id	sid	tid	token	lemma	pos
ejpam-4976	1	1	european	european	PROPN
ejpam-4976	1	2	journal	journal	PROPN
ejpam-4976	1	3	of	of	ADP
ejpam-4976	1	4	pure	pure	ADJ
ejpam-4976	1	5	and	and	CCONJ
ejpam-4976	1	6	applied	apply	VERB
ejpam-4976	1	7	mathematics	mathematic	NOUN
ejpam-4976	1	8	vol	vol	NOUN
ejpam-4976	1	9	.	.	PROPN
ejpam-4976	2	1	17	17	NUM
ejpam-4976	2	2	,	,	PUNCT
ejpam-4976	2	3	no	no	INTJ
ejpam-4976	2	4	.	.	NOUN
ejpam-4976	2	5	1	1	NUM
ejpam-4976	2	6	,	,	PUNCT
ejpam-4976	2	7	2024	2024	NUM
ejpam-4976	2	8	,	,	PUNCT
ejpam-4976	2	9	416	416	NUM
ejpam-4976	2	10	-	-	SYM
ejpam-4976	2	11	425	425	NUM
ejpam-4976	2	12	issn	issn	PROPN
ejpam-4976	2	13	1307	1307	NUM
ejpam-4976	2	14	-	-	SYM
ejpam-4976	2	15	5543	5543	NUM
ejpam-4976	2	16	–	–	PUNCT
ejpam-4976	3	1	ejpam.com	ejpam.com	X
ejpam-4976	3	2	published	publish	VERB
ejpam-4976	3	3	by	by	ADP
ejpam-4976	3	4	new	new	PROPN
ejpam-4976	3	5	york	york	PROPN
ejpam-4976	3	6	business	business	PROPN
ejpam-4976	3	7	global	global	ADJ
ejpam-4976	3	8	on	on	ADP
ejpam-4976	3	9	weakly	weakly	ADJ
ejpam-4976	3	10	(	(	PUNCT
ejpam-4976	3	11	τ1	τ1	NOUN
ejpam-4976	3	12	,	,	PUNCT
ejpam-4976	3	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	3	14	functions	function	NOUN
ejpam-4976	3	15	chawalit	chawalit	VERB
ejpam-4976	3	16	boonpok1	boonpok1	PROPN
ejpam-4976	3	17	,	,	PUNCT
ejpam-4976	3	18	chalongchai	chalongchai	PROPN
ejpam-4976	3	19	klanarong1,∗	klanarong1,∗	PROPN
ejpam-4976	3	20	1	1	NUM
ejpam-4976	3	21	mathematics	mathematic	NOUN
ejpam-4976	3	22	and	and	CCONJ
ejpam-4976	3	23	applied	apply	VERB
ejpam-4976	3	24	mathematics	mathematics	PROPN
ejpam-4976	3	25	research	research	NOUN
ejpam-4976	3	26	unit	unit	NOUN
ejpam-4976	3	27	,	,	PUNCT
ejpam-4976	3	28	department	department	NOUN
ejpam-4976	3	29	of	of	ADP
ejpam-4976	3	30	mathematics	mathematic	NOUN
ejpam-4976	3	31	,	,	PUNCT
ejpam-4976	3	32	faculty	faculty	NOUN
ejpam-4976	3	33	of	of	ADP
ejpam-4976	3	34	science	science	NOUN
ejpam-4976	3	35	,	,	PUNCT
ejpam-4976	3	36	mahasarakham	mahasarakham	PROPN
ejpam-4976	3	37	university	university	PROPN
ejpam-4976	3	38	,	,	PUNCT
ejpam-4976	3	39	maha	maha	PROPN
ejpam-4976	3	40	sarakham	sarakham	PROPN
ejpam-4976	3	41	,	,	PUNCT
ejpam-4976	3	42	44150	44150	NUM
ejpam-4976	3	43	,	,	PUNCT
ejpam-4976	3	44	thailand	thailand	PROPN
ejpam-4976	3	45	abstract	abstract	PROPN
ejpam-4976	3	46	.	.	PUNCT
ejpam-4976	4	1	our	our	PRON
ejpam-4976	4	2	main	main	ADJ
ejpam-4976	4	3	purpose	purpose	NOUN
ejpam-4976	4	4	is	be	AUX
ejpam-4976	4	5	to	to	PART
ejpam-4976	4	6	introduce	introduce	VERB
ejpam-4976	4	7	the	the	DET
ejpam-4976	4	8	concept	concept	NOUN
ejpam-4976	4	9	of	of	ADP
ejpam-4976	4	10	weakly	weakly	ADJ
ejpam-4976	4	11	(	(	PUNCT
ejpam-4976	4	12	τ1	τ1	NOUN
ejpam-4976	4	13	,	,	PUNCT
ejpam-4976	4	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	4	15	functions	function	NOUN
ejpam-4976	4	16	.	.	PUNCT
ejpam-4976	5	1	moreover	moreover	ADV
ejpam-4976	5	2	,	,	PUNCT
ejpam-4976	5	3	several	several	ADJ
ejpam-4976	5	4	characterizations	characterization	NOUN
ejpam-4976	5	5	of	of	ADP
ejpam-4976	5	6	weakly	weakly	ADJ
ejpam-4976	5	7	(	(	PUNCT
ejpam-4976	5	8	τ1	τ1	NOUN
ejpam-4976	5	9	,	,	PUNCT
ejpam-4976	5	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	5	11	functions	function	NOUN
ejpam-4976	5	12	are	be	AUX
ejpam-4976	5	13	considered	consider	VERB
ejpam-4976	5	14	.	.	PUNCT
ejpam-4976	6	1	2020	2020	NUM
ejpam-4976	6	2	mathematics	mathematic	NOUN
ejpam-4976	6	3	subject	subject	NOUN
ejpam-4976	6	4	classifications	classification	NOUN
ejpam-4976	6	5	:	:	PUNCT
ejpam-4976	6	6	54c08	54c08	NUM
ejpam-4976	6	7	,	,	PUNCT
ejpam-4976	6	8	54e55	54e55	NUM
ejpam-4976	6	9	key	key	ADJ
ejpam-4976	6	10	words	word	NOUN
ejpam-4976	6	11	and	and	CCONJ
ejpam-4976	6	12	phrases	phrase	NOUN
ejpam-4976	6	13	:	:	PUNCT
ejpam-4976	6	14	τ1τ2	τ1τ2	ADJ
ejpam-4976	6	15	-	-	ADJ
ejpam-4976	6	16	open	open	ADJ
ejpam-4976	6	17	set	set	NOUN
ejpam-4976	6	18	,	,	PUNCT
ejpam-4976	6	19	weakly	weakly	ADJ
ejpam-4976	6	20	(	(	PUNCT
ejpam-4976	6	21	τ1	τ1	NOUN
ejpam-4976	6	22	,	,	PUNCT
ejpam-4976	6	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	6	24	function	function	NOUN
ejpam-4976	6	25	1	1	NUM
ejpam-4976	6	26	.	.	PUNCT
ejpam-4976	6	27	introduction	introduction	NOUN
ejpam-4976	6	28	in	in	ADP
ejpam-4976	6	29	1961	1961	NUM
ejpam-4976	6	30	,	,	PUNCT
ejpam-4976	6	31	levine	levine	PROPN
ejpam-4976	7	1	[	[	X
ejpam-4976	7	2	10	10	NUM
ejpam-4976	7	3	]	]	PUNCT
ejpam-4976	7	4	introduced	introduce	VERB
ejpam-4976	7	5	the	the	DET
ejpam-4976	7	6	concept	concept	NOUN
ejpam-4976	7	7	of	of	ADP
ejpam-4976	7	8	weakly	weakly	ADJ
ejpam-4976	7	9	continuous	continuous	ADJ
ejpam-4976	7	10	functions	function	NOUN
ejpam-4976	7	11	.	.	PUNCT
ejpam-4976	8	1	moreover	moreover	ADV
ejpam-4976	8	2	,	,	PUNCT
ejpam-4976	8	3	levine	levine	PROPN
ejpam-4976	8	4	[	[	X
ejpam-4976	8	5	11	11	NUM
ejpam-4976	8	6	]	]	PUNCT
ejpam-4976	8	7	introduced	introduce	VERB
ejpam-4976	8	8	the	the	DET
ejpam-4976	8	9	notion	notion	NOUN
ejpam-4976	8	10	of	of	ADP
ejpam-4976	8	11	semi	semi	ADJ
ejpam-4976	8	12	-	-	ADJ
ejpam-4976	8	13	continuous	continuous	ADJ
ejpam-4976	8	14	functions	function	NOUN
ejpam-4976	8	15	.	.	PUNCT
ejpam-4976	9	1	neubrunnová	neubrunnová	PROPN
ejpam-4976	10	1	[	[	X
ejpam-4976	10	2	13	13	NUM
ejpam-4976	10	3	]	]	PUNCT
ejpam-4976	10	4	showed	show	VERB
ejpam-4976	10	5	that	that	SCONJ
ejpam-4976	10	6	semi	semi	ADJ
ejpam-4976	10	7	-	-	NOUN
ejpam-4976	10	8	continuity	continuity	NOUN
ejpam-4976	10	9	is	be	AUX
ejpam-4976	10	10	equivalent	equivalent	ADJ
ejpam-4976	10	11	to	to	ADP
ejpam-4976	10	12	quasi	quasi	NOUN
ejpam-4976	10	13	-	-	NOUN
ejpam-4976	10	14	continuity	continuity	NOUN
ejpam-4976	10	15	due	due	ADP
ejpam-4976	10	16	to	to	ADP
ejpam-4976	10	17	marcus	marcus	PROPN
ejpam-4976	10	18	[	[	X
ejpam-4976	10	19	12	12	NUM
ejpam-4976	10	20	]	]	PUNCT
ejpam-4976	10	21	.	.	PUNCT
ejpam-4976	11	1	in	in	ADP
ejpam-4976	11	2	1973	1973	NUM
ejpam-4976	11	3	,	,	PUNCT
ejpam-4976	11	4	popa	popa	NOUN
ejpam-4976	11	5	and	and	CCONJ
ejpam-4976	11	6	stan	stan	PROPN
ejpam-4976	11	7	[	[	X
ejpam-4976	11	8	17	17	NUM
ejpam-4976	11	9	]	]	PUNCT
ejpam-4976	11	10	introduced	introduce	VERB
ejpam-4976	11	11	and	and	CCONJ
ejpam-4976	11	12	studied	study	VERB
ejpam-4976	11	13	the	the	DET
ejpam-4976	11	14	concept	concept	NOUN
ejpam-4976	11	15	of	of	ADP
ejpam-4976	11	16	weakly	weakly	ADJ
ejpam-4976	11	17	quasi	quasi	ADJ
ejpam-4976	11	18	-	-	ADJ
ejpam-4976	11	19	continuous	continuous	ADJ
ejpam-4976	11	20	functions	function	NOUN
ejpam-4976	11	21	.	.	PUNCT
ejpam-4976	12	1	weak	weak	ADJ
ejpam-4976	12	2	quasi	quasi	NOUN
ejpam-4976	12	3	-	-	NOUN
ejpam-4976	12	4	continuity	continuity	NOUN
ejpam-4976	12	5	is	be	AUX
ejpam-4976	12	6	implied	imply	VERB
ejpam-4976	12	7	by	by	ADP
ejpam-4976	12	8	both	both	DET
ejpam-4976	12	9	quasi	quasi	ADJ
ejpam-4976	12	10	-	-	NOUN
ejpam-4976	12	11	continuity	continuity	NOUN
ejpam-4976	12	12	and	and	CCONJ
ejpam-4976	12	13	weak	weak	ADJ
ejpam-4976	12	14	continuity	continuity	NOUN
ejpam-4976	12	15	which	which	PRON
ejpam-4976	12	16	are	be	AUX
ejpam-4976	12	17	independent	independent	ADJ
ejpam-4976	12	18	of	of	ADP
ejpam-4976	12	19	each	each	DET
ejpam-4976	12	20	other	other	ADJ
ejpam-4976	12	21	.	.	PUNCT
ejpam-4976	13	1	in	in	ADP
ejpam-4976	13	2	1984	1984	NUM
ejpam-4976	13	3	,	,	PUNCT
ejpam-4976	13	4	rose	rise	VERB
ejpam-4976	13	5	[	[	X
ejpam-4976	13	6	18	18	NUM
ejpam-4976	13	7	]	]	PUNCT
ejpam-4976	13	8	introduced	introduce	VERB
ejpam-4976	13	9	the	the	DET
ejpam-4976	13	10	notion	notion	NOUN
ejpam-4976	13	11	of	of	ADP
ejpam-4976	13	12	subweakly	subweakly	ADJ
ejpam-4976	13	13	continuous	continuous	ADJ
ejpam-4976	13	14	functions	function	NOUN
ejpam-4976	13	15	and	and	CCONJ
ejpam-4976	13	16	investigated	investigate	VERB
ejpam-4976	13	17	the	the	DET
ejpam-4976	13	18	relationships	relationship	NOUN
ejpam-4976	13	19	between	between	ADP
ejpam-4976	13	20	subweak	subweak	NOUN
ejpam-4976	13	21	continuity	continuity	NOUN
ejpam-4976	13	22	and	and	CCONJ
ejpam-4976	13	23	weak	weak	ADJ
ejpam-4976	13	24	continuity	continuity	NOUN
ejpam-4976	13	25	.	.	PUNCT
ejpam-4976	14	1	noiri	noiri	PROPN
ejpam-4976	15	1	[	[	X
ejpam-4976	15	2	14	14	NUM
ejpam-4976	15	3	]	]	PUNCT
ejpam-4976	15	4	studied	study	VERB
ejpam-4976	15	5	properties	property	NOUN
ejpam-4976	15	6	of	of	ADP
ejpam-4976	15	7	some	some	DET
ejpam-4976	15	8	weak	weak	ADJ
ejpam-4976	15	9	forms	form	NOUN
ejpam-4976	15	10	of	of	ADP
ejpam-4976	15	11	continuity	continuity	NOUN
ejpam-4976	15	12	.	.	PUNCT
ejpam-4976	16	1	in	in	ADP
ejpam-4976	16	2	2002	2002	NUM
ejpam-4976	16	3	,	,	PUNCT
ejpam-4976	16	4	popa	popa	NOUN
ejpam-4976	16	5	and	and	CCONJ
ejpam-4976	16	6	noiri	noiri	ADV
ejpam-4976	16	7	[	[	X
ejpam-4976	16	8	16	16	NUM
ejpam-4976	16	9	]	]	PUNCT
ejpam-4976	16	10	introduced	introduce	VERB
ejpam-4976	16	11	the	the	DET
ejpam-4976	16	12	concept	concept	NOUN
ejpam-4976	16	13	of	of	ADP
ejpam-4976	16	14	weakly	weakly	ADJ
ejpam-4976	16	15	(	(	PUNCT
ejpam-4976	16	16	τ	τ	PROPN
ejpam-4976	16	17	,	,	PUNCT
ejpam-4976	16	18	m)-continuous	m)-continuous	ADJ
ejpam-4976	16	19	functions	function	NOUN
ejpam-4976	16	20	as	as	ADP
ejpam-4976	16	21	functions	function	NOUN
ejpam-4976	16	22	from	from	ADP
ejpam-4976	16	23	a	a	DET
ejpam-4976	16	24	topological	topological	ADJ
ejpam-4976	16	25	space	space	NOUN
ejpam-4976	16	26	into	into	ADP
ejpam-4976	16	27	a	a	DET
ejpam-4976	16	28	set	set	NOUN
ejpam-4976	16	29	satisfying	satisfy	VERB
ejpam-4976	16	30	some	some	DET
ejpam-4976	16	31	minimal	minimal	ADJ
ejpam-4976	16	32	conditions	condition	NOUN
ejpam-4976	16	33	and	and	CCONJ
ejpam-4976	16	34	investigated	investigate	VERB
ejpam-4976	16	35	several	several	ADJ
ejpam-4976	16	36	characterizations	characterization	NOUN
ejpam-4976	16	37	of	of	ADP
ejpam-4976	16	38	weakly	weakly	ADJ
ejpam-4976	16	39	(	(	PUNCT
ejpam-4976	16	40	τ	τ	PROPN
ejpam-4976	16	41	,	,	PUNCT
ejpam-4976	16	42	m)-continuous	m)-continuous	ADJ
ejpam-4976	16	43	functions	function	NOUN
ejpam-4976	16	44	.	.	PUNCT
ejpam-4976	17	1	popa	popa	NOUN
ejpam-4976	17	2	and	and	CCONJ
ejpam-4976	17	3	noiri	noiri	ADV
ejpam-4976	18	1	[	[	X
ejpam-4976	18	2	15	15	NUM
ejpam-4976	18	3	]	]	PUNCT
ejpam-4976	18	4	introduced	introduce	VERB
ejpam-4976	18	5	and	and	CCONJ
ejpam-4976	18	6	investigated	investigate	VERB
ejpam-4976	18	7	the	the	DET
ejpam-4976	18	8	notion	notion	NOUN
ejpam-4976	18	9	of	of	ADP
ejpam-4976	18	10	weakly	weakly	ADJ
ejpam-4976	18	11	m	m	VERB
ejpam-4976	18	12	-continuous	-continuous	ADJ
ejpam-4976	18	13	functions	function	NOUN
ejpam-4976	18	14	as	as	ADP
ejpam-4976	18	15	functions	function	NOUN
ejpam-4976	18	16	from	from	ADP
ejpam-4976	18	17	a	a	DET
ejpam-4976	18	18	set	set	NOUN
ejpam-4976	18	19	satisfying	satisfy	VERB
ejpam-4976	18	20	some	some	DET
ejpam-4976	18	21	minimal	minimal	ADJ
ejpam-4976	18	22	conditions	condition	NOUN
ejpam-4976	18	23	into	into	ADP
ejpam-4976	18	24	a	a	DET
ejpam-4976	18	25	set	set	NOUN
ejpam-4976	18	26	satisfying	satisfy	VERB
ejpam-4976	18	27	some	some	DET
ejpam-4976	18	28	minimal	minimal	ADJ
ejpam-4976	18	29	conditions	condition	NOUN
ejpam-4976	18	30	.	.	PUNCT
ejpam-4976	19	1	in	in	ADP
ejpam-4976	19	2	2008	2008	NUM
ejpam-4976	19	3	,	,	PUNCT
ejpam-4976	19	4	ekici	ekici	NOUN
ejpam-4976	19	5	et	et	PROPN
ejpam-4976	19	6	al	al	PROPN
ejpam-4976	19	7	.	.	PUNCT
ejpam-4976	20	1	[	[	X
ejpam-4976	20	2	8	8	NUM
ejpam-4976	20	3	]	]	PUNCT
ejpam-4976	20	4	introduced	introduce	VERB
ejpam-4976	20	5	a	a	DET
ejpam-4976	20	6	new	new	ADJ
ejpam-4976	20	7	class	class	NOUN
ejpam-4976	20	8	of	of	ADP
ejpam-4976	20	9	functions	function	NOUN
ejpam-4976	20	10	called	call	VERB
ejpam-4976	20	11	weakly	weakly	ADJ
ejpam-4976	20	12	λ	λ	ADJ
ejpam-4976	20	13	-	-	ADJ
ejpam-4976	20	14	continuous	continuous	ADJ
ejpam-4976	20	15	functions	function	NOUN
ejpam-4976	20	16	which	which	PRON
ejpam-4976	20	17	is	be	AUX
ejpam-4976	20	18	weaker	weak	ADJ
ejpam-4976	20	19	than	than	ADP
ejpam-4976	20	20	λ	λ	NOUN
ejpam-4976	20	21	-	-	ADJ
ejpam-4976	20	22	continuous	continuous	ADJ
ejpam-4976	20	23	functions	function	NOUN
ejpam-4976	20	24	and	and	CCONJ
ejpam-4976	20	25	studied	study	VERB
ejpam-4976	20	26	some	some	DET
ejpam-4976	20	27	fundamental	fundamental	ADJ
ejpam-4976	20	28	properties	property	NOUN
ejpam-4976	20	29	of	of	ADP
ejpam-4976	20	30	weakly	weakly	ADJ
ejpam-4976	20	31	λ	λ	ADJ
ejpam-4976	20	32	-	-	ADJ
ejpam-4976	20	33	continuous	continuous	ADJ
ejpam-4976	20	34	functions	function	NOUN
ejpam-4976	20	35	.	.	PUNCT
ejpam-4976	21	1	in	in	ADP
ejpam-4976	21	2	[	[	X
ejpam-4976	21	3	3	3	NUM
ejpam-4976	21	4	]	]	PUNCT
ejpam-4976	21	5	,	,	PUNCT
ejpam-4976	21	6	the	the	DET
ejpam-4976	21	7	present	present	ADJ
ejpam-4976	21	8	author	author	NOUN
ejpam-4976	21	9	introduced	introduce	VERB
ejpam-4976	21	10	the	the	DET
ejpam-4976	21	11	concept	concept	NOUN
ejpam-4976	21	12	of	of	ADP
ejpam-4976	21	13	weakly	weakly	ADJ
ejpam-4976	21	14	⋆-continuous	⋆-continuous	ADJ
ejpam-4976	21	15	functions	function	NOUN
ejpam-4976	21	16	and	and	CCONJ
ejpam-4976	21	17	established	establish	VERB
ejpam-4976	21	18	the	the	DET
ejpam-4976	21	19	relationships	relationship	NOUN
ejpam-4976	21	20	between	between	ADP
ejpam-4976	21	21	weak	weak	ADJ
ejpam-4976	21	22	⋆-continuity	⋆-continuity	NOUN
ejpam-4976	21	23	and	and	CCONJ
ejpam-4976	21	24	θ(⋆)-continuity	θ(⋆)-continuity	NOUN
ejpam-4976	21	25	.	.	PUNCT
ejpam-4976	22	1	in	in	ADP
ejpam-4976	22	2	2010	2010	NUM
ejpam-4976	22	3	,	,	PUNCT
ejpam-4976	22	4	boonpok	boonpok	X
ejpam-4976	22	5	[	[	X
ejpam-4976	22	6	1	1	NUM
ejpam-4976	22	7	]	]	PUNCT
ejpam-4976	22	8	introduced	introduce	VERB
ejpam-4976	22	9	and	and	CCONJ
ejpam-4976	22	10	studied	study	VERB
ejpam-4976	22	11	the	the	DET
ejpam-4976	22	12	concept	concept	NOUN
ejpam-4976	22	13	of	of	ADP
ejpam-4976	22	14	pairwise	pairwise	NOUN
ejpam-4976	22	15	weakly	weakly	ADJ
ejpam-4976	22	16	m	m	VERB
ejpam-4976	22	17	-continuous	-continuous	ADJ
ejpam-4976	22	18	functions	function	NOUN
ejpam-4976	22	19	in	in	ADP
ejpam-4976	22	20	bimininmal	bimininmal	ADJ
ejpam-4976	22	21	structure	structure	NOUN
ejpam-4976	22	22	spaces	space	NOUN
ejpam-4976	22	23	.	.	PUNCT
ejpam-4976	23	1	viriyapong	viriyapong	PROPN
ejpam-4976	23	2	and	and	CCONJ
ejpam-4976	23	3	boonpok	boonpok	VERB
ejpam-4976	23	4	[	[	X
ejpam-4976	23	5	20	20	NUM
ejpam-4976	23	6	]	]	PUNCT
ejpam-4976	23	7	introduced	introduce	VERB
ejpam-4976	23	8	and	and	CCONJ
ejpam-4976	23	9	investigated	investigate	VERB
ejpam-4976	23	10	the	the	DET
ejpam-4976	23	11	concept	concept	NOUN
ejpam-4976	23	12	of	of	ADP
ejpam-4976	23	13	(	(	PUNCT
ejpam-4976	23	14	λ	λ	PROPN
ejpam-4976	23	15	,	,	PUNCT
ejpam-4976	23	16	sp)-continuous	sp)-continuous	ADJ
ejpam-4976	23	17	functions	function	NOUN
ejpam-4976	23	18	.	.	PUNCT
ejpam-4976	24	1	∗corresponding	∗corresponde	VERB
ejpam-4976	24	2	author	author	NOUN
ejpam-4976	24	3	.	.	PUNCT
ejpam-4976	25	1	doi	doi	NOUN
ejpam-4976	25	2	:	:	PUNCT
ejpam-4976	25	3	https://doi.org/10.29020/nybg.ejpam.v17i1.4976	https://doi.org/10.29020/nybg.ejpam.v17i1.4976	NOUN
ejpam-4976	25	4	email	email	NOUN
ejpam-4976	25	5	addresses	address	NOUN
ejpam-4976	25	6	:	:	PUNCT
ejpam-4976	25	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4976	25	8	(	(	PUNCT
ejpam-4976	25	9	c.	c.	PROPN
ejpam-4976	25	10	boonpok	boonpok	PROPN
ejpam-4976	25	11	)	)	PUNCT
ejpam-4976	25	12	,	,	PUNCT
ejpam-4976	25	13	chalongchai.k@msu.ac.th	chalongchai.k@msu.ac.th	PROPN
ejpam-4976	25	14	(	(	PUNCT
ejpam-4976	25	15	c.	c.	PROPN
ejpam-4976	25	16	klanarong	klanarong	PROPN
ejpam-4976	25	17	)	)	PUNCT
ejpam-4976	25	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4976	26	1	416	416	NUM
ejpam-4976	26	2	©	©	ADP
ejpam-4976	26	3	2024	2024	NUM
ejpam-4976	26	4	ejpam	ejpam	NOUN
ejpam-4976	26	5	all	all	DET
ejpam-4976	26	6	rights	right	NOUN
ejpam-4976	26	7	reserved	reserve	VERB
ejpam-4976	26	8	.	.	PUNCT
ejpam-4976	27	1	c.	c.	PROPN
ejpam-4976	27	2	boonpok	boonpok	PROPN
ejpam-4976	27	3	,	,	PUNCT
ejpam-4976	27	4	c.	c.	PROPN
ejpam-4976	27	5	klanarong	klanarong	PROPN
ejpam-4976	27	6	/	/	SYM
ejpam-4976	27	7	eur	eur	PROPN
ejpam-4976	27	8	.	.	PUNCT
ejpam-4976	28	1	j.	j.	PROPN
ejpam-4976	28	2	pure	pure	PROPN
ejpam-4976	28	3	appl	appl	PROPN
ejpam-4976	28	4	.	.	PROPN
ejpam-4976	28	5	math	math	PROPN
ejpam-4976	28	6	,	,	PUNCT
ejpam-4976	28	7	17	17	NUM
ejpam-4976	28	8	(	(	PUNCT
ejpam-4976	28	9	1	1	NUM
ejpam-4976	28	10	)	)	PUNCT
ejpam-4976	28	11	(	(	PUNCT
ejpam-4976	28	12	2024	2024	NUM
ejpam-4976	28	13	)	)	PUNCT
ejpam-4976	28	14	,	,	PUNCT
ejpam-4976	28	15	416	416	NUM
ejpam-4976	28	16	-	-	SYM
ejpam-4976	28	17	425	425	NUM
ejpam-4976	28	18	417	417	NUM
ejpam-4976	28	19	moreover	moreover	ADV
ejpam-4976	28	20	,	,	PUNCT
ejpam-4976	28	21	some	some	DET
ejpam-4976	28	22	characterizations	characterization	NOUN
ejpam-4976	28	23	of	of	ADP
ejpam-4976	28	24	almost	almost	ADV
ejpam-4976	28	25	(	(	PUNCT
ejpam-4976	28	26	λ	λ	PROPN
ejpam-4976	28	27	,	,	PUNCT
ejpam-4976	28	28	s)-continuous	s)-continuous	ADJ
ejpam-4976	28	29	functions	function	NOUN
ejpam-4976	28	30	were	be	AUX
ejpam-4976	28	31	presented	present	VERB
ejpam-4976	28	32	in	in	ADP
ejpam-4976	28	33	[	[	X
ejpam-4976	28	34	6	6	NUM
ejpam-4976	28	35	]	]	PUNCT
ejpam-4976	28	36	.	.	PUNCT
ejpam-4976	29	1	in	in	ADP
ejpam-4976	29	2	[	[	X
ejpam-4976	29	3	5	5	NUM
ejpam-4976	29	4	]	]	PUNCT
ejpam-4976	29	5	,	,	PUNCT
ejpam-4976	29	6	the	the	DET
ejpam-4976	29	7	authors	author	NOUN
ejpam-4976	29	8	introduced	introduce	VERB
ejpam-4976	29	9	and	and	CCONJ
ejpam-4976	29	10	studied	study	VERB
ejpam-4976	29	11	the	the	DET
ejpam-4976	29	12	notion	notion	NOUN
ejpam-4976	29	13	of	of	ADP
ejpam-4976	29	14	weakly	weakly	ADJ
ejpam-4976	29	15	(	(	PUNCT
ejpam-4976	29	16	λ	λ	PROPN
ejpam-4976	29	17	,	,	PUNCT
ejpam-4976	29	18	p)-continuous	p)-continuous	ADJ
ejpam-4976	29	19	functions	function	NOUN
ejpam-4976	29	20	.	.	PUNCT
ejpam-4976	30	1	laprom	laprom	ADP
ejpam-4976	30	2	et	et	PROPN
ejpam-4976	30	3	al	al	PROPN
ejpam-4976	30	4	.	.	PUNCT
ejpam-4976	31	1	[	[	X
ejpam-4976	31	2	9	9	NUM
ejpam-4976	31	3	]	]	PUNCT
ejpam-4976	31	4	studied	study	VERB
ejpam-4976	31	5	the	the	DET
ejpam-4976	31	6	concept	concept	NOUN
ejpam-4976	31	7	of	of	ADP
ejpam-4976	31	8	β(τ1	β(τ1	NOUN
ejpam-4976	31	9	,	,	PUNCT
ejpam-4976	31	10	τ2)-continuity	τ2)-continuity	NOUN
ejpam-4976	31	11	for	for	ADP
ejpam-4976	31	12	multifunctions	multifunction	NOUN
ejpam-4976	31	13	.	.	PUNCT
ejpam-4976	32	1	in	in	ADP
ejpam-4976	32	2	addition	addition	NOUN
ejpam-4976	32	3	,	,	PUNCT
ejpam-4976	32	4	some	some	DET
ejpam-4976	32	5	characterizations	characterization	NOUN
ejpam-4976	32	6	of	of	ADP
ejpam-4976	32	7	almost	almost	ADV
ejpam-4976	32	8	weak	weak	ADJ
ejpam-4976	32	9	(	(	PUNCT
ejpam-4976	32	10	τ1	τ1	NOUN
ejpam-4976	32	11	,	,	PUNCT
ejpam-4976	32	12	τ2)-continuity	τ2)-continuity	NOUN
ejpam-4976	32	13	for	for	ADP
ejpam-4976	32	14	multifunctions	multifunction	NOUN
ejpam-4976	32	15	were	be	AUX
ejpam-4976	32	16	established	establish	VERB
ejpam-4976	32	17	in	in	ADP
ejpam-4976	32	18	[	[	X
ejpam-4976	32	19	4	4	NUM
ejpam-4976	32	20	]	]	PUNCT
ejpam-4976	32	21	.	.	PUNCT
ejpam-4976	33	1	in	in	ADP
ejpam-4976	33	2	this	this	DET
ejpam-4976	33	3	paper	paper	NOUN
ejpam-4976	33	4	,	,	PUNCT
ejpam-4976	33	5	we	we	PRON
ejpam-4976	33	6	introduce	introduce	VERB
ejpam-4976	33	7	the	the	DET
ejpam-4976	33	8	concept	concept	NOUN
ejpam-4976	33	9	of	of	ADP
ejpam-4976	33	10	weakly	weakly	ADJ
ejpam-4976	33	11	(	(	PUNCT
ejpam-4976	33	12	τ1	τ1	NOUN
ejpam-4976	33	13	,	,	PUNCT
ejpam-4976	33	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	33	15	functions	function	NOUN
ejpam-4976	33	16	.	.	PUNCT
ejpam-4976	34	1	furthermore	furthermore	ADV
ejpam-4976	34	2	,	,	PUNCT
ejpam-4976	34	3	several	several	ADJ
ejpam-4976	34	4	characterizations	characterization	NOUN
ejpam-4976	34	5	of	of	ADP
ejpam-4976	34	6	weakly	weakly	ADJ
ejpam-4976	34	7	(	(	PUNCT
ejpam-4976	34	8	τ1	τ1	NOUN
ejpam-4976	34	9	,	,	PUNCT
ejpam-4976	34	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	34	11	functions	function	NOUN
ejpam-4976	34	12	are	be	AUX
ejpam-4976	34	13	discussed	discuss	VERB
ejpam-4976	34	14	.	.	PUNCT
ejpam-4976	35	1	2	2	X
ejpam-4976	35	2	.	.	X
ejpam-4976	35	3	preliminaries	preliminary	NOUN
ejpam-4976	35	4	throughout	throughout	ADP
ejpam-4976	35	5	the	the	DET
ejpam-4976	35	6	present	present	ADJ
ejpam-4976	35	7	paper	paper	NOUN
ejpam-4976	35	8	,	,	PUNCT
ejpam-4976	35	9	spaces	space	NOUN
ejpam-4976	35	10	(	(	PUNCT
ejpam-4976	35	11	x	x	NOUN
ejpam-4976	35	12	,	,	PUNCT
ejpam-4976	35	13	τ1	τ1	NOUN
ejpam-4976	35	14	,	,	PUNCT
ejpam-4976	35	15	τ2	τ2	NOUN
ejpam-4976	35	16	)	)	PUNCT
ejpam-4976	35	17	and	and	CCONJ
ejpam-4976	35	18	(	(	PUNCT
ejpam-4976	35	19	y	y	PROPN
ejpam-4976	35	20	,	,	PUNCT
ejpam-4976	35	21	σ1	σ1	PROPN
ejpam-4976	35	22	,	,	PUNCT
ejpam-4976	35	23	σ2	σ2	NOUN
ejpam-4976	35	24	)	)	PUNCT
ejpam-4976	35	25	(	(	PUNCT
ejpam-4976	35	26	or	or	CCONJ
ejpam-4976	35	27	simply	simply	ADV
ejpam-4976	35	28	x	x	X
ejpam-4976	35	29	and	and	CCONJ
ejpam-4976	35	30	y	y	PROPN
ejpam-4976	35	31	)	)	PUNCT
ejpam-4976	35	32	always	always	ADV
ejpam-4976	35	33	mean	mean	VERB
ejpam-4976	35	34	bitopological	bitopological	ADJ
ejpam-4976	35	35	spaces	space	NOUN
ejpam-4976	35	36	on	on	ADP
ejpam-4976	35	37	which	which	PRON
ejpam-4976	35	38	no	no	DET
ejpam-4976	35	39	separation	separation	NOUN
ejpam-4976	35	40	axioms	axiom	NOUN
ejpam-4976	35	41	are	be	AUX
ejpam-4976	35	42	assumed	assume	VERB
ejpam-4976	35	43	unless	unless	SCONJ
ejpam-4976	35	44	explicitly	explicitly	ADV
ejpam-4976	35	45	stated	state	VERB
ejpam-4976	35	46	.	.	PUNCT
ejpam-4976	36	1	let	let	VERB
ejpam-4976	36	2	a	a	DET
ejpam-4976	36	3	be	be	AUX
ejpam-4976	36	4	a	a	DET
ejpam-4976	36	5	subset	subset	NOUN
ejpam-4976	36	6	of	of	ADP
ejpam-4976	36	7	a	a	DET
ejpam-4976	36	8	bitopological	bitopological	ADJ
ejpam-4976	36	9	space	space	NOUN
ejpam-4976	36	10	(	(	PUNCT
ejpam-4976	36	11	x	x	NOUN
ejpam-4976	36	12	,	,	PUNCT
ejpam-4976	36	13	τ1	τ1	NOUN
ejpam-4976	36	14	,	,	PUNCT
ejpam-4976	36	15	τ2	τ2	NOUN
ejpam-4976	36	16	)	)	PUNCT
ejpam-4976	36	17	.	.	PUNCT
ejpam-4976	37	1	the	the	DET
ejpam-4976	37	2	closure	closure	NOUN
ejpam-4976	37	3	of	of	ADP
ejpam-4976	37	4	a	a	PRON
ejpam-4976	37	5	and	and	CCONJ
ejpam-4976	37	6	the	the	DET
ejpam-4976	37	7	interior	interior	NOUN
ejpam-4976	37	8	of	of	ADP
ejpam-4976	37	9	a	a	PRON
ejpam-4976	37	10	with	with	ADP
ejpam-4976	37	11	respect	respect	NOUN
ejpam-4976	37	12	to	to	ADP
ejpam-4976	37	13	τi	τi	PROPN
ejpam-4976	37	14	are	be	AUX
ejpam-4976	37	15	denoted	denote	VERB
ejpam-4976	37	16	by	by	ADP
ejpam-4976	37	17	τi	τi	NOUN
ejpam-4976	37	18	-	-	PUNCT
ejpam-4976	37	19	cl(a	cl(a	NUM
ejpam-4976	37	20	)	)	PUNCT
ejpam-4976	37	21	and	and	CCONJ
ejpam-4976	37	22	τi	τi	NOUN
ejpam-4976	37	23	-	-	PUNCT
ejpam-4976	37	24	int(a	int(a	NOUN
ejpam-4976	37	25	)	)	PUNCT
ejpam-4976	37	26	,	,	PUNCT
ejpam-4976	37	27	respectively	respectively	ADV
ejpam-4976	37	28	,	,	PUNCT
ejpam-4976	37	29	for	for	ADP
ejpam-4976	37	30	i	i	PROPN
ejpam-4976	37	31	=	=	SYM
ejpam-4976	37	32	1	1	NUM
ejpam-4976	37	33	,	,	PUNCT
ejpam-4976	37	34	2	2	NUM
ejpam-4976	37	35	.	.	X
ejpam-4976	37	36	a	a	DET
ejpam-4976	37	37	subset	subset	NOUN
ejpam-4976	37	38	a	a	PRON
ejpam-4976	37	39	of	of	ADP
ejpam-4976	37	40	a	a	DET
ejpam-4976	37	41	bitopological	bitopological	ADJ
ejpam-4976	37	42	space	space	NOUN
ejpam-4976	37	43	(	(	PUNCT
ejpam-4976	37	44	x	x	NOUN
ejpam-4976	37	45	,	,	PUNCT
ejpam-4976	37	46	τ1	τ1	NOUN
ejpam-4976	37	47	,	,	PUNCT
ejpam-4976	37	48	τ2	τ2	NOUN
ejpam-4976	37	49	)	)	PUNCT
ejpam-4976	37	50	is	be	AUX
ejpam-4976	37	51	called	call	VERB
ejpam-4976	37	52	τ1τ2	τ1τ2	VERB
ejpam-4976	37	53	-	-	ADJ
ejpam-4976	37	54	closed	closed	ADJ
ejpam-4976	37	55	[	[	X
ejpam-4976	37	56	7	7	X
ejpam-4976	37	57	]	]	X
ejpam-4976	37	58	if	if	SCONJ
ejpam-4976	37	59	a	a	DET
ejpam-4976	37	60	=	=	NOUN
ejpam-4976	37	61	τ1	τ1	NOUN
ejpam-4976	37	62	-	-	PUNCT
ejpam-4976	37	63	cl(τ2	cl(τ2	NOUN
ejpam-4976	37	64	-	-	PUNCT
ejpam-4976	37	65	cl(a	cl(a	NUM
ejpam-4976	37	66	)	)	PUNCT
ejpam-4976	37	67	)	)	PUNCT
ejpam-4976	37	68	.	.	PUNCT
ejpam-4976	38	1	the	the	DET
ejpam-4976	38	2	complement	complement	NOUN
ejpam-4976	38	3	of	of	ADP
ejpam-4976	38	4	a	a	DET
ejpam-4976	38	5	τ1τ2	τ1τ2	ADJ
ejpam-4976	38	6	-	-	ADJ
ejpam-4976	38	7	closed	closed	ADJ
ejpam-4976	38	8	set	set	NOUN
ejpam-4976	38	9	is	be	AUX
ejpam-4976	38	10	called	call	VERB
ejpam-4976	38	11	τ1τ2	τ1τ2	NOUN
ejpam-4976	38	12	-	-	ADJ
ejpam-4976	38	13	open	open	ADJ
ejpam-4976	38	14	.	.	PUNCT
ejpam-4976	39	1	the	the	DET
ejpam-4976	39	2	intersection	intersection	NOUN
ejpam-4976	39	3	of	of	ADP
ejpam-4976	39	4	all	all	DET
ejpam-4976	39	5	τ1τ2	τ1τ2	ADJ
ejpam-4976	39	6	-	-	ADJ
ejpam-4976	39	7	closed	closed	ADJ
ejpam-4976	39	8	sets	set	NOUN
ejpam-4976	39	9	of	of	ADP
ejpam-4976	39	10	x	x	PUNCT
ejpam-4976	39	11	containing	contain	VERB
ejpam-4976	39	12	a	a	PRON
ejpam-4976	39	13	is	be	AUX
ejpam-4976	39	14	called	call	VERB
ejpam-4976	39	15	the	the	DET
ejpam-4976	39	16	τ1τ2	τ1τ2	NOUN
ejpam-4976	39	17	-	-	NOUN
ejpam-4976	39	18	closure	closure	NOUN
ejpam-4976	39	19	[	[	X
ejpam-4976	39	20	7	7	NUM
ejpam-4976	39	21	]	]	PUNCT
ejpam-4976	39	22	of	of	ADP
ejpam-4976	39	23	a	a	PRON
ejpam-4976	39	24	and	and	CCONJ
ejpam-4976	39	25	is	be	AUX
ejpam-4976	39	26	denoted	denote	VERB
ejpam-4976	39	27	by	by	ADP
ejpam-4976	39	28	τ1τ2	τ1τ2	NOUN
ejpam-4976	39	29	-	-	NUM
ejpam-4976	39	30	cl(a	cl(a	NUM
ejpam-4976	39	31	)	)	PUNCT
ejpam-4976	39	32	.	.	PUNCT
ejpam-4976	40	1	the	the	DET
ejpam-4976	40	2	union	union	NOUN
ejpam-4976	40	3	of	of	ADP
ejpam-4976	40	4	all	all	DET
ejpam-4976	40	5	τ1τ2	τ1τ2	ADJ
ejpam-4976	40	6	-	-	ADJ
ejpam-4976	40	7	open	open	ADJ
ejpam-4976	40	8	sets	set	NOUN
ejpam-4976	40	9	of	of	ADP
ejpam-4976	40	10	x	x	PUNCT
ejpam-4976	40	11	contained	contain	VERB
ejpam-4976	40	12	in	in	ADP
ejpam-4976	40	13	a	a	PRON
ejpam-4976	40	14	is	be	AUX
ejpam-4976	40	15	called	call	VERB
ejpam-4976	40	16	the	the	DET
ejpam-4976	40	17	τ1τ2	τ1τ2	NOUN
ejpam-4976	40	18	-	-	ADJ
ejpam-4976	40	19	interior	interior	ADJ
ejpam-4976	40	20	[	[	X
ejpam-4976	40	21	7	7	NUM
ejpam-4976	40	22	]	]	PUNCT
ejpam-4976	40	23	of	of	ADP
ejpam-4976	40	24	a	a	PRON
ejpam-4976	40	25	and	and	CCONJ
ejpam-4976	40	26	is	be	AUX
ejpam-4976	40	27	denoted	denote	VERB
ejpam-4976	40	28	by	by	ADP
ejpam-4976	40	29	τ1τ2	τ1τ2	NOUN
ejpam-4976	40	30	-	-	ADJ
ejpam-4976	40	31	int(a	int(a	NOUN
ejpam-4976	40	32	)	)	PUNCT
ejpam-4976	40	33	.	.	PUNCT
ejpam-4976	41	1	a	a	DET
ejpam-4976	41	2	subset	subset	NOUN
ejpam-4976	41	3	a	a	PRON
ejpam-4976	41	4	of	of	ADP
ejpam-4976	41	5	a	a	DET
ejpam-4976	41	6	bitopological	bitopological	ADJ
ejpam-4976	41	7	space	space	NOUN
ejpam-4976	41	8	(	(	PUNCT
ejpam-4976	41	9	x	x	NOUN
ejpam-4976	41	10	,	,	PUNCT
ejpam-4976	41	11	τ1	τ1	NOUN
ejpam-4976	41	12	,	,	PUNCT
ejpam-4976	41	13	τ2	τ2	NOUN
ejpam-4976	41	14	)	)	PUNCT
ejpam-4976	41	15	is	be	AUX
ejpam-4976	41	16	said	say	VERB
ejpam-4976	41	17	to	to	PART
ejpam-4976	41	18	be	be	AUX
ejpam-4976	41	19	(	(	PUNCT
ejpam-4976	41	20	τ1	τ1	NOUN
ejpam-4976	41	21	,	,	PUNCT
ejpam-4976	41	22	τ2)r	τ2)r	NOUN
ejpam-4976	41	23	-	-	PUNCT
ejpam-4976	41	24	open	open	NOUN
ejpam-4976	42	1	[	[	X
ejpam-4976	42	2	19	19	NUM
ejpam-4976	42	3	]	]	PUNCT
ejpam-4976	42	4	(	(	PUNCT
ejpam-4976	42	5	resp	resp	NOUN
ejpam-4976	42	6	.	.	PUNCT
ejpam-4976	43	1	(	(	PUNCT
ejpam-4976	43	2	τ1	τ1	NOUN
ejpam-4976	43	3	,	,	PUNCT
ejpam-4976	43	4	τ2)s	τ2)s	NOUN
ejpam-4976	43	5	-	-	PUNCT
ejpam-4976	43	6	open	open	ADJ
ejpam-4976	43	7	[	[	X
ejpam-4976	43	8	2	2	NUM
ejpam-4976	43	9	]	]	PUNCT
ejpam-4976	43	10	,	,	PUNCT
ejpam-4976	43	11	(	(	PUNCT
ejpam-4976	43	12	τ1	τ1	NOUN
ejpam-4976	43	13	,	,	PUNCT
ejpam-4976	43	14	τ2)p	τ2)p	NOUN
ejpam-4976	43	15	-	-	ADJ
ejpam-4976	43	16	open	open	ADJ
ejpam-4976	43	17	[	[	X
ejpam-4976	43	18	2	2	NUM
ejpam-4976	43	19	]	]	PUNCT
ejpam-4976	43	20	,	,	PUNCT
ejpam-4976	43	21	(	(	PUNCT
ejpam-4976	43	22	τ1	τ1	NOUN
ejpam-4976	43	23	,	,	PUNCT
ejpam-4976	43	24	τ2)β	τ2)β	ADJ
ejpam-4976	43	25	-	-	PUNCT
ejpam-4976	43	26	open	open	ADJ
ejpam-4976	43	27	[	[	X
ejpam-4976	43	28	2	2	NUM
ejpam-4976	43	29	]	]	PUNCT
ejpam-4976	43	30	)	)	PUNCT
ejpam-4976	43	31	if	if	SCONJ
ejpam-4976	43	32	a	a	DET
ejpam-4976	43	33	=	=	PUNCT
ejpam-4976	43	34	τ1τ2	τ1τ2	NOUN
ejpam-4976	43	35	-	-	NOUN
ejpam-4976	43	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-4976	43	37	-	-	PUNCT
ejpam-4976	43	38	cl(a	cl(a	NUM
ejpam-4976	43	39	)	)	PUNCT
ejpam-4976	43	40	)	)	PUNCT
ejpam-4976	43	41	(	(	PUNCT
ejpam-4976	43	42	resp	resp	NOUN
ejpam-4976	43	43	.	.	PUNCT
ejpam-4976	44	1	a	a	DET
ejpam-4976	44	2	⊆	⊆	NUM
ejpam-4976	44	3	τ1τ2	τ1τ2	NOUN
ejpam-4976	44	4	-	-	ADJ
ejpam-4976	44	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-4976	44	6	-	-	PUNCT
ejpam-4976	44	7	int(a	int(a	NOUN
ejpam-4976	44	8	)	)	PUNCT
ejpam-4976	44	9	)	)	PUNCT
ejpam-4976	44	10	,	,	PUNCT
ejpam-4976	44	11	a	a	DET
ejpam-4976	44	12	⊆	⊆	NUM
ejpam-4976	44	13	τ1τ2	τ1τ2	NOUN
ejpam-4976	44	14	-	-	NOUN
ejpam-4976	44	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-4976	44	16	-	-	PUNCT
ejpam-4976	44	17	cl(a	cl(a	NUM
ejpam-4976	44	18	)	)	PUNCT
ejpam-4976	44	19	)	)	PUNCT
ejpam-4976	44	20	,	,	PUNCT
ejpam-4976	44	21	a	a	DET
ejpam-4976	44	22	⊆	⊆	NUM
ejpam-4976	44	23	τ1τ2	τ1τ2	NOUN
ejpam-4976	44	24	-	-	PUNCT
ejpam-4976	44	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-4976	44	26	-	-	PUNCT
ejpam-4976	44	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-4976	44	28	-	-	PUNCT
ejpam-4976	44	29	cl(a	cl(a	NUM
ejpam-4976	44	30	)	)	PUNCT
ejpam-4976	44	31	)	)	PUNCT
ejpam-4976	44	32	)	)	PUNCT
ejpam-4976	44	33	)	)	PUNCT
ejpam-4976	44	34	.	.	PUNCT
ejpam-4976	45	1	the	the	DET
ejpam-4976	45	2	complement	complement	NOUN
ejpam-4976	45	3	of	of	ADP
ejpam-4976	45	4	a	a	DET
ejpam-4976	45	5	(	(	PUNCT
ejpam-4976	45	6	τ1	τ1	NOUN
ejpam-4976	45	7	,	,	PUNCT
ejpam-4976	45	8	τ2)r	τ2)r	NOUN
ejpam-4976	45	9	-	-	PUNCT
ejpam-4976	45	10	open	open	ADJ
ejpam-4976	45	11	(	(	PUNCT
ejpam-4976	45	12	resp	resp	NOUN
ejpam-4976	45	13	.	.	PUNCT
ejpam-4976	46	1	(	(	PUNCT
ejpam-4976	46	2	τ1	τ1	NOUN
ejpam-4976	46	3	,	,	PUNCT
ejpam-4976	46	4	τ2)sopen	τ2)sopen	ADJ
ejpam-4976	46	5	,	,	PUNCT
ejpam-4976	46	6	(	(	PUNCT
ejpam-4976	46	7	τ1	τ1	NOUN
ejpam-4976	46	8	,	,	PUNCT
ejpam-4976	46	9	τ2)p	τ2)p	NOUN
ejpam-4976	46	10	-	-	ADJ
ejpam-4976	46	11	open	open	ADJ
ejpam-4976	46	12	,	,	PUNCT
ejpam-4976	46	13	(	(	PUNCT
ejpam-4976	46	14	τ1	τ1	NOUN
ejpam-4976	46	15	,	,	PUNCT
ejpam-4976	46	16	τ2)β	τ2)β	ADJ
ejpam-4976	46	17	-	-	PUNCT
ejpam-4976	46	18	open	open	ADJ
ejpam-4976	46	19	)	)	PUNCT
ejpam-4976	46	20	set	set	NOUN
ejpam-4976	46	21	is	be	AUX
ejpam-4976	46	22	called	call	VERB
ejpam-4976	46	23	(	(	PUNCT
ejpam-4976	46	24	τ1	τ1	NOUN
ejpam-4976	46	25	,	,	PUNCT
ejpam-4976	46	26	τ2)r	τ2)r	NOUN
ejpam-4976	46	27	-	-	PUNCT
ejpam-4976	46	28	closed	closed	ADJ
ejpam-4976	46	29	,	,	PUNCT
ejpam-4976	46	30	(	(	PUNCT
ejpam-4976	46	31	τ1	τ1	NOUN
ejpam-4976	46	32	,	,	PUNCT
ejpam-4976	46	33	τ2)s	τ2)s	NOUN
ejpam-4976	46	34	-	-	PUNCT
ejpam-4976	46	35	closed	closed	ADJ
ejpam-4976	46	36	,	,	PUNCT
ejpam-4976	46	37	(	(	PUNCT
ejpam-4976	46	38	τ1	τ1	NOUN
ejpam-4976	46	39	,	,	PUNCT
ejpam-4976	46	40	τ2)pclosed	τ2)pclose	VERB
ejpam-4976	46	41	,	,	PUNCT
ejpam-4976	46	42	(	(	PUNCT
ejpam-4976	46	43	τ1	τ1	NOUN
ejpam-4976	46	44	,	,	PUNCT
ejpam-4976	46	45	τ2)β	τ2)β	NOUN
ejpam-4976	46	46	-	-	PUNCT
ejpam-4976	46	47	closed	closed	ADJ
ejpam-4976	46	48	.	.	PUNCT
ejpam-4976	47	1	let	let	VERB
ejpam-4976	47	2	a	a	DET
ejpam-4976	47	3	be	be	AUX
ejpam-4976	47	4	a	a	DET
ejpam-4976	47	5	subset	subset	NOUN
ejpam-4976	47	6	of	of	ADP
ejpam-4976	47	7	a	a	DET
ejpam-4976	47	8	bitopological	bitopological	ADJ
ejpam-4976	47	9	space	space	NOUN
ejpam-4976	47	10	(	(	PUNCT
ejpam-4976	47	11	x	x	NOUN
ejpam-4976	47	12	,	,	PUNCT
ejpam-4976	47	13	τ1	τ1	NOUN
ejpam-4976	47	14	,	,	PUNCT
ejpam-4976	47	15	τ2	τ2	NOUN
ejpam-4976	47	16	)	)	PUNCT
ejpam-4976	47	17	.	.	PUNCT
ejpam-4976	48	1	a	a	DET
ejpam-4976	48	2	point	point	NOUN
ejpam-4976	48	3	x	x	X
ejpam-4976	48	4	∈	∈	NOUN
ejpam-4976	48	5	x	x	PUNCT
ejpam-4976	48	6	is	be	AUX
ejpam-4976	48	7	called	call	VERB
ejpam-4976	48	8	a	a	DET
ejpam-4976	48	9	(	(	PUNCT
ejpam-4976	48	10	τ1	τ1	NOUN
ejpam-4976	48	11	,	,	PUNCT
ejpam-4976	48	12	τ2)θ	τ2)θ	ADJ
ejpam-4976	48	13	-	-	PUNCT
ejpam-4976	48	14	cluster	cluster	NOUN
ejpam-4976	48	15	point	point	NOUN
ejpam-4976	48	16	[	[	X
ejpam-4976	48	17	19	19	NUM
ejpam-4976	48	18	]	]	PUNCT
ejpam-4976	48	19	of	of	ADP
ejpam-4976	48	20	a	a	DET
ejpam-4976	48	21	if	if	SCONJ
ejpam-4976	48	22	τ1τ2	τ1τ2	ADJ
ejpam-4976	48	23	-	-	ADJ
ejpam-4976	48	24	cl(u)∩a	cl(u)∩a	ADJ
ejpam-4976	48	25	̸=	̸=	PROPN
ejpam-4976	48	26	∅	∅	NOUN
ejpam-4976	48	27	for	for	ADP
ejpam-4976	48	28	every	every	DET
ejpam-4976	48	29	τ1τ2	τ1τ2	ADJ
ejpam-4976	48	30	-	-	ADJ
ejpam-4976	48	31	open	open	ADJ
ejpam-4976	48	32	set	set	NOUN
ejpam-4976	48	33	u	u	NOUN
ejpam-4976	48	34	containing	contain	VERB
ejpam-4976	48	35	x.	x.	NOUN
ejpam-4976	48	36	the	the	DET
ejpam-4976	48	37	set	set	NOUN
ejpam-4976	48	38	of	of	ADP
ejpam-4976	48	39	all	all	DET
ejpam-4976	48	40	(	(	PUNCT
ejpam-4976	48	41	τ1	τ1	NOUN
ejpam-4976	48	42	,	,	PUNCT
ejpam-4976	48	43	τ2)θ	τ2)θ	ADJ
ejpam-4976	48	44	-	-	PUNCT
ejpam-4976	48	45	cluster	cluster	NOUN
ejpam-4976	48	46	points	point	NOUN
ejpam-4976	48	47	of	of	ADP
ejpam-4976	48	48	a	a	PRON
ejpam-4976	48	49	is	be	AUX
ejpam-4976	48	50	called	call	VERB
ejpam-4976	48	51	the	the	DET
ejpam-4976	48	52	(	(	PUNCT
ejpam-4976	48	53	τ1	τ1	NOUN
ejpam-4976	48	54	,	,	PUNCT
ejpam-4976	48	55	τ2)θ	τ2)θ	ADJ
ejpam-4976	48	56	-	-	PUNCT
ejpam-4976	48	57	closure	closure	NOUN
ejpam-4976	48	58	[	[	X
ejpam-4976	48	59	19	19	NUM
ejpam-4976	48	60	]	]	PUNCT
ejpam-4976	48	61	of	of	ADP
ejpam-4976	48	62	a	a	PRON
ejpam-4976	48	63	and	and	CCONJ
ejpam-4976	48	64	is	be	AUX
ejpam-4976	48	65	denoted	denote	VERB
ejpam-4976	48	66	by	by	ADP
ejpam-4976	48	67	(	(	PUNCT
ejpam-4976	48	68	τ1	τ1	NOUN
ejpam-4976	48	69	,	,	PUNCT
ejpam-4976	48	70	τ2)θ	τ2)θ	NOUN
ejpam-4976	48	71	-	-	PUNCT
ejpam-4976	48	72	cl(a	cl(a	NUM
ejpam-4976	48	73	)	)	PUNCT
ejpam-4976	48	74	.	.	PUNCT
ejpam-4976	49	1	a	a	DET
ejpam-4976	49	2	subset	subset	NOUN
ejpam-4976	49	3	a	a	PRON
ejpam-4976	49	4	of	of	ADP
ejpam-4976	49	5	a	a	DET
ejpam-4976	49	6	bitopological	bitopological	ADJ
ejpam-4976	49	7	space	space	NOUN
ejpam-4976	49	8	(	(	PUNCT
ejpam-4976	49	9	x	x	NOUN
ejpam-4976	49	10	,	,	PUNCT
ejpam-4976	49	11	τ1	τ1	NOUN
ejpam-4976	49	12	,	,	PUNCT
ejpam-4976	49	13	τ2	τ2	NOUN
ejpam-4976	49	14	)	)	PUNCT
ejpam-4976	49	15	is	be	AUX
ejpam-4976	49	16	said	say	VERB
ejpam-4976	49	17	to	to	PART
ejpam-4976	49	18	be	be	AUX
ejpam-4976	49	19	(	(	PUNCT
ejpam-4976	49	20	τ1	τ1	NOUN
ejpam-4976	49	21	,	,	PUNCT
ejpam-4976	49	22	τ2)θ	τ2)θ	NOUN
ejpam-4976	49	23	-	-	PUNCT
ejpam-4976	49	24	closed	closed	ADJ
ejpam-4976	49	25	[	[	X
ejpam-4976	49	26	19	19	NUM
ejpam-4976	49	27	]	]	X
ejpam-4976	49	28	if	if	SCONJ
ejpam-4976	49	29	(	(	PUNCT
ejpam-4976	49	30	τ1	τ1	NOUN
ejpam-4976	49	31	,	,	PUNCT
ejpam-4976	49	32	τ2)θ	τ2)θ	NOUN
ejpam-4976	49	33	-	-	PUNCT
ejpam-4976	49	34	cl(a	cl(a	NUM
ejpam-4976	49	35	)	)	PUNCT
ejpam-4976	50	1	=	=	PUNCT
ejpam-4976	50	2	a.	a.	NOUN
ejpam-4976	50	3	the	the	DET
ejpam-4976	50	4	complement	complement	NOUN
ejpam-4976	50	5	of	of	ADP
ejpam-4976	50	6	a	a	DET
ejpam-4976	50	7	(	(	PUNCT
ejpam-4976	50	8	τ1	τ1	NOUN
ejpam-4976	50	9	,	,	PUNCT
ejpam-4976	50	10	τ2)θclosed	τ2)θclose	VERB
ejpam-4976	50	11	set	set	NOUN
ejpam-4976	50	12	is	be	AUX
ejpam-4976	50	13	said	say	VERB
ejpam-4976	50	14	to	to	PART
ejpam-4976	50	15	be	be	AUX
ejpam-4976	50	16	(	(	PUNCT
ejpam-4976	50	17	τ1	τ1	NOUN
ejpam-4976	50	18	,	,	PUNCT
ejpam-4976	50	19	τ2)θ	τ2)θ	NOUN
ejpam-4976	50	20	-	-	PUNCT
ejpam-4976	50	21	open	open	ADJ
ejpam-4976	50	22	.	.	PUNCT
ejpam-4976	51	1	the	the	DET
ejpam-4976	51	2	union	union	NOUN
ejpam-4976	51	3	of	of	ADP
ejpam-4976	51	4	all	all	DET
ejpam-4976	51	5	(	(	PUNCT
ejpam-4976	51	6	τ1	τ1	NOUN
ejpam-4976	51	7	,	,	PUNCT
ejpam-4976	51	8	τ2)θ	τ2)θ	ADJ
ejpam-4976	51	9	-	-	PUNCT
ejpam-4976	51	10	open	open	ADJ
ejpam-4976	51	11	sets	set	NOUN
ejpam-4976	51	12	of	of	ADP
ejpam-4976	51	13	x	x	PUNCT
ejpam-4976	51	14	contained	contain	VERB
ejpam-4976	51	15	in	in	ADP
ejpam-4976	51	16	a	a	PRON
ejpam-4976	51	17	is	be	AUX
ejpam-4976	51	18	called	call	VERB
ejpam-4976	51	19	the	the	DET
ejpam-4976	51	20	(	(	PUNCT
ejpam-4976	51	21	τ1	τ1	NOUN
ejpam-4976	51	22	,	,	PUNCT
ejpam-4976	51	23	τ2)θ	τ2)θ	ADJ
ejpam-4976	51	24	-	-	PUNCT
ejpam-4976	51	25	interior	interior	NOUN
ejpam-4976	51	26	[	[	X
ejpam-4976	51	27	19	19	NUM
ejpam-4976	51	28	]	]	PUNCT
ejpam-4976	51	29	of	of	ADP
ejpam-4976	51	30	a	a	PRON
ejpam-4976	51	31	and	and	CCONJ
ejpam-4976	51	32	is	be	AUX
ejpam-4976	51	33	denoted	denote	VERB
ejpam-4976	51	34	by	by	ADP
ejpam-4976	51	35	(	(	PUNCT
ejpam-4976	51	36	τ1	τ1	NOUN
ejpam-4976	51	37	,	,	PUNCT
ejpam-4976	51	38	τ2)θ	τ2)θ	NOUN
ejpam-4976	51	39	-	-	PUNCT
ejpam-4976	51	40	int(a	int(a	NOUN
ejpam-4976	51	41	)	)	PUNCT
ejpam-4976	51	42	.	.	PUNCT
ejpam-4976	52	1	3	3	X
ejpam-4976	52	2	.	.	X
ejpam-4976	52	3	characterizations	characterization	NOUN
ejpam-4976	52	4	of	of	ADP
ejpam-4976	52	5	weakly	weakly	ADJ
ejpam-4976	52	6	(	(	PUNCT
ejpam-4976	52	7	τ1	τ1	NOUN
ejpam-4976	52	8	,	,	PUNCT
ejpam-4976	52	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	52	10	functions	function	NOUN
ejpam-4976	52	11	in	in	ADP
ejpam-4976	52	12	this	this	DET
ejpam-4976	52	13	section	section	NOUN
ejpam-4976	52	14	,	,	PUNCT
ejpam-4976	52	15	we	we	PRON
ejpam-4976	52	16	introduce	introduce	VERB
ejpam-4976	52	17	the	the	DET
ejpam-4976	52	18	notion	notion	NOUN
ejpam-4976	52	19	of	of	ADP
ejpam-4976	52	20	weakly	weakly	ADJ
ejpam-4976	52	21	(	(	PUNCT
ejpam-4976	52	22	τ1	τ1	NOUN
ejpam-4976	52	23	,	,	PUNCT
ejpam-4976	52	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	52	25	functions	function	NOUN
ejpam-4976	52	26	.	.	PUNCT
ejpam-4976	53	1	moreover	moreover	ADV
ejpam-4976	53	2	,	,	PUNCT
ejpam-4976	53	3	some	some	DET
ejpam-4976	53	4	characterizations	characterization	NOUN
ejpam-4976	53	5	of	of	ADP
ejpam-4976	53	6	weakly	weakly	ADJ
ejpam-4976	53	7	(	(	PUNCT
ejpam-4976	53	8	τ1	τ1	NOUN
ejpam-4976	53	9	,	,	PUNCT
ejpam-4976	53	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	53	11	functions	function	NOUN
ejpam-4976	53	12	are	be	AUX
ejpam-4976	53	13	discussed	discuss	VERB
ejpam-4976	53	14	.	.	PUNCT
ejpam-4976	54	1	definition	definition	NOUN
ejpam-4976	54	2	1	1	NUM
ejpam-4976	54	3	.	.	PUNCT
ejpam-4976	55	1	a	a	DET
ejpam-4976	55	2	function	function	NOUN
ejpam-4976	55	3	f	f	NOUN
ejpam-4976	55	4	:	:	PUNCT
ejpam-4976	55	5	(	(	PUNCT
ejpam-4976	55	6	x	x	NOUN
ejpam-4976	55	7	,	,	PUNCT
ejpam-4976	55	8	τ1	τ1	NOUN
ejpam-4976	55	9	,	,	PUNCT
ejpam-4976	55	10	τ2	τ2	NOUN
ejpam-4976	55	11	)	)	PUNCT
ejpam-4976	55	12	→	→	SYM
ejpam-4976	55	13	(	(	PUNCT
ejpam-4976	55	14	y	y	PROPN
ejpam-4976	55	15	,	,	PUNCT
ejpam-4976	55	16	σ1	σ1	PROPN
ejpam-4976	55	17	,	,	PUNCT
ejpam-4976	55	18	σ2	σ2	PROPN
ejpam-4976	55	19	)	)	PUNCT
ejpam-4976	55	20	is	be	AUX
ejpam-4976	55	21	said	say	VERB
ejpam-4976	55	22	to	to	PART
ejpam-4976	55	23	be	be	AUX
ejpam-4976	55	24	weakly	weakly	ADJ
ejpam-4976	55	25	(	(	PUNCT
ejpam-4976	55	26	τ1	τ1	NOUN
ejpam-4976	55	27	,	,	PUNCT
ejpam-4976	55	28	τ2)continuous	τ2)continuous	ADJ
ejpam-4976	55	29	at	at	ADP
ejpam-4976	55	30	a	a	DET
ejpam-4976	55	31	point	point	NOUN
ejpam-4976	55	32	x	x	SYM
ejpam-4976	55	33	∈	∈	NOUN
ejpam-4976	55	34	x	x	PUNCT
ejpam-4976	55	35	if	if	SCONJ
ejpam-4976	55	36	for	for	ADP
ejpam-4976	55	37	each	each	DET
ejpam-4976	55	38	τ1τ2	τ1τ2	ADJ
ejpam-4976	55	39	-	-	ADJ
ejpam-4976	55	40	open	open	ADJ
ejpam-4976	55	41	set	set	VERB
ejpam-4976	55	42	v	v	NOUN
ejpam-4976	55	43	of	of	ADP
ejpam-4976	55	44	y	y	NOUN
ejpam-4976	55	45	containing	contain	VERB
ejpam-4976	55	46	f(x	f(x	PROPN
ejpam-4976	55	47	)	)	PUNCT
ejpam-4976	55	48	,	,	PUNCT
ejpam-4976	55	49	there	there	PRON
ejpam-4976	55	50	exists	exist	VERB
ejpam-4976	55	51	a	a	DET
ejpam-4976	55	52	τ1τ2	τ1τ2	NOUN
ejpam-4976	55	53	-	-	ADJ
ejpam-4976	55	54	open	open	ADJ
ejpam-4976	55	55	set	set	ADJ
ejpam-4976	55	56	u	u	NOUN
ejpam-4976	55	57	of	of	ADP
ejpam-4976	55	58	x	x	PUNCT
ejpam-4976	55	59	containing	contain	VERB
ejpam-4976	55	60	x	x	PUNCT
ejpam-4976	55	61	such	such	ADJ
ejpam-4976	55	62	that	that	DET
ejpam-4976	55	63	f(u	f(u	PROPN
ejpam-4976	55	64	)	)	PUNCT
ejpam-4976	55	65	⊆	⊆	NUM
ejpam-4976	55	66	σ1σ2	σ1σ2	NOUN
ejpam-4976	55	67	-	-	NUM
ejpam-4976	55	68	cl(v	cl(v	NOUN
ejpam-4976	55	69	)	)	PUNCT
ejpam-4976	55	70	.	.	PUNCT
ejpam-4976	56	1	a	a	DET
ejpam-4976	56	2	function	function	NOUN
ejpam-4976	56	3	f	f	NOUN
ejpam-4976	56	4	:	:	PUNCT
ejpam-4976	56	5	(	(	PUNCT
ejpam-4976	56	6	x	x	NOUN
ejpam-4976	56	7	,	,	PUNCT
ejpam-4976	56	8	τ1	τ1	NOUN
ejpam-4976	56	9	,	,	PUNCT
ejpam-4976	56	10	τ2	τ2	NOUN
ejpam-4976	56	11	)	)	PUNCT
ejpam-4976	56	12	→	→	SYM
ejpam-4976	56	13	(	(	PUNCT
ejpam-4976	56	14	y	y	PROPN
ejpam-4976	56	15	,	,	PUNCT
ejpam-4976	56	16	σ1	σ1	PROPN
ejpam-4976	56	17	,	,	PUNCT
ejpam-4976	56	18	σ2	σ2	PROPN
ejpam-4976	56	19	)	)	PUNCT
ejpam-4976	56	20	is	be	AUX
ejpam-4976	56	21	said	say	VERB
ejpam-4976	56	22	to	to	PART
ejpam-4976	56	23	be	be	AUX
ejpam-4976	56	24	weakly	weakly	ADJ
ejpam-4976	56	25	(	(	PUNCT
ejpam-4976	56	26	τ1	τ1	NOUN
ejpam-4976	56	27	,	,	PUNCT
ejpam-4976	56	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	56	29	if	if	SCONJ
ejpam-4976	56	30	f	f	PROPN
ejpam-4976	56	31	has	have	VERB
ejpam-4976	56	32	this	this	DET
ejpam-4976	56	33	property	property	NOUN
ejpam-4976	56	34	at	at	ADP
ejpam-4976	56	35	each	each	DET
ejpam-4976	56	36	point	point	NOUN
ejpam-4976	56	37	of	of	ADP
ejpam-4976	56	38	x.	x.	PROPN
ejpam-4976	56	39	c.	c.	PROPN
ejpam-4976	56	40	boonpok	boonpok	PROPN
ejpam-4976	56	41	,	,	PUNCT
ejpam-4976	56	42	c.	c.	PROPN
ejpam-4976	56	43	klanarong	klanarong	PROPN
ejpam-4976	56	44	/	/	SYM
ejpam-4976	56	45	eur	eur	PROPN
ejpam-4976	56	46	.	.	PUNCT
ejpam-4976	57	1	j.	j.	PROPN
ejpam-4976	57	2	pure	pure	PROPN
ejpam-4976	57	3	appl	appl	PROPN
ejpam-4976	57	4	.	.	PROPN
ejpam-4976	57	5	math	math	PROPN
ejpam-4976	57	6	,	,	PUNCT
ejpam-4976	57	7	17	17	NUM
ejpam-4976	57	8	(	(	PUNCT
ejpam-4976	57	9	1	1	NUM
ejpam-4976	57	10	)	)	PUNCT
ejpam-4976	57	11	(	(	PUNCT
ejpam-4976	57	12	2024	2024	NUM
ejpam-4976	57	13	)	)	PUNCT
ejpam-4976	57	14	,	,	PUNCT
ejpam-4976	57	15	416	416	NUM
ejpam-4976	57	16	-	-	SYM
ejpam-4976	57	17	425	425	NUM
ejpam-4976	57	18	418	418	NUM
ejpam-4976	57	19	theorem	theorem	NOUN
ejpam-4976	57	20	1	1	NUM
ejpam-4976	57	21	.	.	PUNCT
ejpam-4976	58	1	a	a	DET
ejpam-4976	58	2	function	function	NOUN
ejpam-4976	58	3	f	f	NOUN
ejpam-4976	58	4	:	:	PUNCT
ejpam-4976	58	5	(	(	PUNCT
ejpam-4976	58	6	x	x	NOUN
ejpam-4976	58	7	,	,	PUNCT
ejpam-4976	58	8	τ1	τ1	NOUN
ejpam-4976	58	9	,	,	PUNCT
ejpam-4976	58	10	τ2	τ2	NOUN
ejpam-4976	58	11	)	)	PUNCT
ejpam-4976	58	12	→	→	SYM
ejpam-4976	58	13	(	(	PUNCT
ejpam-4976	58	14	y	y	PROPN
ejpam-4976	58	15	,	,	PUNCT
ejpam-4976	58	16	σ1	σ1	PROPN
ejpam-4976	58	17	,	,	PUNCT
ejpam-4976	58	18	σ2	σ2	NOUN
ejpam-4976	58	19	)	)	PUNCT
ejpam-4976	58	20	is	be	AUX
ejpam-4976	58	21	weakly	weakly	ADJ
ejpam-4976	58	22	(	(	PUNCT
ejpam-4976	58	23	τ1	τ1	NOUN
ejpam-4976	58	24	,	,	PUNCT
ejpam-4976	58	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	58	26	at	at	ADP
ejpam-4976	58	27	x	x	X
ejpam-4976	58	28	∈	∈	PROPN
ejpam-4976	58	29	x	x	SYM
ejpam-4976	58	30	if	if	SCONJ
ejpam-4976	58	31	and	and	CCONJ
ejpam-4976	58	32	only	only	ADV
ejpam-4976	58	33	if	if	SCONJ
ejpam-4976	58	34	x	x	X
ejpam-4976	58	35	∈	∈	PROPN
ejpam-4976	58	36	τ1τ2	τ1τ2	NOUN
ejpam-4976	58	37	-	-	NUM
ejpam-4976	58	38	int(f	int(f	VERB
ejpam-4976	58	39	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	58	40	-	-	PUNCT
ejpam-4976	58	41	cl(v	cl(v	NOUN
ejpam-4976	58	42	)	)	PUNCT
ejpam-4976	58	43	)	)	PUNCT
ejpam-4976	58	44	)	)	PUNCT
ejpam-4976	59	1	for	for	ADP
ejpam-4976	59	2	every	every	DET
ejpam-4976	59	3	σ1σ2	σ1σ2	NOUN
ejpam-4976	59	4	-	-	ADJ
ejpam-4976	59	5	open	open	ADJ
ejpam-4976	59	6	set	set	NOUN
ejpam-4976	59	7	v	v	NOUN
ejpam-4976	59	8	of	of	ADP
ejpam-4976	59	9	y	y	NOUN
ejpam-4976	59	10	containing	contain	VERB
ejpam-4976	59	11	f(x	f(x	PROPN
ejpam-4976	59	12	)	)	PUNCT
ejpam-4976	59	13	.	.	PUNCT
ejpam-4976	60	1	proof	proof	NOUN
ejpam-4976	60	2	.	.	PUNCT
ejpam-4976	61	1	let	let	VERB
ejpam-4976	61	2	x	x	PUNCT
ejpam-4976	61	3	∈	∈	PROPN
ejpam-4976	61	4	x	x	X
ejpam-4976	61	5	and	and	CCONJ
ejpam-4976	61	6	v	v	X
ejpam-4976	61	7	be	be	AUX
ejpam-4976	61	8	any	any	DET
ejpam-4976	61	9	σ1σ2	σ1σ2	NOUN
ejpam-4976	61	10	-	-	ADJ
ejpam-4976	61	11	open	open	ADJ
ejpam-4976	61	12	set	set	NOUN
ejpam-4976	61	13	of	of	ADP
ejpam-4976	61	14	y	y	PROPN
ejpam-4976	61	15	containing	contain	VERB
ejpam-4976	61	16	f(x	f(x	PROPN
ejpam-4976	61	17	)	)	PUNCT
ejpam-4976	61	18	.	.	PUNCT
ejpam-4976	62	1	then	then	ADV
ejpam-4976	62	2	,	,	PUNCT
ejpam-4976	62	3	there	there	PRON
ejpam-4976	62	4	exists	exist	VERB
ejpam-4976	62	5	a	a	DET
ejpam-4976	62	6	τ1τ2	τ1τ2	NOUN
ejpam-4976	62	7	-	-	ADJ
ejpam-4976	62	8	open	open	ADJ
ejpam-4976	62	9	set	set	ADJ
ejpam-4976	62	10	u	u	NOUN
ejpam-4976	62	11	of	of	ADP
ejpam-4976	62	12	x	x	PUNCT
ejpam-4976	62	13	containing	contain	VERB
ejpam-4976	62	14	x	x	PUNCT
ejpam-4976	62	15	such	such	ADJ
ejpam-4976	62	16	that	that	DET
ejpam-4976	62	17	f(u	f(u	PROPN
ejpam-4976	62	18	)	)	PUNCT
ejpam-4976	62	19	⊆	⊆	NUM
ejpam-4976	62	20	σ1σ2	σ1σ2	NOUN
ejpam-4976	62	21	-	-	NUM
ejpam-4976	62	22	cl(v	cl(v	NOUN
ejpam-4976	62	23	)	)	PUNCT
ejpam-4976	62	24	.	.	PUNCT
ejpam-4976	63	1	thus	thus	ADV
ejpam-4976	63	2	,	,	PUNCT
ejpam-4976	63	3	x	x	PUNCT
ejpam-4976	63	4	∈	∈	PROPN
ejpam-4976	63	5	u	u	NOUN
ejpam-4976	63	6	⊆	⊆	NUM
ejpam-4976	63	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	63	8	-	-	PUNCT
ejpam-4976	63	9	cl(v	cl(v	NOUN
ejpam-4976	63	10	)	)	PUNCT
ejpam-4976	63	11	)	)	PUNCT
ejpam-4976	63	12	and	and	CCONJ
ejpam-4976	63	13	hence	hence	ADV
ejpam-4976	63	14	x	x	X
ejpam-4976	63	15	∈	∈	PRON
ejpam-4976	63	16	τ1τ2	τ1τ2	PUNCT
ejpam-4976	63	17	-	-	NUM
ejpam-4976	63	18	int(f	int(f	VERB
ejpam-4976	63	19	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	63	20	-	-	PUNCT
ejpam-4976	63	21	cl(v	cl(v	NOUN
ejpam-4976	63	22	)	)	PUNCT
ejpam-4976	63	23	)	)	PUNCT
ejpam-4976	63	24	)	)	PUNCT
ejpam-4976	63	25	.	.	PUNCT
ejpam-4976	64	1	conversely	conversely	ADV
ejpam-4976	64	2	,	,	PUNCT
ejpam-4976	64	3	let	let	VERB
ejpam-4976	64	4	v	v	PART
ejpam-4976	64	5	be	be	AUX
ejpam-4976	64	6	any	any	DET
ejpam-4976	64	7	σ1σ2	σ1σ2	NOUN
ejpam-4976	64	8	-	-	ADJ
ejpam-4976	64	9	open	open	ADJ
ejpam-4976	64	10	set	set	NOUN
ejpam-4976	64	11	of	of	ADP
ejpam-4976	64	12	y	y	PROPN
ejpam-4976	64	13	containing	contain	VERB
ejpam-4976	64	14	f(x	f(x	PROPN
ejpam-4976	64	15	)	)	PUNCT
ejpam-4976	64	16	.	.	PUNCT
ejpam-4976	65	1	by	by	ADP
ejpam-4976	65	2	the	the	DET
ejpam-4976	65	3	hypothesis	hypothesis	NOUN
ejpam-4976	65	4	,	,	PUNCT
ejpam-4976	65	5	x	x	X
ejpam-4976	65	6	∈	∈	PROPN
ejpam-4976	65	7	τ1τ2	τ1τ2	PUNCT
ejpam-4976	65	8	-	-	NUM
ejpam-4976	65	9	int(f	int(f	VERB
ejpam-4976	65	10	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	65	11	-	-	PUNCT
ejpam-4976	65	12	cl(v	cl(v	NOUN
ejpam-4976	65	13	)	)	PUNCT
ejpam-4976	65	14	)	)	PUNCT
ejpam-4976	65	15	)	)	PUNCT
ejpam-4976	65	16	.	.	PUNCT
ejpam-4976	66	1	then	then	ADV
ejpam-4976	66	2	,	,	PUNCT
ejpam-4976	66	3	there	there	PRON
ejpam-4976	66	4	exists	exist	VERB
ejpam-4976	66	5	a	a	DET
ejpam-4976	66	6	τ1τ2	τ1τ2	NOUN
ejpam-4976	66	7	-	-	ADJ
ejpam-4976	66	8	open	open	ADJ
ejpam-4976	66	9	set	set	ADJ
ejpam-4976	66	10	u	u	NOUN
ejpam-4976	66	11	of	of	ADP
ejpam-4976	66	12	x	x	SYM
ejpam-4976	66	13	such	such	ADJ
ejpam-4976	66	14	that	that	SCONJ
ejpam-4976	66	15	x	x	SYM
ejpam-4976	66	16	∈	∈	PROPN
ejpam-4976	66	17	u	u	NOUN
ejpam-4976	66	18	⊆	⊆	NUM
ejpam-4976	66	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	66	20	-	-	PUNCT
ejpam-4976	66	21	cl(v	cl(v	NOUN
ejpam-4976	66	22	)	)	PUNCT
ejpam-4976	66	23	)	)	PUNCT
ejpam-4976	66	24	.	.	PUNCT
ejpam-4976	67	1	thus	thus	ADV
ejpam-4976	67	2	,	,	PUNCT
ejpam-4976	67	3	f(u	f(u	PROPN
ejpam-4976	67	4	)	)	PUNCT
ejpam-4976	67	5	⊆	⊆	NUM
ejpam-4976	67	6	σ1σ2	σ1σ2	NOUN
ejpam-4976	67	7	-	-	NUM
ejpam-4976	67	8	cl(v	cl(v	NOUN
ejpam-4976	67	9	)	)	PUNCT
ejpam-4976	67	10	and	and	CCONJ
ejpam-4976	67	11	hence	hence	ADV
ejpam-4976	67	12	f	f	PROPN
ejpam-4976	67	13	is	be	AUX
ejpam-4976	67	14	weakly	weakly	ADJ
ejpam-4976	67	15	(	(	PUNCT
ejpam-4976	67	16	τ1	τ1	NOUN
ejpam-4976	67	17	,	,	PUNCT
ejpam-4976	67	18	τ2)continuous	τ2)continuous	ADJ
ejpam-4976	67	19	at	at	ADP
ejpam-4976	67	20	x.	x.	NOUN
ejpam-4976	67	21	theorem	theorem	VERB
ejpam-4976	67	22	2	2	NUM
ejpam-4976	67	23	.	.	PUNCT
ejpam-4976	68	1	a	a	DET
ejpam-4976	68	2	function	function	NOUN
ejpam-4976	68	3	f	f	NOUN
ejpam-4976	68	4	:	:	PUNCT
ejpam-4976	68	5	(	(	PUNCT
ejpam-4976	68	6	x	x	NOUN
ejpam-4976	68	7	,	,	PUNCT
ejpam-4976	68	8	τ1	τ1	NOUN
ejpam-4976	68	9	,	,	PUNCT
ejpam-4976	68	10	τ2	τ2	NOUN
ejpam-4976	68	11	)	)	PUNCT
ejpam-4976	68	12	→	→	SYM
ejpam-4976	68	13	(	(	PUNCT
ejpam-4976	68	14	y	y	PROPN
ejpam-4976	68	15	,	,	PUNCT
ejpam-4976	68	16	σ1	σ1	PROPN
ejpam-4976	68	17	,	,	PUNCT
ejpam-4976	68	18	σ2	σ2	NOUN
ejpam-4976	68	19	)	)	PUNCT
ejpam-4976	68	20	is	be	AUX
ejpam-4976	68	21	weakly	weakly	ADJ
ejpam-4976	68	22	(	(	PUNCT
ejpam-4976	68	23	τ1	τ1	NOUN
ejpam-4976	68	24	,	,	PUNCT
ejpam-4976	68	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	68	26	if	if	SCONJ
ejpam-4976	68	27	and	and	CCONJ
ejpam-4976	68	28	only	only	ADV
ejpam-4976	68	29	if	if	SCONJ
ejpam-4976	68	30	f−1(v	f−1(v	PROPN
ejpam-4976	68	31	)	)	PUNCT
ejpam-4976	69	1	⊆	⊆	X
ejpam-4976	69	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	69	3	-	-	NUM
ejpam-4976	69	4	int(f	int(f	PRON
ejpam-4976	69	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	69	6	-	-	PUNCT
ejpam-4976	69	7	cl(v	cl(v	NOUN
ejpam-4976	69	8	)	)	PUNCT
ejpam-4976	69	9	)	)	PUNCT
ejpam-4976	69	10	)	)	PUNCT
ejpam-4976	69	11	for	for	ADP
ejpam-4976	69	12	every	every	DET
ejpam-4976	69	13	σ1σ2	σ1σ2	NOUN
ejpam-4976	69	14	-	-	ADJ
ejpam-4976	69	15	open	open	ADJ
ejpam-4976	69	16	set	set	NOUN
ejpam-4976	69	17	v	v	NOUN
ejpam-4976	69	18	of	of	ADP
ejpam-4976	69	19	y	y	PROPN
ejpam-4976	69	20	.	.	PUNCT
ejpam-4976	70	1	proof	proof	NOUN
ejpam-4976	70	2	.	.	PUNCT
ejpam-4976	71	1	let	let	VERB
ejpam-4976	71	2	v	v	PART
ejpam-4976	71	3	be	be	AUX
ejpam-4976	71	4	any	any	DET
ejpam-4976	71	5	σ1σ2	σ1σ2	NOUN
ejpam-4976	71	6	-	-	ADJ
ejpam-4976	71	7	open	open	ADJ
ejpam-4976	71	8	set	set	NOUN
ejpam-4976	71	9	of	of	ADP
ejpam-4976	71	10	y	y	PROPN
ejpam-4976	71	11	and	and	CCONJ
ejpam-4976	71	12	x	x	PROPN
ejpam-4976	71	13	∈	∈	PROPN
ejpam-4976	71	14	f−1(v	f−1(v	NOUN
ejpam-4976	71	15	)	)	PUNCT
ejpam-4976	71	16	.	.	PUNCT
ejpam-4976	72	1	then	then	ADV
ejpam-4976	72	2	,	,	PUNCT
ejpam-4976	72	3	f(x	f(x	PROPN
ejpam-4976	72	4	)	)	PUNCT
ejpam-4976	72	5	∈	∈	PROPN
ejpam-4976	72	6	v	v	NOUN
ejpam-4976	72	7	.	.	PUNCT
ejpam-4976	73	1	since	since	SCONJ
ejpam-4976	73	2	f	f	PROPN
ejpam-4976	73	3	is	be	AUX
ejpam-4976	73	4	weakly	weakly	ADJ
ejpam-4976	73	5	(	(	PUNCT
ejpam-4976	73	6	τ1	τ1	NOUN
ejpam-4976	73	7	,	,	PUNCT
ejpam-4976	73	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	73	9	at	at	ADP
ejpam-4976	73	10	x	x	X
ejpam-4976	73	11	,	,	PUNCT
ejpam-4976	73	12	by	by	ADP
ejpam-4976	73	13	theorem	theorem	NOUN
ejpam-4976	73	14	1	1	NUM
ejpam-4976	73	15	we	we	PRON
ejpam-4976	73	16	have	have	VERB
ejpam-4976	73	17	x	x	PART
ejpam-4976	73	18	∈	∈	PROPN
ejpam-4976	73	19	τ1τ2	τ1τ2	PROPN
ejpam-4976	73	20	-	-	NUM
ejpam-4976	73	21	int(f	int(f	VERB
ejpam-4976	73	22	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	73	23	-	-	PUNCT
ejpam-4976	73	24	cl(v	cl(v	NOUN
ejpam-4976	73	25	)	)	PUNCT
ejpam-4976	73	26	)	)	PUNCT
ejpam-4976	73	27	)	)	PUNCT
ejpam-4976	73	28	and	and	CCONJ
ejpam-4976	73	29	hence	hence	ADV
ejpam-4976	73	30	f−1(v	f−1(v	NOUN
ejpam-4976	73	31	)	)	PUNCT
ejpam-4976	74	1	⊆	⊆	X
ejpam-4976	74	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	74	3	-	-	NUM
ejpam-4976	74	4	int(f	int(f	PRON
ejpam-4976	74	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	74	6	-	-	PUNCT
ejpam-4976	74	7	cl(v	cl(v	NOUN
ejpam-4976	74	8	)	)	PUNCT
ejpam-4976	74	9	)	)	PUNCT
ejpam-4976	74	10	)	)	PUNCT
ejpam-4976	74	11	.	.	PUNCT
ejpam-4976	75	1	conversely	conversely	ADV
ejpam-4976	75	2	,	,	PUNCT
ejpam-4976	75	3	let	let	VERB
ejpam-4976	75	4	x	x	X
ejpam-4976	75	5	∈	∈	PROPN
ejpam-4976	75	6	x	x	X
ejpam-4976	75	7	and	and	CCONJ
ejpam-4976	75	8	v	v	AUX
ejpam-4976	75	9	be	be	AUX
ejpam-4976	75	10	any	any	DET
ejpam-4976	75	11	σ1σ2	σ1σ2	NOUN
ejpam-4976	75	12	-	-	ADJ
ejpam-4976	75	13	open	open	ADJ
ejpam-4976	75	14	set	set	NOUN
ejpam-4976	75	15	of	of	ADP
ejpam-4976	75	16	y	y	PROPN
ejpam-4976	75	17	containing	contain	VERB
ejpam-4976	75	18	f(x	f(x	PROPN
ejpam-4976	75	19	)	)	PUNCT
ejpam-4976	75	20	.	.	PUNCT
ejpam-4976	76	1	then	then	ADV
ejpam-4976	76	2	,	,	PUNCT
ejpam-4976	76	3	we	we	PRON
ejpam-4976	76	4	have	have	VERB
ejpam-4976	76	5	x	x	X
ejpam-4976	76	6	∈	∈	PROPN
ejpam-4976	76	7	f−1(v	f−1(v	NOUN
ejpam-4976	76	8	)	)	PUNCT
ejpam-4976	77	1	⊆	⊆	X
ejpam-4976	77	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	77	3	-	-	NUM
ejpam-4976	77	4	int(f	int(f	PRON
ejpam-4976	77	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	77	6	-	-	PUNCT
ejpam-4976	77	7	cl(v	cl(v	NOUN
ejpam-4976	77	8	)	)	PUNCT
ejpam-4976	77	9	)	)	PUNCT
ejpam-4976	77	10	)	)	PUNCT
ejpam-4976	77	11	.	.	PUNCT
ejpam-4976	78	1	by	by	ADP
ejpam-4976	78	2	theorem	theorem	NOUN
ejpam-4976	78	3	1	1	NUM
ejpam-4976	78	4	,	,	PUNCT
ejpam-4976	78	5	f	f	PROPN
ejpam-4976	78	6	is	be	AUX
ejpam-4976	78	7	weakly	weakly	ADJ
ejpam-4976	78	8	(	(	PUNCT
ejpam-4976	78	9	τ1	τ1	NOUN
ejpam-4976	78	10	,	,	PUNCT
ejpam-4976	78	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	78	12	at	at	ADP
ejpam-4976	78	13	x.	x.	NOUN
ejpam-4976	78	14	this	this	PRON
ejpam-4976	78	15	shows	show	VERB
ejpam-4976	78	16	that	that	SCONJ
ejpam-4976	78	17	f	f	PROPN
ejpam-4976	78	18	is	be	AUX
ejpam-4976	78	19	weakly	weakly	ADJ
ejpam-4976	78	20	(	(	PUNCT
ejpam-4976	78	21	τ1	τ1	NOUN
ejpam-4976	78	22	,	,	PUNCT
ejpam-4976	78	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	78	24	.	.	PUNCT
ejpam-4976	79	1	theorem	theorem	NOUN
ejpam-4976	79	2	3	3	NUM
ejpam-4976	79	3	.	.	PUNCT
ejpam-4976	80	1	a	a	DET
ejpam-4976	80	2	function	function	NOUN
ejpam-4976	80	3	f	f	NOUN
ejpam-4976	80	4	:	:	PUNCT
ejpam-4976	80	5	(	(	PUNCT
ejpam-4976	80	6	x	x	NOUN
ejpam-4976	80	7	,	,	PUNCT
ejpam-4976	80	8	τ1	τ1	NOUN
ejpam-4976	80	9	,	,	PUNCT
ejpam-4976	80	10	τ2	τ2	NOUN
ejpam-4976	80	11	)	)	PUNCT
ejpam-4976	80	12	→	→	SYM
ejpam-4976	80	13	(	(	PUNCT
ejpam-4976	80	14	y	y	PROPN
ejpam-4976	80	15	,	,	PUNCT
ejpam-4976	80	16	σ1	σ1	PROPN
ejpam-4976	80	17	,	,	PUNCT
ejpam-4976	80	18	σ2	σ2	NOUN
ejpam-4976	80	19	)	)	PUNCT
ejpam-4976	80	20	is	be	AUX
ejpam-4976	80	21	weakly	weakly	ADJ
ejpam-4976	80	22	(	(	PUNCT
ejpam-4976	80	23	τ1	τ1	NOUN
ejpam-4976	80	24	,	,	PUNCT
ejpam-4976	80	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	80	26	if	if	SCONJ
ejpam-4976	80	27	and	and	CCONJ
ejpam-4976	80	28	only	only	ADV
ejpam-4976	80	29	if	if	SCONJ
ejpam-4976	80	30	τ1τ2	τ1τ2	NOUN
ejpam-4976	80	31	-	-	NOUN
ejpam-4976	80	32	cl(f	cl(f	PRON
ejpam-4976	80	33	−1(v	−1(v	NOUN
ejpam-4976	80	34	)	)	PUNCT
ejpam-4976	80	35	)	)	PUNCT
ejpam-4976	81	1	⊆	⊆	NUM
ejpam-4976	81	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	81	3	-	-	PUNCT
ejpam-4976	81	4	cl(v	cl(v	NOUN
ejpam-4976	81	5	)	)	PUNCT
ejpam-4976	81	6	)	)	PUNCT
ejpam-4976	81	7	for	for	ADP
ejpam-4976	81	8	every	every	DET
ejpam-4976	81	9	σ1σ2	σ1σ2	NOUN
ejpam-4976	81	10	-	-	ADJ
ejpam-4976	81	11	open	open	ADJ
ejpam-4976	81	12	set	set	NOUN
ejpam-4976	81	13	v	v	NOUN
ejpam-4976	81	14	of	of	ADP
ejpam-4976	81	15	y	y	PROPN
ejpam-4976	81	16	.	.	PUNCT
ejpam-4976	82	1	proof	proof	NOUN
ejpam-4976	82	2	.	.	PUNCT
ejpam-4976	83	1	let	let	VERB
ejpam-4976	83	2	v	v	PART
ejpam-4976	83	3	be	be	AUX
ejpam-4976	83	4	any	any	DET
ejpam-4976	83	5	σ1σ2	σ1σ2	NOUN
ejpam-4976	83	6	-	-	ADJ
ejpam-4976	83	7	open	open	ADJ
ejpam-4976	83	8	set	set	NOUN
ejpam-4976	83	9	of	of	ADP
ejpam-4976	83	10	y	y	PROPN
ejpam-4976	83	11	.	.	PUNCT
ejpam-4976	84	1	suppose	suppose	VERB
ejpam-4976	84	2	that	that	SCONJ
ejpam-4976	84	3	τ1τ2	τ1τ2	NOUN
ejpam-4976	84	4	-	-	PROPN
ejpam-4976	84	5	cl(f	cl(f	PRON
ejpam-4976	84	6	−1(v	−1(v	NOUN
ejpam-4976	84	7	)	)	PUNCT
ejpam-4976	84	8	)	)	PUNCT
ejpam-4976	84	9	⊈	⊈	PROPN
ejpam-4976	84	10	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	84	11	-	-	PUNCT
ejpam-4976	84	12	cl(v	cl(v	NOUN
ejpam-4976	84	13	)	)	PUNCT
ejpam-4976	84	14	)	)	PUNCT
ejpam-4976	84	15	.	.	PUNCT
ejpam-4976	85	1	there	there	PRON
ejpam-4976	85	2	exists	exist	VERB
ejpam-4976	85	3	x	x	X
ejpam-4976	85	4	∈	∈	PROPN
ejpam-4976	85	5	τ1τ2	τ1τ2	PROPN
ejpam-4976	85	6	-	-	NOUN
ejpam-4976	85	7	cl(f	cl(f	PRON
ejpam-4976	85	8	−1(v	−1(v	NOUN
ejpam-4976	85	9	)	)	PUNCT
ejpam-4976	85	10	)	)	PUNCT
ejpam-4976	85	11	,	,	PUNCT
ejpam-4976	85	12	but	but	CCONJ
ejpam-4976	85	13	x	x	X
ejpam-4976	85	14	̸∈	̸∈	PROPN
ejpam-4976	85	15	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	85	16	-	-	PUNCT
ejpam-4976	85	17	cl(v	cl(v	NOUN
ejpam-4976	85	18	)	)	PUNCT
ejpam-4976	85	19	)	)	PUNCT
ejpam-4976	85	20	.	.	PUNCT
ejpam-4976	86	1	then	then	ADV
ejpam-4976	86	2	,	,	PUNCT
ejpam-4976	86	3	f(x	f(x	PROPN
ejpam-4976	86	4	)	)	PUNCT
ejpam-4976	86	5	̸∈	̸∈	PROPN
ejpam-4976	86	6	σ1σ2	σ1σ2	NOUN
ejpam-4976	86	7	-	-	NUM
ejpam-4976	86	8	cl(v	cl(v	NOUN
ejpam-4976	86	9	)	)	PUNCT
ejpam-4976	86	10	and	and	CCONJ
ejpam-4976	86	11	there	there	PRON
ejpam-4976	86	12	exists	exist	VERB
ejpam-4976	86	13	a	a	DET
ejpam-4976	86	14	σ1σ2	σ1σ2	NUM
ejpam-4976	86	15	-	-	ADJ
ejpam-4976	86	16	open	open	ADJ
ejpam-4976	86	17	set	set	NOUN
ejpam-4976	86	18	w	w	PROPN
ejpam-4976	86	19	of	of	ADP
ejpam-4976	86	20	y	y	PROPN
ejpam-4976	86	21	containing	contain	VERB
ejpam-4976	86	22	f(x	f(x	PROPN
ejpam-4976	86	23	)	)	PUNCT
ejpam-4976	86	24	such	such	ADJ
ejpam-4976	86	25	that	that	SCONJ
ejpam-4976	86	26	w	w	PROPN
ejpam-4976	86	27	∩	∩	NOUN
ejpam-4976	86	28	v	v	NOUN
ejpam-4976	86	29	=	=	PUNCT
ejpam-4976	86	30	∅.	∅.	ADP
ejpam-4976	86	31	thus	thus	ADV
ejpam-4976	86	32	,	,	PUNCT
ejpam-4976	86	33	σ1σ2	σ1σ2	NOUN
ejpam-4976	86	34	-	-	PUNCT
ejpam-4976	86	35	cl(w	cl(w	NOUN
ejpam-4976	86	36	)	)	PUNCT
ejpam-4976	86	37	∩	∩	NOUN
ejpam-4976	86	38	v	v	NOUN
ejpam-4976	86	39	=	=	PUNCT
ejpam-4976	86	40	∅.	∅.	NOUN
ejpam-4976	86	41	since	since	SCONJ
ejpam-4976	86	42	f	f	PROPN
ejpam-4976	86	43	is	be	AUX
ejpam-4976	86	44	weakly	weakly	ADJ
ejpam-4976	86	45	(	(	PUNCT
ejpam-4976	86	46	τ1	τ1	NOUN
ejpam-4976	86	47	,	,	PUNCT
ejpam-4976	86	48	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	86	49	at	at	ADP
ejpam-4976	86	50	x	x	X
ejpam-4976	86	51	,	,	PUNCT
ejpam-4976	86	52	there	there	PRON
ejpam-4976	86	53	exists	exist	VERB
ejpam-4976	86	54	a	a	DET
ejpam-4976	86	55	τ1τ2	τ1τ2	NOUN
ejpam-4976	86	56	-	-	ADJ
ejpam-4976	86	57	open	open	ADJ
ejpam-4976	86	58	set	set	ADJ
ejpam-4976	86	59	u	u	NOUN
ejpam-4976	86	60	of	of	ADP
ejpam-4976	86	61	x	x	PUNCT
ejpam-4976	86	62	containing	contain	VERB
ejpam-4976	86	63	x	x	PUNCT
ejpam-4976	86	64	such	such	ADJ
ejpam-4976	86	65	that	that	DET
ejpam-4976	86	66	f(u	f(u	PROPN
ejpam-4976	86	67	)	)	PUNCT
ejpam-4976	86	68	⊆	⊆	NUM
ejpam-4976	86	69	σ1σ2	σ1σ2	NOUN
ejpam-4976	86	70	-	-	PUNCT
ejpam-4976	86	71	cl(w	cl(w	NOUN
ejpam-4976	86	72	)	)	PUNCT
ejpam-4976	86	73	.	.	PUNCT
ejpam-4976	87	1	therefore	therefore	ADV
ejpam-4976	87	2	,	,	PUNCT
ejpam-4976	87	3	f(u	f(u	PROPN
ejpam-4976	87	4	)	)	PUNCT
ejpam-4976	87	5	∩	∩	NOUN
ejpam-4976	87	6	v	v	NOUN
ejpam-4976	87	7	=	=	PUNCT
ejpam-4976	87	8	∅.	∅.	NOUN
ejpam-4976	87	9	since	since	SCONJ
ejpam-4976	87	10	x	x	PROPN
ejpam-4976	87	11	∈	∈	PROPN
ejpam-4976	87	12	τ1τ2	τ1τ2	PROPN
ejpam-4976	87	13	-	-	NOUN
ejpam-4976	87	14	cl(f	cl(f	PRON
ejpam-4976	87	15	−1(v	−1(v	NOUN
ejpam-4976	87	16	)	)	PUNCT
ejpam-4976	87	17	)	)	PUNCT
ejpam-4976	87	18	,	,	PUNCT
ejpam-4976	87	19	u	u	NOUN
ejpam-4976	87	20	∩	∩	ADJ
ejpam-4976	87	21	f−1(v	f−1(v	NOUN
ejpam-4976	87	22	)	)	PUNCT
ejpam-4976	88	1	̸=	̸=	PROPN
ejpam-4976	88	2	∅	∅	NOUN
ejpam-4976	88	3	and	and	CCONJ
ejpam-4976	88	4	f(u	f(u	PROPN
ejpam-4976	88	5	)	)	PUNCT
ejpam-4976	88	6	∩	∩	NOUN
ejpam-4976	88	7	v	v	ADP
ejpam-4976	88	8	̸=	̸=	PROPN
ejpam-4976	88	9	∅	∅	NOUN
ejpam-4976	88	10	,	,	PUNCT
ejpam-4976	88	11	which	which	PRON
ejpam-4976	88	12	is	be	AUX
ejpam-4976	88	13	a	a	DET
ejpam-4976	88	14	contradiction	contradiction	NOUN
ejpam-4976	88	15	.	.	PUNCT
ejpam-4976	89	1	this	this	PRON
ejpam-4976	89	2	shows	show	VERB
ejpam-4976	89	3	that	that	SCONJ
ejpam-4976	89	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	89	5	-	-	PROPN
ejpam-4976	89	6	cl(f	cl(f	PRON
ejpam-4976	89	7	−1(v	−1(v	NOUN
ejpam-4976	89	8	)	)	PUNCT
ejpam-4976	89	9	)	)	PUNCT
ejpam-4976	89	10	⊆	⊆	NUM
ejpam-4976	89	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	89	12	-	-	PUNCT
ejpam-4976	89	13	cl(v	cl(v	NOUN
ejpam-4976	89	14	)	)	PUNCT
ejpam-4976	89	15	)	)	PUNCT
ejpam-4976	89	16	.	.	PUNCT
ejpam-4976	90	1	conversely	conversely	ADV
ejpam-4976	90	2	,	,	PUNCT
ejpam-4976	90	3	let	let	VERB
ejpam-4976	90	4	v	v	PART
ejpam-4976	90	5	be	be	AUX
ejpam-4976	90	6	any	any	DET
ejpam-4976	90	7	σ1σ2	σ1σ2	NOUN
ejpam-4976	90	8	-	-	ADJ
ejpam-4976	90	9	open	open	ADJ
ejpam-4976	90	10	set	set	NOUN
ejpam-4976	90	11	of	of	ADP
ejpam-4976	90	12	y	y	PROPN
ejpam-4976	90	13	.	.	PUNCT
ejpam-4976	91	1	then	then	ADV
ejpam-4976	91	2	,	,	PUNCT
ejpam-4976	91	3	y	y	PROPN
ejpam-4976	91	4	−σ1σ2	−σ1σ2	PROPN
ejpam-4976	91	5	-	-	PUNCT
ejpam-4976	91	6	cl(v	cl(v	NOUN
ejpam-4976	91	7	)	)	PUNCT
ejpam-4976	91	8	is	be	AUX
ejpam-4976	91	9	σ1σ2	σ1σ2	NOUN
ejpam-4976	91	10	-	-	ADJ
ejpam-4976	91	11	open	open	ADJ
ejpam-4976	91	12	in	in	ADP
ejpam-4976	91	13	y	y	PROPN
ejpam-4976	91	14	.	.	PUNCT
ejpam-4976	92	1	by	by	ADP
ejpam-4976	92	2	the	the	DET
ejpam-4976	92	3	hypothesis	hypothesis	NOUN
ejpam-4976	92	4	,	,	PUNCT
ejpam-4976	92	5	τ1τ2	τ1τ2	NOUN
ejpam-4976	92	6	-	-	NOUN
ejpam-4976	92	7	cl(f	cl(f	NOUN
ejpam-4976	92	8	−1(y	−1(y	PUNCT
ejpam-4976	92	9	−σ1σ2	−σ1σ2	NOUN
ejpam-4976	92	10	-	-	NOUN
ejpam-4976	92	11	cl(v	cl(v	NOUN
ejpam-4976	92	12	)	)	PUNCT
ejpam-4976	92	13	)	)	PUNCT
ejpam-4976	92	14	)	)	PUNCT
ejpam-4976	93	1	⊆	⊆	NUM
ejpam-4976	93	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	93	3	-	-	PUNCT
ejpam-4976	93	4	cl(y	cl(y	NOUN
ejpam-4976	93	5	−σ1σ2	−σ1σ2	NOUN
ejpam-4976	93	6	-	-	NOUN
ejpam-4976	93	7	cl(v	cl(v	NOUN
ejpam-4976	93	8	)	)	PUNCT
ejpam-4976	93	9	)	)	PUNCT
ejpam-4976	93	10	)	)	PUNCT
ejpam-4976	93	11	.	.	PUNCT
ejpam-4976	94	1	thus	thus	ADV
ejpam-4976	94	2	,	,	PUNCT
ejpam-4976	94	3	x−τ1τ2	x−τ1τ2	PROPN
ejpam-4976	94	4	-	-	PUNCT
ejpam-4976	94	5	int(f	int(f	PROPN
ejpam-4976	94	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	94	7	-	-	PUNCT
ejpam-4976	94	8	cl(v	cl(v	NOUN
ejpam-4976	94	9	)	)	PUNCT
ejpam-4976	94	10	)	)	PUNCT
ejpam-4976	94	11	)	)	PUNCT
ejpam-4976	95	1	⊆	⊆	NUM
ejpam-4976	95	2	x−f−1(σ1σ2	x−f−1(σ1σ2	NOUN
ejpam-4976	95	3	-	-	PUNCT
ejpam-4976	95	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	95	5	-	-	PUNCT
ejpam-4976	95	6	cl(v	cl(v	NOUN
ejpam-4976	95	7	)	)	PUNCT
ejpam-4976	95	8	)	)	PUNCT
ejpam-4976	95	9	)	)	PUNCT
ejpam-4976	96	1	⊆	⊆	NUM
ejpam-4976	96	2	x−f−1(v	x−f−1(v	PUNCT
ejpam-4976	96	3	)	)	PUNCT
ejpam-4976	96	4	and	and	CCONJ
ejpam-4976	96	5	hence	hence	ADV
ejpam-4976	96	6	f−1(v	f−1(v	NOUN
ejpam-4976	96	7	)	)	PUNCT
ejpam-4976	96	8	⊆	⊆	X
ejpam-4976	96	9	τ1τ2	τ1τ2	NOUN
ejpam-4976	96	10	-	-	NUM
ejpam-4976	96	11	int(f	int(f	PRON
ejpam-4976	96	12	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	96	13	-	-	PUNCT
ejpam-4976	96	14	cl(v	cl(v	NOUN
ejpam-4976	96	15	)	)	PUNCT
ejpam-4976	96	16	)	)	PUNCT
ejpam-4976	96	17	)	)	PUNCT
ejpam-4976	96	18	.	.	PUNCT
ejpam-4976	97	1	by	by	ADP
ejpam-4976	97	2	theorem	theorem	NOUN
ejpam-4976	97	3	2	2	NUM
ejpam-4976	97	4	,	,	PUNCT
ejpam-4976	97	5	f	f	PROPN
ejpam-4976	97	6	is	be	AUX
ejpam-4976	97	7	weakly	weakly	ADJ
ejpam-4976	97	8	(	(	PUNCT
ejpam-4976	97	9	τ1	τ1	NOUN
ejpam-4976	97	10	,	,	PUNCT
ejpam-4976	97	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	97	12	.	.	PUNCT
ejpam-4976	98	1	theorem	theorem	NOUN
ejpam-4976	98	2	4	4	NUM
ejpam-4976	98	3	.	.	X
ejpam-4976	98	4	for	for	ADP
ejpam-4976	98	5	a	a	DET
ejpam-4976	98	6	function	function	NOUN
ejpam-4976	98	7	(	(	PUNCT
ejpam-4976	98	8	x	x	NOUN
ejpam-4976	98	9	,	,	PUNCT
ejpam-4976	98	10	τ1	τ1	NOUN
ejpam-4976	98	11	,	,	PUNCT
ejpam-4976	98	12	τ2	τ2	NOUN
ejpam-4976	98	13	)	)	PUNCT
ejpam-4976	98	14	→	→	SYM
ejpam-4976	98	15	(	(	PUNCT
ejpam-4976	98	16	y	y	PROPN
ejpam-4976	98	17	,	,	PUNCT
ejpam-4976	98	18	σ1	σ1	PROPN
ejpam-4976	98	19	,	,	PUNCT
ejpam-4976	98	20	σ2	σ2	NOUN
ejpam-4976	98	21	)	)	PUNCT
ejpam-4976	98	22	,	,	PUNCT
ejpam-4976	98	23	the	the	DET
ejpam-4976	98	24	following	follow	VERB
ejpam-4976	98	25	properties	property	NOUN
ejpam-4976	98	26	are	be	AUX
ejpam-4976	98	27	equivalent	equivalent	ADJ
ejpam-4976	98	28	:	:	PUNCT
ejpam-4976	98	29	c.	c.	PROPN
ejpam-4976	98	30	boonpok	boonpok	PROPN
ejpam-4976	98	31	,	,	PUNCT
ejpam-4976	98	32	c.	c.	PROPN
ejpam-4976	98	33	klanarong	klanarong	PROPN
ejpam-4976	98	34	/	/	SYM
ejpam-4976	98	35	eur	eur	PROPN
ejpam-4976	98	36	.	.	PUNCT
ejpam-4976	99	1	j.	j.	PROPN
ejpam-4976	99	2	pure	pure	PROPN
ejpam-4976	99	3	appl	appl	PROPN
ejpam-4976	99	4	.	.	PROPN
ejpam-4976	99	5	math	math	PROPN
ejpam-4976	99	6	,	,	PUNCT
ejpam-4976	99	7	17	17	NUM
ejpam-4976	99	8	(	(	PUNCT
ejpam-4976	99	9	1	1	NUM
ejpam-4976	99	10	)	)	PUNCT
ejpam-4976	99	11	(	(	PUNCT
ejpam-4976	99	12	2024	2024	NUM
ejpam-4976	99	13	)	)	PUNCT
ejpam-4976	99	14	,	,	PUNCT
ejpam-4976	99	15	416	416	NUM
ejpam-4976	99	16	-	-	SYM
ejpam-4976	99	17	425	425	NUM
ejpam-4976	99	18	419	419	NUM
ejpam-4976	99	19	(	(	PUNCT
ejpam-4976	99	20	1	1	NUM
ejpam-4976	99	21	)	)	PUNCT
ejpam-4976	99	22	f	f	PROPN
ejpam-4976	99	23	is	be	AUX
ejpam-4976	99	24	weakly	weakly	ADJ
ejpam-4976	99	25	(	(	PUNCT
ejpam-4976	99	26	τ1	τ1	NOUN
ejpam-4976	99	27	,	,	PUNCT
ejpam-4976	99	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	99	29	;	;	PUNCT
ejpam-4976	99	30	(	(	PUNCT
ejpam-4976	99	31	2	2	X
ejpam-4976	99	32	)	)	PUNCT
ejpam-4976	99	33	f−1(v	f−1(v	NOUN
ejpam-4976	99	34	)	)	PUNCT
ejpam-4976	100	1	⊆	⊆	X
ejpam-4976	100	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	100	3	-	-	NUM
ejpam-4976	100	4	int(f	int(f	PRON
ejpam-4976	100	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	100	6	-	-	PUNCT
ejpam-4976	100	7	cl(v	cl(v	NOUN
ejpam-4976	100	8	)	)	PUNCT
ejpam-4976	100	9	)	)	PUNCT
ejpam-4976	100	10	)	)	PUNCT
ejpam-4976	100	11	for	for	ADP
ejpam-4976	100	12	every	every	DET
ejpam-4976	100	13	σ1σ2	σ1σ2	NOUN
ejpam-4976	100	14	-	-	ADJ
ejpam-4976	100	15	open	open	ADJ
ejpam-4976	100	16	set	set	NOUN
ejpam-4976	100	17	v	v	NOUN
ejpam-4976	100	18	of	of	ADP
ejpam-4976	100	19	y	y	PROPN
ejpam-4976	100	20	;	;	PUNCT
ejpam-4976	100	21	(	(	PUNCT
ejpam-4976	100	22	3	3	X
ejpam-4976	100	23	)	)	PUNCT
ejpam-4976	100	24	τ1τ2	τ1τ2	NOUN
ejpam-4976	100	25	-	-	NOUN
ejpam-4976	100	26	cl(f	cl(f	NOUN
ejpam-4976	100	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	100	28	-	-	PUNCT
ejpam-4976	100	29	int(k	int(k	NUM
ejpam-4976	100	30	)	)	PUNCT
ejpam-4976	100	31	)	)	PUNCT
ejpam-4976	100	32	)	)	PUNCT
ejpam-4976	101	1	⊆	⊆	NUM
ejpam-4976	101	2	f−1(k	f−1(k	PROPN
ejpam-4976	101	3	)	)	PUNCT
ejpam-4976	101	4	for	for	ADP
ejpam-4976	101	5	every	every	DET
ejpam-4976	101	6	σ1σ2	σ1σ2	NUM
ejpam-4976	101	7	-	-	PUNCT
ejpam-4976	101	8	closed	closed	ADJ
ejpam-4976	101	9	set	set	NOUN
ejpam-4976	101	10	k	k	PROPN
ejpam-4976	101	11	of	of	ADP
ejpam-4976	101	12	y	y	PROPN
ejpam-4976	101	13	;	;	PUNCT
ejpam-4976	101	14	(	(	PUNCT
ejpam-4976	101	15	4	4	X
ejpam-4976	101	16	)	)	PUNCT
ejpam-4976	101	17	τ1τ2	τ1τ2	NOUN
ejpam-4976	101	18	-	-	NOUN
ejpam-4976	101	19	cl(f	cl(f	NOUN
ejpam-4976	101	20	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	101	21	-	-	PUNCT
ejpam-4976	101	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	101	23	-	-	PUNCT
ejpam-4976	101	24	cl(b	cl(b	NOUN
ejpam-4976	101	25	)	)	PUNCT
ejpam-4976	101	26	)	)	PUNCT
ejpam-4976	101	27	)	)	PUNCT
ejpam-4976	101	28	)	)	PUNCT
ejpam-4976	102	1	⊆	⊆	NUM
ejpam-4976	102	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	102	3	-	-	PUNCT
ejpam-4976	102	4	cl(b	cl(b	NOUN
ejpam-4976	102	5	)	)	PUNCT
ejpam-4976	102	6	)	)	PUNCT
ejpam-4976	102	7	for	for	ADP
ejpam-4976	102	8	every	every	DET
ejpam-4976	102	9	subset	subset	NOUN
ejpam-4976	102	10	b	b	PROPN
ejpam-4976	102	11	of	of	ADP
ejpam-4976	102	12	y	y	PROPN
ejpam-4976	102	13	;	;	PUNCT
ejpam-4976	102	14	(	(	PUNCT
ejpam-4976	102	15	5	5	X
ejpam-4976	102	16	)	)	PUNCT
ejpam-4976	102	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	102	18	-	-	PUNCT
ejpam-4976	102	19	int(b	int(b	NOUN
ejpam-4976	102	20	)	)	PUNCT
ejpam-4976	102	21	)	)	PUNCT
ejpam-4976	103	1	⊆	⊆	X
ejpam-4976	103	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	103	3	-	-	NUM
ejpam-4976	103	4	int(f	int(f	VERB
ejpam-4976	103	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	103	6	-	-	PUNCT
ejpam-4976	103	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	103	8	-	-	PUNCT
ejpam-4976	103	9	int(b	int(b	NOUN
ejpam-4976	103	10	)	)	PUNCT
ejpam-4976	103	11	)	)	PUNCT
ejpam-4976	103	12	)	)	PUNCT
ejpam-4976	103	13	)	)	PUNCT
ejpam-4976	103	14	for	for	ADP
ejpam-4976	103	15	every	every	DET
ejpam-4976	103	16	subset	subset	NOUN
ejpam-4976	103	17	b	b	PROPN
ejpam-4976	103	18	of	of	ADP
ejpam-4976	103	19	y	y	PROPN
ejpam-4976	103	20	;	;	PUNCT
ejpam-4976	103	21	(	(	PUNCT
ejpam-4976	103	22	6	6	X
ejpam-4976	103	23	)	)	PUNCT
ejpam-4976	103	24	τ1τ2	τ1τ2	NOUN
ejpam-4976	103	25	-	-	NOUN
ejpam-4976	103	26	cl(f	cl(f	PRON
ejpam-4976	103	27	−1(v	−1(v	NOUN
ejpam-4976	103	28	)	)	PUNCT
ejpam-4976	103	29	)	)	PUNCT
ejpam-4976	104	1	⊆	⊆	NUM
ejpam-4976	104	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	104	3	-	-	PUNCT
ejpam-4976	104	4	cl(v	cl(v	NOUN
ejpam-4976	104	5	)	)	PUNCT
ejpam-4976	104	6	)	)	PUNCT
ejpam-4976	104	7	for	for	ADP
ejpam-4976	104	8	every	every	DET
ejpam-4976	104	9	σ1σ2	σ1σ2	NOUN
ejpam-4976	104	10	-	-	ADJ
ejpam-4976	104	11	open	open	ADJ
ejpam-4976	104	12	set	set	NOUN
ejpam-4976	104	13	v	v	NOUN
ejpam-4976	104	14	of	of	ADP
ejpam-4976	104	15	y	y	PROPN
ejpam-4976	104	16	.	.	PUNCT
ejpam-4976	105	1	proof	proof	NOUN
ejpam-4976	105	2	.	.	PUNCT
ejpam-4976	106	1	(	(	PUNCT
ejpam-4976	106	2	1	1	X
ejpam-4976	106	3	)	)	PUNCT
ejpam-4976	106	4	⇒	⇒	NOUN
ejpam-4976	106	5	(	(	PUNCT
ejpam-4976	106	6	2	2	NUM
ejpam-4976	106	7	):	):	PUNCT
ejpam-4976	106	8	by	by	ADP
ejpam-4976	106	9	theorem	theorem	NOUN
ejpam-4976	106	10	2	2	NUM
ejpam-4976	106	11	.	.	PUNCT
ejpam-4976	106	12	(	(	PUNCT
ejpam-4976	106	13	2	2	X
ejpam-4976	106	14	)	)	PUNCT
ejpam-4976	106	15	⇒	⇒	NOUN
ejpam-4976	106	16	(	(	PUNCT
ejpam-4976	106	17	3	3	NUM
ejpam-4976	106	18	):	):	PUNCT
ejpam-4976	106	19	let	let	VERB
ejpam-4976	106	20	k	k	PRON
ejpam-4976	106	21	be	be	AUX
ejpam-4976	106	22	any	any	DET
ejpam-4976	106	23	σ1σ2	σ1σ2	NUM
ejpam-4976	106	24	-	-	PUNCT
ejpam-4976	106	25	closed	closed	ADJ
ejpam-4976	106	26	set	set	NOUN
ejpam-4976	106	27	of	of	ADP
ejpam-4976	106	28	y	y	PROPN
ejpam-4976	106	29	.	.	PUNCT
ejpam-4976	107	1	then	then	ADV
ejpam-4976	107	2	,	,	PUNCT
ejpam-4976	107	3	y	y	PROPN
ejpam-4976	107	4	−k	−k	PROPN
ejpam-4976	107	5	is	be	AUX
ejpam-4976	107	6	σ1σ2	σ1σ2	NOUN
ejpam-4976	107	7	-	-	ADJ
ejpam-4976	107	8	open	open	ADJ
ejpam-4976	107	9	in	in	ADP
ejpam-4976	107	10	y	y	PROPN
ejpam-4976	107	11	and	and	CCONJ
ejpam-4976	107	12	by	by	ADP
ejpam-4976	107	13	(	(	PUNCT
ejpam-4976	107	14	2	2	NUM
ejpam-4976	107	15	)	)	PUNCT
ejpam-4976	107	16	,	,	PUNCT
ejpam-4976	107	17	x	x	PUNCT
ejpam-4976	107	18	−	−	PROPN
ejpam-4976	107	19	f−1(k	f−1(k	PROPN
ejpam-4976	107	20	)	)	PUNCT
ejpam-4976	107	21	=	=	SYM
ejpam-4976	107	22	f−1(y	f−1(y	PROPN
ejpam-4976	107	23	−k	−k	PROPN
ejpam-4976	107	24	)	)	PUNCT
ejpam-4976	107	25	⊆	⊆	NUM
ejpam-4976	107	26	τ1τ2	τ1τ2	NOUN
ejpam-4976	107	27	-	-	NUM
ejpam-4976	107	28	int(f	int(f	PRON
ejpam-4976	107	29	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	107	30	-	-	PUNCT
ejpam-4976	107	31	cl(y	cl(y	NOUN
ejpam-4976	107	32	−k	−k	NOUN
ejpam-4976	107	33	)	)	PUNCT
ejpam-4976	107	34	)	)	PUNCT
ejpam-4976	107	35	)	)	PUNCT
ejpam-4976	108	1	=	=	PUNCT
ejpam-4976	108	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	108	3	-	-	NUM
ejpam-4976	108	4	int(f	int(f	VERB
ejpam-4976	108	5	−1(y	−1(y	VERB
ejpam-4976	108	6	−	−	ADP
ejpam-4976	108	7	σ1σ2	σ1σ2	NUM
ejpam-4976	108	8	-	-	PUNCT
ejpam-4976	108	9	int(k	int(k	NOUN
ejpam-4976	108	10	)	)	PUNCT
ejpam-4976	108	11	)	)	PUNCT
ejpam-4976	108	12	)	)	PUNCT
ejpam-4976	109	1	=	=	PUNCT
ejpam-4976	110	1	x	x	X
ejpam-4976	110	2	−	−	ADP
ejpam-4976	110	3	τ1τ2	τ1τ2	NOUN
ejpam-4976	110	4	-	-	NOUN
ejpam-4976	110	5	cl(f	cl(f	NOUN
ejpam-4976	110	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	110	7	-	-	PUNCT
ejpam-4976	110	8	int(k	int(k	NUM
ejpam-4976	110	9	)	)	PUNCT
ejpam-4976	110	10	)	)	PUNCT
ejpam-4976	110	11	)	)	PUNCT
ejpam-4976	110	12	.	.	PUNCT
ejpam-4976	111	1	thus	thus	ADV
ejpam-4976	111	2	,	,	PUNCT
ejpam-4976	111	3	τ1τ2	τ1τ2	NOUN
ejpam-4976	111	4	-	-	ADJ
ejpam-4976	111	5	cl(f	cl(f	NOUN
ejpam-4976	111	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	111	7	-	-	PUNCT
ejpam-4976	111	8	int(k	int(k	NUM
ejpam-4976	111	9	)	)	PUNCT
ejpam-4976	111	10	)	)	PUNCT
ejpam-4976	111	11	)	)	PUNCT
ejpam-4976	112	1	⊆	⊆	NUM
ejpam-4976	112	2	f−1(k	f−1(k	NOUN
ejpam-4976	112	3	)	)	PUNCT
ejpam-4976	112	4	.	.	PUNCT
ejpam-4976	113	1	(	(	PUNCT
ejpam-4976	113	2	3	3	X
ejpam-4976	113	3	)	)	PUNCT
ejpam-4976	113	4	⇒	⇒	NOUN
ejpam-4976	113	5	(	(	PUNCT
ejpam-4976	113	6	4	4	NUM
ejpam-4976	113	7	):	):	PUNCT
ejpam-4976	113	8	let	let	VERB
ejpam-4976	113	9	b	b	X
ejpam-4976	113	10	be	be	AUX
ejpam-4976	113	11	any	any	DET
ejpam-4976	113	12	subset	subset	NOUN
ejpam-4976	113	13	of	of	ADP
ejpam-4976	113	14	y	y	PROPN
ejpam-4976	113	15	.	.	PUNCT
ejpam-4976	114	1	then	then	ADV
ejpam-4976	114	2	,	,	PUNCT
ejpam-4976	114	3	σ1σ2	σ1σ2	X
ejpam-4976	114	4	-	-	PUNCT
ejpam-4976	114	5	int(b	int(b	NOUN
ejpam-4976	114	6	)	)	PUNCT
ejpam-4976	114	7	is	be	AUX
ejpam-4976	114	8	σ1σ2	σ1σ2	NOUN
ejpam-4976	114	9	-	-	ADJ
ejpam-4976	114	10	closed	closed	ADJ
ejpam-4976	114	11	in	in	ADP
ejpam-4976	114	12	y	y	PROPN
ejpam-4976	114	13	.	.	PUNCT
ejpam-4976	115	1	by	by	ADP
ejpam-4976	115	2	(	(	PUNCT
ejpam-4976	115	3	3	3	NUM
ejpam-4976	115	4	)	)	PUNCT
ejpam-4976	115	5	,	,	PUNCT
ejpam-4976	115	6	τ1τ2	τ1τ2	NOUN
ejpam-4976	115	7	-	-	ADJ
ejpam-4976	115	8	cl(f	cl(f	NOUN
ejpam-4976	115	9	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	115	10	-	-	PUNCT
ejpam-4976	115	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	115	12	-	-	PUNCT
ejpam-4976	115	13	cl(b	cl(b	NOUN
ejpam-4976	115	14	)	)	PUNCT
ejpam-4976	115	15	)	)	PUNCT
ejpam-4976	115	16	)	)	PUNCT
ejpam-4976	115	17	)	)	PUNCT
ejpam-4976	116	1	⊆	⊆	NUM
ejpam-4976	116	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	116	3	-	-	PUNCT
ejpam-4976	116	4	cl(b	cl(b	NOUN
ejpam-4976	116	5	)	)	PUNCT
ejpam-4976	116	6	)	)	PUNCT
ejpam-4976	116	7	.	.	PUNCT
ejpam-4976	117	1	(	(	PUNCT
ejpam-4976	117	2	4	4	X
ejpam-4976	117	3	)	)	PUNCT
ejpam-4976	117	4	⇒	⇒	NOUN
ejpam-4976	117	5	(	(	PUNCT
ejpam-4976	117	6	5	5	NUM
ejpam-4976	117	7	):	):	PUNCT
ejpam-4976	117	8	let	let	VERB
ejpam-4976	117	9	b	b	X
ejpam-4976	117	10	be	be	AUX
ejpam-4976	117	11	any	any	DET
ejpam-4976	117	12	subset	subset	NOUN
ejpam-4976	117	13	of	of	ADP
ejpam-4976	117	14	y	y	PROPN
ejpam-4976	117	15	.	.	PUNCT
ejpam-4976	118	1	by	by	ADP
ejpam-4976	118	2	(	(	PUNCT
ejpam-4976	118	3	4	4	NUM
ejpam-4976	118	4	)	)	PUNCT
ejpam-4976	118	5	,	,	PUNCT
ejpam-4976	118	6	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	118	7	-	-	PUNCT
ejpam-4976	118	8	int(b	int(b	NOUN
ejpam-4976	118	9	)	)	PUNCT
ejpam-4976	118	10	)	)	PUNCT
ejpam-4976	118	11	=	=	PUNCT
ejpam-4976	119	1	x	x	X
ejpam-4976	119	2	−	−	PRON
ejpam-4976	119	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	119	4	-	-	PUNCT
ejpam-4976	119	5	cl(y	cl(y	NOUN
ejpam-4976	119	6	−b	−b	NOUN
ejpam-4976	119	7	)	)	PUNCT
ejpam-4976	119	8	)	)	PUNCT
ejpam-4976	120	1	⊆	⊆	NUM
ejpam-4976	120	2	x	x	SYM
ejpam-4976	120	3	−	−	NUM
ejpam-4976	120	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	120	5	-	-	NOUN
ejpam-4976	120	6	cl(f	cl(f	NOUN
ejpam-4976	120	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	120	8	-	-	PUNCT
ejpam-4976	120	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	120	10	-	-	PUNCT
ejpam-4976	120	11	cl(y	cl(y	NOUN
ejpam-4976	120	12	−b	−b	NOUN
ejpam-4976	120	13	)	)	PUNCT
ejpam-4976	120	14	)	)	PUNCT
ejpam-4976	120	15	)	)	PUNCT
ejpam-4976	120	16	)	)	PUNCT
ejpam-4976	121	1	=	=	PUNCT
ejpam-4976	121	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	121	3	-	-	PUNCT
ejpam-4976	121	4	int(f	int(f	VERB
ejpam-4976	121	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	121	6	-	-	PUNCT
ejpam-4976	121	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	121	8	-	-	PUNCT
ejpam-4976	121	9	int(b	int(b	NOUN
ejpam-4976	121	10	)	)	PUNCT
ejpam-4976	121	11	)	)	PUNCT
ejpam-4976	121	12	)	)	PUNCT
ejpam-4976	121	13	)	)	PUNCT
ejpam-4976	121	14	.	.	PUNCT
ejpam-4976	122	1	(	(	PUNCT
ejpam-4976	122	2	5	5	X
ejpam-4976	122	3	)	)	PUNCT
ejpam-4976	122	4	⇒	⇒	NOUN
ejpam-4976	122	5	(	(	PUNCT
ejpam-4976	122	6	6	6	NUM
ejpam-4976	122	7	):	):	PUNCT
ejpam-4976	122	8	let	let	VERB
ejpam-4976	122	9	v	v	PART
ejpam-4976	122	10	be	be	AUX
ejpam-4976	122	11	any	any	DET
ejpam-4976	122	12	σ1σ2	σ1σ2	NOUN
ejpam-4976	122	13	-	-	ADJ
ejpam-4976	122	14	open	open	ADJ
ejpam-4976	122	15	set	set	NOUN
ejpam-4976	122	16	of	of	ADP
ejpam-4976	122	17	y	y	PROPN
ejpam-4976	122	18	and	and	CCONJ
ejpam-4976	122	19	x	x	PROPN
ejpam-4976	122	20	̸∈	̸∈	PROPN
ejpam-4976	122	21	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	122	22	-	-	PUNCT
ejpam-4976	122	23	cl(v	cl(v	NOUN
ejpam-4976	122	24	)	)	PUNCT
ejpam-4976	122	25	)	)	PUNCT
ejpam-4976	122	26	.	.	PUNCT
ejpam-4976	123	1	then	then	ADV
ejpam-4976	123	2	,	,	PUNCT
ejpam-4976	123	3	there	there	PRON
ejpam-4976	123	4	exists	exist	VERB
ejpam-4976	123	5	a	a	DET
ejpam-4976	123	6	σ1σ2	σ1σ2	NUM
ejpam-4976	123	7	-	-	ADJ
ejpam-4976	123	8	open	open	ADJ
ejpam-4976	123	9	set	set	NOUN
ejpam-4976	123	10	u	u	NOUN
ejpam-4976	123	11	of	of	ADP
ejpam-4976	123	12	y	y	PROPN
ejpam-4976	123	13	containing	contain	VERB
ejpam-4976	123	14	f(x	f(x	PROPN
ejpam-4976	123	15	)	)	PUNCT
ejpam-4976	123	16	such	such	ADJ
ejpam-4976	123	17	that	that	SCONJ
ejpam-4976	123	18	u	u	PROPN
ejpam-4976	123	19	∩	∩	NOUN
ejpam-4976	123	20	v	v	ADJ
ejpam-4976	123	21	=	=	PUNCT
ejpam-4976	123	22	∅.	∅.	X
ejpam-4976	123	23	by	by	ADP
ejpam-4976	123	24	(	(	PUNCT
ejpam-4976	123	25	5	5	NUM
ejpam-4976	123	26	)	)	PUNCT
ejpam-4976	123	27	,	,	PUNCT
ejpam-4976	123	28	x	x	PUNCT
ejpam-4976	123	29	∈	∈	PROPN
ejpam-4976	123	30	f−1(u	f−1(u	PROPN
ejpam-4976	123	31	)	)	PUNCT
ejpam-4976	123	32	⊆	⊆	NUM
ejpam-4976	123	33	τ1τ2	τ1τ2	NOUN
ejpam-4976	123	34	-	-	NUM
ejpam-4976	123	35	int(f	int(f	PRON
ejpam-4976	123	36	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	123	37	-	-	PUNCT
ejpam-4976	123	38	cl(u	cl(u	NOUN
ejpam-4976	123	39	)	)	PUNCT
ejpam-4976	123	40	)	)	PUNCT
ejpam-4976	123	41	)	)	PUNCT
ejpam-4976	123	42	and	and	CCONJ
ejpam-4976	123	43	there	there	PRON
ejpam-4976	123	44	exists	exist	VERB
ejpam-4976	123	45	a	a	DET
ejpam-4976	123	46	τ1τ2	τ1τ2	NOUN
ejpam-4976	123	47	-	-	ADJ
ejpam-4976	123	48	open	open	ADJ
ejpam-4976	123	49	set	set	NOUN
ejpam-4976	123	50	g	g	NOUN
ejpam-4976	123	51	of	of	ADP
ejpam-4976	123	52	x	x	PUNCT
ejpam-4976	123	53	containing	contain	VERB
ejpam-4976	123	54	x	x	PUNCT
ejpam-4976	123	55	such	such	ADJ
ejpam-4976	123	56	that	that	DET
ejpam-4976	123	57	f(g	f(g	NOUN
ejpam-4976	123	58	)	)	PUNCT
ejpam-4976	123	59	⊆	⊆	NUM
ejpam-4976	123	60	σ1σ2	σ1σ2	NOUN
ejpam-4976	123	61	-	-	PUNCT
ejpam-4976	123	62	cl(u	cl(u	NUM
ejpam-4976	123	63	)	)	PUNCT
ejpam-4976	123	64	.	.	PUNCT
ejpam-4976	124	1	thus	thus	ADV
ejpam-4976	124	2	,	,	PUNCT
ejpam-4976	124	3	g	g	PROPN
ejpam-4976	124	4	∩	∩	ADJ
ejpam-4976	124	5	f−1(v	f−1(v	NOUN
ejpam-4976	124	6	)	)	PUNCT
ejpam-4976	125	1	=	=	NOUN
ejpam-4976	125	2	∅	∅	NOUN
ejpam-4976	125	3	and	and	CCONJ
ejpam-4976	125	4	hence	hence	ADV
ejpam-4976	125	5	x	x	X
ejpam-4976	125	6	̸∈	̸∈	PROPN
ejpam-4976	125	7	τ1τ2	τ1τ2	PROPN
ejpam-4976	125	8	-	-	PROPN
ejpam-4976	125	9	cl(f	cl(f	PRON
ejpam-4976	125	10	−1(v	−1(v	PROPN
ejpam-4976	125	11	)	)	PUNCT
ejpam-4976	125	12	)	)	PUNCT
ejpam-4976	125	13	.	.	PUNCT
ejpam-4976	126	1	this	this	PRON
ejpam-4976	126	2	shows	show	VERB
ejpam-4976	126	3	that	that	SCONJ
ejpam-4976	126	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	126	5	-	-	PROPN
ejpam-4976	126	6	cl(f	cl(f	PRON
ejpam-4976	126	7	−1(v	−1(v	NOUN
ejpam-4976	126	8	)	)	PUNCT
ejpam-4976	126	9	)	)	PUNCT
ejpam-4976	126	10	⊆	⊆	NUM
ejpam-4976	126	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	126	12	-	-	PUNCT
ejpam-4976	126	13	cl(v	cl(v	NOUN
ejpam-4976	126	14	)	)	PUNCT
ejpam-4976	126	15	)	)	PUNCT
ejpam-4976	126	16	.	.	PUNCT
ejpam-4976	127	1	(	(	PUNCT
ejpam-4976	127	2	6	6	X
ejpam-4976	127	3	)	)	PUNCT
ejpam-4976	127	4	⇒	⇒	NOUN
ejpam-4976	127	5	(	(	PUNCT
ejpam-4976	127	6	1	1	NUM
ejpam-4976	127	7	):	):	PUNCT
ejpam-4976	127	8	let	let	VERB
ejpam-4976	127	9	x	x	PUNCT
ejpam-4976	127	10	∈	∈	PROPN
ejpam-4976	127	11	x	x	X
ejpam-4976	127	12	and	and	CCONJ
ejpam-4976	127	13	v	v	X
ejpam-4976	127	14	be	be	AUX
ejpam-4976	127	15	any	any	DET
ejpam-4976	127	16	be	be	AUX
ejpam-4976	127	17	any	any	DET
ejpam-4976	127	18	σ1σ2	σ1σ2	NOUN
ejpam-4976	127	19	-	-	ADJ
ejpam-4976	127	20	open	open	ADJ
ejpam-4976	127	21	set	set	NOUN
ejpam-4976	127	22	of	of	ADP
ejpam-4976	127	23	y	y	PROPN
ejpam-4976	127	24	containing	contain	VERB
ejpam-4976	127	25	f(x	f(x	PROPN
ejpam-4976	127	26	)	)	PUNCT
ejpam-4976	127	27	.	.	PUNCT
ejpam-4976	128	1	since	since	SCONJ
ejpam-4976	128	2	v	v	NUM
ejpam-4976	128	3	=	=	SYM
ejpam-4976	128	4	σ1σ2	σ1σ2	NUM
ejpam-4976	128	5	-	-	PUNCT
ejpam-4976	128	6	int(v	int(v	NOUN
ejpam-4976	128	7	)	)	PUNCT
ejpam-4976	128	8	⊆	⊆	NUM
ejpam-4976	128	9	σ1σ2	σ1σ2	X
ejpam-4976	128	10	-	-	PUNCT
ejpam-4976	128	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	128	12	-	-	PUNCT
ejpam-4976	128	13	cl(v	cl(v	NOUN
ejpam-4976	128	14	)	)	PUNCT
ejpam-4976	128	15	)	)	PUNCT
ejpam-4976	128	16	,	,	PUNCT
ejpam-4976	128	17	by	by	ADP
ejpam-4976	128	18	(	(	PUNCT
ejpam-4976	128	19	6	6	X
ejpam-4976	128	20	)	)	PUNCT
ejpam-4976	128	21	we	we	PRON
ejpam-4976	128	22	have	have	VERB
ejpam-4976	128	23	x	x	X
ejpam-4976	128	24	∈	∈	PROPN
ejpam-4976	128	25	f−1(v	f−1(v	NOUN
ejpam-4976	128	26	)	)	PUNCT
ejpam-4976	128	27	⊆	⊆	NUM
ejpam-4976	128	28	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	128	29	-	-	PUNCT
ejpam-4976	128	30	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	128	31	-	-	PUNCT
ejpam-4976	128	32	cl(v	cl(v	NOUN
ejpam-4976	128	33	)	)	PUNCT
ejpam-4976	128	34	)	)	PUNCT
ejpam-4976	128	35	)	)	PUNCT
ejpam-4976	129	1	=	=	PUNCT
ejpam-4976	130	1	x	x	X
ejpam-4976	130	2	−	−	PRON
ejpam-4976	130	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	130	4	-	-	PUNCT
ejpam-4976	130	5	cl(y	cl(y	NOUN
ejpam-4976	130	6	−	−	NOUN
ejpam-4976	130	7	σ1σ2	σ1σ2	NOUN
ejpam-4976	130	8	-	-	NUM
ejpam-4976	130	9	cl(v	cl(v	NOUN
ejpam-4976	130	10	)	)	PUNCT
ejpam-4976	130	11	)	)	PUNCT
ejpam-4976	130	12	)	)	PUNCT
ejpam-4976	131	1	c.	c.	PROPN
ejpam-4976	131	2	boonpok	boonpok	PROPN
ejpam-4976	131	3	,	,	PUNCT
ejpam-4976	131	4	c.	c.	PROPN
ejpam-4976	131	5	klanarong	klanarong	PROPN
ejpam-4976	131	6	/	/	SYM
ejpam-4976	131	7	eur	eur	PROPN
ejpam-4976	131	8	.	.	PUNCT
ejpam-4976	132	1	j.	j.	PROPN
ejpam-4976	132	2	pure	pure	PROPN
ejpam-4976	132	3	appl	appl	PROPN
ejpam-4976	132	4	.	.	PROPN
ejpam-4976	132	5	math	math	PROPN
ejpam-4976	132	6	,	,	PUNCT
ejpam-4976	132	7	17	17	NUM
ejpam-4976	132	8	(	(	PUNCT
ejpam-4976	132	9	1	1	NUM
ejpam-4976	132	10	)	)	PUNCT
ejpam-4976	132	11	(	(	PUNCT
ejpam-4976	132	12	2024	2024	NUM
ejpam-4976	132	13	)	)	PUNCT
ejpam-4976	132	14	,	,	PUNCT
ejpam-4976	132	15	416	416	NUM
ejpam-4976	132	16	-	-	SYM
ejpam-4976	132	17	425	425	NUM
ejpam-4976	132	18	420	420	NUM
ejpam-4976	132	19	⊆	⊆	NUM
ejpam-4976	132	20	x	x	SYM
ejpam-4976	132	21	−	−	PRON
ejpam-4976	132	22	τ1τ2	τ1τ2	NOUN
ejpam-4976	132	23	-	-	NOUN
ejpam-4976	132	24	cl(f	cl(f	NOUN
ejpam-4976	132	25	−1(y	−1(y	ADP
ejpam-4976	132	26	−	−	PUNCT
ejpam-4976	132	27	σ1σ2	σ1σ2	NOUN
ejpam-4976	132	28	-	-	NUM
ejpam-4976	132	29	cl(v	cl(v	NOUN
ejpam-4976	132	30	)	)	PUNCT
ejpam-4976	132	31	)	)	PUNCT
ejpam-4976	132	32	)	)	PUNCT
ejpam-4976	133	1	=	=	PUNCT
ejpam-4976	133	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	133	3	-	-	PUNCT
ejpam-4976	133	4	int(f	int(f	VERB
ejpam-4976	133	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	133	6	-	-	PUNCT
ejpam-4976	133	7	cl(v	cl(v	NOUN
ejpam-4976	133	8	)	)	PUNCT
ejpam-4976	133	9	)	)	PUNCT
ejpam-4976	133	10	)	)	PUNCT
ejpam-4976	133	11	.	.	PUNCT
ejpam-4976	134	1	there	there	PRON
ejpam-4976	134	2	exists	exist	VERB
ejpam-4976	134	3	a	a	DET
ejpam-4976	134	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	134	5	-	-	ADJ
ejpam-4976	134	6	open	open	ADJ
ejpam-4976	134	7	set	set	ADJ
ejpam-4976	134	8	u	u	NOUN
ejpam-4976	134	9	of	of	ADP
ejpam-4976	134	10	x	x	PUNCT
ejpam-4976	134	11	containing	contain	VERB
ejpam-4976	134	12	x	x	PUNCT
ejpam-4976	134	13	such	such	ADJ
ejpam-4976	134	14	that	that	SCONJ
ejpam-4976	134	15	u	u	PROPN
ejpam-4976	134	16	⊆	⊆	NUM
ejpam-4976	134	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	134	18	-	-	PUNCT
ejpam-4976	134	19	cl(v	cl(v	NOUN
ejpam-4976	134	20	)	)	PUNCT
ejpam-4976	134	21	)	)	PUNCT
ejpam-4976	134	22	;	;	PUNCT
ejpam-4976	134	23	hence	hence	ADV
ejpam-4976	134	24	f(u	f(u	PROPN
ejpam-4976	134	25	)	)	PUNCT
ejpam-4976	135	1	⊆	⊆	NUM
ejpam-4976	135	2	σ1σ2	σ1σ2	NOUN
ejpam-4976	135	3	-	-	NUM
ejpam-4976	135	4	cl(v	cl(v	NOUN
ejpam-4976	135	5	)	)	PUNCT
ejpam-4976	135	6	.	.	PUNCT
ejpam-4976	136	1	thus	thus	ADV
ejpam-4976	136	2	,	,	PUNCT
ejpam-4976	136	3	f	f	PROPN
ejpam-4976	136	4	is	be	AUX
ejpam-4976	136	5	weakly	weakly	ADJ
ejpam-4976	136	6	(	(	PUNCT
ejpam-4976	136	7	τ1	τ1	NOUN
ejpam-4976	136	8	,	,	PUNCT
ejpam-4976	136	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	136	10	at	at	ADP
ejpam-4976	136	11	x.	x.	NOUN
ejpam-4976	136	12	this	this	PRON
ejpam-4976	136	13	shows	show	VERB
ejpam-4976	136	14	that	that	SCONJ
ejpam-4976	136	15	f	f	PROPN
ejpam-4976	136	16	is	be	AUX
ejpam-4976	136	17	weakly	weakly	ADJ
ejpam-4976	136	18	(	(	PUNCT
ejpam-4976	136	19	τ1	τ1	NOUN
ejpam-4976	136	20	,	,	PUNCT
ejpam-4976	136	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	136	22	.	.	PUNCT
ejpam-4976	137	1	theorem	theorem	NOUN
ejpam-4976	137	2	5	5	NUM
ejpam-4976	137	3	.	.	X
ejpam-4976	137	4	for	for	ADP
ejpam-4976	137	5	a	a	DET
ejpam-4976	137	6	function	function	NOUN
ejpam-4976	137	7	(	(	PUNCT
ejpam-4976	137	8	x	x	NOUN
ejpam-4976	137	9	,	,	PUNCT
ejpam-4976	137	10	τ1	τ1	NOUN
ejpam-4976	137	11	,	,	PUNCT
ejpam-4976	137	12	τ2	τ2	NOUN
ejpam-4976	137	13	)	)	PUNCT
ejpam-4976	137	14	→	→	SYM
ejpam-4976	137	15	(	(	PUNCT
ejpam-4976	137	16	y	y	PROPN
ejpam-4976	137	17	,	,	PUNCT
ejpam-4976	137	18	σ1	σ1	PROPN
ejpam-4976	137	19	,	,	PUNCT
ejpam-4976	137	20	σ2	σ2	NOUN
ejpam-4976	137	21	)	)	PUNCT
ejpam-4976	137	22	,	,	PUNCT
ejpam-4976	137	23	the	the	DET
ejpam-4976	137	24	following	follow	VERB
ejpam-4976	137	25	properties	property	NOUN
ejpam-4976	137	26	are	be	AUX
ejpam-4976	137	27	equivalent	equivalent	ADJ
ejpam-4976	137	28	:	:	PUNCT
ejpam-4976	137	29	(	(	PUNCT
ejpam-4976	137	30	1	1	X
ejpam-4976	137	31	)	)	PUNCT
ejpam-4976	137	32	f	f	PROPN
ejpam-4976	137	33	is	be	AUX
ejpam-4976	137	34	weakly	weakly	ADJ
ejpam-4976	137	35	(	(	PUNCT
ejpam-4976	137	36	τ1	τ1	NOUN
ejpam-4976	137	37	,	,	PUNCT
ejpam-4976	137	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	137	39	;	;	PUNCT
ejpam-4976	137	40	(	(	PUNCT
ejpam-4976	137	41	2	2	X
ejpam-4976	137	42	)	)	PUNCT
ejpam-4976	137	43	τ1τ2	τ1τ2	NOUN
ejpam-4976	137	44	-	-	NOUN
ejpam-4976	137	45	cl(f	cl(f	NOUN
ejpam-4976	137	46	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	137	47	-	-	PUNCT
ejpam-4976	137	48	int(k	int(k	NUM
ejpam-4976	137	49	)	)	PUNCT
ejpam-4976	137	50	)	)	PUNCT
ejpam-4976	137	51	)	)	PUNCT
ejpam-4976	138	1	⊆	⊆	NUM
ejpam-4976	138	2	f−1(k	f−1(k	PROPN
ejpam-4976	138	3	)	)	PUNCT
ejpam-4976	138	4	for	for	ADP
ejpam-4976	138	5	every	every	DET
ejpam-4976	138	6	(	(	PUNCT
ejpam-4976	138	7	σ1	σ1	PROPN
ejpam-4976	138	8	,	,	PUNCT
ejpam-4976	138	9	σ2)r	σ2)r	NOUN
ejpam-4976	138	10	-	-	PUNCT
ejpam-4976	138	11	closed	close	VERB
ejpam-4976	138	12	set	set	ADJ
ejpam-4976	138	13	k	k	PROPN
ejpam-4976	138	14	of	of	ADP
ejpam-4976	138	15	y	y	PROPN
ejpam-4976	138	16	;	;	PUNCT
ejpam-4976	138	17	(	(	PUNCT
ejpam-4976	138	18	3	3	X
ejpam-4976	138	19	)	)	PUNCT
ejpam-4976	138	20	τ1τ2	τ1τ2	NOUN
ejpam-4976	138	21	-	-	NOUN
ejpam-4976	138	22	cl(f	cl(f	NOUN
ejpam-4976	138	23	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	138	24	-	-	PUNCT
ejpam-4976	138	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	138	26	-	-	PUNCT
ejpam-4976	138	27	cl(v	cl(v	NOUN
ejpam-4976	138	28	)	)	PUNCT
ejpam-4976	138	29	)	)	PUNCT
ejpam-4976	138	30	)	)	PUNCT
ejpam-4976	138	31	)	)	PUNCT
ejpam-4976	139	1	⊆	⊆	NUM
ejpam-4976	139	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	139	3	-	-	PUNCT
ejpam-4976	139	4	cl(v	cl(v	NOUN
ejpam-4976	139	5	)	)	PUNCT
ejpam-4976	139	6	)	)	PUNCT
ejpam-4976	139	7	for	for	ADP
ejpam-4976	139	8	every	every	DET
ejpam-4976	139	9	(	(	PUNCT
ejpam-4976	139	10	σ1	σ1	PROPN
ejpam-4976	139	11	,	,	PUNCT
ejpam-4976	139	12	σ2)β	σ2)β	NOUN
ejpam-4976	139	13	-	-	PUNCT
ejpam-4976	139	14	open	open	NOUN
ejpam-4976	139	15	set	set	NOUN
ejpam-4976	139	16	v	v	NOUN
ejpam-4976	139	17	of	of	ADP
ejpam-4976	139	18	y	y	PROPN
ejpam-4976	139	19	;	;	PUNCT
ejpam-4976	139	20	(	(	PUNCT
ejpam-4976	139	21	4	4	X
ejpam-4976	139	22	)	)	PUNCT
ejpam-4976	139	23	τ1τ2	τ1τ2	NOUN
ejpam-4976	139	24	-	-	NOUN
ejpam-4976	139	25	cl(f	cl(f	NOUN
ejpam-4976	139	26	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	139	27	-	-	PUNCT
ejpam-4976	139	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	139	29	-	-	PUNCT
ejpam-4976	139	30	cl(v	cl(v	NOUN
ejpam-4976	139	31	)	)	PUNCT
ejpam-4976	139	32	)	)	PUNCT
ejpam-4976	139	33	)	)	PUNCT
ejpam-4976	139	34	)	)	PUNCT
ejpam-4976	140	1	⊆	⊆	NUM
ejpam-4976	140	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	140	3	-	-	PUNCT
ejpam-4976	140	4	cl(v	cl(v	NOUN
ejpam-4976	140	5	)	)	PUNCT
ejpam-4976	140	6	)	)	PUNCT
ejpam-4976	140	7	for	for	ADP
ejpam-4976	140	8	every	every	DET
ejpam-4976	140	9	(	(	PUNCT
ejpam-4976	140	10	σ1	σ1	PROPN
ejpam-4976	140	11	,	,	PUNCT
ejpam-4976	140	12	σ2)s	σ2)s	NOUN
ejpam-4976	140	13	-	-	PUNCT
ejpam-4976	140	14	open	open	NOUN
ejpam-4976	140	15	set	set	NOUN
ejpam-4976	140	16	v	v	NOUN
ejpam-4976	140	17	of	of	ADP
ejpam-4976	140	18	y	y	PROPN
ejpam-4976	140	19	.	.	PUNCT
ejpam-4976	141	1	proof	proof	NOUN
ejpam-4976	141	2	.	.	PUNCT
ejpam-4976	142	1	(	(	PUNCT
ejpam-4976	142	2	1	1	X
ejpam-4976	142	3	)	)	PUNCT
ejpam-4976	142	4	⇒	⇒	NOUN
ejpam-4976	142	5	(	(	PUNCT
ejpam-4976	142	6	2	2	NUM
ejpam-4976	142	7	):	):	PUNCT
ejpam-4976	142	8	let	let	VERB
ejpam-4976	142	9	k	k	PRON
ejpam-4976	142	10	be	be	AUX
ejpam-4976	142	11	any	any	DET
ejpam-4976	142	12	(	(	PUNCT
ejpam-4976	142	13	σ1	σ1	NOUN
ejpam-4976	142	14	,	,	PUNCT
ejpam-4976	142	15	σ2)r	σ2)r	NOUN
ejpam-4976	142	16	-	-	PUNCT
ejpam-4976	142	17	closed	close	VERB
ejpam-4976	142	18	set	set	NOUN
ejpam-4976	142	19	of	of	ADP
ejpam-4976	142	20	y	y	PROPN
ejpam-4976	142	21	.	.	PUNCT
ejpam-4976	143	1	then	then	ADV
ejpam-4976	143	2	,	,	PUNCT
ejpam-4976	143	3	σ1σ2	σ1σ2	NOUN
ejpam-4976	143	4	-	-	PUNCT
ejpam-4976	143	5	int(k	int(k	NUM
ejpam-4976	143	6	)	)	PUNCT
ejpam-4976	143	7	is	be	AUX
ejpam-4976	143	8	σ1σ2	σ1σ2	NOUN
ejpam-4976	143	9	-	-	ADJ
ejpam-4976	143	10	open	open	ADJ
ejpam-4976	143	11	in	in	ADP
ejpam-4976	143	12	y	y	PROPN
ejpam-4976	143	13	,	,	PUNCT
ejpam-4976	143	14	by	by	ADP
ejpam-4976	143	15	theorem	theorem	NOUN
ejpam-4976	143	16	4	4	NUM
ejpam-4976	143	17	(	(	PUNCT
ejpam-4976	143	18	6	6	NUM
ejpam-4976	143	19	)	)	PUNCT
ejpam-4976	143	20	we	we	PRON
ejpam-4976	143	21	have	have	VERB
ejpam-4976	143	22	τ1τ2	τ1τ2	NOUN
ejpam-4976	143	23	-	-	ADJ
ejpam-4976	143	24	cl(f	cl(f	NOUN
ejpam-4976	143	25	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	143	26	-	-	PUNCT
ejpam-4976	143	27	int(k	int(k	NUM
ejpam-4976	143	28	)	)	PUNCT
ejpam-4976	143	29	)	)	PUNCT
ejpam-4976	143	30	)	)	PUNCT
ejpam-4976	144	1	⊆	⊆	NUM
ejpam-4976	144	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	144	3	-	-	PUNCT
ejpam-4976	144	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	144	5	-	-	PUNCT
ejpam-4976	144	6	cl(k	cl(k	NUM
ejpam-4976	144	7	)	)	PUNCT
ejpam-4976	144	8	)	)	PUNCT
ejpam-4976	144	9	)	)	PUNCT
ejpam-4976	145	1	=	=	PUNCT
ejpam-4976	145	2	f−1(k	f−1(k	PROPN
ejpam-4976	145	3	)	)	PUNCT
ejpam-4976	145	4	.	.	PUNCT
ejpam-4976	146	1	(	(	PUNCT
ejpam-4976	146	2	2	2	X
ejpam-4976	146	3	)	)	PUNCT
ejpam-4976	146	4	⇒	⇒	NOUN
ejpam-4976	146	5	(	(	PUNCT
ejpam-4976	146	6	3	3	NUM
ejpam-4976	146	7	):	):	PUNCT
ejpam-4976	146	8	let	let	VERB
ejpam-4976	146	9	v	v	PART
ejpam-4976	146	10	be	be	AUX
ejpam-4976	146	11	any	any	DET
ejpam-4976	146	12	(	(	PUNCT
ejpam-4976	146	13	σ1	σ1	PROPN
ejpam-4976	146	14	,	,	PUNCT
ejpam-4976	146	15	σ2)β	σ2)β	NOUN
ejpam-4976	146	16	-	-	PUNCT
ejpam-4976	146	17	open	open	ADJ
ejpam-4976	146	18	set	set	NOUN
ejpam-4976	146	19	of	of	ADP
ejpam-4976	146	20	y	y	PROPN
ejpam-4976	146	21	.	.	PUNCT
ejpam-4976	147	1	then	then	ADV
ejpam-4976	147	2	,	,	PUNCT
ejpam-4976	147	3	we	we	PRON
ejpam-4976	147	4	have	have	VERB
ejpam-4976	147	5	σ1σ2	σ1σ2	NOUN
ejpam-4976	147	6	-	-	NUM
ejpam-4976	147	7	cl(v	cl(v	NOUN
ejpam-4976	147	8	)	)	PUNCT
ejpam-4976	148	1	⊆	⊆	NUM
ejpam-4976	148	2	σ1σ2	σ1σ2	X
ejpam-4976	148	3	-	-	PUNCT
ejpam-4976	148	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	148	5	-	-	PUNCT
ejpam-4976	148	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	148	7	-	-	PUNCT
ejpam-4976	148	8	cl(v	cl(v	NOUN
ejpam-4976	148	9	)	)	PUNCT
ejpam-4976	148	10	)	)	PUNCT
ejpam-4976	148	11	)	)	PUNCT
ejpam-4976	149	1	⊆	⊆	X
ejpam-4976	149	2	σ1σ2	σ1σ2	NOUN
ejpam-4976	149	3	-	-	NUM
ejpam-4976	149	4	cl(v	cl(v	NOUN
ejpam-4976	149	5	)	)	PUNCT
ejpam-4976	149	6	and	and	CCONJ
ejpam-4976	149	7	hence	hence	ADV
ejpam-4976	149	8	σ1σ2	σ1σ2	NOUN
ejpam-4976	149	9	-	-	NOUN
ejpam-4976	149	10	cl(v	cl(v	NOUN
ejpam-4976	149	11	)	)	PUNCT
ejpam-4976	149	12	is	be	AUX
ejpam-4976	149	13	(	(	PUNCT
ejpam-4976	149	14	σ1	σ1	NOUN
ejpam-4976	149	15	,	,	PUNCT
ejpam-4976	149	16	σ2)r	σ2)r	NOUN
ejpam-4976	149	17	-	-	PUNCT
ejpam-4976	149	18	closed	closed	ADJ
ejpam-4976	149	19	.	.	PUNCT
ejpam-4976	150	1	by	by	ADP
ejpam-4976	150	2	(	(	PUNCT
ejpam-4976	150	3	2	2	NUM
ejpam-4976	150	4	)	)	PUNCT
ejpam-4976	150	5	,	,	PUNCT
ejpam-4976	150	6	τ1τ2	τ1τ2	NOUN
ejpam-4976	150	7	-	-	ADJ
ejpam-4976	150	8	cl(f	cl(f	NOUN
ejpam-4976	150	9	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	150	10	-	-	PUNCT
ejpam-4976	150	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	150	12	-	-	PUNCT
ejpam-4976	150	13	cl(v	cl(v	NOUN
ejpam-4976	150	14	)	)	PUNCT
ejpam-4976	150	15	)	)	PUNCT
ejpam-4976	150	16	)	)	PUNCT
ejpam-4976	150	17	)	)	PUNCT
ejpam-4976	151	1	⊆	⊆	NUM
ejpam-4976	151	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	151	3	-	-	PUNCT
ejpam-4976	151	4	cl(v	cl(v	NOUN
ejpam-4976	151	5	)	)	PUNCT
ejpam-4976	151	6	)	)	PUNCT
ejpam-4976	151	7	.	.	PUNCT
ejpam-4976	152	1	(	(	PUNCT
ejpam-4976	152	2	3	3	X
ejpam-4976	152	3	)	)	PUNCT
ejpam-4976	152	4	⇒	⇒	NOUN
ejpam-4976	152	5	(	(	PUNCT
ejpam-4976	152	6	4	4	NUM
ejpam-4976	152	7	):	):	PUNCT
ejpam-4976	152	8	this	this	PRON
ejpam-4976	152	9	is	be	AUX
ejpam-4976	152	10	obvious	obvious	ADJ
ejpam-4976	152	11	.	.	PUNCT
ejpam-4976	153	1	(	(	PUNCT
ejpam-4976	153	2	4	4	X
ejpam-4976	153	3	)	)	PUNCT
ejpam-4976	153	4	⇒	⇒	NOUN
ejpam-4976	153	5	(	(	PUNCT
ejpam-4976	153	6	1	1	NUM
ejpam-4976	153	7	):	):	PUNCT
ejpam-4976	153	8	let	let	VERB
ejpam-4976	153	9	v	v	PART
ejpam-4976	153	10	be	be	AUX
ejpam-4976	153	11	any	any	DET
ejpam-4976	153	12	σ1σ2	σ1σ2	NOUN
ejpam-4976	153	13	-	-	ADJ
ejpam-4976	153	14	open	open	ADJ
ejpam-4976	153	15	set	set	NOUN
ejpam-4976	153	16	of	of	ADP
ejpam-4976	153	17	y	y	PROPN
ejpam-4976	153	18	.	.	PUNCT
ejpam-4976	154	1	then	then	ADV
ejpam-4976	154	2	,	,	PUNCT
ejpam-4976	154	3	we	we	PRON
ejpam-4976	154	4	have	have	VERB
ejpam-4976	154	5	v	v	NOUN
ejpam-4976	154	6	is	be	AUX
ejpam-4976	154	7	(	(	PUNCT
ejpam-4976	154	8	σ1	σ1	PROPN
ejpam-4976	154	9	,	,	PUNCT
ejpam-4976	154	10	σ2)s	σ2)s	NOUN
ejpam-4976	154	11	-	-	PUNCT
ejpam-4976	154	12	open	open	ADJ
ejpam-4976	154	13	in	in	ADP
ejpam-4976	154	14	y	y	PROPN
ejpam-4976	154	15	.	.	PUNCT
ejpam-4976	155	1	by	by	ADP
ejpam-4976	155	2	(	(	PUNCT
ejpam-4976	155	3	4	4	NUM
ejpam-4976	155	4	)	)	PUNCT
ejpam-4976	155	5	,	,	PUNCT
ejpam-4976	155	6	τ1τ2	τ1τ2	NOUN
ejpam-4976	155	7	-	-	NOUN
ejpam-4976	155	8	cl(f	cl(f	PRON
ejpam-4976	155	9	−1(v	−1(v	NOUN
ejpam-4976	155	10	)	)	PUNCT
ejpam-4976	155	11	)	)	PUNCT
ejpam-4976	156	1	⊆	⊆	X
ejpam-4976	156	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	156	3	-	-	ADJ
ejpam-4976	156	4	cl(f	cl(f	NOUN
ejpam-4976	156	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	156	6	-	-	PUNCT
ejpam-4976	156	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	156	8	-	-	PUNCT
ejpam-4976	156	9	cl(v	cl(v	NOUN
ejpam-4976	156	10	)	)	PUNCT
ejpam-4976	156	11	)	)	PUNCT
ejpam-4976	156	12	)	)	PUNCT
ejpam-4976	156	13	)	)	PUNCT
ejpam-4976	156	14	⊆	⊆	NUM
ejpam-4976	156	15	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	156	16	-	-	PUNCT
ejpam-4976	156	17	cl(v	cl(v	NOUN
ejpam-4976	156	18	)	)	PUNCT
ejpam-4976	156	19	)	)	PUNCT
ejpam-4976	156	20	and	and	CCONJ
ejpam-4976	156	21	by	by	ADP
ejpam-4976	156	22	theorem	theorem	NOUN
ejpam-4976	156	23	4	4	NUM
ejpam-4976	156	24	(	(	PUNCT
ejpam-4976	156	25	6	6	NUM
ejpam-4976	156	26	)	)	PUNCT
ejpam-4976	156	27	,	,	PUNCT
ejpam-4976	156	28	f	f	PROPN
ejpam-4976	156	29	is	be	AUX
ejpam-4976	156	30	weakly	weakly	ADJ
ejpam-4976	156	31	(	(	PUNCT
ejpam-4976	156	32	τ1	τ1	NOUN
ejpam-4976	156	33	,	,	PUNCT
ejpam-4976	156	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	156	35	.	.	PUNCT
ejpam-4976	157	1	theorem	theorem	VERB
ejpam-4976	157	2	6	6	NUM
ejpam-4976	157	3	.	.	PUNCT
ejpam-4976	157	4	for	for	ADP
ejpam-4976	157	5	a	a	DET
ejpam-4976	157	6	function	function	NOUN
ejpam-4976	157	7	(	(	PUNCT
ejpam-4976	157	8	x	x	NOUN
ejpam-4976	157	9	,	,	PUNCT
ejpam-4976	157	10	τ1	τ1	NOUN
ejpam-4976	157	11	,	,	PUNCT
ejpam-4976	157	12	τ2	τ2	NOUN
ejpam-4976	157	13	)	)	PUNCT
ejpam-4976	157	14	→	→	SYM
ejpam-4976	157	15	(	(	PUNCT
ejpam-4976	157	16	y	y	PROPN
ejpam-4976	157	17	,	,	PUNCT
ejpam-4976	157	18	σ1	σ1	PROPN
ejpam-4976	157	19	,	,	PUNCT
ejpam-4976	157	20	σ2	σ2	NOUN
ejpam-4976	157	21	)	)	PUNCT
ejpam-4976	157	22	,	,	PUNCT
ejpam-4976	157	23	the	the	DET
ejpam-4976	157	24	following	follow	VERB
ejpam-4976	157	25	properties	property	NOUN
ejpam-4976	157	26	are	be	AUX
ejpam-4976	157	27	equivalent	equivalent	ADJ
ejpam-4976	157	28	:	:	PUNCT
ejpam-4976	157	29	(	(	PUNCT
ejpam-4976	157	30	1	1	X
ejpam-4976	157	31	)	)	PUNCT
ejpam-4976	157	32	f	f	PROPN
ejpam-4976	157	33	is	be	AUX
ejpam-4976	157	34	weakly	weakly	ADJ
ejpam-4976	157	35	(	(	PUNCT
ejpam-4976	157	36	τ1	τ1	NOUN
ejpam-4976	157	37	,	,	PUNCT
ejpam-4976	157	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	157	39	;	;	PUNCT
ejpam-4976	157	40	(	(	PUNCT
ejpam-4976	157	41	2	2	X
ejpam-4976	157	42	)	)	PUNCT
ejpam-4976	157	43	τ1τ2	τ1τ2	NOUN
ejpam-4976	157	44	-	-	NOUN
ejpam-4976	157	45	cl(f	cl(f	NOUN
ejpam-4976	157	46	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	157	47	-	-	PUNCT
ejpam-4976	157	48	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	157	49	-	-	PUNCT
ejpam-4976	157	50	cl(v	cl(v	NOUN
ejpam-4976	157	51	)	)	PUNCT
ejpam-4976	157	52	)	)	PUNCT
ejpam-4976	157	53	)	)	PUNCT
ejpam-4976	157	54	)	)	PUNCT
ejpam-4976	158	1	⊆	⊆	NUM
ejpam-4976	158	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	158	3	-	-	PUNCT
ejpam-4976	158	4	cl(v	cl(v	NOUN
ejpam-4976	158	5	)	)	PUNCT
ejpam-4976	158	6	)	)	PUNCT
ejpam-4976	158	7	for	for	ADP
ejpam-4976	158	8	every	every	DET
ejpam-4976	158	9	(	(	PUNCT
ejpam-4976	158	10	σ1	σ1	PROPN
ejpam-4976	158	11	,	,	PUNCT
ejpam-4976	158	12	σ2)p	σ2)p	NOUN
ejpam-4976	158	13	-	-	PUNCT
ejpam-4976	158	14	open	open	NOUN
ejpam-4976	158	15	set	set	NOUN
ejpam-4976	158	16	v	v	NOUN
ejpam-4976	158	17	of	of	ADP
ejpam-4976	158	18	y	y	PROPN
ejpam-4976	158	19	;	;	PUNCT
ejpam-4976	158	20	(	(	PUNCT
ejpam-4976	158	21	3	3	X
ejpam-4976	158	22	)	)	PUNCT
ejpam-4976	158	23	τ1τ2	τ1τ2	NOUN
ejpam-4976	158	24	-	-	NOUN
ejpam-4976	158	25	cl(f	cl(f	PRON
ejpam-4976	158	26	−1(v	−1(v	NOUN
ejpam-4976	158	27	)	)	PUNCT
ejpam-4976	158	28	)	)	PUNCT
ejpam-4976	159	1	⊆	⊆	NUM
ejpam-4976	159	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	159	3	-	-	PUNCT
ejpam-4976	159	4	cl(v	cl(v	NOUN
ejpam-4976	159	5	)	)	PUNCT
ejpam-4976	159	6	)	)	PUNCT
ejpam-4976	159	7	for	for	ADP
ejpam-4976	159	8	every	every	DET
ejpam-4976	159	9	(	(	PUNCT
ejpam-4976	159	10	σ1	σ1	PROPN
ejpam-4976	159	11	,	,	PUNCT
ejpam-4976	159	12	σ2)p	σ2)p	NOUN
ejpam-4976	159	13	-	-	PUNCT
ejpam-4976	159	14	open	open	NOUN
ejpam-4976	159	15	set	set	NOUN
ejpam-4976	159	16	v	v	NOUN
ejpam-4976	159	17	of	of	ADP
ejpam-4976	159	18	y	y	PROPN
ejpam-4976	159	19	;	;	PUNCT
ejpam-4976	159	20	c.	c.	PROPN
ejpam-4976	159	21	boonpok	boonpok	PROPN
ejpam-4976	159	22	,	,	PUNCT
ejpam-4976	159	23	c.	c.	PROPN
ejpam-4976	159	24	klanarong	klanarong	PROPN
ejpam-4976	159	25	/	/	SYM
ejpam-4976	159	26	eur	eur	PROPN
ejpam-4976	159	27	.	.	PUNCT
ejpam-4976	160	1	j.	j.	PROPN
ejpam-4976	160	2	pure	pure	PROPN
ejpam-4976	160	3	appl	appl	PROPN
ejpam-4976	160	4	.	.	PROPN
ejpam-4976	160	5	math	math	PROPN
ejpam-4976	160	6	,	,	PUNCT
ejpam-4976	160	7	17	17	NUM
ejpam-4976	160	8	(	(	PUNCT
ejpam-4976	160	9	1	1	NUM
ejpam-4976	160	10	)	)	PUNCT
ejpam-4976	160	11	(	(	PUNCT
ejpam-4976	160	12	2024	2024	NUM
ejpam-4976	160	13	)	)	PUNCT
ejpam-4976	160	14	,	,	PUNCT
ejpam-4976	160	15	416	416	NUM
ejpam-4976	160	16	-	-	SYM
ejpam-4976	160	17	425	425	NUM
ejpam-4976	160	18	421	421	NUM
ejpam-4976	160	19	(	(	PUNCT
ejpam-4976	160	20	4	4	X
ejpam-4976	160	21	)	)	PUNCT
ejpam-4976	160	22	f−1(v	f−1(v	NOUN
ejpam-4976	160	23	)	)	PUNCT
ejpam-4976	161	1	⊆	⊆	X
ejpam-4976	161	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	161	3	-	-	NUM
ejpam-4976	161	4	int(f	int(f	PRON
ejpam-4976	161	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	161	6	-	-	PUNCT
ejpam-4976	161	7	cl(v	cl(v	NOUN
ejpam-4976	161	8	)	)	PUNCT
ejpam-4976	161	9	)	)	PUNCT
ejpam-4976	161	10	)	)	PUNCT
ejpam-4976	161	11	for	for	ADP
ejpam-4976	161	12	every	every	DET
ejpam-4976	161	13	(	(	PUNCT
ejpam-4976	161	14	σ1	σ1	PROPN
ejpam-4976	161	15	,	,	PUNCT
ejpam-4976	161	16	σ2)p	σ2)p	NOUN
ejpam-4976	161	17	-	-	PUNCT
ejpam-4976	161	18	open	open	NOUN
ejpam-4976	161	19	set	set	NOUN
ejpam-4976	161	20	v	v	NOUN
ejpam-4976	161	21	of	of	ADP
ejpam-4976	161	22	y	y	PROPN
ejpam-4976	161	23	.	.	PUNCT
ejpam-4976	162	1	proof	proof	NOUN
ejpam-4976	162	2	.	.	PUNCT
ejpam-4976	163	1	(	(	PUNCT
ejpam-4976	163	2	1	1	X
ejpam-4976	163	3	)	)	PUNCT
ejpam-4976	163	4	⇒	⇒	NOUN
ejpam-4976	163	5	(	(	PUNCT
ejpam-4976	163	6	2	2	NUM
ejpam-4976	163	7	):	):	PUNCT
ejpam-4976	163	8	let	let	VERB
ejpam-4976	163	9	v	v	PART
ejpam-4976	163	10	be	be	AUX
ejpam-4976	163	11	any	any	DET
ejpam-4976	163	12	(	(	PUNCT
ejpam-4976	163	13	σ1	σ1	PROPN
ejpam-4976	163	14	,	,	PUNCT
ejpam-4976	163	15	σ2)p	σ2)p	NOUN
ejpam-4976	163	16	-	-	PUNCT
ejpam-4976	163	17	open	open	ADJ
ejpam-4976	163	18	set	set	NOUN
ejpam-4976	163	19	of	of	ADP
ejpam-4976	163	20	y	y	PROPN
ejpam-4976	163	21	.	.	PUNCT
ejpam-4976	164	1	then	then	ADV
ejpam-4976	164	2	,	,	PUNCT
ejpam-4976	164	3	we	we	PRON
ejpam-4976	164	4	have	have	VERB
ejpam-4976	164	5	σ1σ2	σ1σ2	NOUN
ejpam-4976	164	6	-	-	NUM
ejpam-4976	164	7	cl(v	cl(v	NOUN
ejpam-4976	164	8	)	)	PUNCT
ejpam-4976	165	1	⊆	⊆	NUM
ejpam-4976	165	2	σ1σ2	σ1σ2	X
ejpam-4976	165	3	-	-	PUNCT
ejpam-4976	165	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	165	5	-	-	PUNCT
ejpam-4976	165	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	165	7	-	-	PUNCT
ejpam-4976	165	8	cl(v	cl(v	NOUN
ejpam-4976	165	9	)	)	PUNCT
ejpam-4976	165	10	)	)	PUNCT
ejpam-4976	165	11	)	)	PUNCT
ejpam-4976	165	12	and	and	CCONJ
ejpam-4976	165	13	hence	hence	ADV
ejpam-4976	165	14	σ1σ2	σ1σ2	NOUN
ejpam-4976	165	15	-	-	NOUN
ejpam-4976	165	16	cl(v	cl(v	NOUN
ejpam-4976	165	17	)	)	PUNCT
ejpam-4976	165	18	is	be	AUX
ejpam-4976	165	19	(	(	PUNCT
ejpam-4976	165	20	σ1	σ1	NOUN
ejpam-4976	165	21	,	,	PUNCT
ejpam-4976	165	22	σ2)r	σ2)r	NOUN
ejpam-4976	165	23	-	-	PUNCT
ejpam-4976	165	24	closed	closed	ADJ
ejpam-4976	165	25	in	in	ADP
ejpam-4976	165	26	y	y	PROPN
ejpam-4976	165	27	.	.	PUNCT
ejpam-4976	166	1	thus	thus	ADV
ejpam-4976	166	2	,	,	PUNCT
ejpam-4976	166	3	by	by	ADP
ejpam-4976	166	4	theorem	theorem	NOUN
ejpam-4976	166	5	5	5	NUM
ejpam-4976	166	6	(	(	PUNCT
ejpam-4976	166	7	2	2	NUM
ejpam-4976	166	8	)	)	PUNCT
ejpam-4976	166	9	,	,	PUNCT
ejpam-4976	166	10	τ1τ2	τ1τ2	NOUN
ejpam-4976	166	11	-	-	ADJ
ejpam-4976	166	12	cl(f	cl(f	NOUN
ejpam-4976	166	13	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	166	14	-	-	PUNCT
ejpam-4976	166	15	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	166	16	-	-	PUNCT
ejpam-4976	166	17	cl(v	cl(v	NOUN
ejpam-4976	166	18	)	)	PUNCT
ejpam-4976	166	19	)	)	PUNCT
ejpam-4976	166	20	)	)	PUNCT
ejpam-4976	166	21	)	)	PUNCT
ejpam-4976	167	1	⊆	⊆	NUM
ejpam-4976	167	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	167	3	-	-	PUNCT
ejpam-4976	167	4	cl(v	cl(v	NOUN
ejpam-4976	167	5	)	)	PUNCT
ejpam-4976	167	6	)	)	PUNCT
ejpam-4976	167	7	.	.	PUNCT
ejpam-4976	168	1	(	(	PUNCT
ejpam-4976	168	2	2	2	X
ejpam-4976	168	3	)	)	PUNCT
ejpam-4976	168	4	⇒	⇒	NOUN
ejpam-4976	168	5	(	(	PUNCT
ejpam-4976	168	6	3	3	NUM
ejpam-4976	168	7	):	):	PUNCT
ejpam-4976	168	8	the	the	DET
ejpam-4976	168	9	proof	proof	NOUN
ejpam-4976	168	10	is	be	AUX
ejpam-4976	168	11	obvious	obvious	ADJ
ejpam-4976	168	12	.	.	PUNCT
ejpam-4976	169	1	(	(	PUNCT
ejpam-4976	169	2	3	3	X
ejpam-4976	169	3	)	)	PUNCT
ejpam-4976	169	4	⇒	⇒	NOUN
ejpam-4976	169	5	(	(	PUNCT
ejpam-4976	169	6	4	4	NUM
ejpam-4976	169	7	):	):	PUNCT
ejpam-4976	169	8	let	let	VERB
ejpam-4976	169	9	v	v	PART
ejpam-4976	169	10	be	be	AUX
ejpam-4976	169	11	any	any	DET
ejpam-4976	169	12	(	(	PUNCT
ejpam-4976	169	13	σ1	σ1	PROPN
ejpam-4976	169	14	,	,	PUNCT
ejpam-4976	169	15	σ2)p	σ2)p	NOUN
ejpam-4976	169	16	-	-	PUNCT
ejpam-4976	169	17	open	open	ADJ
ejpam-4976	169	18	set	set	NOUN
ejpam-4976	169	19	of	of	ADP
ejpam-4976	169	20	y	y	PROPN
ejpam-4976	169	21	.	.	PUNCT
ejpam-4976	170	1	by	by	ADP
ejpam-4976	170	2	(	(	PUNCT
ejpam-4976	170	3	3	3	NUM
ejpam-4976	170	4	)	)	PUNCT
ejpam-4976	170	5	,	,	PUNCT
ejpam-4976	170	6	f−1(v	f−1(v	PROPN
ejpam-4976	170	7	)	)	PUNCT
ejpam-4976	170	8	⊆	⊆	NUM
ejpam-4976	170	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	170	10	-	-	PUNCT
ejpam-4976	170	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	170	12	-	-	PUNCT
ejpam-4976	170	13	cl(v	cl(v	NOUN
ejpam-4976	170	14	)	)	PUNCT
ejpam-4976	170	15	)	)	PUNCT
ejpam-4976	170	16	)	)	PUNCT
ejpam-4976	171	1	=	=	PUNCT
ejpam-4976	172	1	x	x	X
ejpam-4976	172	2	−	−	PRON
ejpam-4976	172	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	172	4	-	-	PUNCT
ejpam-4976	172	5	cl(y	cl(y	NOUN
ejpam-4976	172	6	−	−	NOUN
ejpam-4976	172	7	σ1σ2	σ1σ2	NOUN
ejpam-4976	172	8	-	-	NUM
ejpam-4976	172	9	cl(v	cl(v	NOUN
ejpam-4976	172	10	)	)	PUNCT
ejpam-4976	172	11	)	)	PUNCT
ejpam-4976	172	12	)	)	PUNCT
ejpam-4976	173	1	⊆	⊆	NUM
ejpam-4976	173	2	x	x	SYM
ejpam-4976	173	3	−	−	PRON
ejpam-4976	173	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	173	5	-	-	NOUN
ejpam-4976	173	6	cl(f	cl(f	NOUN
ejpam-4976	173	7	−1(y	−1(y	ADP
ejpam-4976	173	8	−	−	PUNCT
ejpam-4976	173	9	σ1σ2	σ1σ2	NOUN
ejpam-4976	173	10	-	-	NUM
ejpam-4976	173	11	cl(v	cl(v	NOUN
ejpam-4976	173	12	)	)	PUNCT
ejpam-4976	173	13	)	)	PUNCT
ejpam-4976	173	14	)	)	PUNCT
ejpam-4976	174	1	=	=	PUNCT
ejpam-4976	174	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	174	3	-	-	PUNCT
ejpam-4976	174	4	int(f	int(f	VERB
ejpam-4976	174	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	174	6	-	-	PUNCT
ejpam-4976	174	7	cl(v	cl(v	NOUN
ejpam-4976	174	8	)	)	PUNCT
ejpam-4976	174	9	)	)	PUNCT
ejpam-4976	174	10	)	)	PUNCT
ejpam-4976	174	11	.	.	PUNCT
ejpam-4976	175	1	(	(	PUNCT
ejpam-4976	175	2	4	4	X
ejpam-4976	175	3	)	)	PUNCT
ejpam-4976	175	4	⇒	⇒	NOUN
ejpam-4976	175	5	(	(	PUNCT
ejpam-4976	175	6	1	1	NUM
ejpam-4976	175	7	):	):	PUNCT
ejpam-4976	175	8	let	let	VERB
ejpam-4976	175	9	v	v	PART
ejpam-4976	175	10	be	be	AUX
ejpam-4976	175	11	any	any	DET
ejpam-4976	175	12	σ1σ2	σ1σ2	NOUN
ejpam-4976	175	13	-	-	ADJ
ejpam-4976	175	14	open	open	ADJ
ejpam-4976	175	15	set	set	NOUN
ejpam-4976	175	16	of	of	ADP
ejpam-4976	175	17	y	y	PROPN
ejpam-4976	175	18	.	.	PUNCT
ejpam-4976	176	1	then	then	ADV
ejpam-4976	176	2	,	,	PUNCT
ejpam-4976	176	3	v	v	NOUN
ejpam-4976	176	4	is	be	AUX
ejpam-4976	176	5	(	(	PUNCT
ejpam-4976	176	6	σ1	σ1	PROPN
ejpam-4976	176	7	,	,	PUNCT
ejpam-4976	176	8	σ2)p	σ2)p	NOUN
ejpam-4976	176	9	-	-	PUNCT
ejpam-4976	176	10	open	open	ADJ
ejpam-4976	176	11	in	in	ADP
ejpam-4976	176	12	y	y	PROPN
ejpam-4976	176	13	.	.	PUNCT
ejpam-4976	177	1	thus	thus	ADV
ejpam-4976	177	2	by	by	ADP
ejpam-4976	177	3	(	(	PUNCT
ejpam-4976	177	4	4	4	NUM
ejpam-4976	177	5	)	)	PUNCT
ejpam-4976	177	6	and	and	CCONJ
ejpam-4976	177	7	theorem	theorem	VERB
ejpam-4976	177	8	4	4	NUM
ejpam-4976	177	9	(	(	PUNCT
ejpam-4976	177	10	2	2	NUM
ejpam-4976	177	11	)	)	PUNCT
ejpam-4976	177	12	,	,	PUNCT
ejpam-4976	177	13	f	f	PROPN
ejpam-4976	177	14	is	be	AUX
ejpam-4976	177	15	weakly	weakly	ADJ
ejpam-4976	177	16	(	(	PUNCT
ejpam-4976	177	17	τ1	τ1	NOUN
ejpam-4976	177	18	,	,	PUNCT
ejpam-4976	177	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	177	20	.	.	PUNCT
ejpam-4976	178	1	theorem	theorem	VERB
ejpam-4976	178	2	7	7	NUM
ejpam-4976	178	3	.	.	X
ejpam-4976	178	4	for	for	ADP
ejpam-4976	178	5	a	a	DET
ejpam-4976	178	6	function	function	NOUN
ejpam-4976	178	7	(	(	PUNCT
ejpam-4976	178	8	x	x	NOUN
ejpam-4976	178	9	,	,	PUNCT
ejpam-4976	178	10	τ1	τ1	NOUN
ejpam-4976	178	11	,	,	PUNCT
ejpam-4976	178	12	τ2	τ2	NOUN
ejpam-4976	178	13	)	)	PUNCT
ejpam-4976	178	14	→	→	SYM
ejpam-4976	178	15	(	(	PUNCT
ejpam-4976	178	16	y	y	PROPN
ejpam-4976	178	17	,	,	PUNCT
ejpam-4976	178	18	σ1	σ1	PROPN
ejpam-4976	178	19	,	,	PUNCT
ejpam-4976	178	20	σ2	σ2	NOUN
ejpam-4976	178	21	)	)	PUNCT
ejpam-4976	178	22	,	,	PUNCT
ejpam-4976	178	23	the	the	DET
ejpam-4976	178	24	following	follow	VERB
ejpam-4976	178	25	properties	property	NOUN
ejpam-4976	178	26	are	be	AUX
ejpam-4976	178	27	equivalent	equivalent	ADJ
ejpam-4976	178	28	:	:	PUNCT
ejpam-4976	178	29	(	(	PUNCT
ejpam-4976	178	30	1	1	X
ejpam-4976	178	31	)	)	PUNCT
ejpam-4976	178	32	f	f	PROPN
ejpam-4976	178	33	is	be	AUX
ejpam-4976	178	34	weakly	weakly	ADJ
ejpam-4976	178	35	(	(	PUNCT
ejpam-4976	178	36	τ1	τ1	NOUN
ejpam-4976	178	37	,	,	PUNCT
ejpam-4976	178	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	178	39	;	;	PUNCT
ejpam-4976	178	40	(	(	PUNCT
ejpam-4976	178	41	2	2	X
ejpam-4976	178	42	)	)	PUNCT
ejpam-4976	178	43	τ1τ2	τ1τ2	NOUN
ejpam-4976	178	44	-	-	NOUN
ejpam-4976	178	45	cl(f	cl(f	NOUN
ejpam-4976	178	46	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	178	47	-	-	PUNCT
ejpam-4976	178	48	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	178	49	-	-	PUNCT
ejpam-4976	178	50	cl(b	cl(b	NOUN
ejpam-4976	178	51	)	)	PUNCT
ejpam-4976	178	52	)	)	PUNCT
ejpam-4976	178	53	)	)	PUNCT
ejpam-4976	178	54	)	)	PUNCT
ejpam-4976	179	1	⊆	⊆	NUM
ejpam-4976	179	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	179	3	-	-	PUNCT
ejpam-4976	179	4	cl(b	cl(b	NOUN
ejpam-4976	179	5	)	)	PUNCT
ejpam-4976	179	6	)	)	PUNCT
ejpam-4976	179	7	for	for	ADP
ejpam-4976	179	8	every	every	DET
ejpam-4976	179	9	subset	subset	NOUN
ejpam-4976	179	10	b	b	PROPN
ejpam-4976	179	11	of	of	ADP
ejpam-4976	179	12	y	y	PROPN
ejpam-4976	179	13	;	;	PUNCT
ejpam-4976	179	14	(	(	PUNCT
ejpam-4976	179	15	3	3	X
ejpam-4976	179	16	)	)	PUNCT
ejpam-4976	179	17	τ1τ2	τ1τ2	NOUN
ejpam-4976	179	18	-	-	NOUN
ejpam-4976	179	19	cl(f	cl(f	NOUN
ejpam-4976	179	20	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	179	21	-	-	PUNCT
ejpam-4976	179	22	int(k	int(k	NUM
ejpam-4976	179	23	)	)	PUNCT
ejpam-4976	179	24	)	)	PUNCT
ejpam-4976	179	25	)	)	PUNCT
ejpam-4976	180	1	⊆	⊆	NUM
ejpam-4976	180	2	f−1(k	f−1(k	PROPN
ejpam-4976	180	3	)	)	PUNCT
ejpam-4976	180	4	for	for	ADP
ejpam-4976	180	5	every	every	DET
ejpam-4976	180	6	(	(	PUNCT
ejpam-4976	180	7	σ1	σ1	PROPN
ejpam-4976	180	8	,	,	PUNCT
ejpam-4976	180	9	σ2)r	σ2)r	NOUN
ejpam-4976	180	10	-	-	PUNCT
ejpam-4976	180	11	closed	close	VERB
ejpam-4976	180	12	set	set	ADJ
ejpam-4976	180	13	k	k	PROPN
ejpam-4976	180	14	of	of	ADP
ejpam-4976	180	15	y	y	PROPN
ejpam-4976	180	16	;	;	PUNCT
ejpam-4976	180	17	(	(	PUNCT
ejpam-4976	180	18	4	4	X
ejpam-4976	180	19	)	)	PUNCT
ejpam-4976	180	20	τ1τ2	τ1τ2	NOUN
ejpam-4976	180	21	-	-	NOUN
ejpam-4976	180	22	cl(f	cl(f	PRON
ejpam-4976	180	23	−1(v	−1(v	NOUN
ejpam-4976	180	24	)	)	PUNCT
ejpam-4976	180	25	)	)	PUNCT
ejpam-4976	181	1	⊆	⊆	NUM
ejpam-4976	181	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	181	3	-	-	PUNCT
ejpam-4976	181	4	cl(v	cl(v	NOUN
ejpam-4976	181	5	)	)	PUNCT
ejpam-4976	181	6	)	)	PUNCT
ejpam-4976	181	7	for	for	ADP
ejpam-4976	181	8	every	every	DET
ejpam-4976	181	9	σ1σ2	σ1σ2	NOUN
ejpam-4976	181	10	-	-	ADJ
ejpam-4976	181	11	open	open	ADJ
ejpam-4976	181	12	set	set	NOUN
ejpam-4976	181	13	v	v	NOUN
ejpam-4976	181	14	of	of	ADP
ejpam-4976	181	15	y	y	PROPN
ejpam-4976	181	16	;	;	PUNCT
ejpam-4976	181	17	(	(	PUNCT
ejpam-4976	181	18	5	5	X
ejpam-4976	181	19	)	)	PUNCT
ejpam-4976	181	20	f−1(v	f−1(v	NOUN
ejpam-4976	181	21	)	)	PUNCT
ejpam-4976	182	1	⊆	⊆	X
ejpam-4976	182	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	182	3	-	-	NUM
ejpam-4976	182	4	int(f	int(f	PRON
ejpam-4976	182	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	182	6	-	-	PUNCT
ejpam-4976	182	7	cl(v	cl(v	NOUN
ejpam-4976	182	8	)	)	PUNCT
ejpam-4976	182	9	)	)	PUNCT
ejpam-4976	182	10	)	)	PUNCT
ejpam-4976	182	11	for	for	ADP
ejpam-4976	182	12	every	every	DET
ejpam-4976	182	13	σ1σ2	σ1σ2	NOUN
ejpam-4976	182	14	-	-	ADJ
ejpam-4976	182	15	open	open	ADJ
ejpam-4976	182	16	set	set	NOUN
ejpam-4976	182	17	v	v	NOUN
ejpam-4976	182	18	of	of	ADP
ejpam-4976	182	19	y	y	PROPN
ejpam-4976	182	20	;	;	PUNCT
ejpam-4976	182	21	(	(	PUNCT
ejpam-4976	182	22	6	6	X
ejpam-4976	182	23	)	)	PUNCT
ejpam-4976	182	24	τ1τ2	τ1τ2	NOUN
ejpam-4976	182	25	-	-	NOUN
ejpam-4976	182	26	cl(f	cl(f	PRON
ejpam-4976	182	27	−1(v	−1(v	NOUN
ejpam-4976	182	28	)	)	PUNCT
ejpam-4976	182	29	)	)	PUNCT
ejpam-4976	182	30	⊆	⊆	NUM
ejpam-4976	182	31	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	182	32	-	-	PUNCT
ejpam-4976	182	33	cl(v	cl(v	NOUN
ejpam-4976	182	34	)	)	PUNCT
ejpam-4976	182	35	)	)	PUNCT
ejpam-4976	182	36	for	for	ADP
ejpam-4976	182	37	every	every	DET
ejpam-4976	182	38	(	(	PUNCT
ejpam-4976	182	39	σ1	σ1	PROPN
ejpam-4976	182	40	,	,	PUNCT
ejpam-4976	182	41	σ2)p	σ2)p	NOUN
ejpam-4976	182	42	-	-	PUNCT
ejpam-4976	182	43	open	open	NOUN
ejpam-4976	182	44	set	set	NOUN
ejpam-4976	182	45	v	v	NOUN
ejpam-4976	182	46	of	of	ADP
ejpam-4976	182	47	y	y	PROPN
ejpam-4976	182	48	;	;	PUNCT
ejpam-4976	182	49	(	(	PUNCT
ejpam-4976	182	50	7	7	X
ejpam-4976	182	51	)	)	PUNCT
ejpam-4976	182	52	f−1(v	f−1(v	NOUN
ejpam-4976	182	53	)	)	PUNCT
ejpam-4976	182	54	⊆	⊆	X
ejpam-4976	182	55	τ1τ2	τ1τ2	NOUN
ejpam-4976	182	56	-	-	NUM
ejpam-4976	182	57	int(f	int(f	PRON
ejpam-4976	182	58	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	182	59	-	-	PUNCT
ejpam-4976	182	60	cl(v	cl(v	NOUN
ejpam-4976	182	61	)	)	PUNCT
ejpam-4976	182	62	)	)	PUNCT
ejpam-4976	182	63	)	)	PUNCT
ejpam-4976	182	64	for	for	ADP
ejpam-4976	182	65	every	every	DET
ejpam-4976	182	66	(	(	PUNCT
ejpam-4976	182	67	σ1	σ1	PROPN
ejpam-4976	182	68	,	,	PUNCT
ejpam-4976	182	69	σ2)p	σ2)p	NOUN
ejpam-4976	182	70	-	-	PUNCT
ejpam-4976	182	71	open	open	NOUN
ejpam-4976	182	72	set	set	NOUN
ejpam-4976	182	73	v	v	NOUN
ejpam-4976	182	74	of	of	ADP
ejpam-4976	182	75	y	y	PROPN
ejpam-4976	182	76	.	.	PUNCT
ejpam-4976	183	1	proof	proof	NOUN
ejpam-4976	183	2	.	.	PUNCT
ejpam-4976	184	1	(	(	PUNCT
ejpam-4976	184	2	1	1	X
ejpam-4976	184	3	)	)	PUNCT
ejpam-4976	184	4	⇒	⇒	NOUN
ejpam-4976	184	5	(	(	PUNCT
ejpam-4976	184	6	2	2	NUM
ejpam-4976	184	7	):	):	PUNCT
ejpam-4976	184	8	let	let	VERB
ejpam-4976	184	9	b	b	X
ejpam-4976	184	10	be	be	AUX
ejpam-4976	184	11	any	any	DET
ejpam-4976	184	12	subset	subset	NOUN
ejpam-4976	184	13	of	of	ADP
ejpam-4976	184	14	y	y	PROPN
ejpam-4976	184	15	and	and	CCONJ
ejpam-4976	184	16	x	x	PROPN
ejpam-4976	184	17	̸∈	̸∈	PROPN
ejpam-4976	184	18	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	184	19	-	-	PUNCT
ejpam-4976	184	20	cl(b	cl(b	NOUN
ejpam-4976	184	21	)	)	PUNCT
ejpam-4976	184	22	)	)	PUNCT
ejpam-4976	184	23	.	.	PUNCT
ejpam-4976	185	1	then	then	ADV
ejpam-4976	185	2	,	,	PUNCT
ejpam-4976	185	3	we	we	PRON
ejpam-4976	185	4	have	have	VERB
ejpam-4976	185	5	f(x	f(x	PROPN
ejpam-4976	185	6	)	)	PUNCT
ejpam-4976	185	7	̸∈	̸∈	PROPN
ejpam-4976	185	8	σ1σ2	σ1σ2	NOUN
ejpam-4976	185	9	-	-	NUM
ejpam-4976	185	10	cl(b	cl(b	NOUN
ejpam-4976	185	11	)	)	PUNCT
ejpam-4976	185	12	and	and	CCONJ
ejpam-4976	185	13	there	there	PRON
ejpam-4976	185	14	exists	exist	VERB
ejpam-4976	185	15	a	a	DET
ejpam-4976	185	16	σ1σ2	σ1σ2	NUM
ejpam-4976	185	17	-	-	ADJ
ejpam-4976	185	18	open	open	ADJ
ejpam-4976	185	19	set	set	NOUN
ejpam-4976	185	20	u	u	NOUN
ejpam-4976	185	21	of	of	ADP
ejpam-4976	185	22	y	y	PROPN
ejpam-4976	185	23	containing	contain	VERB
ejpam-4976	185	24	f(x	f(x	PROPN
ejpam-4976	185	25	)	)	PUNCT
ejpam-4976	186	1	such	such	ADJ
ejpam-4976	186	2	that	that	SCONJ
ejpam-4976	186	3	u	u	PROPN
ejpam-4976	186	4	∩	∩	NOUN
ejpam-4976	186	5	b	b	NOUN
ejpam-4976	186	6	=	=	X
ejpam-4976	186	7	∅.	∅.	VERB
ejpam-4976	186	8	therefore	therefore	ADV
ejpam-4976	186	9	,	,	PUNCT
ejpam-4976	186	10	σ1σ2	σ1σ2	NOUN
ejpam-4976	186	11	-	-	PUNCT
ejpam-4976	186	12	cl(u	cl(u	NOUN
ejpam-4976	186	13	)	)	PUNCT
ejpam-4976	186	14	∩	∩	NOUN
ejpam-4976	186	15	σ1σ2	σ1σ2	X
ejpam-4976	186	16	-	-	PUNCT
ejpam-4976	186	17	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	186	18	-	-	PUNCT
ejpam-4976	186	19	cl(b	cl(b	NOUN
ejpam-4976	186	20	)	)	PUNCT
ejpam-4976	186	21	)	)	PUNCT
ejpam-4976	187	1	=	=	PUNCT
ejpam-4976	187	2	∅.	∅.	NOUN
ejpam-4976	187	3	since	since	SCONJ
ejpam-4976	187	4	f	f	PROPN
ejpam-4976	187	5	is	be	AUX
ejpam-4976	187	6	weakly	weakly	ADJ
ejpam-4976	187	7	(	(	PUNCT
ejpam-4976	187	8	τ1	τ1	NOUN
ejpam-4976	187	9	,	,	PUNCT
ejpam-4976	187	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	187	11	at	at	ADP
ejpam-4976	187	12	x	x	X
ejpam-4976	187	13	,	,	PUNCT
ejpam-4976	187	14	there	there	PRON
ejpam-4976	187	15	exists	exist	VERB
ejpam-4976	187	16	a	a	DET
ejpam-4976	187	17	τ1τ2	τ1τ2	NOUN
ejpam-4976	187	18	-	-	ADJ
ejpam-4976	187	19	open	open	ADJ
ejpam-4976	187	20	set	set	NOUN
ejpam-4976	187	21	w	w	PROPN
ejpam-4976	187	22	of	of	ADP
ejpam-4976	187	23	x	x	PUNCT
ejpam-4976	187	24	containing	contain	VERB
ejpam-4976	187	25	x	x	PUNCT
ejpam-4976	187	26	such	such	ADJ
ejpam-4976	187	27	that	that	SCONJ
ejpam-4976	187	28	f(w	f(w	PROPN
ejpam-4976	187	29	)	)	PUNCT
ejpam-4976	188	1	⊆	⊆	NUM
ejpam-4976	188	2	σ1σ2	σ1σ2	NOUN
ejpam-4976	188	3	-	-	PUNCT
ejpam-4976	188	4	cl(u	cl(u	NUM
ejpam-4976	188	5	)	)	PUNCT
ejpam-4976	188	6	.	.	PUNCT
ejpam-4976	189	1	thus	thus	ADV
ejpam-4976	189	2	,	,	PUNCT
ejpam-4976	189	3	w	w	PROPN
ejpam-4976	189	4	∩	∩	NOUN
ejpam-4976	189	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	189	6	-	-	PUNCT
ejpam-4976	189	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	189	8	-	-	PUNCT
ejpam-4976	189	9	cl(b	cl(b	NOUN
ejpam-4976	189	10	)	)	PUNCT
ejpam-4976	189	11	)	)	PUNCT
ejpam-4976	189	12	)	)	PUNCT
ejpam-4976	190	1	=	=	NOUN
ejpam-4976	190	2	∅	∅	NOUN
ejpam-4976	190	3	and	and	CCONJ
ejpam-4976	190	4	hence	hence	ADV
ejpam-4976	190	5	x	x	X
ejpam-4976	190	6	̸∈	̸∈	PROPN
ejpam-4976	190	7	τ1τ2	τ1τ2	PROPN
ejpam-4976	190	8	-	-	PROPN
ejpam-4976	190	9	cl(f	cl(f	NOUN
ejpam-4976	190	10	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	190	11	-	-	PUNCT
ejpam-4976	190	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	190	13	-	-	PUNCT
ejpam-4976	190	14	cl(b	cl(b	NOUN
ejpam-4976	190	15	)	)	PUNCT
ejpam-4976	190	16	)	)	PUNCT
ejpam-4976	190	17	)	)	PUNCT
ejpam-4976	190	18	)	)	PUNCT
ejpam-4976	190	19	.	.	PUNCT
ejpam-4976	191	1	this	this	PRON
ejpam-4976	191	2	shows	show	VERB
ejpam-4976	191	3	that	that	SCONJ
ejpam-4976	191	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	191	5	-	-	ADJ
ejpam-4976	191	6	cl(f	cl(f	NOUN
ejpam-4976	191	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	191	8	-	-	PUNCT
ejpam-4976	191	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	191	10	-	-	PUNCT
ejpam-4976	191	11	cl(b	cl(b	NOUN
ejpam-4976	191	12	)	)	PUNCT
ejpam-4976	191	13	)	)	PUNCT
ejpam-4976	191	14	)	)	PUNCT
ejpam-4976	191	15	)	)	PUNCT
ejpam-4976	191	16	⊆	⊆	NUM
ejpam-4976	191	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	191	18	-	-	PUNCT
ejpam-4976	191	19	cl(b	cl(b	NOUN
ejpam-4976	191	20	)	)	PUNCT
ejpam-4976	191	21	)	)	PUNCT
ejpam-4976	191	22	.	.	PUNCT
ejpam-4976	192	1	c.	c.	PROPN
ejpam-4976	192	2	boonpok	boonpok	PROPN
ejpam-4976	192	3	,	,	PUNCT
ejpam-4976	192	4	c.	c.	PROPN
ejpam-4976	192	5	klanarong	klanarong	PROPN
ejpam-4976	192	6	/	/	SYM
ejpam-4976	192	7	eur	eur	PROPN
ejpam-4976	192	8	.	.	PUNCT
ejpam-4976	193	1	j.	j.	PROPN
ejpam-4976	193	2	pure	pure	PROPN
ejpam-4976	193	3	appl	appl	PROPN
ejpam-4976	193	4	.	.	PROPN
ejpam-4976	193	5	math	math	PROPN
ejpam-4976	193	6	,	,	PUNCT
ejpam-4976	193	7	17	17	NUM
ejpam-4976	193	8	(	(	PUNCT
ejpam-4976	193	9	1	1	NUM
ejpam-4976	193	10	)	)	PUNCT
ejpam-4976	193	11	(	(	PUNCT
ejpam-4976	193	12	2024	2024	NUM
ejpam-4976	193	13	)	)	PUNCT
ejpam-4976	193	14	,	,	PUNCT
ejpam-4976	193	15	416	416	NUM
ejpam-4976	193	16	-	-	SYM
ejpam-4976	193	17	425	425	NUM
ejpam-4976	193	18	422	422	NUM
ejpam-4976	193	19	(	(	PUNCT
ejpam-4976	193	20	2	2	NUM
ejpam-4976	193	21	)	)	PUNCT
ejpam-4976	193	22	⇒	⇒	NOUN
ejpam-4976	193	23	(	(	PUNCT
ejpam-4976	193	24	3	3	NUM
ejpam-4976	193	25	):	):	PUNCT
ejpam-4976	193	26	let	let	VERB
ejpam-4976	193	27	k	k	PRON
ejpam-4976	193	28	be	be	AUX
ejpam-4976	193	29	any	any	DET
ejpam-4976	193	30	(	(	PUNCT
ejpam-4976	193	31	σ1	σ1	NOUN
ejpam-4976	193	32	,	,	PUNCT
ejpam-4976	193	33	σ2)r	σ2)r	NOUN
ejpam-4976	193	34	-	-	PUNCT
ejpam-4976	193	35	closed	close	VERB
ejpam-4976	193	36	set	set	NOUN
ejpam-4976	193	37	of	of	ADP
ejpam-4976	193	38	y	y	PROPN
ejpam-4976	193	39	.	.	PUNCT
ejpam-4976	194	1	then	then	ADV
ejpam-4976	194	2	by	by	ADP
ejpam-4976	194	3	(	(	PUNCT
ejpam-4976	194	4	2	2	NUM
ejpam-4976	194	5	)	)	PUNCT
ejpam-4976	194	6	,	,	PUNCT
ejpam-4976	194	7	we	we	PRON
ejpam-4976	194	8	have	have	VERB
ejpam-4976	194	9	τ1τ2	τ1τ2	NOUN
ejpam-4976	194	10	-	-	ADJ
ejpam-4976	194	11	cl(f	cl(f	NOUN
ejpam-4976	194	12	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	194	13	-	-	PUNCT
ejpam-4976	194	14	int(k	int(k	NUM
ejpam-4976	194	15	)	)	PUNCT
ejpam-4976	194	16	)	)	PUNCT
ejpam-4976	194	17	)	)	PUNCT
ejpam-4976	195	1	=	=	PUNCT
ejpam-4976	195	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	195	3	-	-	ADJ
ejpam-4976	195	4	cl(f	cl(f	NOUN
ejpam-4976	195	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	195	6	-	-	PUNCT
ejpam-4976	195	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-4976	195	8	-	-	PUNCT
ejpam-4976	195	9	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-4976	195	10	-	-	PUNCT
ejpam-4976	195	11	int(k	int(k	NOUN
ejpam-4976	195	12	)	)	PUNCT
ejpam-4976	195	13	)	)	PUNCT
ejpam-4976	195	14	)	)	PUNCT
ejpam-4976	195	15	)	)	PUNCT
ejpam-4976	195	16	)	)	PUNCT
ejpam-4976	196	1	⊆	⊆	NUM
ejpam-4976	196	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	196	3	-	-	PUNCT
ejpam-4976	196	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-4976	196	5	-	-	PUNCT
ejpam-4976	196	6	int(k	int(k	NOUN
ejpam-4976	196	7	)	)	PUNCT
ejpam-4976	196	8	)	)	PUNCT
ejpam-4976	196	9	)	)	PUNCT
ejpam-4976	197	1	=	=	PUNCT
ejpam-4976	197	2	f−1(k	f−1(k	PROPN
ejpam-4976	197	3	)	)	PUNCT
ejpam-4976	197	4	.	.	PUNCT
ejpam-4976	198	1	(	(	PUNCT
ejpam-4976	198	2	3	3	X
ejpam-4976	198	3	)	)	PUNCT
ejpam-4976	198	4	⇒	⇒	NOUN
ejpam-4976	198	5	(	(	PUNCT
ejpam-4976	198	6	4	4	NUM
ejpam-4976	198	7	):	):	PUNCT
ejpam-4976	198	8	let	let	VERB
ejpam-4976	198	9	v	v	PART
ejpam-4976	198	10	be	be	AUX
ejpam-4976	198	11	any	any	DET
ejpam-4976	198	12	σ1σ2	σ1σ2	NOUN
ejpam-4976	198	13	-	-	ADJ
ejpam-4976	198	14	open	open	ADJ
ejpam-4976	198	15	set	set	NOUN
ejpam-4976	198	16	of	of	ADP
ejpam-4976	198	17	y	y	PROPN
ejpam-4976	198	18	.	.	PUNCT
ejpam-4976	199	1	then	then	ADV
ejpam-4976	199	2	,	,	PUNCT
ejpam-4976	199	3	σ1σ2	σ1σ2	NOUN
ejpam-4976	199	4	-	-	NUM
ejpam-4976	199	5	cl(v	cl(v	NOUN
ejpam-4976	199	6	)	)	PUNCT
ejpam-4976	199	7	is	be	AUX
ejpam-4976	199	8	(	(	PUNCT
ejpam-4976	199	9	σ1	σ1	NOUN
ejpam-4976	199	10	,	,	PUNCT
ejpam-4976	199	11	σ2)r	σ2)r	NOUN
ejpam-4976	199	12	-	-	PUNCT
ejpam-4976	199	13	closed	closed	ADJ
ejpam-4976	199	14	in	in	ADP
ejpam-4976	199	15	y	y	PROPN
ejpam-4976	199	16	.	.	PUNCT
ejpam-4976	200	1	by	by	ADP
ejpam-4976	200	2	(	(	PUNCT
ejpam-4976	200	3	3	3	NUM
ejpam-4976	200	4	)	)	PUNCT
ejpam-4976	200	5	,	,	PUNCT
ejpam-4976	200	6	τ1τ2	τ1τ2	NOUN
ejpam-4976	200	7	-	-	NOUN
ejpam-4976	200	8	cl(f	cl(f	PRON
ejpam-4976	200	9	−1(v	−1(v	NOUN
ejpam-4976	200	10	)	)	PUNCT
ejpam-4976	200	11	)	)	PUNCT
ejpam-4976	201	1	⊆	⊆	X
ejpam-4976	201	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	201	3	-	-	ADJ
ejpam-4976	201	4	cl(f	cl(f	NOUN
ejpam-4976	201	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	201	6	-	-	PUNCT
ejpam-4976	201	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	201	8	-	-	PUNCT
ejpam-4976	201	9	cl(v	cl(v	NOUN
ejpam-4976	201	10	)	)	PUNCT
ejpam-4976	201	11	)	)	PUNCT
ejpam-4976	201	12	)	)	PUNCT
ejpam-4976	201	13	)	)	PUNCT
ejpam-4976	201	14	⊆	⊆	NUM
ejpam-4976	201	15	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	201	16	-	-	PUNCT
ejpam-4976	201	17	cl(v	cl(v	NOUN
ejpam-4976	201	18	)	)	PUNCT
ejpam-4976	201	19	)	)	PUNCT
ejpam-4976	201	20	.	.	PUNCT
ejpam-4976	202	1	(	(	PUNCT
ejpam-4976	202	2	4	4	X
ejpam-4976	202	3	)	)	PUNCT
ejpam-4976	202	4	⇒	⇒	NOUN
ejpam-4976	202	5	(	(	PUNCT
ejpam-4976	202	6	5	5	NUM
ejpam-4976	202	7	):	):	PUNCT
ejpam-4976	202	8	let	let	VERB
ejpam-4976	202	9	v	v	PART
ejpam-4976	202	10	be	be	AUX
ejpam-4976	202	11	any	any	DET
ejpam-4976	202	12	σ1σ2	σ1σ2	NOUN
ejpam-4976	202	13	-	-	ADJ
ejpam-4976	202	14	open	open	ADJ
ejpam-4976	202	15	set	set	NOUN
ejpam-4976	202	16	of	of	ADP
ejpam-4976	202	17	y	y	PROPN
ejpam-4976	202	18	.	.	PUNCT
ejpam-4976	203	1	since	since	SCONJ
ejpam-4976	203	2	y	y	PROPN
ejpam-4976	203	3	−σ1σ2	−σ1σ2	PROPN
ejpam-4976	203	4	-	-	PUNCT
ejpam-4976	203	5	cl(v	cl(v	NOUN
ejpam-4976	203	6	)	)	PUNCT
ejpam-4976	203	7	is	be	AUX
ejpam-4976	203	8	σ1σ2	σ1σ2	NOUN
ejpam-4976	203	9	-	-	ADJ
ejpam-4976	203	10	open	open	ADJ
ejpam-4976	203	11	in	in	ADP
ejpam-4976	203	12	y	y	PROPN
ejpam-4976	203	13	,	,	PUNCT
ejpam-4976	203	14	by	by	ADP
ejpam-4976	203	15	(	(	PUNCT
ejpam-4976	203	16	4	4	X
ejpam-4976	203	17	)	)	PUNCT
ejpam-4976	203	18	we	we	PRON
ejpam-4976	203	19	have	have	VERB
ejpam-4976	203	20	x	x	INTJ
ejpam-4976	203	21	−	−	ADP
ejpam-4976	203	22	τ1τ2	τ1τ2	NOUN
ejpam-4976	203	23	-	-	NUM
ejpam-4976	203	24	int(f	int(f	PRON
ejpam-4976	203	25	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	203	26	-	-	PUNCT
ejpam-4976	203	27	cl(v	cl(v	NOUN
ejpam-4976	203	28	)	)	PUNCT
ejpam-4976	203	29	)	)	PUNCT
ejpam-4976	203	30	)	)	PUNCT
ejpam-4976	204	1	=	=	PUNCT
ejpam-4976	204	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	204	3	-	-	PROPN
ejpam-4976	204	4	cl(f	cl(f	NOUN
ejpam-4976	204	5	−1(y	−1(y	ADP
ejpam-4976	204	6	−	−	PUNCT
ejpam-4976	204	7	σ1σ2	σ1σ2	NOUN
ejpam-4976	204	8	-	-	NUM
ejpam-4976	204	9	cl(v	cl(v	NOUN
ejpam-4976	204	10	)	)	PUNCT
ejpam-4976	204	11	)	)	PUNCT
ejpam-4976	204	12	)	)	PUNCT
ejpam-4976	204	13	⊆	⊆	NUM
ejpam-4976	204	14	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	204	15	-	-	PUNCT
ejpam-4976	204	16	cl(y	cl(y	NOUN
ejpam-4976	204	17	−	−	NOUN
ejpam-4976	204	18	σ1σ2	σ1σ2	NOUN
ejpam-4976	204	19	-	-	NUM
ejpam-4976	204	20	cl(v	cl(v	NOUN
ejpam-4976	204	21	)	)	PUNCT
ejpam-4976	204	22	)	)	PUNCT
ejpam-4976	204	23	)	)	PUNCT
ejpam-4976	204	24	⊆	⊆	NUM
ejpam-4976	204	25	x	x	SYM
ejpam-4976	204	26	−	−	PROPN
ejpam-4976	204	27	f−1(v	f−1(v	PROPN
ejpam-4976	204	28	)	)	PUNCT
ejpam-4976	204	29	and	and	CCONJ
ejpam-4976	204	30	hence	hence	ADV
ejpam-4976	204	31	f−1(v	f−1(v	NOUN
ejpam-4976	204	32	)	)	PUNCT
ejpam-4976	204	33	⊆	⊆	X
ejpam-4976	204	34	τ1τ2	τ1τ2	NOUN
ejpam-4976	204	35	-	-	NUM
ejpam-4976	204	36	int(f	int(f	PRON
ejpam-4976	204	37	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	204	38	-	-	PUNCT
ejpam-4976	204	39	cl(v	cl(v	NOUN
ejpam-4976	204	40	)	)	PUNCT
ejpam-4976	204	41	)	)	PUNCT
ejpam-4976	204	42	)	)	PUNCT
ejpam-4976	204	43	.	.	PUNCT
ejpam-4976	205	1	(	(	PUNCT
ejpam-4976	205	2	5	5	X
ejpam-4976	205	3	)	)	PUNCT
ejpam-4976	205	4	⇒	⇒	NOUN
ejpam-4976	205	5	(	(	PUNCT
ejpam-4976	205	6	1	1	NUM
ejpam-4976	205	7	):	):	PUNCT
ejpam-4976	205	8	let	let	VERB
ejpam-4976	205	9	x	x	PUNCT
ejpam-4976	205	10	∈	∈	PROPN
ejpam-4976	205	11	x	x	X
ejpam-4976	205	12	and	and	CCONJ
ejpam-4976	205	13	v	v	X
ejpam-4976	205	14	be	be	AUX
ejpam-4976	205	15	any	any	DET
ejpam-4976	205	16	σ1σ2	σ1σ2	NOUN
ejpam-4976	205	17	-	-	ADJ
ejpam-4976	205	18	open	open	ADJ
ejpam-4976	205	19	set	set	NOUN
ejpam-4976	205	20	of	of	ADP
ejpam-4976	205	21	y	y	PROPN
ejpam-4976	205	22	containing	contain	VERB
ejpam-4976	205	23	f(x	f(x	PROPN
ejpam-4976	205	24	)	)	PUNCT
ejpam-4976	205	25	.	.	PUNCT
ejpam-4976	206	1	by	by	ADP
ejpam-4976	206	2	(	(	PUNCT
ejpam-4976	206	3	5	5	NUM
ejpam-4976	206	4	)	)	PUNCT
ejpam-4976	206	5	,	,	PUNCT
ejpam-4976	206	6	x	x	PUNCT
ejpam-4976	206	7	∈	∈	PROPN
ejpam-4976	206	8	f−1(v	f−1(v	NOUN
ejpam-4976	206	9	)	)	PUNCT
ejpam-4976	207	1	⊆	⊆	X
ejpam-4976	207	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	207	3	-	-	NUM
ejpam-4976	207	4	int(f	int(f	PRON
ejpam-4976	207	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	207	6	-	-	PUNCT
ejpam-4976	207	7	cl(v	cl(v	NOUN
ejpam-4976	207	8	)	)	PUNCT
ejpam-4976	207	9	)	)	PUNCT
ejpam-4976	207	10	)	)	PUNCT
ejpam-4976	207	11	.	.	PUNCT
ejpam-4976	208	1	put	put	VERB
ejpam-4976	208	2	w	w	NOUN
ejpam-4976	208	3	=	=	PUNCT
ejpam-4976	208	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	208	5	-	-	NUM
ejpam-4976	208	6	int(f	int(f	VERB
ejpam-4976	208	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	208	8	-	-	PUNCT
ejpam-4976	208	9	cl(v	cl(v	NOUN
ejpam-4976	208	10	)	)	PUNCT
ejpam-4976	208	11	)	)	PUNCT
ejpam-4976	208	12	)	)	PUNCT
ejpam-4976	208	13	.	.	PUNCT
ejpam-4976	209	1	then	then	ADV
ejpam-4976	209	2	,	,	PUNCT
ejpam-4976	209	3	w	w	PROPN
ejpam-4976	209	4	is	be	AUX
ejpam-4976	209	5	τ1τ2	τ1τ2	ADJ
ejpam-4976	209	6	-	-	ADJ
ejpam-4976	209	7	open	open	ADJ
ejpam-4976	209	8	set	set	NOUN
ejpam-4976	209	9	of	of	ADP
ejpam-4976	209	10	x	x	PUNCT
ejpam-4976	209	11	containing	contain	VERB
ejpam-4976	209	12	x	x	PUNCT
ejpam-4976	210	1	such	such	ADJ
ejpam-4976	210	2	that	that	SCONJ
ejpam-4976	210	3	f(w	f(w	PROPN
ejpam-4976	210	4	)	)	PUNCT
ejpam-4976	210	5	⊆	⊆	X
ejpam-4976	210	6	σ1σ2	σ1σ2	NOUN
ejpam-4976	210	7	-	-	NUM
ejpam-4976	210	8	cl(v	cl(v	NOUN
ejpam-4976	210	9	)	)	PUNCT
ejpam-4976	210	10	.	.	PUNCT
ejpam-4976	211	1	thus	thus	ADV
ejpam-4976	211	2	,	,	PUNCT
ejpam-4976	211	3	f	f	PROPN
ejpam-4976	211	4	is	be	AUX
ejpam-4976	211	5	weakly	weakly	ADJ
ejpam-4976	211	6	(	(	PUNCT
ejpam-4976	211	7	τ1	τ1	NOUN
ejpam-4976	211	8	,	,	PUNCT
ejpam-4976	211	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	211	10	at	at	ADP
ejpam-4976	211	11	x.	x.	NOUN
ejpam-4976	211	12	this	this	PRON
ejpam-4976	211	13	shows	show	VERB
ejpam-4976	211	14	that	that	SCONJ
ejpam-4976	211	15	f	f	PROPN
ejpam-4976	211	16	is	be	AUX
ejpam-4976	211	17	weakly	weakly	ADJ
ejpam-4976	211	18	(	(	PUNCT
ejpam-4976	211	19	τ1	τ1	NOUN
ejpam-4976	211	20	,	,	PUNCT
ejpam-4976	211	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	211	22	.	.	PUNCT
ejpam-4976	212	1	(	(	PUNCT
ejpam-4976	212	2	1	1	X
ejpam-4976	212	3	)	)	PUNCT
ejpam-4976	212	4	⇒	⇒	NOUN
ejpam-4976	212	5	(	(	PUNCT
ejpam-4976	212	6	6	6	NUM
ejpam-4976	212	7	):	):	PUNCT
ejpam-4976	212	8	let	let	VERB
ejpam-4976	212	9	v	v	PART
ejpam-4976	212	10	be	be	AUX
ejpam-4976	212	11	any	any	DET
ejpam-4976	212	12	(	(	PUNCT
ejpam-4976	212	13	σ1	σ1	PROPN
ejpam-4976	212	14	,	,	PUNCT
ejpam-4976	212	15	σ2)p	σ2)p	NOUN
ejpam-4976	212	16	-	-	PUNCT
ejpam-4976	212	17	open	open	ADJ
ejpam-4976	212	18	set	set	NOUN
ejpam-4976	212	19	of	of	ADP
ejpam-4976	212	20	y	y	PROPN
ejpam-4976	212	21	and	and	CCONJ
ejpam-4976	212	22	x	x	PROPN
ejpam-4976	212	23	̸∈	̸∈	PROPN
ejpam-4976	212	24	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	212	25	-	-	PUNCT
ejpam-4976	212	26	cl(v	cl(v	NOUN
ejpam-4976	212	27	)	)	PUNCT
ejpam-4976	212	28	)	)	PUNCT
ejpam-4976	212	29	.	.	PUNCT
ejpam-4976	213	1	then	then	ADV
ejpam-4976	213	2	,	,	PUNCT
ejpam-4976	213	3	f(x	f(x	PROPN
ejpam-4976	213	4	)	)	PUNCT
ejpam-4976	213	5	̸∈	̸∈	PROPN
ejpam-4976	213	6	σ1σ2	σ1σ2	NOUN
ejpam-4976	213	7	-	-	NUM
ejpam-4976	213	8	cl(v	cl(v	NOUN
ejpam-4976	213	9	)	)	PUNCT
ejpam-4976	213	10	and	and	CCONJ
ejpam-4976	213	11	there	there	PRON
ejpam-4976	213	12	exists	exist	VERB
ejpam-4976	213	13	a	a	DET
ejpam-4976	213	14	σ1σ2	σ1σ2	NUM
ejpam-4976	213	15	-	-	ADJ
ejpam-4976	213	16	open	open	ADJ
ejpam-4976	213	17	set	set	NOUN
ejpam-4976	213	18	g	g	NOUN
ejpam-4976	213	19	of	of	ADP
ejpam-4976	213	20	y	y	PROPN
ejpam-4976	213	21	containing	contain	VERB
ejpam-4976	213	22	f(x	f(x	PROPN
ejpam-4976	213	23	)	)	PUNCT
ejpam-4976	213	24	such	such	ADJ
ejpam-4976	213	25	that	that	SCONJ
ejpam-4976	213	26	g	g	PROPN
ejpam-4976	213	27	∩	∩	NOUN
ejpam-4976	213	28	v	v	X
ejpam-4976	213	29	=	=	PUNCT
ejpam-4976	213	30	∅.	∅.	NOUN
ejpam-4976	213	31	since	since	SCONJ
ejpam-4976	213	32	v	v	NOUN
ejpam-4976	213	33	is	be	AUX
ejpam-4976	213	34	(	(	PUNCT
ejpam-4976	213	35	σ1	σ1	PROPN
ejpam-4976	213	36	,	,	PUNCT
ejpam-4976	213	37	σ2)p	σ2)p	NOUN
ejpam-4976	213	38	-	-	PUNCT
ejpam-4976	213	39	open	open	ADJ
ejpam-4976	213	40	,	,	PUNCT
ejpam-4976	213	41	we	we	PRON
ejpam-4976	213	42	have	have	VERB
ejpam-4976	213	43	v	v	NUM
ejpam-4976	213	44	∩	∩	ADJ
ejpam-4976	213	45	σ1σ2	σ1σ2	NOUN
ejpam-4976	213	46	-	-	PUNCT
ejpam-4976	213	47	cl(g	cl(g	ADJ
ejpam-4976	213	48	)	)	PUNCT
ejpam-4976	213	49	⊆	⊆	NUM
ejpam-4976	213	50	σ1σ2	σ1σ2	X
ejpam-4976	213	51	-	-	PUNCT
ejpam-4976	213	52	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	213	53	-	-	PUNCT
ejpam-4976	213	54	cl(v	cl(v	NOUN
ejpam-4976	213	55	)	)	PUNCT
ejpam-4976	213	56	)	)	PUNCT
ejpam-4976	214	1	∩	∩	NOUN
ejpam-4976	214	2	σ1σ2	σ1σ2	NOUN
ejpam-4976	214	3	-	-	PUNCT
ejpam-4976	214	4	cl(g	cl(g	ADJ
ejpam-4976	214	5	)	)	PUNCT
ejpam-4976	214	6	⊆	⊆	NUM
ejpam-4976	214	7	σ1σ2	σ1σ2	X
ejpam-4976	214	8	-	-	PUNCT
ejpam-4976	214	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	214	10	-	-	PUNCT
ejpam-4976	214	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	214	12	-	-	PUNCT
ejpam-4976	214	13	cl(v	cl(v	NOUN
ejpam-4976	214	14	)	)	PUNCT
ejpam-4976	214	15	)	)	PUNCT
ejpam-4976	214	16	∩g	∩g	PROPN
ejpam-4976	214	17	)	)	PUNCT
ejpam-4976	214	18	⊆	⊆	NUM
ejpam-4976	214	19	σ1σ2	σ1σ2	X
ejpam-4976	214	20	-	-	PUNCT
ejpam-4976	214	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	214	22	-	-	PUNCT
ejpam-4976	214	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	214	24	-	-	PUNCT
ejpam-4976	214	25	cl(v	cl(v	NOUN
ejpam-4976	214	26	)	)	PUNCT
ejpam-4976	214	27	∩g	∩g	PROPN
ejpam-4976	214	28	)	)	PUNCT
ejpam-4976	214	29	)	)	PUNCT
ejpam-4976	215	1	⊆	⊆	X
ejpam-4976	215	2	σ1σ2	σ1σ2	X
ejpam-4976	215	3	-	-	PUNCT
ejpam-4976	215	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-4976	215	5	-	-	PUNCT
ejpam-4976	215	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	215	7	-	-	PUNCT
ejpam-4976	215	8	cl(v	cl(v	NOUN
ejpam-4976	215	9	∩g	∩g	NOUN
ejpam-4976	215	10	)	)	PUNCT
ejpam-4976	215	11	)	)	PUNCT
ejpam-4976	215	12	)	)	PUNCT
ejpam-4976	215	13	⊆	⊆	X
ejpam-4976	215	14	σ1σ2	σ1σ2	NUM
ejpam-4976	215	15	-	-	PUNCT
ejpam-4976	215	16	cl(v	cl(v	NOUN
ejpam-4976	215	17	∩g	∩g	NOUN
ejpam-4976	215	18	)	)	PUNCT
ejpam-4976	215	19	=	=	PUNCT
ejpam-4976	215	20	∅.	∅.	NOUN
ejpam-4976	215	21	since	since	SCONJ
ejpam-4976	215	22	f	f	PROPN
ejpam-4976	215	23	is	be	AUX
ejpam-4976	215	24	weakly	weakly	ADJ
ejpam-4976	215	25	(	(	PUNCT
ejpam-4976	215	26	τ1	τ1	NOUN
ejpam-4976	215	27	,	,	PUNCT
ejpam-4976	215	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	215	29	at	at	ADP
ejpam-4976	215	30	x	x	X
ejpam-4976	215	31	,	,	PUNCT
ejpam-4976	215	32	there	there	PRON
ejpam-4976	215	33	exists	exist	VERB
ejpam-4976	215	34	a	a	DET
ejpam-4976	215	35	τ1τ2	τ1τ2	NOUN
ejpam-4976	215	36	-	-	ADJ
ejpam-4976	215	37	open	open	ADJ
ejpam-4976	215	38	set	set	NOUN
ejpam-4976	215	39	w	w	PROPN
ejpam-4976	215	40	of	of	ADP
ejpam-4976	215	41	x	x	PUNCT
ejpam-4976	215	42	containing	contain	VERB
ejpam-4976	215	43	x	x	PUNCT
ejpam-4976	215	44	such	such	ADJ
ejpam-4976	215	45	that	that	SCONJ
ejpam-4976	215	46	f(w	f(w	PROPN
ejpam-4976	215	47	)	)	PUNCT
ejpam-4976	216	1	⊆	⊆	X
ejpam-4976	216	2	σ1σ2	σ1σ2	NOUN
ejpam-4976	216	3	-	-	PUNCT
ejpam-4976	216	4	cl(g	cl(g	NUM
ejpam-4976	216	5	)	)	PUNCT
ejpam-4976	216	6	.	.	PUNCT
ejpam-4976	217	1	thus	thus	ADV
ejpam-4976	217	2	,	,	PUNCT
ejpam-4976	217	3	f(w	f(w	PROPN
ejpam-4976	217	4	)	)	PUNCT
ejpam-4976	217	5	∩	∩	NOUN
ejpam-4976	217	6	v	v	NOUN
ejpam-4976	217	7	=	=	NOUN
ejpam-4976	217	8	∅	∅	NOUN
ejpam-4976	217	9	and	and	CCONJ
ejpam-4976	217	10	hence	hence	ADV
ejpam-4976	217	11	w	w	ADP
ejpam-4976	217	12	∩	∩	ADJ
ejpam-4976	217	13	f−1(v	f−1(v	NOUN
ejpam-4976	217	14	)	)	PUNCT
ejpam-4976	218	1	=	=	PUNCT
ejpam-4976	218	2	∅.	∅.	VERB
ejpam-4976	218	3	therefore	therefore	ADV
ejpam-4976	218	4	,	,	PUNCT
ejpam-4976	218	5	x	x	PROPN
ejpam-4976	218	6	̸∈	̸∈	PROPN
ejpam-4976	218	7	τ1τ2	τ1τ2	PROPN
ejpam-4976	218	8	-	-	PROPN
ejpam-4976	218	9	cl(f	cl(f	PRON
ejpam-4976	218	10	−1(v	−1(v	NOUN
ejpam-4976	218	11	)	)	PUNCT
ejpam-4976	218	12	.	.	PUNCT
ejpam-4976	219	1	this	this	PRON
ejpam-4976	219	2	shows	show	VERB
ejpam-4976	219	3	that	that	SCONJ
ejpam-4976	219	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	219	5	-	-	PROPN
ejpam-4976	219	6	cl(f	cl(f	PRON
ejpam-4976	219	7	−1(v	−1(v	NOUN
ejpam-4976	219	8	)	)	PUNCT
ejpam-4976	219	9	)	)	PUNCT
ejpam-4976	219	10	⊆	⊆	NUM
ejpam-4976	219	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	219	12	-	-	PUNCT
ejpam-4976	219	13	cl(v	cl(v	NOUN
ejpam-4976	219	14	)	)	PUNCT
ejpam-4976	219	15	)	)	PUNCT
ejpam-4976	219	16	.	.	PUNCT
ejpam-4976	220	1	(	(	PUNCT
ejpam-4976	220	2	6	6	X
ejpam-4976	220	3	)	)	PUNCT
ejpam-4976	220	4	⇒	⇒	NOUN
ejpam-4976	220	5	(	(	PUNCT
ejpam-4976	220	6	7	7	NUM
ejpam-4976	220	7	):	):	PUNCT
ejpam-4976	220	8	let	let	VERB
ejpam-4976	220	9	v	v	PART
ejpam-4976	220	10	be	be	AUX
ejpam-4976	220	11	any	any	DET
ejpam-4976	220	12	(	(	PUNCT
ejpam-4976	220	13	σ1	σ1	PROPN
ejpam-4976	220	14	,	,	PUNCT
ejpam-4976	220	15	σ2)p	σ2)p	NOUN
ejpam-4976	220	16	-	-	PUNCT
ejpam-4976	220	17	open	open	ADJ
ejpam-4976	220	18	set	set	NOUN
ejpam-4976	220	19	of	of	ADP
ejpam-4976	220	20	y	y	PROPN
ejpam-4976	220	21	.	.	PUNCT
ejpam-4976	221	1	then	then	ADV
ejpam-4976	221	2	,	,	PUNCT
ejpam-4976	221	3	y	y	PROPN
ejpam-4976	221	4	−	−	NUM
ejpam-4976	221	5	σ1σ2	σ1σ2	NOUN
ejpam-4976	221	6	-	-	NUM
ejpam-4976	221	7	cl(v	cl(v	NOUN
ejpam-4976	221	8	)	)	PUNCT
ejpam-4976	221	9	is	be	AUX
ejpam-4976	221	10	σ1σ2	σ1σ2	NOUN
ejpam-4976	221	11	-	-	ADJ
ejpam-4976	221	12	open	open	ADJ
ejpam-4976	221	13	and	and	CCONJ
ejpam-4976	221	14	hence	hence	ADV
ejpam-4976	221	15	y	y	NOUN
ejpam-4976	221	16	−	−	NUM
ejpam-4976	221	17	σ1σ2	σ1σ2	NOUN
ejpam-4976	221	18	-	-	NUM
ejpam-4976	221	19	cl(v	cl(v	NOUN
ejpam-4976	221	20	)	)	PUNCT
ejpam-4976	221	21	is	be	AUX
ejpam-4976	221	22	(	(	PUNCT
ejpam-4976	221	23	σ1	σ1	PROPN
ejpam-4976	221	24	,	,	PUNCT
ejpam-4976	221	25	σ2)p	σ2)p	NOUN
ejpam-4976	221	26	-	-	PUNCT
ejpam-4976	221	27	open	open	ADJ
ejpam-4976	221	28	in	in	ADP
ejpam-4976	221	29	y	y	PROPN
ejpam-4976	221	30	.	.	PUNCT
ejpam-4976	222	1	then	then	ADV
ejpam-4976	222	2	by	by	ADP
ejpam-4976	222	3	(	(	PUNCT
ejpam-4976	222	4	6	6	NUM
ejpam-4976	222	5	)	)	PUNCT
ejpam-4976	222	6	,	,	PUNCT
ejpam-4976	222	7	we	we	PRON
ejpam-4976	222	8	have	have	VERB
ejpam-4976	222	9	x	x	INTJ
ejpam-4976	222	10	−	−	ADP
ejpam-4976	222	11	τ1τ2	τ1τ2	NOUN
ejpam-4976	222	12	-	-	NUM
ejpam-4976	222	13	int(f	int(f	PRON
ejpam-4976	222	14	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	222	15	-	-	PUNCT
ejpam-4976	222	16	cl(v	cl(v	NOUN
ejpam-4976	222	17	)	)	PUNCT
ejpam-4976	222	18	)	)	PUNCT
ejpam-4976	222	19	)	)	PUNCT
ejpam-4976	223	1	=	=	PUNCT
ejpam-4976	223	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	223	3	-	-	NOUN
ejpam-4976	223	4	cl(x	cl(x	PUNCT
ejpam-4976	223	5	−	−	PRON
ejpam-4976	223	6	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	223	7	-	-	PUNCT
ejpam-4976	223	8	cl(v	cl(v	NOUN
ejpam-4976	223	9	)	)	PUNCT
ejpam-4976	223	10	)	)	PUNCT
ejpam-4976	223	11	)	)	PUNCT
ejpam-4976	224	1	=	=	PUNCT
ejpam-4976	224	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	224	3	-	-	PROPN
ejpam-4976	224	4	cl(f	cl(f	NOUN
ejpam-4976	224	5	−1(y	−1(y	ADP
ejpam-4976	224	6	−	−	PUNCT
ejpam-4976	224	7	σ1σ2	σ1σ2	NOUN
ejpam-4976	224	8	-	-	NUM
ejpam-4976	224	9	cl(v	cl(v	NOUN
ejpam-4976	224	10	)	)	PUNCT
ejpam-4976	224	11	)	)	PUNCT
ejpam-4976	224	12	)	)	PUNCT
ejpam-4976	224	13	⊆	⊆	NUM
ejpam-4976	224	14	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-4976	224	15	-	-	PUNCT
ejpam-4976	224	16	cl(y	cl(y	NOUN
ejpam-4976	224	17	−	−	NOUN
ejpam-4976	224	18	σ1σ2	σ1σ2	NOUN
ejpam-4976	224	19	-	-	NUM
ejpam-4976	224	20	cl(v	cl(v	NOUN
ejpam-4976	224	21	)	)	PUNCT
ejpam-4976	224	22	)	)	PUNCT
ejpam-4976	224	23	)	)	PUNCT
ejpam-4976	224	24	=	=	PUNCT
ejpam-4976	224	25	f−1(y	f−1(y	PROPN
ejpam-4976	225	1	−	−	NOUN
ejpam-4976	225	2	σ1σ2	σ1σ2	X
ejpam-4976	225	3	-	-	PUNCT
ejpam-4976	225	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	225	5	-	-	PUNCT
ejpam-4976	225	6	cl(v	cl(v	NOUN
ejpam-4976	225	7	)	)	PUNCT
ejpam-4976	225	8	)	)	PUNCT
ejpam-4976	225	9	)	)	PUNCT
ejpam-4976	226	1	=	=	PUNCT
ejpam-4976	226	2	x	x	X
ejpam-4976	226	3	−	−	PRON
ejpam-4976	226	4	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-4976	226	5	-	-	PUNCT
ejpam-4976	226	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-4976	226	7	-	-	PUNCT
ejpam-4976	226	8	cl(v	cl(v	NOUN
ejpam-4976	226	9	)	)	PUNCT
ejpam-4976	226	10	)	)	PUNCT
ejpam-4976	226	11	)	)	PUNCT
ejpam-4976	226	12	c.	c.	PROPN
ejpam-4976	226	13	boonpok	boonpok	PROPN
ejpam-4976	226	14	,	,	PUNCT
ejpam-4976	226	15	c.	c.	PROPN
ejpam-4976	226	16	klanarong	klanarong	PROPN
ejpam-4976	226	17	/	/	SYM
ejpam-4976	226	18	eur	eur	PROPN
ejpam-4976	226	19	.	.	PUNCT
ejpam-4976	227	1	j.	j.	PROPN
ejpam-4976	227	2	pure	pure	PROPN
ejpam-4976	227	3	appl	appl	PROPN
ejpam-4976	227	4	.	.	PROPN
ejpam-4976	227	5	math	math	PROPN
ejpam-4976	227	6	,	,	PUNCT
ejpam-4976	227	7	17	17	NUM
ejpam-4976	227	8	(	(	PUNCT
ejpam-4976	227	9	1	1	NUM
ejpam-4976	227	10	)	)	PUNCT
ejpam-4976	227	11	(	(	PUNCT
ejpam-4976	227	12	2024	2024	NUM
ejpam-4976	227	13	)	)	PUNCT
ejpam-4976	227	14	,	,	PUNCT
ejpam-4976	227	15	416	416	NUM
ejpam-4976	227	16	-	-	SYM
ejpam-4976	227	17	425	425	NUM
ejpam-4976	227	18	423	423	NUM
ejpam-4976	227	19	⊆	⊆	NUM
ejpam-4976	227	20	x	x	SYM
ejpam-4976	227	21	−	−	PROPN
ejpam-4976	227	22	f−1(v	f−1(v	PROPN
ejpam-4976	227	23	)	)	PUNCT
ejpam-4976	227	24	and	and	CCONJ
ejpam-4976	227	25	hence	hence	ADV
ejpam-4976	227	26	f−1(v	f−1(v	NOUN
ejpam-4976	227	27	)	)	PUNCT
ejpam-4976	228	1	⊆	⊆	X
ejpam-4976	228	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	228	3	-	-	NUM
ejpam-4976	228	4	int(f	int(f	PRON
ejpam-4976	228	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	228	6	-	-	PUNCT
ejpam-4976	228	7	cl(v	cl(v	NOUN
ejpam-4976	228	8	)	)	PUNCT
ejpam-4976	228	9	)	)	PUNCT
ejpam-4976	228	10	)	)	PUNCT
ejpam-4976	228	11	.	.	PUNCT
ejpam-4976	229	1	(	(	PUNCT
ejpam-4976	229	2	7	7	X
ejpam-4976	229	3	)	)	PUNCT
ejpam-4976	229	4	⇒	⇒	NOUN
ejpam-4976	229	5	(	(	PUNCT
ejpam-4976	229	6	1	1	NUM
ejpam-4976	229	7	):	):	PUNCT
ejpam-4976	229	8	let	let	VERB
ejpam-4976	229	9	x	x	PUNCT
ejpam-4976	229	10	∈	∈	PROPN
ejpam-4976	229	11	x	x	X
ejpam-4976	229	12	and	and	CCONJ
ejpam-4976	229	13	v	v	X
ejpam-4976	229	14	be	be	AUX
ejpam-4976	229	15	any	any	DET
ejpam-4976	229	16	σ1σ2	σ1σ2	NOUN
ejpam-4976	229	17	-	-	ADJ
ejpam-4976	229	18	open	open	ADJ
ejpam-4976	229	19	set	set	NOUN
ejpam-4976	229	20	of	of	ADP
ejpam-4976	229	21	y	y	PROPN
ejpam-4976	229	22	containing	contain	VERB
ejpam-4976	229	23	f(x	f(x	PROPN
ejpam-4976	229	24	)	)	PUNCT
ejpam-4976	229	25	.	.	PUNCT
ejpam-4976	230	1	since	since	SCONJ
ejpam-4976	230	2	v	v	NOUN
ejpam-4976	230	3	is	be	AUX
ejpam-4976	230	4	(	(	PUNCT
ejpam-4976	230	5	σ1	σ1	PROPN
ejpam-4976	230	6	,	,	PUNCT
ejpam-4976	230	7	σ2)p	σ2)p	NOUN
ejpam-4976	230	8	-	-	PUNCT
ejpam-4976	230	9	open	open	ADJ
ejpam-4976	230	10	in	in	ADP
ejpam-4976	230	11	y	y	PROPN
ejpam-4976	230	12	and	and	CCONJ
ejpam-4976	230	13	by	by	ADP
ejpam-4976	230	14	(	(	PUNCT
ejpam-4976	230	15	7	7	NUM
ejpam-4976	230	16	)	)	PUNCT
ejpam-4976	230	17	,	,	PUNCT
ejpam-4976	230	18	x	x	PUNCT
ejpam-4976	230	19	∈	∈	PROPN
ejpam-4976	230	20	f−1(v	f−1(v	NOUN
ejpam-4976	230	21	)	)	PUNCT
ejpam-4976	231	1	⊆	⊆	X
ejpam-4976	231	2	τ1τ2	τ1τ2	NOUN
ejpam-4976	231	3	-	-	NUM
ejpam-4976	231	4	int(f	int(f	PRON
ejpam-4976	231	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	231	6	-	-	PUNCT
ejpam-4976	231	7	cl(v	cl(v	NOUN
ejpam-4976	231	8	)	)	PUNCT
ejpam-4976	231	9	)	)	PUNCT
ejpam-4976	231	10	)	)	PUNCT
ejpam-4976	231	11	.	.	PUNCT
ejpam-4976	232	1	put	put	VERB
ejpam-4976	232	2	u	u	NOUN
ejpam-4976	232	3	=	=	PUNCT
ejpam-4976	232	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	232	5	-	-	NUM
ejpam-4976	232	6	int(f	int(f	VERB
ejpam-4976	232	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-4976	232	8	-	-	PUNCT
ejpam-4976	232	9	cl(v	cl(v	NOUN
ejpam-4976	232	10	)	)	PUNCT
ejpam-4976	232	11	)	)	PUNCT
ejpam-4976	232	12	)	)	PUNCT
ejpam-4976	232	13	.	.	PUNCT
ejpam-4976	233	1	then	then	ADV
ejpam-4976	233	2	,	,	PUNCT
ejpam-4976	233	3	u	u	NOUN
ejpam-4976	233	4	is	be	AUX
ejpam-4976	233	5	a	a	DET
ejpam-4976	233	6	τ1τ2	τ1τ2	ADJ
ejpam-4976	233	7	-	-	ADJ
ejpam-4976	233	8	open	open	ADJ
ejpam-4976	233	9	set	set	NOUN
ejpam-4976	233	10	of	of	ADP
ejpam-4976	233	11	x	x	PUNCT
ejpam-4976	233	12	containing	contain	VERB
ejpam-4976	233	13	x	x	PUNCT
ejpam-4976	233	14	such	such	ADJ
ejpam-4976	233	15	that	that	DET
ejpam-4976	233	16	f(u	f(u	PROPN
ejpam-4976	233	17	)	)	PUNCT
ejpam-4976	233	18	⊆	⊆	NUM
ejpam-4976	233	19	σ1σ2	σ1σ2	NOUN
ejpam-4976	233	20	-	-	NUM
ejpam-4976	233	21	cl(v	cl(v	NOUN
ejpam-4976	233	22	)	)	PUNCT
ejpam-4976	233	23	.	.	PUNCT
ejpam-4976	234	1	thus	thus	ADV
ejpam-4976	234	2	,	,	PUNCT
ejpam-4976	234	3	f	f	PROPN
ejpam-4976	234	4	is	be	AUX
ejpam-4976	234	5	weakly	weakly	ADJ
ejpam-4976	234	6	(	(	PUNCT
ejpam-4976	234	7	τ1	τ1	NOUN
ejpam-4976	234	8	,	,	PUNCT
ejpam-4976	234	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	234	10	at	at	ADP
ejpam-4976	234	11	x.	x.	NOUN
ejpam-4976	234	12	this	this	PRON
ejpam-4976	234	13	shows	show	VERB
ejpam-4976	234	14	that	that	SCONJ
ejpam-4976	234	15	f	f	PROPN
ejpam-4976	234	16	is	be	AUX
ejpam-4976	234	17	weakly	weakly	ADJ
ejpam-4976	234	18	(	(	PUNCT
ejpam-4976	234	19	τ1	τ1	NOUN
ejpam-4976	234	20	,	,	PUNCT
ejpam-4976	234	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	234	22	.	.	PUNCT
ejpam-4976	235	1	theorem	theorem	VERB
ejpam-4976	235	2	8	8	NUM
ejpam-4976	235	3	.	.	PUNCT
ejpam-4976	236	1	for	for	ADP
ejpam-4976	236	2	a	a	DET
ejpam-4976	236	3	function	function	NOUN
ejpam-4976	236	4	(	(	PUNCT
ejpam-4976	236	5	x	x	NOUN
ejpam-4976	236	6	,	,	PUNCT
ejpam-4976	236	7	τ1	τ1	NOUN
ejpam-4976	236	8	,	,	PUNCT
ejpam-4976	236	9	τ2	τ2	NOUN
ejpam-4976	236	10	)	)	PUNCT
ejpam-4976	236	11	→	→	SYM
ejpam-4976	236	12	(	(	PUNCT
ejpam-4976	236	13	y	y	PROPN
ejpam-4976	236	14	,	,	PUNCT
ejpam-4976	236	15	σ1	σ1	PROPN
ejpam-4976	236	16	,	,	PUNCT
ejpam-4976	236	17	σ2	σ2	NOUN
ejpam-4976	236	18	)	)	PUNCT
ejpam-4976	236	19	,	,	PUNCT
ejpam-4976	236	20	the	the	DET
ejpam-4976	236	21	following	follow	VERB
ejpam-4976	236	22	properties	property	NOUN
ejpam-4976	236	23	are	be	AUX
ejpam-4976	236	24	equivalent	equivalent	ADJ
ejpam-4976	236	25	:	:	PUNCT
ejpam-4976	236	26	(	(	PUNCT
ejpam-4976	236	27	1	1	X
ejpam-4976	236	28	)	)	PUNCT
ejpam-4976	236	29	f	f	PROPN
ejpam-4976	236	30	is	be	AUX
ejpam-4976	236	31	weakly	weakly	ADJ
ejpam-4976	236	32	(	(	PUNCT
ejpam-4976	236	33	τ1	τ1	NOUN
ejpam-4976	236	34	,	,	PUNCT
ejpam-4976	236	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	236	36	;	;	PUNCT
ejpam-4976	236	37	(	(	PUNCT
ejpam-4976	236	38	2	2	X
ejpam-4976	236	39	)	)	PUNCT
ejpam-4976	236	40	f(τ1τ2	f(τ1τ2	NOUN
ejpam-4976	236	41	-	-	PUNCT
ejpam-4976	236	42	cl(a	cl(a	NUM
ejpam-4976	236	43	)	)	PUNCT
ejpam-4976	236	44	)	)	PUNCT
ejpam-4976	237	1	⊆	⊆	NUM
ejpam-4976	237	2	(	(	PUNCT
ejpam-4976	237	3	σ1	σ1	PROPN
ejpam-4976	237	4	,	,	PUNCT
ejpam-4976	237	5	σ2)θ	σ2)θ	NOUN
ejpam-4976	237	6	-	-	PUNCT
ejpam-4976	237	7	cl(f(a	cl(f(a	NOUN
ejpam-4976	237	8	)	)	PUNCT
ejpam-4976	237	9	)	)	PUNCT
ejpam-4976	237	10	for	for	ADP
ejpam-4976	237	11	every	every	DET
ejpam-4976	237	12	subset	subset	NOUN
ejpam-4976	237	13	a	a	PRON
ejpam-4976	237	14	of	of	ADP
ejpam-4976	237	15	x	x	PRON
ejpam-4976	237	16	;	;	PUNCT
ejpam-4976	237	17	(	(	PUNCT
ejpam-4976	237	18	3	3	X
ejpam-4976	237	19	)	)	PUNCT
ejpam-4976	237	20	τ1τ2	τ1τ2	NOUN
ejpam-4976	237	21	-	-	NOUN
ejpam-4976	237	22	cl(f	cl(f	NOUN
ejpam-4976	237	23	−1(b	−1(b	NOUN
ejpam-4976	237	24	)	)	PUNCT
ejpam-4976	237	25	)	)	PUNCT
ejpam-4976	238	1	⊆	⊆	NUM
ejpam-4976	238	2	f−1((σ1	f−1((σ1	NOUN
ejpam-4976	238	3	,	,	PUNCT
ejpam-4976	238	4	σ2)θ	σ2)θ	NOUN
ejpam-4976	238	5	-	-	PUNCT
ejpam-4976	238	6	cl(b	cl(b	NOUN
ejpam-4976	238	7	)	)	PUNCT
ejpam-4976	238	8	)	)	PUNCT
ejpam-4976	238	9	for	for	ADP
ejpam-4976	238	10	every	every	DET
ejpam-4976	238	11	subset	subset	NOUN
ejpam-4976	238	12	b	b	PROPN
ejpam-4976	238	13	of	of	ADP
ejpam-4976	238	14	y	y	PROPN
ejpam-4976	238	15	.	.	PUNCT
ejpam-4976	239	1	proof	proof	NOUN
ejpam-4976	239	2	.	.	PUNCT
ejpam-4976	240	1	(	(	PUNCT
ejpam-4976	240	2	1	1	X
ejpam-4976	240	3	)	)	PUNCT
ejpam-4976	240	4	⇒	⇒	NOUN
ejpam-4976	240	5	(	(	PUNCT
ejpam-4976	240	6	2	2	NUM
ejpam-4976	240	7	):	):	PUNCT
ejpam-4976	240	8	let	let	VERB
ejpam-4976	240	9	a	a	DET
ejpam-4976	240	10	be	be	AUX
ejpam-4976	240	11	any	any	DET
ejpam-4976	240	12	subset	subset	NOUN
ejpam-4976	240	13	of	of	ADP
ejpam-4976	240	14	x.	x.	PROPN
ejpam-4976	240	15	suppose	suppose	VERB
ejpam-4976	240	16	that	that	SCONJ
ejpam-4976	240	17	x	x	PUNCT
ejpam-4976	240	18	∈	∈	PROPN
ejpam-4976	240	19	τ1τ2	τ1τ2	NOUN
ejpam-4976	240	20	-	-	NUM
ejpam-4976	240	21	cl(a	cl(a	NUM
ejpam-4976	240	22	)	)	PUNCT
ejpam-4976	240	23	and	and	CCONJ
ejpam-4976	240	24	g	g	PROPN
ejpam-4976	240	25	is	be	AUX
ejpam-4976	240	26	any	any	DET
ejpam-4976	240	27	σ1σ2	σ1σ2	NOUN
ejpam-4976	240	28	-	-	ADJ
ejpam-4976	240	29	open	open	ADJ
ejpam-4976	240	30	set	set	NOUN
ejpam-4976	240	31	of	of	ADP
ejpam-4976	240	32	y	y	PROPN
ejpam-4976	240	33	containing	contain	VERB
ejpam-4976	240	34	f(x	f(x	PROPN
ejpam-4976	240	35	)	)	PUNCT
ejpam-4976	240	36	.	.	PUNCT
ejpam-4976	241	1	since	since	SCONJ
ejpam-4976	241	2	f	f	PROPN
ejpam-4976	241	3	is	be	AUX
ejpam-4976	241	4	weakly	weakly	ADJ
ejpam-4976	241	5	(	(	PUNCT
ejpam-4976	241	6	τ1	τ1	NOUN
ejpam-4976	241	7	,	,	PUNCT
ejpam-4976	241	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	241	9	,	,	PUNCT
ejpam-4976	241	10	there	there	PRON
ejpam-4976	241	11	exists	exist	VERB
ejpam-4976	241	12	a	a	DET
ejpam-4976	241	13	τ1τ2	τ1τ2	NOUN
ejpam-4976	241	14	-	-	ADJ
ejpam-4976	241	15	open	open	ADJ
ejpam-4976	241	16	set	set	ADJ
ejpam-4976	241	17	u	u	NOUN
ejpam-4976	241	18	of	of	ADP
ejpam-4976	241	19	x	x	PUNCT
ejpam-4976	241	20	containing	contain	VERB
ejpam-4976	241	21	x	x	PUNCT
ejpam-4976	241	22	such	such	ADJ
ejpam-4976	241	23	that	that	DET
ejpam-4976	241	24	f(u	f(u	PROPN
ejpam-4976	241	25	)	)	PUNCT
ejpam-4976	241	26	⊆	⊆	NUM
ejpam-4976	241	27	σ1σ2	σ1σ2	NOUN
ejpam-4976	241	28	-	-	PUNCT
ejpam-4976	241	29	cl(g	cl(g	NUM
ejpam-4976	241	30	)	)	PUNCT
ejpam-4976	241	31	.	.	PUNCT
ejpam-4976	242	1	since	since	SCONJ
ejpam-4976	242	2	x	x	PROPN
ejpam-4976	242	3	∈	∈	PROPN
ejpam-4976	242	4	τ1τ2	τ1τ2	NOUN
ejpam-4976	242	5	-	-	NUM
ejpam-4976	242	6	cl(a	cl(a	NUM
ejpam-4976	242	7	)	)	PUNCT
ejpam-4976	242	8	,	,	PUNCT
ejpam-4976	242	9	we	we	PRON
ejpam-4976	242	10	have	have	VERB
ejpam-4976	242	11	u	u	NOUN
ejpam-4976	242	12	∩	∩	NOUN
ejpam-4976	242	13	a	a	DET
ejpam-4976	242	14	̸=	̸=	PROPN
ejpam-4976	242	15	∅.	∅.	NOUN
ejpam-4976	242	16	it	it	PRON
ejpam-4976	242	17	follows	follow	VERB
ejpam-4976	242	18	that	that	SCONJ
ejpam-4976	242	19	∅	∅	NOUN
ejpam-4976	242	20	̸=	̸=	PROPN
ejpam-4976	242	21	f(u	f(u	PROPN
ejpam-4976	242	22	)	)	PUNCT
ejpam-4976	242	23	∩	∩	NOUN
ejpam-4976	242	24	f(a	f(a	NOUN
ejpam-4976	242	25	)	)	PUNCT
ejpam-4976	242	26	⊆	⊆	NUM
ejpam-4976	242	27	σ1σ2	σ1σ2	NOUN
ejpam-4976	242	28	-	-	PUNCT
ejpam-4976	242	29	cl(g	cl(g	ADJ
ejpam-4976	242	30	)	)	PUNCT
ejpam-4976	242	31	∩	∩	ADJ
ejpam-4976	242	32	f(a	f(a	NOUN
ejpam-4976	242	33	)	)	PUNCT
ejpam-4976	242	34	.	.	PUNCT
ejpam-4976	243	1	thus	thus	ADV
ejpam-4976	243	2	,	,	PUNCT
ejpam-4976	243	3	f(x	f(x	PROPN
ejpam-4976	243	4	)	)	PUNCT
ejpam-4976	243	5	∈	∈	PROPN
ejpam-4976	243	6	(	(	PUNCT
ejpam-4976	243	7	σ1	σ1	PROPN
ejpam-4976	243	8	,	,	PUNCT
ejpam-4976	243	9	σ2)θ	σ2)θ	NOUN
ejpam-4976	243	10	-	-	PUNCT
ejpam-4976	243	11	cl(f(a	cl(f(a	NOUN
ejpam-4976	243	12	)	)	PUNCT
ejpam-4976	243	13	)	)	PUNCT
ejpam-4976	243	14	and	and	CCONJ
ejpam-4976	243	15	hence	hence	ADV
ejpam-4976	243	16	f(τ1τ2	f(τ1τ2	NOUN
ejpam-4976	243	17	-	-	PUNCT
ejpam-4976	243	18	cl(a	cl(a	NUM
ejpam-4976	243	19	)	)	PUNCT
ejpam-4976	243	20	)	)	PUNCT
ejpam-4976	244	1	⊆	⊆	NUM
ejpam-4976	244	2	(	(	PUNCT
ejpam-4976	244	3	σ1	σ1	PROPN
ejpam-4976	244	4	,	,	PUNCT
ejpam-4976	244	5	σ2)θ	σ2)θ	NOUN
ejpam-4976	244	6	-	-	PUNCT
ejpam-4976	244	7	cl(f(a	cl(f(a	NOUN
ejpam-4976	244	8	)	)	PUNCT
ejpam-4976	244	9	)	)	PUNCT
ejpam-4976	244	10	.	.	PUNCT
ejpam-4976	245	1	(	(	PUNCT
ejpam-4976	245	2	2	2	X
ejpam-4976	245	3	)	)	PUNCT
ejpam-4976	245	4	⇒	⇒	NOUN
ejpam-4976	245	5	(	(	PUNCT
ejpam-4976	245	6	3	3	NUM
ejpam-4976	245	7	):	):	PUNCT
ejpam-4976	245	8	let	let	VERB
ejpam-4976	245	9	b	b	X
ejpam-4976	245	10	be	be	AUX
ejpam-4976	245	11	any	any	DET
ejpam-4976	245	12	subset	subset	NOUN
ejpam-4976	245	13	of	of	ADP
ejpam-4976	245	14	y	y	PROPN
ejpam-4976	245	15	.	.	PUNCT
ejpam-4976	246	1	then	then	ADV
ejpam-4976	246	2	,	,	PUNCT
ejpam-4976	246	3	f(τ1τ2	f(τ1τ2	PROPN
ejpam-4976	246	4	-	-	PUNCT
ejpam-4976	246	5	cl(f	cl(f	NOUN
ejpam-4976	246	6	−1(b	−1(b	NOUN
ejpam-4976	246	7	)	)	PUNCT
ejpam-4976	246	8	)	)	PUNCT
ejpam-4976	246	9	)	)	PUNCT
ejpam-4976	247	1	⊆	⊆	X
ejpam-4976	247	2	(	(	PUNCT
ejpam-4976	247	3	σ1	σ1	PROPN
ejpam-4976	247	4	,	,	PUNCT
ejpam-4976	247	5	σ2)θ	σ2)θ	ADJ
ejpam-4976	247	6	-	-	PUNCT
ejpam-4976	247	7	cl(f(f	cl(f(f	ADJ
ejpam-4976	247	8	−1(b	−1(b	NOUN
ejpam-4976	247	9	)	)	PUNCT
ejpam-4976	247	10	)	)	PUNCT
ejpam-4976	247	11	)	)	PUNCT
ejpam-4976	248	1	⊆	⊆	X
ejpam-4976	248	2	(	(	PUNCT
ejpam-4976	248	3	σ1	σ1	PROPN
ejpam-4976	248	4	,	,	PUNCT
ejpam-4976	248	5	σ2)θ	σ2)θ	NOUN
ejpam-4976	248	6	-	-	PUNCT
ejpam-4976	248	7	cl(b	cl(b	NOUN
ejpam-4976	248	8	)	)	PUNCT
ejpam-4976	248	9	and	and	CCONJ
ejpam-4976	248	10	hence	hence	ADV
ejpam-4976	248	11	τ1τ2	τ1τ2	NOUN
ejpam-4976	248	12	-	-	NOUN
ejpam-4976	248	13	cl(f	cl(f	NOUN
ejpam-4976	248	14	−1(b	−1(b	NOUN
ejpam-4976	248	15	)	)	PUNCT
ejpam-4976	248	16	)	)	PUNCT
ejpam-4976	248	17	⊆	⊆	NUM
ejpam-4976	248	18	f−1((σ1	f−1((σ1	NOUN
ejpam-4976	248	19	,	,	PUNCT
ejpam-4976	248	20	σ2)θ	σ2)θ	NOUN
ejpam-4976	248	21	-	-	PUNCT
ejpam-4976	248	22	cl(b	cl(b	NOUN
ejpam-4976	248	23	)	)	PUNCT
ejpam-4976	248	24	)	)	PUNCT
ejpam-4976	248	25	.	.	PUNCT
ejpam-4976	249	1	(	(	PUNCT
ejpam-4976	249	2	3	3	X
ejpam-4976	249	3	)	)	PUNCT
ejpam-4976	249	4	⇒	⇒	NOUN
ejpam-4976	249	5	(	(	PUNCT
ejpam-4976	249	6	1	1	NUM
ejpam-4976	249	7	):	):	PUNCT
ejpam-4976	249	8	let	let	VERB
ejpam-4976	249	9	x	x	PUNCT
ejpam-4976	249	10	∈	∈	PROPN
ejpam-4976	249	11	x	x	X
ejpam-4976	249	12	and	and	CCONJ
ejpam-4976	249	13	v	v	X
ejpam-4976	249	14	be	be	AUX
ejpam-4976	249	15	any	any	DET
ejpam-4976	249	16	σ1σ2	σ1σ2	NOUN
ejpam-4976	249	17	-	-	ADJ
ejpam-4976	249	18	open	open	ADJ
ejpam-4976	249	19	set	set	NOUN
ejpam-4976	249	20	of	of	ADP
ejpam-4976	249	21	y	y	PROPN
ejpam-4976	249	22	containing	contain	VERB
ejpam-4976	249	23	f(x	f(x	PROPN
ejpam-4976	249	24	)	)	PUNCT
ejpam-4976	249	25	.	.	PUNCT
ejpam-4976	250	1	since	since	SCONJ
ejpam-4976	250	2	σ1σ2	σ1σ2	NOUN
ejpam-4976	250	3	-	-	PUNCT
ejpam-4976	250	4	cl(v	cl(v	NOUN
ejpam-4976	250	5	)	)	PUNCT
ejpam-4976	250	6	∩	∩	NOUN
ejpam-4976	250	7	(	(	PUNCT
ejpam-4976	250	8	y	y	PROPN
ejpam-4976	250	9	−	−	PROPN
ejpam-4976	250	10	σ1σ2	σ1σ2	NOUN
ejpam-4976	250	11	-	-	NUM
ejpam-4976	250	12	cl(v	cl(v	NOUN
ejpam-4976	250	13	)	)	PUNCT
ejpam-4976	250	14	)	)	PUNCT
ejpam-4976	250	15	=	=	NOUN
ejpam-4976	250	16	∅	∅	NOUN
ejpam-4976	250	17	,	,	PUNCT
ejpam-4976	250	18	f(x	f(x	PROPN
ejpam-4976	250	19	)	)	PUNCT
ejpam-4976	250	20	̸∈	̸∈	PROPN
ejpam-4976	250	21	(	(	PUNCT
ejpam-4976	250	22	σ1	σ1	PROPN
ejpam-4976	250	23	,	,	PUNCT
ejpam-4976	250	24	σ2)θ	σ2)θ	ADJ
ejpam-4976	250	25	-	-	PUNCT
ejpam-4976	250	26	cl(y	cl(y	PRON
ejpam-4976	250	27	−σ1σ2	−σ1σ2	NOUN
ejpam-4976	250	28	-	-	NOUN
ejpam-4976	250	29	cl(v	cl(v	NOUN
ejpam-4976	250	30	)	)	PUNCT
ejpam-4976	250	31	)	)	PUNCT
ejpam-4976	250	32	and	and	CCONJ
ejpam-4976	250	33	hence	hence	ADV
ejpam-4976	250	34	x	x	X
ejpam-4976	250	35	̸∈	̸∈	PROPN
ejpam-4976	250	36	f−1((σ1	f−1((σ1	NOUN
ejpam-4976	250	37	,	,	PUNCT
ejpam-4976	250	38	σ2)θ	σ2)θ	ADJ
ejpam-4976	250	39	-	-	PUNCT
ejpam-4976	250	40	cl(y	cl(y	NOUN
ejpam-4976	250	41	−σ1σ2	−σ1σ2	NOUN
ejpam-4976	250	42	-	-	NOUN
ejpam-4976	250	43	cl(v	cl(v	NOUN
ejpam-4976	250	44	)	)	PUNCT
ejpam-4976	250	45	)	)	PUNCT
ejpam-4976	250	46	)	)	PUNCT
ejpam-4976	250	47	.	.	PUNCT
ejpam-4976	251	1	by	by	ADP
ejpam-4976	251	2	(	(	PUNCT
ejpam-4976	251	3	3	3	NUM
ejpam-4976	251	4	)	)	PUNCT
ejpam-4976	251	5	,	,	PUNCT
ejpam-4976	251	6	x	x	PROPN
ejpam-4976	251	7	̸∈	̸∈	PROPN
ejpam-4976	251	8	τ1τ2	τ1τ2	PROPN
ejpam-4976	251	9	-	-	PROPN
ejpam-4976	251	10	cl(f	cl(f	NOUN
ejpam-4976	251	11	−1(y	−1(y	PUNCT
ejpam-4976	251	12	−σ1σ2	−σ1σ2	NOUN
ejpam-4976	251	13	-	-	NOUN
ejpam-4976	251	14	cl(v	cl(v	NOUN
ejpam-4976	251	15	)	)	PUNCT
ejpam-4976	251	16	)	)	PUNCT
ejpam-4976	251	17	)	)	PUNCT
ejpam-4976	252	1	and	and	CCONJ
ejpam-4976	252	2	there	there	PRON
ejpam-4976	252	3	exists	exist	VERB
ejpam-4976	252	4	a	a	DET
ejpam-4976	252	5	τ1τ2	τ1τ2	NOUN
ejpam-4976	252	6	-	-	ADJ
ejpam-4976	252	7	open	open	ADJ
ejpam-4976	252	8	set	set	ADJ
ejpam-4976	252	9	u	u	NOUN
ejpam-4976	252	10	of	of	ADP
ejpam-4976	252	11	x	x	PUNCT
ejpam-4976	252	12	containing	contain	VERB
ejpam-4976	252	13	x	x	PUNCT
ejpam-4976	252	14	such	such	ADJ
ejpam-4976	252	15	that	that	SCONJ
ejpam-4976	252	16	u	u	PROPN
ejpam-4976	252	17	∩	∩	NOUN
ejpam-4976	252	18	f−1(y	f−1(y	NOUN
ejpam-4976	252	19	−	−	PUNCT
ejpam-4976	252	20	σ1σ2	σ1σ2	NOUN
ejpam-4976	252	21	-	-	NUM
ejpam-4976	252	22	cl(v	cl(v	NOUN
ejpam-4976	252	23	)	)	PUNCT
ejpam-4976	252	24	)	)	PUNCT
ejpam-4976	253	1	=	=	NOUN
ejpam-4976	253	2	∅	∅	NOUN
ejpam-4976	253	3	;	;	PUNCT
ejpam-4976	253	4	hence	hence	ADV
ejpam-4976	253	5	f(u	f(u	ADJ
ejpam-4976	253	6	)	)	PUNCT
ejpam-4976	253	7	∩	∩	NOUN
ejpam-4976	253	8	(	(	PUNCT
ejpam-4976	253	9	y	y	PROPN
ejpam-4976	253	10	−	−	PROPN
ejpam-4976	253	11	σ1σ2	σ1σ2	NOUN
ejpam-4976	253	12	-	-	NUM
ejpam-4976	253	13	cl(v	cl(v	NOUN
ejpam-4976	253	14	)	)	PUNCT
ejpam-4976	253	15	)	)	PUNCT
ejpam-4976	254	1	=	=	PUNCT
ejpam-4976	254	2	∅.	∅.	ADP
ejpam-4976	254	3	thus	thus	ADV
ejpam-4976	254	4	,	,	PUNCT
ejpam-4976	254	5	f(u	f(u	PROPN
ejpam-4976	254	6	)	)	PUNCT
ejpam-4976	254	7	⊆	⊆	NUM
ejpam-4976	254	8	σ1σ2	σ1σ2	NOUN
ejpam-4976	254	9	-	-	NUM
ejpam-4976	254	10	cl(v	cl(v	NOUN
ejpam-4976	254	11	)	)	PUNCT
ejpam-4976	254	12	and	and	CCONJ
ejpam-4976	254	13	hence	hence	ADV
ejpam-4976	254	14	f	f	PROPN
ejpam-4976	254	15	is	be	AUX
ejpam-4976	254	16	weakly	weakly	ADJ
ejpam-4976	254	17	(	(	PUNCT
ejpam-4976	254	18	τ1	τ1	NOUN
ejpam-4976	254	19	,	,	PUNCT
ejpam-4976	254	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	254	21	at	at	ADP
ejpam-4976	254	22	x.	x.	NOUN
ejpam-4976	254	23	this	this	PRON
ejpam-4976	254	24	shows	show	VERB
ejpam-4976	254	25	that	that	SCONJ
ejpam-4976	254	26	f	f	PROPN
ejpam-4976	254	27	is	be	AUX
ejpam-4976	254	28	weakly	weakly	ADJ
ejpam-4976	254	29	(	(	PUNCT
ejpam-4976	254	30	τ1	τ1	NOUN
ejpam-4976	254	31	,	,	PUNCT
ejpam-4976	254	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	254	33	.	.	PUNCT
ejpam-4976	255	1	acknowledgements	acknowledgement	NOUN
ejpam-4976	255	2	this	this	DET
ejpam-4976	255	3	research	research	NOUN
ejpam-4976	255	4	project	project	NOUN
ejpam-4976	255	5	was	be	AUX
ejpam-4976	255	6	financially	financially	ADV
ejpam-4976	255	7	supported	support	VERB
ejpam-4976	255	8	by	by	ADP
ejpam-4976	255	9	mahasarakham	mahasarakham	PROPN
ejpam-4976	255	10	university	university	PROPN
ejpam-4976	255	11	.	.	PUNCT
ejpam-4976	256	1	references	reference	VERB
ejpam-4976	256	2	424	424	NUM
ejpam-4976	256	3	references	reference	NOUN
ejpam-4976	256	4	[	[	X
ejpam-4976	256	5	1	1	NUM
ejpam-4976	256	6	]	]	PUNCT
ejpam-4976	256	7	c.	c.	PROPN
ejpam-4976	256	8	boonpok	boonpok	PROPN
ejpam-4976	256	9	.	.	PUNCT
ejpam-4976	257	1	m	m	VERB
ejpam-4976	257	2	-continuous	-continuous	ADJ
ejpam-4976	257	3	functions	function	NOUN
ejpam-4976	257	4	in	in	ADP
ejpam-4976	257	5	biminimal	biminimal	NOUN
ejpam-4976	257	6	structure	structure	NOUN
ejpam-4976	257	7	spaces	space	NOUN
ejpam-4976	257	8	.	.	PUNCT
ejpam-4976	258	1	far	far	PROPN
ejpam-4976	258	2	east	east	PROPN
ejpam-4976	258	3	journal	journal	PROPN
ejpam-4976	258	4	of	of	ADP
ejpam-4976	258	5	mathematical	mathematical	ADJ
ejpam-4976	258	6	sciences	science	NOUN
ejpam-4976	258	7	,	,	PUNCT
ejpam-4976	258	8	43(1):41–58	43(1):41–58	NUM
ejpam-4976	258	9	,	,	PUNCT
ejpam-4976	258	10	2010	2010	NUM
ejpam-4976	258	11	.	.	PUNCT
ejpam-4976	259	1	[	[	X
ejpam-4976	259	2	2	2	NUM
ejpam-4976	259	3	]	]	PUNCT
ejpam-4976	259	4	c.	c.	PROPN
ejpam-4976	259	5	boonpok	boonpok	PROPN
ejpam-4976	259	6	.	.	PUNCT
ejpam-4976	260	1	(	(	PUNCT
ejpam-4976	260	2	τ1	τ1	NOUN
ejpam-4976	260	3	,	,	PUNCT
ejpam-4976	260	4	τ2)δ	τ2)δ	ADJ
ejpam-4976	260	5	-	-	PUNCT
ejpam-4976	260	6	semicontinuous	semicontinuous	ADJ
ejpam-4976	260	7	multifunctions	multifunction	NOUN
ejpam-4976	260	8	.	.	PUNCT
ejpam-4976	261	1	heliyon	heliyon	NOUN
ejpam-4976	261	2	,	,	PUNCT
ejpam-4976	261	3	6	6	NUM
ejpam-4976	261	4	:	:	SYM
ejpam-4976	261	5	e05367	e05367	PROPN
ejpam-4976	261	6	,	,	PUNCT
ejpam-4976	261	7	2020	2020	NUM
ejpam-4976	261	8	.	.	PUNCT
ejpam-4976	262	1	[	[	X
ejpam-4976	262	2	3	3	X
ejpam-4976	262	3	]	]	PUNCT
ejpam-4976	262	4	c.	c.	PROPN
ejpam-4976	262	5	boonpok	boonpok	PROPN
ejpam-4976	262	6	.	.	PUNCT
ejpam-4976	263	1	weak	weak	ADJ
ejpam-4976	263	2	openness	openness	NOUN
ejpam-4976	263	3	and	and	CCONJ
ejpam-4976	263	4	weak	weak	ADJ
ejpam-4976	263	5	continuity	continuity	NOUN
ejpam-4976	263	6	.	.	PUNCT
ejpam-4976	264	1	mathematica	mathematica	PROPN
ejpam-4976	264	2	,	,	PUNCT
ejpam-4976	264	3	64(2):173–185	64(2):173–185	NOUN
ejpam-4976	264	4	,	,	PUNCT
ejpam-4976	264	5	2022	2022	NUM
ejpam-4976	264	6	.	.	PUNCT
ejpam-4976	265	1	[	[	X
ejpam-4976	265	2	4	4	NUM
ejpam-4976	265	3	]	]	PUNCT
ejpam-4976	265	4	c.	c.	PROPN
ejpam-4976	265	5	boonpok	boonpok	PROPN
ejpam-4976	265	6	and	and	CCONJ
ejpam-4976	265	7	c.	c.	PROPN
ejpam-4976	265	8	viriyapong	viriyapong	PROPN
ejpam-4976	265	9	.	.	PUNCT
ejpam-4976	266	1	upper	upper	ADJ
ejpam-4976	266	2	and	and	CCONJ
ejpam-4976	266	3	lower	low	ADJ
ejpam-4976	266	4	almost	almost	ADV
ejpam-4976	266	5	weak	weak	ADJ
ejpam-4976	266	6	(	(	PUNCT
ejpam-4976	266	7	τ1	τ1	NOUN
ejpam-4976	266	8	,	,	PUNCT
ejpam-4976	266	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-4976	266	10	.	.	PUNCT
ejpam-4976	267	1	european	european	PROPN
ejpam-4976	267	2	journal	journal	PROPN
ejpam-4976	267	3	of	of	ADP
ejpam-4976	267	4	pure	pure	ADJ
ejpam-4976	267	5	and	and	CCONJ
ejpam-4976	267	6	applied	applied	ADJ
ejpam-4976	267	7	mathematics	mathematic	NOUN
ejpam-4976	267	8	,	,	PUNCT
ejpam-4976	267	9	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-4976	267	10	,	,	PUNCT
ejpam-4976	267	11	2021	2021	NUM
ejpam-4976	267	12	.	.	PUNCT
ejpam-4976	268	1	[	[	X
ejpam-4976	268	2	5	5	X
ejpam-4976	268	3	]	]	PUNCT
ejpam-4976	268	4	c.	c.	PROPN
ejpam-4976	268	5	boonpok	boonpok	PROPN
ejpam-4976	268	6	and	and	CCONJ
ejpam-4976	268	7	c.	c.	PROPN
ejpam-4976	268	8	viriyapong	viriyapong	PROPN
ejpam-4976	268	9	.	.	PUNCT
ejpam-4976	269	1	on	on	ADP
ejpam-4976	269	2	(	(	PUNCT
ejpam-4976	269	3	λ	λ	PROPN
ejpam-4976	269	4	,	,	PUNCT
ejpam-4976	269	5	p)-closed	p)-close	VERB
ejpam-4976	269	6	sets	set	NOUN
ejpam-4976	269	7	and	and	CCONJ
ejpam-4976	269	8	the	the	DET
ejpam-4976	269	9	related	related	ADJ
ejpam-4976	269	10	notions	notion	NOUN
ejpam-4976	269	11	in	in	ADP
ejpam-4976	269	12	topological	topological	ADJ
ejpam-4976	269	13	spaces	space	NOUN
ejpam-4976	269	14	.	.	PUNCT
ejpam-4976	270	1	european	european	ADJ
ejpam-4976	270	2	journal	journal	PROPN
ejpam-4976	270	3	of	of	ADP
ejpam-4976	270	4	pure	pure	ADJ
ejpam-4976	270	5	and	and	CCONJ
ejpam-4976	270	6	applied	applied	ADJ
ejpam-4976	270	7	mathematics	mathematic	NOUN
ejpam-4976	270	8	,	,	PUNCT
ejpam-4976	270	9	15(2):415	15(2):415	PROPN
ejpam-4976	270	10	–	–	PUNCT
ejpam-4976	270	11	436	436	NUM
ejpam-4976	270	12	,	,	PUNCT
ejpam-4976	270	13	2022	2022	NUM
ejpam-4976	270	14	.	.	PUNCT
ejpam-4976	271	1	[	[	X
ejpam-4976	271	2	6	6	NUM
ejpam-4976	271	3	]	]	PUNCT
ejpam-4976	271	4	c.	c.	PROPN
ejpam-4976	271	5	boonpok	boonpok	PROPN
ejpam-4976	271	6	and	and	CCONJ
ejpam-4976	271	7	c.	c.	PROPN
ejpam-4976	271	8	viriyapong	viriyapong	PROPN
ejpam-4976	271	9	.	.	PUNCT
ejpam-4976	272	1	on	on	ADP
ejpam-4976	272	2	some	some	DET
ejpam-4976	272	3	forms	form	NOUN
ejpam-4976	272	4	of	of	ADP
ejpam-4976	272	5	closed	closed	ADJ
ejpam-4976	272	6	sets	set	NOUN
ejpam-4976	272	7	and	and	CCONJ
ejpam-4976	272	8	related	related	ADJ
ejpam-4976	272	9	topics	topic	NOUN
ejpam-4976	272	10	.	.	PUNCT
ejpam-4976	273	1	european	european	ADJ
ejpam-4976	273	2	journal	journal	PROPN
ejpam-4976	273	3	of	of	ADP
ejpam-4976	273	4	pure	pure	ADJ
ejpam-4976	273	5	and	and	CCONJ
ejpam-4976	273	6	applied	applied	ADJ
ejpam-4976	273	7	mathematics	mathematic	NOUN
ejpam-4976	273	8	,	,	PUNCT
ejpam-4976	273	9	16(1):336–362	16(1):336–362	NUM
ejpam-4976	273	10	,	,	PUNCT
ejpam-4976	273	11	2023	2023	NUM
ejpam-4976	273	12	.	.	PUNCT
ejpam-4976	274	1	[	[	X
ejpam-4976	274	2	7	7	X
ejpam-4976	274	3	]	]	X
ejpam-4976	274	4	c.	c.	PROPN
ejpam-4976	274	5	boonpok	boonpok	PROPN
ejpam-4976	274	6	,	,	PUNCT
ejpam-4976	274	7	c.	c.	PROPN
ejpam-4976	274	8	viriyapong	viriyapong	PROPN
ejpam-4976	274	9	,	,	PUNCT
ejpam-4976	274	10	and	and	CCONJ
ejpam-4976	274	11	m.	m.	NOUN
ejpam-4976	274	12	thongmoon	thongmoon	NOUN
ejpam-4976	274	13	.	.	PUNCT
ejpam-4976	275	1	on	on	ADP
ejpam-4976	275	2	upper	upper	ADJ
ejpam-4976	275	3	and	and	CCONJ
ejpam-4976	275	4	lower	low	ADJ
ejpam-4976	275	5	(	(	PUNCT
ejpam-4976	275	6	τ1	τ1	NOUN
ejpam-4976	275	7	,	,	PUNCT
ejpam-4976	275	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-4976	275	9	multifunctions	multifunction	NOUN
ejpam-4976	275	10	,	,	PUNCT
ejpam-4976	275	11	.	.	PUNCT
ejpam-4976	276	1	journal	journal	PROPN
ejpam-4976	276	2	of	of	ADP
ejpam-4976	276	3	mathematics	mathematics	PROPN
ejpam-4976	276	4	and	and	CCONJ
ejpam-4976	276	5	computer	computer	NOUN
ejpam-4976	276	6	science	science	NOUN
ejpam-4976	276	7	,	,	PUNCT
ejpam-4976	276	8	18:282–293	18:282–293	NUM
ejpam-4976	276	9	,	,	PUNCT
ejpam-4976	276	10	2018	2018	NUM
ejpam-4976	276	11	.	.	PUNCT
ejpam-4976	277	1	[	[	X
ejpam-4976	277	2	8	8	NUM
ejpam-4976	277	3	]	]	X
ejpam-4976	277	4	e.	e.	PROPN
ejpam-4976	277	5	ekici	ekici	PROPN
ejpam-4976	277	6	,	,	PUNCT
ejpam-4976	277	7	s.	s.	PROPN
ejpam-4976	277	8	jafari	jafari	PROPN
ejpam-4976	277	9	,	,	PUNCT
ejpam-4976	277	10	m.	m.	PROPN
ejpam-4976	277	11	caldas	caldas	PROPN
ejpam-4976	277	12	,	,	PUNCT
ejpam-4976	277	13	and	and	CCONJ
ejpam-4976	277	14	t.	t.	PROPN
ejpam-4976	277	15	noiri	noiri	PROPN
ejpam-4976	277	16	.	.	PUNCT
ejpam-4976	278	1	weakly	weakly	ADJ
ejpam-4976	278	2	λ	λ	ADJ
ejpam-4976	278	3	-	-	ADJ
ejpam-4976	278	4	continuous	continuous	ADJ
ejpam-4976	278	5	functions	function	NOUN
ejpam-4976	278	6	.	.	PUNCT
ejpam-4976	279	1	novi	novi	PROPN
ejpam-4976	279	2	sad	sad	PROPN
ejpam-4976	279	3	journal	journal	PROPN
ejpam-4976	279	4	of	of	ADP
ejpam-4976	279	5	mathematics	mathematic	NOUN
ejpam-4976	279	6	,	,	PUNCT
ejpam-4976	279	7	38:47–56	38:47–56	NUM
ejpam-4976	279	8	,	,	PUNCT
ejpam-4976	279	9	2008	2008	NUM
ejpam-4976	279	10	.	.	PUNCT
ejpam-4976	280	1	[	[	X
ejpam-4976	280	2	9	9	NUM
ejpam-4976	280	3	]	]	PUNCT
ejpam-4976	280	4	k.	k.	PROPN
ejpam-4976	280	5	laprom	laprom	PROPN
ejpam-4976	280	6	,	,	PUNCT
ejpam-4976	280	7	c.	c.	PROPN
ejpam-4976	280	8	boonpok	boonpok	PROPN
ejpam-4976	280	9	,	,	PUNCT
ejpam-4976	280	10	and	and	CCONJ
ejpam-4976	280	11	c.	c.	PROPN
ejpam-4976	280	12	viriyapong	viriyapong	PROPN
ejpam-4976	280	13	.	.	PUNCT
ejpam-4976	281	1	β(τ1	β(τ1	PROPN
ejpam-4976	281	2	,	,	PUNCT
ejpam-4976	281	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-4976	281	4	multifunctions	multifunction	NOUN
ejpam-4976	281	5	on	on	ADP
ejpam-4976	281	6	bitopological	bitopological	ADJ
ejpam-4976	281	7	spaces	space	NOUN
ejpam-4976	281	8	.	.	PUNCT
ejpam-4976	282	1	journal	journal	NOUN
ejpam-4976	282	2	of	of	ADP
ejpam-4976	282	3	mathematics	mathematic	NOUN
ejpam-4976	282	4	,	,	PUNCT
ejpam-4976	282	5	2020:4020971	2020:4020971	NUM
ejpam-4976	282	6	,	,	PUNCT
ejpam-4976	282	7	2020	2020	NUM
ejpam-4976	282	8	.	.	PUNCT
ejpam-4976	283	1	[	[	X
ejpam-4976	283	2	10	10	NUM
ejpam-4976	283	3	]	]	X
ejpam-4976	283	4	n.	n.	PROPN
ejpam-4976	283	5	levine	levine	PROPN
ejpam-4976	283	6	.	.	PUNCT
ejpam-4976	284	1	a	a	DET
ejpam-4976	284	2	decomposition	decomposition	NOUN
ejpam-4976	284	3	of	of	ADP
ejpam-4976	284	4	continuity	continuity	NOUN
ejpam-4976	284	5	in	in	ADP
ejpam-4976	284	6	topological	topological	ADJ
ejpam-4976	284	7	spaces	space	NOUN
ejpam-4976	284	8	.	.	PUNCT
ejpam-4976	285	1	the	the	DET
ejpam-4976	285	2	american	american	PROPN
ejpam-4976	285	3	mathematical	mathematical	PROPN
ejpam-4976	285	4	monthly	monthly	ADV
ejpam-4976	285	5	,	,	PUNCT
ejpam-4976	285	6	68:44–46	68:44–46	NUM
ejpam-4976	285	7	,	,	PUNCT
ejpam-4976	285	8	1961	1961	NUM
ejpam-4976	285	9	.	.	PUNCT
ejpam-4976	286	1	[	[	X
ejpam-4976	286	2	11	11	NUM
ejpam-4976	286	3	]	]	X
ejpam-4976	286	4	n.	n.	PROPN
ejpam-4976	286	5	levine	levine	PROPN
ejpam-4976	286	6	.	.	PUNCT
ejpam-4976	287	1	semi	semi	ADJ
ejpam-4976	287	2	-	-	ADJ
ejpam-4976	287	3	open	open	ADJ
ejpam-4976	287	4	sets	set	NOUN
ejpam-4976	287	5	and	and	CCONJ
ejpam-4976	287	6	semi	semi	ADJ
ejpam-4976	287	7	-	-	NOUN
ejpam-4976	287	8	continuity	continuity	NOUN
ejpam-4976	287	9	in	in	ADP
ejpam-4976	287	10	topological	topological	ADJ
ejpam-4976	287	11	spaces	space	NOUN
ejpam-4976	287	12	.	.	PUNCT
ejpam-4976	288	1	the	the	DET
ejpam-4976	288	2	american	american	PROPN
ejpam-4976	288	3	mathematical	mathematical	PROPN
ejpam-4976	288	4	monthly	monthly	ADV
ejpam-4976	288	5	,	,	PUNCT
ejpam-4976	288	6	70(1):36–41	70(1):36–41	NUM
ejpam-4976	288	7	,	,	PUNCT
ejpam-4976	288	8	1963	1963	NUM
ejpam-4976	288	9	.	.	PUNCT
ejpam-4976	289	1	[	[	X
ejpam-4976	289	2	12	12	NUM
ejpam-4976	289	3	]	]	PUNCT
ejpam-4976	289	4	s.	s.	PROPN
ejpam-4976	289	5	marcus	marcus	PROPN
ejpam-4976	289	6	.	.	PUNCT
ejpam-4976	290	1	sur	sur	PROPN
ejpam-4976	290	2	les	les	PROPN
ejpam-4976	290	3	fonctions	fonctions	PROPN
ejpam-4976	290	4	quasicontinues	quasicontinue	NOUN
ejpam-4976	290	5	au	au	PROPN
ejpam-4976	290	6	sens	sens	X
ejpam-4976	290	7	de	de	PROPN
ejpam-4976	290	8	s.	s.	PROPN
ejpam-4976	290	9	kempisty	kempisty	PROPN
ejpam-4976	290	10	.	.	PUNCT
ejpam-4976	291	1	colloquium	colloquium	NOUN
ejpam-4976	291	2	mathematicum	mathematicum	PROPN
ejpam-4976	291	3	,	,	PUNCT
ejpam-4976	291	4	8:47–53	8:47–53	NUM
ejpam-4976	291	5	,	,	PUNCT
ejpam-4976	291	6	1961	1961	NUM
ejpam-4976	291	7	.	.	PUNCT
ejpam-4976	292	1	[	[	X
ejpam-4976	292	2	13	13	NUM
ejpam-4976	292	3	]	]	PUNCT
ejpam-4976	292	4	a.	a.	NOUN
ejpam-4976	292	5	neubrunnová.	neubrunnová.	PROPN
ejpam-4976	292	6	on	on	ADP
ejpam-4976	292	7	certain	certain	ADJ
ejpam-4976	292	8	generalizations	generalization	NOUN
ejpam-4976	292	9	of	of	ADP
ejpam-4976	292	10	the	the	DET
ejpam-4976	292	11	notions	notion	NOUN
ejpam-4976	292	12	of	of	ADP
ejpam-4976	292	13	continuity	continuity	NOUN
ejpam-4976	292	14	.	.	PUNCT
ejpam-4976	293	1	matematiký	matematiký	ADJ
ejpam-4976	293	2	časopis	časopis	PROPN
ejpam-4976	293	3	,	,	PUNCT
ejpam-4976	293	4	23:374–380	23:374–380	NUM
ejpam-4976	293	5	,	,	PUNCT
ejpam-4976	293	6	1973	1973	NUM
ejpam-4976	293	7	.	.	PUNCT
ejpam-4976	294	1	[	[	X
ejpam-4976	294	2	14	14	NUM
ejpam-4976	294	3	]	]	PUNCT
ejpam-4976	294	4	t.	t.	PROPN
ejpam-4976	294	5	noiri	noiri	PROPN
ejpam-4976	294	6	.	.	PUNCT
ejpam-4976	295	1	properties	property	NOUN
ejpam-4976	295	2	of	of	ADP
ejpam-4976	295	3	some	some	DET
ejpam-4976	295	4	weak	weak	ADJ
ejpam-4976	295	5	forms	form	NOUN
ejpam-4976	295	6	of	of	ADP
ejpam-4976	295	7	continuity	continuity	NOUN
ejpam-4976	295	8	.	.	PUNCT
ejpam-4976	296	1	international	international	ADJ
ejpam-4976	296	2	journal	journal	PROPN
ejpam-4976	296	3	of	of	ADP
ejpam-4976	296	4	mathematics	mathematics	PROPN
ejpam-4976	296	5	and	and	CCONJ
ejpam-4976	296	6	mathematical	mathematical	ADJ
ejpam-4976	296	7	sciences	science	NOUN
ejpam-4976	296	8	,	,	PUNCT
ejpam-4976	296	9	10(1):97–111	10(1):97–111	NUM
ejpam-4976	296	10	,	,	PUNCT
ejpam-4976	296	11	1987	1987	NUM
ejpam-4976	296	12	.	.	PUNCT
ejpam-4976	297	1	[	[	X
ejpam-4976	297	2	15	15	NUM
ejpam-4976	297	3	]	]	X
ejpam-4976	297	4	v.	v.	CCONJ
ejpam-4976	297	5	popa	popa	NOUN
ejpam-4976	297	6	and	and	CCONJ
ejpam-4976	297	7	t.	t.	PROPN
ejpam-4976	297	8	noiri	noiri	PROPN
ejpam-4976	297	9	.	.	PUNCT
ejpam-4976	298	1	a	a	DET
ejpam-4976	298	2	unified	unified	ADJ
ejpam-4976	298	3	theory	theory	NOUN
ejpam-4976	298	4	of	of	ADP
ejpam-4976	298	5	weak	weak	ADJ
ejpam-4976	298	6	continuity	continuity	NOUN
ejpam-4976	298	7	for	for	ADP
ejpam-4976	298	8	functions	function	NOUN
ejpam-4976	298	9	.	.	PUNCT
ejpam-4976	299	1	rendiconti	rendiconti	ADJ
ejpam-4976	299	2	del	del	PROPN
ejpam-4976	299	3	circolo	circolo	PROPN
ejpam-4976	299	4	matematico	matematico	NOUN
ejpam-4976	299	5	di	di	X
ejpam-4976	299	6	palermo	palermo	NOUN
ejpam-4976	299	7	(	(	PUNCT
ejpam-4976	299	8	2	2	NUM
ejpam-4976	299	9	)	)	PUNCT
ejpam-4976	299	10	,	,	PUNCT
ejpam-4976	299	11	51:439–464	51:439–464	PROPN
ejpam-4976	299	12	,	,	PUNCT
ejpam-4976	299	13	2002	2002	NUM
ejpam-4976	299	14	.	.	PUNCT
ejpam-4976	300	1	[	[	X
ejpam-4976	300	2	16	16	NUM
ejpam-4976	300	3	]	]	PUNCT
ejpam-4976	300	4	v.	v.	CCONJ
ejpam-4976	300	5	popa	popa	NOUN
ejpam-4976	300	6	and	and	CCONJ
ejpam-4976	300	7	t.	t.	PROPN
ejpam-4976	300	8	noiri	noiri	PROPN
ejpam-4976	300	9	.	.	PUNCT
ejpam-4976	301	1	on	on	ADP
ejpam-4976	301	2	weakly	weakly	ADJ
ejpam-4976	301	3	(	(	PUNCT
ejpam-4976	301	4	τ	τ	PROPN
ejpam-4976	301	5	,	,	PUNCT
ejpam-4976	301	6	m)-continuous	m)-continuous	ADJ
ejpam-4976	301	7	functions	function	NOUN
ejpam-4976	301	8	.	.	PUNCT
ejpam-4976	302	1	rendiconti	rendiconti	ADJ
ejpam-4976	302	2	del	del	PROPN
ejpam-4976	302	3	circolo	circolo	PROPN
ejpam-4976	302	4	matematico	matematico	NOUN
ejpam-4976	302	5	di	di	X
ejpam-4976	302	6	palermo	palermo	NOUN
ejpam-4976	302	7	(	(	PUNCT
ejpam-4976	302	8	2	2	NUM
ejpam-4976	302	9	)	)	PUNCT
ejpam-4976	302	10	,	,	PUNCT
ejpam-4976	302	11	51:295–316	51:295–316	PROPN
ejpam-4976	302	12	,	,	PUNCT
ejpam-4976	302	13	2002	2002	NUM
ejpam-4976	302	14	.	.	PUNCT
ejpam-4976	303	1	references	reference	NOUN
ejpam-4976	303	2	425	425	NUM
ejpam-4976	304	1	[	[	X
ejpam-4976	304	2	17	17	NUM
ejpam-4976	304	3	]	]	PUNCT
ejpam-4976	304	4	v.	v.	CCONJ
ejpam-4976	304	5	popa	popa	NOUN
ejpam-4976	304	6	and	and	CCONJ
ejpam-4976	304	7	c.	c.	PROPN
ejpam-4976	304	8	stan	stan	PROPN
ejpam-4976	304	9	.	.	PUNCT
ejpam-4976	305	1	on	on	ADP
ejpam-4976	305	2	a	a	DET
ejpam-4976	305	3	decomposition	decomposition	NOUN
ejpam-4976	305	4	of	of	ADP
ejpam-4976	305	5	quasicontinuity	quasicontinuity	NOUN
ejpam-4976	305	6	in	in	ADP
ejpam-4976	305	7	topological	topological	ADJ
ejpam-4976	305	8	spaces	space	NOUN
ejpam-4976	305	9	.	.	PUNCT
ejpam-4976	306	1	studii	studii	PROPN
ejpam-4976	306	2	şi	şi	PROPN
ejpam-4976	306	3	cercetǎri	cercetǎri	PROPN
ejpam-4976	306	4	de	de	X
ejpam-4976	306	5	matematicǎ	matematicǎ	NOUN
ejpam-4976	306	6	,	,	PUNCT
ejpam-4976	306	7	25:41–43	25:41–43	NUM
ejpam-4976	306	8	,	,	PUNCT
ejpam-4976	306	9	1973	1973	NUM
ejpam-4976	306	10	.	.	PUNCT
ejpam-4976	307	1	[	[	X
ejpam-4976	307	2	18	18	NUM
ejpam-4976	307	3	]	]	X
ejpam-4976	307	4	d.	d.	PROPN
ejpam-4976	307	5	a.	a.	PROPN
ejpam-4976	307	6	rose	rise	VERB
ejpam-4976	307	7	.	.	PUNCT
ejpam-4976	308	1	weak	weak	ADJ
ejpam-4976	308	2	continuity	continuity	NOUN
ejpam-4976	308	3	and	and	CCONJ
ejpam-4976	308	4	almost	almost	ADV
ejpam-4976	308	5	continuity	continuity	NOUN
ejpam-4976	308	6	.	.	PUNCT
ejpam-4976	309	1	international	international	ADJ
ejpam-4976	309	2	journal	journal	PROPN
ejpam-4976	309	3	of	of	ADP
ejpam-4976	309	4	mathematics	mathematics	PROPN
ejpam-4976	309	5	and	and	CCONJ
ejpam-4976	309	6	mathematical	mathematical	ADJ
ejpam-4976	309	7	sciences	science	NOUN
ejpam-4976	309	8	,	,	PUNCT
ejpam-4976	309	9	7:311–318	7:311–318	PROPN
ejpam-4976	309	10	,	,	PUNCT
ejpam-4976	309	11	1984	1984	NUM
ejpam-4976	309	12	.	.	PUNCT
ejpam-4976	310	1	[	[	X
ejpam-4976	310	2	19	19	NUM
ejpam-4976	310	3	]	]	X
ejpam-4976	310	4	c.	c.	PROPN
ejpam-4976	310	5	viriyapong	viriyapong	PROPN
ejpam-4976	310	6	and	and	CCONJ
ejpam-4976	310	7	c.	c.	PROPN
ejpam-4976	310	8	boonpok	boonpok	PROPN
ejpam-4976	310	9	.	.	PUNCT
ejpam-4976	311	1	(	(	PUNCT
ejpam-4976	311	2	τ1	τ1	NOUN
ejpam-4976	311	3	,	,	PUNCT
ejpam-4976	311	4	τ2)α	τ2)α	NOUN
ejpam-4976	311	5	-	-	PUNCT
ejpam-4976	311	6	continuity	continuity	NOUN
ejpam-4976	311	7	for	for	ADP
ejpam-4976	311	8	multifunctions	multifunction	NOUN
ejpam-4976	311	9	.	.	PUNCT
ejpam-4976	312	1	journal	journal	PROPN
ejpam-4976	312	2	of	of	ADP
ejpam-4976	312	3	mathematics	mathematic	NOUN
ejpam-4976	312	4	,	,	PUNCT
ejpam-4976	312	5	2020:6285763	2020:6285763	NUM
ejpam-4976	312	6	,	,	PUNCT
ejpam-4976	312	7	2020	2020	NUM
ejpam-4976	312	8	.	.	PUNCT
ejpam-4976	313	1	[	[	X
ejpam-4976	313	2	20	20	NUM
ejpam-4976	313	3	]	]	X
ejpam-4976	313	4	c.	c.	PROPN
ejpam-4976	313	5	viriyapong	viriyapong	PROPN
ejpam-4976	313	6	and	and	CCONJ
ejpam-4976	313	7	c.	c.	PROPN
ejpam-4976	313	8	boonpok	boonpok	PROPN
ejpam-4976	313	9	.	.	PUNCT
ejpam-4976	314	1	(	(	PUNCT
ejpam-4976	314	2	λ	λ	X
ejpam-4976	314	3	,	,	PUNCT
ejpam-4976	314	4	sp)-continuous	sp)-continuous	ADJ
ejpam-4976	314	5	functions	function	NOUN
ejpam-4976	314	6	.	.	PUNCT
ejpam-4976	315	1	wseas	wseas	VERB
ejpam-4976	315	2	transactions	transaction	NOUN
ejpam-4976	315	3	on	on	ADP
ejpam-4976	315	4	mathematics	mathematic	NOUN
ejpam-4976	315	5	,	,	PUNCT
ejpam-4976	315	6	21:380–385	21:380–385	NUM
ejpam-4976	315	7	,	,	PUNCT
ejpam-4976	315	8	2022	2022	NUM
ejpam-4976	315	9	.	.	PUNCT