id	sid	tid	token	lemma	pos
ejpam-4572	1	1	european	european	PROPN
ejpam-4572	1	2	journal	journal	PROPN
ejpam-4572	1	3	of	of	ADP
ejpam-4572	1	4	pure	pure	ADJ
ejpam-4572	1	5	and	and	CCONJ
ejpam-4572	1	6	applied	apply	VERB
ejpam-4572	1	7	mathematics	mathematic	NOUN
ejpam-4572	1	8	vol	vol	NOUN
ejpam-4572	1	9	.	.	PUNCT
ejpam-4572	2	1	16	16	NUM
ejpam-4572	2	2	,	,	PUNCT
ejpam-4572	2	3	no	no	INTJ
ejpam-4572	2	4	.	.	NOUN
ejpam-4572	2	5	1	1	NUM
ejpam-4572	2	6	,	,	PUNCT
ejpam-4572	2	7	2023	2023	NUM
ejpam-4572	2	8	,	,	PUNCT
ejpam-4572	2	9	465	465	NUM
ejpam-4572	2	10	-	-	SYM
ejpam-4572	2	11	478	478	NUM
ejpam-4572	2	12	issn	issn	PROPN
ejpam-4572	2	13	1307	1307	NUM
ejpam-4572	2	14	-	-	SYM
ejpam-4572	2	15	5543	5543	NUM
ejpam-4572	2	16	–	–	PUNCT
ejpam-4572	2	17	ejpam.com	ejpam.com	X
ejpam-4572	2	18	published	publish	VERB
ejpam-4572	2	19	by	by	ADP
ejpam-4572	2	20	new	new	PROPN
ejpam-4572	2	21	york	york	PROPN
ejpam-4572	2	22	business	business	PROPN
ejpam-4572	2	23	global	global	PROPN
ejpam-4572	2	24	weak	weak	ADJ
ejpam-4572	2	25	α(λ	α(λ	PROPN
ejpam-4572	2	26	,	,	PUNCT
ejpam-4572	2	27	sp)-continuity	sp)-continuity	NOUN
ejpam-4572	2	28	for	for	ADP
ejpam-4572	2	29	multifunctions	multifunction	NOUN
ejpam-4572	2	30	chawalit	chawalit	VERB
ejpam-4572	2	31	boonpok1	boonpok1	PROPN
ejpam-4572	2	32	,	,	PUNCT
ejpam-4572	2	33	montri	montri	PROPN
ejpam-4572	2	34	thongmoon1,∗	thongmoon1,∗	NOUN
ejpam-4572	2	35	1	1	NUM
ejpam-4572	2	36	mathematics	mathematic	NOUN
ejpam-4572	2	37	and	and	CCONJ
ejpam-4572	2	38	applied	apply	VERB
ejpam-4572	2	39	mathematics	mathematics	PROPN
ejpam-4572	2	40	research	research	NOUN
ejpam-4572	2	41	unit	unit	NOUN
ejpam-4572	2	42	,	,	PUNCT
ejpam-4572	2	43	department	department	NOUN
ejpam-4572	2	44	of	of	ADP
ejpam-4572	2	45	mathematics	mathematic	NOUN
ejpam-4572	2	46	,	,	PUNCT
ejpam-4572	2	47	faculty	faculty	NOUN
ejpam-4572	2	48	of	of	ADP
ejpam-4572	2	49	science	science	NOUN
ejpam-4572	2	50	,	,	PUNCT
ejpam-4572	2	51	mahasarakham	mahasarakham	PROPN
ejpam-4572	2	52	university	university	PROPN
ejpam-4572	2	53	,	,	PUNCT
ejpam-4572	2	54	maha	maha	PROPN
ejpam-4572	2	55	sarakham	sarakham	PROPN
ejpam-4572	2	56	,	,	PUNCT
ejpam-4572	2	57	44150	44150	NUM
ejpam-4572	2	58	,	,	PUNCT
ejpam-4572	2	59	thailand	thailand	PROPN
ejpam-4572	2	60	abstract	abstract	PROPN
ejpam-4572	2	61	.	.	PUNCT
ejpam-4572	3	1	this	this	DET
ejpam-4572	3	2	paper	paper	NOUN
ejpam-4572	3	3	deals	deal	NOUN
ejpam-4572	3	4	with	with	ADP
ejpam-4572	3	5	the	the	DET
ejpam-4572	3	6	concepts	concept	NOUN
ejpam-4572	3	7	of	of	ADP
ejpam-4572	3	8	upper	upper	ADJ
ejpam-4572	3	9	and	and	CCONJ
ejpam-4572	3	10	lower	low	ADJ
ejpam-4572	3	11	weakly	weakly	ADJ
ejpam-4572	3	12	α(λ	α(λ	PROPN
ejpam-4572	3	13	,	,	PUNCT
ejpam-4572	3	14	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	3	15	multifunctions	multifunction	NOUN
ejpam-4572	3	16	.	.	PUNCT
ejpam-4572	4	1	moreover	moreover	ADV
ejpam-4572	4	2	,	,	PUNCT
ejpam-4572	4	3	several	several	ADJ
ejpam-4572	4	4	characterizations	characterization	NOUN
ejpam-4572	4	5	of	of	ADP
ejpam-4572	4	6	upper	upper	ADJ
ejpam-4572	4	7	and	and	CCONJ
ejpam-4572	4	8	lower	low	ADJ
ejpam-4572	4	9	weakly	weakly	ADJ
ejpam-4572	4	10	α(λ	α(λ	PROPN
ejpam-4572	4	11	,	,	PUNCT
ejpam-4572	4	12	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	4	13	multifunctions	multifunction	NOUN
ejpam-4572	4	14	are	be	AUX
ejpam-4572	4	15	established	establish	VERB
ejpam-4572	4	16	.	.	PUNCT
ejpam-4572	5	1	2020	2020	NUM
ejpam-4572	5	2	mathematics	mathematics	PROPN
ejpam-4572	5	3	subject	subject	NOUN
ejpam-4572	5	4	classifications	classification	NOUN
ejpam-4572	5	5	:	:	PUNCT
ejpam-4572	5	6	54c08	54c08	NUM
ejpam-4572	5	7	,	,	PUNCT
ejpam-4572	5	8	54c60	54c60	NUM
ejpam-4572	5	9	key	key	ADJ
ejpam-4572	5	10	words	word	NOUN
ejpam-4572	5	11	and	and	CCONJ
ejpam-4572	5	12	phrases	phrase	NOUN
ejpam-4572	5	13	:	:	PUNCT
ejpam-4572	5	14	α(λ	α(λ	NOUN
ejpam-4572	5	15	,	,	PUNCT
ejpam-4572	5	16	sp)-open	sp)-open	ADJ
ejpam-4572	5	17	set	set	NOUN
ejpam-4572	5	18	,	,	PUNCT
ejpam-4572	5	19	upper	upper	ADJ
ejpam-4572	5	20	weakly	weakly	ADJ
ejpam-4572	5	21	α(λ	α(λ	PROPN
ejpam-4572	5	22	,	,	PUNCT
ejpam-4572	5	23	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	5	24	multifunction	multifunction	NOUN
ejpam-4572	5	25	,	,	PUNCT
ejpam-4572	5	26	lower	low	ADJ
ejpam-4572	5	27	weakly	weakly	ADJ
ejpam-4572	5	28	α(λ	α(λ	PROPN
ejpam-4572	5	29	,	,	PUNCT
ejpam-4572	5	30	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	5	31	multifunction	multifunction	NOUN
ejpam-4572	5	32	1	1	NUM
ejpam-4572	5	33	.	.	PUNCT
ejpam-4572	6	1	introduction	introduction	NOUN
ejpam-4572	6	2	weaker	weak	ADJ
ejpam-4572	6	3	and	and	CCONJ
ejpam-4572	6	4	stronger	strong	ADJ
ejpam-4572	6	5	forms	form	NOUN
ejpam-4572	6	6	of	of	ADP
ejpam-4572	6	7	open	open	ADJ
ejpam-4572	6	8	sets	set	NOUN
ejpam-4572	6	9	play	play	VERB
ejpam-4572	6	10	an	an	DET
ejpam-4572	6	11	important	important	ADJ
ejpam-4572	6	12	role	role	NOUN
ejpam-4572	6	13	in	in	ADP
ejpam-4572	6	14	the	the	DET
ejpam-4572	6	15	generalization	generalization	NOUN
ejpam-4572	6	16	of	of	ADP
ejpam-4572	6	17	different	different	ADJ
ejpam-4572	6	18	forms	form	NOUN
ejpam-4572	6	19	of	of	ADP
ejpam-4572	6	20	continuity	continuity	NOUN
ejpam-4572	6	21	.	.	PUNCT
ejpam-4572	7	1	using	use	VERB
ejpam-4572	7	2	different	different	ADJ
ejpam-4572	7	3	forms	form	NOUN
ejpam-4572	7	4	of	of	ADP
ejpam-4572	7	5	open	open	ADJ
ejpam-4572	7	6	sets	set	NOUN
ejpam-4572	7	7	,	,	PUNCT
ejpam-4572	7	8	several	several	ADJ
ejpam-4572	7	9	authors	author	NOUN
ejpam-4572	7	10	have	have	AUX
ejpam-4572	7	11	introduced	introduce	VERB
ejpam-4572	7	12	and	and	CCONJ
ejpam-4572	7	13	investigated	investigate	VERB
ejpam-4572	7	14	various	various	ADJ
ejpam-4572	7	15	types	type	NOUN
ejpam-4572	7	16	of	of	ADP
ejpam-4572	7	17	continuity	continuity	NOUN
ejpam-4572	7	18	for	for	ADP
ejpam-4572	7	19	functions	function	NOUN
ejpam-4572	7	20	and	and	CCONJ
ejpam-4572	7	21	multifunctions	multifunction	NOUN
ejpam-4572	7	22	.	.	PUNCT
ejpam-4572	8	1	in	in	ADP
ejpam-4572	8	2	1987	1987	NUM
ejpam-4572	8	3	,	,	PUNCT
ejpam-4572	8	4	noiri	noiri	ADV
ejpam-4572	8	5	[	[	X
ejpam-4572	8	6	8	8	NUM
ejpam-4572	8	7	]	]	PUNCT
ejpam-4572	8	8	introduced	introduce	VERB
ejpam-4572	8	9	the	the	DET
ejpam-4572	8	10	class	class	NOUN
ejpam-4572	8	11	of	of	ADP
ejpam-4572	8	12	functions	function	NOUN
ejpam-4572	8	13	called	call	VERB
ejpam-4572	8	14	weakly	weakly	ADJ
ejpam-4572	8	15	α	α	ADJ
ejpam-4572	8	16	-	-	ADJ
ejpam-4572	8	17	continuous	continuous	ADJ
ejpam-4572	8	18	functions	function	NOUN
ejpam-4572	8	19	.	.	PUNCT
ejpam-4572	9	1	popa	popa	NOUN
ejpam-4572	9	2	and	and	CCONJ
ejpam-4572	9	3	noiri	noiri	ADV
ejpam-4572	10	1	[	[	X
ejpam-4572	10	2	11	11	NUM
ejpam-4572	10	3	]	]	PUNCT
ejpam-4572	10	4	,	,	PUNCT
ejpam-4572	10	5	rose	rise	VERB
ejpam-4572	10	6	[	[	X
ejpam-4572	10	7	14	14	NUM
ejpam-4572	10	8	]	]	PUNCT
ejpam-4572	10	9	and	and	CCONJ
ejpam-4572	10	10	sen	sen	PROPN
ejpam-4572	10	11	and	and	CCONJ
ejpam-4572	10	12	bhattacharyya	bhattacharyya	ADJ
ejpam-4572	11	1	[	[	X
ejpam-4572	11	2	15	15	NUM
ejpam-4572	11	3	]	]	PUNCT
ejpam-4572	11	4	studied	study	VERB
ejpam-4572	11	5	some	some	DET
ejpam-4572	11	6	properties	property	NOUN
ejpam-4572	11	7	of	of	ADP
ejpam-4572	11	8	weakly	weakly	ADJ
ejpam-4572	11	9	α	α	ADJ
ejpam-4572	11	10	-	-	ADJ
ejpam-4572	11	11	continuous	continuous	ADJ
ejpam-4572	11	12	functions	function	NOUN
ejpam-4572	11	13	.	.	PUNCT
ejpam-4572	12	1	several	several	ADJ
ejpam-4572	12	2	different	different	ADJ
ejpam-4572	12	3	forms	form	NOUN
ejpam-4572	12	4	of	of	ADP
ejpam-4572	12	5	continuous	continuous	ADJ
ejpam-4572	12	6	multifunctions	multifunction	NOUN
ejpam-4572	12	7	have	have	AUX
ejpam-4572	12	8	been	be	AUX
ejpam-4572	12	9	introduced	introduce	VERB
ejpam-4572	12	10	and	and	CCONJ
ejpam-4572	12	11	studied	study	VERB
ejpam-4572	12	12	over	over	ADP
ejpam-4572	12	13	the	the	DET
ejpam-4572	12	14	years	year	NOUN
ejpam-4572	12	15	.	.	PUNCT
ejpam-4572	13	1	many	many	ADJ
ejpam-4572	13	2	authors	author	NOUN
ejpam-4572	13	3	have	have	AUX
ejpam-4572	13	4	researched	research	VERB
ejpam-4572	13	5	and	and	CCONJ
ejpam-4572	13	6	studied	study	VERB
ejpam-4572	13	7	several	several	ADJ
ejpam-4572	13	8	stronger	strong	ADJ
ejpam-4572	13	9	and	and	CCONJ
ejpam-4572	13	10	weaker	weak	ADJ
ejpam-4572	13	11	forms	form	NOUN
ejpam-4572	13	12	of	of	ADP
ejpam-4572	13	13	continuous	continuous	ADJ
ejpam-4572	13	14	functions	function	NOUN
ejpam-4572	13	15	and	and	CCONJ
ejpam-4572	13	16	multifunctions	multifunction	NOUN
ejpam-4572	13	17	.	.	PUNCT
ejpam-4572	14	1	neubrunn	neubrunn	NOUN
ejpam-4572	15	1	[	[	X
ejpam-4572	15	2	7	7	X
ejpam-4572	15	3	]	]	PUNCT
ejpam-4572	15	4	introduced	introduce	VERB
ejpam-4572	15	5	and	and	CCONJ
ejpam-4572	15	6	investigated	investigate	VERB
ejpam-4572	15	7	the	the	DET
ejpam-4572	15	8	notions	notion	NOUN
ejpam-4572	15	9	of	of	ADP
ejpam-4572	15	10	upper	upper	ADJ
ejpam-4572	15	11	and	and	CCONJ
ejpam-4572	15	12	lower	low	ADJ
ejpam-4572	15	13	α	α	ADJ
ejpam-4572	15	14	-	-	ADJ
ejpam-4572	15	15	continuous	continuous	ADJ
ejpam-4572	15	16	multifunctions	multifunction	NOUN
ejpam-4572	15	17	.	.	PUNCT
ejpam-4572	16	1	popa	popa	NOUN
ejpam-4572	16	2	and	and	CCONJ
ejpam-4572	16	3	noiri	noiri	ADV
ejpam-4572	17	1	[	[	X
ejpam-4572	17	2	12	12	NUM
ejpam-4572	17	3	]	]	PUNCT
ejpam-4572	17	4	studied	study	VERB
ejpam-4572	17	5	some	some	DET
ejpam-4572	17	6	characterizations	characterization	NOUN
ejpam-4572	17	7	of	of	ADP
ejpam-4572	17	8	upper	upper	ADJ
ejpam-4572	17	9	and	and	CCONJ
ejpam-4572	17	10	lower	low	ADJ
ejpam-4572	17	11	α	α	ADJ
ejpam-4572	17	12	-	-	ADJ
ejpam-4572	17	13	continuous	continuous	ADJ
ejpam-4572	17	14	multifunctions	multifunction	NOUN
ejpam-4572	17	15	.	.	PUNCT
ejpam-4572	18	1	moreover	moreover	ADV
ejpam-4572	18	2	,	,	PUNCT
ejpam-4572	18	3	popa	popa	NOUN
ejpam-4572	18	4	and	and	CCONJ
ejpam-4572	18	5	noiri	noiri	ADV
ejpam-4572	19	1	[	[	X
ejpam-4572	19	2	13	13	NUM
ejpam-4572	19	3	]	]	PUNCT
ejpam-4572	19	4	defined	define	VERB
ejpam-4572	19	5	a	a	DET
ejpam-4572	19	6	class	class	NOUN
ejpam-4572	19	7	of	of	ADP
ejpam-4572	19	8	multifunctions	multifunction	NOUN
ejpam-4572	19	9	called	call	VERB
ejpam-4572	19	10	weakly	weakly	ADJ
ejpam-4572	19	11	α	α	ADJ
ejpam-4572	19	12	-	-	ADJ
ejpam-4572	19	13	continuous	continuous	ADJ
ejpam-4572	19	14	multifunctions	multifunction	NOUN
ejpam-4572	19	15	.	.	PUNCT
ejpam-4572	20	1	cao	cao	PROPN
ejpam-4572	20	2	and	and	CCONJ
ejpam-4572	20	3	dontchev	dontchev	ADJ
ejpam-4572	20	4	[	[	X
ejpam-4572	20	5	4	4	NUM
ejpam-4572	20	6	]	]	PUNCT
ejpam-4572	20	7	and	and	CCONJ
ejpam-4572	20	8	popa	popa	NOUN
ejpam-4572	20	9	and	and	CCONJ
ejpam-4572	20	10	noiri	noiri	ADV
ejpam-4572	20	11	[	[	X
ejpam-4572	20	12	13	13	NUM
ejpam-4572	20	13	]	]	PUNCT
ejpam-4572	20	14	investigated	investigate	VERB
ejpam-4572	20	15	several	several	ADJ
ejpam-4572	20	16	characterizations	characterization	NOUN
ejpam-4572	20	17	of	of	ADP
ejpam-4572	20	18	weakly	weakly	ADJ
ejpam-4572	20	19	α	α	ADJ
ejpam-4572	20	20	-	-	ADJ
ejpam-4572	20	21	continuous	continuous	ADJ
ejpam-4572	20	22	multifunctions	multifunction	NOUN
ejpam-4572	20	23	.	.	PUNCT
ejpam-4572	21	1	in	in	ADP
ejpam-4572	21	2	[	[	X
ejpam-4572	21	3	3	3	NUM
ejpam-4572	21	4	]	]	PUNCT
ejpam-4572	21	5	,	,	PUNCT
ejpam-4572	21	6	the	the	DET
ejpam-4572	21	7	present	present	ADJ
ejpam-4572	21	8	authors	author	NOUN
ejpam-4572	21	9	introduced	introduce	VERB
ejpam-4572	21	10	and	and	CCONJ
ejpam-4572	21	11	studied	study	VERB
ejpam-4572	21	12	the	the	DET
ejpam-4572	21	13	notions	notion	NOUN
ejpam-4572	21	14	of	of	ADP
ejpam-4572	21	15	upper	upper	ADJ
ejpam-4572	21	16	and	and	CCONJ
ejpam-4572	21	17	lower	low	ADJ
ejpam-4572	21	18	(	(	PUNCT
ejpam-4572	21	19	τ1	τ1	NOUN
ejpam-4572	21	20	,	,	PUNCT
ejpam-4572	21	21	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-4572	21	22	multifunctions	multifunction	NOUN
ejpam-4572	21	23	.	.	PUNCT
ejpam-4572	22	1	viriyapong	viriyapong	PROPN
ejpam-4572	22	2	and	and	CCONJ
ejpam-4572	22	3	boonpok	boonpok	VERB
ejpam-4572	23	1	[	[	X
ejpam-4572	23	2	17	17	NUM
ejpam-4572	23	3	]	]	PUNCT
ejpam-4572	23	4	introduced	introduce	VERB
ejpam-4572	23	5	and	and	CCONJ
ejpam-4572	23	6	investigated	investigate	VERB
ejpam-4572	23	7	the	the	DET
ejpam-4572	23	8	concepts	concept	NOUN
ejpam-4572	23	9	of	of	ADP
ejpam-4572	23	10	upper	upper	ADJ
ejpam-4572	23	11	and	and	CCONJ
ejpam-4572	23	12	lower	low	ADJ
ejpam-4572	23	13	weakly	weakly	ADJ
ejpam-4572	23	14	(	(	PUNCT
ejpam-4572	23	15	τ1	τ1	NOUN
ejpam-4572	23	16	,	,	PUNCT
ejpam-4572	23	17	τ2)α	τ2)α	ADJ
ejpam-4572	23	18	-	-	PUNCT
ejpam-4572	23	19	continuous	continuous	ADJ
ejpam-4572	23	20	multifunctions	multifunction	NOUN
ejpam-4572	23	21	.	.	PUNCT
ejpam-4572	24	1	noiri	noiri	PROPN
ejpam-4572	24	2	and	and	CCONJ
ejpam-4572	24	3	hatir	hatir	NOUN
ejpam-4572	25	1	[	[	X
ejpam-4572	25	2	9	9	NUM
ejpam-4572	25	3	]	]	PUNCT
ejpam-4572	25	4	introduced	introduce	VERB
ejpam-4572	25	5	the	the	DET
ejpam-4572	25	6	notions	notion	NOUN
ejpam-4572	25	7	of	of	ADP
ejpam-4572	25	8	λsp	λsp	NOUN
ejpam-4572	25	9	-	-	PUNCT
ejpam-4572	25	10	closed	close	VERB
ejpam-4572	25	11	and	and	CCONJ
ejpam-4572	25	12	spg	spg	ADJ
ejpam-4572	25	13	-	-	PUNCT
ejpam-4572	25	14	closed	closed	ADJ
ejpam-4572	25	15	sets	set	NOUN
ejpam-4572	25	16	and	and	CCONJ
ejpam-4572	25	17	investigated	investigate	VERB
ejpam-4572	25	18	properties	property	NOUN
ejpam-4572	25	19	of	of	ADP
ejpam-4572	25	20	these	these	DET
ejpam-4572	25	21	sets	set	NOUN
ejpam-4572	25	22	.	.	PUNCT
ejpam-4572	26	1	in	in	ADP
ejpam-4572	26	2	[	[	X
ejpam-4572	26	3	2	2	NUM
ejpam-4572	26	4	]	]	PUNCT
ejpam-4572	26	5	by	by	ADP
ejpam-4572	26	6	considering	consider	VERB
ejpam-4572	26	7	the	the	DET
ejpam-4572	26	8	notion	notion	NOUN
ejpam-4572	26	9	of	of	ADP
ejpam-4572	26	10	λsp	λsp	NOUN
ejpam-4572	26	11	-	-	PUNCT
ejpam-4572	26	12	sets	set	NOUN
ejpam-4572	26	13	,	,	PUNCT
ejpam-4572	26	14	introduced	introduce	VERB
ejpam-4572	26	15	and	and	CCONJ
ejpam-4572	26	16	investigated	investigate	VERB
ejpam-4572	26	17	∗corresponding	∗corresponde	VERB
ejpam-4572	26	18	author	author	NOUN
ejpam-4572	26	19	.	.	PUNCT
ejpam-4572	27	1	doi	doi	NOUN
ejpam-4572	27	2	:	:	PUNCT
ejpam-4572	27	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4572	https://doi.org/10.29020/nybg.ejpam.v16i1.4572	NOUN
ejpam-4572	27	4	email	email	NOUN
ejpam-4572	27	5	addresses	address	NOUN
ejpam-4572	27	6	:	:	PUNCT
ejpam-4572	27	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4572	27	8	(	(	PUNCT
ejpam-4572	27	9	c.	c.	PROPN
ejpam-4572	27	10	boonpok	boonpok	PROPN
ejpam-4572	27	11	)	)	PUNCT
ejpam-4572	27	12	,	,	PUNCT
ejpam-4572	27	13	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-4572	27	14	(	(	PUNCT
ejpam-4572	27	15	m.	m.	NOUN
ejpam-4572	27	16	thongmoon	thongmoon	PROPN
ejpam-4572	27	17	)	)	PUNCT
ejpam-4572	27	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4572	28	1	465	465	NUM
ejpam-4572	29	1	©	©	ADP
ejpam-4572	29	2	2023	2023	NUM
ejpam-4572	29	3	ejpam	ejpam	NOUN
ejpam-4572	29	4	all	all	DET
ejpam-4572	29	5	rights	right	NOUN
ejpam-4572	29	6	reserved	reserve	VERB
ejpam-4572	29	7	.	.	PUNCT
ejpam-4572	30	1	c.	c.	PROPN
ejpam-4572	30	2	boonpok	boonpok	PROPN
ejpam-4572	30	3	,	,	PUNCT
ejpam-4572	30	4	m.	m.	NOUN
ejpam-4572	30	5	thongmoon	thongmoon	PROPN
ejpam-4572	30	6	/	/	SYM
ejpam-4572	30	7	eur	eur	PROPN
ejpam-4572	30	8	.	.	PUNCT
ejpam-4572	31	1	j.	j.	PROPN
ejpam-4572	31	2	pure	pure	PROPN
ejpam-4572	31	3	appl	appl	PROPN
ejpam-4572	31	4	.	.	PROPN
ejpam-4572	31	5	math	math	PROPN
ejpam-4572	31	6	,	,	PUNCT
ejpam-4572	31	7	16	16	NUM
ejpam-4572	31	8	(	(	PUNCT
ejpam-4572	31	9	1	1	NUM
ejpam-4572	31	10	)	)	PUNCT
ejpam-4572	31	11	(	(	PUNCT
ejpam-4572	31	12	2023	2023	NUM
ejpam-4572	31	13	)	)	PUNCT
ejpam-4572	31	14	,	,	PUNCT
ejpam-4572	31	15	465	465	NUM
ejpam-4572	31	16	-	-	SYM
ejpam-4572	31	17	478	478	NUM
ejpam-4572	31	18	466	466	NUM
ejpam-4572	31	19	(	(	PUNCT
ejpam-4572	31	20	λ	λ	PROPN
ejpam-4572	31	21	,	,	PUNCT
ejpam-4572	31	22	sp)-closed	sp)-close	VERB
ejpam-4572	31	23	sets	set	NOUN
ejpam-4572	31	24	,	,	PUNCT
ejpam-4572	31	25	(	(	PUNCT
ejpam-4572	31	26	λ	λ	NOUN
ejpam-4572	31	27	,	,	PUNCT
ejpam-4572	31	28	sp)-open	sp)-open	ADJ
ejpam-4572	31	29	sets	set	NOUN
ejpam-4572	31	30	and	and	CCONJ
ejpam-4572	31	31	the	the	DET
ejpam-4572	31	32	(	(	PUNCT
ejpam-4572	31	33	λ	λ	PROPN
ejpam-4572	31	34	,	,	PUNCT
ejpam-4572	31	35	sp)-closure	sp)-closure	NOUN
ejpam-4572	31	36	operator	operator	NOUN
ejpam-4572	31	37	.	.	PUNCT
ejpam-4572	32	1	the	the	DET
ejpam-4572	32	2	purpose	purpose	NOUN
ejpam-4572	32	3	of	of	ADP
ejpam-4572	32	4	the	the	DET
ejpam-4572	32	5	present	present	ADJ
ejpam-4572	32	6	paper	paper	NOUN
ejpam-4572	32	7	is	be	AUX
ejpam-4572	32	8	to	to	PART
ejpam-4572	32	9	introduce	introduce	VERB
ejpam-4572	32	10	the	the	DET
ejpam-4572	32	11	concepts	concept	NOUN
ejpam-4572	32	12	of	of	ADP
ejpam-4572	32	13	upper	upper	ADJ
ejpam-4572	32	14	and	and	CCONJ
ejpam-4572	32	15	lower	low	ADJ
ejpam-4572	32	16	weakly	weakly	ADJ
ejpam-4572	32	17	α(λ	α(λ	PROPN
ejpam-4572	32	18	,	,	PUNCT
ejpam-4572	32	19	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	32	20	multifunctions	multifunction	NOUN
ejpam-4572	32	21	.	.	PUNCT
ejpam-4572	33	1	moreover	moreover	ADV
ejpam-4572	33	2	,	,	PUNCT
ejpam-4572	33	3	several	several	ADJ
ejpam-4572	33	4	characterizations	characterization	NOUN
ejpam-4572	33	5	of	of	ADP
ejpam-4572	33	6	upper	upper	ADJ
ejpam-4572	33	7	and	and	CCONJ
ejpam-4572	33	8	lower	low	ADJ
ejpam-4572	33	9	weakly	weakly	ADJ
ejpam-4572	33	10	α(λ	α(λ	PROPN
ejpam-4572	33	11	,	,	PUNCT
ejpam-4572	33	12	sp)continuous	sp)continuous	ADJ
ejpam-4572	33	13	multifunctions	multifunction	NOUN
ejpam-4572	33	14	are	be	AUX
ejpam-4572	33	15	discussed	discuss	VERB
ejpam-4572	33	16	.	.	PUNCT
ejpam-4572	34	1	2	2	X
ejpam-4572	34	2	.	.	X
ejpam-4572	34	3	preliminaries	preliminary	NOUN
ejpam-4572	34	4	throughout	throughout	ADP
ejpam-4572	34	5	this	this	DET
ejpam-4572	34	6	paper	paper	NOUN
ejpam-4572	34	7	,	,	PUNCT
ejpam-4572	34	8	spaces	space	NOUN
ejpam-4572	34	9	(	(	PUNCT
ejpam-4572	34	10	x	x	X
ejpam-4572	34	11	,	,	PUNCT
ejpam-4572	34	12	τ	τ	X
ejpam-4572	34	13	)	)	PUNCT
ejpam-4572	34	14	and	and	CCONJ
ejpam-4572	34	15	(	(	PUNCT
ejpam-4572	34	16	y	y	PROPN
ejpam-4572	34	17	,	,	PUNCT
ejpam-4572	34	18	σ	σ	PROPN
ejpam-4572	34	19	)	)	PUNCT
ejpam-4572	34	20	(	(	PUNCT
ejpam-4572	34	21	or	or	CCONJ
ejpam-4572	34	22	simply	simply	ADV
ejpam-4572	34	23	x	x	X
ejpam-4572	34	24	and	and	CCONJ
ejpam-4572	34	25	y	y	PROPN
ejpam-4572	34	26	)	)	PUNCT
ejpam-4572	34	27	always	always	ADV
ejpam-4572	34	28	mean	mean	VERB
ejpam-4572	34	29	topological	topological	ADJ
ejpam-4572	34	30	spaces	space	NOUN
ejpam-4572	34	31	on	on	ADP
ejpam-4572	34	32	which	which	PRON
ejpam-4572	34	33	no	no	DET
ejpam-4572	34	34	separation	separation	NOUN
ejpam-4572	34	35	axioms	axiom	NOUN
ejpam-4572	34	36	are	be	AUX
ejpam-4572	34	37	assumed	assume	VERB
ejpam-4572	34	38	unless	unless	SCONJ
ejpam-4572	34	39	explicitly	explicitly	ADV
ejpam-4572	34	40	stated	state	VERB
ejpam-4572	34	41	.	.	PUNCT
ejpam-4572	35	1	let	let	VERB
ejpam-4572	35	2	a	a	DET
ejpam-4572	35	3	be	be	AUX
ejpam-4572	35	4	a	a	DET
ejpam-4572	35	5	subset	subset	NOUN
ejpam-4572	35	6	of	of	ADP
ejpam-4572	35	7	a	a	DET
ejpam-4572	35	8	topological	topological	ADJ
ejpam-4572	35	9	space	space	NOUN
ejpam-4572	35	10	(	(	PUNCT
ejpam-4572	35	11	x	x	X
ejpam-4572	35	12	,	,	PUNCT
ejpam-4572	35	13	τ	τ	PROPN
ejpam-4572	35	14	)	)	PUNCT
ejpam-4572	35	15	.	.	PUNCT
ejpam-4572	36	1	the	the	DET
ejpam-4572	36	2	closure	closure	NOUN
ejpam-4572	36	3	of	of	ADP
ejpam-4572	36	4	a	a	PRON
ejpam-4572	36	5	and	and	CCONJ
ejpam-4572	36	6	the	the	DET
ejpam-4572	36	7	interior	interior	NOUN
ejpam-4572	36	8	of	of	ADP
ejpam-4572	36	9	a	a	PRON
ejpam-4572	36	10	are	be	AUX
ejpam-4572	36	11	denoted	denote	VERB
ejpam-4572	36	12	by	by	ADP
ejpam-4572	36	13	cl(a	cl(a	NOUN
ejpam-4572	36	14	)	)	PUNCT
ejpam-4572	36	15	and	and	CCONJ
ejpam-4572	36	16	int(a	int(a	PROPN
ejpam-4572	36	17	)	)	PUNCT
ejpam-4572	36	18	,	,	PUNCT
ejpam-4572	36	19	respectively	respectively	ADV
ejpam-4572	36	20	.	.	PUNCT
ejpam-4572	37	1	a	a	DET
ejpam-4572	37	2	subset	subset	NOUN
ejpam-4572	37	3	a	a	PRON
ejpam-4572	37	4	of	of	ADP
ejpam-4572	37	5	a	a	DET
ejpam-4572	37	6	topological	topological	ADJ
ejpam-4572	37	7	space	space	NOUN
ejpam-4572	37	8	(	(	PUNCT
ejpam-4572	37	9	x	x	X
ejpam-4572	37	10	,	,	PUNCT
ejpam-4572	37	11	τ	τ	X
ejpam-4572	37	12	)	)	PUNCT
ejpam-4572	37	13	is	be	AUX
ejpam-4572	37	14	said	say	VERB
ejpam-4572	37	15	to	to	PART
ejpam-4572	37	16	be	be	AUX
ejpam-4572	37	17	β	β	X
ejpam-4572	37	18	-	-	ADJ
ejpam-4572	37	19	open	open	ADJ
ejpam-4572	37	20	[	[	X
ejpam-4572	37	21	5	5	NUM
ejpam-4572	37	22	]	]	PUNCT
ejpam-4572	37	23	if	if	SCONJ
ejpam-4572	37	24	a	a	DET
ejpam-4572	37	25	⊆	⊆	NUM
ejpam-4572	37	26	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4572	37	27	)	)	PUNCT
ejpam-4572	37	28	)	)	PUNCT
ejpam-4572	37	29	)	)	PUNCT
ejpam-4572	37	30	.	.	PUNCT
ejpam-4572	38	1	the	the	DET
ejpam-4572	38	2	complement	complement	NOUN
ejpam-4572	38	3	of	of	ADP
ejpam-4572	38	4	a	a	DET
ejpam-4572	38	5	β	β	X
ejpam-4572	38	6	-	-	ADJ
ejpam-4572	38	7	open	open	ADJ
ejpam-4572	38	8	set	set	NOUN
ejpam-4572	38	9	is	be	AUX
ejpam-4572	38	10	called	call	VERB
ejpam-4572	38	11	β	β	NOUN
ejpam-4572	38	12	-	-	VERB
ejpam-4572	38	13	closed	closed	ADJ
ejpam-4572	38	14	.	.	PUNCT
ejpam-4572	39	1	the	the	DET
ejpam-4572	39	2	family	family	NOUN
ejpam-4572	39	3	of	of	ADP
ejpam-4572	39	4	all	all	DET
ejpam-4572	39	5	β	β	ADJ
ejpam-4572	39	6	-	-	ADJ
ejpam-4572	39	7	open	open	ADJ
ejpam-4572	39	8	sets	set	NOUN
ejpam-4572	39	9	of	of	ADP
ejpam-4572	39	10	a	a	DET
ejpam-4572	39	11	topological	topological	ADJ
ejpam-4572	39	12	space	space	NOUN
ejpam-4572	39	13	(	(	PUNCT
ejpam-4572	39	14	x	x	X
ejpam-4572	39	15	,	,	PUNCT
ejpam-4572	39	16	τ	τ	X
ejpam-4572	39	17	)	)	PUNCT
ejpam-4572	39	18	is	be	AUX
ejpam-4572	39	19	denoted	denote	VERB
ejpam-4572	39	20	by	by	ADP
ejpam-4572	39	21	β(x	β(x	PROPN
ejpam-4572	39	22	,	,	PUNCT
ejpam-4572	39	23	τ	τ	PROPN
ejpam-4572	39	24	)	)	PUNCT
ejpam-4572	39	25	.	.	PUNCT
ejpam-4572	40	1	let	let	VERB
ejpam-4572	40	2	a	a	DET
ejpam-4572	40	3	be	be	AUX
ejpam-4572	40	4	a	a	DET
ejpam-4572	40	5	subset	subset	NOUN
ejpam-4572	40	6	of	of	ADP
ejpam-4572	40	7	a	a	DET
ejpam-4572	40	8	topological	topological	ADJ
ejpam-4572	40	9	space	space	NOUN
ejpam-4572	40	10	(	(	PUNCT
ejpam-4572	40	11	x	x	X
ejpam-4572	40	12	,	,	PUNCT
ejpam-4572	40	13	τ	τ	PROPN
ejpam-4572	40	14	)	)	PUNCT
ejpam-4572	40	15	.	.	PUNCT
ejpam-4572	41	1	a	a	DET
ejpam-4572	41	2	subset	subset	NOUN
ejpam-4572	41	3	λsp(a	λsp(a	NOUN
ejpam-4572	41	4	)	)	PUNCT
ejpam-4572	42	1	[	[	X
ejpam-4572	42	2	9	9	NUM
ejpam-4572	42	3	]	]	PUNCT
ejpam-4572	42	4	is	be	AUX
ejpam-4572	42	5	defined	define	VERB
ejpam-4572	42	6	as	as	SCONJ
ejpam-4572	42	7	follows	follow	VERB
ejpam-4572	42	8	:	:	PUNCT
ejpam-4572	42	9	λsp(a	λsp(a	NUM
ejpam-4572	42	10	)	)	PUNCT
ejpam-4572	42	11	=	=	PUNCT
ejpam-4572	43	1	∩{u	∩{u	PROPN
ejpam-4572	43	2	|	|	ADV
ejpam-4572	43	3	a	a	DET
ejpam-4572	43	4	⊆	⊆	NUM
ejpam-4572	43	5	u	u	NOUN
ejpam-4572	43	6	,	,	PUNCT
ejpam-4572	43	7	u	u	NOUN
ejpam-4572	43	8	∈	∈	PROPN
ejpam-4572	43	9	β(x	β(x	PROPN
ejpam-4572	43	10	,	,	PUNCT
ejpam-4572	43	11	τ	τ	X
ejpam-4572	43	12	)	)	PUNCT
ejpam-4572	43	13	}	}	PUNCT
ejpam-4572	43	14	.	.	PUNCT
ejpam-4572	44	1	lemma	lemma	PROPN
ejpam-4572	44	2	1	1	NUM
ejpam-4572	44	3	.	.	PUNCT
ejpam-4572	45	1	[	[	X
ejpam-4572	45	2	9	9	NUM
ejpam-4572	45	3	]	]	PUNCT
ejpam-4572	45	4	for	for	ADP
ejpam-4572	45	5	subsets	subset	NOUN
ejpam-4572	45	6	a	a	PRON
ejpam-4572	45	7	,	,	PUNCT
ejpam-4572	45	8	b	b	PROPN
ejpam-4572	45	9	and	and	CCONJ
ejpam-4572	45	10	aα(α	aα(α	NOUN
ejpam-4572	45	11	∈	∈	PROPN
ejpam-4572	45	12	∇	∇	NOUN
ejpam-4572	45	13	)	)	PUNCT
ejpam-4572	45	14	of	of	ADP
ejpam-4572	45	15	a	a	DET
ejpam-4572	45	16	topological	topological	ADJ
ejpam-4572	45	17	space	space	NOUN
ejpam-4572	45	18	(	(	PUNCT
ejpam-4572	45	19	x	x	X
ejpam-4572	45	20	,	,	PUNCT
ejpam-4572	45	21	τ	τ	PROPN
ejpam-4572	45	22	)	)	PUNCT
ejpam-4572	45	23	,	,	PUNCT
ejpam-4572	45	24	the	the	DET
ejpam-4572	45	25	following	follow	VERB
ejpam-4572	45	26	properties	property	NOUN
ejpam-4572	45	27	hold	hold	VERB
ejpam-4572	45	28	:	:	PUNCT
ejpam-4572	45	29	(	(	PUNCT
ejpam-4572	45	30	1	1	X
ejpam-4572	45	31	)	)	PUNCT
ejpam-4572	45	32	a	a	DET
ejpam-4572	45	33	⊆	⊆	NUM
ejpam-4572	45	34	λsp(a	λsp(a	NOUN
ejpam-4572	45	35	)	)	PUNCT
ejpam-4572	45	36	.	.	PUNCT
ejpam-4572	46	1	(	(	PUNCT
ejpam-4572	46	2	2	2	X
ejpam-4572	46	3	)	)	PUNCT
ejpam-4572	46	4	if	if	SCONJ
ejpam-4572	46	5	a	a	DET
ejpam-4572	46	6	⊆	⊆	NUM
ejpam-4572	46	7	b	b	NOUN
ejpam-4572	46	8	,	,	PUNCT
ejpam-4572	46	9	then	then	ADV
ejpam-4572	46	10	λsp(a	λsp(a	PROPN
ejpam-4572	46	11	)	)	PUNCT
ejpam-4572	46	12	⊆	⊆	NUM
ejpam-4572	46	13	λsp(b	λsp(b	PROPN
ejpam-4572	46	14	)	)	PUNCT
ejpam-4572	46	15	.	.	PUNCT
ejpam-4572	47	1	(	(	PUNCT
ejpam-4572	47	2	3	3	X
ejpam-4572	47	3	)	)	PUNCT
ejpam-4572	47	4	λsp(λsp(a	λsp(λsp(a	NUM
ejpam-4572	47	5	)	)	PUNCT
ejpam-4572	47	6	)	)	PUNCT
ejpam-4572	48	1	=	=	SYM
ejpam-4572	48	2	λsp(a	λsp(a	PROPN
ejpam-4572	48	3	)	)	PUNCT
ejpam-4572	48	4	.	.	PUNCT
ejpam-4572	49	1	(	(	PUNCT
ejpam-4572	49	2	4	4	X
ejpam-4572	49	3	)	)	PUNCT
ejpam-4572	49	4	if	if	SCONJ
ejpam-4572	49	5	u	u	PROPN
ejpam-4572	49	6	∈	∈	PROPN
ejpam-4572	49	7	β(x	β(x	PROPN
ejpam-4572	49	8	,	,	PUNCT
ejpam-4572	49	9	τ	τ	PROPN
ejpam-4572	49	10	)	)	PUNCT
ejpam-4572	49	11	,	,	PUNCT
ejpam-4572	49	12	then	then	ADV
ejpam-4572	49	13	λsp(u	λsp(u	X
ejpam-4572	49	14	)	)	PUNCT
ejpam-4572	49	15	=	=	SYM
ejpam-4572	49	16	u	u	NOUN
ejpam-4572	49	17	(	(	PUNCT
ejpam-4572	49	18	5	5	NUM
ejpam-4572	49	19	)	)	PUNCT
ejpam-4572	49	20	λsp(∩{aα|α	λsp(∩{aα|α	ADV
ejpam-4572	49	21	∈	∈	NOUN
ejpam-4572	49	22	∇	∇	NOUN
ejpam-4572	49	23	}	}	PUNCT
ejpam-4572	49	24	)	)	PUNCT
ejpam-4572	49	25	⊆	⊆	NUM
ejpam-4572	49	26	∩{λsp(aα)|α	∩{λsp(aα)|α	PROPN
ejpam-4572	49	27	∈	∈	NOUN
ejpam-4572	49	28	∇	∇	X
ejpam-4572	49	29	}	}	PUNCT
ejpam-4572	49	30	.	.	PUNCT
ejpam-4572	50	1	(	(	PUNCT
ejpam-4572	50	2	6	6	NUM
ejpam-4572	50	3	)	)	PUNCT
ejpam-4572	50	4	λsp(∪{aα|α	λsp(∪{aα|α	PROPN
ejpam-4572	50	5	∈	∈	PROPN
ejpam-4572	50	6	∇	∇	X
ejpam-4572	50	7	}	}	PUNCT
ejpam-4572	50	8	)	)	PUNCT
ejpam-4572	50	9	=	=	SYM
ejpam-4572	51	1	∪{λsp(aα)|α	∪{λsp(aα)|α	PROPN
ejpam-4572	51	2	∈	∈	NOUN
ejpam-4572	51	3	∇	∇	NOUN
ejpam-4572	51	4	}	}	PUNCT
ejpam-4572	51	5	.	.	PUNCT
ejpam-4572	52	1	a	a	DET
ejpam-4572	52	2	subset	subset	NOUN
ejpam-4572	52	3	a	a	PRON
ejpam-4572	52	4	of	of	ADP
ejpam-4572	52	5	a	a	DET
ejpam-4572	52	6	topological	topological	ADJ
ejpam-4572	52	7	space	space	NOUN
ejpam-4572	52	8	(	(	PUNCT
ejpam-4572	52	9	x	x	X
ejpam-4572	52	10	,	,	PUNCT
ejpam-4572	52	11	τ	τ	X
ejpam-4572	52	12	)	)	PUNCT
ejpam-4572	52	13	is	be	AUX
ejpam-4572	52	14	called	call	VERB
ejpam-4572	52	15	a	a	DET
ejpam-4572	52	16	λsp	λsp	NOUN
ejpam-4572	52	17	-	-	PUNCT
ejpam-4572	52	18	set	set	VERB
ejpam-4572	52	19	[	[	X
ejpam-4572	52	20	9	9	NUM
ejpam-4572	52	21	]	]	X
ejpam-4572	52	22	if	if	SCONJ
ejpam-4572	52	23	a	a	DET
ejpam-4572	52	24	=	=	NOUN
ejpam-4572	52	25	λsp(a	λsp(a	NOUN
ejpam-4572	52	26	)	)	PUNCT
ejpam-4572	52	27	.	.	PUNCT
ejpam-4572	53	1	lemma	lemma	PROPN
ejpam-4572	53	2	2	2	X
ejpam-4572	53	3	.	.	PUNCT
ejpam-4572	54	1	[	[	X
ejpam-4572	54	2	9	9	NUM
ejpam-4572	54	3	]	]	PUNCT
ejpam-4572	54	4	for	for	ADP
ejpam-4572	54	5	subsets	subset	NOUN
ejpam-4572	54	6	a	a	DET
ejpam-4572	54	7	and	and	CCONJ
ejpam-4572	54	8	aα(α	aα(α	NOUN
ejpam-4572	54	9	∈	∈	NOUN
ejpam-4572	54	10	∇	∇	NOUN
ejpam-4572	54	11	)	)	PUNCT
ejpam-4572	54	12	of	of	ADP
ejpam-4572	54	13	a	a	DET
ejpam-4572	54	14	topological	topological	ADJ
ejpam-4572	54	15	space	space	NOUN
ejpam-4572	54	16	(	(	PUNCT
ejpam-4572	54	17	x	x	X
ejpam-4572	54	18	,	,	PUNCT
ejpam-4572	54	19	τ	τ	PROPN
ejpam-4572	54	20	)	)	PUNCT
ejpam-4572	54	21	,	,	PUNCT
ejpam-4572	54	22	the	the	DET
ejpam-4572	54	23	following	follow	VERB
ejpam-4572	54	24	properties	property	NOUN
ejpam-4572	54	25	hold	hold	VERB
ejpam-4572	54	26	:	:	PUNCT
ejpam-4572	54	27	(	(	PUNCT
ejpam-4572	54	28	1	1	X
ejpam-4572	54	29	)	)	PUNCT
ejpam-4572	54	30	λsp(a	λsp(a	NOUN
ejpam-4572	54	31	)	)	PUNCT
ejpam-4572	54	32	is	be	AUX
ejpam-4572	54	33	a	a	DET
ejpam-4572	54	34	λsp	λsp	NOUN
ejpam-4572	54	35	-	-	PUNCT
ejpam-4572	54	36	set	set	NOUN
ejpam-4572	54	37	.	.	PUNCT
ejpam-4572	55	1	(	(	PUNCT
ejpam-4572	55	2	2	2	X
ejpam-4572	55	3	)	)	PUNCT
ejpam-4572	55	4	if	if	SCONJ
ejpam-4572	55	5	a	a	PRON
ejpam-4572	55	6	is	be	AUX
ejpam-4572	55	7	β	β	NOUN
ejpam-4572	55	8	-	-	ADJ
ejpam-4572	55	9	open	open	ADJ
ejpam-4572	55	10	,	,	PUNCT
ejpam-4572	55	11	then	then	ADV
ejpam-4572	55	12	a	a	PRON
ejpam-4572	55	13	is	be	AUX
ejpam-4572	55	14	a	a	DET
ejpam-4572	55	15	λsp	λsp	NOUN
ejpam-4572	55	16	-	-	PUNCT
ejpam-4572	55	17	set	set	NOUN
ejpam-4572	55	18	.	.	PUNCT
ejpam-4572	56	1	(	(	PUNCT
ejpam-4572	56	2	3	3	X
ejpam-4572	56	3	)	)	PUNCT
ejpam-4572	56	4	if	if	SCONJ
ejpam-4572	56	5	aα	aα	NOUN
ejpam-4572	56	6	is	be	AUX
ejpam-4572	56	7	a	a	DET
ejpam-4572	56	8	λsp	λsp	NOUN
ejpam-4572	56	9	-	-	PUNCT
ejpam-4572	56	10	set	set	VERB
ejpam-4572	56	11	for	for	ADP
ejpam-4572	56	12	each	each	DET
ejpam-4572	56	13	α	α	PROPN
ejpam-4572	56	14	∈	∈	PROPN
ejpam-4572	56	15	∇	∇	NOUN
ejpam-4572	56	16	,	,	PUNCT
ejpam-4572	56	17	then	then	ADV
ejpam-4572	56	18	∩α∈∇aα	∩α∈∇aα	PROPN
ejpam-4572	56	19	is	be	AUX
ejpam-4572	56	20	a	a	DET
ejpam-4572	56	21	λsp	λsp	NOUN
ejpam-4572	56	22	-	-	PUNCT
ejpam-4572	56	23	set	set	NOUN
ejpam-4572	56	24	.	.	PUNCT
ejpam-4572	57	1	(	(	PUNCT
ejpam-4572	57	2	4	4	X
ejpam-4572	57	3	)	)	PUNCT
ejpam-4572	57	4	if	if	SCONJ
ejpam-4572	57	5	aα	aα	NOUN
ejpam-4572	57	6	is	be	AUX
ejpam-4572	57	7	a	a	DET
ejpam-4572	57	8	λsp	λsp	NOUN
ejpam-4572	57	9	-	-	PUNCT
ejpam-4572	57	10	set	set	VERB
ejpam-4572	57	11	for	for	ADP
ejpam-4572	57	12	each	each	DET
ejpam-4572	57	13	α	α	PROPN
ejpam-4572	57	14	∈	∈	PROPN
ejpam-4572	57	15	∇	∇	NOUN
ejpam-4572	57	16	,	,	PUNCT
ejpam-4572	57	17	then	then	ADV
ejpam-4572	57	18	∪α∈∇aα	∪α∈∇aα	VERB
ejpam-4572	57	19	is	be	AUX
ejpam-4572	57	20	a	a	DET
ejpam-4572	57	21	λsp	λsp	NOUN
ejpam-4572	57	22	-	-	PUNCT
ejpam-4572	57	23	set	set	NOUN
ejpam-4572	57	24	.	.	PUNCT
ejpam-4572	58	1	a	a	DET
ejpam-4572	58	2	subset	subset	NOUN
ejpam-4572	58	3	a	a	PRON
ejpam-4572	58	4	of	of	ADP
ejpam-4572	58	5	a	a	DET
ejpam-4572	58	6	topological	topological	ADJ
ejpam-4572	58	7	space	space	NOUN
ejpam-4572	58	8	(	(	PUNCT
ejpam-4572	58	9	x	x	X
ejpam-4572	58	10	,	,	PUNCT
ejpam-4572	58	11	τ	τ	X
ejpam-4572	58	12	)	)	PUNCT
ejpam-4572	58	13	is	be	AUX
ejpam-4572	58	14	called	call	VERB
ejpam-4572	58	15	(	(	PUNCT
ejpam-4572	58	16	λ	λ	X
ejpam-4572	58	17	,	,	PUNCT
ejpam-4572	58	18	sp)-closed	sp)-close	VERB
ejpam-4572	58	19	[	[	PUNCT
ejpam-4572	58	20	2	2	X
ejpam-4572	58	21	]	]	X
ejpam-4572	58	22	if	if	SCONJ
ejpam-4572	58	23	a	a	DET
ejpam-4572	58	24	=	=	X
ejpam-4572	58	25	t	t	NOUN
ejpam-4572	58	26	∩c	∩c	NOUN
ejpam-4572	58	27	,	,	PUNCT
ejpam-4572	58	28	where	where	SCONJ
ejpam-4572	58	29	t	t	PROPN
ejpam-4572	58	30	is	be	AUX
ejpam-4572	58	31	a	a	DET
ejpam-4572	58	32	λsp	λsp	NOUN
ejpam-4572	58	33	-	-	PUNCT
ejpam-4572	58	34	set	set	VERB
ejpam-4572	58	35	and	and	CCONJ
ejpam-4572	58	36	c	c	NOUN
ejpam-4572	58	37	is	be	AUX
ejpam-4572	58	38	a	a	DET
ejpam-4572	58	39	β	β	NOUN
ejpam-4572	58	40	-	-	ADJ
ejpam-4572	58	41	closed	closed	ADJ
ejpam-4572	58	42	set	set	NOUN
ejpam-4572	58	43	.	.	PUNCT
ejpam-4572	59	1	the	the	DET
ejpam-4572	59	2	complement	complement	NOUN
ejpam-4572	59	3	of	of	ADP
ejpam-4572	59	4	a	a	DET
ejpam-4572	59	5	(	(	PUNCT
ejpam-4572	59	6	λ	λ	PROPN
ejpam-4572	59	7	,	,	PUNCT
ejpam-4572	59	8	sp)-closed	sp)-close	VERB
ejpam-4572	59	9	set	set	VERB
ejpam-4572	59	10	is	be	AUX
ejpam-4572	59	11	called	call	VERB
ejpam-4572	59	12	(	(	PUNCT
ejpam-4572	59	13	λ	λ	NOUN
ejpam-4572	59	14	,	,	PUNCT
ejpam-4572	59	15	sp)-open	sp)-open	NOUN
ejpam-4572	59	16	.	.	PUNCT
ejpam-4572	60	1	let	let	VERB
ejpam-4572	60	2	a	a	DET
ejpam-4572	60	3	be	be	AUX
ejpam-4572	60	4	a	a	DET
ejpam-4572	60	5	subset	subset	NOUN
ejpam-4572	60	6	of	of	ADP
ejpam-4572	60	7	a	a	DET
ejpam-4572	60	8	topological	topological	ADJ
ejpam-4572	60	9	space	space	NOUN
ejpam-4572	60	10	(	(	PUNCT
ejpam-4572	60	11	x	x	X
ejpam-4572	60	12	,	,	PUNCT
ejpam-4572	60	13	τ	τ	PROPN
ejpam-4572	60	14	)	)	PUNCT
ejpam-4572	60	15	.	.	PUNCT
ejpam-4572	61	1	a	a	DET
ejpam-4572	61	2	point	point	NOUN
ejpam-4572	61	3	x	x	X
ejpam-4572	61	4	∈	∈	NOUN
ejpam-4572	61	5	x	x	PUNCT
ejpam-4572	61	6	is	be	AUX
ejpam-4572	61	7	called	call	VERB
ejpam-4572	61	8	a	a	DET
ejpam-4572	61	9	(	(	PUNCT
ejpam-4572	61	10	λ	λ	NOUN
ejpam-4572	61	11	,	,	PUNCT
ejpam-4572	61	12	sp)-cluster	sp)-cluster	NOUN
ejpam-4572	61	13	point	point	NOUN
ejpam-4572	61	14	[	[	X
ejpam-4572	61	15	2	2	X
ejpam-4572	61	16	]	]	PUNCT
ejpam-4572	61	17	of	of	ADP
ejpam-4572	61	18	a	a	PRON
ejpam-4572	61	19	if	if	SCONJ
ejpam-4572	61	20	a	a	DET
ejpam-4572	61	21	∩	∩	ADJ
ejpam-4572	61	22	u	u	ADJ
ejpam-4572	61	23	̸=	̸=	PROPN
ejpam-4572	61	24	∅	∅	NOUN
ejpam-4572	61	25	for	for	ADP
ejpam-4572	61	26	every	every	DET
ejpam-4572	61	27	(	(	PUNCT
ejpam-4572	61	28	λ	λ	NOUN
ejpam-4572	61	29	,	,	PUNCT
ejpam-4572	61	30	sp)-open	sp)-open	NOUN
ejpam-4572	61	31	set	set	VERB
ejpam-4572	61	32	u	u	NOUN
ejpam-4572	61	33	of	of	ADP
ejpam-4572	61	34	x	x	SYM
ejpam-4572	61	35	containing	contain	VERB
ejpam-4572	61	36	x.	x.	NOUN
ejpam-4572	61	37	the	the	DET
ejpam-4572	61	38	set	set	NOUN
ejpam-4572	61	39	of	of	ADP
ejpam-4572	61	40	all	all	DET
ejpam-4572	61	41	(	(	PUNCT
ejpam-4572	61	42	λ	λ	PROPN
ejpam-4572	61	43	,	,	PUNCT
ejpam-4572	61	44	sp)-cluster	sp)-cluster	NOUN
ejpam-4572	61	45	points	point	NOUN
ejpam-4572	61	46	of	of	ADP
ejpam-4572	61	47	a	a	PRON
ejpam-4572	61	48	is	be	AUX
ejpam-4572	61	49	called	call	VERB
ejpam-4572	61	50	the	the	DET
ejpam-4572	61	51	(	(	PUNCT
ejpam-4572	61	52	λ	λ	PROPN
ejpam-4572	61	53	,	,	PUNCT
ejpam-4572	61	54	sp)-closure	sp)-closure	NOUN
ejpam-4572	61	55	[	[	X
ejpam-4572	61	56	2	2	NUM
ejpam-4572	61	57	]	]	PUNCT
ejpam-4572	61	58	of	of	ADP
ejpam-4572	61	59	a	a	PRON
ejpam-4572	61	60	and	and	CCONJ
ejpam-4572	61	61	is	be	AUX
ejpam-4572	61	62	denoted	denote	VERB
ejpam-4572	61	63	by	by	ADP
ejpam-4572	61	64	a(λ	a(λ	ADV
ejpam-4572	61	65	,	,	PUNCT
ejpam-4572	61	66	sp	sp	NOUN
ejpam-4572	61	67	)	)	PUNCT
ejpam-4572	61	68	.	.	PUNCT
ejpam-4572	62	1	c.	c.	PROPN
ejpam-4572	62	2	boonpok	boonpok	PROPN
ejpam-4572	62	3	,	,	PUNCT
ejpam-4572	62	4	m.	m.	NOUN
ejpam-4572	62	5	thongmoon	thongmoon	PROPN
ejpam-4572	62	6	/	/	SYM
ejpam-4572	62	7	eur	eur	PROPN
ejpam-4572	62	8	.	.	PUNCT
ejpam-4572	63	1	j.	j.	PROPN
ejpam-4572	63	2	pure	pure	PROPN
ejpam-4572	63	3	appl	appl	PROPN
ejpam-4572	63	4	.	.	PROPN
ejpam-4572	63	5	math	math	PROPN
ejpam-4572	63	6	,	,	PUNCT
ejpam-4572	63	7	16	16	NUM
ejpam-4572	63	8	(	(	PUNCT
ejpam-4572	63	9	1	1	NUM
ejpam-4572	63	10	)	)	PUNCT
ejpam-4572	63	11	(	(	PUNCT
ejpam-4572	63	12	2023	2023	NUM
ejpam-4572	63	13	)	)	PUNCT
ejpam-4572	63	14	,	,	PUNCT
ejpam-4572	63	15	465	465	NUM
ejpam-4572	63	16	-	-	SYM
ejpam-4572	63	17	478	478	NUM
ejpam-4572	63	18	467	467	NUM
ejpam-4572	63	19	lemma	lemma	PROPN
ejpam-4572	63	20	3	3	NUM
ejpam-4572	63	21	.	.	PUNCT
ejpam-4572	64	1	[	[	X
ejpam-4572	64	2	2	2	X
ejpam-4572	64	3	]	]	PUNCT
ejpam-4572	64	4	let	let	VERB
ejpam-4572	64	5	a	a	PRON
ejpam-4572	64	6	and	and	CCONJ
ejpam-4572	64	7	b	b	NOUN
ejpam-4572	64	8	be	be	AUX
ejpam-4572	64	9	subsets	subset	NOUN
ejpam-4572	64	10	of	of	ADP
ejpam-4572	64	11	a	a	DET
ejpam-4572	64	12	topological	topological	ADJ
ejpam-4572	64	13	space	space	NOUN
ejpam-4572	64	14	(	(	PUNCT
ejpam-4572	64	15	x	x	X
ejpam-4572	64	16	,	,	PUNCT
ejpam-4572	64	17	τ	τ	PROPN
ejpam-4572	64	18	)	)	PUNCT
ejpam-4572	64	19	.	.	PUNCT
ejpam-4572	65	1	for	for	ADP
ejpam-4572	65	2	the	the	DET
ejpam-4572	65	3	(	(	PUNCT
ejpam-4572	65	4	λ	λ	PROPN
ejpam-4572	65	5	,	,	PUNCT
ejpam-4572	65	6	sp)-closure	sp)-closure	NOUN
ejpam-4572	65	7	,	,	PUNCT
ejpam-4572	65	8	the	the	DET
ejpam-4572	65	9	following	follow	VERB
ejpam-4572	65	10	properties	property	NOUN
ejpam-4572	65	11	hold	hold	VERB
ejpam-4572	65	12	:	:	PUNCT
ejpam-4572	65	13	(	(	PUNCT
ejpam-4572	65	14	1	1	X
ejpam-4572	65	15	)	)	PUNCT
ejpam-4572	65	16	a	a	DET
ejpam-4572	65	17	⊆	⊆	NUM
ejpam-4572	65	18	a(λ	a(λ	ADJ
ejpam-4572	65	19	,	,	PUNCT
ejpam-4572	65	20	sp	sp	NOUN
ejpam-4572	65	21	)	)	PUNCT
ejpam-4572	65	22	and	and	CCONJ
ejpam-4572	65	23	[	[	X
ejpam-4572	65	24	a(λ	a(λ	ADV
ejpam-4572	65	25	,	,	PUNCT
ejpam-4572	65	26	sp)](λ	sp)](λ	PROPN
ejpam-4572	65	27	,	,	PUNCT
ejpam-4572	65	28	sp	sp	NOUN
ejpam-4572	65	29	)	)	PUNCT
ejpam-4572	65	30	=	=	PUNCT
ejpam-4572	65	31	a(λ	a(λ	ADV
ejpam-4572	65	32	,	,	PUNCT
ejpam-4572	65	33	sp	sp	NOUN
ejpam-4572	65	34	)	)	PUNCT
ejpam-4572	65	35	.	.	PUNCT
ejpam-4572	66	1	(	(	PUNCT
ejpam-4572	66	2	2	2	X
ejpam-4572	66	3	)	)	PUNCT
ejpam-4572	66	4	if	if	SCONJ
ejpam-4572	66	5	a	a	DET
ejpam-4572	66	6	⊆	⊆	NUM
ejpam-4572	66	7	b	b	NOUN
ejpam-4572	66	8	,	,	PUNCT
ejpam-4572	66	9	then	then	ADV
ejpam-4572	66	10	a(λ	a(λ	ADV
ejpam-4572	66	11	,	,	PUNCT
ejpam-4572	66	12	sp	sp	NOUN
ejpam-4572	66	13	)	)	PUNCT
ejpam-4572	66	14	⊆	⊆	NUM
ejpam-4572	66	15	b(λ	b(λ	NOUN
ejpam-4572	66	16	,	,	PUNCT
ejpam-4572	66	17	sp	sp	NOUN
ejpam-4572	66	18	)	)	PUNCT
ejpam-4572	66	19	.	.	PUNCT
ejpam-4572	67	1	(	(	PUNCT
ejpam-4572	67	2	3	3	X
ejpam-4572	67	3	)	)	PUNCT
ejpam-4572	67	4	a(λ	a(λ	ADV
ejpam-4572	67	5	,	,	PUNCT
ejpam-4572	67	6	sp	sp	NOUN
ejpam-4572	67	7	)	)	PUNCT
ejpam-4572	67	8	=	=	SYM
ejpam-4572	67	9	∩{f	∩{f	NOUN
ejpam-4572	67	10	|a	|a	VERB
ejpam-4572	67	11	⊆	⊆	NUM
ejpam-4572	67	12	f	f	PROPN
ejpam-4572	67	13	and	and	CCONJ
ejpam-4572	67	14	f	f	PROPN
ejpam-4572	67	15	is	be	AUX
ejpam-4572	67	16	(	(	PUNCT
ejpam-4572	67	17	λ	λ	X
ejpam-4572	67	18	,	,	PUNCT
ejpam-4572	67	19	sp)-closed	sp)-close	VERB
ejpam-4572	67	20	}	}	PUNCT
ejpam-4572	67	21	.	.	PUNCT
ejpam-4572	68	1	(	(	PUNCT
ejpam-4572	68	2	4	4	NUM
ejpam-4572	68	3	)	)	PUNCT
ejpam-4572	68	4	a(λ	a(λ	ADV
ejpam-4572	68	5	,	,	PUNCT
ejpam-4572	68	6	sp	sp	NOUN
ejpam-4572	68	7	)	)	PUNCT
ejpam-4572	68	8	is	be	AUX
ejpam-4572	68	9	(	(	PUNCT
ejpam-4572	68	10	λ	λ	X
ejpam-4572	68	11	,	,	PUNCT
ejpam-4572	68	12	sp)-closed	sp)-close	VERB
ejpam-4572	68	13	.	.	PUNCT
ejpam-4572	69	1	(	(	PUNCT
ejpam-4572	69	2	5	5	X
ejpam-4572	69	3	)	)	PUNCT
ejpam-4572	69	4	a	a	PRON
ejpam-4572	69	5	is	be	AUX
ejpam-4572	69	6	(	(	PUNCT
ejpam-4572	69	7	λ	λ	X
ejpam-4572	69	8	,	,	PUNCT
ejpam-4572	69	9	sp)-closed	sp)-close	VERB
ejpam-4572	69	10	if	if	SCONJ
ejpam-4572	69	11	and	and	CCONJ
ejpam-4572	69	12	only	only	ADV
ejpam-4572	69	13	if	if	SCONJ
ejpam-4572	69	14	a	a	DET
ejpam-4572	69	15	=	=	X
ejpam-4572	69	16	a(λ	a(λ	ADV
ejpam-4572	69	17	,	,	PUNCT
ejpam-4572	69	18	sp	sp	NOUN
ejpam-4572	69	19	)	)	PUNCT
ejpam-4572	69	20	.	.	PUNCT
ejpam-4572	70	1	the	the	DET
ejpam-4572	70	2	union	union	NOUN
ejpam-4572	70	3	of	of	ADP
ejpam-4572	70	4	all	all	DET
ejpam-4572	70	5	(	(	PUNCT
ejpam-4572	70	6	λ	λ	NOUN
ejpam-4572	70	7	,	,	PUNCT
ejpam-4572	70	8	sp)-open	sp)-open	ADJ
ejpam-4572	70	9	sets	set	NOUN
ejpam-4572	70	10	contained	contain	VERB
ejpam-4572	70	11	in	in	ADP
ejpam-4572	70	12	a	a	PRON
ejpam-4572	70	13	is	be	AUX
ejpam-4572	70	14	called	call	VERB
ejpam-4572	70	15	the	the	DET
ejpam-4572	70	16	(	(	PUNCT
ejpam-4572	70	17	λ	λ	PROPN
ejpam-4572	70	18	,	,	PUNCT
ejpam-4572	70	19	sp)-interior	sp)-interior	NOUN
ejpam-4572	70	20	[	[	X
ejpam-4572	70	21	2	2	NUM
ejpam-4572	70	22	]	]	PUNCT
ejpam-4572	70	23	of	of	ADP
ejpam-4572	70	24	a	a	PRON
ejpam-4572	70	25	and	and	CCONJ
ejpam-4572	70	26	is	be	AUX
ejpam-4572	70	27	denoted	denote	VERB
ejpam-4572	70	28	by	by	ADP
ejpam-4572	70	29	a(λ	a(λ	ADV
ejpam-4572	70	30	,	,	PUNCT
ejpam-4572	70	31	sp	sp	NOUN
ejpam-4572	70	32	)	)	PUNCT
ejpam-4572	70	33	.	.	PUNCT
ejpam-4572	71	1	lemma	lemma	PROPN
ejpam-4572	71	2	4	4	NUM
ejpam-4572	71	3	.	.	PUNCT
ejpam-4572	72	1	[	[	X
ejpam-4572	72	2	2	2	X
ejpam-4572	72	3	]	]	PUNCT
ejpam-4572	72	4	let	let	VERB
ejpam-4572	72	5	a	a	PRON
ejpam-4572	72	6	and	and	CCONJ
ejpam-4572	72	7	b	b	NOUN
ejpam-4572	72	8	be	be	AUX
ejpam-4572	72	9	subsets	subset	NOUN
ejpam-4572	72	10	of	of	ADP
ejpam-4572	72	11	a	a	DET
ejpam-4572	72	12	topological	topological	ADJ
ejpam-4572	72	13	space	space	NOUN
ejpam-4572	72	14	(	(	PUNCT
ejpam-4572	72	15	x	x	X
ejpam-4572	72	16	,	,	PUNCT
ejpam-4572	72	17	τ	τ	PROPN
ejpam-4572	72	18	)	)	PUNCT
ejpam-4572	72	19	.	.	PUNCT
ejpam-4572	73	1	for	for	ADP
ejpam-4572	73	2	the	the	DET
ejpam-4572	73	3	(	(	PUNCT
ejpam-4572	73	4	λ	λ	PROPN
ejpam-4572	73	5	,	,	PUNCT
ejpam-4572	73	6	sp)interior	sp)interior	PROPN
ejpam-4572	73	7	,	,	PUNCT
ejpam-4572	73	8	the	the	DET
ejpam-4572	73	9	following	follow	VERB
ejpam-4572	73	10	properties	property	NOUN
ejpam-4572	73	11	hold	hold	VERB
ejpam-4572	73	12	:	:	PUNCT
ejpam-4572	73	13	(	(	PUNCT
ejpam-4572	73	14	1	1	X
ejpam-4572	73	15	)	)	PUNCT
ejpam-4572	73	16	a(λ	a(λ	ADV
ejpam-4572	73	17	,	,	PUNCT
ejpam-4572	73	18	sp	sp	NOUN
ejpam-4572	73	19	)	)	PUNCT
ejpam-4572	73	20	⊆	⊆	NUM
ejpam-4572	73	21	a	a	DET
ejpam-4572	73	22	and	and	CCONJ
ejpam-4572	73	23	[	[	X
ejpam-4572	73	24	a(λ	a(λ	ADV
ejpam-4572	73	25	,	,	PUNCT
ejpam-4572	73	26	sp)](λ	sp)](λ	PROPN
ejpam-4572	73	27	,	,	PUNCT
ejpam-4572	73	28	sp	sp	NOUN
ejpam-4572	73	29	)	)	PUNCT
ejpam-4572	73	30	=	=	PUNCT
ejpam-4572	73	31	a(λ	a(λ	ADV
ejpam-4572	73	32	,	,	PUNCT
ejpam-4572	73	33	sp	sp	NOUN
ejpam-4572	73	34	)	)	PUNCT
ejpam-4572	73	35	.	.	PUNCT
ejpam-4572	74	1	(	(	PUNCT
ejpam-4572	74	2	2	2	X
ejpam-4572	74	3	)	)	PUNCT
ejpam-4572	74	4	if	if	SCONJ
ejpam-4572	74	5	a	a	DET
ejpam-4572	74	6	⊆	⊆	NUM
ejpam-4572	74	7	b	b	NOUN
ejpam-4572	74	8	,	,	PUNCT
ejpam-4572	74	9	then	then	ADV
ejpam-4572	74	10	a(λ	a(λ	ADV
ejpam-4572	74	11	,	,	PUNCT
ejpam-4572	74	12	sp	sp	NOUN
ejpam-4572	74	13	)	)	PUNCT
ejpam-4572	74	14	⊆	⊆	NUM
ejpam-4572	74	15	b(λ	b(λ	NOUN
ejpam-4572	74	16	,	,	PUNCT
ejpam-4572	74	17	sp	sp	NOUN
ejpam-4572	74	18	)	)	PUNCT
ejpam-4572	74	19	.	.	PUNCT
ejpam-4572	75	1	(	(	PUNCT
ejpam-4572	75	2	3	3	X
ejpam-4572	75	3	)	)	PUNCT
ejpam-4572	75	4	a(λ	a(λ	ADV
ejpam-4572	75	5	,	,	PUNCT
ejpam-4572	75	6	sp	sp	NOUN
ejpam-4572	75	7	)	)	PUNCT
ejpam-4572	75	8	is	be	AUX
ejpam-4572	75	9	(	(	PUNCT
ejpam-4572	75	10	λ	λ	INTJ
ejpam-4572	75	11	,	,	PUNCT
ejpam-4572	75	12	sp)-open	sp)-open	NOUN
ejpam-4572	75	13	.	.	PUNCT
ejpam-4572	76	1	(	(	PUNCT
ejpam-4572	76	2	4	4	X
ejpam-4572	76	3	)	)	PUNCT
ejpam-4572	76	4	a	a	DET
ejpam-4572	76	5	is	be	AUX
ejpam-4572	76	6	(	(	PUNCT
ejpam-4572	76	7	λ	λ	NOUN
ejpam-4572	76	8	,	,	PUNCT
ejpam-4572	76	9	sp)-open	sp)-open	ADJ
ejpam-4572	76	10	if	if	SCONJ
ejpam-4572	76	11	and	and	CCONJ
ejpam-4572	76	12	only	only	ADV
ejpam-4572	76	13	if	if	SCONJ
ejpam-4572	76	14	a(λ	a(λ	ADV
ejpam-4572	76	15	,	,	PUNCT
ejpam-4572	76	16	sp	sp	NOUN
ejpam-4572	76	17	)	)	PUNCT
ejpam-4572	76	18	=	=	SYM
ejpam-4572	76	19	a.	a.	NOUN
ejpam-4572	76	20	(	(	PUNCT
ejpam-4572	76	21	5	5	NUM
ejpam-4572	76	22	)	)	PUNCT
ejpam-4572	77	1	[	[	X
ejpam-4572	77	2	x	x	X
ejpam-4572	77	3	−a](λ	−a](λ	PROPN
ejpam-4572	77	4	,	,	PUNCT
ejpam-4572	77	5	sp	sp	NOUN
ejpam-4572	77	6	)	)	PUNCT
ejpam-4572	77	7	=	=	SYM
ejpam-4572	77	8	x	x	SYM
ejpam-4572	77	9	−a(λ	−a(λ	NOUN
ejpam-4572	77	10	,	,	PUNCT
ejpam-4572	77	11	sp	sp	NOUN
ejpam-4572	77	12	)	)	PUNCT
ejpam-4572	77	13	.	.	PUNCT
ejpam-4572	78	1	(	(	PUNCT
ejpam-4572	78	2	6	6	NUM
ejpam-4572	78	3	)	)	PUNCT
ejpam-4572	79	1	[	[	X
ejpam-4572	79	2	x	x	X
ejpam-4572	79	3	−a](λ	−a](λ	PROPN
ejpam-4572	79	4	,	,	PUNCT
ejpam-4572	79	5	sp	sp	NOUN
ejpam-4572	79	6	)	)	PUNCT
ejpam-4572	79	7	=	=	SYM
ejpam-4572	79	8	x	x	SYM
ejpam-4572	79	9	−a(λ	−a(λ	NOUN
ejpam-4572	79	10	,	,	PUNCT
ejpam-4572	79	11	sp	sp	NOUN
ejpam-4572	79	12	)	)	PUNCT
ejpam-4572	79	13	.	.	PUNCT
ejpam-4572	80	1	a	a	DET
ejpam-4572	80	2	subset	subset	NOUN
ejpam-4572	80	3	a	a	PRON
ejpam-4572	80	4	of	of	ADP
ejpam-4572	80	5	a	a	DET
ejpam-4572	80	6	topological	topological	ADJ
ejpam-4572	80	7	space	space	NOUN
ejpam-4572	80	8	(	(	PUNCT
ejpam-4572	80	9	x	x	X
ejpam-4572	80	10	,	,	PUNCT
ejpam-4572	80	11	τ	τ	X
ejpam-4572	80	12	)	)	PUNCT
ejpam-4572	80	13	is	be	AUX
ejpam-4572	80	14	called	call	VERB
ejpam-4572	80	15	α(λ	α(λ	PROPN
ejpam-4572	80	16	,	,	PUNCT
ejpam-4572	80	17	sp)-open	sp)-open	ADJ
ejpam-4572	80	18	(	(	PUNCT
ejpam-4572	80	19	resp	resp	NOUN
ejpam-4572	80	20	.	.	PUNCT
ejpam-4572	81	1	s(λ	s(λ	NOUN
ejpam-4572	81	2	,	,	PUNCT
ejpam-4572	81	3	sp)-open	sp)-open	NOUN
ejpam-4572	81	4	)	)	PUNCT
ejpam-4572	81	5	if	if	SCONJ
ejpam-4572	81	6	a	a	DET
ejpam-4572	81	7	⊆	⊆	NUM
ejpam-4572	81	8	[	[	X
ejpam-4572	81	9	[	[	X
ejpam-4572	81	10	a(λ	a(λ	ADJ
ejpam-4572	81	11	,	,	PUNCT
ejpam-4572	81	12	sp	sp	NOUN
ejpam-4572	81	13	)	)	PUNCT
ejpam-4572	81	14	]	]	PUNCT
ejpam-4572	81	15	(	(	PUNCT
ejpam-4572	81	16	λ	λ	X
ejpam-4572	81	17	,	,	PUNCT
ejpam-4572	81	18	sp)](λ	sp)](λ	PROPN
ejpam-4572	81	19	,	,	PUNCT
ejpam-4572	81	20	sp	sp	NOUN
ejpam-4572	81	21	)	)	PUNCT
ejpam-4572	81	22	(	(	PUNCT
ejpam-4572	81	23	resp	resp	NOUN
ejpam-4572	81	24	.	.	PUNCT
ejpam-4572	82	1	a	a	DET
ejpam-4572	82	2	⊆	⊆	NUM
ejpam-4572	82	3	[	[	X
ejpam-4572	82	4	a(λ	a(λ	ADV
ejpam-4572	82	5	,	,	PUNCT
ejpam-4572	82	6	sp	sp	NOUN
ejpam-4572	82	7	)	)	PUNCT
ejpam-4572	82	8	]	]	PUNCT
ejpam-4572	83	1	(	(	PUNCT
ejpam-4572	83	2	λ	λ	NOUN
ejpam-4572	83	3	,	,	PUNCT
ejpam-4572	83	4	sp	sp	NOUN
ejpam-4572	83	5	)	)	PUNCT
ejpam-4572	83	6	)	)	PUNCT
ejpam-4572	84	1	[	[	X
ejpam-4572	84	2	2	2	NUM
ejpam-4572	84	3	]	]	PUNCT
ejpam-4572	84	4	.	.	PUNCT
ejpam-4572	85	1	the	the	DET
ejpam-4572	85	2	complement	complement	NOUN
ejpam-4572	85	3	of	of	ADP
ejpam-4572	85	4	an	an	DET
ejpam-4572	85	5	α(λ	α(λ	PROPN
ejpam-4572	85	6	,	,	PUNCT
ejpam-4572	85	7	sp)open	sp)open	NOUN
ejpam-4572	85	8	(	(	PUNCT
ejpam-4572	85	9	resp	resp	NOUN
ejpam-4572	85	10	.	.	PUNCT
ejpam-4572	86	1	s(λ	s(λ	NOUN
ejpam-4572	86	2	,	,	PUNCT
ejpam-4572	86	3	sp)-open	sp)-open	NOUN
ejpam-4572	86	4	)	)	PUNCT
ejpam-4572	86	5	set	set	NOUN
ejpam-4572	86	6	is	be	AUX
ejpam-4572	86	7	called	call	VERB
ejpam-4572	86	8	α(λ	α(λ	PROPN
ejpam-4572	86	9	,	,	PUNCT
ejpam-4572	86	10	sp)-closed	sp)-close	VERB
ejpam-4572	86	11	(	(	PUNCT
ejpam-4572	86	12	resp	resp	NOUN
ejpam-4572	86	13	.	.	PUNCT
ejpam-4572	87	1	s(λ	s(λ	NOUN
ejpam-4572	87	2	,	,	PUNCT
ejpam-4572	87	3	sp)-closed	sp)-close	VERB
ejpam-4572	87	4	)	)	PUNCT
ejpam-4572	87	5	.	.	PUNCT
ejpam-4572	88	1	the	the	DET
ejpam-4572	88	2	family	family	NOUN
ejpam-4572	88	3	of	of	ADP
ejpam-4572	88	4	all	all	DET
ejpam-4572	88	5	α(λ	α(λ	PROPN
ejpam-4572	88	6	,	,	PUNCT
ejpam-4572	88	7	sp)-open	sp)-open	ADJ
ejpam-4572	88	8	(	(	PUNCT
ejpam-4572	88	9	resp	resp	NOUN
ejpam-4572	88	10	.	.	PUNCT
ejpam-4572	89	1	s(λ	s(λ	NOUN
ejpam-4572	89	2	,	,	PUNCT
ejpam-4572	89	3	sp)-open	sp)-open	NOUN
ejpam-4572	89	4	)	)	PUNCT
ejpam-4572	89	5	sets	set	NOUN
ejpam-4572	89	6	in	in	ADP
ejpam-4572	89	7	a	a	DET
ejpam-4572	89	8	topological	topological	ADJ
ejpam-4572	89	9	space	space	NOUN
ejpam-4572	89	10	(	(	PUNCT
ejpam-4572	89	11	x	x	X
ejpam-4572	89	12	,	,	PUNCT
ejpam-4572	89	13	τ	τ	X
ejpam-4572	89	14	)	)	PUNCT
ejpam-4572	89	15	is	be	AUX
ejpam-4572	89	16	denoted	denote	VERB
ejpam-4572	89	17	by	by	ADP
ejpam-4572	89	18	αλspo(x	αλspo(x	NOUN
ejpam-4572	89	19	,	,	PUNCT
ejpam-4572	89	20	τ	τ	PROPN
ejpam-4572	89	21	)	)	PUNCT
ejpam-4572	89	22	(	(	PUNCT
ejpam-4572	89	23	resp	resp	NOUN
ejpam-4572	89	24	.	.	PUNCT
ejpam-4572	90	1	sλspo(x	sλspo(x	PROPN
ejpam-4572	90	2	,	,	PUNCT
ejpam-4572	90	3	τ	τ	PROPN
ejpam-4572	90	4	)	)	PUNCT
ejpam-4572	90	5	)	)	PUNCT
ejpam-4572	90	6	.	.	PUNCT
ejpam-4572	91	1	let	let	VERB
ejpam-4572	91	2	a	a	DET
ejpam-4572	91	3	be	be	AUX
ejpam-4572	91	4	a	a	DET
ejpam-4572	91	5	subset	subset	NOUN
ejpam-4572	91	6	of	of	ADP
ejpam-4572	91	7	a	a	DET
ejpam-4572	91	8	topological	topological	ADJ
ejpam-4572	91	9	space	space	NOUN
ejpam-4572	91	10	(	(	PUNCT
ejpam-4572	91	11	x	x	X
ejpam-4572	91	12	,	,	PUNCT
ejpam-4572	91	13	τ	τ	PROPN
ejpam-4572	91	14	)	)	PUNCT
ejpam-4572	91	15	.	.	PUNCT
ejpam-4572	92	1	the	the	DET
ejpam-4572	92	2	intersection	intersection	NOUN
ejpam-4572	92	3	of	of	ADP
ejpam-4572	92	4	all	all	DET
ejpam-4572	92	5	α(λ	α(λ	PROPN
ejpam-4572	92	6	,	,	PUNCT
ejpam-4572	92	7	sp)-closed	sp)-close	VERB
ejpam-4572	92	8	(	(	PUNCT
ejpam-4572	92	9	resp	resp	NOUN
ejpam-4572	92	10	.	.	PUNCT
ejpam-4572	93	1	s(λ	s(λ	NOUN
ejpam-4572	93	2	,	,	PUNCT
ejpam-4572	93	3	sp)-closed	sp)-closed	ADJ
ejpam-4572	93	4	)	)	PUNCT
ejpam-4572	93	5	sets	set	NOUN
ejpam-4572	93	6	of	of	ADP
ejpam-4572	93	7	x	x	PUNCT
ejpam-4572	93	8	containing	contain	VERB
ejpam-4572	93	9	a	a	PRON
ejpam-4572	93	10	is	be	AUX
ejpam-4572	93	11	called	call	VERB
ejpam-4572	93	12	the	the	DET
ejpam-4572	93	13	α(λ	α(λ	PROPN
ejpam-4572	93	14	,	,	PUNCT
ejpam-4572	93	15	sp)-closure	sp)-closure	NOUN
ejpam-4572	93	16	(	(	PUNCT
ejpam-4572	93	17	resp	resp	NOUN
ejpam-4572	93	18	.	.	PUNCT
ejpam-4572	94	1	s(λ	s(λ	NOUN
ejpam-4572	94	2	,	,	PUNCT
ejpam-4572	94	3	sp)-closure	sp)-closure	NOUN
ejpam-4572	94	4	)	)	PUNCT
ejpam-4572	94	5	of	of	ADP
ejpam-4572	94	6	a	a	PRON
ejpam-4572	94	7	and	and	CCONJ
ejpam-4572	94	8	is	be	AUX
ejpam-4572	94	9	denoted	denote	VERB
ejpam-4572	94	10	by	by	ADP
ejpam-4572	94	11	aα(λ	aα(λ	PROPN
ejpam-4572	94	12	,	,	PUNCT
ejpam-4572	94	13	sp	sp	NOUN
ejpam-4572	94	14	)	)	PUNCT
ejpam-4572	94	15	(	(	PUNCT
ejpam-4572	94	16	resp	resp	NOUN
ejpam-4572	94	17	.	.	PUNCT
ejpam-4572	95	1	as(λ	as(λ	PROPN
ejpam-4572	95	2	,	,	PUNCT
ejpam-4572	95	3	sp	sp	NOUN
ejpam-4572	95	4	)	)	PUNCT
ejpam-4572	95	5	)	)	PUNCT
ejpam-4572	95	6	.	.	PUNCT
ejpam-4572	96	1	the	the	DET
ejpam-4572	96	2	union	union	NOUN
ejpam-4572	96	3	of	of	ADP
ejpam-4572	96	4	all	all	DET
ejpam-4572	96	5	α(λ	α(λ	PROPN
ejpam-4572	96	6	,	,	PUNCT
ejpam-4572	96	7	sp)-open	sp)-open	ADJ
ejpam-4572	96	8	(	(	PUNCT
ejpam-4572	96	9	resp	resp	NOUN
ejpam-4572	96	10	.	.	PUNCT
ejpam-4572	97	1	s(λ	s(λ	NOUN
ejpam-4572	97	2	,	,	PUNCT
ejpam-4572	97	3	sp)-open	sp)-open	NOUN
ejpam-4572	97	4	)	)	PUNCT
ejpam-4572	97	5	sets	set	NOUN
ejpam-4572	97	6	of	of	ADP
ejpam-4572	97	7	x	x	PUNCT
ejpam-4572	97	8	contained	contain	VERB
ejpam-4572	97	9	in	in	ADP
ejpam-4572	97	10	a	a	PRON
ejpam-4572	97	11	is	be	AUX
ejpam-4572	97	12	called	call	VERB
ejpam-4572	97	13	the	the	DET
ejpam-4572	97	14	α(λ	α(λ	PROPN
ejpam-4572	97	15	,	,	PUNCT
ejpam-4572	97	16	sp)-interior	sp)-interior	NOUN
ejpam-4572	97	17	(	(	PUNCT
ejpam-4572	97	18	resp	resp	NOUN
ejpam-4572	97	19	.	.	PUNCT
ejpam-4572	98	1	s(λ	s(λ	NOUN
ejpam-4572	98	2	,	,	PUNCT
ejpam-4572	98	3	sp)-interior	sp)-interior	NOUN
ejpam-4572	98	4	)	)	PUNCT
ejpam-4572	98	5	of	of	ADP
ejpam-4572	98	6	a	a	PRON
ejpam-4572	98	7	and	and	CCONJ
ejpam-4572	98	8	is	be	AUX
ejpam-4572	98	9	denoted	denote	VERB
ejpam-4572	98	10	by	by	ADP
ejpam-4572	98	11	aα(λ	aα(λ	PROPN
ejpam-4572	98	12	,	,	PUNCT
ejpam-4572	98	13	sp	sp	NOUN
ejpam-4572	98	14	)	)	PUNCT
ejpam-4572	98	15	(	(	PUNCT
ejpam-4572	98	16	resp	resp	NOUN
ejpam-4572	98	17	.	.	PUNCT
ejpam-4572	99	1	as(λ	as(λ	PROPN
ejpam-4572	99	2	,	,	PUNCT
ejpam-4572	99	3	sp	sp	NOUN
ejpam-4572	99	4	)	)	PUNCT
ejpam-4572	99	5	)	)	PUNCT
ejpam-4572	99	6	.	.	PUNCT
ejpam-4572	100	1	lemma	lemma	PROPN
ejpam-4572	100	2	5	5	X
ejpam-4572	100	3	.	.	PUNCT
ejpam-4572	101	1	let	let	VERB
ejpam-4572	101	2	a	a	DET
ejpam-4572	101	3	be	be	AUX
ejpam-4572	101	4	a	a	DET
ejpam-4572	101	5	subset	subset	NOUN
ejpam-4572	101	6	of	of	ADP
ejpam-4572	101	7	a	a	DET
ejpam-4572	101	8	topological	topological	ADJ
ejpam-4572	101	9	space	space	NOUN
ejpam-4572	101	10	(	(	PUNCT
ejpam-4572	101	11	x	x	X
ejpam-4572	101	12	,	,	PUNCT
ejpam-4572	101	13	τ	τ	PROPN
ejpam-4572	101	14	)	)	PUNCT
ejpam-4572	101	15	.	.	PUNCT
ejpam-4572	102	1	then	then	ADV
ejpam-4572	102	2	,	,	PUNCT
ejpam-4572	102	3	x	x	PUNCT
ejpam-4572	102	4	∈	∈	NOUN
ejpam-4572	102	5	as(λ	as(λ	NOUN
ejpam-4572	102	6	,	,	PUNCT
ejpam-4572	102	7	sp	sp	NOUN
ejpam-4572	102	8	)	)	PUNCT
ejpam-4572	102	9	if	if	SCONJ
ejpam-4572	102	10	and	and	CCONJ
ejpam-4572	102	11	only	only	ADV
ejpam-4572	102	12	if	if	SCONJ
ejpam-4572	102	13	u	u	PROPN
ejpam-4572	102	14	∩a	∩a	PROPN
ejpam-4572	102	15	̸=	̸=	PROPN
ejpam-4572	102	16	∅	∅	NOUN
ejpam-4572	102	17	for	for	ADP
ejpam-4572	102	18	every	every	DET
ejpam-4572	102	19	u	u	PROPN
ejpam-4572	102	20	∈	∈	PROPN
ejpam-4572	102	21	sλspo(x	sλspo(x	PROPN
ejpam-4572	102	22	,	,	PUNCT
ejpam-4572	102	23	τ	τ	X
ejpam-4572	102	24	)	)	PUNCT
ejpam-4572	102	25	containing	contain	VERB
ejpam-4572	102	26	x.	x.	NOUN
ejpam-4572	102	27	lemma	lemma	PROPN
ejpam-4572	102	28	6	6	NUM
ejpam-4572	102	29	.	.	PUNCT
ejpam-4572	102	30	for	for	ADP
ejpam-4572	102	31	a	a	DET
ejpam-4572	102	32	subset	subset	NOUN
ejpam-4572	102	33	a	a	PRON
ejpam-4572	102	34	of	of	ADP
ejpam-4572	102	35	a	a	DET
ejpam-4572	102	36	topological	topological	ADJ
ejpam-4572	102	37	space	space	NOUN
ejpam-4572	102	38	(	(	PUNCT
ejpam-4572	102	39	x	x	X
ejpam-4572	102	40	,	,	PUNCT
ejpam-4572	102	41	τ	τ	PROPN
ejpam-4572	102	42	)	)	PUNCT
ejpam-4572	102	43	,	,	PUNCT
ejpam-4572	102	44	the	the	DET
ejpam-4572	102	45	following	follow	VERB
ejpam-4572	102	46	properties	property	NOUN
ejpam-4572	102	47	hold	hold	VERB
ejpam-4572	102	48	:	:	PUNCT
ejpam-4572	102	49	(	(	PUNCT
ejpam-4572	102	50	1	1	X
ejpam-4572	102	51	)	)	PUNCT
ejpam-4572	102	52	if	if	SCONJ
ejpam-4572	102	53	a	a	PRON
ejpam-4572	102	54	is	be	AUX
ejpam-4572	102	55	(	(	PUNCT
ejpam-4572	102	56	λ	λ	NOUN
ejpam-4572	102	57	,	,	PUNCT
ejpam-4572	102	58	sp)-open	sp)-open	ADJ
ejpam-4572	102	59	in	in	ADP
ejpam-4572	102	60	x	x	NOUN
ejpam-4572	102	61	,	,	PUNCT
ejpam-4572	102	62	then	then	ADV
ejpam-4572	102	63	as(λ	as(λ	NUM
ejpam-4572	102	64	,	,	PUNCT
ejpam-4572	102	65	sp	sp	NOUN
ejpam-4572	102	66	)	)	PUNCT
ejpam-4572	102	67	=	=	PUNCT
ejpam-4572	103	1	[	[	X
ejpam-4572	103	2	a(λ	a(λ	ADV
ejpam-4572	103	3	,	,	PUNCT
ejpam-4572	103	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	103	5	,	,	PUNCT
ejpam-4572	103	6	sp	sp	NOUN
ejpam-4572	103	7	)	)	PUNCT
ejpam-4572	103	8	.	.	PUNCT
ejpam-4572	104	1	c.	c.	PROPN
ejpam-4572	104	2	boonpok	boonpok	PROPN
ejpam-4572	104	3	,	,	PUNCT
ejpam-4572	104	4	m.	m.	NOUN
ejpam-4572	104	5	thongmoon	thongmoon	PROPN
ejpam-4572	104	6	/	/	SYM
ejpam-4572	104	7	eur	eur	PROPN
ejpam-4572	104	8	.	.	PUNCT
ejpam-4572	105	1	j.	j.	PROPN
ejpam-4572	105	2	pure	pure	PROPN
ejpam-4572	105	3	appl	appl	PROPN
ejpam-4572	105	4	.	.	PROPN
ejpam-4572	105	5	math	math	PROPN
ejpam-4572	105	6	,	,	PUNCT
ejpam-4572	105	7	16	16	NUM
ejpam-4572	105	8	(	(	PUNCT
ejpam-4572	105	9	1	1	NUM
ejpam-4572	105	10	)	)	PUNCT
ejpam-4572	105	11	(	(	PUNCT
ejpam-4572	105	12	2023	2023	NUM
ejpam-4572	105	13	)	)	PUNCT
ejpam-4572	105	14	,	,	PUNCT
ejpam-4572	105	15	465	465	NUM
ejpam-4572	105	16	-	-	SYM
ejpam-4572	105	17	478	478	NUM
ejpam-4572	105	18	468	468	NUM
ejpam-4572	105	19	(	(	PUNCT
ejpam-4572	105	20	2	2	NUM
ejpam-4572	105	21	)	)	PUNCT
ejpam-4572	105	22	a	a	PRON
ejpam-4572	105	23	is	be	AUX
ejpam-4572	105	24	α(λ	α(λ	PROPN
ejpam-4572	105	25	,	,	PUNCT
ejpam-4572	105	26	sp)-open	sp)-open	VERB
ejpam-4572	105	27	in	in	ADP
ejpam-4572	105	28	x	x	PUNCT
ejpam-4572	105	29	if	if	SCONJ
ejpam-4572	106	1	and	and	CCONJ
ejpam-4572	106	2	only	only	ADV
ejpam-4572	106	3	if	if	SCONJ
ejpam-4572	106	4	u	u	PROPN
ejpam-4572	106	5	⊆	⊆	NUM
ejpam-4572	106	6	a	a	DET
ejpam-4572	106	7	⊆	⊆	NUM
ejpam-4572	106	8	u	u	SYM
ejpam-4572	106	9	s(λ	s(λ	PROPN
ejpam-4572	106	10	,	,	PUNCT
ejpam-4572	106	11	sp	sp	NOUN
ejpam-4572	106	12	)	)	PUNCT
ejpam-4572	106	13	for	for	ADP
ejpam-4572	106	14	some	some	DET
ejpam-4572	106	15	(	(	PUNCT
ejpam-4572	106	16	λ	λ	NOUN
ejpam-4572	106	17	,	,	PUNCT
ejpam-4572	106	18	sp)-open	sp)-open	NOUN
ejpam-4572	106	19	set	set	VERB
ejpam-4572	106	20	u	u	NOUN
ejpam-4572	106	21	of	of	ADP
ejpam-4572	106	22	x.	x.	PROPN
ejpam-4572	106	23	(	(	PUNCT
ejpam-4572	106	24	3	3	NUM
ejpam-4572	106	25	)	)	PUNCT
ejpam-4572	106	26	aα(λ	aα(λ	NOUN
ejpam-4572	106	27	,	,	PUNCT
ejpam-4572	106	28	sp	sp	NOUN
ejpam-4572	106	29	)	)	PUNCT
ejpam-4572	106	30	=	=	NOUN
ejpam-4572	106	31	a	a	DET
ejpam-4572	106	32	∪	∪	ADJ
ejpam-4572	106	33	[	[	X
ejpam-4572	106	34	[	[	X
ejpam-4572	106	35	a(λ	a(λ	ADJ
ejpam-4572	106	36	,	,	PUNCT
ejpam-4572	106	37	sp)](λ	sp)](λ	PROPN
ejpam-4572	106	38	,	,	PUNCT
ejpam-4572	106	39	sp	sp	NOUN
ejpam-4572	106	40	)	)	PUNCT
ejpam-4572	106	41	]	]	PUNCT
ejpam-4572	107	1	(	(	PUNCT
ejpam-4572	107	2	λ	λ	NOUN
ejpam-4572	107	3	,	,	PUNCT
ejpam-4572	107	4	sp	sp	NOUN
ejpam-4572	107	5	)	)	PUNCT
ejpam-4572	107	6	.	.	PUNCT
ejpam-4572	108	1	proof	proof	NOUN
ejpam-4572	108	2	.	.	PUNCT
ejpam-4572	109	1	(	(	PUNCT
ejpam-4572	109	2	1	1	X
ejpam-4572	109	3	)	)	PUNCT
ejpam-4572	109	4	let	let	VERB
ejpam-4572	109	5	x	x	PUNCT
ejpam-4572	109	6	∈	∈	PROPN
ejpam-4572	109	7	[	[	X
ejpam-4572	109	8	a(λ	a(λ	PROPN
ejpam-4572	109	9	,	,	PUNCT
ejpam-4572	109	10	sp)](λ	sp)](λ	PROPN
ejpam-4572	109	11	,	,	PUNCT
ejpam-4572	109	12	sp	sp	NOUN
ejpam-4572	109	13	)	)	PUNCT
ejpam-4572	109	14	and	and	CCONJ
ejpam-4572	109	15	g	g	PROPN
ejpam-4572	109	16	be	be	AUX
ejpam-4572	109	17	any	any	DET
ejpam-4572	109	18	s(λ	s(λ	NOUN
ejpam-4572	109	19	,	,	PUNCT
ejpam-4572	109	20	sp)-open	sp)-open	ADJ
ejpam-4572	109	21	set	set	NOUN
ejpam-4572	109	22	of	of	ADP
ejpam-4572	109	23	x	x	PUNCT
ejpam-4572	109	24	containing	contain	VERB
ejpam-4572	109	25	x.	x.	NOUN
ejpam-4572	109	26	then	then	ADV
ejpam-4572	109	27	,	,	PUNCT
ejpam-4572	109	28	there	there	PRON
ejpam-4572	109	29	exists	exist	VERB
ejpam-4572	109	30	a	a	DET
ejpam-4572	109	31	(	(	PUNCT
ejpam-4572	109	32	λ	λ	NOUN
ejpam-4572	109	33	,	,	PUNCT
ejpam-4572	109	34	sp)-open	sp)-open	NOUN
ejpam-4572	109	35	set	set	VERB
ejpam-4572	109	36	u	u	NOUN
ejpam-4572	109	37	of	of	ADP
ejpam-4572	109	38	x	x	SYM
ejpam-4572	109	39	such	such	ADJ
ejpam-4572	109	40	that	that	SCONJ
ejpam-4572	109	41	u	u	NOUN
ejpam-4572	109	42	⊆	⊆	NUM
ejpam-4572	109	43	g	g	ADP
ejpam-4572	109	44	⊆	⊆	NUM
ejpam-4572	109	45	u	u	NOUN
ejpam-4572	109	46	(	(	PUNCT
ejpam-4572	109	47	λ	λ	PROPN
ejpam-4572	109	48	,	,	PUNCT
ejpam-4572	109	49	sp	sp	NOUN
ejpam-4572	109	50	)	)	PUNCT
ejpam-4572	109	51	.	.	PUNCT
ejpam-4572	110	1	since	since	SCONJ
ejpam-4572	110	2	x	x	PROPN
ejpam-4572	110	3	∈	∈	PROPN
ejpam-4572	110	4	g	g	NOUN
ejpam-4572	110	5	⊆	⊆	NUM
ejpam-4572	110	6	u	u	PROPN
ejpam-4572	110	7	(	(	PUNCT
ejpam-4572	110	8	λ	λ	PROPN
ejpam-4572	110	9	,	,	PUNCT
ejpam-4572	110	10	sp	sp	NOUN
ejpam-4572	110	11	)	)	PUNCT
ejpam-4572	110	12	and	and	CCONJ
ejpam-4572	110	13	x	x	PUNCT
ejpam-4572	110	14	∈	∈	PROPN
ejpam-4572	110	15	[	[	X
ejpam-4572	110	16	a(λ	a(λ	PROPN
ejpam-4572	110	17	,	,	PUNCT
ejpam-4572	110	18	sp)](λ	sp)](λ	PROPN
ejpam-4572	110	19	,	,	PUNCT
ejpam-4572	110	20	sp	sp	NOUN
ejpam-4572	110	21	)	)	PUNCT
ejpam-4572	110	22	,	,	PUNCT
ejpam-4572	110	23	∅	∅	NOUN
ejpam-4572	110	24	=	=	SYM
ejpam-4572	110	25	̸	̸	NUM
ejpam-4572	110	26	u	u	NOUN
ejpam-4572	110	27	∩	∩	NOUN
ejpam-4572	110	28	[	[	X
ejpam-4572	110	29	a(λ	a(λ	PROPN
ejpam-4572	110	30	,	,	PUNCT
ejpam-4572	110	31	sp)](λ	sp)](λ	PROPN
ejpam-4572	110	32	,	,	PUNCT
ejpam-4572	110	33	sp	sp	NOUN
ejpam-4572	110	34	)	)	PUNCT
ejpam-4572	110	35	⊆	⊆	NUM
ejpam-4572	110	36	u	u	NOUN
ejpam-4572	110	37	∩a(λ	∩a(λ	NOUN
ejpam-4572	110	38	,	,	PUNCT
ejpam-4572	110	39	sp	sp	NOUN
ejpam-4572	110	40	)	)	PUNCT
ejpam-4572	110	41	⊆	⊆	NUM
ejpam-4572	111	1	[	[	X
ejpam-4572	111	2	u	u	NOUN
ejpam-4572	111	3	∩a](λ	∩a](λ	PROPN
ejpam-4572	111	4	,	,	PUNCT
ejpam-4572	111	5	sp	sp	NOUN
ejpam-4572	111	6	)	)	PUNCT
ejpam-4572	111	7	.	.	PUNCT
ejpam-4572	112	1	thus	thus	ADV
ejpam-4572	112	2	,	,	PUNCT
ejpam-4572	112	3	a	a	DET
ejpam-4572	112	4	∩	∩	ADJ
ejpam-4572	112	5	u	u	ADJ
ejpam-4572	112	6	̸=	̸=	PROPN
ejpam-4572	112	7	∅	∅	NOUN
ejpam-4572	112	8	and	and	CCONJ
ejpam-4572	112	9	hence	hence	ADV
ejpam-4572	112	10	a	a	DET
ejpam-4572	112	11	∩g	∩g	ADJ
ejpam-4572	112	12	̸=	̸=	PROPN
ejpam-4572	112	13	∅.	∅.	ADP
ejpam-4572	112	14	this	this	PRON
ejpam-4572	112	15	shows	show	VERB
ejpam-4572	112	16	that	that	SCONJ
ejpam-4572	112	17	x	x	PUNCT
ejpam-4572	112	18	∈	∈	NOUN
ejpam-4572	112	19	as(λ	as(λ	NOUN
ejpam-4572	112	20	,	,	PUNCT
ejpam-4572	112	21	sp	sp	NOUN
ejpam-4572	112	22	)	)	PUNCT
ejpam-4572	112	23	.	.	PUNCT
ejpam-4572	113	1	on	on	ADP
ejpam-4572	113	2	the	the	DET
ejpam-4572	113	3	other	other	ADJ
ejpam-4572	113	4	hand	hand	NOUN
ejpam-4572	113	5	,	,	PUNCT
ejpam-4572	113	6	assume	assume	VERB
ejpam-4572	113	7	that	that	SCONJ
ejpam-4572	113	8	x	x	PROPN
ejpam-4572	113	9	̸∈	̸∈	PROPN
ejpam-4572	113	10	[	[	X
ejpam-4572	113	11	a(λ	a(λ	PROPN
ejpam-4572	113	12	,	,	PUNCT
ejpam-4572	113	13	sp)](λ	sp)](λ	PROPN
ejpam-4572	113	14	,	,	PUNCT
ejpam-4572	113	15	sp	sp	NOUN
ejpam-4572	113	16	)	)	PUNCT
ejpam-4572	113	17	.	.	PUNCT
ejpam-4572	114	1	then	then	ADV
ejpam-4572	114	2	,	,	PUNCT
ejpam-4572	114	3	x	x	PUNCT
ejpam-4572	114	4	∈	∈	NOUN
ejpam-4572	115	1	[	[	X
ejpam-4572	115	2	[	[	X
ejpam-4572	115	3	x	x	X
ejpam-4572	115	4	−	−	NOUN
ejpam-4572	115	5	a](λ	a](λ	NOUN
ejpam-4572	115	6	,	,	PUNCT
ejpam-4572	115	7	sp	sp	NOUN
ejpam-4572	115	8	)	)	PUNCT
ejpam-4572	115	9	]	]	PUNCT
ejpam-4572	115	10	(	(	PUNCT
ejpam-4572	115	11	λ	λ	NOUN
ejpam-4572	115	12	,	,	PUNCT
ejpam-4572	115	13	sp	sp	NOUN
ejpam-4572	115	14	)	)	PUNCT
ejpam-4572	115	15	∈	∈	PROPN
ejpam-4572	115	16	sλspo(x	sλspo(x	PROPN
ejpam-4572	115	17	,	,	PUNCT
ejpam-4572	115	18	τ	τ	PROPN
ejpam-4572	115	19	)	)	PUNCT
ejpam-4572	115	20	.	.	PUNCT
ejpam-4572	116	1	since	since	SCONJ
ejpam-4572	116	2	a	a	DET
ejpam-4572	116	3	is	is	NOUN
ejpam-4572	116	4	(	(	PUNCT
ejpam-4572	116	5	λ	λ	NOUN
ejpam-4572	116	6	,	,	PUNCT
ejpam-4572	116	7	sp)-open	sp)-open	ADJ
ejpam-4572	116	8	,	,	PUNCT
ejpam-4572	116	9	we	we	PRON
ejpam-4572	116	10	have	have	VERB
ejpam-4572	116	11	a	a	DET
ejpam-4572	116	12	⊆	⊆	NUM
ejpam-4572	116	13	[	[	X
ejpam-4572	116	14	a(λ	a(λ	ADJ
ejpam-4572	116	15	,	,	PUNCT
ejpam-4572	116	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	116	17	,	,	PUNCT
ejpam-4572	116	18	sp	sp	NOUN
ejpam-4572	116	19	)	)	PUNCT
ejpam-4572	116	20	and	and	CCONJ
ejpam-4572	116	21	a∩	a∩	PROPN
ejpam-4572	117	1	[	[	X
ejpam-4572	117	2	[	[	X
ejpam-4572	117	3	x	x	X
ejpam-4572	117	4	−a](λ	−a](λ	PROPN
ejpam-4572	117	5	,	,	PUNCT
ejpam-4572	117	6	sp	sp	NOUN
ejpam-4572	117	7	)	)	PUNCT
ejpam-4572	117	8	]	]	PUNCT
ejpam-4572	117	9	(	(	PUNCT
ejpam-4572	117	10	λ	λ	NOUN
ejpam-4572	117	11	,	,	PUNCT
ejpam-4572	117	12	sp	sp	NOUN
ejpam-4572	117	13	)	)	PUNCT
ejpam-4572	117	14	=	=	PUNCT
ejpam-4572	117	15	∅.	∅.	ADP
ejpam-4572	117	16	this	this	PRON
ejpam-4572	117	17	shows	show	VERB
ejpam-4572	117	18	that	that	SCONJ
ejpam-4572	117	19	x	x	PROPN
ejpam-4572	117	20	̸∈	̸∈	PROPN
ejpam-4572	117	21	as(λ	as(λ	NUM
ejpam-4572	117	22	,	,	PUNCT
ejpam-4572	117	23	sp	sp	NOUN
ejpam-4572	117	24	)	)	PUNCT
ejpam-4572	117	25	.	.	PUNCT
ejpam-4572	118	1	therefore	therefore	ADV
ejpam-4572	118	2	,	,	PUNCT
ejpam-4572	118	3	as(λ	as(λ	ADV
ejpam-4572	118	4	,	,	PUNCT
ejpam-4572	118	5	sp	sp	NOUN
ejpam-4572	118	6	)	)	PUNCT
ejpam-4572	118	7	=	=	PUNCT
ejpam-4572	119	1	[	[	X
ejpam-4572	119	2	a(λ	a(λ	ADV
ejpam-4572	119	3	,	,	PUNCT
ejpam-4572	119	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	119	5	,	,	PUNCT
ejpam-4572	119	6	sp	sp	NOUN
ejpam-4572	119	7	)	)	PUNCT
ejpam-4572	119	8	.	.	PUNCT
ejpam-4572	120	1	(	(	PUNCT
ejpam-4572	120	2	2	2	X
ejpam-4572	120	3	)	)	PUNCT
ejpam-4572	120	4	suppose	suppose	VERB
ejpam-4572	120	5	that	that	SCONJ
ejpam-4572	120	6	a	a	PRON
ejpam-4572	120	7	is	be	AUX
ejpam-4572	120	8	α(λ	α(λ	PROPN
ejpam-4572	120	9	,	,	PUNCT
ejpam-4572	120	10	sp)-open	sp)-open	NOUN
ejpam-4572	120	11	.	.	PUNCT
ejpam-4572	121	1	then	then	ADV
ejpam-4572	121	2	,	,	PUNCT
ejpam-4572	121	3	a	a	DET
ejpam-4572	121	4	⊆	⊆	NUM
ejpam-4572	121	5	[	[	X
ejpam-4572	121	6	[	[	X
ejpam-4572	121	7	a(λ	a(λ	ADJ
ejpam-4572	121	8	,	,	PUNCT
ejpam-4572	121	9	sp	sp	NOUN
ejpam-4572	121	10	)	)	PUNCT
ejpam-4572	121	11	]	]	PUNCT
ejpam-4572	122	1	(	(	PUNCT
ejpam-4572	122	2	λ	λ	X
ejpam-4572	122	3	,	,	PUNCT
ejpam-4572	122	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	122	5	,	,	PUNCT
ejpam-4572	122	6	sp	sp	NOUN
ejpam-4572	122	7	)	)	PUNCT
ejpam-4572	122	8	.	.	PUNCT
ejpam-4572	123	1	let	let	VERB
ejpam-4572	123	2	u	u	PRON
ejpam-4572	123	3	=	=	PUNCT
ejpam-4572	123	4	a(λ	a(λ	ADV
ejpam-4572	123	5	,	,	PUNCT
ejpam-4572	123	6	sp	sp	NOUN
ejpam-4572	123	7	)	)	PUNCT
ejpam-4572	123	8	.	.	PUNCT
ejpam-4572	124	1	thus	thus	ADV
ejpam-4572	124	2	,	,	PUNCT
ejpam-4572	124	3	u	u	NOUN
ejpam-4572	124	4	⊆	⊆	NUM
ejpam-4572	124	5	a	a	DET
ejpam-4572	124	6	⊆	⊆	NUM
ejpam-4572	124	7	[	[	X
ejpam-4572	124	8	u	u	X
ejpam-4572	124	9	(	(	PUNCT
ejpam-4572	124	10	λ	λ	PROPN
ejpam-4572	124	11	,	,	PUNCT
ejpam-4572	124	12	sp)](λ	sp)](λ	PROPN
ejpam-4572	124	13	,	,	PUNCT
ejpam-4572	124	14	sp	sp	NOUN
ejpam-4572	124	15	)	)	PUNCT
ejpam-4572	124	16	and	and	CCONJ
ejpam-4572	124	17	by	by	ADP
ejpam-4572	124	18	(	(	PUNCT
ejpam-4572	124	19	1	1	NUM
ejpam-4572	124	20	)	)	PUNCT
ejpam-4572	124	21	,	,	PUNCT
ejpam-4572	124	22	u	u	NOUN
ejpam-4572	124	23	⊆	⊆	NUM
ejpam-4572	124	24	a	a	DET
ejpam-4572	124	25	⊆	⊆	NUM
ejpam-4572	124	26	u	u	SYM
ejpam-4572	124	27	s(λ	s(λ	PROPN
ejpam-4572	124	28	,	,	PUNCT
ejpam-4572	124	29	sp	sp	NOUN
ejpam-4572	124	30	)	)	PUNCT
ejpam-4572	124	31	.	.	PUNCT
ejpam-4572	125	1	conversely	conversely	ADV
ejpam-4572	125	2	,	,	PUNCT
ejpam-4572	125	3	assume	assume	VERB
ejpam-4572	125	4	that	that	SCONJ
ejpam-4572	125	5	there	there	PRON
ejpam-4572	125	6	exists	exist	VERB
ejpam-4572	125	7	a	a	DET
ejpam-4572	125	8	(	(	PUNCT
ejpam-4572	125	9	λ	λ	NOUN
ejpam-4572	125	10	,	,	PUNCT
ejpam-4572	125	11	sp)-open	sp)-open	NOUN
ejpam-4572	125	12	set	set	VERB
ejpam-4572	125	13	u	u	PRON
ejpam-4572	125	14	such	such	ADJ
ejpam-4572	125	15	that	that	SCONJ
ejpam-4572	125	16	u	u	NOUN
ejpam-4572	125	17	⊆	⊆	NUM
ejpam-4572	125	18	a	a	DET
ejpam-4572	125	19	⊆	⊆	NUM
ejpam-4572	125	20	u	u	SYM
ejpam-4572	125	21	s(λ	s(λ	PROPN
ejpam-4572	125	22	,	,	PUNCT
ejpam-4572	125	23	sp	sp	NOUN
ejpam-4572	125	24	)	)	PUNCT
ejpam-4572	125	25	.	.	PUNCT
ejpam-4572	126	1	by	by	ADP
ejpam-4572	126	2	(	(	PUNCT
ejpam-4572	126	3	1	1	NUM
ejpam-4572	126	4	)	)	PUNCT
ejpam-4572	126	5	,	,	PUNCT
ejpam-4572	126	6	we	we	PRON
ejpam-4572	126	7	have	have	VERB
ejpam-4572	126	8	u	u	NOUN
ejpam-4572	126	9	⊆	⊆	NUM
ejpam-4572	126	10	a	a	DET
ejpam-4572	126	11	⊆	⊆	NUM
ejpam-4572	126	12	[	[	X
ejpam-4572	126	13	u	u	X
ejpam-4572	126	14	(	(	PUNCT
ejpam-4572	126	15	λ	λ	PROPN
ejpam-4572	126	16	,	,	PUNCT
ejpam-4572	126	17	sp)](λ	sp)](λ	PROPN
ejpam-4572	126	18	,	,	PUNCT
ejpam-4572	126	19	sp	sp	NOUN
ejpam-4572	126	20	)	)	PUNCT
ejpam-4572	126	21	.	.	PUNCT
ejpam-4572	127	1	thus	thus	ADV
ejpam-4572	127	2	,	,	PUNCT
ejpam-4572	127	3	u	u	NOUN
ejpam-4572	127	4	⊆	⊆	NUM
ejpam-4572	127	5	a(λ	a(λ	ADV
ejpam-4572	127	6	,	,	PUNCT
ejpam-4572	127	7	sp	sp	NOUN
ejpam-4572	127	8	)	)	PUNCT
ejpam-4572	127	9	and	and	CCONJ
ejpam-4572	127	10	hence	hence	ADV
ejpam-4572	127	11	[	[	X
ejpam-4572	127	12	u	u	X
ejpam-4572	127	13	(	(	PUNCT
ejpam-4572	127	14	λ	λ	PROPN
ejpam-4572	127	15	,	,	PUNCT
ejpam-4572	127	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	127	17	,	,	PUNCT
ejpam-4572	127	18	sp	sp	NOUN
ejpam-4572	127	19	)	)	PUNCT
ejpam-4572	127	20	⊆	⊆	NUM
ejpam-4572	128	1	[	[	X
ejpam-4572	128	2	[	[	X
ejpam-4572	128	3	a(λ	a(λ	ADJ
ejpam-4572	128	4	,	,	PUNCT
ejpam-4572	128	5	sp	sp	NOUN
ejpam-4572	128	6	)	)	PUNCT
ejpam-4572	128	7	]	]	PUNCT
ejpam-4572	128	8	(	(	PUNCT
ejpam-4572	128	9	λ	λ	X
ejpam-4572	128	10	,	,	PUNCT
ejpam-4572	128	11	sp)](λ	sp)](λ	PROPN
ejpam-4572	128	12	,	,	PUNCT
ejpam-4572	128	13	sp	sp	NOUN
ejpam-4572	128	14	)	)	PUNCT
ejpam-4572	128	15	.	.	PUNCT
ejpam-4572	129	1	since	since	SCONJ
ejpam-4572	129	2	a	a	DET
ejpam-4572	129	3	⊆	⊆	NUM
ejpam-4572	129	4	[	[	X
ejpam-4572	129	5	u	u	X
ejpam-4572	129	6	(	(	PUNCT
ejpam-4572	129	7	λ	λ	PROPN
ejpam-4572	129	8	,	,	PUNCT
ejpam-4572	129	9	sp)](λ	sp)](λ	PROPN
ejpam-4572	129	10	,	,	PUNCT
ejpam-4572	129	11	sp	sp	NOUN
ejpam-4572	129	12	)	)	PUNCT
ejpam-4572	129	13	,	,	PUNCT
ejpam-4572	129	14	a	a	DET
ejpam-4572	129	15	⊆	⊆	NUM
ejpam-4572	129	16	[	[	X
ejpam-4572	129	17	[	[	X
ejpam-4572	129	18	a(λ	a(λ	ADJ
ejpam-4572	129	19	,	,	PUNCT
ejpam-4572	129	20	sp	sp	NOUN
ejpam-4572	129	21	)	)	PUNCT
ejpam-4572	129	22	]	]	PUNCT
ejpam-4572	129	23	(	(	PUNCT
ejpam-4572	129	24	λ	λ	X
ejpam-4572	129	25	,	,	PUNCT
ejpam-4572	129	26	sp)](λ	sp)](λ	PROPN
ejpam-4572	129	27	,	,	PUNCT
ejpam-4572	129	28	sp	sp	NOUN
ejpam-4572	129	29	)	)	PUNCT
ejpam-4572	129	30	.	.	PUNCT
ejpam-4572	130	1	this	this	PRON
ejpam-4572	130	2	shows	show	VERB
ejpam-4572	130	3	that	that	SCONJ
ejpam-4572	130	4	a	a	PRON
ejpam-4572	130	5	is	be	AUX
ejpam-4572	130	6	α(λ	α(λ	PROPN
ejpam-4572	130	7	,	,	PUNCT
ejpam-4572	130	8	sp)-open	sp)-open	NOUN
ejpam-4572	130	9	.	.	PUNCT
ejpam-4572	131	1	(	(	PUNCT
ejpam-4572	131	2	3	3	X
ejpam-4572	131	3	)	)	PUNCT
ejpam-4572	131	4	we	we	PRON
ejpam-4572	131	5	observe	observe	VERB
ejpam-4572	131	6	that	that	SCONJ
ejpam-4572	131	7	[	[	X
ejpam-4572	131	8	[	[	X
ejpam-4572	131	9	[	[	X
ejpam-4572	131	10	a	a	DET
ejpam-4572	131	11	∪	∪	ADJ
ejpam-4572	131	12	[	[	X
ejpam-4572	131	13	[	[	X
ejpam-4572	131	14	a(λ	a(λ	ADJ
ejpam-4572	131	15	,	,	PUNCT
ejpam-4572	131	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	131	17	,	,	PUNCT
ejpam-4572	131	18	sp	sp	NOUN
ejpam-4572	131	19	)	)	PUNCT
ejpam-4572	131	20	]	]	PUNCT
ejpam-4572	131	21	(	(	PUNCT
ejpam-4572	131	22	λ	λ	X
ejpam-4572	131	23	,	,	PUNCT
ejpam-4572	131	24	sp)](λ	sp)](λ	PROPN
ejpam-4572	131	25	,	,	PUNCT
ejpam-4572	131	26	sp)](λ	sp)](λ	PROPN
ejpam-4572	131	27	,	,	PUNCT
ejpam-4572	131	28	sp	sp	NOUN
ejpam-4572	131	29	)	)	PUNCT
ejpam-4572	131	30	]	]	PUNCT
ejpam-4572	131	31	(	(	PUNCT
ejpam-4572	131	32	λ	λ	NOUN
ejpam-4572	131	33	,	,	PUNCT
ejpam-4572	131	34	sp	sp	NOUN
ejpam-4572	131	35	)	)	PUNCT
ejpam-4572	131	36	⊆	⊆	NUM
ejpam-4572	132	1	[	[	X
ejpam-4572	132	2	[	[	X
ejpam-4572	132	3	[	[	X
ejpam-4572	132	4	a	a	DET
ejpam-4572	132	5	∪	∪	ADJ
ejpam-4572	132	6	[	[	X
ejpam-4572	132	7	[	[	X
ejpam-4572	132	8	a(λ	a(λ	ADJ
ejpam-4572	132	9	,	,	PUNCT
ejpam-4572	132	10	sp)](λ	sp)](λ	PROPN
ejpam-4572	132	11	,	,	PUNCT
ejpam-4572	132	12	sp	sp	NOUN
ejpam-4572	132	13	)	)	PUNCT
ejpam-4572	132	14	]	]	PUNCT
ejpam-4572	132	15	(	(	PUNCT
ejpam-4572	132	16	λ	λ	X
ejpam-4572	132	17	,	,	PUNCT
ejpam-4572	132	18	sp)](λ	sp)](λ	PROPN
ejpam-4572	132	19	,	,	PUNCT
ejpam-4572	132	20	sp	sp	NOUN
ejpam-4572	132	21	)	)	PUNCT
ejpam-4572	132	22	]	]	PUNCT
ejpam-4572	133	1	(	(	PUNCT
ejpam-4572	133	2	λ	λ	NOUN
ejpam-4572	133	3	,	,	PUNCT
ejpam-4572	133	4	sp	sp	NOUN
ejpam-4572	133	5	)	)	PUNCT
ejpam-4572	133	6	⊆	⊆	NUM
ejpam-4572	134	1	[	[	X
ejpam-4572	134	2	[	[	X
ejpam-4572	134	3	a(λ	a(λ	ADJ
ejpam-4572	134	4	,	,	PUNCT
ejpam-4572	134	5	sp	sp	NOUN
ejpam-4572	134	6	)	)	PUNCT
ejpam-4572	134	7	∪	∪	ADP
ejpam-4572	134	8	[	[	X
ejpam-4572	134	9	a(λ	a(λ	ADJ
ejpam-4572	134	10	,	,	PUNCT
ejpam-4572	134	11	sp)](λ	sp)](λ	PROPN
ejpam-4572	134	12	,	,	PUNCT
ejpam-4572	134	13	sp)](λ	sp)](λ	PROPN
ejpam-4572	134	14	,	,	PUNCT
ejpam-4572	134	15	sp	sp	NOUN
ejpam-4572	134	16	)	)	PUNCT
ejpam-4572	134	17	]	]	PUNCT
ejpam-4572	135	1	(	(	PUNCT
ejpam-4572	135	2	λ	λ	NOUN
ejpam-4572	135	3	,	,	PUNCT
ejpam-4572	135	4	sp	sp	NOUN
ejpam-4572	135	5	)	)	PUNCT
ejpam-4572	135	6	=	=	PUNCT
ejpam-4572	136	1	[	[	X
ejpam-4572	136	2	[	[	X
ejpam-4572	136	3	a(λ	a(λ	ADJ
ejpam-4572	136	4	,	,	PUNCT
ejpam-4572	136	5	sp)](λ	sp)](λ	PROPN
ejpam-4572	136	6	,	,	PUNCT
ejpam-4572	136	7	sp	sp	NOUN
ejpam-4572	136	8	)	)	PUNCT
ejpam-4572	136	9	]	]	PUNCT
ejpam-4572	137	1	(	(	PUNCT
ejpam-4572	137	2	λ	λ	NOUN
ejpam-4572	137	3	,	,	PUNCT
ejpam-4572	137	4	sp	sp	NOUN
ejpam-4572	137	5	)	)	PUNCT
ejpam-4572	137	6	⊆	⊆	NUM
ejpam-4572	137	7	a	a	DET
ejpam-4572	137	8	∪	∪	NOUN
ejpam-4572	137	9	[	[	X
ejpam-4572	137	10	[	[	X
ejpam-4572	137	11	a(λ	a(λ	ADJ
ejpam-4572	137	12	,	,	PUNCT
ejpam-4572	137	13	sp)](λ	sp)](λ	PROPN
ejpam-4572	137	14	,	,	PUNCT
ejpam-4572	137	15	sp	sp	NOUN
ejpam-4572	137	16	)	)	PUNCT
ejpam-4572	137	17	]	]	PUNCT
ejpam-4572	137	18	(	(	PUNCT
ejpam-4572	137	19	λ	λ	NOUN
ejpam-4572	137	20	,	,	PUNCT
ejpam-4572	137	21	sp	sp	NOUN
ejpam-4572	137	22	)	)	PUNCT
ejpam-4572	137	23	.	.	PUNCT
ejpam-4572	138	1	thus	thus	ADV
ejpam-4572	138	2	,	,	PUNCT
ejpam-4572	138	3	a∪[[a(λ	a∪[[a(λ	PROPN
ejpam-4572	138	4	,	,	PUNCT
ejpam-4572	138	5	sp)](λ	sp)](λ	PROPN
ejpam-4572	138	6	,	,	PUNCT
ejpam-4572	138	7	sp	sp	NOUN
ejpam-4572	138	8	)	)	PUNCT
ejpam-4572	138	9	]	]	PUNCT
ejpam-4572	138	10	(	(	PUNCT
ejpam-4572	138	11	λ	λ	NOUN
ejpam-4572	138	12	,	,	PUNCT
ejpam-4572	138	13	sp	sp	NOUN
ejpam-4572	138	14	)	)	PUNCT
ejpam-4572	138	15	is	be	AUX
ejpam-4572	138	16	α(λ	α(λ	PROPN
ejpam-4572	138	17	,	,	PUNCT
ejpam-4572	138	18	sp)-closed	sp)-close	VERB
ejpam-4572	138	19	and	and	CCONJ
ejpam-4572	138	20	henceaα(λ	henceaα(λ	PROPN
ejpam-4572	138	21	,	,	PUNCT
ejpam-4572	138	22	sp	sp	NOUN
ejpam-4572	138	23	)	)	PUNCT
ejpam-4572	138	24	⊆	⊆	NUM
ejpam-4572	138	25	a∪[[a(λ	a∪[[a(λ	PROPN
ejpam-4572	138	26	,	,	PUNCT
ejpam-4572	138	27	sp)](λ	sp)](λ	PROPN
ejpam-4572	138	28	,	,	PUNCT
ejpam-4572	138	29	sp	sp	NOUN
ejpam-4572	138	30	)	)	PUNCT
ejpam-4572	138	31	]	]	PUNCT
ejpam-4572	138	32	(	(	PUNCT
ejpam-4572	138	33	λ	λ	NOUN
ejpam-4572	138	34	,	,	PUNCT
ejpam-4572	138	35	sp	sp	NOUN
ejpam-4572	138	36	)	)	PUNCT
ejpam-4572	138	37	.	.	PUNCT
ejpam-4572	139	1	on	on	ADP
ejpam-4572	139	2	the	the	DET
ejpam-4572	139	3	other	other	ADJ
ejpam-4572	139	4	hand	hand	NOUN
ejpam-4572	139	5	,	,	PUNCT
ejpam-4572	139	6	since	since	SCONJ
ejpam-4572	139	7	aα(λ	aα(λ	PROPN
ejpam-4572	139	8	,	,	PUNCT
ejpam-4572	139	9	sp	sp	NOUN
ejpam-4572	139	10	)	)	PUNCT
ejpam-4572	139	11	is	be	AUX
ejpam-4572	139	12	α(λ	α(λ	PROPN
ejpam-4572	139	13	,	,	PUNCT
ejpam-4572	139	14	sp)-closed	sp)-close	VERB
ejpam-4572	139	15	,	,	PUNCT
ejpam-4572	139	16	we	we	PRON
ejpam-4572	139	17	have	have	VERB
ejpam-4572	139	18	[	[	X
ejpam-4572	139	19	[	[	X
ejpam-4572	139	20	a(λ	a(λ	ADJ
ejpam-4572	139	21	,	,	PUNCT
ejpam-4572	139	22	sp)](λ	sp)](λ	PROPN
ejpam-4572	139	23	,	,	PUNCT
ejpam-4572	139	24	sp	sp	NOUN
ejpam-4572	139	25	)	)	PUNCT
ejpam-4572	139	26	]	]	PUNCT
ejpam-4572	140	1	(	(	PUNCT
ejpam-4572	140	2	λ	λ	NOUN
ejpam-4572	140	3	,	,	PUNCT
ejpam-4572	140	4	sp	sp	NOUN
ejpam-4572	140	5	)	)	PUNCT
ejpam-4572	140	6	⊆	⊆	NUM
ejpam-4572	141	1	[	[	X
ejpam-4572	141	2	[	[	X
ejpam-4572	141	3	[	[	X
ejpam-4572	141	4	aα(λ	aα(λ	ADJ
ejpam-4572	141	5	,	,	PUNCT
ejpam-4572	141	6	sp	sp	NOUN
ejpam-4572	141	7	)	)	PUNCT
ejpam-4572	141	8	]	]	PUNCT
ejpam-4572	141	9	(	(	PUNCT
ejpam-4572	141	10	λ	λ	X
ejpam-4572	141	11	,	,	PUNCT
ejpam-4572	141	12	sp)](λ	sp)](λ	PROPN
ejpam-4572	141	13	,	,	PUNCT
ejpam-4572	141	14	sp	sp	NOUN
ejpam-4572	141	15	)	)	PUNCT
ejpam-4572	141	16	]	]	PUNCT
ejpam-4572	141	17	(	(	PUNCT
ejpam-4572	141	18	λ	λ	NOUN
ejpam-4572	141	19	,	,	PUNCT
ejpam-4572	141	20	sp	sp	NOUN
ejpam-4572	141	21	)	)	PUNCT
ejpam-4572	141	22	⊆	⊆	NUM
ejpam-4572	141	23	aα(λ	aα(λ	NUM
ejpam-4572	141	24	,	,	PUNCT
ejpam-4572	141	25	sp	sp	NOUN
ejpam-4572	141	26	)	)	PUNCT
ejpam-4572	141	27	.	.	PUNCT
ejpam-4572	142	1	therefore	therefore	ADV
ejpam-4572	142	2	,	,	PUNCT
ejpam-4572	142	3	a	a	DET
ejpam-4572	142	4	∪	∪	NOUN
ejpam-4572	142	5	[	[	X
ejpam-4572	142	6	[	[	X
ejpam-4572	142	7	a(λ	a(λ	ADJ
ejpam-4572	142	8	,	,	PUNCT
ejpam-4572	142	9	sp)](λ	sp)](λ	PROPN
ejpam-4572	142	10	,	,	PUNCT
ejpam-4572	142	11	sp	sp	NOUN
ejpam-4572	142	12	)	)	PUNCT
ejpam-4572	142	13	]	]	PUNCT
ejpam-4572	143	1	(	(	PUNCT
ejpam-4572	143	2	λ	λ	NOUN
ejpam-4572	143	3	,	,	PUNCT
ejpam-4572	143	4	sp	sp	NOUN
ejpam-4572	143	5	)	)	PUNCT
ejpam-4572	143	6	⊆	⊆	NUM
ejpam-4572	143	7	aα(λ	aα(λ	NUM
ejpam-4572	143	8	,	,	PUNCT
ejpam-4572	143	9	sp	sp	NOUN
ejpam-4572	143	10	)	)	PUNCT
ejpam-4572	143	11	.	.	PUNCT
ejpam-4572	144	1	thus	thus	ADV
ejpam-4572	144	2	,	,	PUNCT
ejpam-4572	144	3	a	a	DET
ejpam-4572	144	4	α(λ	α(λ	PROPN
ejpam-4572	144	5	,	,	PUNCT
ejpam-4572	144	6	sp	sp	NOUN
ejpam-4572	144	7	)	)	PUNCT
ejpam-4572	144	8	=	=	NOUN
ejpam-4572	144	9	a	a	DET
ejpam-4572	144	10	∪	∪	ADJ
ejpam-4572	144	11	[	[	X
ejpam-4572	144	12	[	[	X
ejpam-4572	144	13	a(λ	a(λ	ADJ
ejpam-4572	144	14	,	,	PUNCT
ejpam-4572	144	15	sp)](λ	sp)](λ	PROPN
ejpam-4572	144	16	,	,	PUNCT
ejpam-4572	144	17	sp	sp	NOUN
ejpam-4572	144	18	)	)	PUNCT
ejpam-4572	144	19	]	]	PUNCT
ejpam-4572	144	20	(	(	PUNCT
ejpam-4572	144	21	λ	λ	NOUN
ejpam-4572	144	22	,	,	PUNCT
ejpam-4572	144	23	sp	sp	NOUN
ejpam-4572	144	24	)	)	PUNCT
ejpam-4572	144	25	.	.	PUNCT
ejpam-4572	145	1	by	by	ADP
ejpam-4572	145	2	a	a	DET
ejpam-4572	145	3	multifunction	multifunction	NOUN
ejpam-4572	145	4	f	f	NOUN
ejpam-4572	145	5	:	:	PUNCT
ejpam-4572	145	6	x	x	X
ejpam-4572	145	7	→	→	SYM
ejpam-4572	145	8	y	y	PROPN
ejpam-4572	145	9	,	,	PUNCT
ejpam-4572	145	10	we	we	PRON
ejpam-4572	145	11	mean	mean	VERB
ejpam-4572	145	12	a	a	DET
ejpam-4572	145	13	point	point	NOUN
ejpam-4572	145	14	-	-	PUNCT
ejpam-4572	145	15	to	to	ADP
ejpam-4572	145	16	-	-	PUNCT
ejpam-4572	145	17	set	set	VERB
ejpam-4572	145	18	correspondence	correspondence	NOUN
ejpam-4572	145	19	from	from	ADP
ejpam-4572	145	20	x	x	PUNCT
ejpam-4572	145	21	into	into	ADP
ejpam-4572	145	22	y	y	PROPN
ejpam-4572	145	23	,	,	PUNCT
ejpam-4572	145	24	and	and	CCONJ
ejpam-4572	145	25	always	always	ADV
ejpam-4572	145	26	assume	assume	VERB
ejpam-4572	145	27	that	that	SCONJ
ejpam-4572	146	1	f	f	PROPN
ejpam-4572	146	2	(	(	PUNCT
ejpam-4572	146	3	x	x	X
ejpam-4572	146	4	)	)	PUNCT
ejpam-4572	146	5	̸=	̸=	NOUN
ejpam-4572	146	6	∅	∅	NOUN
ejpam-4572	146	7	for	for	ADP
ejpam-4572	146	8	all	all	PRON
ejpam-4572	146	9	x	x	SYM
ejpam-4572	146	10	∈	∈	ADJ
ejpam-4572	146	11	x.	x.	NOUN
ejpam-4572	146	12	for	for	ADP
ejpam-4572	146	13	a	a	DET
ejpam-4572	146	14	multifunction	multifunction	NOUN
ejpam-4572	146	15	f	f	NOUN
ejpam-4572	146	16	:	:	PUNCT
ejpam-4572	146	17	x	x	X
ejpam-4572	146	18	→	→	SYM
ejpam-4572	146	19	y	y	PROPN
ejpam-4572	146	20	,	,	PUNCT
ejpam-4572	146	21	following	follow	VERB
ejpam-4572	146	22	[	[	X
ejpam-4572	146	23	1	1	X
ejpam-4572	146	24	]	]	PUNCT
ejpam-4572	146	25	we	we	PRON
ejpam-4572	146	26	shall	shall	AUX
ejpam-4572	146	27	denote	denote	VERB
ejpam-4572	146	28	the	the	DET
ejpam-4572	146	29	upper	upper	ADJ
ejpam-4572	146	30	and	and	CCONJ
ejpam-4572	146	31	lower	low	ADJ
ejpam-4572	146	32	inverse	inverse	NOUN
ejpam-4572	146	33	of	of	ADP
ejpam-4572	146	34	a	a	DET
ejpam-4572	146	35	set	set	NOUN
ejpam-4572	146	36	b	b	PROPN
ejpam-4572	146	37	of	of	ADP
ejpam-4572	146	38	y	y	PROPN
ejpam-4572	146	39	by	by	ADP
ejpam-4572	146	40	f+(b	f+(b	NOUN
ejpam-4572	146	41	)	)	PUNCT
ejpam-4572	146	42	and	and	CCONJ
ejpam-4572	146	43	f−(b	f−(b	NOUN
ejpam-4572	146	44	)	)	PUNCT
ejpam-4572	146	45	,	,	PUNCT
ejpam-4572	146	46	respectively	respectively	ADV
ejpam-4572	146	47	,	,	PUNCT
ejpam-4572	146	48	that	that	ADV
ejpam-4572	146	49	is	is	ADV
ejpam-4572	146	50	,	,	PUNCT
ejpam-4572	146	51	f+(b	f+(b	NOUN
ejpam-4572	146	52	)	)	PUNCT
ejpam-4572	146	53	=	=	PRON
ejpam-4572	147	1	{	{	PUNCT
ejpam-4572	147	2	x	x	PUNCT
ejpam-4572	147	3	∈	∈	PROPN
ejpam-4572	147	4	x	x	INTJ
ejpam-4572	148	1	|	|	NOUN
ejpam-4572	148	2	f	f	X
ejpam-4572	148	3	(	(	PUNCT
ejpam-4572	148	4	x	x	NOUN
ejpam-4572	148	5	)	)	PUNCT
ejpam-4572	148	6	⊆	⊆	NUM
ejpam-4572	148	7	b	b	NOUN
ejpam-4572	148	8	}	}	PUNCT
ejpam-4572	148	9	and	and	CCONJ
ejpam-4572	148	10	f−(b	f−(b	PROPN
ejpam-4572	148	11	)	)	PUNCT
ejpam-4572	148	12	=	=	PRON
ejpam-4572	149	1	{	{	PUNCT
ejpam-4572	149	2	x	x	PUNCT
ejpam-4572	149	3	∈	∈	PROPN
ejpam-4572	149	4	x	x	INTJ
ejpam-4572	150	1	|	|	NOUN
ejpam-4572	150	2	f	f	X
ejpam-4572	150	3	(	(	PUNCT
ejpam-4572	150	4	x)∩b	x)∩b	PROPN
ejpam-4572	150	5	̸=	̸=	PROPN
ejpam-4572	150	6	∅	∅	NOUN
ejpam-4572	150	7	}	}	PUNCT
ejpam-4572	150	8	.	.	PUNCT
ejpam-4572	151	1	in	in	ADP
ejpam-4572	151	2	particular	particular	ADJ
ejpam-4572	151	3	,	,	PUNCT
ejpam-4572	151	4	f−(y	f−(y	NOUN
ejpam-4572	151	5	)	)	PUNCT
ejpam-4572	151	6	=	=	SYM
ejpam-4572	152	1	{	{	PUNCT
ejpam-4572	152	2	x	x	PUNCT
ejpam-4572	152	3	∈	∈	PROPN
ejpam-4572	152	4	x	x	INTJ
ejpam-4572	153	1	|	|	ADV
ejpam-4572	153	2	y	y	PROPN
ejpam-4572	153	3	∈	∈	PROPN
ejpam-4572	153	4	f	f	X
ejpam-4572	153	5	(	(	PUNCT
ejpam-4572	153	6	x	x	NOUN
ejpam-4572	153	7	)	)	PUNCT
ejpam-4572	153	8	}	}	PUNCT
ejpam-4572	153	9	for	for	ADP
ejpam-4572	153	10	each	each	DET
ejpam-4572	153	11	point	point	NOUN
ejpam-4572	153	12	y	y	PROPN
ejpam-4572	153	13	∈	∈	PROPN
ejpam-4572	153	14	y	y	PROPN
ejpam-4572	153	15	.	.	PUNCT
ejpam-4572	154	1	for	for	ADP
ejpam-4572	154	2	each	each	PRON
ejpam-4572	154	3	a	a	DET
ejpam-4572	154	4	⊆	⊆	NUM
ejpam-4572	154	5	x	x	SYM
ejpam-4572	154	6	,	,	PUNCT
ejpam-4572	154	7	f	f	PROPN
ejpam-4572	154	8	(	(	PUNCT
ejpam-4572	154	9	a	a	NOUN
ejpam-4572	154	10	)	)	PUNCT
ejpam-4572	154	11	=	=	SYM
ejpam-4572	154	12	∪x∈af	∪x∈af	NOUN
ejpam-4572	154	13	(	(	PUNCT
ejpam-4572	154	14	x	x	NOUN
ejpam-4572	154	15	)	)	PUNCT
ejpam-4572	154	16	.	.	PUNCT
ejpam-4572	155	1	then	then	ADV
ejpam-4572	155	2	,	,	PUNCT
ejpam-4572	155	3	f	f	PROPN
ejpam-4572	155	4	is	be	AUX
ejpam-4572	155	5	said	say	VERB
ejpam-4572	155	6	to	to	PART
ejpam-4572	155	7	be	be	AUX
ejpam-4572	155	8	a	a	DET
ejpam-4572	155	9	surjection	surjection	NOUN
ejpam-4572	155	10	if	if	SCONJ
ejpam-4572	155	11	f	f	PROPN
ejpam-4572	155	12	(	(	PUNCT
ejpam-4572	155	13	x	x	X
ejpam-4572	155	14	)	)	PUNCT
ejpam-4572	155	15	=	=	SYM
ejpam-4572	155	16	y	y	PROPN
ejpam-4572	155	17	,	,	PUNCT
ejpam-4572	155	18	or	or	CCONJ
ejpam-4572	155	19	equivalently	equivalently	ADV
ejpam-4572	155	20	,	,	PUNCT
ejpam-4572	155	21	if	if	SCONJ
ejpam-4572	155	22	for	for	ADP
ejpam-4572	155	23	each	each	DET
ejpam-4572	155	24	y	y	PROPN
ejpam-4572	155	25	∈	∈	PROPN
ejpam-4572	155	26	y	y	PROPN
ejpam-4572	155	27	,	,	PUNCT
ejpam-4572	155	28	there	there	PRON
ejpam-4572	155	29	exists	exist	VERB
ejpam-4572	155	30	an	an	DET
ejpam-4572	155	31	x	x	SYM
ejpam-4572	155	32	∈	∈	PROPN
ejpam-4572	155	33	x	x	X
ejpam-4572	155	34	such	such	ADJ
ejpam-4572	155	35	that	that	SCONJ
ejpam-4572	155	36	y	y	PROPN
ejpam-4572	155	37	∈	∈	PROPN
ejpam-4572	155	38	f	f	X
ejpam-4572	155	39	(	(	PUNCT
ejpam-4572	155	40	x	x	NOUN
ejpam-4572	155	41	)	)	PUNCT
ejpam-4572	155	42	.	.	PUNCT
ejpam-4572	156	1	moreover	moreover	ADV
ejpam-4572	156	2	,	,	PUNCT
ejpam-4572	156	3	f	f	X
ejpam-4572	156	4	:	:	PUNCT
ejpam-4572	156	5	x	x	X
ejpam-4572	156	6	→	→	SYM
ejpam-4572	156	7	y	y	PROPN
ejpam-4572	156	8	is	be	AUX
ejpam-4572	156	9	called	call	VERB
ejpam-4572	156	10	upper	upper	ADJ
ejpam-4572	156	11	semi	semi	ADJ
ejpam-4572	156	12	-	-	ADJ
ejpam-4572	156	13	continuous	continuous	ADJ
ejpam-4572	156	14	(	(	PUNCT
ejpam-4572	156	15	resp	resp	NOUN
ejpam-4572	156	16	.	.	PUNCT
ejpam-4572	157	1	lower	low	ADJ
ejpam-4572	157	2	semi	semi	ADJ
ejpam-4572	157	3	-	-	ADJ
ejpam-4572	157	4	continuous	continuous	ADJ
ejpam-4572	157	5	)	)	PUNCT
ejpam-4572	157	6	if	if	SCONJ
ejpam-4572	157	7	f+(v	f+(v	PROPN
ejpam-4572	157	8	)	)	PUNCT
ejpam-4572	157	9	(	(	PUNCT
ejpam-4572	157	10	resp	resp	NOUN
ejpam-4572	157	11	.	.	PUNCT
ejpam-4572	158	1	f−(v	f−(v	NOUN
ejpam-4572	158	2	)	)	PUNCT
ejpam-4572	158	3	)	)	PUNCT
ejpam-4572	159	1	is	be	AUX
ejpam-4572	159	2	open	open	ADJ
ejpam-4572	159	3	in	in	ADP
ejpam-4572	159	4	x	x	PUNCT
ejpam-4572	159	5	for	for	ADP
ejpam-4572	159	6	every	every	DET
ejpam-4572	159	7	open	open	ADJ
ejpam-4572	159	8	set	set	VERB
ejpam-4572	159	9	v	v	NOUN
ejpam-4572	159	10	of	of	ADP
ejpam-4572	159	11	y	y	PROPN
ejpam-4572	160	1	[	[	X
ejpam-4572	160	2	10	10	NUM
ejpam-4572	160	3	]	]	PUNCT
ejpam-4572	160	4	.	.	PUNCT
ejpam-4572	161	1	c.	c.	PROPN
ejpam-4572	161	2	boonpok	boonpok	PROPN
ejpam-4572	161	3	,	,	PUNCT
ejpam-4572	161	4	m.	m.	NOUN
ejpam-4572	161	5	thongmoon	thongmoon	PROPN
ejpam-4572	161	6	/	/	SYM
ejpam-4572	161	7	eur	eur	PROPN
ejpam-4572	161	8	.	.	PUNCT
ejpam-4572	162	1	j.	j.	PROPN
ejpam-4572	162	2	pure	pure	PROPN
ejpam-4572	162	3	appl	appl	PROPN
ejpam-4572	162	4	.	.	PROPN
ejpam-4572	162	5	math	math	PROPN
ejpam-4572	162	6	,	,	PUNCT
ejpam-4572	162	7	16	16	NUM
ejpam-4572	162	8	(	(	PUNCT
ejpam-4572	162	9	1	1	NUM
ejpam-4572	162	10	)	)	PUNCT
ejpam-4572	162	11	(	(	PUNCT
ejpam-4572	162	12	2023	2023	NUM
ejpam-4572	162	13	)	)	PUNCT
ejpam-4572	162	14	,	,	PUNCT
ejpam-4572	162	15	465	465	NUM
ejpam-4572	162	16	-	-	SYM
ejpam-4572	162	17	478	478	NUM
ejpam-4572	162	18	469	469	NUM
ejpam-4572	162	19	3	3	NUM
ejpam-4572	162	20	.	.	PUNCT
ejpam-4572	162	21	characterizations	characterization	NOUN
ejpam-4572	162	22	of	of	ADP
ejpam-4572	162	23	upper	upper	ADJ
ejpam-4572	162	24	and	and	CCONJ
ejpam-4572	162	25	lower	low	ADJ
ejpam-4572	162	26	weakly	weakly	ADJ
ejpam-4572	162	27	α(λ	α(λ	PROPN
ejpam-4572	162	28	,	,	PUNCT
ejpam-4572	162	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	162	30	multifunctions	multifunction	NOUN
ejpam-4572	162	31	we	we	PRON
ejpam-4572	162	32	begin	begin	VERB
ejpam-4572	162	33	this	this	DET
ejpam-4572	162	34	section	section	NOUN
ejpam-4572	162	35	by	by	ADP
ejpam-4572	162	36	introducing	introduce	VERB
ejpam-4572	162	37	the	the	DET
ejpam-4572	162	38	notions	notion	NOUN
ejpam-4572	162	39	of	of	ADP
ejpam-4572	162	40	upper	upper	ADJ
ejpam-4572	162	41	and	and	CCONJ
ejpam-4572	162	42	lower	low	ADJ
ejpam-4572	162	43	weakly	weakly	ADJ
ejpam-4572	162	44	α(λ	α(λ	PROPN
ejpam-4572	162	45	,	,	PUNCT
ejpam-4572	162	46	sp)continuous	sp)continuous	ADJ
ejpam-4572	162	47	multifunctions	multifunction	NOUN
ejpam-4572	162	48	.	.	PUNCT
ejpam-4572	163	1	definition	definition	NOUN
ejpam-4572	163	2	1	1	NUM
ejpam-4572	163	3	.	.	PUNCT
ejpam-4572	164	1	a	a	DET
ejpam-4572	164	2	multifunction	multifunction	NOUN
ejpam-4572	164	3	f	f	NOUN
ejpam-4572	164	4	:	:	PUNCT
ejpam-4572	164	5	(	(	PUNCT
ejpam-4572	164	6	x	x	X
ejpam-4572	164	7	,	,	PUNCT
ejpam-4572	164	8	τ	τ	X
ejpam-4572	164	9	)	)	PUNCT
ejpam-4572	164	10	→	→	SYM
ejpam-4572	164	11	(	(	PUNCT
ejpam-4572	164	12	y	y	PROPN
ejpam-4572	164	13	,	,	PUNCT
ejpam-4572	164	14	σ	σ	PROPN
ejpam-4572	164	15	)	)	PUNCT
ejpam-4572	164	16	is	be	AUX
ejpam-4572	164	17	said	say	VERB
ejpam-4572	164	18	to	to	PART
ejpam-4572	164	19	be	be	AUX
ejpam-4572	164	20	:	:	PUNCT
ejpam-4572	164	21	(	(	PUNCT
ejpam-4572	164	22	1	1	X
ejpam-4572	164	23	)	)	PUNCT
ejpam-4572	164	24	upper	upper	ADJ
ejpam-4572	164	25	weakly	weakly	ADJ
ejpam-4572	164	26	α(λ	α(λ	PROPN
ejpam-4572	164	27	,	,	PUNCT
ejpam-4572	164	28	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	164	29	at	at	ADP
ejpam-4572	164	30	x	x	X
ejpam-4572	164	31	∈	∈	PROPN
ejpam-4572	164	32	x	x	INTJ
ejpam-4572	164	33	if	if	SCONJ
ejpam-4572	164	34	,	,	PUNCT
ejpam-4572	164	35	for	for	ADP
ejpam-4572	164	36	each	each	DET
ejpam-4572	164	37	s(λ	s(λ	PROPN
ejpam-4572	164	38	,	,	PUNCT
ejpam-4572	164	39	sp)-open	sp)-open	VERB
ejpam-4572	164	40	set	set	VERB
ejpam-4572	164	41	u	u	NOUN
ejpam-4572	164	42	of	of	ADP
ejpam-4572	164	43	x	x	PUNCT
ejpam-4572	164	44	containing	contain	VERB
ejpam-4572	164	45	x	x	PUNCT
ejpam-4572	164	46	and	and	CCONJ
ejpam-4572	164	47	each	each	DET
ejpam-4572	164	48	(	(	PUNCT
ejpam-4572	164	49	λ	λ	PROPN
ejpam-4572	164	50	,	,	PUNCT
ejpam-4572	164	51	sp)-open	sp)-open	NOUN
ejpam-4572	164	52	set	set	VERB
ejpam-4572	164	53	v	v	NUM
ejpam-4572	164	54	of	of	ADP
ejpam-4572	164	55	y	y	PROPN
ejpam-4572	164	56	containing	contain	VERB
ejpam-4572	164	57	f	f	PROPN
ejpam-4572	164	58	(	(	PUNCT
ejpam-4572	164	59	x	x	NOUN
ejpam-4572	164	60	)	)	PUNCT
ejpam-4572	164	61	,	,	PUNCT
ejpam-4572	164	62	there	there	PRON
ejpam-4572	164	63	exists	exist	VERB
ejpam-4572	164	64	a	a	DET
ejpam-4572	164	65	nonempty	nonempty	ADJ
ejpam-4572	164	66	(	(	PUNCT
ejpam-4572	164	67	λ	λ	NOUN
ejpam-4572	164	68	,	,	PUNCT
ejpam-4572	164	69	sp)-open	sp)-open	NOUN
ejpam-4572	164	70	set	set	VERB
ejpam-4572	164	71	g	g	NOUN
ejpam-4572	164	72	of	of	ADP
ejpam-4572	164	73	x	x	SYM
ejpam-4572	164	74	such	such	ADJ
ejpam-4572	164	75	that	that	SCONJ
ejpam-4572	164	76	g	g	PROPN
ejpam-4572	164	77	⊆	⊆	NUM
ejpam-4572	164	78	u	u	NOUN
ejpam-4572	164	79	and	and	CCONJ
ejpam-4572	164	80	f	f	PROPN
ejpam-4572	164	81	(	(	PUNCT
ejpam-4572	164	82	g	g	NOUN
ejpam-4572	164	83	)	)	PUNCT
ejpam-4572	164	84	⊆	⊆	NUM
ejpam-4572	164	85	v	v	ADP
ejpam-4572	164	86	s(λ	s(λ	PROPN
ejpam-4572	164	87	,	,	PUNCT
ejpam-4572	164	88	sp	sp	NOUN
ejpam-4572	164	89	)	)	PUNCT
ejpam-4572	164	90	;	;	PUNCT
ejpam-4572	164	91	(	(	PUNCT
ejpam-4572	164	92	2	2	X
ejpam-4572	164	93	)	)	PUNCT
ejpam-4572	164	94	lower	low	ADJ
ejpam-4572	164	95	weakly	weakly	ADJ
ejpam-4572	164	96	α(λ	α(λ	PROPN
ejpam-4572	164	97	,	,	PUNCT
ejpam-4572	164	98	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	164	99	at	at	ADP
ejpam-4572	164	100	x	x	X
ejpam-4572	164	101	∈	∈	PROPN
ejpam-4572	164	102	x	x	INTJ
ejpam-4572	164	103	if	if	SCONJ
ejpam-4572	164	104	,	,	PUNCT
ejpam-4572	164	105	for	for	ADP
ejpam-4572	164	106	each	each	DET
ejpam-4572	164	107	s(λ	s(λ	PROPN
ejpam-4572	164	108	,	,	PUNCT
ejpam-4572	164	109	sp)-open	sp)-open	VERB
ejpam-4572	164	110	set	set	VERB
ejpam-4572	164	111	u	u	NOUN
ejpam-4572	164	112	of	of	ADP
ejpam-4572	164	113	x	x	PUNCT
ejpam-4572	164	114	containing	contain	VERB
ejpam-4572	164	115	x	x	PUNCT
ejpam-4572	164	116	and	and	CCONJ
ejpam-4572	164	117	each	each	DET
ejpam-4572	164	118	(	(	PUNCT
ejpam-4572	164	119	λ	λ	PROPN
ejpam-4572	164	120	,	,	PUNCT
ejpam-4572	164	121	sp)-open	sp)-open	NOUN
ejpam-4572	164	122	set	set	VERB
ejpam-4572	164	123	v	v	NUM
ejpam-4572	164	124	of	of	ADP
ejpam-4572	164	125	y	y	PRON
ejpam-4572	164	126	such	such	ADJ
ejpam-4572	164	127	that	that	SCONJ
ejpam-4572	164	128	f	f	PROPN
ejpam-4572	164	129	(	(	PUNCT
ejpam-4572	164	130	x	x	NOUN
ejpam-4572	164	131	)	)	PUNCT
ejpam-4572	164	132	∩	∩	NOUN
ejpam-4572	164	133	v	v	ADP
ejpam-4572	164	134	̸=	̸=	PROPN
ejpam-4572	164	135	∅	∅	NOUN
ejpam-4572	164	136	,	,	PUNCT
ejpam-4572	164	137	there	there	PRON
ejpam-4572	164	138	exists	exist	VERB
ejpam-4572	164	139	a	a	DET
ejpam-4572	164	140	nonempty	nonempty	ADJ
ejpam-4572	164	141	(	(	PUNCT
ejpam-4572	164	142	λ	λ	NOUN
ejpam-4572	164	143	,	,	PUNCT
ejpam-4572	164	144	sp)-open	sp)-open	NOUN
ejpam-4572	164	145	set	set	VERB
ejpam-4572	164	146	g	g	NOUN
ejpam-4572	164	147	of	of	ADP
ejpam-4572	164	148	x	x	SYM
ejpam-4572	164	149	such	such	ADJ
ejpam-4572	164	150	that	that	SCONJ
ejpam-4572	164	151	g	g	PROPN
ejpam-4572	164	152	⊆	⊆	NUM
ejpam-4572	164	153	u	u	NOUN
ejpam-4572	164	154	and	and	CCONJ
ejpam-4572	164	155	f	f	PROPN
ejpam-4572	164	156	(	(	PUNCT
ejpam-4572	164	157	z	z	NOUN
ejpam-4572	164	158	)	)	PUNCT
ejpam-4572	164	159	∩	∩	PROPN
ejpam-4572	164	160	v	v	ADP
ejpam-4572	164	161	s(λ	s(λ	PROPN
ejpam-4572	164	162	,	,	PUNCT
ejpam-4572	164	163	sp	sp	NOUN
ejpam-4572	164	164	)	)	PUNCT
ejpam-4572	164	165	̸=	̸=	NOUN
ejpam-4572	164	166	∅	∅	NOUN
ejpam-4572	164	167	for	for	ADP
ejpam-4572	164	168	every	every	DET
ejpam-4572	164	169	z	z	PROPN
ejpam-4572	164	170	∈	∈	PROPN
ejpam-4572	164	171	g	g	NOUN
ejpam-4572	164	172	;	;	PUNCT
ejpam-4572	164	173	(	(	PUNCT
ejpam-4572	164	174	3	3	X
ejpam-4572	164	175	)	)	PUNCT
ejpam-4572	164	176	upper	upper	ADJ
ejpam-4572	164	177	(	(	PUNCT
ejpam-4572	164	178	lower	low	ADJ
ejpam-4572	164	179	)	)	PUNCT
ejpam-4572	164	180	weakly	weakly	ADJ
ejpam-4572	164	181	α(λ	α(λ	PROPN
ejpam-4572	164	182	,	,	PUNCT
ejpam-4572	164	183	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	164	184	if	if	SCONJ
ejpam-4572	164	185	f	f	PROPN
ejpam-4572	164	186	has	have	VERB
ejpam-4572	164	187	this	this	DET
ejpam-4572	164	188	property	property	NOUN
ejpam-4572	164	189	at	at	ADP
ejpam-4572	164	190	each	each	DET
ejpam-4572	164	191	point	point	NOUN
ejpam-4572	164	192	of	of	ADP
ejpam-4572	164	193	x.	x.	NOUN
ejpam-4572	164	194	theorem	theorem	VERB
ejpam-4572	164	195	1	1	NUM
ejpam-4572	164	196	.	.	X
ejpam-4572	164	197	for	for	ADP
ejpam-4572	164	198	a	a	DET
ejpam-4572	164	199	multifunction	multifunction	NOUN
ejpam-4572	164	200	f	f	NOUN
ejpam-4572	164	201	:	:	PUNCT
ejpam-4572	164	202	(	(	PUNCT
ejpam-4572	164	203	x	x	X
ejpam-4572	164	204	,	,	PUNCT
ejpam-4572	164	205	τ	τ	X
ejpam-4572	164	206	)	)	PUNCT
ejpam-4572	164	207	→	→	SYM
ejpam-4572	164	208	(	(	PUNCT
ejpam-4572	164	209	y	y	PROPN
ejpam-4572	164	210	,	,	PUNCT
ejpam-4572	164	211	σ	σ	PROPN
ejpam-4572	164	212	)	)	PUNCT
ejpam-4572	164	213	,	,	PUNCT
ejpam-4572	164	214	the	the	DET
ejpam-4572	164	215	following	follow	VERB
ejpam-4572	164	216	properties	property	NOUN
ejpam-4572	164	217	are	be	AUX
ejpam-4572	164	218	equivalent	equivalent	ADJ
ejpam-4572	164	219	:	:	PUNCT
ejpam-4572	164	220	(	(	PUNCT
ejpam-4572	164	221	1	1	X
ejpam-4572	164	222	)	)	PUNCT
ejpam-4572	164	223	f	f	PROPN
ejpam-4572	164	224	is	be	AUX
ejpam-4572	164	225	upper	upper	ADJ
ejpam-4572	164	226	weakly	weakly	ADJ
ejpam-4572	164	227	α(λ	α(λ	PROPN
ejpam-4572	164	228	,	,	PUNCT
ejpam-4572	164	229	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	164	230	at	at	ADP
ejpam-4572	164	231	a	a	DET
ejpam-4572	164	232	point	point	NOUN
ejpam-4572	164	233	x	x	X
ejpam-4572	164	234	∈	∈	NOUN
ejpam-4572	164	235	x	x	X
ejpam-4572	164	236	;	;	PUNCT
ejpam-4572	164	237	(	(	PUNCT
ejpam-4572	164	238	2	2	X
ejpam-4572	164	239	)	)	PUNCT
ejpam-4572	164	240	for	for	ADP
ejpam-4572	164	241	any	any	DET
ejpam-4572	164	242	(	(	PUNCT
ejpam-4572	164	243	λ	λ	NOUN
ejpam-4572	164	244	,	,	PUNCT
ejpam-4572	164	245	sp)-open	sp)-open	NOUN
ejpam-4572	164	246	set	set	VERB
ejpam-4572	164	247	v	v	NUM
ejpam-4572	164	248	of	of	ADP
ejpam-4572	164	249	y	y	PROPN
ejpam-4572	164	250	containing	contain	VERB
ejpam-4572	164	251	f	f	PROPN
ejpam-4572	164	252	(	(	PUNCT
ejpam-4572	164	253	x	x	NOUN
ejpam-4572	164	254	)	)	PUNCT
ejpam-4572	164	255	,	,	PUNCT
ejpam-4572	164	256	there	there	PRON
ejpam-4572	164	257	exists	exist	VERB
ejpam-4572	164	258	an	an	DET
ejpam-4572	164	259	α(λ	α(λ	PROPN
ejpam-4572	164	260	,	,	PUNCT
ejpam-4572	164	261	sp)-open	sp)-open	VERB
ejpam-4572	164	262	set	set	VERB
ejpam-4572	164	263	u	u	NOUN
ejpam-4572	164	264	of	of	ADP
ejpam-4572	164	265	x	x	PUNCT
ejpam-4572	164	266	containing	contain	VERB
ejpam-4572	164	267	x	x	PUNCT
ejpam-4572	165	1	such	such	ADJ
ejpam-4572	165	2	that	that	SCONJ
ejpam-4572	165	3	f	f	PROPN
ejpam-4572	165	4	(	(	PUNCT
ejpam-4572	165	5	u	u	NOUN
ejpam-4572	165	6	)	)	PUNCT
ejpam-4572	165	7	⊆	⊆	NUM
ejpam-4572	165	8	v	v	NOUN
ejpam-4572	165	9	(	(	PUNCT
ejpam-4572	165	10	λ	λ	NOUN
ejpam-4572	165	11	,	,	PUNCT
ejpam-4572	165	12	sp	sp	NOUN
ejpam-4572	165	13	)	)	PUNCT
ejpam-4572	165	14	;	;	PUNCT
ejpam-4572	165	15	(	(	PUNCT
ejpam-4572	165	16	3	3	X
ejpam-4572	165	17	)	)	PUNCT
ejpam-4572	165	18	x	x	SYM
ejpam-4572	165	19	∈	∈	PROPN
ejpam-4572	166	1	[	[	X
ejpam-4572	166	2	f+(v	f+(v	NOUN
ejpam-4572	166	3	(	(	PUNCT
ejpam-4572	166	4	λ	λ	PROPN
ejpam-4572	166	5	,	,	PUNCT
ejpam-4572	166	6	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	166	7	,	,	PUNCT
ejpam-4572	166	8	sp	sp	NOUN
ejpam-4572	166	9	)	)	PUNCT
ejpam-4572	166	10	for	for	ADP
ejpam-4572	166	11	every	every	DET
ejpam-4572	166	12	(	(	PUNCT
ejpam-4572	166	13	λ	λ	NOUN
ejpam-4572	166	14	,	,	PUNCT
ejpam-4572	166	15	sp)-open	sp)-open	NOUN
ejpam-4572	166	16	set	set	VERB
ejpam-4572	166	17	v	v	NUM
ejpam-4572	166	18	of	of	ADP
ejpam-4572	166	19	y	y	PROPN
ejpam-4572	166	20	containing	contain	VERB
ejpam-4572	166	21	f	f	PROPN
ejpam-4572	166	22	(	(	PUNCT
ejpam-4572	166	23	x	x	NOUN
ejpam-4572	166	24	)	)	PUNCT
ejpam-4572	166	25	;	;	PUNCT
ejpam-4572	166	26	(	(	PUNCT
ejpam-4572	166	27	4	4	X
ejpam-4572	166	28	)	)	PUNCT
ejpam-4572	166	29	x	x	SYM
ejpam-4572	166	30	∈	∈	PROPN
ejpam-4572	167	1	[	[	X
ejpam-4572	167	2	[	[	X
ejpam-4572	167	3	[	[	X
ejpam-4572	167	4	f+(v	f+(v	PROPN
ejpam-4572	167	5	(	(	PUNCT
ejpam-4572	167	6	λ	λ	PROPN
ejpam-4572	167	7	,	,	PUNCT
ejpam-4572	167	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	167	9	,	,	PUNCT
ejpam-4572	167	10	sp	sp	NOUN
ejpam-4572	167	11	)	)	PUNCT
ejpam-4572	167	12	]	]	PUNCT
ejpam-4572	167	13	(	(	PUNCT
ejpam-4572	167	14	λ	λ	X
ejpam-4572	167	15	,	,	PUNCT
ejpam-4572	167	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	167	17	,	,	PUNCT
ejpam-4572	167	18	sp	sp	NOUN
ejpam-4572	167	19	)	)	PUNCT
ejpam-4572	167	20	for	for	ADP
ejpam-4572	167	21	every	every	DET
ejpam-4572	167	22	(	(	PUNCT
ejpam-4572	167	23	λ	λ	NOUN
ejpam-4572	167	24	,	,	PUNCT
ejpam-4572	167	25	sp)-open	sp)-open	NOUN
ejpam-4572	167	26	set	set	VERB
ejpam-4572	167	27	v	v	NUM
ejpam-4572	167	28	of	of	ADP
ejpam-4572	167	29	y	y	PROPN
ejpam-4572	167	30	containing	contain	VERB
ejpam-4572	167	31	f	f	PROPN
ejpam-4572	167	32	(	(	PUNCT
ejpam-4572	167	33	x	x	NOUN
ejpam-4572	167	34	)	)	PUNCT
ejpam-4572	167	35	.	.	PUNCT
ejpam-4572	168	1	proof	proof	NOUN
ejpam-4572	168	2	.	.	PUNCT
ejpam-4572	169	1	(	(	PUNCT
ejpam-4572	169	2	1	1	X
ejpam-4572	169	3	)	)	PUNCT
ejpam-4572	169	4	⇒	⇒	NOUN
ejpam-4572	169	5	(	(	PUNCT
ejpam-4572	169	6	2	2	NUM
ejpam-4572	169	7	):	):	PUNCT
ejpam-4572	169	8	let	let	VERB
ejpam-4572	169	9	v	v	PART
ejpam-4572	169	10	be	be	AUX
ejpam-4572	169	11	any	any	DET
ejpam-4572	169	12	(	(	PUNCT
ejpam-4572	169	13	λ	λ	NOUN
ejpam-4572	169	14	,	,	PUNCT
ejpam-4572	169	15	sp)-open	sp)-open	ADJ
ejpam-4572	169	16	set	set	NOUN
ejpam-4572	169	17	of	of	ADP
ejpam-4572	169	18	y	y	PROPN
ejpam-4572	169	19	containing	contain	VERB
ejpam-4572	169	20	f	f	PROPN
ejpam-4572	169	21	(	(	PUNCT
ejpam-4572	169	22	x	x	NOUN
ejpam-4572	169	23	)	)	PUNCT
ejpam-4572	169	24	.	.	PUNCT
ejpam-4572	170	1	by	by	ADP
ejpam-4572	170	2	sλspo(x	sλspo(x	PROPN
ejpam-4572	170	3	,	,	PUNCT
ejpam-4572	170	4	x	x	NOUN
ejpam-4572	170	5	)	)	PUNCT
ejpam-4572	170	6	,	,	PUNCT
ejpam-4572	170	7	we	we	PRON
ejpam-4572	170	8	denote	denote	VERB
ejpam-4572	170	9	the	the	DET
ejpam-4572	170	10	family	family	NOUN
ejpam-4572	170	11	of	of	ADP
ejpam-4572	170	12	all	all	DET
ejpam-4572	170	13	s(λ	s(λ	NOUN
ejpam-4572	170	14	,	,	PUNCT
ejpam-4572	170	15	sp)-open	sp)-open	ADJ
ejpam-4572	170	16	set	set	NOUN
ejpam-4572	170	17	of	of	ADP
ejpam-4572	170	18	x	x	PUNCT
ejpam-4572	170	19	containing	contain	VERB
ejpam-4572	170	20	x.	x.	NOUN
ejpam-4572	170	21	for	for	ADP
ejpam-4572	170	22	each	each	DET
ejpam-4572	170	23	s(λ	s(λ	PROPN
ejpam-4572	170	24	,	,	PUNCT
ejpam-4572	170	25	sp)-open	sp)-open	VERB
ejpam-4572	170	26	set	set	VERB
ejpam-4572	170	27	u	u	NOUN
ejpam-4572	170	28	of	of	ADP
ejpam-4572	170	29	x	x	SYM
ejpam-4572	170	30	containing	contain	VERB
ejpam-4572	170	31	x	x	PRON
ejpam-4572	170	32	,	,	PUNCT
ejpam-4572	170	33	there	there	PRON
ejpam-4572	170	34	exists	exist	VERB
ejpam-4572	170	35	a	a	DET
ejpam-4572	170	36	nonempty	nonempty	ADJ
ejpam-4572	170	37	(	(	PUNCT
ejpam-4572	170	38	λ	λ	NOUN
ejpam-4572	170	39	,	,	PUNCT
ejpam-4572	170	40	sp)-open	sp)-open	ADJ
ejpam-4572	170	41	set	set	NOUN
ejpam-4572	170	42	gu	gu	NOUN
ejpam-4572	170	43	of	of	ADP
ejpam-4572	170	44	x	x	SYM
ejpam-4572	170	45	such	such	ADJ
ejpam-4572	170	46	that	that	DET
ejpam-4572	170	47	gu	gu	NOUN
ejpam-4572	170	48	⊆	⊆	NUM
ejpam-4572	170	49	u	u	NOUN
ejpam-4572	170	50	and	and	CCONJ
ejpam-4572	170	51	f	f	PROPN
ejpam-4572	170	52	(	(	PUNCT
ejpam-4572	170	53	gu	gu	NOUN
ejpam-4572	170	54	)	)	PUNCT
ejpam-4572	170	55	⊆	⊆	NUM
ejpam-4572	170	56	v	v	NOUN
ejpam-4572	170	57	(	(	PUNCT
ejpam-4572	170	58	λ	λ	NOUN
ejpam-4572	170	59	,	,	PUNCT
ejpam-4572	170	60	sp	sp	NOUN
ejpam-4572	170	61	)	)	PUNCT
ejpam-4572	170	62	.	.	PUNCT
ejpam-4572	171	1	let	let	VERB
ejpam-4572	171	2	w	w	NOUN
ejpam-4572	171	3	=	=	SYM
ejpam-4572	172	1	∪{gu	∪{gu	PROPN
ejpam-4572	172	2	|	|	ADV
ejpam-4572	172	3	u	u	PROPN
ejpam-4572	172	4	∈	∈	PROPN
ejpam-4572	172	5	sλspo(x	sλspo(x	PROPN
ejpam-4572	172	6	,	,	PUNCT
ejpam-4572	172	7	x	x	NOUN
ejpam-4572	172	8	)	)	PUNCT
ejpam-4572	172	9	}	}	PUNCT
ejpam-4572	172	10	.	.	PUNCT
ejpam-4572	173	1	put	put	VERB
ejpam-4572	173	2	s	s	PART
ejpam-4572	173	3	=	=	X
ejpam-4572	173	4	w	w	NOUN
ejpam-4572	173	5	∪	∪	X
ejpam-4572	173	6	{	{	PUNCT
ejpam-4572	173	7	x	x	NOUN
ejpam-4572	173	8	}	}	PUNCT
ejpam-4572	173	9	,	,	PUNCT
ejpam-4572	173	10	then	then	ADV
ejpam-4572	173	11	w	w	NOUN
ejpam-4572	173	12	is	be	AUX
ejpam-4572	173	13	(	(	PUNCT
ejpam-4572	173	14	λ	λ	INTJ
ejpam-4572	173	15	,	,	PUNCT
ejpam-4572	173	16	sp)-open	sp)-open	ADJ
ejpam-4572	173	17	in	in	ADP
ejpam-4572	173	18	x	x	X
ejpam-4572	173	19	,	,	PUNCT
ejpam-4572	173	20	x	x	SYM
ejpam-4572	173	21	∈	∈	PROPN
ejpam-4572	173	22	w	w	PROPN
ejpam-4572	173	23	s(λ	s(λ	PROPN
ejpam-4572	173	24	,	,	PUNCT
ejpam-4572	173	25	sp	sp	NOUN
ejpam-4572	173	26	)	)	PUNCT
ejpam-4572	173	27	and	and	CCONJ
ejpam-4572	173	28	f	f	PROPN
ejpam-4572	173	29	(	(	PUNCT
ejpam-4572	173	30	w	w	PROPN
ejpam-4572	173	31	)	)	PUNCT
ejpam-4572	173	32	⊆	⊆	NUM
ejpam-4572	173	33	v	v	NOUN
ejpam-4572	173	34	(	(	PUNCT
ejpam-4572	173	35	λ	λ	NOUN
ejpam-4572	173	36	,	,	PUNCT
ejpam-4572	173	37	sp	sp	NOUN
ejpam-4572	173	38	)	)	PUNCT
ejpam-4572	173	39	.	.	PUNCT
ejpam-4572	174	1	therefore	therefore	ADV
ejpam-4572	174	2	,	,	PUNCT
ejpam-4572	174	3	we	we	PRON
ejpam-4572	174	4	have	have	VERB
ejpam-4572	174	5	s	s	PROPN
ejpam-4572	174	6	is	be	AUX
ejpam-4572	174	7	an	an	DET
ejpam-4572	174	8	α(λ	α(λ	PROPN
ejpam-4572	174	9	,	,	PUNCT
ejpam-4572	174	10	sp)-open	sp)-open	ADJ
ejpam-4572	174	11	set	set	NOUN
ejpam-4572	174	12	of	of	ADP
ejpam-4572	174	13	x	x	PUNCT
ejpam-4572	174	14	containing	contain	VERB
ejpam-4572	174	15	x	x	PUNCT
ejpam-4572	174	16	by	by	ADP
ejpam-4572	174	17	lemma	lemma	PROPN
ejpam-4572	174	18	6	6	NUM
ejpam-4572	174	19	and	and	CCONJ
ejpam-4572	174	20	f	f	PROPN
ejpam-4572	174	21	(	(	PUNCT
ejpam-4572	174	22	s	s	PROPN
ejpam-4572	174	23	)	)	PUNCT
ejpam-4572	174	24	⊆	⊆	NUM
ejpam-4572	174	25	v	v	NOUN
ejpam-4572	174	26	(	(	PUNCT
ejpam-4572	174	27	λ	λ	NOUN
ejpam-4572	174	28	,	,	PUNCT
ejpam-4572	174	29	sp	sp	NOUN
ejpam-4572	174	30	)	)	PUNCT
ejpam-4572	174	31	.	.	PUNCT
ejpam-4572	175	1	(	(	PUNCT
ejpam-4572	175	2	2	2	X
ejpam-4572	175	3	)	)	PUNCT
ejpam-4572	175	4	⇒	⇒	NOUN
ejpam-4572	175	5	(	(	PUNCT
ejpam-4572	175	6	3	3	NUM
ejpam-4572	175	7	):	):	PUNCT
ejpam-4572	175	8	let	let	VERB
ejpam-4572	175	9	v	v	PART
ejpam-4572	175	10	be	be	AUX
ejpam-4572	175	11	any	any	DET
ejpam-4572	175	12	(	(	PUNCT
ejpam-4572	175	13	λ	λ	NOUN
ejpam-4572	175	14	,	,	PUNCT
ejpam-4572	175	15	sp)-open	sp)-open	ADJ
ejpam-4572	175	16	set	set	NOUN
ejpam-4572	175	17	of	of	ADP
ejpam-4572	175	18	y	y	PROPN
ejpam-4572	175	19	containing	contain	VERB
ejpam-4572	175	20	f	f	PROPN
ejpam-4572	175	21	(	(	PUNCT
ejpam-4572	175	22	x	x	NOUN
ejpam-4572	175	23	)	)	PUNCT
ejpam-4572	175	24	.	.	PUNCT
ejpam-4572	176	1	then	then	ADV
ejpam-4572	176	2	,	,	PUNCT
ejpam-4572	176	3	there	there	PRON
ejpam-4572	176	4	exists	exist	VERB
ejpam-4572	176	5	an	an	DET
ejpam-4572	176	6	α(λ	α(λ	PROPN
ejpam-4572	176	7	,	,	PUNCT
ejpam-4572	176	8	sp)-open	sp)-open	VERB
ejpam-4572	176	9	set	set	NOUN
ejpam-4572	176	10	s	s	PRON
ejpam-4572	176	11	of	of	ADP
ejpam-4572	176	12	x	x	PUNCT
ejpam-4572	176	13	containing	contain	VERB
ejpam-4572	176	14	x	x	PUNCT
ejpam-4572	176	15	such	such	ADJ
ejpam-4572	176	16	that	that	SCONJ
ejpam-4572	176	17	f	f	PROPN
ejpam-4572	176	18	(	(	PUNCT
ejpam-4572	176	19	s	s	PROPN
ejpam-4572	176	20	)	)	PUNCT
ejpam-4572	176	21	⊆	⊆	NUM
ejpam-4572	176	22	v	v	NOUN
ejpam-4572	176	23	(	(	PUNCT
ejpam-4572	176	24	λ	λ	NOUN
ejpam-4572	176	25	,	,	PUNCT
ejpam-4572	176	26	sp	sp	NOUN
ejpam-4572	176	27	)	)	PUNCT
ejpam-4572	176	28	.	.	PUNCT
ejpam-4572	177	1	thus	thus	ADV
ejpam-4572	177	2	,	,	PUNCT
ejpam-4572	177	3	x	x	PUNCT
ejpam-4572	177	4	∈	∈	PROPN
ejpam-4572	177	5	s	s	VERB
ejpam-4572	177	6	⊆	⊆	NUM
ejpam-4572	177	7	f+(v	f+(v	NOUN
ejpam-4572	177	8	(	(	PUNCT
ejpam-4572	177	9	λ	λ	NOUN
ejpam-4572	177	10	,	,	PUNCT
ejpam-4572	177	11	sp	sp	NOUN
ejpam-4572	177	12	)	)	PUNCT
ejpam-4572	177	13	)	)	PUNCT
ejpam-4572	177	14	and	and	CCONJ
ejpam-4572	177	15	hence	hence	ADV
ejpam-4572	177	16	x	x	X
ejpam-4572	177	17	∈	∈	PROPN
ejpam-4572	177	18	[	[	X
ejpam-4572	177	19	f+(v	f+(v	NOUN
ejpam-4572	177	20	(	(	PUNCT
ejpam-4572	177	21	λ	λ	PROPN
ejpam-4572	177	22	,	,	PUNCT
ejpam-4572	177	23	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	177	24	,	,	PUNCT
ejpam-4572	177	25	sp	sp	NOUN
ejpam-4572	177	26	)	)	PUNCT
ejpam-4572	177	27	.	.	PUNCT
ejpam-4572	178	1	(	(	PUNCT
ejpam-4572	178	2	3	3	X
ejpam-4572	178	3	)	)	PUNCT
ejpam-4572	178	4	⇒	⇒	NOUN
ejpam-4572	178	5	(	(	PUNCT
ejpam-4572	178	6	4	4	NUM
ejpam-4572	178	7	):	):	PUNCT
ejpam-4572	178	8	let	let	VERB
ejpam-4572	178	9	v	v	PART
ejpam-4572	178	10	be	be	AUX
ejpam-4572	178	11	any	any	DET
ejpam-4572	178	12	(	(	PUNCT
ejpam-4572	178	13	λ	λ	NOUN
ejpam-4572	178	14	,	,	PUNCT
ejpam-4572	178	15	sp)-open	sp)-open	ADJ
ejpam-4572	178	16	set	set	NOUN
ejpam-4572	178	17	of	of	ADP
ejpam-4572	178	18	y	y	PROPN
ejpam-4572	178	19	containing	contain	VERB
ejpam-4572	178	20	f	f	PROPN
ejpam-4572	178	21	(	(	PUNCT
ejpam-4572	178	22	x	x	NOUN
ejpam-4572	178	23	)	)	PUNCT
ejpam-4572	178	24	.	.	PUNCT
ejpam-4572	179	1	now	now	ADV
ejpam-4572	179	2	put	put	VERB
ejpam-4572	179	3	u	u	NOUN
ejpam-4572	179	4	=	=	PUNCT
ejpam-4572	180	1	[	[	X
ejpam-4572	180	2	f+(v	f+(v	PROPN
ejpam-4572	180	3	(	(	PUNCT
ejpam-4572	180	4	λ	λ	PROPN
ejpam-4572	180	5	,	,	PUNCT
ejpam-4572	180	6	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	180	7	,	,	PUNCT
ejpam-4572	180	8	sp	sp	NOUN
ejpam-4572	180	9	)	)	PUNCT
ejpam-4572	180	10	.	.	PUNCT
ejpam-4572	181	1	c.	c.	PROPN
ejpam-4572	181	2	boonpok	boonpok	PROPN
ejpam-4572	181	3	,	,	PUNCT
ejpam-4572	181	4	m.	m.	NOUN
ejpam-4572	181	5	thongmoon	thongmoon	PROPN
ejpam-4572	181	6	/	/	SYM
ejpam-4572	181	7	eur	eur	PROPN
ejpam-4572	181	8	.	.	PUNCT
ejpam-4572	182	1	j.	j.	PROPN
ejpam-4572	182	2	pure	pure	PROPN
ejpam-4572	182	3	appl	appl	PROPN
ejpam-4572	182	4	.	.	PROPN
ejpam-4572	182	5	math	math	PROPN
ejpam-4572	182	6	,	,	PUNCT
ejpam-4572	182	7	16	16	NUM
ejpam-4572	182	8	(	(	PUNCT
ejpam-4572	182	9	1	1	NUM
ejpam-4572	182	10	)	)	PUNCT
ejpam-4572	182	11	(	(	PUNCT
ejpam-4572	182	12	2023	2023	NUM
ejpam-4572	182	13	)	)	PUNCT
ejpam-4572	182	14	,	,	PUNCT
ejpam-4572	182	15	465	465	NUM
ejpam-4572	182	16	-	-	SYM
ejpam-4572	182	17	478	478	NUM
ejpam-4572	182	18	470	470	NUM
ejpam-4572	182	19	then	then	ADV
ejpam-4572	182	20	,	,	PUNCT
ejpam-4572	182	21	u	u	PROPN
ejpam-4572	182	22	∈	∈	PROPN
ejpam-4572	182	23	αλspo(x	αλspo(x	PROPN
ejpam-4572	182	24	,	,	PUNCT
ejpam-4572	182	25	τ	τ	PROPN
ejpam-4572	182	26	)	)	PUNCT
ejpam-4572	182	27	and	and	CCONJ
ejpam-4572	182	28	x	x	PUNCT
ejpam-4572	182	29	∈	∈	PROPN
ejpam-4572	182	30	u	u	NOUN
ejpam-4572	182	31	⊆	⊆	NUM
ejpam-4572	182	32	f+(v	f+(v	NOUN
ejpam-4572	182	33	(	(	PUNCT
ejpam-4572	182	34	λ	λ	NOUN
ejpam-4572	182	35	,	,	PUNCT
ejpam-4572	182	36	sp	sp	NOUN
ejpam-4572	182	37	)	)	PUNCT
ejpam-4572	182	38	)	)	PUNCT
ejpam-4572	182	39	.	.	PUNCT
ejpam-4572	183	1	this	this	PRON
ejpam-4572	183	2	shows	show	VERB
ejpam-4572	183	3	that	that	SCONJ
ejpam-4572	183	4	x	x	X
ejpam-4572	183	5	∈	∈	PROPN
ejpam-4572	184	1	[	[	X
ejpam-4572	184	2	[	[	X
ejpam-4572	184	3	[	[	X
ejpam-4572	184	4	f+(v	f+(v	PROPN
ejpam-4572	184	5	(	(	PUNCT
ejpam-4572	184	6	λ	λ	PROPN
ejpam-4572	184	7	,	,	PUNCT
ejpam-4572	184	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	184	9	,	,	PUNCT
ejpam-4572	184	10	sp	sp	NOUN
ejpam-4572	184	11	)	)	PUNCT
ejpam-4572	184	12	]	]	PUNCT
ejpam-4572	184	13	(	(	PUNCT
ejpam-4572	184	14	λ	λ	X
ejpam-4572	184	15	,	,	PUNCT
ejpam-4572	184	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	184	17	,	,	PUNCT
ejpam-4572	184	18	sp	sp	NOUN
ejpam-4572	184	19	)	)	PUNCT
ejpam-4572	184	20	.	.	PUNCT
ejpam-4572	185	1	(	(	PUNCT
ejpam-4572	185	2	4	4	X
ejpam-4572	185	3	)	)	PUNCT
ejpam-4572	185	4	⇒	⇒	NOUN
ejpam-4572	185	5	(	(	PUNCT
ejpam-4572	185	6	1	1	NUM
ejpam-4572	185	7	):	):	PUNCT
ejpam-4572	185	8	let	let	VERB
ejpam-4572	185	9	u	u	PRON
ejpam-4572	185	10	be	be	AUX
ejpam-4572	185	11	any	any	DET
ejpam-4572	185	12	s(λ	s(λ	NOUN
ejpam-4572	185	13	,	,	PUNCT
ejpam-4572	185	14	sp)-open	sp)-open	ADJ
ejpam-4572	185	15	set	set	NOUN
ejpam-4572	185	16	of	of	ADP
ejpam-4572	185	17	x	x	PUNCT
ejpam-4572	185	18	containing	contain	VERB
ejpam-4572	185	19	x	x	PUNCT
ejpam-4572	185	20	and	and	CCONJ
ejpam-4572	185	21	let	let	VERB
ejpam-4572	185	22	v	v	PART
ejpam-4572	185	23	be	be	AUX
ejpam-4572	185	24	any	any	DET
ejpam-4572	185	25	(	(	PUNCT
ejpam-4572	185	26	λ	λ	NOUN
ejpam-4572	185	27	,	,	PUNCT
ejpam-4572	185	28	sp)open	sp)open	VERB
ejpam-4572	185	29	set	set	NOUN
ejpam-4572	185	30	of	of	ADP
ejpam-4572	185	31	y	y	PROPN
ejpam-4572	185	32	containing	contain	VERB
ejpam-4572	185	33	f	f	PROPN
ejpam-4572	185	34	(	(	PUNCT
ejpam-4572	185	35	x	x	NOUN
ejpam-4572	185	36	)	)	PUNCT
ejpam-4572	185	37	.	.	PUNCT
ejpam-4572	186	1	then	then	ADV
ejpam-4572	186	2	,	,	PUNCT
ejpam-4572	186	3	we	we	PRON
ejpam-4572	186	4	have	have	VERB
ejpam-4572	186	5	x	x	X
ejpam-4572	186	6	∈	∈	PROPN
ejpam-4572	187	1	[	[	X
ejpam-4572	187	2	[	[	X
ejpam-4572	187	3	[	[	X
ejpam-4572	187	4	f+(v	f+(v	PROPN
ejpam-4572	187	5	(	(	PUNCT
ejpam-4572	187	6	λ	λ	PROPN
ejpam-4572	187	7	,	,	PUNCT
ejpam-4572	187	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	187	9	,	,	PUNCT
ejpam-4572	187	10	sp	sp	NOUN
ejpam-4572	187	11	)	)	PUNCT
ejpam-4572	187	12	]	]	PUNCT
ejpam-4572	187	13	(	(	PUNCT
ejpam-4572	187	14	λ	λ	X
ejpam-4572	187	15	,	,	PUNCT
ejpam-4572	187	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	187	17	,	,	PUNCT
ejpam-4572	187	18	sp	sp	NOUN
ejpam-4572	187	19	)	)	PUNCT
ejpam-4572	187	20	=	=	PUNCT
ejpam-4572	188	1	[	[	X
ejpam-4572	188	2	[	[	X
ejpam-4572	188	3	f+(v	f+(v	PROPN
ejpam-4572	188	4	(	(	PUNCT
ejpam-4572	188	5	λ	λ	PROPN
ejpam-4572	188	6	,	,	PUNCT
ejpam-4572	188	7	sp))](λ	sp))](λ	PROPN
ejpam-4572	188	8	,	,	PUNCT
ejpam-4572	188	9	sp	sp	NOUN
ejpam-4572	188	10	)	)	PUNCT
ejpam-4572	188	11	]	]	PUNCT
ejpam-4572	189	1	s(λ	s(λ	PROPN
ejpam-4572	189	2	,	,	PUNCT
ejpam-4572	189	3	sp	sp	NOUN
ejpam-4572	189	4	)	)	PUNCT
ejpam-4572	189	5	.	.	PUNCT
ejpam-4572	190	1	it	it	PRON
ejpam-4572	190	2	follows	follow	VERB
ejpam-4572	190	3	from	from	ADP
ejpam-4572	190	4	lemma	lemma	PROPN
ejpam-4572	190	5	5	5	NUM
ejpam-4572	190	6	that	that	DET
ejpam-4572	190	7	∅	∅	NOUN
ejpam-4572	190	8	=	=	NOUN
ejpam-4572	190	9	̸	̸	NUM
ejpam-4572	190	10	u	u	NOUN
ejpam-4572	190	11	∩	∩	NOUN
ejpam-4572	190	12	[	[	X
ejpam-4572	190	13	f+(v	f+(v	PROPN
ejpam-4572	190	14	(	(	PUNCT
ejpam-4572	190	15	λ	λ	PROPN
ejpam-4572	190	16	,	,	PUNCT
ejpam-4572	190	17	sp))](λ	sp))](λ	PROPN
ejpam-4572	190	18	,	,	PUNCT
ejpam-4572	190	19	sp	sp	NOUN
ejpam-4572	190	20	)	)	PUNCT
ejpam-4572	190	21	∈	∈	PROPN
ejpam-4572	190	22	sλspo(x	sλspo(x	PROPN
ejpam-4572	190	23	,	,	PUNCT
ejpam-4572	190	24	τ	τ	PROPN
ejpam-4572	190	25	)	)	PUNCT
ejpam-4572	190	26	.	.	PUNCT
ejpam-4572	191	1	put	put	VERB
ejpam-4572	191	2	g	g	NOUN
ejpam-4572	191	3	=	=	PUNCT
ejpam-4572	192	1	[	[	X
ejpam-4572	192	2	u	u	NOUN
ejpam-4572	192	3	∩	∩	NOUN
ejpam-4572	192	4	[	[	X
ejpam-4572	192	5	f+(v	f+(v	PROPN
ejpam-4572	192	6	(	(	PUNCT
ejpam-4572	192	7	λ	λ	PROPN
ejpam-4572	192	8	,	,	PUNCT
ejpam-4572	192	9	sp))](λ	sp))](λ	PROPN
ejpam-4572	192	10	,	,	PUNCT
ejpam-4572	192	11	sp)](λ	sp)](λ	PROPN
ejpam-4572	192	12	,	,	PUNCT
ejpam-4572	192	13	sp	sp	NOUN
ejpam-4572	192	14	)	)	PUNCT
ejpam-4572	192	15	.	.	PUNCT
ejpam-4572	193	1	then	then	ADV
ejpam-4572	193	2	,	,	PUNCT
ejpam-4572	193	3	g	g	PROPN
ejpam-4572	193	4	is	be	AUX
ejpam-4572	193	5	a	a	DET
ejpam-4572	193	6	nonempty	nonempty	ADJ
ejpam-4572	193	7	(	(	PUNCT
ejpam-4572	193	8	λ	λ	NOUN
ejpam-4572	193	9	,	,	PUNCT
ejpam-4572	193	10	sp)-open	sp)-open	ADJ
ejpam-4572	193	11	set	set	NOUN
ejpam-4572	193	12	of	of	ADP
ejpam-4572	193	13	y	y	PROPN
ejpam-4572	193	14	,	,	PUNCT
ejpam-4572	193	15	g	g	PROPN
ejpam-4572	193	16	⊆	⊆	NUM
ejpam-4572	193	17	u	u	NOUN
ejpam-4572	193	18	and	and	CCONJ
ejpam-4572	193	19	f	f	PROPN
ejpam-4572	193	20	(	(	PUNCT
ejpam-4572	193	21	g	g	NOUN
ejpam-4572	193	22	)	)	PUNCT
ejpam-4572	193	23	⊆	⊆	NUM
ejpam-4572	193	24	v	v	NOUN
ejpam-4572	193	25	(	(	PUNCT
ejpam-4572	193	26	λ	λ	NOUN
ejpam-4572	193	27	,	,	PUNCT
ejpam-4572	193	28	sp	sp	NOUN
ejpam-4572	193	29	)	)	PUNCT
ejpam-4572	193	30	.	.	PUNCT
ejpam-4572	194	1	theorem	theorem	NOUN
ejpam-4572	194	2	2	2	NUM
ejpam-4572	194	3	.	.	X
ejpam-4572	194	4	for	for	ADP
ejpam-4572	194	5	a	a	DET
ejpam-4572	194	6	multifunction	multifunction	NOUN
ejpam-4572	195	1	f	f	NOUN
ejpam-4572	195	2	:	:	PUNCT
ejpam-4572	195	3	(	(	PUNCT
ejpam-4572	195	4	x	x	X
ejpam-4572	195	5	,	,	PUNCT
ejpam-4572	195	6	τ	τ	X
ejpam-4572	195	7	)	)	PUNCT
ejpam-4572	195	8	→	→	SYM
ejpam-4572	195	9	(	(	PUNCT
ejpam-4572	195	10	y	y	PROPN
ejpam-4572	195	11	,	,	PUNCT
ejpam-4572	195	12	σ	σ	PROPN
ejpam-4572	195	13	)	)	PUNCT
ejpam-4572	195	14	,	,	PUNCT
ejpam-4572	195	15	the	the	DET
ejpam-4572	195	16	following	follow	VERB
ejpam-4572	195	17	properties	property	NOUN
ejpam-4572	195	18	are	be	AUX
ejpam-4572	195	19	equivalent	equivalent	ADJ
ejpam-4572	195	20	:	:	PUNCT
ejpam-4572	195	21	(	(	PUNCT
ejpam-4572	195	22	1	1	X
ejpam-4572	195	23	)	)	PUNCT
ejpam-4572	195	24	f	f	PROPN
ejpam-4572	195	25	is	be	AUX
ejpam-4572	195	26	lower	low	ADJ
ejpam-4572	195	27	weakly	weakly	ADJ
ejpam-4572	195	28	α(λ	α(λ	PROPN
ejpam-4572	195	29	,	,	PUNCT
ejpam-4572	195	30	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	195	31	at	at	ADP
ejpam-4572	195	32	a	a	DET
ejpam-4572	195	33	point	point	NOUN
ejpam-4572	195	34	x	x	X
ejpam-4572	195	35	∈	∈	NOUN
ejpam-4572	195	36	x	x	X
ejpam-4572	195	37	;	;	PUNCT
ejpam-4572	195	38	(	(	PUNCT
ejpam-4572	195	39	2	2	X
ejpam-4572	195	40	)	)	PUNCT
ejpam-4572	195	41	for	for	ADP
ejpam-4572	195	42	any	any	DET
ejpam-4572	195	43	(	(	PUNCT
ejpam-4572	195	44	λ	λ	NOUN
ejpam-4572	195	45	,	,	PUNCT
ejpam-4572	195	46	sp)-open	sp)-open	NOUN
ejpam-4572	195	47	set	set	VERB
ejpam-4572	195	48	v	v	NUM
ejpam-4572	195	49	of	of	ADP
ejpam-4572	195	50	y	y	PRON
ejpam-4572	195	51	such	such	ADJ
ejpam-4572	195	52	that	that	SCONJ
ejpam-4572	195	53	f	f	PROPN
ejpam-4572	195	54	(	(	PUNCT
ejpam-4572	195	55	x)∩v	x)∩v	PROPN
ejpam-4572	195	56	̸=	̸=	PROPN
ejpam-4572	195	57	∅	∅	NOUN
ejpam-4572	195	58	,	,	PUNCT
ejpam-4572	195	59	there	there	PRON
ejpam-4572	195	60	exists	exist	VERB
ejpam-4572	195	61	an	an	DET
ejpam-4572	195	62	α(λ	α(λ	PROPN
ejpam-4572	195	63	,	,	PUNCT
ejpam-4572	195	64	sp)-open	sp)-open	VERB
ejpam-4572	195	65	set	set	VERB
ejpam-4572	195	66	u	u	NOUN
ejpam-4572	195	67	of	of	ADP
ejpam-4572	195	68	x	x	PUNCT
ejpam-4572	195	69	containing	contain	VERB
ejpam-4572	195	70	x	x	PUNCT
ejpam-4572	196	1	such	such	ADJ
ejpam-4572	196	2	that	that	SCONJ
ejpam-4572	196	3	f	f	PROPN
ejpam-4572	196	4	(	(	PUNCT
ejpam-4572	196	5	z	z	NOUN
ejpam-4572	196	6	)	)	PUNCT
ejpam-4572	196	7	⊆	⊆	NUM
ejpam-4572	196	8	v	v	NOUN
ejpam-4572	196	9	(	(	PUNCT
ejpam-4572	196	10	λ	λ	NOUN
ejpam-4572	196	11	,	,	PUNCT
ejpam-4572	196	12	sp	sp	NOUN
ejpam-4572	196	13	)	)	PUNCT
ejpam-4572	196	14	̸=	̸=	NOUN
ejpam-4572	196	15	∅	∅	NOUN
ejpam-4572	196	16	for	for	ADP
ejpam-4572	196	17	every	every	DET
ejpam-4572	196	18	z	z	NOUN
ejpam-4572	196	19	∈	∈	PROPN
ejpam-4572	196	20	u	u	NOUN
ejpam-4572	196	21	;	;	PUNCT
ejpam-4572	196	22	(	(	PUNCT
ejpam-4572	196	23	3	3	X
ejpam-4572	196	24	)	)	PUNCT
ejpam-4572	196	25	x	x	SYM
ejpam-4572	196	26	∈	∈	PROPN
ejpam-4572	197	1	[	[	X
ejpam-4572	197	2	f−(v	f−(v	ADJ
ejpam-4572	197	3	(	(	PUNCT
ejpam-4572	197	4	λ	λ	PROPN
ejpam-4572	197	5	,	,	PUNCT
ejpam-4572	197	6	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	197	7	,	,	PUNCT
ejpam-4572	197	8	sp	sp	NOUN
ejpam-4572	197	9	)	)	PUNCT
ejpam-4572	197	10	for	for	ADP
ejpam-4572	197	11	every	every	DET
ejpam-4572	197	12	(	(	PUNCT
ejpam-4572	197	13	λ	λ	NOUN
ejpam-4572	197	14	,	,	PUNCT
ejpam-4572	197	15	sp)-open	sp)-open	NOUN
ejpam-4572	197	16	set	set	VERB
ejpam-4572	197	17	v	v	NUM
ejpam-4572	197	18	of	of	ADP
ejpam-4572	197	19	y	y	PRON
ejpam-4572	197	20	such	such	ADJ
ejpam-4572	197	21	that	that	SCONJ
ejpam-4572	197	22	f	f	PROPN
ejpam-4572	197	23	(	(	PUNCT
ejpam-4572	197	24	x	x	NOUN
ejpam-4572	197	25	)	)	PUNCT
ejpam-4572	197	26	∩	∩	NOUN
ejpam-4572	197	27	v	v	ADP
ejpam-4572	197	28	̸=	̸=	PROPN
ejpam-4572	197	29	∅	∅	NOUN
ejpam-4572	197	30	;	;	PUNCT
ejpam-4572	197	31	(	(	PUNCT
ejpam-4572	197	32	4	4	X
ejpam-4572	197	33	)	)	PUNCT
ejpam-4572	197	34	x	x	SYM
ejpam-4572	198	1	∈	∈	PROPN
ejpam-4572	199	1	[	[	X
ejpam-4572	199	2	[	[	X
ejpam-4572	199	3	[	[	X
ejpam-4572	199	4	f−(v	f−(v	ADJ
ejpam-4572	199	5	(	(	PUNCT
ejpam-4572	199	6	λ	λ	PROPN
ejpam-4572	199	7	,	,	PUNCT
ejpam-4572	199	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	199	9	,	,	PUNCT
ejpam-4572	199	10	sp	sp	NOUN
ejpam-4572	199	11	)	)	PUNCT
ejpam-4572	199	12	]	]	PUNCT
ejpam-4572	199	13	(	(	PUNCT
ejpam-4572	199	14	λ	λ	X
ejpam-4572	199	15	,	,	PUNCT
ejpam-4572	199	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	199	17	,	,	PUNCT
ejpam-4572	199	18	sp	sp	NOUN
ejpam-4572	199	19	)	)	PUNCT
ejpam-4572	199	20	for	for	ADP
ejpam-4572	199	21	every	every	DET
ejpam-4572	199	22	(	(	PUNCT
ejpam-4572	199	23	λ	λ	NOUN
ejpam-4572	199	24	,	,	PUNCT
ejpam-4572	199	25	sp)-open	sp)-open	NOUN
ejpam-4572	199	26	set	set	VERB
ejpam-4572	199	27	v	v	NUM
ejpam-4572	199	28	of	of	ADP
ejpam-4572	199	29	y	y	PRON
ejpam-4572	199	30	such	such	ADJ
ejpam-4572	199	31	that	that	SCONJ
ejpam-4572	199	32	f	f	PROPN
ejpam-4572	199	33	(	(	PUNCT
ejpam-4572	199	34	x	x	NOUN
ejpam-4572	199	35	)	)	PUNCT
ejpam-4572	199	36	∩	∩	NOUN
ejpam-4572	199	37	v	v	ADP
ejpam-4572	199	38	̸=	̸=	PROPN
ejpam-4572	199	39	∅.	∅.	ADP
ejpam-4572	199	40	proof	proof	NOUN
ejpam-4572	199	41	.	.	PUNCT
ejpam-4572	200	1	the	the	DET
ejpam-4572	200	2	proof	proof	NOUN
ejpam-4572	200	3	is	be	AUX
ejpam-4572	200	4	similar	similar	ADJ
ejpam-4572	200	5	to	to	ADP
ejpam-4572	200	6	that	that	PRON
ejpam-4572	200	7	of	of	ADP
ejpam-4572	200	8	theorem	theorem	NOUN
ejpam-4572	200	9	1	1	NUM
ejpam-4572	200	10	.	.	PUNCT
ejpam-4572	200	11	definition	definition	NOUN
ejpam-4572	200	12	2	2	NUM
ejpam-4572	200	13	.	.	PUNCT
ejpam-4572	201	1	[	[	X
ejpam-4572	201	2	2	2	X
ejpam-4572	201	3	]	]	PUNCT
ejpam-4572	201	4	let	let	VERB
ejpam-4572	201	5	a	a	PRON
ejpam-4572	201	6	be	be	AUX
ejpam-4572	201	7	a	a	DET
ejpam-4572	201	8	subset	subset	NOUN
ejpam-4572	201	9	of	of	ADP
ejpam-4572	201	10	a	a	DET
ejpam-4572	201	11	topological	topological	ADJ
ejpam-4572	201	12	space	space	NOUN
ejpam-4572	201	13	(	(	PUNCT
ejpam-4572	201	14	x	x	X
ejpam-4572	201	15	,	,	PUNCT
ejpam-4572	201	16	τ	τ	PROPN
ejpam-4572	201	17	)	)	PUNCT
ejpam-4572	201	18	.	.	PUNCT
ejpam-4572	202	1	the	the	DET
ejpam-4572	202	2	θ(λ	θ(λ	PROPN
ejpam-4572	202	3	,	,	PUNCT
ejpam-4572	202	4	sp)-closure	sp)-closure	NOUN
ejpam-4572	202	5	of	of	ADP
ejpam-4572	202	6	a	a	DET
ejpam-4572	202	7	,	,	PUNCT
ejpam-4572	202	8	aθ(λ	aθ(λ	NOUN
ejpam-4572	202	9	,	,	PUNCT
ejpam-4572	202	10	sp	sp	NOUN
ejpam-4572	202	11	)	)	PUNCT
ejpam-4572	202	12	,	,	PUNCT
ejpam-4572	202	13	is	be	AUX
ejpam-4572	202	14	defined	define	VERB
ejpam-4572	202	15	as	as	SCONJ
ejpam-4572	202	16	follows	follow	VERB
ejpam-4572	202	17	:	:	PUNCT
ejpam-4572	202	18	aθ(λ	aθ(λ	NOUN
ejpam-4572	202	19	,	,	PUNCT
ejpam-4572	202	20	sp	sp	NOUN
ejpam-4572	202	21	)	)	PUNCT
ejpam-4572	202	22	=	=	PRON
ejpam-4572	203	1	{	{	PUNCT
ejpam-4572	203	2	x	x	PUNCT
ejpam-4572	203	3	∈	∈	NOUN
ejpam-4572	203	4	x	x	PUNCT
ejpam-4572	203	5	|	|	ADV
ejpam-4572	203	6	a	a	DET
ejpam-4572	203	7	∩	∩	ADJ
ejpam-4572	203	8	u	u	NOUN
ejpam-4572	203	9	(	(	PUNCT
ejpam-4572	203	10	λ	λ	PROPN
ejpam-4572	203	11	,	,	PUNCT
ejpam-4572	203	12	sp	sp	NOUN
ejpam-4572	203	13	)	)	PUNCT
ejpam-4572	203	14	̸=	̸=	NOUN
ejpam-4572	203	15	∅	∅	NOUN
ejpam-4572	203	16	for	for	ADP
ejpam-4572	203	17	each	each	DET
ejpam-4572	203	18	u	u	PROPN
ejpam-4572	203	19	∈	∈	PROPN
ejpam-4572	203	20	λspo(x	λspo(x	PROPN
ejpam-4572	203	21	,	,	PUNCT
ejpam-4572	203	22	τ	τ	X
ejpam-4572	203	23	)	)	PUNCT
ejpam-4572	203	24	containing	contain	VERB
ejpam-4572	203	25	x	x	X
ejpam-4572	203	26	}	}	PUNCT
ejpam-4572	203	27	.	.	PUNCT
ejpam-4572	204	1	lemma	lemma	PROPN
ejpam-4572	204	2	7	7	NUM
ejpam-4572	204	3	.	.	PUNCT
ejpam-4572	205	1	[	[	X
ejpam-4572	205	2	2	2	X
ejpam-4572	205	3	]	]	PUNCT
ejpam-4572	205	4	let	let	VERB
ejpam-4572	205	5	a	a	PRON
ejpam-4572	205	6	be	be	AUX
ejpam-4572	205	7	a	a	DET
ejpam-4572	205	8	subset	subset	NOUN
ejpam-4572	205	9	of	of	ADP
ejpam-4572	205	10	a	a	DET
ejpam-4572	205	11	topological	topological	ADJ
ejpam-4572	205	12	space	space	NOUN
ejpam-4572	205	13	(	(	PUNCT
ejpam-4572	205	14	x	x	X
ejpam-4572	205	15	,	,	PUNCT
ejpam-4572	205	16	τ	τ	PROPN
ejpam-4572	205	17	)	)	PUNCT
ejpam-4572	205	18	.	.	PUNCT
ejpam-4572	206	1	then	then	ADV
ejpam-4572	206	2	,	,	PUNCT
ejpam-4572	206	3	x	x	PUNCT
ejpam-4572	206	4	∈	∈	PROPN
ejpam-4572	206	5	a(λ	a(λ	ADV
ejpam-4572	206	6	,	,	PUNCT
ejpam-4572	206	7	sp	sp	NOUN
ejpam-4572	206	8	)	)	PUNCT
ejpam-4572	206	9	if	if	SCONJ
ejpam-4572	207	1	and	and	CCONJ
ejpam-4572	207	2	only	only	ADV
ejpam-4572	207	3	if	if	SCONJ
ejpam-4572	207	4	u	u	PROPN
ejpam-4572	207	5	∩a	∩a	PROPN
ejpam-4572	207	6	̸=	̸=	PROPN
ejpam-4572	207	7	∅	∅	NOUN
ejpam-4572	207	8	for	for	ADP
ejpam-4572	207	9	every	every	DET
ejpam-4572	207	10	u	u	PROPN
ejpam-4572	207	11	∈	∈	PROPN
ejpam-4572	207	12	λspo(x	λspo(x	PROPN
ejpam-4572	207	13	,	,	PUNCT
ejpam-4572	207	14	τ	τ	X
ejpam-4572	207	15	)	)	PUNCT
ejpam-4572	207	16	containing	contain	VERB
ejpam-4572	207	17	x.	x.	PROPN
ejpam-4572	207	18	lemma	lemma	PROPN
ejpam-4572	207	19	8	8	NUM
ejpam-4572	208	1	.	.	PUNCT
ejpam-4572	209	1	[	[	X
ejpam-4572	209	2	2	2	X
ejpam-4572	209	3	]	]	PUNCT
ejpam-4572	209	4	for	for	ADP
ejpam-4572	209	5	a	a	DET
ejpam-4572	209	6	subset	subset	NOUN
ejpam-4572	209	7	a	a	PRON
ejpam-4572	209	8	of	of	ADP
ejpam-4572	209	9	a	a	DET
ejpam-4572	209	10	topological	topological	ADJ
ejpam-4572	209	11	space	space	NOUN
ejpam-4572	209	12	(	(	PUNCT
ejpam-4572	209	13	x	x	X
ejpam-4572	209	14	,	,	PUNCT
ejpam-4572	209	15	τ	τ	PROPN
ejpam-4572	209	16	)	)	PUNCT
ejpam-4572	209	17	,	,	PUNCT
ejpam-4572	209	18	the	the	DET
ejpam-4572	209	19	following	follow	VERB
ejpam-4572	209	20	properties	property	NOUN
ejpam-4572	209	21	hold	hold	VERB
ejpam-4572	209	22	:	:	PUNCT
ejpam-4572	209	23	(	(	PUNCT
ejpam-4572	209	24	1	1	X
ejpam-4572	209	25	)	)	PUNCT
ejpam-4572	209	26	if	if	SCONJ
ejpam-4572	209	27	a	a	PRON
ejpam-4572	209	28	is	be	AUX
ejpam-4572	209	29	(	(	PUNCT
ejpam-4572	209	30	λ	λ	NOUN
ejpam-4572	209	31	,	,	PUNCT
ejpam-4572	209	32	sp)-open	sp)-open	ADJ
ejpam-4572	209	33	in	in	ADP
ejpam-4572	209	34	(	(	PUNCT
ejpam-4572	209	35	x	x	NOUN
ejpam-4572	209	36	,	,	PUNCT
ejpam-4572	209	37	τ	τ	PROPN
ejpam-4572	209	38	)	)	PUNCT
ejpam-4572	209	39	,	,	PUNCT
ejpam-4572	209	40	then	then	ADV
ejpam-4572	209	41	a(λ	a(λ	ADV
ejpam-4572	209	42	,	,	PUNCT
ejpam-4572	209	43	sp	sp	NOUN
ejpam-4572	209	44	)	)	PUNCT
ejpam-4572	209	45	=	=	SYM
ejpam-4572	209	46	aθ(λ	aθ(λ	NOUN
ejpam-4572	209	47	,	,	PUNCT
ejpam-4572	209	48	sp	sp	NOUN
ejpam-4572	209	49	)	)	PUNCT
ejpam-4572	209	50	.	.	PUNCT
ejpam-4572	210	1	(	(	PUNCT
ejpam-4572	210	2	2	2	X
ejpam-4572	210	3	)	)	PUNCT
ejpam-4572	210	4	aθ(λ	aθ(λ	NOUN
ejpam-4572	210	5	,	,	PUNCT
ejpam-4572	210	6	sp	sp	NOUN
ejpam-4572	210	7	)	)	PUNCT
ejpam-4572	210	8	is	be	AUX
ejpam-4572	210	9	(	(	PUNCT
ejpam-4572	210	10	λ	λ	X
ejpam-4572	210	11	,	,	PUNCT
ejpam-4572	210	12	sp)-closed	sp)-close	VERB
ejpam-4572	210	13	for	for	ADP
ejpam-4572	210	14	every	every	DET
ejpam-4572	210	15	subset	subset	NOUN
ejpam-4572	210	16	a	a	PRON
ejpam-4572	210	17	of	of	ADP
ejpam-4572	210	18	x.	x.	NOUN
ejpam-4572	210	19	theorem	theorem	NOUN
ejpam-4572	210	20	3	3	NUM
ejpam-4572	210	21	.	.	X
ejpam-4572	210	22	for	for	ADP
ejpam-4572	210	23	a	a	DET
ejpam-4572	210	24	multifunction	multifunction	NOUN
ejpam-4572	211	1	f	f	NOUN
ejpam-4572	211	2	:	:	PUNCT
ejpam-4572	211	3	(	(	PUNCT
ejpam-4572	211	4	x	x	X
ejpam-4572	211	5	,	,	PUNCT
ejpam-4572	211	6	τ	τ	X
ejpam-4572	211	7	)	)	PUNCT
ejpam-4572	211	8	→	→	SYM
ejpam-4572	211	9	(	(	PUNCT
ejpam-4572	211	10	y	y	PROPN
ejpam-4572	211	11	,	,	PUNCT
ejpam-4572	211	12	σ	σ	PROPN
ejpam-4572	211	13	)	)	PUNCT
ejpam-4572	211	14	,	,	PUNCT
ejpam-4572	211	15	the	the	DET
ejpam-4572	211	16	following	follow	VERB
ejpam-4572	211	17	properties	property	NOUN
ejpam-4572	211	18	are	be	AUX
ejpam-4572	211	19	equivalent	equivalent	ADJ
ejpam-4572	211	20	:	:	PUNCT
ejpam-4572	211	21	c.	c.	PROPN
ejpam-4572	211	22	boonpok	boonpok	PROPN
ejpam-4572	211	23	,	,	PUNCT
ejpam-4572	211	24	m.	m.	NOUN
ejpam-4572	211	25	thongmoon	thongmoon	PROPN
ejpam-4572	211	26	/	/	SYM
ejpam-4572	211	27	eur	eur	PROPN
ejpam-4572	211	28	.	.	PUNCT
ejpam-4572	212	1	j.	j.	PROPN
ejpam-4572	212	2	pure	pure	PROPN
ejpam-4572	212	3	appl	appl	PROPN
ejpam-4572	212	4	.	.	PROPN
ejpam-4572	212	5	math	math	PROPN
ejpam-4572	212	6	,	,	PUNCT
ejpam-4572	212	7	16	16	NUM
ejpam-4572	212	8	(	(	PUNCT
ejpam-4572	212	9	1	1	NUM
ejpam-4572	212	10	)	)	PUNCT
ejpam-4572	212	11	(	(	PUNCT
ejpam-4572	212	12	2023	2023	NUM
ejpam-4572	212	13	)	)	PUNCT
ejpam-4572	212	14	,	,	PUNCT
ejpam-4572	212	15	465	465	NUM
ejpam-4572	212	16	-	-	SYM
ejpam-4572	212	17	478	478	NUM
ejpam-4572	212	18	471	471	NUM
ejpam-4572	212	19	(	(	PUNCT
ejpam-4572	212	20	1	1	NUM
ejpam-4572	212	21	)	)	PUNCT
ejpam-4572	212	22	f	f	PROPN
ejpam-4572	212	23	is	be	AUX
ejpam-4572	212	24	upper	upper	ADJ
ejpam-4572	212	25	weakly	weakly	ADJ
ejpam-4572	212	26	α(λ	α(λ	PROPN
ejpam-4572	212	27	,	,	PUNCT
ejpam-4572	212	28	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	212	29	;	;	PUNCT
ejpam-4572	212	30	(	(	PUNCT
ejpam-4572	212	31	2	2	X
ejpam-4572	212	32	)	)	PUNCT
ejpam-4572	212	33	for	for	ADP
ejpam-4572	212	34	each	each	DET
ejpam-4572	212	35	x	x	SYM
ejpam-4572	212	36	∈	∈	PROPN
ejpam-4572	212	37	x	x	X
ejpam-4572	212	38	and	and	CCONJ
ejpam-4572	212	39	each	each	DET
ejpam-4572	212	40	(	(	PUNCT
ejpam-4572	212	41	λ	λ	PROPN
ejpam-4572	212	42	,	,	PUNCT
ejpam-4572	212	43	sp)-open	sp)-open	NOUN
ejpam-4572	212	44	set	set	VERB
ejpam-4572	212	45	v	v	NUM
ejpam-4572	212	46	of	of	ADP
ejpam-4572	212	47	y	y	PROPN
ejpam-4572	212	48	containing	contain	VERB
ejpam-4572	212	49	f	f	PROPN
ejpam-4572	212	50	(	(	PUNCT
ejpam-4572	212	51	x	x	NOUN
ejpam-4572	212	52	)	)	PUNCT
ejpam-4572	212	53	,	,	PUNCT
ejpam-4572	212	54	there	there	PRON
ejpam-4572	212	55	exists	exist	VERB
ejpam-4572	212	56	an	an	DET
ejpam-4572	212	57	α(λ	α(λ	PROPN
ejpam-4572	212	58	,	,	PUNCT
ejpam-4572	212	59	sp)-open	sp)-open	VERB
ejpam-4572	212	60	set	set	VERB
ejpam-4572	212	61	u	u	NOUN
ejpam-4572	212	62	of	of	ADP
ejpam-4572	212	63	x	x	PUNCT
ejpam-4572	212	64	containing	contain	VERB
ejpam-4572	212	65	x	x	PUNCT
ejpam-4572	213	1	such	such	ADJ
ejpam-4572	213	2	that	that	SCONJ
ejpam-4572	213	3	f	f	PROPN
ejpam-4572	213	4	(	(	PUNCT
ejpam-4572	213	5	u	u	NOUN
ejpam-4572	213	6	)	)	PUNCT
ejpam-4572	213	7	⊆	⊆	NUM
ejpam-4572	213	8	v	v	NOUN
ejpam-4572	213	9	(	(	PUNCT
ejpam-4572	213	10	λ	λ	NOUN
ejpam-4572	213	11	,	,	PUNCT
ejpam-4572	213	12	sp	sp	NOUN
ejpam-4572	213	13	)	)	PUNCT
ejpam-4572	213	14	;	;	PUNCT
ejpam-4572	213	15	(	(	PUNCT
ejpam-4572	213	16	3	3	X
ejpam-4572	213	17	)	)	PUNCT
ejpam-4572	213	18	f+(v	f+(v	NOUN
ejpam-4572	213	19	)	)	PUNCT
ejpam-4572	213	20	⊆	⊆	NUM
ejpam-4572	214	1	[	[	X
ejpam-4572	214	2	[	[	X
ejpam-4572	214	3	[	[	X
ejpam-4572	214	4	f+(v	f+(v	PROPN
ejpam-4572	214	5	(	(	PUNCT
ejpam-4572	214	6	λ	λ	PROPN
ejpam-4572	214	7	,	,	PUNCT
ejpam-4572	214	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	214	9	,	,	PUNCT
ejpam-4572	214	10	sp	sp	NOUN
ejpam-4572	214	11	)	)	PUNCT
ejpam-4572	214	12	]	]	PUNCT
ejpam-4572	214	13	(	(	PUNCT
ejpam-4572	214	14	λ	λ	X
ejpam-4572	214	15	,	,	PUNCT
ejpam-4572	214	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	214	17	,	,	PUNCT
ejpam-4572	214	18	sp	sp	NOUN
ejpam-4572	214	19	)	)	PUNCT
ejpam-4572	214	20	for	for	ADP
ejpam-4572	214	21	every	every	DET
ejpam-4572	214	22	(	(	PUNCT
ejpam-4572	214	23	λ	λ	NOUN
ejpam-4572	214	24	,	,	PUNCT
ejpam-4572	214	25	sp)-open	sp)-open	NOUN
ejpam-4572	214	26	set	set	VERB
ejpam-4572	214	27	v	v	NOUN
ejpam-4572	214	28	of	of	ADP
ejpam-4572	214	29	y	y	PROPN
ejpam-4572	214	30	;	;	PUNCT
ejpam-4572	214	31	(	(	PUNCT
ejpam-4572	214	32	4	4	X
ejpam-4572	214	33	)	)	PUNCT
ejpam-4572	215	1	[	[	X
ejpam-4572	215	2	[	[	X
ejpam-4572	215	3	[	[	X
ejpam-4572	215	4	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	215	5	,	,	PUNCT
ejpam-4572	215	6	sp	sp	NOUN
ejpam-4572	215	7	)	)	PUNCT
ejpam-4572	215	8	)	)	PUNCT
ejpam-4572	215	9	]	]	PUNCT
ejpam-4572	216	1	(	(	PUNCT
ejpam-4572	216	2	λ	λ	X
ejpam-4572	216	3	,	,	PUNCT
ejpam-4572	216	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	216	5	,	,	PUNCT
ejpam-4572	216	6	sp	sp	NOUN
ejpam-4572	216	7	)	)	PUNCT
ejpam-4572	216	8	]	]	PUNCT
ejpam-4572	216	9	(	(	PUNCT
ejpam-4572	216	10	λ	λ	NOUN
ejpam-4572	216	11	,	,	PUNCT
ejpam-4572	216	12	sp	sp	NOUN
ejpam-4572	216	13	)	)	PUNCT
ejpam-4572	216	14	⊆	⊆	NUM
ejpam-4572	216	15	f−(k	f−(k	PROPN
ejpam-4572	216	16	)	)	PUNCT
ejpam-4572	216	17	for	for	SCONJ
ejpam-4572	216	18	every	every	DET
ejpam-4572	216	19	(	(	PUNCT
ejpam-4572	216	20	λ	λ	PROPN
ejpam-4572	216	21	,	,	PUNCT
ejpam-4572	216	22	sp)-closed	sp)-close	VERB
ejpam-4572	216	23	set	set	VERB
ejpam-4572	216	24	k	k	PROPN
ejpam-4572	216	25	of	of	ADP
ejpam-4572	216	26	y	y	PROPN
ejpam-4572	216	27	;	;	PUNCT
ejpam-4572	216	28	(	(	PUNCT
ejpam-4572	216	29	5	5	X
ejpam-4572	216	30	)	)	PUNCT
ejpam-4572	217	1	[	[	X
ejpam-4572	217	2	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	217	3	,	,	PUNCT
ejpam-4572	217	4	sp	sp	NOUN
ejpam-4572	217	5	)	)	PUNCT
ejpam-4572	217	6	)	)	PUNCT
ejpam-4572	217	7	]	]	PUNCT
ejpam-4572	218	1	α(λ	α(λ	PROPN
ejpam-4572	218	2	,	,	PUNCT
ejpam-4572	218	3	sp	sp	NOUN
ejpam-4572	218	4	)	)	PUNCT
ejpam-4572	218	5	⊆	⊆	NUM
ejpam-4572	218	6	f−(k	f−(k	PROPN
ejpam-4572	218	7	)	)	PUNCT
ejpam-4572	218	8	for	for	ADP
ejpam-4572	218	9	every	every	DET
ejpam-4572	218	10	(	(	PUNCT
ejpam-4572	218	11	λ	λ	PROPN
ejpam-4572	218	12	,	,	PUNCT
ejpam-4572	218	13	sp)-closed	sp)-close	VERB
ejpam-4572	218	14	set	set	VERB
ejpam-4572	218	15	k	k	PROPN
ejpam-4572	218	16	of	of	ADP
ejpam-4572	218	17	y	y	PROPN
ejpam-4572	218	18	;	;	PUNCT
ejpam-4572	218	19	(	(	PUNCT
ejpam-4572	218	20	6	6	X
ejpam-4572	218	21	)	)	PUNCT
ejpam-4572	219	1	[	[	X
ejpam-4572	219	2	f−([b(λ	f−([b(λ	PROPN
ejpam-4572	219	3	,	,	PUNCT
ejpam-4572	219	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	219	5	,	,	PUNCT
ejpam-4572	219	6	sp	sp	NOUN
ejpam-4572	219	7	)	)	PUNCT
ejpam-4572	219	8	)	)	PUNCT
ejpam-4572	219	9	]	]	PUNCT
ejpam-4572	220	1	α(λ	α(λ	PROPN
ejpam-4572	220	2	,	,	PUNCT
ejpam-4572	220	3	sp	sp	NOUN
ejpam-4572	220	4	)	)	PUNCT
ejpam-4572	220	5	⊆	⊆	NUM
ejpam-4572	220	6	f−(b(λ	f−(b(λ	NOUN
ejpam-4572	220	7	,	,	PUNCT
ejpam-4572	220	8	sp	sp	NOUN
ejpam-4572	220	9	)	)	PUNCT
ejpam-4572	220	10	)	)	PUNCT
ejpam-4572	220	11	for	for	ADP
ejpam-4572	220	12	every	every	DET
ejpam-4572	220	13	subset	subset	NOUN
ejpam-4572	220	14	b	b	PROPN
ejpam-4572	220	15	of	of	ADP
ejpam-4572	220	16	y	y	PROPN
ejpam-4572	220	17	;	;	PUNCT
ejpam-4572	220	18	(	(	PUNCT
ejpam-4572	220	19	7	7	X
ejpam-4572	220	20	)	)	PUNCT
ejpam-4572	220	21	f+(b(λ	f+(b(λ	PROPN
ejpam-4572	220	22	,	,	PUNCT
ejpam-4572	220	23	sp	sp	NOUN
ejpam-4572	220	24	)	)	PUNCT
ejpam-4572	220	25	)	)	PUNCT
ejpam-4572	221	1	⊆	⊆	NUM
ejpam-4572	221	2	[	[	X
ejpam-4572	221	3	f+([b(λ	f+([b(λ	PROPN
ejpam-4572	221	4	,	,	PUNCT
ejpam-4572	221	5	sp	sp	NOUN
ejpam-4572	221	6	)	)	PUNCT
ejpam-4572	221	7	]	]	PUNCT
ejpam-4572	221	8	(	(	PUNCT
ejpam-4572	221	9	λ	λ	NOUN
ejpam-4572	221	10	,	,	PUNCT
ejpam-4572	221	11	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	221	12	,	,	PUNCT
ejpam-4572	221	13	sp	sp	NOUN
ejpam-4572	221	14	)	)	PUNCT
ejpam-4572	221	15	for	for	ADP
ejpam-4572	221	16	every	every	DET
ejpam-4572	221	17	subset	subset	NOUN
ejpam-4572	221	18	b	b	PROPN
ejpam-4572	221	19	of	of	ADP
ejpam-4572	221	20	y	y	PROPN
ejpam-4572	221	21	;	;	PUNCT
ejpam-4572	221	22	(	(	PUNCT
ejpam-4572	221	23	8)	8)	NUM
ejpam-4572	221	24	f+(v	f+(v	NOUN
ejpam-4572	221	25	)	)	PUNCT
ejpam-4572	222	1	⊆	⊆	NUM
ejpam-4572	222	2	[	[	X
ejpam-4572	222	3	f+(v	f+(v	NOUN
ejpam-4572	222	4	(	(	PUNCT
ejpam-4572	222	5	λ	λ	PROPN
ejpam-4572	222	6	,	,	PUNCT
ejpam-4572	222	7	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	222	8	,	,	PUNCT
ejpam-4572	222	9	sp	sp	NOUN
ejpam-4572	222	10	)	)	PUNCT
ejpam-4572	222	11	for	for	ADP
ejpam-4572	222	12	every	every	DET
ejpam-4572	222	13	(	(	PUNCT
ejpam-4572	222	14	λ	λ	NOUN
ejpam-4572	222	15	,	,	PUNCT
ejpam-4572	222	16	sp)-open	sp)-open	NOUN
ejpam-4572	222	17	set	set	VERB
ejpam-4572	222	18	v	v	NOUN
ejpam-4572	222	19	of	of	ADP
ejpam-4572	222	20	y	y	PROPN
ejpam-4572	222	21	;	;	PUNCT
ejpam-4572	222	22	(	(	PUNCT
ejpam-4572	222	23	9	9	X
ejpam-4572	222	24	)	)	PUNCT
ejpam-4572	223	1	[	[	X
ejpam-4572	223	2	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	223	3	,	,	PUNCT
ejpam-4572	223	4	sp	sp	NOUN
ejpam-4572	223	5	)	)	PUNCT
ejpam-4572	223	6	)	)	PUNCT
ejpam-4572	223	7	]	]	PUNCT
ejpam-4572	224	1	α(λ	α(λ	PROPN
ejpam-4572	224	2	,	,	PUNCT
ejpam-4572	224	3	sp	sp	NOUN
ejpam-4572	224	4	)	)	PUNCT
ejpam-4572	224	5	⊆	⊆	NUM
ejpam-4572	224	6	f−(k	f−(k	PROPN
ejpam-4572	224	7	)	)	PUNCT
ejpam-4572	224	8	for	for	ADP
ejpam-4572	224	9	every	every	DET
ejpam-4572	224	10	r(λ	r(λ	NOUN
ejpam-4572	224	11	,	,	PUNCT
ejpam-4572	224	12	sp)-closed	sp)-close	VERB
ejpam-4572	224	13	set	set	VERB
ejpam-4572	224	14	k	k	PROPN
ejpam-4572	224	15	of	of	ADP
ejpam-4572	224	16	y	y	PROPN
ejpam-4572	224	17	;	;	PUNCT
ejpam-4572	224	18	(	(	PUNCT
ejpam-4572	224	19	10	10	NUM
ejpam-4572	224	20	)	)	PUNCT
ejpam-4572	225	1	[	[	X
ejpam-4572	225	2	f−(v	f−(v	NOUN
ejpam-4572	225	3	)	)	PUNCT
ejpam-4572	225	4	]	]	PUNCT
ejpam-4572	226	1	α(λ	α(λ	PROPN
ejpam-4572	226	2	,	,	PUNCT
ejpam-4572	226	3	sp	sp	NOUN
ejpam-4572	226	4	)	)	PUNCT
ejpam-4572	226	5	⊆	⊆	NUM
ejpam-4572	226	6	f−(v	f−(v	NOUN
ejpam-4572	226	7	(	(	PUNCT
ejpam-4572	226	8	λ	λ	NOUN
ejpam-4572	226	9	,	,	PUNCT
ejpam-4572	226	10	sp	sp	NOUN
ejpam-4572	226	11	)	)	PUNCT
ejpam-4572	226	12	)	)	PUNCT
ejpam-4572	226	13	for	for	ADP
ejpam-4572	226	14	every	every	DET
ejpam-4572	226	15	(	(	PUNCT
ejpam-4572	226	16	λ	λ	NOUN
ejpam-4572	226	17	,	,	PUNCT
ejpam-4572	226	18	sp)-open	sp)-open	NOUN
ejpam-4572	226	19	set	set	VERB
ejpam-4572	226	20	v	v	NOUN
ejpam-4572	226	21	of	of	ADP
ejpam-4572	226	22	y	y	PROPN
ejpam-4572	226	23	;	;	PUNCT
ejpam-4572	226	24	(	(	PUNCT
ejpam-4572	226	25	11	11	NUM
ejpam-4572	226	26	)	)	PUNCT
ejpam-4572	227	1	[	[	X
ejpam-4572	227	2	f−(bθ(λ	f−(bθ(λ	PROPN
ejpam-4572	227	3	,	,	PUNCT
ejpam-4572	227	4	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	227	5	,	,	PUNCT
ejpam-4572	227	6	sp	sp	NOUN
ejpam-4572	227	7	)	)	PUNCT
ejpam-4572	227	8	⊆	⊆	NUM
ejpam-4572	227	9	f−(bθ(λ	f−(bθ(λ	PROPN
ejpam-4572	227	10	,	,	PUNCT
ejpam-4572	227	11	sp	sp	NOUN
ejpam-4572	227	12	)	)	PUNCT
ejpam-4572	227	13	)	)	PUNCT
ejpam-4572	227	14	for	for	ADP
ejpam-4572	227	15	every	every	DET
ejpam-4572	227	16	subset	subset	NOUN
ejpam-4572	227	17	b	b	PROPN
ejpam-4572	227	18	of	of	ADP
ejpam-4572	227	19	y	y	PROPN
ejpam-4572	227	20	.	.	PUNCT
ejpam-4572	228	1	proof	proof	NOUN
ejpam-4572	228	2	.	.	PUNCT
ejpam-4572	229	1	(	(	PUNCT
ejpam-4572	229	2	1	1	X
ejpam-4572	229	3	)	)	PUNCT
ejpam-4572	229	4	⇒	⇒	NOUN
ejpam-4572	229	5	(	(	PUNCT
ejpam-4572	229	6	2	2	NUM
ejpam-4572	229	7	):	):	PUNCT
ejpam-4572	229	8	the	the	DET
ejpam-4572	229	9	proof	proof	NOUN
ejpam-4572	229	10	follows	follow	VERB
ejpam-4572	229	11	immediately	immediately	ADV
ejpam-4572	229	12	from	from	ADP
ejpam-4572	229	13	theorem	theorem	ADJ
ejpam-4572	229	14	1	1	NUM
ejpam-4572	229	15	.	.	PUNCT
ejpam-4572	229	16	(	(	PUNCT
ejpam-4572	229	17	2	2	X
ejpam-4572	229	18	)	)	PUNCT
ejpam-4572	229	19	⇒	⇒	NOUN
ejpam-4572	229	20	(	(	PUNCT
ejpam-4572	229	21	3	3	NUM
ejpam-4572	229	22	):	):	PUNCT
ejpam-4572	229	23	let	let	VERB
ejpam-4572	229	24	v	v	PART
ejpam-4572	229	25	be	be	AUX
ejpam-4572	229	26	any	any	DET
ejpam-4572	229	27	(	(	PUNCT
ejpam-4572	229	28	λ	λ	NOUN
ejpam-4572	229	29	,	,	PUNCT
ejpam-4572	229	30	sp)-open	sp)-open	ADJ
ejpam-4572	229	31	set	set	NOUN
ejpam-4572	229	32	of	of	ADP
ejpam-4572	229	33	y	y	PROPN
ejpam-4572	229	34	and	and	CCONJ
ejpam-4572	229	35	x	x	PROPN
ejpam-4572	229	36	∈	∈	PROPN
ejpam-4572	229	37	f+(v	f+(v	NOUN
ejpam-4572	229	38	)	)	PUNCT
ejpam-4572	229	39	.	.	PUNCT
ejpam-4572	230	1	then	then	ADV
ejpam-4572	230	2	,	,	PUNCT
ejpam-4572	230	3	f	f	PROPN
ejpam-4572	230	4	(	(	PUNCT
ejpam-4572	230	5	x	x	X
ejpam-4572	230	6	)	)	PUNCT
ejpam-4572	230	7	⊆	⊆	NUM
ejpam-4572	230	8	v	v	NOUN
ejpam-4572	230	9	and	and	CCONJ
ejpam-4572	230	10	there	there	PRON
ejpam-4572	230	11	exists	exist	VERB
ejpam-4572	230	12	an	an	DET
ejpam-4572	230	13	α(λ	α(λ	PROPN
ejpam-4572	230	14	,	,	PUNCT
ejpam-4572	230	15	sp)-open	sp)-open	VERB
ejpam-4572	230	16	set	set	VERB
ejpam-4572	230	17	u	u	NOUN
ejpam-4572	230	18	of	of	ADP
ejpam-4572	230	19	x	x	PUNCT
ejpam-4572	230	20	containing	contain	VERB
ejpam-4572	230	21	x	x	PUNCT
ejpam-4572	230	22	such	such	ADJ
ejpam-4572	230	23	that	that	SCONJ
ejpam-4572	230	24	f	f	PROPN
ejpam-4572	230	25	(	(	PUNCT
ejpam-4572	230	26	u	u	NOUN
ejpam-4572	230	27	)	)	PUNCT
ejpam-4572	230	28	⊆	⊆	NUM
ejpam-4572	230	29	v	v	NOUN
ejpam-4572	230	30	(	(	PUNCT
ejpam-4572	230	31	λ	λ	NOUN
ejpam-4572	230	32	,	,	PUNCT
ejpam-4572	230	33	sp	sp	NOUN
ejpam-4572	230	34	)	)	PUNCT
ejpam-4572	230	35	.	.	PUNCT
ejpam-4572	231	1	therefore	therefore	ADV
ejpam-4572	231	2	,	,	PUNCT
ejpam-4572	231	3	x	x	PUNCT
ejpam-4572	231	4	∈	∈	PROPN
ejpam-4572	231	5	u	u	NOUN
ejpam-4572	231	6	⊆	⊆	NUM
ejpam-4572	231	7	f+(v	f+(v	NOUN
ejpam-4572	231	8	(	(	PUNCT
ejpam-4572	231	9	λ	λ	NOUN
ejpam-4572	231	10	,	,	PUNCT
ejpam-4572	231	11	sp	sp	NOUN
ejpam-4572	231	12	)	)	PUNCT
ejpam-4572	231	13	)	)	PUNCT
ejpam-4572	231	14	.	.	PUNCT
ejpam-4572	232	1	since	since	SCONJ
ejpam-4572	232	2	u	u	NOUN
ejpam-4572	232	3	is	be	AUX
ejpam-4572	232	4	an	an	DET
ejpam-4572	232	5	α(λ	α(λ	PROPN
ejpam-4572	232	6	,	,	PUNCT
ejpam-4572	232	7	sp)-open	sp)-open	ADJ
ejpam-4572	232	8	set	set	NOUN
ejpam-4572	232	9	containing	contain	VERB
ejpam-4572	232	10	x	x	SYM
ejpam-4572	232	11	,	,	PUNCT
ejpam-4572	232	12	we	we	PRON
ejpam-4572	232	13	have	have	VERB
ejpam-4572	232	14	x	x	X
ejpam-4572	232	15	∈	∈	PROPN
ejpam-4572	232	16	u	u	NOUN
ejpam-4572	232	17	⊆	⊆	NUM
ejpam-4572	232	18	[	[	X
ejpam-4572	232	19	[	[	X
ejpam-4572	232	20	[	[	X
ejpam-4572	232	21	f+(v	f+(v	PROPN
ejpam-4572	232	22	(	(	PUNCT
ejpam-4572	232	23	λ	λ	PROPN
ejpam-4572	232	24	,	,	PUNCT
ejpam-4572	232	25	sp))](λ	sp))](λ	PROPN
ejpam-4572	232	26	,	,	PUNCT
ejpam-4572	232	27	sp	sp	NOUN
ejpam-4572	232	28	)	)	PUNCT
ejpam-4572	232	29	]	]	PUNCT
ejpam-4572	233	1	(	(	PUNCT
ejpam-4572	233	2	λ	λ	X
ejpam-4572	233	3	,	,	PUNCT
ejpam-4572	233	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	233	5	,	,	PUNCT
ejpam-4572	233	6	sp	sp	NOUN
ejpam-4572	233	7	)	)	PUNCT
ejpam-4572	233	8	.	.	PUNCT
ejpam-4572	234	1	(	(	PUNCT
ejpam-4572	234	2	3	3	X
ejpam-4572	234	3	)	)	PUNCT
ejpam-4572	234	4	⇒	⇒	NOUN
ejpam-4572	234	5	(	(	PUNCT
ejpam-4572	234	6	4	4	NUM
ejpam-4572	234	7	):	):	PUNCT
ejpam-4572	234	8	let	let	VERB
ejpam-4572	234	9	k	k	PRON
ejpam-4572	234	10	be	be	AUX
ejpam-4572	234	11	any	any	DET
ejpam-4572	234	12	(	(	PUNCT
ejpam-4572	234	13	λ	λ	PROPN
ejpam-4572	234	14	,	,	PUNCT
ejpam-4572	234	15	sp)-closed	sp)-close	VERB
ejpam-4572	234	16	set	set	NOUN
ejpam-4572	234	17	of	of	ADP
ejpam-4572	234	18	y	y	PROPN
ejpam-4572	234	19	.	.	PUNCT
ejpam-4572	235	1	then	then	ADV
ejpam-4572	235	2	,	,	PUNCT
ejpam-4572	235	3	y	y	PROPN
ejpam-4572	235	4	−	−	PROPN
ejpam-4572	235	5	k	k	PROPN
ejpam-4572	235	6	is	be	AUX
ejpam-4572	235	7	(	(	PUNCT
ejpam-4572	235	8	λ	λ	INTJ
ejpam-4572	235	9	,	,	PUNCT
ejpam-4572	235	10	sp)-open	sp)-open	ADJ
ejpam-4572	235	11	in	in	ADP
ejpam-4572	235	12	y	y	PROPN
ejpam-4572	235	13	and	and	CCONJ
ejpam-4572	235	14	by	by	ADP
ejpam-4572	235	15	(	(	PUNCT
ejpam-4572	235	16	3	3	NUM
ejpam-4572	235	17	)	)	PUNCT
ejpam-4572	235	18	,	,	PUNCT
ejpam-4572	235	19	f+(y	f+(y	PROPN
ejpam-4572	235	20	−k	−k	PROPN
ejpam-4572	235	21	)	)	PUNCT
ejpam-4572	235	22	⊆	⊆	NUM
ejpam-4572	236	1	[	[	X
ejpam-4572	236	2	[	[	X
ejpam-4572	236	3	[	[	X
ejpam-4572	236	4	f+([y	f+([y	ADJ
ejpam-4572	236	5	−k](λ	−k](λ	NUM
ejpam-4572	236	6	,	,	PUNCT
ejpam-4572	236	7	sp))](λ	sp))](λ	PROPN
ejpam-4572	236	8	,	,	PUNCT
ejpam-4572	236	9	sp	sp	NOUN
ejpam-4572	236	10	)	)	PUNCT
ejpam-4572	236	11	]	]	PUNCT
ejpam-4572	236	12	(	(	PUNCT
ejpam-4572	236	13	λ	λ	X
ejpam-4572	236	14	,	,	PUNCT
ejpam-4572	236	15	sp)](λ	sp)](λ	PROPN
ejpam-4572	236	16	,	,	PUNCT
ejpam-4572	236	17	sp	sp	NOUN
ejpam-4572	236	18	)	)	PUNCT
ejpam-4572	236	19	.	.	PUNCT
ejpam-4572	237	1	by	by	ADP
ejpam-4572	237	2	the	the	DET
ejpam-4572	237	3	straightforward	straightforward	ADJ
ejpam-4572	237	4	calculations	calculation	NOUN
ejpam-4572	237	5	,	,	PUNCT
ejpam-4572	237	6	we	we	PRON
ejpam-4572	237	7	obtain	obtain	VERB
ejpam-4572	237	8	[	[	X
ejpam-4572	237	9	[	[	X
ejpam-4572	237	10	[	[	X
ejpam-4572	237	11	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	237	12	,	,	PUNCT
ejpam-4572	237	13	sp	sp	NOUN
ejpam-4572	237	14	)	)	PUNCT
ejpam-4572	237	15	)	)	PUNCT
ejpam-4572	237	16	]	]	PUNCT
ejpam-4572	238	1	(	(	PUNCT
ejpam-4572	238	2	λ	λ	X
ejpam-4572	238	3	,	,	PUNCT
ejpam-4572	238	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	238	5	,	,	PUNCT
ejpam-4572	238	6	sp	sp	NOUN
ejpam-4572	238	7	)	)	PUNCT
ejpam-4572	238	8	]	]	PUNCT
ejpam-4572	238	9	(	(	PUNCT
ejpam-4572	238	10	λ	λ	NOUN
ejpam-4572	238	11	,	,	PUNCT
ejpam-4572	238	12	sp	sp	NOUN
ejpam-4572	238	13	)	)	PUNCT
ejpam-4572	238	14	⊆	⊆	NUM
ejpam-4572	238	15	f−(k	f−(k	PROPN
ejpam-4572	238	16	)	)	PUNCT
ejpam-4572	238	17	.	.	PUNCT
ejpam-4572	239	1	(	(	PUNCT
ejpam-4572	239	2	4	4	X
ejpam-4572	239	3	)	)	PUNCT
ejpam-4572	239	4	⇒	⇒	NOUN
ejpam-4572	239	5	(	(	PUNCT
ejpam-4572	239	6	5	5	NUM
ejpam-4572	239	7	):	):	PUNCT
ejpam-4572	239	8	let	let	VERB
ejpam-4572	239	9	k	k	PRON
ejpam-4572	239	10	be	be	AUX
ejpam-4572	239	11	any	any	DET
ejpam-4572	239	12	(	(	PUNCT
ejpam-4572	239	13	λ	λ	PROPN
ejpam-4572	239	14	,	,	PUNCT
ejpam-4572	239	15	sp)-closed	sp)-close	VERB
ejpam-4572	239	16	set	set	NOUN
ejpam-4572	239	17	of	of	ADP
ejpam-4572	239	18	y	y	PROPN
ejpam-4572	239	19	.	.	PUNCT
ejpam-4572	240	1	then	then	ADV
ejpam-4572	240	2	,	,	PUNCT
ejpam-4572	240	3	we	we	PRON
ejpam-4572	240	4	have	have	VERB
ejpam-4572	240	5	[	[	X
ejpam-4572	240	6	[	[	X
ejpam-4572	240	7	[	[	X
ejpam-4572	240	8	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	240	9	,	,	PUNCT
ejpam-4572	240	10	sp	sp	NOUN
ejpam-4572	240	11	)	)	PUNCT
ejpam-4572	240	12	)	)	PUNCT
ejpam-4572	240	13	]	]	PUNCT
ejpam-4572	241	1	(	(	PUNCT
ejpam-4572	241	2	λ	λ	X
ejpam-4572	241	3	,	,	PUNCT
ejpam-4572	241	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	241	5	,	,	PUNCT
ejpam-4572	241	6	sp	sp	NOUN
ejpam-4572	241	7	)	)	PUNCT
ejpam-4572	241	8	]	]	PUNCT
ejpam-4572	241	9	(	(	PUNCT
ejpam-4572	241	10	λ	λ	NOUN
ejpam-4572	241	11	,	,	PUNCT
ejpam-4572	241	12	sp	sp	NOUN
ejpam-4572	241	13	)	)	PUNCT
ejpam-4572	241	14	⊆	⊆	NUM
ejpam-4572	241	15	f−(k	f−(k	PROPN
ejpam-4572	241	16	)	)	PUNCT
ejpam-4572	241	17	and	and	CCONJ
ejpam-4572	241	18	hence	hence	ADV
ejpam-4572	241	19	[	[	X
ejpam-4572	241	20	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	241	21	,	,	PUNCT
ejpam-4572	241	22	sp	sp	NOUN
ejpam-4572	241	23	)	)	PUNCT
ejpam-4572	241	24	)	)	PUNCT
ejpam-4572	241	25	]	]	PUNCT
ejpam-4572	242	1	α(λ	α(λ	PROPN
ejpam-4572	242	2	,	,	PUNCT
ejpam-4572	242	3	sp	sp	NOUN
ejpam-4572	242	4	)	)	PUNCT
ejpam-4572	242	5	⊆	⊆	NUM
ejpam-4572	242	6	f−(k	f−(k	PROPN
ejpam-4572	242	7	)	)	PUNCT
ejpam-4572	242	8	by	by	ADP
ejpam-4572	242	9	lemma	lemma	PROPN
ejpam-4572	242	10	6	6	NUM
ejpam-4572	242	11	.	.	PUNCT
ejpam-4572	242	12	(	(	PUNCT
ejpam-4572	242	13	5	5	X
ejpam-4572	242	14	)	)	PUNCT
ejpam-4572	242	15	⇒	⇒	NOUN
ejpam-4572	242	16	(	(	PUNCT
ejpam-4572	242	17	6	6	NUM
ejpam-4572	242	18	):	):	PUNCT
ejpam-4572	242	19	let	let	VERB
ejpam-4572	242	20	b	b	X
ejpam-4572	242	21	be	be	AUX
ejpam-4572	242	22	any	any	DET
ejpam-4572	242	23	subset	subset	NOUN
ejpam-4572	242	24	of	of	ADP
ejpam-4572	242	25	y	y	PROPN
ejpam-4572	242	26	.	.	PUNCT
ejpam-4572	243	1	then	then	ADV
ejpam-4572	243	2	,	,	PUNCT
ejpam-4572	243	3	b(λ	b(λ	PROPN
ejpam-4572	243	4	,	,	PUNCT
ejpam-4572	243	5	sp	sp	NOUN
ejpam-4572	243	6	)	)	PUNCT
ejpam-4572	243	7	is	be	AUX
ejpam-4572	243	8	(	(	PUNCT
ejpam-4572	243	9	λ	λ	X
ejpam-4572	243	10	,	,	PUNCT
ejpam-4572	243	11	sp)-closed	sp)-close	VERB
ejpam-4572	243	12	in	in	ADP
ejpam-4572	243	13	y	y	PROPN
ejpam-4572	243	14	.	.	PUNCT
ejpam-4572	244	1	thus	thus	ADV
ejpam-4572	244	2	,	,	PUNCT
ejpam-4572	244	3	by	by	ADP
ejpam-4572	244	4	(	(	PUNCT
ejpam-4572	244	5	5	5	NUM
ejpam-4572	244	6	)	)	PUNCT
ejpam-4572	244	7	,	,	PUNCT
ejpam-4572	244	8	we	we	PRON
ejpam-4572	244	9	have	have	VERB
ejpam-4572	244	10	[	[	X
ejpam-4572	244	11	f−([b(λ	f−([b(λ	PROPN
ejpam-4572	244	12	,	,	PUNCT
ejpam-4572	244	13	sp)](λ	sp)](λ	PROPN
ejpam-4572	244	14	,	,	PUNCT
ejpam-4572	244	15	sp	sp	NOUN
ejpam-4572	244	16	)	)	PUNCT
ejpam-4572	244	17	)	)	PUNCT
ejpam-4572	244	18	]	]	PUNCT
ejpam-4572	245	1	α(λ	α(λ	PROPN
ejpam-4572	245	2	,	,	PUNCT
ejpam-4572	245	3	sp	sp	NOUN
ejpam-4572	245	4	)	)	PUNCT
ejpam-4572	245	5	⊆	⊆	NUM
ejpam-4572	245	6	f−(b(λ	f−(b(λ	NOUN
ejpam-4572	245	7	,	,	PUNCT
ejpam-4572	245	8	sp	sp	NOUN
ejpam-4572	245	9	)	)	PUNCT
ejpam-4572	245	10	)	)	PUNCT
ejpam-4572	245	11	.	.	PUNCT
ejpam-4572	246	1	(	(	PUNCT
ejpam-4572	246	2	6	6	X
ejpam-4572	246	3	)	)	PUNCT
ejpam-4572	246	4	⇒	⇒	NOUN
ejpam-4572	246	5	(	(	PUNCT
ejpam-4572	246	6	7	7	NUM
ejpam-4572	246	7	):	):	PUNCT
ejpam-4572	246	8	let	let	VERB
ejpam-4572	246	9	b	b	X
ejpam-4572	246	10	be	be	AUX
ejpam-4572	246	11	any	any	DET
ejpam-4572	246	12	subset	subset	NOUN
ejpam-4572	246	13	of	of	ADP
ejpam-4572	246	14	y	y	PROPN
ejpam-4572	246	15	.	.	PUNCT
ejpam-4572	247	1	then	then	ADV
ejpam-4572	247	2	,	,	PUNCT
ejpam-4572	247	3	x	x	PUNCT
ejpam-4572	247	4	−	−	PROPN
ejpam-4572	247	5	f+(b(λ	f+(b(λ	PROPN
ejpam-4572	247	6	,	,	PUNCT
ejpam-4572	247	7	sp	sp	NOUN
ejpam-4572	247	8	)	)	PUNCT
ejpam-4572	247	9	)	)	PUNCT
ejpam-4572	248	1	=	=	PUNCT
ejpam-4572	249	1	f−([y	f−([y	NOUN
ejpam-4572	249	2	−b](λ	−b](λ	PROPN
ejpam-4572	249	3	,	,	PUNCT
ejpam-4572	249	4	sp	sp	NOUN
ejpam-4572	249	5	)	)	PUNCT
ejpam-4572	249	6	)	)	PUNCT
ejpam-4572	249	7	⊇	⊇	NOUN
ejpam-4572	250	1	[	[	X
ejpam-4572	250	2	f−([[y	f−([[y	NUM
ejpam-4572	250	3	−b](λ	−b](λ	PROPN
ejpam-4572	250	4	,	,	PUNCT
ejpam-4572	250	5	sp)](λ	sp)](λ	PROPN
ejpam-4572	250	6	,	,	PUNCT
ejpam-4572	250	7	sp	sp	NOUN
ejpam-4572	250	8	)	)	PUNCT
ejpam-4572	250	9	)	)	PUNCT
ejpam-4572	250	10	]	]	PUNCT
ejpam-4572	251	1	α(λ	α(λ	PROPN
ejpam-4572	251	2	,	,	PUNCT
ejpam-4572	251	3	sp	sp	NOUN
ejpam-4572	251	4	)	)	PUNCT
ejpam-4572	251	5	=	=	PUNCT
ejpam-4572	252	1	[	[	X
ejpam-4572	252	2	f−([y	f−([y	INTJ
ejpam-4572	252	3	−	−	NOUN
ejpam-4572	252	4	[	[	X
ejpam-4572	252	5	b(λ	b(λ	NOUN
ejpam-4572	252	6	,	,	PUNCT
ejpam-4572	252	7	sp	sp	NOUN
ejpam-4572	252	8	)	)	PUNCT
ejpam-4572	252	9	]	]	PUNCT
ejpam-4572	252	10	(	(	PUNCT
ejpam-4572	252	11	λ	λ	INTJ
ejpam-4572	252	12	,	,	PUNCT
ejpam-4572	252	13	sp)])]α(λ	sp)])]α(λ	NOUN
ejpam-4572	252	14	,	,	PUNCT
ejpam-4572	252	15	sp	sp	NOUN
ejpam-4572	252	16	)	)	PUNCT
ejpam-4572	252	17	=	=	PUNCT
ejpam-4572	253	1	[	[	X
ejpam-4572	253	2	x	x	X
ejpam-4572	253	3	−	−	PROPN
ejpam-4572	253	4	f+([b(λ	f+([b(λ	PROPN
ejpam-4572	253	5	,	,	PUNCT
ejpam-4572	253	6	sp	sp	NOUN
ejpam-4572	253	7	)	)	PUNCT
ejpam-4572	253	8	]	]	PUNCT
ejpam-4572	253	9	(	(	PUNCT
ejpam-4572	253	10	λ	λ	NOUN
ejpam-4572	253	11	,	,	PUNCT
ejpam-4572	253	12	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	253	13	,	,	PUNCT
ejpam-4572	253	14	sp	sp	NOUN
ejpam-4572	253	15	)	)	PUNCT
ejpam-4572	253	16	=	=	PUNCT
ejpam-4572	253	17	x	x	X
ejpam-4572	254	1	−	−	PROPN
ejpam-4572	255	1	[	[	X
ejpam-4572	255	2	f+([b(λ	f+([b(λ	PROPN
ejpam-4572	255	3	,	,	PUNCT
ejpam-4572	255	4	sp	sp	NOUN
ejpam-4572	255	5	)	)	PUNCT
ejpam-4572	255	6	]	]	PUNCT
ejpam-4572	255	7	(	(	PUNCT
ejpam-4572	255	8	λ	λ	NOUN
ejpam-4572	255	9	,	,	PUNCT
ejpam-4572	255	10	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	255	11	,	,	PUNCT
ejpam-4572	255	12	sp	sp	NOUN
ejpam-4572	255	13	)	)	PUNCT
ejpam-4572	255	14	.	.	PUNCT
ejpam-4572	256	1	c.	c.	PROPN
ejpam-4572	256	2	boonpok	boonpok	PROPN
ejpam-4572	256	3	,	,	PUNCT
ejpam-4572	256	4	m.	m.	NOUN
ejpam-4572	256	5	thongmoon	thongmoon	PROPN
ejpam-4572	256	6	/	/	SYM
ejpam-4572	256	7	eur	eur	PROPN
ejpam-4572	256	8	.	.	PUNCT
ejpam-4572	257	1	j.	j.	PROPN
ejpam-4572	257	2	pure	pure	PROPN
ejpam-4572	257	3	appl	appl	PROPN
ejpam-4572	257	4	.	.	PROPN
ejpam-4572	257	5	math	math	PROPN
ejpam-4572	257	6	,	,	PUNCT
ejpam-4572	257	7	16	16	NUM
ejpam-4572	257	8	(	(	PUNCT
ejpam-4572	257	9	1	1	NUM
ejpam-4572	257	10	)	)	PUNCT
ejpam-4572	257	11	(	(	PUNCT
ejpam-4572	257	12	2023	2023	NUM
ejpam-4572	257	13	)	)	PUNCT
ejpam-4572	257	14	,	,	PUNCT
ejpam-4572	257	15	465	465	NUM
ejpam-4572	257	16	-	-	SYM
ejpam-4572	257	17	478	478	NUM
ejpam-4572	257	18	472	472	NUM
ejpam-4572	257	19	thus	thus	ADV
ejpam-4572	257	20	,	,	PUNCT
ejpam-4572	257	21	f+(b(λ	f+(b(λ	PROPN
ejpam-4572	257	22	,	,	PUNCT
ejpam-4572	257	23	sp	sp	NOUN
ejpam-4572	257	24	)	)	PUNCT
ejpam-4572	257	25	)	)	PUNCT
ejpam-4572	258	1	⊆	⊆	NUM
ejpam-4572	258	2	[	[	X
ejpam-4572	258	3	f+([b(λ	f+([b(λ	PROPN
ejpam-4572	258	4	,	,	PUNCT
ejpam-4572	258	5	sp	sp	NOUN
ejpam-4572	258	6	)	)	PUNCT
ejpam-4572	258	7	]	]	PUNCT
ejpam-4572	258	8	(	(	PUNCT
ejpam-4572	258	9	λ	λ	NOUN
ejpam-4572	258	10	,	,	PUNCT
ejpam-4572	258	11	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	258	12	,	,	PUNCT
ejpam-4572	258	13	sp	sp	NOUN
ejpam-4572	258	14	)	)	PUNCT
ejpam-4572	258	15	.	.	PUNCT
ejpam-4572	259	1	(	(	PUNCT
ejpam-4572	259	2	7	7	X
ejpam-4572	259	3	)	)	PUNCT
ejpam-4572	259	4	⇒	⇒	NOUN
ejpam-4572	259	5	(	(	PUNCT
ejpam-4572	259	6	8)	8)	NUM
ejpam-4572	259	7	:	:	PUNCT
ejpam-4572	259	8	the	the	DET
ejpam-4572	259	9	proof	proof	NOUN
ejpam-4572	259	10	is	be	AUX
ejpam-4572	259	11	obvious	obvious	ADJ
ejpam-4572	259	12	.	.	PUNCT
ejpam-4572	260	1	(	(	PUNCT
ejpam-4572	260	2	8)	8)	NUM
ejpam-4572	260	3	⇒	⇒	NOUN
ejpam-4572	260	4	(	(	PUNCT
ejpam-4572	260	5	1	1	NUM
ejpam-4572	260	6	):	):	PUNCT
ejpam-4572	260	7	let	let	VERB
ejpam-4572	260	8	x	x	PRON
ejpam-4572	260	9	be	be	AUX
ejpam-4572	260	10	any	any	DET
ejpam-4572	260	11	point	point	NOUN
ejpam-4572	260	12	of	of	ADP
ejpam-4572	260	13	x	x	PUNCT
ejpam-4572	260	14	and	and	CCONJ
ejpam-4572	260	15	let	let	VERB
ejpam-4572	260	16	v	v	PART
ejpam-4572	260	17	be	be	AUX
ejpam-4572	260	18	any	any	DET
ejpam-4572	260	19	(	(	PUNCT
ejpam-4572	260	20	λ	λ	NOUN
ejpam-4572	260	21	,	,	PUNCT
ejpam-4572	260	22	sp)-open	sp)-open	ADJ
ejpam-4572	260	23	set	set	NOUN
ejpam-4572	260	24	of	of	ADP
ejpam-4572	260	25	y	y	PROPN
ejpam-4572	260	26	containing	contain	VERB
ejpam-4572	260	27	f	f	PROPN
ejpam-4572	260	28	(	(	PUNCT
ejpam-4572	260	29	x	x	NOUN
ejpam-4572	260	30	)	)	PUNCT
ejpam-4572	260	31	,	,	PUNCT
ejpam-4572	260	32	by	by	ADP
ejpam-4572	260	33	lemma	lemma	PROPN
ejpam-4572	260	34	6	6	NUM
ejpam-4572	260	35	,	,	PUNCT
ejpam-4572	260	36	x	x	SYM
ejpam-4572	260	37	∈	∈	NOUN
ejpam-4572	260	38	f+(v	f+(v	NOUN
ejpam-4572	260	39	)	)	PUNCT
ejpam-4572	261	1	⊆	⊆	NUM
ejpam-4572	261	2	[	[	X
ejpam-4572	261	3	f+(v	f+(v	NOUN
ejpam-4572	261	4	(	(	PUNCT
ejpam-4572	261	5	λ	λ	PROPN
ejpam-4572	261	6	,	,	PUNCT
ejpam-4572	261	7	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	261	8	,	,	PUNCT
ejpam-4572	261	9	sp	sp	NOUN
ejpam-4572	261	10	)	)	PUNCT
ejpam-4572	261	11	⊆	⊆	NUM
ejpam-4572	262	1	[	[	X
ejpam-4572	262	2	[	[	X
ejpam-4572	262	3	[	[	X
ejpam-4572	262	4	f+(v	f+(v	PROPN
ejpam-4572	262	5	(	(	PUNCT
ejpam-4572	262	6	λ	λ	PROPN
ejpam-4572	262	7	,	,	PUNCT
ejpam-4572	262	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	262	9	,	,	PUNCT
ejpam-4572	262	10	sp	sp	NOUN
ejpam-4572	262	11	)	)	PUNCT
ejpam-4572	262	12	]	]	PUNCT
ejpam-4572	262	13	(	(	PUNCT
ejpam-4572	262	14	λ	λ	X
ejpam-4572	262	15	,	,	PUNCT
ejpam-4572	262	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	262	17	,	,	PUNCT
ejpam-4572	262	18	sp	sp	NOUN
ejpam-4572	262	19	)	)	PUNCT
ejpam-4572	262	20	and	and	CCONJ
ejpam-4572	262	21	hence	hence	ADV
ejpam-4572	262	22	f	f	PROPN
ejpam-4572	262	23	is	be	AUX
ejpam-4572	262	24	upper	upper	ADJ
ejpam-4572	262	25	weakly	weakly	ADJ
ejpam-4572	262	26	α(λ	α(λ	PROPN
ejpam-4572	262	27	,	,	PUNCT
ejpam-4572	262	28	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	262	29	at	at	ADP
ejpam-4572	262	30	x	x	PUNCT
ejpam-4572	262	31	by	by	ADP
ejpam-4572	262	32	theorem	theorem	NOUN
ejpam-4572	262	33	1	1	NUM
ejpam-4572	262	34	.	.	PUNCT
ejpam-4572	262	35	(	(	PUNCT
ejpam-4572	262	36	5	5	X
ejpam-4572	262	37	)	)	PUNCT
ejpam-4572	262	38	⇒	⇒	NOUN
ejpam-4572	262	39	(	(	PUNCT
ejpam-4572	262	40	9	9	NUM
ejpam-4572	262	41	):	):	PUNCT
ejpam-4572	262	42	the	the	DET
ejpam-4572	262	43	proof	proof	NOUN
ejpam-4572	262	44	is	be	AUX
ejpam-4572	262	45	obvious	obvious	ADJ
ejpam-4572	262	46	.	.	PUNCT
ejpam-4572	263	1	(	(	PUNCT
ejpam-4572	263	2	9	9	X
ejpam-4572	263	3	)	)	PUNCT
ejpam-4572	263	4	⇒	⇒	NOUN
ejpam-4572	263	5	(	(	PUNCT
ejpam-4572	263	6	10	10	NUM
ejpam-4572	263	7	):	):	PUNCT
ejpam-4572	263	8	let	let	VERB
ejpam-4572	263	9	v	v	PART
ejpam-4572	263	10	be	be	AUX
ejpam-4572	263	11	any	any	DET
ejpam-4572	263	12	(	(	PUNCT
ejpam-4572	263	13	λ	λ	NOUN
ejpam-4572	263	14	,	,	PUNCT
ejpam-4572	263	15	sp)-open	sp)-open	ADJ
ejpam-4572	263	16	set	set	NOUN
ejpam-4572	263	17	of	of	ADP
ejpam-4572	263	18	y	y	PROPN
ejpam-4572	263	19	.	.	PUNCT
ejpam-4572	264	1	then	then	ADV
ejpam-4572	264	2	,	,	PUNCT
ejpam-4572	264	3	v	v	INTJ
ejpam-4572	264	4	(	(	PUNCT
ejpam-4572	264	5	λ	λ	NOUN
ejpam-4572	264	6	,	,	PUNCT
ejpam-4572	264	7	sp	sp	NOUN
ejpam-4572	264	8	)	)	PUNCT
ejpam-4572	264	9	is	be	AUX
ejpam-4572	264	10	r(λ	r(λ	NOUN
ejpam-4572	264	11	,	,	PUNCT
ejpam-4572	264	12	sp)-closed	sp)-close	VERB
ejpam-4572	264	13	in	in	ADP
ejpam-4572	264	14	y	y	PROPN
ejpam-4572	264	15	and	and	CCONJ
ejpam-4572	264	16	hence	hence	ADV
ejpam-4572	265	1	[	[	X
ejpam-4572	265	2	f−(v	f−(v	ADJ
ejpam-4572	265	3	)	)	PUNCT
ejpam-4572	265	4	]	]	PUNCT
ejpam-4572	265	5	α(λ	α(λ	PROPN
ejpam-4572	265	6	,	,	PUNCT
ejpam-4572	265	7	sp	sp	NOUN
ejpam-4572	265	8	)	)	PUNCT
ejpam-4572	265	9	⊆	⊆	NUM
ejpam-4572	265	10	[	[	X
ejpam-4572	265	11	f−([v	f−([v	ADJ
ejpam-4572	265	12	(	(	PUNCT
ejpam-4572	265	13	λ	λ	PROPN
ejpam-4572	265	14	,	,	PUNCT
ejpam-4572	265	15	sp)](λ	sp)](λ	PROPN
ejpam-4572	265	16	,	,	PUNCT
ejpam-4572	265	17	sp	sp	NOUN
ejpam-4572	265	18	)	)	PUNCT
ejpam-4572	265	19	)	)	PUNCT
ejpam-4572	265	20	]	]	PUNCT
ejpam-4572	266	1	α(λ	α(λ	PROPN
ejpam-4572	266	2	,	,	PUNCT
ejpam-4572	266	3	sp	sp	NOUN
ejpam-4572	266	4	)	)	PUNCT
ejpam-4572	266	5	⊆	⊆	NUM
ejpam-4572	266	6	f−(v	f−(v	NOUN
ejpam-4572	266	7	(	(	PUNCT
ejpam-4572	266	8	λ	λ	NOUN
ejpam-4572	266	9	,	,	PUNCT
ejpam-4572	266	10	sp	sp	NOUN
ejpam-4572	266	11	)	)	PUNCT
ejpam-4572	266	12	)	)	PUNCT
ejpam-4572	266	13	.	.	PUNCT
ejpam-4572	267	1	(	(	PUNCT
ejpam-4572	267	2	10	10	NUM
ejpam-4572	267	3	)	)	PUNCT
ejpam-4572	267	4	⇒	⇒	NOUN
ejpam-4572	267	5	(	(	PUNCT
ejpam-4572	267	6	8)	8)	NUM
ejpam-4572	267	7	:	:	PUNCT
ejpam-4572	267	8	let	let	VERB
ejpam-4572	267	9	v	v	PART
ejpam-4572	267	10	be	be	AUX
ejpam-4572	267	11	any	any	DET
ejpam-4572	267	12	(	(	PUNCT
ejpam-4572	267	13	λ	λ	NOUN
ejpam-4572	267	14	,	,	PUNCT
ejpam-4572	267	15	sp)-open	sp)-open	ADJ
ejpam-4572	267	16	set	set	NOUN
ejpam-4572	267	17	of	of	ADP
ejpam-4572	267	18	y	y	PROPN
ejpam-4572	267	19	.	.	PUNCT
ejpam-4572	268	1	thus	thus	ADV
ejpam-4572	268	2	,	,	PUNCT
ejpam-4572	268	3	x	x	PUNCT
ejpam-4572	268	4	−	−	PROPN
ejpam-4572	269	1	[	[	X
ejpam-4572	269	2	f+(v	f+(v	INTJ
ejpam-4572	269	3	(	(	PUNCT
ejpam-4572	269	4	λ	λ	PROPN
ejpam-4572	269	5	,	,	PUNCT
ejpam-4572	269	6	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	269	7	,	,	PUNCT
ejpam-4572	269	8	sp	sp	NOUN
ejpam-4572	269	9	)	)	PUNCT
ejpam-4572	269	10	=	=	PUNCT
ejpam-4572	270	1	[	[	X
ejpam-4572	270	2	x	x	X
ejpam-4572	270	3	−	−	PROPN
ejpam-4572	270	4	f+(v	f+(v	PROPN
ejpam-4572	270	5	(	(	PUNCT
ejpam-4572	270	6	λ	λ	PROPN
ejpam-4572	270	7	,	,	PUNCT
ejpam-4572	270	8	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	270	9	,	,	PUNCT
ejpam-4572	270	10	sp	sp	NOUN
ejpam-4572	270	11	)	)	PUNCT
ejpam-4572	270	12	=	=	PUNCT
ejpam-4572	271	1	[	[	X
ejpam-4572	271	2	f−(y	f−(y	NOUN
ejpam-4572	271	3	−	−	NOUN
ejpam-4572	271	4	v	v	NOUN
ejpam-4572	271	5	(	(	PUNCT
ejpam-4572	271	6	λ	λ	PROPN
ejpam-4572	271	7	,	,	PUNCT
ejpam-4572	271	8	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	271	9	,	,	PUNCT
ejpam-4572	271	10	sp	sp	NOUN
ejpam-4572	271	11	)	)	PUNCT
ejpam-4572	271	12	⊆	⊆	NUM
ejpam-4572	271	13	f−([y	f−([y	PROPN
ejpam-4572	271	14	−	−	PROPN
ejpam-4572	271	15	v	v	NOUN
ejpam-4572	271	16	(	(	PUNCT
ejpam-4572	271	17	λ	λ	PROPN
ejpam-4572	271	18	,	,	PUNCT
ejpam-4572	271	19	sp)](λ	sp)](λ	PROPN
ejpam-4572	271	20	,	,	PUNCT
ejpam-4572	271	21	sp	sp	NOUN
ejpam-4572	271	22	)	)	PUNCT
ejpam-4572	271	23	)	)	PUNCT
ejpam-4572	272	1	=	=	PUNCT
ejpam-4572	273	1	x	x	PUNCT
ejpam-4572	273	2	−	−	PROPN
ejpam-4572	273	3	f+([v	f+([v	NOUN
ejpam-4572	273	4	(	(	PUNCT
ejpam-4572	273	5	λ	λ	PROPN
ejpam-4572	273	6	,	,	PUNCT
ejpam-4572	273	7	sp)](λ	sp)](λ	PROPN
ejpam-4572	273	8	,	,	PUNCT
ejpam-4572	273	9	sp	sp	NOUN
ejpam-4572	273	10	)	)	PUNCT
ejpam-4572	273	11	)	)	PUNCT
ejpam-4572	273	12	and	and	CCONJ
ejpam-4572	273	13	hence	hence	ADV
ejpam-4572	273	14	f+(v	f+(v	ADV
ejpam-4572	273	15	)	)	PUNCT
ejpam-4572	274	1	⊆	⊆	NUM
ejpam-4572	274	2	f+([v	f+([v	NOUN
ejpam-4572	274	3	(	(	PUNCT
ejpam-4572	274	4	λ	λ	PROPN
ejpam-4572	274	5	,	,	PUNCT
ejpam-4572	274	6	sp)](λ	sp)](λ	PROPN
ejpam-4572	274	7	,	,	PUNCT
ejpam-4572	274	8	sp	sp	NOUN
ejpam-4572	274	9	)	)	PUNCT
ejpam-4572	274	10	)	)	PUNCT
ejpam-4572	275	1	⊆	⊆	NUM
ejpam-4572	275	2	[	[	X
ejpam-4572	275	3	f+(v	f+(v	NOUN
ejpam-4572	275	4	(	(	PUNCT
ejpam-4572	275	5	λ	λ	PROPN
ejpam-4572	275	6	,	,	PUNCT
ejpam-4572	275	7	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	275	8	,	,	PUNCT
ejpam-4572	275	9	sp	sp	NOUN
ejpam-4572	275	10	)	)	PUNCT
ejpam-4572	275	11	.	.	PUNCT
ejpam-4572	276	1	(	(	PUNCT
ejpam-4572	276	2	10	10	NUM
ejpam-4572	276	3	)	)	PUNCT
ejpam-4572	276	4	⇒	⇒	NOUN
ejpam-4572	276	5	(	(	PUNCT
ejpam-4572	276	6	11	11	NUM
ejpam-4572	276	7	):	):	PUNCT
ejpam-4572	276	8	let	let	VERB
ejpam-4572	276	9	b	b	X
ejpam-4572	276	10	be	be	AUX
ejpam-4572	276	11	any	any	DET
ejpam-4572	276	12	subset	subset	NOUN
ejpam-4572	276	13	of	of	ADP
ejpam-4572	276	14	y	y	PROPN
ejpam-4572	276	15	.	.	PUNCT
ejpam-4572	277	1	put	put	VERB
ejpam-4572	277	2	v	v	NOUN
ejpam-4572	277	3	=	=	SYM
ejpam-4572	277	4	[	[	X
ejpam-4572	277	5	bθ(λ	bθ(λ	X
ejpam-4572	277	6	,	,	PUNCT
ejpam-4572	277	7	sp)](λ	sp)](λ	PROPN
ejpam-4572	277	8	,	,	PUNCT
ejpam-4572	277	9	sp	sp	NOUN
ejpam-4572	277	10	)	)	PUNCT
ejpam-4572	277	11	.	.	PUNCT
ejpam-4572	278	1	by	by	ADP
ejpam-4572	278	2	lemma	lemma	PROPN
ejpam-4572	278	3	8	8	NUM
ejpam-4572	278	4	,	,	PUNCT
ejpam-4572	278	5	we	we	PRON
ejpam-4572	278	6	have	have	VERB
ejpam-4572	278	7	bθ(λ	bθ(λ	NOUN
ejpam-4572	278	8	,	,	PUNCT
ejpam-4572	278	9	sp	sp	NOUN
ejpam-4572	278	10	)	)	PUNCT
ejpam-4572	278	11	is	be	AUX
ejpam-4572	278	12	(	(	PUNCT
ejpam-4572	278	13	λ	λ	X
ejpam-4572	278	14	,	,	PUNCT
ejpam-4572	278	15	sp)-closed	sp)-close	VERB
ejpam-4572	278	16	in	in	ADP
ejpam-4572	278	17	y	y	PROPN
ejpam-4572	278	18	and	and	CCONJ
ejpam-4572	278	19	by	by	ADP
ejpam-4572	278	20	(	(	PUNCT
ejpam-4572	278	21	10	10	NUM
ejpam-4572	278	22	)	)	PUNCT
ejpam-4572	278	23	,	,	PUNCT
ejpam-4572	279	1	[	[	X
ejpam-4572	279	2	f−([bθ(λ	f−([bθ(λ	X
ejpam-4572	279	3	,	,	PUNCT
ejpam-4572	279	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	279	5	,	,	PUNCT
ejpam-4572	279	6	sp	sp	NOUN
ejpam-4572	279	7	)	)	PUNCT
ejpam-4572	279	8	)	)	PUNCT
ejpam-4572	279	9	]	]	PUNCT
ejpam-4572	280	1	α(λ	α(λ	PROPN
ejpam-4572	280	2	,	,	PUNCT
ejpam-4572	280	3	sp	sp	NOUN
ejpam-4572	280	4	)	)	PUNCT
ejpam-4572	280	5	⊆	⊆	NUM
ejpam-4572	280	6	f−[bθ(λ	f−[bθ(λ	NOUN
ejpam-4572	280	7	,	,	PUNCT
ejpam-4572	280	8	sp	sp	NOUN
ejpam-4572	280	9	)	)	PUNCT
ejpam-4572	280	10	]	]	PUNCT
ejpam-4572	280	11	.	.	PUNCT
ejpam-4572	281	1	(	(	PUNCT
ejpam-4572	281	2	11	11	NUM
ejpam-4572	281	3	)	)	PUNCT
ejpam-4572	281	4	⇒	⇒	NOUN
ejpam-4572	281	5	(	(	PUNCT
ejpam-4572	281	6	9	9	NUM
ejpam-4572	281	7	):	):	PUNCT
ejpam-4572	281	8	let	let	VERB
ejpam-4572	281	9	k	k	PRON
ejpam-4572	281	10	be	be	AUX
ejpam-4572	281	11	any	any	DET
ejpam-4572	281	12	r(λ	r(λ	NOUN
ejpam-4572	281	13	,	,	PUNCT
ejpam-4572	281	14	sp)-closed	sp)-close	VERB
ejpam-4572	281	15	set	set	NOUN
ejpam-4572	281	16	of	of	ADP
ejpam-4572	281	17	y	y	PROPN
ejpam-4572	281	18	.	.	PUNCT
ejpam-4572	282	1	by	by	ADP
ejpam-4572	282	2	(	(	PUNCT
ejpam-4572	282	3	11	11	NUM
ejpam-4572	282	4	)	)	PUNCT
ejpam-4572	282	5	and	and	CCONJ
ejpam-4572	282	6	lemma	lemma	PROPN
ejpam-4572	282	7	8	8	NUM
ejpam-4572	282	8	,	,	PUNCT
ejpam-4572	282	9	[	[	X
ejpam-4572	282	10	f−(k(λ	f−(k(λ	NOUN
ejpam-4572	282	11	,	,	PUNCT
ejpam-4572	282	12	sp	sp	NOUN
ejpam-4572	282	13	)	)	PUNCT
ejpam-4572	282	14	)	)	PUNCT
ejpam-4572	282	15	]	]	PUNCT
ejpam-4572	283	1	α(λ	α(λ	PROPN
ejpam-4572	283	2	,	,	PUNCT
ejpam-4572	283	3	sp	sp	NOUN
ejpam-4572	283	4	)	)	PUNCT
ejpam-4572	283	5	=	=	NOUN
ejpam-4572	284	1	[	[	X
ejpam-4572	284	2	f−([k(λ	f−([k(λ	NOUN
ejpam-4572	284	3	,	,	PUNCT
ejpam-4572	284	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	284	5	,	,	PUNCT
ejpam-4572	284	6	sp	sp	NOUN
ejpam-4572	284	7	)	)	PUNCT
ejpam-4572	284	8	)	)	PUNCT
ejpam-4572	284	9	]	]	PUNCT
ejpam-4572	285	1	α(λ	α(λ	PROPN
ejpam-4572	285	2	,	,	PUNCT
ejpam-4572	285	3	sp	sp	NOUN
ejpam-4572	285	4	)	)	PUNCT
ejpam-4572	285	5	=	=	PUNCT
ejpam-4572	286	1	[	[	X
ejpam-4572	286	2	f−([[k(λ	f−([[k(λ	X
ejpam-4572	286	3	,	,	PUNCT
ejpam-4572	286	4	sp	sp	NOUN
ejpam-4572	286	5	)	)	PUNCT
ejpam-4572	286	6	]	]	PUNCT
ejpam-4572	286	7	θ(λ	θ(λ	PROPN
ejpam-4572	286	8	,	,	PUNCT
ejpam-4572	286	9	sp)](λ	sp)](λ	PROPN
ejpam-4572	286	10	,	,	PUNCT
ejpam-4572	286	11	sp	sp	NOUN
ejpam-4572	286	12	)	)	PUNCT
ejpam-4572	286	13	)	)	PUNCT
ejpam-4572	286	14	]	]	PUNCT
ejpam-4572	287	1	α(λ	α(λ	PROPN
ejpam-4572	287	2	,	,	PUNCT
ejpam-4572	287	3	sp	sp	NOUN
ejpam-4572	287	4	)	)	PUNCT
ejpam-4572	287	5	⊆	⊆	NUM
ejpam-4572	287	6	f−([k(λ	f−([k(λ	NOUN
ejpam-4572	287	7	,	,	PUNCT
ejpam-4572	287	8	sp	sp	NOUN
ejpam-4572	287	9	)	)	PUNCT
ejpam-4572	287	10	]	]	PUNCT
ejpam-4572	288	1	θ(λ	θ(λ	PROPN
ejpam-4572	288	2	,	,	PUNCT
ejpam-4572	288	3	sp	sp	NOUN
ejpam-4572	288	4	)	)	PUNCT
ejpam-4572	288	5	)	)	PUNCT
ejpam-4572	289	1	=	=	SYM
ejpam-4572	290	1	f−([k(λ	f−([k(λ	NOUN
ejpam-4572	290	2	,	,	PUNCT
ejpam-4572	290	3	sp	sp	NOUN
ejpam-4572	290	4	)	)	PUNCT
ejpam-4572	290	5	]	]	PUNCT
ejpam-4572	290	6	(	(	PUNCT
ejpam-4572	290	7	λ	λ	NOUN
ejpam-4572	290	8	,	,	PUNCT
ejpam-4572	290	9	sp	sp	NOUN
ejpam-4572	290	10	)	)	PUNCT
ejpam-4572	290	11	)	)	PUNCT
ejpam-4572	291	1	=	=	SYM
ejpam-4572	291	2	f−(k	f−(k	PROPN
ejpam-4572	291	3	)	)	PUNCT
ejpam-4572	291	4	.	.	PUNCT
ejpam-4572	292	1	theorem	theorem	ADJ
ejpam-4572	292	2	4	4	NUM
ejpam-4572	292	3	.	.	X
ejpam-4572	292	4	for	for	ADP
ejpam-4572	292	5	a	a	DET
ejpam-4572	292	6	multifunction	multifunction	NOUN
ejpam-4572	293	1	f	f	NOUN
ejpam-4572	293	2	:	:	PUNCT
ejpam-4572	293	3	(	(	PUNCT
ejpam-4572	293	4	x	x	X
ejpam-4572	293	5	,	,	PUNCT
ejpam-4572	293	6	τ	τ	X
ejpam-4572	293	7	)	)	PUNCT
ejpam-4572	293	8	→	→	SYM
ejpam-4572	293	9	(	(	PUNCT
ejpam-4572	293	10	y	y	PROPN
ejpam-4572	293	11	,	,	PUNCT
ejpam-4572	293	12	σ	σ	PROPN
ejpam-4572	293	13	)	)	PUNCT
ejpam-4572	293	14	,	,	PUNCT
ejpam-4572	293	15	the	the	DET
ejpam-4572	293	16	following	follow	VERB
ejpam-4572	293	17	properties	property	NOUN
ejpam-4572	293	18	are	be	AUX
ejpam-4572	293	19	equivalent	equivalent	ADJ
ejpam-4572	293	20	:	:	PUNCT
ejpam-4572	293	21	(	(	PUNCT
ejpam-4572	293	22	1	1	X
ejpam-4572	293	23	)	)	PUNCT
ejpam-4572	293	24	f	f	PROPN
ejpam-4572	293	25	is	be	AUX
ejpam-4572	293	26	lower	low	ADJ
ejpam-4572	293	27	weakly	weakly	ADJ
ejpam-4572	293	28	α(λ	α(λ	PROPN
ejpam-4572	293	29	,	,	PUNCT
ejpam-4572	293	30	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	293	31	;	;	PUNCT
ejpam-4572	293	32	(	(	PUNCT
ejpam-4572	293	33	2	2	X
ejpam-4572	293	34	)	)	PUNCT
ejpam-4572	293	35	for	for	ADP
ejpam-4572	293	36	each	each	DET
ejpam-4572	293	37	x	x	SYM
ejpam-4572	293	38	∈	∈	PROPN
ejpam-4572	293	39	x	x	X
ejpam-4572	293	40	and	and	CCONJ
ejpam-4572	293	41	each	each	DET
ejpam-4572	293	42	(	(	PUNCT
ejpam-4572	293	43	λ	λ	PROPN
ejpam-4572	293	44	,	,	PUNCT
ejpam-4572	293	45	sp)-open	sp)-open	NOUN
ejpam-4572	293	46	set	set	VERB
ejpam-4572	293	47	v	v	NUM
ejpam-4572	293	48	of	of	ADP
ejpam-4572	293	49	y	y	PRON
ejpam-4572	293	50	such	such	ADJ
ejpam-4572	293	51	that	that	SCONJ
ejpam-4572	293	52	f	f	PROPN
ejpam-4572	293	53	(	(	PUNCT
ejpam-4572	293	54	x	x	NOUN
ejpam-4572	293	55	)	)	PUNCT
ejpam-4572	293	56	∩	∩	NOUN
ejpam-4572	293	57	v	v	ADP
ejpam-4572	293	58	̸=	̸=	PROPN
ejpam-4572	293	59	∅	∅	NOUN
ejpam-4572	293	60	,	,	PUNCT
ejpam-4572	293	61	there	there	PRON
ejpam-4572	293	62	exists	exist	VERB
ejpam-4572	293	63	an	an	DET
ejpam-4572	293	64	α(λ	α(λ	PROPN
ejpam-4572	293	65	,	,	PUNCT
ejpam-4572	293	66	sp)-open	sp)-open	VERB
ejpam-4572	293	67	set	set	VERB
ejpam-4572	293	68	u	u	NOUN
ejpam-4572	293	69	of	of	ADP
ejpam-4572	293	70	x	x	PUNCT
ejpam-4572	293	71	containing	contain	VERB
ejpam-4572	293	72	x	x	PUNCT
ejpam-4572	293	73	such	such	ADJ
ejpam-4572	293	74	that	that	SCONJ
ejpam-4572	293	75	u	u	PROPN
ejpam-4572	293	76	⊆	⊆	NUM
ejpam-4572	293	77	f−(v	f−(v	NOUN
ejpam-4572	293	78	(	(	PUNCT
ejpam-4572	293	79	λ	λ	NOUN
ejpam-4572	293	80	,	,	PUNCT
ejpam-4572	293	81	sp	sp	NOUN
ejpam-4572	293	82	)	)	PUNCT
ejpam-4572	293	83	)	)	PUNCT
ejpam-4572	293	84	;	;	PUNCT
ejpam-4572	293	85	(	(	PUNCT
ejpam-4572	293	86	3	3	X
ejpam-4572	293	87	)	)	PUNCT
ejpam-4572	293	88	f−(v	f−(v	NOUN
ejpam-4572	293	89	)	)	PUNCT
ejpam-4572	293	90	⊆	⊆	NUM
ejpam-4572	294	1	[	[	X
ejpam-4572	294	2	[	[	X
ejpam-4572	294	3	[	[	X
ejpam-4572	294	4	f−(v	f−(v	ADJ
ejpam-4572	294	5	(	(	PUNCT
ejpam-4572	294	6	λ	λ	PROPN
ejpam-4572	294	7	,	,	PUNCT
ejpam-4572	294	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	294	9	,	,	PUNCT
ejpam-4572	294	10	sp	sp	NOUN
ejpam-4572	294	11	)	)	PUNCT
ejpam-4572	294	12	]	]	PUNCT
ejpam-4572	294	13	(	(	PUNCT
ejpam-4572	294	14	λ	λ	X
ejpam-4572	294	15	,	,	PUNCT
ejpam-4572	294	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	294	17	,	,	PUNCT
ejpam-4572	294	18	sp	sp	NOUN
ejpam-4572	294	19	)	)	PUNCT
ejpam-4572	294	20	for	for	ADP
ejpam-4572	294	21	every	every	DET
ejpam-4572	294	22	(	(	PUNCT
ejpam-4572	294	23	λ	λ	NOUN
ejpam-4572	294	24	,	,	PUNCT
ejpam-4572	294	25	sp)-open	sp)-open	NOUN
ejpam-4572	294	26	set	set	VERB
ejpam-4572	294	27	v	v	NOUN
ejpam-4572	294	28	of	of	ADP
ejpam-4572	294	29	y	y	PROPN
ejpam-4572	294	30	;	;	PUNCT
ejpam-4572	294	31	(	(	PUNCT
ejpam-4572	294	32	4	4	X
ejpam-4572	294	33	)	)	PUNCT
ejpam-4572	295	1	[	[	X
ejpam-4572	295	2	[	[	X
ejpam-4572	295	3	[	[	X
ejpam-4572	295	4	f+(k(λ	f+(k(λ	NOUN
ejpam-4572	295	5	,	,	PUNCT
ejpam-4572	295	6	sp	sp	NOUN
ejpam-4572	295	7	)	)	PUNCT
ejpam-4572	295	8	)	)	PUNCT
ejpam-4572	295	9	]	]	PUNCT
ejpam-4572	296	1	(	(	PUNCT
ejpam-4572	296	2	λ	λ	X
ejpam-4572	296	3	,	,	PUNCT
ejpam-4572	296	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	296	5	,	,	PUNCT
ejpam-4572	296	6	sp	sp	NOUN
ejpam-4572	296	7	)	)	PUNCT
ejpam-4572	296	8	]	]	PUNCT
ejpam-4572	296	9	(	(	PUNCT
ejpam-4572	296	10	λ	λ	NOUN
ejpam-4572	296	11	,	,	PUNCT
ejpam-4572	296	12	sp	sp	NOUN
ejpam-4572	296	13	)	)	PUNCT
ejpam-4572	296	14	⊆	⊆	NUM
ejpam-4572	296	15	f+(k	f+(k	NOUN
ejpam-4572	296	16	)	)	PUNCT
ejpam-4572	296	17	for	for	SCONJ
ejpam-4572	296	18	every	every	DET
ejpam-4572	296	19	(	(	PUNCT
ejpam-4572	296	20	λ	λ	PROPN
ejpam-4572	296	21	,	,	PUNCT
ejpam-4572	296	22	sp)-closed	sp)-close	VERB
ejpam-4572	296	23	set	set	VERB
ejpam-4572	296	24	k	k	PROPN
ejpam-4572	296	25	of	of	ADP
ejpam-4572	296	26	y	y	PROPN
ejpam-4572	296	27	;	;	PUNCT
ejpam-4572	296	28	(	(	PUNCT
ejpam-4572	296	29	5	5	X
ejpam-4572	296	30	)	)	PUNCT
ejpam-4572	296	31	[	[	X
ejpam-4572	296	32	f+(k(λ	f+(k(λ	NOUN
ejpam-4572	296	33	,	,	PUNCT
ejpam-4572	296	34	sp	sp	NOUN
ejpam-4572	296	35	)	)	PUNCT
ejpam-4572	296	36	)	)	PUNCT
ejpam-4572	296	37	]	]	PUNCT
ejpam-4572	297	1	α(λ	α(λ	PROPN
ejpam-4572	297	2	,	,	PUNCT
ejpam-4572	297	3	sp	sp	NOUN
ejpam-4572	297	4	)	)	PUNCT
ejpam-4572	297	5	⊆	⊆	NUM
ejpam-4572	297	6	f+(k	f+(k	NOUN
ejpam-4572	297	7	)	)	PUNCT
ejpam-4572	297	8	for	for	ADP
ejpam-4572	297	9	every	every	DET
ejpam-4572	297	10	(	(	PUNCT
ejpam-4572	297	11	λ	λ	PROPN
ejpam-4572	297	12	,	,	PUNCT
ejpam-4572	297	13	sp)-closed	sp)-close	VERB
ejpam-4572	297	14	set	set	VERB
ejpam-4572	297	15	k	k	PROPN
ejpam-4572	297	16	of	of	ADP
ejpam-4572	297	17	y	y	PROPN
ejpam-4572	297	18	;	;	PUNCT
ejpam-4572	297	19	(	(	PUNCT
ejpam-4572	297	20	6	6	X
ejpam-4572	297	21	)	)	PUNCT
ejpam-4572	298	1	[	[	X
ejpam-4572	298	2	f+([b(λ	f+([b(λ	PROPN
ejpam-4572	298	3	,	,	PUNCT
ejpam-4572	298	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	298	5	,	,	PUNCT
ejpam-4572	298	6	sp	sp	NOUN
ejpam-4572	298	7	)	)	PUNCT
ejpam-4572	298	8	)	)	PUNCT
ejpam-4572	298	9	]	]	PUNCT
ejpam-4572	299	1	α(λ	α(λ	PROPN
ejpam-4572	299	2	,	,	PUNCT
ejpam-4572	299	3	sp	sp	NOUN
ejpam-4572	299	4	)	)	PUNCT
ejpam-4572	299	5	⊆	⊆	NUM
ejpam-4572	299	6	f+(b(λ	f+(b(λ	PROPN
ejpam-4572	299	7	,	,	PUNCT
ejpam-4572	299	8	sp	sp	NOUN
ejpam-4572	299	9	)	)	PUNCT
ejpam-4572	299	10	)	)	PUNCT
ejpam-4572	299	11	for	for	ADP
ejpam-4572	299	12	every	every	DET
ejpam-4572	299	13	subset	subset	NOUN
ejpam-4572	299	14	b	b	PROPN
ejpam-4572	299	15	of	of	ADP
ejpam-4572	299	16	y	y	PROPN
ejpam-4572	299	17	;	;	PUNCT
ejpam-4572	299	18	c.	c.	PROPN
ejpam-4572	299	19	boonpok	boonpok	PROPN
ejpam-4572	299	20	,	,	PUNCT
ejpam-4572	299	21	m.	m.	NOUN
ejpam-4572	299	22	thongmoon	thongmoon	PROPN
ejpam-4572	299	23	/	/	SYM
ejpam-4572	299	24	eur	eur	PROPN
ejpam-4572	299	25	.	.	PUNCT
ejpam-4572	300	1	j.	j.	PROPN
ejpam-4572	300	2	pure	pure	PROPN
ejpam-4572	300	3	appl	appl	PROPN
ejpam-4572	300	4	.	.	PROPN
ejpam-4572	300	5	math	math	PROPN
ejpam-4572	300	6	,	,	PUNCT
ejpam-4572	300	7	16	16	NUM
ejpam-4572	300	8	(	(	PUNCT
ejpam-4572	300	9	1	1	NUM
ejpam-4572	300	10	)	)	PUNCT
ejpam-4572	300	11	(	(	PUNCT
ejpam-4572	300	12	2023	2023	NUM
ejpam-4572	300	13	)	)	PUNCT
ejpam-4572	300	14	,	,	PUNCT
ejpam-4572	300	15	465	465	NUM
ejpam-4572	300	16	-	-	SYM
ejpam-4572	300	17	478	478	NUM
ejpam-4572	300	18	473	473	NUM
ejpam-4572	300	19	(	(	PUNCT
ejpam-4572	300	20	7	7	NUM
ejpam-4572	300	21	)	)	PUNCT
ejpam-4572	300	22	f−(b(λ	f−(b(λ	NOUN
ejpam-4572	300	23	,	,	PUNCT
ejpam-4572	300	24	sp	sp	NOUN
ejpam-4572	300	25	)	)	PUNCT
ejpam-4572	300	26	)	)	PUNCT
ejpam-4572	301	1	⊆	⊆	NUM
ejpam-4572	301	2	[	[	X
ejpam-4572	301	3	f−([b(λ	f−([b(λ	PROPN
ejpam-4572	301	4	,	,	PUNCT
ejpam-4572	301	5	sp	sp	NOUN
ejpam-4572	301	6	)	)	PUNCT
ejpam-4572	301	7	]	]	PUNCT
ejpam-4572	301	8	(	(	PUNCT
ejpam-4572	301	9	λ	λ	NOUN
ejpam-4572	301	10	,	,	PUNCT
ejpam-4572	301	11	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	301	12	,	,	PUNCT
ejpam-4572	301	13	sp	sp	NOUN
ejpam-4572	301	14	)	)	PUNCT
ejpam-4572	301	15	for	for	ADP
ejpam-4572	301	16	every	every	DET
ejpam-4572	301	17	subset	subset	NOUN
ejpam-4572	301	18	b	b	PROPN
ejpam-4572	301	19	of	of	ADP
ejpam-4572	301	20	y	y	PROPN
ejpam-4572	301	21	;	;	PUNCT
ejpam-4572	301	22	(	(	PUNCT
ejpam-4572	301	23	8)	8)	NUM
ejpam-4572	301	24	f−(v	f−(v	NOUN
ejpam-4572	301	25	)	)	PUNCT
ejpam-4572	301	26	⊆	⊆	NUM
ejpam-4572	302	1	[	[	X
ejpam-4572	302	2	f−(v	f−(v	ADJ
ejpam-4572	302	3	(	(	PUNCT
ejpam-4572	302	4	λ	λ	PROPN
ejpam-4572	302	5	,	,	PUNCT
ejpam-4572	302	6	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	302	7	,	,	PUNCT
ejpam-4572	302	8	sp	sp	NOUN
ejpam-4572	302	9	)	)	PUNCT
ejpam-4572	302	10	for	for	ADP
ejpam-4572	302	11	every	every	DET
ejpam-4572	302	12	(	(	PUNCT
ejpam-4572	302	13	λ	λ	NOUN
ejpam-4572	302	14	,	,	PUNCT
ejpam-4572	302	15	sp)-open	sp)-open	NOUN
ejpam-4572	302	16	set	set	VERB
ejpam-4572	302	17	v	v	NOUN
ejpam-4572	302	18	of	of	ADP
ejpam-4572	302	19	y	y	PROPN
ejpam-4572	302	20	;	;	PUNCT
ejpam-4572	302	21	(	(	PUNCT
ejpam-4572	302	22	9	9	X
ejpam-4572	302	23	)	)	PUNCT
ejpam-4572	303	1	[	[	X
ejpam-4572	303	2	f+(k(λ	f+(k(λ	NOUN
ejpam-4572	303	3	,	,	PUNCT
ejpam-4572	303	4	sp	sp	NOUN
ejpam-4572	303	5	)	)	PUNCT
ejpam-4572	303	6	)	)	PUNCT
ejpam-4572	303	7	]	]	PUNCT
ejpam-4572	304	1	α(λ	α(λ	PROPN
ejpam-4572	304	2	,	,	PUNCT
ejpam-4572	304	3	sp	sp	NOUN
ejpam-4572	304	4	)	)	PUNCT
ejpam-4572	304	5	⊆	⊆	NUM
ejpam-4572	304	6	f+(k	f+(k	NOUN
ejpam-4572	304	7	)	)	PUNCT
ejpam-4572	304	8	for	for	ADP
ejpam-4572	304	9	every	every	DET
ejpam-4572	304	10	r(λ	r(λ	NOUN
ejpam-4572	304	11	,	,	PUNCT
ejpam-4572	304	12	sp)-closed	sp)-close	VERB
ejpam-4572	304	13	set	set	VERB
ejpam-4572	304	14	k	k	PROPN
ejpam-4572	304	15	of	of	ADP
ejpam-4572	304	16	y	y	PROPN
ejpam-4572	304	17	;	;	PUNCT
ejpam-4572	304	18	(	(	PUNCT
ejpam-4572	304	19	10	10	NUM
ejpam-4572	304	20	)	)	PUNCT
ejpam-4572	305	1	[	[	X
ejpam-4572	305	2	f+(v	f+(v	NOUN
ejpam-4572	305	3	)	)	PUNCT
ejpam-4572	306	1	]	]	PUNCT
ejpam-4572	306	2	α(λ	α(λ	PROPN
ejpam-4572	306	3	,	,	PUNCT
ejpam-4572	306	4	sp	sp	NOUN
ejpam-4572	306	5	)	)	PUNCT
ejpam-4572	306	6	⊆	⊆	NUM
ejpam-4572	306	7	f+(v	f+(v	NUM
ejpam-4572	306	8	(	(	PUNCT
ejpam-4572	306	9	λ	λ	NOUN
ejpam-4572	306	10	,	,	PUNCT
ejpam-4572	306	11	sp	sp	NOUN
ejpam-4572	306	12	)	)	PUNCT
ejpam-4572	306	13	)	)	PUNCT
ejpam-4572	306	14	for	for	ADP
ejpam-4572	306	15	every	every	DET
ejpam-4572	306	16	(	(	PUNCT
ejpam-4572	306	17	λ	λ	NOUN
ejpam-4572	306	18	,	,	PUNCT
ejpam-4572	306	19	sp)-open	sp)-open	NOUN
ejpam-4572	306	20	set	set	VERB
ejpam-4572	306	21	v	v	NOUN
ejpam-4572	306	22	of	of	ADP
ejpam-4572	306	23	y	y	PROPN
ejpam-4572	306	24	;	;	PUNCT
ejpam-4572	306	25	(	(	PUNCT
ejpam-4572	306	26	11	11	NUM
ejpam-4572	306	27	)	)	PUNCT
ejpam-4572	307	1	[	[	X
ejpam-4572	307	2	f+([bθ(λ	f+([bθ(λ	PROPN
ejpam-4572	307	3	,	,	PUNCT
ejpam-4572	307	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	307	5	,	,	PUNCT
ejpam-4572	307	6	sp	sp	NOUN
ejpam-4572	307	7	)	)	PUNCT
ejpam-4572	307	8	)	)	PUNCT
ejpam-4572	307	9	]	]	PUNCT
ejpam-4572	308	1	α(λ	α(λ	PROPN
ejpam-4572	308	2	,	,	PUNCT
ejpam-4572	308	3	sp	sp	NOUN
ejpam-4572	308	4	)	)	PUNCT
ejpam-4572	308	5	⊆	⊆	NUM
ejpam-4572	308	6	f+(bθ(λ	f+(bθ(λ	PROPN
ejpam-4572	308	7	,	,	PUNCT
ejpam-4572	308	8	sp	sp	NOUN
ejpam-4572	308	9	)	)	PUNCT
ejpam-4572	308	10	)	)	PUNCT
ejpam-4572	308	11	for	for	ADP
ejpam-4572	308	12	every	every	DET
ejpam-4572	308	13	subset	subset	NOUN
ejpam-4572	308	14	b	b	PROPN
ejpam-4572	308	15	of	of	ADP
ejpam-4572	308	16	y	y	PROPN
ejpam-4572	308	17	.	.	PUNCT
ejpam-4572	309	1	proof	proof	NOUN
ejpam-4572	309	2	.	.	PUNCT
ejpam-4572	310	1	the	the	DET
ejpam-4572	310	2	proof	proof	NOUN
ejpam-4572	310	3	is	be	AUX
ejpam-4572	310	4	similar	similar	ADJ
ejpam-4572	310	5	to	to	ADP
ejpam-4572	310	6	that	that	PRON
ejpam-4572	310	7	of	of	ADP
ejpam-4572	310	8	theorem	theorem	NOUN
ejpam-4572	310	9	3	3	NUM
ejpam-4572	310	10	.	.	PUNCT
ejpam-4572	310	11	lemma	lemma	PROPN
ejpam-4572	310	12	9	9	NUM
ejpam-4572	310	13	.	.	PUNCT
ejpam-4572	311	1	if	if	SCONJ
ejpam-4572	311	2	f	f	PROPN
ejpam-4572	311	3	:	:	PUNCT
ejpam-4572	311	4	(	(	PUNCT
ejpam-4572	311	5	x	x	X
ejpam-4572	311	6	,	,	PUNCT
ejpam-4572	311	7	τ	τ	X
ejpam-4572	311	8	)	)	PUNCT
ejpam-4572	311	9	→	→	SYM
ejpam-4572	311	10	(	(	PUNCT
ejpam-4572	311	11	y	y	PROPN
ejpam-4572	311	12	,	,	PUNCT
ejpam-4572	311	13	σ	σ	PROPN
ejpam-4572	311	14	)	)	PUNCT
ejpam-4572	311	15	is	be	AUX
ejpam-4572	311	16	lower	low	ADJ
ejpam-4572	311	17	weakly	weakly	ADJ
ejpam-4572	311	18	α(λ	α(λ	PROPN
ejpam-4572	311	19	,	,	PUNCT
ejpam-4572	311	20	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	311	21	,	,	PUNCT
ejpam-4572	311	22	then	then	ADV
ejpam-4572	311	23	for	for	ADP
ejpam-4572	311	24	each	each	DET
ejpam-4572	311	25	x	x	SYM
ejpam-4572	311	26	∈	∈	PROPN
ejpam-4572	311	27	x	x	X
ejpam-4572	311	28	and	and	CCONJ
ejpam-4572	311	29	each	each	DET
ejpam-4572	311	30	subset	subset	NOUN
ejpam-4572	311	31	b	b	PROPN
ejpam-4572	311	32	of	of	ADP
ejpam-4572	311	33	y	y	PROPN
ejpam-4572	311	34	with	with	ADP
ejpam-4572	311	35	f	f	PROPN
ejpam-4572	311	36	(	(	PUNCT
ejpam-4572	311	37	x	x	NOUN
ejpam-4572	311	38	)	)	PUNCT
ejpam-4572	311	39	∩bθ(λ	∩bθ(λ	PROPN
ejpam-4572	311	40	,	,	PUNCT
ejpam-4572	311	41	sp	sp	NOUN
ejpam-4572	311	42	)	)	PUNCT
ejpam-4572	311	43	̸=	̸=	NOUN
ejpam-4572	311	44	∅	∅	NOUN
ejpam-4572	311	45	,	,	PUNCT
ejpam-4572	311	46	there	there	PRON
ejpam-4572	311	47	exists	exist	VERB
ejpam-4572	311	48	an	an	DET
ejpam-4572	311	49	α(λ	α(λ	PROPN
ejpam-4572	311	50	,	,	PUNCT
ejpam-4572	311	51	sp)-open	sp)-open	VERB
ejpam-4572	311	52	set	set	VERB
ejpam-4572	311	53	u	u	NOUN
ejpam-4572	311	54	of	of	ADP
ejpam-4572	311	55	x	x	PUNCT
ejpam-4572	311	56	containing	contain	VERB
ejpam-4572	311	57	x	x	PUNCT
ejpam-4572	311	58	such	such	ADJ
ejpam-4572	311	59	that	that	SCONJ
ejpam-4572	311	60	u	u	PROPN
ejpam-4572	311	61	⊆	⊆	NUM
ejpam-4572	311	62	f−(b	f−(b	NOUN
ejpam-4572	311	63	)	)	PUNCT
ejpam-4572	311	64	.	.	PUNCT
ejpam-4572	312	1	proof	proof	NOUN
ejpam-4572	312	2	.	.	PUNCT
ejpam-4572	313	1	since	since	SCONJ
ejpam-4572	313	2	f	f	PROPN
ejpam-4572	313	3	(	(	PUNCT
ejpam-4572	313	4	x)∩bθ(λ	x)∩bθ(λ	PROPN
ejpam-4572	313	5	,	,	PUNCT
ejpam-4572	313	6	sp	sp	NOUN
ejpam-4572	313	7	)	)	PUNCT
ejpam-4572	313	8	̸=	̸=	NOUN
ejpam-4572	313	9	∅	∅	NOUN
ejpam-4572	313	10	,	,	PUNCT
ejpam-4572	313	11	there	there	PRON
ejpam-4572	313	12	exists	exist	VERB
ejpam-4572	313	13	a	a	DET
ejpam-4572	313	14	nonempty	nonempty	ADJ
ejpam-4572	313	15	(	(	PUNCT
ejpam-4572	313	16	λ	λ	NOUN
ejpam-4572	313	17	,	,	PUNCT
ejpam-4572	313	18	sp)-open	sp)-open	NOUN
ejpam-4572	313	19	set	set	VERB
ejpam-4572	313	20	u	u	NOUN
ejpam-4572	313	21	of	of	ADP
ejpam-4572	313	22	y	y	PRON
ejpam-4572	313	23	such	such	ADJ
ejpam-4572	313	24	that	that	PRON
ejpam-4572	313	25	v	v	ADP
ejpam-4572	313	26	⊆	⊆	NUM
ejpam-4572	313	27	v	v	NOUN
ejpam-4572	313	28	(	(	PUNCT
ejpam-4572	313	29	λ	λ	NOUN
ejpam-4572	313	30	,	,	PUNCT
ejpam-4572	313	31	sp	sp	NOUN
ejpam-4572	313	32	)	)	PUNCT
ejpam-4572	313	33	⊆	⊆	NUM
ejpam-4572	313	34	b	b	NOUN
ejpam-4572	313	35	and	and	CCONJ
ejpam-4572	313	36	f	f	PROPN
ejpam-4572	313	37	(	(	PUNCT
ejpam-4572	313	38	x	x	NOUN
ejpam-4572	313	39	)	)	PUNCT
ejpam-4572	313	40	∩	∩	NOUN
ejpam-4572	313	41	v	v	ADP
ejpam-4572	313	42	̸=	̸=	PROPN
ejpam-4572	313	43	∅.	∅.	NOUN
ejpam-4572	313	44	since	since	SCONJ
ejpam-4572	313	45	f	f	PROPN
ejpam-4572	313	46	is	be	AUX
ejpam-4572	313	47	lower	low	ADJ
ejpam-4572	313	48	weakly	weakly	ADJ
ejpam-4572	313	49	α(λ	α(λ	PROPN
ejpam-4572	313	50	,	,	PUNCT
ejpam-4572	313	51	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	313	52	,	,	PUNCT
ejpam-4572	313	53	there	there	PRON
ejpam-4572	313	54	exists	exist	VERB
ejpam-4572	313	55	an	an	DET
ejpam-4572	313	56	α(λ	α(λ	PROPN
ejpam-4572	313	57	,	,	PUNCT
ejpam-4572	313	58	sp)-open	sp)-open	VERB
ejpam-4572	313	59	set	set	VERB
ejpam-4572	313	60	u	u	NOUN
ejpam-4572	313	61	of	of	ADP
ejpam-4572	313	62	x	x	PUNCT
ejpam-4572	313	63	containing	contain	VERB
ejpam-4572	313	64	x	x	PUNCT
ejpam-4572	313	65	such	such	ADJ
ejpam-4572	313	66	that	that	SCONJ
ejpam-4572	313	67	f	f	PROPN
ejpam-4572	313	68	(	(	PUNCT
ejpam-4572	313	69	z	z	NOUN
ejpam-4572	313	70	)	)	PUNCT
ejpam-4572	313	71	∩	∩	ADJ
ejpam-4572	313	72	v	v	X
ejpam-4572	313	73	(	(	PUNCT
ejpam-4572	313	74	λ	λ	PROPN
ejpam-4572	313	75	,	,	PUNCT
ejpam-4572	313	76	sp	sp	NOUN
ejpam-4572	313	77	)	)	PUNCT
ejpam-4572	313	78	̸=	̸=	NOUN
ejpam-4572	313	79	∅	∅	NOUN
ejpam-4572	313	80	for	for	ADP
ejpam-4572	313	81	every	every	DET
ejpam-4572	313	82	z	z	NOUN
ejpam-4572	313	83	∈	∈	PROPN
ejpam-4572	313	84	u	u	NOUN
ejpam-4572	313	85	.	.	PUNCT
ejpam-4572	314	1	thus	thus	ADV
ejpam-4572	314	2	,	,	PUNCT
ejpam-4572	314	3	u	u	PROPN
ejpam-4572	314	4	⊆	⊆	NUM
ejpam-4572	314	5	f−(b	f−(b	NOUN
ejpam-4572	314	6	)	)	PUNCT
ejpam-4572	314	7	.	.	PUNCT
ejpam-4572	315	1	theorem	theorem	NOUN
ejpam-4572	315	2	5	5	NUM
ejpam-4572	315	3	.	.	X
ejpam-4572	315	4	for	for	ADP
ejpam-4572	315	5	a	a	DET
ejpam-4572	315	6	multifunction	multifunction	NOUN
ejpam-4572	316	1	f	f	NOUN
ejpam-4572	316	2	:	:	PUNCT
ejpam-4572	316	3	(	(	PUNCT
ejpam-4572	316	4	x	x	X
ejpam-4572	316	5	,	,	PUNCT
ejpam-4572	316	6	τ	τ	X
ejpam-4572	316	7	)	)	PUNCT
ejpam-4572	316	8	→	→	SYM
ejpam-4572	316	9	(	(	PUNCT
ejpam-4572	316	10	y	y	PROPN
ejpam-4572	316	11	,	,	PUNCT
ejpam-4572	316	12	σ	σ	PROPN
ejpam-4572	316	13	)	)	PUNCT
ejpam-4572	316	14	,	,	PUNCT
ejpam-4572	316	15	the	the	DET
ejpam-4572	316	16	following	follow	VERB
ejpam-4572	316	17	properties	property	NOUN
ejpam-4572	316	18	are	be	AUX
ejpam-4572	316	19	equivalent	equivalent	ADJ
ejpam-4572	316	20	:	:	PUNCT
ejpam-4572	316	21	(	(	PUNCT
ejpam-4572	316	22	1	1	X
ejpam-4572	316	23	)	)	PUNCT
ejpam-4572	316	24	f	f	PROPN
ejpam-4572	316	25	is	be	AUX
ejpam-4572	316	26	lower	low	ADJ
ejpam-4572	316	27	weakly	weakly	ADJ
ejpam-4572	316	28	α(λ	α(λ	PROPN
ejpam-4572	316	29	,	,	PUNCT
ejpam-4572	316	30	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	316	31	;	;	PUNCT
ejpam-4572	316	32	(	(	PUNCT
ejpam-4572	316	33	2	2	X
ejpam-4572	316	34	)	)	PUNCT
ejpam-4572	317	1	[	[	X
ejpam-4572	317	2	f+(b)]α(λ	f+(b)]α(λ	NOUN
ejpam-4572	317	3	,	,	PUNCT
ejpam-4572	317	4	sp	sp	NOUN
ejpam-4572	317	5	)	)	PUNCT
ejpam-4572	317	6	⊆	⊆	NUM
ejpam-4572	317	7	f+(bθ(λ	f+(bθ(λ	PROPN
ejpam-4572	317	8	,	,	PUNCT
ejpam-4572	317	9	sp	sp	NOUN
ejpam-4572	317	10	)	)	PUNCT
ejpam-4572	317	11	)	)	PUNCT
ejpam-4572	317	12	for	for	ADP
ejpam-4572	317	13	every	every	DET
ejpam-4572	317	14	subset	subset	NOUN
ejpam-4572	317	15	b	b	PROPN
ejpam-4572	317	16	of	of	ADP
ejpam-4572	317	17	y	y	PROPN
ejpam-4572	317	18	;	;	PUNCT
ejpam-4572	317	19	(	(	PUNCT
ejpam-4572	317	20	3	3	X
ejpam-4572	317	21	)	)	PUNCT
ejpam-4572	317	22	f	f	NOUN
ejpam-4572	317	23	(	(	PUNCT
ejpam-4572	317	24	aα(λ	aα(λ	PROPN
ejpam-4572	317	25	,	,	PUNCT
ejpam-4572	317	26	sp	sp	NOUN
ejpam-4572	317	27	)	)	PUNCT
ejpam-4572	317	28	)	)	PUNCT
ejpam-4572	318	1	⊆	⊆	NUM
ejpam-4572	319	1	[	[	X
ejpam-4572	319	2	f	f	X
ejpam-4572	319	3	(	(	PUNCT
ejpam-4572	319	4	a)]θ(λ	a)]θ(λ	PROPN
ejpam-4572	319	5	,	,	PUNCT
ejpam-4572	319	6	sp	sp	NOUN
ejpam-4572	319	7	)	)	PUNCT
ejpam-4572	319	8	for	for	ADP
ejpam-4572	319	9	every	every	DET
ejpam-4572	319	10	subset	subset	NOUN
ejpam-4572	319	11	a	a	PRON
ejpam-4572	319	12	of	of	ADP
ejpam-4572	319	13	x.	x.	NOUN
ejpam-4572	319	14	proof	proof	NOUN
ejpam-4572	319	15	.	.	PUNCT
ejpam-4572	320	1	(	(	PUNCT
ejpam-4572	320	2	1	1	X
ejpam-4572	320	3	)	)	PUNCT
ejpam-4572	320	4	⇒	⇒	NOUN
ejpam-4572	320	5	(	(	PUNCT
ejpam-4572	320	6	2	2	NUM
ejpam-4572	320	7	):	):	PUNCT
ejpam-4572	320	8	let	let	VERB
ejpam-4572	320	9	b	b	X
ejpam-4572	320	10	be	be	AUX
ejpam-4572	320	11	any	any	DET
ejpam-4572	320	12	subset	subset	NOUN
ejpam-4572	320	13	of	of	ADP
ejpam-4572	320	14	y	y	PROPN
ejpam-4572	320	15	.	.	PUNCT
ejpam-4572	320	16	suppose	suppose	VERB
ejpam-4572	320	17	that	that	SCONJ
ejpam-4572	320	18	x	x	PUNCT
ejpam-4572	320	19	∈	∈	PROPN
ejpam-4572	320	20	f−(y	f−(y	NOUN
ejpam-4572	320	21	−bθ(λ	−bθ(λ	NOUN
ejpam-4572	320	22	,	,	PUNCT
ejpam-4572	320	23	sp	sp	NOUN
ejpam-4572	320	24	)	)	PUNCT
ejpam-4572	320	25	)	)	PUNCT
ejpam-4572	321	1	=	=	PUNCT
ejpam-4572	321	2	f−([y	f−([y	ADJ
ejpam-4572	321	3	−b]θ(λ	−b]θ(λ	NOUN
ejpam-4572	321	4	,	,	PUNCT
ejpam-4572	321	5	sp	sp	NOUN
ejpam-4572	321	6	)	)	PUNCT
ejpam-4572	321	7	)	)	PUNCT
ejpam-4572	321	8	.	.	PUNCT
ejpam-4572	322	1	by	by	ADP
ejpam-4572	322	2	lemma	lemma	PROPN
ejpam-4572	322	3	9	9	NUM
ejpam-4572	322	4	,	,	PUNCT
ejpam-4572	322	5	there	there	PRON
ejpam-4572	322	6	exists	exist	VERB
ejpam-4572	322	7	an	an	DET
ejpam-4572	322	8	α(λ	α(λ	PROPN
ejpam-4572	322	9	,	,	PUNCT
ejpam-4572	322	10	sp)-open	sp)-open	VERB
ejpam-4572	322	11	set	set	VERB
ejpam-4572	322	12	u	u	NOUN
ejpam-4572	322	13	of	of	ADP
ejpam-4572	322	14	x	x	PUNCT
ejpam-4572	322	15	containing	contain	VERB
ejpam-4572	322	16	x	x	PUNCT
ejpam-4572	322	17	such	such	ADJ
ejpam-4572	322	18	that	that	SCONJ
ejpam-4572	322	19	u	u	NOUN
ejpam-4572	322	20	⊆	⊆	NUM
ejpam-4572	322	21	f−(y	f−(y	NOUN
ejpam-4572	322	22	−b	−b	NOUN
ejpam-4572	322	23	)	)	PUNCT
ejpam-4572	323	1	=	=	PUNCT
ejpam-4572	323	2	x	x	X
ejpam-4572	324	1	−	−	NOUN
ejpam-4572	324	2	f+(b	f+(b	NOUN
ejpam-4572	324	3	)	)	PUNCT
ejpam-4572	324	4	.	.	PUNCT
ejpam-4572	325	1	this	this	PRON
ejpam-4572	325	2	shows	show	VERB
ejpam-4572	325	3	that	that	SCONJ
ejpam-4572	325	4	u	u	NOUN
ejpam-4572	325	5	∩	∩	NOUN
ejpam-4572	325	6	f+(b	f+(b	NOUN
ejpam-4572	325	7	)	)	PUNCT
ejpam-4572	325	8	=	=	PUNCT
ejpam-4572	325	9	∅.	∅.	VERB
ejpam-4572	325	10	therefore	therefore	ADV
ejpam-4572	325	11	,	,	PUNCT
ejpam-4572	325	12	x	x	PUNCT
ejpam-4572	325	13	∈	∈	NOUN
ejpam-4572	325	14	x	x	X
ejpam-4572	325	15	−	−	PROPN
ejpam-4572	326	1	[	[	X
ejpam-4572	326	2	f+(b)]α(λ	f+(b)]α(λ	NOUN
ejpam-4572	326	3	,	,	PUNCT
ejpam-4572	326	4	sp	sp	NOUN
ejpam-4572	326	5	)	)	PUNCT
ejpam-4572	326	6	.	.	PUNCT
ejpam-4572	327	1	(	(	PUNCT
ejpam-4572	327	2	2	2	X
ejpam-4572	327	3	)	)	PUNCT
ejpam-4572	327	4	⇒	⇒	NOUN
ejpam-4572	327	5	(	(	PUNCT
ejpam-4572	327	6	1	1	NUM
ejpam-4572	327	7	):	):	PUNCT
ejpam-4572	327	8	let	let	VERB
ejpam-4572	327	9	v	v	PART
ejpam-4572	327	10	be	be	AUX
ejpam-4572	327	11	any	any	DET
ejpam-4572	327	12	(	(	PUNCT
ejpam-4572	327	13	λ	λ	NOUN
ejpam-4572	327	14	,	,	PUNCT
ejpam-4572	327	15	sp)-open	sp)-open	ADJ
ejpam-4572	327	16	set	set	NOUN
ejpam-4572	327	17	of	of	ADP
ejpam-4572	327	18	y	y	PROPN
ejpam-4572	327	19	.	.	PUNCT
ejpam-4572	328	1	since	since	SCONJ
ejpam-4572	328	2	v	v	NOUN
ejpam-4572	328	3	(	(	PUNCT
ejpam-4572	328	4	λ	λ	NOUN
ejpam-4572	328	5	,	,	PUNCT
ejpam-4572	328	6	sp	sp	NOUN
ejpam-4572	328	7	)	)	PUNCT
ejpam-4572	328	8	=	=	SYM
ejpam-4572	328	9	v	v	ADP
ejpam-4572	328	10	θ(λ	θ(λ	PROPN
ejpam-4572	328	11	,	,	PUNCT
ejpam-4572	328	12	sp	sp	NOUN
ejpam-4572	328	13	)	)	PUNCT
ejpam-4572	328	14	for	for	ADP
ejpam-4572	328	15	every	every	DET
ejpam-4572	328	16	(	(	PUNCT
ejpam-4572	328	17	λ	λ	NOUN
ejpam-4572	328	18	,	,	PUNCT
ejpam-4572	328	19	sp)-open	sp)-open	NOUN
ejpam-4572	328	20	set	set	VERB
ejpam-4572	328	21	v	v	NOUN
ejpam-4572	328	22	of	of	ADP
ejpam-4572	328	23	y	y	PROPN
ejpam-4572	328	24	,	,	PUNCT
ejpam-4572	328	25	we	we	PRON
ejpam-4572	328	26	have	have	VERB
ejpam-4572	328	27	[	[	X
ejpam-4572	328	28	f+(v	f+(v	PROPN
ejpam-4572	328	29	)	)	PUNCT
ejpam-4572	329	1	]	]	PUNCT
ejpam-4572	329	2	α(λ	α(λ	PROPN
ejpam-4572	329	3	,	,	PUNCT
ejpam-4572	329	4	sp	sp	NOUN
ejpam-4572	329	5	)	)	PUNCT
ejpam-4572	329	6	⊆	⊆	NUM
ejpam-4572	329	7	f+(v	f+(v	NUM
ejpam-4572	329	8	(	(	PUNCT
ejpam-4572	329	9	λ	λ	NOUN
ejpam-4572	329	10	,	,	PUNCT
ejpam-4572	329	11	sp	sp	NOUN
ejpam-4572	329	12	)	)	PUNCT
ejpam-4572	329	13	)	)	PUNCT
ejpam-4572	329	14	and	and	CCONJ
ejpam-4572	329	15	by	by	ADP
ejpam-4572	329	16	theorem	theorem	NOUN
ejpam-4572	329	17	4	4	NUM
ejpam-4572	329	18	,	,	PUNCT
ejpam-4572	329	19	f	f	PROPN
ejpam-4572	329	20	is	be	AUX
ejpam-4572	329	21	lower	low	ADJ
ejpam-4572	329	22	weakly	weakly	ADJ
ejpam-4572	329	23	α(λ	α(λ	PROPN
ejpam-4572	329	24	,	,	PUNCT
ejpam-4572	329	25	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	329	26	.	.	PUNCT
ejpam-4572	330	1	(	(	PUNCT
ejpam-4572	330	2	2	2	X
ejpam-4572	330	3	)	)	PUNCT
ejpam-4572	330	4	⇒	⇒	NOUN
ejpam-4572	330	5	(	(	PUNCT
ejpam-4572	330	6	3	3	NUM
ejpam-4572	330	7	):	):	PUNCT
ejpam-4572	330	8	let	let	VERB
ejpam-4572	330	9	a	a	DET
ejpam-4572	330	10	be	be	AUX
ejpam-4572	330	11	any	any	DET
ejpam-4572	330	12	subset	subset	NOUN
ejpam-4572	330	13	of	of	ADP
ejpam-4572	330	14	x.	x.	NOUN
ejpam-4572	330	15	by	by	ADP
ejpam-4572	330	16	(	(	PUNCT
ejpam-4572	330	17	2	2	NUM
ejpam-4572	330	18	)	)	PUNCT
ejpam-4572	330	19	,	,	PUNCT
ejpam-4572	330	20	we	we	PRON
ejpam-4572	330	21	have	have	VERB
ejpam-4572	330	22	aα(λ	aα(λ	NOUN
ejpam-4572	330	23	,	,	PUNCT
ejpam-4572	330	24	sp	sp	NOUN
ejpam-4572	330	25	)	)	PUNCT
ejpam-4572	330	26	⊆	⊆	NUM
ejpam-4572	331	1	[	[	X
ejpam-4572	331	2	f+(f	f+(f	PROPN
ejpam-4572	331	3	(	(	PUNCT
ejpam-4572	331	4	a))]α(λ	a))]α(λ	PROPN
ejpam-4572	331	5	,	,	PUNCT
ejpam-4572	331	6	sp	sp	NOUN
ejpam-4572	331	7	)	)	PUNCT
ejpam-4572	331	8	⊆	⊆	NUM
ejpam-4572	331	9	f+([f	f+([f	X
ejpam-4572	331	10	(	(	PUNCT
ejpam-4572	331	11	a)]θ(λ	a)]θ(λ	PROPN
ejpam-4572	331	12	,	,	PUNCT
ejpam-4572	331	13	sp	sp	NOUN
ejpam-4572	331	14	)	)	PUNCT
ejpam-4572	331	15	)	)	PUNCT
ejpam-4572	331	16	and	and	CCONJ
ejpam-4572	331	17	hence	hence	ADV
ejpam-4572	331	18	f	f	PROPN
ejpam-4572	331	19	(	(	PUNCT
ejpam-4572	331	20	aα(λ	aα(λ	PROPN
ejpam-4572	331	21	,	,	PUNCT
ejpam-4572	331	22	sp	sp	NOUN
ejpam-4572	331	23	)	)	PUNCT
ejpam-4572	331	24	)	)	PUNCT
ejpam-4572	331	25	⊆	⊆	NUM
ejpam-4572	332	1	[	[	X
ejpam-4572	332	2	f	f	X
ejpam-4572	332	3	(	(	PUNCT
ejpam-4572	332	4	a)]θ(λ	a)]θ(λ	PROPN
ejpam-4572	332	5	,	,	PUNCT
ejpam-4572	332	6	sp	sp	NOUN
ejpam-4572	332	7	)	)	PUNCT
ejpam-4572	332	8	.	.	PUNCT
ejpam-4572	333	1	(	(	PUNCT
ejpam-4572	333	2	3	3	X
ejpam-4572	333	3	)	)	PUNCT
ejpam-4572	333	4	⇒	⇒	NOUN
ejpam-4572	333	5	(	(	PUNCT
ejpam-4572	333	6	2	2	NUM
ejpam-4572	333	7	):	):	PUNCT
ejpam-4572	333	8	let	let	VERB
ejpam-4572	333	9	b	b	X
ejpam-4572	333	10	be	be	AUX
ejpam-4572	333	11	any	any	DET
ejpam-4572	333	12	subset	subset	NOUN
ejpam-4572	333	13	of	of	ADP
ejpam-4572	333	14	y	y	PROPN
ejpam-4572	333	15	.	.	PUNCT
ejpam-4572	334	1	by	by	ADP
ejpam-4572	334	2	(	(	PUNCT
ejpam-4572	334	3	3	3	NUM
ejpam-4572	334	4	)	)	PUNCT
ejpam-4572	334	5	,	,	PUNCT
ejpam-4572	334	6	f	f	PROPN
ejpam-4572	334	7	(	(	PUNCT
ejpam-4572	334	8	[	[	X
ejpam-4572	334	9	f+(b)]α(λ	f+(b)]α(λ	NOUN
ejpam-4572	334	10	,	,	PUNCT
ejpam-4572	334	11	sp	sp	NOUN
ejpam-4572	334	12	)	)	PUNCT
ejpam-4572	334	13	)	)	PUNCT
ejpam-4572	334	14	⊆	⊆	NUM
ejpam-4572	335	1	[	[	X
ejpam-4572	335	2	f	f	X
ejpam-4572	335	3	(	(	PUNCT
ejpam-4572	335	4	f+(b))]θ(λ	f+(b))]θ(λ	NOUN
ejpam-4572	335	5	,	,	PUNCT
ejpam-4572	335	6	sp	sp	NOUN
ejpam-4572	335	7	)	)	PUNCT
ejpam-4572	335	8	⊆	⊆	NUM
ejpam-4572	335	9	bθ(λ	bθ(λ	NOUN
ejpam-4572	335	10	,	,	PUNCT
ejpam-4572	335	11	sp	sp	NOUN
ejpam-4572	335	12	)	)	PUNCT
ejpam-4572	335	13	.	.	PUNCT
ejpam-4572	336	1	thus	thus	ADV
ejpam-4572	336	2	,	,	PUNCT
ejpam-4572	336	3	[	[	X
ejpam-4572	336	4	f+(b)]α(λ	f+(b)]α(λ	NOUN
ejpam-4572	336	5	,	,	PUNCT
ejpam-4572	336	6	sp	sp	NOUN
ejpam-4572	336	7	)	)	PUNCT
ejpam-4572	336	8	⊆	⊆	NUM
ejpam-4572	336	9	f+(bθ(λ	f+(bθ(λ	PROPN
ejpam-4572	336	10	,	,	PUNCT
ejpam-4572	336	11	sp	sp	NOUN
ejpam-4572	336	12	)	)	PUNCT
ejpam-4572	336	13	)	)	PUNCT
ejpam-4572	336	14	.	.	PUNCT
ejpam-4572	337	1	c.	c.	PROPN
ejpam-4572	337	2	boonpok	boonpok	PROPN
ejpam-4572	337	3	,	,	PUNCT
ejpam-4572	337	4	m.	m.	NOUN
ejpam-4572	337	5	thongmoon	thongmoon	PROPN
ejpam-4572	337	6	/	/	SYM
ejpam-4572	337	7	eur	eur	PROPN
ejpam-4572	337	8	.	.	PUNCT
ejpam-4572	338	1	j.	j.	PROPN
ejpam-4572	338	2	pure	pure	PROPN
ejpam-4572	338	3	appl	appl	PROPN
ejpam-4572	338	4	.	.	PROPN
ejpam-4572	338	5	math	math	PROPN
ejpam-4572	338	6	,	,	PUNCT
ejpam-4572	338	7	16	16	NUM
ejpam-4572	338	8	(	(	PUNCT
ejpam-4572	338	9	1	1	NUM
ejpam-4572	338	10	)	)	PUNCT
ejpam-4572	338	11	(	(	PUNCT
ejpam-4572	338	12	2023	2023	NUM
ejpam-4572	338	13	)	)	PUNCT
ejpam-4572	338	14	,	,	PUNCT
ejpam-4572	338	15	465	465	NUM
ejpam-4572	338	16	-	-	SYM
ejpam-4572	338	17	478	478	NUM
ejpam-4572	338	18	474	474	NUM
ejpam-4572	338	19	definition	definition	NOUN
ejpam-4572	338	20	3	3	NUM
ejpam-4572	338	21	.	.	PUNCT
ejpam-4572	339	1	a	a	DET
ejpam-4572	339	2	function	function	NOUN
ejpam-4572	339	3	f	f	NOUN
ejpam-4572	339	4	:	:	PUNCT
ejpam-4572	339	5	(	(	PUNCT
ejpam-4572	339	6	x	x	X
ejpam-4572	339	7	,	,	PUNCT
ejpam-4572	339	8	τ	τ	PROPN
ejpam-4572	339	9	→	→	SYM
ejpam-4572	339	10	(	(	PUNCT
ejpam-4572	339	11	y	y	PROPN
ejpam-4572	339	12	,	,	PUNCT
ejpam-4572	339	13	σ	σ	PROPN
ejpam-4572	339	14	)	)	PUNCT
ejpam-4572	339	15	)	)	PUNCT
ejpam-4572	339	16	is	be	AUX
ejpam-4572	339	17	said	say	VERB
ejpam-4572	339	18	to	to	PART
ejpam-4572	339	19	be	be	AUX
ejpam-4572	339	20	weakly	weakly	ADJ
ejpam-4572	339	21	α(λ	α(λ	NOUN
ejpam-4572	339	22	,	,	PUNCT
ejpam-4572	339	23	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	339	24	if	if	SCONJ
ejpam-4572	339	25	,	,	PUNCT
ejpam-4572	339	26	for	for	SCONJ
ejpam-4572	339	27	each	each	DET
ejpam-4572	339	28	x	x	SYM
ejpam-4572	339	29	∈	∈	PROPN
ejpam-4572	339	30	x	x	X
ejpam-4572	339	31	and	and	CCONJ
ejpam-4572	339	32	each	each	DET
ejpam-4572	339	33	(	(	PUNCT
ejpam-4572	339	34	λ	λ	PROPN
ejpam-4572	339	35	,	,	PUNCT
ejpam-4572	339	36	sp)-open	sp)-open	ADJ
ejpam-4572	339	37	set	set	VERB
ejpam-4572	339	38	v	v	NOUN
ejpam-4572	339	39	containing	contain	VERB
ejpam-4572	339	40	f(x	f(x	PROPN
ejpam-4572	339	41	)	)	PUNCT
ejpam-4572	339	42	,	,	PUNCT
ejpam-4572	339	43	there	there	PRON
ejpam-4572	339	44	exists	exist	VERB
ejpam-4572	339	45	an	an	DET
ejpam-4572	339	46	α(λ	α(λ	PROPN
ejpam-4572	339	47	,	,	PUNCT
ejpam-4572	339	48	sp)-open	sp)-open	VERB
ejpam-4572	339	49	set	set	VERB
ejpam-4572	339	50	u	u	NOUN
ejpam-4572	339	51	of	of	ADP
ejpam-4572	339	52	x	x	PUNCT
ejpam-4572	339	53	containing	contain	VERB
ejpam-4572	339	54	x	x	PUNCT
ejpam-4572	339	55	such	such	ADJ
ejpam-4572	339	56	that	that	DET
ejpam-4572	339	57	f(u	f(u	PROPN
ejpam-4572	339	58	)	)	PUNCT
ejpam-4572	339	59	⊆	⊆	NUM
ejpam-4572	339	60	v	v	NOUN
ejpam-4572	339	61	(	(	PUNCT
ejpam-4572	339	62	λ	λ	NOUN
ejpam-4572	339	63	,	,	PUNCT
ejpam-4572	339	64	sp	sp	NOUN
ejpam-4572	339	65	)	)	PUNCT
ejpam-4572	339	66	.	.	PUNCT
ejpam-4572	340	1	theorem	theorem	VERB
ejpam-4572	340	2	6	6	NUM
ejpam-4572	340	3	.	.	PUNCT
ejpam-4572	340	4	for	for	ADP
ejpam-4572	340	5	a	a	DET
ejpam-4572	340	6	function	function	NOUN
ejpam-4572	340	7	f	f	NOUN
ejpam-4572	340	8	:	:	PUNCT
ejpam-4572	340	9	(	(	PUNCT
ejpam-4572	340	10	x	x	X
ejpam-4572	340	11	,	,	PUNCT
ejpam-4572	340	12	τ	τ	X
ejpam-4572	340	13	)	)	PUNCT
ejpam-4572	340	14	→	→	SYM
ejpam-4572	340	15	(	(	PUNCT
ejpam-4572	340	16	y	y	PROPN
ejpam-4572	340	17	,	,	PUNCT
ejpam-4572	340	18	σ	σ	PROPN
ejpam-4572	340	19	)	)	PUNCT
ejpam-4572	340	20	,	,	PUNCT
ejpam-4572	340	21	the	the	DET
ejpam-4572	340	22	following	follow	VERB
ejpam-4572	340	23	properties	property	NOUN
ejpam-4572	340	24	are	be	AUX
ejpam-4572	340	25	equivalent	equivalent	ADJ
ejpam-4572	340	26	:	:	PUNCT
ejpam-4572	340	27	(	(	PUNCT
ejpam-4572	340	28	1	1	X
ejpam-4572	340	29	)	)	PUNCT
ejpam-4572	340	30	f	f	PROPN
ejpam-4572	340	31	is	be	AUX
ejpam-4572	340	32	weakly	weakly	ADJ
ejpam-4572	340	33	α(λ	α(λ	NOUN
ejpam-4572	340	34	,	,	PUNCT
ejpam-4572	340	35	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	340	36	;	;	PUNCT
ejpam-4572	340	37	(	(	PUNCT
ejpam-4572	340	38	2	2	X
ejpam-4572	340	39	)	)	PUNCT
ejpam-4572	340	40	f−1(v	f−1(v	NOUN
ejpam-4572	340	41	)	)	PUNCT
ejpam-4572	341	1	⊆	⊆	NUM
ejpam-4572	341	2	[	[	X
ejpam-4572	341	3	f−1(v	f−1(v	NOUN
ejpam-4572	341	4	(	(	PUNCT
ejpam-4572	341	5	λ	λ	PROPN
ejpam-4572	341	6	,	,	PUNCT
ejpam-4572	341	7	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	341	8	,	,	PUNCT
ejpam-4572	341	9	sp	sp	NOUN
ejpam-4572	341	10	)	)	PUNCT
ejpam-4572	341	11	for	for	ADP
ejpam-4572	341	12	every	every	DET
ejpam-4572	341	13	(	(	PUNCT
ejpam-4572	341	14	λ	λ	NOUN
ejpam-4572	341	15	,	,	PUNCT
ejpam-4572	341	16	sp)-open	sp)-open	NOUN
ejpam-4572	341	17	set	set	VERB
ejpam-4572	341	18	v	v	NOUN
ejpam-4572	341	19	of	of	ADP
ejpam-4572	341	20	y	y	PROPN
ejpam-4572	341	21	;	;	PUNCT
ejpam-4572	341	22	(	(	PUNCT
ejpam-4572	341	23	3	3	X
ejpam-4572	341	24	)	)	PUNCT
ejpam-4572	341	25	[	[	X
ejpam-4572	341	26	f−1(k(λ	f−1(k(λ	NOUN
ejpam-4572	341	27	,	,	PUNCT
ejpam-4572	341	28	sp	sp	NOUN
ejpam-4572	341	29	)	)	PUNCT
ejpam-4572	341	30	)	)	PUNCT
ejpam-4572	341	31	]	]	PUNCT
ejpam-4572	342	1	α(λ	α(λ	PROPN
ejpam-4572	342	2	,	,	PUNCT
ejpam-4572	342	3	sp	sp	NOUN
ejpam-4572	342	4	)	)	PUNCT
ejpam-4572	342	5	⊆	⊆	NUM
ejpam-4572	342	6	f−1(k	f−1(k	PROPN
ejpam-4572	342	7	)	)	PUNCT
ejpam-4572	342	8	for	for	ADP
ejpam-4572	342	9	every	every	DET
ejpam-4572	342	10	r(λ	r(λ	NOUN
ejpam-4572	342	11	,	,	PUNCT
ejpam-4572	342	12	sp)-closed	sp)-close	VERB
ejpam-4572	342	13	set	set	VERB
ejpam-4572	342	14	k	k	PROPN
ejpam-4572	342	15	of	of	ADP
ejpam-4572	342	16	y	y	PROPN
ejpam-4572	342	17	;	;	PUNCT
ejpam-4572	342	18	(	(	PUNCT
ejpam-4572	342	19	4	4	X
ejpam-4572	342	20	)	)	PUNCT
ejpam-4572	343	1	[	[	X
ejpam-4572	343	2	f−1(v	f−1(v	NOUN
ejpam-4572	343	3	)	)	PUNCT
ejpam-4572	343	4	]	]	PUNCT
ejpam-4572	343	5	α(λ	α(λ	PROPN
ejpam-4572	343	6	,	,	PUNCT
ejpam-4572	343	7	sp	sp	NOUN
ejpam-4572	343	8	)	)	PUNCT
ejpam-4572	343	9	⊆	⊆	NUM
ejpam-4572	343	10	f−1(v	f−1(v	NOUN
ejpam-4572	343	11	(	(	PUNCT
ejpam-4572	343	12	λ	λ	PROPN
ejpam-4572	343	13	,	,	PUNCT
ejpam-4572	343	14	sp	sp	NOUN
ejpam-4572	343	15	)	)	PUNCT
ejpam-4572	343	16	)	)	PUNCT
ejpam-4572	343	17	for	for	ADP
ejpam-4572	343	18	every	every	DET
ejpam-4572	343	19	(	(	PUNCT
ejpam-4572	343	20	λ	λ	NOUN
ejpam-4572	343	21	,	,	PUNCT
ejpam-4572	343	22	sp)-open	sp)-open	NOUN
ejpam-4572	343	23	set	set	VERB
ejpam-4572	343	24	v	v	NOUN
ejpam-4572	343	25	of	of	ADP
ejpam-4572	343	26	y	y	PROPN
ejpam-4572	343	27	;	;	PUNCT
ejpam-4572	343	28	(	(	PUNCT
ejpam-4572	343	29	5	5	X
ejpam-4572	343	30	)	)	PUNCT
ejpam-4572	343	31	[	[	X
ejpam-4572	343	32	f−1([bθ(λ	f−1([bθ(λ	NOUN
ejpam-4572	343	33	,	,	PUNCT
ejpam-4572	343	34	sp)](λ	sp)](λ	PROPN
ejpam-4572	343	35	,	,	PUNCT
ejpam-4572	343	36	sp	sp	NOUN
ejpam-4572	343	37	)	)	PUNCT
ejpam-4572	343	38	)	)	PUNCT
ejpam-4572	343	39	]	]	PUNCT
ejpam-4572	344	1	α(λ	α(λ	PROPN
ejpam-4572	344	2	,	,	PUNCT
ejpam-4572	344	3	sp	sp	NOUN
ejpam-4572	344	4	)	)	PUNCT
ejpam-4572	344	5	⊆	⊆	NUM
ejpam-4572	344	6	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4572	344	7	,	,	PUNCT
ejpam-4572	344	8	sp	sp	NOUN
ejpam-4572	344	9	)	)	PUNCT
ejpam-4572	344	10	)	)	PUNCT
ejpam-4572	344	11	for	for	ADP
ejpam-4572	344	12	every	every	DET
ejpam-4572	344	13	subset	subset	NOUN
ejpam-4572	344	14	b	b	PROPN
ejpam-4572	344	15	of	of	ADP
ejpam-4572	344	16	y	y	PROPN
ejpam-4572	344	17	;	;	PUNCT
ejpam-4572	344	18	(	(	PUNCT
ejpam-4572	344	19	6	6	X
ejpam-4572	344	20	)	)	PUNCT
ejpam-4572	345	1	[	[	X
ejpam-4572	345	2	[	[	X
ejpam-4572	345	3	[	[	X
ejpam-4572	345	4	f−1(v	f−1(v	NOUN
ejpam-4572	345	5	)	)	PUNCT
ejpam-4572	345	6	]	]	PUNCT
ejpam-4572	345	7	(	(	PUNCT
ejpam-4572	345	8	λ	λ	PROPN
ejpam-4572	345	9	,	,	PUNCT
ejpam-4572	345	10	sp)](λ	sp)](λ	PROPN
ejpam-4572	345	11	,	,	PUNCT
ejpam-4572	345	12	sp	sp	NOUN
ejpam-4572	345	13	)	)	PUNCT
ejpam-4572	345	14	]	]	PUNCT
ejpam-4572	345	15	(	(	PUNCT
ejpam-4572	345	16	λ	λ	NOUN
ejpam-4572	345	17	,	,	PUNCT
ejpam-4572	345	18	sp	sp	NOUN
ejpam-4572	345	19	)	)	PUNCT
ejpam-4572	345	20	⊆	⊆	NUM
ejpam-4572	345	21	f−1(v	f−1(v	NOUN
ejpam-4572	345	22	(	(	PUNCT
ejpam-4572	345	23	λ	λ	PROPN
ejpam-4572	345	24	,	,	PUNCT
ejpam-4572	345	25	sp	sp	NOUN
ejpam-4572	345	26	)	)	PUNCT
ejpam-4572	345	27	)	)	PUNCT
ejpam-4572	345	28	for	for	ADP
ejpam-4572	345	29	every	every	DET
ejpam-4572	345	30	(	(	PUNCT
ejpam-4572	345	31	λ	λ	NOUN
ejpam-4572	345	32	,	,	PUNCT
ejpam-4572	345	33	sp)-open	sp)-open	NOUN
ejpam-4572	345	34	set	set	VERB
ejpam-4572	345	35	v	v	NOUN
ejpam-4572	345	36	of	of	ADP
ejpam-4572	345	37	y	y	PROPN
ejpam-4572	345	38	;	;	PUNCT
ejpam-4572	345	39	(	(	PUNCT
ejpam-4572	345	40	7	7	X
ejpam-4572	345	41	)	)	PUNCT
ejpam-4572	345	42	f−1(v	f−1(v	NOUN
ejpam-4572	345	43	)	)	PUNCT
ejpam-4572	345	44	⊆	⊆	NUM
ejpam-4572	346	1	[	[	X
ejpam-4572	346	2	[	[	X
ejpam-4572	346	3	[	[	X
ejpam-4572	346	4	f−1(v	f−1(v	NOUN
ejpam-4572	346	5	(	(	PUNCT
ejpam-4572	346	6	λ	λ	PROPN
ejpam-4572	346	7	,	,	PUNCT
ejpam-4572	346	8	sp))](λ	sp))](λ	PROPN
ejpam-4572	346	9	,	,	PUNCT
ejpam-4572	346	10	sp	sp	NOUN
ejpam-4572	346	11	)	)	PUNCT
ejpam-4572	346	12	]	]	PUNCT
ejpam-4572	346	13	(	(	PUNCT
ejpam-4572	346	14	λ	λ	X
ejpam-4572	346	15	,	,	PUNCT
ejpam-4572	346	16	sp)](λ	sp)](λ	PROPN
ejpam-4572	346	17	,	,	PUNCT
ejpam-4572	346	18	sp	sp	NOUN
ejpam-4572	346	19	)	)	PUNCT
ejpam-4572	346	20	for	for	ADP
ejpam-4572	346	21	every	every	DET
ejpam-4572	346	22	(	(	PUNCT
ejpam-4572	346	23	λ	λ	NOUN
ejpam-4572	346	24	,	,	PUNCT
ejpam-4572	346	25	sp)-open	sp)-open	NOUN
ejpam-4572	346	26	set	set	VERB
ejpam-4572	346	27	v	v	NOUN
ejpam-4572	346	28	of	of	ADP
ejpam-4572	346	29	y	y	PROPN
ejpam-4572	346	30	;	;	PUNCT
ejpam-4572	346	31	(	(	PUNCT
ejpam-4572	346	32	8)	8)	NUM
ejpam-4572	346	33	f([[a(λ	f([[a(λ	NOUN
ejpam-4572	346	34	,	,	PUNCT
ejpam-4572	346	35	sp)](λ	sp)](λ	PROPN
ejpam-4572	346	36	,	,	PUNCT
ejpam-4572	346	37	sp	sp	NOUN
ejpam-4572	346	38	)	)	PUNCT
ejpam-4572	346	39	]	]	PUNCT
ejpam-4572	346	40	(	(	PUNCT
ejpam-4572	346	41	λ	λ	NOUN
ejpam-4572	346	42	,	,	PUNCT
ejpam-4572	346	43	sp	sp	NOUN
ejpam-4572	346	44	)	)	PUNCT
ejpam-4572	346	45	)	)	PUNCT
ejpam-4572	347	1	⊆	⊆	NUM
ejpam-4572	347	2	[	[	X
ejpam-4572	347	3	f(a)]θ(λ	f(a)]θ(λ	X
ejpam-4572	347	4	,	,	PUNCT
ejpam-4572	347	5	sp	sp	NOUN
ejpam-4572	347	6	)	)	PUNCT
ejpam-4572	347	7	for	for	ADP
ejpam-4572	347	8	every	every	DET
ejpam-4572	347	9	subset	subset	NOUN
ejpam-4572	347	10	a	a	PRON
ejpam-4572	347	11	of	of	ADP
ejpam-4572	347	12	x	x	PRON
ejpam-4572	347	13	;	;	PUNCT
ejpam-4572	347	14	(	(	PUNCT
ejpam-4572	347	15	9	9	X
ejpam-4572	347	16	)	)	PUNCT
ejpam-4572	348	1	[	[	X
ejpam-4572	348	2	[	[	X
ejpam-4572	348	3	[	[	X
ejpam-4572	348	4	f−1(b)](λ	f−1(b)](λ	X
ejpam-4572	348	5	,	,	PUNCT
ejpam-4572	348	6	sp)](λ	sp)](λ	PROPN
ejpam-4572	348	7	,	,	PUNCT
ejpam-4572	348	8	sp	sp	NOUN
ejpam-4572	348	9	)	)	PUNCT
ejpam-4572	348	10	]	]	PUNCT
ejpam-4572	348	11	(	(	PUNCT
ejpam-4572	348	12	λ	λ	NOUN
ejpam-4572	348	13	,	,	PUNCT
ejpam-4572	348	14	sp	sp	NOUN
ejpam-4572	348	15	)	)	PUNCT
ejpam-4572	348	16	⊆	⊆	NUM
ejpam-4572	348	17	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4572	348	18	,	,	PUNCT
ejpam-4572	348	19	sp	sp	NOUN
ejpam-4572	348	20	)	)	PUNCT
ejpam-4572	348	21	)	)	PUNCT
ejpam-4572	348	22	for	for	ADP
ejpam-4572	348	23	every	every	DET
ejpam-4572	348	24	subset	subset	NOUN
ejpam-4572	348	25	b	b	PROPN
ejpam-4572	348	26	of	of	ADP
ejpam-4572	348	27	y	y	PROPN
ejpam-4572	348	28	.	.	PUNCT
ejpam-4572	349	1	for	for	ADP
ejpam-4572	349	2	a	a	DET
ejpam-4572	349	3	multifunction	multifunction	NOUN
ejpam-4572	349	4	f	f	NOUN
ejpam-4572	349	5	:	:	PUNCT
ejpam-4572	349	6	(	(	PUNCT
ejpam-4572	349	7	x	x	X
ejpam-4572	349	8	,	,	PUNCT
ejpam-4572	349	9	τ	τ	X
ejpam-4572	349	10	)	)	PUNCT
ejpam-4572	349	11	→	→	SYM
ejpam-4572	349	12	(	(	PUNCT
ejpam-4572	349	13	y	y	PROPN
ejpam-4572	349	14	,	,	PUNCT
ejpam-4572	349	15	σ	σ	PROPN
ejpam-4572	349	16	)	)	PUNCT
ejpam-4572	349	17	,	,	PUNCT
ejpam-4572	349	18	by	by	ADP
ejpam-4572	349	19	f	f	PROPN
ejpam-4572	349	20	(	(	PUNCT
ejpam-4572	349	21	λ	λ	PROPN
ejpam-4572	349	22	,	,	PUNCT
ejpam-4572	349	23	sp	sp	NOUN
ejpam-4572	349	24	)	)	PUNCT
ejpam-4572	349	25	:	:	PUNCT
ejpam-4572	349	26	(	(	PUNCT
ejpam-4572	349	27	x	x	X
ejpam-4572	349	28	,	,	PUNCT
ejpam-4572	349	29	τ	τ	X
ejpam-4572	349	30	)	)	PUNCT
ejpam-4572	349	31	→	→	SYM
ejpam-4572	349	32	(	(	PUNCT
ejpam-4572	349	33	y	y	PROPN
ejpam-4572	349	34	,	,	PUNCT
ejpam-4572	349	35	σ	σ	PROPN
ejpam-4572	349	36	)	)	PUNCT
ejpam-4572	349	37	(	(	PUNCT
ejpam-4572	349	38	resp	resp	NOUN
ejpam-4572	349	39	.	.	PUNCT
ejpam-4572	350	1	fα(λ	fα(λ	PROPN
ejpam-4572	350	2	,	,	PUNCT
ejpam-4572	350	3	sp	sp	NOUN
ejpam-4572	350	4	)	)	PUNCT
ejpam-4572	350	5	:	:	PUNCT
ejpam-4572	350	6	(	(	PUNCT
ejpam-4572	350	7	x	x	X
ejpam-4572	350	8	,	,	PUNCT
ejpam-4572	350	9	τ	τ	X
ejpam-4572	350	10	)	)	PUNCT
ejpam-4572	350	11	→	→	SYM
ejpam-4572	350	12	(	(	PUNCT
ejpam-4572	350	13	y	y	PROPN
ejpam-4572	350	14	,	,	PUNCT
ejpam-4572	350	15	σ	σ	PROPN
ejpam-4572	350	16	)	)	PUNCT
ejpam-4572	351	1	[	[	X
ejpam-4572	351	2	6	6	NUM
ejpam-4572	351	3	]	]	PUNCT
ejpam-4572	351	4	)	)	PUNCT
ejpam-4572	351	5	we	we	PRON
ejpam-4572	351	6	denote	denote	VERB
ejpam-4572	351	7	a	a	DET
ejpam-4572	351	8	multifunction	multifunction	NOUN
ejpam-4572	351	9	defined	define	VERB
ejpam-4572	351	10	as	as	SCONJ
ejpam-4572	351	11	follows	follow	VERB
ejpam-4572	351	12	:	:	PUNCT
ejpam-4572	351	13	f	f	PROPN
ejpam-4572	351	14	(	(	PUNCT
ejpam-4572	351	15	λ	λ	PROPN
ejpam-4572	351	16	,	,	PUNCT
ejpam-4572	351	17	sp)(x	sp)(x	PROPN
ejpam-4572	351	18	)	)	PUNCT
ejpam-4572	351	19	=	=	PUNCT
ejpam-4572	352	1	[	[	X
ejpam-4572	352	2	f	f	X
ejpam-4572	352	3	(	(	PUNCT
ejpam-4572	352	4	x)](λ	x)](λ	PROPN
ejpam-4572	352	5	,	,	PUNCT
ejpam-4572	352	6	sp	sp	NOUN
ejpam-4572	352	7	)	)	PUNCT
ejpam-4572	352	8	(	(	PUNCT
ejpam-4572	352	9	resp	resp	NOUN
ejpam-4572	352	10	.	.	PUNCT
ejpam-4572	353	1	fα(λ	fα(λ	PROPN
ejpam-4572	353	2	,	,	PUNCT
ejpam-4572	353	3	sp)(x	sp)(x	PROPN
ejpam-4572	353	4	)	)	PUNCT
ejpam-4572	354	1	=	=	PUNCT
ejpam-4572	355	1	[	[	X
ejpam-4572	355	2	f	f	X
ejpam-4572	355	3	(	(	PUNCT
ejpam-4572	355	4	x)]α(λ	x)]α(λ	PROPN
ejpam-4572	355	5	,	,	PUNCT
ejpam-4572	355	6	sp	sp	NOUN
ejpam-4572	355	7	)	)	PUNCT
ejpam-4572	355	8	)	)	PUNCT
ejpam-4572	355	9	for	for	ADP
ejpam-4572	355	10	each	each	DET
ejpam-4572	355	11	x	x	SYM
ejpam-4572	355	12	∈	∈	PROPN
ejpam-4572	355	13	x.	x.	NOUN
ejpam-4572	355	14	definition	definition	NOUN
ejpam-4572	355	15	4	4	NUM
ejpam-4572	355	16	.	.	PUNCT
ejpam-4572	356	1	[	[	X
ejpam-4572	356	2	6	6	NUM
ejpam-4572	356	3	]	]	PUNCT
ejpam-4572	356	4	a	a	DET
ejpam-4572	356	5	subset	subset	NOUN
ejpam-4572	356	6	a	a	PRON
ejpam-4572	356	7	of	of	ADP
ejpam-4572	356	8	a	a	DET
ejpam-4572	356	9	topological	topological	ADJ
ejpam-4572	356	10	space	space	NOUN
ejpam-4572	356	11	(	(	PUNCT
ejpam-4572	356	12	x	x	X
ejpam-4572	356	13	,	,	PUNCT
ejpam-4572	356	14	τ	τ	X
ejpam-4572	356	15	)	)	PUNCT
ejpam-4572	356	16	is	be	AUX
ejpam-4572	356	17	said	say	VERB
ejpam-4572	356	18	to	to	PART
ejpam-4572	356	19	be	be	AUX
ejpam-4572	356	20	:	:	PUNCT
ejpam-4572	356	21	(	(	PUNCT
ejpam-4572	356	22	1	1	X
ejpam-4572	356	23	)	)	PUNCT
ejpam-4572	356	24	(	(	PUNCT
ejpam-4572	356	25	λ	λ	NOUN
ejpam-4572	356	26	,	,	PUNCT
ejpam-4572	356	27	sp)-paracompact	sp)-paracompact	ADJ
ejpam-4572	356	28	if	if	SCONJ
ejpam-4572	356	29	every	every	DET
ejpam-4572	356	30	cover	cover	NOUN
ejpam-4572	356	31	of	of	ADP
ejpam-4572	356	32	a	a	DET
ejpam-4572	356	33	by	by	ADP
ejpam-4572	356	34	(	(	PUNCT
ejpam-4572	356	35	λ	λ	PROPN
ejpam-4572	356	36	,	,	PUNCT
ejpam-4572	356	37	sp)-open	sp)-open	ADJ
ejpam-4572	356	38	sets	set	NOUN
ejpam-4572	356	39	of	of	ADP
ejpam-4572	356	40	x	x	PUNCT
ejpam-4572	356	41	is	be	AUX
ejpam-4572	356	42	refined	refine	VERB
ejpam-4572	356	43	by	by	ADP
ejpam-4572	356	44	a	a	DET
ejpam-4572	356	45	cover	cover	NOUN
ejpam-4572	356	46	of	of	ADP
ejpam-4572	356	47	a	a	PRON
ejpam-4572	356	48	which	which	PRON
ejpam-4572	356	49	consists	consist	VERB
ejpam-4572	356	50	of	of	ADP
ejpam-4572	356	51	(	(	PUNCT
ejpam-4572	356	52	λ	λ	PROPN
ejpam-4572	356	53	,	,	PUNCT
ejpam-4572	356	54	sp)-open	sp)-open	ADJ
ejpam-4572	356	55	sets	set	NOUN
ejpam-4572	356	56	of	of	ADP
ejpam-4572	356	57	x	x	PUNCT
ejpam-4572	356	58	and	and	CCONJ
ejpam-4572	356	59	is	be	AUX
ejpam-4572	356	60	locally	locally	ADV
ejpam-4572	356	61	finite	finite	ADJ
ejpam-4572	356	62	in	in	ADP
ejpam-4572	356	63	x	x	PRON
ejpam-4572	356	64	;	;	PUNCT
ejpam-4572	356	65	(	(	PUNCT
ejpam-4572	356	66	2	2	X
ejpam-4572	356	67	)	)	PUNCT
ejpam-4572	356	68	(	(	PUNCT
ejpam-4572	356	69	λ	λ	NOUN
ejpam-4572	356	70	,	,	PUNCT
ejpam-4572	356	71	sp)-regular	sp)-regular	ADJ
ejpam-4572	356	72	if	if	SCONJ
ejpam-4572	356	73	for	for	ADP
ejpam-4572	356	74	each	each	DET
ejpam-4572	356	75	x	x	SYM
ejpam-4572	356	76	∈	∈	PROPN
ejpam-4572	356	77	a	a	PRON
ejpam-4572	356	78	and	and	CCONJ
ejpam-4572	356	79	each	each	DET
ejpam-4572	356	80	(	(	PUNCT
ejpam-4572	356	81	λ	λ	PROPN
ejpam-4572	356	82	,	,	PUNCT
ejpam-4572	356	83	sp)-open	sp)-open	NOUN
ejpam-4572	356	84	set	set	VERB
ejpam-4572	356	85	u	u	NOUN
ejpam-4572	356	86	of	of	ADP
ejpam-4572	356	87	x	x	SYM
ejpam-4572	356	88	containing	contain	VERB
ejpam-4572	356	89	x	x	PRON
ejpam-4572	356	90	,	,	PUNCT
ejpam-4572	356	91	there	there	PRON
ejpam-4572	356	92	exists	exist	VERB
ejpam-4572	356	93	a	a	DET
ejpam-4572	356	94	(	(	PUNCT
ejpam-4572	356	95	λ	λ	NOUN
ejpam-4572	356	96	,	,	PUNCT
ejpam-4572	356	97	sp)-open	sp)-open	NOUN
ejpam-4572	356	98	set	set	VERB
ejpam-4572	356	99	v	v	NOUN
ejpam-4572	356	100	of	of	ADP
ejpam-4572	356	101	x	x	PUNCT
ejpam-4572	356	102	such	such	ADJ
ejpam-4572	356	103	that	that	SCONJ
ejpam-4572	356	104	x	x	SYM
ejpam-4572	356	105	∈	∈	NOUN
ejpam-4572	356	106	v	v	ADP
ejpam-4572	356	107	⊆	⊆	NUM
ejpam-4572	356	108	v	v	NOUN
ejpam-4572	356	109	(	(	PUNCT
ejpam-4572	356	110	λ	λ	NOUN
ejpam-4572	356	111	,	,	PUNCT
ejpam-4572	356	112	sp	sp	NOUN
ejpam-4572	356	113	)	)	PUNCT
ejpam-4572	356	114	⊆	⊆	NUM
ejpam-4572	356	115	u	u	NOUN
ejpam-4572	356	116	.	.	PUNCT
ejpam-4572	357	1	lemma	lemma	PROPN
ejpam-4572	357	2	10	10	NUM
ejpam-4572	357	3	.	.	PUNCT
ejpam-4572	358	1	[	[	X
ejpam-4572	358	2	6	6	NUM
ejpam-4572	358	3	]	]	X
ejpam-4572	358	4	if	if	SCONJ
ejpam-4572	358	5	a	a	PRON
ejpam-4572	358	6	is	be	AUX
ejpam-4572	358	7	a	a	DET
ejpam-4572	358	8	(	(	PUNCT
ejpam-4572	358	9	λ	λ	NOUN
ejpam-4572	358	10	,	,	PUNCT
ejpam-4572	358	11	sp)-regular	sp)-regular	ADJ
ejpam-4572	358	12	(	(	PUNCT
ejpam-4572	358	13	λ	λ	PROPN
ejpam-4572	358	14	,	,	PUNCT
ejpam-4572	358	15	sp)-paracompact	sp)-paracompact	NOUN
ejpam-4572	358	16	set	set	VERB
ejpam-4572	358	17	of	of	ADP
ejpam-4572	358	18	a	a	DET
ejpam-4572	358	19	topological	topological	ADJ
ejpam-4572	358	20	space	space	NOUN
ejpam-4572	358	21	(	(	PUNCT
ejpam-4572	358	22	x	x	X
ejpam-4572	358	23	,	,	PUNCT
ejpam-4572	358	24	τ	τ	X
ejpam-4572	358	25	)	)	PUNCT
ejpam-4572	358	26	and	and	CCONJ
ejpam-4572	358	27	u	u	NOUN
ejpam-4572	358	28	is	be	AUX
ejpam-4572	358	29	a	a	DET
ejpam-4572	358	30	(	(	PUNCT
ejpam-4572	358	31	λ	λ	NOUN
ejpam-4572	358	32	,	,	PUNCT
ejpam-4572	358	33	sp)-open	sp)-open	ADJ
ejpam-4572	358	34	neighbourhood	neighbourhood	NOUN
ejpam-4572	358	35	of	of	ADP
ejpam-4572	358	36	a	a	PRON
ejpam-4572	358	37	,	,	PUNCT
ejpam-4572	358	38	then	then	ADV
ejpam-4572	358	39	there	there	PRON
ejpam-4572	358	40	exists	exist	VERB
ejpam-4572	358	41	a	a	DET
ejpam-4572	358	42	(	(	PUNCT
ejpam-4572	358	43	λ	λ	NOUN
ejpam-4572	358	44	,	,	PUNCT
ejpam-4572	358	45	sp)-open	sp)-open	NOUN
ejpam-4572	358	46	set	set	VERB
ejpam-4572	358	47	v	v	NOUN
ejpam-4572	358	48	of	of	ADP
ejpam-4572	358	49	x	x	PUNCT
ejpam-4572	358	50	such	such	ADJ
ejpam-4572	358	51	that	that	SCONJ
ejpam-4572	358	52	a	a	DET
ejpam-4572	358	53	⊆	⊆	NUM
ejpam-4572	358	54	v	v	ADP
ejpam-4572	358	55	⊆	⊆	NUM
ejpam-4572	358	56	v	v	NOUN
ejpam-4572	358	57	(	(	PUNCT
ejpam-4572	358	58	λ	λ	NOUN
ejpam-4572	358	59	,	,	PUNCT
ejpam-4572	358	60	sp	sp	NOUN
ejpam-4572	358	61	)	)	PUNCT
ejpam-4572	358	62	⊆	⊆	NUM
ejpam-4572	358	63	u	u	NOUN
ejpam-4572	358	64	.	.	PUNCT
ejpam-4572	359	1	lemma	lemma	PROPN
ejpam-4572	359	2	11	11	NUM
ejpam-4572	359	3	.	.	PUNCT
ejpam-4572	360	1	[	[	X
ejpam-4572	360	2	6	6	NUM
ejpam-4572	360	3	]	]	PUNCT
ejpam-4572	360	4	if	if	SCONJ
ejpam-4572	360	5	f	f	PROPN
ejpam-4572	360	6	:	:	PUNCT
ejpam-4572	360	7	(	(	PUNCT
ejpam-4572	360	8	x	x	X
ejpam-4572	360	9	,	,	PUNCT
ejpam-4572	360	10	τ	τ	X
ejpam-4572	360	11	)	)	PUNCT
ejpam-4572	360	12	→	→	SYM
ejpam-4572	360	13	(	(	PUNCT
ejpam-4572	360	14	y	y	PROPN
ejpam-4572	360	15	,	,	PUNCT
ejpam-4572	360	16	σ	σ	PROPN
ejpam-4572	360	17	)	)	PUNCT
ejpam-4572	360	18	is	be	AUX
ejpam-4572	360	19	(	(	PUNCT
ejpam-4572	360	20	λ	λ	INTJ
ejpam-4572	360	21	,	,	PUNCT
ejpam-4572	360	22	sp)-regular	sp)-regular	ADJ
ejpam-4572	360	23	and	and	CCONJ
ejpam-4572	360	24	(	(	PUNCT
ejpam-4572	360	25	λ	λ	PROPN
ejpam-4572	360	26	,	,	PUNCT
ejpam-4572	360	27	sp)-paracompact	sp)-paracompact	NOUN
ejpam-4572	360	28	,	,	PUNCT
ejpam-4572	360	29	then	then	ADV
ejpam-4572	360	30	[	[	X
ejpam-4572	360	31	fα(λ	fα(λ	NOUN
ejpam-4572	360	32	,	,	PUNCT
ejpam-4572	360	33	sp)]+(v	sp)]+(v	X
ejpam-4572	360	34	)	)	PUNCT
ejpam-4572	360	35	=	=	PUNCT
ejpam-4572	361	1	f+(v	f+(v	NOUN
ejpam-4572	361	2	)	)	PUNCT
ejpam-4572	361	3	for	for	ADP
ejpam-4572	361	4	every	every	DET
ejpam-4572	361	5	(	(	PUNCT
ejpam-4572	361	6	λ	λ	NOUN
ejpam-4572	361	7	,	,	PUNCT
ejpam-4572	361	8	sp)-open	sp)-open	NOUN
ejpam-4572	361	9	set	set	VERB
ejpam-4572	361	10	v	v	NOUN
ejpam-4572	361	11	of	of	ADP
ejpam-4572	361	12	y	y	PROPN
ejpam-4572	361	13	.	.	PUNCT
ejpam-4572	362	1	theorem	theorem	ADJ
ejpam-4572	362	2	7	7	NUM
ejpam-4572	362	3	.	.	PUNCT
ejpam-4572	363	1	let	let	VERB
ejpam-4572	363	2	f	f	NOUN
ejpam-4572	363	3	:	:	PUNCT
ejpam-4572	363	4	(	(	PUNCT
ejpam-4572	363	5	x	x	X
ejpam-4572	363	6	,	,	PUNCT
ejpam-4572	363	7	τ	τ	X
ejpam-4572	363	8	)	)	PUNCT
ejpam-4572	363	9	→	→	SYM
ejpam-4572	363	10	(	(	PUNCT
ejpam-4572	363	11	y	y	PROPN
ejpam-4572	363	12	,	,	PUNCT
ejpam-4572	363	13	σ	σ	PROPN
ejpam-4572	363	14	)	)	PUNCT
ejpam-4572	363	15	be	be	VERB
ejpam-4572	363	16	a	a	DET
ejpam-4572	363	17	multifunction	multifunction	NOUN
ejpam-4572	363	18	such	such	ADJ
ejpam-4572	363	19	that	that	SCONJ
ejpam-4572	363	20	f	f	PROPN
ejpam-4572	363	21	(	(	PUNCT
ejpam-4572	363	22	x	x	X
ejpam-4572	363	23	)	)	PUNCT
ejpam-4572	363	24	is	be	AUX
ejpam-4572	363	25	(	(	PUNCT
ejpam-4572	363	26	λ	λ	INTJ
ejpam-4572	363	27	,	,	PUNCT
ejpam-4572	363	28	sp)paracompact	sp)paracompact	ADJ
ejpam-4572	363	29	and	and	CCONJ
ejpam-4572	363	30	(	(	PUNCT
ejpam-4572	363	31	λ	λ	PROPN
ejpam-4572	363	32	,	,	PUNCT
ejpam-4572	363	33	sp)-regular	sp)-regular	NOUN
ejpam-4572	363	34	for	for	ADP
ejpam-4572	363	35	each	each	DET
ejpam-4572	363	36	x	x	SYM
ejpam-4572	363	37	∈	∈	PROPN
ejpam-4572	363	38	x.	x.	NOUN
ejpam-4572	363	39	then	then	ADV
ejpam-4572	363	40	,	,	PUNCT
ejpam-4572	363	41	the	the	DET
ejpam-4572	363	42	following	follow	VERB
ejpam-4572	363	43	properties	property	NOUN
ejpam-4572	363	44	are	be	AUX
ejpam-4572	363	45	equivalent	equivalent	ADJ
ejpam-4572	363	46	:	:	PUNCT
ejpam-4572	363	47	(	(	PUNCT
ejpam-4572	363	48	1	1	X
ejpam-4572	363	49	)	)	PUNCT
ejpam-4572	363	50	f	f	PROPN
ejpam-4572	363	51	is	be	AUX
ejpam-4572	363	52	upper	upper	ADJ
ejpam-4572	363	53	weakly	weakly	ADJ
ejpam-4572	363	54	α(λ	α(λ	PROPN
ejpam-4572	363	55	,	,	PUNCT
ejpam-4572	363	56	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	363	57	;	;	PUNCT
ejpam-4572	363	58	c.	c.	PROPN
ejpam-4572	363	59	boonpok	boonpok	PROPN
ejpam-4572	363	60	,	,	PUNCT
ejpam-4572	363	61	m.	m.	NOUN
ejpam-4572	363	62	thongmoon	thongmoon	PROPN
ejpam-4572	363	63	/	/	SYM
ejpam-4572	363	64	eur	eur	PROPN
ejpam-4572	363	65	.	.	PUNCT
ejpam-4572	364	1	j.	j.	PROPN
ejpam-4572	364	2	pure	pure	PROPN
ejpam-4572	364	3	appl	appl	PROPN
ejpam-4572	364	4	.	.	PROPN
ejpam-4572	364	5	math	math	PROPN
ejpam-4572	364	6	,	,	PUNCT
ejpam-4572	364	7	16	16	NUM
ejpam-4572	364	8	(	(	PUNCT
ejpam-4572	364	9	1	1	NUM
ejpam-4572	364	10	)	)	PUNCT
ejpam-4572	364	11	(	(	PUNCT
ejpam-4572	364	12	2023	2023	NUM
ejpam-4572	364	13	)	)	PUNCT
ejpam-4572	364	14	,	,	PUNCT
ejpam-4572	364	15	465	465	NUM
ejpam-4572	364	16	-	-	SYM
ejpam-4572	364	17	478	478	NUM
ejpam-4572	364	18	475	475	NUM
ejpam-4572	364	19	(	(	PUNCT
ejpam-4572	364	20	2	2	NUM
ejpam-4572	364	21	)	)	PUNCT
ejpam-4572	364	22	fα(λ	fα(λ	NOUN
ejpam-4572	364	23	,	,	PUNCT
ejpam-4572	364	24	sp	sp	NOUN
ejpam-4572	364	25	)	)	PUNCT
ejpam-4572	364	26	is	be	AUX
ejpam-4572	364	27	upper	upper	ADJ
ejpam-4572	364	28	weakly	weakly	ADJ
ejpam-4572	364	29	α(λ	α(λ	PROPN
ejpam-4572	364	30	,	,	PUNCT
ejpam-4572	364	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	364	32	;	;	PUNCT
ejpam-4572	364	33	(	(	PUNCT
ejpam-4572	364	34	3	3	X
ejpam-4572	364	35	)	)	PUNCT
ejpam-4572	364	36	f	f	NOUN
ejpam-4572	364	37	(	(	PUNCT
ejpam-4572	364	38	λ	λ	NOUN
ejpam-4572	364	39	,	,	PUNCT
ejpam-4572	364	40	sp	sp	NOUN
ejpam-4572	364	41	)	)	PUNCT
ejpam-4572	364	42	is	be	AUX
ejpam-4572	364	43	upper	upper	ADJ
ejpam-4572	364	44	weakly	weakly	ADJ
ejpam-4572	364	45	α(λ	α(λ	PROPN
ejpam-4572	364	46	,	,	PUNCT
ejpam-4572	364	47	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	364	48	.	.	PUNCT
ejpam-4572	365	1	proof	proof	NOUN
ejpam-4572	365	2	.	.	PUNCT
ejpam-4572	366	1	similarly	similarly	ADV
ejpam-4572	366	2	to	to	PART
ejpam-4572	366	3	lemma	lemma	PROPN
ejpam-4572	366	4	11	11	NUM
ejpam-4572	366	5	,	,	PUNCT
ejpam-4572	366	6	we	we	PRON
ejpam-4572	366	7	put	put	VERB
ejpam-4572	366	8	g	g	NOUN
ejpam-4572	366	9	=	=	SYM
ejpam-4572	366	10	fα(λ	fα(λ	NOUN
ejpam-4572	366	11	,	,	PUNCT
ejpam-4572	366	12	sp	sp	NOUN
ejpam-4572	366	13	)	)	PUNCT
ejpam-4572	366	14	or	or	CCONJ
ejpam-4572	366	15	f	f	PROPN
ejpam-4572	366	16	(	(	PUNCT
ejpam-4572	366	17	λ	λ	PROPN
ejpam-4572	366	18	,	,	PUNCT
ejpam-4572	366	19	sp	sp	NOUN
ejpam-4572	366	20	)	)	PUNCT
ejpam-4572	366	21	.	.	PUNCT
ejpam-4572	367	1	first	first	ADV
ejpam-4572	367	2	,	,	PUNCT
ejpam-4572	367	3	suppose	suppose	VERB
ejpam-4572	367	4	that	that	SCONJ
ejpam-4572	367	5	f	f	PROPN
ejpam-4572	367	6	is	be	AUX
ejpam-4572	367	7	upper	upper	ADJ
ejpam-4572	367	8	weakly	weakly	ADJ
ejpam-4572	367	9	α(λ	α(λ	PROPN
ejpam-4572	367	10	,	,	PUNCT
ejpam-4572	367	11	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	367	12	.	.	PUNCT
ejpam-4572	368	1	let	let	VERB
ejpam-4572	368	2	x	x	PUNCT
ejpam-4572	368	3	∈	∈	PROPN
ejpam-4572	368	4	x	x	X
ejpam-4572	368	5	and	and	CCONJ
ejpam-4572	368	6	v	v	X
ejpam-4572	368	7	be	be	AUX
ejpam-4572	368	8	any	any	DET
ejpam-4572	368	9	(	(	PUNCT
ejpam-4572	368	10	λ	λ	NOUN
ejpam-4572	368	11	,	,	PUNCT
ejpam-4572	368	12	sp)-open	sp)-open	ADJ
ejpam-4572	368	13	set	set	NOUN
ejpam-4572	368	14	of	of	ADP
ejpam-4572	368	15	y	y	NOUN
ejpam-4572	368	16	containing	contain	VERB
ejpam-4572	368	17	g(x	g(x	NOUN
ejpam-4572	368	18	)	)	PUNCT
ejpam-4572	368	19	.	.	PUNCT
ejpam-4572	369	1	by	by	ADP
ejpam-4572	369	2	lemma	lemma	PROPN
ejpam-4572	369	3	11	11	NUM
ejpam-4572	369	4	,	,	PUNCT
ejpam-4572	369	5	x	x	SYM
ejpam-4572	369	6	∈	∈	PROPN
ejpam-4572	369	7	g+(v	g+(v	PROPN
ejpam-4572	369	8	)	)	PUNCT
ejpam-4572	369	9	=	=	PUNCT
ejpam-4572	370	1	f+(v	f+(v	NOUN
ejpam-4572	370	2	)	)	PUNCT
ejpam-4572	371	1	and	and	CCONJ
ejpam-4572	371	2	there	there	PRON
ejpam-4572	371	3	exists	exist	VERB
ejpam-4572	371	4	an	an	DET
ejpam-4572	371	5	α(λ	α(λ	PROPN
ejpam-4572	371	6	,	,	PUNCT
ejpam-4572	371	7	sp)-open	sp)-open	VERB
ejpam-4572	371	8	set	set	VERB
ejpam-4572	371	9	u	u	NOUN
ejpam-4572	371	10	of	of	ADP
ejpam-4572	371	11	x	x	PUNCT
ejpam-4572	371	12	containing	contain	VERB
ejpam-4572	371	13	x	x	PUNCT
ejpam-4572	371	14	such	such	ADJ
ejpam-4572	371	15	that	that	SCONJ
ejpam-4572	371	16	f	f	PROPN
ejpam-4572	371	17	(	(	PUNCT
ejpam-4572	371	18	z	z	NOUN
ejpam-4572	371	19	)	)	PUNCT
ejpam-4572	371	20	⊆	⊆	NUM
ejpam-4572	371	21	v	v	NOUN
ejpam-4572	371	22	(	(	PUNCT
ejpam-4572	371	23	λ	λ	NOUN
ejpam-4572	371	24	,	,	PUNCT
ejpam-4572	371	25	sp	sp	NOUN
ejpam-4572	371	26	)	)	PUNCT
ejpam-4572	371	27	for	for	ADP
ejpam-4572	371	28	each	each	DET
ejpam-4572	371	29	z	z	NOUN
ejpam-4572	371	30	∈	∈	PROPN
ejpam-4572	371	31	u	u	NOUN
ejpam-4572	371	32	.	.	PUNCT
ejpam-4572	372	1	therefore	therefore	ADV
ejpam-4572	372	2	,	,	PUNCT
ejpam-4572	372	3	we	we	PRON
ejpam-4572	372	4	have	have	VERB
ejpam-4572	372	5	fα(λ	fα(λ	NUM
ejpam-4572	372	6	,	,	PUNCT
ejpam-4572	372	7	sp)(z	sp)(z	NOUN
ejpam-4572	372	8	)	)	PUNCT
ejpam-4572	372	9	⊆	⊆	NUM
ejpam-4572	372	10	f	f	X
ejpam-4572	372	11	(	(	PUNCT
ejpam-4572	372	12	λ	λ	PROPN
ejpam-4572	372	13	,	,	PUNCT
ejpam-4572	372	14	sp)(z	sp)(z	NOUN
ejpam-4572	372	15	)	)	PUNCT
ejpam-4572	372	16	⊆	⊆	NUM
ejpam-4572	372	17	v	v	X
ejpam-4572	372	18	(	(	PUNCT
ejpam-4572	372	19	λ	λ	NOUN
ejpam-4572	372	20	,	,	PUNCT
ejpam-4572	372	21	sp	sp	NOUN
ejpam-4572	372	22	)	)	PUNCT
ejpam-4572	372	23	;	;	PUNCT
ejpam-4572	372	24	hence	hence	ADV
ejpam-4572	372	25	g(z	g(z	ADJ
ejpam-4572	372	26	)	)	PUNCT
ejpam-4572	372	27	⊆	⊆	NUM
ejpam-4572	372	28	v	v	NOUN
ejpam-4572	372	29	(	(	PUNCT
ejpam-4572	372	30	λ	λ	NOUN
ejpam-4572	372	31	,	,	PUNCT
ejpam-4572	372	32	sp	sp	NOUN
ejpam-4572	372	33	)	)	PUNCT
ejpam-4572	372	34	for	for	ADP
ejpam-4572	372	35	each	each	DET
ejpam-4572	372	36	z	z	NOUN
ejpam-4572	372	37	∈	∈	PROPN
ejpam-4572	372	38	u	u	NOUN
ejpam-4572	372	39	.	.	PUNCT
ejpam-4572	373	1	this	this	PRON
ejpam-4572	373	2	shows	show	VERB
ejpam-4572	373	3	that	that	SCONJ
ejpam-4572	373	4	g	g	PROPN
ejpam-4572	373	5	is	be	AUX
ejpam-4572	373	6	weakly	weakly	ADJ
ejpam-4572	373	7	α(λ	α(λ	NOUN
ejpam-4572	373	8	,	,	PUNCT
ejpam-4572	373	9	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	373	10	.	.	PUNCT
ejpam-4572	374	1	conversely	conversely	ADV
ejpam-4572	374	2	,	,	PUNCT
ejpam-4572	374	3	suppose	suppose	VERB
ejpam-4572	374	4	that	that	SCONJ
ejpam-4572	374	5	g	g	PROPN
ejpam-4572	374	6	is	be	AUX
ejpam-4572	374	7	upper	upper	ADJ
ejpam-4572	374	8	weakly	weakly	ADJ
ejpam-4572	374	9	α(λ	α(λ	PROPN
ejpam-4572	374	10	,	,	PUNCT
ejpam-4572	374	11	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	374	12	.	.	PUNCT
ejpam-4572	375	1	let	let	VERB
ejpam-4572	375	2	x	x	PUNCT
ejpam-4572	375	3	∈	∈	PROPN
ejpam-4572	375	4	x	x	X
ejpam-4572	375	5	and	and	CCONJ
ejpam-4572	375	6	v	v	X
ejpam-4572	375	7	be	be	AUX
ejpam-4572	375	8	any	any	DET
ejpam-4572	375	9	(	(	PUNCT
ejpam-4572	375	10	λ	λ	NOUN
ejpam-4572	375	11	,	,	PUNCT
ejpam-4572	375	12	sp)-open	sp)-open	ADJ
ejpam-4572	375	13	set	set	NOUN
ejpam-4572	375	14	of	of	ADP
ejpam-4572	375	15	y	y	PROPN
ejpam-4572	375	16	containing	contain	VERB
ejpam-4572	375	17	f	f	PROPN
ejpam-4572	375	18	(	(	PUNCT
ejpam-4572	375	19	x	x	NOUN
ejpam-4572	375	20	)	)	PUNCT
ejpam-4572	375	21	.	.	PUNCT
ejpam-4572	376	1	by	by	ADP
ejpam-4572	376	2	lemma	lemma	PROPN
ejpam-4572	376	3	11	11	NUM
ejpam-4572	376	4	,	,	PUNCT
ejpam-4572	376	5	x	x	SYM
ejpam-4572	376	6	∈	∈	NOUN
ejpam-4572	376	7	f+(v	f+(v	NOUN
ejpam-4572	376	8	)	)	PUNCT
ejpam-4572	376	9	=	=	PUNCT
ejpam-4572	376	10	g+(v	g+(v	PROPN
ejpam-4572	376	11	)	)	PUNCT
ejpam-4572	376	12	and	and	CCONJ
ejpam-4572	376	13	hence	hence	ADV
ejpam-4572	376	14	g(x	g(x	NOUN
ejpam-4572	376	15	)	)	PUNCT
ejpam-4572	376	16	⊆	⊆	NUM
ejpam-4572	376	17	v	v	NOUN
ejpam-4572	376	18	.	.	PUNCT
ejpam-4572	377	1	there	there	PRON
ejpam-4572	377	2	exists	exist	VERB
ejpam-4572	377	3	an	an	DET
ejpam-4572	377	4	α(λ	α(λ	PROPN
ejpam-4572	377	5	,	,	PUNCT
ejpam-4572	377	6	sp)-open	sp)-open	VERB
ejpam-4572	377	7	set	set	VERB
ejpam-4572	377	8	u	u	NOUN
ejpam-4572	377	9	of	of	ADP
ejpam-4572	377	10	x	x	PUNCT
ejpam-4572	377	11	containing	contain	VERB
ejpam-4572	377	12	x	x	PUNCT
ejpam-4572	377	13	such	such	ADJ
ejpam-4572	377	14	that	that	SCONJ
ejpam-4572	377	15	g(u	g(u	PROPN
ejpam-4572	377	16	)	)	PUNCT
ejpam-4572	377	17	⊆	⊆	NUM
ejpam-4572	377	18	v	v	NOUN
ejpam-4572	377	19	(	(	PUNCT
ejpam-4572	377	20	λ	λ	NOUN
ejpam-4572	377	21	,	,	PUNCT
ejpam-4572	377	22	sp	sp	NOUN
ejpam-4572	377	23	)	)	PUNCT
ejpam-4572	377	24	.	.	PUNCT
ejpam-4572	378	1	thus	thus	ADV
ejpam-4572	378	2	,	,	PUNCT
ejpam-4572	378	3	f	f	PROPN
ejpam-4572	378	4	(	(	PUNCT
ejpam-4572	378	5	u	u	NOUN
ejpam-4572	378	6	)	)	PUNCT
ejpam-4572	378	7	⊆	⊆	NUM
ejpam-4572	378	8	v	v	NOUN
ejpam-4572	378	9	(	(	PUNCT
ejpam-4572	378	10	λ	λ	NOUN
ejpam-4572	378	11	,	,	PUNCT
ejpam-4572	378	12	sp	sp	NOUN
ejpam-4572	378	13	)	)	PUNCT
ejpam-4572	378	14	and	and	CCONJ
ejpam-4572	378	15	hence	hence	ADV
ejpam-4572	378	16	f	f	PROPN
ejpam-4572	378	17	is	be	AUX
ejpam-4572	378	18	upper	upper	ADJ
ejpam-4572	378	19	weakly	weakly	ADJ
ejpam-4572	378	20	β(λ	β(λ	NOUN
ejpam-4572	378	21	,	,	PUNCT
ejpam-4572	378	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	378	23	.	.	PUNCT
ejpam-4572	379	1	lemma	lemma	PROPN
ejpam-4572	379	2	12	12	NUM
ejpam-4572	379	3	.	.	PUNCT
ejpam-4572	380	1	[	[	X
ejpam-4572	380	2	6	6	NUM
ejpam-4572	380	3	]	]	PUNCT
ejpam-4572	380	4	for	for	ADP
ejpam-4572	380	5	a	a	DET
ejpam-4572	380	6	multifunction	multifunction	NOUN
ejpam-4572	380	7	f	f	NOUN
ejpam-4572	380	8	:	:	PUNCT
ejpam-4572	380	9	(	(	PUNCT
ejpam-4572	380	10	x	x	X
ejpam-4572	380	11	,	,	PUNCT
ejpam-4572	380	12	τ	τ	X
ejpam-4572	380	13	)	)	PUNCT
ejpam-4572	380	14	→	→	SYM
ejpam-4572	380	15	(	(	PUNCT
ejpam-4572	380	16	y	y	PROPN
ejpam-4572	380	17	,	,	PUNCT
ejpam-4572	380	18	σ	σ	PROPN
ejpam-4572	380	19	)	)	PUNCT
ejpam-4572	380	20	,	,	PUNCT
ejpam-4572	380	21	it	it	PRON
ejpam-4572	380	22	follows	follow	VERB
ejpam-4572	380	23	that	that	SCONJ
ejpam-4572	380	24	for	for	ADP
ejpam-4572	380	25	each	each	DET
ejpam-4572	380	26	α(λ	α(λ	PROPN
ejpam-4572	380	27	,	,	PUNCT
ejpam-4572	380	28	sp)open	sp)open	VERB
ejpam-4572	380	29	set	set	VERB
ejpam-4572	380	30	v	v	NOUN
ejpam-4572	380	31	of	of	ADP
ejpam-4572	380	32	y	y	PROPN
ejpam-4572	381	1	[	[	X
ejpam-4572	381	2	fα(λ	fα(λ	NUM
ejpam-4572	381	3	,	,	PUNCT
ejpam-4572	381	4	sp)]−(v	sp)]−(v	NOUN
ejpam-4572	381	5	)	)	PUNCT
ejpam-4572	381	6	=	=	SYM
ejpam-4572	381	7	f−(v	f−(v	ADJ
ejpam-4572	381	8	)	)	PUNCT
ejpam-4572	381	9	,	,	PUNCT
ejpam-4572	381	10	where	where	SCONJ
ejpam-4572	381	11	g	g	PROPN
ejpam-4572	381	12	denotes	denote	NOUN
ejpam-4572	381	13	fα(λ	fα(λ	PROPN
ejpam-4572	381	14	,	,	PUNCT
ejpam-4572	381	15	sp	sp	NOUN
ejpam-4572	381	16	)	)	PUNCT
ejpam-4572	381	17	or	or	CCONJ
ejpam-4572	381	18	f	f	PROPN
ejpam-4572	381	19	(	(	PUNCT
ejpam-4572	381	20	λ	λ	PROPN
ejpam-4572	381	21	,	,	PUNCT
ejpam-4572	381	22	sp	sp	NOUN
ejpam-4572	381	23	)	)	PUNCT
ejpam-4572	381	24	.	.	PUNCT
ejpam-4572	382	1	theorem	theorem	ADJ
ejpam-4572	382	2	8	8	NUM
ejpam-4572	382	3	.	.	PUNCT
ejpam-4572	383	1	for	for	ADP
ejpam-4572	383	2	a	a	DET
ejpam-4572	383	3	multifunction	multifunction	NOUN
ejpam-4572	383	4	f	f	NOUN
ejpam-4572	383	5	:	:	PUNCT
ejpam-4572	383	6	(	(	PUNCT
ejpam-4572	383	7	x	x	X
ejpam-4572	383	8	,	,	PUNCT
ejpam-4572	383	9	τ	τ	X
ejpam-4572	383	10	)	)	PUNCT
ejpam-4572	383	11	→	→	SYM
ejpam-4572	383	12	(	(	PUNCT
ejpam-4572	383	13	y	y	PROPN
ejpam-4572	383	14	,	,	PUNCT
ejpam-4572	383	15	σ	σ	PROPN
ejpam-4572	383	16	)	)	PUNCT
ejpam-4572	383	17	,	,	PUNCT
ejpam-4572	383	18	the	the	DET
ejpam-4572	383	19	following	follow	VERB
ejpam-4572	383	20	properties	property	NOUN
ejpam-4572	383	21	are	be	AUX
ejpam-4572	383	22	equivalent	equivalent	ADJ
ejpam-4572	383	23	:	:	PUNCT
ejpam-4572	383	24	(	(	PUNCT
ejpam-4572	383	25	1	1	X
ejpam-4572	383	26	)	)	PUNCT
ejpam-4572	383	27	f	f	PROPN
ejpam-4572	383	28	is	be	AUX
ejpam-4572	383	29	lower	low	ADJ
ejpam-4572	383	30	weakly	weakly	ADJ
ejpam-4572	383	31	α(λ	α(λ	PROPN
ejpam-4572	383	32	,	,	PUNCT
ejpam-4572	383	33	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	383	34	;	;	PUNCT
ejpam-4572	383	35	(	(	PUNCT
ejpam-4572	383	36	2	2	NUM
ejpam-4572	383	37	)	)	PUNCT
ejpam-4572	383	38	fα(λ	fα(λ	NOUN
ejpam-4572	383	39	,	,	PUNCT
ejpam-4572	383	40	sp	sp	NOUN
ejpam-4572	383	41	)	)	PUNCT
ejpam-4572	383	42	is	be	AUX
ejpam-4572	383	43	lower	low	ADJ
ejpam-4572	383	44	weakly	weakly	ADJ
ejpam-4572	383	45	α(λ	α(λ	PROPN
ejpam-4572	383	46	,	,	PUNCT
ejpam-4572	383	47	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	383	48	;	;	PUNCT
ejpam-4572	383	49	(	(	PUNCT
ejpam-4572	383	50	3	3	X
ejpam-4572	383	51	)	)	PUNCT
ejpam-4572	383	52	f	f	NOUN
ejpam-4572	383	53	(	(	PUNCT
ejpam-4572	383	54	λ	λ	NOUN
ejpam-4572	383	55	,	,	PUNCT
ejpam-4572	383	56	sp	sp	NOUN
ejpam-4572	383	57	)	)	PUNCT
ejpam-4572	383	58	is	be	AUX
ejpam-4572	383	59	lower	low	ADJ
ejpam-4572	383	60	weakly	weakly	ADJ
ejpam-4572	383	61	α(λ	α(λ	PROPN
ejpam-4572	383	62	,	,	PUNCT
ejpam-4572	383	63	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	383	64	.	.	PUNCT
ejpam-4572	384	1	proof	proof	NOUN
ejpam-4572	384	2	.	.	PUNCT
ejpam-4572	385	1	by	by	ADP
ejpam-4572	385	2	utilizing	utilize	VERB
ejpam-4572	385	3	lemma	lemma	PROPN
ejpam-4572	385	4	12	12	NUM
ejpam-4572	385	5	,	,	PUNCT
ejpam-4572	385	6	this	this	PRON
ejpam-4572	385	7	can	can	AUX
ejpam-4572	385	8	be	be	AUX
ejpam-4572	385	9	proved	prove	VERB
ejpam-4572	385	10	in	in	ADP
ejpam-4572	385	11	a	a	DET
ejpam-4572	385	12	similar	similar	ADJ
ejpam-4572	385	13	way	way	NOUN
ejpam-4572	385	14	as	as	SCONJ
ejpam-4572	385	15	theorem	theorem	ADJ
ejpam-4572	385	16	7	7	NUM
ejpam-4572	385	17	.	.	PUNCT
ejpam-4572	385	18	definition	definition	NOUN
ejpam-4572	385	19	5	5	NUM
ejpam-4572	385	20	.	.	PUNCT
ejpam-4572	386	1	a	a	DET
ejpam-4572	386	2	subset	subset	NOUN
ejpam-4572	386	3	a	a	PRON
ejpam-4572	386	4	of	of	ADP
ejpam-4572	386	5	a	a	DET
ejpam-4572	386	6	topological	topological	ADJ
ejpam-4572	386	7	space	space	NOUN
ejpam-4572	386	8	(	(	PUNCT
ejpam-4572	386	9	x	x	X
ejpam-4572	386	10	,	,	PUNCT
ejpam-4572	386	11	τ	τ	X
ejpam-4572	386	12	)	)	PUNCT
ejpam-4572	386	13	is	be	AUX
ejpam-4572	386	14	called	call	VERB
ejpam-4572	386	15	(	(	PUNCT
ejpam-4572	386	16	λ	λ	PROPN
ejpam-4572	386	17	,	,	PUNCT
ejpam-4572	386	18	sp)-clopen	sp)-clopen	ADJ
ejpam-4572	386	19	if	if	SCONJ
ejpam-4572	386	20	a	a	PRON
ejpam-4572	386	21	is	be	AUX
ejpam-4572	386	22	both	both	PRON
ejpam-4572	386	23	(	(	PUNCT
ejpam-4572	386	24	λ	λ	NOUN
ejpam-4572	386	25	,	,	PUNCT
ejpam-4572	386	26	sp)-closed	sp)-close	VERB
ejpam-4572	386	27	and	and	CCONJ
ejpam-4572	386	28	(	(	PUNCT
ejpam-4572	386	29	λ	λ	NOUN
ejpam-4572	386	30	,	,	PUNCT
ejpam-4572	386	31	sp)-open	sp)-open	NOUN
ejpam-4572	386	32	.	.	PUNCT
ejpam-4572	387	1	lemma	lemma	PROPN
ejpam-4572	387	2	13	13	NUM
ejpam-4572	387	3	.	.	PUNCT
ejpam-4572	388	1	if	if	SCONJ
ejpam-4572	388	2	a	a	PRON
ejpam-4572	388	3	is	be	AUX
ejpam-4572	388	4	α(λ	α(λ	PROPN
ejpam-4572	388	5	,	,	PUNCT
ejpam-4572	388	6	sp)-open	sp)-open	NOUN
ejpam-4572	388	7	and	and	CCONJ
ejpam-4572	388	8	α(λ	α(λ	PROPN
ejpam-4572	388	9	,	,	PUNCT
ejpam-4572	388	10	sp)-closed	sp)-close	VERB
ejpam-4572	388	11	in	in	ADP
ejpam-4572	388	12	a	a	DET
ejpam-4572	388	13	topological	topological	ADJ
ejpam-4572	388	14	space	space	NOUN
ejpam-4572	388	15	(	(	PUNCT
ejpam-4572	388	16	x	x	X
ejpam-4572	388	17	,	,	PUNCT
ejpam-4572	388	18	τ	τ	PROPN
ejpam-4572	388	19	)	)	PUNCT
ejpam-4572	388	20	,	,	PUNCT
ejpam-4572	388	21	then	then	ADV
ejpam-4572	388	22	a	a	PRON
ejpam-4572	388	23	is	be	AUX
ejpam-4572	388	24	(	(	PUNCT
ejpam-4572	388	25	λ	λ	PROPN
ejpam-4572	388	26	,	,	PUNCT
ejpam-4572	388	27	sp)-clopen	sp)-clopen	NOUN
ejpam-4572	388	28	.	.	PUNCT
ejpam-4572	389	1	proof	proof	NOUN
ejpam-4572	389	2	.	.	PUNCT
ejpam-4572	390	1	let	let	VERB
ejpam-4572	390	2	a	a	DET
ejpam-4572	390	3	be	be	AUX
ejpam-4572	390	4	α(λ	α(λ	PROPN
ejpam-4572	390	5	,	,	PUNCT
ejpam-4572	390	6	sp)-open	sp)-open	NOUN
ejpam-4572	390	7	and	and	CCONJ
ejpam-4572	390	8	α(λ	α(λ	PROPN
ejpam-4572	390	9	,	,	PUNCT
ejpam-4572	390	10	sp)-closed	sp)-close	VERB
ejpam-4572	390	11	.	.	PUNCT
ejpam-4572	391	1	then	then	ADV
ejpam-4572	391	2	,	,	PUNCT
ejpam-4572	391	3	a	a	DET
ejpam-4572	391	4	⊆	⊆	NUM
ejpam-4572	391	5	[	[	X
ejpam-4572	391	6	[	[	X
ejpam-4572	391	7	a(λ	a(λ	ADJ
ejpam-4572	391	8	,	,	PUNCT
ejpam-4572	391	9	sp	sp	NOUN
ejpam-4572	391	10	)	)	PUNCT
ejpam-4572	391	11	]	]	PUNCT
ejpam-4572	392	1	(	(	PUNCT
ejpam-4572	392	2	λ	λ	X
ejpam-4572	392	3	,	,	PUNCT
ejpam-4572	392	4	sp)](λ	sp)](λ	PROPN
ejpam-4572	392	5	,	,	PUNCT
ejpam-4572	392	6	sp	sp	NOUN
ejpam-4572	392	7	)	)	PUNCT
ejpam-4572	392	8	and	and	CCONJ
ejpam-4572	392	9	[	[	X
ejpam-4572	392	10	[	[	X
ejpam-4572	392	11	a(λ	a(λ	ADJ
ejpam-4572	392	12	,	,	PUNCT
ejpam-4572	392	13	sp)](λ	sp)](λ	PROPN
ejpam-4572	392	14	,	,	PUNCT
ejpam-4572	392	15	sp	sp	NOUN
ejpam-4572	392	16	)	)	PUNCT
ejpam-4572	392	17	]	]	PUNCT
ejpam-4572	392	18	(	(	PUNCT
ejpam-4572	392	19	λ	λ	NOUN
ejpam-4572	392	20	,	,	PUNCT
ejpam-4572	392	21	sp	sp	NOUN
ejpam-4572	392	22	)	)	PUNCT
ejpam-4572	392	23	⊆	⊆	NUM
ejpam-4572	392	24	a.	a.	NOUN
ejpam-4572	392	25	thus	thus	ADV
ejpam-4572	392	26	,	,	PUNCT
ejpam-4572	392	27	a(λ	a(λ	ADV
ejpam-4572	392	28	,	,	PUNCT
ejpam-4572	392	29	sp	sp	NOUN
ejpam-4572	392	30	)	)	PUNCT
ejpam-4572	392	31	=	=	PUNCT
ejpam-4572	393	1	[	[	X
ejpam-4572	393	2	[	[	X
ejpam-4572	393	3	[	[	X
ejpam-4572	393	4	a(λ	a(λ	ADJ
ejpam-4572	393	5	,	,	PUNCT
ejpam-4572	393	6	sp	sp	NOUN
ejpam-4572	393	7	)	)	PUNCT
ejpam-4572	393	8	]	]	PUNCT
ejpam-4572	393	9	(	(	PUNCT
ejpam-4572	393	10	λ	λ	X
ejpam-4572	393	11	,	,	PUNCT
ejpam-4572	393	12	sp)](λ	sp)](λ	PROPN
ejpam-4572	393	13	,	,	PUNCT
ejpam-4572	393	14	sp	sp	NOUN
ejpam-4572	393	15	)	)	PUNCT
ejpam-4572	393	16	]	]	PUNCT
ejpam-4572	393	17	(	(	PUNCT
ejpam-4572	393	18	λ	λ	NOUN
ejpam-4572	393	19	,	,	PUNCT
ejpam-4572	393	20	sp	sp	NOUN
ejpam-4572	393	21	)	)	PUNCT
ejpam-4572	393	22	=	=	PUNCT
ejpam-4572	394	1	[	[	X
ejpam-4572	394	2	a(λ	a(λ	ADV
ejpam-4572	394	3	,	,	PUNCT
ejpam-4572	394	4	sp	sp	NOUN
ejpam-4572	394	5	)	)	PUNCT
ejpam-4572	394	6	]	]	PUNCT
ejpam-4572	394	7	(	(	PUNCT
ejpam-4572	394	8	λ	λ	NOUN
ejpam-4572	394	9	,	,	PUNCT
ejpam-4572	394	10	sp	sp	NOUN
ejpam-4572	394	11	)	)	PUNCT
ejpam-4572	394	12	and	and	CCONJ
ejpam-4572	394	13	hence	hence	ADV
ejpam-4572	394	14	a(λ	a(λ	ADV
ejpam-4572	394	15	,	,	PUNCT
ejpam-4572	394	16	sp	sp	NOUN
ejpam-4572	394	17	)	)	PUNCT
ejpam-4572	394	18	⊆	⊆	NUM
ejpam-4572	395	1	[	[	X
ejpam-4572	395	2	[	[	X
ejpam-4572	395	3	a(λ	a(λ	ADJ
ejpam-4572	395	4	,	,	PUNCT
ejpam-4572	395	5	sp)](λ	sp)](λ	PROPN
ejpam-4572	395	6	,	,	PUNCT
ejpam-4572	395	7	sp	sp	NOUN
ejpam-4572	395	8	)	)	PUNCT
ejpam-4572	395	9	]	]	PUNCT
ejpam-4572	396	1	(	(	PUNCT
ejpam-4572	396	2	λ	λ	NOUN
ejpam-4572	396	3	,	,	PUNCT
ejpam-4572	396	4	sp	sp	NOUN
ejpam-4572	396	5	)	)	PUNCT
ejpam-4572	396	6	⊆	⊆	NUM
ejpam-4572	396	7	a.	a.	NOUN
ejpam-4572	396	8	this	this	PRON
ejpam-4572	396	9	shows	show	VERB
ejpam-4572	396	10	that	that	SCONJ
ejpam-4572	396	11	a	a	PRON
ejpam-4572	396	12	is	be	AUX
ejpam-4572	396	13	(	(	PUNCT
ejpam-4572	396	14	λ	λ	NOUN
ejpam-4572	396	15	,	,	PUNCT
ejpam-4572	396	16	sp)-closed	sp)-close	VERB
ejpam-4572	396	17	.	.	PUNCT
ejpam-4572	397	1	therefore	therefore	ADV
ejpam-4572	397	2	,	,	PUNCT
ejpam-4572	397	3	a	a	DET
ejpam-4572	397	4	⊆	⊆	NUM
ejpam-4572	397	5	[	[	X
ejpam-4572	397	6	[	[	X
ejpam-4572	397	7	a(λ	a(λ	ADJ
ejpam-4572	397	8	,	,	PUNCT
ejpam-4572	397	9	sp	sp	NOUN
ejpam-4572	397	10	)	)	PUNCT
ejpam-4572	397	11	]	]	PUNCT
ejpam-4572	397	12	(	(	PUNCT
ejpam-4572	397	13	λ	λ	X
ejpam-4572	397	14	,	,	PUNCT
ejpam-4572	397	15	sp)](λ	sp)](λ	PROPN
ejpam-4572	397	16	,	,	PUNCT
ejpam-4572	397	17	sp	sp	NOUN
ejpam-4572	397	18	)	)	PUNCT
ejpam-4572	397	19	⊆	⊆	NUM
ejpam-4572	397	20	[	[	X
ejpam-4572	397	21	a(λ	a(λ	ADV
ejpam-4572	397	22	,	,	PUNCT
ejpam-4572	397	23	sp)](λ	sp)](λ	PROPN
ejpam-4572	397	24	,	,	PUNCT
ejpam-4572	397	25	sp	sp	NOUN
ejpam-4572	397	26	)	)	PUNCT
ejpam-4572	397	27	=	=	PUNCT
ejpam-4572	398	1	a(λ	a(λ	ADV
ejpam-4572	398	2	,	,	PUNCT
ejpam-4572	398	3	sp	sp	NOUN
ejpam-4572	398	4	)	)	PUNCT
ejpam-4572	398	5	and	and	CCONJ
ejpam-4572	398	6	hence	hence	ADV
ejpam-4572	398	7	a	a	PRON
ejpam-4572	398	8	is	be	AUX
ejpam-4572	398	9	(	(	PUNCT
ejpam-4572	398	10	λ	λ	NOUN
ejpam-4572	398	11	,	,	PUNCT
ejpam-4572	398	12	sp)-open	sp)-open	NOUN
ejpam-4572	398	13	.	.	PUNCT
ejpam-4572	399	1	consequently	consequently	ADV
ejpam-4572	399	2	,	,	PUNCT
ejpam-4572	399	3	we	we	PRON
ejpam-4572	399	4	obtain	obtain	VERB
ejpam-4572	399	5	a	a	DET
ejpam-4572	399	6	is	be	AUX
ejpam-4572	399	7	(	(	PUNCT
ejpam-4572	399	8	λ	λ	PROPN
ejpam-4572	399	9	,	,	PUNCT
ejpam-4572	399	10	sp)-clopen	sp)-clopen	NOUN
ejpam-4572	399	11	.	.	PUNCT
ejpam-4572	400	1	lemma	lemma	PROPN
ejpam-4572	400	2	14	14	NUM
ejpam-4572	400	3	.	.	PUNCT
ejpam-4572	401	1	if	if	SCONJ
ejpam-4572	401	2	a	a	DET
ejpam-4572	401	3	multifunction	multifunction	NOUN
ejpam-4572	401	4	f	f	NOUN
ejpam-4572	401	5	:	:	PUNCT
ejpam-4572	401	6	(	(	PUNCT
ejpam-4572	401	7	x	x	X
ejpam-4572	401	8	,	,	PUNCT
ejpam-4572	401	9	τ	τ	X
ejpam-4572	401	10	)	)	PUNCT
ejpam-4572	401	11	→	→	SYM
ejpam-4572	401	12	(	(	PUNCT
ejpam-4572	401	13	y	y	PROPN
ejpam-4572	401	14	,	,	PUNCT
ejpam-4572	401	15	σ	σ	PROPN
ejpam-4572	401	16	)	)	PUNCT
ejpam-4572	401	17	is	be	AUX
ejpam-4572	401	18	upper	upper	ADJ
ejpam-4572	401	19	weakly	weakly	ADJ
ejpam-4572	401	20	α(λ	α(λ	PROPN
ejpam-4572	401	21	,	,	PUNCT
ejpam-4572	401	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	401	23	and	and	CCONJ
ejpam-4572	401	24	lower	low	ADJ
ejpam-4572	401	25	weakly	weakly	ADJ
ejpam-4572	401	26	α(λ	α(λ	PROPN
ejpam-4572	401	27	,	,	PUNCT
ejpam-4572	401	28	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	401	29	,	,	PUNCT
ejpam-4572	401	30	then	then	ADV
ejpam-4572	401	31	f+(v	f+(v	PROPN
ejpam-4572	401	32	)	)	PUNCT
ejpam-4572	402	1	is	be	AUX
ejpam-4572	402	2	(	(	PUNCT
ejpam-4572	402	3	λ	λ	PROPN
ejpam-4572	402	4	,	,	PUNCT
ejpam-4572	402	5	sp)-clopen	sp)-clopen	ADJ
ejpam-4572	402	6	in	in	ADP
ejpam-4572	402	7	x	x	PUNCT
ejpam-4572	402	8	for	for	SCONJ
ejpam-4572	402	9	every	every	DET
ejpam-4572	402	10	(	(	PUNCT
ejpam-4572	402	11	λ	λ	NOUN
ejpam-4572	402	12	,	,	PUNCT
ejpam-4572	402	13	sp)clopen	sp)clopen	NOUN
ejpam-4572	402	14	set	set	VERB
ejpam-4572	402	15	v	v	NOUN
ejpam-4572	402	16	of	of	ADP
ejpam-4572	402	17	y	y	PROPN
ejpam-4572	402	18	.	.	PUNCT
ejpam-4572	403	1	c.	c.	PROPN
ejpam-4572	403	2	boonpok	boonpok	PROPN
ejpam-4572	403	3	,	,	PUNCT
ejpam-4572	403	4	m.	m.	NOUN
ejpam-4572	403	5	thongmoon	thongmoon	PROPN
ejpam-4572	403	6	/	/	SYM
ejpam-4572	403	7	eur	eur	PROPN
ejpam-4572	403	8	.	.	PUNCT
ejpam-4572	404	1	j.	j.	PROPN
ejpam-4572	404	2	pure	pure	PROPN
ejpam-4572	404	3	appl	appl	PROPN
ejpam-4572	404	4	.	.	PROPN
ejpam-4572	404	5	math	math	PROPN
ejpam-4572	404	6	,	,	PUNCT
ejpam-4572	404	7	16	16	NUM
ejpam-4572	404	8	(	(	PUNCT
ejpam-4572	404	9	1	1	NUM
ejpam-4572	404	10	)	)	PUNCT
ejpam-4572	404	11	(	(	PUNCT
ejpam-4572	404	12	2023	2023	NUM
ejpam-4572	404	13	)	)	PUNCT
ejpam-4572	404	14	,	,	PUNCT
ejpam-4572	404	15	465	465	NUM
ejpam-4572	404	16	-	-	SYM
ejpam-4572	404	17	478	478	NUM
ejpam-4572	404	18	476	476	NUM
ejpam-4572	404	19	proof	proof	NOUN
ejpam-4572	404	20	.	.	PUNCT
ejpam-4572	405	1	let	let	VERB
ejpam-4572	405	2	v	v	PART
ejpam-4572	405	3	be	be	AUX
ejpam-4572	405	4	any	any	DET
ejpam-4572	405	5	(	(	PUNCT
ejpam-4572	405	6	λ	λ	PROPN
ejpam-4572	405	7	,	,	PUNCT
ejpam-4572	405	8	sp)-clopen	sp)-clopen	ADJ
ejpam-4572	405	9	set	set	NOUN
ejpam-4572	405	10	of	of	ADP
ejpam-4572	405	11	y	y	PROPN
ejpam-4572	405	12	.	.	PUNCT
ejpam-4572	406	1	it	it	PRON
ejpam-4572	406	2	follows	follow	VERB
ejpam-4572	406	3	from	from	ADP
ejpam-4572	406	4	theorem	theorem	ADJ
ejpam-4572	406	5	3	3	NUM
ejpam-4572	406	6	that	that	SCONJ
ejpam-4572	406	7	f+(v	f+(v	AUX
ejpam-4572	406	8	)	)	PUNCT
ejpam-4572	407	1	⊆	⊆	NUM
ejpam-4572	407	2	[	[	X
ejpam-4572	407	3	f+(v	f+(v	NOUN
ejpam-4572	407	4	(	(	PUNCT
ejpam-4572	407	5	λ	λ	PROPN
ejpam-4572	407	6	,	,	PUNCT
ejpam-4572	407	7	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	407	8	,	,	PUNCT
ejpam-4572	407	9	sp	sp	NOUN
ejpam-4572	407	10	)	)	PUNCT
ejpam-4572	407	11	=	=	PUNCT
ejpam-4572	408	1	[	[	X
ejpam-4572	408	2	f+(v	f+(v	NOUN
ejpam-4572	408	3	)	)	PUNCT
ejpam-4572	409	1	]	]	PUNCT
ejpam-4572	409	2	α(λ	α(λ	PROPN
ejpam-4572	409	3	,	,	PUNCT
ejpam-4572	409	4	sp	sp	NOUN
ejpam-4572	409	5	)	)	PUNCT
ejpam-4572	409	6	.	.	PUNCT
ejpam-4572	410	1	this	this	PRON
ejpam-4572	410	2	shows	show	VERB
ejpam-4572	410	3	that	that	SCONJ
ejpam-4572	410	4	f+(v	f+(v	PROPN
ejpam-4572	410	5	)	)	PUNCT
ejpam-4572	410	6	is	be	AUX
ejpam-4572	410	7	α(λ	α(λ	PROPN
ejpam-4572	410	8	,	,	PUNCT
ejpam-4572	410	9	sp)-open	sp)-open	ADJ
ejpam-4572	410	10	in	in	ADP
ejpam-4572	410	11	x.	x.	NOUN
ejpam-4572	410	12	furthermore	furthermore	ADV
ejpam-4572	410	13	,	,	PUNCT
ejpam-4572	410	14	since	since	SCONJ
ejpam-4572	410	15	v	v	NOUN
ejpam-4572	410	16	is	be	AUX
ejpam-4572	410	17	(	(	PUNCT
ejpam-4572	410	18	λ	λ	NOUN
ejpam-4572	410	19	,	,	PUNCT
ejpam-4572	410	20	sp)-open	sp)-open	ADJ
ejpam-4572	410	21	,	,	PUNCT
ejpam-4572	410	22	it	it	PRON
ejpam-4572	410	23	follows	follow	VERB
ejpam-4572	410	24	from	from	ADP
ejpam-4572	410	25	theorem	theorem	ADJ
ejpam-4572	410	26	4	4	NUM
ejpam-4572	410	27	that	that	PRON
ejpam-4572	410	28	[	[	X
ejpam-4572	410	29	f+(v	f+(v	NOUN
ejpam-4572	410	30	)	)	PUNCT
ejpam-4572	411	1	]	]	PUNCT
ejpam-4572	411	2	α(λ	α(λ	PROPN
ejpam-4572	411	3	,	,	PUNCT
ejpam-4572	411	4	sp	sp	NOUN
ejpam-4572	411	5	)	)	PUNCT
ejpam-4572	411	6	⊆	⊆	NUM
ejpam-4572	411	7	f+(v	f+(v	NUM
ejpam-4572	411	8	(	(	PUNCT
ejpam-4572	411	9	λ	λ	NOUN
ejpam-4572	411	10	,	,	PUNCT
ejpam-4572	411	11	sp	sp	NOUN
ejpam-4572	411	12	)	)	PUNCT
ejpam-4572	411	13	)	)	PUNCT
ejpam-4572	411	14	=	=	PUNCT
ejpam-4572	411	15	f+(v	f+(v	NOUN
ejpam-4572	411	16	)	)	PUNCT
ejpam-4572	411	17	.	.	PUNCT
ejpam-4572	412	1	thus	thus	ADV
ejpam-4572	412	2	,	,	PUNCT
ejpam-4572	412	3	f+(v	f+(v	PROPN
ejpam-4572	412	4	)	)	PUNCT
ejpam-4572	412	5	is	be	AUX
ejpam-4572	412	6	α(λ	α(λ	PROPN
ejpam-4572	412	7	,	,	PUNCT
ejpam-4572	412	8	sp)-closed	sp)-close	VERB
ejpam-4572	412	9	and	and	CCONJ
ejpam-4572	412	10	by	by	ADP
ejpam-4572	412	11	lemma	lemma	PROPN
ejpam-4572	412	12	13	13	NUM
ejpam-4572	412	13	,	,	PUNCT
ejpam-4572	412	14	f+(v	f+(v	PROPN
ejpam-4572	412	15	)	)	PUNCT
ejpam-4572	412	16	is	be	AUX
ejpam-4572	412	17	(	(	PUNCT
ejpam-4572	412	18	λ	λ	PROPN
ejpam-4572	412	19	,	,	PUNCT
ejpam-4572	412	20	sp)-clopen	sp)-clopen	ADJ
ejpam-4572	412	21	in	in	ADP
ejpam-4572	412	22	x.	x.	NOUN
ejpam-4572	412	23	definition	definition	NOUN
ejpam-4572	412	24	6	6	NUM
ejpam-4572	412	25	.	.	PUNCT
ejpam-4572	413	1	[	[	X
ejpam-4572	413	2	16	16	NUM
ejpam-4572	413	3	]	]	PUNCT
ejpam-4572	413	4	a	a	DET
ejpam-4572	413	5	topological	topological	ADJ
ejpam-4572	413	6	space	space	NOUN
ejpam-4572	413	7	(	(	PUNCT
ejpam-4572	413	8	x	x	X
ejpam-4572	413	9	,	,	PUNCT
ejpam-4572	413	10	τ	τ	X
ejpam-4572	413	11	)	)	PUNCT
ejpam-4572	413	12	is	be	AUX
ejpam-4572	413	13	called	call	VERB
ejpam-4572	413	14	λsp	λsp	ADV
ejpam-4572	413	15	-	-	PUNCT
ejpam-4572	413	16	connected	connect	VERB
ejpam-4572	413	17	if	if	SCONJ
ejpam-4572	413	18	x	x	PRON
ejpam-4572	413	19	can	can	AUX
ejpam-4572	413	20	not	not	PART
ejpam-4572	413	21	be	be	AUX
ejpam-4572	413	22	written	write	VERB
ejpam-4572	413	23	as	as	ADP
ejpam-4572	413	24	a	a	DET
ejpam-4572	413	25	disjoint	disjoint	NOUN
ejpam-4572	413	26	union	union	NOUN
ejpam-4572	413	27	of	of	ADP
ejpam-4572	413	28	two	two	NUM
ejpam-4572	413	29	nonempty	nonempty	ADJ
ejpam-4572	413	30	(	(	PUNCT
ejpam-4572	413	31	λ	λ	NOUN
ejpam-4572	413	32	,	,	PUNCT
ejpam-4572	413	33	sp)-open	sp)-open	ADJ
ejpam-4572	413	34	sets	set	NOUN
ejpam-4572	413	35	.	.	PUNCT
ejpam-4572	414	1	theorem	theorem	NOUN
ejpam-4572	414	2	9	9	NUM
ejpam-4572	414	3	.	.	PUNCT
ejpam-4572	415	1	let	let	VERB
ejpam-4572	415	2	f	f	NOUN
ejpam-4572	415	3	:	:	PUNCT
ejpam-4572	415	4	(	(	PUNCT
ejpam-4572	415	5	x	x	X
ejpam-4572	415	6	,	,	PUNCT
ejpam-4572	415	7	τ	τ	X
ejpam-4572	415	8	)	)	PUNCT
ejpam-4572	415	9	→	→	SYM
ejpam-4572	415	10	(	(	PUNCT
ejpam-4572	415	11	y	y	PROPN
ejpam-4572	415	12	,	,	PUNCT
ejpam-4572	415	13	σ	σ	PROPN
ejpam-4572	415	14	)	)	PUNCT
ejpam-4572	415	15	be	be	VERB
ejpam-4572	415	16	an	an	DET
ejpam-4572	415	17	upper	upper	ADJ
ejpam-4572	415	18	weakly	weakly	ADJ
ejpam-4572	415	19	α(λ	α(λ	PROPN
ejpam-4572	415	20	,	,	PUNCT
ejpam-4572	415	21	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	415	22	and	and	CCONJ
ejpam-4572	415	23	lower	low	ADJ
ejpam-4572	415	24	weakly	weakly	ADJ
ejpam-4572	415	25	α(λ	α(λ	PROPN
ejpam-4572	415	26	,	,	PUNCT
ejpam-4572	415	27	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	415	28	surjective	surjective	ADJ
ejpam-4572	415	29	multifunction	multifunction	NOUN
ejpam-4572	415	30	.	.	PUNCT
ejpam-4572	416	1	if	if	SCONJ
ejpam-4572	416	2	(	(	PUNCT
ejpam-4572	416	3	x	x	NOUN
ejpam-4572	416	4	,	,	PUNCT
ejpam-4572	416	5	τ	τ	X
ejpam-4572	416	6	)	)	PUNCT
ejpam-4572	416	7	is	be	AUX
ejpam-4572	416	8	λsp	λsp	VERB
ejpam-4572	416	9	-	-	PUNCT
ejpam-4572	416	10	connected	connect	VERB
ejpam-4572	416	11	and	and	CCONJ
ejpam-4572	416	12	f	f	PROPN
ejpam-4572	416	13	(	(	PUNCT
ejpam-4572	416	14	x	x	X
ejpam-4572	416	15	)	)	PUNCT
ejpam-4572	416	16	is	be	AUX
ejpam-4572	416	17	λsp	λsp	VERB
ejpam-4572	416	18	-	-	PUNCT
ejpam-4572	416	19	connected	connect	VERB
ejpam-4572	416	20	for	for	ADP
ejpam-4572	416	21	each	each	DET
ejpam-4572	416	22	x	x	SYM
ejpam-4572	416	23	∈	∈	PROPN
ejpam-4572	416	24	x	x	NOUN
ejpam-4572	416	25	,	,	PUNCT
ejpam-4572	416	26	then	then	ADV
ejpam-4572	416	27	(	(	PUNCT
ejpam-4572	416	28	y	y	PROPN
ejpam-4572	416	29	,	,	PUNCT
ejpam-4572	416	30	σ	σ	PROPN
ejpam-4572	416	31	)	)	PUNCT
ejpam-4572	416	32	is	be	AUX
ejpam-4572	416	33	λsp	λsp	VERB
ejpam-4572	416	34	-	-	PUNCT
ejpam-4572	416	35	connected	connect	VERB
ejpam-4572	416	36	.	.	PUNCT
ejpam-4572	417	1	proof	proof	NOUN
ejpam-4572	417	2	.	.	PUNCT
ejpam-4572	418	1	suppose	suppose	VERB
ejpam-4572	418	2	that	that	SCONJ
ejpam-4572	418	3	(	(	PUNCT
ejpam-4572	418	4	y	y	PROPN
ejpam-4572	418	5	,	,	PUNCT
ejpam-4572	418	6	σ	σ	PROPN
ejpam-4572	418	7	)	)	PUNCT
ejpam-4572	418	8	is	be	AUX
ejpam-4572	418	9	not	not	PART
ejpam-4572	418	10	λsp	λsp	ADV
ejpam-4572	418	11	-	-	PUNCT
ejpam-4572	418	12	connected	connect	VERB
ejpam-4572	418	13	.	.	PUNCT
ejpam-4572	419	1	there	there	PRON
ejpam-4572	419	2	exist	exist	VERB
ejpam-4572	419	3	nonempty	nonempty	ADJ
ejpam-4572	419	4	(	(	PUNCT
ejpam-4572	419	5	λ	λ	NOUN
ejpam-4572	419	6	,	,	PUNCT
ejpam-4572	419	7	sp)-open	sp)-open	NOUN
ejpam-4572	419	8	sets	set	VERB
ejpam-4572	419	9	u	u	NOUN
ejpam-4572	419	10	and	and	CCONJ
ejpam-4572	419	11	v	v	NOUN
ejpam-4572	419	12	of	of	ADP
ejpam-4572	419	13	y	y	PRON
ejpam-4572	419	14	such	such	ADJ
ejpam-4572	419	15	that	that	SCONJ
ejpam-4572	419	16	u	u	PROPN
ejpam-4572	419	17	∪	∪	VERB
ejpam-4572	419	18	v	v	NOUN
ejpam-4572	419	19	=	=	SYM
ejpam-4572	419	20	y	y	PROPN
ejpam-4572	419	21	and	and	CCONJ
ejpam-4572	419	22	u	u	PROPN
ejpam-4572	419	23	∩	∩	NOUN
ejpam-4572	419	24	v	v	NOUN
ejpam-4572	419	25	=	=	PUNCT
ejpam-4572	419	26	∅.	∅.	NOUN
ejpam-4572	419	27	since	since	SCONJ
ejpam-4572	419	28	f	f	PROPN
ejpam-4572	419	29	(	(	PUNCT
ejpam-4572	419	30	x	x	X
ejpam-4572	419	31	)	)	PUNCT
ejpam-4572	419	32	is	be	AUX
ejpam-4572	419	33	λsp	λsp	VERB
ejpam-4572	419	34	-	-	PUNCT
ejpam-4572	419	35	connected	connect	VERB
ejpam-4572	419	36	for	for	ADP
ejpam-4572	419	37	each	each	DET
ejpam-4572	419	38	x	x	SYM
ejpam-4572	419	39	∈	∈	PROPN
ejpam-4572	419	40	x	x	X
ejpam-4572	419	41	,	,	PUNCT
ejpam-4572	419	42	we	we	PRON
ejpam-4572	419	43	have	have	VERB
ejpam-4572	419	44	either	either	CCONJ
ejpam-4572	419	45	f	f	PROPN
ejpam-4572	419	46	(	(	PUNCT
ejpam-4572	419	47	x	x	NOUN
ejpam-4572	419	48	)	)	PUNCT
ejpam-4572	419	49	⊆	⊆	NUM
ejpam-4572	419	50	u	u	NOUN
ejpam-4572	419	51	or	or	CCONJ
ejpam-4572	419	52	f	f	PROPN
ejpam-4572	419	53	(	(	PUNCT
ejpam-4572	419	54	x	x	NOUN
ejpam-4572	419	55	)	)	PUNCT
ejpam-4572	419	56	⊆	⊆	NUM
ejpam-4572	419	57	v	v	NOUN
ejpam-4572	419	58	.	.	PUNCT
ejpam-4572	420	1	if	if	SCONJ
ejpam-4572	420	2	x	x	SYM
ejpam-4572	420	3	∈	∈	NOUN
ejpam-4572	420	4	f+(u	f+(u	PUNCT
ejpam-4572	420	5	∪v	∪v	NUM
ejpam-4572	420	6	)	)	PUNCT
ejpam-4572	420	7	,	,	PUNCT
ejpam-4572	420	8	then	then	ADV
ejpam-4572	420	9	f	f	X
ejpam-4572	420	10	(	(	PUNCT
ejpam-4572	420	11	x	x	NOUN
ejpam-4572	420	12	)	)	PUNCT
ejpam-4572	420	13	⊆	⊆	NUM
ejpam-4572	420	14	u	u	NOUN
ejpam-4572	420	15	∪v	∪v	PUNCT
ejpam-4572	420	16	and	and	CCONJ
ejpam-4572	420	17	hence	hence	ADV
ejpam-4572	420	18	x	x	X
ejpam-4572	420	19	∈	∈	PROPN
ejpam-4572	420	20	f+(u	f+(u	NUM
ejpam-4572	420	21	)	)	PUNCT
ejpam-4572	420	22	∪	∪	ADP
ejpam-4572	420	23	f+(v	f+(v	NOUN
ejpam-4572	420	24	)	)	PUNCT
ejpam-4572	420	25	.	.	PUNCT
ejpam-4572	421	1	moreover	moreover	ADV
ejpam-4572	421	2	,	,	PUNCT
ejpam-4572	421	3	since	since	SCONJ
ejpam-4572	421	4	f	f	PROPN
ejpam-4572	421	5	is	be	AUX
ejpam-4572	421	6	surjective	surjective	ADJ
ejpam-4572	421	7	,	,	PUNCT
ejpam-4572	421	8	there	there	PRON
ejpam-4572	421	9	exist	exist	VERB
ejpam-4572	421	10	x	x	PUNCT
ejpam-4572	421	11	and	and	CCONJ
ejpam-4572	421	12	y	y	PROPN
ejpam-4572	421	13	in	in	ADP
ejpam-4572	421	14	x	x	PUNCT
ejpam-4572	421	15	such	such	ADJ
ejpam-4572	421	16	that	that	SCONJ
ejpam-4572	421	17	f	f	PROPN
ejpam-4572	421	18	(	(	PUNCT
ejpam-4572	421	19	x	x	X
ejpam-4572	421	20	)	)	PUNCT
ejpam-4572	421	21	⊆	⊆	NUM
ejpam-4572	421	22	u	u	NOUN
ejpam-4572	421	23	and	and	CCONJ
ejpam-4572	421	24	f	f	PROPN
ejpam-4572	421	25	(	(	PUNCT
ejpam-4572	421	26	y	y	PROPN
ejpam-4572	421	27	)	)	PUNCT
ejpam-4572	421	28	⊆	⊆	NUM
ejpam-4572	421	29	v	v	NOUN
ejpam-4572	421	30	;	;	PUNCT
ejpam-4572	421	31	x	x	SYM
ejpam-4572	421	32	∈	∈	PROPN
ejpam-4572	421	33	f+(u	f+(u	NUM
ejpam-4572	421	34	)	)	PUNCT
ejpam-4572	421	35	and	and	CCONJ
ejpam-4572	421	36	y	y	PROPN
ejpam-4572	421	37	∈	∈	PROPN
ejpam-4572	421	38	f+(v	f+(v	PROPN
ejpam-4572	421	39	)	)	PUNCT
ejpam-4572	421	40	.	.	PUNCT
ejpam-4572	422	1	thus	thus	ADV
ejpam-4572	422	2	,	,	PUNCT
ejpam-4572	422	3	(	(	PUNCT
ejpam-4572	422	4	1	1	X
ejpam-4572	422	5	)	)	PUNCT
ejpam-4572	422	6	f+(u	f+(u	NUM
ejpam-4572	422	7	)	)	PUNCT
ejpam-4572	422	8	∪	∪	ADP
ejpam-4572	422	9	f+(v	f+(v	NOUN
ejpam-4572	422	10	)	)	PUNCT
ejpam-4572	423	1	=	=	PUNCT
ejpam-4572	424	1	f+(u	f+(u	PUNCT
ejpam-4572	424	2	∪	∪	ADP
ejpam-4572	424	3	v	v	NOUN
ejpam-4572	424	4	)	)	PUNCT
ejpam-4572	424	5	=	=	SYM
ejpam-4572	425	1	x	x	X
ejpam-4572	425	2	;	;	PUNCT
ejpam-4572	425	3	(	(	PUNCT
ejpam-4572	425	4	2	2	X
ejpam-4572	425	5	)	)	PUNCT
ejpam-4572	425	6	f+(u	f+(u	NUM
ejpam-4572	425	7	)	)	PUNCT
ejpam-4572	425	8	∩	∩	NOUN
ejpam-4572	425	9	f+(v	f+(v	NOUN
ejpam-4572	425	10	)	)	PUNCT
ejpam-4572	425	11	=	=	SYM
ejpam-4572	426	1	f+(u	f+(u	NUM
ejpam-4572	426	2	∩	∩	NOUN
ejpam-4572	426	3	v	v	NOUN
ejpam-4572	426	4	)	)	PUNCT
ejpam-4572	426	5	=	=	NOUN
ejpam-4572	426	6	∅	∅	NOUN
ejpam-4572	426	7	;	;	PUNCT
ejpam-4572	426	8	(	(	PUNCT
ejpam-4572	426	9	3	3	X
ejpam-4572	426	10	)	)	PUNCT
ejpam-4572	426	11	f+(u	f+(u	NUM
ejpam-4572	426	12	)	)	PUNCT
ejpam-4572	426	13	̸=	̸=	PROPN
ejpam-4572	426	14	∅	∅	NOUN
ejpam-4572	426	15	and	and	CCONJ
ejpam-4572	426	16	f+(v	f+(v	NUM
ejpam-4572	426	17	)	)	PUNCT
ejpam-4572	427	1	̸=	̸=	PROPN
ejpam-4572	427	2	∅.	∅.	PRON
ejpam-4572	427	3	by	by	ADP
ejpam-4572	427	4	lemma	lemma	PROPN
ejpam-4572	427	5	14	14	NUM
ejpam-4572	427	6	,	,	PUNCT
ejpam-4572	427	7	f+(u	f+(u	NUM
ejpam-4572	427	8	)	)	PUNCT
ejpam-4572	427	9	and	and	CCONJ
ejpam-4572	427	10	f+(v	f+(v	NUM
ejpam-4572	427	11	)	)	PUNCT
ejpam-4572	427	12	are	be	AUX
ejpam-4572	427	13	(	(	PUNCT
ejpam-4572	427	14	λ	λ	X
ejpam-4572	427	15	,	,	PUNCT
ejpam-4572	427	16	sp)-clopen	sp)-clopen	NOUN
ejpam-4572	427	17	.	.	PUNCT
ejpam-4572	428	1	this	this	PRON
ejpam-4572	428	2	shows	show	VERB
ejpam-4572	428	3	that	that	SCONJ
ejpam-4572	428	4	(	(	PUNCT
ejpam-4572	428	5	x	x	X
ejpam-4572	428	6	,	,	PUNCT
ejpam-4572	428	7	τ	τ	X
ejpam-4572	428	8	)	)	PUNCT
ejpam-4572	428	9	is	be	AUX
ejpam-4572	428	10	not	not	PART
ejpam-4572	428	11	λspconnected	λspconnecte	VERB
ejpam-4572	428	12	.	.	PUNCT
ejpam-4572	429	1	definition	definition	NOUN
ejpam-4572	429	2	7	7	NUM
ejpam-4572	429	3	.	.	PUNCT
ejpam-4572	430	1	let	let	VERB
ejpam-4572	430	2	a	a	DET
ejpam-4572	430	3	be	be	AUX
ejpam-4572	430	4	a	a	DET
ejpam-4572	430	5	subset	subset	NOUN
ejpam-4572	430	6	of	of	ADP
ejpam-4572	430	7	a	a	DET
ejpam-4572	430	8	topological	topological	ADJ
ejpam-4572	430	9	space	space	NOUN
ejpam-4572	430	10	(	(	PUNCT
ejpam-4572	430	11	x	x	X
ejpam-4572	430	12	,	,	PUNCT
ejpam-4572	430	13	τ	τ	PROPN
ejpam-4572	430	14	)	)	PUNCT
ejpam-4572	430	15	.	.	PUNCT
ejpam-4572	431	1	the	the	DET
ejpam-4572	431	2	α(λ	α(λ	PROPN
ejpam-4572	431	3	,	,	PUNCT
ejpam-4572	431	4	sp)-frontier	sp)-frontier	NOUN
ejpam-4572	431	5	of	of	ADP
ejpam-4572	431	6	a	a	DET
ejpam-4572	431	7	,	,	PUNCT
ejpam-4572	431	8	αλspfr(a	αλspfr(a	NUM
ejpam-4572	431	9	)	)	PUNCT
ejpam-4572	431	10	,	,	PUNCT
ejpam-4572	431	11	is	be	AUX
ejpam-4572	431	12	defined	define	VERB
ejpam-4572	431	13	as	as	SCONJ
ejpam-4572	431	14	follows	follow	VERB
ejpam-4572	431	15	:	:	PUNCT
ejpam-4572	431	16	αλspfr(a	αλspfr(a	NUM
ejpam-4572	431	17	)	)	PUNCT
ejpam-4572	431	18	=	=	SYM
ejpam-4572	431	19	aα(λ	aα(λ	PROPN
ejpam-4572	431	20	,	,	PUNCT
ejpam-4572	431	21	sp	sp	NOUN
ejpam-4572	431	22	)	)	PUNCT
ejpam-4572	431	23	∩	∩	NOUN
ejpam-4572	431	24	[	[	X
ejpam-4572	431	25	x	x	X
ejpam-4572	431	26	−a]α(λ	−a]α(λ	ADJ
ejpam-4572	431	27	,	,	PUNCT
ejpam-4572	431	28	sp	sp	NOUN
ejpam-4572	431	29	)	)	PUNCT
ejpam-4572	431	30	=	=	SYM
ejpam-4572	431	31	aα(λ	aα(λ	PROPN
ejpam-4572	431	32	,	,	PUNCT
ejpam-4572	431	33	sp	sp	NOUN
ejpam-4572	431	34	)	)	PUNCT
ejpam-4572	431	35	−aα(λ	−aα(λ	NOUN
ejpam-4572	431	36	,	,	PUNCT
ejpam-4572	431	37	sp	sp	NOUN
ejpam-4572	431	38	)	)	PUNCT
ejpam-4572	431	39	.	.	PUNCT
ejpam-4572	432	1	theorem	theorem	ADJ
ejpam-4572	432	2	10	10	NUM
ejpam-4572	432	3	.	.	PUNCT
ejpam-4572	433	1	the	the	DET
ejpam-4572	433	2	set	set	NOUN
ejpam-4572	433	3	of	of	ADP
ejpam-4572	433	4	all	all	DET
ejpam-4572	433	5	points	point	NOUN
ejpam-4572	433	6	of	of	ADP
ejpam-4572	433	7	x	x	PUNCT
ejpam-4572	433	8	at	at	ADP
ejpam-4572	433	9	which	which	PRON
ejpam-4572	433	10	a	a	DET
ejpam-4572	433	11	multifunction	multifunction	NOUN
ejpam-4572	433	12	f	f	NOUN
ejpam-4572	433	13	:	:	PUNCT
ejpam-4572	433	14	(	(	PUNCT
ejpam-4572	433	15	x	x	X
ejpam-4572	433	16	,	,	PUNCT
ejpam-4572	433	17	τ	τ	X
ejpam-4572	433	18	)	)	PUNCT
ejpam-4572	433	19	→	→	SYM
ejpam-4572	433	20	(	(	PUNCT
ejpam-4572	433	21	y	y	PROPN
ejpam-4572	433	22	,	,	PUNCT
ejpam-4572	433	23	σ	σ	PROPN
ejpam-4572	433	24	)	)	PUNCT
ejpam-4572	433	25	is	be	AUX
ejpam-4572	433	26	not	not	PART
ejpam-4572	433	27	upper	upper	ADJ
ejpam-4572	433	28	weakly	weakly	ADJ
ejpam-4572	433	29	α(λ	α(λ	PROPN
ejpam-4572	433	30	,	,	PUNCT
ejpam-4572	433	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	433	32	is	be	AUX
ejpam-4572	433	33	identical	identical	ADJ
ejpam-4572	433	34	with	with	ADP
ejpam-4572	433	35	the	the	DET
ejpam-4572	433	36	union	union	NOUN
ejpam-4572	433	37	of	of	ADP
ejpam-4572	433	38	the	the	DET
ejpam-4572	433	39	α(λ	α(λ	PROPN
ejpam-4572	433	40	,	,	PUNCT
ejpam-4572	433	41	sp)-frontier	sp)-frontier	NOUN
ejpam-4572	433	42	of	of	ADP
ejpam-4572	433	43	the	the	DET
ejpam-4572	433	44	upper	upper	ADJ
ejpam-4572	433	45	inverse	inverse	NOUN
ejpam-4572	433	46	images	image	NOUN
ejpam-4572	433	47	of	of	ADP
ejpam-4572	433	48	the	the	DET
ejpam-4572	433	49	(	(	PUNCT
ejpam-4572	433	50	λ	λ	PROPN
ejpam-4572	433	51	,	,	PUNCT
ejpam-4572	433	52	sp)-closures	sp)-closure	NOUN
ejpam-4572	433	53	of	of	ADP
ejpam-4572	433	54	(	(	PUNCT
ejpam-4572	433	55	λ	λ	PROPN
ejpam-4572	433	56	,	,	PUNCT
ejpam-4572	433	57	sp)-open	sp)-open	ADJ
ejpam-4572	433	58	sets	set	NOUN
ejpam-4572	433	59	of	of	ADP
ejpam-4572	433	60	y	y	PROPN
ejpam-4572	433	61	containing	contain	VERB
ejpam-4572	433	62	f	f	PROPN
ejpam-4572	433	63	(	(	PUNCT
ejpam-4572	433	64	x	x	NOUN
ejpam-4572	433	65	)	)	PUNCT
ejpam-4572	433	66	.	.	PUNCT
ejpam-4572	434	1	proof	proof	NOUN
ejpam-4572	434	2	.	.	PUNCT
ejpam-4572	435	1	let	let	VERB
ejpam-4572	435	2	x	x	PRON
ejpam-4572	435	3	be	be	AUX
ejpam-4572	435	4	a	a	DET
ejpam-4572	435	5	point	point	NOUN
ejpam-4572	435	6	ofx	ofx	NOUN
ejpam-4572	435	7	at	at	ADP
ejpam-4572	435	8	which	which	PRON
ejpam-4572	435	9	f	f	NOUN
ejpam-4572	435	10	is	be	AUX
ejpam-4572	435	11	not	not	PART
ejpam-4572	435	12	upper	upper	ADJ
ejpam-4572	435	13	weakly	weakly	ADJ
ejpam-4572	435	14	β(λ	β(λ	NOUN
ejpam-4572	435	15	,	,	PUNCT
ejpam-4572	435	16	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	435	17	.	.	PUNCT
ejpam-4572	436	1	then	then	ADV
ejpam-4572	436	2	,	,	PUNCT
ejpam-4572	436	3	there	there	PRON
ejpam-4572	436	4	exists	exist	VERB
ejpam-4572	436	5	a	a	DET
ejpam-4572	436	6	(	(	PUNCT
ejpam-4572	436	7	λ	λ	NOUN
ejpam-4572	436	8	,	,	PUNCT
ejpam-4572	436	9	sp)-open	sp)-open	ADJ
ejpam-4572	436	10	set	set	VERB
ejpam-4572	436	11	v	v	NOUN
ejpam-4572	436	12	containing	contain	VERB
ejpam-4572	436	13	f	f	X
ejpam-4572	436	14	(	(	PUNCT
ejpam-4572	436	15	x	x	X
ejpam-4572	436	16	)	)	PUNCT
ejpam-4572	437	1	such	such	ADJ
ejpam-4572	437	2	that	that	SCONJ
ejpam-4572	437	3	u	u	PROPN
ejpam-4572	437	4	∩	∩	NOUN
ejpam-4572	437	5	(	(	PUNCT
ejpam-4572	437	6	x	x	SYM
ejpam-4572	437	7	−	−	PROPN
ejpam-4572	437	8	f+(v	f+(v	PROPN
ejpam-4572	437	9	(	(	PUNCT
ejpam-4572	437	10	λ	λ	NOUN
ejpam-4572	437	11	,	,	PUNCT
ejpam-4572	437	12	sp	sp	NOUN
ejpam-4572	437	13	)	)	PUNCT
ejpam-4572	437	14	)	)	PUNCT
ejpam-4572	437	15	)	)	PUNCT
ejpam-4572	437	16	̸=	̸=	NOUN
ejpam-4572	437	17	∅	∅	NOUN
ejpam-4572	437	18	for	for	ADP
ejpam-4572	437	19	every	every	DET
ejpam-4572	437	20	α(λ	α(λ	PROPN
ejpam-4572	437	21	,	,	PUNCT
ejpam-4572	437	22	sp)-open	sp)-open	VERB
ejpam-4572	437	23	set	set	VERB
ejpam-4572	437	24	u	u	NOUN
ejpam-4572	437	25	containing	contain	VERB
ejpam-4572	437	26	x.	x.	NOUN
ejpam-4572	437	27	then	then	ADV
ejpam-4572	437	28	,	,	PUNCT
ejpam-4572	437	29	we	we	PRON
ejpam-4572	437	30	have	have	VERB
ejpam-4572	437	31	x	x	X
ejpam-4572	437	32	∈	∈	PROPN
ejpam-4572	438	1	[	[	X
ejpam-4572	438	2	x	x	X
ejpam-4572	438	3	−	−	PROPN
ejpam-4572	438	4	f+(v	f+(v	PROPN
ejpam-4572	438	5	(	(	PUNCT
ejpam-4572	438	6	λ	λ	PROPN
ejpam-4572	438	7	,	,	PUNCT
ejpam-4572	438	8	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	438	9	,	,	PUNCT
ejpam-4572	438	10	sp	sp	NOUN
ejpam-4572	438	11	)	)	PUNCT
ejpam-4572	438	12	.	.	PUNCT
ejpam-4572	439	1	since	since	SCONJ
ejpam-4572	439	2	x	x	PROPN
ejpam-4572	439	3	∈	∈	PROPN
ejpam-4572	439	4	f+(v	f+(v	NOUN
ejpam-4572	439	5	)	)	PUNCT
ejpam-4572	439	6	,	,	PUNCT
ejpam-4572	439	7	x	x	PUNCT
ejpam-4572	439	8	∈	∈	PROPN
ejpam-4572	439	9	[	[	X
ejpam-4572	439	10	f+(v	f+(v	NOUN
ejpam-4572	439	11	(	(	PUNCT
ejpam-4572	439	12	λ	λ	PROPN
ejpam-4572	439	13	,	,	PUNCT
ejpam-4572	439	14	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	439	15	,	,	PUNCT
ejpam-4572	439	16	sp	sp	NOUN
ejpam-4572	439	17	)	)	PUNCT
ejpam-4572	439	18	and	and	CCONJ
ejpam-4572	439	19	hence	hence	ADV
ejpam-4572	439	20	x	x	PART
ejpam-4572	439	21	∈	∈	NOUN
ejpam-4572	439	22	αλspfr(f+(v	αλspfr(f+(v	NOUN
ejpam-4572	439	23	(	(	PUNCT
ejpam-4572	439	24	λ	λ	NOUN
ejpam-4572	439	25	,	,	PUNCT
ejpam-4572	439	26	sp	sp	NOUN
ejpam-4572	439	27	)	)	PUNCT
ejpam-4572	439	28	)	)	PUNCT
ejpam-4572	439	29	)	)	PUNCT
ejpam-4572	439	30	.	.	PUNCT
ejpam-4572	440	1	if	if	SCONJ
ejpam-4572	440	2	f	f	PROPN
ejpam-4572	440	3	is	be	AUX
ejpam-4572	440	4	upper	upper	ADJ
ejpam-4572	440	5	weakly	weakly	ADJ
ejpam-4572	440	6	β(λ	β(λ	NOUN
ejpam-4572	440	7	,	,	PUNCT
ejpam-4572	440	8	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	440	9	at	at	ADP
ejpam-4572	440	10	x	x	X
ejpam-4572	440	11	,	,	PUNCT
ejpam-4572	440	12	then	then	ADV
ejpam-4572	440	13	there	there	PRON
ejpam-4572	440	14	exists	exist	VERB
ejpam-4572	440	15	a	a	DET
ejpam-4572	440	16	α(λ	α(λ	PROPN
ejpam-4572	440	17	,	,	PUNCT
ejpam-4572	440	18	sp)-open	sp)-open	VERB
ejpam-4572	440	19	set	set	VERB
ejpam-4572	440	20	u	u	PRON
ejpam-4572	440	21	containing	contain	VERB
ejpam-4572	440	22	x	x	PUNCT
ejpam-4572	440	23	such	such	ADJ
ejpam-4572	440	24	that	that	SCONJ
ejpam-4572	440	25	f	f	PROPN
ejpam-4572	440	26	(	(	PUNCT
ejpam-4572	440	27	u	u	NOUN
ejpam-4572	440	28	)	)	PUNCT
ejpam-4572	440	29	⊆	⊆	NUM
ejpam-4572	440	30	v	v	NOUN
ejpam-4572	440	31	(	(	PUNCT
ejpam-4572	440	32	λ	λ	NOUN
ejpam-4572	440	33	,	,	PUNCT
ejpam-4572	440	34	sp	sp	NOUN
ejpam-4572	440	35	)	)	PUNCT
ejpam-4572	440	36	;	;	PUNCT
ejpam-4572	440	37	hence	hence	ADV
ejpam-4572	440	38	u	u	NOUN
ejpam-4572	440	39	⊆	⊆	NUM
ejpam-4572	440	40	f+[v	f+[v	NOUN
ejpam-4572	440	41	(	(	PUNCT
ejpam-4572	440	42	λ	λ	NOUN
ejpam-4572	440	43	,	,	PUNCT
ejpam-4572	440	44	sp	sp	NOUN
ejpam-4572	440	45	)	)	PUNCT
ejpam-4572	440	46	]	]	PUNCT
ejpam-4572	440	47	.	.	PUNCT
ejpam-4572	441	1	this	this	PRON
ejpam-4572	441	2	shows	show	VERB
ejpam-4572	441	3	that	that	SCONJ
ejpam-4572	441	4	x	x	PUNCT
ejpam-4572	441	5	∈	∈	PROPN
ejpam-4572	441	6	u	u	NOUN
ejpam-4572	441	7	⊆	⊆	NUM
ejpam-4572	441	8	[	[	X
ejpam-4572	441	9	f+(v	f+(v	NOUN
ejpam-4572	441	10	(	(	PUNCT
ejpam-4572	441	11	λ	λ	PROPN
ejpam-4572	441	12	,	,	PUNCT
ejpam-4572	441	13	sp))]α(λ	sp))]α(λ	NOUN
ejpam-4572	441	14	,	,	PUNCT
ejpam-4572	441	15	sp	sp	NOUN
ejpam-4572	441	16	)	)	PUNCT
ejpam-4572	441	17	.	.	PUNCT
ejpam-4572	442	1	this	this	PRON
ejpam-4572	442	2	contradicts	contradict	VERB
ejpam-4572	442	3	that	that	SCONJ
ejpam-4572	442	4	x	x	SYM
ejpam-4572	442	5	∈	∈	PROPN
ejpam-4572	442	6	αλspfr(f+(v	αλspfr(f+(v	NOUN
ejpam-4572	442	7	(	(	PUNCT
ejpam-4572	442	8	λ	λ	NOUN
ejpam-4572	442	9	,	,	PUNCT
ejpam-4572	442	10	sp	sp	NOUN
ejpam-4572	442	11	)	)	PUNCT
ejpam-4572	442	12	)	)	PUNCT
ejpam-4572	442	13	)	)	PUNCT
ejpam-4572	442	14	.	.	PUNCT
ejpam-4572	443	1	thus	thus	ADV
ejpam-4572	443	2	,	,	PUNCT
ejpam-4572	443	3	f	f	PROPN
ejpam-4572	443	4	is	be	AUX
ejpam-4572	443	5	not	not	PART
ejpam-4572	443	6	upper	upper	ADJ
ejpam-4572	443	7	weakly	weakly	ADJ
ejpam-4572	443	8	α(λ	α(λ	PROPN
ejpam-4572	443	9	,	,	PUNCT
ejpam-4572	443	10	sp)continuous	sp)continuous	ADJ
ejpam-4572	443	11	at	at	ADP
ejpam-4572	443	12	x.	x.	PROPN
ejpam-4572	443	13	references	reference	NOUN
ejpam-4572	443	14	477	477	NUM
ejpam-4572	443	15	theorem	theorem	VERB
ejpam-4572	443	16	11	11	NUM
ejpam-4572	443	17	.	.	PUNCT
ejpam-4572	444	1	the	the	DET
ejpam-4572	444	2	set	set	NOUN
ejpam-4572	444	3	of	of	ADP
ejpam-4572	444	4	all	all	DET
ejpam-4572	444	5	points	point	NOUN
ejpam-4572	444	6	of	of	ADP
ejpam-4572	444	7	x	x	PUNCT
ejpam-4572	444	8	at	at	ADP
ejpam-4572	444	9	which	which	PRON
ejpam-4572	444	10	a	a	DET
ejpam-4572	444	11	multifunction	multifunction	NOUN
ejpam-4572	444	12	f	f	NOUN
ejpam-4572	444	13	:	:	PUNCT
ejpam-4572	444	14	(	(	PUNCT
ejpam-4572	444	15	x	x	X
ejpam-4572	444	16	,	,	PUNCT
ejpam-4572	444	17	τ	τ	X
ejpam-4572	444	18	)	)	PUNCT
ejpam-4572	444	19	→	→	SYM
ejpam-4572	444	20	(	(	PUNCT
ejpam-4572	444	21	y	y	PROPN
ejpam-4572	444	22	,	,	PUNCT
ejpam-4572	444	23	σ	σ	PROPN
ejpam-4572	444	24	)	)	PUNCT
ejpam-4572	444	25	is	be	AUX
ejpam-4572	444	26	not	not	PART
ejpam-4572	444	27	lower	low	ADJ
ejpam-4572	444	28	weakly	weakly	ADJ
ejpam-4572	444	29	α(λ	α(λ	PROPN
ejpam-4572	444	30	,	,	PUNCT
ejpam-4572	444	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	444	32	is	be	AUX
ejpam-4572	444	33	identical	identical	ADJ
ejpam-4572	444	34	with	with	ADP
ejpam-4572	444	35	the	the	DET
ejpam-4572	444	36	union	union	NOUN
ejpam-4572	444	37	of	of	ADP
ejpam-4572	444	38	the	the	DET
ejpam-4572	444	39	α(λ	α(λ	PROPN
ejpam-4572	444	40	,	,	PUNCT
ejpam-4572	444	41	sp)-frontier	sp)-frontier	NOUN
ejpam-4572	444	42	of	of	ADP
ejpam-4572	444	43	the	the	DET
ejpam-4572	444	44	lower	low	ADJ
ejpam-4572	444	45	inverse	inverse	NOUN
ejpam-4572	444	46	images	image	NOUN
ejpam-4572	444	47	of	of	ADP
ejpam-4572	444	48	the	the	DET
ejpam-4572	444	49	(	(	PUNCT
ejpam-4572	444	50	λ	λ	PROPN
ejpam-4572	444	51	,	,	PUNCT
ejpam-4572	444	52	sp)-closures	sp)-closure	NOUN
ejpam-4572	444	53	of	of	ADP
ejpam-4572	444	54	(	(	PUNCT
ejpam-4572	444	55	λ	λ	PROPN
ejpam-4572	444	56	,	,	PUNCT
ejpam-4572	444	57	sp)-open	sp)-open	ADJ
ejpam-4572	444	58	sets	set	NOUN
ejpam-4572	444	59	of	of	ADP
ejpam-4572	444	60	y	y	PROPN
ejpam-4572	444	61	meeting	meet	VERB
ejpam-4572	444	62	f	f	PROPN
ejpam-4572	444	63	(	(	PUNCT
ejpam-4572	444	64	x	x	NOUN
ejpam-4572	444	65	)	)	PUNCT
ejpam-4572	444	66	.	.	PUNCT
ejpam-4572	445	1	proof	proof	NOUN
ejpam-4572	445	2	.	.	PUNCT
ejpam-4572	446	1	the	the	DET
ejpam-4572	446	2	proof	proof	NOUN
ejpam-4572	446	3	is	be	AUX
ejpam-4572	446	4	similar	similar	ADJ
ejpam-4572	446	5	to	to	ADP
ejpam-4572	446	6	that	that	PRON
ejpam-4572	446	7	of	of	ADP
ejpam-4572	446	8	theorem	theorem	ADJ
ejpam-4572	446	9	10	10	NUM
ejpam-4572	446	10	.	.	NOUN
ejpam-4572	446	11	4	4	NUM
ejpam-4572	446	12	.	.	X
ejpam-4572	446	13	conclusion	conclusion	NOUN
ejpam-4572	446	14	topology	topology	NOUN
ejpam-4572	446	15	is	be	AUX
ejpam-4572	446	16	concerned	concern	VERB
ejpam-4572	446	17	with	with	ADP
ejpam-4572	446	18	all	all	DET
ejpam-4572	446	19	questions	question	NOUN
ejpam-4572	446	20	directly	directly	ADV
ejpam-4572	446	21	or	or	CCONJ
ejpam-4572	446	22	indirectly	indirectly	ADV
ejpam-4572	446	23	related	relate	VERB
ejpam-4572	446	24	to	to	ADP
ejpam-4572	446	25	continuity	continuity	NOUN
ejpam-4572	446	26	.	.	PUNCT
ejpam-4572	447	1	the	the	DET
ejpam-4572	447	2	concept	concept	NOUN
ejpam-4572	447	3	of	of	ADP
ejpam-4572	447	4	continuity	continuity	NOUN
ejpam-4572	447	5	is	be	AUX
ejpam-4572	447	6	one	one	NUM
ejpam-4572	447	7	of	of	ADP
ejpam-4572	447	8	the	the	DET
ejpam-4572	447	9	most	most	ADV
ejpam-4572	447	10	important	important	ADJ
ejpam-4572	447	11	tools	tool	NOUN
ejpam-4572	447	12	in	in	ADP
ejpam-4572	447	13	mathematics	mathematic	NOUN
ejpam-4572	447	14	and	and	CCONJ
ejpam-4572	447	15	many	many	ADJ
ejpam-4572	447	16	different	different	ADJ
ejpam-4572	447	17	forms	form	NOUN
ejpam-4572	447	18	of	of	ADP
ejpam-4572	447	19	generalizations	generalization	NOUN
ejpam-4572	447	20	of	of	ADP
ejpam-4572	447	21	continuity	continuity	NOUN
ejpam-4572	447	22	have	have	AUX
ejpam-4572	447	23	been	be	AUX
ejpam-4572	447	24	introduced	introduce	VERB
ejpam-4572	447	25	and	and	CCONJ
ejpam-4572	447	26	studied	study	VERB
ejpam-4572	447	27	.	.	PUNCT
ejpam-4572	448	1	this	this	DET
ejpam-4572	448	2	paper	paper	NOUN
ejpam-4572	448	3	deals	deal	NOUN
ejpam-4572	448	4	with	with	ADP
ejpam-4572	448	5	the	the	DET
ejpam-4572	448	6	concept	concept	NOUN
ejpam-4572	448	7	of	of	ADP
ejpam-4572	448	8	upper	upper	ADJ
ejpam-4572	448	9	(	(	PUNCT
ejpam-4572	448	10	resp	resp	NOUN
ejpam-4572	448	11	.	.	PUNCT
ejpam-4572	449	1	lower	low	ADJ
ejpam-4572	449	2	)	)	PUNCT
ejpam-4572	449	3	weakly	weakly	ADJ
ejpam-4572	449	4	α(λ	α(λ	PROPN
ejpam-4572	449	5	,	,	PUNCT
ejpam-4572	449	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	449	7	multifunctions	multifunction	NOUN
ejpam-4572	449	8	.	.	PUNCT
ejpam-4572	450	1	a	a	DET
ejpam-4572	450	2	multifunction	multifunction	NOUN
ejpam-4572	450	3	f	f	NOUN
ejpam-4572	450	4	:	:	PUNCT
ejpam-4572	450	5	(	(	PUNCT
ejpam-4572	450	6	x	x	X
ejpam-4572	450	7	,	,	PUNCT
ejpam-4572	450	8	τ	τ	X
ejpam-4572	450	9	)	)	PUNCT
ejpam-4572	450	10	→	→	SYM
ejpam-4572	450	11	(	(	PUNCT
ejpam-4572	450	12	y	y	PROPN
ejpam-4572	450	13	,	,	PUNCT
ejpam-4572	450	14	σ	σ	PROPN
ejpam-4572	450	15	)	)	PUNCT
ejpam-4572	450	16	is	be	AUX
ejpam-4572	450	17	called	call	VERB
ejpam-4572	450	18	upper	upper	ADJ
ejpam-4572	450	19	(	(	PUNCT
ejpam-4572	450	20	resp	resp	NOUN
ejpam-4572	450	21	.	.	PUNCT
ejpam-4572	451	1	lower	low	ADJ
ejpam-4572	451	2	)	)	PUNCT
ejpam-4572	451	3	weakly	weakly	ADJ
ejpam-4572	451	4	α(λ	α(λ	PROPN
ejpam-4572	451	5	,	,	PUNCT
ejpam-4572	451	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	451	7	multifunctions	multifunction	NOUN
ejpam-4572	451	8	at	at	ADP
ejpam-4572	451	9	x	x	PROPN
ejpam-4572	451	10	∈	∈	PROPN
ejpam-4572	451	11	x	x	INTJ
ejpam-4572	451	12	if	if	SCONJ
ejpam-4572	451	13	,	,	PUNCT
ejpam-4572	451	14	for	for	ADP
ejpam-4572	451	15	each	each	DET
ejpam-4572	451	16	s(λ	s(λ	PROPN
ejpam-4572	451	17	,	,	PUNCT
ejpam-4572	451	18	sp)-open	sp)-open	VERB
ejpam-4572	451	19	set	set	VERB
ejpam-4572	451	20	u	u	NOUN
ejpam-4572	451	21	of	of	ADP
ejpam-4572	451	22	x	x	PUNCT
ejpam-4572	451	23	containing	contain	VERB
ejpam-4572	451	24	x	x	PUNCT
ejpam-4572	451	25	and	and	CCONJ
ejpam-4572	451	26	each	each	DET
ejpam-4572	451	27	(	(	PUNCT
ejpam-4572	451	28	λ	λ	PROPN
ejpam-4572	451	29	,	,	PUNCT
ejpam-4572	451	30	sp)-open	sp)-open	NOUN
ejpam-4572	451	31	set	set	VERB
ejpam-4572	451	32	v	v	NUM
ejpam-4572	451	33	of	of	ADP
ejpam-4572	451	34	y	y	PRON
ejpam-4572	451	35	such	such	ADJ
ejpam-4572	451	36	that	that	SCONJ
ejpam-4572	451	37	f	f	PROPN
ejpam-4572	451	38	(	(	PUNCT
ejpam-4572	451	39	x	x	X
ejpam-4572	451	40	)	)	PUNCT
ejpam-4572	451	41	⊆	⊆	NUM
ejpam-4572	451	42	v	v	NOUN
ejpam-4572	451	43	(	(	PUNCT
ejpam-4572	451	44	resp	resp	NOUN
ejpam-4572	451	45	.	.	PUNCT
ejpam-4572	452	1	f	f	X
ejpam-4572	452	2	(	(	PUNCT
ejpam-4572	452	3	x	x	NOUN
ejpam-4572	452	4	)	)	PUNCT
ejpam-4572	452	5	∩	∩	NOUN
ejpam-4572	452	6	v	v	ADP
ejpam-4572	452	7	̸=	̸=	PROPN
ejpam-4572	452	8	∅	∅	NOUN
ejpam-4572	452	9	)	)	PUNCT
ejpam-4572	452	10	,	,	PUNCT
ejpam-4572	452	11	there	there	PRON
ejpam-4572	452	12	exists	exist	VERB
ejpam-4572	452	13	a	a	DET
ejpam-4572	452	14	nonempty	nonempty	ADJ
ejpam-4572	452	15	(	(	PUNCT
ejpam-4572	452	16	λ	λ	NOUN
ejpam-4572	452	17	,	,	PUNCT
ejpam-4572	452	18	sp)-open	sp)-open	NOUN
ejpam-4572	452	19	set	set	VERB
ejpam-4572	452	20	g	g	PRON
ejpam-4572	452	21	such	such	ADJ
ejpam-4572	452	22	that	that	SCONJ
ejpam-4572	452	23	g	g	PROPN
ejpam-4572	452	24	⊆	⊆	NUM
ejpam-4572	452	25	u	u	NOUN
ejpam-4572	452	26	and	and	CCONJ
ejpam-4572	452	27	g	g	PROPN
ejpam-4572	452	28	⊆	⊆	NUM
ejpam-4572	452	29	f+(v	f+(v	PROPN
ejpam-4572	452	30	s(λ	s(λ	PROPN
ejpam-4572	452	31	,	,	PUNCT
ejpam-4572	452	32	sp	sp	NOUN
ejpam-4572	452	33	)	)	PUNCT
ejpam-4572	452	34	)	)	PUNCT
ejpam-4572	452	35	(	(	PUNCT
ejpam-4572	452	36	resp	resp	VERB
ejpam-4572	452	37	g	g	PROPN
ejpam-4572	452	38	⊆	⊆	NUM
ejpam-4572	452	39	f−(v	f−(v	PROPN
ejpam-4572	452	40	s(λ	s(λ	PROPN
ejpam-4572	452	41	,	,	PUNCT
ejpam-4572	452	42	sp	sp	NOUN
ejpam-4572	452	43	)	)	PUNCT
ejpam-4572	452	44	)	)	PUNCT
ejpam-4572	452	45	)	)	PUNCT
ejpam-4572	452	46	.	.	PUNCT
ejpam-4572	453	1	a	a	DET
ejpam-4572	453	2	multifunction	multifunction	NOUN
ejpam-4572	453	3	f	f	NOUN
ejpam-4572	453	4	:	:	PUNCT
ejpam-4572	453	5	(	(	PUNCT
ejpam-4572	453	6	x	x	X
ejpam-4572	453	7	,	,	PUNCT
ejpam-4572	453	8	τ	τ	X
ejpam-4572	453	9	)	)	PUNCT
ejpam-4572	453	10	→	→	SYM
ejpam-4572	453	11	(	(	PUNCT
ejpam-4572	453	12	y	y	PROPN
ejpam-4572	453	13	,	,	PUNCT
ejpam-4572	453	14	σ	σ	PROPN
ejpam-4572	453	15	)	)	PUNCT
ejpam-4572	453	16	is	be	AUX
ejpam-4572	453	17	called	call	VERB
ejpam-4572	453	18	upper	upper	ADJ
ejpam-4572	453	19	(	(	PUNCT
ejpam-4572	453	20	resp	resp	NOUN
ejpam-4572	453	21	.	.	PUNCT
ejpam-4572	454	1	lower	low	ADJ
ejpam-4572	454	2	)	)	PUNCT
ejpam-4572	454	3	weakly	weakly	ADJ
ejpam-4572	454	4	α(λ	α(λ	PROPN
ejpam-4572	454	5	,	,	PUNCT
ejpam-4572	454	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	454	7	multifunctions	multifunction	NOUN
ejpam-4572	454	8	if	if	SCONJ
ejpam-4572	454	9	f	f	PROPN
ejpam-4572	454	10	has	have	VERB
ejpam-4572	454	11	this	this	DET
ejpam-4572	454	12	property	property	NOUN
ejpam-4572	454	13	at	at	ADP
ejpam-4572	454	14	each	each	DET
ejpam-4572	454	15	point	point	NOUN
ejpam-4572	454	16	of	of	ADP
ejpam-4572	454	17	x.	x.	NOUN
ejpam-4572	454	18	some	some	DET
ejpam-4572	454	19	characterizations	characterization	NOUN
ejpam-4572	454	20	and	and	CCONJ
ejpam-4572	454	21	several	several	ADJ
ejpam-4572	454	22	properties	property	NOUN
ejpam-4572	454	23	concerning	concern	VERB
ejpam-4572	454	24	upper	upper	ADJ
ejpam-4572	454	25	(	(	PUNCT
ejpam-4572	454	26	resp	resp	NOUN
ejpam-4572	454	27	.	.	PUNCT
ejpam-4572	455	1	lower	low	ADJ
ejpam-4572	455	2	)	)	PUNCT
ejpam-4572	455	3	weakly	weakly	ADJ
ejpam-4572	455	4	α(λ	α(λ	PROPN
ejpam-4572	455	5	,	,	PUNCT
ejpam-4572	455	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	455	7	multifunctions	multifunction	NOUN
ejpam-4572	455	8	are	be	AUX
ejpam-4572	455	9	established	establish	VERB
ejpam-4572	455	10	.	.	PUNCT
ejpam-4572	456	1	the	the	DET
ejpam-4572	456	2	ideas	idea	NOUN
ejpam-4572	456	3	and	and	CCONJ
ejpam-4572	456	4	results	result	NOUN
ejpam-4572	456	5	of	of	ADP
ejpam-4572	456	6	this	this	DET
ejpam-4572	456	7	paper	paper	NOUN
ejpam-4572	456	8	may	may	AUX
ejpam-4572	456	9	motivate	motivate	VERB
ejpam-4572	456	10	further	further	ADJ
ejpam-4572	456	11	research	research	NOUN
ejpam-4572	456	12	.	.	PUNCT
ejpam-4572	457	1	acknowledgements	acknowledgement	NOUN
ejpam-4572	457	2	this	this	DET
ejpam-4572	457	3	research	research	NOUN
ejpam-4572	457	4	project	project	NOUN
ejpam-4572	457	5	was	be	AUX
ejpam-4572	457	6	financially	financially	ADV
ejpam-4572	457	7	supported	support	VERB
ejpam-4572	457	8	by	by	ADP
ejpam-4572	457	9	mahasarakham	mahasarakham	PROPN
ejpam-4572	457	10	university	university	PROPN
ejpam-4572	457	11	.	.	PUNCT
ejpam-4572	458	1	references	reference	NOUN
ejpam-4572	458	2	[	[	X
ejpam-4572	458	3	1	1	NUM
ejpam-4572	458	4	]	]	PUNCT
ejpam-4572	458	5	c.	c.	PROPN
ejpam-4572	458	6	berge	berge	PROPN
ejpam-4572	458	7	.	.	PUNCT
ejpam-4572	459	1	espaces	espace	VERB
ejpam-4572	459	2	topologiques	topologique	NOUN
ejpam-4572	459	3	fonctions	fonction	NOUN
ejpam-4572	459	4	multivoques	multivoque	NOUN
ejpam-4572	459	5	.	.	PUNCT
ejpam-4572	460	1	dunod	dunod	PROPN
ejpam-4572	460	2	,	,	PUNCT
ejpam-4572	460	3	paris	paris	PROPN
ejpam-4572	460	4	,	,	PUNCT
ejpam-4572	460	5	1959	1959	NUM
ejpam-4572	460	6	.	.	PUNCT
ejpam-4572	461	1	[	[	X
ejpam-4572	461	2	2	2	NUM
ejpam-4572	461	3	]	]	PUNCT
ejpam-4572	461	4	c.	c.	PROPN
ejpam-4572	461	5	boonpok	boonpok	PROPN
ejpam-4572	461	6	.	.	PUNCT
ejpam-4572	462	1	(	(	PUNCT
ejpam-4572	462	2	λ	λ	NOUN
ejpam-4572	462	3	,	,	PUNCT
ejpam-4572	462	4	sp)-closed	sp)-close	VERB
ejpam-4572	462	5	sets	set	NOUN
ejpam-4572	462	6	and	and	CCONJ
ejpam-4572	462	7	related	related	ADJ
ejpam-4572	462	8	topics	topic	NOUN
ejpam-4572	462	9	in	in	ADP
ejpam-4572	462	10	topological	topological	ADJ
ejpam-4572	462	11	spaces	space	NOUN
ejpam-4572	462	12	.	.	PUNCT
ejpam-4572	463	1	wseas	wseas	VERB
ejpam-4572	463	2	transactions	transaction	NOUN
ejpam-4572	463	3	on	on	ADP
ejpam-4572	463	4	mathematics	mathematic	NOUN
ejpam-4572	463	5	,	,	PUNCT
ejpam-4572	463	6	19:312–322	19:312–322	PROPN
ejpam-4572	463	7	,	,	PUNCT
ejpam-4572	463	8	2020	2020	NUM
ejpam-4572	463	9	.	.	PUNCT
ejpam-4572	464	1	[	[	X
ejpam-4572	464	2	3	3	X
ejpam-4572	464	3	]	]	PUNCT
ejpam-4572	464	4	c.	c.	PROPN
ejpam-4572	464	5	boonpok	boonpok	PROPN
ejpam-4572	464	6	,	,	PUNCT
ejpam-4572	464	7	c.	c.	PROPN
ejpam-4572	464	8	viriyapong	viriyapong	PROPN
ejpam-4572	464	9	,	,	PUNCT
ejpam-4572	464	10	and	and	CCONJ
ejpam-4572	464	11	m.	m.	NOUN
ejpam-4572	464	12	thongmoon	thongmoon	NOUN
ejpam-4572	464	13	.	.	PUNCT
ejpam-4572	465	1	on	on	ADP
ejpam-4572	465	2	upper	upper	ADJ
ejpam-4572	465	3	and	and	CCONJ
ejpam-4572	465	4	lower	low	ADJ
ejpam-4572	465	5	(	(	PUNCT
ejpam-4572	465	6	τ1	τ1	NOUN
ejpam-4572	465	7	,	,	PUNCT
ejpam-4572	465	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-4572	465	9	multifunctions	multifunction	NOUN
ejpam-4572	465	10	.	.	PUNCT
ejpam-4572	466	1	journal	journal	PROPN
ejpam-4572	466	2	of	of	ADP
ejpam-4572	466	3	mathematics	mathematics	PROPN
ejpam-4572	466	4	and	and	CCONJ
ejpam-4572	466	5	computer	computer	NOUN
ejpam-4572	466	6	science	science	NOUN
ejpam-4572	466	7	,	,	PUNCT
ejpam-4572	466	8	18:282–293	18:282–293	NUM
ejpam-4572	466	9	,	,	PUNCT
ejpam-4572	466	10	2018	2018	NUM
ejpam-4572	466	11	.	.	PUNCT
ejpam-4572	467	1	[	[	X
ejpam-4572	467	2	4	4	X
ejpam-4572	467	3	]	]	PUNCT
ejpam-4572	467	4	j.	j.	PROPN
ejpam-4572	467	5	cao	cao	PROPN
ejpam-4572	467	6	and	and	CCONJ
ejpam-4572	467	7	j.	j.	PROPN
ejpam-4572	467	8	dontchev	dontchev	PROPN
ejpam-4572	467	9	.	.	PUNCT
ejpam-4572	468	1	on	on	ADP
ejpam-4572	468	2	seme	seme	NOUN
ejpam-4572	468	3	weaker	weak	ADJ
ejpam-4572	468	4	forms	form	NOUN
ejpam-4572	468	5	of	of	ADP
ejpam-4572	468	6	continuity	continuity	NOUN
ejpam-4572	468	7	for	for	ADP
ejpam-4572	468	8	multifunctions	multifunction	NOUN
ejpam-4572	468	9	.	.	PUNCT
ejpam-4572	469	1	real	real	ADJ
ejpam-4572	469	2	analysis	analysis	NOUN
ejpam-4572	469	3	exchange	exchange	NOUN
ejpam-4572	469	4	,	,	PUNCT
ejpam-4572	469	5	22:842–852	22:842–852	PROPN
ejpam-4572	469	6	,	,	PUNCT
ejpam-4572	469	7	1996/97	1996/97	NUM
ejpam-4572	469	8	.	.	PUNCT
ejpam-4572	470	1	[	[	X
ejpam-4572	470	2	5	5	NUM
ejpam-4572	470	3	]	]	PUNCT
ejpam-4572	470	4	m.	m.	NOUN
ejpam-4572	470	5	e.	e.	PROPN
ejpam-4572	470	6	abd	abd	PROPN
ejpam-4572	471	1	el	el	PROPN
ejpam-4572	471	2	-	-	PROPN
ejpam-4572	471	3	monsef	monsef	PROPN
ejpam-4572	471	4	,	,	PUNCT
ejpam-4572	471	5	s.	s.	PROPN
ejpam-4572	471	6	n.	n.	PROPN
ejpam-4572	471	7	el	el	PROPN
ejpam-4572	471	8	-	-	PROPN
ejpam-4572	471	9	deeb	deeb	PROPN
ejpam-4572	471	10	,	,	PUNCT
ejpam-4572	471	11	and	and	CCONJ
ejpam-4572	471	12	r.	r.	PROPN
ejpam-4572	471	13	a.	a.	PROPN
ejpam-4572	471	14	mahmoud	mahmoud	PROPN
ejpam-4572	471	15	.	.	PUNCT
ejpam-4572	472	1	β	β	X
ejpam-4572	472	2	-	-	ADJ
ejpam-4572	472	3	open	open	ADJ
ejpam-4572	472	4	sets	set	NOUN
ejpam-4572	472	5	and	and	CCONJ
ejpam-4572	472	6	βcontinuous	βcontinuous	ADJ
ejpam-4572	472	7	mappings	mapping	NOUN
ejpam-4572	472	8	.	.	PUNCT
ejpam-4572	473	1	bulletin	bulletin	NOUN
ejpam-4572	473	2	of	of	ADP
ejpam-4572	473	3	the	the	DET
ejpam-4572	473	4	faculty	faculty	NOUN
ejpam-4572	473	5	of	of	ADP
ejpam-4572	473	6	science	science	NOUN
ejpam-4572	473	7	.	.	PUNCT
ejpam-4572	474	1	assiut	assiut	PROPN
ejpam-4572	474	2	university	university	PROPN
ejpam-4572	474	3	.	.	PUNCT
ejpam-4572	474	4	,	,	PUNCT
ejpam-4572	474	5	12:77–90	12:77–90	NUM
ejpam-4572	474	6	,	,	PUNCT
ejpam-4572	474	7	1983	1983	NUM
ejpam-4572	474	8	.	.	PUNCT
ejpam-4572	475	1	references	reference	NOUN
ejpam-4572	475	2	478	478	NUM
ejpam-4572	475	3	[	[	X
ejpam-4572	475	4	6	6	NUM
ejpam-4572	475	5	]	]	PUNCT
ejpam-4572	475	6	j.	j.	PROPN
ejpam-4572	475	7	khampakdee	khampakdee	PROPN
ejpam-4572	475	8	and	and	CCONJ
ejpam-4572	475	9	c.	c.	PROPN
ejpam-4572	475	10	boonpok	boonpok	PROPN
ejpam-4572	475	11	.	.	PUNCT
ejpam-4572	476	1	upper	upper	ADJ
ejpam-4572	476	2	and	and	CCONJ
ejpam-4572	476	3	lower	low	ADJ
ejpam-4572	476	4	α(λ	α(λ	PROPN
ejpam-4572	476	5	,	,	PUNCT
ejpam-4572	476	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	476	7	multifunctions	multifunction	NOUN
ejpam-4572	476	8	.	.	PUNCT
ejpam-4572	477	1	wseas	wseas	VERB
ejpam-4572	477	2	transactions	transaction	NOUN
ejpam-4572	477	3	on	on	ADP
ejpam-4572	477	4	mathematics	mathematic	NOUN
ejpam-4572	477	5	,	,	PUNCT
ejpam-4572	477	6	21:684–690	21:684–690	NUM
ejpam-4572	477	7	,	,	PUNCT
ejpam-4572	477	8	2022	2022	NUM
ejpam-4572	477	9	.	.	PUNCT
ejpam-4572	478	1	[	[	X
ejpam-4572	478	2	7	7	X
ejpam-4572	478	3	]	]	PUNCT
ejpam-4572	478	4	t.	t.	NOUN
ejpam-4572	478	5	neubrunn	neubrunn	PROPN
ejpam-4572	478	6	.	.	PUNCT
ejpam-4572	479	1	strongly	strongly	ADV
ejpam-4572	479	2	quasi	quasi	ADJ
ejpam-4572	479	3	-	-	ADJ
ejpam-4572	479	4	continuous	continuous	ADJ
ejpam-4572	479	5	multivalued	multivalued	ADJ
ejpam-4572	479	6	mappings	mapping	NOUN
ejpam-4572	479	7	.	.	PUNCT
ejpam-4572	480	1	general	general	ADJ
ejpam-4572	480	2	topology	topology	NOUN
ejpam-4572	480	3	and	and	CCONJ
ejpam-4572	480	4	its	its	PRON
ejpam-4572	480	5	relations	relation	NOUN
ejpam-4572	480	6	to	to	ADP
ejpam-4572	480	7	modern	modern	ADJ
ejpam-4572	480	8	analysis	analysis	NOUN
ejpam-4572	480	9	and	and	CCONJ
ejpam-4572	480	10	algebra	algebra	NOUN
ejpam-4572	480	11	vi	vi	PROPN
ejpam-4572	480	12	(	(	PUNCT
ejpam-4572	480	13	prague	prague	NOUN
ejpam-4572	480	14	1986	1986	NUM
ejpam-4572	480	15	)	)	PUNCT
ejpam-4572	480	16	heldermann	heldermann	PROPN
ejpam-4572	480	17	,	,	PUNCT
ejpam-4572	480	18	berlin	berlin	PROPN
ejpam-4572	480	19	,	,	PUNCT
ejpam-4572	480	20	pages	page	NOUN
ejpam-4572	480	21	351–359	351–359	NUM
ejpam-4572	480	22	,	,	PUNCT
ejpam-4572	480	23	1988	1988	NUM
ejpam-4572	480	24	.	.	PUNCT
ejpam-4572	481	1	[	[	X
ejpam-4572	481	2	8	8	X
ejpam-4572	481	3	]	]	PUNCT
ejpam-4572	481	4	t.	t.	PROPN
ejpam-4572	481	5	noiri	noiri	PROPN
ejpam-4572	481	6	.	.	PUNCT
ejpam-4572	482	1	on	on	ADP
ejpam-4572	482	2	α	α	NUM
ejpam-4572	482	3	-	-	ADJ
ejpam-4572	482	4	continuous	continuous	ADJ
ejpam-4572	482	5	functions	function	NOUN
ejpam-4572	482	6	.	.	PUNCT
ejpam-4572	483	1	international	international	ADJ
ejpam-4572	483	2	journal	journal	PROPN
ejpam-4572	483	3	of	of	ADP
ejpam-4572	483	4	mathematics	mathematics	PROPN
ejpam-4572	483	5	and	and	CCONJ
ejpam-4572	483	6	mathematical	mathematical	ADJ
ejpam-4572	483	7	sciences	science	NOUN
ejpam-4572	483	8	,	,	PUNCT
ejpam-4572	483	9	10:483–490	10:483–490	NUM
ejpam-4572	483	10	,	,	PUNCT
ejpam-4572	483	11	1987	1987	NUM
ejpam-4572	483	12	.	.	PUNCT
ejpam-4572	484	1	[	[	X
ejpam-4572	484	2	9	9	NUM
ejpam-4572	484	3	]	]	PUNCT
ejpam-4572	484	4	t.	t.	PROPN
ejpam-4572	484	5	noiri	noiri	PROPN
ejpam-4572	484	6	and	and	CCONJ
ejpam-4572	484	7	e.	e.	PROPN
ejpam-4572	484	8	hatir	hatir	PROPN
ejpam-4572	484	9	.	.	PUNCT
ejpam-4572	485	1	λsp	λsp	NOUN
ejpam-4572	485	2	-	-	PUNCT
ejpam-4572	485	3	sets	set	NOUN
ejpam-4572	485	4	and	and	CCONJ
ejpam-4572	485	5	some	some	DET
ejpam-4572	485	6	weak	weak	ADJ
ejpam-4572	485	7	separation	separation	NOUN
ejpam-4572	485	8	axioms	axiom	NOUN
ejpam-4572	485	9	.	.	PUNCT
ejpam-4572	486	1	acta	acta	PROPN
ejpam-4572	486	2	mathematica	mathematica	PROPN
ejpam-4572	486	3	hungarica	hungarica	PROPN
ejpam-4572	486	4	,	,	PUNCT
ejpam-4572	486	5	103(3):225–232	103(3):225–232	NUM
ejpam-4572	486	6	,	,	PUNCT
ejpam-4572	486	7	2004	2004	NUM
ejpam-4572	486	8	.	.	PUNCT
ejpam-4572	487	1	[	[	X
ejpam-4572	487	2	10	10	NUM
ejpam-4572	487	3	]	]	X
ejpam-4572	487	4	v.	v.	PROPN
ejpam-4572	487	5	i.	i.	PROPN
ejpam-4572	487	6	ponomarev	ponomarev	PROPN
ejpam-4572	487	7	.	.	PUNCT
ejpam-4572	488	1	properties	property	NOUN
ejpam-4572	488	2	of	of	ADP
ejpam-4572	488	3	topological	topological	ADJ
ejpam-4572	488	4	spaces	space	NOUN
ejpam-4572	488	5	preserved	preserve	VERB
ejpam-4572	488	6	under	under	ADP
ejpam-4572	488	7	multivalued	multivalued	ADJ
ejpam-4572	488	8	continuous	continuous	ADJ
ejpam-4572	488	9	mappings	mapping	NOUN
ejpam-4572	488	10	on	on	ADP
ejpam-4572	488	11	compacta	compacta	NOUN
ejpam-4572	488	12	.	.	PUNCT
ejpam-4572	489	1	american	american	PROPN
ejpam-4572	489	2	mathematical	mathematical	ADJ
ejpam-4572	489	3	society	society	NOUN
ejpam-4572	489	4	translations	translation	NOUN
ejpam-4572	489	5	,	,	PUNCT
ejpam-4572	489	6	38(2):119–140	38(2):119–140	NUM
ejpam-4572	489	7	,	,	PUNCT
ejpam-4572	489	8	1964	1964	NUM
ejpam-4572	489	9	.	.	PUNCT
ejpam-4572	490	1	[	[	X
ejpam-4572	490	2	11	11	NUM
ejpam-4572	490	3	]	]	PUNCT
ejpam-4572	490	4	v.	v.	CCONJ
ejpam-4572	490	5	popa	popa	NOUN
ejpam-4572	490	6	and	and	CCONJ
ejpam-4572	490	7	t.	t.	PROPN
ejpam-4572	490	8	noiri	noiri	PROPN
ejpam-4572	490	9	.	.	PUNCT
ejpam-4572	491	1	almost	almost	ADV
ejpam-4572	491	2	weakly	weakly	ADJ
ejpam-4572	491	3	continuous	continuous	ADJ
ejpam-4572	491	4	functions	function	NOUN
ejpam-4572	491	5	.	.	PUNCT
ejpam-4572	492	1	demonstratio	demonstratio	PROPN
ejpam-4572	492	2	mathematica	mathematica	PROPN
ejpam-4572	492	3	,	,	PUNCT
ejpam-4572	492	4	25:241–251	25:241–251	PROPN
ejpam-4572	492	5	,	,	PUNCT
ejpam-4572	492	6	1992	1992	NUM
ejpam-4572	492	7	.	.	PUNCT
ejpam-4572	493	1	[	[	X
ejpam-4572	493	2	12	12	NUM
ejpam-4572	493	3	]	]	PUNCT
ejpam-4572	493	4	v.	v.	CCONJ
ejpam-4572	493	5	popa	popa	NOUN
ejpam-4572	493	6	and	and	CCONJ
ejpam-4572	493	7	t.	t.	PROPN
ejpam-4572	493	8	noiri	noiri	PROPN
ejpam-4572	493	9	.	.	PUNCT
ejpam-4572	494	1	on	on	ADP
ejpam-4572	494	2	upper	upper	ADJ
ejpam-4572	494	3	and	and	CCONJ
ejpam-4572	494	4	lower	low	ADJ
ejpam-4572	494	5	α	α	ADJ
ejpam-4572	494	6	-	-	ADJ
ejpam-4572	494	7	continuous	continuous	ADJ
ejpam-4572	494	8	multifunctions	multifunction	NOUN
ejpam-4572	494	9	.	.	PUNCT
ejpam-4572	495	1	mathematica	mathematica	PROPN
ejpam-4572	495	2	slovaca	slovaca	PROPN
ejpam-4572	495	3	,	,	PUNCT
ejpam-4572	495	4	43:477–491	43:477–491	NUM
ejpam-4572	495	5	,	,	PUNCT
ejpam-4572	495	6	1993	1993	NUM
ejpam-4572	495	7	.	.	PUNCT
ejpam-4572	496	1	[	[	X
ejpam-4572	496	2	13	13	NUM
ejpam-4572	496	3	]	]	PUNCT
ejpam-4572	496	4	v.	v.	CCONJ
ejpam-4572	496	5	popa	popa	NOUN
ejpam-4572	496	6	and	and	CCONJ
ejpam-4572	496	7	t.	t.	PROPN
ejpam-4572	496	8	noiri	noiri	PROPN
ejpam-4572	496	9	.	.	PUNCT
ejpam-4572	497	1	on	on	ADP
ejpam-4572	497	2	upper	upper	ADJ
ejpam-4572	497	3	and	and	CCONJ
ejpam-4572	497	4	lower	low	ADJ
ejpam-4572	497	5	almost	almost	ADV
ejpam-4572	497	6	α	α	ADJ
ejpam-4572	497	7	-	-	ADJ
ejpam-4572	497	8	continuous	continuous	ADJ
ejpam-4572	497	9	multifunctions	multifunction	NOUN
ejpam-4572	497	10	.	.	PUNCT
ejpam-4572	498	1	demonstratio	demonstratio	PROPN
ejpam-4572	498	2	mathematica	mathematica	PROPN
ejpam-4572	498	3	,	,	PUNCT
ejpam-4572	498	4	29:381–396	29:381–396	PROPN
ejpam-4572	498	5	,	,	PUNCT
ejpam-4572	498	6	1996	1996	NUM
ejpam-4572	498	7	.	.	PUNCT
ejpam-4572	499	1	[	[	X
ejpam-4572	499	2	14	14	NUM
ejpam-4572	499	3	]	]	X
ejpam-4572	499	4	d.	d.	PROPN
ejpam-4572	499	5	a.	a.	PROPN
ejpam-4572	499	6	rose	rise	VERB
ejpam-4572	499	7	.	.	PUNCT
ejpam-4572	500	1	a	a	DET
ejpam-4572	500	2	note	note	NOUN
ejpam-4572	500	3	on	on	ADP
ejpam-4572	500	4	levine	levine	PROPN
ejpam-4572	500	5	’s	’s	PART
ejpam-4572	500	6	decomposition	decomposition	NOUN
ejpam-4572	500	7	of	of	ADP
ejpam-4572	500	8	continuity	continuity	NOUN
ejpam-4572	500	9	.	.	PUNCT
ejpam-4572	501	1	indian	indian	ADJ
ejpam-4572	501	2	journal	journal	PROPN
ejpam-4572	501	3	of	of	ADP
ejpam-4572	501	4	pure	pure	ADJ
ejpam-4572	501	5	and	and	CCONJ
ejpam-4572	501	6	applied	applied	ADJ
ejpam-4572	501	7	mathematics	mathematic	NOUN
ejpam-4572	501	8	,	,	PUNCT
ejpam-4572	501	9	21:985–987	21:985–987	NUM
ejpam-4572	501	10	,	,	PUNCT
ejpam-4572	501	11	1990	1990	NUM
ejpam-4572	501	12	.	.	PUNCT
ejpam-4572	502	1	[	[	X
ejpam-4572	502	2	15	15	NUM
ejpam-4572	502	3	]	]	PUNCT
ejpam-4572	502	4	a.	a.	NOUN
ejpam-4572	502	5	k.	k.	PROPN
ejpam-4572	502	6	sen	sen	PROPN
ejpam-4572	502	7	and	and	CCONJ
ejpam-4572	502	8	p.	p.	PROPN
ejpam-4572	502	9	bhattacharyya	bhattacharyya	PROPN
ejpam-4572	502	10	.	.	PUNCT
ejpam-4572	503	1	on	on	ADP
ejpam-4572	503	2	weakly	weakly	ADJ
ejpam-4572	503	3	α	α	NOUN
ejpam-4572	503	4	-	-	ADJ
ejpam-4572	503	5	continuous	continuous	ADJ
ejpam-4572	503	6	functions	function	NOUN
ejpam-4572	503	7	.	.	PUNCT
ejpam-4572	504	1	tamkang	tamkang	PROPN
ejpam-4572	504	2	journal	journal	PROPN
ejpam-4572	504	3	of	of	ADP
ejpam-4572	504	4	mathematics	mathematic	NOUN
ejpam-4572	504	5	,	,	PUNCT
ejpam-4572	504	6	24:445–460	24:445–460	NUM
ejpam-4572	504	7	,	,	PUNCT
ejpam-4572	504	8	1993	1993	NUM
ejpam-4572	504	9	.	.	PUNCT
ejpam-4572	505	1	[	[	X
ejpam-4572	505	2	16	16	NUM
ejpam-4572	505	3	]	]	X
ejpam-4572	505	4	c.	c.	PROPN
ejpam-4572	505	5	viriyapong	viriyapong	PROPN
ejpam-4572	505	6	and	and	CCONJ
ejpam-4572	505	7	c.	c.	PROPN
ejpam-4572	505	8	boonpok	boonpok	PROPN
ejpam-4572	505	9	.	.	PUNCT
ejpam-4572	506	1	(	(	PUNCT
ejpam-4572	506	2	λ	λ	X
ejpam-4572	506	3	,	,	PUNCT
ejpam-4572	506	4	sp)-continuous	sp)-continuous	ADJ
ejpam-4572	506	5	functions	function	NOUN
ejpam-4572	506	6	.	.	PUNCT
ejpam-4572	507	1	wseas	wseas	VERB
ejpam-4572	507	2	transactions	transaction	NOUN
ejpam-4572	507	3	on	on	ADP
ejpam-4572	507	4	mathematics	mathematic	NOUN
ejpam-4572	507	5	,	,	PUNCT
ejpam-4572	507	6	21:380–385	21:380–385	NUM
ejpam-4572	507	7	,	,	PUNCT
ejpam-4572	507	8	2022	2022	NUM
ejpam-4572	507	9	.	.	PUNCT
ejpam-4572	508	1	[	[	X
ejpam-4572	508	2	17	17	NUM
ejpam-4572	508	3	]	]	X
ejpam-4572	508	4	c.	c.	PROPN
ejpam-4572	508	5	viriyapong	viriyapong	PROPN
ejpam-4572	508	6	and	and	CCONJ
ejpam-4572	508	7	c.	c.	PROPN
ejpam-4572	508	8	boonpok	boonpok	PROPN
ejpam-4572	508	9	.	.	PUNCT
ejpam-4572	509	1	(	(	PUNCT
ejpam-4572	509	2	τ1	τ1	NOUN
ejpam-4572	509	3	,	,	PUNCT
ejpam-4572	509	4	τ2)α	τ2)α	NOUN
ejpam-4572	509	5	-	-	PUNCT
ejpam-4572	509	6	continuity	continuity	NOUN
ejpam-4572	509	7	for	for	ADP
ejpam-4572	509	8	multifunctions	multifunction	NOUN
ejpam-4572	509	9	.	.	PUNCT
ejpam-4572	510	1	journal	journal	PROPN
ejpam-4572	510	2	of	of	ADP
ejpam-4572	510	3	mathematics	mathematic	NOUN
ejpam-4572	510	4	,	,	PUNCT
ejpam-4572	510	5	2022:6285763	2022:6285763	NUM
ejpam-4572	510	6	,	,	PUNCT
ejpam-4572	510	7	2022	2022	NUM
ejpam-4572	510	8	.	.	PUNCT