id	sid	tid	token	lemma	pos
ejpam-5717	1	1	european	european	PROPN
ejpam-5717	1	2	journal	journal	PROPN
ejpam-5717	1	3	of	of	ADP
ejpam-5717	1	4	pure	pure	ADJ
ejpam-5717	1	5	and	and	CCONJ
ejpam-5717	1	6	applied	applied	ADJ
ejpam-5717	1	7	mathematics	mathematic	NOUN
ejpam-5717	1	8	2025	2025	NUM
ejpam-5717	1	9	,	,	PUNCT
ejpam-5717	1	10	vol	vol	NOUN
ejpam-5717	1	11	.	.	PROPN
ejpam-5717	1	12	18	18	NUM
ejpam-5717	1	13	,	,	PUNCT
ejpam-5717	1	14	issue	issue	NOUN
ejpam-5717	1	15	1	1	NUM
ejpam-5717	1	16	,	,	PUNCT
ejpam-5717	1	17	article	article	NOUN
ejpam-5717	1	18	number	number	NOUN
ejpam-5717	1	19	5717	5717	NUM
ejpam-5717	1	20	issn	issn	VERB
ejpam-5717	1	21	1307	1307	NUM
ejpam-5717	1	22	-	-	SYM
ejpam-5717	1	23	5543	5543	NUM
ejpam-5717	1	24	–	–	PUNCT
ejpam-5717	1	25	ejpam.com	ejpam.com	X
ejpam-5717	1	26	published	publish	VERB
ejpam-5717	1	27	by	by	ADP
ejpam-5717	1	28	new	new	PROPN
ejpam-5717	1	29	york	york	PROPN
ejpam-5717	1	30	business	business	PROPN
ejpam-5717	1	31	global	global	PROPN
ejpam-5717	1	32	quasi	quasi	PROPN
ejpam-5717	1	33	θ(τ1	θ(τ1	PROPN
ejpam-5717	1	34	,	,	PUNCT
ejpam-5717	1	35	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5717	1	36	for	for	ADP
ejpam-5717	1	37	multifunctions	multifunction	NOUN
ejpam-5717	1	38	prapart	prapart	VERB
ejpam-5717	1	39	pue	pue	PROPN
ejpam-5717	1	40	-	-	PUNCT
ejpam-5717	1	41	on1	on1	PROPN
ejpam-5717	1	42	,	,	PUNCT
ejpam-5717	1	43	areeyuth	areeyuth	NOUN
ejpam-5717	1	44	sama	sama	NOUN
ejpam-5717	1	45	-	-	PUNCT
ejpam-5717	1	46	ae2	ae2	PROPN
ejpam-5717	1	47	,	,	PUNCT
ejpam-5717	1	48	chawalit	chawalit	VERB
ejpam-5717	1	49	boonpok1,∗	boonpok1,∗	NOUN
ejpam-5717	1	50	1	1	NUM
ejpam-5717	1	51	mathematics	mathematic	NOUN
ejpam-5717	1	52	and	and	CCONJ
ejpam-5717	1	53	applied	apply	VERB
ejpam-5717	1	54	mathematics	mathematics	PROPN
ejpam-5717	1	55	research	research	NOUN
ejpam-5717	1	56	unit	unit	NOUN
ejpam-5717	1	57	,	,	PUNCT
ejpam-5717	1	58	department	department	NOUN
ejpam-5717	1	59	of	of	ADP
ejpam-5717	1	60	mathematics	mathematic	NOUN
ejpam-5717	1	61	,	,	PUNCT
ejpam-5717	1	62	faculty	faculty	NOUN
ejpam-5717	1	63	of	of	ADP
ejpam-5717	1	64	science	science	NOUN
ejpam-5717	1	65	,	,	PUNCT
ejpam-5717	1	66	mahasarakham	mahasarakham	PROPN
ejpam-5717	1	67	university	university	PROPN
ejpam-5717	1	68	,	,	PUNCT
ejpam-5717	1	69	maha	maha	PROPN
ejpam-5717	1	70	sarakham	sarakham	PROPN
ejpam-5717	1	71	,	,	PUNCT
ejpam-5717	1	72	44150	44150	NUM
ejpam-5717	1	73	,	,	PUNCT
ejpam-5717	1	74	thailand	thailand	PROPN
ejpam-5717	1	75	2	2	NUM
ejpam-5717	1	76	department	department	NOUN
ejpam-5717	1	77	of	of	ADP
ejpam-5717	1	78	mathematics	mathematic	NOUN
ejpam-5717	1	79	and	and	CCONJ
ejpam-5717	1	80	computer	computer	NOUN
ejpam-5717	1	81	science	science	NOUN
ejpam-5717	1	82	,	,	PUNCT
ejpam-5717	1	83	faculty	faculty	NOUN
ejpam-5717	1	84	of	of	ADP
ejpam-5717	1	85	science	science	NOUN
ejpam-5717	1	86	and	and	CCONJ
ejpam-5717	1	87	technology	technology	NOUN
ejpam-5717	1	88	,	,	PUNCT
ejpam-5717	1	89	prince	prince	NOUN
ejpam-5717	1	90	of	of	ADP
ejpam-5717	1	91	songkla	songkla	PROPN
ejpam-5717	1	92	university	university	PROPN
ejpam-5717	1	93	,	,	PUNCT
ejpam-5717	1	94	pattani	pattani	NOUN
ejpam-5717	1	95	campus	campus	NOUN
ejpam-5717	1	96	,	,	PUNCT
ejpam-5717	1	97	pattani	pattani	NOUN
ejpam-5717	1	98	,	,	PUNCT
ejpam-5717	1	99	94000	94000	NUM
ejpam-5717	1	100	,	,	PUNCT
ejpam-5717	1	101	thailand	thailand	PROPN
ejpam-5717	1	102	abstract	abstract	PROPN
ejpam-5717	1	103	.	.	PUNCT
ejpam-5717	2	1	this	this	DET
ejpam-5717	2	2	paper	paper	NOUN
ejpam-5717	2	3	presents	present	VERB
ejpam-5717	2	4	new	new	ADJ
ejpam-5717	2	5	classes	class	NOUN
ejpam-5717	2	6	of	of	ADP
ejpam-5717	2	7	multifunctions	multifunction	NOUN
ejpam-5717	2	8	called	call	VERB
ejpam-5717	2	9	upper	upper	ADJ
ejpam-5717	2	10	quasi	quasi	NOUN
ejpam-5717	2	11	θ(τ1	θ(τ1	NOUN
ejpam-5717	2	12	,	,	PUNCT
ejpam-5717	2	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	2	14	multifunctions	multifunction	NOUN
ejpam-5717	2	15	and	and	CCONJ
ejpam-5717	2	16	lower	low	ADJ
ejpam-5717	2	17	quasi	quasi	NOUN
ejpam-5717	2	18	θ(τ1	θ(τ1	NOUN
ejpam-5717	2	19	,	,	PUNCT
ejpam-5717	2	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	2	21	multifunctions	multifunction	NOUN
ejpam-5717	2	22	.	.	PUNCT
ejpam-5717	3	1	furthermore	furthermore	ADV
ejpam-5717	3	2	,	,	PUNCT
ejpam-5717	3	3	several	several	ADJ
ejpam-5717	3	4	characterizations	characterization	NOUN
ejpam-5717	3	5	and	and	CCONJ
ejpam-5717	3	6	some	some	DET
ejpam-5717	3	7	properties	property	NOUN
ejpam-5717	3	8	concerning	concern	VERB
ejpam-5717	3	9	upper	upper	ADJ
ejpam-5717	3	10	quasi	quasi	NOUN
ejpam-5717	3	11	θ(τ1	θ(τ1	NOUN
ejpam-5717	3	12	,	,	PUNCT
ejpam-5717	3	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	3	14	multifunctions	multifunction	NOUN
ejpam-5717	3	15	and	and	CCONJ
ejpam-5717	3	16	lower	low	ADJ
ejpam-5717	3	17	quasi	quasi	NOUN
ejpam-5717	3	18	θ(τ1	θ(τ1	NOUN
ejpam-5717	3	19	,	,	PUNCT
ejpam-5717	3	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	3	21	multifunctions	multifunction	NOUN
ejpam-5717	3	22	are	be	AUX
ejpam-5717	3	23	established	establish	VERB
ejpam-5717	3	24	.	.	PUNCT
ejpam-5717	4	1	2020	2020	NUM
ejpam-5717	4	2	mathematics	mathematics	PROPN
ejpam-5717	4	3	subject	subject	NOUN
ejpam-5717	4	4	classifications	classification	NOUN
ejpam-5717	4	5	:	:	PUNCT
ejpam-5717	4	6	54c08	54c08	NUM
ejpam-5717	4	7	,	,	PUNCT
ejpam-5717	4	8	54c60	54c60	NUM
ejpam-5717	4	9	key	key	ADJ
ejpam-5717	4	10	words	word	NOUN
ejpam-5717	4	11	and	and	CCONJ
ejpam-5717	4	12	phrases	phrase	NOUN
ejpam-5717	4	13	:	:	PUNCT
ejpam-5717	4	14	upper	upper	ADJ
ejpam-5717	4	15	quasi	quasi	NOUN
ejpam-5717	4	16	θ(τ1	θ(τ1	NOUN
ejpam-5717	4	17	,	,	PUNCT
ejpam-5717	4	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	4	19	multifunction	multifunction	NOUN
ejpam-5717	4	20	,	,	PUNCT
ejpam-5717	4	21	lower	low	ADJ
ejpam-5717	4	22	quasi	quasi	NOUN
ejpam-5717	4	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	4	24	,	,	PUNCT
ejpam-5717	4	25	τ2)continuous	τ2)continuous	ADJ
ejpam-5717	4	26	multifunction	multifunction	NOUN
ejpam-5717	4	27	1	1	NUM
ejpam-5717	4	28	.	.	PUNCT
ejpam-5717	5	1	introduction	introduction	NOUN
ejpam-5717	5	2	stronger	strong	ADJ
ejpam-5717	5	3	and	and	CCONJ
ejpam-5717	5	4	weaker	weak	ADJ
ejpam-5717	5	5	forms	form	NOUN
ejpam-5717	5	6	of	of	ADP
ejpam-5717	5	7	open	open	ADJ
ejpam-5717	5	8	sets	set	NOUN
ejpam-5717	5	9	in	in	ADP
ejpam-5717	5	10	topological	topological	ADJ
ejpam-5717	5	11	spaces	space	NOUN
ejpam-5717	5	12	such	such	ADJ
ejpam-5717	5	13	as	as	ADP
ejpam-5717	5	14	semi	semi	ADJ
ejpam-5717	5	15	-	-	ADJ
ejpam-5717	5	16	open	open	ADJ
ejpam-5717	5	17	sets	set	NOUN
ejpam-5717	5	18	[	[	X
ejpam-5717	5	19	42	42	NUM
ejpam-5717	5	20	]	]	PUNCT
ejpam-5717	5	21	,	,	PUNCT
ejpam-5717	5	22	preopen	preopen	ADJ
ejpam-5717	5	23	sets	set	NOUN
ejpam-5717	5	24	[	[	X
ejpam-5717	5	25	44	44	NUM
ejpam-5717	5	26	]	]	PUNCT
ejpam-5717	5	27	,	,	PUNCT
ejpam-5717	5	28	α	α	X
ejpam-5717	5	29	-	-	ADJ
ejpam-5717	5	30	open	open	ADJ
ejpam-5717	5	31	sets	set	NOUN
ejpam-5717	5	32	[	[	X
ejpam-5717	5	33	46	46	NUM
ejpam-5717	5	34	]	]	PUNCT
ejpam-5717	5	35	,	,	PUNCT
ejpam-5717	5	36	β	β	X
ejpam-5717	5	37	-	-	ADJ
ejpam-5717	5	38	open	open	ADJ
ejpam-5717	5	39	sets	set	NOUN
ejpam-5717	5	40	[	[	X
ejpam-5717	5	41	35	35	NUM
ejpam-5717	5	42	]	]	PUNCT
ejpam-5717	5	43	and	and	CCONJ
ejpam-5717	5	44	θ	θ	ADJ
ejpam-5717	5	45	-	-	ADJ
ejpam-5717	5	46	open	open	ADJ
ejpam-5717	5	47	sets	set	NOUN
ejpam-5717	5	48	[	[	X
ejpam-5717	5	49	65	65	NUM
ejpam-5717	5	50	]	]	PUNCT
ejpam-5717	5	51	play	play	VERB
ejpam-5717	5	52	an	an	DET
ejpam-5717	5	53	important	important	ADJ
ejpam-5717	5	54	role	role	NOUN
ejpam-5717	5	55	in	in	ADP
ejpam-5717	5	56	the	the	DET
ejpam-5717	5	57	research	research	NOUN
ejpam-5717	5	58	of	of	ADP
ejpam-5717	5	59	generalizations	generalization	NOUN
ejpam-5717	5	60	of	of	ADP
ejpam-5717	5	61	continuity	continuity	NOUN
ejpam-5717	5	62	.	.	PUNCT
ejpam-5717	6	1	using	use	VERB
ejpam-5717	6	2	these	these	DET
ejpam-5717	6	3	notions	notion	NOUN
ejpam-5717	6	4	many	many	ADJ
ejpam-5717	6	5	authors	author	NOUN
ejpam-5717	6	6	introduced	introduce	VERB
ejpam-5717	6	7	and	and	CCONJ
ejpam-5717	6	8	studied	study	VERB
ejpam-5717	6	9	various	various	ADJ
ejpam-5717	6	10	types	type	NOUN
ejpam-5717	6	11	of	of	ADP
ejpam-5717	6	12	generalizations	generalization	NOUN
ejpam-5717	6	13	of	of	ADP
ejpam-5717	6	14	continuity	continuity	NOUN
ejpam-5717	6	15	for	for	ADP
ejpam-5717	6	16	functions	function	NOUN
ejpam-5717	6	17	and	and	CCONJ
ejpam-5717	6	18	multifunctions	multifunction	NOUN
ejpam-5717	6	19	.	.	PUNCT
ejpam-5717	7	1	levine	levine	PROPN
ejpam-5717	7	2	[	[	X
ejpam-5717	7	3	42	42	NUM
ejpam-5717	7	4	]	]	PUNCT
ejpam-5717	7	5	introduced	introduce	VERB
ejpam-5717	7	6	and	and	CCONJ
ejpam-5717	7	7	studied	study	VERB
ejpam-5717	7	8	the	the	DET
ejpam-5717	7	9	notion	notion	NOUN
ejpam-5717	7	10	of	of	ADP
ejpam-5717	7	11	semi	semi	ADJ
ejpam-5717	7	12	-	-	ADJ
ejpam-5717	7	13	continuous	continuous	ADJ
ejpam-5717	7	14	functions	function	NOUN
ejpam-5717	7	15	.	.	PUNCT
ejpam-5717	8	1	arya	arya	PROPN
ejpam-5717	8	2	and	and	CCONJ
ejpam-5717	8	3	bhamini	bhamini	PROPN
ejpam-5717	9	1	[	[	X
ejpam-5717	9	2	1	1	NUM
ejpam-5717	9	3	]	]	PUNCT
ejpam-5717	9	4	introduced	introduce	VERB
ejpam-5717	9	5	the	the	DET
ejpam-5717	9	6	concept	concept	NOUN
ejpam-5717	9	7	of	of	ADP
ejpam-5717	9	8	θ	θ	NOUN
ejpam-5717	9	9	-	-	PUNCT
ejpam-5717	9	10	semi	semi	NOUN
ejpam-5717	9	11	-	-	NOUN
ejpam-5717	9	12	continuity	continuity	NOUN
ejpam-5717	9	13	as	as	ADP
ejpam-5717	9	14	a	a	DET
ejpam-5717	9	15	generalization	generalization	NOUN
ejpam-5717	9	16	of	of	ADP
ejpam-5717	9	17	semi	semi	NOUN
ejpam-5717	9	18	-	-	NOUN
ejpam-5717	9	19	continuity	continuity	NOUN
ejpam-5717	9	20	.	.	PUNCT
ejpam-5717	10	1	noiri	noiri	PROPN
ejpam-5717	11	1	[	[	X
ejpam-5717	11	2	47	47	NUM
ejpam-5717	11	3	]	]	PUNCT
ejpam-5717	11	4	and	and	CCONJ
ejpam-5717	11	5	jafari	jafari	ADJ
ejpam-5717	11	6	and	and	CCONJ
ejpam-5717	11	7	noiri	noiri	ADV
ejpam-5717	11	8	[	[	X
ejpam-5717	11	9	36	36	NUM
ejpam-5717	11	10	]	]	PUNCT
ejpam-5717	11	11	have	have	AUX
ejpam-5717	11	12	further	far	ADV
ejpam-5717	11	13	investigated	investigate	VERB
ejpam-5717	11	14	some	some	DET
ejpam-5717	11	15	characterizations	characterization	NOUN
ejpam-5717	11	16	of	of	ADP
ejpam-5717	11	17	θ	θ	NOUN
ejpam-5717	11	18	-	-	PUNCT
ejpam-5717	11	19	semi	semi	ADJ
ejpam-5717	11	20	-	-	ADJ
ejpam-5717	11	21	continuous	continuous	ADJ
ejpam-5717	11	22	functions	function	NOUN
ejpam-5717	11	23	.	.	PUNCT
ejpam-5717	12	1	marcus	marcus	PROPN
ejpam-5717	13	1	[	[	X
ejpam-5717	13	2	43	43	NUM
ejpam-5717	13	3	]	]	PUNCT
ejpam-5717	13	4	introduced	introduce	VERB
ejpam-5717	13	5	and	and	CCONJ
ejpam-5717	13	6	investigated	investigate	VERB
ejpam-5717	13	7	the	the	DET
ejpam-5717	13	8	notion	notion	NOUN
ejpam-5717	13	9	of	of	ADP
ejpam-5717	13	10	quasi	quasi	ADJ
ejpam-5717	13	11	continuous	continuous	ADJ
ejpam-5717	13	12	functions	function	NOUN
ejpam-5717	13	13	.	.	PUNCT
ejpam-5717	14	1	popa	popa	NOUN
ejpam-5717	15	1	[	[	X
ejpam-5717	15	2	51	51	NUM
ejpam-5717	15	3	]	]	PUNCT
ejpam-5717	15	4	introduced	introduce	VERB
ejpam-5717	15	5	and	and	CCONJ
ejpam-5717	15	6	studied	study	VERB
ejpam-5717	15	7	the	the	DET
ejpam-5717	15	8	notion	notion	NOUN
ejpam-5717	15	9	of	of	ADP
ejpam-5717	15	10	almost	almost	ADV
ejpam-5717	15	11	quasi	quasi	ADJ
ejpam-5717	15	12	continuous	continuous	ADJ
ejpam-5717	15	13	functions	function	NOUN
ejpam-5717	15	14	.	.	PUNCT
ejpam-5717	16	1	neubrunnovaá	neubrunnovaá	PUNCT
ejpam-5717	17	1	[	[	X
ejpam-5717	17	2	45	45	NUM
ejpam-5717	17	3	]	]	PUNCT
ejpam-5717	17	4	showed	show	VERB
ejpam-5717	17	5	that	that	SCONJ
ejpam-5717	17	6	quasi	quasi	NOUN
ejpam-5717	17	7	continuity	continuity	NOUN
ejpam-5717	17	8	is	be	AUX
ejpam-5717	17	9	equivalent	equivalent	ADJ
ejpam-5717	17	10	to	to	ADP
ejpam-5717	17	11	semi	semi	ADJ
ejpam-5717	17	12	-	-	NOUN
ejpam-5717	17	13	continuity	continuity	NOUN
ejpam-5717	17	14	due	due	ADP
ejpam-5717	17	15	to	to	ADP
ejpam-5717	17	16	levine	levine	PROPN
ejpam-5717	17	17	[	[	X
ejpam-5717	17	18	42	42	NUM
ejpam-5717	17	19	]	]	PUNCT
ejpam-5717	17	20	.	.	PUNCT
ejpam-5717	18	1	popa	popa	NOUN
ejpam-5717	18	2	and	and	CCONJ
ejpam-5717	18	3	stan	stan	PROPN
ejpam-5717	19	1	[	[	X
ejpam-5717	19	2	54	54	NUM
ejpam-5717	19	3	]	]	PUNCT
ejpam-5717	19	4	introduced	introduce	VERB
ejpam-5717	19	5	and	and	CCONJ
ejpam-5717	19	6	investigated	investigate	VERB
ejpam-5717	19	7	the	the	DET
ejpam-5717	19	8	notion	notion	NOUN
ejpam-5717	19	9	of	of	ADP
ejpam-5717	19	10	weakly	weakly	ADJ
ejpam-5717	19	11	quasi	quasi	ADJ
ejpam-5717	19	12	continuous	continuous	ADJ
ejpam-5717	19	13	functions	function	NOUN
ejpam-5717	19	14	.	.	PUNCT
ejpam-5717	20	1	weak	weak	ADJ
ejpam-5717	20	2	quasi	quasi	NOUN
ejpam-5717	20	3	continuity	continuity	NOUN
ejpam-5717	20	4	is	be	AUX
ejpam-5717	20	5	implied	imply	VERB
ejpam-5717	20	6	by	by	ADP
ejpam-5717	20	7	quasi	quasi	NOUN
ejpam-5717	20	8	continuity	continuity	NOUN
ejpam-5717	20	9	and	and	CCONJ
ejpam-5717	20	10	weak	weak	ADJ
ejpam-5717	20	11	continuity	continuity	NOUN
ejpam-5717	20	12	[	[	X
ejpam-5717	20	13	41	41	NUM
ejpam-5717	20	14	]	]	PUNCT
ejpam-5717	20	15	which	which	PRON
ejpam-5717	20	16	are	be	AUX
ejpam-5717	20	17	independent	independent	ADJ
ejpam-5717	20	18	of	of	ADP
ejpam-5717	20	19	each	each	DET
ejpam-5717	20	20	other	other	ADJ
ejpam-5717	20	21	.	.	PUNCT
ejpam-5717	21	1	viriyapong	viriyapong	PROPN
ejpam-5717	22	1	and	and	CCONJ
ejpam-5717	22	2	boonpok	boonpok	VERB
ejpam-5717	23	1	[	[	X
ejpam-5717	23	2	67	67	NUM
ejpam-5717	23	3	]	]	PUNCT
ejpam-5717	23	4	investigated	investigate	VERB
ejpam-5717	23	5	some	some	DET
ejpam-5717	23	6	characterizations	characterization	NOUN
ejpam-5717	23	7	of	of	ADP
ejpam-5717	23	8	(	(	PUNCT
ejpam-5717	23	9	λ	λ	PROPN
ejpam-5717	23	10	,	,	PUNCT
ejpam-5717	23	11	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	23	12	∗corresponding	∗corresponding	NOUN
ejpam-5717	23	13	author	author	NOUN
ejpam-5717	23	14	.	.	PUNCT
ejpam-5717	24	1	doi	doi	NOUN
ejpam-5717	24	2	:	:	PUNCT
ejpam-5717	24	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5717	https://doi.org/10.29020/nybg.ejpam.v18i1.5717	DET
ejpam-5717	24	4	email	email	NOUN
ejpam-5717	24	5	addresses	address	NOUN
ejpam-5717	24	6	:	:	PUNCT
ejpam-5717	24	7	prapart.p@msu.ac.th	prapart.p@msu.ac.th	PROPN
ejpam-5717	24	8	(	(	PUNCT
ejpam-5717	24	9	p.	p.	NOUN
ejpam-5717	24	10	pue	pue	NOUN
ejpam-5717	24	11	-	-	PUNCT
ejpam-5717	24	12	on	on	ADP
ejpam-5717	24	13	)	)	PUNCT
ejpam-5717	24	14	,	,	PUNCT
ejpam-5717	24	15	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-5717	24	16	(	(	PUNCT
ejpam-5717	24	17	a.	a.	PROPN
ejpam-5717	24	18	sama	sama	PROPN
ejpam-5717	24	19	-	-	PUNCT
ejpam-5717	24	20	ae	ae	PROPN
ejpam-5717	24	21	)	)	PUNCT
ejpam-5717	24	22	,	,	PUNCT
ejpam-5717	24	23	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5717	24	24	(	(	PUNCT
ejpam-5717	24	25	c.	c.	PROPN
ejpam-5717	24	26	boonpok	boonpok	PROPN
ejpam-5717	24	27	)	)	PUNCT
ejpam-5717	24	28	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5717	25	1	1	1	NUM
ejpam-5717	25	2	copyright	copyright	NOUN
ejpam-5717	25	3	:	:	PUNCT
ejpam-5717	25	4	©	©	PROPN
ejpam-5717	25	5	2025	2025	NUM
ejpam-5717	25	6	the	the	DET
ejpam-5717	25	7	author(s	author(s	NOUN
ejpam-5717	25	8	)	)	PUNCT
ejpam-5717	25	9	.	.	PUNCT
ejpam-5717	26	1	(	(	PUNCT
ejpam-5717	26	2	cc	cc	NOUN
ejpam-5717	26	3	by	by	ADP
ejpam-5717	26	4	-	-	PUNCT
ejpam-5717	26	5	nc	nc	PROPN
ejpam-5717	26	6	4.0	4.0	NUM
ejpam-5717	26	7	)	)	PUNCT
ejpam-5717	26	8	p.	p.	NOUN
ejpam-5717	26	9	pue	pue	NOUN
ejpam-5717	26	10	-	-	PUNCT
ejpam-5717	26	11	on	on	ADP
ejpam-5717	26	12	,	,	PUNCT
ejpam-5717	26	13	a.	a.	PROPN
ejpam-5717	26	14	sama	sama	PROPN
ejpam-5717	26	15	-	-	PUNCT
ejpam-5717	26	16	ae	ae	PROPN
ejpam-5717	26	17	,	,	PUNCT
ejpam-5717	26	18	c.	c.	PROPN
ejpam-5717	26	19	boonpok	boonpok	PROPN
ejpam-5717	26	20	/	/	SYM
ejpam-5717	26	21	eur	eur	PROPN
ejpam-5717	26	22	.	.	PUNCT
ejpam-5717	27	1	j.	j.	PROPN
ejpam-5717	27	2	pure	pure	PROPN
ejpam-5717	27	3	appl	appl	PROPN
ejpam-5717	27	4	.	.	PROPN
ejpam-5717	27	5	math	math	PROPN
ejpam-5717	27	6	,	,	PUNCT
ejpam-5717	27	7	18	18	NUM
ejpam-5717	27	8	(	(	PUNCT
ejpam-5717	27	9	1	1	NUM
ejpam-5717	27	10	)	)	PUNCT
ejpam-5717	27	11	(	(	PUNCT
ejpam-5717	27	12	2025	2025	NUM
ejpam-5717	27	13	)	)	PUNCT
ejpam-5717	27	14	,	,	PUNCT
ejpam-5717	27	15	5717	5717	NUM
ejpam-5717	27	16	2	2	NUM
ejpam-5717	27	17	of	of	ADP
ejpam-5717	27	18	16	16	NUM
ejpam-5717	27	19	functions	function	NOUN
ejpam-5717	27	20	by	by	ADP
ejpam-5717	27	21	utilizing	utilize	VERB
ejpam-5717	27	22	the	the	DET
ejpam-5717	27	23	notions	notion	NOUN
ejpam-5717	27	24	of	of	ADP
ejpam-5717	27	25	(	(	PUNCT
ejpam-5717	27	26	λ	λ	PROPN
ejpam-5717	27	27	,	,	PUNCT
ejpam-5717	27	28	sp)-open	sp)-open	ADJ
ejpam-5717	27	29	sets	set	NOUN
ejpam-5717	27	30	and	and	CCONJ
ejpam-5717	27	31	(	(	PUNCT
ejpam-5717	27	32	λ	λ	PROPN
ejpam-5717	27	33	,	,	PUNCT
ejpam-5717	27	34	sp)-closed	sp)-close	VERB
ejpam-5717	27	35	sets	set	NOUN
ejpam-5717	27	36	due	due	ADP
ejpam-5717	27	37	to	to	ADP
ejpam-5717	27	38	boonpok	boonpok	NOUN
ejpam-5717	27	39	and	and	CCONJ
ejpam-5717	27	40	khampakdee	khampakdee	NOUN
ejpam-5717	27	41	[	[	X
ejpam-5717	27	42	13	13	NUM
ejpam-5717	27	43	]	]	PUNCT
ejpam-5717	27	44	.	.	PUNCT
ejpam-5717	28	1	dungthaisong	dungthaisong	NOUN
ejpam-5717	28	2	et	et	PROPN
ejpam-5717	28	3	al	al	PROPN
ejpam-5717	28	4	.	.	PUNCT
ejpam-5717	29	1	[	[	X
ejpam-5717	29	2	34	34	NUM
ejpam-5717	29	3	]	]	PUNCT
ejpam-5717	29	4	introduced	introduce	VERB
ejpam-5717	29	5	and	and	CCONJ
ejpam-5717	29	6	studied	study	VERB
ejpam-5717	29	7	the	the	DET
ejpam-5717	29	8	concept	concept	NOUN
ejpam-5717	29	9	of	of	ADP
ejpam-5717	29	10	g(m	g(m	ADJ
ejpam-5717	29	11	,	,	PUNCT
ejpam-5717	29	12	n)-continuous	n)-continuous	ADJ
ejpam-5717	29	13	functions	function	NOUN
ejpam-5717	29	14	.	.	PUNCT
ejpam-5717	30	1	duangphui	duangphui	NOUN
ejpam-5717	30	2	et	et	PROPN
ejpam-5717	30	3	al	al	PROPN
ejpam-5717	30	4	.	.	PUNCT
ejpam-5717	31	1	[	[	X
ejpam-5717	31	2	33	33	NUM
ejpam-5717	31	3	]	]	PUNCT
ejpam-5717	31	4	introduced	introduce	VERB
ejpam-5717	31	5	and	and	CCONJ
ejpam-5717	31	6	investigated	investigate	VERB
ejpam-5717	31	7	the	the	DET
ejpam-5717	31	8	notion	notion	NOUN
ejpam-5717	31	9	of	of	ADP
ejpam-5717	31	10	(	(	PUNCT
ejpam-5717	31	11	µ	µ	NOUN
ejpam-5717	31	12	,	,	PUNCT
ejpam-5717	31	13	µ′)(m	µ′)(m	VERB
ejpam-5717	31	14	,	,	PUNCT
ejpam-5717	31	15	n)-continuous	n)-continuous	ADJ
ejpam-5717	31	16	functions	function	NOUN
ejpam-5717	31	17	.	.	PUNCT
ejpam-5717	32	1	moreover	moreover	ADV
ejpam-5717	32	2	,	,	PUNCT
ejpam-5717	32	3	several	several	ADJ
ejpam-5717	32	4	characterizations	characterization	NOUN
ejpam-5717	32	5	of	of	ADP
ejpam-5717	32	6	almost	almost	ADV
ejpam-5717	32	7	(	(	PUNCT
ejpam-5717	32	8	λ	λ	PROPN
ejpam-5717	32	9	,	,	PUNCT
ejpam-5717	32	10	p)-continuous	p)-continuous	ADJ
ejpam-5717	32	11	functions	function	NOUN
ejpam-5717	32	12	,	,	PUNCT
ejpam-5717	32	13	strongly	strongly	ADV
ejpam-5717	32	14	θ(λ	θ(λ	PROPN
ejpam-5717	32	15	,	,	PUNCT
ejpam-5717	32	16	p)-continuous	p)-continuous	ADJ
ejpam-5717	32	17	functions	function	NOUN
ejpam-5717	32	18	,	,	PUNCT
ejpam-5717	32	19	almost	almost	ADV
ejpam-5717	32	20	strongly	strongly	ADV
ejpam-5717	32	21	θ(λ	θ(λ	VERB
ejpam-5717	32	22	,	,	PUNCT
ejpam-5717	32	23	p)continuous	p)continuous	ADJ
ejpam-5717	32	24	functions	function	NOUN
ejpam-5717	32	25	,	,	PUNCT
ejpam-5717	32	26	θ(λ	θ(λ	PROPN
ejpam-5717	32	27	,	,	PUNCT
ejpam-5717	32	28	p)-continuous	p)-continuous	ADJ
ejpam-5717	32	29	functions	function	NOUN
ejpam-5717	32	30	,	,	PUNCT
ejpam-5717	32	31	weakly	weakly	ADJ
ejpam-5717	32	32	(	(	PUNCT
ejpam-5717	32	33	λ	λ	PROPN
ejpam-5717	32	34	,	,	PUNCT
ejpam-5717	32	35	b)-continuous	b)-continuous	ADJ
ejpam-5717	32	36	functions	function	NOUN
ejpam-5717	32	37	,	,	PUNCT
ejpam-5717	32	38	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5717	32	39	functions	function	NOUN
ejpam-5717	32	40	,	,	PUNCT
ejpam-5717	32	41	(	(	PUNCT
ejpam-5717	32	42	λ	λ	NOUN
ejpam-5717	32	43	,	,	PUNCT
ejpam-5717	32	44	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-5717	32	45	functions	function	NOUN
ejpam-5717	32	46	,	,	PUNCT
ejpam-5717	32	47	⋆-continuous	⋆-continuous	ADJ
ejpam-5717	32	48	functions	function	NOUN
ejpam-5717	32	49	,	,	PUNCT
ejpam-5717	32	50	θ	θ	PROPN
ejpam-5717	32	51	-	-	ADJ
ejpam-5717	32	52	i	i	VERB
ejpam-5717	32	53	continuous	continuous	ADJ
ejpam-5717	32	54	functions	function	NOUN
ejpam-5717	32	55	,	,	PUNCT
ejpam-5717	32	56	almost	almost	ADV
ejpam-5717	32	57	(	(	PUNCT
ejpam-5717	32	58	g	g	NOUN
ejpam-5717	32	59	,	,	PUNCT
ejpam-5717	32	60	m)-continuous	m)-continuous	ADJ
ejpam-5717	32	61	functions	function	NOUN
ejpam-5717	32	62	,	,	PUNCT
ejpam-5717	32	63	pairwise	pairwise	NOUN
ejpam-5717	32	64	almost	almost	ADV
ejpam-5717	32	65	m	m	VERB
ejpam-5717	32	66	-continuous	-continuous	ADJ
ejpam-5717	32	67	functions	function	NOUN
ejpam-5717	32	68	,	,	PUNCT
ejpam-5717	32	69	(	(	PUNCT
ejpam-5717	32	70	τ1	τ1	NOUN
ejpam-5717	32	71	,	,	PUNCT
ejpam-5717	32	72	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	32	73	functions	function	NOUN
ejpam-5717	32	74	,	,	PUNCT
ejpam-5717	32	75	almost	almost	ADV
ejpam-5717	32	76	(	(	PUNCT
ejpam-5717	32	77	τ1	τ1	NOUN
ejpam-5717	32	78	,	,	PUNCT
ejpam-5717	32	79	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	32	80	functions	function	NOUN
ejpam-5717	32	81	and	and	CCONJ
ejpam-5717	32	82	weakly	weakly	ADJ
ejpam-5717	32	83	(	(	PUNCT
ejpam-5717	32	84	τ1	τ1	NOUN
ejpam-5717	32	85	,	,	PUNCT
ejpam-5717	32	86	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	32	87	functions	function	NOUN
ejpam-5717	32	88	were	be	AUX
ejpam-5717	32	89	presented	present	VERB
ejpam-5717	32	90	in	in	ADP
ejpam-5717	32	91	[	[	X
ejpam-5717	32	92	60	60	NUM
ejpam-5717	32	93	]	]	PUNCT
ejpam-5717	32	94	,	,	PUNCT
ejpam-5717	32	95	[	[	X
ejpam-5717	32	96	63	63	NUM
ejpam-5717	32	97	]	]	PUNCT
ejpam-5717	32	98	,	,	PUNCT
ejpam-5717	32	99	[	[	X
ejpam-5717	32	100	17	17	NUM
ejpam-5717	32	101	]	]	PUNCT
ejpam-5717	32	102	,	,	PUNCT
ejpam-5717	32	103	[	[	X
ejpam-5717	32	104	55	55	NUM
ejpam-5717	32	105	]	]	PUNCT
ejpam-5717	32	106	,	,	PUNCT
ejpam-5717	32	107	[	[	X
ejpam-5717	32	108	26	26	NUM
ejpam-5717	32	109	]	]	PUNCT
ejpam-5717	32	110	,	,	PUNCT
ejpam-5717	33	1	[	[	X
ejpam-5717	33	2	12	12	NUM
ejpam-5717	33	3	]	]	PUNCT
ejpam-5717	33	4	,	,	PUNCT
ejpam-5717	34	1	[	[	X
ejpam-5717	34	2	9	9	NUM
ejpam-5717	34	3	]	]	PUNCT
ejpam-5717	34	4	,	,	PUNCT
ejpam-5717	34	5	[	[	X
ejpam-5717	34	6	11	11	NUM
ejpam-5717	34	7	]	]	PUNCT
ejpam-5717	34	8	,	,	PUNCT
ejpam-5717	34	9	[	[	X
ejpam-5717	34	10	5	5	NUM
ejpam-5717	34	11	]	]	PUNCT
ejpam-5717	34	12	,	,	PUNCT
ejpam-5717	34	13	[	[	X
ejpam-5717	34	14	2	2	NUM
ejpam-5717	34	15	]	]	PUNCT
ejpam-5717	34	16	,	,	PUNCT
ejpam-5717	34	17	[	[	X
ejpam-5717	34	18	3	3	NUM
ejpam-5717	34	19	]	]	PUNCT
ejpam-5717	34	20	,	,	PUNCT
ejpam-5717	34	21	[	[	X
ejpam-5717	34	22	27	27	NUM
ejpam-5717	34	23	]	]	PUNCT
ejpam-5717	34	24	,	,	PUNCT
ejpam-5717	34	25	[	[	X
ejpam-5717	34	26	24	24	NUM
ejpam-5717	34	27	]	]	PUNCT
ejpam-5717	34	28	and	and	CCONJ
ejpam-5717	34	29	[	[	X
ejpam-5717	34	30	19	19	NUM
ejpam-5717	34	31	]	]	PUNCT
ejpam-5717	34	32	,	,	PUNCT
ejpam-5717	34	33	respectively	respectively	ADV
ejpam-5717	34	34	.	.	PUNCT
ejpam-5717	35	1	srisarakham	srisarakham	PROPN
ejpam-5717	35	2	et	et	PROPN
ejpam-5717	35	3	al	al	PROPN
ejpam-5717	35	4	.	.	PUNCT
ejpam-5717	36	1	[	[	X
ejpam-5717	36	2	61	61	NUM
ejpam-5717	36	3	]	]	PUNCT
ejpam-5717	36	4	introduced	introduce	VERB
ejpam-5717	36	5	and	and	CCONJ
ejpam-5717	36	6	studied	study	VERB
ejpam-5717	36	7	the	the	DET
ejpam-5717	36	8	concept	concept	NOUN
ejpam-5717	36	9	of	of	ADP
ejpam-5717	36	10	faintly	faintly	ADV
ejpam-5717	36	11	(	(	PUNCT
ejpam-5717	36	12	τ1	τ1	PROPN
ejpam-5717	36	13	,	,	PUNCT
ejpam-5717	36	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	36	15	functions	function	NOUN
ejpam-5717	36	16	.	.	PUNCT
ejpam-5717	37	1	kong	kong	PROPN
ejpam-5717	37	2	-	-	PUNCT
ejpam-5717	37	3	ied	ied	PROPN
ejpam-5717	37	4	et	et	PROPN
ejpam-5717	37	5	al	al	PROPN
ejpam-5717	37	6	.	.	PUNCT
ejpam-5717	38	1	[	[	X
ejpam-5717	38	2	40	40	NUM
ejpam-5717	38	3	]	]	PUNCT
ejpam-5717	38	4	introduced	introduce	VERB
ejpam-5717	38	5	and	and	CCONJ
ejpam-5717	38	6	investigated	investigate	VERB
ejpam-5717	38	7	the	the	DET
ejpam-5717	38	8	notion	notion	NOUN
ejpam-5717	38	9	of	of	ADP
ejpam-5717	38	10	almost	almost	ADV
ejpam-5717	38	11	quasi	quasi	X
ejpam-5717	38	12	(	(	PUNCT
ejpam-5717	38	13	τ1	τ1	NOUN
ejpam-5717	38	14	,	,	PUNCT
ejpam-5717	38	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	38	16	functions	function	NOUN
ejpam-5717	38	17	.	.	PUNCT
ejpam-5717	39	1	chiangpradit	chiangpradit	NOUN
ejpam-5717	39	2	et	et	PROPN
ejpam-5717	39	3	al	al	PROPN
ejpam-5717	39	4	.	.	PUNCT
ejpam-5717	40	1	[	[	X
ejpam-5717	40	2	32	32	NUM
ejpam-5717	40	3	]	]	PUNCT
ejpam-5717	40	4	introduced	introduce	VERB
ejpam-5717	40	5	and	and	CCONJ
ejpam-5717	40	6	studied	study	VERB
ejpam-5717	40	7	the	the	DET
ejpam-5717	40	8	concept	concept	NOUN
ejpam-5717	40	9	of	of	ADP
ejpam-5717	40	10	weakly	weakly	ADJ
ejpam-5717	40	11	quasi	quasi	NOUN
ejpam-5717	40	12	(	(	PUNCT
ejpam-5717	40	13	τ1	τ1	PROPN
ejpam-5717	40	14	,	,	PUNCT
ejpam-5717	40	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	40	16	functions	function	NOUN
ejpam-5717	40	17	.	.	PUNCT
ejpam-5717	41	1	in	in	ADP
ejpam-5717	41	2	1975	1975	NUM
ejpam-5717	41	3	,	,	PUNCT
ejpam-5717	41	4	popa	popa	NOUN
ejpam-5717	41	5	[	[	X
ejpam-5717	41	6	50	50	NUM
ejpam-5717	41	7	]	]	PUNCT
ejpam-5717	41	8	extended	extend	VERB
ejpam-5717	41	9	the	the	DET
ejpam-5717	41	10	concept	concept	NOUN
ejpam-5717	41	11	of	of	ADP
ejpam-5717	41	12	quasicontinuous	quasicontinuous	ADJ
ejpam-5717	41	13	functions	function	NOUN
ejpam-5717	41	14	to	to	ADP
ejpam-5717	41	15	the	the	DET
ejpam-5717	41	16	setting	setting	NOUN
ejpam-5717	41	17	of	of	ADP
ejpam-5717	41	18	multifunctions	multifunction	NOUN
ejpam-5717	41	19	.	.	PUNCT
ejpam-5717	42	1	furthermore	furthermore	ADV
ejpam-5717	42	2	,	,	PUNCT
ejpam-5717	42	3	popa	popa	NOUN
ejpam-5717	42	4	and	and	CCONJ
ejpam-5717	42	5	noiri	noiri	ADV
ejpam-5717	43	1	[	[	X
ejpam-5717	43	2	53	53	NUM
ejpam-5717	43	3	]	]	PUNCT
ejpam-5717	43	4	introduced	introduce	VERB
ejpam-5717	43	5	the	the	DET
ejpam-5717	43	6	concept	concept	NOUN
ejpam-5717	43	7	of	of	ADP
ejpam-5717	43	8	almost	almost	ADV
ejpam-5717	43	9	quasi	quasi	ADJ
ejpam-5717	43	10	continuous	continuous	ADJ
ejpam-5717	43	11	multifunctions	multifunction	NOUN
ejpam-5717	43	12	and	and	CCONJ
ejpam-5717	43	13	investigated	investigate	VERB
ejpam-5717	43	14	some	some	DET
ejpam-5717	43	15	characterizations	characterization	NOUN
ejpam-5717	43	16	of	of	ADP
ejpam-5717	43	17	such	such	ADJ
ejpam-5717	43	18	multifunctions	multifunction	NOUN
ejpam-5717	43	19	.	.	PUNCT
ejpam-5717	44	1	noiri	noiri	PROPN
ejpam-5717	44	2	and	and	CCONJ
ejpam-5717	44	3	popa	popa	NOUN
ejpam-5717	45	1	[	[	X
ejpam-5717	45	2	48	48	NUM
ejpam-5717	45	3	]	]	PUNCT
ejpam-5717	45	4	introduced	introduce	VERB
ejpam-5717	45	5	and	and	CCONJ
ejpam-5717	45	6	studied	study	VERB
ejpam-5717	45	7	the	the	DET
ejpam-5717	45	8	notion	notion	NOUN
ejpam-5717	45	9	of	of	ADP
ejpam-5717	45	10	weakly	weakly	ADJ
ejpam-5717	45	11	quasi	quasi	ADJ
ejpam-5717	45	12	continuous	continuous	ADJ
ejpam-5717	45	13	multifunctions	multifunction	NOUN
ejpam-5717	45	14	.	.	PUNCT
ejpam-5717	46	1	popa	popa	NOUN
ejpam-5717	46	2	and	and	CCONJ
ejpam-5717	46	3	noiri	noiri	ADV
ejpam-5717	47	1	[	[	X
ejpam-5717	47	2	52	52	NUM
ejpam-5717	47	3	]	]	PUNCT
ejpam-5717	47	4	introduced	introduce	VERB
ejpam-5717	47	5	the	the	DET
ejpam-5717	47	6	notion	notion	NOUN
ejpam-5717	47	7	of	of	ADP
ejpam-5717	47	8	θ	θ	ADJ
ejpam-5717	47	9	-	-	ADJ
ejpam-5717	47	10	quasicontinuous	quasicontinuous	ADJ
ejpam-5717	47	11	multifunctions	multifunction	NOUN
ejpam-5717	47	12	and	and	CCONJ
ejpam-5717	47	13	investigated	investigate	VERB
ejpam-5717	47	14	several	several	ADJ
ejpam-5717	47	15	further	further	ADJ
ejpam-5717	47	16	properties	property	NOUN
ejpam-5717	47	17	of	of	ADP
ejpam-5717	47	18	such	such	ADJ
ejpam-5717	47	19	multifunctions	multifunction	NOUN
ejpam-5717	47	20	.	.	PUNCT
ejpam-5717	48	1	moreover	moreover	ADV
ejpam-5717	48	2	,	,	PUNCT
ejpam-5717	48	3	several	several	ADJ
ejpam-5717	48	4	characterizations	characterization	NOUN
ejpam-5717	48	5	and	and	CCONJ
ejpam-5717	48	6	some	some	DET
ejpam-5717	48	7	properties	property	NOUN
ejpam-5717	48	8	concerning	concern	VERB
ejpam-5717	48	9	(	(	PUNCT
ejpam-5717	48	10	τ1	τ1	NOUN
ejpam-5717	48	11	,	,	PUNCT
ejpam-5717	48	12	τ2)δ	τ2)δ	ADJ
ejpam-5717	48	13	-	-	PUNCT
ejpam-5717	48	14	semicontinuous	semicontinuous	ADJ
ejpam-5717	48	15	multifunctions	multifunction	NOUN
ejpam-5717	48	16	,	,	PUNCT
ejpam-5717	48	17	almost	almost	ADV
ejpam-5717	48	18	weakly	weakly	ADJ
ejpam-5717	48	19	(	(	PUNCT
ejpam-5717	48	20	τ1	τ1	NOUN
ejpam-5717	48	21	,	,	PUNCT
ejpam-5717	48	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	23	multifunctions	multifunction	NOUN
ejpam-5717	48	24	,	,	PUNCT
ejpam-5717	48	25	weakly	weakly	ADJ
ejpam-5717	48	26	quasi	quasi	NOUN
ejpam-5717	48	27	(	(	PUNCT
ejpam-5717	48	28	λ	λ	PROPN
ejpam-5717	48	29	,	,	PUNCT
ejpam-5717	48	30	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	31	multifunctions	multifunction	NOUN
ejpam-5717	48	32	,	,	PUNCT
ejpam-5717	48	33	⋆-continuous	⋆-continuous	ADJ
ejpam-5717	48	34	multifunctions	multifunction	NOUN
ejpam-5717	48	35	,	,	PUNCT
ejpam-5717	48	36	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-5717	48	37	multifunctions	multifunction	NOUN
ejpam-5717	48	38	,	,	PUNCT
ejpam-5717	48	39	α-⋆-continuous	α-⋆-continuous	PROPN
ejpam-5717	48	40	multifunctions	multifunction	NOUN
ejpam-5717	48	41	,	,	PUNCT
ejpam-5717	48	42	almost	almost	ADV
ejpam-5717	48	43	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5717	48	44	multifunctions	multifunction	NOUN
ejpam-5717	48	45	,	,	PUNCT
ejpam-5717	48	46	almost	almost	ADV
ejpam-5717	48	47	quasi	quasi	VERB
ejpam-5717	48	48	⋆-continuous	⋆-continuous	ADJ
ejpam-5717	48	49	multifunctions	multifunction	NOUN
ejpam-5717	48	50	,	,	PUNCT
ejpam-5717	48	51	weakly	weakly	ADJ
ejpam-5717	48	52	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5717	48	53	multifunctions	multifunction	NOUN
ejpam-5717	48	54	,	,	PUNCT
ejpam-5717	48	55	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-5717	48	56	multifunctions	multifunction	NOUN
ejpam-5717	48	57	,	,	PUNCT
ejpam-5717	48	58	weakly	weakly	ADJ
ejpam-5717	48	59	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-5717	48	60	multifunctions	multifunction	NOUN
ejpam-5717	48	61	,	,	PUNCT
ejpam-5717	48	62	θ(⋆)-quasi	θ(⋆)-quasi	NUM
ejpam-5717	48	63	continuous	continuous	ADJ
ejpam-5717	48	64	multifunctions	multifunction	NOUN
ejpam-5717	48	65	,	,	PUNCT
ejpam-5717	48	66	almost	almost	ADV
ejpam-5717	48	67	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-5717	48	68	multifunctions	multifunction	NOUN
ejpam-5717	48	69	,	,	PUNCT
ejpam-5717	48	70	weakly	weakly	ADJ
ejpam-5717	48	71	(	(	PUNCT
ejpam-5717	48	72	λ	λ	NOUN
ejpam-5717	48	73	,	,	PUNCT
ejpam-5717	48	74	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	75	multifunctions	multifunction	NOUN
ejpam-5717	48	76	,	,	PUNCT
ejpam-5717	48	77	α(λ	α(λ	PROPN
ejpam-5717	48	78	,	,	PUNCT
ejpam-5717	48	79	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	80	multifunctions	multifunction	NOUN
ejpam-5717	48	81	,	,	PUNCT
ejpam-5717	48	82	almost	almost	ADV
ejpam-5717	48	83	α(λ	α(λ	PROPN
ejpam-5717	48	84	,	,	PUNCT
ejpam-5717	48	85	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	86	multifunctions	multifunction	NOUN
ejpam-5717	48	87	,	,	PUNCT
ejpam-5717	48	88	weakly	weakly	ADJ
ejpam-5717	48	89	α(λ	α(λ	PROPN
ejpam-5717	48	90	,	,	PUNCT
ejpam-5717	48	91	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	92	multifunctions	multifunction	NOUN
ejpam-5717	48	93	,	,	PUNCT
ejpam-5717	48	94	almost	almost	ADV
ejpam-5717	48	95	β(λ	β(λ	NOUN
ejpam-5717	48	96	,	,	PUNCT
ejpam-5717	48	97	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	98	multifunctions	multifunction	NOUN
ejpam-5717	48	99	,	,	PUNCT
ejpam-5717	48	100	slightly	slightly	ADV
ejpam-5717	48	101	(	(	PUNCT
ejpam-5717	48	102	λ	λ	NOUN
ejpam-5717	48	103	,	,	PUNCT
ejpam-5717	48	104	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	48	105	multifunctions	multifunction	NOUN
ejpam-5717	48	106	,	,	PUNCT
ejpam-5717	48	107	(	(	PUNCT
ejpam-5717	48	108	τ1	τ1	NOUN
ejpam-5717	48	109	,	,	PUNCT
ejpam-5717	48	110	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	111	multifunctions	multifunction	NOUN
ejpam-5717	48	112	,	,	PUNCT
ejpam-5717	48	113	almost	almost	ADV
ejpam-5717	48	114	(	(	PUNCT
ejpam-5717	48	115	τ1	τ1	NOUN
ejpam-5717	48	116	,	,	PUNCT
ejpam-5717	48	117	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	118	multifunctions	multifunction	NOUN
ejpam-5717	48	119	,	,	PUNCT
ejpam-5717	48	120	weakly	weakly	ADJ
ejpam-5717	48	121	(	(	PUNCT
ejpam-5717	48	122	τ1	τ1	NOUN
ejpam-5717	48	123	,	,	PUNCT
ejpam-5717	48	124	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	125	multifunctions	multifunction	NOUN
ejpam-5717	48	126	,	,	PUNCT
ejpam-5717	48	127	weakly	weakly	ADJ
ejpam-5717	48	128	quasi	quasi	NOUN
ejpam-5717	48	129	(	(	PUNCT
ejpam-5717	48	130	τ1	τ1	PROPN
ejpam-5717	48	131	,	,	PUNCT
ejpam-5717	48	132	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	133	multifunctions	multifunction	NOUN
ejpam-5717	48	134	,	,	PUNCT
ejpam-5717	48	135	almost	almost	ADV
ejpam-5717	48	136	quasi	quasi	NOUN
ejpam-5717	48	137	(	(	PUNCT
ejpam-5717	48	138	τ1	τ1	NOUN
ejpam-5717	48	139	,	,	PUNCT
ejpam-5717	48	140	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	141	multifunctions	multifunction	NOUN
ejpam-5717	48	142	,	,	PUNCT
ejpam-5717	48	143	c-(τ1	c-(τ1	PROPN
ejpam-5717	48	144	,	,	PUNCT
ejpam-5717	48	145	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	48	146	multifunctions	multifunction	NOUN
ejpam-5717	48	147	and	and	CCONJ
ejpam-5717	48	148	slightly	slightly	ADV
ejpam-5717	48	149	(	(	PUNCT
ejpam-5717	48	150	τ1	τ1	NOUN
ejpam-5717	48	151	,	,	PUNCT
ejpam-5717	48	152	τ2)p	τ2)p	ADJ
ejpam-5717	48	153	-	-	PUNCT
ejpam-5717	48	154	continuous	continuous	ADJ
ejpam-5717	48	155	multifunctions	multifunction	NOUN
ejpam-5717	48	156	were	be	AUX
ejpam-5717	48	157	established	establish	VERB
ejpam-5717	48	158	in	in	ADP
ejpam-5717	48	159	[	[	X
ejpam-5717	48	160	6	6	NUM
ejpam-5717	48	161	]	]	PUNCT
ejpam-5717	48	162	,	,	PUNCT
ejpam-5717	49	1	[	[	X
ejpam-5717	49	2	29	29	NUM
ejpam-5717	49	3	]	]	PUNCT
ejpam-5717	49	4	,	,	PUNCT
ejpam-5717	50	1	[	[	X
ejpam-5717	50	2	68	68	NUM
ejpam-5717	50	3	]	]	PUNCT
ejpam-5717	50	4	,	,	PUNCT
ejpam-5717	50	5	[	[	X
ejpam-5717	50	6	4	4	NUM
ejpam-5717	50	7	]	]	PUNCT
ejpam-5717	50	8	,	,	PUNCT
ejpam-5717	50	9	[	[	X
ejpam-5717	50	10	8	8	NUM
ejpam-5717	50	11	]	]	PUNCT
ejpam-5717	50	12	,	,	PUNCT
ejpam-5717	51	1	[	[	X
ejpam-5717	51	2	18	18	NUM
ejpam-5717	51	3	]	]	PUNCT
ejpam-5717	51	4	,	,	PUNCT
ejpam-5717	51	5	[	[	X
ejpam-5717	51	6	25	25	NUM
ejpam-5717	51	7	]	]	PUNCT
ejpam-5717	51	8	,	,	PUNCT
ejpam-5717	52	1	[	[	X
ejpam-5717	52	2	7	7	NUM
ejpam-5717	52	3	]	]	PUNCT
ejpam-5717	52	4	,	,	PUNCT
ejpam-5717	52	5	[	[	X
ejpam-5717	52	6	22	22	NUM
ejpam-5717	52	7	]	]	PUNCT
ejpam-5717	52	8	,	,	PUNCT
ejpam-5717	53	1	[	[	X
ejpam-5717	53	2	21	21	NUM
ejpam-5717	53	3	]	]	PUNCT
ejpam-5717	53	4	,	,	PUNCT
ejpam-5717	53	5	[	[	X
ejpam-5717	53	6	16	16	NUM
ejpam-5717	53	7	]	]	PUNCT
ejpam-5717	53	8	,	,	PUNCT
ejpam-5717	53	9	[	[	X
ejpam-5717	53	10	10	10	NUM
ejpam-5717	53	11	]	]	PUNCT
ejpam-5717	53	12	,	,	PUNCT
ejpam-5717	53	13	[	[	X
ejpam-5717	53	14	20	20	NUM
ejpam-5717	53	15	]	]	PUNCT
ejpam-5717	53	16	,	,	PUNCT
ejpam-5717	53	17	[	[	X
ejpam-5717	53	18	23	23	NUM
ejpam-5717	53	19	]	]	PUNCT
ejpam-5717	53	20	,	,	PUNCT
ejpam-5717	53	21	[	[	X
ejpam-5717	53	22	37	37	NUM
ejpam-5717	53	23	]	]	PUNCT
ejpam-5717	53	24	,	,	PUNCT
ejpam-5717	54	1	[	[	X
ejpam-5717	54	2	14	14	NUM
ejpam-5717	54	3	]	]	PUNCT
ejpam-5717	54	4	,	,	PUNCT
ejpam-5717	54	5	[	[	X
ejpam-5717	54	6	28	28	NUM
ejpam-5717	54	7	]	]	PUNCT
ejpam-5717	54	8	,	,	PUNCT
ejpam-5717	54	9	[	[	X
ejpam-5717	54	10	62	62	NUM
ejpam-5717	54	11	]	]	PUNCT
ejpam-5717	54	12	,	,	PUNCT
ejpam-5717	54	13	[	[	X
ejpam-5717	54	14	15	15	NUM
ejpam-5717	54	15	]	]	PUNCT
ejpam-5717	54	16	,	,	PUNCT
ejpam-5717	54	17	[	[	X
ejpam-5717	54	18	58	58	NUM
ejpam-5717	54	19	]	]	PUNCT
ejpam-5717	54	20	,	,	PUNCT
ejpam-5717	54	21	[	[	X
ejpam-5717	54	22	39	39	NUM
ejpam-5717	54	23	]	]	PUNCT
ejpam-5717	54	24	,	,	PUNCT
ejpam-5717	54	25	[	[	X
ejpam-5717	54	26	64	64	NUM
ejpam-5717	54	27	]	]	PUNCT
ejpam-5717	54	28	,	,	PUNCT
ejpam-5717	54	29	[	[	X
ejpam-5717	54	30	59	59	NUM
ejpam-5717	54	31	]	]	PUNCT
ejpam-5717	54	32	,	,	PUNCT
ejpam-5717	54	33	[	[	X
ejpam-5717	54	34	57	57	NUM
ejpam-5717	54	35	]	]	PUNCT
ejpam-5717	54	36	,	,	PUNCT
ejpam-5717	55	1	[	[	X
ejpam-5717	55	2	38	38	NUM
ejpam-5717	55	3	]	]	PUNCT
ejpam-5717	55	4	and	and	CCONJ
ejpam-5717	55	5	[	[	X
ejpam-5717	56	1	70	70	NUM
ejpam-5717	56	2	]	]	PUNCT
ejpam-5717	56	3	,	,	PUNCT
ejpam-5717	56	4	respectively	respectively	ADV
ejpam-5717	56	5	.	.	PUNCT
ejpam-5717	57	1	noiri	noiri	PROPN
ejpam-5717	57	2	and	and	CCONJ
ejpam-5717	57	3	popa	popa	NOUN
ejpam-5717	58	1	[	[	X
ejpam-5717	58	2	49	49	NUM
ejpam-5717	58	3	]	]	PUNCT
ejpam-5717	58	4	investigated	investigate	VERB
ejpam-5717	58	5	some	some	DET
ejpam-5717	58	6	characterizations	characterization	NOUN
ejpam-5717	58	7	of	of	ADP
ejpam-5717	58	8	upper	upper	ADJ
ejpam-5717	58	9	and	and	CCONJ
ejpam-5717	58	10	lower	low	ADJ
ejpam-5717	58	11	θ	θ	ADJ
ejpam-5717	58	12	-	-	ADJ
ejpam-5717	58	13	quasicontinuous	quasicontinuous	ADJ
ejpam-5717	58	14	multifunctions	multifunction	NOUN
ejpam-5717	58	15	.	.	PUNCT
ejpam-5717	59	1	pue	pue	NOUN
ejpam-5717	59	2	-	-	PUNCT
ejpam-5717	59	3	on	on	NOUN
ejpam-5717	59	4	et	et	PROPN
ejpam-5717	59	5	al	al	PROPN
ejpam-5717	59	6	.	.	PUNCT
ejpam-5717	60	1	[	[	X
ejpam-5717	60	2	56	56	NUM
ejpam-5717	60	3	]	]	PUNCT
ejpam-5717	60	4	introduced	introduce	VERB
ejpam-5717	60	5	and	and	CCONJ
ejpam-5717	60	6	studied	study	VERB
ejpam-5717	60	7	the	the	DET
ejpam-5717	60	8	concept	concept	NOUN
ejpam-5717	60	9	of	of	ADP
ejpam-5717	60	10	c	c	NOUN
ejpam-5717	60	11	-	-	PUNCT
ejpam-5717	60	12	quasi	quasi	NOUN
ejpam-5717	60	13	(	(	PUNCT
ejpam-5717	60	14	τ1	τ1	PROPN
ejpam-5717	60	15	,	,	PUNCT
ejpam-5717	60	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	60	17	multifunctions	multifunction	NOUN
ejpam-5717	60	18	.	.	PUNCT
ejpam-5717	61	1	viriyapong	viriyapong	PROPN
ejpam-5717	61	2	et	et	PROPN
ejpam-5717	61	3	al	al	PROPN
ejpam-5717	61	4	.	.	PUNCT
ejpam-5717	62	1	[	[	X
ejpam-5717	62	2	72	72	NUM
ejpam-5717	62	3	]	]	PUNCT
ejpam-5717	62	4	introduced	introduce	VERB
ejpam-5717	62	5	and	and	CCONJ
ejpam-5717	62	6	investigated	investigate	VERB
ejpam-5717	62	7	the	the	DET
ejpam-5717	62	8	notion	notion	NOUN
ejpam-5717	62	9	of	of	ADP
ejpam-5717	62	10	s-(τ1	s-(τ1	PROPN
ejpam-5717	62	11	,	,	PUNCT
ejpam-5717	62	12	τ2)p	τ2)p	ADJ
ejpam-5717	62	13	-	-	PUNCT
ejpam-5717	62	14	continuous	continuous	ADJ
ejpam-5717	62	15	multifunctions	multifunction	NOUN
ejpam-5717	62	16	.	.	PUNCT
ejpam-5717	63	1	furthermore	furthermore	ADV
ejpam-5717	63	2	,	,	PUNCT
ejpam-5717	63	3	viriyapong	viriyapong	PROPN
ejpam-5717	63	4	et	et	PROPN
ejpam-5717	63	5	al	al	PROPN
ejpam-5717	63	6	.	.	PUNCT
ejpam-5717	64	1	[	[	X
ejpam-5717	64	2	69	69	NUM
ejpam-5717	64	3	]	]	PUNCT
ejpam-5717	64	4	introduced	introduce	VERB
ejpam-5717	64	5	and	and	CCONJ
ejpam-5717	64	6	studied	study	VERB
ejpam-5717	64	7	the	the	DET
ejpam-5717	64	8	concept	concept	NOUN
ejpam-5717	64	9	of	of	ADP
ejpam-5717	64	10	slightly	slightly	ADV
ejpam-5717	64	11	(	(	PUNCT
ejpam-5717	64	12	τ1	τ1	NOUN
ejpam-5717	64	13	,	,	PUNCT
ejpam-5717	64	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	64	15	multifunctions	multifunction	NOUN
ejpam-5717	64	16	.	.	PUNCT
ejpam-5717	65	1	in	in	ADP
ejpam-5717	65	2	this	this	DET
ejpam-5717	65	3	paper	paper	NOUN
ejpam-5717	65	4	,	,	PUNCT
ejpam-5717	65	5	we	we	PRON
ejpam-5717	65	6	introduce	introduce	VERB
ejpam-5717	65	7	the	the	DET
ejpam-5717	65	8	notions	notion	NOUN
ejpam-5717	65	9	of	of	ADP
ejpam-5717	65	10	upper	upper	ADJ
ejpam-5717	65	11	quasi	quasi	NOUN
ejpam-5717	65	12	θ(τ1	θ(τ1	NOUN
ejpam-5717	65	13	,	,	PUNCT
ejpam-5717	65	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	65	15	multifunctions	multifunction	NOUN
ejpam-5717	65	16	and	and	CCONJ
ejpam-5717	65	17	lower	low	ADJ
ejpam-5717	65	18	quasi	quasi	NOUN
ejpam-5717	65	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	65	20	,	,	PUNCT
ejpam-5717	65	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	65	22	multifunctions	multifunction	NOUN
ejpam-5717	65	23	.	.	PUNCT
ejpam-5717	66	1	we	we	PRON
ejpam-5717	66	2	also	also	ADV
ejpam-5717	66	3	investigate	investigate	VERB
ejpam-5717	66	4	several	several	ADJ
ejpam-5717	66	5	characterizations	characterization	NOUN
ejpam-5717	66	6	of	of	ADP
ejpam-5717	66	7	upper	upper	ADJ
ejpam-5717	66	8	quasi	quasi	NOUN
ejpam-5717	66	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	66	10	,	,	PUNCT
ejpam-5717	66	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	66	12	multifunctions	multifunction	NOUN
ejpam-5717	66	13	and	and	CCONJ
ejpam-5717	66	14	lower	low	ADJ
ejpam-5717	66	15	quasi	quasi	NOUN
ejpam-5717	66	16	θ(τ1	θ(τ1	NOUN
ejpam-5717	66	17	,	,	PUNCT
ejpam-5717	66	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	66	19	multifunctions	multifunction	NOUN
ejpam-5717	66	20	.	.	PUNCT
ejpam-5717	67	1	p.	p.	NOUN
ejpam-5717	67	2	pue	pue	NOUN
ejpam-5717	67	3	-	-	PUNCT
ejpam-5717	67	4	on	on	ADP
ejpam-5717	67	5	,	,	PUNCT
ejpam-5717	67	6	a.	a.	PROPN
ejpam-5717	67	7	sama	sama	PROPN
ejpam-5717	67	8	-	-	PUNCT
ejpam-5717	67	9	ae	ae	PROPN
ejpam-5717	67	10	,	,	PUNCT
ejpam-5717	67	11	c.	c.	PROPN
ejpam-5717	67	12	boonpok	boonpok	PROPN
ejpam-5717	67	13	/	/	SYM
ejpam-5717	67	14	eur	eur	PROPN
ejpam-5717	67	15	.	.	PUNCT
ejpam-5717	68	1	j.	j.	PROPN
ejpam-5717	68	2	pure	pure	PROPN
ejpam-5717	68	3	appl	appl	PROPN
ejpam-5717	68	4	.	.	PROPN
ejpam-5717	68	5	math	math	PROPN
ejpam-5717	68	6	,	,	PUNCT
ejpam-5717	68	7	18	18	NUM
ejpam-5717	68	8	(	(	PUNCT
ejpam-5717	68	9	1	1	NUM
ejpam-5717	68	10	)	)	PUNCT
ejpam-5717	68	11	(	(	PUNCT
ejpam-5717	68	12	2025	2025	NUM
ejpam-5717	68	13	)	)	PUNCT
ejpam-5717	68	14	,	,	PUNCT
ejpam-5717	68	15	5717	5717	NUM
ejpam-5717	68	16	3	3	NUM
ejpam-5717	68	17	of	of	ADP
ejpam-5717	68	18	16	16	NUM
ejpam-5717	68	19	2	2	NUM
ejpam-5717	68	20	.	.	PUNCT
ejpam-5717	68	21	preliminaries	preliminary	NOUN
ejpam-5717	68	22	throughout	throughout	ADP
ejpam-5717	68	23	the	the	DET
ejpam-5717	68	24	present	present	ADJ
ejpam-5717	68	25	paper	paper	NOUN
ejpam-5717	68	26	,	,	PUNCT
ejpam-5717	68	27	spaces	space	NOUN
ejpam-5717	68	28	(	(	PUNCT
ejpam-5717	68	29	x	x	NOUN
ejpam-5717	68	30	,	,	PUNCT
ejpam-5717	68	31	τ1	τ1	NOUN
ejpam-5717	68	32	,	,	PUNCT
ejpam-5717	68	33	τ2	τ2	NOUN
ejpam-5717	68	34	)	)	PUNCT
ejpam-5717	68	35	and	and	CCONJ
ejpam-5717	68	36	(	(	PUNCT
ejpam-5717	68	37	y	y	PROPN
ejpam-5717	68	38	,	,	PUNCT
ejpam-5717	68	39	σ1	σ1	PROPN
ejpam-5717	68	40	,	,	PUNCT
ejpam-5717	68	41	σ2	σ2	NOUN
ejpam-5717	68	42	)	)	PUNCT
ejpam-5717	68	43	(	(	PUNCT
ejpam-5717	68	44	or	or	CCONJ
ejpam-5717	68	45	simply	simply	ADV
ejpam-5717	68	46	x	x	X
ejpam-5717	68	47	and	and	CCONJ
ejpam-5717	68	48	y	y	PROPN
ejpam-5717	68	49	)	)	PUNCT
ejpam-5717	68	50	always	always	ADV
ejpam-5717	68	51	mean	mean	VERB
ejpam-5717	68	52	bitopological	bitopological	ADJ
ejpam-5717	68	53	spaces	space	NOUN
ejpam-5717	68	54	on	on	ADP
ejpam-5717	68	55	which	which	PRON
ejpam-5717	68	56	no	no	DET
ejpam-5717	68	57	separation	separation	NOUN
ejpam-5717	68	58	axioms	axiom	NOUN
ejpam-5717	68	59	are	be	AUX
ejpam-5717	68	60	assumed	assume	VERB
ejpam-5717	68	61	unless	unless	SCONJ
ejpam-5717	68	62	explicitly	explicitly	ADV
ejpam-5717	68	63	stated	state	VERB
ejpam-5717	68	64	.	.	PUNCT
ejpam-5717	69	1	let	let	VERB
ejpam-5717	69	2	a	a	DET
ejpam-5717	69	3	be	be	AUX
ejpam-5717	69	4	a	a	DET
ejpam-5717	69	5	subset	subset	NOUN
ejpam-5717	69	6	of	of	ADP
ejpam-5717	69	7	a	a	DET
ejpam-5717	69	8	bitopological	bitopological	ADJ
ejpam-5717	69	9	space	space	NOUN
ejpam-5717	69	10	(	(	PUNCT
ejpam-5717	69	11	x	x	NOUN
ejpam-5717	69	12	,	,	PUNCT
ejpam-5717	69	13	τ1	τ1	NOUN
ejpam-5717	69	14	,	,	PUNCT
ejpam-5717	69	15	τ2	τ2	NOUN
ejpam-5717	69	16	)	)	PUNCT
ejpam-5717	69	17	.	.	PUNCT
ejpam-5717	70	1	the	the	DET
ejpam-5717	70	2	closure	closure	NOUN
ejpam-5717	70	3	of	of	ADP
ejpam-5717	70	4	a	a	PRON
ejpam-5717	70	5	and	and	CCONJ
ejpam-5717	70	6	the	the	DET
ejpam-5717	70	7	interior	interior	NOUN
ejpam-5717	70	8	of	of	ADP
ejpam-5717	70	9	a	a	PRON
ejpam-5717	70	10	with	with	ADP
ejpam-5717	70	11	respect	respect	NOUN
ejpam-5717	70	12	to	to	ADP
ejpam-5717	70	13	τi	τi	PROPN
ejpam-5717	70	14	are	be	AUX
ejpam-5717	70	15	denoted	denote	VERB
ejpam-5717	70	16	by	by	ADP
ejpam-5717	70	17	τi	τi	NOUN
ejpam-5717	70	18	-	-	PUNCT
ejpam-5717	70	19	cl(a	cl(a	NUM
ejpam-5717	70	20	)	)	PUNCT
ejpam-5717	70	21	and	and	CCONJ
ejpam-5717	70	22	τi	τi	NOUN
ejpam-5717	70	23	-	-	PUNCT
ejpam-5717	70	24	int(a	int(a	NOUN
ejpam-5717	70	25	)	)	PUNCT
ejpam-5717	70	26	,	,	PUNCT
ejpam-5717	70	27	respectively	respectively	ADV
ejpam-5717	70	28	,	,	PUNCT
ejpam-5717	70	29	for	for	ADP
ejpam-5717	70	30	i	i	PROPN
ejpam-5717	70	31	=	=	SYM
ejpam-5717	70	32	1	1	NUM
ejpam-5717	70	33	,	,	PUNCT
ejpam-5717	70	34	2	2	NUM
ejpam-5717	70	35	.	.	X
ejpam-5717	70	36	a	a	DET
ejpam-5717	70	37	subset	subset	NOUN
ejpam-5717	70	38	a	a	PRON
ejpam-5717	70	39	of	of	ADP
ejpam-5717	70	40	a	a	DET
ejpam-5717	70	41	bitopological	bitopological	ADJ
ejpam-5717	70	42	space	space	NOUN
ejpam-5717	70	43	(	(	PUNCT
ejpam-5717	70	44	x	x	NOUN
ejpam-5717	70	45	,	,	PUNCT
ejpam-5717	70	46	τ1	τ1	NOUN
ejpam-5717	70	47	,	,	PUNCT
ejpam-5717	70	48	τ2	τ2	NOUN
ejpam-5717	70	49	)	)	PUNCT
ejpam-5717	70	50	is	be	AUX
ejpam-5717	70	51	called	call	VERB
ejpam-5717	70	52	τ1τ2	τ1τ2	VERB
ejpam-5717	70	53	-	-	ADJ
ejpam-5717	70	54	closed	closed	ADJ
ejpam-5717	70	55	[	[	X
ejpam-5717	70	56	30	30	NUM
ejpam-5717	70	57	]	]	X
ejpam-5717	70	58	if	if	SCONJ
ejpam-5717	70	59	a	a	DET
ejpam-5717	70	60	=	=	NOUN
ejpam-5717	70	61	τ1	τ1	NOUN
ejpam-5717	70	62	-	-	PUNCT
ejpam-5717	70	63	cl(τ2	cl(τ2	NOUN
ejpam-5717	70	64	-	-	PUNCT
ejpam-5717	70	65	cl(a	cl(a	NUM
ejpam-5717	70	66	)	)	PUNCT
ejpam-5717	70	67	)	)	PUNCT
ejpam-5717	70	68	.	.	PUNCT
ejpam-5717	71	1	the	the	DET
ejpam-5717	71	2	complement	complement	NOUN
ejpam-5717	71	3	of	of	ADP
ejpam-5717	71	4	a	a	DET
ejpam-5717	71	5	τ1τ2	τ1τ2	ADJ
ejpam-5717	71	6	-	-	ADJ
ejpam-5717	71	7	closed	closed	ADJ
ejpam-5717	71	8	set	set	NOUN
ejpam-5717	71	9	is	be	AUX
ejpam-5717	71	10	called	call	VERB
ejpam-5717	71	11	τ1τ2	τ1τ2	NOUN
ejpam-5717	71	12	-	-	ADJ
ejpam-5717	71	13	open	open	ADJ
ejpam-5717	71	14	.	.	PUNCT
ejpam-5717	72	1	let	let	VERB
ejpam-5717	72	2	a	a	DET
ejpam-5717	72	3	be	be	AUX
ejpam-5717	72	4	a	a	DET
ejpam-5717	72	5	subset	subset	NOUN
ejpam-5717	72	6	of	of	ADP
ejpam-5717	72	7	a	a	DET
ejpam-5717	72	8	bitopological	bitopological	ADJ
ejpam-5717	72	9	space	space	NOUN
ejpam-5717	72	10	(	(	PUNCT
ejpam-5717	72	11	x	x	NOUN
ejpam-5717	72	12	,	,	PUNCT
ejpam-5717	72	13	τ1	τ1	NOUN
ejpam-5717	72	14	,	,	PUNCT
ejpam-5717	72	15	τ2	τ2	NOUN
ejpam-5717	72	16	)	)	PUNCT
ejpam-5717	72	17	.	.	PUNCT
ejpam-5717	73	1	the	the	DET
ejpam-5717	73	2	intersection	intersection	NOUN
ejpam-5717	73	3	of	of	ADP
ejpam-5717	73	4	all	all	DET
ejpam-5717	73	5	τ1τ2	τ1τ2	ADJ
ejpam-5717	73	6	-	-	ADJ
ejpam-5717	73	7	closed	closed	ADJ
ejpam-5717	73	8	sets	set	NOUN
ejpam-5717	73	9	of	of	ADP
ejpam-5717	73	10	x	x	PUNCT
ejpam-5717	73	11	containing	contain	VERB
ejpam-5717	73	12	a	a	PRON
ejpam-5717	73	13	is	be	AUX
ejpam-5717	73	14	called	call	VERB
ejpam-5717	73	15	the	the	DET
ejpam-5717	73	16	τ1τ2	τ1τ2	NOUN
ejpam-5717	73	17	-	-	NOUN
ejpam-5717	73	18	closure	closure	NOUN
ejpam-5717	73	19	[	[	X
ejpam-5717	73	20	30	30	NUM
ejpam-5717	73	21	]	]	PUNCT
ejpam-5717	73	22	of	of	ADP
ejpam-5717	73	23	a	a	PRON
ejpam-5717	73	24	and	and	CCONJ
ejpam-5717	73	25	is	be	AUX
ejpam-5717	73	26	denoted	denote	VERB
ejpam-5717	73	27	by	by	ADP
ejpam-5717	73	28	τ1τ2	τ1τ2	NOUN
ejpam-5717	73	29	-	-	NUM
ejpam-5717	73	30	cl(a	cl(a	NUM
ejpam-5717	73	31	)	)	PUNCT
ejpam-5717	73	32	.	.	PUNCT
ejpam-5717	74	1	the	the	DET
ejpam-5717	74	2	union	union	NOUN
ejpam-5717	74	3	of	of	ADP
ejpam-5717	74	4	all	all	DET
ejpam-5717	74	5	τ1τ2	τ1τ2	ADJ
ejpam-5717	74	6	-	-	ADJ
ejpam-5717	74	7	open	open	ADJ
ejpam-5717	74	8	sets	set	NOUN
ejpam-5717	74	9	of	of	ADP
ejpam-5717	74	10	x	x	PUNCT
ejpam-5717	74	11	contained	contain	VERB
ejpam-5717	74	12	in	in	ADP
ejpam-5717	74	13	a	a	PRON
ejpam-5717	74	14	is	be	AUX
ejpam-5717	74	15	called	call	VERB
ejpam-5717	74	16	the	the	DET
ejpam-5717	74	17	τ1τ2	τ1τ2	NOUN
ejpam-5717	74	18	-	-	ADJ
ejpam-5717	74	19	interior	interior	ADJ
ejpam-5717	74	20	[	[	X
ejpam-5717	74	21	30	30	NUM
ejpam-5717	74	22	]	]	PUNCT
ejpam-5717	74	23	of	of	ADP
ejpam-5717	74	24	a	a	PRON
ejpam-5717	74	25	and	and	CCONJ
ejpam-5717	74	26	is	be	AUX
ejpam-5717	74	27	denoted	denote	VERB
ejpam-5717	74	28	by	by	ADP
ejpam-5717	74	29	τ1τ2	τ1τ2	NOUN
ejpam-5717	74	30	-	-	ADJ
ejpam-5717	74	31	int(a	int(a	NOUN
ejpam-5717	74	32	)	)	PUNCT
ejpam-5717	74	33	.	.	PUNCT
ejpam-5717	75	1	a	a	DET
ejpam-5717	75	2	subset	subset	NOUN
ejpam-5717	75	3	a	a	PRON
ejpam-5717	75	4	of	of	ADP
ejpam-5717	75	5	a	a	DET
ejpam-5717	75	6	bitopological	bitopological	ADJ
ejpam-5717	75	7	space	space	NOUN
ejpam-5717	75	8	(	(	PUNCT
ejpam-5717	75	9	x	x	NOUN
ejpam-5717	75	10	,	,	PUNCT
ejpam-5717	75	11	τ1	τ1	NOUN
ejpam-5717	75	12	,	,	PUNCT
ejpam-5717	75	13	τ2	τ2	NOUN
ejpam-5717	75	14	)	)	PUNCT
ejpam-5717	75	15	is	be	AUX
ejpam-5717	75	16	said	say	VERB
ejpam-5717	75	17	to	to	PART
ejpam-5717	75	18	be	be	AUX
ejpam-5717	75	19	τ1τ2	τ1τ2	NOUN
ejpam-5717	75	20	-	-	ADJ
ejpam-5717	75	21	clopen	clopen	ADJ
ejpam-5717	76	1	[	[	X
ejpam-5717	76	2	30	30	NUM
ejpam-5717	76	3	]	]	X
ejpam-5717	76	4	if	if	SCONJ
ejpam-5717	76	5	a	a	PRON
ejpam-5717	76	6	is	be	AUX
ejpam-5717	76	7	both	both	PRON
ejpam-5717	76	8	τ1τ2	τ1τ2	ADJ
ejpam-5717	76	9	-	-	ADJ
ejpam-5717	76	10	open	open	ADJ
ejpam-5717	76	11	and	and	CCONJ
ejpam-5717	76	12	τ1τ2	τ1τ2	NOUN
ejpam-5717	76	13	-	-	ADJ
ejpam-5717	76	14	closed	closed	ADJ
ejpam-5717	76	15	.	.	PUNCT
ejpam-5717	77	1	a	a	DET
ejpam-5717	77	2	subset	subset	NOUN
ejpam-5717	77	3	a	a	PRON
ejpam-5717	77	4	of	of	ADP
ejpam-5717	77	5	a	a	DET
ejpam-5717	77	6	bitopological	bitopological	ADJ
ejpam-5717	77	7	space	space	NOUN
ejpam-5717	77	8	(	(	PUNCT
ejpam-5717	77	9	x	x	NOUN
ejpam-5717	77	10	,	,	PUNCT
ejpam-5717	77	11	τ1	τ1	NOUN
ejpam-5717	77	12	,	,	PUNCT
ejpam-5717	77	13	τ2	τ2	NOUN
ejpam-5717	77	14	)	)	PUNCT
ejpam-5717	77	15	is	be	AUX
ejpam-5717	77	16	said	say	VERB
ejpam-5717	77	17	to	to	PART
ejpam-5717	77	18	be	be	AUX
ejpam-5717	77	19	(	(	PUNCT
ejpam-5717	77	20	τ1	τ1	NOUN
ejpam-5717	77	21	,	,	PUNCT
ejpam-5717	77	22	τ2)r	τ2)r	NOUN
ejpam-5717	77	23	-	-	PUNCT
ejpam-5717	77	24	open	open	ADJ
ejpam-5717	78	1	[	[	X
ejpam-5717	78	2	66	66	NUM
ejpam-5717	78	3	]	]	PUNCT
ejpam-5717	78	4	(	(	PUNCT
ejpam-5717	78	5	resp	resp	NOUN
ejpam-5717	78	6	.	.	PUNCT
ejpam-5717	79	1	(	(	PUNCT
ejpam-5717	79	2	τ1	τ1	NOUN
ejpam-5717	79	3	,	,	PUNCT
ejpam-5717	79	4	τ2)s	τ2)s	NOUN
ejpam-5717	79	5	-	-	PUNCT
ejpam-5717	79	6	open	open	ADJ
ejpam-5717	79	7	[	[	X
ejpam-5717	79	8	6	6	NUM
ejpam-5717	79	9	]	]	PUNCT
ejpam-5717	79	10	,	,	PUNCT
ejpam-5717	79	11	(	(	PUNCT
ejpam-5717	79	12	τ1	τ1	NOUN
ejpam-5717	79	13	,	,	PUNCT
ejpam-5717	79	14	τ2)p	τ2)p	NOUN
ejpam-5717	79	15	-	-	ADJ
ejpam-5717	79	16	open	open	ADJ
ejpam-5717	79	17	[	[	X
ejpam-5717	79	18	6	6	NUM
ejpam-5717	79	19	]	]	PUNCT
ejpam-5717	79	20	,	,	PUNCT
ejpam-5717	79	21	(	(	PUNCT
ejpam-5717	79	22	τ1	τ1	NOUN
ejpam-5717	79	23	,	,	PUNCT
ejpam-5717	79	24	τ2)β	τ2)β	ADJ
ejpam-5717	79	25	-	-	PUNCT
ejpam-5717	79	26	open	open	NOUN
ejpam-5717	80	1	[	[	X
ejpam-5717	80	2	6	6	NUM
ejpam-5717	80	3	]	]	PUNCT
ejpam-5717	80	4	)	)	PUNCT
ejpam-5717	80	5	if	if	SCONJ
ejpam-5717	80	6	a	a	DET
ejpam-5717	80	7	=	=	PUNCT
ejpam-5717	80	8	τ1τ2	τ1τ2	NOUN
ejpam-5717	80	9	-	-	NOUN
ejpam-5717	80	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5717	80	11	-	-	PUNCT
ejpam-5717	80	12	cl(a	cl(a	NUM
ejpam-5717	80	13	)	)	PUNCT
ejpam-5717	80	14	)	)	PUNCT
ejpam-5717	80	15	(	(	PUNCT
ejpam-5717	80	16	resp	resp	NOUN
ejpam-5717	80	17	.	.	PUNCT
ejpam-5717	81	1	a	a	DET
ejpam-5717	81	2	⊆	⊆	NUM
ejpam-5717	81	3	τ1τ2	τ1τ2	NOUN
ejpam-5717	81	4	-	-	ADJ
ejpam-5717	81	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5717	81	6	-	-	PUNCT
ejpam-5717	81	7	int(a	int(a	NOUN
ejpam-5717	81	8	)	)	PUNCT
ejpam-5717	81	9	)	)	PUNCT
ejpam-5717	81	10	,	,	PUNCT
ejpam-5717	81	11	a	a	DET
ejpam-5717	81	12	⊆	⊆	NUM
ejpam-5717	81	13	τ1τ2	τ1τ2	NOUN
ejpam-5717	81	14	-	-	NOUN
ejpam-5717	81	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5717	81	16	-	-	PUNCT
ejpam-5717	81	17	cl(a	cl(a	NUM
ejpam-5717	81	18	)	)	PUNCT
ejpam-5717	81	19	)	)	PUNCT
ejpam-5717	81	20	,	,	PUNCT
ejpam-5717	81	21	a	a	DET
ejpam-5717	81	22	⊆	⊆	NUM
ejpam-5717	81	23	τ1τ2	τ1τ2	NOUN
ejpam-5717	81	24	-	-	PUNCT
ejpam-5717	81	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5717	81	26	-	-	PUNCT
ejpam-5717	81	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5717	81	28	-	-	PUNCT
ejpam-5717	81	29	cl(a	cl(a	NUM
ejpam-5717	81	30	)	)	PUNCT
ejpam-5717	81	31	)	)	PUNCT
ejpam-5717	81	32	)	)	PUNCT
ejpam-5717	81	33	)	)	PUNCT
ejpam-5717	81	34	.	.	PUNCT
ejpam-5717	82	1	the	the	DET
ejpam-5717	82	2	complement	complement	NOUN
ejpam-5717	82	3	of	of	ADP
ejpam-5717	82	4	a	a	DET
ejpam-5717	82	5	(	(	PUNCT
ejpam-5717	82	6	τ1	τ1	NOUN
ejpam-5717	82	7	,	,	PUNCT
ejpam-5717	82	8	τ2)r	τ2)r	NOUN
ejpam-5717	82	9	-	-	PUNCT
ejpam-5717	82	10	open	open	ADJ
ejpam-5717	82	11	(	(	PUNCT
ejpam-5717	82	12	resp	resp	NOUN
ejpam-5717	82	13	.	.	PUNCT
ejpam-5717	83	1	(	(	PUNCT
ejpam-5717	83	2	τ1	τ1	NOUN
ejpam-5717	83	3	,	,	PUNCT
ejpam-5717	83	4	τ2)sopen	τ2)sopen	ADJ
ejpam-5717	83	5	,	,	PUNCT
ejpam-5717	83	6	(	(	PUNCT
ejpam-5717	83	7	τ1	τ1	NOUN
ejpam-5717	83	8	,	,	PUNCT
ejpam-5717	83	9	τ2)p	τ2)p	NOUN
ejpam-5717	83	10	-	-	ADJ
ejpam-5717	83	11	open	open	ADJ
ejpam-5717	83	12	,	,	PUNCT
ejpam-5717	83	13	(	(	PUNCT
ejpam-5717	83	14	τ1	τ1	NOUN
ejpam-5717	83	15	,	,	PUNCT
ejpam-5717	83	16	τ2)β	τ2)β	ADJ
ejpam-5717	83	17	-	-	PUNCT
ejpam-5717	83	18	open	open	ADJ
ejpam-5717	83	19	)	)	PUNCT
ejpam-5717	83	20	set	set	NOUN
ejpam-5717	83	21	is	be	AUX
ejpam-5717	83	22	called	call	VERB
ejpam-5717	83	23	(	(	PUNCT
ejpam-5717	83	24	τ1	τ1	NOUN
ejpam-5717	83	25	,	,	PUNCT
ejpam-5717	83	26	τ2)r	τ2)r	NOUN
ejpam-5717	83	27	-	-	PUNCT
ejpam-5717	83	28	closed	closed	ADJ
ejpam-5717	83	29	(	(	PUNCT
ejpam-5717	83	30	resp	resp	NOUN
ejpam-5717	83	31	.	.	PUNCT
ejpam-5717	84	1	(	(	PUNCT
ejpam-5717	84	2	τ1	τ1	NOUN
ejpam-5717	84	3	,	,	PUNCT
ejpam-5717	84	4	τ2)s	τ2)s	NOUN
ejpam-5717	84	5	-	-	PUNCT
ejpam-5717	84	6	closed	closed	ADJ
ejpam-5717	84	7	,	,	PUNCT
ejpam-5717	84	8	(	(	PUNCT
ejpam-5717	84	9	τ1	τ1	NOUN
ejpam-5717	84	10	,	,	PUNCT
ejpam-5717	84	11	τ2)p	τ2)p	NOUN
ejpam-5717	84	12	-	-	PUNCT
ejpam-5717	84	13	closed	closed	ADJ
ejpam-5717	84	14	,	,	PUNCT
ejpam-5717	84	15	(	(	PUNCT
ejpam-5717	84	16	τ1	τ1	NOUN
ejpam-5717	84	17	,	,	PUNCT
ejpam-5717	84	18	τ2)β	τ2)β	ADJ
ejpam-5717	84	19	-	-	PUNCT
ejpam-5717	84	20	closed	closed	ADJ
ejpam-5717	84	21	)	)	PUNCT
ejpam-5717	84	22	.	.	PUNCT
ejpam-5717	85	1	a	a	DET
ejpam-5717	85	2	subset	subset	NOUN
ejpam-5717	85	3	a	a	PRON
ejpam-5717	85	4	of	of	ADP
ejpam-5717	85	5	a	a	DET
ejpam-5717	85	6	bitopological	bitopological	ADJ
ejpam-5717	85	7	space	space	NOUN
ejpam-5717	85	8	(	(	PUNCT
ejpam-5717	85	9	x	x	NOUN
ejpam-5717	85	10	,	,	PUNCT
ejpam-5717	85	11	τ1	τ1	NOUN
ejpam-5717	85	12	,	,	PUNCT
ejpam-5717	85	13	τ2	τ2	NOUN
ejpam-5717	85	14	)	)	PUNCT
ejpam-5717	85	15	is	be	AUX
ejpam-5717	85	16	said	say	VERB
ejpam-5717	85	17	to	to	PART
ejpam-5717	85	18	be	be	AUX
ejpam-5717	85	19	α(τ1	α(τ1	NOUN
ejpam-5717	85	20	,	,	PUNCT
ejpam-5717	85	21	τ2)-open	τ2)-open	ADJ
ejpam-5717	85	22	[	[	X
ejpam-5717	85	23	71	71	NUM
ejpam-5717	85	24	]	]	PUNCT
ejpam-5717	85	25	if	if	SCONJ
ejpam-5717	85	26	a	a	DET
ejpam-5717	85	27	⊆	⊆	NUM
ejpam-5717	85	28	τ1τ2	τ1τ2	NOUN
ejpam-5717	85	29	-	-	PUNCT
ejpam-5717	85	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5717	85	31	-	-	PUNCT
ejpam-5717	85	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5717	85	33	-	-	PUNCT
ejpam-5717	85	34	int(a	int(a	NOUN
ejpam-5717	85	35	)	)	PUNCT
ejpam-5717	85	36	)	)	PUNCT
ejpam-5717	85	37	)	)	PUNCT
ejpam-5717	85	38	.	.	PUNCT
ejpam-5717	86	1	the	the	DET
ejpam-5717	86	2	complement	complement	NOUN
ejpam-5717	86	3	of	of	ADP
ejpam-5717	86	4	an	an	DET
ejpam-5717	86	5	α(τ1	α(τ1	NOUN
ejpam-5717	86	6	,	,	PUNCT
ejpam-5717	86	7	τ2)-open	τ2)-open	ADJ
ejpam-5717	86	8	set	set	NOUN
ejpam-5717	86	9	is	be	AUX
ejpam-5717	86	10	said	say	VERB
ejpam-5717	86	11	to	to	PART
ejpam-5717	86	12	be	be	AUX
ejpam-5717	86	13	α(τ1	α(τ1	NOUN
ejpam-5717	86	14	,	,	PUNCT
ejpam-5717	86	15	τ2)-closed	τ2)-close	VERB
ejpam-5717	86	16	.	.	PUNCT
ejpam-5717	87	1	let	let	VERB
ejpam-5717	87	2	a	a	DET
ejpam-5717	87	3	be	be	AUX
ejpam-5717	87	4	a	a	DET
ejpam-5717	87	5	subset	subset	NOUN
ejpam-5717	87	6	of	of	ADP
ejpam-5717	87	7	a	a	DET
ejpam-5717	87	8	bitopological	bitopological	ADJ
ejpam-5717	87	9	space	space	NOUN
ejpam-5717	87	10	(	(	PUNCT
ejpam-5717	87	11	x	x	NOUN
ejpam-5717	87	12	,	,	PUNCT
ejpam-5717	87	13	τ1	τ1	NOUN
ejpam-5717	87	14	,	,	PUNCT
ejpam-5717	87	15	τ2	τ2	NOUN
ejpam-5717	87	16	)	)	PUNCT
ejpam-5717	87	17	.	.	PUNCT
ejpam-5717	88	1	a	a	DET
ejpam-5717	88	2	point	point	NOUN
ejpam-5717	88	3	x	x	X
ejpam-5717	88	4	∈	∈	NOUN
ejpam-5717	88	5	x	x	PUNCT
ejpam-5717	88	6	is	be	AUX
ejpam-5717	88	7	called	call	VERB
ejpam-5717	88	8	a	a	DET
ejpam-5717	88	9	(	(	PUNCT
ejpam-5717	88	10	τ1	τ1	NOUN
ejpam-5717	88	11	,	,	PUNCT
ejpam-5717	88	12	τ2)θ	τ2)θ	ADJ
ejpam-5717	88	13	-	-	PUNCT
ejpam-5717	88	14	cluster	cluster	NOUN
ejpam-5717	88	15	point	point	NOUN
ejpam-5717	88	16	[	[	X
ejpam-5717	88	17	66	66	NUM
ejpam-5717	88	18	]	]	PUNCT
ejpam-5717	88	19	of	of	ADP
ejpam-5717	88	20	a	a	DET
ejpam-5717	88	21	if	if	SCONJ
ejpam-5717	88	22	τ1τ2	τ1τ2	ADJ
ejpam-5717	88	23	-	-	ADJ
ejpam-5717	88	24	cl(u)∩a	cl(u)∩a	ADJ
ejpam-5717	88	25	̸=	̸=	PROPN
ejpam-5717	88	26	∅	∅	NOUN
ejpam-5717	88	27	for	for	ADP
ejpam-5717	88	28	every	every	DET
ejpam-5717	88	29	τ1τ2	τ1τ2	ADJ
ejpam-5717	88	30	-	-	ADJ
ejpam-5717	88	31	open	open	ADJ
ejpam-5717	88	32	set	set	NOUN
ejpam-5717	88	33	u	u	NOUN
ejpam-5717	88	34	containing	contain	VERB
ejpam-5717	88	35	x.	x.	NOUN
ejpam-5717	88	36	the	the	DET
ejpam-5717	88	37	set	set	NOUN
ejpam-5717	88	38	of	of	ADP
ejpam-5717	88	39	all	all	DET
ejpam-5717	88	40	(	(	PUNCT
ejpam-5717	88	41	τ1	τ1	NOUN
ejpam-5717	88	42	,	,	PUNCT
ejpam-5717	88	43	τ2)θ	τ2)θ	ADJ
ejpam-5717	88	44	-	-	PUNCT
ejpam-5717	88	45	cluster	cluster	NOUN
ejpam-5717	88	46	points	point	NOUN
ejpam-5717	88	47	of	of	ADP
ejpam-5717	88	48	a	a	PRON
ejpam-5717	88	49	is	be	AUX
ejpam-5717	88	50	called	call	VERB
ejpam-5717	88	51	the	the	DET
ejpam-5717	88	52	(	(	PUNCT
ejpam-5717	88	53	τ1	τ1	NOUN
ejpam-5717	88	54	,	,	PUNCT
ejpam-5717	88	55	τ2)θ	τ2)θ	ADJ
ejpam-5717	88	56	-	-	PUNCT
ejpam-5717	88	57	closure	closure	NOUN
ejpam-5717	88	58	[	[	X
ejpam-5717	88	59	66	66	NUM
ejpam-5717	88	60	]	]	PUNCT
ejpam-5717	88	61	of	of	ADP
ejpam-5717	88	62	a	a	PRON
ejpam-5717	88	63	and	and	CCONJ
ejpam-5717	88	64	is	be	AUX
ejpam-5717	88	65	denoted	denote	VERB
ejpam-5717	88	66	by	by	ADP
ejpam-5717	88	67	(	(	PUNCT
ejpam-5717	88	68	τ1	τ1	NOUN
ejpam-5717	88	69	,	,	PUNCT
ejpam-5717	88	70	τ2)θ	τ2)θ	NOUN
ejpam-5717	88	71	-	-	PUNCT
ejpam-5717	88	72	cl(a	cl(a	NUM
ejpam-5717	88	73	)	)	PUNCT
ejpam-5717	88	74	.	.	PUNCT
ejpam-5717	89	1	a	a	DET
ejpam-5717	89	2	subset	subset	NOUN
ejpam-5717	89	3	a	a	PRON
ejpam-5717	89	4	of	of	ADP
ejpam-5717	89	5	a	a	DET
ejpam-5717	89	6	bitopological	bitopological	ADJ
ejpam-5717	89	7	space	space	NOUN
ejpam-5717	89	8	(	(	PUNCT
ejpam-5717	89	9	x	x	NOUN
ejpam-5717	89	10	,	,	PUNCT
ejpam-5717	89	11	τ1	τ1	NOUN
ejpam-5717	89	12	,	,	PUNCT
ejpam-5717	89	13	τ2	τ2	NOUN
ejpam-5717	89	14	)	)	PUNCT
ejpam-5717	89	15	is	be	AUX
ejpam-5717	89	16	said	say	VERB
ejpam-5717	89	17	to	to	PART
ejpam-5717	89	18	be	be	AUX
ejpam-5717	89	19	(	(	PUNCT
ejpam-5717	89	20	τ1	τ1	NOUN
ejpam-5717	89	21	,	,	PUNCT
ejpam-5717	89	22	τ2)θ	τ2)θ	NOUN
ejpam-5717	89	23	-	-	PUNCT
ejpam-5717	89	24	closed	closed	ADJ
ejpam-5717	89	25	[	[	X
ejpam-5717	89	26	66	66	NUM
ejpam-5717	89	27	]	]	PUNCT
ejpam-5717	89	28	if	if	SCONJ
ejpam-5717	89	29	(	(	PUNCT
ejpam-5717	89	30	τ1	τ1	NOUN
ejpam-5717	89	31	,	,	PUNCT
ejpam-5717	89	32	τ2)θ	τ2)θ	NOUN
ejpam-5717	89	33	-	-	PUNCT
ejpam-5717	89	34	cl(a	cl(a	NUM
ejpam-5717	89	35	)	)	PUNCT
ejpam-5717	90	1	=	=	PUNCT
ejpam-5717	90	2	a.	a.	NOUN
ejpam-5717	90	3	the	the	DET
ejpam-5717	90	4	complement	complement	NOUN
ejpam-5717	90	5	of	of	ADP
ejpam-5717	90	6	a	a	DET
ejpam-5717	90	7	(	(	PUNCT
ejpam-5717	90	8	τ1	τ1	NOUN
ejpam-5717	90	9	,	,	PUNCT
ejpam-5717	90	10	τ2)θ	τ2)θ	ADJ
ejpam-5717	90	11	-	-	PUNCT
ejpam-5717	90	12	closed	close	VERB
ejpam-5717	90	13	set	set	NOUN
ejpam-5717	90	14	is	be	AUX
ejpam-5717	90	15	said	say	VERB
ejpam-5717	90	16	to	to	PART
ejpam-5717	90	17	be	be	AUX
ejpam-5717	90	18	(	(	PUNCT
ejpam-5717	90	19	τ1	τ1	NOUN
ejpam-5717	90	20	,	,	PUNCT
ejpam-5717	90	21	τ2)θ	τ2)θ	NOUN
ejpam-5717	90	22	-	-	PUNCT
ejpam-5717	90	23	open	open	ADJ
ejpam-5717	90	24	.	.	PUNCT
ejpam-5717	91	1	the	the	DET
ejpam-5717	91	2	union	union	NOUN
ejpam-5717	91	3	of	of	ADP
ejpam-5717	91	4	all	all	DET
ejpam-5717	91	5	(	(	PUNCT
ejpam-5717	91	6	τ1	τ1	NOUN
ejpam-5717	91	7	,	,	PUNCT
ejpam-5717	91	8	τ2)θ	τ2)θ	ADJ
ejpam-5717	91	9	-	-	PUNCT
ejpam-5717	91	10	open	open	ADJ
ejpam-5717	91	11	sets	set	NOUN
ejpam-5717	91	12	of	of	ADP
ejpam-5717	91	13	x	x	PUNCT
ejpam-5717	91	14	contained	contain	VERB
ejpam-5717	91	15	in	in	ADP
ejpam-5717	91	16	a	a	PRON
ejpam-5717	91	17	is	be	AUX
ejpam-5717	91	18	called	call	VERB
ejpam-5717	91	19	the	the	DET
ejpam-5717	91	20	(	(	PUNCT
ejpam-5717	91	21	τ1	τ1	NOUN
ejpam-5717	91	22	,	,	PUNCT
ejpam-5717	91	23	τ2)θ	τ2)θ	ADJ
ejpam-5717	91	24	-	-	PUNCT
ejpam-5717	91	25	interior	interior	NOUN
ejpam-5717	91	26	[	[	X
ejpam-5717	91	27	66	66	NUM
ejpam-5717	91	28	]	]	PUNCT
ejpam-5717	91	29	of	of	ADP
ejpam-5717	91	30	a	a	PRON
ejpam-5717	91	31	and	and	CCONJ
ejpam-5717	91	32	is	be	AUX
ejpam-5717	91	33	denoted	denote	VERB
ejpam-5717	91	34	by	by	ADP
ejpam-5717	91	35	(	(	PUNCT
ejpam-5717	91	36	τ1	τ1	NOUN
ejpam-5717	91	37	,	,	PUNCT
ejpam-5717	91	38	τ2)θ	τ2)θ	NOUN
ejpam-5717	91	39	-	-	PUNCT
ejpam-5717	91	40	int(a	int(a	NOUN
ejpam-5717	91	41	)	)	PUNCT
ejpam-5717	91	42	.	.	PUNCT
ejpam-5717	92	1	lemma	lemma	PROPN
ejpam-5717	92	2	1	1	NUM
ejpam-5717	92	3	.	.	PUNCT
ejpam-5717	93	1	[	[	X
ejpam-5717	93	2	66	66	NUM
ejpam-5717	93	3	]	]	PUNCT
ejpam-5717	93	4	for	for	ADP
ejpam-5717	93	5	a	a	DET
ejpam-5717	93	6	subset	subset	NOUN
ejpam-5717	93	7	a	a	PRON
ejpam-5717	93	8	of	of	ADP
ejpam-5717	93	9	a	a	DET
ejpam-5717	93	10	bitopological	bitopological	ADJ
ejpam-5717	93	11	space	space	NOUN
ejpam-5717	93	12	(	(	PUNCT
ejpam-5717	93	13	x	x	NOUN
ejpam-5717	93	14	,	,	PUNCT
ejpam-5717	93	15	τ1	τ1	NOUN
ejpam-5717	93	16	,	,	PUNCT
ejpam-5717	93	17	τ2	τ2	NOUN
ejpam-5717	93	18	)	)	PUNCT
ejpam-5717	93	19	,	,	PUNCT
ejpam-5717	93	20	the	the	DET
ejpam-5717	93	21	following	follow	VERB
ejpam-5717	93	22	properties	property	NOUN
ejpam-5717	93	23	hold	hold	VERB
ejpam-5717	93	24	:	:	PUNCT
ejpam-5717	93	25	(	(	PUNCT
ejpam-5717	93	26	1	1	X
ejpam-5717	93	27	)	)	PUNCT
ejpam-5717	93	28	if	if	SCONJ
ejpam-5717	93	29	a	a	PRON
ejpam-5717	93	30	is	be	AUX
ejpam-5717	93	31	τ1τ2	τ1τ2	NOUN
ejpam-5717	93	32	-	-	ADJ
ejpam-5717	93	33	open	open	ADJ
ejpam-5717	93	34	in	in	ADP
ejpam-5717	93	35	x	x	NOUN
ejpam-5717	93	36	,	,	PUNCT
ejpam-5717	93	37	then	then	ADV
ejpam-5717	93	38	τ1τ2	τ1τ2	NOUN
ejpam-5717	93	39	-	-	NUM
ejpam-5717	93	40	cl(a	cl(a	NUM
ejpam-5717	93	41	)	)	PUNCT
ejpam-5717	93	42	=	=	PUNCT
ejpam-5717	93	43	(	(	PUNCT
ejpam-5717	93	44	τ1	τ1	NOUN
ejpam-5717	93	45	,	,	PUNCT
ejpam-5717	93	46	τ2)θ	τ2)θ	NOUN
ejpam-5717	93	47	-	-	PUNCT
ejpam-5717	93	48	cl(a	cl(a	NUM
ejpam-5717	93	49	)	)	PUNCT
ejpam-5717	93	50	.	.	PUNCT
ejpam-5717	94	1	(	(	PUNCT
ejpam-5717	94	2	2	2	X
ejpam-5717	94	3	)	)	PUNCT
ejpam-5717	94	4	(	(	PUNCT
ejpam-5717	94	5	τ1	τ1	NOUN
ejpam-5717	94	6	,	,	PUNCT
ejpam-5717	94	7	τ2)θ	τ2)θ	NOUN
ejpam-5717	94	8	-	-	PUNCT
ejpam-5717	94	9	cl(a	cl(a	NUM
ejpam-5717	94	10	)	)	PUNCT
ejpam-5717	94	11	is	be	AUX
ejpam-5717	94	12	τ1τ2	τ1τ2	NOUN
ejpam-5717	94	13	-	-	ADJ
ejpam-5717	94	14	closed	closed	ADJ
ejpam-5717	94	15	in	in	ADP
ejpam-5717	94	16	x.	x.	NOUN
ejpam-5717	94	17	let	let	VERB
ejpam-5717	94	18	a	a	PRON
ejpam-5717	94	19	be	be	AUX
ejpam-5717	94	20	a	a	DET
ejpam-5717	94	21	subset	subset	NOUN
ejpam-5717	94	22	of	of	ADP
ejpam-5717	94	23	a	a	DET
ejpam-5717	94	24	bitopological	bitopological	ADJ
ejpam-5717	94	25	space	space	NOUN
ejpam-5717	94	26	(	(	PUNCT
ejpam-5717	94	27	x	x	NOUN
ejpam-5717	94	28	,	,	PUNCT
ejpam-5717	94	29	τ1	τ1	NOUN
ejpam-5717	94	30	,	,	PUNCT
ejpam-5717	94	31	τ2	τ2	NOUN
ejpam-5717	94	32	)	)	PUNCT
ejpam-5717	94	33	.	.	PUNCT
ejpam-5717	95	1	a	a	DET
ejpam-5717	95	2	point	point	NOUN
ejpam-5717	95	3	x	x	X
ejpam-5717	95	4	∈	∈	NOUN
ejpam-5717	95	5	x	x	PUNCT
ejpam-5717	95	6	is	be	AUX
ejpam-5717	95	7	called	call	VERB
ejpam-5717	95	8	a	a	DET
ejpam-5717	95	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	95	10	,	,	PUNCT
ejpam-5717	95	11	τ2)s	τ2)s	NOUN
ejpam-5717	95	12	-	-	PUNCT
ejpam-5717	95	13	cluster	cluster	NOUN
ejpam-5717	95	14	point	point	NOUN
ejpam-5717	95	15	of	of	ADP
ejpam-5717	95	16	a	a	DET
ejpam-5717	95	17	if	if	NOUN
ejpam-5717	95	18	(	(	PUNCT
ejpam-5717	95	19	τ1	τ1	NOUN
ejpam-5717	95	20	,	,	PUNCT
ejpam-5717	95	21	τ2)-scl(u	τ2)-scl(u	ADJ
ejpam-5717	95	22	)	)	PUNCT
ejpam-5717	95	23	∩	∩	NOUN
ejpam-5717	95	24	a	a	DET
ejpam-5717	95	25	̸=	̸=	PROPN
ejpam-5717	95	26	∅	∅	NOUN
ejpam-5717	95	27	for	for	ADP
ejpam-5717	95	28	every	every	DET
ejpam-5717	95	29	(	(	PUNCT
ejpam-5717	95	30	τ1	τ1	NOUN
ejpam-5717	95	31	,	,	PUNCT
ejpam-5717	95	32	τ2)s	τ2)s	NOUN
ejpam-5717	95	33	-	-	PUNCT
ejpam-5717	95	34	open	open	ADJ
ejpam-5717	95	35	set	set	NOUN
ejpam-5717	95	36	u	u	NOUN
ejpam-5717	95	37	containing	contain	VERB
ejpam-5717	95	38	x.	x.	NOUN
ejpam-5717	95	39	the	the	DET
ejpam-5717	95	40	set	set	NOUN
ejpam-5717	95	41	of	of	ADP
ejpam-5717	95	42	all	all	DET
ejpam-5717	95	43	θ(τ1	θ(τ1	NOUN
ejpam-5717	95	44	,	,	PUNCT
ejpam-5717	95	45	τ2)s	τ2)s	ADJ
ejpam-5717	95	46	-	-	PUNCT
ejpam-5717	95	47	cluster	cluster	NOUN
ejpam-5717	95	48	points	point	NOUN
ejpam-5717	95	49	of	of	ADP
ejpam-5717	95	50	a	a	PRON
ejpam-5717	95	51	is	be	AUX
ejpam-5717	95	52	called	call	VERB
ejpam-5717	95	53	the	the	DET
ejpam-5717	95	54	θ(τ1	θ(τ1	NOUN
ejpam-5717	95	55	,	,	PUNCT
ejpam-5717	95	56	τ2)s	τ2)s	NOUN
ejpam-5717	95	57	-	-	PUNCT
ejpam-5717	95	58	closure	closure	NOUN
ejpam-5717	95	59	of	of	ADP
ejpam-5717	95	60	a	a	PRON
ejpam-5717	95	61	and	and	CCONJ
ejpam-5717	95	62	is	be	AUX
ejpam-5717	95	63	denoted	denote	VERB
ejpam-5717	95	64	by	by	ADP
ejpam-5717	95	65	θ(τ1	θ(τ1	NOUN
ejpam-5717	95	66	,	,	PUNCT
ejpam-5717	95	67	τ2)-scl(a	τ2)-scl(a	PROPN
ejpam-5717	95	68	)	)	PUNCT
ejpam-5717	95	69	.	.	PUNCT
ejpam-5717	96	1	a	a	DET
ejpam-5717	96	2	subset	subset	NOUN
ejpam-5717	96	3	a	a	PRON
ejpam-5717	96	4	of	of	ADP
ejpam-5717	96	5	a	a	DET
ejpam-5717	96	6	bitopological	bitopological	ADJ
ejpam-5717	96	7	space	space	NOUN
ejpam-5717	96	8	(	(	PUNCT
ejpam-5717	96	9	x	x	NOUN
ejpam-5717	96	10	,	,	PUNCT
ejpam-5717	96	11	τ1	τ1	NOUN
ejpam-5717	96	12	,	,	PUNCT
ejpam-5717	96	13	τ2	τ2	NOUN
ejpam-5717	96	14	)	)	PUNCT
ejpam-5717	96	15	is	be	AUX
ejpam-5717	96	16	said	say	VERB
ejpam-5717	96	17	to	to	PART
ejpam-5717	96	18	be	be	AUX
ejpam-5717	96	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	96	20	,	,	PUNCT
ejpam-5717	96	21	τ2)s	τ2)s	NOUN
ejpam-5717	96	22	-	-	PUNCT
ejpam-5717	96	23	closed	close	VERB
ejpam-5717	96	24	if	if	SCONJ
ejpam-5717	96	25	θ(τ1	θ(τ1	NOUN
ejpam-5717	96	26	,	,	PUNCT
ejpam-5717	96	27	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-5717	96	28	)	)	PUNCT
ejpam-5717	96	29	=	=	VERB
ejpam-5717	97	1	a.	a.	NOUN
ejpam-5717	97	2	the	the	DET
ejpam-5717	97	3	complement	complement	NOUN
ejpam-5717	97	4	of	of	ADP
ejpam-5717	97	5	a	a	DET
ejpam-5717	97	6	θ(τ1	θ(τ1	NOUN
ejpam-5717	97	7	,	,	PUNCT
ejpam-5717	97	8	τ2)s	τ2)s	NOUN
ejpam-5717	97	9	-	-	PUNCT
ejpam-5717	97	10	closed	close	VERB
ejpam-5717	97	11	set	set	NOUN
ejpam-5717	97	12	is	be	AUX
ejpam-5717	97	13	said	say	VERB
ejpam-5717	97	14	to	to	PART
ejpam-5717	97	15	be	be	AUX
ejpam-5717	97	16	θ(τ1	θ(τ1	NOUN
ejpam-5717	97	17	,	,	PUNCT
ejpam-5717	97	18	τ2)s	τ2)s	NOUN
ejpam-5717	97	19	-	-	PUNCT
ejpam-5717	97	20	open	open	ADJ
ejpam-5717	97	21	.	.	PUNCT
ejpam-5717	98	1	the	the	DET
ejpam-5717	98	2	union	union	NOUN
ejpam-5717	98	3	of	of	ADP
ejpam-5717	98	4	all	all	DET
ejpam-5717	98	5	θ(τ1	θ(τ1	NOUN
ejpam-5717	98	6	,	,	PUNCT
ejpam-5717	98	7	τ2)s	τ2)s	NOUN
ejpam-5717	98	8	-	-	PUNCT
ejpam-5717	98	9	open	open	ADJ
ejpam-5717	98	10	sets	set	NOUN
ejpam-5717	98	11	of	of	ADP
ejpam-5717	98	12	x	x	PUNCT
ejpam-5717	98	13	contained	contain	VERB
ejpam-5717	98	14	in	in	ADP
ejpam-5717	98	15	a	a	PRON
ejpam-5717	98	16	is	be	AUX
ejpam-5717	98	17	called	call	VERB
ejpam-5717	98	18	the	the	DET
ejpam-5717	98	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	98	20	,	,	PUNCT
ejpam-5717	98	21	τ2)s	τ2)s	NOUN
ejpam-5717	98	22	-	-	NOUN
ejpam-5717	98	23	interior	interior	NOUN
ejpam-5717	98	24	of	of	ADP
ejpam-5717	98	25	a	a	PRON
ejpam-5717	98	26	and	and	CCONJ
ejpam-5717	98	27	is	be	AUX
ejpam-5717	98	28	denoted	denote	VERB
ejpam-5717	98	29	by	by	ADP
ejpam-5717	98	30	θ(τ1	θ(τ1	NOUN
ejpam-5717	98	31	,	,	PUNCT
ejpam-5717	98	32	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-5717	98	33	)	)	PUNCT
ejpam-5717	98	34	.	.	PUNCT
ejpam-5717	99	1	by	by	ADP
ejpam-5717	99	2	a	a	DET
ejpam-5717	99	3	multifunction	multifunction	NOUN
ejpam-5717	99	4	f	f	NOUN
ejpam-5717	99	5	:	:	PUNCT
ejpam-5717	99	6	x	x	X
ejpam-5717	99	7	→	→	SYM
ejpam-5717	99	8	y	y	PROPN
ejpam-5717	99	9	,	,	PUNCT
ejpam-5717	99	10	we	we	PRON
ejpam-5717	99	11	mean	mean	VERB
ejpam-5717	99	12	a	a	DET
ejpam-5717	99	13	point	point	NOUN
ejpam-5717	99	14	-	-	PUNCT
ejpam-5717	99	15	to	to	ADP
ejpam-5717	99	16	-	-	PUNCT
ejpam-5717	99	17	set	set	VERB
ejpam-5717	99	18	correspondence	correspondence	NOUN
ejpam-5717	99	19	from	from	ADP
ejpam-5717	99	20	x	x	PUNCT
ejpam-5717	99	21	into	into	ADP
ejpam-5717	99	22	y	y	PROPN
ejpam-5717	99	23	,	,	PUNCT
ejpam-5717	99	24	and	and	CCONJ
ejpam-5717	99	25	always	always	ADV
ejpam-5717	99	26	assume	assume	VERB
ejpam-5717	99	27	that	that	SCONJ
ejpam-5717	100	1	f	f	PROPN
ejpam-5717	100	2	(	(	PUNCT
ejpam-5717	100	3	x	x	X
ejpam-5717	100	4	)	)	PUNCT
ejpam-5717	100	5	̸=	̸=	NOUN
ejpam-5717	100	6	∅	∅	NOUN
ejpam-5717	100	7	for	for	ADP
ejpam-5717	100	8	all	all	PRON
ejpam-5717	100	9	x	x	SYM
ejpam-5717	100	10	∈	∈	ADJ
ejpam-5717	100	11	x.	x.	NOUN
ejpam-5717	100	12	for	for	ADP
ejpam-5717	100	13	a	a	DET
ejpam-5717	100	14	multifunction	multifunction	NOUN
ejpam-5717	100	15	f	f	NOUN
ejpam-5717	100	16	:	:	PUNCT
ejpam-5717	100	17	x	x	X
ejpam-5717	100	18	→	→	SYM
ejpam-5717	100	19	y	y	PROPN
ejpam-5717	100	20	,	,	PUNCT
ejpam-5717	100	21	we	we	PRON
ejpam-5717	100	22	shall	shall	AUX
ejpam-5717	100	23	denote	denote	VERB
ejpam-5717	100	24	the	the	DET
ejpam-5717	100	25	upper	upper	ADJ
ejpam-5717	100	26	and	and	CCONJ
ejpam-5717	100	27	lower	low	ADJ
ejpam-5717	100	28	inverse	inverse	NOUN
ejpam-5717	100	29	of	of	ADP
ejpam-5717	100	30	a	a	DET
ejpam-5717	100	31	set	set	NOUN
ejpam-5717	100	32	b	b	PROPN
ejpam-5717	100	33	of	of	ADP
ejpam-5717	100	34	y	y	PROPN
ejpam-5717	100	35	by	by	ADP
ejpam-5717	100	36	f+(b	f+(b	NOUN
ejpam-5717	100	37	)	)	PUNCT
ejpam-5717	100	38	and	and	CCONJ
ejpam-5717	100	39	f−(b	f−(b	NOUN
ejpam-5717	100	40	)	)	PUNCT
ejpam-5717	100	41	,	,	PUNCT
ejpam-5717	100	42	respectively	respectively	ADV
ejpam-5717	100	43	,	,	PUNCT
ejpam-5717	100	44	that	that	ADV
ejpam-5717	100	45	is	is	ADV
ejpam-5717	100	46	,	,	PUNCT
ejpam-5717	100	47	f+(b	f+(b	NOUN
ejpam-5717	100	48	)	)	PUNCT
ejpam-5717	100	49	=	=	PRON
ejpam-5717	101	1	{	{	PUNCT
ejpam-5717	101	2	x	x	PUNCT
ejpam-5717	101	3	∈	∈	PROPN
ejpam-5717	101	4	x	x	INTJ
ejpam-5717	102	1	|	|	NOUN
ejpam-5717	102	2	f	f	X
ejpam-5717	102	3	(	(	PUNCT
ejpam-5717	102	4	x	x	NOUN
ejpam-5717	102	5	)	)	PUNCT
ejpam-5717	102	6	⊆	⊆	NUM
ejpam-5717	102	7	b	b	NOUN
ejpam-5717	102	8	}	}	PUNCT
ejpam-5717	102	9	and	and	CCONJ
ejpam-5717	102	10	f−(b	f−(b	PROPN
ejpam-5717	102	11	)	)	PUNCT
ejpam-5717	102	12	=	=	PRON
ejpam-5717	103	1	{	{	PUNCT
ejpam-5717	103	2	x	x	PUNCT
ejpam-5717	103	3	∈	∈	PROPN
ejpam-5717	103	4	x	x	INTJ
ejpam-5717	104	1	|	|	NOUN
ejpam-5717	104	2	f	f	X
ejpam-5717	104	3	(	(	PUNCT
ejpam-5717	104	4	x	x	NOUN
ejpam-5717	104	5	)	)	PUNCT
ejpam-5717	104	6	∩b	∩b	NOUN
ejpam-5717	104	7	̸=	̸=	PROPN
ejpam-5717	104	8	∅	∅	NOUN
ejpam-5717	104	9	}	}	PUNCT
ejpam-5717	104	10	.	.	PUNCT
ejpam-5717	105	1	p.	p.	NOUN
ejpam-5717	105	2	pue	pue	NOUN
ejpam-5717	105	3	-	-	PUNCT
ejpam-5717	105	4	on	on	ADP
ejpam-5717	105	5	,	,	PUNCT
ejpam-5717	105	6	a.	a.	PROPN
ejpam-5717	105	7	sama	sama	PROPN
ejpam-5717	105	8	-	-	PUNCT
ejpam-5717	105	9	ae	ae	PROPN
ejpam-5717	105	10	,	,	PUNCT
ejpam-5717	105	11	c.	c.	PROPN
ejpam-5717	105	12	boonpok	boonpok	PROPN
ejpam-5717	105	13	/	/	SYM
ejpam-5717	105	14	eur	eur	PROPN
ejpam-5717	105	15	.	.	PUNCT
ejpam-5717	106	1	j.	j.	PROPN
ejpam-5717	106	2	pure	pure	PROPN
ejpam-5717	106	3	appl	appl	PROPN
ejpam-5717	106	4	.	.	PROPN
ejpam-5717	106	5	math	math	PROPN
ejpam-5717	106	6	,	,	PUNCT
ejpam-5717	106	7	18	18	NUM
ejpam-5717	106	8	(	(	PUNCT
ejpam-5717	106	9	1	1	NUM
ejpam-5717	106	10	)	)	PUNCT
ejpam-5717	106	11	(	(	PUNCT
ejpam-5717	106	12	2025	2025	NUM
ejpam-5717	106	13	)	)	PUNCT
ejpam-5717	106	14	,	,	PUNCT
ejpam-5717	106	15	5717	5717	NUM
ejpam-5717	106	16	4	4	NUM
ejpam-5717	106	17	of	of	ADP
ejpam-5717	106	18	16	16	NUM
ejpam-5717	106	19	3	3	NUM
ejpam-5717	106	20	.	.	PUNCT
ejpam-5717	106	21	upper	upper	ADJ
ejpam-5717	106	22	and	and	CCONJ
ejpam-5717	106	23	lower	low	ADJ
ejpam-5717	106	24	quasi	quasi	NOUN
ejpam-5717	106	25	θ(τ1	θ(τ1	NOUN
ejpam-5717	106	26	,	,	PUNCT
ejpam-5717	106	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	106	28	multifunctions	multifunction	NOUN
ejpam-5717	106	29	in	in	ADP
ejpam-5717	106	30	this	this	DET
ejpam-5717	106	31	section	section	NOUN
ejpam-5717	106	32	,	,	PUNCT
ejpam-5717	106	33	we	we	PRON
ejpam-5717	106	34	introduce	introduce	VERB
ejpam-5717	106	35	the	the	DET
ejpam-5717	106	36	notions	notion	NOUN
ejpam-5717	106	37	of	of	ADP
ejpam-5717	106	38	upper	upper	ADJ
ejpam-5717	106	39	quasi	quasi	NOUN
ejpam-5717	106	40	θ(τ1	θ(τ1	NOUN
ejpam-5717	106	41	,	,	PUNCT
ejpam-5717	106	42	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	106	43	multifunctions	multifunction	NOUN
ejpam-5717	106	44	and	and	CCONJ
ejpam-5717	106	45	lower	low	ADJ
ejpam-5717	106	46	quasi	quasi	NOUN
ejpam-5717	106	47	θ(τ1	θ(τ1	NOUN
ejpam-5717	106	48	,	,	PUNCT
ejpam-5717	106	49	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	106	50	multifunctions	multifunction	NOUN
ejpam-5717	106	51	.	.	PUNCT
ejpam-5717	107	1	moreover	moreover	ADV
ejpam-5717	107	2	,	,	PUNCT
ejpam-5717	107	3	several	several	ADJ
ejpam-5717	107	4	characterizations	characterization	NOUN
ejpam-5717	107	5	of	of	ADP
ejpam-5717	107	6	upper	upper	ADJ
ejpam-5717	107	7	quasi	quasi	NOUN
ejpam-5717	107	8	θ(τ1	θ(τ1	NOUN
ejpam-5717	107	9	,	,	PUNCT
ejpam-5717	107	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	107	11	multifunctions	multifunction	NOUN
ejpam-5717	107	12	and	and	CCONJ
ejpam-5717	107	13	lower	low	ADJ
ejpam-5717	107	14	quasi	quasi	NOUN
ejpam-5717	107	15	θ(τ1	θ(τ1	NOUN
ejpam-5717	107	16	,	,	PUNCT
ejpam-5717	107	17	τ2)continuous	τ2)continuous	ADJ
ejpam-5717	107	18	multifunctions	multifunction	NOUN
ejpam-5717	107	19	are	be	AUX
ejpam-5717	107	20	discussed	discuss	VERB
ejpam-5717	107	21	.	.	PUNCT
ejpam-5717	108	1	definition	definition	NOUN
ejpam-5717	108	2	1	1	NUM
ejpam-5717	108	3	.	.	PUNCT
ejpam-5717	109	1	a	a	DET
ejpam-5717	109	2	multifunction	multifunction	NOUN
ejpam-5717	109	3	f	f	NOUN
ejpam-5717	109	4	:	:	PUNCT
ejpam-5717	109	5	(	(	PUNCT
ejpam-5717	109	6	x	x	NOUN
ejpam-5717	109	7	,	,	PUNCT
ejpam-5717	109	8	τ1	τ1	NOUN
ejpam-5717	109	9	,	,	PUNCT
ejpam-5717	109	10	τ2	τ2	NOUN
ejpam-5717	109	11	)	)	PUNCT
ejpam-5717	109	12	→	→	SYM
ejpam-5717	109	13	(	(	PUNCT
ejpam-5717	109	14	y	y	PROPN
ejpam-5717	109	15	,	,	PUNCT
ejpam-5717	109	16	σ1	σ1	PROPN
ejpam-5717	109	17	,	,	PUNCT
ejpam-5717	109	18	σ2	σ2	PROPN
ejpam-5717	109	19	)	)	PUNCT
ejpam-5717	109	20	is	be	AUX
ejpam-5717	109	21	said	say	VERB
ejpam-5717	109	22	to	to	PART
ejpam-5717	109	23	be	be	AUX
ejpam-5717	109	24	upper	upper	ADJ
ejpam-5717	109	25	quasi	quasi	NOUN
ejpam-5717	109	26	θ(τ1	θ(τ1	NOUN
ejpam-5717	109	27	,	,	PUNCT
ejpam-5717	109	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	109	29	if	if	SCONJ
ejpam-5717	109	30	for	for	ADP
ejpam-5717	109	31	each	each	DET
ejpam-5717	109	32	x	x	SYM
ejpam-5717	109	33	∈	∈	PROPN
ejpam-5717	109	34	x	x	X
ejpam-5717	109	35	and	and	CCONJ
ejpam-5717	109	36	each	each	DET
ejpam-5717	109	37	σ1σ2	σ1σ2	VERB
ejpam-5717	109	38	-	-	ADJ
ejpam-5717	109	39	open	open	ADJ
ejpam-5717	109	40	set	set	NOUN
ejpam-5717	109	41	v	v	NOUN
ejpam-5717	109	42	of	of	ADP
ejpam-5717	109	43	y	y	PROPN
ejpam-5717	109	44	containing	contain	VERB
ejpam-5717	109	45	f	f	PROPN
ejpam-5717	109	46	(	(	PUNCT
ejpam-5717	109	47	x	x	NOUN
ejpam-5717	109	48	)	)	PUNCT
ejpam-5717	109	49	,	,	PUNCT
ejpam-5717	109	50	there	there	PRON
ejpam-5717	109	51	exists	exist	VERB
ejpam-5717	109	52	a	a	DET
ejpam-5717	109	53	(	(	PUNCT
ejpam-5717	109	54	τ1	τ1	NOUN
ejpam-5717	109	55	,	,	PUNCT
ejpam-5717	109	56	τ2)s	τ2)s	NOUN
ejpam-5717	109	57	-	-	PUNCT
ejpam-5717	109	58	open	open	ADJ
ejpam-5717	109	59	set	set	NOUN
ejpam-5717	109	60	u	u	NOUN
ejpam-5717	109	61	of	of	ADP
ejpam-5717	109	62	x	x	PUNCT
ejpam-5717	109	63	containing	contain	VERB
ejpam-5717	109	64	x	x	PUNCT
ejpam-5717	109	65	such	such	ADJ
ejpam-5717	109	66	that	that	SCONJ
ejpam-5717	109	67	f	f	PROPN
ejpam-5717	109	68	(	(	PUNCT
ejpam-5717	109	69	(	(	PUNCT
ejpam-5717	109	70	τ1	τ1	NOUN
ejpam-5717	109	71	,	,	PUNCT
ejpam-5717	109	72	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	109	73	)	)	PUNCT
ejpam-5717	109	74	)	)	PUNCT
ejpam-5717	110	1	⊆	⊆	X
ejpam-5717	110	2	σ1σ2	σ1σ2	NOUN
ejpam-5717	110	3	-	-	NUM
ejpam-5717	110	4	cl(v	cl(v	NOUN
ejpam-5717	110	5	)	)	PUNCT
ejpam-5717	110	6	.	.	PUNCT
ejpam-5717	111	1	theorem	theorem	NOUN
ejpam-5717	111	2	1	1	NUM
ejpam-5717	111	3	.	.	X
ejpam-5717	111	4	for	for	ADP
ejpam-5717	111	5	a	a	DET
ejpam-5717	111	6	multifunction	multifunction	NOUN
ejpam-5717	112	1	f	f	NOUN
ejpam-5717	112	2	:	:	PUNCT
ejpam-5717	112	3	(	(	PUNCT
ejpam-5717	112	4	x	x	NOUN
ejpam-5717	112	5	,	,	PUNCT
ejpam-5717	112	6	τ1	τ1	NOUN
ejpam-5717	112	7	,	,	PUNCT
ejpam-5717	112	8	τ2	τ2	NOUN
ejpam-5717	112	9	)	)	PUNCT
ejpam-5717	112	10	→	→	SYM
ejpam-5717	112	11	(	(	PUNCT
ejpam-5717	112	12	y	y	PROPN
ejpam-5717	112	13	,	,	PUNCT
ejpam-5717	112	14	σ1	σ1	PROPN
ejpam-5717	112	15	,	,	PUNCT
ejpam-5717	112	16	σ2	σ2	NOUN
ejpam-5717	112	17	)	)	PUNCT
ejpam-5717	112	18	,	,	PUNCT
ejpam-5717	112	19	the	the	DET
ejpam-5717	112	20	following	follow	VERB
ejpam-5717	112	21	properties	property	NOUN
ejpam-5717	112	22	are	be	AUX
ejpam-5717	112	23	equivalent	equivalent	ADJ
ejpam-5717	112	24	:	:	PUNCT
ejpam-5717	112	25	(	(	PUNCT
ejpam-5717	112	26	1	1	X
ejpam-5717	112	27	)	)	PUNCT
ejpam-5717	112	28	f	f	PROPN
ejpam-5717	112	29	is	be	AUX
ejpam-5717	112	30	upper	upper	ADJ
ejpam-5717	112	31	quasi	quasi	NOUN
ejpam-5717	112	32	θ(τ1	θ(τ1	NOUN
ejpam-5717	112	33	,	,	PUNCT
ejpam-5717	112	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	112	35	;	;	PUNCT
ejpam-5717	112	36	(	(	PUNCT
ejpam-5717	112	37	2	2	X
ejpam-5717	112	38	)	)	PUNCT
ejpam-5717	112	39	θ(τ1	θ(τ1	NOUN
ejpam-5717	112	40	,	,	PUNCT
ejpam-5717	112	41	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	112	42	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	112	43	-	-	PUNCT
ejpam-5717	112	44	int((σ1	int((σ1	ADJ
ejpam-5717	112	45	,	,	PUNCT
ejpam-5717	112	46	σ2)θ	σ2)θ	ADJ
ejpam-5717	112	47	-	-	PUNCT
ejpam-5717	112	48	cl(b	cl(b	NOUN
ejpam-5717	112	49	)	)	PUNCT
ejpam-5717	112	50	)	)	PUNCT
ejpam-5717	112	51	)	)	PUNCT
ejpam-5717	112	52	)	)	PUNCT
ejpam-5717	113	1	⊆	⊆	NUM
ejpam-5717	113	2	f−((σ1	f−((σ1	NOUN
ejpam-5717	113	3	,	,	PUNCT
ejpam-5717	113	4	σ2)θ	σ2)θ	ADJ
ejpam-5717	113	5	-	-	PUNCT
ejpam-5717	113	6	cl(b	cl(b	NOUN
ejpam-5717	113	7	)	)	PUNCT
ejpam-5717	113	8	)	)	PUNCT
ejpam-5717	113	9	for	for	ADP
ejpam-5717	113	10	every	every	DET
ejpam-5717	113	11	subset	subset	NOUN
ejpam-5717	113	12	b	b	PROPN
ejpam-5717	113	13	of	of	ADP
ejpam-5717	113	14	y	y	PROPN
ejpam-5717	113	15	;	;	PUNCT
ejpam-5717	113	16	(	(	PUNCT
ejpam-5717	113	17	3	3	X
ejpam-5717	113	18	)	)	PUNCT
ejpam-5717	113	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	113	20	,	,	PUNCT
ejpam-5717	113	21	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	113	22	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	113	23	-	-	PUNCT
ejpam-5717	113	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	113	25	-	-	PUNCT
ejpam-5717	113	26	cl(v	cl(v	NOUN
ejpam-5717	113	27	)	)	PUNCT
ejpam-5717	113	28	)	)	PUNCT
ejpam-5717	113	29	)	)	PUNCT
ejpam-5717	113	30	)	)	PUNCT
ejpam-5717	114	1	⊆	⊆	X
ejpam-5717	114	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	114	3	-	-	PUNCT
ejpam-5717	114	4	cl(v	cl(v	NOUN
ejpam-5717	114	5	)	)	PUNCT
ejpam-5717	114	6	)	)	PUNCT
ejpam-5717	114	7	for	for	ADP
ejpam-5717	114	8	every	every	DET
ejpam-5717	114	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	114	10	-	-	ADJ
ejpam-5717	114	11	open	open	ADJ
ejpam-5717	114	12	set	set	NOUN
ejpam-5717	114	13	v	v	NOUN
ejpam-5717	114	14	of	of	ADP
ejpam-5717	114	15	y	y	PROPN
ejpam-5717	114	16	;	;	PUNCT
ejpam-5717	114	17	(	(	PUNCT
ejpam-5717	114	18	4	4	X
ejpam-5717	114	19	)	)	PUNCT
ejpam-5717	114	20	θ(τ1	θ(τ1	NOUN
ejpam-5717	114	21	,	,	PUNCT
ejpam-5717	114	22	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	114	23	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	114	24	-	-	PUNCT
ejpam-5717	114	25	int(k	int(k	NUM
ejpam-5717	114	26	)	)	PUNCT
ejpam-5717	114	27	)	)	PUNCT
ejpam-5717	114	28	)	)	PUNCT
ejpam-5717	115	1	⊆	⊆	X
ejpam-5717	115	2	f−(k	f−(k	PROPN
ejpam-5717	115	3	)	)	PUNCT
ejpam-5717	115	4	for	for	ADP
ejpam-5717	115	5	every	every	DET
ejpam-5717	115	6	(	(	PUNCT
ejpam-5717	115	7	σ1	σ1	PROPN
ejpam-5717	115	8	,	,	PUNCT
ejpam-5717	115	9	σ2)r	σ2)r	NOUN
ejpam-5717	115	10	-	-	PUNCT
ejpam-5717	115	11	closed	close	VERB
ejpam-5717	115	12	set	set	ADJ
ejpam-5717	115	13	k	k	PROPN
ejpam-5717	115	14	of	of	ADP
ejpam-5717	115	15	y	y	PROPN
ejpam-5717	115	16	;	;	PUNCT
ejpam-5717	115	17	(	(	PUNCT
ejpam-5717	115	18	5	5	NUM
ejpam-5717	115	19	)	)	PUNCT
ejpam-5717	115	20	f+(v	f+(v	NOUN
ejpam-5717	115	21	)	)	PUNCT
ejpam-5717	116	1	⊆	⊆	NUM
ejpam-5717	116	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	116	3	,	,	PUNCT
ejpam-5717	116	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	117	1	+	+	ADJ
ejpam-5717	117	2	(	(	PUNCT
ejpam-5717	117	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	117	4	-	-	NUM
ejpam-5717	117	5	cl(v	cl(v	NOUN
ejpam-5717	117	6	)	)	PUNCT
ejpam-5717	117	7	)	)	PUNCT
ejpam-5717	117	8	)	)	PUNCT
ejpam-5717	117	9	for	for	ADP
ejpam-5717	117	10	every	every	DET
ejpam-5717	117	11	σ1σ2	σ1σ2	NOUN
ejpam-5717	117	12	-	-	ADJ
ejpam-5717	117	13	open	open	ADJ
ejpam-5717	117	14	set	set	NOUN
ejpam-5717	117	15	v	v	NOUN
ejpam-5717	117	16	of	of	ADP
ejpam-5717	117	17	y	y	PROPN
ejpam-5717	117	18	;	;	PUNCT
ejpam-5717	117	19	(	(	PUNCT
ejpam-5717	117	20	6	6	NUM
ejpam-5717	117	21	)	)	PUNCT
ejpam-5717	117	22	θ(τ1	θ(τ1	NOUN
ejpam-5717	117	23	,	,	PUNCT
ejpam-5717	117	24	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	117	25	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	117	26	-	-	PUNCT
ejpam-5717	117	27	int(k	int(k	NUM
ejpam-5717	117	28	)	)	PUNCT
ejpam-5717	117	29	)	)	PUNCT
ejpam-5717	117	30	)	)	PUNCT
ejpam-5717	118	1	⊆	⊆	X
ejpam-5717	118	2	f−(k	f−(k	PROPN
ejpam-5717	118	3	)	)	PUNCT
ejpam-5717	118	4	for	for	ADP
ejpam-5717	118	5	every	every	DET
ejpam-5717	118	6	σ1σ2	σ1σ2	NUM
ejpam-5717	118	7	-	-	PUNCT
ejpam-5717	118	8	closed	closed	ADJ
ejpam-5717	118	9	set	set	NOUN
ejpam-5717	118	10	k	k	PROPN
ejpam-5717	118	11	of	of	ADP
ejpam-5717	118	12	y	y	PROPN
ejpam-5717	118	13	;	;	PUNCT
ejpam-5717	118	14	(	(	PUNCT
ejpam-5717	118	15	7	7	X
ejpam-5717	118	16	)	)	PUNCT
ejpam-5717	118	17	θ(τ1	θ(τ1	NOUN
ejpam-5717	118	18	,	,	PUNCT
ejpam-5717	118	19	τ2)-scl(f	τ2)-scl(f	ADV
ejpam-5717	118	20	−(v	−(v	NOUN
ejpam-5717	118	21	)	)	PUNCT
ejpam-5717	118	22	)	)	PUNCT
ejpam-5717	119	1	⊆	⊆	X
ejpam-5717	119	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	119	3	-	-	PUNCT
ejpam-5717	119	4	cl(v	cl(v	NOUN
ejpam-5717	119	5	)	)	PUNCT
ejpam-5717	119	6	)	)	PUNCT
ejpam-5717	119	7	for	for	ADP
ejpam-5717	119	8	every	every	DET
ejpam-5717	119	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	119	10	-	-	ADJ
ejpam-5717	119	11	open	open	ADJ
ejpam-5717	119	12	set	set	NOUN
ejpam-5717	119	13	v	v	NOUN
ejpam-5717	119	14	of	of	ADP
ejpam-5717	119	15	y	y	PROPN
ejpam-5717	119	16	.	.	PUNCT
ejpam-5717	120	1	proof	proof	NOUN
ejpam-5717	120	2	.	.	PUNCT
ejpam-5717	121	1	(	(	PUNCT
ejpam-5717	121	2	1	1	X
ejpam-5717	121	3	)	)	PUNCT
ejpam-5717	121	4	⇒	⇒	NOUN
ejpam-5717	121	5	(	(	PUNCT
ejpam-5717	121	6	2	2	NUM
ejpam-5717	121	7	):	):	PUNCT
ejpam-5717	121	8	let	let	VERB
ejpam-5717	121	9	b	b	X
ejpam-5717	121	10	be	be	AUX
ejpam-5717	121	11	any	any	DET
ejpam-5717	121	12	subset	subset	NOUN
ejpam-5717	121	13	of	of	ADP
ejpam-5717	121	14	y	y	PROPN
ejpam-5717	121	15	.	.	PUNCT
ejpam-5717	121	16	suppose	suppose	VERB
ejpam-5717	121	17	that	that	SCONJ
ejpam-5717	121	18	x	x	PRON
ejpam-5717	121	19	̸∈	̸∈	PROPN
ejpam-5717	121	20	f−((σ1	f−((σ1	VERB
ejpam-5717	121	21	,	,	PUNCT
ejpam-5717	121	22	σ2)θ	σ2)θ	ADJ
ejpam-5717	121	23	-	-	PUNCT
ejpam-5717	121	24	cl(b	cl(b	NOUN
ejpam-5717	121	25	)	)	PUNCT
ejpam-5717	121	26	)	)	PUNCT
ejpam-5717	121	27	.	.	PUNCT
ejpam-5717	122	1	then	then	ADV
ejpam-5717	122	2	,	,	PUNCT
ejpam-5717	122	3	x	x	X
ejpam-5717	122	4	∈	∈	NOUN
ejpam-5717	122	5	x−f−((σ1	x−f−((σ1	PUNCT
ejpam-5717	122	6	,	,	PUNCT
ejpam-5717	122	7	σ2)θ	σ2)θ	ADJ
ejpam-5717	122	8	-	-	PUNCT
ejpam-5717	122	9	cl(b	cl(b	NOUN
ejpam-5717	122	10	)	)	PUNCT
ejpam-5717	122	11	)	)	PUNCT
ejpam-5717	122	12	and	and	CCONJ
ejpam-5717	122	13	f	f	PROPN
ejpam-5717	122	14	(	(	PUNCT
ejpam-5717	122	15	x	x	X
ejpam-5717	122	16	)	)	PUNCT
ejpam-5717	122	17	⊆	⊆	NUM
ejpam-5717	122	18	y	y	PROPN
ejpam-5717	122	19	−(σ1	−(σ1	ADJ
ejpam-5717	122	20	,	,	PUNCT
ejpam-5717	122	21	σ2)θ	σ2)θ	ADJ
ejpam-5717	122	22	-	-	PUNCT
ejpam-5717	122	23	cl(b	cl(b	NOUN
ejpam-5717	122	24	)	)	PUNCT
ejpam-5717	122	25	.	.	PUNCT
ejpam-5717	123	1	since	since	SCONJ
ejpam-5717	123	2	(	(	PUNCT
ejpam-5717	123	3	σ1	σ1	PROPN
ejpam-5717	123	4	,	,	PUNCT
ejpam-5717	123	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	123	6	-	-	PUNCT
ejpam-5717	123	7	cl(b	cl(b	NOUN
ejpam-5717	123	8	)	)	PUNCT
ejpam-5717	123	9	is	be	AUX
ejpam-5717	123	10	σ1σ2	σ1σ2	NOUN
ejpam-5717	123	11	-	-	ADJ
ejpam-5717	123	12	closed	closed	ADJ
ejpam-5717	123	13	in	in	ADP
ejpam-5717	123	14	y	y	PROPN
ejpam-5717	123	15	,	,	PUNCT
ejpam-5717	123	16	there	there	PRON
ejpam-5717	123	17	exists	exist	VERB
ejpam-5717	123	18	a	a	DET
ejpam-5717	123	19	(	(	PUNCT
ejpam-5717	123	20	τ1	τ1	NOUN
ejpam-5717	123	21	,	,	PUNCT
ejpam-5717	123	22	τ2)s	τ2)s	NOUN
ejpam-5717	123	23	-	-	PUNCT
ejpam-5717	123	24	open	open	ADJ
ejpam-5717	123	25	set	set	NOUN
ejpam-5717	123	26	u	u	NOUN
ejpam-5717	123	27	of	of	ADP
ejpam-5717	123	28	x	x	PUNCT
ejpam-5717	123	29	containing	contain	VERB
ejpam-5717	123	30	x	x	PUNCT
ejpam-5717	123	31	such	such	ADJ
ejpam-5717	123	32	that	that	SCONJ
ejpam-5717	123	33	f	f	PROPN
ejpam-5717	123	34	(	(	PUNCT
ejpam-5717	123	35	(	(	PUNCT
ejpam-5717	123	36	τ1	τ1	NOUN
ejpam-5717	123	37	,	,	PUNCT
ejpam-5717	123	38	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	123	39	)	)	PUNCT
ejpam-5717	123	40	)	)	PUNCT
ejpam-5717	124	1	⊆	⊆	X
ejpam-5717	124	2	σ1σ2	σ1σ2	NUM
ejpam-5717	124	3	-	-	PUNCT
ejpam-5717	124	4	cl(y	cl(y	NOUN
ejpam-5717	124	5	−(σ1	−(σ1	SYM
ejpam-5717	124	6	,	,	PUNCT
ejpam-5717	124	7	σ2)θ	σ2)θ	ADJ
ejpam-5717	124	8	-	-	PUNCT
ejpam-5717	124	9	cl(b	cl(b	NOUN
ejpam-5717	124	10	)	)	PUNCT
ejpam-5717	124	11	)	)	PUNCT
ejpam-5717	125	1	=	=	PUNCT
ejpam-5717	125	2	y	y	PROPN
ejpam-5717	125	3	−σ1σ2	−σ1σ2	PROPN
ejpam-5717	125	4	-	-	PUNCT
ejpam-5717	125	5	int((σ1	int((σ1	PROPN
ejpam-5717	125	6	,	,	PUNCT
ejpam-5717	125	7	σ2)θ	σ2)θ	ADJ
ejpam-5717	125	8	-	-	PUNCT
ejpam-5717	125	9	cl(b	cl(b	NOUN
ejpam-5717	125	10	)	)	PUNCT
ejpam-5717	125	11	)	)	PUNCT
ejpam-5717	125	12	.	.	PUNCT
ejpam-5717	126	1	thus	thus	ADV
ejpam-5717	126	2	,	,	PUNCT
ejpam-5717	126	3	we	we	PRON
ejpam-5717	126	4	have	have	VERB
ejpam-5717	126	5	f	f	PROPN
ejpam-5717	126	6	(	(	PUNCT
ejpam-5717	126	7	(	(	PUNCT
ejpam-5717	126	8	τ1	τ1	NOUN
ejpam-5717	126	9	,	,	PUNCT
ejpam-5717	126	10	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	126	11	)	)	PUNCT
ejpam-5717	126	12	)	)	PUNCT
ejpam-5717	126	13	∩	∩	NOUN
ejpam-5717	126	14	σ1σ2	σ1σ2	NOUN
ejpam-5717	126	15	-	-	PUNCT
ejpam-5717	126	16	int((σ1	int((σ1	ADJ
ejpam-5717	126	17	,	,	PUNCT
ejpam-5717	126	18	σ2)θ	σ2)θ	ADJ
ejpam-5717	126	19	-	-	PUNCT
ejpam-5717	126	20	cl(b	cl(b	NOUN
ejpam-5717	126	21	)	)	PUNCT
ejpam-5717	126	22	)	)	PUNCT
ejpam-5717	127	1	=	=	PUNCT
ejpam-5717	127	2	∅	∅	NOUN
ejpam-5717	127	3	and	and	CCONJ
ejpam-5717	127	4	(	(	PUNCT
ejpam-5717	127	5	τ1	τ1	NOUN
ejpam-5717	127	6	,	,	PUNCT
ejpam-5717	127	7	τ2)-scl(u	τ2)-scl(u	ADJ
ejpam-5717	127	8	)	)	PUNCT
ejpam-5717	127	9	∩	∩	ADJ
ejpam-5717	127	10	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	127	11	-	-	PUNCT
ejpam-5717	127	12	int((σ1	int((σ1	PROPN
ejpam-5717	127	13	,	,	PUNCT
ejpam-5717	127	14	σ2)θ	σ2)θ	ADJ
ejpam-5717	127	15	-	-	PUNCT
ejpam-5717	127	16	cl(b	cl(b	NOUN
ejpam-5717	127	17	)	)	PUNCT
ejpam-5717	127	18	)	)	PUNCT
ejpam-5717	127	19	)	)	PUNCT
ejpam-5717	128	1	=	=	PUNCT
ejpam-5717	128	2	∅.	∅.	ADP
ejpam-5717	128	3	this	this	PRON
ejpam-5717	128	4	shows	show	VERB
ejpam-5717	128	5	that	that	SCONJ
ejpam-5717	128	6	x	x	PROPN
ejpam-5717	128	7	̸∈	̸∈	PROPN
ejpam-5717	128	8	θ(τ1	θ(τ1	NOUN
ejpam-5717	128	9	,	,	PUNCT
ejpam-5717	128	10	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	128	11	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	128	12	-	-	PUNCT
ejpam-5717	128	13	int((σ1	int((σ1	ADJ
ejpam-5717	128	14	,	,	PUNCT
ejpam-5717	128	15	σ2)θ	σ2)θ	ADJ
ejpam-5717	128	16	-	-	PUNCT
ejpam-5717	128	17	cl(b	cl(b	NOUN
ejpam-5717	128	18	)	)	PUNCT
ejpam-5717	128	19	)	)	PUNCT
ejpam-5717	128	20	)	)	PUNCT
ejpam-5717	128	21	)	)	PUNCT
ejpam-5717	128	22	.	.	PUNCT
ejpam-5717	129	1	thus	thus	ADV
ejpam-5717	129	2	,	,	PUNCT
ejpam-5717	129	3	θ(τ1	θ(τ1	NOUN
ejpam-5717	129	4	,	,	PUNCT
ejpam-5717	129	5	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	129	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	129	7	-	-	PUNCT
ejpam-5717	129	8	int((σ1	int((σ1	ADJ
ejpam-5717	129	9	,	,	PUNCT
ejpam-5717	129	10	σ2)θ	σ2)θ	ADJ
ejpam-5717	129	11	-	-	PUNCT
ejpam-5717	129	12	cl(b	cl(b	NOUN
ejpam-5717	129	13	)	)	PUNCT
ejpam-5717	129	14	)	)	PUNCT
ejpam-5717	129	15	)	)	PUNCT
ejpam-5717	129	16	)	)	PUNCT
ejpam-5717	130	1	⊆	⊆	NUM
ejpam-5717	130	2	f−((σ1	f−((σ1	NOUN
ejpam-5717	130	3	,	,	PUNCT
ejpam-5717	130	4	σ2)θ	σ2)θ	ADJ
ejpam-5717	130	5	-	-	PUNCT
ejpam-5717	130	6	cl(b	cl(b	NOUN
ejpam-5717	130	7	)	)	PUNCT
ejpam-5717	130	8	)	)	PUNCT
ejpam-5717	130	9	.	.	PUNCT
ejpam-5717	131	1	(	(	PUNCT
ejpam-5717	131	2	2	2	X
ejpam-5717	131	3	)	)	PUNCT
ejpam-5717	131	4	⇒	⇒	NOUN
ejpam-5717	131	5	(	(	PUNCT
ejpam-5717	131	6	3	3	NUM
ejpam-5717	131	7	):	):	PUNCT
ejpam-5717	131	8	this	this	PRON
ejpam-5717	131	9	is	be	AUX
ejpam-5717	131	10	obvious	obvious	ADJ
ejpam-5717	131	11	since	since	SCONJ
ejpam-5717	131	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	131	13	-	-	NOUN
ejpam-5717	131	14	cl(v	cl(v	X
ejpam-5717	131	15	)	)	PUNCT
ejpam-5717	132	1	=	=	SYM
ejpam-5717	132	2	(	(	PUNCT
ejpam-5717	132	3	σ1	σ1	PROPN
ejpam-5717	132	4	,	,	PUNCT
ejpam-5717	132	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	132	6	-	-	PUNCT
ejpam-5717	132	7	cl(v	cl(v	NOUN
ejpam-5717	132	8	)	)	PUNCT
ejpam-5717	132	9	for	for	ADP
ejpam-5717	132	10	every	every	DET
ejpam-5717	132	11	σ1σ2	σ1σ2	NOUN
ejpam-5717	132	12	-	-	ADJ
ejpam-5717	132	13	open	open	ADJ
ejpam-5717	132	14	set	set	NOUN
ejpam-5717	132	15	v	v	NOUN
ejpam-5717	132	16	of	of	ADP
ejpam-5717	132	17	y	y	PROPN
ejpam-5717	132	18	.	.	PUNCT
ejpam-5717	133	1	(	(	PUNCT
ejpam-5717	133	2	3	3	X
ejpam-5717	133	3	)	)	PUNCT
ejpam-5717	133	4	⇒	⇒	NOUN
ejpam-5717	133	5	(	(	PUNCT
ejpam-5717	133	6	4	4	NUM
ejpam-5717	133	7	):	):	PUNCT
ejpam-5717	133	8	let	let	VERB
ejpam-5717	133	9	k	k	PRON
ejpam-5717	133	10	be	be	AUX
ejpam-5717	133	11	any	any	DET
ejpam-5717	133	12	(	(	PUNCT
ejpam-5717	133	13	σ1	σ1	NOUN
ejpam-5717	133	14	,	,	PUNCT
ejpam-5717	133	15	σ2)r	σ2)r	NOUN
ejpam-5717	133	16	-	-	PUNCT
ejpam-5717	133	17	closed	close	VERB
ejpam-5717	133	18	set	set	NOUN
ejpam-5717	133	19	of	of	ADP
ejpam-5717	133	20	y	y	PROPN
ejpam-5717	133	21	.	.	PUNCT
ejpam-5717	134	1	by	by	ADP
ejpam-5717	134	2	(	(	PUNCT
ejpam-5717	134	3	3	3	NUM
ejpam-5717	134	4	)	)	PUNCT
ejpam-5717	134	5	,	,	PUNCT
ejpam-5717	134	6	we	we	PRON
ejpam-5717	134	7	have	have	VERB
ejpam-5717	134	8	θ(τ1	θ(τ1	NOUN
ejpam-5717	134	9	,	,	PUNCT
ejpam-5717	135	1	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	135	2	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	135	3	-	-	PUNCT
ejpam-5717	135	4	int(k	int(k	NUM
ejpam-5717	135	5	)	)	PUNCT
ejpam-5717	135	6	)	)	PUNCT
ejpam-5717	135	7	)	)	PUNCT
ejpam-5717	136	1	=	=	SYM
ejpam-5717	136	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	136	3	,	,	PUNCT
ejpam-5717	136	4	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	136	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	136	6	-	-	PUNCT
ejpam-5717	136	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-5717	136	8	-	-	PUNCT
ejpam-5717	136	9	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5717	136	10	-	-	PUNCT
ejpam-5717	136	11	int(k	int(k	NOUN
ejpam-5717	136	12	)	)	PUNCT
ejpam-5717	136	13	)	)	PUNCT
ejpam-5717	136	14	)	)	PUNCT
ejpam-5717	136	15	)	)	PUNCT
ejpam-5717	136	16	)	)	PUNCT
ejpam-5717	137	1	p.	p.	NOUN
ejpam-5717	137	2	pue	pue	NOUN
ejpam-5717	137	3	-	-	PUNCT
ejpam-5717	137	4	on	on	ADP
ejpam-5717	137	5	,	,	PUNCT
ejpam-5717	137	6	a.	a.	PROPN
ejpam-5717	137	7	sama	sama	PROPN
ejpam-5717	137	8	-	-	PUNCT
ejpam-5717	137	9	ae	ae	PROPN
ejpam-5717	137	10	,	,	PUNCT
ejpam-5717	137	11	c.	c.	PROPN
ejpam-5717	137	12	boonpok	boonpok	PROPN
ejpam-5717	137	13	/	/	SYM
ejpam-5717	137	14	eur	eur	PROPN
ejpam-5717	137	15	.	.	PUNCT
ejpam-5717	138	1	j.	j.	PROPN
ejpam-5717	138	2	pure	pure	PROPN
ejpam-5717	138	3	appl	appl	PROPN
ejpam-5717	138	4	.	.	PROPN
ejpam-5717	138	5	math	math	PROPN
ejpam-5717	138	6	,	,	PUNCT
ejpam-5717	138	7	18	18	NUM
ejpam-5717	138	8	(	(	PUNCT
ejpam-5717	138	9	1	1	NUM
ejpam-5717	138	10	)	)	PUNCT
ejpam-5717	138	11	(	(	PUNCT
ejpam-5717	138	12	2025	2025	NUM
ejpam-5717	138	13	)	)	PUNCT
ejpam-5717	138	14	,	,	PUNCT
ejpam-5717	138	15	5717	5717	NUM
ejpam-5717	138	16	5	5	NUM
ejpam-5717	138	17	of	of	ADP
ejpam-5717	138	18	16	16	NUM
ejpam-5717	138	19	⊆	⊆	NUM
ejpam-5717	138	20	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	138	21	-	-	PUNCT
ejpam-5717	138	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	138	23	-	-	PUNCT
ejpam-5717	138	24	int(k	int(k	NOUN
ejpam-5717	138	25	)	)	PUNCT
ejpam-5717	138	26	)	)	PUNCT
ejpam-5717	138	27	)	)	PUNCT
ejpam-5717	139	1	=	=	SYM
ejpam-5717	139	2	f−(k	f−(k	PROPN
ejpam-5717	139	3	)	)	PUNCT
ejpam-5717	139	4	.	.	PUNCT
ejpam-5717	140	1	(	(	PUNCT
ejpam-5717	140	2	4	4	X
ejpam-5717	140	3	)	)	PUNCT
ejpam-5717	140	4	⇒	⇒	NOUN
ejpam-5717	140	5	(	(	PUNCT
ejpam-5717	140	6	5	5	NUM
ejpam-5717	140	7	):	):	PUNCT
ejpam-5717	140	8	let	let	VERB
ejpam-5717	140	9	v	v	PART
ejpam-5717	140	10	be	be	AUX
ejpam-5717	140	11	any	any	DET
ejpam-5717	140	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	140	13	-	-	ADJ
ejpam-5717	140	14	open	open	ADJ
ejpam-5717	140	15	set	set	NOUN
ejpam-5717	140	16	of	of	ADP
ejpam-5717	140	17	y	y	PROPN
ejpam-5717	140	18	.	.	PUNCT
ejpam-5717	141	1	then	then	ADV
ejpam-5717	141	2	,	,	PUNCT
ejpam-5717	141	3	we	we	PRON
ejpam-5717	141	4	have	have	VERB
ejpam-5717	141	5	x	x	PART
ejpam-5717	141	6	−	−	NOUN
ejpam-5717	141	7	θ(τ1	θ(τ1	NOUN
ejpam-5717	141	8	,	,	PUNCT
ejpam-5717	141	9	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	142	1	+	+	ADJ
ejpam-5717	142	2	(	(	PUNCT
ejpam-5717	142	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	142	4	-	-	NUM
ejpam-5717	142	5	cl(v	cl(v	NOUN
ejpam-5717	142	6	)	)	PUNCT
ejpam-5717	142	7	)	)	PUNCT
ejpam-5717	142	8	)	)	PUNCT
ejpam-5717	143	1	=	=	SYM
ejpam-5717	143	2	θ(τ1	θ(τ1	PROPN
ejpam-5717	143	3	,	,	PUNCT
ejpam-5717	143	4	τ2)-scl(x	τ2)-scl(x	NOUN
ejpam-5717	143	5	−	−	ADP
ejpam-5717	143	6	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	143	7	-	-	PUNCT
ejpam-5717	143	8	cl(v	cl(v	NOUN
ejpam-5717	143	9	)	)	PUNCT
ejpam-5717	143	10	)	)	PUNCT
ejpam-5717	143	11	)	)	PUNCT
ejpam-5717	144	1	=	=	SYM
ejpam-5717	144	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	144	3	,	,	PUNCT
ejpam-5717	144	4	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	145	1	−(y	−(y	NOUN
ejpam-5717	145	2	−	−	NOUN
ejpam-5717	145	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	145	4	-	-	NUM
ejpam-5717	145	5	cl(v	cl(v	NOUN
ejpam-5717	145	6	)	)	PUNCT
ejpam-5717	145	7	)	)	PUNCT
ejpam-5717	145	8	)	)	PUNCT
ejpam-5717	145	9	,	,	PUNCT
ejpam-5717	145	10	y	y	PROPN
ejpam-5717	145	11	−	−	NUM
ejpam-5717	145	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	145	13	-	-	NUM
ejpam-5717	145	14	cl(v	cl(v	X
ejpam-5717	145	15	)	)	PUNCT
ejpam-5717	146	1	=	=	SYM
ejpam-5717	146	2	σ1σ2	σ1σ2	X
ejpam-5717	146	3	-	-	PUNCT
ejpam-5717	146	4	int(y	int(y	ADJ
ejpam-5717	146	5	−	−	NOUN
ejpam-5717	146	6	σ1σ2	σ1σ2	NOUN
ejpam-5717	146	7	-	-	NUM
ejpam-5717	146	8	cl(v	cl(v	NOUN
ejpam-5717	146	9	)	)	PUNCT
ejpam-5717	146	10	)	)	PUNCT
ejpam-5717	147	1	⊆	⊆	X
ejpam-5717	147	2	σ1σ2	σ1σ2	X
ejpam-5717	147	3	-	-	PUNCT
ejpam-5717	147	4	int(y	int(y	ADJ
ejpam-5717	147	5	−	−	NOUN
ejpam-5717	147	6	σ1σ2	σ1σ2	NOUN
ejpam-5717	147	7	-	-	PUNCT
ejpam-5717	147	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	147	9	-	-	PUNCT
ejpam-5717	147	10	cl(v	cl(v	NOUN
ejpam-5717	147	11	)	)	PUNCT
ejpam-5717	147	12	)	)	PUNCT
ejpam-5717	147	13	)	)	PUNCT
ejpam-5717	147	14	and	and	CCONJ
ejpam-5717	147	15	y	y	PROPN
ejpam-5717	147	16	−	−	PROPN
ejpam-5717	147	17	σ1σ2	σ1σ2	ADV
ejpam-5717	147	18	-	-	PUNCT
ejpam-5717	147	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	147	20	-	-	PUNCT
ejpam-5717	147	21	cl(v	cl(v	NOUN
ejpam-5717	147	22	)	)	PUNCT
ejpam-5717	147	23	)	)	PUNCT
ejpam-5717	147	24	is	be	AUX
ejpam-5717	147	25	(	(	PUNCT
ejpam-5717	147	26	σ1	σ1	NOUN
ejpam-5717	147	27	,	,	PUNCT
ejpam-5717	147	28	σ2)r	σ2)r	NOUN
ejpam-5717	147	29	-	-	PUNCT
ejpam-5717	147	30	closed	closed	ADJ
ejpam-5717	147	31	in	in	ADP
ejpam-5717	147	32	y	y	PROPN
ejpam-5717	147	33	.	.	PUNCT
ejpam-5717	148	1	thus	thus	ADV
ejpam-5717	148	2	by	by	ADP
ejpam-5717	148	3	(	(	PUNCT
ejpam-5717	148	4	4	4	NUM
ejpam-5717	148	5	)	)	PUNCT
ejpam-5717	148	6	,	,	PUNCT
ejpam-5717	148	7	θ(τ1	θ(τ1	VERB
ejpam-5717	148	8	,	,	PUNCT
ejpam-5717	148	9	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	148	10	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	148	11	-	-	PUNCT
ejpam-5717	148	12	int(y	int(y	ADJ
ejpam-5717	148	13	−	−	NOUN
ejpam-5717	148	14	σ1σ2	σ1σ2	NOUN
ejpam-5717	148	15	-	-	PUNCT
ejpam-5717	148	16	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	148	17	-	-	PUNCT
ejpam-5717	148	18	cl(v	cl(v	NOUN
ejpam-5717	148	19	)	)	PUNCT
ejpam-5717	148	20	)	)	PUNCT
ejpam-5717	148	21	)	)	PUNCT
ejpam-5717	148	22	)	)	PUNCT
ejpam-5717	148	23	)	)	PUNCT
ejpam-5717	149	1	⊆	⊆	NUM
ejpam-5717	149	2	f−(y	f−(y	NOUN
ejpam-5717	149	3	−	−	NOUN
ejpam-5717	149	4	σ1σ2	σ1σ2	NOUN
ejpam-5717	149	5	-	-	PUNCT
ejpam-5717	149	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	149	7	-	-	PUNCT
ejpam-5717	149	8	cl(v	cl(v	NOUN
ejpam-5717	149	9	)	)	PUNCT
ejpam-5717	149	10	)	)	PUNCT
ejpam-5717	149	11	)	)	PUNCT
ejpam-5717	150	1	=	=	PUNCT
ejpam-5717	150	2	x	x	X
ejpam-5717	150	3	−	−	ADP
ejpam-5717	150	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-5717	150	5	-	-	PUNCT
ejpam-5717	150	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	150	7	-	-	PUNCT
ejpam-5717	150	8	cl(v	cl(v	NOUN
ejpam-5717	150	9	)	)	PUNCT
ejpam-5717	150	10	)	)	PUNCT
ejpam-5717	150	11	)	)	PUNCT
ejpam-5717	151	1	⊆	⊆	NUM
ejpam-5717	151	2	x	x	SYM
ejpam-5717	151	3	−	−	NOUN
ejpam-5717	151	4	f+(v	f+(v	NOUN
ejpam-5717	151	5	)	)	PUNCT
ejpam-5717	151	6	and	and	CCONJ
ejpam-5717	151	7	hence	hence	ADV
ejpam-5717	151	8	f+(v	f+(v	NOUN
ejpam-5717	151	9	)	)	PUNCT
ejpam-5717	152	1	⊆	⊆	NUM
ejpam-5717	152	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	152	3	,	,	PUNCT
ejpam-5717	152	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	153	1	+	+	ADJ
ejpam-5717	153	2	(	(	PUNCT
ejpam-5717	153	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	153	4	-	-	NUM
ejpam-5717	153	5	cl(v	cl(v	NOUN
ejpam-5717	153	6	)	)	PUNCT
ejpam-5717	153	7	)	)	PUNCT
ejpam-5717	153	8	)	)	PUNCT
ejpam-5717	153	9	.	.	PUNCT
ejpam-5717	154	1	(	(	PUNCT
ejpam-5717	154	2	5	5	X
ejpam-5717	154	3	)	)	PUNCT
ejpam-5717	154	4	⇒	⇒	NOUN
ejpam-5717	154	5	(	(	PUNCT
ejpam-5717	154	6	6	6	NUM
ejpam-5717	154	7	):	):	PUNCT
ejpam-5717	154	8	let	let	VERB
ejpam-5717	154	9	k	k	PRON
ejpam-5717	154	10	be	be	AUX
ejpam-5717	154	11	any	any	DET
ejpam-5717	154	12	σ1σ2	σ1σ2	NUM
ejpam-5717	154	13	-	-	PUNCT
ejpam-5717	154	14	closed	closed	ADJ
ejpam-5717	154	15	set	set	NOUN
ejpam-5717	154	16	of	of	ADP
ejpam-5717	154	17	y	y	PROPN
ejpam-5717	154	18	.	.	PUNCT
ejpam-5717	155	1	then	then	ADV
ejpam-5717	155	2	by	by	ADP
ejpam-5717	155	3	(	(	PUNCT
ejpam-5717	155	4	5	5	NUM
ejpam-5717	155	5	)	)	PUNCT
ejpam-5717	155	6	,	,	PUNCT
ejpam-5717	155	7	we	we	PRON
ejpam-5717	155	8	have	have	VERB
ejpam-5717	155	9	x	x	INTJ
ejpam-5717	155	10	−	−	DET
ejpam-5717	155	11	f−(k	f−(k	PROPN
ejpam-5717	155	12	)	)	PUNCT
ejpam-5717	155	13	=	=	PUNCT
ejpam-5717	156	1	f+(y	f+(y	PROPN
ejpam-5717	156	2	−k	−k	PROPN
ejpam-5717	156	3	)	)	PUNCT
ejpam-5717	156	4	⊆	⊆	NUM
ejpam-5717	156	5	θ(τ1	θ(τ1	NOUN
ejpam-5717	156	6	,	,	PUNCT
ejpam-5717	156	7	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	156	8	+	+	ADJ
ejpam-5717	156	9	(	(	PUNCT
ejpam-5717	156	10	σ1σ2	σ1σ2	NUM
ejpam-5717	156	11	-	-	PUNCT
ejpam-5717	156	12	cl(y	cl(y	NOUN
ejpam-5717	156	13	−k	−k	NOUN
ejpam-5717	156	14	)	)	PUNCT
ejpam-5717	156	15	)	)	PUNCT
ejpam-5717	156	16	)	)	PUNCT
ejpam-5717	157	1	=	=	SYM
ejpam-5717	157	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	157	3	,	,	PUNCT
ejpam-5717	157	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	158	1	+	+	ADJ
ejpam-5717	158	2	(	(	PUNCT
ejpam-5717	158	3	y	y	PROPN
ejpam-5717	158	4	−	−	PROPN
ejpam-5717	158	5	σ1σ2	σ1σ2	NUM
ejpam-5717	158	6	-	-	PUNCT
ejpam-5717	158	7	int(k	int(k	NOUN
ejpam-5717	158	8	)	)	PUNCT
ejpam-5717	158	9	)	)	PUNCT
ejpam-5717	158	10	)	)	PUNCT
ejpam-5717	159	1	=	=	SYM
ejpam-5717	159	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	159	3	,	,	PUNCT
ejpam-5717	159	4	τ2)-sint(x	τ2)-sint(x	PUNCT
ejpam-5717	159	5	−	−	NOUN
ejpam-5717	159	6	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	159	7	-	-	PUNCT
ejpam-5717	159	8	int(k	int(k	NOUN
ejpam-5717	159	9	)	)	PUNCT
ejpam-5717	159	10	)	)	PUNCT
ejpam-5717	159	11	)	)	PUNCT
ejpam-5717	160	1	=	=	PUNCT
ejpam-5717	160	2	x	x	SYM
ejpam-5717	160	3	−	−	NOUN
ejpam-5717	160	4	θ(τ1	θ(τ1	NOUN
ejpam-5717	160	5	,	,	PUNCT
ejpam-5717	160	6	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	160	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	160	8	-	-	PUNCT
ejpam-5717	160	9	int(k	int(k	NUM
ejpam-5717	160	10	)	)	PUNCT
ejpam-5717	160	11	)	)	PUNCT
ejpam-5717	160	12	)	)	PUNCT
ejpam-5717	160	13	.	.	PUNCT
ejpam-5717	161	1	thus	thus	ADV
ejpam-5717	161	2	,	,	PUNCT
ejpam-5717	161	3	θ(τ1	θ(τ1	NOUN
ejpam-5717	161	4	,	,	PUNCT
ejpam-5717	161	5	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	161	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	161	7	-	-	PUNCT
ejpam-5717	161	8	int(k	int(k	NUM
ejpam-5717	161	9	)	)	PUNCT
ejpam-5717	161	10	)	)	PUNCT
ejpam-5717	161	11	)	)	PUNCT
ejpam-5717	162	1	⊆	⊆	NUM
ejpam-5717	162	2	f−(k	f−(k	PROPN
ejpam-5717	162	3	)	)	PUNCT
ejpam-5717	162	4	.	.	PUNCT
ejpam-5717	163	1	(	(	PUNCT
ejpam-5717	163	2	6	6	X
ejpam-5717	163	3	)	)	PUNCT
ejpam-5717	163	4	⇒	⇒	NOUN
ejpam-5717	163	5	(	(	PUNCT
ejpam-5717	163	6	7	7	NUM
ejpam-5717	163	7	):	):	PUNCT
ejpam-5717	163	8	let	let	VERB
ejpam-5717	163	9	v	v	PART
ejpam-5717	163	10	be	be	AUX
ejpam-5717	163	11	any	any	DET
ejpam-5717	163	12	σ1σ2	σ1σ2	NUM
ejpam-5717	163	13	-	-	PUNCT
ejpam-5717	163	14	closed	closed	ADJ
ejpam-5717	163	15	set	set	NOUN
ejpam-5717	163	16	of	of	ADP
ejpam-5717	163	17	y	y	PROPN
ejpam-5717	163	18	.	.	PUNCT
ejpam-5717	164	1	then	then	ADV
ejpam-5717	164	2	,	,	PUNCT
ejpam-5717	164	3	we	we	PRON
ejpam-5717	164	4	have	have	VERB
ejpam-5717	164	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	164	6	-	-	NUM
ejpam-5717	164	7	cl(v	cl(v	NOUN
ejpam-5717	164	8	)	)	PUNCT
ejpam-5717	164	9	is	be	AUX
ejpam-5717	164	10	σ1σ2	σ1σ2	NOUN
ejpam-5717	164	11	-	-	ADJ
ejpam-5717	164	12	closed	closed	ADJ
ejpam-5717	164	13	in	in	ADP
ejpam-5717	164	14	y	y	PROPN
ejpam-5717	164	15	and	and	CCONJ
ejpam-5717	164	16	by	by	ADP
ejpam-5717	164	17	(	(	PUNCT
ejpam-5717	164	18	6	6	NUM
ejpam-5717	164	19	)	)	PUNCT
ejpam-5717	164	20	,	,	PUNCT
ejpam-5717	164	21	θ(τ1	θ(τ1	VERB
ejpam-5717	164	22	,	,	PUNCT
ejpam-5717	164	23	τ2)-scl(f	τ2)-scl(f	ADV
ejpam-5717	164	24	−(v	−(v	NOUN
ejpam-5717	164	25	)	)	PUNCT
ejpam-5717	164	26	)	)	PUNCT
ejpam-5717	165	1	⊆	⊆	NUM
ejpam-5717	165	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	165	3	,	,	PUNCT
ejpam-5717	165	4	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	165	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	165	6	-	-	PUNCT
ejpam-5717	165	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	165	8	-	-	PUNCT
ejpam-5717	165	9	cl(v	cl(v	NOUN
ejpam-5717	165	10	)	)	PUNCT
ejpam-5717	165	11	)	)	PUNCT
ejpam-5717	165	12	)	)	PUNCT
ejpam-5717	165	13	)	)	PUNCT
ejpam-5717	166	1	⊆	⊆	X
ejpam-5717	166	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	166	3	-	-	PUNCT
ejpam-5717	166	4	cl(v	cl(v	NOUN
ejpam-5717	166	5	)	)	PUNCT
ejpam-5717	166	6	)	)	PUNCT
ejpam-5717	166	7	.	.	PUNCT
ejpam-5717	167	1	(	(	PUNCT
ejpam-5717	167	2	7	7	X
ejpam-5717	167	3	)	)	PUNCT
ejpam-5717	167	4	⇒	⇒	NOUN
ejpam-5717	167	5	(	(	PUNCT
ejpam-5717	167	6	1	1	NUM
ejpam-5717	167	7	):	):	PUNCT
ejpam-5717	167	8	let	let	VERB
ejpam-5717	167	9	x	x	PUNCT
ejpam-5717	167	10	∈	∈	PROPN
ejpam-5717	167	11	x	x	X
ejpam-5717	167	12	and	and	CCONJ
ejpam-5717	167	13	v	v	X
ejpam-5717	167	14	be	be	AUX
ejpam-5717	167	15	any	any	DET
ejpam-5717	167	16	σ1σ2	σ1σ2	NOUN
ejpam-5717	167	17	-	-	ADJ
ejpam-5717	167	18	open	open	ADJ
ejpam-5717	167	19	set	set	NOUN
ejpam-5717	167	20	of	of	ADP
ejpam-5717	167	21	y	y	PROPN
ejpam-5717	167	22	containing	contain	VERB
ejpam-5717	167	23	f	f	PROPN
ejpam-5717	167	24	(	(	PUNCT
ejpam-5717	167	25	x	x	NOUN
ejpam-5717	167	26	)	)	PUNCT
ejpam-5717	167	27	.	.	PUNCT
ejpam-5717	168	1	then	then	ADV
ejpam-5717	168	2	,	,	PUNCT
ejpam-5717	168	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	168	4	-	-	PUNCT
ejpam-5717	168	5	cl(y	cl(y	NOUN
ejpam-5717	168	6	−	−	NOUN
ejpam-5717	168	7	σ1σ2	σ1σ2	NOUN
ejpam-5717	168	8	-	-	NUM
ejpam-5717	168	9	cl(v	cl(v	NOUN
ejpam-5717	168	10	)	)	PUNCT
ejpam-5717	168	11	)	)	PUNCT
ejpam-5717	168	12	∩	∩	PROPN
ejpam-5717	168	13	f	f	X
ejpam-5717	168	14	(	(	PUNCT
ejpam-5717	168	15	x	x	X
ejpam-5717	168	16	)	)	PUNCT
ejpam-5717	168	17	=	=	SYM
ejpam-5717	168	18	∅	∅	NOUN
ejpam-5717	168	19	and	and	CCONJ
ejpam-5717	168	20	x	x	PART
ejpam-5717	168	21	̸∈	̸∈	PROPN
ejpam-5717	168	22	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5717	168	23	-	-	PUNCT
ejpam-5717	168	24	cl(y	cl(y	NOUN
ejpam-5717	168	25	−	−	NOUN
ejpam-5717	168	26	σ1σ2	σ1σ2	NOUN
ejpam-5717	168	27	-	-	NUM
ejpam-5717	168	28	cl(v	cl(v	NOUN
ejpam-5717	168	29	)	)	PUNCT
ejpam-5717	168	30	)	)	PUNCT
ejpam-5717	168	31	)	)	PUNCT
ejpam-5717	168	32	.	.	PUNCT
ejpam-5717	169	1	it	it	PRON
ejpam-5717	169	2	follows	follow	VERB
ejpam-5717	169	3	from	from	ADP
ejpam-5717	169	4	(	(	PUNCT
ejpam-5717	169	5	7	7	NUM
ejpam-5717	169	6	)	)	PUNCT
ejpam-5717	169	7	that	that	PRON
ejpam-5717	169	8	x	x	SYM
ejpam-5717	169	9	̸∈	̸∈	PROPN
ejpam-5717	169	10	θ(τ1	θ(τ1	VERB
ejpam-5717	169	11	,	,	PUNCT
ejpam-5717	170	1	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	171	1	−(y	−(y	NOUN
ejpam-5717	171	2	−	−	NOUN
ejpam-5717	171	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	171	4	-	-	NUM
ejpam-5717	171	5	cl(v	cl(v	NOUN
ejpam-5717	171	6	)	)	PUNCT
ejpam-5717	171	7	)	)	PUNCT
ejpam-5717	171	8	)	)	PUNCT
ejpam-5717	171	9	.	.	PUNCT
ejpam-5717	172	1	then	then	ADV
ejpam-5717	172	2	,	,	PUNCT
ejpam-5717	172	3	there	there	PRON
ejpam-5717	172	4	exists	exist	VERB
ejpam-5717	172	5	a	a	DET
ejpam-5717	172	6	(	(	PUNCT
ejpam-5717	172	7	τ1	τ1	NOUN
ejpam-5717	172	8	,	,	PUNCT
ejpam-5717	172	9	τ2)s	τ2)s	NOUN
ejpam-5717	172	10	-	-	PUNCT
ejpam-5717	172	11	open	open	ADJ
ejpam-5717	172	12	set	set	NOUN
ejpam-5717	172	13	u	u	NOUN
ejpam-5717	172	14	of	of	ADP
ejpam-5717	172	15	x	x	PUNCT
ejpam-5717	172	16	containing	contain	VERB
ejpam-5717	172	17	x	x	PUNCT
ejpam-5717	172	18	such	such	ADJ
ejpam-5717	172	19	that	that	SCONJ
ejpam-5717	172	20	(	(	PUNCT
ejpam-5717	172	21	τ1	τ1	NOUN
ejpam-5717	172	22	,	,	PUNCT
ejpam-5717	172	23	τ2)-scl(u	τ2)-scl(u	ADJ
ejpam-5717	172	24	)	)	PUNCT
ejpam-5717	172	25	∩	∩	NOUN
ejpam-5717	172	26	f−(y	f−(y	NOUN
ejpam-5717	172	27	−	−	NUM
ejpam-5717	172	28	σ1σ2	σ1σ2	NOUN
ejpam-5717	172	29	-	-	NUM
ejpam-5717	172	30	cl(v	cl(v	NOUN
ejpam-5717	172	31	)	)	PUNCT
ejpam-5717	172	32	)	)	PUNCT
ejpam-5717	173	1	=	=	NOUN
ejpam-5717	173	2	∅	∅	NOUN
ejpam-5717	173	3	;	;	PUNCT
ejpam-5717	173	4	hence	hence	ADV
ejpam-5717	173	5	f	f	X
ejpam-5717	173	6	(	(	PUNCT
ejpam-5717	173	7	(	(	PUNCT
ejpam-5717	173	8	τ1	τ1	NOUN
ejpam-5717	173	9	,	,	PUNCT
ejpam-5717	173	10	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	173	11	)	)	PUNCT
ejpam-5717	173	12	)	)	PUNCT
ejpam-5717	174	1	⊆	⊆	X
ejpam-5717	174	2	σ1σ2	σ1σ2	NOUN
ejpam-5717	174	3	-	-	NUM
ejpam-5717	174	4	cl(v	cl(v	NOUN
ejpam-5717	174	5	)	)	PUNCT
ejpam-5717	174	6	.	.	PUNCT
ejpam-5717	175	1	this	this	PRON
ejpam-5717	175	2	shows	show	VERB
ejpam-5717	175	3	that	that	SCONJ
ejpam-5717	175	4	f	f	PROPN
ejpam-5717	175	5	is	be	AUX
ejpam-5717	175	6	upper	upper	ADJ
ejpam-5717	175	7	quasi	quasi	NOUN
ejpam-5717	175	8	θ(τ1	θ(τ1	NOUN
ejpam-5717	175	9	,	,	PUNCT
ejpam-5717	175	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	175	11	.	.	PUNCT
ejpam-5717	176	1	definition	definition	NOUN
ejpam-5717	176	2	2	2	NUM
ejpam-5717	176	3	.	.	PUNCT
ejpam-5717	176	4	a	a	DET
ejpam-5717	176	5	multifunction	multifunction	NOUN
ejpam-5717	176	6	f	f	NOUN
ejpam-5717	176	7	:	:	PUNCT
ejpam-5717	176	8	(	(	PUNCT
ejpam-5717	176	9	x	x	NOUN
ejpam-5717	176	10	,	,	PUNCT
ejpam-5717	176	11	τ1	τ1	NOUN
ejpam-5717	176	12	,	,	PUNCT
ejpam-5717	176	13	τ2	τ2	NOUN
ejpam-5717	176	14	)	)	PUNCT
ejpam-5717	176	15	→	→	SYM
ejpam-5717	176	16	(	(	PUNCT
ejpam-5717	176	17	y	y	PROPN
ejpam-5717	176	18	,	,	PUNCT
ejpam-5717	176	19	σ1	σ1	PROPN
ejpam-5717	176	20	,	,	PUNCT
ejpam-5717	176	21	σ2	σ2	PROPN
ejpam-5717	176	22	)	)	PUNCT
ejpam-5717	176	23	is	be	AUX
ejpam-5717	176	24	said	say	VERB
ejpam-5717	176	25	to	to	PART
ejpam-5717	176	26	be	be	AUX
ejpam-5717	176	27	lower	low	ADJ
ejpam-5717	176	28	quasi	quasi	NOUN
ejpam-5717	176	29	θ(τ1	θ(τ1	NOUN
ejpam-5717	176	30	,	,	PUNCT
ejpam-5717	176	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	176	32	if	if	SCONJ
ejpam-5717	176	33	for	for	ADP
ejpam-5717	176	34	each	each	DET
ejpam-5717	176	35	x	x	SYM
ejpam-5717	176	36	∈	∈	PROPN
ejpam-5717	176	37	x	x	X
ejpam-5717	176	38	and	and	CCONJ
ejpam-5717	176	39	each	each	DET
ejpam-5717	176	40	σ1σ2	σ1σ2	VERB
ejpam-5717	176	41	-	-	ADJ
ejpam-5717	176	42	open	open	ADJ
ejpam-5717	176	43	set	set	NOUN
ejpam-5717	176	44	v	v	NOUN
ejpam-5717	176	45	of	of	ADP
ejpam-5717	176	46	y	y	PRON
ejpam-5717	176	47	such	such	ADJ
ejpam-5717	176	48	that	that	SCONJ
ejpam-5717	176	49	f	f	PROPN
ejpam-5717	176	50	(	(	PUNCT
ejpam-5717	176	51	x)∩v	x)∩v	PROPN
ejpam-5717	176	52	̸=	̸=	PROPN
ejpam-5717	176	53	∅	∅	NOUN
ejpam-5717	176	54	,	,	PUNCT
ejpam-5717	176	55	there	there	PRON
ejpam-5717	176	56	exists	exist	VERB
ejpam-5717	176	57	a	a	DET
ejpam-5717	176	58	(	(	PUNCT
ejpam-5717	176	59	τ1	τ1	NOUN
ejpam-5717	176	60	,	,	PUNCT
ejpam-5717	176	61	τ2)s	τ2)s	NOUN
ejpam-5717	176	62	-	-	PUNCT
ejpam-5717	176	63	open	open	ADJ
ejpam-5717	176	64	set	set	NOUN
ejpam-5717	176	65	u	u	NOUN
ejpam-5717	176	66	of	of	ADP
ejpam-5717	176	67	x	x	PUNCT
ejpam-5717	176	68	containing	contain	VERB
ejpam-5717	176	69	x	x	PUNCT
ejpam-5717	176	70	such	such	ADJ
ejpam-5717	176	71	that	that	SCONJ
ejpam-5717	176	72	σ1σ2	σ1σ2	NOUN
ejpam-5717	176	73	-	-	PUNCT
ejpam-5717	176	74	cl(v	cl(v	NOUN
ejpam-5717	176	75	)	)	PUNCT
ejpam-5717	176	76	∩	∩	PROPN
ejpam-5717	176	77	f	f	X
ejpam-5717	176	78	(	(	PUNCT
ejpam-5717	176	79	z	z	NOUN
ejpam-5717	176	80	)	)	PUNCT
ejpam-5717	176	81	̸=	̸=	NOUN
ejpam-5717	176	82	∅	∅	NOUN
ejpam-5717	176	83	for	for	ADP
ejpam-5717	176	84	every	every	DET
ejpam-5717	176	85	z	z	NOUN
ejpam-5717	176	86	∈	∈	PROPN
ejpam-5717	176	87	(	(	PUNCT
ejpam-5717	176	88	τ1	τ1	NOUN
ejpam-5717	176	89	,	,	PUNCT
ejpam-5717	176	90	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	176	91	)	)	PUNCT
ejpam-5717	176	92	.	.	PUNCT
ejpam-5717	177	1	lemma	lemma	PROPN
ejpam-5717	178	1	2	2	X
ejpam-5717	178	2	.	.	PUNCT
ejpam-5717	179	1	if	if	SCONJ
ejpam-5717	179	2	f	f	PROPN
ejpam-5717	179	3	:	:	PUNCT
ejpam-5717	179	4	(	(	PUNCT
ejpam-5717	179	5	x	x	NOUN
ejpam-5717	179	6	,	,	PUNCT
ejpam-5717	179	7	τ1	τ1	NOUN
ejpam-5717	179	8	,	,	PUNCT
ejpam-5717	179	9	τ2	τ2	NOUN
ejpam-5717	179	10	)	)	PUNCT
ejpam-5717	179	11	→	→	SYM
ejpam-5717	179	12	(	(	PUNCT
ejpam-5717	179	13	y	y	PROPN
ejpam-5717	179	14	,	,	PUNCT
ejpam-5717	179	15	σ1	σ1	PROPN
ejpam-5717	179	16	,	,	PUNCT
ejpam-5717	179	17	σ2	σ2	NOUN
ejpam-5717	179	18	)	)	PUNCT
ejpam-5717	179	19	is	be	AUX
ejpam-5717	179	20	lower	low	ADJ
ejpam-5717	179	21	quasi	quasi	NOUN
ejpam-5717	179	22	θ(τ1	θ(τ1	NOUN
ejpam-5717	179	23	,	,	PUNCT
ejpam-5717	179	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	179	25	,	,	PUNCT
ejpam-5717	179	26	then	then	ADV
ejpam-5717	179	27	for	for	ADP
ejpam-5717	179	28	each	each	DET
ejpam-5717	179	29	x	x	SYM
ejpam-5717	179	30	∈	∈	PROPN
ejpam-5717	179	31	x	x	X
ejpam-5717	179	32	and	and	CCONJ
ejpam-5717	179	33	each	each	DET
ejpam-5717	179	34	subset	subset	NOUN
ejpam-5717	179	35	b	b	PROPN
ejpam-5717	179	36	of	of	ADP
ejpam-5717	179	37	y	y	PROPN
ejpam-5717	179	38	with	with	ADP
ejpam-5717	179	39	(	(	PUNCT
ejpam-5717	179	40	σ1	σ1	PROPN
ejpam-5717	179	41	,	,	PUNCT
ejpam-5717	179	42	σ2)θ	σ2)θ	NOUN
ejpam-5717	179	43	-	-	PUNCT
ejpam-5717	179	44	int(b)∩f	int(b)∩f	NOUN
ejpam-5717	179	45	(	(	PUNCT
ejpam-5717	179	46	x	x	X
ejpam-5717	179	47	)	)	PUNCT
ejpam-5717	179	48	̸=	̸=	PROPN
ejpam-5717	179	49	∅	∅	NOUN
ejpam-5717	179	50	there	there	ADV
ejpam-5717	179	51	exists	exist	VERB
ejpam-5717	179	52	a	a	DET
ejpam-5717	179	53	(	(	PUNCT
ejpam-5717	179	54	τ1	τ1	NOUN
ejpam-5717	179	55	,	,	PUNCT
ejpam-5717	179	56	τ2)s	τ2)s	NOUN
ejpam-5717	179	57	-	-	PUNCT
ejpam-5717	179	58	open	open	ADJ
ejpam-5717	179	59	set	set	NOUN
ejpam-5717	179	60	u	u	NOUN
ejpam-5717	179	61	of	of	ADP
ejpam-5717	179	62	x	x	PUNCT
ejpam-5717	179	63	containing	contain	VERB
ejpam-5717	179	64	x	x	PUNCT
ejpam-5717	179	65	such	such	ADJ
ejpam-5717	179	66	that	that	SCONJ
ejpam-5717	179	67	(	(	PUNCT
ejpam-5717	179	68	τ1	τ1	NOUN
ejpam-5717	179	69	,	,	PUNCT
ejpam-5717	179	70	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	179	71	)	)	PUNCT
ejpam-5717	179	72	⊆	⊆	NUM
ejpam-5717	179	73	f−(b	f−(b	NOUN
ejpam-5717	179	74	)	)	PUNCT
ejpam-5717	179	75	.	.	PUNCT
ejpam-5717	180	1	p.	p.	NOUN
ejpam-5717	180	2	pue	pue	NOUN
ejpam-5717	180	3	-	-	PUNCT
ejpam-5717	180	4	on	on	ADP
ejpam-5717	180	5	,	,	PUNCT
ejpam-5717	180	6	a.	a.	PROPN
ejpam-5717	180	7	sama	sama	PROPN
ejpam-5717	180	8	-	-	PUNCT
ejpam-5717	180	9	ae	ae	PROPN
ejpam-5717	180	10	,	,	PUNCT
ejpam-5717	180	11	c.	c.	PROPN
ejpam-5717	180	12	boonpok	boonpok	PROPN
ejpam-5717	180	13	/	/	SYM
ejpam-5717	180	14	eur	eur	PROPN
ejpam-5717	180	15	.	.	PUNCT
ejpam-5717	181	1	j.	j.	PROPN
ejpam-5717	181	2	pure	pure	PROPN
ejpam-5717	181	3	appl	appl	PROPN
ejpam-5717	181	4	.	.	PROPN
ejpam-5717	181	5	math	math	PROPN
ejpam-5717	181	6	,	,	PUNCT
ejpam-5717	181	7	18	18	NUM
ejpam-5717	181	8	(	(	PUNCT
ejpam-5717	181	9	1	1	NUM
ejpam-5717	181	10	)	)	PUNCT
ejpam-5717	181	11	(	(	PUNCT
ejpam-5717	181	12	2025	2025	NUM
ejpam-5717	181	13	)	)	PUNCT
ejpam-5717	181	14	,	,	PUNCT
ejpam-5717	181	15	5717	5717	NUM
ejpam-5717	181	16	6	6	NUM
ejpam-5717	181	17	of	of	ADP
ejpam-5717	181	18	16	16	NUM
ejpam-5717	181	19	proof	proof	NOUN
ejpam-5717	181	20	.	.	PUNCT
ejpam-5717	182	1	since	since	SCONJ
ejpam-5717	182	2	(	(	PUNCT
ejpam-5717	182	3	σ1	σ1	PROPN
ejpam-5717	182	4	,	,	PUNCT
ejpam-5717	182	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	182	6	-	-	PUNCT
ejpam-5717	182	7	int(b	int(b	NOUN
ejpam-5717	182	8	)	)	PUNCT
ejpam-5717	182	9	∩	∩	ADJ
ejpam-5717	182	10	f	f	PROPN
ejpam-5717	182	11	(	(	PUNCT
ejpam-5717	182	12	x	x	X
ejpam-5717	182	13	)	)	PUNCT
ejpam-5717	182	14	̸=	̸=	NOUN
ejpam-5717	182	15	∅	∅	NOUN
ejpam-5717	182	16	,	,	PUNCT
ejpam-5717	182	17	there	there	PRON
ejpam-5717	182	18	exists	exist	VERB
ejpam-5717	182	19	a	a	DET
ejpam-5717	182	20	σ1σ2	σ1σ2	NUM
ejpam-5717	182	21	-	-	ADJ
ejpam-5717	182	22	open	open	ADJ
ejpam-5717	182	23	set	set	NOUN
ejpam-5717	182	24	v	v	NOUN
ejpam-5717	182	25	of	of	ADP
ejpam-5717	182	26	y	y	PRON
ejpam-5717	183	1	such	such	ADJ
ejpam-5717	183	2	that	that	SCONJ
ejpam-5717	183	3	v	v	ADP
ejpam-5717	183	4	⊆	⊆	NUM
ejpam-5717	183	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	183	6	-	-	PUNCT
ejpam-5717	183	7	cl(v	cl(v	NOUN
ejpam-5717	183	8	)	)	PUNCT
ejpam-5717	183	9	⊆	⊆	NUM
ejpam-5717	183	10	b	b	NOUN
ejpam-5717	183	11	and	and	CCONJ
ejpam-5717	183	12	f	f	PROPN
ejpam-5717	183	13	(	(	PUNCT
ejpam-5717	183	14	x	x	NOUN
ejpam-5717	183	15	)	)	PUNCT
ejpam-5717	183	16	∩	∩	NOUN
ejpam-5717	183	17	v	v	ADP
ejpam-5717	183	18	̸=	̸=	PROPN
ejpam-5717	183	19	∅.	∅.	NOUN
ejpam-5717	183	20	since	since	SCONJ
ejpam-5717	183	21	f	f	PROPN
ejpam-5717	183	22	is	be	AUX
ejpam-5717	183	23	lower	low	ADJ
ejpam-5717	183	24	quasi	quasi	NOUN
ejpam-5717	183	25	θ(τ1	θ(τ1	NOUN
ejpam-5717	183	26	,	,	PUNCT
ejpam-5717	183	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	183	28	,	,	PUNCT
ejpam-5717	183	29	there	there	PRON
ejpam-5717	183	30	exists	exist	VERB
ejpam-5717	183	31	a	a	DET
ejpam-5717	183	32	(	(	PUNCT
ejpam-5717	183	33	τ1	τ1	NOUN
ejpam-5717	183	34	,	,	PUNCT
ejpam-5717	183	35	τ2)s	τ2)s	NOUN
ejpam-5717	183	36	-	-	PUNCT
ejpam-5717	183	37	open	open	ADJ
ejpam-5717	183	38	set	set	NOUN
ejpam-5717	183	39	u	u	NOUN
ejpam-5717	183	40	of	of	ADP
ejpam-5717	183	41	x	x	PUNCT
ejpam-5717	183	42	containing	contain	VERB
ejpam-5717	183	43	x	x	PUNCT
ejpam-5717	183	44	such	such	ADJ
ejpam-5717	183	45	that	that	SCONJ
ejpam-5717	183	46	σ1σ2	σ1σ2	NOUN
ejpam-5717	183	47	-	-	PUNCT
ejpam-5717	183	48	cl(v	cl(v	NOUN
ejpam-5717	183	49	)	)	PUNCT
ejpam-5717	183	50	∩	∩	PROPN
ejpam-5717	183	51	f	f	X
ejpam-5717	183	52	(	(	PUNCT
ejpam-5717	183	53	z	z	NOUN
ejpam-5717	183	54	)	)	PUNCT
ejpam-5717	183	55	̸=	̸=	NOUN
ejpam-5717	183	56	∅	∅	NOUN
ejpam-5717	183	57	for	for	ADP
ejpam-5717	183	58	every	every	DET
ejpam-5717	183	59	z	z	NOUN
ejpam-5717	183	60	∈	∈	PROPN
ejpam-5717	183	61	(	(	PUNCT
ejpam-5717	183	62	τ1	τ1	NOUN
ejpam-5717	183	63	,	,	PUNCT
ejpam-5717	183	64	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	183	65	)	)	PUNCT
ejpam-5717	183	66	and	and	CCONJ
ejpam-5717	183	67	hence	hence	ADV
ejpam-5717	183	68	(	(	PUNCT
ejpam-5717	183	69	τ1	τ1	NOUN
ejpam-5717	183	70	,	,	PUNCT
ejpam-5717	183	71	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	183	72	)	)	PUNCT
ejpam-5717	183	73	⊆	⊆	NUM
ejpam-5717	183	74	f−(b	f−(b	NOUN
ejpam-5717	183	75	)	)	PUNCT
ejpam-5717	183	76	.	.	PUNCT
ejpam-5717	184	1	theorem	theorem	NOUN
ejpam-5717	184	2	2	2	NUM
ejpam-5717	184	3	.	.	X
ejpam-5717	184	4	for	for	ADP
ejpam-5717	184	5	a	a	DET
ejpam-5717	184	6	multifunction	multifunction	NOUN
ejpam-5717	185	1	f	f	NOUN
ejpam-5717	185	2	:	:	PUNCT
ejpam-5717	185	3	(	(	PUNCT
ejpam-5717	185	4	x	x	NOUN
ejpam-5717	185	5	,	,	PUNCT
ejpam-5717	185	6	τ1	τ1	NOUN
ejpam-5717	185	7	,	,	PUNCT
ejpam-5717	185	8	τ2	τ2	NOUN
ejpam-5717	185	9	)	)	PUNCT
ejpam-5717	185	10	→	→	SYM
ejpam-5717	185	11	(	(	PUNCT
ejpam-5717	185	12	y	y	PROPN
ejpam-5717	185	13	,	,	PUNCT
ejpam-5717	185	14	σ1	σ1	PROPN
ejpam-5717	185	15	,	,	PUNCT
ejpam-5717	185	16	σ2	σ2	NOUN
ejpam-5717	185	17	)	)	PUNCT
ejpam-5717	185	18	,	,	PUNCT
ejpam-5717	185	19	the	the	DET
ejpam-5717	185	20	following	follow	VERB
ejpam-5717	185	21	properties	property	NOUN
ejpam-5717	185	22	are	be	AUX
ejpam-5717	185	23	equivalent	equivalent	ADJ
ejpam-5717	185	24	:	:	PUNCT
ejpam-5717	185	25	(	(	PUNCT
ejpam-5717	185	26	1	1	X
ejpam-5717	185	27	)	)	PUNCT
ejpam-5717	185	28	f	f	PROPN
ejpam-5717	185	29	is	be	AUX
ejpam-5717	185	30	lower	low	ADJ
ejpam-5717	185	31	quasi	quasi	NOUN
ejpam-5717	185	32	θ(τ1	θ(τ1	NOUN
ejpam-5717	185	33	,	,	PUNCT
ejpam-5717	185	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	185	35	;	;	PUNCT
ejpam-5717	185	36	(	(	PUNCT
ejpam-5717	185	37	2	2	X
ejpam-5717	185	38	)	)	PUNCT
ejpam-5717	185	39	θ(τ1	θ(τ1	NOUN
ejpam-5717	185	40	,	,	PUNCT
ejpam-5717	185	41	τ2)-scl(f	τ2)-scl(f	X
ejpam-5717	186	1	+	+	ADJ
ejpam-5717	186	2	(	(	PUNCT
ejpam-5717	186	3	b	b	NOUN
ejpam-5717	186	4	)	)	PUNCT
ejpam-5717	186	5	)	)	PUNCT
ejpam-5717	186	6	⊆	⊆	NUM
ejpam-5717	186	7	f+((σ1	f+((σ1	NOUN
ejpam-5717	186	8	,	,	PUNCT
ejpam-5717	186	9	σ2)θ	σ2)θ	ADJ
ejpam-5717	186	10	-	-	PUNCT
ejpam-5717	186	11	cl(b	cl(b	NOUN
ejpam-5717	186	12	)	)	PUNCT
ejpam-5717	186	13	)	)	PUNCT
ejpam-5717	187	1	for	for	ADP
ejpam-5717	187	2	every	every	DET
ejpam-5717	187	3	subset	subset	NOUN
ejpam-5717	187	4	b	b	PROPN
ejpam-5717	187	5	of	of	ADP
ejpam-5717	187	6	y	y	PROPN
ejpam-5717	187	7	;	;	PUNCT
ejpam-5717	187	8	(	(	PUNCT
ejpam-5717	187	9	3	3	X
ejpam-5717	187	10	)	)	PUNCT
ejpam-5717	187	11	θ(τ1	θ(τ1	NOUN
ejpam-5717	187	12	,	,	PUNCT
ejpam-5717	187	13	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	188	1	+	+	ADJ
ejpam-5717	188	2	(	(	PUNCT
ejpam-5717	188	3	v	v	NOUN
ejpam-5717	188	4	)	)	PUNCT
ejpam-5717	188	5	)	)	PUNCT
ejpam-5717	189	1	⊆	⊆	NUM
ejpam-5717	189	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	189	3	-	-	PUNCT
ejpam-5717	189	4	cl(v	cl(v	NOUN
ejpam-5717	189	5	)	)	PUNCT
ejpam-5717	189	6	)	)	PUNCT
ejpam-5717	189	7	for	for	ADP
ejpam-5717	189	8	every	every	DET
ejpam-5717	189	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	189	10	-	-	ADJ
ejpam-5717	189	11	open	open	ADJ
ejpam-5717	189	12	set	set	NOUN
ejpam-5717	189	13	v	v	NOUN
ejpam-5717	189	14	of	of	ADP
ejpam-5717	189	15	y	y	PROPN
ejpam-5717	189	16	;	;	PUNCT
ejpam-5717	189	17	(	(	PUNCT
ejpam-5717	189	18	4	4	X
ejpam-5717	189	19	)	)	PUNCT
ejpam-5717	189	20	f−(v	f−(v	ADJ
ejpam-5717	189	21	)	)	PUNCT
ejpam-5717	189	22	⊆	⊆	NUM
ejpam-5717	189	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	189	24	,	,	PUNCT
ejpam-5717	189	25	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	189	26	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	189	27	-	-	NOUN
ejpam-5717	189	28	cl(v	cl(v	NOUN
ejpam-5717	189	29	)	)	PUNCT
ejpam-5717	189	30	)	)	PUNCT
ejpam-5717	189	31	)	)	PUNCT
ejpam-5717	189	32	for	for	ADP
ejpam-5717	189	33	every	every	DET
ejpam-5717	189	34	σ1σ2	σ1σ2	NOUN
ejpam-5717	189	35	-	-	ADJ
ejpam-5717	189	36	open	open	ADJ
ejpam-5717	189	37	set	set	NOUN
ejpam-5717	189	38	v	v	NOUN
ejpam-5717	189	39	of	of	ADP
ejpam-5717	189	40	y	y	PROPN
ejpam-5717	189	41	;	;	PUNCT
ejpam-5717	189	42	(	(	PUNCT
ejpam-5717	189	43	5	5	X
ejpam-5717	189	44	)	)	PUNCT
ejpam-5717	189	45	f	f	NOUN
ejpam-5717	189	46	(	(	PUNCT
ejpam-5717	189	47	θ(τ1	θ(τ1	NOUN
ejpam-5717	189	48	,	,	PUNCT
ejpam-5717	189	49	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-5717	189	50	)	)	PUNCT
ejpam-5717	189	51	)	)	PUNCT
ejpam-5717	190	1	⊆	⊆	NUM
ejpam-5717	190	2	(	(	PUNCT
ejpam-5717	190	3	σ1	σ1	PROPN
ejpam-5717	190	4	,	,	PUNCT
ejpam-5717	190	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	190	6	-	-	PUNCT
ejpam-5717	190	7	cl(f	cl(f	PROPN
ejpam-5717	190	8	(	(	PUNCT
ejpam-5717	190	9	a	a	NOUN
ejpam-5717	190	10	)	)	PUNCT
ejpam-5717	190	11	)	)	PUNCT
ejpam-5717	190	12	for	for	ADP
ejpam-5717	190	13	every	every	DET
ejpam-5717	190	14	subset	subset	NOUN
ejpam-5717	190	15	a	a	PRON
ejpam-5717	190	16	of	of	ADP
ejpam-5717	190	17	x	x	PRON
ejpam-5717	190	18	;	;	PUNCT
ejpam-5717	190	19	(	(	PUNCT
ejpam-5717	190	20	6	6	NUM
ejpam-5717	190	21	)	)	PUNCT
ejpam-5717	190	22	θ(τ1	θ(τ1	NOUN
ejpam-5717	190	23	,	,	PUNCT
ejpam-5717	190	24	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	191	1	+	+	ADJ
ejpam-5717	191	2	(	(	PUNCT
ejpam-5717	191	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	191	4	-	-	PUNCT
ejpam-5717	191	5	int((σ1	int((σ1	ADJ
ejpam-5717	191	6	,	,	PUNCT
ejpam-5717	191	7	σ2)θ	σ2)θ	ADJ
ejpam-5717	191	8	-	-	PUNCT
ejpam-5717	191	9	cl(b	cl(b	NOUN
ejpam-5717	191	10	)	)	PUNCT
ejpam-5717	191	11	)	)	PUNCT
ejpam-5717	191	12	)	)	PUNCT
ejpam-5717	191	13	)	)	PUNCT
ejpam-5717	192	1	⊆	⊆	NUM
ejpam-5717	192	2	f+((σ1	f+((σ1	NOUN
ejpam-5717	192	3	,	,	PUNCT
ejpam-5717	192	4	σ2)θ	σ2)θ	ADJ
ejpam-5717	192	5	-	-	PUNCT
ejpam-5717	192	6	cl(b	cl(b	NOUN
ejpam-5717	192	7	)	)	PUNCT
ejpam-5717	192	8	)	)	PUNCT
ejpam-5717	192	9	for	for	ADP
ejpam-5717	192	10	every	every	DET
ejpam-5717	192	11	subset	subset	NOUN
ejpam-5717	192	12	b	b	PROPN
ejpam-5717	192	13	of	of	ADP
ejpam-5717	192	14	y	y	PROPN
ejpam-5717	192	15	;	;	PUNCT
ejpam-5717	192	16	(	(	PUNCT
ejpam-5717	192	17	7	7	X
ejpam-5717	192	18	)	)	PUNCT
ejpam-5717	192	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	192	20	,	,	PUNCT
ejpam-5717	192	21	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	193	1	+	+	ADJ
ejpam-5717	193	2	(	(	PUNCT
ejpam-5717	193	3	σ1σ2	σ1σ2	NUM
ejpam-5717	193	4	-	-	PUNCT
ejpam-5717	193	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	193	6	-	-	PUNCT
ejpam-5717	193	7	cl(v	cl(v	NOUN
ejpam-5717	193	8	)	)	PUNCT
ejpam-5717	193	9	)	)	PUNCT
ejpam-5717	193	10	)	)	PUNCT
ejpam-5717	193	11	)	)	PUNCT
ejpam-5717	194	1	⊆	⊆	X
ejpam-5717	194	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	194	3	-	-	PUNCT
ejpam-5717	194	4	cl(v	cl(v	NOUN
ejpam-5717	194	5	)	)	PUNCT
ejpam-5717	194	6	)	)	PUNCT
ejpam-5717	194	7	for	for	ADP
ejpam-5717	194	8	every	every	DET
ejpam-5717	194	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	194	10	-	-	ADJ
ejpam-5717	194	11	open	open	ADJ
ejpam-5717	194	12	set	set	NOUN
ejpam-5717	194	13	v	v	NOUN
ejpam-5717	194	14	of	of	ADP
ejpam-5717	194	15	y	y	PROPN
ejpam-5717	194	16	;	;	PUNCT
ejpam-5717	194	17	(	(	PUNCT
ejpam-5717	194	18	8)	8)	NUM
ejpam-5717	194	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	194	20	,	,	PUNCT
ejpam-5717	194	21	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	195	1	+	+	ADJ
ejpam-5717	195	2	(	(	PUNCT
ejpam-5717	195	3	σ1σ2	σ1σ2	NUM
ejpam-5717	195	4	-	-	PUNCT
ejpam-5717	195	5	int(k	int(k	NUM
ejpam-5717	195	6	)	)	PUNCT
ejpam-5717	195	7	)	)	PUNCT
ejpam-5717	195	8	)	)	PUNCT
ejpam-5717	196	1	⊆	⊆	NUM
ejpam-5717	196	2	f+(k	f+(k	NOUN
ejpam-5717	196	3	)	)	PUNCT
ejpam-5717	196	4	for	for	ADP
ejpam-5717	196	5	every	every	DET
ejpam-5717	196	6	(	(	PUNCT
ejpam-5717	196	7	σ1	σ1	PROPN
ejpam-5717	196	8	,	,	PUNCT
ejpam-5717	196	9	σ2)r	σ2)r	NOUN
ejpam-5717	196	10	-	-	PUNCT
ejpam-5717	196	11	closed	close	VERB
ejpam-5717	196	12	set	set	ADJ
ejpam-5717	196	13	k	k	PROPN
ejpam-5717	196	14	of	of	ADP
ejpam-5717	196	15	y	y	PROPN
ejpam-5717	196	16	;	;	PUNCT
ejpam-5717	196	17	(	(	PUNCT
ejpam-5717	196	18	9	9	X
ejpam-5717	196	19	)	)	PUNCT
ejpam-5717	196	20	θ(τ1	θ(τ1	NOUN
ejpam-5717	196	21	,	,	PUNCT
ejpam-5717	196	22	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	197	1	+	+	ADJ
ejpam-5717	197	2	(	(	PUNCT
ejpam-5717	197	3	σ1σ2	σ1σ2	NUM
ejpam-5717	197	4	-	-	PUNCT
ejpam-5717	197	5	int(k	int(k	NUM
ejpam-5717	197	6	)	)	PUNCT
ejpam-5717	197	7	)	)	PUNCT
ejpam-5717	197	8	)	)	PUNCT
ejpam-5717	198	1	⊆	⊆	NUM
ejpam-5717	198	2	f+(k	f+(k	NOUN
ejpam-5717	198	3	)	)	PUNCT
ejpam-5717	198	4	for	for	ADP
ejpam-5717	198	5	every	every	DET
ejpam-5717	198	6	σ1σ2	σ1σ2	NUM
ejpam-5717	198	7	-	-	PUNCT
ejpam-5717	198	8	closed	closed	ADJ
ejpam-5717	198	9	set	set	NOUN
ejpam-5717	198	10	k	k	PROPN
ejpam-5717	198	11	of	of	ADP
ejpam-5717	198	12	y	y	PROPN
ejpam-5717	198	13	.	.	PUNCT
ejpam-5717	199	1	proof	proof	NOUN
ejpam-5717	199	2	.	.	PUNCT
ejpam-5717	200	1	(	(	PUNCT
ejpam-5717	200	2	1	1	X
ejpam-5717	200	3	)	)	PUNCT
ejpam-5717	200	4	⇒	⇒	NOUN
ejpam-5717	200	5	(	(	PUNCT
ejpam-5717	200	6	2	2	NUM
ejpam-5717	200	7	):	):	PUNCT
ejpam-5717	200	8	let	let	VERB
ejpam-5717	200	9	b	b	X
ejpam-5717	200	10	be	be	AUX
ejpam-5717	200	11	any	any	DET
ejpam-5717	200	12	subset	subset	NOUN
ejpam-5717	200	13	of	of	ADP
ejpam-5717	200	14	y	y	PROPN
ejpam-5717	200	15	.	.	PUNCT
ejpam-5717	200	16	suppose	suppose	VERB
ejpam-5717	200	17	that	that	SCONJ
ejpam-5717	200	18	x	x	PROPN
ejpam-5717	200	19	̸∈	̸∈	PROPN
ejpam-5717	200	20	f+((σ1	f+((σ1	ADV
ejpam-5717	200	21	,	,	PUNCT
ejpam-5717	200	22	σ2)θ	σ2)θ	ADJ
ejpam-5717	200	23	-	-	PUNCT
ejpam-5717	200	24	cl(b	cl(b	NOUN
ejpam-5717	200	25	)	)	PUNCT
ejpam-5717	200	26	)	)	PUNCT
ejpam-5717	200	27	.	.	PUNCT
ejpam-5717	201	1	then	then	ADV
ejpam-5717	201	2	,	,	PUNCT
ejpam-5717	201	3	x	x	PUNCT
ejpam-5717	201	4	∈	∈	NOUN
ejpam-5717	201	5	f−(y	f−(y	NOUN
ejpam-5717	201	6	−	−	PROPN
ejpam-5717	201	7	(	(	PUNCT
ejpam-5717	201	8	σ1	σ1	PROPN
ejpam-5717	201	9	,	,	PUNCT
ejpam-5717	201	10	σ2)θ	σ2)θ	NOUN
ejpam-5717	201	11	-	-	PUNCT
ejpam-5717	201	12	cl(b	cl(b	NOUN
ejpam-5717	201	13	)	)	PUNCT
ejpam-5717	201	14	)	)	PUNCT
ejpam-5717	202	1	=	=	SYM
ejpam-5717	202	2	f−((σ1	f−((σ1	NOUN
ejpam-5717	202	3	,	,	PUNCT
ejpam-5717	202	4	σ2)θ	σ2)θ	ADJ
ejpam-5717	202	5	-	-	PUNCT
ejpam-5717	202	6	int(y	int(y	PROPN
ejpam-5717	202	7	−	−	PROPN
ejpam-5717	202	8	b	b	NOUN
ejpam-5717	202	9	)	)	PUNCT
ejpam-5717	202	10	)	)	PUNCT
ejpam-5717	202	11	.	.	PUNCT
ejpam-5717	203	1	since	since	SCONJ
ejpam-5717	203	2	f	f	PROPN
ejpam-5717	203	3	is	be	AUX
ejpam-5717	203	4	lower	low	ADJ
ejpam-5717	203	5	quasi	quasi	NOUN
ejpam-5717	203	6	θ(τ1	θ(τ1	NOUN
ejpam-5717	203	7	,	,	PUNCT
ejpam-5717	203	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	203	9	,	,	PUNCT
ejpam-5717	203	10	by	by	ADP
ejpam-5717	203	11	lemma	lemma	PROPN
ejpam-5717	203	12	2	2	NUM
ejpam-5717	203	13	there	there	PRON
ejpam-5717	203	14	exists	exist	VERB
ejpam-5717	203	15	a	a	DET
ejpam-5717	203	16	(	(	PUNCT
ejpam-5717	203	17	τ1	τ1	NOUN
ejpam-5717	203	18	,	,	PUNCT
ejpam-5717	203	19	τ2)s	τ2)s	NOUN
ejpam-5717	203	20	-	-	PUNCT
ejpam-5717	203	21	open	open	ADJ
ejpam-5717	203	22	set	set	NOUN
ejpam-5717	203	23	u	u	NOUN
ejpam-5717	203	24	of	of	ADP
ejpam-5717	203	25	x	x	PUNCT
ejpam-5717	203	26	containing	contain	VERB
ejpam-5717	203	27	x	x	PUNCT
ejpam-5717	203	28	such	such	ADJ
ejpam-5717	203	29	that	that	SCONJ
ejpam-5717	203	30	(	(	PUNCT
ejpam-5717	203	31	τ1	τ1	NOUN
ejpam-5717	203	32	,	,	PUNCT
ejpam-5717	203	33	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	203	34	)	)	PUNCT
ejpam-5717	203	35	⊆	⊆	NUM
ejpam-5717	203	36	f−(y	f−(y	NOUN
ejpam-5717	203	37	−b	−b	NOUN
ejpam-5717	203	38	)	)	PUNCT
ejpam-5717	204	1	=	=	PUNCT
ejpam-5717	204	2	x	x	X
ejpam-5717	205	1	−	−	NOUN
ejpam-5717	205	2	f+(b	f+(b	NOUN
ejpam-5717	205	3	)	)	PUNCT
ejpam-5717	205	4	.	.	PUNCT
ejpam-5717	206	1	thus	thus	ADV
ejpam-5717	206	2	,	,	PUNCT
ejpam-5717	206	3	we	we	PRON
ejpam-5717	206	4	have	have	AUX
ejpam-5717	206	5	(	(	PUNCT
ejpam-5717	206	6	τ1	τ1	NOUN
ejpam-5717	206	7	,	,	PUNCT
ejpam-5717	206	8	τ2)-scl(u	τ2)-scl(u	ADJ
ejpam-5717	206	9	)	)	PUNCT
ejpam-5717	206	10	∩	∩	ADJ
ejpam-5717	206	11	f+(b	f+(b	NOUN
ejpam-5717	206	12	)	)	PUNCT
ejpam-5717	206	13	=	=	SYM
ejpam-5717	206	14	∅	∅	NOUN
ejpam-5717	206	15	and	and	CCONJ
ejpam-5717	206	16	hence	hence	ADV
ejpam-5717	206	17	x	x	X
ejpam-5717	206	18	̸∈	̸∈	PROPN
ejpam-5717	206	19	θ(τ1	θ(τ1	VERB
ejpam-5717	206	20	,	,	PUNCT
ejpam-5717	206	21	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	207	1	+	+	ADJ
ejpam-5717	207	2	(	(	PUNCT
ejpam-5717	207	3	b	b	NOUN
ejpam-5717	207	4	)	)	PUNCT
ejpam-5717	207	5	)	)	PUNCT
ejpam-5717	207	6	.	.	PUNCT
ejpam-5717	208	1	(	(	PUNCT
ejpam-5717	208	2	2	2	X
ejpam-5717	208	3	)	)	PUNCT
ejpam-5717	208	4	⇒	⇒	NOUN
ejpam-5717	208	5	(	(	PUNCT
ejpam-5717	208	6	3	3	NUM
ejpam-5717	208	7	):	):	PUNCT
ejpam-5717	208	8	this	this	PRON
ejpam-5717	208	9	is	be	AUX
ejpam-5717	208	10	obvious	obvious	ADJ
ejpam-5717	208	11	since	since	SCONJ
ejpam-5717	208	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	208	13	-	-	NOUN
ejpam-5717	208	14	cl(v	cl(v	X
ejpam-5717	208	15	)	)	PUNCT
ejpam-5717	209	1	=	=	SYM
ejpam-5717	209	2	(	(	PUNCT
ejpam-5717	209	3	σ1	σ1	PROPN
ejpam-5717	209	4	,	,	PUNCT
ejpam-5717	209	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	209	6	-	-	PUNCT
ejpam-5717	209	7	cl(v	cl(v	NOUN
ejpam-5717	209	8	)	)	PUNCT
ejpam-5717	209	9	for	for	ADP
ejpam-5717	209	10	every	every	DET
ejpam-5717	209	11	σ1σ2	σ1σ2	NOUN
ejpam-5717	209	12	-	-	ADJ
ejpam-5717	209	13	open	open	ADJ
ejpam-5717	209	14	set	set	NOUN
ejpam-5717	209	15	v	v	NOUN
ejpam-5717	209	16	of	of	ADP
ejpam-5717	209	17	y	y	PROPN
ejpam-5717	209	18	.	.	PUNCT
ejpam-5717	210	1	(	(	PUNCT
ejpam-5717	210	2	3	3	X
ejpam-5717	210	3	)	)	PUNCT
ejpam-5717	210	4	⇒	⇒	NOUN
ejpam-5717	210	5	(	(	PUNCT
ejpam-5717	210	6	4	4	NUM
ejpam-5717	210	7	):	):	PUNCT
ejpam-5717	210	8	let	let	VERB
ejpam-5717	210	9	v	v	PART
ejpam-5717	210	10	be	be	AUX
ejpam-5717	210	11	any	any	DET
ejpam-5717	210	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	210	13	-	-	ADJ
ejpam-5717	210	14	open	open	ADJ
ejpam-5717	210	15	set	set	NOUN
ejpam-5717	210	16	of	of	ADP
ejpam-5717	210	17	y	y	PROPN
ejpam-5717	210	18	.	.	PUNCT
ejpam-5717	211	1	then	then	ADV
ejpam-5717	211	2	by	by	ADP
ejpam-5717	211	3	(	(	PUNCT
ejpam-5717	211	4	3	3	NUM
ejpam-5717	211	5	)	)	PUNCT
ejpam-5717	211	6	,	,	PUNCT
ejpam-5717	211	7	we	we	PRON
ejpam-5717	211	8	have	have	VERB
ejpam-5717	211	9	x	x	PART
ejpam-5717	211	10	−	−	NOUN
ejpam-5717	211	11	θ(τ1	θ(τ1	NOUN
ejpam-5717	211	12	,	,	PUNCT
ejpam-5717	211	13	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	211	14	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	211	15	-	-	NOUN
ejpam-5717	211	16	cl(v	cl(v	NOUN
ejpam-5717	211	17	)	)	PUNCT
ejpam-5717	211	18	)	)	PUNCT
ejpam-5717	211	19	)	)	PUNCT
ejpam-5717	212	1	=	=	SYM
ejpam-5717	212	2	θ(τ1	θ(τ1	PROPN
ejpam-5717	212	3	,	,	PUNCT
ejpam-5717	212	4	τ2)-scl(x	τ2)-scl(x	NOUN
ejpam-5717	212	5	−	−	ADP
ejpam-5717	212	6	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	212	7	-	-	PUNCT
ejpam-5717	212	8	cl(v	cl(v	NOUN
ejpam-5717	212	9	)	)	PUNCT
ejpam-5717	212	10	)	)	PUNCT
ejpam-5717	212	11	)	)	PUNCT
ejpam-5717	213	1	=	=	SYM
ejpam-5717	213	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	213	3	,	,	PUNCT
ejpam-5717	213	4	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	214	1	+	+	PROPN
ejpam-5717	214	2	(	(	PUNCT
ejpam-5717	214	3	y	y	PROPN
ejpam-5717	214	4	−	−	PROPN
ejpam-5717	214	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	214	6	-	-	NUM
ejpam-5717	214	7	cl(v	cl(v	NOUN
ejpam-5717	214	8	)	)	PUNCT
ejpam-5717	214	9	)	)	PUNCT
ejpam-5717	214	10	)	)	PUNCT
ejpam-5717	215	1	⊆	⊆	X
ejpam-5717	215	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	215	3	-	-	PUNCT
ejpam-5717	215	4	cl(y	cl(y	NOUN
ejpam-5717	215	5	−	−	NOUN
ejpam-5717	215	6	σ1σ2	σ1σ2	NOUN
ejpam-5717	215	7	-	-	NUM
ejpam-5717	215	8	cl(v	cl(v	NOUN
ejpam-5717	215	9	)	)	PUNCT
ejpam-5717	215	10	)	)	PUNCT
ejpam-5717	215	11	)	)	PUNCT
ejpam-5717	215	12	⊆	⊆	X
ejpam-5717	215	13	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	215	14	-	-	PUNCT
ejpam-5717	215	15	cl(y	cl(y	NOUN
ejpam-5717	215	16	−	−	PROPN
ejpam-5717	215	17	v	v	NOUN
ejpam-5717	215	18	)	)	PUNCT
ejpam-5717	215	19	)	)	PUNCT
ejpam-5717	216	1	=	=	PUNCT
ejpam-5717	217	1	f+(y	f+(y	NOUN
ejpam-5717	217	2	−	−	PROPN
ejpam-5717	217	3	v	v	NOUN
ejpam-5717	217	4	)	)	PUNCT
ejpam-5717	217	5	=	=	PUNCT
ejpam-5717	218	1	x	x	SYM
ejpam-5717	218	2	−	−	PROPN
ejpam-5717	218	3	f−(v	f−(v	PROPN
ejpam-5717	218	4	)	)	PUNCT
ejpam-5717	219	1	p.	p.	NOUN
ejpam-5717	219	2	pue	pue	NOUN
ejpam-5717	219	3	-	-	PUNCT
ejpam-5717	219	4	on	on	ADP
ejpam-5717	219	5	,	,	PUNCT
ejpam-5717	219	6	a.	a.	PROPN
ejpam-5717	219	7	sama	sama	PROPN
ejpam-5717	219	8	-	-	PUNCT
ejpam-5717	219	9	ae	ae	PROPN
ejpam-5717	219	10	,	,	PUNCT
ejpam-5717	219	11	c.	c.	PROPN
ejpam-5717	219	12	boonpok	boonpok	PROPN
ejpam-5717	219	13	/	/	SYM
ejpam-5717	219	14	eur	eur	PROPN
ejpam-5717	219	15	.	.	PUNCT
ejpam-5717	220	1	j.	j.	PROPN
ejpam-5717	220	2	pure	pure	PROPN
ejpam-5717	220	3	appl	appl	PROPN
ejpam-5717	220	4	.	.	PROPN
ejpam-5717	220	5	math	math	PROPN
ejpam-5717	220	6	,	,	PUNCT
ejpam-5717	220	7	18	18	NUM
ejpam-5717	220	8	(	(	PUNCT
ejpam-5717	220	9	1	1	NUM
ejpam-5717	220	10	)	)	PUNCT
ejpam-5717	220	11	(	(	PUNCT
ejpam-5717	220	12	2025	2025	NUM
ejpam-5717	220	13	)	)	PUNCT
ejpam-5717	220	14	,	,	PUNCT
ejpam-5717	220	15	5717	5717	NUM
ejpam-5717	220	16	7	7	NUM
ejpam-5717	220	17	of	of	ADP
ejpam-5717	220	18	16	16	NUM
ejpam-5717	220	19	and	and	CCONJ
ejpam-5717	220	20	hence	hence	ADV
ejpam-5717	220	21	f−(v	f−(v	ADJ
ejpam-5717	220	22	)	)	PUNCT
ejpam-5717	220	23	⊆	⊆	NUM
ejpam-5717	220	24	θ(τ1	θ(τ1	NOUN
ejpam-5717	220	25	,	,	PUNCT
ejpam-5717	220	26	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	220	27	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	220	28	-	-	NOUN
ejpam-5717	220	29	cl(v	cl(v	NOUN
ejpam-5717	220	30	)	)	PUNCT
ejpam-5717	220	31	)	)	PUNCT
ejpam-5717	220	32	)	)	PUNCT
ejpam-5717	220	33	.	.	PUNCT
ejpam-5717	221	1	(	(	PUNCT
ejpam-5717	221	2	4	4	X
ejpam-5717	221	3	)	)	PUNCT
ejpam-5717	221	4	⇒	⇒	NOUN
ejpam-5717	221	5	(	(	PUNCT
ejpam-5717	221	6	1	1	NUM
ejpam-5717	221	7	):	):	PUNCT
ejpam-5717	221	8	let	let	VERB
ejpam-5717	221	9	x	x	PUNCT
ejpam-5717	221	10	∈	∈	PROPN
ejpam-5717	221	11	x	x	X
ejpam-5717	221	12	and	and	CCONJ
ejpam-5717	221	13	v	v	X
ejpam-5717	221	14	be	be	AUX
ejpam-5717	221	15	any	any	DET
ejpam-5717	221	16	σ1σ2	σ1σ2	NOUN
ejpam-5717	221	17	-	-	ADJ
ejpam-5717	221	18	open	open	ADJ
ejpam-5717	221	19	set	set	NOUN
ejpam-5717	221	20	of	of	ADP
ejpam-5717	221	21	y	y	PRON
ejpam-5717	221	22	such	such	ADJ
ejpam-5717	221	23	that	that	SCONJ
ejpam-5717	221	24	f	f	PROPN
ejpam-5717	221	25	(	(	PUNCT
ejpam-5717	221	26	x	x	NOUN
ejpam-5717	221	27	)	)	PUNCT
ejpam-5717	221	28	∩	∩	NOUN
ejpam-5717	221	29	v	v	ADP
ejpam-5717	221	30	̸=	̸=	PROPN
ejpam-5717	221	31	∅.	∅.	ADV
ejpam-5717	221	32	by	by	ADP
ejpam-5717	221	33	(	(	PUNCT
ejpam-5717	221	34	4	4	NUM
ejpam-5717	221	35	)	)	PUNCT
ejpam-5717	221	36	,	,	PUNCT
ejpam-5717	221	37	x	x	PUNCT
ejpam-5717	221	38	∈	∈	PROPN
ejpam-5717	221	39	f−(v	f−(v	NOUN
ejpam-5717	221	40	)	)	PUNCT
ejpam-5717	221	41	⊆	⊆	NUM
ejpam-5717	221	42	θ(τ1	θ(τ1	NOUN
ejpam-5717	221	43	,	,	PUNCT
ejpam-5717	221	44	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	221	45	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	221	46	-	-	NOUN
ejpam-5717	221	47	cl(v	cl(v	NOUN
ejpam-5717	221	48	)	)	PUNCT
ejpam-5717	221	49	)	)	PUNCT
ejpam-5717	221	50	)	)	PUNCT
ejpam-5717	221	51	.	.	PUNCT
ejpam-5717	222	1	then	then	ADV
ejpam-5717	222	2	,	,	PUNCT
ejpam-5717	222	3	there	there	PRON
ejpam-5717	222	4	exists	exist	VERB
ejpam-5717	222	5	a	a	DET
ejpam-5717	222	6	(	(	PUNCT
ejpam-5717	222	7	τ1	τ1	NOUN
ejpam-5717	222	8	,	,	PUNCT
ejpam-5717	222	9	τ2)s	τ2)s	NOUN
ejpam-5717	222	10	-	-	PUNCT
ejpam-5717	222	11	open	open	ADJ
ejpam-5717	222	12	set	set	NOUN
ejpam-5717	222	13	u	u	NOUN
ejpam-5717	222	14	of	of	ADP
ejpam-5717	222	15	x	x	PUNCT
ejpam-5717	222	16	containing	contain	VERB
ejpam-5717	222	17	x	x	PUNCT
ejpam-5717	222	18	such	such	ADJ
ejpam-5717	222	19	that	that	SCONJ
ejpam-5717	222	20	(	(	PUNCT
ejpam-5717	222	21	τ1	τ1	NOUN
ejpam-5717	222	22	,	,	PUNCT
ejpam-5717	222	23	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	222	24	)	)	PUNCT
ejpam-5717	222	25	⊆	⊆	NUM
ejpam-5717	222	26	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	222	27	-	-	PUNCT
ejpam-5717	222	28	cl(v	cl(v	NOUN
ejpam-5717	222	29	)	)	PUNCT
ejpam-5717	222	30	)	)	PUNCT
ejpam-5717	222	31	;	;	PUNCT
ejpam-5717	222	32	hence	hence	ADV
ejpam-5717	222	33	σ1σ2	σ1σ2	NOUN
ejpam-5717	222	34	-	-	PUNCT
ejpam-5717	222	35	cl(v	cl(v	NOUN
ejpam-5717	222	36	)	)	PUNCT
ejpam-5717	222	37	∩	∩	PROPN
ejpam-5717	222	38	f	f	X
ejpam-5717	222	39	(	(	PUNCT
ejpam-5717	222	40	z	z	NOUN
ejpam-5717	222	41	)	)	PUNCT
ejpam-5717	222	42	̸=	̸=	NOUN
ejpam-5717	222	43	∅	∅	NOUN
ejpam-5717	222	44	for	for	ADP
ejpam-5717	222	45	every	every	DET
ejpam-5717	222	46	z	z	NOUN
ejpam-5717	222	47	∈	∈	PROPN
ejpam-5717	222	48	(	(	PUNCT
ejpam-5717	222	49	τ1	τ1	NOUN
ejpam-5717	222	50	,	,	PUNCT
ejpam-5717	222	51	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	222	52	)	)	PUNCT
ejpam-5717	222	53	.	.	PUNCT
ejpam-5717	223	1	this	this	PRON
ejpam-5717	223	2	shows	show	VERB
ejpam-5717	223	3	that	that	SCONJ
ejpam-5717	223	4	f	f	PROPN
ejpam-5717	223	5	is	be	AUX
ejpam-5717	223	6	lower	low	ADJ
ejpam-5717	223	7	quasi	quasi	NOUN
ejpam-5717	223	8	θ(τ1	θ(τ1	NOUN
ejpam-5717	223	9	,	,	PUNCT
ejpam-5717	223	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	223	11	.	.	PUNCT
ejpam-5717	224	1	(	(	PUNCT
ejpam-5717	224	2	2	2	X
ejpam-5717	224	3	)	)	PUNCT
ejpam-5717	224	4	⇒	⇒	NOUN
ejpam-5717	224	5	(	(	PUNCT
ejpam-5717	224	6	5	5	NUM
ejpam-5717	224	7	):	):	PUNCT
ejpam-5717	224	8	let	let	VERB
ejpam-5717	224	9	a	a	PRON
ejpam-5717	224	10	be	be	AUX
ejpam-5717	224	11	any	any	DET
ejpam-5717	224	12	subset	subset	NOUN
ejpam-5717	224	13	of	of	ADP
ejpam-5717	224	14	x.	x.	NOUN
ejpam-5717	224	15	by	by	ADP
ejpam-5717	224	16	replacing	replace	VERB
ejpam-5717	224	17	b	b	NOUN
ejpam-5717	224	18	in	in	ADP
ejpam-5717	224	19	(	(	PUNCT
ejpam-5717	224	20	2	2	NUM
ejpam-5717	224	21	)	)	PUNCT
ejpam-5717	224	22	by	by	ADP
ejpam-5717	224	23	f	f	PROPN
ejpam-5717	224	24	(	(	PUNCT
ejpam-5717	224	25	a	a	PROPN
ejpam-5717	224	26	)	)	PUNCT
ejpam-5717	224	27	,	,	PUNCT
ejpam-5717	224	28	we	we	PRON
ejpam-5717	224	29	have	have	VERB
ejpam-5717	224	30	θ(τ1	θ(τ1	NOUN
ejpam-5717	224	31	,	,	PUNCT
ejpam-5717	224	32	τ2)-scl(a	τ2)-scl(a	ADJ
ejpam-5717	224	33	)	)	PUNCT
ejpam-5717	224	34	⊆	⊆	NUM
ejpam-5717	224	35	θ(τ1	θ(τ1	NOUN
ejpam-5717	224	36	,	,	PUNCT
ejpam-5717	224	37	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	225	1	+	+	PROPN
ejpam-5717	225	2	(	(	PUNCT
ejpam-5717	225	3	f	f	X
ejpam-5717	225	4	(	(	PUNCT
ejpam-5717	225	5	a	a	NOUN
ejpam-5717	225	6	)	)	PUNCT
ejpam-5717	225	7	)	)	PUNCT
ejpam-5717	225	8	)	)	PUNCT
ejpam-5717	225	9	⊆	⊆	NUM
ejpam-5717	225	10	f+((σ1	f+((σ1	NOUN
ejpam-5717	225	11	,	,	PUNCT
ejpam-5717	225	12	σ2)θ	σ2)θ	NOUN
ejpam-5717	225	13	-	-	PUNCT
ejpam-5717	225	14	cl(f	cl(f	PROPN
ejpam-5717	225	15	(	(	PUNCT
ejpam-5717	225	16	a	a	NOUN
ejpam-5717	225	17	)	)	PUNCT
ejpam-5717	225	18	)	)	PUNCT
ejpam-5717	225	19	)	)	PUNCT
ejpam-5717	225	20	.	.	PUNCT
ejpam-5717	226	1	thus	thus	ADV
ejpam-5717	226	2	,	,	PUNCT
ejpam-5717	226	3	f	f	PROPN
ejpam-5717	226	4	(	(	PUNCT
ejpam-5717	226	5	θ(τ1	θ(τ1	PROPN
ejpam-5717	226	6	,	,	PUNCT
ejpam-5717	226	7	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-5717	226	8	)	)	PUNCT
ejpam-5717	226	9	)	)	PUNCT
ejpam-5717	226	10	⊆	⊆	NUM
ejpam-5717	226	11	(	(	PUNCT
ejpam-5717	226	12	σ1	σ1	PROPN
ejpam-5717	226	13	,	,	PUNCT
ejpam-5717	226	14	σ2)θ	σ2)θ	NOUN
ejpam-5717	226	15	-	-	PUNCT
ejpam-5717	226	16	cl(f	cl(f	PROPN
ejpam-5717	226	17	(	(	PUNCT
ejpam-5717	226	18	a	a	NOUN
ejpam-5717	226	19	)	)	PUNCT
ejpam-5717	226	20	)	)	PUNCT
ejpam-5717	226	21	.	.	PUNCT
ejpam-5717	227	1	(	(	PUNCT
ejpam-5717	227	2	5	5	X
ejpam-5717	227	3	)	)	PUNCT
ejpam-5717	227	4	⇒	⇒	NOUN
ejpam-5717	227	5	(	(	PUNCT
ejpam-5717	227	6	2	2	NUM
ejpam-5717	227	7	):	):	PUNCT
ejpam-5717	227	8	let	let	VERB
ejpam-5717	227	9	b	b	X
ejpam-5717	227	10	be	be	AUX
ejpam-5717	227	11	any	any	DET
ejpam-5717	227	12	subset	subset	NOUN
ejpam-5717	227	13	of	of	ADP
ejpam-5717	227	14	y	y	PROPN
ejpam-5717	227	15	.	.	PUNCT
ejpam-5717	228	1	replacing	replace	VERB
ejpam-5717	228	2	a	a	DET
ejpam-5717	228	3	in	in	ADP
ejpam-5717	228	4	(	(	PUNCT
ejpam-5717	228	5	5	5	NUM
ejpam-5717	228	6	)	)	PUNCT
ejpam-5717	228	7	by	by	ADP
ejpam-5717	228	8	f+(b	f+(b	NOUN
ejpam-5717	228	9	)	)	PUNCT
ejpam-5717	228	10	,	,	PUNCT
ejpam-5717	228	11	we	we	PRON
ejpam-5717	228	12	have	have	VERB
ejpam-5717	228	13	f	f	PROPN
ejpam-5717	228	14	(	(	PUNCT
ejpam-5717	228	15	θ(τ1	θ(τ1	NOUN
ejpam-5717	228	16	,	,	PUNCT
ejpam-5717	228	17	τ2)-scl(f	τ2)-scl(f	X
ejpam-5717	229	1	+	+	ADJ
ejpam-5717	229	2	(	(	PUNCT
ejpam-5717	229	3	b	b	NOUN
ejpam-5717	229	4	)	)	PUNCT
ejpam-5717	229	5	)	)	PUNCT
ejpam-5717	229	6	)	)	PUNCT
ejpam-5717	230	1	⊆	⊆	X
ejpam-5717	230	2	(	(	PUNCT
ejpam-5717	230	3	σ1	σ1	PROPN
ejpam-5717	230	4	,	,	PUNCT
ejpam-5717	230	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	230	6	-	-	PUNCT
ejpam-5717	230	7	cl(f	cl(f	PROPN
ejpam-5717	230	8	(	(	PUNCT
ejpam-5717	230	9	f+(b	f+(b	PROPN
ejpam-5717	230	10	)	)	PUNCT
ejpam-5717	230	11	)	)	PUNCT
ejpam-5717	230	12	)	)	PUNCT
ejpam-5717	231	1	⊆	⊆	X
ejpam-5717	231	2	(	(	PUNCT
ejpam-5717	231	3	σ1	σ1	PROPN
ejpam-5717	231	4	,	,	PUNCT
ejpam-5717	231	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	231	6	-	-	PUNCT
ejpam-5717	231	7	cl(b	cl(b	NOUN
ejpam-5717	231	8	)	)	PUNCT
ejpam-5717	231	9	and	and	CCONJ
ejpam-5717	231	10	hence	hence	ADV
ejpam-5717	231	11	θ(τ1	θ(τ1	VERB
ejpam-5717	231	12	,	,	PUNCT
ejpam-5717	231	13	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	231	14	+	+	ADJ
ejpam-5717	231	15	(	(	PUNCT
ejpam-5717	231	16	b	b	NOUN
ejpam-5717	231	17	)	)	PUNCT
ejpam-5717	231	18	)	)	PUNCT
ejpam-5717	231	19	⊆	⊆	NUM
ejpam-5717	231	20	f+((σ1	f+((σ1	NOUN
ejpam-5717	231	21	,	,	PUNCT
ejpam-5717	231	22	σ2)θ	σ2)θ	ADJ
ejpam-5717	231	23	-	-	PUNCT
ejpam-5717	231	24	cl(b	cl(b	NOUN
ejpam-5717	231	25	)	)	PUNCT
ejpam-5717	231	26	)	)	PUNCT
ejpam-5717	231	27	.	.	PUNCT
ejpam-5717	232	1	(	(	PUNCT
ejpam-5717	232	2	3	3	X
ejpam-5717	232	3	)	)	PUNCT
ejpam-5717	232	4	⇒	⇒	NOUN
ejpam-5717	232	5	(	(	PUNCT
ejpam-5717	232	6	6	6	NUM
ejpam-5717	232	7	):	):	PUNCT
ejpam-5717	232	8	let	let	VERB
ejpam-5717	232	9	b	b	X
ejpam-5717	232	10	be	be	AUX
ejpam-5717	232	11	any	any	DET
ejpam-5717	232	12	subset	subset	NOUN
ejpam-5717	232	13	of	of	ADP
ejpam-5717	232	14	y	y	PROPN
ejpam-5717	232	15	.	.	PUNCT
ejpam-5717	233	1	put	put	VERB
ejpam-5717	233	2	v	v	NOUN
ejpam-5717	233	3	=	=	SYM
ejpam-5717	233	4	σ1σ2	σ1σ2	NOUN
ejpam-5717	233	5	-	-	PUNCT
ejpam-5717	233	6	int((σ1	int((σ1	ADJ
ejpam-5717	233	7	,	,	PUNCT
ejpam-5717	233	8	σ2)θ	σ2)θ	ADJ
ejpam-5717	233	9	-	-	PUNCT
ejpam-5717	233	10	cl(b	cl(b	NOUN
ejpam-5717	233	11	)	)	PUNCT
ejpam-5717	233	12	)	)	PUNCT
ejpam-5717	233	13	in	in	ADP
ejpam-5717	233	14	(	(	PUNCT
ejpam-5717	233	15	3	3	NUM
ejpam-5717	233	16	)	)	PUNCT
ejpam-5717	233	17	.	.	PUNCT
ejpam-5717	234	1	then	then	ADV
ejpam-5717	234	2	,	,	PUNCT
ejpam-5717	234	3	since	since	SCONJ
ejpam-5717	234	4	(	(	PUNCT
ejpam-5717	234	5	σ1	σ1	PROPN
ejpam-5717	234	6	,	,	PUNCT
ejpam-5717	234	7	σ2)θ	σ2)θ	NOUN
ejpam-5717	234	8	-	-	PUNCT
ejpam-5717	234	9	cl(b	cl(b	NOUN
ejpam-5717	234	10	)	)	PUNCT
ejpam-5717	234	11	is	be	AUX
ejpam-5717	234	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	234	13	-	-	ADJ
ejpam-5717	234	14	closed	closed	ADJ
ejpam-5717	234	15	in	in	ADP
ejpam-5717	234	16	y	y	PROPN
ejpam-5717	234	17	,	,	PUNCT
ejpam-5717	234	18	we	we	PRON
ejpam-5717	234	19	have	have	AUX
ejpam-5717	234	20	θ(τ1	θ(τ1	NOUN
ejpam-5717	234	21	,	,	PUNCT
ejpam-5717	234	22	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	235	1	+	+	ADJ
ejpam-5717	235	2	(	(	PUNCT
ejpam-5717	235	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	235	4	-	-	PUNCT
ejpam-5717	235	5	int((σ1	int((σ1	ADJ
ejpam-5717	235	6	,	,	PUNCT
ejpam-5717	235	7	σ2)θ	σ2)θ	ADJ
ejpam-5717	235	8	-	-	PUNCT
ejpam-5717	235	9	cl(b	cl(b	NOUN
ejpam-5717	235	10	)	)	PUNCT
ejpam-5717	235	11	)	)	PUNCT
ejpam-5717	235	12	)	)	PUNCT
ejpam-5717	235	13	)	)	PUNCT
ejpam-5717	236	1	⊆	⊆	X
ejpam-5717	236	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	236	3	-	-	PUNCT
ejpam-5717	236	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5717	236	5	-	-	PUNCT
ejpam-5717	236	6	int((σ1	int((σ1	ADJ
ejpam-5717	236	7	,	,	PUNCT
ejpam-5717	236	8	σ2)θ	σ2)θ	ADJ
ejpam-5717	236	9	-	-	PUNCT
ejpam-5717	236	10	cl(b	cl(b	NOUN
ejpam-5717	236	11	)	)	PUNCT
ejpam-5717	236	12	)	)	PUNCT
ejpam-5717	236	13	)	)	PUNCT
ejpam-5717	236	14	)	)	PUNCT
ejpam-5717	237	1	⊆	⊆	NUM
ejpam-5717	237	2	f+((σ1	f+((σ1	NOUN
ejpam-5717	237	3	,	,	PUNCT
ejpam-5717	237	4	σ2)θ	σ2)θ	ADJ
ejpam-5717	237	5	-	-	PUNCT
ejpam-5717	237	6	cl(b	cl(b	NOUN
ejpam-5717	237	7	)	)	PUNCT
ejpam-5717	237	8	)	)	PUNCT
ejpam-5717	237	9	.	.	PUNCT
ejpam-5717	238	1	(	(	PUNCT
ejpam-5717	238	2	6	6	X
ejpam-5717	238	3	)	)	PUNCT
ejpam-5717	238	4	⇒	⇒	NOUN
ejpam-5717	238	5	(	(	PUNCT
ejpam-5717	238	6	7	7	NUM
ejpam-5717	238	7	):	):	PUNCT
ejpam-5717	238	8	this	this	PRON
ejpam-5717	238	9	is	be	AUX
ejpam-5717	238	10	obvious	obvious	ADJ
ejpam-5717	238	11	since	since	SCONJ
ejpam-5717	238	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	238	13	-	-	NOUN
ejpam-5717	238	14	cl(v	cl(v	X
ejpam-5717	238	15	)	)	PUNCT
ejpam-5717	239	1	=	=	SYM
ejpam-5717	239	2	(	(	PUNCT
ejpam-5717	239	3	σ1	σ1	PROPN
ejpam-5717	239	4	,	,	PUNCT
ejpam-5717	239	5	σ2)θ	σ2)θ	NOUN
ejpam-5717	239	6	-	-	PUNCT
ejpam-5717	239	7	cl(v	cl(v	NOUN
ejpam-5717	239	8	)	)	PUNCT
ejpam-5717	239	9	for	for	ADP
ejpam-5717	239	10	every	every	DET
ejpam-5717	239	11	σ1σ2	σ1σ2	NOUN
ejpam-5717	239	12	-	-	ADJ
ejpam-5717	239	13	open	open	ADJ
ejpam-5717	239	14	set	set	NOUN
ejpam-5717	239	15	v	v	NOUN
ejpam-5717	239	16	of	of	ADP
ejpam-5717	239	17	y	y	PROPN
ejpam-5717	239	18	.	.	PUNCT
ejpam-5717	240	1	(	(	PUNCT
ejpam-5717	240	2	7	7	X
ejpam-5717	240	3	)	)	PUNCT
ejpam-5717	240	4	⇒	⇒	NOUN
ejpam-5717	240	5	(	(	PUNCT
ejpam-5717	240	6	8)	8)	NUM
ejpam-5717	240	7	:	:	PUNCT
ejpam-5717	240	8	let	let	VERB
ejpam-5717	240	9	k	k	X
ejpam-5717	240	10	be	be	AUX
ejpam-5717	240	11	any	any	DET
ejpam-5717	240	12	(	(	PUNCT
ejpam-5717	240	13	σ1	σ1	NOUN
ejpam-5717	240	14	,	,	PUNCT
ejpam-5717	240	15	σ2)r	σ2)r	NOUN
ejpam-5717	240	16	-	-	PUNCT
ejpam-5717	240	17	closed	close	VERB
ejpam-5717	240	18	set	set	NOUN
ejpam-5717	240	19	of	of	ADP
ejpam-5717	240	20	y	y	PROPN
ejpam-5717	240	21	.	.	PUNCT
ejpam-5717	241	1	then	then	ADV
ejpam-5717	241	2	by	by	ADP
ejpam-5717	241	3	(	(	PUNCT
ejpam-5717	241	4	7	7	NUM
ejpam-5717	241	5	)	)	PUNCT
ejpam-5717	241	6	,	,	PUNCT
ejpam-5717	241	7	we	we	PRON
ejpam-5717	241	8	have	have	VERB
ejpam-5717	241	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	241	10	,	,	PUNCT
ejpam-5717	241	11	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	242	1	+	+	ADJ
ejpam-5717	242	2	(	(	PUNCT
ejpam-5717	242	3	σ1σ2	σ1σ2	NUM
ejpam-5717	242	4	-	-	PUNCT
ejpam-5717	242	5	int(k	int(k	NOUN
ejpam-5717	242	6	)	)	PUNCT
ejpam-5717	242	7	)	)	PUNCT
ejpam-5717	242	8	)	)	PUNCT
ejpam-5717	243	1	=	=	SYM
ejpam-5717	243	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	243	3	,	,	PUNCT
ejpam-5717	243	4	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	244	1	+	+	ADJ
ejpam-5717	244	2	(	(	PUNCT
ejpam-5717	244	3	σ1σ2	σ1σ2	ADJ
ejpam-5717	244	4	-	-	PUNCT
ejpam-5717	244	5	int(σ1σ2	int(σ1σ2	ADV
ejpam-5717	244	6	-	-	PUNCT
ejpam-5717	244	7	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5717	244	8	-	-	PUNCT
ejpam-5717	244	9	int(k	int(k	NOUN
ejpam-5717	244	10	)	)	PUNCT
ejpam-5717	244	11	)	)	PUNCT
ejpam-5717	244	12	)	)	PUNCT
ejpam-5717	244	13	)	)	PUNCT
ejpam-5717	244	14	)	)	PUNCT
ejpam-5717	245	1	⊆	⊆	X
ejpam-5717	245	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	245	3	-	-	PUNCT
ejpam-5717	245	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	245	5	-	-	PUNCT
ejpam-5717	245	6	int(k	int(k	NOUN
ejpam-5717	245	7	)	)	PUNCT
ejpam-5717	245	8	)	)	PUNCT
ejpam-5717	245	9	)	)	PUNCT
ejpam-5717	246	1	=	=	SYM
ejpam-5717	246	2	f+(k	f+(k	NOUN
ejpam-5717	246	3	)	)	PUNCT
ejpam-5717	246	4	.	.	PUNCT
ejpam-5717	247	1	(	(	PUNCT
ejpam-5717	247	2	8)	8)	NUM
ejpam-5717	247	3	⇒	⇒	NOUN
ejpam-5717	247	4	(	(	PUNCT
ejpam-5717	247	5	9	9	NUM
ejpam-5717	247	6	):	):	PUNCT
ejpam-5717	247	7	letk	letk	ADJ
ejpam-5717	247	8	be	be	AUX
ejpam-5717	247	9	any	any	DET
ejpam-5717	247	10	σ1σ2	σ1σ2	NUM
ejpam-5717	247	11	-	-	PUNCT
ejpam-5717	247	12	closed	closed	ADJ
ejpam-5717	247	13	set	set	NOUN
ejpam-5717	247	14	of	of	ADP
ejpam-5717	247	15	y	y	PROPN
ejpam-5717	247	16	.	.	PUNCT
ejpam-5717	248	1	then	then	ADV
ejpam-5717	248	2	,	,	PUNCT
ejpam-5717	248	3	σ1σ2	σ1σ2	X
ejpam-5717	248	4	-	-	PUNCT
ejpam-5717	248	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	248	6	-	-	PUNCT
ejpam-5717	248	7	int(k	int(k	NOUN
ejpam-5717	248	8	)	)	PUNCT
ejpam-5717	248	9	)	)	PUNCT
ejpam-5717	248	10	is	be	AUX
ejpam-5717	248	11	(	(	PUNCT
ejpam-5717	248	12	σ1	σ1	PROPN
ejpam-5717	248	13	,	,	PUNCT
ejpam-5717	248	14	σ2)rclosed	σ2)rclose	VERB
ejpam-5717	248	15	in	in	ADP
ejpam-5717	248	16	y	y	PROPN
ejpam-5717	248	17	and	and	CCONJ
ejpam-5717	248	18	by	by	ADP
ejpam-5717	248	19	(	(	PUNCT
ejpam-5717	248	20	8)	8)	NUM
ejpam-5717	248	21	,	,	PUNCT
ejpam-5717	248	22	θ(τ1	θ(τ1	NOUN
ejpam-5717	248	23	,	,	PUNCT
ejpam-5717	248	24	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	249	1	+	+	ADJ
ejpam-5717	249	2	(	(	PUNCT
ejpam-5717	249	3	σ1σ2	σ1σ2	NUM
ejpam-5717	249	4	-	-	PUNCT
ejpam-5717	249	5	int(k	int(k	NOUN
ejpam-5717	249	6	)	)	PUNCT
ejpam-5717	249	7	)	)	PUNCT
ejpam-5717	249	8	)	)	PUNCT
ejpam-5717	250	1	=	=	SYM
ejpam-5717	250	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	250	3	,	,	PUNCT
ejpam-5717	250	4	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	251	1	+	+	ADJ
ejpam-5717	251	2	(	(	PUNCT
ejpam-5717	251	3	σ1σ2	σ1σ2	ADJ
ejpam-5717	251	4	-	-	PUNCT
ejpam-5717	251	5	int(σ1σ2	int(σ1σ2	ADV
ejpam-5717	251	6	-	-	PUNCT
ejpam-5717	251	7	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5717	251	8	-	-	PUNCT
ejpam-5717	251	9	int(k	int(k	NOUN
ejpam-5717	251	10	)	)	PUNCT
ejpam-5717	251	11	)	)	PUNCT
ejpam-5717	251	12	)	)	PUNCT
ejpam-5717	251	13	)	)	PUNCT
ejpam-5717	251	14	)	)	PUNCT
ejpam-5717	252	1	⊆	⊆	X
ejpam-5717	252	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	252	3	-	-	PUNCT
ejpam-5717	252	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	252	5	-	-	PUNCT
ejpam-5717	252	6	int(k	int(k	NOUN
ejpam-5717	252	7	)	)	PUNCT
ejpam-5717	252	8	)	)	PUNCT
ejpam-5717	252	9	)	)	PUNCT
ejpam-5717	253	1	⊆	⊆	NUM
ejpam-5717	253	2	f+(k	f+(k	NOUN
ejpam-5717	253	3	)	)	PUNCT
ejpam-5717	253	4	.	.	PUNCT
ejpam-5717	254	1	(	(	PUNCT
ejpam-5717	254	2	9	9	X
ejpam-5717	254	3	)	)	PUNCT
ejpam-5717	254	4	⇒	⇒	NOUN
ejpam-5717	254	5	(	(	PUNCT
ejpam-5717	254	6	4	4	NUM
ejpam-5717	254	7	):	):	PUNCT
ejpam-5717	254	8	let	let	VERB
ejpam-5717	254	9	v	v	PART
ejpam-5717	254	10	be	be	AUX
ejpam-5717	254	11	any	any	DET
ejpam-5717	254	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	254	13	-	-	ADJ
ejpam-5717	254	14	open	open	ADJ
ejpam-5717	254	15	set	set	NOUN
ejpam-5717	254	16	of	of	ADP
ejpam-5717	254	17	y	y	PROPN
ejpam-5717	254	18	.	.	PUNCT
ejpam-5717	255	1	then	then	ADV
ejpam-5717	255	2	,	,	PUNCT
ejpam-5717	255	3	y	y	PROPN
ejpam-5717	255	4	−	−	PROPN
ejpam-5717	255	5	v	v	NOUN
ejpam-5717	255	6	is	be	AUX
ejpam-5717	255	7	σ1σ2	σ1σ2	NOUN
ejpam-5717	255	8	-	-	ADJ
ejpam-5717	255	9	closed	closed	ADJ
ejpam-5717	255	10	in	in	ADP
ejpam-5717	255	11	y	y	PROPN
ejpam-5717	255	12	and	and	CCONJ
ejpam-5717	255	13	by	by	ADP
ejpam-5717	255	14	(	(	PUNCT
ejpam-5717	255	15	9	9	NUM
ejpam-5717	255	16	)	)	PUNCT
ejpam-5717	255	17	,	,	PUNCT
ejpam-5717	255	18	θ(τ1	θ(τ1	VERB
ejpam-5717	255	19	,	,	PUNCT
ejpam-5717	255	20	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	256	1	+	+	ADJ
ejpam-5717	256	2	(	(	PUNCT
ejpam-5717	256	3	σ1σ2	σ1σ2	X
ejpam-5717	256	4	-	-	PUNCT
ejpam-5717	256	5	int(y	int(y	ADJ
ejpam-5717	256	6	−	−	PROPN
ejpam-5717	256	7	v	v	NOUN
ejpam-5717	256	8	)	)	PUNCT
ejpam-5717	256	9	)	)	PUNCT
ejpam-5717	256	10	)	)	PUNCT
ejpam-5717	257	1	⊆	⊆	NUM
ejpam-5717	257	2	f+(y	f+(y	ADP
ejpam-5717	257	3	−	−	PROPN
ejpam-5717	257	4	v	v	NOUN
ejpam-5717	257	5	)	)	PUNCT
ejpam-5717	257	6	=	=	PUNCT
ejpam-5717	257	7	x	x	SYM
ejpam-5717	257	8	−	−	NOUN
ejpam-5717	257	9	f−(v	f−(v	NOUN
ejpam-5717	257	10	)	)	PUNCT
ejpam-5717	257	11	.	.	PUNCT
ejpam-5717	258	1	moreover	moreover	ADV
ejpam-5717	258	2	,	,	PUNCT
ejpam-5717	258	3	we	we	PRON
ejpam-5717	258	4	have	have	VERB
ejpam-5717	258	5	θ(τ1	θ(τ1	NOUN
ejpam-5717	258	6	,	,	PUNCT
ejpam-5717	258	7	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	259	1	+	+	ADJ
ejpam-5717	259	2	(	(	PUNCT
ejpam-5717	259	3	σ1σ2	σ1σ2	X
ejpam-5717	259	4	-	-	PUNCT
ejpam-5717	259	5	int(y	int(y	ADJ
ejpam-5717	259	6	−	−	PROPN
ejpam-5717	259	7	v	v	NOUN
ejpam-5717	259	8	)	)	PUNCT
ejpam-5717	259	9	)	)	PUNCT
ejpam-5717	259	10	)	)	PUNCT
ejpam-5717	260	1	=	=	SYM
ejpam-5717	260	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	260	3	,	,	PUNCT
ejpam-5717	260	4	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	261	1	+	+	PROPN
ejpam-5717	261	2	(	(	PUNCT
ejpam-5717	261	3	y	y	PROPN
ejpam-5717	261	4	−	−	PROPN
ejpam-5717	261	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	261	6	-	-	NUM
ejpam-5717	261	7	cl(v	cl(v	NOUN
ejpam-5717	261	8	)	)	PUNCT
ejpam-5717	261	9	)	)	PUNCT
ejpam-5717	261	10	)	)	PUNCT
ejpam-5717	262	1	=	=	SYM
ejpam-5717	262	2	θ(τ1	θ(τ1	PROPN
ejpam-5717	262	3	,	,	PUNCT
ejpam-5717	262	4	τ2)-scl(x	τ2)-scl(x	NOUN
ejpam-5717	262	5	−	−	ADP
ejpam-5717	262	6	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	262	7	-	-	PUNCT
ejpam-5717	262	8	cl(v	cl(v	NOUN
ejpam-5717	262	9	)	)	PUNCT
ejpam-5717	262	10	)	)	PUNCT
ejpam-5717	262	11	)	)	PUNCT
ejpam-5717	263	1	=	=	PUNCT
ejpam-5717	263	2	x	x	SYM
ejpam-5717	263	3	−	−	NOUN
ejpam-5717	263	4	θ(τ1	θ(τ1	NOUN
ejpam-5717	263	5	,	,	PUNCT
ejpam-5717	263	6	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	263	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	263	8	-	-	NOUN
ejpam-5717	263	9	cl(v	cl(v	NOUN
ejpam-5717	263	10	)	)	PUNCT
ejpam-5717	263	11	)	)	PUNCT
ejpam-5717	263	12	)	)	PUNCT
ejpam-5717	263	13	.	.	PUNCT
ejpam-5717	264	1	thus	thus	ADV
ejpam-5717	264	2	,	,	PUNCT
ejpam-5717	264	3	f−(v	f−(v	ADJ
ejpam-5717	264	4	)	)	PUNCT
ejpam-5717	264	5	⊆	⊆	NUM
ejpam-5717	264	6	θ(τ1	θ(τ1	NOUN
ejpam-5717	264	7	,	,	PUNCT
ejpam-5717	264	8	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	264	9	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	264	10	-	-	NOUN
ejpam-5717	264	11	cl(v	cl(v	NOUN
ejpam-5717	264	12	)	)	PUNCT
ejpam-5717	264	13	)	)	PUNCT
ejpam-5717	264	14	)	)	PUNCT
ejpam-5717	264	15	.	.	PUNCT
ejpam-5717	265	1	p.	p.	NOUN
ejpam-5717	265	2	pue	pue	NOUN
ejpam-5717	265	3	-	-	PUNCT
ejpam-5717	265	4	on	on	ADP
ejpam-5717	265	5	,	,	PUNCT
ejpam-5717	265	6	a.	a.	PROPN
ejpam-5717	265	7	sama	sama	PROPN
ejpam-5717	265	8	-	-	PUNCT
ejpam-5717	265	9	ae	ae	PROPN
ejpam-5717	265	10	,	,	PUNCT
ejpam-5717	265	11	c.	c.	PROPN
ejpam-5717	265	12	boonpok	boonpok	PROPN
ejpam-5717	265	13	/	/	SYM
ejpam-5717	265	14	eur	eur	PROPN
ejpam-5717	265	15	.	.	PUNCT
ejpam-5717	266	1	j.	j.	PROPN
ejpam-5717	266	2	pure	pure	PROPN
ejpam-5717	266	3	appl	appl	PROPN
ejpam-5717	266	4	.	.	PROPN
ejpam-5717	266	5	math	math	PROPN
ejpam-5717	266	6	,	,	PUNCT
ejpam-5717	266	7	18	18	NUM
ejpam-5717	266	8	(	(	PUNCT
ejpam-5717	266	9	1	1	NUM
ejpam-5717	266	10	)	)	PUNCT
ejpam-5717	266	11	(	(	PUNCT
ejpam-5717	266	12	2025	2025	NUM
ejpam-5717	266	13	)	)	PUNCT
ejpam-5717	266	14	,	,	PUNCT
ejpam-5717	266	15	5717	5717	NUM
ejpam-5717	266	16	8	8	NUM
ejpam-5717	266	17	of	of	ADP
ejpam-5717	266	18	16	16	NUM
ejpam-5717	266	19	theorem	theorem	NOUN
ejpam-5717	266	20	3	3	NUM
ejpam-5717	266	21	.	.	X
ejpam-5717	266	22	for	for	ADP
ejpam-5717	266	23	a	a	DET
ejpam-5717	266	24	multifunction	multifunction	NOUN
ejpam-5717	266	25	f	f	NOUN
ejpam-5717	266	26	:	:	PUNCT
ejpam-5717	266	27	(	(	PUNCT
ejpam-5717	266	28	x	x	NOUN
ejpam-5717	266	29	,	,	PUNCT
ejpam-5717	266	30	τ1	τ1	NOUN
ejpam-5717	266	31	,	,	PUNCT
ejpam-5717	266	32	τ2	τ2	NOUN
ejpam-5717	266	33	)	)	PUNCT
ejpam-5717	266	34	→	→	SYM
ejpam-5717	266	35	(	(	PUNCT
ejpam-5717	266	36	y	y	PROPN
ejpam-5717	266	37	,	,	PUNCT
ejpam-5717	266	38	σ1	σ1	PROPN
ejpam-5717	266	39	,	,	PUNCT
ejpam-5717	266	40	σ2	σ2	NOUN
ejpam-5717	266	41	)	)	PUNCT
ejpam-5717	266	42	,	,	PUNCT
ejpam-5717	266	43	the	the	DET
ejpam-5717	266	44	following	follow	VERB
ejpam-5717	266	45	properties	property	NOUN
ejpam-5717	266	46	are	be	AUX
ejpam-5717	266	47	equivalent	equivalent	ADJ
ejpam-5717	266	48	:	:	PUNCT
ejpam-5717	266	49	(	(	PUNCT
ejpam-5717	266	50	1	1	X
ejpam-5717	266	51	)	)	PUNCT
ejpam-5717	266	52	f	f	PROPN
ejpam-5717	266	53	is	be	AUX
ejpam-5717	266	54	upper	upper	ADJ
ejpam-5717	266	55	quasi	quasi	NOUN
ejpam-5717	266	56	θ(τ1	θ(τ1	NOUN
ejpam-5717	266	57	,	,	PUNCT
ejpam-5717	266	58	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	266	59	;	;	PUNCT
ejpam-5717	266	60	(	(	PUNCT
ejpam-5717	266	61	2	2	X
ejpam-5717	266	62	)	)	PUNCT
ejpam-5717	266	63	θ(τ1	θ(τ1	NOUN
ejpam-5717	266	64	,	,	PUNCT
ejpam-5717	267	1	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	267	2	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	267	3	-	-	PUNCT
ejpam-5717	267	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	267	5	-	-	PUNCT
ejpam-5717	267	6	cl(v	cl(v	NOUN
ejpam-5717	267	7	)	)	PUNCT
ejpam-5717	267	8	)	)	PUNCT
ejpam-5717	267	9	)	)	PUNCT
ejpam-5717	267	10	)	)	PUNCT
ejpam-5717	268	1	⊆	⊆	X
ejpam-5717	268	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	268	3	-	-	PUNCT
ejpam-5717	268	4	cl(v	cl(v	NOUN
ejpam-5717	268	5	)	)	PUNCT
ejpam-5717	268	6	)	)	PUNCT
ejpam-5717	268	7	for	for	ADP
ejpam-5717	268	8	every	every	DET
ejpam-5717	268	9	(	(	PUNCT
ejpam-5717	268	10	σ1	σ1	PROPN
ejpam-5717	268	11	,	,	PUNCT
ejpam-5717	268	12	σ2)β	σ2)β	NOUN
ejpam-5717	268	13	-	-	PUNCT
ejpam-5717	268	14	open	open	NOUN
ejpam-5717	268	15	set	set	NOUN
ejpam-5717	268	16	v	v	NOUN
ejpam-5717	268	17	of	of	ADP
ejpam-5717	268	18	y	y	PROPN
ejpam-5717	268	19	;	;	PUNCT
ejpam-5717	268	20	(	(	PUNCT
ejpam-5717	268	21	3	3	X
ejpam-5717	268	22	)	)	PUNCT
ejpam-5717	268	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	268	24	,	,	PUNCT
ejpam-5717	268	25	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	268	26	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	268	27	-	-	PUNCT
ejpam-5717	268	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	268	29	-	-	PUNCT
ejpam-5717	268	30	cl(v	cl(v	NOUN
ejpam-5717	268	31	)	)	PUNCT
ejpam-5717	268	32	)	)	PUNCT
ejpam-5717	268	33	)	)	PUNCT
ejpam-5717	268	34	)	)	PUNCT
ejpam-5717	269	1	⊆	⊆	X
ejpam-5717	269	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	269	3	-	-	PUNCT
ejpam-5717	269	4	cl(v	cl(v	NOUN
ejpam-5717	269	5	)	)	PUNCT
ejpam-5717	269	6	)	)	PUNCT
ejpam-5717	269	7	for	for	ADP
ejpam-5717	269	8	every	every	DET
ejpam-5717	269	9	(	(	PUNCT
ejpam-5717	269	10	σ1	σ1	PROPN
ejpam-5717	269	11	,	,	PUNCT
ejpam-5717	269	12	σ2)s	σ2)s	NOUN
ejpam-5717	269	13	-	-	PUNCT
ejpam-5717	269	14	open	open	NOUN
ejpam-5717	269	15	set	set	NOUN
ejpam-5717	269	16	v	v	NOUN
ejpam-5717	269	17	of	of	ADP
ejpam-5717	269	18	y	y	PROPN
ejpam-5717	269	19	.	.	PUNCT
ejpam-5717	270	1	proof	proof	NOUN
ejpam-5717	270	2	.	.	PUNCT
ejpam-5717	271	1	(	(	PUNCT
ejpam-5717	271	2	1	1	X
ejpam-5717	271	3	)	)	PUNCT
ejpam-5717	271	4	⇒	⇒	NOUN
ejpam-5717	271	5	(	(	PUNCT
ejpam-5717	271	6	2	2	NUM
ejpam-5717	271	7	):	):	PUNCT
ejpam-5717	271	8	let	let	VERB
ejpam-5717	271	9	v	v	PART
ejpam-5717	271	10	be	be	AUX
ejpam-5717	271	11	any	any	DET
ejpam-5717	271	12	(	(	PUNCT
ejpam-5717	271	13	σ1	σ1	PROPN
ejpam-5717	271	14	,	,	PUNCT
ejpam-5717	271	15	σ2)β	σ2)β	NOUN
ejpam-5717	271	16	-	-	PUNCT
ejpam-5717	271	17	open	open	ADJ
ejpam-5717	271	18	set	set	NOUN
ejpam-5717	271	19	of	of	ADP
ejpam-5717	271	20	y	y	PROPN
ejpam-5717	271	21	.	.	PUNCT
ejpam-5717	272	1	then	then	ADV
ejpam-5717	272	2	,	,	PUNCT
ejpam-5717	272	3	v	v	ADP
ejpam-5717	272	4	⊆	⊆	NUM
ejpam-5717	272	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	272	6	-	-	PUNCT
ejpam-5717	272	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	272	8	-	-	PUNCT
ejpam-5717	272	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	272	10	-	-	PUNCT
ejpam-5717	272	11	cl(v	cl(v	NOUN
ejpam-5717	272	12	)	)	PUNCT
ejpam-5717	272	13	)	)	PUNCT
ejpam-5717	272	14	)	)	PUNCT
ejpam-5717	272	15	and	and	CCONJ
ejpam-5717	272	16	hence	hence	ADV
ejpam-5717	272	17	σ1σ2	σ1σ2	NOUN
ejpam-5717	272	18	-	-	NOUN
ejpam-5717	272	19	cl(v	cl(v	NOUN
ejpam-5717	272	20	)	)	PUNCT
ejpam-5717	273	1	=	=	SYM
ejpam-5717	273	2	σ1σ2	σ1σ2	X
ejpam-5717	273	3	-	-	PUNCT
ejpam-5717	273	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	273	5	-	-	PUNCT
ejpam-5717	273	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	273	7	-	-	PUNCT
ejpam-5717	273	8	cl(v	cl(v	NOUN
ejpam-5717	273	9	)	)	PUNCT
ejpam-5717	273	10	)	)	PUNCT
ejpam-5717	273	11	)	)	PUNCT
ejpam-5717	273	12	.	.	PUNCT
ejpam-5717	274	1	since	since	SCONJ
ejpam-5717	274	2	σ1σ2	σ1σ2	NOUN
ejpam-5717	274	3	-	-	NOUN
ejpam-5717	274	4	cl(v	cl(v	NOUN
ejpam-5717	274	5	)	)	PUNCT
ejpam-5717	274	6	is	be	AUX
ejpam-5717	274	7	(	(	PUNCT
ejpam-5717	274	8	σ1	σ1	PROPN
ejpam-5717	274	9	,	,	PUNCT
ejpam-5717	274	10	σ2)rclosed	σ2)rclose	VERB
ejpam-5717	274	11	in	in	ADP
ejpam-5717	274	12	y	y	PROPN
ejpam-5717	274	13	,	,	PUNCT
ejpam-5717	274	14	by	by	ADP
ejpam-5717	274	15	theorem	theorem	NOUN
ejpam-5717	274	16	1	1	NUM
ejpam-5717	274	17	we	we	PRON
ejpam-5717	274	18	have	have	AUX
ejpam-5717	274	19	θ(τ1	θ(τ1	NOUN
ejpam-5717	274	20	,	,	PUNCT
ejpam-5717	274	21	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	274	22	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	274	23	-	-	PUNCT
ejpam-5717	274	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	274	25	-	-	PUNCT
ejpam-5717	274	26	cl(v	cl(v	NOUN
ejpam-5717	274	27	)	)	PUNCT
ejpam-5717	274	28	)	)	PUNCT
ejpam-5717	274	29	)	)	PUNCT
ejpam-5717	274	30	)	)	PUNCT
ejpam-5717	275	1	⊆	⊆	X
ejpam-5717	275	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	275	3	-	-	PUNCT
ejpam-5717	275	4	cl(v	cl(v	NOUN
ejpam-5717	275	5	)	)	PUNCT
ejpam-5717	275	6	)	)	PUNCT
ejpam-5717	275	7	.	.	PUNCT
ejpam-5717	276	1	(	(	PUNCT
ejpam-5717	276	2	2	2	X
ejpam-5717	276	3	)	)	PUNCT
ejpam-5717	276	4	⇒	⇒	NOUN
ejpam-5717	276	5	(	(	PUNCT
ejpam-5717	276	6	3	3	NUM
ejpam-5717	276	7	):	):	PUNCT
ejpam-5717	276	8	the	the	DET
ejpam-5717	276	9	proof	proof	NOUN
ejpam-5717	276	10	is	be	AUX
ejpam-5717	276	11	obvious	obvious	ADJ
ejpam-5717	276	12	.	.	PUNCT
ejpam-5717	277	1	(	(	PUNCT
ejpam-5717	277	2	3	3	X
ejpam-5717	277	3	)	)	PUNCT
ejpam-5717	277	4	⇒	⇒	NOUN
ejpam-5717	277	5	(	(	PUNCT
ejpam-5717	277	6	1	1	NUM
ejpam-5717	277	7	):	):	PUNCT
ejpam-5717	277	8	let	let	VERB
ejpam-5717	277	9	v	v	PART
ejpam-5717	277	10	be	be	AUX
ejpam-5717	277	11	any	any	DET
ejpam-5717	277	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	277	13	-	-	ADJ
ejpam-5717	277	14	open	open	ADJ
ejpam-5717	277	15	set	set	NOUN
ejpam-5717	277	16	of	of	ADP
ejpam-5717	277	17	y	y	PROPN
ejpam-5717	277	18	.	.	PUNCT
ejpam-5717	278	1	then	then	ADV
ejpam-5717	278	2	,	,	PUNCT
ejpam-5717	278	3	v	v	NOUN
ejpam-5717	278	4	is	be	AUX
ejpam-5717	278	5	(	(	PUNCT
ejpam-5717	278	6	σ1	σ1	PROPN
ejpam-5717	278	7	,	,	PUNCT
ejpam-5717	278	8	σ2)s	σ2)s	NOUN
ejpam-5717	278	9	-	-	PUNCT
ejpam-5717	278	10	open	open	ADJ
ejpam-5717	278	11	in	in	ADP
ejpam-5717	278	12	y	y	PROPN
ejpam-5717	278	13	and	and	CCONJ
ejpam-5717	278	14	by	by	ADP
ejpam-5717	278	15	(	(	PUNCT
ejpam-5717	278	16	3	3	NUM
ejpam-5717	278	17	)	)	PUNCT
ejpam-5717	278	18	,	,	PUNCT
ejpam-5717	278	19	θ(τ1	θ(τ1	VERB
ejpam-5717	278	20	,	,	PUNCT
ejpam-5717	278	21	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	278	22	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	278	23	-	-	PUNCT
ejpam-5717	278	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	278	25	-	-	PUNCT
ejpam-5717	278	26	cl(v	cl(v	NOUN
ejpam-5717	278	27	)	)	PUNCT
ejpam-5717	278	28	)	)	PUNCT
ejpam-5717	278	29	)	)	PUNCT
ejpam-5717	278	30	)	)	PUNCT
ejpam-5717	279	1	⊆	⊆	X
ejpam-5717	279	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	279	3	-	-	PUNCT
ejpam-5717	279	4	cl(v	cl(v	NOUN
ejpam-5717	279	5	)	)	PUNCT
ejpam-5717	279	6	)	)	PUNCT
ejpam-5717	279	7	.	.	PUNCT
ejpam-5717	280	1	thus	thus	ADV
ejpam-5717	280	2	by	by	ADP
ejpam-5717	280	3	theorem	theorem	NOUN
ejpam-5717	280	4	1	1	NUM
ejpam-5717	280	5	,	,	PUNCT
ejpam-5717	280	6	f	f	PROPN
ejpam-5717	280	7	is	be	AUX
ejpam-5717	280	8	upper	upper	ADJ
ejpam-5717	280	9	quasi	quasi	NOUN
ejpam-5717	280	10	θ(τ1	θ(τ1	NOUN
ejpam-5717	280	11	,	,	PUNCT
ejpam-5717	280	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	280	13	.	.	PUNCT
ejpam-5717	281	1	theorem	theorem	NOUN
ejpam-5717	281	2	4	4	NUM
ejpam-5717	281	3	.	.	X
ejpam-5717	281	4	for	for	ADP
ejpam-5717	281	5	a	a	DET
ejpam-5717	281	6	multifunction	multifunction	NOUN
ejpam-5717	282	1	f	f	NOUN
ejpam-5717	282	2	:	:	PUNCT
ejpam-5717	282	3	(	(	PUNCT
ejpam-5717	282	4	x	x	NOUN
ejpam-5717	282	5	,	,	PUNCT
ejpam-5717	282	6	τ1	τ1	NOUN
ejpam-5717	282	7	,	,	PUNCT
ejpam-5717	282	8	τ2	τ2	NOUN
ejpam-5717	282	9	)	)	PUNCT
ejpam-5717	282	10	→	→	SYM
ejpam-5717	282	11	(	(	PUNCT
ejpam-5717	282	12	y	y	PROPN
ejpam-5717	282	13	,	,	PUNCT
ejpam-5717	282	14	σ1	σ1	PROPN
ejpam-5717	282	15	,	,	PUNCT
ejpam-5717	282	16	σ2	σ2	NOUN
ejpam-5717	282	17	)	)	PUNCT
ejpam-5717	282	18	,	,	PUNCT
ejpam-5717	282	19	the	the	DET
ejpam-5717	282	20	following	follow	VERB
ejpam-5717	282	21	properties	property	NOUN
ejpam-5717	282	22	are	be	AUX
ejpam-5717	282	23	equivalent	equivalent	ADJ
ejpam-5717	282	24	:	:	PUNCT
ejpam-5717	282	25	(	(	PUNCT
ejpam-5717	282	26	1	1	X
ejpam-5717	282	27	)	)	PUNCT
ejpam-5717	282	28	f	f	PROPN
ejpam-5717	282	29	is	be	AUX
ejpam-5717	282	30	lower	low	ADJ
ejpam-5717	282	31	quasi	quasi	NOUN
ejpam-5717	282	32	θ(τ1	θ(τ1	NOUN
ejpam-5717	282	33	,	,	PUNCT
ejpam-5717	282	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	282	35	;	;	PUNCT
ejpam-5717	282	36	(	(	PUNCT
ejpam-5717	282	37	2	2	X
ejpam-5717	282	38	)	)	PUNCT
ejpam-5717	282	39	θ(τ1	θ(τ1	NOUN
ejpam-5717	282	40	,	,	PUNCT
ejpam-5717	282	41	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	283	1	+	+	ADJ
ejpam-5717	283	2	(	(	PUNCT
ejpam-5717	283	3	σ1σ2	σ1σ2	NUM
ejpam-5717	283	4	-	-	PUNCT
ejpam-5717	283	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	283	6	-	-	PUNCT
ejpam-5717	283	7	cl(v	cl(v	NOUN
ejpam-5717	283	8	)	)	PUNCT
ejpam-5717	283	9	)	)	PUNCT
ejpam-5717	283	10	)	)	PUNCT
ejpam-5717	283	11	)	)	PUNCT
ejpam-5717	284	1	⊆	⊆	X
ejpam-5717	284	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	284	3	-	-	PUNCT
ejpam-5717	284	4	cl(v	cl(v	NOUN
ejpam-5717	284	5	)	)	PUNCT
ejpam-5717	284	6	)	)	PUNCT
ejpam-5717	284	7	for	for	ADP
ejpam-5717	284	8	every	every	DET
ejpam-5717	284	9	(	(	PUNCT
ejpam-5717	284	10	σ1	σ1	PROPN
ejpam-5717	284	11	,	,	PUNCT
ejpam-5717	284	12	σ2)β	σ2)β	NOUN
ejpam-5717	284	13	-	-	PUNCT
ejpam-5717	284	14	open	open	NOUN
ejpam-5717	284	15	set	set	NOUN
ejpam-5717	284	16	v	v	NOUN
ejpam-5717	284	17	of	of	ADP
ejpam-5717	284	18	y	y	PROPN
ejpam-5717	284	19	;	;	PUNCT
ejpam-5717	284	20	(	(	PUNCT
ejpam-5717	284	21	3	3	X
ejpam-5717	284	22	)	)	PUNCT
ejpam-5717	284	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	284	24	,	,	PUNCT
ejpam-5717	284	25	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	285	1	+	+	ADJ
ejpam-5717	285	2	(	(	PUNCT
ejpam-5717	285	3	σ1σ2	σ1σ2	NUM
ejpam-5717	285	4	-	-	PUNCT
ejpam-5717	285	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	285	6	-	-	PUNCT
ejpam-5717	285	7	cl(v	cl(v	NOUN
ejpam-5717	285	8	)	)	PUNCT
ejpam-5717	285	9	)	)	PUNCT
ejpam-5717	285	10	)	)	PUNCT
ejpam-5717	285	11	)	)	PUNCT
ejpam-5717	286	1	⊆	⊆	X
ejpam-5717	286	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	286	3	-	-	PUNCT
ejpam-5717	286	4	cl(v	cl(v	NOUN
ejpam-5717	286	5	)	)	PUNCT
ejpam-5717	286	6	)	)	PUNCT
ejpam-5717	286	7	for	for	SCONJ
ejpam-5717	286	8	every	every	DET
ejpam-5717	286	9	(	(	PUNCT
ejpam-5717	286	10	σ1	σ1	PROPN
ejpam-5717	286	11	,	,	PUNCT
ejpam-5717	286	12	σ2)s	σ2)s	NOUN
ejpam-5717	286	13	-	-	PUNCT
ejpam-5717	286	14	open	open	NOUN
ejpam-5717	286	15	set	set	NOUN
ejpam-5717	286	16	v	v	NOUN
ejpam-5717	286	17	of	of	ADP
ejpam-5717	286	18	y	y	PROPN
ejpam-5717	286	19	.	.	PUNCT
ejpam-5717	287	1	proof	proof	NOUN
ejpam-5717	287	2	.	.	PUNCT
ejpam-5717	288	1	the	the	DET
ejpam-5717	288	2	proof	proof	NOUN
ejpam-5717	288	3	is	be	AUX
ejpam-5717	288	4	similar	similar	ADJ
ejpam-5717	288	5	to	to	ADP
ejpam-5717	288	6	that	that	PRON
ejpam-5717	288	7	of	of	ADP
ejpam-5717	288	8	theorem	theorem	ADJ
ejpam-5717	288	9	3	3	NUM
ejpam-5717	288	10	.	.	PUNCT
ejpam-5717	288	11	theorem	theorem	NOUN
ejpam-5717	288	12	5	5	NUM
ejpam-5717	288	13	.	.	X
ejpam-5717	288	14	for	for	ADP
ejpam-5717	288	15	a	a	DET
ejpam-5717	288	16	multifunction	multifunction	NOUN
ejpam-5717	288	17	f	f	NOUN
ejpam-5717	288	18	:	:	PUNCT
ejpam-5717	288	19	(	(	PUNCT
ejpam-5717	288	20	x	x	NOUN
ejpam-5717	288	21	,	,	PUNCT
ejpam-5717	288	22	τ1	τ1	NOUN
ejpam-5717	288	23	,	,	PUNCT
ejpam-5717	288	24	τ2	τ2	NOUN
ejpam-5717	288	25	)	)	PUNCT
ejpam-5717	288	26	→	→	SYM
ejpam-5717	288	27	(	(	PUNCT
ejpam-5717	288	28	y	y	PROPN
ejpam-5717	288	29	,	,	PUNCT
ejpam-5717	288	30	σ1	σ1	PROPN
ejpam-5717	288	31	,	,	PUNCT
ejpam-5717	288	32	σ2	σ2	NOUN
ejpam-5717	288	33	)	)	PUNCT
ejpam-5717	288	34	,	,	PUNCT
ejpam-5717	288	35	the	the	DET
ejpam-5717	288	36	following	follow	VERB
ejpam-5717	288	37	properties	property	NOUN
ejpam-5717	288	38	are	be	AUX
ejpam-5717	288	39	equivalent	equivalent	ADJ
ejpam-5717	288	40	:	:	PUNCT
ejpam-5717	288	41	(	(	PUNCT
ejpam-5717	288	42	1	1	X
ejpam-5717	288	43	)	)	PUNCT
ejpam-5717	288	44	f	f	PROPN
ejpam-5717	288	45	is	be	AUX
ejpam-5717	288	46	upper	upper	ADJ
ejpam-5717	288	47	quasi	quasi	NOUN
ejpam-5717	288	48	θ(τ1	θ(τ1	NOUN
ejpam-5717	288	49	,	,	PUNCT
ejpam-5717	288	50	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	288	51	;	;	PUNCT
ejpam-5717	288	52	(	(	PUNCT
ejpam-5717	288	53	2	2	X
ejpam-5717	288	54	)	)	PUNCT
ejpam-5717	288	55	θ(τ1	θ(τ1	NOUN
ejpam-5717	288	56	,	,	PUNCT
ejpam-5717	288	57	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	288	58	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	288	59	-	-	PUNCT
ejpam-5717	288	60	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	288	61	-	-	PUNCT
ejpam-5717	288	62	cl(v	cl(v	NOUN
ejpam-5717	288	63	)	)	PUNCT
ejpam-5717	288	64	)	)	PUNCT
ejpam-5717	288	65	)	)	PUNCT
ejpam-5717	288	66	)	)	PUNCT
ejpam-5717	289	1	⊆	⊆	X
ejpam-5717	289	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	289	3	-	-	PUNCT
ejpam-5717	289	4	cl(v	cl(v	NOUN
ejpam-5717	289	5	)	)	PUNCT
ejpam-5717	289	6	)	)	PUNCT
ejpam-5717	289	7	for	for	ADP
ejpam-5717	289	8	every	every	DET
ejpam-5717	289	9	(	(	PUNCT
ejpam-5717	289	10	σ1	σ1	PROPN
ejpam-5717	289	11	,	,	PUNCT
ejpam-5717	289	12	σ2)p	σ2)p	NOUN
ejpam-5717	289	13	-	-	PUNCT
ejpam-5717	289	14	open	open	NOUN
ejpam-5717	289	15	set	set	NOUN
ejpam-5717	289	16	v	v	NOUN
ejpam-5717	289	17	of	of	ADP
ejpam-5717	289	18	y	y	PROPN
ejpam-5717	289	19	;	;	PUNCT
ejpam-5717	289	20	(	(	PUNCT
ejpam-5717	289	21	3	3	X
ejpam-5717	289	22	)	)	PUNCT
ejpam-5717	289	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	289	24	,	,	PUNCT
ejpam-5717	289	25	τ2)-scl(f	τ2)-scl(f	ADV
ejpam-5717	289	26	−(v	−(v	NOUN
ejpam-5717	289	27	)	)	PUNCT
ejpam-5717	289	28	)	)	PUNCT
ejpam-5717	290	1	⊆	⊆	X
ejpam-5717	290	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	290	3	-	-	PUNCT
ejpam-5717	290	4	cl(v	cl(v	NOUN
ejpam-5717	290	5	)	)	PUNCT
ejpam-5717	290	6	)	)	PUNCT
ejpam-5717	290	7	for	for	ADP
ejpam-5717	290	8	every	every	DET
ejpam-5717	290	9	(	(	PUNCT
ejpam-5717	290	10	σ1	σ1	PROPN
ejpam-5717	290	11	,	,	PUNCT
ejpam-5717	290	12	σ2)p	σ2)p	NOUN
ejpam-5717	290	13	-	-	PUNCT
ejpam-5717	290	14	open	open	NOUN
ejpam-5717	290	15	set	set	NOUN
ejpam-5717	290	16	v	v	NOUN
ejpam-5717	290	17	of	of	ADP
ejpam-5717	290	18	y	y	PROPN
ejpam-5717	290	19	;	;	PUNCT
ejpam-5717	290	20	p.	p.	NOUN
ejpam-5717	290	21	pue	pue	PROPN
ejpam-5717	290	22	-	-	PUNCT
ejpam-5717	290	23	on	on	ADP
ejpam-5717	290	24	,	,	PUNCT
ejpam-5717	290	25	a.	a.	PROPN
ejpam-5717	290	26	sama	sama	PROPN
ejpam-5717	290	27	-	-	PUNCT
ejpam-5717	290	28	ae	ae	PROPN
ejpam-5717	290	29	,	,	PUNCT
ejpam-5717	290	30	c.	c.	PROPN
ejpam-5717	290	31	boonpok	boonpok	PROPN
ejpam-5717	290	32	/	/	SYM
ejpam-5717	290	33	eur	eur	PROPN
ejpam-5717	290	34	.	.	PUNCT
ejpam-5717	291	1	j.	j.	PROPN
ejpam-5717	291	2	pure	pure	PROPN
ejpam-5717	291	3	appl	appl	PROPN
ejpam-5717	291	4	.	.	PROPN
ejpam-5717	291	5	math	math	PROPN
ejpam-5717	291	6	,	,	PUNCT
ejpam-5717	291	7	18	18	NUM
ejpam-5717	291	8	(	(	PUNCT
ejpam-5717	291	9	1	1	NUM
ejpam-5717	291	10	)	)	PUNCT
ejpam-5717	291	11	(	(	PUNCT
ejpam-5717	291	12	2025	2025	NUM
ejpam-5717	291	13	)	)	PUNCT
ejpam-5717	291	14	,	,	PUNCT
ejpam-5717	291	15	5717	5717	NUM
ejpam-5717	291	16	9	9	NUM
ejpam-5717	291	17	of	of	ADP
ejpam-5717	291	18	16	16	NUM
ejpam-5717	291	19	(	(	PUNCT
ejpam-5717	291	20	4	4	NUM
ejpam-5717	291	21	)	)	PUNCT
ejpam-5717	291	22	f+(v	f+(v	NOUN
ejpam-5717	291	23	)	)	PUNCT
ejpam-5717	292	1	⊆	⊆	NUM
ejpam-5717	292	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	292	3	,	,	PUNCT
ejpam-5717	292	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	293	1	+	+	ADJ
ejpam-5717	293	2	(	(	PUNCT
ejpam-5717	293	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	293	4	-	-	NUM
ejpam-5717	293	5	cl(v	cl(v	NOUN
ejpam-5717	293	6	)	)	PUNCT
ejpam-5717	293	7	)	)	PUNCT
ejpam-5717	293	8	)	)	PUNCT
ejpam-5717	293	9	for	for	ADP
ejpam-5717	293	10	every	every	DET
ejpam-5717	293	11	(	(	PUNCT
ejpam-5717	293	12	σ1	σ1	PROPN
ejpam-5717	293	13	,	,	PUNCT
ejpam-5717	293	14	σ2)p	σ2)p	NOUN
ejpam-5717	293	15	-	-	PUNCT
ejpam-5717	293	16	open	open	NOUN
ejpam-5717	293	17	set	set	NOUN
ejpam-5717	293	18	v	v	NOUN
ejpam-5717	293	19	of	of	ADP
ejpam-5717	293	20	y	y	PROPN
ejpam-5717	293	21	.	.	PUNCT
ejpam-5717	294	1	proof	proof	NOUN
ejpam-5717	294	2	.	.	PUNCT
ejpam-5717	295	1	(	(	PUNCT
ejpam-5717	295	2	1	1	X
ejpam-5717	295	3	)	)	PUNCT
ejpam-5717	295	4	⇒	⇒	NOUN
ejpam-5717	295	5	(	(	PUNCT
ejpam-5717	295	6	2	2	NUM
ejpam-5717	295	7	):	):	PUNCT
ejpam-5717	295	8	let	let	VERB
ejpam-5717	295	9	v	v	PART
ejpam-5717	295	10	be	be	AUX
ejpam-5717	295	11	any	any	DET
ejpam-5717	295	12	(	(	PUNCT
ejpam-5717	295	13	σ1	σ1	PROPN
ejpam-5717	295	14	,	,	PUNCT
ejpam-5717	295	15	σ2)p	σ2)p	NOUN
ejpam-5717	295	16	-	-	PUNCT
ejpam-5717	295	17	open	open	ADJ
ejpam-5717	295	18	set	set	NOUN
ejpam-5717	295	19	of	of	ADP
ejpam-5717	295	20	y	y	PROPN
ejpam-5717	295	21	.	.	PUNCT
ejpam-5717	296	1	since	since	SCONJ
ejpam-5717	296	2	σ1σ2	σ1σ2	ADV
ejpam-5717	296	3	-	-	PUNCT
ejpam-5717	296	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	296	5	-	-	PUNCT
ejpam-5717	296	6	cl(v	cl(v	NOUN
ejpam-5717	296	7	)	)	PUNCT
ejpam-5717	296	8	)	)	PUNCT
ejpam-5717	296	9	is	be	AUX
ejpam-5717	296	10	a	a	DET
ejpam-5717	296	11	σ1σ2	σ1σ2	NUM
ejpam-5717	296	12	-	-	ADJ
ejpam-5717	296	13	open	open	ADJ
ejpam-5717	296	14	set	set	NOUN
ejpam-5717	296	15	of	of	ADP
ejpam-5717	296	16	y	y	PROPN
ejpam-5717	296	17	,	,	PUNCT
ejpam-5717	296	18	by	by	ADP
ejpam-5717	296	19	theorem	theorem	NOUN
ejpam-5717	296	20	3	3	NUM
ejpam-5717	296	21	we	we	PRON
ejpam-5717	296	22	have	have	AUX
ejpam-5717	296	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	296	24	,	,	PUNCT
ejpam-5717	296	25	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	296	26	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	296	27	-	-	PUNCT
ejpam-5717	296	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	296	29	-	-	PUNCT
ejpam-5717	296	30	cl(v	cl(v	NOUN
ejpam-5717	296	31	)	)	PUNCT
ejpam-5717	296	32	)	)	PUNCT
ejpam-5717	296	33	)	)	PUNCT
ejpam-5717	296	34	)	)	PUNCT
ejpam-5717	297	1	⊆	⊆	X
ejpam-5717	297	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-5717	297	3	-	-	PUNCT
ejpam-5717	297	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5717	297	5	-	-	PUNCT
ejpam-5717	297	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	297	7	-	-	PUNCT
ejpam-5717	297	8	cl(v	cl(v	NOUN
ejpam-5717	297	9	)	)	PUNCT
ejpam-5717	297	10	)	)	PUNCT
ejpam-5717	297	11	)	)	PUNCT
ejpam-5717	297	12	)	)	PUNCT
ejpam-5717	298	1	=	=	PUNCT
ejpam-5717	298	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	298	3	-	-	PUNCT
ejpam-5717	298	4	cl(v	cl(v	NOUN
ejpam-5717	298	5	)	)	PUNCT
ejpam-5717	298	6	)	)	PUNCT
ejpam-5717	298	7	.	.	PUNCT
ejpam-5717	299	1	(	(	PUNCT
ejpam-5717	299	2	2	2	X
ejpam-5717	299	3	)	)	PUNCT
ejpam-5717	299	4	⇒	⇒	NOUN
ejpam-5717	299	5	(	(	PUNCT
ejpam-5717	299	6	3	3	NUM
ejpam-5717	299	7	):	):	PUNCT
ejpam-5717	299	8	let	let	VERB
ejpam-5717	299	9	v	v	PART
ejpam-5717	299	10	be	be	AUX
ejpam-5717	299	11	any	any	DET
ejpam-5717	299	12	(	(	PUNCT
ejpam-5717	299	13	σ1	σ1	PROPN
ejpam-5717	299	14	,	,	PUNCT
ejpam-5717	299	15	σ2)p	σ2)p	NOUN
ejpam-5717	299	16	-	-	PUNCT
ejpam-5717	299	17	open	open	ADJ
ejpam-5717	299	18	set	set	NOUN
ejpam-5717	299	19	of	of	ADP
ejpam-5717	299	20	y	y	PROPN
ejpam-5717	299	21	.	.	PUNCT
ejpam-5717	300	1	then	then	ADV
ejpam-5717	300	2	,	,	PUNCT
ejpam-5717	300	3	v	v	ADP
ejpam-5717	300	4	⊆	⊆	NUM
ejpam-5717	300	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	300	6	-	-	PUNCT
ejpam-5717	300	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	300	8	-	-	PUNCT
ejpam-5717	300	9	cl(v	cl(v	NOUN
ejpam-5717	300	10	)	)	PUNCT
ejpam-5717	300	11	)	)	PUNCT
ejpam-5717	300	12	and	and	CCONJ
ejpam-5717	300	13	by	by	ADP
ejpam-5717	300	14	(	(	PUNCT
ejpam-5717	300	15	2	2	NUM
ejpam-5717	300	16	)	)	PUNCT
ejpam-5717	300	17	,	,	PUNCT
ejpam-5717	300	18	θ(τ1	θ(τ1	VERB
ejpam-5717	300	19	,	,	PUNCT
ejpam-5717	300	20	τ2)-scl(f	τ2)-scl(f	ADV
ejpam-5717	300	21	−(v	−(v	NOUN
ejpam-5717	300	22	)	)	PUNCT
ejpam-5717	300	23	)	)	PUNCT
ejpam-5717	301	1	⊆	⊆	NUM
ejpam-5717	301	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	301	3	,	,	PUNCT
ejpam-5717	301	4	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	301	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	301	6	-	-	PUNCT
ejpam-5717	301	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	301	8	-	-	PUNCT
ejpam-5717	301	9	cl(v	cl(v	NOUN
ejpam-5717	301	10	)	)	PUNCT
ejpam-5717	301	11	)	)	PUNCT
ejpam-5717	301	12	)	)	PUNCT
ejpam-5717	301	13	)	)	PUNCT
ejpam-5717	302	1	⊆	⊆	X
ejpam-5717	302	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	302	3	-	-	PUNCT
ejpam-5717	302	4	cl(v	cl(v	NOUN
ejpam-5717	302	5	)	)	PUNCT
ejpam-5717	302	6	)	)	PUNCT
ejpam-5717	302	7	.	.	PUNCT
ejpam-5717	303	1	(	(	PUNCT
ejpam-5717	303	2	3	3	X
ejpam-5717	303	3	)	)	PUNCT
ejpam-5717	303	4	⇒	⇒	NOUN
ejpam-5717	303	5	(	(	PUNCT
ejpam-5717	303	6	4	4	NUM
ejpam-5717	303	7	):	):	PUNCT
ejpam-5717	303	8	let	let	VERB
ejpam-5717	303	9	v	v	PART
ejpam-5717	303	10	be	be	AUX
ejpam-5717	303	11	any	any	DET
ejpam-5717	303	12	(	(	PUNCT
ejpam-5717	303	13	σ1	σ1	PROPN
ejpam-5717	303	14	,	,	PUNCT
ejpam-5717	303	15	σ2)p	σ2)p	NOUN
ejpam-5717	303	16	-	-	PUNCT
ejpam-5717	303	17	open	open	ADJ
ejpam-5717	303	18	set	set	NOUN
ejpam-5717	303	19	of	of	ADP
ejpam-5717	303	20	y	y	PROPN
ejpam-5717	303	21	.	.	PUNCT
ejpam-5717	304	1	then	then	ADV
ejpam-5717	304	2	by	by	ADP
ejpam-5717	304	3	(	(	PUNCT
ejpam-5717	304	4	3	3	NUM
ejpam-5717	304	5	)	)	PUNCT
ejpam-5717	304	6	,	,	PUNCT
ejpam-5717	304	7	we	we	PRON
ejpam-5717	304	8	have	have	VERB
ejpam-5717	304	9	x	x	PART
ejpam-5717	304	10	−	−	NOUN
ejpam-5717	304	11	θ(τ1	θ(τ1	NOUN
ejpam-5717	304	12	,	,	PUNCT
ejpam-5717	304	13	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	305	1	+	+	ADJ
ejpam-5717	305	2	(	(	PUNCT
ejpam-5717	305	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	305	4	-	-	NUM
ejpam-5717	305	5	cl(v	cl(v	NOUN
ejpam-5717	305	6	)	)	PUNCT
ejpam-5717	305	7	)	)	PUNCT
ejpam-5717	305	8	)	)	PUNCT
ejpam-5717	306	1	=	=	SYM
ejpam-5717	306	2	θ(τ1	θ(τ1	PROPN
ejpam-5717	306	3	,	,	PUNCT
ejpam-5717	306	4	τ2)-scl(x	τ2)-scl(x	NOUN
ejpam-5717	306	5	−	−	ADP
ejpam-5717	306	6	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	306	7	-	-	PUNCT
ejpam-5717	306	8	cl(v	cl(v	NOUN
ejpam-5717	306	9	)	)	PUNCT
ejpam-5717	306	10	)	)	PUNCT
ejpam-5717	306	11	)	)	PUNCT
ejpam-5717	307	1	=	=	SYM
ejpam-5717	307	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	307	3	,	,	PUNCT
ejpam-5717	307	4	τ2)-scl(f	τ2)-scl(f	NOUN
ejpam-5717	308	1	−(y	−(y	NOUN
ejpam-5717	308	2	−	−	NOUN
ejpam-5717	308	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	308	4	-	-	NUM
ejpam-5717	308	5	cl(v	cl(v	NOUN
ejpam-5717	308	6	)	)	PUNCT
ejpam-5717	308	7	)	)	PUNCT
ejpam-5717	308	8	)	)	PUNCT
ejpam-5717	309	1	⊆	⊆	X
ejpam-5717	309	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5717	309	3	-	-	PUNCT
ejpam-5717	309	4	cl(y	cl(y	NOUN
ejpam-5717	309	5	−	−	NOUN
ejpam-5717	309	6	σ1σ2	σ1σ2	NOUN
ejpam-5717	309	7	-	-	NUM
ejpam-5717	309	8	cl(v	cl(v	NOUN
ejpam-5717	309	9	)	)	PUNCT
ejpam-5717	309	10	)	)	PUNCT
ejpam-5717	309	11	)	)	PUNCT
ejpam-5717	310	1	=	=	PUNCT
ejpam-5717	310	2	x	x	X
ejpam-5717	310	3	−	−	ADP
ejpam-5717	310	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-5717	310	5	-	-	PUNCT
ejpam-5717	310	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	310	7	-	-	PUNCT
ejpam-5717	310	8	cl(v	cl(v	NOUN
ejpam-5717	310	9	)	)	PUNCT
ejpam-5717	310	10	)	)	PUNCT
ejpam-5717	310	11	)	)	PUNCT
ejpam-5717	311	1	⊆	⊆	NUM
ejpam-5717	311	2	x	x	SYM
ejpam-5717	311	3	−	−	NOUN
ejpam-5717	311	4	f+(v	f+(v	NOUN
ejpam-5717	311	5	)	)	PUNCT
ejpam-5717	311	6	and	and	CCONJ
ejpam-5717	311	7	hence	hence	ADV
ejpam-5717	311	8	f+(v	f+(v	NOUN
ejpam-5717	311	9	)	)	PUNCT
ejpam-5717	312	1	⊆	⊆	NUM
ejpam-5717	312	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	312	3	,	,	PUNCT
ejpam-5717	312	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	313	1	+	+	ADJ
ejpam-5717	313	2	(	(	PUNCT
ejpam-5717	313	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	313	4	-	-	NUM
ejpam-5717	313	5	cl(v	cl(v	NOUN
ejpam-5717	313	6	)	)	PUNCT
ejpam-5717	313	7	)	)	PUNCT
ejpam-5717	313	8	)	)	PUNCT
ejpam-5717	313	9	.	.	PUNCT
ejpam-5717	314	1	(	(	PUNCT
ejpam-5717	314	2	4	4	X
ejpam-5717	314	3	)	)	PUNCT
ejpam-5717	314	4	⇒	⇒	NOUN
ejpam-5717	314	5	(	(	PUNCT
ejpam-5717	314	6	1	1	NUM
ejpam-5717	314	7	):	):	PUNCT
ejpam-5717	314	8	let	let	VERB
ejpam-5717	314	9	v	v	PART
ejpam-5717	314	10	be	be	AUX
ejpam-5717	314	11	any	any	DET
ejpam-5717	314	12	σ1σ2	σ1σ2	NOUN
ejpam-5717	314	13	-	-	ADJ
ejpam-5717	314	14	open	open	ADJ
ejpam-5717	314	15	set	set	NOUN
ejpam-5717	314	16	of	of	ADP
ejpam-5717	314	17	y	y	PROPN
ejpam-5717	314	18	.	.	PUNCT
ejpam-5717	315	1	then	then	ADV
ejpam-5717	315	2	,	,	PUNCT
ejpam-5717	315	3	v	v	NOUN
ejpam-5717	315	4	is	be	AUX
ejpam-5717	315	5	(	(	PUNCT
ejpam-5717	315	6	σ1	σ1	PROPN
ejpam-5717	315	7	,	,	PUNCT
ejpam-5717	315	8	σ2)p	σ2)p	NOUN
ejpam-5717	315	9	-	-	PUNCT
ejpam-5717	315	10	open	open	ADJ
ejpam-5717	315	11	in	in	ADP
ejpam-5717	315	12	y	y	PROPN
ejpam-5717	315	13	and	and	CCONJ
ejpam-5717	315	14	by	by	ADP
ejpam-5717	315	15	(	(	PUNCT
ejpam-5717	315	16	4	4	NUM
ejpam-5717	315	17	)	)	PUNCT
ejpam-5717	315	18	,	,	PUNCT
ejpam-5717	315	19	we	we	PRON
ejpam-5717	315	20	have	have	VERB
ejpam-5717	315	21	f+(v	f+(v	NOUN
ejpam-5717	315	22	)	)	PUNCT
ejpam-5717	316	1	⊆	⊆	NUM
ejpam-5717	316	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	316	3	,	,	PUNCT
ejpam-5717	316	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	317	1	+	+	ADJ
ejpam-5717	317	2	(	(	PUNCT
ejpam-5717	317	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	317	4	-	-	NUM
ejpam-5717	317	5	cl(v	cl(v	NOUN
ejpam-5717	317	6	)	)	PUNCT
ejpam-5717	317	7	)	)	PUNCT
ejpam-5717	317	8	)	)	PUNCT
ejpam-5717	317	9	.	.	PUNCT
ejpam-5717	318	1	by	by	ADP
ejpam-5717	318	2	theorem	theorem	NOUN
ejpam-5717	318	3	1	1	NUM
ejpam-5717	318	4	,	,	PUNCT
ejpam-5717	318	5	f	f	PROPN
ejpam-5717	318	6	is	be	AUX
ejpam-5717	318	7	upper	upper	ADJ
ejpam-5717	318	8	quasi	quasi	NOUN
ejpam-5717	318	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	318	10	,	,	PUNCT
ejpam-5717	318	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	318	12	.	.	PUNCT
ejpam-5717	319	1	theorem	theorem	VERB
ejpam-5717	319	2	6	6	NUM
ejpam-5717	319	3	.	.	PUNCT
ejpam-5717	319	4	for	for	ADP
ejpam-5717	319	5	a	a	DET
ejpam-5717	319	6	multifunction	multifunction	NOUN
ejpam-5717	320	1	f	f	NOUN
ejpam-5717	320	2	:	:	PUNCT
ejpam-5717	320	3	(	(	PUNCT
ejpam-5717	320	4	x	x	NOUN
ejpam-5717	320	5	,	,	PUNCT
ejpam-5717	320	6	τ1	τ1	NOUN
ejpam-5717	320	7	,	,	PUNCT
ejpam-5717	320	8	τ2	τ2	NOUN
ejpam-5717	320	9	)	)	PUNCT
ejpam-5717	320	10	→	→	SYM
ejpam-5717	320	11	(	(	PUNCT
ejpam-5717	320	12	y	y	PROPN
ejpam-5717	320	13	,	,	PUNCT
ejpam-5717	320	14	σ1	σ1	PROPN
ejpam-5717	320	15	,	,	PUNCT
ejpam-5717	320	16	σ2	σ2	NOUN
ejpam-5717	320	17	)	)	PUNCT
ejpam-5717	320	18	,	,	PUNCT
ejpam-5717	320	19	the	the	DET
ejpam-5717	320	20	following	follow	VERB
ejpam-5717	320	21	properties	property	NOUN
ejpam-5717	320	22	are	be	AUX
ejpam-5717	320	23	equivalent	equivalent	ADJ
ejpam-5717	320	24	:	:	PUNCT
ejpam-5717	320	25	(	(	PUNCT
ejpam-5717	320	26	1	1	X
ejpam-5717	320	27	)	)	PUNCT
ejpam-5717	320	28	f	f	PROPN
ejpam-5717	320	29	is	be	AUX
ejpam-5717	320	30	lower	low	ADJ
ejpam-5717	320	31	quasi	quasi	NOUN
ejpam-5717	320	32	θ(τ1	θ(τ1	NOUN
ejpam-5717	320	33	,	,	PUNCT
ejpam-5717	320	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	320	35	;	;	PUNCT
ejpam-5717	320	36	(	(	PUNCT
ejpam-5717	320	37	2	2	X
ejpam-5717	320	38	)	)	PUNCT
ejpam-5717	320	39	θ(τ1	θ(τ1	NOUN
ejpam-5717	320	40	,	,	PUNCT
ejpam-5717	320	41	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	321	1	+	+	ADJ
ejpam-5717	321	2	(	(	PUNCT
ejpam-5717	321	3	σ1σ2	σ1σ2	NUM
ejpam-5717	321	4	-	-	PUNCT
ejpam-5717	321	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5717	321	6	-	-	PUNCT
ejpam-5717	321	7	cl(v	cl(v	NOUN
ejpam-5717	321	8	)	)	PUNCT
ejpam-5717	321	9	)	)	PUNCT
ejpam-5717	321	10	)	)	PUNCT
ejpam-5717	321	11	)	)	PUNCT
ejpam-5717	322	1	⊆	⊆	X
ejpam-5717	322	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	322	3	-	-	PUNCT
ejpam-5717	322	4	cl(v	cl(v	NOUN
ejpam-5717	322	5	)	)	PUNCT
ejpam-5717	322	6	)	)	PUNCT
ejpam-5717	322	7	for	for	ADP
ejpam-5717	322	8	every	every	DET
ejpam-5717	322	9	(	(	PUNCT
ejpam-5717	322	10	σ1	σ1	PROPN
ejpam-5717	322	11	,	,	PUNCT
ejpam-5717	322	12	σ2)p	σ2)p	NOUN
ejpam-5717	322	13	-	-	PUNCT
ejpam-5717	322	14	open	open	NOUN
ejpam-5717	322	15	set	set	NOUN
ejpam-5717	322	16	v	v	NOUN
ejpam-5717	322	17	of	of	ADP
ejpam-5717	322	18	y	y	PROPN
ejpam-5717	322	19	;	;	PUNCT
ejpam-5717	322	20	(	(	PUNCT
ejpam-5717	322	21	3	3	X
ejpam-5717	322	22	)	)	PUNCT
ejpam-5717	322	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	322	24	,	,	PUNCT
ejpam-5717	322	25	τ2)-scl(f	τ2)-scl(f	PROPN
ejpam-5717	323	1	+	+	ADJ
ejpam-5717	323	2	(	(	PUNCT
ejpam-5717	323	3	v	v	NOUN
ejpam-5717	323	4	)	)	PUNCT
ejpam-5717	323	5	)	)	PUNCT
ejpam-5717	324	1	⊆	⊆	NUM
ejpam-5717	324	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5717	324	3	-	-	PUNCT
ejpam-5717	324	4	cl(v	cl(v	NOUN
ejpam-5717	324	5	)	)	PUNCT
ejpam-5717	324	6	)	)	PUNCT
ejpam-5717	324	7	for	for	ADP
ejpam-5717	324	8	every	every	DET
ejpam-5717	324	9	(	(	PUNCT
ejpam-5717	324	10	σ1	σ1	PROPN
ejpam-5717	324	11	,	,	PUNCT
ejpam-5717	324	12	σ2)p	σ2)p	NOUN
ejpam-5717	324	13	-	-	PUNCT
ejpam-5717	324	14	open	open	NOUN
ejpam-5717	324	15	set	set	NOUN
ejpam-5717	324	16	v	v	NOUN
ejpam-5717	324	17	of	of	ADP
ejpam-5717	324	18	y	y	PROPN
ejpam-5717	324	19	;	;	PUNCT
ejpam-5717	324	20	(	(	PUNCT
ejpam-5717	324	21	4	4	X
ejpam-5717	324	22	)	)	PUNCT
ejpam-5717	324	23	f−(v	f−(v	ADJ
ejpam-5717	324	24	)	)	PUNCT
ejpam-5717	324	25	⊆	⊆	NUM
ejpam-5717	324	26	θ(τ1	θ(τ1	NOUN
ejpam-5717	324	27	,	,	PUNCT
ejpam-5717	324	28	τ2)-sint(f	τ2)-sint(f	ADP
ejpam-5717	324	29	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5717	324	30	-	-	NOUN
ejpam-5717	324	31	cl(v	cl(v	NOUN
ejpam-5717	324	32	)	)	PUNCT
ejpam-5717	324	33	)	)	PUNCT
ejpam-5717	324	34	)	)	PUNCT
ejpam-5717	324	35	for	for	ADP
ejpam-5717	324	36	every	every	DET
ejpam-5717	324	37	(	(	PUNCT
ejpam-5717	324	38	σ1	σ1	PROPN
ejpam-5717	324	39	,	,	PUNCT
ejpam-5717	324	40	σ2)p	σ2)p	NOUN
ejpam-5717	324	41	-	-	PUNCT
ejpam-5717	324	42	open	open	NOUN
ejpam-5717	324	43	set	set	NOUN
ejpam-5717	324	44	v	v	NOUN
ejpam-5717	324	45	of	of	ADP
ejpam-5717	324	46	y	y	PROPN
ejpam-5717	324	47	.	.	PUNCT
ejpam-5717	325	1	proof	proof	NOUN
ejpam-5717	325	2	.	.	PUNCT
ejpam-5717	326	1	the	the	DET
ejpam-5717	326	2	proof	proof	NOUN
ejpam-5717	326	3	is	be	AUX
ejpam-5717	326	4	similar	similar	ADJ
ejpam-5717	326	5	to	to	ADP
ejpam-5717	326	6	that	that	PRON
ejpam-5717	326	7	of	of	ADP
ejpam-5717	326	8	theorem	theorem	NOUN
ejpam-5717	326	9	5	5	NUM
ejpam-5717	326	10	.	.	PUNCT
ejpam-5717	326	11	recall	recall	VERB
ejpam-5717	326	12	that	that	SCONJ
ejpam-5717	326	13	a	a	DET
ejpam-5717	326	14	bitopological	bitopological	ADJ
ejpam-5717	326	15	space	space	NOUN
ejpam-5717	326	16	(	(	PUNCT
ejpam-5717	326	17	x	x	NOUN
ejpam-5717	326	18	,	,	PUNCT
ejpam-5717	326	19	τ1	τ1	NOUN
ejpam-5717	326	20	,	,	PUNCT
ejpam-5717	326	21	τ2	τ2	NOUN
ejpam-5717	326	22	)	)	PUNCT
ejpam-5717	326	23	is	be	AUX
ejpam-5717	326	24	said	say	VERB
ejpam-5717	326	25	to	to	PART
ejpam-5717	326	26	be	be	AUX
ejpam-5717	326	27	τ1τ2	τ1τ2	NOUN
ejpam-5717	326	28	-	-	ADJ
ejpam-5717	326	29	compact	compact	ADJ
ejpam-5717	326	30	[	[	X
ejpam-5717	326	31	30	30	NUM
ejpam-5717	326	32	]	]	PUNCT
ejpam-5717	326	33	if	if	SCONJ
ejpam-5717	326	34	every	every	DET
ejpam-5717	326	35	cover	cover	NOUN
ejpam-5717	326	36	of	of	ADP
ejpam-5717	326	37	x	x	PUNCT
ejpam-5717	326	38	by	by	ADP
ejpam-5717	326	39	τ1τ2	τ1τ2	ADJ
ejpam-5717	326	40	-	-	ADJ
ejpam-5717	326	41	open	open	ADJ
ejpam-5717	326	42	sets	set	NOUN
ejpam-5717	326	43	of	of	ADP
ejpam-5717	326	44	x	x	PUNCT
ejpam-5717	326	45	has	have	VERB
ejpam-5717	326	46	a	a	DET
ejpam-5717	326	47	finite	finite	ADJ
ejpam-5717	326	48	subcover	subcover	PROPN
ejpam-5717	326	49	.	.	PUNCT
ejpam-5717	327	1	a	a	DET
ejpam-5717	327	2	bitopological	bitopological	ADJ
ejpam-5717	327	3	space	space	NOUN
ejpam-5717	327	4	(	(	PUNCT
ejpam-5717	327	5	x	x	NOUN
ejpam-5717	327	6	,	,	PUNCT
ejpam-5717	327	7	τ1	τ1	NOUN
ejpam-5717	327	8	,	,	PUNCT
ejpam-5717	327	9	τ2	τ2	NOUN
ejpam-5717	327	10	)	)	PUNCT
ejpam-5717	327	11	is	be	AUX
ejpam-5717	327	12	said	say	VERB
ejpam-5717	327	13	to	to	PART
ejpam-5717	327	14	be	be	AUX
ejpam-5717	327	15	quasi	quasi	X
ejpam-5717	327	16	(	(	PUNCT
ejpam-5717	327	17	τ1	τ1	NOUN
ejpam-5717	327	18	,	,	PUNCT
ejpam-5717	327	19	τ2)-h	τ2)-h	PUNCT
ejpam-5717	327	20	-closed	-closed	ADJ
ejpam-5717	327	21	[	[	PUNCT
ejpam-5717	327	22	64	64	NUM
ejpam-5717	327	23	]	]	PUNCT
ejpam-5717	327	24	if	if	SCONJ
ejpam-5717	327	25	every	every	DET
ejpam-5717	327	26	τ1τ2	τ1τ2	ADJ
ejpam-5717	327	27	-	-	ADJ
ejpam-5717	327	28	open	open	ADJ
ejpam-5717	327	29	cover	cover	NOUN
ejpam-5717	327	30	{	{	PUNCT
ejpam-5717	327	31	uγ	uγ	ADV
ejpam-5717	327	32	|	|	ADV
ejpam-5717	327	33	γ	γ	X
ejpam-5717	327	34	∈	∈	PROPN
ejpam-5717	327	35	γ	γ	X
ejpam-5717	327	36	}	}	PUNCT
ejpam-5717	327	37	,	,	PUNCT
ejpam-5717	327	38	there	there	PRON
ejpam-5717	327	39	exists	exist	VERB
ejpam-5717	327	40	a	a	DET
ejpam-5717	327	41	finite	finite	NOUN
ejpam-5717	327	42	subset	subset	NOUN
ejpam-5717	327	43	γ0	γ0	NOUN
ejpam-5717	327	44	of	of	ADP
ejpam-5717	327	45	γ	γ	NOUN
ejpam-5717	327	46	such	such	ADJ
ejpam-5717	327	47	that	that	SCONJ
ejpam-5717	327	48	x	x	X
ejpam-5717	328	1	=	=	PUNCT
ejpam-5717	328	2	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-5717	328	3	-	-	NOUN
ejpam-5717	328	4	cl(uγ	cl(uγ	NOUN
ejpam-5717	328	5	)	)	PUNCT
ejpam-5717	328	6	|	|	ADV
ejpam-5717	328	7	γ	γ	PROPN
ejpam-5717	328	8	∈	∈	PROPN
ejpam-5717	328	9	γ0	γ0	NOUN
ejpam-5717	328	10	}	}	PUNCT
ejpam-5717	328	11	.	.	PUNCT
ejpam-5717	329	1	p.	p.	NOUN
ejpam-5717	329	2	pue	pue	NOUN
ejpam-5717	329	3	-	-	PUNCT
ejpam-5717	329	4	on	on	ADP
ejpam-5717	329	5	,	,	PUNCT
ejpam-5717	329	6	a.	a.	PROPN
ejpam-5717	329	7	sama	sama	PROPN
ejpam-5717	329	8	-	-	PUNCT
ejpam-5717	329	9	ae	ae	PROPN
ejpam-5717	329	10	,	,	PUNCT
ejpam-5717	329	11	c.	c.	PROPN
ejpam-5717	329	12	boonpok	boonpok	PROPN
ejpam-5717	329	13	/	/	SYM
ejpam-5717	329	14	eur	eur	PROPN
ejpam-5717	329	15	.	.	PUNCT
ejpam-5717	330	1	j.	j.	PROPN
ejpam-5717	330	2	pure	pure	PROPN
ejpam-5717	330	3	appl	appl	PROPN
ejpam-5717	330	4	.	.	PROPN
ejpam-5717	330	5	math	math	PROPN
ejpam-5717	330	6	,	,	PUNCT
ejpam-5717	330	7	18	18	NUM
ejpam-5717	330	8	(	(	PUNCT
ejpam-5717	330	9	1	1	NUM
ejpam-5717	330	10	)	)	PUNCT
ejpam-5717	330	11	(	(	PUNCT
ejpam-5717	330	12	2025	2025	NUM
ejpam-5717	330	13	)	)	PUNCT
ejpam-5717	330	14	,	,	PUNCT
ejpam-5717	330	15	5717	5717	NUM
ejpam-5717	330	16	10	10	NUM
ejpam-5717	330	17	of	of	ADP
ejpam-5717	330	18	16	16	NUM
ejpam-5717	330	19	definition	definition	NOUN
ejpam-5717	330	20	3	3	NUM
ejpam-5717	330	21	.	.	PUNCT
ejpam-5717	331	1	a	a	DET
ejpam-5717	331	2	bitopological	bitopological	ADJ
ejpam-5717	331	3	space	space	NOUN
ejpam-5717	331	4	(	(	PUNCT
ejpam-5717	331	5	x	x	NOUN
ejpam-5717	331	6	,	,	PUNCT
ejpam-5717	331	7	τ1	τ1	NOUN
ejpam-5717	331	8	,	,	PUNCT
ejpam-5717	331	9	τ2	τ2	NOUN
ejpam-5717	331	10	)	)	PUNCT
ejpam-5717	331	11	is	be	AUX
ejpam-5717	331	12	called	call	VERB
ejpam-5717	331	13	s-(τ1	s-(τ1	PROPN
ejpam-5717	331	14	,	,	PUNCT
ejpam-5717	331	15	τ2)-closed	τ2)-close	VERB
ejpam-5717	331	16	if	if	SCONJ
ejpam-5717	331	17	every	every	DET
ejpam-5717	331	18	(	(	PUNCT
ejpam-5717	331	19	τ1	τ1	NOUN
ejpam-5717	331	20	,	,	PUNCT
ejpam-5717	331	21	τ2)sopen	τ2)sopen	ADJ
ejpam-5717	331	22	cover	cover	NOUN
ejpam-5717	331	23	{	{	PUNCT
ejpam-5717	331	24	uγ	uγ	ADV
ejpam-5717	331	25	|	|	ADV
ejpam-5717	331	26	γ	γ	X
ejpam-5717	331	27	∈	∈	PROPN
ejpam-5717	331	28	γ	γ	X
ejpam-5717	331	29	}	}	PUNCT
ejpam-5717	331	30	,	,	PUNCT
ejpam-5717	331	31	there	there	PRON
ejpam-5717	331	32	exists	exist	VERB
ejpam-5717	331	33	a	a	DET
ejpam-5717	331	34	finite	finite	NOUN
ejpam-5717	331	35	subset	subset	NOUN
ejpam-5717	331	36	γ0	γ0	NOUN
ejpam-5717	331	37	of	of	ADP
ejpam-5717	331	38	γ	γ	NOUN
ejpam-5717	331	39	such	such	ADJ
ejpam-5717	331	40	that	that	SCONJ
ejpam-5717	331	41	x	x	X
ejpam-5717	331	42	=	=	SYM
ejpam-5717	331	43	∪{(τ1	∪{(τ1	PROPN
ejpam-5717	331	44	,	,	PUNCT
ejpam-5717	331	45	τ2)-scl(uγ	τ2)-scl(uγ	NOUN
ejpam-5717	331	46	)	)	PUNCT
ejpam-5717	331	47	|	|	ADV
ejpam-5717	331	48	γ	γ	PROPN
ejpam-5717	331	49	∈	∈	PROPN
ejpam-5717	331	50	γ0	γ0	PROPN
ejpam-5717	331	51	}	}	PUNCT
ejpam-5717	331	52	.	.	PUNCT
ejpam-5717	332	1	theorem	theorem	VERB
ejpam-5717	332	2	7	7	NUM
ejpam-5717	332	3	.	.	PUNCT
ejpam-5717	333	1	let	let	VERB
ejpam-5717	333	2	f	f	NOUN
ejpam-5717	333	3	:	:	PUNCT
ejpam-5717	333	4	(	(	PUNCT
ejpam-5717	333	5	x	x	NOUN
ejpam-5717	333	6	,	,	PUNCT
ejpam-5717	333	7	τ1	τ1	NOUN
ejpam-5717	333	8	,	,	PUNCT
ejpam-5717	333	9	τ2	τ2	NOUN
ejpam-5717	333	10	)	)	PUNCT
ejpam-5717	333	11	→	→	SYM
ejpam-5717	333	12	(	(	PUNCT
ejpam-5717	333	13	y	y	PROPN
ejpam-5717	333	14	,	,	PUNCT
ejpam-5717	333	15	σ1	σ1	PROPN
ejpam-5717	333	16	,	,	PUNCT
ejpam-5717	333	17	σ2	σ2	PROPN
ejpam-5717	333	18	)	)	PUNCT
ejpam-5717	333	19	be	be	VERB
ejpam-5717	333	20	an	an	DET
ejpam-5717	333	21	upper	upper	ADJ
ejpam-5717	333	22	quasi	quasi	NOUN
ejpam-5717	333	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	333	24	,	,	PUNCT
ejpam-5717	333	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	333	26	surjective	surjective	ADJ
ejpam-5717	333	27	multifunction	multifunction	NOUN
ejpam-5717	333	28	such	such	ADJ
ejpam-5717	333	29	that	that	SCONJ
ejpam-5717	333	30	f	f	PROPN
ejpam-5717	333	31	(	(	PUNCT
ejpam-5717	333	32	x	x	X
ejpam-5717	333	33	)	)	PUNCT
ejpam-5717	333	34	is	be	AUX
ejpam-5717	333	35	σ1σ2	σ1σ2	NOUN
ejpam-5717	333	36	-	-	ADJ
ejpam-5717	333	37	compact	compact	ADJ
ejpam-5717	333	38	for	for	ADP
ejpam-5717	333	39	each	each	DET
ejpam-5717	333	40	x	x	SYM
ejpam-5717	333	41	∈	∈	PROPN
ejpam-5717	333	42	x.	x.	NOUN
ejpam-5717	334	1	if	if	SCONJ
ejpam-5717	334	2	(	(	PUNCT
ejpam-5717	334	3	x	x	NOUN
ejpam-5717	334	4	,	,	PUNCT
ejpam-5717	334	5	τ1	τ1	NOUN
ejpam-5717	334	6	,	,	PUNCT
ejpam-5717	334	7	τ2	τ2	NOUN
ejpam-5717	334	8	)	)	PUNCT
ejpam-5717	334	9	is	be	AUX
ejpam-5717	334	10	s-(τ1	s-(τ1	PROPN
ejpam-5717	334	11	,	,	PUNCT
ejpam-5717	334	12	τ2)-closed	τ2)-closed	ADJ
ejpam-5717	334	13	,	,	PUNCT
ejpam-5717	334	14	then	then	ADV
ejpam-5717	334	15	(	(	PUNCT
ejpam-5717	334	16	y	y	PROPN
ejpam-5717	334	17	,	,	PUNCT
ejpam-5717	334	18	σ1	σ1	PROPN
ejpam-5717	334	19	,	,	PUNCT
ejpam-5717	334	20	σ2	σ2	PROPN
ejpam-5717	334	21	)	)	PUNCT
ejpam-5717	334	22	is	be	AUX
ejpam-5717	334	23	quasi	quasi	X
ejpam-5717	334	24	(	(	PUNCT
ejpam-5717	334	25	σ1	σ1	PROPN
ejpam-5717	334	26	,	,	PUNCT
ejpam-5717	334	27	σ2)-h	σ2)-h	PROPN
ejpam-5717	334	28	-closed	-closed	ADJ
ejpam-5717	334	29	.	.	PUNCT
ejpam-5717	335	1	proof	proof	NOUN
ejpam-5717	335	2	.	.	PUNCT
ejpam-5717	336	1	let	let	VERB
ejpam-5717	336	2	{	{	PUNCT
ejpam-5717	336	3	vγ	vγ	VERB
ejpam-5717	336	4	|	|	ADV
ejpam-5717	336	5	γ	γ	PROPN
ejpam-5717	336	6	∈	∈	PROPN
ejpam-5717	336	7	γ	γ	AUX
ejpam-5717	336	8	}	}	PUNCT
ejpam-5717	336	9	be	be	VERB
ejpam-5717	336	10	any	any	DET
ejpam-5717	336	11	σ1σ2	σ1σ2	NOUN
ejpam-5717	336	12	-	-	PUNCT
ejpam-5717	336	13	open	open	ADJ
ejpam-5717	336	14	cover	cover	NOUN
ejpam-5717	336	15	of	of	ADP
ejpam-5717	336	16	y	y	PROPN
ejpam-5717	336	17	.	.	PUNCT
ejpam-5717	337	1	for	for	ADP
ejpam-5717	337	2	each	each	DET
ejpam-5717	337	3	x	x	SYM
ejpam-5717	337	4	∈	∈	PROPN
ejpam-5717	337	5	x	x	X
ejpam-5717	337	6	,	,	PUNCT
ejpam-5717	337	7	f	f	PROPN
ejpam-5717	337	8	(	(	PUNCT
ejpam-5717	337	9	x	x	X
ejpam-5717	337	10	)	)	PUNCT
ejpam-5717	337	11	is	be	AUX
ejpam-5717	337	12	σ1σ2compact	σ1σ2compact	PUNCT
ejpam-5717	337	13	and	and	CCONJ
ejpam-5717	337	14	there	there	PRON
ejpam-5717	337	15	exists	exist	VERB
ejpam-5717	337	16	a	a	DET
ejpam-5717	337	17	finite	finite	NOUN
ejpam-5717	337	18	subset	subset	NOUN
ejpam-5717	337	19	γ(x	γ(x	NOUN
ejpam-5717	337	20	)	)	PUNCT
ejpam-5717	337	21	of	of	ADP
ejpam-5717	337	22	γ	γ	PRON
ejpam-5717	337	23	such	such	ADJ
ejpam-5717	337	24	that	that	SCONJ
ejpam-5717	337	25	f	f	PROPN
ejpam-5717	337	26	(	(	PUNCT
ejpam-5717	337	27	x	x	X
ejpam-5717	337	28	)	)	PUNCT
ejpam-5717	337	29	⊆	⊆	NUM
ejpam-5717	337	30	∪{vγ	∪{vγ	PROPN
ejpam-5717	337	31	|	|	ADV
ejpam-5717	337	32	γ	γ	X
ejpam-5717	337	33	∈	∈	PROPN
ejpam-5717	337	34	γ(x	γ(x	PROPN
ejpam-5717	337	35	)	)	PUNCT
ejpam-5717	337	36	}	}	PUNCT
ejpam-5717	337	37	.	.	PUNCT
ejpam-5717	338	1	put	put	VERB
ejpam-5717	338	2	v	v	NOUN
ejpam-5717	338	3	(	(	PUNCT
ejpam-5717	338	4	x	x	NOUN
ejpam-5717	338	5	)	)	PUNCT
ejpam-5717	338	6	=	=	SYM
ejpam-5717	338	7	∪{vγ	∪{vγ	PROPN
ejpam-5717	338	8	|	|	ADV
ejpam-5717	338	9	γ	γ	X
ejpam-5717	338	10	∈	∈	PROPN
ejpam-5717	338	11	γ(x	γ(x	PROPN
ejpam-5717	338	12	)	)	PUNCT
ejpam-5717	338	13	}	}	PUNCT
ejpam-5717	338	14	.	.	PUNCT
ejpam-5717	339	1	then	then	ADV
ejpam-5717	339	2	,	,	PUNCT
ejpam-5717	339	3	f	f	PROPN
ejpam-5717	339	4	(	(	PUNCT
ejpam-5717	339	5	x	x	X
ejpam-5717	339	6	)	)	PUNCT
ejpam-5717	339	7	⊆	⊆	NUM
ejpam-5717	339	8	v	v	NOUN
ejpam-5717	339	9	(	(	PUNCT
ejpam-5717	339	10	x	x	NOUN
ejpam-5717	339	11	)	)	PUNCT
ejpam-5717	339	12	and	and	CCONJ
ejpam-5717	339	13	v	v	NOUN
ejpam-5717	339	14	(	(	PUNCT
ejpam-5717	339	15	x	x	X
ejpam-5717	339	16	)	)	PUNCT
ejpam-5717	339	17	is	be	AUX
ejpam-5717	339	18	σ1σ2	σ1σ2	NOUN
ejpam-5717	339	19	-	-	ADJ
ejpam-5717	339	20	open	open	ADJ
ejpam-5717	339	21	in	in	ADP
ejpam-5717	339	22	y	y	PROPN
ejpam-5717	339	23	.	.	PUNCT
ejpam-5717	340	1	since	since	SCONJ
ejpam-5717	340	2	f	f	PROPN
ejpam-5717	340	3	is	be	AUX
ejpam-5717	340	4	upper	upper	ADJ
ejpam-5717	340	5	quasi	quasi	NOUN
ejpam-5717	340	6	θ(τ1	θ(τ1	NOUN
ejpam-5717	340	7	,	,	PUNCT
ejpam-5717	340	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	340	9	,	,	PUNCT
ejpam-5717	340	10	there	there	PRON
ejpam-5717	340	11	exists	exist	VERB
ejpam-5717	340	12	a	a	DET
ejpam-5717	340	13	(	(	PUNCT
ejpam-5717	340	14	τ1	τ1	NOUN
ejpam-5717	340	15	,	,	PUNCT
ejpam-5717	340	16	τ2)s	τ2)s	NOUN
ejpam-5717	340	17	-	-	PUNCT
ejpam-5717	340	18	open	open	ADJ
ejpam-5717	340	19	set	set	VERB
ejpam-5717	340	20	u(x	u(x	NOUN
ejpam-5717	340	21	)	)	PUNCT
ejpam-5717	340	22	of	of	ADP
ejpam-5717	340	23	x	x	SYM
ejpam-5717	340	24	containing	contain	VERB
ejpam-5717	340	25	x	x	PUNCT
ejpam-5717	340	26	such	such	ADJ
ejpam-5717	340	27	that	that	SCONJ
ejpam-5717	340	28	f	f	PROPN
ejpam-5717	340	29	(	(	PUNCT
ejpam-5717	340	30	(	(	PUNCT
ejpam-5717	340	31	τ1	τ1	PROPN
ejpam-5717	340	32	,	,	PUNCT
ejpam-5717	340	33	τ2)-scl(u(x	τ2)-scl(u(x	PROPN
ejpam-5717	340	34	)	)	PUNCT
ejpam-5717	340	35	)	)	PUNCT
ejpam-5717	340	36	)	)	PUNCT
ejpam-5717	341	1	⊆	⊆	X
ejpam-5717	341	2	σ1σ2	σ1σ2	NOUN
ejpam-5717	341	3	-	-	PUNCT
ejpam-5717	341	4	cl(v	cl(v	PRON
ejpam-5717	341	5	(	(	PUNCT
ejpam-5717	341	6	x	x	NOUN
ejpam-5717	341	7	)	)	PUNCT
ejpam-5717	341	8	)	)	PUNCT
ejpam-5717	341	9	.	.	PUNCT
ejpam-5717	342	1	the	the	DET
ejpam-5717	342	2	family	family	NOUN
ejpam-5717	342	3	{	{	PUNCT
ejpam-5717	342	4	u(x	u(x	PROPN
ejpam-5717	342	5	)	)	PUNCT
ejpam-5717	342	6	|	|	ADV
ejpam-5717	342	7	x	x	SYM
ejpam-5717	342	8	∈	∈	NOUN
ejpam-5717	342	9	x	x	X
ejpam-5717	342	10	}	}	PUNCT
ejpam-5717	342	11	is	be	AUX
ejpam-5717	342	12	a	a	DET
ejpam-5717	342	13	(	(	PUNCT
ejpam-5717	342	14	τ1	τ1	NOUN
ejpam-5717	342	15	,	,	PUNCT
ejpam-5717	342	16	τ2)s	τ2)s	NOUN
ejpam-5717	342	17	-	-	PUNCT
ejpam-5717	342	18	open	open	ADJ
ejpam-5717	342	19	cover	cover	NOUN
ejpam-5717	342	20	of	of	ADP
ejpam-5717	342	21	x.	x.	NOUN
ejpam-5717	342	22	since	since	SCONJ
ejpam-5717	342	23	(	(	PUNCT
ejpam-5717	342	24	x	x	NOUN
ejpam-5717	342	25	,	,	PUNCT
ejpam-5717	342	26	τ1	τ1	NOUN
ejpam-5717	342	27	,	,	PUNCT
ejpam-5717	342	28	τ2	τ2	NOUN
ejpam-5717	342	29	)	)	PUNCT
ejpam-5717	342	30	is	be	AUX
ejpam-5717	342	31	s-(τ1	s-(τ1	PROPN
ejpam-5717	342	32	,	,	PUNCT
ejpam-5717	343	1	τ2)-closed	τ2)-close	VERB
ejpam-5717	343	2	,	,	PUNCT
ejpam-5717	343	3	there	there	PRON
ejpam-5717	343	4	exists	exist	VERB
ejpam-5717	343	5	a	a	DET
ejpam-5717	343	6	finite	finite	ADJ
ejpam-5717	343	7	number	number	NOUN
ejpam-5717	343	8	of	of	ADP
ejpam-5717	343	9	points	point	NOUN
ejpam-5717	343	10	,	,	PUNCT
ejpam-5717	343	11	says	say	VERB
ejpam-5717	343	12	,	,	PUNCT
ejpam-5717	343	13	x1	x1	PROPN
ejpam-5717	343	14	,	,	PUNCT
ejpam-5717	343	15	x2	x2	PROPN
ejpam-5717	343	16	,	,	PUNCT
ejpam-5717	343	17	...	...	PUNCT
ejpam-5717	343	18	,	,	PUNCT
ejpam-5717	343	19	xn	xn	PROPN
ejpam-5717	344	1	in	in	ADP
ejpam-5717	344	2	x	x	X
ejpam-5717	344	3	such	such	ADJ
ejpam-5717	344	4	that	that	SCONJ
ejpam-5717	344	5	x	x	X
ejpam-5717	344	6	=	=	SYM
ejpam-5717	344	7	∪{(τ1	∪{(τ1	PROPN
ejpam-5717	344	8	,	,	PUNCT
ejpam-5717	344	9	τ2)-scl(u(xi	τ2)-scl(u(xi	PROPN
ejpam-5717	344	10	)	)	PUNCT
ejpam-5717	344	11	)	)	PUNCT
ejpam-5717	345	1	|	|	ADV
ejpam-5717	345	2	i	i	PRON
ejpam-5717	345	3	=	=	NOUN
ejpam-5717	345	4	1	1	NUM
ejpam-5717	345	5	,	,	PUNCT
ejpam-5717	345	6	2	2	NUM
ejpam-5717	345	7	,	,	PUNCT
ejpam-5717	345	8	...	...	PUNCT
ejpam-5717	345	9	,	,	PUNCT
ejpam-5717	345	10	n	n	CCONJ
ejpam-5717	345	11	}	}	PUNCT
ejpam-5717	345	12	.	.	PUNCT
ejpam-5717	346	1	since	since	SCONJ
ejpam-5717	346	2	f	f	PROPN
ejpam-5717	346	3	is	be	AUX
ejpam-5717	346	4	surjective	surjective	ADJ
ejpam-5717	346	5	,	,	PUNCT
ejpam-5717	346	6	y	y	PROPN
ejpam-5717	346	7	=	=	SYM
ejpam-5717	346	8	f	f	PROPN
ejpam-5717	346	9	(	(	PUNCT
ejpam-5717	346	10	x	x	X
ejpam-5717	346	11	)	)	PUNCT
ejpam-5717	347	1	=	=	SYM
ejpam-5717	347	2	f	f	PROPN
ejpam-5717	347	3	(	(	PUNCT
ejpam-5717	347	4	n	n	CCONJ
ejpam-5717	347	5	∪	∪	VERB
ejpam-5717	347	6	i=1	i=1	PROPN
ejpam-5717	347	7	(	(	PUNCT
ejpam-5717	347	8	τ1	τ1	NOUN
ejpam-5717	347	9	,	,	PUNCT
ejpam-5717	347	10	τ2)-scl(u(xi	τ2)-scl(u(xi	PROPN
ejpam-5717	347	11	)	)	PUNCT
ejpam-5717	347	12	)	)	PUNCT
ejpam-5717	347	13	)	)	PUNCT
ejpam-5717	348	1	=	=	PRON
ejpam-5717	349	1	n	n	PRON
ejpam-5717	349	2	∪	∪	VERB
ejpam-5717	349	3	i=1	i=1	PROPN
ejpam-5717	349	4	f	f	PROPN
ejpam-5717	349	5	(	(	PUNCT
ejpam-5717	349	6	(	(	PUNCT
ejpam-5717	349	7	τ1	τ1	NOUN
ejpam-5717	349	8	,	,	PUNCT
ejpam-5717	349	9	τ2)-scl(u(xi	τ2)-scl(u(xi	PROPN
ejpam-5717	349	10	)	)	PUNCT
ejpam-5717	349	11	)	)	PUNCT
ejpam-5717	349	12	)	)	PUNCT
ejpam-5717	350	1	⊆	⊆	NUM
ejpam-5717	350	2	n	n	PRON
ejpam-5717	350	3	∪	∪	VERB
ejpam-5717	350	4	i=1	i=1	PRON
ejpam-5717	350	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	350	6	-	-	PUNCT
ejpam-5717	350	7	cl(v	cl(v	PRON
ejpam-5717	350	8	(	(	PUNCT
ejpam-5717	350	9	xi	xi	NOUN
ejpam-5717	350	10	)	)	PUNCT
ejpam-5717	350	11	)	)	PUNCT
ejpam-5717	351	1	=	=	SYM
ejpam-5717	351	2	n	n	PRON
ejpam-5717	351	3	∪	∪	VERB
ejpam-5717	351	4	i=1	i=1	PROPN
ejpam-5717	351	5	∪γ∈γ(xi	∪γ∈γ(xi	X
ejpam-5717	351	6	)	)	PUNCT
ejpam-5717	351	7	σ1σ2	σ1σ2	X
ejpam-5717	351	8	-	-	PUNCT
ejpam-5717	351	9	cl(vγ	cl(vγ	ADJ
ejpam-5717	351	10	)	)	PUNCT
ejpam-5717	351	11	.	.	PUNCT
ejpam-5717	352	1	this	this	PRON
ejpam-5717	352	2	shows	show	VERB
ejpam-5717	352	3	that	that	SCONJ
ejpam-5717	352	4	(	(	PUNCT
ejpam-5717	352	5	y	y	PROPN
ejpam-5717	352	6	,	,	PUNCT
ejpam-5717	352	7	σ1	σ1	PROPN
ejpam-5717	352	8	,	,	PUNCT
ejpam-5717	352	9	σ2	σ2	PROPN
ejpam-5717	352	10	)	)	PUNCT
ejpam-5717	352	11	is	be	AUX
ejpam-5717	352	12	quasi	quasi	X
ejpam-5717	352	13	(	(	PUNCT
ejpam-5717	352	14	σ1	σ1	PROPN
ejpam-5717	352	15	,	,	PUNCT
ejpam-5717	352	16	σ2)-h	σ2)-h	PROPN
ejpam-5717	352	17	-closed	-closed	ADJ
ejpam-5717	352	18	.	.	PUNCT
ejpam-5717	353	1	for	for	ADP
ejpam-5717	353	2	a	a	DET
ejpam-5717	353	3	multifunction	multifunction	NOUN
ejpam-5717	353	4	f	f	NOUN
ejpam-5717	353	5	:	:	PUNCT
ejpam-5717	353	6	(	(	PUNCT
ejpam-5717	353	7	x	x	NOUN
ejpam-5717	353	8	,	,	PUNCT
ejpam-5717	353	9	τ1	τ1	NOUN
ejpam-5717	353	10	,	,	PUNCT
ejpam-5717	353	11	τ2	τ2	NOUN
ejpam-5717	353	12	)	)	PUNCT
ejpam-5717	353	13	→	→	SYM
ejpam-5717	353	14	(	(	PUNCT
ejpam-5717	353	15	y	y	PROPN
ejpam-5717	353	16	,	,	PUNCT
ejpam-5717	353	17	σ1	σ1	PROPN
ejpam-5717	353	18	,	,	PUNCT
ejpam-5717	353	19	σ2	σ2	PROPN
ejpam-5717	353	20	)	)	PUNCT
ejpam-5717	353	21	,	,	PUNCT
ejpam-5717	353	22	a	a	DET
ejpam-5717	353	23	multifunction	multifunction	NOUN
ejpam-5717	353	24	sclf⊛	sclf⊛	PROPN
ejpam-5717	353	25	:	:	PUNCT
ejpam-5717	353	26	(	(	PUNCT
ejpam-5717	353	27	x	x	NOUN
ejpam-5717	353	28	,	,	PUNCT
ejpam-5717	353	29	τ1	τ1	NOUN
ejpam-5717	353	30	,	,	PUNCT
ejpam-5717	353	31	τ2	τ2	NOUN
ejpam-5717	353	32	)	)	PUNCT
ejpam-5717	353	33	→	→	SYM
ejpam-5717	353	34	(	(	PUNCT
ejpam-5717	353	35	y	y	PROPN
ejpam-5717	353	36	,	,	PUNCT
ejpam-5717	353	37	σ1	σ1	PROPN
ejpam-5717	353	38	,	,	PUNCT
ejpam-5717	353	39	σ2	σ2	PROPN
ejpam-5717	353	40	)	)	PUNCT
ejpam-5717	353	41	is	be	AUX
ejpam-5717	353	42	defined	define	VERB
ejpam-5717	353	43	in	in	ADP
ejpam-5717	353	44	[	[	X
ejpam-5717	353	45	31	31	NUM
ejpam-5717	353	46	]	]	PUNCT
ejpam-5717	353	47	as	as	SCONJ
ejpam-5717	353	48	follows	follow	VERB
ejpam-5717	353	49	:	:	PUNCT
ejpam-5717	353	50	sclf⊛(x	sclf⊛(x	NUM
ejpam-5717	353	51	)	)	PUNCT
ejpam-5717	353	52	=	=	SYM
ejpam-5717	353	53	(	(	PUNCT
ejpam-5717	353	54	σ1	σ1	PROPN
ejpam-5717	353	55	,	,	PUNCT
ejpam-5717	353	56	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-5717	353	57	(	(	PUNCT
ejpam-5717	353	58	x	x	NOUN
ejpam-5717	353	59	)	)	PUNCT
ejpam-5717	353	60	)	)	PUNCT
ejpam-5717	353	61	for	for	ADP
ejpam-5717	353	62	each	each	DET
ejpam-5717	353	63	x	x	SYM
ejpam-5717	353	64	∈	∈	PROPN
ejpam-5717	353	65	x.	x.	NOUN
ejpam-5717	353	66	lemma	lemma	PROPN
ejpam-5717	354	1	3	3	X
ejpam-5717	354	2	.	.	PUNCT
ejpam-5717	355	1	let	let	VERB
ejpam-5717	355	2	f	f	NOUN
ejpam-5717	355	3	:	:	PUNCT
ejpam-5717	355	4	(	(	PUNCT
ejpam-5717	355	5	x	x	NOUN
ejpam-5717	355	6	,	,	PUNCT
ejpam-5717	355	7	τ1	τ1	NOUN
ejpam-5717	355	8	,	,	PUNCT
ejpam-5717	355	9	τ2	τ2	NOUN
ejpam-5717	355	10	)	)	PUNCT
ejpam-5717	355	11	→	→	SYM
ejpam-5717	355	12	(	(	PUNCT
ejpam-5717	355	13	y	y	PROPN
ejpam-5717	355	14	,	,	PUNCT
ejpam-5717	355	15	σ1	σ1	PROPN
ejpam-5717	355	16	,	,	PUNCT
ejpam-5717	355	17	σ2	σ2	PROPN
ejpam-5717	355	18	)	)	PUNCT
ejpam-5717	355	19	be	be	AUX
ejpam-5717	355	20	a	a	DET
ejpam-5717	355	21	multifunction	multifunction	NOUN
ejpam-5717	355	22	.	.	PUNCT
ejpam-5717	356	1	then	then	ADV
ejpam-5717	356	2	,	,	PUNCT
ejpam-5717	356	3	sclf−	sclf−	PROPN
ejpam-5717	356	4	⊛	⊛	NUM
ejpam-5717	356	5	(	(	PUNCT
ejpam-5717	356	6	v	v	NOUN
ejpam-5717	356	7	)	)	PUNCT
ejpam-5717	356	8	=	=	SYM
ejpam-5717	356	9	f−(v	f−(v	ADJ
ejpam-5717	356	10	)	)	PUNCT
ejpam-5717	356	11	for	for	ADP
ejpam-5717	356	12	ever	ever	ADV
ejpam-5717	356	13	(	(	PUNCT
ejpam-5717	356	14	σ1	σ1	PROPN
ejpam-5717	356	15	,	,	PUNCT
ejpam-5717	356	16	σ2)s	σ2)s	NOUN
ejpam-5717	356	17	-	-	PUNCT
ejpam-5717	356	18	open	open	NOUN
ejpam-5717	356	19	set	set	NOUN
ejpam-5717	356	20	v	v	NOUN
ejpam-5717	356	21	of	of	ADP
ejpam-5717	356	22	y	y	PROPN
ejpam-5717	356	23	.	.	PUNCT
ejpam-5717	357	1	proof	proof	NOUN
ejpam-5717	357	2	.	.	PUNCT
ejpam-5717	358	1	let	let	VERB
ejpam-5717	358	2	v	v	PART
ejpam-5717	358	3	be	be	AUX
ejpam-5717	358	4	any	any	DET
ejpam-5717	358	5	(	(	PUNCT
ejpam-5717	358	6	σ1	σ1	NOUN
ejpam-5717	358	7	,	,	PUNCT
ejpam-5717	358	8	σ2)s	σ2)s	NOUN
ejpam-5717	358	9	-	-	PUNCT
ejpam-5717	358	10	open	open	ADJ
ejpam-5717	358	11	set	set	NOUN
ejpam-5717	358	12	of	of	ADP
ejpam-5717	358	13	y	y	PROPN
ejpam-5717	358	14	.	.	PUNCT
ejpam-5717	359	1	let	let	VERB
ejpam-5717	359	2	x	x	X
ejpam-5717	359	3	∈	∈	PROPN
ejpam-5717	359	4	sclf−	sclf−	NOUN
ejpam-5717	359	5	⊛	⊛	NUM
ejpam-5717	359	6	(	(	PUNCT
ejpam-5717	359	7	v	v	NOUN
ejpam-5717	359	8	)	)	PUNCT
ejpam-5717	359	9	.	.	PUNCT
ejpam-5717	360	1	then	then	ADV
ejpam-5717	360	2	,	,	PUNCT
ejpam-5717	360	3	(	(	PUNCT
ejpam-5717	360	4	σ1	σ1	PROPN
ejpam-5717	360	5	,	,	PUNCT
ejpam-5717	360	6	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-5717	360	7	(	(	PUNCT
ejpam-5717	360	8	x	x	NOUN
ejpam-5717	360	9	)	)	PUNCT
ejpam-5717	360	10	)	)	PUNCT
ejpam-5717	360	11	∩	∩	NOUN
ejpam-5717	360	12	v	v	ADP
ejpam-5717	360	13	=	=	SYM
ejpam-5717	360	14	sclf⊛(x	sclf⊛(x	PROPN
ejpam-5717	360	15	)	)	PUNCT
ejpam-5717	360	16	∩	∩	NOUN
ejpam-5717	360	17	v	v	ADP
ejpam-5717	360	18	̸=	̸=	PROPN
ejpam-5717	360	19	∅.	∅.	NOUN
ejpam-5717	360	20	since	since	SCONJ
ejpam-5717	360	21	v	v	NOUN
ejpam-5717	360	22	is	be	AUX
ejpam-5717	360	23	(	(	PUNCT
ejpam-5717	360	24	σ1	σ1	PROPN
ejpam-5717	360	25	,	,	PUNCT
ejpam-5717	360	26	σ2)s	σ2)s	NOUN
ejpam-5717	360	27	-	-	PUNCT
ejpam-5717	360	28	open	open	ADJ
ejpam-5717	360	29	in	in	ADP
ejpam-5717	360	30	y	y	PROPN
ejpam-5717	360	31	,	,	PUNCT
ejpam-5717	360	32	we	we	PRON
ejpam-5717	360	33	have	have	VERB
ejpam-5717	360	34	v	v	NUM
ejpam-5717	360	35	∩	∩	ADJ
ejpam-5717	360	36	f	f	X
ejpam-5717	360	37	(	(	PUNCT
ejpam-5717	360	38	x	x	X
ejpam-5717	360	39	)	)	PUNCT
ejpam-5717	360	40	̸=	̸=	PROPN
ejpam-5717	360	41	∅	∅	NOUN
ejpam-5717	360	42	and	and	CCONJ
ejpam-5717	360	43	hence	hence	ADV
ejpam-5717	360	44	x	x	PART
ejpam-5717	360	45	∈	∈	PROPN
ejpam-5717	360	46	f−(v	f−(v	NOUN
ejpam-5717	360	47	)	)	PUNCT
ejpam-5717	360	48	.	.	PUNCT
ejpam-5717	361	1	thus	thus	ADV
ejpam-5717	361	2	,	,	PUNCT
ejpam-5717	361	3	sclf−	sclf−	PROPN
ejpam-5717	361	4	⊛	⊛	NUM
ejpam-5717	361	5	(	(	PUNCT
ejpam-5717	361	6	v	v	NOUN
ejpam-5717	361	7	)	)	PUNCT
ejpam-5717	361	8	⊆	⊆	NUM
ejpam-5717	361	9	f−(v	f−(v	NOUN
ejpam-5717	361	10	)	)	PUNCT
ejpam-5717	361	11	.	.	PUNCT
ejpam-5717	362	1	on	on	ADP
ejpam-5717	362	2	the	the	DET
ejpam-5717	362	3	other	other	ADJ
ejpam-5717	362	4	hand	hand	NOUN
ejpam-5717	362	5	,	,	PUNCT
ejpam-5717	362	6	let	let	VERB
ejpam-5717	362	7	x	x	PUNCT
ejpam-5717	362	8	∈	∈	PROPN
ejpam-5717	362	9	f−(v	f−(v	NOUN
ejpam-5717	362	10	)	)	PUNCT
ejpam-5717	362	11	.	.	PUNCT
ejpam-5717	363	1	then	then	ADV
ejpam-5717	363	2	,	,	PUNCT
ejpam-5717	363	3	∅	∅	NOUN
ejpam-5717	363	4	=	=	NOUN
ejpam-5717	363	5	̸	̸	NUM
ejpam-5717	363	6	f	f	NOUN
ejpam-5717	363	7	(	(	PUNCT
ejpam-5717	363	8	x	x	NOUN
ejpam-5717	363	9	)	)	PUNCT
ejpam-5717	363	10	∩	∩	NOUN
ejpam-5717	363	11	v	v	ADP
ejpam-5717	363	12	⊆	⊆	NUM
ejpam-5717	363	13	(	(	PUNCT
ejpam-5717	363	14	σ1	σ1	PROPN
ejpam-5717	363	15	,	,	PUNCT
ejpam-5717	363	16	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-5717	363	17	(	(	PUNCT
ejpam-5717	363	18	x	x	NOUN
ejpam-5717	363	19	)	)	PUNCT
ejpam-5717	363	20	)	)	PUNCT
ejpam-5717	363	21	∩	∩	PROPN
ejpam-5717	363	22	v	v	NOUN
ejpam-5717	363	23	and	and	CCONJ
ejpam-5717	363	24	so	so	ADV
ejpam-5717	363	25	x	x	SYM
ejpam-5717	363	26	∈	∈	PROPN
ejpam-5717	363	27	sclf−	sclf−	NOUN
ejpam-5717	363	28	⊛	⊛	NUM
ejpam-5717	363	29	(	(	PUNCT
ejpam-5717	363	30	v	v	NOUN
ejpam-5717	363	31	)	)	PUNCT
ejpam-5717	363	32	.	.	PUNCT
ejpam-5717	364	1	consequently	consequently	ADV
ejpam-5717	364	2	,	,	PUNCT
ejpam-5717	364	3	we	we	PRON
ejpam-5717	364	4	obtain	obtain	VERB
ejpam-5717	364	5	sclf−	sclf−	NOUN
ejpam-5717	364	6	⊛	⊛	NUM
ejpam-5717	364	7	(	(	PUNCT
ejpam-5717	364	8	v	v	NOUN
ejpam-5717	364	9	)	)	PUNCT
ejpam-5717	364	10	=	=	SYM
ejpam-5717	364	11	f−(v	f−(v	ADJ
ejpam-5717	364	12	)	)	PUNCT
ejpam-5717	364	13	.	.	PUNCT
ejpam-5717	365	1	p.	p.	NOUN
ejpam-5717	365	2	pue	pue	NOUN
ejpam-5717	365	3	-	-	PUNCT
ejpam-5717	365	4	on	on	ADP
ejpam-5717	365	5	,	,	PUNCT
ejpam-5717	365	6	a.	a.	PROPN
ejpam-5717	365	7	sama	sama	PROPN
ejpam-5717	365	8	-	-	PUNCT
ejpam-5717	365	9	ae	ae	PROPN
ejpam-5717	365	10	,	,	PUNCT
ejpam-5717	365	11	c.	c.	PROPN
ejpam-5717	365	12	boonpok	boonpok	PROPN
ejpam-5717	365	13	/	/	SYM
ejpam-5717	365	14	eur	eur	PROPN
ejpam-5717	365	15	.	.	PUNCT
ejpam-5717	366	1	j.	j.	PROPN
ejpam-5717	366	2	pure	pure	PROPN
ejpam-5717	366	3	appl	appl	PROPN
ejpam-5717	366	4	.	.	PROPN
ejpam-5717	366	5	math	math	PROPN
ejpam-5717	366	6	,	,	PUNCT
ejpam-5717	366	7	18	18	NUM
ejpam-5717	366	8	(	(	PUNCT
ejpam-5717	366	9	1	1	NUM
ejpam-5717	366	10	)	)	PUNCT
ejpam-5717	366	11	(	(	PUNCT
ejpam-5717	366	12	2025	2025	NUM
ejpam-5717	366	13	)	)	PUNCT
ejpam-5717	366	14	,	,	PUNCT
ejpam-5717	366	15	5717	5717	NUM
ejpam-5717	366	16	11	11	NUM
ejpam-5717	366	17	of	of	ADP
ejpam-5717	366	18	16	16	NUM
ejpam-5717	366	19	theorem	theorem	NOUN
ejpam-5717	366	20	8	8	NUM
ejpam-5717	366	21	.	.	PUNCT
ejpam-5717	367	1	a	a	DET
ejpam-5717	367	2	multifunction	multifunction	NOUN
ejpam-5717	367	3	f	f	NOUN
ejpam-5717	367	4	:	:	PUNCT
ejpam-5717	367	5	(	(	PUNCT
ejpam-5717	367	6	x	x	NOUN
ejpam-5717	367	7	,	,	PUNCT
ejpam-5717	367	8	τ1	τ1	NOUN
ejpam-5717	367	9	,	,	PUNCT
ejpam-5717	367	10	τ2	τ2	NOUN
ejpam-5717	367	11	)	)	PUNCT
ejpam-5717	367	12	→	→	SYM
ejpam-5717	367	13	(	(	PUNCT
ejpam-5717	367	14	y	y	PROPN
ejpam-5717	367	15	,	,	PUNCT
ejpam-5717	367	16	σ1	σ1	PROPN
ejpam-5717	367	17	,	,	PUNCT
ejpam-5717	367	18	σ2	σ2	NOUN
ejpam-5717	367	19	)	)	PUNCT
ejpam-5717	367	20	is	be	AUX
ejpam-5717	367	21	lower	low	ADJ
ejpam-5717	367	22	quasi	quasi	NOUN
ejpam-5717	367	23	θ(τ1	θ(τ1	NOUN
ejpam-5717	367	24	,	,	PUNCT
ejpam-5717	367	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	367	26	if	if	SCONJ
ejpam-5717	367	27	and	and	CCONJ
ejpam-5717	367	28	only	only	ADV
ejpam-5717	367	29	if	if	SCONJ
ejpam-5717	367	30	sclf⊛	sclf⊛	PRON
ejpam-5717	367	31	:	:	PUNCT
ejpam-5717	367	32	(	(	PUNCT
ejpam-5717	367	33	x	x	NOUN
ejpam-5717	367	34	,	,	PUNCT
ejpam-5717	367	35	τ1	τ1	NOUN
ejpam-5717	367	36	,	,	PUNCT
ejpam-5717	367	37	τ2	τ2	NOUN
ejpam-5717	367	38	)	)	PUNCT
ejpam-5717	367	39	→	→	SYM
ejpam-5717	367	40	(	(	PUNCT
ejpam-5717	367	41	y	y	PROPN
ejpam-5717	367	42	,	,	PUNCT
ejpam-5717	367	43	σ1	σ1	PROPN
ejpam-5717	367	44	,	,	PUNCT
ejpam-5717	367	45	σ2	σ2	NOUN
ejpam-5717	367	46	)	)	PUNCT
ejpam-5717	367	47	is	be	AUX
ejpam-5717	367	48	lower	low	ADJ
ejpam-5717	367	49	quasi	quasi	NOUN
ejpam-5717	367	50	θ(τ1	θ(τ1	NOUN
ejpam-5717	367	51	,	,	PUNCT
ejpam-5717	367	52	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	367	53	.	.	PUNCT
ejpam-5717	368	1	proof	proof	NOUN
ejpam-5717	368	2	.	.	PUNCT
ejpam-5717	369	1	suppose	suppose	VERB
ejpam-5717	369	2	that	that	SCONJ
ejpam-5717	369	3	f	f	PROPN
ejpam-5717	369	4	is	be	AUX
ejpam-5717	369	5	lower	low	ADJ
ejpam-5717	369	6	quasi	quasi	NOUN
ejpam-5717	369	7	θ(τ1	θ(τ1	NOUN
ejpam-5717	369	8	,	,	PUNCT
ejpam-5717	369	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	369	10	.	.	PUNCT
ejpam-5717	370	1	let	let	VERB
ejpam-5717	370	2	x	x	PUNCT
ejpam-5717	370	3	∈	∈	PROPN
ejpam-5717	370	4	x	x	X
ejpam-5717	370	5	and	and	CCONJ
ejpam-5717	370	6	v	v	X
ejpam-5717	370	7	be	be	AUX
ejpam-5717	370	8	any	any	DET
ejpam-5717	370	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	370	10	-	-	ADJ
ejpam-5717	370	11	open	open	ADJ
ejpam-5717	370	12	set	set	NOUN
ejpam-5717	370	13	of	of	ADP
ejpam-5717	370	14	y	y	PRON
ejpam-5717	370	15	such	such	ADJ
ejpam-5717	370	16	that	that	SCONJ
ejpam-5717	370	17	sclf⊛(x)∩v	sclf⊛(x)∩v	PROPN
ejpam-5717	370	18	̸=	̸=	PROPN
ejpam-5717	370	19	∅.	∅.	ADV
ejpam-5717	370	20	by	by	ADP
ejpam-5717	370	21	lemma	lemma	PROPN
ejpam-5717	370	22	3	3	NUM
ejpam-5717	370	23	,	,	PUNCT
ejpam-5717	370	24	we	we	PRON
ejpam-5717	370	25	have	have	VERB
ejpam-5717	370	26	f	f	PROPN
ejpam-5717	370	27	(	(	PUNCT
ejpam-5717	370	28	x)∩v	x)∩v	PROPN
ejpam-5717	370	29	̸=	̸=	PROPN
ejpam-5717	370	30	∅.	∅.	ADV
ejpam-5717	370	31	since	since	SCONJ
ejpam-5717	370	32	f	f	PROPN
ejpam-5717	370	33	is	be	AUX
ejpam-5717	370	34	lower	low	ADJ
ejpam-5717	370	35	quasi	quasi	NOUN
ejpam-5717	370	36	θ(τ1	θ(τ1	NOUN
ejpam-5717	370	37	,	,	PUNCT
ejpam-5717	370	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	370	39	,	,	PUNCT
ejpam-5717	370	40	there	there	PRON
ejpam-5717	370	41	exists	exist	VERB
ejpam-5717	370	42	a	a	DET
ejpam-5717	370	43	(	(	PUNCT
ejpam-5717	370	44	τ1	τ1	NOUN
ejpam-5717	370	45	,	,	PUNCT
ejpam-5717	370	46	τ2)s	τ2)s	NOUN
ejpam-5717	370	47	-	-	PUNCT
ejpam-5717	370	48	open	open	ADJ
ejpam-5717	370	49	set	set	VERB
ejpam-5717	370	50	ofx	ofx	NOUN
ejpam-5717	370	51	containing	contain	VERB
ejpam-5717	370	52	x	x	PUNCT
ejpam-5717	370	53	such	such	ADJ
ejpam-5717	370	54	that	that	SCONJ
ejpam-5717	370	55	σ1σ2	σ1σ2	NOUN
ejpam-5717	370	56	-	-	PUNCT
ejpam-5717	370	57	cl(v	cl(v	NOUN
ejpam-5717	370	58	)	)	PUNCT
ejpam-5717	370	59	∩	∩	PROPN
ejpam-5717	370	60	f	f	X
ejpam-5717	370	61	(	(	PUNCT
ejpam-5717	370	62	z	z	NOUN
ejpam-5717	370	63	)	)	PUNCT
ejpam-5717	370	64	̸=	̸=	NOUN
ejpam-5717	370	65	∅	∅	NOUN
ejpam-5717	370	66	for	for	ADP
ejpam-5717	370	67	every	every	DET
ejpam-5717	370	68	z	z	NOUN
ejpam-5717	370	69	∈	∈	PROPN
ejpam-5717	370	70	(	(	PUNCT
ejpam-5717	370	71	τ1	τ1	NOUN
ejpam-5717	370	72	,	,	PUNCT
ejpam-5717	370	73	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	370	74	)	)	PUNCT
ejpam-5717	370	75	.	.	PUNCT
ejpam-5717	371	1	since	since	SCONJ
ejpam-5717	371	2	σ1σ2	σ1σ2	NOUN
ejpam-5717	371	3	-	-	NOUN
ejpam-5717	371	4	cl(v	cl(v	NOUN
ejpam-5717	371	5	)	)	PUNCT
ejpam-5717	371	6	is	be	AUX
ejpam-5717	371	7	(	(	PUNCT
ejpam-5717	371	8	σ1	σ1	PROPN
ejpam-5717	371	9	,	,	PUNCT
ejpam-5717	371	10	σ2)sopen	σ2)sopen	VERB
ejpam-5717	371	11	in	in	ADP
ejpam-5717	371	12	y	y	PROPN
ejpam-5717	371	13	,	,	PUNCT
ejpam-5717	371	14	by	by	ADP
ejpam-5717	371	15	lemma	lemma	PROPN
ejpam-5717	371	16	3	3	NUM
ejpam-5717	371	17	we	we	PRON
ejpam-5717	371	18	have	have	AUX
ejpam-5717	371	19	(	(	PUNCT
ejpam-5717	371	20	τ1	τ1	NOUN
ejpam-5717	371	21	,	,	PUNCT
ejpam-5717	371	22	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	371	23	)	)	PUNCT
ejpam-5717	371	24	⊆	⊆	NUM
ejpam-5717	371	25	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	371	26	-	-	PUNCT
ejpam-5717	371	27	cl(v	cl(v	NOUN
ejpam-5717	371	28	)	)	PUNCT
ejpam-5717	371	29	)	)	PUNCT
ejpam-5717	372	1	=	=	PRON
ejpam-5717	372	2	sclf−	sclf−	NOUN
ejpam-5717	372	3	⊛	⊛	NUM
ejpam-5717	372	4	(	(	PUNCT
ejpam-5717	372	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	372	6	-	-	NUM
ejpam-5717	372	7	cl(v	cl(v	NOUN
ejpam-5717	372	8	)	)	PUNCT
ejpam-5717	372	9	)	)	PUNCT
ejpam-5717	372	10	and	and	CCONJ
ejpam-5717	372	11	hence	hence	ADV
ejpam-5717	372	12	sclf⊛(z)∩σ1σ2	sclf⊛(z)∩σ1σ2	NOUN
ejpam-5717	372	13	-	-	PUNCT
ejpam-5717	372	14	cl(v	cl(v	NOUN
ejpam-5717	372	15	)	)	PUNCT
ejpam-5717	372	16	̸=	̸=	NOUN
ejpam-5717	372	17	∅	∅	NOUN
ejpam-5717	372	18	for	for	ADP
ejpam-5717	372	19	every	every	DET
ejpam-5717	372	20	z	z	NOUN
ejpam-5717	372	21	∈	∈	PROPN
ejpam-5717	372	22	(	(	PUNCT
ejpam-5717	372	23	τ1	τ1	NOUN
ejpam-5717	372	24	,	,	PUNCT
ejpam-5717	372	25	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	372	26	)	)	PUNCT
ejpam-5717	372	27	.	.	PUNCT
ejpam-5717	373	1	this	this	PRON
ejpam-5717	373	2	shows	show	VERB
ejpam-5717	373	3	that	that	SCONJ
ejpam-5717	373	4	sclf⊛	sclf⊛	PROPN
ejpam-5717	373	5	is	be	AUX
ejpam-5717	373	6	lower	low	ADJ
ejpam-5717	373	7	quasi	quasi	NOUN
ejpam-5717	373	8	θ(τ1	θ(τ1	NOUN
ejpam-5717	373	9	,	,	PUNCT
ejpam-5717	373	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	373	11	.	.	PUNCT
ejpam-5717	374	1	conversely	conversely	ADV
ejpam-5717	374	2	,	,	PUNCT
ejpam-5717	374	3	suppose	suppose	VERB
ejpam-5717	374	4	that	that	SCONJ
ejpam-5717	374	5	sclf⊛	sclf⊛	PROPN
ejpam-5717	374	6	is	be	AUX
ejpam-5717	374	7	lower	low	ADJ
ejpam-5717	374	8	quasi	quasi	NOUN
ejpam-5717	374	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	374	10	,	,	PUNCT
ejpam-5717	374	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	374	12	.	.	PUNCT
ejpam-5717	375	1	let	let	VERB
ejpam-5717	375	2	x	x	PUNCT
ejpam-5717	375	3	∈	∈	PROPN
ejpam-5717	375	4	x	x	X
ejpam-5717	375	5	and	and	CCONJ
ejpam-5717	375	6	v	v	X
ejpam-5717	375	7	be	be	AUX
ejpam-5717	375	8	any	any	DET
ejpam-5717	375	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	375	10	-	-	ADJ
ejpam-5717	375	11	open	open	ADJ
ejpam-5717	375	12	set	set	NOUN
ejpam-5717	375	13	of	of	ADP
ejpam-5717	375	14	y	y	PRON
ejpam-5717	375	15	such	such	ADJ
ejpam-5717	375	16	that	that	SCONJ
ejpam-5717	375	17	f	f	PROPN
ejpam-5717	375	18	(	(	PUNCT
ejpam-5717	375	19	x)∩v	x)∩v	PROPN
ejpam-5717	375	20	̸=	̸=	PROPN
ejpam-5717	375	21	∅.	∅.	NOUN
ejpam-5717	375	22	then	then	ADV
ejpam-5717	375	23	,	,	PUNCT
ejpam-5717	375	24	(	(	PUNCT
ejpam-5717	375	25	σ1	σ1	PROPN
ejpam-5717	375	26	,	,	PUNCT
ejpam-5717	375	27	σ2)-scl(f	σ2)-scl(f	PROPN
ejpam-5717	375	28	(	(	PUNCT
ejpam-5717	375	29	x))∩v	x))∩v	NOUN
ejpam-5717	375	30	̸=	̸=	PROPN
ejpam-5717	375	31	∅.	∅.	NOUN
ejpam-5717	375	32	since	since	SCONJ
ejpam-5717	375	33	sclf⊛	sclf⊛	PROPN
ejpam-5717	375	34	is	be	AUX
ejpam-5717	375	35	lower	low	ADJ
ejpam-5717	375	36	quasi	quasi	NOUN
ejpam-5717	375	37	θ(τ1	θ(τ1	NOUN
ejpam-5717	375	38	,	,	PUNCT
ejpam-5717	375	39	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	375	40	,	,	PUNCT
ejpam-5717	375	41	there	there	PRON
ejpam-5717	375	42	exists	exist	VERB
ejpam-5717	375	43	a	a	DET
ejpam-5717	375	44	(	(	PUNCT
ejpam-5717	375	45	τ1	τ1	NOUN
ejpam-5717	375	46	,	,	PUNCT
ejpam-5717	375	47	τ2)s	τ2)s	NOUN
ejpam-5717	375	48	-	-	PUNCT
ejpam-5717	375	49	open	open	ADJ
ejpam-5717	375	50	set	set	NOUN
ejpam-5717	375	51	of	of	ADP
ejpam-5717	375	52	x	x	PUNCT
ejpam-5717	375	53	containing	contain	VERB
ejpam-5717	375	54	x	x	PUNCT
ejpam-5717	375	55	such	such	ADJ
ejpam-5717	375	56	that	that	DET
ejpam-5717	375	57	sclf⊛(z	sclf⊛(z	NOUN
ejpam-5717	375	58	)	)	PUNCT
ejpam-5717	375	59	∩	∩	NOUN
ejpam-5717	376	1	σ1σ2	σ1σ2	NOUN
ejpam-5717	376	2	-	-	NOUN
ejpam-5717	376	3	cl(v	cl(v	NOUN
ejpam-5717	376	4	)	)	PUNCT
ejpam-5717	376	5	̸=	̸=	NOUN
ejpam-5717	376	6	∅	∅	NOUN
ejpam-5717	376	7	for	for	ADP
ejpam-5717	376	8	every	every	DET
ejpam-5717	376	9	z	z	NOUN
ejpam-5717	376	10	∈	∈	PROPN
ejpam-5717	376	11	(	(	PUNCT
ejpam-5717	376	12	τ1	τ1	NOUN
ejpam-5717	376	13	,	,	PUNCT
ejpam-5717	376	14	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	376	15	)	)	PUNCT
ejpam-5717	376	16	.	.	PUNCT
ejpam-5717	377	1	since	since	SCONJ
ejpam-5717	377	2	σ1σ2	σ1σ2	NOUN
ejpam-5717	377	3	-	-	NOUN
ejpam-5717	377	4	cl(v	cl(v	NOUN
ejpam-5717	377	5	)	)	PUNCT
ejpam-5717	377	6	is	be	AUX
ejpam-5717	377	7	(	(	PUNCT
ejpam-5717	377	8	σ1	σ1	PROPN
ejpam-5717	377	9	,	,	PUNCT
ejpam-5717	377	10	σ2)s	σ2)s	NOUN
ejpam-5717	377	11	-	-	PUNCT
ejpam-5717	377	12	open	open	ADJ
ejpam-5717	377	13	in	in	ADP
ejpam-5717	377	14	y	y	PROPN
ejpam-5717	377	15	,	,	PUNCT
ejpam-5717	377	16	by	by	ADP
ejpam-5717	377	17	lemma	lemma	PROPN
ejpam-5717	377	18	3	3	NUM
ejpam-5717	377	19	(	(	PUNCT
ejpam-5717	377	20	τ1	τ1	NOUN
ejpam-5717	377	21	,	,	PUNCT
ejpam-5717	377	22	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	377	23	)	)	PUNCT
ejpam-5717	377	24	⊆	⊆	NUM
ejpam-5717	377	25	sclf−	sclf−	NOUN
ejpam-5717	377	26	⊛	⊛	NUM
ejpam-5717	377	27	(	(	PUNCT
ejpam-5717	377	28	σ1σ2	σ1σ2	NOUN
ejpam-5717	377	29	-	-	NUM
ejpam-5717	377	30	cl(v	cl(v	NOUN
ejpam-5717	377	31	)	)	PUNCT
ejpam-5717	377	32	)	)	PUNCT
ejpam-5717	378	1	=	=	PUNCT
ejpam-5717	378	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5717	378	3	-	-	PUNCT
ejpam-5717	378	4	cl(v	cl(v	NOUN
ejpam-5717	378	5	)	)	PUNCT
ejpam-5717	378	6	)	)	PUNCT
ejpam-5717	378	7	and	and	CCONJ
ejpam-5717	378	8	hence	hence	ADV
ejpam-5717	378	9	σ1σ2	σ1σ2	NOUN
ejpam-5717	378	10	-	-	PUNCT
ejpam-5717	378	11	cl(v	cl(v	NOUN
ejpam-5717	378	12	)	)	PUNCT
ejpam-5717	378	13	∩	∩	PROPN
ejpam-5717	378	14	f	f	X
ejpam-5717	378	15	(	(	PUNCT
ejpam-5717	378	16	z	z	NOUN
ejpam-5717	378	17	)	)	PUNCT
ejpam-5717	378	18	̸=	̸=	NOUN
ejpam-5717	378	19	∅	∅	NOUN
ejpam-5717	378	20	for	for	ADP
ejpam-5717	378	21	every	every	DET
ejpam-5717	378	22	z	z	NOUN
ejpam-5717	378	23	∈	∈	PROPN
ejpam-5717	378	24	(	(	PUNCT
ejpam-5717	378	25	τ1	τ1	NOUN
ejpam-5717	378	26	,	,	PUNCT
ejpam-5717	378	27	τ2)-scl(u	τ2)-scl(u	NOUN
ejpam-5717	378	28	)	)	PUNCT
ejpam-5717	378	29	.	.	PUNCT
ejpam-5717	379	1	thus	thus	ADV
ejpam-5717	379	2	,	,	PUNCT
ejpam-5717	379	3	f	f	PROPN
ejpam-5717	379	4	is	be	AUX
ejpam-5717	379	5	lower	low	ADJ
ejpam-5717	379	6	quasi	quasi	NOUN
ejpam-5717	379	7	θ(τ1	θ(τ1	NOUN
ejpam-5717	379	8	,	,	PUNCT
ejpam-5717	379	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	379	10	.	.	PUNCT
ejpam-5717	380	1	definition	definition	NOUN
ejpam-5717	380	2	4	4	NUM
ejpam-5717	380	3	.	.	PUNCT
ejpam-5717	381	1	[	[	X
ejpam-5717	381	2	30	30	NUM
ejpam-5717	381	3	]	]	X
ejpam-5717	381	4	a	a	DET
ejpam-5717	381	5	subset	subset	NOUN
ejpam-5717	381	6	a	a	PRON
ejpam-5717	381	7	of	of	ADP
ejpam-5717	381	8	a	a	DET
ejpam-5717	381	9	bitopological	bitopological	ADJ
ejpam-5717	381	10	space	space	NOUN
ejpam-5717	381	11	(	(	PUNCT
ejpam-5717	381	12	x	x	NOUN
ejpam-5717	381	13	,	,	PUNCT
ejpam-5717	381	14	τ1	τ1	NOUN
ejpam-5717	381	15	,	,	PUNCT
ejpam-5717	381	16	τ2	τ2	NOUN
ejpam-5717	381	17	)	)	PUNCT
ejpam-5717	381	18	is	be	AUX
ejpam-5717	381	19	said	say	VERB
ejpam-5717	381	20	to	to	PART
ejpam-5717	381	21	be	be	AUX
ejpam-5717	381	22	:	:	PUNCT
ejpam-5717	381	23	(	(	PUNCT
ejpam-5717	381	24	1	1	X
ejpam-5717	381	25	)	)	PUNCT
ejpam-5717	381	26	τ1τ2	τ1τ2	NOUN
ejpam-5717	381	27	-	-	NOUN
ejpam-5717	381	28	paracompact	paracompact	ADJ
ejpam-5717	381	29	if	if	SCONJ
ejpam-5717	381	30	every	every	DET
ejpam-5717	381	31	cover	cover	NOUN
ejpam-5717	381	32	of	of	ADP
ejpam-5717	381	33	a	a	PRON
ejpam-5717	381	34	by	by	ADP
ejpam-5717	381	35	τ1τ2	τ1τ2	ADJ
ejpam-5717	381	36	-	-	ADJ
ejpam-5717	381	37	open	open	ADJ
ejpam-5717	381	38	sets	set	NOUN
ejpam-5717	381	39	of	of	ADP
ejpam-5717	381	40	x	x	VERB
ejpam-5717	381	41	is	be	AUX
ejpam-5717	381	42	refined	refine	VERB
ejpam-5717	381	43	by	by	ADP
ejpam-5717	381	44	a	a	DET
ejpam-5717	381	45	cover	cover	NOUN
ejpam-5717	381	46	of	of	ADP
ejpam-5717	381	47	a	a	PRON
ejpam-5717	381	48	which	which	PRON
ejpam-5717	381	49	consists	consist	VERB
ejpam-5717	381	50	of	of	ADP
ejpam-5717	381	51	τ1τ2	τ1τ2	ADJ
ejpam-5717	381	52	-	-	ADJ
ejpam-5717	381	53	open	open	ADJ
ejpam-5717	381	54	sets	set	NOUN
ejpam-5717	381	55	of	of	ADP
ejpam-5717	381	56	x	x	PUNCT
ejpam-5717	381	57	and	and	CCONJ
ejpam-5717	381	58	is	be	AUX
ejpam-5717	381	59	τ1τ2	τ1τ2	NOUN
ejpam-5717	381	60	-	-	ADJ
ejpam-5717	381	61	locally	locally	ADV
ejpam-5717	381	62	finite	finite	NOUN
ejpam-5717	381	63	in	in	ADP
ejpam-5717	381	64	x	x	PRON
ejpam-5717	381	65	;	;	PUNCT
ejpam-5717	381	66	(	(	PUNCT
ejpam-5717	381	67	2	2	X
ejpam-5717	381	68	)	)	PUNCT
ejpam-5717	381	69	τ1τ2	τ1τ2	NOUN
ejpam-5717	381	70	-	-	NOUN
ejpam-5717	381	71	regular	regular	ADJ
ejpam-5717	381	72	if	if	SCONJ
ejpam-5717	381	73	for	for	ADP
ejpam-5717	381	74	each	each	DET
ejpam-5717	381	75	x	x	SYM
ejpam-5717	381	76	∈	∈	PROPN
ejpam-5717	381	77	a	a	PRON
ejpam-5717	381	78	and	and	CCONJ
ejpam-5717	381	79	each	each	DET
ejpam-5717	381	80	τ1τ2	τ1τ2	ADJ
ejpam-5717	381	81	-	-	ADJ
ejpam-5717	381	82	open	open	ADJ
ejpam-5717	381	83	set	set	ADJ
ejpam-5717	381	84	u	u	NOUN
ejpam-5717	381	85	of	of	ADP
ejpam-5717	381	86	x	x	PUNCT
ejpam-5717	381	87	containing	contain	VERB
ejpam-5717	381	88	x	x	PRON
ejpam-5717	381	89	,	,	PUNCT
ejpam-5717	381	90	there	there	PRON
ejpam-5717	381	91	exists	exist	VERB
ejpam-5717	381	92	a	a	DET
ejpam-5717	381	93	τ1τ2	τ1τ2	NOUN
ejpam-5717	381	94	-	-	ADJ
ejpam-5717	381	95	open	open	ADJ
ejpam-5717	381	96	set	set	NOUN
ejpam-5717	381	97	v	v	NOUN
ejpam-5717	381	98	of	of	ADP
ejpam-5717	381	99	x	x	PUNCT
ejpam-5717	381	100	such	such	ADJ
ejpam-5717	381	101	that	that	SCONJ
ejpam-5717	381	102	x	x	SYM
ejpam-5717	381	103	∈	∈	NOUN
ejpam-5717	381	104	v	v	ADP
ejpam-5717	381	105	⊆	⊆	NUM
ejpam-5717	381	106	τ1τ2	τ1τ2	NOUN
ejpam-5717	381	107	-	-	NOUN
ejpam-5717	381	108	cl(v	cl(v	X
ejpam-5717	381	109	)	)	PUNCT
ejpam-5717	381	110	⊆	⊆	NUM
ejpam-5717	381	111	u	u	NOUN
ejpam-5717	381	112	.	.	PUNCT
ejpam-5717	382	1	lemma	lemma	PROPN
ejpam-5717	382	2	4	4	NUM
ejpam-5717	382	3	.	.	PUNCT
ejpam-5717	383	1	[	[	X
ejpam-5717	383	2	30	30	NUM
ejpam-5717	383	3	]	]	X
ejpam-5717	383	4	if	if	SCONJ
ejpam-5717	383	5	a	a	PRON
ejpam-5717	383	6	is	be	AUX
ejpam-5717	383	7	a	a	DET
ejpam-5717	383	8	τ1τ2	τ1τ2	ADJ
ejpam-5717	383	9	-	-	ADJ
ejpam-5717	383	10	regular	regular	ADJ
ejpam-5717	383	11	τ1τ2	τ1τ2	NOUN
ejpam-5717	383	12	-	-	ADJ
ejpam-5717	383	13	paracompact	paracompact	ADJ
ejpam-5717	383	14	set	set	NOUN
ejpam-5717	383	15	of	of	ADP
ejpam-5717	383	16	a	a	DET
ejpam-5717	383	17	bitopological	bitopological	ADJ
ejpam-5717	383	18	space	space	NOUN
ejpam-5717	383	19	(	(	PUNCT
ejpam-5717	383	20	x	x	NOUN
ejpam-5717	383	21	,	,	PUNCT
ejpam-5717	383	22	τ1	τ1	NOUN
ejpam-5717	383	23	,	,	PUNCT
ejpam-5717	383	24	τ2	τ2	NOUN
ejpam-5717	383	25	)	)	PUNCT
ejpam-5717	383	26	and	and	CCONJ
ejpam-5717	383	27	u	u	NOUN
ejpam-5717	383	28	is	be	AUX
ejpam-5717	383	29	a	a	DET
ejpam-5717	383	30	τ1τ2	τ1τ2	ADJ
ejpam-5717	383	31	-	-	ADJ
ejpam-5717	383	32	open	open	ADJ
ejpam-5717	383	33	neighborhood	neighborhood	NOUN
ejpam-5717	383	34	of	of	ADP
ejpam-5717	383	35	a	a	PRON
ejpam-5717	383	36	,	,	PUNCT
ejpam-5717	383	37	then	then	ADV
ejpam-5717	383	38	there	there	PRON
ejpam-5717	383	39	exists	exist	VERB
ejpam-5717	383	40	a	a	DET
ejpam-5717	383	41	τ1τ2	τ1τ2	NOUN
ejpam-5717	383	42	-	-	ADJ
ejpam-5717	383	43	open	open	ADJ
ejpam-5717	383	44	set	set	NOUN
ejpam-5717	383	45	v	v	NOUN
ejpam-5717	383	46	of	of	ADP
ejpam-5717	383	47	x	x	PUNCT
ejpam-5717	383	48	such	such	ADJ
ejpam-5717	383	49	that	that	SCONJ
ejpam-5717	383	50	a	a	DET
ejpam-5717	383	51	⊆	⊆	NUM
ejpam-5717	383	52	v	v	ADP
ejpam-5717	383	53	⊆	⊆	NUM
ejpam-5717	383	54	τ1τ2	τ1τ2	NOUN
ejpam-5717	383	55	-	-	NOUN
ejpam-5717	383	56	cl(v	cl(v	X
ejpam-5717	383	57	)	)	PUNCT
ejpam-5717	383	58	⊆	⊆	NUM
ejpam-5717	383	59	u	u	NOUN
ejpam-5717	383	60	.	.	PUNCT
ejpam-5717	384	1	lemma	lemma	PROPN
ejpam-5717	384	2	5	5	NUM
ejpam-5717	384	3	.	.	PUNCT
ejpam-5717	385	1	if	if	SCONJ
ejpam-5717	385	2	f	f	PROPN
ejpam-5717	385	3	:	:	PUNCT
ejpam-5717	385	4	(	(	PUNCT
ejpam-5717	385	5	x	x	NOUN
ejpam-5717	385	6	,	,	PUNCT
ejpam-5717	385	7	τ1	τ1	NOUN
ejpam-5717	385	8	,	,	PUNCT
ejpam-5717	385	9	τ2	τ2	NOUN
ejpam-5717	385	10	)	)	PUNCT
ejpam-5717	385	11	→	→	SYM
ejpam-5717	385	12	(	(	PUNCT
ejpam-5717	385	13	y	y	PROPN
ejpam-5717	385	14	,	,	PUNCT
ejpam-5717	385	15	σ1	σ1	PROPN
ejpam-5717	385	16	,	,	PUNCT
ejpam-5717	385	17	σ2	σ2	PROPN
ejpam-5717	385	18	)	)	PUNCT
ejpam-5717	385	19	is	be	AUX
ejpam-5717	385	20	a	a	DET
ejpam-5717	385	21	multifunction	multifunction	NOUN
ejpam-5717	385	22	such	such	ADJ
ejpam-5717	385	23	that	that	SCONJ
ejpam-5717	385	24	f	f	PROPN
ejpam-5717	385	25	(	(	PUNCT
ejpam-5717	385	26	x	x	X
ejpam-5717	385	27	)	)	PUNCT
ejpam-5717	385	28	is	be	AUX
ejpam-5717	385	29	τ1τ2	τ1τ2	NOUN
ejpam-5717	385	30	-	-	ADJ
ejpam-5717	385	31	regular	regular	ADJ
ejpam-5717	385	32	and	and	CCONJ
ejpam-5717	385	33	τ1τ2	τ1τ2	NOUN
ejpam-5717	385	34	-	-	NOUN
ejpam-5717	385	35	paracompact	paracompact	NOUN
ejpam-5717	385	36	for	for	ADP
ejpam-5717	385	37	each	each	DET
ejpam-5717	385	38	x	x	SYM
ejpam-5717	385	39	∈	∈	PROPN
ejpam-5717	385	40	x	x	NOUN
ejpam-5717	385	41	,	,	PUNCT
ejpam-5717	385	42	then	then	ADV
ejpam-5717	385	43	sclf+	sclf+	ADP
ejpam-5717	385	44	⊛	⊛	NUM
ejpam-5717	385	45	(	(	PUNCT
ejpam-5717	385	46	v	v	NOUN
ejpam-5717	385	47	)	)	PUNCT
ejpam-5717	385	48	=	=	PUNCT
ejpam-5717	386	1	f+(v	f+(v	NOUN
ejpam-5717	386	2	)	)	PUNCT
ejpam-5717	386	3	for	for	ADP
ejpam-5717	386	4	each	each	DET
ejpam-5717	386	5	σ1σ2	σ1σ2	VERB
ejpam-5717	386	6	-	-	ADJ
ejpam-5717	386	7	open	open	ADJ
ejpam-5717	386	8	set	set	NOUN
ejpam-5717	386	9	v	v	NOUN
ejpam-5717	386	10	of	of	ADP
ejpam-5717	386	11	y	y	PROPN
ejpam-5717	386	12	.	.	PUNCT
ejpam-5717	387	1	proof	proof	NOUN
ejpam-5717	387	2	.	.	PUNCT
ejpam-5717	388	1	let	let	VERB
ejpam-5717	388	2	v	v	PART
ejpam-5717	388	3	be	be	AUX
ejpam-5717	388	4	any	any	DET
ejpam-5717	388	5	σ1σ2	σ1σ2	NOUN
ejpam-5717	388	6	-	-	ADJ
ejpam-5717	388	7	open	open	ADJ
ejpam-5717	388	8	set	set	NOUN
ejpam-5717	388	9	of	of	ADP
ejpam-5717	388	10	y	y	PROPN
ejpam-5717	388	11	and	and	CCONJ
ejpam-5717	388	12	x	x	PROPN
ejpam-5717	388	13	∈	∈	PROPN
ejpam-5717	388	14	sclf+	sclf+	ADP
ejpam-5717	388	15	⊛	⊛	NUM
ejpam-5717	388	16	(	(	PUNCT
ejpam-5717	388	17	v	v	NOUN
ejpam-5717	388	18	)	)	PUNCT
ejpam-5717	388	19	.	.	PUNCT
ejpam-5717	389	1	then	then	ADV
ejpam-5717	389	2	,	,	PUNCT
ejpam-5717	389	3	sclf+	sclf+	ADP
ejpam-5717	389	4	⊛	⊛	NUM
ejpam-5717	389	5	(	(	PUNCT
ejpam-5717	389	6	x	x	X
ejpam-5717	389	7	)	)	PUNCT
ejpam-5717	389	8	⊆	⊆	NUM
ejpam-5717	389	9	v	v	NOUN
ejpam-5717	389	10	and	and	CCONJ
ejpam-5717	389	11	f	f	PROPN
ejpam-5717	389	12	(	(	PUNCT
ejpam-5717	389	13	x	x	X
ejpam-5717	389	14	)	)	PUNCT
ejpam-5717	389	15	⊆	⊆	NUM
ejpam-5717	389	16	(	(	PUNCT
ejpam-5717	389	17	σ1	σ1	PROPN
ejpam-5717	389	18	,	,	PUNCT
ejpam-5717	389	19	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-5717	389	20	(	(	PUNCT
ejpam-5717	389	21	x	x	NOUN
ejpam-5717	389	22	)	)	PUNCT
ejpam-5717	389	23	)	)	PUNCT
ejpam-5717	390	1	=	=	PUNCT
ejpam-5717	390	2	sclf+	sclf+	ADP
ejpam-5717	390	3	⊛	⊛	NUM
ejpam-5717	390	4	(	(	PUNCT
ejpam-5717	390	5	x	x	X
ejpam-5717	390	6	)	)	PUNCT
ejpam-5717	390	7	⊆	⊆	NUM
ejpam-5717	390	8	v	v	NOUN
ejpam-5717	390	9	.	.	PUNCT
ejpam-5717	391	1	thus	thus	ADV
ejpam-5717	391	2	,	,	PUNCT
ejpam-5717	391	3	x	x	SYM
ejpam-5717	391	4	∈	∈	PROPN
ejpam-5717	391	5	f+(v	f+(v	NOUN
ejpam-5717	391	6	)	)	PUNCT
ejpam-5717	391	7	and	and	CCONJ
ejpam-5717	391	8	hence	hence	ADV
ejpam-5717	391	9	sclf+	sclf+	ADP
ejpam-5717	391	10	⊛	⊛	NUM
ejpam-5717	391	11	(	(	PUNCT
ejpam-5717	391	12	v	v	NOUN
ejpam-5717	391	13	)	)	PUNCT
ejpam-5717	391	14	⊆	⊆	NUM
ejpam-5717	391	15	f+(v	f+(v	NOUN
ejpam-5717	391	16	)	)	PUNCT
ejpam-5717	391	17	.	.	PUNCT
ejpam-5717	392	1	on	on	ADP
ejpam-5717	392	2	the	the	DET
ejpam-5717	392	3	other	other	ADJ
ejpam-5717	392	4	hand	hand	NOUN
ejpam-5717	392	5	,	,	PUNCT
ejpam-5717	392	6	let	let	VERB
ejpam-5717	392	7	x	x	X
ejpam-5717	392	8	∈	∈	PROPN
ejpam-5717	392	9	f+(v	f+(v	NOUN
ejpam-5717	392	10	)	)	PUNCT
ejpam-5717	392	11	.	.	PUNCT
ejpam-5717	393	1	then	then	ADV
ejpam-5717	393	2	,	,	PUNCT
ejpam-5717	393	3	f	f	PROPN
ejpam-5717	393	4	(	(	PUNCT
ejpam-5717	393	5	x	x	X
ejpam-5717	393	6	)	)	PUNCT
ejpam-5717	393	7	⊆	⊆	NUM
ejpam-5717	393	8	v	v	NOUN
ejpam-5717	393	9	and	and	CCONJ
ejpam-5717	393	10	by	by	ADP
ejpam-5717	393	11	lemma	lemma	PROPN
ejpam-5717	393	12	5	5	NUM
ejpam-5717	393	13	,	,	PUNCT
ejpam-5717	393	14	there	there	PRON
ejpam-5717	393	15	exists	exist	VERB
ejpam-5717	393	16	a	a	DET
ejpam-5717	393	17	σ1σ2	σ1σ2	NUM
ejpam-5717	393	18	-	-	ADJ
ejpam-5717	393	19	open	open	ADJ
ejpam-5717	393	20	set	set	NOUN
ejpam-5717	393	21	w	w	PROPN
ejpam-5717	393	22	of	of	ADP
ejpam-5717	393	23	y	y	PRON
ejpam-5717	393	24	such	such	ADJ
ejpam-5717	393	25	that	that	SCONJ
ejpam-5717	393	26	f	f	PROPN
ejpam-5717	393	27	(	(	PUNCT
ejpam-5717	393	28	x	x	X
ejpam-5717	393	29	)	)	PUNCT
ejpam-5717	393	30	⊆	⊆	NUM
ejpam-5717	393	31	w	w	ADP
ejpam-5717	393	32	⊆	⊆	NUM
ejpam-5717	393	33	σ1σ2	σ1σ2	NOUN
ejpam-5717	393	34	-	-	PUNCT
ejpam-5717	393	35	cl(w	cl(w	NOUN
ejpam-5717	393	36	)	)	PUNCT
ejpam-5717	393	37	⊆	⊆	NUM
ejpam-5717	393	38	v	v	NOUN
ejpam-5717	393	39	;	;	PUNCT
ejpam-5717	393	40	hence	hence	ADV
ejpam-5717	393	41	sclf+	sclf+	ADP
ejpam-5717	393	42	⊛	⊛	PROPN
ejpam-5717	393	43	(	(	PUNCT
ejpam-5717	393	44	x	x	X
ejpam-5717	393	45	)	)	PUNCT
ejpam-5717	393	46	=	=	SYM
ejpam-5717	393	47	(	(	PUNCT
ejpam-5717	393	48	σ1	σ1	PROPN
ejpam-5717	393	49	,	,	PUNCT
ejpam-5717	393	50	σ2)-scl(f	σ2)-scl(f	NOUN
ejpam-5717	393	51	(	(	PUNCT
ejpam-5717	393	52	x	x	NOUN
ejpam-5717	393	53	)	)	PUNCT
ejpam-5717	393	54	)	)	PUNCT
ejpam-5717	393	55	⊆	⊆	X
ejpam-5717	393	56	σ1σ2	σ1σ2	NOUN
ejpam-5717	393	57	-	-	PUNCT
ejpam-5717	393	58	cl(w	cl(w	NOUN
ejpam-5717	393	59	)	)	PUNCT
ejpam-5717	393	60	⊆	⊆	PROPN
ejpam-5717	393	61	v.	v.	ADP
ejpam-5717	393	62	thus	thus	ADV
ejpam-5717	393	63	,	,	PUNCT
ejpam-5717	393	64	x	x	PROPN
ejpam-5717	393	65	∈	∈	PROPN
ejpam-5717	393	66	sclf+	sclf+	ADP
ejpam-5717	393	67	⊛	⊛	NUM
ejpam-5717	393	68	(	(	PUNCT
ejpam-5717	393	69	v	v	NOUN
ejpam-5717	393	70	)	)	PUNCT
ejpam-5717	393	71	and	and	CCONJ
ejpam-5717	393	72	so	so	ADV
ejpam-5717	393	73	f+(v	f+(v	PROPN
ejpam-5717	393	74	)	)	PUNCT
ejpam-5717	394	1	⊆	⊆	NUM
ejpam-5717	394	2	sclf+	sclf+	ADP
ejpam-5717	394	3	⊛	⊛	NUM
ejpam-5717	394	4	(	(	PUNCT
ejpam-5717	394	5	v	v	NOUN
ejpam-5717	394	6	)	)	PUNCT
ejpam-5717	394	7	.	.	PUNCT
ejpam-5717	395	1	therefore	therefore	ADV
ejpam-5717	395	2	,	,	PUNCT
ejpam-5717	395	3	f+(v	f+(v	PROPN
ejpam-5717	395	4	)	)	PUNCT
ejpam-5717	396	1	=	=	PUNCT
ejpam-5717	396	2	sclf+	sclf+	ADP
ejpam-5717	396	3	⊛	⊛	NUM
ejpam-5717	396	4	(	(	PUNCT
ejpam-5717	396	5	v	v	NOUN
ejpam-5717	396	6	)	)	PUNCT
ejpam-5717	396	7	.	.	PUNCT
ejpam-5717	397	1	p.	p.	NOUN
ejpam-5717	397	2	pue	pue	NOUN
ejpam-5717	397	3	-	-	PUNCT
ejpam-5717	397	4	on	on	ADP
ejpam-5717	397	5	,	,	PUNCT
ejpam-5717	397	6	a.	a.	PROPN
ejpam-5717	397	7	sama	sama	PROPN
ejpam-5717	397	8	-	-	PUNCT
ejpam-5717	397	9	ae	ae	PROPN
ejpam-5717	397	10	,	,	PUNCT
ejpam-5717	397	11	c.	c.	PROPN
ejpam-5717	397	12	boonpok	boonpok	PROPN
ejpam-5717	397	13	/	/	SYM
ejpam-5717	397	14	eur	eur	PROPN
ejpam-5717	397	15	.	.	PUNCT
ejpam-5717	398	1	j.	j.	PROPN
ejpam-5717	398	2	pure	pure	PROPN
ejpam-5717	398	3	appl	appl	PROPN
ejpam-5717	398	4	.	.	PROPN
ejpam-5717	398	5	math	math	PROPN
ejpam-5717	398	6	,	,	PUNCT
ejpam-5717	398	7	18	18	NUM
ejpam-5717	398	8	(	(	PUNCT
ejpam-5717	398	9	1	1	NUM
ejpam-5717	398	10	)	)	PUNCT
ejpam-5717	398	11	(	(	PUNCT
ejpam-5717	398	12	2025	2025	NUM
ejpam-5717	398	13	)	)	PUNCT
ejpam-5717	398	14	,	,	PUNCT
ejpam-5717	398	15	5717	5717	NUM
ejpam-5717	398	16	12	12	NUM
ejpam-5717	398	17	of	of	ADP
ejpam-5717	398	18	16	16	NUM
ejpam-5717	398	19	theorem	theorem	NOUN
ejpam-5717	398	20	9	9	NUM
ejpam-5717	398	21	.	.	PUNCT
ejpam-5717	399	1	let	let	VERB
ejpam-5717	399	2	f	f	NOUN
ejpam-5717	399	3	:	:	PUNCT
ejpam-5717	399	4	(	(	PUNCT
ejpam-5717	399	5	x	x	NOUN
ejpam-5717	399	6	,	,	PUNCT
ejpam-5717	399	7	τ1	τ1	NOUN
ejpam-5717	399	8	,	,	PUNCT
ejpam-5717	399	9	τ2	τ2	NOUN
ejpam-5717	399	10	)	)	PUNCT
ejpam-5717	399	11	→	→	SYM
ejpam-5717	399	12	(	(	PUNCT
ejpam-5717	399	13	y	y	PROPN
ejpam-5717	399	14	,	,	PUNCT
ejpam-5717	399	15	σ1	σ1	PROPN
ejpam-5717	399	16	,	,	PUNCT
ejpam-5717	399	17	σ2	σ2	PROPN
ejpam-5717	399	18	)	)	PUNCT
ejpam-5717	399	19	be	be	VERB
ejpam-5717	399	20	a	a	DET
ejpam-5717	399	21	multifunction	multifunction	NOUN
ejpam-5717	399	22	such	such	ADJ
ejpam-5717	399	23	that	that	SCONJ
ejpam-5717	399	24	f	f	PROPN
ejpam-5717	399	25	(	(	PUNCT
ejpam-5717	399	26	x	x	X
ejpam-5717	399	27	)	)	PUNCT
ejpam-5717	399	28	is	be	AUX
ejpam-5717	399	29	σ1σ2paracompact	σ1σ2paracompact	NUM
ejpam-5717	399	30	and	and	CCONJ
ejpam-5717	399	31	σ1σ2	σ1σ2	NOUN
ejpam-5717	399	32	-	-	ADJ
ejpam-5717	399	33	regular	regular	ADJ
ejpam-5717	399	34	for	for	ADP
ejpam-5717	399	35	each	each	DET
ejpam-5717	399	36	x	x	SYM
ejpam-5717	399	37	∈	∈	PROPN
ejpam-5717	399	38	x.	x.	NOUN
ejpam-5717	399	39	then	then	ADV
ejpam-5717	399	40	,	,	PUNCT
ejpam-5717	399	41	f	f	PROPN
ejpam-5717	399	42	is	be	AUX
ejpam-5717	399	43	upper	upper	ADJ
ejpam-5717	399	44	quasi	quasi	NOUN
ejpam-5717	399	45	θ(τ1	θ(τ1	NOUN
ejpam-5717	399	46	,	,	PUNCT
ejpam-5717	399	47	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	400	1	if	if	SCONJ
ejpam-5717	400	2	and	and	CCONJ
ejpam-5717	400	3	only	only	ADV
ejpam-5717	400	4	if	if	SCONJ
ejpam-5717	400	5	sclf⊛	sclf⊛	PRON
ejpam-5717	400	6	:	:	PUNCT
ejpam-5717	400	7	(	(	PUNCT
ejpam-5717	400	8	x	x	NOUN
ejpam-5717	400	9	,	,	PUNCT
ejpam-5717	400	10	τ1	τ1	NOUN
ejpam-5717	400	11	,	,	PUNCT
ejpam-5717	400	12	τ2	τ2	NOUN
ejpam-5717	400	13	)	)	PUNCT
ejpam-5717	400	14	→	→	SYM
ejpam-5717	400	15	(	(	PUNCT
ejpam-5717	400	16	y	y	PROPN
ejpam-5717	400	17	,	,	PUNCT
ejpam-5717	400	18	σ1	σ1	PROPN
ejpam-5717	400	19	,	,	PUNCT
ejpam-5717	400	20	σ2	σ2	PROPN
ejpam-5717	400	21	)	)	PUNCT
ejpam-5717	400	22	is	be	AUX
ejpam-5717	400	23	upper	upper	ADJ
ejpam-5717	400	24	quasi	quasi	NOUN
ejpam-5717	400	25	θ(τ1	θ(τ1	NOUN
ejpam-5717	400	26	,	,	PUNCT
ejpam-5717	400	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	400	28	.	.	PUNCT
ejpam-5717	401	1	proof	proof	NOUN
ejpam-5717	401	2	.	.	PUNCT
ejpam-5717	402	1	suppose	suppose	VERB
ejpam-5717	402	2	that	that	SCONJ
ejpam-5717	402	3	f	f	PROPN
ejpam-5717	402	4	is	be	AUX
ejpam-5717	402	5	upper	upper	ADJ
ejpam-5717	402	6	quasi	quasi	NOUN
ejpam-5717	402	7	θ(τ1	θ(τ1	NOUN
ejpam-5717	402	8	,	,	PUNCT
ejpam-5717	402	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	402	10	.	.	PUNCT
ejpam-5717	403	1	it	it	PRON
ejpam-5717	403	2	follows	follow	VERB
ejpam-5717	403	3	from	from	ADP
ejpam-5717	403	4	theorem	theorem	ADJ
ejpam-5717	403	5	1	1	NUM
ejpam-5717	403	6	and	and	CCONJ
ejpam-5717	403	7	lemma	lemma	PROPN
ejpam-5717	403	8	5	5	NUM
ejpam-5717	403	9	that	that	PRON
ejpam-5717	403	10	for	for	ADP
ejpam-5717	403	11	every	every	DET
ejpam-5717	403	12	σ1σ2	σ1σ2	NUM
ejpam-5717	403	13	-	-	ADJ
ejpam-5717	403	14	open	open	ADJ
ejpam-5717	403	15	set	set	NOUN
ejpam-5717	403	16	v	v	NOUN
ejpam-5717	403	17	of	of	ADP
ejpam-5717	403	18	y	y	PROPN
ejpam-5717	403	19	,	,	PUNCT
ejpam-5717	403	20	sclf+	sclf+	ADP
ejpam-5717	403	21	⊛	⊛	NUM
ejpam-5717	403	22	(	(	PUNCT
ejpam-5717	403	23	v	v	NOUN
ejpam-5717	403	24	)	)	PUNCT
ejpam-5717	403	25	=	=	PUNCT
ejpam-5717	404	1	f+(v	f+(v	NOUN
ejpam-5717	404	2	)	)	PUNCT
ejpam-5717	405	1	⊆	⊆	NUM
ejpam-5717	405	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	405	3	,	,	PUNCT
ejpam-5717	405	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	406	1	+	+	ADJ
ejpam-5717	406	2	(	(	PUNCT
ejpam-5717	406	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	406	4	-	-	NUM
ejpam-5717	406	5	cl(v	cl(v	NOUN
ejpam-5717	406	6	)	)	PUNCT
ejpam-5717	406	7	)	)	PUNCT
ejpam-5717	406	8	)	)	PUNCT
ejpam-5717	407	1	=	=	SYM
ejpam-5717	407	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	407	3	,	,	PUNCT
ejpam-5717	407	4	τ2)-sint(sclf	τ2)-sint(sclf	PUNCT
ejpam-5717	407	5	+	+	NUM
ejpam-5717	407	6	⊛	⊛	NUM
ejpam-5717	407	7	(	(	PUNCT
ejpam-5717	407	8	σ1σ2	σ1σ2	NOUN
ejpam-5717	407	9	-	-	NUM
ejpam-5717	407	10	cl(v	cl(v	NOUN
ejpam-5717	407	11	)	)	PUNCT
ejpam-5717	407	12	)	)	PUNCT
ejpam-5717	407	13	)	)	PUNCT
ejpam-5717	407	14	.	.	PUNCT
ejpam-5717	408	1	by	by	ADP
ejpam-5717	408	2	theorem	theorem	NOUN
ejpam-5717	408	3	1	1	NUM
ejpam-5717	408	4	,	,	PUNCT
ejpam-5717	408	5	sclf⊛	sclf⊛	PROPN
ejpam-5717	408	6	is	be	AUX
ejpam-5717	408	7	upper	upper	ADJ
ejpam-5717	408	8	quasi	quasi	NOUN
ejpam-5717	408	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	408	10	,	,	PUNCT
ejpam-5717	408	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	408	12	.	.	PUNCT
ejpam-5717	409	1	conversely	conversely	ADV
ejpam-5717	409	2	,	,	PUNCT
ejpam-5717	409	3	suppose	suppose	VERB
ejpam-5717	409	4	that	that	SCONJ
ejpam-5717	409	5	sclf⊛	sclf⊛	PROPN
ejpam-5717	409	6	is	be	AUX
ejpam-5717	409	7	upper	upper	ADJ
ejpam-5717	409	8	quasi	quasi	NOUN
ejpam-5717	409	9	θ(τ1	θ(τ1	NOUN
ejpam-5717	409	10	,	,	PUNCT
ejpam-5717	409	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	409	12	.	.	PUNCT
ejpam-5717	410	1	it	it	PRON
ejpam-5717	410	2	follows	follow	VERB
ejpam-5717	410	3	from	from	ADP
ejpam-5717	410	4	theorem	theorem	ADJ
ejpam-5717	410	5	1	1	NUM
ejpam-5717	410	6	and	and	CCONJ
ejpam-5717	410	7	lemma	lemma	PROPN
ejpam-5717	410	8	5	5	NUM
ejpam-5717	410	9	that	that	PRON
ejpam-5717	410	10	for	for	ADP
ejpam-5717	410	11	every	every	DET
ejpam-5717	410	12	σ1σ2	σ1σ2	NUM
ejpam-5717	410	13	-	-	ADJ
ejpam-5717	410	14	open	open	ADJ
ejpam-5717	410	15	set	set	NOUN
ejpam-5717	410	16	v	v	NOUN
ejpam-5717	410	17	of	of	ADP
ejpam-5717	410	18	y	y	PROPN
ejpam-5717	410	19	,	,	PUNCT
ejpam-5717	410	20	f+(v	f+(v	PROPN
ejpam-5717	410	21	)	)	PUNCT
ejpam-5717	411	1	=	=	PUNCT
ejpam-5717	412	1	sclf+	sclf+	ADP
ejpam-5717	412	2	⊛	⊛	NUM
ejpam-5717	412	3	(	(	PUNCT
ejpam-5717	412	4	v	v	NOUN
ejpam-5717	412	5	)	)	PUNCT
ejpam-5717	412	6	⊆	⊆	NUM
ejpam-5717	412	7	θ(τ1	θ(τ1	NOUN
ejpam-5717	412	8	,	,	PUNCT
ejpam-5717	412	9	τ2)-sint(sclf	τ2)-sint(sclf	PUNCT
ejpam-5717	412	10	+	+	NUM
ejpam-5717	412	11	⊛	⊛	NUM
ejpam-5717	412	12	(	(	PUNCT
ejpam-5717	412	13	σ1σ2	σ1σ2	NOUN
ejpam-5717	412	14	-	-	NUM
ejpam-5717	412	15	cl(v	cl(v	NOUN
ejpam-5717	412	16	)	)	PUNCT
ejpam-5717	412	17	)	)	PUNCT
ejpam-5717	412	18	)	)	PUNCT
ejpam-5717	413	1	=	=	SYM
ejpam-5717	413	2	θ(τ1	θ(τ1	NOUN
ejpam-5717	413	3	,	,	PUNCT
ejpam-5717	413	4	τ2)-sint(f	τ2)-sint(f	PUNCT
ejpam-5717	414	1	+	+	ADJ
ejpam-5717	414	2	(	(	PUNCT
ejpam-5717	414	3	σ1σ2	σ1σ2	NOUN
ejpam-5717	414	4	-	-	NUM
ejpam-5717	414	5	cl(v	cl(v	NOUN
ejpam-5717	414	6	)	)	PUNCT
ejpam-5717	414	7	)	)	PUNCT
ejpam-5717	414	8	)	)	PUNCT
ejpam-5717	414	9	.	.	PUNCT
ejpam-5717	415	1	thus	thus	ADV
ejpam-5717	415	2	by	by	ADP
ejpam-5717	415	3	theorem	theorem	NOUN
ejpam-5717	415	4	1	1	NUM
ejpam-5717	415	5	,	,	PUNCT
ejpam-5717	415	6	f	f	PROPN
ejpam-5717	415	7	is	be	AUX
ejpam-5717	415	8	upper	upper	ADJ
ejpam-5717	415	9	quasi	quasi	NOUN
ejpam-5717	415	10	θ(τ1	θ(τ1	NOUN
ejpam-5717	415	11	,	,	PUNCT
ejpam-5717	415	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	415	13	.	.	PUNCT
ejpam-5717	416	1	acknowledgements	acknowledgement	NOUN
ejpam-5717	416	2	this	this	DET
ejpam-5717	416	3	research	research	NOUN
ejpam-5717	416	4	project	project	NOUN
ejpam-5717	416	5	was	be	AUX
ejpam-5717	416	6	financially	financially	ADV
ejpam-5717	416	7	supported	support	VERB
ejpam-5717	416	8	by	by	ADP
ejpam-5717	416	9	mahasarakham	mahasarakham	PROPN
ejpam-5717	416	10	university	university	PROPN
ejpam-5717	416	11	.	.	PUNCT
ejpam-5717	417	1	references	reference	NOUN
ejpam-5717	417	2	[	[	X
ejpam-5717	417	3	1	1	X
ejpam-5717	417	4	]	]	PUNCT
ejpam-5717	417	5	s.	s.	PROPN
ejpam-5717	417	6	p.	p.	PROPN
ejpam-5717	417	7	arya	arya	PROPN
ejpam-5717	417	8	and	and	CCONJ
ejpam-5717	417	9	m.	m.	PROPN
ejpam-5717	417	10	p.	p.	PROPN
ejpam-5717	417	11	bhamini	bhamini	PROPN
ejpam-5717	417	12	.	.	PUNCT
ejpam-5717	418	1	some	some	DET
ejpam-5717	418	2	weaker	weak	ADJ
ejpam-5717	418	3	forms	form	NOUN
ejpam-5717	418	4	of	of	ADP
ejpam-5717	418	5	semi	semi	ADJ
ejpam-5717	418	6	-	-	ADJ
ejpam-5717	418	7	continuous	continuous	ADJ
ejpam-5717	418	8	functions	function	NOUN
ejpam-5717	418	9	.	.	PUNCT
ejpam-5717	419	1	ganita	ganita	NOUN
ejpam-5717	419	2	,	,	PUNCT
ejpam-5717	419	3	33:124–134	33:124–134	NUM
ejpam-5717	419	4	,	,	PUNCT
ejpam-5717	419	5	1982	1982	NUM
ejpam-5717	419	6	.	.	PUNCT
ejpam-5717	420	1	[	[	X
ejpam-5717	420	2	2	2	NUM
ejpam-5717	420	3	]	]	PUNCT
ejpam-5717	420	4	c.	c.	PROPN
ejpam-5717	420	5	boonpok	boonpok	PROPN
ejpam-5717	420	6	.	.	PUNCT
ejpam-5717	421	1	almost	almost	ADV
ejpam-5717	421	2	(	(	PUNCT
ejpam-5717	421	3	g	g	NOUN
ejpam-5717	421	4	,	,	PUNCT
ejpam-5717	421	5	m)-continuous	m)-continuous	ADJ
ejpam-5717	421	6	functions	function	NOUN
ejpam-5717	421	7	.	.	PUNCT
ejpam-5717	422	1	international	international	ADJ
ejpam-5717	422	2	journal	journal	PROPN
ejpam-5717	422	3	of	of	ADP
ejpam-5717	422	4	mathematical	mathematical	ADJ
ejpam-5717	422	5	analysis	analysis	NOUN
ejpam-5717	422	6	,	,	PUNCT
ejpam-5717	422	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-5717	422	8	,	,	PUNCT
ejpam-5717	422	9	2010	2010	NUM
ejpam-5717	422	10	.	.	PUNCT
ejpam-5717	423	1	[	[	X
ejpam-5717	423	2	3	3	X
ejpam-5717	423	3	]	]	PUNCT
ejpam-5717	423	4	c.	c.	PROPN
ejpam-5717	423	5	boonpok	boonpok	PROPN
ejpam-5717	423	6	.	.	PUNCT
ejpam-5717	424	1	m	m	VERB
ejpam-5717	424	2	-continuous	-continuous	ADJ
ejpam-5717	424	3	functions	function	NOUN
ejpam-5717	424	4	in	in	ADP
ejpam-5717	424	5	biminimal	biminimal	NOUN
ejpam-5717	424	6	structure	structure	NOUN
ejpam-5717	424	7	spaces	space	NOUN
ejpam-5717	424	8	.	.	PUNCT
ejpam-5717	425	1	far	far	PROPN
ejpam-5717	425	2	east	east	PROPN
ejpam-5717	425	3	journal	journal	PROPN
ejpam-5717	425	4	of	of	ADP
ejpam-5717	425	5	mathematical	mathematical	ADJ
ejpam-5717	425	6	sciences	science	NOUN
ejpam-5717	425	7	,	,	PUNCT
ejpam-5717	425	8	43(1):41–58	43(1):41–58	NUM
ejpam-5717	425	9	,	,	PUNCT
ejpam-5717	425	10	2010	2010	NUM
ejpam-5717	425	11	.	.	PUNCT
ejpam-5717	426	1	[	[	X
ejpam-5717	426	2	4	4	NUM
ejpam-5717	426	3	]	]	PUNCT
ejpam-5717	426	4	c.	c.	PROPN
ejpam-5717	426	5	boonpok	boonpok	PROPN
ejpam-5717	426	6	.	.	PUNCT
ejpam-5717	427	1	on	on	ADP
ejpam-5717	427	2	continuous	continuous	ADJ
ejpam-5717	427	3	multifunctions	multifunction	NOUN
ejpam-5717	427	4	in	in	ADP
ejpam-5717	427	5	ideal	ideal	ADJ
ejpam-5717	427	6	topological	topological	ADJ
ejpam-5717	427	7	spaces	space	NOUN
ejpam-5717	427	8	.	.	PUNCT
ejpam-5717	428	1	lobachevskii	lobachevskii	PROPN
ejpam-5717	428	2	journal	journal	PROPN
ejpam-5717	428	3	of	of	ADP
ejpam-5717	428	4	mathematics	mathematic	NOUN
ejpam-5717	428	5	,	,	PUNCT
ejpam-5717	428	6	40(1):24–35	40(1):24–35	NUM
ejpam-5717	428	7	,	,	PUNCT
ejpam-5717	428	8	2019	2019	NUM
ejpam-5717	428	9	.	.	PUNCT
ejpam-5717	429	1	[	[	X
ejpam-5717	429	2	5	5	X
ejpam-5717	429	3	]	]	PUNCT
ejpam-5717	429	4	c.	c.	PROPN
ejpam-5717	429	5	boonpok	boonpok	PROPN
ejpam-5717	429	6	.	.	PUNCT
ejpam-5717	430	1	on	on	ADP
ejpam-5717	430	2	characterizations	characterization	NOUN
ejpam-5717	430	3	of	of	ADP
ejpam-5717	430	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5717	430	5	ideal	ideal	ADJ
ejpam-5717	430	6	topological	topological	ADJ
ejpam-5717	430	7	spaces	space	NOUN
ejpam-5717	430	8	.	.	PUNCT
ejpam-5717	431	1	journal	journal	NOUN
ejpam-5717	431	2	of	of	ADP
ejpam-5717	431	3	mathematics	mathematic	NOUN
ejpam-5717	431	4	,	,	PUNCT
ejpam-5717	431	5	2020:9387601	2020:9387601	NUM
ejpam-5717	431	6	,	,	PUNCT
ejpam-5717	431	7	2020	2020	NUM
ejpam-5717	431	8	.	.	PUNCT
ejpam-5717	432	1	[	[	X
ejpam-5717	432	2	6	6	NUM
ejpam-5717	432	3	]	]	PUNCT
ejpam-5717	432	4	c.	c.	PROPN
ejpam-5717	432	5	boonpok	boonpok	PROPN
ejpam-5717	432	6	.	.	PUNCT
ejpam-5717	433	1	(	(	PUNCT
ejpam-5717	433	2	τ1	τ1	NOUN
ejpam-5717	433	3	,	,	PUNCT
ejpam-5717	433	4	τ2)δ	τ2)δ	ADJ
ejpam-5717	433	5	-	-	PUNCT
ejpam-5717	433	6	semicontinuous	semicontinuous	ADJ
ejpam-5717	433	7	multifunctions	multifunction	NOUN
ejpam-5717	433	8	.	.	PUNCT
ejpam-5717	434	1	heliyon	heliyon	NOUN
ejpam-5717	434	2	,	,	PUNCT
ejpam-5717	434	3	6	6	NUM
ejpam-5717	434	4	:	:	SYM
ejpam-5717	434	5	e05367	e05367	PROPN
ejpam-5717	434	6	,	,	PUNCT
ejpam-5717	434	7	2020	2020	NUM
ejpam-5717	434	8	.	.	PUNCT
ejpam-5717	435	1	[	[	X
ejpam-5717	435	2	7	7	X
ejpam-5717	435	3	]	]	X
ejpam-5717	435	4	c.	c.	PROPN
ejpam-5717	435	5	boonpok	boonpok	PROPN
ejpam-5717	435	6	.	.	PUNCT
ejpam-5717	436	1	weak	weak	ADJ
ejpam-5717	436	2	quasi	quasi	ADJ
ejpam-5717	436	3	continuity	continuity	NOUN
ejpam-5717	436	4	for	for	ADP
ejpam-5717	436	5	multifunctions	multifunction	NOUN
ejpam-5717	436	6	in	in	ADP
ejpam-5717	436	7	ideal	ideal	ADJ
ejpam-5717	436	8	topological	topological	ADJ
ejpam-5717	436	9	spaces	space	NOUN
ejpam-5717	436	10	.	.	PUNCT
ejpam-5717	437	1	advances	advance	NOUN
ejpam-5717	437	2	in	in	ADP
ejpam-5717	437	3	mathematics	mathematic	NOUN
ejpam-5717	437	4	:	:	PUNCT
ejpam-5717	437	5	scientific	scientific	ADJ
ejpam-5717	437	6	journal	journal	NOUN
ejpam-5717	437	7	,	,	PUNCT
ejpam-5717	437	8	9(1):339–355	9(1):339–355	NUM
ejpam-5717	437	9	,	,	PUNCT
ejpam-5717	437	10	2020	2020	NUM
ejpam-5717	437	11	.	.	PUNCT
ejpam-5717	438	1	[	[	X
ejpam-5717	438	2	8	8	NUM
ejpam-5717	438	3	]	]	X
ejpam-5717	438	4	c.	c.	PROPN
ejpam-5717	438	5	boonpok	boonpok	PROPN
ejpam-5717	438	6	.	.	PUNCT
ejpam-5717	439	1	upper	upper	ADJ
ejpam-5717	439	2	and	and	CCONJ
ejpam-5717	439	3	lower	low	ADJ
ejpam-5717	439	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-5717	439	5	.	.	PUNCT
ejpam-5717	439	6	heliyon	heliyon	NOUN
ejpam-5717	439	7	,	,	PUNCT
ejpam-5717	439	8	7	7	NUM
ejpam-5717	439	9	:	:	PUNCT
ejpam-5717	439	10	e05986	e05986	PROPN
ejpam-5717	439	11	,	,	PUNCT
ejpam-5717	439	12	2021	2021	NUM
ejpam-5717	439	13	.	.	PUNCT
ejpam-5717	440	1	[	[	X
ejpam-5717	440	2	9	9	NUM
ejpam-5717	440	3	]	]	PUNCT
ejpam-5717	440	4	c.	c.	PROPN
ejpam-5717	440	5	boonpok	boonpok	PROPN
ejpam-5717	440	6	.	.	PUNCT
ejpam-5717	441	1	on	on	ADP
ejpam-5717	441	2	some	some	DET
ejpam-5717	441	3	closed	closed	ADJ
ejpam-5717	441	4	sets	set	NOUN
ejpam-5717	441	5	and	and	CCONJ
ejpam-5717	441	6	low	low	ADJ
ejpam-5717	441	7	separation	separation	NOUN
ejpam-5717	441	8	axioms	axiom	NOUN
ejpam-5717	441	9	via	via	ADP
ejpam-5717	441	10	topological	topological	ADJ
ejpam-5717	441	11	ideals	ideal	NOUN
ejpam-5717	441	12	.	.	PUNCT
ejpam-5717	442	1	european	european	ADJ
ejpam-5717	442	2	journal	journal	PROPN
ejpam-5717	442	3	of	of	ADP
ejpam-5717	442	4	pure	pure	ADJ
ejpam-5717	442	5	and	and	CCONJ
ejpam-5717	442	6	applied	applied	ADJ
ejpam-5717	442	7	mathematics	mathematic	NOUN
ejpam-5717	442	8	,	,	PUNCT
ejpam-5717	442	9	15(3):1023–1046	15(3):1023–1046	NUM
ejpam-5717	442	10	,	,	PUNCT
ejpam-5717	442	11	2022	2022	NUM
ejpam-5717	442	12	.	.	PUNCT
ejpam-5717	443	1	[	[	X
ejpam-5717	443	2	10	10	NUM
ejpam-5717	443	3	]	]	X
ejpam-5717	443	4	c.	c.	PROPN
ejpam-5717	443	5	boonpok	boonpok	PROPN
ejpam-5717	443	6	.	.	PUNCT
ejpam-5717	444	1	θ(⋆)-quasi	θ(⋆)-quasi	DET
ejpam-5717	444	2	continuity	continuity	NOUN
ejpam-5717	444	3	for	for	ADP
ejpam-5717	444	4	multifunctions	multifunction	NOUN
ejpam-5717	444	5	.	.	PUNCT
ejpam-5717	445	1	wseas	wseas	PROPN
ejpam-5717	445	2	transactions	transaction	NOUN
ejpam-5717	445	3	on	on	ADP
ejpam-5717	445	4	mathematics	mathematic	NOUN
ejpam-5717	445	5	,	,	PUNCT
ejpam-5717	445	6	21:245–251	21:245–251	NUM
ejpam-5717	445	7	,	,	PUNCT
ejpam-5717	445	8	2022	2022	NUM
ejpam-5717	445	9	.	.	PUNCT
ejpam-5717	446	1	p.	p.	NOUN
ejpam-5717	446	2	pue	pue	NOUN
ejpam-5717	446	3	-	-	PUNCT
ejpam-5717	446	4	on	on	ADP
ejpam-5717	446	5	,	,	PUNCT
ejpam-5717	446	6	a.	a.	PROPN
ejpam-5717	446	7	sama	sama	PROPN
ejpam-5717	446	8	-	-	PUNCT
ejpam-5717	446	9	ae	ae	PROPN
ejpam-5717	446	10	,	,	PUNCT
ejpam-5717	446	11	c.	c.	PROPN
ejpam-5717	446	12	boonpok	boonpok	PROPN
ejpam-5717	446	13	/	/	SYM
ejpam-5717	446	14	eur	eur	PROPN
ejpam-5717	446	15	.	.	PUNCT
ejpam-5717	447	1	j.	j.	PROPN
ejpam-5717	447	2	pure	pure	PROPN
ejpam-5717	447	3	appl	appl	PROPN
ejpam-5717	447	4	.	.	PROPN
ejpam-5717	447	5	math	math	PROPN
ejpam-5717	447	6	,	,	PUNCT
ejpam-5717	447	7	18	18	NUM
ejpam-5717	447	8	(	(	PUNCT
ejpam-5717	447	9	1	1	NUM
ejpam-5717	447	10	)	)	PUNCT
ejpam-5717	447	11	(	(	PUNCT
ejpam-5717	447	12	2025	2025	NUM
ejpam-5717	447	13	)	)	PUNCT
ejpam-5717	447	14	,	,	PUNCT
ejpam-5717	447	15	5717	5717	NUM
ejpam-5717	447	16	13	13	NUM
ejpam-5717	447	17	of	of	ADP
ejpam-5717	447	18	16	16	NUM
ejpam-5717	448	1	[	[	X
ejpam-5717	448	2	11	11	NUM
ejpam-5717	448	3	]	]	PUNCT
ejpam-5717	448	4	c.	c.	PROPN
ejpam-5717	448	5	boonpok	boonpok	PROPN
ejpam-5717	448	6	.	.	PUNCT
ejpam-5717	449	1	on	on	ADP
ejpam-5717	449	2	some	some	DET
ejpam-5717	449	3	spaces	space	NOUN
ejpam-5717	449	4	via	via	ADP
ejpam-5717	449	5	topological	topological	ADJ
ejpam-5717	449	6	ideals	ideal	NOUN
ejpam-5717	449	7	.	.	PUNCT
ejpam-5717	450	1	open	open	ADJ
ejpam-5717	450	2	mathematics	mathematic	NOUN
ejpam-5717	450	3	,	,	PUNCT
ejpam-5717	450	4	21:20230118	21:20230118	NUM
ejpam-5717	450	5	,	,	PUNCT
ejpam-5717	450	6	2023	2023	NUM
ejpam-5717	450	7	.	.	PUNCT
ejpam-5717	451	1	[	[	X
ejpam-5717	451	2	12	12	NUM
ejpam-5717	451	3	]	]	PUNCT
ejpam-5717	451	4	c.	c.	PROPN
ejpam-5717	451	5	boonpok	boonpok	PROPN
ejpam-5717	451	6	.	.	PUNCT
ejpam-5717	452	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5717	452	2	.	.	PUNCT
ejpam-5717	453	1	mathematica	mathematica	PROPN
ejpam-5717	453	2	,	,	PUNCT
ejpam-5717	453	3	65(1):31–42	65(1):31–42	NUM
ejpam-5717	453	4	,	,	PUNCT
ejpam-5717	453	5	2023	2023	NUM
ejpam-5717	453	6	.	.	PUNCT
ejpam-5717	454	1	[	[	X
ejpam-5717	454	2	13	13	NUM
ejpam-5717	454	3	]	]	PUNCT
ejpam-5717	454	4	c.	c.	PROPN
ejpam-5717	454	5	boonpok	boonpok	PROPN
ejpam-5717	454	6	and	and	CCONJ
ejpam-5717	454	7	j.	j.	PROPN
ejpam-5717	454	8	khampakdee	khampakdee	PROPN
ejpam-5717	454	9	.	.	PUNCT
ejpam-5717	455	1	(	(	PUNCT
ejpam-5717	455	2	λ	λ	NOUN
ejpam-5717	455	3	,	,	PUNCT
ejpam-5717	455	4	sp)-open	sp)-open	ADJ
ejpam-5717	455	5	sets	set	NOUN
ejpam-5717	455	6	in	in	ADP
ejpam-5717	455	7	topological	topological	ADJ
ejpam-5717	455	8	spaces	space	NOUN
ejpam-5717	455	9	.	.	PUNCT
ejpam-5717	456	1	european	european	ADJ
ejpam-5717	456	2	journal	journal	PROPN
ejpam-5717	456	3	of	of	ADP
ejpam-5717	456	4	pure	pure	ADJ
ejpam-5717	456	5	and	and	CCONJ
ejpam-5717	456	6	applied	applied	ADJ
ejpam-5717	456	7	mathematics	mathematic	NOUN
ejpam-5717	456	8	,	,	PUNCT
ejpam-5717	456	9	15(2):572–588	15(2):572–588	NUM
ejpam-5717	456	10	,	,	PUNCT
ejpam-5717	456	11	2022	2022	NUM
ejpam-5717	456	12	.	.	PUNCT
ejpam-5717	457	1	[	[	X
ejpam-5717	457	2	14	14	NUM
ejpam-5717	457	3	]	]	X
ejpam-5717	457	4	c.	c.	PROPN
ejpam-5717	457	5	boonpok	boonpok	PROPN
ejpam-5717	457	6	and	and	CCONJ
ejpam-5717	457	7	j.	j.	PROPN
ejpam-5717	457	8	khampakdee	khampakdee	PROPN
ejpam-5717	457	9	.	.	PUNCT
ejpam-5717	458	1	on	on	ADP
ejpam-5717	458	2	almost	almost	ADV
ejpam-5717	458	3	α(λ	α(λ	PROPN
ejpam-5717	458	4	,	,	PUNCT
ejpam-5717	458	5	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	458	6	multifunctions	multifunction	NOUN
ejpam-5717	458	7	.	.	PUNCT
ejpam-5717	459	1	european	european	PROPN
ejpam-5717	459	2	journal	journal	PROPN
ejpam-5717	459	3	of	of	ADP
ejpam-5717	459	4	pure	pure	ADJ
ejpam-5717	459	5	and	and	CCONJ
ejpam-5717	459	6	applied	applied	ADJ
ejpam-5717	459	7	mathematics	mathematic	NOUN
ejpam-5717	459	8	,	,	PUNCT
ejpam-5717	459	9	15(2):626–634	15(2):626–634	PROPN
ejpam-5717	459	10	,	,	PUNCT
ejpam-5717	459	11	2022	2022	NUM
ejpam-5717	459	12	.	.	PUNCT
ejpam-5717	460	1	[	[	X
ejpam-5717	460	2	15	15	NUM
ejpam-5717	460	3	]	]	X
ejpam-5717	460	4	c.	c.	PROPN
ejpam-5717	460	5	boonpok	boonpok	PROPN
ejpam-5717	460	6	and	and	CCONJ
ejpam-5717	460	7	j.	j.	PROPN
ejpam-5717	460	8	khampakdee	khampakdee	PROPN
ejpam-5717	460	9	.	.	PUNCT
ejpam-5717	461	1	slight	slight	PROPN
ejpam-5717	461	2	(	(	PUNCT
ejpam-5717	461	3	λ	λ	NOUN
ejpam-5717	461	4	,	,	PUNCT
ejpam-5717	461	5	sp)-continuity	sp)-continuity	NOUN
ejpam-5717	461	6	and	and	CCONJ
ejpam-5717	461	7	λsp	λsp	NOUN
ejpam-5717	461	8	-	-	PUNCT
ejpam-5717	461	9	extremally	extremally	ADV
ejpam-5717	461	10	disconnectedness	disconnectedness	NOUN
ejpam-5717	461	11	.	.	PUNCT
ejpam-5717	462	1	european	european	ADJ
ejpam-5717	462	2	journal	journal	PROPN
ejpam-5717	462	3	of	of	ADP
ejpam-5717	462	4	pure	pure	ADJ
ejpam-5717	462	5	and	and	CCONJ
ejpam-5717	462	6	applied	applied	ADJ
ejpam-5717	462	7	mathematics	mathematic	NOUN
ejpam-5717	462	8	,	,	PUNCT
ejpam-5717	462	9	15(3):1180–1188	15(3):1180–1188	NUM
ejpam-5717	462	10	,	,	PUNCT
ejpam-5717	462	11	2022	2022	NUM
ejpam-5717	462	12	.	.	PUNCT
ejpam-5717	463	1	[	[	X
ejpam-5717	463	2	16	16	NUM
ejpam-5717	463	3	]	]	X
ejpam-5717	463	4	c.	c.	PROPN
ejpam-5717	463	5	boonpok	boonpok	PROPN
ejpam-5717	463	6	and	and	CCONJ
ejpam-5717	463	7	j.	j.	PROPN
ejpam-5717	463	8	khampakdee	khampakdee	PROPN
ejpam-5717	463	9	.	.	PUNCT
ejpam-5717	464	1	upper	upper	ADJ
ejpam-5717	464	2	and	and	CCONJ
ejpam-5717	464	3	lower	low	ADJ
ejpam-5717	464	4	weak	weak	ADJ
ejpam-5717	464	5	sβ(⋆)-continuity	sβ(⋆)-continuity	NOUN
ejpam-5717	464	6	.	.	PUNCT
ejpam-5717	465	1	european	european	PROPN
ejpam-5717	465	2	journal	journal	PROPN
ejpam-5717	465	3	of	of	ADP
ejpam-5717	465	4	pure	pure	ADJ
ejpam-5717	465	5	and	and	CCONJ
ejpam-5717	465	6	applied	applied	ADJ
ejpam-5717	465	7	mathematics	mathematic	NOUN
ejpam-5717	465	8	,	,	PUNCT
ejpam-5717	465	9	16(4):2544–2556	16(4):2544–2556	NUM
ejpam-5717	465	10	,	,	PUNCT
ejpam-5717	465	11	2023	2023	NUM
ejpam-5717	465	12	.	.	PUNCT
ejpam-5717	466	1	[	[	X
ejpam-5717	466	2	17	17	NUM
ejpam-5717	466	3	]	]	X
ejpam-5717	466	4	c.	c.	PROPN
ejpam-5717	466	5	boonpok	boonpok	PROPN
ejpam-5717	466	6	and	and	CCONJ
ejpam-5717	466	7	j.	j.	PROPN
ejpam-5717	466	8	khampakdee	khampakdee	PROPN
ejpam-5717	466	9	.	.	PUNCT
ejpam-5717	467	1	almost	almost	ADV
ejpam-5717	467	2	strong	strong	ADJ
ejpam-5717	467	3	θ(λ	θ(λ	PROPN
ejpam-5717	467	4	,	,	PUNCT
ejpam-5717	467	5	p)-continuity	p)-continuity	NOUN
ejpam-5717	467	6	for	for	ADP
ejpam-5717	467	7	functions	function	NOUN
ejpam-5717	467	8	.	.	PUNCT
ejpam-5717	468	1	european	european	ADJ
ejpam-5717	468	2	journal	journal	PROPN
ejpam-5717	468	3	of	of	ADP
ejpam-5717	468	4	pure	pure	ADJ
ejpam-5717	468	5	and	and	CCONJ
ejpam-5717	468	6	applied	applied	ADJ
ejpam-5717	468	7	mathematics	mathematic	NOUN
ejpam-5717	468	8	,	,	PUNCT
ejpam-5717	468	9	17(1):300–309	17(1):300–309	PROPN
ejpam-5717	468	10	,	,	PUNCT
ejpam-5717	468	11	2024	2024	NUM
ejpam-5717	468	12	.	.	PUNCT
ejpam-5717	469	1	[	[	X
ejpam-5717	469	2	18	18	NUM
ejpam-5717	469	3	]	]	PUNCT
ejpam-5717	469	4	c.	c.	PROPN
ejpam-5717	469	5	boonpok	boonpok	PROPN
ejpam-5717	469	6	and	and	CCONJ
ejpam-5717	469	7	j.	j.	PROPN
ejpam-5717	469	8	khampakdee	khampakdee	PROPN
ejpam-5717	469	9	.	.	PUNCT
ejpam-5717	470	1	upper	upper	ADJ
ejpam-5717	470	2	and	and	CCONJ
ejpam-5717	470	3	lower	low	ADJ
ejpam-5717	470	4	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-5717	470	5	.	.	PUNCT
ejpam-5717	470	6	european	european	PROPN
ejpam-5717	470	7	journal	journal	PROPN
ejpam-5717	470	8	of	of	ADP
ejpam-5717	470	9	pure	pure	ADJ
ejpam-5717	470	10	and	and	CCONJ
ejpam-5717	470	11	applied	applied	ADJ
ejpam-5717	470	12	mathematics	mathematic	NOUN
ejpam-5717	470	13	,	,	PUNCT
ejpam-5717	470	14	17(1):201–211	17(1):201–211	NUM
ejpam-5717	470	15	,	,	PUNCT
ejpam-5717	470	16	2024	2024	NUM
ejpam-5717	470	17	.	.	PUNCT
ejpam-5717	471	1	[	[	X
ejpam-5717	471	2	19	19	NUM
ejpam-5717	471	3	]	]	X
ejpam-5717	471	4	c.	c.	PROPN
ejpam-5717	471	5	boonpok	boonpok	PROPN
ejpam-5717	471	6	and	and	CCONJ
ejpam-5717	471	7	c.	c.	PROPN
ejpam-5717	471	8	klanarong	klanarong	PROPN
ejpam-5717	471	9	.	.	PUNCT
ejpam-5717	472	1	on	on	ADP
ejpam-5717	472	2	weakly	weakly	ADJ
ejpam-5717	472	3	(	(	PUNCT
ejpam-5717	472	4	τ1	τ1	NOUN
ejpam-5717	472	5	,	,	PUNCT
ejpam-5717	472	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	472	7	functions	function	NOUN
ejpam-5717	472	8	.	.	PUNCT
ejpam-5717	473	1	european	european	ADJ
ejpam-5717	473	2	journal	journal	PROPN
ejpam-5717	473	3	of	of	ADP
ejpam-5717	473	4	pure	pure	ADJ
ejpam-5717	473	5	and	and	CCONJ
ejpam-5717	473	6	applied	applied	ADJ
ejpam-5717	473	7	mathematics	mathematic	NOUN
ejpam-5717	473	8	,	,	PUNCT
ejpam-5717	473	9	17(1):416–425	17(1):416–425	NUM
ejpam-5717	473	10	,	,	PUNCT
ejpam-5717	473	11	2024	2024	NUM
ejpam-5717	473	12	.	.	PUNCT
ejpam-5717	474	1	[	[	X
ejpam-5717	474	2	20	20	NUM
ejpam-5717	474	3	]	]	PUNCT
ejpam-5717	474	4	c.	c.	PROPN
ejpam-5717	474	5	boonpok	boonpok	PROPN
ejpam-5717	474	6	and	and	CCONJ
ejpam-5717	474	7	p.	p.	NOUN
ejpam-5717	474	8	pue	pue	NOUN
ejpam-5717	474	9	-	-	PUNCT
ejpam-5717	474	10	on	on	ADP
ejpam-5717	474	11	.	.	PUNCT
ejpam-5717	475	1	continuity	continuity	NOUN
ejpam-5717	475	2	for	for	ADP
ejpam-5717	475	3	multifunctions	multifunction	NOUN
ejpam-5717	475	4	in	in	ADP
ejpam-5717	475	5	ideal	ideal	ADJ
ejpam-5717	475	6	topological	topological	ADJ
ejpam-5717	475	7	spaces	space	NOUN
ejpam-5717	475	8	.	.	PUNCT
ejpam-5717	476	1	wseas	wseas	VERB
ejpam-5717	476	2	transactions	transaction	NOUN
ejpam-5717	476	3	on	on	ADP
ejpam-5717	476	4	mathematics	mathematic	NOUN
ejpam-5717	476	5	,	,	PUNCT
ejpam-5717	476	6	19:624–631	19:624–631	NUM
ejpam-5717	476	7	,	,	PUNCT
ejpam-5717	476	8	2020	2020	NUM
ejpam-5717	476	9	.	.	PUNCT
ejpam-5717	477	1	[	[	X
ejpam-5717	477	2	21	21	NUM
ejpam-5717	477	3	]	]	X
ejpam-5717	477	4	c.	c.	PROPN
ejpam-5717	477	5	boonpok	boonpok	PROPN
ejpam-5717	477	6	and	and	CCONJ
ejpam-5717	477	7	p.	p.	NOUN
ejpam-5717	477	8	pue	pue	NOUN
ejpam-5717	477	9	-	-	PUNCT
ejpam-5717	477	10	on	on	ADP
ejpam-5717	477	11	.	.	PUNCT
ejpam-5717	478	1	upper	upper	ADJ
ejpam-5717	478	2	and	and	CCONJ
ejpam-5717	478	3	lower	low	ADJ
ejpam-5717	478	4	sβ(⋆)-continuous	sβ(⋆)-continuous	ADJ
ejpam-5717	478	5	multifunctions	multifunction	NOUN
ejpam-5717	478	6	.	.	PUNCT
ejpam-5717	479	1	european	european	ADJ
ejpam-5717	479	2	journal	journal	PROPN
ejpam-5717	479	3	of	of	ADP
ejpam-5717	479	4	pure	pure	ADJ
ejpam-5717	479	5	and	and	CCONJ
ejpam-5717	479	6	applied	applied	ADJ
ejpam-5717	479	7	mathematics	mathematic	NOUN
ejpam-5717	479	8	,	,	PUNCT
ejpam-5717	479	9	16(3):1634–1646	16(3):1634–1646	NUM
ejpam-5717	479	10	,	,	PUNCT
ejpam-5717	479	11	2023	2023	NUM
ejpam-5717	479	12	.	.	PUNCT
ejpam-5717	480	1	[	[	X
ejpam-5717	480	2	22	22	NUM
ejpam-5717	480	3	]	]	PUNCT
ejpam-5717	480	4	c.	c.	PROPN
ejpam-5717	480	5	boonpok	boonpok	PROPN
ejpam-5717	480	6	and	and	CCONJ
ejpam-5717	480	7	p.	p.	NOUN
ejpam-5717	480	8	pue	pue	NOUN
ejpam-5717	480	9	-	-	PUNCT
ejpam-5717	480	10	on	on	ADP
ejpam-5717	480	11	.	.	PUNCT
ejpam-5717	481	1	upper	upper	ADJ
ejpam-5717	481	2	and	and	CCONJ
ejpam-5717	481	3	lower	low	ADJ
ejpam-5717	481	4	weakly	weakly	ADJ
ejpam-5717	481	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5717	481	6	multifunctions	multifunction	NOUN
ejpam-5717	481	7	.	.	PUNCT
ejpam-5717	482	1	international	international	ADJ
ejpam-5717	482	2	journal	journal	NOUN
ejpam-5717	482	3	of	of	ADP
ejpam-5717	482	4	analysis	analysis	NOUN
ejpam-5717	482	5	and	and	CCONJ
ejpam-5717	482	6	applications	application	NOUN
ejpam-5717	482	7	,	,	PUNCT
ejpam-5717	482	8	21:90	21:90	NUM
ejpam-5717	482	9	,	,	PUNCT
ejpam-5717	482	10	2023	2023	NUM
ejpam-5717	482	11	.	.	PUNCT
ejpam-5717	483	1	[	[	X
ejpam-5717	483	2	23	23	NUM
ejpam-5717	483	3	]	]	X
ejpam-5717	483	4	c.	c.	PROPN
ejpam-5717	483	5	boonpok	boonpok	PROPN
ejpam-5717	483	6	and	and	CCONJ
ejpam-5717	483	7	p.	p.	NOUN
ejpam-5717	483	8	pue	pue	NOUN
ejpam-5717	483	9	-	-	PUNCT
ejpam-5717	483	10	on	on	ADP
ejpam-5717	483	11	.	.	PUNCT
ejpam-5717	484	1	upper	upper	ADJ
ejpam-5717	484	2	and	and	CCONJ
ejpam-5717	484	3	lower	low	ADJ
ejpam-5717	484	4	weakly	weakly	ADJ
ejpam-5717	484	5	(	(	PUNCT
ejpam-5717	484	6	λ	λ	NOUN
ejpam-5717	484	7	,	,	PUNCT
ejpam-5717	484	8	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	484	9	multifunctions	multifunction	NOUN
ejpam-5717	484	10	.	.	PUNCT
ejpam-5717	485	1	european	european	PROPN
ejpam-5717	485	2	journal	journal	PROPN
ejpam-5717	485	3	of	of	ADP
ejpam-5717	485	4	pure	pure	ADJ
ejpam-5717	485	5	and	and	CCONJ
ejpam-5717	485	6	applied	applied	ADJ
ejpam-5717	485	7	mathematics	mathematic	NOUN
ejpam-5717	485	8	,	,	PUNCT
ejpam-5717	485	9	16(2):1047–1058	16(2):1047–1058	NUM
ejpam-5717	485	10	,	,	PUNCT
ejpam-5717	485	11	2023	2023	NUM
ejpam-5717	485	12	.	.	PUNCT
ejpam-5717	486	1	[	[	X
ejpam-5717	486	2	24	24	NUM
ejpam-5717	486	3	]	]	PUNCT
ejpam-5717	486	4	c.	c.	PROPN
ejpam-5717	486	5	boonpok	boonpok	PROPN
ejpam-5717	486	6	and	and	CCONJ
ejpam-5717	486	7	p.	p.	NOUN
ejpam-5717	486	8	pue	pue	NOUN
ejpam-5717	486	9	-	-	PUNCT
ejpam-5717	486	10	on	on	ADP
ejpam-5717	486	11	.	.	PUNCT
ejpam-5717	487	1	characterizations	characterization	NOUN
ejpam-5717	487	2	of	of	ADP
ejpam-5717	487	3	almost	almost	ADV
ejpam-5717	487	4	(	(	PUNCT
ejpam-5717	487	5	τ1	τ1	NOUN
ejpam-5717	487	6	,	,	PUNCT
ejpam-5717	487	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	487	8	functions	function	NOUN
ejpam-5717	487	9	.	.	PUNCT
ejpam-5717	488	1	international	international	ADJ
ejpam-5717	488	2	journal	journal	NOUN
ejpam-5717	488	3	of	of	ADP
ejpam-5717	488	4	analysis	analysis	NOUN
ejpam-5717	488	5	and	and	CCONJ
ejpam-5717	488	6	applications	application	NOUN
ejpam-5717	488	7	,	,	PUNCT
ejpam-5717	488	8	22:33	22:33	NUM
ejpam-5717	488	9	,	,	PUNCT
ejpam-5717	488	10	2024	2024	NUM
ejpam-5717	488	11	.	.	PUNCT
ejpam-5717	489	1	[	[	X
ejpam-5717	489	2	25	25	NUM
ejpam-5717	489	3	]	]	PUNCT
ejpam-5717	489	4	c.	c.	PROPN
ejpam-5717	489	5	boonpok	boonpok	PROPN
ejpam-5717	489	6	and	and	CCONJ
ejpam-5717	489	7	n.	n.	PROPN
ejpam-5717	489	8	srisarakham	srisarakham	PROPN
ejpam-5717	489	9	.	.	PUNCT
ejpam-5717	490	1	almost	almost	ADV
ejpam-5717	490	2	α-⋆-continuity	α-⋆-continuity	NUM
ejpam-5717	490	3	for	for	ADP
ejpam-5717	490	4	multifunctions	multifunction	NOUN
ejpam-5717	490	5	.	.	PUNCT
ejpam-5717	491	1	international	international	ADJ
ejpam-5717	491	2	journal	journal	NOUN
ejpam-5717	491	3	of	of	ADP
ejpam-5717	491	4	analysis	analysis	NOUN
ejpam-5717	491	5	and	and	CCONJ
ejpam-5717	491	6	applications	application	NOUN
ejpam-5717	491	7	,	,	PUNCT
ejpam-5717	491	8	21:107	21:107	NUM
ejpam-5717	491	9	,	,	PUNCT
ejpam-5717	491	10	2023	2023	NUM
ejpam-5717	491	11	.	.	PUNCT
ejpam-5717	492	1	[	[	X
ejpam-5717	492	2	26	26	NUM
ejpam-5717	492	3	]	]	X
ejpam-5717	492	4	c.	c.	PROPN
ejpam-5717	492	5	boonpok	boonpok	PROPN
ejpam-5717	492	6	and	and	CCONJ
ejpam-5717	492	7	n.	n.	PROPN
ejpam-5717	492	8	srisarakham	srisarakham	PROPN
ejpam-5717	492	9	.	.	PUNCT
ejpam-5717	493	1	weak	weak	ADJ
ejpam-5717	493	2	forms	form	NOUN
ejpam-5717	493	3	of	of	ADP
ejpam-5717	493	4	(	(	PUNCT
ejpam-5717	493	5	λ	λ	PROPN
ejpam-5717	493	6	,	,	PUNCT
ejpam-5717	493	7	b)-open	b)-open	VERB
ejpam-5717	493	8	sets	set	NOUN
ejpam-5717	493	9	and	and	CCONJ
ejpam-5717	493	10	weak	weak	ADJ
ejpam-5717	493	11	(	(	PUNCT
ejpam-5717	493	12	λ	λ	NOUN
ejpam-5717	493	13	,	,	PUNCT
ejpam-5717	493	14	b)continuity	b)continuity	NOUN
ejpam-5717	493	15	.	.	PUNCT
ejpam-5717	494	1	european	european	PROPN
ejpam-5717	494	2	journal	journal	PROPN
ejpam-5717	494	3	of	of	ADP
ejpam-5717	494	4	pure	pure	ADJ
ejpam-5717	494	5	and	and	CCONJ
ejpam-5717	494	6	applied	applied	ADJ
ejpam-5717	494	7	mathematics	mathematic	NOUN
ejpam-5717	494	8	,	,	PUNCT
ejpam-5717	494	9	16(1):29–43	16(1):29–43	NUM
ejpam-5717	494	10	,	,	PUNCT
ejpam-5717	494	11	2023	2023	NUM
ejpam-5717	494	12	.	.	PUNCT
ejpam-5717	495	1	[	[	X
ejpam-5717	495	2	27	27	NUM
ejpam-5717	495	3	]	]	X
ejpam-5717	495	4	c.	c.	PROPN
ejpam-5717	495	5	boonpok	boonpok	PROPN
ejpam-5717	495	6	and	and	CCONJ
ejpam-5717	495	7	n.	n.	PROPN
ejpam-5717	495	8	srisarakham	srisarakham	PROPN
ejpam-5717	495	9	.	.	PUNCT
ejpam-5717	496	1	(	(	PUNCT
ejpam-5717	496	2	τ1	τ1	NOUN
ejpam-5717	496	3	,	,	PUNCT
ejpam-5717	496	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5717	496	5	for	for	ADP
ejpam-5717	496	6	functions	function	NOUN
ejpam-5717	496	7	.	.	PUNCT
ejpam-5717	497	1	asia	asia	PROPN
ejpam-5717	497	2	pacific	pacific	PROPN
ejpam-5717	497	3	journal	journal	PROPN
ejpam-5717	497	4	of	of	ADP
ejpam-5717	497	5	mathematics	mathematic	NOUN
ejpam-5717	497	6	,	,	PUNCT
ejpam-5717	497	7	11:21	11:21	NUM
ejpam-5717	497	8	,	,	PUNCT
ejpam-5717	497	9	2024	2024	NUM
ejpam-5717	497	10	.	.	PUNCT
ejpam-5717	498	1	[	[	X
ejpam-5717	498	2	28	28	NUM
ejpam-5717	498	3	]	]	X
ejpam-5717	498	4	c.	c.	PROPN
ejpam-5717	498	5	boonpok	boonpok	PROPN
ejpam-5717	498	6	and	and	CCONJ
ejpam-5717	498	7	m.	m.	NOUN
ejpam-5717	498	8	thongmoon	thongmoon	NOUN
ejpam-5717	498	9	.	.	PUNCT
ejpam-5717	499	1	weak	weak	ADJ
ejpam-5717	499	2	α(λ	α(λ	PROPN
ejpam-5717	499	3	,	,	PUNCT
ejpam-5717	499	4	sp)-continuity	sp)-continuity	NOUN
ejpam-5717	499	5	for	for	ADP
ejpam-5717	499	6	multifunctions	multifunction	NOUN
ejpam-5717	499	7	.	.	PUNCT
ejpam-5717	500	1	european	european	ADJ
ejpam-5717	500	2	journal	journal	PROPN
ejpam-5717	500	3	of	of	ADP
ejpam-5717	500	4	pure	pure	ADJ
ejpam-5717	500	5	and	and	CCONJ
ejpam-5717	500	6	applied	applied	ADJ
ejpam-5717	500	7	mathematics	mathematic	NOUN
ejpam-5717	500	8	,	,	PUNCT
ejpam-5717	500	9	16(1):465–478	16(1):465–478	NUM
ejpam-5717	500	10	,	,	PUNCT
ejpam-5717	500	11	2023	2023	NUM
ejpam-5717	500	12	.	.	PUNCT
ejpam-5717	501	1	[	[	X
ejpam-5717	501	2	29	29	NUM
ejpam-5717	501	3	]	]	X
ejpam-5717	501	4	c.	c.	PROPN
ejpam-5717	501	5	boonpok	boonpok	PROPN
ejpam-5717	501	6	and	and	CCONJ
ejpam-5717	501	7	c.	c.	PROPN
ejpam-5717	501	8	viriyapong	viriyapong	PROPN
ejpam-5717	501	9	.	.	PUNCT
ejpam-5717	502	1	upper	upper	ADJ
ejpam-5717	502	2	and	and	CCONJ
ejpam-5717	502	3	lower	low	ADJ
ejpam-5717	502	4	almost	almost	ADV
ejpam-5717	502	5	weak	weak	ADJ
ejpam-5717	502	6	(	(	PUNCT
ejpam-5717	502	7	τ1	τ1	NOUN
ejpam-5717	502	8	,	,	PUNCT
ejpam-5717	502	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5717	502	10	.	.	PUNCT
ejpam-5717	503	1	european	european	PROPN
ejpam-5717	503	2	journal	journal	PROPN
ejpam-5717	503	3	of	of	ADP
ejpam-5717	503	4	pure	pure	ADJ
ejpam-5717	503	5	and	and	CCONJ
ejpam-5717	503	6	applied	applied	ADJ
ejpam-5717	503	7	mathematics	mathematic	NOUN
ejpam-5717	503	8	,	,	PUNCT
ejpam-5717	503	9	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-5717	503	10	,	,	PUNCT
ejpam-5717	503	11	2021	2021	NUM
ejpam-5717	503	12	.	.	PUNCT
ejpam-5717	504	1	[	[	X
ejpam-5717	504	2	30	30	NUM
ejpam-5717	504	3	]	]	X
ejpam-5717	504	4	c.	c.	PROPN
ejpam-5717	504	5	boonpok	boonpok	PROPN
ejpam-5717	504	6	,	,	PUNCT
ejpam-5717	504	7	c.	c.	PROPN
ejpam-5717	504	8	viriyapong	viriyapong	PROPN
ejpam-5717	504	9	,	,	PUNCT
ejpam-5717	504	10	and	and	CCONJ
ejpam-5717	504	11	m.	m.	NOUN
ejpam-5717	504	12	thongmoon	thongmoon	NOUN
ejpam-5717	504	13	.	.	PUNCT
ejpam-5717	505	1	on	on	ADP
ejpam-5717	505	2	upper	upper	ADJ
ejpam-5717	505	3	and	and	CCONJ
ejpam-5717	505	4	lower	low	ADJ
ejpam-5717	505	5	(	(	PUNCT
ejpam-5717	505	6	τ1	τ1	NOUN
ejpam-5717	505	7	,	,	PUNCT
ejpam-5717	505	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-5717	505	9	multifunctions	multifunction	NOUN
ejpam-5717	505	10	.	.	PUNCT
ejpam-5717	506	1	journal	journal	PROPN
ejpam-5717	506	2	of	of	ADP
ejpam-5717	506	3	mathematics	mathematics	PROPN
ejpam-5717	506	4	and	and	CCONJ
ejpam-5717	506	5	computer	computer	NOUN
ejpam-5717	506	6	science	science	NOUN
ejpam-5717	506	7	,	,	PUNCT
ejpam-5717	506	8	18:282–293	18:282–293	NUM
ejpam-5717	506	9	,	,	PUNCT
ejpam-5717	506	10	2018	2018	NUM
ejpam-5717	506	11	.	.	PUNCT
ejpam-5717	507	1	[	[	X
ejpam-5717	507	2	31	31	NUM
ejpam-5717	507	3	]	]	PUNCT
ejpam-5717	507	4	m.	m.	NOUN
ejpam-5717	507	5	chiangpradit	chiangpradit	NOUN
ejpam-5717	507	6	,	,	PUNCT
ejpam-5717	507	7	a.	a.	PROPN
ejpam-5717	507	8	sama	sama	PROPN
ejpam-5717	507	9	-	-	PUNCT
ejpam-5717	507	10	ae	ae	PROPN
ejpam-5717	507	11	,	,	PUNCT
ejpam-5717	507	12	and	and	CCONJ
ejpam-5717	507	13	c.	c.	PROPN
ejpam-5717	507	14	boonpok	boonpok	PROPN
ejpam-5717	507	15	.	.	PUNCT
ejpam-5717	508	1	almost	almost	ADV
ejpam-5717	508	2	nearly	nearly	ADV
ejpam-5717	508	3	quasi	quasi	NOUN
ejpam-5717	508	4	(	(	PUNCT
ejpam-5717	508	5	τ1	τ1	NOUN
ejpam-5717	508	6	,	,	PUNCT
ejpam-5717	508	7	τ2)continuous	τ2)continuous	ADJ
ejpam-5717	508	8	multifunctions	multifunction	NOUN
ejpam-5717	508	9	.	.	PUNCT
ejpam-5717	509	1	(	(	PUNCT
ejpam-5717	509	2	accepted	accept	VERB
ejpam-5717	509	3	)	)	PUNCT
ejpam-5717	509	4	.	.	PUNCT
ejpam-5717	510	1	p.	p.	NOUN
ejpam-5717	510	2	pue	pue	NOUN
ejpam-5717	510	3	-	-	PUNCT
ejpam-5717	510	4	on	on	ADP
ejpam-5717	510	5	,	,	PUNCT
ejpam-5717	510	6	a.	a.	PROPN
ejpam-5717	510	7	sama	sama	PROPN
ejpam-5717	510	8	-	-	PUNCT
ejpam-5717	510	9	ae	ae	PROPN
ejpam-5717	510	10	,	,	PUNCT
ejpam-5717	510	11	c.	c.	PROPN
ejpam-5717	510	12	boonpok	boonpok	PROPN
ejpam-5717	510	13	/	/	SYM
ejpam-5717	510	14	eur	eur	PROPN
ejpam-5717	510	15	.	.	PUNCT
ejpam-5717	511	1	j.	j.	PROPN
ejpam-5717	511	2	pure	pure	PROPN
ejpam-5717	511	3	appl	appl	PROPN
ejpam-5717	511	4	.	.	PROPN
ejpam-5717	511	5	math	math	PROPN
ejpam-5717	511	6	,	,	PUNCT
ejpam-5717	511	7	18	18	NUM
ejpam-5717	511	8	(	(	PUNCT
ejpam-5717	511	9	1	1	NUM
ejpam-5717	511	10	)	)	PUNCT
ejpam-5717	511	11	(	(	PUNCT
ejpam-5717	511	12	2025	2025	NUM
ejpam-5717	511	13	)	)	PUNCT
ejpam-5717	511	14	,	,	PUNCT
ejpam-5717	511	15	5717	5717	NUM
ejpam-5717	511	16	14	14	NUM
ejpam-5717	511	17	of	of	ADP
ejpam-5717	511	18	16	16	NUM
ejpam-5717	511	19	[	[	X
ejpam-5717	511	20	32	32	NUM
ejpam-5717	511	21	]	]	PUNCT
ejpam-5717	511	22	m.	m.	NOUN
ejpam-5717	511	23	chiangpradit	chiangpradit	NOUN
ejpam-5717	511	24	,	,	PUNCT
ejpam-5717	511	25	s.	s.	PROPN
ejpam-5717	511	26	sompong	sompong	PROPN
ejpam-5717	511	27	,	,	PUNCT
ejpam-5717	511	28	and	and	CCONJ
ejpam-5717	511	29	c.	c.	PROPN
ejpam-5717	511	30	boonpok	boonpok	PROPN
ejpam-5717	511	31	.	.	PUNCT
ejpam-5717	512	1	weakly	weakly	ADJ
ejpam-5717	512	2	quasi	quasi	NOUN
ejpam-5717	512	3	(	(	PUNCT
ejpam-5717	512	4	τ1	τ1	PROPN
ejpam-5717	512	5	,	,	PUNCT
ejpam-5717	512	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	512	7	functions	function	NOUN
ejpam-5717	512	8	.	.	PUNCT
ejpam-5717	513	1	international	international	ADJ
ejpam-5717	513	2	journal	journal	NOUN
ejpam-5717	513	3	of	of	ADP
ejpam-5717	513	4	analysis	analysis	NOUN
ejpam-5717	513	5	and	and	CCONJ
ejpam-5717	513	6	applications	application	NOUN
ejpam-5717	513	7	,	,	PUNCT
ejpam-5717	513	8	22:125	22:125	NUM
ejpam-5717	513	9	,	,	PUNCT
ejpam-5717	513	10	2024	2024	NUM
ejpam-5717	513	11	.	.	PUNCT
ejpam-5717	514	1	[	[	X
ejpam-5717	514	2	33	33	NUM
ejpam-5717	514	3	]	]	PUNCT
ejpam-5717	514	4	t.	t.	PROPN
ejpam-5717	514	5	duangphui	duangphui	PROPN
ejpam-5717	514	6	,	,	PUNCT
ejpam-5717	514	7	c.	c.	PROPN
ejpam-5717	514	8	boonpok	boonpok	PROPN
ejpam-5717	514	9	,	,	PUNCT
ejpam-5717	514	10	and	and	CCONJ
ejpam-5717	514	11	c.	c.	PROPN
ejpam-5717	514	12	viriyapong	viriyapong	PROPN
ejpam-5717	514	13	.	.	PUNCT
ejpam-5717	515	1	continuous	continuous	ADJ
ejpam-5717	515	2	functions	function	NOUN
ejpam-5717	515	3	on	on	ADP
ejpam-5717	515	4	bigeneralized	bigeneralize	VERB
ejpam-5717	515	5	topological	topological	ADJ
ejpam-5717	515	6	spaces	space	NOUN
ejpam-5717	515	7	.	.	PUNCT
ejpam-5717	516	1	international	international	ADJ
ejpam-5717	516	2	journal	journal	PROPN
ejpam-5717	516	3	of	of	ADP
ejpam-5717	516	4	mathematical	mathematical	ADJ
ejpam-5717	516	5	analysis	analysis	NOUN
ejpam-5717	516	6	,	,	PUNCT
ejpam-5717	516	7	5(24):1165	5(24):1165	NUM
ejpam-5717	516	8	–	–	PUNCT
ejpam-5717	516	9	1174	1174	NUM
ejpam-5717	516	10	,	,	PUNCT
ejpam-5717	516	11	2011	2011	NUM
ejpam-5717	516	12	.	.	PUNCT
ejpam-5717	517	1	[	[	X
ejpam-5717	517	2	34	34	NUM
ejpam-5717	517	3	]	]	PUNCT
ejpam-5717	517	4	t.	t.	NOUN
ejpam-5717	517	5	dungthaisong	dungthaisong	PROPN
ejpam-5717	517	6	,	,	PUNCT
ejpam-5717	517	7	c.	c.	PROPN
ejpam-5717	517	8	boonpok	boonpok	PROPN
ejpam-5717	517	9	,	,	PUNCT
ejpam-5717	517	10	and	and	CCONJ
ejpam-5717	517	11	c.	c.	PROPN
ejpam-5717	517	12	viriyapong	viriyapong	PROPN
ejpam-5717	517	13	.	.	PUNCT
ejpam-5717	518	1	generalized	generalize	VERB
ejpam-5717	518	2	closed	close	VERB
ejpam-5717	518	3	sets	set	NOUN
ejpam-5717	518	4	in	in	ADP
ejpam-5717	518	5	bigeneralized	bigeneralize	VERB
ejpam-5717	518	6	topological	topological	ADJ
ejpam-5717	518	7	spaces	space	NOUN
ejpam-5717	518	8	.	.	PUNCT
ejpam-5717	519	1	international	international	ADJ
ejpam-5717	519	2	journal	journal	PROPN
ejpam-5717	519	3	of	of	ADP
ejpam-5717	519	4	mathematical	mathematical	ADJ
ejpam-5717	519	5	analysis	analysis	NOUN
ejpam-5717	519	6	,	,	PUNCT
ejpam-5717	519	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-5717	519	8	,	,	PUNCT
ejpam-5717	519	9	2011	2011	NUM
ejpam-5717	519	10	.	.	PUNCT
ejpam-5717	520	1	[	[	X
ejpam-5717	520	2	35	35	NUM
ejpam-5717	520	3	]	]	PUNCT
ejpam-5717	520	4	m.	m.	NOUN
ejpam-5717	520	5	e.	e.	PROPN
ejpam-5717	520	6	abd	abd	PROPN
ejpam-5717	521	1	el	el	PROPN
ejpam-5717	521	2	-	-	PROPN
ejpam-5717	521	3	monsef	monsef	PROPN
ejpam-5717	521	4	,	,	PUNCT
ejpam-5717	521	5	s.	s.	PROPN
ejpam-5717	521	6	n.	n.	PROPN
ejpam-5717	521	7	el	el	PROPN
ejpam-5717	521	8	-	-	PROPN
ejpam-5717	521	9	deeb	deeb	PROPN
ejpam-5717	521	10	,	,	PUNCT
ejpam-5717	521	11	and	and	CCONJ
ejpam-5717	521	12	r.	r.	PROPN
ejpam-5717	521	13	a.	a.	PROPN
ejpam-5717	521	14	mahmoud	mahmoud	PROPN
ejpam-5717	521	15	.	.	PUNCT
ejpam-5717	522	1	β	β	X
ejpam-5717	522	2	-	-	ADJ
ejpam-5717	522	3	open	open	ADJ
ejpam-5717	522	4	sets	set	NOUN
ejpam-5717	522	5	and	and	CCONJ
ejpam-5717	522	6	βcontinuous	βcontinuous	ADJ
ejpam-5717	522	7	mappings	mapping	NOUN
ejpam-5717	522	8	.	.	PUNCT
ejpam-5717	523	1	bulletin	bulletin	NOUN
ejpam-5717	523	2	of	of	ADP
ejpam-5717	523	3	the	the	DET
ejpam-5717	523	4	faculty	faculty	NOUN
ejpam-5717	523	5	of	of	ADP
ejpam-5717	523	6	science	science	NOUN
ejpam-5717	523	7	,	,	PUNCT
ejpam-5717	523	8	assiut	assiut	NOUN
ejpam-5717	523	9	university	university	NOUN
ejpam-5717	523	10	,	,	PUNCT
ejpam-5717	523	11	12:77–90	12:77–90	NUM
ejpam-5717	523	12	,	,	PUNCT
ejpam-5717	523	13	1983	1983	NUM
ejpam-5717	523	14	.	.	PUNCT
ejpam-5717	524	1	[	[	X
ejpam-5717	524	2	36	36	NUM
ejpam-5717	524	3	]	]	X
ejpam-5717	524	4	s.	s.	PROPN
ejpam-5717	524	5	jafari	jafari	PROPN
ejpam-5717	524	6	and	and	CCONJ
ejpam-5717	524	7	t.	t.	PROPN
ejpam-5717	524	8	noiri	noiri	PROPN
ejpam-5717	524	9	.	.	PUNCT
ejpam-5717	525	1	properties	property	NOUN
ejpam-5717	525	2	of	of	ADP
ejpam-5717	525	3	θ	θ	ADJ
ejpam-5717	525	4	-	-	ADJ
ejpam-5717	525	5	continuous	continuous	ADJ
ejpam-5717	525	6	functions	function	NOUN
ejpam-5717	525	7	.	.	PUNCT
ejpam-5717	526	1	journal	journal	PROPN
ejpam-5717	526	2	of	of	ADP
ejpam-5717	526	3	institute	institute	PROPN
ejpam-5717	526	4	of	of	ADP
ejpam-5717	526	5	mathematics	mathematics	PROPN
ejpam-5717	526	6	and	and	CCONJ
ejpam-5717	526	7	computer	computer	NOUN
ejpam-5717	526	8	science	science	NOUN
ejpam-5717	526	9	,	,	PUNCT
ejpam-5717	526	10	mathematics	mathematics	NOUN
ejpam-5717	526	11	series	series	NOUN
ejpam-5717	526	12	,	,	PUNCT
ejpam-5717	526	13	13:123–128	13:123–128	NUM
ejpam-5717	526	14	,	,	PUNCT
ejpam-5717	526	15	2000	2000	NUM
ejpam-5717	526	16	.	.	PUNCT
ejpam-5717	527	1	[	[	X
ejpam-5717	527	2	37	37	NUM
ejpam-5717	527	3	]	]	X
ejpam-5717	527	4	j.	j.	PROPN
ejpam-5717	527	5	khampakdee	khampakdee	PROPN
ejpam-5717	527	6	and	and	CCONJ
ejpam-5717	527	7	c.	c.	PROPN
ejpam-5717	527	8	boonpok	boonpok	PROPN
ejpam-5717	527	9	.	.	PUNCT
ejpam-5717	528	1	upper	upper	ADJ
ejpam-5717	528	2	and	and	CCONJ
ejpam-5717	528	3	lower	low	ADJ
ejpam-5717	528	4	α(λ	α(λ	PROPN
ejpam-5717	528	5	,	,	PUNCT
ejpam-5717	528	6	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	528	7	multifunctions	multifunction	NOUN
ejpam-5717	528	8	.	.	PUNCT
ejpam-5717	529	1	wseas	wseas	VERB
ejpam-5717	529	2	transactions	transaction	NOUN
ejpam-5717	529	3	on	on	ADP
ejpam-5717	529	4	mathematics	mathematic	NOUN
ejpam-5717	529	5	,	,	PUNCT
ejpam-5717	529	6	21:684–690	21:684–690	NUM
ejpam-5717	529	7	,	,	PUNCT
ejpam-5717	529	8	2022	2022	NUM
ejpam-5717	529	9	.	.	PUNCT
ejpam-5717	530	1	[	[	X
ejpam-5717	530	2	38	38	NUM
ejpam-5717	530	3	]	]	PUNCT
ejpam-5717	530	4	j.	j.	PROPN
ejpam-5717	530	5	khampakdee	khampakdee	PROPN
ejpam-5717	530	6	,	,	PUNCT
ejpam-5717	530	7	s.	s.	PROPN
ejpam-5717	530	8	sompong	sompong	PROPN
ejpam-5717	530	9	,	,	PUNCT
ejpam-5717	530	10	and	and	CCONJ
ejpam-5717	530	11	c.	c.	PROPN
ejpam-5717	530	12	boonpok	boonpok	PROPN
ejpam-5717	530	13	.	.	PUNCT
ejpam-5717	531	1	c-(τ1	c-(τ1	PROPN
ejpam-5717	531	2	,	,	PUNCT
ejpam-5717	531	3	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5717	531	4	for	for	ADP
ejpam-5717	531	5	multifunctions	multifunction	NOUN
ejpam-5717	531	6	.	.	PUNCT
ejpam-5717	532	1	european	european	ADJ
ejpam-5717	532	2	journal	journal	PROPN
ejpam-5717	532	3	of	of	ADP
ejpam-5717	532	4	pure	pure	ADJ
ejpam-5717	532	5	and	and	CCONJ
ejpam-5717	532	6	applied	applied	ADJ
ejpam-5717	532	7	mathematics	mathematic	NOUN
ejpam-5717	532	8	,	,	PUNCT
ejpam-5717	532	9	17(3):2289–2299	17(3):2289–2299	NUM
ejpam-5717	532	10	,	,	PUNCT
ejpam-5717	532	11	2024	2024	NUM
ejpam-5717	532	12	.	.	PUNCT
ejpam-5717	533	1	[	[	X
ejpam-5717	533	2	39	39	NUM
ejpam-5717	533	3	]	]	PUNCT
ejpam-5717	533	4	c.	c.	PROPN
ejpam-5717	533	5	klanarong	klanarong	PROPN
ejpam-5717	533	6	,	,	PUNCT
ejpam-5717	533	7	s.	s.	PROPN
ejpam-5717	533	8	sompong	sompong	PROPN
ejpam-5717	533	9	,	,	PUNCT
ejpam-5717	533	10	and	and	CCONJ
ejpam-5717	533	11	c.	c.	PROPN
ejpam-5717	533	12	boonpok	boonpok	PROPN
ejpam-5717	533	13	.	.	PUNCT
ejpam-5717	534	1	upper	upper	ADJ
ejpam-5717	534	2	and	and	CCONJ
ejpam-5717	534	3	lower	low	ADJ
ejpam-5717	534	4	almost	almost	ADV
ejpam-5717	534	5	(	(	PUNCT
ejpam-5717	534	6	τ1	τ1	NOUN
ejpam-5717	534	7	,	,	PUNCT
ejpam-5717	534	8	τ2)continuous	τ2)continuous	ADJ
ejpam-5717	534	9	multifunctions	multifunction	NOUN
ejpam-5717	534	10	.	.	PUNCT
ejpam-5717	535	1	european	european	ADJ
ejpam-5717	535	2	journal	journal	PROPN
ejpam-5717	535	3	of	of	ADP
ejpam-5717	535	4	pure	pure	ADJ
ejpam-5717	535	5	and	and	CCONJ
ejpam-5717	535	6	applied	applied	ADJ
ejpam-5717	535	7	mathematics	mathematic	NOUN
ejpam-5717	535	8	,	,	PUNCT
ejpam-5717	535	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-5717	535	10	,	,	PUNCT
ejpam-5717	535	11	2024	2024	NUM
ejpam-5717	535	12	.	.	PUNCT
ejpam-5717	536	1	[	[	X
ejpam-5717	536	2	40	40	NUM
ejpam-5717	536	3	]	]	PUNCT
ejpam-5717	536	4	b.	b.	PROPN
ejpam-5717	536	5	kong	kong	PROPN
ejpam-5717	536	6	-	-	PUNCT
ejpam-5717	536	7	ied	ied	PROPN
ejpam-5717	536	8	,	,	PUNCT
ejpam-5717	536	9	s.	s.	PROPN
ejpam-5717	536	10	sompong	sompong	PROPN
ejpam-5717	536	11	,	,	PUNCT
ejpam-5717	536	12	and	and	CCONJ
ejpam-5717	536	13	c.	c.	PROPN
ejpam-5717	536	14	boonpok	boonpok	PROPN
ejpam-5717	536	15	.	.	PUNCT
ejpam-5717	537	1	almost	almost	ADV
ejpam-5717	537	2	quasi	quasi	X
ejpam-5717	537	3	(	(	PUNCT
ejpam-5717	537	4	τ1	τ1	NOUN
ejpam-5717	537	5	,	,	PUNCT
ejpam-5717	537	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	537	7	functions	function	NOUN
ejpam-5717	537	8	.	.	PUNCT
ejpam-5717	538	1	asia	asia	PROPN
ejpam-5717	538	2	pacific	pacific	PROPN
ejpam-5717	538	3	journal	journal	PROPN
ejpam-5717	538	4	of	of	ADP
ejpam-5717	538	5	mathematics	mathematic	NOUN
ejpam-5717	538	6	,	,	PUNCT
ejpam-5717	538	7	11:64	11:64	NUM
ejpam-5717	538	8	,	,	PUNCT
ejpam-5717	538	9	2024	2024	NUM
ejpam-5717	538	10	.	.	PUNCT
ejpam-5717	539	1	[	[	X
ejpam-5717	539	2	41	41	NUM
ejpam-5717	539	3	]	]	X
ejpam-5717	539	4	n.	n.	PROPN
ejpam-5717	539	5	levine	levine	PROPN
ejpam-5717	539	6	.	.	PUNCT
ejpam-5717	540	1	a	a	DET
ejpam-5717	540	2	decomposition	decomposition	NOUN
ejpam-5717	540	3	of	of	ADP
ejpam-5717	540	4	continuity	continuity	NOUN
ejpam-5717	540	5	in	in	ADP
ejpam-5717	540	6	topological	topological	ADJ
ejpam-5717	540	7	spaces	space	NOUN
ejpam-5717	540	8	.	.	PUNCT
ejpam-5717	541	1	the	the	DET
ejpam-5717	541	2	american	american	PROPN
ejpam-5717	541	3	mathematical	mathematical	PROPN
ejpam-5717	541	4	monthly	monthly	ADV
ejpam-5717	541	5	,	,	PUNCT
ejpam-5717	541	6	60:44–46	60:44–46	NUM
ejpam-5717	541	7	,	,	PUNCT
ejpam-5717	541	8	1961	1961	NUM
ejpam-5717	541	9	.	.	PUNCT
ejpam-5717	542	1	[	[	X
ejpam-5717	542	2	42	42	NUM
ejpam-5717	542	3	]	]	X
ejpam-5717	542	4	n.	n.	PROPN
ejpam-5717	542	5	levine	levine	PROPN
ejpam-5717	542	6	.	.	PUNCT
ejpam-5717	543	1	semi	semi	ADJ
ejpam-5717	543	2	-	-	ADJ
ejpam-5717	543	3	open	open	ADJ
ejpam-5717	543	4	sets	set	NOUN
ejpam-5717	543	5	and	and	CCONJ
ejpam-5717	543	6	semi	semi	ADJ
ejpam-5717	543	7	-	-	NOUN
ejpam-5717	543	8	continuity	continuity	NOUN
ejpam-5717	543	9	in	in	ADP
ejpam-5717	543	10	topological	topological	ADJ
ejpam-5717	543	11	spaces	space	NOUN
ejpam-5717	543	12	.	.	PUNCT
ejpam-5717	544	1	the	the	DET
ejpam-5717	544	2	american	american	PROPN
ejpam-5717	544	3	mathematical	mathematical	PROPN
ejpam-5717	544	4	monthly	monthly	ADV
ejpam-5717	544	5	,	,	PUNCT
ejpam-5717	544	6	70:36–41	70:36–41	NUM
ejpam-5717	544	7	,	,	PUNCT
ejpam-5717	544	8	1963	1963	NUM
ejpam-5717	544	9	.	.	PUNCT
ejpam-5717	545	1	[	[	X
ejpam-5717	545	2	43	43	NUM
ejpam-5717	545	3	]	]	PUNCT
ejpam-5717	545	4	s.	s.	PROPN
ejpam-5717	545	5	marcus	marcus	PROPN
ejpam-5717	545	6	.	.	PUNCT
ejpam-5717	546	1	sur	sur	PROPN
ejpam-5717	546	2	les	les	PROPN
ejpam-5717	546	3	fonctions	fonctions	PROPN
ejpam-5717	546	4	quasicontinues	quasicontinue	NOUN
ejpam-5717	546	5	au	au	PROPN
ejpam-5717	546	6	sens	sens	X
ejpam-5717	546	7	de	de	PROPN
ejpam-5717	546	8	s.	s.	PROPN
ejpam-5717	546	9	kempisty	kempisty	PROPN
ejpam-5717	546	10	.	.	PUNCT
ejpam-5717	547	1	colloquium	colloquium	NOUN
ejpam-5717	547	2	mathematicum	mathematicum	PROPN
ejpam-5717	547	3	,	,	PUNCT
ejpam-5717	547	4	8:47–53	8:47–53	NUM
ejpam-5717	547	5	,	,	PUNCT
ejpam-5717	547	6	1961	1961	NUM
ejpam-5717	547	7	.	.	PUNCT
ejpam-5717	548	1	[	[	X
ejpam-5717	548	2	44	44	NUM
ejpam-5717	548	3	]	]	PUNCT
ejpam-5717	548	4	a.	a.	PROPN
ejpam-5717	548	5	s.	s.	PROPN
ejpam-5717	548	6	mashhour	mashhour	PROPN
ejpam-5717	548	7	,	,	PUNCT
ejpam-5717	548	8	m.	m.	PROPN
ejpam-5717	548	9	e.	e.	PROPN
ejpam-5717	548	10	abd	abd	PROPN
ejpam-5717	549	1	el	el	PROPN
ejpam-5717	549	2	-	-	PROPN
ejpam-5717	549	3	monsef	monsef	ADJ
ejpam-5717	549	4	,	,	PUNCT
ejpam-5717	549	5	and	and	CCONJ
ejpam-5717	549	6	s.	s.	PROPN
ejpam-5717	549	7	n.	n.	PROPN
ejpam-5717	549	8	el	el	PROPN
ejpam-5717	549	9	-	-	PROPN
ejpam-5717	549	10	deeb	deeb	PROPN
ejpam-5717	549	11	.	.	PUNCT
ejpam-5717	550	1	on	on	ADP
ejpam-5717	550	2	precontinuous	precontinuous	ADJ
ejpam-5717	550	3	and	and	CCONJ
ejpam-5717	550	4	weak	weak	ADJ
ejpam-5717	550	5	precontinuous	precontinuous	ADJ
ejpam-5717	550	6	mappings	mapping	NOUN
ejpam-5717	550	7	.	.	PUNCT
ejpam-5717	551	1	proceedings	proceeding	NOUN
ejpam-5717	551	2	of	of	ADP
ejpam-5717	551	3	the	the	DET
ejpam-5717	551	4	mathematical	mathematical	ADJ
ejpam-5717	551	5	and	and	CCONJ
ejpam-5717	551	6	physical	physical	ADJ
ejpam-5717	551	7	society	society	NOUN
ejpam-5717	551	8	of	of	ADP
ejpam-5717	551	9	egypt	egypt	PROPN
ejpam-5717	551	10	,	,	PUNCT
ejpam-5717	551	11	53:47–53	53:47–53	NUM
ejpam-5717	551	12	,	,	PUNCT
ejpam-5717	551	13	1982	1982	NUM
ejpam-5717	551	14	.	.	PUNCT
ejpam-5717	552	1	[	[	X
ejpam-5717	552	2	45	45	NUM
ejpam-5717	552	3	]	]	PUNCT
ejpam-5717	552	4	a.	a.	NOUN
ejpam-5717	552	5	neubrunnová.	neubrunnová.	PROPN
ejpam-5717	552	6	on	on	ADP
ejpam-5717	552	7	certain	certain	ADJ
ejpam-5717	552	8	generalizations	generalization	NOUN
ejpam-5717	552	9	of	of	ADP
ejpam-5717	552	10	the	the	DET
ejpam-5717	552	11	notion	notion	NOUN
ejpam-5717	552	12	of	of	ADP
ejpam-5717	552	13	continuity	continuity	NOUN
ejpam-5717	552	14	.	.	PUNCT
ejpam-5717	553	1	matematický	matematický	ADJ
ejpam-5717	553	2	časopis	časopis	PROPN
ejpam-5717	553	3	,	,	PUNCT
ejpam-5717	553	4	23:374–380	23:374–380	NUM
ejpam-5717	553	5	,	,	PUNCT
ejpam-5717	553	6	1973	1973	NUM
ejpam-5717	553	7	.	.	PUNCT
ejpam-5717	554	1	[	[	X
ejpam-5717	554	2	46	46	NUM
ejpam-5717	554	3	]	]	X
ejpam-5717	554	4	o.	o.	NOUN
ejpam-5717	554	5	nj̊astad	nj̊astad	NOUN
ejpam-5717	554	6	.	.	PUNCT
ejpam-5717	555	1	on	on	ADP
ejpam-5717	555	2	some	some	DET
ejpam-5717	555	3	classes	class	NOUN
ejpam-5717	555	4	of	of	ADP
ejpam-5717	555	5	nearly	nearly	ADV
ejpam-5717	555	6	open	open	ADJ
ejpam-5717	555	7	sets	set	NOUN
ejpam-5717	555	8	.	.	PUNCT
ejpam-5717	556	1	pacific	pacific	PROPN
ejpam-5717	556	2	journal	journal	PROPN
ejpam-5717	556	3	of	of	ADP
ejpam-5717	556	4	mathematics	mathematic	NOUN
ejpam-5717	556	5	,	,	PUNCT
ejpam-5717	556	6	15:961–970	15:961–970	PROPN
ejpam-5717	556	7	,	,	PUNCT
ejpam-5717	556	8	1965	1965	NUM
ejpam-5717	556	9	.	.	PUNCT
ejpam-5717	557	1	[	[	X
ejpam-5717	557	2	47	47	NUM
ejpam-5717	557	3	]	]	PUNCT
ejpam-5717	557	4	t.	t.	PROPN
ejpam-5717	557	5	noiri	noiri	PROPN
ejpam-5717	557	6	.	.	PUNCT
ejpam-5717	558	1	on	on	ADP
ejpam-5717	558	2	θ	θ	ADJ
ejpam-5717	558	3	-	-	ADJ
ejpam-5717	558	4	continuous	continuous	ADJ
ejpam-5717	558	5	functions	function	NOUN
ejpam-5717	558	6	.	.	PUNCT
ejpam-5717	559	1	indian	indian	ADJ
ejpam-5717	559	2	journal	journal	PROPN
ejpam-5717	559	3	of	of	ADP
ejpam-5717	559	4	pure	pure	ADJ
ejpam-5717	559	5	and	and	CCONJ
ejpam-5717	559	6	applied	applied	ADJ
ejpam-5717	559	7	mathematics	mathematic	NOUN
ejpam-5717	559	8	,	,	PUNCT
ejpam-5717	559	9	21:410–415	21:410–415	NUM
ejpam-5717	559	10	,	,	PUNCT
ejpam-5717	559	11	1990	1990	NUM
ejpam-5717	559	12	.	.	PUNCT
ejpam-5717	560	1	[	[	X
ejpam-5717	560	2	48	48	NUM
ejpam-5717	560	3	]	]	PUNCT
ejpam-5717	560	4	t.	t.	PROPN
ejpam-5717	560	5	noiri	noiri	PROPN
ejpam-5717	560	6	and	and	CCONJ
ejpam-5717	560	7	v.	v.	ADP
ejpam-5717	560	8	popa	popa	NOUN
ejpam-5717	560	9	.	.	PUNCT
ejpam-5717	561	1	weakly	weakly	ADJ
ejpam-5717	561	2	quasi	quasi	ADJ
ejpam-5717	561	3	continuous	continuous	ADJ
ejpam-5717	561	4	multifunctions	multifunction	NOUN
ejpam-5717	561	5	.	.	PUNCT
ejpam-5717	562	1	analele	analele	ADP
ejpam-5717	562	2	universită	universită	PROPN
ejpam-5717	562	3	ţii	ţii	PROPN
ejpam-5717	562	4	din	din	PROPN
ejpam-5717	562	5	timişoara	timişoara	NOUN
ejpam-5717	562	6	,	,	PUNCT
ejpam-5717	562	7	seria	seria	PROPN
ejpam-5717	562	8	ştiinţe	ştiinţe	PROPN
ejpam-5717	562	9	matematice	matematice	NOUN
ejpam-5717	562	10	,	,	PUNCT
ejpam-5717	562	11	26:33–38	26:33–38	NUM
ejpam-5717	562	12	,	,	PUNCT
ejpam-5717	562	13	1988	1988	NUM
ejpam-5717	562	14	.	.	PUNCT
ejpam-5717	563	1	[	[	X
ejpam-5717	563	2	49	49	NUM
ejpam-5717	563	3	]	]	PUNCT
ejpam-5717	563	4	t.	t.	PROPN
ejpam-5717	563	5	noiri	noiri	PROPN
ejpam-5717	563	6	and	and	CCONJ
ejpam-5717	563	7	v.	v.	ADP
ejpam-5717	563	8	popa	popa	NOUN
ejpam-5717	563	9	.	.	PUNCT
ejpam-5717	564	1	some	some	DET
ejpam-5717	564	2	properties	property	NOUN
ejpam-5717	564	3	of	of	ADP
ejpam-5717	564	4	upper	upper	ADJ
ejpam-5717	564	5	and	and	CCONJ
ejpam-5717	564	6	lower	low	ADJ
ejpam-5717	564	7	θ	θ	ADJ
ejpam-5717	564	8	-	-	ADJ
ejpam-5717	564	9	quasicontinuous	quasicontinuous	ADJ
ejpam-5717	564	10	multifunctions	multifunction	NOUN
ejpam-5717	564	11	.	.	PUNCT
ejpam-5717	565	1	demonstratio	demonstratio	PROPN
ejpam-5717	565	2	mathematica	mathematica	PROPN
ejpam-5717	565	3	,	,	PUNCT
ejpam-5717	565	4	38(1):223–234	38(1):223–234	PROPN
ejpam-5717	565	5	,	,	PUNCT
ejpam-5717	565	6	2005	2005	NUM
ejpam-5717	565	7	.	.	PUNCT
ejpam-5717	566	1	[	[	X
ejpam-5717	566	2	50	50	NUM
ejpam-5717	566	3	]	]	PUNCT
ejpam-5717	566	4	v.	v.	CCONJ
ejpam-5717	566	5	popa	popa	NOUN
ejpam-5717	566	6	.	.	PUNCT
ejpam-5717	567	1	on	on	ADP
ejpam-5717	567	2	a	a	DET
ejpam-5717	567	3	decomposition	decomposition	NOUN
ejpam-5717	567	4	of	of	ADP
ejpam-5717	567	5	quasicontinuity	quasicontinuity	NOUN
ejpam-5717	567	6	for	for	ADP
ejpam-5717	567	7	multifunctions	multifunction	NOUN
ejpam-5717	567	8	.	.	PUNCT
ejpam-5717	568	1	studii	studii	PROPN
ejpam-5717	568	2	şi	şi	PROPN
ejpam-5717	568	3	cercetǎri	cercetǎri	NOUN
ejpam-5717	568	4	matematicǎ	matematicǎ	VERB
ejpam-5717	568	5	,	,	PUNCT
ejpam-5717	568	6	27:323–328	27:323–328	PROPN
ejpam-5717	568	7	,	,	PUNCT
ejpam-5717	568	8	1975	1975	NUM
ejpam-5717	568	9	.	.	PUNCT
ejpam-5717	569	1	p.	p.	NOUN
ejpam-5717	569	2	pue	pue	NOUN
ejpam-5717	569	3	-	-	PUNCT
ejpam-5717	569	4	on	on	ADP
ejpam-5717	569	5	,	,	PUNCT
ejpam-5717	569	6	a.	a.	PROPN
ejpam-5717	569	7	sama	sama	PROPN
ejpam-5717	569	8	-	-	PUNCT
ejpam-5717	569	9	ae	ae	PROPN
ejpam-5717	569	10	,	,	PUNCT
ejpam-5717	569	11	c.	c.	PROPN
ejpam-5717	569	12	boonpok	boonpok	PROPN
ejpam-5717	569	13	/	/	SYM
ejpam-5717	569	14	eur	eur	PROPN
ejpam-5717	569	15	.	.	PUNCT
ejpam-5717	570	1	j.	j.	PROPN
ejpam-5717	570	2	pure	pure	PROPN
ejpam-5717	570	3	appl	appl	PROPN
ejpam-5717	570	4	.	.	PROPN
ejpam-5717	570	5	math	math	PROPN
ejpam-5717	570	6	,	,	PUNCT
ejpam-5717	570	7	18	18	NUM
ejpam-5717	570	8	(	(	PUNCT
ejpam-5717	570	9	1	1	NUM
ejpam-5717	570	10	)	)	PUNCT
ejpam-5717	570	11	(	(	PUNCT
ejpam-5717	570	12	2025	2025	NUM
ejpam-5717	570	13	)	)	PUNCT
ejpam-5717	570	14	,	,	PUNCT
ejpam-5717	570	15	5717	5717	NUM
ejpam-5717	570	16	15	15	NUM
ejpam-5717	570	17	of	of	ADP
ejpam-5717	570	18	16	16	NUM
ejpam-5717	571	1	[	[	X
ejpam-5717	571	2	51	51	NUM
ejpam-5717	571	3	]	]	PUNCT
ejpam-5717	571	4	v.	v.	CCONJ
ejpam-5717	571	5	popa	popa	NOUN
ejpam-5717	571	6	.	.	PUNCT
ejpam-5717	572	1	on	on	ADP
ejpam-5717	572	2	the	the	DET
ejpam-5717	572	3	decompositions	decomposition	NOUN
ejpam-5717	572	4	of	of	ADP
ejpam-5717	572	5	the	the	DET
ejpam-5717	572	6	quasicontinuity	quasicontinuity	NOUN
ejpam-5717	572	7	in	in	ADP
ejpam-5717	572	8	topological	topological	ADJ
ejpam-5717	572	9	spaces	space	NOUN
ejpam-5717	572	10	.	.	PUNCT
ejpam-5717	573	1	studii	studii	PROPN
ejpam-5717	573	2	şi	şi	PROPN
ejpam-5717	573	3	cercetǎri	cercetǎri	NOUN
ejpam-5717	573	4	matematicǎ	matematicǎ	VERB
ejpam-5717	573	5	,	,	PUNCT
ejpam-5717	573	6	30:31–35	30:31–35	NUM
ejpam-5717	573	7	,	,	PUNCT
ejpam-5717	573	8	1978	1978	NUM
ejpam-5717	573	9	.	.	PUNCT
ejpam-5717	574	1	[	[	X
ejpam-5717	574	2	52	52	NUM
ejpam-5717	574	3	]	]	PUNCT
ejpam-5717	574	4	v.	v.	CCONJ
ejpam-5717	574	5	popa	popa	NOUN
ejpam-5717	574	6	and	and	CCONJ
ejpam-5717	574	7	t.	t.	PROPN
ejpam-5717	574	8	noiri	noiri	PROPN
ejpam-5717	574	9	.	.	PUNCT
ejpam-5717	575	1	on	on	ADP
ejpam-5717	575	2	θ	θ	ADJ
ejpam-5717	575	3	-	-	ADJ
ejpam-5717	575	4	quasicontinuous	quasicontinuous	ADJ
ejpam-5717	575	5	multifunctions	multifunction	NOUN
ejpam-5717	575	6	.	.	PUNCT
ejpam-5717	576	1	demonstratio	demonstratio	PROPN
ejpam-5717	576	2	mathematica	mathematica	PROPN
ejpam-5717	576	3	,	,	PUNCT
ejpam-5717	576	4	28:111–122	28:111–122	PROPN
ejpam-5717	576	5	,	,	PUNCT
ejpam-5717	576	6	1995	1995	NUM
ejpam-5717	576	7	.	.	PUNCT
ejpam-5717	577	1	[	[	X
ejpam-5717	577	2	53	53	NUM
ejpam-5717	577	3	]	]	PUNCT
ejpam-5717	577	4	v.	v.	CCONJ
ejpam-5717	577	5	popa	popa	NOUN
ejpam-5717	577	6	and	and	CCONJ
ejpam-5717	577	7	t.	t.	NOUN
ejpam-5717	577	8	noiri	noiri	PROPN
ejpam-5717	577	9	.	.	PUNCT
ejpam-5717	578	1	almost	almost	ADV
ejpam-5717	578	2	quasi	quasi	VERB
ejpam-5717	578	3	continuous	continuous	ADJ
ejpam-5717	578	4	multifunctions	multifunction	NOUN
ejpam-5717	578	5	.	.	PUNCT
ejpam-5717	579	1	tatra	tatra	PROPN
ejpam-5717	579	2	mountains	mountains	PROPN
ejpam-5717	579	3	mathematical	mathematical	ADJ
ejpam-5717	579	4	publications	publication	NOUN
ejpam-5717	579	5	,	,	PUNCT
ejpam-5717	579	6	14:81–90	14:81–90	PROPN
ejpam-5717	579	7	,	,	PUNCT
ejpam-5717	579	8	1998	1998	NUM
ejpam-5717	579	9	.	.	PUNCT
ejpam-5717	580	1	[	[	X
ejpam-5717	580	2	54	54	NUM
ejpam-5717	580	3	]	]	PUNCT
ejpam-5717	580	4	v.	v.	ADP
ejpam-5717	580	5	popa	popa	NOUN
ejpam-5717	580	6	and	and	CCONJ
ejpam-5717	580	7	c.	c.	PROPN
ejpam-5717	580	8	stan	stan	PROPN
ejpam-5717	580	9	.	.	PUNCT
ejpam-5717	581	1	on	on	ADP
ejpam-5717	581	2	a	a	DET
ejpam-5717	581	3	decomposition	decomposition	NOUN
ejpam-5717	581	4	of	of	ADP
ejpam-5717	581	5	quasicontinuity	quasicontinuity	NOUN
ejpam-5717	581	6	in	in	ADP
ejpam-5717	581	7	topological	topological	ADJ
ejpam-5717	581	8	spaces	space	NOUN
ejpam-5717	581	9	.	.	PUNCT
ejpam-5717	582	1	studii	studii	PROPN
ejpam-5717	582	2	şi	şi	PROPN
ejpam-5717	582	3	cercetǎri	cercetǎri	NOUN
ejpam-5717	582	4	matematicǎ	matematicǎ	VERB
ejpam-5717	582	5	,	,	PUNCT
ejpam-5717	582	6	25:41–43	25:41–43	NUM
ejpam-5717	582	7	,	,	PUNCT
ejpam-5717	582	8	1973	1973	NUM
ejpam-5717	582	9	.	.	PUNCT
ejpam-5717	583	1	[	[	X
ejpam-5717	583	2	55	55	NUM
ejpam-5717	583	3	]	]	X
ejpam-5717	583	4	p.	p.	NOUN
ejpam-5717	583	5	pue	pue	NOUN
ejpam-5717	583	6	-	-	PUNCT
ejpam-5717	583	7	on	on	ADP
ejpam-5717	583	8	and	and	CCONJ
ejpam-5717	583	9	c.	c.	PROPN
ejpam-5717	583	10	boonpok	boonpok	PROPN
ejpam-5717	583	11	.	.	PUNCT
ejpam-5717	584	1	θ(λ	θ(λ	PROPN
ejpam-5717	584	2	,	,	PUNCT
ejpam-5717	584	3	p)-continuity	p)-continuity	NOUN
ejpam-5717	584	4	for	for	ADP
ejpam-5717	584	5	functions	function	NOUN
ejpam-5717	584	6	.	.	PUNCT
ejpam-5717	585	1	international	international	ADJ
ejpam-5717	585	2	journal	journal	NOUN
ejpam-5717	585	3	of	of	ADP
ejpam-5717	585	4	mathematics	mathematic	NOUN
ejpam-5717	585	5	and	and	CCONJ
ejpam-5717	585	6	computer	computer	NOUN
ejpam-5717	585	7	science	science	NOUN
ejpam-5717	585	8	,	,	PUNCT
ejpam-5717	585	9	19(2):491–495	19(2):491–495	NUM
ejpam-5717	585	10	,	,	PUNCT
ejpam-5717	585	11	2024	2024	NUM
ejpam-5717	585	12	.	.	PUNCT
ejpam-5717	586	1	[	[	X
ejpam-5717	586	2	56	56	NUM
ejpam-5717	586	3	]	]	X
ejpam-5717	586	4	p.	p.	NOUN
ejpam-5717	586	5	pue	pue	NOUN
ejpam-5717	586	6	-	-	PUNCT
ejpam-5717	586	7	on	on	ADP
ejpam-5717	586	8	,	,	PUNCT
ejpam-5717	586	9	a.	a.	PROPN
ejpam-5717	586	10	sama	sama	PROPN
ejpam-5717	586	11	-	-	PUNCT
ejpam-5717	586	12	ae	ae	PROPN
ejpam-5717	586	13	,	,	PUNCT
ejpam-5717	586	14	and	and	CCONJ
ejpam-5717	586	15	c.	c.	PROPN
ejpam-5717	586	16	boonpok	boonpok	PROPN
ejpam-5717	586	17	.	.	PUNCT
ejpam-5717	587	1	c	c	X
ejpam-5717	587	2	-	-	PUNCT
ejpam-5717	587	3	quasi	quasi	X
ejpam-5717	587	4	(	(	PUNCT
ejpam-5717	587	5	τ1	τ1	PROPN
ejpam-5717	587	6	,	,	PUNCT
ejpam-5717	587	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	587	8	multifunctions	multifunction	NOUN
ejpam-5717	587	9	.	.	PUNCT
ejpam-5717	588	1	european	european	ADJ
ejpam-5717	588	2	journal	journal	PROPN
ejpam-5717	588	3	of	of	ADP
ejpam-5717	588	4	pure	pure	ADJ
ejpam-5717	588	5	and	and	CCONJ
ejpam-5717	588	6	applied	applied	ADJ
ejpam-5717	588	7	mathematics	mathematic	NOUN
ejpam-5717	588	8	,	,	PUNCT
ejpam-5717	588	9	17(4):3242–3253	17(4):3242–3253	NUM
ejpam-5717	588	10	,	,	PUNCT
ejpam-5717	588	11	2024	2024	NUM
ejpam-5717	588	12	.	.	PUNCT
ejpam-5717	589	1	[	[	X
ejpam-5717	589	2	57	57	NUM
ejpam-5717	589	3	]	]	X
ejpam-5717	589	4	p.	p.	NOUN
ejpam-5717	589	5	pue	pue	NOUN
ejpam-5717	589	6	-	-	PUNCT
ejpam-5717	589	7	on	on	ADP
ejpam-5717	589	8	,	,	PUNCT
ejpam-5717	589	9	s.	s.	PROPN
ejpam-5717	589	10	sompong	sompong	PROPN
ejpam-5717	589	11	,	,	PUNCT
ejpam-5717	589	12	and	and	CCONJ
ejpam-5717	589	13	c.	c.	PROPN
ejpam-5717	589	14	boonpok	boonpok	PROPN
ejpam-5717	589	15	.	.	PUNCT
ejpam-5717	590	1	almost	almost	ADV
ejpam-5717	590	2	quasi	quasi	X
ejpam-5717	590	3	(	(	PUNCT
ejpam-5717	590	4	τ1	τ1	NOUN
ejpam-5717	590	5	,	,	PUNCT
ejpam-5717	590	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5717	590	7	for	for	ADP
ejpam-5717	590	8	multifunctions	multifunction	NOUN
ejpam-5717	590	9	.	.	PUNCT
ejpam-5717	591	1	international	international	ADJ
ejpam-5717	591	2	journal	journal	NOUN
ejpam-5717	591	3	of	of	ADP
ejpam-5717	591	4	analysis	analysis	NOUN
ejpam-5717	591	5	and	and	CCONJ
ejpam-5717	591	6	applications	application	NOUN
ejpam-5717	591	7	,	,	PUNCT
ejpam-5717	591	8	22:97	22:97	NUM
ejpam-5717	591	9	,	,	PUNCT
ejpam-5717	591	10	2024	2024	NUM
ejpam-5717	591	11	.	.	PUNCT
ejpam-5717	592	1	[	[	X
ejpam-5717	592	2	58	58	NUM
ejpam-5717	592	3	]	]	PUNCT
ejpam-5717	592	4	p.	p.	NOUN
ejpam-5717	592	5	pue	pue	NOUN
ejpam-5717	592	6	-	-	PUNCT
ejpam-5717	592	7	on	on	ADP
ejpam-5717	592	8	,	,	PUNCT
ejpam-5717	592	9	s.	s.	PROPN
ejpam-5717	592	10	sompong	sompong	PROPN
ejpam-5717	592	11	,	,	PUNCT
ejpam-5717	592	12	and	and	CCONJ
ejpam-5717	592	13	c.	c.	PROPN
ejpam-5717	592	14	boonpok	boonpok	PROPN
ejpam-5717	592	15	.	.	PUNCT
ejpam-5717	593	1	upper	upper	ADJ
ejpam-5717	593	2	and	and	CCONJ
ejpam-5717	593	3	lower	low	ADJ
ejpam-5717	593	4	(	(	PUNCT
ejpam-5717	593	5	τ1	τ1	NOUN
ejpam-5717	593	6	,	,	PUNCT
ejpam-5717	593	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	593	8	mulfunctions	mulfunction	NOUN
ejpam-5717	593	9	.	.	PUNCT
ejpam-5717	594	1	international	international	ADJ
ejpam-5717	594	2	journal	journal	NOUN
ejpam-5717	594	3	of	of	ADP
ejpam-5717	594	4	mathematics	mathematic	NOUN
ejpam-5717	594	5	and	and	CCONJ
ejpam-5717	594	6	computer	computer	NOUN
ejpam-5717	594	7	science	science	NOUN
ejpam-5717	594	8	,	,	PUNCT
ejpam-5717	594	9	19(4):1305	19(4):1305	NUM
ejpam-5717	594	10	–	–	PUNCT
ejpam-5717	594	11	1310	1310	NUM
ejpam-5717	594	12	,	,	PUNCT
ejpam-5717	594	13	2024	2024	NUM
ejpam-5717	594	14	.	.	PUNCT
ejpam-5717	595	1	[	[	X
ejpam-5717	595	2	59	59	NUM
ejpam-5717	595	3	]	]	X
ejpam-5717	595	4	p.	p.	NOUN
ejpam-5717	595	5	pue	pue	NOUN
ejpam-5717	595	6	-	-	PUNCT
ejpam-5717	595	7	on	on	ADP
ejpam-5717	595	8	,	,	PUNCT
ejpam-5717	595	9	s.	s.	PROPN
ejpam-5717	595	10	sompong	sompong	PROPN
ejpam-5717	595	11	,	,	PUNCT
ejpam-5717	595	12	and	and	CCONJ
ejpam-5717	595	13	c.	c.	PROPN
ejpam-5717	595	14	boonpok	boonpok	PROPN
ejpam-5717	595	15	.	.	PUNCT
ejpam-5717	596	1	weakly	weakly	ADJ
ejpam-5717	596	2	quasi	quasi	NOUN
ejpam-5717	596	3	(	(	PUNCT
ejpam-5717	596	4	τ1	τ1	PROPN
ejpam-5717	596	5	,	,	PUNCT
ejpam-5717	596	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	596	7	multifunctions	multifunction	NOUN
ejpam-5717	596	8	.	.	PUNCT
ejpam-5717	597	1	european	european	ADJ
ejpam-5717	597	2	journal	journal	PROPN
ejpam-5717	597	3	of	of	ADP
ejpam-5717	597	4	pure	pure	ADJ
ejpam-5717	597	5	and	and	CCONJ
ejpam-5717	597	6	applied	applied	ADJ
ejpam-5717	597	7	mathematics	mathematic	NOUN
ejpam-5717	597	8	,	,	PUNCT
ejpam-5717	597	9	17(3):1553–1564	17(3):1553–1564	NUM
ejpam-5717	597	10	,	,	PUNCT
ejpam-5717	597	11	2024	2024	NUM
ejpam-5717	597	12	.	.	PUNCT
ejpam-5717	598	1	[	[	X
ejpam-5717	598	2	60	60	NUM
ejpam-5717	598	3	]	]	X
ejpam-5717	598	4	n.	n.	NOUN
ejpam-5717	598	5	srisarakham	srisarakham	PROPN
ejpam-5717	598	6	and	and	CCONJ
ejpam-5717	598	7	c.	c.	PROPN
ejpam-5717	598	8	boonpok	boonpok	PROPN
ejpam-5717	598	9	.	.	PUNCT
ejpam-5717	599	1	almost	almost	ADV
ejpam-5717	599	2	(	(	PUNCT
ejpam-5717	599	3	λ	λ	NOUN
ejpam-5717	599	4	,	,	PUNCT
ejpam-5717	599	5	p)-continuous	p)-continuous	ADJ
ejpam-5717	599	6	functions	function	NOUN
ejpam-5717	599	7	.	.	PUNCT
ejpam-5717	600	1	international	international	ADJ
ejpam-5717	600	2	journal	journal	PROPN
ejpam-5717	600	3	of	of	ADP
ejpam-5717	600	4	mathematics	mathematic	NOUN
ejpam-5717	600	5	and	and	CCONJ
ejpam-5717	600	6	computer	computer	NOUN
ejpam-5717	600	7	science	science	NOUN
ejpam-5717	600	8	,	,	PUNCT
ejpam-5717	600	9	18(2):255–259	18(2):255–259	NUM
ejpam-5717	600	10	,	,	PUNCT
ejpam-5717	600	11	2023	2023	NUM
ejpam-5717	600	12	.	.	PUNCT
ejpam-5717	601	1	[	[	X
ejpam-5717	601	2	61	61	NUM
ejpam-5717	601	3	]	]	X
ejpam-5717	601	4	n.	n.	NOUN
ejpam-5717	601	5	srisarakham	srisarakham	PROPN
ejpam-5717	601	6	,	,	PUNCT
ejpam-5717	601	7	a.	a.	PROPN
ejpam-5717	601	8	sama	sama	PROPN
ejpam-5717	601	9	-	-	PUNCT
ejpam-5717	601	10	ae	ae	PROPN
ejpam-5717	601	11	,	,	PUNCT
ejpam-5717	601	12	and	and	CCONJ
ejpam-5717	601	13	c.	c.	PROPN
ejpam-5717	601	14	boonpok	boonpok	PROPN
ejpam-5717	601	15	.	.	PUNCT
ejpam-5717	602	1	characterizations	characterization	NOUN
ejpam-5717	602	2	of	of	ADP
ejpam-5717	602	3	faintly	faintly	ADV
ejpam-5717	602	4	(	(	PUNCT
ejpam-5717	602	5	τ1	τ1	PROPN
ejpam-5717	602	6	,	,	PUNCT
ejpam-5717	602	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5717	602	8	functions	function	NOUN
ejpam-5717	602	9	.	.	PUNCT
ejpam-5717	603	1	european	european	ADJ
ejpam-5717	603	2	journal	journal	PROPN
ejpam-5717	603	3	of	of	ADP
ejpam-5717	603	4	pure	pure	ADJ
ejpam-5717	603	5	and	and	CCONJ
ejpam-5717	603	6	applied	applied	ADJ
ejpam-5717	603	7	mathematics	mathematic	NOUN
ejpam-5717	603	8	,	,	PUNCT
ejpam-5717	603	9	17(4):2753–2762	17(4):2753–2762	NUM
ejpam-5717	603	10	,	,	PUNCT
ejpam-5717	603	11	2024	2024	NUM
ejpam-5717	603	12	.	.	PUNCT
ejpam-5717	604	1	[	[	X
ejpam-5717	604	2	62	62	NUM
ejpam-5717	604	3	]	]	PUNCT
ejpam-5717	604	4	m.	m.	NOUN
ejpam-5717	604	5	thongmoon	thongmoon	NOUN
ejpam-5717	604	6	and	and	CCONJ
ejpam-5717	604	7	c.	c.	PROPN
ejpam-5717	604	8	boonpok	boonpok	PROPN
ejpam-5717	604	9	.	.	PUNCT
ejpam-5717	605	1	upper	upper	ADJ
ejpam-5717	605	2	and	and	CCONJ
ejpam-5717	605	3	lower	low	ADJ
ejpam-5717	605	4	almost	almost	ADV
ejpam-5717	605	5	β(λ	β(λ	NOUN
ejpam-5717	605	6	,	,	PUNCT
ejpam-5717	605	7	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	605	8	multifunctions	multifunction	NOUN
ejpam-5717	605	9	.	.	PUNCT
ejpam-5717	606	1	wseas	wseas	VERB
ejpam-5717	606	2	transactions	transaction	NOUN
ejpam-5717	606	3	on	on	ADP
ejpam-5717	606	4	mathematics	mathematic	NOUN
ejpam-5717	606	5	,	,	PUNCT
ejpam-5717	606	6	21:844–853	21:844–853	NUM
ejpam-5717	606	7	,	,	PUNCT
ejpam-5717	606	8	2022	2022	NUM
ejpam-5717	606	9	.	.	PUNCT
ejpam-5717	607	1	[	[	X
ejpam-5717	607	2	63	63	NUM
ejpam-5717	607	3	]	]	PUNCT
ejpam-5717	607	4	m.	m.	NOUN
ejpam-5717	607	5	thongmoon	thongmoon	NOUN
ejpam-5717	607	6	and	and	CCONJ
ejpam-5717	607	7	c.	c.	PROPN
ejpam-5717	607	8	boonpok	boonpok	PROPN
ejpam-5717	607	9	.	.	PUNCT
ejpam-5717	608	1	strongly	strongly	ADV
ejpam-5717	608	2	θ(λ	θ(λ	PROPN
ejpam-5717	608	3	,	,	PUNCT
ejpam-5717	608	4	p)-continuous	p)-continuous	ADJ
ejpam-5717	608	5	functions	function	NOUN
ejpam-5717	608	6	.	.	PUNCT
ejpam-5717	609	1	international	international	ADJ
ejpam-5717	609	2	journal	journal	PROPN
ejpam-5717	609	3	of	of	ADP
ejpam-5717	609	4	mathematics	mathematic	NOUN
ejpam-5717	609	5	and	and	CCONJ
ejpam-5717	609	6	computer	computer	NOUN
ejpam-5717	609	7	science	science	NOUN
ejpam-5717	609	8	,	,	PUNCT
ejpam-5717	609	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5717	609	10	,	,	PUNCT
ejpam-5717	609	11	2024	2024	NUM
ejpam-5717	609	12	.	.	PUNCT
ejpam-5717	610	1	[	[	X
ejpam-5717	610	2	64	64	NUM
ejpam-5717	610	3	]	]	PUNCT
ejpam-5717	610	4	m.	m.	NOUN
ejpam-5717	610	5	thongmoon	thongmoon	NOUN
ejpam-5717	610	6	,	,	PUNCT
ejpam-5717	610	7	s.	s.	PROPN
ejpam-5717	610	8	sompong	sompong	PROPN
ejpam-5717	610	9	,	,	PUNCT
ejpam-5717	610	10	and	and	CCONJ
ejpam-5717	610	11	c.	c.	PROPN
ejpam-5717	610	12	boonpok	boonpok	PROPN
ejpam-5717	610	13	.	.	PUNCT
ejpam-5717	611	1	upper	upper	ADJ
ejpam-5717	611	2	and	and	CCONJ
ejpam-5717	611	3	lower	low	ADJ
ejpam-5717	611	4	weak	weak	ADJ
ejpam-5717	611	5	(	(	PUNCT
ejpam-5717	611	6	τ1	τ1	NOUN
ejpam-5717	611	7	,	,	PUNCT
ejpam-5717	611	8	τ2)continuity	τ2)continuity	PROPN
ejpam-5717	611	9	.	.	PUNCT
ejpam-5717	612	1	european	european	PROPN
ejpam-5717	612	2	journal	journal	PROPN
ejpam-5717	612	3	of	of	ADP
ejpam-5717	612	4	pure	pure	ADJ
ejpam-5717	612	5	and	and	CCONJ
ejpam-5717	612	6	applied	applied	ADJ
ejpam-5717	612	7	mathematics	mathematic	NOUN
ejpam-5717	612	8	,	,	PUNCT
ejpam-5717	612	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-5717	612	10	,	,	PUNCT
ejpam-5717	612	11	2024	2024	NUM
ejpam-5717	612	12	.	.	PUNCT
ejpam-5717	613	1	[	[	X
ejpam-5717	613	2	65	65	NUM
ejpam-5717	613	3	]	]	X
ejpam-5717	613	4	n.	n.	PROPN
ejpam-5717	613	5	v.	v.	PROPN
ejpam-5717	613	6	veličko	veličko	PROPN
ejpam-5717	613	7	.	.	PUNCT
ejpam-5717	614	1	h	h	NOUN
ejpam-5717	614	2	-	-	PUNCT
ejpam-5717	614	3	closed	close	VERB
ejpam-5717	614	4	topological	topological	ADJ
ejpam-5717	614	5	spaces	space	NOUN
ejpam-5717	614	6	.	.	PUNCT
ejpam-5717	615	1	american	american	PROPN
ejpam-5717	615	2	mathematical	mathematical	ADJ
ejpam-5717	615	3	society	society	NOUN
ejpam-5717	615	4	translations	translation	NOUN
ejpam-5717	615	5	,	,	PUNCT
ejpam-5717	615	6	78(2):102–118	78(2):102–118	NUM
ejpam-5717	615	7	,	,	PUNCT
ejpam-5717	615	8	1968	1968	NUM
ejpam-5717	615	9	.	.	PUNCT
ejpam-5717	616	1	[	[	X
ejpam-5717	616	2	66	66	NUM
ejpam-5717	616	3	]	]	PUNCT
ejpam-5717	616	4	c.	c.	PROPN
ejpam-5717	616	5	viriyapong	viriyapong	PROPN
ejpam-5717	616	6	and	and	CCONJ
ejpam-5717	616	7	c.	c.	PROPN
ejpam-5717	616	8	boonpok	boonpok	PROPN
ejpam-5717	616	9	.	.	PUNCT
ejpam-5717	617	1	(	(	PUNCT
ejpam-5717	617	2	τ1	τ1	NOUN
ejpam-5717	617	3	,	,	PUNCT
ejpam-5717	617	4	τ2)α	τ2)α	NOUN
ejpam-5717	617	5	-	-	PUNCT
ejpam-5717	617	6	continuity	continuity	NOUN
ejpam-5717	617	7	for	for	ADP
ejpam-5717	617	8	multifunctions	multifunction	NOUN
ejpam-5717	617	9	.	.	PUNCT
ejpam-5717	618	1	journal	journal	PROPN
ejpam-5717	618	2	of	of	ADP
ejpam-5717	618	3	mathematics	mathematic	NOUN
ejpam-5717	618	4	,	,	PUNCT
ejpam-5717	618	5	2020:6285763	2020:6285763	NUM
ejpam-5717	618	6	,	,	PUNCT
ejpam-5717	618	7	2020	2020	NUM
ejpam-5717	618	8	.	.	PUNCT
ejpam-5717	619	1	[	[	X
ejpam-5717	619	2	67	67	NUM
ejpam-5717	619	3	]	]	X
ejpam-5717	619	4	c.	c.	PROPN
ejpam-5717	619	5	viriyapong	viriyapong	PROPN
ejpam-5717	619	6	and	and	CCONJ
ejpam-5717	619	7	c.	c.	PROPN
ejpam-5717	619	8	boonpok	boonpok	PROPN
ejpam-5717	619	9	.	.	PUNCT
ejpam-5717	620	1	(	(	PUNCT
ejpam-5717	620	2	λ	λ	X
ejpam-5717	620	3	,	,	PUNCT
ejpam-5717	620	4	sp)-continuous	sp)-continuous	ADJ
ejpam-5717	620	5	functions	function	NOUN
ejpam-5717	620	6	.	.	PUNCT
ejpam-5717	621	1	wseas	wseas	VERB
ejpam-5717	621	2	transactions	transaction	NOUN
ejpam-5717	621	3	on	on	ADP
ejpam-5717	621	4	mathematics	mathematic	NOUN
ejpam-5717	621	5	,	,	PUNCT
ejpam-5717	621	6	21:380–385	21:380–385	NUM
ejpam-5717	621	7	,	,	PUNCT
ejpam-5717	621	8	2022	2022	NUM
ejpam-5717	621	9	.	.	PUNCT
ejpam-5717	622	1	[	[	X
ejpam-5717	622	2	68	68	NUM
ejpam-5717	622	3	]	]	X
ejpam-5717	622	4	c.	c.	PROPN
ejpam-5717	622	5	viriyapong	viriyapong	PROPN
ejpam-5717	622	6	and	and	CCONJ
ejpam-5717	622	7	c.	c.	PROPN
ejpam-5717	622	8	boonpok	boonpok	PROPN
ejpam-5717	622	9	.	.	PUNCT
ejpam-5717	623	1	weak	weak	ADJ
ejpam-5717	623	2	quasi	quasi	NOUN
ejpam-5717	623	3	(	(	PUNCT
ejpam-5717	623	4	λ	λ	PROPN
ejpam-5717	623	5	,	,	PUNCT
ejpam-5717	623	6	sp)-continuity	sp)-continuity	NOUN
ejpam-5717	623	7	for	for	ADP
ejpam-5717	623	8	multifunctions	multifunction	NOUN
ejpam-5717	623	9	.	.	PUNCT
ejpam-5717	624	1	international	international	ADJ
ejpam-5717	624	2	journal	journal	PROPN
ejpam-5717	624	3	of	of	ADP
ejpam-5717	624	4	mathematics	mathematic	NOUN
ejpam-5717	624	5	and	and	CCONJ
ejpam-5717	624	6	computer	computer	NOUN
ejpam-5717	624	7	science	science	NOUN
ejpam-5717	624	8	,	,	PUNCT
ejpam-5717	624	9	17(3):1201–1209	17(3):1201–1209	NUM
ejpam-5717	624	10	,	,	PUNCT
ejpam-5717	624	11	2022	2022	NUM
ejpam-5717	624	12	.	.	PUNCT
ejpam-5717	625	1	[	[	X
ejpam-5717	625	2	69	69	NUM
ejpam-5717	625	3	]	]	X
ejpam-5717	625	4	c.	c.	PROPN
ejpam-5717	625	5	viriyapong	viriyapong	PROPN
ejpam-5717	625	6	,	,	PUNCT
ejpam-5717	625	7	s.	s.	PROPN
ejpam-5717	625	8	sompong	sompong	PROPN
ejpam-5717	625	9	,	,	PUNCT
ejpam-5717	625	10	and	and	CCONJ
ejpam-5717	625	11	c.	c.	PROPN
ejpam-5717	625	12	boonpok	boonpok	PROPN
ejpam-5717	625	13	.	.	PUNCT
ejpam-5717	626	1	upper	upper	ADJ
ejpam-5717	626	2	and	and	CCONJ
ejpam-5717	626	3	lower	low	ADJ
ejpam-5717	626	4	slight	slight	ADJ
ejpam-5717	626	5	(	(	PUNCT
ejpam-5717	626	6	τ1	τ1	NOUN
ejpam-5717	626	7	,	,	PUNCT
ejpam-5717	626	8	τ2)continuity	τ2)continuity	PROPN
ejpam-5717	626	9	.	.	PUNCT
ejpam-5717	627	1	european	european	PROPN
ejpam-5717	627	2	journal	journal	PROPN
ejpam-5717	627	3	of	of	ADP
ejpam-5717	627	4	pure	pure	ADJ
ejpam-5717	627	5	and	and	CCONJ
ejpam-5717	627	6	applied	applied	ADJ
ejpam-5717	627	7	mathematics	mathematic	NOUN
ejpam-5717	627	8	,	,	PUNCT
ejpam-5717	627	9	17(3):2142–2154	17(3):2142–2154	NUM
ejpam-5717	627	10	,	,	PUNCT
ejpam-5717	627	11	2024	2024	NUM
ejpam-5717	627	12	.	.	PUNCT
ejpam-5717	628	1	[	[	X
ejpam-5717	628	2	70	70	NUM
ejpam-5717	628	3	]	]	X
ejpam-5717	628	4	n.	n.	PROPN
ejpam-5717	628	5	viriyapong	viriyapong	PROPN
ejpam-5717	628	6	,	,	PUNCT
ejpam-5717	628	7	s.	s.	PROPN
ejpam-5717	628	8	sompong	sompong	PROPN
ejpam-5717	628	9	,	,	PUNCT
ejpam-5717	628	10	and	and	CCONJ
ejpam-5717	628	11	c.	c.	PROPN
ejpam-5717	628	12	boonpok	boonpok	PROPN
ejpam-5717	628	13	.	.	PUNCT
ejpam-5717	629	1	slightly	slightly	ADV
ejpam-5717	629	2	(	(	PUNCT
ejpam-5717	629	3	τ1	τ1	NOUN
ejpam-5717	629	4	,	,	PUNCT
ejpam-5717	629	5	τ2)p	τ2)p	ADJ
ejpam-5717	629	6	-	-	ADJ
ejpam-5717	629	7	continuous	continuous	ADJ
ejpam-5717	629	8	multifuncp	multifuncp	NOUN
ejpam-5717	629	9	.	.	PUNCT
ejpam-5717	630	1	pue	pue	NOUN
ejpam-5717	630	2	-	-	PUNCT
ejpam-5717	630	3	on	on	ADP
ejpam-5717	630	4	,	,	PUNCT
ejpam-5717	630	5	a.	a.	PROPN
ejpam-5717	630	6	sama	sama	PROPN
ejpam-5717	630	7	-	-	PUNCT
ejpam-5717	630	8	ae	ae	PROPN
ejpam-5717	630	9	,	,	PUNCT
ejpam-5717	630	10	c.	c.	PROPN
ejpam-5717	630	11	boonpok	boonpok	PROPN
ejpam-5717	630	12	/	/	SYM
ejpam-5717	630	13	eur	eur	PROPN
ejpam-5717	630	14	.	.	PUNCT
ejpam-5717	631	1	j.	j.	PROPN
ejpam-5717	631	2	pure	pure	PROPN
ejpam-5717	631	3	appl	appl	PROPN
ejpam-5717	631	4	.	.	PROPN
ejpam-5717	631	5	math	math	PROPN
ejpam-5717	631	6	,	,	PUNCT
ejpam-5717	631	7	18	18	NUM
ejpam-5717	631	8	(	(	PUNCT
ejpam-5717	631	9	1	1	NUM
ejpam-5717	631	10	)	)	PUNCT
ejpam-5717	631	11	(	(	PUNCT
ejpam-5717	631	12	2025	2025	NUM
ejpam-5717	631	13	)	)	PUNCT
ejpam-5717	631	14	,	,	PUNCT
ejpam-5717	631	15	5717	5717	NUM
ejpam-5717	631	16	16	16	NUM
ejpam-5717	631	17	of	of	ADP
ejpam-5717	631	18	16	16	NUM
ejpam-5717	631	19	tions	tion	NOUN
ejpam-5717	631	20	.	.	PUNCT
ejpam-5717	632	1	international	international	ADJ
ejpam-5717	632	2	journal	journal	NOUN
ejpam-5717	632	3	of	of	ADP
ejpam-5717	632	4	analysis	analysis	NOUN
ejpam-5717	632	5	and	and	CCONJ
ejpam-5717	632	6	applications	application	NOUN
ejpam-5717	632	7	,	,	PUNCT
ejpam-5717	632	8	22:152	22:152	NUM
ejpam-5717	632	9	,	,	PUNCT
ejpam-5717	632	10	2024	2024	NUM
ejpam-5717	632	11	.	.	PUNCT
ejpam-5717	633	1	[	[	X
ejpam-5717	633	2	71	71	NUM
ejpam-5717	633	3	]	]	X
ejpam-5717	633	4	n.	n.	PROPN
ejpam-5717	633	5	viriyapong	viriyapong	PROPN
ejpam-5717	633	6	,	,	PUNCT
ejpam-5717	633	7	s.	s.	PROPN
ejpam-5717	633	8	sompong	sompong	PROPN
ejpam-5717	633	9	,	,	PUNCT
ejpam-5717	633	10	and	and	CCONJ
ejpam-5717	633	11	c.	c.	PROPN
ejpam-5717	633	12	boonpok	boonpok	PROPN
ejpam-5717	633	13	.	.	PUNCT
ejpam-5717	634	1	(	(	PUNCT
ejpam-5717	634	2	τ1	τ1	NOUN
ejpam-5717	634	3	,	,	PUNCT
ejpam-5717	634	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5717	634	5	disconnectedness	disconnectedness	NOUN
ejpam-5717	634	6	in	in	ADP
ejpam-5717	634	7	bitopological	bitopological	ADJ
ejpam-5717	634	8	spaces	space	NOUN
ejpam-5717	634	9	.	.	PUNCT
ejpam-5717	635	1	international	international	ADJ
ejpam-5717	635	2	journal	journal	PROPN
ejpam-5717	635	3	of	of	ADP
ejpam-5717	635	4	mathematics	mathematic	NOUN
ejpam-5717	635	5	and	and	CCONJ
ejpam-5717	635	6	computer	computer	NOUN
ejpam-5717	635	7	science	science	NOUN
ejpam-5717	635	8	,	,	PUNCT
ejpam-5717	635	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5717	635	10	,	,	PUNCT
ejpam-5717	635	11	2024	2024	NUM
ejpam-5717	635	12	.	.	PUNCT
ejpam-5717	636	1	[	[	X
ejpam-5717	636	2	72	72	NUM
ejpam-5717	636	3	]	]	X
ejpam-5717	636	4	n.	n.	PROPN
ejpam-5717	636	5	viriyapong	viriyapong	PROPN
ejpam-5717	636	6	,	,	PUNCT
ejpam-5717	636	7	s.	s.	PROPN
ejpam-5717	636	8	sompong	sompong	PROPN
ejpam-5717	636	9	,	,	PUNCT
ejpam-5717	636	10	and	and	CCONJ
ejpam-5717	636	11	c.	c.	PROPN
ejpam-5717	636	12	boonpok	boonpok	PROPN
ejpam-5717	636	13	.	.	PUNCT
ejpam-5717	637	1	upper	upper	ADJ
ejpam-5717	637	2	and	and	CCONJ
ejpam-5717	637	3	lower	low	ADJ
ejpam-5717	637	4	s-(τ1	s-(τ1	NOUN
ejpam-5717	637	5	,	,	PUNCT
ejpam-5717	637	6	τ2)p	τ2)p	ADJ
ejpam-5717	637	7	-	-	PUNCT
ejpam-5717	637	8	continuous	continuous	ADJ
ejpam-5717	637	9	multifunctions	multifunction	NOUN
ejpam-5717	637	10	.	.	PUNCT
ejpam-5717	638	1	european	european	ADJ
ejpam-5717	638	2	journal	journal	PROPN
ejpam-5717	638	3	of	of	ADP
ejpam-5717	638	4	pure	pure	ADJ
ejpam-5717	638	5	and	and	CCONJ
ejpam-5717	638	6	applied	applied	ADJ
ejpam-5717	638	7	mathematics	mathematic	NOUN
ejpam-5717	638	8	,	,	PUNCT
ejpam-5717	638	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-5717	638	10	,	,	PUNCT
ejpam-5717	638	11	2024	2024	NUM
ejpam-5717	638	12	.	.	PUNCT