12	2	5	5	NUM
ejpam-6569	12	3	]	]	PUNCT
ejpam-6569	12	4	introduced	introduce	VERB
ejpam-6569	12	5	and	and	CCONJ
ejpam-6569	12	6	investigated	investigate	VERB
ejpam-6569	12	7	the	the	DET
ejpam-6569	12	8	notion	notion	NOUN
ejpam-6569	12	9	of	of	ADP
ejpam-6569	12	10	quasi	quasi	ADJ
ejpam-6569	12	11	continuous	continuous	ADJ
ejpam-6569	12	12	functions	function	NOUN
ejpam-6569	12	13	.	.	PUNCT
ejpam-6569	13	1	popa	popa	NOUN
ejpam-6569	14	1	[	[	X
ejpam-6569	14	2	6	6	NUM
ejpam-6569	14	3	]	]	PUNCT
ejpam-6569	14	4	introduced	introduce	VERB
ejpam-6569	14	5	and	and	CCONJ
ejpam-6569	14	6	studied	study	VERB
ejpam-6569	14	7	the	the	DET
ejpam-6569	14	8	notion	notion	NOUN
ejpam-6569	14	9	of	of	ADP
ejpam-6569	14	10	almost	almost	ADV
ejpam-6569	14	11	quasi	quasi	ADJ
ejpam-6569	14	12	continuous	continuous	ADJ
ejpam-6569	14	13	functions	function	NOUN
ejpam-6569	14	14	.	.	PUNCT
ejpam-6569	15	1	neubrunnovaá	neubrunnovaá	PUNCT
ejpam-6569	16	1	[	[	X
ejpam-6569	16	2	7	7	NUM
ejpam-6569	16	3	]	]	PUNCT
ejpam-6569	16	4	showed	show	VERB
ejpam-6569	16	5	that	that	SCONJ
ejpam-6569	16	6	quasi	quasi	NOUN
ejpam-6569	16	7	continuity	continuity	NOUN
ejpam-6569	16	8	is	be	AUX
ejpam-6569	16	9	equivalent	equivalent	ADJ
ejpam-6569	16	10	to	to	ADP
ejpam-6569	16	11	semi	semi	ADJ
ejpam-6569	16	12	-	-	NOUN
ejpam-6569	16	13	continuity	continuity	NOUN
ejpam-6569	16	14	due	due	ADP
ejpam-6569	16	15	to	to	ADP
ejpam-6569	16	16	levine	levine	PROPN
ejpam-6569	16	17	[	[	X
ejpam-6569	16	18	1	1	NUM
ejpam-6569	16	19	]	]	PUNCT
ejpam-6569	16	20	.	.	PUNCT
ejpam-6569	17	1	popa	popa	NOUN
ejpam-6569	17	2	and	and	CCONJ
ejpam-6569	17	3	stan	stan	PROPN
ejpam-6569	18	1	[	[	X
ejpam-6569	18	2	8	8	NUM
ejpam-6569	18	3	]	]	PUNCT
ejpam-6569	18	4	introduced	introduce	VERB
ejpam-6569	18	5	and	and	CCONJ
ejpam-6569	18	6	investigated	investigate	VERB
ejpam-6569	18	7	the	the	DET
ejpam-6569	18	8	notion	notion	NOUN
ejpam-6569	18	9	of	of	ADP
ejpam-6569	18	10	weakly	weakly	ADJ
ejpam-6569	18	11	quasi	quasi	ADJ
ejpam-6569	18	12	continuous	continuous	ADJ
ejpam-6569	18	13	functions	function	NOUN
ejpam-6569	18	14	.	.	PUNCT
ejpam-6569	19	1	weak	weak	ADJ
ejpam-6569	19	2	quasi	quasi	NOUN
ejpam-6569	19	3	continuity	continuity	NOUN
ejpam-6569	19	4	is	be	AUX
ejpam-6569	19	5	implied	imply	VERB
ejpam-6569	19	6	by	by	ADP
ejpam-6569	19	7	quasi	quasi	NOUN
ejpam-6569	19	8	continuity	continuity	NOUN
ejpam-6569	19	9	and	and	CCONJ
ejpam-6569	19	10	weak	weak	ADJ
ejpam-6569	19	11	continuity	continuity	NOUN
ejpam-6569	19	12	[	[	X
ejpam-6569	19	13	9	9	NUM
ejpam-6569	19	14	]	]	PUNCT
ejpam-6569	19	15	which	which	PRON
ejpam-6569	19	16	are	be	AUX
ejpam-6569	19	17	independent	independent	ADJ
ejpam-6569	19	18	of	of	ADP
ejpam-6569	19	19	each	each	DET
ejpam-6569	19	20	other	other	ADJ
ejpam-6569	19	21	.	.	PUNCT
ejpam-6569	20	1	popa	popa	NOUN
ejpam-6569	21	1	[	[	X
ejpam-6569	21	2	10	10	NUM
ejpam-6569	21	3	]	]	PUNCT
ejpam-6569	21	4	extended	extend	VERB
ejpam-6569	21	5	the	the	DET
ejpam-6569	21	6	concept	concept	NOUN
ejpam-6569	21	7	of	of	ADP
ejpam-6569	21	8	quasicontinuous	quasicontinuous	ADJ
ejpam-6569	21	9	functions	function	NOUN
ejpam-6569	21	10	to	to	ADP
ejpam-6569	21	11	the	the	DET
ejpam-6569	21	12	setting	setting	NOUN
ejpam-6569	21	13	of	of	ADP
ejpam-6569	21	14	multifunctions	multifunction	NOUN
ejpam-6569	21	15	.	.	PUNCT
ejpam-6569	22	1	popa	popa	NOUN
ejpam-6569	22	2	and	and	CCONJ
ejpam-6569	22	3	noiri	noiri	ADV
ejpam-6569	23	1	[	[	X
ejpam-6569	23	2	11	11	NUM
ejpam-6569	23	3	]	]	PUNCT
ejpam-6569	23	4	introduced	introduce	VERB
ejpam-6569	23	5	the	the	DET
ejpam-6569	23	6	concept	concept	NOUN
ejpam-6569	23	7	of	of	ADP
ejpam-6569	23	8	almost	almost	ADV
ejpam-6569	23	9	quasi	quasi	ADJ
ejpam-6569	23	10	continuous	continuous	ADJ
ejpam-6569	23	11	multifunctions	multifunction	NOUN
ejpam-6569	23	12	and	and	CCONJ
ejpam-6569	23	13	investigated	investigate	VERB
ejpam-6569	23	14	some	some	DET
ejpam-6569	23	15	characterizations	characterization	NOUN
ejpam-6569	23	16	of	of	ADP
ejpam-6569	23	17	such	such	ADJ
ejpam-6569	23	18	multifunctions	multifunction	NOUN
ejpam-6569	23	19	.	.	PUNCT
ejpam-6569	24	1	noiri	noiri	PROPN
ejpam-6569	24	2	and	and	CCONJ
ejpam-6569	24	3	popa	popa	NOUN
ejpam-6569	24	4	[	[	X
ejpam-6569	24	5	12	12	NUM
ejpam-6569	24	6	]	]	PUNCT
ejpam-6569	24	7	introduced	introduce	VERB
ejpam-6569	24	8	and	and	CCONJ
ejpam-6569	24	9	studied	study	VERB
ejpam-6569	24	10	the	the	DET
ejpam-6569	24	11	notion	notion	NOUN
ejpam-6569	24	12	of	of	ADP
ejpam-6569	24	13	weakly	weakly	ADJ
ejpam-6569	24	14	quasi	quasi	ADJ
ejpam-6569	24	15	continuous	continuous	ADJ
ejpam-6569	24	16	multifunctions	multifunction	NOUN
ejpam-6569	24	17	.	.	PUNCT
ejpam-6569	25	1	popa	popa	NOUN
ejpam-6569	25	2	and	and	CCONJ
ejpam-6569	25	3	noiri	noiri	ADV
ejpam-6569	26	1	[	[	X
ejpam-6569	26	2	13	13	NUM
ejpam-6569	26	3	]	]	PUNCT
ejpam-6569	26	4	introduced	introduce	VERB
ejpam-6569	26	5	the	the	DET
ejpam-6569	26	6	notion	notion	NOUN
ejpam-6569	26	7	of	of	ADP
ejpam-6569	26	8	θ	θ	ADJ
ejpam-6569	26	9	-	-	ADJ
ejpam-6569	26	10	quasicontinuous	quasicontinuous	ADJ
ejpam-6569	26	11	multifunctions	multifunction	NOUN
ejpam-6569	26	12	and	and	CCONJ
ejpam-6569	26	13	investigated	investigate	VERB
ejpam-6569	26	14	several	several	ADJ
ejpam-6569	26	15	further	further	ADJ
ejpam-6569	26	16	properties	property	NOUN
ejpam-6569	26	17	of	of	ADP
ejpam-6569	26	18	such	such	ADJ
ejpam-6569	26	19	multifunctions	multifunction	NOUN
ejpam-6569	26	20	.	.	PUNCT
ejpam-6569	27	1	moreover	moreover	ADV
ejpam-6569	27	2	,	,	PUNCT
ejpam-6569	27	3	some	some	DET
ejpam-6569	27	4	characterizations	characterization	NOUN
ejpam-6569	27	5	of	of	ADP
ejpam-6569	27	6	upper	upper	ADJ
ejpam-6569	27	7	and	and	CCONJ
ejpam-6569	27	8	∗corresponding	∗corresponde	VERB
ejpam-6569	27	9	author	author	NOUN
ejpam-6569	27	10	.	.	PUNCT
ejpam-6569	28	1	doi	doi	NOUN
ejpam-6569	28	2	:	:	PUNCT
ejpam-6569	28	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6569	https://doi.org/10.29020/nybg.ejpam.v18i3.6569	PROPN
ejpam-6569	28	4	email	email	NOUN
ejpam-6569	28	5	addresses	address	VERB
ejpam-6569	28	6	:	:	PUNCT
ejpam-6569	28	7	napassanan.sri@msu.ac.th	napassanan.sri@msu.ac.th	PRON
ejpam-6569	28	8	(	(	PUNCT
ejpam-6569	28	9	n.	n.	NOUN
ejpam-6569	28	10	srisarakham	srisarakham	PROPN
ejpam-6569	28	11	)	)	PUNCT
ejpam-6569	28	12	,	,	PUNCT
ejpam-6569	28	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6569	28	14	(	(	PUNCT
ejpam-6569	28	15	a.	a.	PROPN
ejpam-6569	28	16	sama	sama	PROPN
ejpam-6569	28	17	-	-	PUNCT
ejpam-6569	28	18	ae	ae	PROPN
ejpam-6569	28	19	)	)	PUNCT
ejpam-6569	28	20	,	,	PUNCT
ejpam-6569	28	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6569	28	22	(	(	PUNCT
ejpam-6569	28	23	c.	c.	PROPN
ejpam-6569	28	24	boonpok	boonpok	PROPN
ejpam-6569	28	25	)	)	PUNCT
ejpam-6569	28	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6569	29	1	1	1	NUM
ejpam-6569	29	2	copyright	copyright	NOUN
ejpam-6569	29	3	:	:	PUNCT
ejpam-6569	29	4	©	©	PROPN
ejpam-6569	29	5	2025	2025	NUM
ejpam-6569	29	6	the	the	DET
ejpam-6569	29	7	author(s	author(s	NOUN
ejpam-6569	29	8	)	)	PUNCT
ejpam-6569	29	9	.	.	PUNCT
ejpam-6569	30	1	(	(	PUNCT
ejpam-6569	30	2	cc	cc	NOUN
ejpam-6569	30	3	by	by	ADP
ejpam-6569	30	4	-	-	PUNCT
ejpam-6569	30	5	nc	nc	PROPN
ejpam-6569	30	6	4.0	4.0	NUM
ejpam-6569	30	7	)	)	PUNCT
ejpam-6569	30	8	n.	n.	NOUN
ejpam-6569	30	9	srisarakham	srisarakham	PROPN
ejpam-6569	30	10	,	,	PUNCT
ejpam-6569	30	11	a.	a.	PROPN
ejpam-6569	30	12	sama	sama	PROPN
ejpam-6569	30	13	-	-	PUNCT
ejpam-6569	30	14	ae	ae	PROPN
ejpam-6569	30	15	,	,	PUNCT
ejpam-6569	30	16	c.	c.	PROPN
ejpam-6569	30	17	boonpok	boonpok	PROPN
ejpam-6569	30	18	/	/	SYM
ejpam-6569	30	19	eur	eur	PROPN
ejpam-6569	30	20	.	.	PUNCT
ejpam-6569	31	1	j.	j.	PROPN
ejpam-6569	31	2	pure	pure	PROPN
ejpam-6569	31	3	appl	appl	PROPN
ejpam-6569	31	4	.	.	PROPN
ejpam-6569	31	5	math	math	PROPN
ejpam-6569	31	6	,	,	PUNCT
ejpam-6569	31	7	18	18	NUM
ejpam-6569	31	8	(	(	PUNCT
ejpam-6569	31	9	3	3	NUM
ejpam-6569	31	10	)	)	PUNCT
ejpam-6569	31	11	(	(	PUNCT
ejpam-6569	31	12	2025	2025	NUM
ejpam-6569	31	13	)	)	PUNCT
ejpam-6569	31	14	,	,	PUNCT
ejpam-6569	31	15	6569	6569	NUM
ejpam-6569	31	16	2	2	NUM
ejpam-6569	31	17	of	of	ADP
ejpam-6569	31	18	13	13	NUM
ejpam-6569	31	19	lower	low	ADJ
ejpam-6569	31	20	θ	θ	ADJ
ejpam-6569	31	21	-	-	ADJ
ejpam-6569	31	22	quasicontinuous	quasicontinuous	ADJ
ejpam-6569	31	23	multifunctions	multifunction	NOUN
ejpam-6569	31	24	were	be	AUX
ejpam-6569	31	25	presented	present	VERB
ejpam-6569	31	26	in	in	ADP
ejpam-6569	31	27	[	[	X
ejpam-6569	31	28	14	14	NUM
ejpam-6569	31	29	]	]	PUNCT
ejpam-6569	31	30	.	.	PUNCT
ejpam-6569	32	1	semi	semi	ADJ
ejpam-6569	32	2	-	-	ADJ
ejpam-6569	32	3	i	i	ADJ
ejpam-6569	32	4	-open	-open	NOUN
ejpam-6569	32	5	sets	set	NOUN
ejpam-6569	32	6	,	,	PUNCT
ejpam-6569	32	7	prei	prei	NOUN
ejpam-6569	32	8	-open	-open	NOUN
ejpam-6569	32	9	sets	set	NOUN
ejpam-6569	32	10	,	,	PUNCT
ejpam-6569	32	11	α	α	X
ejpam-6569	32	12	-	-	PUNCT
ejpam-6569	32	13	i	i	PRON
ejpam-6569	32	14	-open	-open	NOUN
ejpam-6569	32	15	sets	set	NOUN
ejpam-6569	32	16	,	,	PUNCT
ejpam-6569	32	17	β	β	X
ejpam-6569	32	18	-	-	ADJ
ejpam-6569	32	19	i	i	PRON
ejpam-6569	32	20	-open	-open	NOUN
ejpam-6569	32	21	sets	set	NOUN
ejpam-6569	32	22	and	and	CCONJ
ejpam-6569	32	23	δ	δ	PROPN
ejpam-6569	32	24	-	-	PUNCT
ejpam-6569	32	25	i	i	PRON
ejpam-6569	32	26	-open	-open	NOUN
ejpam-6569	32	27	sets	set	NOUN
ejpam-6569	32	28	play	play	VERB
ejpam-6569	32	29	an	an	DET
ejpam-6569	32	30	important	important	ADJ
ejpam-6569	32	31	role	role	NOUN
ejpam-6569	32	32	in	in	ADP
ejpam-6569	32	33	the	the	DET
ejpam-6569	32	34	research	research	NOUN
ejpam-6569	32	35	of	of	ADP
ejpam-6569	32	36	generalizations	generalization	NOUN
ejpam-6569	32	37	of	of	ADP
ejpam-6569	32	38	continuity	continuity	NOUN
ejpam-6569	32	39	in	in	ADP
ejpam-6569	32	40	ideal	ideal	ADJ
ejpam-6569	32	41	topological	topological	ADJ
ejpam-6569	32	42	spaces	space	NOUN
ejpam-6569	32	43	.	.	PUNCT
ejpam-6569	33	1	hatir	hatir	PROPN
ejpam-6569	33	2	and	and	CCONJ
ejpam-6569	33	3	noiri	noiri	ADV
ejpam-6569	34	1	[	[	X
ejpam-6569	34	2	15	15	NUM
ejpam-6569	34	3	]	]	PUNCT
ejpam-6569	34	4	introduced	introduce	VERB
ejpam-6569	34	5	and	and	CCONJ
ejpam-6569	34	6	investigated	investigate	VERB
ejpam-6569	34	7	the	the	DET
ejpam-6569	34	8	notions	notion	NOUN
ejpam-6569	34	9	of	of	ADP
ejpam-6569	34	10	weakly	weakly	ADJ
ejpam-6569	34	11	pre	pre	ADJ
ejpam-6569	34	12	-	-	ADJ
ejpam-6569	34	13	i	i	PRON
ejpam-6569	34	14	-open	-open	NOUN
ejpam-6569	34	15	sets	set	NOUN
ejpam-6569	34	16	and	and	CCONJ
ejpam-6569	34	17	weakly	weakly	ADJ
ejpam-6569	34	18	pre	pre	ADJ
ejpam-6569	34	19	-	-	ADJ
ejpam-6569	34	20	i	i	ADJ
ejpam-6569	34	21	-continuous	-continuous	ADJ
ejpam-6569	34	22	functions	function	NOUN
ejpam-6569	34	23	.	.	PUNCT
ejpam-6569	35	1	furthermore	furthermore	ADV
ejpam-6569	35	2	,	,	PUNCT
ejpam-6569	35	3	hatir	hatir	PROPN
ejpam-6569	35	4	and	and	CCONJ
ejpam-6569	35	5	noiri	noiri	ADV
ejpam-6569	36	1	[	[	X
ejpam-6569	36	2	16	16	NUM
ejpam-6569	36	3	]	]	PUNCT
ejpam-6569	36	4	investigated	investigate	VERB
ejpam-6569	36	5	further	further	ADJ
ejpam-6569	36	6	properties	property	NOUN
ejpam-6569	36	7	of	of	ADP
ejpam-6569	36	8	semi	semi	ADJ
ejpam-6569	36	9	-	-	ADJ
ejpam-6569	36	10	i	i	PRON
ejpam-6569	36	11	-open	-open	NOUN
ejpam-6569	36	12	sets	set	NOUN
ejpam-6569	36	13	and	and	CCONJ
ejpam-6569	36	14	semi	semi	ADJ
ejpam-6569	36	15	-	-	ADJ
ejpam-6569	36	16	i	i	ADV
ejpam-6569	36	17	-continuous	-continuous	ADJ
ejpam-6569	36	18	functions	function	NOUN
ejpam-6569	36	19	.	.	PUNCT
ejpam-6569	37	1	in	in	ADP
ejpam-6569	37	2	2019	2019	NUM
ejpam-6569	37	3	,	,	PUNCT
ejpam-6569	37	4	the	the	DET
ejpam-6569	37	5	present	present	ADJ
ejpam-6569	37	6	author	author	NOUN
ejpam-6569	37	7	[	[	X
ejpam-6569	37	8	17	17	NUM
ejpam-6569	37	9	]	]	PUNCT
ejpam-6569	37	10	introduced	introduce	VERB
ejpam-6569	37	11	new	new	ADJ
ejpam-6569	37	12	classes	class	NOUN
ejpam-6569	37	13	of	of	ADP
ejpam-6569	37	14	multifunctions	multifunction	NOUN
ejpam-6569	37	15	between	between	ADP
ejpam-6569	37	16	ideal	ideal	ADJ
ejpam-6569	37	17	topological	topological	ADJ
ejpam-6569	37	18	spaces	space	NOUN
ejpam-6569	37	19	,	,	PUNCT
ejpam-6569	37	20	namely	namely	ADV
ejpam-6569	37	21	upper	upper	ADJ
ejpam-6569	37	22	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	37	23	multifunctions	multifunction	NOUN
ejpam-6569	37	24	and	and	CCONJ
ejpam-6569	37	25	lower	low	ADJ
ejpam-6569	37	26	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	37	27	multifunctions	multifunction	NOUN
ejpam-6569	37	28	.	.	PUNCT
ejpam-6569	38	1	in	in	ADP
ejpam-6569	38	2	particular	particular	ADJ
ejpam-6569	38	3	,	,	PUNCT
ejpam-6569	38	4	several	several	ADJ
ejpam-6569	38	5	characterizations	characterization	NOUN
ejpam-6569	38	6	of	of	ADP
ejpam-6569	38	7	upper	upper	ADJ
ejpam-6569	38	8	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	38	9	multifunctions	multifunction	NOUN
ejpam-6569	38	10	,	,	PUNCT
ejpam-6569	38	11	lower	low	ADJ
ejpam-6569	38	12	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	38	13	multifunctions	multifunction	NOUN
ejpam-6569	38	14	,	,	PUNCT
ejpam-6569	38	15	upper	upper	ADJ
ejpam-6569	38	16	almost	almost	ADV
ejpam-6569	38	17	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	38	18	multifunctions	multifunction	NOUN
ejpam-6569	38	19	,	,	PUNCT
ejpam-6569	38	20	lower	low	ADJ
ejpam-6569	38	21	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	38	22	multifunctions	multifunction	NOUN
ejpam-6569	38	23	,	,	PUNCT
ejpam-6569	38	24	upper	upper	ADJ
ejpam-6569	38	25	weakly	weakly	ADJ
ejpam-6569	38	26	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	38	27	multifunctions	multifunction	NOUN
ejpam-6569	38	28	and	and	CCONJ
ejpam-6569	38	29	lower	low	ADJ
ejpam-6569	38	30	weakly	weakly	ADJ
ejpam-6569	38	31	⋆-continuous	⋆-continuous	ADJ
ejpam-6569	38	32	multifunctions	multifunction	NOUN
ejpam-6569	38	33	were	be	AUX
ejpam-6569	38	34	considered	consider	VERB
ejpam-6569	38	35	in	in	ADP
ejpam-6569	38	36	[	[	X
ejpam-6569	38	37	17	17	NUM
ejpam-6569	38	38	]	]	PUNCT
ejpam-6569	38	39	.	.	PUNCT
ejpam-6569	39	1	on	on	ADP
ejpam-6569	39	2	the	the	DET
ejpam-6569	39	3	other	other	ADJ
ejpam-6569	39	4	hand	hand	NOUN
ejpam-6569	39	5	,	,	PUNCT
ejpam-6569	39	6	the	the	DET
ejpam-6569	39	7	present	present	ADJ
ejpam-6569	39	8	author	author	NOUN
ejpam-6569	39	9	introduced	introduce	VERB
ejpam-6569	39	10	and	and	CCONJ
ejpam-6569	39	11	investigated	investigate	VERB
ejpam-6569	39	12	the	the	DET
ejpam-6569	39	13	notions	notion	NOUN
ejpam-6569	39	14	of	of	ADP
ejpam-6569	39	15	pı	pı	ADJ
ejpam-6569	39	16	-	-	ADJ
ejpam-6569	39	17	continuous	continuous	ADJ
ejpam-6569	39	18	multifunctions	multifunction	NOUN
ejpam-6569	40	1	[	[	X
ejpam-6569	40	2	18	18	NUM
ejpam-6569	40	3	]	]	PUNCT
ejpam-6569	40	4	and	and	CCONJ
ejpam-6569	40	5	weakly	weakly	ADJ
ejpam-6569	40	6	pı	pı	ADJ
ejpam-6569	40	7	-	-	ADJ
ejpam-6569	40	8	continuous	continuous	ADJ
ejpam-6569	40	9	multifunctions	multifunction	NOUN
ejpam-6569	41	1	[	[	X
ejpam-6569	41	2	18	18	NUM
ejpam-6569	41	3	]	]	PUNCT
ejpam-6569	41	4	.	.	PUNCT
ejpam-6569	42	1	pue	pue	NOUN
ejpam-6569	42	2	-	-	PUNCT
ejpam-6569	42	3	on	on	NOUN
ejpam-6569	42	4	et	et	PROPN
ejpam-6569	42	5	al	al	PROPN
ejpam-6569	42	6	.	.	PUNCT
ejpam-6569	43	1	[	[	X
ejpam-6569	43	2	19	19	NUM
ejpam-6569	43	3	]	]	PUNCT
ejpam-6569	43	4	introduced	introduce	VERB
ejpam-6569	43	5	and	and	CCONJ
ejpam-6569	43	6	studied	study	VERB
ejpam-6569	43	7	the	the	DET
ejpam-6569	43	8	concepts	concept	NOUN
ejpam-6569	43	9	of	of	ADP
ejpam-6569	43	10	upper	upper	ADJ
ejpam-6569	43	11	(	(	PUNCT
ejpam-6569	43	12	τ1	τ1	NOUN
ejpam-6569	43	13	,	,	PUNCT
ejpam-6569	43	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	43	15	multifunctions	multifunction	NOUN
ejpam-6569	43	16	and	and	CCONJ
ejpam-6569	43	17	lower	low	ADJ
ejpam-6569	43	18	(	(	PUNCT
ejpam-6569	43	19	τ1	τ1	NOUN
ejpam-6569	43	20	,	,	PUNCT
ejpam-6569	43	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	43	22	multifunctions	multifunction	NOUN
ejpam-6569	43	23	.	.	PUNCT
ejpam-6569	44	1	klanarong	klanarong	NOUN
ejpam-6569	44	2	et	et	PROPN
ejpam-6569	44	3	al	al	PROPN
ejpam-6569	44	4	.	.	PUNCT
ejpam-6569	45	1	[	[	X
ejpam-6569	45	2	20	20	NUM
ejpam-6569	45	3	]	]	PUNCT
ejpam-6569	45	4	introduced	introduce	VERB
ejpam-6569	45	5	and	and	CCONJ
ejpam-6569	45	6	investigated	investigate	VERB
ejpam-6569	45	7	the	the	DET
ejpam-6569	45	8	notions	notion	NOUN
ejpam-6569	45	9	of	of	ADP
ejpam-6569	45	10	upper	upper	ADJ
ejpam-6569	45	11	almost	almost	ADV
ejpam-6569	45	12	(	(	PUNCT
ejpam-6569	45	13	τ1	τ1	NOUN
ejpam-6569	45	14	,	,	PUNCT
ejpam-6569	45	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	45	16	multifunctions	multifunction	NOUN
ejpam-6569	45	17	and	and	CCONJ
ejpam-6569	45	18	lower	low	ADJ
ejpam-6569	45	19	almost	almost	ADV
ejpam-6569	45	20	(	(	PUNCT
ejpam-6569	45	21	τ1	τ1	NOUN
ejpam-6569	45	22	,	,	PUNCT
ejpam-6569	45	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	45	24	multifunctions	multifunction	NOUN
ejpam-6569	45	25	.	.	PUNCT
ejpam-6569	46	1	thongmoon	thongmoon	NOUN
ejpam-6569	46	2	et	et	PROPN
ejpam-6569	46	3	al	al	PROPN
ejpam-6569	46	4	.	.	PUNCT
ejpam-6569	47	1	[	[	X
ejpam-6569	47	2	21	21	NUM
ejpam-6569	47	3	]	]	PUNCT
ejpam-6569	47	4	introduced	introduce	VERB
ejpam-6569	47	5	and	and	CCONJ
ejpam-6569	47	6	studied	study	VERB
ejpam-6569	47	7	the	the	DET
ejpam-6569	47	8	concepts	concept	NOUN
ejpam-6569	47	9	of	of	ADP
ejpam-6569	47	10	upper	upper	ADJ
ejpam-6569	47	11	weakly	weakly	ADJ
ejpam-6569	47	12	(	(	PUNCT
ejpam-6569	47	13	τ1	τ1	NOUN
ejpam-6569	47	14	,	,	PUNCT
ejpam-6569	47	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	47	16	multifunctions	multifunction	NOUN
ejpam-6569	47	17	and	and	CCONJ
ejpam-6569	47	18	lower	low	ADJ
ejpam-6569	47	19	weakly	weakly	ADJ
ejpam-6569	47	20	(	(	PUNCT
ejpam-6569	47	21	τ1	τ1	NOUN
ejpam-6569	47	22	,	,	PUNCT
ejpam-6569	47	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	47	24	multifunctions	multifunction	NOUN
ejpam-6569	47	25	.	.	PUNCT
ejpam-6569	48	1	in	in	ADP
ejpam-6569	48	2	this	this	DET
ejpam-6569	48	3	paper	paper	NOUN
ejpam-6569	48	4	,	,	PUNCT
ejpam-6569	48	5	we	we	PRON
ejpam-6569	48	6	introduce	introduce	VERB
ejpam-6569	48	7	the	the	DET
ejpam-6569	48	8	notions	notion	NOUN
ejpam-6569	48	9	of	of	ADP
ejpam-6569	48	10	upper	upper	ADJ
ejpam-6569	48	11	quasi	quasi	NOUN
ejpam-6569	48	12	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	48	13	,	,	PUNCT
ejpam-6569	48	14	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	48	15	multifunctions	multifunction	NOUN
ejpam-6569	48	16	and	and	CCONJ
ejpam-6569	48	17	lower	low	ADJ
ejpam-6569	48	18	quasi	quasi	NOUN
ejpam-6569	48	19	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	48	20	,	,	PUNCT
ejpam-6569	48	21	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	48	22	multifunctions	multifunction	NOUN
ejpam-6569	48	23	.	.	PUNCT
ejpam-6569	49	1	we	we	PRON
ejpam-6569	49	2	also	also	ADV
ejpam-6569	49	3	investigate	investigate	VERB
ejpam-6569	49	4	several	several	ADJ
ejpam-6569	49	5	characterizations	characterization	NOUN
ejpam-6569	49	6	of	of	ADP
ejpam-6569	49	7	upper	upper	ADJ
ejpam-6569	49	8	quasi	quasi	NOUN
ejpam-6569	49	9	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	49	10	,	,	PUNCT
ejpam-6569	49	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	49	12	multifunctions	multifunction	NOUN
ejpam-6569	49	13	and	and	CCONJ
ejpam-6569	49	14	lower	low	ADJ
ejpam-6569	49	15	quasi	quasi	NOUN
ejpam-6569	49	16	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	49	17	,	,	PUNCT
ejpam-6569	49	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	49	19	multifunctions	multifunction	NOUN
ejpam-6569	49	20	.	.	PUNCT
ejpam-6569	50	1	2	2	X
ejpam-6569	50	2	.	.	X
ejpam-6569	50	3	preliminaries	preliminary	NOUN
ejpam-6569	50	4	throughout	throughout	ADP
ejpam-6569	50	5	the	the	DET
ejpam-6569	50	6	present	present	ADJ
ejpam-6569	50	7	paper	paper	NOUN
ejpam-6569	50	8	,	,	PUNCT
ejpam-6569	50	9	spaces	space	NOUN
ejpam-6569	50	10	(	(	PUNCT
ejpam-6569	50	11	x	x	NOUN
ejpam-6569	50	12	,	,	PUNCT
ejpam-6569	50	13	τ1	τ1	NOUN
ejpam-6569	50	14	,	,	PUNCT
ejpam-6569	50	15	τ2	τ2	NOUN
ejpam-6569	50	16	)	)	PUNCT
ejpam-6569	50	17	and	and	CCONJ
ejpam-6569	50	18	(	(	PUNCT
ejpam-6569	50	19	y	y	PROPN
ejpam-6569	50	20	,	,	PUNCT
ejpam-6569	50	21	σ1	σ1	PROPN
ejpam-6569	50	22	,	,	PUNCT
ejpam-6569	50	23	σ2	σ2	NOUN
ejpam-6569	50	24	)	)	PUNCT
ejpam-6569	50	25	(	(	PUNCT
ejpam-6569	50	26	or	or	CCONJ
ejpam-6569	50	27	simply	simply	ADV
ejpam-6569	50	28	x	x	X
ejpam-6569	50	29	and	and	CCONJ
ejpam-6569	50	30	y	y	PROPN
ejpam-6569	50	31	)	)	PUNCT
ejpam-6569	50	32	always	always	ADV
ejpam-6569	50	33	mean	mean	VERB
ejpam-6569	50	34	bitopological	bitopological	ADJ
ejpam-6569	50	35	spaces	space	NOUN
ejpam-6569	50	36	on	on	ADP
ejpam-6569	50	37	which	which	PRON
ejpam-6569	50	38	no	no	DET
ejpam-6569	50	39	separation	separation	NOUN
ejpam-6569	50	40	axioms	axiom	NOUN
ejpam-6569	50	41	are	be	AUX
ejpam-6569	50	42	assumed	assume	VERB
ejpam-6569	50	43	unless	unless	SCONJ
ejpam-6569	50	44	explicitly	explicitly	ADV
ejpam-6569	50	45	stated	state	VERB
ejpam-6569	50	46	.	.	PUNCT
ejpam-6569	51	1	let	let	VERB
ejpam-6569	51	2	a	a	DET
ejpam-6569	51	3	be	be	AUX
ejpam-6569	51	4	a	a	DET
ejpam-6569	51	5	subset	subset	NOUN
ejpam-6569	51	6	of	of	ADP
ejpam-6569	51	7	a	a	DET
ejpam-6569	51	8	bitopological	bitopological	ADJ
ejpam-6569	51	9	space	space	NOUN
ejpam-6569	51	10	(	(	PUNCT
ejpam-6569	51	11	x	x	NOUN
ejpam-6569	51	12	,	,	PUNCT
ejpam-6569	51	13	τ1	τ1	NOUN
ejpam-6569	51	14	,	,	PUNCT
ejpam-6569	51	15	τ2	τ2	NOUN
ejpam-6569	51	16	)	)	PUNCT
ejpam-6569	51	17	.	.	PUNCT
ejpam-6569	52	1	the	the	DET
ejpam-6569	52	2	closure	closure	NOUN
ejpam-6569	52	3	of	of	ADP
ejpam-6569	52	4	a	a	PRON
ejpam-6569	52	5	and	and	CCONJ
ejpam-6569	52	6	the	the	DET
ejpam-6569	52	7	interior	interior	NOUN
ejpam-6569	52	8	of	of	ADP
ejpam-6569	52	9	a	a	PRON
ejpam-6569	52	10	with	with	ADP
ejpam-6569	52	11	respect	respect	NOUN
ejpam-6569	52	12	to	to	ADP
ejpam-6569	52	13	τi	τi	PROPN
ejpam-6569	52	14	are	be	AUX
ejpam-6569	52	15	denoted	denote	VERB
ejpam-6569	52	16	by	by	ADP
ejpam-6569	52	17	τi	τi	NOUN
ejpam-6569	52	18	-	-	PUNCT
ejpam-6569	52	19	cl(a	cl(a	NUM
ejpam-6569	52	20	)	)	PUNCT
ejpam-6569	52	21	and	and	CCONJ
ejpam-6569	52	22	τi	τi	NOUN
ejpam-6569	52	23	-	-	PUNCT
ejpam-6569	52	24	int(a	int(a	NOUN
ejpam-6569	52	25	)	)	PUNCT
ejpam-6569	52	26	,	,	PUNCT
ejpam-6569	52	27	respectively	respectively	ADV
ejpam-6569	52	28	,	,	PUNCT
ejpam-6569	52	29	for	for	ADP
ejpam-6569	52	30	i	i	PROPN
ejpam-6569	52	31	=	=	SYM
ejpam-6569	52	32	1	1	NUM
ejpam-6569	52	33	,	,	PUNCT
ejpam-6569	52	34	2	2	NUM
ejpam-6569	52	35	.	.	X
ejpam-6569	52	36	a	a	DET
ejpam-6569	52	37	subset	subset	NOUN
ejpam-6569	52	38	a	a	PRON
ejpam-6569	52	39	of	of	ADP
ejpam-6569	52	40	a	a	DET
ejpam-6569	52	41	bitopological	bitopological	ADJ
ejpam-6569	52	42	space	space	NOUN
ejpam-6569	52	43	(	(	PUNCT
ejpam-6569	52	44	x	x	NOUN
ejpam-6569	52	45	,	,	PUNCT
ejpam-6569	52	46	τ1	τ1	NOUN
ejpam-6569	52	47	,	,	PUNCT
ejpam-6569	52	48	τ2	τ2	NOUN
ejpam-6569	52	49	)	)	PUNCT
ejpam-6569	52	50	is	be	AUX
ejpam-6569	52	51	called	call	VERB
ejpam-6569	52	52	τ1τ2	τ1τ2	VERB
ejpam-6569	52	53	-	-	ADJ
ejpam-6569	52	54	closed	closed	ADJ
ejpam-6569	52	55	[	[	X
ejpam-6569	52	56	22	22	NUM
ejpam-6569	52	57	]	]	PUNCT
ejpam-6569	52	58	if	if	SCONJ
ejpam-6569	52	59	a	a	DET
ejpam-6569	52	60	=	=	NOUN
ejpam-6569	52	61	τ1	τ1	NOUN
ejpam-6569	52	62	-	-	PUNCT
ejpam-6569	52	63	cl(τ2	cl(τ2	NOUN
ejpam-6569	52	64	-	-	PUNCT
ejpam-6569	52	65	cl(a	cl(a	NUM
ejpam-6569	52	66	)	)	PUNCT
ejpam-6569	52	67	)	)	PUNCT
ejpam-6569	52	68	.	.	PUNCT
ejpam-6569	53	1	the	the	DET
ejpam-6569	53	2	complement	complement	NOUN
ejpam-6569	53	3	of	of	ADP
ejpam-6569	53	4	a	a	DET
ejpam-6569	53	5	τ1τ2	τ1τ2	ADJ
ejpam-6569	53	6	-	-	ADJ
ejpam-6569	53	7	closed	closed	ADJ
ejpam-6569	53	8	set	set	NOUN
ejpam-6569	53	9	is	be	AUX
ejpam-6569	53	10	called	call	VERB
ejpam-6569	53	11	τ1τ2	τ1τ2	NOUN
ejpam-6569	53	12	-	-	ADJ
ejpam-6569	53	13	open	open	ADJ
ejpam-6569	53	14	.	.	PUNCT
ejpam-6569	54	1	let	let	VERB
ejpam-6569	54	2	a	a	DET
ejpam-6569	54	3	be	be	AUX
ejpam-6569	54	4	a	a	DET
ejpam-6569	54	5	subset	subset	NOUN
ejpam-6569	54	6	of	of	ADP
ejpam-6569	54	7	a	a	DET
ejpam-6569	54	8	bitopological	bitopological	ADJ
ejpam-6569	54	9	space	space	NOUN
ejpam-6569	54	10	(	(	PUNCT
ejpam-6569	54	11	x	x	NOUN
ejpam-6569	54	12	,	,	PUNCT
ejpam-6569	54	13	τ1	τ1	NOUN
ejpam-6569	54	14	,	,	PUNCT
ejpam-6569	54	15	τ2	τ2	NOUN
ejpam-6569	54	16	)	)	PUNCT
ejpam-6569	54	17	.	.	PUNCT
ejpam-6569	55	1	the	the	DET
ejpam-6569	55	2	intersection	intersection	NOUN
ejpam-6569	55	3	of	of	ADP
ejpam-6569	55	4	all	all	DET
ejpam-6569	55	5	τ1τ2	τ1τ2	ADJ
ejpam-6569	55	6	-	-	ADJ
ejpam-6569	55	7	closed	closed	ADJ
ejpam-6569	55	8	sets	set	NOUN
ejpam-6569	55	9	of	of	ADP
ejpam-6569	55	10	x	x	PUNCT
ejpam-6569	55	11	containing	contain	VERB
ejpam-6569	55	12	a	a	PRON
ejpam-6569	55	13	is	be	AUX
ejpam-6569	55	14	called	call	VERB
ejpam-6569	55	15	the	the	DET
ejpam-6569	55	16	τ1τ2	τ1τ2	NOUN
ejpam-6569	55	17	-	-	NOUN
ejpam-6569	55	18	closure	closure	NOUN
ejpam-6569	55	19	[	[	X
ejpam-6569	55	20	22	22	NUM
ejpam-6569	55	21	]	]	PUNCT
ejpam-6569	55	22	of	of	ADP
ejpam-6569	55	23	a	a	PRON
ejpam-6569	55	24	and	and	CCONJ
ejpam-6569	55	25	is	be	AUX
ejpam-6569	55	26	denoted	denote	VERB
ejpam-6569	55	27	by	by	ADP
ejpam-6569	55	28	τ1τ2	τ1τ2	NOUN
ejpam-6569	55	29	-	-	NUM
ejpam-6569	55	30	cl(a	cl(a	NUM
ejpam-6569	55	31	)	)	PUNCT
ejpam-6569	55	32	.	.	PUNCT
ejpam-6569	56	1	the	the	DET
ejpam-6569	56	2	union	union	NOUN
ejpam-6569	56	3	of	of	ADP
ejpam-6569	56	4	all	all	DET
ejpam-6569	56	5	τ1τ2	τ1τ2	ADJ
ejpam-6569	56	6	-	-	ADJ
ejpam-6569	56	7	open	open	ADJ
ejpam-6569	56	8	sets	set	NOUN
ejpam-6569	56	9	of	of	ADP
ejpam-6569	56	10	x	x	PUNCT
ejpam-6569	56	11	contained	contain	VERB
ejpam-6569	56	12	in	in	ADP
ejpam-6569	56	13	a	a	PRON
ejpam-6569	56	14	is	be	AUX
ejpam-6569	56	15	called	call	VERB
ejpam-6569	56	16	the	the	DET
ejpam-6569	56	17	τ1τ2	τ1τ2	NOUN
ejpam-6569	56	18	-	-	ADJ
ejpam-6569	56	19	interior	interior	ADJ
ejpam-6569	56	20	[	[	X
ejpam-6569	56	21	22	22	NUM
ejpam-6569	56	22	]	]	PUNCT
ejpam-6569	56	23	of	of	ADP
ejpam-6569	56	24	a	a	PRON
ejpam-6569	56	25	and	and	CCONJ
ejpam-6569	56	26	is	be	AUX
ejpam-6569	56	27	denoted	denote	VERB
ejpam-6569	56	28	by	by	ADP
ejpam-6569	56	29	τ1τ2	τ1τ2	NOUN
ejpam-6569	56	30	-	-	ADJ
ejpam-6569	56	31	int(a	int(a	NOUN
ejpam-6569	56	32	)	)	PUNCT
ejpam-6569	56	33	.	.	PUNCT
ejpam-6569	57	1	a	a	DET
ejpam-6569	57	2	subset	subset	NOUN
ejpam-6569	57	3	a	a	PRON
ejpam-6569	57	4	of	of	ADP
ejpam-6569	57	5	a	a	DET
ejpam-6569	57	6	bitopological	bitopological	ADJ
ejpam-6569	57	7	space	space	NOUN
ejpam-6569	57	8	(	(	PUNCT
ejpam-6569	57	9	x	x	NOUN
ejpam-6569	57	10	,	,	PUNCT
ejpam-6569	57	11	τ1	τ1	NOUN
ejpam-6569	57	12	,	,	PUNCT
ejpam-6569	57	13	τ2	τ2	NOUN
ejpam-6569	57	14	)	)	PUNCT
ejpam-6569	57	15	is	be	AUX
ejpam-6569	57	16	said	say	VERB
ejpam-6569	57	17	to	to	PART
ejpam-6569	57	18	be	be	AUX
ejpam-6569	57	19	τ1τ2	τ1τ2	NOUN
ejpam-6569	57	20	-	-	ADJ
ejpam-6569	57	21	clopen	clopen	ADJ
ejpam-6569	57	22	[	[	X
ejpam-6569	57	23	22	22	NUM
ejpam-6569	57	24	]	]	X
ejpam-6569	57	25	if	if	SCONJ
ejpam-6569	57	26	a	a	PRON
ejpam-6569	57	27	is	be	AUX
ejpam-6569	57	28	both	both	PRON
ejpam-6569	57	29	τ1τ2	τ1τ2	ADJ
ejpam-6569	57	30	-	-	ADJ
ejpam-6569	57	31	open	open	ADJ
ejpam-6569	57	32	and	and	CCONJ
ejpam-6569	57	33	τ1τ2	τ1τ2	NOUN
ejpam-6569	57	34	-	-	ADJ
ejpam-6569	57	35	closed	closed	ADJ
ejpam-6569	57	36	.	.	PUNCT
ejpam-6569	58	1	a	a	DET
ejpam-6569	58	2	subset	subset	NOUN
ejpam-6569	58	3	a	a	PRON
ejpam-6569	58	4	of	of	ADP
ejpam-6569	58	5	a	a	DET
ejpam-6569	58	6	bitopological	bitopological	ADJ
ejpam-6569	58	7	space	space	NOUN
ejpam-6569	58	8	(	(	PUNCT
ejpam-6569	58	9	x	x	NOUN
ejpam-6569	58	10	,	,	PUNCT
ejpam-6569	58	11	τ1	τ1	NOUN
ejpam-6569	58	12	,	,	PUNCT
ejpam-6569	58	13	τ2	τ2	NOUN
ejpam-6569	58	14	)	)	PUNCT
ejpam-6569	58	15	is	be	AUX
ejpam-6569	58	16	said	say	VERB
ejpam-6569	58	17	to	to	PART
ejpam-6569	58	18	be	be	AUX
ejpam-6569	58	19	(	(	PUNCT
ejpam-6569	58	20	τ1	τ1	NOUN
ejpam-6569	58	21	,	,	PUNCT
ejpam-6569	58	22	τ2)r	τ2)r	NOUN
ejpam-6569	58	23	-	-	PUNCT
ejpam-6569	58	24	open	open	NOUN
ejpam-6569	59	1	[	[	X
ejpam-6569	59	2	23	23	NUM
ejpam-6569	59	3	]	]	PUNCT
ejpam-6569	59	4	(	(	PUNCT
ejpam-6569	59	5	resp	resp	NOUN
ejpam-6569	59	6	.	.	PUNCT
ejpam-6569	60	1	(	(	PUNCT
ejpam-6569	60	2	τ1	τ1	NOUN
ejpam-6569	60	3	,	,	PUNCT
ejpam-6569	60	4	τ2)s	τ2)s	NOUN
ejpam-6569	60	5	-	-	PUNCT
ejpam-6569	60	6	open	open	ADJ
ejpam-6569	60	7	[	[	X
ejpam-6569	60	8	24	24	NUM
ejpam-6569	60	9	]	]	PUNCT
ejpam-6569	60	10	,	,	PUNCT
ejpam-6569	60	11	(	(	PUNCT
ejpam-6569	60	12	τ1	τ1	NOUN
ejpam-6569	60	13	,	,	PUNCT
ejpam-6569	60	14	τ2)p	τ2)p	NOUN
ejpam-6569	60	15	-	-	ADJ
ejpam-6569	60	16	open	open	ADJ
ejpam-6569	60	17	[	[	X
ejpam-6569	60	18	24	24	NUM
ejpam-6569	60	19	]	]	PUNCT
ejpam-6569	60	20	,	,	PUNCT
ejpam-6569	60	21	(	(	PUNCT
ejpam-6569	60	22	τ1	τ1	NOUN
ejpam-6569	60	23	,	,	PUNCT
ejpam-6569	60	24	τ2)β	τ2)β	ADJ
ejpam-6569	60	25	-	-	PUNCT
ejpam-6569	60	26	open	open	NOUN
ejpam-6569	60	27	[	[	X
ejpam-6569	60	28	24	24	NUM
ejpam-6569	60	29	]	]	PUNCT
ejpam-6569	60	30	)	)	PUNCT
ejpam-6569	60	31	if	if	SCONJ
ejpam-6569	60	32	a	a	DET
ejpam-6569	60	33	=	=	PUNCT
ejpam-6569	60	34	τ1τ2	τ1τ2	NOUN
ejpam-6569	60	35	-	-	NOUN
ejpam-6569	60	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6569	60	37	-	-	PUNCT
ejpam-6569	60	38	cl(a	cl(a	NUM
ejpam-6569	60	39	)	)	PUNCT
ejpam-6569	60	40	)	)	PUNCT
ejpam-6569	60	41	(	(	PUNCT
ejpam-6569	60	42	resp	resp	NOUN
ejpam-6569	60	43	.	.	PUNCT
ejpam-6569	61	1	a	a	DET
ejpam-6569	61	2	⊆	⊆	NUM
ejpam-6569	61	3	τ1τ2	τ1τ2	NOUN
ejpam-6569	61	4	-	-	ADJ
ejpam-6569	61	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6569	61	6	-	-	PUNCT
ejpam-6569	61	7	int(a	int(a	NOUN
ejpam-6569	61	8	)	)	PUNCT
ejpam-6569	61	9	)	)	PUNCT
ejpam-6569	61	10	,	,	PUNCT
ejpam-6569	61	11	a	a	DET
ejpam-6569	61	12	⊆	⊆	NUM
ejpam-6569	61	13	τ1τ2	τ1τ2	NOUN
ejpam-6569	61	14	-	-	NOUN
ejpam-6569	61	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6569	61	16	-	-	PUNCT
ejpam-6569	61	17	cl(a	cl(a	NUM
ejpam-6569	61	18	)	)	PUNCT
ejpam-6569	61	19	)	)	PUNCT
ejpam-6569	61	20	,	,	PUNCT
ejpam-6569	61	21	a	a	DET
ejpam-6569	61	22	⊆	⊆	NUM
ejpam-6569	61	23	τ1τ2	τ1τ2	NOUN
ejpam-6569	61	24	-	-	PUNCT
ejpam-6569	61	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6569	61	26	-	-	PUNCT
ejpam-6569	61	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6569	61	28	-	-	PUNCT
ejpam-6569	61	29	cl(a	cl(a	NUM
ejpam-6569	61	30	)	)	PUNCT
ejpam-6569	61	31	)	)	PUNCT
ejpam-6569	61	32	)	)	PUNCT
ejpam-6569	61	33	)	)	PUNCT
ejpam-6569	61	34	.	.	PUNCT
ejpam-6569	62	1	the	the	DET
ejpam-6569	62	2	complement	complement	NOUN
ejpam-6569	62	3	of	of	ADP
ejpam-6569	62	4	a	a	DET
ejpam-6569	62	5	(	(	PUNCT
ejpam-6569	62	6	τ1	τ1	NOUN
ejpam-6569	62	7	,	,	PUNCT
ejpam-6569	62	8	τ2)r	τ2)r	NOUN
ejpam-6569	62	9	-	-	PUNCT
ejpam-6569	62	10	open	open	ADJ
ejpam-6569	62	11	(	(	PUNCT
ejpam-6569	62	12	resp	resp	NOUN
ejpam-6569	62	13	.	.	PUNCT
ejpam-6569	63	1	(	(	PUNCT
ejpam-6569	63	2	τ1	τ1	NOUN
ejpam-6569	63	3	,	,	PUNCT
ejpam-6569	63	4	τ2)sopen	τ2)sopen	ADJ
ejpam-6569	63	5	,	,	PUNCT
ejpam-6569	63	6	(	(	PUNCT
ejpam-6569	63	7	τ1	τ1	NOUN
ejpam-6569	63	8	,	,	PUNCT
ejpam-6569	63	9	τ2)p	τ2)p	NOUN
ejpam-6569	63	10	-	-	ADJ
ejpam-6569	63	11	open	open	ADJ
ejpam-6569	63	12	,	,	PUNCT
ejpam-6569	63	13	(	(	PUNCT
ejpam-6569	63	14	τ1	τ1	NOUN
ejpam-6569	63	15	,	,	PUNCT
ejpam-6569	63	16	τ2)β	τ2)β	ADJ
ejpam-6569	63	17	-	-	PUNCT
ejpam-6569	63	18	open	open	ADJ
ejpam-6569	63	19	)	)	PUNCT
ejpam-6569	63	20	set	set	NOUN
ejpam-6569	63	21	is	be	AUX
ejpam-6569	63	22	called	call	VERB
ejpam-6569	63	23	(	(	PUNCT
ejpam-6569	63	24	τ1	τ1	NOUN
ejpam-6569	63	25	,	,	PUNCT
ejpam-6569	63	26	τ2)r	τ2)r	NOUN
ejpam-6569	63	27	-	-	PUNCT
ejpam-6569	63	28	closed	closed	ADJ
ejpam-6569	63	29	(	(	PUNCT
ejpam-6569	63	30	resp	resp	NOUN
ejpam-6569	63	31	.	.	PUNCT
ejpam-6569	64	1	(	(	PUNCT
ejpam-6569	64	2	τ1	τ1	NOUN
ejpam-6569	64	3	,	,	PUNCT
ejpam-6569	64	4	τ2)s	τ2)s	NOUN
ejpam-6569	64	5	-	-	PUNCT
ejpam-6569	64	6	closed	closed	ADJ
ejpam-6569	64	7	,	,	PUNCT
ejpam-6569	64	8	(	(	PUNCT
ejpam-6569	64	9	τ1	τ1	NOUN
ejpam-6569	64	10	,	,	PUNCT
ejpam-6569	64	11	τ2)p	τ2)p	NOUN
ejpam-6569	64	12	-	-	PUNCT
ejpam-6569	64	13	closed	closed	ADJ
ejpam-6569	64	14	,	,	PUNCT
ejpam-6569	64	15	(	(	PUNCT
ejpam-6569	64	16	τ1	τ1	NOUN
ejpam-6569	64	17	,	,	PUNCT
ejpam-6569	64	18	τ2)β	τ2)β	ADJ
ejpam-6569	64	19	-	-	PUNCT
ejpam-6569	64	20	closed	closed	ADJ
ejpam-6569	64	21	)	)	PUNCT
ejpam-6569	64	22	.	.	PUNCT
ejpam-6569	65	1	a	a	DET
ejpam-6569	65	2	subset	subset	NOUN
ejpam-6569	65	3	a	a	PRON
ejpam-6569	65	4	of	of	ADP
ejpam-6569	65	5	a	a	DET
ejpam-6569	65	6	bitopological	bitopological	ADJ
ejpam-6569	65	7	space	space	NOUN
ejpam-6569	65	8	(	(	PUNCT
ejpam-6569	65	9	x	x	NOUN
ejpam-6569	65	10	,	,	PUNCT
ejpam-6569	65	11	τ1	τ1	NOUN
ejpam-6569	65	12	,	,	PUNCT
ejpam-6569	65	13	τ2	τ2	NOUN
ejpam-6569	65	14	)	)	PUNCT
ejpam-6569	65	15	is	be	AUX
ejpam-6569	65	16	said	say	VERB
ejpam-6569	65	17	to	to	PART
ejpam-6569	65	18	be	be	AUX
ejpam-6569	65	19	α(τ1	α(τ1	NOUN
ejpam-6569	65	20	,	,	PUNCT
ejpam-6569	65	21	τ2)-open	τ2)-open	ADJ
ejpam-6569	65	22	[	[	X
ejpam-6569	65	23	25	25	NUM
ejpam-6569	65	24	]	]	PUNCT
ejpam-6569	65	25	if	if	SCONJ
ejpam-6569	65	26	a	a	DET
ejpam-6569	65	27	⊆	⊆	NUM
ejpam-6569	65	28	τ1τ2	τ1τ2	NOUN
ejpam-6569	65	29	-	-	PUNCT
ejpam-6569	65	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6569	65	31	-	-	PUNCT
ejpam-6569	65	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6569	65	33	-	-	PUNCT
ejpam-6569	65	34	int(a	int(a	NOUN
ejpam-6569	65	35	)	)	PUNCT
ejpam-6569	65	36	)	)	PUNCT
ejpam-6569	65	37	)	)	PUNCT
ejpam-6569	65	38	.	.	PUNCT
ejpam-6569	66	1	the	the	DET
ejpam-6569	66	2	complement	complement	NOUN
ejpam-6569	66	3	of	of	ADP
ejpam-6569	66	4	an	an	DET
ejpam-6569	66	5	α(τ1	α(τ1	NOUN
ejpam-6569	66	6	,	,	PUNCT
ejpam-6569	66	7	τ2)-open	τ2)-open	ADJ
ejpam-6569	66	8	set	set	NOUN
ejpam-6569	66	9	is	be	AUX
ejpam-6569	66	10	said	say	VERB
ejpam-6569	66	11	to	to	PART
ejpam-6569	66	12	be	be	AUX
ejpam-6569	66	13	α(τ1	α(τ1	NOUN
ejpam-6569	66	14	,	,	PUNCT
ejpam-6569	66	15	τ2)-closed	τ2)-closed	PROPN
ejpam-6569	66	16	.	.	PUNCT
ejpam-6569	67	1	n.	n.	PROPN
ejpam-6569	67	2	srisarakham	srisarakham	PROPN
ejpam-6569	67	3	,	,	PUNCT
ejpam-6569	67	4	a.	a.	PROPN
ejpam-6569	67	5	sama	sama	PROPN
ejpam-6569	67	6	-	-	PUNCT
ejpam-6569	67	7	ae	ae	PROPN
ejpam-6569	67	8	,	,	PUNCT
ejpam-6569	67	9	c.	c.	PROPN
ejpam-6569	67	10	boonpok	boonpok	PROPN
ejpam-6569	67	11	/	/	SYM
ejpam-6569	67	12	eur	eur	PROPN
ejpam-6569	67	13	.	.	PUNCT
ejpam-6569	68	1	j.	j.	PROPN
ejpam-6569	68	2	pure	pure	PROPN
ejpam-6569	68	3	appl	appl	PROPN
ejpam-6569	68	4	.	.	PROPN
ejpam-6569	68	5	math	math	PROPN
ejpam-6569	68	6	,	,	PUNCT
ejpam-6569	68	7	18	18	NUM
ejpam-6569	68	8	(	(	PUNCT
ejpam-6569	68	9	3	3	NUM
ejpam-6569	68	10	)	)	PUNCT
ejpam-6569	68	11	(	(	PUNCT
ejpam-6569	68	12	2025	2025	NUM
ejpam-6569	68	13	)	)	PUNCT
ejpam-6569	68	14	,	,	PUNCT
ejpam-6569	68	15	6569	6569	NUM
ejpam-6569	68	16	3	3	NUM
ejpam-6569	68	17	of	of	ADP
ejpam-6569	68	18	13	13	NUM
ejpam-6569	68	19	let	let	VERB
ejpam-6569	68	20	a	a	PRON
ejpam-6569	68	21	be	be	AUX
ejpam-6569	68	22	a	a	DET
ejpam-6569	68	23	subset	subset	NOUN
ejpam-6569	68	24	of	of	ADP
ejpam-6569	68	25	a	a	DET
ejpam-6569	68	26	bitopological	bitopological	ADJ
ejpam-6569	68	27	space	space	NOUN
ejpam-6569	68	28	(	(	PUNCT
ejpam-6569	68	29	x	x	NOUN
ejpam-6569	68	30	,	,	PUNCT
ejpam-6569	68	31	τ1	τ1	NOUN
ejpam-6569	68	32	,	,	PUNCT
ejpam-6569	68	33	τ2	τ2	NOUN
ejpam-6569	68	34	)	)	PUNCT
ejpam-6569	68	35	.	.	PUNCT
ejpam-6569	69	1	a	a	DET
ejpam-6569	69	2	point	point	NOUN
ejpam-6569	69	3	x	x	X
ejpam-6569	69	4	∈	∈	NOUN
ejpam-6569	69	5	x	x	PUNCT
ejpam-6569	69	6	is	be	AUX
ejpam-6569	69	7	called	call	VERB
ejpam-6569	69	8	a	a	DET
ejpam-6569	69	9	(	(	PUNCT
ejpam-6569	69	10	τ1	τ1	NOUN
ejpam-6569	69	11	,	,	PUNCT
ejpam-6569	69	12	τ2)θ	τ2)θ	ADJ
ejpam-6569	69	13	-	-	PUNCT
ejpam-6569	69	14	cluster	cluster	NOUN
ejpam-6569	69	15	point	point	NOUN
ejpam-6569	69	16	[	[	X
ejpam-6569	69	17	23	23	NUM
ejpam-6569	69	18	]	]	PUNCT
ejpam-6569	69	19	of	of	ADP
ejpam-6569	69	20	a	a	DET
ejpam-6569	69	21	if	if	SCONJ
ejpam-6569	69	22	τ1τ2	τ1τ2	ADJ
ejpam-6569	69	23	-	-	ADJ
ejpam-6569	69	24	cl(u)∩a	cl(u)∩a	ADJ
ejpam-6569	69	25	̸=	̸=	PROPN
ejpam-6569	69	26	∅	∅	NOUN
ejpam-6569	69	27	for	for	ADP
ejpam-6569	69	28	every	every	DET
ejpam-6569	69	29	τ1τ2	τ1τ2	ADJ
ejpam-6569	69	30	-	-	ADJ
ejpam-6569	69	31	open	open	ADJ
ejpam-6569	69	32	set	set	NOUN
ejpam-6569	69	33	u	u	NOUN
ejpam-6569	69	34	containing	contain	VERB
ejpam-6569	69	35	x.	x.	NOUN
ejpam-6569	69	36	the	the	DET
ejpam-6569	69	37	set	set	NOUN
ejpam-6569	69	38	of	of	ADP
ejpam-6569	69	39	all	all	DET
ejpam-6569	69	40	(	(	PUNCT
ejpam-6569	69	41	τ1	τ1	NOUN
ejpam-6569	69	42	,	,	PUNCT
ejpam-6569	69	43	τ2)θ	τ2)θ	ADJ
ejpam-6569	69	44	-	-	PUNCT
ejpam-6569	69	45	cluster	cluster	NOUN
ejpam-6569	69	46	points	point	NOUN
ejpam-6569	69	47	of	of	ADP
ejpam-6569	69	48	a	a	PRON
ejpam-6569	69	49	is	be	AUX
ejpam-6569	69	50	called	call	VERB
ejpam-6569	69	51	the	the	DET
ejpam-6569	69	52	(	(	PUNCT
ejpam-6569	69	53	τ1	τ1	NOUN
ejpam-6569	69	54	,	,	PUNCT
ejpam-6569	69	55	τ2)θ	τ2)θ	ADJ
ejpam-6569	69	56	-	-	PUNCT
ejpam-6569	69	57	closure	closure	NOUN
ejpam-6569	69	58	[	[	X
ejpam-6569	69	59	23	23	NUM
ejpam-6569	69	60	]	]	PUNCT
ejpam-6569	69	61	of	of	ADP
ejpam-6569	69	62	a	a	PRON
ejpam-6569	69	63	and	and	CCONJ
ejpam-6569	69	64	is	be	AUX
ejpam-6569	69	65	denoted	denote	VERB
ejpam-6569	69	66	by	by	ADP
ejpam-6569	69	67	(	(	PUNCT
ejpam-6569	69	68	τ1	τ1	NOUN
ejpam-6569	69	69	,	,	PUNCT
ejpam-6569	69	70	τ2)θ	τ2)θ	NOUN
ejpam-6569	69	71	-	-	PUNCT
ejpam-6569	69	72	cl(a	cl(a	NUM
ejpam-6569	69	73	)	)	PUNCT
ejpam-6569	69	74	.	.	PUNCT
ejpam-6569	70	1	a	a	DET
ejpam-6569	70	2	subset	subset	NOUN
ejpam-6569	70	3	a	a	PRON
ejpam-6569	70	4	of	of	ADP
ejpam-6569	70	5	a	a	DET
ejpam-6569	70	6	bitopological	bitopological	ADJ
ejpam-6569	70	7	space	space	NOUN
ejpam-6569	70	8	(	(	PUNCT
ejpam-6569	70	9	x	x	NOUN
ejpam-6569	70	10	,	,	PUNCT
ejpam-6569	70	11	τ1	τ1	NOUN
ejpam-6569	70	12	,	,	PUNCT
ejpam-6569	70	13	τ2	τ2	NOUN
ejpam-6569	70	14	)	)	PUNCT
ejpam-6569	70	15	is	be	AUX
ejpam-6569	70	16	said	say	VERB
ejpam-6569	70	17	to	to	PART
ejpam-6569	70	18	be	be	AUX
ejpam-6569	70	19	(	(	PUNCT
ejpam-6569	70	20	τ1	τ1	NOUN
ejpam-6569	70	21	,	,	PUNCT
ejpam-6569	70	22	τ2)θ	τ2)θ	NOUN
ejpam-6569	70	23	-	-	PUNCT
ejpam-6569	70	24	closed	closed	ADJ
ejpam-6569	70	25	[	[	X
ejpam-6569	70	26	23	23	NUM
ejpam-6569	70	27	]	]	X
ejpam-6569	70	28	if	if	SCONJ
ejpam-6569	70	29	(	(	PUNCT
ejpam-6569	70	30	τ1	τ1	NOUN
ejpam-6569	70	31	,	,	PUNCT
ejpam-6569	70	32	τ2)θ	τ2)θ	NOUN
ejpam-6569	70	33	-	-	PUNCT
ejpam-6569	70	34	cl(a	cl(a	NUM
ejpam-6569	70	35	)	)	PUNCT
ejpam-6569	71	1	=	=	PUNCT
ejpam-6569	71	2	a.	a.	NOUN
ejpam-6569	71	3	the	the	DET
ejpam-6569	71	4	complement	complement	NOUN
ejpam-6569	71	5	of	of	ADP
ejpam-6569	71	6	a	a	DET
ejpam-6569	71	7	(	(	PUNCT
ejpam-6569	71	8	τ1	τ1	NOUN
ejpam-6569	71	9	,	,	PUNCT
ejpam-6569	71	10	τ2)θ	τ2)θ	ADJ
ejpam-6569	71	11	-	-	PUNCT
ejpam-6569	71	12	closed	close	VERB
ejpam-6569	71	13	set	set	NOUN
ejpam-6569	71	14	is	be	AUX
ejpam-6569	71	15	said	say	VERB
ejpam-6569	71	16	to	to	PART
ejpam-6569	71	17	be	be	AUX
ejpam-6569	71	18	(	(	PUNCT
ejpam-6569	71	19	τ1	τ1	NOUN
ejpam-6569	71	20	,	,	PUNCT
ejpam-6569	71	21	τ2)θ	τ2)θ	NOUN
ejpam-6569	71	22	-	-	PUNCT
ejpam-6569	71	23	open	open	ADJ
ejpam-6569	71	24	.	.	PUNCT
ejpam-6569	72	1	the	the	DET
ejpam-6569	72	2	union	union	NOUN
ejpam-6569	72	3	of	of	ADP
ejpam-6569	72	4	all	all	DET
ejpam-6569	72	5	(	(	PUNCT
ejpam-6569	72	6	τ1	τ1	NOUN
ejpam-6569	72	7	,	,	PUNCT
ejpam-6569	72	8	τ2)θ	τ2)θ	ADJ
ejpam-6569	72	9	-	-	PUNCT
ejpam-6569	72	10	open	open	ADJ
ejpam-6569	72	11	sets	set	NOUN
ejpam-6569	72	12	of	of	ADP
ejpam-6569	72	13	x	x	PUNCT
ejpam-6569	72	14	contained	contain	VERB
ejpam-6569	72	15	in	in	ADP
ejpam-6569	72	16	a	a	PRON
ejpam-6569	72	17	is	be	AUX
ejpam-6569	72	18	called	call	VERB
ejpam-6569	72	19	the	the	DET
ejpam-6569	72	20	(	(	PUNCT
ejpam-6569	72	21	τ1	τ1	NOUN
ejpam-6569	72	22	,	,	PUNCT
ejpam-6569	72	23	τ2)θ	τ2)θ	ADJ
ejpam-6569	72	24	-	-	PUNCT
ejpam-6569	72	25	interior	interior	NOUN
ejpam-6569	72	26	[	[	X
ejpam-6569	72	27	23	23	NUM
ejpam-6569	72	28	]	]	PUNCT
ejpam-6569	72	29	of	of	ADP
ejpam-6569	72	30	a	a	PRON
ejpam-6569	72	31	and	and	CCONJ
ejpam-6569	72	32	is	be	AUX
ejpam-6569	72	33	denoted	denote	VERB
ejpam-6569	72	34	by	by	ADP
ejpam-6569	72	35	(	(	PUNCT
ejpam-6569	72	36	τ1	τ1	NOUN
ejpam-6569	72	37	,	,	PUNCT
ejpam-6569	72	38	τ2)θ	τ2)θ	NOUN
ejpam-6569	72	39	-	-	PUNCT
ejpam-6569	72	40	int(a	int(a	NOUN
ejpam-6569	72	41	)	)	PUNCT
ejpam-6569	72	42	.	.	PUNCT
ejpam-6569	73	1	lemma	lemma	PROPN
ejpam-6569	73	2	1	1	NUM
ejpam-6569	73	3	.	.	PUNCT
ejpam-6569	74	1	[	[	X
ejpam-6569	74	2	23	23	NUM
ejpam-6569	74	3	]	]	PUNCT
ejpam-6569	74	4	for	for	ADP
ejpam-6569	74	5	a	a	DET
ejpam-6569	74	6	subset	subset	NOUN
ejpam-6569	74	7	a	a	PRON
ejpam-6569	74	8	of	of	ADP
ejpam-6569	74	9	a	a	DET
ejpam-6569	74	10	bitopological	bitopological	ADJ
ejpam-6569	74	11	space	space	NOUN
ejpam-6569	74	12	(	(	PUNCT
ejpam-6569	74	13	x	x	NOUN
ejpam-6569	74	14	,	,	PUNCT
ejpam-6569	74	15	τ1	τ1	NOUN
ejpam-6569	74	16	,	,	PUNCT
ejpam-6569	74	17	τ2	τ2	NOUN
ejpam-6569	74	18	)	)	PUNCT
ejpam-6569	74	19	,	,	PUNCT
ejpam-6569	74	20	the	the	DET
ejpam-6569	74	21	following	follow	VERB
ejpam-6569	74	22	properties	property	NOUN
ejpam-6569	74	23	hold	hold	VERB
ejpam-6569	74	24	:	:	PUNCT
ejpam-6569	74	25	(	(	PUNCT
ejpam-6569	74	26	1	1	X
ejpam-6569	74	27	)	)	PUNCT
ejpam-6569	74	28	if	if	SCONJ
ejpam-6569	74	29	a	a	PRON
ejpam-6569	74	30	is	be	AUX
ejpam-6569	74	31	τ1τ2	τ1τ2	NOUN
ejpam-6569	74	32	-	-	ADJ
ejpam-6569	74	33	open	open	ADJ
ejpam-6569	74	34	in	in	ADP
ejpam-6569	74	35	x	x	NOUN
ejpam-6569	74	36	,	,	PUNCT
ejpam-6569	74	37	then	then	ADV
ejpam-6569	74	38	τ1τ2	τ1τ2	NOUN
ejpam-6569	74	39	-	-	NUM
ejpam-6569	74	40	cl(a	cl(a	NUM
ejpam-6569	74	41	)	)	PUNCT
ejpam-6569	74	42	=	=	PUNCT
ejpam-6569	74	43	(	(	PUNCT
ejpam-6569	74	44	τ1	τ1	NOUN
ejpam-6569	74	45	,	,	PUNCT
ejpam-6569	74	46	τ2)θ	τ2)θ	NOUN
ejpam-6569	74	47	-	-	PUNCT
ejpam-6569	74	48	cl(a	cl(a	NUM
ejpam-6569	74	49	)	)	PUNCT
ejpam-6569	74	50	.	.	PUNCT
ejpam-6569	75	1	(	(	PUNCT
ejpam-6569	75	2	2	2	X
ejpam-6569	75	3	)	)	PUNCT
ejpam-6569	75	4	(	(	PUNCT
ejpam-6569	75	5	τ1	τ1	NOUN
ejpam-6569	75	6	,	,	PUNCT
ejpam-6569	75	7	τ2)θ	τ2)θ	NOUN
ejpam-6569	75	8	-	-	PUNCT
ejpam-6569	75	9	cl(a	cl(a	NUM
ejpam-6569	75	10	)	)	PUNCT
ejpam-6569	75	11	is	be	AUX
ejpam-6569	75	12	τ1τ2	τ1τ2	NOUN
ejpam-6569	75	13	-	-	ADJ
ejpam-6569	75	14	closed	closed	ADJ
ejpam-6569	75	15	in	in	ADP
ejpam-6569	75	16	x.	x.	NOUN
ejpam-6569	75	17	an	an	DET
ejpam-6569	75	18	ideal	ideal	NOUN
ejpam-6569	75	19	i	i	PRON
ejpam-6569	75	20	on	on	ADP
ejpam-6569	75	21	a	a	DET
ejpam-6569	75	22	topological	topological	ADJ
ejpam-6569	75	23	space	space	NOUN
ejpam-6569	75	24	(	(	PUNCT
ejpam-6569	75	25	x	x	X
ejpam-6569	75	26	,	,	PUNCT
ejpam-6569	75	27	τ	τ	X
ejpam-6569	75	28	)	)	PUNCT
ejpam-6569	75	29	is	be	AUX
ejpam-6569	75	30	a	a	DET
ejpam-6569	75	31	nonempty	nonempty	ADJ
ejpam-6569	75	32	collection	collection	NOUN
ejpam-6569	75	33	of	of	ADP
ejpam-6569	75	34	subsets	subset	NOUN
ejpam-6569	75	35	of	of	ADP
ejpam-6569	75	36	x	x	PUNCT
ejpam-6569	75	37	satisfying	satisfy	VERB
ejpam-6569	75	38	the	the	DET
ejpam-6569	75	39	following	follow	VERB
ejpam-6569	75	40	properties	property	NOUN
ejpam-6569	75	41	:	:	PUNCT
ejpam-6569	75	42	(	(	PUNCT
ejpam-6569	75	43	1	1	X
ejpam-6569	75	44	)	)	PUNCT
ejpam-6569	75	45	a	a	DET
ejpam-6569	75	46	∈	∈	NOUN
ejpam-6569	76	1	i	i	PRON
ejpam-6569	76	2	and	and	CCONJ
ejpam-6569	76	3	b	b	X
ejpam-6569	76	4	⊆	⊆	NUM
ejpam-6569	76	5	a	a	DET
ejpam-6569	76	6	imply	imply	NOUN
ejpam-6569	76	7	b	b	X
ejpam-6569	76	8	∈	∈	PROPN
ejpam-6569	76	9	i	i	PRON
ejpam-6569	76	10	;	;	PUNCT
ejpam-6569	76	11	(	(	PUNCT
ejpam-6569	76	12	2	2	X
ejpam-6569	76	13	)	)	PUNCT
ejpam-6569	77	1	a	a	DET
ejpam-6569	77	2	∈	∈	NOUN
ejpam-6569	77	3	i	i	PRON
ejpam-6569	77	4	and	and	CCONJ
ejpam-6569	77	5	b	b	X
ejpam-6569	77	6	∈	∈	NOUN
ejpam-6569	78	1	i	i	PRON
ejpam-6569	78	2	imply	imply	VERB
ejpam-6569	78	3	a	a	DET
ejpam-6569	78	4	∪	∪	X
ejpam-6569	78	5	b	b	NOUN
ejpam-6569	78	6	∈	∈	NOUN
ejpam-6569	79	1	i	i	PRON
ejpam-6569	79	2	.	.	PUNCT
ejpam-6569	80	1	a	a	DET
ejpam-6569	80	2	topological	topological	ADJ
ejpam-6569	80	3	space	space	NOUN
ejpam-6569	80	4	(	(	PUNCT
ejpam-6569	80	5	x	x	X
ejpam-6569	80	6	,	,	PUNCT
ejpam-6569	80	7	τ	τ	X
ejpam-6569	80	8	)	)	PUNCT
ejpam-6569	80	9	with	with	ADP
ejpam-6569	80	10	an	an	DET
ejpam-6569	80	11	ideal	ideal	ADJ
ejpam-6569	80	12	i	i	PRON
ejpam-6569	80	13	on	on	ADP
ejpam-6569	80	14	x	x	SYM
ejpam-6569	80	15	is	be	AUX
ejpam-6569	80	16	called	call	VERB
ejpam-6569	80	17	an	an	DET
ejpam-6569	80	18	ideal	ideal	ADJ
ejpam-6569	80	19	topological	topological	ADJ
ejpam-6569	80	20	space	space	NOUN
ejpam-6569	80	21	and	and	CCONJ
ejpam-6569	80	22	is	be	AUX
ejpam-6569	80	23	denoted	denote	VERB
ejpam-6569	80	24	by	by	ADP
ejpam-6569	80	25	(	(	PUNCT
ejpam-6569	80	26	x	x	X
ejpam-6569	80	27	,	,	PUNCT
ejpam-6569	80	28	τ	τ	PROPN
ejpam-6569	80	29	,	,	PUNCT
ejpam-6569	80	30	i	i	NOUN
ejpam-6569	80	31	)	)	PUNCT
ejpam-6569	80	32	.	.	PUNCT
ejpam-6569	81	1	for	for	ADP
ejpam-6569	81	2	an	an	DET
ejpam-6569	81	3	ideal	ideal	ADJ
ejpam-6569	81	4	topological	topological	ADJ
ejpam-6569	81	5	space	space	NOUN
ejpam-6569	81	6	(	(	PUNCT
ejpam-6569	81	7	x	x	X
ejpam-6569	81	8	,	,	PUNCT
ejpam-6569	81	9	τ	τ	PROPN
ejpam-6569	81	10	,	,	PUNCT
ejpam-6569	81	11	i	i	PROPN
ejpam-6569	81	12	)	)	PUNCT
ejpam-6569	81	13	and	and	CCONJ
ejpam-6569	81	14	a	a	DET
ejpam-6569	81	15	subset	subset	NOUN
ejpam-6569	81	16	a	a	PRON
ejpam-6569	81	17	of	of	ADP
ejpam-6569	81	18	x	x	PRON
ejpam-6569	81	19	,	,	PUNCT
ejpam-6569	81	20	a⋆(i	a⋆(i	PROPN
ejpam-6569	81	21	)	)	PUNCT
ejpam-6569	81	22	is	be	AUX
ejpam-6569	81	23	defined	define	VERB
ejpam-6569	81	24	as	as	SCONJ
ejpam-6569	81	25	follows	follow	VERB
ejpam-6569	81	26	:	:	PUNCT
ejpam-6569	81	27	a⋆(i	a⋆(i	NOUN
ejpam-6569	81	28	)	)	PUNCT
ejpam-6569	82	1	=	=	PUNCT
ejpam-6569	82	2	{	{	PUNCT
ejpam-6569	82	3	x	x	PUNCT
ejpam-6569	82	4	∈	∈	PROPN
ejpam-6569	82	5	x	x	X
ejpam-6569	82	6	:	:	PUNCT
ejpam-6569	82	7	u	u	X
ejpam-6569	82	8	∩a	∩a	PROPN
ejpam-6569	82	9	̸∈	̸∈	PROPN
ejpam-6569	82	10	i	i	PRON
ejpam-6569	82	11	for	for	ADP
ejpam-6569	82	12	every	every	DET
ejpam-6569	82	13	open	open	ADJ
ejpam-6569	82	14	neighbourhood	neighbourhood	NOUN
ejpam-6569	82	15	u	u	NOUN
ejpam-6569	82	16	of	of	ADP
ejpam-6569	82	17	x	x	NOUN
ejpam-6569	82	18	}	}	PUNCT
ejpam-6569	82	19	.	.	PUNCT
ejpam-6569	83	1	in	in	ADP
ejpam-6569	83	2	case	case	NOUN
ejpam-6569	83	3	there	there	PRON
ejpam-6569	83	4	is	be	VERB
ejpam-6569	83	5	no	no	DET
ejpam-6569	83	6	chance	chance	NOUN
ejpam-6569	83	7	for	for	ADP
ejpam-6569	83	8	confusion	confusion	NOUN
ejpam-6569	83	9	,	,	PUNCT
ejpam-6569	83	10	a⋆(i	a⋆(i	NOUN
ejpam-6569	83	11	)	)	PUNCT
ejpam-6569	83	12	is	be	AUX
ejpam-6569	83	13	simply	simply	ADV
ejpam-6569	83	14	written	write	VERB
ejpam-6569	83	15	as	as	ADP
ejpam-6569	83	16	a⋆.	a⋆.	NOUN
ejpam-6569	83	17	in	in	ADP
ejpam-6569	83	18	[	[	X
ejpam-6569	83	19	26	26	NUM
ejpam-6569	83	20	]	]	PUNCT
ejpam-6569	83	21	,	,	PUNCT
ejpam-6569	83	22	a⋆	a⋆	ADV
ejpam-6569	83	23	is	be	AUX
ejpam-6569	83	24	called	call	VERB
ejpam-6569	83	25	the	the	DET
ejpam-6569	83	26	local	local	ADJ
ejpam-6569	83	27	function	function	NOUN
ejpam-6569	83	28	of	of	ADP
ejpam-6569	83	29	a	a	PRON
ejpam-6569	83	30	with	with	ADP
ejpam-6569	83	31	respect	respect	NOUN
ejpam-6569	83	32	to	to	ADP
ejpam-6569	83	33	i	i	PRON
ejpam-6569	83	34	and	and	CCONJ
ejpam-6569	83	35	τ	τ	PROPN
ejpam-6569	83	36	and	and	CCONJ
ejpam-6569	83	37	cl⋆(a	cl⋆(a	PROPN
ejpam-6569	83	38	)	)	PUNCT
ejpam-6569	83	39	=	=	NOUN
ejpam-6569	83	40	a⋆∪a	a⋆∪a	NOUN
ejpam-6569	83	41	defines	define	VERB
ejpam-6569	83	42	a	a	DET
ejpam-6569	83	43	kuratowski	kuratowski	ADJ
ejpam-6569	83	44	closure	closure	NOUN
ejpam-6569	83	45	operator	operator	NOUN
ejpam-6569	83	46	for	for	ADP
ejpam-6569	83	47	a	a	DET
ejpam-6569	83	48	topology	topology	NOUN
ejpam-6569	83	49	τ⋆(i	τ⋆(i	NOUN
ejpam-6569	83	50	)	)	PUNCT
ejpam-6569	83	51	finer	fine	ADJ
ejpam-6569	83	52	than	than	ADP
ejpam-6569	83	53	τ	τ	PROPN
ejpam-6569	83	54	.	.	PUNCT
ejpam-6569	84	1	a	a	DET
ejpam-6569	84	2	subset	subset	NOUN
ejpam-6569	84	3	a	a	PRON
ejpam-6569	84	4	is	be	AUX
ejpam-6569	84	5	said	say	VERB
ejpam-6569	84	6	to	to	PART
ejpam-6569	84	7	be	be	AUX
ejpam-6569	84	8	⋆-closed	⋆-close	VERB
ejpam-6569	84	9	[	[	X
ejpam-6569	84	10	27	27	NUM
ejpam-6569	84	11	]	]	X
ejpam-6569	84	12	if	if	SCONJ
ejpam-6569	84	13	a⋆	a⋆	ADJ
ejpam-6569	84	14	⊆	⊆	NUM
ejpam-6569	84	15	a.	a.	NOUN
ejpam-6569	84	16	the	the	DET
ejpam-6569	84	17	interior	interior	NOUN
ejpam-6569	84	18	of	of	ADP
ejpam-6569	84	19	a	a	DET
ejpam-6569	84	20	subset	subset	NOUN
ejpam-6569	84	21	a	a	DET
ejpam-6569	84	22	in	in	ADP
ejpam-6569	84	23	(	(	PUNCT
ejpam-6569	84	24	x	x	X
ejpam-6569	84	25	,	,	PUNCT
ejpam-6569	84	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6569	84	27	)	)	PUNCT
ejpam-6569	84	28	)	)	PUNCT
ejpam-6569	84	29	is	be	AUX
ejpam-6569	84	30	denoted	denote	VERB
ejpam-6569	84	31	by	by	ADP
ejpam-6569	84	32	int⋆(a	int⋆(a	NOUN
ejpam-6569	84	33	)	)	PUNCT
ejpam-6569	84	34	.	.	PUNCT
ejpam-6569	85	1	a	a	DET
ejpam-6569	85	2	subset	subset	NOUN
ejpam-6569	85	3	a	a	PRON
ejpam-6569	85	4	of	of	ADP
ejpam-6569	85	5	an	an	DET
ejpam-6569	85	6	ideal	ideal	ADJ
ejpam-6569	85	7	topological	topological	ADJ
ejpam-6569	85	8	space	space	NOUN
ejpam-6569	85	9	(	(	PUNCT
ejpam-6569	85	10	x	x	X
ejpam-6569	85	11	,	,	PUNCT
ejpam-6569	85	12	τ	τ	PROPN
ejpam-6569	85	13	,	,	PUNCT
ejpam-6569	85	14	i	i	PROPN
ejpam-6569	85	15	)	)	PUNCT
ejpam-6569	85	16	is	be	AUX
ejpam-6569	85	17	said	say	VERB
ejpam-6569	85	18	to	to	PART
ejpam-6569	85	19	be	be	AUX
ejpam-6569	85	20	semi⋆-i	semi⋆-i	X
ejpam-6569	85	21	-open	-open	VERB
ejpam-6569	85	22	[	[	X
ejpam-6569	85	23	28	28	NUM
ejpam-6569	85	24	]	]	X
ejpam-6569	85	25	(	(	PUNCT
ejpam-6569	85	26	resp	resp	NOUN
ejpam-6569	85	27	.	.	PUNCT
ejpam-6569	86	1	semi	semi	ADJ
ejpam-6569	86	2	-	-	VERB
ejpam-6569	86	3	i	i	PRON
ejpam-6569	86	4	-open	-open	NOUN
ejpam-6569	87	1	[	[	X
ejpam-6569	87	2	16	16	NUM
ejpam-6569	87	3	]	]	SYM
ejpam-6569	87	4	)	)	PUNCT
ejpam-6569	87	5	if	if	SCONJ
ejpam-6569	87	6	a	a	DET
ejpam-6569	87	7	⊆	⊆	NUM
ejpam-6569	87	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6569	87	9	)	)	PUNCT
ejpam-6569	87	10	)	)	PUNCT
ejpam-6569	87	11	(	(	PUNCT
ejpam-6569	87	12	resp	resp	NOUN
ejpam-6569	87	13	.	.	PUNCT
ejpam-6569	88	1	a	a	DET
ejpam-6569	88	2	⊆	⊆	NUM
ejpam-6569	88	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6569	88	4	)	)	PUNCT
ejpam-6569	88	5	)	)	PUNCT
ejpam-6569	88	6	)	)	PUNCT
ejpam-6569	88	7	.	.	PUNCT
ejpam-6569	89	1	the	the	DET
ejpam-6569	89	2	complement	complement	NOUN
ejpam-6569	89	3	of	of	ADP
ejpam-6569	89	4	a	a	DET
ejpam-6569	89	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6569	89	6	-open	-open	ADJ
ejpam-6569	89	7	(	(	PUNCT
ejpam-6569	89	8	resp	resp	NOUN
ejpam-6569	89	9	.	.	PUNCT
ejpam-6569	90	1	semi	semi	ADJ
ejpam-6569	90	2	-	-	VERB
ejpam-6569	90	3	i	i	PRON
ejpam-6569	90	4	-open	-open	NOUN
ejpam-6569	90	5	)	)	PUNCT
ejpam-6569	91	1	set	set	NOUN
ejpam-6569	91	2	is	be	AUX
ejpam-6569	91	3	said	say	VERB
ejpam-6569	91	4	to	to	PART
ejpam-6569	91	5	be	be	AUX
ejpam-6569	91	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6569	91	7	-closed	-close	VERB
ejpam-6569	91	8	[	[	X
ejpam-6569	91	9	28	28	NUM
ejpam-6569	91	10	]	]	X
ejpam-6569	91	11	(	(	PUNCT
ejpam-6569	91	12	resp	resp	NOUN
ejpam-6569	91	13	.	.	PUNCT
ejpam-6569	92	1	semi	semi	ADJ
ejpam-6569	92	2	-	-	VERB
ejpam-6569	92	3	i	i	PRON
ejpam-6569	92	4	-closed	-close	VERB
ejpam-6569	92	5	[	[	X
ejpam-6569	92	6	16	16	NUM
ejpam-6569	92	7	]	]	PUNCT
ejpam-6569	92	8	)	)	PUNCT
ejpam-6569	92	9	.	.	PUNCT
ejpam-6569	93	1	a	a	DET
ejpam-6569	93	2	subset	subset	NOUN
ejpam-6569	93	3	a	a	PRON
ejpam-6569	93	4	of	of	ADP
ejpam-6569	93	5	an	an	DET
ejpam-6569	93	6	ideal	ideal	ADJ
ejpam-6569	93	7	topological	topological	ADJ
ejpam-6569	93	8	space	space	NOUN
ejpam-6569	93	9	(	(	PUNCT
ejpam-6569	93	10	x	x	X
ejpam-6569	93	11	,	,	PUNCT
ejpam-6569	93	12	τ	τ	PROPN
ejpam-6569	93	13	,	,	PUNCT
ejpam-6569	93	14	i	i	PROPN
ejpam-6569	93	15	)	)	PUNCT
ejpam-6569	93	16	is	be	AUX
ejpam-6569	93	17	called	call	VERB
ejpam-6569	93	18	semi	semi	ADJ
ejpam-6569	93	19	-	-	ADJ
ejpam-6569	93	20	i	i	PRON
ejpam-6569	93	21	⋆-open	⋆-open	VERB
ejpam-6569	94	1	[	[	X
ejpam-6569	94	2	29	29	NUM
ejpam-6569	94	3	]	]	X
ejpam-6569	94	4	if	if	SCONJ
ejpam-6569	94	5	a	a	DET
ejpam-6569	94	6	⊆	⊆	NUM
ejpam-6569	94	7	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-6569	94	8	)	)	PUNCT
ejpam-6569	94	9	)	)	PUNCT
ejpam-6569	94	10	.	.	PUNCT
ejpam-6569	95	1	the	the	DET
ejpam-6569	95	2	complement	complement	NOUN
ejpam-6569	95	3	of	of	ADP
ejpam-6569	95	4	a	a	DET
ejpam-6569	95	5	semi	semi	NOUN
ejpam-6569	95	6	-	-	ADJ
ejpam-6569	95	7	i	i	PRON
ejpam-6569	95	8	⋆-open	⋆-open	ADJ
ejpam-6569	95	9	set	set	VERB
ejpam-6569	95	10	is	be	AUX
ejpam-6569	95	11	called	call	VERB
ejpam-6569	95	12	semi	semi	ADJ
ejpam-6569	95	13	-	-	ADJ
ejpam-6569	95	14	i	i	PRON
ejpam-6569	95	15	⋆-closed	⋆-close	VERB
ejpam-6569	95	16	.	.	PUNCT
ejpam-6569	96	1	for	for	ADP
ejpam-6569	96	2	a	a	DET
ejpam-6569	96	3	subset	subset	NOUN
ejpam-6569	96	4	a	a	PRON
ejpam-6569	96	5	of	of	ADP
ejpam-6569	96	6	an	an	DET
ejpam-6569	96	7	ideal	ideal	ADJ
ejpam-6569	96	8	topological	topological	ADJ
ejpam-6569	96	9	space	space	NOUN
ejpam-6569	96	10	(	(	PUNCT
ejpam-6569	96	11	x	x	X
ejpam-6569	96	12	,	,	PUNCT
ejpam-6569	96	13	τ	τ	PROPN
ejpam-6569	96	14	,	,	PUNCT
ejpam-6569	96	15	i	i	NOUN
ejpam-6569	96	16	)	)	PUNCT
ejpam-6569	96	17	,	,	PUNCT
ejpam-6569	96	18	the	the	DET
ejpam-6569	96	19	intersection	intersection	NOUN
ejpam-6569	96	20	of	of	ADP
ejpam-6569	96	21	all	all	PRON
ejpam-6569	96	22	semi	semi	NOUN
ejpam-6569	96	23	-	-	ADJ
ejpam-6569	96	24	i	i	PRON
ejpam-6569	96	25	⋆-closed	⋆-close	VERB
ejpam-6569	96	26	sets	set	NOUN
ejpam-6569	96	27	containing	contain	VERB
ejpam-6569	96	28	a	a	PRON
ejpam-6569	96	29	is	be	AUX
ejpam-6569	96	30	called	call	VERB
ejpam-6569	96	31	the	the	DET
ejpam-6569	96	32	semi	semi	NOUN
ejpam-6569	96	33	-	-	ADJ
ejpam-6569	96	34	i	i	PRON
ejpam-6569	96	35	⋆-closure	⋆-closure	NOUN
ejpam-6569	97	1	[	[	X
ejpam-6569	97	2	29	29	NUM
ejpam-6569	97	3	]	]	PUNCT
ejpam-6569	97	4	of	of	ADP
ejpam-6569	97	5	a	a	PRON
ejpam-6569	97	6	and	and	CCONJ
ejpam-6569	97	7	is	be	AUX
ejpam-6569	97	8	denoted	denote	VERB
ejpam-6569	97	9	by	by	ADP
ejpam-6569	97	10	scl⋆(a	scl⋆(a	NOUN
ejpam-6569	97	11	)	)	PUNCT
ejpam-6569	97	12	(	(	PUNCT
ejpam-6569	97	13	scli	scli	PROPN
ejpam-6569	97	14	⋆(a	⋆(a	NUM
ejpam-6569	97	15	)	)	PUNCT
ejpam-6569	98	1	[	[	X
ejpam-6569	98	2	29	29	NUM
ejpam-6569	98	3	]	]	PUNCT
ejpam-6569	98	4	)	)	PUNCT
ejpam-6569	98	5	.	.	PUNCT
ejpam-6569	99	1	the	the	DET
ejpam-6569	99	2	union	union	NOUN
ejpam-6569	99	3	of	of	ADP
ejpam-6569	99	4	all	all	PRON
ejpam-6569	99	5	semi	semi	ADJ
ejpam-6569	99	6	-	-	ADJ
ejpam-6569	99	7	i	i	PRON
ejpam-6569	99	8	⋆-open	⋆-open	ADJ
ejpam-6569	99	9	sets	set	NOUN
ejpam-6569	99	10	contained	contain	VERB
ejpam-6569	99	11	in	in	ADP
ejpam-6569	99	12	a	a	PRON
ejpam-6569	99	13	is	be	AUX
ejpam-6569	99	14	called	call	VERB
ejpam-6569	99	15	the	the	DET
ejpam-6569	99	16	semi	semi	NOUN
ejpam-6569	99	17	-	-	ADJ
ejpam-6569	99	18	i	i	PRON
ejpam-6569	99	19	⋆-interior	⋆-interior	PUNCT
ejpam-6569	100	1	[	[	X
ejpam-6569	100	2	29	29	NUM
ejpam-6569	100	3	]	]	PUNCT
ejpam-6569	100	4	of	of	ADP
ejpam-6569	100	5	a	a	PRON
ejpam-6569	100	6	and	and	CCONJ
ejpam-6569	100	7	is	be	AUX
ejpam-6569	100	8	denoted	denote	VERB
ejpam-6569	100	9	by	by	ADP
ejpam-6569	100	10	sint⋆(a	sint⋆(a	PROPN
ejpam-6569	100	11	)	)	PUNCT
ejpam-6569	101	1	(	(	PUNCT
ejpam-6569	101	2	sinti	sinti	PROPN
ejpam-6569	101	3	⋆(a	⋆(a	NOUN
ejpam-6569	101	4	)	)	PUNCT
ejpam-6569	102	1	[	[	X
ejpam-6569	102	2	29	29	NUM
ejpam-6569	102	3	]	]	PUNCT
ejpam-6569	102	4	)	)	PUNCT
ejpam-6569	102	5	.	.	PUNCT
ejpam-6569	103	1	the	the	DET
ejpam-6569	103	2	family	family	NOUN
ejpam-6569	103	3	of	of	ADP
ejpam-6569	103	4	all	all	PRON
ejpam-6569	103	5	semi	semi	ADJ
ejpam-6569	103	6	-	-	ADJ
ejpam-6569	103	7	i	i	PRON
ejpam-6569	103	8	⋆-open	⋆-open	ADJ
ejpam-6569	103	9	sets	set	NOUN
ejpam-6569	103	10	of	of	ADP
ejpam-6569	103	11	an	an	DET
ejpam-6569	103	12	ideal	ideal	ADJ
ejpam-6569	103	13	topological	topological	ADJ
ejpam-6569	103	14	space	space	NOUN
ejpam-6569	103	15	(	(	PUNCT
ejpam-6569	103	16	x	x	X
ejpam-6569	103	17	,	,	PUNCT
ejpam-6569	103	18	τ	τ	PROPN
ejpam-6569	103	19	,	,	PUNCT
ejpam-6569	103	20	i	i	PROPN
ejpam-6569	103	21	)	)	PUNCT
ejpam-6569	103	22	is	be	AUX
ejpam-6569	103	23	denoted	denote	VERB
ejpam-6569	103	24	by	by	ADP
ejpam-6569	103	25	si	si	X
ejpam-6569	103	26	⋆o(x	⋆o(x	NUM
ejpam-6569	103	27	)	)	PUNCT
ejpam-6569	103	28	.	.	PUNCT
ejpam-6569	104	1	let	let	VERB
ejpam-6569	104	2	a	a	DET
ejpam-6569	104	3	be	be	AUX
ejpam-6569	104	4	a	a	DET
ejpam-6569	104	5	subset	subset	NOUN
ejpam-6569	104	6	of	of	ADP
ejpam-6569	104	7	an	an	DET
ejpam-6569	104	8	ideal	ideal	ADJ
ejpam-6569	104	9	topological	topological	ADJ
ejpam-6569	104	10	space	space	NOUN
ejpam-6569	104	11	(	(	PUNCT
ejpam-6569	104	12	x	x	X
ejpam-6569	104	13	,	,	PUNCT
ejpam-6569	104	14	τ	τ	PROPN
ejpam-6569	104	15	,	,	PUNCT
ejpam-6569	104	16	i	i	NOUN
ejpam-6569	104	17	)	)	PUNCT
ejpam-6569	104	18	.	.	PUNCT
ejpam-6569	105	1	the	the	DET
ejpam-6569	105	2	semi	semi	ADJ
ejpam-6569	105	3	-	-	ADJ
ejpam-6569	105	4	θ(⋆)-closure	θ(⋆)-closure	ADJ
ejpam-6569	105	5	[	[	X
ejpam-6569	105	6	30	30	NUM
ejpam-6569	105	7	]	]	PUNCT
ejpam-6569	105	8	of	of	ADP
ejpam-6569	105	9	a	a	DET
ejpam-6569	105	10	,	,	PUNCT
ejpam-6569	105	11	⋆θscl(a	⋆θscl(a	PROPN
ejpam-6569	105	12	)	)	PUNCT
ejpam-6569	105	13	and	and	CCONJ
ejpam-6569	105	14	the	the	DET
ejpam-6569	105	15	semi	semi	ADJ
ejpam-6569	105	16	-	-	ADJ
ejpam-6569	105	17	θ(⋆)-interior	θ(⋆)-interior	ADJ
ejpam-6569	105	18	[	[	X
ejpam-6569	105	19	30	30	NUM
ejpam-6569	105	20	]	]	PUNCT
ejpam-6569	105	21	of	of	ADP
ejpam-6569	105	22	a	a	DET
ejpam-6569	105	23	,	,	PUNCT
ejpam-6569	105	24	⋆θsint(a	⋆θsint(a	PROPN
ejpam-6569	105	25	)	)	PUNCT
ejpam-6569	105	26	are	be	AUX
ejpam-6569	105	27	defined	define	VERB
ejpam-6569	105	28	as	as	SCONJ
ejpam-6569	105	29	follows	follow	VERB
ejpam-6569	105	30	:	:	PUNCT
ejpam-6569	105	31	⋆θ	⋆θ	PROPN
ejpam-6569	105	32	scl(a	scl(a	X
ejpam-6569	105	33	)	)	PUNCT
ejpam-6569	105	34	=	=	PRON
ejpam-6569	106	1	{	{	PUNCT
ejpam-6569	106	2	x	x	PUNCT
ejpam-6569	106	3	∈	∈	NOUN
ejpam-6569	106	4	x	x	PUNCT
ejpam-6569	106	5	|	|	ADV
ejpam-6569	106	6	a	a	DET
ejpam-6569	106	7	∩	∩	NOUN
ejpam-6569	106	8	scl⋆(u	scl⋆(u	X
ejpam-6569	106	9	)	)	PUNCT
ejpam-6569	106	10	̸=	̸=	NOUN
ejpam-6569	106	11	∅	∅	NOUN
ejpam-6569	106	12	for	for	ADP
ejpam-6569	106	13	every	every	DET
ejpam-6569	106	14	u	u	PROPN
ejpam-6569	106	15	∈	∈	PROPN
ejpam-6569	106	16	si	si	X
ejpam-6569	106	17	⋆o(x	⋆o(x	PROPN
ejpam-6569	106	18	,	,	PUNCT
ejpam-6569	106	19	x	x	NOUN
ejpam-6569	106	20	)	)	PUNCT
ejpam-6569	106	21	}	}	PUNCT
ejpam-6569	106	22	,	,	PUNCT
ejpam-6569	106	23	⋆θ	⋆θ	PROPN
ejpam-6569	106	24	sint(a	sint(a	NOUN
ejpam-6569	106	25	)	)	PUNCT
ejpam-6569	106	26	=	=	SYM
ejpam-6569	106	27	{	{	PUNCT
ejpam-6569	106	28	x	x	PUNCT
ejpam-6569	106	29	∈	∈	PROPN
ejpam-6569	106	30	x	x	X
ejpam-6569	106	31	|	|	ADV
ejpam-6569	106	32	scl⋆(u	scl⋆(u	ADV
ejpam-6569	106	33	)	)	PUNCT
ejpam-6569	107	1	⊆	⊆	NUM
ejpam-6569	107	2	a	a	PRON
ejpam-6569	107	3	for	for	ADP
ejpam-6569	107	4	some	some	DET
ejpam-6569	107	5	u	u	NOUN
ejpam-6569	107	6	∈	∈	NOUN
ejpam-6569	107	7	si	si	X
ejpam-6569	107	8	⋆o(x	⋆o(x	PROPN
ejpam-6569	107	9	,	,	PUNCT
ejpam-6569	107	10	x	x	NOUN
ejpam-6569	107	11	)	)	PUNCT
ejpam-6569	107	12	}	}	PUNCT
ejpam-6569	107	13	,	,	PUNCT
ejpam-6569	107	14	where	where	SCONJ
ejpam-6569	107	15	si	si	PROPN
ejpam-6569	107	16	⋆o(x	⋆o(x	PROPN
ejpam-6569	107	17	,	,	PUNCT
ejpam-6569	107	18	x	x	X
ejpam-6569	107	19	)	)	PUNCT
ejpam-6569	107	20	=	=	SYM
ejpam-6569	107	21	{	{	PUNCT
ejpam-6569	107	22	u	u	NOUN
ejpam-6569	107	23	|	|	NOUN
ejpam-6569	107	24	x	x	SYM
ejpam-6569	107	25	∈	∈	PROPN
ejpam-6569	107	26	u	u	NOUN
ejpam-6569	107	27	and	and	CCONJ
ejpam-6569	107	28	u	u	PROPN
ejpam-6569	107	29	∈	∈	PROPN
ejpam-6569	107	30	si	si	X
ejpam-6569	107	31	⋆o(x	⋆o(x	NUM
ejpam-6569	107	32	)	)	PUNCT
ejpam-6569	107	33	}	}	PUNCT
ejpam-6569	107	34	.	.	PUNCT
ejpam-6569	108	1	by	by	ADP
ejpam-6569	108	2	a	a	DET
ejpam-6569	108	3	multifunction	multifunction	NOUN
ejpam-6569	108	4	f	f	NOUN
ejpam-6569	108	5	:	:	PUNCT
ejpam-6569	108	6	x	x	X
ejpam-6569	108	7	→	→	SYM
ejpam-6569	108	8	y	y	PROPN
ejpam-6569	108	9	,	,	PUNCT
ejpam-6569	108	10	we	we	PRON
ejpam-6569	108	11	mean	mean	VERB
ejpam-6569	108	12	a	a	DET
ejpam-6569	108	13	point	point	NOUN
ejpam-6569	108	14	-	-	PUNCT
ejpam-6569	108	15	to	to	ADP
ejpam-6569	108	16	-	-	PUNCT
ejpam-6569	108	17	set	set	VERB
ejpam-6569	108	18	correspondence	correspondence	NOUN
ejpam-6569	108	19	from	from	ADP
ejpam-6569	108	20	x	x	PUNCT
ejpam-6569	108	21	into	into	ADP
ejpam-6569	108	22	y	y	PROPN
ejpam-6569	108	23	,	,	PUNCT
ejpam-6569	108	24	and	and	CCONJ
ejpam-6569	108	25	always	always	ADV
ejpam-6569	108	26	assume	assume	VERB
ejpam-6569	108	27	that	that	SCONJ
ejpam-6569	109	1	f	f	PROPN
ejpam-6569	109	2	(	(	PUNCT
ejpam-6569	109	3	x	x	X
ejpam-6569	109	4	)	)	PUNCT
ejpam-6569	109	5	̸=	̸=	NOUN
ejpam-6569	109	6	∅	∅	NOUN
ejpam-6569	109	7	for	for	ADP
ejpam-6569	109	8	all	all	PRON
ejpam-6569	109	9	x	x	SYM
ejpam-6569	109	10	∈	∈	ADJ
ejpam-6569	109	11	x.	x.	NOUN
ejpam-6569	109	12	for	for	ADP
ejpam-6569	109	13	a	a	DET
ejpam-6569	109	14	multifunction	multifunction	NOUN
ejpam-6569	109	15	f	f	NOUN
ejpam-6569	109	16	:	:	PUNCT
ejpam-6569	109	17	x	x	X
ejpam-6569	109	18	→	→	SYM
ejpam-6569	109	19	y	y	PROPN
ejpam-6569	109	20	,	,	PUNCT
ejpam-6569	109	21	we	we	PRON
ejpam-6569	109	22	shall	shall	AUX
ejpam-6569	109	23	denote	denote	VERB
ejpam-6569	109	24	the	the	DET
ejpam-6569	109	25	upper	upper	ADJ
ejpam-6569	109	26	and	and	CCONJ
ejpam-6569	109	27	lower	low	ADJ
ejpam-6569	109	28	inverse	inverse	NOUN
ejpam-6569	109	29	of	of	ADP
ejpam-6569	109	30	a	a	DET
ejpam-6569	109	31	set	set	NOUN
ejpam-6569	109	32	b	b	PROPN
ejpam-6569	109	33	of	of	ADP
ejpam-6569	109	34	y	y	PROPN
ejpam-6569	109	35	by	by	ADP
ejpam-6569	109	36	f+(b	f+(b	NOUN
ejpam-6569	109	37	)	)	PUNCT
ejpam-6569	109	38	and	and	CCONJ
ejpam-6569	109	39	f−(b	f−(b	NOUN
ejpam-6569	109	40	)	)	PUNCT
ejpam-6569	109	41	,	,	PUNCT
ejpam-6569	109	42	respectively	respectively	ADV
ejpam-6569	109	43	,	,	PUNCT
ejpam-6569	109	44	that	that	ADV
ejpam-6569	109	45	is	is	ADV
ejpam-6569	109	46	,	,	PUNCT
ejpam-6569	109	47	f+(b	f+(b	NOUN
ejpam-6569	109	48	)	)	PUNCT
ejpam-6569	109	49	=	=	PRON
ejpam-6569	110	1	{	{	PUNCT
ejpam-6569	110	2	x	x	PUNCT
ejpam-6569	110	3	∈	∈	PROPN
ejpam-6569	110	4	x	x	INTJ
ejpam-6569	111	1	|	|	NOUN
ejpam-6569	111	2	f	f	X
ejpam-6569	111	3	(	(	PUNCT
ejpam-6569	111	4	x	x	NOUN
ejpam-6569	111	5	)	)	PUNCT
ejpam-6569	111	6	⊆	⊆	NUM
ejpam-6569	111	7	b	b	NOUN
ejpam-6569	111	8	}	}	PUNCT
ejpam-6569	111	9	and	and	CCONJ
ejpam-6569	111	10	f−(b	f−(b	PROPN
ejpam-6569	111	11	)	)	PUNCT
ejpam-6569	111	12	=	=	PRON
ejpam-6569	112	1	{	{	PUNCT
ejpam-6569	112	2	x	x	PUNCT
ejpam-6569	112	3	∈	∈	PROPN
ejpam-6569	112	4	x	x	INTJ
ejpam-6569	113	1	|	|	NOUN
ejpam-6569	113	2	f	f	X
ejpam-6569	113	3	(	(	PUNCT
ejpam-6569	113	4	x	x	NOUN
ejpam-6569	113	5	)	)	PUNCT
ejpam-6569	113	6	∩b	∩b	NOUN
ejpam-6569	113	7	̸=	̸=	PROPN
ejpam-6569	113	8	∅	∅	NOUN
ejpam-6569	113	9	}	}	PUNCT
ejpam-6569	113	10	.	.	PUNCT
ejpam-6569	114	1	n.	n.	PROPN
ejpam-6569	114	2	srisarakham	srisarakham	PROPN
ejpam-6569	114	3	,	,	PUNCT
ejpam-6569	114	4	a.	a.	PROPN
ejpam-6569	114	5	sama	sama	PROPN
ejpam-6569	114	6	-	-	PUNCT
ejpam-6569	114	7	ae	ae	PROPN
ejpam-6569	114	8	,	,	PUNCT
ejpam-6569	114	9	c.	c.	PROPN
ejpam-6569	114	10	boonpok	boonpok	PROPN
ejpam-6569	114	11	/	/	SYM
ejpam-6569	114	12	eur	eur	PROPN
ejpam-6569	114	13	.	.	PUNCT
ejpam-6569	115	1	j.	j.	PROPN
ejpam-6569	115	2	pure	pure	PROPN
ejpam-6569	115	3	appl	appl	PROPN
ejpam-6569	115	4	.	.	PROPN
ejpam-6569	115	5	math	math	PROPN
ejpam-6569	115	6	,	,	PUNCT
ejpam-6569	115	7	18	18	NUM
ejpam-6569	115	8	(	(	PUNCT
ejpam-6569	115	9	3	3	NUM
ejpam-6569	115	10	)	)	PUNCT
ejpam-6569	115	11	(	(	PUNCT
ejpam-6569	115	12	2025	2025	NUM
ejpam-6569	115	13	)	)	PUNCT
ejpam-6569	115	14	,	,	PUNCT
ejpam-6569	115	15	6569	6569	NUM
ejpam-6569	115	16	4	4	NUM
ejpam-6569	115	17	of	of	ADP
ejpam-6569	115	18	13	13	NUM
ejpam-6569	115	19	3	3	NUM
ejpam-6569	115	20	.	.	PUNCT
ejpam-6569	115	21	upper	upper	ADJ
ejpam-6569	115	22	and	and	CCONJ
ejpam-6569	115	23	lower	low	ADJ
ejpam-6569	115	24	quasi	quasi	NOUN
ejpam-6569	115	25	θτ	θτ	ADP
ejpam-6569	115	26	⋆(σ1	⋆(σ1	PROPN
ejpam-6569	115	27	,	,	PUNCT
ejpam-6569	115	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	115	29	multifunctions	multifunction	NOUN
ejpam-6569	115	30	in	in	ADP
ejpam-6569	115	31	this	this	DET
ejpam-6569	115	32	section	section	NOUN
ejpam-6569	115	33	,	,	PUNCT
ejpam-6569	115	34	we	we	PRON
ejpam-6569	115	35	introduce	introduce	VERB
ejpam-6569	115	36	the	the	DET
ejpam-6569	115	37	notions	notion	NOUN
ejpam-6569	115	38	of	of	ADP
ejpam-6569	115	39	upper	upper	ADJ
ejpam-6569	115	40	quasi	quasi	NOUN
ejpam-6569	115	41	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	115	42	,	,	PUNCT
ejpam-6569	115	43	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	115	44	multifunctions	multifunction	NOUN
ejpam-6569	115	45	and	and	CCONJ
ejpam-6569	115	46	lower	low	ADJ
ejpam-6569	115	47	quasi	quasi	NOUN
ejpam-6569	115	48	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	115	49	,	,	PUNCT
ejpam-6569	115	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	115	51	multifunctions	multifunction	NOUN
ejpam-6569	115	52	.	.	PUNCT
ejpam-6569	116	1	moreover	moreover	ADV
ejpam-6569	116	2	,	,	PUNCT
ejpam-6569	116	3	several	several	ADJ
ejpam-6569	116	4	characterizations	characterization	NOUN
ejpam-6569	116	5	of	of	ADP
ejpam-6569	116	6	upper	upper	ADJ
ejpam-6569	116	7	quasi	quasi	NOUN
ejpam-6569	116	8	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	116	9	,	,	PUNCT
ejpam-6569	116	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	116	11	multifunctions	multifunction	NOUN
ejpam-6569	116	12	and	and	CCONJ
ejpam-6569	116	13	lower	low	ADJ
ejpam-6569	116	14	quasi	quasi	NOUN
ejpam-6569	116	15	θτ⋆(σ1	θτ⋆(σ1	PROPN
ejpam-6569	116	16	,	,	PUNCT
ejpam-6569	116	17	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	116	18	multifunctions	multifunction	NOUN
ejpam-6569	116	19	are	be	AUX
ejpam-6569	116	20	discussed	discuss	VERB
ejpam-6569	116	21	.	.	PUNCT
ejpam-6569	117	1	definition	definition	NOUN
ejpam-6569	117	2	1	1	NUM
ejpam-6569	117	3	.	.	PUNCT
ejpam-6569	118	1	a	a	DET
ejpam-6569	118	2	multifunction	multifunction	NOUN
ejpam-6569	118	3	f	f	NOUN
ejpam-6569	118	4	:	:	PUNCT
ejpam-6569	118	5	(	(	PUNCT
ejpam-6569	118	6	x	x	X
ejpam-6569	118	7	,	,	PUNCT
ejpam-6569	118	8	τ	τ	PROPN
ejpam-6569	118	9	,	,	PUNCT
ejpam-6569	118	10	i	i	NOUN
ejpam-6569	118	11	)	)	PUNCT
ejpam-6569	118	12	→	→	PUNCT
ejpam-6569	118	13	(	(	PUNCT
ejpam-6569	118	14	y	y	PROPN
ejpam-6569	118	15	,	,	PUNCT
ejpam-6569	118	16	σ1	σ1	PROPN
ejpam-6569	118	17	,	,	PUNCT
ejpam-6569	118	18	σ2	σ2	PROPN
ejpam-6569	118	19	)	)	PUNCT
ejpam-6569	118	20	is	be	AUX
ejpam-6569	118	21	said	say	VERB
ejpam-6569	118	22	to	to	PART
ejpam-6569	118	23	be	be	AUX
ejpam-6569	118	24	upper	upper	ADJ
ejpam-6569	118	25	quasi	quasi	NOUN
ejpam-6569	118	26	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	118	27	,	,	PUNCT
ejpam-6569	118	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	118	29	if	if	SCONJ
ejpam-6569	118	30	for	for	SCONJ
ejpam-6569	118	31	each	each	DET
ejpam-6569	118	32	x	x	SYM
ejpam-6569	118	33	∈	∈	PROPN
ejpam-6569	118	34	x	x	X
ejpam-6569	118	35	and	and	CCONJ
ejpam-6569	118	36	each	each	DET
ejpam-6569	118	37	σ1σ2	σ1σ2	VERB
ejpam-6569	118	38	-	-	ADJ
ejpam-6569	118	39	open	open	ADJ
ejpam-6569	118	40	set	set	NOUN
ejpam-6569	118	41	v	v	NOUN
ejpam-6569	118	42	of	of	ADP
ejpam-6569	118	43	y	y	PROPN
ejpam-6569	118	44	containing	contain	VERB
ejpam-6569	118	45	f	f	PROPN
ejpam-6569	118	46	(	(	PUNCT
ejpam-6569	118	47	x	x	NOUN
ejpam-6569	118	48	)	)	PUNCT
ejpam-6569	118	49	,	,	PUNCT
ejpam-6569	118	50	there	there	PRON
ejpam-6569	118	51	exists	exist	VERB
ejpam-6569	118	52	a	a	DET
ejpam-6569	118	53	semi	semi	NOUN
ejpam-6569	118	54	-	-	ADJ
ejpam-6569	118	55	i	i	PRON
ejpam-6569	118	56	⋆-open	⋆-open	VERB
ejpam-6569	118	57	set	set	VERB
ejpam-6569	118	58	u	u	NOUN
ejpam-6569	118	59	of	of	ADP
ejpam-6569	118	60	x	x	PUNCT
ejpam-6569	118	61	containing	contain	VERB
ejpam-6569	118	62	x	x	PUNCT
ejpam-6569	119	1	such	such	ADJ
ejpam-6569	119	2	that	that	SCONJ
ejpam-6569	119	3	f	f	PROPN
ejpam-6569	119	4	(	(	PUNCT
ejpam-6569	119	5	scl⋆(u	scl⋆(u	NUM
ejpam-6569	119	6	)	)	PUNCT
ejpam-6569	119	7	)	)	PUNCT
ejpam-6569	119	8	⊆	⊆	X
ejpam-6569	119	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	119	10	-	-	NUM
ejpam-6569	119	11	cl(v	cl(v	NOUN
ejpam-6569	119	12	)	)	PUNCT
ejpam-6569	119	13	.	.	PUNCT
ejpam-6569	120	1	theorem	theorem	NOUN
ejpam-6569	120	2	1	1	NUM
ejpam-6569	120	3	.	.	X
ejpam-6569	120	4	for	for	ADP
ejpam-6569	120	5	a	a	DET
ejpam-6569	120	6	multifunction	multifunction	NOUN
ejpam-6569	121	1	f	f	NOUN
ejpam-6569	121	2	:	:	PUNCT
ejpam-6569	121	3	(	(	PUNCT
ejpam-6569	121	4	x	x	X
ejpam-6569	121	5	,	,	PUNCT
ejpam-6569	121	6	τ	τ	PROPN
ejpam-6569	121	7	,	,	PUNCT
ejpam-6569	121	8	i	i	NOUN
ejpam-6569	121	9	)	)	PUNCT
ejpam-6569	121	10	→	→	PUNCT
ejpam-6569	121	11	(	(	PUNCT
ejpam-6569	121	12	y	y	PROPN
ejpam-6569	121	13	,	,	PUNCT
ejpam-6569	121	14	σ1	σ1	PROPN
ejpam-6569	121	15	,	,	PUNCT
ejpam-6569	121	16	σ2	σ2	NOUN
ejpam-6569	121	17	)	)	PUNCT
ejpam-6569	121	18	,	,	PUNCT
ejpam-6569	121	19	the	the	DET
ejpam-6569	121	20	following	follow	VERB
ejpam-6569	121	21	properties	property	NOUN
ejpam-6569	121	22	are	be	AUX
ejpam-6569	121	23	equivalent	equivalent	ADJ
ejpam-6569	121	24	:	:	PUNCT
ejpam-6569	121	25	(	(	PUNCT
ejpam-6569	121	26	1	1	X
ejpam-6569	121	27	)	)	PUNCT
ejpam-6569	121	28	f	f	PROPN
ejpam-6569	121	29	is	be	AUX
ejpam-6569	121	30	upper	upper	ADJ
ejpam-6569	121	31	quasi	quasi	NOUN
ejpam-6569	121	32	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	121	33	,	,	PUNCT
ejpam-6569	121	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	121	35	;	;	PUNCT
ejpam-6569	121	36	(	(	PUNCT
ejpam-6569	121	37	2	2	X
ejpam-6569	121	38	)	)	PUNCT
ejpam-6569	121	39	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	121	40	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	121	41	-	-	PUNCT
ejpam-6569	121	42	int((σ1	int((σ1	ADJ
ejpam-6569	121	43	,	,	PUNCT
ejpam-6569	121	44	σ2)θ	σ2)θ	ADJ
ejpam-6569	121	45	-	-	PUNCT
ejpam-6569	121	46	cl(b	cl(b	NOUN
ejpam-6569	121	47	)	)	PUNCT
ejpam-6569	121	48	)	)	PUNCT
ejpam-6569	121	49	)	)	PUNCT
ejpam-6569	121	50	)	)	PUNCT
ejpam-6569	122	1	⊆	⊆	NUM
ejpam-6569	122	2	f−((σ1	f−((σ1	NOUN
ejpam-6569	122	3	,	,	PUNCT
ejpam-6569	122	4	σ2)θ	σ2)θ	ADJ
ejpam-6569	122	5	-	-	PUNCT
ejpam-6569	122	6	cl(b	cl(b	NOUN
ejpam-6569	122	7	)	)	PUNCT
ejpam-6569	122	8	)	)	PUNCT
ejpam-6569	122	9	for	for	ADP
ejpam-6569	122	10	every	every	DET
ejpam-6569	122	11	subset	subset	NOUN
ejpam-6569	122	12	b	b	PROPN
ejpam-6569	122	13	of	of	ADP
ejpam-6569	122	14	y	y	PROPN
ejpam-6569	122	15	;	;	PUNCT
ejpam-6569	122	16	(	(	PUNCT
ejpam-6569	122	17	3	3	X
ejpam-6569	122	18	)	)	PUNCT
ejpam-6569	122	19	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	122	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	122	21	-	-	PUNCT
ejpam-6569	122	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	122	23	-	-	PUNCT
ejpam-6569	122	24	cl(v	cl(v	NOUN
ejpam-6569	122	25	)	)	PUNCT
ejpam-6569	122	26	)	)	PUNCT
ejpam-6569	122	27	)	)	PUNCT
ejpam-6569	122	28	)	)	PUNCT
ejpam-6569	122	29	⊆	⊆	X
ejpam-6569	122	30	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	122	31	-	-	PUNCT
ejpam-6569	122	32	cl(v	cl(v	NOUN
ejpam-6569	122	33	)	)	PUNCT
ejpam-6569	122	34	)	)	PUNCT
ejpam-6569	122	35	for	for	ADP
ejpam-6569	122	36	every	every	DET
ejpam-6569	122	37	σ1σ2	σ1σ2	NOUN
ejpam-6569	122	38	-	-	ADJ
ejpam-6569	122	39	open	open	ADJ
ejpam-6569	122	40	set	set	NOUN
ejpam-6569	122	41	v	v	NOUN
ejpam-6569	122	42	of	of	ADP
ejpam-6569	122	43	y	y	PROPN
ejpam-6569	122	44	;	;	PUNCT
ejpam-6569	122	45	(	(	PUNCT
ejpam-6569	122	46	4	4	X
ejpam-6569	122	47	)	)	PUNCT
ejpam-6569	122	48	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	122	49	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	122	50	-	-	PUNCT
ejpam-6569	122	51	int(k	int(k	NUM
ejpam-6569	122	52	)	)	PUNCT
ejpam-6569	122	53	)	)	PUNCT
ejpam-6569	122	54	)	)	PUNCT
ejpam-6569	122	55	⊆	⊆	X
ejpam-6569	122	56	f−(k	f−(k	PROPN
ejpam-6569	122	57	)	)	PUNCT
ejpam-6569	122	58	for	for	ADP
ejpam-6569	122	59	every	every	DET
ejpam-6569	122	60	(	(	PUNCT
ejpam-6569	122	61	σ1	σ1	PROPN
ejpam-6569	122	62	,	,	PUNCT
ejpam-6569	122	63	σ2)r	σ2)r	NOUN
ejpam-6569	122	64	-	-	PUNCT
ejpam-6569	122	65	closed	close	VERB
ejpam-6569	122	66	set	set	ADJ
ejpam-6569	122	67	k	k	PROPN
ejpam-6569	122	68	of	of	ADP
ejpam-6569	122	69	y	y	PROPN
ejpam-6569	122	70	;	;	PUNCT
ejpam-6569	122	71	(	(	PUNCT
ejpam-6569	122	72	5	5	NUM
ejpam-6569	122	73	)	)	PUNCT
ejpam-6569	122	74	f+(v	f+(v	NOUN
ejpam-6569	122	75	)	)	PUNCT
ejpam-6569	123	1	⊆	⊆	NUM
ejpam-6569	123	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	123	3	+	+	ADJ
ejpam-6569	123	4	(	(	PUNCT
ejpam-6569	123	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	123	6	-	-	NUM
ejpam-6569	123	7	cl(v	cl(v	NOUN
ejpam-6569	123	8	)	)	PUNCT
ejpam-6569	123	9	)	)	PUNCT
ejpam-6569	123	10	)	)	PUNCT
ejpam-6569	123	11	for	for	ADP
ejpam-6569	123	12	every	every	DET
ejpam-6569	123	13	σ1σ2	σ1σ2	NOUN
ejpam-6569	123	14	-	-	ADJ
ejpam-6569	123	15	open	open	ADJ
ejpam-6569	123	16	set	set	NOUN
ejpam-6569	123	17	v	v	NOUN
ejpam-6569	123	18	of	of	ADP
ejpam-6569	123	19	y	y	PROPN
ejpam-6569	123	20	;	;	PUNCT
ejpam-6569	123	21	(	(	PUNCT
ejpam-6569	123	22	6	6	X
ejpam-6569	123	23	)	)	PUNCT
ejpam-6569	123	24	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	123	25	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	123	26	-	-	PUNCT
ejpam-6569	123	27	int(k	int(k	NUM
ejpam-6569	123	28	)	)	PUNCT
ejpam-6569	123	29	)	)	PUNCT
ejpam-6569	123	30	)	)	PUNCT
ejpam-6569	124	1	⊆	⊆	X
ejpam-6569	124	2	f−(k	f−(k	PROPN
ejpam-6569	124	3	)	)	PUNCT
ejpam-6569	124	4	for	for	ADP
ejpam-6569	124	5	every	every	DET
ejpam-6569	124	6	σ1σ2	σ1σ2	NUM
ejpam-6569	124	7	-	-	PUNCT
ejpam-6569	124	8	closed	closed	ADJ
ejpam-6569	124	9	set	set	NOUN
ejpam-6569	124	10	k	k	PROPN
ejpam-6569	124	11	of	of	ADP
ejpam-6569	124	12	y	y	PROPN
ejpam-6569	124	13	;	;	PUNCT
ejpam-6569	124	14	(	(	PUNCT
ejpam-6569	124	15	7	7	X
ejpam-6569	124	16	)	)	PUNCT
ejpam-6569	124	17	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	124	18	−(v	−(v	NOUN
ejpam-6569	124	19	)	)	PUNCT
ejpam-6569	124	20	)	)	PUNCT
ejpam-6569	125	1	⊆	⊆	X
ejpam-6569	125	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	125	3	-	-	PUNCT
ejpam-6569	125	4	cl(v	cl(v	NOUN
ejpam-6569	125	5	)	)	PUNCT
ejpam-6569	125	6	)	)	PUNCT
ejpam-6569	125	7	for	for	ADP
ejpam-6569	125	8	every	every	DET
ejpam-6569	125	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	125	10	-	-	ADJ
ejpam-6569	125	11	open	open	ADJ
ejpam-6569	125	12	set	set	NOUN
ejpam-6569	125	13	v	v	NOUN
ejpam-6569	125	14	of	of	ADP
ejpam-6569	125	15	y	y	PROPN
ejpam-6569	125	16	.	.	PUNCT
ejpam-6569	126	1	proof	proof	NOUN
ejpam-6569	126	2	.	.	PUNCT
ejpam-6569	127	1	(	(	PUNCT
ejpam-6569	127	2	1	1	X
ejpam-6569	127	3	)	)	PUNCT
ejpam-6569	127	4	⇒	⇒	NOUN
ejpam-6569	127	5	(	(	PUNCT
ejpam-6569	127	6	2	2	NUM
ejpam-6569	127	7	):	):	PUNCT
ejpam-6569	127	8	let	let	VERB
ejpam-6569	127	9	b	b	X
ejpam-6569	127	10	be	be	AUX
ejpam-6569	127	11	any	any	DET
ejpam-6569	127	12	subset	subset	NOUN
ejpam-6569	127	13	of	of	ADP
ejpam-6569	127	14	y	y	PROPN
ejpam-6569	127	15	.	.	PUNCT
ejpam-6569	127	16	suppose	suppose	VERB
ejpam-6569	127	17	that	that	SCONJ
ejpam-6569	127	18	x	x	PRON
ejpam-6569	127	19	̸∈	̸∈	PROPN
ejpam-6569	127	20	f−((σ1	f−((σ1	VERB
ejpam-6569	127	21	,	,	PUNCT
ejpam-6569	127	22	σ2)θ	σ2)θ	ADJ
ejpam-6569	127	23	-	-	PUNCT
ejpam-6569	127	24	cl(b	cl(b	NOUN
ejpam-6569	127	25	)	)	PUNCT
ejpam-6569	127	26	)	)	PUNCT
ejpam-6569	127	27	.	.	PUNCT
ejpam-6569	128	1	then	then	ADV
ejpam-6569	128	2	,	,	PUNCT
ejpam-6569	128	3	x	x	X
ejpam-6569	128	4	∈	∈	NOUN
ejpam-6569	128	5	x−f−((σ1	x−f−((σ1	PUNCT
ejpam-6569	128	6	,	,	PUNCT
ejpam-6569	128	7	σ2)θ	σ2)θ	ADJ
ejpam-6569	128	8	-	-	PUNCT
ejpam-6569	128	9	cl(b	cl(b	NOUN
ejpam-6569	128	10	)	)	PUNCT
ejpam-6569	128	11	)	)	PUNCT
ejpam-6569	128	12	and	and	CCONJ
ejpam-6569	128	13	f	f	PROPN
ejpam-6569	128	14	(	(	PUNCT
ejpam-6569	128	15	x	x	X
ejpam-6569	128	16	)	)	PUNCT
ejpam-6569	128	17	⊆	⊆	NUM
ejpam-6569	128	18	y	y	PROPN
ejpam-6569	128	19	−(σ1	−(σ1	ADJ
ejpam-6569	128	20	,	,	PUNCT
ejpam-6569	128	21	σ2)θ	σ2)θ	ADJ
ejpam-6569	128	22	-	-	PUNCT
ejpam-6569	128	23	cl(b	cl(b	NOUN
ejpam-6569	128	24	)	)	PUNCT
ejpam-6569	128	25	.	.	PUNCT
ejpam-6569	129	1	since	since	SCONJ
ejpam-6569	129	2	(	(	PUNCT
ejpam-6569	129	3	σ1	σ1	PROPN
ejpam-6569	129	4	,	,	PUNCT
ejpam-6569	129	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	129	6	-	-	PUNCT
ejpam-6569	129	7	cl(b	cl(b	NOUN
ejpam-6569	129	8	)	)	PUNCT
ejpam-6569	129	9	is	be	AUX
ejpam-6569	129	10	σ1σ2	σ1σ2	NOUN
ejpam-6569	129	11	-	-	ADJ
ejpam-6569	129	12	closed	closed	ADJ
ejpam-6569	129	13	in	in	ADP
ejpam-6569	129	14	y	y	PROPN
ejpam-6569	129	15	,	,	PUNCT
ejpam-6569	129	16	by	by	ADP
ejpam-6569	129	17	(	(	PUNCT
ejpam-6569	129	18	1	1	X
ejpam-6569	129	19	)	)	PUNCT
ejpam-6569	129	20	there	there	PRON
ejpam-6569	129	21	exists	exist	VERB
ejpam-6569	129	22	a	a	DET
ejpam-6569	129	23	semi	semi	NOUN
ejpam-6569	129	24	-	-	ADJ
ejpam-6569	129	25	i	i	PRON
ejpam-6569	129	26	⋆-open	⋆-open	VERB
ejpam-6569	129	27	set	set	VERB
ejpam-6569	129	28	u	u	NOUN
ejpam-6569	129	29	of	of	ADP
ejpam-6569	129	30	x	x	PUNCT
ejpam-6569	129	31	containing	contain	VERB
ejpam-6569	129	32	x	x	PUNCT
ejpam-6569	129	33	such	such	ADJ
ejpam-6569	129	34	that	that	SCONJ
ejpam-6569	129	35	f	f	PROPN
ejpam-6569	129	36	(	(	PUNCT
ejpam-6569	129	37	scl⋆(u	scl⋆(u	NUM
ejpam-6569	129	38	)	)	PUNCT
ejpam-6569	129	39	)	)	PUNCT
ejpam-6569	129	40	⊆	⊆	X
ejpam-6569	129	41	σ1σ2	σ1σ2	NUM
ejpam-6569	129	42	-	-	PUNCT
ejpam-6569	129	43	cl(y	cl(y	NOUN
ejpam-6569	129	44	−	−	PROPN
ejpam-6569	129	45	(	(	PUNCT
ejpam-6569	129	46	σ1	σ1	PROPN
ejpam-6569	129	47	,	,	PUNCT
ejpam-6569	129	48	σ2)θ	σ2)θ	NOUN
ejpam-6569	129	49	-	-	PUNCT
ejpam-6569	129	50	cl(b	cl(b	NOUN
ejpam-6569	129	51	)	)	PUNCT
ejpam-6569	129	52	)	)	PUNCT
ejpam-6569	130	1	=	=	PUNCT
ejpam-6569	130	2	y	y	PROPN
ejpam-6569	130	3	−	−	ADP
ejpam-6569	130	4	σ1σ2	σ1σ2	NOUN
ejpam-6569	130	5	-	-	PUNCT
ejpam-6569	130	6	int((σ1	int((σ1	ADJ
ejpam-6569	130	7	,	,	PUNCT
ejpam-6569	130	8	σ2)θ	σ2)θ	ADJ
ejpam-6569	130	9	-	-	PUNCT
ejpam-6569	130	10	cl(b	cl(b	NOUN
ejpam-6569	130	11	)	)	PUNCT
ejpam-6569	130	12	)	)	PUNCT
ejpam-6569	130	13	.	.	PUNCT
ejpam-6569	131	1	thus	thus	ADV
ejpam-6569	131	2	,	,	PUNCT
ejpam-6569	131	3	we	we	PRON
ejpam-6569	131	4	have	have	VERB
ejpam-6569	131	5	f	f	PROPN
ejpam-6569	131	6	(	(	PUNCT
ejpam-6569	131	7	scl⋆(u	scl⋆(u	NUM
ejpam-6569	131	8	)	)	PUNCT
ejpam-6569	131	9	)	)	PUNCT
ejpam-6569	132	1	∩	∩	NOUN
ejpam-6569	132	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	132	3	-	-	PUNCT
ejpam-6569	132	4	int((σ1	int((σ1	ADJ
ejpam-6569	132	5	,	,	PUNCT
ejpam-6569	132	6	σ2)θ	σ2)θ	ADJ
ejpam-6569	132	7	-	-	PUNCT
ejpam-6569	132	8	cl(b	cl(b	NOUN
ejpam-6569	132	9	)	)	PUNCT
ejpam-6569	132	10	)	)	PUNCT
ejpam-6569	133	1	=	=	NOUN
ejpam-6569	133	2	∅	∅	NOUN
ejpam-6569	133	3	and	and	CCONJ
ejpam-6569	133	4	scl⋆(u	scl⋆(u	NUM
ejpam-6569	133	5	)	)	PUNCT
ejpam-6569	133	6	∩	∩	NOUN
ejpam-6569	133	7	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	133	8	-	-	PUNCT
ejpam-6569	133	9	int((σ1	int((σ1	PROPN
ejpam-6569	133	10	,	,	PUNCT
ejpam-6569	133	11	σ2)θ	σ2)θ	ADJ
ejpam-6569	133	12	-	-	PUNCT
ejpam-6569	133	13	cl(b	cl(b	NOUN
ejpam-6569	133	14	)	)	PUNCT
ejpam-6569	133	15	)	)	PUNCT
ejpam-6569	133	16	)	)	PUNCT
ejpam-6569	134	1	=	=	PUNCT
ejpam-6569	134	2	∅.	∅.	ADP
ejpam-6569	134	3	this	this	PRON
ejpam-6569	134	4	shows	show	VERB
ejpam-6569	134	5	that	that	SCONJ
ejpam-6569	134	6	x	x	PROPN
ejpam-6569	134	7	̸∈	̸∈	PROPN
ejpam-6569	134	8	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	134	9	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	134	10	-	-	PUNCT
ejpam-6569	134	11	int((σ1	int((σ1	ADJ
ejpam-6569	134	12	,	,	PUNCT
ejpam-6569	134	13	σ2)θ	σ2)θ	ADJ
ejpam-6569	134	14	-	-	PUNCT
ejpam-6569	134	15	cl(b	cl(b	NOUN
ejpam-6569	134	16	)	)	PUNCT
ejpam-6569	134	17	)	)	PUNCT
ejpam-6569	134	18	)	)	PUNCT
ejpam-6569	134	19	)	)	PUNCT
ejpam-6569	134	20	.	.	PUNCT
ejpam-6569	135	1	thus	thus	ADV
ejpam-6569	135	2	,	,	PUNCT
ejpam-6569	135	3	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	135	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	135	5	-	-	PUNCT
ejpam-6569	135	6	int((σ1	int((σ1	ADJ
ejpam-6569	135	7	,	,	PUNCT
ejpam-6569	135	8	σ2)θ	σ2)θ	ADJ
ejpam-6569	135	9	-	-	PUNCT
ejpam-6569	135	10	cl(b	cl(b	NOUN
ejpam-6569	135	11	)	)	PUNCT
ejpam-6569	135	12	)	)	PUNCT
ejpam-6569	135	13	)	)	PUNCT
ejpam-6569	135	14	)	)	PUNCT
ejpam-6569	135	15	⊆	⊆	NUM
ejpam-6569	135	16	f−((σ1	f−((σ1	NOUN
ejpam-6569	135	17	,	,	PUNCT
ejpam-6569	135	18	σ2)θ	σ2)θ	ADJ
ejpam-6569	135	19	-	-	PUNCT
ejpam-6569	135	20	cl(b	cl(b	NOUN
ejpam-6569	135	21	)	)	PUNCT
ejpam-6569	135	22	)	)	PUNCT
ejpam-6569	135	23	.	.	PUNCT
ejpam-6569	136	1	(	(	PUNCT
ejpam-6569	136	2	2	2	X
ejpam-6569	136	3	)	)	PUNCT
ejpam-6569	136	4	⇒	⇒	NOUN
ejpam-6569	136	5	(	(	PUNCT
ejpam-6569	136	6	3	3	NUM
ejpam-6569	136	7	):	):	PUNCT
ejpam-6569	136	8	this	this	PRON
ejpam-6569	136	9	is	be	AUX
ejpam-6569	136	10	obvious	obvious	ADJ
ejpam-6569	136	11	since	since	SCONJ
ejpam-6569	136	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	136	13	-	-	NOUN
ejpam-6569	136	14	cl(v	cl(v	X
ejpam-6569	136	15	)	)	PUNCT
ejpam-6569	137	1	=	=	SYM
ejpam-6569	137	2	(	(	PUNCT
ejpam-6569	137	3	σ1	σ1	PROPN
ejpam-6569	137	4	,	,	PUNCT
ejpam-6569	137	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	137	6	-	-	PUNCT
ejpam-6569	137	7	cl(v	cl(v	NOUN
ejpam-6569	137	8	)	)	PUNCT
ejpam-6569	137	9	for	for	ADP
ejpam-6569	137	10	every	every	DET
ejpam-6569	137	11	σ1σ2	σ1σ2	NOUN
ejpam-6569	137	12	-	-	ADJ
ejpam-6569	137	13	open	open	ADJ
ejpam-6569	137	14	set	set	NOUN
ejpam-6569	137	15	v	v	NOUN
ejpam-6569	137	16	of	of	ADP
ejpam-6569	137	17	y	y	PROPN
ejpam-6569	137	18	.	.	PUNCT
ejpam-6569	138	1	(	(	PUNCT
ejpam-6569	138	2	3	3	X
ejpam-6569	138	3	)	)	PUNCT
ejpam-6569	138	4	⇒	⇒	NOUN
ejpam-6569	138	5	(	(	PUNCT
ejpam-6569	138	6	4	4	NUM
ejpam-6569	138	7	):	):	PUNCT
ejpam-6569	138	8	let	let	VERB
ejpam-6569	138	9	k	k	PRON
ejpam-6569	138	10	be	be	AUX
ejpam-6569	138	11	any	any	DET
ejpam-6569	138	12	(	(	PUNCT
ejpam-6569	138	13	σ1	σ1	NOUN
ejpam-6569	138	14	,	,	PUNCT
ejpam-6569	138	15	σ2)r	σ2)r	NOUN
ejpam-6569	138	16	-	-	PUNCT
ejpam-6569	138	17	closed	close	VERB
ejpam-6569	138	18	set	set	NOUN
ejpam-6569	138	19	of	of	ADP
ejpam-6569	138	20	y	y	PROPN
ejpam-6569	138	21	.	.	PUNCT
ejpam-6569	139	1	thus	thus	ADV
ejpam-6569	139	2	by	by	ADP
ejpam-6569	139	3	(	(	PUNCT
ejpam-6569	139	4	3	3	NUM
ejpam-6569	139	5	)	)	PUNCT
ejpam-6569	139	6	,	,	PUNCT
ejpam-6569	139	7	we	we	PRON
ejpam-6569	139	8	have	have	VERB
ejpam-6569	139	9	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	139	10	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	139	11	-	-	PUNCT
ejpam-6569	139	12	int(k	int(k	NUM
ejpam-6569	139	13	)	)	PUNCT
ejpam-6569	139	14	)	)	PUNCT
ejpam-6569	139	15	)	)	PUNCT
ejpam-6569	140	1	=	=	PUNCT
ejpam-6569	140	2	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	140	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	140	4	-	-	PUNCT
ejpam-6569	140	5	int(σ1σ2	int(σ1σ2	ADV
ejpam-6569	140	6	-	-	PUNCT
ejpam-6569	140	7	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6569	140	8	-	-	PUNCT
ejpam-6569	140	9	int(k	int(k	NOUN
ejpam-6569	140	10	)	)	PUNCT
ejpam-6569	140	11	)	)	PUNCT
ejpam-6569	140	12	)	)	PUNCT
ejpam-6569	140	13	)	)	PUNCT
ejpam-6569	140	14	)	)	PUNCT
ejpam-6569	141	1	⊆	⊆	X
ejpam-6569	141	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6569	141	3	-	-	PUNCT
ejpam-6569	141	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	141	5	-	-	PUNCT
ejpam-6569	141	6	int(k	int(k	NOUN
ejpam-6569	141	7	)	)	PUNCT
ejpam-6569	141	8	)	)	PUNCT
ejpam-6569	141	9	)	)	PUNCT
ejpam-6569	142	1	=	=	SYM
ejpam-6569	142	2	f−(k	f−(k	PROPN
ejpam-6569	142	3	)	)	PUNCT
ejpam-6569	142	4	.	.	PUNCT
ejpam-6569	143	1	n.	n.	PROPN
ejpam-6569	143	2	srisarakham	srisarakham	PROPN
ejpam-6569	143	3	,	,	PUNCT
ejpam-6569	143	4	a.	a.	PROPN
ejpam-6569	143	5	sama	sama	PROPN
ejpam-6569	143	6	-	-	PUNCT
ejpam-6569	143	7	ae	ae	PROPN
ejpam-6569	143	8	,	,	PUNCT
ejpam-6569	143	9	c.	c.	PROPN
ejpam-6569	143	10	boonpok	boonpok	PROPN
ejpam-6569	143	11	/	/	SYM
ejpam-6569	143	12	eur	eur	PROPN
ejpam-6569	143	13	.	.	PUNCT
ejpam-6569	144	1	j.	j.	PROPN
ejpam-6569	144	2	pure	pure	PROPN
ejpam-6569	144	3	appl	appl	PROPN
ejpam-6569	144	4	.	.	PROPN
ejpam-6569	144	5	math	math	PROPN
ejpam-6569	144	6	,	,	PUNCT
ejpam-6569	144	7	18	18	NUM
ejpam-6569	144	8	(	(	PUNCT
ejpam-6569	144	9	3	3	NUM
ejpam-6569	144	10	)	)	PUNCT
ejpam-6569	144	11	(	(	PUNCT
ejpam-6569	144	12	2025	2025	NUM
ejpam-6569	144	13	)	)	PUNCT
ejpam-6569	144	14	,	,	PUNCT
ejpam-6569	144	15	6569	6569	NUM
ejpam-6569	144	16	5	5	NUM
ejpam-6569	144	17	of	of	ADP
ejpam-6569	144	18	13	13	NUM
ejpam-6569	144	19	(	(	PUNCT
ejpam-6569	144	20	4	4	NUM
ejpam-6569	144	21	)	)	PUNCT
ejpam-6569	144	22	⇒	⇒	NOUN
ejpam-6569	144	23	(	(	PUNCT
ejpam-6569	144	24	5	5	NUM
ejpam-6569	144	25	):	):	PUNCT
ejpam-6569	144	26	let	let	VERB
ejpam-6569	144	27	v	v	PART
ejpam-6569	144	28	be	be	AUX
ejpam-6569	144	29	any	any	DET
ejpam-6569	144	30	σ1σ2	σ1σ2	NOUN
ejpam-6569	144	31	-	-	ADJ
ejpam-6569	144	32	open	open	ADJ
ejpam-6569	144	33	set	set	NOUN
ejpam-6569	144	34	of	of	ADP
ejpam-6569	144	35	y	y	PROPN
ejpam-6569	144	36	.	.	PUNCT
ejpam-6569	145	1	then	then	ADV
ejpam-6569	145	2	,	,	PUNCT
ejpam-6569	145	3	we	we	PRON
ejpam-6569	145	4	have	have	VERB
ejpam-6569	145	5	x	x	X
ejpam-6569	145	6	−	−	X
ejpam-6569	145	7	⋆θsint(f	⋆θsint(f	SYM
ejpam-6569	146	1	+	+	ADJ
ejpam-6569	146	2	(	(	PUNCT
ejpam-6569	146	3	σ1σ2	σ1σ2	NOUN
ejpam-6569	146	4	-	-	NUM
ejpam-6569	146	5	cl(v	cl(v	NOUN
ejpam-6569	146	6	)	)	PUNCT
ejpam-6569	146	7	)	)	PUNCT
ejpam-6569	146	8	)	)	PUNCT
ejpam-6569	147	1	=	=	SYM
ejpam-6569	147	2	⋆θscl(x	⋆θscl(x	PROPN
ejpam-6569	147	3	−	−	NUM
ejpam-6569	147	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	147	5	-	-	PUNCT
ejpam-6569	147	6	cl(v	cl(v	NOUN
ejpam-6569	147	7	)	)	PUNCT
ejpam-6569	147	8	)	)	PUNCT
ejpam-6569	147	9	)	)	PUNCT
ejpam-6569	148	1	=	=	PUNCT
ejpam-6569	148	2	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	148	3	−(y	−(y	NOUN
ejpam-6569	148	4	−	−	NOUN
ejpam-6569	148	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	148	6	-	-	NUM
ejpam-6569	148	7	cl(v	cl(v	NOUN
ejpam-6569	148	8	)	)	PUNCT
ejpam-6569	148	9	)	)	PUNCT
ejpam-6569	148	10	)	)	PUNCT
ejpam-6569	148	11	,	,	PUNCT
ejpam-6569	148	12	y	y	PROPN
ejpam-6569	148	13	−	−	NUM
ejpam-6569	148	14	σ1σ2	σ1σ2	NOUN
ejpam-6569	148	15	-	-	NUM
ejpam-6569	148	16	cl(v	cl(v	X
ejpam-6569	148	17	)	)	PUNCT
ejpam-6569	148	18	=	=	SYM
ejpam-6569	148	19	σ1σ2	σ1σ2	X
ejpam-6569	148	20	-	-	PUNCT
ejpam-6569	148	21	int(y	int(y	ADJ
ejpam-6569	148	22	−	−	NOUN
ejpam-6569	148	23	σ1σ2	σ1σ2	NOUN
ejpam-6569	148	24	-	-	NUM
ejpam-6569	148	25	cl(v	cl(v	NOUN
ejpam-6569	148	26	)	)	PUNCT
ejpam-6569	148	27	)	)	PUNCT
ejpam-6569	149	1	⊆	⊆	X
ejpam-6569	149	2	σ1σ2	σ1σ2	X
ejpam-6569	149	3	-	-	PUNCT
ejpam-6569	149	4	int(y	int(y	ADJ
ejpam-6569	149	5	−	−	NOUN
ejpam-6569	149	6	σ1σ2	σ1σ2	NOUN
ejpam-6569	149	7	-	-	PUNCT
ejpam-6569	149	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	149	9	-	-	PUNCT
ejpam-6569	149	10	cl(v	cl(v	NOUN
ejpam-6569	149	11	)	)	PUNCT
ejpam-6569	149	12	)	)	PUNCT
ejpam-6569	149	13	)	)	PUNCT
ejpam-6569	149	14	and	and	CCONJ
ejpam-6569	149	15	y	y	PROPN
ejpam-6569	149	16	−	−	PROPN
ejpam-6569	149	17	σ1σ2	σ1σ2	ADV
ejpam-6569	149	18	-	-	PUNCT
ejpam-6569	149	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	149	20	-	-	PUNCT
ejpam-6569	149	21	cl(v	cl(v	NOUN
ejpam-6569	149	22	)	)	PUNCT
ejpam-6569	149	23	)	)	PUNCT
ejpam-6569	149	24	is	be	AUX
ejpam-6569	149	25	(	(	PUNCT
ejpam-6569	149	26	σ1	σ1	NOUN
ejpam-6569	149	27	,	,	PUNCT
ejpam-6569	149	28	σ2)r	σ2)r	NOUN
ejpam-6569	149	29	-	-	PUNCT
ejpam-6569	149	30	closed	closed	ADJ
ejpam-6569	149	31	in	in	ADP
ejpam-6569	149	32	y	y	PROPN
ejpam-6569	149	33	.	.	PUNCT
ejpam-6569	150	1	thus	thus	ADV
ejpam-6569	150	2	by	by	ADP
ejpam-6569	150	3	(	(	PUNCT
ejpam-6569	150	4	4	4	NUM
ejpam-6569	150	5	)	)	PUNCT
ejpam-6569	150	6	,	,	PUNCT
ejpam-6569	150	7	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	150	8	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	150	9	-	-	PUNCT
ejpam-6569	150	10	int(y	int(y	ADJ
ejpam-6569	150	11	−	−	NOUN
ejpam-6569	150	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	150	13	-	-	PUNCT
ejpam-6569	150	14	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	150	15	-	-	PUNCT
ejpam-6569	150	16	cl(v	cl(v	NOUN
ejpam-6569	150	17	)	)	PUNCT
ejpam-6569	150	18	)	)	PUNCT
ejpam-6569	150	19	)	)	PUNCT
ejpam-6569	150	20	)	)	PUNCT
ejpam-6569	150	21	)	)	PUNCT
ejpam-6569	151	1	⊆	⊆	NUM
ejpam-6569	151	2	f−(y	f−(y	NOUN
ejpam-6569	151	3	−	−	NOUN
ejpam-6569	151	4	σ1σ2	σ1σ2	NOUN
ejpam-6569	151	5	-	-	PUNCT
ejpam-6569	151	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	151	7	-	-	PUNCT
ejpam-6569	151	8	cl(v	cl(v	NOUN
ejpam-6569	151	9	)	)	PUNCT
ejpam-6569	151	10	)	)	PUNCT
ejpam-6569	151	11	)	)	PUNCT
ejpam-6569	152	1	=	=	PUNCT
ejpam-6569	152	2	x	x	X
ejpam-6569	152	3	−	−	ADP
ejpam-6569	152	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6569	152	5	-	-	PUNCT
ejpam-6569	152	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	152	7	-	-	PUNCT
ejpam-6569	152	8	cl(v	cl(v	NOUN
ejpam-6569	152	9	)	)	PUNCT
ejpam-6569	152	10	)	)	PUNCT
ejpam-6569	152	11	)	)	PUNCT
ejpam-6569	153	1	⊆	⊆	NUM
ejpam-6569	153	2	x	x	SYM
ejpam-6569	153	3	−	−	NOUN
ejpam-6569	153	4	f+(v	f+(v	NOUN
ejpam-6569	153	5	)	)	PUNCT
ejpam-6569	153	6	and	and	CCONJ
ejpam-6569	153	7	hence	hence	ADV
ejpam-6569	153	8	f+(v	f+(v	NOUN
ejpam-6569	153	9	)	)	PUNCT
ejpam-6569	154	1	⊆	⊆	NUM
ejpam-6569	154	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	154	3	+	+	ADJ
ejpam-6569	154	4	(	(	PUNCT
ejpam-6569	154	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	154	6	-	-	NUM
ejpam-6569	154	7	cl(v	cl(v	NOUN
ejpam-6569	154	8	)	)	PUNCT
ejpam-6569	154	9	)	)	PUNCT
ejpam-6569	154	10	)	)	PUNCT
ejpam-6569	154	11	.	.	PUNCT
ejpam-6569	155	1	(	(	PUNCT
ejpam-6569	155	2	5	5	X
ejpam-6569	155	3	)	)	PUNCT
ejpam-6569	155	4	⇒	⇒	NOUN
ejpam-6569	155	5	(	(	PUNCT
ejpam-6569	155	6	6	6	NUM
ejpam-6569	155	7	):	):	PUNCT
ejpam-6569	155	8	let	let	VERB
ejpam-6569	155	9	k	k	PRON
ejpam-6569	155	10	be	be	AUX
ejpam-6569	155	11	any	any	DET
ejpam-6569	155	12	σ1σ2	σ1σ2	NUM
ejpam-6569	155	13	-	-	PUNCT
ejpam-6569	155	14	closed	closed	ADJ
ejpam-6569	155	15	set	set	NOUN
ejpam-6569	155	16	of	of	ADP
ejpam-6569	155	17	y	y	PROPN
ejpam-6569	155	18	.	.	PUNCT
ejpam-6569	156	1	then	then	ADV
ejpam-6569	156	2	by	by	ADP
ejpam-6569	156	3	(	(	PUNCT
ejpam-6569	156	4	5	5	NUM
ejpam-6569	156	5	)	)	PUNCT
ejpam-6569	156	6	,	,	PUNCT
ejpam-6569	156	7	we	we	PRON
ejpam-6569	156	8	have	have	VERB
ejpam-6569	156	9	x	x	INTJ
ejpam-6569	156	10	−	−	DET
ejpam-6569	156	11	f−(k	f−(k	PROPN
ejpam-6569	156	12	)	)	PUNCT
ejpam-6569	156	13	=	=	PUNCT
ejpam-6569	157	1	f+(y	f+(y	PROPN
ejpam-6569	157	2	−k	−k	PROPN
ejpam-6569	157	3	)	)	PUNCT
ejpam-6569	157	4	⊆	⊆	NUM
ejpam-6569	157	5	⋆θsint(f	⋆θsint(f	X
ejpam-6569	157	6	+	+	ADJ
ejpam-6569	157	7	(	(	PUNCT
ejpam-6569	157	8	σ1σ2	σ1σ2	NUM
ejpam-6569	157	9	-	-	PUNCT
ejpam-6569	157	10	cl(y	cl(y	NOUN
ejpam-6569	157	11	−k	−k	NOUN
ejpam-6569	157	12	)	)	PUNCT
ejpam-6569	157	13	)	)	PUNCT
ejpam-6569	157	14	)	)	PUNCT
ejpam-6569	158	1	=	=	PUNCT
ejpam-6569	158	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	159	1	+	+	ADJ
ejpam-6569	159	2	(	(	PUNCT
ejpam-6569	159	3	y	y	PROPN
ejpam-6569	159	4	−	−	PROPN
ejpam-6569	159	5	σ1σ2	σ1σ2	NUM
ejpam-6569	159	6	-	-	PUNCT
ejpam-6569	159	7	int(k	int(k	NOUN
ejpam-6569	159	8	)	)	PUNCT
ejpam-6569	159	9	)	)	PUNCT
ejpam-6569	159	10	)	)	PUNCT
ejpam-6569	160	1	=	=	PUNCT
ejpam-6569	161	1	⋆θsint(x	⋆θsint(x	NUM
ejpam-6569	161	2	−	−	NOUN
ejpam-6569	161	3	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	161	4	-	-	PUNCT
ejpam-6569	161	5	int(k	int(k	NOUN
ejpam-6569	161	6	)	)	PUNCT
ejpam-6569	161	7	)	)	PUNCT
ejpam-6569	161	8	)	)	PUNCT
ejpam-6569	162	1	=	=	PUNCT
ejpam-6569	162	2	x	x	PUNCT
ejpam-6569	163	1	−	−	PROPN
ejpam-6569	163	2	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	163	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	163	4	-	-	PUNCT
ejpam-6569	163	5	int(k	int(k	NUM
ejpam-6569	163	6	)	)	PUNCT
ejpam-6569	163	7	)	)	PUNCT
ejpam-6569	163	8	)	)	PUNCT
ejpam-6569	163	9	.	.	PUNCT
ejpam-6569	164	1	thus	thus	ADV
ejpam-6569	164	2	,	,	PUNCT
ejpam-6569	164	3	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	164	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	164	5	-	-	PUNCT
ejpam-6569	164	6	int(k	int(k	NUM
ejpam-6569	164	7	)	)	PUNCT
ejpam-6569	164	8	)	)	PUNCT
ejpam-6569	164	9	)	)	PUNCT
ejpam-6569	164	10	⊆	⊆	NUM
ejpam-6569	164	11	f−(k	f−(k	PROPN
ejpam-6569	164	12	)	)	PUNCT
ejpam-6569	164	13	.	.	PUNCT
ejpam-6569	165	1	(	(	PUNCT
ejpam-6569	165	2	6	6	X
ejpam-6569	165	3	)	)	PUNCT
ejpam-6569	165	4	⇒	⇒	NOUN
ejpam-6569	165	5	(	(	PUNCT
ejpam-6569	165	6	7	7	NUM
ejpam-6569	165	7	):	):	PUNCT
ejpam-6569	165	8	let	let	VERB
ejpam-6569	165	9	v	v	PART
ejpam-6569	165	10	be	be	AUX
ejpam-6569	165	11	any	any	DET
ejpam-6569	165	12	σ1σ2	σ1σ2	NUM
ejpam-6569	165	13	-	-	PUNCT
ejpam-6569	165	14	closed	closed	ADJ
ejpam-6569	165	15	set	set	NOUN
ejpam-6569	165	16	of	of	ADP
ejpam-6569	165	17	y	y	PROPN
ejpam-6569	165	18	.	.	PUNCT
ejpam-6569	166	1	then	then	ADV
ejpam-6569	166	2	,	,	PUNCT
ejpam-6569	166	3	we	we	PRON
ejpam-6569	166	4	have	have	VERB
ejpam-6569	166	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	166	6	-	-	NUM
ejpam-6569	166	7	cl(v	cl(v	NOUN
ejpam-6569	166	8	)	)	PUNCT
ejpam-6569	166	9	is	be	AUX
ejpam-6569	166	10	σ1σ2	σ1σ2	NOUN
ejpam-6569	166	11	-	-	ADJ
ejpam-6569	166	12	closed	closed	ADJ
ejpam-6569	166	13	in	in	ADP
ejpam-6569	166	14	y	y	PROPN
ejpam-6569	166	15	and	and	CCONJ
ejpam-6569	166	16	by	by	ADP
ejpam-6569	166	17	(	(	PUNCT
ejpam-6569	166	18	6	6	NUM
ejpam-6569	166	19	)	)	PUNCT
ejpam-6569	166	20	,	,	PUNCT
ejpam-6569	166	21	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	166	22	−(v	−(v	NOUN
ejpam-6569	166	23	)	)	PUNCT
ejpam-6569	166	24	)	)	PUNCT
ejpam-6569	167	1	⊆	⊆	NUM
ejpam-6569	167	2	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	167	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	167	4	-	-	PUNCT
ejpam-6569	167	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	167	6	-	-	PUNCT
ejpam-6569	167	7	cl(v	cl(v	NOUN
ejpam-6569	167	8	)	)	PUNCT
ejpam-6569	167	9	)	)	PUNCT
ejpam-6569	167	10	)	)	PUNCT
ejpam-6569	167	11	)	)	PUNCT
ejpam-6569	168	1	⊆	⊆	X
ejpam-6569	168	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	168	3	-	-	PUNCT
ejpam-6569	168	4	cl(v	cl(v	NOUN
ejpam-6569	168	5	)	)	PUNCT
ejpam-6569	168	6	)	)	PUNCT
ejpam-6569	168	7	.	.	PUNCT
ejpam-6569	169	1	(	(	PUNCT
ejpam-6569	169	2	7	7	X
ejpam-6569	169	3	)	)	PUNCT
ejpam-6569	169	4	⇒	⇒	NOUN
ejpam-6569	169	5	(	(	PUNCT
ejpam-6569	169	6	1	1	NUM
ejpam-6569	169	7	):	):	PUNCT
ejpam-6569	169	8	let	let	VERB
ejpam-6569	169	9	x	x	PUNCT
ejpam-6569	169	10	∈	∈	PROPN
ejpam-6569	169	11	x	x	X
ejpam-6569	169	12	and	and	CCONJ
ejpam-6569	169	13	v	v	X
ejpam-6569	169	14	be	be	AUX
ejpam-6569	169	15	any	any	DET
ejpam-6569	169	16	σ1σ2	σ1σ2	NOUN
ejpam-6569	169	17	-	-	ADJ
ejpam-6569	169	18	open	open	ADJ
ejpam-6569	169	19	set	set	NOUN
ejpam-6569	169	20	of	of	ADP
ejpam-6569	169	21	y	y	PROPN
ejpam-6569	169	22	containing	contain	VERB
ejpam-6569	169	23	f	f	PROPN
ejpam-6569	169	24	(	(	PUNCT
ejpam-6569	169	25	x	x	NOUN
ejpam-6569	169	26	)	)	PUNCT
ejpam-6569	169	27	.	.	PUNCT
ejpam-6569	170	1	then	then	ADV
ejpam-6569	170	2	,	,	PUNCT
ejpam-6569	170	3	σ1σ2	σ1σ2	NOUN
ejpam-6569	170	4	-	-	PUNCT
ejpam-6569	170	5	cl(y	cl(y	NOUN
ejpam-6569	170	6	−	−	NOUN
ejpam-6569	170	7	σ1σ2	σ1σ2	NOUN
ejpam-6569	170	8	-	-	NUM
ejpam-6569	170	9	cl(v	cl(v	NOUN
ejpam-6569	170	10	)	)	PUNCT
ejpam-6569	170	11	)	)	PUNCT
ejpam-6569	170	12	∩	∩	PROPN
ejpam-6569	170	13	f	f	X
ejpam-6569	170	14	(	(	PUNCT
ejpam-6569	170	15	x	x	X
ejpam-6569	170	16	)	)	PUNCT
ejpam-6569	170	17	=	=	SYM
ejpam-6569	170	18	∅	∅	NOUN
ejpam-6569	170	19	and	and	CCONJ
ejpam-6569	170	20	x	x	PART
ejpam-6569	170	21	̸∈	̸∈	PROPN
ejpam-6569	170	22	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6569	170	23	-	-	PUNCT
ejpam-6569	170	24	cl(y	cl(y	NOUN
ejpam-6569	170	25	−	−	NOUN
ejpam-6569	170	26	σ1σ2	σ1σ2	NOUN
ejpam-6569	170	27	-	-	NUM
ejpam-6569	170	28	cl(v	cl(v	NOUN
ejpam-6569	170	29	)	)	PUNCT
ejpam-6569	170	30	)	)	PUNCT
ejpam-6569	170	31	)	)	PUNCT
ejpam-6569	170	32	.	.	PUNCT
ejpam-6569	171	1	it	it	PRON
ejpam-6569	171	2	follows	follow	VERB
ejpam-6569	171	3	from	from	ADP
ejpam-6569	171	4	(	(	PUNCT
ejpam-6569	171	5	7	7	NUM
ejpam-6569	171	6	)	)	PUNCT
ejpam-6569	171	7	that	that	PRON
ejpam-6569	171	8	x	x	X
ejpam-6569	171	9	̸∈	̸∈	PROPN
ejpam-6569	171	10	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	171	11	−(y	−(y	NOUN
ejpam-6569	171	12	−	−	NOUN
ejpam-6569	171	13	σ1σ2	σ1σ2	NOUN
ejpam-6569	171	14	-	-	NUM
ejpam-6569	171	15	cl(v	cl(v	NOUN
ejpam-6569	171	16	)	)	PUNCT
ejpam-6569	171	17	)	)	PUNCT
ejpam-6569	171	18	)	)	PUNCT
ejpam-6569	171	19	.	.	PUNCT
ejpam-6569	172	1	then	then	ADV
ejpam-6569	172	2	,	,	PUNCT
ejpam-6569	172	3	there	there	PRON
ejpam-6569	172	4	exists	exist	VERB
ejpam-6569	172	5	a	a	DET
ejpam-6569	172	6	semi	semi	NOUN
ejpam-6569	172	7	-	-	ADJ
ejpam-6569	172	8	i	i	PRON
ejpam-6569	172	9	⋆-open	⋆-open	VERB
ejpam-6569	172	10	set	set	VERB
ejpam-6569	172	11	u	u	NOUN
ejpam-6569	172	12	of	of	ADP
ejpam-6569	172	13	x	x	PUNCT
ejpam-6569	172	14	containing	contain	VERB
ejpam-6569	172	15	x	x	PUNCT
ejpam-6569	172	16	such	such	ADJ
ejpam-6569	172	17	that	that	SCONJ
ejpam-6569	172	18	scl⋆(u	scl⋆(u	ADJ
ejpam-6569	172	19	)	)	PUNCT
ejpam-6569	172	20	∩	∩	NOUN
ejpam-6569	172	21	f−(y	f−(y	NOUN
ejpam-6569	172	22	−	−	NUM
ejpam-6569	172	23	σ1σ2	σ1σ2	NOUN
ejpam-6569	172	24	-	-	NUM
ejpam-6569	172	25	cl(v	cl(v	NOUN
ejpam-6569	172	26	)	)	PUNCT
ejpam-6569	172	27	)	)	PUNCT
ejpam-6569	173	1	=	=	NOUN
ejpam-6569	173	2	∅	∅	NOUN
ejpam-6569	173	3	;	;	PUNCT
ejpam-6569	173	4	hence	hence	ADV
ejpam-6569	173	5	f	f	X
ejpam-6569	173	6	(	(	PUNCT
ejpam-6569	173	7	scl⋆(u	scl⋆(u	NUM
ejpam-6569	173	8	)	)	PUNCT
ejpam-6569	173	9	)	)	PUNCT
ejpam-6569	174	1	⊆	⊆	X
ejpam-6569	174	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	174	3	-	-	NUM
ejpam-6569	174	4	cl(v	cl(v	NOUN
ejpam-6569	174	5	)	)	PUNCT
ejpam-6569	174	6	.	.	PUNCT
ejpam-6569	175	1	this	this	PRON
ejpam-6569	175	2	shows	show	VERB
ejpam-6569	175	3	that	that	SCONJ
ejpam-6569	175	4	f	f	PROPN
ejpam-6569	175	5	is	be	AUX
ejpam-6569	175	6	upper	upper	ADJ
ejpam-6569	175	7	quasi	quasi	NOUN
ejpam-6569	175	8	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	175	9	,	,	PUNCT
ejpam-6569	175	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	175	11	.	.	NOUN
ejpam-6569	175	12	definition	definition	NOUN
ejpam-6569	175	13	2	2	NUM
ejpam-6569	175	14	.	.	PUNCT
ejpam-6569	175	15	a	a	DET
ejpam-6569	175	16	multifunction	multifunction	NOUN
ejpam-6569	176	1	f	f	NOUN
ejpam-6569	176	2	:	:	PUNCT
ejpam-6569	176	3	(	(	PUNCT
ejpam-6569	176	4	x	x	X
ejpam-6569	176	5	,	,	PUNCT
ejpam-6569	176	6	τ	τ	PROPN
ejpam-6569	176	7	,	,	PUNCT
ejpam-6569	176	8	i	i	NOUN
ejpam-6569	176	9	)	)	PUNCT
ejpam-6569	176	10	→	→	PUNCT
ejpam-6569	176	11	(	(	PUNCT
ejpam-6569	176	12	y	y	PROPN
ejpam-6569	176	13	,	,	PUNCT
ejpam-6569	176	14	σ1	σ1	PROPN
ejpam-6569	176	15	,	,	PUNCT
ejpam-6569	176	16	σ2	σ2	PROPN
ejpam-6569	176	17	)	)	PUNCT
ejpam-6569	176	18	is	be	AUX
ejpam-6569	176	19	said	say	VERB
ejpam-6569	176	20	to	to	PART
ejpam-6569	176	21	be	be	AUX
ejpam-6569	176	22	lower	low	ADJ
ejpam-6569	176	23	quasi	quasi	NOUN
ejpam-6569	176	24	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	176	25	,	,	PUNCT
ejpam-6569	176	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	176	27	if	if	SCONJ
ejpam-6569	176	28	for	for	ADP
ejpam-6569	176	29	each	each	DET
ejpam-6569	176	30	x	x	SYM
ejpam-6569	176	31	∈	∈	PROPN
ejpam-6569	176	32	x	x	X
ejpam-6569	176	33	and	and	CCONJ
ejpam-6569	176	34	each	each	DET
ejpam-6569	176	35	σ1σ2	σ1σ2	VERB
ejpam-6569	176	36	-	-	ADJ
ejpam-6569	176	37	open	open	ADJ
ejpam-6569	176	38	set	set	NOUN
ejpam-6569	176	39	v	v	NOUN
ejpam-6569	176	40	of	of	ADP
ejpam-6569	176	41	y	y	PRON
ejpam-6569	176	42	such	such	ADJ
ejpam-6569	176	43	that	that	PRON
ejpam-6569	176	44	v	v	ADP
ejpam-6569	176	45	∩	∩	ADJ
ejpam-6569	176	46	f	f	X
ejpam-6569	176	47	(	(	PUNCT
ejpam-6569	176	48	x	x	X
ejpam-6569	176	49	)	)	PUNCT
ejpam-6569	176	50	̸=	̸=	NOUN
ejpam-6569	176	51	∅	∅	NOUN
ejpam-6569	176	52	,	,	PUNCT
ejpam-6569	176	53	there	there	PRON
ejpam-6569	176	54	exists	exist	VERB
ejpam-6569	176	55	a	a	DET
ejpam-6569	176	56	semi	semi	NOUN
ejpam-6569	176	57	-	-	ADJ
ejpam-6569	176	58	i	i	PRON
ejpam-6569	176	59	⋆-open	⋆-open	VERB
ejpam-6569	176	60	set	set	VERB
ejpam-6569	176	61	u	u	NOUN
ejpam-6569	176	62	of	of	ADP
ejpam-6569	176	63	x	x	PUNCT
ejpam-6569	176	64	containing	contain	VERB
ejpam-6569	176	65	x	x	PUNCT
ejpam-6569	176	66	such	such	ADJ
ejpam-6569	176	67	that	that	SCONJ
ejpam-6569	176	68	σ1σ2	σ1σ2	NOUN
ejpam-6569	176	69	-	-	NUM
ejpam-6569	176	70	cl(v	cl(v	PUNCT
ejpam-6569	176	71	)	)	PUNCT
ejpam-6569	176	72	∩f	∩f	NOUN
ejpam-6569	176	73	(	(	PUNCT
ejpam-6569	176	74	z	z	X
ejpam-6569	176	75	)	)	PUNCT
ejpam-6569	176	76	̸=	̸=	NOUN
ejpam-6569	176	77	∅	∅	NOUN
ejpam-6569	176	78	for	for	ADP
ejpam-6569	176	79	every	every	DET
ejpam-6569	176	80	z	z	PROPN
ejpam-6569	176	81	∈	∈	PROPN
ejpam-6569	176	82	scl⋆(u	scl⋆(u	NUM
ejpam-6569	176	83	)	)	PUNCT
ejpam-6569	176	84	.	.	PUNCT
ejpam-6569	177	1	lemma	lemma	PROPN
ejpam-6569	177	2	2	2	X
ejpam-6569	177	3	.	.	PUNCT
ejpam-6569	178	1	if	if	SCONJ
ejpam-6569	178	2	f	f	PROPN
ejpam-6569	178	3	:	:	PUNCT
ejpam-6569	178	4	(	(	PUNCT
ejpam-6569	178	5	x	x	X
ejpam-6569	178	6	,	,	PUNCT
ejpam-6569	178	7	τ	τ	PROPN
ejpam-6569	178	8	,	,	PUNCT
ejpam-6569	178	9	i	i	NOUN
ejpam-6569	178	10	)	)	PUNCT
ejpam-6569	178	11	→	→	PUNCT
ejpam-6569	178	12	(	(	PUNCT
ejpam-6569	178	13	y	y	PROPN
ejpam-6569	178	14	,	,	PUNCT
ejpam-6569	178	15	σ1	σ1	PROPN
ejpam-6569	178	16	,	,	PUNCT
ejpam-6569	178	17	σ2	σ2	NOUN
ejpam-6569	178	18	)	)	PUNCT
ejpam-6569	178	19	is	be	AUX
ejpam-6569	178	20	lower	low	ADJ
ejpam-6569	178	21	quasi	quasi	NOUN
ejpam-6569	178	22	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	178	23	,	,	PUNCT
ejpam-6569	178	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	178	25	,	,	PUNCT
ejpam-6569	178	26	then	then	ADV
ejpam-6569	178	27	for	for	ADP
ejpam-6569	178	28	each	each	DET
ejpam-6569	178	29	x	x	SYM
ejpam-6569	178	30	∈	∈	PROPN
ejpam-6569	178	31	x	x	X
ejpam-6569	178	32	and	and	CCONJ
ejpam-6569	178	33	each	each	DET
ejpam-6569	178	34	subset	subset	NOUN
ejpam-6569	178	35	b	b	PROPN
ejpam-6569	178	36	of	of	ADP
ejpam-6569	178	37	y	y	PROPN
ejpam-6569	178	38	with	with	ADP
ejpam-6569	178	39	(	(	PUNCT
ejpam-6569	178	40	σ1	σ1	PROPN
ejpam-6569	178	41	,	,	PUNCT
ejpam-6569	178	42	σ2)θ	σ2)θ	NOUN
ejpam-6569	178	43	-	-	PUNCT
ejpam-6569	178	44	int(b	int(b	NOUN
ejpam-6569	178	45	)	)	PUNCT
ejpam-6569	178	46	∩	∩	ADJ
ejpam-6569	179	1	f	f	PROPN
ejpam-6569	179	2	(	(	PUNCT
ejpam-6569	179	3	x	x	X
ejpam-6569	179	4	)	)	PUNCT
ejpam-6569	179	5	̸=	̸=	PROPN
ejpam-6569	179	6	∅	∅	NOUN
ejpam-6569	179	7	there	there	ADV
ejpam-6569	179	8	exists	exist	VERB
ejpam-6569	179	9	a	a	DET
ejpam-6569	179	10	semi	semi	NOUN
ejpam-6569	179	11	-	-	ADJ
ejpam-6569	179	12	i	i	PRON
ejpam-6569	179	13	⋆-open	⋆-open	VERB
ejpam-6569	179	14	set	set	VERB
ejpam-6569	179	15	u	u	NOUN
ejpam-6569	179	16	of	of	ADP
ejpam-6569	179	17	x	x	PUNCT
ejpam-6569	179	18	containing	contain	VERB
ejpam-6569	179	19	x	x	PUNCT
ejpam-6569	179	20	such	such	ADJ
ejpam-6569	179	21	that	that	SCONJ
ejpam-6569	179	22	scl⋆(u	scl⋆(u	NUM
ejpam-6569	179	23	)	)	PUNCT
ejpam-6569	179	24	⊆	⊆	NUM
ejpam-6569	179	25	f−(b	f−(b	NOUN
ejpam-6569	179	26	)	)	PUNCT
ejpam-6569	179	27	.	.	PUNCT
ejpam-6569	180	1	proof	proof	NOUN
ejpam-6569	180	2	.	.	PUNCT
ejpam-6569	181	1	since	since	SCONJ
ejpam-6569	181	2	(	(	PUNCT
ejpam-6569	181	3	σ1	σ1	PROPN
ejpam-6569	181	4	,	,	PUNCT
ejpam-6569	181	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	181	6	-	-	PUNCT
ejpam-6569	181	7	int(b)∩f	int(b)∩f	NOUN
ejpam-6569	181	8	(	(	PUNCT
ejpam-6569	181	9	x	x	X
ejpam-6569	181	10	)	)	PUNCT
ejpam-6569	181	11	̸=	̸=	NOUN
ejpam-6569	181	12	∅	∅	NOUN
ejpam-6569	181	13	,	,	PUNCT
ejpam-6569	181	14	there	there	PRON
ejpam-6569	181	15	exists	exist	VERB
ejpam-6569	181	16	a	a	DET
ejpam-6569	181	17	σ1σ2	σ1σ2	NUM
ejpam-6569	181	18	-	-	ADJ
ejpam-6569	181	19	open	open	ADJ
ejpam-6569	181	20	set	set	NOUN
ejpam-6569	181	21	v	v	NOUN
ejpam-6569	181	22	of	of	ADP
ejpam-6569	181	23	y	y	PRON
ejpam-6569	182	1	such	such	ADJ
ejpam-6569	182	2	that	that	SCONJ
ejpam-6569	182	3	v	v	ADP
ejpam-6569	182	4	⊆	⊆	NUM
ejpam-6569	182	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	182	6	-	-	PUNCT
ejpam-6569	182	7	cl(v	cl(v	NOUN
ejpam-6569	182	8	)	)	PUNCT
ejpam-6569	182	9	⊆	⊆	NUM
ejpam-6569	182	10	b	b	NOUN
ejpam-6569	182	11	and	and	CCONJ
ejpam-6569	182	12	v	v	NOUN
ejpam-6569	182	13	∩	∩	ADJ
ejpam-6569	182	14	f	f	X
ejpam-6569	182	15	(	(	PUNCT
ejpam-6569	182	16	x	x	X
ejpam-6569	182	17	)	)	PUNCT
ejpam-6569	182	18	̸=	̸=	PROPN
ejpam-6569	182	19	∅.	∅.	ADV
ejpam-6569	182	20	since	since	SCONJ
ejpam-6569	182	21	f	f	PROPN
ejpam-6569	182	22	is	be	AUX
ejpam-6569	182	23	lower	low	ADJ
ejpam-6569	182	24	quasi	quasi	NOUN
ejpam-6569	182	25	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	182	26	,	,	PUNCT
ejpam-6569	182	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	182	28	,	,	PUNCT
ejpam-6569	182	29	there	there	PRON
ejpam-6569	182	30	exists	exist	VERB
ejpam-6569	182	31	a	a	DET
ejpam-6569	182	32	semi	semi	NOUN
ejpam-6569	182	33	-	-	ADJ
ejpam-6569	182	34	i	i	PRON
ejpam-6569	182	35	⋆-open	⋆-open	VERB
ejpam-6569	182	36	set	set	VERB
ejpam-6569	182	37	u	u	NOUN
ejpam-6569	182	38	of	of	ADP
ejpam-6569	182	39	x	x	PUNCT
ejpam-6569	182	40	containing	contain	VERB
ejpam-6569	182	41	x	x	PUNCT
ejpam-6569	182	42	such	such	ADJ
ejpam-6569	182	43	that	that	SCONJ
ejpam-6569	182	44	σ1σ2	σ1σ2	NOUN
ejpam-6569	182	45	-	-	NUM
ejpam-6569	182	46	cl(v	cl(v	PUNCT
ejpam-6569	182	47	)	)	PUNCT
ejpam-6569	182	48	∩f	∩f	NOUN
ejpam-6569	182	49	(	(	PUNCT
ejpam-6569	182	50	z	z	X
ejpam-6569	182	51	)	)	PUNCT
ejpam-6569	182	52	̸=	̸=	NOUN
ejpam-6569	182	53	∅	∅	NOUN
ejpam-6569	182	54	for	for	ADP
ejpam-6569	182	55	every	every	DET
ejpam-6569	182	56	z	z	PROPN
ejpam-6569	182	57	∈	∈	PROPN
ejpam-6569	182	58	scl⋆(u	scl⋆(u	NOUN
ejpam-6569	182	59	)	)	PUNCT
ejpam-6569	182	60	and	and	CCONJ
ejpam-6569	182	61	hence	hence	ADV
ejpam-6569	182	62	scl⋆(u	scl⋆(u	NUM
ejpam-6569	182	63	)	)	PUNCT
ejpam-6569	182	64	⊆	⊆	NUM
ejpam-6569	182	65	f−(b	f−(b	NOUN
ejpam-6569	182	66	)	)	PUNCT
ejpam-6569	182	67	.	.	PUNCT
ejpam-6569	183	1	n.	n.	PROPN
ejpam-6569	183	2	srisarakham	srisarakham	PROPN
ejpam-6569	183	3	,	,	PUNCT
ejpam-6569	183	4	a.	a.	PROPN
ejpam-6569	183	5	sama	sama	PROPN
ejpam-6569	183	6	-	-	PUNCT
ejpam-6569	183	7	ae	ae	PROPN
ejpam-6569	183	8	,	,	PUNCT
ejpam-6569	183	9	c.	c.	PROPN
ejpam-6569	183	10	boonpok	boonpok	PROPN
ejpam-6569	183	11	/	/	SYM
ejpam-6569	183	12	eur	eur	PROPN
ejpam-6569	183	13	.	.	PUNCT
ejpam-6569	184	1	j.	j.	PROPN
ejpam-6569	184	2	pure	pure	PROPN
ejpam-6569	184	3	appl	appl	PROPN
ejpam-6569	184	4	.	.	PROPN
ejpam-6569	184	5	math	math	PROPN
ejpam-6569	184	6	,	,	PUNCT
ejpam-6569	184	7	18	18	NUM
ejpam-6569	184	8	(	(	PUNCT
ejpam-6569	184	9	3	3	NUM
ejpam-6569	184	10	)	)	PUNCT
ejpam-6569	184	11	(	(	PUNCT
ejpam-6569	184	12	2025	2025	NUM
ejpam-6569	184	13	)	)	PUNCT
ejpam-6569	184	14	,	,	PUNCT
ejpam-6569	184	15	6569	6569	NUM
ejpam-6569	184	16	6	6	NUM
ejpam-6569	184	17	of	of	ADP
ejpam-6569	184	18	13	13	NUM
ejpam-6569	184	19	theorem	theorem	NOUN
ejpam-6569	184	20	2	2	NUM
ejpam-6569	184	21	.	.	X
ejpam-6569	184	22	for	for	ADP
ejpam-6569	184	23	a	a	DET
ejpam-6569	184	24	multifunction	multifunction	NOUN
ejpam-6569	184	25	f	f	NOUN
ejpam-6569	184	26	:	:	PUNCT
ejpam-6569	184	27	(	(	PUNCT
ejpam-6569	184	28	x	x	X
ejpam-6569	184	29	,	,	PUNCT
ejpam-6569	184	30	τ	τ	PROPN
ejpam-6569	184	31	,	,	PUNCT
ejpam-6569	184	32	i	i	NOUN
ejpam-6569	184	33	)	)	PUNCT
ejpam-6569	184	34	→	→	PUNCT
ejpam-6569	184	35	(	(	PUNCT
ejpam-6569	184	36	y	y	PROPN
ejpam-6569	184	37	,	,	PUNCT
ejpam-6569	184	38	σ1	σ1	PROPN
ejpam-6569	184	39	,	,	PUNCT
ejpam-6569	184	40	σ2	σ2	NOUN
ejpam-6569	184	41	)	)	PUNCT
ejpam-6569	184	42	,	,	PUNCT
ejpam-6569	184	43	the	the	DET
ejpam-6569	184	44	following	follow	VERB
ejpam-6569	184	45	properties	property	NOUN
ejpam-6569	184	46	are	be	AUX
ejpam-6569	184	47	equivalent	equivalent	ADJ
ejpam-6569	184	48	:	:	PUNCT
ejpam-6569	184	49	(	(	PUNCT
ejpam-6569	184	50	1	1	X
ejpam-6569	184	51	)	)	PUNCT
ejpam-6569	184	52	f	f	PROPN
ejpam-6569	184	53	is	be	AUX
ejpam-6569	184	54	lower	low	ADJ
ejpam-6569	184	55	quasi	quasi	NOUN
ejpam-6569	184	56	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	184	57	,	,	PUNCT
ejpam-6569	184	58	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	184	59	;	;	PUNCT
ejpam-6569	184	60	(	(	PUNCT
ejpam-6569	184	61	2	2	X
ejpam-6569	184	62	)	)	PUNCT
ejpam-6569	184	63	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	184	64	+	+	PROPN
ejpam-6569	184	65	(	(	PUNCT
ejpam-6569	184	66	b	b	NOUN
ejpam-6569	184	67	)	)	PUNCT
ejpam-6569	184	68	)	)	PUNCT
ejpam-6569	185	1	⊆	⊆	NUM
ejpam-6569	185	2	f+((σ1	f+((σ1	NOUN
ejpam-6569	185	3	,	,	PUNCT
ejpam-6569	185	4	σ2)θ	σ2)θ	ADJ
ejpam-6569	185	5	-	-	PUNCT
ejpam-6569	185	6	cl(b	cl(b	NOUN
ejpam-6569	185	7	)	)	PUNCT
ejpam-6569	185	8	)	)	PUNCT
ejpam-6569	185	9	for	for	ADP
ejpam-6569	185	10	every	every	DET
ejpam-6569	185	11	subset	subset	NOUN
ejpam-6569	185	12	b	b	PROPN
ejpam-6569	185	13	of	of	ADP
ejpam-6569	185	14	y	y	PROPN
ejpam-6569	185	15	;	;	PUNCT
ejpam-6569	185	16	(	(	PUNCT
ejpam-6569	185	17	3	3	X
ejpam-6569	185	18	)	)	PUNCT
ejpam-6569	185	19	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	186	1	+	+	NOUN
ejpam-6569	186	2	(	(	PUNCT
ejpam-6569	186	3	v	v	NOUN
ejpam-6569	186	4	)	)	PUNCT
ejpam-6569	186	5	)	)	PUNCT
ejpam-6569	187	1	⊆	⊆	NUM
ejpam-6569	187	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	187	3	-	-	PUNCT
ejpam-6569	187	4	cl(v	cl(v	NOUN
ejpam-6569	187	5	)	)	PUNCT
ejpam-6569	187	6	)	)	PUNCT
ejpam-6569	187	7	for	for	ADP
ejpam-6569	187	8	every	every	DET
ejpam-6569	187	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	187	10	-	-	ADJ
ejpam-6569	187	11	open	open	ADJ
ejpam-6569	187	12	set	set	NOUN
ejpam-6569	187	13	v	v	NOUN
ejpam-6569	187	14	of	of	ADP
ejpam-6569	187	15	y	y	PROPN
ejpam-6569	187	16	;	;	PUNCT
ejpam-6569	187	17	(	(	PUNCT
ejpam-6569	187	18	4	4	X
ejpam-6569	187	19	)	)	PUNCT
ejpam-6569	187	20	f−(v	f−(v	NOUN
ejpam-6569	187	21	)	)	PUNCT
ejpam-6569	187	22	⊆	⊆	NUM
ejpam-6569	187	23	⋆θsint(f	⋆θsint(f	NUM
ejpam-6569	187	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	187	25	-	-	NOUN
ejpam-6569	187	26	cl(v	cl(v	NOUN
ejpam-6569	187	27	)	)	PUNCT
ejpam-6569	187	28	)	)	PUNCT
ejpam-6569	187	29	)	)	PUNCT
ejpam-6569	188	1	for	for	ADP
ejpam-6569	188	2	every	every	DET
ejpam-6569	188	3	σ1σ2	σ1σ2	NOUN
ejpam-6569	188	4	-	-	ADJ
ejpam-6569	188	5	open	open	ADJ
ejpam-6569	188	6	set	set	NOUN
ejpam-6569	188	7	v	v	NOUN
ejpam-6569	188	8	of	of	ADP
ejpam-6569	188	9	y	y	PROPN
ejpam-6569	188	10	;	;	PUNCT
ejpam-6569	188	11	(	(	PUNCT
ejpam-6569	188	12	5	5	X
ejpam-6569	188	13	)	)	PUNCT
ejpam-6569	188	14	f	f	NOUN
ejpam-6569	188	15	(	(	PUNCT
ejpam-6569	188	16	⋆θscl(a	⋆θscl(a	NOUN
ejpam-6569	188	17	)	)	PUNCT
ejpam-6569	188	18	)	)	PUNCT
ejpam-6569	189	1	⊆	⊆	NUM
ejpam-6569	189	2	(	(	PUNCT
ejpam-6569	189	3	σ1	σ1	PROPN
ejpam-6569	189	4	,	,	PUNCT
ejpam-6569	189	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	189	6	-	-	PUNCT
ejpam-6569	189	7	cl(f	cl(f	PROPN
ejpam-6569	189	8	(	(	PUNCT
ejpam-6569	189	9	a	a	NOUN
ejpam-6569	189	10	)	)	PUNCT
ejpam-6569	189	11	)	)	PUNCT
ejpam-6569	189	12	for	for	ADP
ejpam-6569	189	13	every	every	DET
ejpam-6569	189	14	subset	subset	NOUN
ejpam-6569	189	15	a	a	PRON
ejpam-6569	189	16	of	of	ADP
ejpam-6569	189	17	x	x	PRON
ejpam-6569	189	18	;	;	PUNCT
ejpam-6569	189	19	(	(	PUNCT
ejpam-6569	189	20	6	6	NUM
ejpam-6569	189	21	)	)	PUNCT
ejpam-6569	189	22	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	190	1	+	+	PROPN
ejpam-6569	190	2	(	(	PUNCT
ejpam-6569	190	3	σ1σ2	σ1σ2	NOUN
ejpam-6569	190	4	-	-	PUNCT
ejpam-6569	190	5	int((σ1	int((σ1	ADJ
ejpam-6569	190	6	,	,	PUNCT
ejpam-6569	190	7	σ2)θ	σ2)θ	ADJ
ejpam-6569	190	8	-	-	PUNCT
ejpam-6569	190	9	cl(b	cl(b	NOUN
ejpam-6569	190	10	)	)	PUNCT
ejpam-6569	190	11	)	)	PUNCT
ejpam-6569	190	12	)	)	PUNCT
ejpam-6569	190	13	)	)	PUNCT
ejpam-6569	191	1	⊆	⊆	NUM
ejpam-6569	191	2	f+((σ1	f+((σ1	NOUN
ejpam-6569	191	3	,	,	PUNCT
ejpam-6569	191	4	σ2)θ	σ2)θ	ADJ
ejpam-6569	191	5	-	-	PUNCT
ejpam-6569	191	6	cl(b	cl(b	NOUN
ejpam-6569	191	7	)	)	PUNCT
ejpam-6569	191	8	)	)	PUNCT
ejpam-6569	191	9	for	for	ADP
ejpam-6569	191	10	every	every	DET
ejpam-6569	191	11	subset	subset	NOUN
ejpam-6569	191	12	b	b	PROPN
ejpam-6569	191	13	of	of	ADP
ejpam-6569	191	14	y	y	PROPN
ejpam-6569	191	15	;	;	PUNCT
ejpam-6569	191	16	(	(	PUNCT
ejpam-6569	191	17	7	7	X
ejpam-6569	191	18	)	)	PUNCT
ejpam-6569	191	19	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	192	1	+	+	PROPN
ejpam-6569	192	2	(	(	PUNCT
ejpam-6569	192	3	σ1σ2	σ1σ2	NUM
ejpam-6569	192	4	-	-	PUNCT
ejpam-6569	192	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	192	6	-	-	PUNCT
ejpam-6569	192	7	cl(v	cl(v	NOUN
ejpam-6569	192	8	)	)	PUNCT
ejpam-6569	192	9	)	)	PUNCT
ejpam-6569	192	10	)	)	PUNCT
ejpam-6569	192	11	)	)	PUNCT
ejpam-6569	193	1	⊆	⊆	X
ejpam-6569	193	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	193	3	-	-	PUNCT
ejpam-6569	193	4	cl(v	cl(v	NOUN
ejpam-6569	193	5	)	)	PUNCT
ejpam-6569	193	6	)	)	PUNCT
ejpam-6569	193	7	for	for	SCONJ
ejpam-6569	193	8	every	every	DET
ejpam-6569	193	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	193	10	-	-	ADJ
ejpam-6569	193	11	open	open	ADJ
ejpam-6569	193	12	set	set	NOUN
ejpam-6569	193	13	v	v	NOUN
ejpam-6569	193	14	of	of	ADP
ejpam-6569	193	15	y	y	PROPN
ejpam-6569	193	16	;	;	PUNCT
ejpam-6569	193	17	(	(	PUNCT
ejpam-6569	193	18	8)	8)	NUM
ejpam-6569	193	19	⋆θscl(f	⋆θscl(f	ADJ
ejpam-6569	193	20	+	+	PROPN
ejpam-6569	193	21	(	(	PUNCT
ejpam-6569	193	22	σ1σ2	σ1σ2	NUM
ejpam-6569	193	23	-	-	PUNCT
ejpam-6569	193	24	int(k	int(k	NUM
ejpam-6569	193	25	)	)	PUNCT
ejpam-6569	193	26	)	)	PUNCT
ejpam-6569	193	27	)	)	PUNCT
ejpam-6569	193	28	⊆	⊆	NUM
ejpam-6569	193	29	f+(k	f+(k	NOUN
ejpam-6569	193	30	)	)	PUNCT
ejpam-6569	193	31	for	for	ADP
ejpam-6569	193	32	every	every	DET
ejpam-6569	193	33	(	(	PUNCT
ejpam-6569	193	34	σ1	σ1	PROPN
ejpam-6569	193	35	,	,	PUNCT
ejpam-6569	193	36	σ2)r	σ2)r	NOUN
ejpam-6569	193	37	-	-	PUNCT
ejpam-6569	193	38	closed	close	VERB
ejpam-6569	193	39	set	set	ADJ
ejpam-6569	193	40	k	k	PROPN
ejpam-6569	193	41	of	of	ADP
ejpam-6569	193	42	y	y	PROPN
ejpam-6569	193	43	;	;	PUNCT
ejpam-6569	193	44	(	(	PUNCT
ejpam-6569	193	45	9	9	X
ejpam-6569	193	46	)	)	PUNCT
ejpam-6569	193	47	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	194	1	+	+	PROPN
ejpam-6569	194	2	(	(	PUNCT
ejpam-6569	194	3	σ1σ2	σ1σ2	NUM
ejpam-6569	194	4	-	-	PUNCT
ejpam-6569	194	5	int(k	int(k	NUM
ejpam-6569	194	6	)	)	PUNCT
ejpam-6569	194	7	)	)	PUNCT
ejpam-6569	194	8	)	)	PUNCT
ejpam-6569	195	1	⊆	⊆	NUM
ejpam-6569	195	2	f+(k	f+(k	NOUN
ejpam-6569	195	3	)	)	PUNCT
ejpam-6569	195	4	for	for	ADP
ejpam-6569	195	5	every	every	DET
ejpam-6569	195	6	σ1σ2	σ1σ2	NUM
ejpam-6569	195	7	-	-	PUNCT
ejpam-6569	195	8	closed	closed	ADJ
ejpam-6569	195	9	set	set	NOUN
ejpam-6569	195	10	k	k	PROPN
ejpam-6569	195	11	of	of	ADP
ejpam-6569	195	12	y	y	PROPN
ejpam-6569	195	13	.	.	PUNCT
ejpam-6569	196	1	proof	proof	NOUN
ejpam-6569	196	2	.	.	PUNCT
ejpam-6569	197	1	(	(	PUNCT
ejpam-6569	197	2	1	1	X
ejpam-6569	197	3	)	)	PUNCT
ejpam-6569	197	4	⇒	⇒	NOUN
ejpam-6569	197	5	(	(	PUNCT
ejpam-6569	197	6	2	2	NUM
ejpam-6569	197	7	):	):	PUNCT
ejpam-6569	197	8	let	let	VERB
ejpam-6569	197	9	b	b	X
ejpam-6569	197	10	be	be	AUX
ejpam-6569	197	11	any	any	DET
ejpam-6569	197	12	subset	subset	NOUN
ejpam-6569	197	13	of	of	ADP
ejpam-6569	197	14	y	y	PROPN
ejpam-6569	197	15	.	.	PUNCT
ejpam-6569	197	16	suppose	suppose	VERB
ejpam-6569	197	17	that	that	SCONJ
ejpam-6569	197	18	x	x	PROPN
ejpam-6569	197	19	̸∈	̸∈	PROPN
ejpam-6569	197	20	f+((σ1	f+((σ1	ADV
ejpam-6569	197	21	,	,	PUNCT
ejpam-6569	197	22	σ2)θ	σ2)θ	ADJ
ejpam-6569	197	23	-	-	PUNCT
ejpam-6569	197	24	cl(b	cl(b	NOUN
ejpam-6569	197	25	)	)	PUNCT
ejpam-6569	197	26	)	)	PUNCT
ejpam-6569	197	27	.	.	PUNCT
ejpam-6569	198	1	then	then	ADV
ejpam-6569	198	2	,	,	PUNCT
ejpam-6569	198	3	x	x	PUNCT
ejpam-6569	198	4	∈	∈	NOUN
ejpam-6569	198	5	f−(y	f−(y	NOUN
ejpam-6569	198	6	−	−	PROPN
ejpam-6569	198	7	(	(	PUNCT
ejpam-6569	198	8	σ1	σ1	PROPN
ejpam-6569	198	9	,	,	PUNCT
ejpam-6569	198	10	σ2)θ	σ2)θ	NOUN
ejpam-6569	198	11	-	-	PUNCT
ejpam-6569	198	12	cl(b	cl(b	NOUN
ejpam-6569	198	13	)	)	PUNCT
ejpam-6569	198	14	)	)	PUNCT
ejpam-6569	199	1	=	=	SYM
ejpam-6569	199	2	f−((σ1	f−((σ1	NOUN
ejpam-6569	199	3	,	,	PUNCT
ejpam-6569	199	4	σ2)θ	σ2)θ	ADJ
ejpam-6569	199	5	-	-	PUNCT
ejpam-6569	199	6	int(y	int(y	PROPN
ejpam-6569	199	7	−	−	PROPN
ejpam-6569	199	8	b	b	NOUN
ejpam-6569	199	9	)	)	PUNCT
ejpam-6569	199	10	)	)	PUNCT
ejpam-6569	199	11	.	.	PUNCT
ejpam-6569	200	1	since	since	SCONJ
ejpam-6569	200	2	f	f	PROPN
ejpam-6569	200	3	is	be	AUX
ejpam-6569	200	4	lower	low	ADJ
ejpam-6569	200	5	quasi	quasi	NOUN
ejpam-6569	200	6	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	200	7	,	,	PUNCT
ejpam-6569	200	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	200	9	,	,	PUNCT
ejpam-6569	200	10	by	by	ADP
ejpam-6569	200	11	lemma	lemma	PROPN
ejpam-6569	200	12	2	2	NUM
ejpam-6569	200	13	there	there	PRON
ejpam-6569	200	14	exists	exist	VERB
ejpam-6569	200	15	a	a	DET
ejpam-6569	200	16	semi	semi	NOUN
ejpam-6569	200	17	-	-	ADJ
ejpam-6569	200	18	i	i	PRON
ejpam-6569	200	19	⋆-open	⋆-open	VERB
ejpam-6569	200	20	set	set	VERB
ejpam-6569	200	21	u	u	NOUN
ejpam-6569	200	22	of	of	ADP
ejpam-6569	200	23	x	x	PUNCT
ejpam-6569	200	24	containing	contain	VERB
ejpam-6569	200	25	x	x	PUNCT
ejpam-6569	200	26	such	such	ADJ
ejpam-6569	200	27	that	that	SCONJ
ejpam-6569	200	28	scl⋆(u	scl⋆(u	NUM
ejpam-6569	200	29	)	)	PUNCT
ejpam-6569	200	30	⊆	⊆	NUM
ejpam-6569	200	31	f−(y	f−(y	NOUN
ejpam-6569	200	32	−	−	NOUN
ejpam-6569	200	33	b	b	NOUN
ejpam-6569	200	34	)	)	PUNCT
ejpam-6569	200	35	=	=	PUNCT
ejpam-6569	200	36	x	x	X
ejpam-6569	200	37	−	−	NOUN
ejpam-6569	200	38	f+(b	f+(b	NOUN
ejpam-6569	200	39	)	)	PUNCT
ejpam-6569	200	40	.	.	PUNCT
ejpam-6569	201	1	thus	thus	ADV
ejpam-6569	201	2	,	,	PUNCT
ejpam-6569	201	3	scl⋆(u	scl⋆(u	X
ejpam-6569	201	4	)	)	PUNCT
ejpam-6569	201	5	∩	∩	ADJ
ejpam-6569	201	6	f+(b	f+(b	NOUN
ejpam-6569	201	7	)	)	PUNCT
ejpam-6569	201	8	=	=	SYM
ejpam-6569	201	9	∅	∅	NOUN
ejpam-6569	201	10	and	and	CCONJ
ejpam-6569	201	11	hence	hence	ADV
ejpam-6569	201	12	x	x	X
ejpam-6569	201	13	̸∈	̸∈	PROPN
ejpam-6569	201	14	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	201	15	+	+	PROPN
ejpam-6569	201	16	(	(	PUNCT
ejpam-6569	201	17	b	b	NOUN
ejpam-6569	201	18	)	)	PUNCT
ejpam-6569	201	19	)	)	PUNCT
ejpam-6569	201	20	.	.	PUNCT
ejpam-6569	202	1	(	(	PUNCT
ejpam-6569	202	2	2	2	X
ejpam-6569	202	3	)	)	PUNCT
ejpam-6569	202	4	⇒	⇒	NOUN
ejpam-6569	202	5	(	(	PUNCT
ejpam-6569	202	6	3	3	NUM
ejpam-6569	202	7	):	):	PUNCT
ejpam-6569	202	8	this	this	PRON
ejpam-6569	202	9	is	be	AUX
ejpam-6569	202	10	obvious	obvious	ADJ
ejpam-6569	202	11	since	since	SCONJ
ejpam-6569	202	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	202	13	-	-	NOUN
ejpam-6569	202	14	cl(v	cl(v	X
ejpam-6569	202	15	)	)	PUNCT
ejpam-6569	203	1	=	=	SYM
ejpam-6569	203	2	(	(	PUNCT
ejpam-6569	203	3	σ1	σ1	PROPN
ejpam-6569	203	4	,	,	PUNCT
ejpam-6569	203	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	203	6	-	-	PUNCT
ejpam-6569	203	7	cl(v	cl(v	NOUN
ejpam-6569	203	8	)	)	PUNCT
ejpam-6569	203	9	for	for	ADP
ejpam-6569	203	10	every	every	DET
ejpam-6569	203	11	σ1σ2	σ1σ2	NOUN
ejpam-6569	203	12	-	-	ADJ
ejpam-6569	203	13	open	open	ADJ
ejpam-6569	203	14	set	set	NOUN
ejpam-6569	203	15	v	v	NOUN
ejpam-6569	203	16	of	of	ADP
ejpam-6569	203	17	y	y	PROPN
ejpam-6569	203	18	.	.	PUNCT
ejpam-6569	204	1	(	(	PUNCT
ejpam-6569	204	2	3	3	X
ejpam-6569	204	3	)	)	PUNCT
ejpam-6569	204	4	⇒	⇒	NOUN
ejpam-6569	204	5	(	(	PUNCT
ejpam-6569	204	6	4	4	NUM
ejpam-6569	204	7	):	):	PUNCT
ejpam-6569	204	8	let	let	VERB
ejpam-6569	204	9	v	v	PART
ejpam-6569	204	10	be	be	AUX
ejpam-6569	204	11	any	any	DET
ejpam-6569	204	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	204	13	-	-	ADJ
ejpam-6569	204	14	open	open	ADJ
ejpam-6569	204	15	set	set	NOUN
ejpam-6569	204	16	of	of	ADP
ejpam-6569	204	17	y	y	PROPN
ejpam-6569	204	18	.	.	PUNCT
ejpam-6569	205	1	then	then	ADV
ejpam-6569	205	2	by	by	ADP
ejpam-6569	205	3	(	(	PUNCT
ejpam-6569	205	4	3	3	NUM
ejpam-6569	205	5	)	)	PUNCT
ejpam-6569	205	6	,	,	PUNCT
ejpam-6569	205	7	we	we	PRON
ejpam-6569	205	8	have	have	VERB
ejpam-6569	205	9	x	x	PART
ejpam-6569	205	10	−	−	PROPN
ejpam-6569	205	11	⋆θsint(f	⋆θsint(f	NOUN
ejpam-6569	205	12	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	205	13	-	-	NOUN
ejpam-6569	205	14	cl(v	cl(v	NOUN
ejpam-6569	205	15	)	)	PUNCT
ejpam-6569	205	16	)	)	PUNCT
ejpam-6569	205	17	)	)	PUNCT
ejpam-6569	206	1	=	=	SYM
ejpam-6569	206	2	⋆θscl(x	⋆θscl(x	PROPN
ejpam-6569	206	3	−	−	PROPN
ejpam-6569	206	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	206	5	-	-	PUNCT
ejpam-6569	206	6	cl(v	cl(v	NOUN
ejpam-6569	206	7	)	)	PUNCT
ejpam-6569	206	8	)	)	PUNCT
ejpam-6569	206	9	)	)	PUNCT
ejpam-6569	207	1	=	=	PUNCT
ejpam-6569	208	1	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	208	2	+	+	PROPN
ejpam-6569	208	3	(	(	PUNCT
ejpam-6569	208	4	y	y	PROPN
ejpam-6569	208	5	−	−	PROPN
ejpam-6569	208	6	σ1σ2	σ1σ2	NOUN
ejpam-6569	208	7	-	-	NUM
ejpam-6569	208	8	cl(v	cl(v	NOUN
ejpam-6569	208	9	)	)	PUNCT
ejpam-6569	208	10	)	)	PUNCT
ejpam-6569	208	11	)	)	PUNCT
ejpam-6569	209	1	⊆	⊆	X
ejpam-6569	209	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	209	3	-	-	PUNCT
ejpam-6569	209	4	cl(y	cl(y	NOUN
ejpam-6569	209	5	−	−	NOUN
ejpam-6569	209	6	σ1σ2	σ1σ2	NOUN
ejpam-6569	209	7	-	-	NUM
ejpam-6569	209	8	cl(v	cl(v	NOUN
ejpam-6569	209	9	)	)	PUNCT
ejpam-6569	209	10	)	)	PUNCT
ejpam-6569	209	11	)	)	PUNCT
ejpam-6569	209	12	⊆	⊆	X
ejpam-6569	209	13	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	209	14	-	-	PUNCT
ejpam-6569	209	15	cl(y	cl(y	NOUN
ejpam-6569	209	16	−	−	PROPN
ejpam-6569	209	17	v	v	NOUN
ejpam-6569	209	18	)	)	PUNCT
ejpam-6569	209	19	)	)	PUNCT
ejpam-6569	210	1	=	=	PUNCT
ejpam-6569	211	1	f+(y	f+(y	NOUN
ejpam-6569	211	2	−	−	PROPN
ejpam-6569	211	3	v	v	NOUN
ejpam-6569	211	4	)	)	PUNCT
ejpam-6569	211	5	=	=	PUNCT
ejpam-6569	212	1	x	x	SYM
ejpam-6569	212	2	−	−	NOUN
ejpam-6569	212	3	f−(v	f−(v	NOUN
ejpam-6569	212	4	)	)	PUNCT
ejpam-6569	212	5	and	and	CCONJ
ejpam-6569	212	6	hence	hence	ADV
ejpam-6569	212	7	f−(v	f−(v	ADJ
ejpam-6569	212	8	)	)	PUNCT
ejpam-6569	212	9	⊆	⊆	NUM
ejpam-6569	212	10	⋆θsint(f	⋆θsint(f	NOUN
ejpam-6569	212	11	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	212	12	-	-	NOUN
ejpam-6569	212	13	cl(v	cl(v	NOUN
ejpam-6569	212	14	)	)	PUNCT
ejpam-6569	212	15	)	)	PUNCT
ejpam-6569	212	16	)	)	PUNCT
ejpam-6569	212	17	.	.	PUNCT
ejpam-6569	213	1	(	(	PUNCT
ejpam-6569	213	2	4	4	X
ejpam-6569	213	3	)	)	PUNCT
ejpam-6569	213	4	⇒	⇒	NOUN
ejpam-6569	213	5	(	(	PUNCT
ejpam-6569	213	6	1	1	NUM
ejpam-6569	213	7	):	):	PUNCT
ejpam-6569	213	8	let	let	VERB
ejpam-6569	213	9	x	x	PUNCT
ejpam-6569	213	10	∈	∈	PROPN
ejpam-6569	213	11	x	x	X
ejpam-6569	213	12	and	and	CCONJ
ejpam-6569	213	13	v	v	X
ejpam-6569	213	14	be	be	AUX
ejpam-6569	213	15	any	any	DET
ejpam-6569	213	16	σ1σ2	σ1σ2	NOUN
ejpam-6569	213	17	-	-	ADJ
ejpam-6569	213	18	open	open	ADJ
ejpam-6569	213	19	set	set	NOUN
ejpam-6569	213	20	of	of	ADP
ejpam-6569	213	21	y	y	PRON
ejpam-6569	213	22	such	such	ADJ
ejpam-6569	213	23	that	that	SCONJ
ejpam-6569	213	24	f	f	PROPN
ejpam-6569	213	25	(	(	PUNCT
ejpam-6569	213	26	x	x	NOUN
ejpam-6569	213	27	)	)	PUNCT
ejpam-6569	213	28	∩	∩	NOUN
ejpam-6569	213	29	v	v	ADP
ejpam-6569	213	30	̸=	̸=	PROPN
ejpam-6569	213	31	∅.	∅.	ADV
ejpam-6569	213	32	by	by	ADP
ejpam-6569	213	33	(	(	PUNCT
ejpam-6569	213	34	4	4	NUM
ejpam-6569	213	35	)	)	PUNCT
ejpam-6569	213	36	,	,	PUNCT
ejpam-6569	213	37	x	x	PUNCT
ejpam-6569	213	38	∈	∈	PROPN
ejpam-6569	213	39	f−(v	f−(v	NOUN
ejpam-6569	213	40	)	)	PUNCT
ejpam-6569	213	41	⊆	⊆	NUM
ejpam-6569	213	42	⋆θsint(f	⋆θsint(f	NOUN
ejpam-6569	213	43	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	213	44	-	-	NOUN
ejpam-6569	213	45	cl(v	cl(v	NOUN
ejpam-6569	213	46	)	)	PUNCT
ejpam-6569	213	47	)	)	PUNCT
ejpam-6569	213	48	)	)	PUNCT
ejpam-6569	213	49	.	.	PUNCT
ejpam-6569	214	1	then	then	ADV
ejpam-6569	214	2	,	,	PUNCT
ejpam-6569	214	3	there	there	PRON
ejpam-6569	214	4	exists	exist	VERB
ejpam-6569	214	5	a	a	DET
ejpam-6569	214	6	semi	semi	NOUN
ejpam-6569	214	7	-	-	ADJ
ejpam-6569	214	8	i	i	PRON
ejpam-6569	214	9	⋆-open	⋆-open	VERB
ejpam-6569	214	10	set	set	VERB
ejpam-6569	214	11	u	u	NOUN
ejpam-6569	214	12	of	of	ADP
ejpam-6569	214	13	x	x	PUNCT
ejpam-6569	214	14	containing	contain	VERB
ejpam-6569	214	15	x	x	PUNCT
ejpam-6569	214	16	such	such	ADJ
ejpam-6569	214	17	that	that	SCONJ
ejpam-6569	214	18	scl⋆(u	scl⋆(u	NUM
ejpam-6569	214	19	)	)	PUNCT
ejpam-6569	214	20	⊆	⊆	NUM
ejpam-6569	214	21	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	214	22	-	-	PUNCT
ejpam-6569	214	23	cl(v	cl(v	NOUN
ejpam-6569	214	24	)	)	PUNCT
ejpam-6569	214	25	)	)	PUNCT
ejpam-6569	214	26	;	;	PUNCT
ejpam-6569	214	27	hence	hence	ADV
ejpam-6569	214	28	σ1σ2	σ1σ2	NOUN
ejpam-6569	214	29	-	-	PUNCT
ejpam-6569	214	30	cl(v	cl(v	NOUN
ejpam-6569	214	31	)	)	PUNCT
ejpam-6569	214	32	∩	∩	PROPN
ejpam-6569	214	33	f	f	X
ejpam-6569	214	34	(	(	PUNCT
ejpam-6569	214	35	z	z	NOUN
ejpam-6569	214	36	)	)	PUNCT
ejpam-6569	214	37	̸=	̸=	NOUN
ejpam-6569	214	38	∅	∅	NOUN
ejpam-6569	214	39	for	for	ADP
ejpam-6569	214	40	every	every	DET
ejpam-6569	214	41	z	z	PROPN
ejpam-6569	214	42	∈	∈	PROPN
ejpam-6569	214	43	scl⋆(u	scl⋆(u	NUM
ejpam-6569	214	44	)	)	PUNCT
ejpam-6569	214	45	.	.	PUNCT
ejpam-6569	215	1	this	this	PRON
ejpam-6569	215	2	shows	show	VERB
ejpam-6569	215	3	that	that	SCONJ
ejpam-6569	215	4	f	f	PROPN
ejpam-6569	215	5	is	be	AUX
ejpam-6569	215	6	lower	low	ADJ
ejpam-6569	215	7	quasi	quasi	NOUN
ejpam-6569	215	8	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	215	9	,	,	PUNCT
ejpam-6569	215	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	215	11	.	.	PUNCT
ejpam-6569	216	1	(	(	PUNCT
ejpam-6569	216	2	2	2	X
ejpam-6569	216	3	)	)	PUNCT
ejpam-6569	216	4	⇒	⇒	NOUN
ejpam-6569	216	5	(	(	PUNCT
ejpam-6569	216	6	5	5	NUM
ejpam-6569	216	7	):	):	PUNCT
ejpam-6569	216	8	let	let	VERB
ejpam-6569	216	9	a	a	PRON
ejpam-6569	216	10	be	be	AUX
ejpam-6569	216	11	any	any	DET
ejpam-6569	216	12	subset	subset	NOUN
ejpam-6569	216	13	of	of	ADP
ejpam-6569	216	14	x.	x.	NOUN
ejpam-6569	216	15	by	by	ADP
ejpam-6569	216	16	replacing	replace	VERB
ejpam-6569	216	17	b	b	NOUN
ejpam-6569	216	18	in	in	ADP
ejpam-6569	216	19	(	(	PUNCT
ejpam-6569	216	20	2	2	NUM
ejpam-6569	216	21	)	)	PUNCT
ejpam-6569	216	22	by	by	ADP
ejpam-6569	216	23	f	f	PROPN
ejpam-6569	216	24	(	(	PUNCT
ejpam-6569	216	25	a	a	PROPN
ejpam-6569	216	26	)	)	PUNCT
ejpam-6569	216	27	,	,	PUNCT
ejpam-6569	216	28	we	we	PRON
ejpam-6569	216	29	have	have	VERB
ejpam-6569	216	30	⋆θscl(a	⋆θscl(a	NUM
ejpam-6569	216	31	)	)	PUNCT
ejpam-6569	216	32	⊆	⊆	NUM
ejpam-6569	216	33	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	216	34	+	+	PROPN
ejpam-6569	216	35	(	(	PUNCT
ejpam-6569	216	36	f	f	X
ejpam-6569	216	37	(	(	PUNCT
ejpam-6569	216	38	a	a	NOUN
ejpam-6569	216	39	)	)	PUNCT
ejpam-6569	216	40	)	)	PUNCT
ejpam-6569	216	41	)	)	PUNCT
ejpam-6569	216	42	⊆	⊆	NUM
ejpam-6569	216	43	f+((σ1	f+((σ1	NOUN
ejpam-6569	216	44	,	,	PUNCT
ejpam-6569	216	45	σ2)θ	σ2)θ	NOUN
ejpam-6569	216	46	-	-	PUNCT
ejpam-6569	216	47	cl(f	cl(f	PROPN
ejpam-6569	216	48	(	(	PUNCT
ejpam-6569	216	49	a	a	NOUN
ejpam-6569	216	50	)	)	PUNCT
ejpam-6569	216	51	)	)	PUNCT
ejpam-6569	216	52	)	)	PUNCT
ejpam-6569	216	53	.	.	PUNCT
ejpam-6569	217	1	thus	thus	ADV
ejpam-6569	217	2	,	,	PUNCT
ejpam-6569	217	3	f	f	PROPN
ejpam-6569	217	4	(	(	PUNCT
ejpam-6569	217	5	⋆θscl(a	⋆θscl(a	PROPN
ejpam-6569	217	6	)	)	PUNCT
ejpam-6569	217	7	)	)	PUNCT
ejpam-6569	217	8	⊆	⊆	NUM
ejpam-6569	217	9	(	(	PUNCT
ejpam-6569	217	10	σ1	σ1	PROPN
ejpam-6569	217	11	,	,	PUNCT
ejpam-6569	217	12	σ2)θ	σ2)θ	NOUN
ejpam-6569	217	13	-	-	PUNCT
ejpam-6569	217	14	cl(f	cl(f	PROPN
ejpam-6569	217	15	(	(	PUNCT
ejpam-6569	217	16	a	a	NOUN
ejpam-6569	217	17	)	)	PUNCT
ejpam-6569	217	18	)	)	PUNCT
ejpam-6569	217	19	.	.	PUNCT
ejpam-6569	218	1	n.	n.	PROPN
ejpam-6569	218	2	srisarakham	srisarakham	PROPN
ejpam-6569	218	3	,	,	PUNCT
ejpam-6569	218	4	a.	a.	PROPN
ejpam-6569	218	5	sama	sama	PROPN
ejpam-6569	218	6	-	-	PUNCT
ejpam-6569	218	7	ae	ae	PROPN
ejpam-6569	218	8	,	,	PUNCT
ejpam-6569	218	9	c.	c.	PROPN
ejpam-6569	218	10	boonpok	boonpok	PROPN
ejpam-6569	218	11	/	/	SYM
ejpam-6569	218	12	eur	eur	PROPN
ejpam-6569	218	13	.	.	PUNCT
ejpam-6569	219	1	j.	j.	PROPN
ejpam-6569	219	2	pure	pure	PROPN
ejpam-6569	219	3	appl	appl	PROPN
ejpam-6569	219	4	.	.	PROPN
ejpam-6569	219	5	math	math	PROPN
ejpam-6569	219	6	,	,	PUNCT
ejpam-6569	219	7	18	18	NUM
ejpam-6569	219	8	(	(	PUNCT
ejpam-6569	219	9	3	3	NUM
ejpam-6569	219	10	)	)	PUNCT
ejpam-6569	219	11	(	(	PUNCT
ejpam-6569	219	12	2025	2025	NUM
ejpam-6569	219	13	)	)	PUNCT
ejpam-6569	219	14	,	,	PUNCT
ejpam-6569	219	15	6569	6569	NUM
ejpam-6569	219	16	7	7	NUM
ejpam-6569	219	17	of	of	ADP
ejpam-6569	219	18	13	13	NUM
ejpam-6569	219	19	(	(	PUNCT
ejpam-6569	219	20	5	5	NUM
ejpam-6569	219	21	)	)	PUNCT
ejpam-6569	219	22	⇒	⇒	NOUN
ejpam-6569	219	23	(	(	PUNCT
ejpam-6569	219	24	2	2	NUM
ejpam-6569	219	25	):	):	PUNCT
ejpam-6569	219	26	let	let	VERB
ejpam-6569	219	27	b	b	X
ejpam-6569	219	28	be	be	AUX
ejpam-6569	219	29	any	any	DET
ejpam-6569	219	30	subset	subset	NOUN
ejpam-6569	219	31	of	of	ADP
ejpam-6569	219	32	y	y	PROPN
ejpam-6569	219	33	.	.	PUNCT
ejpam-6569	220	1	replacing	replace	VERB
ejpam-6569	220	2	a	a	DET
ejpam-6569	220	3	in	in	ADP
ejpam-6569	220	4	(	(	PUNCT
ejpam-6569	220	5	5	5	NUM
ejpam-6569	220	6	)	)	PUNCT
ejpam-6569	220	7	by	by	ADP
ejpam-6569	220	8	f+(b	f+(b	NOUN
ejpam-6569	220	9	)	)	PUNCT
ejpam-6569	220	10	,	,	PUNCT
ejpam-6569	220	11	we	we	PRON
ejpam-6569	220	12	have	have	VERB
ejpam-6569	220	13	f	f	PROPN
ejpam-6569	220	14	(	(	PUNCT
ejpam-6569	220	15	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	220	16	+	+	PROPN
ejpam-6569	220	17	(	(	PUNCT
ejpam-6569	220	18	b	b	NOUN
ejpam-6569	220	19	)	)	PUNCT
ejpam-6569	220	20	)	)	PUNCT
ejpam-6569	220	21	)	)	PUNCT
ejpam-6569	221	1	⊆	⊆	X
ejpam-6569	221	2	(	(	PUNCT
ejpam-6569	221	3	σ1	σ1	PROPN
ejpam-6569	221	4	,	,	PUNCT
ejpam-6569	221	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	221	6	-	-	PUNCT
ejpam-6569	221	7	cl(f	cl(f	PROPN
ejpam-6569	221	8	(	(	PUNCT
ejpam-6569	221	9	f+(b	f+(b	PROPN
ejpam-6569	221	10	)	)	PUNCT
ejpam-6569	221	11	)	)	PUNCT
ejpam-6569	221	12	)	)	PUNCT
ejpam-6569	222	1	⊆	⊆	X
ejpam-6569	222	2	(	(	PUNCT
ejpam-6569	222	3	σ1	σ1	PROPN
ejpam-6569	222	4	,	,	PUNCT
ejpam-6569	222	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	222	6	-	-	PUNCT
ejpam-6569	222	7	cl(b	cl(b	NOUN
ejpam-6569	222	8	)	)	PUNCT
ejpam-6569	222	9	and	and	CCONJ
ejpam-6569	222	10	hence	hence	ADV
ejpam-6569	222	11	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	222	12	+	+	PROPN
ejpam-6569	222	13	(	(	PUNCT
ejpam-6569	222	14	b	b	NOUN
ejpam-6569	222	15	)	)	PUNCT
ejpam-6569	222	16	)	)	PUNCT
ejpam-6569	223	1	⊆	⊆	NUM
ejpam-6569	223	2	f+((σ1	f+((σ1	NOUN
ejpam-6569	223	3	,	,	PUNCT
ejpam-6569	223	4	σ2)θ	σ2)θ	ADJ
ejpam-6569	223	5	-	-	PUNCT
ejpam-6569	223	6	cl(b	cl(b	NOUN
ejpam-6569	223	7	)	)	PUNCT
ejpam-6569	223	8	)	)	PUNCT
ejpam-6569	223	9	.	.	PUNCT
ejpam-6569	224	1	(	(	PUNCT
ejpam-6569	224	2	3	3	X
ejpam-6569	224	3	)	)	PUNCT
ejpam-6569	224	4	⇒	⇒	NOUN
ejpam-6569	224	5	(	(	PUNCT
ejpam-6569	224	6	6	6	NUM
ejpam-6569	224	7	):	):	PUNCT
ejpam-6569	224	8	let	let	VERB
ejpam-6569	224	9	b	b	X
ejpam-6569	224	10	be	be	AUX
ejpam-6569	224	11	any	any	DET
ejpam-6569	224	12	subset	subset	NOUN
ejpam-6569	224	13	of	of	ADP
ejpam-6569	224	14	y	y	PROPN
ejpam-6569	224	15	.	.	PUNCT
ejpam-6569	225	1	put	put	VERB
ejpam-6569	225	2	v	v	NOUN
ejpam-6569	225	3	=	=	SYM
ejpam-6569	225	4	σ1σ2	σ1σ2	NOUN
ejpam-6569	225	5	-	-	PUNCT
ejpam-6569	225	6	int((σ1	int((σ1	ADJ
ejpam-6569	225	7	,	,	PUNCT
ejpam-6569	225	8	σ2)θ	σ2)θ	ADJ
ejpam-6569	225	9	-	-	PUNCT
ejpam-6569	225	10	cl(b	cl(b	NOUN
ejpam-6569	225	11	)	)	PUNCT
ejpam-6569	225	12	)	)	PUNCT
ejpam-6569	225	13	in	in	ADP
ejpam-6569	225	14	(	(	PUNCT
ejpam-6569	225	15	3	3	NUM
ejpam-6569	225	16	)	)	PUNCT
ejpam-6569	225	17	.	.	PUNCT
ejpam-6569	226	1	then	then	ADV
ejpam-6569	226	2	,	,	PUNCT
ejpam-6569	226	3	since	since	SCONJ
ejpam-6569	226	4	(	(	PUNCT
ejpam-6569	226	5	σ1	σ1	PROPN
ejpam-6569	226	6	,	,	PUNCT
ejpam-6569	226	7	σ2)θ	σ2)θ	NOUN
ejpam-6569	226	8	-	-	PUNCT
ejpam-6569	226	9	cl(b	cl(b	NOUN
ejpam-6569	226	10	)	)	PUNCT
ejpam-6569	226	11	is	be	AUX
ejpam-6569	226	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	226	13	-	-	ADJ
ejpam-6569	226	14	closed	closed	ADJ
ejpam-6569	226	15	in	in	ADP
ejpam-6569	226	16	y	y	PROPN
ejpam-6569	226	17	,	,	PUNCT
ejpam-6569	226	18	we	we	PRON
ejpam-6569	226	19	have	have	VERB
ejpam-6569	226	20	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	226	21	+	+	ADJ
ejpam-6569	226	22	(	(	PUNCT
ejpam-6569	226	23	σ1σ2	σ1σ2	NOUN
ejpam-6569	226	24	-	-	PUNCT
ejpam-6569	226	25	int((σ1	int((σ1	ADJ
ejpam-6569	226	26	,	,	PUNCT
ejpam-6569	226	27	σ2)θ	σ2)θ	ADJ
ejpam-6569	226	28	-	-	PUNCT
ejpam-6569	226	29	cl(b	cl(b	NOUN
ejpam-6569	226	30	)	)	PUNCT
ejpam-6569	226	31	)	)	PUNCT
ejpam-6569	226	32	)	)	PUNCT
ejpam-6569	226	33	)	)	PUNCT
ejpam-6569	227	1	⊆	⊆	X
ejpam-6569	227	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	227	3	-	-	PUNCT
ejpam-6569	227	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6569	227	5	-	-	PUNCT
ejpam-6569	227	6	int((σ1	int((σ1	ADJ
ejpam-6569	227	7	,	,	PUNCT
ejpam-6569	227	8	σ2)θ	σ2)θ	ADJ
ejpam-6569	227	9	-	-	PUNCT
ejpam-6569	227	10	cl(b	cl(b	NOUN
ejpam-6569	227	11	)	)	PUNCT
ejpam-6569	227	12	)	)	PUNCT
ejpam-6569	227	13	)	)	PUNCT
ejpam-6569	227	14	)	)	PUNCT
ejpam-6569	228	1	⊆	⊆	NUM
ejpam-6569	228	2	f+((σ1	f+((σ1	NOUN
ejpam-6569	228	3	,	,	PUNCT
ejpam-6569	228	4	σ2)θ	σ2)θ	ADJ
ejpam-6569	228	5	-	-	PUNCT
ejpam-6569	228	6	cl(b	cl(b	NOUN
ejpam-6569	228	7	)	)	PUNCT
ejpam-6569	228	8	)	)	PUNCT
ejpam-6569	228	9	.	.	PUNCT
ejpam-6569	229	1	(	(	PUNCT
ejpam-6569	229	2	6	6	X
ejpam-6569	229	3	)	)	PUNCT
ejpam-6569	229	4	⇒	⇒	NOUN
ejpam-6569	229	5	(	(	PUNCT
ejpam-6569	229	6	7	7	NUM
ejpam-6569	229	7	):	):	PUNCT
ejpam-6569	229	8	this	this	PRON
ejpam-6569	229	9	is	be	AUX
ejpam-6569	229	10	obvious	obvious	ADJ
ejpam-6569	229	11	since	since	SCONJ
ejpam-6569	229	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	229	13	-	-	NOUN
ejpam-6569	229	14	cl(v	cl(v	X
ejpam-6569	229	15	)	)	PUNCT
ejpam-6569	230	1	=	=	SYM
ejpam-6569	230	2	(	(	PUNCT
ejpam-6569	230	3	σ1	σ1	PROPN
ejpam-6569	230	4	,	,	PUNCT
ejpam-6569	230	5	σ2)θ	σ2)θ	NOUN
ejpam-6569	230	6	-	-	PUNCT
ejpam-6569	230	7	cl(v	cl(v	NOUN
ejpam-6569	230	8	)	)	PUNCT
ejpam-6569	230	9	for	for	ADP
ejpam-6569	230	10	every	every	DET
ejpam-6569	230	11	σ1σ2	σ1σ2	NOUN
ejpam-6569	230	12	-	-	ADJ
ejpam-6569	230	13	open	open	ADJ
ejpam-6569	230	14	set	set	NOUN
ejpam-6569	230	15	v	v	NOUN
ejpam-6569	230	16	of	of	ADP
ejpam-6569	230	17	y	y	PROPN
ejpam-6569	230	18	.	.	PUNCT
ejpam-6569	231	1	(	(	PUNCT
ejpam-6569	231	2	7	7	X
ejpam-6569	231	3	)	)	PUNCT
ejpam-6569	231	4	⇒	⇒	NOUN
ejpam-6569	231	5	(	(	PUNCT
ejpam-6569	231	6	8)	8)	NUM
ejpam-6569	231	7	:	:	PUNCT
ejpam-6569	231	8	let	let	VERB
ejpam-6569	231	9	k	k	X
ejpam-6569	231	10	be	be	AUX
ejpam-6569	231	11	any	any	DET
ejpam-6569	231	12	(	(	PUNCT
ejpam-6569	231	13	σ1	σ1	NOUN
ejpam-6569	231	14	,	,	PUNCT
ejpam-6569	231	15	σ2)r	σ2)r	NOUN
ejpam-6569	231	16	-	-	PUNCT
ejpam-6569	231	17	closed	close	VERB
ejpam-6569	231	18	set	set	NOUN
ejpam-6569	231	19	of	of	ADP
ejpam-6569	231	20	y	y	PROPN
ejpam-6569	231	21	.	.	PUNCT
ejpam-6569	232	1	then	then	ADV
ejpam-6569	232	2	by	by	ADP
ejpam-6569	232	3	(	(	PUNCT
ejpam-6569	232	4	7	7	NUM
ejpam-6569	232	5	)	)	PUNCT
ejpam-6569	232	6	,	,	PUNCT
ejpam-6569	232	7	we	we	PRON
ejpam-6569	232	8	have	have	VERB
ejpam-6569	232	9	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	232	10	+	+	ADJ
ejpam-6569	232	11	(	(	PUNCT
ejpam-6569	232	12	σ1σ2	σ1σ2	NUM
ejpam-6569	232	13	-	-	PUNCT
ejpam-6569	232	14	int(k	int(k	NOUN
ejpam-6569	232	15	)	)	PUNCT
ejpam-6569	232	16	)	)	PUNCT
ejpam-6569	232	17	)	)	PUNCT
ejpam-6569	233	1	=	=	PUNCT
ejpam-6569	234	1	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	234	2	+	+	ADJ
ejpam-6569	234	3	(	(	PUNCT
ejpam-6569	234	4	σ1σ2	σ1σ2	NUM
ejpam-6569	234	5	-	-	PUNCT
ejpam-6569	234	6	int(σ1σ2	int(σ1σ2	ADV
ejpam-6569	234	7	-	-	PUNCT
ejpam-6569	234	8	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6569	234	9	-	-	PUNCT
ejpam-6569	234	10	int(k	int(k	NOUN
ejpam-6569	234	11	)	)	PUNCT
ejpam-6569	234	12	)	)	PUNCT
ejpam-6569	234	13	)	)	PUNCT
ejpam-6569	234	14	)	)	PUNCT
ejpam-6569	234	15	)	)	PUNCT
ejpam-6569	235	1	⊆	⊆	X
ejpam-6569	235	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	235	3	-	-	PUNCT
ejpam-6569	235	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	235	5	-	-	PUNCT
ejpam-6569	235	6	int(k	int(k	NOUN
ejpam-6569	235	7	)	)	PUNCT
ejpam-6569	235	8	)	)	PUNCT
ejpam-6569	235	9	)	)	PUNCT
ejpam-6569	236	1	=	=	SYM
ejpam-6569	236	2	f+(k	f+(k	NOUN
ejpam-6569	236	3	)	)	PUNCT
ejpam-6569	236	4	.	.	PUNCT
ejpam-6569	237	1	(	(	PUNCT
ejpam-6569	237	2	8)	8)	NUM
ejpam-6569	237	3	⇒	⇒	NOUN
ejpam-6569	237	4	(	(	PUNCT
ejpam-6569	237	5	9	9	NUM
ejpam-6569	237	6	):	):	PUNCT
ejpam-6569	237	7	letk	letk	ADJ
ejpam-6569	237	8	be	be	AUX
ejpam-6569	237	9	any	any	DET
ejpam-6569	237	10	σ1σ2	σ1σ2	NUM
ejpam-6569	237	11	-	-	PUNCT
ejpam-6569	237	12	closed	closed	ADJ
ejpam-6569	237	13	set	set	NOUN
ejpam-6569	237	14	of	of	ADP
ejpam-6569	237	15	y	y	PROPN
ejpam-6569	237	16	.	.	PUNCT
ejpam-6569	238	1	then	then	ADV
ejpam-6569	238	2	,	,	PUNCT
ejpam-6569	238	3	σ1σ2	σ1σ2	X
ejpam-6569	238	4	-	-	PUNCT
ejpam-6569	238	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	238	6	-	-	PUNCT
ejpam-6569	238	7	int(k	int(k	NOUN
ejpam-6569	238	8	)	)	PUNCT
ejpam-6569	238	9	)	)	PUNCT
ejpam-6569	238	10	is	be	AUX
ejpam-6569	238	11	(	(	PUNCT
ejpam-6569	238	12	σ1	σ1	PROPN
ejpam-6569	238	13	,	,	PUNCT
ejpam-6569	238	14	σ2)rclosed	σ2)rclose	VERB
ejpam-6569	238	15	in	in	ADP
ejpam-6569	238	16	y	y	PROPN
ejpam-6569	238	17	and	and	CCONJ
ejpam-6569	238	18	by	by	ADP
ejpam-6569	238	19	(	(	PUNCT
ejpam-6569	238	20	8)	8)	NUM
ejpam-6569	238	21	,	,	PUNCT
ejpam-6569	238	22	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	239	1	+	+	ADJ
ejpam-6569	239	2	(	(	PUNCT
ejpam-6569	239	3	σ1σ2	σ1σ2	NUM
ejpam-6569	239	4	-	-	PUNCT
ejpam-6569	239	5	int(k	int(k	NOUN
ejpam-6569	239	6	)	)	PUNCT
ejpam-6569	239	7	)	)	PUNCT
ejpam-6569	239	8	)	)	PUNCT
ejpam-6569	240	1	=	=	PUNCT
ejpam-6569	241	1	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	241	2	+	+	ADJ
ejpam-6569	241	3	(	(	PUNCT
ejpam-6569	241	4	σ1σ2	σ1σ2	NUM
ejpam-6569	241	5	-	-	PUNCT
ejpam-6569	241	6	int(σ1σ2	int(σ1σ2	ADV
ejpam-6569	241	7	-	-	PUNCT
ejpam-6569	241	8	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6569	241	9	-	-	PUNCT
ejpam-6569	241	10	int(k	int(k	NOUN
ejpam-6569	241	11	)	)	PUNCT
ejpam-6569	241	12	)	)	PUNCT
ejpam-6569	241	13	)	)	PUNCT
ejpam-6569	241	14	)	)	PUNCT
ejpam-6569	241	15	)	)	PUNCT
ejpam-6569	242	1	⊆	⊆	X
ejpam-6569	242	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	242	3	-	-	PUNCT
ejpam-6569	242	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	242	5	-	-	PUNCT
ejpam-6569	242	6	int(k	int(k	NOUN
ejpam-6569	242	7	)	)	PUNCT
ejpam-6569	242	8	)	)	PUNCT
ejpam-6569	242	9	)	)	PUNCT
ejpam-6569	243	1	⊆	⊆	NUM
ejpam-6569	243	2	f+(k	f+(k	NOUN
ejpam-6569	243	3	)	)	PUNCT
ejpam-6569	243	4	.	.	PUNCT
ejpam-6569	244	1	(	(	PUNCT
ejpam-6569	244	2	9	9	X
ejpam-6569	244	3	)	)	PUNCT
ejpam-6569	244	4	⇒	⇒	NOUN
ejpam-6569	244	5	(	(	PUNCT
ejpam-6569	244	6	4	4	NUM
ejpam-6569	244	7	):	):	PUNCT
ejpam-6569	244	8	let	let	VERB
ejpam-6569	244	9	v	v	PART
ejpam-6569	244	10	be	be	AUX
ejpam-6569	244	11	any	any	DET
ejpam-6569	244	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	244	13	-	-	ADJ
ejpam-6569	244	14	open	open	ADJ
ejpam-6569	244	15	set	set	NOUN
ejpam-6569	244	16	of	of	ADP
ejpam-6569	244	17	y	y	PROPN
ejpam-6569	244	18	.	.	PUNCT
ejpam-6569	245	1	then	then	ADV
ejpam-6569	245	2	,	,	PUNCT
ejpam-6569	245	3	y	y	PROPN
ejpam-6569	245	4	−	−	PROPN
ejpam-6569	245	5	v	v	NOUN
ejpam-6569	245	6	is	be	AUX
ejpam-6569	245	7	σ1σ2	σ1σ2	NOUN
ejpam-6569	245	8	-	-	ADJ
ejpam-6569	245	9	closed	closed	ADJ
ejpam-6569	245	10	in	in	ADP
ejpam-6569	245	11	y	y	PROPN
ejpam-6569	245	12	and	and	CCONJ
ejpam-6569	245	13	by	by	ADP
ejpam-6569	245	14	(	(	PUNCT
ejpam-6569	245	15	9	9	NUM
ejpam-6569	245	16	)	)	PUNCT
ejpam-6569	245	17	,	,	PUNCT
ejpam-6569	245	18	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	245	19	+	+	ADJ
ejpam-6569	245	20	(	(	PUNCT
ejpam-6569	245	21	σ1σ2	σ1σ2	NUM
ejpam-6569	245	22	-	-	PUNCT
ejpam-6569	245	23	int(y	int(y	ADJ
ejpam-6569	245	24	−	−	PROPN
ejpam-6569	245	25	v	v	NOUN
ejpam-6569	245	26	)	)	PUNCT
ejpam-6569	245	27	)	)	PUNCT
ejpam-6569	245	28	)	)	PUNCT
ejpam-6569	246	1	⊆	⊆	NUM
ejpam-6569	246	2	f+(y	f+(y	ADP
ejpam-6569	246	3	−	−	PROPN
ejpam-6569	246	4	v	v	NOUN
ejpam-6569	246	5	)	)	PUNCT
ejpam-6569	246	6	=	=	PUNCT
ejpam-6569	246	7	x	x	SYM
ejpam-6569	246	8	−	−	NOUN
ejpam-6569	246	9	f−(v	f−(v	NOUN
ejpam-6569	246	10	)	)	PUNCT
ejpam-6569	246	11	.	.	PUNCT
ejpam-6569	247	1	moreover	moreover	ADV
ejpam-6569	247	2	,	,	PUNCT
ejpam-6569	247	3	we	we	PRON
ejpam-6569	247	4	have	have	VERB
ejpam-6569	247	5	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	248	1	+	+	ADJ
ejpam-6569	248	2	(	(	PUNCT
ejpam-6569	248	3	σ1σ2	σ1σ2	NUM
ejpam-6569	248	4	-	-	PUNCT
ejpam-6569	248	5	int(y	int(y	ADJ
ejpam-6569	248	6	−	−	PROPN
ejpam-6569	248	7	v	v	NOUN
ejpam-6569	248	8	)	)	PUNCT
ejpam-6569	248	9	)	)	PUNCT
ejpam-6569	248	10	)	)	PUNCT
ejpam-6569	249	1	=	=	PUNCT
ejpam-6569	250	1	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	250	2	+	+	PROPN
ejpam-6569	250	3	(	(	PUNCT
ejpam-6569	250	4	y	y	PROPN
ejpam-6569	250	5	−	−	PROPN
ejpam-6569	250	6	σ1σ2	σ1σ2	NOUN
ejpam-6569	250	7	-	-	NUM
ejpam-6569	250	8	cl(v	cl(v	NOUN
ejpam-6569	250	9	)	)	PUNCT
ejpam-6569	250	10	)	)	PUNCT
ejpam-6569	250	11	)	)	PUNCT
ejpam-6569	251	1	=	=	SYM
ejpam-6569	251	2	⋆θscl(x	⋆θscl(x	PROPN
ejpam-6569	251	3	−	−	PROPN
ejpam-6569	251	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	251	5	-	-	PUNCT
ejpam-6569	251	6	cl(v	cl(v	NOUN
ejpam-6569	251	7	)	)	PUNCT
ejpam-6569	251	8	)	)	PUNCT
ejpam-6569	251	9	)	)	PUNCT
ejpam-6569	252	1	=	=	PUNCT
ejpam-6569	252	2	x	x	PUNCT
ejpam-6569	252	3	−	−	PROPN
ejpam-6569	252	4	⋆θsint(f	⋆θsint(f	NUM
ejpam-6569	252	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	252	6	-	-	NOUN
ejpam-6569	252	7	cl(v	cl(v	NOUN
ejpam-6569	252	8	)	)	PUNCT
ejpam-6569	252	9	)	)	PUNCT
ejpam-6569	252	10	)	)	PUNCT
ejpam-6569	252	11	.	.	PUNCT
ejpam-6569	253	1	thus	thus	ADV
ejpam-6569	253	2	,	,	PUNCT
ejpam-6569	253	3	f−(v	f−(v	ADJ
ejpam-6569	253	4	)	)	PUNCT
ejpam-6569	253	5	⊆	⊆	NUM
ejpam-6569	253	6	⋆θsint(f	⋆θsint(f	NOUN
ejpam-6569	253	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	253	8	-	-	NOUN
ejpam-6569	253	9	cl(v	cl(v	NOUN
ejpam-6569	253	10	)	)	PUNCT
ejpam-6569	253	11	)	)	PUNCT
ejpam-6569	253	12	)	)	PUNCT
ejpam-6569	253	13	.	.	PUNCT
ejpam-6569	254	1	theorem	theorem	NOUN
ejpam-6569	254	2	3	3	NUM
ejpam-6569	254	3	.	.	X
ejpam-6569	254	4	for	for	ADP
ejpam-6569	254	5	a	a	DET
ejpam-6569	254	6	multifunction	multifunction	NOUN
ejpam-6569	255	1	f	f	NOUN
ejpam-6569	255	2	:	:	PUNCT
ejpam-6569	255	3	(	(	PUNCT
ejpam-6569	255	4	x	x	X
ejpam-6569	255	5	,	,	PUNCT
ejpam-6569	255	6	τ	τ	PROPN
ejpam-6569	255	7	,	,	PUNCT
ejpam-6569	255	8	i	i	NOUN
ejpam-6569	255	9	)	)	PUNCT
ejpam-6569	255	10	→	→	PUNCT
ejpam-6569	255	11	(	(	PUNCT
ejpam-6569	255	12	y	y	PROPN
ejpam-6569	255	13	,	,	PUNCT
ejpam-6569	255	14	σ1	σ1	PROPN
ejpam-6569	255	15	,	,	PUNCT
ejpam-6569	255	16	σ2	σ2	NOUN
ejpam-6569	255	17	)	)	PUNCT
ejpam-6569	255	18	,	,	PUNCT
ejpam-6569	255	19	the	the	DET
ejpam-6569	255	20	following	follow	VERB
ejpam-6569	255	21	properties	property	NOUN
ejpam-6569	255	22	are	be	AUX
ejpam-6569	255	23	equivalent	equivalent	ADJ
ejpam-6569	255	24	:	:	PUNCT
ejpam-6569	255	25	(	(	PUNCT
ejpam-6569	255	26	1	1	X
ejpam-6569	255	27	)	)	PUNCT
ejpam-6569	255	28	f	f	PROPN
ejpam-6569	255	29	is	be	AUX
ejpam-6569	255	30	upper	upper	ADJ
ejpam-6569	255	31	quasi	quasi	NOUN
ejpam-6569	255	32	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	255	33	,	,	PUNCT
ejpam-6569	255	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	255	35	;	;	PUNCT
ejpam-6569	255	36	(	(	PUNCT
ejpam-6569	255	37	2	2	X
ejpam-6569	255	38	)	)	PUNCT
ejpam-6569	255	39	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	255	40	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	255	41	-	-	PUNCT
ejpam-6569	255	42	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	255	43	-	-	PUNCT
ejpam-6569	255	44	cl(v	cl(v	NOUN
ejpam-6569	255	45	)	)	PUNCT
ejpam-6569	255	46	)	)	PUNCT
ejpam-6569	255	47	)	)	PUNCT
ejpam-6569	255	48	)	)	PUNCT
ejpam-6569	256	1	⊆	⊆	X
ejpam-6569	256	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	256	3	-	-	PUNCT
ejpam-6569	256	4	cl(v	cl(v	NOUN
ejpam-6569	256	5	)	)	PUNCT
ejpam-6569	256	6	)	)	PUNCT
ejpam-6569	256	7	for	for	ADP
ejpam-6569	256	8	every	every	DET
ejpam-6569	256	9	(	(	PUNCT
ejpam-6569	256	10	σ1	σ1	PROPN
ejpam-6569	256	11	,	,	PUNCT
ejpam-6569	256	12	σ2)β	σ2)β	NOUN
ejpam-6569	256	13	-	-	PUNCT
ejpam-6569	256	14	open	open	NOUN
ejpam-6569	256	15	set	set	NOUN
ejpam-6569	256	16	v	v	NOUN
ejpam-6569	256	17	of	of	ADP
ejpam-6569	256	18	y	y	PROPN
ejpam-6569	256	19	;	;	PUNCT
ejpam-6569	256	20	(	(	PUNCT
ejpam-6569	256	21	3	3	X
ejpam-6569	256	22	)	)	PUNCT
ejpam-6569	256	23	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	256	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	256	25	-	-	PUNCT
ejpam-6569	256	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	256	27	-	-	PUNCT
ejpam-6569	256	28	cl(v	cl(v	NOUN
ejpam-6569	256	29	)	)	PUNCT
ejpam-6569	256	30	)	)	PUNCT
ejpam-6569	256	31	)	)	PUNCT
ejpam-6569	256	32	)	)	PUNCT
ejpam-6569	257	1	⊆	⊆	X
ejpam-6569	257	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	257	3	-	-	PUNCT
ejpam-6569	257	4	cl(v	cl(v	NOUN
ejpam-6569	257	5	)	)	PUNCT
ejpam-6569	257	6	)	)	PUNCT
ejpam-6569	257	7	for	for	ADP
ejpam-6569	257	8	every	every	DET
ejpam-6569	257	9	(	(	PUNCT
ejpam-6569	257	10	σ1	σ1	PROPN
ejpam-6569	257	11	,	,	PUNCT
ejpam-6569	257	12	σ2)s	σ2)s	NOUN
ejpam-6569	257	13	-	-	PUNCT
ejpam-6569	257	14	open	open	NOUN
ejpam-6569	257	15	set	set	NOUN
ejpam-6569	257	16	v	v	NOUN
ejpam-6569	257	17	of	of	ADP
ejpam-6569	257	18	y	y	PROPN
ejpam-6569	257	19	.	.	PUNCT
ejpam-6569	258	1	n.	n.	PROPN
ejpam-6569	258	2	srisarakham	srisarakham	PROPN
ejpam-6569	258	3	,	,	PUNCT
ejpam-6569	258	4	a.	a.	PROPN
ejpam-6569	258	5	sama	sama	PROPN
ejpam-6569	258	6	-	-	PUNCT
ejpam-6569	258	7	ae	ae	PROPN
ejpam-6569	258	8	,	,	PUNCT
ejpam-6569	258	9	c.	c.	PROPN
ejpam-6569	258	10	boonpok	boonpok	PROPN
ejpam-6569	258	11	/	/	SYM
ejpam-6569	258	12	eur	eur	PROPN
ejpam-6569	258	13	.	.	PUNCT
ejpam-6569	259	1	j.	j.	PROPN
ejpam-6569	259	2	pure	pure	PROPN
ejpam-6569	259	3	appl	appl	PROPN
ejpam-6569	259	4	.	.	PROPN
ejpam-6569	259	5	math	math	PROPN
ejpam-6569	259	6	,	,	PUNCT
ejpam-6569	259	7	18	18	NUM
ejpam-6569	259	8	(	(	PUNCT
ejpam-6569	259	9	3	3	NUM
ejpam-6569	259	10	)	)	PUNCT
ejpam-6569	259	11	(	(	PUNCT
ejpam-6569	259	12	2025	2025	NUM
ejpam-6569	259	13	)	)	PUNCT
ejpam-6569	259	14	,	,	PUNCT
ejpam-6569	259	15	6569	6569	NUM
ejpam-6569	259	16	8	8	NUM
ejpam-6569	259	17	of	of	ADP
ejpam-6569	259	18	13	13	NUM
ejpam-6569	259	19	proof	proof	NOUN
ejpam-6569	259	20	.	.	PUNCT
ejpam-6569	260	1	(	(	PUNCT
ejpam-6569	260	2	1	1	X
ejpam-6569	260	3	)	)	PUNCT
ejpam-6569	260	4	⇒	⇒	NOUN
ejpam-6569	260	5	(	(	PUNCT
ejpam-6569	260	6	2	2	NUM
ejpam-6569	260	7	):	):	PUNCT
ejpam-6569	260	8	let	let	VERB
ejpam-6569	260	9	v	v	PART
ejpam-6569	260	10	be	be	AUX
ejpam-6569	260	11	any	any	DET
ejpam-6569	260	12	(	(	PUNCT
ejpam-6569	260	13	σ1	σ1	PROPN
ejpam-6569	260	14	,	,	PUNCT
ejpam-6569	260	15	σ2)β	σ2)β	NOUN
ejpam-6569	260	16	-	-	PUNCT
ejpam-6569	260	17	open	open	ADJ
ejpam-6569	260	18	set	set	NOUN
ejpam-6569	260	19	of	of	ADP
ejpam-6569	260	20	y	y	PROPN
ejpam-6569	260	21	.	.	PUNCT
ejpam-6569	261	1	then	then	ADV
ejpam-6569	261	2	,	,	PUNCT
ejpam-6569	261	3	v	v	ADP
ejpam-6569	261	4	⊆	⊆	NUM
ejpam-6569	261	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	261	6	-	-	PUNCT
ejpam-6569	261	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	261	8	-	-	PUNCT
ejpam-6569	261	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	261	10	-	-	PUNCT
ejpam-6569	261	11	cl(v	cl(v	NOUN
ejpam-6569	261	12	)	)	PUNCT
ejpam-6569	261	13	)	)	PUNCT
ejpam-6569	261	14	)	)	PUNCT
ejpam-6569	261	15	and	and	CCONJ
ejpam-6569	261	16	hence	hence	ADV
ejpam-6569	261	17	σ1σ2	σ1σ2	NOUN
ejpam-6569	261	18	-	-	NOUN
ejpam-6569	261	19	cl(v	cl(v	NOUN
ejpam-6569	261	20	)	)	PUNCT
ejpam-6569	262	1	=	=	SYM
ejpam-6569	262	2	σ1σ2	σ1σ2	X
ejpam-6569	262	3	-	-	PUNCT
ejpam-6569	262	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	262	5	-	-	PUNCT
ejpam-6569	262	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	262	7	-	-	PUNCT
ejpam-6569	262	8	cl(v	cl(v	NOUN
ejpam-6569	262	9	)	)	PUNCT
ejpam-6569	262	10	)	)	PUNCT
ejpam-6569	262	11	)	)	PUNCT
ejpam-6569	262	12	.	.	PUNCT
ejpam-6569	263	1	since	since	SCONJ
ejpam-6569	263	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	263	3	-	-	NOUN
ejpam-6569	263	4	cl(v	cl(v	NOUN
ejpam-6569	263	5	)	)	PUNCT
ejpam-6569	263	6	is	be	AUX
ejpam-6569	263	7	(	(	PUNCT
ejpam-6569	263	8	σ1	σ1	PROPN
ejpam-6569	263	9	,	,	PUNCT
ejpam-6569	263	10	σ2)rclosed	σ2)rclose	VERB
ejpam-6569	263	11	in	in	ADP
ejpam-6569	263	12	y	y	PROPN
ejpam-6569	263	13	,	,	PUNCT
ejpam-6569	263	14	by	by	ADP
ejpam-6569	263	15	theorem	theorem	NOUN
ejpam-6569	263	16	1	1	NUM
ejpam-6569	263	17	we	we	PRON
ejpam-6569	263	18	have	have	AUX
ejpam-6569	263	19	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	263	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	263	21	-	-	PUNCT
ejpam-6569	263	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	263	23	-	-	PUNCT
ejpam-6569	263	24	cl(v	cl(v	NOUN
ejpam-6569	263	25	)	)	PUNCT
ejpam-6569	263	26	)	)	PUNCT
ejpam-6569	263	27	)	)	PUNCT
ejpam-6569	263	28	)	)	PUNCT
ejpam-6569	264	1	⊆	⊆	X
ejpam-6569	264	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	264	3	-	-	PUNCT
ejpam-6569	264	4	cl(v	cl(v	NOUN
ejpam-6569	264	5	)	)	PUNCT
ejpam-6569	264	6	)	)	PUNCT
ejpam-6569	264	7	.	.	PUNCT
ejpam-6569	265	1	(	(	PUNCT
ejpam-6569	265	2	2	2	X
ejpam-6569	265	3	)	)	PUNCT
ejpam-6569	265	4	⇒	⇒	NOUN
ejpam-6569	265	5	(	(	PUNCT
ejpam-6569	265	6	3	3	NUM
ejpam-6569	265	7	):	):	PUNCT
ejpam-6569	265	8	the	the	DET
ejpam-6569	265	9	proof	proof	NOUN
ejpam-6569	265	10	is	be	AUX
ejpam-6569	265	11	obvious	obvious	ADJ
ejpam-6569	265	12	.	.	PUNCT
ejpam-6569	266	1	(	(	PUNCT
ejpam-6569	266	2	3	3	X
ejpam-6569	266	3	)	)	PUNCT
ejpam-6569	266	4	⇒	⇒	NOUN
ejpam-6569	266	5	(	(	PUNCT
ejpam-6569	266	6	1	1	NUM
ejpam-6569	266	7	):	):	PUNCT
ejpam-6569	266	8	let	let	VERB
ejpam-6569	266	9	v	v	PART
ejpam-6569	266	10	be	be	AUX
ejpam-6569	266	11	any	any	DET
ejpam-6569	266	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	266	13	-	-	ADJ
ejpam-6569	266	14	open	open	ADJ
ejpam-6569	266	15	set	set	NOUN
ejpam-6569	266	16	of	of	ADP
ejpam-6569	266	17	y	y	PROPN
ejpam-6569	266	18	.	.	PUNCT
ejpam-6569	267	1	then	then	ADV
ejpam-6569	267	2	,	,	PUNCT
ejpam-6569	267	3	v	v	NOUN
ejpam-6569	267	4	is	be	AUX
ejpam-6569	267	5	(	(	PUNCT
ejpam-6569	267	6	σ1	σ1	PROPN
ejpam-6569	267	7	,	,	PUNCT
ejpam-6569	267	8	σ2)s	σ2)s	NOUN
ejpam-6569	267	9	-	-	PUNCT
ejpam-6569	267	10	open	open	ADJ
ejpam-6569	267	11	in	in	ADP
ejpam-6569	267	12	y	y	PROPN
ejpam-6569	267	13	and	and	CCONJ
ejpam-6569	267	14	by	by	ADP
ejpam-6569	267	15	(	(	PUNCT
ejpam-6569	267	16	3	3	NUM
ejpam-6569	267	17	)	)	PUNCT
ejpam-6569	267	18	,	,	PUNCT
ejpam-6569	267	19	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	267	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	267	21	-	-	PUNCT
ejpam-6569	267	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	267	23	-	-	PUNCT
ejpam-6569	267	24	cl(v	cl(v	NOUN
ejpam-6569	267	25	)	)	PUNCT
ejpam-6569	267	26	)	)	PUNCT
ejpam-6569	267	27	)	)	PUNCT
ejpam-6569	267	28	)	)	PUNCT
ejpam-6569	268	1	⊆	⊆	X
ejpam-6569	268	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	268	3	-	-	PUNCT
ejpam-6569	268	4	cl(v	cl(v	NOUN
ejpam-6569	268	5	)	)	PUNCT
ejpam-6569	268	6	)	)	PUNCT
ejpam-6569	268	7	.	.	PUNCT
ejpam-6569	269	1	thus	thus	ADV
ejpam-6569	269	2	by	by	ADP
ejpam-6569	269	3	theorem	theorem	NOUN
ejpam-6569	269	4	1	1	NUM
ejpam-6569	269	5	,	,	PUNCT
ejpam-6569	269	6	f	f	PROPN
ejpam-6569	269	7	is	be	AUX
ejpam-6569	269	8	upper	upper	ADJ
ejpam-6569	269	9	quasi	quasi	ADJ
ejpam-6569	269	10	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	269	11	,	,	PUNCT
ejpam-6569	269	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	269	13	.	.	X
ejpam-6569	269	14	theorem	theorem	NOUN
ejpam-6569	269	15	4	4	NUM
ejpam-6569	269	16	.	.	X
ejpam-6569	269	17	for	for	ADP
ejpam-6569	269	18	a	a	DET
ejpam-6569	269	19	multifunction	multifunction	NOUN
ejpam-6569	269	20	f	f	NOUN
ejpam-6569	269	21	:	:	PUNCT
ejpam-6569	269	22	(	(	PUNCT
ejpam-6569	269	23	x	x	X
ejpam-6569	269	24	,	,	PUNCT
ejpam-6569	269	25	τ	τ	PROPN
ejpam-6569	269	26	,	,	PUNCT
ejpam-6569	269	27	i	i	NOUN
ejpam-6569	269	28	)	)	PUNCT
ejpam-6569	269	29	→	→	PUNCT
ejpam-6569	269	30	(	(	PUNCT
ejpam-6569	269	31	y	y	PROPN
ejpam-6569	269	32	,	,	PUNCT
ejpam-6569	269	33	σ1	σ1	PROPN
ejpam-6569	269	34	,	,	PUNCT
ejpam-6569	269	35	σ2	σ2	NOUN
ejpam-6569	269	36	)	)	PUNCT
ejpam-6569	269	37	,	,	PUNCT
ejpam-6569	269	38	the	the	DET
ejpam-6569	269	39	following	follow	VERB
ejpam-6569	269	40	properties	property	NOUN
ejpam-6569	269	41	are	be	AUX
ejpam-6569	269	42	equivalent	equivalent	ADJ
ejpam-6569	269	43	:	:	PUNCT
ejpam-6569	269	44	(	(	PUNCT
ejpam-6569	269	45	1	1	X
ejpam-6569	269	46	)	)	PUNCT
ejpam-6569	269	47	f	f	PROPN
ejpam-6569	269	48	is	be	AUX
ejpam-6569	269	49	lower	low	ADJ
ejpam-6569	269	50	quasi	quasi	NOUN
ejpam-6569	269	51	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	269	52	,	,	PUNCT
ejpam-6569	269	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	269	54	;	;	PUNCT
ejpam-6569	269	55	(	(	PUNCT
ejpam-6569	269	56	2	2	X
ejpam-6569	269	57	)	)	PUNCT
ejpam-6569	269	58	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	270	1	+	+	PROPN
ejpam-6569	270	2	(	(	PUNCT
ejpam-6569	270	3	σ1σ2	σ1σ2	NUM
ejpam-6569	270	4	-	-	PUNCT
ejpam-6569	270	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	270	6	-	-	PUNCT
ejpam-6569	270	7	cl(v	cl(v	NOUN
ejpam-6569	270	8	)	)	PUNCT
ejpam-6569	270	9	)	)	PUNCT
ejpam-6569	270	10	)	)	PUNCT
ejpam-6569	270	11	)	)	PUNCT
ejpam-6569	271	1	⊆	⊆	X
ejpam-6569	271	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	271	3	-	-	PUNCT
ejpam-6569	271	4	cl(v	cl(v	NOUN
ejpam-6569	271	5	)	)	PUNCT
ejpam-6569	271	6	)	)	PUNCT
ejpam-6569	271	7	for	for	ADP
ejpam-6569	271	8	every	every	DET
ejpam-6569	271	9	(	(	PUNCT
ejpam-6569	271	10	σ1	σ1	PROPN
ejpam-6569	271	11	,	,	PUNCT
ejpam-6569	271	12	σ2)β	σ2)β	NOUN
ejpam-6569	271	13	-	-	PUNCT
ejpam-6569	271	14	open	open	NOUN
ejpam-6569	271	15	set	set	NOUN
ejpam-6569	271	16	v	v	NOUN
ejpam-6569	271	17	of	of	ADP
ejpam-6569	271	18	y	y	PROPN
ejpam-6569	271	19	;	;	PUNCT
ejpam-6569	271	20	(	(	PUNCT
ejpam-6569	271	21	3	3	X
ejpam-6569	271	22	)	)	PUNCT
ejpam-6569	271	23	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	272	1	+	+	PROPN
ejpam-6569	272	2	(	(	PUNCT
ejpam-6569	272	3	σ1σ2	σ1σ2	NUM
ejpam-6569	272	4	-	-	PUNCT
ejpam-6569	272	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	272	6	-	-	PUNCT
ejpam-6569	272	7	cl(v	cl(v	NOUN
ejpam-6569	272	8	)	)	PUNCT
ejpam-6569	272	9	)	)	PUNCT
ejpam-6569	272	10	)	)	PUNCT
ejpam-6569	272	11	)	)	PUNCT
ejpam-6569	273	1	⊆	⊆	X
ejpam-6569	273	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	273	3	-	-	PUNCT
ejpam-6569	273	4	cl(v	cl(v	NOUN
ejpam-6569	273	5	)	)	PUNCT
ejpam-6569	273	6	)	)	PUNCT
ejpam-6569	273	7	for	for	SCONJ
ejpam-6569	273	8	every	every	DET
ejpam-6569	273	9	(	(	PUNCT
ejpam-6569	273	10	σ1	σ1	PROPN
ejpam-6569	273	11	,	,	PUNCT
ejpam-6569	273	12	σ2)s	σ2)s	NOUN
ejpam-6569	273	13	-	-	PUNCT
ejpam-6569	273	14	open	open	NOUN
ejpam-6569	273	15	set	set	NOUN
ejpam-6569	273	16	v	v	NOUN
ejpam-6569	273	17	of	of	ADP
ejpam-6569	273	18	y	y	PROPN
ejpam-6569	273	19	.	.	PUNCT
ejpam-6569	274	1	proof	proof	NOUN
ejpam-6569	274	2	.	.	PUNCT
ejpam-6569	275	1	the	the	DET
ejpam-6569	275	2	proof	proof	NOUN
ejpam-6569	275	3	is	be	AUX
ejpam-6569	275	4	similar	similar	ADJ
ejpam-6569	275	5	to	to	ADP
ejpam-6569	275	6	that	that	PRON
ejpam-6569	275	7	of	of	ADP
ejpam-6569	275	8	theorem	theorem	ADJ
ejpam-6569	275	9	3	3	NUM
ejpam-6569	275	10	.	.	PUNCT
ejpam-6569	275	11	theorem	theorem	NOUN
ejpam-6569	275	12	5	5	NUM
ejpam-6569	275	13	.	.	X
ejpam-6569	275	14	for	for	ADP
ejpam-6569	275	15	a	a	DET
ejpam-6569	275	16	multifunction	multifunction	NOUN
ejpam-6569	275	17	f	f	NOUN
ejpam-6569	275	18	:	:	PUNCT
ejpam-6569	275	19	(	(	PUNCT
ejpam-6569	275	20	x	x	X
ejpam-6569	275	21	,	,	PUNCT
ejpam-6569	275	22	τ	τ	PROPN
ejpam-6569	275	23	,	,	PUNCT
ejpam-6569	275	24	i	i	NOUN
ejpam-6569	275	25	)	)	PUNCT
ejpam-6569	275	26	→	→	PUNCT
ejpam-6569	275	27	(	(	PUNCT
ejpam-6569	275	28	y	y	PROPN
ejpam-6569	275	29	,	,	PUNCT
ejpam-6569	275	30	σ1	σ1	PROPN
ejpam-6569	275	31	,	,	PUNCT
ejpam-6569	275	32	σ2	σ2	NOUN
ejpam-6569	275	33	)	)	PUNCT
ejpam-6569	275	34	,	,	PUNCT
ejpam-6569	275	35	the	the	DET
ejpam-6569	275	36	following	follow	VERB
ejpam-6569	275	37	properties	property	NOUN
ejpam-6569	275	38	are	be	AUX
ejpam-6569	275	39	equivalent	equivalent	ADJ
ejpam-6569	275	40	:	:	PUNCT
ejpam-6569	275	41	(	(	PUNCT
ejpam-6569	275	42	1	1	X
ejpam-6569	275	43	)	)	PUNCT
ejpam-6569	275	44	f	f	PROPN
ejpam-6569	275	45	is	be	AUX
ejpam-6569	275	46	upper	upper	ADJ
ejpam-6569	275	47	quasi	quasi	NOUN
ejpam-6569	275	48	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	275	49	,	,	PUNCT
ejpam-6569	275	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	275	51	;	;	PUNCT
ejpam-6569	275	52	(	(	PUNCT
ejpam-6569	275	53	2	2	X
ejpam-6569	275	54	)	)	PUNCT
ejpam-6569	275	55	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	275	56	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	275	57	-	-	PUNCT
ejpam-6569	275	58	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	275	59	-	-	PUNCT
ejpam-6569	275	60	cl(v	cl(v	NOUN
ejpam-6569	275	61	)	)	PUNCT
ejpam-6569	275	62	)	)	PUNCT
ejpam-6569	275	63	)	)	PUNCT
ejpam-6569	275	64	)	)	PUNCT
ejpam-6569	276	1	⊆	⊆	X
ejpam-6569	276	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	276	3	-	-	PUNCT
ejpam-6569	276	4	cl(v	cl(v	NOUN
ejpam-6569	276	5	)	)	PUNCT
ejpam-6569	276	6	)	)	PUNCT
ejpam-6569	276	7	for	for	ADP
ejpam-6569	276	8	every	every	DET
ejpam-6569	276	9	(	(	PUNCT
ejpam-6569	276	10	σ1	σ1	PROPN
ejpam-6569	276	11	,	,	PUNCT
ejpam-6569	276	12	σ2)p	σ2)p	NOUN
ejpam-6569	276	13	-	-	PUNCT
ejpam-6569	276	14	open	open	NOUN
ejpam-6569	276	15	set	set	NOUN
ejpam-6569	276	16	v	v	NOUN
ejpam-6569	276	17	of	of	ADP
ejpam-6569	276	18	y	y	PROPN
ejpam-6569	276	19	;	;	PUNCT
ejpam-6569	276	20	(	(	PUNCT
ejpam-6569	276	21	3	3	X
ejpam-6569	276	22	)	)	PUNCT
ejpam-6569	276	23	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	276	24	−(v	−(v	NOUN
ejpam-6569	276	25	)	)	PUNCT
ejpam-6569	276	26	)	)	PUNCT
ejpam-6569	277	1	⊆	⊆	X
ejpam-6569	277	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	277	3	-	-	PUNCT
ejpam-6569	277	4	cl(v	cl(v	NOUN
ejpam-6569	277	5	)	)	PUNCT
ejpam-6569	277	6	)	)	PUNCT
ejpam-6569	277	7	for	for	ADP
ejpam-6569	277	8	every	every	DET
ejpam-6569	277	9	(	(	PUNCT
ejpam-6569	277	10	σ1	σ1	PROPN
ejpam-6569	277	11	,	,	PUNCT
ejpam-6569	277	12	σ2)p	σ2)p	NOUN
ejpam-6569	277	13	-	-	PUNCT
ejpam-6569	277	14	open	open	NOUN
ejpam-6569	277	15	set	set	NOUN
ejpam-6569	277	16	v	v	NOUN
ejpam-6569	277	17	of	of	ADP
ejpam-6569	277	18	y	y	PROPN
ejpam-6569	277	19	;	;	PUNCT
ejpam-6569	277	20	(	(	PUNCT
ejpam-6569	277	21	4	4	X
ejpam-6569	277	22	)	)	PUNCT
ejpam-6569	277	23	f+(v	f+(v	NOUN
ejpam-6569	277	24	)	)	PUNCT
ejpam-6569	278	1	⊆	⊆	NUM
ejpam-6569	278	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	278	3	+	+	ADJ
ejpam-6569	278	4	(	(	PUNCT
ejpam-6569	278	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	278	6	-	-	NUM
ejpam-6569	278	7	cl(v	cl(v	NOUN
ejpam-6569	278	8	)	)	PUNCT
ejpam-6569	278	9	)	)	PUNCT
ejpam-6569	278	10	)	)	PUNCT
ejpam-6569	278	11	for	for	ADP
ejpam-6569	278	12	every	every	DET
ejpam-6569	278	13	(	(	PUNCT
ejpam-6569	278	14	σ1	σ1	PROPN
ejpam-6569	278	15	,	,	PUNCT
ejpam-6569	278	16	σ2)p	σ2)p	NOUN
ejpam-6569	278	17	-	-	PUNCT
ejpam-6569	278	18	open	open	NOUN
ejpam-6569	278	19	set	set	NOUN
ejpam-6569	278	20	v	v	NOUN
ejpam-6569	278	21	of	of	ADP
ejpam-6569	278	22	y	y	PROPN
ejpam-6569	278	23	.	.	PUNCT
ejpam-6569	279	1	proof	proof	NOUN
ejpam-6569	279	2	.	.	PUNCT
ejpam-6569	280	1	(	(	PUNCT
ejpam-6569	280	2	1	1	X
ejpam-6569	280	3	)	)	PUNCT
ejpam-6569	280	4	⇒	⇒	NOUN
ejpam-6569	280	5	(	(	PUNCT
ejpam-6569	280	6	2	2	NUM
ejpam-6569	280	7	):	):	PUNCT
ejpam-6569	280	8	let	let	VERB
ejpam-6569	280	9	v	v	PART
ejpam-6569	280	10	be	be	AUX
ejpam-6569	280	11	any	any	DET
ejpam-6569	280	12	(	(	PUNCT
ejpam-6569	280	13	σ1	σ1	PROPN
ejpam-6569	280	14	,	,	PUNCT
ejpam-6569	280	15	σ2)p	σ2)p	NOUN
ejpam-6569	280	16	-	-	PUNCT
ejpam-6569	280	17	open	open	ADJ
ejpam-6569	280	18	set	set	NOUN
ejpam-6569	280	19	of	of	ADP
ejpam-6569	280	20	y	y	PROPN
ejpam-6569	280	21	.	.	PUNCT
ejpam-6569	281	1	since	since	SCONJ
ejpam-6569	281	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	281	3	-	-	NOUN
ejpam-6569	281	4	cl(v	cl(v	NOUN
ejpam-6569	281	5	)	)	PUNCT
ejpam-6569	281	6	is	be	AUX
ejpam-6569	281	7	a	a	DET
ejpam-6569	281	8	σ1σ2	σ1σ2	NUM
ejpam-6569	281	9	-	-	ADJ
ejpam-6569	281	10	open	open	ADJ
ejpam-6569	281	11	set	set	NOUN
ejpam-6569	281	12	of	of	ADP
ejpam-6569	281	13	y	y	PROPN
ejpam-6569	281	14	,	,	PUNCT
ejpam-6569	281	15	by	by	ADP
ejpam-6569	281	16	theorem	theorem	NOUN
ejpam-6569	281	17	3	3	NUM
ejpam-6569	281	18	we	we	PRON
ejpam-6569	281	19	have	have	AUX
ejpam-6569	281	20	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	281	21	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	281	22	-	-	PUNCT
ejpam-6569	281	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	281	24	-	-	PUNCT
ejpam-6569	281	25	cl(v	cl(v	NOUN
ejpam-6569	281	26	)	)	PUNCT
ejpam-6569	281	27	)	)	PUNCT
ejpam-6569	281	28	)	)	PUNCT
ejpam-6569	281	29	)	)	PUNCT
ejpam-6569	282	1	⊆	⊆	X
ejpam-6569	282	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6569	282	3	-	-	PUNCT
ejpam-6569	282	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6569	282	5	-	-	PUNCT
ejpam-6569	282	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	282	7	-	-	PUNCT
ejpam-6569	282	8	cl(v	cl(v	NOUN
ejpam-6569	282	9	)	)	PUNCT
ejpam-6569	282	10	)	)	PUNCT
ejpam-6569	282	11	)	)	PUNCT
ejpam-6569	282	12	)	)	PUNCT
ejpam-6569	283	1	=	=	PUNCT
ejpam-6569	283	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	283	3	-	-	PUNCT
ejpam-6569	283	4	cl(v	cl(v	NOUN
ejpam-6569	283	5	)	)	PUNCT
ejpam-6569	283	6	)	)	PUNCT
ejpam-6569	283	7	.	.	PUNCT
ejpam-6569	284	1	(	(	PUNCT
ejpam-6569	284	2	2	2	X
ejpam-6569	284	3	)	)	PUNCT
ejpam-6569	284	4	⇒	⇒	NOUN
ejpam-6569	284	5	(	(	PUNCT
ejpam-6569	284	6	3	3	NUM
ejpam-6569	284	7	):	):	PUNCT
ejpam-6569	284	8	let	let	VERB
ejpam-6569	284	9	v	v	PART
ejpam-6569	284	10	be	be	AUX
ejpam-6569	284	11	any	any	DET
ejpam-6569	284	12	(	(	PUNCT
ejpam-6569	284	13	σ1	σ1	PROPN
ejpam-6569	284	14	,	,	PUNCT
ejpam-6569	284	15	σ2)p	σ2)p	NOUN
ejpam-6569	284	16	-	-	PUNCT
ejpam-6569	284	17	open	open	ADJ
ejpam-6569	284	18	set	set	NOUN
ejpam-6569	284	19	of	of	ADP
ejpam-6569	284	20	y	y	PROPN
ejpam-6569	284	21	.	.	PUNCT
ejpam-6569	285	1	then	then	ADV
ejpam-6569	285	2	,	,	PUNCT
ejpam-6569	285	3	v	v	ADP
ejpam-6569	285	4	⊆	⊆	NUM
ejpam-6569	285	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	285	6	-	-	PUNCT
ejpam-6569	285	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	285	8	-	-	PUNCT
ejpam-6569	285	9	cl(v	cl(v	NOUN
ejpam-6569	285	10	)	)	PUNCT
ejpam-6569	285	11	)	)	PUNCT
ejpam-6569	285	12	and	and	CCONJ
ejpam-6569	285	13	by	by	ADP
ejpam-6569	285	14	(	(	PUNCT
ejpam-6569	285	15	2	2	NUM
ejpam-6569	285	16	)	)	PUNCT
ejpam-6569	285	17	,	,	PUNCT
ejpam-6569	285	18	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	285	19	−(v	−(v	NOUN
ejpam-6569	285	20	)	)	PUNCT
ejpam-6569	285	21	)	)	PUNCT
ejpam-6569	286	1	⊆	⊆	NUM
ejpam-6569	286	2	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	286	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	286	4	-	-	PUNCT
ejpam-6569	286	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	286	6	-	-	PUNCT
ejpam-6569	286	7	cl(v	cl(v	NOUN
ejpam-6569	286	8	)	)	PUNCT
ejpam-6569	286	9	)	)	PUNCT
ejpam-6569	286	10	)	)	PUNCT
ejpam-6569	286	11	)	)	PUNCT
ejpam-6569	287	1	n.	n.	PROPN
ejpam-6569	287	2	srisarakham	srisarakham	PROPN
ejpam-6569	287	3	,	,	PUNCT
ejpam-6569	287	4	a.	a.	PROPN
ejpam-6569	287	5	sama	sama	PROPN
ejpam-6569	287	6	-	-	PUNCT
ejpam-6569	287	7	ae	ae	PROPN
ejpam-6569	287	8	,	,	PUNCT
ejpam-6569	287	9	c.	c.	PROPN
ejpam-6569	287	10	boonpok	boonpok	PROPN
ejpam-6569	287	11	/	/	SYM
ejpam-6569	287	12	eur	eur	PROPN
ejpam-6569	287	13	.	.	PUNCT
ejpam-6569	288	1	j.	j.	PROPN
ejpam-6569	288	2	pure	pure	PROPN
ejpam-6569	288	3	appl	appl	PROPN
ejpam-6569	288	4	.	.	PROPN
ejpam-6569	288	5	math	math	PROPN
ejpam-6569	288	6	,	,	PUNCT
ejpam-6569	288	7	18	18	NUM
ejpam-6569	288	8	(	(	PUNCT
ejpam-6569	288	9	3	3	NUM
ejpam-6569	288	10	)	)	PUNCT
ejpam-6569	288	11	(	(	PUNCT
ejpam-6569	288	12	2025	2025	NUM
ejpam-6569	288	13	)	)	PUNCT
ejpam-6569	288	14	,	,	PUNCT
ejpam-6569	288	15	6569	6569	NUM
ejpam-6569	288	16	9	9	NUM
ejpam-6569	288	17	of	of	ADP
ejpam-6569	288	18	13	13	NUM
ejpam-6569	288	19	⊆	⊆	NUM
ejpam-6569	288	20	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	288	21	-	-	PUNCT
ejpam-6569	288	22	cl(v	cl(v	NOUN
ejpam-6569	288	23	)	)	PUNCT
ejpam-6569	288	24	)	)	PUNCT
ejpam-6569	288	25	.	.	PUNCT
ejpam-6569	289	1	(	(	PUNCT
ejpam-6569	289	2	3	3	X
ejpam-6569	289	3	)	)	PUNCT
ejpam-6569	289	4	⇒	⇒	NOUN
ejpam-6569	289	5	(	(	PUNCT
ejpam-6569	289	6	4	4	NUM
ejpam-6569	289	7	):	):	PUNCT
ejpam-6569	289	8	let	let	VERB
ejpam-6569	289	9	v	v	PART
ejpam-6569	289	10	be	be	AUX
ejpam-6569	289	11	any	any	DET
ejpam-6569	289	12	(	(	PUNCT
ejpam-6569	289	13	σ1	σ1	PROPN
ejpam-6569	289	14	,	,	PUNCT
ejpam-6569	289	15	σ2)p	σ2)p	NOUN
ejpam-6569	289	16	-	-	PUNCT
ejpam-6569	289	17	open	open	ADJ
ejpam-6569	289	18	set	set	NOUN
ejpam-6569	289	19	of	of	ADP
ejpam-6569	289	20	y	y	PROPN
ejpam-6569	289	21	.	.	PUNCT
ejpam-6569	290	1	then	then	ADV
ejpam-6569	290	2	by	by	ADP
ejpam-6569	290	3	(	(	PUNCT
ejpam-6569	290	4	3	3	NUM
ejpam-6569	290	5	)	)	PUNCT
ejpam-6569	290	6	,	,	PUNCT
ejpam-6569	290	7	we	we	PRON
ejpam-6569	290	8	have	have	VERB
ejpam-6569	290	9	x	x	X
ejpam-6569	290	10	−	−	X
ejpam-6569	290	11	⋆θsint(f	⋆θsint(f	SYM
ejpam-6569	291	1	+	+	ADJ
ejpam-6569	291	2	(	(	PUNCT
ejpam-6569	291	3	σ1σ2	σ1σ2	NOUN
ejpam-6569	291	4	-	-	NUM
ejpam-6569	291	5	cl(v	cl(v	NOUN
ejpam-6569	291	6	)	)	PUNCT
ejpam-6569	291	7	)	)	PUNCT
ejpam-6569	291	8	)	)	PUNCT
ejpam-6569	292	1	=	=	SYM
ejpam-6569	292	2	⋆θscl(x	⋆θscl(x	PROPN
ejpam-6569	292	3	−	−	NUM
ejpam-6569	292	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	292	5	-	-	PUNCT
ejpam-6569	292	6	cl(v	cl(v	NOUN
ejpam-6569	292	7	)	)	PUNCT
ejpam-6569	292	8	)	)	PUNCT
ejpam-6569	292	9	)	)	PUNCT
ejpam-6569	293	1	=	=	PUNCT
ejpam-6569	293	2	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	293	3	−(y	−(y	NOUN
ejpam-6569	293	4	−	−	NOUN
ejpam-6569	293	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	293	6	-	-	NUM
ejpam-6569	293	7	cl(v	cl(v	NOUN
ejpam-6569	293	8	)	)	PUNCT
ejpam-6569	293	9	)	)	PUNCT
ejpam-6569	293	10	)	)	PUNCT
ejpam-6569	294	1	⊆	⊆	X
ejpam-6569	294	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6569	294	3	-	-	PUNCT
ejpam-6569	294	4	cl(y	cl(y	NOUN
ejpam-6569	294	5	−	−	NOUN
ejpam-6569	294	6	σ1σ2	σ1σ2	NOUN
ejpam-6569	294	7	-	-	NUM
ejpam-6569	294	8	cl(v	cl(v	NOUN
ejpam-6569	294	9	)	)	PUNCT
ejpam-6569	294	10	)	)	PUNCT
ejpam-6569	294	11	)	)	PUNCT
ejpam-6569	295	1	=	=	PUNCT
ejpam-6569	295	2	x	x	X
ejpam-6569	295	3	−	−	ADP
ejpam-6569	295	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6569	295	5	-	-	PUNCT
ejpam-6569	295	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	295	7	-	-	PUNCT
ejpam-6569	295	8	cl(v	cl(v	NOUN
ejpam-6569	295	9	)	)	PUNCT
ejpam-6569	295	10	)	)	PUNCT
ejpam-6569	295	11	)	)	PUNCT
ejpam-6569	296	1	⊆	⊆	NUM
ejpam-6569	296	2	x	x	SYM
ejpam-6569	296	3	−	−	NOUN
ejpam-6569	296	4	f+(v	f+(v	NOUN
ejpam-6569	296	5	)	)	PUNCT
ejpam-6569	296	6	and	and	CCONJ
ejpam-6569	296	7	hence	hence	ADV
ejpam-6569	296	8	f+(v	f+(v	NOUN
ejpam-6569	296	9	)	)	PUNCT
ejpam-6569	297	1	⊆	⊆	NUM
ejpam-6569	297	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	297	3	+	+	ADJ
ejpam-6569	297	4	(	(	PUNCT
ejpam-6569	297	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	297	6	-	-	NUM
ejpam-6569	297	7	cl(v	cl(v	NOUN
ejpam-6569	297	8	)	)	PUNCT
ejpam-6569	297	9	)	)	PUNCT
ejpam-6569	297	10	)	)	PUNCT
ejpam-6569	297	11	.	.	PUNCT
ejpam-6569	298	1	(	(	PUNCT
ejpam-6569	298	2	4	4	X
ejpam-6569	298	3	)	)	PUNCT
ejpam-6569	298	4	⇒	⇒	NOUN
ejpam-6569	298	5	(	(	PUNCT
ejpam-6569	298	6	1	1	NUM
ejpam-6569	298	7	):	):	PUNCT
ejpam-6569	298	8	let	let	VERB
ejpam-6569	298	9	v	v	PART
ejpam-6569	298	10	be	be	AUX
ejpam-6569	298	11	any	any	DET
ejpam-6569	298	12	σ1σ2	σ1σ2	NOUN
ejpam-6569	298	13	-	-	ADJ
ejpam-6569	298	14	open	open	ADJ
ejpam-6569	298	15	set	set	NOUN
ejpam-6569	298	16	of	of	ADP
ejpam-6569	298	17	y	y	PROPN
ejpam-6569	298	18	.	.	PUNCT
ejpam-6569	299	1	then	then	ADV
ejpam-6569	299	2	,	,	PUNCT
ejpam-6569	299	3	v	v	NOUN
ejpam-6569	299	4	is	be	AUX
ejpam-6569	299	5	(	(	PUNCT
ejpam-6569	299	6	σ1	σ1	PROPN
ejpam-6569	299	7	,	,	PUNCT
ejpam-6569	299	8	σ2)p	σ2)p	NOUN
ejpam-6569	299	9	-	-	PUNCT
ejpam-6569	299	10	open	open	ADJ
ejpam-6569	299	11	in	in	ADP
ejpam-6569	299	12	y	y	PROPN
ejpam-6569	299	13	and	and	CCONJ
ejpam-6569	299	14	by	by	ADP
ejpam-6569	299	15	(	(	PUNCT
ejpam-6569	299	16	4	4	NUM
ejpam-6569	299	17	)	)	PUNCT
ejpam-6569	299	18	,	,	PUNCT
ejpam-6569	299	19	we	we	PRON
ejpam-6569	299	20	have	have	VERB
ejpam-6569	299	21	f+(v	f+(v	NOUN
ejpam-6569	299	22	)	)	PUNCT
ejpam-6569	300	1	⊆	⊆	NUM
ejpam-6569	300	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	300	3	+	+	ADJ
ejpam-6569	300	4	(	(	PUNCT
ejpam-6569	300	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	300	6	-	-	NUM
ejpam-6569	300	7	cl(v	cl(v	NOUN
ejpam-6569	300	8	)	)	PUNCT
ejpam-6569	300	9	)	)	PUNCT
ejpam-6569	300	10	)	)	PUNCT
ejpam-6569	300	11	.	.	PUNCT
ejpam-6569	301	1	by	by	ADP
ejpam-6569	301	2	theorem	theorem	NOUN
ejpam-6569	301	3	1	1	NUM
ejpam-6569	301	4	,	,	PUNCT
ejpam-6569	301	5	f	f	PROPN
ejpam-6569	301	6	is	be	AUX
ejpam-6569	301	7	upper	upper	ADJ
ejpam-6569	301	8	quasi	quasi	ADJ
ejpam-6569	301	9	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	301	10	,	,	PUNCT
ejpam-6569	301	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	301	12	.	.	X
ejpam-6569	301	13	theorem	theorem	VERB
ejpam-6569	301	14	6	6	NUM
ejpam-6569	301	15	.	.	PUNCT
ejpam-6569	301	16	for	for	ADP
ejpam-6569	301	17	a	a	DET
ejpam-6569	301	18	multifunction	multifunction	NOUN
ejpam-6569	301	19	f	f	NOUN
ejpam-6569	301	20	:	:	PUNCT
ejpam-6569	301	21	(	(	PUNCT
ejpam-6569	301	22	x	x	X
ejpam-6569	301	23	,	,	PUNCT
ejpam-6569	301	24	τ	τ	PROPN
ejpam-6569	301	25	,	,	PUNCT
ejpam-6569	301	26	i	i	NOUN
ejpam-6569	301	27	)	)	PUNCT
ejpam-6569	301	28	→	→	PUNCT
ejpam-6569	301	29	(	(	PUNCT
ejpam-6569	301	30	y	y	PROPN
ejpam-6569	301	31	,	,	PUNCT
ejpam-6569	301	32	σ1	σ1	PROPN
ejpam-6569	301	33	,	,	PUNCT
ejpam-6569	301	34	σ2	σ2	NOUN
ejpam-6569	301	35	)	)	PUNCT
ejpam-6569	301	36	,	,	PUNCT
ejpam-6569	301	37	the	the	DET
ejpam-6569	301	38	following	follow	VERB
ejpam-6569	301	39	properties	property	NOUN
ejpam-6569	301	40	are	be	AUX
ejpam-6569	301	41	equivalent	equivalent	ADJ
ejpam-6569	301	42	:	:	PUNCT
ejpam-6569	301	43	(	(	PUNCT
ejpam-6569	301	44	1	1	X
ejpam-6569	301	45	)	)	PUNCT
ejpam-6569	301	46	f	f	PROPN
ejpam-6569	301	47	is	be	AUX
ejpam-6569	301	48	lower	low	ADJ
ejpam-6569	301	49	quasi	quasi	NOUN
ejpam-6569	301	50	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	301	51	,	,	PUNCT
ejpam-6569	301	52	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	301	53	;	;	PUNCT
ejpam-6569	301	54	(	(	PUNCT
ejpam-6569	301	55	2	2	X
ejpam-6569	301	56	)	)	PUNCT
ejpam-6569	301	57	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	302	1	+	+	PROPN
ejpam-6569	302	2	(	(	PUNCT
ejpam-6569	302	3	σ1σ2	σ1σ2	NUM
ejpam-6569	302	4	-	-	PUNCT
ejpam-6569	302	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6569	302	6	-	-	PUNCT
ejpam-6569	302	7	cl(v	cl(v	NOUN
ejpam-6569	302	8	)	)	PUNCT
ejpam-6569	302	9	)	)	PUNCT
ejpam-6569	302	10	)	)	PUNCT
ejpam-6569	302	11	)	)	PUNCT
ejpam-6569	303	1	⊆	⊆	X
ejpam-6569	303	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	303	3	-	-	PUNCT
ejpam-6569	303	4	cl(v	cl(v	NOUN
ejpam-6569	303	5	)	)	PUNCT
ejpam-6569	303	6	)	)	PUNCT
ejpam-6569	303	7	for	for	ADP
ejpam-6569	303	8	every	every	DET
ejpam-6569	303	9	(	(	PUNCT
ejpam-6569	303	10	σ1	σ1	PROPN
ejpam-6569	303	11	,	,	PUNCT
ejpam-6569	303	12	σ2)p	σ2)p	NOUN
ejpam-6569	303	13	-	-	PUNCT
ejpam-6569	303	14	open	open	NOUN
ejpam-6569	303	15	set	set	NOUN
ejpam-6569	303	16	v	v	NOUN
ejpam-6569	303	17	of	of	ADP
ejpam-6569	303	18	y	y	PROPN
ejpam-6569	303	19	;	;	PUNCT
ejpam-6569	303	20	(	(	PUNCT
ejpam-6569	303	21	3	3	X
ejpam-6569	303	22	)	)	PUNCT
ejpam-6569	303	23	⋆θscl(f	⋆θscl(f	PROPN
ejpam-6569	304	1	+	+	NOUN
ejpam-6569	304	2	(	(	PUNCT
ejpam-6569	304	3	v	v	NOUN
ejpam-6569	304	4	)	)	PUNCT
ejpam-6569	304	5	)	)	PUNCT
ejpam-6569	305	1	⊆	⊆	NUM
ejpam-6569	305	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6569	305	3	-	-	PUNCT
ejpam-6569	305	4	cl(v	cl(v	NOUN
ejpam-6569	305	5	)	)	PUNCT
ejpam-6569	305	6	)	)	PUNCT
ejpam-6569	305	7	for	for	ADP
ejpam-6569	305	8	every	every	DET
ejpam-6569	305	9	(	(	PUNCT
ejpam-6569	305	10	σ1	σ1	PROPN
ejpam-6569	305	11	,	,	PUNCT
ejpam-6569	305	12	σ2)p	σ2)p	NOUN
ejpam-6569	305	13	-	-	PUNCT
ejpam-6569	305	14	open	open	NOUN
ejpam-6569	305	15	set	set	NOUN
ejpam-6569	305	16	v	v	NOUN
ejpam-6569	305	17	of	of	ADP
ejpam-6569	305	18	y	y	PROPN
ejpam-6569	305	19	;	;	PUNCT
ejpam-6569	305	20	(	(	PUNCT
ejpam-6569	305	21	4	4	X
ejpam-6569	305	22	)	)	PUNCT
ejpam-6569	305	23	f−(v	f−(v	NOUN
ejpam-6569	305	24	)	)	PUNCT
ejpam-6569	305	25	⊆	⊆	NUM
ejpam-6569	305	26	⋆θsint(f	⋆θsint(f	NUM
ejpam-6569	305	27	−(σ1σ2	−(σ1σ2	NOUN
ejpam-6569	305	28	-	-	NOUN
ejpam-6569	305	29	cl(v	cl(v	NOUN
ejpam-6569	305	30	)	)	PUNCT
ejpam-6569	305	31	)	)	PUNCT
ejpam-6569	305	32	)	)	PUNCT
ejpam-6569	306	1	for	for	ADP
ejpam-6569	306	2	every	every	DET
ejpam-6569	306	3	(	(	PUNCT
ejpam-6569	306	4	σ1	σ1	PROPN
ejpam-6569	306	5	,	,	PUNCT
ejpam-6569	306	6	σ2)p	σ2)p	NOUN
ejpam-6569	306	7	-	-	PUNCT
ejpam-6569	306	8	open	open	NOUN
ejpam-6569	306	9	set	set	NOUN
ejpam-6569	306	10	v	v	NOUN
ejpam-6569	306	11	of	of	ADP
ejpam-6569	306	12	y	y	PROPN
ejpam-6569	306	13	.	.	PUNCT
ejpam-6569	307	1	proof	proof	NOUN
ejpam-6569	307	2	.	.	PUNCT
ejpam-6569	308	1	the	the	DET
ejpam-6569	308	2	proof	proof	NOUN
ejpam-6569	308	3	is	be	AUX
ejpam-6569	308	4	similar	similar	ADJ
ejpam-6569	308	5	to	to	ADP
ejpam-6569	308	6	that	that	PRON
ejpam-6569	308	7	of	of	ADP
ejpam-6569	308	8	theorem	theorem	NOUN
ejpam-6569	308	9	5	5	NUM
ejpam-6569	308	10	.	.	PUNCT
ejpam-6569	308	11	recall	recall	VERB
ejpam-6569	308	12	that	that	SCONJ
ejpam-6569	308	13	a	a	DET
ejpam-6569	308	14	bitopological	bitopological	ADJ
ejpam-6569	308	15	space	space	NOUN
ejpam-6569	308	16	(	(	PUNCT
ejpam-6569	308	17	x	x	NOUN
ejpam-6569	308	18	,	,	PUNCT
ejpam-6569	308	19	τ1	τ1	NOUN
ejpam-6569	308	20	,	,	PUNCT
ejpam-6569	308	21	τ2	τ2	NOUN
ejpam-6569	308	22	)	)	PUNCT
ejpam-6569	308	23	is	be	AUX
ejpam-6569	308	24	said	say	VERB
ejpam-6569	308	25	to	to	PART
ejpam-6569	308	26	be	be	AUX
ejpam-6569	308	27	τ1τ2	τ1τ2	NOUN
ejpam-6569	308	28	-	-	ADJ
ejpam-6569	309	1	compact	compact	ADJ
ejpam-6569	309	2	[	[	X
ejpam-6569	309	3	22	22	NUM
ejpam-6569	309	4	]	]	X
ejpam-6569	309	5	if	if	SCONJ
ejpam-6569	309	6	for	for	ADP
ejpam-6569	309	7	every	every	DET
ejpam-6569	309	8	cover	cover	NOUN
ejpam-6569	309	9	of	of	ADP
ejpam-6569	309	10	x	x	PUNCT
ejpam-6569	309	11	by	by	ADP
ejpam-6569	309	12	τ1τ2	τ1τ2	ADJ
ejpam-6569	309	13	-	-	ADJ
ejpam-6569	309	14	open	open	ADJ
ejpam-6569	309	15	sets	set	NOUN
ejpam-6569	309	16	of	of	ADP
ejpam-6569	309	17	x	x	PUNCT
ejpam-6569	309	18	has	have	VERB
ejpam-6569	309	19	a	a	DET
ejpam-6569	309	20	finite	finite	ADJ
ejpam-6569	309	21	subcover	subcover	PROPN
ejpam-6569	309	22	.	.	PUNCT
ejpam-6569	310	1	a	a	DET
ejpam-6569	310	2	bitopological	bitopological	ADJ
ejpam-6569	310	3	space	space	NOUN
ejpam-6569	310	4	(	(	PUNCT
ejpam-6569	310	5	x	x	NOUN
ejpam-6569	310	6	,	,	PUNCT
ejpam-6569	310	7	τ1	τ1	NOUN
ejpam-6569	310	8	,	,	PUNCT
ejpam-6569	310	9	τ2	τ2	NOUN
ejpam-6569	310	10	)	)	PUNCT
ejpam-6569	310	11	is	be	AUX
ejpam-6569	310	12	said	say	VERB
ejpam-6569	310	13	to	to	PART
ejpam-6569	310	14	be	be	AUX
ejpam-6569	310	15	quasi	quasi	X
ejpam-6569	310	16	(	(	PUNCT
ejpam-6569	310	17	τ1	τ1	NOUN
ejpam-6569	310	18	,	,	PUNCT
ejpam-6569	310	19	τ2)-h	τ2)-h	PUNCT
ejpam-6569	310	20	-closed	-closed	ADJ
ejpam-6569	310	21	[	[	X
ejpam-6569	310	22	21	21	NUM
ejpam-6569	310	23	]	]	X
ejpam-6569	310	24	if	if	SCONJ
ejpam-6569	310	25	for	for	ADP
ejpam-6569	310	26	every	every	DET
ejpam-6569	310	27	τ1τ2	τ1τ2	ADJ
ejpam-6569	310	28	-	-	ADJ
ejpam-6569	310	29	open	open	ADJ
ejpam-6569	310	30	cover	cover	NOUN
ejpam-6569	310	31	{	{	PUNCT
ejpam-6569	310	32	uγ	uγ	ADV
ejpam-6569	310	33	|	|	ADV
ejpam-6569	310	34	γ	γ	X
ejpam-6569	310	35	∈	∈	PROPN
ejpam-6569	310	36	γ	γ	X
ejpam-6569	310	37	}	}	PUNCT
ejpam-6569	310	38	,	,	PUNCT
ejpam-6569	310	39	there	there	PRON
ejpam-6569	310	40	exists	exist	VERB
ejpam-6569	310	41	a	a	DET
ejpam-6569	310	42	finite	finite	NOUN
ejpam-6569	310	43	subset	subset	NOUN
ejpam-6569	310	44	γ0	γ0	NOUN
ejpam-6569	310	45	of	of	ADP
ejpam-6569	310	46	γ	γ	NOUN
ejpam-6569	311	1	such	such	ADJ
ejpam-6569	311	2	that	that	SCONJ
ejpam-6569	311	3	x	x	X
ejpam-6569	311	4	=	=	PUNCT
ejpam-6569	311	5	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-6569	311	6	-	-	NOUN
ejpam-6569	311	7	cl(uγ	cl(uγ	NOUN
ejpam-6569	311	8	)	)	PUNCT
ejpam-6569	311	9	|	|	ADV
ejpam-6569	311	10	γ	γ	PROPN
ejpam-6569	311	11	∈	∈	PROPN
ejpam-6569	311	12	γ0	γ0	PROPN
ejpam-6569	311	13	}	}	PUNCT
ejpam-6569	311	14	.	.	PUNCT
ejpam-6569	312	1	an	an	DET
ejpam-6569	312	2	ideal	ideal	ADJ
ejpam-6569	312	3	topological	topological	ADJ
ejpam-6569	312	4	space	space	NOUN
ejpam-6569	312	5	(	(	PUNCT
ejpam-6569	312	6	x	x	X
ejpam-6569	312	7	,	,	PUNCT
ejpam-6569	312	8	τ	τ	PROPN
ejpam-6569	312	9	,	,	PUNCT
ejpam-6569	312	10	i	i	PROPN
ejpam-6569	312	11	)	)	PUNCT
ejpam-6569	312	12	is	be	AUX
ejpam-6569	312	13	called	call	VERB
ejpam-6569	312	14	s⋆-closed	s⋆-closed	ADJ
ejpam-6569	312	15	[	[	PUNCT
ejpam-6569	312	16	29	29	NUM
ejpam-6569	312	17	]	]	X
ejpam-6569	312	18	if	if	SCONJ
ejpam-6569	312	19	for	for	ADP
ejpam-6569	312	20	every	every	DET
ejpam-6569	312	21	semi	semi	NOUN
ejpam-6569	312	22	-	-	ADJ
ejpam-6569	312	23	i	i	PRON
ejpam-6569	312	24	⋆-open	⋆-open	ADJ
ejpam-6569	312	25	cover	cover	VERB
ejpam-6569	312	26	{	{	PUNCT
ejpam-6569	313	1	vα	vα	INTJ
ejpam-6569	313	2	|	|	ADV
ejpam-6569	313	3	α	α	NUM
ejpam-6569	313	4	∈	∈	NOUN
ejpam-6569	313	5	∇	∇	NOUN
ejpam-6569	313	6	}	}	PUNCT
ejpam-6569	313	7	of	of	ADP
ejpam-6569	313	8	x	x	NOUN
ejpam-6569	313	9	,	,	PUNCT
ejpam-6569	313	10	there	there	PRON
ejpam-6569	313	11	exists	exist	VERB
ejpam-6569	313	12	a	a	DET
ejpam-6569	313	13	finite	finite	NOUN
ejpam-6569	313	14	subset	subset	NOUN
ejpam-6569	313	15	∇0	∇0	NUM
ejpam-6569	313	16	of	of	ADP
ejpam-6569	313	17	∇	∇	NOUN
ejpam-6569	313	18	such	such	ADJ
ejpam-6569	313	19	that	that	SCONJ
ejpam-6569	313	20	x	x	X
ejpam-6569	313	21	=	=	SYM
ejpam-6569	313	22	∪{scl⋆(vα	∪{scl⋆(vα	NOUN
ejpam-6569	313	23	)	)	PUNCT
ejpam-6569	313	24	|	|	ADV
ejpam-6569	313	25	α	α	NUM
ejpam-6569	313	26	∈	∈	NOUN
ejpam-6569	313	27	∇0	∇0	NOUN
ejpam-6569	313	28	}	}	PUNCT
ejpam-6569	313	29	.	.	PUNCT
ejpam-6569	314	1	theorem	theorem	VERB
ejpam-6569	314	2	7	7	NUM
ejpam-6569	314	3	.	.	PUNCT
ejpam-6569	315	1	let	let	VERB
ejpam-6569	315	2	f	f	NOUN
ejpam-6569	315	3	:	:	PUNCT
ejpam-6569	315	4	(	(	PUNCT
ejpam-6569	315	5	x	x	X
ejpam-6569	315	6	,	,	PUNCT
ejpam-6569	315	7	τ	τ	PROPN
ejpam-6569	315	8	,	,	PUNCT
ejpam-6569	315	9	i	i	NOUN
ejpam-6569	315	10	)	)	PUNCT
ejpam-6569	315	11	→	→	PUNCT
ejpam-6569	315	12	(	(	PUNCT
ejpam-6569	315	13	y	y	PROPN
ejpam-6569	315	14	,	,	PUNCT
ejpam-6569	315	15	σ1	σ1	PROPN
ejpam-6569	315	16	,	,	PUNCT
ejpam-6569	315	17	σ2	σ2	PROPN
ejpam-6569	315	18	)	)	PUNCT
ejpam-6569	315	19	be	be	VERB
ejpam-6569	315	20	an	an	DET
ejpam-6569	315	21	upper	upper	ADJ
ejpam-6569	315	22	quasi	quasi	NOUN
ejpam-6569	315	23	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	315	24	,	,	PUNCT
ejpam-6569	315	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	315	26	surjective	surjective	ADJ
ejpam-6569	315	27	multifunction	multifunction	NOUN
ejpam-6569	315	28	such	such	ADJ
ejpam-6569	315	29	that	that	SCONJ
ejpam-6569	315	30	f	f	PROPN
ejpam-6569	315	31	(	(	PUNCT
ejpam-6569	315	32	x	x	X
ejpam-6569	315	33	)	)	PUNCT
ejpam-6569	315	34	is	be	AUX
ejpam-6569	315	35	σ1σ2	σ1σ2	NOUN
ejpam-6569	315	36	-	-	ADJ
ejpam-6569	315	37	compact	compact	ADJ
ejpam-6569	315	38	for	for	ADP
ejpam-6569	315	39	each	each	DET
ejpam-6569	315	40	x	x	SYM
ejpam-6569	315	41	∈	∈	PROPN
ejpam-6569	315	42	x.	x.	NOUN
ejpam-6569	316	1	if	if	SCONJ
ejpam-6569	316	2	(	(	PUNCT
ejpam-6569	316	3	x	x	NOUN
ejpam-6569	316	4	,	,	PUNCT
ejpam-6569	316	5	τ	τ	PROPN
ejpam-6569	316	6	,	,	PUNCT
ejpam-6569	316	7	i	i	PROPN
ejpam-6569	316	8	)	)	PUNCT
ejpam-6569	316	9	is	be	AUX
ejpam-6569	316	10	s⋆-closed	s⋆-close	VERB
ejpam-6569	316	11	,	,	PUNCT
ejpam-6569	316	12	then	then	ADV
ejpam-6569	316	13	(	(	PUNCT
ejpam-6569	316	14	y	y	PROPN
ejpam-6569	316	15	,	,	PUNCT
ejpam-6569	316	16	σ1	σ1	PROPN
ejpam-6569	316	17	,	,	PUNCT
ejpam-6569	316	18	σ2	σ2	PROPN
ejpam-6569	316	19	)	)	PUNCT
ejpam-6569	316	20	is	be	AUX
ejpam-6569	316	21	quasi	quasi	X
ejpam-6569	316	22	(	(	PUNCT
ejpam-6569	316	23	σ1	σ1	PROPN
ejpam-6569	316	24	,	,	PUNCT
ejpam-6569	316	25	σ2)-h	σ2)-h	PROPN
ejpam-6569	316	26	-closed	-closed	ADJ
ejpam-6569	316	27	.	.	PUNCT
ejpam-6569	317	1	proof	proof	NOUN
ejpam-6569	317	2	.	.	PUNCT
ejpam-6569	318	1	let	let	VERB
ejpam-6569	318	2	{	{	PUNCT
ejpam-6569	318	3	vγ	vγ	VERB
ejpam-6569	318	4	|	|	ADV
ejpam-6569	318	5	γ	γ	PROPN
ejpam-6569	318	6	∈	∈	PROPN
ejpam-6569	318	7	γ	γ	AUX
ejpam-6569	318	8	}	}	PUNCT
ejpam-6569	318	9	be	be	VERB
ejpam-6569	318	10	any	any	DET
ejpam-6569	318	11	σ1σ2	σ1σ2	NOUN
ejpam-6569	318	12	-	-	PUNCT
ejpam-6569	318	13	open	open	ADJ
ejpam-6569	318	14	cover	cover	NOUN
ejpam-6569	318	15	of	of	ADP
ejpam-6569	318	16	y	y	PROPN
ejpam-6569	318	17	.	.	PUNCT
ejpam-6569	319	1	for	for	ADP
ejpam-6569	319	2	each	each	DET
ejpam-6569	319	3	x	x	SYM
ejpam-6569	319	4	∈	∈	PROPN
ejpam-6569	319	5	x	x	X
ejpam-6569	319	6	,	,	PUNCT
ejpam-6569	319	7	f	f	PROPN
ejpam-6569	319	8	(	(	PUNCT
ejpam-6569	319	9	x	x	X
ejpam-6569	319	10	)	)	PUNCT
ejpam-6569	319	11	is	be	AUX
ejpam-6569	319	12	σ1σ2compact	σ1σ2compact	PUNCT
ejpam-6569	319	13	and	and	CCONJ
ejpam-6569	319	14	there	there	PRON
ejpam-6569	319	15	exists	exist	VERB
ejpam-6569	319	16	a	a	DET
ejpam-6569	319	17	finite	finite	NOUN
ejpam-6569	319	18	subset	subset	NOUN
ejpam-6569	319	19	γ(x	γ(x	NOUN
ejpam-6569	319	20	)	)	PUNCT
ejpam-6569	319	21	of	of	ADP
ejpam-6569	319	22	γ	γ	PRON
ejpam-6569	319	23	such	such	ADJ
ejpam-6569	319	24	that	that	SCONJ
ejpam-6569	319	25	f	f	PROPN
ejpam-6569	319	26	(	(	PUNCT
ejpam-6569	319	27	x	x	X
ejpam-6569	319	28	)	)	PUNCT
ejpam-6569	319	29	⊆	⊆	NUM
ejpam-6569	319	30	∪{vγ	∪{vγ	PROPN
ejpam-6569	319	31	|	|	ADV
ejpam-6569	319	32	γ	γ	X
ejpam-6569	319	33	∈	∈	PROPN
ejpam-6569	319	34	γ(x	γ(x	PROPN
ejpam-6569	319	35	)	)	PUNCT
ejpam-6569	319	36	}	}	PUNCT
ejpam-6569	319	37	.	.	PUNCT
ejpam-6569	320	1	put	put	VERB
ejpam-6569	320	2	v	v	NOUN
ejpam-6569	320	3	(	(	PUNCT
ejpam-6569	320	4	x	x	NOUN
ejpam-6569	320	5	)	)	PUNCT
ejpam-6569	320	6	=	=	SYM
ejpam-6569	320	7	∪{vγ	∪{vγ	PROPN
ejpam-6569	320	8	|	|	ADV
ejpam-6569	320	9	γ	γ	X
ejpam-6569	320	10	∈	∈	PROPN
ejpam-6569	320	11	γ(x	γ(x	PROPN
ejpam-6569	320	12	)	)	PUNCT
ejpam-6569	320	13	}	}	PUNCT
ejpam-6569	320	14	.	.	PUNCT
ejpam-6569	321	1	then	then	ADV
ejpam-6569	321	2	,	,	PUNCT
ejpam-6569	321	3	f	f	PROPN
ejpam-6569	321	4	(	(	PUNCT
ejpam-6569	321	5	x	x	X
ejpam-6569	321	6	)	)	PUNCT
ejpam-6569	321	7	⊆	⊆	NUM
ejpam-6569	321	8	v	v	NOUN
ejpam-6569	321	9	(	(	PUNCT
ejpam-6569	321	10	x	x	NOUN
ejpam-6569	321	11	)	)	PUNCT
ejpam-6569	321	12	and	and	CCONJ
ejpam-6569	321	13	v	v	NOUN
ejpam-6569	321	14	(	(	PUNCT
ejpam-6569	321	15	x	x	X
ejpam-6569	321	16	)	)	PUNCT
ejpam-6569	321	17	is	be	AUX
ejpam-6569	321	18	σ1σ2	σ1σ2	NOUN
ejpam-6569	321	19	-	-	ADJ
ejpam-6569	321	20	open	open	ADJ
ejpam-6569	321	21	in	in	ADP
ejpam-6569	321	22	y	y	PROPN
ejpam-6569	321	23	.	.	PUNCT
ejpam-6569	322	1	since	since	SCONJ
ejpam-6569	322	2	f	f	PROPN
ejpam-6569	322	3	is	be	AUX
ejpam-6569	322	4	upper	upper	ADJ
ejpam-6569	322	5	quasi	quasi	NOUN
ejpam-6569	322	6	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	322	7	,	,	PUNCT
ejpam-6569	322	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	322	9	,	,	PUNCT
ejpam-6569	322	10	there	there	PRON
ejpam-6569	322	11	exists	exist	VERB
ejpam-6569	322	12	a	a	DET
ejpam-6569	322	13	semi	semi	NOUN
ejpam-6569	322	14	-	-	ADJ
ejpam-6569	322	15	i	i	PRON
ejpam-6569	322	16	⋆-open	⋆-open	VERB
ejpam-6569	322	17	set	set	VERB
ejpam-6569	322	18	u(x	u(x	NOUN
ejpam-6569	322	19	)	)	PUNCT
ejpam-6569	322	20	of	of	ADP
ejpam-6569	322	21	x	x	SYM
ejpam-6569	322	22	containing	contain	VERB
ejpam-6569	322	23	n.	n.	PROPN
ejpam-6569	322	24	srisarakham	srisarakham	PROPN
ejpam-6569	322	25	,	,	PUNCT
ejpam-6569	322	26	a.	a.	PROPN
ejpam-6569	322	27	sama	sama	PROPN
ejpam-6569	322	28	-	-	PUNCT
ejpam-6569	322	29	ae	ae	PROPN
ejpam-6569	322	30	,	,	PUNCT
ejpam-6569	322	31	c.	c.	PROPN
ejpam-6569	322	32	boonpok	boonpok	PROPN
ejpam-6569	322	33	/	/	SYM
ejpam-6569	322	34	eur	eur	PROPN
ejpam-6569	322	35	.	.	PUNCT
ejpam-6569	323	1	j.	j.	PROPN
ejpam-6569	323	2	pure	pure	PROPN
ejpam-6569	323	3	appl	appl	PROPN
ejpam-6569	323	4	.	.	PROPN
ejpam-6569	323	5	math	math	PROPN
ejpam-6569	323	6	,	,	PUNCT
ejpam-6569	323	7	18	18	NUM
ejpam-6569	323	8	(	(	PUNCT
ejpam-6569	323	9	3	3	NUM
ejpam-6569	323	10	)	)	PUNCT
ejpam-6569	323	11	(	(	PUNCT
ejpam-6569	323	12	2025	2025	NUM
ejpam-6569	323	13	)	)	PUNCT
ejpam-6569	323	14	,	,	PUNCT
ejpam-6569	323	15	6569	6569	NUM
ejpam-6569	323	16	10	10	NUM
ejpam-6569	323	17	of	of	ADP
ejpam-6569	323	18	13	13	NUM
ejpam-6569	323	19	x	x	NOUN
ejpam-6569	323	20	such	such	ADJ
ejpam-6569	323	21	that	that	SCONJ
ejpam-6569	323	22	f	f	PROPN
ejpam-6569	323	23	(	(	PUNCT
ejpam-6569	323	24	scl⋆(u(x	scl⋆(u(x	NOUN
ejpam-6569	323	25	)	)	PUNCT
ejpam-6569	323	26	)	)	PUNCT
ejpam-6569	323	27	)	)	PUNCT
ejpam-6569	324	1	⊆	⊆	X
ejpam-6569	324	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	324	3	-	-	PUNCT
ejpam-6569	324	4	cl(v	cl(v	PRON
ejpam-6569	324	5	(	(	PUNCT
ejpam-6569	324	6	x	x	NOUN
ejpam-6569	324	7	)	)	PUNCT
ejpam-6569	324	8	)	)	PUNCT
ejpam-6569	324	9	.	.	PUNCT
ejpam-6569	325	1	the	the	DET
ejpam-6569	325	2	family	family	NOUN
ejpam-6569	325	3	{	{	PUNCT
ejpam-6569	325	4	u(x	u(x	PROPN
ejpam-6569	325	5	)	)	PUNCT
ejpam-6569	325	6	|	|	ADV
ejpam-6569	325	7	x	x	SYM
ejpam-6569	325	8	∈	∈	NOUN
ejpam-6569	325	9	x	x	X
ejpam-6569	325	10	}	}	PUNCT
ejpam-6569	325	11	is	be	AUX
ejpam-6569	325	12	a	a	DET
ejpam-6569	325	13	semi	semi	ADJ
ejpam-6569	325	14	-	-	ADJ
ejpam-6569	325	15	i	i	PRON
ejpam-6569	325	16	⋆open	⋆open	VERB
ejpam-6569	325	17	cover	cover	NOUN
ejpam-6569	325	18	of	of	ADP
ejpam-6569	325	19	x.	x.	NOUN
ejpam-6569	325	20	since	since	SCONJ
ejpam-6569	325	21	(	(	PUNCT
ejpam-6569	325	22	x	x	X
ejpam-6569	325	23	,	,	PUNCT
ejpam-6569	325	24	τ	τ	PROPN
ejpam-6569	325	25	,	,	PUNCT
ejpam-6569	325	26	i	i	PROPN
ejpam-6569	325	27	)	)	PUNCT
ejpam-6569	325	28	is	be	AUX
ejpam-6569	325	29	s⋆-closed	s⋆-close	VERB
ejpam-6569	325	30	,	,	PUNCT
ejpam-6569	325	31	there	there	PRON
ejpam-6569	325	32	exists	exist	VERB
ejpam-6569	325	33	a	a	DET
ejpam-6569	325	34	finite	finite	ADJ
ejpam-6569	325	35	number	number	NOUN
ejpam-6569	325	36	of	of	ADP
ejpam-6569	325	37	points	point	NOUN
ejpam-6569	325	38	,	,	PUNCT
ejpam-6569	325	39	say	say	INTJ
ejpam-6569	325	40	,	,	PUNCT
ejpam-6569	325	41	x1	x1	PROPN
ejpam-6569	325	42	,	,	PUNCT
ejpam-6569	325	43	x2	x2	PROPN
ejpam-6569	325	44	,	,	PUNCT
ejpam-6569	325	45	...	...	PUNCT
ejpam-6569	325	46	,	,	PUNCT
ejpam-6569	325	47	xn	xn	PROPN
ejpam-6569	326	1	in	in	ADP
ejpam-6569	326	2	x	x	X
ejpam-6569	326	3	such	such	ADJ
ejpam-6569	326	4	that	that	SCONJ
ejpam-6569	326	5	x	x	X
ejpam-6569	326	6	=	=	SYM
ejpam-6569	326	7	∪{scl⋆(u(xi	∪{scl⋆(u(xi	PROPN
ejpam-6569	326	8	)	)	PUNCT
ejpam-6569	326	9	)	)	PUNCT
ejpam-6569	327	1	|	|	ADV
ejpam-6569	327	2	i	i	PRON
ejpam-6569	327	3	=	=	NOUN
ejpam-6569	327	4	1	1	NUM
ejpam-6569	327	5	,	,	PUNCT
ejpam-6569	327	6	2	2	NUM
ejpam-6569	327	7	,	,	PUNCT
ejpam-6569	327	8	...	...	PUNCT
ejpam-6569	327	9	,	,	PUNCT
ejpam-6569	327	10	n	n	CCONJ
ejpam-6569	327	11	}	}	PUNCT
ejpam-6569	327	12	.	.	PUNCT
ejpam-6569	328	1	since	since	SCONJ
ejpam-6569	328	2	f	f	PROPN
ejpam-6569	328	3	is	be	AUX
ejpam-6569	328	4	surjective	surjective	ADJ
ejpam-6569	328	5	,	,	PUNCT
ejpam-6569	328	6	y	y	PROPN
ejpam-6569	328	7	=	=	SYM
ejpam-6569	328	8	f	f	PROPN
ejpam-6569	328	9	(	(	PUNCT
ejpam-6569	328	10	x	x	X
ejpam-6569	328	11	)	)	PUNCT
ejpam-6569	329	1	=	=	SYM
ejpam-6569	329	2	f	f	PROPN
ejpam-6569	329	3	(	(	PUNCT
ejpam-6569	329	4	n	n	CCONJ
ejpam-6569	329	5	∪	∪	VERB
ejpam-6569	329	6	i=1	i=1	PROPN
ejpam-6569	329	7	scl⋆(u(xi	scl⋆(u(xi	PROPN
ejpam-6569	329	8	)	)	PUNCT
ejpam-6569	329	9	)	)	PUNCT
ejpam-6569	329	10	)	)	PUNCT
ejpam-6569	330	1	=	=	PRON
ejpam-6569	331	1	n	n	PRON
ejpam-6569	331	2	∪	∪	VERB
ejpam-6569	331	3	i=1	i=1	PROPN
ejpam-6569	331	4	f	f	PROPN
ejpam-6569	331	5	(	(	PUNCT
ejpam-6569	331	6	scl⋆(u(xi	scl⋆(u(xi	PROPN
ejpam-6569	331	7	)	)	PUNCT
ejpam-6569	331	8	)	)	PUNCT
ejpam-6569	331	9	)	)	PUNCT
ejpam-6569	332	1	⊆	⊆	NUM
ejpam-6569	332	2	n	n	PRON
ejpam-6569	332	3	∪	∪	VERB
ejpam-6569	332	4	i=1	i=1	PRON
ejpam-6569	332	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	332	6	-	-	PUNCT
ejpam-6569	332	7	cl(v	cl(v	PRON
ejpam-6569	332	8	(	(	PUNCT
ejpam-6569	332	9	xi	xi	NOUN
ejpam-6569	332	10	)	)	PUNCT
ejpam-6569	332	11	)	)	PUNCT
ejpam-6569	333	1	=	=	SYM
ejpam-6569	333	2	n	n	PRON
ejpam-6569	333	3	∪	∪	VERB
ejpam-6569	333	4	i=1	i=1	PROPN
ejpam-6569	333	5	∪γ∈γ(xi	∪γ∈γ(xi	X
ejpam-6569	333	6	)	)	PUNCT
ejpam-6569	333	7	σ1σ2	σ1σ2	X
ejpam-6569	333	8	-	-	PUNCT
ejpam-6569	333	9	cl(vγ	cl(vγ	ADJ
ejpam-6569	333	10	)	)	PUNCT
ejpam-6569	333	11	.	.	PUNCT
ejpam-6569	334	1	this	this	PRON
ejpam-6569	334	2	shows	show	VERB
ejpam-6569	334	3	that	that	SCONJ
ejpam-6569	334	4	(	(	PUNCT
ejpam-6569	334	5	y	y	PROPN
ejpam-6569	334	6	,	,	PUNCT
ejpam-6569	334	7	σ1	σ1	PROPN
ejpam-6569	334	8	,	,	PUNCT
ejpam-6569	334	9	σ2	σ2	PROPN
ejpam-6569	334	10	)	)	PUNCT
ejpam-6569	334	11	is	be	AUX
ejpam-6569	334	12	quasi	quasi	X
ejpam-6569	334	13	(	(	PUNCT
ejpam-6569	334	14	σ1	σ1	PROPN
ejpam-6569	334	15	,	,	PUNCT
ejpam-6569	334	16	σ2)-h	σ2)-h	PROPN
ejpam-6569	334	17	-closed	-closed	ADJ
ejpam-6569	334	18	.	.	PUNCT
ejpam-6569	335	1	for	for	ADP
ejpam-6569	335	2	a	a	DET
ejpam-6569	335	3	multifunction	multifunction	NOUN
ejpam-6569	335	4	f	f	NOUN
ejpam-6569	335	5	:	:	PUNCT
ejpam-6569	335	6	(	(	PUNCT
ejpam-6569	335	7	x	x	X
ejpam-6569	335	8	,	,	PUNCT
ejpam-6569	335	9	τ	τ	PROPN
ejpam-6569	335	10	,	,	PUNCT
ejpam-6569	335	11	i	i	NOUN
ejpam-6569	335	12	)	)	PUNCT
ejpam-6569	335	13	→	→	PUNCT
ejpam-6569	335	14	(	(	PUNCT
ejpam-6569	335	15	y	y	PROPN
ejpam-6569	335	16	,	,	PUNCT
ejpam-6569	335	17	σ1	σ1	PROPN
ejpam-6569	335	18	,	,	PUNCT
ejpam-6569	335	19	σ2	σ2	PROPN
ejpam-6569	335	20	)	)	PUNCT
ejpam-6569	335	21	,	,	PUNCT
ejpam-6569	335	22	a	a	DET
ejpam-6569	335	23	multifunction	multifunction	NOUN
ejpam-6569	335	24	sclf⊛	sclf⊛	PROPN
ejpam-6569	335	25	:	:	PUNCT
ejpam-6569	335	26	(	(	PUNCT
ejpam-6569	335	27	x	x	X
ejpam-6569	335	28	,	,	PUNCT
ejpam-6569	335	29	τ	τ	PROPN
ejpam-6569	335	30	,	,	PUNCT
ejpam-6569	335	31	i	i	NOUN
ejpam-6569	335	32	)	)	PUNCT
ejpam-6569	335	33	→	→	PUNCT
ejpam-6569	335	34	(	(	PUNCT
ejpam-6569	335	35	y	y	PROPN
ejpam-6569	335	36	,	,	PUNCT
ejpam-6569	335	37	σ1	σ1	PROPN
ejpam-6569	335	38	,	,	PUNCT
ejpam-6569	335	39	σ2	σ2	PROPN
ejpam-6569	335	40	)	)	PUNCT
ejpam-6569	335	41	is	be	AUX
ejpam-6569	335	42	defined	define	VERB
ejpam-6569	335	43	as	as	SCONJ
ejpam-6569	335	44	follows	follow	VERB
ejpam-6569	335	45	:	:	PUNCT
ejpam-6569	335	46	sclf⊛(x	sclf⊛(x	NUM
ejpam-6569	335	47	)	)	PUNCT
ejpam-6569	335	48	=	=	SYM
ejpam-6569	335	49	(	(	PUNCT
ejpam-6569	335	50	σ1	σ1	PROPN
ejpam-6569	335	51	,	,	PUNCT
ejpam-6569	335	52	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-6569	335	53	(	(	PUNCT
ejpam-6569	335	54	x	x	NOUN
ejpam-6569	335	55	)	)	PUNCT
ejpam-6569	335	56	)	)	PUNCT
ejpam-6569	335	57	for	for	ADP
ejpam-6569	335	58	each	each	DET
ejpam-6569	335	59	x	x	SYM
ejpam-6569	335	60	∈	∈	PROPN
ejpam-6569	335	61	x.	x.	NOUN
ejpam-6569	335	62	lemma	lemma	PROPN
ejpam-6569	336	1	3	3	X
ejpam-6569	336	2	.	.	PUNCT
ejpam-6569	337	1	let	let	VERB
ejpam-6569	337	2	f	f	NOUN
ejpam-6569	337	3	:	:	PUNCT
ejpam-6569	337	4	(	(	PUNCT
ejpam-6569	337	5	x	x	X
ejpam-6569	337	6	,	,	PUNCT
ejpam-6569	337	7	τ	τ	PROPN
ejpam-6569	337	8	,	,	PUNCT
ejpam-6569	337	9	i	i	NOUN
ejpam-6569	337	10	)	)	PUNCT
ejpam-6569	337	11	→	→	PUNCT
ejpam-6569	337	12	(	(	PUNCT
ejpam-6569	337	13	y	y	PROPN
ejpam-6569	337	14	,	,	PUNCT
ejpam-6569	337	15	σ1	σ1	PROPN
ejpam-6569	337	16	,	,	PUNCT
ejpam-6569	337	17	σ2	σ2	PROPN
ejpam-6569	337	18	)	)	PUNCT
ejpam-6569	337	19	be	be	AUX
ejpam-6569	337	20	a	a	DET
ejpam-6569	337	21	multifunction	multifunction	NOUN
ejpam-6569	337	22	.	.	PUNCT
ejpam-6569	338	1	then	then	ADV
ejpam-6569	338	2	,	,	PUNCT
ejpam-6569	338	3	sclf−	sclf−	PROPN
ejpam-6569	338	4	⊛	⊛	NUM
ejpam-6569	338	5	(	(	PUNCT
ejpam-6569	338	6	v	v	NOUN
ejpam-6569	338	7	)	)	PUNCT
ejpam-6569	338	8	=	=	SYM
ejpam-6569	338	9	f−(v	f−(v	ADJ
ejpam-6569	338	10	)	)	PUNCT
ejpam-6569	338	11	for	for	ADP
ejpam-6569	338	12	ever	ever	ADV
ejpam-6569	338	13	(	(	PUNCT
ejpam-6569	338	14	σ1	σ1	PROPN
ejpam-6569	338	15	,	,	PUNCT
ejpam-6569	338	16	σ2)s	σ2)s	NOUN
ejpam-6569	338	17	-	-	PUNCT
ejpam-6569	338	18	open	open	NOUN
ejpam-6569	338	19	set	set	NOUN
ejpam-6569	338	20	v	v	NOUN
ejpam-6569	338	21	of	of	ADP
ejpam-6569	338	22	y	y	PROPN
ejpam-6569	338	23	.	.	PUNCT
ejpam-6569	339	1	proof	proof	NOUN
ejpam-6569	339	2	.	.	PUNCT
ejpam-6569	340	1	let	let	VERB
ejpam-6569	340	2	v	v	PART
ejpam-6569	340	3	be	be	AUX
ejpam-6569	340	4	any	any	DET
ejpam-6569	340	5	(	(	PUNCT
ejpam-6569	340	6	σ1	σ1	NOUN
ejpam-6569	340	7	,	,	PUNCT
ejpam-6569	340	8	σ2)s	σ2)s	NOUN
ejpam-6569	340	9	-	-	PUNCT
ejpam-6569	340	10	open	open	ADJ
ejpam-6569	340	11	set	set	NOUN
ejpam-6569	340	12	of	of	ADP
ejpam-6569	340	13	y	y	PROPN
ejpam-6569	340	14	.	.	PUNCT
ejpam-6569	341	1	let	let	VERB
ejpam-6569	341	2	x	x	X
ejpam-6569	341	3	∈	∈	PROPN
ejpam-6569	341	4	sclf−	sclf−	NOUN
ejpam-6569	341	5	⊛	⊛	NUM
ejpam-6569	341	6	(	(	PUNCT
ejpam-6569	341	7	v	v	NOUN
ejpam-6569	341	8	)	)	PUNCT
ejpam-6569	341	9	.	.	PUNCT
ejpam-6569	342	1	then	then	ADV
ejpam-6569	342	2	,	,	PUNCT
ejpam-6569	342	3	(	(	PUNCT
ejpam-6569	342	4	σ1	σ1	PROPN
ejpam-6569	342	5	,	,	PUNCT
ejpam-6569	342	6	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-6569	342	7	(	(	PUNCT
ejpam-6569	342	8	x	x	NOUN
ejpam-6569	342	9	)	)	PUNCT
ejpam-6569	342	10	)	)	PUNCT
ejpam-6569	342	11	∩	∩	NOUN
ejpam-6569	342	12	v	v	ADP
ejpam-6569	342	13	=	=	SYM
ejpam-6569	342	14	sclf⊛(x	sclf⊛(x	PROPN
ejpam-6569	342	15	)	)	PUNCT
ejpam-6569	342	16	∩	∩	NOUN
ejpam-6569	342	17	v	v	ADP
ejpam-6569	342	18	̸=	̸=	PROPN
ejpam-6569	342	19	∅.	∅.	NOUN
ejpam-6569	342	20	since	since	SCONJ
ejpam-6569	342	21	v	v	NOUN
ejpam-6569	342	22	is	be	AUX
ejpam-6569	342	23	(	(	PUNCT
ejpam-6569	342	24	σ1	σ1	PROPN
ejpam-6569	342	25	,	,	PUNCT
ejpam-6569	342	26	σ2)s	σ2)s	NOUN
ejpam-6569	342	27	-	-	PUNCT
ejpam-6569	342	28	open	open	ADJ
ejpam-6569	342	29	in	in	ADP
ejpam-6569	342	30	y	y	PROPN
ejpam-6569	342	31	,	,	PUNCT
ejpam-6569	342	32	we	we	PRON
ejpam-6569	342	33	have	have	VERB
ejpam-6569	342	34	v	v	NUM
ejpam-6569	342	35	∩	∩	ADJ
ejpam-6569	342	36	f	f	X
ejpam-6569	342	37	(	(	PUNCT
ejpam-6569	342	38	x	x	X
ejpam-6569	342	39	)	)	PUNCT
ejpam-6569	342	40	̸=	̸=	PROPN
ejpam-6569	342	41	∅	∅	NOUN
ejpam-6569	342	42	and	and	CCONJ
ejpam-6569	342	43	hence	hence	ADV
ejpam-6569	342	44	x	x	PART
ejpam-6569	342	45	∈	∈	PROPN
ejpam-6569	342	46	f−(v	f−(v	NOUN
ejpam-6569	342	47	)	)	PUNCT
ejpam-6569	342	48	.	.	PUNCT
ejpam-6569	343	1	this	this	PRON
ejpam-6569	343	2	shows	show	VERB
ejpam-6569	343	3	that	that	SCONJ
ejpam-6569	343	4	sclf−	sclf−	NOUN
ejpam-6569	343	5	⊛	⊛	NUM
ejpam-6569	343	6	(	(	PUNCT
ejpam-6569	343	7	v	v	NOUN
ejpam-6569	343	8	)	)	PUNCT
ejpam-6569	343	9	⊆	⊆	NUM
ejpam-6569	343	10	f−(v	f−(v	NOUN
ejpam-6569	343	11	)	)	PUNCT
ejpam-6569	343	12	.	.	PUNCT
ejpam-6569	344	1	on	on	ADP
ejpam-6569	344	2	the	the	DET
ejpam-6569	344	3	other	other	ADJ
ejpam-6569	344	4	hand	hand	NOUN
ejpam-6569	344	5	,	,	PUNCT
ejpam-6569	344	6	let	let	VERB
ejpam-6569	344	7	x	x	PUNCT
ejpam-6569	344	8	∈	∈	PROPN
ejpam-6569	344	9	f−(v	f−(v	NOUN
ejpam-6569	344	10	)	)	PUNCT
ejpam-6569	344	11	.	.	PUNCT
ejpam-6569	345	1	then	then	ADV
ejpam-6569	345	2	,	,	PUNCT
ejpam-6569	345	3	∅	∅	NOUN
ejpam-6569	345	4	=	=	NOUN
ejpam-6569	345	5	̸	̸	NUM
ejpam-6569	345	6	f	f	NOUN
ejpam-6569	345	7	(	(	PUNCT
ejpam-6569	345	8	x	x	NOUN
ejpam-6569	345	9	)	)	PUNCT
ejpam-6569	345	10	∩	∩	NOUN
ejpam-6569	345	11	v	v	ADP
ejpam-6569	345	12	⊆	⊆	NUM
ejpam-6569	345	13	(	(	PUNCT
ejpam-6569	345	14	σ1	σ1	PROPN
ejpam-6569	345	15	,	,	PUNCT
ejpam-6569	345	16	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-6569	345	17	(	(	PUNCT
ejpam-6569	345	18	x	x	NOUN
ejpam-6569	345	19	)	)	PUNCT
ejpam-6569	345	20	)	)	PUNCT
ejpam-6569	345	21	∩	∩	PROPN
ejpam-6569	345	22	v.	v.	ADP
ejpam-6569	345	23	thus	thus	ADV
ejpam-6569	345	24	,	,	PUNCT
ejpam-6569	345	25	x	x	PROPN
ejpam-6569	345	26	∈	∈	PROPN
ejpam-6569	345	27	sclf−	sclf−	NOUN
ejpam-6569	345	28	⊛	⊛	NUM
ejpam-6569	345	29	(	(	PUNCT
ejpam-6569	345	30	v	v	NOUN
ejpam-6569	345	31	)	)	PUNCT
ejpam-6569	345	32	and	and	CCONJ
ejpam-6569	345	33	so	so	ADV
ejpam-6569	345	34	sclf−	sclf−	PROPN
ejpam-6569	345	35	⊛	⊛	NUM
ejpam-6569	345	36	(	(	PUNCT
ejpam-6569	345	37	v	v	NOUN
ejpam-6569	345	38	)	)	PUNCT
ejpam-6569	345	39	=	=	SYM
ejpam-6569	345	40	f−(v	f−(v	ADJ
ejpam-6569	345	41	)	)	PUNCT
ejpam-6569	345	42	.	.	PUNCT
ejpam-6569	346	1	theorem	theorem	VERB
ejpam-6569	346	2	8	8	NUM
ejpam-6569	346	3	.	.	PUNCT
ejpam-6569	347	1	a	a	DET
ejpam-6569	347	2	multifunction	multifunction	NOUN
ejpam-6569	347	3	f	f	NOUN
ejpam-6569	347	4	:	:	PUNCT
ejpam-6569	347	5	(	(	PUNCT
ejpam-6569	347	6	x	x	X
ejpam-6569	347	7	,	,	PUNCT
ejpam-6569	347	8	τ	τ	PROPN
ejpam-6569	347	9	,	,	PUNCT
ejpam-6569	347	10	i	i	NOUN
ejpam-6569	347	11	)	)	PUNCT
ejpam-6569	347	12	→	→	PUNCT
ejpam-6569	347	13	(	(	PUNCT
ejpam-6569	347	14	y	y	PROPN
ejpam-6569	347	15	,	,	PUNCT
ejpam-6569	347	16	σ1	σ1	PROPN
ejpam-6569	347	17	,	,	PUNCT
ejpam-6569	347	18	σ2	σ2	NOUN
ejpam-6569	347	19	)	)	PUNCT
ejpam-6569	347	20	is	be	AUX
ejpam-6569	347	21	lower	low	ADJ
ejpam-6569	347	22	quasi	quasi	NOUN
ejpam-6569	347	23	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	347	24	,	,	PUNCT
ejpam-6569	347	25	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	347	26	if	if	SCONJ
ejpam-6569	347	27	and	and	CCONJ
ejpam-6569	347	28	only	only	ADV
ejpam-6569	347	29	if	if	SCONJ
ejpam-6569	347	30	sclf⊛	sclf⊛	PRON
ejpam-6569	347	31	:	:	PUNCT
ejpam-6569	347	32	(	(	PUNCT
ejpam-6569	347	33	x	x	X
ejpam-6569	347	34	,	,	PUNCT
ejpam-6569	347	35	τ	τ	PROPN
ejpam-6569	347	36	,	,	PUNCT
ejpam-6569	347	37	i	i	NOUN
ejpam-6569	347	38	)	)	PUNCT
ejpam-6569	347	39	→	→	PUNCT
ejpam-6569	347	40	(	(	PUNCT
ejpam-6569	347	41	y	y	PROPN
ejpam-6569	347	42	,	,	PUNCT
ejpam-6569	347	43	σ1	σ1	PROPN
ejpam-6569	347	44	,	,	PUNCT
ejpam-6569	347	45	σ2	σ2	NOUN
ejpam-6569	347	46	)	)	PUNCT
ejpam-6569	347	47	is	be	AUX
ejpam-6569	347	48	lower	low	ADJ
ejpam-6569	347	49	quasi	quasi	NOUN
ejpam-6569	347	50	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	347	51	,	,	PUNCT
ejpam-6569	347	52	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	347	53	.	.	PUNCT
ejpam-6569	348	1	proof	proof	NOUN
ejpam-6569	348	2	.	.	PUNCT
ejpam-6569	349	1	suppose	suppose	VERB
ejpam-6569	349	2	that	that	SCONJ
ejpam-6569	349	3	f	f	PROPN
ejpam-6569	349	4	is	be	AUX
ejpam-6569	349	5	lower	low	ADJ
ejpam-6569	349	6	quasi	quasi	NOUN
ejpam-6569	349	7	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	349	8	,	,	PUNCT
ejpam-6569	349	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	349	10	.	.	PUNCT
ejpam-6569	350	1	let	let	VERB
ejpam-6569	350	2	x	x	SYM
ejpam-6569	350	3	∈	∈	PROPN
ejpam-6569	350	4	x	x	X
ejpam-6569	350	5	and	and	CCONJ
ejpam-6569	350	6	v	v	X
ejpam-6569	350	7	be	be	AUX
ejpam-6569	350	8	any	any	DET
ejpam-6569	350	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	350	10	-	-	ADJ
ejpam-6569	350	11	open	open	ADJ
ejpam-6569	350	12	set	set	NOUN
ejpam-6569	350	13	of	of	ADP
ejpam-6569	350	14	y	y	PRON
ejpam-6569	350	15	such	such	ADJ
ejpam-6569	350	16	that	that	SCONJ
ejpam-6569	350	17	sclf⊛(x)∩v	sclf⊛(x)∩v	PROPN
ejpam-6569	350	18	̸=	̸=	PROPN
ejpam-6569	350	19	∅.	∅.	ADV
ejpam-6569	350	20	by	by	ADP
ejpam-6569	350	21	lemma	lemma	PROPN
ejpam-6569	350	22	3	3	NUM
ejpam-6569	350	23	,	,	PUNCT
ejpam-6569	350	24	we	we	PRON
ejpam-6569	350	25	have	have	VERB
ejpam-6569	350	26	f	f	PROPN
ejpam-6569	350	27	(	(	PUNCT
ejpam-6569	350	28	x)∩v	x)∩v	PROPN
ejpam-6569	350	29	̸=	̸=	PROPN
ejpam-6569	350	30	∅.	∅.	ADV
ejpam-6569	350	31	since	since	SCONJ
ejpam-6569	350	32	f	f	PROPN
ejpam-6569	350	33	is	be	AUX
ejpam-6569	350	34	lower	low	ADJ
ejpam-6569	350	35	quasi	quasi	NOUN
ejpam-6569	350	36	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	350	37	,	,	PUNCT
ejpam-6569	350	38	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	350	39	,	,	PUNCT
ejpam-6569	350	40	there	there	PRON
ejpam-6569	350	41	exists	exist	VERB
ejpam-6569	350	42	a	a	DET
ejpam-6569	350	43	semi	semi	NOUN
ejpam-6569	350	44	-	-	ADJ
ejpam-6569	350	45	i	i	PRON
ejpam-6569	350	46	⋆-open	⋆-open	VERB
ejpam-6569	350	47	set	set	VERB
ejpam-6569	350	48	u	u	NOUN
ejpam-6569	350	49	of	of	ADP
ejpam-6569	350	50	x	x	PUNCT
ejpam-6569	350	51	containing	contain	VERB
ejpam-6569	350	52	x	x	PUNCT
ejpam-6569	350	53	such	such	ADJ
ejpam-6569	350	54	that	that	SCONJ
ejpam-6569	350	55	σ1σ2	σ1σ2	NOUN
ejpam-6569	350	56	-	-	PUNCT
ejpam-6569	350	57	cl(v	cl(v	NOUN
ejpam-6569	350	58	)	)	PUNCT
ejpam-6569	350	59	∩	∩	PROPN
ejpam-6569	350	60	f	f	X
ejpam-6569	350	61	(	(	PUNCT
ejpam-6569	350	62	z	z	NOUN
ejpam-6569	350	63	)	)	PUNCT
ejpam-6569	350	64	̸=	̸=	NOUN
ejpam-6569	350	65	∅	∅	NOUN
ejpam-6569	350	66	for	for	ADP
ejpam-6569	350	67	every	every	DET
ejpam-6569	350	68	z	z	PROPN
ejpam-6569	350	69	∈	∈	PROPN
ejpam-6569	350	70	scl⋆(u	scl⋆(u	NUM
ejpam-6569	350	71	)	)	PUNCT
ejpam-6569	350	72	.	.	PUNCT
ejpam-6569	351	1	since	since	SCONJ
ejpam-6569	351	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	351	3	-	-	NOUN
ejpam-6569	351	4	cl(v	cl(v	NOUN
ejpam-6569	351	5	)	)	PUNCT
ejpam-6569	351	6	is	be	AUX
ejpam-6569	351	7	(	(	PUNCT
ejpam-6569	351	8	σ1	σ1	PROPN
ejpam-6569	351	9	,	,	PUNCT
ejpam-6569	351	10	σ2)sopen	σ2)sopen	VERB
ejpam-6569	351	11	in	in	ADP
ejpam-6569	351	12	y	y	PROPN
ejpam-6569	351	13	,	,	PUNCT
ejpam-6569	351	14	by	by	ADP
ejpam-6569	351	15	lemma	lemma	PROPN
ejpam-6569	351	16	3	3	NUM
ejpam-6569	351	17	we	we	PRON
ejpam-6569	351	18	have	have	VERB
ejpam-6569	351	19	scl⋆(u	scl⋆(u	NUM
ejpam-6569	351	20	)	)	PUNCT
ejpam-6569	352	1	⊆	⊆	NUM
ejpam-6569	352	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	352	3	-	-	PUNCT
ejpam-6569	352	4	cl(v	cl(v	NOUN
ejpam-6569	352	5	)	)	PUNCT
ejpam-6569	352	6	)	)	PUNCT
ejpam-6569	353	1	=	=	PRON
ejpam-6569	353	2	sclf−	sclf−	NOUN
ejpam-6569	353	3	⊛	⊛	NUM
ejpam-6569	353	4	(	(	PUNCT
ejpam-6569	353	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	353	6	-	-	NUM
ejpam-6569	353	7	cl(v	cl(v	NOUN
ejpam-6569	353	8	)	)	PUNCT
ejpam-6569	353	9	)	)	PUNCT
ejpam-6569	353	10	and	and	CCONJ
ejpam-6569	353	11	hence	hence	ADV
ejpam-6569	353	12	sclf⊛(z	sclf⊛(z	VERB
ejpam-6569	353	13	)	)	PUNCT
ejpam-6569	353	14	∩	∩	NOUN
ejpam-6569	353	15	σ1σ2	σ1σ2	NOUN
ejpam-6569	353	16	-	-	NOUN
ejpam-6569	353	17	cl(v	cl(v	NOUN
ejpam-6569	353	18	)	)	PUNCT
ejpam-6569	353	19	̸=	̸=	NOUN
ejpam-6569	353	20	∅	∅	NOUN
ejpam-6569	353	21	for	for	ADP
ejpam-6569	353	22	every	every	DET
ejpam-6569	353	23	z	z	PROPN
ejpam-6569	353	24	∈	∈	PROPN
ejpam-6569	353	25	scl⋆(u	scl⋆(u	NUM
ejpam-6569	353	26	)	)	PUNCT
ejpam-6569	353	27	.	.	PUNCT
ejpam-6569	354	1	this	this	PRON
ejpam-6569	354	2	shows	show	VERB
ejpam-6569	354	3	that	that	SCONJ
ejpam-6569	354	4	sclf⊛	sclf⊛	PROPN
ejpam-6569	354	5	is	be	AUX
ejpam-6569	354	6	lower	low	ADJ
ejpam-6569	354	7	quasi	quasi	NOUN
ejpam-6569	354	8	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	354	9	,	,	PUNCT
ejpam-6569	354	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	354	11	.	.	PUNCT
ejpam-6569	354	12	conversely	conversely	ADV
ejpam-6569	354	13	,	,	PUNCT
ejpam-6569	354	14	suppose	suppose	VERB
ejpam-6569	354	15	that	that	SCONJ
ejpam-6569	354	16	sclf⊛	sclf⊛	PROPN
ejpam-6569	354	17	is	be	AUX
ejpam-6569	354	18	lower	low	ADJ
ejpam-6569	354	19	quasi	quasi	NOUN
ejpam-6569	354	20	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	354	21	,	,	PUNCT
ejpam-6569	354	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	354	23	.	.	PUNCT
ejpam-6569	355	1	let	let	VERB
ejpam-6569	355	2	x	x	SYM
ejpam-6569	355	3	∈	∈	PROPN
ejpam-6569	355	4	x	x	X
ejpam-6569	355	5	and	and	CCONJ
ejpam-6569	355	6	v	v	X
ejpam-6569	355	7	be	be	AUX
ejpam-6569	355	8	any	any	DET
ejpam-6569	355	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	355	10	-	-	ADJ
ejpam-6569	355	11	open	open	ADJ
ejpam-6569	355	12	set	set	NOUN
ejpam-6569	355	13	of	of	ADP
ejpam-6569	355	14	y	y	PRON
ejpam-6569	355	15	such	such	ADJ
ejpam-6569	355	16	that	that	SCONJ
ejpam-6569	355	17	f	f	PROPN
ejpam-6569	355	18	(	(	PUNCT
ejpam-6569	355	19	x	x	NOUN
ejpam-6569	355	20	)	)	PUNCT
ejpam-6569	355	21	∩	∩	NOUN
ejpam-6569	355	22	v	v	ADP
ejpam-6569	355	23	̸=	̸=	PROPN
ejpam-6569	355	24	∅.	∅.	NOUN
ejpam-6569	355	25	then	then	ADV
ejpam-6569	355	26	,	,	PUNCT
ejpam-6569	355	27	(	(	PUNCT
ejpam-6569	355	28	σ1	σ1	PROPN
ejpam-6569	355	29	,	,	PUNCT
ejpam-6569	355	30	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-6569	355	31	(	(	PUNCT
ejpam-6569	355	32	x	x	NOUN
ejpam-6569	355	33	)	)	PUNCT
ejpam-6569	355	34	)	)	PUNCT
ejpam-6569	355	35	∩	∩	NOUN
ejpam-6569	355	36	v	v	ADP
ejpam-6569	355	37	̸=	̸=	PROPN
ejpam-6569	355	38	∅.	∅.	NOUN
ejpam-6569	355	39	since	since	SCONJ
ejpam-6569	355	40	sclf⊛	sclf⊛	PROPN
ejpam-6569	355	41	is	be	AUX
ejpam-6569	355	42	lower	low	ADJ
ejpam-6569	355	43	quasi	quasi	NOUN
ejpam-6569	355	44	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	355	45	,	,	PUNCT
ejpam-6569	355	46	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	355	47	,	,	PUNCT
ejpam-6569	355	48	there	there	PRON
ejpam-6569	355	49	exists	exist	VERB
ejpam-6569	355	50	a	a	DET
ejpam-6569	355	51	semi	semi	NOUN
ejpam-6569	355	52	-	-	ADJ
ejpam-6569	355	53	i	i	PRON
ejpam-6569	355	54	⋆-open	⋆-open	VERB
ejpam-6569	355	55	set	set	VERB
ejpam-6569	355	56	of	of	ADP
ejpam-6569	355	57	x	x	PUNCT
ejpam-6569	355	58	containing	contain	VERB
ejpam-6569	355	59	x	x	PUNCT
ejpam-6569	355	60	such	such	ADJ
ejpam-6569	355	61	that	that	SCONJ
ejpam-6569	355	62	sclf⊛(z)∩σ1σ2	sclf⊛(z)∩σ1σ2	NOUN
ejpam-6569	355	63	-	-	PUNCT
ejpam-6569	355	64	cl(v	cl(v	NOUN
ejpam-6569	355	65	)	)	PUNCT
ejpam-6569	355	66	̸=	̸=	NOUN
ejpam-6569	355	67	∅	∅	NOUN
ejpam-6569	355	68	for	for	ADP
ejpam-6569	355	69	every	every	DET
ejpam-6569	355	70	z	z	PROPN
ejpam-6569	355	71	∈	∈	PROPN
ejpam-6569	355	72	scl⋆(u	scl⋆(u	NUM
ejpam-6569	355	73	)	)	PUNCT
ejpam-6569	355	74	.	.	PUNCT
ejpam-6569	356	1	since	since	SCONJ
ejpam-6569	356	2	σ1σ2	σ1σ2	NOUN
ejpam-6569	356	3	-	-	NOUN
ejpam-6569	356	4	cl(v	cl(v	NOUN
ejpam-6569	356	5	)	)	PUNCT
ejpam-6569	356	6	is	be	AUX
ejpam-6569	356	7	(	(	PUNCT
ejpam-6569	356	8	σ1	σ1	PROPN
ejpam-6569	356	9	,	,	PUNCT
ejpam-6569	356	10	σ2)s	σ2)s	NOUN
ejpam-6569	356	11	-	-	PUNCT
ejpam-6569	356	12	open	open	ADJ
ejpam-6569	356	13	in	in	ADP
ejpam-6569	356	14	y	y	PROPN
ejpam-6569	356	15	and	and	CCONJ
ejpam-6569	356	16	by	by	ADP
ejpam-6569	356	17	lemma	lemma	PROPN
ejpam-6569	356	18	3	3	NUM
ejpam-6569	356	19	,	,	PUNCT
ejpam-6569	356	20	scl⋆(u	scl⋆(u	NUM
ejpam-6569	356	21	)	)	PUNCT
ejpam-6569	356	22	⊆	⊆	NUM
ejpam-6569	356	23	sclf−	sclf−	NOUN
ejpam-6569	356	24	⊛	⊛	NUM
ejpam-6569	356	25	(	(	PUNCT
ejpam-6569	356	26	σ1σ2	σ1σ2	NOUN
ejpam-6569	356	27	-	-	NUM
ejpam-6569	356	28	cl(v	cl(v	NOUN
ejpam-6569	356	29	)	)	PUNCT
ejpam-6569	356	30	)	)	PUNCT
ejpam-6569	357	1	=	=	PUNCT
ejpam-6569	357	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6569	357	3	-	-	PUNCT
ejpam-6569	357	4	cl(v	cl(v	NOUN
ejpam-6569	357	5	)	)	PUNCT
ejpam-6569	357	6	)	)	PUNCT
ejpam-6569	357	7	and	and	CCONJ
ejpam-6569	357	8	hence	hence	ADV
ejpam-6569	357	9	σ1σ2	σ1σ2	NOUN
ejpam-6569	357	10	-	-	NUM
ejpam-6569	357	11	cl(v	cl(v	PUNCT
ejpam-6569	357	12	)	)	PUNCT
ejpam-6569	357	13	∩f	∩f	NOUN
ejpam-6569	357	14	(	(	PUNCT
ejpam-6569	357	15	z	z	X
ejpam-6569	357	16	)	)	PUNCT
ejpam-6569	357	17	̸=	̸=	NOUN
ejpam-6569	357	18	∅	∅	NOUN
ejpam-6569	357	19	for	for	ADP
ejpam-6569	357	20	every	every	DET
ejpam-6569	357	21	z	z	PROPN
ejpam-6569	357	22	∈	∈	PROPN
ejpam-6569	357	23	scl⋆(u	scl⋆(u	NUM
ejpam-6569	357	24	)	)	PUNCT
ejpam-6569	357	25	.	.	PUNCT
ejpam-6569	358	1	thus	thus	ADV
ejpam-6569	358	2	,	,	PUNCT
ejpam-6569	358	3	f	f	PROPN
ejpam-6569	358	4	is	be	AUX
ejpam-6569	358	5	lower	low	ADJ
ejpam-6569	358	6	quasi	quasi	NOUN
ejpam-6569	358	7	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	358	8	,	,	PUNCT
ejpam-6569	358	9	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	358	10	.	.	PUNCT
ejpam-6569	359	1	n.	n.	PROPN
ejpam-6569	359	2	srisarakham	srisarakham	PROPN
ejpam-6569	359	3	,	,	PUNCT
ejpam-6569	359	4	a.	a.	PROPN
ejpam-6569	359	5	sama	sama	PROPN
ejpam-6569	359	6	-	-	PUNCT
ejpam-6569	359	7	ae	ae	PROPN
ejpam-6569	359	8	,	,	PUNCT
ejpam-6569	359	9	c.	c.	PROPN
ejpam-6569	359	10	boonpok	boonpok	PROPN
ejpam-6569	359	11	/	/	SYM
ejpam-6569	359	12	eur	eur	PROPN
ejpam-6569	359	13	.	.	PUNCT
ejpam-6569	360	1	j.	j.	PROPN
ejpam-6569	360	2	pure	pure	PROPN
ejpam-6569	360	3	appl	appl	PROPN
ejpam-6569	360	4	.	.	PROPN
ejpam-6569	360	5	math	math	PROPN
ejpam-6569	360	6	,	,	PUNCT
ejpam-6569	360	7	18	18	NUM
ejpam-6569	360	8	(	(	PUNCT
ejpam-6569	360	9	3	3	NUM
ejpam-6569	360	10	)	)	PUNCT
ejpam-6569	360	11	(	(	PUNCT
ejpam-6569	360	12	2025	2025	NUM
ejpam-6569	360	13	)	)	PUNCT
ejpam-6569	360	14	,	,	PUNCT
ejpam-6569	360	15	6569	6569	NUM
ejpam-6569	360	16	11	11	NUM
ejpam-6569	360	17	of	of	ADP
ejpam-6569	360	18	13	13	NUM
ejpam-6569	360	19	definition	definition	NOUN
ejpam-6569	360	20	3	3	NUM
ejpam-6569	360	21	.	.	PUNCT
ejpam-6569	361	1	[	[	X
ejpam-6569	361	2	22	22	NUM
ejpam-6569	361	3	]	]	PUNCT
ejpam-6569	361	4	a	a	DET
ejpam-6569	361	5	subset	subset	NOUN
ejpam-6569	361	6	a	a	PRON
ejpam-6569	361	7	of	of	ADP
ejpam-6569	361	8	a	a	DET
ejpam-6569	361	9	bitopological	bitopological	ADJ
ejpam-6569	361	10	space	space	NOUN
ejpam-6569	361	11	(	(	PUNCT
ejpam-6569	361	12	x	x	NOUN
ejpam-6569	361	13	,	,	PUNCT
ejpam-6569	361	14	τ1	τ1	NOUN
ejpam-6569	361	15	,	,	PUNCT
ejpam-6569	361	16	τ2	τ2	NOUN
ejpam-6569	361	17	)	)	PUNCT
ejpam-6569	361	18	is	be	AUX
ejpam-6569	361	19	said	say	VERB
ejpam-6569	361	20	to	to	PART
ejpam-6569	361	21	be	be	AUX
ejpam-6569	361	22	:	:	PUNCT
ejpam-6569	361	23	(	(	PUNCT
ejpam-6569	361	24	1	1	X
ejpam-6569	361	25	)	)	PUNCT
ejpam-6569	361	26	τ1τ2	τ1τ2	NOUN
ejpam-6569	361	27	-	-	NOUN
ejpam-6569	361	28	paracompact	paracompact	ADJ
ejpam-6569	361	29	if	if	SCONJ
ejpam-6569	361	30	every	every	DET
ejpam-6569	361	31	cover	cover	NOUN
ejpam-6569	361	32	of	of	ADP
ejpam-6569	361	33	a	a	PRON
ejpam-6569	361	34	by	by	ADP
ejpam-6569	361	35	τ1τ2	τ1τ2	ADJ
ejpam-6569	361	36	-	-	ADJ
ejpam-6569	361	37	open	open	ADJ
ejpam-6569	361	38	sets	set	NOUN
ejpam-6569	361	39	of	of	ADP
ejpam-6569	361	40	x	x	VERB
ejpam-6569	361	41	is	be	AUX
ejpam-6569	361	42	refined	refine	VERB
ejpam-6569	361	43	by	by	ADP
ejpam-6569	361	44	a	a	DET
ejpam-6569	361	45	cover	cover	NOUN
ejpam-6569	361	46	of	of	ADP
ejpam-6569	361	47	a	a	PRON
ejpam-6569	361	48	which	which	PRON
ejpam-6569	361	49	consists	consist	VERB
ejpam-6569	361	50	of	of	ADP
ejpam-6569	361	51	τ1τ2	τ1τ2	ADJ
ejpam-6569	361	52	-	-	ADJ
ejpam-6569	361	53	open	open	ADJ
ejpam-6569	361	54	sets	set	NOUN
ejpam-6569	361	55	of	of	ADP
ejpam-6569	361	56	x	x	PUNCT
ejpam-6569	361	57	and	and	CCONJ
ejpam-6569	361	58	is	be	AUX
ejpam-6569	361	59	τ1τ2	τ1τ2	NOUN
ejpam-6569	361	60	-	-	ADJ
ejpam-6569	361	61	locally	locally	ADV
ejpam-6569	361	62	finite	finite	NOUN
ejpam-6569	361	63	in	in	ADP
ejpam-6569	361	64	x	x	PRON
ejpam-6569	361	65	;	;	PUNCT
ejpam-6569	361	66	(	(	PUNCT
ejpam-6569	361	67	2	2	X
ejpam-6569	361	68	)	)	PUNCT
ejpam-6569	361	69	τ1τ2	τ1τ2	NOUN
ejpam-6569	361	70	-	-	NOUN
ejpam-6569	361	71	regular	regular	ADJ
ejpam-6569	361	72	if	if	SCONJ
ejpam-6569	361	73	for	for	ADP
ejpam-6569	361	74	each	each	DET
ejpam-6569	361	75	x	x	SYM
ejpam-6569	361	76	∈	∈	PROPN
ejpam-6569	361	77	a	a	PRON
ejpam-6569	361	78	and	and	CCONJ
ejpam-6569	361	79	each	each	DET
ejpam-6569	361	80	τ1τ2	τ1τ2	ADJ
ejpam-6569	361	81	-	-	ADJ
ejpam-6569	361	82	open	open	ADJ
ejpam-6569	361	83	set	set	ADJ
ejpam-6569	361	84	u	u	NOUN
ejpam-6569	361	85	of	of	ADP
ejpam-6569	361	86	x	x	PUNCT
ejpam-6569	361	87	containing	contain	VERB
ejpam-6569	361	88	x	x	PRON
ejpam-6569	361	89	,	,	PUNCT
ejpam-6569	361	90	there	there	PRON
ejpam-6569	361	91	exists	exist	VERB
ejpam-6569	361	92	a	a	DET
ejpam-6569	361	93	τ1τ2	τ1τ2	NOUN
ejpam-6569	361	94	-	-	ADJ
ejpam-6569	361	95	open	open	ADJ
ejpam-6569	361	96	set	set	NOUN
ejpam-6569	361	97	v	v	NOUN
ejpam-6569	361	98	of	of	ADP
ejpam-6569	361	99	x	x	PUNCT
ejpam-6569	361	100	such	such	ADJ
ejpam-6569	361	101	that	that	SCONJ
ejpam-6569	361	102	x	x	SYM
ejpam-6569	361	103	∈	∈	NOUN
ejpam-6569	361	104	v	v	ADP
ejpam-6569	361	105	⊆	⊆	NUM
ejpam-6569	361	106	τ1τ2	τ1τ2	NOUN
ejpam-6569	361	107	-	-	NOUN
ejpam-6569	361	108	cl(v	cl(v	X
ejpam-6569	361	109	)	)	PUNCT
ejpam-6569	361	110	⊆	⊆	NUM
ejpam-6569	361	111	u	u	NOUN
ejpam-6569	361	112	.	.	PUNCT
ejpam-6569	362	1	lemma	lemma	PROPN
ejpam-6569	362	2	4	4	NUM
ejpam-6569	362	3	.	.	PUNCT
ejpam-6569	363	1	[	[	X
ejpam-6569	363	2	22	22	NUM
ejpam-6569	363	3	]	]	X
ejpam-6569	363	4	if	if	SCONJ
ejpam-6569	363	5	a	a	PRON
ejpam-6569	363	6	is	be	AUX
ejpam-6569	363	7	a	a	DET
ejpam-6569	363	8	τ1τ2	τ1τ2	ADJ
ejpam-6569	363	9	-	-	ADJ
ejpam-6569	363	10	regular	regular	ADJ
ejpam-6569	363	11	τ1τ2	τ1τ2	NOUN
ejpam-6569	363	12	-	-	ADJ
ejpam-6569	363	13	paracompact	paracompact	ADJ
ejpam-6569	363	14	set	set	NOUN
ejpam-6569	363	15	of	of	ADP
ejpam-6569	363	16	a	a	DET
ejpam-6569	363	17	bitopological	bitopological	ADJ
ejpam-6569	363	18	space	space	NOUN
ejpam-6569	363	19	(	(	PUNCT
ejpam-6569	363	20	x	x	NOUN
ejpam-6569	363	21	,	,	PUNCT
ejpam-6569	363	22	τ1	τ1	NOUN
ejpam-6569	363	23	,	,	PUNCT
ejpam-6569	363	24	τ2	τ2	NOUN
ejpam-6569	363	25	)	)	PUNCT
ejpam-6569	363	26	and	and	CCONJ
ejpam-6569	363	27	u	u	NOUN
ejpam-6569	363	28	is	be	AUX
ejpam-6569	363	29	a	a	DET
ejpam-6569	363	30	τ1τ2	τ1τ2	ADJ
ejpam-6569	363	31	-	-	ADJ
ejpam-6569	363	32	open	open	ADJ
ejpam-6569	363	33	neighborhood	neighborhood	NOUN
ejpam-6569	363	34	of	of	ADP
ejpam-6569	363	35	a	a	PRON
ejpam-6569	363	36	,	,	PUNCT
ejpam-6569	363	37	then	then	ADV
ejpam-6569	363	38	there	there	PRON
ejpam-6569	363	39	exists	exist	VERB
ejpam-6569	363	40	a	a	DET
ejpam-6569	363	41	τ1τ2	τ1τ2	NOUN
ejpam-6569	363	42	-	-	ADJ
ejpam-6569	363	43	open	open	ADJ
ejpam-6569	363	44	set	set	NOUN
ejpam-6569	363	45	v	v	NOUN
ejpam-6569	363	46	of	of	ADP
ejpam-6569	363	47	x	x	PUNCT
ejpam-6569	363	48	such	such	ADJ
ejpam-6569	363	49	that	that	SCONJ
ejpam-6569	363	50	a	a	DET
ejpam-6569	363	51	⊆	⊆	NUM
ejpam-6569	363	52	v	v	ADP
ejpam-6569	363	53	⊆	⊆	NUM
ejpam-6569	363	54	τ1τ2	τ1τ2	NOUN
ejpam-6569	363	55	-	-	NOUN
ejpam-6569	363	56	cl(v	cl(v	X
ejpam-6569	363	57	)	)	PUNCT
ejpam-6569	363	58	⊆	⊆	NUM
ejpam-6569	363	59	u	u	NOUN
ejpam-6569	363	60	.	.	PUNCT
ejpam-6569	364	1	lemma	lemma	PROPN
ejpam-6569	364	2	5	5	NUM
ejpam-6569	364	3	.	.	PUNCT
ejpam-6569	365	1	if	if	SCONJ
ejpam-6569	365	2	f	f	PROPN
ejpam-6569	365	3	:	:	PUNCT
ejpam-6569	365	4	(	(	PUNCT
ejpam-6569	365	5	x	x	X
ejpam-6569	365	6	,	,	PUNCT
ejpam-6569	365	7	τ	τ	PROPN
ejpam-6569	365	8	,	,	PUNCT
ejpam-6569	365	9	i	i	NOUN
ejpam-6569	365	10	)	)	PUNCT
ejpam-6569	365	11	→	→	PUNCT
ejpam-6569	365	12	(	(	PUNCT
ejpam-6569	365	13	y	y	PROPN
ejpam-6569	365	14	,	,	PUNCT
ejpam-6569	365	15	σ1	σ1	PROPN
ejpam-6569	365	16	,	,	PUNCT
ejpam-6569	365	17	σ2	σ2	PROPN
ejpam-6569	365	18	)	)	PUNCT
ejpam-6569	365	19	is	be	AUX
ejpam-6569	365	20	a	a	DET
ejpam-6569	365	21	multifunction	multifunction	NOUN
ejpam-6569	365	22	such	such	ADJ
ejpam-6569	365	23	that	that	SCONJ
ejpam-6569	365	24	f	f	PROPN
ejpam-6569	365	25	(	(	PUNCT
ejpam-6569	365	26	x	x	X
ejpam-6569	365	27	)	)	PUNCT
ejpam-6569	365	28	is	be	AUX
ejpam-6569	365	29	σ1σ2	σ1σ2	NOUN
ejpam-6569	365	30	-	-	ADJ
ejpam-6569	365	31	regular	regular	ADJ
ejpam-6569	365	32	and	and	CCONJ
ejpam-6569	365	33	σ1σ2	σ1σ2	NOUN
ejpam-6569	365	34	-	-	ADJ
ejpam-6569	365	35	paracompact	paracompact	NOUN
ejpam-6569	365	36	for	for	ADP
ejpam-6569	365	37	each	each	DET
ejpam-6569	365	38	x	x	SYM
ejpam-6569	365	39	∈	∈	PROPN
ejpam-6569	365	40	x	x	NOUN
ejpam-6569	365	41	,	,	PUNCT
ejpam-6569	365	42	then	then	ADV
ejpam-6569	365	43	sclf+	sclf+	ADP
ejpam-6569	365	44	⊛	⊛	NUM
ejpam-6569	365	45	(	(	PUNCT
ejpam-6569	365	46	v	v	NOUN
ejpam-6569	365	47	)	)	PUNCT
ejpam-6569	365	48	=	=	PUNCT
ejpam-6569	365	49	f+(v	f+(v	NOUN
ejpam-6569	365	50	)	)	PUNCT
ejpam-6569	365	51	for	for	ADP
ejpam-6569	365	52	each	each	DET
ejpam-6569	365	53	σ1σ2	σ1σ2	VERB
ejpam-6569	365	54	-	-	ADJ
ejpam-6569	365	55	open	open	ADJ
ejpam-6569	365	56	set	set	NOUN
ejpam-6569	365	57	v	v	NOUN
ejpam-6569	365	58	of	of	ADP
ejpam-6569	365	59	y	y	PROPN
ejpam-6569	365	60	.	.	PUNCT
ejpam-6569	366	1	proof	proof	NOUN
ejpam-6569	366	2	.	.	PUNCT
ejpam-6569	367	1	let	let	VERB
ejpam-6569	367	2	v	v	PART
ejpam-6569	367	3	be	be	AUX
ejpam-6569	367	4	any	any	DET
ejpam-6569	367	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	367	6	-	-	ADJ
ejpam-6569	367	7	open	open	ADJ
ejpam-6569	367	8	set	set	NOUN
ejpam-6569	367	9	of	of	ADP
ejpam-6569	367	10	y	y	PROPN
ejpam-6569	367	11	and	and	CCONJ
ejpam-6569	367	12	x	x	PROPN
ejpam-6569	367	13	∈	∈	PROPN
ejpam-6569	367	14	sclf+	sclf+	ADP
ejpam-6569	367	15	⊛	⊛	NUM
ejpam-6569	367	16	(	(	PUNCT
ejpam-6569	367	17	v	v	NOUN
ejpam-6569	367	18	)	)	PUNCT
ejpam-6569	367	19	.	.	PUNCT
ejpam-6569	368	1	then	then	ADV
ejpam-6569	368	2	,	,	PUNCT
ejpam-6569	368	3	sclf+	sclf+	ADP
ejpam-6569	368	4	⊛	⊛	NUM
ejpam-6569	368	5	(	(	PUNCT
ejpam-6569	368	6	x	x	X
ejpam-6569	368	7	)	)	PUNCT
ejpam-6569	368	8	⊆	⊆	NUM
ejpam-6569	368	9	v	v	NOUN
ejpam-6569	368	10	and	and	CCONJ
ejpam-6569	368	11	f	f	PROPN
ejpam-6569	368	12	(	(	PUNCT
ejpam-6569	368	13	x	x	X
ejpam-6569	368	14	)	)	PUNCT
ejpam-6569	368	15	⊆	⊆	NUM
ejpam-6569	368	16	(	(	PUNCT
ejpam-6569	368	17	σ1	σ1	PROPN
ejpam-6569	368	18	,	,	PUNCT
ejpam-6569	368	19	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-6569	368	20	(	(	PUNCT
ejpam-6569	368	21	x	x	NOUN
ejpam-6569	368	22	)	)	PUNCT
ejpam-6569	368	23	)	)	PUNCT
ejpam-6569	369	1	=	=	PUNCT
ejpam-6569	369	2	sclf+	sclf+	ADP
ejpam-6569	369	3	⊛	⊛	NUM
ejpam-6569	369	4	(	(	PUNCT
ejpam-6569	369	5	x	x	X
ejpam-6569	369	6	)	)	PUNCT
ejpam-6569	369	7	⊆	⊆	NUM
ejpam-6569	369	8	v	v	NOUN
ejpam-6569	369	9	.	.	PUNCT
ejpam-6569	370	1	thus	thus	ADV
ejpam-6569	370	2	,	,	PUNCT
ejpam-6569	370	3	x	x	SYM
ejpam-6569	370	4	∈	∈	PROPN
ejpam-6569	370	5	f+(v	f+(v	NOUN
ejpam-6569	370	6	)	)	PUNCT
ejpam-6569	371	1	and	and	CCONJ
ejpam-6569	371	2	so	so	ADV
ejpam-6569	371	3	sclf+	sclf+	ADP
ejpam-6569	371	4	⊛	⊛	NUM
ejpam-6569	371	5	(	(	PUNCT
ejpam-6569	371	6	v	v	NOUN
ejpam-6569	371	7	)	)	PUNCT
ejpam-6569	371	8	⊆	⊆	NUM
ejpam-6569	371	9	f+(v	f+(v	NOUN
ejpam-6569	371	10	)	)	PUNCT
ejpam-6569	371	11	.	.	PUNCT
ejpam-6569	372	1	on	on	ADP
ejpam-6569	372	2	the	the	DET
ejpam-6569	372	3	other	other	ADJ
ejpam-6569	372	4	hand	hand	NOUN
ejpam-6569	372	5	,	,	PUNCT
ejpam-6569	372	6	let	let	VERB
ejpam-6569	372	7	x	x	X
ejpam-6569	372	8	∈	∈	PROPN
ejpam-6569	372	9	f+(v	f+(v	NOUN
ejpam-6569	372	10	)	)	PUNCT
ejpam-6569	372	11	.	.	PUNCT
ejpam-6569	373	1	then	then	ADV
ejpam-6569	373	2	,	,	PUNCT
ejpam-6569	373	3	f	f	PROPN
ejpam-6569	373	4	(	(	PUNCT
ejpam-6569	373	5	x	x	X
ejpam-6569	373	6	)	)	PUNCT
ejpam-6569	373	7	⊆	⊆	NUM
ejpam-6569	373	8	v	v	NOUN
ejpam-6569	373	9	and	and	CCONJ
ejpam-6569	373	10	by	by	ADP
ejpam-6569	373	11	lemma	lemma	PROPN
ejpam-6569	373	12	4	4	NUM
ejpam-6569	373	13	,	,	PUNCT
ejpam-6569	373	14	there	there	PRON
ejpam-6569	373	15	exists	exist	VERB
ejpam-6569	373	16	a	a	DET
ejpam-6569	373	17	σ1σ2	σ1σ2	NUM
ejpam-6569	373	18	-	-	ADJ
ejpam-6569	373	19	open	open	ADJ
ejpam-6569	373	20	set	set	NOUN
ejpam-6569	373	21	w	w	PROPN
ejpam-6569	373	22	of	of	ADP
ejpam-6569	373	23	y	y	PRON
ejpam-6569	373	24	such	such	ADJ
ejpam-6569	373	25	that	that	SCONJ
ejpam-6569	373	26	f	f	PROPN
ejpam-6569	373	27	(	(	PUNCT
ejpam-6569	373	28	x	x	X
ejpam-6569	373	29	)	)	PUNCT
ejpam-6569	373	30	⊆	⊆	NUM
ejpam-6569	373	31	w	w	ADP
ejpam-6569	373	32	⊆	⊆	NUM
ejpam-6569	373	33	σ1σ2	σ1σ2	NOUN
ejpam-6569	373	34	-	-	PUNCT
ejpam-6569	373	35	cl(w	cl(w	NOUN
ejpam-6569	373	36	)	)	PUNCT
ejpam-6569	373	37	⊆	⊆	NUM
ejpam-6569	373	38	v	v	NOUN
ejpam-6569	373	39	;	;	PUNCT
ejpam-6569	373	40	hence	hence	ADV
ejpam-6569	373	41	sclf+	sclf+	ADP
ejpam-6569	373	42	⊛	⊛	PROPN
ejpam-6569	373	43	(	(	PUNCT
ejpam-6569	373	44	x	x	X
ejpam-6569	373	45	)	)	PUNCT
ejpam-6569	373	46	=	=	SYM
ejpam-6569	373	47	(	(	PUNCT
ejpam-6569	373	48	σ1	σ1	PROPN
ejpam-6569	373	49	,	,	PUNCT
ejpam-6569	373	50	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-6569	373	51	(	(	PUNCT
ejpam-6569	373	52	x	x	NOUN
ejpam-6569	373	53	)	)	PUNCT
ejpam-6569	373	54	)	)	PUNCT
ejpam-6569	373	55	⊆	⊆	X
ejpam-6569	373	56	σ1σ2	σ1σ2	NOUN
ejpam-6569	373	57	-	-	PUNCT
ejpam-6569	373	58	cl(w	cl(w	NOUN
ejpam-6569	373	59	)	)	PUNCT
ejpam-6569	373	60	⊆	⊆	PROPN
ejpam-6569	373	61	v.	v.	ADP
ejpam-6569	373	62	thus	thus	ADV
ejpam-6569	373	63	,	,	PUNCT
ejpam-6569	373	64	x	x	PROPN
ejpam-6569	373	65	∈	∈	PROPN
ejpam-6569	373	66	sclf+	sclf+	ADP
ejpam-6569	373	67	⊛	⊛	NUM
ejpam-6569	373	68	(	(	PUNCT
ejpam-6569	373	69	v	v	NOUN
ejpam-6569	373	70	)	)	PUNCT
ejpam-6569	373	71	and	and	CCONJ
ejpam-6569	373	72	hence	hence	ADV
ejpam-6569	373	73	f+(v	f+(v	PROPN
ejpam-6569	373	74	)	)	PUNCT
ejpam-6569	374	1	⊆	⊆	NUM
ejpam-6569	374	2	sclf+	sclf+	ADP
ejpam-6569	374	3	⊛	⊛	NUM
ejpam-6569	374	4	(	(	PUNCT
ejpam-6569	374	5	v	v	NOUN
ejpam-6569	374	6	)	)	PUNCT
ejpam-6569	374	7	.	.	PUNCT
ejpam-6569	375	1	therefore	therefore	ADV
ejpam-6569	375	2	,	,	PUNCT
ejpam-6569	375	3	f+(v	f+(v	PROPN
ejpam-6569	375	4	)	)	PUNCT
ejpam-6569	376	1	=	=	PUNCT
ejpam-6569	376	2	sclf+	sclf+	ADP
ejpam-6569	376	3	⊛	⊛	NUM
ejpam-6569	376	4	(	(	PUNCT
ejpam-6569	376	5	v	v	NOUN
ejpam-6569	376	6	)	)	PUNCT
ejpam-6569	376	7	.	.	PUNCT
ejpam-6569	377	1	theorem	theorem	NOUN
ejpam-6569	377	2	9	9	NUM
ejpam-6569	377	3	.	.	PUNCT
ejpam-6569	378	1	let	let	VERB
ejpam-6569	378	2	f	f	NOUN
ejpam-6569	378	3	:	:	PUNCT
ejpam-6569	378	4	(	(	PUNCT
ejpam-6569	378	5	x	x	X
ejpam-6569	378	6	,	,	PUNCT
ejpam-6569	378	7	τ	τ	PROPN
ejpam-6569	378	8	,	,	PUNCT
ejpam-6569	378	9	i	i	NOUN
ejpam-6569	378	10	)	)	PUNCT
ejpam-6569	378	11	→	→	PUNCT
ejpam-6569	378	12	(	(	PUNCT
ejpam-6569	378	13	y	y	PROPN
ejpam-6569	378	14	,	,	PUNCT
ejpam-6569	378	15	σ1	σ1	PROPN
ejpam-6569	378	16	,	,	PUNCT
ejpam-6569	378	17	σ2	σ2	PROPN
ejpam-6569	378	18	)	)	PUNCT
ejpam-6569	378	19	be	be	VERB
ejpam-6569	378	20	a	a	DET
ejpam-6569	378	21	multifunction	multifunction	NOUN
ejpam-6569	378	22	such	such	ADJ
ejpam-6569	378	23	that	that	SCONJ
ejpam-6569	378	24	f	f	PROPN
ejpam-6569	378	25	(	(	PUNCT
ejpam-6569	378	26	x	x	X
ejpam-6569	378	27	)	)	PUNCT
ejpam-6569	378	28	is	be	AUX
ejpam-6569	378	29	σ1σ2paracompact	σ1σ2paracompact	NUM
ejpam-6569	378	30	and	and	CCONJ
ejpam-6569	378	31	σ1σ2	σ1σ2	NOUN
ejpam-6569	378	32	-	-	ADJ
ejpam-6569	378	33	regular	regular	ADJ
ejpam-6569	378	34	for	for	ADP
ejpam-6569	378	35	each	each	DET
ejpam-6569	378	36	x	x	SYM
ejpam-6569	378	37	∈	∈	PROPN
ejpam-6569	378	38	x.	x.	NOUN
ejpam-6569	378	39	then	then	ADV
ejpam-6569	378	40	,	,	PUNCT
ejpam-6569	378	41	f	f	PROPN
ejpam-6569	378	42	is	be	AUX
ejpam-6569	378	43	upper	upper	ADJ
ejpam-6569	378	44	quasi	quasi	NOUN
ejpam-6569	378	45	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	378	46	,	,	PUNCT
ejpam-6569	378	47	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	378	48	if	if	SCONJ
ejpam-6569	378	49	and	and	CCONJ
ejpam-6569	378	50	only	only	ADV
ejpam-6569	378	51	if	if	SCONJ
ejpam-6569	378	52	sclf⊛	sclf⊛	PRON
ejpam-6569	378	53	:	:	PUNCT
ejpam-6569	378	54	(	(	PUNCT
ejpam-6569	378	55	x	x	X
ejpam-6569	378	56	,	,	PUNCT
ejpam-6569	378	57	τ	τ	PROPN
ejpam-6569	378	58	,	,	PUNCT
ejpam-6569	378	59	i	i	NOUN
ejpam-6569	378	60	)	)	PUNCT
ejpam-6569	379	1	→	→	PUNCT
ejpam-6569	379	2	(	(	PUNCT
ejpam-6569	379	3	y	y	PROPN
ejpam-6569	379	4	,	,	PUNCT
ejpam-6569	379	5	σ1	σ1	PROPN
ejpam-6569	379	6	,	,	PUNCT
ejpam-6569	379	7	σ2	σ2	PROPN
ejpam-6569	379	8	)	)	PUNCT
ejpam-6569	379	9	is	be	AUX
ejpam-6569	379	10	upper	upper	ADJ
ejpam-6569	379	11	quasi	quasi	NOUN
ejpam-6569	379	12	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	379	13	,	,	PUNCT
ejpam-6569	379	14	σ2)continuous	σ2)continuous	ADJ
ejpam-6569	379	15	.	.	PUNCT
ejpam-6569	380	1	proof	proof	NOUN
ejpam-6569	380	2	.	.	PUNCT
ejpam-6569	381	1	suppose	suppose	VERB
ejpam-6569	381	2	that	that	SCONJ
ejpam-6569	381	3	f	f	PROPN
ejpam-6569	381	4	is	be	AUX
ejpam-6569	381	5	upper	upper	ADJ
ejpam-6569	381	6	quasi	quasi	NOUN
ejpam-6569	381	7	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	381	8	,	,	PUNCT
ejpam-6569	381	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	381	10	.	.	PUNCT
ejpam-6569	382	1	it	it	PRON
ejpam-6569	382	2	follows	follow	VERB
ejpam-6569	382	3	from	from	ADP
ejpam-6569	382	4	theorem	theorem	ADJ
ejpam-6569	382	5	1	1	NUM
ejpam-6569	382	6	and	and	CCONJ
ejpam-6569	382	7	lemma	lemma	PROPN
ejpam-6569	382	8	5	5	NUM
ejpam-6569	382	9	that	that	PRON
ejpam-6569	382	10	for	for	ADP
ejpam-6569	382	11	every	every	DET
ejpam-6569	382	12	σ1σ2	σ1σ2	NUM
ejpam-6569	382	13	-	-	ADJ
ejpam-6569	382	14	open	open	ADJ
ejpam-6569	382	15	set	set	NOUN
ejpam-6569	382	16	v	v	NOUN
ejpam-6569	382	17	of	of	ADP
ejpam-6569	382	18	y	y	PROPN
ejpam-6569	382	19	,	,	PUNCT
ejpam-6569	382	20	sclf+	sclf+	ADP
ejpam-6569	382	21	⊛	⊛	NUM
ejpam-6569	382	22	(	(	PUNCT
ejpam-6569	382	23	v	v	NOUN
ejpam-6569	382	24	)	)	PUNCT
ejpam-6569	382	25	=	=	PUNCT
ejpam-6569	382	26	f+(v	f+(v	NOUN
ejpam-6569	382	27	)	)	PUNCT
ejpam-6569	383	1	⊆	⊆	NUM
ejpam-6569	383	2	⋆θsint(f	⋆θsint(f	X
ejpam-6569	383	3	+	+	ADJ
ejpam-6569	383	4	(	(	PUNCT
ejpam-6569	383	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	383	6	-	-	NUM
ejpam-6569	383	7	cl(v	cl(v	NOUN
ejpam-6569	383	8	)	)	PUNCT
ejpam-6569	383	9	)	)	PUNCT
ejpam-6569	383	10	)	)	PUNCT
ejpam-6569	384	1	=	=	PUNCT
ejpam-6569	385	1	⋆θsint(sclf	⋆θsint(sclf	X
ejpam-6569	385	2	+	+	NUM
ejpam-6569	385	3	⊛	⊛	NUM
ejpam-6569	385	4	(	(	PUNCT
ejpam-6569	385	5	σ1σ2	σ1σ2	NOUN
ejpam-6569	385	6	-	-	NUM
ejpam-6569	385	7	cl(v	cl(v	NOUN
ejpam-6569	385	8	)	)	PUNCT
ejpam-6569	385	9	)	)	PUNCT
ejpam-6569	385	10	)	)	PUNCT
ejpam-6569	385	11	.	.	PUNCT
ejpam-6569	386	1	thus	thus	ADV
ejpam-6569	386	2	by	by	ADP
ejpam-6569	386	3	theorem	theorem	NOUN
ejpam-6569	386	4	1	1	NUM
ejpam-6569	386	5	,	,	PUNCT
ejpam-6569	386	6	sclf⊛	sclf⊛	PROPN
ejpam-6569	386	7	is	be	AUX
ejpam-6569	386	8	upper	upper	ADJ
ejpam-6569	386	9	quasi	quasi	NOUN
ejpam-6569	386	10	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	386	11	,	,	PUNCT
ejpam-6569	386	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	386	13	.	.	PUNCT
ejpam-6569	386	14	conversely	conversely	ADV
ejpam-6569	386	15	,	,	PUNCT
ejpam-6569	386	16	suppose	suppose	VERB
ejpam-6569	386	17	that	that	SCONJ
ejpam-6569	386	18	sclf⊛	sclf⊛	PROPN
ejpam-6569	386	19	is	be	AUX
ejpam-6569	386	20	upper	upper	ADJ
ejpam-6569	386	21	quasi	quasi	ADJ
ejpam-6569	386	22	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	386	23	,	,	PUNCT
ejpam-6569	386	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	386	25	.	.	PUNCT
ejpam-6569	387	1	it	it	PRON
ejpam-6569	387	2	follows	follow	VERB
ejpam-6569	387	3	from	from	ADP
ejpam-6569	387	4	theorem	theorem	ADJ
ejpam-6569	387	5	1	1	NUM
ejpam-6569	387	6	and	and	CCONJ
ejpam-6569	387	7	lemma	lemma	PROPN
ejpam-6569	387	8	5	5	NUM
ejpam-6569	387	9	that	that	PRON
ejpam-6569	387	10	for	for	ADP
ejpam-6569	387	11	every	every	DET
ejpam-6569	387	12	σ1σ2	σ1σ2	NUM
ejpam-6569	387	13	-	-	ADJ
ejpam-6569	387	14	open	open	ADJ
ejpam-6569	387	15	set	set	NOUN
ejpam-6569	387	16	v	v	NOUN
ejpam-6569	387	17	of	of	ADP
ejpam-6569	387	18	y	y	PROPN
ejpam-6569	387	19	,	,	PUNCT
ejpam-6569	387	20	f+(v	f+(v	PROPN
ejpam-6569	387	21	)	)	PUNCT
ejpam-6569	388	1	=	=	PUNCT
ejpam-6569	389	1	sclf+	sclf+	ADP
ejpam-6569	389	2	⊛	⊛	NUM
ejpam-6569	389	3	(	(	PUNCT
ejpam-6569	389	4	v	v	NOUN
ejpam-6569	389	5	)	)	PUNCT
ejpam-6569	389	6	⊆	⊆	NUM
ejpam-6569	389	7	⋆θsint(sclf	⋆θsint(sclf	X
ejpam-6569	389	8	+	+	NUM
ejpam-6569	389	9	⊛	⊛	NUM
ejpam-6569	389	10	(	(	PUNCT
ejpam-6569	389	11	σ1σ2	σ1σ2	NOUN
ejpam-6569	389	12	-	-	NUM
ejpam-6569	389	13	cl(v	cl(v	NOUN
ejpam-6569	389	14	)	)	PUNCT
ejpam-6569	389	15	)	)	PUNCT
ejpam-6569	389	16	)	)	PUNCT
ejpam-6569	390	1	=	=	PUNCT
ejpam-6569	391	1	⋆θsint(f	⋆θsint(f	X
ejpam-6569	391	2	+	+	ADJ
ejpam-6569	391	3	(	(	PUNCT
ejpam-6569	391	4	σ1σ2	σ1σ2	NOUN
ejpam-6569	391	5	-	-	NUM
ejpam-6569	391	6	cl(v	cl(v	NOUN
ejpam-6569	391	7	)	)	PUNCT
ejpam-6569	391	8	)	)	PUNCT
ejpam-6569	391	9	)	)	PUNCT
ejpam-6569	391	10	.	.	PUNCT
ejpam-6569	392	1	by	by	ADP
ejpam-6569	392	2	theorem	theorem	NOUN
ejpam-6569	392	3	1	1	NUM
ejpam-6569	392	4	,	,	PUNCT
ejpam-6569	392	5	f	f	PROPN
ejpam-6569	392	6	is	be	AUX
ejpam-6569	392	7	upper	upper	ADJ
ejpam-6569	392	8	quasi	quasi	ADJ
ejpam-6569	392	9	θτ⋆(σ1	θτ⋆(σ1	NOUN
ejpam-6569	392	10	,	,	PUNCT
ejpam-6569	392	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6569	392	12	.	.	PUNCT
ejpam-6569	393	1	acknowledgements	acknowledgement	NOUN
ejpam-6569	393	2	this	this	DET
ejpam-6569	393	3	research	research	NOUN
ejpam-6569	393	4	project	project	NOUN
ejpam-6569	393	5	was	be	AUX
ejpam-6569	393	6	financially	financially	ADV
ejpam-6569	393	7	supported	support	VERB
ejpam-6569	393	8	by	by	ADP
ejpam-6569	393	9	mahasarakham	mahasarakham	PROPN
ejpam-6569	393	10	university	university	PROPN
ejpam-6569	393	11	.	.	PUNCT
ejpam-6569	394	1	n.	n.	PROPN
ejpam-6569	394	2	srisarakham	srisarakham	PROPN
ejpam-6569	394	3	,	,	PUNCT
ejpam-6569	394	4	a.	a.	PROPN
ejpam-6569	394	5	sama	sama	PROPN
ejpam-6569	394	6	-	-	PUNCT
ejpam-6569	394	7	ae	ae	PROPN
ejpam-6569	394	8	,	,	PUNCT
ejpam-6569	394	9	c.	c.	PROPN
ejpam-6569	394	10	boonpok	boonpok	PROPN
ejpam-6569	394	11	/	/	SYM
ejpam-6569	394	12	eur	eur	PROPN
ejpam-6569	394	13	.	.	PUNCT
ejpam-6569	395	1	j.	j.	PROPN
ejpam-6569	395	2	pure	pure	PROPN
ejpam-6569	395	3	appl	appl	PROPN
ejpam-6569	395	4	.	.	PROPN
ejpam-6569	395	5	math	math	PROPN
ejpam-6569	395	6	,	,	PUNCT
ejpam-6569	395	7	18	18	NUM
ejpam-6569	395	8	(	(	PUNCT
ejpam-6569	395	9	3	3	NUM
ejpam-6569	395	10	)	)	PUNCT
ejpam-6569	395	11	(	(	PUNCT
ejpam-6569	395	12	2025	2025	NUM
ejpam-6569	395	13	)	)	PUNCT
ejpam-6569	395	14	,	,	PUNCT
ejpam-6569	395	15	6569	6569	NUM
ejpam-6569	395	16	12	12	NUM
ejpam-6569	395	17	of	of	ADP
ejpam-6569	395	18	13	13	NUM
ejpam-6569	395	19	references	reference	NOUN
ejpam-6569	395	20	[	[	X
ejpam-6569	395	21	1	1	NUM
ejpam-6569	395	22	]	]	X
ejpam-6569	395	23	n.	n.	PROPN
ejpam-6569	395	24	levine	levine	PROPN
ejpam-6569	395	25	.	.	PUNCT
ejpam-6569	396	1	semi	semi	ADJ
ejpam-6569	396	2	-	-	ADJ
ejpam-6569	396	3	open	open	ADJ
ejpam-6569	396	4	sets	set	NOUN
ejpam-6569	396	5	and	and	CCONJ
ejpam-6569	396	6	semi	semi	ADJ
ejpam-6569	396	7	-	-	NOUN
ejpam-6569	396	8	continuity	continuity	NOUN
ejpam-6569	396	9	in	in	ADP
ejpam-6569	396	10	topological	topological	ADJ
ejpam-6569	396	11	spaces	space	NOUN
ejpam-6569	396	12	.	.	PUNCT
ejpam-6569	397	1	the	the	DET
ejpam-6569	397	2	american	american	PROPN
ejpam-6569	397	3	mathematical	mathematical	PROPN
ejpam-6569	397	4	monthly	monthly	ADV
ejpam-6569	397	5	,	,	PUNCT
ejpam-6569	397	6	70:36–41	70:36–41	NUM
ejpam-6569	397	7	,	,	PUNCT
ejpam-6569	397	8	1963	1963	NUM
ejpam-6569	397	9	.	.	PUNCT
ejpam-6569	398	1	[	[	X
ejpam-6569	398	2	2	2	X
ejpam-6569	398	3	]	]	PUNCT
ejpam-6569	398	4	s.	s.	PROPN
ejpam-6569	398	5	p.	p.	PROPN
ejpam-6569	398	6	arya	arya	PROPN
ejpam-6569	398	7	and	and	CCONJ
ejpam-6569	398	8	m.	m.	PROPN
ejpam-6569	398	9	p.	p.	PROPN
ejpam-6569	398	10	bhamini	bhamini	PROPN
ejpam-6569	398	11	.	.	PUNCT
ejpam-6569	399	1	some	some	DET
ejpam-6569	399	2	weaker	weak	ADJ
ejpam-6569	399	3	forms	form	NOUN
ejpam-6569	399	4	of	of	ADP
ejpam-6569	399	5	semi	semi	ADJ
ejpam-6569	399	6	-	-	ADJ
ejpam-6569	399	7	continuous	continuous	ADJ
ejpam-6569	399	8	functions	function	NOUN
ejpam-6569	399	9	.	.	PUNCT
ejpam-6569	400	1	ganita	ganita	NOUN
ejpam-6569	400	2	,	,	PUNCT
ejpam-6569	400	3	33:124–134	33:124–134	NUM
ejpam-6569	400	4	,	,	PUNCT
ejpam-6569	400	5	1982	1982	NUM
ejpam-6569	400	6	.	.	PUNCT
ejpam-6569	401	1	[	[	X
ejpam-6569	401	2	3	3	X
ejpam-6569	401	3	]	]	PUNCT
ejpam-6569	401	4	t.	t.	PROPN
ejpam-6569	401	5	noiri	noiri	PROPN
ejpam-6569	401	6	.	.	PUNCT
ejpam-6569	402	1	on	on	ADP
ejpam-6569	402	2	θ	θ	ADJ
ejpam-6569	402	3	-	-	ADJ
ejpam-6569	402	4	continuous	continuous	ADJ
ejpam-6569	402	5	functions	function	NOUN
ejpam-6569	402	6	.	.	PUNCT
ejpam-6569	403	1	indian	indian	ADJ
ejpam-6569	403	2	journal	journal	PROPN
ejpam-6569	403	3	of	of	ADP
ejpam-6569	403	4	pure	pure	ADJ
ejpam-6569	403	5	and	and	CCONJ
ejpam-6569	403	6	applied	applied	ADJ
ejpam-6569	403	7	mathematics	mathematic	NOUN
ejpam-6569	403	8	,	,	PUNCT
ejpam-6569	403	9	21:410–415	21:410–415	NUM
ejpam-6569	403	10	,	,	PUNCT
ejpam-6569	403	11	1990	1990	NUM
ejpam-6569	403	12	.	.	PUNCT
ejpam-6569	404	1	[	[	X
ejpam-6569	404	2	4	4	X
ejpam-6569	404	3	]	]	PUNCT
ejpam-6569	404	4	s.	s.	PROPN
ejpam-6569	404	5	jafari	jafari	PROPN
ejpam-6569	404	6	and	and	CCONJ
ejpam-6569	404	7	t.	t.	PROPN
ejpam-6569	404	8	noiri	noiri	PROPN
ejpam-6569	404	9	.	.	PUNCT
ejpam-6569	405	1	properties	property	NOUN
ejpam-6569	405	2	of	of	ADP
ejpam-6569	405	3	θ	θ	ADJ
ejpam-6569	405	4	-	-	ADJ
ejpam-6569	405	5	continuous	continuous	ADJ
ejpam-6569	405	6	functions	function	NOUN
ejpam-6569	405	7	.	.	PUNCT
ejpam-6569	406	1	journal	journal	PROPN
ejpam-6569	406	2	of	of	ADP
ejpam-6569	406	3	institute	institute	PROPN
ejpam-6569	406	4	of	of	ADP
ejpam-6569	406	5	mathematics	mathematics	PROPN
ejpam-6569	406	6	and	and	CCONJ
ejpam-6569	406	7	computer	computer	NOUN
ejpam-6569	406	8	sciences	science	NOUN
ejpam-6569	406	9	,	,	PUNCT
ejpam-6569	406	10	mathematics	mathematic	NOUN
ejpam-6569	406	11	series	series	NOUN
ejpam-6569	406	12	,	,	PUNCT
ejpam-6569	406	13	13:123–128	13:123–128	NUM
ejpam-6569	406	14	,	,	PUNCT
ejpam-6569	406	15	2000	2000	NUM
ejpam-6569	406	16	.	.	PUNCT
ejpam-6569	407	1	[	[	X
ejpam-6569	407	2	5	5	X
ejpam-6569	407	3	]	]	PUNCT
ejpam-6569	407	4	s.	s.	PROPN
ejpam-6569	407	5	marcus	marcus	PROPN
ejpam-6569	407	6	.	.	PUNCT
ejpam-6569	408	1	sur	sur	PROPN
ejpam-6569	408	2	les	les	PROPN
ejpam-6569	408	3	fonctions	fonctions	PROPN
ejpam-6569	408	4	quasicontinues	quasicontinue	NOUN
ejpam-6569	408	5	au	au	ADP
ejpam-6569	408	6	sense	sense	NOUN
ejpam-6569	408	7	de	de	X
ejpam-6569	408	8	s.	s.	PROPN
ejpam-6569	408	9	kempisty	kempisty	PROPN
ejpam-6569	408	10	.	.	PUNCT
ejpam-6569	409	1	colloquium	colloquium	NOUN
ejpam-6569	409	2	mathematicum	mathematicum	PROPN
ejpam-6569	409	3	,	,	PUNCT
ejpam-6569	409	4	8:47–53	8:47–53	NUM
ejpam-6569	409	5	,	,	PUNCT
ejpam-6569	409	6	1961	1961	NUM
ejpam-6569	409	7	.	.	PUNCT
ejpam-6569	410	1	[	[	X
ejpam-6569	410	2	6	6	NUM
ejpam-6569	410	3	]	]	PUNCT
ejpam-6569	410	4	v.	v.	CCONJ
ejpam-6569	410	5	popa	popa	NOUN
ejpam-6569	410	6	.	.	PUNCT
ejpam-6569	411	1	on	on	ADP
ejpam-6569	411	2	the	the	DET
ejpam-6569	411	3	decomposition	decomposition	NOUN
ejpam-6569	411	4	of	of	ADP
ejpam-6569	411	5	the	the	DET
ejpam-6569	411	6	quasi	quasi	NOUN
ejpam-6569	411	7	-	-	NOUN
ejpam-6569	411	8	continuity	continuity	NOUN
ejpam-6569	411	9	in	in	ADP
ejpam-6569	411	10	topological	topological	ADJ
ejpam-6569	411	11	spaces	space	NOUN
ejpam-6569	411	12	(	(	PUNCT
ejpam-6569	411	13	romanian	romanian	ADJ
ejpam-6569	411	14	)	)	PUNCT
ejpam-6569	411	15	.	.	PUNCT
ejpam-6569	412	1	studii	studii	PROPN
ejpam-6569	412	2	şi	şi	PROPN
ejpam-6569	412	3	cercetǎri	cercetǎri	NOUN
ejpam-6569	412	4	de	de	X
ejpam-6569	412	5	matematicǎ	matematicǎ	NOUN
ejpam-6569	412	6	,	,	PUNCT
ejpam-6569	412	7	30:31–35	30:31–35	NUM
ejpam-6569	412	8	,	,	PUNCT
ejpam-6569	412	9	1978	1978	NUM
ejpam-6569	412	10	.	.	PUNCT
ejpam-6569	413	1	[	[	X
ejpam-6569	413	2	7	7	NUM
ejpam-6569	413	3	]	]	PUNCT
ejpam-6569	413	4	a.	a.	NOUN
ejpam-6569	413	5	neubrunnová.	neubrunnová.	PROPN
ejpam-6569	413	6	on	on	ADP
ejpam-6569	413	7	certain	certain	ADJ
ejpam-6569	413	8	generalizations	generalization	NOUN
ejpam-6569	413	9	of	of	ADP
ejpam-6569	413	10	the	the	DET
ejpam-6569	413	11	notion	notion	NOUN
ejpam-6569	413	12	of	of	ADP
ejpam-6569	413	13	continuity	continuity	NOUN
ejpam-6569	413	14	.	.	PUNCT
ejpam-6569	414	1	matematický	matematický	ADJ
ejpam-6569	414	2	c̆asopis	c̆asopis	PROPN
ejpam-6569	414	3	,	,	PUNCT
ejpam-6569	414	4	23:374–380	23:374–380	NUM
ejpam-6569	414	5	,	,	PUNCT
ejpam-6569	414	6	1973	1973	NUM
ejpam-6569	414	7	.	.	PUNCT
ejpam-6569	415	1	[	[	X
ejpam-6569	415	2	8	8	NUM
ejpam-6569	415	3	]	]	PUNCT
ejpam-6569	415	4	v.	v.	CCONJ
ejpam-6569	415	5	popa	popa	NOUN
ejpam-6569	415	6	and	and	CCONJ
ejpam-6569	415	7	c.	c.	PROPN
ejpam-6569	415	8	stan	stan	PROPN
ejpam-6569	415	9	.	.	PUNCT
ejpam-6569	416	1	on	on	ADP
ejpam-6569	416	2	a	a	DET
ejpam-6569	416	3	decomposition	decomposition	NOUN
ejpam-6569	416	4	of	of	ADP
ejpam-6569	416	5	quasicontinuity	quasicontinuity	NOUN
ejpam-6569	416	6	in	in	ADP
ejpam-6569	416	7	topological	topological	ADJ
ejpam-6569	416	8	spaces	space	NOUN
ejpam-6569	416	9	.	.	PUNCT
ejpam-6569	417	1	studii	studii	PROPN
ejpam-6569	417	2	şi	şi	PROPN
ejpam-6569	417	3	cercetǎri	cercetǎri	PROPN
ejpam-6569	417	4	de	de	X
ejpam-6569	417	5	matematicǎ	matematicǎ	NOUN
ejpam-6569	417	6	,	,	PUNCT
ejpam-6569	417	7	25:41–43	25:41–43	NUM
ejpam-6569	417	8	,	,	PUNCT
ejpam-6569	417	9	1973	1973	NUM
ejpam-6569	417	10	.	.	PUNCT
ejpam-6569	418	1	[	[	X
ejpam-6569	418	2	9	9	NUM
ejpam-6569	418	3	]	]	X
ejpam-6569	418	4	n.	n.	PROPN
ejpam-6569	418	5	levine	levine	PROPN
ejpam-6569	418	6	.	.	PUNCT
ejpam-6569	419	1	a	a	DET
ejpam-6569	419	2	decomposition	decomposition	NOUN
ejpam-6569	419	3	of	of	ADP
ejpam-6569	419	4	continuity	continuity	NOUN
ejpam-6569	419	5	in	in	ADP
ejpam-6569	419	6	topological	topological	ADJ
ejpam-6569	419	7	spaces	space	NOUN
ejpam-6569	419	8	.	.	PUNCT
ejpam-6569	420	1	the	the	DET
ejpam-6569	420	2	american	american	PROPN
ejpam-6569	420	3	mathematical	mathematical	PROPN
ejpam-6569	420	4	monthly	monthly	ADV
ejpam-6569	420	5	,	,	PUNCT
ejpam-6569	420	6	68:44–46	68:44–46	NUM
ejpam-6569	420	7	,	,	PUNCT
ejpam-6569	420	8	1961	1961	NUM
ejpam-6569	420	9	.	.	PUNCT
ejpam-6569	421	1	[	[	X
ejpam-6569	421	2	10	10	NUM
ejpam-6569	421	3	]	]	X
ejpam-6569	421	4	v.	v.	CCONJ
ejpam-6569	421	5	popa	popa	NOUN
ejpam-6569	421	6	.	.	PUNCT
ejpam-6569	422	1	on	on	ADP
ejpam-6569	422	2	some	some	DET
ejpam-6569	422	3	decomposition	decomposition	NOUN
ejpam-6569	422	4	of	of	ADP
ejpam-6569	422	5	quasicontinuity	quasicontinuity	NOUN
ejpam-6569	422	6	of	of	ADP
ejpam-6569	422	7	multifunctions	multifunction	NOUN
ejpam-6569	422	8	.	.	PUNCT
ejpam-6569	423	1	studii	studii	PROPN
ejpam-6569	423	2	şi	şi	PROPN
ejpam-6569	423	3	cercetǎri	cercetǎri	PROPN
ejpam-6569	423	4	de	de	X
ejpam-6569	423	5	matematicǎ	matematicǎ	NOUN
ejpam-6569	423	6	,	,	PUNCT
ejpam-6569	423	7	27:322–328	27:322–328	PROPN
ejpam-6569	423	8	,	,	PUNCT
ejpam-6569	423	9	1975	1975	NUM
ejpam-6569	423	10	.	.	PUNCT
ejpam-6569	424	1	[	[	X
ejpam-6569	424	2	11	11	NUM
ejpam-6569	424	3	]	]	PUNCT
ejpam-6569	424	4	v.	v.	CCONJ
ejpam-6569	424	5	popa	popa	NOUN
ejpam-6569	424	6	and	and	CCONJ
ejpam-6569	424	7	t.	t.	NOUN
ejpam-6569	424	8	noiri	noiri	PROPN
ejpam-6569	424	9	.	.	PUNCT
ejpam-6569	425	1	almost	almost	ADV
ejpam-6569	425	2	quasi	quasi	VERB
ejpam-6569	425	3	continuous	continuous	ADJ
ejpam-6569	425	4	multifunctions	multifunction	NOUN
ejpam-6569	425	5	.	.	PUNCT
ejpam-6569	426	1	tatra	tatra	PROPN
ejpam-6569	426	2	mountains	mountains	PROPN
ejpam-6569	426	3	mathematical	mathematical	ADJ
ejpam-6569	426	4	publications	publication	NOUN
ejpam-6569	426	5	,	,	PUNCT
ejpam-6569	426	6	14:81–90	14:81–90	PROPN
ejpam-6569	426	7	,	,	PUNCT
ejpam-6569	426	8	1998	1998	NUM
ejpam-6569	426	9	.	.	PUNCT
ejpam-6569	427	1	[	[	X
ejpam-6569	427	2	12	12	NUM
ejpam-6569	427	3	]	]	PUNCT
ejpam-6569	427	4	t.	t.	PROPN
ejpam-6569	427	5	noiri	noiri	PROPN
ejpam-6569	427	6	and	and	CCONJ
ejpam-6569	427	7	v.	v.	ADP
ejpam-6569	427	8	popa	popa	NOUN
ejpam-6569	427	9	.	.	PUNCT
ejpam-6569	428	1	weakly	weakly	ADJ
ejpam-6569	428	2	quasi	quasi	ADJ
ejpam-6569	428	3	continuous	continuous	ADJ
ejpam-6569	428	4	multifunctions	multifunction	NOUN
ejpam-6569	428	5	.	.	PUNCT
ejpam-6569	429	1	analele	analele	AUX
ejpam-6569	429	2	universitǎţii	universitǎţii	PROPN
ejpam-6569	429	3	din	din	VERB
ejpam-6569	429	4	timişoara	timişoara	NOUN
ejpam-6569	429	5	,	,	PUNCT
ejpam-6569	429	6	seria	seria	PROPN
ejpam-6569	429	7	ştiinţe	ştiinţe	PROPN
ejpam-6569	429	8	matematice	matematice	NOUN
ejpam-6569	429	9	,	,	PUNCT
ejpam-6569	429	10	26:33–38	26:33–38	NUM
ejpam-6569	429	11	,	,	PUNCT
ejpam-6569	429	12	1988	1988	NUM
ejpam-6569	429	13	.	.	PUNCT
ejpam-6569	430	1	[	[	X
ejpam-6569	430	2	13	13	NUM
ejpam-6569	430	3	]	]	PUNCT
ejpam-6569	430	4	v.	v.	CCONJ
ejpam-6569	430	5	popa	popa	NOUN
ejpam-6569	430	6	and	and	CCONJ
ejpam-6569	430	7	t.	t.	NOUN
ejpam-6569	430	8	noiri	noiri	PROPN
ejpam-6569	430	9	.	.	PUNCT
ejpam-6569	431	1	θ	θ	X
ejpam-6569	431	2	-	-	ADJ
ejpam-6569	431	3	quasicontinuous	quasicontinuous	ADJ
ejpam-6569	431	4	multifunctions	multifunction	NOUN
ejpam-6569	431	5	.	.	PUNCT
ejpam-6569	432	1	demonstratio	demonstratio	PROPN
ejpam-6569	432	2	mathematica	mathematica	PROPN
ejpam-6569	432	3	,	,	PUNCT
ejpam-6569	432	4	28:111–122	28:111–122	PROPN
ejpam-6569	432	5	,	,	PUNCT
ejpam-6569	432	6	1995	1995	NUM
ejpam-6569	432	7	.	.	PUNCT
ejpam-6569	433	1	[	[	X
ejpam-6569	433	2	14	14	NUM
ejpam-6569	433	3	]	]	PUNCT
ejpam-6569	433	4	t.	t.	PROPN
ejpam-6569	433	5	noiri	noiri	PROPN
ejpam-6569	433	6	and	and	CCONJ
ejpam-6569	433	7	v.	v.	ADP
ejpam-6569	433	8	popa	popa	NOUN
ejpam-6569	433	9	.	.	PUNCT
ejpam-6569	434	1	some	some	DET
ejpam-6569	434	2	properties	property	NOUN
ejpam-6569	434	3	of	of	ADP
ejpam-6569	434	4	upper	upper	ADJ
ejpam-6569	434	5	and	and	CCONJ
ejpam-6569	434	6	lower	low	ADJ
ejpam-6569	434	7	θ	θ	ADJ
ejpam-6569	434	8	-	-	ADJ
ejpam-6569	434	9	quasicontinuous	quasicontinuous	ADJ
ejpam-6569	434	10	multifunctions	multifunction	NOUN
ejpam-6569	434	11	.	.	PUNCT
ejpam-6569	435	1	demonstratio	demonstratio	PROPN
ejpam-6569	435	2	mathematica	mathematica	PROPN
ejpam-6569	435	3	,	,	PUNCT
ejpam-6569	435	4	38(1):223–234	38(1):223–234	PROPN
ejpam-6569	435	5	,	,	PUNCT
ejpam-6569	435	6	2005	2005	NUM
ejpam-6569	435	7	.	.	PUNCT
ejpam-6569	436	1	[	[	X
ejpam-6569	436	2	15	15	NUM
ejpam-6569	436	3	]	]	X
ejpam-6569	436	4	e.	e.	PROPN
ejpam-6569	436	5	hatir	hatir	PROPN
ejpam-6569	436	6	and	and	CCONJ
ejpam-6569	436	7	t.	t.	PROPN
ejpam-6569	436	8	noiri	noiri	PROPN
ejpam-6569	436	9	.	.	PUNCT
ejpam-6569	437	1	weakly	weakly	ADJ
ejpam-6569	437	2	pre	pre	ADJ
ejpam-6569	437	3	-	-	ADJ
ejpam-6569	437	4	i	i	PRON
ejpam-6569	437	5	-	-	PUNCT
ejpam-6569	437	6	open	open	ADJ
ejpam-6569	437	7	sets	set	NOUN
ejpam-6569	437	8	and	and	CCONJ
ejpam-6569	437	9	decomposition	decomposition	NOUN
ejpam-6569	437	10	of	of	ADP
ejpam-6569	437	11	continuity	continuity	NOUN
ejpam-6569	437	12	.	.	PUNCT
ejpam-6569	438	1	acta	acta	PROPN
ejpam-6569	438	2	mathematica	mathematica	PROPN
ejpam-6569	438	3	hungarica	hungarica	PROPN
ejpam-6569	438	4	,	,	PUNCT
ejpam-6569	438	5	106(3):227–238	106(3):227–238	NUM
ejpam-6569	438	6	,	,	PUNCT
ejpam-6569	438	7	2005	2005	NUM
ejpam-6569	438	8	.	.	PUNCT
ejpam-6569	439	1	[	[	X
ejpam-6569	439	2	16	16	NUM
ejpam-6569	439	3	]	]	X
ejpam-6569	439	4	e.	e.	PROPN
ejpam-6569	439	5	hatir	hatir	PROPN
ejpam-6569	439	6	and	and	CCONJ
ejpam-6569	439	7	t.	t.	PROPN
ejpam-6569	439	8	noiri	noiri	PROPN
ejpam-6569	439	9	.	.	PUNCT
ejpam-6569	440	1	on	on	ADP
ejpam-6569	440	2	decompositions	decomposition	NOUN
ejpam-6569	440	3	of	of	ADP
ejpam-6569	440	4	continuity	continuity	NOUN
ejpam-6569	440	5	via	via	ADP
ejpam-6569	440	6	idealization	idealization	NOUN
ejpam-6569	440	7	.	.	PUNCT
ejpam-6569	441	1	acta	acta	PROPN
ejpam-6569	441	2	mathematica	mathematica	PROPN
ejpam-6569	441	3	hungarica	hungarica	PROPN
ejpam-6569	441	4	,	,	PUNCT
ejpam-6569	441	5	96:341–349	96:341–349	PROPN
ejpam-6569	441	6	,	,	PUNCT
ejpam-6569	441	7	2002	2002	NUM
ejpam-6569	441	8	.	.	PUNCT
ejpam-6569	442	1	[	[	X
ejpam-6569	442	2	17	17	NUM
ejpam-6569	442	3	]	]	PUNCT
ejpam-6569	442	4	c.	c.	PROPN
ejpam-6569	442	5	boonpok	boonpok	PROPN
ejpam-6569	442	6	.	.	PUNCT
ejpam-6569	443	1	on	on	ADP
ejpam-6569	443	2	continuous	continuous	ADJ
ejpam-6569	443	3	multifunctions	multifunction	NOUN
ejpam-6569	443	4	in	in	ADP
ejpam-6569	443	5	ideal	ideal	ADJ
ejpam-6569	443	6	topological	topological	ADJ
ejpam-6569	443	7	spaces	space	NOUN
ejpam-6569	443	8	.	.	PUNCT
ejpam-6569	444	1	lobachevskii	lobachevskii	PROPN
ejpam-6569	444	2	journal	journal	PROPN
ejpam-6569	444	3	of	of	ADP
ejpam-6569	444	4	mathematics	mathematic	NOUN
ejpam-6569	444	5	,	,	PUNCT
ejpam-6569	444	6	40(1):24–35	40(1):24–35	NUM
ejpam-6569	444	7	,	,	PUNCT
ejpam-6569	444	8	2019	2019	NUM
ejpam-6569	444	9	.	.	PUNCT
ejpam-6569	445	1	[	[	X
ejpam-6569	445	2	18	18	NUM
ejpam-6569	445	3	]	]	PUNCT
ejpam-6569	445	4	c.	c.	PROPN
ejpam-6569	445	5	boonpok	boonpok	PROPN
ejpam-6569	445	6	.	.	PUNCT
ejpam-6569	446	1	pı	pı	NOUN
ejpam-6569	446	2	-	-	NOUN
ejpam-6569	446	3	continuity	continuity	NOUN
ejpam-6569	446	4	and	and	CCONJ
ejpam-6569	446	5	weak	weak	ADJ
ejpam-6569	446	6	pı	pı	NOUN
ejpam-6569	446	7	-	-	NOUN
ejpam-6569	446	8	continuity	continuity	NOUN
ejpam-6569	446	9	.	.	PUNCT
ejpam-6569	447	1	carpathian	carpathian	ADJ
ejpam-6569	447	2	mathematical	mathematical	ADJ
ejpam-6569	447	3	publications	publication	NOUN
ejpam-6569	447	4	,	,	PUNCT
ejpam-6569	447	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6569	447	6	,	,	PUNCT
ejpam-6569	447	7	2025	2025	NUM
ejpam-6569	447	8	.	.	PUNCT
ejpam-6569	448	1	[	[	X
ejpam-6569	448	2	19	19	NUM
ejpam-6569	448	3	]	]	X
ejpam-6569	448	4	p.	p.	NOUN
ejpam-6569	448	5	pue	pue	NOUN
ejpam-6569	448	6	-	-	PUNCT
ejpam-6569	448	7	on	on	ADP
ejpam-6569	448	8	,	,	PUNCT
ejpam-6569	448	9	s.	s.	PROPN
ejpam-6569	448	10	sompong	sompong	PROPN
ejpam-6569	448	11	,	,	PUNCT
ejpam-6569	448	12	and	and	CCONJ
ejpam-6569	448	13	c.	c.	PROPN
ejpam-6569	448	14	boonpok	boonpok	PROPN
ejpam-6569	448	15	.	.	PUNCT
ejpam-6569	449	1	upper	upper	ADJ
ejpam-6569	449	2	and	and	CCONJ
ejpam-6569	449	3	lower	low	ADJ
ejpam-6569	449	4	(	(	PUNCT
ejpam-6569	449	5	τ1	τ1	NOUN
ejpam-6569	449	6	,	,	PUNCT
ejpam-6569	449	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6569	449	8	multifunctions	multifunction	NOUN
ejpam-6569	449	9	.	.	PUNCT
ejpam-6569	450	1	international	international	ADJ
ejpam-6569	450	2	journal	journal	PROPN
ejpam-6569	450	3	of	of	ADP
ejpam-6569	450	4	mathematics	mathematic	NOUN
ejpam-6569	450	5	and	and	CCONJ
ejpam-6569	450	6	computer	computer	NOUN
ejpam-6569	450	7	science	science	NOUN
ejpam-6569	450	8	,	,	PUNCT
ejpam-6569	450	9	19(4):1305	19(4):1305	NUM
ejpam-6569	450	10	–	–	PUNCT
ejpam-6569	450	11	1310	1310	NUM
ejpam-6569	450	12	,	,	PUNCT
ejpam-6569	450	13	2024	2024	NUM
ejpam-6569	450	14	.	.	PUNCT
ejpam-6569	451	1	[	[	X
ejpam-6569	451	2	20	20	NUM
ejpam-6569	451	3	]	]	X
ejpam-6569	451	4	c.	c.	PROPN
ejpam-6569	451	5	klanarong	klanarong	PROPN
ejpam-6569	451	6	,	,	PUNCT
ejpam-6569	451	7	s.	s.	PROPN
ejpam-6569	451	8	sompong	sompong	PROPN
ejpam-6569	451	9	,	,	PUNCT
ejpam-6569	451	10	and	and	CCONJ
ejpam-6569	451	11	c.	c.	PROPN
ejpam-6569	451	12	boonpok	boonpok	PROPN
ejpam-6569	451	13	.	.	PUNCT
ejpam-6569	452	1	upper	upper	ADJ
ejpam-6569	452	2	and	and	CCONJ
ejpam-6569	452	3	lower	low	ADJ
ejpam-6569	452	4	almost	almost	ADV
ejpam-6569	452	5	(	(	PUNCT
ejpam-6569	452	6	τ1	τ1	NOUN
ejpam-6569	452	7	,	,	PUNCT
ejpam-6569	452	8	τ2)continuous	τ2)continuous	ADJ
ejpam-6569	452	9	multifunctions	multifunction	NOUN
ejpam-6569	452	10	.	.	PUNCT
ejpam-6569	453	1	european	european	ADJ
ejpam-6569	453	2	journal	journal	PROPN
ejpam-6569	453	3	of	of	ADP
ejpam-6569	453	4	pure	pure	ADJ
ejpam-6569	453	5	and	and	CCONJ
ejpam-6569	453	6	applied	applied	ADJ
ejpam-6569	453	7	mathematics	mathematic	NOUN
ejpam-6569	453	8	,	,	PUNCT
ejpam-6569	453	9	n.	n.	PROPN
ejpam-6569	453	10	srisarakham	srisarakham	PROPN
ejpam-6569	453	11	,	,	PUNCT
ejpam-6569	453	12	a.	a.	PROPN
ejpam-6569	453	13	sama	sama	PROPN
ejpam-6569	453	14	-	-	PUNCT
ejpam-6569	453	15	ae	ae	PROPN
ejpam-6569	453	16	,	,	PUNCT
ejpam-6569	453	17	c.	c.	PROPN
ejpam-6569	453	18	boonpok	boonpok	PROPN
ejpam-6569	453	19	/	/	SYM
ejpam-6569	453	20	eur	eur	PROPN
ejpam-6569	453	21	.	.	PUNCT
ejpam-6569	454	1	j.	j.	PROPN
ejpam-6569	454	2	pure	pure	PROPN
ejpam-6569	454	3	appl	appl	PROPN
ejpam-6569	454	4	.	.	PROPN
ejpam-6569	454	5	math	math	PROPN
ejpam-6569	454	6	,	,	PUNCT
ejpam-6569	454	7	18	18	NUM
ejpam-6569	454	8	(	(	PUNCT
ejpam-6569	454	9	3	3	NUM
ejpam-6569	454	10	)	)	PUNCT
ejpam-6569	454	11	(	(	PUNCT
ejpam-6569	454	12	2025	2025	NUM
ejpam-6569	454	13	)	)	PUNCT
ejpam-6569	454	14	,	,	PUNCT
ejpam-6569	454	15	6569	6569	NUM
ejpam-6569	454	16	13	13	NUM
ejpam-6569	454	17	of	of	ADP
ejpam-6569	454	18	13	13	NUM
ejpam-6569	454	19	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-6569	454	20	,	,	PUNCT
ejpam-6569	454	21	2024	2024	NUM
ejpam-6569	454	22	.	.	PUNCT
ejpam-6569	455	1	[	[	X
ejpam-6569	455	2	21	21	NUM
ejpam-6569	455	3	]	]	PUNCT
ejpam-6569	455	4	m.	m.	NOUN
ejpam-6569	455	5	thongmoon	thongmoon	NOUN
ejpam-6569	455	6	,	,	PUNCT
ejpam-6569	455	7	s.	s.	PROPN
ejpam-6569	455	8	sompong	sompong	PROPN
ejpam-6569	455	9	,	,	PUNCT
ejpam-6569	455	10	and	and	CCONJ
ejpam-6569	455	11	c.	c.	PROPN
ejpam-6569	455	12	boonpok	boonpok	PROPN
ejpam-6569	455	13	.	.	PUNCT
ejpam-6569	456	1	upper	upper	ADJ
ejpam-6569	456	2	and	and	CCONJ
ejpam-6569	456	3	lower	low	ADJ
ejpam-6569	456	4	weak	weak	ADJ
ejpam-6569	456	5	(	(	PUNCT
ejpam-6569	456	6	τ1	τ1	NOUN
ejpam-6569	456	7	,	,	PUNCT
ejpam-6569	456	8	τ2)continuity	τ2)continuity	PROPN
ejpam-6569	456	9	.	.	PUNCT
ejpam-6569	457	1	european	european	PROPN
ejpam-6569	457	2	journal	journal	PROPN
ejpam-6569	457	3	of	of	ADP
ejpam-6569	457	4	pure	pure	ADJ
ejpam-6569	457	5	and	and	CCONJ
ejpam-6569	457	6	applied	applied	ADJ
ejpam-6569	457	7	mathematics	mathematic	NOUN
ejpam-6569	457	8	,	,	PUNCT
ejpam-6569	457	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-6569	457	10	,	,	PUNCT
ejpam-6569	457	11	2024	2024	NUM
ejpam-6569	457	12	.	.	PUNCT
ejpam-6569	458	1	[	[	X
ejpam-6569	458	2	22	22	NUM
ejpam-6569	458	3	]	]	PUNCT
ejpam-6569	458	4	c.	c.	PROPN
ejpam-6569	458	5	boonpok	boonpok	PROPN
ejpam-6569	458	6	,	,	PUNCT
ejpam-6569	458	7	c.	c.	PROPN
ejpam-6569	458	8	viriyapong	viriyapong	PROPN
ejpam-6569	458	9	,	,	PUNCT
ejpam-6569	458	10	and	and	CCONJ
ejpam-6569	458	11	m.	m.	NOUN
ejpam-6569	458	12	thongmoon	thongmoon	NOUN
ejpam-6569	458	13	.	.	PUNCT
ejpam-6569	459	1	on	on	ADP
ejpam-6569	459	2	upper	upper	ADJ
ejpam-6569	459	3	and	and	CCONJ
ejpam-6569	459	4	lower	low	ADJ
ejpam-6569	459	5	(	(	PUNCT
ejpam-6569	459	6	τ1	τ1	NOUN
ejpam-6569	459	7	,	,	PUNCT
ejpam-6569	459	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6569	459	9	multifunctions	multifunction	NOUN
ejpam-6569	459	10	.	.	PUNCT
ejpam-6569	460	1	journal	journal	PROPN
ejpam-6569	460	2	of	of	ADP
ejpam-6569	460	3	mathematics	mathematics	PROPN
ejpam-6569	460	4	and	and	CCONJ
ejpam-6569	460	5	computer	computer	NOUN
ejpam-6569	460	6	science	science	NOUN
ejpam-6569	460	7	,	,	PUNCT
ejpam-6569	460	8	18:282–293	18:282–293	NUM
ejpam-6569	460	9	,	,	PUNCT
ejpam-6569	460	10	2018	2018	NUM
ejpam-6569	460	11	.	.	PUNCT
ejpam-6569	461	1	[	[	X
ejpam-6569	461	2	23	23	NUM
ejpam-6569	461	3	]	]	X
ejpam-6569	461	4	c.	c.	PROPN
ejpam-6569	461	5	viriyapong	viriyapong	PROPN
ejpam-6569	461	6	and	and	CCONJ
ejpam-6569	461	7	c.	c.	PROPN
ejpam-6569	461	8	boonpok	boonpok	PROPN
ejpam-6569	461	9	.	.	PUNCT
ejpam-6569	462	1	(	(	PUNCT
ejpam-6569	462	2	τ1	τ1	NOUN
ejpam-6569	462	3	,	,	PUNCT
ejpam-6569	462	4	τ2)α	τ2)α	NOUN
ejpam-6569	462	5	-	-	PUNCT
ejpam-6569	462	6	continuity	continuity	NOUN
ejpam-6569	462	7	for	for	ADP
ejpam-6569	462	8	multifunctions	multifunction	NOUN
ejpam-6569	462	9	.	.	PUNCT
ejpam-6569	463	1	journal	journal	PROPN
ejpam-6569	463	2	of	of	ADP
ejpam-6569	463	3	mathematics	mathematic	NOUN
ejpam-6569	463	4	,	,	PUNCT
ejpam-6569	463	5	2020:6285763	2020:6285763	NUM
ejpam-6569	463	6	,	,	PUNCT
ejpam-6569	463	7	2020	2020	NUM
ejpam-6569	463	8	.	.	PUNCT
ejpam-6569	464	1	[	[	X
ejpam-6569	464	2	24	24	NUM
ejpam-6569	464	3	]	]	PUNCT
ejpam-6569	464	4	c.	c.	PROPN
ejpam-6569	464	5	boonpok	boonpok	PROPN
ejpam-6569	464	6	.	.	PUNCT
ejpam-6569	465	1	(	(	PUNCT
ejpam-6569	465	2	τ1	τ1	NOUN
ejpam-6569	465	3	,	,	PUNCT
ejpam-6569	465	4	τ2)δ	τ2)δ	ADJ
ejpam-6569	465	5	-	-	PUNCT
ejpam-6569	465	6	semicontinuous	semicontinuous	ADJ
ejpam-6569	465	7	multifunctions	multifunction	NOUN
ejpam-6569	465	8	.	.	PUNCT
ejpam-6569	466	1	heliyon	heliyon	NOUN
ejpam-6569	466	2	,	,	PUNCT
ejpam-6569	466	3	6	6	NUM
ejpam-6569	466	4	:	:	SYM
ejpam-6569	466	5	e05367	e05367	PROPN
ejpam-6569	466	6	,	,	PUNCT
ejpam-6569	466	7	2020	2020	NUM
ejpam-6569	466	8	.	.	PUNCT
ejpam-6569	467	1	[	[	X
ejpam-6569	467	2	25	25	NUM
ejpam-6569	467	3	]	]	X
ejpam-6569	467	4	n.	n.	PROPN
ejpam-6569	467	5	viriyapong	viriyapong	PROPN
ejpam-6569	467	6	,	,	PUNCT
ejpam-6569	467	7	s.	s.	PROPN
ejpam-6569	467	8	sompong	sompong	PROPN
ejpam-6569	467	9	,	,	PUNCT
ejpam-6569	467	10	and	and	CCONJ
ejpam-6569	467	11	c.	c.	PROPN
ejpam-6569	467	12	boonpok	boonpok	PROPN
ejpam-6569	467	13	.	.	PUNCT
ejpam-6569	468	1	(	(	PUNCT
ejpam-6569	468	2	τ1	τ1	NOUN
ejpam-6569	468	3	,	,	PUNCT
ejpam-6569	468	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6569	468	5	disconnectedness	disconnectedness	NOUN
ejpam-6569	468	6	in	in	ADP
ejpam-6569	468	7	bitopological	bitopological	ADJ
ejpam-6569	468	8	spaces	space	NOUN
ejpam-6569	468	9	.	.	PUNCT
ejpam-6569	469	1	international	international	ADJ
ejpam-6569	469	2	journal	journal	PROPN
ejpam-6569	469	3	of	of	ADP
ejpam-6569	469	4	mathematics	mathematic	NOUN
ejpam-6569	469	5	and	and	CCONJ
ejpam-6569	469	6	computer	computer	NOUN
ejpam-6569	469	7	science	science	NOUN
ejpam-6569	469	8	,	,	PUNCT
ejpam-6569	469	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6569	469	10	,	,	PUNCT
ejpam-6569	469	11	2024	2024	NUM
ejpam-6569	469	12	.	.	PUNCT
ejpam-6569	470	1	[	[	X
ejpam-6569	470	2	26	26	NUM
ejpam-6569	470	3	]	]	PUNCT
ejpam-6569	470	4	k.	k.	PROPN
ejpam-6569	470	5	kuratowski	kuratowski	PROPN
ejpam-6569	470	6	.	.	PUNCT
ejpam-6569	471	1	topology	topology	PROPN
ejpam-6569	471	2	,	,	PUNCT
ejpam-6569	471	3	vol	vol	NOUN
ejpam-6569	471	4	.	.	PUNCT
ejpam-6569	471	5	i.	i.	PROPN
ejpam-6569	471	6	academic	academic	PROPN
ejpam-6569	471	7	press	press	PROPN
ejpam-6569	471	8	,	,	PUNCT
ejpam-6569	471	9	new	new	PROPN
ejpam-6569	471	10	york	york	PROPN
ejpam-6569	471	11	,	,	PUNCT
ejpam-6569	471	12	1966	1966	NUM
ejpam-6569	471	13	.	.	PUNCT
ejpam-6569	472	1	[	[	X
ejpam-6569	472	2	27	27	NUM
ejpam-6569	472	3	]	]	X
ejpam-6569	472	4	d.	d.	PROPN
ejpam-6569	472	5	janković	janković	PROPN
ejpam-6569	472	6	and	and	CCONJ
ejpam-6569	472	7	t.	t.	PROPN
ejpam-6569	472	8	r.	r.	PROPN
ejpam-6569	472	9	hamlett	hamlett	PROPN
ejpam-6569	472	10	.	.	PUNCT
ejpam-6569	473	1	new	new	ADJ
ejpam-6569	473	2	topologies	topology	NOUN
ejpam-6569	473	3	from	from	ADP
ejpam-6569	473	4	old	old	ADJ
ejpam-6569	473	5	via	via	ADP
ejpam-6569	473	6	ideals	ideal	NOUN
ejpam-6569	473	7	.	.	PUNCT
ejpam-6569	474	1	the	the	DET
ejpam-6569	474	2	american	american	PROPN
ejpam-6569	474	3	mathematical	mathematical	PROPN
ejpam-6569	474	4	monthly	monthly	ADV
ejpam-6569	474	5	,	,	PUNCT
ejpam-6569	474	6	97:295–310	97:295–310	PROPN
ejpam-6569	474	7	,	,	PUNCT
ejpam-6569	474	8	1990	1990	NUM
ejpam-6569	474	9	.	.	PUNCT
ejpam-6569	475	1	[	[	X
ejpam-6569	475	2	28	28	NUM
ejpam-6569	475	3	]	]	X
ejpam-6569	475	4	e.	e.	PROPN
ejpam-6569	475	5	ekici	ekici	PROPN
ejpam-6569	475	6	and	and	CCONJ
ejpam-6569	475	7	t.	t.	PROPN
ejpam-6569	475	8	noiri	noiri	PROPN
ejpam-6569	475	9	.	.	PUNCT
ejpam-6569	476	1	⋆-extremally	⋆-extremally	ADV
ejpam-6569	476	2	disconnected	disconnect	VERB
ejpam-6569	476	3	ideal	ideal	ADJ
ejpam-6569	476	4	topological	topological	ADJ
ejpam-6569	476	5	spaces	space	NOUN
ejpam-6569	476	6	.	.	PUNCT
ejpam-6569	477	1	acta	acta	PROPN
ejpam-6569	477	2	mathematica	mathematica	PROPN
ejpam-6569	477	3	hungarica	hungarica	PROPN
ejpam-6569	477	4	,	,	PUNCT
ejpam-6569	477	5	122:81–90	122:81–90	NUM
ejpam-6569	477	6	,	,	PUNCT
ejpam-6569	477	7	2009	2009	NUM
ejpam-6569	477	8	.	.	PUNCT
ejpam-6569	478	1	[	[	X
ejpam-6569	478	2	29	29	NUM
ejpam-6569	478	3	]	]	X
ejpam-6569	478	4	c.	c.	PROPN
ejpam-6569	478	5	boonpok	boonpok	PROPN
ejpam-6569	478	6	.	.	PUNCT
ejpam-6569	479	1	weak	weak	ADJ
ejpam-6569	479	2	quasi	quasi	ADJ
ejpam-6569	479	3	continuity	continuity	NOUN
ejpam-6569	479	4	for	for	ADP
ejpam-6569	479	5	multifunctions	multifunction	NOUN
ejpam-6569	479	6	in	in	ADP
ejpam-6569	479	7	ideal	ideal	ADJ
ejpam-6569	479	8	topological	topological	ADJ
ejpam-6569	479	9	spaces	space	NOUN
ejpam-6569	479	10	.	.	PUNCT
ejpam-6569	480	1	advances	advance	NOUN
ejpam-6569	480	2	in	in	ADP
ejpam-6569	480	3	mathematics	mathematic	NOUN
ejpam-6569	480	4	:	:	PUNCT
ejpam-6569	480	5	scientific	scientific	ADJ
ejpam-6569	480	6	journal	journal	NOUN
ejpam-6569	480	7	,	,	PUNCT
ejpam-6569	480	8	9(1):339–355	9(1):339–355	NUM
ejpam-6569	480	9	,	,	PUNCT
ejpam-6569	480	10	2020	2020	NUM
ejpam-6569	480	11	.	.	PUNCT
ejpam-6569	481	1	[	[	X
ejpam-6569	481	2	30	30	NUM
ejpam-6569	481	3	]	]	X
ejpam-6569	481	4	c.	c.	PROPN
ejpam-6569	481	5	boonpok	boonpok	PROPN
ejpam-6569	481	6	.	.	PUNCT
ejpam-6569	482	1	θ(⋆)-quasi	θ(⋆)-quasi	DET
ejpam-6569	482	2	continuity	continuity	NOUN
ejpam-6569	482	3	for	for	ADP
ejpam-6569	482	4	multifunctions	multifunction	NOUN
ejpam-6569	482	5	.	.	PUNCT
ejpam-6569	483	1	wseas	wseas	PROPN
ejpam-6569	483	2	transactions	transaction	NOUN
ejpam-6569	483	3	on	on	ADP
ejpam-6569	483	4	mathematics	mathematic	NOUN
ejpam-6569	483	5	,	,	PUNCT
ejpam-6569	483	6	21:245–251	21:245–251	NUM
ejpam-6569	483	7	,	,	PUNCT
ejpam-6569	483	8	2022	2022	NUM
ejpam-6569	483	9	.	.	PUNCT