id	sid	tid	token	lemma	pos
ejpam-4072	1	1	european	european	PROPN
ejpam-4072	1	2	journal	journal	PROPN
ejpam-4072	1	3	of	of	ADP
ejpam-4072	1	4	pure	pure	ADJ
ejpam-4072	1	5	and	and	CCONJ
ejpam-4072	1	6	applied	apply	VERB
ejpam-4072	1	7	mathematics	mathematic	NOUN
ejpam-4072	1	8	vol	vol	NOUN
ejpam-4072	1	9	.	.	PUNCT
ejpam-4072	2	1	14	14	NUM
ejpam-4072	2	2	,	,	PUNCT
ejpam-4072	2	3	no	no	INTJ
ejpam-4072	2	4	.	.	NOUN
ejpam-4072	2	5	4	4	NUM
ejpam-4072	2	6	,	,	PUNCT
ejpam-4072	2	7	2021	2021	NUM
ejpam-4072	2	8	,	,	PUNCT
ejpam-4072	2	9	1212	1212	NUM
ejpam-4072	2	10	-	-	SYM
ejpam-4072	2	11	1225	1225	NUM
ejpam-4072	2	12	issn	issn	PROPN
ejpam-4072	2	13	1307	1307	NUM
ejpam-4072	2	14	-	-	SYM
ejpam-4072	2	15	5543	5543	NUM
ejpam-4072	2	16	–	–	PUNCT
ejpam-4072	3	1	ejpam.com	ejpam.com	X
ejpam-4072	3	2	published	publish	VERB
ejpam-4072	3	3	by	by	ADP
ejpam-4072	3	4	new	new	PROPN
ejpam-4072	3	5	york	york	PROPN
ejpam-4072	3	6	business	business	PROPN
ejpam-4072	3	7	global	global	PROPN
ejpam-4072	3	8	upper	upper	ADJ
ejpam-4072	3	9	and	and	CCONJ
ejpam-4072	3	10	lower	low	ADJ
ejpam-4072	3	11	almost	almost	ADV
ejpam-4072	3	12	weak	weak	ADJ
ejpam-4072	3	13	(	(	PUNCT
ejpam-4072	3	14	τ1	τ1	NOUN
ejpam-4072	3	15	,	,	PUNCT
ejpam-4072	3	16	τ2)-continuity	τ2)-continuity	NOUN
ejpam-4072	3	17	chawalit	chawalit	VERB
ejpam-4072	3	18	boonpok1,∗	boonpok1,∗	NOUN
ejpam-4072	3	19	,	,	PUNCT
ejpam-4072	3	20	chokchai	chokchai	ADJ
ejpam-4072	3	21	viriyapong1	viriyapong1	NOUN
ejpam-4072	3	22	1	1	NUM
ejpam-4072	3	23	mathematics	mathematic	NOUN
ejpam-4072	3	24	and	and	CCONJ
ejpam-4072	3	25	applied	apply	VERB
ejpam-4072	3	26	mathematics	mathematics	PROPN
ejpam-4072	3	27	research	research	NOUN
ejpam-4072	3	28	unit	unit	NOUN
ejpam-4072	3	29	,	,	PUNCT
ejpam-4072	3	30	department	department	NOUN
ejpam-4072	3	31	of	of	ADP
ejpam-4072	3	32	mathematics	mathematic	NOUN
ejpam-4072	3	33	,	,	PUNCT
ejpam-4072	3	34	faculty	faculty	NOUN
ejpam-4072	3	35	of	of	ADP
ejpam-4072	3	36	science	science	NOUN
ejpam-4072	3	37	,	,	PUNCT
ejpam-4072	3	38	mahasarakham	mahasarakham	PROPN
ejpam-4072	3	39	university	university	PROPN
ejpam-4072	3	40	,	,	PUNCT
ejpam-4072	3	41	maha	maha	PROPN
ejpam-4072	3	42	sarakham	sarakham	PROPN
ejpam-4072	3	43	,	,	PUNCT
ejpam-4072	3	44	44150	44150	NUM
ejpam-4072	3	45	,	,	PUNCT
ejpam-4072	3	46	thailand	thailand	PROPN
ejpam-4072	3	47	abstract	abstract	PROPN
ejpam-4072	3	48	.	.	PUNCT
ejpam-4072	4	1	the	the	DET
ejpam-4072	4	2	purpose	purpose	NOUN
ejpam-4072	4	3	of	of	ADP
ejpam-4072	4	4	the	the	DET
ejpam-4072	4	5	present	present	ADJ
ejpam-4072	4	6	paper	paper	NOUN
ejpam-4072	4	7	is	be	AUX
ejpam-4072	4	8	to	to	PART
ejpam-4072	4	9	introduce	introduce	VERB
ejpam-4072	4	10	the	the	DET
ejpam-4072	4	11	notions	notion	NOUN
ejpam-4072	4	12	of	of	ADP
ejpam-4072	4	13	upper	upper	ADJ
ejpam-4072	4	14	and	and	CCONJ
ejpam-4072	4	15	lower	low	ADJ
ejpam-4072	4	16	almost	almost	ADV
ejpam-4072	4	17	weakly	weakly	ADJ
ejpam-4072	4	18	(	(	PUNCT
ejpam-4072	4	19	τ1	τ1	NOUN
ejpam-4072	4	20	,	,	PUNCT
ejpam-4072	4	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	4	22	multifunctions	multifunction	NOUN
ejpam-4072	4	23	.	.	PUNCT
ejpam-4072	5	1	several	several	ADJ
ejpam-4072	5	2	characterizations	characterization	NOUN
ejpam-4072	5	3	of	of	ADP
ejpam-4072	5	4	upper	upper	ADJ
ejpam-4072	5	5	and	and	CCONJ
ejpam-4072	5	6	lower	low	ADJ
ejpam-4072	5	7	almost	almost	ADV
ejpam-4072	5	8	weakly	weakly	ADJ
ejpam-4072	5	9	(	(	PUNCT
ejpam-4072	5	10	τ1	τ1	NOUN
ejpam-4072	5	11	,	,	PUNCT
ejpam-4072	5	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	5	13	multifunctions	multifunction	NOUN
ejpam-4072	5	14	are	be	AUX
ejpam-4072	5	15	investigated	investigate	VERB
ejpam-4072	5	16	.	.	PUNCT
ejpam-4072	6	1	2020	2020	NUM
ejpam-4072	6	2	mathematics	mathematic	NOUN
ejpam-4072	6	3	subject	subject	NOUN
ejpam-4072	6	4	classifications	classification	NOUN
ejpam-4072	6	5	:	:	PUNCT
ejpam-4072	6	6	54c08	54c08	NUM
ejpam-4072	6	7	,	,	PUNCT
ejpam-4072	6	8	54c60	54c60	NUM
ejpam-4072	6	9	,	,	PUNCT
ejpam-4072	6	10	54e55	54e55	NUM
ejpam-4072	6	11	key	key	ADJ
ejpam-4072	6	12	words	word	NOUN
ejpam-4072	6	13	and	and	CCONJ
ejpam-4072	6	14	phrases	phrase	NOUN
ejpam-4072	6	15	:	:	PUNCT
ejpam-4072	6	16	upper	upper	ADJ
ejpam-4072	6	17	almost	almost	ADV
ejpam-4072	6	18	weakly	weakly	ADJ
ejpam-4072	6	19	(	(	PUNCT
ejpam-4072	6	20	τ1	τ1	NOUN
ejpam-4072	6	21	,	,	PUNCT
ejpam-4072	6	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	6	23	multifunction	multifunction	NOUN
ejpam-4072	6	24	,	,	PUNCT
ejpam-4072	6	25	lower	low	ADJ
ejpam-4072	6	26	almost	almost	ADV
ejpam-4072	6	27	weakly	weakly	ADJ
ejpam-4072	6	28	(	(	PUNCT
ejpam-4072	6	29	τ1	τ1	NOUN
ejpam-4072	6	30	,	,	PUNCT
ejpam-4072	6	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	6	32	multifunction	multifunction	NOUN
ejpam-4072	6	33	1	1	NUM
ejpam-4072	6	34	.	.	PUNCT
ejpam-4072	6	35	introduction	introduction	NOUN
ejpam-4072	6	36	topology	topology	NOUN
ejpam-4072	6	37	as	as	ADP
ejpam-4072	6	38	a	a	DET
ejpam-4072	6	39	field	field	NOUN
ejpam-4072	6	40	of	of	ADP
ejpam-4072	6	41	mathematics	mathematic	NOUN
ejpam-4072	6	42	is	be	AUX
ejpam-4072	6	43	concerned	concern	VERB
ejpam-4072	6	44	with	with	ADP
ejpam-4072	6	45	all	all	DET
ejpam-4072	6	46	questions	question	NOUN
ejpam-4072	6	47	directly	directly	ADV
ejpam-4072	6	48	or	or	CCONJ
ejpam-4072	6	49	indirectly	indirectly	ADV
ejpam-4072	6	50	related	relate	VERB
ejpam-4072	6	51	to	to	ADP
ejpam-4072	6	52	continuity	continuity	NOUN
ejpam-4072	6	53	.	.	PUNCT
ejpam-4072	7	1	continuity	continuity	NOUN
ejpam-4072	7	2	of	of	ADP
ejpam-4072	7	3	functions	function	NOUN
ejpam-4072	7	4	in	in	ADP
ejpam-4072	7	5	topological	topological	ADJ
ejpam-4072	7	6	spaces	space	NOUN
ejpam-4072	7	7	has	have	AUX
ejpam-4072	7	8	been	be	AUX
ejpam-4072	7	9	investigated	investigate	VERB
ejpam-4072	7	10	by	by	ADP
ejpam-4072	7	11	many	many	ADJ
ejpam-4072	7	12	mathematicians	mathematician	NOUN
ejpam-4072	7	13	.	.	PUNCT
ejpam-4072	8	1	this	this	DET
ejpam-4072	8	2	concept	concept	NOUN
ejpam-4072	8	3	has	have	AUX
ejpam-4072	8	4	been	be	AUX
ejpam-4072	8	5	extended	extend	VERB
ejpam-4072	8	6	to	to	ADP
ejpam-4072	8	7	the	the	DET
ejpam-4072	8	8	setting	set	VERB
ejpam-4072	8	9	multifunctions	multifunction	NOUN
ejpam-4072	8	10	and	and	CCONJ
ejpam-4072	8	11	has	have	AUX
ejpam-4072	8	12	been	be	AUX
ejpam-4072	8	13	generalized	generalize	VERB
ejpam-4072	8	14	by	by	ADP
ejpam-4072	8	15	weaker	weak	ADJ
ejpam-4072	8	16	forms	form	NOUN
ejpam-4072	8	17	of	of	ADP
ejpam-4072	8	18	open	open	ADJ
ejpam-4072	8	19	sets	set	NOUN
ejpam-4072	8	20	.	.	PUNCT
ejpam-4072	9	1	semi	semi	ADJ
ejpam-4072	9	2	-	-	ADJ
ejpam-4072	9	3	open	open	ADJ
ejpam-4072	9	4	sets	set	NOUN
ejpam-4072	9	5	[	[	X
ejpam-4072	9	6	18	18	NUM
ejpam-4072	9	7	]	]	PUNCT
ejpam-4072	9	8	,	,	PUNCT
ejpam-4072	9	9	preopen	preopen	ADJ
ejpam-4072	9	10	sets	set	NOUN
ejpam-4072	9	11	[	[	X
ejpam-4072	9	12	19	19	NUM
ejpam-4072	9	13	]	]	PUNCT
ejpam-4072	9	14	,	,	PUNCT
ejpam-4072	9	15	α	α	X
ejpam-4072	9	16	-	-	ADJ
ejpam-4072	9	17	open	open	ADJ
ejpam-4072	9	18	sets	set	NOUN
ejpam-4072	9	19	[	[	X
ejpam-4072	9	20	20	20	NUM
ejpam-4072	9	21	]	]	PUNCT
ejpam-4072	9	22	and	and	CCONJ
ejpam-4072	9	23	β	β	X
ejpam-4072	9	24	-	-	ADJ
ejpam-4072	9	25	open	open	ADJ
ejpam-4072	9	26	sets	set	NOUN
ejpam-4072	9	27	[	[	X
ejpam-4072	9	28	10	10	NUM
ejpam-4072	9	29	]	]	PUNCT
ejpam-4072	9	30	play	play	VERB
ejpam-4072	9	31	an	an	DET
ejpam-4072	9	32	important	important	ADJ
ejpam-4072	9	33	role	role	NOUN
ejpam-4072	9	34	in	in	ADP
ejpam-4072	9	35	the	the	DET
ejpam-4072	9	36	researching	researching	NOUN
ejpam-4072	9	37	of	of	ADP
ejpam-4072	9	38	generalizations	generalization	NOUN
ejpam-4072	9	39	of	of	ADP
ejpam-4072	9	40	continuity	continuity	NOUN
ejpam-4072	9	41	in	in	ADP
ejpam-4072	9	42	topological	topological	ADJ
ejpam-4072	9	43	spaces	space	NOUN
ejpam-4072	9	44	.	.	PUNCT
ejpam-4072	10	1	by	by	ADP
ejpam-4072	10	2	using	use	VERB
ejpam-4072	10	3	these	these	DET
ejpam-4072	10	4	sets	set	NOUN
ejpam-4072	10	5	many	many	ADJ
ejpam-4072	10	6	authors	author	NOUN
ejpam-4072	10	7	introduced	introduce	VERB
ejpam-4072	10	8	and	and	CCONJ
ejpam-4072	10	9	studied	study	VERB
ejpam-4072	10	10	various	various	ADJ
ejpam-4072	10	11	types	type	NOUN
ejpam-4072	10	12	of	of	ADP
ejpam-4072	10	13	weak	weak	ADJ
ejpam-4072	10	14	forms	form	NOUN
ejpam-4072	10	15	of	of	ADP
ejpam-4072	10	16	continuity	continuity	NOUN
ejpam-4072	10	17	for	for	ADP
ejpam-4072	10	18	functions	function	NOUN
ejpam-4072	10	19	and	and	CCONJ
ejpam-4072	10	20	multifunctions	multifunction	NOUN
ejpam-4072	10	21	.	.	PUNCT
ejpam-4072	11	1	in	in	ADP
ejpam-4072	11	2	1961	1961	NUM
ejpam-4072	11	3	,	,	PUNCT
ejpam-4072	11	4	levine	levine	PROPN
ejpam-4072	12	1	[	[	X
ejpam-4072	12	2	17	17	NUM
ejpam-4072	12	3	]	]	PUNCT
ejpam-4072	12	4	introduced	introduce	VERB
ejpam-4072	12	5	the	the	DET
ejpam-4072	12	6	concept	concept	NOUN
ejpam-4072	12	7	of	of	ADP
ejpam-4072	12	8	weakly	weakly	ADJ
ejpam-4072	12	9	continuous	continuous	ADJ
ejpam-4072	12	10	functions	function	NOUN
ejpam-4072	12	11	in	in	ADP
ejpam-4072	12	12	topological	topological	ADJ
ejpam-4072	12	13	spaces	space	NOUN
ejpam-4072	12	14	.	.	PUNCT
ejpam-4072	13	1	husain	husain	PROPN
ejpam-4072	14	1	[	[	X
ejpam-4072	14	2	11	11	NUM
ejpam-4072	14	3	]	]	PUNCT
ejpam-4072	14	4	introduced	introduce	VERB
ejpam-4072	14	5	the	the	DET
ejpam-4072	14	6	concept	concept	NOUN
ejpam-4072	14	7	of	of	ADP
ejpam-4072	14	8	almost	almost	ADV
ejpam-4072	14	9	continuous	continuous	ADJ
ejpam-4072	14	10	functions	function	NOUN
ejpam-4072	14	11	.	.	PUNCT
ejpam-4072	15	1	janković	janković	PUNCT
ejpam-4072	16	1	[	[	X
ejpam-4072	16	2	12	12	NUM
ejpam-4072	16	3	]	]	PUNCT
ejpam-4072	16	4	defined	define	VERB
ejpam-4072	16	5	almost	almost	ADV
ejpam-4072	16	6	weakly	weakly	ADJ
ejpam-4072	16	7	continuous	continuous	ADJ
ejpam-4072	16	8	functions	function	NOUN
ejpam-4072	16	9	as	as	ADP
ejpam-4072	16	10	a	a	DET
ejpam-4072	16	11	generalization	generalization	NOUN
ejpam-4072	16	12	of	of	ADP
ejpam-4072	16	13	both	both	CCONJ
ejpam-4072	16	14	weakly	weakly	ADJ
ejpam-4072	16	15	continuous	continuous	ADJ
ejpam-4072	16	16	functions	function	NOUN
ejpam-4072	16	17	due	due	ADP
ejpam-4072	16	18	to	to	ADP
ejpam-4072	16	19	levine	levine	PROPN
ejpam-4072	16	20	[	[	X
ejpam-4072	16	21	17	17	NUM
ejpam-4072	16	22	]	]	PUNCT
ejpam-4072	16	23	and	and	CCONJ
ejpam-4072	16	24	almost	almost	ADV
ejpam-4072	16	25	continuous	continuous	ADJ
ejpam-4072	16	26	functions	function	NOUN
ejpam-4072	16	27	in	in	ADP
ejpam-4072	16	28	the	the	DET
ejpam-4072	16	29	sense	sense	NOUN
ejpam-4072	16	30	of	of	ADP
ejpam-4072	16	31	husain	husain	NOUN
ejpam-4072	17	1	[	[	X
ejpam-4072	17	2	11	11	NUM
ejpam-4072	17	3	]	]	PUNCT
ejpam-4072	17	4	.	.	PUNCT
ejpam-4072	18	1	noiri	noiri	PROPN
ejpam-4072	18	2	and	and	CCONJ
ejpam-4072	18	3	popa	popa	NOUN
ejpam-4072	18	4	[	[	X
ejpam-4072	18	5	21	21	NUM
ejpam-4072	18	6	,	,	PUNCT
ejpam-4072	18	7	25	25	NUM
ejpam-4072	18	8	]	]	PUNCT
ejpam-4072	18	9	investigated	investigate	VERB
ejpam-4072	18	10	further	further	ADJ
ejpam-4072	18	11	characterizations	characterization	NOUN
ejpam-4072	18	12	of	of	ADP
ejpam-4072	18	13	almost	almost	ADV
ejpam-4072	18	14	weakly	weakly	ADJ
ejpam-4072	18	15	continuous	continuous	ADJ
ejpam-4072	18	16	functions	function	NOUN
ejpam-4072	18	17	.	.	PUNCT
ejpam-4072	19	1	smithson	smithson	PROPN
ejpam-4072	20	1	[	[	X
ejpam-4072	20	2	27	27	NUM
ejpam-4072	20	3	]	]	PUNCT
ejpam-4072	20	4	and	and	CCONJ
ejpam-4072	20	5	popa	popa	NOUN
ejpam-4072	21	1	[	[	X
ejpam-4072	21	2	23	23	NUM
ejpam-4072	21	3	,	,	PUNCT
ejpam-4072	21	4	24	24	NUM
ejpam-4072	21	5	]	]	PUNCT
ejpam-4072	21	6	extended	extend	VERB
ejpam-4072	21	7	independently	independently	ADV
ejpam-4072	21	8	these	these	DET
ejpam-4072	21	9	concepts	concept	NOUN
ejpam-4072	21	10	to	to	ADP
ejpam-4072	21	11	multifunctions	multifunction	NOUN
ejpam-4072	21	12	by	by	ADP
ejpam-4072	21	13	introducing	introduce	VERB
ejpam-4072	21	14	and	and	CCONJ
ejpam-4072	21	15	characterizing	characterize	VERB
ejpam-4072	21	16	the	the	DET
ejpam-4072	21	17	notions	notion	NOUN
ejpam-4072	21	18	of	of	ADP
ejpam-4072	21	19	almost	almost	ADV
ejpam-4072	21	20	continuous	continuous	ADJ
ejpam-4072	21	21	multifunctions	multifunction	NOUN
ejpam-4072	21	22	and	and	CCONJ
ejpam-4072	21	23	weakly	weakly	ADJ
ejpam-4072	21	24	continuous	continuous	ADJ
ejpam-4072	21	25	multifunctions	multifunction	NOUN
ejpam-4072	21	26	.	.	PUNCT
ejpam-4072	22	1	ekici	ekici	NOUN
ejpam-4072	22	2	and	and	CCONJ
ejpam-4072	22	3	park	park	NOUN
ejpam-4072	23	1	[	[	X
ejpam-4072	23	2	9	9	NUM
ejpam-4072	23	3	]	]	PUNCT
ejpam-4072	23	4	introduced	introduce	VERB
ejpam-4072	23	5	and	and	CCONJ
ejpam-4072	23	6	studied	study	VERB
ejpam-4072	23	7	upper	upper	ADJ
ejpam-4072	23	8	and	and	CCONJ
ejpam-4072	23	9	lower	low	ADJ
ejpam-4072	23	10	almost	almost	ADV
ejpam-4072	23	11	γ	γ	ADJ
ejpam-4072	23	12	-	-	ADJ
ejpam-4072	23	13	continuous	continuous	ADJ
ejpam-4072	23	14	multifunctions	multifunction	NOUN
ejpam-4072	23	15	as	as	ADP
ejpam-4072	23	16	a	a	DET
ejpam-4072	23	17	generalization	generalization	NOUN
ejpam-4072	23	18	of	of	ADP
ejpam-4072	23	19	some	some	DET
ejpam-4072	23	20	types	type	NOUN
ejpam-4072	23	21	of	of	ADP
ejpam-4072	23	22	continuous	continuous	ADJ
ejpam-4072	23	23	multifunctions	multifunction	NOUN
ejpam-4072	23	24	including	include	VERB
ejpam-4072	23	25	almost	almost	ADV
ejpam-4072	23	26	continuity	continuity	NOUN
ejpam-4072	23	27	,	,	PUNCT
ejpam-4072	23	28	almost	almost	ADV
ejpam-4072	23	29	α	α	NOUN
ejpam-4072	23	30	-	-	NOUN
ejpam-4072	23	31	continuity	continuity	NOUN
ejpam-4072	23	32	,	,	PUNCT
ejpam-4072	23	33	almost	almost	ADV
ejpam-4072	23	34	precontinuity	precontinuity	NOUN
ejpam-4072	23	35	,	,	PUNCT
ejpam-4072	23	36	almost	almost	ADV
ejpam-4072	23	37	quasi	quasi	ADJ
ejpam-4072	23	38	-	-	NOUN
ejpam-4072	23	39	continuity	continuity	NOUN
ejpam-4072	23	40	and	and	CCONJ
ejpam-4072	23	41	γ	γ	NOUN
ejpam-4072	23	42	-	-	NOUN
ejpam-4072	23	43	continuity	continuity	NOUN
ejpam-4072	23	44	.	.	PUNCT
ejpam-4072	24	1	the	the	DET
ejpam-4072	24	2	concept	concept	NOUN
ejpam-4072	24	3	∗corresponding	∗corresponde	VERB
ejpam-4072	24	4	author	author	NOUN
ejpam-4072	24	5	.	.	PUNCT
ejpam-4072	25	1	doi	doi	NOUN
ejpam-4072	25	2	:	:	PUNCT
ejpam-4072	25	3	https://doi.org/10.29020/nybg.ejpam.v14i4.4072	https://doi.org/10.29020/nybg.ejpam.v14i4.4072	NUM
ejpam-4072	25	4	email	email	NOUN
ejpam-4072	25	5	addresses	address	NOUN
ejpam-4072	25	6	:	:	PUNCT
ejpam-4072	26	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4072	26	2	(	(	PUNCT
ejpam-4072	26	3	c.	c.	PROPN
ejpam-4072	26	4	boonpok	boonpok	PROPN
ejpam-4072	26	5	)	)	PUNCT
ejpam-4072	26	6	,	,	PUNCT
ejpam-4072	26	7	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-4072	26	8	(	(	PUNCT
ejpam-4072	26	9	c.	c.	PROPN
ejpam-4072	26	10	viriyapong	viriyapong	PROPN
ejpam-4072	26	11	)	)	PUNCT
ejpam-4072	26	12	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4072	26	13	1212	1212	NUM
ejpam-4072	27	1	©	©	PROPN
ejpam-4072	27	2	2021	2021	NUM
ejpam-4072	27	3	ejpam	ejpam	VERB
ejpam-4072	27	4	all	all	DET
ejpam-4072	27	5	rights	right	NOUN
ejpam-4072	27	6	reserved	reserve	VERB
ejpam-4072	27	7	.	.	PUNCT
ejpam-4072	28	1	c.	c.	PROPN
ejpam-4072	28	2	boonpok	boonpok	PROPN
ejpam-4072	28	3	,	,	PUNCT
ejpam-4072	28	4	c.	c.	PROPN
ejpam-4072	28	5	viriyapong	viriyapong	PROPN
ejpam-4072	28	6	/	/	SYM
ejpam-4072	28	7	eur	eur	PROPN
ejpam-4072	28	8	.	.	PUNCT
ejpam-4072	29	1	j.	j.	PROPN
ejpam-4072	29	2	pure	pure	PROPN
ejpam-4072	29	3	appl	appl	PROPN
ejpam-4072	29	4	.	.	PROPN
ejpam-4072	29	5	math	math	PROPN
ejpam-4072	29	6	,	,	PUNCT
ejpam-4072	29	7	14	14	NUM
ejpam-4072	29	8	(	(	PUNCT
ejpam-4072	29	9	4	4	NUM
ejpam-4072	29	10	)	)	PUNCT
ejpam-4072	29	11	(	(	PUNCT
ejpam-4072	29	12	2021	2021	NUM
ejpam-4072	29	13	)	)	PUNCT
ejpam-4072	29	14	,	,	PUNCT
ejpam-4072	29	15	1212	1212	NUM
ejpam-4072	29	16	-	-	SYM
ejpam-4072	29	17	1225	1225	NUM
ejpam-4072	29	18	1213	1213	NUM
ejpam-4072	29	19	of	of	ADP
ejpam-4072	29	20	bitopological	bitopological	ADJ
ejpam-4072	29	21	spaces	space	NOUN
ejpam-4072	29	22	was	be	AUX
ejpam-4072	29	23	first	first	ADV
ejpam-4072	29	24	introduced	introduce	VERB
ejpam-4072	29	25	by	by	ADP
ejpam-4072	29	26	kelly	kelly	PROPN
ejpam-4072	30	1	[	[	X
ejpam-4072	30	2	14	14	NUM
ejpam-4072	30	3	]	]	PUNCT
ejpam-4072	30	4	.	.	PUNCT
ejpam-4072	31	1	şenel	şenel	NOUN
ejpam-4072	31	2	and	and	CCONJ
ejpam-4072	31	3	çağman	çağman	NOUN
ejpam-4072	32	1	[	[	X
ejpam-4072	32	2	8	8	NUM
ejpam-4072	32	3	]	]	PUNCT
ejpam-4072	32	4	extended	extend	VERB
ejpam-4072	32	5	the	the	DET
ejpam-4072	32	6	notion	notion	NOUN
ejpam-4072	32	7	of	of	ADP
ejpam-4072	32	8	bitopological	bitopological	ADJ
ejpam-4072	32	9	spaces	space	NOUN
ejpam-4072	32	10	to	to	ADP
ejpam-4072	32	11	soft	soft	ADJ
ejpam-4072	32	12	bitopological	bitopological	ADJ
ejpam-4072	32	13	spaces	space	NOUN
ejpam-4072	32	14	.	.	PUNCT
ejpam-4072	33	1	şenel	şenel	VERB
ejpam-4072	34	1	[	[	X
ejpam-4072	34	2	7	7	X
ejpam-4072	34	3	]	]	PUNCT
ejpam-4072	34	4	presented	present	VERB
ejpam-4072	34	5	the	the	DET
ejpam-4072	34	6	concept	concept	NOUN
ejpam-4072	34	7	of	of	ADP
ejpam-4072	34	8	soft	soft	ADJ
ejpam-4072	34	9	bitopological	bitopological	ADJ
ejpam-4072	34	10	hausdorff	hausdorff	NOUN
ejpam-4072	34	11	spaces	space	NOUN
ejpam-4072	34	12	and	and	CCONJ
ejpam-4072	34	13	introduced	introduce	VERB
ejpam-4072	34	14	some	some	DET
ejpam-4072	34	15	new	new	ADJ
ejpam-4072	34	16	notions	notion	NOUN
ejpam-4072	34	17	in	in	ADP
ejpam-4072	34	18	soft	soft	ADJ
ejpam-4072	34	19	bitopological	bitopological	ADJ
ejpam-4072	34	20	spaces	space	NOUN
ejpam-4072	34	21	such	such	ADJ
ejpam-4072	34	22	as	as	ADP
ejpam-4072	34	23	sbt	sbt	PROPN
ejpam-4072	34	24	points	point	NOUN
ejpam-4072	34	25	,	,	PUNCT
ejpam-4072	34	26	sbt	sbt	PROPN
ejpam-4072	34	27	continuous	continuous	ADJ
ejpam-4072	34	28	functions	function	NOUN
ejpam-4072	34	29	and	and	CCONJ
ejpam-4072	34	30	sbt	sbt	PROPN
ejpam-4072	34	31	homeomorphisms	homeomorphisms	PROPN
ejpam-4072	34	32	.	.	PUNCT
ejpam-4072	35	1	khedr	khedr	PROPN
ejpam-4072	35	2	et	et	PROPN
ejpam-4072	35	3	al	al	PROPN
ejpam-4072	35	4	.	.	PUNCT
ejpam-4072	36	1	[	[	X
ejpam-4072	36	2	15	15	NUM
ejpam-4072	36	3	]	]	PUNCT
ejpam-4072	36	4	investigated	investigate	VERB
ejpam-4072	36	5	the	the	DET
ejpam-4072	36	6	notions	notion	NOUN
ejpam-4072	36	7	of	of	ADP
ejpam-4072	36	8	β	β	ADJ
ejpam-4072	36	9	-	-	ADJ
ejpam-4072	36	10	open	open	ADJ
ejpam-4072	36	11	sets	set	NOUN
ejpam-4072	36	12	and	and	CCONJ
ejpam-4072	36	13	β	β	NOUN
ejpam-4072	36	14	-	-	NOUN
ejpam-4072	36	15	continuity	continuity	NOUN
ejpam-4072	36	16	in	in	ADP
ejpam-4072	36	17	bitopological	bitopological	ADJ
ejpam-4072	36	18	spaces	space	NOUN
ejpam-4072	36	19	.	.	PUNCT
ejpam-4072	37	1	in	in	ADP
ejpam-4072	37	2	2020	2020	NUM
ejpam-4072	37	3	,	,	PUNCT
ejpam-4072	37	4	laprom	laprom	ADP
ejpam-4072	37	5	et	et	PROPN
ejpam-4072	37	6	al	al	PROPN
ejpam-4072	37	7	.	.	PUNCT
ejpam-4072	38	1	[	[	X
ejpam-4072	38	2	16	16	NUM
ejpam-4072	38	3	]	]	PUNCT
ejpam-4072	38	4	introduced	introduce	VERB
ejpam-4072	38	5	and	and	CCONJ
ejpam-4072	38	6	investigated	investigate	VERB
ejpam-4072	38	7	the	the	DET
ejpam-4072	38	8	notions	notion	NOUN
ejpam-4072	38	9	of	of	ADP
ejpam-4072	38	10	β(τ1	β(τ1	NOUN
ejpam-4072	38	11	,	,	PUNCT
ejpam-4072	38	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	38	13	multifunctions	multifunction	NOUN
ejpam-4072	38	14	and	and	CCONJ
ejpam-4072	38	15	almost	almost	ADV
ejpam-4072	38	16	β(τ1	β(τ1	NOUN
ejpam-4072	38	17	,	,	PUNCT
ejpam-4072	38	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	38	19	multifunctions	multifunction	NOUN
ejpam-4072	38	20	.	.	PUNCT
ejpam-4072	39	1	in	in	ADP
ejpam-4072	39	2	this	this	DET
ejpam-4072	39	3	paper	paper	NOUN
ejpam-4072	39	4	,	,	PUNCT
ejpam-4072	39	5	we	we	PRON
ejpam-4072	39	6	introduce	introduce	VERB
ejpam-4072	39	7	the	the	DET
ejpam-4072	39	8	concepts	concept	NOUN
ejpam-4072	39	9	of	of	ADP
ejpam-4072	39	10	upper	upper	ADJ
ejpam-4072	39	11	and	and	CCONJ
ejpam-4072	39	12	lower	low	ADJ
ejpam-4072	39	13	almost	almost	ADV
ejpam-4072	39	14	weakly	weakly	ADJ
ejpam-4072	39	15	(	(	PUNCT
ejpam-4072	39	16	τ1	τ1	NOUN
ejpam-4072	39	17	,	,	PUNCT
ejpam-4072	39	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	39	19	multifunctions	multifunction	NOUN
ejpam-4072	39	20	.	.	PUNCT
ejpam-4072	40	1	furthermore	furthermore	ADV
ejpam-4072	40	2	,	,	PUNCT
ejpam-4072	40	3	several	several	ADJ
ejpam-4072	40	4	characterizations	characterization	NOUN
ejpam-4072	40	5	of	of	ADP
ejpam-4072	40	6	upper	upper	ADJ
ejpam-4072	40	7	and	and	CCONJ
ejpam-4072	40	8	lower	low	ADJ
ejpam-4072	40	9	almost	almost	ADV
ejpam-4072	40	10	weakly	weakly	ADJ
ejpam-4072	40	11	(	(	PUNCT
ejpam-4072	40	12	τ1	τ1	NOUN
ejpam-4072	40	13	,	,	PUNCT
ejpam-4072	40	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	40	15	multifunctions	multifunction	NOUN
ejpam-4072	40	16	are	be	AUX
ejpam-4072	40	17	discussed	discuss	VERB
ejpam-4072	40	18	.	.	PUNCT
ejpam-4072	41	1	2	2	X
ejpam-4072	41	2	.	.	X
ejpam-4072	41	3	preliminaries	preliminary	NOUN
ejpam-4072	41	4	throughout	throughout	ADP
ejpam-4072	41	5	the	the	DET
ejpam-4072	41	6	present	present	ADJ
ejpam-4072	41	7	paper	paper	NOUN
ejpam-4072	41	8	,	,	PUNCT
ejpam-4072	41	9	spaces	space	NOUN
ejpam-4072	41	10	(	(	PUNCT
ejpam-4072	41	11	x	x	NOUN
ejpam-4072	41	12	,	,	PUNCT
ejpam-4072	41	13	τ1	τ1	NOUN
ejpam-4072	41	14	,	,	PUNCT
ejpam-4072	41	15	τ2	τ2	NOUN
ejpam-4072	41	16	)	)	PUNCT
ejpam-4072	41	17	and	and	CCONJ
ejpam-4072	41	18	(	(	PUNCT
ejpam-4072	41	19	y	y	PROPN
ejpam-4072	41	20	,	,	PUNCT
ejpam-4072	41	21	σ1	σ1	PROPN
ejpam-4072	41	22	,	,	PUNCT
ejpam-4072	41	23	σ2	σ2	NOUN
ejpam-4072	41	24	)	)	PUNCT
ejpam-4072	41	25	(	(	PUNCT
ejpam-4072	41	26	or	or	CCONJ
ejpam-4072	41	27	simply	simply	ADV
ejpam-4072	41	28	x	x	X
ejpam-4072	41	29	and	and	CCONJ
ejpam-4072	41	30	y	y	PROPN
ejpam-4072	41	31	)	)	PUNCT
ejpam-4072	41	32	always	always	ADV
ejpam-4072	41	33	mean	mean	VERB
ejpam-4072	41	34	bitopological	bitopological	ADJ
ejpam-4072	41	35	spaces	space	NOUN
ejpam-4072	41	36	on	on	ADP
ejpam-4072	41	37	which	which	PRON
ejpam-4072	41	38	no	no	DET
ejpam-4072	41	39	separation	separation	NOUN
ejpam-4072	41	40	axioms	axiom	NOUN
ejpam-4072	41	41	are	be	AUX
ejpam-4072	41	42	assumed	assume	VERB
ejpam-4072	41	43	unless	unless	SCONJ
ejpam-4072	41	44	explicitly	explicitly	ADV
ejpam-4072	41	45	stated	state	VERB
ejpam-4072	41	46	.	.	PUNCT
ejpam-4072	42	1	let	let	AUX
ejpam-4072	42	2	(	(	PUNCT
ejpam-4072	42	3	x	x	NOUN
ejpam-4072	42	4	,	,	PUNCT
ejpam-4072	42	5	τ1	τ1	NOUN
ejpam-4072	42	6	,	,	PUNCT
ejpam-4072	42	7	τ2	τ2	PROPN
ejpam-4072	42	8	)	)	PUNCT
ejpam-4072	42	9	be	be	VERB
ejpam-4072	42	10	a	a	DET
ejpam-4072	42	11	bitopological	bitopological	ADJ
ejpam-4072	42	12	space	space	NOUN
ejpam-4072	42	13	and	and	CCONJ
ejpam-4072	42	14	let	let	VERB
ejpam-4072	42	15	a	a	PRON
ejpam-4072	42	16	be	be	AUX
ejpam-4072	42	17	a	a	DET
ejpam-4072	42	18	subset	subset	NOUN
ejpam-4072	42	19	of	of	ADP
ejpam-4072	42	20	x.	x.	NOUN
ejpam-4072	42	21	the	the	DET
ejpam-4072	42	22	closure	closure	NOUN
ejpam-4072	42	23	of	of	ADP
ejpam-4072	42	24	a	a	PRON
ejpam-4072	42	25	and	and	CCONJ
ejpam-4072	42	26	the	the	DET
ejpam-4072	42	27	interior	interior	NOUN
ejpam-4072	42	28	of	of	ADP
ejpam-4072	42	29	a	a	PRON
ejpam-4072	42	30	with	with	ADP
ejpam-4072	42	31	respect	respect	NOUN
ejpam-4072	42	32	to	to	ADP
ejpam-4072	42	33	the	the	DET
ejpam-4072	42	34	topology	topology	NOUN
ejpam-4072	42	35	τi	τi	NOUN
ejpam-4072	42	36	are	be	AUX
ejpam-4072	42	37	denoted	denote	VERB
ejpam-4072	42	38	by	by	ADP
ejpam-4072	42	39	τi	τi	NOUN
ejpam-4072	42	40	-	-	PUNCT
ejpam-4072	42	41	cl(a	cl(a	NUM
ejpam-4072	42	42	)	)	PUNCT
ejpam-4072	42	43	and	and	CCONJ
ejpam-4072	42	44	τi	τi	NOUN
ejpam-4072	42	45	-	-	PUNCT
ejpam-4072	42	46	int(a	int(a	NOUN
ejpam-4072	42	47	)	)	PUNCT
ejpam-4072	42	48	,	,	PUNCT
ejpam-4072	42	49	respectively	respectively	ADV
ejpam-4072	42	50	,	,	PUNCT
ejpam-4072	42	51	for	for	ADP
ejpam-4072	42	52	i	i	PROPN
ejpam-4072	42	53	=	=	SYM
ejpam-4072	42	54	1	1	NUM
ejpam-4072	42	55	,	,	PUNCT
ejpam-4072	42	56	2	2	NUM
ejpam-4072	42	57	.	.	X
ejpam-4072	42	58	a	a	DET
ejpam-4072	42	59	subset	subset	NOUN
ejpam-4072	42	60	a	a	PRON
ejpam-4072	42	61	of	of	ADP
ejpam-4072	42	62	a	a	DET
ejpam-4072	42	63	bitopological	bitopological	ADJ
ejpam-4072	42	64	space	space	NOUN
ejpam-4072	42	65	(	(	PUNCT
ejpam-4072	42	66	x	x	NOUN
ejpam-4072	42	67	,	,	PUNCT
ejpam-4072	42	68	τ1	τ1	NOUN
ejpam-4072	42	69	,	,	PUNCT
ejpam-4072	42	70	τ2	τ2	NOUN
ejpam-4072	42	71	)	)	PUNCT
ejpam-4072	42	72	is	be	AUX
ejpam-4072	42	73	called	call	VERB
ejpam-4072	42	74	τ1τ2	τ1τ2	NOUN
ejpam-4072	42	75	-	-	ADJ
ejpam-4072	42	76	semi	semi	ADJ
ejpam-4072	42	77	-	-	ADJ
ejpam-4072	42	78	open	open	ADJ
ejpam-4072	42	79	[	[	X
ejpam-4072	42	80	5	5	NUM
ejpam-4072	42	81	]	]	PUNCT
ejpam-4072	42	82	(	(	PUNCT
ejpam-4072	42	83	resp	resp	NOUN
ejpam-4072	42	84	.	.	PUNCT
ejpam-4072	43	1	τ1τ2	τ1τ2	ADJ
ejpam-4072	43	2	-	-	ADJ
ejpam-4072	43	3	regular	regular	ADJ
ejpam-4072	43	4	open	open	ADJ
ejpam-4072	43	5	[	[	X
ejpam-4072	43	6	2	2	NUM
ejpam-4072	43	7	]	]	PUNCT
ejpam-4072	43	8	,	,	PUNCT
ejpam-4072	43	9	τ1τ2	τ1τ2	ADJ
ejpam-4072	43	10	-	-	ADJ
ejpam-4072	43	11	regular	regular	ADJ
ejpam-4072	43	12	closed	closed	ADJ
ejpam-4072	43	13	[	[	X
ejpam-4072	43	14	6	6	NUM
ejpam-4072	43	15	]	]	PUNCT
ejpam-4072	43	16	,	,	PUNCT
ejpam-4072	43	17	τ1τ2	τ1τ2	NOUN
ejpam-4072	43	18	-	-	ADJ
ejpam-4072	43	19	preopen	preopen	ADJ
ejpam-4072	43	20	[	[	X
ejpam-4072	43	21	13	13	NUM
ejpam-4072	43	22	]	]	SYM
ejpam-4072	43	23	)	)	PUNCT
ejpam-4072	43	24	if	if	SCONJ
ejpam-4072	43	25	a	a	DET
ejpam-4072	43	26	⊆	⊆	NUM
ejpam-4072	43	27	τ1	τ1	NOUN
ejpam-4072	43	28	-	-	PUNCT
ejpam-4072	43	29	cl(τ2	cl(τ2	NOUN
ejpam-4072	43	30	-	-	PUNCT
ejpam-4072	43	31	int(a	int(a	NOUN
ejpam-4072	43	32	)	)	PUNCT
ejpam-4072	43	33	)	)	PUNCT
ejpam-4072	43	34	(	(	PUNCT
ejpam-4072	43	35	resp	resp	NOUN
ejpam-4072	43	36	.	.	PUNCT
ejpam-4072	44	1	a	a	DET
ejpam-4072	44	2	=	=	NOUN
ejpam-4072	44	3	τ1	τ1	NOUN
ejpam-4072	44	4	-	-	PUNCT
ejpam-4072	44	5	int(τ2	int(τ2	NOUN
ejpam-4072	44	6	-	-	PUNCT
ejpam-4072	44	7	cl(a	cl(a	NUM
ejpam-4072	44	8	)	)	PUNCT
ejpam-4072	44	9	)	)	PUNCT
ejpam-4072	44	10	,	,	PUNCT
ejpam-4072	44	11	a	a	DET
ejpam-4072	44	12	=	=	NOUN
ejpam-4072	44	13	τ1	τ1	NOUN
ejpam-4072	44	14	-	-	PUNCT
ejpam-4072	44	15	cl(τ2	cl(τ2	NOUN
ejpam-4072	44	16	-	-	PUNCT
ejpam-4072	44	17	int(a	int(a	NOUN
ejpam-4072	44	18	)	)	PUNCT
ejpam-4072	44	19	)	)	PUNCT
ejpam-4072	44	20	,	,	PUNCT
ejpam-4072	44	21	a	a	DET
ejpam-4072	44	22	⊆	⊆	NUM
ejpam-4072	44	23	τ1	τ1	NOUN
ejpam-4072	44	24	-	-	PUNCT
ejpam-4072	44	25	int(τ2	int(τ2	NOUN
ejpam-4072	44	26	-	-	PUNCT
ejpam-4072	44	27	cl(a	cl(a	NUM
ejpam-4072	44	28	)	)	PUNCT
ejpam-4072	44	29	)	)	PUNCT
ejpam-4072	44	30	)	)	PUNCT
ejpam-4072	44	31	.	.	PUNCT
ejpam-4072	45	1	the	the	DET
ejpam-4072	45	2	complement	complement	NOUN
ejpam-4072	45	3	of	of	ADP
ejpam-4072	45	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	45	5	-	-	ADJ
ejpam-4072	45	6	semi	semi	ADJ
ejpam-4072	45	7	-	-	ADJ
ejpam-4072	45	8	open	open	ADJ
ejpam-4072	45	9	(	(	PUNCT
ejpam-4072	45	10	resp	resp	NOUN
ejpam-4072	45	11	.	.	PUNCT
ejpam-4072	46	1	τ1τ2	τ1τ2	ADJ
ejpam-4072	46	2	-	-	ADJ
ejpam-4072	46	3	preopen	preopen	ADJ
ejpam-4072	46	4	)	)	PUNCT
ejpam-4072	46	5	set	set	NOUN
ejpam-4072	46	6	is	be	AUX
ejpam-4072	46	7	said	say	VERB
ejpam-4072	46	8	to	to	PART
ejpam-4072	46	9	be	be	AUX
ejpam-4072	46	10	τ1τ2	τ1τ2	NOUN
ejpam-4072	46	11	-	-	ADJ
ejpam-4072	46	12	semi	semi	ADV
ejpam-4072	46	13	-	-	ADJ
ejpam-4072	46	14	closed	closed	ADJ
ejpam-4072	46	15	(	(	PUNCT
ejpam-4072	46	16	resp	resp	NOUN
ejpam-4072	46	17	.	.	PUNCT
ejpam-4072	47	1	τ1τ2	τ1τ2	VERB
ejpam-4072	47	2	-	-	ADJ
ejpam-4072	47	3	preclosed	preclose	VERB
ejpam-4072	47	4	)	)	PUNCT
ejpam-4072	47	5	.	.	PUNCT
ejpam-4072	48	1	the	the	DET
ejpam-4072	48	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	48	3	-	-	ADJ
ejpam-4072	48	4	semi	semi	NOUN
ejpam-4072	48	5	-	-	NOUN
ejpam-4072	48	6	closure	closure	NOUN
ejpam-4072	48	7	[	[	X
ejpam-4072	48	8	5	5	NUM
ejpam-4072	48	9	]	]	PUNCT
ejpam-4072	48	10	(	(	PUNCT
ejpam-4072	48	11	resp	resp	NOUN
ejpam-4072	48	12	.	.	PUNCT
ejpam-4072	49	1	τ1τ2	τ1τ2	VERB
ejpam-4072	49	2	-	-	NOUN
ejpam-4072	49	3	preclosure	preclosure	ADJ
ejpam-4072	49	4	[	[	X
ejpam-4072	49	5	15	15	NUM
ejpam-4072	49	6	]	]	PUNCT
ejpam-4072	49	7	)	)	PUNCT
ejpam-4072	49	8	of	of	ADP
ejpam-4072	49	9	a	a	PRON
ejpam-4072	49	10	is	be	AUX
ejpam-4072	49	11	defined	define	VERB
ejpam-4072	49	12	by	by	ADP
ejpam-4072	49	13	the	the	DET
ejpam-4072	49	14	intersection	intersection	NOUN
ejpam-4072	49	15	of	of	ADP
ejpam-4072	49	16	τ1τ2	τ1τ2	NOUN
ejpam-4072	49	17	-	-	ADJ
ejpam-4072	49	18	semi	semi	ADV
ejpam-4072	49	19	-	-	ADJ
ejpam-4072	49	20	closed	closed	ADJ
ejpam-4072	49	21	(	(	PUNCT
ejpam-4072	49	22	resp	resp	NOUN
ejpam-4072	49	23	.	.	PUNCT
ejpam-4072	50	1	τ1τ2	τ1τ2	VERB
ejpam-4072	50	2	-	-	ADJ
ejpam-4072	50	3	preclosed	preclose	VERB
ejpam-4072	50	4	)	)	PUNCT
ejpam-4072	50	5	sets	set	NOUN
ejpam-4072	50	6	containing	contain	VERB
ejpam-4072	50	7	a	a	PRON
ejpam-4072	50	8	and	and	CCONJ
ejpam-4072	50	9	is	be	AUX
ejpam-4072	50	10	denoted	denote	VERB
ejpam-4072	50	11	by	by	ADP
ejpam-4072	50	12	τ1τ2	τ1τ2	NOUN
ejpam-4072	50	13	-	-	ADJ
ejpam-4072	50	14	scl(a	scl(a	NOUN
ejpam-4072	50	15	)	)	PUNCT
ejpam-4072	50	16	(	(	PUNCT
ejpam-4072	50	17	resp	resp	NOUN
ejpam-4072	50	18	.	.	PUNCT
ejpam-4072	51	1	τ1τ2	τ1τ2	NOUN
ejpam-4072	51	2	-	-	PUNCT
ejpam-4072	51	3	pcl(a	pcl(a	NUM
ejpam-4072	51	4	)	)	PUNCT
ejpam-4072	51	5	)	)	PUNCT
ejpam-4072	51	6	.	.	PUNCT
ejpam-4072	52	1	the	the	DET
ejpam-4072	52	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	52	3	-	-	ADJ
ejpam-4072	52	4	semi	semi	ADJ
ejpam-4072	52	5	-	-	ADJ
ejpam-4072	52	6	interior	interior	ADJ
ejpam-4072	52	7	[	[	X
ejpam-4072	52	8	5	5	NUM
ejpam-4072	52	9	]	]	PUNCT
ejpam-4072	52	10	(	(	PUNCT
ejpam-4072	52	11	resp	resp	NOUN
ejpam-4072	52	12	.	.	PUNCT
ejpam-4072	53	1	τ1τ2	τ1τ2	VERB
ejpam-4072	53	2	-	-	ADJ
ejpam-4072	53	3	preinterior	preinterior	ADJ
ejpam-4072	53	4	[	[	X
ejpam-4072	53	5	22	22	NUM
ejpam-4072	53	6	]	]	PUNCT
ejpam-4072	53	7	)	)	PUNCT
ejpam-4072	53	8	of	of	ADP
ejpam-4072	53	9	a	a	PRON
ejpam-4072	53	10	is	be	AUX
ejpam-4072	53	11	defined	define	VERB
ejpam-4072	53	12	by	by	ADP
ejpam-4072	53	13	the	the	DET
ejpam-4072	53	14	union	union	NOUN
ejpam-4072	53	15	of	of	ADP
ejpam-4072	53	16	τ1τ2	τ1τ2	NOUN
ejpam-4072	53	17	-	-	ADJ
ejpam-4072	53	18	semi	semi	ADJ
ejpam-4072	53	19	-	-	ADJ
ejpam-4072	53	20	open	open	ADJ
ejpam-4072	53	21	(	(	PUNCT
ejpam-4072	53	22	resp	resp	NOUN
ejpam-4072	53	23	.	.	PUNCT
ejpam-4072	54	1	τ1τ2preopen	τ1τ2preopen	PUNCT
ejpam-4072	54	2	)	)	PUNCT
ejpam-4072	55	1	sets	set	NOUN
ejpam-4072	55	2	contained	contain	VERB
ejpam-4072	55	3	in	in	ADP
ejpam-4072	55	4	a	a	PRON
ejpam-4072	55	5	and	and	CCONJ
ejpam-4072	55	6	is	be	AUX
ejpam-4072	55	7	denoted	denote	VERB
ejpam-4072	55	8	by	by	ADP
ejpam-4072	55	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	55	10	-	-	NOUN
ejpam-4072	55	11	sint(a	sint(a	NOUN
ejpam-4072	55	12	)	)	PUNCT
ejpam-4072	55	13	(	(	PUNCT
ejpam-4072	55	14	resp	resp	NOUN
ejpam-4072	55	15	.	.	PUNCT
ejpam-4072	56	1	τ1τ2	τ1τ2	NOUN
ejpam-4072	56	2	-	-	NOUN
ejpam-4072	56	3	pint(a	pint(a	NOUN
ejpam-4072	56	4	)	)	PUNCT
ejpam-4072	56	5	)	)	PUNCT
ejpam-4072	56	6	.	.	PUNCT
ejpam-4072	57	1	by	by	ADP
ejpam-4072	57	2	a	a	DET
ejpam-4072	57	3	multifunction	multifunction	NOUN
ejpam-4072	57	4	f	f	NOUN
ejpam-4072	57	5	:	:	PUNCT
ejpam-4072	57	6	x	x	X
ejpam-4072	57	7	→	→	SYM
ejpam-4072	57	8	y	y	PROPN
ejpam-4072	57	9	,	,	PUNCT
ejpam-4072	57	10	we	we	PRON
ejpam-4072	57	11	mean	mean	VERB
ejpam-4072	57	12	a	a	DET
ejpam-4072	57	13	point	point	NOUN
ejpam-4072	57	14	-	-	PUNCT
ejpam-4072	57	15	to	to	ADP
ejpam-4072	57	16	-	-	PUNCT
ejpam-4072	57	17	set	set	VERB
ejpam-4072	57	18	correspondence	correspondence	NOUN
ejpam-4072	57	19	from	from	ADP
ejpam-4072	57	20	x	x	PUNCT
ejpam-4072	57	21	into	into	ADP
ejpam-4072	57	22	y	y	PROPN
ejpam-4072	57	23	,	,	PUNCT
ejpam-4072	57	24	and	and	CCONJ
ejpam-4072	57	25	we	we	PRON
ejpam-4072	57	26	always	always	ADV
ejpam-4072	57	27	assume	assume	VERB
ejpam-4072	57	28	that	that	SCONJ
ejpam-4072	57	29	f	f	PROPN
ejpam-4072	57	30	(	(	PUNCT
ejpam-4072	57	31	x	x	X
ejpam-4072	57	32	)	)	PUNCT
ejpam-4072	57	33	̸=	̸=	NOUN
ejpam-4072	57	34	∅	∅	NOUN
ejpam-4072	57	35	for	for	ADP
ejpam-4072	57	36	all	all	PRON
ejpam-4072	57	37	x	x	SYM
ejpam-4072	57	38	∈	∈	ADJ
ejpam-4072	57	39	x.	x.	NOUN
ejpam-4072	57	40	for	for	ADP
ejpam-4072	57	41	a	a	DET
ejpam-4072	57	42	multifunction	multifunction	NOUN
ejpam-4072	57	43	f	f	NOUN
ejpam-4072	58	1	:	:	PUNCT
ejpam-4072	58	2	x	x	X
ejpam-4072	58	3	→	→	SYM
ejpam-4072	58	4	y	y	PROPN
ejpam-4072	58	5	,	,	PUNCT
ejpam-4072	58	6	following	follow	VERB
ejpam-4072	58	7	[	[	X
ejpam-4072	58	8	3	3	NUM
ejpam-4072	58	9	]	]	PUNCT
ejpam-4072	58	10	,	,	PUNCT
ejpam-4072	58	11	we	we	PRON
ejpam-4072	58	12	shall	shall	AUX
ejpam-4072	58	13	denote	denote	VERB
ejpam-4072	58	14	the	the	DET
ejpam-4072	58	15	upper	upper	ADJ
ejpam-4072	58	16	and	and	CCONJ
ejpam-4072	58	17	lower	low	ADJ
ejpam-4072	58	18	inverse	inverse	NOUN
ejpam-4072	58	19	of	of	ADP
ejpam-4072	58	20	a	a	DET
ejpam-4072	58	21	set	set	NOUN
ejpam-4072	58	22	b	b	PROPN
ejpam-4072	58	23	of	of	ADP
ejpam-4072	58	24	y	y	PROPN
ejpam-4072	58	25	by	by	ADP
ejpam-4072	58	26	f+(b	f+(b	NOUN
ejpam-4072	58	27	)	)	PUNCT
ejpam-4072	58	28	and	and	CCONJ
ejpam-4072	58	29	f−(b	f−(b	NOUN
ejpam-4072	58	30	)	)	PUNCT
ejpam-4072	58	31	,	,	PUNCT
ejpam-4072	58	32	respectively	respectively	ADV
ejpam-4072	58	33	,	,	PUNCT
ejpam-4072	58	34	that	that	ADV
ejpam-4072	58	35	is	is	ADV
ejpam-4072	58	36	,	,	PUNCT
ejpam-4072	58	37	f+(b	f+(b	NOUN
ejpam-4072	58	38	)	)	PUNCT
ejpam-4072	58	39	=	=	PRON
ejpam-4072	59	1	{	{	PUNCT
ejpam-4072	59	2	x	x	PUNCT
ejpam-4072	59	3	∈	∈	PROPN
ejpam-4072	59	4	x	x	INTJ
ejpam-4072	60	1	|	|	NOUN
ejpam-4072	60	2	f	f	X
ejpam-4072	60	3	(	(	PUNCT
ejpam-4072	60	4	x	x	NOUN
ejpam-4072	60	5	)	)	PUNCT
ejpam-4072	60	6	⊆	⊆	NUM
ejpam-4072	60	7	b	b	NOUN
ejpam-4072	60	8	}	}	PUNCT
ejpam-4072	60	9	and	and	CCONJ
ejpam-4072	60	10	f−(b	f−(b	PROPN
ejpam-4072	60	11	)	)	PUNCT
ejpam-4072	60	12	=	=	PRON
ejpam-4072	61	1	{	{	PUNCT
ejpam-4072	61	2	x	x	PUNCT
ejpam-4072	61	3	∈	∈	PROPN
ejpam-4072	61	4	x	x	INTJ
ejpam-4072	62	1	|	|	NOUN
ejpam-4072	62	2	f	f	X
ejpam-4072	62	3	(	(	PUNCT
ejpam-4072	62	4	x	x	NOUN
ejpam-4072	62	5	)	)	PUNCT
ejpam-4072	62	6	∩b	∩b	NOUN
ejpam-4072	62	7	̸=	̸=	PROPN
ejpam-4072	62	8	∅	∅	NOUN
ejpam-4072	62	9	}	}	PUNCT
ejpam-4072	62	10	.	.	PUNCT
ejpam-4072	63	1	in	in	ADP
ejpam-4072	63	2	particular	particular	ADJ
ejpam-4072	63	3	,	,	PUNCT
ejpam-4072	63	4	f−(y	f−(y	NOUN
ejpam-4072	63	5	)	)	PUNCT
ejpam-4072	63	6	=	=	SYM
ejpam-4072	64	1	{	{	PUNCT
ejpam-4072	64	2	x	x	PUNCT
ejpam-4072	64	3	∈	∈	PROPN
ejpam-4072	64	4	x	x	INTJ
ejpam-4072	65	1	|	|	ADV
ejpam-4072	65	2	y	y	PROPN
ejpam-4072	65	3	∈	∈	PROPN
ejpam-4072	65	4	f	f	X
ejpam-4072	65	5	(	(	PUNCT
ejpam-4072	65	6	x	x	NOUN
ejpam-4072	65	7	)	)	PUNCT
ejpam-4072	65	8	}	}	PUNCT
ejpam-4072	65	9	for	for	ADP
ejpam-4072	65	10	each	each	DET
ejpam-4072	65	11	point	point	NOUN
ejpam-4072	65	12	y	y	PROPN
ejpam-4072	65	13	∈	∈	PROPN
ejpam-4072	65	14	y	y	PROPN
ejpam-4072	65	15	.	.	PUNCT
ejpam-4072	66	1	lemma	lemma	PROPN
ejpam-4072	66	2	1	1	NUM
ejpam-4072	66	3	.	.	PUNCT
ejpam-4072	67	1	[	[	X
ejpam-4072	67	2	22	22	NUM
ejpam-4072	67	3	]	]	PUNCT
ejpam-4072	67	4	for	for	ADP
ejpam-4072	67	5	a	a	DET
ejpam-4072	67	6	subset	subset	NOUN
ejpam-4072	67	7	a	a	PRON
ejpam-4072	67	8	of	of	ADP
ejpam-4072	67	9	a	a	DET
ejpam-4072	67	10	bitopological	bitopological	ADJ
ejpam-4072	67	11	space	space	NOUN
ejpam-4072	67	12	(	(	PUNCT
ejpam-4072	67	13	x	x	NOUN
ejpam-4072	67	14	,	,	PUNCT
ejpam-4072	67	15	τ1	τ1	NOUN
ejpam-4072	67	16	,	,	PUNCT
ejpam-4072	67	17	τ2	τ2	NOUN
ejpam-4072	67	18	)	)	PUNCT
ejpam-4072	67	19	,	,	PUNCT
ejpam-4072	67	20	the	the	DET
ejpam-4072	67	21	following	follow	VERB
ejpam-4072	67	22	properties	property	NOUN
ejpam-4072	67	23	are	be	AUX
ejpam-4072	67	24	hold	hold	ADJ
ejpam-4072	67	25	:	:	PUNCT
ejpam-4072	67	26	(	(	PUNCT
ejpam-4072	67	27	1	1	X
ejpam-4072	67	28	)	)	PUNCT
ejpam-4072	67	29	τ1τ2	τ1τ2	NOUN
ejpam-4072	67	30	-	-	NOUN
ejpam-4072	67	31	pint(a	pint(a	NOUN
ejpam-4072	67	32	)	)	PUNCT
ejpam-4072	67	33	is	be	AUX
ejpam-4072	67	34	τ1τ2	τ1τ2	NOUN
ejpam-4072	67	35	-	-	ADJ
ejpam-4072	67	36	preopen	preopen	ADJ
ejpam-4072	67	37	.	.	PUNCT
ejpam-4072	68	1	(	(	PUNCT
ejpam-4072	68	2	2	2	X
ejpam-4072	68	3	)	)	PUNCT
ejpam-4072	68	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	68	5	-	-	PUNCT
ejpam-4072	68	6	pcl(a	pcl(a	NOUN
ejpam-4072	68	7	)	)	PUNCT
ejpam-4072	68	8	is	be	AUX
ejpam-4072	68	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	68	10	-	-	ADJ
ejpam-4072	68	11	preclosed	preclose	VERB
ejpam-4072	68	12	.	.	PUNCT
ejpam-4072	69	1	lemma	lemma	PROPN
ejpam-4072	69	2	2	2	NUM
ejpam-4072	69	3	.	.	PUNCT
ejpam-4072	70	1	[	[	X
ejpam-4072	70	2	22	22	NUM
ejpam-4072	70	3	]	]	PUNCT
ejpam-4072	70	4	for	for	ADP
ejpam-4072	70	5	a	a	DET
ejpam-4072	70	6	subset	subset	NOUN
ejpam-4072	70	7	a	a	PRON
ejpam-4072	70	8	of	of	ADP
ejpam-4072	70	9	a	a	DET
ejpam-4072	70	10	bitopological	bitopological	ADJ
ejpam-4072	70	11	space	space	NOUN
ejpam-4072	70	12	(	(	PUNCT
ejpam-4072	70	13	x	x	NOUN
ejpam-4072	70	14	,	,	PUNCT
ejpam-4072	70	15	τ1	τ1	NOUN
ejpam-4072	70	16	,	,	PUNCT
ejpam-4072	70	17	τ2	τ2	NOUN
ejpam-4072	70	18	)	)	PUNCT
ejpam-4072	70	19	,	,	PUNCT
ejpam-4072	70	20	x	x	PUNCT
ejpam-4072	70	21	∈	∈	PROPN
ejpam-4072	70	22	τ1τ2	τ1τ2	NOUN
ejpam-4072	70	23	-	-	ADJ
ejpam-4072	70	24	pcl(a	pcl(a	ADJ
ejpam-4072	70	25	)	)	PUNCT
ejpam-4072	70	26	if	if	SCONJ
ejpam-4072	70	27	and	and	CCONJ
ejpam-4072	70	28	only	only	ADV
ejpam-4072	70	29	if	if	SCONJ
ejpam-4072	70	30	u	u	PROPN
ejpam-4072	70	31	∩a	∩a	PROPN
ejpam-4072	70	32	̸=	̸=	PROPN
ejpam-4072	70	33	∅	∅	NOUN
ejpam-4072	70	34	for	for	ADP
ejpam-4072	70	35	every	every	DET
ejpam-4072	70	36	τ1τ2	τ1τ2	NOUN
ejpam-4072	70	37	-	-	ADJ
ejpam-4072	70	38	preopen	preopen	ADJ
ejpam-4072	70	39	set	set	NOUN
ejpam-4072	70	40	u	u	NOUN
ejpam-4072	70	41	containing	contain	VERB
ejpam-4072	70	42	x.	x.	PROPN
ejpam-4072	70	43	c.	c.	PROPN
ejpam-4072	70	44	boonpok	boonpok	PROPN
ejpam-4072	70	45	,	,	PUNCT
ejpam-4072	70	46	c.	c.	PROPN
ejpam-4072	70	47	viriyapong	viriyapong	PROPN
ejpam-4072	70	48	/	/	SYM
ejpam-4072	70	49	eur	eur	PROPN
ejpam-4072	70	50	.	.	PUNCT
ejpam-4072	71	1	j.	j.	PROPN
ejpam-4072	71	2	pure	pure	PROPN
ejpam-4072	71	3	appl	appl	PROPN
ejpam-4072	71	4	.	.	PROPN
ejpam-4072	71	5	math	math	PROPN
ejpam-4072	71	6	,	,	PUNCT
ejpam-4072	71	7	14	14	NUM
ejpam-4072	71	8	(	(	PUNCT
ejpam-4072	71	9	4	4	NUM
ejpam-4072	71	10	)	)	PUNCT
ejpam-4072	71	11	(	(	PUNCT
ejpam-4072	71	12	2021	2021	NUM
ejpam-4072	71	13	)	)	PUNCT
ejpam-4072	71	14	,	,	PUNCT
ejpam-4072	71	15	1212	1212	NUM
ejpam-4072	71	16	-	-	SYM
ejpam-4072	71	17	1225	1225	NUM
ejpam-4072	71	18	1214	1214	NUM
ejpam-4072	71	19	lemma	lemma	PROPN
ejpam-4072	71	20	3	3	X
ejpam-4072	71	21	.	.	PUNCT
ejpam-4072	72	1	[	[	X
ejpam-4072	72	2	22	22	NUM
ejpam-4072	72	3	]	]	PUNCT
ejpam-4072	72	4	for	for	ADP
ejpam-4072	72	5	a	a	DET
ejpam-4072	72	6	subset	subset	NOUN
ejpam-4072	72	7	a	a	PRON
ejpam-4072	72	8	of	of	ADP
ejpam-4072	72	9	a	a	DET
ejpam-4072	72	10	bitopological	bitopological	ADJ
ejpam-4072	72	11	space	space	NOUN
ejpam-4072	72	12	(	(	PUNCT
ejpam-4072	72	13	x	x	NOUN
ejpam-4072	72	14	,	,	PUNCT
ejpam-4072	72	15	τ1	τ1	NOUN
ejpam-4072	72	16	,	,	PUNCT
ejpam-4072	72	17	τ2	τ2	NOUN
ejpam-4072	72	18	)	)	PUNCT
ejpam-4072	72	19	,	,	PUNCT
ejpam-4072	72	20	the	the	DET
ejpam-4072	72	21	following	follow	VERB
ejpam-4072	72	22	properties	property	NOUN
ejpam-4072	72	23	are	be	AUX
ejpam-4072	72	24	hold	hold	ADJ
ejpam-4072	72	25	:	:	PUNCT
ejpam-4072	72	26	(	(	PUNCT
ejpam-4072	72	27	1	1	X
ejpam-4072	72	28	)	)	PUNCT
ejpam-4072	72	29	x	x	NOUN
ejpam-4072	73	1	−	−	ADP
ejpam-4072	73	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	73	3	-	-	NOUN
ejpam-4072	73	4	pint(a	pint(a	NOUN
ejpam-4072	73	5	)	)	PUNCT
ejpam-4072	73	6	=	=	PUNCT
ejpam-4072	73	7	τ1τ2	τ1τ2	X
ejpam-4072	73	8	-	-	ADJ
ejpam-4072	73	9	pcl(x	pcl(x	ADJ
ejpam-4072	73	10	−a	−a	NOUN
ejpam-4072	73	11	)	)	PUNCT
ejpam-4072	73	12	.	.	PUNCT
ejpam-4072	74	1	(	(	PUNCT
ejpam-4072	74	2	2	2	X
ejpam-4072	74	3	)	)	PUNCT
ejpam-4072	74	4	x	x	NOUN
ejpam-4072	75	1	−	−	ADP
ejpam-4072	75	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	75	3	-	-	ADJ
ejpam-4072	75	4	pcl(a	pcl(a	ADJ
ejpam-4072	75	5	)	)	PUNCT
ejpam-4072	75	6	=	=	PUNCT
ejpam-4072	76	1	τ1τ2	τ1τ2	ADJ
ejpam-4072	76	2	-	-	ADJ
ejpam-4072	76	3	pint(x	pint(x	ADJ
ejpam-4072	76	4	−a	−a	NOUN
ejpam-4072	76	5	)	)	PUNCT
ejpam-4072	76	6	.	.	PUNCT
ejpam-4072	77	1	a	a	DET
ejpam-4072	77	2	subset	subset	NOUN
ejpam-4072	77	3	a	a	PRON
ejpam-4072	77	4	of	of	ADP
ejpam-4072	77	5	a	a	DET
ejpam-4072	77	6	bitopological	bitopological	ADJ
ejpam-4072	77	7	space	space	NOUN
ejpam-4072	77	8	(	(	PUNCT
ejpam-4072	77	9	x	x	NOUN
ejpam-4072	77	10	,	,	PUNCT
ejpam-4072	77	11	τ1	τ1	NOUN
ejpam-4072	77	12	,	,	PUNCT
ejpam-4072	77	13	τ2	τ2	NOUN
ejpam-4072	77	14	)	)	PUNCT
ejpam-4072	77	15	is	be	AUX
ejpam-4072	77	16	said	say	VERB
ejpam-4072	77	17	to	to	PART
ejpam-4072	77	18	be	be	AUX
ejpam-4072	77	19	τ1τ2	τ1τ2	NOUN
ejpam-4072	77	20	-	-	ADJ
ejpam-4072	77	21	closed	closed	ADJ
ejpam-4072	77	22	[	[	X
ejpam-4072	77	23	5	5	NUM
ejpam-4072	77	24	]	]	PUNCT
ejpam-4072	77	25	if	if	SCONJ
ejpam-4072	77	26	a	a	DET
ejpam-4072	77	27	=	=	NOUN
ejpam-4072	77	28	τ1	τ1	NOUN
ejpam-4072	77	29	-	-	PUNCT
ejpam-4072	77	30	cl(τ2	cl(τ2	NOUN
ejpam-4072	77	31	-	-	PUNCT
ejpam-4072	77	32	cl(a	cl(a	NUM
ejpam-4072	77	33	)	)	PUNCT
ejpam-4072	77	34	)	)	PUNCT
ejpam-4072	77	35	.	.	PUNCT
ejpam-4072	78	1	the	the	DET
ejpam-4072	78	2	complement	complement	NOUN
ejpam-4072	78	3	of	of	ADP
ejpam-4072	78	4	a	a	DET
ejpam-4072	78	5	τ1τ2	τ1τ2	ADJ
ejpam-4072	78	6	-	-	ADJ
ejpam-4072	78	7	closed	closed	ADJ
ejpam-4072	78	8	set	set	NOUN
ejpam-4072	78	9	is	be	AUX
ejpam-4072	78	10	said	say	VERB
ejpam-4072	78	11	to	to	PART
ejpam-4072	78	12	be	be	AUX
ejpam-4072	78	13	τ1τ2	τ1τ2	NOUN
ejpam-4072	78	14	-	-	ADJ
ejpam-4072	78	15	open	open	ADJ
ejpam-4072	78	16	.	.	PUNCT
ejpam-4072	79	1	the	the	DET
ejpam-4072	79	2	intersection	intersection	NOUN
ejpam-4072	79	3	of	of	ADP
ejpam-4072	79	4	all	all	DET
ejpam-4072	79	5	τ1τ2	τ1τ2	ADJ
ejpam-4072	79	6	-	-	ADJ
ejpam-4072	79	7	closed	closed	ADJ
ejpam-4072	79	8	sets	set	NOUN
ejpam-4072	79	9	containing	contain	VERB
ejpam-4072	79	10	a	a	PRON
ejpam-4072	79	11	is	be	AUX
ejpam-4072	79	12	called	call	VERB
ejpam-4072	79	13	the	the	DET
ejpam-4072	79	14	τ1τ2	τ1τ2	NOUN
ejpam-4072	79	15	-	-	NOUN
ejpam-4072	79	16	closure	closure	NOUN
ejpam-4072	79	17	[	[	X
ejpam-4072	79	18	5	5	NUM
ejpam-4072	79	19	]	]	PUNCT
ejpam-4072	79	20	of	of	ADP
ejpam-4072	79	21	a	a	PRON
ejpam-4072	79	22	and	and	CCONJ
ejpam-4072	79	23	denoted	denote	VERB
ejpam-4072	79	24	by	by	ADP
ejpam-4072	79	25	τ1τ2	τ1τ2	NOUN
ejpam-4072	79	26	-	-	NUM
ejpam-4072	79	27	cl(a	cl(a	NUM
ejpam-4072	79	28	)	)	PUNCT
ejpam-4072	79	29	.	.	PUNCT
ejpam-4072	80	1	the	the	DET
ejpam-4072	80	2	union	union	NOUN
ejpam-4072	80	3	of	of	ADP
ejpam-4072	80	4	all	all	DET
ejpam-4072	80	5	τ1τ2	τ1τ2	ADJ
ejpam-4072	80	6	-	-	ADJ
ejpam-4072	80	7	open	open	ADJ
ejpam-4072	80	8	sets	set	NOUN
ejpam-4072	80	9	contained	contain	VERB
ejpam-4072	80	10	in	in	ADP
ejpam-4072	80	11	a	a	PRON
ejpam-4072	80	12	is	be	AUX
ejpam-4072	80	13	called	call	VERB
ejpam-4072	80	14	the	the	DET
ejpam-4072	80	15	τ1τ2	τ1τ2	NOUN
ejpam-4072	80	16	-	-	ADJ
ejpam-4072	80	17	interior	interior	ADJ
ejpam-4072	80	18	[	[	X
ejpam-4072	80	19	5	5	NUM
ejpam-4072	80	20	]	]	PUNCT
ejpam-4072	80	21	of	of	ADP
ejpam-4072	80	22	a	a	PRON
ejpam-4072	80	23	and	and	CCONJ
ejpam-4072	80	24	denoted	denote	VERB
ejpam-4072	80	25	by	by	ADP
ejpam-4072	80	26	τ1τ2	τ1τ2	NOUN
ejpam-4072	80	27	-	-	ADJ
ejpam-4072	80	28	int(a	int(a	NOUN
ejpam-4072	80	29	)	)	PUNCT
ejpam-4072	80	30	.	.	PUNCT
ejpam-4072	81	1	a	a	DET
ejpam-4072	81	2	subset	subset	NOUN
ejpam-4072	81	3	n	n	NOUN
ejpam-4072	81	4	of	of	ADP
ejpam-4072	81	5	a	a	DET
ejpam-4072	81	6	bitopological	bitopological	ADJ
ejpam-4072	81	7	space	space	NOUN
ejpam-4072	81	8	(	(	PUNCT
ejpam-4072	81	9	x	x	NOUN
ejpam-4072	81	10	,	,	PUNCT
ejpam-4072	81	11	τ1	τ1	NOUN
ejpam-4072	81	12	,	,	PUNCT
ejpam-4072	81	13	τ2	τ2	NOUN
ejpam-4072	81	14	)	)	PUNCT
ejpam-4072	81	15	is	be	AUX
ejpam-4072	81	16	said	say	VERB
ejpam-4072	81	17	to	to	PART
ejpam-4072	81	18	be	be	AUX
ejpam-4072	81	19	a	a	DET
ejpam-4072	81	20	τ1τ2	τ1τ2	NOUN
ejpam-4072	81	21	-	-	NOUN
ejpam-4072	81	22	neighbourhood	neighbourhood	NOUN
ejpam-4072	81	23	[	[	X
ejpam-4072	81	24	5	5	NUM
ejpam-4072	81	25	]	]	PUNCT
ejpam-4072	81	26	(	(	PUNCT
ejpam-4072	81	27	resp	resp	NOUN
ejpam-4072	81	28	.	.	PUNCT
ejpam-4072	82	1	τ1τ2	τ1τ2	NOUN
ejpam-4072	82	2	-	-	NOUN
ejpam-4072	82	3	preneighbourhood	preneighbourhood	ADJ
ejpam-4072	82	4	[	[	X
ejpam-4072	82	5	5	5	NUM
ejpam-4072	82	6	]	]	PUNCT
ejpam-4072	82	7	)	)	PUNCT
ejpam-4072	82	8	of	of	ADP
ejpam-4072	82	9	x	x	SYM
ejpam-4072	82	10	∈	∈	PROPN
ejpam-4072	82	11	x	x	INTJ
ejpam-4072	82	12	if	if	SCONJ
ejpam-4072	82	13	there	there	PRON
ejpam-4072	82	14	exists	exist	VERB
ejpam-4072	82	15	a	a	DET
ejpam-4072	82	16	τ1τ2	τ1τ2	NOUN
ejpam-4072	82	17	-	-	ADJ
ejpam-4072	82	18	open	open	ADJ
ejpam-4072	82	19	(	(	PUNCT
ejpam-4072	82	20	resp	resp	NOUN
ejpam-4072	82	21	.	.	PUNCT
ejpam-4072	83	1	τ1τ2	τ1τ2	ADJ
ejpam-4072	83	2	-	-	ADJ
ejpam-4072	83	3	preopen	preopen	ADJ
ejpam-4072	83	4	)	)	PUNCT
ejpam-4072	83	5	set	set	VERB
ejpam-4072	83	6	v	v	NUM
ejpam-4072	83	7	of	of	ADP
ejpam-4072	83	8	(	(	PUNCT
ejpam-4072	83	9	x	x	NOUN
ejpam-4072	83	10	,	,	PUNCT
ejpam-4072	83	11	τ1	τ1	NOUN
ejpam-4072	83	12	,	,	PUNCT
ejpam-4072	83	13	τ2	τ2	NOUN
ejpam-4072	83	14	)	)	PUNCT
ejpam-4072	83	15	such	such	ADJ
ejpam-4072	83	16	that	that	SCONJ
ejpam-4072	83	17	x	x	SYM
ejpam-4072	83	18	∈	∈	NOUN
ejpam-4072	83	19	v	v	ADP
ejpam-4072	83	20	⊆	⊆	NUM
ejpam-4072	83	21	n	n	NOUN
ejpam-4072	83	22	.	.	PUNCT
ejpam-4072	84	1	lemma	lemma	PROPN
ejpam-4072	84	2	4	4	NUM
ejpam-4072	84	3	.	.	PUNCT
ejpam-4072	85	1	[	[	X
ejpam-4072	85	2	5	5	X
ejpam-4072	85	3	]	]	PUNCT
ejpam-4072	85	4	let	let	VERB
ejpam-4072	85	5	a	a	PRON
ejpam-4072	85	6	and	and	CCONJ
ejpam-4072	85	7	b	b	NOUN
ejpam-4072	85	8	be	be	AUX
ejpam-4072	85	9	subsets	subset	NOUN
ejpam-4072	85	10	of	of	ADP
ejpam-4072	85	11	a	a	DET
ejpam-4072	85	12	bitopological	bitopological	ADJ
ejpam-4072	85	13	space	space	NOUN
ejpam-4072	85	14	(	(	PUNCT
ejpam-4072	85	15	x	x	NOUN
ejpam-4072	85	16	,	,	PUNCT
ejpam-4072	85	17	τ1	τ1	NOUN
ejpam-4072	85	18	,	,	PUNCT
ejpam-4072	85	19	τ2	τ2	NOUN
ejpam-4072	85	20	)	)	PUNCT
ejpam-4072	85	21	.	.	PUNCT
ejpam-4072	86	1	for	for	ADP
ejpam-4072	86	2	the	the	DET
ejpam-4072	86	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-4072	86	4	,	,	PUNCT
ejpam-4072	86	5	the	the	DET
ejpam-4072	86	6	following	follow	VERB
ejpam-4072	86	7	properties	property	NOUN
ejpam-4072	86	8	hold	hold	VERB
ejpam-4072	86	9	:	:	PUNCT
ejpam-4072	86	10	(	(	PUNCT
ejpam-4072	86	11	1	1	X
ejpam-4072	86	12	)	)	PUNCT
ejpam-4072	86	13	a	a	DET
ejpam-4072	86	14	⊆	⊆	NUM
ejpam-4072	86	15	τ1τ2	τ1τ2	NOUN
ejpam-4072	86	16	-	-	NUM
ejpam-4072	86	17	cl(a	cl(a	NUM
ejpam-4072	86	18	)	)	PUNCT
ejpam-4072	86	19	and	and	CCONJ
ejpam-4072	86	20	τ1τ2	τ1τ2	NOUN
ejpam-4072	86	21	-	-	ADJ
ejpam-4072	86	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-4072	86	23	-	-	PUNCT
ejpam-4072	86	24	cl(a	cl(a	NUM
ejpam-4072	86	25	)	)	PUNCT
ejpam-4072	86	26	)	)	PUNCT
ejpam-4072	87	1	=	=	PUNCT
ejpam-4072	87	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	87	3	-	-	NUM
ejpam-4072	87	4	cl(a	cl(a	NUM
ejpam-4072	87	5	)	)	PUNCT
ejpam-4072	87	6	.	.	PUNCT
ejpam-4072	88	1	(	(	PUNCT
ejpam-4072	88	2	2	2	X
ejpam-4072	88	3	)	)	PUNCT
ejpam-4072	88	4	if	if	SCONJ
ejpam-4072	88	5	a	a	DET
ejpam-4072	88	6	⊆	⊆	NUM
ejpam-4072	88	7	b	b	NOUN
ejpam-4072	88	8	,	,	PUNCT
ejpam-4072	88	9	then	then	ADV
ejpam-4072	88	10	τ1τ2	τ1τ2	NOUN
ejpam-4072	88	11	-	-	NUM
ejpam-4072	88	12	cl(a	cl(a	NUM
ejpam-4072	88	13	)	)	PUNCT
ejpam-4072	88	14	⊆	⊆	NUM
ejpam-4072	88	15	τ1τ2	τ1τ2	NOUN
ejpam-4072	88	16	-	-	NOUN
ejpam-4072	88	17	cl(b	cl(b	NOUN
ejpam-4072	88	18	)	)	PUNCT
ejpam-4072	88	19	.	.	PUNCT
ejpam-4072	89	1	(	(	PUNCT
ejpam-4072	89	2	3	3	X
ejpam-4072	89	3	)	)	PUNCT
ejpam-4072	89	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	89	5	-	-	NUM
ejpam-4072	89	6	cl(a	cl(a	NUM
ejpam-4072	89	7	)	)	PUNCT
ejpam-4072	89	8	is	be	AUX
ejpam-4072	89	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	89	10	-	-	ADJ
ejpam-4072	89	11	closed	closed	ADJ
ejpam-4072	89	12	.	.	PUNCT
ejpam-4072	90	1	(	(	PUNCT
ejpam-4072	90	2	4	4	X
ejpam-4072	90	3	)	)	PUNCT
ejpam-4072	90	4	a	a	PRON
ejpam-4072	90	5	is	be	AUX
ejpam-4072	90	6	τ1τ2	τ1τ2	NOUN
ejpam-4072	90	7	-	-	ADJ
ejpam-4072	90	8	closed	closed	ADJ
ejpam-4072	90	9	if	if	SCONJ
ejpam-4072	90	10	and	and	CCONJ
ejpam-4072	90	11	only	only	ADV
ejpam-4072	90	12	if	if	SCONJ
ejpam-4072	90	13	a	a	DET
ejpam-4072	90	14	=	=	PUNCT
ejpam-4072	90	15	τ1τ2	τ1τ2	NOUN
ejpam-4072	90	16	-	-	NUM
ejpam-4072	90	17	cl(a	cl(a	NUM
ejpam-4072	90	18	)	)	PUNCT
ejpam-4072	90	19	.	.	PUNCT
ejpam-4072	91	1	(	(	PUNCT
ejpam-4072	91	2	5	5	X
ejpam-4072	91	3	)	)	PUNCT
ejpam-4072	91	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	91	5	-	-	NOUN
ejpam-4072	91	6	cl(x	cl(x	X
ejpam-4072	91	7	−a	−a	NOUN
ejpam-4072	91	8	)	)	PUNCT
ejpam-4072	92	1	=	=	PUNCT
ejpam-4072	92	2	x	x	X
ejpam-4072	93	1	−	−	ADP
ejpam-4072	93	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	93	3	-	-	ADJ
ejpam-4072	93	4	int(a	int(a	NOUN
ejpam-4072	93	5	)	)	PUNCT
ejpam-4072	93	6	.	.	PUNCT
ejpam-4072	94	1	lemma	lemma	PROPN
ejpam-4072	94	2	5	5	NUM
ejpam-4072	94	3	.	.	PUNCT
ejpam-4072	95	1	[	[	X
ejpam-4072	95	2	1	1	X
ejpam-4072	95	3	]	]	PUNCT
ejpam-4072	95	4	for	for	ADP
ejpam-4072	95	5	a	a	DET
ejpam-4072	95	6	subset	subset	NOUN
ejpam-4072	95	7	a	a	PRON
ejpam-4072	95	8	of	of	ADP
ejpam-4072	95	9	a	a	DET
ejpam-4072	95	10	topological	topological	ADJ
ejpam-4072	95	11	space	space	NOUN
ejpam-4072	95	12	(	(	PUNCT
ejpam-4072	95	13	x	x	X
ejpam-4072	95	14	,	,	PUNCT
ejpam-4072	95	15	τ	τ	PROPN
ejpam-4072	95	16	)	)	PUNCT
ejpam-4072	95	17	,	,	PUNCT
ejpam-4072	95	18	the	the	DET
ejpam-4072	95	19	following	follow	VERB
ejpam-4072	95	20	properties	property	NOUN
ejpam-4072	95	21	hold	hold	VERB
ejpam-4072	95	22	:	:	PUNCT
ejpam-4072	95	23	(	(	PUNCT
ejpam-4072	95	24	1	1	NUM
ejpam-4072	95	25	)	)	PUNCT
ejpam-4072	95	26	cl(a	cl(a	NUM
ejpam-4072	95	27	)	)	PUNCT
ejpam-4072	95	28	∩g	∩g	NOUN
ejpam-4072	95	29	⊆	⊆	NUM
ejpam-4072	95	30	cl(a	cl(a	X
ejpam-4072	95	31	∩g	∩g	NOUN
ejpam-4072	95	32	)	)	PUNCT
ejpam-4072	95	33	for	for	ADP
ejpam-4072	95	34	every	every	DET
ejpam-4072	95	35	open	open	ADJ
ejpam-4072	95	36	set	set	VERB
ejpam-4072	95	37	g.	g.	NOUN
ejpam-4072	95	38	(	(	PUNCT
ejpam-4072	95	39	2	2	NUM
ejpam-4072	95	40	)	)	PUNCT
ejpam-4072	95	41	int(a	int(a	NOUN
ejpam-4072	95	42	∪	∪	PROPN
ejpam-4072	95	43	f	f	PROPN
ejpam-4072	95	44	)	)	PUNCT
ejpam-4072	95	45	⊆	⊆	NUM
ejpam-4072	95	46	int(a	int(a	NOUN
ejpam-4072	95	47	)	)	PUNCT
ejpam-4072	95	48	∪	∪	NOUN
ejpam-4072	95	49	f	f	PROPN
ejpam-4072	95	50	for	for	ADP
ejpam-4072	95	51	every	every	DET
ejpam-4072	95	52	closed	close	VERB
ejpam-4072	95	53	set	set	VERB
ejpam-4072	95	54	f	f	PROPN
ejpam-4072	95	55	.	.	PUNCT
ejpam-4072	96	1	lemma	lemma	PROPN
ejpam-4072	96	2	6	6	NUM
ejpam-4072	96	3	.	.	PUNCT
ejpam-4072	97	1	for	for	ADP
ejpam-4072	97	2	a	a	DET
ejpam-4072	97	3	subset	subset	NOUN
ejpam-4072	97	4	a	a	PRON
ejpam-4072	97	5	of	of	ADP
ejpam-4072	97	6	a	a	DET
ejpam-4072	97	7	bitopological	bitopological	ADJ
ejpam-4072	97	8	space	space	NOUN
ejpam-4072	97	9	(	(	PUNCT
ejpam-4072	97	10	x	x	NOUN
ejpam-4072	97	11	,	,	PUNCT
ejpam-4072	97	12	τ1	τ1	NOUN
ejpam-4072	97	13	,	,	PUNCT
ejpam-4072	97	14	τ2	τ2	NOUN
ejpam-4072	97	15	)	)	PUNCT
ejpam-4072	97	16	,	,	PUNCT
ejpam-4072	97	17	the	the	DET
ejpam-4072	97	18	following	follow	VERB
ejpam-4072	97	19	properties	property	NOUN
ejpam-4072	97	20	hold	hold	VERB
ejpam-4072	97	21	:	:	PUNCT
ejpam-4072	97	22	(	(	PUNCT
ejpam-4072	97	23	1	1	X
ejpam-4072	97	24	)	)	PUNCT
ejpam-4072	97	25	τ1τ2	τ1τ2	NOUN
ejpam-4072	97	26	-	-	ADJ
ejpam-4072	97	27	pcl(a	pcl(a	ADJ
ejpam-4072	97	28	)	)	PUNCT
ejpam-4072	97	29	=	=	NOUN
ejpam-4072	97	30	a	a	DET
ejpam-4072	97	31	∪	∪	ADJ
ejpam-4072	97	32	τ1	τ1	NOUN
ejpam-4072	97	33	-	-	PUNCT
ejpam-4072	97	34	cl(τ2	cl(τ2	NOUN
ejpam-4072	97	35	-	-	PUNCT
ejpam-4072	97	36	int(a	int(a	NOUN
ejpam-4072	97	37	)	)	PUNCT
ejpam-4072	97	38	)	)	PUNCT
ejpam-4072	97	39	.	.	PUNCT
ejpam-4072	98	1	(	(	PUNCT
ejpam-4072	98	2	2	2	X
ejpam-4072	98	3	)	)	PUNCT
ejpam-4072	98	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	98	5	-	-	NOUN
ejpam-4072	98	6	pint(a	pint(a	NOUN
ejpam-4072	98	7	)	)	PUNCT
ejpam-4072	98	8	=	=	PUNCT
ejpam-4072	99	1	a	a	DET
ejpam-4072	99	2	∩	∩	ADJ
ejpam-4072	99	3	τ1	τ1	NOUN
ejpam-4072	99	4	-	-	PUNCT
ejpam-4072	99	5	int(τ2	int(τ2	NOUN
ejpam-4072	99	6	-	-	PUNCT
ejpam-4072	99	7	cl(a	cl(a	NUM
ejpam-4072	99	8	)	)	PUNCT
ejpam-4072	99	9	)	)	PUNCT
ejpam-4072	99	10	.	.	PUNCT
ejpam-4072	100	1	proof	proof	NOUN
ejpam-4072	100	2	.	.	PUNCT
ejpam-4072	101	1	(	(	PUNCT
ejpam-4072	101	2	1	1	X
ejpam-4072	101	3	)	)	PUNCT
ejpam-4072	101	4	to	to	PART
ejpam-4072	101	5	begin	begin	VERB
ejpam-4072	101	6	with	with	ADP
ejpam-4072	101	7	,	,	PUNCT
ejpam-4072	101	8	observe	observe	VERB
ejpam-4072	101	9	that	that	SCONJ
ejpam-4072	101	10	τ1	τ1	NOUN
ejpam-4072	101	11	-	-	PUNCT
ejpam-4072	101	12	cl(τ2	cl(τ2	NOUN
ejpam-4072	101	13	-	-	PUNCT
ejpam-4072	101	14	int(a	int(a	NOUN
ejpam-4072	101	15	∪	∪	NOUN
ejpam-4072	101	16	τ1	τ1	NOUN
ejpam-4072	101	17	-	-	PUNCT
ejpam-4072	101	18	cl(τ2	cl(τ2	NOUN
ejpam-4072	101	19	-	-	PUNCT
ejpam-4072	101	20	int(a	int(a	NOUN
ejpam-4072	101	21	)	)	PUNCT
ejpam-4072	101	22	)	)	PUNCT
ejpam-4072	101	23	)	)	PUNCT
ejpam-4072	101	24	)	)	PUNCT
ejpam-4072	102	1	⊆	⊆	NUM
ejpam-4072	102	2	τ1	τ1	NOUN
ejpam-4072	102	3	-	-	PUNCT
ejpam-4072	102	4	cl(τ2	cl(τ2	NOUN
ejpam-4072	102	5	-	-	PUNCT
ejpam-4072	102	6	int(a	int(a	NOUN
ejpam-4072	102	7	)	)	PUNCT
ejpam-4072	102	8	∪	∪	ADP
ejpam-4072	102	9	τ1	τ1	NOUN
ejpam-4072	102	10	-	-	PUNCT
ejpam-4072	102	11	cl(τ2	cl(τ2	NOUN
ejpam-4072	102	12	-	-	PUNCT
ejpam-4072	102	13	int(a	int(a	NOUN
ejpam-4072	102	14	)	)	PUNCT
ejpam-4072	102	15	)	)	PUNCT
ejpam-4072	102	16	)	)	PUNCT
ejpam-4072	103	1	=	=	SYM
ejpam-4072	103	2	τ1	τ1	NOUN
ejpam-4072	103	3	-	-	PUNCT
ejpam-4072	103	4	cl(τ1	cl(τ1	NOUN
ejpam-4072	103	5	-	-	PUNCT
ejpam-4072	103	6	cl(τ2	cl(τ2	NOUN
ejpam-4072	103	7	-	-	PUNCT
ejpam-4072	103	8	int(a	int(a	NOUN
ejpam-4072	103	9	)	)	PUNCT
ejpam-4072	103	10	)	)	PUNCT
ejpam-4072	103	11	)	)	PUNCT
ejpam-4072	104	1	=	=	SYM
ejpam-4072	104	2	τ1	τ1	NOUN
ejpam-4072	104	3	-	-	PUNCT
ejpam-4072	104	4	cl(τ2	cl(τ2	NOUN
ejpam-4072	104	5	-	-	PUNCT
ejpam-4072	104	6	int(a	int(a	NOUN
ejpam-4072	104	7	)	)	PUNCT
ejpam-4072	104	8	)	)	PUNCT
ejpam-4072	105	1	⊆	⊆	ADP
ejpam-4072	105	2	a	a	DET
ejpam-4072	105	3	∪	∪	ADJ
ejpam-4072	105	4	τ1	τ1	NOUN
ejpam-4072	105	5	-	-	PUNCT
ejpam-4072	105	6	cl(τ2	cl(τ2	NOUN
ejpam-4072	105	7	-	-	PUNCT
ejpam-4072	105	8	int(a	int(a	NOUN
ejpam-4072	105	9	)	)	PUNCT
ejpam-4072	105	10	)	)	PUNCT
ejpam-4072	105	11	c.	c.	PROPN
ejpam-4072	105	12	boonpok	boonpok	PROPN
ejpam-4072	105	13	,	,	PUNCT
ejpam-4072	105	14	c.	c.	PROPN
ejpam-4072	105	15	viriyapong	viriyapong	PROPN
ejpam-4072	105	16	/	/	SYM
ejpam-4072	105	17	eur	eur	PROPN
ejpam-4072	105	18	.	.	PUNCT
ejpam-4072	106	1	j.	j.	PROPN
ejpam-4072	106	2	pure	pure	PROPN
ejpam-4072	106	3	appl	appl	PROPN
ejpam-4072	106	4	.	.	PROPN
ejpam-4072	106	5	math	math	PROPN
ejpam-4072	106	6	,	,	PUNCT
ejpam-4072	106	7	14	14	NUM
ejpam-4072	106	8	(	(	PUNCT
ejpam-4072	106	9	4	4	NUM
ejpam-4072	106	10	)	)	PUNCT
ejpam-4072	106	11	(	(	PUNCT
ejpam-4072	106	12	2021	2021	NUM
ejpam-4072	106	13	)	)	PUNCT
ejpam-4072	106	14	,	,	PUNCT
ejpam-4072	106	15	1212	1212	NUM
ejpam-4072	106	16	-	-	SYM
ejpam-4072	106	17	1225	1225	NUM
ejpam-4072	106	18	1215	1215	NUM
ejpam-4072	106	19	by	by	ADP
ejpam-4072	106	20	lemma	lemma	PROPN
ejpam-4072	106	21	5(2	5(2	NUM
ejpam-4072	106	22	)	)	PUNCT
ejpam-4072	106	23	.	.	PUNCT
ejpam-4072	107	1	hence	hence	ADV
ejpam-4072	107	2	,	,	PUNCT
ejpam-4072	107	3	a	a	DET
ejpam-4072	107	4	∪	∪	ADJ
ejpam-4072	107	5	τ1	τ1	NOUN
ejpam-4072	107	6	-	-	PUNCT
ejpam-4072	107	7	cl(τ2	cl(τ2	NOUN
ejpam-4072	107	8	-	-	PUNCT
ejpam-4072	107	9	int(a	int(a	NOUN
ejpam-4072	107	10	)	)	PUNCT
ejpam-4072	107	11	)	)	PUNCT
ejpam-4072	107	12	is	be	AUX
ejpam-4072	107	13	τ1τ2	τ1τ2	VERB
ejpam-4072	107	14	-	-	ADJ
ejpam-4072	107	15	preclosed	preclose	VERB
ejpam-4072	107	16	and	and	CCONJ
ejpam-4072	107	17	thus	thus	ADV
ejpam-4072	107	18	τ1τ2	τ1τ2	NOUN
ejpam-4072	107	19	-	-	ADJ
ejpam-4072	107	20	pcl(a	pcl(a	ADJ
ejpam-4072	107	21	)	)	PUNCT
ejpam-4072	107	22	⊆	⊆	NUM
ejpam-4072	107	23	a	a	DET
ejpam-4072	107	24	∪	∪	ADJ
ejpam-4072	107	25	τ1	τ1	NOUN
ejpam-4072	107	26	-	-	PUNCT
ejpam-4072	107	27	cl(τ2	cl(τ2	NOUN
ejpam-4072	107	28	-	-	PUNCT
ejpam-4072	107	29	int(a	int(a	NOUN
ejpam-4072	107	30	)	)	PUNCT
ejpam-4072	107	31	)	)	PUNCT
ejpam-4072	107	32	.	.	PUNCT
ejpam-4072	108	1	on	on	ADP
ejpam-4072	108	2	the	the	DET
ejpam-4072	108	3	other	other	ADJ
ejpam-4072	108	4	hand	hand	NOUN
ejpam-4072	108	5	,	,	PUNCT
ejpam-4072	108	6	since	since	SCONJ
ejpam-4072	108	7	τ1τ2	τ1τ2	NOUN
ejpam-4072	108	8	-	-	ADJ
ejpam-4072	108	9	pcl(a	pcl(a	ADJ
ejpam-4072	108	10	)	)	PUNCT
ejpam-4072	108	11	is	be	AUX
ejpam-4072	108	12	τ1τ2	τ1τ2	NOUN
ejpam-4072	108	13	-	-	ADJ
ejpam-4072	108	14	preclosed	preclosed	ADJ
ejpam-4072	108	15	,	,	PUNCT
ejpam-4072	108	16	we	we	PRON
ejpam-4072	108	17	have	have	VERB
ejpam-4072	108	18	τ1	τ1	NOUN
ejpam-4072	108	19	-	-	PUNCT
ejpam-4072	108	20	cl(τ2	cl(τ2	NOUN
ejpam-4072	108	21	-	-	PUNCT
ejpam-4072	108	22	int(a	int(a	NOUN
ejpam-4072	108	23	)	)	PUNCT
ejpam-4072	108	24	)	)	PUNCT
ejpam-4072	109	1	⊆	⊆	NUM
ejpam-4072	109	2	τ1	τ1	NOUN
ejpam-4072	109	3	-	-	PUNCT
ejpam-4072	109	4	cl(τ2	cl(τ2	NOUN
ejpam-4072	109	5	-	-	PUNCT
ejpam-4072	109	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-4072	109	7	-	-	PUNCT
ejpam-4072	109	8	pcl(a	pcl(a	NOUN
ejpam-4072	109	9	)	)	PUNCT
ejpam-4072	109	10	)	)	PUNCT
ejpam-4072	109	11	)	)	PUNCT
ejpam-4072	110	1	⊆	⊆	X
ejpam-4072	110	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	110	3	-	-	ADJ
ejpam-4072	110	4	pcl(a	pcl(a	ADJ
ejpam-4072	110	5	)	)	PUNCT
ejpam-4072	110	6	and	and	CCONJ
ejpam-4072	110	7	so	so	ADV
ejpam-4072	110	8	a	a	DET
ejpam-4072	110	9	∪	∪	ADJ
ejpam-4072	110	10	τ1	τ1	NOUN
ejpam-4072	110	11	-	-	PUNCT
ejpam-4072	110	12	cl(τ2	cl(τ2	NOUN
ejpam-4072	110	13	-	-	PUNCT
ejpam-4072	110	14	int(a	int(a	NOUN
ejpam-4072	110	15	)	)	PUNCT
ejpam-4072	110	16	)	)	PUNCT
ejpam-4072	111	1	⊆	⊆	X
ejpam-4072	111	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	111	3	-	-	ADJ
ejpam-4072	111	4	pcl(a	pcl(a	NOUN
ejpam-4072	111	5	)	)	PUNCT
ejpam-4072	111	6	.	.	PUNCT
ejpam-4072	112	1	consequently	consequently	ADV
ejpam-4072	112	2	,	,	PUNCT
ejpam-4072	112	3	we	we	PRON
ejpam-4072	112	4	obtain	obtain	VERB
ejpam-4072	112	5	a	a	DET
ejpam-4072	112	6	∪	∪	ADJ
ejpam-4072	112	7	τ1	τ1	NOUN
ejpam-4072	112	8	-	-	PUNCT
ejpam-4072	112	9	cl(τ2	cl(τ2	NOUN
ejpam-4072	112	10	-	-	PUNCT
ejpam-4072	112	11	int(a	int(a	NOUN
ejpam-4072	112	12	)	)	PUNCT
ejpam-4072	112	13	)	)	PUNCT
ejpam-4072	113	1	=	=	PUNCT
ejpam-4072	113	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	113	3	-	-	ADJ
ejpam-4072	113	4	pcl(a	pcl(a	NOUN
ejpam-4072	113	5	)	)	PUNCT
ejpam-4072	113	6	.	.	PUNCT
ejpam-4072	114	1	(	(	PUNCT
ejpam-4072	114	2	2	2	X
ejpam-4072	114	3	)	)	PUNCT
ejpam-4072	114	4	this	this	PRON
ejpam-4072	114	5	follows	follow	VERB
ejpam-4072	114	6	from	from	ADP
ejpam-4072	114	7	(	(	PUNCT
ejpam-4072	114	8	1	1	NUM
ejpam-4072	114	9	)	)	PUNCT
ejpam-4072	114	10	.	.	PUNCT
ejpam-4072	115	1	3	3	X
ejpam-4072	115	2	.	.	X
ejpam-4072	115	3	characterizations	characterization	NOUN
ejpam-4072	115	4	in	in	ADP
ejpam-4072	115	5	this	this	DET
ejpam-4072	115	6	section	section	NOUN
ejpam-4072	115	7	,	,	PUNCT
ejpam-4072	115	8	we	we	PRON
ejpam-4072	115	9	introduce	introduce	VERB
ejpam-4072	115	10	the	the	DET
ejpam-4072	115	11	notions	notion	NOUN
ejpam-4072	115	12	of	of	ADP
ejpam-4072	115	13	upper	upper	ADJ
ejpam-4072	115	14	and	and	CCONJ
ejpam-4072	115	15	lower	low	ADJ
ejpam-4072	115	16	almost	almost	ADV
ejpam-4072	115	17	weakly	weakly	ADJ
ejpam-4072	115	18	(	(	PUNCT
ejpam-4072	115	19	τ1	τ1	NOUN
ejpam-4072	115	20	,	,	PUNCT
ejpam-4072	115	21	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	115	22	multifunctions	multifunction	NOUN
ejpam-4072	115	23	.	.	PUNCT
ejpam-4072	116	1	moreover	moreover	ADV
ejpam-4072	116	2	,	,	PUNCT
ejpam-4072	116	3	some	some	DET
ejpam-4072	116	4	characterizations	characterization	NOUN
ejpam-4072	116	5	of	of	ADP
ejpam-4072	116	6	upper	upper	ADJ
ejpam-4072	116	7	and	and	CCONJ
ejpam-4072	116	8	lower	low	ADJ
ejpam-4072	116	9	almost	almost	ADV
ejpam-4072	116	10	weakly	weakly	ADJ
ejpam-4072	116	11	(	(	PUNCT
ejpam-4072	116	12	τ1	τ1	NOUN
ejpam-4072	116	13	,	,	PUNCT
ejpam-4072	116	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	116	15	multifunctions	multifunction	NOUN
ejpam-4072	116	16	are	be	AUX
ejpam-4072	116	17	discussed	discuss	VERB
ejpam-4072	116	18	.	.	PUNCT
ejpam-4072	117	1	definition	definition	NOUN
ejpam-4072	117	2	1	1	NUM
ejpam-4072	117	3	.	.	PUNCT
ejpam-4072	118	1	a	a	DET
ejpam-4072	118	2	multifunction	multifunction	NOUN
ejpam-4072	118	3	f	f	NOUN
ejpam-4072	118	4	:	:	PUNCT
ejpam-4072	118	5	(	(	PUNCT
ejpam-4072	118	6	x	x	NOUN
ejpam-4072	118	7	,	,	PUNCT
ejpam-4072	118	8	τ1	τ1	NOUN
ejpam-4072	118	9	,	,	PUNCT
ejpam-4072	118	10	τ2	τ2	NOUN
ejpam-4072	118	11	)	)	PUNCT
ejpam-4072	118	12	→	→	SYM
ejpam-4072	118	13	(	(	PUNCT
ejpam-4072	118	14	y	y	PROPN
ejpam-4072	118	15	,	,	PUNCT
ejpam-4072	118	16	σ1	σ1	PROPN
ejpam-4072	118	17	,	,	PUNCT
ejpam-4072	118	18	σ2	σ2	PROPN
ejpam-4072	118	19	)	)	PUNCT
ejpam-4072	118	20	is	be	AUX
ejpam-4072	118	21	said	say	VERB
ejpam-4072	118	22	to	to	PART
ejpam-4072	118	23	be	be	AUX
ejpam-4072	118	24	:	:	PUNCT
ejpam-4072	118	25	(	(	PUNCT
ejpam-4072	118	26	1	1	X
ejpam-4072	118	27	)	)	PUNCT
ejpam-4072	118	28	upper	upper	ADJ
ejpam-4072	118	29	almost	almost	ADV
ejpam-4072	118	30	weakly	weakly	ADJ
ejpam-4072	118	31	(	(	PUNCT
ejpam-4072	118	32	τ1	τ1	NOUN
ejpam-4072	118	33	,	,	PUNCT
ejpam-4072	118	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	118	35	if	if	SCONJ
ejpam-4072	118	36	for	for	ADP
ejpam-4072	118	37	each	each	DET
ejpam-4072	118	38	x	x	SYM
ejpam-4072	118	39	∈	∈	PROPN
ejpam-4072	118	40	x	x	X
ejpam-4072	118	41	and	and	CCONJ
ejpam-4072	118	42	each	each	DET
ejpam-4072	118	43	σ1σ2	σ1σ2	VERB
ejpam-4072	118	44	-	-	ADJ
ejpam-4072	118	45	open	open	ADJ
ejpam-4072	118	46	set	set	NOUN
ejpam-4072	118	47	v	v	NOUN
ejpam-4072	118	48	of	of	ADP
ejpam-4072	118	49	y	y	PRON
ejpam-4072	118	50	such	such	ADJ
ejpam-4072	118	51	that	that	SCONJ
ejpam-4072	118	52	f	f	PROPN
ejpam-4072	118	53	(	(	PUNCT
ejpam-4072	118	54	x	x	X
ejpam-4072	118	55	)	)	PUNCT
ejpam-4072	118	56	⊆	⊆	NUM
ejpam-4072	118	57	v	v	NOUN
ejpam-4072	118	58	,	,	PUNCT
ejpam-4072	118	59	x	x	SYM
ejpam-4072	118	60	∈	∈	PROPN
ejpam-4072	118	61	τ1	τ1	NOUN
ejpam-4072	118	62	-	-	PUNCT
ejpam-4072	118	63	int(τ2	int(τ2	NOUN
ejpam-4072	118	64	-	-	PUNCT
ejpam-4072	118	65	cl(f	cl(f	NOUN
ejpam-4072	118	66	+	+	NOUN
ejpam-4072	118	67	(	(	PUNCT
ejpam-4072	118	68	σ1σ2	σ1σ2	NOUN
ejpam-4072	118	69	-	-	NUM
ejpam-4072	118	70	cl(v	cl(v	NOUN
ejpam-4072	118	71	)	)	PUNCT
ejpam-4072	118	72	)	)	PUNCT
ejpam-4072	118	73	)	)	PUNCT
ejpam-4072	118	74	)	)	PUNCT
ejpam-4072	118	75	;	;	PUNCT
ejpam-4072	118	76	(	(	PUNCT
ejpam-4072	118	77	2	2	X
ejpam-4072	118	78	)	)	PUNCT
ejpam-4072	118	79	lower	low	ADJ
ejpam-4072	118	80	almost	almost	ADV
ejpam-4072	118	81	weakly	weakly	ADJ
ejpam-4072	118	82	(	(	PUNCT
ejpam-4072	118	83	τ1	τ1	NOUN
ejpam-4072	118	84	,	,	PUNCT
ejpam-4072	118	85	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	118	86	if	if	SCONJ
ejpam-4072	118	87	for	for	ADP
ejpam-4072	118	88	each	each	DET
ejpam-4072	118	89	x	x	SYM
ejpam-4072	118	90	∈	∈	PROPN
ejpam-4072	118	91	x	x	X
ejpam-4072	118	92	and	and	CCONJ
ejpam-4072	118	93	each	each	DET
ejpam-4072	118	94	σ1σ2	σ1σ2	VERB
ejpam-4072	118	95	-	-	ADJ
ejpam-4072	118	96	open	open	ADJ
ejpam-4072	118	97	set	set	NOUN
ejpam-4072	118	98	v	v	NOUN
ejpam-4072	118	99	of	of	ADP
ejpam-4072	118	100	y	y	PRON
ejpam-4072	118	101	such	such	ADJ
ejpam-4072	118	102	that	that	SCONJ
ejpam-4072	118	103	f	f	PROPN
ejpam-4072	118	104	(	(	PUNCT
ejpam-4072	118	105	x	x	NOUN
ejpam-4072	118	106	)	)	PUNCT
ejpam-4072	118	107	∩	∩	NOUN
ejpam-4072	118	108	v	v	ADP
ejpam-4072	118	109	̸=	̸=	PROPN
ejpam-4072	118	110	∅	∅	NOUN
ejpam-4072	118	111	,	,	PUNCT
ejpam-4072	118	112	x	x	PROPN
ejpam-4072	118	113	∈	∈	PROPN
ejpam-4072	118	114	τ1	τ1	NOUN
ejpam-4072	118	115	-	-	PUNCT
ejpam-4072	118	116	int(τ2	int(τ2	NOUN
ejpam-4072	118	117	-	-	PUNCT
ejpam-4072	118	118	cl(f	cl(f	NOUN
ejpam-4072	118	119	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	118	120	-	-	NOUN
ejpam-4072	118	121	cl(v	cl(v	NOUN
ejpam-4072	118	122	)	)	PUNCT
ejpam-4072	118	123	)	)	PUNCT
ejpam-4072	118	124	)	)	PUNCT
ejpam-4072	118	125	)	)	PUNCT
ejpam-4072	118	126	.	.	PUNCT
ejpam-4072	119	1	theorem	theorem	NOUN
ejpam-4072	119	2	1	1	NUM
ejpam-4072	119	3	.	.	X
ejpam-4072	119	4	for	for	ADP
ejpam-4072	119	5	a	a	DET
ejpam-4072	119	6	multifunction	multifunction	NOUN
ejpam-4072	120	1	f	f	NOUN
ejpam-4072	120	2	:	:	PUNCT
ejpam-4072	120	3	(	(	PUNCT
ejpam-4072	120	4	x	x	NOUN
ejpam-4072	120	5	,	,	PUNCT
ejpam-4072	120	6	τ1	τ1	NOUN
ejpam-4072	120	7	,	,	PUNCT
ejpam-4072	120	8	τ2	τ2	NOUN
ejpam-4072	120	9	)	)	PUNCT
ejpam-4072	120	10	→	→	SYM
ejpam-4072	120	11	(	(	PUNCT
ejpam-4072	120	12	y	y	PROPN
ejpam-4072	120	13	,	,	PUNCT
ejpam-4072	120	14	σ1	σ1	PROPN
ejpam-4072	120	15	,	,	PUNCT
ejpam-4072	120	16	σ2	σ2	NOUN
ejpam-4072	120	17	)	)	PUNCT
ejpam-4072	120	18	,	,	PUNCT
ejpam-4072	120	19	the	the	DET
ejpam-4072	120	20	following	follow	VERB
ejpam-4072	120	21	properties	property	NOUN
ejpam-4072	120	22	are	be	AUX
ejpam-4072	120	23	equivalent	equivalent	ADJ
ejpam-4072	120	24	:	:	PUNCT
ejpam-4072	120	25	(	(	PUNCT
ejpam-4072	120	26	1	1	X
ejpam-4072	120	27	)	)	PUNCT
ejpam-4072	120	28	f	f	PROPN
ejpam-4072	120	29	is	be	AUX
ejpam-4072	120	30	upper	upper	ADJ
ejpam-4072	120	31	almost	almost	ADV
ejpam-4072	120	32	weakly	weakly	ADJ
ejpam-4072	120	33	(	(	PUNCT
ejpam-4072	120	34	τ1	τ1	NOUN
ejpam-4072	120	35	,	,	PUNCT
ejpam-4072	120	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	120	37	;	;	PUNCT
ejpam-4072	120	38	(	(	PUNCT
ejpam-4072	120	39	2	2	NUM
ejpam-4072	120	40	)	)	PUNCT
ejpam-4072	120	41	f+(v	f+(v	NOUN
ejpam-4072	120	42	)	)	PUNCT
ejpam-4072	121	1	⊆	⊆	NUM
ejpam-4072	121	2	τ1	τ1	NOUN
ejpam-4072	121	3	-	-	PUNCT
ejpam-4072	121	4	int(τ2	int(τ2	NOUN
ejpam-4072	121	5	-	-	PUNCT
ejpam-4072	121	6	cl(f	cl(f	NOUN
ejpam-4072	121	7	+	+	NOUN
ejpam-4072	121	8	(	(	PUNCT
ejpam-4072	121	9	σ1σ2	σ1σ2	NOUN
ejpam-4072	121	10	-	-	NUM
ejpam-4072	121	11	cl(v	cl(v	NOUN
ejpam-4072	121	12	)	)	PUNCT
ejpam-4072	121	13	)	)	PUNCT
ejpam-4072	121	14	)	)	PUNCT
ejpam-4072	121	15	)	)	PUNCT
ejpam-4072	122	1	for	for	ADP
ejpam-4072	122	2	every	every	DET
ejpam-4072	122	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	122	4	-	-	ADJ
ejpam-4072	122	5	open	open	ADJ
ejpam-4072	122	6	set	set	NOUN
ejpam-4072	122	7	v	v	NOUN
ejpam-4072	122	8	of	of	ADP
ejpam-4072	122	9	y	y	PROPN
ejpam-4072	122	10	;	;	PUNCT
ejpam-4072	122	11	(	(	PUNCT
ejpam-4072	122	12	3	3	X
ejpam-4072	122	13	)	)	PUNCT
ejpam-4072	122	14	τ1	τ1	NOUN
ejpam-4072	122	15	-	-	PUNCT
ejpam-4072	122	16	cl(τ2	cl(τ2	NOUN
ejpam-4072	122	17	-	-	PUNCT
ejpam-4072	122	18	int(f	int(f	PROPN
ejpam-4072	122	19	−(v	−(v	NOUN
ejpam-4072	122	20	)	)	PUNCT
ejpam-4072	122	21	)	)	PUNCT
ejpam-4072	122	22	)	)	PUNCT
ejpam-4072	123	1	⊆	⊆	NUM
ejpam-4072	123	2	f−(σ1σ2cl(v	f−(σ1σ2cl(v	NOUN
ejpam-4072	123	3	)	)	PUNCT
ejpam-4072	123	4	)	)	PUNCT
ejpam-4072	123	5	for	for	ADP
ejpam-4072	123	6	every	every	DET
ejpam-4072	123	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	123	8	-	-	ADJ
ejpam-4072	123	9	open	open	ADJ
ejpam-4072	123	10	set	set	NOUN
ejpam-4072	123	11	v	v	NOUN
ejpam-4072	123	12	of	of	ADP
ejpam-4072	123	13	y	y	PROPN
ejpam-4072	123	14	;	;	PUNCT
ejpam-4072	123	15	(	(	PUNCT
ejpam-4072	123	16	4	4	X
ejpam-4072	123	17	)	)	PUNCT
ejpam-4072	123	18	τ1τ2	τ1τ2	NOUN
ejpam-4072	123	19	-	-	PROPN
ejpam-4072	123	20	pcl(f	pcl(f	PROPN
ejpam-4072	123	21	−(v	−(v	NOUN
ejpam-4072	123	22	)	)	PUNCT
ejpam-4072	123	23	)	)	PUNCT
ejpam-4072	124	1	⊆	⊆	X
ejpam-4072	124	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	124	3	-	-	PUNCT
ejpam-4072	124	4	cl(v	cl(v	NOUN
ejpam-4072	124	5	)	)	PUNCT
ejpam-4072	124	6	)	)	PUNCT
ejpam-4072	124	7	for	for	ADP
ejpam-4072	124	8	every	every	DET
ejpam-4072	124	9	σ1σ2	σ1σ2	NOUN
ejpam-4072	124	10	-	-	ADJ
ejpam-4072	124	11	open	open	ADJ
ejpam-4072	124	12	set	set	NOUN
ejpam-4072	124	13	v	v	NOUN
ejpam-4072	124	14	of	of	ADP
ejpam-4072	124	15	y	y	PROPN
ejpam-4072	124	16	;	;	PUNCT
ejpam-4072	124	17	(	(	PUNCT
ejpam-4072	124	18	5	5	NUM
ejpam-4072	124	19	)	)	PUNCT
ejpam-4072	124	20	f+(v	f+(v	NOUN
ejpam-4072	124	21	)	)	PUNCT
ejpam-4072	125	1	⊆	⊆	X
ejpam-4072	125	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	125	3	-	-	NOUN
ejpam-4072	125	4	pint(f	pint(f	ADJ
ejpam-4072	125	5	+	+	PROPN
ejpam-4072	125	6	(	(	PUNCT
ejpam-4072	125	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	125	8	-	-	NUM
ejpam-4072	125	9	cl(v	cl(v	NOUN
ejpam-4072	125	10	)	)	PUNCT
ejpam-4072	125	11	)	)	PUNCT
ejpam-4072	125	12	)	)	PUNCT
ejpam-4072	125	13	for	for	ADP
ejpam-4072	125	14	every	every	DET
ejpam-4072	125	15	σ1σ2	σ1σ2	NOUN
ejpam-4072	125	16	-	-	ADJ
ejpam-4072	125	17	open	open	ADJ
ejpam-4072	125	18	set	set	NOUN
ejpam-4072	125	19	v	v	NOUN
ejpam-4072	125	20	of	of	ADP
ejpam-4072	125	21	y	y	PROPN
ejpam-4072	125	22	;	;	PUNCT
ejpam-4072	125	23	(	(	PUNCT
ejpam-4072	125	24	6	6	NUM
ejpam-4072	125	25	)	)	PUNCT
ejpam-4072	125	26	for	for	ADP
ejpam-4072	125	27	each	each	DET
ejpam-4072	125	28	x	x	SYM
ejpam-4072	125	29	∈	∈	PROPN
ejpam-4072	125	30	x	x	X
ejpam-4072	125	31	and	and	CCONJ
ejpam-4072	125	32	each	each	DET
ejpam-4072	125	33	σ1σ2	σ1σ2	VERB
ejpam-4072	125	34	-	-	ADJ
ejpam-4072	125	35	open	open	ADJ
ejpam-4072	125	36	set	set	NOUN
ejpam-4072	125	37	v	v	NOUN
ejpam-4072	125	38	of	of	ADP
ejpam-4072	125	39	y	y	PROPN
ejpam-4072	125	40	containing	contain	VERB
ejpam-4072	125	41	f	f	PROPN
ejpam-4072	125	42	(	(	PUNCT
ejpam-4072	125	43	x	x	NOUN
ejpam-4072	125	44	)	)	PUNCT
ejpam-4072	125	45	,	,	PUNCT
ejpam-4072	125	46	there	there	PRON
ejpam-4072	125	47	exists	exist	VERB
ejpam-4072	125	48	a	a	DET
ejpam-4072	125	49	τ1τ2	τ1τ2	NOUN
ejpam-4072	125	50	-	-	ADJ
ejpam-4072	125	51	preopen	preopen	ADJ
ejpam-4072	125	52	set	set	NOUN
ejpam-4072	125	53	u	u	NOUN
ejpam-4072	125	54	of	of	ADP
ejpam-4072	125	55	x	x	PUNCT
ejpam-4072	125	56	containing	contain	VERB
ejpam-4072	125	57	x	x	PUNCT
ejpam-4072	125	58	such	such	ADJ
ejpam-4072	125	59	that	that	SCONJ
ejpam-4072	125	60	f	f	PROPN
ejpam-4072	125	61	(	(	PUNCT
ejpam-4072	125	62	u	u	NOUN
ejpam-4072	125	63	)	)	PUNCT
ejpam-4072	125	64	⊆	⊆	NUM
ejpam-4072	125	65	σ1σ2	σ1σ2	NOUN
ejpam-4072	125	66	-	-	NUM
ejpam-4072	125	67	cl(v	cl(v	NOUN
ejpam-4072	125	68	)	)	PUNCT
ejpam-4072	125	69	.	.	PUNCT
ejpam-4072	126	1	proof	proof	NOUN
ejpam-4072	126	2	.	.	PUNCT
ejpam-4072	127	1	(	(	PUNCT
ejpam-4072	127	2	1	1	X
ejpam-4072	127	3	)	)	PUNCT
ejpam-4072	127	4	⇒	⇒	NOUN
ejpam-4072	127	5	(	(	PUNCT
ejpam-4072	127	6	2	2	NUM
ejpam-4072	127	7	):	):	PUNCT
ejpam-4072	127	8	let	let	VERB
ejpam-4072	127	9	v	v	PART
ejpam-4072	127	10	be	be	AUX
ejpam-4072	127	11	any	any	DET
ejpam-4072	127	12	σ1σ2	σ1σ2	NOUN
ejpam-4072	127	13	-	-	ADJ
ejpam-4072	127	14	open	open	ADJ
ejpam-4072	127	15	set	set	NOUN
ejpam-4072	127	16	of	of	ADP
ejpam-4072	127	17	y	y	PROPN
ejpam-4072	127	18	and	and	CCONJ
ejpam-4072	127	19	x	x	PROPN
ejpam-4072	127	20	∈	∈	PROPN
ejpam-4072	127	21	f+(v	f+(v	NOUN
ejpam-4072	127	22	)	)	PUNCT
ejpam-4072	127	23	.	.	PUNCT
ejpam-4072	128	1	then	then	ADV
ejpam-4072	128	2	,	,	PUNCT
ejpam-4072	128	3	f	f	PROPN
ejpam-4072	128	4	(	(	PUNCT
ejpam-4072	128	5	x	x	X
ejpam-4072	128	6	)	)	PUNCT
ejpam-4072	128	7	⊆	⊆	NUM
ejpam-4072	128	8	v	v	NOUN
ejpam-4072	128	9	and	and	CCONJ
ejpam-4072	128	10	by	by	ADP
ejpam-4072	128	11	(	(	PUNCT
ejpam-4072	128	12	1	1	NUM
ejpam-4072	128	13	)	)	PUNCT
ejpam-4072	128	14	,	,	PUNCT
ejpam-4072	128	15	we	we	PRON
ejpam-4072	128	16	have	have	VERB
ejpam-4072	128	17	x	x	NOUN
ejpam-4072	128	18	∈	∈	PROPN
ejpam-4072	128	19	τ1	τ1	NOUN
ejpam-4072	128	20	-	-	PUNCT
ejpam-4072	128	21	int(τ2	int(τ2	NOUN
ejpam-4072	128	22	-	-	PUNCT
ejpam-4072	128	23	cl(f	cl(f	NOUN
ejpam-4072	128	24	+	+	NOUN
ejpam-4072	128	25	(	(	PUNCT
ejpam-4072	128	26	σ1σ2	σ1σ2	NOUN
ejpam-4072	128	27	-	-	NUM
ejpam-4072	128	28	cl(v	cl(v	NOUN
ejpam-4072	128	29	)	)	PUNCT
ejpam-4072	128	30	)	)	PUNCT
ejpam-4072	128	31	)	)	PUNCT
ejpam-4072	128	32	)	)	PUNCT
ejpam-4072	128	33	.	.	PUNCT
ejpam-4072	129	1	therefore	therefore	ADV
ejpam-4072	129	2	,	,	PUNCT
ejpam-4072	129	3	f+(v	f+(v	PROPN
ejpam-4072	129	4	)	)	PUNCT
ejpam-4072	129	5	⊆	⊆	NUM
ejpam-4072	129	6	τ1	τ1	NOUN
ejpam-4072	129	7	-	-	PUNCT
ejpam-4072	129	8	int(τ2	int(τ2	NOUN
ejpam-4072	129	9	-	-	PUNCT
ejpam-4072	129	10	cl(f	cl(f	NOUN
ejpam-4072	129	11	+	+	NOUN
ejpam-4072	129	12	(	(	PUNCT
ejpam-4072	129	13	σ1σ2	σ1σ2	NOUN
ejpam-4072	129	14	-	-	NUM
ejpam-4072	129	15	cl(v	cl(v	NOUN
ejpam-4072	129	16	)	)	PUNCT
ejpam-4072	129	17	)	)	PUNCT
ejpam-4072	129	18	)	)	PUNCT
ejpam-4072	129	19	)	)	PUNCT
ejpam-4072	129	20	.	.	PUNCT
ejpam-4072	130	1	c.	c.	PROPN
ejpam-4072	130	2	boonpok	boonpok	PROPN
ejpam-4072	130	3	,	,	PUNCT
ejpam-4072	130	4	c.	c.	PROPN
ejpam-4072	130	5	viriyapong	viriyapong	PROPN
ejpam-4072	130	6	/	/	SYM
ejpam-4072	130	7	eur	eur	PROPN
ejpam-4072	130	8	.	.	PUNCT
ejpam-4072	131	1	j.	j.	PROPN
ejpam-4072	131	2	pure	pure	PROPN
ejpam-4072	131	3	appl	appl	PROPN
ejpam-4072	131	4	.	.	PROPN
ejpam-4072	131	5	math	math	PROPN
ejpam-4072	131	6	,	,	PUNCT
ejpam-4072	131	7	14	14	NUM
ejpam-4072	131	8	(	(	PUNCT
ejpam-4072	131	9	4	4	NUM
ejpam-4072	131	10	)	)	PUNCT
ejpam-4072	131	11	(	(	PUNCT
ejpam-4072	131	12	2021	2021	NUM
ejpam-4072	131	13	)	)	PUNCT
ejpam-4072	131	14	,	,	PUNCT
ejpam-4072	131	15	1212	1212	NUM
ejpam-4072	131	16	-	-	SYM
ejpam-4072	131	17	1225	1225	NUM
ejpam-4072	131	18	1216	1216	NUM
ejpam-4072	131	19	(	(	PUNCT
ejpam-4072	131	20	2	2	NUM
ejpam-4072	131	21	)	)	PUNCT
ejpam-4072	131	22	⇒	⇒	NOUN
ejpam-4072	131	23	(	(	PUNCT
ejpam-4072	131	24	3	3	NUM
ejpam-4072	131	25	):	):	PUNCT
ejpam-4072	131	26	let	let	VERB
ejpam-4072	131	27	v	v	PART
ejpam-4072	131	28	be	be	AUX
ejpam-4072	131	29	any	any	DET
ejpam-4072	131	30	σ1σ2	σ1σ2	NOUN
ejpam-4072	131	31	-	-	ADJ
ejpam-4072	131	32	open	open	ADJ
ejpam-4072	131	33	set	set	NOUN
ejpam-4072	131	34	of	of	ADP
ejpam-4072	131	35	y	y	PROPN
ejpam-4072	131	36	.	.	PUNCT
ejpam-4072	132	1	since	since	SCONJ
ejpam-4072	132	2	y	y	PROPN
ejpam-4072	132	3	−	−	PROPN
ejpam-4072	132	4	σ1σ2	σ1σ2	NOUN
ejpam-4072	132	5	-	-	NUM
ejpam-4072	132	6	cl(v	cl(v	NOUN
ejpam-4072	132	7	)	)	PUNCT
ejpam-4072	132	8	is	be	AUX
ejpam-4072	132	9	σ1σ2	σ1σ2	NOUN
ejpam-4072	132	10	-	-	ADJ
ejpam-4072	132	11	open	open	ADJ
ejpam-4072	132	12	and	and	CCONJ
ejpam-4072	132	13	by	by	ADP
ejpam-4072	132	14	(	(	PUNCT
ejpam-4072	132	15	2	2	NUM
ejpam-4072	132	16	)	)	PUNCT
ejpam-4072	132	17	,	,	PUNCT
ejpam-4072	132	18	x	x	PUNCT
ejpam-4072	132	19	−	−	ADP
ejpam-4072	132	20	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	132	21	-	-	PUNCT
ejpam-4072	132	22	cl(v	cl(v	NOUN
ejpam-4072	132	23	)	)	PUNCT
ejpam-4072	132	24	)	)	PUNCT
ejpam-4072	133	1	=	=	PUNCT
ejpam-4072	134	1	f+(y	f+(y	NOUN
ejpam-4072	134	2	−	−	NUM
ejpam-4072	134	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	134	4	-	-	NUM
ejpam-4072	134	5	cl(v	cl(v	NOUN
ejpam-4072	134	6	)	)	PUNCT
ejpam-4072	134	7	)	)	PUNCT
ejpam-4072	135	1	⊆	⊆	NUM
ejpam-4072	135	2	τ1	τ1	NOUN
ejpam-4072	135	3	-	-	PUNCT
ejpam-4072	135	4	int(τ2	int(τ2	NOUN
ejpam-4072	135	5	-	-	PUNCT
ejpam-4072	135	6	cl(f	cl(f	NOUN
ejpam-4072	135	7	+	+	NOUN
ejpam-4072	135	8	(	(	PUNCT
ejpam-4072	135	9	σ1σ2	σ1σ2	NUM
ejpam-4072	135	10	-	-	PUNCT
ejpam-4072	135	11	cl(y	cl(y	NOUN
ejpam-4072	135	12	−	−	NOUN
ejpam-4072	135	13	σ1σ2	σ1σ2	NOUN
ejpam-4072	135	14	-	-	NUM
ejpam-4072	135	15	cl(v	cl(v	NOUN
ejpam-4072	135	16	)	)	PUNCT
ejpam-4072	135	17	)	)	PUNCT
ejpam-4072	135	18	)	)	PUNCT
ejpam-4072	135	19	)	)	PUNCT
ejpam-4072	135	20	)	)	PUNCT
ejpam-4072	136	1	⊆	⊆	NUM
ejpam-4072	136	2	τ1	τ1	NOUN
ejpam-4072	136	3	-	-	PUNCT
ejpam-4072	136	4	int(τ2	int(τ2	NOUN
ejpam-4072	136	5	-	-	PUNCT
ejpam-4072	136	6	cl(f	cl(f	NOUN
ejpam-4072	136	7	+	+	PROPN
ejpam-4072	136	8	(	(	PUNCT
ejpam-4072	136	9	y	y	PROPN
ejpam-4072	136	10	−	−	PROPN
ejpam-4072	136	11	v	v	NOUN
ejpam-4072	136	12	)	)	PUNCT
ejpam-4072	136	13	)	)	PUNCT
ejpam-4072	136	14	)	)	PUNCT
ejpam-4072	137	1	=	=	SYM
ejpam-4072	137	2	τ1	τ1	NOUN
ejpam-4072	137	3	-	-	PUNCT
ejpam-4072	137	4	int(τ2	int(τ2	NOUN
ejpam-4072	137	5	-	-	PUNCT
ejpam-4072	137	6	cl(x	cl(x	PUNCT
ejpam-4072	137	7	−	−	PROPN
ejpam-4072	137	8	f−(v	f−(v	PROPN
ejpam-4072	137	9	)	)	PUNCT
ejpam-4072	137	10	)	)	PUNCT
ejpam-4072	137	11	)	)	PUNCT
ejpam-4072	138	1	=	=	PUNCT
ejpam-4072	138	2	x	x	X
ejpam-4072	139	1	−	−	NOUN
ejpam-4072	139	2	τ1	τ1	NOUN
ejpam-4072	139	3	-	-	PUNCT
ejpam-4072	139	4	cl(τ2	cl(τ2	NOUN
ejpam-4072	139	5	-	-	PUNCT
ejpam-4072	139	6	int(f	int(f	PROPN
ejpam-4072	139	7	−(v	−(v	NOUN
ejpam-4072	139	8	)	)	PUNCT
ejpam-4072	139	9	)	)	PUNCT
ejpam-4072	139	10	)	)	PUNCT
ejpam-4072	139	11	.	.	PUNCT
ejpam-4072	140	1	thus	thus	ADV
ejpam-4072	140	2	,	,	PUNCT
ejpam-4072	140	3	τ1	τ1	NOUN
ejpam-4072	140	4	-	-	PUNCT
ejpam-4072	140	5	cl(τ2	cl(τ2	NOUN
ejpam-4072	140	6	-	-	PUNCT
ejpam-4072	140	7	int(f	int(f	PROPN
ejpam-4072	140	8	−(v	−(v	NOUN
ejpam-4072	140	9	)	)	PUNCT
ejpam-4072	140	10	)	)	PUNCT
ejpam-4072	140	11	)	)	PUNCT
ejpam-4072	141	1	⊆	⊆	X
ejpam-4072	141	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	141	3	-	-	PUNCT
ejpam-4072	141	4	cl(v	cl(v	NOUN
ejpam-4072	141	5	)	)	PUNCT
ejpam-4072	141	6	)	)	PUNCT
ejpam-4072	141	7	.	.	PUNCT
ejpam-4072	142	1	(	(	PUNCT
ejpam-4072	142	2	3	3	X
ejpam-4072	142	3	)	)	PUNCT
ejpam-4072	142	4	⇒	⇒	NOUN
ejpam-4072	142	5	(	(	PUNCT
ejpam-4072	142	6	4	4	NUM
ejpam-4072	142	7	):	):	PUNCT
ejpam-4072	142	8	let	let	VERB
ejpam-4072	142	9	v	v	PART
ejpam-4072	142	10	be	be	AUX
ejpam-4072	142	11	any	any	DET
ejpam-4072	142	12	σ1σ2	σ1σ2	NOUN
ejpam-4072	142	13	-	-	ADJ
ejpam-4072	142	14	open	open	ADJ
ejpam-4072	142	15	set	set	NOUN
ejpam-4072	142	16	of	of	ADP
ejpam-4072	142	17	y	y	PROPN
ejpam-4072	142	18	.	.	PUNCT
ejpam-4072	143	1	by	by	ADP
ejpam-4072	143	2	(	(	PUNCT
ejpam-4072	143	3	3	3	NUM
ejpam-4072	143	4	)	)	PUNCT
ejpam-4072	143	5	and	and	CCONJ
ejpam-4072	143	6	lemma	lemma	PROPN
ejpam-4072	143	7	6(1	6(1	NUM
ejpam-4072	143	8	)	)	PUNCT
ejpam-4072	143	9	,	,	PUNCT
ejpam-4072	143	10	τ1τ2	τ1τ2	NOUN
ejpam-4072	143	11	-	-	PROPN
ejpam-4072	143	12	pcl(f	pcl(f	PROPN
ejpam-4072	143	13	−(v	−(v	NOUN
ejpam-4072	143	14	)	)	PUNCT
ejpam-4072	143	15	)	)	PUNCT
ejpam-4072	144	1	=	=	SYM
ejpam-4072	144	2	f−(v	f−(v	ADJ
ejpam-4072	144	3	)	)	PUNCT
ejpam-4072	144	4	∪	∪	ADP
ejpam-4072	144	5	τ1	τ1	NOUN
ejpam-4072	144	6	-	-	PUNCT
ejpam-4072	144	7	cl(τ2	cl(τ2	NOUN
ejpam-4072	144	8	-	-	PUNCT
ejpam-4072	144	9	int(f	int(f	PROPN
ejpam-4072	144	10	−(v	−(v	NOUN
ejpam-4072	144	11	)	)	PUNCT
ejpam-4072	144	12	)	)	PUNCT
ejpam-4072	145	1	⊆	⊆	X
ejpam-4072	145	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	145	3	-	-	PUNCT
ejpam-4072	145	4	cl(v	cl(v	NOUN
ejpam-4072	145	5	)	)	PUNCT
ejpam-4072	145	6	)	)	PUNCT
ejpam-4072	145	7	.	.	PUNCT
ejpam-4072	146	1	(	(	PUNCT
ejpam-4072	146	2	4	4	X
ejpam-4072	146	3	)	)	PUNCT
ejpam-4072	146	4	⇒	⇒	NOUN
ejpam-4072	146	5	(	(	PUNCT
ejpam-4072	146	6	5	5	NUM
ejpam-4072	146	7	):	):	PUNCT
ejpam-4072	146	8	let	let	VERB
ejpam-4072	146	9	v	v	PART
ejpam-4072	146	10	be	be	AUX
ejpam-4072	146	11	any	any	DET
ejpam-4072	146	12	σ1σ2	σ1σ2	NOUN
ejpam-4072	146	13	-	-	ADJ
ejpam-4072	146	14	open	open	ADJ
ejpam-4072	146	15	set	set	NOUN
ejpam-4072	146	16	of	of	ADP
ejpam-4072	146	17	y	y	PROPN
ejpam-4072	146	18	.	.	PUNCT
ejpam-4072	147	1	then	then	ADV
ejpam-4072	147	2	,	,	PUNCT
ejpam-4072	147	3	y	y	PROPN
ejpam-4072	147	4	−	−	NUM
ejpam-4072	147	5	σ1σ2	σ1σ2	NOUN
ejpam-4072	147	6	-	-	NUM
ejpam-4072	147	7	cl(v	cl(v	NOUN
ejpam-4072	147	8	)	)	PUNCT
ejpam-4072	147	9	is	be	AUX
ejpam-4072	147	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	147	11	-	-	ADJ
ejpam-4072	147	12	open	open	ADJ
ejpam-4072	147	13	and	and	CCONJ
ejpam-4072	147	14	by	by	ADP
ejpam-4072	147	15	(	(	PUNCT
ejpam-4072	147	16	4	4	NUM
ejpam-4072	147	17	)	)	PUNCT
ejpam-4072	147	18	,	,	PUNCT
ejpam-4072	147	19	x	x	X
ejpam-4072	147	20	−	−	ADP
ejpam-4072	147	21	τ1τ2	τ1τ2	NOUN
ejpam-4072	147	22	-	-	NOUN
ejpam-4072	147	23	pint(f	pint(f	ADJ
ejpam-4072	147	24	+	+	PROPN
ejpam-4072	147	25	(	(	PUNCT
ejpam-4072	147	26	σ1σ2	σ1σ2	NOUN
ejpam-4072	147	27	-	-	NUM
ejpam-4072	147	28	cl(v	cl(v	NOUN
ejpam-4072	147	29	)	)	PUNCT
ejpam-4072	147	30	)	)	PUNCT
ejpam-4072	147	31	)	)	PUNCT
ejpam-4072	148	1	=	=	PUNCT
ejpam-4072	148	2	τ1τ2	τ1τ2	X
ejpam-4072	148	3	-	-	ADJ
ejpam-4072	148	4	pcl(x	pcl(x	ADJ
ejpam-4072	148	5	−	−	NOUN
ejpam-4072	148	6	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	148	7	-	-	PUNCT
ejpam-4072	148	8	cl(v	cl(v	NOUN
ejpam-4072	148	9	)	)	PUNCT
ejpam-4072	148	10	)	)	PUNCT
ejpam-4072	148	11	)	)	PUNCT
ejpam-4072	149	1	=	=	PUNCT
ejpam-4072	150	1	τ1τ2	τ1τ2	ADJ
ejpam-4072	150	2	-	-	PROPN
ejpam-4072	150	3	pcl(f	pcl(f	PROPN
ejpam-4072	150	4	−(y	−(y	NOUN
ejpam-4072	150	5	−	−	NOUN
ejpam-4072	150	6	σ1σ2	σ1σ2	NOUN
ejpam-4072	150	7	-	-	NUM
ejpam-4072	150	8	cl(v	cl(v	NOUN
ejpam-4072	150	9	)	)	PUNCT
ejpam-4072	150	10	)	)	PUNCT
ejpam-4072	150	11	)	)	PUNCT
ejpam-4072	151	1	⊆	⊆	X
ejpam-4072	151	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-4072	151	3	-	-	PUNCT
ejpam-4072	151	4	cl(y	cl(y	NOUN
ejpam-4072	151	5	−	−	NOUN
ejpam-4072	151	6	σ1σ2	σ1σ2	NOUN
ejpam-4072	151	7	-	-	NUM
ejpam-4072	151	8	cl(v	cl(v	NOUN
ejpam-4072	151	9	)	)	PUNCT
ejpam-4072	151	10	)	)	PUNCT
ejpam-4072	151	11	)	)	PUNCT
ejpam-4072	152	1	⊆	⊆	NUM
ejpam-4072	152	2	f−(y	f−(y	NOUN
ejpam-4072	152	3	−	−	NOUN
ejpam-4072	152	4	v	v	NOUN
ejpam-4072	152	5	)	)	PUNCT
ejpam-4072	152	6	=	=	PUNCT
ejpam-4072	152	7	x	x	X
ejpam-4072	152	8	−	−	PROPN
ejpam-4072	152	9	f+(v	f+(v	NOUN
ejpam-4072	152	10	)	)	PUNCT
ejpam-4072	152	11	.	.	PUNCT
ejpam-4072	153	1	thus	thus	ADV
ejpam-4072	153	2	,	,	PUNCT
ejpam-4072	153	3	f+(v	f+(v	PROPN
ejpam-4072	153	4	)	)	PUNCT
ejpam-4072	154	1	⊆	⊆	X
ejpam-4072	154	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	154	3	-	-	NOUN
ejpam-4072	154	4	pint(f	pint(f	ADJ
ejpam-4072	154	5	+	+	PROPN
ejpam-4072	154	6	(	(	PUNCT
ejpam-4072	154	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	154	8	-	-	NUM
ejpam-4072	154	9	cl(v	cl(v	NOUN
ejpam-4072	154	10	)	)	PUNCT
ejpam-4072	154	11	)	)	PUNCT
ejpam-4072	154	12	)	)	PUNCT
ejpam-4072	154	13	.	.	PUNCT
ejpam-4072	155	1	(	(	PUNCT
ejpam-4072	155	2	5	5	X
ejpam-4072	155	3	)	)	PUNCT
ejpam-4072	155	4	⇒	⇒	NOUN
ejpam-4072	155	5	(	(	PUNCT
ejpam-4072	155	6	6	6	NUM
ejpam-4072	155	7	):	):	PUNCT
ejpam-4072	155	8	let	let	VERB
ejpam-4072	155	9	x	x	PUNCT
ejpam-4072	155	10	∈	∈	PROPN
ejpam-4072	155	11	x	x	PUNCT
ejpam-4072	155	12	and	and	CCONJ
ejpam-4072	155	13	let	let	VERB
ejpam-4072	155	14	v	v	PART
ejpam-4072	155	15	be	be	AUX
ejpam-4072	155	16	any	any	DET
ejpam-4072	155	17	σ1σ2	σ1σ2	NOUN
ejpam-4072	155	18	-	-	ADJ
ejpam-4072	155	19	open	open	ADJ
ejpam-4072	155	20	set	set	NOUN
ejpam-4072	155	21	of	of	ADP
ejpam-4072	155	22	y	y	PROPN
ejpam-4072	155	23	containing	contain	VERB
ejpam-4072	155	24	f	f	PROPN
ejpam-4072	155	25	(	(	PUNCT
ejpam-4072	155	26	x	x	NOUN
ejpam-4072	155	27	)	)	PUNCT
ejpam-4072	155	28	.	.	PUNCT
ejpam-4072	156	1	by	by	ADP
ejpam-4072	156	2	(	(	PUNCT
ejpam-4072	156	3	5	5	NUM
ejpam-4072	156	4	)	)	PUNCT
ejpam-4072	156	5	,	,	PUNCT
ejpam-4072	156	6	x	x	PUNCT
ejpam-4072	156	7	∈	∈	PROPN
ejpam-4072	156	8	f+(v	f+(v	NOUN
ejpam-4072	156	9	)	)	PUNCT
ejpam-4072	157	1	⊆	⊆	X
ejpam-4072	157	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	157	3	-	-	NOUN
ejpam-4072	157	4	pint(f	pint(f	ADJ
ejpam-4072	157	5	+	+	PROPN
ejpam-4072	157	6	(	(	PUNCT
ejpam-4072	157	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	157	8	-	-	NUM
ejpam-4072	157	9	cl(v	cl(v	NOUN
ejpam-4072	157	10	)	)	PUNCT
ejpam-4072	157	11	)	)	PUNCT
ejpam-4072	157	12	)	)	PUNCT
ejpam-4072	157	13	and	and	CCONJ
ejpam-4072	157	14	there	there	PRON
ejpam-4072	157	15	exists	exist	VERB
ejpam-4072	157	16	a	a	DET
ejpam-4072	157	17	τ1τ2	τ1τ2	NOUN
ejpam-4072	157	18	-	-	ADJ
ejpam-4072	157	19	preopen	preopen	ADJ
ejpam-4072	157	20	set	set	NOUN
ejpam-4072	157	21	u	u	NOUN
ejpam-4072	157	22	of	of	ADP
ejpam-4072	157	23	x	x	PUNCT
ejpam-4072	157	24	containing	contain	VERB
ejpam-4072	157	25	x	x	PUNCT
ejpam-4072	157	26	such	such	ADJ
ejpam-4072	157	27	that	that	SCONJ
ejpam-4072	157	28	f	f	PROPN
ejpam-4072	157	29	(	(	PUNCT
ejpam-4072	157	30	u	u	NOUN
ejpam-4072	157	31	)	)	PUNCT
ejpam-4072	157	32	⊆	⊆	NUM
ejpam-4072	157	33	σ1σ2	σ1σ2	NOUN
ejpam-4072	157	34	-	-	NUM
ejpam-4072	157	35	cl(v	cl(v	NOUN
ejpam-4072	157	36	)	)	PUNCT
ejpam-4072	157	37	.	.	PUNCT
ejpam-4072	158	1	(	(	PUNCT
ejpam-4072	158	2	6	6	X
ejpam-4072	158	3	)	)	PUNCT
ejpam-4072	158	4	⇒	⇒	NOUN
ejpam-4072	158	5	(	(	PUNCT
ejpam-4072	158	6	1	1	NUM
ejpam-4072	158	7	):	):	PUNCT
ejpam-4072	158	8	let	let	VERB
ejpam-4072	158	9	x	x	PUNCT
ejpam-4072	158	10	∈	∈	PROPN
ejpam-4072	158	11	x	x	PUNCT
ejpam-4072	158	12	and	and	CCONJ
ejpam-4072	158	13	let	let	VERB
ejpam-4072	158	14	v	v	PART
ejpam-4072	158	15	be	be	AUX
ejpam-4072	158	16	any	any	DET
ejpam-4072	158	17	σ1σ2	σ1σ2	NOUN
ejpam-4072	158	18	-	-	ADJ
ejpam-4072	158	19	open	open	ADJ
ejpam-4072	158	20	set	set	NOUN
ejpam-4072	158	21	of	of	ADP
ejpam-4072	158	22	y	y	PROPN
ejpam-4072	158	23	containing	contain	VERB
ejpam-4072	158	24	f	f	PROPN
ejpam-4072	158	25	(	(	PUNCT
ejpam-4072	158	26	x	x	NOUN
ejpam-4072	158	27	)	)	PUNCT
ejpam-4072	158	28	.	.	PUNCT
ejpam-4072	159	1	by	by	ADP
ejpam-4072	159	2	(	(	PUNCT
ejpam-4072	159	3	6	6	NUM
ejpam-4072	159	4	)	)	PUNCT
ejpam-4072	159	5	,	,	PUNCT
ejpam-4072	159	6	there	there	PRON
ejpam-4072	159	7	exists	exist	VERB
ejpam-4072	159	8	a	a	DET
ejpam-4072	159	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	159	10	-	-	ADJ
ejpam-4072	159	11	preopen	preopen	ADJ
ejpam-4072	159	12	set	set	NOUN
ejpam-4072	159	13	u	u	NOUN
ejpam-4072	159	14	of	of	ADP
ejpam-4072	159	15	x	x	PUNCT
ejpam-4072	159	16	containing	contain	VERB
ejpam-4072	159	17	x	x	PUNCT
ejpam-4072	159	18	such	such	ADJ
ejpam-4072	159	19	that	that	SCONJ
ejpam-4072	159	20	f	f	PROPN
ejpam-4072	159	21	(	(	PUNCT
ejpam-4072	159	22	u	u	NOUN
ejpam-4072	159	23	)	)	PUNCT
ejpam-4072	159	24	⊆	⊆	NUM
ejpam-4072	159	25	σ1σ2	σ1σ2	NOUN
ejpam-4072	159	26	-	-	NUM
ejpam-4072	159	27	cl(v	cl(v	NOUN
ejpam-4072	159	28	)	)	PUNCT
ejpam-4072	159	29	;	;	PUNCT
ejpam-4072	159	30	hence	hence	ADV
ejpam-4072	159	31	u	u	NOUN
ejpam-4072	159	32	⊆	⊆	NUM
ejpam-4072	159	33	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	159	34	-	-	PUNCT
ejpam-4072	159	35	cl(v	cl(v	NOUN
ejpam-4072	159	36	)	)	PUNCT
ejpam-4072	159	37	)	)	PUNCT
ejpam-4072	159	38	.	.	PUNCT
ejpam-4072	160	1	thus	thus	ADV
ejpam-4072	160	2	,	,	PUNCT
ejpam-4072	160	3	x	x	PUNCT
ejpam-4072	160	4	∈	∈	PROPN
ejpam-4072	160	5	u	u	NOUN
ejpam-4072	160	6	⊆	⊆	NUM
ejpam-4072	160	7	τ1	τ1	NOUN
ejpam-4072	160	8	-	-	PUNCT
ejpam-4072	160	9	int(τ2	int(τ2	NOUN
ejpam-4072	160	10	-	-	PUNCT
ejpam-4072	160	11	cl(u	cl(u	NOUN
ejpam-4072	160	12	)	)	PUNCT
ejpam-4072	160	13	)	)	PUNCT
ejpam-4072	161	1	⊆	⊆	NUM
ejpam-4072	161	2	τ1	τ1	NOUN
ejpam-4072	161	3	-	-	PUNCT
ejpam-4072	161	4	int(τ2	int(τ2	NOUN
ejpam-4072	161	5	-	-	PUNCT
ejpam-4072	161	6	cl(f	cl(f	NOUN
ejpam-4072	161	7	+	+	NOUN
ejpam-4072	161	8	(	(	PUNCT
ejpam-4072	161	9	σ1σ2	σ1σ2	NOUN
ejpam-4072	161	10	-	-	NUM
ejpam-4072	161	11	cl(v	cl(v	NOUN
ejpam-4072	161	12	)	)	PUNCT
ejpam-4072	161	13	)	)	PUNCT
ejpam-4072	161	14	)	)	PUNCT
ejpam-4072	161	15	)	)	PUNCT
ejpam-4072	161	16	.	.	PUNCT
ejpam-4072	162	1	this	this	PRON
ejpam-4072	162	2	shows	show	VERB
ejpam-4072	162	3	that	that	SCONJ
ejpam-4072	162	4	f	f	PROPN
ejpam-4072	162	5	is	be	AUX
ejpam-4072	162	6	upper	upper	ADJ
ejpam-4072	162	7	almost	almost	ADV
ejpam-4072	162	8	weakly	weakly	ADJ
ejpam-4072	162	9	(	(	PUNCT
ejpam-4072	162	10	τ1	τ1	NOUN
ejpam-4072	162	11	,	,	PUNCT
ejpam-4072	162	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	162	13	.	.	PUNCT
ejpam-4072	162	14	theorem	theorem	NOUN
ejpam-4072	162	15	2	2	NUM
ejpam-4072	162	16	.	.	X
ejpam-4072	162	17	for	for	ADP
ejpam-4072	162	18	a	a	DET
ejpam-4072	162	19	multifunction	multifunction	NOUN
ejpam-4072	162	20	f	f	NOUN
ejpam-4072	162	21	:	:	PUNCT
ejpam-4072	162	22	(	(	PUNCT
ejpam-4072	162	23	x	x	NOUN
ejpam-4072	162	24	,	,	PUNCT
ejpam-4072	162	25	τ1	τ1	NOUN
ejpam-4072	162	26	,	,	PUNCT
ejpam-4072	162	27	τ2	τ2	NOUN
ejpam-4072	162	28	)	)	PUNCT
ejpam-4072	162	29	→	→	SYM
ejpam-4072	162	30	(	(	PUNCT
ejpam-4072	162	31	y	y	PROPN
ejpam-4072	162	32	,	,	PUNCT
ejpam-4072	162	33	σ1	σ1	PROPN
ejpam-4072	162	34	,	,	PUNCT
ejpam-4072	162	35	σ2	σ2	NOUN
ejpam-4072	162	36	)	)	PUNCT
ejpam-4072	162	37	,	,	PUNCT
ejpam-4072	162	38	the	the	DET
ejpam-4072	162	39	following	follow	VERB
ejpam-4072	162	40	properties	property	NOUN
ejpam-4072	162	41	are	be	AUX
ejpam-4072	162	42	equivalent	equivalent	ADJ
ejpam-4072	162	43	:	:	PUNCT
ejpam-4072	162	44	(	(	PUNCT
ejpam-4072	162	45	1	1	X
ejpam-4072	162	46	)	)	PUNCT
ejpam-4072	162	47	f	f	PROPN
ejpam-4072	162	48	is	be	AUX
ejpam-4072	162	49	lower	low	ADJ
ejpam-4072	162	50	almost	almost	ADV
ejpam-4072	162	51	weakly	weakly	ADJ
ejpam-4072	162	52	(	(	PUNCT
ejpam-4072	162	53	τ1	τ1	NOUN
ejpam-4072	162	54	,	,	PUNCT
ejpam-4072	162	55	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	162	56	;	;	PUNCT
ejpam-4072	162	57	(	(	PUNCT
ejpam-4072	162	58	2	2	X
ejpam-4072	162	59	)	)	PUNCT
ejpam-4072	162	60	f−(v	f−(v	NOUN
ejpam-4072	162	61	)	)	PUNCT
ejpam-4072	163	1	⊆	⊆	NUM
ejpam-4072	163	2	τ1	τ1	NOUN
ejpam-4072	163	3	-	-	PUNCT
ejpam-4072	163	4	int(τ2	int(τ2	NOUN
ejpam-4072	163	5	-	-	PUNCT
ejpam-4072	163	6	cl(f	cl(f	NOUN
ejpam-4072	163	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	163	8	-	-	NOUN
ejpam-4072	163	9	cl(v	cl(v	NOUN
ejpam-4072	163	10	)	)	PUNCT
ejpam-4072	163	11	)	)	PUNCT
ejpam-4072	163	12	)	)	PUNCT
ejpam-4072	163	13	)	)	PUNCT
ejpam-4072	164	1	for	for	ADP
ejpam-4072	164	2	every	every	DET
ejpam-4072	164	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	164	4	-	-	ADJ
ejpam-4072	164	5	open	open	ADJ
ejpam-4072	164	6	set	set	NOUN
ejpam-4072	164	7	v	v	NOUN
ejpam-4072	164	8	of	of	ADP
ejpam-4072	164	9	y	y	PROPN
ejpam-4072	164	10	;	;	PUNCT
ejpam-4072	164	11	(	(	PUNCT
ejpam-4072	164	12	3	3	X
ejpam-4072	164	13	)	)	PUNCT
ejpam-4072	164	14	τ1	τ1	NOUN
ejpam-4072	164	15	-	-	PUNCT
ejpam-4072	164	16	cl(τ2	cl(τ2	NOUN
ejpam-4072	164	17	-	-	PUNCT
ejpam-4072	164	18	int(f	int(f	VERB
ejpam-4072	164	19	+	+	ADJ
ejpam-4072	164	20	(	(	PUNCT
ejpam-4072	164	21	v	v	NOUN
ejpam-4072	164	22	)	)	PUNCT
ejpam-4072	164	23	)	)	PUNCT
ejpam-4072	164	24	)	)	PUNCT
ejpam-4072	164	25	⊆	⊆	NUM
ejpam-4072	164	26	f+(σ1σ2cl(v	f+(σ1σ2cl(v	NOUN
ejpam-4072	164	27	)	)	PUNCT
ejpam-4072	164	28	)	)	PUNCT
ejpam-4072	164	29	for	for	ADP
ejpam-4072	164	30	every	every	DET
ejpam-4072	164	31	σ1σ2	σ1σ2	NOUN
ejpam-4072	164	32	-	-	ADJ
ejpam-4072	164	33	open	open	ADJ
ejpam-4072	164	34	set	set	NOUN
ejpam-4072	164	35	v	v	NOUN
ejpam-4072	164	36	of	of	ADP
ejpam-4072	164	37	y	y	PROPN
ejpam-4072	164	38	;	;	PUNCT
ejpam-4072	164	39	(	(	PUNCT
ejpam-4072	164	40	4	4	X
ejpam-4072	164	41	)	)	PUNCT
ejpam-4072	164	42	τ1τ2	τ1τ2	NOUN
ejpam-4072	164	43	-	-	PROPN
ejpam-4072	164	44	pcl(f	pcl(f	PROPN
ejpam-4072	164	45	+	+	ADJ
ejpam-4072	164	46	(	(	PUNCT
ejpam-4072	164	47	v	v	NOUN
ejpam-4072	164	48	)	)	PUNCT
ejpam-4072	164	49	)	)	PUNCT
ejpam-4072	164	50	⊆	⊆	NUM
ejpam-4072	164	51	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	164	52	-	-	PUNCT
ejpam-4072	164	53	cl(v	cl(v	NOUN
ejpam-4072	164	54	)	)	PUNCT
ejpam-4072	164	55	)	)	PUNCT
ejpam-4072	164	56	for	for	ADP
ejpam-4072	164	57	every	every	DET
ejpam-4072	164	58	σ1σ2	σ1σ2	NOUN
ejpam-4072	164	59	-	-	ADJ
ejpam-4072	164	60	open	open	ADJ
ejpam-4072	164	61	set	set	NOUN
ejpam-4072	164	62	v	v	NOUN
ejpam-4072	164	63	of	of	ADP
ejpam-4072	164	64	y	y	PROPN
ejpam-4072	164	65	;	;	PUNCT
ejpam-4072	164	66	(	(	PUNCT
ejpam-4072	164	67	5	5	X
ejpam-4072	164	68	)	)	PUNCT
ejpam-4072	164	69	f−(v	f−(v	NOUN
ejpam-4072	164	70	)	)	PUNCT
ejpam-4072	164	71	⊆	⊆	NUM
ejpam-4072	164	72	τ1τ2	τ1τ2	NOUN
ejpam-4072	164	73	-	-	ADJ
ejpam-4072	164	74	pint(f	pint(f	ADJ
ejpam-4072	164	75	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	164	76	-	-	NOUN
ejpam-4072	164	77	cl(v	cl(v	NOUN
ejpam-4072	164	78	)	)	PUNCT
ejpam-4072	164	79	)	)	PUNCT
ejpam-4072	164	80	)	)	PUNCT
ejpam-4072	165	1	for	for	ADP
ejpam-4072	165	2	every	every	DET
ejpam-4072	165	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	165	4	-	-	ADJ
ejpam-4072	165	5	open	open	ADJ
ejpam-4072	165	6	set	set	NOUN
ejpam-4072	165	7	v	v	NOUN
ejpam-4072	165	8	of	of	ADP
ejpam-4072	165	9	y	y	PROPN
ejpam-4072	165	10	;	;	PUNCT
ejpam-4072	165	11	c.	c.	PROPN
ejpam-4072	165	12	boonpok	boonpok	PROPN
ejpam-4072	165	13	,	,	PUNCT
ejpam-4072	165	14	c.	c.	PROPN
ejpam-4072	165	15	viriyapong	viriyapong	PROPN
ejpam-4072	165	16	/	/	SYM
ejpam-4072	165	17	eur	eur	PROPN
ejpam-4072	165	18	.	.	PUNCT
ejpam-4072	166	1	j.	j.	PROPN
ejpam-4072	166	2	pure	pure	PROPN
ejpam-4072	166	3	appl	appl	PROPN
ejpam-4072	166	4	.	.	PROPN
ejpam-4072	166	5	math	math	PROPN
ejpam-4072	166	6	,	,	PUNCT
ejpam-4072	166	7	14	14	NUM
ejpam-4072	166	8	(	(	PUNCT
ejpam-4072	166	9	4	4	NUM
ejpam-4072	166	10	)	)	PUNCT
ejpam-4072	166	11	(	(	PUNCT
ejpam-4072	166	12	2021	2021	NUM
ejpam-4072	166	13	)	)	PUNCT
ejpam-4072	166	14	,	,	PUNCT
ejpam-4072	166	15	1212	1212	NUM
ejpam-4072	166	16	-	-	SYM
ejpam-4072	166	17	1225	1225	NUM
ejpam-4072	166	18	1217	1217	NUM
ejpam-4072	166	19	(	(	PUNCT
ejpam-4072	166	20	6	6	NUM
ejpam-4072	166	21	)	)	PUNCT
ejpam-4072	166	22	for	for	ADP
ejpam-4072	166	23	each	each	DET
ejpam-4072	166	24	x	x	SYM
ejpam-4072	166	25	∈	∈	PROPN
ejpam-4072	166	26	x	x	X
ejpam-4072	166	27	and	and	CCONJ
ejpam-4072	166	28	each	each	DET
ejpam-4072	166	29	σ1σ2	σ1σ2	VERB
ejpam-4072	166	30	-	-	ADJ
ejpam-4072	166	31	open	open	ADJ
ejpam-4072	166	32	set	set	NOUN
ejpam-4072	166	33	v	v	NOUN
ejpam-4072	166	34	of	of	ADP
ejpam-4072	166	35	y	y	PRON
ejpam-4072	166	36	such	such	ADJ
ejpam-4072	166	37	that	that	SCONJ
ejpam-4072	166	38	f	f	PROPN
ejpam-4072	166	39	(	(	PUNCT
ejpam-4072	166	40	x	x	NOUN
ejpam-4072	166	41	)	)	PUNCT
ejpam-4072	166	42	∩	∩	NOUN
ejpam-4072	166	43	v	v	ADP
ejpam-4072	166	44	̸=	̸=	PROPN
ejpam-4072	166	45	∅	∅	NOUN
ejpam-4072	166	46	,	,	PUNCT
ejpam-4072	166	47	there	there	PRON
ejpam-4072	166	48	exists	exist	VERB
ejpam-4072	166	49	a	a	DET
ejpam-4072	166	50	τ1τ2	τ1τ2	NOUN
ejpam-4072	166	51	-	-	ADJ
ejpam-4072	166	52	preopen	preopen	ADJ
ejpam-4072	166	53	set	set	NOUN
ejpam-4072	166	54	u	u	NOUN
ejpam-4072	166	55	of	of	ADP
ejpam-4072	166	56	x	x	PUNCT
ejpam-4072	166	57	containing	contain	VERB
ejpam-4072	166	58	x	x	PUNCT
ejpam-4072	166	59	such	such	ADJ
ejpam-4072	166	60	that	that	SCONJ
ejpam-4072	166	61	f	f	PROPN
ejpam-4072	166	62	(	(	PUNCT
ejpam-4072	166	63	z	z	NOUN
ejpam-4072	166	64	)	)	PUNCT
ejpam-4072	166	65	∩	∩	NOUN
ejpam-4072	166	66	σ1σ2	σ1σ2	NOUN
ejpam-4072	166	67	-	-	NUM
ejpam-4072	166	68	cl(v	cl(v	NOUN
ejpam-4072	166	69	)	)	PUNCT
ejpam-4072	166	70	̸=	̸=	NOUN
ejpam-4072	166	71	∅	∅	NOUN
ejpam-4072	166	72	for	for	ADP
ejpam-4072	166	73	each	each	DET
ejpam-4072	166	74	z	z	NOUN
ejpam-4072	166	75	∈	∈	PROPN
ejpam-4072	166	76	u	u	NOUN
ejpam-4072	166	77	.	.	PUNCT
ejpam-4072	167	1	proof	proof	NOUN
ejpam-4072	167	2	.	.	PUNCT
ejpam-4072	168	1	the	the	DET
ejpam-4072	168	2	proof	proof	NOUN
ejpam-4072	168	3	is	be	AUX
ejpam-4072	168	4	similar	similar	ADJ
ejpam-4072	168	5	to	to	ADP
ejpam-4072	168	6	that	that	PRON
ejpam-4072	168	7	of	of	ADP
ejpam-4072	168	8	theorem	theorem	ADJ
ejpam-4072	168	9	1	1	NUM
ejpam-4072	168	10	.	.	PUNCT
ejpam-4072	168	11	theorem	theorem	NOUN
ejpam-4072	168	12	3	3	NUM
ejpam-4072	168	13	.	.	X
ejpam-4072	168	14	for	for	ADP
ejpam-4072	168	15	a	a	DET
ejpam-4072	168	16	multifunction	multifunction	NOUN
ejpam-4072	168	17	f	f	NOUN
ejpam-4072	168	18	:	:	PUNCT
ejpam-4072	168	19	(	(	PUNCT
ejpam-4072	168	20	x	x	NOUN
ejpam-4072	168	21	,	,	PUNCT
ejpam-4072	168	22	τ1	τ1	NOUN
ejpam-4072	168	23	,	,	PUNCT
ejpam-4072	168	24	τ2	τ2	NOUN
ejpam-4072	168	25	)	)	PUNCT
ejpam-4072	168	26	→	→	SYM
ejpam-4072	168	27	(	(	PUNCT
ejpam-4072	168	28	y	y	PROPN
ejpam-4072	168	29	,	,	PUNCT
ejpam-4072	168	30	σ1	σ1	PROPN
ejpam-4072	168	31	,	,	PUNCT
ejpam-4072	168	32	σ2	σ2	NOUN
ejpam-4072	168	33	)	)	PUNCT
ejpam-4072	168	34	,	,	PUNCT
ejpam-4072	168	35	the	the	DET
ejpam-4072	168	36	following	follow	VERB
ejpam-4072	168	37	properties	property	NOUN
ejpam-4072	168	38	are	be	AUX
ejpam-4072	168	39	equivalent	equivalent	ADJ
ejpam-4072	168	40	:	:	PUNCT
ejpam-4072	168	41	(	(	PUNCT
ejpam-4072	168	42	1	1	X
ejpam-4072	168	43	)	)	PUNCT
ejpam-4072	168	44	f	f	PROPN
ejpam-4072	168	45	is	be	AUX
ejpam-4072	168	46	upper	upper	ADJ
ejpam-4072	168	47	almost	almost	ADV
ejpam-4072	168	48	weakly	weakly	ADJ
ejpam-4072	168	49	(	(	PUNCT
ejpam-4072	168	50	τ1	τ1	NOUN
ejpam-4072	168	51	,	,	PUNCT
ejpam-4072	168	52	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	168	53	;	;	PUNCT
ejpam-4072	168	54	(	(	PUNCT
ejpam-4072	168	55	2	2	X
ejpam-4072	168	56	)	)	PUNCT
ejpam-4072	168	57	τ1	τ1	NOUN
ejpam-4072	168	58	-	-	PUNCT
ejpam-4072	168	59	cl(τ2	cl(τ2	NOUN
ejpam-4072	168	60	-	-	PUNCT
ejpam-4072	168	61	int(f	int(f	PRON
ejpam-4072	168	62	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	168	63	-	-	PUNCT
ejpam-4072	168	64	int(h	int(h	NOUN
ejpam-4072	168	65	)	)	PUNCT
ejpam-4072	168	66	)	)	PUNCT
ejpam-4072	168	67	)	)	PUNCT
ejpam-4072	168	68	)	)	PUNCT
ejpam-4072	169	1	⊆	⊆	NUM
ejpam-4072	169	2	f−(h	f−(h	NOUN
ejpam-4072	169	3	)	)	PUNCT
ejpam-4072	169	4	for	for	ADP
ejpam-4072	169	5	every	every	DET
ejpam-4072	169	6	σ1σ2	σ1σ2	NUM
ejpam-4072	169	7	-	-	ADJ
ejpam-4072	169	8	closed	closed	ADJ
ejpam-4072	169	9	set	set	ADJ
ejpam-4072	169	10	h	h	NOUN
ejpam-4072	169	11	of	of	ADP
ejpam-4072	169	12	y	y	PROPN
ejpam-4072	169	13	;	;	PUNCT
ejpam-4072	169	14	(	(	PUNCT
ejpam-4072	169	15	3	3	X
ejpam-4072	169	16	)	)	PUNCT
ejpam-4072	169	17	τ1τ2	τ1τ2	NOUN
ejpam-4072	169	18	-	-	PROPN
ejpam-4072	169	19	pcl(f	pcl(f	PROPN
ejpam-4072	169	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	169	21	-	-	PUNCT
ejpam-4072	169	22	int(h	int(h	NOUN
ejpam-4072	169	23	)	)	PUNCT
ejpam-4072	169	24	)	)	PUNCT
ejpam-4072	169	25	)	)	PUNCT
ejpam-4072	170	1	⊆	⊆	NUM
ejpam-4072	170	2	f−(h	f−(h	NOUN
ejpam-4072	170	3	)	)	PUNCT
ejpam-4072	170	4	for	for	ADP
ejpam-4072	170	5	every	every	DET
ejpam-4072	170	6	σ1σ2	σ1σ2	NUM
ejpam-4072	170	7	-	-	ADJ
ejpam-4072	170	8	closed	closed	ADJ
ejpam-4072	170	9	set	set	ADJ
ejpam-4072	170	10	h	h	NOUN
ejpam-4072	170	11	of	of	ADP
ejpam-4072	170	12	y	y	PROPN
ejpam-4072	170	13	;	;	PUNCT
ejpam-4072	170	14	(	(	PUNCT
ejpam-4072	170	15	4	4	X
ejpam-4072	170	16	)	)	PUNCT
ejpam-4072	170	17	τ1τ2	τ1τ2	NOUN
ejpam-4072	170	18	-	-	PROPN
ejpam-4072	170	19	pcl(f	pcl(f	PROPN
ejpam-4072	170	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	170	21	-	-	PUNCT
ejpam-4072	170	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	170	23	-	-	PUNCT
ejpam-4072	170	24	cl(b	cl(b	NOUN
ejpam-4072	170	25	)	)	PUNCT
ejpam-4072	170	26	)	)	PUNCT
ejpam-4072	170	27	)	)	PUNCT
ejpam-4072	170	28	)	)	PUNCT
ejpam-4072	171	1	⊆	⊆	X
ejpam-4072	171	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-4072	171	3	-	-	PUNCT
ejpam-4072	171	4	cl(b	cl(b	NOUN
ejpam-4072	171	5	)	)	PUNCT
ejpam-4072	171	6	)	)	PUNCT
ejpam-4072	172	1	for	for	ADP
ejpam-4072	172	2	every	every	DET
ejpam-4072	172	3	subset	subset	NOUN
ejpam-4072	172	4	b	b	PROPN
ejpam-4072	172	5	of	of	ADP
ejpam-4072	172	6	y	y	PROPN
ejpam-4072	172	7	;	;	PUNCT
ejpam-4072	172	8	(	(	PUNCT
ejpam-4072	172	9	5	5	X
ejpam-4072	172	10	)	)	PUNCT
ejpam-4072	172	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	172	12	-	-	PUNCT
ejpam-4072	172	13	int(b	int(b	NOUN
ejpam-4072	172	14	)	)	PUNCT
ejpam-4072	172	15	)	)	PUNCT
ejpam-4072	173	1	⊆	⊆	X
ejpam-4072	173	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	173	3	-	-	NOUN
ejpam-4072	173	4	pint(f	pint(f	ADJ
ejpam-4072	173	5	+	+	PROPN
ejpam-4072	173	6	(	(	PUNCT
ejpam-4072	173	7	σ1σ2	σ1σ2	NUM
ejpam-4072	173	8	-	-	PUNCT
ejpam-4072	173	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	173	10	-	-	PUNCT
ejpam-4072	173	11	int(b	int(b	NOUN
ejpam-4072	173	12	)	)	PUNCT
ejpam-4072	173	13	)	)	PUNCT
ejpam-4072	173	14	)	)	PUNCT
ejpam-4072	173	15	)	)	PUNCT
ejpam-4072	173	16	for	for	ADP
ejpam-4072	173	17	every	every	DET
ejpam-4072	173	18	subset	subset	NOUN
ejpam-4072	173	19	b	b	PROPN
ejpam-4072	173	20	of	of	ADP
ejpam-4072	173	21	y	y	PROPN
ejpam-4072	173	22	.	.	PUNCT
ejpam-4072	174	1	proof	proof	NOUN
ejpam-4072	174	2	.	.	PUNCT
ejpam-4072	175	1	(	(	PUNCT
ejpam-4072	175	2	1	1	X
ejpam-4072	175	3	)	)	PUNCT
ejpam-4072	175	4	⇒	⇒	NOUN
ejpam-4072	175	5	(	(	PUNCT
ejpam-4072	175	6	2	2	NUM
ejpam-4072	175	7	):	):	PUNCT
ejpam-4072	175	8	let	let	VERB
ejpam-4072	175	9	h	h	PRON
ejpam-4072	175	10	be	be	AUX
ejpam-4072	175	11	any	any	DET
ejpam-4072	175	12	σ1σ2	σ1σ2	NUM
ejpam-4072	175	13	-	-	PUNCT
ejpam-4072	175	14	closed	closed	ADJ
ejpam-4072	175	15	set	set	NOUN
ejpam-4072	175	16	of	of	ADP
ejpam-4072	175	17	y	y	PROPN
ejpam-4072	175	18	.	.	PUNCT
ejpam-4072	176	1	then	then	ADV
ejpam-4072	176	2	,	,	PUNCT
ejpam-4072	176	3	y	y	PROPN
ejpam-4072	176	4	−h	−h	ADV
ejpam-4072	176	5	is	be	AUX
ejpam-4072	176	6	σ1σ2	σ1σ2	NOUN
ejpam-4072	176	7	-	-	ADJ
ejpam-4072	176	8	open	open	ADJ
ejpam-4072	176	9	in	in	ADP
ejpam-4072	176	10	y	y	PROPN
ejpam-4072	176	11	,	,	PUNCT
ejpam-4072	176	12	by	by	ADP
ejpam-4072	176	13	theorem	theorem	NOUN
ejpam-4072	176	14	1	1	NUM
ejpam-4072	176	15	,	,	PUNCT
ejpam-4072	176	16	x	x	PUNCT
ejpam-4072	176	17	−	−	NOUN
ejpam-4072	176	18	f−(h	f−(h	NOUN
ejpam-4072	176	19	)	)	PUNCT
ejpam-4072	176	20	=	=	PUNCT
ejpam-4072	177	1	f+(y	f+(y	NOUN
ejpam-4072	177	2	−h	−h	ADV
ejpam-4072	177	3	)	)	PUNCT
ejpam-4072	178	1	⊆	⊆	NUM
ejpam-4072	178	2	τ1	τ1	NOUN
ejpam-4072	178	3	-	-	PUNCT
ejpam-4072	178	4	int(τ2	int(τ2	NOUN
ejpam-4072	178	5	-	-	PUNCT
ejpam-4072	178	6	cl(f	cl(f	NOUN
ejpam-4072	179	1	+	+	NOUN
ejpam-4072	179	2	(	(	PUNCT
ejpam-4072	179	3	σ1σ2	σ1σ2	NUM
ejpam-4072	179	4	-	-	PUNCT
ejpam-4072	179	5	cl(y	cl(y	NOUN
ejpam-4072	179	6	−h	−h	NOUN
ejpam-4072	179	7	)	)	PUNCT
ejpam-4072	179	8	)	)	PUNCT
ejpam-4072	179	9	)	)	PUNCT
ejpam-4072	179	10	)	)	PUNCT
ejpam-4072	180	1	=	=	SYM
ejpam-4072	180	2	τ1	τ1	NOUN
ejpam-4072	180	3	-	-	PUNCT
ejpam-4072	180	4	int(τ2	int(τ2	NOUN
ejpam-4072	180	5	-	-	PUNCT
ejpam-4072	180	6	cl(f	cl(f	NOUN
ejpam-4072	180	7	+	+	PROPN
ejpam-4072	180	8	(	(	PUNCT
ejpam-4072	180	9	y	y	PROPN
ejpam-4072	180	10	−	−	PROPN
ejpam-4072	180	11	σ1σ2	σ1σ2	PROPN
ejpam-4072	180	12	-	-	PUNCT
ejpam-4072	180	13	int(h	int(h	NOUN
ejpam-4072	180	14	)	)	PUNCT
ejpam-4072	180	15	)	)	PUNCT
ejpam-4072	180	16	)	)	PUNCT
ejpam-4072	180	17	)	)	PUNCT
ejpam-4072	181	1	=	=	SYM
ejpam-4072	181	2	τ1	τ1	NOUN
ejpam-4072	181	3	-	-	PUNCT
ejpam-4072	181	4	int(τ2	int(τ2	NOUN
ejpam-4072	181	5	-	-	PUNCT
ejpam-4072	181	6	cl(x	cl(x	NOUN
ejpam-4072	181	7	−	−	NOUN
ejpam-4072	181	8	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	181	9	-	-	PUNCT
ejpam-4072	181	10	int(h	int(h	NOUN
ejpam-4072	181	11	)	)	PUNCT
ejpam-4072	181	12	)	)	PUNCT
ejpam-4072	181	13	)	)	PUNCT
ejpam-4072	181	14	)	)	PUNCT
ejpam-4072	182	1	=	=	PUNCT
ejpam-4072	182	2	x	x	X
ejpam-4072	183	1	−	−	NOUN
ejpam-4072	183	2	τ1	τ1	NOUN
ejpam-4072	183	3	-	-	PUNCT
ejpam-4072	183	4	cl(τ2	cl(τ2	NOUN
ejpam-4072	183	5	-	-	PUNCT
ejpam-4072	183	6	int(f	int(f	PRON
ejpam-4072	183	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	183	8	-	-	PUNCT
ejpam-4072	183	9	int(h	int(h	NOUN
ejpam-4072	183	10	)	)	PUNCT
ejpam-4072	183	11	)	)	PUNCT
ejpam-4072	183	12	)	)	PUNCT
ejpam-4072	183	13	)	)	PUNCT
ejpam-4072	184	1	and	and	CCONJ
ejpam-4072	184	2	hence	hence	ADV
ejpam-4072	184	3	τ1	τ1	NOUN
ejpam-4072	184	4	-	-	PUNCT
ejpam-4072	184	5	cl(τ2	cl(τ2	NOUN
ejpam-4072	184	6	-	-	PUNCT
ejpam-4072	184	7	int(f	int(f	PRON
ejpam-4072	184	8	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	184	9	-	-	PUNCT
ejpam-4072	184	10	int(h	int(h	NOUN
ejpam-4072	184	11	)	)	PUNCT
ejpam-4072	184	12	)	)	PUNCT
ejpam-4072	184	13	)	)	PUNCT
ejpam-4072	184	14	)	)	PUNCT
ejpam-4072	185	1	⊆	⊆	NUM
ejpam-4072	185	2	f−(h	f−(h	NOUN
ejpam-4072	185	3	)	)	PUNCT
ejpam-4072	185	4	.	.	PUNCT
ejpam-4072	186	1	(	(	PUNCT
ejpam-4072	186	2	2	2	X
ejpam-4072	186	3	)	)	PUNCT
ejpam-4072	186	4	⇒	⇒	NOUN
ejpam-4072	186	5	(	(	PUNCT
ejpam-4072	186	6	3	3	NUM
ejpam-4072	186	7	):	):	PUNCT
ejpam-4072	186	8	let	let	VERB
ejpam-4072	186	9	h	h	PRON
ejpam-4072	186	10	be	be	AUX
ejpam-4072	186	11	any	any	DET
ejpam-4072	186	12	σ1σ2	σ1σ2	NUM
ejpam-4072	186	13	-	-	PUNCT
ejpam-4072	186	14	closed	closed	ADJ
ejpam-4072	186	15	set	set	NOUN
ejpam-4072	186	16	of	of	ADP
ejpam-4072	186	17	y	y	PROPN
ejpam-4072	186	18	.	.	PUNCT
ejpam-4072	187	1	by	by	ADP
ejpam-4072	187	2	lemma	lemma	PROPN
ejpam-4072	187	3	6(1	6(1	NUM
ejpam-4072	187	4	)	)	PUNCT
ejpam-4072	187	5	,	,	PUNCT
ejpam-4072	187	6	we	we	PRON
ejpam-4072	187	7	have	have	VERB
ejpam-4072	187	8	τ1τ2	τ1τ2	NOUN
ejpam-4072	187	9	-	-	PROPN
ejpam-4072	187	10	pcl(f	pcl(f	PROPN
ejpam-4072	187	11	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	187	12	-	-	PUNCT
ejpam-4072	187	13	int(h	int(h	NOUN
ejpam-4072	187	14	)	)	PUNCT
ejpam-4072	187	15	)	)	PUNCT
ejpam-4072	187	16	)	)	PUNCT
ejpam-4072	188	1	=	=	PUNCT
ejpam-4072	188	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	188	3	-	-	PUNCT
ejpam-4072	188	4	int(h	int(h	NOUN
ejpam-4072	188	5	)	)	PUNCT
ejpam-4072	188	6	)	)	PUNCT
ejpam-4072	188	7	∪	∪	ADP
ejpam-4072	188	8	τ1	τ1	NOUN
ejpam-4072	188	9	-	-	PUNCT
ejpam-4072	188	10	cl(τ2	cl(τ2	NOUN
ejpam-4072	188	11	-	-	PUNCT
ejpam-4072	188	12	int(f	int(f	PRON
ejpam-4072	188	13	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	188	14	-	-	PUNCT
ejpam-4072	188	15	int(h	int(h	NOUN
ejpam-4072	188	16	)	)	PUNCT
ejpam-4072	188	17	)	)	PUNCT
ejpam-4072	188	18	)	)	PUNCT
ejpam-4072	188	19	)	)	PUNCT
ejpam-4072	189	1	⊆	⊆	NUM
ejpam-4072	189	2	f−(h	f−(h	NOUN
ejpam-4072	189	3	)	)	PUNCT
ejpam-4072	189	4	.	.	PUNCT
ejpam-4072	190	1	(	(	PUNCT
ejpam-4072	190	2	3	3	X
ejpam-4072	190	3	)	)	PUNCT
ejpam-4072	190	4	⇒	⇒	NOUN
ejpam-4072	190	5	(	(	PUNCT
ejpam-4072	190	6	4	4	NUM
ejpam-4072	190	7	):	):	PUNCT
ejpam-4072	190	8	this	this	PRON
ejpam-4072	190	9	is	be	AUX
ejpam-4072	190	10	obvious	obvious	ADJ
ejpam-4072	190	11	.	.	PUNCT
ejpam-4072	191	1	(	(	PUNCT
ejpam-4072	191	2	4	4	X
ejpam-4072	191	3	)	)	PUNCT
ejpam-4072	191	4	⇒	⇒	NOUN
ejpam-4072	191	5	(	(	PUNCT
ejpam-4072	191	6	5	5	NUM
ejpam-4072	191	7	):	):	PUNCT
ejpam-4072	191	8	let	let	VERB
ejpam-4072	191	9	b	b	X
ejpam-4072	191	10	be	be	AUX
ejpam-4072	191	11	any	any	DET
ejpam-4072	191	12	subset	subset	NOUN
ejpam-4072	191	13	of	of	ADP
ejpam-4072	191	14	y	y	PROPN
ejpam-4072	191	15	.	.	PUNCT
ejpam-4072	192	1	by	by	ADP
ejpam-4072	192	2	(	(	PUNCT
ejpam-4072	192	3	4	4	NUM
ejpam-4072	192	4	)	)	PUNCT
ejpam-4072	192	5	,	,	PUNCT
ejpam-4072	192	6	we	we	PRON
ejpam-4072	192	7	have	have	VERB
ejpam-4072	192	8	x	x	INTJ
ejpam-4072	192	9	−	−	ADP
ejpam-4072	192	10	τ1τ2	τ1τ2	NOUN
ejpam-4072	192	11	-	-	NOUN
ejpam-4072	192	12	pint(f	pint(f	ADJ
ejpam-4072	192	13	+	+	PROPN
ejpam-4072	192	14	(	(	PUNCT
ejpam-4072	192	15	σ1σ2	σ1σ2	NUM
ejpam-4072	192	16	-	-	PUNCT
ejpam-4072	192	17	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	192	18	-	-	PUNCT
ejpam-4072	192	19	int(b	int(b	NOUN
ejpam-4072	192	20	)	)	PUNCT
ejpam-4072	192	21	)	)	PUNCT
ejpam-4072	192	22	)	)	PUNCT
ejpam-4072	192	23	)	)	PUNCT
ejpam-4072	193	1	=	=	PUNCT
ejpam-4072	193	2	τ1τ2	τ1τ2	X
ejpam-4072	193	3	-	-	ADJ
ejpam-4072	193	4	pcl(x	pcl(x	ADJ
ejpam-4072	193	5	−	−	ADP
ejpam-4072	193	6	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-4072	193	7	-	-	PUNCT
ejpam-4072	193	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	193	9	-	-	PUNCT
ejpam-4072	193	10	int(b	int(b	NOUN
ejpam-4072	193	11	)	)	PUNCT
ejpam-4072	193	12	)	)	PUNCT
ejpam-4072	193	13	)	)	PUNCT
ejpam-4072	193	14	)	)	PUNCT
ejpam-4072	194	1	=	=	PUNCT
ejpam-4072	195	1	τ1τ2	τ1τ2	ADJ
ejpam-4072	195	2	-	-	PROPN
ejpam-4072	195	3	pcl(f	pcl(f	PROPN
ejpam-4072	195	4	−(y	−(y	NOUN
ejpam-4072	195	5	−	−	ADP
ejpam-4072	195	6	σ1σ2	σ1σ2	SYM
ejpam-4072	195	7	-	-	PUNCT
ejpam-4072	195	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	195	9	-	-	PUNCT
ejpam-4072	195	10	int(b	int(b	NOUN
ejpam-4072	195	11	)	)	PUNCT
ejpam-4072	195	12	)	)	PUNCT
ejpam-4072	195	13	)	)	PUNCT
ejpam-4072	195	14	)	)	PUNCT
ejpam-4072	196	1	=	=	PUNCT
ejpam-4072	197	1	τ1τ2	τ1τ2	PROPN
ejpam-4072	197	2	-	-	PROPN
ejpam-4072	197	3	pcl(f	pcl(f	PROPN
ejpam-4072	197	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	197	5	-	-	PUNCT
ejpam-4072	197	6	int(σ1σ2	int(σ1σ2	VERB
ejpam-4072	197	7	-	-	PUNCT
ejpam-4072	197	8	cl(y	cl(y	NOUN
ejpam-4072	197	9	−b	−b	NOUN
ejpam-4072	197	10	)	)	PUNCT
ejpam-4072	197	11	)	)	PUNCT
ejpam-4072	197	12	)	)	PUNCT
ejpam-4072	197	13	)	)	PUNCT
ejpam-4072	198	1	⊆	⊆	X
ejpam-4072	198	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-4072	198	3	-	-	PUNCT
ejpam-4072	198	4	cl(y	cl(y	NOUN
ejpam-4072	198	5	−b	−b	NOUN
ejpam-4072	198	6	)	)	PUNCT
ejpam-4072	198	7	)	)	PUNCT
ejpam-4072	199	1	=	=	PUNCT
ejpam-4072	199	2	x	x	X
ejpam-4072	200	1	−	−	ADP
ejpam-4072	200	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	200	3	-	-	PUNCT
ejpam-4072	200	4	int(b	int(b	NOUN
ejpam-4072	200	5	)	)	PUNCT
ejpam-4072	200	6	)	)	PUNCT
ejpam-4072	200	7	.	.	PUNCT
ejpam-4072	201	1	thus	thus	ADV
ejpam-4072	201	2	,	,	PUNCT
ejpam-4072	201	3	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	201	4	-	-	PUNCT
ejpam-4072	201	5	int(b	int(b	NOUN
ejpam-4072	201	6	)	)	PUNCT
ejpam-4072	201	7	)	)	PUNCT
ejpam-4072	202	1	⊆	⊆	X
ejpam-4072	202	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	202	3	-	-	NOUN
ejpam-4072	202	4	pint(f	pint(f	ADJ
ejpam-4072	202	5	+	+	PROPN
ejpam-4072	202	6	(	(	PUNCT
ejpam-4072	202	7	σ1σ2	σ1σ2	NUM
ejpam-4072	202	8	-	-	PUNCT
ejpam-4072	202	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	202	10	-	-	PUNCT
ejpam-4072	202	11	int(b	int(b	NOUN
ejpam-4072	202	12	)	)	PUNCT
ejpam-4072	202	13	)	)	PUNCT
ejpam-4072	202	14	)	)	PUNCT
ejpam-4072	202	15	)	)	PUNCT
ejpam-4072	202	16	.	.	PUNCT
ejpam-4072	203	1	(	(	PUNCT
ejpam-4072	203	2	5	5	X
ejpam-4072	203	3	)	)	PUNCT
ejpam-4072	203	4	⇒	⇒	NOUN
ejpam-4072	203	5	(	(	PUNCT
ejpam-4072	203	6	1	1	NUM
ejpam-4072	203	7	):	):	PUNCT
ejpam-4072	203	8	let	let	VERB
ejpam-4072	203	9	v	v	PART
ejpam-4072	203	10	be	be	AUX
ejpam-4072	203	11	any	any	DET
ejpam-4072	203	12	σ1σ2	σ1σ2	NOUN
ejpam-4072	203	13	-	-	ADJ
ejpam-4072	203	14	open	open	ADJ
ejpam-4072	203	15	set	set	NOUN
ejpam-4072	203	16	of	of	ADP
ejpam-4072	203	17	y	y	PROPN
ejpam-4072	203	18	.	.	PUNCT
ejpam-4072	204	1	by	by	ADP
ejpam-4072	204	2	(	(	PUNCT
ejpam-4072	204	3	5	5	NUM
ejpam-4072	204	4	)	)	PUNCT
ejpam-4072	204	5	,	,	PUNCT
ejpam-4072	204	6	we	we	PRON
ejpam-4072	204	7	have	have	VERB
ejpam-4072	204	8	f+(v	f+(v	NOUN
ejpam-4072	204	9	)	)	PUNCT
ejpam-4072	205	1	⊆	⊆	X
ejpam-4072	205	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	205	3	-	-	NOUN
ejpam-4072	205	4	pint(f	pint(f	ADJ
ejpam-4072	205	5	+	+	PROPN
ejpam-4072	205	6	(	(	PUNCT
ejpam-4072	205	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	205	8	-	-	NUM
ejpam-4072	205	9	cl(v	cl(v	NOUN
ejpam-4072	205	10	)	)	PUNCT
ejpam-4072	205	11	)	)	PUNCT
ejpam-4072	205	12	)	)	PUNCT
ejpam-4072	206	1	and	and	CCONJ
ejpam-4072	206	2	hence	hence	ADV
ejpam-4072	206	3	f	f	PROPN
ejpam-4072	206	4	is	be	AUX
ejpam-4072	206	5	upper	upper	ADJ
ejpam-4072	206	6	almost	almost	ADV
ejpam-4072	206	7	weakly	weakly	ADJ
ejpam-4072	206	8	(	(	PUNCT
ejpam-4072	206	9	τ1	τ1	NOUN
ejpam-4072	206	10	,	,	PUNCT
ejpam-4072	206	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	206	12	by	by	ADP
ejpam-4072	206	13	theorem	theorem	NOUN
ejpam-4072	206	14	1	1	NUM
ejpam-4072	206	15	.	.	PUNCT
ejpam-4072	206	16	c.	c.	PROPN
ejpam-4072	206	17	boonpok	boonpok	PROPN
ejpam-4072	206	18	,	,	PUNCT
ejpam-4072	206	19	c.	c.	PROPN
ejpam-4072	206	20	viriyapong	viriyapong	PROPN
ejpam-4072	206	21	/	/	SYM
ejpam-4072	206	22	eur	eur	PROPN
ejpam-4072	206	23	.	.	PUNCT
ejpam-4072	207	1	j.	j.	PROPN
ejpam-4072	207	2	pure	pure	PROPN
ejpam-4072	207	3	appl	appl	PROPN
ejpam-4072	207	4	.	.	PROPN
ejpam-4072	207	5	math	math	PROPN
ejpam-4072	207	6	,	,	PUNCT
ejpam-4072	207	7	14	14	NUM
ejpam-4072	207	8	(	(	PUNCT
ejpam-4072	207	9	4	4	NUM
ejpam-4072	207	10	)	)	PUNCT
ejpam-4072	207	11	(	(	PUNCT
ejpam-4072	207	12	2021	2021	NUM
ejpam-4072	207	13	)	)	PUNCT
ejpam-4072	207	14	,	,	PUNCT
ejpam-4072	207	15	1212	1212	NUM
ejpam-4072	207	16	-	-	SYM
ejpam-4072	207	17	1225	1225	NUM
ejpam-4072	207	18	1218	1218	NUM
ejpam-4072	207	19	theorem	theorem	VERB
ejpam-4072	207	20	4	4	NUM
ejpam-4072	207	21	.	.	X
ejpam-4072	207	22	for	for	ADP
ejpam-4072	207	23	a	a	DET
ejpam-4072	207	24	multifunction	multifunction	NOUN
ejpam-4072	207	25	f	f	NOUN
ejpam-4072	207	26	:	:	PUNCT
ejpam-4072	207	27	(	(	PUNCT
ejpam-4072	207	28	x	x	NOUN
ejpam-4072	207	29	,	,	PUNCT
ejpam-4072	207	30	τ1	τ1	NOUN
ejpam-4072	207	31	,	,	PUNCT
ejpam-4072	207	32	τ2	τ2	NOUN
ejpam-4072	207	33	)	)	PUNCT
ejpam-4072	207	34	→	→	SYM
ejpam-4072	207	35	(	(	PUNCT
ejpam-4072	207	36	y	y	PROPN
ejpam-4072	207	37	,	,	PUNCT
ejpam-4072	207	38	σ1	σ1	PROPN
ejpam-4072	207	39	,	,	PUNCT
ejpam-4072	207	40	σ2	σ2	NOUN
ejpam-4072	207	41	)	)	PUNCT
ejpam-4072	207	42	,	,	PUNCT
ejpam-4072	207	43	the	the	DET
ejpam-4072	207	44	following	follow	VERB
ejpam-4072	207	45	properties	property	NOUN
ejpam-4072	207	46	are	be	AUX
ejpam-4072	207	47	equivalent	equivalent	ADJ
ejpam-4072	207	48	:	:	PUNCT
ejpam-4072	207	49	(	(	PUNCT
ejpam-4072	207	50	1	1	X
ejpam-4072	207	51	)	)	PUNCT
ejpam-4072	207	52	f	f	PROPN
ejpam-4072	207	53	is	be	AUX
ejpam-4072	207	54	lower	low	ADJ
ejpam-4072	207	55	almost	almost	ADV
ejpam-4072	207	56	weakly	weakly	ADJ
ejpam-4072	207	57	(	(	PUNCT
ejpam-4072	207	58	τ1	τ1	NOUN
ejpam-4072	207	59	,	,	PUNCT
ejpam-4072	207	60	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	207	61	;	;	PUNCT
ejpam-4072	207	62	(	(	PUNCT
ejpam-4072	207	63	2	2	X
ejpam-4072	207	64	)	)	PUNCT
ejpam-4072	207	65	τ1	τ1	NOUN
ejpam-4072	207	66	-	-	PUNCT
ejpam-4072	207	67	cl(τ2	cl(τ2	NOUN
ejpam-4072	207	68	-	-	PUNCT
ejpam-4072	207	69	int(f	int(f	VERB
ejpam-4072	207	70	+	+	ADJ
ejpam-4072	207	71	(	(	PUNCT
ejpam-4072	207	72	σ1σ2	σ1σ2	X
ejpam-4072	207	73	-	-	PUNCT
ejpam-4072	207	74	int(h	int(h	NOUN
ejpam-4072	207	75	)	)	PUNCT
ejpam-4072	207	76	)	)	PUNCT
ejpam-4072	207	77	)	)	PUNCT
ejpam-4072	207	78	)	)	PUNCT
ejpam-4072	208	1	⊆	⊆	NUM
ejpam-4072	208	2	f+(h	f+(h	PROPN
ejpam-4072	208	3	)	)	PUNCT
ejpam-4072	208	4	for	for	ADP
ejpam-4072	208	5	every	every	DET
ejpam-4072	208	6	σ1σ2	σ1σ2	NUM
ejpam-4072	208	7	-	-	ADJ
ejpam-4072	208	8	closed	closed	ADJ
ejpam-4072	208	9	set	set	ADJ
ejpam-4072	208	10	h	h	NOUN
ejpam-4072	208	11	of	of	ADP
ejpam-4072	208	12	y	y	PROPN
ejpam-4072	208	13	;	;	PUNCT
ejpam-4072	208	14	(	(	PUNCT
ejpam-4072	208	15	3	3	X
ejpam-4072	208	16	)	)	PUNCT
ejpam-4072	208	17	τ1τ2	τ1τ2	NOUN
ejpam-4072	208	18	-	-	PROPN
ejpam-4072	208	19	pcl(f	pcl(f	PROPN
ejpam-4072	208	20	+	+	ADJ
ejpam-4072	208	21	(	(	PUNCT
ejpam-4072	208	22	σ1σ2	σ1σ2	X
ejpam-4072	208	23	-	-	PUNCT
ejpam-4072	208	24	int(h	int(h	NOUN
ejpam-4072	208	25	)	)	PUNCT
ejpam-4072	208	26	)	)	PUNCT
ejpam-4072	208	27	)	)	PUNCT
ejpam-4072	209	1	⊆	⊆	NUM
ejpam-4072	209	2	f+(h	f+(h	PROPN
ejpam-4072	209	3	)	)	PUNCT
ejpam-4072	209	4	for	for	ADP
ejpam-4072	209	5	every	every	DET
ejpam-4072	209	6	σ1σ2	σ1σ2	NUM
ejpam-4072	209	7	-	-	ADJ
ejpam-4072	209	8	closed	closed	ADJ
ejpam-4072	209	9	set	set	ADJ
ejpam-4072	209	10	h	h	NOUN
ejpam-4072	209	11	of	of	ADP
ejpam-4072	209	12	y	y	PROPN
ejpam-4072	209	13	;	;	PUNCT
ejpam-4072	209	14	(	(	PUNCT
ejpam-4072	209	15	4	4	X
ejpam-4072	209	16	)	)	PUNCT
ejpam-4072	209	17	τ1τ2	τ1τ2	NOUN
ejpam-4072	209	18	-	-	PROPN
ejpam-4072	209	19	pcl(f	pcl(f	PROPN
ejpam-4072	209	20	+	+	ADJ
ejpam-4072	209	21	(	(	PUNCT
ejpam-4072	209	22	σ1σ2	σ1σ2	NUM
ejpam-4072	209	23	-	-	PUNCT
ejpam-4072	209	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	209	25	-	-	PUNCT
ejpam-4072	209	26	cl(b	cl(b	NOUN
ejpam-4072	209	27	)	)	PUNCT
ejpam-4072	209	28	)	)	PUNCT
ejpam-4072	209	29	)	)	PUNCT
ejpam-4072	209	30	)	)	PUNCT
ejpam-4072	210	1	⊆	⊆	X
ejpam-4072	210	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	210	3	-	-	PUNCT
ejpam-4072	210	4	cl(b	cl(b	NOUN
ejpam-4072	210	5	)	)	PUNCT
ejpam-4072	210	6	)	)	PUNCT
ejpam-4072	210	7	for	for	ADP
ejpam-4072	210	8	every	every	DET
ejpam-4072	210	9	subset	subset	NOUN
ejpam-4072	210	10	b	b	PROPN
ejpam-4072	210	11	of	of	ADP
ejpam-4072	210	12	y	y	PROPN
ejpam-4072	210	13	;	;	PUNCT
ejpam-4072	210	14	(	(	PUNCT
ejpam-4072	210	15	5	5	X
ejpam-4072	210	16	)	)	PUNCT
ejpam-4072	210	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	210	18	-	-	PUNCT
ejpam-4072	210	19	int(b	int(b	NOUN
ejpam-4072	210	20	)	)	PUNCT
ejpam-4072	210	21	)	)	PUNCT
ejpam-4072	211	1	⊆	⊆	X
ejpam-4072	211	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	211	3	-	-	ADJ
ejpam-4072	211	4	pint(f	pint(f	ADJ
ejpam-4072	211	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	211	6	-	-	PUNCT
ejpam-4072	211	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	211	8	-	-	PUNCT
ejpam-4072	211	9	int(b	int(b	NOUN
ejpam-4072	211	10	)	)	PUNCT
ejpam-4072	211	11	)	)	PUNCT
ejpam-4072	211	12	)	)	PUNCT
ejpam-4072	211	13	)	)	PUNCT
ejpam-4072	212	1	for	for	ADP
ejpam-4072	212	2	every	every	DET
ejpam-4072	212	3	subset	subset	NOUN
ejpam-4072	212	4	b	b	PROPN
ejpam-4072	212	5	of	of	ADP
ejpam-4072	212	6	y	y	PROPN
ejpam-4072	212	7	.	.	PUNCT
ejpam-4072	213	1	proof	proof	NOUN
ejpam-4072	213	2	.	.	PUNCT
ejpam-4072	214	1	the	the	DET
ejpam-4072	214	2	proof	proof	NOUN
ejpam-4072	214	3	is	be	AUX
ejpam-4072	214	4	similar	similar	ADJ
ejpam-4072	214	5	to	to	ADP
ejpam-4072	214	6	that	that	PRON
ejpam-4072	214	7	of	of	ADP
ejpam-4072	214	8	theorem	theorem	ADJ
ejpam-4072	214	9	3	3	NUM
ejpam-4072	214	10	.	.	PUNCT
ejpam-4072	214	11	definition	definition	NOUN
ejpam-4072	214	12	2	2	NUM
ejpam-4072	214	13	.	.	PUNCT
ejpam-4072	215	1	[	[	X
ejpam-4072	215	2	28	28	NUM
ejpam-4072	215	3	]	]	PUNCT
ejpam-4072	215	4	let	let	VERB
ejpam-4072	215	5	a	a	PRON
ejpam-4072	215	6	be	be	AUX
ejpam-4072	215	7	a	a	DET
ejpam-4072	215	8	subset	subset	NOUN
ejpam-4072	215	9	of	of	ADP
ejpam-4072	215	10	a	a	DET
ejpam-4072	215	11	bitopological	bitopological	ADJ
ejpam-4072	215	12	space	space	NOUN
ejpam-4072	215	13	(	(	PUNCT
ejpam-4072	215	14	x	x	NOUN
ejpam-4072	215	15	,	,	PUNCT
ejpam-4072	215	16	τ1	τ1	NOUN
ejpam-4072	215	17	,	,	PUNCT
ejpam-4072	215	18	τ2	τ2	NOUN
ejpam-4072	215	19	)	)	PUNCT
ejpam-4072	215	20	.	.	PUNCT
ejpam-4072	216	1	a	a	DET
ejpam-4072	216	2	point	point	NOUN
ejpam-4072	216	3	x	x	X
ejpam-4072	216	4	∈	∈	NOUN
ejpam-4072	216	5	x	x	PUNCT
ejpam-4072	216	6	is	be	AUX
ejpam-4072	216	7	called	call	VERB
ejpam-4072	216	8	(	(	PUNCT
ejpam-4072	216	9	τ1	τ1	NOUN
ejpam-4072	216	10	,	,	PUNCT
ejpam-4072	216	11	τ2)θ	τ2)θ	ADJ
ejpam-4072	216	12	-	-	PUNCT
ejpam-4072	216	13	cluster	cluster	NOUN
ejpam-4072	216	14	point	point	NOUN
ejpam-4072	216	15	of	of	ADP
ejpam-4072	216	16	a	a	DET
ejpam-4072	216	17	if	if	SCONJ
ejpam-4072	216	18	τ1τ2	τ1τ2	NOUN
ejpam-4072	216	19	-	-	NOUN
ejpam-4072	216	20	cl(u	cl(u	NOUN
ejpam-4072	216	21	)	)	PUNCT
ejpam-4072	216	22	∩	∩	NOUN
ejpam-4072	216	23	a	a	DET
ejpam-4072	216	24	̸=	̸=	PROPN
ejpam-4072	216	25	∅	∅	NOUN
ejpam-4072	216	26	for	for	ADP
ejpam-4072	216	27	every	every	DET
ejpam-4072	216	28	τ1τ2	τ1τ2	ADJ
ejpam-4072	216	29	-	-	ADJ
ejpam-4072	216	30	open	open	ADJ
ejpam-4072	216	31	set	set	NOUN
ejpam-4072	216	32	u	u	NOUN
ejpam-4072	216	33	containing	contain	VERB
ejpam-4072	216	34	x.	x.	NOUN
ejpam-4072	216	35	the	the	DET
ejpam-4072	216	36	set	set	NOUN
ejpam-4072	216	37	of	of	ADP
ejpam-4072	216	38	all	all	DET
ejpam-4072	216	39	(	(	PUNCT
ejpam-4072	216	40	τ1	τ1	NOUN
ejpam-4072	216	41	,	,	PUNCT
ejpam-4072	216	42	τ2)θ	τ2)θ	ADJ
ejpam-4072	216	43	-	-	PUNCT
ejpam-4072	216	44	cluster	cluster	NOUN
ejpam-4072	216	45	points	point	NOUN
ejpam-4072	216	46	of	of	ADP
ejpam-4072	216	47	a	a	PRON
ejpam-4072	216	48	is	be	AUX
ejpam-4072	216	49	called	call	VERB
ejpam-4072	216	50	the	the	DET
ejpam-4072	216	51	(	(	PUNCT
ejpam-4072	216	52	τ1	τ1	NOUN
ejpam-4072	216	53	,	,	PUNCT
ejpam-4072	216	54	τ2)θ	τ2)θ	NOUN
ejpam-4072	216	55	-	-	PUNCT
ejpam-4072	216	56	closure	closure	NOUN
ejpam-4072	216	57	of	of	ADP
ejpam-4072	216	58	a	a	PRON
ejpam-4072	216	59	and	and	CCONJ
ejpam-4072	216	60	is	be	AUX
ejpam-4072	216	61	denoted	denote	VERB
ejpam-4072	216	62	by	by	ADP
ejpam-4072	216	63	(	(	PUNCT
ejpam-4072	216	64	τ1	τ1	NOUN
ejpam-4072	216	65	,	,	PUNCT
ejpam-4072	216	66	τ2)θ	τ2)θ	NOUN
ejpam-4072	216	67	-	-	PUNCT
ejpam-4072	216	68	cl(a	cl(a	NUM
ejpam-4072	216	69	)	)	PUNCT
ejpam-4072	216	70	.	.	PUNCT
ejpam-4072	217	1	a	a	DET
ejpam-4072	217	2	subset	subset	NOUN
ejpam-4072	217	3	a	a	PRON
ejpam-4072	217	4	of	of	ADP
ejpam-4072	217	5	a	a	DET
ejpam-4072	217	6	bitopological	bitopological	ADJ
ejpam-4072	217	7	space	space	NOUN
ejpam-4072	217	8	(	(	PUNCT
ejpam-4072	217	9	x	x	NOUN
ejpam-4072	217	10	,	,	PUNCT
ejpam-4072	217	11	τ1	τ1	NOUN
ejpam-4072	217	12	,	,	PUNCT
ejpam-4072	217	13	τ2	τ2	NOUN
ejpam-4072	217	14	)	)	PUNCT
ejpam-4072	217	15	is	be	AUX
ejpam-4072	217	16	said	say	VERB
ejpam-4072	217	17	to	to	PART
ejpam-4072	217	18	be	be	AUX
ejpam-4072	217	19	(	(	PUNCT
ejpam-4072	217	20	τ1	τ1	NOUN
ejpam-4072	217	21	,	,	PUNCT
ejpam-4072	217	22	τ2)θ	τ2)θ	NOUN
ejpam-4072	217	23	-	-	PUNCT
ejpam-4072	217	24	closed	closed	ADJ
ejpam-4072	217	25	[	[	X
ejpam-4072	217	26	28	28	NUM
ejpam-4072	217	27	]	]	X
ejpam-4072	217	28	if	if	SCONJ
ejpam-4072	217	29	a	a	PRON
ejpam-4072	217	30	=	=	X
ejpam-4072	217	31	(	(	PUNCT
ejpam-4072	217	32	τ1	τ1	NOUN
ejpam-4072	217	33	,	,	PUNCT
ejpam-4072	217	34	τ2)θ	τ2)θ	NOUN
ejpam-4072	217	35	-	-	PUNCT
ejpam-4072	217	36	cl(a	cl(a	NUM
ejpam-4072	217	37	)	)	PUNCT
ejpam-4072	217	38	.	.	PUNCT
ejpam-4072	218	1	the	the	DET
ejpam-4072	218	2	complement	complement	NOUN
ejpam-4072	218	3	of	of	ADP
ejpam-4072	218	4	a	a	DET
ejpam-4072	218	5	(	(	PUNCT
ejpam-4072	218	6	τ1	τ1	NOUN
ejpam-4072	218	7	,	,	PUNCT
ejpam-4072	218	8	τ2)θ	τ2)θ	ADJ
ejpam-4072	218	9	-	-	PUNCT
ejpam-4072	218	10	closed	close	VERB
ejpam-4072	218	11	set	set	NOUN
ejpam-4072	218	12	is	be	AUX
ejpam-4072	218	13	said	say	VERB
ejpam-4072	218	14	to	to	PART
ejpam-4072	218	15	be	be	AUX
ejpam-4072	218	16	(	(	PUNCT
ejpam-4072	218	17	τ1	τ1	NOUN
ejpam-4072	218	18	,	,	PUNCT
ejpam-4072	218	19	τ2)θ	τ2)θ	NOUN
ejpam-4072	218	20	-	-	PUNCT
ejpam-4072	218	21	open	open	ADJ
ejpam-4072	218	22	.	.	PUNCT
ejpam-4072	219	1	the	the	DET
ejpam-4072	219	2	union	union	NOUN
ejpam-4072	219	3	of	of	ADP
ejpam-4072	219	4	all	all	DET
ejpam-4072	219	5	(	(	PUNCT
ejpam-4072	219	6	τ1	τ1	NOUN
ejpam-4072	219	7	,	,	PUNCT
ejpam-4072	219	8	τ2)θ	τ2)θ	ADJ
ejpam-4072	219	9	-	-	PUNCT
ejpam-4072	219	10	open	open	ADJ
ejpam-4072	219	11	sets	set	NOUN
ejpam-4072	219	12	contained	contain	VERB
ejpam-4072	219	13	in	in	ADP
ejpam-4072	219	14	a	a	PRON
ejpam-4072	219	15	is	be	AUX
ejpam-4072	219	16	called	call	VERB
ejpam-4072	219	17	the	the	DET
ejpam-4072	219	18	(	(	PUNCT
ejpam-4072	219	19	τ1	τ1	NOUN
ejpam-4072	219	20	,	,	PUNCT
ejpam-4072	219	21	τ2)θ	τ2)θ	ADJ
ejpam-4072	219	22	-	-	PUNCT
ejpam-4072	219	23	interior	interior	NOUN
ejpam-4072	219	24	[	[	X
ejpam-4072	219	25	28	28	NUM
ejpam-4072	219	26	]	]	PUNCT
ejpam-4072	219	27	of	of	ADP
ejpam-4072	219	28	a	a	PRON
ejpam-4072	219	29	and	and	CCONJ
ejpam-4072	219	30	is	be	AUX
ejpam-4072	219	31	denoted	denote	VERB
ejpam-4072	219	32	by	by	ADP
ejpam-4072	219	33	(	(	PUNCT
ejpam-4072	219	34	τ1	τ1	NOUN
ejpam-4072	219	35	,	,	PUNCT
ejpam-4072	219	36	τ2)θ	τ2)θ	NOUN
ejpam-4072	219	37	-	-	PUNCT
ejpam-4072	219	38	int(a	int(a	NOUN
ejpam-4072	219	39	)	)	PUNCT
ejpam-4072	219	40	.	.	PUNCT
ejpam-4072	220	1	lemma	lemma	PROPN
ejpam-4072	220	2	7	7	NUM
ejpam-4072	220	3	.	.	PUNCT
ejpam-4072	221	1	[	[	X
ejpam-4072	221	2	28	28	NUM
ejpam-4072	221	3	]	]	PUNCT
ejpam-4072	221	4	for	for	ADP
ejpam-4072	221	5	a	a	DET
ejpam-4072	221	6	subset	subset	NOUN
ejpam-4072	221	7	a	a	PRON
ejpam-4072	221	8	of	of	ADP
ejpam-4072	221	9	a	a	DET
ejpam-4072	221	10	bitopological	bitopological	ADJ
ejpam-4072	221	11	space	space	NOUN
ejpam-4072	221	12	(	(	PUNCT
ejpam-4072	221	13	x	x	NOUN
ejpam-4072	221	14	,	,	PUNCT
ejpam-4072	221	15	τ1	τ1	NOUN
ejpam-4072	221	16	,	,	PUNCT
ejpam-4072	221	17	τ2	τ2	NOUN
ejpam-4072	221	18	)	)	PUNCT
ejpam-4072	221	19	,	,	PUNCT
ejpam-4072	221	20	the	the	DET
ejpam-4072	221	21	following	follow	VERB
ejpam-4072	221	22	properties	property	NOUN
ejpam-4072	221	23	hold	hold	VERB
ejpam-4072	221	24	:	:	PUNCT
ejpam-4072	221	25	(	(	PUNCT
ejpam-4072	221	26	1	1	X
ejpam-4072	221	27	)	)	PUNCT
ejpam-4072	221	28	if	if	SCONJ
ejpam-4072	221	29	a	a	PRON
ejpam-4072	221	30	is	be	AUX
ejpam-4072	221	31	τ2τ2	τ2τ2	VERB
ejpam-4072	221	32	-	-	VERB
ejpam-4072	221	33	open	open	ADJ
ejpam-4072	221	34	in	in	ADP
ejpam-4072	221	35	x	x	NOUN
ejpam-4072	221	36	,	,	PUNCT
ejpam-4072	221	37	then	then	ADV
ejpam-4072	221	38	τ1τ2	τ1τ2	NOUN
ejpam-4072	221	39	-	-	NUM
ejpam-4072	221	40	cl(a	cl(a	NUM
ejpam-4072	221	41	)	)	PUNCT
ejpam-4072	221	42	=	=	PUNCT
ejpam-4072	221	43	(	(	PUNCT
ejpam-4072	221	44	τ1	τ1	NOUN
ejpam-4072	221	45	,	,	PUNCT
ejpam-4072	221	46	τ2)θ	τ2)θ	NOUN
ejpam-4072	221	47	-	-	PUNCT
ejpam-4072	221	48	cl(a	cl(a	NUM
ejpam-4072	221	49	)	)	PUNCT
ejpam-4072	221	50	.	.	PUNCT
ejpam-4072	222	1	(	(	PUNCT
ejpam-4072	222	2	2	2	X
ejpam-4072	222	3	)	)	PUNCT
ejpam-4072	222	4	(	(	PUNCT
ejpam-4072	222	5	τ1	τ1	NOUN
ejpam-4072	222	6	,	,	PUNCT
ejpam-4072	222	7	τ2)θ	τ2)θ	NOUN
ejpam-4072	222	8	-	-	PUNCT
ejpam-4072	222	9	cl(a	cl(a	NUM
ejpam-4072	222	10	)	)	PUNCT
ejpam-4072	222	11	is	be	AUX
ejpam-4072	222	12	τ1τ2	τ1τ2	NOUN
ejpam-4072	222	13	-	-	ADJ
ejpam-4072	222	14	closed	closed	ADJ
ejpam-4072	222	15	in	in	ADP
ejpam-4072	222	16	x.	x.	NOUN
ejpam-4072	222	17	definition	definition	NOUN
ejpam-4072	222	18	3	3	NUM
ejpam-4072	222	19	.	.	PUNCT
ejpam-4072	223	1	a	a	DET
ejpam-4072	223	2	subset	subset	NOUN
ejpam-4072	223	3	a	a	PRON
ejpam-4072	223	4	of	of	ADP
ejpam-4072	223	5	a	a	DET
ejpam-4072	223	6	bitopological	bitopological	ADJ
ejpam-4072	223	7	space	space	NOUN
ejpam-4072	223	8	(	(	PUNCT
ejpam-4072	223	9	x	x	NOUN
ejpam-4072	223	10	,	,	PUNCT
ejpam-4072	223	11	τ1	τ1	NOUN
ejpam-4072	223	12	,	,	PUNCT
ejpam-4072	223	13	τ2	τ2	NOUN
ejpam-4072	223	14	)	)	PUNCT
ejpam-4072	223	15	is	be	AUX
ejpam-4072	223	16	said	say	VERB
ejpam-4072	223	17	to	to	PART
ejpam-4072	223	18	be	be	AUX
ejpam-4072	223	19	(	(	PUNCT
ejpam-4072	223	20	τ1	τ1	NOUN
ejpam-4072	223	21	,	,	PUNCT
ejpam-4072	223	22	τ2)r	τ2)r	NOUN
ejpam-4072	223	23	-	-	PUNCT
ejpam-4072	223	24	closed	closed	ADJ
ejpam-4072	223	25	[	[	X
ejpam-4072	223	26	28	28	NUM
ejpam-4072	223	27	]	]	X
ejpam-4072	223	28	(	(	PUNCT
ejpam-4072	223	29	resp	resp	NOUN
ejpam-4072	223	30	.	.	PUNCT
ejpam-4072	224	1	(	(	PUNCT
ejpam-4072	224	2	τ1	τ1	NOUN
ejpam-4072	224	3	,	,	PUNCT
ejpam-4072	224	4	τ2)p	τ2)p	NOUN
ejpam-4072	224	5	-	-	ADJ
ejpam-4072	224	6	open	open	ADJ
ejpam-4072	224	7	[	[	X
ejpam-4072	224	8	4	4	NUM
ejpam-4072	224	9	]	]	PUNCT
ejpam-4072	224	10	)	)	PUNCT
ejpam-4072	224	11	if	if	SCONJ
ejpam-4072	224	12	a	a	DET
ejpam-4072	224	13	=	=	PUNCT
ejpam-4072	224	14	τ1τ2	τ1τ2	NOUN
ejpam-4072	224	15	-	-	ADJ
ejpam-4072	224	16	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-4072	224	17	-	-	PUNCT
ejpam-4072	224	18	int(a	int(a	NOUN
ejpam-4072	224	19	)	)	PUNCT
ejpam-4072	224	20	)	)	PUNCT
ejpam-4072	224	21	(	(	PUNCT
ejpam-4072	224	22	resp	resp	NOUN
ejpam-4072	224	23	.	.	PUNCT
ejpam-4072	225	1	a	a	DET
ejpam-4072	225	2	⊆	⊆	NUM
ejpam-4072	225	3	τ1τ2	τ1τ2	NOUN
ejpam-4072	225	4	-	-	NOUN
ejpam-4072	225	5	int(τ1τ2	int(τ1τ2	NOUN
ejpam-4072	225	6	-	-	PUNCT
ejpam-4072	225	7	cl(a	cl(a	NUM
ejpam-4072	225	8	)	)	PUNCT
ejpam-4072	225	9	)	)	PUNCT
ejpam-4072	225	10	)	)	PUNCT
ejpam-4072	225	11	.	.	PUNCT
ejpam-4072	226	1	theorem	theorem	NOUN
ejpam-4072	226	2	5	5	NUM
ejpam-4072	226	3	.	.	X
ejpam-4072	226	4	for	for	ADP
ejpam-4072	226	5	a	a	DET
ejpam-4072	226	6	multifunction	multifunction	NOUN
ejpam-4072	227	1	f	f	NOUN
ejpam-4072	227	2	:	:	PUNCT
ejpam-4072	227	3	(	(	PUNCT
ejpam-4072	227	4	x	x	NOUN
ejpam-4072	227	5	,	,	PUNCT
ejpam-4072	227	6	τ1	τ1	NOUN
ejpam-4072	227	7	,	,	PUNCT
ejpam-4072	227	8	τ2	τ2	NOUN
ejpam-4072	227	9	)	)	PUNCT
ejpam-4072	227	10	→	→	SYM
ejpam-4072	227	11	(	(	PUNCT
ejpam-4072	227	12	y	y	PROPN
ejpam-4072	227	13	,	,	PUNCT
ejpam-4072	227	14	σ1	σ1	PROPN
ejpam-4072	227	15	,	,	PUNCT
ejpam-4072	227	16	σ2	σ2	NOUN
ejpam-4072	227	17	)	)	PUNCT
ejpam-4072	227	18	,	,	PUNCT
ejpam-4072	227	19	the	the	DET
ejpam-4072	227	20	following	follow	VERB
ejpam-4072	227	21	properties	property	NOUN
ejpam-4072	227	22	are	be	AUX
ejpam-4072	227	23	equivalent	equivalent	ADJ
ejpam-4072	227	24	:	:	PUNCT
ejpam-4072	227	25	(	(	PUNCT
ejpam-4072	227	26	1	1	X
ejpam-4072	227	27	)	)	PUNCT
ejpam-4072	227	28	f	f	PROPN
ejpam-4072	227	29	is	be	AUX
ejpam-4072	227	30	upper	upper	ADJ
ejpam-4072	227	31	almost	almost	ADV
ejpam-4072	227	32	weakly	weakly	ADJ
ejpam-4072	227	33	(	(	PUNCT
ejpam-4072	227	34	τ1	τ1	NOUN
ejpam-4072	227	35	,	,	PUNCT
ejpam-4072	227	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	227	37	;	;	PUNCT
ejpam-4072	227	38	(	(	PUNCT
ejpam-4072	227	39	2	2	X
ejpam-4072	227	40	)	)	PUNCT
ejpam-4072	227	41	τ1τ2	τ1τ2	NOUN
ejpam-4072	227	42	-	-	PROPN
ejpam-4072	227	43	pcl(f	pcl(f	PROPN
ejpam-4072	227	44	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	227	45	-	-	PUNCT
ejpam-4072	227	46	int((σ1	int((σ1	ADJ
ejpam-4072	227	47	,	,	PUNCT
ejpam-4072	227	48	σ2)θ	σ2)θ	ADJ
ejpam-4072	227	49	-	-	PUNCT
ejpam-4072	227	50	cl(b	cl(b	NOUN
ejpam-4072	227	51	)	)	PUNCT
ejpam-4072	227	52	)	)	PUNCT
ejpam-4072	227	53	)	)	PUNCT
ejpam-4072	227	54	)	)	PUNCT
ejpam-4072	228	1	⊆	⊆	NUM
ejpam-4072	228	2	f−((σ1	f−((σ1	NOUN
ejpam-4072	228	3	,	,	PUNCT
ejpam-4072	228	4	σ2)θ	σ2)θ	ADJ
ejpam-4072	228	5	-	-	PUNCT
ejpam-4072	228	6	cl(b	cl(b	NOUN
ejpam-4072	228	7	)	)	PUNCT
ejpam-4072	228	8	)	)	PUNCT
ejpam-4072	228	9	for	for	ADP
ejpam-4072	228	10	every	every	DET
ejpam-4072	228	11	subset	subset	NOUN
ejpam-4072	228	12	b	b	PROPN
ejpam-4072	228	13	of	of	ADP
ejpam-4072	228	14	y	y	PROPN
ejpam-4072	228	15	;	;	PUNCT
ejpam-4072	228	16	(	(	PUNCT
ejpam-4072	228	17	3	3	X
ejpam-4072	228	18	)	)	PUNCT
ejpam-4072	228	19	τ1τ2	τ1τ2	NOUN
ejpam-4072	228	20	-	-	PROPN
ejpam-4072	228	21	pcl(f	pcl(f	PROPN
ejpam-4072	228	22	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	228	23	-	-	PUNCT
ejpam-4072	228	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	228	25	-	-	PUNCT
ejpam-4072	228	26	cl(v	cl(v	NOUN
ejpam-4072	228	27	)	)	PUNCT
ejpam-4072	228	28	)	)	PUNCT
ejpam-4072	228	29	)	)	PUNCT
ejpam-4072	228	30	)	)	PUNCT
ejpam-4072	229	1	⊆	⊆	X
ejpam-4072	229	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	229	3	-	-	PUNCT
ejpam-4072	229	4	cl(v	cl(v	NOUN
ejpam-4072	229	5	)	)	PUNCT
ejpam-4072	229	6	)	)	PUNCT
ejpam-4072	229	7	for	for	ADP
ejpam-4072	229	8	every	every	DET
ejpam-4072	229	9	σ1σ2	σ1σ2	NOUN
ejpam-4072	229	10	-	-	ADJ
ejpam-4072	229	11	open	open	ADJ
ejpam-4072	229	12	set	set	NOUN
ejpam-4072	229	13	v	v	NOUN
ejpam-4072	229	14	of	of	ADP
ejpam-4072	229	15	y	y	PROPN
ejpam-4072	229	16	;	;	PUNCT
ejpam-4072	229	17	(	(	PUNCT
ejpam-4072	229	18	4	4	X
ejpam-4072	229	19	)	)	PUNCT
ejpam-4072	229	20	τ1τ2	τ1τ2	NOUN
ejpam-4072	229	21	-	-	PROPN
ejpam-4072	229	22	pcl(f	pcl(f	PROPN
ejpam-4072	229	23	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	229	24	-	-	PUNCT
ejpam-4072	229	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	229	26	-	-	PUNCT
ejpam-4072	229	27	cl(v	cl(v	NOUN
ejpam-4072	229	28	)	)	PUNCT
ejpam-4072	229	29	)	)	PUNCT
ejpam-4072	229	30	)	)	PUNCT
ejpam-4072	229	31	)	)	PUNCT
ejpam-4072	230	1	⊆	⊆	X
ejpam-4072	230	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	230	3	-	-	PUNCT
ejpam-4072	230	4	cl(v	cl(v	NOUN
ejpam-4072	230	5	)	)	PUNCT
ejpam-4072	230	6	)	)	PUNCT
ejpam-4072	230	7	for	for	ADP
ejpam-4072	230	8	every	every	DET
ejpam-4072	230	9	(	(	PUNCT
ejpam-4072	230	10	σ1	σ1	PROPN
ejpam-4072	230	11	,	,	PUNCT
ejpam-4072	230	12	σ2)p	σ2)p	NOUN
ejpam-4072	230	13	-	-	PUNCT
ejpam-4072	230	14	open	open	NOUN
ejpam-4072	230	15	set	set	NOUN
ejpam-4072	230	16	v	v	NOUN
ejpam-4072	230	17	of	of	ADP
ejpam-4072	230	18	y	y	PROPN
ejpam-4072	230	19	;	;	PUNCT
ejpam-4072	230	20	c.	c.	PROPN
ejpam-4072	230	21	boonpok	boonpok	PROPN
ejpam-4072	230	22	,	,	PUNCT
ejpam-4072	230	23	c.	c.	PROPN
ejpam-4072	230	24	viriyapong	viriyapong	PROPN
ejpam-4072	230	25	/	/	SYM
ejpam-4072	230	26	eur	eur	PROPN
ejpam-4072	230	27	.	.	PUNCT
ejpam-4072	231	1	j.	j.	PROPN
ejpam-4072	231	2	pure	pure	PROPN
ejpam-4072	231	3	appl	appl	PROPN
ejpam-4072	231	4	.	.	PROPN
ejpam-4072	231	5	math	math	PROPN
ejpam-4072	231	6	,	,	PUNCT
ejpam-4072	231	7	14	14	NUM
ejpam-4072	231	8	(	(	PUNCT
ejpam-4072	231	9	4	4	NUM
ejpam-4072	231	10	)	)	PUNCT
ejpam-4072	231	11	(	(	PUNCT
ejpam-4072	231	12	2021	2021	NUM
ejpam-4072	231	13	)	)	PUNCT
ejpam-4072	231	14	,	,	PUNCT
ejpam-4072	231	15	1212	1212	NUM
ejpam-4072	231	16	-	-	SYM
ejpam-4072	231	17	1225	1225	NUM
ejpam-4072	231	18	1219	1219	NUM
ejpam-4072	231	19	(	(	PUNCT
ejpam-4072	231	20	5	5	NUM
ejpam-4072	231	21	)	)	PUNCT
ejpam-4072	231	22	τ1τ2	τ1τ2	NOUN
ejpam-4072	231	23	-	-	PROPN
ejpam-4072	231	24	pcl(f	pcl(f	PROPN
ejpam-4072	231	25	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	231	26	-	-	PUNCT
ejpam-4072	231	27	int(h	int(h	NOUN
ejpam-4072	231	28	)	)	PUNCT
ejpam-4072	231	29	)	)	PUNCT
ejpam-4072	231	30	)	)	PUNCT
ejpam-4072	232	1	⊆	⊆	NUM
ejpam-4072	232	2	f−(h	f−(h	NOUN
ejpam-4072	232	3	)	)	PUNCT
ejpam-4072	232	4	for	for	ADP
ejpam-4072	232	5	every	every	DET
ejpam-4072	232	6	(	(	PUNCT
ejpam-4072	232	7	σ1	σ1	PROPN
ejpam-4072	232	8	,	,	PUNCT
ejpam-4072	232	9	σ2)r	σ2)r	NOUN
ejpam-4072	232	10	-	-	PUNCT
ejpam-4072	232	11	closed	close	VERB
ejpam-4072	232	12	set	set	ADJ
ejpam-4072	232	13	h	h	NOUN
ejpam-4072	232	14	of	of	ADP
ejpam-4072	232	15	y	y	PROPN
ejpam-4072	232	16	.	.	PUNCT
ejpam-4072	233	1	proof	proof	NOUN
ejpam-4072	233	2	.	.	PUNCT
ejpam-4072	234	1	(	(	PUNCT
ejpam-4072	234	2	1	1	X
ejpam-4072	234	3	)	)	PUNCT
ejpam-4072	234	4	⇒	⇒	NOUN
ejpam-4072	234	5	(	(	PUNCT
ejpam-4072	234	6	2	2	NUM
ejpam-4072	234	7	):	):	PUNCT
ejpam-4072	234	8	let	let	VERB
ejpam-4072	234	9	b	b	X
ejpam-4072	234	10	be	be	AUX
ejpam-4072	234	11	any	any	DET
ejpam-4072	234	12	subset	subset	NOUN
ejpam-4072	234	13	of	of	ADP
ejpam-4072	234	14	y	y	PROPN
ejpam-4072	234	15	.	.	PUNCT
ejpam-4072	235	1	let	let	VERB
ejpam-4072	235	2	x	x	PUNCT
ejpam-4072	235	3	∈	∈	PROPN
ejpam-4072	235	4	x	x	PUNCT
ejpam-4072	235	5	−f−((σ1	−f−((σ1	ADV
ejpam-4072	235	6	,	,	PUNCT
ejpam-4072	235	7	σ2)θ	σ2)θ	ADJ
ejpam-4072	235	8	-	-	PUNCT
ejpam-4072	235	9	cl(b	cl(b	NOUN
ejpam-4072	235	10	)	)	PUNCT
ejpam-4072	235	11	)	)	PUNCT
ejpam-4072	235	12	.	.	PUNCT
ejpam-4072	236	1	then	then	ADV
ejpam-4072	236	2	,	,	PUNCT
ejpam-4072	236	3	x	x	PUNCT
ejpam-4072	236	4	∈	∈	PROPN
ejpam-4072	236	5	f+(y	f+(y	X
ejpam-4072	236	6	−	−	PROPN
ejpam-4072	236	7	(	(	PUNCT
ejpam-4072	236	8	σ1	σ1	PROPN
ejpam-4072	236	9	,	,	PUNCT
ejpam-4072	236	10	σ2)θ	σ2)θ	NOUN
ejpam-4072	236	11	-	-	PUNCT
ejpam-4072	236	12	cl(b	cl(b	NOUN
ejpam-4072	236	13	)	)	PUNCT
ejpam-4072	236	14	)	)	PUNCT
ejpam-4072	236	15	and	and	CCONJ
ejpam-4072	236	16	(	(	PUNCT
ejpam-4072	236	17	σ1	σ1	PROPN
ejpam-4072	236	18	,	,	PUNCT
ejpam-4072	236	19	σ2)θ	σ2)θ	NOUN
ejpam-4072	236	20	-	-	PUNCT
ejpam-4072	236	21	cl(b	cl(b	NOUN
ejpam-4072	236	22	)	)	PUNCT
ejpam-4072	236	23	is	be	AUX
ejpam-4072	236	24	σ1σ2	σ1σ2	NOUN
ejpam-4072	236	25	-	-	ADJ
ejpam-4072	236	26	closed	closed	ADJ
ejpam-4072	236	27	in	in	ADP
ejpam-4072	236	28	y	y	PROPN
ejpam-4072	236	29	.	.	PUNCT
ejpam-4072	237	1	by	by	ADP
ejpam-4072	237	2	theorem	theorem	NOUN
ejpam-4072	237	3	1	1	NUM
ejpam-4072	237	4	,	,	PUNCT
ejpam-4072	237	5	there	there	PRON
ejpam-4072	237	6	exists	exist	VERB
ejpam-4072	237	7	a	a	DET
ejpam-4072	237	8	τ1τ2	τ1τ2	NOUN
ejpam-4072	237	9	-	-	ADJ
ejpam-4072	237	10	preopen	preopen	ADJ
ejpam-4072	237	11	set	set	NOUN
ejpam-4072	237	12	u	u	NOUN
ejpam-4072	237	13	of	of	ADP
ejpam-4072	237	14	x	x	PUNCT
ejpam-4072	237	15	containing	contain	VERB
ejpam-4072	237	16	x	x	PUNCT
ejpam-4072	237	17	such	such	ADJ
ejpam-4072	237	18	that	that	SCONJ
ejpam-4072	237	19	u	u	NOUN
ejpam-4072	237	20	⊆	⊆	NUM
ejpam-4072	237	21	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	237	22	-	-	PUNCT
ejpam-4072	237	23	cl(y	cl(y	NOUN
ejpam-4072	237	24	−	−	PROPN
ejpam-4072	237	25	(	(	PUNCT
ejpam-4072	237	26	σ1	σ1	PROPN
ejpam-4072	237	27	,	,	PUNCT
ejpam-4072	237	28	σ2)θ	σ2)θ	NOUN
ejpam-4072	237	29	-	-	PUNCT
ejpam-4072	237	30	cl(b	cl(b	NOUN
ejpam-4072	237	31	)	)	PUNCT
ejpam-4072	237	32	)	)	PUNCT
ejpam-4072	237	33	)	)	PUNCT
ejpam-4072	238	1	=	=	PUNCT
ejpam-4072	239	1	f+(y	f+(y	NOUN
ejpam-4072	239	2	−	−	NUM
ejpam-4072	239	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	239	4	-	-	PUNCT
ejpam-4072	239	5	int((σ1	int((σ1	ADJ
ejpam-4072	239	6	,	,	PUNCT
ejpam-4072	239	7	σ2)θ	σ2)θ	ADJ
ejpam-4072	239	8	-	-	PUNCT
ejpam-4072	239	9	cl(b	cl(b	NOUN
ejpam-4072	239	10	)	)	PUNCT
ejpam-4072	239	11	)	)	PUNCT
ejpam-4072	239	12	)	)	PUNCT
ejpam-4072	240	1	=	=	PUNCT
ejpam-4072	240	2	x	x	X
ejpam-4072	240	3	−	−	NOUN
ejpam-4072	240	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	240	5	-	-	PUNCT
ejpam-4072	240	6	int((σ1	int((σ1	ADJ
ejpam-4072	240	7	,	,	PUNCT
ejpam-4072	240	8	σ2)θ	σ2)θ	ADJ
ejpam-4072	240	9	-	-	PUNCT
ejpam-4072	240	10	cl(b	cl(b	NOUN
ejpam-4072	240	11	)	)	PUNCT
ejpam-4072	240	12	)	)	PUNCT
ejpam-4072	240	13	)	)	PUNCT
ejpam-4072	240	14	.	.	PUNCT
ejpam-4072	241	1	thus	thus	ADV
ejpam-4072	241	2	,	,	PUNCT
ejpam-4072	241	3	u	u	PROPN
ejpam-4072	241	4	∩	∩	NOUN
ejpam-4072	241	5	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	241	6	-	-	PUNCT
ejpam-4072	241	7	int((σ1	int((σ1	PROPN
ejpam-4072	241	8	,	,	PUNCT
ejpam-4072	241	9	σ2)θ	σ2)θ	ADJ
ejpam-4072	241	10	-	-	PUNCT
ejpam-4072	241	11	cl(b	cl(b	NOUN
ejpam-4072	241	12	)	)	PUNCT
ejpam-4072	241	13	)	)	PUNCT
ejpam-4072	241	14	)	)	PUNCT
ejpam-4072	242	1	=	=	NOUN
ejpam-4072	242	2	∅	∅	NOUN
ejpam-4072	242	3	and	and	CCONJ
ejpam-4072	242	4	hence	hence	ADV
ejpam-4072	242	5	x	x	X
ejpam-4072	242	6	∈	∈	NOUN
ejpam-4072	242	7	x	x	INTJ
ejpam-4072	242	8	−	−	PUNCT
ejpam-4072	242	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	242	10	-	-	PROPN
ejpam-4072	242	11	pcl(f	pcl(f	PROPN
ejpam-4072	242	12	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	242	13	-	-	PUNCT
ejpam-4072	242	14	int((σ1	int((σ1	ADJ
ejpam-4072	242	15	,	,	PUNCT
ejpam-4072	242	16	σ2)θ	σ2)θ	ADJ
ejpam-4072	242	17	-	-	PUNCT
ejpam-4072	242	18	cl(b	cl(b	NOUN
ejpam-4072	242	19	)	)	PUNCT
ejpam-4072	242	20	)	)	PUNCT
ejpam-4072	242	21	)	)	PUNCT
ejpam-4072	242	22	)	)	PUNCT
ejpam-4072	242	23	.	.	PUNCT
ejpam-4072	243	1	therefore	therefore	ADV
ejpam-4072	243	2	,	,	PUNCT
ejpam-4072	243	3	τ1τ2	τ1τ2	NOUN
ejpam-4072	243	4	-	-	PROPN
ejpam-4072	243	5	pcl(f	pcl(f	PROPN
ejpam-4072	243	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	243	7	-	-	PUNCT
ejpam-4072	243	8	int((σ1	int((σ1	ADJ
ejpam-4072	243	9	,	,	PUNCT
ejpam-4072	243	10	σ2)θ	σ2)θ	ADJ
ejpam-4072	243	11	-	-	PUNCT
ejpam-4072	243	12	cl(b	cl(b	NOUN
ejpam-4072	243	13	)	)	PUNCT
ejpam-4072	243	14	)	)	PUNCT
ejpam-4072	243	15	)	)	PUNCT
ejpam-4072	243	16	)	)	PUNCT
ejpam-4072	244	1	⊆	⊆	NUM
ejpam-4072	244	2	f−((σ1	f−((σ1	NOUN
ejpam-4072	244	3	,	,	PUNCT
ejpam-4072	244	4	σ2)θ	σ2)θ	ADJ
ejpam-4072	244	5	-	-	PUNCT
ejpam-4072	244	6	cl(b	cl(b	NOUN
ejpam-4072	244	7	)	)	PUNCT
ejpam-4072	244	8	)	)	PUNCT
ejpam-4072	244	9	.	.	PUNCT
ejpam-4072	245	1	(	(	PUNCT
ejpam-4072	245	2	2	2	X
ejpam-4072	245	3	)	)	PUNCT
ejpam-4072	245	4	⇒	⇒	NOUN
ejpam-4072	245	5	(	(	PUNCT
ejpam-4072	245	6	3	3	NUM
ejpam-4072	245	7	):	):	PUNCT
ejpam-4072	245	8	the	the	DET
ejpam-4072	245	9	proof	proof	NOUN
ejpam-4072	245	10	is	be	AUX
ejpam-4072	245	11	obvious	obvious	ADJ
ejpam-4072	245	12	since	since	SCONJ
ejpam-4072	245	13	(	(	PUNCT
ejpam-4072	245	14	σ1	σ1	PROPN
ejpam-4072	245	15	,	,	PUNCT
ejpam-4072	245	16	σ2)θ	σ2)θ	NOUN
ejpam-4072	245	17	-	-	PUNCT
ejpam-4072	245	18	cl(v	cl(v	NOUN
ejpam-4072	245	19	)	)	PUNCT
ejpam-4072	245	20	=	=	SYM
ejpam-4072	245	21	σ1σ2	σ1σ2	NOUN
ejpam-4072	245	22	-	-	NUM
ejpam-4072	245	23	cl(v	cl(v	NOUN
ejpam-4072	245	24	)	)	PUNCT
ejpam-4072	245	25	for	for	ADP
ejpam-4072	245	26	every	every	DET
ejpam-4072	245	27	σ1σ2open	σ1σ2open	PUNCT
ejpam-4072	245	28	set	set	VERB
ejpam-4072	245	29	v	v	NOUN
ejpam-4072	245	30	of	of	ADP
ejpam-4072	245	31	y	y	PROPN
ejpam-4072	245	32	.	.	PUNCT
ejpam-4072	246	1	(	(	PUNCT
ejpam-4072	246	2	3	3	X
ejpam-4072	246	3	)	)	PUNCT
ejpam-4072	246	4	⇒	⇒	NOUN
ejpam-4072	246	5	(	(	PUNCT
ejpam-4072	246	6	4	4	NUM
ejpam-4072	246	7	):	):	PUNCT
ejpam-4072	246	8	let	let	VERB
ejpam-4072	246	9	v	v	PART
ejpam-4072	246	10	be	be	AUX
ejpam-4072	246	11	any	any	DET
ejpam-4072	246	12	(	(	PUNCT
ejpam-4072	246	13	σ1	σ1	PROPN
ejpam-4072	246	14	,	,	PUNCT
ejpam-4072	246	15	σ2)p	σ2)p	NOUN
ejpam-4072	246	16	-	-	PUNCT
ejpam-4072	246	17	open	open	ADJ
ejpam-4072	246	18	set	set	NOUN
ejpam-4072	246	19	of	of	ADP
ejpam-4072	246	20	y	y	PROPN
ejpam-4072	246	21	.	.	PUNCT
ejpam-4072	247	1	then	then	ADV
ejpam-4072	247	2	,	,	PUNCT
ejpam-4072	247	3	v	v	ADP
ejpam-4072	247	4	⊆	⊆	NUM
ejpam-4072	247	5	σ1σ2	σ1σ2	NOUN
ejpam-4072	247	6	-	-	PUNCT
ejpam-4072	247	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	247	8	-	-	PUNCT
ejpam-4072	247	9	cl(v	cl(v	NOUN
ejpam-4072	247	10	)	)	PUNCT
ejpam-4072	247	11	)	)	PUNCT
ejpam-4072	247	12	and	and	CCONJ
ejpam-4072	247	13	by	by	ADP
ejpam-4072	247	14	(	(	PUNCT
ejpam-4072	247	15	3	3	NUM
ejpam-4072	247	16	)	)	PUNCT
ejpam-4072	247	17	,	,	PUNCT
ejpam-4072	247	18	τ1τ2	τ1τ2	PROPN
ejpam-4072	247	19	-	-	PROPN
ejpam-4072	247	20	pcl(f	pcl(f	PROPN
ejpam-4072	247	21	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	247	22	-	-	PUNCT
ejpam-4072	247	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	247	24	-	-	PUNCT
ejpam-4072	247	25	cl(v	cl(v	NOUN
ejpam-4072	247	26	)	)	PUNCT
ejpam-4072	247	27	)	)	PUNCT
ejpam-4072	247	28	)	)	PUNCT
ejpam-4072	247	29	)	)	PUNCT
ejpam-4072	248	1	=	=	PUNCT
ejpam-4072	249	1	τ1τ2	τ1τ2	PROPN
ejpam-4072	249	2	-	-	PROPN
ejpam-4072	249	3	pcl(f	pcl(f	PROPN
ejpam-4072	249	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	249	5	-	-	PUNCT
ejpam-4072	249	6	int(σ1σ2	int(σ1σ2	ADV
ejpam-4072	249	7	-	-	PUNCT
ejpam-4072	249	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	249	9	-	-	PUNCT
ejpam-4072	249	10	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	249	11	-	-	PUNCT
ejpam-4072	249	12	cl(v	cl(v	NOUN
ejpam-4072	249	13	)	)	PUNCT
ejpam-4072	249	14	)	)	PUNCT
ejpam-4072	249	15	)	)	PUNCT
ejpam-4072	249	16	)	)	PUNCT
ejpam-4072	249	17	)	)	PUNCT
ejpam-4072	249	18	)	)	PUNCT
ejpam-4072	250	1	⊆	⊆	X
ejpam-4072	250	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-4072	250	3	-	-	PUNCT
ejpam-4072	250	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	250	5	-	-	PUNCT
ejpam-4072	250	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	250	7	-	-	PUNCT
ejpam-4072	250	8	int(v	int(v	NOUN
ejpam-4072	250	9	)	)	PUNCT
ejpam-4072	250	10	)	)	PUNCT
ejpam-4072	250	11	)	)	PUNCT
ejpam-4072	250	12	)	)	PUNCT
ejpam-4072	251	1	=	=	PUNCT
ejpam-4072	251	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	251	3	-	-	PUNCT
ejpam-4072	251	4	cl(v	cl(v	NOUN
ejpam-4072	251	5	)	)	PUNCT
ejpam-4072	251	6	)	)	PUNCT
ejpam-4072	251	7	.	.	PUNCT
ejpam-4072	252	1	(	(	PUNCT
ejpam-4072	252	2	4	4	X
ejpam-4072	252	3	)	)	PUNCT
ejpam-4072	252	4	⇒	⇒	NOUN
ejpam-4072	252	5	(	(	PUNCT
ejpam-4072	252	6	5	5	NUM
ejpam-4072	252	7	):	):	PUNCT
ejpam-4072	252	8	let	let	VERB
ejpam-4072	252	9	h	h	PRON
ejpam-4072	252	10	be	be	AUX
ejpam-4072	252	11	any	any	DET
ejpam-4072	252	12	(	(	PUNCT
ejpam-4072	252	13	σ1	σ1	NOUN
ejpam-4072	252	14	,	,	PUNCT
ejpam-4072	252	15	σ2)r	σ2)r	NOUN
ejpam-4072	252	16	-	-	PUNCT
ejpam-4072	252	17	closed	close	VERB
ejpam-4072	252	18	set	set	NOUN
ejpam-4072	252	19	of	of	ADP
ejpam-4072	252	20	y	y	PROPN
ejpam-4072	252	21	.	.	PUNCT
ejpam-4072	253	1	then	then	ADV
ejpam-4072	253	2	,	,	PUNCT
ejpam-4072	253	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	253	4	-	-	PUNCT
ejpam-4072	253	5	int(h	int(h	ADV
ejpam-4072	253	6	)	)	PUNCT
ejpam-4072	253	7	is	be	AUX
ejpam-4072	253	8	(	(	PUNCT
ejpam-4072	253	9	σ1	σ1	PROPN
ejpam-4072	253	10	,	,	PUNCT
ejpam-4072	253	11	σ2)p	σ2)p	NOUN
ejpam-4072	253	12	-	-	PUNCT
ejpam-4072	253	13	open	open	ADJ
ejpam-4072	253	14	in	in	ADP
ejpam-4072	253	15	y	y	PROPN
ejpam-4072	253	16	and	and	CCONJ
ejpam-4072	253	17	by	by	ADP
ejpam-4072	253	18	(	(	PUNCT
ejpam-4072	253	19	4	4	NUM
ejpam-4072	253	20	)	)	PUNCT
ejpam-4072	253	21	,	,	PUNCT
ejpam-4072	253	22	τ1τ2	τ1τ2	PROPN
ejpam-4072	253	23	-	-	PROPN
ejpam-4072	253	24	pcl(f	pcl(f	PROPN
ejpam-4072	253	25	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	253	26	-	-	PUNCT
ejpam-4072	253	27	int(h	int(h	NOUN
ejpam-4072	253	28	)	)	PUNCT
ejpam-4072	253	29	)	)	PUNCT
ejpam-4072	253	30	)	)	PUNCT
ejpam-4072	254	1	=	=	PUNCT
ejpam-4072	255	1	τ1τ2	τ1τ2	PROPN
ejpam-4072	255	2	-	-	PROPN
ejpam-4072	255	3	pcl(f	pcl(f	PROPN
ejpam-4072	255	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	255	5	-	-	PUNCT
ejpam-4072	255	6	int(σ1σ2	int(σ1σ2	ADV
ejpam-4072	255	7	-	-	PUNCT
ejpam-4072	255	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	255	9	-	-	PUNCT
ejpam-4072	255	10	int(h	int(h	NOUN
ejpam-4072	255	11	)	)	PUNCT
ejpam-4072	255	12	)	)	PUNCT
ejpam-4072	255	13	)	)	PUNCT
ejpam-4072	255	14	)	)	PUNCT
ejpam-4072	255	15	)	)	PUNCT
ejpam-4072	256	1	⊆	⊆	X
ejpam-4072	256	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-4072	256	3	-	-	PUNCT
ejpam-4072	256	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4072	256	5	-	-	PUNCT
ejpam-4072	256	6	int(h	int(h	NOUN
ejpam-4072	256	7	)	)	PUNCT
ejpam-4072	256	8	)	)	PUNCT
ejpam-4072	256	9	)	)	PUNCT
ejpam-4072	257	1	=	=	SYM
ejpam-4072	257	2	f−(h	f−(h	PROPN
ejpam-4072	257	3	)	)	PUNCT
ejpam-4072	257	4	.	.	PUNCT
ejpam-4072	258	1	(	(	PUNCT
ejpam-4072	258	2	5	5	X
ejpam-4072	258	3	)	)	PUNCT
ejpam-4072	258	4	⇒	⇒	NOUN
ejpam-4072	258	5	(	(	PUNCT
ejpam-4072	258	6	1	1	NUM
ejpam-4072	258	7	):	):	PUNCT
ejpam-4072	258	8	let	let	VERB
ejpam-4072	258	9	v	v	PART
ejpam-4072	258	10	be	be	AUX
ejpam-4072	258	11	any	any	DET
ejpam-4072	258	12	σ1σ2	σ1σ2	NOUN
ejpam-4072	258	13	-	-	ADJ
ejpam-4072	258	14	open	open	ADJ
ejpam-4072	258	15	set	set	NOUN
ejpam-4072	258	16	of	of	ADP
ejpam-4072	258	17	y	y	PROPN
ejpam-4072	258	18	.	.	PUNCT
ejpam-4072	259	1	then	then	ADV
ejpam-4072	259	2	,	,	PUNCT
ejpam-4072	259	3	σ1σ2	σ1σ2	NOUN
ejpam-4072	259	4	-	-	NUM
ejpam-4072	259	5	cl(v	cl(v	NOUN
ejpam-4072	259	6	)	)	PUNCT
ejpam-4072	259	7	is	be	AUX
ejpam-4072	259	8	(	(	PUNCT
ejpam-4072	259	9	σ1	σ1	NOUN
ejpam-4072	259	10	,	,	PUNCT
ejpam-4072	259	11	σ2)r	σ2)r	NOUN
ejpam-4072	259	12	-	-	PUNCT
ejpam-4072	259	13	closed	closed	ADJ
ejpam-4072	259	14	in	in	ADP
ejpam-4072	259	15	y	y	PROPN
ejpam-4072	259	16	and	and	CCONJ
ejpam-4072	259	17	by	by	ADP
ejpam-4072	259	18	(	(	PUNCT
ejpam-4072	259	19	5	5	NUM
ejpam-4072	259	20	)	)	PUNCT
ejpam-4072	259	21	,	,	PUNCT
ejpam-4072	259	22	τ1τ2	τ1τ2	PROPN
ejpam-4072	259	23	-	-	PROPN
ejpam-4072	259	24	pcl(f	pcl(f	PROPN
ejpam-4072	259	25	−(v	−(v	NOUN
ejpam-4072	259	26	)	)	PUNCT
ejpam-4072	259	27	)	)	PUNCT
ejpam-4072	260	1	⊆	⊆	X
ejpam-4072	260	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	260	3	-	-	PROPN
ejpam-4072	260	4	pcl(f	pcl(f	PROPN
ejpam-4072	260	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-4072	260	6	-	-	PUNCT
ejpam-4072	260	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	260	8	-	-	PUNCT
ejpam-4072	260	9	cl(v	cl(v	NOUN
ejpam-4072	260	10	)	)	PUNCT
ejpam-4072	260	11	)	)	PUNCT
ejpam-4072	260	12	)	)	PUNCT
ejpam-4072	260	13	)	)	PUNCT
ejpam-4072	261	1	⊆	⊆	X
ejpam-4072	261	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	261	3	-	-	PUNCT
ejpam-4072	261	4	cl(v	cl(v	NOUN
ejpam-4072	261	5	)	)	PUNCT
ejpam-4072	261	6	)	)	PUNCT
ejpam-4072	261	7	.	.	PUNCT
ejpam-4072	262	1	it	it	PRON
ejpam-4072	262	2	follows	follow	VERB
ejpam-4072	262	3	from	from	ADP
ejpam-4072	262	4	theorem	theorem	ADJ
ejpam-4072	262	5	1	1	NUM
ejpam-4072	262	6	that	that	SCONJ
ejpam-4072	262	7	f	f	PROPN
ejpam-4072	262	8	is	be	AUX
ejpam-4072	262	9	upper	upper	ADJ
ejpam-4072	262	10	almost	almost	ADV
ejpam-4072	262	11	weakly	weakly	ADJ
ejpam-4072	262	12	(	(	PUNCT
ejpam-4072	262	13	τ1	τ1	NOUN
ejpam-4072	262	14	,	,	PUNCT
ejpam-4072	262	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	262	16	.	.	PUNCT
ejpam-4072	263	1	theorem	theorem	VERB
ejpam-4072	263	2	6	6	NUM
ejpam-4072	263	3	.	.	PUNCT
ejpam-4072	263	4	for	for	ADP
ejpam-4072	263	5	a	a	DET
ejpam-4072	263	6	multifunction	multifunction	NOUN
ejpam-4072	264	1	f	f	NOUN
ejpam-4072	264	2	:	:	PUNCT
ejpam-4072	264	3	(	(	PUNCT
ejpam-4072	264	4	x	x	NOUN
ejpam-4072	264	5	,	,	PUNCT
ejpam-4072	264	6	τ1	τ1	NOUN
ejpam-4072	264	7	,	,	PUNCT
ejpam-4072	264	8	τ2	τ2	NOUN
ejpam-4072	264	9	)	)	PUNCT
ejpam-4072	264	10	→	→	SYM
ejpam-4072	264	11	(	(	PUNCT
ejpam-4072	264	12	y	y	PROPN
ejpam-4072	264	13	,	,	PUNCT
ejpam-4072	264	14	σ1	σ1	PROPN
ejpam-4072	264	15	,	,	PUNCT
ejpam-4072	264	16	σ2	σ2	NOUN
ejpam-4072	264	17	)	)	PUNCT
ejpam-4072	264	18	,	,	PUNCT
ejpam-4072	264	19	the	the	DET
ejpam-4072	264	20	following	follow	VERB
ejpam-4072	264	21	properties	property	NOUN
ejpam-4072	264	22	are	be	AUX
ejpam-4072	264	23	equivalent	equivalent	ADJ
ejpam-4072	264	24	:	:	PUNCT
ejpam-4072	264	25	(	(	PUNCT
ejpam-4072	264	26	1	1	X
ejpam-4072	264	27	)	)	PUNCT
ejpam-4072	264	28	f	f	PROPN
ejpam-4072	264	29	is	be	AUX
ejpam-4072	264	30	lower	low	ADJ
ejpam-4072	264	31	almost	almost	ADV
ejpam-4072	264	32	weakly	weakly	ADJ
ejpam-4072	264	33	(	(	PUNCT
ejpam-4072	264	34	τ1	τ1	NOUN
ejpam-4072	264	35	,	,	PUNCT
ejpam-4072	264	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	264	37	;	;	PUNCT
ejpam-4072	264	38	(	(	PUNCT
ejpam-4072	264	39	2	2	X
ejpam-4072	264	40	)	)	PUNCT
ejpam-4072	264	41	τ1τ2	τ1τ2	NOUN
ejpam-4072	264	42	-	-	PROPN
ejpam-4072	264	43	pcl(f	pcl(f	PROPN
ejpam-4072	264	44	+	+	ADJ
ejpam-4072	264	45	(	(	PUNCT
ejpam-4072	264	46	σ1σ2	σ1σ2	NOUN
ejpam-4072	264	47	-	-	PUNCT
ejpam-4072	264	48	int((σ1	int((σ1	ADJ
ejpam-4072	264	49	,	,	PUNCT
ejpam-4072	264	50	σ2)θ	σ2)θ	ADJ
ejpam-4072	264	51	-	-	PUNCT
ejpam-4072	264	52	cl(b	cl(b	NOUN
ejpam-4072	264	53	)	)	PUNCT
ejpam-4072	264	54	)	)	PUNCT
ejpam-4072	264	55	)	)	PUNCT
ejpam-4072	264	56	)	)	PUNCT
ejpam-4072	265	1	⊆	⊆	NUM
ejpam-4072	265	2	f+((σ1	f+((σ1	NOUN
ejpam-4072	265	3	,	,	PUNCT
ejpam-4072	265	4	σ2)θ	σ2)θ	ADJ
ejpam-4072	265	5	-	-	PUNCT
ejpam-4072	265	6	cl(b	cl(b	NOUN
ejpam-4072	265	7	)	)	PUNCT
ejpam-4072	265	8	)	)	PUNCT
ejpam-4072	265	9	for	for	ADP
ejpam-4072	265	10	every	every	DET
ejpam-4072	265	11	subset	subset	NOUN
ejpam-4072	265	12	b	b	PROPN
ejpam-4072	265	13	of	of	ADP
ejpam-4072	265	14	y	y	PROPN
ejpam-4072	265	15	;	;	PUNCT
ejpam-4072	265	16	c.	c.	PROPN
ejpam-4072	265	17	boonpok	boonpok	PROPN
ejpam-4072	265	18	,	,	PUNCT
ejpam-4072	265	19	c.	c.	PROPN
ejpam-4072	265	20	viriyapong	viriyapong	PROPN
ejpam-4072	265	21	/	/	SYM
ejpam-4072	265	22	eur	eur	PROPN
ejpam-4072	265	23	.	.	PUNCT
ejpam-4072	266	1	j.	j.	PROPN
ejpam-4072	266	2	pure	pure	PROPN
ejpam-4072	266	3	appl	appl	PROPN
ejpam-4072	266	4	.	.	PROPN
ejpam-4072	266	5	math	math	PROPN
ejpam-4072	266	6	,	,	PUNCT
ejpam-4072	266	7	14	14	NUM
ejpam-4072	266	8	(	(	PUNCT
ejpam-4072	266	9	4	4	NUM
ejpam-4072	266	10	)	)	PUNCT
ejpam-4072	266	11	(	(	PUNCT
ejpam-4072	266	12	2021	2021	NUM
ejpam-4072	266	13	)	)	PUNCT
ejpam-4072	266	14	,	,	PUNCT
ejpam-4072	266	15	1212	1212	NUM
ejpam-4072	266	16	-	-	SYM
ejpam-4072	266	17	1225	1225	NUM
ejpam-4072	266	18	1220	1220	NUM
ejpam-4072	266	19	(	(	PUNCT
ejpam-4072	266	20	3	3	NUM
ejpam-4072	266	21	)	)	PUNCT
ejpam-4072	266	22	τ1τ2	τ1τ2	NOUN
ejpam-4072	266	23	-	-	PROPN
ejpam-4072	266	24	pcl(f	pcl(f	PROPN
ejpam-4072	266	25	+	+	ADJ
ejpam-4072	266	26	(	(	PUNCT
ejpam-4072	266	27	σ1σ2	σ1σ2	NUM
ejpam-4072	266	28	-	-	PUNCT
ejpam-4072	266	29	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	266	30	-	-	PUNCT
ejpam-4072	266	31	cl(v	cl(v	NOUN
ejpam-4072	266	32	)	)	PUNCT
ejpam-4072	266	33	)	)	PUNCT
ejpam-4072	266	34	)	)	PUNCT
ejpam-4072	266	35	)	)	PUNCT
ejpam-4072	267	1	⊆	⊆	X
ejpam-4072	267	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	267	3	-	-	PUNCT
ejpam-4072	267	4	cl(v	cl(v	NOUN
ejpam-4072	267	5	)	)	PUNCT
ejpam-4072	267	6	)	)	PUNCT
ejpam-4072	267	7	for	for	ADP
ejpam-4072	267	8	every	every	DET
ejpam-4072	267	9	σ1σ2	σ1σ2	NOUN
ejpam-4072	267	10	-	-	ADJ
ejpam-4072	267	11	open	open	ADJ
ejpam-4072	267	12	set	set	NOUN
ejpam-4072	267	13	v	v	NOUN
ejpam-4072	267	14	of	of	ADP
ejpam-4072	267	15	y	y	PROPN
ejpam-4072	267	16	;	;	PUNCT
ejpam-4072	267	17	(	(	PUNCT
ejpam-4072	267	18	4	4	X
ejpam-4072	267	19	)	)	PUNCT
ejpam-4072	267	20	τ1τ2	τ1τ2	NOUN
ejpam-4072	267	21	-	-	PROPN
ejpam-4072	267	22	pcl(f	pcl(f	PROPN
ejpam-4072	267	23	+	+	ADJ
ejpam-4072	267	24	(	(	PUNCT
ejpam-4072	267	25	σ1σ2	σ1σ2	NUM
ejpam-4072	267	26	-	-	PUNCT
ejpam-4072	267	27	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4072	267	28	-	-	PUNCT
ejpam-4072	267	29	cl(v	cl(v	NOUN
ejpam-4072	267	30	)	)	PUNCT
ejpam-4072	267	31	)	)	PUNCT
ejpam-4072	267	32	)	)	PUNCT
ejpam-4072	267	33	)	)	PUNCT
ejpam-4072	268	1	⊆	⊆	X
ejpam-4072	268	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	268	3	-	-	PUNCT
ejpam-4072	268	4	cl(v	cl(v	NOUN
ejpam-4072	268	5	)	)	PUNCT
ejpam-4072	268	6	)	)	PUNCT
ejpam-4072	268	7	for	for	ADP
ejpam-4072	268	8	every	every	DET
ejpam-4072	268	9	(	(	PUNCT
ejpam-4072	268	10	σ1	σ1	PROPN
ejpam-4072	268	11	,	,	PUNCT
ejpam-4072	268	12	σ2)p	σ2)p	NOUN
ejpam-4072	268	13	-	-	PUNCT
ejpam-4072	268	14	open	open	NOUN
ejpam-4072	268	15	set	set	NOUN
ejpam-4072	268	16	v	v	NOUN
ejpam-4072	268	17	of	of	ADP
ejpam-4072	268	18	y	y	PROPN
ejpam-4072	268	19	;	;	PUNCT
ejpam-4072	268	20	(	(	PUNCT
ejpam-4072	268	21	5	5	X
ejpam-4072	268	22	)	)	PUNCT
ejpam-4072	268	23	τ1τ2	τ1τ2	NOUN
ejpam-4072	268	24	-	-	PROPN
ejpam-4072	268	25	pcl(f	pcl(f	PROPN
ejpam-4072	268	26	+	+	ADJ
ejpam-4072	268	27	(	(	PUNCT
ejpam-4072	268	28	σ1σ2	σ1σ2	X
ejpam-4072	268	29	-	-	PUNCT
ejpam-4072	268	30	int(h	int(h	NOUN
ejpam-4072	268	31	)	)	PUNCT
ejpam-4072	268	32	)	)	PUNCT
ejpam-4072	268	33	)	)	PUNCT
ejpam-4072	269	1	⊆	⊆	NUM
ejpam-4072	269	2	f+(h	f+(h	PROPN
ejpam-4072	269	3	)	)	PUNCT
ejpam-4072	269	4	for	for	ADP
ejpam-4072	269	5	every	every	DET
ejpam-4072	269	6	(	(	PUNCT
ejpam-4072	269	7	σ1	σ1	PROPN
ejpam-4072	269	8	,	,	PUNCT
ejpam-4072	269	9	σ2)r	σ2)r	NOUN
ejpam-4072	269	10	-	-	PUNCT
ejpam-4072	269	11	closed	close	VERB
ejpam-4072	269	12	set	set	ADJ
ejpam-4072	269	13	h	h	NOUN
ejpam-4072	269	14	of	of	ADP
ejpam-4072	269	15	y	y	PROPN
ejpam-4072	269	16	.	.	PUNCT
ejpam-4072	270	1	proof	proof	NOUN
ejpam-4072	270	2	.	.	PUNCT
ejpam-4072	271	1	the	the	DET
ejpam-4072	271	2	proof	proof	NOUN
ejpam-4072	271	3	is	be	AUX
ejpam-4072	271	4	similar	similar	ADJ
ejpam-4072	271	5	to	to	ADP
ejpam-4072	271	6	that	that	PRON
ejpam-4072	271	7	of	of	ADP
ejpam-4072	271	8	theorem	theorem	NOUN
ejpam-4072	271	9	5	5	NUM
ejpam-4072	271	10	.	.	PUNCT
ejpam-4072	272	1	in	in	ADP
ejpam-4072	272	2	order	order	NOUN
ejpam-4072	272	3	to	to	PART
ejpam-4072	272	4	obtain	obtain	VERB
ejpam-4072	272	5	further	further	ADJ
ejpam-4072	272	6	characterizations	characterization	NOUN
ejpam-4072	272	7	of	of	ADP
ejpam-4072	272	8	upper	upper	ADJ
ejpam-4072	272	9	and	and	CCONJ
ejpam-4072	272	10	lower	low	ADJ
ejpam-4072	272	11	almost	almost	ADV
ejpam-4072	272	12	weakly	weakly	ADJ
ejpam-4072	272	13	(	(	PUNCT
ejpam-4072	272	14	τ1	τ1	NOUN
ejpam-4072	272	15	,	,	PUNCT
ejpam-4072	272	16	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	272	17	multifunctions	multifunction	NOUN
ejpam-4072	272	18	,	,	PUNCT
ejpam-4072	272	19	we	we	PRON
ejpam-4072	272	20	recall	recall	VERB
ejpam-4072	272	21	some	some	DET
ejpam-4072	272	22	definitions	definition	NOUN
ejpam-4072	272	23	.	.	PUNCT
ejpam-4072	273	1	for	for	ADP
ejpam-4072	273	2	a	a	DET
ejpam-4072	273	3	multifunction	multifunction	NOUN
ejpam-4072	273	4	f	f	NOUN
ejpam-4072	273	5	:	:	PUNCT
ejpam-4072	273	6	(	(	PUNCT
ejpam-4072	273	7	x	x	NOUN
ejpam-4072	273	8	,	,	PUNCT
ejpam-4072	273	9	τ1	τ1	NOUN
ejpam-4072	273	10	,	,	PUNCT
ejpam-4072	273	11	τ2	τ2	NOUN
ejpam-4072	273	12	)	)	PUNCT
ejpam-4072	273	13	→	→	SYM
ejpam-4072	273	14	(	(	PUNCT
ejpam-4072	273	15	y	y	PROPN
ejpam-4072	273	16	,	,	PUNCT
ejpam-4072	273	17	σ1	σ1	PROPN
ejpam-4072	273	18	,	,	PUNCT
ejpam-4072	273	19	σ2	σ2	NOUN
ejpam-4072	273	20	)	)	PUNCT
ejpam-4072	273	21	,	,	PUNCT
ejpam-4072	273	22	by	by	ADP
ejpam-4072	273	23	clf⊛	clf⊛	PROPN
ejpam-4072	273	24	:	:	PUNCT
ejpam-4072	273	25	(	(	PUNCT
ejpam-4072	273	26	x	x	NOUN
ejpam-4072	273	27	,	,	PUNCT
ejpam-4072	273	28	τ1	τ1	NOUN
ejpam-4072	273	29	,	,	PUNCT
ejpam-4072	273	30	τ2	τ2	NOUN
ejpam-4072	273	31	)	)	PUNCT
ejpam-4072	273	32	→	→	SYM
ejpam-4072	273	33	(	(	PUNCT
ejpam-4072	273	34	y	y	PROPN
ejpam-4072	273	35	,	,	PUNCT
ejpam-4072	273	36	σ1	σ1	PROPN
ejpam-4072	273	37	,	,	PUNCT
ejpam-4072	273	38	σ2	σ2	NOUN
ejpam-4072	273	39	)	)	PUNCT
ejpam-4072	274	1	[	[	X
ejpam-4072	274	2	5	5	NUM
ejpam-4072	274	3	]	]	PUNCT
ejpam-4072	274	4	(	(	PUNCT
ejpam-4072	274	5	resp	resp	NOUN
ejpam-4072	274	6	.	.	PUNCT
ejpam-4072	275	1	pclf⊛	pclf⊛	NOUN
ejpam-4072	275	2	:	:	PUNCT
ejpam-4072	275	3	(	(	PUNCT
ejpam-4072	275	4	x	x	NOUN
ejpam-4072	275	5	,	,	PUNCT
ejpam-4072	275	6	τ1	τ1	NOUN
ejpam-4072	275	7	,	,	PUNCT
ejpam-4072	275	8	τ2	τ2	NOUN
ejpam-4072	275	9	)	)	PUNCT
ejpam-4072	275	10	→	→	SYM
ejpam-4072	275	11	(	(	PUNCT
ejpam-4072	275	12	y	y	PROPN
ejpam-4072	275	13	,	,	PUNCT
ejpam-4072	275	14	σ1	σ1	PROPN
ejpam-4072	275	15	,	,	PUNCT
ejpam-4072	275	16	σ2	σ2	NOUN
ejpam-4072	275	17	)	)	PUNCT
ejpam-4072	275	18	)	)	PUNCT
ejpam-4072	276	1	we	we	PRON
ejpam-4072	276	2	denote	denote	VERB
ejpam-4072	276	3	a	a	DET
ejpam-4072	276	4	multifunction	multifunction	NOUN
ejpam-4072	276	5	defined	define	VERB
ejpam-4072	276	6	as	as	SCONJ
ejpam-4072	276	7	follows	follow	VERB
ejpam-4072	276	8	:	:	PUNCT
ejpam-4072	276	9	clf⊛(x	clf⊛(x	PROPN
ejpam-4072	276	10	)	)	PUNCT
ejpam-4072	276	11	=	=	PUNCT
ejpam-4072	276	12	σ1σ2	σ1σ2	X
ejpam-4072	276	13	-	-	NUM
ejpam-4072	276	14	cl(f	cl(f	NOUN
ejpam-4072	276	15	(	(	PUNCT
ejpam-4072	276	16	x	x	NOUN
ejpam-4072	276	17	)	)	PUNCT
ejpam-4072	276	18	)	)	PUNCT
ejpam-4072	276	19	(	(	PUNCT
ejpam-4072	276	20	resp	resp	NOUN
ejpam-4072	276	21	.	.	PUNCT
ejpam-4072	276	22	pclf⊛(x	pclf⊛(x	PART
ejpam-4072	276	23	)	)	PUNCT
ejpam-4072	276	24	=	=	PUNCT
ejpam-4072	276	25	σ1σ2	σ1σ2	X
ejpam-4072	276	26	-	-	PROPN
ejpam-4072	276	27	pcl(f	pcl(f	PROPN
ejpam-4072	276	28	(	(	PUNCT
ejpam-4072	276	29	x	x	NOUN
ejpam-4072	276	30	)	)	PUNCT
ejpam-4072	276	31	)	)	PUNCT
ejpam-4072	276	32	)	)	PUNCT
ejpam-4072	276	33	for	for	ADP
ejpam-4072	276	34	each	each	DET
ejpam-4072	276	35	x	x	SYM
ejpam-4072	276	36	∈	∈	PROPN
ejpam-4072	276	37	x.	x.	NOUN
ejpam-4072	276	38	definition	definition	NOUN
ejpam-4072	276	39	4	4	NUM
ejpam-4072	276	40	.	.	PUNCT
ejpam-4072	277	1	[	[	X
ejpam-4072	277	2	5	5	NUM
ejpam-4072	277	3	]	]	PUNCT
ejpam-4072	277	4	a	a	DET
ejpam-4072	277	5	subset	subset	NOUN
ejpam-4072	277	6	a	a	PRON
ejpam-4072	277	7	of	of	ADP
ejpam-4072	277	8	a	a	DET
ejpam-4072	277	9	bitopological	bitopological	ADJ
ejpam-4072	277	10	space	space	NOUN
ejpam-4072	277	11	(	(	PUNCT
ejpam-4072	277	12	x	x	NOUN
ejpam-4072	277	13	,	,	PUNCT
ejpam-4072	277	14	τ1	τ1	NOUN
ejpam-4072	277	15	,	,	PUNCT
ejpam-4072	277	16	τ2	τ2	NOUN
ejpam-4072	277	17	)	)	PUNCT
ejpam-4072	277	18	is	be	AUX
ejpam-4072	277	19	said	say	VERB
ejpam-4072	277	20	to	to	PART
ejpam-4072	277	21	be	be	AUX
ejpam-4072	277	22	:	:	PUNCT
ejpam-4072	277	23	(	(	PUNCT
ejpam-4072	277	24	1	1	X
ejpam-4072	277	25	)	)	PUNCT
ejpam-4072	277	26	τ1τ2	τ1τ2	NOUN
ejpam-4072	277	27	-	-	NOUN
ejpam-4072	277	28	paracompact	paracompact	ADJ
ejpam-4072	277	29	if	if	SCONJ
ejpam-4072	277	30	every	every	DET
ejpam-4072	277	31	cover	cover	NOUN
ejpam-4072	277	32	of	of	ADP
ejpam-4072	277	33	a	a	PRON
ejpam-4072	277	34	by	by	ADP
ejpam-4072	277	35	τ1τ2	τ1τ2	ADJ
ejpam-4072	277	36	-	-	ADJ
ejpam-4072	277	37	open	open	ADJ
ejpam-4072	277	38	sets	set	NOUN
ejpam-4072	277	39	of	of	ADP
ejpam-4072	277	40	x	x	VERB
ejpam-4072	277	41	is	be	AUX
ejpam-4072	277	42	refined	refine	VERB
ejpam-4072	277	43	by	by	ADP
ejpam-4072	277	44	a	a	DET
ejpam-4072	277	45	cover	cover	NOUN
ejpam-4072	277	46	of	of	ADP
ejpam-4072	277	47	a	a	PRON
ejpam-4072	277	48	which	which	PRON
ejpam-4072	277	49	consists	consist	VERB
ejpam-4072	277	50	of	of	ADP
ejpam-4072	277	51	τ1τ2	τ1τ2	ADJ
ejpam-4072	277	52	-	-	ADJ
ejpam-4072	277	53	open	open	ADJ
ejpam-4072	277	54	sets	set	NOUN
ejpam-4072	277	55	of	of	ADP
ejpam-4072	277	56	x	x	PUNCT
ejpam-4072	277	57	and	and	CCONJ
ejpam-4072	277	58	is	be	AUX
ejpam-4072	277	59	τ1τ2	τ1τ2	NOUN
ejpam-4072	277	60	-	-	ADJ
ejpam-4072	277	61	locally	locally	ADV
ejpam-4072	277	62	finite	finite	NOUN
ejpam-4072	277	63	in	in	ADP
ejpam-4072	277	64	x	x	PRON
ejpam-4072	277	65	;	;	PUNCT
ejpam-4072	277	66	(	(	PUNCT
ejpam-4072	277	67	2	2	X
ejpam-4072	277	68	)	)	PUNCT
ejpam-4072	277	69	τ1τ2	τ1τ2	NOUN
ejpam-4072	277	70	-	-	NOUN
ejpam-4072	277	71	regular	regular	ADJ
ejpam-4072	277	72	if	if	SCONJ
ejpam-4072	277	73	for	for	ADP
ejpam-4072	277	74	each	each	DET
ejpam-4072	277	75	x	x	SYM
ejpam-4072	277	76	∈	∈	PROPN
ejpam-4072	277	77	a	a	PRON
ejpam-4072	277	78	and	and	CCONJ
ejpam-4072	277	79	each	each	DET
ejpam-4072	277	80	τ1τ2	τ1τ2	ADJ
ejpam-4072	277	81	-	-	ADJ
ejpam-4072	277	82	open	open	ADJ
ejpam-4072	277	83	set	set	ADJ
ejpam-4072	277	84	u	u	NOUN
ejpam-4072	277	85	of	of	ADP
ejpam-4072	277	86	x	x	PUNCT
ejpam-4072	277	87	containing	contain	VERB
ejpam-4072	277	88	x	x	PRON
ejpam-4072	277	89	,	,	PUNCT
ejpam-4072	277	90	there	there	PRON
ejpam-4072	277	91	exists	exist	VERB
ejpam-4072	277	92	a	a	DET
ejpam-4072	277	93	τ1τ2	τ1τ2	NOUN
ejpam-4072	277	94	-	-	ADJ
ejpam-4072	277	95	open	open	ADJ
ejpam-4072	277	96	set	set	NOUN
ejpam-4072	277	97	v	v	NOUN
ejpam-4072	277	98	of	of	ADP
ejpam-4072	277	99	x	x	PUNCT
ejpam-4072	277	100	such	such	ADJ
ejpam-4072	277	101	that	that	SCONJ
ejpam-4072	277	102	x	x	SYM
ejpam-4072	277	103	∈	∈	NOUN
ejpam-4072	277	104	v	v	ADP
ejpam-4072	277	105	⊆	⊆	NUM
ejpam-4072	277	106	τ1τ2	τ1τ2	NOUN
ejpam-4072	277	107	-	-	NOUN
ejpam-4072	277	108	cl(v	cl(v	X
ejpam-4072	277	109	)	)	PUNCT
ejpam-4072	277	110	⊆	⊆	NUM
ejpam-4072	277	111	u	u	NOUN
ejpam-4072	277	112	.	.	PUNCT
ejpam-4072	278	1	lemma	lemma	PROPN
ejpam-4072	278	2	8	8	NUM
ejpam-4072	278	3	.	.	PUNCT
ejpam-4072	279	1	[	[	X
ejpam-4072	279	2	5	5	X
ejpam-4072	279	3	]	]	X
ejpam-4072	279	4	if	if	SCONJ
ejpam-4072	279	5	a	a	PRON
ejpam-4072	279	6	is	be	AUX
ejpam-4072	279	7	a	a	DET
ejpam-4072	279	8	τ1τ2	τ1τ2	ADJ
ejpam-4072	279	9	-	-	ADJ
ejpam-4072	279	10	regular	regular	ADJ
ejpam-4072	279	11	τ1τ2	τ1τ2	NOUN
ejpam-4072	279	12	-	-	ADJ
ejpam-4072	279	13	paracompact	paracompact	ADJ
ejpam-4072	279	14	set	set	NOUN
ejpam-4072	279	15	of	of	ADP
ejpam-4072	279	16	a	a	DET
ejpam-4072	279	17	bitopological	bitopological	ADJ
ejpam-4072	279	18	space	space	NOUN
ejpam-4072	279	19	(	(	PUNCT
ejpam-4072	279	20	x	x	NOUN
ejpam-4072	279	21	,	,	PUNCT
ejpam-4072	279	22	τ1	τ1	NOUN
ejpam-4072	279	23	,	,	PUNCT
ejpam-4072	279	24	τ2	τ2	NOUN
ejpam-4072	279	25	)	)	PUNCT
ejpam-4072	279	26	and	and	CCONJ
ejpam-4072	279	27	u	u	NOUN
ejpam-4072	279	28	is	be	AUX
ejpam-4072	279	29	a	a	DET
ejpam-4072	279	30	τ1τ2	τ1τ2	ADJ
ejpam-4072	279	31	-	-	ADJ
ejpam-4072	279	32	open	open	ADJ
ejpam-4072	279	33	neighbourhood	neighbourhood	NOUN
ejpam-4072	279	34	of	of	ADP
ejpam-4072	279	35	a	a	PRON
ejpam-4072	279	36	,	,	PUNCT
ejpam-4072	279	37	then	then	ADV
ejpam-4072	279	38	there	there	PRON
ejpam-4072	279	39	exists	exist	VERB
ejpam-4072	279	40	a	a	DET
ejpam-4072	279	41	τ1τ2	τ1τ2	NOUN
ejpam-4072	279	42	-	-	ADJ
ejpam-4072	279	43	open	open	ADJ
ejpam-4072	279	44	set	set	NOUN
ejpam-4072	279	45	v	v	NOUN
ejpam-4072	279	46	of	of	ADP
ejpam-4072	279	47	x	x	PUNCT
ejpam-4072	279	48	such	such	ADJ
ejpam-4072	279	49	that	that	SCONJ
ejpam-4072	279	50	a	a	DET
ejpam-4072	279	51	⊆	⊆	NUM
ejpam-4072	279	52	v	v	ADP
ejpam-4072	279	53	⊆	⊆	NUM
ejpam-4072	279	54	τ1τ2	τ1τ2	NOUN
ejpam-4072	279	55	-	-	NOUN
ejpam-4072	279	56	cl(v	cl(v	X
ejpam-4072	279	57	)	)	PUNCT
ejpam-4072	279	58	⊆	⊆	NUM
ejpam-4072	279	59	u	u	NOUN
ejpam-4072	279	60	.	.	PUNCT
ejpam-4072	280	1	lemma	lemma	PROPN
ejpam-4072	280	2	9	9	NUM
ejpam-4072	280	3	.	.	PUNCT
ejpam-4072	281	1	[	[	X
ejpam-4072	281	2	5	5	X
ejpam-4072	281	3	]	]	PUNCT
ejpam-4072	281	4	if	if	SCONJ
ejpam-4072	281	5	f	f	PROPN
ejpam-4072	281	6	:	:	PUNCT
ejpam-4072	281	7	(	(	PUNCT
ejpam-4072	281	8	x	x	NOUN
ejpam-4072	281	9	,	,	PUNCT
ejpam-4072	281	10	τ1	τ1	NOUN
ejpam-4072	281	11	,	,	PUNCT
ejpam-4072	281	12	τ2	τ2	NOUN
ejpam-4072	281	13	)	)	PUNCT
ejpam-4072	281	14	→	→	SYM
ejpam-4072	281	15	(	(	PUNCT
ejpam-4072	281	16	y	y	PROPN
ejpam-4072	281	17	,	,	PUNCT
ejpam-4072	281	18	σ1	σ1	PROPN
ejpam-4072	281	19	,	,	PUNCT
ejpam-4072	281	20	σ2	σ2	PROPN
ejpam-4072	281	21	)	)	PUNCT
ejpam-4072	281	22	is	be	AUX
ejpam-4072	281	23	a	a	DET
ejpam-4072	281	24	multifunction	multifunction	NOUN
ejpam-4072	281	25	such	such	ADJ
ejpam-4072	281	26	that	that	SCONJ
ejpam-4072	281	27	f	f	PROPN
ejpam-4072	281	28	(	(	PUNCT
ejpam-4072	281	29	x	x	X
ejpam-4072	281	30	)	)	PUNCT
ejpam-4072	281	31	is	be	AUX
ejpam-4072	281	32	τ1τ2regular	τ1τ2regular	NUM
ejpam-4072	281	33	and	and	CCONJ
ejpam-4072	281	34	τ1τ2	τ1τ2	NOUN
ejpam-4072	281	35	-	-	ADJ
ejpam-4072	281	36	paracompact	paracompact	ADJ
ejpam-4072	281	37	for	for	SCONJ
ejpam-4072	281	38	each	each	DET
ejpam-4072	281	39	x	x	SYM
ejpam-4072	281	40	∈	∈	PROPN
ejpam-4072	281	41	x	x	NOUN
ejpam-4072	281	42	,	,	PUNCT
ejpam-4072	281	43	then	then	ADV
ejpam-4072	281	44	clf+	clf+	PROPN
ejpam-4072	281	45	⊛	⊛	X
ejpam-4072	281	46	(	(	PUNCT
ejpam-4072	281	47	v	v	NOUN
ejpam-4072	281	48	)	)	PUNCT
ejpam-4072	281	49	=	=	NOUN
ejpam-4072	281	50	pclf+	pclf+	NOUN
ejpam-4072	281	51	⊛	⊛	NUM
ejpam-4072	281	52	(	(	PUNCT
ejpam-4072	281	53	v	v	NOUN
ejpam-4072	281	54	)	)	PUNCT
ejpam-4072	281	55	=	=	PUNCT
ejpam-4072	281	56	f+(v	f+(v	NOUN
ejpam-4072	281	57	)	)	PUNCT
ejpam-4072	281	58	for	for	ADP
ejpam-4072	281	59	each	each	DET
ejpam-4072	281	60	σ1σ2	σ1σ2	VERB
ejpam-4072	281	61	-	-	ADJ
ejpam-4072	281	62	open	open	ADJ
ejpam-4072	281	63	set	set	NOUN
ejpam-4072	281	64	v	v	NOUN
ejpam-4072	281	65	of	of	ADP
ejpam-4072	281	66	y	y	PROPN
ejpam-4072	281	67	.	.	PUNCT
ejpam-4072	282	1	theorem	theorem	ADJ
ejpam-4072	282	2	7	7	NUM
ejpam-4072	282	3	.	.	PUNCT
ejpam-4072	283	1	let	let	VERB
ejpam-4072	283	2	f	f	NOUN
ejpam-4072	283	3	:	:	PUNCT
ejpam-4072	283	4	(	(	PUNCT
ejpam-4072	283	5	x	x	NOUN
ejpam-4072	283	6	,	,	PUNCT
ejpam-4072	283	7	τ1	τ1	NOUN
ejpam-4072	283	8	,	,	PUNCT
ejpam-4072	283	9	τ2	τ2	NOUN
ejpam-4072	283	10	)	)	PUNCT
ejpam-4072	283	11	→	→	SYM
ejpam-4072	283	12	(	(	PUNCT
ejpam-4072	283	13	y	y	PROPN
ejpam-4072	283	14	,	,	PUNCT
ejpam-4072	283	15	σ1	σ1	PROPN
ejpam-4072	283	16	,	,	PUNCT
ejpam-4072	283	17	σ2	σ2	PROPN
ejpam-4072	283	18	)	)	PUNCT
ejpam-4072	283	19	be	be	VERB
ejpam-4072	283	20	a	a	DET
ejpam-4072	283	21	multifunction	multifunction	NOUN
ejpam-4072	283	22	such	such	ADJ
ejpam-4072	283	23	that	that	SCONJ
ejpam-4072	283	24	f	f	PROPN
ejpam-4072	283	25	(	(	PUNCT
ejpam-4072	283	26	x	x	X
ejpam-4072	283	27	)	)	PUNCT
ejpam-4072	283	28	is	be	AUX
ejpam-4072	283	29	σ1σ2paracompact	σ1σ2paracompact	NUM
ejpam-4072	283	30	and	and	CCONJ
ejpam-4072	283	31	σ1σ2	σ1σ2	NOUN
ejpam-4072	283	32	-	-	ADJ
ejpam-4072	283	33	regular	regular	ADJ
ejpam-4072	283	34	for	for	ADP
ejpam-4072	283	35	each	each	DET
ejpam-4072	283	36	x	x	SYM
ejpam-4072	283	37	∈	∈	PROPN
ejpam-4072	283	38	x.	x.	NOUN
ejpam-4072	283	39	then	then	ADV
ejpam-4072	283	40	the	the	DET
ejpam-4072	283	41	following	follow	VERB
ejpam-4072	283	42	properties	property	NOUN
ejpam-4072	283	43	are	be	AUX
ejpam-4072	283	44	equivalent	equivalent	ADJ
ejpam-4072	283	45	:	:	PUNCT
ejpam-4072	283	46	(	(	PUNCT
ejpam-4072	283	47	1	1	X
ejpam-4072	283	48	)	)	PUNCT
ejpam-4072	283	49	f	f	PROPN
ejpam-4072	283	50	is	be	AUX
ejpam-4072	283	51	upper	upper	ADJ
ejpam-4072	283	52	almost	almost	ADV
ejpam-4072	283	53	weakly	weakly	ADJ
ejpam-4072	283	54	(	(	PUNCT
ejpam-4072	283	55	τ1	τ1	NOUN
ejpam-4072	283	56	,	,	PUNCT
ejpam-4072	283	57	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	283	58	;	;	PUNCT
ejpam-4072	283	59	(	(	PUNCT
ejpam-4072	283	60	2	2	X
ejpam-4072	283	61	)	)	PUNCT
ejpam-4072	283	62	pclf⊛	pclf⊛	NOUN
ejpam-4072	283	63	is	be	AUX
ejpam-4072	283	64	upper	upper	ADJ
ejpam-4072	283	65	almost	almost	ADV
ejpam-4072	283	66	weakly	weakly	ADJ
ejpam-4072	283	67	(	(	PUNCT
ejpam-4072	283	68	τ1	τ1	NOUN
ejpam-4072	283	69	,	,	PUNCT
ejpam-4072	283	70	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	283	71	;	;	PUNCT
ejpam-4072	283	72	(	(	PUNCT
ejpam-4072	283	73	3	3	X
ejpam-4072	283	74	)	)	PUNCT
ejpam-4072	283	75	clf⊛	clf⊛	PROPN
ejpam-4072	283	76	is	be	AUX
ejpam-4072	283	77	upper	upper	ADJ
ejpam-4072	283	78	almost	almost	ADV
ejpam-4072	283	79	weakly	weakly	ADJ
ejpam-4072	283	80	(	(	PUNCT
ejpam-4072	283	81	τ1	τ1	NOUN
ejpam-4072	283	82	,	,	PUNCT
ejpam-4072	283	83	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	283	84	.	.	PUNCT
ejpam-4072	284	1	proof	proof	NOUN
ejpam-4072	284	2	.	.	PUNCT
ejpam-4072	285	1	we	we	PRON
ejpam-4072	285	2	put	put	VERB
ejpam-4072	285	3	g	g	PROPN
ejpam-4072	285	4	=	=	PROPN
ejpam-4072	285	5	clf⊛	clf⊛	PROPN
ejpam-4072	285	6	or	or	CCONJ
ejpam-4072	285	7	pclf⊛	pclf⊛	NOUN
ejpam-4072	285	8	in	in	ADP
ejpam-4072	285	9	the	the	DET
ejpam-4072	285	10	sequel	sequel	NOUN
ejpam-4072	285	11	.	.	PUNCT
ejpam-4072	286	1	suppose	suppose	VERB
ejpam-4072	286	2	that	that	SCONJ
ejpam-4072	286	3	f	f	PROPN
ejpam-4072	286	4	is	be	AUX
ejpam-4072	286	5	upper	upper	ADJ
ejpam-4072	286	6	almost	almost	ADV
ejpam-4072	286	7	weakly	weakly	ADJ
ejpam-4072	286	8	(	(	PUNCT
ejpam-4072	286	9	τ1	τ1	NOUN
ejpam-4072	286	10	,	,	PUNCT
ejpam-4072	286	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	286	12	.	.	PUNCT
ejpam-4072	287	1	let	let	VERB
ejpam-4072	287	2	x	x	PUNCT
ejpam-4072	287	3	∈	∈	PROPN
ejpam-4072	287	4	x	x	PUNCT
ejpam-4072	287	5	and	and	CCONJ
ejpam-4072	287	6	let	let	VERB
ejpam-4072	287	7	v	v	PART
ejpam-4072	287	8	be	be	AUX
ejpam-4072	287	9	any	any	DET
ejpam-4072	287	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	287	11	-	-	ADJ
ejpam-4072	287	12	open	open	ADJ
ejpam-4072	287	13	set	set	NOUN
ejpam-4072	287	14	of	of	ADP
ejpam-4072	287	15	y	y	NOUN
ejpam-4072	287	16	containing	contain	VERB
ejpam-4072	287	17	g(x	g(x	NOUN
ejpam-4072	287	18	)	)	PUNCT
ejpam-4072	287	19	.	.	PUNCT
ejpam-4072	288	1	by	by	ADP
ejpam-4072	288	2	lemma	lemma	PROPN
ejpam-4072	288	3	9	9	NUM
ejpam-4072	288	4	,	,	PUNCT
ejpam-4072	288	5	we	we	PRON
ejpam-4072	288	6	have	have	VERB
ejpam-4072	288	7	x	x	X
ejpam-4072	288	8	∈	∈	PROPN
ejpam-4072	288	9	g+(v	g+(v	PROPN
ejpam-4072	288	10	)	)	PUNCT
ejpam-4072	288	11	=	=	PUNCT
ejpam-4072	289	1	f+(v	f+(v	NOUN
ejpam-4072	289	2	)	)	PUNCT
ejpam-4072	290	1	and	and	CCONJ
ejpam-4072	290	2	hence	hence	ADV
ejpam-4072	290	3	,	,	PUNCT
ejpam-4072	290	4	there	there	PRON
ejpam-4072	290	5	exists	exist	VERB
ejpam-4072	290	6	a	a	DET
ejpam-4072	290	7	τ1τ2	τ1τ2	NOUN
ejpam-4072	290	8	-	-	ADJ
ejpam-4072	290	9	open	open	ADJ
ejpam-4072	290	10	set	set	NOUN
ejpam-4072	290	11	u	u	NOUN
ejpam-4072	290	12	containing	contain	VERB
ejpam-4072	290	13	x	x	PUNCT
ejpam-4072	290	14	such	such	ADJ
ejpam-4072	290	15	that	that	SCONJ
ejpam-4072	290	16	f	f	PROPN
ejpam-4072	290	17	(	(	PUNCT
ejpam-4072	290	18	u	u	NOUN
ejpam-4072	290	19	)	)	PUNCT
ejpam-4072	290	20	⊆	⊆	NUM
ejpam-4072	290	21	σ1σ2	σ1σ2	NOUN
ejpam-4072	290	22	-	-	NUM
ejpam-4072	290	23	cl(v	cl(v	NOUN
ejpam-4072	290	24	)	)	PUNCT
ejpam-4072	290	25	.	.	PUNCT
ejpam-4072	291	1	since	since	SCONJ
ejpam-4072	291	2	f	f	PROPN
ejpam-4072	291	3	(	(	PUNCT
ejpam-4072	291	4	z	z	NOUN
ejpam-4072	291	5	)	)	PUNCT
ejpam-4072	291	6	is	be	AUX
ejpam-4072	291	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	291	8	-	-	PUNCT
ejpam-4072	291	9	paracompact	paracompact	ADJ
ejpam-4072	291	10	and	and	CCONJ
ejpam-4072	291	11	c.	c.	PROPN
ejpam-4072	291	12	boonpok	boonpok	PROPN
ejpam-4072	291	13	,	,	PUNCT
ejpam-4072	291	14	c.	c.	PROPN
ejpam-4072	291	15	viriyapong	viriyapong	PROPN
ejpam-4072	291	16	/	/	SYM
ejpam-4072	291	17	eur	eur	PROPN
ejpam-4072	291	18	.	.	PUNCT
ejpam-4072	292	1	j.	j.	PROPN
ejpam-4072	292	2	pure	pure	PROPN
ejpam-4072	292	3	appl	appl	PROPN
ejpam-4072	292	4	.	.	PROPN
ejpam-4072	292	5	math	math	PROPN
ejpam-4072	292	6	,	,	PUNCT
ejpam-4072	292	7	14	14	NUM
ejpam-4072	292	8	(	(	PUNCT
ejpam-4072	292	9	4	4	NUM
ejpam-4072	292	10	)	)	PUNCT
ejpam-4072	292	11	(	(	PUNCT
ejpam-4072	292	12	2021	2021	NUM
ejpam-4072	292	13	)	)	PUNCT
ejpam-4072	292	14	,	,	PUNCT
ejpam-4072	292	15	1212	1212	NUM
ejpam-4072	292	16	-	-	SYM
ejpam-4072	292	17	1225	1225	NUM
ejpam-4072	292	18	1221	1221	NUM
ejpam-4072	292	19	σ1σ2	σ1σ2	VERB
ejpam-4072	292	20	-	-	NOUN
ejpam-4072	292	21	regular	regular	ADJ
ejpam-4072	292	22	for	for	ADP
ejpam-4072	292	23	each	each	DET
ejpam-4072	292	24	z	z	NOUN
ejpam-4072	292	25	∈	∈	PROPN
ejpam-4072	292	26	u	u	NOUN
ejpam-4072	292	27	,	,	PUNCT
ejpam-4072	292	28	by	by	ADP
ejpam-4072	292	29	lemma	lemma	PROPN
ejpam-4072	292	30	8	8	NUM
ejpam-4072	292	31	,	,	PUNCT
ejpam-4072	292	32	there	there	PRON
ejpam-4072	292	33	exists	exist	VERB
ejpam-4072	292	34	a	a	DET
ejpam-4072	292	35	τ1τ2	τ1τ2	NOUN
ejpam-4072	292	36	-	-	ADJ
ejpam-4072	292	37	open	open	ADJ
ejpam-4072	292	38	set	set	NOUN
ejpam-4072	292	39	w	w	ADP
ejpam-4072	292	40	such	such	ADJ
ejpam-4072	292	41	that	that	SCONJ
ejpam-4072	292	42	f	f	PROPN
ejpam-4072	292	43	(	(	PUNCT
ejpam-4072	292	44	z	z	NOUN
ejpam-4072	292	45	)	)	PUNCT
ejpam-4072	292	46	⊆	⊆	NUM
ejpam-4072	292	47	w	w	ADP
ejpam-4072	292	48	⊆	⊆	NUM
ejpam-4072	292	49	σ1σ2	σ1σ2	NOUN
ejpam-4072	292	50	-	-	PUNCT
ejpam-4072	292	51	cl(w	cl(w	NOUN
ejpam-4072	292	52	)	)	PUNCT
ejpam-4072	292	53	⊆	⊆	NUM
ejpam-4072	292	54	v	v	NOUN
ejpam-4072	292	55	;	;	PUNCT
ejpam-4072	292	56	hence	hence	ADV
ejpam-4072	292	57	g(z	g(z	ADJ
ejpam-4072	292	58	)	)	PUNCT
ejpam-4072	292	59	⊆	⊆	NUM
ejpam-4072	292	60	σ1σ2	σ1σ2	NOUN
ejpam-4072	292	61	-	-	PUNCT
ejpam-4072	292	62	cl(w	cl(w	NOUN
ejpam-4072	292	63	)	)	PUNCT
ejpam-4072	292	64	⊆	⊆	NUM
ejpam-4072	292	65	σ1σ2	σ1σ2	NOUN
ejpam-4072	292	66	-	-	NUM
ejpam-4072	292	67	cl(v	cl(v	NOUN
ejpam-4072	292	68	)	)	PUNCT
ejpam-4072	292	69	for	for	ADP
ejpam-4072	292	70	each	each	DET
ejpam-4072	292	71	z	z	NOUN
ejpam-4072	292	72	∈	∈	PROPN
ejpam-4072	292	73	u	u	NOUN
ejpam-4072	292	74	.	.	PUNCT
ejpam-4072	293	1	thus	thus	ADV
ejpam-4072	293	2	,	,	PUNCT
ejpam-4072	293	3	g(u	g(u	PROPN
ejpam-4072	293	4	)	)	PUNCT
ejpam-4072	293	5	⊆	⊆	NUM
ejpam-4072	293	6	σ1σ2	σ1σ2	NOUN
ejpam-4072	293	7	-	-	NUM
ejpam-4072	293	8	cl(v	cl(v	NOUN
ejpam-4072	293	9	)	)	PUNCT
ejpam-4072	293	10	and	and	CCONJ
ejpam-4072	293	11	hence	hence	ADV
ejpam-4072	293	12	g	g	PROPN
ejpam-4072	293	13	is	be	AUX
ejpam-4072	293	14	upper	upper	ADJ
ejpam-4072	293	15	almost	almost	ADV
ejpam-4072	293	16	weakly	weakly	ADJ
ejpam-4072	293	17	(	(	PUNCT
ejpam-4072	293	18	τ1	τ1	NOUN
ejpam-4072	293	19	,	,	PUNCT
ejpam-4072	293	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	293	21	.	.	PUNCT
ejpam-4072	294	1	conversely	conversely	ADV
ejpam-4072	294	2	,	,	PUNCT
ejpam-4072	294	3	suppose	suppose	VERB
ejpam-4072	294	4	that	that	SCONJ
ejpam-4072	294	5	g	g	PROPN
ejpam-4072	294	6	is	be	AUX
ejpam-4072	294	7	upper	upper	ADJ
ejpam-4072	294	8	almost	almost	ADV
ejpam-4072	294	9	weakly	weakly	ADJ
ejpam-4072	294	10	(	(	PUNCT
ejpam-4072	294	11	τ1	τ1	NOUN
ejpam-4072	294	12	,	,	PUNCT
ejpam-4072	294	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	294	14	.	.	PUNCT
ejpam-4072	295	1	let	let	VERB
ejpam-4072	295	2	x	x	PUNCT
ejpam-4072	295	3	∈	∈	PROPN
ejpam-4072	295	4	x	x	PUNCT
ejpam-4072	295	5	and	and	CCONJ
ejpam-4072	295	6	let	let	VERB
ejpam-4072	295	7	v	v	PART
ejpam-4072	295	8	be	be	AUX
ejpam-4072	295	9	any	any	DET
ejpam-4072	295	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	295	11	-	-	ADJ
ejpam-4072	295	12	open	open	ADJ
ejpam-4072	295	13	set	set	NOUN
ejpam-4072	295	14	of	of	ADP
ejpam-4072	295	15	y	y	NOUN
ejpam-4072	295	16	containing	contain	VERB
ejpam-4072	295	17	g(x	g(x	NOUN
ejpam-4072	295	18	)	)	PUNCT
ejpam-4072	295	19	.	.	PUNCT
ejpam-4072	296	1	by	by	ADP
ejpam-4072	296	2	lemma	lemma	PROPN
ejpam-4072	296	3	9	9	NUM
ejpam-4072	296	4	,	,	PUNCT
ejpam-4072	296	5	we	we	PRON
ejpam-4072	296	6	have	have	VERB
ejpam-4072	296	7	x	x	X
ejpam-4072	296	8	∈	∈	NOUN
ejpam-4072	296	9	f+(v	f+(v	NOUN
ejpam-4072	296	10	)	)	PUNCT
ejpam-4072	297	1	=	=	PUNCT
ejpam-4072	297	2	g+(v	g+(v	PROPN
ejpam-4072	297	3	)	)	PUNCT
ejpam-4072	297	4	and	and	CCONJ
ejpam-4072	297	5	hence	hence	ADV
ejpam-4072	297	6	g(x	g(x	NOUN
ejpam-4072	297	7	)	)	PUNCT
ejpam-4072	297	8	⊆	⊆	NUM
ejpam-4072	297	9	v	v	NOUN
ejpam-4072	297	10	.	.	PUNCT
ejpam-4072	298	1	there	there	PRON
ejpam-4072	298	2	exists	exist	VERB
ejpam-4072	298	3	a	a	DET
ejpam-4072	298	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	298	5	-	-	ADJ
ejpam-4072	298	6	open	open	ADJ
ejpam-4072	298	7	set	set	NOUN
ejpam-4072	298	8	u	u	NOUN
ejpam-4072	298	9	containing	contain	VERB
ejpam-4072	298	10	x	x	PUNCT
ejpam-4072	298	11	such	such	ADJ
ejpam-4072	298	12	that	that	SCONJ
ejpam-4072	298	13	f	f	PROPN
ejpam-4072	298	14	(	(	PUNCT
ejpam-4072	298	15	u	u	NOUN
ejpam-4072	298	16	)	)	PUNCT
ejpam-4072	298	17	⊆	⊆	NUM
ejpam-4072	298	18	σ1σ2	σ1σ2	NOUN
ejpam-4072	298	19	-	-	NUM
ejpam-4072	298	20	cl(v	cl(v	NOUN
ejpam-4072	298	21	)	)	PUNCT
ejpam-4072	298	22	.	.	PUNCT
ejpam-4072	299	1	thus	thus	ADV
ejpam-4072	299	2	,	,	PUNCT
ejpam-4072	299	3	u	u	PROPN
ejpam-4072	299	4	⊆	⊆	NUM
ejpam-4072	299	5	g+(v	g+(v	PROPN
ejpam-4072	299	6	)	)	PUNCT
ejpam-4072	299	7	=	=	PUNCT
ejpam-4072	300	1	f+(v	f+(v	NOUN
ejpam-4072	300	2	)	)	PUNCT
ejpam-4072	301	1	and	and	CCONJ
ejpam-4072	301	2	so	so	ADV
ejpam-4072	301	3	f	f	PROPN
ejpam-4072	301	4	(	(	PUNCT
ejpam-4072	301	5	u	u	NOUN
ejpam-4072	301	6	)	)	PUNCT
ejpam-4072	301	7	⊆	⊆	NUM
ejpam-4072	301	8	σ1σ2	σ1σ2	NOUN
ejpam-4072	301	9	-	-	NUM
ejpam-4072	301	10	cl(v	cl(v	NOUN
ejpam-4072	301	11	)	)	PUNCT
ejpam-4072	301	12	.	.	PUNCT
ejpam-4072	302	1	this	this	PRON
ejpam-4072	302	2	shows	show	VERB
ejpam-4072	302	3	that	that	SCONJ
ejpam-4072	302	4	f	f	PROPN
ejpam-4072	302	5	is	be	AUX
ejpam-4072	302	6	upper	upper	ADJ
ejpam-4072	302	7	almost	almost	ADV
ejpam-4072	302	8	weakly	weakly	ADJ
ejpam-4072	302	9	(	(	PUNCT
ejpam-4072	302	10	τ1	τ1	NOUN
ejpam-4072	302	11	,	,	PUNCT
ejpam-4072	302	12	τ2)-continuous	τ2)-continuous	PROPN
ejpam-4072	302	13	.	.	PUNCT
ejpam-4072	303	1	lemma	lemma	PROPN
ejpam-4072	303	2	10	10	NUM
ejpam-4072	303	3	.	.	PUNCT
ejpam-4072	304	1	[	[	X
ejpam-4072	304	2	5	5	NUM
ejpam-4072	304	3	]	]	PUNCT
ejpam-4072	304	4	for	for	ADP
ejpam-4072	304	5	a	a	DET
ejpam-4072	304	6	multifunction	multifunction	NOUN
ejpam-4072	304	7	f	f	NOUN
ejpam-4072	304	8	:	:	PUNCT
ejpam-4072	304	9	(	(	PUNCT
ejpam-4072	304	10	x	x	NOUN
ejpam-4072	304	11	,	,	PUNCT
ejpam-4072	304	12	τ1	τ1	NOUN
ejpam-4072	304	13	,	,	PUNCT
ejpam-4072	304	14	τ2	τ2	NOUN
ejpam-4072	304	15	)	)	PUNCT
ejpam-4072	304	16	→	→	SYM
ejpam-4072	304	17	(	(	PUNCT
ejpam-4072	304	18	y	y	PROPN
ejpam-4072	304	19	,	,	PUNCT
ejpam-4072	304	20	σ1	σ1	PROPN
ejpam-4072	304	21	,	,	PUNCT
ejpam-4072	304	22	σ2	σ2	NOUN
ejpam-4072	304	23	)	)	PUNCT
ejpam-4072	304	24	,	,	PUNCT
ejpam-4072	304	25	clf	clf	PROPN
ejpam-4072	304	26	−	−	PROPN
ejpam-4072	304	27	⊛	⊛	NUM
ejpam-4072	304	28	(	(	PUNCT
ejpam-4072	304	29	v	v	NOUN
ejpam-4072	304	30	)	)	PUNCT
ejpam-4072	304	31	=	=	SYM
ejpam-4072	304	32	pclf−	pclf−	NOUN
ejpam-4072	304	33	⊛	⊛	NUM
ejpam-4072	304	34	(	(	PUNCT
ejpam-4072	304	35	v	v	NOUN
ejpam-4072	304	36	)	)	PUNCT
ejpam-4072	304	37	=	=	SYM
ejpam-4072	304	38	f−(v	f−(v	ADJ
ejpam-4072	304	39	)	)	PUNCT
ejpam-4072	304	40	for	for	ADP
ejpam-4072	304	41	each	each	DET
ejpam-4072	304	42	σ1σ2	σ1σ2	VERB
ejpam-4072	304	43	-	-	ADJ
ejpam-4072	304	44	open	open	ADJ
ejpam-4072	304	45	set	set	NOUN
ejpam-4072	304	46	v	v	NOUN
ejpam-4072	304	47	of	of	ADP
ejpam-4072	304	48	y	y	PROPN
ejpam-4072	304	49	.	.	PUNCT
ejpam-4072	305	1	theorem	theorem	ADJ
ejpam-4072	305	2	8	8	NUM
ejpam-4072	305	3	.	.	PUNCT
ejpam-4072	306	1	for	for	ADP
ejpam-4072	306	2	a	a	DET
ejpam-4072	306	3	multifunction	multifunction	NOUN
ejpam-4072	306	4	f	f	NOUN
ejpam-4072	306	5	:	:	PUNCT
ejpam-4072	306	6	(	(	PUNCT
ejpam-4072	306	7	x	x	NOUN
ejpam-4072	306	8	,	,	PUNCT
ejpam-4072	306	9	τ1	τ1	NOUN
ejpam-4072	306	10	,	,	PUNCT
ejpam-4072	306	11	τ2	τ2	NOUN
ejpam-4072	306	12	)	)	PUNCT
ejpam-4072	306	13	→	→	SYM
ejpam-4072	306	14	(	(	PUNCT
ejpam-4072	306	15	y	y	PROPN
ejpam-4072	306	16	,	,	PUNCT
ejpam-4072	306	17	σ1	σ1	PROPN
ejpam-4072	306	18	,	,	PUNCT
ejpam-4072	306	19	σ2	σ2	NOUN
ejpam-4072	306	20	)	)	PUNCT
ejpam-4072	306	21	,	,	PUNCT
ejpam-4072	306	22	the	the	DET
ejpam-4072	306	23	following	follow	VERB
ejpam-4072	306	24	properties	property	NOUN
ejpam-4072	306	25	are	be	AUX
ejpam-4072	306	26	equivalent	equivalent	ADJ
ejpam-4072	306	27	:	:	PUNCT
ejpam-4072	306	28	(	(	PUNCT
ejpam-4072	306	29	1	1	X
ejpam-4072	306	30	)	)	PUNCT
ejpam-4072	306	31	f	f	PROPN
ejpam-4072	306	32	is	be	AUX
ejpam-4072	306	33	lower	low	ADJ
ejpam-4072	306	34	almost	almost	ADV
ejpam-4072	306	35	weakly	weakly	ADJ
ejpam-4072	306	36	(	(	PUNCT
ejpam-4072	306	37	τ1	τ1	NOUN
ejpam-4072	306	38	,	,	PUNCT
ejpam-4072	306	39	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	306	40	;	;	PUNCT
ejpam-4072	306	41	(	(	PUNCT
ejpam-4072	306	42	2	2	X
ejpam-4072	306	43	)	)	PUNCT
ejpam-4072	306	44	pclf⊛	pclf⊛	NOUN
ejpam-4072	306	45	is	be	AUX
ejpam-4072	306	46	lower	low	ADJ
ejpam-4072	306	47	almost	almost	ADV
ejpam-4072	306	48	weakly	weakly	ADJ
ejpam-4072	306	49	(	(	PUNCT
ejpam-4072	306	50	τ1	τ1	NOUN
ejpam-4072	306	51	,	,	PUNCT
ejpam-4072	306	52	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	306	53	;	;	PUNCT
ejpam-4072	306	54	(	(	PUNCT
ejpam-4072	306	55	3	3	X
ejpam-4072	306	56	)	)	PUNCT
ejpam-4072	306	57	clf⊛	clf⊛	PROPN
ejpam-4072	306	58	is	be	AUX
ejpam-4072	306	59	lower	low	ADJ
ejpam-4072	306	60	almost	almost	ADV
ejpam-4072	306	61	weakly	weakly	ADJ
ejpam-4072	306	62	(	(	PUNCT
ejpam-4072	306	63	τ1	τ1	NOUN
ejpam-4072	306	64	,	,	PUNCT
ejpam-4072	306	65	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	306	66	.	.	PUNCT
ejpam-4072	307	1	proof	proof	NOUN
ejpam-4072	307	2	.	.	PUNCT
ejpam-4072	308	1	by	by	ADP
ejpam-4072	308	2	using	use	VERB
ejpam-4072	308	3	lemma	lemma	PROPN
ejpam-4072	308	4	10	10	NUM
ejpam-4072	308	5	this	this	PRON
ejpam-4072	308	6	can	can	AUX
ejpam-4072	308	7	be	be	AUX
ejpam-4072	308	8	shown	show	VERB
ejpam-4072	308	9	similarly	similarly	ADV
ejpam-4072	308	10	to	to	ADP
ejpam-4072	308	11	that	that	PRON
ejpam-4072	308	12	of	of	ADP
ejpam-4072	308	13	theorem	theorem	NOUN
ejpam-4072	308	14	7	7	NUM
ejpam-4072	308	15	.	.	PUNCT
ejpam-4072	309	1	the	the	DET
ejpam-4072	309	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	309	3	-	-	NOUN
ejpam-4072	309	4	prefrontier	prefronti	ADJ
ejpam-4072	309	5	[	[	X
ejpam-4072	309	6	5	5	NUM
ejpam-4072	309	7	]	]	PUNCT
ejpam-4072	309	8	of	of	ADP
ejpam-4072	309	9	a	a	DET
ejpam-4072	309	10	subset	subset	NOUN
ejpam-4072	309	11	a	a	PRON
ejpam-4072	309	12	of	of	ADP
ejpam-4072	309	13	a	a	DET
ejpam-4072	309	14	bitopological	bitopological	ADJ
ejpam-4072	309	15	space	space	NOUN
ejpam-4072	309	16	(	(	PUNCT
ejpam-4072	309	17	x	x	NOUN
ejpam-4072	309	18	,	,	PUNCT
ejpam-4072	309	19	τ1	τ1	NOUN
ejpam-4072	309	20	,	,	PUNCT
ejpam-4072	309	21	τ2	τ2	PROPN
ejpam-4072	309	22	)	)	PUNCT
ejpam-4072	309	23	,	,	PUNCT
ejpam-4072	309	24	denoted	denote	VERB
ejpam-4072	309	25	by	by	ADP
ejpam-4072	309	26	τ1τ2	τ1τ2	NOUN
ejpam-4072	309	27	-	-	ADJ
ejpam-4072	309	28	pfr(a	pfr(a	NOUN
ejpam-4072	309	29	)	)	PUNCT
ejpam-4072	309	30	,	,	PUNCT
ejpam-4072	309	31	is	be	AUX
ejpam-4072	309	32	defined	define	VERB
ejpam-4072	309	33	by	by	ADP
ejpam-4072	309	34	τ1τ2	τ1τ2	NOUN
ejpam-4072	309	35	-	-	ADJ
ejpam-4072	309	36	pfr(a	pfr(a	ADJ
ejpam-4072	309	37	)	)	PUNCT
ejpam-4072	310	1	=	=	PUNCT
ejpam-4072	310	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	310	3	-	-	ADJ
ejpam-4072	310	4	pcl(a	pcl(a	ADJ
ejpam-4072	310	5	)	)	PUNCT
ejpam-4072	310	6	∩	∩	NOUN
ejpam-4072	310	7	τ1τ2	τ1τ2	NOUN
ejpam-4072	310	8	-	-	ADJ
ejpam-4072	310	9	pcl(x	pcl(x	ADJ
ejpam-4072	310	10	−a	−a	NOUN
ejpam-4072	310	11	)	)	PUNCT
ejpam-4072	310	12	=	=	PUNCT
ejpam-4072	311	1	τ1τ2	τ1τ2	NOUN
ejpam-4072	311	2	-	-	ADJ
ejpam-4072	311	3	pcl(a)−	pcl(a)−	ADJ
ejpam-4072	311	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	311	5	-	-	NOUN
ejpam-4072	311	6	pint(a	pint(a	NOUN
ejpam-4072	311	7	)	)	PUNCT
ejpam-4072	311	8	.	.	PUNCT
ejpam-4072	312	1	theorem	theorem	VERB
ejpam-4072	312	2	9	9	NUM
ejpam-4072	312	3	.	.	PUNCT
ejpam-4072	313	1	the	the	DET
ejpam-4072	313	2	set	set	NOUN
ejpam-4072	313	3	of	of	ADP
ejpam-4072	313	4	all	all	DET
ejpam-4072	313	5	points	point	NOUN
ejpam-4072	313	6	x	x	PUNCT
ejpam-4072	313	7	of	of	ADP
ejpam-4072	313	8	x	x	SYM
ejpam-4072	313	9	at	at	ADP
ejpam-4072	313	10	which	which	PRON
ejpam-4072	313	11	a	a	DET
ejpam-4072	313	12	multifunction	multifunction	NOUN
ejpam-4072	314	1	f	f	NOUN
ejpam-4072	314	2	:	:	PUNCT
ejpam-4072	314	3	(	(	PUNCT
ejpam-4072	314	4	x	x	NOUN
ejpam-4072	314	5	,	,	PUNCT
ejpam-4072	314	6	τ1	τ1	NOUN
ejpam-4072	314	7	,	,	PUNCT
ejpam-4072	314	8	τ2	τ2	NOUN
ejpam-4072	314	9	)	)	PUNCT
ejpam-4072	314	10	→	→	SYM
ejpam-4072	314	11	(	(	PUNCT
ejpam-4072	314	12	y	y	PROPN
ejpam-4072	314	13	,	,	PUNCT
ejpam-4072	314	14	σ1	σ1	PROPN
ejpam-4072	314	15	,	,	PUNCT
ejpam-4072	314	16	σ2	σ2	PROPN
ejpam-4072	314	17	)	)	PUNCT
ejpam-4072	314	18	is	be	AUX
ejpam-4072	314	19	not	not	PART
ejpam-4072	314	20	upper	upper	ADJ
ejpam-4072	314	21	almost	almost	ADV
ejpam-4072	314	22	weakly	weakly	ADJ
ejpam-4072	314	23	(	(	PUNCT
ejpam-4072	314	24	τ1	τ1	NOUN
ejpam-4072	314	25	,	,	PUNCT
ejpam-4072	314	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	314	27	is	be	AUX
ejpam-4072	314	28	identical	identical	ADJ
ejpam-4072	314	29	with	with	ADP
ejpam-4072	314	30	the	the	DET
ejpam-4072	314	31	union	union	NOUN
ejpam-4072	314	32	of	of	ADP
ejpam-4072	314	33	the	the	DET
ejpam-4072	314	34	τ1τ2prefrontier	τ1τ2prefrontier	PUNCT
ejpam-4072	314	35	of	of	ADP
ejpam-4072	314	36	the	the	DET
ejpam-4072	314	37	upper	upper	ADJ
ejpam-4072	314	38	inverse	inverse	NOUN
ejpam-4072	314	39	images	image	NOUN
ejpam-4072	314	40	of	of	ADP
ejpam-4072	314	41	the	the	DET
ejpam-4072	314	42	σ1σ2	σ1σ2	NOUN
ejpam-4072	314	43	-	-	NOUN
ejpam-4072	314	44	closure	closure	NOUN
ejpam-4072	314	45	of	of	ADP
ejpam-4072	314	46	σ1σ2	σ1σ2	NOUN
ejpam-4072	314	47	-	-	PUNCT
ejpam-4072	314	48	open	open	ADJ
ejpam-4072	314	49	sets	set	NOUN
ejpam-4072	314	50	containing	contain	VERB
ejpam-4072	314	51	f	f	X
ejpam-4072	314	52	(	(	PUNCT
ejpam-4072	314	53	x	x	NOUN
ejpam-4072	314	54	)	)	PUNCT
ejpam-4072	314	55	.	.	PUNCT
ejpam-4072	315	1	proof	proof	NOUN
ejpam-4072	315	2	.	.	PUNCT
ejpam-4072	316	1	let	let	VERB
ejpam-4072	316	2	x	x	PUNCT
ejpam-4072	316	3	∈	∈	PROPN
ejpam-4072	316	4	x	x	PUNCT
ejpam-4072	316	5	at	at	ADP
ejpam-4072	316	6	which	which	PRON
ejpam-4072	316	7	f	f	NOUN
ejpam-4072	316	8	is	be	AUX
ejpam-4072	316	9	not	not	PART
ejpam-4072	316	10	upper	upper	ADJ
ejpam-4072	316	11	almost	almost	ADV
ejpam-4072	316	12	weakly	weakly	ADJ
ejpam-4072	316	13	(	(	PUNCT
ejpam-4072	316	14	τ1	τ1	NOUN
ejpam-4072	316	15	,	,	PUNCT
ejpam-4072	316	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	316	17	.	.	PUNCT
ejpam-4072	317	1	there	there	PRON
ejpam-4072	317	2	exists	exist	VERB
ejpam-4072	317	3	a	a	DET
ejpam-4072	317	4	σ1σ2	σ1σ2	NUM
ejpam-4072	317	5	-	-	ADJ
ejpam-4072	317	6	open	open	ADJ
ejpam-4072	317	7	set	set	NOUN
ejpam-4072	317	8	v	v	NOUN
ejpam-4072	317	9	of	of	ADP
ejpam-4072	317	10	y	y	PROPN
ejpam-4072	317	11	containing	contain	VERB
ejpam-4072	317	12	f	f	PROPN
ejpam-4072	317	13	(	(	PUNCT
ejpam-4072	317	14	x	x	X
ejpam-4072	317	15	)	)	PUNCT
ejpam-4072	317	16	such	such	ADJ
ejpam-4072	317	17	that	that	SCONJ
ejpam-4072	317	18	u	u	PROPN
ejpam-4072	317	19	∩	∩	NOUN
ejpam-4072	317	20	(	(	PUNCT
ejpam-4072	317	21	x	x	NOUN
ejpam-4072	317	22	−	−	PROPN
ejpam-4072	317	23	f+(v	f+(v	NOUN
ejpam-4072	317	24	)	)	PUNCT
ejpam-4072	317	25	)	)	PUNCT
ejpam-4072	318	1	̸=	̸=	NOUN
ejpam-4072	318	2	∅	∅	NOUN
ejpam-4072	318	3	for	for	ADP
ejpam-4072	318	4	every	every	DET
ejpam-4072	318	5	τ1τ2	τ1τ2	ADJ
ejpam-4072	318	6	-	-	ADJ
ejpam-4072	318	7	open	open	ADJ
ejpam-4072	318	8	set	set	NOUN
ejpam-4072	318	9	u	u	NOUN
ejpam-4072	318	10	containing	contain	VERB
ejpam-4072	318	11	x.	x.	NOUN
ejpam-4072	318	12	thus	thus	ADV
ejpam-4072	318	13	,	,	PUNCT
ejpam-4072	318	14	x	x	SYM
ejpam-4072	318	15	∈	∈	PROPN
ejpam-4072	318	16	τ1τ2	τ1τ2	NOUN
ejpam-4072	318	17	-	-	ADJ
ejpam-4072	318	18	pcl(x	pcl(x	ADJ
ejpam-4072	318	19	−	−	NOUN
ejpam-4072	318	20	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	318	21	-	-	PUNCT
ejpam-4072	318	22	cl(v	cl(v	NOUN
ejpam-4072	318	23	)	)	PUNCT
ejpam-4072	318	24	)	)	PUNCT
ejpam-4072	318	25	)	)	PUNCT
ejpam-4072	319	1	=	=	PUNCT
ejpam-4072	319	2	x	x	X
ejpam-4072	320	1	−	−	ADP
ejpam-4072	320	2	τ1τ2	τ1τ2	NOUN
ejpam-4072	320	3	-	-	NOUN
ejpam-4072	320	4	pint(f	pint(f	ADJ
ejpam-4072	320	5	+	+	PROPN
ejpam-4072	320	6	(	(	PUNCT
ejpam-4072	320	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	320	8	-	-	NUM
ejpam-4072	320	9	cl(v	cl(v	NOUN
ejpam-4072	320	10	)	)	PUNCT
ejpam-4072	320	11	)	)	PUNCT
ejpam-4072	320	12	)	)	PUNCT
ejpam-4072	320	13	.	.	PUNCT
ejpam-4072	321	1	since	since	SCONJ
ejpam-4072	321	2	x	x	PROPN
ejpam-4072	321	3	∈	∈	PROPN
ejpam-4072	321	4	f+(v	f+(v	NOUN
ejpam-4072	321	5	)	)	PUNCT
ejpam-4072	321	6	,	,	PUNCT
ejpam-4072	321	7	we	we	PRON
ejpam-4072	321	8	have	have	VERB
ejpam-4072	321	9	x	x	PART
ejpam-4072	321	10	∈	∈	PRON
ejpam-4072	321	11	τ1τ2	τ1τ2	NOUN
ejpam-4072	321	12	-	-	PROPN
ejpam-4072	321	13	pcl(f	pcl(f	PROPN
ejpam-4072	321	14	+	+	ADJ
ejpam-4072	321	15	(	(	PUNCT
ejpam-4072	321	16	σ1σ2	σ1σ2	NOUN
ejpam-4072	321	17	-	-	NUM
ejpam-4072	321	18	cl(v	cl(v	NOUN
ejpam-4072	321	19	)	)	PUNCT
ejpam-4072	321	20	)	)	PUNCT
ejpam-4072	321	21	)	)	PUNCT
ejpam-4072	322	1	and	and	CCONJ
ejpam-4072	322	2	hence	hence	ADV
ejpam-4072	322	3	x	x	X
ejpam-4072	322	4	∈	∈	PRON
ejpam-4072	322	5	τ1τ2	τ1τ2	NOUN
ejpam-4072	322	6	-	-	NOUN
ejpam-4072	322	7	pfr(f	pfr(f	ADJ
ejpam-4072	322	8	+	+	NOUN
ejpam-4072	322	9	(	(	PUNCT
ejpam-4072	322	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	322	11	-	-	NUM
ejpam-4072	322	12	cl(v	cl(v	NOUN
ejpam-4072	322	13	)	)	PUNCT
ejpam-4072	322	14	)	)	PUNCT
ejpam-4072	322	15	)	)	PUNCT
ejpam-4072	322	16	.	.	PUNCT
ejpam-4072	323	1	c.	c.	PROPN
ejpam-4072	323	2	boonpok	boonpok	PROPN
ejpam-4072	323	3	,	,	PUNCT
ejpam-4072	323	4	c.	c.	PROPN
ejpam-4072	323	5	viriyapong	viriyapong	PROPN
ejpam-4072	323	6	/	/	SYM
ejpam-4072	323	7	eur	eur	PROPN
ejpam-4072	323	8	.	.	PUNCT
ejpam-4072	324	1	j.	j.	PROPN
ejpam-4072	324	2	pure	pure	PROPN
ejpam-4072	324	3	appl	appl	PROPN
ejpam-4072	324	4	.	.	PROPN
ejpam-4072	324	5	math	math	PROPN
ejpam-4072	324	6	,	,	PUNCT
ejpam-4072	324	7	14	14	NUM
ejpam-4072	324	8	(	(	PUNCT
ejpam-4072	324	9	4	4	NUM
ejpam-4072	324	10	)	)	PUNCT
ejpam-4072	324	11	(	(	PUNCT
ejpam-4072	324	12	2021	2021	NUM
ejpam-4072	324	13	)	)	PUNCT
ejpam-4072	324	14	,	,	PUNCT
ejpam-4072	324	15	1212	1212	NUM
ejpam-4072	324	16	-	-	SYM
ejpam-4072	324	17	1225	1225	NUM
ejpam-4072	324	18	1222	1222	NUM
ejpam-4072	324	19	conversely	conversely	ADV
ejpam-4072	324	20	,	,	PUNCT
ejpam-4072	324	21	if	if	SCONJ
ejpam-4072	324	22	f	f	PROPN
ejpam-4072	324	23	is	be	AUX
ejpam-4072	324	24	upper	upper	ADJ
ejpam-4072	324	25	almost	almost	ADV
ejpam-4072	324	26	weakly	weakly	ADJ
ejpam-4072	324	27	(	(	PUNCT
ejpam-4072	324	28	τ1	τ1	NOUN
ejpam-4072	324	29	,	,	PUNCT
ejpam-4072	324	30	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	324	31	,	,	PUNCT
ejpam-4072	324	32	then	then	ADV
ejpam-4072	324	33	for	for	ADP
ejpam-4072	324	34	any	any	DET
ejpam-4072	324	35	σ1σ2	σ1σ2	NOUN
ejpam-4072	324	36	-	-	ADJ
ejpam-4072	324	37	open	open	ADJ
ejpam-4072	324	38	set	set	NOUN
ejpam-4072	324	39	v	v	NOUN
ejpam-4072	324	40	of	of	ADP
ejpam-4072	324	41	y	y	PROPN
ejpam-4072	324	42	containing	contain	VERB
ejpam-4072	324	43	f	f	PROPN
ejpam-4072	324	44	(	(	PUNCT
ejpam-4072	324	45	x	x	X
ejpam-4072	324	46	)	)	PUNCT
ejpam-4072	324	47	there	there	PRON
ejpam-4072	324	48	exists	exist	VERB
ejpam-4072	324	49	a	a	DET
ejpam-4072	324	50	τ1τ2	τ1τ2	NOUN
ejpam-4072	324	51	-	-	ADJ
ejpam-4072	324	52	open	open	ADJ
ejpam-4072	324	53	set	set	NOUN
ejpam-4072	324	54	u	u	NOUN
ejpam-4072	324	55	containing	contain	VERB
ejpam-4072	324	56	x	x	PUNCT
ejpam-4072	324	57	such	such	ADJ
ejpam-4072	324	58	that	that	SCONJ
ejpam-4072	324	59	f	f	PROPN
ejpam-4072	324	60	(	(	PUNCT
ejpam-4072	324	61	u	u	NOUN
ejpam-4072	324	62	)	)	PUNCT
ejpam-4072	324	63	⊆	⊆	NUM
ejpam-4072	324	64	σ1σ2	σ1σ2	NOUN
ejpam-4072	324	65	-	-	NUM
ejpam-4072	324	66	cl(v	cl(v	NOUN
ejpam-4072	324	67	)	)	PUNCT
ejpam-4072	324	68	;	;	PUNCT
ejpam-4072	324	69	hence	hence	ADV
ejpam-4072	324	70	u	u	NOUN
ejpam-4072	324	71	⊆	⊆	NUM
ejpam-4072	324	72	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	324	73	-	-	PUNCT
ejpam-4072	324	74	cl(v	cl(v	NOUN
ejpam-4072	324	75	)	)	PUNCT
ejpam-4072	324	76	)	)	PUNCT
ejpam-4072	324	77	.	.	PUNCT
ejpam-4072	325	1	thus	thus	ADV
ejpam-4072	325	2	,	,	PUNCT
ejpam-4072	325	3	x	x	SYM
ejpam-4072	325	4	∈	∈	PROPN
ejpam-4072	325	5	τ1τ2	τ1τ2	NOUN
ejpam-4072	325	6	-	-	NOUN
ejpam-4072	325	7	pint(f	pint(f	ADJ
ejpam-4072	325	8	+	+	PROPN
ejpam-4072	325	9	(	(	PUNCT
ejpam-4072	325	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	325	11	-	-	NUM
ejpam-4072	325	12	cl(v	cl(v	NOUN
ejpam-4072	325	13	)	)	PUNCT
ejpam-4072	325	14	)	)	PUNCT
ejpam-4072	325	15	)	)	PUNCT
ejpam-4072	325	16	.	.	PUNCT
ejpam-4072	326	1	this	this	PRON
ejpam-4072	326	2	contradicts	contradict	VERB
ejpam-4072	326	3	with	with	ADP
ejpam-4072	326	4	the	the	DET
ejpam-4072	326	5	fact	fact	NOUN
ejpam-4072	326	6	that	that	SCONJ
ejpam-4072	326	7	x	x	PUNCT
ejpam-4072	326	8	∈	∈	PROPN
ejpam-4072	326	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	326	10	-	-	NOUN
ejpam-4072	326	11	pfr(f	pfr(f	ADJ
ejpam-4072	326	12	+	+	NOUN
ejpam-4072	326	13	(	(	PUNCT
ejpam-4072	326	14	σ1σ2	σ1σ2	NOUN
ejpam-4072	326	15	-	-	NUM
ejpam-4072	326	16	cl(v	cl(v	NOUN
ejpam-4072	326	17	)	)	PUNCT
ejpam-4072	326	18	)	)	PUNCT
ejpam-4072	326	19	)	)	PUNCT
ejpam-4072	326	20	.	.	PUNCT
ejpam-4072	327	1	thus	thus	ADV
ejpam-4072	327	2	,	,	PUNCT
ejpam-4072	327	3	f	f	PROPN
ejpam-4072	327	4	is	be	AUX
ejpam-4072	327	5	not	not	PART
ejpam-4072	327	6	upper	upper	ADJ
ejpam-4072	327	7	almost	almost	ADV
ejpam-4072	327	8	weakly	weakly	ADJ
ejpam-4072	327	9	(	(	PUNCT
ejpam-4072	327	10	τ1	τ1	NOUN
ejpam-4072	327	11	,	,	PUNCT
ejpam-4072	327	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	327	13	at	at	ADP
ejpam-4072	327	14	x.	x.	NOUN
ejpam-4072	327	15	theorem	theorem	VERB
ejpam-4072	327	16	10	10	NUM
ejpam-4072	327	17	.	.	PUNCT
ejpam-4072	328	1	the	the	DET
ejpam-4072	328	2	set	set	NOUN
ejpam-4072	328	3	of	of	ADP
ejpam-4072	328	4	all	all	DET
ejpam-4072	328	5	points	point	NOUN
ejpam-4072	328	6	x	x	PUNCT
ejpam-4072	328	7	of	of	ADP
ejpam-4072	328	8	x	x	SYM
ejpam-4072	328	9	at	at	ADP
ejpam-4072	328	10	which	which	PRON
ejpam-4072	328	11	a	a	DET
ejpam-4072	328	12	multifunction	multifunction	NOUN
ejpam-4072	329	1	f	f	NOUN
ejpam-4072	329	2	:	:	PUNCT
ejpam-4072	329	3	(	(	PUNCT
ejpam-4072	329	4	x	x	NOUN
ejpam-4072	329	5	,	,	PUNCT
ejpam-4072	329	6	τ1	τ1	NOUN
ejpam-4072	329	7	,	,	PUNCT
ejpam-4072	329	8	τ2	τ2	NOUN
ejpam-4072	329	9	)	)	PUNCT
ejpam-4072	329	10	→	→	SYM
ejpam-4072	329	11	(	(	PUNCT
ejpam-4072	329	12	y	y	PROPN
ejpam-4072	329	13	,	,	PUNCT
ejpam-4072	329	14	σ1	σ1	PROPN
ejpam-4072	329	15	,	,	PUNCT
ejpam-4072	329	16	σ2	σ2	PROPN
ejpam-4072	329	17	)	)	PUNCT
ejpam-4072	329	18	is	be	AUX
ejpam-4072	329	19	not	not	PART
ejpam-4072	329	20	lower	low	ADJ
ejpam-4072	329	21	almost	almost	ADV
ejpam-4072	329	22	weakly	weakly	ADJ
ejpam-4072	329	23	(	(	PUNCT
ejpam-4072	329	24	τ1	τ1	NOUN
ejpam-4072	329	25	,	,	PUNCT
ejpam-4072	329	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	329	27	is	be	AUX
ejpam-4072	329	28	identical	identical	ADJ
ejpam-4072	329	29	with	with	ADP
ejpam-4072	329	30	the	the	DET
ejpam-4072	329	31	union	union	NOUN
ejpam-4072	329	32	of	of	ADP
ejpam-4072	329	33	the	the	DET
ejpam-4072	329	34	τ1τ2prefrontier	τ1τ2prefrontier	PUNCT
ejpam-4072	329	35	of	of	ADP
ejpam-4072	329	36	the	the	DET
ejpam-4072	329	37	lower	low	ADJ
ejpam-4072	329	38	inverse	inverse	NOUN
ejpam-4072	329	39	images	image	NOUN
ejpam-4072	329	40	of	of	ADP
ejpam-4072	329	41	σ1σ2	σ1σ2	NOUN
ejpam-4072	329	42	-	-	PUNCT
ejpam-4072	329	43	closure	closure	NOUN
ejpam-4072	329	44	of	of	ADP
ejpam-4072	329	45	σ1σ2	σ1σ2	NOUN
ejpam-4072	329	46	-	-	PUNCT
ejpam-4072	329	47	open	open	ADJ
ejpam-4072	329	48	sets	set	NOUN
ejpam-4072	329	49	meeting	meet	VERB
ejpam-4072	329	50	f	f	X
ejpam-4072	329	51	(	(	PUNCT
ejpam-4072	329	52	x	x	NOUN
ejpam-4072	329	53	)	)	PUNCT
ejpam-4072	329	54	.	.	PUNCT
ejpam-4072	330	1	proof	proof	NOUN
ejpam-4072	330	2	.	.	PUNCT
ejpam-4072	331	1	the	the	DET
ejpam-4072	331	2	proof	proof	NOUN
ejpam-4072	331	3	is	be	AUX
ejpam-4072	331	4	similar	similar	ADJ
ejpam-4072	331	5	to	to	ADP
ejpam-4072	331	6	that	that	PRON
ejpam-4072	331	7	of	of	ADP
ejpam-4072	331	8	theorem	theorem	NOUN
ejpam-4072	331	9	9	9	NUM
ejpam-4072	331	10	.	.	PUNCT
ejpam-4072	332	1	lemma	lemma	PROPN
ejpam-4072	332	2	11	11	NUM
ejpam-4072	332	3	.	.	PUNCT
ejpam-4072	333	1	[	[	X
ejpam-4072	333	2	26	26	NUM
ejpam-4072	333	3	]	]	X
ejpam-4072	333	4	the	the	DET
ejpam-4072	333	5	following	follow	VERB
ejpam-4072	333	6	hold	hold	NOUN
ejpam-4072	333	7	for	for	ADP
ejpam-4072	333	8	a	a	DET
ejpam-4072	333	9	multifunction	multifunction	NOUN
ejpam-4072	333	10	f	f	NOUN
ejpam-4072	333	11	:	:	PUNCT
ejpam-4072	333	12	x	x	X
ejpam-4072	333	13	→	→	SYM
ejpam-4072	333	14	y	y	PROPN
ejpam-4072	333	15	:	:	PUNCT
ejpam-4072	333	16	(	(	PUNCT
ejpam-4072	333	17	i	i	NOUN
ejpam-4072	333	18	)	)	PUNCT
ejpam-4072	333	19	g+	g+	PROPN
ejpam-4072	333	20	f	f	PROPN
ejpam-4072	333	21	(	(	PUNCT
ejpam-4072	333	22	a×b	a×b	PROPN
ejpam-4072	333	23	)	)	PUNCT
ejpam-4072	333	24	=	=	PUNCT
ejpam-4072	333	25	a	a	DET
ejpam-4072	333	26	∩	∩	ADJ
ejpam-4072	333	27	f+(b	f+(b	NOUN
ejpam-4072	333	28	)	)	PUNCT
ejpam-4072	333	29	,	,	PUNCT
ejpam-4072	333	30	(	(	PUNCT
ejpam-4072	333	31	ii	ii	X
ejpam-4072	333	32	)	)	PUNCT
ejpam-4072	333	33	g−	g−	PROPN
ejpam-4072	333	34	f	f	PROPN
ejpam-4072	333	35	(	(	PUNCT
ejpam-4072	333	36	a×b	a×b	PROPN
ejpam-4072	333	37	)	)	PUNCT
ejpam-4072	333	38	=	=	PUNCT
ejpam-4072	333	39	a	a	DET
ejpam-4072	333	40	∩	∩	ADJ
ejpam-4072	333	41	f−(b	f−(b	NOUN
ejpam-4072	333	42	)	)	PUNCT
ejpam-4072	333	43	,	,	PUNCT
ejpam-4072	333	44	for	for	ADP
ejpam-4072	333	45	any	any	DET
ejpam-4072	333	46	subsets	subset	NOUN
ejpam-4072	333	47	a	a	DET
ejpam-4072	333	48	⊆	⊆	NUM
ejpam-4072	333	49	x	x	NOUN
ejpam-4072	333	50	and	and	CCONJ
ejpam-4072	333	51	b	b	PROPN
ejpam-4072	333	52	⊆	⊆	NUM
ejpam-4072	333	53	y	y	PROPN
ejpam-4072	333	54	.	.	PUNCT
ejpam-4072	334	1	lemma	lemma	PROPN
ejpam-4072	334	2	12	12	NUM
ejpam-4072	334	3	.	.	PUNCT
ejpam-4072	335	1	let	let	AUX
ejpam-4072	335	2	(	(	PUNCT
ejpam-4072	335	3	x	x	NOUN
ejpam-4072	335	4	,	,	PUNCT
ejpam-4072	335	5	τ1	τ1	NOUN
ejpam-4072	335	6	,	,	PUNCT
ejpam-4072	335	7	τ2	τ2	PROPN
ejpam-4072	335	8	)	)	PUNCT
ejpam-4072	335	9	be	be	VERB
ejpam-4072	335	10	a	a	DET
ejpam-4072	335	11	bitopological	bitopological	ADJ
ejpam-4072	335	12	space	space	NOUN
ejpam-4072	335	13	.	.	PUNCT
ejpam-4072	336	1	if	if	SCONJ
ejpam-4072	336	2	a	a	PRON
ejpam-4072	336	3	is	be	AUX
ejpam-4072	336	4	τ1τ2	τ1τ2	NOUN
ejpam-4072	336	5	-	-	ADJ
ejpam-4072	336	6	preopen	preopen	ADJ
ejpam-4072	336	7	and	and	CCONJ
ejpam-4072	336	8	b	b	NOUN
ejpam-4072	336	9	is	be	AUX
ejpam-4072	336	10	τ1τ2	τ1τ2	VERB
ejpam-4072	336	11	-	-	ADJ
ejpam-4072	336	12	open	open	ADJ
ejpam-4072	336	13	in	in	ADP
ejpam-4072	336	14	x	x	NOUN
ejpam-4072	336	15	,	,	PUNCT
ejpam-4072	336	16	then	then	ADV
ejpam-4072	336	17	a	a	DET
ejpam-4072	336	18	∩b	∩b	NOUN
ejpam-4072	336	19	is	be	AUX
ejpam-4072	336	20	τ1τ2	τ1τ2	NOUN
ejpam-4072	336	21	-	-	ADJ
ejpam-4072	336	22	preopen	preopen	ADJ
ejpam-4072	336	23	.	.	PUNCT
ejpam-4072	337	1	proof	proof	NOUN
ejpam-4072	337	2	.	.	PUNCT
ejpam-4072	338	1	suppose	suppose	VERB
ejpam-4072	338	2	that	that	SCONJ
ejpam-4072	338	3	a	a	PRON
ejpam-4072	338	4	is	be	AUX
ejpam-4072	338	5	τ1τ2	τ1τ2	NOUN
ejpam-4072	338	6	-	-	ADJ
ejpam-4072	338	7	preopen	preopen	ADJ
ejpam-4072	338	8	and	and	CCONJ
ejpam-4072	338	9	b	b	NOUN
ejpam-4072	338	10	is	be	AUX
ejpam-4072	338	11	τ1τ2	τ1τ2	VERB
ejpam-4072	338	12	-	-	ADJ
ejpam-4072	338	13	open	open	ADJ
ejpam-4072	338	14	in	in	ADP
ejpam-4072	338	15	x.	x.	NOUN
ejpam-4072	338	16	then	then	ADV
ejpam-4072	338	17	,	,	PUNCT
ejpam-4072	338	18	a	a	DET
ejpam-4072	338	19	⊆	⊆	NUM
ejpam-4072	338	20	τ1	τ1	NOUN
ejpam-4072	338	21	-	-	PUNCT
ejpam-4072	338	22	int(τ2	int(τ2	NOUN
ejpam-4072	338	23	-	-	PUNCT
ejpam-4072	338	24	cl(a	cl(a	NUM
ejpam-4072	338	25	)	)	PUNCT
ejpam-4072	338	26	)	)	PUNCT
ejpam-4072	339	1	and	and	CCONJ
ejpam-4072	339	2	b	b	X
ejpam-4072	339	3	=	=	SYM
ejpam-4072	339	4	τ1	τ1	NOUN
ejpam-4072	339	5	-	-	PUNCT
ejpam-4072	339	6	int(b	int(b	NOUN
ejpam-4072	339	7	)	)	PUNCT
ejpam-4072	339	8	=	=	SYM
ejpam-4072	339	9	τ2	τ2	NOUN
ejpam-4072	339	10	-	-	PUNCT
ejpam-4072	339	11	int(b	int(b	NOUN
ejpam-4072	339	12	)	)	PUNCT
ejpam-4072	339	13	.	.	PUNCT
ejpam-4072	340	1	by	by	ADP
ejpam-4072	340	2	lemma	lemma	PROPN
ejpam-4072	340	3	5(1	5(1	NUM
ejpam-4072	340	4	)	)	PUNCT
ejpam-4072	340	5	,	,	PUNCT
ejpam-4072	340	6	a	a	DET
ejpam-4072	340	7	∩b	∩b	NOUN
ejpam-4072	340	8	⊆	⊆	NUM
ejpam-4072	340	9	τ1	τ1	NOUN
ejpam-4072	340	10	-	-	PUNCT
ejpam-4072	340	11	int(τ2	int(τ2	NOUN
ejpam-4072	340	12	-	-	PUNCT
ejpam-4072	340	13	cl(a	cl(a	NUM
ejpam-4072	340	14	)	)	PUNCT
ejpam-4072	340	15	)	)	PUNCT
ejpam-4072	340	16	∩b	∩b	NOUN
ejpam-4072	340	17	=	=	SYM
ejpam-4072	340	18	τ1	τ1	NOUN
ejpam-4072	340	19	-	-	PUNCT
ejpam-4072	340	20	int(τ2	int(τ2	NOUN
ejpam-4072	340	21	-	-	PUNCT
ejpam-4072	340	22	cl(a	cl(a	NUM
ejpam-4072	340	23	)	)	PUNCT
ejpam-4072	340	24	∩b	∩b	NOUN
ejpam-4072	340	25	)	)	PUNCT
ejpam-4072	340	26	⊆	⊆	NUM
ejpam-4072	340	27	τ1	τ1	NOUN
ejpam-4072	340	28	-	-	PUNCT
ejpam-4072	340	29	int(τ2	int(τ2	NOUN
ejpam-4072	340	30	-	-	PUNCT
ejpam-4072	340	31	cl(a	cl(a	VERB
ejpam-4072	340	32	∩b	∩b	NOUN
ejpam-4072	340	33	)	)	PUNCT
ejpam-4072	340	34	)	)	PUNCT
ejpam-4072	340	35	.	.	PUNCT
ejpam-4072	341	1	thus	thus	ADV
ejpam-4072	341	2	,	,	PUNCT
ejpam-4072	341	3	a	a	DET
ejpam-4072	341	4	∩b	∩b	NOUN
ejpam-4072	341	5	is	be	AUX
ejpam-4072	341	6	τ1τ2	τ1τ2	NOUN
ejpam-4072	341	7	-	-	ADJ
ejpam-4072	341	8	preopen	preopen	ADJ
ejpam-4072	341	9	.	.	PUNCT
ejpam-4072	342	1	definition	definition	NOUN
ejpam-4072	342	2	5	5	NUM
ejpam-4072	342	3	.	.	PUNCT
ejpam-4072	343	1	[	[	X
ejpam-4072	343	2	5	5	NUM
ejpam-4072	343	3	]	]	PUNCT
ejpam-4072	343	4	a	a	DET
ejpam-4072	343	5	bitopological	bitopological	ADJ
ejpam-4072	343	6	space	space	NOUN
ejpam-4072	343	7	(	(	PUNCT
ejpam-4072	343	8	x	x	NOUN
ejpam-4072	343	9	,	,	PUNCT
ejpam-4072	343	10	τ1	τ1	NOUN
ejpam-4072	343	11	,	,	PUNCT
ejpam-4072	343	12	τ2	τ2	NOUN
ejpam-4072	343	13	)	)	PUNCT
ejpam-4072	343	14	is	be	AUX
ejpam-4072	343	15	said	say	VERB
ejpam-4072	343	16	to	to	PART
ejpam-4072	343	17	be	be	AUX
ejpam-4072	343	18	τ1τ2	τ1τ2	NOUN
ejpam-4072	343	19	-	-	ADJ
ejpam-4072	343	20	compact	compact	ADJ
ejpam-4072	343	21	if	if	SCONJ
ejpam-4072	343	22	every	every	DET
ejpam-4072	343	23	cover	cover	NOUN
ejpam-4072	343	24	of	of	ADP
ejpam-4072	343	25	x	x	PUNCT
ejpam-4072	343	26	by	by	ADP
ejpam-4072	343	27	τ1τ2	τ1τ2	ADJ
ejpam-4072	343	28	-	-	ADJ
ejpam-4072	343	29	open	open	ADJ
ejpam-4072	343	30	sets	set	NOUN
ejpam-4072	343	31	of	of	ADP
ejpam-4072	343	32	x	x	PUNCT
ejpam-4072	343	33	has	have	VERB
ejpam-4072	343	34	a	a	DET
ejpam-4072	343	35	finite	finite	ADJ
ejpam-4072	343	36	subcover	subcover	PROPN
ejpam-4072	343	37	.	.	PUNCT
ejpam-4072	344	1	by	by	ADP
ejpam-4072	344	2	ρi	ρi	PROPN
ejpam-4072	344	3	,	,	PUNCT
ejpam-4072	344	4	we	we	PRON
ejpam-4072	344	5	denote	denote	VERB
ejpam-4072	344	6	the	the	DET
ejpam-4072	344	7	product	product	NOUN
ejpam-4072	344	8	topology	topology	NOUN
ejpam-4072	344	9	τi	τi	ADP
ejpam-4072	344	10	×	×	PROPN
ejpam-4072	344	11	σi	σi	NOUN
ejpam-4072	344	12	for	for	ADP
ejpam-4072	344	13	i	i	PROPN
ejpam-4072	344	14	=	=	NOUN
ejpam-4072	344	15	1	1	NUM
ejpam-4072	344	16	,	,	PUNCT
ejpam-4072	344	17	2	2	NUM
ejpam-4072	344	18	.	.	X
ejpam-4072	344	19	theorem	theorem	NOUN
ejpam-4072	344	20	11	11	NUM
ejpam-4072	344	21	.	.	PUNCT
ejpam-4072	345	1	let	let	VERB
ejpam-4072	345	2	f	f	NOUN
ejpam-4072	345	3	:	:	PUNCT
ejpam-4072	345	4	(	(	PUNCT
ejpam-4072	345	5	x	x	NOUN
ejpam-4072	345	6	,	,	PUNCT
ejpam-4072	345	7	τ1	τ1	NOUN
ejpam-4072	345	8	,	,	PUNCT
ejpam-4072	345	9	τ2	τ2	NOUN
ejpam-4072	345	10	)	)	PUNCT
ejpam-4072	345	11	→	→	SYM
ejpam-4072	345	12	(	(	PUNCT
ejpam-4072	345	13	y	y	PROPN
ejpam-4072	345	14	,	,	PUNCT
ejpam-4072	345	15	σ1	σ1	PROPN
ejpam-4072	345	16	,	,	PUNCT
ejpam-4072	345	17	σ2	σ2	PROPN
ejpam-4072	345	18	)	)	PUNCT
ejpam-4072	345	19	be	be	VERB
ejpam-4072	345	20	a	a	DET
ejpam-4072	345	21	multifunction	multifunction	NOUN
ejpam-4072	345	22	such	such	ADJ
ejpam-4072	345	23	that	that	SCONJ
ejpam-4072	345	24	f	f	PROPN
ejpam-4072	345	25	(	(	PUNCT
ejpam-4072	345	26	x	x	X
ejpam-4072	345	27	)	)	PUNCT
ejpam-4072	345	28	is	be	AUX
ejpam-4072	345	29	σ1σ2compact	σ1σ2compact	VERB
ejpam-4072	345	30	for	for	ADP
ejpam-4072	345	31	each	each	DET
ejpam-4072	345	32	x	x	SYM
ejpam-4072	345	33	∈	∈	PROPN
ejpam-4072	345	34	x.	x.	NOUN
ejpam-4072	346	1	then	then	ADV
ejpam-4072	346	2	f	f	PROPN
ejpam-4072	346	3	is	be	AUX
ejpam-4072	346	4	upper	upper	ADJ
ejpam-4072	346	5	almost	almost	ADV
ejpam-4072	346	6	weakly	weakly	ADJ
ejpam-4072	346	7	(	(	PUNCT
ejpam-4072	346	8	τ1	τ1	NOUN
ejpam-4072	346	9	,	,	PUNCT
ejpam-4072	346	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	346	11	if	if	SCONJ
ejpam-4072	346	12	and	and	CCONJ
ejpam-4072	346	13	only	only	ADV
ejpam-4072	346	14	if	if	SCONJ
ejpam-4072	346	15	gf	gf	NOUN
ejpam-4072	346	16	:	:	PUNCT
ejpam-4072	346	17	(	(	PUNCT
ejpam-4072	346	18	x	x	NOUN
ejpam-4072	346	19	,	,	PUNCT
ejpam-4072	346	20	τ1	τ1	NOUN
ejpam-4072	346	21	,	,	PUNCT
ejpam-4072	346	22	τ2	τ2	NOUN
ejpam-4072	346	23	)	)	PUNCT
ejpam-4072	346	24	→	→	PUNCT
ejpam-4072	346	25	(	(	PUNCT
ejpam-4072	346	26	x	x	SYM
ejpam-4072	346	27	×	×	PROPN
ejpam-4072	346	28	y	y	PROPN
ejpam-4072	346	29	,	,	PUNCT
ejpam-4072	346	30	ρ1	ρ1	NOUN
ejpam-4072	346	31	,	,	PUNCT
ejpam-4072	346	32	ρ2	ρ2	NOUN
ejpam-4072	346	33	)	)	PUNCT
ejpam-4072	346	34	is	be	AUX
ejpam-4072	346	35	upper	upper	ADJ
ejpam-4072	346	36	almost	almost	ADV
ejpam-4072	346	37	weakly	weakly	ADJ
ejpam-4072	346	38	(	(	PUNCT
ejpam-4072	346	39	τ1	τ1	NOUN
ejpam-4072	346	40	,	,	PUNCT
ejpam-4072	346	41	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	346	42	.	.	PUNCT
ejpam-4072	347	1	c.	c.	PROPN
ejpam-4072	347	2	boonpok	boonpok	PROPN
ejpam-4072	347	3	,	,	PUNCT
ejpam-4072	347	4	c.	c.	PROPN
ejpam-4072	347	5	viriyapong	viriyapong	PROPN
ejpam-4072	347	6	/	/	SYM
ejpam-4072	347	7	eur	eur	PROPN
ejpam-4072	347	8	.	.	PUNCT
ejpam-4072	348	1	j.	j.	PROPN
ejpam-4072	348	2	pure	pure	PROPN
ejpam-4072	348	3	appl	appl	PROPN
ejpam-4072	348	4	.	.	PROPN
ejpam-4072	348	5	math	math	PROPN
ejpam-4072	348	6	,	,	PUNCT
ejpam-4072	348	7	14	14	NUM
ejpam-4072	348	8	(	(	PUNCT
ejpam-4072	348	9	4	4	NUM
ejpam-4072	348	10	)	)	PUNCT
ejpam-4072	348	11	(	(	PUNCT
ejpam-4072	348	12	2021	2021	NUM
ejpam-4072	348	13	)	)	PUNCT
ejpam-4072	348	14	,	,	PUNCT
ejpam-4072	348	15	1212	1212	NUM
ejpam-4072	348	16	-	-	SYM
ejpam-4072	348	17	1225	1225	NUM
ejpam-4072	348	18	1223	1223	NUM
ejpam-4072	348	19	proof	proof	NOUN
ejpam-4072	348	20	.	.	PUNCT
ejpam-4072	348	21	suppose	suppose	VERB
ejpam-4072	348	22	that	that	SCONJ
ejpam-4072	348	23	f	f	X
ejpam-4072	348	24	:	:	PUNCT
ejpam-4072	348	25	(	(	PUNCT
ejpam-4072	348	26	x	x	NOUN
ejpam-4072	348	27	,	,	PUNCT
ejpam-4072	348	28	τ1	τ1	NOUN
ejpam-4072	348	29	,	,	PUNCT
ejpam-4072	348	30	τ2	τ2	NOUN
ejpam-4072	348	31	)	)	PUNCT
ejpam-4072	348	32	→	→	SYM
ejpam-4072	348	33	(	(	PUNCT
ejpam-4072	348	34	y	y	PROPN
ejpam-4072	348	35	,	,	PUNCT
ejpam-4072	348	36	σ1	σ1	PROPN
ejpam-4072	348	37	,	,	PUNCT
ejpam-4072	348	38	σ2	σ2	PROPN
ejpam-4072	348	39	)	)	PUNCT
ejpam-4072	348	40	is	be	AUX
ejpam-4072	348	41	upper	upper	ADJ
ejpam-4072	348	42	almost	almost	ADV
ejpam-4072	348	43	weakly	weakly	ADJ
ejpam-4072	348	44	(	(	PUNCT
ejpam-4072	348	45	τ1	τ1	NOUN
ejpam-4072	348	46	,	,	PUNCT
ejpam-4072	348	47	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	348	48	.	.	PUNCT
ejpam-4072	349	1	let	let	VERB
ejpam-4072	349	2	x	x	PUNCT
ejpam-4072	349	3	∈	∈	PROPN
ejpam-4072	349	4	x	x	PUNCT
ejpam-4072	349	5	and	and	CCONJ
ejpam-4072	349	6	let	let	VERB
ejpam-4072	349	7	w	w	NOUN
ejpam-4072	349	8	be	be	AUX
ejpam-4072	349	9	any	any	DET
ejpam-4072	349	10	ρ1ρ2	ρ1ρ2	NOUN
ejpam-4072	349	11	-	-	PUNCT
ejpam-4072	349	12	open	open	ADJ
ejpam-4072	349	13	set	set	NOUN
ejpam-4072	349	14	of	of	ADP
ejpam-4072	349	15	x	x	SYM
ejpam-4072	349	16	×	×	PROPN
ejpam-4072	349	17	y	y	NOUN
ejpam-4072	349	18	containing	contain	VERB
ejpam-4072	349	19	gf	gf	X
ejpam-4072	349	20	(	(	PUNCT
ejpam-4072	349	21	x	x	NOUN
ejpam-4072	349	22	)	)	PUNCT
ejpam-4072	349	23	.	.	PUNCT
ejpam-4072	350	1	for	for	ADP
ejpam-4072	350	2	each	each	DET
ejpam-4072	350	3	y	y	PROPN
ejpam-4072	350	4	∈	∈	PROPN
ejpam-4072	350	5	f	f	X
ejpam-4072	350	6	(	(	PUNCT
ejpam-4072	350	7	x	x	NOUN
ejpam-4072	350	8	)	)	PUNCT
ejpam-4072	350	9	,	,	PUNCT
ejpam-4072	350	10	there	there	PRON
ejpam-4072	350	11	exist	exist	VERB
ejpam-4072	350	12	τ1τ2	τ1τ2	ADJ
ejpam-4072	350	13	-	-	ADJ
ejpam-4072	350	14	open	open	ADJ
ejpam-4072	350	15	set	set	NOUN
ejpam-4072	350	16	u(y	u(y	NOUN
ejpam-4072	350	17	)	)	PUNCT
ejpam-4072	350	18	of	of	ADP
ejpam-4072	350	19	x	x	X
ejpam-4072	350	20	and	and	CCONJ
ejpam-4072	350	21	σ1σ2	σ1σ2	NOUN
ejpam-4072	350	22	-	-	ADJ
ejpam-4072	350	23	open	open	ADJ
ejpam-4072	350	24	set	set	NOUN
ejpam-4072	350	25	v	v	NOUN
ejpam-4072	350	26	(	(	PUNCT
ejpam-4072	350	27	y	y	NOUN
ejpam-4072	350	28	)	)	PUNCT
ejpam-4072	350	29	of	of	ADP
ejpam-4072	350	30	y	y	PRON
ejpam-4072	350	31	such	such	ADJ
ejpam-4072	350	32	that	that	SCONJ
ejpam-4072	350	33	(	(	PUNCT
ejpam-4072	350	34	x	x	NOUN
ejpam-4072	350	35	,	,	PUNCT
ejpam-4072	350	36	y	y	PROPN
ejpam-4072	350	37	)	)	PUNCT
ejpam-4072	350	38	∈	∈	PROPN
ejpam-4072	350	39	u(y	u(y	PROPN
ejpam-4072	350	40	)	)	PUNCT
ejpam-4072	350	41	×	×	NOUN
ejpam-4072	350	42	v	v	NOUN
ejpam-4072	350	43	(	(	PUNCT
ejpam-4072	350	44	y	y	NOUN
ejpam-4072	350	45	)	)	PUNCT
ejpam-4072	350	46	⊆	⊆	NUM
ejpam-4072	350	47	w	w	NOUN
ejpam-4072	350	48	.	.	PUNCT
ejpam-4072	351	1	the	the	DET
ejpam-4072	351	2	family	family	NOUN
ejpam-4072	351	3	{	{	PUNCT
ejpam-4072	351	4	v	v	PROPN
ejpam-4072	351	5	(	(	PUNCT
ejpam-4072	351	6	y	y	NOUN
ejpam-4072	351	7	)	)	PUNCT
ejpam-4072	352	1	|	|	ADV
ejpam-4072	352	2	y	y	PROPN
ejpam-4072	352	3	∈	∈	PROPN
ejpam-4072	352	4	f	f	X
ejpam-4072	352	5	(	(	PUNCT
ejpam-4072	352	6	x	x	NOUN
ejpam-4072	352	7	)	)	PUNCT
ejpam-4072	352	8	}	}	PUNCT
ejpam-4072	352	9	is	be	AUX
ejpam-4072	352	10	a	a	DET
ejpam-4072	352	11	σ1σ2	σ1σ2	NUM
ejpam-4072	352	12	-	-	PUNCT
ejpam-4072	352	13	open	open	ADJ
ejpam-4072	352	14	cover	cover	NOUN
ejpam-4072	352	15	of	of	ADP
ejpam-4072	352	16	f	f	PROPN
ejpam-4072	352	17	(	(	PUNCT
ejpam-4072	352	18	x	x	NOUN
ejpam-4072	352	19	)	)	PUNCT
ejpam-4072	352	20	and	and	CCONJ
ejpam-4072	352	21	there	there	PRON
ejpam-4072	352	22	exists	exist	VERB
ejpam-4072	352	23	a	a	DET
ejpam-4072	352	24	finite	finite	ADJ
ejpam-4072	352	25	number	number	NOUN
ejpam-4072	352	26	of	of	ADP
ejpam-4072	352	27	points	point	NOUN
ejpam-4072	352	28	,	,	PUNCT
ejpam-4072	352	29	say	say	INTJ
ejpam-4072	352	30	,	,	PUNCT
ejpam-4072	352	31	y1	y1	PROPN
ejpam-4072	352	32	,	,	PUNCT
ejpam-4072	352	33	y2	y2	PROPN
ejpam-4072	352	34	,	,	PUNCT
ejpam-4072	352	35	...	...	PUNCT
ejpam-4072	352	36	,	,	PUNCT
ejpam-4072	352	37	yn	yn	PROPN
ejpam-4072	352	38	in	in	ADP
ejpam-4072	352	39	f	f	PROPN
ejpam-4072	352	40	(	(	PUNCT
ejpam-4072	352	41	x	x	X
ejpam-4072	352	42	)	)	PUNCT
ejpam-4072	352	43	such	such	ADJ
ejpam-4072	352	44	that	that	SCONJ
ejpam-4072	352	45	f	f	PROPN
ejpam-4072	352	46	(	(	PUNCT
ejpam-4072	352	47	x	x	X
ejpam-4072	352	48	)	)	PUNCT
ejpam-4072	352	49	⊆	⊆	NUM
ejpam-4072	352	50	∪{v	∪{v	PROPN
ejpam-4072	352	51	(	(	PUNCT
ejpam-4072	352	52	yi	yi	NOUN
ejpam-4072	352	53	)	)	PUNCT
ejpam-4072	352	54	|	|	ADV
ejpam-4072	352	55	1	1	NUM
ejpam-4072	352	56	≤	≤	NUM
ejpam-4072	352	57	i	i	PRON
ejpam-4072	352	58	≤	≤	NOUN
ejpam-4072	352	59	n	n	CCONJ
ejpam-4072	352	60	}	}	PUNCT
ejpam-4072	352	61	.	.	PUNCT
ejpam-4072	353	1	put	put	VERB
ejpam-4072	353	2	u	u	NOUN
ejpam-4072	353	3	=	=	PROPN
ejpam-4072	353	4	∩{u(yi	∩{u(yi	PROPN
ejpam-4072	353	5	)	)	PUNCT
ejpam-4072	354	1	|	|	ADV
ejpam-4072	354	2	i	i	PRON
ejpam-4072	354	3	=	=	NOUN
ejpam-4072	354	4	1	1	NUM
ejpam-4072	354	5	,	,	PUNCT
ejpam-4072	354	6	2	2	NUM
ejpam-4072	354	7	,	,	PUNCT
ejpam-4072	354	8	...	...	PUNCT
ejpam-4072	354	9	,	,	PUNCT
ejpam-4072	354	10	n	n	CCONJ
ejpam-4072	354	11	}	}	PUNCT
ejpam-4072	354	12	and	and	CCONJ
ejpam-4072	354	13	v	v	NOUN
ejpam-4072	354	14	=	=	SYM
ejpam-4072	354	15	∪{v	∪{v	PROPN
ejpam-4072	354	16	(	(	PUNCT
ejpam-4072	354	17	yi	yi	NOUN
ejpam-4072	354	18	)	)	PUNCT
ejpam-4072	355	1	|	|	ADV
ejpam-4072	355	2	i	i	PRON
ejpam-4072	355	3	=	=	NOUN
ejpam-4072	355	4	1	1	NUM
ejpam-4072	355	5	,	,	PUNCT
ejpam-4072	355	6	2	2	NUM
ejpam-4072	355	7	,	,	PUNCT
ejpam-4072	355	8	...	...	PUNCT
ejpam-4072	355	9	,	,	PUNCT
ejpam-4072	355	10	n	n	CCONJ
ejpam-4072	355	11	}	}	PUNCT
ejpam-4072	355	12	.	.	PUNCT
ejpam-4072	356	1	then	then	ADV
ejpam-4072	356	2	,	,	PUNCT
ejpam-4072	356	3	u	u	NOUN
ejpam-4072	356	4	is	be	AUX
ejpam-4072	356	5	τ1τ2	τ1τ2	NOUN
ejpam-4072	356	6	-	-	ADJ
ejpam-4072	356	7	open	open	ADJ
ejpam-4072	356	8	in	in	ADP
ejpam-4072	356	9	x	x	PUNCT
ejpam-4072	356	10	and	and	CCONJ
ejpam-4072	356	11	v	v	NOUN
ejpam-4072	356	12	is	be	AUX
ejpam-4072	356	13	σ1σ2	σ1σ2	NOUN
ejpam-4072	356	14	-	-	ADJ
ejpam-4072	356	15	open	open	ADJ
ejpam-4072	356	16	in	in	ADP
ejpam-4072	356	17	y	y	PROPN
ejpam-4072	356	18	such	such	ADJ
ejpam-4072	356	19	that	that	SCONJ
ejpam-4072	356	20	{	{	PUNCT
ejpam-4072	356	21	x	x	NOUN
ejpam-4072	356	22	}	}	PUNCT
ejpam-4072	356	23	×	×	PROPN
ejpam-4072	356	24	f	f	X
ejpam-4072	356	25	(	(	PUNCT
ejpam-4072	356	26	x	x	NOUN
ejpam-4072	356	27	)	)	PUNCT
ejpam-4072	356	28	⊆	⊆	NUM
ejpam-4072	356	29	u	u	NOUN
ejpam-4072	356	30	×	×	NOUN
ejpam-4072	356	31	v	v	ADP
ejpam-4072	356	32	⊆	⊆	NUM
ejpam-4072	356	33	w	w	NOUN
ejpam-4072	356	34	.	.	PUNCT
ejpam-4072	357	1	since	since	SCONJ
ejpam-4072	357	2	f	f	PROPN
ejpam-4072	357	3	is	be	AUX
ejpam-4072	357	4	upper	upper	ADJ
ejpam-4072	357	5	almost	almost	ADV
ejpam-4072	357	6	weakly	weakly	ADJ
ejpam-4072	357	7	(	(	PUNCT
ejpam-4072	357	8	τ1	τ1	NOUN
ejpam-4072	357	9	,	,	PUNCT
ejpam-4072	357	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	357	11	,	,	PUNCT
ejpam-4072	357	12	there	there	PRON
ejpam-4072	357	13	exists	exist	VERB
ejpam-4072	357	14	a	a	DET
ejpam-4072	357	15	τ1τ2	τ1τ2	NOUN
ejpam-4072	357	16	-	-	ADJ
ejpam-4072	357	17	preopen	preopen	ADJ
ejpam-4072	357	18	set	set	VERB
ejpam-4072	357	19	g	g	NOUN
ejpam-4072	357	20	containing	contain	VERB
ejpam-4072	357	21	x	x	PUNCT
ejpam-4072	357	22	such	such	ADJ
ejpam-4072	357	23	that	that	SCONJ
ejpam-4072	357	24	f	f	PROPN
ejpam-4072	357	25	(	(	PUNCT
ejpam-4072	357	26	g	g	NOUN
ejpam-4072	357	27	)	)	PUNCT
ejpam-4072	357	28	⊆	⊆	NUM
ejpam-4072	357	29	σ1σ2	σ1σ2	NOUN
ejpam-4072	357	30	-	-	NUM
ejpam-4072	357	31	cl(v	cl(v	NOUN
ejpam-4072	357	32	)	)	PUNCT
ejpam-4072	357	33	.	.	PUNCT
ejpam-4072	358	1	by	by	ADP
ejpam-4072	358	2	lemma	lemma	PROPN
ejpam-4072	358	3	11	11	NUM
ejpam-4072	358	4	,	,	PUNCT
ejpam-4072	358	5	u∩g	u∩g	VERB
ejpam-4072	358	6	⊆	⊆	NUM
ejpam-4072	358	7	u∩f+(σ1σ2	u∩f+(σ1σ2	NOUN
ejpam-4072	358	8	-	-	NOUN
ejpam-4072	358	9	cl(v	cl(v	NOUN
ejpam-4072	358	10	)	)	PUNCT
ejpam-4072	358	11	)	)	PUNCT
ejpam-4072	359	1	=	=	SYM
ejpam-4072	359	2	g+	g+	X
ejpam-4072	359	3	f	f	X
ejpam-4072	359	4	(	(	PUNCT
ejpam-4072	359	5	u	u	NOUN
ejpam-4072	359	6	×	×	NOUN
ejpam-4072	359	7	σ1σ2	σ1σ2	NOUN
ejpam-4072	359	8	-	-	NUM
ejpam-4072	359	9	cl(v	cl(v	NOUN
ejpam-4072	359	10	)	)	PUNCT
ejpam-4072	359	11	)	)	PUNCT
ejpam-4072	360	1	⊆	⊆	NUM
ejpam-4072	360	2	g+	g+	NOUN
ejpam-4072	360	3	f	f	X
ejpam-4072	360	4	(	(	PUNCT
ejpam-4072	360	5	σ1σ2	σ1σ2	NOUN
ejpam-4072	360	6	-	-	PUNCT
ejpam-4072	360	7	cl(w	cl(w	NOUN
ejpam-4072	360	8	)	)	PUNCT
ejpam-4072	360	9	)	)	PUNCT
ejpam-4072	360	10	.	.	PUNCT
ejpam-4072	361	1	by	by	ADP
ejpam-4072	361	2	lemma	lemma	PROPN
ejpam-4072	361	3	12	12	NUM
ejpam-4072	361	4	,	,	PUNCT
ejpam-4072	361	5	u	u	PROPN
ejpam-4072	361	6	∩	∩	NOUN
ejpam-4072	361	7	g	g	PROPN
ejpam-4072	361	8	is	be	AUX
ejpam-4072	361	9	τ1τ2	τ1τ2	NOUN
ejpam-4072	361	10	-	-	ADJ
ejpam-4072	361	11	preopen	preopen	ADJ
ejpam-4072	361	12	in	in	ADP
ejpam-4072	361	13	x	x	PUNCT
ejpam-4072	361	14	containing	contain	VERB
ejpam-4072	361	15	x	x	X
ejpam-4072	361	16	and	and	CCONJ
ejpam-4072	361	17	gf	gf	PROPN
ejpam-4072	361	18	(	(	PUNCT
ejpam-4072	361	19	u	u	NOUN
ejpam-4072	361	20	∩g	∩g	PROPN
ejpam-4072	361	21	)	)	PUNCT
ejpam-4072	362	1	⊆	⊆	NUM
ejpam-4072	362	2	σ1σ2	σ1σ2	NOUN
ejpam-4072	362	3	-	-	PUNCT
ejpam-4072	362	4	cl(w	cl(w	NOUN
ejpam-4072	362	5	)	)	PUNCT
ejpam-4072	362	6	.	.	PUNCT
ejpam-4072	363	1	this	this	PRON
ejpam-4072	363	2	shows	show	VERB
ejpam-4072	363	3	that	that	SCONJ
ejpam-4072	363	4	gf	gf	PROPN
ejpam-4072	363	5	is	be	AUX
ejpam-4072	363	6	upper	upper	ADJ
ejpam-4072	363	7	almost	almost	ADV
ejpam-4072	363	8	weakly	weakly	ADJ
ejpam-4072	363	9	(	(	PUNCT
ejpam-4072	363	10	τ1	τ1	NOUN
ejpam-4072	363	11	,	,	PUNCT
ejpam-4072	363	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	363	13	.	.	PUNCT
ejpam-4072	364	1	conversely	conversely	ADV
ejpam-4072	364	2	,	,	PUNCT
ejpam-4072	364	3	suppose	suppose	VERB
ejpam-4072	364	4	that	that	SCONJ
ejpam-4072	364	5	gf	gf	NOUN
ejpam-4072	364	6	:	:	PUNCT
ejpam-4072	364	7	(	(	PUNCT
ejpam-4072	364	8	x	x	NOUN
ejpam-4072	364	9	,	,	PUNCT
ejpam-4072	364	10	τ1	τ1	NOUN
ejpam-4072	364	11	,	,	PUNCT
ejpam-4072	364	12	τ2	τ2	NOUN
ejpam-4072	364	13	)	)	PUNCT
ejpam-4072	364	14	→	→	PUNCT
ejpam-4072	364	15	(	(	PUNCT
ejpam-4072	364	16	x	x	SYM
ejpam-4072	364	17	×	×	PROPN
ejpam-4072	364	18	y	y	PROPN
ejpam-4072	364	19	,	,	PUNCT
ejpam-4072	364	20	ρ1	ρ1	NOUN
ejpam-4072	364	21	,	,	PUNCT
ejpam-4072	364	22	ρ2	ρ2	NOUN
ejpam-4072	364	23	)	)	PUNCT
ejpam-4072	364	24	is	be	AUX
ejpam-4072	364	25	upper	upper	ADJ
ejpam-4072	364	26	almost	almost	ADV
ejpam-4072	364	27	weakly	weakly	ADJ
ejpam-4072	364	28	(	(	PUNCT
ejpam-4072	364	29	τ1	τ1	NOUN
ejpam-4072	364	30	,	,	PUNCT
ejpam-4072	364	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	364	32	.	.	PUNCT
ejpam-4072	365	1	let	let	VERB
ejpam-4072	365	2	x	x	PUNCT
ejpam-4072	365	3	∈	∈	PROPN
ejpam-4072	365	4	x	x	PUNCT
ejpam-4072	365	5	and	and	CCONJ
ejpam-4072	365	6	let	let	VERB
ejpam-4072	365	7	v	v	PART
ejpam-4072	365	8	be	be	AUX
ejpam-4072	365	9	any	any	DET
ejpam-4072	365	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	365	11	-	-	ADJ
ejpam-4072	365	12	open	open	ADJ
ejpam-4072	365	13	set	set	NOUN
ejpam-4072	365	14	of	of	ADP
ejpam-4072	365	15	y	y	PROPN
ejpam-4072	365	16	containing	contain	VERB
ejpam-4072	365	17	f	f	PROPN
ejpam-4072	365	18	(	(	PUNCT
ejpam-4072	365	19	x	x	NOUN
ejpam-4072	365	20	)	)	PUNCT
ejpam-4072	365	21	.	.	PUNCT
ejpam-4072	366	1	since	since	SCONJ
ejpam-4072	366	2	x	x	SYM
ejpam-4072	366	3	×	×	NOUN
ejpam-4072	366	4	v	v	NOUN
ejpam-4072	366	5	is	be	AUX
ejpam-4072	366	6	ρ1ρ2	ρ1ρ2	VERB
ejpam-4072	366	7	-	-	PUNCT
ejpam-4072	366	8	open	open	ADJ
ejpam-4072	366	9	in	in	ADP
ejpam-4072	366	10	x	x	SYM
ejpam-4072	366	11	×	×	PROPN
ejpam-4072	366	12	y	y	PROPN
ejpam-4072	366	13	and	and	CCONJ
ejpam-4072	366	14	gf	gf	PROPN
ejpam-4072	366	15	(	(	PUNCT
ejpam-4072	366	16	x	x	X
ejpam-4072	366	17	)	)	PUNCT
ejpam-4072	366	18	⊆	⊆	NUM
ejpam-4072	366	19	x	x	SYM
ejpam-4072	366	20	×	×	PROPN
ejpam-4072	366	21	y	y	PROPN
ejpam-4072	366	22	,	,	PUNCT
ejpam-4072	366	23	by	by	ADP
ejpam-4072	366	24	theorem	theorem	NOUN
ejpam-4072	366	25	1	1	NUM
ejpam-4072	366	26	,	,	PUNCT
ejpam-4072	366	27	there	there	PRON
ejpam-4072	366	28	exists	exist	VERB
ejpam-4072	366	29	a	a	DET
ejpam-4072	366	30	τ1τ2	τ1τ2	NOUN
ejpam-4072	366	31	-	-	ADJ
ejpam-4072	366	32	preopen	preopen	ADJ
ejpam-4072	366	33	set	set	NOUN
ejpam-4072	366	34	u	u	NOUN
ejpam-4072	366	35	containing	contain	VERB
ejpam-4072	366	36	x	x	PUNCT
ejpam-4072	366	37	such	such	ADJ
ejpam-4072	366	38	that	that	DET
ejpam-4072	366	39	gf	gf	PROPN
ejpam-4072	366	40	(	(	PUNCT
ejpam-4072	366	41	u	u	NOUN
ejpam-4072	366	42	)	)	PUNCT
ejpam-4072	367	1	⊆	⊆	NUM
ejpam-4072	367	2	x	x	SYM
ejpam-4072	367	3	×	×	NOUN
ejpam-4072	367	4	σ1σ2	σ1σ2	NOUN
ejpam-4072	367	5	-	-	NUM
ejpam-4072	367	6	cl(v	cl(v	NOUN
ejpam-4072	367	7	)	)	PUNCT
ejpam-4072	367	8	.	.	PUNCT
ejpam-4072	368	1	therefore	therefore	ADV
ejpam-4072	368	2	,	,	PUNCT
ejpam-4072	368	3	by	by	ADP
ejpam-4072	368	4	lemma	lemma	PROPN
ejpam-4072	368	5	11	11	NUM
ejpam-4072	368	6	,	,	PUNCT
ejpam-4072	368	7	u	u	NOUN
ejpam-4072	368	8	⊆	⊆	NUM
ejpam-4072	368	9	g+	g+	NOUN
ejpam-4072	368	10	f	f	X
ejpam-4072	368	11	(	(	PUNCT
ejpam-4072	368	12	x	x	PROPN
ejpam-4072	368	13	×	×	NOUN
ejpam-4072	368	14	σ1σ2	σ1σ2	NOUN
ejpam-4072	368	15	-	-	NUM
ejpam-4072	368	16	cl(v	cl(v	NOUN
ejpam-4072	368	17	)	)	PUNCT
ejpam-4072	368	18	)	)	PUNCT
ejpam-4072	369	1	=	=	SYM
ejpam-4072	369	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-4072	369	3	-	-	NUM
ejpam-4072	369	4	cl(v	cl(v	NOUN
ejpam-4072	369	5	)	)	PUNCT
ejpam-4072	369	6	)	)	PUNCT
ejpam-4072	370	1	and	and	CCONJ
ejpam-4072	370	2	hence	hence	ADV
ejpam-4072	370	3	f	f	PROPN
ejpam-4072	370	4	(	(	PUNCT
ejpam-4072	370	5	u	u	NOUN
ejpam-4072	370	6	)	)	PUNCT
ejpam-4072	370	7	⊆	⊆	NUM
ejpam-4072	370	8	σ1σ2	σ1σ2	NOUN
ejpam-4072	370	9	-	-	NUM
ejpam-4072	370	10	cl(v	cl(v	NOUN
ejpam-4072	370	11	)	)	PUNCT
ejpam-4072	370	12	.	.	PUNCT
ejpam-4072	371	1	this	this	PRON
ejpam-4072	371	2	shows	show	VERB
ejpam-4072	371	3	that	that	SCONJ
ejpam-4072	371	4	f	f	PROPN
ejpam-4072	371	5	is	be	AUX
ejpam-4072	371	6	upper	upper	ADJ
ejpam-4072	371	7	almost	almost	ADV
ejpam-4072	371	8	weakly	weakly	ADJ
ejpam-4072	371	9	(	(	PUNCT
ejpam-4072	371	10	τ1	τ1	NOUN
ejpam-4072	371	11	,	,	PUNCT
ejpam-4072	371	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	371	13	.	.	PUNCT
ejpam-4072	372	1	theorem	theorem	NOUN
ejpam-4072	372	2	12	12	NUM
ejpam-4072	372	3	.	.	PUNCT
ejpam-4072	373	1	a	a	DET
ejpam-4072	373	2	multifunction	multifunction	NOUN
ejpam-4072	373	3	f	f	NOUN
ejpam-4072	373	4	:	:	PUNCT
ejpam-4072	373	5	(	(	PUNCT
ejpam-4072	373	6	x	x	NOUN
ejpam-4072	373	7	,	,	PUNCT
ejpam-4072	373	8	τ1	τ1	NOUN
ejpam-4072	373	9	,	,	PUNCT
ejpam-4072	373	10	τ2	τ2	NOUN
ejpam-4072	373	11	)	)	PUNCT
ejpam-4072	373	12	→	→	SYM
ejpam-4072	373	13	(	(	PUNCT
ejpam-4072	373	14	y	y	PROPN
ejpam-4072	373	15	,	,	PUNCT
ejpam-4072	373	16	σ1	σ1	PROPN
ejpam-4072	373	17	,	,	PUNCT
ejpam-4072	373	18	σ2	σ2	NOUN
ejpam-4072	373	19	)	)	PUNCT
ejpam-4072	373	20	is	be	AUX
ejpam-4072	373	21	lower	low	ADJ
ejpam-4072	373	22	almost	almost	ADV
ejpam-4072	373	23	weakly	weakly	ADJ
ejpam-4072	373	24	(	(	PUNCT
ejpam-4072	373	25	τ1	τ1	NOUN
ejpam-4072	373	26	,	,	PUNCT
ejpam-4072	373	27	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	373	28	if	if	SCONJ
ejpam-4072	373	29	and	and	CCONJ
ejpam-4072	373	30	only	only	ADV
ejpam-4072	373	31	if	if	SCONJ
ejpam-4072	373	32	gf	gf	NOUN
ejpam-4072	373	33	:	:	PUNCT
ejpam-4072	373	34	(	(	PUNCT
ejpam-4072	373	35	x	x	NOUN
ejpam-4072	373	36	,	,	PUNCT
ejpam-4072	373	37	τ1	τ1	NOUN
ejpam-4072	373	38	,	,	PUNCT
ejpam-4072	373	39	τ2	τ2	NOUN
ejpam-4072	373	40	)	)	PUNCT
ejpam-4072	373	41	→	→	PUNCT
ejpam-4072	373	42	(	(	PUNCT
ejpam-4072	373	43	x	x	SYM
ejpam-4072	373	44	×	×	PROPN
ejpam-4072	373	45	y	y	PROPN
ejpam-4072	373	46	,	,	PUNCT
ejpam-4072	373	47	ρ1	ρ1	NOUN
ejpam-4072	373	48	,	,	PUNCT
ejpam-4072	373	49	ρ2	ρ2	NOUN
ejpam-4072	373	50	)	)	PUNCT
ejpam-4072	373	51	is	be	AUX
ejpam-4072	373	52	lower	low	ADJ
ejpam-4072	373	53	almost	almost	ADV
ejpam-4072	373	54	weakly	weakly	ADJ
ejpam-4072	373	55	(	(	PUNCT
ejpam-4072	373	56	τ1	τ1	NOUN
ejpam-4072	373	57	,	,	PUNCT
ejpam-4072	373	58	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	373	59	.	.	PUNCT
ejpam-4072	374	1	proof	proof	NOUN
ejpam-4072	374	2	.	.	PUNCT
ejpam-4072	375	1	suppose	suppose	VERB
ejpam-4072	375	2	that	that	SCONJ
ejpam-4072	375	3	f	f	X
ejpam-4072	375	4	:	:	PUNCT
ejpam-4072	375	5	(	(	PUNCT
ejpam-4072	375	6	x	x	NOUN
ejpam-4072	375	7	,	,	PUNCT
ejpam-4072	375	8	τ1	τ1	NOUN
ejpam-4072	375	9	,	,	PUNCT
ejpam-4072	375	10	τ2	τ2	NOUN
ejpam-4072	375	11	)	)	PUNCT
ejpam-4072	375	12	→	→	SYM
ejpam-4072	375	13	(	(	PUNCT
ejpam-4072	375	14	y	y	PROPN
ejpam-4072	375	15	,	,	PUNCT
ejpam-4072	375	16	σ1	σ1	PROPN
ejpam-4072	375	17	,	,	PUNCT
ejpam-4072	375	18	σ2	σ2	NOUN
ejpam-4072	375	19	)	)	PUNCT
ejpam-4072	375	20	is	be	AUX
ejpam-4072	375	21	lower	low	ADJ
ejpam-4072	375	22	almost	almost	ADV
ejpam-4072	375	23	weakly	weakly	ADJ
ejpam-4072	375	24	(	(	PUNCT
ejpam-4072	375	25	τ1	τ1	NOUN
ejpam-4072	375	26	,	,	PUNCT
ejpam-4072	375	27	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	375	28	.	.	PUNCT
ejpam-4072	376	1	let	let	VERB
ejpam-4072	376	2	x	x	PUNCT
ejpam-4072	376	3	∈	∈	PROPN
ejpam-4072	376	4	x	x	PUNCT
ejpam-4072	376	5	and	and	CCONJ
ejpam-4072	376	6	let	let	VERB
ejpam-4072	376	7	w	w	NOUN
ejpam-4072	376	8	be	be	AUX
ejpam-4072	376	9	any	any	DET
ejpam-4072	376	10	ρ1ρ2	ρ1ρ2	NOUN
ejpam-4072	376	11	-	-	PUNCT
ejpam-4072	376	12	open	open	ADJ
ejpam-4072	376	13	set	set	NOUN
ejpam-4072	376	14	of	of	ADP
ejpam-4072	376	15	x×y	x×y	PUNCT
ejpam-4072	376	16	such	such	ADJ
ejpam-4072	376	17	that	that	PRON
ejpam-4072	376	18	gf	gf	PROPN
ejpam-4072	376	19	(	(	PUNCT
ejpam-4072	376	20	x)∩w	x)∩w	PROPN
ejpam-4072	376	21	̸=	̸=	PROPN
ejpam-4072	376	22	∅.	∅.	NOUN
ejpam-4072	376	23	then	then	ADV
ejpam-4072	376	24	,	,	PUNCT
ejpam-4072	376	25	there	there	PRON
ejpam-4072	376	26	exists	exist	VERB
ejpam-4072	376	27	y	y	PROPN
ejpam-4072	376	28	∈	∈	PROPN
ejpam-4072	376	29	f	f	X
ejpam-4072	376	30	(	(	PUNCT
ejpam-4072	376	31	x	x	X
ejpam-4072	376	32	)	)	PUNCT
ejpam-4072	376	33	such	such	ADJ
ejpam-4072	376	34	that	that	SCONJ
ejpam-4072	376	35	(	(	PUNCT
ejpam-4072	376	36	x	x	NOUN
ejpam-4072	376	37	,	,	PUNCT
ejpam-4072	376	38	y	y	NOUN
ejpam-4072	376	39	)	)	PUNCT
ejpam-4072	376	40	∈	∈	PROPN
ejpam-4072	376	41	w	w	NOUN
ejpam-4072	376	42	and	and	CCONJ
ejpam-4072	376	43	hence	hence	ADV
ejpam-4072	376	44	(	(	PUNCT
ejpam-4072	376	45	x	x	X
ejpam-4072	376	46	,	,	PUNCT
ejpam-4072	376	47	y	y	NOUN
ejpam-4072	376	48	)	)	PUNCT
ejpam-4072	376	49	∈	∈	NOUN
ejpam-4072	376	50	u	u	NOUN
ejpam-4072	376	51	×v	×v	VERB
ejpam-4072	376	52	⊆	⊆	NUM
ejpam-4072	376	53	w	w	NOUN
ejpam-4072	376	54	for	for	ADP
ejpam-4072	376	55	some	some	DET
ejpam-4072	376	56	τ1τ2	τ1τ2	ADJ
ejpam-4072	376	57	-	-	ADJ
ejpam-4072	376	58	open	open	ADJ
ejpam-4072	376	59	set	set	ADJ
ejpam-4072	376	60	u	u	NOUN
ejpam-4072	376	61	of	of	ADP
ejpam-4072	376	62	x	x	X
ejpam-4072	376	63	and	and	CCONJ
ejpam-4072	376	64	σ1σ2	σ1σ2	NOUN
ejpam-4072	376	65	-	-	ADJ
ejpam-4072	376	66	open	open	ADJ
ejpam-4072	376	67	set	set	NOUN
ejpam-4072	376	68	v	v	NOUN
ejpam-4072	376	69	of	of	ADP
ejpam-4072	376	70	y	y	PROPN
ejpam-4072	376	71	.	.	PUNCT
ejpam-4072	377	1	since	since	SCONJ
ejpam-4072	377	2	f	f	PROPN
ejpam-4072	377	3	is	be	AUX
ejpam-4072	377	4	lower	low	ADJ
ejpam-4072	377	5	almost	almost	ADV
ejpam-4072	377	6	weakly	weakly	ADJ
ejpam-4072	377	7	(	(	PUNCT
ejpam-4072	377	8	τ1	τ1	NOUN
ejpam-4072	377	9	,	,	PUNCT
ejpam-4072	377	10	τ2)continuous	τ2)continuous	ADJ
ejpam-4072	377	11	and	and	CCONJ
ejpam-4072	377	12	y	y	PROPN
ejpam-4072	377	13	∈	∈	PROPN
ejpam-4072	377	14	f	f	X
ejpam-4072	377	15	(	(	PUNCT
ejpam-4072	377	16	x	x	NOUN
ejpam-4072	377	17	)	)	PUNCT
ejpam-4072	377	18	∩	∩	ADJ
ejpam-4072	377	19	v	v	X
ejpam-4072	377	20	,	,	PUNCT
ejpam-4072	377	21	there	there	PRON
ejpam-4072	377	22	exists	exist	VERB
ejpam-4072	377	23	a	a	DET
ejpam-4072	377	24	τ1τ2	τ1τ2	NOUN
ejpam-4072	377	25	-	-	ADJ
ejpam-4072	377	26	preopen	preopen	ADJ
ejpam-4072	377	27	set	set	VERB
ejpam-4072	377	28	g	g	NOUN
ejpam-4072	377	29	of	of	ADP
ejpam-4072	377	30	x	x	PUNCT
ejpam-4072	377	31	containing	contain	VERB
ejpam-4072	377	32	x	x	PUNCT
ejpam-4072	377	33	such	such	ADJ
ejpam-4072	377	34	that	that	SCONJ
ejpam-4072	377	35	f	f	PROPN
ejpam-4072	377	36	(	(	PUNCT
ejpam-4072	377	37	z	z	NOUN
ejpam-4072	377	38	)	)	PUNCT
ejpam-4072	377	39	∩	∩	NOUN
ejpam-4072	377	40	σ1σ2	σ1σ2	NOUN
ejpam-4072	377	41	-	-	NUM
ejpam-4072	377	42	cl(v	cl(v	NOUN
ejpam-4072	377	43	)	)	PUNCT
ejpam-4072	377	44	̸=	̸=	NOUN
ejpam-4072	377	45	∅	∅	NOUN
ejpam-4072	377	46	for	for	ADP
ejpam-4072	377	47	each	each	DET
ejpam-4072	377	48	z	z	NOUN
ejpam-4072	377	49	∈	∈	PROPN
ejpam-4072	377	50	g	g	NOUN
ejpam-4072	377	51	;	;	PUNCT
ejpam-4072	377	52	hence	hence	ADV
ejpam-4072	377	53	g	g	PROPN
ejpam-4072	377	54	⊆	⊆	NUM
ejpam-4072	377	55	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	377	56	-	-	PUNCT
ejpam-4072	377	57	cl(v	cl(v	NOUN
ejpam-4072	377	58	)	)	PUNCT
ejpam-4072	377	59	)	)	PUNCT
ejpam-4072	377	60	.	.	PUNCT
ejpam-4072	378	1	by	by	ADP
ejpam-4072	378	2	lemma	lemma	PROPN
ejpam-4072	378	3	11	11	NUM
ejpam-4072	378	4	,	,	PUNCT
ejpam-4072	378	5	we	we	PRON
ejpam-4072	378	6	have	have	VERB
ejpam-4072	378	7	u	u	NOUN
ejpam-4072	378	8	∩	∩	NOUN
ejpam-4072	378	9	g	g	PROPN
ejpam-4072	378	10	⊆	⊆	NUM
ejpam-4072	378	11	u	u	NOUN
ejpam-4072	378	12	∩	∩	NOUN
ejpam-4072	378	13	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	378	14	-	-	PUNCT
ejpam-4072	378	15	cl(v	cl(v	NOUN
ejpam-4072	378	16	)	)	PUNCT
ejpam-4072	378	17	)	)	PUNCT
ejpam-4072	379	1	=	=	PUNCT
ejpam-4072	380	1	g−	g−	PRON
ejpam-4072	380	2	f	f	X
ejpam-4072	380	3	(	(	PUNCT
ejpam-4072	380	4	u	u	NOUN
ejpam-4072	380	5	×	×	NOUN
ejpam-4072	380	6	σ1σ2	σ1σ2	NOUN
ejpam-4072	380	7	-	-	NUM
ejpam-4072	380	8	cl(v	cl(v	NOUN
ejpam-4072	380	9	)	)	PUNCT
ejpam-4072	380	10	)	)	PUNCT
ejpam-4072	381	1	⊆	⊆	NUM
ejpam-4072	381	2	g−	g−	PROPN
ejpam-4072	381	3	f	f	X
ejpam-4072	381	4	(	(	PUNCT
ejpam-4072	381	5	σ1σ2	σ1σ2	NOUN
ejpam-4072	381	6	-	-	PUNCT
ejpam-4072	381	7	cl(w	cl(w	NOUN
ejpam-4072	381	8	)	)	PUNCT
ejpam-4072	381	9	)	)	PUNCT
ejpam-4072	381	10	.	.	PUNCT
ejpam-4072	382	1	moreover	moreover	ADV
ejpam-4072	382	2	,	,	PUNCT
ejpam-4072	382	3	u	u	NOUN
ejpam-4072	382	4	∩g	∩g	NOUN
ejpam-4072	382	5	is	be	AUX
ejpam-4072	382	6	a	a	DET
ejpam-4072	382	7	τ1τ2	τ1τ2	ADJ
ejpam-4072	382	8	-	-	ADJ
ejpam-4072	382	9	preopen	preopen	ADJ
ejpam-4072	382	10	set	set	NOUN
ejpam-4072	382	11	containing	contain	VERB
ejpam-4072	382	12	x	x	PUNCT
ejpam-4072	382	13	and	and	CCONJ
ejpam-4072	382	14	hence	hence	ADV
ejpam-4072	382	15	gf	gf	PROPN
ejpam-4072	382	16	is	be	AUX
ejpam-4072	382	17	lower	low	ADJ
ejpam-4072	382	18	almost	almost	ADV
ejpam-4072	382	19	weakly	weakly	ADJ
ejpam-4072	382	20	(	(	PUNCT
ejpam-4072	382	21	τ1	τ1	NOUN
ejpam-4072	382	22	,	,	PUNCT
ejpam-4072	382	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	382	24	.	.	PUNCT
ejpam-4072	383	1	conversely	conversely	ADV
ejpam-4072	383	2	,	,	PUNCT
ejpam-4072	383	3	suppose	suppose	VERB
ejpam-4072	383	4	that	that	SCONJ
ejpam-4072	383	5	gf	gf	NOUN
ejpam-4072	383	6	:	:	PUNCT
ejpam-4072	383	7	(	(	PUNCT
ejpam-4072	383	8	x	x	NOUN
ejpam-4072	383	9	,	,	PUNCT
ejpam-4072	383	10	τ1	τ1	NOUN
ejpam-4072	383	11	,	,	PUNCT
ejpam-4072	383	12	τ2	τ2	NOUN
ejpam-4072	383	13	)	)	PUNCT
ejpam-4072	383	14	→	→	PUNCT
ejpam-4072	383	15	(	(	PUNCT
ejpam-4072	383	16	x	x	SYM
ejpam-4072	383	17	×	×	PROPN
ejpam-4072	383	18	y	y	PROPN
ejpam-4072	383	19	,	,	PUNCT
ejpam-4072	383	20	ρ1	ρ1	NOUN
ejpam-4072	383	21	,	,	PUNCT
ejpam-4072	383	22	ρ2	ρ2	NOUN
ejpam-4072	383	23	)	)	PUNCT
ejpam-4072	383	24	is	be	AUX
ejpam-4072	383	25	lower	low	ADJ
ejpam-4072	383	26	almost	almost	ADV
ejpam-4072	383	27	weakly	weakly	ADJ
ejpam-4072	383	28	(	(	PUNCT
ejpam-4072	383	29	τ1	τ1	NOUN
ejpam-4072	383	30	,	,	PUNCT
ejpam-4072	383	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	383	32	.	.	PUNCT
ejpam-4072	384	1	let	let	VERB
ejpam-4072	384	2	x	x	PUNCT
ejpam-4072	384	3	∈	∈	PROPN
ejpam-4072	384	4	x	x	PUNCT
ejpam-4072	384	5	and	and	CCONJ
ejpam-4072	384	6	let	let	VERB
ejpam-4072	384	7	v	v	PART
ejpam-4072	384	8	be	be	AUX
ejpam-4072	384	9	any	any	DET
ejpam-4072	384	10	σ1σ2	σ1σ2	NOUN
ejpam-4072	384	11	-	-	ADJ
ejpam-4072	384	12	open	open	ADJ
ejpam-4072	384	13	set	set	NOUN
ejpam-4072	384	14	of	of	ADP
ejpam-4072	384	15	y	y	PRON
ejpam-4072	384	16	such	such	ADJ
ejpam-4072	384	17	that	that	SCONJ
ejpam-4072	384	18	f	f	PROPN
ejpam-4072	384	19	(	(	PUNCT
ejpam-4072	384	20	x)∩v	x)∩v	PROPN
ejpam-4072	384	21	̸=	̸=	PROPN
ejpam-4072	384	22	∅.	∅.	NOUN
ejpam-4072	384	23	then	then	ADV
ejpam-4072	384	24	,	,	PUNCT
ejpam-4072	384	25	x	x	PUNCT
ejpam-4072	384	26	×	×	NOUN
ejpam-4072	384	27	y	y	PROPN
ejpam-4072	384	28	is	be	AUX
ejpam-4072	384	29	ρ1ρ2	ρ1ρ2	VERB
ejpam-4072	384	30	-	-	PUNCT
ejpam-4072	384	31	open	open	ADJ
ejpam-4072	384	32	and	and	CCONJ
ejpam-4072	384	33	gf	gf	ADJ
ejpam-4072	384	34	(	(	PUNCT
ejpam-4072	384	35	x	x	NOUN
ejpam-4072	384	36	)	)	PUNCT
ejpam-4072	384	37	∩	∩	NOUN
ejpam-4072	384	38	(	(	PUNCT
ejpam-4072	384	39	x	x	SYM
ejpam-4072	384	40	×	×	NOUN
ejpam-4072	384	41	v	v	NOUN
ejpam-4072	384	42	)	)	PUNCT
ejpam-4072	384	43	=	=	SYM
ejpam-4072	384	44	(	(	PUNCT
ejpam-4072	384	45	{	{	PUNCT
ejpam-4072	384	46	x	x	NOUN
ejpam-4072	384	47	}	}	PUNCT
ejpam-4072	384	48	×	×	PROPN
ejpam-4072	384	49	f	f	X
ejpam-4072	384	50	(	(	PUNCT
ejpam-4072	384	51	x	x	NOUN
ejpam-4072	384	52	)	)	PUNCT
ejpam-4072	384	53	)	)	PUNCT
ejpam-4072	384	54	∩	∩	NOUN
ejpam-4072	384	55	(	(	PUNCT
ejpam-4072	384	56	x	x	SYM
ejpam-4072	384	57	×	×	NOUN
ejpam-4072	384	58	v	v	NOUN
ejpam-4072	384	59	)	)	PUNCT
ejpam-4072	384	60	=	=	SYM
ejpam-4072	384	61	{	{	PUNCT
ejpam-4072	384	62	x	x	NOUN
ejpam-4072	384	63	}	}	PUNCT
ejpam-4072	384	64	×	×	NOUN
ejpam-4072	384	65	(	(	PUNCT
ejpam-4072	384	66	f	f	PROPN
ejpam-4072	384	67	(	(	PUNCT
ejpam-4072	384	68	x	x	NOUN
ejpam-4072	384	69	)	)	PUNCT
ejpam-4072	384	70	∩	∩	ADJ
ejpam-4072	384	71	v	v	NOUN
ejpam-4072	384	72	)	)	PUNCT
ejpam-4072	384	73	̸=	̸=	PROPN
ejpam-4072	384	74	∅.	∅.	NOUN
ejpam-4072	384	75	there	there	ADV
ejpam-4072	384	76	exists	exist	VERB
ejpam-4072	384	77	a	a	DET
ejpam-4072	384	78	τ1τ2	τ1τ2	NOUN
ejpam-4072	384	79	-	-	ADJ
ejpam-4072	384	80	preopen	preopen	ADJ
ejpam-4072	384	81	set	set	NOUN
ejpam-4072	384	82	u	u	NOUN
ejpam-4072	384	83	containing	contain	VERB
ejpam-4072	384	84	x	x	PUNCT
ejpam-4072	384	85	such	such	ADJ
ejpam-4072	384	86	that	that	DET
ejpam-4072	384	87	gf	gf	NOUN
ejpam-4072	384	88	(	(	PUNCT
ejpam-4072	384	89	z	z	NOUN
ejpam-4072	384	90	)	)	PUNCT
ejpam-4072	384	91	∩	∩	NOUN
ejpam-4072	384	92	ρ1ρ2	ρ1ρ2	PUNCT
ejpam-4072	384	93	-	-	PUNCT
ejpam-4072	384	94	cl((x	cl((x	NUM
ejpam-4072	384	95	×	×	NOUN
ejpam-4072	384	96	v	v	NOUN
ejpam-4072	384	97	)	)	PUNCT
ejpam-4072	384	98	)	)	PUNCT
ejpam-4072	384	99	̸=	̸=	NOUN
ejpam-4072	384	100	∅	∅	NOUN
ejpam-4072	384	101	for	for	ADP
ejpam-4072	384	102	each	each	DET
ejpam-4072	384	103	z	z	NOUN
ejpam-4072	384	104	∈	∈	PROPN
ejpam-4072	384	105	u	u	NOUN
ejpam-4072	384	106	.	.	PUNCT
ejpam-4072	385	1	by	by	ADP
ejpam-4072	385	2	lemma	lemma	PROPN
ejpam-4072	385	3	11	11	NUM
ejpam-4072	385	4	,	,	PUNCT
ejpam-4072	385	5	u	u	NOUN
ejpam-4072	385	6	⊆	⊆	NUM
ejpam-4072	385	7	g−	g−	PROPN
ejpam-4072	385	8	f	f	X
ejpam-4072	385	9	(	(	PUNCT
ejpam-4072	385	10	ρ1ρ2	ρ1ρ2	NOUN
ejpam-4072	385	11	-	-	PUNCT
ejpam-4072	385	12	cl(x	cl(x	NUM
ejpam-4072	385	13	×	×	NOUN
ejpam-4072	385	14	v	v	NOUN
ejpam-4072	385	15	)	)	PUNCT
ejpam-4072	385	16	)	)	PUNCT
ejpam-4072	386	1	=	=	SYM
ejpam-4072	386	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-4072	386	3	-	-	PUNCT
ejpam-4072	386	4	cl(v	cl(v	NOUN
ejpam-4072	386	5	)	)	PUNCT
ejpam-4072	386	6	)	)	PUNCT
ejpam-4072	386	7	.	.	PUNCT
ejpam-4072	387	1	this	this	PRON
ejpam-4072	387	2	shows	show	VERB
ejpam-4072	387	3	that	that	SCONJ
ejpam-4072	387	4	f	f	PROPN
ejpam-4072	387	5	is	be	AUX
ejpam-4072	387	6	lower	low	ADJ
ejpam-4072	387	7	almost	almost	ADV
ejpam-4072	387	8	weakly	weakly	ADJ
ejpam-4072	387	9	(	(	PUNCT
ejpam-4072	387	10	τ1	τ1	NOUN
ejpam-4072	387	11	,	,	PUNCT
ejpam-4072	387	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	387	13	.	.	PUNCT
ejpam-4072	388	1	references	reference	NOUN
ejpam-4072	388	2	1224	1224	NUM
ejpam-4072	388	3	4	4	NUM
ejpam-4072	388	4	.	.	PUNCT
ejpam-4072	388	5	conclusion	conclusion	VERB
ejpam-4072	388	6	the	the	DET
ejpam-4072	388	7	concepts	concept	NOUN
ejpam-4072	388	8	of	of	ADP
ejpam-4072	388	9	openness	openness	NOUN
ejpam-4072	388	10	and	and	CCONJ
ejpam-4072	388	11	continuity	continuity	NOUN
ejpam-4072	388	12	are	be	AUX
ejpam-4072	388	13	extensively	extensively	ADV
ejpam-4072	388	14	developed	develop	VERB
ejpam-4072	388	15	and	and	CCONJ
ejpam-4072	388	16	used	use	VERB
ejpam-4072	388	17	in	in	ADP
ejpam-4072	388	18	many	many	ADJ
ejpam-4072	388	19	fields	field	NOUN
ejpam-4072	388	20	of	of	ADP
ejpam-4072	388	21	applications	application	NOUN
ejpam-4072	388	22	such	such	ADJ
ejpam-4072	388	23	as	as	ADP
ejpam-4072	388	24	data	datum	NOUN
ejpam-4072	388	25	mining	mining	NOUN
ejpam-4072	388	26	,	,	PUNCT
ejpam-4072	388	27	computational	computational	ADJ
ejpam-4072	388	28	topology	topology	NOUN
ejpam-4072	388	29	for	for	ADP
ejpam-4072	388	30	geometric	geometric	ADJ
ejpam-4072	388	31	design	design	NOUN
ejpam-4072	388	32	and	and	CCONJ
ejpam-4072	388	33	molecular	molecular	ADJ
ejpam-4072	388	34	design	design	NOUN
ejpam-4072	388	35	,	,	PUNCT
ejpam-4072	388	36	information	information	NOUN
ejpam-4072	388	37	systems	system	NOUN
ejpam-4072	388	38	,	,	PUNCT
ejpam-4072	388	39	digital	digital	ADJ
ejpam-4072	388	40	topology	topology	NOUN
ejpam-4072	388	41	and	and	CCONJ
ejpam-4072	388	42	computer	computer	NOUN
ejpam-4072	388	43	graphics	graphic	NOUN
ejpam-4072	388	44	.	.	PUNCT
ejpam-4072	389	1	continuity	continuity	NOUN
ejpam-4072	389	2	of	of	ADP
ejpam-4072	389	3	functions	function	NOUN
ejpam-4072	389	4	and	and	CCONJ
ejpam-4072	389	5	multifunctions	multifunction	NOUN
ejpam-4072	389	6	in	in	ADP
ejpam-4072	389	7	topological	topological	ADJ
ejpam-4072	389	8	spaces	space	NOUN
ejpam-4072	389	9	and	and	CCONJ
ejpam-4072	389	10	bitopological	bitopological	ADJ
ejpam-4072	389	11	spaces	space	NOUN
ejpam-4072	389	12	have	have	AUX
ejpam-4072	389	13	been	be	AUX
ejpam-4072	389	14	researched	research	VERB
ejpam-4072	389	15	by	by	ADP
ejpam-4072	389	16	many	many	ADJ
ejpam-4072	389	17	mathematicians	mathematician	NOUN
ejpam-4072	389	18	.	.	PUNCT
ejpam-4072	390	1	several	several	ADJ
ejpam-4072	390	2	investigations	investigation	NOUN
ejpam-4072	390	3	related	relate	VERB
ejpam-4072	390	4	to	to	PART
ejpam-4072	390	5	open	open	ADJ
ejpam-4072	390	6	sets	set	NOUN
ejpam-4072	390	7	have	have	AUX
ejpam-4072	390	8	been	be	AUX
ejpam-4072	390	9	published	publish	VERB
ejpam-4072	390	10	and	and	CCONJ
ejpam-4072	390	11	various	various	ADJ
ejpam-4072	390	12	forms	form	NOUN
ejpam-4072	390	13	of	of	ADP
ejpam-4072	390	14	continuity	continuity	NOUN
ejpam-4072	390	15	types	type	NOUN
ejpam-4072	390	16	have	have	AUX
ejpam-4072	390	17	been	be	AUX
ejpam-4072	390	18	introduced	introduce	VERB
ejpam-4072	390	19	.	.	PUNCT
ejpam-4072	391	1	this	this	DET
ejpam-4072	391	2	paper	paper	NOUN
ejpam-4072	391	3	deals	deal	NOUN
ejpam-4072	391	4	with	with	ADP
ejpam-4072	391	5	the	the	DET
ejpam-4072	391	6	notions	notion	NOUN
ejpam-4072	391	7	of	of	ADP
ejpam-4072	391	8	upper	upper	ADJ
ejpam-4072	391	9	and	and	CCONJ
ejpam-4072	391	10	lower	low	ADJ
ejpam-4072	391	11	almost	almost	ADV
ejpam-4072	391	12	weakly	weakly	ADJ
ejpam-4072	391	13	(	(	PUNCT
ejpam-4072	391	14	τ1	τ1	NOUN
ejpam-4072	391	15	,	,	PUNCT
ejpam-4072	391	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	391	17	multifunctions	multifunction	NOUN
ejpam-4072	391	18	.	.	PUNCT
ejpam-4072	392	1	some	some	DET
ejpam-4072	392	2	characterizations	characterization	NOUN
ejpam-4072	392	3	of	of	ADP
ejpam-4072	392	4	upper	upper	ADJ
ejpam-4072	392	5	and	and	CCONJ
ejpam-4072	392	6	lower	low	ADJ
ejpam-4072	392	7	almost	almost	ADV
ejpam-4072	392	8	weakly	weakly	ADJ
ejpam-4072	392	9	(	(	PUNCT
ejpam-4072	392	10	τ1	τ1	NOUN
ejpam-4072	392	11	,	,	PUNCT
ejpam-4072	392	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	392	13	multifunctions	multifunction	NOUN
ejpam-4072	392	14	are	be	AUX
ejpam-4072	392	15	established	establish	VERB
ejpam-4072	392	16	.	.	PUNCT
ejpam-4072	393	1	the	the	DET
ejpam-4072	393	2	ideas	idea	NOUN
ejpam-4072	393	3	and	and	CCONJ
ejpam-4072	393	4	results	result	NOUN
ejpam-4072	393	5	of	of	ADP
ejpam-4072	393	6	this	this	DET
ejpam-4072	393	7	paper	paper	NOUN
ejpam-4072	393	8	may	may	AUX
ejpam-4072	393	9	motivate	motivate	VERB
ejpam-4072	393	10	further	further	ADJ
ejpam-4072	393	11	research	research	NOUN
ejpam-4072	393	12	.	.	PUNCT
ejpam-4072	394	1	acknowledgements	acknowledgement	NOUN
ejpam-4072	394	2	this	this	DET
ejpam-4072	394	3	research	research	NOUN
ejpam-4072	394	4	project	project	NOUN
ejpam-4072	394	5	was	be	AUX
ejpam-4072	394	6	financially	financially	ADV
ejpam-4072	394	7	supported	support	VERB
ejpam-4072	394	8	by	by	ADP
ejpam-4072	394	9	mahasarakham	mahasarakham	PROPN
ejpam-4072	394	10	university	university	PROPN
ejpam-4072	394	11	.	.	PUNCT
ejpam-4072	395	1	references	reference	NOUN
ejpam-4072	395	2	[	[	X
ejpam-4072	395	3	1	1	NUM
ejpam-4072	395	4	]	]	PUNCT
ejpam-4072	395	5	d.	d.	PROPN
ejpam-4072	395	6	andrijević.	andrijević.	PROPN
ejpam-4072	395	7	semi	semi	ADJ
ejpam-4072	395	8	-	-	ADJ
ejpam-4072	395	9	preopen	preopen	ADJ
ejpam-4072	395	10	sets	set	NOUN
ejpam-4072	395	11	.	.	PUNCT
ejpam-4072	396	1	mat	mat	X
ejpam-4072	396	2	.	.	PROPN
ejpam-4072	396	3	vesnik	vesnik	PROPN
ejpam-4072	396	4	,	,	PUNCT
ejpam-4072	396	5	38:24–32	38:24–32	NUM
ejpam-4072	396	6	,	,	PUNCT
ejpam-4072	396	7	1986	1986	NUM
ejpam-4072	396	8	.	.	PUNCT
ejpam-4072	397	1	[	[	X
ejpam-4072	397	2	2	2	X
ejpam-4072	397	3	]	]	PUNCT
ejpam-4072	397	4	g.	g.	PROPN
ejpam-4072	397	5	k.	k.	PROPN
ejpam-4072	397	6	banerjee	banerjee	PROPN
ejpam-4072	397	7	.	.	PUNCT
ejpam-4072	398	1	on	on	ADP
ejpam-4072	398	2	pairwise	pairwise	NOUN
ejpam-4072	398	3	almost	almost	ADV
ejpam-4072	398	4	strongly	strongly	ADV
ejpam-4072	398	5	θ	θ	ADJ
ejpam-4072	398	6	-	-	ADJ
ejpam-4072	398	7	continuous	continuous	ADJ
ejpam-4072	398	8	mappings	mapping	NOUN
ejpam-4072	398	9	.	.	PUNCT
ejpam-4072	399	1	bull	bull	NOUN
ejpam-4072	399	2	.	.	PUNCT
ejpam-4072	400	1	calcutta	calcutta	PROPN
ejpam-4072	400	2	math	math	PROPN
ejpam-4072	400	3	.	.	PUNCT
ejpam-4072	401	1	soc	soc	PROPN
ejpam-4072	401	2	.	.	PUNCT
ejpam-4072	401	3	,	,	PUNCT
ejpam-4072	401	4	79:314–320	79:314–320	NUM
ejpam-4072	401	5	,	,	PUNCT
ejpam-4072	401	6	1987	1987	NUM
ejpam-4072	401	7	.	.	PUNCT
ejpam-4072	402	1	[	[	X
ejpam-4072	402	2	3	3	X
ejpam-4072	402	3	]	]	X
ejpam-4072	402	4	c.	c.	PROPN
ejpam-4072	402	5	berge	berge	PROPN
ejpam-4072	402	6	.	.	PUNCT
ejpam-4072	402	7	espaces	espace	VERB
ejpam-4072	402	8	topologiques	topologique	NOUN
ejpam-4072	402	9	fonctions	fonction	NOUN
ejpam-4072	402	10	multivoques	multivoque	NOUN
ejpam-4072	402	11	.	.	PUNCT
ejpam-4072	403	1	dunod	dunod	PROPN
ejpam-4072	403	2	,	,	PUNCT
ejpam-4072	403	3	paris	paris	PROPN
ejpam-4072	403	4	,	,	PUNCT
ejpam-4072	403	5	1959	1959	NUM
ejpam-4072	403	6	.	.	PUNCT
ejpam-4072	404	1	[	[	X
ejpam-4072	404	2	4	4	NUM
ejpam-4072	404	3	]	]	PUNCT
ejpam-4072	404	4	c.	c.	PROPN
ejpam-4072	404	5	boonpok	boonpok	PROPN
ejpam-4072	404	6	.	.	PUNCT
ejpam-4072	405	1	(	(	PUNCT
ejpam-4072	405	2	τ1	τ1	NOUN
ejpam-4072	405	3	,	,	PUNCT
ejpam-4072	405	4	τ2)δ	τ2)δ	ADJ
ejpam-4072	405	5	-	-	PUNCT
ejpam-4072	405	6	semicontinuous	semicontinuous	ADJ
ejpam-4072	405	7	multifunctions	multifunction	NOUN
ejpam-4072	405	8	.	.	PUNCT
ejpam-4072	406	1	heliyon	heliyon	NOUN
ejpam-4072	406	2	,	,	PUNCT
ejpam-4072	406	3	6	6	NUM
ejpam-4072	406	4	:	:	SYM
ejpam-4072	406	5	e05367	e05367	PROPN
ejpam-4072	406	6	,	,	PUNCT
ejpam-4072	406	7	2020	2020	NUM
ejpam-4072	406	8	.	.	PUNCT
ejpam-4072	407	1	[	[	X
ejpam-4072	407	2	5	5	X
ejpam-4072	407	3	]	]	PUNCT
ejpam-4072	407	4	c.	c.	PROPN
ejpam-4072	407	5	boonpok	boonpok	PROPN
ejpam-4072	407	6	,	,	PUNCT
ejpam-4072	407	7	c.	c.	PROPN
ejpam-4072	407	8	viriyapong	viriyapong	PROPN
ejpam-4072	407	9	,	,	PUNCT
ejpam-4072	407	10	and	and	CCONJ
ejpam-4072	407	11	m.	m.	NOUN
ejpam-4072	407	12	thongmoon	thongmoon	NOUN
ejpam-4072	407	13	.	.	PUNCT
ejpam-4072	408	1	on	on	ADP
ejpam-4072	408	2	upper	upper	ADJ
ejpam-4072	408	3	and	and	CCONJ
ejpam-4072	408	4	lower	low	ADJ
ejpam-4072	408	5	(	(	PUNCT
ejpam-4072	408	6	τ1	τ1	NOUN
ejpam-4072	408	7	,	,	PUNCT
ejpam-4072	408	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-4072	408	9	multifunctions	multifunction	NOUN
ejpam-4072	408	10	.	.	PUNCT
ejpam-4072	409	1	j.	j.	PROPN
ejpam-4072	409	2	math	math	PROPN
ejpam-4072	409	3	.	.	PUNCT
ejpam-4072	410	1	computer	computer	PROPN
ejpam-4072	410	2	sci	sci	PROPN
ejpam-4072	410	3	.	.	PROPN
ejpam-4072	410	4	,	,	PUNCT
ejpam-4072	410	5	18(3):282–293	18(3):282–293	NUM
ejpam-4072	410	6	,	,	PUNCT
ejpam-4072	410	7	2018	2018	NUM
ejpam-4072	410	8	.	.	PUNCT
ejpam-4072	411	1	[	[	X
ejpam-4072	411	2	6	6	NUM
ejpam-4072	411	3	]	]	PUNCT
ejpam-4072	411	4	s.	s.	PROPN
ejpam-4072	411	5	bose	bose	PROPN
ejpam-4072	411	6	and	and	CCONJ
ejpam-4072	411	7	d.	d.	PROPN
ejpam-4072	411	8	sinha	sinha	PROPN
ejpam-4072	411	9	.	.	PUNCT
ejpam-4072	412	1	almost	almost	ADV
ejpam-4072	412	2	open	open	ADJ
ejpam-4072	412	3	,	,	PUNCT
ejpam-4072	412	4	almost	almost	ADV
ejpam-4072	412	5	closed	closed	ADJ
ejpam-4072	412	6	,	,	PUNCT
ejpam-4072	412	7	θ	θ	NOUN
ejpam-4072	412	8	-	-	ADJ
ejpam-4072	412	9	continuous	continuous	ADJ
ejpam-4072	412	10	and	and	CCONJ
ejpam-4072	412	11	almost	almost	ADV
ejpam-4072	412	12	compact	compact	ADJ
ejpam-4072	412	13	mappings	mapping	NOUN
ejpam-4072	412	14	in	in	ADP
ejpam-4072	412	15	bitopological	bitopological	ADJ
ejpam-4072	412	16	spaces	space	NOUN
ejpam-4072	412	17	.	.	PUNCT
ejpam-4072	413	1	bull	bull	NOUN
ejpam-4072	413	2	.	.	PUNCT
ejpam-4072	414	1	calcutta	calcutta	PROPN
ejpam-4072	414	2	math	math	PROPN
ejpam-4072	414	3	.	.	PUNCT
ejpam-4072	415	1	soc	soc	PROPN
ejpam-4072	415	2	.	.	PUNCT
ejpam-4072	415	3	,	,	PUNCT
ejpam-4072	415	4	73:345–354	73:345–354	NUM
ejpam-4072	415	5	,	,	PUNCT
ejpam-4072	415	6	1981	1981	NUM
ejpam-4072	415	7	.	.	PUNCT
ejpam-4072	416	1	[	[	X
ejpam-4072	416	2	7	7	X
ejpam-4072	416	3	]	]	X
ejpam-4072	416	4	g.	g.	PROPN
ejpam-4072	416	5	şenel	şenel	PROPN
ejpam-4072	416	6	.	.	PUNCT
ejpam-4072	417	1	a	a	DET
ejpam-4072	417	2	new	new	ADJ
ejpam-4072	417	3	approach	approach	NOUN
ejpam-4072	417	4	to	to	ADP
ejpam-4072	417	5	hausdorff	hausdorff	NOUN
ejpam-4072	417	6	space	space	NOUN
ejpam-4072	417	7	theory	theory	NOUN
ejpam-4072	417	8	via	via	ADP
ejpam-4072	417	9	the	the	DET
ejpam-4072	417	10	soft	soft	ADJ
ejpam-4072	417	11	sets	set	NOUN
ejpam-4072	417	12	.	.	PUNCT
ejpam-4072	418	1	math	math	NOUN
ejpam-4072	418	2	.	.	PUNCT
ejpam-4072	419	1	probl	probl	PROPN
ejpam-4072	419	2	.	.	PUNCT
ejpam-4072	420	1	eng	eng	PROPN
ejpam-4072	420	2	.	.	PROPN
ejpam-4072	420	3	,	,	PUNCT
ejpam-4072	420	4	2016:2196743	2016:2196743	NUM
ejpam-4072	420	5	,	,	PUNCT
ejpam-4072	420	6	2016	2016	NUM
ejpam-4072	420	7	.	.	PUNCT
ejpam-4072	421	1	[	[	X
ejpam-4072	421	2	8	8	NUM
ejpam-4072	421	3	]	]	X
ejpam-4072	421	4	g.	g.	NOUN
ejpam-4072	421	5	şenel	şenel	PROPN
ejpam-4072	421	6	and	and	CCONJ
ejpam-4072	421	7	n.	n.	PROPN
ejpam-4072	421	8	çağman	çağman	PROPN
ejpam-4072	421	9	.	.	PUNCT
ejpam-4072	421	10	soft	soft	ADJ
ejpam-4072	421	11	topological	topological	ADJ
ejpam-4072	421	12	subspaces	subspace	NOUN
ejpam-4072	421	13	.	.	PUNCT
ejpam-4072	422	1	ann	ann	PROPN
ejpam-4072	422	2	.	.	PUNCT
ejpam-4072	422	3	fuzzy	fuzzy	ADJ
ejpam-4072	422	4	math	math	NOUN
ejpam-4072	422	5	.	.	PUNCT
ejpam-4072	423	1	inform	inform	NOUN
ejpam-4072	423	2	.	.	PUNCT
ejpam-4072	423	3	,	,	PUNCT
ejpam-4072	423	4	10(4):525–535	10(4):525–535	NUM
ejpam-4072	423	5	,	,	PUNCT
ejpam-4072	423	6	2015	2015	NUM
ejpam-4072	423	7	.	.	PUNCT
ejpam-4072	424	1	[	[	X
ejpam-4072	424	2	9	9	NUM
ejpam-4072	424	3	]	]	X
ejpam-4072	424	4	e.	e.	PROPN
ejpam-4072	424	5	ekici	ekici	PROPN
ejpam-4072	424	6	and	and	CCONJ
ejpam-4072	424	7	j.	j.	PROPN
ejpam-4072	424	8	h.	h.	PROPN
ejpam-4072	424	9	park	park	PROPN
ejpam-4072	424	10	.	.	PUNCT
ejpam-4072	425	1	a	a	DET
ejpam-4072	425	2	weak	weak	ADJ
ejpam-4072	425	3	form	form	NOUN
ejpam-4072	425	4	of	of	ADP
ejpam-4072	425	5	some	some	DET
ejpam-4072	425	6	types	type	NOUN
ejpam-4072	425	7	of	of	ADP
ejpam-4072	425	8	continuous	continuous	ADJ
ejpam-4072	425	9	multifunctions	multifunction	NOUN
ejpam-4072	425	10	.	.	PUNCT
ejpam-4072	426	1	filomat	filomat	NOUN
ejpam-4072	426	2	,	,	PUNCT
ejpam-4072	426	3	20(2):13–32	20(2):13–32	NUM
ejpam-4072	426	4	,	,	PUNCT
ejpam-4072	426	5	2006	2006	NUM
ejpam-4072	426	6	.	.	PUNCT
ejpam-4072	427	1	[	[	X
ejpam-4072	427	2	10	10	NUM
ejpam-4072	427	3	]	]	PUNCT
ejpam-4072	427	4	m.	m.	NOUN
ejpam-4072	427	5	e.	e.	PROPN
ejpam-4072	427	6	abd	abd	PROPN
ejpam-4072	428	1	el	el	PROPN
ejpam-4072	428	2	-	-	PROPN
ejpam-4072	428	3	monsef	monsef	PROPN
ejpam-4072	428	4	,	,	PUNCT
ejpam-4072	428	5	s.	s.	PROPN
ejpam-4072	428	6	n.	n.	PROPN
ejpam-4072	428	7	el	el	PROPN
ejpam-4072	428	8	-	-	PROPN
ejpam-4072	428	9	deeb	deeb	PROPN
ejpam-4072	428	10	,	,	PUNCT
ejpam-4072	428	11	and	and	CCONJ
ejpam-4072	428	12	r.	r.	PROPN
ejpam-4072	428	13	a.	a.	PROPN
ejpam-4072	428	14	mahmoud	mahmoud	PROPN
ejpam-4072	428	15	.	.	PUNCT
ejpam-4072	429	1	β	β	X
ejpam-4072	429	2	-	-	ADJ
ejpam-4072	429	3	open	open	ADJ
ejpam-4072	429	4	sets	set	NOUN
ejpam-4072	429	5	and	and	CCONJ
ejpam-4072	429	6	βcontinuous	βcontinuous	ADJ
ejpam-4072	429	7	mappings	mapping	NOUN
ejpam-4072	429	8	.	.	PUNCT
ejpam-4072	430	1	bull	bull	NOUN
ejpam-4072	430	2	.	.	PUNCT
ejpam-4072	431	1	fac	fac	PROPN
ejpam-4072	431	2	.	.	PUNCT
ejpam-4072	432	1	sci	sci	PROPN
ejpam-4072	432	2	.	.	PUNCT
ejpam-4072	432	3	assiut	assiut	PROPN
ejpam-4072	432	4	univ	univ	PROPN
ejpam-4072	432	5	.	.	PROPN
ejpam-4072	432	6	,	,	PUNCT
ejpam-4072	432	7	12:77–90	12:77–90	NUM
ejpam-4072	432	8	,	,	PUNCT
ejpam-4072	432	9	1983	1983	NUM
ejpam-4072	432	10	.	.	PUNCT
ejpam-4072	433	1	[	[	X
ejpam-4072	433	2	11	11	NUM
ejpam-4072	433	3	]	]	PUNCT
ejpam-4072	433	4	t.	t.	PROPN
ejpam-4072	433	5	husain	husain	PROPN
ejpam-4072	433	6	.	.	PUNCT
ejpam-4072	434	1	almost	almost	ADV
ejpam-4072	434	2	continuous	continuous	ADJ
ejpam-4072	434	3	mappings	mapping	NOUN
ejpam-4072	434	4	.	.	PUNCT
ejpam-4072	435	1	prace	prace	PROPN
ejpam-4072	435	2	mat	mat	PROPN
ejpam-4072	435	3	.	.	PROPN
ejpam-4072	435	4	,	,	PUNCT
ejpam-4072	435	5	10(1):1–7	10(1):1–7	NUM
ejpam-4072	435	6	,	,	PUNCT
ejpam-4072	435	7	1966	1966	NUM
ejpam-4072	435	8	.	.	PUNCT
ejpam-4072	436	1	references	reference	NOUN
ejpam-4072	436	2	1225	1225	NUM
ejpam-4072	437	1	[	[	X
ejpam-4072	437	2	12	12	NUM
ejpam-4072	437	3	]	]	X
ejpam-4072	437	4	d.	d.	PROPN
ejpam-4072	437	5	s.	s.	PROPN
ejpam-4072	437	6	janković.	janković.	PROPN
ejpam-4072	437	7	θ	θ	PROPN
ejpam-4072	437	8	-	-	ADJ
ejpam-4072	437	9	regular	regular	ADJ
ejpam-4072	437	10	spaces	space	NOUN
ejpam-4072	437	11	.	.	PUNCT
ejpam-4072	438	1	internat	internat	PROPN
ejpam-4072	438	2	.	.	PUNCT
ejpam-4072	439	1	j.	j.	PROPN
ejpam-4072	439	2	math	math	PROPN
ejpam-4072	439	3	.	.	PUNCT
ejpam-4072	440	1	math	math	NOUN
ejpam-4072	440	2	.	.	PUNCT
ejpam-4072	441	1	sci	sci	PROPN
ejpam-4072	441	2	.	.	PROPN
ejpam-4072	441	3	,	,	PUNCT
ejpam-4072	441	4	8(3):615–619	8(3):615–619	NUM
ejpam-4072	441	5	,	,	PUNCT
ejpam-4072	441	6	1985	1985	NUM
ejpam-4072	441	7	.	.	PUNCT
ejpam-4072	442	1	[	[	X
ejpam-4072	442	2	13	13	NUM
ejpam-4072	442	3	]	]	PUNCT
ejpam-4072	442	4	m.	m.	NOUN
ejpam-4072	442	5	jelić.	jelić.	PROPN
ejpam-4072	442	6	a	a	DET
ejpam-4072	442	7	decomposition	decomposition	NOUN
ejpam-4072	442	8	of	of	ADP
ejpam-4072	442	9	pairwise	pairwise	NOUN
ejpam-4072	442	10	continuity	continuity	NOUN
ejpam-4072	442	11	.	.	PUNCT
ejpam-4072	443	1	j.	j.	PROPN
ejpam-4072	443	2	inst	inst	PROPN
ejpam-4072	443	3	.	.	PUNCT
ejpam-4072	443	4	math	math	NOUN
ejpam-4072	443	5	.	.	PUNCT
ejpam-4072	444	1	comput	comput	NOUN
ejpam-4072	444	2	.	.	PUNCT
ejpam-4072	445	1	sci	sci	PROPN
ejpam-4072	445	2	.	.	PROPN
ejpam-4072	445	3	math	math	PROPN
ejpam-4072	445	4	.	.	PUNCT
ejpam-4072	446	1	ser	ser	PROPN
ejpam-4072	446	2	.	.	PROPN
ejpam-4072	446	3	,	,	PUNCT
ejpam-4072	446	4	3(1):25–29	3(1):25–29	NUM
ejpam-4072	446	5	,	,	PUNCT
ejpam-4072	446	6	1990	1990	NUM
ejpam-4072	446	7	.	.	PUNCT
ejpam-4072	447	1	[	[	X
ejpam-4072	447	2	14	14	NUM
ejpam-4072	447	3	]	]	X
ejpam-4072	447	4	j.	j.	PROPN
ejpam-4072	447	5	c.	c.	PROPN
ejpam-4072	447	6	kelly	kelly	PROPN
ejpam-4072	447	7	.	.	PUNCT
ejpam-4072	448	1	bitopological	bitopological	ADJ
ejpam-4072	448	2	spaces	space	NOUN
ejpam-4072	448	3	.	.	PUNCT
ejpam-4072	449	1	proc	proc	NOUN
ejpam-4072	449	2	.	.	PUNCT
ejpam-4072	450	1	london	london	PROPN
ejpam-4072	450	2	math	math	PROPN
ejpam-4072	450	3	.	.	PUNCT
ejpam-4072	451	1	soc	soc	PROPN
ejpam-4072	451	2	.	.	PUNCT
ejpam-4072	451	3	,	,	PUNCT
ejpam-4072	451	4	3(13):71–89	3(13):71–89	NUM
ejpam-4072	451	5	,	,	PUNCT
ejpam-4072	451	6	1963	1963	NUM
ejpam-4072	451	7	.	.	PUNCT
ejpam-4072	452	1	[	[	X
ejpam-4072	452	2	15	15	NUM
ejpam-4072	452	3	]	]	X
ejpam-4072	452	4	f.	f.	PROPN
ejpam-4072	452	5	h.	h.	PROPN
ejpam-4072	452	6	khedr	khedr	PROPN
ejpam-4072	452	7	,	,	PUNCT
ejpam-4072	452	8	s.	s.	PROPN
ejpam-4072	452	9	m.	m.	PROPN
ejpam-4072	452	10	al	al	PROPN
ejpam-4072	452	11	-	-	PUNCT
ejpam-4072	452	12	areefi	areefi	PROPN
ejpam-4072	452	13	,	,	PUNCT
ejpam-4072	452	14	and	and	CCONJ
ejpam-4072	452	15	t.	t.	PROPN
ejpam-4072	452	16	noiri	noiri	PROPN
ejpam-4072	452	17	.	.	PUNCT
ejpam-4072	453	1	precontinuity	precontinuity	NOUN
ejpam-4072	453	2	and	and	CCONJ
ejpam-4072	453	3	semi	semi	NOUN
ejpam-4072	453	4	-	-	NOUN
ejpam-4072	453	5	precontinuity	precontinuity	NOUN
ejpam-4072	453	6	in	in	ADP
ejpam-4072	453	7	bitopological	bitopological	ADJ
ejpam-4072	453	8	spaces	space	NOUN
ejpam-4072	453	9	.	.	PUNCT
ejpam-4072	454	1	indian	indian	PROPN
ejpam-4072	454	2	j.	j.	PROPN
ejpam-4072	454	3	pure	pure	PROPN
ejpam-4072	454	4	appl	appl	PROPN
ejpam-4072	454	5	.	.	PUNCT
ejpam-4072	454	6	math	math	PROPN
ejpam-4072	454	7	.	.	PUNCT
ejpam-4072	454	8	,	,	PUNCT
ejpam-4072	455	1	23(9):625–633	23(9):625–633	PROPN
ejpam-4072	455	2	,	,	PUNCT
ejpam-4072	455	3	1992	1992	NUM
ejpam-4072	455	4	.	.	PUNCT
ejpam-4072	456	1	[	[	X
ejpam-4072	456	2	16	16	NUM
ejpam-4072	456	3	]	]	PUNCT
ejpam-4072	456	4	k.	k.	PROPN
ejpam-4072	457	1	laprom	laprom	PROPN
ejpam-4072	457	2	,	,	PUNCT
ejpam-4072	457	3	c.	c.	PROPN
ejpam-4072	457	4	boonpok	boonpok	PROPN
ejpam-4072	457	5	,	,	PUNCT
ejpam-4072	457	6	and	and	CCONJ
ejpam-4072	457	7	c.	c.	PROPN
ejpam-4072	457	8	viriyapong	viriyapong	PROPN
ejpam-4072	457	9	.	.	PUNCT
ejpam-4072	458	1	β(τ1	β(τ1	PROPN
ejpam-4072	458	2	,	,	PUNCT
ejpam-4072	458	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4072	458	4	multifunctions	multifunction	NOUN
ejpam-4072	458	5	on	on	ADP
ejpam-4072	458	6	bitopological	bitopological	ADJ
ejpam-4072	458	7	spaces	space	NOUN
ejpam-4072	458	8	.	.	PUNCT
ejpam-4072	459	1	j.	j.	PROPN
ejpam-4072	459	2	math	math	PROPN
ejpam-4072	459	3	.	.	PROPN
ejpam-4072	459	4	,	,	PUNCT
ejpam-4072	459	5	2020:4020971	2020:4020971	NUM
ejpam-4072	459	6	,	,	PUNCT
ejpam-4072	459	7	2020	2020	NUM
ejpam-4072	459	8	.	.	PUNCT
ejpam-4072	460	1	[	[	X
ejpam-4072	460	2	17	17	NUM
ejpam-4072	460	3	]	]	X
ejpam-4072	460	4	n.	n.	PROPN
ejpam-4072	460	5	levine	levine	PROPN
ejpam-4072	460	6	.	.	PUNCT
ejpam-4072	461	1	a	a	DET
ejpam-4072	461	2	decomposition	decomposition	NOUN
ejpam-4072	461	3	of	of	ADP
ejpam-4072	461	4	continuity	continuity	NOUN
ejpam-4072	461	5	in	in	ADP
ejpam-4072	461	6	topological	topological	ADJ
ejpam-4072	461	7	spaces	space	NOUN
ejpam-4072	461	8	.	.	PUNCT
ejpam-4072	462	1	amer	amer	PROPN
ejpam-4072	462	2	.	.	PUNCT
ejpam-4072	462	3	math	math	PROPN
ejpam-4072	462	4	.	.	PUNCT
ejpam-4072	463	1	monthly	monthly	ADJ
ejpam-4072	463	2	,	,	PUNCT
ejpam-4072	463	3	68:44–46	68:44–46	PROPN
ejpam-4072	463	4	,	,	PUNCT
ejpam-4072	463	5	1961	1961	NUM
ejpam-4072	463	6	.	.	PUNCT
ejpam-4072	464	1	[	[	X
ejpam-4072	464	2	18	18	NUM
ejpam-4072	464	3	]	]	X
ejpam-4072	464	4	n.	n.	PROPN
ejpam-4072	464	5	levine	levine	PROPN
ejpam-4072	464	6	.	.	PUNCT
ejpam-4072	465	1	semi	semi	ADJ
ejpam-4072	465	2	-	-	ADJ
ejpam-4072	465	3	open	open	ADJ
ejpam-4072	465	4	sets	set	NOUN
ejpam-4072	465	5	and	and	CCONJ
ejpam-4072	465	6	semi	semi	ADJ
ejpam-4072	465	7	-	-	NOUN
ejpam-4072	465	8	continuity	continuity	NOUN
ejpam-4072	465	9	in	in	ADP
ejpam-4072	465	10	topological	topological	ADJ
ejpam-4072	465	11	spaces	space	NOUN
ejpam-4072	465	12	.	.	PUNCT
ejpam-4072	466	1	amer	amer	PROPN
ejpam-4072	466	2	.	.	PUNCT
ejpam-4072	466	3	math	math	PROPN
ejpam-4072	466	4	.	.	PUNCT
ejpam-4072	467	1	monthly	monthly	ADJ
ejpam-4072	467	2	,	,	PUNCT
ejpam-4072	467	3	70:36–41	70:36–41	NUM
ejpam-4072	467	4	,	,	PUNCT
ejpam-4072	467	5	1963	1963	NUM
ejpam-4072	467	6	.	.	PUNCT
ejpam-4072	468	1	[	[	X
ejpam-4072	468	2	19	19	NUM
ejpam-4072	468	3	]	]	PUNCT
ejpam-4072	468	4	a.	a.	NOUN
ejpam-4072	468	5	s.	s.	PROPN
ejpam-4072	468	6	mashhour	mashhour	PROPN
ejpam-4072	468	7	,	,	PUNCT
ejpam-4072	468	8	m.	m.	PROPN
ejpam-4072	468	9	e.	e.	PROPN
ejpam-4072	468	10	abd	abd	PROPN
ejpam-4072	468	11	el	el	PROPN
ejpam-4072	468	12	-	-	PROPN
ejpam-4072	468	13	monsef	monsef	ADJ
ejpam-4072	468	14	,	,	PUNCT
ejpam-4072	468	15	and	and	CCONJ
ejpam-4072	468	16	s.	s.	PROPN
ejpam-4072	468	17	n.	n.	PROPN
ejpam-4072	468	18	el	el	PROPN
ejpam-4072	468	19	-	-	PROPN
ejpam-4072	468	20	deeb	deeb	PROPN
ejpam-4072	468	21	.	.	PUNCT
ejpam-4072	469	1	on	on	ADP
ejpam-4072	469	2	precontinuous	precontinuous	ADJ
ejpam-4072	469	3	and	and	CCONJ
ejpam-4072	469	4	weak	weak	ADJ
ejpam-4072	469	5	precontinuous	precontinuous	ADJ
ejpam-4072	469	6	mappings	mapping	NOUN
ejpam-4072	469	7	.	.	PUNCT
ejpam-4072	470	1	proc	proc	NOUN
ejpam-4072	470	2	.	.	PUNCT
ejpam-4072	471	1	phys	phy	NOUN
ejpam-4072	471	2	.	.	PUNCT
ejpam-4072	472	1	soc	soc	PROPN
ejpam-4072	472	2	.	.	PUNCT
ejpam-4072	473	1	egypt	egypt	PROPN
ejpam-4072	473	2	,	,	PUNCT
ejpam-4072	473	3	53:47–53	53:47–53	NUM
ejpam-4072	473	4	,	,	PUNCT
ejpam-4072	473	5	1982	1982	NUM
ejpam-4072	473	6	.	.	PUNCT
ejpam-4072	474	1	[	[	X
ejpam-4072	474	2	20	20	NUM
ejpam-4072	474	3	]	]	X
ejpam-4072	474	4	o.	o.	NOUN
ejpam-4072	474	5	nj̊astad	nj̊astad	NOUN
ejpam-4072	474	6	.	.	PUNCT
ejpam-4072	475	1	on	on	ADP
ejpam-4072	475	2	some	some	DET
ejpam-4072	475	3	classes	class	NOUN
ejpam-4072	475	4	of	of	ADP
ejpam-4072	475	5	nearly	nearly	ADV
ejpam-4072	475	6	open	open	ADJ
ejpam-4072	475	7	sets	set	NOUN
ejpam-4072	475	8	.	.	PUNCT
ejpam-4072	476	1	pacific	pacific	PROPN
ejpam-4072	476	2	j.	j.	PROPN
ejpam-4072	476	3	math	math	PROPN
ejpam-4072	476	4	.	.	PUNCT
ejpam-4072	476	5	,	,	PUNCT
ejpam-4072	476	6	15(3):961–970	15(3):961–970	PROPN
ejpam-4072	476	7	,	,	PUNCT
ejpam-4072	476	8	1965	1965	NUM
ejpam-4072	476	9	.	.	PUNCT
ejpam-4072	477	1	[	[	X
ejpam-4072	477	2	21	21	NUM
ejpam-4072	477	3	]	]	PUNCT
ejpam-4072	477	4	t.	t.	PROPN
ejpam-4072	477	5	noiri	noiri	PROPN
ejpam-4072	477	6	.	.	PUNCT
ejpam-4072	478	1	properties	property	NOUN
ejpam-4072	478	2	of	of	ADP
ejpam-4072	478	3	some	some	DET
ejpam-4072	478	4	weak	weak	ADJ
ejpam-4072	478	5	forms	form	NOUN
ejpam-4072	478	6	of	of	ADP
ejpam-4072	478	7	continuity	continuity	NOUN
ejpam-4072	478	8	.	.	PUNCT
ejpam-4072	479	1	internat	internat	PROPN
ejpam-4072	479	2	.	.	PUNCT
ejpam-4072	480	1	j.	j.	PROPN
ejpam-4072	480	2	math	math	PROPN
ejpam-4072	480	3	.	.	PUNCT
ejpam-4072	481	1	math	math	NOUN
ejpam-4072	481	2	.	.	PUNCT
ejpam-4072	482	1	sci	sci	PROPN
ejpam-4072	482	2	.	.	PROPN
ejpam-4072	482	3	,	,	PUNCT
ejpam-4072	482	4	10(1):97–111	10(1):97–111	NUM
ejpam-4072	482	5	,	,	PUNCT
ejpam-4072	482	6	1987	1987	NUM
ejpam-4072	482	7	.	.	PUNCT
ejpam-4072	483	1	[	[	X
ejpam-4072	483	2	22	22	NUM
ejpam-4072	483	3	]	]	PUNCT
ejpam-4072	483	4	t.	t.	PROPN
ejpam-4072	483	5	noiri	noiri	PROPN
ejpam-4072	483	6	and	and	CCONJ
ejpam-4072	483	7	v.	v.	ADP
ejpam-4072	483	8	popa	popa	NOUN
ejpam-4072	483	9	.	.	PUNCT
ejpam-4072	484	1	on	on	ADP
ejpam-4072	484	2	weakly	weakly	ADJ
ejpam-4072	484	3	precontinuous	precontinuous	ADJ
ejpam-4072	484	4	functions	function	NOUN
ejpam-4072	484	5	in	in	ADP
ejpam-4072	484	6	bitopological	bitopological	ADJ
ejpam-4072	484	7	spaces	space	NOUN
ejpam-4072	484	8	.	.	PUNCT
ejpam-4072	485	1	soochow	soochow	PROPN
ejpam-4072	485	2	j.	j.	PROPN
ejpam-4072	485	3	of	of	ADP
ejpam-4072	485	4	math	math	PROPN
ejpam-4072	485	5	.	.	PUNCT
ejpam-4072	485	6	,	,	PUNCT
ejpam-4072	485	7	33(1):87–100	33(1):87–100	NUM
ejpam-4072	485	8	,	,	PUNCT
ejpam-4072	485	9	2007	2007	NUM
ejpam-4072	485	10	.	.	PUNCT
ejpam-4072	486	1	[	[	X
ejpam-4072	486	2	23	23	NUM
ejpam-4072	486	3	]	]	X
ejpam-4072	486	4	v.	v.	CCONJ
ejpam-4072	486	5	popa	popa	NOUN
ejpam-4072	486	6	.	.	PUNCT
ejpam-4072	487	1	on	on	ADP
ejpam-4072	487	2	certain	certain	ADJ
ejpam-4072	487	3	properties	property	NOUN
ejpam-4072	487	4	of	of	ADP
ejpam-4072	487	5	quasi	quasi	NOUN
ejpam-4072	487	6	continuous	continuous	ADJ
ejpam-4072	487	7	and	and	CCONJ
ejpam-4072	487	8	almost	almost	ADV
ejpam-4072	487	9	continuous	continuous	ADJ
ejpam-4072	487	10	multifunctions	multifunction	NOUN
ejpam-4072	487	11	(	(	PUNCT
ejpam-4072	487	12	romanian	romanian	ADJ
ejpam-4072	487	13	)	)	PUNCT
ejpam-4072	487	14	.	.	PUNCT
ejpam-4072	488	1	stud	stud	PROPN
ejpam-4072	488	2	.	.	PUNCT
ejpam-4072	489	1	cerc	cerc	PROPN
ejpam-4072	489	2	.	.	PUNCT
ejpam-4072	490	1	mat	mat	NOUN
ejpam-4072	490	2	.	.	PROPN
ejpam-4072	490	3	,	,	PUNCT
ejpam-4072	490	4	30:441–446	30:441–446	NUM
ejpam-4072	490	5	,	,	PUNCT
ejpam-4072	490	6	1978	1978	NUM
ejpam-4072	490	7	.	.	PUNCT
ejpam-4072	491	1	[	[	X
ejpam-4072	491	2	24	24	NUM
ejpam-4072	491	3	]	]	PUNCT
ejpam-4072	491	4	v.	v.	CCONJ
ejpam-4072	491	5	popa	popa	NOUN
ejpam-4072	491	6	.	.	PUNCT
ejpam-4072	492	1	weakly	weakly	ADJ
ejpam-4072	492	2	continuous	continuous	ADJ
ejpam-4072	492	3	multifunctions	multifunction	NOUN
ejpam-4072	492	4	.	.	PUNCT
ejpam-4072	492	5	.	.	PUNCT
ejpam-4072	493	1	boll	boll	PROPN
ejpam-4072	493	2	.	.	PUNCT
ejpam-4072	494	1	un	un	PROPN
ejpam-4072	494	2	.	.	PROPN
ejpam-4072	494	3	mat	mat	PROPN
ejpam-4072	494	4	.	.	PUNCT
ejpam-4072	494	5	ital	ital	PROPN
ejpam-4072	494	6	.	.	PUNCT
ejpam-4072	495	1	(	(	PUNCT
ejpam-4072	495	2	5	5	NUM
ejpam-4072	495	3	)	)	PUNCT
ejpam-4072	495	4	,	,	PUNCT
ejpam-4072	495	5	15	15	NUM
ejpam-4072	495	6	-	-	SYM
ejpam-4072	495	7	a:379–388	a:379–388	PRON
ejpam-4072	495	8	,	,	PUNCT
ejpam-4072	495	9	1978	1978	NUM
ejpam-4072	495	10	.	.	PUNCT
ejpam-4072	496	1	[	[	X
ejpam-4072	496	2	25	25	NUM
ejpam-4072	496	3	]	]	PUNCT
ejpam-4072	496	4	v.	v.	CCONJ
ejpam-4072	496	5	popa	popa	NOUN
ejpam-4072	496	6	and	and	CCONJ
ejpam-4072	496	7	t.	t.	PROPN
ejpam-4072	496	8	noiri	noiri	PROPN
ejpam-4072	496	9	.	.	PUNCT
ejpam-4072	497	1	almost	almost	ADV
ejpam-4072	497	2	weakly	weakly	ADJ
ejpam-4072	497	3	continuous	continuous	ADJ
ejpam-4072	497	4	functions	function	NOUN
ejpam-4072	497	5	.	.	PUNCT
ejpam-4072	498	1	demonstratio	demonstratio	PROPN
ejpam-4072	498	2	math	math	PROPN
ejpam-4072	498	3	.	.	PUNCT
ejpam-4072	498	4	,	,	PUNCT
ejpam-4072	498	5	25:241–251	25:241–251	NUM
ejpam-4072	498	6	,	,	PUNCT
ejpam-4072	498	7	1992	1992	NUM
ejpam-4072	498	8	.	.	PUNCT
ejpam-4072	499	1	[	[	X
ejpam-4072	499	2	26	26	NUM
ejpam-4072	499	3	]	]	PUNCT
ejpam-4072	499	4	v.	v.	CCONJ
ejpam-4072	499	5	popa	popa	NOUN
ejpam-4072	499	6	and	and	CCONJ
ejpam-4072	499	7	t.	t.	PROPN
ejpam-4072	499	8	noiri	noiri	PROPN
ejpam-4072	499	9	.	.	PUNCT
ejpam-4072	500	1	on	on	ADP
ejpam-4072	500	2	upper	upper	ADJ
ejpam-4072	500	3	and	and	CCONJ
ejpam-4072	500	4	lower	low	ADJ
ejpam-4072	500	5	α	α	ADJ
ejpam-4072	500	6	-	-	ADJ
ejpam-4072	500	7	continuous	continuous	ADJ
ejpam-4072	500	8	multifunctions	multifunction	NOUN
ejpam-4072	500	9	.	.	PUNCT
ejpam-4072	500	10	math	math	NOUN
ejpam-4072	500	11	.	.	PUNCT
ejpam-4072	501	1	slovaca	slovaca	PROPN
ejpam-4072	501	2	,	,	PUNCT
ejpam-4072	501	3	43(4):477–491	43(4):477–491	NOUN
ejpam-4072	501	4	,	,	PUNCT
ejpam-4072	501	5	1993	1993	NUM
ejpam-4072	501	6	.	.	PUNCT
ejpam-4072	502	1	[	[	X
ejpam-4072	502	2	27	27	NUM
ejpam-4072	502	3	]	]	X
ejpam-4072	502	4	r.	r.	PROPN
ejpam-4072	502	5	e.	e.	PROPN
ejpam-4072	502	6	smithson	smithson	PROPN
ejpam-4072	502	7	.	.	PUNCT
ejpam-4072	503	1	almost	almost	ADV
ejpam-4072	503	2	and	and	CCONJ
ejpam-4072	503	3	weak	weak	ADJ
ejpam-4072	503	4	continuity	continuity	NOUN
ejpam-4072	503	5	for	for	ADP
ejpam-4072	503	6	multifunctions	multifunction	NOUN
ejpam-4072	503	7	.	.	PUNCT
ejpam-4072	504	1	bull	bull	NOUN
ejpam-4072	504	2	.	.	PUNCT
ejpam-4072	505	1	calcutta	calcutta	PROPN
ejpam-4072	505	2	math	math	PROPN
ejpam-4072	505	3	.	.	PUNCT
ejpam-4072	506	1	soc	soc	PROPN
ejpam-4072	506	2	.	.	PUNCT
ejpam-4072	506	3	,	,	PUNCT
ejpam-4072	506	4	70:383–390	70:383–390	NUM
ejpam-4072	506	5	,	,	PUNCT
ejpam-4072	506	6	1978	1978	NUM
ejpam-4072	506	7	.	.	PUNCT
ejpam-4072	507	1	[	[	X
ejpam-4072	507	2	28	28	NUM
ejpam-4072	507	3	]	]	X
ejpam-4072	507	4	c.	c.	PROPN
ejpam-4072	507	5	viriyapong	viriyapong	PROPN
ejpam-4072	507	6	and	and	CCONJ
ejpam-4072	507	7	c.	c.	PROPN
ejpam-4072	507	8	boonpok	boonpok	PROPN
ejpam-4072	507	9	.	.	PUNCT
ejpam-4072	508	1	(	(	PUNCT
ejpam-4072	508	2	τ1	τ1	NOUN
ejpam-4072	508	3	,	,	PUNCT
ejpam-4072	508	4	τ2)α	τ2)α	NOUN
ejpam-4072	508	5	-	-	PUNCT
ejpam-4072	508	6	continuity	continuity	NOUN
ejpam-4072	508	7	for	for	ADP
ejpam-4072	508	8	multifunctions	multifunction	NOUN
ejpam-4072	508	9	.	.	PUNCT
ejpam-4072	509	1	j.	j.	PROPN
ejpam-4072	509	2	math	math	PROPN
ejpam-4072	509	3	.	.	PROPN
ejpam-4072	509	4	,	,	PUNCT
ejpam-4072	509	5	2020:628763	2020:628763	NUM
ejpam-4072	509	6	,	,	PUNCT
ejpam-4072	509	7	2020	2020	NUM
ejpam-4072	509	8	.	.	PUNCT