id	sid	tid	token	lemma	pos
ejpam-4576	1	1	european	european	PROPN
ejpam-4576	1	2	journal	journal	PROPN
ejpam-4576	1	3	of	of	ADP
ejpam-4576	1	4	pure	pure	ADJ
ejpam-4576	1	5	and	and	CCONJ
ejpam-4576	1	6	applied	apply	VERB
ejpam-4576	1	7	mathematics	mathematic	NOUN
ejpam-4576	1	8	vol	vol	NOUN
ejpam-4576	1	9	.	.	PUNCT
ejpam-4576	2	1	16	16	NUM
ejpam-4576	2	2	,	,	PUNCT
ejpam-4576	2	3	no	no	INTJ
ejpam-4576	2	4	.	.	NOUN
ejpam-4576	2	5	1	1	NUM
ejpam-4576	2	6	,	,	PUNCT
ejpam-4576	2	7	2023	2023	NUM
ejpam-4576	2	8	,	,	PUNCT
ejpam-4576	2	9	84	84	NUM
ejpam-4576	2	10	-	-	SYM
ejpam-4576	2	11	96	96	NUM
ejpam-4576	2	12	issn	issn	PROPN
ejpam-4576	2	13	1307	1307	NUM
ejpam-4576	2	14	-	-	SYM
ejpam-4576	2	15	5543	5543	NUM
ejpam-4576	2	16	–	–	PUNCT
ejpam-4576	2	17	ejpam.com	ejpam.com	X
ejpam-4576	2	18	published	publish	VERB
ejpam-4576	2	19	by	by	ADP
ejpam-4576	2	20	new	new	PROPN
ejpam-4576	2	21	york	york	PROPN
ejpam-4576	2	22	business	business	NOUN
ejpam-4576	2	23	global	global	PROPN
ejpam-4576	2	24	almost	almost	ADV
ejpam-4576	2	25	(	(	PUNCT
ejpam-4576	2	26	λ	λ	PROPN
ejpam-4576	2	27	,	,	PUNCT
ejpam-4576	2	28	sp)-continuity	sp)-continuity	NOUN
ejpam-4576	2	29	for	for	ADP
ejpam-4576	2	30	multifunctions	multifunction	NOUN
ejpam-4576	2	31	chawalit	chawalit	VERB
ejpam-4576	2	32	boonpok1	boonpok1	PROPN
ejpam-4576	2	33	,	,	PUNCT
ejpam-4576	2	34	nongluk	nongluk	PROPN
ejpam-4576	2	35	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-4576	2	36	1	1	NUM
ejpam-4576	2	37	mathematics	mathematic	NOUN
ejpam-4576	2	38	and	and	CCONJ
ejpam-4576	2	39	applied	apply	VERB
ejpam-4576	2	40	mathematics	mathematics	PROPN
ejpam-4576	2	41	research	research	NOUN
ejpam-4576	2	42	unit	unit	NOUN
ejpam-4576	2	43	,	,	PUNCT
ejpam-4576	2	44	department	department	NOUN
ejpam-4576	2	45	of	of	ADP
ejpam-4576	2	46	mathematics	mathematic	NOUN
ejpam-4576	2	47	,	,	PUNCT
ejpam-4576	2	48	faculty	faculty	NOUN
ejpam-4576	2	49	of	of	ADP
ejpam-4576	2	50	science	science	NOUN
ejpam-4576	2	51	,	,	PUNCT
ejpam-4576	2	52	mahasarakham	mahasarakham	PROPN
ejpam-4576	2	53	university	university	PROPN
ejpam-4576	2	54	,	,	PUNCT
ejpam-4576	2	55	maha	maha	PROPN
ejpam-4576	2	56	sarakham	sarakham	PROPN
ejpam-4576	2	57	,	,	PUNCT
ejpam-4576	2	58	44150	44150	NUM
ejpam-4576	2	59	,	,	PUNCT
ejpam-4576	2	60	thailand	thailand	PROPN
ejpam-4576	2	61	abstract	abstract	PROPN
ejpam-4576	2	62	.	.	PUNCT
ejpam-4576	3	1	this	this	DET
ejpam-4576	3	2	paper	paper	NOUN
ejpam-4576	3	3	deals	deal	NOUN
ejpam-4576	3	4	with	with	ADP
ejpam-4576	3	5	the	the	DET
ejpam-4576	3	6	concepts	concept	NOUN
ejpam-4576	3	7	of	of	ADP
ejpam-4576	3	8	upper	upper	ADJ
ejpam-4576	3	9	and	and	CCONJ
ejpam-4576	3	10	lower	low	ADJ
ejpam-4576	3	11	almost	almost	ADV
ejpam-4576	3	12	(	(	PUNCT
ejpam-4576	3	13	λ	λ	NOUN
ejpam-4576	3	14	,	,	PUNCT
ejpam-4576	3	15	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	3	16	multifunctions	multifunction	NOUN
ejpam-4576	3	17	.	.	PUNCT
ejpam-4576	4	1	moreover	moreover	ADV
ejpam-4576	4	2	,	,	PUNCT
ejpam-4576	4	3	several	several	ADJ
ejpam-4576	4	4	characterizations	characterization	NOUN
ejpam-4576	4	5	of	of	ADP
ejpam-4576	4	6	upper	upper	ADJ
ejpam-4576	4	7	and	and	CCONJ
ejpam-4576	4	8	lower	low	ADJ
ejpam-4576	4	9	almost	almost	ADV
ejpam-4576	4	10	(	(	PUNCT
ejpam-4576	4	11	λ	λ	NOUN
ejpam-4576	4	12	,	,	PUNCT
ejpam-4576	4	13	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	4	14	multifunctions	multifunction	NOUN
ejpam-4576	4	15	are	be	AUX
ejpam-4576	4	16	investigated	investigate	VERB
ejpam-4576	4	17	.	.	PUNCT
ejpam-4576	5	1	2020	2020	NUM
ejpam-4576	5	2	mathematics	mathematic	NOUN
ejpam-4576	5	3	subject	subject	NOUN
ejpam-4576	5	4	classifications	classification	NOUN
ejpam-4576	5	5	:	:	PUNCT
ejpam-4576	5	6	54c08	54c08	NUM
ejpam-4576	5	7	,	,	PUNCT
ejpam-4576	5	8	54c60	54c60	NUM
ejpam-4576	5	9	key	key	ADJ
ejpam-4576	5	10	words	word	NOUN
ejpam-4576	5	11	and	and	CCONJ
ejpam-4576	5	12	phrases	phrase	NOUN
ejpam-4576	5	13	:	:	PUNCT
ejpam-4576	5	14	upper	upper	ADJ
ejpam-4576	5	15	almost	almost	ADV
ejpam-4576	5	16	(	(	PUNCT
ejpam-4576	5	17	λ	λ	NOUN
ejpam-4576	5	18	,	,	PUNCT
ejpam-4576	5	19	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	5	20	multifunction	multifunction	NOUN
ejpam-4576	5	21	,	,	PUNCT
ejpam-4576	5	22	lower	low	ADJ
ejpam-4576	5	23	almost	almost	ADV
ejpam-4576	5	24	(	(	PUNCT
ejpam-4576	5	25	λ	λ	NOUN
ejpam-4576	5	26	,	,	PUNCT
ejpam-4576	5	27	sp)continuous	sp)continuous	ADJ
ejpam-4576	5	28	multifunction	multifunction	NOUN
ejpam-4576	5	29	1	1	NUM
ejpam-4576	5	30	.	.	PUNCT
ejpam-4576	6	1	introduction	introduction	NOUN
ejpam-4576	6	2	it	it	PRON
ejpam-4576	6	3	is	be	AUX
ejpam-4576	6	4	well	well	ADV
ejpam-4576	6	5	-	-	PUNCT
ejpam-4576	6	6	known	know	VERB
ejpam-4576	6	7	that	that	SCONJ
ejpam-4576	6	8	the	the	DET
ejpam-4576	6	9	branch	branch	NOUN
ejpam-4576	6	10	of	of	ADP
ejpam-4576	6	11	mathematics	mathematic	NOUN
ejpam-4576	6	12	called	call	VERB
ejpam-4576	6	13	topology	topology	NOUN
ejpam-4576	6	14	is	be	AUX
ejpam-4576	6	15	related	relate	VERB
ejpam-4576	6	16	to	to	ADP
ejpam-4576	6	17	all	all	DET
ejpam-4576	6	18	questions	question	NOUN
ejpam-4576	6	19	directly	directly	ADV
ejpam-4576	6	20	or	or	CCONJ
ejpam-4576	6	21	indirectly	indirectly	ADV
ejpam-4576	6	22	concerned	concerned	ADJ
ejpam-4576	6	23	with	with	ADP
ejpam-4576	6	24	continuity	continuity	NOUN
ejpam-4576	6	25	.	.	PUNCT
ejpam-4576	7	1	semi	semi	ADJ
ejpam-4576	7	2	-	-	ADJ
ejpam-4576	7	3	open	open	ADJ
ejpam-4576	7	4	sets	set	NOUN
ejpam-4576	7	5	,	,	PUNCT
ejpam-4576	7	6	preopen	preopen	ADJ
ejpam-4576	7	7	sets	set	NOUN
ejpam-4576	7	8	,	,	PUNCT
ejpam-4576	7	9	α	α	NOUN
ejpam-4576	7	10	-	-	ADJ
ejpam-4576	7	11	open	open	ADJ
ejpam-4576	7	12	sets	set	NOUN
ejpam-4576	7	13	,	,	PUNCT
ejpam-4576	7	14	β	β	ADJ
ejpam-4576	7	15	-	-	ADJ
ejpam-4576	7	16	open	open	ADJ
ejpam-4576	7	17	sets	set	NOUN
ejpam-4576	7	18	and	and	CCONJ
ejpam-4576	7	19	δ	δ	NOUN
ejpam-4576	7	20	-	-	PUNCT
ejpam-4576	7	21	open	open	ADJ
ejpam-4576	7	22	sets	set	NOUN
ejpam-4576	7	23	play	play	VERB
ejpam-4576	7	24	an	an	DET
ejpam-4576	7	25	important	important	ADJ
ejpam-4576	7	26	role	role	NOUN
ejpam-4576	7	27	in	in	ADP
ejpam-4576	7	28	the	the	DET
ejpam-4576	7	29	researches	research	NOUN
ejpam-4576	7	30	of	of	ADP
ejpam-4576	7	31	generalizations	generalization	NOUN
ejpam-4576	7	32	of	of	ADP
ejpam-4576	7	33	continuity	continuity	NOUN
ejpam-4576	7	34	in	in	ADP
ejpam-4576	7	35	topological	topological	ADJ
ejpam-4576	7	36	spaces	space	NOUN
ejpam-4576	7	37	.	.	PUNCT
ejpam-4576	8	1	by	by	ADP
ejpam-4576	8	2	using	use	VERB
ejpam-4576	8	3	these	these	DET
ejpam-4576	8	4	sets	set	NOUN
ejpam-4576	8	5	many	many	ADJ
ejpam-4576	8	6	authors	author	NOUN
ejpam-4576	8	7	introduced	introduce	VERB
ejpam-4576	8	8	and	and	CCONJ
ejpam-4576	8	9	studied	study	VERB
ejpam-4576	8	10	various	various	ADJ
ejpam-4576	8	11	types	type	NOUN
ejpam-4576	8	12	of	of	ADP
ejpam-4576	8	13	weak	weak	ADJ
ejpam-4576	8	14	forms	form	NOUN
ejpam-4576	8	15	of	of	ADP
ejpam-4576	8	16	continuity	continuity	NOUN
ejpam-4576	8	17	for	for	ADP
ejpam-4576	8	18	functions	function	NOUN
ejpam-4576	8	19	and	and	CCONJ
ejpam-4576	8	20	multifunctions	multifunction	NOUN
ejpam-4576	8	21	.	.	PUNCT
ejpam-4576	9	1	in	in	ADP
ejpam-4576	9	2	1968	1968	NUM
ejpam-4576	9	3	,	,	PUNCT
ejpam-4576	9	4	singal	singal	NOUN
ejpam-4576	9	5	and	and	CCONJ
ejpam-4576	9	6	singal	singal	ADJ
ejpam-4576	9	7	[	[	X
ejpam-4576	9	8	18	18	NUM
ejpam-4576	9	9	]	]	PUNCT
ejpam-4576	9	10	introduced	introduce	VERB
ejpam-4576	9	11	and	and	CCONJ
ejpam-4576	9	12	studied	study	VERB
ejpam-4576	9	13	the	the	DET
ejpam-4576	9	14	notion	notion	NOUN
ejpam-4576	9	15	of	of	ADP
ejpam-4576	9	16	almost	almost	ADV
ejpam-4576	9	17	continuous	continuous	ADJ
ejpam-4576	9	18	functions	function	NOUN
ejpam-4576	9	19	.	.	PUNCT
ejpam-4576	10	1	in	in	ADP
ejpam-4576	10	2	1982	1982	NUM
ejpam-4576	10	3	,	,	PUNCT
ejpam-4576	10	4	popa	popa	NOUN
ejpam-4576	10	5	[	[	X
ejpam-4576	10	6	12	12	NUM
ejpam-4576	10	7	]	]	PUNCT
ejpam-4576	10	8	introduced	introduce	VERB
ejpam-4576	10	9	the	the	DET
ejpam-4576	10	10	concepts	concept	NOUN
ejpam-4576	10	11	of	of	ADP
ejpam-4576	10	12	upper	upper	ADJ
ejpam-4576	10	13	and	and	CCONJ
ejpam-4576	10	14	lower	low	ADJ
ejpam-4576	10	15	almost	almost	ADV
ejpam-4576	10	16	continuous	continuous	ADJ
ejpam-4576	10	17	multifunctions	multifunction	NOUN
ejpam-4576	10	18	.	.	PUNCT
ejpam-4576	11	1	the	the	DET
ejpam-4576	11	2	notion	notion	NOUN
ejpam-4576	11	3	of	of	ADP
ejpam-4576	11	4	almost	almost	ADV
ejpam-4576	11	5	quasi	quasi	ADJ
ejpam-4576	11	6	-	-	ADJ
ejpam-4576	11	7	continuous	continuous	ADJ
ejpam-4576	11	8	multifunctions	multifunction	NOUN
ejpam-4576	11	9	was	be	AUX
ejpam-4576	11	10	introduced	introduce	VERB
ejpam-4576	11	11	by	by	ADP
ejpam-4576	11	12	popa	popa	NOUN
ejpam-4576	11	13	and	and	CCONJ
ejpam-4576	11	14	noiri	noiri	ADV
ejpam-4576	11	15	[	[	X
ejpam-4576	11	16	13	13	NUM
ejpam-4576	11	17	]	]	PUNCT
ejpam-4576	11	18	.	.	PUNCT
ejpam-4576	12	1	noiri	noiri	PROPN
ejpam-4576	12	2	and	and	CCONJ
ejpam-4576	12	3	popa	popa	NOUN
ejpam-4576	12	4	[	[	X
ejpam-4576	12	5	8	8	NUM
ejpam-4576	12	6	]	]	PUNCT
ejpam-4576	12	7	investigated	investigate	VERB
ejpam-4576	12	8	several	several	ADJ
ejpam-4576	12	9	characterizations	characterization	NOUN
ejpam-4576	12	10	of	of	ADP
ejpam-4576	12	11	upper	upper	ADJ
ejpam-4576	12	12	and	and	CCONJ
ejpam-4576	12	13	lower	low	ADJ
ejpam-4576	12	14	almost	almost	ADV
ejpam-4576	12	15	quasi	quasi	ADJ
ejpam-4576	12	16	-	-	ADJ
ejpam-4576	12	17	continuous	continuous	ADJ
ejpam-4576	12	18	multifunctions	multifunction	NOUN
ejpam-4576	12	19	.	.	PUNCT
ejpam-4576	13	1	in	in	ADP
ejpam-4576	13	2	1993	1993	NUM
ejpam-4576	13	3	,	,	PUNCT
ejpam-4576	13	4	popa	popa	NOUN
ejpam-4576	13	5	et	et	PROPN
ejpam-4576	13	6	al	al	PROPN
ejpam-4576	13	7	.	.	PUNCT
ejpam-4576	14	1	[	[	X
ejpam-4576	14	2	16	16	NUM
ejpam-4576	14	3	]	]	PUNCT
ejpam-4576	14	4	introduced	introduce	VERB
ejpam-4576	14	5	the	the	DET
ejpam-4576	14	6	concepts	concept	NOUN
ejpam-4576	14	7	of	of	ADP
ejpam-4576	14	8	upper	upper	ADJ
ejpam-4576	14	9	and	and	CCONJ
ejpam-4576	14	10	lower	low	ADJ
ejpam-4576	14	11	almost	almost	ADV
ejpam-4576	14	12	precontinuous	precontinuous	ADJ
ejpam-4576	14	13	multifunctions	multifunction	NOUN
ejpam-4576	14	14	.	.	PUNCT
ejpam-4576	15	1	moreover	moreover	ADV
ejpam-4576	15	2	,	,	PUNCT
ejpam-4576	15	3	popa	popa	NOUN
ejpam-4576	15	4	et	et	PROPN
ejpam-4576	15	5	al	al	PROPN
ejpam-4576	15	6	.	.	PUNCT
ejpam-4576	16	1	[	[	X
ejpam-4576	16	2	17	17	NUM
ejpam-4576	16	3	]	]	PUNCT
ejpam-4576	16	4	obtained	obtain	VERB
ejpam-4576	16	5	some	some	DET
ejpam-4576	16	6	characterizations	characterization	NOUN
ejpam-4576	16	7	of	of	ADP
ejpam-4576	16	8	upper	upper	ADJ
ejpam-4576	16	9	and	and	CCONJ
ejpam-4576	16	10	lower	low	ADJ
ejpam-4576	16	11	almost	almost	ADV
ejpam-4576	16	12	precontinuous	precontinuous	ADJ
ejpam-4576	16	13	multifunctions	multifunction	NOUN
ejpam-4576	16	14	.	.	PUNCT
ejpam-4576	17	1	in	in	ADP
ejpam-4576	17	2	1996	1996	NUM
ejpam-4576	17	3	,	,	PUNCT
ejpam-4576	17	4	popa	popa	NOUN
ejpam-4576	17	5	and	and	CCONJ
ejpam-4576	17	6	noiri	noiri	ADV
ejpam-4576	17	7	[	[	X
ejpam-4576	17	8	14	14	NUM
ejpam-4576	17	9	]	]	PUNCT
ejpam-4576	17	10	introduced	introduce	VERB
ejpam-4576	17	11	and	and	CCONJ
ejpam-4576	17	12	investigated	investigate	VERB
ejpam-4576	17	13	the	the	DET
ejpam-4576	17	14	notions	notion	NOUN
ejpam-4576	17	15	of	of	ADP
ejpam-4576	17	16	upper	upper	ADJ
ejpam-4576	17	17	and	and	CCONJ
ejpam-4576	17	18	lower	low	ADJ
ejpam-4576	17	19	almost	almost	ADV
ejpam-4576	17	20	α	α	ADJ
ejpam-4576	17	21	-	-	ADJ
ejpam-4576	17	22	continuous	continuous	ADJ
ejpam-4576	17	23	multifunctions	multifunction	NOUN
ejpam-4576	17	24	.	.	PUNCT
ejpam-4576	18	1	in	in	ADP
ejpam-4576	18	2	1999	1999	NUM
ejpam-4576	18	3	,	,	PUNCT
ejpam-4576	18	4	noiri	noiri	PRON
ejpam-4576	18	5	and	and	CCONJ
ejpam-4576	18	6	popa	popa	NOUN
ejpam-4576	18	7	[	[	X
ejpam-4576	18	8	9	9	NUM
ejpam-4576	18	9	]	]	PUNCT
ejpam-4576	18	10	introduced	introduce	VERB
ejpam-4576	18	11	the	the	DET
ejpam-4576	18	12	concepts	concept	NOUN
ejpam-4576	18	13	of	of	ADP
ejpam-4576	18	14	upper	upper	ADJ
ejpam-4576	18	15	and	and	CCONJ
ejpam-4576	18	16	lower	low	ADJ
ejpam-4576	18	17	almost	almost	ADV
ejpam-4576	18	18	β	β	ADJ
ejpam-4576	18	19	-	-	ADJ
ejpam-4576	18	20	continuous	continuous	ADJ
ejpam-4576	18	21	multifunctions	multifunction	NOUN
ejpam-4576	18	22	.	.	PUNCT
ejpam-4576	19	1	popa	popa	NOUN
ejpam-4576	19	2	and	and	CCONJ
ejpam-4576	19	3	noiri	noiri	ADV
ejpam-4576	20	1	[	[	X
ejpam-4576	20	2	15	15	NUM
ejpam-4576	20	3	]	]	PUNCT
ejpam-4576	20	4	further	far	ADV
ejpam-4576	20	5	studied	study	VERB
ejpam-4576	20	6	some	some	DET
ejpam-4576	20	7	characterizations	characterization	NOUN
ejpam-4576	20	8	of	of	ADP
ejpam-4576	20	9	upper	upper	ADJ
ejpam-4576	20	10	and	and	CCONJ
ejpam-4576	20	11	lower	low	ADJ
ejpam-4576	20	12	almost	almost	ADV
ejpam-4576	20	13	β	β	ADJ
ejpam-4576	20	14	-	-	ADJ
ejpam-4576	20	15	continuous	continuous	ADJ
ejpam-4576	20	16	multifunctions	multifunction	NOUN
ejpam-4576	20	17	.	.	PUNCT
ejpam-4576	21	1	in	in	ADP
ejpam-4576	21	2	2006	2006	NUM
ejpam-4576	21	3	,	,	PUNCT
ejpam-4576	21	4	ekici	ekici	NOUN
ejpam-4576	21	5	and	and	CCONJ
ejpam-4576	21	6	park	park	NOUN
ejpam-4576	21	7	[	[	X
ejpam-4576	21	8	5	5	NUM
ejpam-4576	21	9	]	]	PUNCT
ejpam-4576	21	10	introduced	introduce	VERB
ejpam-4576	21	11	and	and	CCONJ
ejpam-4576	21	12	studied	study	VERB
ejpam-4576	21	13	almost	almost	ADV
ejpam-4576	21	14	γ	γ	ADJ
ejpam-4576	21	15	-	-	ADJ
ejpam-4576	21	16	continuous	continuous	ADJ
ejpam-4576	21	17	multifunctions	multifunction	NOUN
ejpam-4576	21	18	.	.	PUNCT
ejpam-4576	22	1	in	in	ADP
ejpam-4576	22	2	2010	2010	NUM
ejpam-4576	22	3	,	,	PUNCT
ejpam-4576	22	4	noiri	noiri	ADV
ejpam-4576	22	5	and	and	CCONJ
ejpam-4576	22	6	popa	popa	NOUN
ejpam-4576	22	7	[	[	X
ejpam-4576	22	8	10	10	NUM
ejpam-4576	22	9	]	]	PUNCT
ejpam-4576	22	10	introduced	introduce	VERB
ejpam-4576	22	11	and	and	CCONJ
ejpam-4576	22	12	studied	study	VERB
ejpam-4576	22	13	the	the	DET
ejpam-4576	22	14	notions	notion	NOUN
ejpam-4576	22	15	of	of	ADP
ejpam-4576	22	16	upper	upper	ADJ
ejpam-4576	22	17	and	and	CCONJ
ejpam-4576	22	18	lower	low	ADJ
ejpam-4576	22	19	almost	almost	ADV
ejpam-4576	22	20	∗corresponding	∗corresponde	VERB
ejpam-4576	22	21	author	author	NOUN
ejpam-4576	22	22	.	.	PUNCT
ejpam-4576	23	1	doi	doi	NOUN
ejpam-4576	23	2	:	:	PUNCT
ejpam-4576	23	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4576	https://doi.org/10.29020/nybg.ejpam.v16i1.4576	PROPN
ejpam-4576	23	4	email	email	NOUN
ejpam-4576	23	5	addresses	address	NOUN
ejpam-4576	23	6	:	:	PUNCT
ejpam-4576	23	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4576	23	8	(	(	PUNCT
ejpam-4576	23	9	c.	c.	PROPN
ejpam-4576	23	10	boonpok	boonpok	PROPN
ejpam-4576	23	11	)	)	PUNCT
ejpam-4576	23	12	,	,	PUNCT
ejpam-4576	23	13	nongluk.h@msu.ac.th	nongluk.h@msu.ac.th	PROPN
ejpam-4576	23	14	(	(	PUNCT
ejpam-4576	23	15	n.	n.	PROPN
ejpam-4576	23	16	viriyapong	viriyapong	PROPN
ejpam-4576	23	17	)	)	PUNCT
ejpam-4576	23	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4576	24	1	84	84	NUM
ejpam-4576	25	1	©	©	ADP
ejpam-4576	25	2	2023	2023	NUM
ejpam-4576	25	3	ejpam	ejpam	NOUN
ejpam-4576	25	4	all	all	DET
ejpam-4576	25	5	rights	right	NOUN
ejpam-4576	25	6	reserved	reserve	VERB
ejpam-4576	25	7	.	.	PUNCT
ejpam-4576	26	1	c.	c.	PROPN
ejpam-4576	26	2	boonpok	boonpok	PROPN
ejpam-4576	26	3	,	,	PUNCT
ejpam-4576	26	4	n.	n.	PROPN
ejpam-4576	26	5	viriyapong	viriyapong	PROPN
ejpam-4576	26	6	/	/	SYM
ejpam-4576	26	7	eur	eur	PROPN
ejpam-4576	26	8	.	.	PUNCT
ejpam-4576	27	1	j.	j.	PROPN
ejpam-4576	27	2	pure	pure	PROPN
ejpam-4576	27	3	appl	appl	PROPN
ejpam-4576	27	4	.	.	PROPN
ejpam-4576	27	5	math	math	PROPN
ejpam-4576	27	6	,	,	PUNCT
ejpam-4576	27	7	16	16	NUM
ejpam-4576	27	8	(	(	PUNCT
ejpam-4576	27	9	1	1	NUM
ejpam-4576	27	10	)	)	PUNCT
ejpam-4576	27	11	(	(	PUNCT
ejpam-4576	27	12	2023	2023	NUM
ejpam-4576	27	13	)	)	PUNCT
ejpam-4576	27	14	,	,	PUNCT
ejpam-4576	27	15	84	84	NUM
ejpam-4576	27	16	-	-	SYM
ejpam-4576	27	17	96	96	NUM
ejpam-4576	27	18	85	85	NUM
ejpam-4576	27	19	m	m	NOUN
ejpam-4576	27	20	-	-	ADJ
ejpam-4576	27	21	continuous	continuous	ADJ
ejpam-4576	27	22	multifunctions	multifunction	NOUN
ejpam-4576	27	23	as	as	ADP
ejpam-4576	27	24	multifunctions	multifunction	NOUN
ejpam-4576	27	25	from	from	ADP
ejpam-4576	27	26	a	a	DET
ejpam-4576	27	27	set	set	NOUN
ejpam-4576	27	28	satisfying	satisfy	VERB
ejpam-4576	27	29	some	some	DET
ejpam-4576	27	30	minimal	minimal	ADJ
ejpam-4576	27	31	conditions	condition	NOUN
ejpam-4576	27	32	into	into	ADP
ejpam-4576	27	33	a	a	DET
ejpam-4576	27	34	topological	topological	ADJ
ejpam-4576	27	35	space	space	NOUN
ejpam-4576	27	36	.	.	PUNCT
ejpam-4576	28	1	in	in	ADP
ejpam-4576	28	2	2018	2018	NUM
ejpam-4576	28	3	,	,	PUNCT
ejpam-4576	28	4	boonpok	boonpok	PROPN
ejpam-4576	28	5	et	et	PROPN
ejpam-4576	28	6	al	al	PROPN
ejpam-4576	28	7	.	.	PUNCT
ejpam-4576	29	1	[	[	X
ejpam-4576	29	2	4	4	X
ejpam-4576	29	3	]	]	PUNCT
ejpam-4576	29	4	introduced	introduce	VERB
ejpam-4576	29	5	and	and	CCONJ
ejpam-4576	29	6	investigated	investigate	VERB
ejpam-4576	29	7	the	the	DET
ejpam-4576	29	8	concepts	concept	NOUN
ejpam-4576	29	9	of	of	ADP
ejpam-4576	29	10	upper	upper	ADJ
ejpam-4576	29	11	and	and	CCONJ
ejpam-4576	29	12	lower	low	ADJ
ejpam-4576	29	13	almost	almost	ADV
ejpam-4576	29	14	(	(	PUNCT
ejpam-4576	29	15	τ1	τ1	NOUN
ejpam-4576	29	16	,	,	PUNCT
ejpam-4576	29	17	τ2)-precontinuous	τ2)-precontinuous	ADJ
ejpam-4576	29	18	multifunctions	multifunction	NOUN
ejpam-4576	29	19	.	.	PUNCT
ejpam-4576	30	1	the	the	DET
ejpam-4576	30	2	concept	concept	NOUN
ejpam-4576	30	3	of	of	ADP
ejpam-4576	30	4	β	β	ADJ
ejpam-4576	30	5	-	-	ADJ
ejpam-4576	30	6	open	open	ADJ
ejpam-4576	30	7	sets	set	NOUN
ejpam-4576	30	8	due	due	ADP
ejpam-4576	30	9	to	to	ADP
ejpam-4576	30	10	abd	abd	PROPN
ejpam-4576	30	11	el	el	PROPN
ejpam-4576	30	12	-	-	PROPN
ejpam-4576	30	13	monsef	monsef	PROPN
ejpam-4576	30	14	et	et	PROPN
ejpam-4576	30	15	al	al	PROPN
ejpam-4576	30	16	.	.	PUNCT
ejpam-4576	31	1	[	[	X
ejpam-4576	31	2	6	6	NUM
ejpam-4576	31	3	]	]	PUNCT
ejpam-4576	31	4	or	or	CCONJ
ejpam-4576	31	5	semi	semi	ADJ
ejpam-4576	31	6	-	-	ADJ
ejpam-4576	31	7	preopen	preopen	ADJ
ejpam-4576	31	8	sets	set	NOUN
ejpam-4576	31	9	in	in	ADP
ejpam-4576	31	10	the	the	DET
ejpam-4576	31	11	sense	sense	NOUN
ejpam-4576	31	12	of	of	ADP
ejpam-4576	31	13	andrijević	andrijević	NOUN
ejpam-4576	31	14	[	[	X
ejpam-4576	31	15	1	1	X
ejpam-4576	31	16	]	]	PUNCT
ejpam-4576	31	17	plays	play	VERB
ejpam-4576	31	18	a	a	DET
ejpam-4576	31	19	significant	significant	ADJ
ejpam-4576	31	20	role	role	NOUN
ejpam-4576	31	21	in	in	ADP
ejpam-4576	31	22	general	general	ADJ
ejpam-4576	31	23	topology	topology	NOUN
ejpam-4576	31	24	.	.	PUNCT
ejpam-4576	32	1	noiri	noiri	PROPN
ejpam-4576	32	2	and	and	CCONJ
ejpam-4576	32	3	hatir	hatir	VERB
ejpam-4576	33	1	[	[	X
ejpam-4576	33	2	7	7	NUM
ejpam-4576	33	3	]	]	PUNCT
ejpam-4576	33	4	introduced	introduce	VERB
ejpam-4576	33	5	the	the	DET
ejpam-4576	33	6	concept	concept	NOUN
ejpam-4576	33	7	of	of	ADP
ejpam-4576	33	8	λsp	λsp	NOUN
ejpam-4576	33	9	-	-	PUNCT
ejpam-4576	33	10	sets	set	NOUN
ejpam-4576	33	11	in	in	ADP
ejpam-4576	33	12	terms	term	NOUN
ejpam-4576	33	13	of	of	ADP
ejpam-4576	33	14	the	the	DET
ejpam-4576	33	15	concept	concept	NOUN
ejpam-4576	33	16	of	of	ADP
ejpam-4576	33	17	β	β	ADJ
ejpam-4576	33	18	-	-	ADJ
ejpam-4576	33	19	open	open	ADJ
ejpam-4576	33	20	sets	set	NOUN
ejpam-4576	33	21	and	and	CCONJ
ejpam-4576	33	22	investigated	investigate	VERB
ejpam-4576	33	23	the	the	DET
ejpam-4576	33	24	notion	notion	NOUN
ejpam-4576	33	25	of	of	ADP
ejpam-4576	33	26	λsp	λsp	NOUN
ejpam-4576	33	27	-	-	PUNCT
ejpam-4576	33	28	closed	close	VERB
ejpam-4576	33	29	sets	set	NOUN
ejpam-4576	33	30	by	by	ADP
ejpam-4576	33	31	using	use	VERB
ejpam-4576	33	32	λsp	λsp	NOUN
ejpam-4576	33	33	-	-	PUNCT
ejpam-4576	33	34	sets	set	NOUN
ejpam-4576	33	35	.	.	PUNCT
ejpam-4576	34	1	in	in	ADP
ejpam-4576	34	2	[	[	X
ejpam-4576	34	3	3	3	NUM
ejpam-4576	34	4	]	]	PUNCT
ejpam-4576	34	5	,	,	PUNCT
ejpam-4576	34	6	the	the	DET
ejpam-4576	34	7	author	author	NOUN
ejpam-4576	34	8	introduced	introduce	VERB
ejpam-4576	34	9	the	the	DET
ejpam-4576	34	10	concepts	concept	NOUN
ejpam-4576	34	11	of	of	ADP
ejpam-4576	34	12	(	(	PUNCT
ejpam-4576	34	13	λ	λ	PROPN
ejpam-4576	34	14	,	,	PUNCT
ejpam-4576	34	15	sp)-open	sp)-open	ADJ
ejpam-4576	34	16	sets	set	NOUN
ejpam-4576	34	17	and	and	CCONJ
ejpam-4576	34	18	(	(	PUNCT
ejpam-4576	34	19	λ	λ	PROPN
ejpam-4576	34	20	,	,	PUNCT
ejpam-4576	34	21	sp)-closed	sp)-close	VERB
ejpam-4576	34	22	sets	set	NOUN
ejpam-4576	34	23	which	which	PRON
ejpam-4576	34	24	are	be	AUX
ejpam-4576	34	25	defined	define	VERB
ejpam-4576	34	26	by	by	ADP
ejpam-4576	34	27	utilizing	utilize	VERB
ejpam-4576	34	28	the	the	DET
ejpam-4576	34	29	notions	notion	NOUN
ejpam-4576	34	30	of	of	ADP
ejpam-4576	34	31	λsp	λsp	NOUN
ejpam-4576	34	32	-	-	PUNCT
ejpam-4576	34	33	sets	set	NOUN
ejpam-4576	34	34	and	and	CCONJ
ejpam-4576	34	35	β	β	NOUN
ejpam-4576	34	36	-	-	ADJ
ejpam-4576	34	37	closed	closed	ADJ
ejpam-4576	34	38	sets	set	NOUN
ejpam-4576	34	39	.	.	PUNCT
ejpam-4576	35	1	the	the	DET
ejpam-4576	35	2	notion	notion	NOUN
ejpam-4576	35	3	of	of	ADP
ejpam-4576	35	4	(	(	PUNCT
ejpam-4576	35	5	λ	λ	PROPN
ejpam-4576	35	6	,	,	PUNCT
ejpam-4576	35	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	35	8	multifunctions	multifunction	NOUN
ejpam-4576	35	9	was	be	AUX
ejpam-4576	35	10	studied	study	VERB
ejpam-4576	35	11	in	in	ADP
ejpam-4576	35	12	[	[	X
ejpam-4576	35	13	3	3	NUM
ejpam-4576	35	14	]	]	PUNCT
ejpam-4576	35	15	.	.	PUNCT
ejpam-4576	36	1	the	the	DET
ejpam-4576	36	2	purpose	purpose	NOUN
ejpam-4576	36	3	of	of	ADP
ejpam-4576	36	4	the	the	DET
ejpam-4576	36	5	present	present	ADJ
ejpam-4576	36	6	paper	paper	NOUN
ejpam-4576	36	7	is	be	AUX
ejpam-4576	36	8	to	to	PART
ejpam-4576	36	9	introduce	introduce	VERB
ejpam-4576	36	10	the	the	DET
ejpam-4576	36	11	concepts	concept	NOUN
ejpam-4576	36	12	of	of	ADP
ejpam-4576	36	13	upper	upper	ADJ
ejpam-4576	36	14	and	and	CCONJ
ejpam-4576	36	15	lower	low	ADJ
ejpam-4576	36	16	almost	almost	ADV
ejpam-4576	36	17	(	(	PUNCT
ejpam-4576	36	18	λ	λ	NOUN
ejpam-4576	36	19	,	,	PUNCT
ejpam-4576	36	20	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	36	21	multifunctions	multifunction	NOUN
ejpam-4576	36	22	.	.	PUNCT
ejpam-4576	37	1	moreover	moreover	ADV
ejpam-4576	37	2	,	,	PUNCT
ejpam-4576	37	3	some	some	DET
ejpam-4576	37	4	characterizations	characterization	NOUN
ejpam-4576	37	5	of	of	ADP
ejpam-4576	37	6	upper	upper	ADJ
ejpam-4576	37	7	and	and	CCONJ
ejpam-4576	37	8	lower	low	ADJ
ejpam-4576	37	9	almost	almost	ADV
ejpam-4576	37	10	(	(	PUNCT
ejpam-4576	37	11	λ	λ	NOUN
ejpam-4576	37	12	,	,	PUNCT
ejpam-4576	37	13	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	37	14	multifunctions	multifunction	NOUN
ejpam-4576	37	15	are	be	AUX
ejpam-4576	37	16	discussed	discuss	VERB
ejpam-4576	37	17	.	.	PUNCT
ejpam-4576	38	1	2	2	X
ejpam-4576	38	2	.	.	X
ejpam-4576	38	3	preliminaries	preliminary	NOUN
ejpam-4576	38	4	throughout	throughout	ADP
ejpam-4576	38	5	this	this	DET
ejpam-4576	38	6	paper	paper	NOUN
ejpam-4576	38	7	,	,	PUNCT
ejpam-4576	38	8	spaces	space	NOUN
ejpam-4576	38	9	(	(	PUNCT
ejpam-4576	38	10	x	x	X
ejpam-4576	38	11	,	,	PUNCT
ejpam-4576	38	12	τ	τ	X
ejpam-4576	38	13	)	)	PUNCT
ejpam-4576	38	14	and	and	CCONJ
ejpam-4576	38	15	(	(	PUNCT
ejpam-4576	38	16	y	y	PROPN
ejpam-4576	38	17	,	,	PUNCT
ejpam-4576	38	18	σ	σ	PROPN
ejpam-4576	38	19	)	)	PUNCT
ejpam-4576	38	20	(	(	PUNCT
ejpam-4576	38	21	or	or	CCONJ
ejpam-4576	38	22	simply	simply	ADV
ejpam-4576	38	23	x	x	X
ejpam-4576	38	24	and	and	CCONJ
ejpam-4576	38	25	y	y	PROPN
ejpam-4576	38	26	)	)	PUNCT
ejpam-4576	38	27	always	always	ADV
ejpam-4576	38	28	mean	mean	VERB
ejpam-4576	38	29	topological	topological	ADJ
ejpam-4576	38	30	spaces	space	NOUN
ejpam-4576	38	31	on	on	ADP
ejpam-4576	38	32	which	which	PRON
ejpam-4576	38	33	no	no	DET
ejpam-4576	38	34	separation	separation	NOUN
ejpam-4576	38	35	axioms	axiom	NOUN
ejpam-4576	38	36	are	be	AUX
ejpam-4576	38	37	assumed	assume	VERB
ejpam-4576	38	38	unless	unless	SCONJ
ejpam-4576	38	39	explicitly	explicitly	ADV
ejpam-4576	38	40	stated	state	VERB
ejpam-4576	38	41	.	.	PUNCT
ejpam-4576	39	1	let	let	VERB
ejpam-4576	39	2	a	a	DET
ejpam-4576	39	3	be	be	AUX
ejpam-4576	39	4	a	a	DET
ejpam-4576	39	5	subset	subset	NOUN
ejpam-4576	39	6	of	of	ADP
ejpam-4576	39	7	a	a	DET
ejpam-4576	39	8	topological	topological	ADJ
ejpam-4576	39	9	space	space	NOUN
ejpam-4576	39	10	(	(	PUNCT
ejpam-4576	39	11	x	x	X
ejpam-4576	39	12	,	,	PUNCT
ejpam-4576	39	13	τ	τ	PROPN
ejpam-4576	39	14	)	)	PUNCT
ejpam-4576	39	15	.	.	PUNCT
ejpam-4576	40	1	the	the	DET
ejpam-4576	40	2	closure	closure	NOUN
ejpam-4576	40	3	of	of	ADP
ejpam-4576	40	4	a	a	PRON
ejpam-4576	40	5	and	and	CCONJ
ejpam-4576	40	6	the	the	DET
ejpam-4576	40	7	interior	interior	NOUN
ejpam-4576	40	8	of	of	ADP
ejpam-4576	40	9	a	a	PRON
ejpam-4576	40	10	are	be	AUX
ejpam-4576	40	11	denoted	denote	VERB
ejpam-4576	40	12	by	by	ADP
ejpam-4576	40	13	cl(a	cl(a	NOUN
ejpam-4576	40	14	)	)	PUNCT
ejpam-4576	40	15	and	and	CCONJ
ejpam-4576	40	16	int(a	int(a	PROPN
ejpam-4576	40	17	)	)	PUNCT
ejpam-4576	40	18	,	,	PUNCT
ejpam-4576	40	19	respectively	respectively	ADV
ejpam-4576	40	20	.	.	PUNCT
ejpam-4576	41	1	a	a	DET
ejpam-4576	41	2	subset	subset	NOUN
ejpam-4576	41	3	a	a	PRON
ejpam-4576	41	4	of	of	ADP
ejpam-4576	41	5	a	a	DET
ejpam-4576	41	6	topological	topological	ADJ
ejpam-4576	41	7	space	space	NOUN
ejpam-4576	41	8	(	(	PUNCT
ejpam-4576	41	9	x	x	X
ejpam-4576	41	10	,	,	PUNCT
ejpam-4576	41	11	τ	τ	X
ejpam-4576	41	12	)	)	PUNCT
ejpam-4576	41	13	is	be	AUX
ejpam-4576	41	14	said	say	VERB
ejpam-4576	41	15	to	to	PART
ejpam-4576	41	16	be	be	AUX
ejpam-4576	41	17	β	β	X
ejpam-4576	41	18	-	-	VERB
ejpam-4576	41	19	open	open	ADJ
ejpam-4576	42	1	[	[	X
ejpam-4576	42	2	6	6	NUM
ejpam-4576	42	3	]	]	PUNCT
ejpam-4576	42	4	if	if	SCONJ
ejpam-4576	42	5	a	a	DET
ejpam-4576	42	6	⊆	⊆	NUM
ejpam-4576	42	7	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4576	42	8	)	)	PUNCT
ejpam-4576	42	9	)	)	PUNCT
ejpam-4576	42	10	)	)	PUNCT
ejpam-4576	42	11	.	.	PUNCT
ejpam-4576	43	1	the	the	DET
ejpam-4576	43	2	complement	complement	NOUN
ejpam-4576	43	3	of	of	ADP
ejpam-4576	43	4	a	a	DET
ejpam-4576	43	5	β	β	X
ejpam-4576	43	6	-	-	ADJ
ejpam-4576	43	7	open	open	ADJ
ejpam-4576	43	8	set	set	NOUN
ejpam-4576	43	9	is	be	AUX
ejpam-4576	43	10	called	call	VERB
ejpam-4576	43	11	β	β	NOUN
ejpam-4576	43	12	-	-	VERB
ejpam-4576	43	13	closed	closed	ADJ
ejpam-4576	43	14	.	.	PUNCT
ejpam-4576	44	1	the	the	DET
ejpam-4576	44	2	family	family	NOUN
ejpam-4576	44	3	of	of	ADP
ejpam-4576	44	4	all	all	DET
ejpam-4576	44	5	β	β	ADJ
ejpam-4576	44	6	-	-	ADJ
ejpam-4576	44	7	open	open	ADJ
ejpam-4576	44	8	sets	set	NOUN
ejpam-4576	44	9	of	of	ADP
ejpam-4576	44	10	a	a	DET
ejpam-4576	44	11	topological	topological	ADJ
ejpam-4576	44	12	space	space	NOUN
ejpam-4576	44	13	(	(	PUNCT
ejpam-4576	44	14	x	x	X
ejpam-4576	44	15	,	,	PUNCT
ejpam-4576	44	16	τ	τ	X
ejpam-4576	44	17	)	)	PUNCT
ejpam-4576	44	18	is	be	AUX
ejpam-4576	44	19	denoted	denote	VERB
ejpam-4576	44	20	by	by	ADP
ejpam-4576	44	21	β(x	β(x	PROPN
ejpam-4576	44	22	,	,	PUNCT
ejpam-4576	44	23	τ	τ	PROPN
ejpam-4576	44	24	)	)	PUNCT
ejpam-4576	44	25	.	.	PUNCT
ejpam-4576	45	1	a	a	DET
ejpam-4576	45	2	subset	subset	NOUN
ejpam-4576	45	3	λsp(a	λsp(a	NOUN
ejpam-4576	45	4	)	)	PUNCT
ejpam-4576	46	1	[	[	X
ejpam-4576	46	2	7	7	X
ejpam-4576	46	3	]	]	PUNCT
ejpam-4576	46	4	is	be	AUX
ejpam-4576	46	5	defined	define	VERB
ejpam-4576	46	6	as	as	SCONJ
ejpam-4576	46	7	follows	follow	VERB
ejpam-4576	46	8	:	:	PUNCT
ejpam-4576	46	9	λsp(a	λsp(a	NUM
ejpam-4576	46	10	)	)	PUNCT
ejpam-4576	46	11	=	=	PUNCT
ejpam-4576	47	1	∩{u	∩{u	PROPN
ejpam-4576	47	2	|	|	ADV
ejpam-4576	47	3	a	a	DET
ejpam-4576	47	4	⊆	⊆	NUM
ejpam-4576	47	5	u	u	NOUN
ejpam-4576	47	6	,	,	PUNCT
ejpam-4576	47	7	u	u	NOUN
ejpam-4576	47	8	∈	∈	PROPN
ejpam-4576	47	9	β(x	β(x	PROPN
ejpam-4576	47	10	,	,	PUNCT
ejpam-4576	47	11	τ	τ	X
ejpam-4576	47	12	)	)	PUNCT
ejpam-4576	47	13	}	}	PUNCT
ejpam-4576	47	14	.	.	PUNCT
ejpam-4576	48	1	a	a	DET
ejpam-4576	48	2	subset	subset	NOUN
ejpam-4576	48	3	a	a	PRON
ejpam-4576	48	4	of	of	ADP
ejpam-4576	48	5	a	a	DET
ejpam-4576	48	6	topological	topological	ADJ
ejpam-4576	48	7	space	space	NOUN
ejpam-4576	48	8	(	(	PUNCT
ejpam-4576	48	9	x	x	X
ejpam-4576	48	10	,	,	PUNCT
ejpam-4576	48	11	τ	τ	X
ejpam-4576	48	12	)	)	PUNCT
ejpam-4576	48	13	is	be	AUX
ejpam-4576	48	14	called	call	VERB
ejpam-4576	48	15	a	a	DET
ejpam-4576	48	16	λsp	λsp	NOUN
ejpam-4576	48	17	-	-	PUNCT
ejpam-4576	48	18	set	set	VERB
ejpam-4576	48	19	[	[	X
ejpam-4576	48	20	7	7	X
ejpam-4576	48	21	]	]	X
ejpam-4576	48	22	if	if	SCONJ
ejpam-4576	48	23	a	a	DET
ejpam-4576	48	24	=	=	NOUN
ejpam-4576	48	25	λsp(a	λsp(a	NOUN
ejpam-4576	48	26	)	)	PUNCT
ejpam-4576	48	27	.	.	PUNCT
ejpam-4576	49	1	a	a	DET
ejpam-4576	49	2	subset	subset	NOUN
ejpam-4576	49	3	a	a	PRON
ejpam-4576	49	4	of	of	ADP
ejpam-4576	49	5	a	a	DET
ejpam-4576	49	6	topological	topological	ADJ
ejpam-4576	49	7	space	space	NOUN
ejpam-4576	49	8	(	(	PUNCT
ejpam-4576	49	9	x	x	X
ejpam-4576	49	10	,	,	PUNCT
ejpam-4576	49	11	τ	τ	X
ejpam-4576	49	12	)	)	PUNCT
ejpam-4576	49	13	is	be	AUX
ejpam-4576	49	14	called	call	VERB
ejpam-4576	49	15	(	(	PUNCT
ejpam-4576	49	16	λ	λ	X
ejpam-4576	49	17	,	,	PUNCT
ejpam-4576	49	18	sp)-closed	sp)-close	VERB
ejpam-4576	49	19	[	[	PUNCT
ejpam-4576	49	20	3	3	X
ejpam-4576	49	21	]	]	X
ejpam-4576	49	22	if	if	SCONJ
ejpam-4576	49	23	a	a	DET
ejpam-4576	49	24	=	=	X
ejpam-4576	49	25	t	t	NOUN
ejpam-4576	49	26	∩c	∩c	NOUN
ejpam-4576	49	27	,	,	PUNCT
ejpam-4576	49	28	where	where	SCONJ
ejpam-4576	49	29	t	t	PROPN
ejpam-4576	49	30	is	be	AUX
ejpam-4576	49	31	a	a	DET
ejpam-4576	49	32	λsp	λsp	NOUN
ejpam-4576	49	33	-	-	PUNCT
ejpam-4576	49	34	set	set	VERB
ejpam-4576	49	35	and	and	CCONJ
ejpam-4576	49	36	c	c	NOUN
ejpam-4576	49	37	is	be	AUX
ejpam-4576	49	38	a	a	DET
ejpam-4576	49	39	β	β	NOUN
ejpam-4576	49	40	-	-	ADJ
ejpam-4576	49	41	closed	closed	ADJ
ejpam-4576	49	42	set	set	NOUN
ejpam-4576	49	43	.	.	PUNCT
ejpam-4576	50	1	the	the	DET
ejpam-4576	50	2	complement	complement	NOUN
ejpam-4576	50	3	of	of	ADP
ejpam-4576	50	4	a	a	DET
ejpam-4576	50	5	(	(	PUNCT
ejpam-4576	50	6	λ	λ	PROPN
ejpam-4576	50	7	,	,	PUNCT
ejpam-4576	50	8	sp)-closed	sp)-close	VERB
ejpam-4576	50	9	set	set	VERB
ejpam-4576	50	10	is	be	AUX
ejpam-4576	50	11	called	call	VERB
ejpam-4576	50	12	(	(	PUNCT
ejpam-4576	50	13	λ	λ	NOUN
ejpam-4576	50	14	,	,	PUNCT
ejpam-4576	50	15	sp)-open	sp)-open	NOUN
ejpam-4576	50	16	.	.	PUNCT
ejpam-4576	51	1	the	the	DET
ejpam-4576	51	2	family	family	NOUN
ejpam-4576	51	3	of	of	ADP
ejpam-4576	51	4	all	all	DET
ejpam-4576	51	5	(	(	PUNCT
ejpam-4576	51	6	λ	λ	NOUN
ejpam-4576	51	7	,	,	PUNCT
ejpam-4576	51	8	sp)-open	sp)-open	ADJ
ejpam-4576	51	9	sets	set	NOUN
ejpam-4576	51	10	in	in	ADP
ejpam-4576	51	11	a	a	DET
ejpam-4576	51	12	topological	topological	ADJ
ejpam-4576	51	13	space	space	NOUN
ejpam-4576	51	14	(	(	PUNCT
ejpam-4576	51	15	x	x	X
ejpam-4576	51	16	,	,	PUNCT
ejpam-4576	51	17	τ	τ	X
ejpam-4576	51	18	)	)	PUNCT
ejpam-4576	51	19	is	be	AUX
ejpam-4576	51	20	denoted	denote	VERB
ejpam-4576	51	21	by	by	ADP
ejpam-4576	51	22	λspo(x	λspo(x	PROPN
ejpam-4576	51	23	,	,	PUNCT
ejpam-4576	51	24	τ	τ	PROPN
ejpam-4576	51	25	)	)	PUNCT
ejpam-4576	51	26	.	.	PUNCT
ejpam-4576	52	1	let	let	VERB
ejpam-4576	52	2	a	a	DET
ejpam-4576	52	3	be	be	AUX
ejpam-4576	52	4	a	a	DET
ejpam-4576	52	5	subset	subset	NOUN
ejpam-4576	52	6	of	of	ADP
ejpam-4576	52	7	a	a	DET
ejpam-4576	52	8	topological	topological	ADJ
ejpam-4576	52	9	space	space	NOUN
ejpam-4576	52	10	(	(	PUNCT
ejpam-4576	52	11	x	x	X
ejpam-4576	52	12	,	,	PUNCT
ejpam-4576	52	13	τ	τ	PROPN
ejpam-4576	52	14	)	)	PUNCT
ejpam-4576	52	15	.	.	PUNCT
ejpam-4576	53	1	a	a	DET
ejpam-4576	53	2	point	point	NOUN
ejpam-4576	53	3	x	x	X
ejpam-4576	53	4	∈	∈	NOUN
ejpam-4576	53	5	x	x	PUNCT
ejpam-4576	53	6	is	be	AUX
ejpam-4576	53	7	called	call	VERB
ejpam-4576	53	8	a	a	DET
ejpam-4576	53	9	(	(	PUNCT
ejpam-4576	53	10	λ	λ	NOUN
ejpam-4576	53	11	,	,	PUNCT
ejpam-4576	53	12	sp)-cluster	sp)-cluster	NOUN
ejpam-4576	53	13	point	point	NOUN
ejpam-4576	53	14	[	[	X
ejpam-4576	53	15	3	3	X
ejpam-4576	53	16	]	]	PUNCT
ejpam-4576	53	17	of	of	ADP
ejpam-4576	53	18	a	a	PRON
ejpam-4576	53	19	if	if	SCONJ
ejpam-4576	53	20	a	a	DET
ejpam-4576	53	21	∩	∩	ADJ
ejpam-4576	53	22	u	u	ADJ
ejpam-4576	53	23	̸=	̸=	PROPN
ejpam-4576	53	24	∅	∅	NOUN
ejpam-4576	53	25	for	for	ADP
ejpam-4576	53	26	every	every	DET
ejpam-4576	53	27	(	(	PUNCT
ejpam-4576	53	28	λ	λ	NOUN
ejpam-4576	53	29	,	,	PUNCT
ejpam-4576	53	30	sp)-open	sp)-open	NOUN
ejpam-4576	53	31	set	set	VERB
ejpam-4576	53	32	u	u	NOUN
ejpam-4576	53	33	of	of	ADP
ejpam-4576	53	34	x	x	SYM
ejpam-4576	53	35	containing	contain	VERB
ejpam-4576	53	36	x.	x.	NOUN
ejpam-4576	53	37	the	the	DET
ejpam-4576	53	38	set	set	NOUN
ejpam-4576	53	39	of	of	ADP
ejpam-4576	53	40	all	all	DET
ejpam-4576	53	41	(	(	PUNCT
ejpam-4576	53	42	λ	λ	PROPN
ejpam-4576	53	43	,	,	PUNCT
ejpam-4576	53	44	sp)-cluster	sp)-cluster	NOUN
ejpam-4576	53	45	points	point	NOUN
ejpam-4576	53	46	of	of	ADP
ejpam-4576	53	47	a	a	PRON
ejpam-4576	53	48	is	be	AUX
ejpam-4576	53	49	called	call	VERB
ejpam-4576	53	50	the	the	DET
ejpam-4576	53	51	(	(	PUNCT
ejpam-4576	53	52	λ	λ	PROPN
ejpam-4576	53	53	,	,	PUNCT
ejpam-4576	53	54	sp)-closure	sp)-closure	NOUN
ejpam-4576	53	55	[	[	X
ejpam-4576	53	56	3	3	NUM
ejpam-4576	53	57	]	]	PUNCT
ejpam-4576	53	58	of	of	ADP
ejpam-4576	53	59	a	a	PRON
ejpam-4576	53	60	and	and	CCONJ
ejpam-4576	53	61	is	be	AUX
ejpam-4576	53	62	denoted	denote	VERB
ejpam-4576	53	63	by	by	ADP
ejpam-4576	53	64	a(λ	a(λ	ADV
ejpam-4576	53	65	,	,	PUNCT
ejpam-4576	53	66	sp	sp	NOUN
ejpam-4576	53	67	)	)	PUNCT
ejpam-4576	53	68	.	.	PUNCT
ejpam-4576	54	1	the	the	DET
ejpam-4576	54	2	union	union	NOUN
ejpam-4576	54	3	of	of	ADP
ejpam-4576	54	4	all	all	DET
ejpam-4576	54	5	(	(	PUNCT
ejpam-4576	54	6	λ	λ	NOUN
ejpam-4576	54	7	,	,	PUNCT
ejpam-4576	54	8	sp)-open	sp)-open	ADJ
ejpam-4576	54	9	sets	set	NOUN
ejpam-4576	54	10	contained	contain	VERB
ejpam-4576	54	11	in	in	ADP
ejpam-4576	54	12	a	a	PRON
ejpam-4576	54	13	is	be	AUX
ejpam-4576	54	14	called	call	VERB
ejpam-4576	54	15	the	the	DET
ejpam-4576	54	16	(	(	PUNCT
ejpam-4576	54	17	λ	λ	PROPN
ejpam-4576	54	18	,	,	PUNCT
ejpam-4576	54	19	sp)-interior	sp)-interior	NOUN
ejpam-4576	54	20	[	[	X
ejpam-4576	54	21	3	3	NUM
ejpam-4576	54	22	]	]	PUNCT
ejpam-4576	54	23	of	of	ADP
ejpam-4576	54	24	a	a	PRON
ejpam-4576	54	25	and	and	CCONJ
ejpam-4576	54	26	is	be	AUX
ejpam-4576	54	27	denoted	denote	VERB
ejpam-4576	54	28	by	by	ADP
ejpam-4576	54	29	a(λ	a(λ	ADV
ejpam-4576	54	30	,	,	PUNCT
ejpam-4576	54	31	sp	sp	NOUN
ejpam-4576	54	32	)	)	PUNCT
ejpam-4576	54	33	.	.	PUNCT
ejpam-4576	55	1	lemma	lemma	PROPN
ejpam-4576	55	2	1	1	NUM
ejpam-4576	55	3	.	.	PUNCT
ejpam-4576	56	1	[	[	X
ejpam-4576	56	2	3	3	X
ejpam-4576	56	3	]	]	PUNCT
ejpam-4576	56	4	let	let	VERB
ejpam-4576	56	5	a	a	PRON
ejpam-4576	56	6	and	and	CCONJ
ejpam-4576	56	7	b	b	NOUN
ejpam-4576	56	8	be	be	AUX
ejpam-4576	56	9	subsets	subset	NOUN
ejpam-4576	56	10	of	of	ADP
ejpam-4576	56	11	a	a	DET
ejpam-4576	56	12	topological	topological	ADJ
ejpam-4576	56	13	space	space	NOUN
ejpam-4576	56	14	(	(	PUNCT
ejpam-4576	56	15	x	x	X
ejpam-4576	56	16	,	,	PUNCT
ejpam-4576	56	17	τ	τ	PROPN
ejpam-4576	56	18	)	)	PUNCT
ejpam-4576	56	19	.	.	PUNCT
ejpam-4576	57	1	for	for	ADP
ejpam-4576	57	2	the	the	DET
ejpam-4576	57	3	(	(	PUNCT
ejpam-4576	57	4	λ	λ	PROPN
ejpam-4576	57	5	,	,	PUNCT
ejpam-4576	57	6	sp)-closure	sp)-closure	NOUN
ejpam-4576	57	7	,	,	PUNCT
ejpam-4576	57	8	the	the	DET
ejpam-4576	57	9	following	follow	VERB
ejpam-4576	57	10	properties	property	NOUN
ejpam-4576	57	11	hold	hold	VERB
ejpam-4576	57	12	:	:	PUNCT
ejpam-4576	57	13	(	(	PUNCT
ejpam-4576	57	14	1	1	X
ejpam-4576	57	15	)	)	PUNCT
ejpam-4576	57	16	a	a	DET
ejpam-4576	57	17	⊆	⊆	NUM
ejpam-4576	57	18	a(λ	a(λ	ADJ
ejpam-4576	57	19	,	,	PUNCT
ejpam-4576	57	20	sp	sp	NOUN
ejpam-4576	57	21	)	)	PUNCT
ejpam-4576	57	22	and	and	CCONJ
ejpam-4576	57	23	[	[	X
ejpam-4576	57	24	a(λ	a(λ	ADV
ejpam-4576	57	25	,	,	PUNCT
ejpam-4576	57	26	sp)](λ	sp)](λ	PROPN
ejpam-4576	57	27	,	,	PUNCT
ejpam-4576	57	28	sp	sp	NOUN
ejpam-4576	57	29	)	)	PUNCT
ejpam-4576	57	30	=	=	PUNCT
ejpam-4576	57	31	a(λ	a(λ	ADV
ejpam-4576	57	32	,	,	PUNCT
ejpam-4576	57	33	sp	sp	NOUN
ejpam-4576	57	34	)	)	PUNCT
ejpam-4576	57	35	.	.	PUNCT
ejpam-4576	58	1	(	(	PUNCT
ejpam-4576	58	2	2	2	X
ejpam-4576	58	3	)	)	PUNCT
ejpam-4576	58	4	if	if	SCONJ
ejpam-4576	58	5	a	a	DET
ejpam-4576	58	6	⊆	⊆	NUM
ejpam-4576	58	7	b	b	NOUN
ejpam-4576	58	8	,	,	PUNCT
ejpam-4576	58	9	then	then	ADV
ejpam-4576	58	10	a(λ	a(λ	ADV
ejpam-4576	58	11	,	,	PUNCT
ejpam-4576	58	12	sp	sp	NOUN
ejpam-4576	58	13	)	)	PUNCT
ejpam-4576	58	14	⊆	⊆	NUM
ejpam-4576	58	15	b(λ	b(λ	NOUN
ejpam-4576	58	16	,	,	PUNCT
ejpam-4576	58	17	sp	sp	NOUN
ejpam-4576	58	18	)	)	PUNCT
ejpam-4576	58	19	.	.	PUNCT
ejpam-4576	59	1	(	(	PUNCT
ejpam-4576	59	2	3	3	X
ejpam-4576	59	3	)	)	PUNCT
ejpam-4576	59	4	a(λ	a(λ	ADV
ejpam-4576	59	5	,	,	PUNCT
ejpam-4576	59	6	sp	sp	NOUN
ejpam-4576	59	7	)	)	PUNCT
ejpam-4576	59	8	is	be	AUX
ejpam-4576	59	9	(	(	PUNCT
ejpam-4576	59	10	λ	λ	X
ejpam-4576	59	11	,	,	PUNCT
ejpam-4576	59	12	sp)-closed	sp)-close	VERB
ejpam-4576	59	13	.	.	PUNCT
ejpam-4576	60	1	(	(	PUNCT
ejpam-4576	60	2	4	4	X
ejpam-4576	60	3	)	)	PUNCT
ejpam-4576	60	4	a	a	PRON
ejpam-4576	60	5	is	be	AUX
ejpam-4576	60	6	(	(	PUNCT
ejpam-4576	60	7	λ	λ	X
ejpam-4576	60	8	,	,	PUNCT
ejpam-4576	60	9	sp)-closed	sp)-close	VERB
ejpam-4576	60	10	if	if	SCONJ
ejpam-4576	60	11	and	and	CCONJ
ejpam-4576	60	12	only	only	ADV
ejpam-4576	60	13	if	if	SCONJ
ejpam-4576	60	14	a	a	DET
ejpam-4576	60	15	=	=	X
ejpam-4576	60	16	a(λ	a(λ	ADV
ejpam-4576	60	17	,	,	PUNCT
ejpam-4576	60	18	sp	sp	NOUN
ejpam-4576	60	19	)	)	PUNCT
ejpam-4576	60	20	.	.	PUNCT
ejpam-4576	61	1	lemma	lemma	PROPN
ejpam-4576	61	2	2	2	NUM
ejpam-4576	61	3	.	.	PUNCT
ejpam-4576	62	1	[	[	X
ejpam-4576	62	2	3	3	X
ejpam-4576	62	3	]	]	PUNCT
ejpam-4576	62	4	for	for	ADP
ejpam-4576	62	5	subsets	subset	NOUN
ejpam-4576	62	6	a	a	PRON
ejpam-4576	62	7	and	and	CCONJ
ejpam-4576	62	8	b	b	NOUN
ejpam-4576	62	9	of	of	ADP
ejpam-4576	62	10	a	a	DET
ejpam-4576	62	11	topological	topological	ADJ
ejpam-4576	62	12	space	space	NOUN
ejpam-4576	62	13	(	(	PUNCT
ejpam-4576	62	14	x	x	X
ejpam-4576	62	15	,	,	PUNCT
ejpam-4576	62	16	τ	τ	PROPN
ejpam-4576	62	17	)	)	PUNCT
ejpam-4576	62	18	,	,	PUNCT
ejpam-4576	62	19	the	the	DET
ejpam-4576	62	20	following	follow	VERB
ejpam-4576	62	21	properties	property	NOUN
ejpam-4576	62	22	hold	hold	VERB
ejpam-4576	62	23	:	:	PUNCT
ejpam-4576	62	24	c.	c.	PROPN
ejpam-4576	62	25	boonpok	boonpok	PROPN
ejpam-4576	62	26	,	,	PUNCT
ejpam-4576	62	27	n.	n.	PROPN
ejpam-4576	62	28	viriyapong	viriyapong	PROPN
ejpam-4576	62	29	/	/	SYM
ejpam-4576	62	30	eur	eur	PROPN
ejpam-4576	62	31	.	.	PUNCT
ejpam-4576	63	1	j.	j.	PROPN
ejpam-4576	63	2	pure	pure	PROPN
ejpam-4576	63	3	appl	appl	PROPN
ejpam-4576	63	4	.	.	PROPN
ejpam-4576	63	5	math	math	PROPN
ejpam-4576	63	6	,	,	PUNCT
ejpam-4576	63	7	16	16	NUM
ejpam-4576	63	8	(	(	PUNCT
ejpam-4576	63	9	1	1	NUM
ejpam-4576	63	10	)	)	PUNCT
ejpam-4576	63	11	(	(	PUNCT
ejpam-4576	63	12	2023	2023	NUM
ejpam-4576	63	13	)	)	PUNCT
ejpam-4576	63	14	,	,	PUNCT
ejpam-4576	63	15	84	84	NUM
ejpam-4576	63	16	-	-	SYM
ejpam-4576	63	17	96	96	NUM
ejpam-4576	63	18	86	86	NUM
ejpam-4576	63	19	(	(	PUNCT
ejpam-4576	63	20	1	1	NUM
ejpam-4576	63	21	)	)	PUNCT
ejpam-4576	63	22	a(λ	a(λ	ADV
ejpam-4576	63	23	,	,	PUNCT
ejpam-4576	63	24	sp	sp	NOUN
ejpam-4576	63	25	)	)	PUNCT
ejpam-4576	63	26	⊆	⊆	NUM
ejpam-4576	63	27	a	a	DET
ejpam-4576	63	28	and	and	CCONJ
ejpam-4576	63	29	[	[	X
ejpam-4576	63	30	a(λ	a(λ	ADV
ejpam-4576	63	31	,	,	PUNCT
ejpam-4576	63	32	sp)](λ	sp)](λ	PROPN
ejpam-4576	63	33	,	,	PUNCT
ejpam-4576	63	34	sp	sp	NOUN
ejpam-4576	63	35	)	)	PUNCT
ejpam-4576	63	36	=	=	PUNCT
ejpam-4576	64	1	a(λ	a(λ	ADV
ejpam-4576	64	2	,	,	PUNCT
ejpam-4576	64	3	sp	sp	NOUN
ejpam-4576	64	4	)	)	PUNCT
ejpam-4576	64	5	.	.	PUNCT
ejpam-4576	65	1	(	(	PUNCT
ejpam-4576	65	2	2	2	X
ejpam-4576	65	3	)	)	PUNCT
ejpam-4576	65	4	if	if	SCONJ
ejpam-4576	65	5	a	a	DET
ejpam-4576	65	6	⊆	⊆	NUM
ejpam-4576	65	7	b	b	NOUN
ejpam-4576	65	8	,	,	PUNCT
ejpam-4576	65	9	then	then	ADV
ejpam-4576	65	10	a(λ	a(λ	ADV
ejpam-4576	65	11	,	,	PUNCT
ejpam-4576	65	12	sp	sp	NOUN
ejpam-4576	65	13	)	)	PUNCT
ejpam-4576	65	14	⊆	⊆	NUM
ejpam-4576	65	15	b(λ	b(λ	NOUN
ejpam-4576	65	16	,	,	PUNCT
ejpam-4576	65	17	sp	sp	NOUN
ejpam-4576	65	18	)	)	PUNCT
ejpam-4576	65	19	.	.	PUNCT
ejpam-4576	66	1	(	(	PUNCT
ejpam-4576	66	2	3	3	X
ejpam-4576	66	3	)	)	PUNCT
ejpam-4576	66	4	a(λ	a(λ	ADV
ejpam-4576	66	5	,	,	PUNCT
ejpam-4576	66	6	sp	sp	NOUN
ejpam-4576	66	7	)	)	PUNCT
ejpam-4576	66	8	is	be	AUX
ejpam-4576	66	9	(	(	PUNCT
ejpam-4576	66	10	λ	λ	INTJ
ejpam-4576	66	11	,	,	PUNCT
ejpam-4576	66	12	sp)-open	sp)-open	NOUN
ejpam-4576	66	13	.	.	PUNCT
ejpam-4576	67	1	(	(	PUNCT
ejpam-4576	67	2	4	4	X
ejpam-4576	67	3	)	)	PUNCT
ejpam-4576	67	4	a	a	DET
ejpam-4576	67	5	is	be	AUX
ejpam-4576	67	6	(	(	PUNCT
ejpam-4576	67	7	λ	λ	NOUN
ejpam-4576	67	8	,	,	PUNCT
ejpam-4576	67	9	sp)-open	sp)-open	ADJ
ejpam-4576	67	10	if	if	SCONJ
ejpam-4576	67	11	and	and	CCONJ
ejpam-4576	67	12	only	only	ADV
ejpam-4576	67	13	if	if	SCONJ
ejpam-4576	67	14	a(λ	a(λ	ADV
ejpam-4576	67	15	,	,	PUNCT
ejpam-4576	67	16	sp	sp	NOUN
ejpam-4576	67	17	)	)	PUNCT
ejpam-4576	67	18	=	=	SYM
ejpam-4576	67	19	a.	a.	NOUN
ejpam-4576	67	20	(	(	PUNCT
ejpam-4576	67	21	5	5	NUM
ejpam-4576	67	22	)	)	PUNCT
ejpam-4576	68	1	[	[	X
ejpam-4576	68	2	x	x	X
ejpam-4576	68	3	−a](λ	−a](λ	PROPN
ejpam-4576	68	4	,	,	PUNCT
ejpam-4576	68	5	sp	sp	NOUN
ejpam-4576	68	6	)	)	PUNCT
ejpam-4576	68	7	=	=	SYM
ejpam-4576	68	8	x	x	SYM
ejpam-4576	68	9	−a(λ	−a(λ	NOUN
ejpam-4576	68	10	,	,	PUNCT
ejpam-4576	68	11	sp	sp	NOUN
ejpam-4576	68	12	)	)	PUNCT
ejpam-4576	68	13	.	.	PUNCT
ejpam-4576	69	1	(	(	PUNCT
ejpam-4576	69	2	6	6	NUM
ejpam-4576	69	3	)	)	PUNCT
ejpam-4576	70	1	[	[	X
ejpam-4576	70	2	x	x	X
ejpam-4576	70	3	−a](λ	−a](λ	PROPN
ejpam-4576	70	4	,	,	PUNCT
ejpam-4576	70	5	sp	sp	NOUN
ejpam-4576	70	6	)	)	PUNCT
ejpam-4576	70	7	=	=	SYM
ejpam-4576	70	8	x	x	SYM
ejpam-4576	70	9	−a(λ	−a(λ	NOUN
ejpam-4576	70	10	,	,	PUNCT
ejpam-4576	70	11	sp	sp	NOUN
ejpam-4576	70	12	)	)	PUNCT
ejpam-4576	70	13	.	.	PUNCT
ejpam-4576	71	1	a	a	DET
ejpam-4576	71	2	subset	subset	NOUN
ejpam-4576	71	3	a	a	PRON
ejpam-4576	71	4	of	of	ADP
ejpam-4576	71	5	a	a	DET
ejpam-4576	71	6	topological	topological	ADJ
ejpam-4576	71	7	space	space	NOUN
ejpam-4576	71	8	(	(	PUNCT
ejpam-4576	71	9	x	x	X
ejpam-4576	71	10	,	,	PUNCT
ejpam-4576	71	11	τ	τ	X
ejpam-4576	71	12	)	)	PUNCT
ejpam-4576	71	13	is	be	AUX
ejpam-4576	71	14	said	say	VERB
ejpam-4576	71	15	to	to	PART
ejpam-4576	71	16	be	be	AUX
ejpam-4576	71	17	s(λ	s(λ	NOUN
ejpam-4576	71	18	,	,	PUNCT
ejpam-4576	71	19	sp)-open	sp)-open	ADJ
ejpam-4576	71	20	(	(	PUNCT
ejpam-4576	71	21	resp	resp	NOUN
ejpam-4576	71	22	.	.	PUNCT
ejpam-4576	72	1	p(λ	p(λ	NOUN
ejpam-4576	72	2	,	,	PUNCT
ejpam-4576	72	3	sp)open	sp)open	VERB
ejpam-4576	72	4	,	,	PUNCT
ejpam-4576	72	5	β(λ	β(λ	X
ejpam-4576	72	6	,	,	PUNCT
ejpam-4576	72	7	sp)-open	sp)-open	NOUN
ejpam-4576	72	8	,	,	PUNCT
ejpam-4576	72	9	r(λ	r(λ	NOUN
ejpam-4576	72	10	,	,	PUNCT
ejpam-4576	72	11	sp)-open	sp)-open	NOUN
ejpam-4576	72	12	)	)	PUNCT
ejpam-4576	72	13	if	if	SCONJ
ejpam-4576	72	14	a	a	DET
ejpam-4576	72	15	⊆	⊆	NUM
ejpam-4576	72	16	[	[	X
ejpam-4576	72	17	a(λ	a(λ	ADV
ejpam-4576	72	18	,	,	PUNCT
ejpam-4576	72	19	sp	sp	NOUN
ejpam-4576	72	20	)	)	PUNCT
ejpam-4576	72	21	]	]	PUNCT
ejpam-4576	72	22	(	(	PUNCT
ejpam-4576	72	23	λ	λ	NOUN
ejpam-4576	72	24	,	,	PUNCT
ejpam-4576	72	25	sp	sp	NOUN
ejpam-4576	72	26	)	)	PUNCT
ejpam-4576	72	27	(	(	PUNCT
ejpam-4576	72	28	resp	resp	NOUN
ejpam-4576	72	29	.	.	PUNCT
ejpam-4576	73	1	a	a	DET
ejpam-4576	73	2	⊆	⊆	NUM
ejpam-4576	73	3	[	[	X
ejpam-4576	73	4	a(λ	a(λ	ADJ
ejpam-4576	73	5	,	,	PUNCT
ejpam-4576	73	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	73	7	,	,	PUNCT
ejpam-4576	73	8	sp	sp	NOUN
ejpam-4576	73	9	)	)	PUNCT
ejpam-4576	73	10	,	,	PUNCT
ejpam-4576	73	11	a	a	PRON
ejpam-4576	73	12	⊆	⊆	NUM
ejpam-4576	74	1	[	[	X
ejpam-4576	74	2	[	[	X
ejpam-4576	74	3	a(λ	a(λ	ADJ
ejpam-4576	74	4	,	,	PUNCT
ejpam-4576	74	5	sp)](λ	sp)](λ	PROPN
ejpam-4576	74	6	,	,	PUNCT
ejpam-4576	74	7	sp	sp	NOUN
ejpam-4576	74	8	)	)	PUNCT
ejpam-4576	74	9	]	]	PUNCT
ejpam-4576	75	1	(	(	PUNCT
ejpam-4576	75	2	λ	λ	NOUN
ejpam-4576	75	3	,	,	PUNCT
ejpam-4576	75	4	sp	sp	NOUN
ejpam-4576	75	5	)	)	PUNCT
ejpam-4576	75	6	,	,	PUNCT
ejpam-4576	75	7	a	a	PRON
ejpam-4576	75	8	=	=	X
ejpam-4576	76	1	[	[	X
ejpam-4576	76	2	a(λ	a(λ	PROPN
ejpam-4576	76	3	,	,	PUNCT
ejpam-4576	76	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	76	5	,	,	PUNCT
ejpam-4576	76	6	sp	sp	NOUN
ejpam-4576	76	7	)	)	PUNCT
ejpam-4576	76	8	)	)	PUNCT
ejpam-4576	77	1	[	[	X
ejpam-4576	77	2	3	3	NUM
ejpam-4576	77	3	]	]	PUNCT
ejpam-4576	77	4	.	.	PUNCT
ejpam-4576	78	1	the	the	DET
ejpam-4576	78	2	complement	complement	NOUN
ejpam-4576	78	3	of	of	ADP
ejpam-4576	78	4	a	a	DET
ejpam-4576	78	5	s(λ	s(λ	PROPN
ejpam-4576	78	6	,	,	PUNCT
ejpam-4576	78	7	sp)-open	sp)-open	ADJ
ejpam-4576	78	8	(	(	PUNCT
ejpam-4576	78	9	resp	resp	NOUN
ejpam-4576	78	10	.	.	PUNCT
ejpam-4576	79	1	p(λ	p(λ	NOUN
ejpam-4576	79	2	,	,	PUNCT
ejpam-4576	79	3	sp)-open	sp)-open	NOUN
ejpam-4576	79	4	,	,	PUNCT
ejpam-4576	79	5	β(λ	β(λ	X
ejpam-4576	79	6	,	,	PUNCT
ejpam-4576	79	7	sp)-open	sp)-open	NOUN
ejpam-4576	79	8	,	,	PUNCT
ejpam-4576	79	9	r(λ	r(λ	NOUN
ejpam-4576	79	10	,	,	PUNCT
ejpam-4576	79	11	sp)-open	sp)-open	NOUN
ejpam-4576	79	12	)	)	PUNCT
ejpam-4576	79	13	set	set	NOUN
ejpam-4576	79	14	is	be	AUX
ejpam-4576	79	15	said	say	VERB
ejpam-4576	79	16	to	to	PART
ejpam-4576	79	17	be	be	AUX
ejpam-4576	79	18	s(λ	s(λ	PROPN
ejpam-4576	79	19	,	,	PUNCT
ejpam-4576	79	20	sp)-closed	sp)-close	VERB
ejpam-4576	79	21	(	(	PUNCT
ejpam-4576	79	22	resp	resp	NOUN
ejpam-4576	79	23	.	.	PUNCT
ejpam-4576	80	1	p(λ	p(λ	NOUN
ejpam-4576	80	2	,	,	PUNCT
ejpam-4576	80	3	sp)closed	sp)close	VERB
ejpam-4576	80	4	,	,	PUNCT
ejpam-4576	80	5	β(λ	β(λ	X
ejpam-4576	80	6	,	,	PUNCT
ejpam-4576	80	7	sp)-closed	sp)-close	VERB
ejpam-4576	80	8	,	,	PUNCT
ejpam-4576	80	9	r(λ	r(λ	NOUN
ejpam-4576	80	10	,	,	PUNCT
ejpam-4576	80	11	sp)-closed	sp)-close	VERB
ejpam-4576	80	12	)	)	PUNCT
ejpam-4576	80	13	.	.	PUNCT
ejpam-4576	81	1	the	the	DET
ejpam-4576	81	2	family	family	NOUN
ejpam-4576	81	3	of	of	ADP
ejpam-4576	81	4	all	all	DET
ejpam-4576	81	5	s(λ	s(λ	NOUN
ejpam-4576	81	6	,	,	PUNCT
ejpam-4576	81	7	sp)-open	sp)-open	ADJ
ejpam-4576	81	8	(	(	PUNCT
ejpam-4576	81	9	resp	resp	NOUN
ejpam-4576	81	10	.	.	PUNCT
ejpam-4576	82	1	p(λ	p(λ	NOUN
ejpam-4576	82	2	,	,	PUNCT
ejpam-4576	82	3	sp)-open	sp)-open	NOUN
ejpam-4576	82	4	,	,	PUNCT
ejpam-4576	82	5	β(λ	β(λ	X
ejpam-4576	82	6	,	,	PUNCT
ejpam-4576	82	7	sp)-open	sp)-open	NOUN
ejpam-4576	82	8	,	,	PUNCT
ejpam-4576	82	9	r(λ	r(λ	NOUN
ejpam-4576	82	10	,	,	PUNCT
ejpam-4576	82	11	sp)-open	sp)-open	NOUN
ejpam-4576	82	12	)	)	PUNCT
ejpam-4576	82	13	sets	set	NOUN
ejpam-4576	82	14	in	in	ADP
ejpam-4576	82	15	a	a	DET
ejpam-4576	82	16	topological	topological	ADJ
ejpam-4576	82	17	space	space	NOUN
ejpam-4576	82	18	(	(	PUNCT
ejpam-4576	82	19	x	x	X
ejpam-4576	82	20	,	,	PUNCT
ejpam-4576	82	21	τ	τ	X
ejpam-4576	82	22	)	)	PUNCT
ejpam-4576	82	23	is	be	AUX
ejpam-4576	82	24	denoted	denote	VERB
ejpam-4576	82	25	by	by	ADP
ejpam-4576	82	26	sλspo(x	sλspo(x	PROPN
ejpam-4576	82	27	,	,	PUNCT
ejpam-4576	82	28	τ	τ	PROPN
ejpam-4576	82	29	)	)	PUNCT
ejpam-4576	82	30	(	(	PUNCT
ejpam-4576	82	31	resp	resp	NOUN
ejpam-4576	82	32	.	.	PUNCT
ejpam-4576	83	1	pλspo(x	pλspo(x	ADJ
ejpam-4576	83	2	,	,	PUNCT
ejpam-4576	83	3	τ	τ	PROPN
ejpam-4576	83	4	)	)	PUNCT
ejpam-4576	83	5	,	,	PUNCT
ejpam-4576	83	6	βλspo(x	βλspo(x	PROPN
ejpam-4576	83	7	,	,	PUNCT
ejpam-4576	83	8	τ	τ	PROPN
ejpam-4576	83	9	)	)	PUNCT
ejpam-4576	83	10	,	,	PUNCT
ejpam-4576	83	11	rλspo(x	rλspo(x	PROPN
ejpam-4576	83	12	,	,	PUNCT
ejpam-4576	83	13	τ	τ	PROPN
ejpam-4576	83	14	)	)	PUNCT
ejpam-4576	83	15	)	)	PUNCT
ejpam-4576	83	16	.	.	PUNCT
ejpam-4576	84	1	let	let	VERB
ejpam-4576	84	2	a	a	DET
ejpam-4576	84	3	be	be	AUX
ejpam-4576	84	4	a	a	DET
ejpam-4576	84	5	subset	subset	NOUN
ejpam-4576	84	6	of	of	ADP
ejpam-4576	84	7	a	a	DET
ejpam-4576	84	8	topological	topological	ADJ
ejpam-4576	84	9	space	space	NOUN
ejpam-4576	84	10	(	(	PUNCT
ejpam-4576	84	11	x	x	X
ejpam-4576	84	12	,	,	PUNCT
ejpam-4576	84	13	τ	τ	PROPN
ejpam-4576	84	14	)	)	PUNCT
ejpam-4576	84	15	.	.	PUNCT
ejpam-4576	85	1	the	the	DET
ejpam-4576	85	2	intersection	intersection	NOUN
ejpam-4576	85	3	of	of	ADP
ejpam-4576	85	4	all	all	DET
ejpam-4576	85	5	s(λ	s(λ	PROPN
ejpam-4576	85	6	,	,	PUNCT
ejpam-4576	85	7	sp)-closed	sp)-close	VERB
ejpam-4576	85	8	sets	set	NOUN
ejpam-4576	85	9	containing	contain	VERB
ejpam-4576	85	10	a	a	PRON
ejpam-4576	85	11	is	be	AUX
ejpam-4576	85	12	called	call	VERB
ejpam-4576	85	13	the	the	DET
ejpam-4576	85	14	s(λ	s(λ	PROPN
ejpam-4576	85	15	,	,	PUNCT
ejpam-4576	85	16	sp)-closure	sp)-closure	NOUN
ejpam-4576	85	17	of	of	ADP
ejpam-4576	85	18	a	a	PRON
ejpam-4576	85	19	and	and	CCONJ
ejpam-4576	85	20	is	be	AUX
ejpam-4576	85	21	denoted	denote	VERB
ejpam-4576	85	22	by	by	ADP
ejpam-4576	85	23	as(λ	as(λ	NUM
ejpam-4576	85	24	,	,	PUNCT
ejpam-4576	85	25	sp	sp	NOUN
ejpam-4576	85	26	)	)	PUNCT
ejpam-4576	85	27	.	.	PUNCT
ejpam-4576	86	1	the	the	DET
ejpam-4576	86	2	union	union	NOUN
ejpam-4576	86	3	of	of	ADP
ejpam-4576	86	4	all	all	DET
ejpam-4576	86	5	s(λ	s(λ	NOUN
ejpam-4576	86	6	,	,	PUNCT
ejpam-4576	86	7	sp)-open	sp)-open	ADJ
ejpam-4576	86	8	sets	set	NOUN
ejpam-4576	86	9	contained	contain	VERB
ejpam-4576	86	10	in	in	ADP
ejpam-4576	86	11	a	a	PRON
ejpam-4576	86	12	is	be	AUX
ejpam-4576	86	13	called	call	VERB
ejpam-4576	86	14	the	the	DET
ejpam-4576	86	15	s(λ	s(λ	PROPN
ejpam-4576	86	16	,	,	PUNCT
ejpam-4576	86	17	sp)-interior	sp)-interior	NOUN
ejpam-4576	86	18	of	of	ADP
ejpam-4576	86	19	a	a	PRON
ejpam-4576	86	20	and	and	CCONJ
ejpam-4576	86	21	is	be	AUX
ejpam-4576	86	22	denoted	denote	VERB
ejpam-4576	86	23	by	by	ADP
ejpam-4576	86	24	as(λ	as(λ	NUM
ejpam-4576	86	25	,	,	PUNCT
ejpam-4576	86	26	sp	sp	NOUN
ejpam-4576	86	27	)	)	PUNCT
ejpam-4576	86	28	.	.	PUNCT
ejpam-4576	87	1	by	by	ADP
ejpam-4576	87	2	a	a	DET
ejpam-4576	87	3	multifunction	multifunction	NOUN
ejpam-4576	87	4	f	f	NOUN
ejpam-4576	87	5	:	:	PUNCT
ejpam-4576	87	6	x	x	X
ejpam-4576	87	7	→	→	SYM
ejpam-4576	87	8	y	y	PROPN
ejpam-4576	87	9	,	,	PUNCT
ejpam-4576	87	10	we	we	PRON
ejpam-4576	87	11	mean	mean	VERB
ejpam-4576	87	12	a	a	DET
ejpam-4576	87	13	point	point	NOUN
ejpam-4576	87	14	-	-	PUNCT
ejpam-4576	87	15	to	to	ADP
ejpam-4576	87	16	-	-	PUNCT
ejpam-4576	87	17	set	set	VERB
ejpam-4576	87	18	correspondence	correspondence	NOUN
ejpam-4576	87	19	from	from	ADP
ejpam-4576	87	20	x	x	PUNCT
ejpam-4576	87	21	into	into	ADP
ejpam-4576	87	22	y	y	PROPN
ejpam-4576	87	23	,	,	PUNCT
ejpam-4576	87	24	and	and	CCONJ
ejpam-4576	87	25	always	always	ADV
ejpam-4576	87	26	assume	assume	VERB
ejpam-4576	87	27	that	that	SCONJ
ejpam-4576	88	1	f	f	PROPN
ejpam-4576	88	2	(	(	PUNCT
ejpam-4576	88	3	x	x	X
ejpam-4576	88	4	)	)	PUNCT
ejpam-4576	88	5	̸=	̸=	NOUN
ejpam-4576	88	6	∅	∅	NOUN
ejpam-4576	88	7	for	for	ADP
ejpam-4576	88	8	all	all	PRON
ejpam-4576	88	9	x	x	SYM
ejpam-4576	88	10	∈	∈	ADJ
ejpam-4576	88	11	x.	x.	NOUN
ejpam-4576	88	12	for	for	ADP
ejpam-4576	88	13	a	a	DET
ejpam-4576	88	14	multifunction	multifunction	NOUN
ejpam-4576	88	15	f	f	NOUN
ejpam-4576	88	16	:	:	PUNCT
ejpam-4576	88	17	x	x	X
ejpam-4576	88	18	→	→	SYM
ejpam-4576	88	19	y	y	PROPN
ejpam-4576	88	20	,	,	PUNCT
ejpam-4576	88	21	following	follow	VERB
ejpam-4576	88	22	[	[	X
ejpam-4576	88	23	2	2	X
ejpam-4576	88	24	]	]	PUNCT
ejpam-4576	88	25	we	we	PRON
ejpam-4576	88	26	shall	shall	AUX
ejpam-4576	88	27	denote	denote	VERB
ejpam-4576	88	28	the	the	DET
ejpam-4576	88	29	upper	upper	ADJ
ejpam-4576	88	30	and	and	CCONJ
ejpam-4576	88	31	lower	low	ADJ
ejpam-4576	88	32	inverse	inverse	NOUN
ejpam-4576	88	33	of	of	ADP
ejpam-4576	88	34	a	a	DET
ejpam-4576	88	35	set	set	NOUN
ejpam-4576	88	36	b	b	PROPN
ejpam-4576	88	37	of	of	ADP
ejpam-4576	88	38	y	y	PROPN
ejpam-4576	88	39	by	by	ADP
ejpam-4576	88	40	f+(b	f+(b	NOUN
ejpam-4576	88	41	)	)	PUNCT
ejpam-4576	88	42	and	and	CCONJ
ejpam-4576	88	43	f−(b	f−(b	NOUN
ejpam-4576	88	44	)	)	PUNCT
ejpam-4576	88	45	,	,	PUNCT
ejpam-4576	88	46	respectively	respectively	ADV
ejpam-4576	88	47	,	,	PUNCT
ejpam-4576	88	48	that	that	ADV
ejpam-4576	88	49	is	is	ADV
ejpam-4576	88	50	,	,	PUNCT
ejpam-4576	88	51	f+(b	f+(b	NOUN
ejpam-4576	88	52	)	)	PUNCT
ejpam-4576	88	53	=	=	PRON
ejpam-4576	89	1	{	{	PUNCT
ejpam-4576	89	2	x	x	PUNCT
ejpam-4576	89	3	∈	∈	PROPN
ejpam-4576	89	4	x	x	INTJ
ejpam-4576	90	1	|	|	NOUN
ejpam-4576	90	2	f	f	X
ejpam-4576	90	3	(	(	PUNCT
ejpam-4576	90	4	x	x	NOUN
ejpam-4576	90	5	)	)	PUNCT
ejpam-4576	90	6	⊆	⊆	NUM
ejpam-4576	90	7	b	b	NOUN
ejpam-4576	90	8	}	}	PUNCT
ejpam-4576	90	9	and	and	CCONJ
ejpam-4576	90	10	f−(b	f−(b	PROPN
ejpam-4576	90	11	)	)	PUNCT
ejpam-4576	90	12	=	=	PRON
ejpam-4576	91	1	{	{	PUNCT
ejpam-4576	91	2	x	x	PUNCT
ejpam-4576	91	3	∈	∈	PROPN
ejpam-4576	91	4	x	x	INTJ
ejpam-4576	92	1	|	|	NOUN
ejpam-4576	92	2	f	f	X
ejpam-4576	92	3	(	(	PUNCT
ejpam-4576	92	4	x)∩b	x)∩b	PROPN
ejpam-4576	92	5	̸=	̸=	PROPN
ejpam-4576	92	6	∅	∅	NOUN
ejpam-4576	92	7	}	}	PUNCT
ejpam-4576	92	8	.	.	PUNCT
ejpam-4576	93	1	in	in	ADP
ejpam-4576	93	2	particular	particular	ADJ
ejpam-4576	93	3	,	,	PUNCT
ejpam-4576	93	4	f−(y	f−(y	NOUN
ejpam-4576	93	5	)	)	PUNCT
ejpam-4576	93	6	=	=	SYM
ejpam-4576	94	1	{	{	PUNCT
ejpam-4576	94	2	x	x	PUNCT
ejpam-4576	94	3	∈	∈	PROPN
ejpam-4576	94	4	x	x	INTJ
ejpam-4576	95	1	|	|	ADV
ejpam-4576	95	2	y	y	PROPN
ejpam-4576	95	3	∈	∈	PROPN
ejpam-4576	95	4	f	f	X
ejpam-4576	95	5	(	(	PUNCT
ejpam-4576	95	6	x	x	NOUN
ejpam-4576	95	7	)	)	PUNCT
ejpam-4576	95	8	}	}	PUNCT
ejpam-4576	95	9	for	for	ADP
ejpam-4576	95	10	each	each	DET
ejpam-4576	95	11	point	point	NOUN
ejpam-4576	95	12	y	y	PROPN
ejpam-4576	95	13	∈	∈	PROPN
ejpam-4576	95	14	y	y	PROPN
ejpam-4576	95	15	.	.	PUNCT
ejpam-4576	96	1	for	for	ADP
ejpam-4576	96	2	each	each	PRON
ejpam-4576	96	3	a	a	DET
ejpam-4576	96	4	⊆	⊆	NUM
ejpam-4576	96	5	x	x	SYM
ejpam-4576	96	6	,	,	PUNCT
ejpam-4576	96	7	f	f	PROPN
ejpam-4576	96	8	(	(	PUNCT
ejpam-4576	96	9	a	a	NOUN
ejpam-4576	96	10	)	)	PUNCT
ejpam-4576	96	11	=	=	SYM
ejpam-4576	96	12	∪x∈af	∪x∈af	NOUN
ejpam-4576	96	13	(	(	PUNCT
ejpam-4576	96	14	x	x	NOUN
ejpam-4576	96	15	)	)	PUNCT
ejpam-4576	96	16	.	.	PUNCT
ejpam-4576	97	1	then	then	ADV
ejpam-4576	97	2	,	,	PUNCT
ejpam-4576	97	3	f	f	PROPN
ejpam-4576	97	4	is	be	AUX
ejpam-4576	97	5	said	say	VERB
ejpam-4576	97	6	to	to	PART
ejpam-4576	97	7	be	be	AUX
ejpam-4576	97	8	a	a	DET
ejpam-4576	97	9	surjection	surjection	NOUN
ejpam-4576	97	10	if	if	SCONJ
ejpam-4576	97	11	f	f	PROPN
ejpam-4576	97	12	(	(	PUNCT
ejpam-4576	97	13	x	x	X
ejpam-4576	97	14	)	)	PUNCT
ejpam-4576	97	15	=	=	SYM
ejpam-4576	97	16	y	y	PROPN
ejpam-4576	97	17	,	,	PUNCT
ejpam-4576	97	18	or	or	CCONJ
ejpam-4576	97	19	equivalently	equivalently	ADV
ejpam-4576	97	20	,	,	PUNCT
ejpam-4576	97	21	if	if	SCONJ
ejpam-4576	97	22	for	for	ADP
ejpam-4576	97	23	each	each	DET
ejpam-4576	97	24	y	y	PROPN
ejpam-4576	97	25	∈	∈	PROPN
ejpam-4576	97	26	y	y	PROPN
ejpam-4576	97	27	,	,	PUNCT
ejpam-4576	97	28	there	there	PRON
ejpam-4576	97	29	exists	exist	VERB
ejpam-4576	97	30	x	x	X
ejpam-4576	97	31	∈	∈	PROPN
ejpam-4576	97	32	x	x	X
ejpam-4576	97	33	such	such	ADJ
ejpam-4576	97	34	that	that	SCONJ
ejpam-4576	97	35	y	y	PROPN
ejpam-4576	97	36	∈	∈	PROPN
ejpam-4576	97	37	f	f	X
ejpam-4576	97	38	(	(	PUNCT
ejpam-4576	97	39	x	x	NOUN
ejpam-4576	97	40	)	)	PUNCT
ejpam-4576	97	41	.	.	PUNCT
ejpam-4576	98	1	moreover	moreover	ADV
ejpam-4576	98	2	,	,	PUNCT
ejpam-4576	98	3	f	f	X
ejpam-4576	98	4	:	:	PUNCT
ejpam-4576	98	5	x	x	X
ejpam-4576	98	6	→	→	SYM
ejpam-4576	98	7	y	y	PROPN
ejpam-4576	98	8	is	be	AUX
ejpam-4576	98	9	called	call	VERB
ejpam-4576	98	10	upper	upper	ADJ
ejpam-4576	98	11	semi	semi	ADJ
ejpam-4576	98	12	-	-	ADJ
ejpam-4576	98	13	continuous	continuous	ADJ
ejpam-4576	98	14	(	(	PUNCT
ejpam-4576	98	15	resp	resp	NOUN
ejpam-4576	98	16	.	.	PUNCT
ejpam-4576	99	1	lower	low	ADJ
ejpam-4576	99	2	semi	semi	ADJ
ejpam-4576	99	3	-	-	ADJ
ejpam-4576	99	4	continuous	continuous	ADJ
ejpam-4576	99	5	)	)	PUNCT
ejpam-4576	99	6	if	if	SCONJ
ejpam-4576	99	7	f+(v	f+(v	PROPN
ejpam-4576	99	8	)	)	PUNCT
ejpam-4576	99	9	(	(	PUNCT
ejpam-4576	99	10	resp	resp	NOUN
ejpam-4576	99	11	.	.	PUNCT
ejpam-4576	100	1	f−(v	f−(v	NOUN
ejpam-4576	100	2	)	)	PUNCT
ejpam-4576	100	3	)	)	PUNCT
ejpam-4576	101	1	is	be	AUX
ejpam-4576	101	2	open	open	ADJ
ejpam-4576	101	3	in	in	ADP
ejpam-4576	101	4	x	x	PUNCT
ejpam-4576	101	5	for	for	ADP
ejpam-4576	101	6	every	every	DET
ejpam-4576	101	7	open	open	ADJ
ejpam-4576	101	8	set	set	VERB
ejpam-4576	101	9	v	v	NOUN
ejpam-4576	101	10	of	of	ADP
ejpam-4576	101	11	y	y	PROPN
ejpam-4576	102	1	[	[	X
ejpam-4576	102	2	11	11	NUM
ejpam-4576	102	3	]	]	PUNCT
ejpam-4576	102	4	.	.	PUNCT
ejpam-4576	103	1	3	3	X
ejpam-4576	103	2	.	.	X
ejpam-4576	103	3	upper	upper	ADJ
ejpam-4576	103	4	and	and	CCONJ
ejpam-4576	103	5	lower	low	ADJ
ejpam-4576	103	6	almost	almost	ADV
ejpam-4576	103	7	(	(	PUNCT
ejpam-4576	103	8	λ	λ	NOUN
ejpam-4576	103	9	,	,	PUNCT
ejpam-4576	103	10	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	103	11	multifunctions	multifunction	NOUN
ejpam-4576	103	12	in	in	ADP
ejpam-4576	103	13	this	this	DET
ejpam-4576	103	14	section	section	NOUN
ejpam-4576	103	15	,	,	PUNCT
ejpam-4576	103	16	we	we	PRON
ejpam-4576	103	17	introduce	introduce	VERB
ejpam-4576	103	18	the	the	DET
ejpam-4576	103	19	notions	notion	NOUN
ejpam-4576	103	20	of	of	ADP
ejpam-4576	103	21	upper	upper	ADJ
ejpam-4576	103	22	and	and	CCONJ
ejpam-4576	103	23	lower	low	ADJ
ejpam-4576	103	24	almost	almost	ADV
ejpam-4576	103	25	(	(	PUNCT
ejpam-4576	103	26	λ	λ	NOUN
ejpam-4576	103	27	,	,	PUNCT
ejpam-4576	103	28	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	103	29	multifunctions	multifunction	NOUN
ejpam-4576	103	30	.	.	PUNCT
ejpam-4576	104	1	in	in	ADP
ejpam-4576	104	2	particular	particular	ADJ
ejpam-4576	104	3	,	,	PUNCT
ejpam-4576	104	4	several	several	ADJ
ejpam-4576	104	5	characterizations	characterization	NOUN
ejpam-4576	104	6	of	of	ADP
ejpam-4576	104	7	upper	upper	ADJ
ejpam-4576	104	8	and	and	CCONJ
ejpam-4576	104	9	lower	low	ADJ
ejpam-4576	104	10	almost	almost	ADV
ejpam-4576	104	11	(	(	PUNCT
ejpam-4576	104	12	λ	λ	NOUN
ejpam-4576	104	13	,	,	PUNCT
ejpam-4576	104	14	sp)continuous	sp)continuous	ADJ
ejpam-4576	104	15	multifunctions	multifunction	NOUN
ejpam-4576	104	16	are	be	AUX
ejpam-4576	104	17	discussed	discuss	VERB
ejpam-4576	104	18	.	.	PUNCT
ejpam-4576	105	1	definition	definition	NOUN
ejpam-4576	105	2	1	1	NUM
ejpam-4576	105	3	.	.	PUNCT
ejpam-4576	106	1	a	a	DET
ejpam-4576	106	2	multifunction	multifunction	NOUN
ejpam-4576	106	3	f	f	NOUN
ejpam-4576	106	4	:	:	PUNCT
ejpam-4576	106	5	(	(	PUNCT
ejpam-4576	106	6	x	x	X
ejpam-4576	106	7	,	,	PUNCT
ejpam-4576	106	8	τ	τ	X
ejpam-4576	106	9	)	)	PUNCT
ejpam-4576	106	10	→	→	SYM
ejpam-4576	106	11	(	(	PUNCT
ejpam-4576	106	12	y	y	PROPN
ejpam-4576	106	13	,	,	PUNCT
ejpam-4576	106	14	σ	σ	PROPN
ejpam-4576	106	15	)	)	PUNCT
ejpam-4576	106	16	is	be	AUX
ejpam-4576	106	17	said	say	VERB
ejpam-4576	106	18	to	to	PART
ejpam-4576	106	19	be	be	AUX
ejpam-4576	106	20	:	:	PUNCT
ejpam-4576	106	21	(	(	PUNCT
ejpam-4576	106	22	i	i	NOUN
ejpam-4576	106	23	)	)	PUNCT
ejpam-4576	106	24	upper	upper	ADJ
ejpam-4576	106	25	almost	almost	ADV
ejpam-4576	106	26	(	(	PUNCT
ejpam-4576	106	27	λ	λ	X
ejpam-4576	106	28	,	,	PUNCT
ejpam-4576	106	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	106	30	at	at	ADP
ejpam-4576	106	31	a	a	DET
ejpam-4576	106	32	point	point	NOUN
ejpam-4576	106	33	x	x	SYM
ejpam-4576	106	34	∈	∈	NOUN
ejpam-4576	106	35	x	x	INTJ
ejpam-4576	106	36	if	if	SCONJ
ejpam-4576	106	37	,	,	PUNCT
ejpam-4576	106	38	for	for	ADP
ejpam-4576	106	39	each	each	DET
ejpam-4576	106	40	(	(	PUNCT
ejpam-4576	106	41	λ	λ	PROPN
ejpam-4576	106	42	,	,	PUNCT
ejpam-4576	106	43	sp)-open	sp)-open	NOUN
ejpam-4576	106	44	set	set	VERB
ejpam-4576	106	45	v	v	NUM
ejpam-4576	106	46	of	of	ADP
ejpam-4576	106	47	y	y	PROPN
ejpam-4576	106	48	containing	contain	VERB
ejpam-4576	106	49	f	f	PROPN
ejpam-4576	106	50	(	(	PUNCT
ejpam-4576	106	51	x	x	NOUN
ejpam-4576	106	52	)	)	PUNCT
ejpam-4576	106	53	,	,	PUNCT
ejpam-4576	106	54	there	there	PRON
ejpam-4576	106	55	exists	exist	VERB
ejpam-4576	106	56	a	a	DET
ejpam-4576	106	57	(	(	PUNCT
ejpam-4576	106	58	λ	λ	NOUN
ejpam-4576	106	59	,	,	PUNCT
ejpam-4576	106	60	sp)-open	sp)-open	NOUN
ejpam-4576	106	61	set	set	VERB
ejpam-4576	106	62	u	u	NOUN
ejpam-4576	106	63	of	of	ADP
ejpam-4576	106	64	x	x	PUNCT
ejpam-4576	106	65	containing	contain	VERB
ejpam-4576	106	66	x	x	PUNCT
ejpam-4576	107	1	such	such	ADJ
ejpam-4576	107	2	that	that	SCONJ
ejpam-4576	107	3	f	f	PROPN
ejpam-4576	107	4	(	(	PUNCT
ejpam-4576	107	5	u	u	NOUN
ejpam-4576	107	6	)	)	PUNCT
ejpam-4576	107	7	⊆	⊆	NUM
ejpam-4576	107	8	[	[	X
ejpam-4576	107	9	v	v	X
ejpam-4576	107	10	(	(	PUNCT
ejpam-4576	107	11	λ	λ	PROPN
ejpam-4576	107	12	,	,	PUNCT
ejpam-4576	107	13	sp)](λ	sp)](λ	PROPN
ejpam-4576	107	14	,	,	PUNCT
ejpam-4576	107	15	sp	sp	NOUN
ejpam-4576	107	16	)	)	PUNCT
ejpam-4576	107	17	;	;	PUNCT
ejpam-4576	107	18	c.	c.	PROPN
ejpam-4576	107	19	boonpok	boonpok	PROPN
ejpam-4576	107	20	,	,	PUNCT
ejpam-4576	107	21	n.	n.	PROPN
ejpam-4576	107	22	viriyapong	viriyapong	PROPN
ejpam-4576	107	23	/	/	SYM
ejpam-4576	107	24	eur	eur	PROPN
ejpam-4576	107	25	.	.	PUNCT
ejpam-4576	108	1	j.	j.	PROPN
ejpam-4576	108	2	pure	pure	PROPN
ejpam-4576	108	3	appl	appl	PROPN
ejpam-4576	108	4	.	.	PROPN
ejpam-4576	108	5	math	math	PROPN
ejpam-4576	108	6	,	,	PUNCT
ejpam-4576	108	7	16	16	NUM
ejpam-4576	108	8	(	(	PUNCT
ejpam-4576	108	9	1	1	NUM
ejpam-4576	108	10	)	)	PUNCT
ejpam-4576	108	11	(	(	PUNCT
ejpam-4576	108	12	2023	2023	NUM
ejpam-4576	108	13	)	)	PUNCT
ejpam-4576	108	14	,	,	PUNCT
ejpam-4576	108	15	84	84	NUM
ejpam-4576	108	16	-	-	SYM
ejpam-4576	108	17	96	96	NUM
ejpam-4576	108	18	87	87	NUM
ejpam-4576	108	19	(	(	PUNCT
ejpam-4576	108	20	ii	ii	NOUN
ejpam-4576	108	21	)	)	PUNCT
ejpam-4576	108	22	lower	low	ADJ
ejpam-4576	108	23	almost	almost	ADV
ejpam-4576	108	24	(	(	PUNCT
ejpam-4576	108	25	λ	λ	NOUN
ejpam-4576	108	26	,	,	PUNCT
ejpam-4576	108	27	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	108	28	at	at	ADP
ejpam-4576	108	29	a	a	DET
ejpam-4576	108	30	point	point	NOUN
ejpam-4576	108	31	x	x	SYM
ejpam-4576	108	32	∈	∈	NOUN
ejpam-4576	108	33	x	x	INTJ
ejpam-4576	108	34	if	if	SCONJ
ejpam-4576	108	35	,	,	PUNCT
ejpam-4576	108	36	for	for	SCONJ
ejpam-4576	108	37	each	each	DET
ejpam-4576	108	38	(	(	PUNCT
ejpam-4576	108	39	λ	λ	PROPN
ejpam-4576	108	40	,	,	PUNCT
ejpam-4576	108	41	sp)-open	sp)-open	NOUN
ejpam-4576	108	42	set	set	VERB
ejpam-4576	108	43	v	v	NUM
ejpam-4576	108	44	of	of	ADP
ejpam-4576	108	45	y	y	PRON
ejpam-4576	108	46	such	such	ADJ
ejpam-4576	108	47	that	that	SCONJ
ejpam-4576	108	48	f	f	PROPN
ejpam-4576	108	49	(	(	PUNCT
ejpam-4576	108	50	x	x	NOUN
ejpam-4576	108	51	)	)	PUNCT
ejpam-4576	108	52	∩	∩	NOUN
ejpam-4576	108	53	v	v	ADP
ejpam-4576	108	54	̸=	̸=	PROPN
ejpam-4576	108	55	∅	∅	NOUN
ejpam-4576	108	56	,	,	PUNCT
ejpam-4576	108	57	there	there	PRON
ejpam-4576	108	58	exists	exist	VERB
ejpam-4576	108	59	a	a	DET
ejpam-4576	108	60	(	(	PUNCT
ejpam-4576	108	61	λ	λ	NOUN
ejpam-4576	108	62	,	,	PUNCT
ejpam-4576	108	63	sp)-open	sp)-open	NOUN
ejpam-4576	108	64	set	set	VERB
ejpam-4576	108	65	u	u	NOUN
ejpam-4576	108	66	of	of	ADP
ejpam-4576	108	67	x	x	PUNCT
ejpam-4576	108	68	containing	contain	VERB
ejpam-4576	108	69	x	x	PUNCT
ejpam-4576	108	70	such	such	ADJ
ejpam-4576	108	71	that	that	SCONJ
ejpam-4576	108	72	f	f	PROPN
ejpam-4576	108	73	(	(	PUNCT
ejpam-4576	108	74	z	z	NOUN
ejpam-4576	108	75	)	)	PUNCT
ejpam-4576	108	76	∩	∩	NOUN
ejpam-4576	109	1	[	[	X
ejpam-4576	109	2	v	v	X
ejpam-4576	109	3	(	(	PUNCT
ejpam-4576	109	4	λ	λ	PROPN
ejpam-4576	109	5	,	,	PUNCT
ejpam-4576	109	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	109	7	,	,	PUNCT
ejpam-4576	109	8	sp	sp	NOUN
ejpam-4576	109	9	)	)	PUNCT
ejpam-4576	109	10	̸=	̸=	PROPN
ejpam-4576	109	11	∅	∅	NOUN
ejpam-4576	109	12	for	for	ADP
ejpam-4576	109	13	each	each	DET
ejpam-4576	109	14	z	z	NOUN
ejpam-4576	109	15	∈	∈	PROPN
ejpam-4576	109	16	u	u	NOUN
ejpam-4576	109	17	;	;	PUNCT
ejpam-4576	109	18	(	(	PUNCT
ejpam-4576	109	19	iii	iii	X
ejpam-4576	109	20	)	)	PUNCT
ejpam-4576	109	21	upper	upper	ADJ
ejpam-4576	109	22	(	(	PUNCT
ejpam-4576	109	23	lower	low	ADJ
ejpam-4576	109	24	)	)	PUNCT
ejpam-4576	109	25	almost	almost	ADV
ejpam-4576	109	26	(	(	PUNCT
ejpam-4576	109	27	λ	λ	NOUN
ejpam-4576	109	28	,	,	PUNCT
ejpam-4576	109	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	109	30	if	if	SCONJ
ejpam-4576	109	31	f	f	PROPN
ejpam-4576	109	32	has	have	VERB
ejpam-4576	109	33	this	this	DET
ejpam-4576	109	34	property	property	NOUN
ejpam-4576	109	35	at	at	ADP
ejpam-4576	109	36	each	each	DET
ejpam-4576	109	37	point	point	NOUN
ejpam-4576	109	38	of	of	ADP
ejpam-4576	109	39	x.	x.	PROPN
ejpam-4576	109	40	lemma	lemma	PROPN
ejpam-4576	109	41	3	3	X
ejpam-4576	109	42	.	.	PUNCT
ejpam-4576	110	1	[	[	X
ejpam-4576	110	2	21	21	NUM
ejpam-4576	110	3	]	]	PUNCT
ejpam-4576	110	4	for	for	ADP
ejpam-4576	110	5	a	a	DET
ejpam-4576	110	6	subset	subset	NOUN
ejpam-4576	110	7	a	a	PRON
ejpam-4576	110	8	of	of	ADP
ejpam-4576	110	9	a	a	DET
ejpam-4576	110	10	topological	topological	ADJ
ejpam-4576	110	11	space	space	NOUN
ejpam-4576	110	12	(	(	PUNCT
ejpam-4576	110	13	x	x	X
ejpam-4576	110	14	,	,	PUNCT
ejpam-4576	110	15	τ	τ	PROPN
ejpam-4576	110	16	)	)	PUNCT
ejpam-4576	110	17	,	,	PUNCT
ejpam-4576	110	18	the	the	DET
ejpam-4576	110	19	following	follow	VERB
ejpam-4576	110	20	properties	property	NOUN
ejpam-4576	110	21	hold	hold	VERB
ejpam-4576	110	22	:	:	PUNCT
ejpam-4576	110	23	(	(	PUNCT
ejpam-4576	110	24	1	1	X
ejpam-4576	110	25	)	)	PUNCT
ejpam-4576	110	26	as(λ	as(λ	NUM
ejpam-4576	110	27	,	,	PUNCT
ejpam-4576	110	28	sp	sp	NOUN
ejpam-4576	110	29	)	)	PUNCT
ejpam-4576	110	30	=	=	NOUN
ejpam-4576	110	31	a	a	DET
ejpam-4576	110	32	∪	∪	ADJ
ejpam-4576	110	33	[	[	X
ejpam-4576	110	34	a(λ	a(λ	ADJ
ejpam-4576	110	35	,	,	PUNCT
ejpam-4576	110	36	sp)](λ	sp)](λ	PROPN
ejpam-4576	110	37	,	,	PUNCT
ejpam-4576	110	38	sp	sp	NOUN
ejpam-4576	110	39	)	)	PUNCT
ejpam-4576	110	40	;	;	PUNCT
ejpam-4576	110	41	(	(	PUNCT
ejpam-4576	110	42	2	2	X
ejpam-4576	110	43	)	)	PUNCT
ejpam-4576	110	44	as(λ	as(λ	NUM
ejpam-4576	110	45	,	,	PUNCT
ejpam-4576	110	46	sp	sp	NOUN
ejpam-4576	110	47	)	)	PUNCT
ejpam-4576	110	48	=	=	PUNCT
ejpam-4576	110	49	a	a	DET
ejpam-4576	110	50	∩	∩	NOUN
ejpam-4576	110	51	[	[	X
ejpam-4576	110	52	a(λ	a(λ	ADV
ejpam-4576	110	53	,	,	PUNCT
ejpam-4576	110	54	sp	sp	NOUN
ejpam-4576	110	55	)	)	PUNCT
ejpam-4576	110	56	]	]	PUNCT
ejpam-4576	110	57	(	(	PUNCT
ejpam-4576	110	58	λ	λ	NOUN
ejpam-4576	110	59	,	,	PUNCT
ejpam-4576	110	60	sp	sp	NOUN
ejpam-4576	110	61	)	)	PUNCT
ejpam-4576	110	62	.	.	PUNCT
ejpam-4576	111	1	lemma	lemma	PROPN
ejpam-4576	111	2	4	4	X
ejpam-4576	111	3	.	.	PUNCT
ejpam-4576	111	4	let	let	VERB
ejpam-4576	111	5	a	a	DET
ejpam-4576	111	6	be	be	AUX
ejpam-4576	111	7	a	a	DET
ejpam-4576	111	8	subset	subset	NOUN
ejpam-4576	111	9	of	of	ADP
ejpam-4576	111	10	a	a	DET
ejpam-4576	111	11	topological	topological	ADJ
ejpam-4576	111	12	space	space	NOUN
ejpam-4576	111	13	(	(	PUNCT
ejpam-4576	111	14	x	x	X
ejpam-4576	111	15	,	,	PUNCT
ejpam-4576	111	16	τ	τ	PROPN
ejpam-4576	111	17	)	)	PUNCT
ejpam-4576	111	18	.	.	PUNCT
ejpam-4576	112	1	if	if	SCONJ
ejpam-4576	112	2	a	a	PRON
ejpam-4576	112	3	is	be	AUX
ejpam-4576	112	4	(	(	PUNCT
ejpam-4576	112	5	λ	λ	NOUN
ejpam-4576	112	6	,	,	PUNCT
ejpam-4576	112	7	sp)-open	sp)-open	ADJ
ejpam-4576	112	8	in	in	ADP
ejpam-4576	112	9	x	x	NOUN
ejpam-4576	112	10	,	,	PUNCT
ejpam-4576	112	11	then	then	ADV
ejpam-4576	112	12	as(λ	as(λ	NUM
ejpam-4576	112	13	,	,	PUNCT
ejpam-4576	112	14	sp	sp	NOUN
ejpam-4576	112	15	)	)	PUNCT
ejpam-4576	112	16	=	=	PUNCT
ejpam-4576	113	1	[	[	X
ejpam-4576	113	2	a(λ	a(λ	ADV
ejpam-4576	113	3	,	,	PUNCT
ejpam-4576	113	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	113	5	,	,	PUNCT
ejpam-4576	113	6	sp	sp	NOUN
ejpam-4576	113	7	)	)	PUNCT
ejpam-4576	113	8	.	.	PUNCT
ejpam-4576	114	1	theorem	theorem	NOUN
ejpam-4576	114	2	1	1	NUM
ejpam-4576	114	3	.	.	X
ejpam-4576	114	4	for	for	ADP
ejpam-4576	114	5	a	a	DET
ejpam-4576	114	6	multifunction	multifunction	NOUN
ejpam-4576	115	1	f	f	NOUN
ejpam-4576	115	2	:	:	PUNCT
ejpam-4576	115	3	(	(	PUNCT
ejpam-4576	115	4	x	x	X
ejpam-4576	115	5	,	,	PUNCT
ejpam-4576	115	6	τ	τ	X
ejpam-4576	115	7	)	)	PUNCT
ejpam-4576	115	8	→	→	SYM
ejpam-4576	115	9	(	(	PUNCT
ejpam-4576	115	10	y	y	PROPN
ejpam-4576	115	11	,	,	PUNCT
ejpam-4576	115	12	σ	σ	PROPN
ejpam-4576	115	13	)	)	PUNCT
ejpam-4576	115	14	,	,	PUNCT
ejpam-4576	115	15	the	the	DET
ejpam-4576	115	16	following	follow	VERB
ejpam-4576	115	17	properties	property	NOUN
ejpam-4576	115	18	are	be	AUX
ejpam-4576	115	19	equivalent	equivalent	ADJ
ejpam-4576	115	20	:	:	PUNCT
ejpam-4576	115	21	(	(	PUNCT
ejpam-4576	115	22	1	1	X
ejpam-4576	115	23	)	)	PUNCT
ejpam-4576	115	24	f	f	PROPN
ejpam-4576	115	25	is	be	AUX
ejpam-4576	115	26	upper	upper	ADJ
ejpam-4576	115	27	almost	almost	ADV
ejpam-4576	115	28	(	(	PUNCT
ejpam-4576	115	29	λ	λ	X
ejpam-4576	115	30	,	,	PUNCT
ejpam-4576	115	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	115	32	at	at	ADP
ejpam-4576	115	33	x	x	X
ejpam-4576	115	34	∈	∈	PROPN
ejpam-4576	115	35	x	x	X
ejpam-4576	115	36	;	;	PUNCT
ejpam-4576	115	37	(	(	PUNCT
ejpam-4576	115	38	2	2	X
ejpam-4576	115	39	)	)	PUNCT
ejpam-4576	115	40	x	x	SYM
ejpam-4576	115	41	∈	∈	PROPN
ejpam-4576	115	42	[	[	X
ejpam-4576	115	43	f+([v	f+([v	X
ejpam-4576	115	44	(	(	PUNCT
ejpam-4576	115	45	λ	λ	PROPN
ejpam-4576	115	46	,	,	PUNCT
ejpam-4576	115	47	sp)](λ	sp)](λ	PROPN
ejpam-4576	115	48	,	,	PUNCT
ejpam-4576	115	49	sp))](λ	sp))](λ	PROPN
ejpam-4576	115	50	,	,	PUNCT
ejpam-4576	115	51	sp	sp	NOUN
ejpam-4576	115	52	)	)	PUNCT
ejpam-4576	115	53	for	for	ADP
ejpam-4576	115	54	every	every	DET
ejpam-4576	115	55	(	(	PUNCT
ejpam-4576	115	56	λ	λ	NOUN
ejpam-4576	115	57	,	,	PUNCT
ejpam-4576	115	58	sp)-open	sp)-open	NOUN
ejpam-4576	115	59	set	set	VERB
ejpam-4576	115	60	v	v	NUM
ejpam-4576	115	61	of	of	ADP
ejpam-4576	115	62	y	y	PROPN
ejpam-4576	115	63	containing	contain	VERB
ejpam-4576	115	64	f	f	PROPN
ejpam-4576	115	65	(	(	PUNCT
ejpam-4576	115	66	x	x	NOUN
ejpam-4576	115	67	)	)	PUNCT
ejpam-4576	115	68	;	;	PUNCT
ejpam-4576	115	69	(	(	PUNCT
ejpam-4576	115	70	3	3	X
ejpam-4576	115	71	)	)	PUNCT
ejpam-4576	115	72	x	x	SYM
ejpam-4576	115	73	∈	∈	PROPN
ejpam-4576	116	1	[	[	X
ejpam-4576	116	2	f+(v	f+(v	PROPN
ejpam-4576	116	3	s(λ	s(λ	PROPN
ejpam-4576	116	4	,	,	PUNCT
ejpam-4576	116	5	sp))](λ	sp))](λ	PROPN
ejpam-4576	116	6	,	,	PUNCT
ejpam-4576	116	7	sp	sp	NOUN
ejpam-4576	116	8	)	)	PUNCT
ejpam-4576	116	9	for	for	ADP
ejpam-4576	116	10	every	every	DET
ejpam-4576	116	11	(	(	PUNCT
ejpam-4576	116	12	λ	λ	NOUN
ejpam-4576	116	13	,	,	PUNCT
ejpam-4576	116	14	sp)-open	sp)-open	NOUN
ejpam-4576	116	15	set	set	VERB
ejpam-4576	116	16	v	v	NUM
ejpam-4576	116	17	of	of	ADP
ejpam-4576	116	18	y	y	PROPN
ejpam-4576	116	19	containing	contain	VERB
ejpam-4576	116	20	f	f	PROPN
ejpam-4576	116	21	(	(	PUNCT
ejpam-4576	116	22	x	x	NOUN
ejpam-4576	116	23	)	)	PUNCT
ejpam-4576	116	24	;	;	PUNCT
ejpam-4576	116	25	(	(	PUNCT
ejpam-4576	116	26	4	4	X
ejpam-4576	116	27	)	)	PUNCT
ejpam-4576	116	28	x	x	SYM
ejpam-4576	116	29	∈	∈	PROPN
ejpam-4576	116	30	[	[	X
ejpam-4576	116	31	f+(v	f+(v	NOUN
ejpam-4576	116	32	)	)	PUNCT
ejpam-4576	116	33	]	]	PUNCT
ejpam-4576	116	34	(	(	PUNCT
ejpam-4576	116	35	λ	λ	NOUN
ejpam-4576	116	36	,	,	PUNCT
ejpam-4576	116	37	sp	sp	NOUN
ejpam-4576	116	38	)	)	PUNCT
ejpam-4576	116	39	for	for	ADP
ejpam-4576	116	40	every	every	DET
ejpam-4576	116	41	r(λ	r(λ	NOUN
ejpam-4576	116	42	,	,	PUNCT
ejpam-4576	116	43	sp)-open	sp)-open	ADJ
ejpam-4576	116	44	set	set	VERB
ejpam-4576	116	45	v	v	NUM
ejpam-4576	116	46	of	of	ADP
ejpam-4576	116	47	y	y	PROPN
ejpam-4576	116	48	containing	contain	VERB
ejpam-4576	116	49	f	f	PROPN
ejpam-4576	116	50	(	(	PUNCT
ejpam-4576	116	51	x	x	NOUN
ejpam-4576	116	52	)	)	PUNCT
ejpam-4576	116	53	;	;	PUNCT
ejpam-4576	116	54	(	(	PUNCT
ejpam-4576	116	55	5	5	X
ejpam-4576	116	56	)	)	PUNCT
ejpam-4576	116	57	for	for	ADP
ejpam-4576	116	58	each	each	DET
ejpam-4576	116	59	r(λ	r(λ	NOUN
ejpam-4576	116	60	,	,	PUNCT
ejpam-4576	116	61	sp)-open	sp)-open	VERB
ejpam-4576	116	62	set	set	VERB
ejpam-4576	116	63	v	v	NUM
ejpam-4576	116	64	of	of	ADP
ejpam-4576	116	65	y	y	PROPN
ejpam-4576	116	66	containing	contain	VERB
ejpam-4576	116	67	f	f	PROPN
ejpam-4576	116	68	(	(	PUNCT
ejpam-4576	116	69	x	x	NOUN
ejpam-4576	116	70	)	)	PUNCT
ejpam-4576	116	71	,	,	PUNCT
ejpam-4576	116	72	there	there	PRON
ejpam-4576	116	73	exists	exist	VERB
ejpam-4576	116	74	a	a	DET
ejpam-4576	116	75	(	(	PUNCT
ejpam-4576	116	76	λ	λ	NOUN
ejpam-4576	116	77	,	,	PUNCT
ejpam-4576	116	78	sp)-open	sp)-open	NOUN
ejpam-4576	116	79	set	set	VERB
ejpam-4576	116	80	u	u	NOUN
ejpam-4576	116	81	of	of	ADP
ejpam-4576	116	82	x	x	PUNCT
ejpam-4576	116	83	containing	contain	VERB
ejpam-4576	116	84	x	x	PUNCT
ejpam-4576	116	85	such	such	ADJ
ejpam-4576	116	86	that	that	SCONJ
ejpam-4576	116	87	f	f	PROPN
ejpam-4576	116	88	(	(	PUNCT
ejpam-4576	116	89	u	u	NOUN
ejpam-4576	116	90	)	)	PUNCT
ejpam-4576	116	91	⊆	⊆	NUM
ejpam-4576	116	92	v	v	NOUN
ejpam-4576	116	93	.	.	PUNCT
ejpam-4576	117	1	proof	proof	NOUN
ejpam-4576	117	2	.	.	PUNCT
ejpam-4576	118	1	(	(	PUNCT
ejpam-4576	118	2	1	1	X
ejpam-4576	118	3	)	)	PUNCT
ejpam-4576	118	4	⇒	⇒	NOUN
ejpam-4576	118	5	(	(	PUNCT
ejpam-4576	118	6	2	2	NUM
ejpam-4576	118	7	):	):	PUNCT
ejpam-4576	118	8	let	let	VERB
ejpam-4576	118	9	v	v	PART
ejpam-4576	118	10	be	be	AUX
ejpam-4576	118	11	any	any	DET
ejpam-4576	118	12	(	(	PUNCT
ejpam-4576	118	13	λ	λ	NOUN
ejpam-4576	118	14	,	,	PUNCT
ejpam-4576	118	15	sp)-open	sp)-open	ADJ
ejpam-4576	118	16	set	set	NOUN
ejpam-4576	118	17	of	of	ADP
ejpam-4576	118	18	y	y	PROPN
ejpam-4576	118	19	containing	contain	VERB
ejpam-4576	118	20	f	f	PROPN
ejpam-4576	118	21	(	(	PUNCT
ejpam-4576	118	22	x	x	NOUN
ejpam-4576	118	23	)	)	PUNCT
ejpam-4576	118	24	.	.	PUNCT
ejpam-4576	119	1	there	there	PRON
ejpam-4576	119	2	exists	exist	VERB
ejpam-4576	119	3	a	a	DET
ejpam-4576	119	4	(	(	PUNCT
ejpam-4576	119	5	λ	λ	NOUN
ejpam-4576	119	6	,	,	PUNCT
ejpam-4576	119	7	sp)-open	sp)-open	NOUN
ejpam-4576	119	8	set	set	VERB
ejpam-4576	119	9	u	u	NOUN
ejpam-4576	119	10	of	of	ADP
ejpam-4576	119	11	x	x	PUNCT
ejpam-4576	119	12	containing	contain	VERB
ejpam-4576	119	13	x	x	PUNCT
ejpam-4576	119	14	such	such	ADJ
ejpam-4576	119	15	that	that	SCONJ
ejpam-4576	119	16	f	f	PROPN
ejpam-4576	119	17	(	(	PUNCT
ejpam-4576	119	18	u	u	NOUN
ejpam-4576	119	19	)	)	PUNCT
ejpam-4576	119	20	⊆	⊆	NUM
ejpam-4576	120	1	[	[	X
ejpam-4576	120	2	v	v	X
ejpam-4576	120	3	(	(	PUNCT
ejpam-4576	120	4	λ	λ	PROPN
ejpam-4576	120	5	,	,	PUNCT
ejpam-4576	120	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	120	7	,	,	PUNCT
ejpam-4576	120	8	sp	sp	NOUN
ejpam-4576	120	9	)	)	PUNCT
ejpam-4576	120	10	.	.	PUNCT
ejpam-4576	121	1	thus	thus	ADV
ejpam-4576	121	2	,	,	PUNCT
ejpam-4576	121	3	x	x	PUNCT
ejpam-4576	121	4	∈	∈	PROPN
ejpam-4576	121	5	u	u	NOUN
ejpam-4576	121	6	⊆	⊆	NUM
ejpam-4576	121	7	f+([v	f+([v	NOUN
ejpam-4576	121	8	(	(	PUNCT
ejpam-4576	121	9	λ	λ	PROPN
ejpam-4576	121	10	,	,	PUNCT
ejpam-4576	121	11	sp)](λ	sp)](λ	PROPN
ejpam-4576	121	12	,	,	PUNCT
ejpam-4576	121	13	sp	sp	NOUN
ejpam-4576	121	14	)	)	PUNCT
ejpam-4576	121	15	)	)	PUNCT
ejpam-4576	121	16	and	and	CCONJ
ejpam-4576	121	17	hence	hence	ADV
ejpam-4576	121	18	x	x	X
ejpam-4576	121	19	∈	∈	PROPN
ejpam-4576	122	1	[	[	X
ejpam-4576	122	2	f+([v	f+([v	X
ejpam-4576	122	3	(	(	PUNCT
ejpam-4576	122	4	λ	λ	PROPN
ejpam-4576	122	5	,	,	PUNCT
ejpam-4576	122	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	122	7	,	,	PUNCT
ejpam-4576	122	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	122	9	,	,	PUNCT
ejpam-4576	122	10	sp	sp	NOUN
ejpam-4576	122	11	)	)	PUNCT
ejpam-4576	122	12	.	.	PUNCT
ejpam-4576	123	1	(	(	PUNCT
ejpam-4576	123	2	2	2	X
ejpam-4576	123	3	)	)	PUNCT
ejpam-4576	123	4	⇒	⇒	NOUN
ejpam-4576	123	5	(	(	PUNCT
ejpam-4576	123	6	3	3	NUM
ejpam-4576	123	7	):	):	PUNCT
ejpam-4576	123	8	this	this	PRON
ejpam-4576	123	9	follows	follow	VERB
ejpam-4576	123	10	from	from	ADP
ejpam-4576	123	11	lemma	lemma	PROPN
ejpam-4576	123	12	4	4	NUM
ejpam-4576	123	13	.	.	PUNCT
ejpam-4576	123	14	(	(	PUNCT
ejpam-4576	123	15	3	3	X
ejpam-4576	123	16	)	)	PUNCT
ejpam-4576	123	17	⇒	⇒	NOUN
ejpam-4576	123	18	(	(	PUNCT
ejpam-4576	123	19	4	4	NUM
ejpam-4576	123	20	):	):	PUNCT
ejpam-4576	123	21	let	let	VERB
ejpam-4576	123	22	v	v	PART
ejpam-4576	123	23	be	be	AUX
ejpam-4576	123	24	any	any	DET
ejpam-4576	123	25	r(λ	r(λ	NOUN
ejpam-4576	123	26	,	,	PUNCT
ejpam-4576	123	27	sp)-open	sp)-open	ADJ
ejpam-4576	123	28	set	set	NOUN
ejpam-4576	123	29	of	of	ADP
ejpam-4576	123	30	y	y	PROPN
ejpam-4576	123	31	containing	contain	VERB
ejpam-4576	123	32	f	f	PROPN
ejpam-4576	123	33	(	(	PUNCT
ejpam-4576	123	34	x	x	NOUN
ejpam-4576	123	35	)	)	PUNCT
ejpam-4576	123	36	.	.	PUNCT
ejpam-4576	124	1	then	then	ADV
ejpam-4576	124	2	,	,	PUNCT
ejpam-4576	124	3	it	it	PRON
ejpam-4576	124	4	follows	follow	VERB
ejpam-4576	124	5	from	from	ADP
ejpam-4576	124	6	lemma	lemma	PROPN
ejpam-4576	124	7	4	4	NUM
ejpam-4576	124	8	that	that	PRON
ejpam-4576	124	9	v	v	NOUN
ejpam-4576	124	10	=	=	PUNCT
ejpam-4576	125	1	[	[	X
ejpam-4576	125	2	v	v	X
ejpam-4576	125	3	(	(	PUNCT
ejpam-4576	125	4	λ	λ	PROPN
ejpam-4576	125	5	,	,	PUNCT
ejpam-4576	125	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	125	7	,	,	PUNCT
ejpam-4576	125	8	sp	sp	NOUN
ejpam-4576	125	9	)	)	PUNCT
ejpam-4576	125	10	=	=	SYM
ejpam-4576	125	11	v	v	ADP
ejpam-4576	125	12	s(λ	s(λ	PROPN
ejpam-4576	125	13	,	,	PUNCT
ejpam-4576	125	14	sp	sp	NOUN
ejpam-4576	125	15	)	)	PUNCT
ejpam-4576	125	16	.	.	PUNCT
ejpam-4576	126	1	(	(	PUNCT
ejpam-4576	126	2	4	4	X
ejpam-4576	126	3	)	)	PUNCT
ejpam-4576	126	4	⇒	⇒	NOUN
ejpam-4576	126	5	(	(	PUNCT
ejpam-4576	126	6	5	5	NUM
ejpam-4576	126	7	):	):	PUNCT
ejpam-4576	126	8	let	let	VERB
ejpam-4576	126	9	v	v	PART
ejpam-4576	126	10	be	be	AUX
ejpam-4576	126	11	any	any	DET
ejpam-4576	126	12	r(λ	r(λ	NOUN
ejpam-4576	126	13	,	,	PUNCT
ejpam-4576	126	14	sp)-open	sp)-open	ADJ
ejpam-4576	126	15	set	set	NOUN
ejpam-4576	126	16	of	of	ADP
ejpam-4576	126	17	y	y	PROPN
ejpam-4576	126	18	containing	contain	VERB
ejpam-4576	126	19	f	f	PROPN
ejpam-4576	126	20	(	(	PUNCT
ejpam-4576	126	21	x	x	NOUN
ejpam-4576	126	22	)	)	PUNCT
ejpam-4576	126	23	.	.	PUNCT
ejpam-4576	127	1	thus	thus	ADV
ejpam-4576	127	2	,	,	PUNCT
ejpam-4576	127	3	by	by	ADP
ejpam-4576	127	4	(	(	PUNCT
ejpam-4576	127	5	4	4	NUM
ejpam-4576	127	6	)	)	PUNCT
ejpam-4576	127	7	,	,	PUNCT
ejpam-4576	127	8	x	x	PUNCT
ejpam-4576	127	9	∈	∈	PROPN
ejpam-4576	127	10	[	[	X
ejpam-4576	127	11	f+(v	f+(v	NOUN
ejpam-4576	127	12	)	)	PUNCT
ejpam-4576	127	13	]	]	PUNCT
ejpam-4576	127	14	(	(	PUNCT
ejpam-4576	127	15	λ	λ	NOUN
ejpam-4576	127	16	,	,	PUNCT
ejpam-4576	127	17	sp	sp	NOUN
ejpam-4576	127	18	)	)	PUNCT
ejpam-4576	127	19	and	and	CCONJ
ejpam-4576	127	20	there	there	PRON
ejpam-4576	127	21	exists	exist	VERB
ejpam-4576	127	22	a	a	DET
ejpam-4576	127	23	(	(	PUNCT
ejpam-4576	127	24	λ	λ	NOUN
ejpam-4576	127	25	,	,	PUNCT
ejpam-4576	127	26	sp)-open	sp)-open	NOUN
ejpam-4576	127	27	set	set	VERB
ejpam-4576	127	28	u	u	NOUN
ejpam-4576	127	29	of	of	ADP
ejpam-4576	127	30	x	x	PUNCT
ejpam-4576	127	31	containing	contain	VERB
ejpam-4576	127	32	x	x	PUNCT
ejpam-4576	127	33	such	such	ADJ
ejpam-4576	127	34	that	that	SCONJ
ejpam-4576	127	35	x	x	SYM
ejpam-4576	127	36	∈	∈	NUM
ejpam-4576	127	37	u	u	NOUN
ejpam-4576	127	38	⊆	⊆	NUM
ejpam-4576	127	39	f+(v	f+(v	NOUN
ejpam-4576	127	40	)	)	PUNCT
ejpam-4576	127	41	;	;	PUNCT
ejpam-4576	127	42	hence	hence	ADV
ejpam-4576	127	43	f	f	PROPN
ejpam-4576	127	44	(	(	PUNCT
ejpam-4576	127	45	u	u	NOUN
ejpam-4576	127	46	)	)	PUNCT
ejpam-4576	127	47	⊆	⊆	NUM
ejpam-4576	127	48	v	v	NOUN
ejpam-4576	127	49	.	.	PUNCT
ejpam-4576	128	1	(	(	PUNCT
ejpam-4576	128	2	5	5	X
ejpam-4576	128	3	)	)	PUNCT
ejpam-4576	128	4	⇒	⇒	NOUN
ejpam-4576	128	5	(	(	PUNCT
ejpam-4576	128	6	1	1	NUM
ejpam-4576	128	7	):	):	PUNCT
ejpam-4576	128	8	let	let	VERB
ejpam-4576	128	9	v	v	PART
ejpam-4576	128	10	be	be	AUX
ejpam-4576	128	11	any	any	DET
ejpam-4576	128	12	(	(	PUNCT
ejpam-4576	128	13	λ	λ	NOUN
ejpam-4576	128	14	,	,	PUNCT
ejpam-4576	128	15	sp)-open	sp)-open	ADJ
ejpam-4576	128	16	set	set	NOUN
ejpam-4576	128	17	of	of	ADP
ejpam-4576	128	18	y	y	PROPN
ejpam-4576	128	19	containing	contain	VERB
ejpam-4576	128	20	f	f	PROPN
ejpam-4576	128	21	(	(	PUNCT
ejpam-4576	128	22	x	x	NOUN
ejpam-4576	128	23	)	)	PUNCT
ejpam-4576	128	24	.	.	PUNCT
ejpam-4576	129	1	since	since	SCONJ
ejpam-4576	129	2	[	[	X
ejpam-4576	129	3	v	v	X
ejpam-4576	129	4	(	(	PUNCT
ejpam-4576	129	5	λ	λ	PROPN
ejpam-4576	129	6	,	,	PUNCT
ejpam-4576	129	7	sp)](λ	sp)](λ	PROPN
ejpam-4576	129	8	,	,	PUNCT
ejpam-4576	129	9	sp	sp	NOUN
ejpam-4576	129	10	)	)	PUNCT
ejpam-4576	129	11	is	be	AUX
ejpam-4576	129	12	r(λ	r(λ	NOUN
ejpam-4576	129	13	,	,	PUNCT
ejpam-4576	129	14	sp)-open	sp)-open	ADJ
ejpam-4576	129	15	,	,	PUNCT
ejpam-4576	129	16	there	there	PRON
ejpam-4576	129	17	exists	exist	VERB
ejpam-4576	129	18	a	a	DET
ejpam-4576	129	19	(	(	PUNCT
ejpam-4576	129	20	λ	λ	NOUN
ejpam-4576	129	21	,	,	PUNCT
ejpam-4576	129	22	sp)-open	sp)-open	NOUN
ejpam-4576	129	23	set	set	VERB
ejpam-4576	129	24	u	u	NOUN
ejpam-4576	129	25	of	of	ADP
ejpam-4576	129	26	x	x	PUNCT
ejpam-4576	129	27	containing	contain	VERB
ejpam-4576	129	28	x	x	PUNCT
ejpam-4576	130	1	such	such	ADJ
ejpam-4576	130	2	that	that	SCONJ
ejpam-4576	130	3	f	f	PROPN
ejpam-4576	130	4	(	(	PUNCT
ejpam-4576	130	5	u	u	NOUN
ejpam-4576	130	6	)	)	PUNCT
ejpam-4576	130	7	⊆	⊆	NUM
ejpam-4576	130	8	[	[	X
ejpam-4576	130	9	v	v	X
ejpam-4576	130	10	(	(	PUNCT
ejpam-4576	130	11	λ	λ	PROPN
ejpam-4576	130	12	,	,	PUNCT
ejpam-4576	130	13	sp)](λ	sp)](λ	PROPN
ejpam-4576	130	14	,	,	PUNCT
ejpam-4576	130	15	sp	sp	NOUN
ejpam-4576	130	16	)	)	PUNCT
ejpam-4576	130	17	.	.	PUNCT
ejpam-4576	131	1	this	this	PRON
ejpam-4576	131	2	shows	show	VERB
ejpam-4576	131	3	that	that	SCONJ
ejpam-4576	131	4	f	f	PROPN
ejpam-4576	131	5	is	be	AUX
ejpam-4576	131	6	upper	upper	ADJ
ejpam-4576	131	7	almost	almost	ADV
ejpam-4576	131	8	(	(	PUNCT
ejpam-4576	131	9	λ	λ	X
ejpam-4576	131	10	,	,	PUNCT
ejpam-4576	131	11	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	131	12	at	at	ADP
ejpam-4576	131	13	x	x	PROPN
ejpam-4576	131	14	∈	∈	PROPN
ejpam-4576	131	15	x.	x.	NOUN
ejpam-4576	131	16	theorem	theorem	VERB
ejpam-4576	131	17	2	2	NUM
ejpam-4576	131	18	.	.	X
ejpam-4576	131	19	for	for	ADP
ejpam-4576	131	20	a	a	DET
ejpam-4576	131	21	multifunction	multifunction	NOUN
ejpam-4576	131	22	f	f	NOUN
ejpam-4576	131	23	:	:	PUNCT
ejpam-4576	131	24	(	(	PUNCT
ejpam-4576	131	25	x	x	X
ejpam-4576	131	26	,	,	PUNCT
ejpam-4576	131	27	τ	τ	X
ejpam-4576	131	28	)	)	PUNCT
ejpam-4576	131	29	→	→	SYM
ejpam-4576	131	30	(	(	PUNCT
ejpam-4576	131	31	y	y	PROPN
ejpam-4576	131	32	,	,	PUNCT
ejpam-4576	131	33	σ	σ	PROPN
ejpam-4576	131	34	)	)	PUNCT
ejpam-4576	131	35	,	,	PUNCT
ejpam-4576	131	36	the	the	DET
ejpam-4576	131	37	following	follow	VERB
ejpam-4576	131	38	properties	property	NOUN
ejpam-4576	131	39	are	be	AUX
ejpam-4576	131	40	equivalent	equivalent	ADJ
ejpam-4576	131	41	:	:	PUNCT
ejpam-4576	131	42	c.	c.	PROPN
ejpam-4576	131	43	boonpok	boonpok	PROPN
ejpam-4576	131	44	,	,	PUNCT
ejpam-4576	131	45	n.	n.	PROPN
ejpam-4576	131	46	viriyapong	viriyapong	PROPN
ejpam-4576	131	47	/	/	SYM
ejpam-4576	131	48	eur	eur	PROPN
ejpam-4576	131	49	.	.	PUNCT
ejpam-4576	132	1	j.	j.	PROPN
ejpam-4576	132	2	pure	pure	PROPN
ejpam-4576	132	3	appl	appl	PROPN
ejpam-4576	132	4	.	.	PROPN
ejpam-4576	132	5	math	math	PROPN
ejpam-4576	132	6	,	,	PUNCT
ejpam-4576	132	7	16	16	NUM
ejpam-4576	132	8	(	(	PUNCT
ejpam-4576	132	9	1	1	NUM
ejpam-4576	132	10	)	)	PUNCT
ejpam-4576	132	11	(	(	PUNCT
ejpam-4576	132	12	2023	2023	NUM
ejpam-4576	132	13	)	)	PUNCT
ejpam-4576	132	14	,	,	PUNCT
ejpam-4576	132	15	84	84	NUM
ejpam-4576	132	16	-	-	SYM
ejpam-4576	132	17	96	96	NUM
ejpam-4576	132	18	88	88	NUM
ejpam-4576	132	19	(	(	PUNCT
ejpam-4576	132	20	1	1	NUM
ejpam-4576	132	21	)	)	PUNCT
ejpam-4576	132	22	f	f	PROPN
ejpam-4576	132	23	is	be	AUX
ejpam-4576	132	24	lower	low	ADJ
ejpam-4576	132	25	almost	almost	ADV
ejpam-4576	132	26	(	(	PUNCT
ejpam-4576	132	27	λ	λ	NOUN
ejpam-4576	132	28	,	,	PUNCT
ejpam-4576	132	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	132	30	at	at	ADP
ejpam-4576	132	31	x	x	X
ejpam-4576	132	32	∈	∈	PROPN
ejpam-4576	132	33	x	x	X
ejpam-4576	132	34	;	;	PUNCT
ejpam-4576	132	35	(	(	PUNCT
ejpam-4576	132	36	2	2	X
ejpam-4576	132	37	)	)	PUNCT
ejpam-4576	133	1	x	x	SYM
ejpam-4576	133	2	∈	∈	PROPN
ejpam-4576	134	1	[	[	X
ejpam-4576	134	2	f−([v	f−([v	ADJ
ejpam-4576	134	3	(	(	PUNCT
ejpam-4576	134	4	λ	λ	PROPN
ejpam-4576	134	5	,	,	PUNCT
ejpam-4576	134	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	134	7	,	,	PUNCT
ejpam-4576	134	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	134	9	,	,	PUNCT
ejpam-4576	134	10	sp	sp	NOUN
ejpam-4576	134	11	)	)	PUNCT
ejpam-4576	134	12	for	for	ADP
ejpam-4576	134	13	every	every	DET
ejpam-4576	134	14	(	(	PUNCT
ejpam-4576	134	15	λ	λ	NOUN
ejpam-4576	134	16	,	,	PUNCT
ejpam-4576	134	17	sp)-open	sp)-open	NOUN
ejpam-4576	134	18	set	set	VERB
ejpam-4576	134	19	v	v	NUM
ejpam-4576	134	20	of	of	ADP
ejpam-4576	134	21	y	y	PRON
ejpam-4576	134	22	such	such	ADJ
ejpam-4576	134	23	that	that	SCONJ
ejpam-4576	134	24	f	f	PROPN
ejpam-4576	134	25	(	(	PUNCT
ejpam-4576	134	26	x	x	NOUN
ejpam-4576	134	27	)	)	PUNCT
ejpam-4576	134	28	∩	∩	NOUN
ejpam-4576	134	29	v	v	ADP
ejpam-4576	134	30	̸=	̸=	PROPN
ejpam-4576	134	31	∅	∅	NOUN
ejpam-4576	134	32	;	;	PUNCT
ejpam-4576	134	33	(	(	PUNCT
ejpam-4576	134	34	3	3	X
ejpam-4576	134	35	)	)	PUNCT
ejpam-4576	135	1	x	x	SYM
ejpam-4576	135	2	∈	∈	PROPN
ejpam-4576	136	1	[	[	X
ejpam-4576	136	2	f−(v	f−(v	PROPN
ejpam-4576	136	3	s(λ	s(λ	PROPN
ejpam-4576	136	4	,	,	PUNCT
ejpam-4576	136	5	sp))](λ	sp))](λ	PROPN
ejpam-4576	136	6	,	,	PUNCT
ejpam-4576	136	7	sp	sp	NOUN
ejpam-4576	136	8	)	)	PUNCT
ejpam-4576	136	9	for	for	ADP
ejpam-4576	136	10	every	every	DET
ejpam-4576	136	11	(	(	PUNCT
ejpam-4576	136	12	λ	λ	NOUN
ejpam-4576	136	13	,	,	PUNCT
ejpam-4576	136	14	sp)-open	sp)-open	NOUN
ejpam-4576	136	15	set	set	VERB
ejpam-4576	136	16	v	v	NUM
ejpam-4576	136	17	of	of	ADP
ejpam-4576	136	18	y	y	PRON
ejpam-4576	136	19	such	such	ADJ
ejpam-4576	136	20	that	that	SCONJ
ejpam-4576	136	21	f	f	PROPN
ejpam-4576	136	22	(	(	PUNCT
ejpam-4576	136	23	x	x	NOUN
ejpam-4576	136	24	)	)	PUNCT
ejpam-4576	136	25	∩	∩	NOUN
ejpam-4576	136	26	v	v	ADP
ejpam-4576	136	27	̸=	̸=	PROPN
ejpam-4576	136	28	∅	∅	NOUN
ejpam-4576	136	29	;	;	PUNCT
ejpam-4576	136	30	(	(	PUNCT
ejpam-4576	136	31	4	4	X
ejpam-4576	136	32	)	)	PUNCT
ejpam-4576	137	1	x	x	SYM
ejpam-4576	137	2	∈	∈	PROPN
ejpam-4576	138	1	[	[	X
ejpam-4576	138	2	f−(v	f−(v	NOUN
ejpam-4576	138	3	)	)	PUNCT
ejpam-4576	138	4	]	]	PUNCT
ejpam-4576	138	5	(	(	PUNCT
ejpam-4576	138	6	λ	λ	NOUN
ejpam-4576	138	7	,	,	PUNCT
ejpam-4576	138	8	sp	sp	NOUN
ejpam-4576	138	9	)	)	PUNCT
ejpam-4576	138	10	for	for	ADP
ejpam-4576	138	11	every	every	DET
ejpam-4576	138	12	r(λ	r(λ	NOUN
ejpam-4576	138	13	,	,	PUNCT
ejpam-4576	138	14	sp)-open	sp)-open	ADJ
ejpam-4576	138	15	set	set	VERB
ejpam-4576	138	16	v	v	NUM
ejpam-4576	138	17	of	of	ADP
ejpam-4576	138	18	y	y	PROPN
ejpam-4576	138	19	containing	contain	VERB
ejpam-4576	138	20	f	f	PROPN
ejpam-4576	138	21	(	(	PUNCT
ejpam-4576	138	22	x	x	NOUN
ejpam-4576	138	23	)	)	PUNCT
ejpam-4576	138	24	;	;	PUNCT
ejpam-4576	138	25	(	(	PUNCT
ejpam-4576	138	26	5	5	X
ejpam-4576	138	27	)	)	PUNCT
ejpam-4576	138	28	for	for	ADP
ejpam-4576	138	29	each	each	DET
ejpam-4576	138	30	r(λ	r(λ	NOUN
ejpam-4576	138	31	,	,	PUNCT
ejpam-4576	138	32	sp)-open	sp)-open	VERB
ejpam-4576	138	33	set	set	VERB
ejpam-4576	138	34	v	v	NUM
ejpam-4576	138	35	of	of	ADP
ejpam-4576	138	36	y	y	PRON
ejpam-4576	138	37	such	such	ADJ
ejpam-4576	138	38	that	that	SCONJ
ejpam-4576	138	39	f	f	PROPN
ejpam-4576	138	40	(	(	PUNCT
ejpam-4576	138	41	x)∩v	x)∩v	PROPN
ejpam-4576	138	42	̸=	̸=	PROPN
ejpam-4576	138	43	∅	∅	NOUN
ejpam-4576	138	44	,	,	PUNCT
ejpam-4576	138	45	there	there	PRON
ejpam-4576	138	46	exists	exist	VERB
ejpam-4576	138	47	a	a	DET
ejpam-4576	138	48	(	(	PUNCT
ejpam-4576	138	49	λ	λ	NOUN
ejpam-4576	138	50	,	,	PUNCT
ejpam-4576	138	51	sp)-open	sp)-open	NOUN
ejpam-4576	138	52	set	set	VERB
ejpam-4576	138	53	u	u	NOUN
ejpam-4576	138	54	of	of	ADP
ejpam-4576	138	55	x	x	PUNCT
ejpam-4576	138	56	containing	contain	VERB
ejpam-4576	138	57	x	x	PUNCT
ejpam-4576	138	58	such	such	ADJ
ejpam-4576	138	59	that	that	SCONJ
ejpam-4576	138	60	u	u	NOUN
ejpam-4576	138	61	⊆	⊆	NUM
ejpam-4576	138	62	f−(v	f−(v	NOUN
ejpam-4576	138	63	)	)	PUNCT
ejpam-4576	138	64	.	.	PUNCT
ejpam-4576	139	1	proof	proof	NOUN
ejpam-4576	139	2	.	.	PUNCT
ejpam-4576	140	1	the	the	DET
ejpam-4576	140	2	proof	proof	NOUN
ejpam-4576	140	3	is	be	AUX
ejpam-4576	140	4	similar	similar	ADJ
ejpam-4576	140	5	to	to	ADP
ejpam-4576	140	6	that	that	PRON
ejpam-4576	140	7	of	of	ADP
ejpam-4576	140	8	theorem	theorem	NOUN
ejpam-4576	140	9	1	1	NUM
ejpam-4576	140	10	.	.	PUNCT
ejpam-4576	140	11	definition	definition	NOUN
ejpam-4576	140	12	2	2	NUM
ejpam-4576	140	13	.	.	PUNCT
ejpam-4576	141	1	a	a	DET
ejpam-4576	141	2	function	function	NOUN
ejpam-4576	141	3	f	f	NOUN
ejpam-4576	141	4	:	:	PUNCT
ejpam-4576	141	5	(	(	PUNCT
ejpam-4576	141	6	x	x	X
ejpam-4576	141	7	,	,	PUNCT
ejpam-4576	141	8	τ	τ	X
ejpam-4576	141	9	)	)	PUNCT
ejpam-4576	141	10	→	→	SYM
ejpam-4576	141	11	(	(	PUNCT
ejpam-4576	141	12	y	y	PROPN
ejpam-4576	141	13	,	,	PUNCT
ejpam-4576	141	14	σ	σ	PROPN
ejpam-4576	141	15	)	)	PUNCT
ejpam-4576	141	16	is	be	AUX
ejpam-4576	141	17	called	call	VERB
ejpam-4576	141	18	almost	almost	ADV
ejpam-4576	141	19	(	(	PUNCT
ejpam-4576	141	20	λ	λ	NOUN
ejpam-4576	141	21	,	,	PUNCT
ejpam-4576	141	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	141	23	at	at	ADP
ejpam-4576	141	24	a	a	DET
ejpam-4576	141	25	point	point	NOUN
ejpam-4576	141	26	x	x	SYM
ejpam-4576	141	27	∈	∈	NOUN
ejpam-4576	141	28	x	x	INTJ
ejpam-4576	141	29	if	if	SCONJ
ejpam-4576	141	30	,	,	PUNCT
ejpam-4576	141	31	for	for	ADP
ejpam-4576	141	32	each	each	DET
ejpam-4576	141	33	(	(	PUNCT
ejpam-4576	141	34	λ	λ	PROPN
ejpam-4576	141	35	,	,	PUNCT
ejpam-4576	141	36	sp)-open	sp)-open	NOUN
ejpam-4576	141	37	set	set	VERB
ejpam-4576	141	38	v	v	NUM
ejpam-4576	141	39	of	of	ADP
ejpam-4576	141	40	y	y	NOUN
ejpam-4576	141	41	containing	contain	VERB
ejpam-4576	141	42	f(x	f(x	PROPN
ejpam-4576	141	43	)	)	PUNCT
ejpam-4576	141	44	,	,	PUNCT
ejpam-4576	141	45	there	there	PRON
ejpam-4576	141	46	exists	exist	VERB
ejpam-4576	141	47	a	a	DET
ejpam-4576	141	48	(	(	PUNCT
ejpam-4576	141	49	λ	λ	NOUN
ejpam-4576	141	50	,	,	PUNCT
ejpam-4576	141	51	sp)-open	sp)-open	NOUN
ejpam-4576	141	52	set	set	VERB
ejpam-4576	141	53	u	u	NOUN
ejpam-4576	141	54	of	of	ADP
ejpam-4576	141	55	x	x	PUNCT
ejpam-4576	141	56	containing	contain	VERB
ejpam-4576	141	57	x	x	PUNCT
ejpam-4576	141	58	such	such	ADJ
ejpam-4576	141	59	that	that	DET
ejpam-4576	141	60	f(u	f(u	PROPN
ejpam-4576	141	61	)	)	PUNCT
ejpam-4576	142	1	⊆	⊆	NUM
ejpam-4576	142	2	[	[	X
ejpam-4576	142	3	v	v	X
ejpam-4576	142	4	(	(	PUNCT
ejpam-4576	142	5	λ	λ	PROPN
ejpam-4576	142	6	,	,	PUNCT
ejpam-4576	142	7	sp)](λ	sp)](λ	PROPN
ejpam-4576	142	8	,	,	PUNCT
ejpam-4576	142	9	sp	sp	NOUN
ejpam-4576	142	10	)	)	PUNCT
ejpam-4576	142	11	.	.	PUNCT
ejpam-4576	143	1	a	a	DET
ejpam-4576	143	2	function	function	NOUN
ejpam-4576	143	3	f	f	NOUN
ejpam-4576	143	4	:	:	PUNCT
ejpam-4576	143	5	(	(	PUNCT
ejpam-4576	143	6	x	x	X
ejpam-4576	143	7	,	,	PUNCT
ejpam-4576	143	8	τ	τ	X
ejpam-4576	143	9	)	)	PUNCT
ejpam-4576	143	10	→	→	SYM
ejpam-4576	143	11	(	(	PUNCT
ejpam-4576	143	12	y	y	PROPN
ejpam-4576	143	13	,	,	PUNCT
ejpam-4576	143	14	σ	σ	PROPN
ejpam-4576	143	15	)	)	PUNCT
ejpam-4576	143	16	is	be	AUX
ejpam-4576	143	17	called	call	VERB
ejpam-4576	143	18	almost	almost	ADV
ejpam-4576	143	19	(	(	PUNCT
ejpam-4576	143	20	λ	λ	NOUN
ejpam-4576	143	21	,	,	PUNCT
ejpam-4576	143	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	143	23	if	if	SCONJ
ejpam-4576	143	24	f	f	PROPN
ejpam-4576	143	25	has	have	VERB
ejpam-4576	143	26	this	this	DET
ejpam-4576	143	27	property	property	NOUN
ejpam-4576	143	28	at	at	ADP
ejpam-4576	143	29	each	each	DET
ejpam-4576	143	30	point	point	NOUN
ejpam-4576	143	31	of	of	ADP
ejpam-4576	143	32	x.	x.	PROPN
ejpam-4576	143	33	corollary	corollary	NOUN
ejpam-4576	143	34	1	1	NUM
ejpam-4576	143	35	.	.	PUNCT
ejpam-4576	144	1	for	for	ADP
ejpam-4576	144	2	a	a	DET
ejpam-4576	144	3	function	function	NOUN
ejpam-4576	144	4	f	f	NOUN
ejpam-4576	144	5	:	:	PUNCT
ejpam-4576	144	6	(	(	PUNCT
ejpam-4576	144	7	x	x	X
ejpam-4576	144	8	,	,	PUNCT
ejpam-4576	144	9	τ	τ	X
ejpam-4576	144	10	)	)	PUNCT
ejpam-4576	144	11	→	→	SYM
ejpam-4576	144	12	(	(	PUNCT
ejpam-4576	144	13	y	y	PROPN
ejpam-4576	144	14	,	,	PUNCT
ejpam-4576	144	15	σ	σ	PROPN
ejpam-4576	144	16	)	)	PUNCT
ejpam-4576	144	17	,	,	PUNCT
ejpam-4576	144	18	the	the	DET
ejpam-4576	144	19	following	follow	VERB
ejpam-4576	144	20	properties	property	NOUN
ejpam-4576	144	21	are	be	AUX
ejpam-4576	144	22	equivalent	equivalent	ADJ
ejpam-4576	144	23	:	:	PUNCT
ejpam-4576	144	24	(	(	PUNCT
ejpam-4576	144	25	1	1	X
ejpam-4576	144	26	)	)	PUNCT
ejpam-4576	144	27	f	f	NOUN
ejpam-4576	144	28	is	be	AUX
ejpam-4576	144	29	almost	almost	ADV
ejpam-4576	144	30	(	(	PUNCT
ejpam-4576	144	31	λ	λ	X
ejpam-4576	144	32	,	,	PUNCT
ejpam-4576	144	33	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	144	34	at	at	ADP
ejpam-4576	144	35	x	x	X
ejpam-4576	144	36	∈	∈	PROPN
ejpam-4576	144	37	x	x	X
ejpam-4576	144	38	;	;	PUNCT
ejpam-4576	144	39	(	(	PUNCT
ejpam-4576	144	40	2	2	X
ejpam-4576	144	41	)	)	PUNCT
ejpam-4576	144	42	x	x	SYM
ejpam-4576	144	43	∈	∈	PROPN
ejpam-4576	145	1	[	[	X
ejpam-4576	145	2	f−1([v	f−1([v	ADJ
ejpam-4576	145	3	(	(	PUNCT
ejpam-4576	145	4	λ	λ	PROPN
ejpam-4576	145	5	,	,	PUNCT
ejpam-4576	145	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	145	7	,	,	PUNCT
ejpam-4576	145	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	145	9	,	,	PUNCT
ejpam-4576	145	10	sp	sp	NOUN
ejpam-4576	145	11	)	)	PUNCT
ejpam-4576	145	12	for	for	ADP
ejpam-4576	145	13	every	every	DET
ejpam-4576	145	14	(	(	PUNCT
ejpam-4576	145	15	λ	λ	NOUN
ejpam-4576	145	16	,	,	PUNCT
ejpam-4576	145	17	sp)-open	sp)-open	NOUN
ejpam-4576	145	18	set	set	VERB
ejpam-4576	145	19	v	v	NUM
ejpam-4576	145	20	of	of	ADP
ejpam-4576	145	21	y	y	NOUN
ejpam-4576	145	22	containing	contain	VERB
ejpam-4576	145	23	f(x	f(x	PROPN
ejpam-4576	145	24	)	)	PUNCT
ejpam-4576	145	25	;	;	PUNCT
ejpam-4576	145	26	(	(	PUNCT
ejpam-4576	145	27	3	3	X
ejpam-4576	145	28	)	)	PUNCT
ejpam-4576	145	29	x	x	SYM
ejpam-4576	145	30	∈	∈	PROPN
ejpam-4576	146	1	[	[	X
ejpam-4576	146	2	f−1(v	f−1(v	NOUN
ejpam-4576	146	3	s(λ	s(λ	PROPN
ejpam-4576	146	4	,	,	PUNCT
ejpam-4576	146	5	sp))](λ	sp))](λ	PROPN
ejpam-4576	146	6	,	,	PUNCT
ejpam-4576	146	7	sp	sp	NOUN
ejpam-4576	146	8	)	)	PUNCT
ejpam-4576	146	9	for	for	ADP
ejpam-4576	146	10	every	every	DET
ejpam-4576	146	11	(	(	PUNCT
ejpam-4576	146	12	λ	λ	NOUN
ejpam-4576	146	13	,	,	PUNCT
ejpam-4576	146	14	sp)-open	sp)-open	NOUN
ejpam-4576	146	15	set	set	VERB
ejpam-4576	146	16	v	v	NUM
ejpam-4576	146	17	of	of	ADP
ejpam-4576	146	18	y	y	NOUN
ejpam-4576	146	19	containing	contain	VERB
ejpam-4576	146	20	f(x	f(x	PROPN
ejpam-4576	146	21	)	)	PUNCT
ejpam-4576	146	22	;	;	PUNCT
ejpam-4576	146	23	(	(	PUNCT
ejpam-4576	146	24	4	4	X
ejpam-4576	146	25	)	)	PUNCT
ejpam-4576	146	26	x	x	SYM
ejpam-4576	146	27	∈	∈	PROPN
ejpam-4576	146	28	[	[	X
ejpam-4576	146	29	f−1(v	f−1(v	NOUN
ejpam-4576	146	30	)	)	PUNCT
ejpam-4576	146	31	]	]	PUNCT
ejpam-4576	146	32	(	(	PUNCT
ejpam-4576	146	33	λ	λ	NOUN
ejpam-4576	146	34	,	,	PUNCT
ejpam-4576	146	35	sp	sp	NOUN
ejpam-4576	146	36	)	)	PUNCT
ejpam-4576	146	37	for	for	ADP
ejpam-4576	146	38	every	every	DET
ejpam-4576	146	39	r(λ	r(λ	NOUN
ejpam-4576	146	40	,	,	PUNCT
ejpam-4576	146	41	sp)-open	sp)-open	ADJ
ejpam-4576	146	42	set	set	VERB
ejpam-4576	146	43	v	v	NUM
ejpam-4576	146	44	of	of	ADP
ejpam-4576	146	45	y	y	NOUN
ejpam-4576	146	46	containing	contain	VERB
ejpam-4576	146	47	f(x	f(x	PROPN
ejpam-4576	146	48	)	)	PUNCT
ejpam-4576	146	49	;	;	PUNCT
ejpam-4576	146	50	(	(	PUNCT
ejpam-4576	146	51	5	5	X
ejpam-4576	146	52	)	)	PUNCT
ejpam-4576	146	53	for	for	ADP
ejpam-4576	146	54	each	each	DET
ejpam-4576	146	55	r(λ	r(λ	NOUN
ejpam-4576	146	56	,	,	PUNCT
ejpam-4576	146	57	sp)-open	sp)-open	VERB
ejpam-4576	146	58	set	set	VERB
ejpam-4576	146	59	v	v	NUM
ejpam-4576	146	60	of	of	ADP
ejpam-4576	146	61	y	y	NOUN
ejpam-4576	146	62	containing	contain	VERB
ejpam-4576	146	63	f(x	f(x	PROPN
ejpam-4576	146	64	)	)	PUNCT
ejpam-4576	146	65	,	,	PUNCT
ejpam-4576	146	66	there	there	PRON
ejpam-4576	146	67	exists	exist	VERB
ejpam-4576	146	68	a	a	DET
ejpam-4576	146	69	(	(	PUNCT
ejpam-4576	146	70	λ	λ	NOUN
ejpam-4576	146	71	,	,	PUNCT
ejpam-4576	146	72	sp)-open	sp)-open	NOUN
ejpam-4576	146	73	set	set	VERB
ejpam-4576	146	74	u	u	NOUN
ejpam-4576	146	75	of	of	ADP
ejpam-4576	146	76	x	x	PUNCT
ejpam-4576	146	77	containing	contain	VERB
ejpam-4576	146	78	x	x	PUNCT
ejpam-4576	146	79	such	such	ADJ
ejpam-4576	146	80	that	that	SCONJ
ejpam-4576	146	81	u	u	PROPN
ejpam-4576	146	82	⊆	⊆	NUM
ejpam-4576	146	83	f−1(v	f−1(v	NOUN
ejpam-4576	146	84	)	)	PUNCT
ejpam-4576	146	85	.	.	PUNCT
ejpam-4576	147	1	theorem	theorem	NOUN
ejpam-4576	147	2	3	3	NUM
ejpam-4576	147	3	.	.	X
ejpam-4576	147	4	for	for	ADP
ejpam-4576	147	5	a	a	DET
ejpam-4576	147	6	multifunction	multifunction	NOUN
ejpam-4576	148	1	f	f	NOUN
ejpam-4576	148	2	:	:	PUNCT
ejpam-4576	148	3	(	(	PUNCT
ejpam-4576	148	4	x	x	X
ejpam-4576	148	5	,	,	PUNCT
ejpam-4576	148	6	τ	τ	X
ejpam-4576	148	7	)	)	PUNCT
ejpam-4576	148	8	→	→	SYM
ejpam-4576	148	9	(	(	PUNCT
ejpam-4576	148	10	y	y	PROPN
ejpam-4576	148	11	,	,	PUNCT
ejpam-4576	148	12	σ	σ	PROPN
ejpam-4576	148	13	)	)	PUNCT
ejpam-4576	148	14	,	,	PUNCT
ejpam-4576	148	15	the	the	DET
ejpam-4576	148	16	following	follow	VERB
ejpam-4576	148	17	properties	property	NOUN
ejpam-4576	148	18	are	be	AUX
ejpam-4576	148	19	equivalent	equivalent	ADJ
ejpam-4576	148	20	:	:	PUNCT
ejpam-4576	148	21	(	(	PUNCT
ejpam-4576	148	22	1	1	X
ejpam-4576	148	23	)	)	PUNCT
ejpam-4576	148	24	f	f	PROPN
ejpam-4576	148	25	is	be	AUX
ejpam-4576	148	26	upper	upper	ADJ
ejpam-4576	148	27	almost	almost	ADV
ejpam-4576	148	28	(	(	PUNCT
ejpam-4576	148	29	λ	λ	NOUN
ejpam-4576	148	30	,	,	PUNCT
ejpam-4576	148	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	148	32	;	;	PUNCT
ejpam-4576	148	33	(	(	PUNCT
ejpam-4576	148	34	2	2	NUM
ejpam-4576	148	35	)	)	PUNCT
ejpam-4576	148	36	f+(v	f+(v	NOUN
ejpam-4576	148	37	)	)	PUNCT
ejpam-4576	149	1	⊆	⊆	NUM
ejpam-4576	150	1	[	[	X
ejpam-4576	150	2	f+([v	f+([v	X
ejpam-4576	150	3	(	(	PUNCT
ejpam-4576	150	4	λ	λ	PROPN
ejpam-4576	150	5	,	,	PUNCT
ejpam-4576	150	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	150	7	,	,	PUNCT
ejpam-4576	150	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	150	9	,	,	PUNCT
ejpam-4576	150	10	sp	sp	NOUN
ejpam-4576	150	11	)	)	PUNCT
ejpam-4576	150	12	for	for	ADP
ejpam-4576	150	13	every	every	DET
ejpam-4576	150	14	(	(	PUNCT
ejpam-4576	150	15	λ	λ	NOUN
ejpam-4576	150	16	,	,	PUNCT
ejpam-4576	150	17	sp)-open	sp)-open	NOUN
ejpam-4576	150	18	set	set	VERB
ejpam-4576	150	19	v	v	NOUN
ejpam-4576	150	20	of	of	ADP
ejpam-4576	150	21	y	y	PROPN
ejpam-4576	150	22	;	;	PUNCT
ejpam-4576	150	23	(	(	PUNCT
ejpam-4576	150	24	3	3	X
ejpam-4576	150	25	)	)	PUNCT
ejpam-4576	150	26	[	[	X
ejpam-4576	150	27	f−([k(λ	f−([k(λ	NOUN
ejpam-4576	150	28	,	,	PUNCT
ejpam-4576	150	29	sp	sp	NOUN
ejpam-4576	150	30	)	)	PUNCT
ejpam-4576	150	31	]	]	PUNCT
ejpam-4576	150	32	(	(	PUNCT
ejpam-4576	150	33	λ	λ	INTJ
ejpam-4576	150	34	,	,	PUNCT
ejpam-4576	150	35	sp))](λ	sp))](λ	PROPN
ejpam-4576	150	36	,	,	PUNCT
ejpam-4576	150	37	sp	sp	NOUN
ejpam-4576	150	38	)	)	PUNCT
ejpam-4576	150	39	⊆	⊆	NUM
ejpam-4576	150	40	f−(k	f−(k	PROPN
ejpam-4576	150	41	)	)	PUNCT
ejpam-4576	150	42	for	for	SCONJ
ejpam-4576	150	43	every	every	DET
ejpam-4576	150	44	(	(	PUNCT
ejpam-4576	150	45	λ	λ	PROPN
ejpam-4576	150	46	,	,	PUNCT
ejpam-4576	150	47	sp)-closed	sp)-close	VERB
ejpam-4576	150	48	set	set	VERB
ejpam-4576	150	49	k	k	PROPN
ejpam-4576	150	50	of	of	ADP
ejpam-4576	150	51	y	y	PROPN
ejpam-4576	150	52	;	;	PUNCT
ejpam-4576	150	53	(	(	PUNCT
ejpam-4576	150	54	4	4	X
ejpam-4576	150	55	)	)	PUNCT
ejpam-4576	151	1	[	[	X
ejpam-4576	151	2	f−([[b(λ	f−([[b(λ	X
ejpam-4576	151	3	,	,	PUNCT
ejpam-4576	151	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	151	5	,	,	PUNCT
ejpam-4576	151	6	sp	sp	NOUN
ejpam-4576	151	7	)	)	PUNCT
ejpam-4576	151	8	]	]	PUNCT
ejpam-4576	151	9	(	(	PUNCT
ejpam-4576	151	10	λ	λ	INTJ
ejpam-4576	151	11	,	,	PUNCT
ejpam-4576	151	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	151	13	,	,	PUNCT
ejpam-4576	151	14	sp	sp	NOUN
ejpam-4576	151	15	)	)	PUNCT
ejpam-4576	151	16	⊆	⊆	NUM
ejpam-4576	151	17	f−(b(λ	f−(b(λ	NOUN
ejpam-4576	151	18	,	,	PUNCT
ejpam-4576	151	19	sp	sp	NOUN
ejpam-4576	151	20	)	)	PUNCT
ejpam-4576	151	21	)	)	PUNCT
ejpam-4576	151	22	for	for	ADP
ejpam-4576	151	23	every	every	DET
ejpam-4576	151	24	subset	subset	NOUN
ejpam-4576	151	25	b	b	PROPN
ejpam-4576	151	26	of	of	ADP
ejpam-4576	151	27	y	y	PROPN
ejpam-4576	151	28	;	;	PUNCT
ejpam-4576	151	29	(	(	PUNCT
ejpam-4576	151	30	5	5	X
ejpam-4576	151	31	)	)	PUNCT
ejpam-4576	151	32	f+(b(λ	f+(b(λ	PROPN
ejpam-4576	151	33	,	,	PUNCT
ejpam-4576	151	34	sp	sp	NOUN
ejpam-4576	151	35	)	)	PUNCT
ejpam-4576	151	36	)	)	PUNCT
ejpam-4576	152	1	⊆	⊆	NUM
ejpam-4576	152	2	[	[	X
ejpam-4576	152	3	f+([[b(λ	f+([[b(λ	NUM
ejpam-4576	152	4	,	,	PUNCT
ejpam-4576	152	5	sp	sp	NOUN
ejpam-4576	152	6	)	)	PUNCT
ejpam-4576	152	7	]	]	PUNCT
ejpam-4576	153	1	(	(	PUNCT
ejpam-4576	153	2	λ	λ	X
ejpam-4576	153	3	,	,	PUNCT
ejpam-4576	153	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	153	5	,	,	PUNCT
ejpam-4576	153	6	sp))](λ	sp))](λ	PROPN
ejpam-4576	153	7	,	,	PUNCT
ejpam-4576	153	8	sp	sp	NOUN
ejpam-4576	153	9	)	)	PUNCT
ejpam-4576	153	10	for	for	ADP
ejpam-4576	153	11	every	every	DET
ejpam-4576	153	12	subset	subset	NOUN
ejpam-4576	153	13	b	b	PROPN
ejpam-4576	153	14	of	of	ADP
ejpam-4576	153	15	y	y	PROPN
ejpam-4576	153	16	;	;	PUNCT
ejpam-4576	153	17	(	(	PUNCT
ejpam-4576	153	18	6	6	NUM
ejpam-4576	153	19	)	)	PUNCT
ejpam-4576	153	20	f+(v	f+(v	NOUN
ejpam-4576	153	21	)	)	PUNCT
ejpam-4576	153	22	is	be	AUX
ejpam-4576	153	23	(	(	PUNCT
ejpam-4576	153	24	λ	λ	INTJ
ejpam-4576	153	25	,	,	PUNCT
ejpam-4576	153	26	sp)-open	sp)-open	ADJ
ejpam-4576	153	27	in	in	ADP
ejpam-4576	153	28	x	x	PUNCT
ejpam-4576	153	29	for	for	ADP
ejpam-4576	153	30	every	every	DET
ejpam-4576	153	31	r(λ	r(λ	NOUN
ejpam-4576	153	32	,	,	PUNCT
ejpam-4576	153	33	sp)-open	sp)-open	ADJ
ejpam-4576	153	34	set	set	VERB
ejpam-4576	153	35	v	v	NOUN
ejpam-4576	153	36	of	of	ADP
ejpam-4576	153	37	y	y	PROPN
ejpam-4576	153	38	;	;	PUNCT
ejpam-4576	153	39	(	(	PUNCT
ejpam-4576	153	40	7	7	X
ejpam-4576	153	41	)	)	PUNCT
ejpam-4576	153	42	f−(k	f−(k	PROPN
ejpam-4576	153	43	)	)	PUNCT
ejpam-4576	153	44	is	be	AUX
ejpam-4576	153	45	(	(	PUNCT
ejpam-4576	153	46	λ	λ	X
ejpam-4576	153	47	,	,	PUNCT
ejpam-4576	153	48	sp)-closed	sp)-close	VERB
ejpam-4576	153	49	in	in	ADP
ejpam-4576	153	50	x	x	PUNCT
ejpam-4576	153	51	for	for	ADP
ejpam-4576	153	52	every	every	DET
ejpam-4576	153	53	r(λ	r(λ	NOUN
ejpam-4576	153	54	,	,	PUNCT
ejpam-4576	153	55	sp)-closed	sp)-close	VERB
ejpam-4576	153	56	set	set	VERB
ejpam-4576	153	57	k	k	PROPN
ejpam-4576	153	58	of	of	ADP
ejpam-4576	153	59	y	y	PROPN
ejpam-4576	153	60	.	.	PUNCT
ejpam-4576	154	1	c.	c.	PROPN
ejpam-4576	154	2	boonpok	boonpok	PROPN
ejpam-4576	154	3	,	,	PUNCT
ejpam-4576	154	4	n.	n.	PROPN
ejpam-4576	154	5	viriyapong	viriyapong	PROPN
ejpam-4576	154	6	/	/	SYM
ejpam-4576	154	7	eur	eur	PROPN
ejpam-4576	154	8	.	.	PUNCT
ejpam-4576	155	1	j.	j.	PROPN
ejpam-4576	155	2	pure	pure	PROPN
ejpam-4576	155	3	appl	appl	PROPN
ejpam-4576	155	4	.	.	PROPN
ejpam-4576	155	5	math	math	PROPN
ejpam-4576	155	6	,	,	PUNCT
ejpam-4576	155	7	16	16	NUM
ejpam-4576	155	8	(	(	PUNCT
ejpam-4576	155	9	1	1	NUM
ejpam-4576	155	10	)	)	PUNCT
ejpam-4576	155	11	(	(	PUNCT
ejpam-4576	155	12	2023	2023	NUM
ejpam-4576	155	13	)	)	PUNCT
ejpam-4576	155	14	,	,	PUNCT
ejpam-4576	155	15	84	84	NUM
ejpam-4576	155	16	-	-	SYM
ejpam-4576	155	17	96	96	NUM
ejpam-4576	155	18	89	89	NUM
ejpam-4576	155	19	proof	proof	NOUN
ejpam-4576	155	20	.	.	PUNCT
ejpam-4576	156	1	(	(	PUNCT
ejpam-4576	156	2	1	1	X
ejpam-4576	156	3	)	)	PUNCT
ejpam-4576	156	4	⇒	⇒	NOUN
ejpam-4576	156	5	(	(	PUNCT
ejpam-4576	156	6	2	2	NUM
ejpam-4576	156	7	):	):	PUNCT
ejpam-4576	156	8	let	let	VERB
ejpam-4576	156	9	v	v	PART
ejpam-4576	156	10	be	be	AUX
ejpam-4576	156	11	any	any	DET
ejpam-4576	156	12	(	(	PUNCT
ejpam-4576	156	13	λ	λ	NOUN
ejpam-4576	156	14	,	,	PUNCT
ejpam-4576	156	15	sp)-open	sp)-open	ADJ
ejpam-4576	156	16	set	set	NOUN
ejpam-4576	156	17	of	of	ADP
ejpam-4576	156	18	y	y	PROPN
ejpam-4576	156	19	and	and	CCONJ
ejpam-4576	156	20	x	x	PROPN
ejpam-4576	156	21	∈	∈	PROPN
ejpam-4576	156	22	f+(v	f+(v	NOUN
ejpam-4576	156	23	)	)	PUNCT
ejpam-4576	156	24	.	.	PUNCT
ejpam-4576	157	1	then	then	ADV
ejpam-4576	157	2	,	,	PUNCT
ejpam-4576	157	3	f	f	PROPN
ejpam-4576	157	4	(	(	PUNCT
ejpam-4576	157	5	x	x	X
ejpam-4576	157	6	)	)	PUNCT
ejpam-4576	157	7	⊆	⊆	NUM
ejpam-4576	157	8	v	v	NOUN
ejpam-4576	157	9	.	.	PUNCT
ejpam-4576	158	1	thus	thus	ADV
ejpam-4576	158	2	,	,	PUNCT
ejpam-4576	158	3	by	by	ADP
ejpam-4576	158	4	theorem	theorem	NOUN
ejpam-4576	158	5	1	1	NUM
ejpam-4576	158	6	,	,	PUNCT
ejpam-4576	158	7	x	x	SYM
ejpam-4576	158	8	∈	∈	PROPN
ejpam-4576	159	1	[	[	X
ejpam-4576	159	2	f+([v	f+([v	X
ejpam-4576	159	3	(	(	PUNCT
ejpam-4576	159	4	λ	λ	PROPN
ejpam-4576	159	5	,	,	PUNCT
ejpam-4576	159	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	159	7	,	,	PUNCT
ejpam-4576	159	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	159	9	,	,	PUNCT
ejpam-4576	159	10	sp	sp	NOUN
ejpam-4576	159	11	)	)	PUNCT
ejpam-4576	159	12	and	and	CCONJ
ejpam-4576	159	13	hence	hence	ADV
ejpam-4576	159	14	f+(v	f+(v	NOUN
ejpam-4576	159	15	)	)	PUNCT
ejpam-4576	159	16	⊆	⊆	NUM
ejpam-4576	160	1	[	[	X
ejpam-4576	160	2	f+([v	f+([v	X
ejpam-4576	160	3	(	(	PUNCT
ejpam-4576	160	4	λ	λ	PROPN
ejpam-4576	160	5	,	,	PUNCT
ejpam-4576	160	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	160	7	,	,	PUNCT
ejpam-4576	160	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	160	9	,	,	PUNCT
ejpam-4576	160	10	sp	sp	NOUN
ejpam-4576	160	11	)	)	PUNCT
ejpam-4576	160	12	.	.	PUNCT
ejpam-4576	161	1	(	(	PUNCT
ejpam-4576	161	2	2	2	X
ejpam-4576	161	3	)	)	PUNCT
ejpam-4576	161	4	⇒	⇒	NOUN
ejpam-4576	161	5	(	(	PUNCT
ejpam-4576	161	6	3	3	NUM
ejpam-4576	161	7	):	):	PUNCT
ejpam-4576	161	8	let	let	VERB
ejpam-4576	161	9	k	k	PRON
ejpam-4576	161	10	be	be	AUX
ejpam-4576	161	11	any	any	DET
ejpam-4576	161	12	(	(	PUNCT
ejpam-4576	161	13	λ	λ	PROPN
ejpam-4576	161	14	,	,	PUNCT
ejpam-4576	161	15	sp)-closed	sp)-close	VERB
ejpam-4576	161	16	set	set	NOUN
ejpam-4576	161	17	of	of	ADP
ejpam-4576	161	18	y	y	PROPN
ejpam-4576	161	19	.	.	PUNCT
ejpam-4576	162	1	then	then	ADV
ejpam-4576	162	2	,	,	PUNCT
ejpam-4576	162	3	y	y	PROPN
ejpam-4576	162	4	−	−	PROPN
ejpam-4576	162	5	k	k	PROPN
ejpam-4576	162	6	is	be	AUX
ejpam-4576	162	7	(	(	PUNCT
ejpam-4576	162	8	λ	λ	INTJ
ejpam-4576	162	9	,	,	PUNCT
ejpam-4576	162	10	sp)-open	sp)-open	ADJ
ejpam-4576	162	11	in	in	ADP
ejpam-4576	162	12	y	y	PROPN
ejpam-4576	162	13	and	and	CCONJ
ejpam-4576	162	14	by	by	ADP
ejpam-4576	162	15	(	(	PUNCT
ejpam-4576	162	16	2	2	NUM
ejpam-4576	162	17	)	)	PUNCT
ejpam-4576	162	18	,	,	PUNCT
ejpam-4576	162	19	x	x	PUNCT
ejpam-4576	162	20	−	−	PRON
ejpam-4576	162	21	f−(k	f−(k	PROPN
ejpam-4576	162	22	)	)	PUNCT
ejpam-4576	162	23	=	=	PUNCT
ejpam-4576	163	1	f+(y	f+(y	PROPN
ejpam-4576	163	2	−k	−k	PROPN
ejpam-4576	163	3	)	)	PUNCT
ejpam-4576	163	4	⊆	⊆	NUM
ejpam-4576	164	1	[	[	X
ejpam-4576	164	2	f+([[y	f+([[y	X
ejpam-4576	164	3	−k](λ	−k](λ	NUM
ejpam-4576	164	4	,	,	PUNCT
ejpam-4576	164	5	sp)](λ	sp)](λ	PROPN
ejpam-4576	164	6	,	,	PUNCT
ejpam-4576	164	7	sp))](λ	sp))](λ	PROPN
ejpam-4576	164	8	,	,	PUNCT
ejpam-4576	164	9	sp	sp	NOUN
ejpam-4576	164	10	)	)	PUNCT
ejpam-4576	164	11	=	=	PUNCT
ejpam-4576	165	1	[	[	X
ejpam-4576	165	2	x	x	X
ejpam-4576	165	3	−	−	X
ejpam-4576	165	4	f−([k(λ	f−([k(λ	NOUN
ejpam-4576	165	5	,	,	PUNCT
ejpam-4576	165	6	sp	sp	NOUN
ejpam-4576	165	7	)	)	PUNCT
ejpam-4576	165	8	]	]	PUNCT
ejpam-4576	165	9	(	(	PUNCT
ejpam-4576	165	10	λ	λ	INTJ
ejpam-4576	165	11	,	,	PUNCT
ejpam-4576	165	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	165	13	,	,	PUNCT
ejpam-4576	165	14	sp	sp	NOUN
ejpam-4576	165	15	)	)	PUNCT
ejpam-4576	165	16	=	=	PUNCT
ejpam-4576	166	1	x	x	X
ejpam-4576	166	2	−	−	PROPN
ejpam-4576	167	1	[	[	X
ejpam-4576	167	2	f−([k(λ	f−([k(λ	NOUN
ejpam-4576	167	3	,	,	PUNCT
ejpam-4576	167	4	sp	sp	NOUN
ejpam-4576	167	5	)	)	PUNCT
ejpam-4576	167	6	]	]	PUNCT
ejpam-4576	167	7	(	(	PUNCT
ejpam-4576	167	8	λ	λ	INTJ
ejpam-4576	167	9	,	,	PUNCT
ejpam-4576	167	10	sp))](λ	sp))](λ	PROPN
ejpam-4576	167	11	,	,	PUNCT
ejpam-4576	167	12	sp	sp	NOUN
ejpam-4576	167	13	)	)	PUNCT
ejpam-4576	167	14	.	.	PUNCT
ejpam-4576	168	1	thus	thus	ADV
ejpam-4576	168	2	,	,	PUNCT
ejpam-4576	168	3	[	[	X
ejpam-4576	168	4	f−([k(λ	f−([k(λ	NOUN
ejpam-4576	168	5	,	,	PUNCT
ejpam-4576	168	6	sp	sp	NOUN
ejpam-4576	168	7	)	)	PUNCT
ejpam-4576	168	8	]	]	PUNCT
ejpam-4576	169	1	(	(	PUNCT
ejpam-4576	169	2	λ	λ	INTJ
ejpam-4576	169	3	,	,	PUNCT
ejpam-4576	169	4	sp))](λ	sp))](λ	PROPN
ejpam-4576	169	5	,	,	PUNCT
ejpam-4576	169	6	sp	sp	NOUN
ejpam-4576	169	7	)	)	PUNCT
ejpam-4576	169	8	⊆	⊆	NUM
ejpam-4576	169	9	f−(k	f−(k	PROPN
ejpam-4576	169	10	)	)	PUNCT
ejpam-4576	169	11	.	.	PUNCT
ejpam-4576	170	1	(	(	PUNCT
ejpam-4576	170	2	3	3	X
ejpam-4576	170	3	)	)	PUNCT
ejpam-4576	170	4	⇒	⇒	NOUN
ejpam-4576	170	5	(	(	PUNCT
ejpam-4576	170	6	4	4	NUM
ejpam-4576	170	7	):	):	PUNCT
ejpam-4576	170	8	let	let	VERB
ejpam-4576	170	9	b	b	X
ejpam-4576	170	10	be	be	AUX
ejpam-4576	170	11	any	any	DET
ejpam-4576	170	12	subset	subset	NOUN
ejpam-4576	170	13	of	of	ADP
ejpam-4576	170	14	y	y	PROPN
ejpam-4576	170	15	.	.	PUNCT
ejpam-4576	171	1	then	then	ADV
ejpam-4576	171	2	,	,	PUNCT
ejpam-4576	171	3	b(λ	b(λ	PROPN
ejpam-4576	171	4	,	,	PUNCT
ejpam-4576	171	5	sp	sp	NOUN
ejpam-4576	171	6	)	)	PUNCT
ejpam-4576	171	7	is	be	AUX
ejpam-4576	171	8	a	a	DET
ejpam-4576	171	9	(	(	PUNCT
ejpam-4576	171	10	λ	λ	PROPN
ejpam-4576	171	11	,	,	PUNCT
ejpam-4576	171	12	sp)-closed	sp)-close	VERB
ejpam-4576	171	13	set	set	NOUN
ejpam-4576	171	14	of	of	ADP
ejpam-4576	171	15	y	y	PROPN
ejpam-4576	171	16	and	and	CCONJ
ejpam-4576	171	17	by	by	ADP
ejpam-4576	171	18	(	(	PUNCT
ejpam-4576	171	19	3	3	NUM
ejpam-4576	171	20	)	)	PUNCT
ejpam-4576	171	21	,	,	PUNCT
ejpam-4576	171	22	[	[	X
ejpam-4576	171	23	f−([[b(λ	f−([[b(λ	X
ejpam-4576	171	24	,	,	PUNCT
ejpam-4576	171	25	sp)](λ	sp)](λ	PROPN
ejpam-4576	171	26	,	,	PUNCT
ejpam-4576	171	27	sp	sp	NOUN
ejpam-4576	171	28	)	)	PUNCT
ejpam-4576	171	29	]	]	PUNCT
ejpam-4576	171	30	(	(	PUNCT
ejpam-4576	171	31	λ	λ	INTJ
ejpam-4576	171	32	,	,	PUNCT
ejpam-4576	171	33	sp))](λ	sp))](λ	PROPN
ejpam-4576	171	34	,	,	PUNCT
ejpam-4576	171	35	sp	sp	NOUN
ejpam-4576	171	36	)	)	PUNCT
ejpam-4576	171	37	⊆	⊆	NUM
ejpam-4576	171	38	f−(b(λ	f−(b(λ	NOUN
ejpam-4576	171	39	,	,	PUNCT
ejpam-4576	171	40	sp	sp	NOUN
ejpam-4576	171	41	)	)	PUNCT
ejpam-4576	171	42	)	)	PUNCT
ejpam-4576	171	43	.	.	PUNCT
ejpam-4576	172	1	(	(	PUNCT
ejpam-4576	172	2	4	4	X
ejpam-4576	172	3	)	)	PUNCT
ejpam-4576	172	4	⇒	⇒	NOUN
ejpam-4576	172	5	(	(	PUNCT
ejpam-4576	172	6	5	5	NUM
ejpam-4576	172	7	):	):	PUNCT
ejpam-4576	172	8	let	let	VERB
ejpam-4576	172	9	b	b	X
ejpam-4576	172	10	be	be	AUX
ejpam-4576	172	11	any	any	DET
ejpam-4576	172	12	subset	subset	NOUN
ejpam-4576	172	13	of	of	ADP
ejpam-4576	172	14	y	y	PROPN
ejpam-4576	172	15	.	.	PUNCT
ejpam-4576	173	1	then	then	ADV
ejpam-4576	173	2	,	,	PUNCT
ejpam-4576	173	3	we	we	PRON
ejpam-4576	173	4	have	have	VERB
ejpam-4576	173	5	f+(b(λ	f+(b(λ	PROPN
ejpam-4576	173	6	,	,	PUNCT
ejpam-4576	173	7	sp	sp	NOUN
ejpam-4576	173	8	)	)	PUNCT
ejpam-4576	173	9	)	)	PUNCT
ejpam-4576	174	1	=	=	PUNCT
ejpam-4576	174	2	x	x	PUNCT
ejpam-4576	175	1	−	−	NOUN
ejpam-4576	175	2	f−([y	f−([y	ADJ
ejpam-4576	175	3	−b](λ	−b](λ	PROPN
ejpam-4576	175	4	,	,	PUNCT
ejpam-4576	175	5	sp	sp	NOUN
ejpam-4576	175	6	)	)	PUNCT
ejpam-4576	175	7	)	)	PUNCT
ejpam-4576	176	1	⊆	⊆	NUM
ejpam-4576	176	2	x	x	SYM
ejpam-4576	176	3	−	−	NOUN
ejpam-4576	177	1	[	[	X
ejpam-4576	177	2	f−([[[y	f−([[[y	VERB
ejpam-4576	177	3	−b](λ	−b](λ	NUM
ejpam-4576	177	4	,	,	PUNCT
ejpam-4576	177	5	sp)](λ	sp)](λ	PROPN
ejpam-4576	177	6	,	,	PUNCT
ejpam-4576	177	7	sp	sp	NOUN
ejpam-4576	177	8	)	)	PUNCT
ejpam-4576	177	9	]	]	PUNCT
ejpam-4576	177	10	(	(	PUNCT
ejpam-4576	177	11	λ	λ	INTJ
ejpam-4576	177	12	,	,	PUNCT
ejpam-4576	177	13	sp))](λ	sp))](λ	PROPN
ejpam-4576	177	14	,	,	PUNCT
ejpam-4576	177	15	sp	sp	NOUN
ejpam-4576	177	16	)	)	PUNCT
ejpam-4576	177	17	=	=	PUNCT
ejpam-4576	177	18	x	x	X
ejpam-4576	177	19	−	−	PUNCT
ejpam-4576	178	1	[	[	X
ejpam-4576	178	2	f−(y	f−(y	NOUN
ejpam-4576	178	3	−	−	PUNCT
ejpam-4576	179	1	[	[	X
ejpam-4576	179	2	[	[	X
ejpam-4576	179	3	b(λ	b(λ	NOUN
ejpam-4576	179	4	,	,	PUNCT
ejpam-4576	179	5	sp	sp	NOUN
ejpam-4576	179	6	)	)	PUNCT
ejpam-4576	179	7	]	]	PUNCT
ejpam-4576	179	8	(	(	PUNCT
ejpam-4576	179	9	λ	λ	X
ejpam-4576	179	10	,	,	PUNCT
ejpam-4576	179	11	sp)](λ	sp)](λ	PROPN
ejpam-4576	179	12	,	,	PUNCT
ejpam-4576	179	13	sp	sp	NOUN
ejpam-4576	179	14	)	)	PUNCT
ejpam-4576	179	15	)	)	PUNCT
ejpam-4576	179	16	]	]	PUNCT
ejpam-4576	180	1	(	(	PUNCT
ejpam-4576	180	2	λ	λ	NOUN
ejpam-4576	180	3	,	,	PUNCT
ejpam-4576	180	4	sp	sp	NOUN
ejpam-4576	180	5	)	)	PUNCT
ejpam-4576	180	6	=	=	PUNCT
ejpam-4576	181	1	[	[	X
ejpam-4576	181	2	f+([[b(λ	f+([[b(λ	ADJ
ejpam-4576	181	3	,	,	PUNCT
ejpam-4576	181	4	sp	sp	NOUN
ejpam-4576	181	5	)	)	PUNCT
ejpam-4576	181	6	]	]	PUNCT
ejpam-4576	181	7	(	(	PUNCT
ejpam-4576	181	8	λ	λ	X
ejpam-4576	181	9	,	,	PUNCT
ejpam-4576	181	10	sp)](λ	sp)](λ	PROPN
ejpam-4576	181	11	,	,	PUNCT
ejpam-4576	181	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	181	13	,	,	PUNCT
ejpam-4576	181	14	sp	sp	NOUN
ejpam-4576	181	15	)	)	PUNCT
ejpam-4576	181	16	.	.	PUNCT
ejpam-4576	182	1	(	(	PUNCT
ejpam-4576	182	2	5	5	X
ejpam-4576	182	3	)	)	PUNCT
ejpam-4576	182	4	⇒	⇒	NOUN
ejpam-4576	182	5	(	(	PUNCT
ejpam-4576	182	6	6	6	NUM
ejpam-4576	182	7	):	):	PUNCT
ejpam-4576	182	8	let	let	VERB
ejpam-4576	182	9	v	v	PART
ejpam-4576	182	10	be	be	AUX
ejpam-4576	182	11	any	any	DET
ejpam-4576	182	12	r(λ	r(λ	NOUN
ejpam-4576	182	13	,	,	PUNCT
ejpam-4576	182	14	sp)-open	sp)-open	ADJ
ejpam-4576	182	15	set	set	NOUN
ejpam-4576	182	16	of	of	ADP
ejpam-4576	182	17	y	y	PROPN
ejpam-4576	182	18	.	.	PUNCT
ejpam-4576	183	1	by	by	ADP
ejpam-4576	183	2	(	(	PUNCT
ejpam-4576	183	3	5	5	NUM
ejpam-4576	183	4	)	)	PUNCT
ejpam-4576	183	5	,	,	PUNCT
ejpam-4576	183	6	we	we	PRON
ejpam-4576	183	7	have	have	VERB
ejpam-4576	183	8	f+(v	f+(v	NOUN
ejpam-4576	183	9	)	)	PUNCT
ejpam-4576	184	1	⊆	⊆	NUM
ejpam-4576	184	2	[	[	X
ejpam-4576	184	3	f+(v	f+(v	NOUN
ejpam-4576	184	4	)	)	PUNCT
ejpam-4576	184	5	]	]	PUNCT
ejpam-4576	184	6	(	(	PUNCT
ejpam-4576	184	7	λ	λ	NOUN
ejpam-4576	184	8	,	,	PUNCT
ejpam-4576	184	9	sp	sp	NOUN
ejpam-4576	184	10	)	)	PUNCT
ejpam-4576	184	11	and	and	CCONJ
ejpam-4576	184	12	hence	hence	ADV
ejpam-4576	184	13	f+(v	f+(v	PROPN
ejpam-4576	184	14	)	)	PUNCT
ejpam-4576	184	15	is	be	AUX
ejpam-4576	184	16	(	(	PUNCT
ejpam-4576	184	17	λ	λ	INTJ
ejpam-4576	184	18	,	,	PUNCT
ejpam-4576	184	19	sp)-open	sp)-open	ADJ
ejpam-4576	184	20	in	in	ADP
ejpam-4576	184	21	x.	x.	NOUN
ejpam-4576	184	22	(	(	PUNCT
ejpam-4576	184	23	6	6	NUM
ejpam-4576	184	24	)	)	PUNCT
ejpam-4576	184	25	⇒	⇒	NOUN
ejpam-4576	184	26	(	(	PUNCT
ejpam-4576	184	27	7	7	NUM
ejpam-4576	184	28	):	):	PUNCT
ejpam-4576	184	29	the	the	DET
ejpam-4576	184	30	proof	proof	NOUN
ejpam-4576	184	31	is	be	AUX
ejpam-4576	184	32	obvious	obvious	ADJ
ejpam-4576	184	33	.	.	PUNCT
ejpam-4576	185	1	(	(	PUNCT
ejpam-4576	185	2	7	7	X
ejpam-4576	185	3	)	)	PUNCT
ejpam-4576	185	4	⇒	⇒	NOUN
ejpam-4576	185	5	(	(	PUNCT
ejpam-4576	185	6	1	1	NUM
ejpam-4576	185	7	):	):	PUNCT
ejpam-4576	185	8	let	let	VERB
ejpam-4576	185	9	x	x	PUNCT
ejpam-4576	185	10	∈	∈	PROPN
ejpam-4576	185	11	x	x	X
ejpam-4576	185	12	and	and	CCONJ
ejpam-4576	185	13	v	v	X
ejpam-4576	185	14	be	be	AUX
ejpam-4576	185	15	any	any	DET
ejpam-4576	185	16	r(λ	r(λ	NOUN
ejpam-4576	185	17	,	,	PUNCT
ejpam-4576	185	18	sp)-open	sp)-open	ADJ
ejpam-4576	185	19	set	set	NOUN
ejpam-4576	185	20	of	of	ADP
ejpam-4576	185	21	y	y	PROPN
ejpam-4576	185	22	containing	contain	VERB
ejpam-4576	185	23	f	f	PROPN
ejpam-4576	185	24	(	(	PUNCT
ejpam-4576	185	25	x	x	NOUN
ejpam-4576	185	26	)	)	PUNCT
ejpam-4576	185	27	.	.	PUNCT
ejpam-4576	186	1	since	since	SCONJ
ejpam-4576	186	2	y	y	PROPN
ejpam-4576	186	3	−v	−v	NOUN
ejpam-4576	186	4	is	be	AUX
ejpam-4576	186	5	r(λ	r(λ	NOUN
ejpam-4576	186	6	,	,	PUNCT
ejpam-4576	186	7	sp)-closed	sp)-close	VERB
ejpam-4576	186	8	and	and	CCONJ
ejpam-4576	186	9	by	by	ADP
ejpam-4576	186	10	(	(	PUNCT
ejpam-4576	186	11	7	7	NUM
ejpam-4576	186	12	)	)	PUNCT
ejpam-4576	186	13	,	,	PUNCT
ejpam-4576	186	14	x−f+(v	x−f+(v	PROPN
ejpam-4576	186	15	)	)	PUNCT
ejpam-4576	187	1	=	=	PUNCT
ejpam-4576	187	2	f−(y	f−(y	NOUN
ejpam-4576	187	3	−v	−v	NOUN
ejpam-4576	187	4	)	)	PUNCT
ejpam-4576	187	5	is	be	AUX
ejpam-4576	187	6	(	(	PUNCT
ejpam-4576	187	7	λ	λ	X
ejpam-4576	187	8	,	,	PUNCT
ejpam-4576	187	9	sp)-closed	sp)-close	VERB
ejpam-4576	187	10	in	in	ADP
ejpam-4576	187	11	x.	x.	NOUN
ejpam-4576	187	12	thus	thus	ADV
ejpam-4576	187	13	,	,	PUNCT
ejpam-4576	187	14	f+(v	f+(v	PROPN
ejpam-4576	187	15	)	)	PUNCT
ejpam-4576	188	1	is	be	AUX
ejpam-4576	188	2	(	(	PUNCT
ejpam-4576	188	3	λ	λ	X
ejpam-4576	188	4	,	,	PUNCT
ejpam-4576	188	5	sp)-open	sp)-open	ADJ
ejpam-4576	188	6	and	and	CCONJ
ejpam-4576	188	7	hence	hence	ADV
ejpam-4576	188	8	x	x	X
ejpam-4576	188	9	∈	∈	PROPN
ejpam-4576	188	10	[	[	X
ejpam-4576	188	11	f+(v	f+(v	NOUN
ejpam-4576	188	12	)	)	PUNCT
ejpam-4576	188	13	]	]	PUNCT
ejpam-4576	188	14	(	(	PUNCT
ejpam-4576	188	15	λ	λ	NOUN
ejpam-4576	188	16	,	,	PUNCT
ejpam-4576	188	17	sp	sp	NOUN
ejpam-4576	188	18	)	)	PUNCT
ejpam-4576	188	19	.	.	PUNCT
ejpam-4576	189	1	then	then	ADV
ejpam-4576	189	2	,	,	PUNCT
ejpam-4576	189	3	there	there	PRON
ejpam-4576	189	4	exists	exist	VERB
ejpam-4576	189	5	a	a	DET
ejpam-4576	189	6	(	(	PUNCT
ejpam-4576	189	7	λ	λ	NOUN
ejpam-4576	189	8	,	,	PUNCT
ejpam-4576	189	9	sp)-open	sp)-open	NOUN
ejpam-4576	189	10	set	set	VERB
ejpam-4576	189	11	u	u	NOUN
ejpam-4576	189	12	of	of	ADP
ejpam-4576	189	13	x	x	PUNCT
ejpam-4576	189	14	containing	contain	VERB
ejpam-4576	189	15	x	x	PUNCT
ejpam-4576	189	16	such	such	ADJ
ejpam-4576	189	17	that	that	SCONJ
ejpam-4576	189	18	f	f	PROPN
ejpam-4576	189	19	(	(	PUNCT
ejpam-4576	189	20	u	u	NOUN
ejpam-4576	189	21	)	)	PUNCT
ejpam-4576	189	22	⊆	⊆	NUM
ejpam-4576	189	23	v	v	NOUN
ejpam-4576	189	24	.	.	PUNCT
ejpam-4576	190	1	it	it	PRON
ejpam-4576	190	2	follows	follow	VERB
ejpam-4576	190	3	from	from	ADP
ejpam-4576	190	4	theorem	theorem	ADJ
ejpam-4576	190	5	1	1	NUM
ejpam-4576	190	6	that	that	SCONJ
ejpam-4576	190	7	f	f	PROPN
ejpam-4576	190	8	is	be	AUX
ejpam-4576	190	9	upper	upper	ADJ
ejpam-4576	190	10	almost	almost	ADV
ejpam-4576	190	11	(	(	PUNCT
ejpam-4576	190	12	λ	λ	NOUN
ejpam-4576	190	13	,	,	PUNCT
ejpam-4576	190	14	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	190	15	.	.	PUNCT
ejpam-4576	191	1	theorem	theorem	VERB
ejpam-4576	191	2	4	4	NUM
ejpam-4576	191	3	.	.	X
ejpam-4576	191	4	for	for	ADP
ejpam-4576	191	5	a	a	DET
ejpam-4576	191	6	multifunction	multifunction	NOUN
ejpam-4576	192	1	f	f	NOUN
ejpam-4576	192	2	:	:	PUNCT
ejpam-4576	192	3	(	(	PUNCT
ejpam-4576	192	4	x	x	X
ejpam-4576	192	5	,	,	PUNCT
ejpam-4576	192	6	τ	τ	X
ejpam-4576	192	7	)	)	PUNCT
ejpam-4576	192	8	→	→	SYM
ejpam-4576	192	9	(	(	PUNCT
ejpam-4576	192	10	y	y	PROPN
ejpam-4576	192	11	,	,	PUNCT
ejpam-4576	192	12	σ	σ	PROPN
ejpam-4576	192	13	)	)	PUNCT
ejpam-4576	192	14	,	,	PUNCT
ejpam-4576	192	15	the	the	DET
ejpam-4576	192	16	following	follow	VERB
ejpam-4576	192	17	properties	property	NOUN
ejpam-4576	192	18	are	be	AUX
ejpam-4576	192	19	equivalent	equivalent	ADJ
ejpam-4576	192	20	:	:	PUNCT
ejpam-4576	192	21	(	(	PUNCT
ejpam-4576	192	22	1	1	X
ejpam-4576	192	23	)	)	PUNCT
ejpam-4576	192	24	f	f	PROPN
ejpam-4576	192	25	is	be	AUX
ejpam-4576	192	26	lower	low	ADJ
ejpam-4576	192	27	almost	almost	ADV
ejpam-4576	192	28	(	(	PUNCT
ejpam-4576	192	29	λ	λ	NOUN
ejpam-4576	192	30	,	,	PUNCT
ejpam-4576	192	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	192	32	;	;	PUNCT
ejpam-4576	192	33	(	(	PUNCT
ejpam-4576	192	34	2	2	X
ejpam-4576	192	35	)	)	PUNCT
ejpam-4576	192	36	f−(v	f−(v	NOUN
ejpam-4576	192	37	)	)	PUNCT
ejpam-4576	192	38	⊆	⊆	NUM
ejpam-4576	193	1	[	[	X
ejpam-4576	193	2	f−([v	f−([v	ADJ
ejpam-4576	193	3	(	(	PUNCT
ejpam-4576	193	4	λ	λ	PROPN
ejpam-4576	193	5	,	,	PUNCT
ejpam-4576	193	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	193	7	,	,	PUNCT
ejpam-4576	193	8	sp))](λ	sp))](λ	PROPN
ejpam-4576	193	9	,	,	PUNCT
ejpam-4576	193	10	sp	sp	NOUN
ejpam-4576	193	11	)	)	PUNCT
ejpam-4576	193	12	for	for	ADP
ejpam-4576	193	13	every	every	DET
ejpam-4576	193	14	(	(	PUNCT
ejpam-4576	193	15	λ	λ	NOUN
ejpam-4576	193	16	,	,	PUNCT
ejpam-4576	193	17	sp)-open	sp)-open	NOUN
ejpam-4576	193	18	set	set	VERB
ejpam-4576	193	19	v	v	NOUN
ejpam-4576	193	20	of	of	ADP
ejpam-4576	193	21	y	y	PROPN
ejpam-4576	193	22	;	;	PUNCT
ejpam-4576	193	23	(	(	PUNCT
ejpam-4576	193	24	3	3	X
ejpam-4576	193	25	)	)	PUNCT
ejpam-4576	194	1	[	[	X
ejpam-4576	194	2	f+([k(λ	f+([k(λ	NOUN
ejpam-4576	194	3	,	,	PUNCT
ejpam-4576	194	4	sp	sp	NOUN
ejpam-4576	194	5	)	)	PUNCT
ejpam-4576	194	6	]	]	PUNCT
ejpam-4576	194	7	(	(	PUNCT
ejpam-4576	194	8	λ	λ	INTJ
ejpam-4576	194	9	,	,	PUNCT
ejpam-4576	194	10	sp))](λ	sp))](λ	PROPN
ejpam-4576	194	11	,	,	PUNCT
ejpam-4576	194	12	sp	sp	NOUN
ejpam-4576	194	13	)	)	PUNCT
ejpam-4576	194	14	⊆	⊆	NUM
ejpam-4576	194	15	f+(k	f+(k	NOUN
ejpam-4576	194	16	)	)	PUNCT
ejpam-4576	194	17	for	for	SCONJ
ejpam-4576	194	18	every	every	DET
ejpam-4576	194	19	(	(	PUNCT
ejpam-4576	194	20	λ	λ	PROPN
ejpam-4576	194	21	,	,	PUNCT
ejpam-4576	194	22	sp)-closed	sp)-close	VERB
ejpam-4576	194	23	set	set	VERB
ejpam-4576	194	24	k	k	PROPN
ejpam-4576	194	25	of	of	ADP
ejpam-4576	194	26	y	y	PROPN
ejpam-4576	194	27	;	;	PUNCT
ejpam-4576	194	28	(	(	PUNCT
ejpam-4576	194	29	4	4	X
ejpam-4576	194	30	)	)	PUNCT
ejpam-4576	195	1	[	[	X
ejpam-4576	195	2	f+([[b(λ	f+([[b(λ	NUM
ejpam-4576	195	3	,	,	PUNCT
ejpam-4576	195	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	195	5	,	,	PUNCT
ejpam-4576	195	6	sp	sp	NOUN
ejpam-4576	195	7	)	)	PUNCT
ejpam-4576	195	8	]	]	PUNCT
ejpam-4576	195	9	(	(	PUNCT
ejpam-4576	195	10	λ	λ	INTJ
ejpam-4576	195	11	,	,	PUNCT
ejpam-4576	195	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	195	13	,	,	PUNCT
ejpam-4576	195	14	sp	sp	NOUN
ejpam-4576	195	15	)	)	PUNCT
ejpam-4576	195	16	⊆	⊆	NUM
ejpam-4576	195	17	f+(b(λ	f+(b(λ	PROPN
ejpam-4576	195	18	,	,	PUNCT
ejpam-4576	195	19	sp	sp	NOUN
ejpam-4576	195	20	)	)	PUNCT
ejpam-4576	195	21	)	)	PUNCT
ejpam-4576	195	22	for	for	ADP
ejpam-4576	195	23	every	every	DET
ejpam-4576	195	24	subset	subset	NOUN
ejpam-4576	195	25	b	b	PROPN
ejpam-4576	195	26	of	of	ADP
ejpam-4576	195	27	y	y	PROPN
ejpam-4576	195	28	;	;	PUNCT
ejpam-4576	195	29	(	(	PUNCT
ejpam-4576	195	30	5	5	X
ejpam-4576	195	31	)	)	PUNCT
ejpam-4576	195	32	f−(b(λ	f−(b(λ	NOUN
ejpam-4576	195	33	,	,	PUNCT
ejpam-4576	195	34	sp	sp	NOUN
ejpam-4576	195	35	)	)	PUNCT
ejpam-4576	195	36	)	)	PUNCT
ejpam-4576	196	1	⊆	⊆	NUM
ejpam-4576	196	2	[	[	X
ejpam-4576	196	3	f−([[b(λ	f−([[b(λ	NOUN
ejpam-4576	196	4	,	,	PUNCT
ejpam-4576	196	5	sp	sp	NOUN
ejpam-4576	196	6	)	)	PUNCT
ejpam-4576	196	7	]	]	PUNCT
ejpam-4576	196	8	(	(	PUNCT
ejpam-4576	196	9	λ	λ	X
ejpam-4576	196	10	,	,	PUNCT
ejpam-4576	196	11	sp)](λ	sp)](λ	PROPN
ejpam-4576	196	12	,	,	PUNCT
ejpam-4576	196	13	sp))](λ	sp))](λ	PROPN
ejpam-4576	196	14	,	,	PUNCT
ejpam-4576	196	15	sp	sp	NOUN
ejpam-4576	196	16	)	)	PUNCT
ejpam-4576	196	17	for	for	ADP
ejpam-4576	196	18	every	every	DET
ejpam-4576	196	19	subset	subset	NOUN
ejpam-4576	196	20	b	b	PROPN
ejpam-4576	196	21	of	of	ADP
ejpam-4576	196	22	y	y	PROPN
ejpam-4576	196	23	;	;	PUNCT
ejpam-4576	196	24	c.	c.	PROPN
ejpam-4576	196	25	boonpok	boonpok	PROPN
ejpam-4576	196	26	,	,	PUNCT
ejpam-4576	196	27	n.	n.	PROPN
ejpam-4576	196	28	viriyapong	viriyapong	PROPN
ejpam-4576	196	29	/	/	SYM
ejpam-4576	196	30	eur	eur	PROPN
ejpam-4576	196	31	.	.	PUNCT
ejpam-4576	197	1	j.	j.	PROPN
ejpam-4576	197	2	pure	pure	PROPN
ejpam-4576	197	3	appl	appl	PROPN
ejpam-4576	197	4	.	.	PROPN
ejpam-4576	197	5	math	math	PROPN
ejpam-4576	197	6	,	,	PUNCT
ejpam-4576	197	7	16	16	NUM
ejpam-4576	197	8	(	(	PUNCT
ejpam-4576	197	9	1	1	NUM
ejpam-4576	197	10	)	)	PUNCT
ejpam-4576	197	11	(	(	PUNCT
ejpam-4576	197	12	2023	2023	NUM
ejpam-4576	197	13	)	)	PUNCT
ejpam-4576	197	14	,	,	PUNCT
ejpam-4576	197	15	84	84	NUM
ejpam-4576	197	16	-	-	SYM
ejpam-4576	197	17	96	96	NUM
ejpam-4576	197	18	90	90	NUM
ejpam-4576	197	19	(	(	PUNCT
ejpam-4576	197	20	6	6	NUM
ejpam-4576	197	21	)	)	PUNCT
ejpam-4576	197	22	f−(v	f−(v	NOUN
ejpam-4576	197	23	)	)	PUNCT
ejpam-4576	197	24	is	be	AUX
ejpam-4576	197	25	(	(	PUNCT
ejpam-4576	197	26	λ	λ	INTJ
ejpam-4576	197	27	,	,	PUNCT
ejpam-4576	197	28	sp)-open	sp)-open	ADJ
ejpam-4576	197	29	in	in	ADP
ejpam-4576	197	30	x	x	PUNCT
ejpam-4576	197	31	for	for	ADP
ejpam-4576	197	32	every	every	DET
ejpam-4576	197	33	r(λ	r(λ	NOUN
ejpam-4576	197	34	,	,	PUNCT
ejpam-4576	197	35	sp)-open	sp)-open	ADJ
ejpam-4576	197	36	set	set	VERB
ejpam-4576	197	37	v	v	NOUN
ejpam-4576	197	38	of	of	ADP
ejpam-4576	197	39	y	y	PROPN
ejpam-4576	197	40	;	;	PUNCT
ejpam-4576	197	41	(	(	PUNCT
ejpam-4576	197	42	7	7	X
ejpam-4576	197	43	)	)	PUNCT
ejpam-4576	197	44	f+(k	f+(k	NUM
ejpam-4576	197	45	)	)	PUNCT
ejpam-4576	198	1	is	be	AUX
ejpam-4576	198	2	(	(	PUNCT
ejpam-4576	198	3	λ	λ	X
ejpam-4576	198	4	,	,	PUNCT
ejpam-4576	198	5	sp)-closed	sp)-close	VERB
ejpam-4576	198	6	in	in	ADP
ejpam-4576	198	7	x	x	PUNCT
ejpam-4576	198	8	for	for	ADP
ejpam-4576	198	9	every	every	DET
ejpam-4576	198	10	r(λ	r(λ	NOUN
ejpam-4576	198	11	,	,	PUNCT
ejpam-4576	198	12	sp)-closed	sp)-close	VERB
ejpam-4576	198	13	set	set	VERB
ejpam-4576	198	14	k	k	PROPN
ejpam-4576	198	15	of	of	ADP
ejpam-4576	198	16	y	y	PROPN
ejpam-4576	198	17	.	.	PUNCT
ejpam-4576	199	1	proof	proof	NOUN
ejpam-4576	199	2	.	.	PUNCT
ejpam-4576	200	1	the	the	DET
ejpam-4576	200	2	proof	proof	NOUN
ejpam-4576	200	3	is	be	AUX
ejpam-4576	200	4	similar	similar	ADJ
ejpam-4576	200	5	to	to	ADP
ejpam-4576	200	6	that	that	PRON
ejpam-4576	200	7	of	of	ADP
ejpam-4576	200	8	theorem	theorem	ADJ
ejpam-4576	200	9	3	3	NUM
ejpam-4576	200	10	.	.	PUNCT
ejpam-4576	200	11	corollary	corollary	ADJ
ejpam-4576	200	12	2	2	NUM
ejpam-4576	200	13	.	.	PUNCT
ejpam-4576	200	14	for	for	ADP
ejpam-4576	200	15	a	a	DET
ejpam-4576	200	16	function	function	NOUN
ejpam-4576	200	17	f	f	NOUN
ejpam-4576	200	18	:	:	PUNCT
ejpam-4576	200	19	(	(	PUNCT
ejpam-4576	200	20	x	x	X
ejpam-4576	200	21	,	,	PUNCT
ejpam-4576	200	22	τ	τ	X
ejpam-4576	200	23	)	)	PUNCT
ejpam-4576	200	24	→	→	SYM
ejpam-4576	200	25	(	(	PUNCT
ejpam-4576	200	26	y	y	PROPN
ejpam-4576	200	27	,	,	PUNCT
ejpam-4576	200	28	σ	σ	PROPN
ejpam-4576	200	29	)	)	PUNCT
ejpam-4576	200	30	,	,	PUNCT
ejpam-4576	200	31	the	the	DET
ejpam-4576	200	32	following	follow	VERB
ejpam-4576	200	33	properties	property	NOUN
ejpam-4576	200	34	are	be	AUX
ejpam-4576	200	35	equivalent	equivalent	ADJ
ejpam-4576	200	36	:	:	PUNCT
ejpam-4576	200	37	(	(	PUNCT
ejpam-4576	200	38	1	1	X
ejpam-4576	200	39	)	)	PUNCT
ejpam-4576	200	40	f	f	NOUN
ejpam-4576	200	41	is	be	AUX
ejpam-4576	200	42	almost	almost	ADV
ejpam-4576	200	43	(	(	PUNCT
ejpam-4576	200	44	λ	λ	NOUN
ejpam-4576	200	45	,	,	PUNCT
ejpam-4576	200	46	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	200	47	;	;	PUNCT
ejpam-4576	200	48	(	(	PUNCT
ejpam-4576	200	49	2	2	X
ejpam-4576	200	50	)	)	PUNCT
ejpam-4576	200	51	f−1(v	f−1(v	NOUN
ejpam-4576	200	52	)	)	PUNCT
ejpam-4576	201	1	⊆	⊆	NUM
ejpam-4576	201	2	[	[	X
ejpam-4576	201	3	f−1([v	f−1([v	ADJ
ejpam-4576	201	4	(	(	PUNCT
ejpam-4576	201	5	λ	λ	NOUN
ejpam-4576	201	6	,	,	PUNCT
ejpam-4576	201	7	sp)](λ	sp)](λ	PROPN
ejpam-4576	201	8	,	,	PUNCT
ejpam-4576	201	9	sp))](λ	sp))](λ	PROPN
ejpam-4576	201	10	,	,	PUNCT
ejpam-4576	201	11	sp	sp	NOUN
ejpam-4576	201	12	)	)	PUNCT
ejpam-4576	201	13	for	for	ADP
ejpam-4576	201	14	every	every	DET
ejpam-4576	201	15	(	(	PUNCT
ejpam-4576	201	16	λ	λ	NOUN
ejpam-4576	201	17	,	,	PUNCT
ejpam-4576	201	18	sp)-open	sp)-open	NOUN
ejpam-4576	201	19	set	set	VERB
ejpam-4576	201	20	v	v	NOUN
ejpam-4576	201	21	of	of	ADP
ejpam-4576	201	22	y	y	PROPN
ejpam-4576	201	23	;	;	PUNCT
ejpam-4576	201	24	(	(	PUNCT
ejpam-4576	201	25	3	3	X
ejpam-4576	201	26	)	)	PUNCT
ejpam-4576	201	27	[	[	X
ejpam-4576	201	28	f−1([k(λ	f−1([k(λ	NOUN
ejpam-4576	201	29	,	,	PUNCT
ejpam-4576	201	30	sp	sp	NOUN
ejpam-4576	201	31	)	)	PUNCT
ejpam-4576	201	32	]	]	PUNCT
ejpam-4576	201	33	(	(	PUNCT
ejpam-4576	201	34	λ	λ	INTJ
ejpam-4576	201	35	,	,	PUNCT
ejpam-4576	201	36	sp))](λ	sp))](λ	PROPN
ejpam-4576	201	37	,	,	PUNCT
ejpam-4576	201	38	sp	sp	NOUN
ejpam-4576	201	39	)	)	PUNCT
ejpam-4576	201	40	⊆	⊆	NUM
ejpam-4576	201	41	f−1(k	f−1(k	PROPN
ejpam-4576	201	42	)	)	PUNCT
ejpam-4576	201	43	for	for	SCONJ
ejpam-4576	201	44	every	every	DET
ejpam-4576	201	45	(	(	PUNCT
ejpam-4576	201	46	λ	λ	PROPN
ejpam-4576	201	47	,	,	PUNCT
ejpam-4576	201	48	sp)-closed	sp)-close	VERB
ejpam-4576	201	49	set	set	VERB
ejpam-4576	201	50	k	k	PROPN
ejpam-4576	201	51	of	of	ADP
ejpam-4576	201	52	y	y	PROPN
ejpam-4576	201	53	;	;	PUNCT
ejpam-4576	201	54	(	(	PUNCT
ejpam-4576	201	55	4	4	X
ejpam-4576	201	56	)	)	PUNCT
ejpam-4576	202	1	[	[	X
ejpam-4576	202	2	f−1([[b(λ	f−1([[b(λ	X
ejpam-4576	202	3	,	,	PUNCT
ejpam-4576	202	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	202	5	,	,	PUNCT
ejpam-4576	202	6	sp	sp	NOUN
ejpam-4576	202	7	)	)	PUNCT
ejpam-4576	202	8	]	]	PUNCT
ejpam-4576	202	9	(	(	PUNCT
ejpam-4576	202	10	λ	λ	INTJ
ejpam-4576	202	11	,	,	PUNCT
ejpam-4576	202	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	202	13	,	,	PUNCT
ejpam-4576	202	14	sp	sp	NOUN
ejpam-4576	202	15	)	)	PUNCT
ejpam-4576	202	16	⊆	⊆	NUM
ejpam-4576	202	17	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4576	202	18	,	,	PUNCT
ejpam-4576	202	19	sp	sp	NOUN
ejpam-4576	202	20	)	)	PUNCT
ejpam-4576	202	21	)	)	PUNCT
ejpam-4576	202	22	for	for	ADP
ejpam-4576	202	23	every	every	DET
ejpam-4576	202	24	subset	subset	NOUN
ejpam-4576	202	25	b	b	PROPN
ejpam-4576	202	26	of	of	ADP
ejpam-4576	202	27	y	y	PROPN
ejpam-4576	202	28	;	;	PUNCT
ejpam-4576	202	29	(	(	PUNCT
ejpam-4576	202	30	5	5	X
ejpam-4576	202	31	)	)	PUNCT
ejpam-4576	202	32	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4576	202	33	,	,	PUNCT
ejpam-4576	202	34	sp	sp	NOUN
ejpam-4576	202	35	)	)	PUNCT
ejpam-4576	202	36	)	)	PUNCT
ejpam-4576	203	1	⊆	⊆	NUM
ejpam-4576	203	2	[	[	X
ejpam-4576	203	3	f−1([[b(λ	f−1([[b(λ	NOUN
ejpam-4576	203	4	,	,	PUNCT
ejpam-4576	203	5	sp	sp	NOUN
ejpam-4576	203	6	)	)	PUNCT
ejpam-4576	203	7	]	]	PUNCT
ejpam-4576	203	8	(	(	PUNCT
ejpam-4576	203	9	λ	λ	X
ejpam-4576	203	10	,	,	PUNCT
ejpam-4576	203	11	sp)](λ	sp)](λ	PROPN
ejpam-4576	203	12	,	,	PUNCT
ejpam-4576	203	13	sp))](λ	sp))](λ	PROPN
ejpam-4576	203	14	,	,	PUNCT
ejpam-4576	203	15	sp	sp	NOUN
ejpam-4576	203	16	)	)	PUNCT
ejpam-4576	203	17	for	for	ADP
ejpam-4576	203	18	every	every	DET
ejpam-4576	203	19	subset	subset	NOUN
ejpam-4576	203	20	b	b	PROPN
ejpam-4576	203	21	of	of	ADP
ejpam-4576	203	22	y	y	PROPN
ejpam-4576	203	23	;	;	PUNCT
ejpam-4576	203	24	(	(	PUNCT
ejpam-4576	203	25	6	6	X
ejpam-4576	203	26	)	)	PUNCT
ejpam-4576	203	27	f−1(v	f−1(v	NOUN
ejpam-4576	203	28	)	)	PUNCT
ejpam-4576	203	29	is	be	AUX
ejpam-4576	203	30	(	(	PUNCT
ejpam-4576	203	31	λ	λ	INTJ
ejpam-4576	203	32	,	,	PUNCT
ejpam-4576	203	33	sp)-open	sp)-open	ADJ
ejpam-4576	203	34	in	in	ADP
ejpam-4576	203	35	x	x	PUNCT
ejpam-4576	203	36	for	for	ADP
ejpam-4576	203	37	every	every	DET
ejpam-4576	203	38	r(λ	r(λ	NOUN
ejpam-4576	203	39	,	,	PUNCT
ejpam-4576	203	40	sp)-open	sp)-open	ADJ
ejpam-4576	203	41	set	set	VERB
ejpam-4576	203	42	v	v	NOUN
ejpam-4576	203	43	of	of	ADP
ejpam-4576	203	44	y	y	PROPN
ejpam-4576	203	45	;	;	PUNCT
ejpam-4576	203	46	(	(	PUNCT
ejpam-4576	203	47	7	7	X
ejpam-4576	203	48	)	)	PUNCT
ejpam-4576	203	49	f−1(k	f−1(k	PROPN
ejpam-4576	203	50	)	)	PUNCT
ejpam-4576	203	51	is	be	AUX
ejpam-4576	203	52	(	(	PUNCT
ejpam-4576	203	53	λ	λ	X
ejpam-4576	203	54	,	,	PUNCT
ejpam-4576	203	55	sp)-closed	sp)-close	VERB
ejpam-4576	203	56	in	in	ADP
ejpam-4576	203	57	x	x	PUNCT
ejpam-4576	203	58	for	for	ADP
ejpam-4576	203	59	every	every	DET
ejpam-4576	203	60	r(λ	r(λ	NOUN
ejpam-4576	203	61	,	,	PUNCT
ejpam-4576	203	62	sp)-closed	sp)-close	VERB
ejpam-4576	203	63	set	set	VERB
ejpam-4576	203	64	k	k	PROPN
ejpam-4576	203	65	of	of	ADP
ejpam-4576	203	66	y	y	PROPN
ejpam-4576	203	67	.	.	PUNCT
ejpam-4576	204	1	theorem	theorem	ADJ
ejpam-4576	204	2	5	5	NUM
ejpam-4576	204	3	.	.	X
ejpam-4576	204	4	for	for	ADP
ejpam-4576	204	5	a	a	DET
ejpam-4576	204	6	multifunction	multifunction	NOUN
ejpam-4576	205	1	f	f	NOUN
ejpam-4576	205	2	:	:	PUNCT
ejpam-4576	205	3	(	(	PUNCT
ejpam-4576	205	4	x	x	X
ejpam-4576	205	5	,	,	PUNCT
ejpam-4576	205	6	τ	τ	X
ejpam-4576	205	7	)	)	PUNCT
ejpam-4576	205	8	→	→	SYM
ejpam-4576	205	9	(	(	PUNCT
ejpam-4576	205	10	y	y	PROPN
ejpam-4576	205	11	,	,	PUNCT
ejpam-4576	205	12	σ	σ	PROPN
ejpam-4576	205	13	)	)	PUNCT
ejpam-4576	205	14	,	,	PUNCT
ejpam-4576	205	15	the	the	DET
ejpam-4576	205	16	following	follow	VERB
ejpam-4576	205	17	properties	property	NOUN
ejpam-4576	205	18	are	be	AUX
ejpam-4576	205	19	equivalent	equivalent	ADJ
ejpam-4576	205	20	:	:	PUNCT
ejpam-4576	205	21	(	(	PUNCT
ejpam-4576	205	22	1	1	X
ejpam-4576	205	23	)	)	PUNCT
ejpam-4576	205	24	f	f	PROPN
ejpam-4576	205	25	is	be	AUX
ejpam-4576	205	26	upper	upper	ADJ
ejpam-4576	205	27	almost	almost	ADV
ejpam-4576	205	28	(	(	PUNCT
ejpam-4576	205	29	λ	λ	NOUN
ejpam-4576	205	30	,	,	PUNCT
ejpam-4576	205	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	205	32	;	;	PUNCT
ejpam-4576	205	33	(	(	PUNCT
ejpam-4576	205	34	2	2	X
ejpam-4576	205	35	)	)	PUNCT
ejpam-4576	206	1	[	[	X
ejpam-4576	206	2	f−(v	f−(v	NOUN
ejpam-4576	206	3	)	)	PUNCT
ejpam-4576	206	4	]	]	PUNCT
ejpam-4576	206	5	(	(	PUNCT
ejpam-4576	206	6	λ	λ	NOUN
ejpam-4576	206	7	,	,	PUNCT
ejpam-4576	206	8	sp	sp	NOUN
ejpam-4576	206	9	)	)	PUNCT
ejpam-4576	206	10	⊆	⊆	NUM
ejpam-4576	206	11	f−(v	f−(v	NOUN
ejpam-4576	206	12	(	(	PUNCT
ejpam-4576	206	13	λ	λ	NOUN
ejpam-4576	206	14	,	,	PUNCT
ejpam-4576	206	15	sp	sp	NOUN
ejpam-4576	206	16	)	)	PUNCT
ejpam-4576	206	17	)	)	PUNCT
ejpam-4576	206	18	for	for	ADP
ejpam-4576	206	19	every	every	DET
ejpam-4576	206	20	β(λ	β(λ	PROPN
ejpam-4576	206	21	,	,	PUNCT
ejpam-4576	206	22	sp)-open	sp)-open	NOUN
ejpam-4576	206	23	set	set	VERB
ejpam-4576	206	24	v	v	NOUN
ejpam-4576	206	25	of	of	ADP
ejpam-4576	206	26	y	y	PROPN
ejpam-4576	206	27	;	;	PUNCT
ejpam-4576	206	28	(	(	PUNCT
ejpam-4576	206	29	3	3	X
ejpam-4576	206	30	)	)	PUNCT
ejpam-4576	207	1	[	[	X
ejpam-4576	207	2	f−(v	f−(v	NOUN
ejpam-4576	207	3	)	)	PUNCT
ejpam-4576	207	4	]	]	PUNCT
ejpam-4576	207	5	(	(	PUNCT
ejpam-4576	207	6	λ	λ	NOUN
ejpam-4576	207	7	,	,	PUNCT
ejpam-4576	207	8	sp	sp	NOUN
ejpam-4576	207	9	)	)	PUNCT
ejpam-4576	207	10	⊆	⊆	NUM
ejpam-4576	207	11	f−(v	f−(v	NOUN
ejpam-4576	207	12	(	(	PUNCT
ejpam-4576	207	13	λ	λ	NOUN
ejpam-4576	207	14	,	,	PUNCT
ejpam-4576	207	15	sp	sp	NOUN
ejpam-4576	207	16	)	)	PUNCT
ejpam-4576	207	17	)	)	PUNCT
ejpam-4576	207	18	for	for	ADP
ejpam-4576	207	19	every	every	DET
ejpam-4576	207	20	s(λ	s(λ	PROPN
ejpam-4576	207	21	,	,	PUNCT
ejpam-4576	207	22	sp)-open	sp)-open	VERB
ejpam-4576	207	23	set	set	VERB
ejpam-4576	207	24	v	v	NOUN
ejpam-4576	207	25	of	of	ADP
ejpam-4576	207	26	y	y	PROPN
ejpam-4576	207	27	.	.	PUNCT
ejpam-4576	208	1	proof	proof	NOUN
ejpam-4576	208	2	.	.	PUNCT
ejpam-4576	209	1	(	(	PUNCT
ejpam-4576	209	2	1	1	X
ejpam-4576	209	3	)	)	PUNCT
ejpam-4576	209	4	⇒	⇒	NOUN
ejpam-4576	209	5	(	(	PUNCT
ejpam-4576	209	6	2	2	NUM
ejpam-4576	209	7	):	):	PUNCT
ejpam-4576	209	8	let	let	VERB
ejpam-4576	209	9	v	v	PART
ejpam-4576	209	10	be	be	AUX
ejpam-4576	209	11	any	any	DET
ejpam-4576	209	12	β(λ	β(λ	NOUN
ejpam-4576	209	13	,	,	PUNCT
ejpam-4576	209	14	sp)-open	sp)-open	ADJ
ejpam-4576	209	15	set	set	NOUN
ejpam-4576	209	16	of	of	ADP
ejpam-4576	209	17	y	y	PROPN
ejpam-4576	209	18	.	.	PUNCT
ejpam-4576	210	1	then	then	ADV
ejpam-4576	210	2	,	,	PUNCT
ejpam-4576	210	3	v	v	INTJ
ejpam-4576	210	4	(	(	PUNCT
ejpam-4576	210	5	λ	λ	NOUN
ejpam-4576	210	6	,	,	PUNCT
ejpam-4576	210	7	sp	sp	NOUN
ejpam-4576	210	8	)	)	PUNCT
ejpam-4576	210	9	is	be	AUX
ejpam-4576	210	10	a	a	DET
ejpam-4576	210	11	r(λ	r(λ	NOUN
ejpam-4576	210	12	,	,	PUNCT
ejpam-4576	210	13	sp)closed	sp)close	VERB
ejpam-4576	210	14	set	set	NOUN
ejpam-4576	210	15	of	of	ADP
ejpam-4576	210	16	y	y	PROPN
ejpam-4576	210	17	.	.	PUNCT
ejpam-4576	211	1	since	since	SCONJ
ejpam-4576	211	2	f	f	PROPN
ejpam-4576	211	3	is	be	AUX
ejpam-4576	211	4	upper	upper	ADJ
ejpam-4576	211	5	almost	almost	ADV
ejpam-4576	211	6	(	(	PUNCT
ejpam-4576	211	7	λ	λ	NOUN
ejpam-4576	211	8	,	,	PUNCT
ejpam-4576	211	9	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	211	10	and	and	CCONJ
ejpam-4576	211	11	by	by	ADP
ejpam-4576	211	12	theorem	theorem	ADJ
ejpam-4576	211	13	3	3	NUM
ejpam-4576	211	14	,	,	PUNCT
ejpam-4576	211	15	f−(v	f−(v	ADJ
ejpam-4576	211	16	(	(	PUNCT
ejpam-4576	211	17	λ	λ	NOUN
ejpam-4576	211	18	,	,	PUNCT
ejpam-4576	211	19	sp	sp	NOUN
ejpam-4576	211	20	)	)	PUNCT
ejpam-4576	211	21	)	)	PUNCT
ejpam-4576	211	22	is	be	AUX
ejpam-4576	211	23	(	(	PUNCT
ejpam-4576	211	24	λ	λ	X
ejpam-4576	211	25	,	,	PUNCT
ejpam-4576	211	26	sp)-closed	sp)-close	VERB
ejpam-4576	211	27	in	in	ADP
ejpam-4576	211	28	x.	x.	NOUN
ejpam-4576	211	29	thus	thus	ADV
ejpam-4576	211	30	,	,	PUNCT
ejpam-4576	211	31	[	[	X
ejpam-4576	211	32	f−(v	f−(v	NOUN
ejpam-4576	211	33	)	)	PUNCT
ejpam-4576	211	34	]	]	PUNCT
ejpam-4576	211	35	(	(	PUNCT
ejpam-4576	211	36	λ	λ	NOUN
ejpam-4576	211	37	,	,	PUNCT
ejpam-4576	211	38	sp	sp	NOUN
ejpam-4576	211	39	)	)	PUNCT
ejpam-4576	211	40	⊆	⊆	NUM
ejpam-4576	211	41	f−(v	f−(v	NOUN
ejpam-4576	211	42	(	(	PUNCT
ejpam-4576	211	43	λ	λ	NOUN
ejpam-4576	211	44	,	,	PUNCT
ejpam-4576	211	45	sp	sp	NOUN
ejpam-4576	211	46	)	)	PUNCT
ejpam-4576	211	47	)	)	PUNCT
ejpam-4576	211	48	.	.	PUNCT
ejpam-4576	212	1	(	(	PUNCT
ejpam-4576	212	2	2	2	X
ejpam-4576	212	3	)	)	PUNCT
ejpam-4576	212	4	⇒	⇒	NOUN
ejpam-4576	212	5	(	(	PUNCT
ejpam-4576	212	6	3	3	NUM
ejpam-4576	212	7	):	):	PUNCT
ejpam-4576	212	8	the	the	DET
ejpam-4576	212	9	proof	proof	NOUN
ejpam-4576	212	10	is	be	AUX
ejpam-4576	212	11	obvious	obvious	ADJ
ejpam-4576	212	12	.	.	PUNCT
ejpam-4576	213	1	(	(	PUNCT
ejpam-4576	213	2	3	3	X
ejpam-4576	213	3	)	)	PUNCT
ejpam-4576	213	4	⇒	⇒	NOUN
ejpam-4576	213	5	(	(	PUNCT
ejpam-4576	213	6	1	1	NUM
ejpam-4576	213	7	):	):	PUNCT
ejpam-4576	213	8	let	let	VERB
ejpam-4576	213	9	k	k	PRON
ejpam-4576	213	10	be	be	AUX
ejpam-4576	213	11	any	any	DET
ejpam-4576	213	12	r(λ	r(λ	NOUN
ejpam-4576	213	13	,	,	PUNCT
ejpam-4576	213	14	sp)-closed	sp)-close	VERB
ejpam-4576	213	15	set	set	NOUN
ejpam-4576	213	16	of	of	ADP
ejpam-4576	213	17	y	y	PROPN
ejpam-4576	213	18	.	.	PUNCT
ejpam-4576	214	1	then	then	ADV
ejpam-4576	214	2	,	,	PUNCT
ejpam-4576	214	3	k	k	PROPN
ejpam-4576	214	4	is	be	AUX
ejpam-4576	214	5	s(λ	s(λ	PROPN
ejpam-4576	214	6	,	,	PUNCT
ejpam-4576	214	7	sp)-open	sp)-open	ADJ
ejpam-4576	214	8	in	in	ADP
ejpam-4576	214	9	y	y	PROPN
ejpam-4576	214	10	.	.	PUNCT
ejpam-4576	215	1	thus	thus	ADV
ejpam-4576	215	2	,	,	PUNCT
ejpam-4576	215	3	by	by	ADP
ejpam-4576	215	4	(	(	PUNCT
ejpam-4576	215	5	3	3	NUM
ejpam-4576	215	6	)	)	PUNCT
ejpam-4576	215	7	,	,	PUNCT
ejpam-4576	215	8	[	[	X
ejpam-4576	215	9	f−(k)](λ	f−(k)](λ	PROPN
ejpam-4576	215	10	,	,	PUNCT
ejpam-4576	215	11	sp	sp	NOUN
ejpam-4576	215	12	)	)	PUNCT
ejpam-4576	215	13	⊆	⊆	NUM
ejpam-4576	215	14	f−(k(λ	f−(k(λ	NOUN
ejpam-4576	215	15	,	,	PUNCT
ejpam-4576	215	16	sp	sp	NOUN
ejpam-4576	215	17	)	)	PUNCT
ejpam-4576	215	18	)	)	PUNCT
ejpam-4576	215	19	=	=	SYM
ejpam-4576	215	20	f−(k	f−(k	PROPN
ejpam-4576	215	21	)	)	PUNCT
ejpam-4576	215	22	and	and	CCONJ
ejpam-4576	215	23	hence	hence	ADV
ejpam-4576	215	24	f−(k	f−(k	PROPN
ejpam-4576	215	25	)	)	PUNCT
ejpam-4576	215	26	is	be	AUX
ejpam-4576	215	27	(	(	PUNCT
ejpam-4576	215	28	λ	λ	X
ejpam-4576	215	29	,	,	PUNCT
ejpam-4576	215	30	sp)-closed	sp)-close	VERB
ejpam-4576	215	31	in	in	ADP
ejpam-4576	215	32	x.	x.	NOUN
ejpam-4576	215	33	by	by	ADP
ejpam-4576	215	34	theorem	theorem	NOUN
ejpam-4576	215	35	3	3	NUM
ejpam-4576	215	36	,	,	PUNCT
ejpam-4576	215	37	f	f	PROPN
ejpam-4576	215	38	is	be	AUX
ejpam-4576	215	39	upper	upper	ADJ
ejpam-4576	215	40	almost	almost	ADV
ejpam-4576	215	41	(	(	PUNCT
ejpam-4576	215	42	λ	λ	NOUN
ejpam-4576	215	43	,	,	PUNCT
ejpam-4576	215	44	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	215	45	.	.	PUNCT
ejpam-4576	216	1	theorem	theorem	VERB
ejpam-4576	216	2	6	6	NUM
ejpam-4576	216	3	.	.	PUNCT
ejpam-4576	216	4	for	for	ADP
ejpam-4576	216	5	a	a	DET
ejpam-4576	216	6	multifunction	multifunction	NOUN
ejpam-4576	217	1	f	f	NOUN
ejpam-4576	217	2	:	:	PUNCT
ejpam-4576	217	3	(	(	PUNCT
ejpam-4576	217	4	x	x	X
ejpam-4576	217	5	,	,	PUNCT
ejpam-4576	217	6	τ	τ	X
ejpam-4576	217	7	)	)	PUNCT
ejpam-4576	217	8	→	→	SYM
ejpam-4576	217	9	(	(	PUNCT
ejpam-4576	217	10	y	y	PROPN
ejpam-4576	217	11	,	,	PUNCT
ejpam-4576	217	12	σ	σ	PROPN
ejpam-4576	217	13	)	)	PUNCT
ejpam-4576	217	14	,	,	PUNCT
ejpam-4576	217	15	the	the	DET
ejpam-4576	217	16	following	follow	VERB
ejpam-4576	217	17	properties	property	NOUN
ejpam-4576	217	18	are	be	AUX
ejpam-4576	217	19	equivalent	equivalent	ADJ
ejpam-4576	217	20	:	:	PUNCT
ejpam-4576	217	21	(	(	PUNCT
ejpam-4576	217	22	1	1	X
ejpam-4576	217	23	)	)	PUNCT
ejpam-4576	217	24	f	f	PROPN
ejpam-4576	217	25	is	be	AUX
ejpam-4576	217	26	lower	low	ADJ
ejpam-4576	217	27	almost	almost	ADV
ejpam-4576	217	28	(	(	PUNCT
ejpam-4576	217	29	λ	λ	NOUN
ejpam-4576	217	30	,	,	PUNCT
ejpam-4576	217	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	217	32	;	;	PUNCT
ejpam-4576	217	33	(	(	PUNCT
ejpam-4576	217	34	2	2	X
ejpam-4576	217	35	)	)	PUNCT
ejpam-4576	217	36	[	[	X
ejpam-4576	217	37	f+(v	f+(v	NOUN
ejpam-4576	217	38	)	)	PUNCT
ejpam-4576	217	39	]	]	PUNCT
ejpam-4576	217	40	(	(	PUNCT
ejpam-4576	217	41	λ	λ	NOUN
ejpam-4576	217	42	,	,	PUNCT
ejpam-4576	217	43	sp	sp	NOUN
ejpam-4576	217	44	)	)	PUNCT
ejpam-4576	217	45	⊆	⊆	NUM
ejpam-4576	217	46	f+(v	f+(v	NUM
ejpam-4576	217	47	(	(	PUNCT
ejpam-4576	217	48	λ	λ	NOUN
ejpam-4576	217	49	,	,	PUNCT
ejpam-4576	217	50	sp	sp	NOUN
ejpam-4576	217	51	)	)	PUNCT
ejpam-4576	217	52	)	)	PUNCT
ejpam-4576	217	53	for	for	ADP
ejpam-4576	217	54	every	every	DET
ejpam-4576	217	55	β(λ	β(λ	PROPN
ejpam-4576	217	56	,	,	PUNCT
ejpam-4576	217	57	sp)-open	sp)-open	NOUN
ejpam-4576	217	58	set	set	VERB
ejpam-4576	217	59	v	v	NOUN
ejpam-4576	217	60	of	of	ADP
ejpam-4576	217	61	y	y	PROPN
ejpam-4576	217	62	;	;	PUNCT
ejpam-4576	217	63	(	(	PUNCT
ejpam-4576	217	64	3	3	X
ejpam-4576	217	65	)	)	PUNCT
ejpam-4576	217	66	[	[	X
ejpam-4576	217	67	f+(v	f+(v	NOUN
ejpam-4576	217	68	)	)	PUNCT
ejpam-4576	217	69	]	]	PUNCT
ejpam-4576	217	70	(	(	PUNCT
ejpam-4576	217	71	λ	λ	NOUN
ejpam-4576	217	72	,	,	PUNCT
ejpam-4576	217	73	sp	sp	NOUN
ejpam-4576	217	74	)	)	PUNCT
ejpam-4576	217	75	⊆	⊆	NUM
ejpam-4576	217	76	f+(v	f+(v	NUM
ejpam-4576	217	77	(	(	PUNCT
ejpam-4576	217	78	λ	λ	NOUN
ejpam-4576	217	79	,	,	PUNCT
ejpam-4576	217	80	sp	sp	NOUN
ejpam-4576	217	81	)	)	PUNCT
ejpam-4576	217	82	)	)	PUNCT
ejpam-4576	217	83	for	for	ADP
ejpam-4576	217	84	every	every	DET
ejpam-4576	217	85	s(λ	s(λ	PROPN
ejpam-4576	217	86	,	,	PUNCT
ejpam-4576	217	87	sp)-open	sp)-open	VERB
ejpam-4576	217	88	set	set	VERB
ejpam-4576	217	89	v	v	NOUN
ejpam-4576	217	90	of	of	ADP
ejpam-4576	217	91	y	y	PROPN
ejpam-4576	217	92	.	.	PUNCT
ejpam-4576	218	1	proof	proof	NOUN
ejpam-4576	218	2	.	.	PUNCT
ejpam-4576	219	1	the	the	DET
ejpam-4576	219	2	proof	proof	NOUN
ejpam-4576	219	3	is	be	AUX
ejpam-4576	219	4	similar	similar	ADJ
ejpam-4576	219	5	to	to	ADP
ejpam-4576	219	6	that	that	PRON
ejpam-4576	219	7	of	of	ADP
ejpam-4576	219	8	theorem	theorem	NOUN
ejpam-4576	219	9	5	5	NUM
ejpam-4576	219	10	.	.	PUNCT
ejpam-4576	219	11	c.	c.	PROPN
ejpam-4576	219	12	boonpok	boonpok	PROPN
ejpam-4576	219	13	,	,	PUNCT
ejpam-4576	219	14	n.	n.	PROPN
ejpam-4576	219	15	viriyapong	viriyapong	PROPN
ejpam-4576	219	16	/	/	SYM
ejpam-4576	219	17	eur	eur	PROPN
ejpam-4576	219	18	.	.	PUNCT
ejpam-4576	220	1	j.	j.	PROPN
ejpam-4576	220	2	pure	pure	PROPN
ejpam-4576	220	3	appl	appl	PROPN
ejpam-4576	220	4	.	.	PROPN
ejpam-4576	220	5	math	math	PROPN
ejpam-4576	220	6	,	,	PUNCT
ejpam-4576	220	7	16	16	NUM
ejpam-4576	220	8	(	(	PUNCT
ejpam-4576	220	9	1	1	NUM
ejpam-4576	220	10	)	)	PUNCT
ejpam-4576	220	11	(	(	PUNCT
ejpam-4576	220	12	2023	2023	NUM
ejpam-4576	220	13	)	)	PUNCT
ejpam-4576	220	14	,	,	PUNCT
ejpam-4576	220	15	84	84	NUM
ejpam-4576	220	16	-	-	SYM
ejpam-4576	220	17	96	96	NUM
ejpam-4576	220	18	91	91	NUM
ejpam-4576	220	19	corollary	corollary	NOUN
ejpam-4576	220	20	3	3	NUM
ejpam-4576	220	21	.	.	PUNCT
ejpam-4576	221	1	for	for	ADP
ejpam-4576	221	2	a	a	DET
ejpam-4576	221	3	function	function	NOUN
ejpam-4576	221	4	f	f	NOUN
ejpam-4576	221	5	:	:	PUNCT
ejpam-4576	221	6	(	(	PUNCT
ejpam-4576	221	7	x	x	X
ejpam-4576	221	8	,	,	PUNCT
ejpam-4576	221	9	τ	τ	X
ejpam-4576	221	10	)	)	PUNCT
ejpam-4576	221	11	→	→	SYM
ejpam-4576	221	12	(	(	PUNCT
ejpam-4576	221	13	y	y	PROPN
ejpam-4576	221	14	,	,	PUNCT
ejpam-4576	221	15	σ	σ	PROPN
ejpam-4576	221	16	)	)	PUNCT
ejpam-4576	221	17	,	,	PUNCT
ejpam-4576	221	18	the	the	DET
ejpam-4576	221	19	following	follow	VERB
ejpam-4576	221	20	properties	property	NOUN
ejpam-4576	221	21	are	be	AUX
ejpam-4576	221	22	equivalent	equivalent	ADJ
ejpam-4576	221	23	:	:	PUNCT
ejpam-4576	221	24	(	(	PUNCT
ejpam-4576	221	25	1	1	X
ejpam-4576	221	26	)	)	PUNCT
ejpam-4576	221	27	f	f	NOUN
ejpam-4576	221	28	is	be	AUX
ejpam-4576	221	29	almost	almost	ADV
ejpam-4576	221	30	(	(	PUNCT
ejpam-4576	221	31	λ	λ	NOUN
ejpam-4576	221	32	,	,	PUNCT
ejpam-4576	221	33	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	221	34	;	;	PUNCT
ejpam-4576	221	35	(	(	PUNCT
ejpam-4576	221	36	2	2	X
ejpam-4576	221	37	)	)	PUNCT
ejpam-4576	222	1	[	[	X
ejpam-4576	222	2	f−1(v	f−1(v	NOUN
ejpam-4576	222	3	)	)	PUNCT
ejpam-4576	222	4	]	]	PUNCT
ejpam-4576	222	5	(	(	PUNCT
ejpam-4576	222	6	λ	λ	NOUN
ejpam-4576	222	7	,	,	PUNCT
ejpam-4576	222	8	sp	sp	NOUN
ejpam-4576	222	9	)	)	PUNCT
ejpam-4576	222	10	⊆	⊆	NUM
ejpam-4576	222	11	f−1(v	f−1(v	NOUN
ejpam-4576	222	12	(	(	PUNCT
ejpam-4576	222	13	λ	λ	PROPN
ejpam-4576	222	14	,	,	PUNCT
ejpam-4576	222	15	sp	sp	NOUN
ejpam-4576	222	16	)	)	PUNCT
ejpam-4576	222	17	)	)	PUNCT
ejpam-4576	222	18	for	for	ADP
ejpam-4576	222	19	every	every	DET
ejpam-4576	222	20	β(λ	β(λ	PROPN
ejpam-4576	222	21	,	,	PUNCT
ejpam-4576	222	22	sp)-open	sp)-open	NOUN
ejpam-4576	222	23	set	set	VERB
ejpam-4576	222	24	v	v	NOUN
ejpam-4576	222	25	of	of	ADP
ejpam-4576	222	26	y	y	PROPN
ejpam-4576	222	27	;	;	PUNCT
ejpam-4576	222	28	(	(	PUNCT
ejpam-4576	222	29	3	3	X
ejpam-4576	222	30	)	)	PUNCT
ejpam-4576	223	1	[	[	X
ejpam-4576	223	2	f−1(v	f−1(v	NOUN
ejpam-4576	223	3	)	)	PUNCT
ejpam-4576	223	4	]	]	PUNCT
ejpam-4576	223	5	(	(	PUNCT
ejpam-4576	223	6	λ	λ	NOUN
ejpam-4576	223	7	,	,	PUNCT
ejpam-4576	223	8	sp	sp	NOUN
ejpam-4576	223	9	)	)	PUNCT
ejpam-4576	223	10	⊆	⊆	NUM
ejpam-4576	223	11	f−1(v	f−1(v	NOUN
ejpam-4576	223	12	(	(	PUNCT
ejpam-4576	223	13	λ	λ	PROPN
ejpam-4576	223	14	,	,	PUNCT
ejpam-4576	223	15	sp	sp	NOUN
ejpam-4576	223	16	)	)	PUNCT
ejpam-4576	223	17	)	)	PUNCT
ejpam-4576	223	18	for	for	ADP
ejpam-4576	223	19	every	every	DET
ejpam-4576	223	20	s(λ	s(λ	PROPN
ejpam-4576	223	21	,	,	PUNCT
ejpam-4576	223	22	sp)-open	sp)-open	VERB
ejpam-4576	223	23	set	set	VERB
ejpam-4576	223	24	v	v	NOUN
ejpam-4576	223	25	of	of	ADP
ejpam-4576	223	26	y	y	PROPN
ejpam-4576	223	27	.	.	PUNCT
ejpam-4576	224	1	let	let	VERB
ejpam-4576	224	2	a	a	DET
ejpam-4576	224	3	be	be	AUX
ejpam-4576	224	4	a	a	DET
ejpam-4576	224	5	subset	subset	NOUN
ejpam-4576	224	6	of	of	ADP
ejpam-4576	224	7	a	a	DET
ejpam-4576	224	8	topological	topological	ADJ
ejpam-4576	224	9	space	space	NOUN
ejpam-4576	224	10	(	(	PUNCT
ejpam-4576	224	11	x	x	X
ejpam-4576	224	12	,	,	PUNCT
ejpam-4576	224	13	τ	τ	PROPN
ejpam-4576	224	14	)	)	PUNCT
ejpam-4576	224	15	.	.	PUNCT
ejpam-4576	225	1	a	a	DET
ejpam-4576	225	2	point	point	NOUN
ejpam-4576	225	3	x	x	X
ejpam-4576	225	4	∈	∈	NOUN
ejpam-4576	225	5	x	x	PUNCT
ejpam-4576	225	6	is	be	AUX
ejpam-4576	225	7	called	call	VERB
ejpam-4576	225	8	a	a	DET
ejpam-4576	225	9	δ(λ	δ(λ	PROPN
ejpam-4576	225	10	,	,	PUNCT
ejpam-4576	225	11	sp)cluster	sp)cluster	NOUN
ejpam-4576	225	12	point	point	NOUN
ejpam-4576	225	13	[	[	X
ejpam-4576	225	14	19	19	NUM
ejpam-4576	225	15	]	]	PUNCT
ejpam-4576	225	16	of	of	ADP
ejpam-4576	225	17	a	a	PRON
ejpam-4576	225	18	if	if	SCONJ
ejpam-4576	225	19	a∩	a∩	PROPN
ejpam-4576	225	20	[	[	X
ejpam-4576	225	21	u	u	X
ejpam-4576	225	22	(	(	PUNCT
ejpam-4576	225	23	λ	λ	PROPN
ejpam-4576	225	24	,	,	PUNCT
ejpam-4576	225	25	sp)](λ	sp)](λ	PROPN
ejpam-4576	225	26	,	,	PUNCT
ejpam-4576	225	27	sp	sp	NOUN
ejpam-4576	225	28	)	)	PUNCT
ejpam-4576	225	29	̸=	̸=	PROPN
ejpam-4576	225	30	∅	∅	NOUN
ejpam-4576	225	31	for	for	ADP
ejpam-4576	225	32	every	every	DET
ejpam-4576	225	33	(	(	PUNCT
ejpam-4576	225	34	λ	λ	NOUN
ejpam-4576	225	35	,	,	PUNCT
ejpam-4576	225	36	sp)-open	sp)-open	NOUN
ejpam-4576	225	37	set	set	VERB
ejpam-4576	225	38	u	u	NOUN
ejpam-4576	225	39	of	of	ADP
ejpam-4576	225	40	x	x	SYM
ejpam-4576	225	41	containing	contain	VERB
ejpam-4576	225	42	x.	x.	NOUN
ejpam-4576	225	43	the	the	DET
ejpam-4576	225	44	set	set	NOUN
ejpam-4576	225	45	of	of	ADP
ejpam-4576	225	46	all	all	DET
ejpam-4576	225	47	δ(λ	δ(λ	PROPN
ejpam-4576	225	48	,	,	PUNCT
ejpam-4576	225	49	sp)-cluster	sp)-cluster	NOUN
ejpam-4576	225	50	points	point	NOUN
ejpam-4576	225	51	of	of	ADP
ejpam-4576	225	52	a	a	PRON
ejpam-4576	225	53	is	be	AUX
ejpam-4576	225	54	called	call	VERB
ejpam-4576	225	55	the	the	DET
ejpam-4576	225	56	δ(λ	δ(λ	PROPN
ejpam-4576	225	57	,	,	PUNCT
ejpam-4576	225	58	sp)-closure	sp)-closure	NOUN
ejpam-4576	226	1	[	[	X
ejpam-4576	226	2	19	19	NUM
ejpam-4576	226	3	]	]	PUNCT
ejpam-4576	226	4	of	of	ADP
ejpam-4576	226	5	a	a	PRON
ejpam-4576	226	6	and	and	CCONJ
ejpam-4576	226	7	is	be	AUX
ejpam-4576	226	8	denoted	denote	VERB
ejpam-4576	226	9	by	by	ADP
ejpam-4576	226	10	aδ(λ	aδ(λ	NUM
ejpam-4576	226	11	,	,	PUNCT
ejpam-4576	226	12	sp	sp	NOUN
ejpam-4576	226	13	)	)	PUNCT
ejpam-4576	226	14	.	.	PUNCT
ejpam-4576	227	1	if	if	SCONJ
ejpam-4576	227	2	a	a	DET
ejpam-4576	227	3	=	=	NOUN
ejpam-4576	227	4	aδ(λ	aδ(λ	NUM
ejpam-4576	227	5	,	,	PUNCT
ejpam-4576	227	6	sp	sp	NOUN
ejpam-4576	227	7	)	)	PUNCT
ejpam-4576	227	8	,	,	PUNCT
ejpam-4576	227	9	then	then	ADV
ejpam-4576	227	10	a	a	PRON
ejpam-4576	227	11	is	be	AUX
ejpam-4576	227	12	said	say	VERB
ejpam-4576	227	13	to	to	PART
ejpam-4576	227	14	be	be	AUX
ejpam-4576	227	15	δ(λ	δ(λ	PROPN
ejpam-4576	227	16	,	,	PUNCT
ejpam-4576	227	17	sp)-closed	sp)-close	VERB
ejpam-4576	227	18	[	[	X
ejpam-4576	227	19	19	19	NUM
ejpam-4576	227	20	]	]	PUNCT
ejpam-4576	227	21	.	.	PUNCT
ejpam-4576	228	1	the	the	DET
ejpam-4576	228	2	complement	complement	NOUN
ejpam-4576	228	3	of	of	ADP
ejpam-4576	228	4	a	a	DET
ejpam-4576	228	5	δ(λ	δ(λ	PROPN
ejpam-4576	228	6	,	,	PUNCT
ejpam-4576	228	7	sp)-closed	sp)-close	VERB
ejpam-4576	228	8	set	set	NOUN
ejpam-4576	228	9	is	be	AUX
ejpam-4576	228	10	said	say	VERB
ejpam-4576	228	11	to	to	PART
ejpam-4576	228	12	be	be	AUX
ejpam-4576	228	13	δ(λ	δ(λ	PROPN
ejpam-4576	228	14	,	,	PUNCT
ejpam-4576	228	15	sp)-open	sp)-open	NOUN
ejpam-4576	228	16	.	.	PUNCT
ejpam-4576	229	1	the	the	DET
ejpam-4576	229	2	union	union	NOUN
ejpam-4576	229	3	of	of	ADP
ejpam-4576	229	4	all	all	DET
ejpam-4576	229	5	δ(λ	δ(λ	PROPN
ejpam-4576	229	6	,	,	PUNCT
ejpam-4576	229	7	sp)open	sp)open	NOUN
ejpam-4576	229	8	sets	set	NOUN
ejpam-4576	229	9	contained	contain	VERB
ejpam-4576	229	10	in	in	ADP
ejpam-4576	229	11	a	a	PRON
ejpam-4576	229	12	is	be	AUX
ejpam-4576	229	13	called	call	VERB
ejpam-4576	229	14	the	the	DET
ejpam-4576	229	15	δ(λ	δ(λ	PROPN
ejpam-4576	229	16	,	,	PUNCT
ejpam-4576	229	17	sp)-interior	sp)-interior	NOUN
ejpam-4576	229	18	[	[	X
ejpam-4576	229	19	19	19	NUM
ejpam-4576	229	20	]	]	PUNCT
ejpam-4576	229	21	of	of	ADP
ejpam-4576	229	22	a	a	PRON
ejpam-4576	229	23	and	and	CCONJ
ejpam-4576	229	24	is	be	AUX
ejpam-4576	229	25	denoted	denote	VERB
ejpam-4576	229	26	by	by	ADP
ejpam-4576	229	27	aδ(λ	aδ(λ	NUM
ejpam-4576	229	28	,	,	PUNCT
ejpam-4576	229	29	sp	sp	NOUN
ejpam-4576	229	30	)	)	PUNCT
ejpam-4576	229	31	.	.	PUNCT
ejpam-4576	230	1	lemma	lemma	PROPN
ejpam-4576	230	2	5	5	NUM
ejpam-4576	230	3	.	.	PUNCT
ejpam-4576	231	1	[	[	X
ejpam-4576	231	2	19	19	NUM
ejpam-4576	231	3	]	]	PUNCT
ejpam-4576	231	4	let	let	VERB
ejpam-4576	231	5	a	a	PRON
ejpam-4576	231	6	be	be	AUX
ejpam-4576	231	7	a	a	DET
ejpam-4576	231	8	subset	subset	NOUN
ejpam-4576	231	9	of	of	ADP
ejpam-4576	231	10	a	a	DET
ejpam-4576	231	11	topological	topological	ADJ
ejpam-4576	231	12	space	space	NOUN
ejpam-4576	231	13	(	(	PUNCT
ejpam-4576	231	14	x	x	X
ejpam-4576	231	15	,	,	PUNCT
ejpam-4576	231	16	τ	τ	PROPN
ejpam-4576	231	17	)	)	PUNCT
ejpam-4576	231	18	.	.	PUNCT
ejpam-4576	232	1	then	then	ADV
ejpam-4576	232	2	,	,	PUNCT
ejpam-4576	232	3	the	the	DET
ejpam-4576	232	4	following	follow	VERB
ejpam-4576	232	5	properties	property	NOUN
ejpam-4576	232	6	hold	hold	VERB
ejpam-4576	232	7	:	:	PUNCT
ejpam-4576	232	8	(	(	PUNCT
ejpam-4576	232	9	1	1	X
ejpam-4576	232	10	)	)	PUNCT
ejpam-4576	232	11	if	if	SCONJ
ejpam-4576	232	12	a	a	PRON
ejpam-4576	232	13	is	be	AUX
ejpam-4576	232	14	(	(	PUNCT
ejpam-4576	232	15	λ	λ	NOUN
ejpam-4576	232	16	,	,	PUNCT
ejpam-4576	232	17	sp)-open	sp)-open	ADJ
ejpam-4576	232	18	in	in	ADP
ejpam-4576	232	19	x	x	NOUN
ejpam-4576	232	20	,	,	PUNCT
ejpam-4576	232	21	then	then	ADV
ejpam-4576	232	22	a(λ	a(λ	ADV
ejpam-4576	232	23	,	,	PUNCT
ejpam-4576	232	24	sp	sp	NOUN
ejpam-4576	232	25	)	)	PUNCT
ejpam-4576	232	26	=	=	SYM
ejpam-4576	232	27	aδ(λ	aδ(λ	NUM
ejpam-4576	232	28	,	,	PUNCT
ejpam-4576	232	29	sp	sp	NOUN
ejpam-4576	232	30	)	)	PUNCT
ejpam-4576	232	31	.	.	PUNCT
ejpam-4576	233	1	(	(	PUNCT
ejpam-4576	233	2	2	2	NUM
ejpam-4576	233	3	)	)	PUNCT
ejpam-4576	233	4	aδ(λ	aδ(λ	NUM
ejpam-4576	233	5	,	,	PUNCT
ejpam-4576	233	6	sp	sp	NOUN
ejpam-4576	233	7	)	)	PUNCT
ejpam-4576	233	8	is	be	AUX
ejpam-4576	233	9	(	(	PUNCT
ejpam-4576	233	10	λ	λ	X
ejpam-4576	233	11	,	,	PUNCT
ejpam-4576	233	12	sp)-closed	sp)-close	VERB
ejpam-4576	233	13	.	.	PUNCT
ejpam-4576	234	1	theorem	theorem	VERB
ejpam-4576	234	2	7	7	NUM
ejpam-4576	234	3	.	.	X
ejpam-4576	234	4	for	for	ADP
ejpam-4576	234	5	a	a	DET
ejpam-4576	234	6	multifunction	multifunction	NOUN
ejpam-4576	235	1	f	f	NOUN
ejpam-4576	235	2	:	:	PUNCT
ejpam-4576	235	3	(	(	PUNCT
ejpam-4576	235	4	x	x	X
ejpam-4576	235	5	,	,	PUNCT
ejpam-4576	235	6	τ	τ	X
ejpam-4576	235	7	)	)	PUNCT
ejpam-4576	235	8	→	→	SYM
ejpam-4576	235	9	(	(	PUNCT
ejpam-4576	235	10	y	y	PROPN
ejpam-4576	235	11	,	,	PUNCT
ejpam-4576	235	12	σ	σ	PROPN
ejpam-4576	235	13	)	)	PUNCT
ejpam-4576	235	14	,	,	PUNCT
ejpam-4576	235	15	the	the	DET
ejpam-4576	235	16	following	follow	VERB
ejpam-4576	235	17	properties	property	NOUN
ejpam-4576	235	18	are	be	AUX
ejpam-4576	235	19	equivalent	equivalent	ADJ
ejpam-4576	235	20	:	:	PUNCT
ejpam-4576	235	21	(	(	PUNCT
ejpam-4576	235	22	1	1	X
ejpam-4576	235	23	)	)	PUNCT
ejpam-4576	235	24	f	f	PROPN
ejpam-4576	235	25	is	be	AUX
ejpam-4576	235	26	upper	upper	ADJ
ejpam-4576	235	27	almost	almost	ADV
ejpam-4576	235	28	(	(	PUNCT
ejpam-4576	235	29	λ	λ	NOUN
ejpam-4576	235	30	,	,	PUNCT
ejpam-4576	235	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	235	32	;	;	PUNCT
ejpam-4576	235	33	(	(	PUNCT
ejpam-4576	235	34	2	2	X
ejpam-4576	235	35	)	)	PUNCT
ejpam-4576	236	1	[	[	X
ejpam-4576	236	2	f−([[bδ(λ	f−([[bδ(λ	PROPN
ejpam-4576	236	3	,	,	PUNCT
ejpam-4576	236	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	236	5	,	,	PUNCT
ejpam-4576	236	6	sp	sp	NOUN
ejpam-4576	236	7	)	)	PUNCT
ejpam-4576	236	8	]	]	PUNCT
ejpam-4576	236	9	(	(	PUNCT
ejpam-4576	236	10	λ	λ	INTJ
ejpam-4576	236	11	,	,	PUNCT
ejpam-4576	236	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	236	13	,	,	PUNCT
ejpam-4576	236	14	sp	sp	NOUN
ejpam-4576	236	15	)	)	PUNCT
ejpam-4576	236	16	⊆	⊆	NUM
ejpam-4576	236	17	f−(bδ(λ	f−(bδ(λ	NOUN
ejpam-4576	236	18	,	,	PUNCT
ejpam-4576	236	19	sp	sp	NOUN
ejpam-4576	236	20	)	)	PUNCT
ejpam-4576	236	21	)	)	PUNCT
ejpam-4576	236	22	for	for	ADP
ejpam-4576	236	23	every	every	DET
ejpam-4576	236	24	subset	subset	NOUN
ejpam-4576	236	25	b	b	PROPN
ejpam-4576	236	26	of	of	ADP
ejpam-4576	236	27	y	y	PROPN
ejpam-4576	236	28	;	;	PUNCT
ejpam-4576	236	29	(	(	PUNCT
ejpam-4576	236	30	3	3	X
ejpam-4576	236	31	)	)	PUNCT
ejpam-4576	236	32	[	[	X
ejpam-4576	236	33	f−([[b(λ	f−([[b(λ	X
ejpam-4576	236	34	,	,	PUNCT
ejpam-4576	236	35	sp)](λ	sp)](λ	PROPN
ejpam-4576	236	36	,	,	PUNCT
ejpam-4576	236	37	sp	sp	NOUN
ejpam-4576	236	38	)	)	PUNCT
ejpam-4576	236	39	]	]	PUNCT
ejpam-4576	236	40	(	(	PUNCT
ejpam-4576	236	41	λ	λ	INTJ
ejpam-4576	236	42	,	,	PUNCT
ejpam-4576	236	43	sp))](λ	sp))](λ	PROPN
ejpam-4576	236	44	,	,	PUNCT
ejpam-4576	236	45	sp	sp	NOUN
ejpam-4576	236	46	)	)	PUNCT
ejpam-4576	236	47	⊆	⊆	NUM
ejpam-4576	236	48	f−(bδ(λ	f−(bδ(λ	NOUN
ejpam-4576	236	49	,	,	PUNCT
ejpam-4576	236	50	sp	sp	NOUN
ejpam-4576	236	51	)	)	PUNCT
ejpam-4576	236	52	)	)	PUNCT
ejpam-4576	236	53	for	for	ADP
ejpam-4576	236	54	every	every	DET
ejpam-4576	236	55	subset	subset	NOUN
ejpam-4576	236	56	b	b	PROPN
ejpam-4576	236	57	of	of	ADP
ejpam-4576	236	58	y	y	PROPN
ejpam-4576	236	59	.	.	PUNCT
ejpam-4576	237	1	proof	proof	NOUN
ejpam-4576	237	2	.	.	PUNCT
ejpam-4576	238	1	(	(	PUNCT
ejpam-4576	238	2	1	1	X
ejpam-4576	238	3	)	)	PUNCT
ejpam-4576	238	4	⇒	⇒	NOUN
ejpam-4576	238	5	(	(	PUNCT
ejpam-4576	238	6	2	2	NUM
ejpam-4576	238	7	):	):	PUNCT
ejpam-4576	238	8	let	let	VERB
ejpam-4576	238	9	b	b	X
ejpam-4576	238	10	be	be	AUX
ejpam-4576	238	11	any	any	DET
ejpam-4576	238	12	subset	subset	NOUN
ejpam-4576	238	13	of	of	ADP
ejpam-4576	238	14	y	y	PROPN
ejpam-4576	238	15	.	.	PUNCT
ejpam-4576	239	1	by	by	ADP
ejpam-4576	239	2	lemma	lemma	PROPN
ejpam-4576	239	3	5	5	NUM
ejpam-4576	239	4	,	,	PUNCT
ejpam-4576	239	5	bδ(λ	bδ(λ	NUM
ejpam-4576	239	6	,	,	PUNCT
ejpam-4576	239	7	sp	sp	NOUN
ejpam-4576	239	8	)	)	PUNCT
ejpam-4576	239	9	is	be	AUX
ejpam-4576	239	10	(	(	PUNCT
ejpam-4576	239	11	λ	λ	X
ejpam-4576	239	12	,	,	PUNCT
ejpam-4576	239	13	sp)-closed	sp)-close	VERB
ejpam-4576	239	14	in	in	ADP
ejpam-4576	239	15	y	y	PROPN
ejpam-4576	239	16	and	and	CCONJ
ejpam-4576	239	17	by	by	ADP
ejpam-4576	239	18	theorem	theorem	NOUN
ejpam-4576	239	19	3	3	NUM
ejpam-4576	239	20	,	,	PUNCT
ejpam-4576	239	21	[	[	X
ejpam-4576	239	22	f−([[bδ(λ	f−([[bδ(λ	PROPN
ejpam-4576	239	23	,	,	PUNCT
ejpam-4576	239	24	sp)](λ	sp)](λ	PROPN
ejpam-4576	239	25	,	,	PUNCT
ejpam-4576	239	26	sp	sp	NOUN
ejpam-4576	239	27	)	)	PUNCT
ejpam-4576	239	28	]	]	PUNCT
ejpam-4576	240	1	(	(	PUNCT
ejpam-4576	240	2	λ	λ	INTJ
ejpam-4576	240	3	,	,	PUNCT
ejpam-4576	240	4	sp))](λ	sp))](λ	PROPN
ejpam-4576	240	5	,	,	PUNCT
ejpam-4576	240	6	sp	sp	NOUN
ejpam-4576	240	7	)	)	PUNCT
ejpam-4576	240	8	⊆	⊆	NUM
ejpam-4576	240	9	f−(bδ(λ	f−(bδ(λ	NOUN
ejpam-4576	240	10	,	,	PUNCT
ejpam-4576	240	11	sp	sp	NOUN
ejpam-4576	240	12	)	)	PUNCT
ejpam-4576	240	13	)	)	PUNCT
ejpam-4576	240	14	.	.	PUNCT
ejpam-4576	241	1	(	(	PUNCT
ejpam-4576	241	2	2	2	X
ejpam-4576	241	3	)	)	PUNCT
ejpam-4576	241	4	⇒	⇒	NOUN
ejpam-4576	241	5	(	(	PUNCT
ejpam-4576	241	6	3	3	NUM
ejpam-4576	241	7	):	):	PUNCT
ejpam-4576	241	8	this	this	PRON
ejpam-4576	241	9	is	be	AUX
ejpam-4576	241	10	obvious	obvious	ADJ
ejpam-4576	241	11	since	since	SCONJ
ejpam-4576	241	12	b(λ	b(λ	PROPN
ejpam-4576	241	13	,	,	PUNCT
ejpam-4576	241	14	sp	sp	NOUN
ejpam-4576	241	15	)	)	PUNCT
ejpam-4576	241	16	⊆	⊆	NUM
ejpam-4576	241	17	bδ(λ	bδ(λ	NUM
ejpam-4576	241	18	,	,	PUNCT
ejpam-4576	241	19	sp	sp	NOUN
ejpam-4576	241	20	)	)	PUNCT
ejpam-4576	241	21	.	.	PUNCT
ejpam-4576	242	1	(	(	PUNCT
ejpam-4576	242	2	3	3	X
ejpam-4576	242	3	)	)	PUNCT
ejpam-4576	242	4	⇒	⇒	NOUN
ejpam-4576	242	5	(	(	PUNCT
ejpam-4576	242	6	1	1	NUM
ejpam-4576	242	7	):	):	PUNCT
ejpam-4576	242	8	letk	letk	ADJ
ejpam-4576	242	9	be	be	VERB
ejpam-4576	242	10	any	any	DET
ejpam-4576	242	11	r(λ	r(λ	NOUN
ejpam-4576	242	12	,	,	PUNCT
ejpam-4576	242	13	sp)-closed	sp)-close	VERB
ejpam-4576	242	14	set	set	NOUN
ejpam-4576	242	15	of	of	ADP
ejpam-4576	242	16	y	y	PROPN
ejpam-4576	242	17	.	.	PUNCT
ejpam-4576	243	1	thus	thus	ADV
ejpam-4576	243	2	,	,	PUNCT
ejpam-4576	243	3	by	by	ADP
ejpam-4576	243	4	(	(	PUNCT
ejpam-4576	243	5	3	3	NUM
ejpam-4576	243	6	)	)	PUNCT
ejpam-4576	243	7	,	,	PUNCT
ejpam-4576	243	8	we	we	PRON
ejpam-4576	243	9	have	have	VERB
ejpam-4576	243	10	[	[	X
ejpam-4576	243	11	f−(k)](λ	f−(k)](λ	PROPN
ejpam-4576	243	12	,	,	PUNCT
ejpam-4576	243	13	sp	sp	NOUN
ejpam-4576	243	14	)	)	PUNCT
ejpam-4576	243	15	=	=	NOUN
ejpam-4576	244	1	[	[	X
ejpam-4576	244	2	f−([k(λ	f−([k(λ	NOUN
ejpam-4576	244	3	,	,	PUNCT
ejpam-4576	244	4	sp	sp	NOUN
ejpam-4576	244	5	)	)	PUNCT
ejpam-4576	244	6	]	]	PUNCT
ejpam-4576	245	1	(	(	PUNCT
ejpam-4576	245	2	λ	λ	INTJ
ejpam-4576	245	3	,	,	PUNCT
ejpam-4576	245	4	sp))](λ	sp))](λ	PROPN
ejpam-4576	245	5	,	,	PUNCT
ejpam-4576	245	6	sp	sp	NOUN
ejpam-4576	245	7	)	)	PUNCT
ejpam-4576	245	8	=	=	PUNCT
ejpam-4576	246	1	[	[	X
ejpam-4576	246	2	f−([[k(λ	f−([[k(λ	X
ejpam-4576	246	3	,	,	PUNCT
ejpam-4576	246	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	246	5	,	,	PUNCT
ejpam-4576	246	6	sp	sp	NOUN
ejpam-4576	246	7	)	)	PUNCT
ejpam-4576	246	8	]	]	PUNCT
ejpam-4576	246	9	(	(	PUNCT
ejpam-4576	246	10	λ	λ	INTJ
ejpam-4576	246	11	,	,	PUNCT
ejpam-4576	246	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	246	13	,	,	PUNCT
ejpam-4576	246	14	sp	sp	NOUN
ejpam-4576	246	15	)	)	PUNCT
ejpam-4576	246	16	⊆	⊆	PROPN
ejpam-4576	246	17	f−(kδ(λ	f−(kδ(λ	PROPN
ejpam-4576	246	18	,	,	PUNCT
ejpam-4576	246	19	sp	sp	NOUN
ejpam-4576	246	20	)	)	PUNCT
ejpam-4576	246	21	)	)	PUNCT
ejpam-4576	246	22	=	=	SYM
ejpam-4576	246	23	f−(k	f−(k	PROPN
ejpam-4576	246	24	)	)	PUNCT
ejpam-4576	246	25	and	and	CCONJ
ejpam-4576	246	26	hence	hence	ADV
ejpam-4576	246	27	f−(k	f−(k	PROPN
ejpam-4576	246	28	)	)	PUNCT
ejpam-4576	246	29	is	be	AUX
ejpam-4576	246	30	(	(	PUNCT
ejpam-4576	246	31	λ	λ	X
ejpam-4576	246	32	,	,	PUNCT
ejpam-4576	246	33	sp)-closed	sp)-close	VERB
ejpam-4576	246	34	in	in	ADP
ejpam-4576	246	35	x.	x.	NOUN
ejpam-4576	246	36	by	by	ADP
ejpam-4576	246	37	theorem	theorem	NOUN
ejpam-4576	246	38	3	3	NUM
ejpam-4576	246	39	,	,	PUNCT
ejpam-4576	246	40	f	f	PROPN
ejpam-4576	246	41	is	be	AUX
ejpam-4576	246	42	upper	upper	ADJ
ejpam-4576	246	43	almost	almost	ADV
ejpam-4576	246	44	(	(	PUNCT
ejpam-4576	246	45	λ	λ	NOUN
ejpam-4576	246	46	,	,	PUNCT
ejpam-4576	246	47	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	246	48	.	.	PUNCT
ejpam-4576	247	1	theorem	theorem	VERB
ejpam-4576	247	2	8	8	NUM
ejpam-4576	247	3	.	.	PUNCT
ejpam-4576	248	1	for	for	ADP
ejpam-4576	248	2	a	a	DET
ejpam-4576	248	3	multifunction	multifunction	NOUN
ejpam-4576	248	4	f	f	NOUN
ejpam-4576	248	5	:	:	PUNCT
ejpam-4576	248	6	(	(	PUNCT
ejpam-4576	248	7	x	x	X
ejpam-4576	248	8	,	,	PUNCT
ejpam-4576	248	9	τ	τ	X
ejpam-4576	248	10	)	)	PUNCT
ejpam-4576	248	11	→	→	SYM
ejpam-4576	248	12	(	(	PUNCT
ejpam-4576	248	13	y	y	PROPN
ejpam-4576	248	14	,	,	PUNCT
ejpam-4576	248	15	σ	σ	PROPN
ejpam-4576	248	16	)	)	PUNCT
ejpam-4576	248	17	,	,	PUNCT
ejpam-4576	248	18	the	the	DET
ejpam-4576	248	19	following	follow	VERB
ejpam-4576	248	20	properties	property	NOUN
ejpam-4576	248	21	are	be	AUX
ejpam-4576	248	22	equivalent	equivalent	ADJ
ejpam-4576	248	23	:	:	PUNCT
ejpam-4576	248	24	(	(	PUNCT
ejpam-4576	248	25	1	1	X
ejpam-4576	248	26	)	)	PUNCT
ejpam-4576	248	27	f	f	PROPN
ejpam-4576	248	28	is	be	AUX
ejpam-4576	248	29	lower	low	ADJ
ejpam-4576	248	30	almost	almost	ADV
ejpam-4576	248	31	(	(	PUNCT
ejpam-4576	248	32	λ	λ	NOUN
ejpam-4576	248	33	,	,	PUNCT
ejpam-4576	248	34	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	248	35	;	;	PUNCT
ejpam-4576	248	36	(	(	PUNCT
ejpam-4576	248	37	2	2	X
ejpam-4576	248	38	)	)	PUNCT
ejpam-4576	249	1	[	[	X
ejpam-4576	249	2	f+([[bδ(λ	f+([[bδ(λ	PROPN
ejpam-4576	249	3	,	,	PUNCT
ejpam-4576	249	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	249	5	,	,	PUNCT
ejpam-4576	249	6	sp	sp	NOUN
ejpam-4576	249	7	)	)	PUNCT
ejpam-4576	249	8	]	]	PUNCT
ejpam-4576	249	9	(	(	PUNCT
ejpam-4576	249	10	λ	λ	INTJ
ejpam-4576	249	11	,	,	PUNCT
ejpam-4576	249	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	249	13	,	,	PUNCT
ejpam-4576	249	14	sp	sp	NOUN
ejpam-4576	249	15	)	)	PUNCT
ejpam-4576	249	16	⊆	⊆	NUM
ejpam-4576	249	17	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	249	18	,	,	PUNCT
ejpam-4576	249	19	sp	sp	NOUN
ejpam-4576	249	20	)	)	PUNCT
ejpam-4576	249	21	)	)	PUNCT
ejpam-4576	249	22	for	for	ADP
ejpam-4576	249	23	every	every	DET
ejpam-4576	249	24	subset	subset	NOUN
ejpam-4576	249	25	b	b	PROPN
ejpam-4576	249	26	of	of	ADP
ejpam-4576	249	27	y	y	PROPN
ejpam-4576	249	28	;	;	PUNCT
ejpam-4576	249	29	(	(	PUNCT
ejpam-4576	249	30	3	3	X
ejpam-4576	249	31	)	)	PUNCT
ejpam-4576	250	1	[	[	X
ejpam-4576	250	2	f+([[b(λ	f+([[b(λ	NUM
ejpam-4576	250	3	,	,	PUNCT
ejpam-4576	250	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	250	5	,	,	PUNCT
ejpam-4576	250	6	sp	sp	NOUN
ejpam-4576	250	7	)	)	PUNCT
ejpam-4576	250	8	]	]	PUNCT
ejpam-4576	250	9	(	(	PUNCT
ejpam-4576	250	10	λ	λ	INTJ
ejpam-4576	250	11	,	,	PUNCT
ejpam-4576	250	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	250	13	,	,	PUNCT
ejpam-4576	250	14	sp	sp	NOUN
ejpam-4576	250	15	)	)	PUNCT
ejpam-4576	250	16	⊆	⊆	NUM
ejpam-4576	250	17	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	250	18	,	,	PUNCT
ejpam-4576	250	19	sp	sp	NOUN
ejpam-4576	250	20	)	)	PUNCT
ejpam-4576	250	21	)	)	PUNCT
ejpam-4576	250	22	for	for	ADP
ejpam-4576	250	23	every	every	DET
ejpam-4576	250	24	subset	subset	NOUN
ejpam-4576	250	25	b	b	PROPN
ejpam-4576	250	26	of	of	ADP
ejpam-4576	250	27	y	y	PROPN
ejpam-4576	250	28	.	.	PUNCT
ejpam-4576	251	1	proof	proof	NOUN
ejpam-4576	251	2	.	.	PUNCT
ejpam-4576	252	1	the	the	DET
ejpam-4576	252	2	proof	proof	NOUN
ejpam-4576	252	3	is	be	AUX
ejpam-4576	252	4	similar	similar	ADJ
ejpam-4576	252	5	to	to	ADP
ejpam-4576	252	6	that	that	PRON
ejpam-4576	252	7	of	of	ADP
ejpam-4576	252	8	theorem	theorem	ADJ
ejpam-4576	252	9	7	7	NUM
ejpam-4576	252	10	.	.	PUNCT
ejpam-4576	252	11	c.	c.	PROPN
ejpam-4576	252	12	boonpok	boonpok	PROPN
ejpam-4576	252	13	,	,	PUNCT
ejpam-4576	252	14	n.	n.	PROPN
ejpam-4576	252	15	viriyapong	viriyapong	PROPN
ejpam-4576	252	16	/	/	SYM
ejpam-4576	252	17	eur	eur	PROPN
ejpam-4576	252	18	.	.	PUNCT
ejpam-4576	253	1	j.	j.	PROPN
ejpam-4576	253	2	pure	pure	PROPN
ejpam-4576	253	3	appl	appl	PROPN
ejpam-4576	253	4	.	.	PROPN
ejpam-4576	253	5	math	math	PROPN
ejpam-4576	253	6	,	,	PUNCT
ejpam-4576	253	7	16	16	NUM
ejpam-4576	253	8	(	(	PUNCT
ejpam-4576	253	9	1	1	NUM
ejpam-4576	253	10	)	)	PUNCT
ejpam-4576	253	11	(	(	PUNCT
ejpam-4576	253	12	2023	2023	NUM
ejpam-4576	253	13	)	)	PUNCT
ejpam-4576	253	14	,	,	PUNCT
ejpam-4576	253	15	84	84	NUM
ejpam-4576	253	16	-	-	SYM
ejpam-4576	253	17	96	96	NUM
ejpam-4576	253	18	92	92	NUM
ejpam-4576	253	19	corollary	corollary	ADJ
ejpam-4576	253	20	4	4	NUM
ejpam-4576	253	21	.	.	PUNCT
ejpam-4576	254	1	for	for	ADP
ejpam-4576	254	2	a	a	DET
ejpam-4576	254	3	function	function	NOUN
ejpam-4576	254	4	f	f	NOUN
ejpam-4576	254	5	:	:	PUNCT
ejpam-4576	254	6	(	(	PUNCT
ejpam-4576	254	7	x	x	X
ejpam-4576	254	8	,	,	PUNCT
ejpam-4576	254	9	τ	τ	X
ejpam-4576	254	10	)	)	PUNCT
ejpam-4576	254	11	→	→	SYM
ejpam-4576	254	12	(	(	PUNCT
ejpam-4576	254	13	y	y	PROPN
ejpam-4576	254	14	,	,	PUNCT
ejpam-4576	254	15	σ	σ	PROPN
ejpam-4576	254	16	)	)	PUNCT
ejpam-4576	254	17	,	,	PUNCT
ejpam-4576	254	18	the	the	DET
ejpam-4576	254	19	following	follow	VERB
ejpam-4576	254	20	properties	property	NOUN
ejpam-4576	254	21	are	be	AUX
ejpam-4576	254	22	equivalent	equivalent	ADJ
ejpam-4576	254	23	:	:	PUNCT
ejpam-4576	254	24	(	(	PUNCT
ejpam-4576	254	25	1	1	X
ejpam-4576	254	26	)	)	PUNCT
ejpam-4576	254	27	f	f	NOUN
ejpam-4576	254	28	is	be	AUX
ejpam-4576	254	29	almost	almost	ADV
ejpam-4576	254	30	(	(	PUNCT
ejpam-4576	254	31	λ	λ	NOUN
ejpam-4576	254	32	,	,	PUNCT
ejpam-4576	254	33	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	254	34	;	;	PUNCT
ejpam-4576	254	35	(	(	PUNCT
ejpam-4576	254	36	2	2	X
ejpam-4576	254	37	)	)	PUNCT
ejpam-4576	254	38	[	[	X
ejpam-4576	254	39	f−1([[bδ(λ	f−1([[bδ(λ	NOUN
ejpam-4576	254	40	,	,	PUNCT
ejpam-4576	254	41	sp)](λ	sp)](λ	PROPN
ejpam-4576	254	42	,	,	PUNCT
ejpam-4576	254	43	sp	sp	NOUN
ejpam-4576	254	44	)	)	PUNCT
ejpam-4576	254	45	]	]	PUNCT
ejpam-4576	255	1	(	(	PUNCT
ejpam-4576	255	2	λ	λ	INTJ
ejpam-4576	255	3	,	,	PUNCT
ejpam-4576	255	4	sp))](λ	sp))](λ	PROPN
ejpam-4576	255	5	,	,	PUNCT
ejpam-4576	255	6	sp	sp	NOUN
ejpam-4576	255	7	)	)	PUNCT
ejpam-4576	255	8	⊆	⊆	NUM
ejpam-4576	255	9	f−1(bδ(λ	f−1(bδ(λ	NOUN
ejpam-4576	255	10	,	,	PUNCT
ejpam-4576	255	11	sp	sp	NOUN
ejpam-4576	255	12	)	)	PUNCT
ejpam-4576	255	13	)	)	PUNCT
ejpam-4576	255	14	for	for	ADP
ejpam-4576	255	15	every	every	DET
ejpam-4576	255	16	subset	subset	NOUN
ejpam-4576	255	17	b	b	PROPN
ejpam-4576	255	18	of	of	ADP
ejpam-4576	255	19	y	y	PROPN
ejpam-4576	255	20	;	;	PUNCT
ejpam-4576	255	21	(	(	PUNCT
ejpam-4576	255	22	3	3	X
ejpam-4576	255	23	)	)	PUNCT
ejpam-4576	256	1	[	[	X
ejpam-4576	256	2	f−1([[b(λ	f−1([[b(λ	X
ejpam-4576	256	3	,	,	PUNCT
ejpam-4576	256	4	sp)](λ	sp)](λ	PROPN
ejpam-4576	256	5	,	,	PUNCT
ejpam-4576	256	6	sp	sp	NOUN
ejpam-4576	256	7	)	)	PUNCT
ejpam-4576	256	8	]	]	PUNCT
ejpam-4576	256	9	(	(	PUNCT
ejpam-4576	256	10	λ	λ	INTJ
ejpam-4576	256	11	,	,	PUNCT
ejpam-4576	256	12	sp))](λ	sp))](λ	PROPN
ejpam-4576	256	13	,	,	PUNCT
ejpam-4576	256	14	sp	sp	NOUN
ejpam-4576	256	15	)	)	PUNCT
ejpam-4576	256	16	⊆	⊆	NUM
ejpam-4576	256	17	f−1(bδ(λ	f−1(bδ(λ	NOUN
ejpam-4576	256	18	,	,	PUNCT
ejpam-4576	256	19	sp	sp	NOUN
ejpam-4576	256	20	)	)	PUNCT
ejpam-4576	256	21	)	)	PUNCT
ejpam-4576	256	22	for	for	ADP
ejpam-4576	256	23	every	every	DET
ejpam-4576	256	24	subset	subset	NOUN
ejpam-4576	256	25	b	b	PROPN
ejpam-4576	256	26	of	of	ADP
ejpam-4576	256	27	y	y	PROPN
ejpam-4576	256	28	.	.	PUNCT
ejpam-4576	257	1	definition	definition	NOUN
ejpam-4576	257	2	3	3	NUM
ejpam-4576	257	3	.	.	PUNCT
ejpam-4576	258	1	[	[	X
ejpam-4576	258	2	3	3	X
ejpam-4576	258	3	]	]	PUNCT
ejpam-4576	258	4	a	a	DET
ejpam-4576	258	5	multifunction	multifunction	NOUN
ejpam-4576	258	6	f	f	NOUN
ejpam-4576	258	7	:	:	PUNCT
ejpam-4576	258	8	(	(	PUNCT
ejpam-4576	258	9	x	x	X
ejpam-4576	258	10	,	,	PUNCT
ejpam-4576	258	11	τ	τ	X
ejpam-4576	258	12	)	)	PUNCT
ejpam-4576	258	13	→	→	SYM
ejpam-4576	258	14	(	(	PUNCT
ejpam-4576	258	15	y	y	PROPN
ejpam-4576	258	16	,	,	PUNCT
ejpam-4576	258	17	σ	σ	PROPN
ejpam-4576	258	18	)	)	PUNCT
ejpam-4576	258	19	is	be	AUX
ejpam-4576	258	20	called	call	VERB
ejpam-4576	258	21	lower	low	ADJ
ejpam-4576	258	22	(	(	PUNCT
ejpam-4576	258	23	λ	λ	X
ejpam-4576	258	24	,	,	PUNCT
ejpam-4576	258	25	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	258	26	at	at	ADP
ejpam-4576	258	27	a	a	DET
ejpam-4576	258	28	point	point	NOUN
ejpam-4576	258	29	x	x	SYM
ejpam-4576	258	30	∈	∈	NOUN
ejpam-4576	258	31	x	x	INTJ
ejpam-4576	258	32	if	if	SCONJ
ejpam-4576	258	33	,	,	PUNCT
ejpam-4576	258	34	for	for	SCONJ
ejpam-4576	258	35	each	each	DET
ejpam-4576	258	36	(	(	PUNCT
ejpam-4576	258	37	λ	λ	PROPN
ejpam-4576	258	38	,	,	PUNCT
ejpam-4576	258	39	sp)-open	sp)-open	NOUN
ejpam-4576	258	40	set	set	VERB
ejpam-4576	258	41	v	v	NUM
ejpam-4576	258	42	of	of	ADP
ejpam-4576	258	43	y	y	PRON
ejpam-4576	258	44	such	such	ADJ
ejpam-4576	258	45	that	that	SCONJ
ejpam-4576	258	46	f	f	PROPN
ejpam-4576	258	47	(	(	PUNCT
ejpam-4576	258	48	x	x	NOUN
ejpam-4576	258	49	)	)	PUNCT
ejpam-4576	258	50	∩	∩	NOUN
ejpam-4576	258	51	v	v	ADP
ejpam-4576	258	52	̸=	̸=	PROPN
ejpam-4576	258	53	∅	∅	NOUN
ejpam-4576	258	54	,	,	PUNCT
ejpam-4576	258	55	there	there	PRON
ejpam-4576	258	56	exists	exist	VERB
ejpam-4576	258	57	a	a	DET
ejpam-4576	258	58	(	(	PUNCT
ejpam-4576	258	59	λ	λ	NOUN
ejpam-4576	258	60	,	,	PUNCT
ejpam-4576	258	61	sp)-open	sp)-open	NOUN
ejpam-4576	258	62	set	set	VERB
ejpam-4576	258	63	u	u	NOUN
ejpam-4576	258	64	of	of	ADP
ejpam-4576	258	65	x	x	PUNCT
ejpam-4576	258	66	containing	contain	VERB
ejpam-4576	258	67	x	x	PUNCT
ejpam-4576	258	68	such	such	ADJ
ejpam-4576	258	69	that	that	SCONJ
ejpam-4576	258	70	f	f	PROPN
ejpam-4576	258	71	(	(	PUNCT
ejpam-4576	258	72	z	z	NOUN
ejpam-4576	258	73	)	)	PUNCT
ejpam-4576	258	74	∩	∩	NOUN
ejpam-4576	258	75	v	v	ADP
ejpam-4576	258	76	̸=	̸=	PROPN
ejpam-4576	258	77	∅	∅	NOUN
ejpam-4576	258	78	for	for	ADP
ejpam-4576	258	79	each	each	DET
ejpam-4576	258	80	z	z	NOUN
ejpam-4576	258	81	∈	∈	PROPN
ejpam-4576	258	82	u	u	NOUN
ejpam-4576	258	83	.	.	PUNCT
ejpam-4576	259	1	a	a	DET
ejpam-4576	259	2	multifunction	multifunction	NOUN
ejpam-4576	259	3	f	f	NOUN
ejpam-4576	259	4	:	:	PUNCT
ejpam-4576	259	5	(	(	PUNCT
ejpam-4576	259	6	x	x	X
ejpam-4576	259	7	,	,	PUNCT
ejpam-4576	259	8	τ	τ	X
ejpam-4576	259	9	)	)	PUNCT
ejpam-4576	259	10	→	→	SYM
ejpam-4576	259	11	(	(	PUNCT
ejpam-4576	259	12	y	y	PROPN
ejpam-4576	259	13	,	,	PUNCT
ejpam-4576	259	14	σ	σ	PROPN
ejpam-4576	259	15	)	)	PUNCT
ejpam-4576	259	16	is	be	AUX
ejpam-4576	259	17	called	call	VERB
ejpam-4576	259	18	lower	low	ADJ
ejpam-4576	259	19	(	(	PUNCT
ejpam-4576	259	20	λ	λ	NOUN
ejpam-4576	259	21	,	,	PUNCT
ejpam-4576	259	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	259	23	if	if	SCONJ
ejpam-4576	259	24	f	f	PROPN
ejpam-4576	259	25	has	have	VERB
ejpam-4576	259	26	this	this	DET
ejpam-4576	259	27	property	property	NOUN
ejpam-4576	259	28	at	at	ADP
ejpam-4576	259	29	each	each	DET
ejpam-4576	259	30	point	point	NOUN
ejpam-4576	259	31	of	of	ADP
ejpam-4576	259	32	x.	x.	PROPN
ejpam-4576	259	33	lemma	lemma	PROPN
ejpam-4576	259	34	6	6	NUM
ejpam-4576	259	35	.	.	PUNCT
ejpam-4576	260	1	if	if	SCONJ
ejpam-4576	260	2	f	f	PROPN
ejpam-4576	260	3	:	:	PUNCT
ejpam-4576	260	4	(	(	PUNCT
ejpam-4576	260	5	x	x	X
ejpam-4576	260	6	,	,	PUNCT
ejpam-4576	260	7	τ	τ	X
ejpam-4576	260	8	)	)	PUNCT
ejpam-4576	260	9	→	→	SYM
ejpam-4576	260	10	(	(	PUNCT
ejpam-4576	260	11	y	y	PROPN
ejpam-4576	260	12	,	,	PUNCT
ejpam-4576	260	13	σ	σ	PROPN
ejpam-4576	260	14	)	)	PUNCT
ejpam-4576	260	15	is	be	AUX
ejpam-4576	260	16	lower	low	ADJ
ejpam-4576	260	17	almost	almost	ADV
ejpam-4576	260	18	(	(	PUNCT
ejpam-4576	260	19	λ	λ	NOUN
ejpam-4576	260	20	,	,	PUNCT
ejpam-4576	260	21	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	260	22	,	,	PUNCT
ejpam-4576	260	23	then	then	ADV
ejpam-4576	260	24	for	for	ADP
ejpam-4576	260	25	each	each	DET
ejpam-4576	260	26	x	x	SYM
ejpam-4576	260	27	∈	∈	PROPN
ejpam-4576	260	28	x	x	X
ejpam-4576	260	29	and	and	CCONJ
ejpam-4576	260	30	each	each	DET
ejpam-4576	260	31	subset	subset	NOUN
ejpam-4576	260	32	b	b	PROPN
ejpam-4576	260	33	of	of	ADP
ejpam-4576	260	34	y	y	PROPN
ejpam-4576	260	35	with	with	ADP
ejpam-4576	260	36	f	f	PROPN
ejpam-4576	260	37	(	(	PUNCT
ejpam-4576	260	38	x	x	X
ejpam-4576	260	39	)	)	PUNCT
ejpam-4576	260	40	∩bδ(λ	∩bδ(λ	PROPN
ejpam-4576	260	41	,	,	PUNCT
ejpam-4576	260	42	sp	sp	NOUN
ejpam-4576	260	43	)	)	PUNCT
ejpam-4576	260	44	̸=	̸=	NOUN
ejpam-4576	260	45	∅	∅	NOUN
ejpam-4576	260	46	,	,	PUNCT
ejpam-4576	260	47	there	there	PRON
ejpam-4576	260	48	exists	exist	VERB
ejpam-4576	260	49	u	u	PROPN
ejpam-4576	260	50	∈	∈	PROPN
ejpam-4576	260	51	λspo(x	λspo(x	PROPN
ejpam-4576	260	52	,	,	PUNCT
ejpam-4576	260	53	τ	τ	X
ejpam-4576	260	54	)	)	PUNCT
ejpam-4576	260	55	containing	contain	VERB
ejpam-4576	260	56	x	x	PUNCT
ejpam-4576	260	57	such	such	ADJ
ejpam-4576	260	58	that	that	SCONJ
ejpam-4576	260	59	u	u	PROPN
ejpam-4576	260	60	⊆	⊆	NUM
ejpam-4576	260	61	f−(b	f−(b	NOUN
ejpam-4576	260	62	)	)	PUNCT
ejpam-4576	260	63	.	.	PUNCT
ejpam-4576	261	1	proof	proof	NOUN
ejpam-4576	261	2	.	.	PUNCT
ejpam-4576	262	1	let	let	VERB
ejpam-4576	262	2	x	x	PUNCT
ejpam-4576	262	3	∈	∈	PROPN
ejpam-4576	262	4	x	x	PUNCT
ejpam-4576	262	5	and	and	CCONJ
ejpam-4576	262	6	let	let	VERB
ejpam-4576	262	7	b	b	X
ejpam-4576	262	8	be	be	AUX
ejpam-4576	262	9	a	a	DET
ejpam-4576	262	10	subset	subset	NOUN
ejpam-4576	262	11	of	of	ADP
ejpam-4576	262	12	y	y	PROPN
ejpam-4576	262	13	with	with	ADP
ejpam-4576	262	14	f	f	PROPN
ejpam-4576	262	15	(	(	PUNCT
ejpam-4576	262	16	x	x	NOUN
ejpam-4576	262	17	)	)	PUNCT
ejpam-4576	262	18	∩	∩	NOUN
ejpam-4576	262	19	bδ(λ	bδ(λ	NUM
ejpam-4576	262	20	,	,	PUNCT
ejpam-4576	262	21	sp	sp	NOUN
ejpam-4576	262	22	)	)	PUNCT
ejpam-4576	262	23	̸=	̸=	PROPN
ejpam-4576	262	24	∅.	∅.	ADV
ejpam-4576	262	25	since	since	SCONJ
ejpam-4576	262	26	f	f	PROPN
ejpam-4576	262	27	(	(	PUNCT
ejpam-4576	262	28	x	x	NOUN
ejpam-4576	262	29	)	)	PUNCT
ejpam-4576	262	30	∩	∩	NOUN
ejpam-4576	262	31	bδ(λ	bδ(λ	NUM
ejpam-4576	262	32	,	,	PUNCT
ejpam-4576	262	33	sp	sp	NOUN
ejpam-4576	262	34	)	)	PUNCT
ejpam-4576	262	35	̸=	̸=	NOUN
ejpam-4576	262	36	∅	∅	NOUN
ejpam-4576	262	37	,	,	PUNCT
ejpam-4576	262	38	there	there	PRON
ejpam-4576	262	39	exists	exist	VERB
ejpam-4576	262	40	a	a	DET
ejpam-4576	262	41	nonempty	nonempty	ADJ
ejpam-4576	262	42	r(λ	r(λ	NOUN
ejpam-4576	262	43	,	,	PUNCT
ejpam-4576	262	44	sp)-open	sp)-open	VERB
ejpam-4576	262	45	set	set	VERB
ejpam-4576	262	46	v	v	NUM
ejpam-4576	262	47	of	of	ADP
ejpam-4576	262	48	y	y	PRON
ejpam-4576	262	49	such	such	ADJ
ejpam-4576	262	50	that	that	PRON
ejpam-4576	262	51	v	v	ADP
ejpam-4576	262	52	⊆	⊆	NUM
ejpam-4576	262	53	b	b	NOUN
ejpam-4576	262	54	and	and	CCONJ
ejpam-4576	262	55	f	f	PROPN
ejpam-4576	262	56	(	(	PUNCT
ejpam-4576	262	57	x)∩v	x)∩v	PROPN
ejpam-4576	262	58	̸=	̸=	PROPN
ejpam-4576	262	59	∅.	∅.	ADV
ejpam-4576	262	60	since	since	SCONJ
ejpam-4576	262	61	f	f	PROPN
ejpam-4576	262	62	is	be	AUX
ejpam-4576	262	63	lower	low	ADJ
ejpam-4576	262	64	almost	almost	ADV
ejpam-4576	262	65	(	(	PUNCT
ejpam-4576	262	66	λ	λ	NOUN
ejpam-4576	262	67	,	,	PUNCT
ejpam-4576	262	68	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	262	69	,	,	PUNCT
ejpam-4576	262	70	there	there	PRON
ejpam-4576	262	71	exists	exist	VERB
ejpam-4576	262	72	u	u	PROPN
ejpam-4576	262	73	∈	∈	PROPN
ejpam-4576	262	74	λspo(x	λspo(x	PROPN
ejpam-4576	262	75	,	,	PUNCT
ejpam-4576	262	76	τ	τ	X
ejpam-4576	262	77	)	)	PUNCT
ejpam-4576	262	78	containing	contain	VERB
ejpam-4576	262	79	x	x	PUNCT
ejpam-4576	262	80	such	such	ADJ
ejpam-4576	262	81	that	that	SCONJ
ejpam-4576	262	82	f	f	PROPN
ejpam-4576	262	83	(	(	PUNCT
ejpam-4576	262	84	z	z	NOUN
ejpam-4576	262	85	)	)	PUNCT
ejpam-4576	262	86	∩	∩	NOUN
ejpam-4576	262	87	v	v	ADP
ejpam-4576	262	88	̸=	̸=	PROPN
ejpam-4576	262	89	∅	∅	NOUN
ejpam-4576	262	90	for	for	ADP
ejpam-4576	262	91	each	each	DET
ejpam-4576	262	92	z	z	NOUN
ejpam-4576	262	93	∈	∈	PROPN
ejpam-4576	262	94	u	u	NOUN
ejpam-4576	262	95	;	;	PUNCT
ejpam-4576	262	96	hence	hence	ADV
ejpam-4576	262	97	u	u	NOUN
ejpam-4576	262	98	⊆	⊆	NUM
ejpam-4576	262	99	f−(b	f−(b	NOUN
ejpam-4576	262	100	)	)	PUNCT
ejpam-4576	262	101	.	.	PUNCT
ejpam-4576	263	1	theorem	theorem	VERB
ejpam-4576	263	2	9	9	NUM
ejpam-4576	263	3	.	.	X
ejpam-4576	263	4	for	for	ADP
ejpam-4576	263	5	a	a	DET
ejpam-4576	263	6	multifunction	multifunction	NOUN
ejpam-4576	264	1	f	f	NOUN
ejpam-4576	264	2	:	:	PUNCT
ejpam-4576	264	3	(	(	PUNCT
ejpam-4576	264	4	x	x	X
ejpam-4576	264	5	,	,	PUNCT
ejpam-4576	264	6	τ	τ	X
ejpam-4576	264	7	)	)	PUNCT
ejpam-4576	264	8	→	→	SYM
ejpam-4576	264	9	(	(	PUNCT
ejpam-4576	264	10	y	y	PROPN
ejpam-4576	264	11	,	,	PUNCT
ejpam-4576	264	12	σ	σ	PROPN
ejpam-4576	264	13	)	)	PUNCT
ejpam-4576	264	14	,	,	PUNCT
ejpam-4576	264	15	the	the	DET
ejpam-4576	264	16	following	follow	VERB
ejpam-4576	264	17	properties	property	NOUN
ejpam-4576	264	18	are	be	AUX
ejpam-4576	264	19	equivalent	equivalent	ADJ
ejpam-4576	264	20	:	:	PUNCT
ejpam-4576	264	21	(	(	PUNCT
ejpam-4576	264	22	1	1	X
ejpam-4576	264	23	)	)	PUNCT
ejpam-4576	264	24	f	f	PROPN
ejpam-4576	264	25	is	be	AUX
ejpam-4576	264	26	lower	low	ADJ
ejpam-4576	264	27	almost	almost	ADV
ejpam-4576	264	28	(	(	PUNCT
ejpam-4576	264	29	λ	λ	NOUN
ejpam-4576	264	30	,	,	PUNCT
ejpam-4576	264	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	264	32	;	;	PUNCT
ejpam-4576	264	33	(	(	PUNCT
ejpam-4576	264	34	2	2	X
ejpam-4576	264	35	)	)	PUNCT
ejpam-4576	264	36	[	[	X
ejpam-4576	264	37	f+(b)](λ	f+(b)](λ	NUM
ejpam-4576	264	38	,	,	PUNCT
ejpam-4576	264	39	sp	sp	NOUN
ejpam-4576	264	40	)	)	PUNCT
ejpam-4576	264	41	⊆	⊆	NUM
ejpam-4576	264	42	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	264	43	,	,	PUNCT
ejpam-4576	264	44	sp	sp	NOUN
ejpam-4576	264	45	)	)	PUNCT
ejpam-4576	264	46	)	)	PUNCT
ejpam-4576	264	47	for	for	ADP
ejpam-4576	264	48	every	every	DET
ejpam-4576	264	49	subset	subset	NOUN
ejpam-4576	264	50	b	b	PROPN
ejpam-4576	264	51	of	of	ADP
ejpam-4576	264	52	y	y	PROPN
ejpam-4576	264	53	;	;	PUNCT
ejpam-4576	264	54	(	(	PUNCT
ejpam-4576	264	55	3	3	X
ejpam-4576	264	56	)	)	PUNCT
ejpam-4576	264	57	f	f	NOUN
ejpam-4576	264	58	(	(	PUNCT
ejpam-4576	264	59	a(λ	a(λ	ADV
ejpam-4576	264	60	,	,	PUNCT
ejpam-4576	264	61	sp	sp	NOUN
ejpam-4576	264	62	)	)	PUNCT
ejpam-4576	264	63	)	)	PUNCT
ejpam-4576	264	64	⊆	⊆	NUM
ejpam-4576	265	1	[	[	X
ejpam-4576	265	2	f	f	X
ejpam-4576	265	3	(	(	PUNCT
ejpam-4576	265	4	a)]δ(λ	a)]δ(λ	NOUN
ejpam-4576	265	5	,	,	PUNCT
ejpam-4576	265	6	sp	sp	NOUN
ejpam-4576	265	7	)	)	PUNCT
ejpam-4576	265	8	for	for	ADP
ejpam-4576	265	9	every	every	DET
ejpam-4576	265	10	subset	subset	NOUN
ejpam-4576	265	11	a	a	PRON
ejpam-4576	265	12	of	of	ADP
ejpam-4576	265	13	x	x	PRON
ejpam-4576	265	14	;	;	PUNCT
ejpam-4576	265	15	(	(	PUNCT
ejpam-4576	265	16	4	4	X
ejpam-4576	265	17	)	)	PUNCT
ejpam-4576	265	18	f+(k	f+(k	NUM
ejpam-4576	265	19	)	)	PUNCT
ejpam-4576	265	20	is	be	AUX
ejpam-4576	265	21	(	(	PUNCT
ejpam-4576	265	22	λ	λ	X
ejpam-4576	265	23	,	,	PUNCT
ejpam-4576	265	24	sp)-closed	sp)-close	VERB
ejpam-4576	265	25	in	in	ADP
ejpam-4576	265	26	x	x	PUNCT
ejpam-4576	265	27	for	for	ADP
ejpam-4576	265	28	every	every	DET
ejpam-4576	265	29	δ(λ	δ(λ	PROPN
ejpam-4576	265	30	,	,	PUNCT
ejpam-4576	265	31	sp)-closed	sp)-close	VERB
ejpam-4576	265	32	set	set	VERB
ejpam-4576	265	33	k	k	PROPN
ejpam-4576	265	34	of	of	ADP
ejpam-4576	265	35	y	y	PROPN
ejpam-4576	265	36	;	;	PUNCT
ejpam-4576	265	37	(	(	PUNCT
ejpam-4576	265	38	5	5	X
ejpam-4576	265	39	)	)	PUNCT
ejpam-4576	265	40	f−(v	f−(v	NOUN
ejpam-4576	265	41	)	)	PUNCT
ejpam-4576	265	42	is	be	AUX
ejpam-4576	265	43	(	(	PUNCT
ejpam-4576	265	44	λ	λ	INTJ
ejpam-4576	265	45	,	,	PUNCT
ejpam-4576	265	46	sp)-open	sp)-open	ADJ
ejpam-4576	265	47	in	in	ADP
ejpam-4576	265	48	x	x	PUNCT
ejpam-4576	265	49	for	for	ADP
ejpam-4576	265	50	every	every	DET
ejpam-4576	265	51	δ(λ	δ(λ	PROPN
ejpam-4576	265	52	,	,	PUNCT
ejpam-4576	265	53	sp)-open	sp)-open	VERB
ejpam-4576	265	54	set	set	VERB
ejpam-4576	265	55	v	v	NOUN
ejpam-4576	265	56	of	of	ADP
ejpam-4576	265	57	y	y	PROPN
ejpam-4576	265	58	;	;	PUNCT
ejpam-4576	265	59	(	(	PUNCT
ejpam-4576	265	60	6	6	X
ejpam-4576	265	61	)	)	PUNCT
ejpam-4576	265	62	f−(bδ(λ	f−(bδ(λ	NOUN
ejpam-4576	265	63	,	,	PUNCT
ejpam-4576	265	64	sp	sp	NOUN
ejpam-4576	265	65	)	)	PUNCT
ejpam-4576	265	66	)	)	PUNCT
ejpam-4576	266	1	⊆	⊆	NUM
ejpam-4576	266	2	[	[	X
ejpam-4576	266	3	f−(b)](λ	f−(b)](λ	PROPN
ejpam-4576	266	4	,	,	PUNCT
ejpam-4576	266	5	sp	sp	NOUN
ejpam-4576	266	6	)	)	PUNCT
ejpam-4576	266	7	for	for	ADP
ejpam-4576	266	8	each	each	DET
ejpam-4576	266	9	subset	subset	NOUN
ejpam-4576	266	10	b	b	PROPN
ejpam-4576	266	11	of	of	ADP
ejpam-4576	266	12	y	y	PROPN
ejpam-4576	266	13	.	.	PUNCT
ejpam-4576	267	1	proof	proof	NOUN
ejpam-4576	267	2	.	.	PUNCT
ejpam-4576	268	1	(	(	PUNCT
ejpam-4576	268	2	1	1	X
ejpam-4576	268	3	)	)	PUNCT
ejpam-4576	268	4	⇒	⇒	NOUN
ejpam-4576	268	5	(	(	PUNCT
ejpam-4576	268	6	2	2	NUM
ejpam-4576	268	7	):	):	PUNCT
ejpam-4576	268	8	let	let	VERB
ejpam-4576	268	9	b	b	X
ejpam-4576	268	10	be	be	AUX
ejpam-4576	268	11	any	any	DET
ejpam-4576	268	12	subset	subset	NOUN
ejpam-4576	268	13	of	of	ADP
ejpam-4576	268	14	y	y	PROPN
ejpam-4576	268	15	.	.	PUNCT
ejpam-4576	268	16	suppose	suppose	VERB
ejpam-4576	268	17	that	that	SCONJ
ejpam-4576	268	18	x	x	PROPN
ejpam-4576	268	19	̸∈	̸∈	PROPN
ejpam-4576	268	20	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	268	21	,	,	PUNCT
ejpam-4576	268	22	sp	sp	NOUN
ejpam-4576	268	23	)	)	PUNCT
ejpam-4576	268	24	)	)	PUNCT
ejpam-4576	268	25	.	.	PUNCT
ejpam-4576	269	1	then	then	ADV
ejpam-4576	269	2	,	,	PUNCT
ejpam-4576	269	3	we	we	PRON
ejpam-4576	269	4	have	have	VERB
ejpam-4576	269	5	x	x	PART
ejpam-4576	269	6	∈	∈	NOUN
ejpam-4576	269	7	f−(y	f−(y	NOUN
ejpam-4576	269	8	−bδ(λ	−bδ(λ	NOUN
ejpam-4576	269	9	,	,	PUNCT
ejpam-4576	269	10	sp	sp	NOUN
ejpam-4576	269	11	)	)	PUNCT
ejpam-4576	269	12	)	)	PUNCT
ejpam-4576	270	1	=	=	PUNCT
ejpam-4576	270	2	f−([y	f−([y	ADJ
ejpam-4576	270	3	−b]δ(λ	−b]δ(λ	ADJ
ejpam-4576	270	4	,	,	PUNCT
ejpam-4576	270	5	sp	sp	NOUN
ejpam-4576	270	6	)	)	PUNCT
ejpam-4576	270	7	)	)	PUNCT
ejpam-4576	270	8	.	.	PUNCT
ejpam-4576	271	1	by	by	ADP
ejpam-4576	271	2	lemma	lemma	PROPN
ejpam-4576	271	3	6	6	NUM
ejpam-4576	271	4	,	,	PUNCT
ejpam-4576	271	5	there	there	PRON
ejpam-4576	271	6	exists	exist	VERB
ejpam-4576	271	7	u	u	PROPN
ejpam-4576	271	8	∈	∈	PROPN
ejpam-4576	271	9	λspo(x	λspo(x	PROPN
ejpam-4576	271	10	,	,	PUNCT
ejpam-4576	271	11	τ	τ	X
ejpam-4576	271	12	)	)	PUNCT
ejpam-4576	271	13	containing	contain	VERB
ejpam-4576	271	14	x	x	PUNCT
ejpam-4576	271	15	such	such	ADJ
ejpam-4576	271	16	that	that	SCONJ
ejpam-4576	271	17	u	u	NOUN
ejpam-4576	271	18	⊆	⊆	NUM
ejpam-4576	271	19	f−(y	f−(y	NOUN
ejpam-4576	271	20	−b	−b	NOUN
ejpam-4576	271	21	)	)	PUNCT
ejpam-4576	272	1	=	=	PUNCT
ejpam-4576	272	2	x	x	X
ejpam-4576	273	1	−	−	NOUN
ejpam-4576	273	2	f+(b	f+(b	NOUN
ejpam-4576	273	3	)	)	PUNCT
ejpam-4576	273	4	.	.	PUNCT
ejpam-4576	274	1	thus	thus	ADV
ejpam-4576	274	2	,	,	PUNCT
ejpam-4576	274	3	u	u	NOUN
ejpam-4576	274	4	∩	∩	NOUN
ejpam-4576	274	5	f+(b	f+(b	NOUN
ejpam-4576	274	6	)	)	PUNCT
ejpam-4576	274	7	=	=	SYM
ejpam-4576	274	8	∅	∅	NOUN
ejpam-4576	274	9	and	and	CCONJ
ejpam-4576	274	10	hence	hence	ADV
ejpam-4576	274	11	x	x	X
ejpam-4576	274	12	∈	∈	NOUN
ejpam-4576	274	13	x	x	X
ejpam-4576	274	14	−	−	PROPN
ejpam-4576	275	1	[	[	X
ejpam-4576	275	2	f+(b)](λ	f+(b)](λ	NUM
ejpam-4576	275	3	,	,	PUNCT
ejpam-4576	275	4	sp	sp	NOUN
ejpam-4576	275	5	)	)	PUNCT
ejpam-4576	275	6	.	.	PUNCT
ejpam-4576	276	1	this	this	PRON
ejpam-4576	276	2	shows	show	VERB
ejpam-4576	276	3	that	that	SCONJ
ejpam-4576	276	4	[	[	X
ejpam-4576	276	5	f+(b)](λ	f+(b)](λ	NUM
ejpam-4576	276	6	,	,	PUNCT
ejpam-4576	276	7	sp	sp	NOUN
ejpam-4576	276	8	)	)	PUNCT
ejpam-4576	276	9	⊆	⊆	NUM
ejpam-4576	276	10	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	276	11	,	,	PUNCT
ejpam-4576	276	12	sp	sp	NOUN
ejpam-4576	276	13	)	)	PUNCT
ejpam-4576	276	14	)	)	PUNCT
ejpam-4576	276	15	.	.	PUNCT
ejpam-4576	277	1	(	(	PUNCT
ejpam-4576	277	2	2	2	X
ejpam-4576	277	3	)	)	PUNCT
ejpam-4576	277	4	⇒	⇒	NOUN
ejpam-4576	277	5	(	(	PUNCT
ejpam-4576	277	6	3	3	NUM
ejpam-4576	277	7	):	):	PUNCT
ejpam-4576	277	8	let	let	VERB
ejpam-4576	277	9	a	a	DET
ejpam-4576	277	10	be	be	AUX
ejpam-4576	277	11	any	any	DET
ejpam-4576	277	12	subset	subset	NOUN
ejpam-4576	277	13	of	of	ADP
ejpam-4576	277	14	x.	x.	NOUN
ejpam-4576	277	15	by	by	ADP
ejpam-4576	277	16	(	(	PUNCT
ejpam-4576	277	17	2	2	NUM
ejpam-4576	277	18	)	)	PUNCT
ejpam-4576	277	19	,	,	PUNCT
ejpam-4576	277	20	we	we	PRON
ejpam-4576	277	21	have	have	VERB
ejpam-4576	277	22	a(λ	a(λ	ADV
ejpam-4576	277	23	,	,	PUNCT
ejpam-4576	277	24	sp	sp	NOUN
ejpam-4576	277	25	)	)	PUNCT
ejpam-4576	277	26	⊆	⊆	NUM
ejpam-4576	278	1	[	[	X
ejpam-4576	278	2	f+(f	f+(f	PROPN
ejpam-4576	278	3	(	(	PUNCT
ejpam-4576	278	4	a))](λ	a))](λ	SYM
ejpam-4576	278	5	,	,	PUNCT
ejpam-4576	278	6	sp	sp	NOUN
ejpam-4576	278	7	)	)	PUNCT
ejpam-4576	278	8	⊆	⊆	NUM
ejpam-4576	278	9	f+([f	f+([f	ADP
ejpam-4576	278	10	(	(	PUNCT
ejpam-4576	278	11	a)]δ(λ	a)]δ(λ	NOUN
ejpam-4576	278	12	,	,	PUNCT
ejpam-4576	278	13	sp	sp	NOUN
ejpam-4576	278	14	)	)	PUNCT
ejpam-4576	278	15	)	)	PUNCT
ejpam-4576	278	16	and	and	CCONJ
ejpam-4576	278	17	hence	hence	ADV
ejpam-4576	278	18	f	f	X
ejpam-4576	278	19	(	(	PUNCT
ejpam-4576	278	20	a(λ	a(λ	ADV
ejpam-4576	278	21	,	,	PUNCT
ejpam-4576	278	22	sp	sp	NOUN
ejpam-4576	278	23	)	)	PUNCT
ejpam-4576	278	24	)	)	PUNCT
ejpam-4576	278	25	⊆	⊆	NUM
ejpam-4576	279	1	[	[	X
ejpam-4576	279	2	f	f	X
ejpam-4576	279	3	(	(	PUNCT
ejpam-4576	279	4	a)]δ(λ	a)]δ(λ	NOUN
ejpam-4576	279	5	,	,	PUNCT
ejpam-4576	279	6	sp	sp	NOUN
ejpam-4576	279	7	)	)	PUNCT
ejpam-4576	279	8	.	.	PUNCT
ejpam-4576	280	1	(	(	PUNCT
ejpam-4576	280	2	3	3	X
ejpam-4576	280	3	)	)	PUNCT
ejpam-4576	280	4	⇒	⇒	NOUN
ejpam-4576	280	5	(	(	PUNCT
ejpam-4576	280	6	1	1	NUM
ejpam-4576	280	7	):	):	PUNCT
ejpam-4576	280	8	let	let	VERB
ejpam-4576	280	9	b	b	X
ejpam-4576	280	10	be	be	AUX
ejpam-4576	280	11	any	any	DET
ejpam-4576	280	12	subset	subset	NOUN
ejpam-4576	280	13	of	of	ADP
ejpam-4576	280	14	y	y	PROPN
ejpam-4576	280	15	.	.	PUNCT
ejpam-4576	281	1	then	then	ADV
ejpam-4576	281	2	,	,	PUNCT
ejpam-4576	281	3	by	by	ADP
ejpam-4576	281	4	the	the	DET
ejpam-4576	281	5	hypothesis	hypothesis	NOUN
ejpam-4576	281	6	and	and	CCONJ
ejpam-4576	281	7	lemma	lemma	PROPN
ejpam-4576	281	8	5	5	NUM
ejpam-4576	281	9	,	,	PUNCT
ejpam-4576	281	10	f	f	X
ejpam-4576	281	11	(	(	PUNCT
ejpam-4576	281	12	[	[	X
ejpam-4576	281	13	f+([[b(λ	f+([[b(λ	NUM
ejpam-4576	281	14	,	,	PUNCT
ejpam-4576	281	15	sp)](λ	sp)](λ	PROPN
ejpam-4576	281	16	,	,	PUNCT
ejpam-4576	281	17	sp	sp	NOUN
ejpam-4576	281	18	)	)	PUNCT
ejpam-4576	281	19	]	]	PUNCT
ejpam-4576	281	20	(	(	PUNCT
ejpam-4576	281	21	λ	λ	INTJ
ejpam-4576	281	22	,	,	PUNCT
ejpam-4576	281	23	sp))](λ	sp))](λ	PROPN
ejpam-4576	281	24	,	,	PUNCT
ejpam-4576	281	25	sp	sp	NOUN
ejpam-4576	281	26	)	)	PUNCT
ejpam-4576	281	27	)	)	PUNCT
ejpam-4576	282	1	⊆	⊆	NUM
ejpam-4576	283	1	[	[	X
ejpam-4576	283	2	f	f	X
ejpam-4576	283	3	(	(	PUNCT
ejpam-4576	283	4	f+([[b(λ	f+([[b(λ	PROPN
ejpam-4576	283	5	,	,	PUNCT
ejpam-4576	283	6	sp)](λ	sp)](λ	PROPN
ejpam-4576	283	7	,	,	PUNCT
ejpam-4576	283	8	sp	sp	NOUN
ejpam-4576	283	9	)	)	PUNCT
ejpam-4576	283	10	]	]	PUNCT
ejpam-4576	283	11	(	(	PUNCT
ejpam-4576	283	12	λ	λ	X
ejpam-4576	283	13	,	,	PUNCT
ejpam-4576	283	14	sp)))]δ(λ	sp)))]δ(λ	PROPN
ejpam-4576	283	15	,	,	PUNCT
ejpam-4576	283	16	sp	sp	NOUN
ejpam-4576	283	17	)	)	PUNCT
ejpam-4576	283	18	c.	c.	NOUN
ejpam-4576	283	19	boonpok	boonpok	PROPN
ejpam-4576	283	20	,	,	PUNCT
ejpam-4576	283	21	n.	n.	PROPN
ejpam-4576	283	22	viriyapong	viriyapong	PROPN
ejpam-4576	283	23	/	/	SYM
ejpam-4576	283	24	eur	eur	PROPN
ejpam-4576	283	25	.	.	PUNCT
ejpam-4576	284	1	j.	j.	PROPN
ejpam-4576	284	2	pure	pure	PROPN
ejpam-4576	284	3	appl	appl	PROPN
ejpam-4576	284	4	.	.	PROPN
ejpam-4576	284	5	math	math	PROPN
ejpam-4576	284	6	,	,	PUNCT
ejpam-4576	284	7	16	16	NUM
ejpam-4576	284	8	(	(	PUNCT
ejpam-4576	284	9	1	1	NUM
ejpam-4576	284	10	)	)	PUNCT
ejpam-4576	284	11	(	(	PUNCT
ejpam-4576	284	12	2023	2023	NUM
ejpam-4576	284	13	)	)	PUNCT
ejpam-4576	284	14	,	,	PUNCT
ejpam-4576	284	15	84	84	NUM
ejpam-4576	284	16	-	-	SYM
ejpam-4576	284	17	96	96	NUM
ejpam-4576	285	1	93	93	NUM
ejpam-4576	285	2	⊆	⊆	NUM
ejpam-4576	286	1	[	[	X
ejpam-4576	286	2	[	[	X
ejpam-4576	286	3	b(λ	b(λ	NOUN
ejpam-4576	286	4	,	,	PUNCT
ejpam-4576	286	5	sp)](λ	sp)](λ	PROPN
ejpam-4576	286	6	,	,	PUNCT
ejpam-4576	286	7	sp	sp	NOUN
ejpam-4576	286	8	)	)	PUNCT
ejpam-4576	286	9	]	]	PUNCT
ejpam-4576	287	1	(	(	PUNCT
ejpam-4576	287	2	λ	λ	NOUN
ejpam-4576	287	3	,	,	PUNCT
ejpam-4576	287	4	sp	sp	NOUN
ejpam-4576	287	5	)	)	PUNCT
ejpam-4576	287	6	⊆	⊆	NUM
ejpam-4576	287	7	b(λ	b(λ	NOUN
ejpam-4576	287	8	,	,	PUNCT
ejpam-4576	287	9	sp	sp	NOUN
ejpam-4576	287	10	)	)	PUNCT
ejpam-4576	287	11	and	and	CCONJ
ejpam-4576	287	12	hence	hence	ADV
ejpam-4576	287	13	[	[	X
ejpam-4576	287	14	f+([[b(λ	f+([[b(λ	NUM
ejpam-4576	287	15	,	,	PUNCT
ejpam-4576	287	16	sp)](λ	sp)](λ	PROPN
ejpam-4576	287	17	,	,	PUNCT
ejpam-4576	287	18	sp	sp	NOUN
ejpam-4576	287	19	)	)	PUNCT
ejpam-4576	287	20	]	]	PUNCT
ejpam-4576	287	21	(	(	PUNCT
ejpam-4576	287	22	λ	λ	INTJ
ejpam-4576	287	23	,	,	PUNCT
ejpam-4576	287	24	sp))](λ	sp))](λ	PROPN
ejpam-4576	287	25	,	,	PUNCT
ejpam-4576	287	26	sp	sp	NOUN
ejpam-4576	287	27	)	)	PUNCT
ejpam-4576	287	28	⊆	⊆	NUM
ejpam-4576	287	29	f+(b(λ	f+(b(λ	PROPN
ejpam-4576	287	30	,	,	PUNCT
ejpam-4576	287	31	sp	sp	NOUN
ejpam-4576	287	32	)	)	PUNCT
ejpam-4576	287	33	)	)	PUNCT
ejpam-4576	287	34	.	.	PUNCT
ejpam-4576	288	1	by	by	ADP
ejpam-4576	288	2	theorem	theorem	NOUN
ejpam-4576	288	3	4	4	NUM
ejpam-4576	288	4	,	,	PUNCT
ejpam-4576	288	5	f	f	PROPN
ejpam-4576	288	6	is	be	AUX
ejpam-4576	288	7	lower	low	ADJ
ejpam-4576	288	8	almost	almost	ADV
ejpam-4576	288	9	(	(	PUNCT
ejpam-4576	288	10	λ	λ	NOUN
ejpam-4576	288	11	,	,	PUNCT
ejpam-4576	288	12	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	288	13	.	.	PUNCT
ejpam-4576	289	1	(	(	PUNCT
ejpam-4576	289	2	2	2	X
ejpam-4576	289	3	)	)	PUNCT
ejpam-4576	289	4	⇒	⇒	NOUN
ejpam-4576	289	5	(	(	PUNCT
ejpam-4576	289	6	4	4	NUM
ejpam-4576	289	7	):	):	PUNCT
ejpam-4576	289	8	let	let	VERB
ejpam-4576	289	9	k	k	PRON
ejpam-4576	289	10	be	be	AUX
ejpam-4576	289	11	any	any	DET
ejpam-4576	289	12	δ(λ	δ(λ	PROPN
ejpam-4576	289	13	,	,	PUNCT
ejpam-4576	289	14	sp)-closed	sp)-close	VERB
ejpam-4576	289	15	set	set	NOUN
ejpam-4576	289	16	of	of	ADP
ejpam-4576	289	17	y	y	PROPN
ejpam-4576	289	18	.	.	PUNCT
ejpam-4576	290	1	then	then	ADV
ejpam-4576	290	2	,	,	PUNCT
ejpam-4576	290	3	kδ(λ	kδ(λ	NOUN
ejpam-4576	290	4	,	,	PUNCT
ejpam-4576	290	5	sp	sp	NOUN
ejpam-4576	290	6	)	)	PUNCT
ejpam-4576	290	7	=	=	SYM
ejpam-4576	290	8	k.	k.	NOUN
ejpam-4576	290	9	by	by	ADP
ejpam-4576	290	10	(	(	PUNCT
ejpam-4576	290	11	2	2	NUM
ejpam-4576	290	12	)	)	PUNCT
ejpam-4576	290	13	,	,	PUNCT
ejpam-4576	290	14	we	we	PRON
ejpam-4576	290	15	have	have	VERB
ejpam-4576	290	16	[	[	X
ejpam-4576	290	17	f+(k)](λ	f+(k)](λ	NUM
ejpam-4576	290	18	,	,	PUNCT
ejpam-4576	290	19	sp	sp	NOUN
ejpam-4576	290	20	)	)	PUNCT
ejpam-4576	290	21	⊆	⊆	NUM
ejpam-4576	290	22	f+(kδ(λ	f+(kδ(λ	PROPN
ejpam-4576	290	23	,	,	PUNCT
ejpam-4576	290	24	sp	sp	NOUN
ejpam-4576	290	25	)	)	PUNCT
ejpam-4576	290	26	)	)	PUNCT
ejpam-4576	291	1	=	=	SYM
ejpam-4576	291	2	f+(k	f+(k	X
ejpam-4576	291	3	)	)	PUNCT
ejpam-4576	291	4	and	and	CCONJ
ejpam-4576	291	5	hence	hence	ADV
ejpam-4576	291	6	f+(k	f+(k	NUM
ejpam-4576	291	7	)	)	PUNCT
ejpam-4576	291	8	is	be	AUX
ejpam-4576	291	9	(	(	PUNCT
ejpam-4576	291	10	λ	λ	X
ejpam-4576	291	11	,	,	PUNCT
ejpam-4576	291	12	sp)-closed	sp)-close	VERB
ejpam-4576	291	13	in	in	ADP
ejpam-4576	291	14	x.	x.	NOUN
ejpam-4576	291	15	(	(	PUNCT
ejpam-4576	291	16	4	4	NUM
ejpam-4576	291	17	)	)	PUNCT
ejpam-4576	291	18	⇒	⇒	NOUN
ejpam-4576	291	19	(	(	PUNCT
ejpam-4576	291	20	5	5	NUM
ejpam-4576	291	21	):	):	PUNCT
ejpam-4576	291	22	the	the	DET
ejpam-4576	291	23	proof	proof	NOUN
ejpam-4576	291	24	is	be	AUX
ejpam-4576	291	25	obvious	obvious	ADJ
ejpam-4576	291	26	.	.	PUNCT
ejpam-4576	292	1	(	(	PUNCT
ejpam-4576	292	2	5	5	X
ejpam-4576	292	3	)	)	PUNCT
ejpam-4576	292	4	⇒	⇒	NOUN
ejpam-4576	292	5	(	(	PUNCT
ejpam-4576	292	6	6	6	NUM
ejpam-4576	292	7	):	):	PUNCT
ejpam-4576	292	8	let	let	VERB
ejpam-4576	292	9	b	b	X
ejpam-4576	292	10	be	be	AUX
ejpam-4576	292	11	any	any	DET
ejpam-4576	292	12	subset	subset	NOUN
ejpam-4576	292	13	of	of	ADP
ejpam-4576	292	14	y	y	PROPN
ejpam-4576	292	15	.	.	PUNCT
ejpam-4576	293	1	by	by	ADP
ejpam-4576	293	2	(	(	PUNCT
ejpam-4576	293	3	5	5	NUM
ejpam-4576	293	4	)	)	PUNCT
ejpam-4576	293	5	,	,	PUNCT
ejpam-4576	293	6	we	we	PRON
ejpam-4576	293	7	have	have	VERB
ejpam-4576	293	8	f−(bδ(λ	f−(bδ(λ	PROPN
ejpam-4576	293	9	,	,	PUNCT
ejpam-4576	293	10	sp	sp	NOUN
ejpam-4576	293	11	)	)	PUNCT
ejpam-4576	293	12	)	)	PUNCT
ejpam-4576	294	1	=	=	PUNCT
ejpam-4576	295	1	[	[	X
ejpam-4576	295	2	f−(bδ(λ	f−(bδ(λ	PROPN
ejpam-4576	295	3	,	,	PUNCT
ejpam-4576	295	4	sp))](λ	sp))](λ	PROPN
ejpam-4576	295	5	,	,	PUNCT
ejpam-4576	295	6	sp	sp	NOUN
ejpam-4576	295	7	)	)	PUNCT
ejpam-4576	295	8	⊆	⊆	NUM
ejpam-4576	295	9	[	[	X
ejpam-4576	295	10	f−(b)](λ	f−(b)](λ	PROPN
ejpam-4576	295	11	,	,	PUNCT
ejpam-4576	295	12	sp	sp	NOUN
ejpam-4576	295	13	)	)	PUNCT
ejpam-4576	295	14	.	.	PUNCT
ejpam-4576	296	1	(	(	PUNCT
ejpam-4576	296	2	6	6	X
ejpam-4576	296	3	)	)	PUNCT
ejpam-4576	296	4	⇒	⇒	NOUN
ejpam-4576	296	5	(	(	PUNCT
ejpam-4576	296	6	1	1	NUM
ejpam-4576	296	7	):	):	PUNCT
ejpam-4576	296	8	let	let	VERB
ejpam-4576	296	9	v	v	PART
ejpam-4576	296	10	be	be	AUX
ejpam-4576	296	11	any	any	DET
ejpam-4576	296	12	r(λ	r(λ	NOUN
ejpam-4576	296	13	,	,	PUNCT
ejpam-4576	296	14	sp)-open	sp)-open	ADJ
ejpam-4576	296	15	set	set	NOUN
ejpam-4576	296	16	of	of	ADP
ejpam-4576	296	17	y	y	PROPN
ejpam-4576	296	18	.	.	PUNCT
ejpam-4576	297	1	then	then	ADV
ejpam-4576	297	2	,	,	PUNCT
ejpam-4576	297	3	we	we	PRON
ejpam-4576	297	4	have	have	VERB
ejpam-4576	297	5	v	v	NOUN
ejpam-4576	297	6	is	be	AUX
ejpam-4576	297	7	δ(λ	δ(λ	PROPN
ejpam-4576	297	8	,	,	PUNCT
ejpam-4576	297	9	sp)-open	sp)-open	ADJ
ejpam-4576	297	10	and	and	CCONJ
ejpam-4576	297	11	vδ(λ	vδ(λ	X
ejpam-4576	297	12	,	,	PUNCT
ejpam-4576	297	13	sp	sp	NOUN
ejpam-4576	297	14	)	)	PUNCT
ejpam-4576	297	15	=	=	SYM
ejpam-4576	297	16	v	v	NOUN
ejpam-4576	297	17	.	.	PUNCT
ejpam-4576	298	1	thus	thus	ADV
ejpam-4576	298	2	,	,	PUNCT
ejpam-4576	298	3	by	by	ADP
ejpam-4576	298	4	(	(	PUNCT
ejpam-4576	298	5	6	6	NUM
ejpam-4576	298	6	)	)	PUNCT
ejpam-4576	298	7	,	,	PUNCT
ejpam-4576	298	8	f−(v	f−(v	ADJ
ejpam-4576	298	9	)	)	PUNCT
ejpam-4576	298	10	⊆	⊆	NUM
ejpam-4576	299	1	[	[	X
ejpam-4576	299	2	f−(v	f−(v	NOUN
ejpam-4576	299	3	)	)	PUNCT
ejpam-4576	299	4	]	]	PUNCT
ejpam-4576	299	5	(	(	PUNCT
ejpam-4576	299	6	λ	λ	NOUN
ejpam-4576	299	7	,	,	PUNCT
ejpam-4576	299	8	sp	sp	NOUN
ejpam-4576	299	9	)	)	PUNCT
ejpam-4576	299	10	and	and	CCONJ
ejpam-4576	299	11	hence	hence	ADV
ejpam-4576	299	12	f−(v	f−(v	ADJ
ejpam-4576	299	13	)	)	PUNCT
ejpam-4576	299	14	is	be	AUX
ejpam-4576	299	15	(	(	PUNCT
ejpam-4576	299	16	λ	λ	INTJ
ejpam-4576	299	17	,	,	PUNCT
ejpam-4576	299	18	sp)-open	sp)-open	ADJ
ejpam-4576	299	19	in	in	ADP
ejpam-4576	299	20	x.	x.	NOUN
ejpam-4576	299	21	by	by	ADP
ejpam-4576	299	22	theorem	theorem	NOUN
ejpam-4576	299	23	4	4	NUM
ejpam-4576	299	24	,	,	PUNCT
ejpam-4576	299	25	f	f	PROPN
ejpam-4576	299	26	is	be	AUX
ejpam-4576	299	27	lower	low	ADJ
ejpam-4576	299	28	almost	almost	ADV
ejpam-4576	299	29	(	(	PUNCT
ejpam-4576	299	30	λ	λ	NOUN
ejpam-4576	299	31	,	,	PUNCT
ejpam-4576	299	32	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	299	33	.	.	PUNCT
ejpam-4576	300	1	definition	definition	NOUN
ejpam-4576	300	2	4	4	NUM
ejpam-4576	300	3	.	.	PUNCT
ejpam-4576	301	1	[	[	X
ejpam-4576	301	2	19	19	NUM
ejpam-4576	301	3	]	]	PUNCT
ejpam-4576	301	4	a	a	DET
ejpam-4576	301	5	topological	topological	ADJ
ejpam-4576	301	6	space	space	NOUN
ejpam-4576	301	7	(	(	PUNCT
ejpam-4576	301	8	x	x	X
ejpam-4576	301	9	,	,	PUNCT
ejpam-4576	301	10	τ	τ	X
ejpam-4576	301	11	)	)	PUNCT
ejpam-4576	301	12	is	be	AUX
ejpam-4576	301	13	said	say	VERB
ejpam-4576	301	14	to	to	PART
ejpam-4576	301	15	be	be	AUX
ejpam-4576	301	16	s(λ	s(λ	PROPN
ejpam-4576	301	17	,	,	PUNCT
ejpam-4576	301	18	sp)-regular	sp)-regular	ADJ
ejpam-4576	301	19	if	if	SCONJ
ejpam-4576	301	20	,	,	PUNCT
ejpam-4576	301	21	for	for	ADP
ejpam-4576	301	22	each	each	DET
ejpam-4576	301	23	s(λ	s(λ	PROPN
ejpam-4576	301	24	,	,	PUNCT
ejpam-4576	301	25	sp)-closed	sp)-close	VERB
ejpam-4576	301	26	set	set	VERB
ejpam-4576	301	27	f	f	PROPN
ejpam-4576	301	28	and	and	CCONJ
ejpam-4576	301	29	each	each	DET
ejpam-4576	301	30	x	x	PROPN
ejpam-4576	301	31	̸∈	̸∈	PROPN
ejpam-4576	301	32	f	f	PROPN
ejpam-4576	301	33	,	,	PUNCT
ejpam-4576	301	34	there	there	PRON
ejpam-4576	301	35	exist	exist	VERB
ejpam-4576	301	36	disjoint	disjoint	NOUN
ejpam-4576	301	37	s(λ	s(λ	NOUN
ejpam-4576	301	38	,	,	PUNCT
ejpam-4576	301	39	sp)-open	sp)-open	NOUN
ejpam-4576	301	40	sets	set	VERB
ejpam-4576	301	41	u	u	NOUN
ejpam-4576	301	42	and	and	CCONJ
ejpam-4576	301	43	v	v	ADP
ejpam-4576	301	44	such	such	ADJ
ejpam-4576	301	45	that	that	SCONJ
ejpam-4576	301	46	x	x	SYM
ejpam-4576	301	47	∈	∈	PROPN
ejpam-4576	301	48	u	u	NOUN
ejpam-4576	301	49	and	and	CCONJ
ejpam-4576	301	50	f	f	PROPN
ejpam-4576	301	51	⊆	⊆	NUM
ejpam-4576	301	52	v	v	NOUN
ejpam-4576	301	53	.	.	PUNCT
ejpam-4576	302	1	lemma	lemma	PROPN
ejpam-4576	302	2	7	7	NUM
ejpam-4576	302	3	.	.	PUNCT
ejpam-4576	303	1	[	[	X
ejpam-4576	303	2	19	19	NUM
ejpam-4576	303	3	]	]	PUNCT
ejpam-4576	303	4	for	for	ADP
ejpam-4576	303	5	a	a	DET
ejpam-4576	303	6	multifunction	multifunction	NOUN
ejpam-4576	303	7	f	f	NOUN
ejpam-4576	303	8	:	:	PUNCT
ejpam-4576	303	9	(	(	PUNCT
ejpam-4576	303	10	x	x	X
ejpam-4576	303	11	,	,	PUNCT
ejpam-4576	303	12	τ	τ	X
ejpam-4576	303	13	)	)	PUNCT
ejpam-4576	303	14	→	→	SYM
ejpam-4576	303	15	(	(	PUNCT
ejpam-4576	303	16	y	y	PROPN
ejpam-4576	303	17	,	,	PUNCT
ejpam-4576	303	18	σ	σ	PROPN
ejpam-4576	303	19	)	)	PUNCT
ejpam-4576	303	20	,	,	PUNCT
ejpam-4576	303	21	where	where	SCONJ
ejpam-4576	303	22	(	(	PUNCT
ejpam-4576	303	23	y	y	PROPN
ejpam-4576	303	24	,	,	PUNCT
ejpam-4576	303	25	σ	σ	PROPN
ejpam-4576	303	26	)	)	PUNCT
ejpam-4576	303	27	is	be	AUX
ejpam-4576	303	28	a	a	DET
ejpam-4576	303	29	s(λ	s(λ	PROPN
ejpam-4576	303	30	,	,	PUNCT
ejpam-4576	303	31	sp)-regular	sp)-regular	ADJ
ejpam-4576	303	32	space	space	NOUN
ejpam-4576	303	33	,	,	PUNCT
ejpam-4576	303	34	the	the	DET
ejpam-4576	303	35	following	follow	VERB
ejpam-4576	303	36	properties	property	NOUN
ejpam-4576	303	37	are	be	AUX
ejpam-4576	303	38	equivalent	equivalent	ADJ
ejpam-4576	303	39	:	:	PUNCT
ejpam-4576	303	40	(	(	PUNCT
ejpam-4576	303	41	1	1	X
ejpam-4576	303	42	)	)	PUNCT
ejpam-4576	303	43	f	f	PROPN
ejpam-4576	303	44	is	be	AUX
ejpam-4576	303	45	lower	low	ADJ
ejpam-4576	303	46	(	(	PUNCT
ejpam-4576	303	47	λ	λ	NOUN
ejpam-4576	303	48	,	,	PUNCT
ejpam-4576	303	49	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	303	50	;	;	PUNCT
ejpam-4576	303	51	(	(	PUNCT
ejpam-4576	303	52	2	2	X
ejpam-4576	303	53	)	)	PUNCT
ejpam-4576	303	54	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	303	55	,	,	PUNCT
ejpam-4576	303	56	sp	sp	NOUN
ejpam-4576	303	57	)	)	PUNCT
ejpam-4576	303	58	)	)	PUNCT
ejpam-4576	304	1	is	be	AUX
ejpam-4576	304	2	(	(	PUNCT
ejpam-4576	304	3	λ	λ	X
ejpam-4576	304	4	,	,	PUNCT
ejpam-4576	304	5	sp)-closed	sp)-close	VERB
ejpam-4576	304	6	in	in	ADP
ejpam-4576	304	7	x	x	PUNCT
ejpam-4576	304	8	for	for	ADP
ejpam-4576	304	9	every	every	DET
ejpam-4576	304	10	subset	subset	NOUN
ejpam-4576	304	11	b	b	PROPN
ejpam-4576	304	12	of	of	ADP
ejpam-4576	304	13	y	y	PROPN
ejpam-4576	304	14	;	;	PUNCT
ejpam-4576	304	15	(	(	PUNCT
ejpam-4576	304	16	3	3	X
ejpam-4576	304	17	)	)	PUNCT
ejpam-4576	304	18	f+(k	f+(k	NUM
ejpam-4576	304	19	)	)	PUNCT
ejpam-4576	305	1	is	be	AUX
ejpam-4576	305	2	(	(	PUNCT
ejpam-4576	305	3	λ	λ	X
ejpam-4576	305	4	,	,	PUNCT
ejpam-4576	305	5	sp)-closed	sp)-close	VERB
ejpam-4576	305	6	in	in	ADP
ejpam-4576	305	7	x	x	PUNCT
ejpam-4576	305	8	for	for	ADP
ejpam-4576	305	9	every	every	DET
ejpam-4576	305	10	δ(λ	δ(λ	PROPN
ejpam-4576	305	11	,	,	PUNCT
ejpam-4576	305	12	sp)-closed	sp)-close	VERB
ejpam-4576	305	13	set	set	VERB
ejpam-4576	305	14	k	k	PROPN
ejpam-4576	305	15	of	of	ADP
ejpam-4576	305	16	y	y	PROPN
ejpam-4576	305	17	;	;	PUNCT
ejpam-4576	305	18	(	(	PUNCT
ejpam-4576	305	19	4	4	X
ejpam-4576	305	20	)	)	PUNCT
ejpam-4576	305	21	f−(v	f−(v	NOUN
ejpam-4576	305	22	)	)	PUNCT
ejpam-4576	305	23	is	be	AUX
ejpam-4576	305	24	(	(	PUNCT
ejpam-4576	305	25	λ	λ	INTJ
ejpam-4576	305	26	,	,	PUNCT
ejpam-4576	305	27	sp)-open	sp)-open	ADJ
ejpam-4576	305	28	in	in	ADP
ejpam-4576	305	29	x	x	PUNCT
ejpam-4576	305	30	for	for	ADP
ejpam-4576	305	31	every	every	DET
ejpam-4576	305	32	δ(λ	δ(λ	PROPN
ejpam-4576	305	33	,	,	PUNCT
ejpam-4576	305	34	sp)-open	sp)-open	VERB
ejpam-4576	305	35	set	set	VERB
ejpam-4576	305	36	v	v	NOUN
ejpam-4576	305	37	of	of	ADP
ejpam-4576	305	38	y	y	PROPN
ejpam-4576	305	39	.	.	PUNCT
ejpam-4576	306	1	lemma	lemma	PROPN
ejpam-4576	306	2	8	8	NUM
ejpam-4576	306	3	.	.	PUNCT
ejpam-4576	307	1	[	[	X
ejpam-4576	307	2	19	19	NUM
ejpam-4576	307	3	]	]	X
ejpam-4576	307	4	let	let	VERB
ejpam-4576	307	5	(	(	PUNCT
ejpam-4576	307	6	x	x	NOUN
ejpam-4576	307	7	,	,	PUNCT
ejpam-4576	307	8	τ	τ	X
ejpam-4576	307	9	)	)	PUNCT
ejpam-4576	307	10	be	be	VERB
ejpam-4576	307	11	a	a	DET
ejpam-4576	307	12	s(λ	s(λ	PROPN
ejpam-4576	307	13	,	,	PUNCT
ejpam-4576	307	14	sp)-regular	sp)-regular	ADJ
ejpam-4576	307	15	space	space	NOUN
ejpam-4576	307	16	.	.	PUNCT
ejpam-4576	308	1	then	then	ADV
ejpam-4576	308	2	,	,	PUNCT
ejpam-4576	308	3	the	the	DET
ejpam-4576	308	4	following	follow	VERB
ejpam-4576	308	5	properties	property	NOUN
ejpam-4576	308	6	hold	hold	VERB
ejpam-4576	308	7	:	:	PUNCT
ejpam-4576	308	8	(	(	PUNCT
ejpam-4576	308	9	1	1	X
ejpam-4576	308	10	)	)	PUNCT
ejpam-4576	308	11	a(λ	a(λ	ADV
ejpam-4576	308	12	,	,	PUNCT
ejpam-4576	308	13	sp	sp	NOUN
ejpam-4576	308	14	)	)	PUNCT
ejpam-4576	308	15	=	=	SYM
ejpam-4576	308	16	aδ(λ	aδ(λ	NUM
ejpam-4576	308	17	,	,	PUNCT
ejpam-4576	308	18	sp	sp	NOUN
ejpam-4576	308	19	)	)	PUNCT
ejpam-4576	308	20	for	for	ADP
ejpam-4576	308	21	every	every	DET
ejpam-4576	308	22	subset	subset	NOUN
ejpam-4576	308	23	a	a	PRON
ejpam-4576	308	24	of	of	ADP
ejpam-4576	308	25	x.	x.	NOUN
ejpam-4576	308	26	(	(	PUNCT
ejpam-4576	308	27	2	2	X
ejpam-4576	308	28	)	)	PUNCT
ejpam-4576	308	29	every	every	DET
ejpam-4576	308	30	(	(	PUNCT
ejpam-4576	308	31	λ	λ	NOUN
ejpam-4576	308	32	,	,	PUNCT
ejpam-4576	308	33	sp)-open	sp)-open	ADJ
ejpam-4576	308	34	set	set	NOUN
ejpam-4576	308	35	is	be	AUX
ejpam-4576	308	36	δ(λ	δ(λ	PROPN
ejpam-4576	308	37	,	,	PUNCT
ejpam-4576	308	38	sp)-open	sp)-open	NOUN
ejpam-4576	308	39	.	.	PUNCT
ejpam-4576	309	1	theorem	theorem	ADJ
ejpam-4576	309	2	10	10	NUM
ejpam-4576	309	3	.	.	PUNCT
ejpam-4576	310	1	for	for	ADP
ejpam-4576	310	2	a	a	DET
ejpam-4576	310	3	multifunction	multifunction	NOUN
ejpam-4576	310	4	f	f	NOUN
ejpam-4576	310	5	:	:	PUNCT
ejpam-4576	310	6	(	(	PUNCT
ejpam-4576	310	7	x	x	X
ejpam-4576	310	8	,	,	PUNCT
ejpam-4576	310	9	τ	τ	X
ejpam-4576	310	10	)	)	PUNCT
ejpam-4576	310	11	→	→	SYM
ejpam-4576	310	12	(	(	PUNCT
ejpam-4576	310	13	y	y	PROPN
ejpam-4576	310	14	,	,	PUNCT
ejpam-4576	310	15	σ	σ	PROPN
ejpam-4576	310	16	)	)	PUNCT
ejpam-4576	310	17	,	,	PUNCT
ejpam-4576	310	18	where	where	SCONJ
ejpam-4576	310	19	(	(	PUNCT
ejpam-4576	310	20	y	y	PROPN
ejpam-4576	310	21	,	,	PUNCT
ejpam-4576	310	22	σ	σ	PROPN
ejpam-4576	310	23	)	)	PUNCT
ejpam-4576	310	24	is	be	AUX
ejpam-4576	310	25	a	a	DET
ejpam-4576	310	26	s(λ	s(λ	PROPN
ejpam-4576	310	27	,	,	PUNCT
ejpam-4576	310	28	sp)-regular	sp)-regular	ADJ
ejpam-4576	310	29	space	space	NOUN
ejpam-4576	310	30	,	,	PUNCT
ejpam-4576	310	31	the	the	DET
ejpam-4576	310	32	following	follow	VERB
ejpam-4576	310	33	properties	property	NOUN
ejpam-4576	310	34	are	be	AUX
ejpam-4576	310	35	equivalent	equivalent	ADJ
ejpam-4576	310	36	:	:	PUNCT
ejpam-4576	310	37	(	(	PUNCT
ejpam-4576	310	38	1	1	X
ejpam-4576	310	39	)	)	PUNCT
ejpam-4576	310	40	f	f	PROPN
ejpam-4576	310	41	is	be	AUX
ejpam-4576	310	42	lower	low	ADJ
ejpam-4576	310	43	(	(	PUNCT
ejpam-4576	310	44	λ	λ	NOUN
ejpam-4576	310	45	,	,	PUNCT
ejpam-4576	310	46	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	310	47	;	;	PUNCT
ejpam-4576	310	48	(	(	PUNCT
ejpam-4576	310	49	2	2	X
ejpam-4576	310	50	)	)	PUNCT
ejpam-4576	310	51	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4576	310	52	,	,	PUNCT
ejpam-4576	310	53	sp	sp	NOUN
ejpam-4576	310	54	)	)	PUNCT
ejpam-4576	310	55	)	)	PUNCT
ejpam-4576	311	1	is	be	AUX
ejpam-4576	311	2	(	(	PUNCT
ejpam-4576	311	3	λ	λ	X
ejpam-4576	311	4	,	,	PUNCT
ejpam-4576	311	5	sp)-closed	sp)-close	VERB
ejpam-4576	311	6	in	in	ADP
ejpam-4576	311	7	x	x	PUNCT
ejpam-4576	311	8	for	for	ADP
ejpam-4576	311	9	every	every	DET
ejpam-4576	311	10	subset	subset	NOUN
ejpam-4576	311	11	b	b	PROPN
ejpam-4576	311	12	of	of	ADP
ejpam-4576	311	13	y	y	PROPN
ejpam-4576	311	14	;	;	PUNCT
ejpam-4576	311	15	(	(	PUNCT
ejpam-4576	311	16	3	3	X
ejpam-4576	311	17	)	)	PUNCT
ejpam-4576	311	18	f+(k	f+(k	NUM
ejpam-4576	311	19	)	)	PUNCT
ejpam-4576	312	1	is	be	AUX
ejpam-4576	312	2	(	(	PUNCT
ejpam-4576	312	3	λ	λ	X
ejpam-4576	312	4	,	,	PUNCT
ejpam-4576	312	5	sp)-closed	sp)-close	VERB
ejpam-4576	312	6	in	in	ADP
ejpam-4576	312	7	x	x	PUNCT
ejpam-4576	312	8	for	for	ADP
ejpam-4576	312	9	every	every	DET
ejpam-4576	312	10	δ(λ	δ(λ	PROPN
ejpam-4576	312	11	,	,	PUNCT
ejpam-4576	312	12	sp)-closed	sp)-close	VERB
ejpam-4576	312	13	set	set	VERB
ejpam-4576	312	14	k	k	PROPN
ejpam-4576	312	15	of	of	ADP
ejpam-4576	312	16	y	y	PROPN
ejpam-4576	312	17	;	;	PUNCT
ejpam-4576	312	18	(	(	PUNCT
ejpam-4576	312	19	4	4	X
ejpam-4576	312	20	)	)	PUNCT
ejpam-4576	312	21	f−(v	f−(v	NOUN
ejpam-4576	312	22	)	)	PUNCT
ejpam-4576	312	23	is	be	AUX
ejpam-4576	312	24	(	(	PUNCT
ejpam-4576	312	25	λ	λ	INTJ
ejpam-4576	312	26	,	,	PUNCT
ejpam-4576	312	27	sp)-open	sp)-open	ADJ
ejpam-4576	312	28	in	in	ADP
ejpam-4576	312	29	x	x	PUNCT
ejpam-4576	312	30	for	for	ADP
ejpam-4576	312	31	every	every	DET
ejpam-4576	312	32	δ(λ	δ(λ	PROPN
ejpam-4576	312	33	,	,	PUNCT
ejpam-4576	312	34	sp)-open	sp)-open	VERB
ejpam-4576	312	35	set	set	VERB
ejpam-4576	312	36	v	v	NOUN
ejpam-4576	312	37	of	of	ADP
ejpam-4576	312	38	y	y	PROPN
ejpam-4576	312	39	;	;	PUNCT
ejpam-4576	312	40	c.	c.	PROPN
ejpam-4576	312	41	boonpok	boonpok	PROPN
ejpam-4576	312	42	,	,	PUNCT
ejpam-4576	312	43	n.	n.	PROPN
ejpam-4576	312	44	viriyapong	viriyapong	PROPN
ejpam-4576	312	45	/	/	SYM
ejpam-4576	312	46	eur	eur	PROPN
ejpam-4576	312	47	.	.	PUNCT
ejpam-4576	313	1	j.	j.	PROPN
ejpam-4576	313	2	pure	pure	PROPN
ejpam-4576	313	3	appl	appl	PROPN
ejpam-4576	313	4	.	.	PROPN
ejpam-4576	313	5	math	math	PROPN
ejpam-4576	313	6	,	,	PUNCT
ejpam-4576	313	7	16	16	NUM
ejpam-4576	313	8	(	(	PUNCT
ejpam-4576	313	9	1	1	NUM
ejpam-4576	313	10	)	)	PUNCT
ejpam-4576	313	11	(	(	PUNCT
ejpam-4576	313	12	2023	2023	NUM
ejpam-4576	313	13	)	)	PUNCT
ejpam-4576	313	14	,	,	PUNCT
ejpam-4576	313	15	84	84	NUM
ejpam-4576	313	16	-	-	SYM
ejpam-4576	313	17	96	96	NUM
ejpam-4576	313	18	94	94	NUM
ejpam-4576	313	19	(	(	PUNCT
ejpam-4576	313	20	5	5	NUM
ejpam-4576	313	21	)	)	PUNCT
ejpam-4576	313	22	f	f	PROPN
ejpam-4576	313	23	is	be	AUX
ejpam-4576	313	24	lower	low	ADJ
ejpam-4576	313	25	almost	almost	ADV
ejpam-4576	313	26	(	(	PUNCT
ejpam-4576	313	27	λ	λ	NOUN
ejpam-4576	313	28	,	,	PUNCT
ejpam-4576	313	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	313	30	.	.	PUNCT
ejpam-4576	314	1	proof	proof	NOUN
ejpam-4576	314	2	.	.	PUNCT
ejpam-4576	315	1	the	the	DET
ejpam-4576	315	2	proofs	proof	NOUN
ejpam-4576	315	3	of	of	ADP
ejpam-4576	315	4	the	the	DET
ejpam-4576	315	5	implications	implication	NOUN
ejpam-4576	315	6	(	(	PUNCT
ejpam-4576	315	7	1	1	X
ejpam-4576	315	8	)	)	PUNCT
ejpam-4576	315	9	⇒	⇒	NOUN
ejpam-4576	315	10	(	(	PUNCT
ejpam-4576	315	11	2	2	NUM
ejpam-4576	315	12	)	)	PUNCT
ejpam-4576	315	13	⇒	⇒	NOUN
ejpam-4576	315	14	(	(	PUNCT
ejpam-4576	315	15	3	3	NUM
ejpam-4576	315	16	)	)	PUNCT
ejpam-4576	315	17	⇒	⇒	NOUN
ejpam-4576	315	18	(	(	PUNCT
ejpam-4576	315	19	4	4	X
ejpam-4576	315	20	)	)	PUNCT
ejpam-4576	315	21	are	be	AUX
ejpam-4576	315	22	similar	similar	ADJ
ejpam-4576	315	23	to	to	ADP
ejpam-4576	315	24	those	those	PRON
ejpam-4576	315	25	in	in	ADP
ejpam-4576	315	26	lemma	lemma	PROPN
ejpam-4576	315	27	7	7	NUM
ejpam-4576	315	28	.	.	PUNCT
ejpam-4576	315	29	(	(	PUNCT
ejpam-4576	315	30	4	4	X
ejpam-4576	315	31	)	)	PUNCT
ejpam-4576	315	32	⇒	⇒	NOUN
ejpam-4576	315	33	(	(	PUNCT
ejpam-4576	315	34	5	5	NUM
ejpam-4576	315	35	):	):	PUNCT
ejpam-4576	315	36	let	let	VERB
ejpam-4576	315	37	v	v	PART
ejpam-4576	315	38	be	be	AUX
ejpam-4576	315	39	any	any	DET
ejpam-4576	315	40	r(λ	r(λ	NOUN
ejpam-4576	315	41	,	,	PUNCT
ejpam-4576	315	42	sp)-open	sp)-open	ADJ
ejpam-4576	315	43	set	set	NOUN
ejpam-4576	315	44	of	of	ADP
ejpam-4576	315	45	y	y	PROPN
ejpam-4576	315	46	.	.	PUNCT
ejpam-4576	316	1	then	then	ADV
ejpam-4576	316	2	,	,	PUNCT
ejpam-4576	316	3	v	v	NOUN
ejpam-4576	316	4	is	be	AUX
ejpam-4576	316	5	(	(	PUNCT
ejpam-4576	316	6	λ	λ	X
ejpam-4576	316	7	,	,	PUNCT
ejpam-4576	316	8	sp)-open	sp)-open	ADJ
ejpam-4576	316	9	in	in	ADP
ejpam-4576	316	10	y	y	PROPN
ejpam-4576	316	11	and	and	CCONJ
ejpam-4576	316	12	by	by	ADP
ejpam-4576	316	13	lemma	lemma	PROPN
ejpam-4576	316	14	8	8	NUM
ejpam-4576	316	15	,	,	PUNCT
ejpam-4576	316	16	v	v	NOUN
ejpam-4576	316	17	is	be	AUX
ejpam-4576	316	18	δ(λ	δ(λ	PROPN
ejpam-4576	316	19	,	,	PUNCT
ejpam-4576	316	20	sp)-open	sp)-open	NOUN
ejpam-4576	316	21	.	.	PUNCT
ejpam-4576	317	1	thus	thus	ADV
ejpam-4576	317	2	,	,	PUNCT
ejpam-4576	317	3	by	by	ADP
ejpam-4576	317	4	(	(	PUNCT
ejpam-4576	317	5	4	4	NUM
ejpam-4576	317	6	)	)	PUNCT
ejpam-4576	317	7	,	,	PUNCT
ejpam-4576	317	8	f−(v	f−(v	ADJ
ejpam-4576	317	9	)	)	PUNCT
ejpam-4576	317	10	is	be	AUX
ejpam-4576	317	11	(	(	PUNCT
ejpam-4576	317	12	λ	λ	INTJ
ejpam-4576	317	13	,	,	PUNCT
ejpam-4576	317	14	sp)-open	sp)-open	ADJ
ejpam-4576	317	15	in	in	ADP
ejpam-4576	317	16	x	x	PUNCT
ejpam-4576	317	17	and	and	CCONJ
ejpam-4576	317	18	by	by	ADP
ejpam-4576	317	19	theorem	theorem	NOUN
ejpam-4576	317	20	4	4	NUM
ejpam-4576	317	21	,	,	PUNCT
ejpam-4576	317	22	f	f	PROPN
ejpam-4576	317	23	is	be	AUX
ejpam-4576	317	24	lower	low	ADJ
ejpam-4576	317	25	almost	almost	ADV
ejpam-4576	317	26	(	(	PUNCT
ejpam-4576	317	27	λ	λ	NOUN
ejpam-4576	317	28	,	,	PUNCT
ejpam-4576	317	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	317	30	.	.	PUNCT
ejpam-4576	318	1	(	(	PUNCT
ejpam-4576	318	2	5	5	X
ejpam-4576	318	3	)	)	PUNCT
ejpam-4576	318	4	⇒	⇒	NOUN
ejpam-4576	318	5	(	(	PUNCT
ejpam-4576	318	6	1	1	NUM
ejpam-4576	318	7	):	):	PUNCT
ejpam-4576	318	8	let	let	VERB
ejpam-4576	318	9	x	x	PUNCT
ejpam-4576	318	10	∈	∈	PROPN
ejpam-4576	318	11	x	x	X
ejpam-4576	318	12	and	and	CCONJ
ejpam-4576	318	13	v	v	AUX
ejpam-4576	318	14	be	be	AUX
ejpam-4576	318	15	any	any	DET
ejpam-4576	318	16	(	(	PUNCT
ejpam-4576	318	17	λ	λ	NOUN
ejpam-4576	318	18	,	,	PUNCT
ejpam-4576	318	19	sp)-open	sp)-open	ADJ
ejpam-4576	318	20	set	set	NOUN
ejpam-4576	318	21	of	of	ADP
ejpam-4576	318	22	y	y	PRON
ejpam-4576	318	23	such	such	ADJ
ejpam-4576	318	24	that	that	SCONJ
ejpam-4576	318	25	f	f	PROPN
ejpam-4576	318	26	(	(	PUNCT
ejpam-4576	318	27	x	x	NOUN
ejpam-4576	318	28	)	)	PUNCT
ejpam-4576	318	29	∩	∩	NOUN
ejpam-4576	318	30	v	v	ADP
ejpam-4576	318	31	̸=	̸=	PROPN
ejpam-4576	318	32	∅.	∅.	NOUN
ejpam-4576	318	33	since	since	SCONJ
ejpam-4576	318	34	(	(	PUNCT
ejpam-4576	318	35	y	y	PROPN
ejpam-4576	318	36	,	,	PUNCT
ejpam-4576	318	37	σ	σ	PROPN
ejpam-4576	318	38	)	)	PUNCT
ejpam-4576	318	39	is	be	AUX
ejpam-4576	318	40	s(λ	s(λ	PROPN
ejpam-4576	318	41	,	,	PUNCT
ejpam-4576	318	42	sp)-regular	sp)-regular	NOUN
ejpam-4576	318	43	,	,	PUNCT
ejpam-4576	318	44	there	there	PRON
ejpam-4576	318	45	exists	exist	VERB
ejpam-4576	318	46	a	a	DET
ejpam-4576	318	47	r(λ	r(λ	NOUN
ejpam-4576	318	48	,	,	PUNCT
ejpam-4576	318	49	sp)-open	sp)-open	NOUN
ejpam-4576	318	50	set	set	VERB
ejpam-4576	318	51	w	w	ADP
ejpam-4576	318	52	such	such	ADJ
ejpam-4576	318	53	that	that	SCONJ
ejpam-4576	318	54	f	f	PROPN
ejpam-4576	318	55	(	(	PUNCT
ejpam-4576	318	56	x)∩w	x)∩w	PROPN
ejpam-4576	318	57	̸=	̸=	PROPN
ejpam-4576	318	58	∅	∅	NOUN
ejpam-4576	318	59	and	and	CCONJ
ejpam-4576	318	60	w	w	ADP
ejpam-4576	318	61	⊆	⊆	NUM
ejpam-4576	318	62	v	v	NOUN
ejpam-4576	318	63	.	.	PUNCT
ejpam-4576	319	1	since	since	SCONJ
ejpam-4576	319	2	f	f	PROPN
ejpam-4576	319	3	is	be	AUX
ejpam-4576	319	4	lower	low	ADJ
ejpam-4576	319	5	almost	almost	ADV
ejpam-4576	319	6	(	(	PUNCT
ejpam-4576	319	7	λ	λ	NOUN
ejpam-4576	319	8	,	,	PUNCT
ejpam-4576	319	9	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	319	10	,	,	PUNCT
ejpam-4576	319	11	there	there	PRON
ejpam-4576	319	12	exists	exist	VERB
ejpam-4576	319	13	u	u	PROPN
ejpam-4576	319	14	∈	∈	PROPN
ejpam-4576	319	15	λspo(x	λspo(x	PROPN
ejpam-4576	319	16	,	,	PUNCT
ejpam-4576	319	17	τ	τ	X
ejpam-4576	319	18	)	)	PUNCT
ejpam-4576	319	19	containing	contain	VERB
ejpam-4576	319	20	x	x	PUNCT
ejpam-4576	319	21	such	such	ADJ
ejpam-4576	319	22	that	that	SCONJ
ejpam-4576	319	23	f	f	PROPN
ejpam-4576	319	24	(	(	PUNCT
ejpam-4576	319	25	z)∩w	z)∩w	PROPN
ejpam-4576	319	26	̸=	̸=	PROPN
ejpam-4576	319	27	∅	∅	NOUN
ejpam-4576	319	28	for	for	ADP
ejpam-4576	319	29	every	every	DET
ejpam-4576	319	30	z	z	NOUN
ejpam-4576	319	31	∈	∈	PROPN
ejpam-4576	319	32	u	u	NOUN
ejpam-4576	319	33	.	.	PUNCT
ejpam-4576	320	1	thus	thus	ADV
ejpam-4576	320	2	,	,	PUNCT
ejpam-4576	320	3	f	f	PROPN
ejpam-4576	320	4	(	(	PUNCT
ejpam-4576	320	5	z)∩v	z)∩v	PROPN
ejpam-4576	320	6	̸=	̸=	PROPN
ejpam-4576	320	7	∅	∅	NOUN
ejpam-4576	320	8	for	for	ADP
ejpam-4576	320	9	every	every	DET
ejpam-4576	320	10	z	z	NOUN
ejpam-4576	320	11	∈	∈	PROPN
ejpam-4576	320	12	u	u	NOUN
ejpam-4576	320	13	.	.	PUNCT
ejpam-4576	321	1	this	this	PRON
ejpam-4576	321	2	shows	show	VERB
ejpam-4576	321	3	that	that	SCONJ
ejpam-4576	321	4	f	f	PROPN
ejpam-4576	321	5	is	be	AUX
ejpam-4576	321	6	lower	low	ADJ
ejpam-4576	321	7	(	(	PUNCT
ejpam-4576	321	8	λ	λ	NOUN
ejpam-4576	321	9	,	,	PUNCT
ejpam-4576	321	10	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	321	11	.	.	PUNCT
ejpam-4576	322	1	definition	definition	NOUN
ejpam-4576	322	2	5	5	NUM
ejpam-4576	322	3	.	.	PUNCT
ejpam-4576	323	1	[	[	X
ejpam-4576	323	2	20	20	NUM
ejpam-4576	323	3	]	]	PUNCT
ejpam-4576	323	4	a	a	DET
ejpam-4576	323	5	function	function	NOUN
ejpam-4576	323	6	f	f	NOUN
ejpam-4576	323	7	:	:	PUNCT
ejpam-4576	323	8	(	(	PUNCT
ejpam-4576	323	9	x	x	X
ejpam-4576	323	10	,	,	PUNCT
ejpam-4576	323	11	τ	τ	X
ejpam-4576	323	12	)	)	PUNCT
ejpam-4576	323	13	→	→	SYM
ejpam-4576	323	14	(	(	PUNCT
ejpam-4576	323	15	y	y	PROPN
ejpam-4576	323	16	,	,	PUNCT
ejpam-4576	323	17	σ	σ	PROPN
ejpam-4576	323	18	)	)	PUNCT
ejpam-4576	323	19	is	be	AUX
ejpam-4576	323	20	said	say	VERB
ejpam-4576	323	21	to	to	PART
ejpam-4576	323	22	be	be	AUX
ejpam-4576	323	23	(	(	PUNCT
ejpam-4576	323	24	λ	λ	X
ejpam-4576	323	25	,	,	PUNCT
ejpam-4576	323	26	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	323	27	at	at	ADP
ejpam-4576	323	28	a	a	DET
ejpam-4576	323	29	point	point	NOUN
ejpam-4576	323	30	x	x	SYM
ejpam-4576	323	31	∈	∈	NOUN
ejpam-4576	323	32	x	x	INTJ
ejpam-4576	323	33	if	if	SCONJ
ejpam-4576	323	34	,	,	PUNCT
ejpam-4576	323	35	for	for	ADP
ejpam-4576	323	36	each	each	DET
ejpam-4576	323	37	(	(	PUNCT
ejpam-4576	323	38	λ	λ	PROPN
ejpam-4576	323	39	,	,	PUNCT
ejpam-4576	323	40	sp)-open	sp)-open	NOUN
ejpam-4576	323	41	set	set	VERB
ejpam-4576	323	42	v	v	NUM
ejpam-4576	323	43	of	of	ADP
ejpam-4576	323	44	y	y	NOUN
ejpam-4576	323	45	containing	contain	VERB
ejpam-4576	323	46	f(x	f(x	PROPN
ejpam-4576	323	47	)	)	PUNCT
ejpam-4576	323	48	,	,	PUNCT
ejpam-4576	323	49	there	there	PRON
ejpam-4576	323	50	exists	exist	VERB
ejpam-4576	323	51	a	a	DET
ejpam-4576	323	52	(	(	PUNCT
ejpam-4576	323	53	λ	λ	NOUN
ejpam-4576	323	54	,	,	PUNCT
ejpam-4576	323	55	sp)open	sp)open	VERB
ejpam-4576	323	56	set	set	VERB
ejpam-4576	323	57	u	u	NOUN
ejpam-4576	323	58	of	of	ADP
ejpam-4576	323	59	x	x	PUNCT
ejpam-4576	323	60	containing	contain	VERB
ejpam-4576	323	61	x	x	PUNCT
ejpam-4576	323	62	such	such	ADJ
ejpam-4576	323	63	that	that	DET
ejpam-4576	323	64	f(u	f(u	PROPN
ejpam-4576	323	65	)	)	PUNCT
ejpam-4576	323	66	⊆	⊆	NUM
ejpam-4576	323	67	v	v	NOUN
ejpam-4576	323	68	.	.	PUNCT
ejpam-4576	324	1	a	a	DET
ejpam-4576	324	2	function	function	NOUN
ejpam-4576	324	3	f	f	NOUN
ejpam-4576	324	4	:	:	PUNCT
ejpam-4576	324	5	(	(	PUNCT
ejpam-4576	324	6	x	x	X
ejpam-4576	324	7	,	,	PUNCT
ejpam-4576	324	8	τ	τ	X
ejpam-4576	324	9	)	)	PUNCT
ejpam-4576	324	10	→	→	SYM
ejpam-4576	324	11	(	(	PUNCT
ejpam-4576	324	12	y	y	PROPN
ejpam-4576	324	13	,	,	PUNCT
ejpam-4576	324	14	σ	σ	PROPN
ejpam-4576	324	15	)	)	PUNCT
ejpam-4576	324	16	is	be	AUX
ejpam-4576	324	17	said	say	VERB
ejpam-4576	324	18	to	to	PART
ejpam-4576	324	19	be	be	AUX
ejpam-4576	324	20	(	(	PUNCT
ejpam-4576	324	21	λ	λ	X
ejpam-4576	324	22	,	,	PUNCT
ejpam-4576	324	23	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	324	24	if	if	SCONJ
ejpam-4576	324	25	f	f	PROPN
ejpam-4576	324	26	has	have	VERB
ejpam-4576	324	27	this	this	DET
ejpam-4576	324	28	property	property	NOUN
ejpam-4576	324	29	at	at	ADP
ejpam-4576	324	30	each	each	DET
ejpam-4576	324	31	point	point	NOUN
ejpam-4576	324	32	x	x	X
ejpam-4576	324	33	∈	∈	PROPN
ejpam-4576	324	34	x.	x.	NOUN
ejpam-4576	324	35	corollary	corollary	NOUN
ejpam-4576	324	36	5	5	NUM
ejpam-4576	324	37	.	.	PUNCT
ejpam-4576	325	1	for	for	ADP
ejpam-4576	325	2	a	a	DET
ejpam-4576	325	3	function	function	NOUN
ejpam-4576	325	4	f	f	NOUN
ejpam-4576	325	5	:	:	PUNCT
ejpam-4576	325	6	(	(	PUNCT
ejpam-4576	325	7	x	x	X
ejpam-4576	325	8	,	,	PUNCT
ejpam-4576	325	9	τ	τ	X
ejpam-4576	325	10	)	)	PUNCT
ejpam-4576	325	11	→	→	SYM
ejpam-4576	325	12	(	(	PUNCT
ejpam-4576	325	13	y	y	PROPN
ejpam-4576	325	14	,	,	PUNCT
ejpam-4576	325	15	σ	σ	PROPN
ejpam-4576	325	16	)	)	PUNCT
ejpam-4576	325	17	,	,	PUNCT
ejpam-4576	325	18	where	where	SCONJ
ejpam-4576	325	19	(	(	PUNCT
ejpam-4576	325	20	y	y	PROPN
ejpam-4576	325	21	,	,	PUNCT
ejpam-4576	325	22	σ	σ	PROPN
ejpam-4576	325	23	)	)	PUNCT
ejpam-4576	325	24	is	be	AUX
ejpam-4576	325	25	a	a	DET
ejpam-4576	325	26	s(λ	s(λ	PROPN
ejpam-4576	325	27	,	,	PUNCT
ejpam-4576	325	28	sp)-regular	sp)-regular	ADJ
ejpam-4576	325	29	space	space	NOUN
ejpam-4576	325	30	,	,	PUNCT
ejpam-4576	325	31	the	the	DET
ejpam-4576	325	32	following	follow	VERB
ejpam-4576	325	33	properties	property	NOUN
ejpam-4576	325	34	are	be	AUX
ejpam-4576	325	35	equivalent	equivalent	ADJ
ejpam-4576	325	36	:	:	PUNCT
ejpam-4576	325	37	(	(	PUNCT
ejpam-4576	325	38	1	1	X
ejpam-4576	325	39	)	)	PUNCT
ejpam-4576	325	40	f	f	PROPN
ejpam-4576	325	41	is	be	AUX
ejpam-4576	325	42	(	(	PUNCT
ejpam-4576	325	43	λ	λ	INTJ
ejpam-4576	325	44	,	,	PUNCT
ejpam-4576	325	45	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	325	46	;	;	PUNCT
ejpam-4576	325	47	(	(	PUNCT
ejpam-4576	325	48	2	2	X
ejpam-4576	325	49	)	)	PUNCT
ejpam-4576	325	50	f−1(bδ(λ	f−1(bδ(λ	NOUN
ejpam-4576	325	51	,	,	PUNCT
ejpam-4576	325	52	sp	sp	NOUN
ejpam-4576	325	53	)	)	PUNCT
ejpam-4576	325	54	)	)	PUNCT
ejpam-4576	326	1	is	be	AUX
ejpam-4576	326	2	(	(	PUNCT
ejpam-4576	326	3	λ	λ	X
ejpam-4576	326	4	,	,	PUNCT
ejpam-4576	326	5	sp)-closed	sp)-close	VERB
ejpam-4576	326	6	in	in	ADP
ejpam-4576	326	7	x	x	PUNCT
ejpam-4576	326	8	for	for	ADP
ejpam-4576	326	9	every	every	DET
ejpam-4576	326	10	subset	subset	NOUN
ejpam-4576	326	11	b	b	PROPN
ejpam-4576	326	12	of	of	ADP
ejpam-4576	326	13	y	y	PROPN
ejpam-4576	326	14	;	;	PUNCT
ejpam-4576	326	15	(	(	PUNCT
ejpam-4576	326	16	3	3	X
ejpam-4576	326	17	)	)	PUNCT
ejpam-4576	326	18	f−1(k	f−1(k	PROPN
ejpam-4576	326	19	)	)	PUNCT
ejpam-4576	327	1	is	be	AUX
ejpam-4576	327	2	(	(	PUNCT
ejpam-4576	327	3	λ	λ	X
ejpam-4576	327	4	,	,	PUNCT
ejpam-4576	327	5	sp)-closed	sp)-close	VERB
ejpam-4576	327	6	in	in	ADP
ejpam-4576	327	7	x	x	PUNCT
ejpam-4576	327	8	for	for	ADP
ejpam-4576	327	9	every	every	DET
ejpam-4576	327	10	δ(λ	δ(λ	PROPN
ejpam-4576	327	11	,	,	PUNCT
ejpam-4576	327	12	sp)-closed	sp)-close	VERB
ejpam-4576	327	13	set	set	VERB
ejpam-4576	327	14	k	k	PROPN
ejpam-4576	327	15	of	of	ADP
ejpam-4576	327	16	y	y	PROPN
ejpam-4576	327	17	;	;	PUNCT
ejpam-4576	327	18	(	(	PUNCT
ejpam-4576	327	19	4	4	X
ejpam-4576	327	20	)	)	PUNCT
ejpam-4576	327	21	f−1(v	f−1(v	NOUN
ejpam-4576	327	22	)	)	PUNCT
ejpam-4576	327	23	is	be	AUX
ejpam-4576	327	24	(	(	PUNCT
ejpam-4576	327	25	λ	λ	INTJ
ejpam-4576	327	26	,	,	PUNCT
ejpam-4576	327	27	sp)-open	sp)-open	ADJ
ejpam-4576	327	28	in	in	ADP
ejpam-4576	327	29	x	x	PUNCT
ejpam-4576	327	30	for	for	ADP
ejpam-4576	327	31	every	every	DET
ejpam-4576	327	32	δ(λ	δ(λ	PROPN
ejpam-4576	327	33	,	,	PUNCT
ejpam-4576	327	34	sp)-open	sp)-open	VERB
ejpam-4576	327	35	set	set	VERB
ejpam-4576	327	36	v	v	NOUN
ejpam-4576	327	37	of	of	ADP
ejpam-4576	327	38	y	y	PROPN
ejpam-4576	327	39	;	;	PUNCT
ejpam-4576	327	40	(	(	PUNCT
ejpam-4576	327	41	5	5	X
ejpam-4576	327	42	)	)	PUNCT
ejpam-4576	327	43	f	f	PROPN
ejpam-4576	327	44	is	be	AUX
ejpam-4576	327	45	almost	almost	ADV
ejpam-4576	327	46	(	(	PUNCT
ejpam-4576	327	47	λ	λ	NOUN
ejpam-4576	327	48	,	,	PUNCT
ejpam-4576	327	49	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	327	50	.	.	PUNCT
ejpam-4576	328	1	4	4	X
ejpam-4576	328	2	.	.	X
ejpam-4576	328	3	conclusion	conclusion	NOUN
ejpam-4576	328	4	in	in	ADP
ejpam-4576	328	5	topology	topology	NOUN
ejpam-4576	329	1	,	,	PUNCT
ejpam-4576	329	2	there	there	PRON
ejpam-4576	329	3	has	have	AUX
ejpam-4576	329	4	been	be	AUX
ejpam-4576	329	5	recently	recently	ADV
ejpam-4576	329	6	significant	significant	ADJ
ejpam-4576	329	7	interest	interest	NOUN
ejpam-4576	329	8	in	in	ADP
ejpam-4576	329	9	characterizing	characterize	VERB
ejpam-4576	329	10	and	and	CCONJ
ejpam-4576	329	11	investigating	investigate	VERB
ejpam-4576	329	12	the	the	DET
ejpam-4576	329	13	properties	property	NOUN
ejpam-4576	329	14	of	of	ADP
ejpam-4576	329	15	several	several	ADJ
ejpam-4576	329	16	weak	weak	ADJ
ejpam-4576	329	17	forms	form	NOUN
ejpam-4576	329	18	of	of	ADP
ejpam-4576	329	19	continuity	continuity	NOUN
ejpam-4576	329	20	for	for	ADP
ejpam-4576	329	21	functions	function	NOUN
ejpam-4576	329	22	and	and	CCONJ
ejpam-4576	329	23	multifunctions	multifunction	NOUN
ejpam-4576	329	24	.	.	PUNCT
ejpam-4576	330	1	the	the	DET
ejpam-4576	330	2	development	development	NOUN
ejpam-4576	330	3	of	of	ADP
ejpam-4576	330	4	such	such	DET
ejpam-4576	330	5	a	a	DET
ejpam-4576	330	6	theory	theory	NOUN
ejpam-4576	330	7	is	be	AUX
ejpam-4576	330	8	in	in	ADP
ejpam-4576	330	9	fact	fact	NOUN
ejpam-4576	330	10	very	very	ADV
ejpam-4576	330	11	well	well	ADV
ejpam-4576	330	12	motivated	motivated	ADJ
ejpam-4576	330	13	.	.	PUNCT
ejpam-4576	331	1	this	this	DET
ejpam-4576	331	2	work	work	NOUN
ejpam-4576	331	3	is	be	AUX
ejpam-4576	331	4	concerned	concern	VERB
ejpam-4576	331	5	with	with	ADP
ejpam-4576	331	6	the	the	DET
ejpam-4576	331	7	concept	concept	NOUN
ejpam-4576	331	8	of	of	ADP
ejpam-4576	331	9	upper	upper	ADJ
ejpam-4576	331	10	(	(	PUNCT
ejpam-4576	331	11	resp	resp	NOUN
ejpam-4576	331	12	.	.	PUNCT
ejpam-4576	332	1	lower	low	ADJ
ejpam-4576	332	2	)	)	PUNCT
ejpam-4576	332	3	almost	almost	ADV
ejpam-4576	332	4	(	(	PUNCT
ejpam-4576	332	5	λ	λ	NOUN
ejpam-4576	332	6	,	,	PUNCT
ejpam-4576	332	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	332	8	multifunctions	multifunction	NOUN
ejpam-4576	332	9	.	.	PUNCT
ejpam-4576	333	1	a	a	DET
ejpam-4576	333	2	multifunction	multifunction	NOUN
ejpam-4576	333	3	f	f	NOUN
ejpam-4576	333	4	:	:	PUNCT
ejpam-4576	333	5	(	(	PUNCT
ejpam-4576	333	6	x	x	X
ejpam-4576	333	7	,	,	PUNCT
ejpam-4576	333	8	τ	τ	X
ejpam-4576	333	9	)	)	PUNCT
ejpam-4576	333	10	→	→	SYM
ejpam-4576	333	11	(	(	PUNCT
ejpam-4576	333	12	y	y	PROPN
ejpam-4576	333	13	,	,	PUNCT
ejpam-4576	333	14	σ	σ	PROPN
ejpam-4576	333	15	)	)	PUNCT
ejpam-4576	333	16	is	be	AUX
ejpam-4576	333	17	called	call	VERB
ejpam-4576	333	18	upper	upper	ADJ
ejpam-4576	333	19	(	(	PUNCT
ejpam-4576	333	20	resp	resp	NOUN
ejpam-4576	333	21	.	.	PUNCT
ejpam-4576	334	1	lower	low	ADJ
ejpam-4576	334	2	)	)	PUNCT
ejpam-4576	334	3	almost	almost	ADV
ejpam-4576	334	4	(	(	PUNCT
ejpam-4576	334	5	λ	λ	NOUN
ejpam-4576	334	6	,	,	PUNCT
ejpam-4576	334	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	334	8	multifunctions	multifunction	NOUN
ejpam-4576	334	9	if	if	SCONJ
ejpam-4576	334	10	,	,	PUNCT
ejpam-4576	334	11	for	for	SCONJ
ejpam-4576	334	12	each	each	DET
ejpam-4576	334	13	x	x	SYM
ejpam-4576	334	14	∈	∈	PROPN
ejpam-4576	334	15	x	x	X
ejpam-4576	334	16	and	and	CCONJ
ejpam-4576	334	17	each	each	DET
ejpam-4576	334	18	(	(	PUNCT
ejpam-4576	334	19	λ	λ	PROPN
ejpam-4576	334	20	,	,	PUNCT
ejpam-4576	334	21	sp)-open	sp)-open	NOUN
ejpam-4576	334	22	set	set	VERB
ejpam-4576	334	23	v	v	NUM
ejpam-4576	334	24	of	of	ADP
ejpam-4576	334	25	y	y	PRON
ejpam-4576	335	1	such	such	ADJ
ejpam-4576	335	2	that	that	SCONJ
ejpam-4576	335	3	f	f	PROPN
ejpam-4576	335	4	(	(	PUNCT
ejpam-4576	335	5	x	x	X
ejpam-4576	335	6	)	)	PUNCT
ejpam-4576	335	7	⊆	⊆	NUM
ejpam-4576	335	8	v	v	NOUN
ejpam-4576	335	9	(	(	PUNCT
ejpam-4576	335	10	resp	resp	NOUN
ejpam-4576	335	11	.	.	PUNCT
ejpam-4576	336	1	f	f	X
ejpam-4576	336	2	(	(	PUNCT
ejpam-4576	336	3	x	x	NOUN
ejpam-4576	336	4	)	)	PUNCT
ejpam-4576	336	5	∩	∩	NOUN
ejpam-4576	336	6	v	v	ADP
ejpam-4576	336	7	̸=	̸=	PROPN
ejpam-4576	336	8	∅	∅	NOUN
ejpam-4576	336	9	)	)	PUNCT
ejpam-4576	336	10	,	,	PUNCT
ejpam-4576	336	11	there	there	PRON
ejpam-4576	336	12	exists	exist	VERB
ejpam-4576	336	13	a	a	DET
ejpam-4576	336	14	(	(	PUNCT
ejpam-4576	336	15	λ	λ	NOUN
ejpam-4576	336	16	,	,	PUNCT
ejpam-4576	336	17	sp)-open	sp)-open	NOUN
ejpam-4576	336	18	set	set	VERB
ejpam-4576	336	19	u	u	NOUN
ejpam-4576	336	20	of	of	ADP
ejpam-4576	336	21	x	x	PUNCT
ejpam-4576	336	22	containing	contain	VERB
ejpam-4576	336	23	x	x	PUNCT
ejpam-4576	336	24	such	such	ADJ
ejpam-4576	336	25	that	that	SCONJ
ejpam-4576	336	26	u	u	PROPN
ejpam-4576	336	27	⊆	⊆	NUM
ejpam-4576	336	28	f+([v	f+([v	NOUN
ejpam-4576	336	29	(	(	PUNCT
ejpam-4576	336	30	λ	λ	PROPN
ejpam-4576	336	31	,	,	PUNCT
ejpam-4576	336	32	sp)](λ	sp)](λ	PROPN
ejpam-4576	336	33	,	,	PUNCT
ejpam-4576	336	34	sp	sp	NOUN
ejpam-4576	336	35	)	)	PUNCT
ejpam-4576	336	36	)	)	PUNCT
ejpam-4576	336	37	(	(	PUNCT
ejpam-4576	336	38	resp	resp	NOUN
ejpam-4576	336	39	.	.	PUNCT
ejpam-4576	337	1	u	u	NOUN
ejpam-4576	337	2	⊆	⊆	NUM
ejpam-4576	337	3	f−([v	f−([v	ADJ
ejpam-4576	337	4	(	(	PUNCT
ejpam-4576	337	5	λ	λ	PROPN
ejpam-4576	337	6	,	,	PUNCT
ejpam-4576	337	7	sp)](λ	sp)](λ	PROPN
ejpam-4576	337	8	,	,	PUNCT
ejpam-4576	337	9	sp	sp	NOUN
ejpam-4576	337	10	)	)	PUNCT
ejpam-4576	337	11	)	)	PUNCT
ejpam-4576	337	12	)	)	PUNCT
ejpam-4576	337	13	.	.	PUNCT
ejpam-4576	338	1	several	several	ADJ
ejpam-4576	338	2	characterizations	characterization	NOUN
ejpam-4576	338	3	and	and	CCONJ
ejpam-4576	338	4	some	some	DET
ejpam-4576	338	5	properties	property	NOUN
ejpam-4576	338	6	concerning	concern	VERB
ejpam-4576	338	7	upper	upper	ADJ
ejpam-4576	338	8	(	(	PUNCT
ejpam-4576	338	9	resp	resp	NOUN
ejpam-4576	338	10	.	.	PUNCT
ejpam-4576	339	1	lower	low	ADJ
ejpam-4576	339	2	)	)	PUNCT
ejpam-4576	339	3	almost	almost	ADV
ejpam-4576	339	4	(	(	PUNCT
ejpam-4576	339	5	λ	λ	NOUN
ejpam-4576	339	6	,	,	PUNCT
ejpam-4576	339	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	339	8	multifunctions	multifunction	NOUN
ejpam-4576	339	9	are	be	AUX
ejpam-4576	339	10	established	establish	VERB
ejpam-4576	339	11	.	.	PUNCT
ejpam-4576	340	1	the	the	DET
ejpam-4576	340	2	ideas	idea	NOUN
ejpam-4576	340	3	and	and	CCONJ
ejpam-4576	340	4	results	result	NOUN
ejpam-4576	340	5	of	of	ADP
ejpam-4576	340	6	this	this	DET
ejpam-4576	340	7	work	work	NOUN
ejpam-4576	340	8	may	may	AUX
ejpam-4576	340	9	motivate	motivate	VERB
ejpam-4576	340	10	further	further	ADJ
ejpam-4576	340	11	research	research	NOUN
ejpam-4576	340	12	.	.	PUNCT
ejpam-4576	341	1	acknowledgements	acknowledgement	NOUN
ejpam-4576	341	2	this	this	DET
ejpam-4576	341	3	research	research	NOUN
ejpam-4576	341	4	project	project	NOUN
ejpam-4576	341	5	was	be	AUX
ejpam-4576	341	6	financially	financially	ADV
ejpam-4576	341	7	supported	support	VERB
ejpam-4576	341	8	by	by	ADP
ejpam-4576	341	9	mahasarakham	mahasarakham	PROPN
ejpam-4576	341	10	university	university	PROPN
ejpam-4576	341	11	.	.	PUNCT
ejpam-4576	342	1	references	reference	NOUN
ejpam-4576	342	2	95	95	NUM
ejpam-4576	342	3	references	reference	NOUN
ejpam-4576	342	4	[	[	X
ejpam-4576	342	5	1	1	NUM
ejpam-4576	342	6	]	]	PUNCT
ejpam-4576	342	7	d.	d.	PROPN
ejpam-4576	342	8	andrijević.	andrijević.	PROPN
ejpam-4576	342	9	on	on	ADP
ejpam-4576	342	10	b	b	X
ejpam-4576	342	11	-	-	PUNCT
ejpam-4576	342	12	open	open	ADJ
ejpam-4576	342	13	sets	set	NOUN
ejpam-4576	342	14	.	.	PUNCT
ejpam-4576	343	1	matematički	matematički	PROPN
ejpam-4576	343	2	vesnik	vesnik	PROPN
ejpam-4576	343	3	,	,	PUNCT
ejpam-4576	343	4	48:59–64	48:59–64	PROPN
ejpam-4576	343	5	,	,	PUNCT
ejpam-4576	343	6	1996	1996	NUM
ejpam-4576	343	7	.	.	PUNCT
ejpam-4576	344	1	[	[	X
ejpam-4576	344	2	2	2	NUM
ejpam-4576	344	3	]	]	PUNCT
ejpam-4576	344	4	c.	c.	PROPN
ejpam-4576	344	5	berge	berge	PROPN
ejpam-4576	344	6	.	.	PUNCT
ejpam-4576	344	7	espaces	espace	VERB
ejpam-4576	344	8	topologiques	topologique	NOUN
ejpam-4576	344	9	fonctions	fonction	NOUN
ejpam-4576	344	10	multivoques	multivoque	NOUN
ejpam-4576	344	11	.	.	PUNCT
ejpam-4576	345	1	dunod	dunod	PROPN
ejpam-4576	345	2	,	,	PUNCT
ejpam-4576	345	3	paris	paris	PROPN
ejpam-4576	345	4	,	,	PUNCT
ejpam-4576	345	5	1959	1959	NUM
ejpam-4576	345	6	.	.	PUNCT
ejpam-4576	346	1	[	[	X
ejpam-4576	346	2	3	3	X
ejpam-4576	346	3	]	]	PUNCT
ejpam-4576	346	4	c.	c.	PROPN
ejpam-4576	346	5	boonpok	boonpok	PROPN
ejpam-4576	346	6	.	.	PUNCT
ejpam-4576	347	1	(	(	PUNCT
ejpam-4576	347	2	λ	λ	NOUN
ejpam-4576	347	3	,	,	PUNCT
ejpam-4576	347	4	sp)-closed	sp)-close	VERB
ejpam-4576	347	5	sets	set	NOUN
ejpam-4576	347	6	and	and	CCONJ
ejpam-4576	347	7	related	related	ADJ
ejpam-4576	347	8	topics	topic	NOUN
ejpam-4576	347	9	in	in	ADP
ejpam-4576	347	10	topological	topological	ADJ
ejpam-4576	347	11	spaces	space	NOUN
ejpam-4576	347	12	.	.	PUNCT
ejpam-4576	348	1	wseas	wseas	VERB
ejpam-4576	348	2	transactions	transaction	NOUN
ejpam-4576	348	3	on	on	ADP
ejpam-4576	348	4	mathematics	mathematic	NOUN
ejpam-4576	348	5	,	,	PUNCT
ejpam-4576	348	6	19:312–322	19:312–322	PROPN
ejpam-4576	348	7	,	,	PUNCT
ejpam-4576	348	8	2020	2020	NUM
ejpam-4576	348	9	.	.	PUNCT
ejpam-4576	349	1	[	[	X
ejpam-4576	349	2	4	4	NUM
ejpam-4576	349	3	]	]	PUNCT
ejpam-4576	349	4	c.	c.	PROPN
ejpam-4576	349	5	boonpok	boonpok	PROPN
ejpam-4576	349	6	,	,	PUNCT
ejpam-4576	349	7	c.	c.	PROPN
ejpam-4576	349	8	viriyapong	viriyapong	PROPN
ejpam-4576	349	9	,	,	PUNCT
ejpam-4576	349	10	and	and	CCONJ
ejpam-4576	349	11	m.	m.	NOUN
ejpam-4576	349	12	thongmoon	thongmoon	NOUN
ejpam-4576	349	13	.	.	PUNCT
ejpam-4576	350	1	on	on	ADP
ejpam-4576	350	2	upper	upper	ADJ
ejpam-4576	350	3	and	and	CCONJ
ejpam-4576	350	4	lower	low	ADJ
ejpam-4576	350	5	(	(	PUNCT
ejpam-4576	350	6	τ1	τ1	NOUN
ejpam-4576	350	7	,	,	PUNCT
ejpam-4576	350	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-4576	350	9	multifunctions	multifunction	NOUN
ejpam-4576	350	10	.	.	PUNCT
ejpam-4576	351	1	journal	journal	PROPN
ejpam-4576	351	2	of	of	ADP
ejpam-4576	351	3	mathematics	mathematics	PROPN
ejpam-4576	351	4	and	and	CCONJ
ejpam-4576	351	5	computer	computer	NOUN
ejpam-4576	351	6	science	science	NOUN
ejpam-4576	351	7	,	,	PUNCT
ejpam-4576	351	8	18:282–293	18:282–293	NUM
ejpam-4576	351	9	,	,	PUNCT
ejpam-4576	351	10	2018	2018	NUM
ejpam-4576	351	11	.	.	PUNCT
ejpam-4576	352	1	[	[	X
ejpam-4576	352	2	5	5	X
ejpam-4576	352	3	]	]	PUNCT
ejpam-4576	352	4	e.	e.	PROPN
ejpam-4576	352	5	ekici	ekici	PROPN
ejpam-4576	352	6	and	and	CCONJ
ejpam-4576	352	7	j.	j.	PROPN
ejpam-4576	352	8	h.	h.	PROPN
ejpam-4576	352	9	park	park	PROPN
ejpam-4576	352	10	.	.	PUNCT
ejpam-4576	353	1	a	a	DET
ejpam-4576	353	2	weak	weak	ADJ
ejpam-4576	353	3	form	form	NOUN
ejpam-4576	353	4	of	of	ADP
ejpam-4576	353	5	some	some	DET
ejpam-4576	353	6	types	type	NOUN
ejpam-4576	353	7	of	of	ADP
ejpam-4576	353	8	continuous	continuous	ADJ
ejpam-4576	353	9	multifunctions	multifunction	NOUN
ejpam-4576	353	10	.	.	PUNCT
ejpam-4576	354	1	filomat	filomat	NOUN
ejpam-4576	354	2	,	,	PUNCT
ejpam-4576	354	3	20(2):13–32	20(2):13–32	NUM
ejpam-4576	354	4	,	,	PUNCT
ejpam-4576	354	5	2006	2006	NUM
ejpam-4576	354	6	.	.	PUNCT
ejpam-4576	355	1	[	[	X
ejpam-4576	355	2	6	6	NUM
ejpam-4576	355	3	]	]	PUNCT
ejpam-4576	355	4	m.	m.	NOUN
ejpam-4576	355	5	e.	e.	PROPN
ejpam-4576	355	6	abd	abd	PROPN
ejpam-4576	355	7	el	el	PROPN
ejpam-4576	355	8	-	-	PROPN
ejpam-4576	355	9	monsef	monsef	PROPN
ejpam-4576	355	10	,	,	PUNCT
ejpam-4576	355	11	s.	s.	PROPN
ejpam-4576	355	12	n.	n.	PROPN
ejpam-4576	355	13	el	el	PROPN
ejpam-4576	355	14	-	-	PROPN
ejpam-4576	355	15	deeb	deeb	PROPN
ejpam-4576	355	16	,	,	PUNCT
ejpam-4576	355	17	and	and	CCONJ
ejpam-4576	355	18	r.	r.	PROPN
ejpam-4576	355	19	a.	a.	PROPN
ejpam-4576	355	20	mahmoud	mahmoud	PROPN
ejpam-4576	355	21	.	.	PUNCT
ejpam-4576	356	1	β	β	X
ejpam-4576	356	2	-	-	ADJ
ejpam-4576	356	3	open	open	ADJ
ejpam-4576	356	4	sets	set	NOUN
ejpam-4576	356	5	and	and	CCONJ
ejpam-4576	356	6	βcontinuous	βcontinuous	ADJ
ejpam-4576	356	7	mappings	mapping	NOUN
ejpam-4576	356	8	.	.	PUNCT
ejpam-4576	357	1	bulletin	bulletin	NOUN
ejpam-4576	357	2	of	of	ADP
ejpam-4576	357	3	the	the	DET
ejpam-4576	357	4	faculty	faculty	NOUN
ejpam-4576	357	5	of	of	ADP
ejpam-4576	357	6	science	science	NOUN
ejpam-4576	357	7	.	.	PUNCT
ejpam-4576	358	1	assiut	assiut	PROPN
ejpam-4576	358	2	university	university	PROPN
ejpam-4576	358	3	.	.	PUNCT
ejpam-4576	358	4	,	,	PUNCT
ejpam-4576	358	5	12:77–90	12:77–90	NUM
ejpam-4576	358	6	,	,	PUNCT
ejpam-4576	358	7	1983	1983	NUM
ejpam-4576	358	8	.	.	PUNCT
ejpam-4576	359	1	[	[	X
ejpam-4576	359	2	7	7	X
ejpam-4576	359	3	]	]	PUNCT
ejpam-4576	359	4	t.	t.	PROPN
ejpam-4576	359	5	noiri	noiri	PROPN
ejpam-4576	359	6	and	and	CCONJ
ejpam-4576	359	7	e.	e.	PROPN
ejpam-4576	359	8	hatir	hatir	PROPN
ejpam-4576	359	9	.	.	PUNCT
ejpam-4576	359	10	λsp	λsp	NOUN
ejpam-4576	359	11	-	-	PUNCT
ejpam-4576	359	12	sets	set	NOUN
ejpam-4576	359	13	and	and	CCONJ
ejpam-4576	359	14	some	some	DET
ejpam-4576	359	15	weak	weak	ADJ
ejpam-4576	359	16	separation	separation	NOUN
ejpam-4576	359	17	axioms	axiom	NOUN
ejpam-4576	359	18	.	.	PUNCT
ejpam-4576	360	1	acta	acta	PROPN
ejpam-4576	360	2	mathematica	mathematica	PROPN
ejpam-4576	360	3	hungarica	hungarica	PROPN
ejpam-4576	360	4	,	,	PUNCT
ejpam-4576	360	5	103(3):225–232	103(3):225–232	NUM
ejpam-4576	360	6	,	,	PUNCT
ejpam-4576	360	7	2004	2004	NUM
ejpam-4576	360	8	.	.	PUNCT
ejpam-4576	361	1	[	[	X
ejpam-4576	361	2	8	8	X
ejpam-4576	361	3	]	]	PUNCT
ejpam-4576	361	4	t.	t.	PROPN
ejpam-4576	361	5	noiri	noiri	PROPN
ejpam-4576	361	6	and	and	CCONJ
ejpam-4576	361	7	v.	v.	ADP
ejpam-4576	361	8	popa	popa	NOUN
ejpam-4576	361	9	.	.	PUNCT
ejpam-4576	362	1	characterizations	characterization	NOUN
ejpam-4576	362	2	of	of	ADP
ejpam-4576	362	3	almost	almost	ADV
ejpam-4576	362	4	quasi	quasi	ADJ
ejpam-4576	362	5	-	-	ADJ
ejpam-4576	362	6	continuous	continuous	ADJ
ejpam-4576	362	7	multifunctions	multifunction	NOUN
ejpam-4576	362	8	.	.	PUNCT
ejpam-4576	363	1	research	research	NOUN
ejpam-4576	363	2	reports	report	NOUN
ejpam-4576	363	3	of	of	ADP
ejpam-4576	363	4	yatsushiro	yatsushiro	PROPN
ejpam-4576	363	5	national	national	PROPN
ejpam-4576	363	6	college	college	PROPN
ejpam-4576	363	7	of	of	ADP
ejpam-4576	363	8	technology	technology	NOUN
ejpam-4576	363	9	,	,	PUNCT
ejpam-4576	363	10	15:97–101	15:97–101	NUM
ejpam-4576	363	11	,	,	PUNCT
ejpam-4576	363	12	1993	1993	NUM
ejpam-4576	363	13	.	.	PUNCT
ejpam-4576	364	1	[	[	X
ejpam-4576	364	2	9	9	NUM
ejpam-4576	364	3	]	]	PUNCT
ejpam-4576	364	4	t.	t.	PROPN
ejpam-4576	364	5	noiri	noiri	PROPN
ejpam-4576	364	6	and	and	CCONJ
ejpam-4576	364	7	v.	v.	ADP
ejpam-4576	364	8	popa	popa	NOUN
ejpam-4576	364	9	.	.	PUNCT
ejpam-4576	365	1	on	on	ADP
ejpam-4576	365	2	upper	upper	ADJ
ejpam-4576	365	3	and	and	CCONJ
ejpam-4576	365	4	lower	low	ADJ
ejpam-4576	365	5	almost	almost	ADV
ejpam-4576	365	6	β	β	ADJ
ejpam-4576	365	7	-	-	ADJ
ejpam-4576	365	8	continuous	continuous	ADJ
ejpam-4576	365	9	multifunctions	multifunction	NOUN
ejpam-4576	365	10	.	.	PUNCT
ejpam-4576	366	1	acta	acta	PROPN
ejpam-4576	366	2	mathematica	mathematica	PROPN
ejpam-4576	366	3	hungarica	hungarica	PROPN
ejpam-4576	366	4	,	,	PUNCT
ejpam-4576	366	5	82:57–73	82:57–73	PROPN
ejpam-4576	366	6	,	,	PUNCT
ejpam-4576	366	7	1999	1999	NUM
ejpam-4576	366	8	.	.	PUNCT
ejpam-4576	367	1	[	[	X
ejpam-4576	367	2	10	10	NUM
ejpam-4576	367	3	]	]	PUNCT
ejpam-4576	367	4	t.	t.	PROPN
ejpam-4576	367	5	noiri	noiri	PROPN
ejpam-4576	367	6	and	and	CCONJ
ejpam-4576	367	7	v.	v.	ADP
ejpam-4576	367	8	popa	popa	NOUN
ejpam-4576	367	9	.	.	PUNCT
ejpam-4576	368	1	a	a	DET
ejpam-4576	368	2	unified	unified	ADJ
ejpam-4576	368	3	theory	theory	NOUN
ejpam-4576	368	4	of	of	ADP
ejpam-4576	368	5	almost	almost	ADV
ejpam-4576	368	6	continuity	continuity	NOUN
ejpam-4576	368	7	for	for	ADP
ejpam-4576	368	8	multifunctions	multifunction	NOUN
ejpam-4576	368	9	.	.	PUNCT
ejpam-4576	369	1	scientific	scientific	ADJ
ejpam-4576	369	2	studies	study	NOUN
ejpam-4576	369	3	and	and	CCONJ
ejpam-4576	369	4	research	research	NOUN
ejpam-4576	369	5	,	,	PUNCT
ejpam-4576	369	6	series	series	NOUN
ejpam-4576	369	7	mathematics	mathematics	PROPN
ejpam-4576	369	8	and	and	CCONJ
ejpam-4576	369	9	informatics	informatic	NOUN
ejpam-4576	369	10	,	,	PUNCT
ejpam-4576	369	11	20(1):185–214	20(1):185–214	PROPN
ejpam-4576	369	12	,	,	PUNCT
ejpam-4576	369	13	2010	2010	NUM
ejpam-4576	369	14	.	.	PUNCT
ejpam-4576	370	1	[	[	X
ejpam-4576	370	2	11	11	NUM
ejpam-4576	370	3	]	]	X
ejpam-4576	370	4	v.	v.	PROPN
ejpam-4576	370	5	i.	i.	PROPN
ejpam-4576	370	6	ponomarev	ponomarev	PROPN
ejpam-4576	370	7	.	.	PUNCT
ejpam-4576	371	1	properties	property	NOUN
ejpam-4576	371	2	of	of	ADP
ejpam-4576	371	3	topological	topological	ADJ
ejpam-4576	371	4	spaces	space	NOUN
ejpam-4576	371	5	preserved	preserve	VERB
ejpam-4576	371	6	under	under	ADP
ejpam-4576	371	7	multivalued	multivalued	ADJ
ejpam-4576	371	8	continuous	continuous	ADJ
ejpam-4576	371	9	mappings	mapping	NOUN
ejpam-4576	371	10	on	on	ADP
ejpam-4576	371	11	compacta	compacta	NOUN
ejpam-4576	371	12	.	.	PUNCT
ejpam-4576	372	1	american	american	PROPN
ejpam-4576	372	2	mathematical	mathematical	ADJ
ejpam-4576	372	3	society	society	NOUN
ejpam-4576	372	4	translations	translation	NOUN
ejpam-4576	372	5	,	,	PUNCT
ejpam-4576	372	6	38(2):119–140	38(2):119–140	NUM
ejpam-4576	372	7	,	,	PUNCT
ejpam-4576	372	8	1964	1964	NUM
ejpam-4576	372	9	.	.	PUNCT
ejpam-4576	373	1	[	[	X
ejpam-4576	373	2	12	12	NUM
ejpam-4576	373	3	]	]	X
ejpam-4576	373	4	v.	v.	CCONJ
ejpam-4576	373	5	popa	popa	NOUN
ejpam-4576	373	6	.	.	PUNCT
ejpam-4576	374	1	almost	almost	ADV
ejpam-4576	374	2	continuous	continuous	ADJ
ejpam-4576	374	3	multifunctions	multifunction	NOUN
ejpam-4576	374	4	.	.	PUNCT
ejpam-4576	375	1	matematički	matematički	PROPN
ejpam-4576	375	2	vesnik	vesnik	PROPN
ejpam-4576	375	3	,	,	PUNCT
ejpam-4576	375	4	6(9)(34):75–84	6(9)(34):75–84	NOUN
ejpam-4576	375	5	,	,	PUNCT
ejpam-4576	375	6	1982	1982	NUM
ejpam-4576	375	7	.	.	PUNCT
ejpam-4576	376	1	[	[	X
ejpam-4576	376	2	13	13	NUM
ejpam-4576	376	3	]	]	PUNCT
ejpam-4576	376	4	v.	v.	CCONJ
ejpam-4576	376	5	popa	popa	NOUN
ejpam-4576	376	6	and	and	CCONJ
ejpam-4576	376	7	t.	t.	PROPN
ejpam-4576	376	8	noiri	noiri	PROPN
ejpam-4576	376	9	.	.	PUNCT
ejpam-4576	377	1	on	on	ADP
ejpam-4576	377	2	upper	upper	ADJ
ejpam-4576	377	3	and	and	CCONJ
ejpam-4576	377	4	lower	low	ADJ
ejpam-4576	377	5	almost	almost	ADV
ejpam-4576	377	6	quasi	quasi	ADJ
ejpam-4576	377	7	-	-	ADJ
ejpam-4576	377	8	continuous	continuous	ADJ
ejpam-4576	377	9	multifunctions	multifunction	NOUN
ejpam-4576	377	10	.	.	PUNCT
ejpam-4576	378	1	bulletin	bulletin	NOUN
ejpam-4576	378	2	of	of	ADP
ejpam-4576	378	3	the	the	DET
ejpam-4576	378	4	institute	institute	NOUN
ejpam-4576	378	5	of	of	ADP
ejpam-4576	378	6	mathematics	mathematics	PROPN
ejpam-4576	378	7	,	,	PUNCT
ejpam-4576	378	8	academia	academia	PROPN
ejpam-4576	378	9	sinica	sinica	PROPN
ejpam-4576	378	10	,	,	PUNCT
ejpam-4576	378	11	21:337–349	21:337–349	NUM
ejpam-4576	378	12	,	,	PUNCT
ejpam-4576	378	13	1993	1993	NUM
ejpam-4576	378	14	.	.	PUNCT
ejpam-4576	379	1	[	[	X
ejpam-4576	379	2	14	14	NUM
ejpam-4576	379	3	]	]	PUNCT
ejpam-4576	379	4	v.	v.	CCONJ
ejpam-4576	379	5	popa	popa	NOUN
ejpam-4576	379	6	and	and	CCONJ
ejpam-4576	379	7	t.	t.	PROPN
ejpam-4576	379	8	noiri	noiri	PROPN
ejpam-4576	379	9	.	.	PUNCT
ejpam-4576	380	1	on	on	ADP
ejpam-4576	380	2	upper	upper	ADJ
ejpam-4576	380	3	and	and	CCONJ
ejpam-4576	380	4	lower	low	ADJ
ejpam-4576	380	5	almost	almost	ADV
ejpam-4576	380	6	α	α	ADJ
ejpam-4576	380	7	-	-	ADJ
ejpam-4576	380	8	continuous	continuous	ADJ
ejpam-4576	380	9	multifunctions	multifunction	NOUN
ejpam-4576	380	10	.	.	PUNCT
ejpam-4576	381	1	demonstratio	demonstratio	PROPN
ejpam-4576	381	2	mathematica	mathematica	PROPN
ejpam-4576	381	3	,	,	PUNCT
ejpam-4576	381	4	29:381–396	29:381–396	PROPN
ejpam-4576	381	5	,	,	PUNCT
ejpam-4576	381	6	1996	1996	NUM
ejpam-4576	381	7	.	.	PUNCT
ejpam-4576	382	1	[	[	X
ejpam-4576	382	2	15	15	NUM
ejpam-4576	382	3	]	]	X
ejpam-4576	382	4	v.	v.	CCONJ
ejpam-4576	382	5	popa	popa	NOUN
ejpam-4576	382	6	and	and	CCONJ
ejpam-4576	382	7	t.	t.	PROPN
ejpam-4576	382	8	noiri	noiri	PROPN
ejpam-4576	382	9	.	.	PUNCT
ejpam-4576	383	1	on	on	ADP
ejpam-4576	383	2	upper	upper	ADJ
ejpam-4576	383	3	and	and	CCONJ
ejpam-4576	383	4	lower	low	ADJ
ejpam-4576	383	5	weakly	weakly	ADJ
ejpam-4576	383	6	β	β	ADJ
ejpam-4576	383	7	-	-	ADJ
ejpam-4576	383	8	continuous	continuous	ADJ
ejpam-4576	383	9	multifunctions	multifunction	NOUN
ejpam-4576	383	10	.	.	PUNCT
ejpam-4576	384	1	annales	annales	PROPN
ejpam-4576	384	2	universitatis	universitatis	PROPN
ejpam-4576	384	3	scientiarum	scientiarum	PROPN
ejpam-4576	384	4	budapestinensis	budapestinensis	PROPN
ejpam-4576	384	5	,	,	PUNCT
ejpam-4576	384	6	43:25–48	43:25–48	PROPN
ejpam-4576	384	7	,	,	PUNCT
ejpam-4576	384	8	2000	2000	NUM
ejpam-4576	384	9	.	.	PUNCT
ejpam-4576	385	1	references	reference	NOUN
ejpam-4576	385	2	96	96	NUM
ejpam-4576	386	1	[	[	X
ejpam-4576	386	2	16	16	NUM
ejpam-4576	386	3	]	]	PUNCT
ejpam-4576	386	4	v.	v.	CCONJ
ejpam-4576	386	5	popa	popa	NOUN
ejpam-4576	386	6	,	,	PUNCT
ejpam-4576	386	7	t.	t.	PROPN
ejpam-4576	386	8	noiri	noiri	PROPN
ejpam-4576	386	9	,	,	PUNCT
ejpam-4576	386	10	and	and	CCONJ
ejpam-4576	386	11	m.	m.	NOUN
ejpam-4576	386	12	ganster	ganster	NOUN
ejpam-4576	386	13	.	.	PUNCT
ejpam-4576	387	1	on	on	ADP
ejpam-4576	387	2	upper	upper	ADJ
ejpam-4576	387	3	and	and	CCONJ
ejpam-4576	387	4	lower	low	ADJ
ejpam-4576	387	5	almost	almost	ADV
ejpam-4576	387	6	precontinuous	precontinuous	ADJ
ejpam-4576	387	7	multifunctions	multifunction	NOUN
ejpam-4576	387	8	.	.	PUNCT
ejpam-4576	388	1	far	far	PROPN
ejpam-4576	388	2	east	east	PROPN
ejpam-4576	388	3	journal	journal	PROPN
ejpam-4576	388	4	of	of	ADP
ejpam-4576	388	5	mathematical	mathematical	ADJ
ejpam-4576	388	6	sciences	science	NOUN
ejpam-4576	388	7	,	,	PUNCT
ejpam-4576	388	8	special	special	ADJ
ejpam-4576	388	9	volume(part	volume(part	NOUN
ejpam-4576	388	10	i):49–68	i):49–68	NOUN
ejpam-4576	388	11	,	,	PUNCT
ejpam-4576	388	12	1997	1997	NUM
ejpam-4576	388	13	.	.	PUNCT
ejpam-4576	389	1	[	[	X
ejpam-4576	389	2	17	17	NUM
ejpam-4576	389	3	]	]	PUNCT
ejpam-4576	389	4	v.	v.	CCONJ
ejpam-4576	389	5	popa	popa	NOUN
ejpam-4576	389	6	,	,	PUNCT
ejpam-4576	389	7	t.	t.	PROPN
ejpam-4576	389	8	noiri	noiri	PROPN
ejpam-4576	389	9	,	,	PUNCT
ejpam-4576	389	10	m.	m.	NOUN
ejpam-4576	389	11	ganster	ganster	NOUN
ejpam-4576	389	12	,	,	PUNCT
ejpam-4576	389	13	and	and	CCONJ
ejpam-4576	389	14	k.	k.	PROPN
ejpam-4576	389	15	dlaska	dlaska	PROPN
ejpam-4576	389	16	.	.	PUNCT
ejpam-4576	390	1	on	on	ADP
ejpam-4576	390	2	upper	upper	ADJ
ejpam-4576	390	3	and	and	CCONJ
ejpam-4576	390	4	lower	low	ADJ
ejpam-4576	390	5	θ	θ	ADJ
ejpam-4576	390	6	-	-	PUNCT
ejpam-4576	390	7	irresolute	irresolute	ADJ
ejpam-4576	390	8	multifunctions	multifunction	NOUN
ejpam-4576	390	9	.	.	PUNCT
ejpam-4576	391	1	journal	journal	PROPN
ejpam-4576	391	2	of	of	ADP
ejpam-4576	391	3	institute	institute	PROPN
ejpam-4576	391	4	of	of	ADP
ejpam-4576	391	5	mathematics	mathematics	PROPN
ejpam-4576	391	6	&	&	CCONJ
ejpam-4576	391	7	computer	computer	PROPN
ejpam-4576	391	8	sciences	sciences	PROPN
ejpam-4576	391	9	.	.	PUNCT
ejpam-4576	392	1	mathematics	mathematic	NOUN
ejpam-4576	392	2	series	series	PROPN
ejpam-4576	392	3	,	,	PUNCT
ejpam-4576	392	4	6:137–149	6:137–149	NOUN
ejpam-4576	392	5	,	,	PUNCT
ejpam-4576	392	6	1993	1993	NUM
ejpam-4576	392	7	.	.	PUNCT
ejpam-4576	393	1	[	[	X
ejpam-4576	393	2	18	18	NUM
ejpam-4576	393	3	]	]	PUNCT
ejpam-4576	393	4	m.	m.	NOUN
ejpam-4576	393	5	k.	k.	PROPN
ejpam-4576	393	6	singal	singal	PROPN
ejpam-4576	393	7	and	and	CCONJ
ejpam-4576	393	8	a.	a.	PROPN
ejpam-4576	393	9	r.	r.	PROPN
ejpam-4576	393	10	singal	singal	PROPN
ejpam-4576	393	11	.	.	PUNCT
ejpam-4576	394	1	almost	almost	ADV
ejpam-4576	394	2	continuous	continuous	ADJ
ejpam-4576	394	3	mappings	mapping	NOUN
ejpam-4576	394	4	.	.	PUNCT
ejpam-4576	395	1	yokohama	yokohama	PROPN
ejpam-4576	395	2	mathematical	mathematical	PROPN
ejpam-4576	395	3	journal	journal	PROPN
ejpam-4576	395	4	,	,	PUNCT
ejpam-4576	395	5	16:63–73	16:63–73	PROPN
ejpam-4576	395	6	,	,	PUNCT
ejpam-4576	395	7	1968	1968	NUM
ejpam-4576	395	8	.	.	PUNCT
ejpam-4576	396	1	[	[	X
ejpam-4576	396	2	19	19	NUM
ejpam-4576	396	3	]	]	X
ejpam-4576	396	4	c.	c.	PROPN
ejpam-4576	396	5	viriyapong	viriyapong	PROPN
ejpam-4576	396	6	and	and	CCONJ
ejpam-4576	396	7	c.	c.	PROPN
ejpam-4576	396	8	boonpok	boonpok	PROPN
ejpam-4576	396	9	.	.	PUNCT
ejpam-4576	397	1	(	(	PUNCT
ejpam-4576	397	2	λ	λ	NOUN
ejpam-4576	397	3	,	,	PUNCT
ejpam-4576	397	4	sp)-continuity	sp)-continuity	NOUN
ejpam-4576	397	5	and	and	CCONJ
ejpam-4576	397	6	δ(λ	δ(λ	PROPN
ejpam-4576	397	7	,	,	PUNCT
ejpam-4576	397	8	sp)-closed	sp)-close	VERB
ejpam-4576	397	9	sets	set	NOUN
ejpam-4576	397	10	.	.	PUNCT
ejpam-4576	398	1	international	international	ADJ
ejpam-4576	398	2	journal	journal	NOUN
ejpam-4576	398	3	of	of	ADP
ejpam-4576	398	4	mathematics	mathematic	NOUN
ejpam-4576	398	5	and	and	CCONJ
ejpam-4576	398	6	computer	computer	NOUN
ejpam-4576	398	7	science	science	NOUN
ejpam-4576	398	8	,	,	PUNCT
ejpam-4576	398	9	17(4):1685–1689	17(4):1685–1689	NUM
ejpam-4576	398	10	,	,	PUNCT
ejpam-4576	398	11	2022	2022	NUM
ejpam-4576	398	12	.	.	PUNCT
ejpam-4576	399	1	[	[	X
ejpam-4576	399	2	20	20	NUM
ejpam-4576	399	3	]	]	X
ejpam-4576	399	4	c.	c.	PROPN
ejpam-4576	399	5	viriyapong	viriyapong	PROPN
ejpam-4576	399	6	and	and	CCONJ
ejpam-4576	399	7	c.	c.	PROPN
ejpam-4576	399	8	boonpok	boonpok	PROPN
ejpam-4576	399	9	.	.	PUNCT
ejpam-4576	400	1	(	(	PUNCT
ejpam-4576	400	2	λ	λ	X
ejpam-4576	400	3	,	,	PUNCT
ejpam-4576	400	4	sp)-continuous	sp)-continuous	ADJ
ejpam-4576	400	5	functions	function	NOUN
ejpam-4576	400	6	.	.	PUNCT
ejpam-4576	401	1	wseas	wseas	VERB
ejpam-4576	401	2	transactions	transaction	NOUN
ejpam-4576	401	3	on	on	ADP
ejpam-4576	401	4	mathematics	mathematic	NOUN
ejpam-4576	401	5	,	,	PUNCT
ejpam-4576	401	6	21:380–385	21:380–385	NUM
ejpam-4576	401	7	,	,	PUNCT
ejpam-4576	401	8	2022	2022	NUM
ejpam-4576	401	9	.	.	PUNCT
ejpam-4576	402	1	[	[	X
ejpam-4576	402	2	21	21	NUM
ejpam-4576	402	3	]	]	X
ejpam-4576	402	4	c.	c.	PROPN
ejpam-4576	402	5	viriyapong	viriyapong	PROPN
ejpam-4576	402	6	and	and	CCONJ
ejpam-4576	402	7	c.	c.	PROPN
ejpam-4576	402	8	boonpok	boonpok	PROPN
ejpam-4576	402	9	.	.	PUNCT
ejpam-4576	403	1	weak	weak	ADJ
ejpam-4576	403	2	quasi	quasi	NOUN
ejpam-4576	403	3	(	(	PUNCT
ejpam-4576	403	4	λ	λ	PROPN
ejpam-4576	403	5	,	,	PUNCT
ejpam-4576	403	6	sp)-continuity	sp)-continuity	NOUN
ejpam-4576	403	7	for	for	ADP
ejpam-4576	403	8	multifunctions	multifunction	NOUN
ejpam-4576	403	9	.	.	PUNCT
ejpam-4576	404	1	international	international	ADJ
ejpam-4576	404	2	journal	journal	PROPN
ejpam-4576	404	3	of	of	ADP
ejpam-4576	404	4	mathematics	mathematic	NOUN
ejpam-4576	404	5	and	and	CCONJ
ejpam-4576	404	6	computer	computer	NOUN
ejpam-4576	404	7	science	science	NOUN
ejpam-4576	404	8	,	,	PUNCT
ejpam-4576	404	9	17(3):1201–1209	17(3):1201–1209	NUM
ejpam-4576	404	10	,	,	PUNCT
ejpam-4576	404	11	2022	2022	NUM
ejpam-4576	404	12	.	.	PUNCT