id	sid	tid	token	lemma	pos
ejpam-4370	1	1	european	european	PROPN
ejpam-4370	1	2	journal	journal	PROPN
ejpam-4370	1	3	of	of	ADP
ejpam-4370	1	4	pure	pure	ADJ
ejpam-4370	1	5	and	and	CCONJ
ejpam-4370	1	6	applied	apply	VERB
ejpam-4370	1	7	mathematics	mathematic	NOUN
ejpam-4370	1	8	vol	vol	NOUN
ejpam-4370	1	9	.	.	PROPN
ejpam-4370	2	1	15	15	NUM
ejpam-4370	2	2	,	,	PUNCT
ejpam-4370	2	3	no	no	INTJ
ejpam-4370	2	4	.	.	NOUN
ejpam-4370	2	5	4	4	NUM
ejpam-4370	2	6	,	,	PUNCT
ejpam-4370	2	7	2022	2022	NUM
ejpam-4370	2	8	,	,	PUNCT
ejpam-4370	2	9	1694	1694	NUM
ejpam-4370	2	10	-	-	SYM
ejpam-4370	2	11	1704	1704	NUM
ejpam-4370	2	12	issn	issn	PROPN
ejpam-4370	2	13	1307	1307	NUM
ejpam-4370	2	14	-	-	SYM
ejpam-4370	2	15	5543	5543	NUM
ejpam-4370	2	16	–	–	PUNCT
ejpam-4370	2	17	ejpam.com	ejpam.com	X
ejpam-4370	2	18	published	publish	VERB
ejpam-4370	2	19	by	by	ADP
ejpam-4370	2	20	new	new	PROPN
ejpam-4370	2	21	york	york	PROPN
ejpam-4370	2	22	business	business	PROPN
ejpam-4370	3	1	global	global	PROPN
ejpam-4370	3	2	contra-(λ	contra-(λ	PROPN
ejpam-4370	3	3	,	,	PUNCT
ejpam-4370	3	4	sp)-continuity	sp)-continuity	NOUN
ejpam-4370	3	5	and	and	CCONJ
ejpam-4370	3	6	δ(λ	δ(λ	PROPN
ejpam-4370	3	7	,	,	PUNCT
ejpam-4370	3	8	sp)-closed	sp)-close	VERB
ejpam-4370	3	9	sets	set	NOUN
ejpam-4370	3	10	chawalit	chawalit	VERB
ejpam-4370	3	11	boonpok1	boonpok1	PROPN
ejpam-4370	3	12	,	,	PUNCT
ejpam-4370	3	13	chokchai	chokchai	ADJ
ejpam-4370	3	14	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-4370	3	15	1	1	NUM
ejpam-4370	3	16	mathematics	mathematic	NOUN
ejpam-4370	3	17	and	and	CCONJ
ejpam-4370	3	18	applied	apply	VERB
ejpam-4370	3	19	mathematics	mathematics	PROPN
ejpam-4370	3	20	research	research	NOUN
ejpam-4370	3	21	unit	unit	NOUN
ejpam-4370	3	22	,	,	PUNCT
ejpam-4370	3	23	department	department	NOUN
ejpam-4370	3	24	of	of	ADP
ejpam-4370	3	25	mathematics	mathematic	NOUN
ejpam-4370	3	26	,	,	PUNCT
ejpam-4370	3	27	faculty	faculty	NOUN
ejpam-4370	3	28	of	of	ADP
ejpam-4370	3	29	science	science	NOUN
ejpam-4370	3	30	,	,	PUNCT
ejpam-4370	3	31	mahasarakham	mahasarakham	PROPN
ejpam-4370	3	32	university	university	PROPN
ejpam-4370	3	33	,	,	PUNCT
ejpam-4370	3	34	maha	maha	PROPN
ejpam-4370	3	35	sarakham	sarakham	PROPN
ejpam-4370	3	36	,	,	PUNCT
ejpam-4370	3	37	44150	44150	NUM
ejpam-4370	3	38	,	,	PUNCT
ejpam-4370	3	39	thailand	thailand	PROPN
ejpam-4370	3	40	abstract	abstract	PROPN
ejpam-4370	3	41	.	.	PUNCT
ejpam-4370	4	1	this	this	DET
ejpam-4370	4	2	paper	paper	NOUN
ejpam-4370	4	3	deals	deal	NOUN
ejpam-4370	4	4	with	with	ADP
ejpam-4370	4	5	the	the	DET
ejpam-4370	4	6	concepts	concept	NOUN
ejpam-4370	4	7	of	of	ADP
ejpam-4370	4	8	upper	upper	ADJ
ejpam-4370	4	9	and	and	CCONJ
ejpam-4370	4	10	lower	low	ADJ
ejpam-4370	4	11	contra-(λ	contra-(λ	PROPN
ejpam-4370	4	12	,	,	PUNCT
ejpam-4370	4	13	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	4	14	multifunctions	multifunction	NOUN
ejpam-4370	4	15	.	.	PUNCT
ejpam-4370	5	1	moreover	moreover	ADV
ejpam-4370	5	2	,	,	PUNCT
ejpam-4370	5	3	some	some	DET
ejpam-4370	5	4	characterizations	characterization	NOUN
ejpam-4370	5	5	of	of	ADP
ejpam-4370	5	6	upper	upper	ADJ
ejpam-4370	5	7	and	and	CCONJ
ejpam-4370	5	8	lower	low	ADJ
ejpam-4370	5	9	contra-(λ	contra-(λ	PROPN
ejpam-4370	5	10	,	,	PUNCT
ejpam-4370	5	11	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	5	12	multifunctions	multifunction	NOUN
ejpam-4370	5	13	are	be	AUX
ejpam-4370	5	14	investigated	investigate	VERB
ejpam-4370	5	15	.	.	PUNCT
ejpam-4370	6	1	2020	2020	NUM
ejpam-4370	6	2	mathematics	mathematic	NOUN
ejpam-4370	6	3	subject	subject	NOUN
ejpam-4370	6	4	classifications	classification	NOUN
ejpam-4370	6	5	:	:	PUNCT
ejpam-4370	6	6	54c08	54c08	NUM
ejpam-4370	6	7	,	,	PUNCT
ejpam-4370	6	8	54c60	54c60	NUM
ejpam-4370	6	9	key	key	ADJ
ejpam-4370	6	10	words	word	NOUN
ejpam-4370	6	11	and	and	CCONJ
ejpam-4370	6	12	phrases	phrase	NOUN
ejpam-4370	6	13	:	:	PUNCT
ejpam-4370	6	14	δ(λ	δ(λ	PROPN
ejpam-4370	6	15	,	,	PUNCT
ejpam-4370	6	16	sp)-closed	sp)-close	VERB
ejpam-4370	6	17	set	set	ADJ
ejpam-4370	6	18	,	,	PUNCT
ejpam-4370	6	19	upper	upper	ADJ
ejpam-4370	6	20	contra-(λ	contra-(λ	PROPN
ejpam-4370	6	21	,	,	PUNCT
ejpam-4370	6	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	6	23	multifunction	multifunction	NOUN
ejpam-4370	6	24	,	,	PUNCT
ejpam-4370	6	25	lower	low	ADJ
ejpam-4370	6	26	contra-(λ	contra-(λ	PROPN
ejpam-4370	6	27	,	,	PUNCT
ejpam-4370	6	28	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	6	29	multifunction	multifunction	NOUN
ejpam-4370	6	30	1	1	NUM
ejpam-4370	6	31	.	.	PUNCT
ejpam-4370	6	32	introduction	introduction	NOUN
ejpam-4370	6	33	the	the	DET
ejpam-4370	6	34	field	field	NOUN
ejpam-4370	6	35	of	of	ADP
ejpam-4370	6	36	the	the	DET
ejpam-4370	6	37	mathematical	mathematical	ADJ
ejpam-4370	6	38	science	science	NOUN
ejpam-4370	6	39	which	which	PRON
ejpam-4370	6	40	goes	go	VERB
ejpam-4370	6	41	under	under	ADP
ejpam-4370	6	42	the	the	DET
ejpam-4370	6	43	name	name	NOUN
ejpam-4370	6	44	of	of	ADP
ejpam-4370	6	45	topology	topology	NOUN
ejpam-4370	6	46	is	be	AUX
ejpam-4370	6	47	concerned	concern	VERB
ejpam-4370	6	48	with	with	ADP
ejpam-4370	6	49	all	all	DET
ejpam-4370	6	50	questions	question	NOUN
ejpam-4370	6	51	directly	directly	ADV
ejpam-4370	6	52	or	or	CCONJ
ejpam-4370	6	53	indirectly	indirectly	ADV
ejpam-4370	6	54	related	relate	VERB
ejpam-4370	6	55	to	to	ADP
ejpam-4370	6	56	continuity	continuity	NOUN
ejpam-4370	6	57	.	.	PUNCT
ejpam-4370	7	1	the	the	DET
ejpam-4370	7	2	concept	concept	NOUN
ejpam-4370	7	3	of	of	ADP
ejpam-4370	7	4	contracontinuity	contracontinuity	NOUN
ejpam-4370	7	5	was	be	AUX
ejpam-4370	7	6	introduced	introduce	VERB
ejpam-4370	7	7	and	and	CCONJ
ejpam-4370	7	8	studied	study	VERB
ejpam-4370	7	9	by	by	ADP
ejpam-4370	7	10	dontchev	dontchev	NOUN
ejpam-4370	7	11	[	[	X
ejpam-4370	7	12	7	7	NUM
ejpam-4370	7	13	]	]	PUNCT
ejpam-4370	7	14	.	.	PUNCT
ejpam-4370	8	1	in	in	ADP
ejpam-4370	8	2	1999	1999	NUM
ejpam-4370	8	3	,	,	PUNCT
ejpam-4370	8	4	dontchev	dontchev	NOUN
ejpam-4370	8	5	and	and	CCONJ
ejpam-4370	8	6	noiri	noiri	ADV
ejpam-4370	8	7	[	[	X
ejpam-4370	8	8	9	9	X
ejpam-4370	8	9	]	]	PUNCT
ejpam-4370	8	10	considered	consider	VERB
ejpam-4370	8	11	a	a	DET
ejpam-4370	8	12	slightly	slightly	ADV
ejpam-4370	8	13	weaker	weak	ADJ
ejpam-4370	8	14	form	form	NOUN
ejpam-4370	8	15	of	of	ADP
ejpam-4370	8	16	contra	contra	PROPN
ejpam-4370	8	17	-	-	ADJ
ejpam-4370	8	18	continuity	continuity	NOUN
ejpam-4370	8	19	called	call	VERB
ejpam-4370	8	20	contra	contra	PROPN
ejpam-4370	8	21	-	-	NOUN
ejpam-4370	8	22	semicontinuity	semicontinuity	NOUN
ejpam-4370	8	23	and	and	CCONJ
ejpam-4370	8	24	investigated	investigate	VERB
ejpam-4370	8	25	the	the	DET
ejpam-4370	8	26	class	class	NOUN
ejpam-4370	8	27	of	of	ADP
ejpam-4370	8	28	strongly	strongly	ADV
ejpam-4370	8	29	s	s	NOUN
ejpam-4370	8	30	-	-	PUNCT
ejpam-4370	8	31	closed	closed	ADJ
ejpam-4370	8	32	spaces	space	NOUN
ejpam-4370	8	33	.	.	PUNCT
ejpam-4370	9	1	in	in	ADP
ejpam-4370	9	2	2001	2001	NUM
ejpam-4370	9	3	,	,	PUNCT
ejpam-4370	9	4	caldas	caldas	PROPN
ejpam-4370	9	5	and	and	CCONJ
ejpam-4370	9	6	jafari	jafari	PROPN
ejpam-4370	10	1	[	[	X
ejpam-4370	10	2	6	6	NUM
ejpam-4370	10	3	]	]	PUNCT
ejpam-4370	10	4	introduced	introduce	VERB
ejpam-4370	10	5	and	and	CCONJ
ejpam-4370	10	6	investigated	investigate	VERB
ejpam-4370	10	7	the	the	DET
ejpam-4370	10	8	concept	concept	NOUN
ejpam-4370	10	9	of	of	ADP
ejpam-4370	10	10	contra	contra	PROPN
ejpam-4370	10	11	-	-	PUNCT
ejpam-4370	10	12	β	β	ADJ
ejpam-4370	10	13	-	-	ADJ
ejpam-4370	10	14	continuous	continuous	ADJ
ejpam-4370	10	15	functions	function	NOUN
ejpam-4370	10	16	.	.	PUNCT
ejpam-4370	11	1	in	in	ADP
ejpam-4370	11	2	2002	2002	NUM
ejpam-4370	11	3	,	,	PUNCT
ejpam-4370	11	4	jafari	jafari	ADJ
ejpam-4370	11	5	and	and	CCONJ
ejpam-4370	11	6	noiri	noiri	ADV
ejpam-4370	11	7	[	[	X
ejpam-4370	11	8	14	14	NUM
ejpam-4370	11	9	]	]	PUNCT
ejpam-4370	11	10	introduced	introduce	VERB
ejpam-4370	11	11	and	and	CCONJ
ejpam-4370	11	12	studied	study	VERB
ejpam-4370	11	13	a	a	DET
ejpam-4370	11	14	new	new	ADJ
ejpam-4370	11	15	form	form	NOUN
ejpam-4370	11	16	of	of	ADP
ejpam-4370	11	17	functions	function	NOUN
ejpam-4370	11	18	called	call	VERB
ejpam-4370	11	19	contra	contra	ADJ
ejpam-4370	11	20	-	-	ADJ
ejpam-4370	11	21	precontinuous	precontinuous	ADJ
ejpam-4370	11	22	functions	function	NOUN
ejpam-4370	11	23	.	.	PUNCT
ejpam-4370	12	1	in	in	ADP
ejpam-4370	12	2	2004	2004	NUM
ejpam-4370	12	3	,	,	PUNCT
ejpam-4370	12	4	ekici	ekici	NOUN
ejpam-4370	12	5	[	[	X
ejpam-4370	12	6	10	10	NUM
ejpam-4370	12	7	]	]	PUNCT
ejpam-4370	12	8	presented	present	VERB
ejpam-4370	12	9	and	and	CCONJ
ejpam-4370	12	10	studied	study	VERB
ejpam-4370	12	11	almost	almost	ADV
ejpam-4370	12	12	contra	contra	NOUN
ejpam-4370	12	13	-	-	NOUN
ejpam-4370	12	14	precontinuity	precontinuity	NOUN
ejpam-4370	12	15	as	as	ADP
ejpam-4370	12	16	a	a	DET
ejpam-4370	12	17	new	new	ADJ
ejpam-4370	12	18	generalization	generalization	NOUN
ejpam-4370	12	19	of	of	ADP
ejpam-4370	12	20	regular	regular	ADJ
ejpam-4370	12	21	set	set	NOUN
ejpam-4370	12	22	-	-	PUNCT
ejpam-4370	12	23	connectedness	connectedness	NOUN
ejpam-4370	12	24	[	[	X
ejpam-4370	12	25	8	8	NUM
ejpam-4370	12	26	]	]	PUNCT
ejpam-4370	12	27	,	,	PUNCT
ejpam-4370	12	28	contra	contra	PROPN
ejpam-4370	12	29	-	-	NOUN
ejpam-4370	12	30	precontinuity	precontinuity	NOUN
ejpam-4370	12	31	[	[	X
ejpam-4370	12	32	14	14	NUM
ejpam-4370	12	33	]	]	PUNCT
ejpam-4370	12	34	,	,	PUNCT
ejpam-4370	12	35	contra	contra	NOUN
ejpam-4370	12	36	-	-	NOUN
ejpam-4370	12	37	continuity	continuity	NOUN
ejpam-4370	12	38	[	[	X
ejpam-4370	12	39	7	7	NUM
ejpam-4370	12	40	]	]	PUNCT
ejpam-4370	12	41	,	,	PUNCT
ejpam-4370	12	42	almost	almost	ADV
ejpam-4370	12	43	s	s	NOUN
ejpam-4370	12	44	-	-	NOUN
ejpam-4370	12	45	continuity	continuity	NOUN
ejpam-4370	12	46	[	[	X
ejpam-4370	12	47	17	17	NUM
ejpam-4370	12	48	]	]	PUNCT
ejpam-4370	12	49	and	and	CCONJ
ejpam-4370	12	50	perfect	perfect	ADJ
ejpam-4370	12	51	continuity	continuity	NOUN
ejpam-4370	12	52	[	[	X
ejpam-4370	12	53	16	16	NUM
ejpam-4370	12	54	]	]	PUNCT
ejpam-4370	12	55	.	.	PUNCT
ejpam-4370	13	1	in	in	ADP
ejpam-4370	13	2	2005	2005	NUM
ejpam-4370	13	3	,	,	PUNCT
ejpam-4370	13	4	nasef	nasef	PROPN
ejpam-4370	13	5	[	[	X
ejpam-4370	13	6	15	15	NUM
ejpam-4370	13	7	]	]	X
ejpam-4370	13	8	defined	define	VERB
ejpam-4370	13	9	a	a	DET
ejpam-4370	13	10	new	new	ADJ
ejpam-4370	13	11	class	class	NOUN
ejpam-4370	13	12	of	of	ADP
ejpam-4370	13	13	functions	function	NOUN
ejpam-4370	13	14	called	call	VERB
ejpam-4370	13	15	contra	contra	PROPN
ejpam-4370	13	16	-	-	PUNCT
ejpam-4370	13	17	γ	γ	ADJ
ejpam-4370	13	18	-	-	ADJ
ejpam-4370	13	19	continuous	continuous	ADJ
ejpam-4370	13	20	functions	function	NOUN
ejpam-4370	13	21	which	which	PRON
ejpam-4370	13	22	lies	lie	VERB
ejpam-4370	13	23	between	between	ADP
ejpam-4370	13	24	classes	class	NOUN
ejpam-4370	13	25	of	of	ADP
ejpam-4370	13	26	contra	contra	ADJ
ejpam-4370	13	27	-	-	ADJ
ejpam-4370	13	28	semicontinuous	semicontinuous	ADJ
ejpam-4370	13	29	functions	function	NOUN
ejpam-4370	13	30	and	and	CCONJ
ejpam-4370	13	31	contra	contra	PROPN
ejpam-4370	13	32	-	-	PUNCT
ejpam-4370	13	33	β	β	ADJ
ejpam-4370	13	34	-	-	ADJ
ejpam-4370	13	35	continuous	continuous	ADJ
ejpam-4370	13	36	functions	function	NOUN
ejpam-4370	13	37	.	.	PUNCT
ejpam-4370	14	1	the	the	DET
ejpam-4370	14	2	first	first	ADJ
ejpam-4370	14	3	initiation	initiation	NOUN
ejpam-4370	14	4	of	of	ADP
ejpam-4370	14	5	the	the	DET
ejpam-4370	14	6	concept	concept	NOUN
ejpam-4370	14	7	of	of	ADP
ejpam-4370	14	8	contra	contra	ADJ
ejpam-4370	14	9	-	-	ADJ
ejpam-4370	14	10	continuous	continuous	ADJ
ejpam-4370	14	11	multifunctions	multifunction	NOUN
ejpam-4370	14	12	has	have	AUX
ejpam-4370	14	13	been	be	AUX
ejpam-4370	14	14	done	do	VERB
ejpam-4370	14	15	by	by	ADP
ejpam-4370	14	16	ekici	ekici	PROPN
ejpam-4370	14	17	et	et	PROPN
ejpam-4370	14	18	al	al	PROPN
ejpam-4370	14	19	.	.	PUNCT
ejpam-4370	15	1	[	[	X
ejpam-4370	15	2	10	10	NUM
ejpam-4370	15	3	]	]	PUNCT
ejpam-4370	15	4	.	.	PUNCT
ejpam-4370	16	1	in	in	ADP
ejpam-4370	16	2	2009	2009	NUM
ejpam-4370	16	3	,	,	PUNCT
ejpam-4370	16	4	ekici	ekici	NOUN
ejpam-4370	16	5	et	et	PROPN
ejpam-4370	16	6	al	al	PROPN
ejpam-4370	16	7	.	.	PUNCT
ejpam-4370	17	1	[	[	X
ejpam-4370	17	2	11	11	NUM
ejpam-4370	17	3	]	]	PUNCT
ejpam-4370	17	4	introduced	introduce	VERB
ejpam-4370	17	5	and	and	CCONJ
ejpam-4370	17	6	studied	study	VERB
ejpam-4370	17	7	a	a	DET
ejpam-4370	17	8	new	new	ADJ
ejpam-4370	17	9	generalization	generalization	NOUN
ejpam-4370	17	10	of	of	ADP
ejpam-4370	17	11	contra	contra	ADJ
ejpam-4370	17	12	-	-	ADJ
ejpam-4370	17	13	continuous	continuous	ADJ
ejpam-4370	17	14	multifunctions	multifunction	NOUN
ejpam-4370	17	15	called	call	VERB
ejpam-4370	17	16	almost	almost	ADV
ejpam-4370	17	17	contra	contra	ADJ
ejpam-4370	17	18	-	-	ADJ
ejpam-4370	17	19	continuous	continuous	ADJ
ejpam-4370	17	20	multifunctions	multifunction	NOUN
ejpam-4370	17	21	.	.	PUNCT
ejpam-4370	18	1	noiri	noiri	PROPN
ejpam-4370	18	2	and	and	CCONJ
ejpam-4370	18	3	popa	popa	NOUN
ejpam-4370	18	4	[	[	X
ejpam-4370	18	5	19	19	NUM
ejpam-4370	18	6	]	]	PUNCT
ejpam-4370	18	7	introduced	introduce	VERB
ejpam-4370	18	8	and	and	CCONJ
ejpam-4370	18	9	investigated	investigate	VERB
ejpam-4370	18	10	the	the	DET
ejpam-4370	18	11	notion	notion	NOUN
ejpam-4370	18	12	of	of	ADP
ejpam-4370	18	13	weakly	weakly	ADJ
ejpam-4370	18	14	precontinuous	precontinuous	ADJ
ejpam-4370	18	15	multifunctions	multifunction	NOUN
ejpam-4370	18	16	.	.	PUNCT
ejpam-4370	19	1	in	in	ADP
ejpam-4370	19	2	2010	2010	NUM
ejpam-4370	19	3	,	,	PUNCT
ejpam-4370	19	4	ekici	ekici	NOUN
ejpam-4370	19	5	et	et	PROPN
ejpam-4370	19	6	al	al	PROPN
ejpam-4370	19	7	.	.	PUNCT
ejpam-4370	20	1	[	[	X
ejpam-4370	20	2	12	12	NUM
ejpam-4370	20	3	]	]	PUNCT
ejpam-4370	20	4	introduced	introduce	VERB
ejpam-4370	20	5	and	and	CCONJ
ejpam-4370	20	6	studied	study	VERB
ejpam-4370	20	7	two	two	NUM
ejpam-4370	20	8	new	new	ADJ
ejpam-4370	20	9	concepts	concept	NOUN
ejpam-4370	20	10	namely	namely	ADV
ejpam-4370	20	11	contra	contra	ADJ
ejpam-4370	20	12	-	-	ADJ
ejpam-4370	20	13	preconrinuous	preconrinuous	ADJ
ejpam-4370	20	14	multifunctions	multifunction	NOUN
ejpam-4370	20	15	and	and	CCONJ
ejpam-4370	20	16	almost	almost	ADV
ejpam-4370	20	17	contra	contra	ADJ
ejpam-4370	20	18	-	-	ADJ
ejpam-4370	20	19	precontinuous	precontinuous	ADJ
ejpam-4370	20	20	multifunctions	multifunction	NOUN
ejpam-4370	20	21	which	which	PRON
ejpam-4370	20	22	are	be	AUX
ejpam-4370	20	23	containing	contain	VERB
ejpam-4370	20	24	the	the	DET
ejpam-4370	20	25	∗corresponding	∗corresponde	VERB
ejpam-4370	20	26	author	author	NOUN
ejpam-4370	20	27	.	.	PUNCT
ejpam-4370	21	1	doi	doi	NOUN
ejpam-4370	21	2	:	:	PUNCT
ejpam-4370	21	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4370	https://doi.org/10.29020/nybg.ejpam.v15i4.4370	NUM
ejpam-4370	21	4	email	email	NOUN
ejpam-4370	21	5	addresses	address	NOUN
ejpam-4370	21	6	:	:	PUNCT
ejpam-4370	22	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4370	22	2	(	(	PUNCT
ejpam-4370	22	3	c.	c.	PROPN
ejpam-4370	22	4	boonpok	boonpok	PROPN
ejpam-4370	22	5	)	)	PUNCT
ejpam-4370	22	6	,	,	PUNCT
ejpam-4370	22	7	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-4370	22	8	(	(	PUNCT
ejpam-4370	22	9	c.	c.	PROPN
ejpam-4370	22	10	viriyapong	viriyapong	PROPN
ejpam-4370	22	11	)	)	PUNCT
ejpam-4370	22	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4370	22	13	1694	1694	NUM
ejpam-4370	23	1	©	©	ADP
ejpam-4370	23	2	2022	2022	NUM
ejpam-4370	23	3	ejpam	ejpam	VERB
ejpam-4370	23	4	all	all	DET
ejpam-4370	23	5	rights	right	NOUN
ejpam-4370	23	6	reserved	reserve	VERB
ejpam-4370	23	7	.	.	PUNCT
ejpam-4370	24	1	c.	c.	PROPN
ejpam-4370	24	2	boonpok	boonpok	PROPN
ejpam-4370	24	3	,	,	PUNCT
ejpam-4370	24	4	c.	c.	PROPN
ejpam-4370	24	5	viriyapong	viriyapong	PROPN
ejpam-4370	24	6	/	/	SYM
ejpam-4370	24	7	eur	eur	PROPN
ejpam-4370	24	8	.	.	PUNCT
ejpam-4370	25	1	j.	j.	PROPN
ejpam-4370	25	2	pure	pure	PROPN
ejpam-4370	25	3	appl	appl	PROPN
ejpam-4370	25	4	.	.	PROPN
ejpam-4370	25	5	math	math	PROPN
ejpam-4370	25	6	,	,	PUNCT
ejpam-4370	25	7	15	15	NUM
ejpam-4370	25	8	(	(	PUNCT
ejpam-4370	25	9	4	4	NUM
ejpam-4370	25	10	)	)	PUNCT
ejpam-4370	25	11	(	(	PUNCT
ejpam-4370	25	12	2022	2022	NUM
ejpam-4370	25	13	)	)	PUNCT
ejpam-4370	25	14	,	,	PUNCT
ejpam-4370	25	15	1694	1694	NUM
ejpam-4370	25	16	-	-	SYM
ejpam-4370	25	17	1704	1704	NUM
ejpam-4370	25	18	1695	1695	NUM
ejpam-4370	25	19	class	class	NOUN
ejpam-4370	25	20	of	of	ADP
ejpam-4370	25	21	contra	contra	ADJ
ejpam-4370	25	22	-	-	ADJ
ejpam-4370	25	23	continuous	continuous	ADJ
ejpam-4370	25	24	multifunctions	multifunction	NOUN
ejpam-4370	25	25	and	and	CCONJ
ejpam-4370	25	26	contained	contain	VERB
ejpam-4370	25	27	in	in	ADP
ejpam-4370	25	28	the	the	DET
ejpam-4370	25	29	class	class	NOUN
ejpam-4370	25	30	of	of	ADP
ejpam-4370	25	31	weakly	weakly	ADJ
ejpam-4370	25	32	precontinuous	precontinuous	ADJ
ejpam-4370	25	33	multifunctions	multifunction	NOUN
ejpam-4370	25	34	.	.	PUNCT
ejpam-4370	26	1	the	the	DET
ejpam-4370	26	2	concept	concept	NOUN
ejpam-4370	26	3	of	of	ADP
ejpam-4370	26	4	β	β	ADJ
ejpam-4370	26	5	-	-	ADJ
ejpam-4370	26	6	open	open	ADJ
ejpam-4370	26	7	sets	set	NOUN
ejpam-4370	26	8	due	due	ADP
ejpam-4370	26	9	to	to	ADP
ejpam-4370	26	10	abd	abd	PROPN
ejpam-4370	26	11	el	el	PROPN
ejpam-4370	26	12	-	-	PROPN
ejpam-4370	26	13	monsef	monsef	PROPN
ejpam-4370	26	14	et	et	PROPN
ejpam-4370	26	15	al	al	PROPN
ejpam-4370	26	16	.	.	PUNCT
ejpam-4370	27	1	[	[	X
ejpam-4370	27	2	13	13	NUM
ejpam-4370	27	3	]	]	PUNCT
ejpam-4370	27	4	or	or	CCONJ
ejpam-4370	27	5	semi	semi	ADJ
ejpam-4370	27	6	-	-	ADJ
ejpam-4370	27	7	preopen	preopen	ADJ
ejpam-4370	27	8	sets	set	NOUN
ejpam-4370	27	9	in	in	ADP
ejpam-4370	27	10	the	the	DET
ejpam-4370	27	11	sense	sense	NOUN
ejpam-4370	27	12	of	of	ADP
ejpam-4370	27	13	andrijević	andrijević	NOUN
ejpam-4370	27	14	[	[	X
ejpam-4370	27	15	1	1	X
ejpam-4370	27	16	]	]	PUNCT
ejpam-4370	27	17	plays	play	VERB
ejpam-4370	27	18	a	a	DET
ejpam-4370	27	19	significant	significant	ADJ
ejpam-4370	27	20	role	role	NOUN
ejpam-4370	27	21	in	in	ADP
ejpam-4370	27	22	general	general	ADJ
ejpam-4370	27	23	topology	topology	NOUN
ejpam-4370	27	24	.	.	PUNCT
ejpam-4370	28	1	noiri	noiri	PROPN
ejpam-4370	28	2	and	and	CCONJ
ejpam-4370	28	3	hatir	hatir	PROPN
ejpam-4370	29	1	[	[	X
ejpam-4370	29	2	18	18	NUM
ejpam-4370	29	3	]	]	PUNCT
ejpam-4370	29	4	introduced	introduce	VERB
ejpam-4370	29	5	the	the	DET
ejpam-4370	29	6	concept	concept	NOUN
ejpam-4370	29	7	of	of	ADP
ejpam-4370	29	8	λsp	λsp	NOUN
ejpam-4370	29	9	-	-	PUNCT
ejpam-4370	29	10	sets	set	NOUN
ejpam-4370	29	11	in	in	ADP
ejpam-4370	29	12	terms	term	NOUN
ejpam-4370	29	13	of	of	ADP
ejpam-4370	29	14	the	the	DET
ejpam-4370	29	15	concept	concept	NOUN
ejpam-4370	29	16	of	of	ADP
ejpam-4370	29	17	β	β	ADJ
ejpam-4370	29	18	-	-	ADJ
ejpam-4370	29	19	open	open	ADJ
ejpam-4370	29	20	sets	set	NOUN
ejpam-4370	29	21	and	and	CCONJ
ejpam-4370	29	22	investigated	investigate	VERB
ejpam-4370	29	23	the	the	DET
ejpam-4370	29	24	notion	notion	NOUN
ejpam-4370	29	25	of	of	ADP
ejpam-4370	29	26	λsp	λsp	NOUN
ejpam-4370	29	27	-	-	PUNCT
ejpam-4370	29	28	closed	close	VERB
ejpam-4370	29	29	sets	set	NOUN
ejpam-4370	29	30	by	by	ADP
ejpam-4370	29	31	using	use	VERB
ejpam-4370	29	32	λsp	λsp	NOUN
ejpam-4370	29	33	-	-	PUNCT
ejpam-4370	29	34	sets	set	NOUN
ejpam-4370	29	35	.	.	PUNCT
ejpam-4370	30	1	in	in	ADP
ejpam-4370	30	2	[	[	X
ejpam-4370	30	3	3	3	NUM
ejpam-4370	30	4	]	]	PUNCT
ejpam-4370	30	5	,	,	PUNCT
ejpam-4370	30	6	the	the	DET
ejpam-4370	30	7	author	author	NOUN
ejpam-4370	30	8	introduced	introduce	VERB
ejpam-4370	30	9	the	the	DET
ejpam-4370	30	10	concepts	concept	NOUN
ejpam-4370	30	11	of	of	ADP
ejpam-4370	30	12	(	(	PUNCT
ejpam-4370	30	13	λ	λ	PROPN
ejpam-4370	30	14	,	,	PUNCT
ejpam-4370	30	15	sp)-open	sp)-open	ADJ
ejpam-4370	30	16	sets	set	NOUN
ejpam-4370	30	17	and	and	CCONJ
ejpam-4370	30	18	(	(	PUNCT
ejpam-4370	30	19	λ	λ	PROPN
ejpam-4370	30	20	,	,	PUNCT
ejpam-4370	30	21	sp)-closed	sp)-close	VERB
ejpam-4370	30	22	sets	set	NOUN
ejpam-4370	30	23	which	which	PRON
ejpam-4370	30	24	are	be	AUX
ejpam-4370	30	25	defined	define	VERB
ejpam-4370	30	26	by	by	ADP
ejpam-4370	30	27	utilizing	utilize	VERB
ejpam-4370	30	28	the	the	DET
ejpam-4370	30	29	notions	notion	NOUN
ejpam-4370	30	30	of	of	ADP
ejpam-4370	30	31	λsp	λsp	NOUN
ejpam-4370	30	32	-	-	PUNCT
ejpam-4370	30	33	sets	set	NOUN
ejpam-4370	30	34	and	and	CCONJ
ejpam-4370	30	35	β	β	NOUN
ejpam-4370	30	36	-	-	ADJ
ejpam-4370	30	37	closed	closed	ADJ
ejpam-4370	30	38	sets	set	NOUN
ejpam-4370	30	39	.	.	PUNCT
ejpam-4370	31	1	moreover	moreover	ADV
ejpam-4370	31	2	,	,	PUNCT
ejpam-4370	31	3	some	some	DET
ejpam-4370	31	4	characterizations	characterization	NOUN
ejpam-4370	31	5	of	of	ADP
ejpam-4370	31	6	λsp	λsp	NOUN
ejpam-4370	31	7	-	-	PUNCT
ejpam-4370	31	8	extremally	extremally	ADV
ejpam-4370	31	9	disconnected	disconnected	ADJ
ejpam-4370	31	10	spaces	space	NOUN
ejpam-4370	31	11	are	be	AUX
ejpam-4370	31	12	investigated	investigate	VERB
ejpam-4370	31	13	in	in	ADP
ejpam-4370	31	14	[	[	X
ejpam-4370	31	15	3	3	NUM
ejpam-4370	31	16	]	]	PUNCT
ejpam-4370	31	17	.	.	PUNCT
ejpam-4370	32	1	the	the	DET
ejpam-4370	32	2	purpose	purpose	NOUN
ejpam-4370	32	3	of	of	ADP
ejpam-4370	32	4	the	the	DET
ejpam-4370	32	5	present	present	ADJ
ejpam-4370	32	6	paper	paper	NOUN
ejpam-4370	32	7	is	be	AUX
ejpam-4370	32	8	to	to	PART
ejpam-4370	32	9	introduce	introduce	VERB
ejpam-4370	32	10	the	the	DET
ejpam-4370	32	11	notions	notion	NOUN
ejpam-4370	32	12	of	of	ADP
ejpam-4370	32	13	upper	upper	ADJ
ejpam-4370	32	14	and	and	CCONJ
ejpam-4370	32	15	lower	low	ADJ
ejpam-4370	32	16	contra-(λ	contra-(λ	PROPN
ejpam-4370	32	17	,	,	PUNCT
ejpam-4370	32	18	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	32	19	multifunctions	multifunction	NOUN
ejpam-4370	32	20	.	.	PUNCT
ejpam-4370	33	1	in	in	ADP
ejpam-4370	33	2	particular	particular	ADJ
ejpam-4370	33	3	,	,	PUNCT
ejpam-4370	33	4	several	several	ADJ
ejpam-4370	33	5	characterizations	characterization	NOUN
ejpam-4370	33	6	of	of	ADP
ejpam-4370	33	7	upper	upper	ADJ
ejpam-4370	33	8	and	and	CCONJ
ejpam-4370	33	9	lower	low	ADJ
ejpam-4370	33	10	contra-(λ	contra-(λ	PROPN
ejpam-4370	33	11	,	,	PUNCT
ejpam-4370	33	12	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	33	13	multifunctions	multifunction	NOUN
ejpam-4370	33	14	are	be	AUX
ejpam-4370	33	15	discussed	discuss	VERB
ejpam-4370	33	16	.	.	PUNCT
ejpam-4370	34	1	2	2	X
ejpam-4370	34	2	.	.	X
ejpam-4370	34	3	preliminaries	preliminary	NOUN
ejpam-4370	34	4	let	let	VERB
ejpam-4370	34	5	a	a	PRON
ejpam-4370	34	6	be	be	AUX
ejpam-4370	34	7	a	a	DET
ejpam-4370	34	8	subset	subset	NOUN
ejpam-4370	34	9	of	of	ADP
ejpam-4370	34	10	a	a	DET
ejpam-4370	34	11	topological	topological	ADJ
ejpam-4370	34	12	space	space	NOUN
ejpam-4370	34	13	(	(	PUNCT
ejpam-4370	34	14	x	x	X
ejpam-4370	34	15	,	,	PUNCT
ejpam-4370	34	16	τ	τ	PROPN
ejpam-4370	34	17	)	)	PUNCT
ejpam-4370	34	18	.	.	PUNCT
ejpam-4370	35	1	the	the	DET
ejpam-4370	35	2	closure	closure	NOUN
ejpam-4370	35	3	of	of	ADP
ejpam-4370	35	4	a	a	PRON
ejpam-4370	35	5	and	and	CCONJ
ejpam-4370	35	6	the	the	DET
ejpam-4370	35	7	interior	interior	NOUN
ejpam-4370	35	8	of	of	ADP
ejpam-4370	35	9	a	a	PRON
ejpam-4370	35	10	are	be	AUX
ejpam-4370	35	11	denoted	denote	VERB
ejpam-4370	35	12	by	by	ADP
ejpam-4370	35	13	cl(a	cl(a	NOUN
ejpam-4370	35	14	)	)	PUNCT
ejpam-4370	35	15	and	and	CCONJ
ejpam-4370	35	16	int(a	int(a	PROPN
ejpam-4370	35	17	)	)	PUNCT
ejpam-4370	35	18	,	,	PUNCT
ejpam-4370	35	19	respectively	respectively	ADV
ejpam-4370	35	20	.	.	PUNCT
ejpam-4370	36	1	a	a	DET
ejpam-4370	36	2	subset	subset	NOUN
ejpam-4370	36	3	a	a	PRON
ejpam-4370	36	4	of	of	ADP
ejpam-4370	36	5	a	a	DET
ejpam-4370	36	6	topological	topological	ADJ
ejpam-4370	36	7	space	space	NOUN
ejpam-4370	36	8	(	(	PUNCT
ejpam-4370	36	9	x	x	X
ejpam-4370	36	10	,	,	PUNCT
ejpam-4370	36	11	τ	τ	X
ejpam-4370	36	12	)	)	PUNCT
ejpam-4370	36	13	is	be	AUX
ejpam-4370	36	14	said	say	VERB
ejpam-4370	36	15	to	to	PART
ejpam-4370	36	16	be	be	AUX
ejpam-4370	36	17	β	β	X
ejpam-4370	36	18	-	-	ADJ
ejpam-4370	36	19	open	open	ADJ
ejpam-4370	37	1	[	[	X
ejpam-4370	37	2	13	13	NUM
ejpam-4370	37	3	]	]	PUNCT
ejpam-4370	37	4	if	if	SCONJ
ejpam-4370	37	5	a	a	DET
ejpam-4370	37	6	⊆	⊆	NUM
ejpam-4370	37	7	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4370	37	8	)	)	PUNCT
ejpam-4370	37	9	)	)	PUNCT
ejpam-4370	37	10	)	)	PUNCT
ejpam-4370	37	11	.	.	PUNCT
ejpam-4370	38	1	the	the	DET
ejpam-4370	38	2	complement	complement	NOUN
ejpam-4370	38	3	of	of	ADP
ejpam-4370	38	4	a	a	DET
ejpam-4370	38	5	β	β	X
ejpam-4370	38	6	-	-	ADJ
ejpam-4370	38	7	open	open	ADJ
ejpam-4370	38	8	set	set	NOUN
ejpam-4370	38	9	is	be	AUX
ejpam-4370	38	10	called	call	VERB
ejpam-4370	38	11	β	β	NOUN
ejpam-4370	38	12	-	-	VERB
ejpam-4370	38	13	closed	closed	ADJ
ejpam-4370	38	14	.	.	PUNCT
ejpam-4370	39	1	the	the	DET
ejpam-4370	39	2	family	family	NOUN
ejpam-4370	39	3	of	of	ADP
ejpam-4370	39	4	all	all	DET
ejpam-4370	39	5	β	β	ADJ
ejpam-4370	39	6	-	-	ADJ
ejpam-4370	39	7	open	open	ADJ
ejpam-4370	39	8	sets	set	NOUN
ejpam-4370	39	9	of	of	ADP
ejpam-4370	39	10	a	a	DET
ejpam-4370	39	11	topological	topological	ADJ
ejpam-4370	39	12	space	space	NOUN
ejpam-4370	39	13	(	(	PUNCT
ejpam-4370	39	14	x	x	X
ejpam-4370	39	15	,	,	PUNCT
ejpam-4370	39	16	τ	τ	X
ejpam-4370	39	17	)	)	PUNCT
ejpam-4370	39	18	is	be	AUX
ejpam-4370	39	19	denoted	denote	VERB
ejpam-4370	39	20	by	by	ADP
ejpam-4370	39	21	β(x	β(x	PROPN
ejpam-4370	39	22	,	,	PUNCT
ejpam-4370	39	23	τ	τ	PROPN
ejpam-4370	39	24	)	)	PUNCT
ejpam-4370	39	25	.	.	PUNCT
ejpam-4370	40	1	a	a	DET
ejpam-4370	40	2	subset	subset	NOUN
ejpam-4370	40	3	λsp(a	λsp(a	NOUN
ejpam-4370	40	4	)	)	PUNCT
ejpam-4370	41	1	[	[	X
ejpam-4370	41	2	18	18	NUM
ejpam-4370	41	3	]	]	PUNCT
ejpam-4370	41	4	is	be	AUX
ejpam-4370	41	5	defined	define	VERB
ejpam-4370	41	6	as	as	SCONJ
ejpam-4370	41	7	follows	follow	VERB
ejpam-4370	41	8	:	:	PUNCT
ejpam-4370	41	9	λsp(a	λsp(a	NUM
ejpam-4370	41	10	)	)	PUNCT
ejpam-4370	41	11	=	=	PUNCT
ejpam-4370	42	1	∩{u	∩{u	PROPN
ejpam-4370	42	2	|	|	ADV
ejpam-4370	42	3	a	a	DET
ejpam-4370	42	4	⊆	⊆	NUM
ejpam-4370	42	5	u	u	NOUN
ejpam-4370	42	6	,	,	PUNCT
ejpam-4370	42	7	u	u	NOUN
ejpam-4370	42	8	∈	∈	PROPN
ejpam-4370	42	9	β(x	β(x	PROPN
ejpam-4370	42	10	,	,	PUNCT
ejpam-4370	42	11	τ	τ	X
ejpam-4370	42	12	)	)	PUNCT
ejpam-4370	42	13	}	}	PUNCT
ejpam-4370	42	14	.	.	PUNCT
ejpam-4370	43	1	a	a	DET
ejpam-4370	43	2	subset	subset	NOUN
ejpam-4370	43	3	a	a	PRON
ejpam-4370	43	4	of	of	ADP
ejpam-4370	43	5	a	a	DET
ejpam-4370	43	6	topological	topological	ADJ
ejpam-4370	43	7	space	space	NOUN
ejpam-4370	43	8	(	(	PUNCT
ejpam-4370	43	9	x	x	X
ejpam-4370	43	10	,	,	PUNCT
ejpam-4370	43	11	τ	τ	X
ejpam-4370	43	12	)	)	PUNCT
ejpam-4370	43	13	is	be	AUX
ejpam-4370	43	14	called	call	VERB
ejpam-4370	43	15	a	a	DET
ejpam-4370	43	16	λsp	λsp	NOUN
ejpam-4370	43	17	-	-	PUNCT
ejpam-4370	43	18	set	set	VERB
ejpam-4370	43	19	[	[	X
ejpam-4370	43	20	18	18	NUM
ejpam-4370	43	21	]	]	X
ejpam-4370	43	22	if	if	SCONJ
ejpam-4370	43	23	a	a	DET
ejpam-4370	43	24	=	=	NOUN
ejpam-4370	43	25	λsp(a	λsp(a	NOUN
ejpam-4370	43	26	)	)	PUNCT
ejpam-4370	43	27	.	.	PUNCT
ejpam-4370	44	1	a	a	DET
ejpam-4370	44	2	subset	subset	NOUN
ejpam-4370	44	3	a	a	PRON
ejpam-4370	44	4	of	of	ADP
ejpam-4370	44	5	a	a	DET
ejpam-4370	44	6	topological	topological	ADJ
ejpam-4370	44	7	space	space	NOUN
ejpam-4370	44	8	(	(	PUNCT
ejpam-4370	44	9	x	x	X
ejpam-4370	44	10	,	,	PUNCT
ejpam-4370	44	11	τ	τ	X
ejpam-4370	44	12	)	)	PUNCT
ejpam-4370	44	13	is	be	AUX
ejpam-4370	44	14	called	call	VERB
ejpam-4370	44	15	(	(	PUNCT
ejpam-4370	44	16	λ	λ	X
ejpam-4370	44	17	,	,	PUNCT
ejpam-4370	44	18	sp)-closed	sp)-close	VERB
ejpam-4370	44	19	[	[	PUNCT
ejpam-4370	44	20	3	3	X
ejpam-4370	44	21	]	]	X
ejpam-4370	44	22	if	if	SCONJ
ejpam-4370	44	23	a	a	DET
ejpam-4370	44	24	=	=	X
ejpam-4370	44	25	t	t	NOUN
ejpam-4370	44	26	∩c	∩c	NOUN
ejpam-4370	44	27	,	,	PUNCT
ejpam-4370	44	28	where	where	SCONJ
ejpam-4370	44	29	t	t	PROPN
ejpam-4370	44	30	is	be	AUX
ejpam-4370	44	31	a	a	DET
ejpam-4370	44	32	λsp	λsp	NOUN
ejpam-4370	44	33	-	-	PUNCT
ejpam-4370	44	34	set	set	VERB
ejpam-4370	44	35	and	and	CCONJ
ejpam-4370	44	36	c	c	NOUN
ejpam-4370	44	37	is	be	AUX
ejpam-4370	44	38	a	a	DET
ejpam-4370	44	39	β	β	NOUN
ejpam-4370	44	40	-	-	ADJ
ejpam-4370	44	41	closed	closed	ADJ
ejpam-4370	44	42	set	set	NOUN
ejpam-4370	44	43	.	.	PUNCT
ejpam-4370	45	1	the	the	DET
ejpam-4370	45	2	complement	complement	NOUN
ejpam-4370	45	3	of	of	ADP
ejpam-4370	45	4	a	a	DET
ejpam-4370	45	5	(	(	PUNCT
ejpam-4370	45	6	λ	λ	PROPN
ejpam-4370	45	7	,	,	PUNCT
ejpam-4370	45	8	sp)-closed	sp)-close	VERB
ejpam-4370	45	9	set	set	VERB
ejpam-4370	45	10	is	be	AUX
ejpam-4370	45	11	called	call	VERB
ejpam-4370	45	12	(	(	PUNCT
ejpam-4370	45	13	λ	λ	NOUN
ejpam-4370	45	14	,	,	PUNCT
ejpam-4370	45	15	sp)-open	sp)-open	NOUN
ejpam-4370	45	16	.	.	PUNCT
ejpam-4370	46	1	the	the	DET
ejpam-4370	46	2	family	family	NOUN
ejpam-4370	46	3	of	of	ADP
ejpam-4370	46	4	all	all	DET
ejpam-4370	46	5	(	(	PUNCT
ejpam-4370	46	6	λ	λ	NOUN
ejpam-4370	46	7	,	,	PUNCT
ejpam-4370	46	8	sp)-open	sp)-open	ADJ
ejpam-4370	46	9	sets	set	NOUN
ejpam-4370	46	10	in	in	ADP
ejpam-4370	46	11	a	a	DET
ejpam-4370	46	12	topological	topological	ADJ
ejpam-4370	46	13	space	space	NOUN
ejpam-4370	46	14	(	(	PUNCT
ejpam-4370	46	15	x	x	X
ejpam-4370	46	16	,	,	PUNCT
ejpam-4370	46	17	τ	τ	X
ejpam-4370	46	18	)	)	PUNCT
ejpam-4370	46	19	is	be	AUX
ejpam-4370	46	20	denoted	denote	VERB
ejpam-4370	46	21	by	by	ADP
ejpam-4370	46	22	λspo(x	λspo(x	PROPN
ejpam-4370	46	23	,	,	PUNCT
ejpam-4370	46	24	τ	τ	PROPN
ejpam-4370	46	25	)	)	PUNCT
ejpam-4370	46	26	.	.	PUNCT
ejpam-4370	47	1	let	let	VERB
ejpam-4370	47	2	a	a	DET
ejpam-4370	47	3	be	be	AUX
ejpam-4370	47	4	a	a	DET
ejpam-4370	47	5	subset	subset	NOUN
ejpam-4370	47	6	of	of	ADP
ejpam-4370	47	7	a	a	DET
ejpam-4370	47	8	topological	topological	ADJ
ejpam-4370	47	9	space	space	NOUN
ejpam-4370	47	10	(	(	PUNCT
ejpam-4370	47	11	x	x	X
ejpam-4370	47	12	,	,	PUNCT
ejpam-4370	47	13	τ	τ	PROPN
ejpam-4370	47	14	)	)	PUNCT
ejpam-4370	47	15	.	.	PUNCT
ejpam-4370	48	1	a	a	DET
ejpam-4370	48	2	point	point	NOUN
ejpam-4370	48	3	x	x	X
ejpam-4370	48	4	∈	∈	NOUN
ejpam-4370	48	5	x	x	PUNCT
ejpam-4370	48	6	is	be	AUX
ejpam-4370	48	7	called	call	VERB
ejpam-4370	48	8	a	a	DET
ejpam-4370	48	9	(	(	PUNCT
ejpam-4370	48	10	λ	λ	NOUN
ejpam-4370	48	11	,	,	PUNCT
ejpam-4370	48	12	sp)-cluster	sp)-cluster	NOUN
ejpam-4370	48	13	point	point	NOUN
ejpam-4370	48	14	[	[	X
ejpam-4370	48	15	3	3	X
ejpam-4370	48	16	]	]	PUNCT
ejpam-4370	48	17	of	of	ADP
ejpam-4370	48	18	a	a	PRON
ejpam-4370	48	19	if	if	SCONJ
ejpam-4370	48	20	a	a	DET
ejpam-4370	48	21	∩	∩	ADJ
ejpam-4370	48	22	u	u	ADJ
ejpam-4370	48	23	̸=	̸=	PROPN
ejpam-4370	48	24	∅	∅	NOUN
ejpam-4370	48	25	for	for	ADP
ejpam-4370	48	26	every	every	DET
ejpam-4370	48	27	(	(	PUNCT
ejpam-4370	48	28	λ	λ	NOUN
ejpam-4370	48	29	,	,	PUNCT
ejpam-4370	48	30	sp)-open	sp)-open	NOUN
ejpam-4370	48	31	set	set	VERB
ejpam-4370	48	32	u	u	NOUN
ejpam-4370	48	33	of	of	ADP
ejpam-4370	48	34	x	x	SYM
ejpam-4370	48	35	containing	contain	VERB
ejpam-4370	48	36	x.	x.	NOUN
ejpam-4370	48	37	the	the	DET
ejpam-4370	48	38	set	set	NOUN
ejpam-4370	48	39	of	of	ADP
ejpam-4370	48	40	all	all	DET
ejpam-4370	48	41	(	(	PUNCT
ejpam-4370	48	42	λ	λ	PROPN
ejpam-4370	48	43	,	,	PUNCT
ejpam-4370	48	44	sp)-cluster	sp)-cluster	NOUN
ejpam-4370	48	45	points	point	NOUN
ejpam-4370	48	46	of	of	ADP
ejpam-4370	48	47	a	a	PRON
ejpam-4370	48	48	is	be	AUX
ejpam-4370	48	49	called	call	VERB
ejpam-4370	48	50	the	the	DET
ejpam-4370	48	51	(	(	PUNCT
ejpam-4370	48	52	λ	λ	PROPN
ejpam-4370	48	53	,	,	PUNCT
ejpam-4370	48	54	sp)-closure	sp)-closure	NOUN
ejpam-4370	48	55	[	[	X
ejpam-4370	48	56	3	3	NUM
ejpam-4370	48	57	]	]	PUNCT
ejpam-4370	48	58	of	of	ADP
ejpam-4370	48	59	a	a	PRON
ejpam-4370	48	60	and	and	CCONJ
ejpam-4370	48	61	is	be	AUX
ejpam-4370	48	62	denoted	denote	VERB
ejpam-4370	48	63	by	by	ADP
ejpam-4370	48	64	a(λ	a(λ	ADV
ejpam-4370	48	65	,	,	PUNCT
ejpam-4370	48	66	sp	sp	NOUN
ejpam-4370	48	67	)	)	PUNCT
ejpam-4370	48	68	.	.	PUNCT
ejpam-4370	49	1	the	the	DET
ejpam-4370	49	2	union	union	NOUN
ejpam-4370	49	3	of	of	ADP
ejpam-4370	49	4	all	all	DET
ejpam-4370	49	5	(	(	PUNCT
ejpam-4370	49	6	λ	λ	NOUN
ejpam-4370	49	7	,	,	PUNCT
ejpam-4370	49	8	sp)-open	sp)-open	ADJ
ejpam-4370	49	9	sets	set	NOUN
ejpam-4370	49	10	contained	contain	VERB
ejpam-4370	49	11	in	in	ADP
ejpam-4370	49	12	a	a	PRON
ejpam-4370	49	13	is	be	AUX
ejpam-4370	49	14	called	call	VERB
ejpam-4370	49	15	the	the	DET
ejpam-4370	49	16	(	(	PUNCT
ejpam-4370	49	17	λ	λ	PROPN
ejpam-4370	49	18	,	,	PUNCT
ejpam-4370	49	19	sp)-interior	sp)-interior	NOUN
ejpam-4370	49	20	[	[	X
ejpam-4370	49	21	3	3	NUM
ejpam-4370	49	22	]	]	PUNCT
ejpam-4370	49	23	of	of	ADP
ejpam-4370	49	24	a	a	PRON
ejpam-4370	49	25	and	and	CCONJ
ejpam-4370	49	26	is	be	AUX
ejpam-4370	49	27	denoted	denote	VERB
ejpam-4370	49	28	by	by	ADP
ejpam-4370	49	29	a(λ	a(λ	ADV
ejpam-4370	49	30	,	,	PUNCT
ejpam-4370	49	31	sp	sp	NOUN
ejpam-4370	49	32	)	)	PUNCT
ejpam-4370	49	33	.	.	PUNCT
ejpam-4370	50	1	lemma	lemma	PROPN
ejpam-4370	50	2	1	1	NUM
ejpam-4370	50	3	.	.	PUNCT
ejpam-4370	51	1	[	[	X
ejpam-4370	51	2	3	3	X
ejpam-4370	51	3	]	]	PUNCT
ejpam-4370	51	4	let	let	VERB
ejpam-4370	51	5	a	a	PRON
ejpam-4370	51	6	and	and	CCONJ
ejpam-4370	51	7	b	b	NOUN
ejpam-4370	51	8	be	be	AUX
ejpam-4370	51	9	subsets	subset	NOUN
ejpam-4370	51	10	of	of	ADP
ejpam-4370	51	11	a	a	DET
ejpam-4370	51	12	topological	topological	ADJ
ejpam-4370	51	13	space	space	NOUN
ejpam-4370	51	14	(	(	PUNCT
ejpam-4370	51	15	x	x	X
ejpam-4370	51	16	,	,	PUNCT
ejpam-4370	51	17	τ	τ	PROPN
ejpam-4370	51	18	)	)	PUNCT
ejpam-4370	51	19	.	.	PUNCT
ejpam-4370	52	1	for	for	ADP
ejpam-4370	52	2	the	the	DET
ejpam-4370	52	3	(	(	PUNCT
ejpam-4370	52	4	λ	λ	PROPN
ejpam-4370	52	5	,	,	PUNCT
ejpam-4370	52	6	sp)-closure	sp)-closure	NOUN
ejpam-4370	52	7	,	,	PUNCT
ejpam-4370	52	8	the	the	DET
ejpam-4370	52	9	following	follow	VERB
ejpam-4370	52	10	properties	property	NOUN
ejpam-4370	52	11	hold	hold	VERB
ejpam-4370	52	12	:	:	PUNCT
ejpam-4370	52	13	(	(	PUNCT
ejpam-4370	52	14	1	1	X
ejpam-4370	52	15	)	)	PUNCT
ejpam-4370	52	16	a	a	DET
ejpam-4370	52	17	⊆	⊆	NUM
ejpam-4370	52	18	a(λ	a(λ	ADJ
ejpam-4370	52	19	,	,	PUNCT
ejpam-4370	52	20	sp	sp	NOUN
ejpam-4370	52	21	)	)	PUNCT
ejpam-4370	52	22	and	and	CCONJ
ejpam-4370	52	23	[	[	X
ejpam-4370	52	24	a(λ	a(λ	ADV
ejpam-4370	52	25	,	,	PUNCT
ejpam-4370	52	26	sp)](λ	sp)](λ	PROPN
ejpam-4370	52	27	,	,	PUNCT
ejpam-4370	52	28	sp	sp	NOUN
ejpam-4370	52	29	)	)	PUNCT
ejpam-4370	52	30	=	=	PUNCT
ejpam-4370	52	31	a(λ	a(λ	ADV
ejpam-4370	52	32	,	,	PUNCT
ejpam-4370	52	33	sp	sp	NOUN
ejpam-4370	52	34	)	)	PUNCT
ejpam-4370	52	35	.	.	PUNCT
ejpam-4370	53	1	(	(	PUNCT
ejpam-4370	53	2	2	2	X
ejpam-4370	53	3	)	)	PUNCT
ejpam-4370	53	4	if	if	SCONJ
ejpam-4370	53	5	a	a	DET
ejpam-4370	53	6	⊆	⊆	NUM
ejpam-4370	53	7	b	b	NOUN
ejpam-4370	53	8	,	,	PUNCT
ejpam-4370	53	9	then	then	ADV
ejpam-4370	53	10	a(λ	a(λ	ADV
ejpam-4370	53	11	,	,	PUNCT
ejpam-4370	53	12	sp	sp	NOUN
ejpam-4370	53	13	)	)	PUNCT
ejpam-4370	53	14	⊆	⊆	NUM
ejpam-4370	53	15	b(λ	b(λ	NOUN
ejpam-4370	53	16	,	,	PUNCT
ejpam-4370	53	17	sp	sp	NOUN
ejpam-4370	53	18	)	)	PUNCT
ejpam-4370	53	19	.	.	PUNCT
ejpam-4370	54	1	(	(	PUNCT
ejpam-4370	54	2	3	3	X
ejpam-4370	54	3	)	)	PUNCT
ejpam-4370	54	4	a(λ	a(λ	ADV
ejpam-4370	54	5	,	,	PUNCT
ejpam-4370	54	6	sp	sp	NOUN
ejpam-4370	54	7	)	)	PUNCT
ejpam-4370	54	8	=	=	SYM
ejpam-4370	54	9	∩{f	∩{f	NOUN
ejpam-4370	54	10	|a	|a	VERB
ejpam-4370	54	11	⊆	⊆	NUM
ejpam-4370	54	12	f	f	PROPN
ejpam-4370	54	13	and	and	CCONJ
ejpam-4370	54	14	f	f	PROPN
ejpam-4370	54	15	is	be	AUX
ejpam-4370	54	16	(	(	PUNCT
ejpam-4370	54	17	λ	λ	X
ejpam-4370	54	18	,	,	PUNCT
ejpam-4370	54	19	sp)-closed	sp)-close	VERB
ejpam-4370	54	20	}	}	PUNCT
ejpam-4370	54	21	.	.	PUNCT
ejpam-4370	55	1	(	(	PUNCT
ejpam-4370	55	2	4	4	NUM
ejpam-4370	55	3	)	)	PUNCT
ejpam-4370	55	4	a(λ	a(λ	ADV
ejpam-4370	55	5	,	,	PUNCT
ejpam-4370	55	6	sp	sp	NOUN
ejpam-4370	55	7	)	)	PUNCT
ejpam-4370	55	8	is	be	AUX
ejpam-4370	55	9	(	(	PUNCT
ejpam-4370	55	10	λ	λ	X
ejpam-4370	55	11	,	,	PUNCT
ejpam-4370	55	12	sp)-closed	sp)-close	VERB
ejpam-4370	55	13	.	.	PUNCT
ejpam-4370	56	1	(	(	PUNCT
ejpam-4370	56	2	5	5	X
ejpam-4370	56	3	)	)	PUNCT
ejpam-4370	56	4	a	a	PRON
ejpam-4370	56	5	is	be	AUX
ejpam-4370	56	6	(	(	PUNCT
ejpam-4370	56	7	λ	λ	X
ejpam-4370	56	8	,	,	PUNCT
ejpam-4370	56	9	sp)-closed	sp)-close	VERB
ejpam-4370	56	10	if	if	SCONJ
ejpam-4370	56	11	and	and	CCONJ
ejpam-4370	56	12	only	only	ADV
ejpam-4370	56	13	if	if	SCONJ
ejpam-4370	56	14	a	a	DET
ejpam-4370	56	15	=	=	X
ejpam-4370	56	16	a(λ	a(λ	ADV
ejpam-4370	56	17	,	,	PUNCT
ejpam-4370	56	18	sp	sp	NOUN
ejpam-4370	56	19	)	)	PUNCT
ejpam-4370	56	20	.	.	PUNCT
ejpam-4370	57	1	lemma	lemma	PROPN
ejpam-4370	57	2	2	2	NUM
ejpam-4370	57	3	.	.	PUNCT
ejpam-4370	58	1	[	[	X
ejpam-4370	58	2	3	3	X
ejpam-4370	58	3	]	]	PUNCT
ejpam-4370	58	4	let	let	VERB
ejpam-4370	58	5	a	a	PRON
ejpam-4370	58	6	and	and	CCONJ
ejpam-4370	58	7	b	b	NOUN
ejpam-4370	58	8	be	be	AUX
ejpam-4370	58	9	subsets	subset	NOUN
ejpam-4370	58	10	of	of	ADP
ejpam-4370	58	11	a	a	DET
ejpam-4370	58	12	topological	topological	ADJ
ejpam-4370	58	13	space	space	NOUN
ejpam-4370	58	14	(	(	PUNCT
ejpam-4370	58	15	x	x	X
ejpam-4370	58	16	,	,	PUNCT
ejpam-4370	58	17	τ	τ	PROPN
ejpam-4370	58	18	)	)	PUNCT
ejpam-4370	58	19	.	.	PUNCT
ejpam-4370	59	1	for	for	ADP
ejpam-4370	59	2	the	the	DET
ejpam-4370	59	3	(	(	PUNCT
ejpam-4370	59	4	λ	λ	PROPN
ejpam-4370	59	5	,	,	PUNCT
ejpam-4370	59	6	sp)interior	sp)interior	PROPN
ejpam-4370	59	7	,	,	PUNCT
ejpam-4370	59	8	the	the	DET
ejpam-4370	59	9	following	follow	VERB
ejpam-4370	59	10	properties	property	NOUN
ejpam-4370	59	11	hold	hold	VERB
ejpam-4370	59	12	:	:	PUNCT
ejpam-4370	59	13	(	(	PUNCT
ejpam-4370	59	14	1	1	X
ejpam-4370	59	15	)	)	PUNCT
ejpam-4370	59	16	a(λ	a(λ	ADV
ejpam-4370	59	17	,	,	PUNCT
ejpam-4370	59	18	sp	sp	NOUN
ejpam-4370	59	19	)	)	PUNCT
ejpam-4370	59	20	⊆	⊆	NUM
ejpam-4370	59	21	a	a	DET
ejpam-4370	59	22	and	and	CCONJ
ejpam-4370	59	23	[	[	X
ejpam-4370	59	24	a(λ	a(λ	ADV
ejpam-4370	59	25	,	,	PUNCT
ejpam-4370	59	26	sp)](λ	sp)](λ	PROPN
ejpam-4370	59	27	,	,	PUNCT
ejpam-4370	59	28	sp	sp	NOUN
ejpam-4370	59	29	)	)	PUNCT
ejpam-4370	59	30	=	=	PUNCT
ejpam-4370	59	31	a(λ	a(λ	ADV
ejpam-4370	59	32	,	,	PUNCT
ejpam-4370	59	33	sp	sp	NOUN
ejpam-4370	59	34	)	)	PUNCT
ejpam-4370	59	35	.	.	PUNCT
ejpam-4370	60	1	c.	c.	PROPN
ejpam-4370	60	2	boonpok	boonpok	PROPN
ejpam-4370	60	3	,	,	PUNCT
ejpam-4370	60	4	c.	c.	PROPN
ejpam-4370	60	5	viriyapong	viriyapong	PROPN
ejpam-4370	60	6	/	/	SYM
ejpam-4370	60	7	eur	eur	PROPN
ejpam-4370	60	8	.	.	PUNCT
ejpam-4370	61	1	j.	j.	PROPN
ejpam-4370	61	2	pure	pure	PROPN
ejpam-4370	61	3	appl	appl	PROPN
ejpam-4370	61	4	.	.	PROPN
ejpam-4370	61	5	math	math	PROPN
ejpam-4370	61	6	,	,	PUNCT
ejpam-4370	61	7	15	15	NUM
ejpam-4370	61	8	(	(	PUNCT
ejpam-4370	61	9	4	4	NUM
ejpam-4370	61	10	)	)	PUNCT
ejpam-4370	61	11	(	(	PUNCT
ejpam-4370	61	12	2022	2022	NUM
ejpam-4370	61	13	)	)	PUNCT
ejpam-4370	61	14	,	,	PUNCT
ejpam-4370	61	15	1694	1694	NUM
ejpam-4370	61	16	-	-	SYM
ejpam-4370	61	17	1704	1704	NUM
ejpam-4370	61	18	1696	1696	NUM
ejpam-4370	61	19	(	(	PUNCT
ejpam-4370	61	20	2	2	NUM
ejpam-4370	61	21	)	)	PUNCT
ejpam-4370	61	22	if	if	SCONJ
ejpam-4370	61	23	a	a	DET
ejpam-4370	61	24	⊆	⊆	NUM
ejpam-4370	61	25	b	b	NOUN
ejpam-4370	61	26	,	,	PUNCT
ejpam-4370	61	27	then	then	ADV
ejpam-4370	61	28	a(λ	a(λ	ADV
ejpam-4370	61	29	,	,	PUNCT
ejpam-4370	61	30	sp	sp	NOUN
ejpam-4370	61	31	)	)	PUNCT
ejpam-4370	61	32	⊆	⊆	NUM
ejpam-4370	61	33	b(λ	b(λ	NOUN
ejpam-4370	61	34	,	,	PUNCT
ejpam-4370	61	35	sp	sp	NOUN
ejpam-4370	61	36	)	)	PUNCT
ejpam-4370	61	37	.	.	PUNCT
ejpam-4370	62	1	(	(	PUNCT
ejpam-4370	62	2	3	3	X
ejpam-4370	62	3	)	)	PUNCT
ejpam-4370	62	4	a(λ	a(λ	ADV
ejpam-4370	62	5	,	,	PUNCT
ejpam-4370	62	6	sp	sp	NOUN
ejpam-4370	62	7	)	)	PUNCT
ejpam-4370	62	8	is	be	AUX
ejpam-4370	62	9	(	(	PUNCT
ejpam-4370	62	10	λ	λ	INTJ
ejpam-4370	62	11	,	,	PUNCT
ejpam-4370	62	12	sp)-open	sp)-open	NOUN
ejpam-4370	62	13	.	.	PUNCT
ejpam-4370	63	1	(	(	PUNCT
ejpam-4370	63	2	4	4	X
ejpam-4370	63	3	)	)	PUNCT
ejpam-4370	63	4	a	a	DET
ejpam-4370	63	5	is	be	AUX
ejpam-4370	63	6	(	(	PUNCT
ejpam-4370	63	7	λ	λ	NOUN
ejpam-4370	63	8	,	,	PUNCT
ejpam-4370	63	9	sp)-open	sp)-open	ADJ
ejpam-4370	63	10	if	if	SCONJ
ejpam-4370	63	11	and	and	CCONJ
ejpam-4370	63	12	only	only	ADV
ejpam-4370	63	13	if	if	SCONJ
ejpam-4370	63	14	a(λ	a(λ	ADV
ejpam-4370	63	15	,	,	PUNCT
ejpam-4370	63	16	sp	sp	NOUN
ejpam-4370	63	17	)	)	PUNCT
ejpam-4370	63	18	=	=	SYM
ejpam-4370	63	19	a.	a.	NOUN
ejpam-4370	63	20	(	(	PUNCT
ejpam-4370	63	21	5	5	NUM
ejpam-4370	63	22	)	)	PUNCT
ejpam-4370	64	1	[	[	X
ejpam-4370	64	2	x	x	X
ejpam-4370	64	3	−a](λ	−a](λ	PROPN
ejpam-4370	64	4	,	,	PUNCT
ejpam-4370	64	5	sp	sp	NOUN
ejpam-4370	64	6	)	)	PUNCT
ejpam-4370	64	7	=	=	SYM
ejpam-4370	64	8	x	x	SYM
ejpam-4370	64	9	−a(λ	−a(λ	NOUN
ejpam-4370	64	10	,	,	PUNCT
ejpam-4370	64	11	sp	sp	NOUN
ejpam-4370	64	12	)	)	PUNCT
ejpam-4370	64	13	.	.	PUNCT
ejpam-4370	65	1	(	(	PUNCT
ejpam-4370	65	2	6	6	NUM
ejpam-4370	65	3	)	)	PUNCT
ejpam-4370	66	1	[	[	X
ejpam-4370	66	2	x	x	X
ejpam-4370	66	3	−a](λ	−a](λ	PROPN
ejpam-4370	66	4	,	,	PUNCT
ejpam-4370	66	5	sp	sp	NOUN
ejpam-4370	66	6	)	)	PUNCT
ejpam-4370	66	7	=	=	SYM
ejpam-4370	66	8	x	x	SYM
ejpam-4370	66	9	−a(λ	−a(λ	NOUN
ejpam-4370	66	10	,	,	PUNCT
ejpam-4370	66	11	sp	sp	NOUN
ejpam-4370	66	12	)	)	PUNCT
ejpam-4370	66	13	.	.	PUNCT
ejpam-4370	67	1	a	a	DET
ejpam-4370	67	2	subset	subset	NOUN
ejpam-4370	67	3	a	a	PRON
ejpam-4370	67	4	of	of	ADP
ejpam-4370	67	5	a	a	DET
ejpam-4370	67	6	topological	topological	ADJ
ejpam-4370	67	7	space	space	NOUN
ejpam-4370	67	8	(	(	PUNCT
ejpam-4370	67	9	x	x	X
ejpam-4370	67	10	,	,	PUNCT
ejpam-4370	67	11	τ	τ	X
ejpam-4370	67	12	)	)	PUNCT
ejpam-4370	67	13	is	be	AUX
ejpam-4370	67	14	called	call	VERB
ejpam-4370	67	15	r(λ	r(λ	NOUN
ejpam-4370	67	16	,	,	PUNCT
ejpam-4370	67	17	sp)-open	sp)-open	ADJ
ejpam-4370	67	18	[	[	X
ejpam-4370	67	19	3	3	X
ejpam-4370	67	20	]	]	X
ejpam-4370	67	21	if	if	SCONJ
ejpam-4370	67	22	a	a	PRON
ejpam-4370	67	23	=	=	X
ejpam-4370	68	1	[	[	X
ejpam-4370	68	2	a(λ	a(λ	PROPN
ejpam-4370	68	3	,	,	PUNCT
ejpam-4370	68	4	sp)](λ	sp)](λ	PROPN
ejpam-4370	68	5	,	,	PUNCT
ejpam-4370	68	6	sp	sp	NOUN
ejpam-4370	68	7	)	)	PUNCT
ejpam-4370	68	8	.	.	PUNCT
ejpam-4370	69	1	the	the	DET
ejpam-4370	69	2	complement	complement	NOUN
ejpam-4370	69	3	of	of	ADP
ejpam-4370	69	4	a	a	DET
ejpam-4370	69	5	r(λ	r(λ	NOUN
ejpam-4370	69	6	,	,	PUNCT
ejpam-4370	69	7	sp)-open	sp)-open	ADJ
ejpam-4370	69	8	set	set	NOUN
ejpam-4370	69	9	is	be	AUX
ejpam-4370	69	10	said	say	VERB
ejpam-4370	69	11	to	to	PART
ejpam-4370	69	12	be	be	AUX
ejpam-4370	69	13	r(λ	r(λ	NOUN
ejpam-4370	69	14	,	,	PUNCT
ejpam-4370	69	15	sp)-closed	sp)-close	VERB
ejpam-4370	69	16	.	.	PUNCT
ejpam-4370	70	1	the	the	DET
ejpam-4370	70	2	family	family	NOUN
ejpam-4370	70	3	of	of	ADP
ejpam-4370	70	4	all	all	DET
ejpam-4370	70	5	r(λ	r(λ	NOUN
ejpam-4370	70	6	,	,	PUNCT
ejpam-4370	70	7	sp)-open	sp)-open	ADJ
ejpam-4370	70	8	sets	set	NOUN
ejpam-4370	70	9	in	in	ADP
ejpam-4370	70	10	a	a	DET
ejpam-4370	70	11	topological	topological	ADJ
ejpam-4370	70	12	space	space	NOUN
ejpam-4370	70	13	(	(	PUNCT
ejpam-4370	70	14	x	x	X
ejpam-4370	70	15	,	,	PUNCT
ejpam-4370	70	16	τ	τ	X
ejpam-4370	70	17	)	)	PUNCT
ejpam-4370	70	18	is	be	AUX
ejpam-4370	70	19	denoted	denote	VERB
ejpam-4370	70	20	by	by	ADP
ejpam-4370	70	21	rλspo(x	rλspo(x	PROPN
ejpam-4370	70	22	,	,	PUNCT
ejpam-4370	70	23	τ	τ	PROPN
ejpam-4370	70	24	)	)	PUNCT
ejpam-4370	70	25	.	.	PUNCT
ejpam-4370	71	1	throughout	throughout	ADP
ejpam-4370	71	2	this	this	DET
ejpam-4370	71	3	paper	paper	NOUN
ejpam-4370	71	4	,	,	PUNCT
ejpam-4370	71	5	(	(	PUNCT
ejpam-4370	71	6	x	x	X
ejpam-4370	71	7	,	,	PUNCT
ejpam-4370	71	8	τ	τ	X
ejpam-4370	71	9	)	)	PUNCT
ejpam-4370	71	10	and	and	CCONJ
ejpam-4370	71	11	(	(	PUNCT
ejpam-4370	71	12	y	y	PROPN
ejpam-4370	71	13	,	,	PUNCT
ejpam-4370	71	14	σ	σ	PROPN
ejpam-4370	71	15	)	)	PUNCT
ejpam-4370	71	16	(	(	PUNCT
ejpam-4370	71	17	or	or	CCONJ
ejpam-4370	71	18	simply	simply	ADV
ejpam-4370	71	19	x	x	X
ejpam-4370	71	20	and	and	CCONJ
ejpam-4370	71	21	y	y	PROPN
ejpam-4370	71	22	)	)	PUNCT
ejpam-4370	71	23	always	always	ADV
ejpam-4370	71	24	mean	mean	VERB
ejpam-4370	71	25	topological	topological	ADJ
ejpam-4370	71	26	spaces	space	NOUN
ejpam-4370	71	27	and	and	CCONJ
ejpam-4370	71	28	f	f	NOUN
ejpam-4370	71	29	:	:	PUNCT
ejpam-4370	71	30	x	x	X
ejpam-4370	71	31	→	→	SYM
ejpam-4370	71	32	y	y	PROPN
ejpam-4370	71	33	(	(	PUNCT
ejpam-4370	71	34	resp	resp	PROPN
ejpam-4370	71	35	.	.	PUNCT
ejpam-4370	72	1	f	f	X
ejpam-4370	72	2	:	:	PUNCT
ejpam-4370	72	3	x	x	X
ejpam-4370	72	4	→	→	SYM
ejpam-4370	72	5	y	y	PROPN
ejpam-4370	72	6	)	)	PUNCT
ejpam-4370	72	7	presents	present	VERB
ejpam-4370	72	8	a	a	DET
ejpam-4370	72	9	multivalued	multivalue	VERB
ejpam-4370	72	10	(	(	PUNCT
ejpam-4370	72	11	resp	resp	NOUN
ejpam-4370	72	12	.	.	PUNCT
ejpam-4370	73	1	single	single	ADJ
ejpam-4370	73	2	valued	value	VERB
ejpam-4370	73	3	)	)	PUNCT
ejpam-4370	73	4	function	function	NOUN
ejpam-4370	73	5	.	.	PUNCT
ejpam-4370	74	1	by	by	ADP
ejpam-4370	74	2	a	a	DET
ejpam-4370	74	3	multifunction	multifunction	NOUN
ejpam-4370	74	4	f	f	NOUN
ejpam-4370	74	5	:	:	PUNCT
ejpam-4370	74	6	x	x	X
ejpam-4370	74	7	→	→	SYM
ejpam-4370	74	8	y	y	PROPN
ejpam-4370	74	9	,	,	PUNCT
ejpam-4370	74	10	we	we	PRON
ejpam-4370	74	11	mean	mean	VERB
ejpam-4370	74	12	a	a	DET
ejpam-4370	74	13	point	point	NOUN
ejpam-4370	74	14	-	-	PUNCT
ejpam-4370	74	15	to	to	ADP
ejpam-4370	74	16	-	-	PUNCT
ejpam-4370	74	17	set	set	VERB
ejpam-4370	74	18	correspondence	correspondence	NOUN
ejpam-4370	74	19	from	from	ADP
ejpam-4370	74	20	x	x	PUNCT
ejpam-4370	74	21	into	into	ADP
ejpam-4370	74	22	y	y	PROPN
ejpam-4370	74	23	,	,	PUNCT
ejpam-4370	74	24	and	and	CCONJ
ejpam-4370	74	25	always	always	ADV
ejpam-4370	74	26	assume	assume	VERB
ejpam-4370	74	27	that	that	SCONJ
ejpam-4370	75	1	f	f	PROPN
ejpam-4370	75	2	(	(	PUNCT
ejpam-4370	75	3	x	x	X
ejpam-4370	75	4	)	)	PUNCT
ejpam-4370	75	5	̸=	̸=	NOUN
ejpam-4370	75	6	∅	∅	NOUN
ejpam-4370	75	7	for	for	ADP
ejpam-4370	75	8	all	all	PRON
ejpam-4370	75	9	x	x	SYM
ejpam-4370	75	10	∈	∈	ADJ
ejpam-4370	75	11	x.	x.	NOUN
ejpam-4370	75	12	for	for	ADP
ejpam-4370	75	13	a	a	DET
ejpam-4370	75	14	multifunction	multifunction	NOUN
ejpam-4370	75	15	f	f	NOUN
ejpam-4370	75	16	:	:	PUNCT
ejpam-4370	75	17	x	x	X
ejpam-4370	75	18	→	→	SYM
ejpam-4370	75	19	y	y	PROPN
ejpam-4370	75	20	,	,	PUNCT
ejpam-4370	75	21	following	follow	VERB
ejpam-4370	75	22	[	[	X
ejpam-4370	75	23	2	2	X
ejpam-4370	75	24	]	]	PUNCT
ejpam-4370	75	25	we	we	PRON
ejpam-4370	75	26	shall	shall	AUX
ejpam-4370	75	27	denote	denote	VERB
ejpam-4370	75	28	the	the	DET
ejpam-4370	75	29	upper	upper	ADJ
ejpam-4370	75	30	and	and	CCONJ
ejpam-4370	75	31	lower	low	ADJ
ejpam-4370	75	32	inverse	inverse	NOUN
ejpam-4370	75	33	of	of	ADP
ejpam-4370	75	34	a	a	DET
ejpam-4370	75	35	set	set	NOUN
ejpam-4370	75	36	b	b	PROPN
ejpam-4370	75	37	of	of	ADP
ejpam-4370	75	38	y	y	PROPN
ejpam-4370	75	39	by	by	ADP
ejpam-4370	75	40	f+(b	f+(b	NOUN
ejpam-4370	75	41	)	)	PUNCT
ejpam-4370	75	42	and	and	CCONJ
ejpam-4370	75	43	f−(b	f−(b	NOUN
ejpam-4370	75	44	)	)	PUNCT
ejpam-4370	75	45	,	,	PUNCT
ejpam-4370	75	46	respectively	respectively	ADV
ejpam-4370	75	47	,	,	PUNCT
ejpam-4370	75	48	that	that	ADV
ejpam-4370	75	49	is	is	ADV
ejpam-4370	75	50	,	,	PUNCT
ejpam-4370	75	51	f+(b	f+(b	NOUN
ejpam-4370	75	52	)	)	PUNCT
ejpam-4370	75	53	=	=	PRON
ejpam-4370	76	1	{	{	PUNCT
ejpam-4370	76	2	x	x	PUNCT
ejpam-4370	76	3	∈	∈	PROPN
ejpam-4370	76	4	x	x	INTJ
ejpam-4370	77	1	|	|	NOUN
ejpam-4370	77	2	f	f	X
ejpam-4370	77	3	(	(	PUNCT
ejpam-4370	77	4	x	x	NOUN
ejpam-4370	77	5	)	)	PUNCT
ejpam-4370	77	6	⊆	⊆	NUM
ejpam-4370	77	7	b	b	NOUN
ejpam-4370	77	8	}	}	PUNCT
ejpam-4370	77	9	and	and	CCONJ
ejpam-4370	77	10	f−(b	f−(b	PROPN
ejpam-4370	77	11	)	)	PUNCT
ejpam-4370	77	12	=	=	PRON
ejpam-4370	78	1	{	{	PUNCT
ejpam-4370	78	2	x	x	PUNCT
ejpam-4370	78	3	∈	∈	PROPN
ejpam-4370	78	4	x	x	INTJ
ejpam-4370	79	1	|	|	NOUN
ejpam-4370	79	2	f	f	X
ejpam-4370	79	3	(	(	PUNCT
ejpam-4370	79	4	x	x	NOUN
ejpam-4370	79	5	)	)	PUNCT
ejpam-4370	79	6	∩b	∩b	NOUN
ejpam-4370	79	7	̸=	̸=	PROPN
ejpam-4370	79	8	∅	∅	NOUN
ejpam-4370	79	9	}	}	PUNCT
ejpam-4370	79	10	.	.	PUNCT
ejpam-4370	80	1	in	in	ADP
ejpam-4370	80	2	particular	particular	ADJ
ejpam-4370	80	3	,	,	PUNCT
ejpam-4370	80	4	f−(y	f−(y	NOUN
ejpam-4370	80	5	)	)	PUNCT
ejpam-4370	80	6	=	=	SYM
ejpam-4370	81	1	{	{	PUNCT
ejpam-4370	81	2	x	x	PUNCT
ejpam-4370	81	3	∈	∈	PROPN
ejpam-4370	81	4	x	x	INTJ
ejpam-4370	82	1	|	|	ADV
ejpam-4370	82	2	y	y	PROPN
ejpam-4370	82	3	∈	∈	PROPN
ejpam-4370	82	4	f	f	X
ejpam-4370	82	5	(	(	PUNCT
ejpam-4370	82	6	x	x	NOUN
ejpam-4370	82	7	)	)	PUNCT
ejpam-4370	82	8	}	}	PUNCT
ejpam-4370	82	9	for	for	ADP
ejpam-4370	82	10	each	each	DET
ejpam-4370	82	11	point	point	NOUN
ejpam-4370	82	12	y	y	PROPN
ejpam-4370	82	13	∈	∈	PROPN
ejpam-4370	82	14	y	y	PROPN
ejpam-4370	82	15	.	.	PUNCT
ejpam-4370	83	1	for	for	ADP
ejpam-4370	83	2	each	each	PRON
ejpam-4370	83	3	a	a	DET
ejpam-4370	83	4	⊆	⊆	NUM
ejpam-4370	83	5	x	x	SYM
ejpam-4370	83	6	,	,	PUNCT
ejpam-4370	83	7	f	f	PROPN
ejpam-4370	83	8	(	(	PUNCT
ejpam-4370	83	9	a	a	NOUN
ejpam-4370	83	10	)	)	PUNCT
ejpam-4370	83	11	=	=	SYM
ejpam-4370	83	12	∪x∈af	∪x∈af	NOUN
ejpam-4370	83	13	(	(	PUNCT
ejpam-4370	83	14	x	x	NOUN
ejpam-4370	83	15	)	)	PUNCT
ejpam-4370	83	16	.	.	PUNCT
ejpam-4370	84	1	then	then	ADV
ejpam-4370	84	2	,	,	PUNCT
ejpam-4370	84	3	f	f	PROPN
ejpam-4370	84	4	is	be	AUX
ejpam-4370	84	5	said	say	VERB
ejpam-4370	84	6	to	to	PART
ejpam-4370	84	7	be	be	AUX
ejpam-4370	84	8	a	a	DET
ejpam-4370	84	9	surjection	surjection	NOUN
ejpam-4370	84	10	if	if	SCONJ
ejpam-4370	84	11	f	f	PROPN
ejpam-4370	84	12	(	(	PUNCT
ejpam-4370	84	13	x	x	X
ejpam-4370	84	14	)	)	PUNCT
ejpam-4370	84	15	=	=	SYM
ejpam-4370	84	16	y	y	PROPN
ejpam-4370	84	17	,	,	PUNCT
ejpam-4370	84	18	or	or	CCONJ
ejpam-4370	84	19	equivalently	equivalently	ADV
ejpam-4370	84	20	,	,	PUNCT
ejpam-4370	84	21	if	if	SCONJ
ejpam-4370	84	22	for	for	ADP
ejpam-4370	84	23	each	each	DET
ejpam-4370	84	24	y	y	PROPN
ejpam-4370	84	25	∈	∈	PROPN
ejpam-4370	84	26	y	y	PROPN
ejpam-4370	84	27	,	,	PUNCT
ejpam-4370	84	28	there	there	PRON
ejpam-4370	84	29	exists	exist	VERB
ejpam-4370	84	30	an	an	DET
ejpam-4370	84	31	x	x	SYM
ejpam-4370	84	32	∈	∈	PROPN
ejpam-4370	84	33	x	x	X
ejpam-4370	84	34	such	such	ADJ
ejpam-4370	84	35	that	that	SCONJ
ejpam-4370	84	36	y	y	PROPN
ejpam-4370	84	37	∈	∈	PROPN
ejpam-4370	84	38	f	f	X
ejpam-4370	84	39	(	(	PUNCT
ejpam-4370	84	40	x	x	NOUN
ejpam-4370	84	41	)	)	PUNCT
ejpam-4370	84	42	.	.	PUNCT
ejpam-4370	85	1	moreover	moreover	ADV
ejpam-4370	85	2	,	,	PUNCT
ejpam-4370	85	3	f	f	X
ejpam-4370	85	4	:	:	PUNCT
ejpam-4370	85	5	x	x	X
ejpam-4370	85	6	→	→	SYM
ejpam-4370	85	7	y	y	PROPN
ejpam-4370	85	8	is	be	AUX
ejpam-4370	85	9	called	call	VERB
ejpam-4370	85	10	upper	upper	ADJ
ejpam-4370	85	11	semi	semi	ADJ
ejpam-4370	85	12	-	-	ADJ
ejpam-4370	85	13	continuous	continuous	ADJ
ejpam-4370	85	14	(	(	PUNCT
ejpam-4370	85	15	resp	resp	NOUN
ejpam-4370	85	16	.	.	PUNCT
ejpam-4370	86	1	lower	low	ADJ
ejpam-4370	86	2	semi	semi	ADJ
ejpam-4370	86	3	-	-	ADJ
ejpam-4370	86	4	continuous	continuous	ADJ
ejpam-4370	86	5	)	)	PUNCT
ejpam-4370	86	6	if	if	SCONJ
ejpam-4370	86	7	f+(v	f+(v	PROPN
ejpam-4370	86	8	)	)	PUNCT
ejpam-4370	86	9	(	(	PUNCT
ejpam-4370	86	10	resp	resp	NOUN
ejpam-4370	86	11	.	.	PUNCT
ejpam-4370	87	1	f−(v	f−(v	NOUN
ejpam-4370	87	2	)	)	PUNCT
ejpam-4370	87	3	)	)	PUNCT
ejpam-4370	88	1	is	be	AUX
ejpam-4370	88	2	open	open	ADJ
ejpam-4370	88	3	in	in	ADP
ejpam-4370	88	4	x	x	PUNCT
ejpam-4370	88	5	for	for	ADP
ejpam-4370	88	6	every	every	DET
ejpam-4370	88	7	open	open	ADJ
ejpam-4370	88	8	set	set	VERB
ejpam-4370	88	9	v	v	NOUN
ejpam-4370	88	10	of	of	ADP
ejpam-4370	88	11	y	y	PROPN
ejpam-4370	89	1	[	[	X
ejpam-4370	89	2	20	20	NUM
ejpam-4370	89	3	]	]	PUNCT
ejpam-4370	89	4	.	.	PUNCT
ejpam-4370	90	1	3	3	X
ejpam-4370	90	2	.	.	X
ejpam-4370	90	3	upper	upper	ADJ
ejpam-4370	90	4	and	and	CCONJ
ejpam-4370	90	5	lower	low	ADJ
ejpam-4370	90	6	contra-(λ	contra-(λ	PROPN
ejpam-4370	90	7	,	,	PUNCT
ejpam-4370	90	8	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	90	9	multifunctions	multifunction	NOUN
ejpam-4370	90	10	in	in	ADP
ejpam-4370	90	11	this	this	DET
ejpam-4370	90	12	section	section	NOUN
ejpam-4370	90	13	,	,	PUNCT
ejpam-4370	90	14	we	we	PRON
ejpam-4370	90	15	introduce	introduce	VERB
ejpam-4370	90	16	the	the	DET
ejpam-4370	90	17	notions	notion	NOUN
ejpam-4370	90	18	of	of	ADP
ejpam-4370	90	19	upper	upper	ADJ
ejpam-4370	90	20	and	and	CCONJ
ejpam-4370	90	21	lower	low	ADJ
ejpam-4370	90	22	contra-(λ	contra-(λ	PROPN
ejpam-4370	90	23	,	,	PUNCT
ejpam-4370	90	24	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	90	25	multifunctions	multifunction	NOUN
ejpam-4370	90	26	.	.	PUNCT
ejpam-4370	91	1	moreover	moreover	ADV
ejpam-4370	91	2	,	,	PUNCT
ejpam-4370	91	3	several	several	ADJ
ejpam-4370	91	4	characterizations	characterization	NOUN
ejpam-4370	91	5	of	of	ADP
ejpam-4370	91	6	upper	upper	ADJ
ejpam-4370	91	7	and	and	CCONJ
ejpam-4370	91	8	lower	low	ADJ
ejpam-4370	91	9	contra-(λ	contra-(λ	PROPN
ejpam-4370	91	10	,	,	PUNCT
ejpam-4370	91	11	sp)continuous	sp)continuous	ADJ
ejpam-4370	91	12	multifunctions	multifunction	NOUN
ejpam-4370	91	13	are	be	AUX
ejpam-4370	91	14	discussed	discuss	VERB
ejpam-4370	91	15	.	.	PUNCT
ejpam-4370	92	1	definition	definition	NOUN
ejpam-4370	92	2	1	1	NUM
ejpam-4370	92	3	.	.	PUNCT
ejpam-4370	93	1	a	a	DET
ejpam-4370	93	2	multifunction	multifunction	NOUN
ejpam-4370	93	3	f	f	NOUN
ejpam-4370	93	4	:	:	PUNCT
ejpam-4370	93	5	(	(	PUNCT
ejpam-4370	93	6	x	x	X
ejpam-4370	93	7	,	,	PUNCT
ejpam-4370	93	8	τ	τ	X
ejpam-4370	93	9	)	)	PUNCT
ejpam-4370	93	10	→	→	SYM
ejpam-4370	93	11	(	(	PUNCT
ejpam-4370	93	12	y	y	PROPN
ejpam-4370	93	13	,	,	PUNCT
ejpam-4370	93	14	σ	σ	PROPN
ejpam-4370	93	15	)	)	PUNCT
ejpam-4370	93	16	is	be	AUX
ejpam-4370	93	17	said	say	VERB
ejpam-4370	93	18	to	to	PART
ejpam-4370	93	19	be	be	AUX
ejpam-4370	93	20	:	:	PUNCT
ejpam-4370	93	21	(	(	PUNCT
ejpam-4370	93	22	1	1	X
ejpam-4370	93	23	)	)	PUNCT
ejpam-4370	93	24	upper	upper	ADJ
ejpam-4370	93	25	contra-(λ	contra-(λ	PROPN
ejpam-4370	93	26	,	,	PUNCT
ejpam-4370	93	27	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	93	28	at	at	ADP
ejpam-4370	93	29	x	x	X
ejpam-4370	93	30	∈	∈	PROPN
ejpam-4370	93	31	x	x	INTJ
ejpam-4370	93	32	if	if	SCONJ
ejpam-4370	93	33	,	,	PUNCT
ejpam-4370	93	34	for	for	ADP
ejpam-4370	93	35	each	each	PRON
ejpam-4370	93	36	(	(	PUNCT
ejpam-4370	93	37	λ	λ	PROPN
ejpam-4370	93	38	,	,	PUNCT
ejpam-4370	93	39	sp)-closed	sp)-close	VERB
ejpam-4370	93	40	set	set	VERB
ejpam-4370	93	41	k	k	PROPN
ejpam-4370	93	42	of	of	ADP
ejpam-4370	93	43	y	y	PROPN
ejpam-4370	93	44	such	such	ADJ
ejpam-4370	93	45	that	that	SCONJ
ejpam-4370	93	46	x	x	SYM
ejpam-4370	93	47	∈	∈	PROPN
ejpam-4370	93	48	f+(k	f+(k	NOUN
ejpam-4370	93	49	)	)	PUNCT
ejpam-4370	93	50	,	,	PUNCT
ejpam-4370	93	51	there	there	PRON
ejpam-4370	93	52	exists	exist	VERB
ejpam-4370	93	53	a	a	DET
ejpam-4370	93	54	(	(	PUNCT
ejpam-4370	93	55	λ	λ	NOUN
ejpam-4370	93	56	,	,	PUNCT
ejpam-4370	93	57	sp)-open	sp)-open	NOUN
ejpam-4370	93	58	set	set	VERB
ejpam-4370	93	59	u	u	NOUN
ejpam-4370	93	60	of	of	ADP
ejpam-4370	93	61	x	x	PUNCT
ejpam-4370	93	62	containing	contain	VERB
ejpam-4370	93	63	x	x	PUNCT
ejpam-4370	93	64	such	such	ADJ
ejpam-4370	93	65	that	that	SCONJ
ejpam-4370	93	66	u	u	PROPN
ejpam-4370	93	67	⊆	⊆	NUM
ejpam-4370	93	68	f+(k	f+(k	NOUN
ejpam-4370	93	69	)	)	PUNCT
ejpam-4370	93	70	;	;	PUNCT
ejpam-4370	93	71	(	(	PUNCT
ejpam-4370	93	72	2	2	X
ejpam-4370	93	73	)	)	PUNCT
ejpam-4370	93	74	lower	low	ADJ
ejpam-4370	93	75	contra-(λ	contra-(λ	PROPN
ejpam-4370	93	76	,	,	PUNCT
ejpam-4370	93	77	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	93	78	at	at	ADP
ejpam-4370	93	79	x	x	X
ejpam-4370	93	80	∈	∈	PROPN
ejpam-4370	93	81	x	x	INTJ
ejpam-4370	93	82	if	if	SCONJ
ejpam-4370	93	83	,	,	PUNCT
ejpam-4370	93	84	for	for	ADP
ejpam-4370	93	85	each	each	PRON
ejpam-4370	93	86	(	(	PUNCT
ejpam-4370	93	87	λ	λ	PROPN
ejpam-4370	93	88	,	,	PUNCT
ejpam-4370	93	89	sp)-closed	sp)-close	VERB
ejpam-4370	93	90	set	set	VERB
ejpam-4370	93	91	k	k	PROPN
ejpam-4370	93	92	of	of	ADP
ejpam-4370	93	93	y	y	PROPN
ejpam-4370	93	94	such	such	ADJ
ejpam-4370	93	95	that	that	SCONJ
ejpam-4370	93	96	x	x	SYM
ejpam-4370	93	97	∈	∈	PROPN
ejpam-4370	93	98	f−(k	f−(k	PROPN
ejpam-4370	93	99	)	)	PUNCT
ejpam-4370	93	100	,	,	PUNCT
ejpam-4370	93	101	there	there	PRON
ejpam-4370	93	102	exists	exist	VERB
ejpam-4370	93	103	a	a	DET
ejpam-4370	93	104	(	(	PUNCT
ejpam-4370	93	105	λ	λ	NOUN
ejpam-4370	93	106	,	,	PUNCT
ejpam-4370	93	107	sp)-open	sp)-open	NOUN
ejpam-4370	93	108	set	set	VERB
ejpam-4370	93	109	u	u	NOUN
ejpam-4370	93	110	of	of	ADP
ejpam-4370	93	111	x	x	PUNCT
ejpam-4370	93	112	containing	contain	VERB
ejpam-4370	93	113	x	x	PUNCT
ejpam-4370	93	114	such	such	ADJ
ejpam-4370	93	115	that	that	SCONJ
ejpam-4370	93	116	u	u	PROPN
ejpam-4370	93	117	⊆	⊆	NUM
ejpam-4370	93	118	f−(k	f−(k	PROPN
ejpam-4370	93	119	)	)	PUNCT
ejpam-4370	93	120	;	;	PUNCT
ejpam-4370	93	121	(	(	PUNCT
ejpam-4370	93	122	3	3	X
ejpam-4370	93	123	)	)	PUNCT
ejpam-4370	93	124	upper	upper	ADJ
ejpam-4370	93	125	(	(	PUNCT
ejpam-4370	93	126	resp	resp	NOUN
ejpam-4370	93	127	.	.	PUNCT
ejpam-4370	94	1	lower	low	ADJ
ejpam-4370	94	2	)	)	PUNCT
ejpam-4370	95	1	contra-(λ	contra-(λ	PROPN
ejpam-4370	95	2	,	,	PUNCT
ejpam-4370	95	3	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	95	4	if	if	SCONJ
ejpam-4370	95	5	f	f	PROPN
ejpam-4370	95	6	has	have	VERB
ejpam-4370	95	7	this	this	DET
ejpam-4370	95	8	property	property	NOUN
ejpam-4370	95	9	at	at	ADP
ejpam-4370	95	10	each	each	DET
ejpam-4370	95	11	point	point	NOUN
ejpam-4370	95	12	of	of	ADP
ejpam-4370	95	13	x.	x.	PROPN
ejpam-4370	95	14	c.	c.	PROPN
ejpam-4370	95	15	boonpok	boonpok	PROPN
ejpam-4370	95	16	,	,	PUNCT
ejpam-4370	95	17	c.	c.	PROPN
ejpam-4370	95	18	viriyapong	viriyapong	PROPN
ejpam-4370	95	19	/	/	SYM
ejpam-4370	95	20	eur	eur	PROPN
ejpam-4370	95	21	.	.	PUNCT
ejpam-4370	96	1	j.	j.	PROPN
ejpam-4370	96	2	pure	pure	PROPN
ejpam-4370	96	3	appl	appl	PROPN
ejpam-4370	96	4	.	.	PROPN
ejpam-4370	96	5	math	math	PROPN
ejpam-4370	96	6	,	,	PUNCT
ejpam-4370	96	7	15	15	NUM
ejpam-4370	96	8	(	(	PUNCT
ejpam-4370	96	9	4	4	NUM
ejpam-4370	96	10	)	)	PUNCT
ejpam-4370	96	11	(	(	PUNCT
ejpam-4370	96	12	2022	2022	NUM
ejpam-4370	96	13	)	)	PUNCT
ejpam-4370	96	14	,	,	PUNCT
ejpam-4370	96	15	1694	1694	NUM
ejpam-4370	96	16	-	-	SYM
ejpam-4370	96	17	1704	1704	NUM
ejpam-4370	96	18	1697	1697	NUM
ejpam-4370	96	19	theorem	theorem	NOUN
ejpam-4370	96	20	1	1	NUM
ejpam-4370	96	21	.	.	X
ejpam-4370	96	22	for	for	ADP
ejpam-4370	96	23	a	a	DET
ejpam-4370	96	24	multifunction	multifunction	NOUN
ejpam-4370	96	25	f	f	NOUN
ejpam-4370	96	26	:	:	PUNCT
ejpam-4370	96	27	(	(	PUNCT
ejpam-4370	96	28	x	x	X
ejpam-4370	96	29	,	,	PUNCT
ejpam-4370	96	30	τ	τ	X
ejpam-4370	96	31	)	)	PUNCT
ejpam-4370	96	32	→	→	SYM
ejpam-4370	96	33	(	(	PUNCT
ejpam-4370	96	34	y	y	PROPN
ejpam-4370	96	35	,	,	PUNCT
ejpam-4370	96	36	σ	σ	PROPN
ejpam-4370	96	37	)	)	PUNCT
ejpam-4370	96	38	,	,	PUNCT
ejpam-4370	96	39	the	the	DET
ejpam-4370	96	40	following	follow	VERB
ejpam-4370	96	41	properties	property	NOUN
ejpam-4370	96	42	are	be	AUX
ejpam-4370	96	43	equivalent	equivalent	ADJ
ejpam-4370	96	44	:	:	PUNCT
ejpam-4370	96	45	(	(	PUNCT
ejpam-4370	96	46	1	1	X
ejpam-4370	96	47	)	)	PUNCT
ejpam-4370	96	48	f	f	PROPN
ejpam-4370	96	49	is	be	AUX
ejpam-4370	96	50	upper	upper	ADJ
ejpam-4370	96	51	contra-(λ	contra-(λ	PROPN
ejpam-4370	96	52	,	,	PUNCT
ejpam-4370	96	53	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	96	54	;	;	PUNCT
ejpam-4370	96	55	(	(	PUNCT
ejpam-4370	96	56	2	2	X
ejpam-4370	96	57	)	)	PUNCT
ejpam-4370	96	58	f+(k	f+(k	NOUN
ejpam-4370	96	59	)	)	PUNCT
ejpam-4370	97	1	is	be	AUX
ejpam-4370	97	2	(	(	PUNCT
ejpam-4370	97	3	λ	λ	INTJ
ejpam-4370	97	4	,	,	PUNCT
ejpam-4370	97	5	sp)-open	sp)-open	ADJ
ejpam-4370	97	6	in	in	ADP
ejpam-4370	97	7	x	x	PUNCT
ejpam-4370	97	8	for	for	SCONJ
ejpam-4370	97	9	every	every	DET
ejpam-4370	97	10	(	(	PUNCT
ejpam-4370	97	11	λ	λ	PROPN
ejpam-4370	97	12	,	,	PUNCT
ejpam-4370	97	13	sp)-closed	sp)-close	VERB
ejpam-4370	97	14	set	set	VERB
ejpam-4370	97	15	k	k	PROPN
ejpam-4370	97	16	of	of	ADP
ejpam-4370	97	17	y	y	PROPN
ejpam-4370	97	18	;	;	PUNCT
ejpam-4370	97	19	(	(	PUNCT
ejpam-4370	97	20	3	3	X
ejpam-4370	97	21	)	)	PUNCT
ejpam-4370	97	22	f−(v	f−(v	NOUN
ejpam-4370	97	23	)	)	PUNCT
ejpam-4370	97	24	is	be	AUX
ejpam-4370	97	25	(	(	PUNCT
ejpam-4370	97	26	λ	λ	X
ejpam-4370	97	27	,	,	PUNCT
ejpam-4370	97	28	sp)-closed	sp)-close	VERB
ejpam-4370	97	29	in	in	ADP
ejpam-4370	97	30	x	x	PUNCT
ejpam-4370	97	31	for	for	ADP
ejpam-4370	97	32	every	every	DET
ejpam-4370	97	33	(	(	PUNCT
ejpam-4370	97	34	λ	λ	NOUN
ejpam-4370	97	35	,	,	PUNCT
ejpam-4370	97	36	sp)-open	sp)-open	NOUN
ejpam-4370	97	37	set	set	VERB
ejpam-4370	97	38	v	v	NOUN
ejpam-4370	97	39	of	of	ADP
ejpam-4370	97	40	y	y	PROPN
ejpam-4370	97	41	;	;	PUNCT
ejpam-4370	97	42	(	(	PUNCT
ejpam-4370	97	43	4	4	X
ejpam-4370	97	44	)	)	PUNCT
ejpam-4370	97	45	for	for	ADP
ejpam-4370	97	46	each	each	DET
ejpam-4370	97	47	x	x	SYM
ejpam-4370	97	48	∈	∈	PROPN
ejpam-4370	97	49	x	x	X
ejpam-4370	97	50	and	and	CCONJ
ejpam-4370	97	51	each	each	PRON
ejpam-4370	97	52	(	(	PUNCT
ejpam-4370	97	53	λ	λ	PROPN
ejpam-4370	97	54	,	,	PUNCT
ejpam-4370	97	55	sp)-closed	sp)-close	VERB
ejpam-4370	97	56	set	set	VERB
ejpam-4370	97	57	k	k	PROPN
ejpam-4370	97	58	of	of	ADP
ejpam-4370	97	59	y	y	PROPN
ejpam-4370	97	60	containing	contain	VERB
ejpam-4370	97	61	f	f	PROPN
ejpam-4370	97	62	(	(	PUNCT
ejpam-4370	97	63	x	x	NOUN
ejpam-4370	97	64	)	)	PUNCT
ejpam-4370	97	65	,	,	PUNCT
ejpam-4370	97	66	there	there	PRON
ejpam-4370	97	67	exists	exist	VERB
ejpam-4370	97	68	u	u	PROPN
ejpam-4370	97	69	∈	∈	PROPN
ejpam-4370	97	70	λspo(x	λspo(x	PROPN
ejpam-4370	97	71	,	,	PUNCT
ejpam-4370	97	72	τ	τ	X
ejpam-4370	97	73	)	)	PUNCT
ejpam-4370	97	74	containing	contain	VERB
ejpam-4370	97	75	x	x	PUNCT
ejpam-4370	97	76	such	such	ADJ
ejpam-4370	97	77	that	that	SCONJ
ejpam-4370	97	78	if	if	SCONJ
ejpam-4370	97	79	y	y	PROPN
ejpam-4370	97	80	∈	∈	PROPN
ejpam-4370	97	81	u	u	PROPN
ejpam-4370	97	82	,	,	PUNCT
ejpam-4370	97	83	then	then	ADV
ejpam-4370	97	84	f	f	PROPN
ejpam-4370	97	85	(	(	PUNCT
ejpam-4370	97	86	y	y	PROPN
ejpam-4370	97	87	)	)	PUNCT
ejpam-4370	97	88	⊆	⊆	NUM
ejpam-4370	97	89	k.	k.	NOUN
ejpam-4370	97	90	proof	proof	NOUN
ejpam-4370	97	91	.	.	PUNCT
ejpam-4370	98	1	(	(	PUNCT
ejpam-4370	98	2	1	1	X
ejpam-4370	98	3	)	)	PUNCT
ejpam-4370	98	4	⇔	⇔	X
ejpam-4370	98	5	(	(	PUNCT
ejpam-4370	98	6	2	2	NUM
ejpam-4370	98	7	):	):	PUNCT
ejpam-4370	98	8	let	let	VERB
ejpam-4370	98	9	k	k	PRON
ejpam-4370	98	10	be	be	AUX
ejpam-4370	98	11	any	any	DET
ejpam-4370	98	12	(	(	PUNCT
ejpam-4370	98	13	λ	λ	PROPN
ejpam-4370	98	14	,	,	PUNCT
ejpam-4370	98	15	sp)-closed	sp)-close	VERB
ejpam-4370	98	16	set	set	NOUN
ejpam-4370	98	17	of	of	ADP
ejpam-4370	98	18	y	y	PROPN
ejpam-4370	98	19	and	and	CCONJ
ejpam-4370	98	20	let	let	VERB
ejpam-4370	98	21	x	x	PUNCT
ejpam-4370	98	22	∈	∈	PROPN
ejpam-4370	98	23	f+(k	f+(k	NOUN
ejpam-4370	98	24	)	)	PUNCT
ejpam-4370	98	25	.	.	PUNCT
ejpam-4370	99	1	since	since	SCONJ
ejpam-4370	99	2	f	f	PROPN
ejpam-4370	99	3	is	be	AUX
ejpam-4370	99	4	upper	upper	ADJ
ejpam-4370	99	5	contra-(λ	contra-(λ	PROPN
ejpam-4370	99	6	,	,	PUNCT
ejpam-4370	99	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	99	8	,	,	PUNCT
ejpam-4370	99	9	there	there	PRON
ejpam-4370	99	10	exists	exist	VERB
ejpam-4370	99	11	u	u	PROPN
ejpam-4370	99	12	∈	∈	PROPN
ejpam-4370	99	13	λspo(x	λspo(x	PROPN
ejpam-4370	99	14	,	,	PUNCT
ejpam-4370	99	15	τ	τ	X
ejpam-4370	99	16	)	)	PUNCT
ejpam-4370	99	17	containing	contain	VERB
ejpam-4370	99	18	x	x	PUNCT
ejpam-4370	99	19	such	such	ADJ
ejpam-4370	99	20	that	that	SCONJ
ejpam-4370	99	21	f	f	PROPN
ejpam-4370	99	22	(	(	PUNCT
ejpam-4370	99	23	u	u	NOUN
ejpam-4370	99	24	)	)	PUNCT
ejpam-4370	99	25	⊆	⊆	NUM
ejpam-4370	99	26	k.	k.	PROPN
ejpam-4370	99	27	thus	thus	ADV
ejpam-4370	99	28	,	,	PUNCT
ejpam-4370	99	29	x	x	PUNCT
ejpam-4370	99	30	∈	∈	PROPN
ejpam-4370	99	31	u	u	NOUN
ejpam-4370	99	32	⊆	⊆	NUM
ejpam-4370	99	33	f+(k	f+(k	NOUN
ejpam-4370	99	34	)	)	PUNCT
ejpam-4370	99	35	and	and	CCONJ
ejpam-4370	99	36	hence	hence	ADV
ejpam-4370	99	37	f+(k	f+(k	NUM
ejpam-4370	99	38	)	)	PUNCT
ejpam-4370	99	39	is	be	AUX
ejpam-4370	99	40	(	(	PUNCT
ejpam-4370	99	41	λ	λ	INTJ
ejpam-4370	99	42	,	,	PUNCT
ejpam-4370	99	43	sp)-open	sp)-open	ADJ
ejpam-4370	99	44	in	in	ADP
ejpam-4370	99	45	x.	x.	NOUN
ejpam-4370	99	46	the	the	DET
ejpam-4370	99	47	converse	converse	NOUN
ejpam-4370	99	48	is	be	AUX
ejpam-4370	99	49	similar	similar	ADJ
ejpam-4370	99	50	.	.	PUNCT
ejpam-4370	100	1	(	(	PUNCT
ejpam-4370	100	2	2	2	X
ejpam-4370	100	3	)	)	PUNCT
ejpam-4370	100	4	⇔	⇔	X
ejpam-4370	100	5	(	(	PUNCT
ejpam-4370	100	6	3	3	NUM
ejpam-4370	100	7	):	):	PUNCT
ejpam-4370	100	8	it	it	PRON
ejpam-4370	100	9	follows	follow	VERB
ejpam-4370	100	10	from	from	ADP
ejpam-4370	100	11	the	the	DET
ejpam-4370	100	12	fact	fact	NOUN
ejpam-4370	100	13	that	that	SCONJ
ejpam-4370	100	14	f+(y	f+(y	PROPN
ejpam-4370	100	15	−b	−b	ADV
ejpam-4370	100	16	)	)	PUNCT
ejpam-4370	100	17	=	=	PUNCT
ejpam-4370	101	1	x	x	X
ejpam-4370	101	2	−	−	PROPN
ejpam-4370	101	3	f−(b	f−(b	PROPN
ejpam-4370	101	4	)	)	PUNCT
ejpam-4370	101	5	for	for	ADP
ejpam-4370	101	6	every	every	DET
ejpam-4370	101	7	subset	subset	NOUN
ejpam-4370	101	8	b	b	PROPN
ejpam-4370	101	9	of	of	ADP
ejpam-4370	101	10	y	y	PROPN
ejpam-4370	101	11	.	.	PUNCT
ejpam-4370	102	1	(	(	PUNCT
ejpam-4370	102	2	1	1	X
ejpam-4370	102	3	)	)	PUNCT
ejpam-4370	102	4	⇔	⇔	X
ejpam-4370	102	5	(	(	PUNCT
ejpam-4370	102	6	4	4	NUM
ejpam-4370	102	7	):	):	PUNCT
ejpam-4370	102	8	this	this	PRON
ejpam-4370	102	9	is	be	AUX
ejpam-4370	102	10	obvious	obvious	ADJ
ejpam-4370	102	11	.	.	PUNCT
ejpam-4370	103	1	theorem	theorem	NOUN
ejpam-4370	103	2	2	2	NUM
ejpam-4370	103	3	.	.	X
ejpam-4370	103	4	for	for	ADP
ejpam-4370	103	5	a	a	DET
ejpam-4370	103	6	multifunction	multifunction	NOUN
ejpam-4370	104	1	f	f	NOUN
ejpam-4370	104	2	:	:	PUNCT
ejpam-4370	104	3	(	(	PUNCT
ejpam-4370	104	4	x	x	X
ejpam-4370	104	5	,	,	PUNCT
ejpam-4370	104	6	τ	τ	X
ejpam-4370	104	7	)	)	PUNCT
ejpam-4370	104	8	→	→	SYM
ejpam-4370	104	9	(	(	PUNCT
ejpam-4370	104	10	y	y	PROPN
ejpam-4370	104	11	,	,	PUNCT
ejpam-4370	104	12	σ	σ	PROPN
ejpam-4370	104	13	)	)	PUNCT
ejpam-4370	104	14	,	,	PUNCT
ejpam-4370	104	15	the	the	DET
ejpam-4370	104	16	following	follow	VERB
ejpam-4370	104	17	properties	property	NOUN
ejpam-4370	104	18	are	be	AUX
ejpam-4370	104	19	equivalent	equivalent	ADJ
ejpam-4370	104	20	:	:	PUNCT
ejpam-4370	104	21	(	(	PUNCT
ejpam-4370	104	22	1	1	X
ejpam-4370	104	23	)	)	PUNCT
ejpam-4370	104	24	f	f	PROPN
ejpam-4370	104	25	is	be	AUX
ejpam-4370	104	26	lower	low	ADJ
ejpam-4370	104	27	contra-(λ	contra-(λ	PROPN
ejpam-4370	104	28	,	,	PUNCT
ejpam-4370	104	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	104	30	;	;	PUNCT
ejpam-4370	104	31	(	(	PUNCT
ejpam-4370	104	32	2	2	X
ejpam-4370	104	33	)	)	PUNCT
ejpam-4370	104	34	f−(k	f−(k	PROPN
ejpam-4370	104	35	)	)	PUNCT
ejpam-4370	104	36	is	be	AUX
ejpam-4370	104	37	(	(	PUNCT
ejpam-4370	104	38	λ	λ	INTJ
ejpam-4370	104	39	,	,	PUNCT
ejpam-4370	104	40	sp)-open	sp)-open	ADJ
ejpam-4370	104	41	in	in	ADP
ejpam-4370	104	42	x	x	PUNCT
ejpam-4370	104	43	for	for	SCONJ
ejpam-4370	104	44	every	every	DET
ejpam-4370	104	45	(	(	PUNCT
ejpam-4370	104	46	λ	λ	PROPN
ejpam-4370	104	47	,	,	PUNCT
ejpam-4370	104	48	sp)-closed	sp)-close	VERB
ejpam-4370	104	49	set	set	VERB
ejpam-4370	104	50	k	k	PROPN
ejpam-4370	104	51	of	of	ADP
ejpam-4370	104	52	y	y	PROPN
ejpam-4370	104	53	;	;	PUNCT
ejpam-4370	104	54	(	(	PUNCT
ejpam-4370	104	55	3	3	X
ejpam-4370	104	56	)	)	PUNCT
ejpam-4370	104	57	f+(v	f+(v	NOUN
ejpam-4370	104	58	)	)	PUNCT
ejpam-4370	105	1	is	be	AUX
ejpam-4370	105	2	(	(	PUNCT
ejpam-4370	105	3	λ	λ	X
ejpam-4370	105	4	,	,	PUNCT
ejpam-4370	105	5	sp)-closed	sp)-close	VERB
ejpam-4370	105	6	in	in	ADP
ejpam-4370	105	7	x	x	PUNCT
ejpam-4370	105	8	for	for	ADP
ejpam-4370	105	9	every	every	DET
ejpam-4370	105	10	(	(	PUNCT
ejpam-4370	105	11	λ	λ	NOUN
ejpam-4370	105	12	,	,	PUNCT
ejpam-4370	105	13	sp)-open	sp)-open	NOUN
ejpam-4370	105	14	set	set	VERB
ejpam-4370	105	15	v	v	NOUN
ejpam-4370	105	16	of	of	ADP
ejpam-4370	105	17	y	y	PROPN
ejpam-4370	105	18	;	;	PUNCT
ejpam-4370	105	19	(	(	PUNCT
ejpam-4370	105	20	4	4	X
ejpam-4370	105	21	)	)	PUNCT
ejpam-4370	105	22	for	for	ADP
ejpam-4370	105	23	each	each	DET
ejpam-4370	105	24	x	x	SYM
ejpam-4370	105	25	∈	∈	PROPN
ejpam-4370	105	26	x	x	X
ejpam-4370	105	27	and	and	CCONJ
ejpam-4370	105	28	each	each	PRON
ejpam-4370	105	29	(	(	PUNCT
ejpam-4370	105	30	λ	λ	PROPN
ejpam-4370	105	31	,	,	PUNCT
ejpam-4370	105	32	sp)-closed	sp)-close	VERB
ejpam-4370	105	33	set	set	VERB
ejpam-4370	105	34	k	k	PROPN
ejpam-4370	105	35	of	of	ADP
ejpam-4370	105	36	y	y	PRON
ejpam-4370	105	37	such	such	ADJ
ejpam-4370	105	38	that	that	SCONJ
ejpam-4370	105	39	f	f	PROPN
ejpam-4370	105	40	(	(	PUNCT
ejpam-4370	105	41	x	x	NOUN
ejpam-4370	105	42	)	)	PUNCT
ejpam-4370	105	43	∩	∩	NOUN
ejpam-4370	105	44	k	k	PROPN
ejpam-4370	105	45	̸=	̸=	PROPN
ejpam-4370	105	46	∅	∅	NOUN
ejpam-4370	105	47	,	,	PUNCT
ejpam-4370	105	48	there	there	PRON
ejpam-4370	105	49	exists	exist	VERB
ejpam-4370	105	50	u	u	PROPN
ejpam-4370	105	51	∈	∈	PROPN
ejpam-4370	105	52	λspo(x	λspo(x	PROPN
ejpam-4370	105	53	,	,	PUNCT
ejpam-4370	105	54	τ	τ	X
ejpam-4370	105	55	)	)	PUNCT
ejpam-4370	105	56	containing	contain	VERB
ejpam-4370	105	57	x	x	PUNCT
ejpam-4370	105	58	such	such	ADJ
ejpam-4370	105	59	that	that	SCONJ
ejpam-4370	105	60	if	if	SCONJ
ejpam-4370	105	61	y	y	PROPN
ejpam-4370	105	62	∈	∈	PROPN
ejpam-4370	105	63	u	u	PROPN
ejpam-4370	105	64	,	,	PUNCT
ejpam-4370	105	65	then	then	ADV
ejpam-4370	105	66	f	f	PROPN
ejpam-4370	105	67	(	(	PUNCT
ejpam-4370	105	68	y	y	NOUN
ejpam-4370	105	69	)	)	PUNCT
ejpam-4370	105	70	∩k	∩k	NOUN
ejpam-4370	105	71	̸=	̸=	PROPN
ejpam-4370	105	72	∅.	∅.	ADP
ejpam-4370	105	73	proof	proof	NOUN
ejpam-4370	105	74	.	.	PUNCT
ejpam-4370	106	1	the	the	DET
ejpam-4370	106	2	proof	proof	NOUN
ejpam-4370	106	3	is	be	AUX
ejpam-4370	106	4	similar	similar	ADJ
ejpam-4370	106	5	to	to	ADP
ejpam-4370	106	6	that	that	PRON
ejpam-4370	106	7	of	of	ADP
ejpam-4370	106	8	theorem	theorem	NOUN
ejpam-4370	106	9	1	1	NUM
ejpam-4370	106	10	.	.	PUNCT
ejpam-4370	106	11	definition	definition	NOUN
ejpam-4370	106	12	2	2	NUM
ejpam-4370	106	13	.	.	PUNCT
ejpam-4370	107	1	a	a	DET
ejpam-4370	107	2	function	function	NOUN
ejpam-4370	107	3	f	f	NOUN
ejpam-4370	107	4	:	:	PUNCT
ejpam-4370	107	5	(	(	PUNCT
ejpam-4370	107	6	x	x	X
ejpam-4370	107	7	,	,	PUNCT
ejpam-4370	107	8	τ	τ	X
ejpam-4370	107	9	)	)	PUNCT
ejpam-4370	107	10	→	→	SYM
ejpam-4370	107	11	(	(	PUNCT
ejpam-4370	107	12	y	y	PROPN
ejpam-4370	107	13	,	,	PUNCT
ejpam-4370	107	14	σ	σ	PROPN
ejpam-4370	107	15	)	)	PUNCT
ejpam-4370	107	16	is	be	AUX
ejpam-4370	107	17	said	say	VERB
ejpam-4370	107	18	to	to	PART
ejpam-4370	107	19	be	be	AUX
ejpam-4370	107	20	contra-(λ	contra-(λ	PROPN
ejpam-4370	107	21	,	,	PUNCT
ejpam-4370	107	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	107	23	if	if	SCONJ
ejpam-4370	107	24	,	,	PUNCT
ejpam-4370	107	25	for	for	SCONJ
ejpam-4370	107	26	each	each	DET
ejpam-4370	107	27	x	x	SYM
ejpam-4370	107	28	∈	∈	PROPN
ejpam-4370	107	29	x	x	X
ejpam-4370	107	30	and	and	CCONJ
ejpam-4370	107	31	each	each	PRON
ejpam-4370	107	32	(	(	PUNCT
ejpam-4370	107	33	λ	λ	PROPN
ejpam-4370	107	34	,	,	PUNCT
ejpam-4370	107	35	sp)-closed	sp)-close	VERB
ejpam-4370	107	36	set	set	VERB
ejpam-4370	107	37	k	k	PROPN
ejpam-4370	107	38	of	of	ADP
ejpam-4370	107	39	y	y	PROPN
ejpam-4370	107	40	containing	contain	VERB
ejpam-4370	107	41	f(x	f(x	PROPN
ejpam-4370	107	42	)	)	PUNCT
ejpam-4370	107	43	,	,	PUNCT
ejpam-4370	107	44	there	there	PRON
ejpam-4370	107	45	exists	exist	VERB
ejpam-4370	107	46	a	a	DET
ejpam-4370	107	47	(	(	PUNCT
ejpam-4370	107	48	λ	λ	NOUN
ejpam-4370	107	49	,	,	PUNCT
ejpam-4370	107	50	sp)-open	sp)-open	NOUN
ejpam-4370	107	51	set	set	VERB
ejpam-4370	107	52	u	u	NOUN
ejpam-4370	107	53	of	of	ADP
ejpam-4370	107	54	x	x	PUNCT
ejpam-4370	107	55	containing	contain	VERB
ejpam-4370	107	56	x	x	PUNCT
ejpam-4370	107	57	such	such	ADJ
ejpam-4370	107	58	that	that	DET
ejpam-4370	107	59	f(u	f(u	PROPN
ejpam-4370	107	60	)	)	PUNCT
ejpam-4370	107	61	⊆	⊆	PROPN
ejpam-4370	107	62	k.	k.	PROPN
ejpam-4370	107	63	corollary	corollary	NOUN
ejpam-4370	107	64	1	1	NUM
ejpam-4370	107	65	.	.	PUNCT
ejpam-4370	108	1	for	for	ADP
ejpam-4370	108	2	a	a	DET
ejpam-4370	108	3	function	function	NOUN
ejpam-4370	108	4	f	f	NOUN
ejpam-4370	108	5	:	:	PUNCT
ejpam-4370	108	6	(	(	PUNCT
ejpam-4370	108	7	x	x	X
ejpam-4370	108	8	,	,	PUNCT
ejpam-4370	108	9	τ	τ	X
ejpam-4370	108	10	)	)	PUNCT
ejpam-4370	108	11	→	→	SYM
ejpam-4370	108	12	(	(	PUNCT
ejpam-4370	108	13	y	y	PROPN
ejpam-4370	108	14	,	,	PUNCT
ejpam-4370	108	15	σ	σ	PROPN
ejpam-4370	108	16	)	)	PUNCT
ejpam-4370	108	17	,	,	PUNCT
ejpam-4370	108	18	the	the	DET
ejpam-4370	108	19	following	follow	VERB
ejpam-4370	108	20	properties	property	NOUN
ejpam-4370	108	21	are	be	AUX
ejpam-4370	108	22	equivalent	equivalent	ADJ
ejpam-4370	108	23	:	:	PUNCT
ejpam-4370	108	24	(	(	PUNCT
ejpam-4370	108	25	1	1	X
ejpam-4370	108	26	)	)	PUNCT
ejpam-4370	108	27	f	f	PROPN
ejpam-4370	108	28	is	be	AUX
ejpam-4370	108	29	contra-(λ	contra-(λ	PROPN
ejpam-4370	108	30	,	,	PUNCT
ejpam-4370	108	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	108	32	;	;	PUNCT
ejpam-4370	108	33	(	(	PUNCT
ejpam-4370	108	34	2	2	X
ejpam-4370	108	35	)	)	PUNCT
ejpam-4370	108	36	f−1(k	f−1(k	PROPN
ejpam-4370	108	37	)	)	PUNCT
ejpam-4370	108	38	is	be	AUX
ejpam-4370	108	39	(	(	PUNCT
ejpam-4370	108	40	λ	λ	INTJ
ejpam-4370	108	41	,	,	PUNCT
ejpam-4370	108	42	sp)-open	sp)-open	ADJ
ejpam-4370	108	43	in	in	ADP
ejpam-4370	108	44	x	x	PUNCT
ejpam-4370	108	45	for	for	SCONJ
ejpam-4370	108	46	every	every	DET
ejpam-4370	108	47	(	(	PUNCT
ejpam-4370	108	48	λ	λ	PROPN
ejpam-4370	108	49	,	,	PUNCT
ejpam-4370	108	50	sp)-closed	sp)-close	VERB
ejpam-4370	108	51	set	set	VERB
ejpam-4370	108	52	k	k	PROPN
ejpam-4370	108	53	of	of	ADP
ejpam-4370	108	54	y	y	PROPN
ejpam-4370	108	55	;	;	PUNCT
ejpam-4370	108	56	(	(	PUNCT
ejpam-4370	108	57	3	3	X
ejpam-4370	108	58	)	)	PUNCT
ejpam-4370	108	59	f−1(v	f−1(v	NOUN
ejpam-4370	108	60	)	)	PUNCT
ejpam-4370	108	61	is	be	AUX
ejpam-4370	108	62	(	(	PUNCT
ejpam-4370	108	63	λ	λ	X
ejpam-4370	108	64	,	,	PUNCT
ejpam-4370	108	65	sp)-closed	sp)-close	VERB
ejpam-4370	108	66	in	in	ADP
ejpam-4370	108	67	x	x	PUNCT
ejpam-4370	108	68	for	for	ADP
ejpam-4370	108	69	every	every	DET
ejpam-4370	108	70	(	(	PUNCT
ejpam-4370	108	71	λ	λ	NOUN
ejpam-4370	108	72	,	,	PUNCT
ejpam-4370	108	73	sp)-open	sp)-open	NOUN
ejpam-4370	108	74	set	set	VERB
ejpam-4370	108	75	v	v	NOUN
ejpam-4370	108	76	of	of	ADP
ejpam-4370	108	77	y	y	PROPN
ejpam-4370	108	78	.	.	PUNCT
ejpam-4370	109	1	c.	c.	PROPN
ejpam-4370	109	2	boonpok	boonpok	PROPN
ejpam-4370	109	3	,	,	PUNCT
ejpam-4370	109	4	c.	c.	PROPN
ejpam-4370	109	5	viriyapong	viriyapong	PROPN
ejpam-4370	109	6	/	/	SYM
ejpam-4370	109	7	eur	eur	PROPN
ejpam-4370	109	8	.	.	PUNCT
ejpam-4370	110	1	j.	j.	PROPN
ejpam-4370	110	2	pure	pure	PROPN
ejpam-4370	110	3	appl	appl	PROPN
ejpam-4370	110	4	.	.	PROPN
ejpam-4370	110	5	math	math	PROPN
ejpam-4370	110	6	,	,	PUNCT
ejpam-4370	110	7	15	15	NUM
ejpam-4370	110	8	(	(	PUNCT
ejpam-4370	110	9	4	4	NUM
ejpam-4370	110	10	)	)	PUNCT
ejpam-4370	110	11	(	(	PUNCT
ejpam-4370	110	12	2022	2022	NUM
ejpam-4370	110	13	)	)	PUNCT
ejpam-4370	110	14	,	,	PUNCT
ejpam-4370	110	15	1694	1694	NUM
ejpam-4370	110	16	-	-	SYM
ejpam-4370	110	17	1704	1704	NUM
ejpam-4370	110	18	1698	1698	NUM
ejpam-4370	110	19	let	let	VERB
ejpam-4370	110	20	a	a	PRON
ejpam-4370	110	21	be	be	AUX
ejpam-4370	110	22	a	a	DET
ejpam-4370	110	23	subset	subset	NOUN
ejpam-4370	110	24	of	of	ADP
ejpam-4370	110	25	a	a	DET
ejpam-4370	110	26	topological	topological	ADJ
ejpam-4370	110	27	space	space	NOUN
ejpam-4370	110	28	(	(	PUNCT
ejpam-4370	110	29	x	x	X
ejpam-4370	110	30	,	,	PUNCT
ejpam-4370	110	31	τ	τ	PROPN
ejpam-4370	110	32	)	)	PUNCT
ejpam-4370	110	33	.	.	PUNCT
ejpam-4370	111	1	a	a	DET
ejpam-4370	111	2	point	point	NOUN
ejpam-4370	111	3	x	x	X
ejpam-4370	111	4	∈	∈	NOUN
ejpam-4370	111	5	x	x	PUNCT
ejpam-4370	111	6	is	be	AUX
ejpam-4370	111	7	called	call	VERB
ejpam-4370	111	8	a	a	DET
ejpam-4370	111	9	δ(λ	δ(λ	PROPN
ejpam-4370	111	10	,	,	PUNCT
ejpam-4370	111	11	sp)cluster	sp)cluster	NOUN
ejpam-4370	111	12	point	point	NOUN
ejpam-4370	111	13	[	[	X
ejpam-4370	111	14	21	21	NUM
ejpam-4370	111	15	]	]	PUNCT
ejpam-4370	111	16	of	of	ADP
ejpam-4370	111	17	a	a	PRON
ejpam-4370	111	18	if	if	SCONJ
ejpam-4370	111	19	a∩	a∩	PROPN
ejpam-4370	111	20	[	[	X
ejpam-4370	111	21	u	u	X
ejpam-4370	111	22	(	(	PUNCT
ejpam-4370	111	23	λ	λ	PROPN
ejpam-4370	111	24	,	,	PUNCT
ejpam-4370	111	25	sp)](λ	sp)](λ	PROPN
ejpam-4370	111	26	,	,	PUNCT
ejpam-4370	111	27	sp	sp	NOUN
ejpam-4370	111	28	)	)	PUNCT
ejpam-4370	111	29	̸=	̸=	PROPN
ejpam-4370	111	30	∅	∅	NOUN
ejpam-4370	111	31	for	for	ADP
ejpam-4370	111	32	every	every	DET
ejpam-4370	111	33	(	(	PUNCT
ejpam-4370	111	34	λ	λ	NOUN
ejpam-4370	111	35	,	,	PUNCT
ejpam-4370	111	36	sp)-open	sp)-open	NOUN
ejpam-4370	111	37	set	set	VERB
ejpam-4370	111	38	u	u	NOUN
ejpam-4370	111	39	of	of	ADP
ejpam-4370	111	40	x	x	SYM
ejpam-4370	111	41	containing	contain	VERB
ejpam-4370	111	42	x.	x.	NOUN
ejpam-4370	111	43	the	the	DET
ejpam-4370	111	44	set	set	NOUN
ejpam-4370	111	45	of	of	ADP
ejpam-4370	111	46	all	all	DET
ejpam-4370	111	47	δ(λ	δ(λ	PROPN
ejpam-4370	111	48	,	,	PUNCT
ejpam-4370	111	49	sp)-cluster	sp)-cluster	NOUN
ejpam-4370	111	50	points	point	NOUN
ejpam-4370	111	51	of	of	ADP
ejpam-4370	111	52	a	a	PRON
ejpam-4370	111	53	is	be	AUX
ejpam-4370	111	54	called	call	VERB
ejpam-4370	111	55	the	the	DET
ejpam-4370	111	56	δ(λ	δ(λ	PROPN
ejpam-4370	111	57	,	,	PUNCT
ejpam-4370	111	58	sp)-closure	sp)-closure	NOUN
ejpam-4370	112	1	[	[	X
ejpam-4370	112	2	21	21	NUM
ejpam-4370	112	3	]	]	PUNCT
ejpam-4370	112	4	of	of	ADP
ejpam-4370	112	5	a	a	PRON
ejpam-4370	112	6	and	and	CCONJ
ejpam-4370	112	7	is	be	AUX
ejpam-4370	112	8	denoted	denote	VERB
ejpam-4370	112	9	by	by	ADP
ejpam-4370	112	10	aδ(λ	aδ(λ	NUM
ejpam-4370	112	11	,	,	PUNCT
ejpam-4370	112	12	sp	sp	NOUN
ejpam-4370	112	13	)	)	PUNCT
ejpam-4370	112	14	.	.	PUNCT
ejpam-4370	113	1	if	if	SCONJ
ejpam-4370	113	2	a	a	DET
ejpam-4370	113	3	=	=	NOUN
ejpam-4370	113	4	aδ(λ	aδ(λ	NUM
ejpam-4370	113	5	,	,	PUNCT
ejpam-4370	113	6	sp	sp	NOUN
ejpam-4370	113	7	)	)	PUNCT
ejpam-4370	113	8	,	,	PUNCT
ejpam-4370	113	9	then	then	ADV
ejpam-4370	113	10	a	a	PRON
ejpam-4370	113	11	is	be	AUX
ejpam-4370	113	12	said	say	VERB
ejpam-4370	113	13	to	to	PART
ejpam-4370	113	14	be	be	AUX
ejpam-4370	113	15	δ(λ	δ(λ	PROPN
ejpam-4370	113	16	,	,	PUNCT
ejpam-4370	113	17	sp)-closed	sp)-close	VERB
ejpam-4370	113	18	[	[	X
ejpam-4370	113	19	21	21	NUM
ejpam-4370	113	20	]	]	PUNCT
ejpam-4370	113	21	.	.	PUNCT
ejpam-4370	114	1	the	the	DET
ejpam-4370	114	2	complement	complement	NOUN
ejpam-4370	114	3	of	of	ADP
ejpam-4370	114	4	a	a	DET
ejpam-4370	114	5	δ(λ	δ(λ	PROPN
ejpam-4370	114	6	,	,	PUNCT
ejpam-4370	114	7	sp)-closed	sp)-close	VERB
ejpam-4370	114	8	set	set	NOUN
ejpam-4370	114	9	is	be	AUX
ejpam-4370	114	10	said	say	VERB
ejpam-4370	114	11	to	to	PART
ejpam-4370	114	12	be	be	AUX
ejpam-4370	114	13	δ(λ	δ(λ	PROPN
ejpam-4370	114	14	,	,	PUNCT
ejpam-4370	114	15	sp)-open	sp)-open	ADJ
ejpam-4370	114	16	[	[	X
ejpam-4370	114	17	21	21	NUM
ejpam-4370	114	18	]	]	PUNCT
ejpam-4370	114	19	.	.	PUNCT
ejpam-4370	115	1	the	the	DET
ejpam-4370	115	2	union	union	NOUN
ejpam-4370	115	3	of	of	ADP
ejpam-4370	115	4	all	all	DET
ejpam-4370	115	5	δ(λ	δ(λ	PROPN
ejpam-4370	115	6	,	,	PUNCT
ejpam-4370	115	7	sp)-open	sp)-open	ADJ
ejpam-4370	115	8	sets	set	NOUN
ejpam-4370	115	9	contained	contain	VERB
ejpam-4370	115	10	in	in	ADP
ejpam-4370	115	11	a	a	PRON
ejpam-4370	115	12	is	be	AUX
ejpam-4370	115	13	called	call	VERB
ejpam-4370	115	14	the	the	DET
ejpam-4370	115	15	δ(λ	δ(λ	PROPN
ejpam-4370	115	16	,	,	PUNCT
ejpam-4370	115	17	sp)-interior	sp)-interior	NOUN
ejpam-4370	115	18	[	[	X
ejpam-4370	115	19	21	21	NUM
ejpam-4370	115	20	]	]	PUNCT
ejpam-4370	115	21	of	of	ADP
ejpam-4370	115	22	a	a	PRON
ejpam-4370	115	23	and	and	CCONJ
ejpam-4370	115	24	is	be	AUX
ejpam-4370	115	25	denoted	denote	VERB
ejpam-4370	115	26	by	by	ADP
ejpam-4370	115	27	aδ(λ	aδ(λ	NUM
ejpam-4370	115	28	,	,	PUNCT
ejpam-4370	115	29	sp	sp	NOUN
ejpam-4370	115	30	)	)	PUNCT
ejpam-4370	115	31	.	.	PUNCT
ejpam-4370	116	1	definition	definition	NOUN
ejpam-4370	116	2	3	3	NUM
ejpam-4370	116	3	.	.	PUNCT
ejpam-4370	117	1	a	a	DET
ejpam-4370	117	2	topological	topological	ADJ
ejpam-4370	117	3	space	space	NOUN
ejpam-4370	117	4	(	(	PUNCT
ejpam-4370	117	5	x	x	X
ejpam-4370	117	6	,	,	PUNCT
ejpam-4370	117	7	τ	τ	X
ejpam-4370	117	8	)	)	PUNCT
ejpam-4370	117	9	is	be	AUX
ejpam-4370	117	10	said	say	VERB
ejpam-4370	117	11	to	to	PART
ejpam-4370	117	12	be	be	AUX
ejpam-4370	117	13	semi-(λ	semi-(λ	PROPN
ejpam-4370	117	14	,	,	PUNCT
ejpam-4370	117	15	sp)-regular	sp)-regular	ADJ
ejpam-4370	117	16	if	if	SCONJ
ejpam-4370	117	17	,	,	PUNCT
ejpam-4370	117	18	for	for	ADP
ejpam-4370	117	19	each	each	DET
ejpam-4370	117	20	(	(	PUNCT
ejpam-4370	117	21	λ	λ	PROPN
ejpam-4370	117	22	,	,	PUNCT
ejpam-4370	117	23	sp)-open	sp)-open	NOUN
ejpam-4370	117	24	set	set	VERB
ejpam-4370	117	25	u	u	NOUN
ejpam-4370	117	26	of	of	ADP
ejpam-4370	117	27	x	x	PUNCT
ejpam-4370	117	28	and	and	CCONJ
ejpam-4370	117	29	each	each	DET
ejpam-4370	117	30	x	x	SYM
ejpam-4370	117	31	∈	∈	PROPN
ejpam-4370	117	32	u	u	NOUN
ejpam-4370	117	33	,	,	PUNCT
ejpam-4370	117	34	there	there	PRON
ejpam-4370	117	35	exists	exist	VERB
ejpam-4370	117	36	a	a	DET
ejpam-4370	117	37	r(λ	r(λ	NOUN
ejpam-4370	117	38	,	,	PUNCT
ejpam-4370	117	39	sp)-open	sp)-open	ADJ
ejpam-4370	117	40	set	set	VERB
ejpam-4370	117	41	v	v	ADP
ejpam-4370	117	42	such	such	ADJ
ejpam-4370	117	43	that	that	SCONJ
ejpam-4370	117	44	x	x	SYM
ejpam-4370	117	45	∈	∈	NOUN
ejpam-4370	117	46	v	v	ADP
ejpam-4370	117	47	⊆	⊆	NUM
ejpam-4370	117	48	u	u	NOUN
ejpam-4370	117	49	.	.	PUNCT
ejpam-4370	118	1	lemma	lemma	PROPN
ejpam-4370	118	2	3	3	X
ejpam-4370	118	3	.	.	PUNCT
ejpam-4370	119	1	let	let	AUX
ejpam-4370	119	2	(	(	PUNCT
ejpam-4370	119	3	x	x	NOUN
ejpam-4370	119	4	,	,	PUNCT
ejpam-4370	119	5	τ	τ	X
ejpam-4370	119	6	)	)	PUNCT
ejpam-4370	119	7	be	be	AUX
ejpam-4370	119	8	a	a	DET
ejpam-4370	119	9	semi-(λ	semi-(λ	PROPN
ejpam-4370	119	10	,	,	PUNCT
ejpam-4370	119	11	sp)-regular	sp)-regular	ADJ
ejpam-4370	119	12	space	space	NOUN
ejpam-4370	119	13	.	.	PUNCT
ejpam-4370	120	1	then	then	ADV
ejpam-4370	120	2	,	,	PUNCT
ejpam-4370	120	3	the	the	DET
ejpam-4370	120	4	following	follow	VERB
ejpam-4370	120	5	properties	property	NOUN
ejpam-4370	120	6	hold	hold	VERB
ejpam-4370	120	7	:	:	PUNCT
ejpam-4370	120	8	(	(	PUNCT
ejpam-4370	120	9	1	1	X
ejpam-4370	120	10	)	)	PUNCT
ejpam-4370	120	11	a(λ	a(λ	ADV
ejpam-4370	120	12	,	,	PUNCT
ejpam-4370	120	13	sp	sp	NOUN
ejpam-4370	120	14	)	)	PUNCT
ejpam-4370	120	15	=	=	SYM
ejpam-4370	120	16	aδ(λ	aδ(λ	NUM
ejpam-4370	120	17	,	,	PUNCT
ejpam-4370	120	18	sp	sp	NOUN
ejpam-4370	120	19	)	)	PUNCT
ejpam-4370	120	20	for	for	ADP
ejpam-4370	120	21	every	every	DET
ejpam-4370	120	22	subset	subset	NOUN
ejpam-4370	120	23	a	a	PRON
ejpam-4370	120	24	of	of	ADP
ejpam-4370	120	25	x.	x.	NOUN
ejpam-4370	120	26	(	(	PUNCT
ejpam-4370	120	27	2	2	X
ejpam-4370	120	28	)	)	PUNCT
ejpam-4370	120	29	every	every	DET
ejpam-4370	120	30	(	(	PUNCT
ejpam-4370	120	31	λ	λ	NOUN
ejpam-4370	120	32	,	,	PUNCT
ejpam-4370	120	33	sp)-open	sp)-open	ADJ
ejpam-4370	120	34	set	set	NOUN
ejpam-4370	120	35	is	be	AUX
ejpam-4370	120	36	δ(λ	δ(λ	PROPN
ejpam-4370	120	37	,	,	PUNCT
ejpam-4370	120	38	sp)-open	sp)-open	NOUN
ejpam-4370	120	39	.	.	PUNCT
ejpam-4370	121	1	theorem	theorem	NOUN
ejpam-4370	121	2	3	3	NUM
ejpam-4370	121	3	.	.	X
ejpam-4370	121	4	for	for	ADP
ejpam-4370	121	5	a	a	DET
ejpam-4370	121	6	multifunction	multifunction	NOUN
ejpam-4370	122	1	f	f	NOUN
ejpam-4370	122	2	:	:	PUNCT
ejpam-4370	122	3	(	(	PUNCT
ejpam-4370	122	4	x	x	X
ejpam-4370	122	5	,	,	PUNCT
ejpam-4370	122	6	τ	τ	X
ejpam-4370	122	7	)	)	PUNCT
ejpam-4370	122	8	→	→	SYM
ejpam-4370	122	9	(	(	PUNCT
ejpam-4370	122	10	y	y	PROPN
ejpam-4370	122	11	,	,	PUNCT
ejpam-4370	122	12	σ	σ	PROPN
ejpam-4370	122	13	)	)	PUNCT
ejpam-4370	122	14	,	,	PUNCT
ejpam-4370	122	15	where	where	SCONJ
ejpam-4370	122	16	(	(	PUNCT
ejpam-4370	122	17	y	y	PROPN
ejpam-4370	122	18	,	,	PUNCT
ejpam-4370	122	19	σ	σ	PROPN
ejpam-4370	122	20	)	)	PUNCT
ejpam-4370	122	21	is	be	AUX
ejpam-4370	122	22	a	a	DET
ejpam-4370	122	23	semi-(λ	semi-(λ	PROPN
ejpam-4370	122	24	,	,	PUNCT
ejpam-4370	122	25	sp)regular	sp)regular	ADJ
ejpam-4370	122	26	space	space	NOUN
ejpam-4370	122	27	,	,	PUNCT
ejpam-4370	122	28	the	the	DET
ejpam-4370	122	29	following	follow	VERB
ejpam-4370	122	30	properties	property	NOUN
ejpam-4370	122	31	are	be	AUX
ejpam-4370	122	32	equivalent	equivalent	ADJ
ejpam-4370	122	33	:	:	PUNCT
ejpam-4370	122	34	(	(	PUNCT
ejpam-4370	122	35	1	1	X
ejpam-4370	122	36	)	)	PUNCT
ejpam-4370	122	37	f	f	PROPN
ejpam-4370	122	38	is	be	AUX
ejpam-4370	122	39	upper	upper	ADJ
ejpam-4370	122	40	contra-(λ	contra-(λ	PROPN
ejpam-4370	122	41	,	,	PUNCT
ejpam-4370	122	42	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	122	43	;	;	PUNCT
ejpam-4370	122	44	(	(	PUNCT
ejpam-4370	122	45	2	2	X
ejpam-4370	122	46	)	)	PUNCT
ejpam-4370	122	47	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4370	122	48	,	,	PUNCT
ejpam-4370	122	49	sp	sp	NOUN
ejpam-4370	122	50	)	)	PUNCT
ejpam-4370	122	51	)	)	PUNCT
ejpam-4370	123	1	is	be	AUX
ejpam-4370	123	2	(	(	PUNCT
ejpam-4370	123	3	λ	λ	INTJ
ejpam-4370	123	4	,	,	PUNCT
ejpam-4370	123	5	sp)-open	sp)-open	ADJ
ejpam-4370	123	6	in	in	ADP
ejpam-4370	123	7	x	x	PUNCT
ejpam-4370	123	8	for	for	ADP
ejpam-4370	123	9	every	every	DET
ejpam-4370	123	10	subset	subset	NOUN
ejpam-4370	123	11	b	b	PROPN
ejpam-4370	123	12	of	of	ADP
ejpam-4370	123	13	y	y	PROPN
ejpam-4370	123	14	;	;	PUNCT
ejpam-4370	123	15	(	(	PUNCT
ejpam-4370	123	16	3	3	X
ejpam-4370	123	17	)	)	PUNCT
ejpam-4370	123	18	f+(k	f+(k	NUM
ejpam-4370	123	19	)	)	PUNCT
ejpam-4370	124	1	is	be	AUX
ejpam-4370	124	2	(	(	PUNCT
ejpam-4370	124	3	λ	λ	INTJ
ejpam-4370	124	4	,	,	PUNCT
ejpam-4370	124	5	sp)-open	sp)-open	ADJ
ejpam-4370	124	6	in	in	ADP
ejpam-4370	124	7	x	x	PUNCT
ejpam-4370	124	8	for	for	ADP
ejpam-4370	124	9	every	every	DET
ejpam-4370	124	10	δ(λ	δ(λ	PROPN
ejpam-4370	124	11	,	,	PUNCT
ejpam-4370	124	12	sp)-closed	sp)-close	VERB
ejpam-4370	124	13	set	set	VERB
ejpam-4370	124	14	k	k	PROPN
ejpam-4370	124	15	of	of	ADP
ejpam-4370	124	16	y	y	PROPN
ejpam-4370	124	17	;	;	PUNCT
ejpam-4370	124	18	(	(	PUNCT
ejpam-4370	124	19	4	4	X
ejpam-4370	124	20	)	)	PUNCT
ejpam-4370	124	21	f−(v	f−(v	NOUN
ejpam-4370	124	22	)	)	PUNCT
ejpam-4370	124	23	is	be	AUX
ejpam-4370	124	24	(	(	PUNCT
ejpam-4370	124	25	λ	λ	X
ejpam-4370	124	26	,	,	PUNCT
ejpam-4370	124	27	sp)-closed	sp)-close	VERB
ejpam-4370	124	28	in	in	ADP
ejpam-4370	124	29	x	x	PUNCT
ejpam-4370	124	30	for	for	ADP
ejpam-4370	124	31	every	every	DET
ejpam-4370	124	32	δ(λ	δ(λ	PROPN
ejpam-4370	124	33	,	,	PUNCT
ejpam-4370	124	34	sp)-open	sp)-open	VERB
ejpam-4370	124	35	set	set	VERB
ejpam-4370	124	36	v	v	NOUN
ejpam-4370	124	37	of	of	ADP
ejpam-4370	124	38	y	y	PROPN
ejpam-4370	124	39	.	.	PUNCT
ejpam-4370	125	1	proof	proof	NOUN
ejpam-4370	125	2	.	.	PUNCT
ejpam-4370	126	1	(	(	PUNCT
ejpam-4370	126	2	1	1	X
ejpam-4370	126	3	)	)	PUNCT
ejpam-4370	126	4	⇒	⇒	NOUN
ejpam-4370	126	5	(	(	PUNCT
ejpam-4370	126	6	2	2	NUM
ejpam-4370	126	7	):	):	PUNCT
ejpam-4370	126	8	let	let	VERB
ejpam-4370	126	9	b	b	X
ejpam-4370	126	10	be	be	AUX
ejpam-4370	126	11	any	any	DET
ejpam-4370	126	12	subset	subset	NOUN
ejpam-4370	126	13	of	of	ADP
ejpam-4370	126	14	y	y	PROPN
ejpam-4370	126	15	.	.	PUNCT
ejpam-4370	127	1	thus	thus	ADV
ejpam-4370	127	2	,	,	PUNCT
ejpam-4370	127	3	by	by	ADP
ejpam-4370	127	4	lemma	lemma	PROPN
ejpam-4370	127	5	3	3	NUM
ejpam-4370	127	6	,	,	PUNCT
ejpam-4370	127	7	bδ(λ	bδ(λ	NUM
ejpam-4370	127	8	,	,	PUNCT
ejpam-4370	127	9	sp	sp	NOUN
ejpam-4370	127	10	)	)	PUNCT
ejpam-4370	127	11	is	be	AUX
ejpam-4370	127	12	a	a	DET
ejpam-4370	127	13	(	(	PUNCT
ejpam-4370	127	14	λ	λ	PROPN
ejpam-4370	127	15	,	,	PUNCT
ejpam-4370	127	16	sp)-closed	sp)-close	VERB
ejpam-4370	127	17	set	set	NOUN
ejpam-4370	127	18	of	of	ADP
ejpam-4370	127	19	y	y	PROPN
ejpam-4370	127	20	and	and	CCONJ
ejpam-4370	127	21	by	by	ADP
ejpam-4370	127	22	theorem	theorem	ADJ
ejpam-4370	127	23	1	1	NUM
ejpam-4370	127	24	,	,	PUNCT
ejpam-4370	127	25	f+(bδ(λ	f+(bδ(λ	PROPN
ejpam-4370	127	26	,	,	PUNCT
ejpam-4370	127	27	sp	sp	NOUN
ejpam-4370	127	28	)	)	PUNCT
ejpam-4370	127	29	)	)	PUNCT
ejpam-4370	128	1	is	be	AUX
ejpam-4370	128	2	(	(	PUNCT
ejpam-4370	128	3	λ	λ	INTJ
ejpam-4370	128	4	,	,	PUNCT
ejpam-4370	128	5	sp)-open	sp)-open	ADJ
ejpam-4370	128	6	in	in	ADP
ejpam-4370	128	7	x.	x.	NOUN
ejpam-4370	128	8	(	(	PUNCT
ejpam-4370	128	9	2	2	NUM
ejpam-4370	128	10	)	)	PUNCT
ejpam-4370	128	11	⇒	⇒	NOUN
ejpam-4370	128	12	(	(	PUNCT
ejpam-4370	128	13	3	3	NUM
ejpam-4370	128	14	):	):	PUNCT
ejpam-4370	128	15	let	let	VERB
ejpam-4370	128	16	k	k	PRON
ejpam-4370	128	17	be	be	AUX
ejpam-4370	128	18	any	any	DET
ejpam-4370	128	19	δ(λ	δ(λ	PROPN
ejpam-4370	128	20	,	,	PUNCT
ejpam-4370	128	21	sp)-closed	sp)-close	VERB
ejpam-4370	128	22	set	set	NOUN
ejpam-4370	128	23	of	of	ADP
ejpam-4370	128	24	y	y	PROPN
ejpam-4370	128	25	.	.	PUNCT
ejpam-4370	129	1	then	then	ADV
ejpam-4370	129	2	,	,	PUNCT
ejpam-4370	129	3	kδ(λ	kδ(λ	NOUN
ejpam-4370	129	4	,	,	PUNCT
ejpam-4370	129	5	sp	sp	NOUN
ejpam-4370	129	6	)	)	PUNCT
ejpam-4370	129	7	=	=	SYM
ejpam-4370	129	8	k.	k.	NOUN
ejpam-4370	129	9	by	by	ADP
ejpam-4370	129	10	(	(	PUNCT
ejpam-4370	129	11	2	2	NUM
ejpam-4370	129	12	)	)	PUNCT
ejpam-4370	129	13	,	,	PUNCT
ejpam-4370	129	14	f+(k	f+(k	X
ejpam-4370	129	15	)	)	PUNCT
ejpam-4370	129	16	is	be	AUX
ejpam-4370	129	17	(	(	PUNCT
ejpam-4370	129	18	λ	λ	INTJ
ejpam-4370	129	19	,	,	PUNCT
ejpam-4370	129	20	sp)-open	sp)-open	ADJ
ejpam-4370	129	21	in	in	ADP
ejpam-4370	129	22	x.	x.	NOUN
ejpam-4370	129	23	(	(	PUNCT
ejpam-4370	129	24	3	3	NUM
ejpam-4370	129	25	)	)	PUNCT
ejpam-4370	129	26	⇒	⇒	NOUN
ejpam-4370	129	27	(	(	PUNCT
ejpam-4370	129	28	4	4	NUM
ejpam-4370	129	29	):	):	PUNCT
ejpam-4370	129	30	let	let	VERB
ejpam-4370	129	31	v	v	PART
ejpam-4370	129	32	be	be	AUX
ejpam-4370	129	33	any	any	DET
ejpam-4370	129	34	δ(λ	δ(λ	PROPN
ejpam-4370	129	35	,	,	PUNCT
ejpam-4370	129	36	sp)-open	sp)-open	ADJ
ejpam-4370	129	37	set	set	NOUN
ejpam-4370	129	38	of	of	ADP
ejpam-4370	129	39	y	y	PROPN
ejpam-4370	129	40	.	.	PUNCT
ejpam-4370	130	1	then	then	ADV
ejpam-4370	130	2	,	,	PUNCT
ejpam-4370	130	3	y	y	PROPN
ejpam-4370	130	4	−	−	PROPN
ejpam-4370	130	5	v	v	NOUN
ejpam-4370	130	6	is	be	AUX
ejpam-4370	130	7	a	a	DET
ejpam-4370	130	8	δ(λ	δ(λ	PROPN
ejpam-4370	130	9	,	,	PUNCT
ejpam-4370	130	10	sp)-closed	sp)-close	VERB
ejpam-4370	130	11	set	set	NOUN
ejpam-4370	130	12	of	of	ADP
ejpam-4370	130	13	y	y	PROPN
ejpam-4370	130	14	.	.	PUNCT
ejpam-4370	131	1	by	by	ADP
ejpam-4370	131	2	(	(	PUNCT
ejpam-4370	131	3	3	3	NUM
ejpam-4370	131	4	)	)	PUNCT
ejpam-4370	131	5	,	,	PUNCT
ejpam-4370	131	6	we	we	PRON
ejpam-4370	131	7	have	have	VERB
ejpam-4370	131	8	x	x	X
ejpam-4370	131	9	−	−	NOUN
ejpam-4370	131	10	f−(v	f−(v	NOUN
ejpam-4370	131	11	)	)	PUNCT
ejpam-4370	131	12	=	=	PUNCT
ejpam-4370	132	1	f+(y	f+(y	NUM
ejpam-4370	132	2	−	−	PROPN
ejpam-4370	132	3	v	v	NOUN
ejpam-4370	132	4	)	)	PUNCT
ejpam-4370	132	5	is	be	AUX
ejpam-4370	132	6	(	(	PUNCT
ejpam-4370	132	7	λ	λ	X
ejpam-4370	132	8	,	,	PUNCT
ejpam-4370	132	9	sp)-open	sp)-open	ADJ
ejpam-4370	132	10	in	in	ADP
ejpam-4370	132	11	x	x	PUNCT
ejpam-4370	132	12	and	and	CCONJ
ejpam-4370	132	13	hence	hence	ADV
ejpam-4370	132	14	f−(v	f−(v	ADJ
ejpam-4370	132	15	)	)	PUNCT
ejpam-4370	133	1	is	be	AUX
ejpam-4370	133	2	(	(	PUNCT
ejpam-4370	133	3	λ	λ	X
ejpam-4370	133	4	,	,	PUNCT
ejpam-4370	133	5	sp)-closed	sp)-close	VERB
ejpam-4370	133	6	.	.	PUNCT
ejpam-4370	134	1	(	(	PUNCT
ejpam-4370	134	2	4	4	X
ejpam-4370	134	3	)	)	PUNCT
ejpam-4370	134	4	⇒	⇒	NOUN
ejpam-4370	134	5	(	(	PUNCT
ejpam-4370	134	6	1	1	NUM
ejpam-4370	134	7	):	):	PUNCT
ejpam-4370	134	8	let	let	VERB
ejpam-4370	134	9	v	v	PART
ejpam-4370	134	10	be	be	AUX
ejpam-4370	134	11	any	any	DET
ejpam-4370	134	12	(	(	PUNCT
ejpam-4370	134	13	λ	λ	NOUN
ejpam-4370	134	14	,	,	PUNCT
ejpam-4370	134	15	sp)-open	sp)-open	ADJ
ejpam-4370	134	16	set	set	NOUN
ejpam-4370	134	17	of	of	ADP
ejpam-4370	134	18	y	y	PROPN
ejpam-4370	134	19	.	.	PUNCT
ejpam-4370	135	1	since	since	SCONJ
ejpam-4370	135	2	(	(	PUNCT
ejpam-4370	135	3	y	y	PROPN
ejpam-4370	135	4	,	,	PUNCT
ejpam-4370	135	5	σ	σ	PROPN
ejpam-4370	135	6	)	)	PUNCT
ejpam-4370	135	7	is	be	AUX
ejpam-4370	135	8	semi-(λ	semi-(λ	PROPN
ejpam-4370	135	9	,	,	PUNCT
ejpam-4370	135	10	sp)-regular	sp)-regular	NOUN
ejpam-4370	135	11	,	,	PUNCT
ejpam-4370	135	12	by	by	ADP
ejpam-4370	135	13	lemma	lemma	PROPN
ejpam-4370	135	14	3	3	NUM
ejpam-4370	135	15	,	,	PUNCT
ejpam-4370	135	16	v	v	NOUN
ejpam-4370	135	17	is	be	AUX
ejpam-4370	135	18	a	a	DET
ejpam-4370	135	19	δ(λ	δ(λ	PROPN
ejpam-4370	135	20	,	,	PUNCT
ejpam-4370	135	21	sp)-open	sp)-open	ADJ
ejpam-4370	135	22	set	set	NOUN
ejpam-4370	135	23	of	of	ADP
ejpam-4370	135	24	y	y	PROPN
ejpam-4370	135	25	.	.	PUNCT
ejpam-4370	136	1	by	by	ADP
ejpam-4370	136	2	(	(	PUNCT
ejpam-4370	136	3	4	4	NUM
ejpam-4370	136	4	)	)	PUNCT
ejpam-4370	136	5	,	,	PUNCT
ejpam-4370	136	6	f−(v	f−(v	ADJ
ejpam-4370	136	7	)	)	PUNCT
ejpam-4370	136	8	is	be	AUX
ejpam-4370	136	9	(	(	PUNCT
ejpam-4370	136	10	λ	λ	X
ejpam-4370	136	11	,	,	PUNCT
ejpam-4370	136	12	sp)-closed	sp)-close	VERB
ejpam-4370	136	13	in	in	ADP
ejpam-4370	136	14	x	x	PUNCT
ejpam-4370	136	15	and	and	CCONJ
ejpam-4370	136	16	by	by	ADP
ejpam-4370	136	17	theorem	theorem	NOUN
ejpam-4370	136	18	1	1	NUM
ejpam-4370	136	19	,	,	PUNCT
ejpam-4370	136	20	f	f	PROPN
ejpam-4370	136	21	is	be	AUX
ejpam-4370	136	22	upper	upper	ADJ
ejpam-4370	136	23	contra-(λ	contra-(λ	PROPN
ejpam-4370	136	24	,	,	PUNCT
ejpam-4370	136	25	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	136	26	.	.	PUNCT
ejpam-4370	137	1	theorem	theorem	VERB
ejpam-4370	137	2	4	4	NUM
ejpam-4370	137	3	.	.	X
ejpam-4370	137	4	for	for	ADP
ejpam-4370	137	5	a	a	DET
ejpam-4370	137	6	multifunction	multifunction	NOUN
ejpam-4370	138	1	f	f	NOUN
ejpam-4370	138	2	:	:	PUNCT
ejpam-4370	138	3	(	(	PUNCT
ejpam-4370	138	4	x	x	X
ejpam-4370	138	5	,	,	PUNCT
ejpam-4370	138	6	τ	τ	X
ejpam-4370	138	7	)	)	PUNCT
ejpam-4370	138	8	→	→	SYM
ejpam-4370	138	9	(	(	PUNCT
ejpam-4370	138	10	y	y	PROPN
ejpam-4370	138	11	,	,	PUNCT
ejpam-4370	138	12	σ	σ	PROPN
ejpam-4370	138	13	)	)	PUNCT
ejpam-4370	138	14	,	,	PUNCT
ejpam-4370	138	15	where	where	SCONJ
ejpam-4370	138	16	(	(	PUNCT
ejpam-4370	138	17	y	y	PROPN
ejpam-4370	138	18	,	,	PUNCT
ejpam-4370	138	19	σ	σ	PROPN
ejpam-4370	138	20	)	)	PUNCT
ejpam-4370	138	21	is	be	AUX
ejpam-4370	138	22	a	a	DET
ejpam-4370	138	23	semi-(λ	semi-(λ	PROPN
ejpam-4370	138	24	,	,	PUNCT
ejpam-4370	138	25	sp)regular	sp)regular	ADJ
ejpam-4370	138	26	space	space	NOUN
ejpam-4370	138	27	,	,	PUNCT
ejpam-4370	138	28	the	the	DET
ejpam-4370	138	29	following	follow	VERB
ejpam-4370	138	30	properties	property	NOUN
ejpam-4370	138	31	are	be	AUX
ejpam-4370	138	32	equivalent	equivalent	ADJ
ejpam-4370	138	33	:	:	PUNCT
ejpam-4370	138	34	(	(	PUNCT
ejpam-4370	138	35	1	1	X
ejpam-4370	138	36	)	)	PUNCT
ejpam-4370	138	37	f	f	PROPN
ejpam-4370	138	38	is	be	AUX
ejpam-4370	138	39	lower	low	ADJ
ejpam-4370	138	40	contra-(λ	contra-(λ	PROPN
ejpam-4370	138	41	,	,	PUNCT
ejpam-4370	138	42	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	138	43	;	;	PUNCT
ejpam-4370	138	44	(	(	PUNCT
ejpam-4370	138	45	2	2	X
ejpam-4370	138	46	)	)	PUNCT
ejpam-4370	138	47	f−(bδ(λ	f−(bδ(λ	NOUN
ejpam-4370	138	48	,	,	PUNCT
ejpam-4370	138	49	sp	sp	NOUN
ejpam-4370	138	50	)	)	PUNCT
ejpam-4370	138	51	)	)	PUNCT
ejpam-4370	139	1	is	be	AUX
ejpam-4370	139	2	(	(	PUNCT
ejpam-4370	139	3	λ	λ	INTJ
ejpam-4370	139	4	,	,	PUNCT
ejpam-4370	139	5	sp)-open	sp)-open	ADJ
ejpam-4370	139	6	in	in	ADP
ejpam-4370	139	7	x	x	PUNCT
ejpam-4370	139	8	for	for	ADP
ejpam-4370	139	9	every	every	DET
ejpam-4370	139	10	subset	subset	NOUN
ejpam-4370	139	11	b	b	PROPN
ejpam-4370	139	12	of	of	ADP
ejpam-4370	139	13	y	y	PROPN
ejpam-4370	139	14	;	;	PUNCT
ejpam-4370	139	15	(	(	PUNCT
ejpam-4370	139	16	3	3	X
ejpam-4370	139	17	)	)	PUNCT
ejpam-4370	139	18	f−(k	f−(k	PROPN
ejpam-4370	139	19	)	)	PUNCT
ejpam-4370	139	20	is	be	AUX
ejpam-4370	139	21	(	(	PUNCT
ejpam-4370	139	22	λ	λ	INTJ
ejpam-4370	139	23	,	,	PUNCT
ejpam-4370	139	24	sp)-open	sp)-open	ADJ
ejpam-4370	139	25	in	in	ADP
ejpam-4370	139	26	x	x	PUNCT
ejpam-4370	139	27	for	for	ADP
ejpam-4370	139	28	every	every	DET
ejpam-4370	139	29	δ(λ	δ(λ	PROPN
ejpam-4370	139	30	,	,	PUNCT
ejpam-4370	139	31	sp)-closed	sp)-close	VERB
ejpam-4370	139	32	set	set	VERB
ejpam-4370	139	33	k	k	PROPN
ejpam-4370	139	34	of	of	ADP
ejpam-4370	139	35	y	y	PROPN
ejpam-4370	139	36	;	;	PUNCT
ejpam-4370	139	37	c.	c.	PROPN
ejpam-4370	139	38	boonpok	boonpok	PROPN
ejpam-4370	139	39	,	,	PUNCT
ejpam-4370	139	40	c.	c.	PROPN
ejpam-4370	139	41	viriyapong	viriyapong	PROPN
ejpam-4370	139	42	/	/	SYM
ejpam-4370	139	43	eur	eur	PROPN
ejpam-4370	139	44	.	.	PUNCT
ejpam-4370	140	1	j.	j.	PROPN
ejpam-4370	140	2	pure	pure	PROPN
ejpam-4370	140	3	appl	appl	PROPN
ejpam-4370	140	4	.	.	PROPN
ejpam-4370	140	5	math	math	PROPN
ejpam-4370	140	6	,	,	PUNCT
ejpam-4370	140	7	15	15	NUM
ejpam-4370	140	8	(	(	PUNCT
ejpam-4370	140	9	4	4	NUM
ejpam-4370	140	10	)	)	PUNCT
ejpam-4370	140	11	(	(	PUNCT
ejpam-4370	140	12	2022	2022	NUM
ejpam-4370	140	13	)	)	PUNCT
ejpam-4370	140	14	,	,	PUNCT
ejpam-4370	140	15	1694	1694	NUM
ejpam-4370	140	16	-	-	SYM
ejpam-4370	140	17	1704	1704	NUM
ejpam-4370	140	18	1699	1699	NUM
ejpam-4370	140	19	(	(	PUNCT
ejpam-4370	140	20	4	4	NUM
ejpam-4370	140	21	)	)	PUNCT
ejpam-4370	140	22	f+(v	f+(v	NOUN
ejpam-4370	140	23	)	)	PUNCT
ejpam-4370	140	24	is	be	AUX
ejpam-4370	140	25	(	(	PUNCT
ejpam-4370	140	26	λ	λ	X
ejpam-4370	140	27	,	,	PUNCT
ejpam-4370	140	28	sp)-closed	sp)-close	VERB
ejpam-4370	140	29	in	in	ADP
ejpam-4370	140	30	x	x	PUNCT
ejpam-4370	140	31	for	for	ADP
ejpam-4370	140	32	every	every	DET
ejpam-4370	140	33	δ(λ	δ(λ	PROPN
ejpam-4370	140	34	,	,	PUNCT
ejpam-4370	140	35	sp)-open	sp)-open	VERB
ejpam-4370	140	36	set	set	VERB
ejpam-4370	140	37	v	v	NOUN
ejpam-4370	140	38	of	of	ADP
ejpam-4370	140	39	y	y	PROPN
ejpam-4370	140	40	.	.	PUNCT
ejpam-4370	141	1	proof	proof	NOUN
ejpam-4370	141	2	.	.	PUNCT
ejpam-4370	142	1	the	the	DET
ejpam-4370	142	2	proof	proof	NOUN
ejpam-4370	142	3	is	be	AUX
ejpam-4370	142	4	similar	similar	ADJ
ejpam-4370	142	5	to	to	ADP
ejpam-4370	142	6	that	that	PRON
ejpam-4370	142	7	of	of	ADP
ejpam-4370	142	8	theorem	theorem	ADJ
ejpam-4370	142	9	3	3	NUM
ejpam-4370	142	10	.	.	PUNCT
ejpam-4370	142	11	corollary	corollary	ADJ
ejpam-4370	142	12	2	2	NUM
ejpam-4370	142	13	.	.	PUNCT
ejpam-4370	142	14	for	for	ADP
ejpam-4370	142	15	a	a	DET
ejpam-4370	142	16	function	function	NOUN
ejpam-4370	142	17	f	f	NOUN
ejpam-4370	142	18	:	:	PUNCT
ejpam-4370	142	19	(	(	PUNCT
ejpam-4370	142	20	x	x	X
ejpam-4370	142	21	,	,	PUNCT
ejpam-4370	142	22	τ	τ	X
ejpam-4370	142	23	)	)	PUNCT
ejpam-4370	142	24	→	→	SYM
ejpam-4370	142	25	(	(	PUNCT
ejpam-4370	142	26	y	y	PROPN
ejpam-4370	142	27	,	,	PUNCT
ejpam-4370	142	28	σ	σ	PROPN
ejpam-4370	142	29	)	)	PUNCT
ejpam-4370	142	30	,	,	PUNCT
ejpam-4370	142	31	where	where	SCONJ
ejpam-4370	142	32	(	(	PUNCT
ejpam-4370	142	33	y	y	PROPN
ejpam-4370	142	34	,	,	PUNCT
ejpam-4370	142	35	σ	σ	PROPN
ejpam-4370	142	36	)	)	PUNCT
ejpam-4370	142	37	is	be	AUX
ejpam-4370	142	38	a	a	DET
ejpam-4370	142	39	semi-(λ	semi-(λ	PROPN
ejpam-4370	142	40	,	,	PUNCT
ejpam-4370	142	41	sp)-regular	sp)-regular	ADJ
ejpam-4370	142	42	space	space	NOUN
ejpam-4370	142	43	,	,	PUNCT
ejpam-4370	142	44	the	the	DET
ejpam-4370	142	45	following	follow	VERB
ejpam-4370	142	46	properties	property	NOUN
ejpam-4370	142	47	are	be	AUX
ejpam-4370	142	48	equivalent	equivalent	ADJ
ejpam-4370	142	49	:	:	PUNCT
ejpam-4370	142	50	(	(	PUNCT
ejpam-4370	142	51	1	1	X
ejpam-4370	142	52	)	)	PUNCT
ejpam-4370	142	53	f	f	PROPN
ejpam-4370	142	54	is	be	AUX
ejpam-4370	142	55	contra-(λ	contra-(λ	PROPN
ejpam-4370	142	56	,	,	PUNCT
ejpam-4370	142	57	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	142	58	;	;	PUNCT
ejpam-4370	142	59	(	(	PUNCT
ejpam-4370	142	60	2	2	X
ejpam-4370	142	61	)	)	PUNCT
ejpam-4370	142	62	f−1(bδ(λ	f−1(bδ(λ	NOUN
ejpam-4370	142	63	,	,	PUNCT
ejpam-4370	142	64	sp	sp	NOUN
ejpam-4370	142	65	)	)	PUNCT
ejpam-4370	142	66	)	)	PUNCT
ejpam-4370	143	1	is	be	AUX
ejpam-4370	143	2	(	(	PUNCT
ejpam-4370	143	3	λ	λ	INTJ
ejpam-4370	143	4	,	,	PUNCT
ejpam-4370	143	5	sp)-open	sp)-open	ADJ
ejpam-4370	143	6	in	in	ADP
ejpam-4370	143	7	x	x	PUNCT
ejpam-4370	143	8	for	for	ADP
ejpam-4370	143	9	every	every	DET
ejpam-4370	143	10	subset	subset	NOUN
ejpam-4370	143	11	b	b	PROPN
ejpam-4370	143	12	of	of	ADP
ejpam-4370	143	13	y	y	PROPN
ejpam-4370	143	14	;	;	PUNCT
ejpam-4370	143	15	(	(	PUNCT
ejpam-4370	143	16	3	3	X
ejpam-4370	143	17	)	)	PUNCT
ejpam-4370	143	18	f−1(k	f−1(k	PROPN
ejpam-4370	143	19	)	)	PUNCT
ejpam-4370	144	1	is	be	AUX
ejpam-4370	144	2	(	(	PUNCT
ejpam-4370	144	3	λ	λ	INTJ
ejpam-4370	144	4	,	,	PUNCT
ejpam-4370	144	5	sp)-open	sp)-open	ADJ
ejpam-4370	144	6	in	in	ADP
ejpam-4370	144	7	x	x	PUNCT
ejpam-4370	144	8	for	for	ADP
ejpam-4370	144	9	every	every	DET
ejpam-4370	144	10	δ(λ	δ(λ	PROPN
ejpam-4370	144	11	,	,	PUNCT
ejpam-4370	144	12	sp)-closed	sp)-close	VERB
ejpam-4370	144	13	set	set	VERB
ejpam-4370	144	14	k	k	PROPN
ejpam-4370	144	15	of	of	ADP
ejpam-4370	144	16	y	y	PROPN
ejpam-4370	144	17	;	;	PUNCT
ejpam-4370	144	18	(	(	PUNCT
ejpam-4370	144	19	4	4	X
ejpam-4370	144	20	)	)	PUNCT
ejpam-4370	144	21	f−1(v	f−1(v	NOUN
ejpam-4370	144	22	)	)	PUNCT
ejpam-4370	144	23	is	be	AUX
ejpam-4370	144	24	(	(	PUNCT
ejpam-4370	144	25	λ	λ	X
ejpam-4370	144	26	,	,	PUNCT
ejpam-4370	144	27	sp)-closed	sp)-close	VERB
ejpam-4370	144	28	in	in	ADP
ejpam-4370	144	29	x	x	PUNCT
ejpam-4370	144	30	for	for	ADP
ejpam-4370	144	31	every	every	DET
ejpam-4370	144	32	δ(λ	δ(λ	PROPN
ejpam-4370	144	33	,	,	PUNCT
ejpam-4370	144	34	sp)-open	sp)-open	VERB
ejpam-4370	144	35	set	set	VERB
ejpam-4370	144	36	v	v	NOUN
ejpam-4370	144	37	of	of	ADP
ejpam-4370	144	38	y	y	PROPN
ejpam-4370	144	39	.	.	PUNCT
ejpam-4370	145	1	definition	definition	NOUN
ejpam-4370	145	2	4	4	NUM
ejpam-4370	145	3	.	.	PUNCT
ejpam-4370	146	1	a	a	DET
ejpam-4370	146	2	subset	subset	NOUN
ejpam-4370	146	3	k	k	PROPN
ejpam-4370	146	4	of	of	ADP
ejpam-4370	146	5	a	a	DET
ejpam-4370	146	6	topological	topological	ADJ
ejpam-4370	146	7	space	space	NOUN
ejpam-4370	146	8	(	(	PUNCT
ejpam-4370	146	9	x	x	X
ejpam-4370	146	10	,	,	PUNCT
ejpam-4370	146	11	τ	τ	X
ejpam-4370	146	12	)	)	PUNCT
ejpam-4370	146	13	is	be	AUX
ejpam-4370	146	14	called	call	VERB
ejpam-4370	146	15	strongly	strongly	ADV
ejpam-4370	146	16	sλsp	sλsp	ADV
ejpam-4370	146	17	-	-	PUNCT
ejpam-4370	146	18	closed	closed	ADJ
ejpam-4370	146	19	(	(	PUNCT
ejpam-4370	146	20	resp	resp	NOUN
ejpam-4370	146	21	.	.	PUNCT
ejpam-4370	147	1	λsp	λsp	PROPN
ejpam-4370	147	2	-	-	PUNCT
ejpam-4370	147	3	compact	compact	ADJ
ejpam-4370	147	4	)	)	PUNCT
ejpam-4370	148	1	relative	relative	ADJ
ejpam-4370	148	2	to	to	ADP
ejpam-4370	148	3	x	x	PRON
ejpam-4370	148	4	if	if	SCONJ
ejpam-4370	148	5	every	every	DET
ejpam-4370	148	6	cover	cover	NOUN
ejpam-4370	148	7	of	of	ADP
ejpam-4370	148	8	k	k	X
ejpam-4370	148	9	by	by	ADP
ejpam-4370	148	10	(	(	PUNCT
ejpam-4370	148	11	λ	λ	PROPN
ejpam-4370	148	12	,	,	PUNCT
ejpam-4370	148	13	sp)-closed	sp)-close	VERB
ejpam-4370	148	14	(	(	PUNCT
ejpam-4370	148	15	resp	resp	NOUN
ejpam-4370	148	16	.	.	PUNCT
ejpam-4370	149	1	(	(	PUNCT
ejpam-4370	149	2	λ	λ	NOUN
ejpam-4370	149	3	,	,	PUNCT
ejpam-4370	149	4	sp)-open	sp)-open	NOUN
ejpam-4370	149	5	)	)	PUNCT
ejpam-4370	149	6	sets	set	NOUN
ejpam-4370	149	7	of	of	ADP
ejpam-4370	149	8	x	x	PUNCT
ejpam-4370	149	9	has	have	VERB
ejpam-4370	149	10	a	a	DET
ejpam-4370	149	11	finite	finite	ADJ
ejpam-4370	149	12	subcover	subcover	PROPN
ejpam-4370	149	13	.	.	PUNCT
ejpam-4370	150	1	a	a	DET
ejpam-4370	150	2	topological	topological	ADJ
ejpam-4370	150	3	space	space	NOUN
ejpam-4370	150	4	(	(	PUNCT
ejpam-4370	150	5	x	x	X
ejpam-4370	150	6	,	,	PUNCT
ejpam-4370	150	7	τ	τ	X
ejpam-4370	150	8	)	)	PUNCT
ejpam-4370	150	9	is	be	AUX
ejpam-4370	150	10	called	call	VERB
ejpam-4370	150	11	strongly	strongly	ADV
ejpam-4370	150	12	sλsp	sλsp	ADV
ejpam-4370	150	13	-	-	PUNCT
ejpam-4370	150	14	closed	closed	ADJ
ejpam-4370	150	15	(	(	PUNCT
ejpam-4370	150	16	resp	resp	NOUN
ejpam-4370	150	17	.	.	PUNCT
ejpam-4370	151	1	λsp	λsp	PROPN
ejpam-4370	151	2	-	-	NOUN
ejpam-4370	151	3	compact	compact	ADJ
ejpam-4370	152	1	[	[	X
ejpam-4370	152	2	22	22	NUM
ejpam-4370	152	3	]	]	SYM
ejpam-4370	152	4	)	)	PUNCT
ejpam-4370	152	5	if	if	SCONJ
ejpam-4370	152	6	x	x	PRON
ejpam-4370	152	7	is	be	AUX
ejpam-4370	152	8	strongly	strongly	ADV
ejpam-4370	152	9	sλsp	sλsp	ADV
ejpam-4370	152	10	-	-	PUNCT
ejpam-4370	152	11	closed	closed	ADJ
ejpam-4370	152	12	(	(	PUNCT
ejpam-4370	152	13	resp	resp	NOUN
ejpam-4370	152	14	.	.	PUNCT
ejpam-4370	153	1	λsp	λsp	PROPN
ejpam-4370	153	2	-	-	PUNCT
ejpam-4370	153	3	compact	compact	ADJ
ejpam-4370	153	4	)	)	PUNCT
ejpam-4370	154	1	relative	relative	ADJ
ejpam-4370	154	2	to	to	ADP
ejpam-4370	154	3	x.	x.	NOUN
ejpam-4370	154	4	theorem	theorem	ADJ
ejpam-4370	154	5	5	5	NUM
ejpam-4370	154	6	.	.	PUNCT
ejpam-4370	155	1	let	let	VERB
ejpam-4370	155	2	f	f	NOUN
ejpam-4370	155	3	:	:	PUNCT
ejpam-4370	155	4	(	(	PUNCT
ejpam-4370	155	5	x	x	X
ejpam-4370	155	6	,	,	PUNCT
ejpam-4370	155	7	τ	τ	X
ejpam-4370	155	8	)	)	PUNCT
ejpam-4370	155	9	→	→	SYM
ejpam-4370	155	10	(	(	PUNCT
ejpam-4370	155	11	y	y	PROPN
ejpam-4370	155	12	,	,	PUNCT
ejpam-4370	155	13	σ	σ	PROPN
ejpam-4370	155	14	)	)	PUNCT
ejpam-4370	155	15	be	be	VERB
ejpam-4370	155	16	an	an	DET
ejpam-4370	155	17	upper	upper	ADJ
ejpam-4370	155	18	contra-(λ	contra-(λ	PROPN
ejpam-4370	155	19	,	,	PUNCT
ejpam-4370	155	20	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	155	21	surjective	surjective	ADJ
ejpam-4370	155	22	multifunction	multifunction	NOUN
ejpam-4370	155	23	such	such	ADJ
ejpam-4370	155	24	that	that	SCONJ
ejpam-4370	155	25	f	f	PROPN
ejpam-4370	155	26	(	(	PUNCT
ejpam-4370	155	27	x	x	X
ejpam-4370	155	28	)	)	PUNCT
ejpam-4370	155	29	is	be	AUX
ejpam-4370	155	30	strongly	strongly	ADV
ejpam-4370	155	31	sλsp	sλsp	ADV
ejpam-4370	155	32	-	-	PUNCT
ejpam-4370	155	33	closed	closed	ADJ
ejpam-4370	155	34	relative	relative	ADJ
ejpam-4370	155	35	to	to	ADP
ejpam-4370	155	36	y	y	PROPN
ejpam-4370	155	37	for	for	ADP
ejpam-4370	155	38	each	each	DET
ejpam-4370	155	39	x	x	SYM
ejpam-4370	155	40	∈	∈	PROPN
ejpam-4370	155	41	x.	x.	NOUN
ejpam-4370	156	1	if	if	SCONJ
ejpam-4370	156	2	a	a	PRON
ejpam-4370	156	3	is	be	AUX
ejpam-4370	156	4	λsp	λsp	ADJ
ejpam-4370	156	5	-	-	ADJ
ejpam-4370	156	6	compact	compact	ADJ
ejpam-4370	156	7	relative	relative	NOUN
ejpam-4370	156	8	to	to	ADP
ejpam-4370	156	9	x	x	PRON
ejpam-4370	156	10	,	,	PUNCT
ejpam-4370	156	11	then	then	ADV
ejpam-4370	156	12	f	f	X
ejpam-4370	156	13	(	(	PUNCT
ejpam-4370	156	14	a	a	X
ejpam-4370	156	15	)	)	PUNCT
ejpam-4370	156	16	is	be	AUX
ejpam-4370	156	17	strongly	strongly	ADV
ejpam-4370	156	18	sλsp	sλsp	ADV
ejpam-4370	156	19	-	-	PUNCT
ejpam-4370	156	20	closed	closed	ADJ
ejpam-4370	156	21	relative	relative	ADJ
ejpam-4370	156	22	to	to	ADP
ejpam-4370	156	23	y	y	PROPN
ejpam-4370	156	24	.	.	PUNCT
ejpam-4370	157	1	proof	proof	NOUN
ejpam-4370	157	2	.	.	PUNCT
ejpam-4370	158	1	let	let	VERB
ejpam-4370	158	2	{	{	PUNCT
ejpam-4370	158	3	vα	vα	VERB
ejpam-4370	158	4	|	|	ADV
ejpam-4370	158	5	α	α	NUM
ejpam-4370	158	6	∈	∈	NOUN
ejpam-4370	158	7	∇	∇	X
ejpam-4370	158	8	}	}	PUNCT
ejpam-4370	158	9	be	be	AUX
ejpam-4370	158	10	any	any	DET
ejpam-4370	158	11	cover	cover	NOUN
ejpam-4370	158	12	of	of	ADP
ejpam-4370	158	13	f	f	PROPN
ejpam-4370	158	14	(	(	PUNCT
ejpam-4370	158	15	a	a	NOUN
ejpam-4370	158	16	)	)	PUNCT
ejpam-4370	158	17	by	by	ADP
ejpam-4370	158	18	(	(	PUNCT
ejpam-4370	158	19	λ	λ	PROPN
ejpam-4370	158	20	,	,	PUNCT
ejpam-4370	158	21	sp)-closed	sp)-close	VERB
ejpam-4370	158	22	sets	set	NOUN
ejpam-4370	158	23	of	of	ADP
ejpam-4370	158	24	y	y	PROPN
ejpam-4370	158	25	.	.	PUNCT
ejpam-4370	159	1	for	for	ADP
ejpam-4370	159	2	each	each	DET
ejpam-4370	159	3	x	x	SYM
ejpam-4370	159	4	∈	∈	PROPN
ejpam-4370	159	5	a	a	X
ejpam-4370	159	6	,	,	PUNCT
ejpam-4370	159	7	f	f	PROPN
ejpam-4370	159	8	(	(	PUNCT
ejpam-4370	159	9	x	x	X
ejpam-4370	159	10	)	)	PUNCT
ejpam-4370	159	11	is	be	AUX
ejpam-4370	159	12	strongly	strongly	ADV
ejpam-4370	159	13	sλsp	sλsp	ADV
ejpam-4370	159	14	-	-	PUNCT
ejpam-4370	159	15	closed	closed	ADJ
ejpam-4370	159	16	relative	relative	ADJ
ejpam-4370	159	17	to	to	ADP
ejpam-4370	159	18	y	y	PROPN
ejpam-4370	159	19	and	and	CCONJ
ejpam-4370	159	20	there	there	PRON
ejpam-4370	159	21	exists	exist	VERB
ejpam-4370	159	22	a	a	DET
ejpam-4370	159	23	finite	finite	NOUN
ejpam-4370	159	24	subset	subset	NOUN
ejpam-4370	159	25	∇(x	∇(x	NOUN
ejpam-4370	159	26	)	)	PUNCT
ejpam-4370	159	27	of	of	ADP
ejpam-4370	159	28	∇	∇	NOUN
ejpam-4370	159	29	such	such	ADJ
ejpam-4370	159	30	that	that	SCONJ
ejpam-4370	159	31	f	f	PROPN
ejpam-4370	159	32	(	(	PUNCT
ejpam-4370	159	33	x	x	X
ejpam-4370	159	34	)	)	PUNCT
ejpam-4370	159	35	⊆	⊆	NUM
ejpam-4370	159	36	∪{vα	∪{vα	NOUN
ejpam-4370	159	37	|	|	ADV
ejpam-4370	159	38	α	α	PROPN
ejpam-4370	159	39	∈	∈	PROPN
ejpam-4370	159	40	∇(x	∇(x	NUM
ejpam-4370	159	41	)	)	PUNCT
ejpam-4370	159	42	}	}	PUNCT
ejpam-4370	159	43	.	.	PUNCT
ejpam-4370	160	1	put	put	VERB
ejpam-4370	160	2	v	v	NOUN
ejpam-4370	160	3	(	(	PUNCT
ejpam-4370	160	4	x	x	NOUN
ejpam-4370	160	5	)	)	PUNCT
ejpam-4370	160	6	=	=	PUNCT
ejpam-4370	161	1	∪{vα	∪{vα	NOUN
ejpam-4370	161	2	|	|	ADV
ejpam-4370	161	3	α	α	PROPN
ejpam-4370	161	4	∈	∈	PROPN
ejpam-4370	161	5	∇(x	∇(x	NUM
ejpam-4370	161	6	)	)	PUNCT
ejpam-4370	161	7	}	}	PUNCT
ejpam-4370	161	8	.	.	PUNCT
ejpam-4370	162	1	then	then	ADV
ejpam-4370	162	2	,	,	PUNCT
ejpam-4370	162	3	f	f	PROPN
ejpam-4370	162	4	(	(	PUNCT
ejpam-4370	162	5	x	x	X
ejpam-4370	162	6	)	)	PUNCT
ejpam-4370	162	7	⊆	⊆	NUM
ejpam-4370	162	8	v	v	NOUN
ejpam-4370	162	9	(	(	PUNCT
ejpam-4370	162	10	x	x	NOUN
ejpam-4370	162	11	)	)	PUNCT
ejpam-4370	162	12	.	.	PUNCT
ejpam-4370	163	1	since	since	SCONJ
ejpam-4370	163	2	f	f	PROPN
ejpam-4370	163	3	is	be	AUX
ejpam-4370	163	4	upper	upper	ADJ
ejpam-4370	163	5	contra-(λ	contra-(λ	PROPN
ejpam-4370	163	6	,	,	PUNCT
ejpam-4370	163	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	163	8	,	,	PUNCT
ejpam-4370	163	9	there	there	PRON
ejpam-4370	163	10	exists	exist	VERB
ejpam-4370	163	11	a	a	DET
ejpam-4370	163	12	(	(	PUNCT
ejpam-4370	163	13	λ	λ	NOUN
ejpam-4370	163	14	,	,	PUNCT
ejpam-4370	163	15	sp)-open	sp)-open	ADJ
ejpam-4370	163	16	u(x	u(x	NOUN
ejpam-4370	163	17	)	)	PUNCT
ejpam-4370	163	18	of	of	ADP
ejpam-4370	163	19	x	x	SYM
ejpam-4370	163	20	containing	contain	VERB
ejpam-4370	163	21	x	x	PUNCT
ejpam-4370	163	22	such	such	ADJ
ejpam-4370	163	23	that	that	SCONJ
ejpam-4370	163	24	f	f	PROPN
ejpam-4370	163	25	(	(	PUNCT
ejpam-4370	163	26	u(x	u(x	PROPN
ejpam-4370	163	27	)	)	PUNCT
ejpam-4370	163	28	)	)	PUNCT
ejpam-4370	164	1	⊆	⊆	NUM
ejpam-4370	164	2	v	v	X
ejpam-4370	164	3	(	(	PUNCT
ejpam-4370	164	4	x	x	NOUN
ejpam-4370	164	5	)	)	PUNCT
ejpam-4370	164	6	.	.	PUNCT
ejpam-4370	165	1	since	since	SCONJ
ejpam-4370	165	2	{	{	PUNCT
ejpam-4370	165	3	u(x	u(x	NOUN
ejpam-4370	165	4	)	)	PUNCT
ejpam-4370	165	5	|	|	ADV
ejpam-4370	165	6	x	x	SYM
ejpam-4370	165	7	∈	∈	PROPN
ejpam-4370	165	8	a	a	PRON
ejpam-4370	165	9	}	}	PUNCT
ejpam-4370	165	10	is	be	AUX
ejpam-4370	165	11	a	a	DET
ejpam-4370	165	12	cover	cover	NOUN
ejpam-4370	165	13	of	of	ADP
ejpam-4370	165	14	a	a	DET
ejpam-4370	165	15	by	by	ADP
ejpam-4370	165	16	(	(	PUNCT
ejpam-4370	165	17	λ	λ	PROPN
ejpam-4370	165	18	,	,	PUNCT
ejpam-4370	165	19	sp)-open	sp)-open	ADJ
ejpam-4370	165	20	sets	set	NOUN
ejpam-4370	165	21	of	of	ADP
ejpam-4370	165	22	x	x	PRON
ejpam-4370	165	23	,	,	PUNCT
ejpam-4370	165	24	there	there	PRON
ejpam-4370	165	25	exists	exist	VERB
ejpam-4370	165	26	a	a	DET
ejpam-4370	165	27	finite	finite	ADJ
ejpam-4370	165	28	number	number	NOUN
ejpam-4370	165	29	of	of	ADP
ejpam-4370	165	30	points	point	NOUN
ejpam-4370	165	31	of	of	ADP
ejpam-4370	165	32	a	a	DET
ejpam-4370	165	33	,	,	PUNCT
ejpam-4370	165	34	say	say	INTJ
ejpam-4370	165	35	,	,	PUNCT
ejpam-4370	165	36	x1	x1	PROPN
ejpam-4370	165	37	,	,	PUNCT
ejpam-4370	165	38	x2	x2	PROPN
ejpam-4370	165	39	,	,	PUNCT
ejpam-4370	165	40	...	...	PUNCT
ejpam-4370	165	41	,	,	PUNCT
ejpam-4370	165	42	xn	xn	PROPN
ejpam-4370	165	43	such	such	ADJ
ejpam-4370	165	44	that	that	SCONJ
ejpam-4370	165	45	a	a	DET
ejpam-4370	165	46	⊆	⊆	NUM
ejpam-4370	165	47	∪{u(xi	∪{u(xi	NUM
ejpam-4370	165	48	)	)	PUNCT
ejpam-4370	165	49	|	|	ADV
ejpam-4370	165	50	1	1	NUM
ejpam-4370	165	51	≤	≤	NUM
ejpam-4370	165	52	i	i	PRON
ejpam-4370	165	53	≤	≤	NOUN
ejpam-4370	165	54	n	n	CCONJ
ejpam-4370	165	55	}	}	PUNCT
ejpam-4370	165	56	.	.	PUNCT
ejpam-4370	166	1	thus	thus	ADV
ejpam-4370	166	2	,	,	PUNCT
ejpam-4370	166	3	f	f	PROPN
ejpam-4370	166	4	(	(	PUNCT
ejpam-4370	166	5	a	a	NOUN
ejpam-4370	166	6	)	)	PUNCT
ejpam-4370	166	7	⊆	⊆	NUM
ejpam-4370	166	8	f	f	X
ejpam-4370	166	9	(	(	PUNCT
ejpam-4370	166	10	n	n	CCONJ
ejpam-4370	166	11	∪	∪	VERB
ejpam-4370	166	12	i=1	i=1	PROPN
ejpam-4370	166	13	u(xi	u(xi	NOUN
ejpam-4370	166	14	)	)	PUNCT
ejpam-4370	166	15	)	)	PUNCT
ejpam-4370	167	1	⊆	⊆	NUM
ejpam-4370	167	2	n	n	NOUN
ejpam-4370	167	3	∪	∪	VERB
ejpam-4370	167	4	i=1	i=1	PROPN
ejpam-4370	167	5	f	f	PROPN
ejpam-4370	167	6	(	(	PUNCT
ejpam-4370	167	7	u(xi	u(xi	PROPN
ejpam-4370	167	8	)	)	PUNCT
ejpam-4370	167	9	)	)	PUNCT
ejpam-4370	168	1	⊆	⊆	NUM
ejpam-4370	168	2	n	n	PRON
ejpam-4370	168	3	∪	∪	VERB
ejpam-4370	168	4	i=1	i=1	PROPN
ejpam-4370	168	5	v	v	PROPN
ejpam-4370	168	6	(	(	PUNCT
ejpam-4370	168	7	xi	xi	PROPN
ejpam-4370	168	8	)	)	PUNCT
ejpam-4370	168	9	⊆	⊆	NUM
ejpam-4370	168	10	n	n	NOUN
ejpam-4370	168	11	∪	∪	VERB
ejpam-4370	168	12	i=1	i=1	PROPN
ejpam-4370	169	1	[	[	X
ejpam-4370	169	2	∪α∈∇(xi)vα	∪α∈∇(xi)vα	X
ejpam-4370	169	3	]	]	X
ejpam-4370	169	4	and	and	CCONJ
ejpam-4370	169	5	hence	hence	ADV
ejpam-4370	169	6	f	f	PROPN
ejpam-4370	169	7	(	(	PUNCT
ejpam-4370	169	8	a	a	X
ejpam-4370	169	9	)	)	PUNCT
ejpam-4370	169	10	is	be	AUX
ejpam-4370	169	11	strongly	strongly	ADV
ejpam-4370	169	12	sλsp	sλsp	ADV
ejpam-4370	169	13	-	-	PUNCT
ejpam-4370	169	14	closed	closed	ADJ
ejpam-4370	169	15	relative	relative	ADJ
ejpam-4370	169	16	to	to	ADP
ejpam-4370	169	17	y	y	PROPN
ejpam-4370	169	18	.	.	PUNCT
ejpam-4370	170	1	corollary	corollary	ADJ
ejpam-4370	170	2	3	3	X
ejpam-4370	170	3	.	.	PUNCT
ejpam-4370	171	1	let	let	VERB
ejpam-4370	171	2	f	f	NOUN
ejpam-4370	171	3	:	:	PUNCT
ejpam-4370	171	4	(	(	PUNCT
ejpam-4370	171	5	x	x	X
ejpam-4370	171	6	,	,	PUNCT
ejpam-4370	171	7	τ	τ	X
ejpam-4370	171	8	)	)	PUNCT
ejpam-4370	171	9	→	→	SYM
ejpam-4370	171	10	(	(	PUNCT
ejpam-4370	171	11	y	y	PROPN
ejpam-4370	171	12	,	,	PUNCT
ejpam-4370	171	13	σ	σ	PROPN
ejpam-4370	171	14	)	)	PUNCT
ejpam-4370	171	15	be	be	VERB
ejpam-4370	171	16	an	an	DET
ejpam-4370	171	17	upper	upper	ADJ
ejpam-4370	171	18	contra-(λ	contra-(λ	PROPN
ejpam-4370	171	19	,	,	PUNCT
ejpam-4370	171	20	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	171	21	surjective	surjective	ADJ
ejpam-4370	171	22	multifunction	multifunction	NOUN
ejpam-4370	171	23	such	such	ADJ
ejpam-4370	171	24	that	that	SCONJ
ejpam-4370	171	25	f	f	PROPN
ejpam-4370	171	26	(	(	PUNCT
ejpam-4370	171	27	x	x	X
ejpam-4370	171	28	)	)	PUNCT
ejpam-4370	171	29	is	be	AUX
ejpam-4370	171	30	strongly	strongly	ADV
ejpam-4370	171	31	sλsp	sλsp	ADV
ejpam-4370	171	32	-	-	PUNCT
ejpam-4370	171	33	closed	closed	ADJ
ejpam-4370	171	34	relative	relative	ADJ
ejpam-4370	171	35	to	to	ADP
ejpam-4370	171	36	y	y	PROPN
ejpam-4370	171	37	for	for	ADP
ejpam-4370	171	38	each	each	DET
ejpam-4370	171	39	x	x	SYM
ejpam-4370	171	40	∈	∈	PROPN
ejpam-4370	171	41	x.	x.	NOUN
ejpam-4370	172	1	if	if	SCONJ
ejpam-4370	172	2	x	x	PRON
ejpam-4370	172	3	is	be	AUX
ejpam-4370	172	4	λsp	λsp	NOUN
ejpam-4370	172	5	-	-	ADJ
ejpam-4370	172	6	compact	compact	ADJ
ejpam-4370	172	7	,	,	PUNCT
ejpam-4370	172	8	then	then	ADV
ejpam-4370	172	9	y	y	PROPN
ejpam-4370	172	10	is	be	AUX
ejpam-4370	172	11	strongly	strongly	ADV
ejpam-4370	172	12	sλsp	sλsp	ADV
ejpam-4370	172	13	-	-	PUNCT
ejpam-4370	172	14	closed	closed	ADJ
ejpam-4370	172	15	.	.	PUNCT
ejpam-4370	173	1	corollary	corollary	ADJ
ejpam-4370	173	2	4	4	NUM
ejpam-4370	173	3	.	.	PUNCT
ejpam-4370	174	1	if	if	SCONJ
ejpam-4370	174	2	f	f	PROPN
ejpam-4370	174	3	:	:	PUNCT
ejpam-4370	174	4	(	(	PUNCT
ejpam-4370	174	5	x	x	X
ejpam-4370	174	6	,	,	PUNCT
ejpam-4370	174	7	τ	τ	X
ejpam-4370	174	8	)	)	PUNCT
ejpam-4370	174	9	→	→	SYM
ejpam-4370	174	10	(	(	PUNCT
ejpam-4370	174	11	y	y	PROPN
ejpam-4370	174	12	,	,	PUNCT
ejpam-4370	174	13	σ	σ	PROPN
ejpam-4370	174	14	)	)	PUNCT
ejpam-4370	174	15	is	be	AUX
ejpam-4370	174	16	a	a	DET
ejpam-4370	174	17	contra-(λ	contra-(λ	PROPN
ejpam-4370	174	18	,	,	PUNCT
ejpam-4370	174	19	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	174	20	surjective	surjective	ADJ
ejpam-4370	174	21	function	function	NOUN
ejpam-4370	174	22	and	and	CCONJ
ejpam-4370	174	23	a	a	PRON
ejpam-4370	174	24	is	be	AUX
ejpam-4370	174	25	λsp	λsp	NOUN
ejpam-4370	174	26	-	-	ADJ
ejpam-4370	174	27	compact	compact	ADJ
ejpam-4370	174	28	relative	relative	NOUN
ejpam-4370	174	29	to	to	ADP
ejpam-4370	174	30	x	x	PRON
ejpam-4370	174	31	,	,	PUNCT
ejpam-4370	174	32	then	then	ADV
ejpam-4370	174	33	f(a	f(a	PROPN
ejpam-4370	174	34	)	)	PUNCT
ejpam-4370	174	35	is	be	AUX
ejpam-4370	174	36	strongly	strongly	ADV
ejpam-4370	174	37	sλsp	sλsp	ADV
ejpam-4370	174	38	-	-	PUNCT
ejpam-4370	174	39	closed	closed	ADJ
ejpam-4370	174	40	relative	relative	ADJ
ejpam-4370	174	41	to	to	ADP
ejpam-4370	174	42	y	y	PROPN
ejpam-4370	174	43	.	.	PUNCT
ejpam-4370	175	1	definition	definition	NOUN
ejpam-4370	175	2	5	5	NUM
ejpam-4370	175	3	.	.	PUNCT
ejpam-4370	176	1	[	[	X
ejpam-4370	176	2	4	4	X
ejpam-4370	176	3	]	]	PUNCT
ejpam-4370	176	4	let	let	VERB
ejpam-4370	176	5	a	a	PRON
ejpam-4370	176	6	be	be	AUX
ejpam-4370	176	7	a	a	DET
ejpam-4370	176	8	subset	subset	NOUN
ejpam-4370	176	9	of	of	ADP
ejpam-4370	176	10	a	a	DET
ejpam-4370	176	11	topological	topological	ADJ
ejpam-4370	176	12	space	space	NOUN
ejpam-4370	176	13	(	(	PUNCT
ejpam-4370	176	14	x	x	X
ejpam-4370	176	15	,	,	PUNCT
ejpam-4370	176	16	τ	τ	PROPN
ejpam-4370	176	17	)	)	PUNCT
ejpam-4370	176	18	.	.	PUNCT
ejpam-4370	177	1	the	the	DET
ejpam-4370	177	2	(	(	PUNCT
ejpam-4370	177	3	λ	λ	NOUN
ejpam-4370	177	4	,	,	PUNCT
ejpam-4370	177	5	sp)-frontier	sp)-fronti	ADJ
ejpam-4370	177	6	of	of	ADP
ejpam-4370	177	7	a	a	PRON
ejpam-4370	177	8	,	,	PUNCT
ejpam-4370	177	9	denoted	denote	VERB
ejpam-4370	177	10	by	by	ADP
ejpam-4370	177	11	(	(	PUNCT
ejpam-4370	177	12	λ	λ	X
ejpam-4370	177	13	,	,	PUNCT
ejpam-4370	177	14	sp)-fr(a	sp)-fr(a	NOUN
ejpam-4370	177	15	)	)	PUNCT
ejpam-4370	177	16	,	,	PUNCT
ejpam-4370	177	17	is	be	AUX
ejpam-4370	177	18	defined	define	VERB
ejpam-4370	177	19	by	by	ADP
ejpam-4370	177	20	(	(	PUNCT
ejpam-4370	177	21	λ	λ	PROPN
ejpam-4370	177	22	,	,	PUNCT
ejpam-4370	177	23	sp)-fr(a	sp)-fr(a	NOUN
ejpam-4370	177	24	)	)	PUNCT
ejpam-4370	177	25	=	=	PUNCT
ejpam-4370	177	26	a(λ	a(λ	ADV
ejpam-4370	177	27	,	,	PUNCT
ejpam-4370	177	28	sp	sp	NOUN
ejpam-4370	177	29	)	)	PUNCT
ejpam-4370	177	30	∩	∩	NOUN
ejpam-4370	177	31	[	[	X
ejpam-4370	177	32	x	x	SYM
ejpam-4370	177	33	−a](λ	−a](λ	PROPN
ejpam-4370	177	34	,	,	PUNCT
ejpam-4370	177	35	sp	sp	NOUN
ejpam-4370	177	36	)	)	PUNCT
ejpam-4370	177	37	=	=	PUNCT
ejpam-4370	178	1	a(λ	a(λ	ADV
ejpam-4370	178	2	,	,	PUNCT
ejpam-4370	178	3	sp	sp	NOUN
ejpam-4370	178	4	)	)	PUNCT
ejpam-4370	178	5	−a(λ	−a(λ	NOUN
ejpam-4370	178	6	,	,	PUNCT
ejpam-4370	178	7	sp	sp	NOUN
ejpam-4370	178	8	)	)	PUNCT
ejpam-4370	178	9	.	.	PUNCT
ejpam-4370	179	1	c.	c.	PROPN
ejpam-4370	179	2	boonpok	boonpok	PROPN
ejpam-4370	179	3	,	,	PUNCT
ejpam-4370	179	4	c.	c.	PROPN
ejpam-4370	179	5	viriyapong	viriyapong	PROPN
ejpam-4370	179	6	/	/	SYM
ejpam-4370	179	7	eur	eur	PROPN
ejpam-4370	179	8	.	.	PUNCT
ejpam-4370	180	1	j.	j.	PROPN
ejpam-4370	180	2	pure	pure	PROPN
ejpam-4370	180	3	appl	appl	PROPN
ejpam-4370	180	4	.	.	PROPN
ejpam-4370	180	5	math	math	PROPN
ejpam-4370	180	6	,	,	PUNCT
ejpam-4370	180	7	15	15	NUM
ejpam-4370	180	8	(	(	PUNCT
ejpam-4370	180	9	4	4	NUM
ejpam-4370	180	10	)	)	PUNCT
ejpam-4370	180	11	(	(	PUNCT
ejpam-4370	180	12	2022	2022	NUM
ejpam-4370	180	13	)	)	PUNCT
ejpam-4370	180	14	,	,	PUNCT
ejpam-4370	180	15	1694	1694	NUM
ejpam-4370	180	16	-	-	SYM
ejpam-4370	180	17	1704	1704	NUM
ejpam-4370	180	18	1700	1700	NUM
ejpam-4370	180	19	theorem	theorem	NOUN
ejpam-4370	180	20	6	6	NUM
ejpam-4370	180	21	.	.	PUNCT
ejpam-4370	181	1	the	the	DET
ejpam-4370	181	2	set	set	NOUN
ejpam-4370	181	3	of	of	ADP
ejpam-4370	181	4	all	all	DET
ejpam-4370	181	5	points	point	NOUN
ejpam-4370	181	6	x	x	PUNCT
ejpam-4370	181	7	of	of	ADP
ejpam-4370	181	8	x	x	SYM
ejpam-4370	181	9	at	at	ADP
ejpam-4370	181	10	which	which	PRON
ejpam-4370	181	11	a	a	DET
ejpam-4370	181	12	multifunction	multifunction	NOUN
ejpam-4370	182	1	f	f	NOUN
ejpam-4370	182	2	:	:	PUNCT
ejpam-4370	182	3	(	(	PUNCT
ejpam-4370	182	4	x	x	X
ejpam-4370	182	5	,	,	PUNCT
ejpam-4370	182	6	τ	τ	X
ejpam-4370	182	7	)	)	PUNCT
ejpam-4370	182	8	→	→	SYM
ejpam-4370	182	9	(	(	PUNCT
ejpam-4370	182	10	y	y	PROPN
ejpam-4370	182	11	,	,	PUNCT
ejpam-4370	182	12	σ	σ	PROPN
ejpam-4370	182	13	)	)	PUNCT
ejpam-4370	182	14	is	be	AUX
ejpam-4370	182	15	not	not	PART
ejpam-4370	182	16	upper	upper	ADJ
ejpam-4370	182	17	contra-(λ	contra-(λ	PROPN
ejpam-4370	182	18	,	,	PUNCT
ejpam-4370	182	19	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	182	20	is	be	AUX
ejpam-4370	182	21	identical	identical	ADJ
ejpam-4370	182	22	with	with	ADP
ejpam-4370	182	23	the	the	DET
ejpam-4370	182	24	union	union	NOUN
ejpam-4370	182	25	of	of	ADP
ejpam-4370	182	26	the	the	DET
ejpam-4370	182	27	(	(	PUNCT
ejpam-4370	182	28	λ	λ	PROPN
ejpam-4370	182	29	,	,	PUNCT
ejpam-4370	182	30	sp)-frontiers	sp)-frontier	NOUN
ejpam-4370	182	31	of	of	ADP
ejpam-4370	182	32	the	the	DET
ejpam-4370	182	33	upper	upper	ADJ
ejpam-4370	182	34	inverse	inverse	NOUN
ejpam-4370	182	35	images	image	NOUN
ejpam-4370	182	36	of	of	ADP
ejpam-4370	182	37	(	(	PUNCT
ejpam-4370	182	38	λ	λ	PROPN
ejpam-4370	182	39	,	,	PUNCT
ejpam-4370	182	40	sp)-closed	sp)-close	VERB
ejpam-4370	182	41	sets	set	NOUN
ejpam-4370	182	42	of	of	ADP
ejpam-4370	182	43	y	y	PROPN
ejpam-4370	182	44	containing	contain	VERB
ejpam-4370	182	45	f	f	PROPN
ejpam-4370	182	46	(	(	PUNCT
ejpam-4370	182	47	x	x	NOUN
ejpam-4370	182	48	)	)	PUNCT
ejpam-4370	182	49	.	.	PUNCT
ejpam-4370	183	1	proof	proof	NOUN
ejpam-4370	183	2	.	.	PUNCT
ejpam-4370	184	1	let	let	VERB
ejpam-4370	184	2	x	x	PUNCT
ejpam-4370	184	3	∈	∈	PROPN
ejpam-4370	184	4	x	x	PUNCT
ejpam-4370	184	5	at	at	ADP
ejpam-4370	184	6	which	which	PRON
ejpam-4370	184	7	f	f	NOUN
ejpam-4370	184	8	is	be	AUX
ejpam-4370	184	9	not	not	PART
ejpam-4370	184	10	upper	upper	ADJ
ejpam-4370	184	11	contra-(λ	contra-(λ	PROPN
ejpam-4370	184	12	,	,	PUNCT
ejpam-4370	184	13	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	184	14	.	.	PUNCT
ejpam-4370	185	1	then	then	ADV
ejpam-4370	185	2	,	,	PUNCT
ejpam-4370	185	3	there	there	PRON
ejpam-4370	185	4	exists	exist	VERB
ejpam-4370	185	5	a	a	DET
ejpam-4370	185	6	(	(	PUNCT
ejpam-4370	185	7	λ	λ	PROPN
ejpam-4370	185	8	,	,	PUNCT
ejpam-4370	185	9	sp)-closed	sp)-close	VERB
ejpam-4370	185	10	set	set	VERB
ejpam-4370	185	11	v	v	NOUN
ejpam-4370	185	12	of	of	ADP
ejpam-4370	185	13	y	y	PROPN
ejpam-4370	185	14	containing	contain	VERB
ejpam-4370	185	15	f	f	PROPN
ejpam-4370	185	16	(	(	PUNCT
ejpam-4370	185	17	x	x	X
ejpam-4370	185	18	)	)	PUNCT
ejpam-4370	185	19	such	such	ADJ
ejpam-4370	185	20	that	that	SCONJ
ejpam-4370	185	21	u	u	PROPN
ejpam-4370	185	22	∩	∩	NOUN
ejpam-4370	185	23	(	(	PUNCT
ejpam-4370	185	24	x	x	NOUN
ejpam-4370	185	25	−	−	PROPN
ejpam-4370	185	26	f+(v	f+(v	NOUN
ejpam-4370	185	27	)	)	PUNCT
ejpam-4370	185	28	)	)	PUNCT
ejpam-4370	186	1	̸=	̸=	NOUN
ejpam-4370	186	2	∅	∅	NOUN
ejpam-4370	186	3	for	for	ADP
ejpam-4370	186	4	every	every	DET
ejpam-4370	186	5	u	u	PROPN
ejpam-4370	186	6	∈	∈	PROPN
ejpam-4370	186	7	λspo(x	λspo(x	PROPN
ejpam-4370	186	8	,	,	PUNCT
ejpam-4370	186	9	τ	τ	X
ejpam-4370	186	10	)	)	PUNCT
ejpam-4370	186	11	containing	contain	VERB
ejpam-4370	186	12	x.	x.	NOUN
ejpam-4370	186	13	thus	thus	ADV
ejpam-4370	186	14	,	,	PUNCT
ejpam-4370	186	15	x	x	PUNCT
ejpam-4370	186	16	∈	∈	PROPN
ejpam-4370	187	1	[	[	X
ejpam-4370	187	2	x	x	X
ejpam-4370	187	3	−	−	PROPN
ejpam-4370	187	4	f+(v	f+(v	NOUN
ejpam-4370	187	5	)	)	PUNCT
ejpam-4370	187	6	]	]	PUNCT
ejpam-4370	187	7	(	(	PUNCT
ejpam-4370	187	8	λ	λ	NOUN
ejpam-4370	187	9	,	,	PUNCT
ejpam-4370	187	10	sp	sp	NOUN
ejpam-4370	187	11	)	)	PUNCT
ejpam-4370	187	12	.	.	PUNCT
ejpam-4370	188	1	on	on	ADP
ejpam-4370	188	2	the	the	DET
ejpam-4370	188	3	other	other	ADJ
ejpam-4370	188	4	hand	hand	NOUN
ejpam-4370	188	5	,	,	PUNCT
ejpam-4370	188	6	we	we	PRON
ejpam-4370	188	7	have	have	VERB
ejpam-4370	188	8	x	x	X
ejpam-4370	188	9	∈	∈	PROPN
ejpam-4370	188	10	f+(v	f+(v	NOUN
ejpam-4370	188	11	)	)	PUNCT
ejpam-4370	189	1	⊆	⊆	NUM
ejpam-4370	189	2	[	[	X
ejpam-4370	189	3	f+(v	f+(v	NOUN
ejpam-4370	189	4	)	)	PUNCT
ejpam-4370	189	5	]	]	PUNCT
ejpam-4370	189	6	(	(	PUNCT
ejpam-4370	189	7	λ	λ	NOUN
ejpam-4370	189	8	,	,	PUNCT
ejpam-4370	189	9	sp	sp	NOUN
ejpam-4370	189	10	)	)	PUNCT
ejpam-4370	189	11	and	and	CCONJ
ejpam-4370	189	12	hence	hence	ADV
ejpam-4370	189	13	x	x	X
ejpam-4370	189	14	∈	∈	PROPN
ejpam-4370	189	15	(	(	PUNCT
ejpam-4370	189	16	λ	λ	NOUN
ejpam-4370	189	17	,	,	PUNCT
ejpam-4370	189	18	sp)-fr(f+(v	sp)-fr(f+(v	PROPN
ejpam-4370	189	19	)	)	PUNCT
ejpam-4370	189	20	)	)	PUNCT
ejpam-4370	189	21	.	.	PUNCT
ejpam-4370	190	1	conversely	conversely	ADV
ejpam-4370	190	2	,	,	PUNCT
ejpam-4370	190	3	let	let	VERB
ejpam-4370	190	4	v	v	PART
ejpam-4370	190	5	be	be	AUX
ejpam-4370	190	6	any	any	DET
ejpam-4370	190	7	(	(	PUNCT
ejpam-4370	190	8	λ	λ	PROPN
ejpam-4370	190	9	,	,	PUNCT
ejpam-4370	190	10	sp)-closed	sp)-close	VERB
ejpam-4370	190	11	set	set	NOUN
ejpam-4370	190	12	of	of	ADP
ejpam-4370	190	13	y	y	PROPN
ejpam-4370	190	14	containing	contain	VERB
ejpam-4370	190	15	f	f	PROPN
ejpam-4370	190	16	(	(	PUNCT
ejpam-4370	190	17	x	x	X
ejpam-4370	190	18	)	)	PUNCT
ejpam-4370	190	19	such	such	ADJ
ejpam-4370	190	20	that	that	SCONJ
ejpam-4370	190	21	x	x	SYM
ejpam-4370	190	22	∈	∈	PROPN
ejpam-4370	190	23	(	(	PUNCT
ejpam-4370	190	24	λ	λ	NOUN
ejpam-4370	190	25	,	,	PUNCT
ejpam-4370	190	26	sp)-fr(f+(v	sp)-fr(f+(v	PROPN
ejpam-4370	190	27	)	)	PUNCT
ejpam-4370	190	28	)	)	PUNCT
ejpam-4370	190	29	.	.	PUNCT
ejpam-4370	191	1	if	if	SCONJ
ejpam-4370	191	2	f	f	PROPN
ejpam-4370	191	3	is	be	AUX
ejpam-4370	191	4	upper	upper	ADJ
ejpam-4370	191	5	contra-(λ	contra-(λ	PROPN
ejpam-4370	191	6	,	,	PUNCT
ejpam-4370	191	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	191	8	at	at	ADP
ejpam-4370	191	9	x	x	X
ejpam-4370	191	10	,	,	PUNCT
ejpam-4370	191	11	then	then	ADV
ejpam-4370	191	12	there	there	PRON
ejpam-4370	191	13	exists	exist	VERB
ejpam-4370	191	14	u	u	PROPN
ejpam-4370	191	15	∈	∈	PROPN
ejpam-4370	191	16	λspo(x	λspo(x	PROPN
ejpam-4370	191	17	,	,	PUNCT
ejpam-4370	191	18	τ	τ	X
ejpam-4370	191	19	)	)	PUNCT
ejpam-4370	191	20	containing	contain	VERB
ejpam-4370	191	21	x	x	PUNCT
ejpam-4370	191	22	such	such	ADJ
ejpam-4370	191	23	that	that	SCONJ
ejpam-4370	191	24	u	u	NOUN
ejpam-4370	191	25	⊆	⊆	NUM
ejpam-4370	191	26	f+(v	f+(v	NOUN
ejpam-4370	191	27	)	)	PUNCT
ejpam-4370	191	28	;	;	PUNCT
ejpam-4370	191	29	hence	hence	ADV
ejpam-4370	191	30	x	x	X
ejpam-4370	191	31	∈	∈	PROPN
ejpam-4370	191	32	[	[	X
ejpam-4370	191	33	f+(v	f+(v	NOUN
ejpam-4370	191	34	)	)	PUNCT
ejpam-4370	191	35	]	]	PUNCT
ejpam-4370	191	36	(	(	PUNCT
ejpam-4370	191	37	λ	λ	NOUN
ejpam-4370	191	38	,	,	PUNCT
ejpam-4370	191	39	sp	sp	NOUN
ejpam-4370	191	40	)	)	PUNCT
ejpam-4370	191	41	.	.	PUNCT
ejpam-4370	192	1	this	this	PRON
ejpam-4370	192	2	is	be	AUX
ejpam-4370	192	3	a	a	DET
ejpam-4370	192	4	contradiction	contradiction	NOUN
ejpam-4370	192	5	and	and	CCONJ
ejpam-4370	192	6	hence	hence	ADV
ejpam-4370	192	7	f	f	PROPN
ejpam-4370	192	8	is	be	AUX
ejpam-4370	192	9	not	not	PART
ejpam-4370	192	10	upper	upper	ADJ
ejpam-4370	192	11	contra-(λ	contra-(λ	PROPN
ejpam-4370	192	12	,	,	PUNCT
ejpam-4370	192	13	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	192	14	at	at	ADP
ejpam-4370	192	15	x.	x.	NOUN
ejpam-4370	192	16	theorem	theorem	VERB
ejpam-4370	192	17	7	7	NUM
ejpam-4370	192	18	.	.	PUNCT
ejpam-4370	193	1	the	the	DET
ejpam-4370	193	2	set	set	NOUN
ejpam-4370	193	3	of	of	ADP
ejpam-4370	193	4	all	all	DET
ejpam-4370	193	5	points	point	NOUN
ejpam-4370	193	6	x	x	PUNCT
ejpam-4370	193	7	of	of	ADP
ejpam-4370	193	8	x	x	SYM
ejpam-4370	193	9	at	at	ADP
ejpam-4370	193	10	which	which	PRON
ejpam-4370	193	11	a	a	DET
ejpam-4370	193	12	multifunction	multifunction	NOUN
ejpam-4370	194	1	f	f	NOUN
ejpam-4370	194	2	:	:	PUNCT
ejpam-4370	194	3	(	(	PUNCT
ejpam-4370	194	4	x	x	X
ejpam-4370	194	5	,	,	PUNCT
ejpam-4370	194	6	τ	τ	X
ejpam-4370	194	7	)	)	PUNCT
ejpam-4370	194	8	→	→	SYM
ejpam-4370	194	9	(	(	PUNCT
ejpam-4370	194	10	y	y	PROPN
ejpam-4370	194	11	,	,	PUNCT
ejpam-4370	194	12	σ	σ	PROPN
ejpam-4370	194	13	)	)	PUNCT
ejpam-4370	194	14	is	be	AUX
ejpam-4370	194	15	not	not	PART
ejpam-4370	194	16	lower	low	ADJ
ejpam-4370	194	17	contra-(λ	contra-(λ	PROPN
ejpam-4370	194	18	,	,	PUNCT
ejpam-4370	194	19	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	194	20	is	be	AUX
ejpam-4370	194	21	identical	identical	ADJ
ejpam-4370	194	22	with	with	ADP
ejpam-4370	194	23	the	the	DET
ejpam-4370	194	24	union	union	NOUN
ejpam-4370	194	25	of	of	ADP
ejpam-4370	194	26	the	the	DET
ejpam-4370	194	27	(	(	PUNCT
ejpam-4370	194	28	λ	λ	PROPN
ejpam-4370	194	29	,	,	PUNCT
ejpam-4370	194	30	sp)-frontiers	sp)-frontier	NOUN
ejpam-4370	194	31	of	of	ADP
ejpam-4370	194	32	the	the	DET
ejpam-4370	194	33	lower	low	ADJ
ejpam-4370	194	34	inverse	inverse	NOUN
ejpam-4370	194	35	images	image	NOUN
ejpam-4370	194	36	of	of	ADP
ejpam-4370	194	37	(	(	PUNCT
ejpam-4370	194	38	λ	λ	PROPN
ejpam-4370	194	39	,	,	PUNCT
ejpam-4370	194	40	sp)-closed	sp)-close	VERB
ejpam-4370	194	41	sets	set	NOUN
ejpam-4370	194	42	of	of	ADP
ejpam-4370	194	43	y	y	PROPN
ejpam-4370	194	44	meeting	meet	VERB
ejpam-4370	194	45	f	f	PROPN
ejpam-4370	194	46	(	(	PUNCT
ejpam-4370	194	47	x	x	NOUN
ejpam-4370	194	48	)	)	PUNCT
ejpam-4370	194	49	.	.	PUNCT
ejpam-4370	195	1	proof	proof	NOUN
ejpam-4370	195	2	.	.	PUNCT
ejpam-4370	196	1	the	the	DET
ejpam-4370	196	2	proof	proof	NOUN
ejpam-4370	196	3	is	be	AUX
ejpam-4370	196	4	similar	similar	ADJ
ejpam-4370	196	5	to	to	ADP
ejpam-4370	196	6	that	that	PRON
ejpam-4370	196	7	of	of	ADP
ejpam-4370	196	8	theorem	theorem	ADJ
ejpam-4370	196	9	6	6	NUM
ejpam-4370	196	10	.	.	PUNCT
ejpam-4370	196	11	corollary	corollary	ADJ
ejpam-4370	196	12	5	5	NUM
ejpam-4370	196	13	.	.	PUNCT
ejpam-4370	197	1	the	the	DET
ejpam-4370	197	2	set	set	NOUN
ejpam-4370	197	3	of	of	ADP
ejpam-4370	197	4	all	all	DET
ejpam-4370	197	5	points	point	NOUN
ejpam-4370	197	6	x	x	PUNCT
ejpam-4370	197	7	of	of	ADP
ejpam-4370	197	8	x	x	SYM
ejpam-4370	197	9	at	at	ADP
ejpam-4370	197	10	which	which	PRON
ejpam-4370	197	11	a	a	DET
ejpam-4370	197	12	function	function	NOUN
ejpam-4370	197	13	f	f	NOUN
ejpam-4370	197	14	:	:	PUNCT
ejpam-4370	197	15	(	(	PUNCT
ejpam-4370	197	16	x	x	X
ejpam-4370	197	17	,	,	PUNCT
ejpam-4370	197	18	τ	τ	X
ejpam-4370	197	19	)	)	PUNCT
ejpam-4370	197	20	→	→	SYM
ejpam-4370	197	21	(	(	PUNCT
ejpam-4370	197	22	y	y	PROPN
ejpam-4370	197	23	,	,	PUNCT
ejpam-4370	197	24	σ	σ	PROPN
ejpam-4370	197	25	)	)	PUNCT
ejpam-4370	197	26	is	be	AUX
ejpam-4370	197	27	not	not	PART
ejpam-4370	197	28	contra-(λ	contra-(λ	PROPN
ejpam-4370	197	29	,	,	PUNCT
ejpam-4370	197	30	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	197	31	is	be	AUX
ejpam-4370	197	32	identical	identical	ADJ
ejpam-4370	197	33	with	with	ADP
ejpam-4370	197	34	the	the	DET
ejpam-4370	197	35	union	union	NOUN
ejpam-4370	197	36	of	of	ADP
ejpam-4370	197	37	the	the	DET
ejpam-4370	197	38	(	(	PUNCT
ejpam-4370	197	39	λ	λ	PROPN
ejpam-4370	197	40	,	,	PUNCT
ejpam-4370	197	41	sp)-frontiers	sp)-frontier	NOUN
ejpam-4370	197	42	of	of	ADP
ejpam-4370	197	43	the	the	DET
ejpam-4370	197	44	inverse	inverse	NOUN
ejpam-4370	197	45	images	image	NOUN
ejpam-4370	197	46	of	of	ADP
ejpam-4370	197	47	(	(	PUNCT
ejpam-4370	197	48	λ	λ	PROPN
ejpam-4370	197	49	,	,	PUNCT
ejpam-4370	197	50	sp)-closed	sp)-close	VERB
ejpam-4370	197	51	sets	set	NOUN
ejpam-4370	197	52	of	of	ADP
ejpam-4370	197	53	y	y	NOUN
ejpam-4370	197	54	containing	contain	VERB
ejpam-4370	197	55	f(x	f(x	PROPN
ejpam-4370	197	56	)	)	PUNCT
ejpam-4370	197	57	.	.	PUNCT
ejpam-4370	198	1	definition	definition	NOUN
ejpam-4370	198	2	6	6	NUM
ejpam-4370	198	3	.	.	PUNCT
ejpam-4370	199	1	[	[	X
ejpam-4370	199	2	5	5	NUM
ejpam-4370	199	3	]	]	PUNCT
ejpam-4370	199	4	let	let	VERB
ejpam-4370	199	5	a	a	PRON
ejpam-4370	199	6	be	be	AUX
ejpam-4370	199	7	a	a	DET
ejpam-4370	199	8	subset	subset	NOUN
ejpam-4370	199	9	of	of	ADP
ejpam-4370	199	10	a	a	DET
ejpam-4370	199	11	topological	topological	ADJ
ejpam-4370	199	12	space	space	NOUN
ejpam-4370	199	13	(	(	PUNCT
ejpam-4370	199	14	x	x	X
ejpam-4370	199	15	,	,	PUNCT
ejpam-4370	199	16	τ	τ	PROPN
ejpam-4370	199	17	)	)	PUNCT
ejpam-4370	199	18	.	.	PUNCT
ejpam-4370	200	1	a	a	DET
ejpam-4370	200	2	subset	subset	NOUN
ejpam-4370	200	3	λ(λ	λ(λ	ADP
ejpam-4370	200	4	,	,	PUNCT
ejpam-4370	200	5	sp)(a	sp)(a	PROPN
ejpam-4370	200	6	)	)	PUNCT
ejpam-4370	200	7	is	be	AUX
ejpam-4370	200	8	defined	define	VERB
ejpam-4370	200	9	as	as	SCONJ
ejpam-4370	200	10	follows	follow	VERB
ejpam-4370	200	11	:	:	PUNCT
ejpam-4370	201	1	λ(λ	λ(λ	ADV
ejpam-4370	201	2	,	,	PUNCT
ejpam-4370	201	3	sp)(a	sp)(a	PROPN
ejpam-4370	201	4	)	)	PUNCT
ejpam-4370	202	1	=	=	PUNCT
ejpam-4370	203	1	∩{u	∩{u	PROPN
ejpam-4370	203	2	|	|	ADV
ejpam-4370	203	3	a	a	DET
ejpam-4370	203	4	⊆	⊆	NUM
ejpam-4370	203	5	u	u	NOUN
ejpam-4370	203	6	,	,	PUNCT
ejpam-4370	203	7	u	u	PROPN
ejpam-4370	203	8	∈	∈	PROPN
ejpam-4370	203	9	λspo(x	λspo(x	PROPN
ejpam-4370	203	10	,	,	PUNCT
ejpam-4370	203	11	τ	τ	PROPN
ejpam-4370	203	12	)	)	PUNCT
ejpam-4370	203	13	}	}	PUNCT
ejpam-4370	203	14	.	.	PUNCT
ejpam-4370	204	1	lemma	lemma	PROPN
ejpam-4370	204	2	4	4	NUM
ejpam-4370	204	3	.	.	PUNCT
ejpam-4370	205	1	[	[	X
ejpam-4370	205	2	5	5	NUM
ejpam-4370	205	3	]	]	PUNCT
ejpam-4370	205	4	for	for	ADP
ejpam-4370	205	5	subsets	subset	NOUN
ejpam-4370	205	6	a	a	DET
ejpam-4370	205	7	,	,	PUNCT
ejpam-4370	205	8	b	b	PROPN
ejpam-4370	205	9	of	of	ADP
ejpam-4370	205	10	a	a	DET
ejpam-4370	205	11	topological	topological	ADJ
ejpam-4370	205	12	space	space	NOUN
ejpam-4370	205	13	(	(	PUNCT
ejpam-4370	205	14	x	x	X
ejpam-4370	205	15	,	,	PUNCT
ejpam-4370	205	16	τ	τ	PROPN
ejpam-4370	205	17	)	)	PUNCT
ejpam-4370	205	18	,	,	PUNCT
ejpam-4370	205	19	the	the	DET
ejpam-4370	205	20	following	follow	VERB
ejpam-4370	205	21	properties	property	NOUN
ejpam-4370	205	22	hold	hold	VERB
ejpam-4370	205	23	:	:	PUNCT
ejpam-4370	205	24	(	(	PUNCT
ejpam-4370	205	25	1	1	X
ejpam-4370	205	26	)	)	PUNCT
ejpam-4370	205	27	a	a	DET
ejpam-4370	205	28	⊆	⊆	NUM
ejpam-4370	205	29	λ(λ	λ(λ	NOUN
ejpam-4370	205	30	,	,	PUNCT
ejpam-4370	205	31	sp)(a	sp)(a	PROPN
ejpam-4370	205	32	)	)	PUNCT
ejpam-4370	205	33	.	.	PUNCT
ejpam-4370	206	1	(	(	PUNCT
ejpam-4370	206	2	2	2	X
ejpam-4370	206	3	)	)	PUNCT
ejpam-4370	206	4	if	if	SCONJ
ejpam-4370	206	5	a	a	DET
ejpam-4370	206	6	⊆	⊆	NUM
ejpam-4370	206	7	b	b	NOUN
ejpam-4370	206	8	,	,	PUNCT
ejpam-4370	206	9	then	then	ADV
ejpam-4370	206	10	λ(λ	λ(λ	PROPN
ejpam-4370	206	11	,	,	PUNCT
ejpam-4370	206	12	sp)(a	sp)(a	PROPN
ejpam-4370	206	13	)	)	PUNCT
ejpam-4370	206	14	⊆	⊆	NUM
ejpam-4370	206	15	λ(λ	λ(λ	NOUN
ejpam-4370	206	16	,	,	PUNCT
ejpam-4370	206	17	sp)(b	sp)(b	PROPN
ejpam-4370	206	18	)	)	PUNCT
ejpam-4370	206	19	.	.	PUNCT
ejpam-4370	207	1	(	(	PUNCT
ejpam-4370	207	2	3	3	X
ejpam-4370	207	3	)	)	PUNCT
ejpam-4370	207	4	λ(λ	λ(λ	ADV
ejpam-4370	207	5	,	,	PUNCT
ejpam-4370	207	6	sp)[λ(λ	sp)[λ(λ	NOUN
ejpam-4370	207	7	,	,	PUNCT
ejpam-4370	207	8	sp)(a	sp)(a	PROPN
ejpam-4370	207	9	)	)	PUNCT
ejpam-4370	207	10	]	]	PUNCT
ejpam-4370	208	1	=	=	PUNCT
ejpam-4370	208	2	λ(λ	λ(λ	PROPN
ejpam-4370	208	3	,	,	PUNCT
ejpam-4370	208	4	sp)(a	sp)(a	PROPN
ejpam-4370	208	5	)	)	PUNCT
ejpam-4370	208	6	.	.	PUNCT
ejpam-4370	209	1	(	(	PUNCT
ejpam-4370	209	2	4	4	X
ejpam-4370	209	3	)	)	PUNCT
ejpam-4370	209	4	if	if	SCONJ
ejpam-4370	209	5	a	a	PRON
ejpam-4370	209	6	is	be	AUX
ejpam-4370	209	7	(	(	PUNCT
ejpam-4370	209	8	λ	λ	NOUN
ejpam-4370	209	9	,	,	PUNCT
ejpam-4370	209	10	sp)-open	sp)-open	ADJ
ejpam-4370	209	11	,	,	PUNCT
ejpam-4370	209	12	λ(λ	λ(λ	ADV
ejpam-4370	209	13	,	,	PUNCT
ejpam-4370	209	14	sp)(a	sp)(a	PROPN
ejpam-4370	209	15	)	)	PUNCT
ejpam-4370	210	1	=	=	SYM
ejpam-4370	210	2	a.	a.	NOUN
ejpam-4370	210	3	theorem	theorem	NOUN
ejpam-4370	210	4	8	8	NUM
ejpam-4370	210	5	.	.	PUNCT
ejpam-4370	211	1	let	let	VERB
ejpam-4370	211	2	f	f	NOUN
ejpam-4370	211	3	:	:	PUNCT
ejpam-4370	211	4	(	(	PUNCT
ejpam-4370	211	5	x	x	X
ejpam-4370	211	6	,	,	PUNCT
ejpam-4370	211	7	τ	τ	X
ejpam-4370	211	8	)	)	PUNCT
ejpam-4370	211	9	→	→	SYM
ejpam-4370	211	10	(	(	PUNCT
ejpam-4370	211	11	y	y	PROPN
ejpam-4370	211	12	,	,	PUNCT
ejpam-4370	211	13	σ	σ	PROPN
ejpam-4370	211	14	)	)	PUNCT
ejpam-4370	211	15	be	be	AUX
ejpam-4370	211	16	a	a	DET
ejpam-4370	211	17	multifunction	multifunction	NOUN
ejpam-4370	211	18	.	.	PUNCT
ejpam-4370	212	1	if	if	SCONJ
ejpam-4370	212	2	[	[	X
ejpam-4370	212	3	f−(b)](λ	f−(b)](λ	PROPN
ejpam-4370	212	4	,	,	PUNCT
ejpam-4370	212	5	sp	sp	NOUN
ejpam-4370	212	6	)	)	PUNCT
ejpam-4370	212	7	⊆	⊆	NUM
ejpam-4370	212	8	f−(λ(λ	f−(λ(λ	NOUN
ejpam-4370	212	9	,	,	PUNCT
ejpam-4370	212	10	sp)(b	sp)(b	PROPN
ejpam-4370	212	11	)	)	PUNCT
ejpam-4370	212	12	)	)	PUNCT
ejpam-4370	212	13	for	for	ADP
ejpam-4370	212	14	every	every	DET
ejpam-4370	212	15	subset	subset	NOUN
ejpam-4370	212	16	b	b	PROPN
ejpam-4370	212	17	of	of	ADP
ejpam-4370	212	18	y	y	PROPN
ejpam-4370	212	19	,	,	PUNCT
ejpam-4370	212	20	then	then	ADV
ejpam-4370	212	21	f	f	PROPN
ejpam-4370	212	22	is	be	AUX
ejpam-4370	212	23	upper	upper	ADJ
ejpam-4370	212	24	contra-(λ	contra-(λ	PROPN
ejpam-4370	212	25	,	,	PUNCT
ejpam-4370	212	26	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	212	27	.	.	PUNCT
ejpam-4370	213	1	proof	proof	NOUN
ejpam-4370	213	2	.	.	PUNCT
ejpam-4370	214	1	let	let	VERB
ejpam-4370	214	2	v	v	PART
ejpam-4370	214	3	be	be	AUX
ejpam-4370	214	4	any	any	DET
ejpam-4370	214	5	(	(	PUNCT
ejpam-4370	214	6	λ	λ	NOUN
ejpam-4370	214	7	,	,	PUNCT
ejpam-4370	214	8	sp)-open	sp)-open	ADJ
ejpam-4370	214	9	set	set	NOUN
ejpam-4370	214	10	of	of	ADP
ejpam-4370	214	11	y	y	PROPN
ejpam-4370	214	12	.	.	PUNCT
ejpam-4370	215	1	by	by	ADP
ejpam-4370	215	2	lemma	lemma	PROPN
ejpam-4370	215	3	4	4	NUM
ejpam-4370	215	4	,	,	PUNCT
ejpam-4370	215	5	[	[	X
ejpam-4370	215	6	f−(v	f−(v	NOUN
ejpam-4370	215	7	)	)	PUNCT
ejpam-4370	215	8	]	]	PUNCT
ejpam-4370	215	9	(	(	PUNCT
ejpam-4370	215	10	λ	λ	NOUN
ejpam-4370	215	11	,	,	PUNCT
ejpam-4370	215	12	sp	sp	NOUN
ejpam-4370	215	13	)	)	PUNCT
ejpam-4370	215	14	⊆	⊆	NUM
ejpam-4370	215	15	f−(λ(λ	f−(λ(λ	NOUN
ejpam-4370	215	16	,	,	PUNCT
ejpam-4370	215	17	sp)(v	sp)(v	PROPN
ejpam-4370	215	18	)	)	PUNCT
ejpam-4370	215	19	)	)	PUNCT
ejpam-4370	216	1	=	=	SYM
ejpam-4370	216	2	f−(v	f−(v	ADJ
ejpam-4370	216	3	)	)	PUNCT
ejpam-4370	216	4	and	and	CCONJ
ejpam-4370	216	5	hence	hence	ADV
ejpam-4370	216	6	[	[	X
ejpam-4370	216	7	f−(v	f−(v	ADJ
ejpam-4370	216	8	)	)	PUNCT
ejpam-4370	216	9	]	]	PUNCT
ejpam-4370	216	10	(	(	PUNCT
ejpam-4370	216	11	λ	λ	NOUN
ejpam-4370	216	12	,	,	PUNCT
ejpam-4370	216	13	sp	sp	NOUN
ejpam-4370	216	14	)	)	PUNCT
ejpam-4370	216	15	=	=	SYM
ejpam-4370	216	16	f−(v	f−(v	NOUN
ejpam-4370	216	17	)	)	PUNCT
ejpam-4370	216	18	.	.	PUNCT
ejpam-4370	217	1	thus	thus	ADV
ejpam-4370	217	2	,	,	PUNCT
ejpam-4370	217	3	f−(v	f−(v	ADJ
ejpam-4370	217	4	)	)	PUNCT
ejpam-4370	217	5	is	be	AUX
ejpam-4370	217	6	(	(	PUNCT
ejpam-4370	217	7	λ	λ	X
ejpam-4370	217	8	,	,	PUNCT
ejpam-4370	217	9	sp)-closed	sp)-close	VERB
ejpam-4370	217	10	in	in	ADP
ejpam-4370	217	11	x	x	PROPN
ejpam-4370	217	12	,	,	PUNCT
ejpam-4370	217	13	by	by	ADP
ejpam-4370	217	14	theorem	theorem	NOUN
ejpam-4370	217	15	1	1	NUM
ejpam-4370	217	16	,	,	PUNCT
ejpam-4370	217	17	f	f	PROPN
ejpam-4370	217	18	is	be	AUX
ejpam-4370	217	19	upper	upper	ADJ
ejpam-4370	217	20	contra-(λ	contra-(λ	PROPN
ejpam-4370	217	21	,	,	PUNCT
ejpam-4370	217	22	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	217	23	.	.	PUNCT
ejpam-4370	218	1	c.	c.	PROPN
ejpam-4370	218	2	boonpok	boonpok	PROPN
ejpam-4370	218	3	,	,	PUNCT
ejpam-4370	218	4	c.	c.	PROPN
ejpam-4370	218	5	viriyapong	viriyapong	PROPN
ejpam-4370	218	6	/	/	SYM
ejpam-4370	218	7	eur	eur	PROPN
ejpam-4370	218	8	.	.	PUNCT
ejpam-4370	219	1	j.	j.	PROPN
ejpam-4370	219	2	pure	pure	PROPN
ejpam-4370	219	3	appl	appl	PROPN
ejpam-4370	219	4	.	.	PROPN
ejpam-4370	219	5	math	math	PROPN
ejpam-4370	219	6	,	,	PUNCT
ejpam-4370	219	7	15	15	NUM
ejpam-4370	219	8	(	(	PUNCT
ejpam-4370	219	9	4	4	NUM
ejpam-4370	219	10	)	)	PUNCT
ejpam-4370	219	11	(	(	PUNCT
ejpam-4370	219	12	2022	2022	NUM
ejpam-4370	219	13	)	)	PUNCT
ejpam-4370	219	14	,	,	PUNCT
ejpam-4370	219	15	1694	1694	NUM
ejpam-4370	219	16	-	-	SYM
ejpam-4370	219	17	1704	1704	NUM
ejpam-4370	219	18	1701	1701	NUM
ejpam-4370	219	19	corollary	corollary	NOUN
ejpam-4370	219	20	6	6	NUM
ejpam-4370	219	21	.	.	PUNCT
ejpam-4370	220	1	let	let	VERB
ejpam-4370	220	2	f	f	NOUN
ejpam-4370	220	3	:	:	PUNCT
ejpam-4370	220	4	(	(	PUNCT
ejpam-4370	220	5	x	x	X
ejpam-4370	220	6	,	,	PUNCT
ejpam-4370	220	7	τ	τ	X
ejpam-4370	220	8	)	)	PUNCT
ejpam-4370	220	9	→	→	SYM
ejpam-4370	220	10	(	(	PUNCT
ejpam-4370	220	11	y	y	PROPN
ejpam-4370	220	12	,	,	PUNCT
ejpam-4370	220	13	σ	σ	PROPN
ejpam-4370	220	14	)	)	PUNCT
ejpam-4370	220	15	be	be	AUX
ejpam-4370	220	16	a	a	DET
ejpam-4370	220	17	function	function	NOUN
ejpam-4370	220	18	.	.	PUNCT
ejpam-4370	221	1	if	if	SCONJ
ejpam-4370	221	2	[	[	X
ejpam-4370	221	3	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4370	221	4	,	,	PUNCT
ejpam-4370	221	5	sp	sp	NOUN
ejpam-4370	221	6	)	)	PUNCT
ejpam-4370	221	7	⊆	⊆	NUM
ejpam-4370	221	8	f−1(λ(λ	f−1(λ(λ	NOUN
ejpam-4370	221	9	,	,	PUNCT
ejpam-4370	221	10	sp)(b	sp)(b	PROPN
ejpam-4370	221	11	)	)	PUNCT
ejpam-4370	221	12	)	)	PUNCT
ejpam-4370	221	13	for	for	ADP
ejpam-4370	221	14	every	every	DET
ejpam-4370	221	15	subset	subset	NOUN
ejpam-4370	221	16	b	b	PROPN
ejpam-4370	221	17	of	of	ADP
ejpam-4370	221	18	y	y	PROPN
ejpam-4370	221	19	,	,	PUNCT
ejpam-4370	221	20	then	then	ADV
ejpam-4370	221	21	f	f	PROPN
ejpam-4370	221	22	is	be	AUX
ejpam-4370	221	23	contra-(λ	contra-(λ	PROPN
ejpam-4370	221	24	,	,	PUNCT
ejpam-4370	221	25	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	221	26	.	.	PUNCT
ejpam-4370	222	1	theorem	theorem	VERB
ejpam-4370	222	2	9	9	NUM
ejpam-4370	222	3	.	.	PUNCT
ejpam-4370	223	1	let	let	VERB
ejpam-4370	223	2	f	f	NOUN
ejpam-4370	223	3	:	:	PUNCT
ejpam-4370	223	4	(	(	PUNCT
ejpam-4370	223	5	x	x	X
ejpam-4370	223	6	,	,	PUNCT
ejpam-4370	223	7	τ	τ	X
ejpam-4370	223	8	)	)	PUNCT
ejpam-4370	223	9	→	→	SYM
ejpam-4370	223	10	(	(	PUNCT
ejpam-4370	223	11	y	y	PROPN
ejpam-4370	223	12	,	,	PUNCT
ejpam-4370	223	13	σ	σ	PROPN
ejpam-4370	223	14	)	)	PUNCT
ejpam-4370	223	15	be	be	AUX
ejpam-4370	223	16	a	a	DET
ejpam-4370	223	17	multifunction	multifunction	NOUN
ejpam-4370	223	18	.	.	PUNCT
ejpam-4370	224	1	if	if	SCONJ
ejpam-4370	224	2	f	f	PROPN
ejpam-4370	224	3	(	(	PUNCT
ejpam-4370	224	4	b(λ	b(λ	PROPN
ejpam-4370	224	5	,	,	PUNCT
ejpam-4370	224	6	sp	sp	NOUN
ejpam-4370	224	7	)	)	PUNCT
ejpam-4370	224	8	)	)	PUNCT
ejpam-4370	224	9	⊆	⊆	NUM
ejpam-4370	224	10	λ(λ	λ(λ	ADP
ejpam-4370	224	11	,	,	PUNCT
ejpam-4370	224	12	sp)(f	sp)(f	PROPN
ejpam-4370	224	13	(	(	PUNCT
ejpam-4370	224	14	b	b	NOUN
ejpam-4370	224	15	)	)	PUNCT
ejpam-4370	224	16	)	)	PUNCT
ejpam-4370	224	17	for	for	ADP
ejpam-4370	224	18	every	every	DET
ejpam-4370	224	19	subset	subset	NOUN
ejpam-4370	224	20	b	b	PROPN
ejpam-4370	224	21	of	of	ADP
ejpam-4370	224	22	y	y	PROPN
ejpam-4370	224	23	,	,	PUNCT
ejpam-4370	224	24	then	then	ADV
ejpam-4370	224	25	f	f	PROPN
ejpam-4370	224	26	is	be	AUX
ejpam-4370	224	27	lower	low	ADJ
ejpam-4370	224	28	contra-(λ	contra-(λ	PROPN
ejpam-4370	224	29	,	,	PUNCT
ejpam-4370	224	30	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	224	31	.	.	PUNCT
ejpam-4370	225	1	proof	proof	NOUN
ejpam-4370	225	2	.	.	PUNCT
ejpam-4370	226	1	let	let	VERB
ejpam-4370	226	2	v	v	PART
ejpam-4370	226	3	be	be	AUX
ejpam-4370	226	4	any	any	DET
ejpam-4370	226	5	(	(	PUNCT
ejpam-4370	226	6	λ	λ	NOUN
ejpam-4370	226	7	,	,	PUNCT
ejpam-4370	226	8	sp)-open	sp)-open	ADJ
ejpam-4370	226	9	set	set	NOUN
ejpam-4370	226	10	of	of	ADP
ejpam-4370	226	11	y	y	PROPN
ejpam-4370	226	12	.	.	PUNCT
ejpam-4370	227	1	then	then	ADV
ejpam-4370	227	2	,	,	PUNCT
ejpam-4370	227	3	f	f	PROPN
ejpam-4370	227	4	(	(	PUNCT
ejpam-4370	227	5	[	[	X
ejpam-4370	227	6	f+(v	f+(v	NOUN
ejpam-4370	227	7	)	)	PUNCT
ejpam-4370	227	8	]	]	PUNCT
ejpam-4370	227	9	(	(	PUNCT
ejpam-4370	227	10	λ	λ	NOUN
ejpam-4370	227	11	,	,	PUNCT
ejpam-4370	227	12	sp	sp	NOUN
ejpam-4370	227	13	)	)	PUNCT
ejpam-4370	227	14	)	)	PUNCT
ejpam-4370	227	15	⊆	⊆	NUM
ejpam-4370	227	16	λ(λ	λ(λ	NOUN
ejpam-4370	227	17	,	,	PUNCT
ejpam-4370	227	18	sp)(v	sp)(v	PROPN
ejpam-4370	227	19	)	)	PUNCT
ejpam-4370	227	20	and	and	CCONJ
ejpam-4370	227	21	[	[	X
ejpam-4370	227	22	f+(v	f+(v	NOUN
ejpam-4370	227	23	)	)	PUNCT
ejpam-4370	227	24	]	]	PUNCT
ejpam-4370	227	25	(	(	PUNCT
ejpam-4370	227	26	λ	λ	NOUN
ejpam-4370	227	27	,	,	PUNCT
ejpam-4370	227	28	sp	sp	NOUN
ejpam-4370	227	29	)	)	PUNCT
ejpam-4370	227	30	⊆	⊆	NUM
ejpam-4370	227	31	f+(λ(λ	f+(λ(λ	NOUN
ejpam-4370	227	32	,	,	PUNCT
ejpam-4370	227	33	sp)(v	sp)(v	PROPN
ejpam-4370	227	34	)	)	PUNCT
ejpam-4370	227	35	)	)	PUNCT
ejpam-4370	227	36	.	.	PUNCT
ejpam-4370	228	1	by	by	ADP
ejpam-4370	228	2	lemma	lemma	PROPN
ejpam-4370	228	3	4	4	NUM
ejpam-4370	228	4	,	,	PUNCT
ejpam-4370	228	5	[	[	X
ejpam-4370	228	6	f+(v	f+(v	NOUN
ejpam-4370	228	7	)	)	PUNCT
ejpam-4370	228	8	]	]	PUNCT
ejpam-4370	228	9	(	(	PUNCT
ejpam-4370	228	10	λ	λ	NOUN
ejpam-4370	228	11	,	,	PUNCT
ejpam-4370	228	12	sp	sp	NOUN
ejpam-4370	228	13	)	)	PUNCT
ejpam-4370	228	14	⊆	⊆	NUM
ejpam-4370	228	15	f+(λ(λ	f+(λ(λ	NOUN
ejpam-4370	228	16	,	,	PUNCT
ejpam-4370	228	17	sp)(v	sp)(v	PROPN
ejpam-4370	228	18	)	)	PUNCT
ejpam-4370	228	19	)	)	PUNCT
ejpam-4370	228	20	=	=	PUNCT
ejpam-4370	228	21	f+(v	f+(v	NOUN
ejpam-4370	228	22	)	)	PUNCT
ejpam-4370	228	23	.	.	PUNCT
ejpam-4370	229	1	thus	thus	ADV
ejpam-4370	229	2	,	,	PUNCT
ejpam-4370	229	3	[	[	X
ejpam-4370	229	4	f+(v	f+(v	NOUN
ejpam-4370	229	5	)	)	PUNCT
ejpam-4370	229	6	]	]	PUNCT
ejpam-4370	229	7	(	(	PUNCT
ejpam-4370	229	8	λ	λ	NOUN
ejpam-4370	229	9	,	,	PUNCT
ejpam-4370	229	10	sp	sp	NOUN
ejpam-4370	229	11	)	)	PUNCT
ejpam-4370	229	12	=	=	SYM
ejpam-4370	229	13	f+(v	f+(v	NOUN
ejpam-4370	229	14	)	)	PUNCT
ejpam-4370	229	15	and	and	CCONJ
ejpam-4370	229	16	hence	hence	ADV
ejpam-4370	229	17	f+(v	f+(v	NOUN
ejpam-4370	229	18	)	)	PUNCT
ejpam-4370	229	19	is	be	AUX
ejpam-4370	229	20	(	(	PUNCT
ejpam-4370	229	21	λ	λ	X
ejpam-4370	229	22	,	,	PUNCT
ejpam-4370	229	23	sp)-closed	sp)-close	VERB
ejpam-4370	229	24	in	in	ADP
ejpam-4370	229	25	x	x	PROPN
ejpam-4370	229	26	,	,	PUNCT
ejpam-4370	229	27	by	by	ADP
ejpam-4370	229	28	theorem	theorem	NOUN
ejpam-4370	229	29	2	2	NUM
ejpam-4370	229	30	,	,	PUNCT
ejpam-4370	229	31	f	f	PROPN
ejpam-4370	229	32	is	be	AUX
ejpam-4370	229	33	lower	low	ADJ
ejpam-4370	229	34	contra-(λ	contra-(λ	PROPN
ejpam-4370	229	35	,	,	PUNCT
ejpam-4370	229	36	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	229	37	.	.	PUNCT
ejpam-4370	230	1	corollary	corollary	ADJ
ejpam-4370	230	2	7	7	NUM
ejpam-4370	230	3	.	.	PUNCT
ejpam-4370	231	1	let	let	VERB
ejpam-4370	231	2	f	f	NOUN
ejpam-4370	231	3	:	:	PUNCT
ejpam-4370	231	4	(	(	PUNCT
ejpam-4370	231	5	x	x	X
ejpam-4370	231	6	,	,	PUNCT
ejpam-4370	231	7	τ	τ	X
ejpam-4370	231	8	)	)	PUNCT
ejpam-4370	231	9	→	→	SYM
ejpam-4370	231	10	(	(	PUNCT
ejpam-4370	231	11	y	y	PROPN
ejpam-4370	231	12	,	,	PUNCT
ejpam-4370	231	13	σ	σ	PROPN
ejpam-4370	231	14	)	)	PUNCT
ejpam-4370	231	15	be	be	AUX
ejpam-4370	231	16	a	a	DET
ejpam-4370	231	17	function	function	NOUN
ejpam-4370	231	18	.	.	PUNCT
ejpam-4370	232	1	if	if	SCONJ
ejpam-4370	232	2	f(b(λ	f(b(λ	PROPN
ejpam-4370	232	3	,	,	PUNCT
ejpam-4370	232	4	sp	sp	NOUN
ejpam-4370	232	5	)	)	PUNCT
ejpam-4370	232	6	)	)	PUNCT
ejpam-4370	233	1	⊆	⊆	NUM
ejpam-4370	233	2	λ(λ	λ(λ	ADP
ejpam-4370	233	3	,	,	PUNCT
ejpam-4370	233	4	sp)(f(b	sp)(f(b	NOUN
ejpam-4370	233	5	)	)	PUNCT
ejpam-4370	233	6	)	)	PUNCT
ejpam-4370	233	7	for	for	ADP
ejpam-4370	233	8	every	every	DET
ejpam-4370	233	9	subset	subset	NOUN
ejpam-4370	233	10	b	b	PROPN
ejpam-4370	233	11	of	of	ADP
ejpam-4370	233	12	y	y	PROPN
ejpam-4370	233	13	,	,	PUNCT
ejpam-4370	233	14	then	then	ADV
ejpam-4370	233	15	f	f	PROPN
ejpam-4370	233	16	is	be	AUX
ejpam-4370	233	17	contra-(λ	contra-(λ	PROPN
ejpam-4370	233	18	,	,	PUNCT
ejpam-4370	233	19	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	233	20	.	.	PUNCT
ejpam-4370	234	1	definition	definition	NOUN
ejpam-4370	234	2	7	7	NUM
ejpam-4370	234	3	.	.	PUNCT
ejpam-4370	235	1	[	[	X
ejpam-4370	235	2	3	3	X
ejpam-4370	235	3	]	]	PUNCT
ejpam-4370	235	4	a	a	DET
ejpam-4370	235	5	multifunction	multifunction	NOUN
ejpam-4370	235	6	f	f	NOUN
ejpam-4370	235	7	:	:	PUNCT
ejpam-4370	235	8	(	(	PUNCT
ejpam-4370	235	9	x	x	X
ejpam-4370	235	10	,	,	PUNCT
ejpam-4370	235	11	τ	τ	X
ejpam-4370	235	12	)	)	PUNCT
ejpam-4370	235	13	→	→	SYM
ejpam-4370	235	14	(	(	PUNCT
ejpam-4370	235	15	y	y	PROPN
ejpam-4370	235	16	,	,	PUNCT
ejpam-4370	235	17	σ	σ	PROPN
ejpam-4370	235	18	)	)	PUNCT
ejpam-4370	235	19	is	be	AUX
ejpam-4370	235	20	said	say	VERB
ejpam-4370	235	21	to	to	PART
ejpam-4370	235	22	be	be	AUX
ejpam-4370	235	23	:	:	PUNCT
ejpam-4370	235	24	(	(	PUNCT
ejpam-4370	235	25	i	i	NOUN
ejpam-4370	235	26	)	)	PUNCT
ejpam-4370	235	27	upper	upper	ADJ
ejpam-4370	235	28	(	(	PUNCT
ejpam-4370	235	29	λ	λ	PROPN
ejpam-4370	235	30	,	,	PUNCT
ejpam-4370	235	31	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	235	32	if	if	SCONJ
ejpam-4370	235	33	,	,	PUNCT
ejpam-4370	235	34	for	for	SCONJ
ejpam-4370	235	35	each	each	DET
ejpam-4370	235	36	x	x	SYM
ejpam-4370	235	37	∈	∈	PROPN
ejpam-4370	235	38	x	x	X
ejpam-4370	235	39	and	and	CCONJ
ejpam-4370	235	40	each	each	DET
ejpam-4370	235	41	(	(	PUNCT
ejpam-4370	235	42	λ	λ	PROPN
ejpam-4370	235	43	,	,	PUNCT
ejpam-4370	235	44	sp)-open	sp)-open	NOUN
ejpam-4370	235	45	set	set	VERB
ejpam-4370	235	46	v	v	NUM
ejpam-4370	235	47	of	of	ADP
ejpam-4370	235	48	y	y	PRON
ejpam-4370	235	49	such	such	ADJ
ejpam-4370	235	50	that	that	SCONJ
ejpam-4370	235	51	f	f	PROPN
ejpam-4370	235	52	(	(	PUNCT
ejpam-4370	235	53	x	x	X
ejpam-4370	235	54	)	)	PUNCT
ejpam-4370	235	55	⊆	⊆	NUM
ejpam-4370	235	56	v	v	NOUN
ejpam-4370	235	57	,	,	PUNCT
ejpam-4370	235	58	there	there	PRON
ejpam-4370	235	59	exists	exist	VERB
ejpam-4370	235	60	a	a	DET
ejpam-4370	235	61	(	(	PUNCT
ejpam-4370	235	62	λ	λ	NOUN
ejpam-4370	235	63	,	,	PUNCT
ejpam-4370	235	64	sp)-open	sp)-open	NOUN
ejpam-4370	235	65	set	set	VERB
ejpam-4370	235	66	u	u	NOUN
ejpam-4370	235	67	of	of	ADP
ejpam-4370	235	68	x	x	PUNCT
ejpam-4370	235	69	containing	contain	VERB
ejpam-4370	235	70	x	x	PUNCT
ejpam-4370	236	1	such	such	ADJ
ejpam-4370	236	2	that	that	SCONJ
ejpam-4370	236	3	f	f	PROPN
ejpam-4370	236	4	(	(	PUNCT
ejpam-4370	236	5	u	u	NOUN
ejpam-4370	236	6	)	)	PUNCT
ejpam-4370	236	7	⊆	⊆	NUM
ejpam-4370	236	8	v	v	NOUN
ejpam-4370	236	9	;	;	PUNCT
ejpam-4370	236	10	(	(	PUNCT
ejpam-4370	236	11	ii	ii	NOUN
ejpam-4370	236	12	)	)	PUNCT
ejpam-4370	236	13	lower	low	ADJ
ejpam-4370	236	14	(	(	PUNCT
ejpam-4370	236	15	λ	λ	NOUN
ejpam-4370	236	16	,	,	PUNCT
ejpam-4370	236	17	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	236	18	if	if	SCONJ
ejpam-4370	236	19	,	,	PUNCT
ejpam-4370	236	20	for	for	SCONJ
ejpam-4370	236	21	each	each	DET
ejpam-4370	236	22	x	x	SYM
ejpam-4370	236	23	∈	∈	PROPN
ejpam-4370	236	24	x	x	X
ejpam-4370	236	25	and	and	CCONJ
ejpam-4370	236	26	each	each	DET
ejpam-4370	236	27	(	(	PUNCT
ejpam-4370	236	28	λ	λ	PROPN
ejpam-4370	236	29	,	,	PUNCT
ejpam-4370	236	30	sp)-open	sp)-open	NOUN
ejpam-4370	236	31	set	set	VERB
ejpam-4370	236	32	v	v	NUM
ejpam-4370	236	33	of	of	ADP
ejpam-4370	236	34	y	y	PRON
ejpam-4370	236	35	such	such	ADJ
ejpam-4370	236	36	that	that	SCONJ
ejpam-4370	236	37	f	f	PROPN
ejpam-4370	236	38	(	(	PUNCT
ejpam-4370	236	39	x	x	NOUN
ejpam-4370	236	40	)	)	PUNCT
ejpam-4370	236	41	∩	∩	NOUN
ejpam-4370	236	42	v	v	ADP
ejpam-4370	236	43	̸=	̸=	PROPN
ejpam-4370	236	44	∅	∅	NOUN
ejpam-4370	236	45	,	,	PUNCT
ejpam-4370	236	46	there	there	PRON
ejpam-4370	236	47	exists	exist	VERB
ejpam-4370	236	48	a	a	DET
ejpam-4370	236	49	(	(	PUNCT
ejpam-4370	236	50	λ	λ	NOUN
ejpam-4370	236	51	,	,	PUNCT
ejpam-4370	236	52	sp)-open	sp)-open	NOUN
ejpam-4370	236	53	set	set	VERB
ejpam-4370	236	54	u	u	NOUN
ejpam-4370	236	55	of	of	ADP
ejpam-4370	236	56	x	x	PUNCT
ejpam-4370	236	57	containing	contain	VERB
ejpam-4370	236	58	x	x	PUNCT
ejpam-4370	236	59	such	such	ADJ
ejpam-4370	236	60	that	that	SCONJ
ejpam-4370	236	61	f	f	PROPN
ejpam-4370	236	62	(	(	PUNCT
ejpam-4370	236	63	z	z	NOUN
ejpam-4370	236	64	)	)	PUNCT
ejpam-4370	236	65	∩	∩	NOUN
ejpam-4370	236	66	v	v	ADP
ejpam-4370	236	67	̸=	̸=	PROPN
ejpam-4370	236	68	∅	∅	NOUN
ejpam-4370	236	69	for	for	ADP
ejpam-4370	236	70	each	each	DET
ejpam-4370	236	71	z	z	NOUN
ejpam-4370	236	72	∈	∈	PROPN
ejpam-4370	236	73	u	u	PROPN
ejpam-4370	236	74	.	.	PUNCT
ejpam-4370	237	1	lemma	lemma	PROPN
ejpam-4370	237	2	5	5	NUM
ejpam-4370	237	3	.	.	PUNCT
ejpam-4370	238	1	[	[	X
ejpam-4370	238	2	3	3	X
ejpam-4370	238	3	]	]	PUNCT
ejpam-4370	238	4	for	for	ADP
ejpam-4370	238	5	a	a	DET
ejpam-4370	238	6	multifunction	multifunction	NOUN
ejpam-4370	238	7	f	f	NOUN
ejpam-4370	238	8	:	:	PUNCT
ejpam-4370	238	9	(	(	PUNCT
ejpam-4370	238	10	x	x	X
ejpam-4370	238	11	,	,	PUNCT
ejpam-4370	238	12	τ	τ	X
ejpam-4370	238	13	)	)	PUNCT
ejpam-4370	238	14	→	→	SYM
ejpam-4370	238	15	(	(	PUNCT
ejpam-4370	238	16	y	y	PROPN
ejpam-4370	238	17	,	,	PUNCT
ejpam-4370	238	18	σ	σ	PROPN
ejpam-4370	238	19	)	)	PUNCT
ejpam-4370	238	20	,	,	PUNCT
ejpam-4370	238	21	the	the	DET
ejpam-4370	238	22	following	follow	VERB
ejpam-4370	238	23	properties	property	NOUN
ejpam-4370	238	24	are	be	AUX
ejpam-4370	238	25	equivalent	equivalent	ADJ
ejpam-4370	238	26	:	:	PUNCT
ejpam-4370	238	27	(	(	PUNCT
ejpam-4370	238	28	1	1	X
ejpam-4370	238	29	)	)	PUNCT
ejpam-4370	238	30	f	f	PROPN
ejpam-4370	238	31	is	be	AUX
ejpam-4370	238	32	upper	upper	ADJ
ejpam-4370	238	33	(	(	PUNCT
ejpam-4370	238	34	λ	λ	NOUN
ejpam-4370	238	35	,	,	PUNCT
ejpam-4370	238	36	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	238	37	;	;	PUNCT
ejpam-4370	238	38	(	(	PUNCT
ejpam-4370	238	39	2	2	NUM
ejpam-4370	238	40	)	)	PUNCT
ejpam-4370	238	41	f+(v	f+(v	NOUN
ejpam-4370	238	42	)	)	PUNCT
ejpam-4370	239	1	is	be	AUX
ejpam-4370	239	2	(	(	PUNCT
ejpam-4370	239	3	λ	λ	INTJ
ejpam-4370	239	4	,	,	PUNCT
ejpam-4370	239	5	sp)-open	sp)-open	ADJ
ejpam-4370	239	6	in	in	ADP
ejpam-4370	239	7	x	x	PUNCT
ejpam-4370	239	8	for	for	ADP
ejpam-4370	239	9	every	every	DET
ejpam-4370	239	10	(	(	PUNCT
ejpam-4370	239	11	λ	λ	NOUN
ejpam-4370	239	12	,	,	PUNCT
ejpam-4370	239	13	sp)-open	sp)-open	NOUN
ejpam-4370	239	14	set	set	VERB
ejpam-4370	239	15	v	v	NOUN
ejpam-4370	239	16	of	of	ADP
ejpam-4370	239	17	y	y	PROPN
ejpam-4370	239	18	;	;	PUNCT
ejpam-4370	239	19	(	(	PUNCT
ejpam-4370	239	20	3	3	X
ejpam-4370	239	21	)	)	PUNCT
ejpam-4370	239	22	f−(k	f−(k	PROPN
ejpam-4370	239	23	)	)	PUNCT
ejpam-4370	239	24	is	be	AUX
ejpam-4370	239	25	(	(	PUNCT
ejpam-4370	239	26	λ	λ	X
ejpam-4370	239	27	,	,	PUNCT
ejpam-4370	239	28	sp)-closed	sp)-close	VERB
ejpam-4370	239	29	in	in	ADP
ejpam-4370	239	30	x	x	PUNCT
ejpam-4370	239	31	for	for	SCONJ
ejpam-4370	239	32	every	every	DET
ejpam-4370	239	33	(	(	PUNCT
ejpam-4370	239	34	λ	λ	PROPN
ejpam-4370	239	35	,	,	PUNCT
ejpam-4370	239	36	sp)-closed	sp)-close	VERB
ejpam-4370	239	37	set	set	VERB
ejpam-4370	239	38	k	k	PROPN
ejpam-4370	239	39	of	of	ADP
ejpam-4370	239	40	y	y	PROPN
ejpam-4370	239	41	;	;	PUNCT
ejpam-4370	239	42	(	(	PUNCT
ejpam-4370	239	43	4	4	X
ejpam-4370	239	44	)	)	PUNCT
ejpam-4370	239	45	[	[	X
ejpam-4370	239	46	f−(b)](λ	f−(b)](λ	PROPN
ejpam-4370	239	47	,	,	PUNCT
ejpam-4370	239	48	sp	sp	NOUN
ejpam-4370	239	49	)	)	PUNCT
ejpam-4370	239	50	⊆	⊆	NUM
ejpam-4370	239	51	f−(b(λ	f−(b(λ	NOUN
ejpam-4370	239	52	,	,	PUNCT
ejpam-4370	239	53	sp	sp	NOUN
ejpam-4370	239	54	)	)	PUNCT
ejpam-4370	239	55	)	)	PUNCT
ejpam-4370	239	56	for	for	ADP
ejpam-4370	239	57	every	every	DET
ejpam-4370	239	58	subset	subset	NOUN
ejpam-4370	239	59	b	b	PROPN
ejpam-4370	239	60	of	of	ADP
ejpam-4370	239	61	y	y	PROPN
ejpam-4370	239	62	;	;	PUNCT
ejpam-4370	239	63	(	(	PUNCT
ejpam-4370	239	64	5	5	X
ejpam-4370	239	65	)	)	PUNCT
ejpam-4370	239	66	f+(b(λ	f+(b(λ	PROPN
ejpam-4370	239	67	,	,	PUNCT
ejpam-4370	239	68	sp	sp	NOUN
ejpam-4370	239	69	)	)	PUNCT
ejpam-4370	239	70	)	)	PUNCT
ejpam-4370	239	71	⊆	⊆	NUM
ejpam-4370	239	72	[	[	X
ejpam-4370	239	73	f+(b)](λ	f+(b)](λ	NUM
ejpam-4370	239	74	,	,	PUNCT
ejpam-4370	239	75	sp	sp	NOUN
ejpam-4370	239	76	)	)	PUNCT
ejpam-4370	239	77	for	for	ADP
ejpam-4370	239	78	every	every	DET
ejpam-4370	239	79	subset	subset	NOUN
ejpam-4370	239	80	b	b	PROPN
ejpam-4370	239	81	of	of	ADP
ejpam-4370	239	82	y	y	PROPN
ejpam-4370	239	83	.	.	PUNCT
ejpam-4370	240	1	theorem	theorem	ADJ
ejpam-4370	240	2	10	10	NUM
ejpam-4370	240	3	.	.	PUNCT
ejpam-4370	241	1	if	if	SCONJ
ejpam-4370	241	2	f	f	PROPN
ejpam-4370	241	3	:	:	PUNCT
ejpam-4370	241	4	(	(	PUNCT
ejpam-4370	241	5	x	x	X
ejpam-4370	241	6	,	,	PUNCT
ejpam-4370	241	7	τ	τ	X
ejpam-4370	241	8	)	)	PUNCT
ejpam-4370	241	9	→	→	SYM
ejpam-4370	241	10	(	(	PUNCT
ejpam-4370	241	11	y	y	PROPN
ejpam-4370	241	12	,	,	PUNCT
ejpam-4370	241	13	σ	σ	PROPN
ejpam-4370	241	14	)	)	PUNCT
ejpam-4370	241	15	is	be	AUX
ejpam-4370	241	16	an	an	DET
ejpam-4370	241	17	upper	upper	ADJ
ejpam-4370	241	18	(	(	PUNCT
ejpam-4370	241	19	λ	λ	NOUN
ejpam-4370	241	20	,	,	PUNCT
ejpam-4370	241	21	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	241	22	multifunction	multifunction	NOUN
ejpam-4370	241	23	and	and	CCONJ
ejpam-4370	241	24	g	g	NOUN
ejpam-4370	241	25	:	:	PUNCT
ejpam-4370	241	26	(	(	PUNCT
ejpam-4370	241	27	y	y	PROPN
ejpam-4370	241	28	,	,	PUNCT
ejpam-4370	241	29	σ	σ	PROPN
ejpam-4370	241	30	)	)	PUNCT
ejpam-4370	241	31	→	→	SYM
ejpam-4370	241	32	(	(	PUNCT
ejpam-4370	241	33	z	z	PROPN
ejpam-4370	241	34	,	,	PUNCT
ejpam-4370	241	35	ρ	ρ	PROPN
ejpam-4370	241	36	)	)	PUNCT
ejpam-4370	242	1	is	be	AUX
ejpam-4370	242	2	an	an	DET
ejpam-4370	242	3	upper	upper	ADJ
ejpam-4370	242	4	contra-(λ	contra-(λ	PROPN
ejpam-4370	242	5	,	,	PUNCT
ejpam-4370	242	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	242	7	multifunction	multifunction	NOUN
ejpam-4370	242	8	,	,	PUNCT
ejpam-4370	242	9	then	then	ADV
ejpam-4370	242	10	g	g	PROPN
ejpam-4370	242	11	◦	◦	NOUN
ejpam-4370	242	12	f	f	X
ejpam-4370	242	13	:	:	PUNCT
ejpam-4370	242	14	(	(	PUNCT
ejpam-4370	242	15	x	x	X
ejpam-4370	242	16	,	,	PUNCT
ejpam-4370	242	17	τ	τ	X
ejpam-4370	242	18	)	)	PUNCT
ejpam-4370	242	19	→	→	SYM
ejpam-4370	242	20	(	(	PUNCT
ejpam-4370	242	21	z	z	PROPN
ejpam-4370	242	22	,	,	PUNCT
ejpam-4370	242	23	ρ	ρ	PROPN
ejpam-4370	242	24	)	)	PUNCT
ejpam-4370	242	25	is	be	AUX
ejpam-4370	242	26	upper	upper	ADJ
ejpam-4370	242	27	contra-(λ	contra-(λ	PROPN
ejpam-4370	242	28	,	,	PUNCT
ejpam-4370	242	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	242	30	.	.	PUNCT
ejpam-4370	243	1	proof	proof	NOUN
ejpam-4370	243	2	.	.	PUNCT
ejpam-4370	244	1	let	let	VERB
ejpam-4370	244	2	v	v	PART
ejpam-4370	244	3	be	be	AUX
ejpam-4370	244	4	any	any	DET
ejpam-4370	244	5	(	(	PUNCT
ejpam-4370	244	6	λ	λ	PROPN
ejpam-4370	244	7	,	,	PUNCT
ejpam-4370	244	8	sp)-closed	sp)-close	VERB
ejpam-4370	244	9	set	set	NOUN
ejpam-4370	244	10	of	of	ADP
ejpam-4370	244	11	z.	z.	PROPN
ejpam-4370	244	12	from	from	ADP
ejpam-4370	244	13	the	the	DET
ejpam-4370	244	14	definition	definition	NOUN
ejpam-4370	244	15	of	of	ADP
ejpam-4370	244	16	g	g	PROPN
ejpam-4370	244	17	◦	◦	NOUN
ejpam-4370	244	18	f	f	PROPN
ejpam-4370	244	19	,	,	PUNCT
ejpam-4370	244	20	we	we	PRON
ejpam-4370	244	21	have	have	VERB
ejpam-4370	244	22	(	(	PUNCT
ejpam-4370	244	23	g	g	NOUN
ejpam-4370	244	24	◦	◦	NOUN
ejpam-4370	244	25	f	f	PROPN
ejpam-4370	244	26	)	)	PUNCT
ejpam-4370	245	1	+	+	ADJ
ejpam-4370	245	2	(	(	PUNCT
ejpam-4370	245	3	v	v	NOUN
ejpam-4370	245	4	)	)	PUNCT
ejpam-4370	245	5	=	=	SYM
ejpam-4370	245	6	f+(g+(v	f+(g+(v	NOUN
ejpam-4370	245	7	)	)	PUNCT
ejpam-4370	245	8	)	)	PUNCT
ejpam-4370	245	9	.	.	PUNCT
ejpam-4370	246	1	since	since	SCONJ
ejpam-4370	246	2	g	g	PROPN
ejpam-4370	246	3	is	be	AUX
ejpam-4370	246	4	upper	upper	ADJ
ejpam-4370	246	5	contra-(λ	contra-(λ	PROPN
ejpam-4370	246	6	,	,	PUNCT
ejpam-4370	246	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	246	8	,	,	PUNCT
ejpam-4370	246	9	by	by	ADP
ejpam-4370	246	10	theorem	theorem	NOUN
ejpam-4370	246	11	1	1	NUM
ejpam-4370	246	12	,	,	PUNCT
ejpam-4370	246	13	g+(v	g+(v	PROPN
ejpam-4370	246	14	)	)	PUNCT
ejpam-4370	246	15	is	be	AUX
ejpam-4370	246	16	(	(	PUNCT
ejpam-4370	246	17	λ	λ	INTJ
ejpam-4370	246	18	,	,	PUNCT
ejpam-4370	246	19	sp)-open	sp)-open	ADJ
ejpam-4370	246	20	in	in	ADP
ejpam-4370	246	21	y	y	PROPN
ejpam-4370	246	22	.	.	PUNCT
ejpam-4370	247	1	since	since	SCONJ
ejpam-4370	247	2	f	f	PROPN
ejpam-4370	247	3	is	be	AUX
ejpam-4370	247	4	upper	upper	ADJ
ejpam-4370	247	5	(	(	PUNCT
ejpam-4370	247	6	λ	λ	PROPN
ejpam-4370	247	7	,	,	PUNCT
ejpam-4370	247	8	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	247	9	,	,	PUNCT
ejpam-4370	247	10	by	by	ADP
ejpam-4370	247	11	lemma	lemma	PROPN
ejpam-4370	247	12	5	5	NUM
ejpam-4370	247	13	,	,	PUNCT
ejpam-4370	247	14	f+(g+(v	f+(g+(v	ADV
ejpam-4370	247	15	)	)	PUNCT
ejpam-4370	247	16	)	)	PUNCT
ejpam-4370	247	17	is	be	AUX
ejpam-4370	247	18	(	(	PUNCT
ejpam-4370	247	19	λ	λ	INTJ
ejpam-4370	247	20	,	,	PUNCT
ejpam-4370	247	21	sp)-open	sp)-open	ADJ
ejpam-4370	247	22	in	in	ADP
ejpam-4370	247	23	x.	x.	NOUN
ejpam-4370	247	24	thus	thus	ADV
ejpam-4370	247	25	,	,	PUNCT
ejpam-4370	247	26	(	(	PUNCT
ejpam-4370	247	27	g	g	PROPN
ejpam-4370	247	28	◦	◦	NOUN
ejpam-4370	247	29	f	f	PROPN
ejpam-4370	247	30	)	)	PUNCT
ejpam-4370	248	1	+	+	ADJ
ejpam-4370	248	2	(	(	PUNCT
ejpam-4370	248	3	v	v	NOUN
ejpam-4370	248	4	)	)	PUNCT
ejpam-4370	248	5	is	be	AUX
ejpam-4370	248	6	(	(	PUNCT
ejpam-4370	248	7	λ	λ	X
ejpam-4370	248	8	,	,	PUNCT
ejpam-4370	248	9	sp)-open	sp)-open	ADJ
ejpam-4370	248	10	in	in	ADP
ejpam-4370	248	11	x	x	PUNCT
ejpam-4370	248	12	and	and	CCONJ
ejpam-4370	248	13	hence	hence	ADV
ejpam-4370	248	14	g	g	PROPN
ejpam-4370	248	15	◦	◦	NOUN
ejpam-4370	248	16	f	f	PROPN
ejpam-4370	248	17	is	be	AUX
ejpam-4370	248	18	upper	upper	ADJ
ejpam-4370	248	19	contra-(λ	contra-(λ	PROPN
ejpam-4370	248	20	,	,	PUNCT
ejpam-4370	248	21	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	248	22	.	.	PUNCT
ejpam-4370	249	1	c.	c.	PROPN
ejpam-4370	249	2	boonpok	boonpok	PROPN
ejpam-4370	249	3	,	,	PUNCT
ejpam-4370	249	4	c.	c.	PROPN
ejpam-4370	249	5	viriyapong	viriyapong	PROPN
ejpam-4370	249	6	/	/	SYM
ejpam-4370	249	7	eur	eur	PROPN
ejpam-4370	249	8	.	.	PUNCT
ejpam-4370	250	1	j.	j.	PROPN
ejpam-4370	250	2	pure	pure	PROPN
ejpam-4370	250	3	appl	appl	PROPN
ejpam-4370	250	4	.	.	PROPN
ejpam-4370	250	5	math	math	PROPN
ejpam-4370	250	6	,	,	PUNCT
ejpam-4370	250	7	15	15	NUM
ejpam-4370	250	8	(	(	PUNCT
ejpam-4370	250	9	4	4	NUM
ejpam-4370	250	10	)	)	PUNCT
ejpam-4370	250	11	(	(	PUNCT
ejpam-4370	250	12	2022	2022	NUM
ejpam-4370	250	13	)	)	PUNCT
ejpam-4370	250	14	,	,	PUNCT
ejpam-4370	250	15	1694	1694	NUM
ejpam-4370	250	16	-	-	SYM
ejpam-4370	250	17	1704	1704	NUM
ejpam-4370	250	18	1702	1702	NUM
ejpam-4370	250	19	lemma	lemma	PROPN
ejpam-4370	250	20	6	6	NUM
ejpam-4370	250	21	.	.	PUNCT
ejpam-4370	251	1	[	[	X
ejpam-4370	251	2	3	3	X
ejpam-4370	251	3	]	]	PUNCT
ejpam-4370	251	4	for	for	ADP
ejpam-4370	251	5	a	a	DET
ejpam-4370	251	6	multifunction	multifunction	NOUN
ejpam-4370	251	7	f	f	NOUN
ejpam-4370	251	8	:	:	PUNCT
ejpam-4370	251	9	(	(	PUNCT
ejpam-4370	251	10	x	x	X
ejpam-4370	251	11	,	,	PUNCT
ejpam-4370	251	12	τ	τ	X
ejpam-4370	251	13	)	)	PUNCT
ejpam-4370	251	14	→	→	SYM
ejpam-4370	251	15	(	(	PUNCT
ejpam-4370	251	16	y	y	PROPN
ejpam-4370	251	17	,	,	PUNCT
ejpam-4370	251	18	σ	σ	PROPN
ejpam-4370	251	19	)	)	PUNCT
ejpam-4370	251	20	,	,	PUNCT
ejpam-4370	251	21	the	the	DET
ejpam-4370	251	22	following	follow	VERB
ejpam-4370	251	23	properties	property	NOUN
ejpam-4370	251	24	are	be	AUX
ejpam-4370	251	25	equivalent	equivalent	ADJ
ejpam-4370	251	26	:	:	PUNCT
ejpam-4370	251	27	(	(	PUNCT
ejpam-4370	251	28	1	1	X
ejpam-4370	251	29	)	)	PUNCT
ejpam-4370	251	30	f	f	PROPN
ejpam-4370	251	31	is	be	AUX
ejpam-4370	251	32	lower	low	ADJ
ejpam-4370	251	33	(	(	PUNCT
ejpam-4370	251	34	λ	λ	NOUN
ejpam-4370	251	35	,	,	PUNCT
ejpam-4370	251	36	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	251	37	;	;	PUNCT
ejpam-4370	251	38	(	(	PUNCT
ejpam-4370	251	39	2	2	X
ejpam-4370	251	40	)	)	PUNCT
ejpam-4370	251	41	f−(v	f−(v	NOUN
ejpam-4370	251	42	)	)	PUNCT
ejpam-4370	251	43	is	be	AUX
ejpam-4370	251	44	(	(	PUNCT
ejpam-4370	251	45	λ	λ	INTJ
ejpam-4370	251	46	,	,	PUNCT
ejpam-4370	251	47	sp)-open	sp)-open	ADJ
ejpam-4370	251	48	in	in	ADP
ejpam-4370	251	49	x	x	PUNCT
ejpam-4370	251	50	for	for	ADP
ejpam-4370	251	51	every	every	DET
ejpam-4370	251	52	(	(	PUNCT
ejpam-4370	251	53	λ	λ	NOUN
ejpam-4370	251	54	,	,	PUNCT
ejpam-4370	251	55	sp)-open	sp)-open	NOUN
ejpam-4370	251	56	set	set	VERB
ejpam-4370	251	57	v	v	NOUN
ejpam-4370	251	58	of	of	ADP
ejpam-4370	251	59	y	y	PROPN
ejpam-4370	251	60	;	;	PUNCT
ejpam-4370	251	61	(	(	PUNCT
ejpam-4370	251	62	3	3	X
ejpam-4370	251	63	)	)	PUNCT
ejpam-4370	251	64	f+(k	f+(k	NUM
ejpam-4370	251	65	)	)	PUNCT
ejpam-4370	252	1	is	be	AUX
ejpam-4370	252	2	(	(	PUNCT
ejpam-4370	252	3	λ	λ	X
ejpam-4370	252	4	,	,	PUNCT
ejpam-4370	252	5	sp)-closed	sp)-close	VERB
ejpam-4370	252	6	in	in	ADP
ejpam-4370	252	7	x	x	PUNCT
ejpam-4370	252	8	for	for	SCONJ
ejpam-4370	252	9	every	every	DET
ejpam-4370	252	10	(	(	PUNCT
ejpam-4370	252	11	λ	λ	PROPN
ejpam-4370	252	12	,	,	PUNCT
ejpam-4370	252	13	sp)-closed	sp)-close	VERB
ejpam-4370	252	14	set	set	VERB
ejpam-4370	252	15	k	k	PROPN
ejpam-4370	252	16	of	of	ADP
ejpam-4370	252	17	y	y	PROPN
ejpam-4370	252	18	;	;	PUNCT
ejpam-4370	252	19	(	(	PUNCT
ejpam-4370	252	20	4	4	X
ejpam-4370	252	21	)	)	PUNCT
ejpam-4370	252	22	[	[	X
ejpam-4370	252	23	f+(b)](λ	f+(b)](λ	NUM
ejpam-4370	252	24	,	,	PUNCT
ejpam-4370	252	25	sp	sp	NOUN
ejpam-4370	252	26	)	)	PUNCT
ejpam-4370	252	27	⊆	⊆	NUM
ejpam-4370	252	28	f+(b(λ	f+(b(λ	PROPN
ejpam-4370	252	29	,	,	PUNCT
ejpam-4370	252	30	sp	sp	NOUN
ejpam-4370	252	31	)	)	PUNCT
ejpam-4370	252	32	)	)	PUNCT
ejpam-4370	252	33	for	for	ADP
ejpam-4370	252	34	every	every	DET
ejpam-4370	252	35	subset	subset	NOUN
ejpam-4370	252	36	b	b	PROPN
ejpam-4370	252	37	of	of	ADP
ejpam-4370	252	38	y	y	PROPN
ejpam-4370	252	39	;	;	PUNCT
ejpam-4370	252	40	(	(	PUNCT
ejpam-4370	252	41	5	5	X
ejpam-4370	252	42	)	)	PUNCT
ejpam-4370	252	43	f	f	NOUN
ejpam-4370	252	44	(	(	PUNCT
ejpam-4370	252	45	a(λ	a(λ	ADV
ejpam-4370	252	46	,	,	PUNCT
ejpam-4370	252	47	sp	sp	NOUN
ejpam-4370	252	48	)	)	PUNCT
ejpam-4370	252	49	)	)	PUNCT
ejpam-4370	253	1	⊆	⊆	NUM
ejpam-4370	254	1	[	[	X
ejpam-4370	254	2	f	f	X
ejpam-4370	254	3	(	(	PUNCT
ejpam-4370	254	4	a)](λ	a)](λ	PROPN
ejpam-4370	254	5	,	,	PUNCT
ejpam-4370	254	6	sp	sp	NOUN
ejpam-4370	254	7	)	)	PUNCT
ejpam-4370	254	8	for	for	ADP
ejpam-4370	254	9	every	every	DET
ejpam-4370	254	10	subset	subset	NOUN
ejpam-4370	254	11	a	a	PRON
ejpam-4370	254	12	of	of	ADP
ejpam-4370	254	13	x	x	PRON
ejpam-4370	254	14	;	;	PUNCT
ejpam-4370	254	15	(	(	PUNCT
ejpam-4370	254	16	6	6	X
ejpam-4370	254	17	)	)	PUNCT
ejpam-4370	254	18	f−(b(λ	f−(b(λ	NOUN
ejpam-4370	254	19	,	,	PUNCT
ejpam-4370	254	20	sp	sp	NOUN
ejpam-4370	254	21	)	)	PUNCT
ejpam-4370	254	22	)	)	PUNCT
ejpam-4370	255	1	⊆	⊆	NUM
ejpam-4370	255	2	[	[	X
ejpam-4370	255	3	f−(b)](λ	f−(b)](λ	PROPN
ejpam-4370	255	4	,	,	PUNCT
ejpam-4370	255	5	sp	sp	NOUN
ejpam-4370	255	6	)	)	PUNCT
ejpam-4370	255	7	for	for	ADP
ejpam-4370	255	8	every	every	DET
ejpam-4370	255	9	subset	subset	NOUN
ejpam-4370	255	10	b	b	PROPN
ejpam-4370	255	11	of	of	ADP
ejpam-4370	255	12	y	y	PROPN
ejpam-4370	255	13	.	.	PUNCT
ejpam-4370	256	1	theorem	theorem	VERB
ejpam-4370	256	2	11	11	NUM
ejpam-4370	256	3	.	.	PUNCT
ejpam-4370	257	1	if	if	SCONJ
ejpam-4370	257	2	f	f	PROPN
ejpam-4370	257	3	:	:	PUNCT
ejpam-4370	257	4	(	(	PUNCT
ejpam-4370	257	5	x	x	X
ejpam-4370	257	6	,	,	PUNCT
ejpam-4370	257	7	τ	τ	X
ejpam-4370	257	8	)	)	PUNCT
ejpam-4370	257	9	→	→	SYM
ejpam-4370	257	10	(	(	PUNCT
ejpam-4370	257	11	y	y	PROPN
ejpam-4370	257	12	,	,	PUNCT
ejpam-4370	257	13	σ	σ	PROPN
ejpam-4370	257	14	)	)	PUNCT
ejpam-4370	257	15	is	be	AUX
ejpam-4370	257	16	a	a	DET
ejpam-4370	257	17	lower	low	ADJ
ejpam-4370	257	18	(	(	PUNCT
ejpam-4370	257	19	λ	λ	NOUN
ejpam-4370	257	20	,	,	PUNCT
ejpam-4370	257	21	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	257	22	multifunction	multifunction	NOUN
ejpam-4370	257	23	and	and	CCONJ
ejpam-4370	257	24	g	g	NOUN
ejpam-4370	257	25	:	:	PUNCT
ejpam-4370	257	26	(	(	PUNCT
ejpam-4370	257	27	y	y	PROPN
ejpam-4370	257	28	,	,	PUNCT
ejpam-4370	257	29	σ	σ	PROPN
ejpam-4370	257	30	)	)	PUNCT
ejpam-4370	257	31	→	→	SYM
ejpam-4370	257	32	(	(	PUNCT
ejpam-4370	257	33	z	z	PROPN
ejpam-4370	257	34	,	,	PUNCT
ejpam-4370	257	35	ρ	ρ	PROPN
ejpam-4370	257	36	)	)	PUNCT
ejpam-4370	258	1	is	be	AUX
ejpam-4370	258	2	a	a	DET
ejpam-4370	258	3	lower	low	ADJ
ejpam-4370	258	4	contra-(λ	contra-(λ	PROPN
ejpam-4370	258	5	,	,	PUNCT
ejpam-4370	258	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	258	7	multifunction	multifunction	NOUN
ejpam-4370	258	8	,	,	PUNCT
ejpam-4370	258	9	then	then	ADV
ejpam-4370	258	10	g	g	PROPN
ejpam-4370	258	11	◦	◦	NOUN
ejpam-4370	258	12	f	f	X
ejpam-4370	258	13	:	:	PUNCT
ejpam-4370	258	14	(	(	PUNCT
ejpam-4370	258	15	x	x	X
ejpam-4370	258	16	,	,	PUNCT
ejpam-4370	258	17	τ	τ	X
ejpam-4370	258	18	)	)	PUNCT
ejpam-4370	258	19	→	→	SYM
ejpam-4370	258	20	(	(	PUNCT
ejpam-4370	258	21	z	z	PROPN
ejpam-4370	258	22	,	,	PUNCT
ejpam-4370	258	23	ρ	ρ	PROPN
ejpam-4370	258	24	)	)	PUNCT
ejpam-4370	258	25	is	be	AUX
ejpam-4370	258	26	lower	low	ADJ
ejpam-4370	258	27	contra-(λ	contra-(λ	PROPN
ejpam-4370	258	28	,	,	PUNCT
ejpam-4370	258	29	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	258	30	.	.	PUNCT
ejpam-4370	259	1	proof	proof	NOUN
ejpam-4370	259	2	.	.	PUNCT
ejpam-4370	260	1	let	let	VERB
ejpam-4370	260	2	v	v	PART
ejpam-4370	260	3	be	be	AUX
ejpam-4370	260	4	any	any	DET
ejpam-4370	260	5	(	(	PUNCT
ejpam-4370	260	6	λ	λ	PROPN
ejpam-4370	260	7	,	,	PUNCT
ejpam-4370	260	8	sp)-closed	sp)-close	VERB
ejpam-4370	260	9	set	set	NOUN
ejpam-4370	260	10	of	of	ADP
ejpam-4370	260	11	z.	z.	PROPN
ejpam-4370	260	12	from	from	ADP
ejpam-4370	260	13	the	the	DET
ejpam-4370	260	14	definition	definition	NOUN
ejpam-4370	260	15	of	of	ADP
ejpam-4370	260	16	g	g	PROPN
ejpam-4370	260	17	◦	◦	NOUN
ejpam-4370	260	18	f	f	PROPN
ejpam-4370	260	19	,	,	PUNCT
ejpam-4370	260	20	we	we	PRON
ejpam-4370	260	21	have	have	VERB
ejpam-4370	260	22	(	(	PUNCT
ejpam-4370	260	23	g	g	NOUN
ejpam-4370	260	24	◦	◦	NOUN
ejpam-4370	260	25	f	f	NOUN
ejpam-4370	260	26	)	)	PUNCT
ejpam-4370	260	27	−(v	−(v	NOUN
ejpam-4370	260	28	)	)	PUNCT
ejpam-4370	261	1	=	=	SYM
ejpam-4370	261	2	f−(g−(v	f−(g−(v	NOUN
ejpam-4370	261	3	)	)	PUNCT
ejpam-4370	261	4	)	)	PUNCT
ejpam-4370	261	5	.	.	PUNCT
ejpam-4370	262	1	since	since	SCONJ
ejpam-4370	262	2	g	g	PROPN
ejpam-4370	262	3	is	be	AUX
ejpam-4370	262	4	lower	low	ADJ
ejpam-4370	262	5	contra-(λ	contra-(λ	PROPN
ejpam-4370	262	6	,	,	PUNCT
ejpam-4370	262	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	262	8	,	,	PUNCT
ejpam-4370	262	9	by	by	ADP
ejpam-4370	262	10	theorem	theorem	NOUN
ejpam-4370	262	11	2	2	NUM
ejpam-4370	262	12	,	,	PUNCT
ejpam-4370	262	13	g−(v	g−(v	PROPN
ejpam-4370	262	14	)	)	PUNCT
ejpam-4370	262	15	is	be	AUX
ejpam-4370	262	16	(	(	PUNCT
ejpam-4370	262	17	λ	λ	X
ejpam-4370	262	18	,	,	PUNCT
ejpam-4370	262	19	sp)-open	sp)-open	ADJ
ejpam-4370	262	20	in	in	ADP
ejpam-4370	262	21	y	y	PROPN
ejpam-4370	262	22	.	.	PUNCT
ejpam-4370	263	1	since	since	SCONJ
ejpam-4370	263	2	f	f	PROPN
ejpam-4370	263	3	is	be	AUX
ejpam-4370	263	4	lower	low	ADJ
ejpam-4370	263	5	(	(	PUNCT
ejpam-4370	263	6	λ	λ	NOUN
ejpam-4370	263	7	,	,	PUNCT
ejpam-4370	263	8	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	263	9	,	,	PUNCT
ejpam-4370	263	10	by	by	ADP
ejpam-4370	263	11	lemma	lemma	PROPN
ejpam-4370	263	12	6	6	NUM
ejpam-4370	263	13	,	,	PUNCT
ejpam-4370	263	14	f−(g−(v	f−(g−(v	NOUN
ejpam-4370	263	15	)	)	PUNCT
ejpam-4370	263	16	)	)	PUNCT
ejpam-4370	263	17	is	be	AUX
ejpam-4370	263	18	(	(	PUNCT
ejpam-4370	263	19	λ	λ	INTJ
ejpam-4370	263	20	,	,	PUNCT
ejpam-4370	263	21	sp)-open	sp)-open	ADJ
ejpam-4370	263	22	in	in	ADP
ejpam-4370	263	23	x.	x.	NOUN
ejpam-4370	263	24	thus	thus	ADV
ejpam-4370	263	25	,	,	PUNCT
ejpam-4370	263	26	(	(	PUNCT
ejpam-4370	263	27	g	g	PROPN
ejpam-4370	263	28	◦	◦	NOUN
ejpam-4370	263	29	f	f	NOUN
ejpam-4370	263	30	)	)	PUNCT
ejpam-4370	263	31	−(v	−(v	NOUN
ejpam-4370	263	32	)	)	PUNCT
ejpam-4370	263	33	is	be	AUX
ejpam-4370	263	34	(	(	PUNCT
ejpam-4370	263	35	λ	λ	INTJ
ejpam-4370	263	36	,	,	PUNCT
ejpam-4370	263	37	sp)-open	sp)-open	ADJ
ejpam-4370	263	38	in	in	ADP
ejpam-4370	263	39	x	x	PUNCT
ejpam-4370	263	40	and	and	CCONJ
ejpam-4370	263	41	hence	hence	ADV
ejpam-4370	263	42	g	g	PROPN
ejpam-4370	263	43	◦	◦	NOUN
ejpam-4370	263	44	f	f	PROPN
ejpam-4370	263	45	is	be	AUX
ejpam-4370	263	46	lower	low	ADJ
ejpam-4370	263	47	contra-(λ	contra-(λ	PROPN
ejpam-4370	263	48	,	,	PUNCT
ejpam-4370	263	49	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	263	50	.	.	PUNCT
ejpam-4370	264	1	definition	definition	NOUN
ejpam-4370	264	2	8	8	NUM
ejpam-4370	264	3	.	.	PUNCT
ejpam-4370	265	1	[	[	X
ejpam-4370	265	2	22	22	NUM
ejpam-4370	265	3	]	]	PUNCT
ejpam-4370	265	4	a	a	DET
ejpam-4370	265	5	function	function	NOUN
ejpam-4370	265	6	f	f	NOUN
ejpam-4370	265	7	:	:	PUNCT
ejpam-4370	265	8	(	(	PUNCT
ejpam-4370	265	9	x	x	X
ejpam-4370	265	10	,	,	PUNCT
ejpam-4370	265	11	τ	τ	X
ejpam-4370	265	12	)	)	PUNCT
ejpam-4370	265	13	→	→	SYM
ejpam-4370	265	14	(	(	PUNCT
ejpam-4370	265	15	y	y	PROPN
ejpam-4370	265	16	,	,	PUNCT
ejpam-4370	265	17	σ	σ	PROPN
ejpam-4370	265	18	)	)	PUNCT
ejpam-4370	265	19	is	be	AUX
ejpam-4370	265	20	said	say	VERB
ejpam-4370	265	21	to	to	PART
ejpam-4370	265	22	be	be	AUX
ejpam-4370	265	23	(	(	PUNCT
ejpam-4370	265	24	λ	λ	X
ejpam-4370	265	25	,	,	PUNCT
ejpam-4370	265	26	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	265	27	at	at	ADP
ejpam-4370	265	28	a	a	DET
ejpam-4370	265	29	point	point	NOUN
ejpam-4370	265	30	x	x	SYM
ejpam-4370	265	31	∈	∈	NOUN
ejpam-4370	265	32	x	x	INTJ
ejpam-4370	265	33	if	if	SCONJ
ejpam-4370	265	34	,	,	PUNCT
ejpam-4370	265	35	for	for	ADP
ejpam-4370	265	36	each	each	DET
ejpam-4370	265	37	(	(	PUNCT
ejpam-4370	265	38	λ	λ	PROPN
ejpam-4370	265	39	,	,	PUNCT
ejpam-4370	265	40	sp)-open	sp)-open	NOUN
ejpam-4370	265	41	set	set	VERB
ejpam-4370	265	42	v	v	NUM
ejpam-4370	265	43	of	of	ADP
ejpam-4370	265	44	y	y	NOUN
ejpam-4370	265	45	containing	contain	VERB
ejpam-4370	265	46	f(x	f(x	PROPN
ejpam-4370	265	47	)	)	PUNCT
ejpam-4370	265	48	,	,	PUNCT
ejpam-4370	265	49	there	there	PRON
ejpam-4370	265	50	exists	exist	VERB
ejpam-4370	265	51	a	a	DET
ejpam-4370	265	52	(	(	PUNCT
ejpam-4370	265	53	λ	λ	NOUN
ejpam-4370	265	54	,	,	PUNCT
ejpam-4370	265	55	sp)open	sp)open	VERB
ejpam-4370	265	56	set	set	VERB
ejpam-4370	265	57	u	u	NOUN
ejpam-4370	265	58	of	of	ADP
ejpam-4370	265	59	x	x	PUNCT
ejpam-4370	265	60	containing	contain	VERB
ejpam-4370	265	61	x	x	PUNCT
ejpam-4370	265	62	such	such	ADJ
ejpam-4370	265	63	that	that	DET
ejpam-4370	265	64	f(u	f(u	PROPN
ejpam-4370	265	65	)	)	PUNCT
ejpam-4370	265	66	⊆	⊆	NUM
ejpam-4370	265	67	v	v	NOUN
ejpam-4370	265	68	.	.	PUNCT
ejpam-4370	266	1	a	a	DET
ejpam-4370	266	2	function	function	NOUN
ejpam-4370	266	3	f	f	NOUN
ejpam-4370	266	4	:	:	PUNCT
ejpam-4370	266	5	(	(	PUNCT
ejpam-4370	266	6	x	x	X
ejpam-4370	266	7	,	,	PUNCT
ejpam-4370	266	8	τ	τ	X
ejpam-4370	266	9	)	)	PUNCT
ejpam-4370	266	10	→	→	SYM
ejpam-4370	266	11	(	(	PUNCT
ejpam-4370	266	12	y	y	PROPN
ejpam-4370	266	13	,	,	PUNCT
ejpam-4370	266	14	σ	σ	PROPN
ejpam-4370	266	15	)	)	PUNCT
ejpam-4370	266	16	is	be	AUX
ejpam-4370	266	17	said	say	VERB
ejpam-4370	266	18	to	to	PART
ejpam-4370	266	19	be	be	AUX
ejpam-4370	266	20	(	(	PUNCT
ejpam-4370	266	21	λ	λ	X
ejpam-4370	266	22	,	,	PUNCT
ejpam-4370	266	23	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	266	24	if	if	SCONJ
ejpam-4370	266	25	f	f	PROPN
ejpam-4370	266	26	has	have	VERB
ejpam-4370	266	27	this	this	DET
ejpam-4370	266	28	property	property	NOUN
ejpam-4370	266	29	at	at	ADP
ejpam-4370	266	30	each	each	DET
ejpam-4370	266	31	point	point	NOUN
ejpam-4370	266	32	of	of	ADP
ejpam-4370	266	33	x.	x.	NOUN
ejpam-4370	266	34	lemma	lemma	PROPN
ejpam-4370	266	35	7	7	NUM
ejpam-4370	266	36	.	.	PUNCT
ejpam-4370	267	1	[	[	X
ejpam-4370	267	2	22	22	NUM
ejpam-4370	267	3	]	]	PUNCT
ejpam-4370	267	4	for	for	ADP
ejpam-4370	267	5	a	a	DET
ejpam-4370	267	6	function	function	NOUN
ejpam-4370	267	7	f	f	NOUN
ejpam-4370	267	8	:	:	PUNCT
ejpam-4370	267	9	(	(	PUNCT
ejpam-4370	267	10	x	x	X
ejpam-4370	267	11	,	,	PUNCT
ejpam-4370	267	12	τ	τ	X
ejpam-4370	267	13	)	)	PUNCT
ejpam-4370	267	14	→	→	SYM
ejpam-4370	267	15	(	(	PUNCT
ejpam-4370	267	16	y	y	PROPN
ejpam-4370	267	17	,	,	PUNCT
ejpam-4370	267	18	σ	σ	PROPN
ejpam-4370	267	19	)	)	PUNCT
ejpam-4370	267	20	,	,	PUNCT
ejpam-4370	267	21	the	the	DET
ejpam-4370	267	22	following	follow	VERB
ejpam-4370	267	23	properties	property	NOUN
ejpam-4370	267	24	are	be	AUX
ejpam-4370	267	25	equivalent	equivalent	ADJ
ejpam-4370	267	26	:	:	PUNCT
ejpam-4370	267	27	(	(	PUNCT
ejpam-4370	267	28	1	1	X
ejpam-4370	267	29	)	)	PUNCT
ejpam-4370	267	30	f	f	PROPN
ejpam-4370	267	31	is	be	AUX
ejpam-4370	267	32	(	(	PUNCT
ejpam-4370	267	33	λ	λ	INTJ
ejpam-4370	267	34	,	,	PUNCT
ejpam-4370	267	35	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	267	36	;	;	PUNCT
ejpam-4370	267	37	(	(	PUNCT
ejpam-4370	267	38	2	2	X
ejpam-4370	267	39	)	)	PUNCT
ejpam-4370	267	40	f−1(v	f−1(v	NOUN
ejpam-4370	267	41	)	)	PUNCT
ejpam-4370	267	42	is	be	AUX
ejpam-4370	267	43	(	(	PUNCT
ejpam-4370	267	44	λ	λ	INTJ
ejpam-4370	267	45	,	,	PUNCT
ejpam-4370	267	46	sp)-open	sp)-open	ADJ
ejpam-4370	267	47	in	in	ADP
ejpam-4370	267	48	x	x	PUNCT
ejpam-4370	267	49	for	for	SCONJ
ejpam-4370	267	50	every	every	DET
ejpam-4370	267	51	(	(	PUNCT
ejpam-4370	267	52	λ	λ	NOUN
ejpam-4370	267	53	,	,	PUNCT
ejpam-4370	267	54	sp)-open	sp)-open	NOUN
ejpam-4370	267	55	set	set	VERB
ejpam-4370	267	56	v	v	NOUN
ejpam-4370	267	57	of	of	ADP
ejpam-4370	267	58	y	y	PROPN
ejpam-4370	267	59	;	;	PUNCT
ejpam-4370	267	60	(	(	PUNCT
ejpam-4370	267	61	3	3	X
ejpam-4370	267	62	)	)	PUNCT
ejpam-4370	267	63	f(a(λ	f(a(λ	NOUN
ejpam-4370	267	64	,	,	PUNCT
ejpam-4370	267	65	sp	sp	NOUN
ejpam-4370	267	66	)	)	PUNCT
ejpam-4370	267	67	)	)	PUNCT
ejpam-4370	267	68	⊆	⊆	NUM
ejpam-4370	268	1	[	[	X
ejpam-4370	268	2	f(a)](λ	f(a)](λ	NUM
ejpam-4370	268	3	,	,	PUNCT
ejpam-4370	268	4	sp	sp	NOUN
ejpam-4370	268	5	)	)	PUNCT
ejpam-4370	268	6	for	for	ADP
ejpam-4370	268	7	every	every	DET
ejpam-4370	268	8	subset	subset	NOUN
ejpam-4370	268	9	a	a	PRON
ejpam-4370	268	10	of	of	ADP
ejpam-4370	268	11	x	x	PRON
ejpam-4370	268	12	;	;	PUNCT
ejpam-4370	268	13	(	(	PUNCT
ejpam-4370	268	14	4	4	X
ejpam-4370	268	15	)	)	PUNCT
ejpam-4370	269	1	[	[	X
ejpam-4370	269	2	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4370	269	3	,	,	PUNCT
ejpam-4370	269	4	sp	sp	NOUN
ejpam-4370	269	5	)	)	PUNCT
ejpam-4370	269	6	⊆	⊆	NUM
ejpam-4370	269	7	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4370	269	8	,	,	PUNCT
ejpam-4370	269	9	sp	sp	NOUN
ejpam-4370	269	10	)	)	PUNCT
ejpam-4370	269	11	)	)	PUNCT
ejpam-4370	269	12	for	for	ADP
ejpam-4370	269	13	every	every	DET
ejpam-4370	269	14	subset	subset	NOUN
ejpam-4370	269	15	b	b	PROPN
ejpam-4370	269	16	of	of	ADP
ejpam-4370	269	17	y	y	PROPN
ejpam-4370	269	18	;	;	PUNCT
ejpam-4370	269	19	(	(	PUNCT
ejpam-4370	269	20	5	5	X
ejpam-4370	269	21	)	)	PUNCT
ejpam-4370	269	22	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4370	269	23	,	,	PUNCT
ejpam-4370	269	24	sp	sp	NOUN
ejpam-4370	269	25	)	)	PUNCT
ejpam-4370	269	26	)	)	PUNCT
ejpam-4370	270	1	⊆	⊆	NUM
ejpam-4370	270	2	[	[	X
ejpam-4370	270	3	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4370	270	4	,	,	PUNCT
ejpam-4370	270	5	sp	sp	NOUN
ejpam-4370	270	6	)	)	PUNCT
ejpam-4370	270	7	for	for	ADP
ejpam-4370	270	8	every	every	DET
ejpam-4370	270	9	subset	subset	NOUN
ejpam-4370	270	10	b	b	PROPN
ejpam-4370	270	11	of	of	ADP
ejpam-4370	270	12	y	y	PROPN
ejpam-4370	270	13	;	;	PUNCT
ejpam-4370	270	14	(	(	PUNCT
ejpam-4370	270	15	6	6	X
ejpam-4370	270	16	)	)	PUNCT
ejpam-4370	270	17	f−1(f	f−1(f	NOUN
ejpam-4370	270	18	)	)	PUNCT
ejpam-4370	270	19	is	be	AUX
ejpam-4370	270	20	(	(	PUNCT
ejpam-4370	270	21	λ	λ	X
ejpam-4370	270	22	,	,	PUNCT
ejpam-4370	270	23	sp)-closed	sp)-close	VERB
ejpam-4370	270	24	in	in	ADP
ejpam-4370	270	25	x	x	PUNCT
ejpam-4370	270	26	for	for	SCONJ
ejpam-4370	270	27	every	every	DET
ejpam-4370	270	28	(	(	PUNCT
ejpam-4370	270	29	λ	λ	PROPN
ejpam-4370	270	30	,	,	PUNCT
ejpam-4370	270	31	sp)-closed	sp)-close	VERB
ejpam-4370	270	32	set	set	VERB
ejpam-4370	270	33	f	f	PROPN
ejpam-4370	270	34	of	of	ADP
ejpam-4370	270	35	y	y	PROPN
ejpam-4370	270	36	.	.	PUNCT
ejpam-4370	271	1	references	reference	NOUN
ejpam-4370	271	2	1703	1703	NUM
ejpam-4370	271	3	corollary	corollary	ADJ
ejpam-4370	271	4	8	8	NUM
ejpam-4370	271	5	.	.	PUNCT
ejpam-4370	272	1	if	if	SCONJ
ejpam-4370	272	2	f	f	PROPN
ejpam-4370	272	3	:	:	PUNCT
ejpam-4370	272	4	(	(	PUNCT
ejpam-4370	272	5	x	x	X
ejpam-4370	272	6	,	,	PUNCT
ejpam-4370	272	7	τ	τ	X
ejpam-4370	272	8	)	)	PUNCT
ejpam-4370	272	9	→	→	SYM
ejpam-4370	272	10	(	(	PUNCT
ejpam-4370	272	11	y	y	PROPN
ejpam-4370	272	12	,	,	PUNCT
ejpam-4370	272	13	σ	σ	PROPN
ejpam-4370	272	14	)	)	PUNCT
ejpam-4370	272	15	is	be	AUX
ejpam-4370	272	16	a	a	DET
ejpam-4370	272	17	(	(	PUNCT
ejpam-4370	272	18	λ	λ	NOUN
ejpam-4370	272	19	,	,	PUNCT
ejpam-4370	272	20	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	272	21	function	function	NOUN
ejpam-4370	272	22	and	and	CCONJ
ejpam-4370	272	23	h	h	NOUN
ejpam-4370	272	24	:	:	PUNCT
ejpam-4370	272	25	(	(	PUNCT
ejpam-4370	272	26	y	y	PROPN
ejpam-4370	272	27	,	,	PUNCT
ejpam-4370	272	28	σ	σ	PROPN
ejpam-4370	272	29	)	)	PUNCT
ejpam-4370	272	30	→	→	SYM
ejpam-4370	272	31	(	(	PUNCT
ejpam-4370	272	32	z	z	PROPN
ejpam-4370	272	33	,	,	PUNCT
ejpam-4370	272	34	ρ	ρ	PROPN
ejpam-4370	272	35	)	)	PUNCT
ejpam-4370	272	36	is	be	AUX
ejpam-4370	272	37	a	a	DET
ejpam-4370	272	38	contra-(λ	contra-(λ	PROPN
ejpam-4370	272	39	,	,	PUNCT
ejpam-4370	272	40	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	272	41	function	function	NOUN
ejpam-4370	272	42	,	,	PUNCT
ejpam-4370	272	43	then	then	ADV
ejpam-4370	273	1	h	h	PROPN
ejpam-4370	273	2	◦	◦	NOUN
ejpam-4370	273	3	f	f	X
ejpam-4370	273	4	:	:	PUNCT
ejpam-4370	273	5	(	(	PUNCT
ejpam-4370	273	6	x	x	X
ejpam-4370	273	7	,	,	PUNCT
ejpam-4370	273	8	τ	τ	X
ejpam-4370	273	9	)	)	PUNCT
ejpam-4370	273	10	→	→	SYM
ejpam-4370	273	11	(	(	PUNCT
ejpam-4370	273	12	z	z	PROPN
ejpam-4370	273	13	,	,	PUNCT
ejpam-4370	273	14	ρ	ρ	PROPN
ejpam-4370	273	15	)	)	PUNCT
ejpam-4370	273	16	is	be	AUX
ejpam-4370	273	17	contra-(λ	contra-(λ	PROPN
ejpam-4370	273	18	,	,	PUNCT
ejpam-4370	273	19	sp)continuous	sp)continuous	ADJ
ejpam-4370	273	20	.	.	PUNCT
ejpam-4370	274	1	proof	proof	NOUN
ejpam-4370	274	2	.	.	PUNCT
ejpam-4370	275	1	let	let	VERB
ejpam-4370	275	2	v	v	PART
ejpam-4370	275	3	be	be	AUX
ejpam-4370	275	4	any	any	DET
ejpam-4370	275	5	(	(	PUNCT
ejpam-4370	275	6	λ	λ	PROPN
ejpam-4370	275	7	,	,	PUNCT
ejpam-4370	275	8	sp)-closed	sp)-close	VERB
ejpam-4370	275	9	set	set	NOUN
ejpam-4370	275	10	of	of	ADP
ejpam-4370	275	11	z.	z.	PROPN
ejpam-4370	275	12	we	we	PRON
ejpam-4370	275	13	have	have	VERB
ejpam-4370	275	14	(	(	PUNCT
ejpam-4370	275	15	h	h	NOUN
ejpam-4370	275	16	◦	◦	NOUN
ejpam-4370	275	17	f)−1(v	f)−1(v	NOUN
ejpam-4370	275	18	)	)	PUNCT
ejpam-4370	276	1	=	=	SYM
ejpam-4370	276	2	f−1(h−1(v	f−1(h−1(v	NOUN
ejpam-4370	276	3	)	)	PUNCT
ejpam-4370	276	4	)	)	PUNCT
ejpam-4370	276	5	.	.	PUNCT
ejpam-4370	277	1	since	since	SCONJ
ejpam-4370	277	2	h	h	PROPN
ejpam-4370	277	3	is	be	AUX
ejpam-4370	277	4	contra-(λ	contra-(λ	PROPN
ejpam-4370	277	5	,	,	PUNCT
ejpam-4370	277	6	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	277	7	,	,	PUNCT
ejpam-4370	277	8	by	by	ADP
ejpam-4370	277	9	corollary	corollary	ADJ
ejpam-4370	277	10	1	1	NUM
ejpam-4370	277	11	,	,	PUNCT
ejpam-4370	277	12	h−1(v	h−1(v	PROPN
ejpam-4370	277	13	)	)	PUNCT
ejpam-4370	277	14	is	be	AUX
ejpam-4370	277	15	(	(	PUNCT
ejpam-4370	277	16	λ	λ	INTJ
ejpam-4370	277	17	,	,	PUNCT
ejpam-4370	277	18	sp)-open	sp)-open	ADJ
ejpam-4370	277	19	in	in	ADP
ejpam-4370	277	20	y	y	PROPN
ejpam-4370	277	21	.	.	PUNCT
ejpam-4370	278	1	since	since	SCONJ
ejpam-4370	278	2	f	f	PROPN
ejpam-4370	278	3	is	be	AUX
ejpam-4370	278	4	(	(	PUNCT
ejpam-4370	278	5	λ	λ	PROPN
ejpam-4370	278	6	,	,	PUNCT
ejpam-4370	278	7	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	278	8	,	,	PUNCT
ejpam-4370	278	9	by	by	ADP
ejpam-4370	278	10	lemma	lemma	PROPN
ejpam-4370	278	11	7	7	NUM
ejpam-4370	278	12	,	,	PUNCT
ejpam-4370	278	13	f−1(h−1(v	f−1(h−1(v	NOUN
ejpam-4370	278	14	)	)	PUNCT
ejpam-4370	278	15	)	)	PUNCT
ejpam-4370	278	16	is	be	AUX
ejpam-4370	278	17	(	(	PUNCT
ejpam-4370	278	18	λ	λ	INTJ
ejpam-4370	278	19	,	,	PUNCT
ejpam-4370	278	20	sp)-open	sp)-open	ADJ
ejpam-4370	278	21	in	in	ADP
ejpam-4370	278	22	x.	x.	NOUN
ejpam-4370	278	23	thus	thus	ADV
ejpam-4370	278	24	,	,	PUNCT
ejpam-4370	278	25	(	(	PUNCT
ejpam-4370	278	26	h	h	NOUN
ejpam-4370	278	27	◦	◦	NOUN
ejpam-4370	278	28	f)−1(v	f)−1(v	NOUN
ejpam-4370	278	29	)	)	PUNCT
ejpam-4370	278	30	is	be	AUX
ejpam-4370	278	31	(	(	PUNCT
ejpam-4370	278	32	λ	λ	INTJ
ejpam-4370	278	33	,	,	PUNCT
ejpam-4370	278	34	sp)-open	sp)-open	ADJ
ejpam-4370	278	35	in	in	ADP
ejpam-4370	278	36	x	x	PUNCT
ejpam-4370	278	37	and	and	CCONJ
ejpam-4370	278	38	hence	hence	ADV
ejpam-4370	278	39	h	h	NOUN
ejpam-4370	278	40	◦	◦	NOUN
ejpam-4370	278	41	f	f	PROPN
ejpam-4370	278	42	is	be	AUX
ejpam-4370	278	43	contra-(λ	contra-(λ	PROPN
ejpam-4370	278	44	,	,	PUNCT
ejpam-4370	278	45	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	278	46	.	.	PUNCT
ejpam-4370	279	1	acknowledgements	acknowledgement	NOUN
ejpam-4370	279	2	this	this	DET
ejpam-4370	279	3	research	research	NOUN
ejpam-4370	279	4	project	project	NOUN
ejpam-4370	279	5	was	be	AUX
ejpam-4370	279	6	financially	financially	ADV
ejpam-4370	279	7	supported	support	VERB
ejpam-4370	279	8	by	by	ADP
ejpam-4370	279	9	mahasarakham	mahasarakham	PROPN
ejpam-4370	279	10	university	university	PROPN
ejpam-4370	279	11	.	.	PUNCT
ejpam-4370	280	1	references	reference	NOUN
ejpam-4370	280	2	[	[	X
ejpam-4370	280	3	1	1	NUM
ejpam-4370	280	4	]	]	PUNCT
ejpam-4370	280	5	d.	d.	PROPN
ejpam-4370	280	6	andrijević.	andrijević.	PROPN
ejpam-4370	280	7	on	on	ADP
ejpam-4370	280	8	b	b	X
ejpam-4370	280	9	-	-	PUNCT
ejpam-4370	280	10	open	open	ADJ
ejpam-4370	280	11	sets	set	NOUN
ejpam-4370	280	12	.	.	PUNCT
ejpam-4370	281	1	matematički	matematički	PROPN
ejpam-4370	281	2	vesnik	vesnik	PROPN
ejpam-4370	281	3	,	,	PUNCT
ejpam-4370	281	4	48:59–64	48:59–64	PROPN
ejpam-4370	281	5	,	,	PUNCT
ejpam-4370	281	6	1996	1996	NUM
ejpam-4370	281	7	.	.	PUNCT
ejpam-4370	282	1	[	[	X
ejpam-4370	282	2	2	2	NUM
ejpam-4370	282	3	]	]	PUNCT
ejpam-4370	282	4	c.	c.	PROPN
ejpam-4370	282	5	berge	berge	PROPN
ejpam-4370	282	6	.	.	PUNCT
ejpam-4370	282	7	espaces	espace	VERB
ejpam-4370	282	8	topologiques	topologique	NOUN
ejpam-4370	282	9	fonctions	fonction	NOUN
ejpam-4370	282	10	multivoques	multivoque	NOUN
ejpam-4370	282	11	.	.	PUNCT
ejpam-4370	283	1	dunod	dunod	PROPN
ejpam-4370	283	2	,	,	PUNCT
ejpam-4370	283	3	paris	paris	PROPN
ejpam-4370	283	4	,	,	PUNCT
ejpam-4370	283	5	1959	1959	NUM
ejpam-4370	283	6	.	.	PUNCT
ejpam-4370	284	1	[	[	X
ejpam-4370	284	2	3	3	X
ejpam-4370	284	3	]	]	PUNCT
ejpam-4370	284	4	c.	c.	PROPN
ejpam-4370	284	5	boonpok	boonpok	PROPN
ejpam-4370	284	6	.	.	PUNCT
ejpam-4370	285	1	(	(	PUNCT
ejpam-4370	285	2	λ	λ	NOUN
ejpam-4370	285	3	,	,	PUNCT
ejpam-4370	285	4	sp)-closed	sp)-close	VERB
ejpam-4370	285	5	sets	set	NOUN
ejpam-4370	285	6	and	and	CCONJ
ejpam-4370	285	7	related	related	ADJ
ejpam-4370	285	8	topics	topic	NOUN
ejpam-4370	285	9	in	in	ADP
ejpam-4370	285	10	topological	topological	ADJ
ejpam-4370	285	11	spaces	space	NOUN
ejpam-4370	285	12	.	.	PUNCT
ejpam-4370	286	1	wseas	wseas	VERB
ejpam-4370	286	2	transactions	transaction	NOUN
ejpam-4370	286	3	on	on	ADP
ejpam-4370	286	4	mathematics	mathematic	NOUN
ejpam-4370	286	5	,	,	PUNCT
ejpam-4370	286	6	19:321–322	19:321–322	PROPN
ejpam-4370	286	7	,	,	PUNCT
ejpam-4370	286	8	2020	2020	NUM
ejpam-4370	286	9	.	.	PUNCT
ejpam-4370	287	1	[	[	X
ejpam-4370	287	2	4	4	NUM
ejpam-4370	287	3	]	]	PUNCT
ejpam-4370	287	4	c.	c.	PROPN
ejpam-4370	287	5	boonpok	boonpok	PROPN
ejpam-4370	287	6	and	and	CCONJ
ejpam-4370	287	7	j.	j.	PROPN
ejpam-4370	287	8	khampakdee	khampakdee	PROPN
ejpam-4370	287	9	.	.	PUNCT
ejpam-4370	288	1	slight	slight	PROPN
ejpam-4370	288	2	(	(	PUNCT
ejpam-4370	288	3	λ	λ	NOUN
ejpam-4370	288	4	,	,	PUNCT
ejpam-4370	288	5	sp)-continuity	sp)-continuity	NOUN
ejpam-4370	288	6	and	and	CCONJ
ejpam-4370	288	7	λsp	λsp	NOUN
ejpam-4370	288	8	-	-	PUNCT
ejpam-4370	288	9	extremally	extremally	ADV
ejpam-4370	288	10	disconnectedness	disconnectedness	NOUN
ejpam-4370	288	11	.	.	PUNCT
ejpam-4370	289	1	european	european	ADJ
ejpam-4370	289	2	journal	journal	PROPN
ejpam-4370	289	3	of	of	ADP
ejpam-4370	289	4	pure	pure	ADJ
ejpam-4370	289	5	and	and	CCONJ
ejpam-4370	289	6	applied	applied	ADJ
ejpam-4370	289	7	mathematics	mathematic	NOUN
ejpam-4370	289	8	,	,	PUNCT
ejpam-4370	289	9	15(3):1180–1188	15(3):1180–1188	NUM
ejpam-4370	289	10	,	,	PUNCT
ejpam-4370	289	11	2022	2022	NUM
ejpam-4370	289	12	.	.	PUNCT
ejpam-4370	290	1	[	[	X
ejpam-4370	290	2	5	5	X
ejpam-4370	290	3	]	]	PUNCT
ejpam-4370	290	4	c.	c.	PROPN
ejpam-4370	290	5	boonpok	boonpok	PROPN
ejpam-4370	290	6	and	and	CCONJ
ejpam-4370	290	7	c.	c.	PROPN
ejpam-4370	290	8	viriyapong	viriyapong	PROPN
ejpam-4370	290	9	.	.	PUNCT
ejpam-4370	291	1	on	on	ADP
ejpam-4370	291	2	generalized	generalized	ADJ
ejpam-4370	291	3	(	(	PUNCT
ejpam-4370	291	4	λ	λ	PROPN
ejpam-4370	291	5	,	,	PUNCT
ejpam-4370	291	6	sp)-closed	sp)-close	VERB
ejpam-4370	291	7	sets	set	NOUN
ejpam-4370	291	8	.	.	PUNCT
ejpam-4370	292	1	(	(	PUNCT
ejpam-4370	292	2	to	to	PART
ejpam-4370	292	3	appear	appear	VERB
ejpam-4370	292	4	)	)	PUNCT
ejpam-4370	292	5	.	.	PUNCT
ejpam-4370	293	1	[	[	X
ejpam-4370	293	2	6	6	NUM
ejpam-4370	293	3	]	]	PUNCT
ejpam-4370	293	4	m.	m.	NOUN
ejpam-4370	293	5	caldas	caldas	PROPN
ejpam-4370	293	6	and	and	CCONJ
ejpam-4370	293	7	j.	j.	PROPN
ejpam-4370	293	8	jafari	jafari	PROPN
ejpam-4370	293	9	.	.	PUNCT
ejpam-4370	294	1	some	some	DET
ejpam-4370	294	2	properties	property	NOUN
ejpam-4370	294	3	of	of	ADP
ejpam-4370	294	4	contra	contra	PROPN
ejpam-4370	294	5	β	β	ADJ
ejpam-4370	294	6	-	-	ADJ
ejpam-4370	294	7	continuous	continuous	ADJ
ejpam-4370	294	8	functions	function	NOUN
ejpam-4370	294	9	.	.	PUNCT
ejpam-4370	295	1	memoirs	memoir	NOUN
ejpam-4370	295	2	of	of	ADP
ejpam-4370	295	3	the	the	DET
ejpam-4370	295	4	faculty	faculty	NOUN
ejpam-4370	295	5	of	of	ADP
ejpam-4370	295	6	science	science	NOUN
ejpam-4370	295	7	,	,	PUNCT
ejpam-4370	295	8	kochi	kochi	PROPN
ejpam-4370	295	9	university	university	PROPN
ejpam-4370	295	10	.	.	PUNCT
ejpam-4370	296	1	series	series	PROPN
ejpam-4370	296	2	a	a	DET
ejpam-4370	296	3	mathematics	mathematics	PROPN
ejpam-4370	296	4	,	,	PUNCT
ejpam-4370	296	5	22:19–28	22:19–28	NUM
ejpam-4370	296	6	,	,	PUNCT
ejpam-4370	296	7	2001	2001	NUM
ejpam-4370	296	8	.	.	PUNCT
ejpam-4370	297	1	[	[	X
ejpam-4370	297	2	7	7	X
ejpam-4370	297	3	]	]	X
ejpam-4370	297	4	j.	j.	PROPN
ejpam-4370	297	5	dontchev	dontchev	PROPN
ejpam-4370	297	6	.	.	PUNCT
ejpam-4370	298	1	contra	contra	ADJ
ejpam-4370	298	2	-	-	ADJ
ejpam-4370	298	3	continuous	continuous	ADJ
ejpam-4370	298	4	functions	function	NOUN
ejpam-4370	298	5	and	and	CCONJ
ejpam-4370	298	6	strongly	strongly	ADV
ejpam-4370	298	7	s	s	NOUN
ejpam-4370	298	8	-	-	PUNCT
ejpam-4370	298	9	closed	closed	ADJ
ejpam-4370	298	10	spaces	space	NOUN
ejpam-4370	298	11	.	.	PUNCT
ejpam-4370	299	1	international	international	ADJ
ejpam-4370	299	2	journal	journal	PROPN
ejpam-4370	299	3	of	of	ADP
ejpam-4370	299	4	mathematics	mathematics	PROPN
ejpam-4370	299	5	and	and	CCONJ
ejpam-4370	299	6	mathematical	mathematical	ADJ
ejpam-4370	299	7	sciences	science	NOUN
ejpam-4370	299	8	,	,	PUNCT
ejpam-4370	299	9	19(2):303–310	19(2):303–310	NUM
ejpam-4370	299	10	,	,	PUNCT
ejpam-4370	299	11	1996	1996	NUM
ejpam-4370	299	12	.	.	PUNCT
ejpam-4370	300	1	[	[	X
ejpam-4370	300	2	8	8	X
ejpam-4370	300	3	]	]	PUNCT
ejpam-4370	300	4	j.	j.	PROPN
ejpam-4370	300	5	dontchev	dontchev	PROPN
ejpam-4370	300	6	,	,	PUNCT
ejpam-4370	300	7	m.	m.	NOUN
ejpam-4370	300	8	ganster	ganster	NOUN
ejpam-4370	300	9	,	,	PUNCT
ejpam-4370	300	10	and	and	CCONJ
ejpam-4370	300	11	i.	i.	PROPN
ejpam-4370	300	12	reilly	reilly	PROPN
ejpam-4370	300	13	.	.	PUNCT
ejpam-4370	301	1	more	more	ADV
ejpam-4370	301	2	on	on	ADP
ejpam-4370	301	3	almost	almost	ADV
ejpam-4370	301	4	s	s	NOUN
ejpam-4370	301	5	-	-	NOUN
ejpam-4370	301	6	continuity	continuity	NOUN
ejpam-4370	301	7	.	.	PUNCT
ejpam-4370	302	1	indian	indian	ADJ
ejpam-4370	302	2	journal	journal	PROPN
ejpam-4370	302	3	of	of	ADP
ejpam-4370	302	4	mathematics	mathematics	PROPN
ejpam-4370	302	5	,	,	PUNCT
ejpam-4370	302	6	41:139–146	41:139–146	PROPN
ejpam-4370	302	7	,	,	PUNCT
ejpam-4370	302	8	1999	1999	NUM
ejpam-4370	302	9	.	.	PUNCT
ejpam-4370	303	1	[	[	X
ejpam-4370	303	2	9	9	NUM
ejpam-4370	303	3	]	]	X
ejpam-4370	303	4	j.	j.	PROPN
ejpam-4370	303	5	dontchev	dontchev	PROPN
ejpam-4370	303	6	and	and	CCONJ
ejpam-4370	303	7	t.	t.	PROPN
ejpam-4370	303	8	noiri	noiri	PROPN
ejpam-4370	303	9	.	.	PUNCT
ejpam-4370	304	1	contra	contra	ADJ
ejpam-4370	304	2	-	-	ADJ
ejpam-4370	304	3	semicontinuous	semicontinuous	ADJ
ejpam-4370	304	4	functions	function	NOUN
ejpam-4370	304	5	.	.	PUNCT
ejpam-4370	305	1	mathematica	mathematica	PROPN
ejpam-4370	305	2	pannonica	pannonica	PROPN
ejpam-4370	305	3	,	,	PUNCT
ejpam-4370	305	4	10(2):159–168	10(2):159–168	PROPN
ejpam-4370	305	5	,	,	PUNCT
ejpam-4370	305	6	1999	1999	NUM
ejpam-4370	305	7	.	.	PUNCT
ejpam-4370	306	1	[	[	X
ejpam-4370	306	2	10	10	NUM
ejpam-4370	306	3	]	]	X
ejpam-4370	306	4	e.	e.	PROPN
ejpam-4370	306	5	ekici	ekici	PROPN
ejpam-4370	306	6	,	,	PUNCT
ejpam-4370	306	7	s.	s.	PROPN
ejpam-4370	306	8	jafari	jafari	PROPN
ejpam-4370	306	9	,	,	PUNCT
ejpam-4370	306	10	and	and	CCONJ
ejpam-4370	306	11	t.	t.	PROPN
ejpam-4370	306	12	noiri	noiri	PROPN
ejpam-4370	306	13	.	.	PUNCT
ejpam-4370	307	1	on	on	ADP
ejpam-4370	307	2	upper	upper	ADJ
ejpam-4370	307	3	and	and	CCONJ
ejpam-4370	307	4	lower	low	ADJ
ejpam-4370	307	5	contra	contra	ADJ
ejpam-4370	307	6	-	-	ADJ
ejpam-4370	307	7	continuous	continuous	ADJ
ejpam-4370	307	8	multifunctions	multifunction	NOUN
ejpam-4370	307	9	.	.	PUNCT
ejpam-4370	308	1	annals	annal	NOUN
ejpam-4370	308	2	of	of	ADP
ejpam-4370	308	3	the	the	DET
ejpam-4370	308	4	”	"	PUNCT
ejpam-4370	308	5	alexandru	alexandru	PROPN
ejpam-4370	308	6	loan	loan	PROPN
ejpam-4370	308	7	cuza	cuza	PROPN
ejpam-4370	308	8	”	"	PUNCT
ejpam-4370	308	9	university	university	NOUN
ejpam-4370	308	10	of	of	ADP
ejpam-4370	308	11	laşi	laşi	NOUN
ejpam-4370	308	12	(	(	PUNCT
ejpam-4370	308	13	new	new	ADJ
ejpam-4370	308	14	series	series	NOUN
ejpam-4370	308	15	)	)	PUNCT
ejpam-4370	308	16	.	.	PUNCT
ejpam-4370	309	1	mathematics	mathematic	NOUN
ejpam-4370	309	2	,	,	PUNCT
ejpam-4370	309	3	54(1):75–85	54(1):75–85	NUM
ejpam-4370	309	4	,	,	PUNCT
ejpam-4370	309	5	2008	2008	NUM
ejpam-4370	309	6	.	.	PUNCT
ejpam-4370	310	1	references	reference	NOUN
ejpam-4370	310	2	1704	1704	NUM
ejpam-4370	310	3	[	[	X
ejpam-4370	310	4	11	11	NUM
ejpam-4370	310	5	]	]	X
ejpam-4370	310	6	e.	e.	PROPN
ejpam-4370	310	7	ekici	ekici	PROPN
ejpam-4370	310	8	,	,	PUNCT
ejpam-4370	310	9	s.	s.	PROPN
ejpam-4370	310	10	jafari	jafari	PROPN
ejpam-4370	310	11	,	,	PUNCT
ejpam-4370	310	12	and	and	CCONJ
ejpam-4370	310	13	v.	v.	ADP
ejpam-4370	310	14	popa	popa	NOUN
ejpam-4370	310	15	.	.	PUNCT
ejpam-4370	311	1	on	on	ADP
ejpam-4370	311	2	almost	almost	ADV
ejpam-4370	311	3	contra	contra	ADJ
ejpam-4370	311	4	-	-	ADJ
ejpam-4370	311	5	continuous	continuous	ADJ
ejpam-4370	311	6	multifunctions	multifunction	NOUN
ejpam-4370	311	7	.	.	PUNCT
ejpam-4370	312	1	lobachevskii	lobachevskii	PROPN
ejpam-4370	312	2	journal	journal	PROPN
ejpam-4370	312	3	of	of	ADP
ejpam-4370	312	4	mathematics	mathematic	NOUN
ejpam-4370	312	5	,	,	PUNCT
ejpam-4370	312	6	30(2):124–131	30(2):124–131	PROPN
ejpam-4370	312	7	,	,	PUNCT
ejpam-4370	312	8	2009	2009	NUM
ejpam-4370	312	9	.	.	PUNCT
ejpam-4370	313	1	[	[	X
ejpam-4370	313	2	12	12	NUM
ejpam-4370	313	3	]	]	X
ejpam-4370	313	4	e.	e.	PROPN
ejpam-4370	313	5	ekici	ekici	PROPN
ejpam-4370	313	6	,	,	PUNCT
ejpam-4370	313	7	s.	s.	PROPN
ejpam-4370	313	8	jafari	jafari	PROPN
ejpam-4370	313	9	,	,	PUNCT
ejpam-4370	313	10	and	and	CCONJ
ejpam-4370	313	11	v.	v.	ADP
ejpam-4370	313	12	popa	popa	NOUN
ejpam-4370	313	13	.	.	PUNCT
ejpam-4370	314	1	on	on	ADP
ejpam-4370	314	2	contra	contra	PROPN
ejpam-4370	314	3	-	-	ADJ
ejpam-4370	314	4	precontinuous	precontinuous	ADJ
ejpam-4370	314	5	and	and	CCONJ
ejpam-4370	314	6	almost	almost	ADV
ejpam-4370	314	7	contraprecontinuous	contraprecontinuous	ADJ
ejpam-4370	314	8	.	.	PUNCT
ejpam-4370	315	1	journal	journal	PROPN
ejpam-4370	315	2	of	of	ADP
ejpam-4370	315	3	advanced	advanced	ADJ
ejpam-4370	315	4	research	research	NOUN
ejpam-4370	315	5	in	in	ADP
ejpam-4370	315	6	pure	pure	ADJ
ejpam-4370	315	7	mathematics	mathematic	NOUN
ejpam-4370	315	8	,	,	PUNCT
ejpam-4370	315	9	2(1):11–25	2(1):11–25	NUM
ejpam-4370	315	10	,	,	PUNCT
ejpam-4370	315	11	2010	2010	NUM
ejpam-4370	315	12	.	.	PUNCT
ejpam-4370	316	1	[	[	X
ejpam-4370	316	2	13	13	NUM
ejpam-4370	316	3	]	]	PUNCT
ejpam-4370	316	4	m.	m.	NOUN
ejpam-4370	316	5	e.	e.	PROPN
ejpam-4370	316	6	abd	abd	PROPN
ejpam-4370	317	1	el	el	PROPN
ejpam-4370	317	2	-	-	PROPN
ejpam-4370	317	3	monsef	monsef	PROPN
ejpam-4370	317	4	,	,	PUNCT
ejpam-4370	317	5	s.	s.	PROPN
ejpam-4370	317	6	n.	n.	PROPN
ejpam-4370	317	7	el	el	PROPN
ejpam-4370	317	8	-	-	PROPN
ejpam-4370	317	9	deeb	deeb	PROPN
ejpam-4370	317	10	,	,	PUNCT
ejpam-4370	317	11	and	and	CCONJ
ejpam-4370	317	12	r.	r.	PROPN
ejpam-4370	317	13	a.	a.	PROPN
ejpam-4370	317	14	mahmoud	mahmoud	PROPN
ejpam-4370	317	15	.	.	PUNCT
ejpam-4370	318	1	β	β	X
ejpam-4370	318	2	-	-	ADJ
ejpam-4370	318	3	open	open	ADJ
ejpam-4370	318	4	sets	set	NOUN
ejpam-4370	318	5	and	and	CCONJ
ejpam-4370	318	6	βcontinuous	βcontinuous	ADJ
ejpam-4370	318	7	mappings	mapping	NOUN
ejpam-4370	318	8	.	.	PUNCT
ejpam-4370	319	1	bulletin	bulletin	NOUN
ejpam-4370	319	2	of	of	ADP
ejpam-4370	319	3	the	the	DET
ejpam-4370	319	4	faculty	faculty	NOUN
ejpam-4370	319	5	of	of	ADP
ejpam-4370	319	6	science	science	NOUN
ejpam-4370	319	7	.	.	PUNCT
ejpam-4370	320	1	assiut	assiut	PROPN
ejpam-4370	320	2	university	university	PROPN
ejpam-4370	320	3	.	.	PUNCT
ejpam-4370	320	4	,	,	PUNCT
ejpam-4370	320	5	12:77–90	12:77–90	NUM
ejpam-4370	320	6	,	,	PUNCT
ejpam-4370	320	7	1983	1983	NUM
ejpam-4370	320	8	.	.	PUNCT
ejpam-4370	321	1	[	[	X
ejpam-4370	321	2	14	14	NUM
ejpam-4370	321	3	]	]	X
ejpam-4370	321	4	s.	s.	PROPN
ejpam-4370	321	5	jafari	jafari	PROPN
ejpam-4370	321	6	and	and	CCONJ
ejpam-4370	321	7	t.	t.	PROPN
ejpam-4370	321	8	noiri	noiri	PROPN
ejpam-4370	321	9	.	.	PUNCT
ejpam-4370	322	1	on	on	ADP
ejpam-4370	322	2	contra	contra	ADJ
ejpam-4370	322	3	-	-	ADJ
ejpam-4370	322	4	precontinuous	precontinuous	ADJ
ejpam-4370	322	5	functions	function	NOUN
ejpam-4370	322	6	.	.	PUNCT
ejpam-4370	323	1	bulletin	bulletin	NOUN
ejpam-4370	323	2	of	of	ADP
ejpam-4370	323	3	the	the	DET
ejpam-4370	323	4	malaysian	malaysian	PROPN
ejpam-4370	323	5	mathematical	mathematical	PROPN
ejpam-4370	323	6	sciences	sciences	PROPN
ejpam-4370	323	7	society	society	NOUN
ejpam-4370	323	8	,	,	PUNCT
ejpam-4370	323	9	25:115–128	25:115–128	PROPN
ejpam-4370	323	10	,	,	PUNCT
ejpam-4370	323	11	2002	2002	NUM
ejpam-4370	323	12	.	.	PUNCT
ejpam-4370	324	1	[	[	X
ejpam-4370	324	2	15	15	NUM
ejpam-4370	324	3	]	]	X
ejpam-4370	324	4	a.	a.	NOUN
ejpam-4370	324	5	a.	a.	NOUN
ejpam-4370	324	6	nasef	nasef	PROPN
ejpam-4370	324	7	.	.	PUNCT
ejpam-4370	325	1	some	some	DET
ejpam-4370	325	2	properties	property	NOUN
ejpam-4370	325	3	of	of	ADP
ejpam-4370	325	4	contra	contra	PROPN
ejpam-4370	325	5	-	-	PUNCT
ejpam-4370	325	6	γ	γ	ADJ
ejpam-4370	325	7	-	-	ADJ
ejpam-4370	325	8	continuous	continuous	ADJ
ejpam-4370	325	9	functions	function	NOUN
ejpam-4370	325	10	.	.	PUNCT
ejpam-4370	326	1	chaos	chaos	NOUN
ejpam-4370	326	2	,	,	PUNCT
ejpam-4370	326	3	solitons	soliton	NOUN
ejpam-4370	326	4	&	&	CCONJ
ejpam-4370	326	5	fractals	fractal	NOUN
ejpam-4370	326	6	,	,	PUNCT
ejpam-4370	326	7	24:471–477	24:471–477	NUM
ejpam-4370	326	8	,	,	PUNCT
ejpam-4370	326	9	2005	2005	NUM
ejpam-4370	326	10	.	.	PUNCT
ejpam-4370	327	1	[	[	X
ejpam-4370	327	2	16	16	NUM
ejpam-4370	327	3	]	]	PUNCT
ejpam-4370	327	4	t.	t.	PROPN
ejpam-4370	327	5	noiri	noiri	PROPN
ejpam-4370	327	6	.	.	PUNCT
ejpam-4370	328	1	super	super	ADJ
ejpam-4370	328	2	-	-	NOUN
ejpam-4370	328	3	continuity	continuity	NOUN
ejpam-4370	328	4	and	and	CCONJ
ejpam-4370	328	5	some	some	DET
ejpam-4370	328	6	strong	strong	ADJ
ejpam-4370	328	7	forms	form	NOUN
ejpam-4370	328	8	of	of	ADP
ejpam-4370	328	9	continuity	continuity	NOUN
ejpam-4370	328	10	.	.	PUNCT
ejpam-4370	329	1	indian	indian	ADJ
ejpam-4370	329	2	journal	journal	PROPN
ejpam-4370	329	3	of	of	ADP
ejpam-4370	329	4	pure	pure	ADJ
ejpam-4370	329	5	and	and	CCONJ
ejpam-4370	329	6	applied	applied	ADJ
ejpam-4370	329	7	mathematics	mathematic	NOUN
ejpam-4370	329	8	,	,	PUNCT
ejpam-4370	329	9	15:241–250	15:241–250	NUM
ejpam-4370	329	10	,	,	PUNCT
ejpam-4370	329	11	1984	1984	NUM
ejpam-4370	329	12	.	.	PUNCT
ejpam-4370	330	1	[	[	X
ejpam-4370	330	2	17	17	NUM
ejpam-4370	330	3	]	]	PUNCT
ejpam-4370	330	4	t.	t.	PROPN
ejpam-4370	330	5	noiri	noiri	PROPN
ejpam-4370	330	6	,	,	PUNCT
ejpam-4370	330	7	b.	b.	PROPN
ejpam-4370	330	8	ahmad	ahmad	PROPN
ejpam-4370	330	9	,	,	PUNCT
ejpam-4370	330	10	and	and	CCONJ
ejpam-4370	330	11	m.	m.	PROPN
ejpam-4370	330	12	khan	khan	PROPN
ejpam-4370	330	13	.	.	PUNCT
ejpam-4370	331	1	almost	almost	ADV
ejpam-4370	331	2	s	s	NOUN
ejpam-4370	331	3	-	-	PUNCT
ejpam-4370	331	4	continuous	continuous	ADJ
ejpam-4370	331	5	functions	function	NOUN
ejpam-4370	331	6	.	.	PUNCT
ejpam-4370	332	1	kyungpook	kyungpook	PROPN
ejpam-4370	332	2	mathematical	mathematical	PROPN
ejpam-4370	332	3	journal	journal	PROPN
ejpam-4370	332	4	,	,	PUNCT
ejpam-4370	332	5	35:311–322	35:311–322	PROPN
ejpam-4370	332	6	,	,	PUNCT
ejpam-4370	332	7	1995	1995	NUM
ejpam-4370	332	8	.	.	PUNCT
ejpam-4370	333	1	[	[	X
ejpam-4370	333	2	18	18	NUM
ejpam-4370	333	3	]	]	PUNCT
ejpam-4370	333	4	t.	t.	PROPN
ejpam-4370	333	5	noiri	noiri	PROPN
ejpam-4370	333	6	and	and	CCONJ
ejpam-4370	333	7	e.	e.	PROPN
ejpam-4370	333	8	hatir	hatir	PROPN
ejpam-4370	333	9	.	.	PUNCT
ejpam-4370	334	1	λsp	λsp	NOUN
ejpam-4370	334	2	-	-	PUNCT
ejpam-4370	334	3	sets	set	NOUN
ejpam-4370	334	4	and	and	CCONJ
ejpam-4370	334	5	some	some	DET
ejpam-4370	334	6	weak	weak	ADJ
ejpam-4370	334	7	separation	separation	NOUN
ejpam-4370	334	8	axioms	axiom	NOUN
ejpam-4370	334	9	.	.	PUNCT
ejpam-4370	335	1	acta	acta	PROPN
ejpam-4370	335	2	mathematica	mathematica	PROPN
ejpam-4370	335	3	hungarica	hungarica	PROPN
ejpam-4370	335	4	,	,	PUNCT
ejpam-4370	335	5	103(3):225–232	103(3):225–232	NUM
ejpam-4370	335	6	,	,	PUNCT
ejpam-4370	335	7	2004	2004	NUM
ejpam-4370	335	8	.	.	PUNCT
ejpam-4370	336	1	[	[	X
ejpam-4370	336	2	19	19	NUM
ejpam-4370	336	3	]	]	X
ejpam-4370	336	4	t.	t.	PROPN
ejpam-4370	336	5	noiri	noiri	PROPN
ejpam-4370	336	6	and	and	CCONJ
ejpam-4370	336	7	v.	v.	ADP
ejpam-4370	336	8	popa	popa	NOUN
ejpam-4370	336	9	.	.	PUNCT
ejpam-4370	337	1	almost	almost	ADV
ejpam-4370	337	2	weakly	weakly	ADJ
ejpam-4370	337	3	continuous	continuous	ADJ
ejpam-4370	337	4	multifunctions	multifunction	NOUN
ejpam-4370	337	5	.	.	PUNCT
ejpam-4370	338	1	demonstratio	demonstratio	PROPN
ejpam-4370	338	2	mathematica	mathematica	PROPN
ejpam-4370	338	3	,	,	PUNCT
ejpam-4370	338	4	26:363–380	26:363–380	PROPN
ejpam-4370	338	5	,	,	PUNCT
ejpam-4370	338	6	1993	1993	NUM
ejpam-4370	338	7	.	.	PUNCT
ejpam-4370	339	1	[	[	X
ejpam-4370	339	2	20	20	NUM
ejpam-4370	339	3	]	]	PUNCT
ejpam-4370	339	4	v.	v.	PROPN
ejpam-4370	339	5	i.	i.	PROPN
ejpam-4370	339	6	ponomarev	ponomarev	PROPN
ejpam-4370	339	7	.	.	PUNCT
ejpam-4370	340	1	properties	property	NOUN
ejpam-4370	340	2	of	of	ADP
ejpam-4370	340	3	topological	topological	ADJ
ejpam-4370	340	4	spaces	space	NOUN
ejpam-4370	340	5	preserved	preserve	VERB
ejpam-4370	340	6	under	under	ADP
ejpam-4370	340	7	multivalued	multivalued	ADJ
ejpam-4370	340	8	continuous	continuous	ADJ
ejpam-4370	340	9	mappings	mapping	NOUN
ejpam-4370	340	10	on	on	ADP
ejpam-4370	340	11	compacta	compacta	NOUN
ejpam-4370	340	12	.	.	PUNCT
ejpam-4370	341	1	american	american	PROPN
ejpam-4370	341	2	mathematical	mathematical	ADJ
ejpam-4370	341	3	society	society	NOUN
ejpam-4370	341	4	translations	translation	NOUN
ejpam-4370	341	5	,	,	PUNCT
ejpam-4370	341	6	38(2):119–140	38(2):119–140	NUM
ejpam-4370	341	7	,	,	PUNCT
ejpam-4370	341	8	1964	1964	NUM
ejpam-4370	341	9	.	.	PUNCT
ejpam-4370	342	1	[	[	X
ejpam-4370	342	2	21	21	NUM
ejpam-4370	342	3	]	]	X
ejpam-4370	342	4	c.	c.	PROPN
ejpam-4370	342	5	viriyapong	viriyapong	PROPN
ejpam-4370	342	6	and	and	CCONJ
ejpam-4370	342	7	c.	c.	PROPN
ejpam-4370	342	8	boonpok	boonpok	PROPN
ejpam-4370	342	9	.	.	PUNCT
ejpam-4370	343	1	(	(	PUNCT
ejpam-4370	343	2	λ	λ	NOUN
ejpam-4370	343	3	,	,	PUNCT
ejpam-4370	343	4	sp)-continuity	sp)-continuity	NOUN
ejpam-4370	343	5	and	and	CCONJ
ejpam-4370	343	6	δ(λ	δ(λ	PROPN
ejpam-4370	343	7	,	,	PUNCT
ejpam-4370	343	8	sp)-closed	sp)-close	VERB
ejpam-4370	343	9	sets	set	NOUN
ejpam-4370	343	10	.	.	PUNCT
ejpam-4370	344	1	international	international	ADJ
ejpam-4370	344	2	journal	journal	NOUN
ejpam-4370	344	3	of	of	ADP
ejpam-4370	344	4	mathematics	mathematic	NOUN
ejpam-4370	344	5	and	and	CCONJ
ejpam-4370	344	6	computer	computer	NOUN
ejpam-4370	344	7	science	science	NOUN
ejpam-4370	344	8	,	,	PUNCT
ejpam-4370	344	9	17(4):1685–1689	17(4):1685–1689	NUM
ejpam-4370	344	10	,	,	PUNCT
ejpam-4370	344	11	2022	2022	NUM
ejpam-4370	344	12	.	.	PUNCT
ejpam-4370	345	1	[	[	X
ejpam-4370	345	2	22	22	NUM
ejpam-4370	345	3	]	]	X
ejpam-4370	345	4	c.	c.	PROPN
ejpam-4370	345	5	viriyapong	viriyapong	PROPN
ejpam-4370	345	6	and	and	CCONJ
ejpam-4370	345	7	c.	c.	PROPN
ejpam-4370	345	8	boonpok	boonpok	PROPN
ejpam-4370	345	9	.	.	PUNCT
ejpam-4370	346	1	(	(	PUNCT
ejpam-4370	346	2	λ	λ	X
ejpam-4370	346	3	,	,	PUNCT
ejpam-4370	346	4	sp)-continuous	sp)-continuous	ADJ
ejpam-4370	346	5	functions	function	NOUN
ejpam-4370	346	6	.	.	PUNCT
ejpam-4370	347	1	wseas	wseas	VERB
ejpam-4370	347	2	transactions	transaction	NOUN
ejpam-4370	347	3	on	on	ADP
ejpam-4370	347	4	mathematics	mathematic	NOUN
ejpam-4370	347	5	,	,	PUNCT
ejpam-4370	347	6	21:380–385	21:380–385	NUM
ejpam-4370	347	7	,	,	PUNCT
ejpam-4370	347	8	2022	2022	NUM
ejpam-4370	347	9	.	.	PUNCT