id	sid	tid	token	lemma	pos
ejpam-5238	1	1	european	european	PROPN
ejpam-5238	1	2	journal	journal	PROPN
ejpam-5238	1	3	of	of	ADP
ejpam-5238	1	4	pure	pure	ADJ
ejpam-5238	1	5	and	and	CCONJ
ejpam-5238	1	6	applied	apply	VERB
ejpam-5238	1	7	mathematics	mathematic	NOUN
ejpam-5238	1	8	vol	vol	NOUN
ejpam-5238	1	9	.	.	PROPN
ejpam-5238	2	1	17	17	NUM
ejpam-5238	2	2	,	,	PUNCT
ejpam-5238	2	3	no	no	INTJ
ejpam-5238	2	4	.	.	NOUN
ejpam-5238	2	5	3	3	NUM
ejpam-5238	2	6	,	,	PUNCT
ejpam-5238	2	7	2024	2024	NUM
ejpam-5238	2	8	,	,	PUNCT
ejpam-5238	2	9	1705	1705	NUM
ejpam-5238	2	10	-	-	SYM
ejpam-5238	2	11	1716	1716	NUM
ejpam-5238	2	12	issn	issn	VERB
ejpam-5238	2	13	1307	1307	NUM
ejpam-5238	2	14	-	-	SYM
ejpam-5238	2	15	5543	5543	NUM
ejpam-5238	2	16	–	–	PUNCT
ejpam-5238	2	17	ejpam.com	ejpam.com	X
ejpam-5238	2	18	published	publish	VERB
ejpam-5238	2	19	by	by	ADP
ejpam-5238	2	20	new	new	PROPN
ejpam-5238	2	21	york	york	PROPN
ejpam-5238	2	22	business	business	PROPN
ejpam-5238	2	23	global	global	PROPN
ejpam-5238	2	24	upper	upper	ADJ
ejpam-5238	2	25	and	and	CCONJ
ejpam-5238	2	26	lower	low	ADJ
ejpam-5238	2	27	weak	weak	ADJ
ejpam-5238	2	28	(	(	PUNCT
ejpam-5238	2	29	τ1	τ1	NOUN
ejpam-5238	2	30	,	,	PUNCT
ejpam-5238	2	31	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5238	2	32	montri	montri	PROPN
ejpam-5238	2	33	thongmoon1	thongmoon1	PROPN
ejpam-5238	2	34	,	,	PUNCT
ejpam-5238	2	35	supannee	supannee	PROPN
ejpam-5238	2	36	sompong2	sompong2	PROPN
ejpam-5238	2	37	,	,	PUNCT
ejpam-5238	2	38	chawalit	chawalit	VERB
ejpam-5238	2	39	boonpok1,∗	boonpok1,∗	NOUN
ejpam-5238	2	40	1	1	NUM
ejpam-5238	2	41	mathematics	mathematic	NOUN
ejpam-5238	2	42	and	and	CCONJ
ejpam-5238	2	43	applied	apply	VERB
ejpam-5238	2	44	mathematics	mathematics	PROPN
ejpam-5238	2	45	research	research	NOUN
ejpam-5238	2	46	unit	unit	NOUN
ejpam-5238	2	47	,	,	PUNCT
ejpam-5238	2	48	department	department	NOUN
ejpam-5238	2	49	of	of	ADP
ejpam-5238	2	50	mathematics	mathematic	NOUN
ejpam-5238	2	51	,	,	PUNCT
ejpam-5238	2	52	faculty	faculty	NOUN
ejpam-5238	2	53	of	of	ADP
ejpam-5238	2	54	science	science	NOUN
ejpam-5238	2	55	,	,	PUNCT
ejpam-5238	2	56	mahasarakham	mahasarakham	PROPN
ejpam-5238	2	57	university	university	PROPN
ejpam-5238	2	58	,	,	PUNCT
ejpam-5238	2	59	maha	maha	PROPN
ejpam-5238	2	60	sarakham	sarakham	PROPN
ejpam-5238	2	61	,	,	PUNCT
ejpam-5238	2	62	44150	44150	NUM
ejpam-5238	2	63	,	,	PUNCT
ejpam-5238	2	64	thailand	thailand	PROPN
ejpam-5238	2	65	2	2	NUM
ejpam-5238	2	66	department	department	NOUN
ejpam-5238	2	67	of	of	ADP
ejpam-5238	2	68	mathematics	mathematic	NOUN
ejpam-5238	2	69	and	and	CCONJ
ejpam-5238	2	70	statistics	statistic	NOUN
ejpam-5238	2	71	,	,	PUNCT
ejpam-5238	2	72	faculty	faculty	NOUN
ejpam-5238	2	73	of	of	ADP
ejpam-5238	2	74	science	science	NOUN
ejpam-5238	2	75	and	and	CCONJ
ejpam-5238	2	76	technology	technology	NOUN
ejpam-5238	2	77	,	,	PUNCT
ejpam-5238	2	78	sakon	sakon	PROPN
ejpam-5238	2	79	nakhon	nakhon	PROPN
ejpam-5238	2	80	rajbhat	rajbhat	PROPN
ejpam-5238	2	81	university	university	PROPN
ejpam-5238	2	82	,	,	PUNCT
ejpam-5238	2	83	sakon	sakon	PROPN
ejpam-5238	2	84	nakhon	nakhon	PROPN
ejpam-5238	2	85	,	,	PUNCT
ejpam-5238	2	86	47000	47000	NUM
ejpam-5238	2	87	,	,	PUNCT
ejpam-5238	2	88	thailand	thailand	PROPN
ejpam-5238	2	89	abstract	abstract	NOUN
ejpam-5238	2	90	.	.	PUNCT
ejpam-5238	3	1	this	this	DET
ejpam-5238	3	2	paper	paper	NOUN
ejpam-5238	3	3	is	be	AUX
ejpam-5238	3	4	concerned	concern	VERB
ejpam-5238	3	5	with	with	ADP
ejpam-5238	3	6	the	the	DET
ejpam-5238	3	7	concepts	concept	NOUN
ejpam-5238	3	8	of	of	ADP
ejpam-5238	3	9	upper	upper	ADJ
ejpam-5238	3	10	and	and	CCONJ
ejpam-5238	3	11	lower	low	ADJ
ejpam-5238	3	12	weakly	weakly	ADJ
ejpam-5238	3	13	(	(	PUNCT
ejpam-5238	3	14	τ1	τ1	NOUN
ejpam-5238	3	15	,	,	PUNCT
ejpam-5238	3	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	3	17	multifunctions	multifunction	NOUN
ejpam-5238	3	18	.	.	PUNCT
ejpam-5238	4	1	moreover	moreover	ADV
ejpam-5238	4	2	,	,	PUNCT
ejpam-5238	4	3	some	some	DET
ejpam-5238	4	4	characterizations	characterization	NOUN
ejpam-5238	4	5	of	of	ADP
ejpam-5238	4	6	upper	upper	ADJ
ejpam-5238	4	7	and	and	CCONJ
ejpam-5238	4	8	lower	low	ADJ
ejpam-5238	4	9	weakly	weakly	ADJ
ejpam-5238	4	10	(	(	PUNCT
ejpam-5238	4	11	τ1	τ1	NOUN
ejpam-5238	4	12	,	,	PUNCT
ejpam-5238	4	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	4	14	multifunctions	multifunction	NOUN
ejpam-5238	4	15	are	be	AUX
ejpam-5238	4	16	investigated	investigate	VERB
ejpam-5238	4	17	.	.	PUNCT
ejpam-5238	5	1	2020	2020	NUM
ejpam-5238	5	2	mathematics	mathematic	NOUN
ejpam-5238	5	3	subject	subject	NOUN
ejpam-5238	5	4	classifications	classification	NOUN
ejpam-5238	5	5	:	:	PUNCT
ejpam-5238	5	6	54c08	54c08	NUM
ejpam-5238	5	7	,	,	PUNCT
ejpam-5238	5	8	54c60	54c60	NUM
ejpam-5238	5	9	,	,	PUNCT
ejpam-5238	5	10	54e55	54e55	NUM
ejpam-5238	5	11	key	key	ADJ
ejpam-5238	5	12	words	word	NOUN
ejpam-5238	5	13	and	and	CCONJ
ejpam-5238	5	14	phrases	phrase	NOUN
ejpam-5238	5	15	:	:	PUNCT
ejpam-5238	5	16	upper	upper	ADJ
ejpam-5238	5	17	weakly	weakly	ADJ
ejpam-5238	5	18	(	(	PUNCT
ejpam-5238	5	19	τ1	τ1	NOUN
ejpam-5238	5	20	,	,	PUNCT
ejpam-5238	5	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	5	22	multifunction	multifunction	NOUN
ejpam-5238	5	23	,	,	PUNCT
ejpam-5238	5	24	lower	low	ADJ
ejpam-5238	5	25	weakly	weakly	ADJ
ejpam-5238	5	26	(	(	PUNCT
ejpam-5238	5	27	τ1	τ1	NOUN
ejpam-5238	5	28	,	,	PUNCT
ejpam-5238	5	29	τ2)continuous	τ2)continuous	ADJ
ejpam-5238	5	30	multifunction	multifunction	NOUN
ejpam-5238	5	31	1	1	NUM
ejpam-5238	5	32	.	.	PUNCT
ejpam-5238	5	33	introduction	introduction	NOUN
ejpam-5238	5	34	the	the	DET
ejpam-5238	5	35	concept	concept	NOUN
ejpam-5238	5	36	of	of	ADP
ejpam-5238	5	37	weakly	weakly	ADJ
ejpam-5238	5	38	continuous	continuous	ADJ
ejpam-5238	5	39	functions	function	NOUN
ejpam-5238	5	40	was	be	AUX
ejpam-5238	5	41	introduced	introduce	VERB
ejpam-5238	5	42	by	by	ADP
ejpam-5238	5	43	levine	levine	PROPN
ejpam-5238	6	1	[	[	X
ejpam-5238	6	2	26	26	NUM
ejpam-5238	6	3	]	]	PUNCT
ejpam-5238	6	4	.	.	PUNCT
ejpam-5238	7	1	husain	husain	PROPN
ejpam-5238	8	1	[	[	X
ejpam-5238	8	2	23	23	NUM
ejpam-5238	8	3	]	]	PUNCT
ejpam-5238	8	4	introduced	introduce	VERB
ejpam-5238	8	5	and	and	CCONJ
ejpam-5238	8	6	studied	study	VERB
ejpam-5238	8	7	the	the	DET
ejpam-5238	8	8	notion	notion	NOUN
ejpam-5238	8	9	of	of	ADP
ejpam-5238	8	10	almost	almost	ADV
ejpam-5238	8	11	continuous	continuous	ADJ
ejpam-5238	8	12	functions	function	NOUN
ejpam-5238	8	13	.	.	PUNCT
ejpam-5238	9	1	janković	janković	PUNCT
ejpam-5238	10	1	[	[	X
ejpam-5238	10	2	24	24	NUM
ejpam-5238	10	3	]	]	PUNCT
ejpam-5238	10	4	introduced	introduce	VERB
ejpam-5238	10	5	almost	almost	ADV
ejpam-5238	10	6	weak	weak	ADJ
ejpam-5238	10	7	continuity	continuity	NOUN
ejpam-5238	10	8	as	as	ADP
ejpam-5238	10	9	a	a	DET
ejpam-5238	10	10	generalization	generalization	NOUN
ejpam-5238	10	11	of	of	ADP
ejpam-5238	10	12	both	both	DET
ejpam-5238	10	13	weak	weak	ADJ
ejpam-5238	10	14	continuity	continuity	NOUN
ejpam-5238	10	15	and	and	CCONJ
ejpam-5238	10	16	almost	almost	ADV
ejpam-5238	10	17	continuity	continuity	NOUN
ejpam-5238	10	18	.	.	PUNCT
ejpam-5238	11	1	noiri	noiri	ADV
ejpam-5238	12	1	[	[	X
ejpam-5238	12	2	27	27	NUM
ejpam-5238	12	3	]	]	PUNCT
ejpam-5238	12	4	investigated	investigate	VERB
ejpam-5238	12	5	several	several	ADJ
ejpam-5238	12	6	characterizations	characterization	NOUN
ejpam-5238	12	7	of	of	ADP
ejpam-5238	12	8	almost	almost	ADV
ejpam-5238	12	9	weakly	weakly	ADJ
ejpam-5238	12	10	continuous	continuous	ADJ
ejpam-5238	12	11	functions	function	NOUN
ejpam-5238	12	12	.	.	PUNCT
ejpam-5238	13	1	rose	rise	VERB
ejpam-5238	14	1	[	[	X
ejpam-5238	14	2	34	34	NUM
ejpam-5238	14	3	]	]	PUNCT
ejpam-5238	14	4	introduced	introduce	VERB
ejpam-5238	14	5	the	the	DET
ejpam-5238	14	6	notion	notion	NOUN
ejpam-5238	14	7	of	of	ADP
ejpam-5238	14	8	subweakly	subweakly	ADJ
ejpam-5238	14	9	continuous	continuous	ADJ
ejpam-5238	14	10	functions	function	NOUN
ejpam-5238	14	11	and	and	CCONJ
ejpam-5238	14	12	investigated	investigate	VERB
ejpam-5238	14	13	the	the	DET
ejpam-5238	14	14	relationships	relationship	NOUN
ejpam-5238	14	15	between	between	ADP
ejpam-5238	14	16	subweak	subweak	NOUN
ejpam-5238	14	17	continuity	continuity	NOUN
ejpam-5238	14	18	and	and	CCONJ
ejpam-5238	14	19	weak	weak	ADJ
ejpam-5238	14	20	continuity	continuity	NOUN
ejpam-5238	14	21	.	.	PUNCT
ejpam-5238	15	1	popa	popa	NOUN
ejpam-5238	15	2	and	and	CCONJ
ejpam-5238	15	3	noiri	noiri	ADV
ejpam-5238	16	1	[	[	X
ejpam-5238	16	2	32	32	NUM
ejpam-5238	16	3	]	]	PUNCT
ejpam-5238	16	4	introduced	introduce	VERB
ejpam-5238	16	5	the	the	DET
ejpam-5238	16	6	concept	concept	NOUN
ejpam-5238	16	7	of	of	ADP
ejpam-5238	16	8	weakly	weakly	ADJ
ejpam-5238	16	9	(	(	PUNCT
ejpam-5238	16	10	τ	τ	PROPN
ejpam-5238	16	11	,	,	PUNCT
ejpam-5238	16	12	m)-continuous	m)-continuous	ADJ
ejpam-5238	16	13	functions	function	NOUN
ejpam-5238	16	14	as	as	ADP
ejpam-5238	16	15	functions	function	NOUN
ejpam-5238	16	16	from	from	ADP
ejpam-5238	16	17	a	a	DET
ejpam-5238	16	18	topological	topological	ADJ
ejpam-5238	16	19	space	space	NOUN
ejpam-5238	16	20	into	into	ADP
ejpam-5238	16	21	a	a	DET
ejpam-5238	16	22	set	set	NOUN
ejpam-5238	16	23	satisfying	satisfy	VERB
ejpam-5238	16	24	some	some	DET
ejpam-5238	16	25	minimal	minimal	ADJ
ejpam-5238	16	26	conditions	condition	NOUN
ejpam-5238	16	27	and	and	CCONJ
ejpam-5238	16	28	investigated	investigate	VERB
ejpam-5238	16	29	several	several	ADJ
ejpam-5238	16	30	characterizations	characterization	NOUN
ejpam-5238	16	31	of	of	ADP
ejpam-5238	16	32	weakly	weakly	ADJ
ejpam-5238	16	33	(	(	PUNCT
ejpam-5238	16	34	τ	τ	PROPN
ejpam-5238	16	35	,	,	PUNCT
ejpam-5238	16	36	m)-continuous	m)-continuous	ADJ
ejpam-5238	16	37	functions	function	NOUN
ejpam-5238	16	38	.	.	PUNCT
ejpam-5238	17	1	ekici	ekici	NOUN
ejpam-5238	17	2	et	et	PROPN
ejpam-5238	17	3	al	al	PROPN
ejpam-5238	17	4	.	.	PUNCT
ejpam-5238	18	1	[	[	X
ejpam-5238	18	2	22	22	NUM
ejpam-5238	18	3	]	]	PUNCT
ejpam-5238	18	4	introduced	introduce	VERB
ejpam-5238	18	5	and	and	CCONJ
ejpam-5238	18	6	studied	study	VERB
ejpam-5238	18	7	the	the	DET
ejpam-5238	18	8	concept	concept	NOUN
ejpam-5238	18	9	of	of	ADP
ejpam-5238	18	10	weakly	weakly	ADJ
ejpam-5238	18	11	λ	λ	ADJ
ejpam-5238	18	12	-	-	ADJ
ejpam-5238	18	13	continuous	continuous	ADJ
ejpam-5238	18	14	functions	function	NOUN
ejpam-5238	18	15	.	.	PUNCT
ejpam-5238	19	1	duangphui	duangphui	NOUN
ejpam-5238	19	2	et	et	PROPN
ejpam-5238	19	3	al	al	PROPN
ejpam-5238	19	4	.	.	PUNCT
ejpam-5238	20	1	[	[	X
ejpam-5238	20	2	21	21	NUM
ejpam-5238	20	3	]	]	PUNCT
ejpam-5238	20	4	introduced	introduce	VERB
ejpam-5238	20	5	and	and	CCONJ
ejpam-5238	20	6	investigated	investigate	VERB
ejpam-5238	20	7	the	the	DET
ejpam-5238	20	8	notion	notion	NOUN
ejpam-5238	20	9	of	of	ADP
ejpam-5238	20	10	weakly	weakly	ADJ
ejpam-5238	20	11	(	(	PUNCT
ejpam-5238	20	12	µ	µ	NOUN
ejpam-5238	20	13	,	,	PUNCT
ejpam-5238	20	14	µ′)(m	µ′)(m	VERB
ejpam-5238	20	15	,	,	PUNCT
ejpam-5238	20	16	n)-continuous	n)-continuous	ADJ
ejpam-5238	20	17	functions	function	NOUN
ejpam-5238	20	18	.	.	PUNCT
ejpam-5238	21	1	moreover	moreover	ADV
ejpam-5238	21	2	,	,	PUNCT
ejpam-5238	21	3	some	some	DET
ejpam-5238	21	4	characterizations	characterization	NOUN
ejpam-5238	21	5	of	of	ADP
ejpam-5238	21	6	almost	almost	ADV
ejpam-5238	21	7	(	(	PUNCT
ejpam-5238	21	8	λ	λ	PROPN
ejpam-5238	21	9	,	,	PUNCT
ejpam-5238	21	10	p)-continuous	p)-continuous	ADJ
ejpam-5238	21	11	functions	function	NOUN
ejpam-5238	21	12	,	,	PUNCT
ejpam-5238	21	13	strongly	strongly	ADV
ejpam-5238	21	14	θ(λ	θ(λ	PROPN
ejpam-5238	21	15	,	,	PUNCT
ejpam-5238	21	16	p)-continuous	p)-continuous	ADJ
ejpam-5238	21	17	functions	function	NOUN
ejpam-5238	21	18	,	,	PUNCT
ejpam-5238	21	19	almost	almost	ADV
ejpam-5238	21	20	strongly	strongly	ADV
ejpam-5238	21	21	θ(λ	θ(λ	VERB
ejpam-5238	21	22	,	,	PUNCT
ejpam-5238	21	23	p)-continuous	p)-continuous	ADJ
ejpam-5238	21	24	functions	function	NOUN
ejpam-5238	21	25	,	,	PUNCT
ejpam-5238	21	26	θ(λ	θ(λ	PROPN
ejpam-5238	21	27	,	,	PUNCT
ejpam-5238	21	28	p)-continuous	p)-continuous	ADJ
ejpam-5238	21	29	functions	function	NOUN
ejpam-5238	21	30	,	,	PUNCT
ejpam-5238	21	31	weakly	weakly	ADJ
ejpam-5238	21	32	(	(	PUNCT
ejpam-5238	21	33	λ	λ	PROPN
ejpam-5238	21	34	,	,	PUNCT
ejpam-5238	21	35	b)-continuous	b)-continuous	ADJ
ejpam-5238	21	36	functions	function	NOUN
ejpam-5238	21	37	,	,	PUNCT
ejpam-5238	21	38	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5238	21	39	functions	function	NOUN
ejpam-5238	21	40	,	,	PUNCT
ejpam-5238	21	41	⋆-continuous	⋆-continuous	ADJ
ejpam-5238	21	42	functions	function	NOUN
ejpam-5238	21	43	,	,	PUNCT
ejpam-5238	21	44	θ	θ	PROPN
ejpam-5238	21	45	-	-	ADJ
ejpam-5238	21	46	i	i	VERB
ejpam-5238	21	47	continuous	continuous	ADJ
ejpam-5238	21	48	functions	function	NOUN
ejpam-5238	21	49	,	,	PUNCT
ejpam-5238	21	50	almost	almost	ADV
ejpam-5238	21	51	(	(	PUNCT
ejpam-5238	21	52	g	g	NOUN
ejpam-5238	21	53	,	,	PUNCT
ejpam-5238	21	54	m)-continuous	m)-continuous	ADJ
ejpam-5238	21	55	functions	function	NOUN
ejpam-5238	21	56	,	,	PUNCT
ejpam-5238	21	57	(	(	PUNCT
ejpam-5238	21	58	λ	λ	NOUN
ejpam-5238	21	59	,	,	PUNCT
ejpam-5238	21	60	sp)-continuous	sp)-continuous	ADJ
ejpam-5238	21	61	functions	function	NOUN
ejpam-5238	21	62	,	,	PUNCT
ejpam-5238	21	63	δp(λ	δp(λ	NOUN
ejpam-5238	21	64	,	,	PUNCT
ejpam-5238	21	65	s)-continuous	s)-continuous	ADJ
ejpam-5238	21	66	functions	function	NOUN
ejpam-5238	21	67	,	,	PUNCT
ejpam-5238	21	68	(	(	PUNCT
ejpam-5238	21	69	λ	λ	NOUN
ejpam-5238	21	70	,	,	PUNCT
ejpam-5238	21	71	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-5238	21	72	functions	function	NOUN
ejpam-5238	21	73	,	,	PUNCT
ejpam-5238	21	74	pairwise	pairwise	NOUN
ejpam-5238	21	75	weaklym	weaklym	PROPN
ejpam-5238	21	76	-continuous	-continuous	ADJ
ejpam-5238	21	77	∗corresponding	∗corresponding	NOUN
ejpam-5238	21	78	author	author	NOUN
ejpam-5238	21	79	.	.	PUNCT
ejpam-5238	22	1	doi	doi	NOUN
ejpam-5238	22	2	:	:	PUNCT
ejpam-5238	22	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5238	https://doi.org/10.29020/nybg.ejpam.v17i3.5238	ADJ
ejpam-5238	22	4	email	email	NOUN
ejpam-5238	22	5	addresses	address	VERB
ejpam-5238	22	6	:	:	PUNCT
ejpam-5238	22	7	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-5238	22	8	(	(	PUNCT
ejpam-5238	22	9	m.	m.	NOUN
ejpam-5238	22	10	thongmoon	thongmoon	PROPN
ejpam-5238	22	11	)	)	PUNCT
ejpam-5238	22	12	,	,	PUNCT
ejpam-5238	22	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-5238	22	14	(	(	PUNCT
ejpam-5238	22	15	s.	s.	PROPN
ejpam-5238	22	16	sompong	sompong	PROPN
ejpam-5238	22	17	)	)	PUNCT
ejpam-5238	22	18	,	,	PUNCT
ejpam-5238	22	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5238	22	20	(	(	PUNCT
ejpam-5238	22	21	c.	c.	PROPN
ejpam-5238	22	22	boonpok	boonpok	PROPN
ejpam-5238	22	23	)	)	PUNCT
ejpam-5238	22	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5238	22	25	1705	1705	NUM
ejpam-5238	23	1	©	©	PROPN
ejpam-5238	23	2	2024	2024	NUM
ejpam-5238	23	3	ejpam	ejpam	NOUN
ejpam-5238	23	4	all	all	DET
ejpam-5238	23	5	rights	right	NOUN
ejpam-5238	23	6	reserved	reserve	VERB
ejpam-5238	23	7	.	.	PUNCT
ejpam-5238	24	1	m.	m.	NOUN
ejpam-5238	24	2	thongmoon	thongmoon	PROPN
ejpam-5238	24	3	,	,	PUNCT
ejpam-5238	24	4	s.	s.	PROPN
ejpam-5238	24	5	sompong	sompong	PROPN
ejpam-5238	24	6	,	,	PUNCT
ejpam-5238	24	7	c.	c.	PROPN
ejpam-5238	24	8	boonpok	boonpok	PROPN
ejpam-5238	24	9	/	/	SYM
ejpam-5238	24	10	eur	eur	PROPN
ejpam-5238	24	11	.	.	PUNCT
ejpam-5238	25	1	j.	j.	PROPN
ejpam-5238	25	2	pure	pure	PROPN
ejpam-5238	25	3	appl	appl	PROPN
ejpam-5238	25	4	.	.	PROPN
ejpam-5238	25	5	math	math	PROPN
ejpam-5238	25	6	,	,	PUNCT
ejpam-5238	25	7	17	17	NUM
ejpam-5238	25	8	(	(	PUNCT
ejpam-5238	25	9	3	3	NUM
ejpam-5238	25	10	)	)	PUNCT
ejpam-5238	25	11	(	(	PUNCT
ejpam-5238	25	12	2024	2024	NUM
ejpam-5238	25	13	)	)	PUNCT
ejpam-5238	25	14	,	,	PUNCT
ejpam-5238	25	15	1705	1705	NUM
ejpam-5238	25	16	-	-	SYM
ejpam-5238	25	17	1716	1716	NUM
ejpam-5238	25	18	1706	1706	NUM
ejpam-5238	25	19	functions	function	NOUN
ejpam-5238	25	20	,	,	PUNCT
ejpam-5238	25	21	(	(	PUNCT
ejpam-5238	25	22	τ1	τ1	NOUN
ejpam-5238	25	23	,	,	PUNCT
ejpam-5238	25	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	25	25	functions	function	NOUN
ejpam-5238	25	26	,	,	PUNCT
ejpam-5238	25	27	almost	almost	ADV
ejpam-5238	25	28	(	(	PUNCT
ejpam-5238	25	29	τ1	τ1	NOUN
ejpam-5238	25	30	,	,	PUNCT
ejpam-5238	25	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	25	32	functions	function	NOUN
ejpam-5238	25	33	and	and	CCONJ
ejpam-5238	25	34	weakly	weakly	ADJ
ejpam-5238	25	35	(	(	PUNCT
ejpam-5238	25	36	τ1	τ1	NOUN
ejpam-5238	25	37	,	,	PUNCT
ejpam-5238	25	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	25	39	functions	function	NOUN
ejpam-5238	25	40	were	be	AUX
ejpam-5238	25	41	presented	present	VERB
ejpam-5238	25	42	in	in	ADP
ejpam-5238	25	43	[	[	X
ejpam-5238	25	44	36	36	NUM
ejpam-5238	25	45	]	]	PUNCT
ejpam-5238	25	46	,	,	PUNCT
ejpam-5238	25	47	[	[	X
ejpam-5238	25	48	38	38	NUM
ejpam-5238	25	49	]	]	PUNCT
ejpam-5238	25	50	,	,	PUNCT
ejpam-5238	25	51	[	[	X
ejpam-5238	25	52	10	10	NUM
ejpam-5238	25	53	]	]	PUNCT
ejpam-5238	25	54	,	,	PUNCT
ejpam-5238	25	55	[	[	X
ejpam-5238	25	56	33	33	NUM
ejpam-5238	25	57	]	]	PUNCT
ejpam-5238	25	58	,	,	PUNCT
ejpam-5238	25	59	[	[	X
ejpam-5238	25	60	16	16	NUM
ejpam-5238	25	61	]	]	PUNCT
ejpam-5238	25	62	,	,	PUNCT
ejpam-5238	25	63	[	[	X
ejpam-5238	25	64	9	9	NUM
ejpam-5238	25	65	]	]	PUNCT
ejpam-5238	25	66	,	,	PUNCT
ejpam-5238	25	67	[	[	X
ejpam-5238	25	68	8	8	NUM
ejpam-5238	25	69	]	]	PUNCT
ejpam-5238	25	70	,	,	PUNCT
ejpam-5238	25	71	[	[	X
ejpam-5238	25	72	5	5	NUM
ejpam-5238	25	73	]	]	PUNCT
ejpam-5238	25	74	,	,	PUNCT
ejpam-5238	25	75	[	[	X
ejpam-5238	25	76	2	2	NUM
ejpam-5238	25	77	]	]	PUNCT
ejpam-5238	25	78	,	,	PUNCT
ejpam-5238	25	79	[	[	X
ejpam-5238	25	80	40	40	NUM
ejpam-5238	25	81	]	]	PUNCT
ejpam-5238	25	82	,	,	PUNCT
ejpam-5238	25	83	[	[	X
ejpam-5238	25	84	37	37	NUM
ejpam-5238	25	85	]	]	PUNCT
ejpam-5238	25	86	,	,	PUNCT
ejpam-5238	25	87	[	[	X
ejpam-5238	25	88	7	7	NUM
ejpam-5238	25	89	]	]	PUNCT
ejpam-5238	25	90	,	,	PUNCT
ejpam-5238	25	91	[	[	X
ejpam-5238	25	92	3	3	NUM
ejpam-5238	25	93	]	]	PUNCT
ejpam-5238	25	94	,	,	PUNCT
ejpam-5238	25	95	[	[	X
ejpam-5238	25	96	17	17	NUM
ejpam-5238	25	97	]	]	PUNCT
ejpam-5238	25	98	,	,	PUNCT
ejpam-5238	25	99	[	[	X
ejpam-5238	25	100	15	15	NUM
ejpam-5238	25	101	]	]	PUNCT
ejpam-5238	25	102	and	and	CCONJ
ejpam-5238	25	103	[	[	X
ejpam-5238	25	104	11	11	NUM
ejpam-5238	25	105	]	]	PUNCT
ejpam-5238	25	106	,	,	PUNCT
ejpam-5238	25	107	respectively	respectively	ADV
ejpam-5238	25	108	.	.	PUNCT
ejpam-5238	26	1	popa	popa	NOUN
ejpam-5238	27	1	[	[	X
ejpam-5238	27	2	29	29	NUM
ejpam-5238	27	3	]	]	PUNCT
ejpam-5238	27	4	and	and	CCONJ
ejpam-5238	27	5	smithson	smithson	PROPN
ejpam-5238	27	6	[	[	X
ejpam-5238	27	7	35	35	NUM
ejpam-5238	27	8	]	]	PUNCT
ejpam-5238	27	9	independently	independently	ADV
ejpam-5238	27	10	introduced	introduce	VERB
ejpam-5238	27	11	the	the	DET
ejpam-5238	27	12	notion	notion	NOUN
ejpam-5238	27	13	of	of	ADP
ejpam-5238	27	14	weakly	weakly	ADJ
ejpam-5238	27	15	continuous	continuous	ADJ
ejpam-5238	27	16	multifunctions	multifunction	NOUN
ejpam-5238	27	17	.	.	PUNCT
ejpam-5238	28	1	popa	popa	NOUN
ejpam-5238	28	2	and	and	CCONJ
ejpam-5238	28	3	noiri	noiri	ADV
ejpam-5238	29	1	[	[	X
ejpam-5238	29	2	31	31	NUM
ejpam-5238	29	3	]	]	PUNCT
ejpam-5238	29	4	introduced	introduce	VERB
ejpam-5238	29	5	a	a	DET
ejpam-5238	29	6	class	class	NOUN
ejpam-5238	29	7	of	of	ADP
ejpam-5238	29	8	multifunctions	multifunction	NOUN
ejpam-5238	29	9	called	call	VERB
ejpam-5238	29	10	weakly	weakly	ADJ
ejpam-5238	29	11	α	α	ADJ
ejpam-5238	29	12	-	-	ADJ
ejpam-5238	29	13	continuous	continuous	ADJ
ejpam-5238	29	14	multifunctions	multifunction	NOUN
ejpam-5238	29	15	.	.	PUNCT
ejpam-5238	30	1	furthermore	furthermore	ADV
ejpam-5238	30	2	,	,	PUNCT
ejpam-5238	30	3	popa	popa	NOUN
ejpam-5238	30	4	and	and	CCONJ
ejpam-5238	30	5	noiri	noiri	ADV
ejpam-5238	31	1	[	[	X
ejpam-5238	31	2	30	30	NUM
ejpam-5238	31	3	]	]	PUNCT
ejpam-5238	31	4	investigated	investigate	VERB
ejpam-5238	31	5	some	some	DET
ejpam-5238	31	6	characterizations	characterization	NOUN
ejpam-5238	31	7	of	of	ADP
ejpam-5238	31	8	upper	upper	ADJ
ejpam-5238	31	9	and	and	CCONJ
ejpam-5238	31	10	lower	low	ADJ
ejpam-5238	31	11	weakly	weakly	ADJ
ejpam-5238	31	12	β	β	ADJ
ejpam-5238	31	13	-	-	ADJ
ejpam-5238	31	14	continuous	continuous	ADJ
ejpam-5238	31	15	multifunctions	multifunction	NOUN
ejpam-5238	31	16	.	.	PUNCT
ejpam-5238	32	1	noiri	noiri	PROPN
ejpam-5238	32	2	and	and	CCONJ
ejpam-5238	32	3	popa	popa	NOUN
ejpam-5238	33	1	[	[	X
ejpam-5238	33	2	28	28	NUM
ejpam-5238	33	3	]	]	PUNCT
ejpam-5238	33	4	introduced	introduce	VERB
ejpam-5238	33	5	and	and	CCONJ
ejpam-5238	33	6	investigated	investigate	VERB
ejpam-5238	33	7	the	the	DET
ejpam-5238	33	8	notion	notion	NOUN
ejpam-5238	33	9	of	of	ADP
ejpam-5238	33	10	weakly	weakly	ADJ
ejpam-5238	33	11	m	m	ADJ
ejpam-5238	33	12	-	-	ADJ
ejpam-5238	33	13	continuous	continuous	ADJ
ejpam-5238	33	14	multifunctions	multifunction	NOUN
ejpam-5238	33	15	as	as	ADP
ejpam-5238	33	16	a	a	DET
ejpam-5238	33	17	multifunction	multifunction	NOUN
ejpam-5238	33	18	from	from	ADP
ejpam-5238	33	19	a	a	DET
ejpam-5238	33	20	set	set	NOUN
ejpam-5238	33	21	satisfying	satisfy	VERB
ejpam-5238	33	22	certain	certain	ADJ
ejpam-5238	33	23	minimal	minimal	ADJ
ejpam-5238	33	24	condition	condition	NOUN
ejpam-5238	33	25	into	into	ADP
ejpam-5238	33	26	a	a	DET
ejpam-5238	33	27	topological	topological	ADJ
ejpam-5238	33	28	space	space	NOUN
ejpam-5238	33	29	.	.	PUNCT
ejpam-5238	34	1	boonpok	boonpok	PROPN
ejpam-5238	34	2	and	and	CCONJ
ejpam-5238	34	3	viriyapong	viriyapong	VERB
ejpam-5238	35	1	[	[	X
ejpam-5238	35	2	19	19	NUM
ejpam-5238	35	3	]	]	PUNCT
ejpam-5238	35	4	introduced	introduce	VERB
ejpam-5238	35	5	and	and	CCONJ
ejpam-5238	35	6	studied	study	VERB
ejpam-5238	35	7	the	the	DET
ejpam-5238	35	8	concepts	concept	NOUN
ejpam-5238	35	9	upper	upper	ADJ
ejpam-5238	35	10	and	and	CCONJ
ejpam-5238	35	11	lower	low	ADJ
ejpam-5238	35	12	almost	almost	ADV
ejpam-5238	35	13	weakly	weakly	ADJ
ejpam-5238	35	14	(	(	PUNCT
ejpam-5238	35	15	τ1	τ1	NOUN
ejpam-5238	35	16	,	,	PUNCT
ejpam-5238	35	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	35	18	multifunctions	multifunction	NOUN
ejpam-5238	35	19	.	.	PUNCT
ejpam-5238	36	1	laprom	laprom	ADP
ejpam-5238	36	2	et	et	PROPN
ejpam-5238	36	3	al	al	PROPN
ejpam-5238	36	4	.	.	PUNCT
ejpam-5238	37	1	[	[	X
ejpam-5238	37	2	25	25	NUM
ejpam-5238	37	3	]	]	PUNCT
ejpam-5238	37	4	introduced	introduce	VERB
ejpam-5238	37	5	and	and	CCONJ
ejpam-5238	37	6	investigated	investigate	VERB
ejpam-5238	37	7	the	the	DET
ejpam-5238	37	8	notions	notion	NOUN
ejpam-5238	37	9	of	of	ADP
ejpam-5238	37	10	upper	upper	ADJ
ejpam-5238	37	11	and	and	CCONJ
ejpam-5238	37	12	lower	low	ADJ
ejpam-5238	37	13	almost	almost	ADV
ejpam-5238	37	14	β(τ1	β(τ1	NOUN
ejpam-5238	37	15	,	,	PUNCT
ejpam-5238	37	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	37	17	multifunctions	multifunction	NOUN
ejpam-5238	37	18	.	.	PUNCT
ejpam-5238	38	1	viriyapong	viriyapong	PROPN
ejpam-5238	38	2	and	and	CCONJ
ejpam-5238	38	3	boonpok	boonpok	VERB
ejpam-5238	39	1	[	[	X
ejpam-5238	39	2	39	39	NUM
ejpam-5238	39	3	]	]	PUNCT
ejpam-5238	39	4	introduced	introduce	VERB
ejpam-5238	39	5	and	and	CCONJ
ejpam-5238	39	6	studied	study	VERB
ejpam-5238	39	7	the	the	DET
ejpam-5238	39	8	concepts	concept	NOUN
ejpam-5238	39	9	of	of	ADP
ejpam-5238	39	10	upper	upper	ADJ
ejpam-5238	39	11	and	and	CCONJ
ejpam-5238	39	12	lower	low	ADJ
ejpam-5238	39	13	weakly	weakly	ADJ
ejpam-5238	39	14	(	(	PUNCT
ejpam-5238	39	15	τ1	τ1	NOUN
ejpam-5238	39	16	,	,	PUNCT
ejpam-5238	39	17	τ2)α	τ2)α	ADJ
ejpam-5238	39	18	-	-	PUNCT
ejpam-5238	39	19	continuous	continuous	ADJ
ejpam-5238	39	20	multifunctions	multifunction	NOUN
ejpam-5238	39	21	.	.	PUNCT
ejpam-5238	40	1	moreover	moreover	ADV
ejpam-5238	40	2	,	,	PUNCT
ejpam-5238	40	3	several	several	ADJ
ejpam-5238	40	4	characterizations	characterization	NOUN
ejpam-5238	40	5	of	of	ADP
ejpam-5238	40	6	weakly	weakly	ADJ
ejpam-5238	40	7	(	(	PUNCT
ejpam-5238	40	8	τ1	τ1	NOUN
ejpam-5238	40	9	,	,	PUNCT
ejpam-5238	40	10	τ2)δ	τ2)δ	ADJ
ejpam-5238	40	11	-	-	PUNCT
ejpam-5238	40	12	semicontinuous	semicontinuous	ADJ
ejpam-5238	40	13	multifunctions	multifunction	NOUN
ejpam-5238	40	14	,	,	PUNCT
ejpam-5238	40	15	almost	almost	ADV
ejpam-5238	40	16	weakly	weakly	ADJ
ejpam-5238	40	17	⋆-continuous	⋆-continuous	ADJ
ejpam-5238	40	18	multifunctions	multifunction	NOUN
ejpam-5238	40	19	,	,	PUNCT
ejpam-5238	40	20	weakly	weakly	ADJ
ejpam-5238	40	21	⋆-continuous	⋆-continuous	ADJ
ejpam-5238	40	22	multifunctions	multifunction	NOUN
ejpam-5238	40	23	,	,	PUNCT
ejpam-5238	40	24	weakly	weakly	ADJ
ejpam-5238	40	25	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5238	40	26	multifunctions	multifunction	NOUN
ejpam-5238	40	27	,	,	PUNCT
ejpam-5238	40	28	weakly	weakly	ADJ
ejpam-5238	40	29	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-5238	40	30	multifunctions	multifunction	NOUN
ejpam-5238	40	31	,	,	PUNCT
ejpam-5238	40	32	weakly	weakly	ADJ
ejpam-5238	40	33	quasi	quasi	NOUN
ejpam-5238	40	34	(	(	PUNCT
ejpam-5238	40	35	λ	λ	PROPN
ejpam-5238	40	36	,	,	PUNCT
ejpam-5238	40	37	sp)-continuous	sp)-continuous	ADJ
ejpam-5238	40	38	multifunctions	multifunction	NOUN
ejpam-5238	40	39	and	and	CCONJ
ejpam-5238	40	40	weakly	weakly	ADJ
ejpam-5238	40	41	(	(	PUNCT
ejpam-5238	40	42	λ	λ	NOUN
ejpam-5238	40	43	,	,	PUNCT
ejpam-5238	40	44	sp)-continuous	sp)-continuous	ADJ
ejpam-5238	40	45	multifunctions	multifunction	NOUN
ejpam-5238	40	46	were	be	AUX
ejpam-5238	40	47	established	establish	VERB
ejpam-5238	40	48	in	in	ADP
ejpam-5238	40	49	[	[	X
ejpam-5238	40	50	6	6	NUM
ejpam-5238	40	51	]	]	PUNCT
ejpam-5238	40	52	,	,	PUNCT
ejpam-5238	40	53	[	[	X
ejpam-5238	40	54	18	18	NUM
ejpam-5238	40	55	]	]	PUNCT
ejpam-5238	40	56	,	,	PUNCT
ejpam-5238	40	57	[	[	X
ejpam-5238	40	58	4	4	NUM
ejpam-5238	40	59	]	]	PUNCT
ejpam-5238	40	60	,	,	PUNCT
ejpam-5238	40	61	[	[	X
ejpam-5238	40	62	13	13	NUM
ejpam-5238	40	63	]	]	PUNCT
ejpam-5238	40	64	,	,	PUNCT
ejpam-5238	40	65	[	[	X
ejpam-5238	40	66	12	12	NUM
ejpam-5238	40	67	]	]	PUNCT
ejpam-5238	40	68	,	,	PUNCT
ejpam-5238	40	69	[	[	X
ejpam-5238	40	70	41	41	NUM
ejpam-5238	40	71	]	]	PUNCT
ejpam-5238	40	72	and	and	CCONJ
ejpam-5238	40	73	[	[	X
ejpam-5238	40	74	14	14	NUM
ejpam-5238	40	75	]	]	X
ejpam-5238	40	76	,	,	PUNCT
ejpam-5238	40	77	respectively	respectively	ADV
ejpam-5238	40	78	.	.	PUNCT
ejpam-5238	41	1	in	in	ADP
ejpam-5238	41	2	this	this	DET
ejpam-5238	41	3	paper	paper	NOUN
ejpam-5238	41	4	,	,	PUNCT
ejpam-5238	41	5	we	we	PRON
ejpam-5238	41	6	introduce	introduce	VERB
ejpam-5238	41	7	the	the	DET
ejpam-5238	41	8	concepts	concept	NOUN
ejpam-5238	41	9	of	of	ADP
ejpam-5238	41	10	upper	upper	ADJ
ejpam-5238	41	11	and	and	CCONJ
ejpam-5238	41	12	lower	low	ADJ
ejpam-5238	41	13	weakly	weakly	ADJ
ejpam-5238	41	14	(	(	PUNCT
ejpam-5238	41	15	τ1	τ1	NOUN
ejpam-5238	41	16	,	,	PUNCT
ejpam-5238	41	17	τ2)continuous	τ2)continuous	ADJ
ejpam-5238	41	18	multifunctions	multifunction	NOUN
ejpam-5238	41	19	.	.	PUNCT
ejpam-5238	42	1	in	in	ADP
ejpam-5238	42	2	particular	particular	ADJ
ejpam-5238	42	3	,	,	PUNCT
ejpam-5238	42	4	some	some	DET
ejpam-5238	42	5	characterizations	characterization	NOUN
ejpam-5238	42	6	of	of	ADP
ejpam-5238	42	7	upper	upper	ADJ
ejpam-5238	42	8	and	and	CCONJ
ejpam-5238	42	9	lower	low	ADJ
ejpam-5238	42	10	weakly	weakly	ADJ
ejpam-5238	42	11	(	(	PUNCT
ejpam-5238	42	12	τ1	τ1	NOUN
ejpam-5238	42	13	,	,	PUNCT
ejpam-5238	42	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	42	15	multifunctions	multifunction	NOUN
ejpam-5238	42	16	are	be	AUX
ejpam-5238	42	17	discussed	discuss	VERB
ejpam-5238	42	18	.	.	PUNCT
ejpam-5238	43	1	2	2	X
ejpam-5238	43	2	.	.	X
ejpam-5238	43	3	preliminaries	preliminary	NOUN
ejpam-5238	43	4	throughout	throughout	ADP
ejpam-5238	43	5	the	the	DET
ejpam-5238	43	6	present	present	ADJ
ejpam-5238	43	7	paper	paper	NOUN
ejpam-5238	43	8	,	,	PUNCT
ejpam-5238	43	9	spaces	space	NOUN
ejpam-5238	43	10	(	(	PUNCT
ejpam-5238	43	11	x	x	NOUN
ejpam-5238	43	12	,	,	PUNCT
ejpam-5238	43	13	τ1	τ1	NOUN
ejpam-5238	43	14	,	,	PUNCT
ejpam-5238	43	15	τ2	τ2	NOUN
ejpam-5238	43	16	)	)	PUNCT
ejpam-5238	43	17	and	and	CCONJ
ejpam-5238	43	18	(	(	PUNCT
ejpam-5238	43	19	y	y	PROPN
ejpam-5238	43	20	,	,	PUNCT
ejpam-5238	43	21	σ1	σ1	PROPN
ejpam-5238	43	22	,	,	PUNCT
ejpam-5238	43	23	σ2	σ2	NOUN
ejpam-5238	43	24	)	)	PUNCT
ejpam-5238	43	25	(	(	PUNCT
ejpam-5238	43	26	or	or	CCONJ
ejpam-5238	43	27	simply	simply	ADV
ejpam-5238	43	28	x	x	X
ejpam-5238	43	29	and	and	CCONJ
ejpam-5238	43	30	y	y	PROPN
ejpam-5238	43	31	)	)	PUNCT
ejpam-5238	43	32	always	always	ADV
ejpam-5238	43	33	mean	mean	VERB
ejpam-5238	43	34	bitopological	bitopological	ADJ
ejpam-5238	43	35	spaces	space	NOUN
ejpam-5238	43	36	on	on	ADP
ejpam-5238	43	37	which	which	PRON
ejpam-5238	43	38	no	no	DET
ejpam-5238	43	39	separation	separation	NOUN
ejpam-5238	43	40	axioms	axiom	NOUN
ejpam-5238	43	41	are	be	AUX
ejpam-5238	43	42	assumed	assume	VERB
ejpam-5238	43	43	unless	unless	SCONJ
ejpam-5238	43	44	explicitly	explicitly	ADV
ejpam-5238	43	45	stated	state	VERB
ejpam-5238	43	46	.	.	PUNCT
ejpam-5238	44	1	let	let	VERB
ejpam-5238	44	2	a	a	DET
ejpam-5238	44	3	be	be	AUX
ejpam-5238	44	4	a	a	DET
ejpam-5238	44	5	subset	subset	NOUN
ejpam-5238	44	6	of	of	ADP
ejpam-5238	44	7	a	a	DET
ejpam-5238	44	8	bitopological	bitopological	ADJ
ejpam-5238	44	9	space	space	NOUN
ejpam-5238	44	10	(	(	PUNCT
ejpam-5238	44	11	x	x	NOUN
ejpam-5238	44	12	,	,	PUNCT
ejpam-5238	44	13	τ1	τ1	NOUN
ejpam-5238	44	14	,	,	PUNCT
ejpam-5238	44	15	τ2	τ2	NOUN
ejpam-5238	44	16	)	)	PUNCT
ejpam-5238	44	17	.	.	PUNCT
ejpam-5238	45	1	the	the	DET
ejpam-5238	45	2	closure	closure	NOUN
ejpam-5238	45	3	of	of	ADP
ejpam-5238	45	4	a	a	PRON
ejpam-5238	45	5	and	and	CCONJ
ejpam-5238	45	6	the	the	DET
ejpam-5238	45	7	interior	interior	NOUN
ejpam-5238	45	8	of	of	ADP
ejpam-5238	45	9	a	a	PRON
ejpam-5238	45	10	with	with	ADP
ejpam-5238	45	11	respect	respect	NOUN
ejpam-5238	45	12	to	to	ADP
ejpam-5238	45	13	τi	τi	PROPN
ejpam-5238	45	14	are	be	AUX
ejpam-5238	45	15	denoted	denote	VERB
ejpam-5238	45	16	by	by	ADP
ejpam-5238	45	17	τi	τi	NOUN
ejpam-5238	45	18	-	-	PUNCT
ejpam-5238	45	19	cl(a	cl(a	NUM
ejpam-5238	45	20	)	)	PUNCT
ejpam-5238	45	21	and	and	CCONJ
ejpam-5238	45	22	τi	τi	NOUN
ejpam-5238	45	23	-	-	PUNCT
ejpam-5238	45	24	int(a	int(a	NOUN
ejpam-5238	45	25	)	)	PUNCT
ejpam-5238	45	26	,	,	PUNCT
ejpam-5238	45	27	respectively	respectively	ADV
ejpam-5238	45	28	,	,	PUNCT
ejpam-5238	45	29	for	for	ADP
ejpam-5238	45	30	i	i	PROPN
ejpam-5238	45	31	=	=	SYM
ejpam-5238	45	32	1	1	NUM
ejpam-5238	45	33	,	,	PUNCT
ejpam-5238	45	34	2	2	NUM
ejpam-5238	45	35	.	.	X
ejpam-5238	45	36	a	a	DET
ejpam-5238	45	37	subset	subset	NOUN
ejpam-5238	45	38	a	a	PRON
ejpam-5238	45	39	of	of	ADP
ejpam-5238	45	40	a	a	DET
ejpam-5238	45	41	bitopological	bitopological	ADJ
ejpam-5238	45	42	space	space	NOUN
ejpam-5238	45	43	(	(	PUNCT
ejpam-5238	45	44	x	x	NOUN
ejpam-5238	45	45	,	,	PUNCT
ejpam-5238	45	46	τ1	τ1	NOUN
ejpam-5238	45	47	,	,	PUNCT
ejpam-5238	45	48	τ2	τ2	NOUN
ejpam-5238	45	49	)	)	PUNCT
ejpam-5238	45	50	is	be	AUX
ejpam-5238	45	51	called	call	VERB
ejpam-5238	45	52	τ1τ2	τ1τ2	VERB
ejpam-5238	45	53	-	-	ADJ
ejpam-5238	45	54	closed	closed	ADJ
ejpam-5238	45	55	[	[	X
ejpam-5238	45	56	20	20	NUM
ejpam-5238	45	57	]	]	PUNCT
ejpam-5238	45	58	if	if	SCONJ
ejpam-5238	45	59	a	a	DET
ejpam-5238	45	60	=	=	NOUN
ejpam-5238	45	61	τ1	τ1	NOUN
ejpam-5238	45	62	-	-	PUNCT
ejpam-5238	45	63	cl(τ2	cl(τ2	NOUN
ejpam-5238	45	64	-	-	PUNCT
ejpam-5238	45	65	cl(a	cl(a	NUM
ejpam-5238	45	66	)	)	PUNCT
ejpam-5238	45	67	)	)	PUNCT
ejpam-5238	45	68	.	.	PUNCT
ejpam-5238	46	1	the	the	DET
ejpam-5238	46	2	complement	complement	NOUN
ejpam-5238	46	3	of	of	ADP
ejpam-5238	46	4	a	a	DET
ejpam-5238	46	5	τ1τ2	τ1τ2	ADJ
ejpam-5238	46	6	-	-	ADJ
ejpam-5238	46	7	closed	closed	ADJ
ejpam-5238	46	8	set	set	NOUN
ejpam-5238	46	9	is	be	AUX
ejpam-5238	46	10	called	call	VERB
ejpam-5238	46	11	τ1τ2	τ1τ2	NOUN
ejpam-5238	46	12	-	-	ADJ
ejpam-5238	46	13	open	open	ADJ
ejpam-5238	46	14	.	.	PUNCT
ejpam-5238	47	1	let	let	VERB
ejpam-5238	47	2	a	a	DET
ejpam-5238	47	3	be	be	AUX
ejpam-5238	47	4	a	a	DET
ejpam-5238	47	5	subset	subset	NOUN
ejpam-5238	47	6	of	of	ADP
ejpam-5238	47	7	a	a	DET
ejpam-5238	47	8	bitopological	bitopological	ADJ
ejpam-5238	47	9	space	space	NOUN
ejpam-5238	47	10	(	(	PUNCT
ejpam-5238	47	11	x	x	NOUN
ejpam-5238	47	12	,	,	PUNCT
ejpam-5238	47	13	τ1	τ1	NOUN
ejpam-5238	47	14	,	,	PUNCT
ejpam-5238	47	15	τ2	τ2	NOUN
ejpam-5238	47	16	)	)	PUNCT
ejpam-5238	47	17	.	.	PUNCT
ejpam-5238	48	1	the	the	DET
ejpam-5238	48	2	intersection	intersection	NOUN
ejpam-5238	48	3	of	of	ADP
ejpam-5238	48	4	all	all	DET
ejpam-5238	48	5	τ1τ2	τ1τ2	ADJ
ejpam-5238	48	6	-	-	ADJ
ejpam-5238	48	7	closed	closed	ADJ
ejpam-5238	48	8	sets	set	NOUN
ejpam-5238	48	9	of	of	ADP
ejpam-5238	48	10	x	x	PUNCT
ejpam-5238	48	11	containing	contain	VERB
ejpam-5238	48	12	a	a	PRON
ejpam-5238	48	13	is	be	AUX
ejpam-5238	48	14	called	call	VERB
ejpam-5238	48	15	the	the	DET
ejpam-5238	48	16	τ1τ2	τ1τ2	NOUN
ejpam-5238	48	17	-	-	NOUN
ejpam-5238	48	18	closure	closure	NOUN
ejpam-5238	48	19	[	[	X
ejpam-5238	48	20	20	20	NUM
ejpam-5238	48	21	]	]	PUNCT
ejpam-5238	48	22	of	of	ADP
ejpam-5238	48	23	a	a	PRON
ejpam-5238	48	24	and	and	CCONJ
ejpam-5238	48	25	is	be	AUX
ejpam-5238	48	26	denoted	denote	VERB
ejpam-5238	48	27	by	by	ADP
ejpam-5238	48	28	τ1τ2	τ1τ2	NOUN
ejpam-5238	48	29	-	-	NUM
ejpam-5238	48	30	cl(a	cl(a	NUM
ejpam-5238	48	31	)	)	PUNCT
ejpam-5238	48	32	.	.	PUNCT
ejpam-5238	49	1	the	the	DET
ejpam-5238	49	2	union	union	NOUN
ejpam-5238	49	3	of	of	ADP
ejpam-5238	49	4	all	all	DET
ejpam-5238	49	5	τ1τ2	τ1τ2	ADJ
ejpam-5238	49	6	-	-	ADJ
ejpam-5238	49	7	open	open	ADJ
ejpam-5238	49	8	sets	set	NOUN
ejpam-5238	49	9	of	of	ADP
ejpam-5238	49	10	x	x	PUNCT
ejpam-5238	49	11	contained	contain	VERB
ejpam-5238	49	12	in	in	ADP
ejpam-5238	49	13	a	a	PRON
ejpam-5238	49	14	is	be	AUX
ejpam-5238	49	15	called	call	VERB
ejpam-5238	49	16	the	the	DET
ejpam-5238	49	17	τ1τ2	τ1τ2	NOUN
ejpam-5238	49	18	-	-	ADJ
ejpam-5238	49	19	interior	interior	ADJ
ejpam-5238	49	20	[	[	X
ejpam-5238	49	21	20	20	NUM
ejpam-5238	49	22	]	]	PUNCT
ejpam-5238	49	23	of	of	ADP
ejpam-5238	49	24	a	a	PRON
ejpam-5238	49	25	and	and	CCONJ
ejpam-5238	49	26	is	be	AUX
ejpam-5238	49	27	denoted	denote	VERB
ejpam-5238	49	28	by	by	ADP
ejpam-5238	49	29	τ1τ2	τ1τ2	NOUN
ejpam-5238	49	30	-	-	ADJ
ejpam-5238	49	31	int(a	int(a	NOUN
ejpam-5238	49	32	)	)	PUNCT
ejpam-5238	49	33	.	.	PUNCT
ejpam-5238	50	1	lemma	lemma	PROPN
ejpam-5238	50	2	1	1	NUM
ejpam-5238	50	3	.	.	PUNCT
ejpam-5238	51	1	[	[	X
ejpam-5238	51	2	20	20	NUM
ejpam-5238	51	3	]	]	PUNCT
ejpam-5238	51	4	let	let	VERB
ejpam-5238	51	5	a	a	PRON
ejpam-5238	51	6	and	and	CCONJ
ejpam-5238	51	7	b	b	NOUN
ejpam-5238	51	8	be	be	AUX
ejpam-5238	51	9	subsets	subset	NOUN
ejpam-5238	51	10	of	of	ADP
ejpam-5238	51	11	a	a	DET
ejpam-5238	51	12	bitopological	bitopological	ADJ
ejpam-5238	51	13	space	space	NOUN
ejpam-5238	51	14	(	(	PUNCT
ejpam-5238	51	15	x	x	NOUN
ejpam-5238	51	16	,	,	PUNCT
ejpam-5238	51	17	τ1	τ1	NOUN
ejpam-5238	51	18	,	,	PUNCT
ejpam-5238	51	19	τ2	τ2	NOUN
ejpam-5238	51	20	)	)	PUNCT
ejpam-5238	51	21	.	.	PUNCT
ejpam-5238	52	1	for	for	ADP
ejpam-5238	52	2	the	the	DET
ejpam-5238	52	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-5238	52	4	,	,	PUNCT
ejpam-5238	52	5	the	the	DET
ejpam-5238	52	6	following	follow	VERB
ejpam-5238	52	7	properties	property	NOUN
ejpam-5238	52	8	hold	hold	VERB
ejpam-5238	52	9	:	:	PUNCT
ejpam-5238	52	10	(	(	PUNCT
ejpam-5238	52	11	1	1	X
ejpam-5238	52	12	)	)	PUNCT
ejpam-5238	52	13	a	a	DET
ejpam-5238	52	14	⊆	⊆	NUM
ejpam-5238	52	15	τ1τ2	τ1τ2	NOUN
ejpam-5238	52	16	-	-	NUM
ejpam-5238	52	17	cl(a	cl(a	NUM
ejpam-5238	52	18	)	)	PUNCT
ejpam-5238	52	19	and	and	CCONJ
ejpam-5238	52	20	τ1τ2	τ1τ2	NOUN
ejpam-5238	52	21	-	-	ADJ
ejpam-5238	52	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5238	52	23	-	-	PUNCT
ejpam-5238	52	24	cl(a	cl(a	NUM
ejpam-5238	52	25	)	)	PUNCT
ejpam-5238	52	26	)	)	PUNCT
ejpam-5238	53	1	=	=	PUNCT
ejpam-5238	53	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	53	3	-	-	NUM
ejpam-5238	53	4	cl(a	cl(a	NUM
ejpam-5238	53	5	)	)	PUNCT
ejpam-5238	53	6	.	.	PUNCT
ejpam-5238	54	1	(	(	PUNCT
ejpam-5238	54	2	2	2	X
ejpam-5238	54	3	)	)	PUNCT
ejpam-5238	54	4	if	if	SCONJ
ejpam-5238	54	5	a	a	DET
ejpam-5238	54	6	⊆	⊆	NUM
ejpam-5238	54	7	b	b	NOUN
ejpam-5238	54	8	,	,	PUNCT
ejpam-5238	54	9	then	then	ADV
ejpam-5238	54	10	τ1τ2	τ1τ2	NOUN
ejpam-5238	54	11	-	-	NUM
ejpam-5238	54	12	cl(a	cl(a	NUM
ejpam-5238	54	13	)	)	PUNCT
ejpam-5238	54	14	⊆	⊆	NUM
ejpam-5238	54	15	τ1τ2	τ1τ2	NOUN
ejpam-5238	54	16	-	-	NOUN
ejpam-5238	54	17	cl(b	cl(b	NOUN
ejpam-5238	54	18	)	)	PUNCT
ejpam-5238	54	19	.	.	PUNCT
ejpam-5238	55	1	(	(	PUNCT
ejpam-5238	55	2	3	3	X
ejpam-5238	55	3	)	)	PUNCT
ejpam-5238	55	4	τ1τ2	τ1τ2	NOUN
ejpam-5238	55	5	-	-	NUM
ejpam-5238	55	6	cl(a	cl(a	NUM
ejpam-5238	55	7	)	)	PUNCT
ejpam-5238	55	8	is	be	AUX
ejpam-5238	55	9	τ1τ2	τ1τ2	NOUN
ejpam-5238	55	10	-	-	ADJ
ejpam-5238	55	11	closed	closed	ADJ
ejpam-5238	55	12	.	.	PUNCT
ejpam-5238	56	1	(	(	PUNCT
ejpam-5238	56	2	4	4	X
ejpam-5238	56	3	)	)	PUNCT
ejpam-5238	56	4	a	a	PRON
ejpam-5238	56	5	is	be	AUX
ejpam-5238	56	6	τ1τ2	τ1τ2	NOUN
ejpam-5238	56	7	-	-	ADJ
ejpam-5238	56	8	closed	closed	ADJ
ejpam-5238	56	9	if	if	SCONJ
ejpam-5238	56	10	and	and	CCONJ
ejpam-5238	56	11	only	only	ADV
ejpam-5238	56	12	if	if	SCONJ
ejpam-5238	56	13	a	a	DET
ejpam-5238	56	14	=	=	PUNCT
ejpam-5238	56	15	τ1τ2	τ1τ2	NOUN
ejpam-5238	56	16	-	-	NUM
ejpam-5238	56	17	cl(a	cl(a	NUM
ejpam-5238	56	18	)	)	PUNCT
ejpam-5238	56	19	.	.	PUNCT
ejpam-5238	57	1	m.	m.	NOUN
ejpam-5238	57	2	thongmoon	thongmoon	PROPN
ejpam-5238	57	3	,	,	PUNCT
ejpam-5238	57	4	s.	s.	PROPN
ejpam-5238	57	5	sompong	sompong	PROPN
ejpam-5238	57	6	,	,	PUNCT
ejpam-5238	57	7	c.	c.	PROPN
ejpam-5238	57	8	boonpok	boonpok	PROPN
ejpam-5238	57	9	/	/	SYM
ejpam-5238	57	10	eur	eur	PROPN
ejpam-5238	57	11	.	.	PUNCT
ejpam-5238	58	1	j.	j.	PROPN
ejpam-5238	58	2	pure	pure	PROPN
ejpam-5238	58	3	appl	appl	PROPN
ejpam-5238	58	4	.	.	PROPN
ejpam-5238	58	5	math	math	PROPN
ejpam-5238	58	6	,	,	PUNCT
ejpam-5238	58	7	17	17	NUM
ejpam-5238	58	8	(	(	PUNCT
ejpam-5238	58	9	3	3	NUM
ejpam-5238	58	10	)	)	PUNCT
ejpam-5238	58	11	(	(	PUNCT
ejpam-5238	58	12	2024	2024	NUM
ejpam-5238	58	13	)	)	PUNCT
ejpam-5238	58	14	,	,	PUNCT
ejpam-5238	58	15	1705	1705	NUM
ejpam-5238	58	16	-	-	SYM
ejpam-5238	58	17	1716	1716	NUM
ejpam-5238	58	18	1707	1707	NUM
ejpam-5238	58	19	(	(	PUNCT
ejpam-5238	58	20	5	5	NUM
ejpam-5238	58	21	)	)	PUNCT
ejpam-5238	58	22	τ1τ2	τ1τ2	NOUN
ejpam-5238	58	23	-	-	NOUN
ejpam-5238	58	24	cl(x	cl(x	X
ejpam-5238	58	25	−a	−a	NOUN
ejpam-5238	58	26	)	)	PUNCT
ejpam-5238	58	27	=	=	PUNCT
ejpam-5238	59	1	x	x	X
ejpam-5238	59	2	−	−	ADP
ejpam-5238	59	3	τ1τ2	τ1τ2	NOUN
ejpam-5238	59	4	-	-	PUNCT
ejpam-5238	59	5	int(a	int(a	NOUN
ejpam-5238	59	6	)	)	PUNCT
ejpam-5238	59	7	.	.	PUNCT
ejpam-5238	60	1	a	a	DET
ejpam-5238	60	2	subseta	subseta	NOUN
ejpam-5238	60	3	of	of	ADP
ejpam-5238	60	4	a	a	DET
ejpam-5238	60	5	bitopological	bitopological	ADJ
ejpam-5238	60	6	space	space	NOUN
ejpam-5238	60	7	(	(	PUNCT
ejpam-5238	60	8	x	x	NOUN
ejpam-5238	60	9	,	,	PUNCT
ejpam-5238	60	10	τ1	τ1	NOUN
ejpam-5238	60	11	,	,	PUNCT
ejpam-5238	60	12	τ2	τ2	NOUN
ejpam-5238	60	13	)	)	PUNCT
ejpam-5238	60	14	is	be	AUX
ejpam-5238	60	15	called	call	VERB
ejpam-5238	60	16	(	(	PUNCT
ejpam-5238	60	17	τ1	τ1	NOUN
ejpam-5238	60	18	,	,	PUNCT
ejpam-5238	60	19	τ2)r	τ2)r	NOUN
ejpam-5238	60	20	-	-	PUNCT
ejpam-5238	60	21	open	open	NOUN
ejpam-5238	60	22	[	[	X
ejpam-5238	60	23	39	39	NUM
ejpam-5238	60	24	]	]	PUNCT
ejpam-5238	60	25	(	(	PUNCT
ejpam-5238	60	26	resp	resp	NOUN
ejpam-5238	60	27	.	.	PUNCT
ejpam-5238	61	1	(	(	PUNCT
ejpam-5238	61	2	τ1	τ1	NOUN
ejpam-5238	61	3	,	,	PUNCT
ejpam-5238	61	4	τ2)sopen	τ2)sopen	VERB
ejpam-5238	61	5	[	[	X
ejpam-5238	61	6	6	6	NUM
ejpam-5238	61	7	]	]	PUNCT
ejpam-5238	61	8	,	,	PUNCT
ejpam-5238	61	9	(	(	PUNCT
ejpam-5238	61	10	τ1	τ1	NOUN
ejpam-5238	61	11	,	,	PUNCT
ejpam-5238	61	12	τ2)p	τ2)p	NOUN
ejpam-5238	61	13	-	-	ADJ
ejpam-5238	61	14	open	open	ADJ
ejpam-5238	61	15	[	[	X
ejpam-5238	61	16	6	6	NUM
ejpam-5238	61	17	]	]	PUNCT
ejpam-5238	61	18	,	,	PUNCT
ejpam-5238	61	19	(	(	PUNCT
ejpam-5238	61	20	τ1	τ1	NOUN
ejpam-5238	61	21	,	,	PUNCT
ejpam-5238	61	22	τ2)β	τ2)β	ADJ
ejpam-5238	61	23	-	-	PUNCT
ejpam-5238	61	24	open	open	NOUN
ejpam-5238	62	1	[	[	X
ejpam-5238	62	2	6	6	NUM
ejpam-5238	62	3	]	]	PUNCT
ejpam-5238	62	4	)	)	PUNCT
ejpam-5238	62	5	if	if	SCONJ
ejpam-5238	62	6	a	a	DET
ejpam-5238	62	7	=	=	PUNCT
ejpam-5238	62	8	τ1τ2	τ1τ2	NOUN
ejpam-5238	62	9	-	-	NOUN
ejpam-5238	62	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5238	62	11	-	-	PUNCT
ejpam-5238	62	12	cl(a	cl(a	NUM
ejpam-5238	62	13	)	)	PUNCT
ejpam-5238	62	14	)	)	PUNCT
ejpam-5238	62	15	(	(	PUNCT
ejpam-5238	62	16	resp	resp	NOUN
ejpam-5238	62	17	.	.	PUNCT
ejpam-5238	63	1	a	a	DET
ejpam-5238	63	2	⊆	⊆	NUM
ejpam-5238	63	3	τ1τ2	τ1τ2	NOUN
ejpam-5238	63	4	-	-	ADJ
ejpam-5238	63	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5238	63	6	-	-	PUNCT
ejpam-5238	63	7	int(a	int(a	NOUN
ejpam-5238	63	8	)	)	PUNCT
ejpam-5238	63	9	)	)	PUNCT
ejpam-5238	63	10	,	,	PUNCT
ejpam-5238	63	11	a	a	DET
ejpam-5238	63	12	⊆	⊆	NUM
ejpam-5238	63	13	τ1τ2	τ1τ2	NOUN
ejpam-5238	63	14	-	-	NOUN
ejpam-5238	63	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5238	63	16	-	-	PUNCT
ejpam-5238	63	17	cl(a	cl(a	NUM
ejpam-5238	63	18	)	)	PUNCT
ejpam-5238	63	19	)	)	PUNCT
ejpam-5238	63	20	,	,	PUNCT
ejpam-5238	63	21	a	a	DET
ejpam-5238	63	22	⊆	⊆	NUM
ejpam-5238	63	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	63	24	-	-	PUNCT
ejpam-5238	63	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5238	63	26	-	-	PUNCT
ejpam-5238	63	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5238	63	28	-	-	PUNCT
ejpam-5238	63	29	cl(a	cl(a	NUM
ejpam-5238	63	30	)	)	PUNCT
ejpam-5238	63	31	)	)	PUNCT
ejpam-5238	63	32	)	)	PUNCT
ejpam-5238	63	33	)	)	PUNCT
ejpam-5238	63	34	.	.	PUNCT
ejpam-5238	64	1	the	the	DET
ejpam-5238	64	2	complement	complement	NOUN
ejpam-5238	64	3	of	of	ADP
ejpam-5238	64	4	a	a	DET
ejpam-5238	64	5	(	(	PUNCT
ejpam-5238	64	6	τ1	τ1	NOUN
ejpam-5238	64	7	,	,	PUNCT
ejpam-5238	64	8	τ2)r	τ2)r	NOUN
ejpam-5238	64	9	-	-	PUNCT
ejpam-5238	64	10	open	open	ADJ
ejpam-5238	64	11	(	(	PUNCT
ejpam-5238	64	12	resp	resp	NOUN
ejpam-5238	64	13	.	.	PUNCT
ejpam-5238	65	1	(	(	PUNCT
ejpam-5238	65	2	τ1	τ1	NOUN
ejpam-5238	65	3	,	,	PUNCT
ejpam-5238	65	4	τ2)s	τ2)s	NOUN
ejpam-5238	65	5	-	-	PUNCT
ejpam-5238	65	6	open	open	ADJ
ejpam-5238	65	7	,	,	PUNCT
ejpam-5238	65	8	(	(	PUNCT
ejpam-5238	65	9	τ1	τ1	NOUN
ejpam-5238	65	10	,	,	PUNCT
ejpam-5238	65	11	τ2)p	τ2)p	NOUN
ejpam-5238	65	12	-	-	ADJ
ejpam-5238	65	13	open	open	ADJ
ejpam-5238	65	14	,	,	PUNCT
ejpam-5238	65	15	(	(	PUNCT
ejpam-5238	65	16	τ1	τ1	NOUN
ejpam-5238	65	17	,	,	PUNCT
ejpam-5238	65	18	τ2)β	τ2)β	ADJ
ejpam-5238	65	19	-	-	PUNCT
ejpam-5238	65	20	open	open	ADJ
ejpam-5238	65	21	)	)	PUNCT
ejpam-5238	65	22	set	set	NOUN
ejpam-5238	65	23	is	be	AUX
ejpam-5238	65	24	called	call	VERB
ejpam-5238	65	25	(	(	PUNCT
ejpam-5238	65	26	τ1	τ1	NOUN
ejpam-5238	65	27	,	,	PUNCT
ejpam-5238	65	28	τ2)r	τ2)r	NOUN
ejpam-5238	65	29	-	-	PUNCT
ejpam-5238	65	30	closed	closed	ADJ
ejpam-5238	65	31	,	,	PUNCT
ejpam-5238	65	32	(	(	PUNCT
ejpam-5238	65	33	τ1	τ1	NOUN
ejpam-5238	65	34	,	,	PUNCT
ejpam-5238	65	35	τ2)s	τ2)s	NOUN
ejpam-5238	65	36	-	-	PUNCT
ejpam-5238	65	37	closed	closed	ADJ
ejpam-5238	65	38	,	,	PUNCT
ejpam-5238	65	39	(	(	PUNCT
ejpam-5238	65	40	τ1	τ1	NOUN
ejpam-5238	65	41	,	,	PUNCT
ejpam-5238	65	42	τ2)p	τ2)p	NOUN
ejpam-5238	65	43	-	-	PUNCT
ejpam-5238	65	44	closed	closed	ADJ
ejpam-5238	65	45	,	,	PUNCT
ejpam-5238	65	46	(	(	PUNCT
ejpam-5238	65	47	τ1	τ1	NOUN
ejpam-5238	65	48	,	,	PUNCT
ejpam-5238	65	49	τ2)β	τ2)β	NOUN
ejpam-5238	65	50	-	-	PUNCT
ejpam-5238	65	51	closed	closed	ADJ
ejpam-5238	65	52	.	.	PUNCT
ejpam-5238	66	1	a	a	DET
ejpam-5238	66	2	subset	subset	NOUN
ejpam-5238	66	3	a	a	PRON
ejpam-5238	66	4	of	of	ADP
ejpam-5238	66	5	a	a	DET
ejpam-5238	66	6	bitopological	bitopological	ADJ
ejpam-5238	66	7	space	space	NOUN
ejpam-5238	66	8	(	(	PUNCT
ejpam-5238	66	9	x	x	NOUN
ejpam-5238	66	10	,	,	PUNCT
ejpam-5238	66	11	τ1	τ1	NOUN
ejpam-5238	66	12	,	,	PUNCT
ejpam-5238	66	13	τ2	τ2	NOUN
ejpam-5238	66	14	)	)	PUNCT
ejpam-5238	66	15	is	be	AUX
ejpam-5238	66	16	called	call	VERB
ejpam-5238	66	17	α(τ1	α(τ1	NOUN
ejpam-5238	66	18	,	,	PUNCT
ejpam-5238	66	19	τ2)-open	τ2)-open	ADJ
ejpam-5238	67	1	[	[	X
ejpam-5238	67	2	42	42	NUM
ejpam-5238	67	3	]	]	PUNCT
ejpam-5238	67	4	ifa	ifa	PROPN
ejpam-5238	67	5	⊆	⊆	NUM
ejpam-5238	67	6	τ1τ2	τ1τ2	NOUN
ejpam-5238	67	7	-	-	PUNCT
ejpam-5238	67	8	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5238	67	9	-	-	PUNCT
ejpam-5238	67	10	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5238	67	11	-	-	PUNCT
ejpam-5238	67	12	int(a	int(a	NOUN
ejpam-5238	67	13	)	)	PUNCT
ejpam-5238	67	14	)	)	PUNCT
ejpam-5238	67	15	)	)	PUNCT
ejpam-5238	67	16	.	.	PUNCT
ejpam-5238	68	1	the	the	DET
ejpam-5238	68	2	complement	complement	NOUN
ejpam-5238	68	3	of	of	ADP
ejpam-5238	68	4	an	an	DET
ejpam-5238	68	5	α(τ1	α(τ1	NOUN
ejpam-5238	68	6	,	,	PUNCT
ejpam-5238	68	7	τ2)-open	τ2)-open	ADJ
ejpam-5238	68	8	set	set	NOUN
ejpam-5238	68	9	is	be	AUX
ejpam-5238	68	10	called	call	VERB
ejpam-5238	68	11	α(τ1	α(τ1	NOUN
ejpam-5238	68	12	,	,	PUNCT
ejpam-5238	68	13	τ2)-closed	τ2)-closed	PROPN
ejpam-5238	68	14	.	.	PUNCT
ejpam-5238	69	1	by	by	ADP
ejpam-5238	69	2	a	a	DET
ejpam-5238	69	3	multifunction	multifunction	NOUN
ejpam-5238	69	4	f	f	NOUN
ejpam-5238	69	5	:	:	PUNCT
ejpam-5238	69	6	x	x	X
ejpam-5238	69	7	→	→	SYM
ejpam-5238	69	8	y	y	PROPN
ejpam-5238	69	9	,	,	PUNCT
ejpam-5238	69	10	we	we	PRON
ejpam-5238	69	11	mean	mean	VERB
ejpam-5238	69	12	a	a	DET
ejpam-5238	69	13	point	point	NOUN
ejpam-5238	69	14	-	-	PUNCT
ejpam-5238	69	15	to	to	ADP
ejpam-5238	69	16	-	-	PUNCT
ejpam-5238	69	17	set	set	VERB
ejpam-5238	69	18	correspondence	correspondence	NOUN
ejpam-5238	69	19	from	from	ADP
ejpam-5238	69	20	x	x	PUNCT
ejpam-5238	69	21	into	into	ADP
ejpam-5238	69	22	y	y	PROPN
ejpam-5238	69	23	,	,	PUNCT
ejpam-5238	69	24	and	and	CCONJ
ejpam-5238	69	25	we	we	PRON
ejpam-5238	69	26	always	always	ADV
ejpam-5238	69	27	assume	assume	VERB
ejpam-5238	69	28	that	that	SCONJ
ejpam-5238	69	29	f	f	PROPN
ejpam-5238	69	30	(	(	PUNCT
ejpam-5238	69	31	x	x	X
ejpam-5238	69	32	)	)	PUNCT
ejpam-5238	69	33	̸=	̸=	NOUN
ejpam-5238	69	34	∅	∅	NOUN
ejpam-5238	69	35	for	for	ADP
ejpam-5238	69	36	all	all	PRON
ejpam-5238	69	37	x	x	SYM
ejpam-5238	69	38	∈	∈	ADJ
ejpam-5238	69	39	x.	x.	NOUN
ejpam-5238	69	40	for	for	ADP
ejpam-5238	69	41	a	a	DET
ejpam-5238	69	42	multifunction	multifunction	NOUN
ejpam-5238	69	43	f	f	NOUN
ejpam-5238	70	1	:	:	PUNCT
ejpam-5238	70	2	x	x	X
ejpam-5238	70	3	→	→	SYM
ejpam-5238	70	4	y	y	PROPN
ejpam-5238	70	5	,	,	PUNCT
ejpam-5238	70	6	following	follow	VERB
ejpam-5238	70	7	[	[	X
ejpam-5238	70	8	1	1	X
ejpam-5238	70	9	]	]	PUNCT
ejpam-5238	70	10	we	we	PRON
ejpam-5238	70	11	shall	shall	AUX
ejpam-5238	70	12	denote	denote	VERB
ejpam-5238	70	13	the	the	DET
ejpam-5238	70	14	upper	upper	ADJ
ejpam-5238	70	15	and	and	CCONJ
ejpam-5238	70	16	lower	low	ADJ
ejpam-5238	70	17	inverse	inverse	NOUN
ejpam-5238	70	18	of	of	ADP
ejpam-5238	70	19	a	a	DET
ejpam-5238	70	20	set	set	NOUN
ejpam-5238	70	21	b	b	PROPN
ejpam-5238	70	22	of	of	ADP
ejpam-5238	70	23	y	y	PROPN
ejpam-5238	70	24	by	by	ADP
ejpam-5238	70	25	f+(b	f+(b	NOUN
ejpam-5238	70	26	)	)	PUNCT
ejpam-5238	70	27	and	and	CCONJ
ejpam-5238	70	28	f−(b	f−(b	NOUN
ejpam-5238	70	29	)	)	PUNCT
ejpam-5238	70	30	,	,	PUNCT
ejpam-5238	70	31	respectively	respectively	ADV
ejpam-5238	70	32	,	,	PUNCT
ejpam-5238	70	33	that	that	ADV
ejpam-5238	70	34	is	is	ADV
ejpam-5238	70	35	,	,	PUNCT
ejpam-5238	70	36	f+(b	f+(b	NOUN
ejpam-5238	70	37	)	)	PUNCT
ejpam-5238	70	38	=	=	PRON
ejpam-5238	71	1	{	{	PUNCT
ejpam-5238	71	2	x	x	PUNCT
ejpam-5238	71	3	∈	∈	PROPN
ejpam-5238	71	4	x	x	INTJ
ejpam-5238	72	1	|	|	NOUN
ejpam-5238	72	2	f	f	X
ejpam-5238	72	3	(	(	PUNCT
ejpam-5238	72	4	x	x	NOUN
ejpam-5238	72	5	)	)	PUNCT
ejpam-5238	72	6	⊆	⊆	NUM
ejpam-5238	72	7	b	b	NOUN
ejpam-5238	72	8	}	}	PUNCT
ejpam-5238	72	9	and	and	CCONJ
ejpam-5238	72	10	f−(b	f−(b	PROPN
ejpam-5238	72	11	)	)	PUNCT
ejpam-5238	72	12	=	=	PRON
ejpam-5238	73	1	{	{	PUNCT
ejpam-5238	73	2	x	x	PUNCT
ejpam-5238	73	3	∈	∈	PROPN
ejpam-5238	73	4	x	x	INTJ
ejpam-5238	74	1	|	|	NOUN
ejpam-5238	74	2	f	f	X
ejpam-5238	74	3	(	(	PUNCT
ejpam-5238	74	4	x	x	NOUN
ejpam-5238	74	5	)	)	PUNCT
ejpam-5238	74	6	∩b	∩b	NOUN
ejpam-5238	74	7	̸=	̸=	PROPN
ejpam-5238	74	8	∅	∅	NOUN
ejpam-5238	74	9	}	}	PUNCT
ejpam-5238	74	10	.	.	PUNCT
ejpam-5238	75	1	in	in	ADP
ejpam-5238	75	2	particular	particular	ADJ
ejpam-5238	75	3	,	,	PUNCT
ejpam-5238	75	4	f−(y	f−(y	NOUN
ejpam-5238	75	5	)	)	PUNCT
ejpam-5238	75	6	=	=	SYM
ejpam-5238	76	1	{	{	PUNCT
ejpam-5238	76	2	x	x	PUNCT
ejpam-5238	76	3	∈	∈	PROPN
ejpam-5238	76	4	x	x	INTJ
ejpam-5238	77	1	|	|	ADV
ejpam-5238	77	2	y	y	PROPN
ejpam-5238	77	3	∈	∈	PROPN
ejpam-5238	77	4	f	f	X
ejpam-5238	77	5	(	(	PUNCT
ejpam-5238	77	6	x	x	NOUN
ejpam-5238	77	7	)	)	PUNCT
ejpam-5238	77	8	}	}	PUNCT
ejpam-5238	77	9	for	for	ADP
ejpam-5238	77	10	each	each	DET
ejpam-5238	77	11	point	point	NOUN
ejpam-5238	77	12	y	y	PROPN
ejpam-5238	77	13	∈	∈	PROPN
ejpam-5238	77	14	y	y	PROPN
ejpam-5238	77	15	.	.	PUNCT
ejpam-5238	78	1	for	for	ADP
ejpam-5238	78	2	each	each	DET
ejpam-5238	78	3	a	a	DET
ejpam-5238	78	4	⊆	⊆	NUM
ejpam-5238	78	5	x	x	SYM
ejpam-5238	78	6	,	,	PUNCT
ejpam-5238	78	7	f	f	PROPN
ejpam-5238	78	8	(	(	PUNCT
ejpam-5238	78	9	a	a	NOUN
ejpam-5238	78	10	)	)	PUNCT
ejpam-5238	78	11	=	=	SYM
ejpam-5238	78	12	∪x∈af	∪x∈af	NOUN
ejpam-5238	78	13	(	(	PUNCT
ejpam-5238	78	14	x	x	NOUN
ejpam-5238	78	15	)	)	PUNCT
ejpam-5238	78	16	.	.	PUNCT
ejpam-5238	79	1	3	3	X
ejpam-5238	79	2	.	.	X
ejpam-5238	79	3	upper	upper	ADJ
ejpam-5238	79	4	and	and	CCONJ
ejpam-5238	79	5	lower	low	ADJ
ejpam-5238	79	6	weakly	weakly	ADJ
ejpam-5238	79	7	(	(	PUNCT
ejpam-5238	79	8	τ1	τ1	NOUN
ejpam-5238	79	9	,	,	PUNCT
ejpam-5238	79	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	79	11	multifunctions	multifunction	NOUN
ejpam-5238	79	12	in	in	ADP
ejpam-5238	79	13	this	this	DET
ejpam-5238	79	14	section	section	NOUN
ejpam-5238	79	15	,	,	PUNCT
ejpam-5238	79	16	we	we	PRON
ejpam-5238	79	17	introduce	introduce	VERB
ejpam-5238	79	18	the	the	DET
ejpam-5238	79	19	notions	notion	NOUN
ejpam-5238	79	20	of	of	ADP
ejpam-5238	79	21	upper	upper	ADJ
ejpam-5238	79	22	and	and	CCONJ
ejpam-5238	79	23	lower	low	ADJ
ejpam-5238	79	24	weakly	weakly	ADJ
ejpam-5238	79	25	(	(	PUNCT
ejpam-5238	79	26	τ1	τ1	NOUN
ejpam-5238	79	27	,	,	PUNCT
ejpam-5238	79	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	79	29	multifunctions	multifunction	NOUN
ejpam-5238	79	30	.	.	PUNCT
ejpam-5238	80	1	moreover	moreover	ADV
ejpam-5238	80	2	,	,	PUNCT
ejpam-5238	80	3	some	some	DET
ejpam-5238	80	4	characterizations	characterization	NOUN
ejpam-5238	80	5	of	of	ADP
ejpam-5238	80	6	upper	upper	ADJ
ejpam-5238	80	7	and	and	CCONJ
ejpam-5238	80	8	lower	low	ADJ
ejpam-5238	80	9	weakly	weakly	ADJ
ejpam-5238	80	10	(	(	PUNCT
ejpam-5238	80	11	τ1	τ1	NOUN
ejpam-5238	80	12	,	,	PUNCT
ejpam-5238	80	13	τ2)continuous	τ2)continuous	ADJ
ejpam-5238	80	14	multifunctions	multifunction	NOUN
ejpam-5238	80	15	are	be	AUX
ejpam-5238	80	16	discussed	discuss	VERB
ejpam-5238	80	17	.	.	PUNCT
ejpam-5238	81	1	definition	definition	NOUN
ejpam-5238	81	2	1	1	NUM
ejpam-5238	81	3	.	.	PUNCT
ejpam-5238	82	1	a	a	DET
ejpam-5238	82	2	multifunction	multifunction	NOUN
ejpam-5238	82	3	f	f	NOUN
ejpam-5238	82	4	:	:	PUNCT
ejpam-5238	82	5	(	(	PUNCT
ejpam-5238	82	6	x	x	NOUN
ejpam-5238	82	7	,	,	PUNCT
ejpam-5238	82	8	τ1	τ1	NOUN
ejpam-5238	82	9	,	,	PUNCT
ejpam-5238	82	10	τ2	τ2	NOUN
ejpam-5238	82	11	)	)	PUNCT
ejpam-5238	82	12	→	→	SYM
ejpam-5238	82	13	(	(	PUNCT
ejpam-5238	82	14	y	y	PROPN
ejpam-5238	82	15	,	,	PUNCT
ejpam-5238	82	16	σ1	σ1	PROPN
ejpam-5238	82	17	,	,	PUNCT
ejpam-5238	82	18	σ2	σ2	PROPN
ejpam-5238	82	19	)	)	PUNCT
ejpam-5238	82	20	is	be	AUX
ejpam-5238	82	21	said	say	VERB
ejpam-5238	82	22	to	to	PART
ejpam-5238	82	23	be	be	AUX
ejpam-5238	82	24	upper	upper	ADJ
ejpam-5238	82	25	weakly	weakly	ADJ
ejpam-5238	82	26	(	(	PUNCT
ejpam-5238	82	27	τ1	τ1	NOUN
ejpam-5238	82	28	,	,	PUNCT
ejpam-5238	82	29	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	82	30	if	if	SCONJ
ejpam-5238	82	31	for	for	ADP
ejpam-5238	82	32	each	each	DET
ejpam-5238	82	33	x	x	SYM
ejpam-5238	82	34	∈	∈	PROPN
ejpam-5238	82	35	x	x	X
ejpam-5238	82	36	and	and	CCONJ
ejpam-5238	82	37	each	each	DET
ejpam-5238	82	38	σ1σ2	σ1σ2	VERB
ejpam-5238	82	39	-	-	ADJ
ejpam-5238	82	40	open	open	ADJ
ejpam-5238	82	41	set	set	NOUN
ejpam-5238	82	42	v	v	NOUN
ejpam-5238	82	43	of	of	ADP
ejpam-5238	82	44	y	y	PROPN
ejpam-5238	82	45	containing	contain	VERB
ejpam-5238	82	46	f	f	PROPN
ejpam-5238	82	47	(	(	PUNCT
ejpam-5238	82	48	x	x	NOUN
ejpam-5238	82	49	)	)	PUNCT
ejpam-5238	82	50	,	,	PUNCT
ejpam-5238	82	51	there	there	PRON
ejpam-5238	82	52	exists	exist	VERB
ejpam-5238	82	53	a	a	DET
ejpam-5238	82	54	τ1τ2	τ1τ2	NOUN
ejpam-5238	82	55	-	-	ADJ
ejpam-5238	82	56	open	open	ADJ
ejpam-5238	82	57	set	set	ADJ
ejpam-5238	82	58	u	u	NOUN
ejpam-5238	82	59	of	of	ADP
ejpam-5238	82	60	x	x	PUNCT
ejpam-5238	82	61	containing	contain	VERB
ejpam-5238	82	62	x	x	PUNCT
ejpam-5238	82	63	such	such	ADJ
ejpam-5238	82	64	that	that	SCONJ
ejpam-5238	82	65	f	f	PROPN
ejpam-5238	82	66	(	(	PUNCT
ejpam-5238	82	67	u	u	NOUN
ejpam-5238	82	68	)	)	PUNCT
ejpam-5238	82	69	⊆	⊆	NUM
ejpam-5238	82	70	σ1σ2	σ1σ2	NOUN
ejpam-5238	82	71	-	-	NUM
ejpam-5238	82	72	cl(v	cl(v	NOUN
ejpam-5238	82	73	)	)	PUNCT
ejpam-5238	82	74	.	.	PUNCT
ejpam-5238	83	1	theorem	theorem	NOUN
ejpam-5238	83	2	1	1	NUM
ejpam-5238	83	3	.	.	X
ejpam-5238	83	4	for	for	ADP
ejpam-5238	83	5	a	a	DET
ejpam-5238	83	6	multifunction	multifunction	NOUN
ejpam-5238	84	1	f	f	NOUN
ejpam-5238	84	2	:	:	PUNCT
ejpam-5238	84	3	(	(	PUNCT
ejpam-5238	84	4	x	x	NOUN
ejpam-5238	84	5	,	,	PUNCT
ejpam-5238	84	6	τ1	τ1	NOUN
ejpam-5238	84	7	,	,	PUNCT
ejpam-5238	84	8	τ2	τ2	NOUN
ejpam-5238	84	9	)	)	PUNCT
ejpam-5238	84	10	→	→	SYM
ejpam-5238	84	11	(	(	PUNCT
ejpam-5238	84	12	y	y	PROPN
ejpam-5238	84	13	,	,	PUNCT
ejpam-5238	84	14	σ1	σ1	PROPN
ejpam-5238	84	15	,	,	PUNCT
ejpam-5238	84	16	σ2	σ2	NOUN
ejpam-5238	84	17	)	)	PUNCT
ejpam-5238	84	18	,	,	PUNCT
ejpam-5238	84	19	the	the	DET
ejpam-5238	84	20	following	follow	VERB
ejpam-5238	84	21	properties	property	NOUN
ejpam-5238	84	22	are	be	AUX
ejpam-5238	84	23	equivalent	equivalent	ADJ
ejpam-5238	84	24	:	:	PUNCT
ejpam-5238	84	25	(	(	PUNCT
ejpam-5238	84	26	1	1	X
ejpam-5238	84	27	)	)	PUNCT
ejpam-5238	84	28	f	f	PROPN
ejpam-5238	84	29	is	be	AUX
ejpam-5238	84	30	upper	upper	ADJ
ejpam-5238	84	31	weakly	weakly	ADJ
ejpam-5238	84	32	(	(	PUNCT
ejpam-5238	84	33	τ1	τ1	NOUN
ejpam-5238	84	34	,	,	PUNCT
ejpam-5238	84	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	84	36	;	;	PUNCT
ejpam-5238	84	37	(	(	PUNCT
ejpam-5238	84	38	2	2	NUM
ejpam-5238	84	39	)	)	PUNCT
ejpam-5238	84	40	f+(v	f+(v	NOUN
ejpam-5238	84	41	)	)	PUNCT
ejpam-5238	85	1	⊆	⊆	X
ejpam-5238	85	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	85	3	-	-	NUM
ejpam-5238	85	4	int(f	int(f	VERB
ejpam-5238	85	5	+	+	ADJ
ejpam-5238	85	6	(	(	PUNCT
ejpam-5238	85	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	85	8	-	-	NUM
ejpam-5238	85	9	cl(v	cl(v	NOUN
ejpam-5238	85	10	)	)	PUNCT
ejpam-5238	85	11	)	)	PUNCT
ejpam-5238	85	12	)	)	PUNCT
ejpam-5238	85	13	for	for	ADP
ejpam-5238	85	14	every	every	DET
ejpam-5238	85	15	σ1σ2	σ1σ2	NOUN
ejpam-5238	85	16	-	-	ADJ
ejpam-5238	85	17	open	open	ADJ
ejpam-5238	85	18	set	set	NOUN
ejpam-5238	85	19	v	v	NOUN
ejpam-5238	85	20	of	of	ADP
ejpam-5238	85	21	y	y	PROPN
ejpam-5238	85	22	;	;	PUNCT
ejpam-5238	85	23	(	(	PUNCT
ejpam-5238	85	24	3	3	X
ejpam-5238	85	25	)	)	PUNCT
ejpam-5238	85	26	τ1τ2	τ1τ2	NOUN
ejpam-5238	85	27	-	-	NOUN
ejpam-5238	85	28	cl(f	cl(f	NOUN
ejpam-5238	85	29	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	85	30	-	-	PUNCT
ejpam-5238	85	31	int(k	int(k	NUM
ejpam-5238	85	32	)	)	PUNCT
ejpam-5238	85	33	)	)	PUNCT
ejpam-5238	85	34	)	)	PUNCT
ejpam-5238	86	1	⊆	⊆	X
ejpam-5238	86	2	f−(k	f−(k	PROPN
ejpam-5238	86	3	)	)	PUNCT
ejpam-5238	86	4	for	for	ADP
ejpam-5238	86	5	every	every	DET
ejpam-5238	86	6	σ1σ2	σ1σ2	NUM
ejpam-5238	86	7	-	-	PUNCT
ejpam-5238	86	8	closed	closed	ADJ
ejpam-5238	86	9	set	set	NOUN
ejpam-5238	86	10	k	k	PROPN
ejpam-5238	86	11	of	of	ADP
ejpam-5238	86	12	y	y	PROPN
ejpam-5238	86	13	;	;	PUNCT
ejpam-5238	86	14	(	(	PUNCT
ejpam-5238	86	15	4	4	X
ejpam-5238	86	16	)	)	PUNCT
ejpam-5238	86	17	τ1τ2	τ1τ2	NOUN
ejpam-5238	86	18	-	-	NOUN
ejpam-5238	86	19	cl(f	cl(f	NOUN
ejpam-5238	86	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	86	21	-	-	PUNCT
ejpam-5238	86	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	86	23	-	-	PUNCT
ejpam-5238	86	24	cl(b	cl(b	NOUN
ejpam-5238	86	25	)	)	PUNCT
ejpam-5238	86	26	)	)	PUNCT
ejpam-5238	86	27	)	)	PUNCT
ejpam-5238	86	28	)	)	PUNCT
ejpam-5238	87	1	⊆	⊆	X
ejpam-5238	87	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5238	87	3	-	-	PUNCT
ejpam-5238	87	4	cl(b	cl(b	NOUN
ejpam-5238	87	5	)	)	PUNCT
ejpam-5238	87	6	)	)	PUNCT
ejpam-5238	88	1	for	for	ADP
ejpam-5238	88	2	every	every	DET
ejpam-5238	88	3	subset	subset	NOUN
ejpam-5238	88	4	b	b	PROPN
ejpam-5238	88	5	of	of	ADP
ejpam-5238	88	6	y	y	PROPN
ejpam-5238	88	7	;	;	PUNCT
ejpam-5238	88	8	(	(	PUNCT
ejpam-5238	88	9	5	5	X
ejpam-5238	88	10	)	)	PUNCT
ejpam-5238	88	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	88	12	-	-	PUNCT
ejpam-5238	88	13	int(b	int(b	NOUN
ejpam-5238	88	14	)	)	PUNCT
ejpam-5238	88	15	)	)	PUNCT
ejpam-5238	89	1	⊆	⊆	X
ejpam-5238	89	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	89	3	-	-	NUM
ejpam-5238	89	4	int(f	int(f	VERB
ejpam-5238	89	5	+	+	ADJ
ejpam-5238	89	6	(	(	PUNCT
ejpam-5238	89	7	σ1σ2	σ1σ2	NUM
ejpam-5238	89	8	-	-	PUNCT
ejpam-5238	89	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	89	10	-	-	PUNCT
ejpam-5238	89	11	int(b	int(b	NOUN
ejpam-5238	89	12	)	)	PUNCT
ejpam-5238	89	13	)	)	PUNCT
ejpam-5238	89	14	)	)	PUNCT
ejpam-5238	89	15	)	)	PUNCT
ejpam-5238	89	16	for	for	ADP
ejpam-5238	89	17	every	every	DET
ejpam-5238	89	18	subset	subset	NOUN
ejpam-5238	89	19	b	b	PROPN
ejpam-5238	89	20	of	of	ADP
ejpam-5238	89	21	y	y	PROPN
ejpam-5238	89	22	;	;	PUNCT
ejpam-5238	89	23	(	(	PUNCT
ejpam-5238	89	24	6	6	X
ejpam-5238	89	25	)	)	PUNCT
ejpam-5238	89	26	τ1τ2	τ1τ2	NOUN
ejpam-5238	89	27	-	-	NOUN
ejpam-5238	89	28	cl(f	cl(f	NOUN
ejpam-5238	89	29	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	89	30	-	-	PUNCT
ejpam-5238	89	31	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	89	32	-	-	PUNCT
ejpam-5238	89	33	cl(v	cl(v	NOUN
ejpam-5238	89	34	)	)	PUNCT
ejpam-5238	89	35	)	)	PUNCT
ejpam-5238	89	36	)	)	PUNCT
ejpam-5238	89	37	)	)	PUNCT
ejpam-5238	90	1	⊆	⊆	X
ejpam-5238	90	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	90	3	-	-	PUNCT
ejpam-5238	90	4	cl(v	cl(v	NOUN
ejpam-5238	90	5	)	)	PUNCT
ejpam-5238	90	6	)	)	PUNCT
ejpam-5238	90	7	for	for	ADP
ejpam-5238	90	8	every	every	DET
ejpam-5238	90	9	σ1σ2	σ1σ2	NOUN
ejpam-5238	90	10	-	-	ADJ
ejpam-5238	90	11	open	open	ADJ
ejpam-5238	90	12	set	set	NOUN
ejpam-5238	90	13	v	v	NOUN
ejpam-5238	90	14	of	of	ADP
ejpam-5238	90	15	y	y	PROPN
ejpam-5238	90	16	;	;	PUNCT
ejpam-5238	90	17	(	(	PUNCT
ejpam-5238	90	18	7	7	X
ejpam-5238	90	19	)	)	PUNCT
ejpam-5238	90	20	τ1τ2	τ1τ2	NOUN
ejpam-5238	90	21	-	-	NOUN
ejpam-5238	90	22	cl(f	cl(f	NUM
ejpam-5238	90	23	−(v	−(v	NOUN
ejpam-5238	90	24	)	)	PUNCT
ejpam-5238	90	25	)	)	PUNCT
ejpam-5238	91	1	⊆	⊆	X
ejpam-5238	91	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	91	3	-	-	PUNCT
ejpam-5238	91	4	cl(v	cl(v	NOUN
ejpam-5238	91	5	)	)	PUNCT
ejpam-5238	91	6	)	)	PUNCT
ejpam-5238	91	7	for	for	ADP
ejpam-5238	91	8	every	every	DET
ejpam-5238	91	9	σ1σ2	σ1σ2	NOUN
ejpam-5238	91	10	-	-	ADJ
ejpam-5238	91	11	open	open	ADJ
ejpam-5238	91	12	set	set	NOUN
ejpam-5238	91	13	v	v	NOUN
ejpam-5238	91	14	of	of	ADP
ejpam-5238	91	15	y	y	PROPN
ejpam-5238	91	16	;	;	PUNCT
ejpam-5238	91	17	(	(	PUNCT
ejpam-5238	91	18	8)	8)	NUM
ejpam-5238	91	19	τ1τ2	τ1τ2	NOUN
ejpam-5238	91	20	-	-	NOUN
ejpam-5238	91	21	cl(f	cl(f	NOUN
ejpam-5238	91	22	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	91	23	-	-	PUNCT
ejpam-5238	91	24	int(k	int(k	NUM
ejpam-5238	91	25	)	)	PUNCT
ejpam-5238	91	26	)	)	PUNCT
ejpam-5238	91	27	)	)	PUNCT
ejpam-5238	91	28	⊆	⊆	X
ejpam-5238	91	29	f−(k	f−(k	PROPN
ejpam-5238	91	30	)	)	PUNCT
ejpam-5238	91	31	for	for	ADP
ejpam-5238	91	32	every	every	DET
ejpam-5238	91	33	(	(	PUNCT
ejpam-5238	91	34	σ1	σ1	PROPN
ejpam-5238	91	35	,	,	PUNCT
ejpam-5238	91	36	σ2)r	σ2)r	NOUN
ejpam-5238	91	37	-	-	PUNCT
ejpam-5238	91	38	closed	close	VERB
ejpam-5238	91	39	set	set	ADJ
ejpam-5238	91	40	k	k	PROPN
ejpam-5238	91	41	of	of	ADP
ejpam-5238	91	42	y	y	PROPN
ejpam-5238	91	43	.	.	PUNCT
ejpam-5238	92	1	m.	m.	NOUN
ejpam-5238	92	2	thongmoon	thongmoon	PROPN
ejpam-5238	92	3	,	,	PUNCT
ejpam-5238	92	4	s.	s.	PROPN
ejpam-5238	92	5	sompong	sompong	PROPN
ejpam-5238	92	6	,	,	PUNCT
ejpam-5238	92	7	c.	c.	PROPN
ejpam-5238	92	8	boonpok	boonpok	PROPN
ejpam-5238	92	9	/	/	SYM
ejpam-5238	92	10	eur	eur	PROPN
ejpam-5238	92	11	.	.	PUNCT
ejpam-5238	93	1	j.	j.	PROPN
ejpam-5238	93	2	pure	pure	PROPN
ejpam-5238	93	3	appl	appl	PROPN
ejpam-5238	93	4	.	.	PROPN
ejpam-5238	93	5	math	math	PROPN
ejpam-5238	93	6	,	,	PUNCT
ejpam-5238	93	7	17	17	NUM
ejpam-5238	93	8	(	(	PUNCT
ejpam-5238	93	9	3	3	NUM
ejpam-5238	93	10	)	)	PUNCT
ejpam-5238	93	11	(	(	PUNCT
ejpam-5238	93	12	2024	2024	NUM
ejpam-5238	93	13	)	)	PUNCT
ejpam-5238	93	14	,	,	PUNCT
ejpam-5238	93	15	1705	1705	NUM
ejpam-5238	93	16	-	-	SYM
ejpam-5238	93	17	1716	1716	NUM
ejpam-5238	93	18	1708	1708	NUM
ejpam-5238	93	19	proof	proof	NOUN
ejpam-5238	93	20	.	.	PUNCT
ejpam-5238	94	1	(	(	PUNCT
ejpam-5238	94	2	1	1	X
ejpam-5238	94	3	)	)	PUNCT
ejpam-5238	94	4	⇒	⇒	NOUN
ejpam-5238	94	5	(	(	PUNCT
ejpam-5238	94	6	2	2	NUM
ejpam-5238	94	7	):	):	PUNCT
ejpam-5238	94	8	let	let	VERB
ejpam-5238	94	9	v	v	PART
ejpam-5238	94	10	be	be	AUX
ejpam-5238	94	11	any	any	DET
ejpam-5238	94	12	σ1σ2	σ1σ2	NOUN
ejpam-5238	94	13	-	-	ADJ
ejpam-5238	94	14	open	open	ADJ
ejpam-5238	94	15	set	set	NOUN
ejpam-5238	94	16	of	of	ADP
ejpam-5238	94	17	y	y	PRON
ejpam-5238	94	18	such	such	ADJ
ejpam-5238	94	19	that	that	SCONJ
ejpam-5238	94	20	x	x	SYM
ejpam-5238	94	21	∈	∈	PROPN
ejpam-5238	94	22	f+(v	f+(v	NOUN
ejpam-5238	94	23	)	)	PUNCT
ejpam-5238	94	24	.	.	PUNCT
ejpam-5238	95	1	then	then	ADV
ejpam-5238	95	2	,	,	PUNCT
ejpam-5238	95	3	f	f	PROPN
ejpam-5238	95	4	(	(	PUNCT
ejpam-5238	95	5	x	x	X
ejpam-5238	95	6	)	)	PUNCT
ejpam-5238	95	7	⊆	⊆	NUM
ejpam-5238	95	8	v	v	NOUN
ejpam-5238	95	9	.	.	PUNCT
ejpam-5238	96	1	there	there	PRON
ejpam-5238	96	2	exists	exist	VERB
ejpam-5238	96	3	a	a	DET
ejpam-5238	96	4	τ1τ2	τ1τ2	NOUN
ejpam-5238	96	5	-	-	ADJ
ejpam-5238	96	6	open	open	ADJ
ejpam-5238	96	7	set	set	NOUN
ejpam-5238	96	8	u	u	PRON
ejpam-5238	96	9	ofx	ofx	NOUN
ejpam-5238	96	10	containing	contain	VERB
ejpam-5238	96	11	x	x	PUNCT
ejpam-5238	96	12	such	such	ADJ
ejpam-5238	96	13	that	that	SCONJ
ejpam-5238	96	14	f	f	PROPN
ejpam-5238	96	15	(	(	PUNCT
ejpam-5238	96	16	u	u	NOUN
ejpam-5238	96	17	)	)	PUNCT
ejpam-5238	96	18	⊆	⊆	NUM
ejpam-5238	96	19	σ1σ2	σ1σ2	NOUN
ejpam-5238	96	20	-	-	NUM
ejpam-5238	96	21	cl(v	cl(v	NOUN
ejpam-5238	96	22	)	)	PUNCT
ejpam-5238	96	23	.	.	PUNCT
ejpam-5238	97	1	thus	thus	ADV
ejpam-5238	97	2	,	,	PUNCT
ejpam-5238	97	3	u	u	NOUN
ejpam-5238	97	4	⊆	⊆	NUM
ejpam-5238	97	5	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	97	6	-	-	PUNCT
ejpam-5238	97	7	cl(v	cl(v	NOUN
ejpam-5238	97	8	)	)	PUNCT
ejpam-5238	97	9	)	)	PUNCT
ejpam-5238	97	10	.	.	PUNCT
ejpam-5238	98	1	since	since	SCONJ
ejpam-5238	98	2	u	u	NOUN
ejpam-5238	98	3	is	be	AUX
ejpam-5238	98	4	τ1τ2	τ1τ2	VERB
ejpam-5238	98	5	-	-	ADJ
ejpam-5238	98	6	open	open	ADJ
ejpam-5238	98	7	,	,	PUNCT
ejpam-5238	98	8	we	we	PRON
ejpam-5238	98	9	have	have	VERB
ejpam-5238	98	10	x	x	PART
ejpam-5238	98	11	∈	∈	PRON
ejpam-5238	98	12	τ1τ2	τ1τ2	NOUN
ejpam-5238	98	13	-	-	NUM
ejpam-5238	98	14	int(f	int(f	VERB
ejpam-5238	98	15	+	+	ADJ
ejpam-5238	98	16	(	(	PUNCT
ejpam-5238	98	17	σ1σ2	σ1σ2	NOUN
ejpam-5238	98	18	-	-	NUM
ejpam-5238	98	19	cl(v	cl(v	NOUN
ejpam-5238	98	20	)	)	PUNCT
ejpam-5238	98	21	)	)	PUNCT
ejpam-5238	98	22	)	)	PUNCT
ejpam-5238	98	23	and	and	CCONJ
ejpam-5238	98	24	hence	hence	ADV
ejpam-5238	98	25	f+(v	f+(v	NOUN
ejpam-5238	98	26	)	)	PUNCT
ejpam-5238	99	1	⊆	⊆	X
ejpam-5238	99	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	99	3	-	-	NUM
ejpam-5238	99	4	int(f	int(f	VERB
ejpam-5238	99	5	+	+	ADJ
ejpam-5238	99	6	(	(	PUNCT
ejpam-5238	99	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	99	8	-	-	NUM
ejpam-5238	99	9	cl(v	cl(v	NOUN
ejpam-5238	99	10	)	)	PUNCT
ejpam-5238	99	11	)	)	PUNCT
ejpam-5238	99	12	)	)	PUNCT
ejpam-5238	99	13	.	.	PUNCT
ejpam-5238	100	1	(	(	PUNCT
ejpam-5238	100	2	2	2	X
ejpam-5238	100	3	)	)	PUNCT
ejpam-5238	100	4	⇒	⇒	NOUN
ejpam-5238	100	5	(	(	PUNCT
ejpam-5238	100	6	3	3	NUM
ejpam-5238	100	7	):	):	PUNCT
ejpam-5238	100	8	let	let	VERB
ejpam-5238	100	9	k	k	PRON
ejpam-5238	100	10	be	be	AUX
ejpam-5238	100	11	any	any	DET
ejpam-5238	100	12	σ1σ2	σ1σ2	NUM
ejpam-5238	100	13	-	-	PUNCT
ejpam-5238	100	14	closed	closed	ADJ
ejpam-5238	100	15	set	set	NOUN
ejpam-5238	100	16	of	of	ADP
ejpam-5238	100	17	y	y	PROPN
ejpam-5238	100	18	.	.	PUNCT
ejpam-5238	101	1	then	then	ADV
ejpam-5238	101	2	,	,	PUNCT
ejpam-5238	101	3	y	y	PROPN
ejpam-5238	101	4	−k	−k	PROPN
ejpam-5238	101	5	is	be	AUX
ejpam-5238	101	6	σ1σ2	σ1σ2	NOUN
ejpam-5238	101	7	-	-	ADJ
ejpam-5238	101	8	open	open	ADJ
ejpam-5238	101	9	in	in	ADP
ejpam-5238	101	10	y	y	PROPN
ejpam-5238	101	11	and	and	CCONJ
ejpam-5238	101	12	by	by	ADP
ejpam-5238	101	13	(	(	PUNCT
ejpam-5238	101	14	2	2	NUM
ejpam-5238	101	15	)	)	PUNCT
ejpam-5238	101	16	,	,	PUNCT
ejpam-5238	101	17	x	x	PUNCT
ejpam-5238	101	18	−	−	DET
ejpam-5238	101	19	f−(k	f−(k	PROPN
ejpam-5238	101	20	)	)	PUNCT
ejpam-5238	101	21	=	=	PUNCT
ejpam-5238	102	1	f+(y	f+(y	PROPN
ejpam-5238	102	2	−k	−k	PROPN
ejpam-5238	102	3	)	)	PUNCT
ejpam-5238	102	4	⊆	⊆	NUM
ejpam-5238	102	5	τ1τ2	τ1τ2	NOUN
ejpam-5238	102	6	-	-	NUM
ejpam-5238	102	7	int(f	int(f	VERB
ejpam-5238	102	8	+	+	ADJ
ejpam-5238	102	9	(	(	PUNCT
ejpam-5238	102	10	σ1σ2	σ1σ2	NUM
ejpam-5238	102	11	-	-	PUNCT
ejpam-5238	102	12	cl(y	cl(y	NOUN
ejpam-5238	102	13	−k	−k	NOUN
ejpam-5238	102	14	)	)	PUNCT
ejpam-5238	102	15	)	)	PUNCT
ejpam-5238	102	16	)	)	PUNCT
ejpam-5238	103	1	=	=	PUNCT
ejpam-5238	104	1	x	x	X
ejpam-5238	104	2	−	−	ADP
ejpam-5238	104	3	τ1τ2	τ1τ2	NOUN
ejpam-5238	104	4	-	-	NOUN
ejpam-5238	104	5	cl(f	cl(f	NOUN
ejpam-5238	104	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	104	7	-	-	PUNCT
ejpam-5238	104	8	int(k	int(k	NUM
ejpam-5238	104	9	)	)	PUNCT
ejpam-5238	104	10	)	)	PUNCT
ejpam-5238	104	11	)	)	PUNCT
ejpam-5238	104	12	.	.	PUNCT
ejpam-5238	105	1	thus	thus	ADV
ejpam-5238	105	2	,	,	PUNCT
ejpam-5238	105	3	τ1τ2	τ1τ2	NOUN
ejpam-5238	105	4	-	-	ADJ
ejpam-5238	105	5	cl(f	cl(f	NOUN
ejpam-5238	105	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	105	7	-	-	PUNCT
ejpam-5238	105	8	int(k	int(k	NUM
ejpam-5238	105	9	)	)	PUNCT
ejpam-5238	105	10	)	)	PUNCT
ejpam-5238	105	11	)	)	PUNCT
ejpam-5238	106	1	⊆	⊆	NUM
ejpam-5238	106	2	f−(k	f−(k	PROPN
ejpam-5238	106	3	)	)	PUNCT
ejpam-5238	106	4	.	.	PUNCT
ejpam-5238	107	1	(	(	PUNCT
ejpam-5238	107	2	3	3	X
ejpam-5238	107	3	)	)	PUNCT
ejpam-5238	107	4	⇒	⇒	NOUN
ejpam-5238	107	5	(	(	PUNCT
ejpam-5238	107	6	4	4	NUM
ejpam-5238	107	7	):	):	PUNCT
ejpam-5238	107	8	let	let	VERB
ejpam-5238	107	9	b	b	X
ejpam-5238	107	10	be	be	AUX
ejpam-5238	107	11	any	any	DET
ejpam-5238	107	12	subset	subset	NOUN
ejpam-5238	107	13	of	of	ADP
ejpam-5238	107	14	y	y	PROPN
ejpam-5238	107	15	.	.	PUNCT
ejpam-5238	108	1	then	then	ADV
ejpam-5238	108	2	,	,	PUNCT
ejpam-5238	108	3	σ1σ2	σ1σ2	NOUN
ejpam-5238	108	4	-	-	NOUN
ejpam-5238	108	5	cl(b	cl(b	NOUN
ejpam-5238	108	6	)	)	PUNCT
ejpam-5238	108	7	is	be	AUX
ejpam-5238	108	8	a	a	DET
ejpam-5238	108	9	σ1σ2	σ1σ2	NUM
ejpam-5238	108	10	-	-	PUNCT
ejpam-5238	108	11	closed	closed	ADJ
ejpam-5238	108	12	set	set	NOUN
ejpam-5238	108	13	of	of	ADP
ejpam-5238	108	14	y	y	PROPN
ejpam-5238	108	15	and	and	CCONJ
ejpam-5238	108	16	by	by	ADP
ejpam-5238	108	17	(	(	PUNCT
ejpam-5238	108	18	3	3	NUM
ejpam-5238	108	19	)	)	PUNCT
ejpam-5238	108	20	,	,	PUNCT
ejpam-5238	108	21	τ1τ2	τ1τ2	NOUN
ejpam-5238	108	22	-	-	ADJ
ejpam-5238	108	23	cl(f	cl(f	NOUN
ejpam-5238	108	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	108	25	-	-	PUNCT
ejpam-5238	108	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	108	27	-	-	PUNCT
ejpam-5238	108	28	cl(b	cl(b	NOUN
ejpam-5238	108	29	)	)	PUNCT
ejpam-5238	108	30	)	)	PUNCT
ejpam-5238	108	31	)	)	PUNCT
ejpam-5238	108	32	)	)	PUNCT
ejpam-5238	109	1	⊆	⊆	X
ejpam-5238	109	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5238	109	3	-	-	PUNCT
ejpam-5238	109	4	cl(b	cl(b	NOUN
ejpam-5238	109	5	)	)	PUNCT
ejpam-5238	109	6	)	)	PUNCT
ejpam-5238	109	7	.	.	PUNCT
ejpam-5238	110	1	(	(	PUNCT
ejpam-5238	110	2	4	4	X
ejpam-5238	110	3	)	)	PUNCT
ejpam-5238	110	4	⇒	⇒	NOUN
ejpam-5238	110	5	(	(	PUNCT
ejpam-5238	110	6	5	5	NUM
ejpam-5238	110	7	):	):	PUNCT
ejpam-5238	110	8	let	let	VERB
ejpam-5238	110	9	b	b	X
ejpam-5238	110	10	be	be	AUX
ejpam-5238	110	11	any	any	DET
ejpam-5238	110	12	subset	subset	NOUN
ejpam-5238	110	13	of	of	ADP
ejpam-5238	110	14	y	y	PROPN
ejpam-5238	110	15	.	.	PUNCT
ejpam-5238	111	1	by	by	ADP
ejpam-5238	111	2	(	(	PUNCT
ejpam-5238	111	3	4	4	NUM
ejpam-5238	111	4	)	)	PUNCT
ejpam-5238	111	5	,	,	PUNCT
ejpam-5238	111	6	we	we	PRON
ejpam-5238	111	7	have	have	VERB
ejpam-5238	111	8	x	x	INTJ
ejpam-5238	111	9	−	−	ADP
ejpam-5238	111	10	τ1τ2	τ1τ2	NOUN
ejpam-5238	111	11	-	-	NUM
ejpam-5238	111	12	int(f	int(f	VERB
ejpam-5238	111	13	+	+	ADJ
ejpam-5238	111	14	(	(	PUNCT
ejpam-5238	111	15	σ1σ2	σ1σ2	NUM
ejpam-5238	111	16	-	-	PUNCT
ejpam-5238	111	17	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	111	18	-	-	PUNCT
ejpam-5238	111	19	int(b	int(b	NOUN
ejpam-5238	111	20	)	)	PUNCT
ejpam-5238	111	21	)	)	PUNCT
ejpam-5238	111	22	)	)	PUNCT
ejpam-5238	111	23	)	)	PUNCT
ejpam-5238	112	1	=	=	PUNCT
ejpam-5238	112	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	112	3	-	-	NOUN
ejpam-5238	112	4	cl(x	cl(x	SYM
ejpam-5238	112	5	−	−	ADP
ejpam-5238	112	6	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-5238	112	7	-	-	PUNCT
ejpam-5238	112	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	112	9	-	-	PUNCT
ejpam-5238	112	10	int(b	int(b	NOUN
ejpam-5238	112	11	)	)	PUNCT
ejpam-5238	112	12	)	)	PUNCT
ejpam-5238	112	13	)	)	PUNCT
ejpam-5238	112	14	)	)	PUNCT
ejpam-5238	113	1	=	=	PUNCT
ejpam-5238	113	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	113	3	-	-	ADJ
ejpam-5238	113	4	cl(f	cl(f	NOUN
ejpam-5238	113	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	113	6	-	-	PUNCT
ejpam-5238	113	7	int(σ1σ2	int(σ1σ2	VERB
ejpam-5238	113	8	-	-	PUNCT
ejpam-5238	113	9	cl(y	cl(y	NOUN
ejpam-5238	113	10	−b	−b	NOUN
ejpam-5238	113	11	)	)	PUNCT
ejpam-5238	113	12	)	)	PUNCT
ejpam-5238	113	13	)	)	PUNCT
ejpam-5238	113	14	)	)	PUNCT
ejpam-5238	114	1	⊆	⊆	X
ejpam-5238	114	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5238	114	3	-	-	PUNCT
ejpam-5238	114	4	cl(y	cl(y	NOUN
ejpam-5238	114	5	−b	−b	NOUN
ejpam-5238	114	6	)	)	PUNCT
ejpam-5238	114	7	)	)	PUNCT
ejpam-5238	115	1	=	=	PUNCT
ejpam-5238	115	2	x	x	X
ejpam-5238	116	1	−	−	ADP
ejpam-5238	116	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	116	3	-	-	PUNCT
ejpam-5238	116	4	int(b	int(b	NOUN
ejpam-5238	116	5	)	)	PUNCT
ejpam-5238	116	6	)	)	PUNCT
ejpam-5238	116	7	and	and	CCONJ
ejpam-5238	116	8	hence	hence	ADV
ejpam-5238	116	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-5238	116	10	-	-	PUNCT
ejpam-5238	116	11	int(b	int(b	NOUN
ejpam-5238	116	12	)	)	PUNCT
ejpam-5238	116	13	)	)	PUNCT
ejpam-5238	117	1	⊆	⊆	X
ejpam-5238	117	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	117	3	-	-	NUM
ejpam-5238	117	4	int(f	int(f	VERB
ejpam-5238	117	5	+	+	ADJ
ejpam-5238	117	6	(	(	PUNCT
ejpam-5238	117	7	σ1σ2	σ1σ2	NUM
ejpam-5238	117	8	-	-	PUNCT
ejpam-5238	117	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	117	10	-	-	PUNCT
ejpam-5238	117	11	int(b	int(b	NOUN
ejpam-5238	117	12	)	)	PUNCT
ejpam-5238	117	13	)	)	PUNCT
ejpam-5238	117	14	)	)	PUNCT
ejpam-5238	117	15	)	)	PUNCT
ejpam-5238	117	16	.	.	PUNCT
ejpam-5238	118	1	(	(	PUNCT
ejpam-5238	118	2	5	5	X
ejpam-5238	118	3	)	)	PUNCT
ejpam-5238	118	4	⇒	⇒	NOUN
ejpam-5238	118	5	(	(	PUNCT
ejpam-5238	118	6	1	1	NUM
ejpam-5238	118	7	):	):	PUNCT
ejpam-5238	118	8	let	let	VERB
ejpam-5238	118	9	x	x	PUNCT
ejpam-5238	118	10	∈	∈	PROPN
ejpam-5238	118	11	x	x	X
ejpam-5238	118	12	and	and	CCONJ
ejpam-5238	118	13	v	v	X
ejpam-5238	118	14	be	be	AUX
ejpam-5238	118	15	any	any	DET
ejpam-5238	118	16	σ1σ2	σ1σ2	NOUN
ejpam-5238	118	17	-	-	ADJ
ejpam-5238	118	18	open	open	ADJ
ejpam-5238	118	19	set	set	NOUN
ejpam-5238	118	20	of	of	ADP
ejpam-5238	118	21	y	y	PRON
ejpam-5238	118	22	such	such	ADJ
ejpam-5238	118	23	that	that	SCONJ
ejpam-5238	118	24	f	f	PROPN
ejpam-5238	118	25	(	(	PUNCT
ejpam-5238	118	26	x	x	X
ejpam-5238	118	27	)	)	PUNCT
ejpam-5238	118	28	⊆	⊆	NUM
ejpam-5238	118	29	v	v	NOUN
ejpam-5238	118	30	.	.	PUNCT
ejpam-5238	119	1	then	then	ADV
ejpam-5238	119	2	,	,	PUNCT
ejpam-5238	119	3	x	x	X
ejpam-5238	119	4	∈	∈	PROPN
ejpam-5238	119	5	f+(v	f+(v	NOUN
ejpam-5238	119	6	)	)	PUNCT
ejpam-5238	120	1	⊆	⊆	X
ejpam-5238	120	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	120	3	-	-	NUM
ejpam-5238	120	4	int(f	int(f	VERB
ejpam-5238	120	5	+	+	ADJ
ejpam-5238	120	6	(	(	PUNCT
ejpam-5238	120	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	120	8	-	-	NUM
ejpam-5238	120	9	cl(v	cl(v	NOUN
ejpam-5238	120	10	)	)	PUNCT
ejpam-5238	120	11	)	)	PUNCT
ejpam-5238	120	12	)	)	PUNCT
ejpam-5238	121	1	and	and	CCONJ
ejpam-5238	121	2	there	there	PRON
ejpam-5238	121	3	exists	exist	VERB
ejpam-5238	121	4	a	a	DET
ejpam-5238	121	5	τ1τ2	τ1τ2	NOUN
ejpam-5238	121	6	-	-	ADJ
ejpam-5238	121	7	open	open	ADJ
ejpam-5238	121	8	set	set	NOUN
ejpam-5238	121	9	u	u	PRON
ejpam-5238	121	10	ofx	ofx	NOUN
ejpam-5238	121	11	containing	contain	VERB
ejpam-5238	121	12	x	x	PUNCT
ejpam-5238	121	13	such	such	ADJ
ejpam-5238	121	14	that	that	SCONJ
ejpam-5238	121	15	u	u	NOUN
ejpam-5238	121	16	⊆	⊆	NUM
ejpam-5238	121	17	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	121	18	-	-	PUNCT
ejpam-5238	121	19	cl(v	cl(v	NOUN
ejpam-5238	121	20	)	)	PUNCT
ejpam-5238	121	21	)	)	PUNCT
ejpam-5238	121	22	.	.	PUNCT
ejpam-5238	122	1	thus	thus	ADV
ejpam-5238	122	2	,	,	PUNCT
ejpam-5238	122	3	f	f	PROPN
ejpam-5238	122	4	(	(	PUNCT
ejpam-5238	122	5	u	u	NOUN
ejpam-5238	122	6	)	)	PUNCT
ejpam-5238	122	7	⊆	⊆	NUM
ejpam-5238	122	8	σ1σ2	σ1σ2	NOUN
ejpam-5238	122	9	-	-	NUM
ejpam-5238	122	10	cl(v	cl(v	NOUN
ejpam-5238	122	11	)	)	PUNCT
ejpam-5238	122	12	and	and	CCONJ
ejpam-5238	122	13	hence	hence	ADV
ejpam-5238	122	14	f	f	PROPN
ejpam-5238	122	15	is	be	AUX
ejpam-5238	122	16	upper	upper	ADJ
ejpam-5238	122	17	weakly	weakly	ADJ
ejpam-5238	122	18	(	(	PUNCT
ejpam-5238	122	19	τ1	τ1	NOUN
ejpam-5238	122	20	,	,	PUNCT
ejpam-5238	122	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	122	22	.	.	PUNCT
ejpam-5238	123	1	(	(	PUNCT
ejpam-5238	123	2	4	4	X
ejpam-5238	123	3	)	)	PUNCT
ejpam-5238	123	4	⇒	⇒	NOUN
ejpam-5238	123	5	(	(	PUNCT
ejpam-5238	123	6	6	6	NUM
ejpam-5238	123	7	)	)	PUNCT
ejpam-5238	123	8	and	and	CCONJ
ejpam-5238	123	9	(	(	PUNCT
ejpam-5238	123	10	6	6	NUM
ejpam-5238	123	11	)	)	PUNCT
ejpam-5238	123	12	⇒	⇒	NOUN
ejpam-5238	123	13	(	(	PUNCT
ejpam-5238	123	14	7	7	NUM
ejpam-5238	123	15	):	):	PUNCT
ejpam-5238	123	16	the	the	DET
ejpam-5238	123	17	proofs	proof	NOUN
ejpam-5238	123	18	are	be	AUX
ejpam-5238	123	19	obvious	obvious	ADJ
ejpam-5238	123	20	.	.	PUNCT
ejpam-5238	124	1	(	(	PUNCT
ejpam-5238	124	2	7	7	X
ejpam-5238	124	3	)	)	PUNCT
ejpam-5238	124	4	⇒	⇒	NOUN
ejpam-5238	124	5	(	(	PUNCT
ejpam-5238	124	6	8)	8)	NUM
ejpam-5238	124	7	:	:	PUNCT
ejpam-5238	124	8	let	let	VERB
ejpam-5238	124	9	k	k	X
ejpam-5238	124	10	be	be	AUX
ejpam-5238	124	11	any	any	DET
ejpam-5238	124	12	(	(	PUNCT
ejpam-5238	124	13	σ1	σ1	NOUN
ejpam-5238	124	14	,	,	PUNCT
ejpam-5238	124	15	σ2)r	σ2)r	NOUN
ejpam-5238	124	16	-	-	PUNCT
ejpam-5238	124	17	closed	close	VERB
ejpam-5238	124	18	set	set	NOUN
ejpam-5238	124	19	of	of	ADP
ejpam-5238	124	20	y	y	PROPN
ejpam-5238	124	21	.	.	PUNCT
ejpam-5238	125	1	thus	thus	ADV
ejpam-5238	125	2	by	by	ADP
ejpam-5238	125	3	(	(	PUNCT
ejpam-5238	125	4	7	7	NUM
ejpam-5238	125	5	)	)	PUNCT
ejpam-5238	125	6	,	,	PUNCT
ejpam-5238	125	7	τ1τ2	τ1τ2	NOUN
ejpam-5238	125	8	-	-	ADJ
ejpam-5238	125	9	cl(f	cl(f	NOUN
ejpam-5238	125	10	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	125	11	-	-	PUNCT
ejpam-5238	125	12	int(k	int(k	NUM
ejpam-5238	125	13	)	)	PUNCT
ejpam-5238	125	14	)	)	PUNCT
ejpam-5238	125	15	)	)	PUNCT
ejpam-5238	126	1	⊆	⊆	X
ejpam-5238	126	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-5238	126	3	-	-	PUNCT
ejpam-5238	126	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	126	5	-	-	PUNCT
ejpam-5238	126	6	int(k	int(k	NOUN
ejpam-5238	126	7	)	)	PUNCT
ejpam-5238	126	8	)	)	PUNCT
ejpam-5238	126	9	)	)	PUNCT
ejpam-5238	127	1	=	=	SYM
ejpam-5238	127	2	f−(k	f−(k	PROPN
ejpam-5238	127	3	)	)	PUNCT
ejpam-5238	127	4	.	.	PUNCT
ejpam-5238	128	1	(	(	PUNCT
ejpam-5238	128	2	8)	8)	NUM
ejpam-5238	128	3	⇒	⇒	NOUN
ejpam-5238	128	4	(	(	PUNCT
ejpam-5238	128	5	3	3	NUM
ejpam-5238	128	6	):	):	PUNCT
ejpam-5238	128	7	letk	letk	ADJ
ejpam-5238	128	8	be	be	AUX
ejpam-5238	128	9	any	any	DET
ejpam-5238	128	10	σ1σ2	σ1σ2	NUM
ejpam-5238	128	11	-	-	PUNCT
ejpam-5238	128	12	closed	closed	ADJ
ejpam-5238	128	13	set	set	NOUN
ejpam-5238	128	14	of	of	ADP
ejpam-5238	128	15	y	y	PROPN
ejpam-5238	128	16	.	.	PUNCT
ejpam-5238	129	1	then	then	ADV
ejpam-5238	129	2	,	,	PUNCT
ejpam-5238	129	3	σ1σ2	σ1σ2	X
ejpam-5238	129	4	-	-	PUNCT
ejpam-5238	129	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	129	6	-	-	PUNCT
ejpam-5238	129	7	int(k	int(k	NOUN
ejpam-5238	129	8	)	)	PUNCT
ejpam-5238	129	9	)	)	PUNCT
ejpam-5238	129	10	is	be	AUX
ejpam-5238	129	11	(	(	PUNCT
ejpam-5238	129	12	σ1	σ1	PROPN
ejpam-5238	129	13	,	,	PUNCT
ejpam-5238	129	14	σ2)rclosed	σ2)rclose	VERB
ejpam-5238	129	15	in	in	ADP
ejpam-5238	129	16	y	y	PROPN
ejpam-5238	129	17	and	and	CCONJ
ejpam-5238	129	18	σ1σ2	σ1σ2	NOUN
ejpam-5238	129	19	-	-	PUNCT
ejpam-5238	129	20	int(σ1σ2	int(σ1σ2	ADV
ejpam-5238	129	21	-	-	PUNCT
ejpam-5238	129	22	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5238	129	23	-	-	PUNCT
ejpam-5238	129	24	int(k	int(k	NOUN
ejpam-5238	129	25	)	)	PUNCT
ejpam-5238	129	26	)	)	PUNCT
ejpam-5238	129	27	)	)	PUNCT
ejpam-5238	130	1	=	=	PUNCT
ejpam-5238	130	2	σ1σ2	σ1σ2	X
ejpam-5238	130	3	-	-	PUNCT
ejpam-5238	130	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	130	5	-	-	PUNCT
ejpam-5238	130	6	cl(k	cl(k	NUM
ejpam-5238	130	7	)	)	PUNCT
ejpam-5238	130	8	)	)	PUNCT
ejpam-5238	131	1	=	=	PUNCT
ejpam-5238	131	2	σ1σ2	σ1σ2	X
ejpam-5238	131	3	-	-	PUNCT
ejpam-5238	131	4	int(k	int(k	NOUN
ejpam-5238	131	5	)	)	PUNCT
ejpam-5238	131	6	.	.	PUNCT
ejpam-5238	132	1	by	by	ADP
ejpam-5238	132	2	(	(	PUNCT
ejpam-5238	132	3	8)	8)	NUM
ejpam-5238	132	4	,	,	PUNCT
ejpam-5238	132	5	τ1τ2	τ1τ2	NOUN
ejpam-5238	132	6	-	-	NOUN
ejpam-5238	132	7	cl(f	cl(f	NOUN
ejpam-5238	132	8	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	132	9	-	-	PUNCT
ejpam-5238	132	10	int(k	int(k	NUM
ejpam-5238	132	11	)	)	PUNCT
ejpam-5238	132	12	)	)	PUNCT
ejpam-5238	132	13	)	)	PUNCT
ejpam-5238	133	1	=	=	PUNCT
ejpam-5238	133	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	133	3	-	-	ADJ
ejpam-5238	133	4	cl(f	cl(f	NOUN
ejpam-5238	133	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	133	6	-	-	PUNCT
ejpam-5238	133	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-5238	133	8	-	-	PUNCT
ejpam-5238	133	9	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5238	133	10	-	-	PUNCT
ejpam-5238	133	11	int(k	int(k	NOUN
ejpam-5238	133	12	)	)	PUNCT
ejpam-5238	133	13	)	)	PUNCT
ejpam-5238	133	14	)	)	PUNCT
ejpam-5238	133	15	)	)	PUNCT
ejpam-5238	133	16	)	)	PUNCT
ejpam-5238	133	17	⊆	⊆	X
ejpam-5238	133	18	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-5238	133	19	-	-	PUNCT
ejpam-5238	133	20	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	133	21	-	-	PUNCT
ejpam-5238	133	22	int(k	int(k	NOUN
ejpam-5238	133	23	)	)	PUNCT
ejpam-5238	133	24	)	)	PUNCT
ejpam-5238	133	25	)	)	PUNCT
ejpam-5238	134	1	⊆	⊆	NUM
ejpam-5238	134	2	f−(k	f−(k	PROPN
ejpam-5238	134	3	)	)	PUNCT
ejpam-5238	134	4	.	.	PUNCT
ejpam-5238	135	1	definition	definition	NOUN
ejpam-5238	135	2	2	2	NUM
ejpam-5238	135	3	.	.	PUNCT
ejpam-5238	135	4	a	a	DET
ejpam-5238	135	5	multifunction	multifunction	NOUN
ejpam-5238	135	6	f	f	NOUN
ejpam-5238	135	7	:	:	PUNCT
ejpam-5238	135	8	(	(	PUNCT
ejpam-5238	135	9	x	x	NOUN
ejpam-5238	135	10	,	,	PUNCT
ejpam-5238	135	11	τ1	τ1	NOUN
ejpam-5238	135	12	,	,	PUNCT
ejpam-5238	135	13	τ2	τ2	NOUN
ejpam-5238	135	14	)	)	PUNCT
ejpam-5238	135	15	→	→	SYM
ejpam-5238	135	16	(	(	PUNCT
ejpam-5238	135	17	y	y	PROPN
ejpam-5238	135	18	,	,	PUNCT
ejpam-5238	135	19	σ1	σ1	PROPN
ejpam-5238	135	20	,	,	PUNCT
ejpam-5238	135	21	σ2	σ2	PROPN
ejpam-5238	135	22	)	)	PUNCT
ejpam-5238	135	23	is	be	AUX
ejpam-5238	135	24	said	say	VERB
ejpam-5238	135	25	to	to	PART
ejpam-5238	135	26	be	be	AUX
ejpam-5238	135	27	lower	low	ADJ
ejpam-5238	135	28	weakly	weakly	ADJ
ejpam-5238	135	29	(	(	PUNCT
ejpam-5238	135	30	τ1	τ1	NOUN
ejpam-5238	135	31	,	,	PUNCT
ejpam-5238	135	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	135	33	if	if	SCONJ
ejpam-5238	135	34	for	for	ADP
ejpam-5238	135	35	each	each	DET
ejpam-5238	135	36	x	x	SYM
ejpam-5238	135	37	∈	∈	PROPN
ejpam-5238	135	38	x	x	X
ejpam-5238	135	39	and	and	CCONJ
ejpam-5238	135	40	each	each	DET
ejpam-5238	135	41	σ1σ2	σ1σ2	VERB
ejpam-5238	135	42	-	-	ADJ
ejpam-5238	135	43	open	open	ADJ
ejpam-5238	135	44	set	set	NOUN
ejpam-5238	135	45	v	v	NOUN
ejpam-5238	135	46	of	of	ADP
ejpam-5238	135	47	y	y	PRON
ejpam-5238	135	48	such	such	ADJ
ejpam-5238	135	49	that	that	SCONJ
ejpam-5238	135	50	f	f	PROPN
ejpam-5238	135	51	(	(	PUNCT
ejpam-5238	135	52	x)∩v	x)∩v	PROPN
ejpam-5238	135	53	̸=	̸=	PROPN
ejpam-5238	135	54	∅	∅	NOUN
ejpam-5238	135	55	,	,	PUNCT
ejpam-5238	135	56	there	there	PRON
ejpam-5238	135	57	exists	exist	VERB
ejpam-5238	135	58	a	a	DET
ejpam-5238	135	59	τ1τ2	τ1τ2	NOUN
ejpam-5238	135	60	-	-	ADJ
ejpam-5238	135	61	open	open	ADJ
ejpam-5238	135	62	set	set	ADJ
ejpam-5238	135	63	u	u	NOUN
ejpam-5238	135	64	of	of	ADP
ejpam-5238	135	65	x	x	PUNCT
ejpam-5238	135	66	containing	contain	VERB
ejpam-5238	135	67	x	x	PUNCT
ejpam-5238	136	1	such	such	ADJ
ejpam-5238	136	2	that	that	SCONJ
ejpam-5238	136	3	σ1σ2	σ1σ2	NOUN
ejpam-5238	136	4	-	-	NUM
ejpam-5238	136	5	cl(v	cl(v	PUNCT
ejpam-5238	136	6	)	)	PUNCT
ejpam-5238	136	7	∩f	∩f	NOUN
ejpam-5238	136	8	(	(	PUNCT
ejpam-5238	136	9	z	z	X
ejpam-5238	136	10	)	)	PUNCT
ejpam-5238	136	11	̸=	̸=	NOUN
ejpam-5238	136	12	∅	∅	NOUN
ejpam-5238	136	13	for	for	ADP
ejpam-5238	136	14	each	each	DET
ejpam-5238	136	15	z	z	NOUN
ejpam-5238	136	16	∈	∈	PROPN
ejpam-5238	136	17	u	u	PROPN
ejpam-5238	136	18	.	.	PUNCT
ejpam-5238	136	19	m.	m.	NOUN
ejpam-5238	136	20	thongmoon	thongmoon	PROPN
ejpam-5238	136	21	,	,	PUNCT
ejpam-5238	136	22	s.	s.	PROPN
ejpam-5238	136	23	sompong	sompong	PROPN
ejpam-5238	136	24	,	,	PUNCT
ejpam-5238	136	25	c.	c.	PROPN
ejpam-5238	136	26	boonpok	boonpok	PROPN
ejpam-5238	136	27	/	/	SYM
ejpam-5238	136	28	eur	eur	PROPN
ejpam-5238	136	29	.	.	PUNCT
ejpam-5238	137	1	j.	j.	PROPN
ejpam-5238	137	2	pure	pure	PROPN
ejpam-5238	137	3	appl	appl	PROPN
ejpam-5238	137	4	.	.	PROPN
ejpam-5238	137	5	math	math	PROPN
ejpam-5238	137	6	,	,	PUNCT
ejpam-5238	137	7	17	17	NUM
ejpam-5238	137	8	(	(	PUNCT
ejpam-5238	137	9	3	3	NUM
ejpam-5238	137	10	)	)	PUNCT
ejpam-5238	137	11	(	(	PUNCT
ejpam-5238	137	12	2024	2024	NUM
ejpam-5238	137	13	)	)	PUNCT
ejpam-5238	137	14	,	,	PUNCT
ejpam-5238	137	15	1705	1705	NUM
ejpam-5238	137	16	-	-	SYM
ejpam-5238	137	17	1716	1716	NUM
ejpam-5238	137	18	1709	1709	NUM
ejpam-5238	137	19	theorem	theorem	NOUN
ejpam-5238	137	20	2	2	NUM
ejpam-5238	137	21	.	.	X
ejpam-5238	137	22	for	for	ADP
ejpam-5238	137	23	a	a	DET
ejpam-5238	137	24	multifunction	multifunction	NOUN
ejpam-5238	137	25	f	f	NOUN
ejpam-5238	137	26	:	:	PUNCT
ejpam-5238	137	27	(	(	PUNCT
ejpam-5238	137	28	x	x	NOUN
ejpam-5238	137	29	,	,	PUNCT
ejpam-5238	137	30	τ1	τ1	NOUN
ejpam-5238	137	31	,	,	PUNCT
ejpam-5238	137	32	τ2	τ2	NOUN
ejpam-5238	137	33	)	)	PUNCT
ejpam-5238	137	34	→	→	SYM
ejpam-5238	137	35	(	(	PUNCT
ejpam-5238	137	36	y	y	PROPN
ejpam-5238	137	37	,	,	PUNCT
ejpam-5238	137	38	σ1	σ1	PROPN
ejpam-5238	137	39	,	,	PUNCT
ejpam-5238	137	40	σ2	σ2	NOUN
ejpam-5238	137	41	)	)	PUNCT
ejpam-5238	137	42	,	,	PUNCT
ejpam-5238	137	43	the	the	DET
ejpam-5238	137	44	following	follow	VERB
ejpam-5238	137	45	properties	property	NOUN
ejpam-5238	137	46	are	be	AUX
ejpam-5238	137	47	equivalent	equivalent	ADJ
ejpam-5238	137	48	:	:	PUNCT
ejpam-5238	137	49	(	(	PUNCT
ejpam-5238	137	50	1	1	X
ejpam-5238	137	51	)	)	PUNCT
ejpam-5238	137	52	f	f	PROPN
ejpam-5238	137	53	is	be	AUX
ejpam-5238	137	54	lower	low	ADJ
ejpam-5238	137	55	weakly	weakly	ADJ
ejpam-5238	137	56	(	(	PUNCT
ejpam-5238	137	57	τ1	τ1	NOUN
ejpam-5238	137	58	,	,	PUNCT
ejpam-5238	137	59	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	137	60	;	;	PUNCT
ejpam-5238	137	61	(	(	PUNCT
ejpam-5238	137	62	2	2	X
ejpam-5238	137	63	)	)	PUNCT
ejpam-5238	137	64	f−(v	f−(v	NOUN
ejpam-5238	137	65	)	)	PUNCT
ejpam-5238	137	66	⊆	⊆	NUM
ejpam-5238	137	67	τ1τ2	τ1τ2	NOUN
ejpam-5238	137	68	-	-	NUM
ejpam-5238	137	69	int(f	int(f	PRON
ejpam-5238	137	70	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	137	71	-	-	NOUN
ejpam-5238	137	72	cl(v	cl(v	NOUN
ejpam-5238	137	73	)	)	PUNCT
ejpam-5238	137	74	)	)	PUNCT
ejpam-5238	137	75	)	)	PUNCT
ejpam-5238	137	76	for	for	ADP
ejpam-5238	137	77	every	every	DET
ejpam-5238	137	78	σ1σ2	σ1σ2	NOUN
ejpam-5238	137	79	-	-	ADJ
ejpam-5238	137	80	open	open	ADJ
ejpam-5238	137	81	set	set	NOUN
ejpam-5238	137	82	v	v	NOUN
ejpam-5238	137	83	of	of	ADP
ejpam-5238	137	84	y	y	PROPN
ejpam-5238	137	85	;	;	PUNCT
ejpam-5238	137	86	(	(	PUNCT
ejpam-5238	137	87	3	3	X
ejpam-5238	137	88	)	)	PUNCT
ejpam-5238	137	89	τ1τ2	τ1τ2	NOUN
ejpam-5238	137	90	-	-	NOUN
ejpam-5238	137	91	cl(f	cl(f	NOUN
ejpam-5238	137	92	+	+	NOUN
ejpam-5238	137	93	(	(	PUNCT
ejpam-5238	137	94	σ1σ2	σ1σ2	NUM
ejpam-5238	137	95	-	-	PUNCT
ejpam-5238	137	96	int(k	int(k	NUM
ejpam-5238	137	97	)	)	PUNCT
ejpam-5238	137	98	)	)	PUNCT
ejpam-5238	137	99	)	)	PUNCT
ejpam-5238	138	1	⊆	⊆	NUM
ejpam-5238	138	2	f+(k	f+(k	NOUN
ejpam-5238	138	3	)	)	PUNCT
ejpam-5238	138	4	for	for	ADP
ejpam-5238	138	5	every	every	DET
ejpam-5238	138	6	σ1σ2	σ1σ2	NUM
ejpam-5238	138	7	-	-	PUNCT
ejpam-5238	138	8	closed	closed	ADJ
ejpam-5238	138	9	set	set	NOUN
ejpam-5238	138	10	k	k	PROPN
ejpam-5238	138	11	of	of	ADP
ejpam-5238	138	12	y	y	PROPN
ejpam-5238	138	13	;	;	PUNCT
ejpam-5238	138	14	(	(	PUNCT
ejpam-5238	138	15	4	4	X
ejpam-5238	138	16	)	)	PUNCT
ejpam-5238	138	17	τ1τ2	τ1τ2	NOUN
ejpam-5238	138	18	-	-	NOUN
ejpam-5238	138	19	cl(f	cl(f	NOUN
ejpam-5238	138	20	+	+	NOUN
ejpam-5238	138	21	(	(	PUNCT
ejpam-5238	138	22	σ1σ2	σ1σ2	NUM
ejpam-5238	138	23	-	-	PUNCT
ejpam-5238	138	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	138	25	-	-	PUNCT
ejpam-5238	138	26	cl(b	cl(b	NOUN
ejpam-5238	138	27	)	)	PUNCT
ejpam-5238	138	28	)	)	PUNCT
ejpam-5238	138	29	)	)	PUNCT
ejpam-5238	138	30	)	)	PUNCT
ejpam-5238	139	1	⊆	⊆	X
ejpam-5238	139	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	139	3	-	-	PUNCT
ejpam-5238	139	4	cl(b	cl(b	NOUN
ejpam-5238	139	5	)	)	PUNCT
ejpam-5238	139	6	)	)	PUNCT
ejpam-5238	139	7	for	for	ADP
ejpam-5238	139	8	every	every	DET
ejpam-5238	139	9	subset	subset	NOUN
ejpam-5238	139	10	b	b	PROPN
ejpam-5238	139	11	of	of	ADP
ejpam-5238	139	12	y	y	PROPN
ejpam-5238	139	13	;	;	PUNCT
ejpam-5238	139	14	(	(	PUNCT
ejpam-5238	139	15	5	5	X
ejpam-5238	139	16	)	)	PUNCT
ejpam-5238	139	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	139	18	-	-	PUNCT
ejpam-5238	139	19	int(b	int(b	NOUN
ejpam-5238	139	20	)	)	PUNCT
ejpam-5238	139	21	)	)	PUNCT
ejpam-5238	140	1	⊆	⊆	X
ejpam-5238	140	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	140	3	-	-	NUM
ejpam-5238	140	4	int(f	int(f	PRON
ejpam-5238	140	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	140	6	-	-	PUNCT
ejpam-5238	140	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	140	8	-	-	PUNCT
ejpam-5238	140	9	int(b	int(b	NOUN
ejpam-5238	140	10	)	)	PUNCT
ejpam-5238	140	11	)	)	PUNCT
ejpam-5238	140	12	)	)	PUNCT
ejpam-5238	140	13	)	)	PUNCT
ejpam-5238	140	14	for	for	ADP
ejpam-5238	140	15	every	every	DET
ejpam-5238	140	16	subset	subset	NOUN
ejpam-5238	140	17	b	b	PROPN
ejpam-5238	140	18	of	of	ADP
ejpam-5238	140	19	y	y	PROPN
ejpam-5238	140	20	;	;	PUNCT
ejpam-5238	140	21	(	(	PUNCT
ejpam-5238	140	22	6	6	X
ejpam-5238	140	23	)	)	PUNCT
ejpam-5238	140	24	τ1τ2	τ1τ2	NOUN
ejpam-5238	140	25	-	-	NOUN
ejpam-5238	140	26	cl(f	cl(f	NOUN
ejpam-5238	140	27	+	+	NOUN
ejpam-5238	140	28	(	(	PUNCT
ejpam-5238	140	29	σ1σ2	σ1σ2	NUM
ejpam-5238	140	30	-	-	PUNCT
ejpam-5238	140	31	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	140	32	-	-	PUNCT
ejpam-5238	140	33	cl(v	cl(v	NOUN
ejpam-5238	140	34	)	)	PUNCT
ejpam-5238	140	35	)	)	PUNCT
ejpam-5238	140	36	)	)	PUNCT
ejpam-5238	140	37	)	)	PUNCT
ejpam-5238	140	38	⊆	⊆	X
ejpam-5238	140	39	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	140	40	-	-	PUNCT
ejpam-5238	140	41	cl(v	cl(v	NOUN
ejpam-5238	140	42	)	)	PUNCT
ejpam-5238	140	43	)	)	PUNCT
ejpam-5238	140	44	for	for	ADP
ejpam-5238	140	45	every	every	DET
ejpam-5238	140	46	σ1σ2	σ1σ2	NOUN
ejpam-5238	140	47	-	-	ADJ
ejpam-5238	140	48	open	open	ADJ
ejpam-5238	140	49	set	set	NOUN
ejpam-5238	140	50	v	v	NOUN
ejpam-5238	140	51	of	of	ADP
ejpam-5238	140	52	y	y	PROPN
ejpam-5238	140	53	;	;	PUNCT
ejpam-5238	140	54	(	(	PUNCT
ejpam-5238	140	55	7	7	X
ejpam-5238	140	56	)	)	PUNCT
ejpam-5238	140	57	τ1τ2	τ1τ2	NOUN
ejpam-5238	140	58	-	-	NOUN
ejpam-5238	140	59	cl(f	cl(f	NOUN
ejpam-5238	140	60	+	+	NOUN
ejpam-5238	140	61	(	(	PUNCT
ejpam-5238	140	62	v	v	NOUN
ejpam-5238	140	63	)	)	PUNCT
ejpam-5238	140	64	)	)	PUNCT
ejpam-5238	140	65	⊆	⊆	NUM
ejpam-5238	140	66	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	140	67	-	-	PUNCT
ejpam-5238	140	68	cl(v	cl(v	NOUN
ejpam-5238	140	69	)	)	PUNCT
ejpam-5238	140	70	)	)	PUNCT
ejpam-5238	140	71	for	for	ADP
ejpam-5238	140	72	every	every	DET
ejpam-5238	140	73	σ1σ2	σ1σ2	NOUN
ejpam-5238	140	74	-	-	ADJ
ejpam-5238	140	75	open	open	ADJ
ejpam-5238	140	76	set	set	NOUN
ejpam-5238	140	77	v	v	NOUN
ejpam-5238	140	78	of	of	ADP
ejpam-5238	140	79	y	y	PROPN
ejpam-5238	140	80	;	;	PUNCT
ejpam-5238	140	81	(	(	PUNCT
ejpam-5238	140	82	8)	8)	NUM
ejpam-5238	140	83	τ1τ2	τ1τ2	NOUN
ejpam-5238	140	84	-	-	NOUN
ejpam-5238	140	85	cl(f	cl(f	NOUN
ejpam-5238	140	86	+	+	NOUN
ejpam-5238	140	87	(	(	PUNCT
ejpam-5238	140	88	σ1σ2	σ1σ2	NUM
ejpam-5238	140	89	-	-	PUNCT
ejpam-5238	140	90	int(k	int(k	NUM
ejpam-5238	140	91	)	)	PUNCT
ejpam-5238	140	92	)	)	PUNCT
ejpam-5238	140	93	)	)	PUNCT
ejpam-5238	140	94	⊆	⊆	NUM
ejpam-5238	140	95	f+(k	f+(k	NOUN
ejpam-5238	140	96	)	)	PUNCT
ejpam-5238	140	97	for	for	ADP
ejpam-5238	140	98	every	every	DET
ejpam-5238	140	99	(	(	PUNCT
ejpam-5238	140	100	σ1	σ1	PROPN
ejpam-5238	140	101	,	,	PUNCT
ejpam-5238	140	102	σ2)r	σ2)r	NOUN
ejpam-5238	140	103	-	-	PUNCT
ejpam-5238	140	104	closed	close	VERB
ejpam-5238	140	105	set	set	ADJ
ejpam-5238	140	106	k	k	PROPN
ejpam-5238	140	107	of	of	ADP
ejpam-5238	140	108	y	y	PROPN
ejpam-5238	140	109	.	.	PUNCT
ejpam-5238	141	1	proof	proof	NOUN
ejpam-5238	141	2	.	.	PUNCT
ejpam-5238	142	1	the	the	DET
ejpam-5238	142	2	proof	proof	NOUN
ejpam-5238	142	3	is	be	AUX
ejpam-5238	142	4	similar	similar	ADJ
ejpam-5238	142	5	to	to	ADP
ejpam-5238	142	6	that	that	PRON
ejpam-5238	142	7	of	of	ADP
ejpam-5238	142	8	theorem	theorem	NOUN
ejpam-5238	142	9	1	1	NUM
ejpam-5238	142	10	.	.	PUNCT
ejpam-5238	142	11	definition	definition	NOUN
ejpam-5238	142	12	3	3	NUM
ejpam-5238	142	13	.	.	PUNCT
ejpam-5238	143	1	[	[	X
ejpam-5238	143	2	11	11	NUM
ejpam-5238	143	3	]	]	PUNCT
ejpam-5238	143	4	a	a	DET
ejpam-5238	143	5	function	function	NOUN
ejpam-5238	143	6	f	f	NOUN
ejpam-5238	143	7	:	:	PUNCT
ejpam-5238	143	8	(	(	PUNCT
ejpam-5238	143	9	x	x	NOUN
ejpam-5238	143	10	,	,	PUNCT
ejpam-5238	143	11	τ1	τ1	NOUN
ejpam-5238	143	12	,	,	PUNCT
ejpam-5238	143	13	τ2	τ2	NOUN
ejpam-5238	143	14	)	)	PUNCT
ejpam-5238	143	15	→	→	SYM
ejpam-5238	143	16	(	(	PUNCT
ejpam-5238	143	17	y	y	PROPN
ejpam-5238	143	18	,	,	PUNCT
ejpam-5238	143	19	σ1	σ1	PROPN
ejpam-5238	143	20	,	,	PUNCT
ejpam-5238	143	21	σ2	σ2	PROPN
ejpam-5238	143	22	)	)	PUNCT
ejpam-5238	143	23	is	be	AUX
ejpam-5238	143	24	said	say	VERB
ejpam-5238	143	25	to	to	PART
ejpam-5238	143	26	be	be	AUX
ejpam-5238	143	27	weakly	weakly	ADJ
ejpam-5238	143	28	(	(	PUNCT
ejpam-5238	143	29	τ1	τ1	NOUN
ejpam-5238	143	30	,	,	PUNCT
ejpam-5238	143	31	τ2)continuous	τ2)continuous	ADJ
ejpam-5238	143	32	at	at	ADP
ejpam-5238	143	33	a	a	DET
ejpam-5238	143	34	point	point	NOUN
ejpam-5238	143	35	x	x	SYM
ejpam-5238	143	36	∈	∈	NOUN
ejpam-5238	143	37	x	x	PUNCT
ejpam-5238	143	38	if	if	SCONJ
ejpam-5238	143	39	for	for	ADP
ejpam-5238	143	40	each	each	DET
ejpam-5238	143	41	σ1σ2	σ1σ2	VERB
ejpam-5238	143	42	-	-	ADJ
ejpam-5238	143	43	open	open	ADJ
ejpam-5238	143	44	set	set	NOUN
ejpam-5238	143	45	v	v	NOUN
ejpam-5238	143	46	of	of	ADP
ejpam-5238	143	47	y	y	NOUN
ejpam-5238	143	48	containing	contain	VERB
ejpam-5238	143	49	f(x	f(x	PROPN
ejpam-5238	143	50	)	)	PUNCT
ejpam-5238	143	51	,	,	PUNCT
ejpam-5238	143	52	there	there	PRON
ejpam-5238	143	53	exists	exist	VERB
ejpam-5238	143	54	a	a	DET
ejpam-5238	143	55	τ1τ2	τ1τ2	NOUN
ejpam-5238	143	56	-	-	ADJ
ejpam-5238	143	57	open	open	ADJ
ejpam-5238	143	58	set	set	ADJ
ejpam-5238	143	59	u	u	NOUN
ejpam-5238	143	60	of	of	ADP
ejpam-5238	143	61	x	x	PUNCT
ejpam-5238	143	62	containing	contain	VERB
ejpam-5238	143	63	x	x	PUNCT
ejpam-5238	143	64	such	such	ADJ
ejpam-5238	143	65	that	that	DET
ejpam-5238	143	66	f(u	f(u	PROPN
ejpam-5238	143	67	)	)	PUNCT
ejpam-5238	143	68	⊆	⊆	NUM
ejpam-5238	143	69	σ1σ2	σ1σ2	NOUN
ejpam-5238	143	70	-	-	NUM
ejpam-5238	143	71	cl(v	cl(v	NOUN
ejpam-5238	143	72	)	)	PUNCT
ejpam-5238	143	73	.	.	PUNCT
ejpam-5238	144	1	a	a	DET
ejpam-5238	144	2	function	function	NOUN
ejpam-5238	144	3	f	f	NOUN
ejpam-5238	144	4	:	:	PUNCT
ejpam-5238	144	5	(	(	PUNCT
ejpam-5238	144	6	x	x	NOUN
ejpam-5238	144	7	,	,	PUNCT
ejpam-5238	144	8	τ1	τ1	NOUN
ejpam-5238	144	9	,	,	PUNCT
ejpam-5238	144	10	τ2	τ2	NOUN
ejpam-5238	144	11	)	)	PUNCT
ejpam-5238	144	12	→	→	SYM
ejpam-5238	144	13	(	(	PUNCT
ejpam-5238	144	14	y	y	PROPN
ejpam-5238	144	15	,	,	PUNCT
ejpam-5238	144	16	σ1	σ1	PROPN
ejpam-5238	144	17	,	,	PUNCT
ejpam-5238	144	18	σ2	σ2	PROPN
ejpam-5238	144	19	)	)	PUNCT
ejpam-5238	144	20	is	be	AUX
ejpam-5238	144	21	said	say	VERB
ejpam-5238	144	22	to	to	PART
ejpam-5238	144	23	be	be	AUX
ejpam-5238	144	24	weakly	weakly	ADJ
ejpam-5238	144	25	(	(	PUNCT
ejpam-5238	144	26	τ1	τ1	NOUN
ejpam-5238	144	27	,	,	PUNCT
ejpam-5238	144	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	144	29	if	if	SCONJ
ejpam-5238	144	30	f	f	PROPN
ejpam-5238	144	31	has	have	VERB
ejpam-5238	144	32	this	this	DET
ejpam-5238	144	33	property	property	NOUN
ejpam-5238	144	34	at	at	ADP
ejpam-5238	144	35	each	each	DET
ejpam-5238	144	36	point	point	NOUN
ejpam-5238	144	37	of	of	ADP
ejpam-5238	144	38	x.	x.	PROPN
ejpam-5238	144	39	corollary	corollary	NOUN
ejpam-5238	144	40	1	1	NUM
ejpam-5238	144	41	.	.	PUNCT
ejpam-5238	145	1	for	for	ADP
ejpam-5238	145	2	a	a	DET
ejpam-5238	145	3	function	function	NOUN
ejpam-5238	145	4	f	f	NOUN
ejpam-5238	145	5	:	:	PUNCT
ejpam-5238	145	6	(	(	PUNCT
ejpam-5238	145	7	x	x	NOUN
ejpam-5238	145	8	,	,	PUNCT
ejpam-5238	145	9	τ1	τ1	NOUN
ejpam-5238	145	10	,	,	PUNCT
ejpam-5238	145	11	τ2	τ2	NOUN
ejpam-5238	145	12	)	)	PUNCT
ejpam-5238	145	13	→	→	SYM
ejpam-5238	145	14	(	(	PUNCT
ejpam-5238	145	15	y	y	PROPN
ejpam-5238	145	16	,	,	PUNCT
ejpam-5238	145	17	σ1	σ1	PROPN
ejpam-5238	145	18	,	,	PUNCT
ejpam-5238	145	19	σ2	σ2	NOUN
ejpam-5238	145	20	)	)	PUNCT
ejpam-5238	145	21	,	,	PUNCT
ejpam-5238	145	22	the	the	DET
ejpam-5238	145	23	following	follow	VERB
ejpam-5238	145	24	properties	property	NOUN
ejpam-5238	145	25	are	be	AUX
ejpam-5238	145	26	equivalent	equivalent	ADJ
ejpam-5238	145	27	:	:	PUNCT
ejpam-5238	145	28	(	(	PUNCT
ejpam-5238	145	29	1	1	X
ejpam-5238	145	30	)	)	PUNCT
ejpam-5238	145	31	f	f	PROPN
ejpam-5238	145	32	is	be	AUX
ejpam-5238	145	33	weakly	weakly	ADJ
ejpam-5238	145	34	(	(	PUNCT
ejpam-5238	145	35	τ1	τ1	NOUN
ejpam-5238	145	36	,	,	PUNCT
ejpam-5238	145	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	145	38	;	;	PUNCT
ejpam-5238	145	39	(	(	PUNCT
ejpam-5238	145	40	2	2	X
ejpam-5238	145	41	)	)	PUNCT
ejpam-5238	145	42	f−1(v	f−1(v	NOUN
ejpam-5238	145	43	)	)	PUNCT
ejpam-5238	146	1	⊆	⊆	X
ejpam-5238	146	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	146	3	-	-	NUM
ejpam-5238	146	4	int(f	int(f	PRON
ejpam-5238	146	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	146	6	-	-	PUNCT
ejpam-5238	146	7	cl(v	cl(v	NOUN
ejpam-5238	146	8	)	)	PUNCT
ejpam-5238	146	9	)	)	PUNCT
ejpam-5238	146	10	)	)	PUNCT
ejpam-5238	146	11	for	for	ADP
ejpam-5238	146	12	every	every	DET
ejpam-5238	146	13	σ1σ2	σ1σ2	NOUN
ejpam-5238	146	14	-	-	ADJ
ejpam-5238	146	15	open	open	ADJ
ejpam-5238	146	16	set	set	NOUN
ejpam-5238	146	17	v	v	NOUN
ejpam-5238	146	18	of	of	ADP
ejpam-5238	146	19	y	y	PROPN
ejpam-5238	146	20	;	;	PUNCT
ejpam-5238	146	21	(	(	PUNCT
ejpam-5238	146	22	3	3	X
ejpam-5238	146	23	)	)	PUNCT
ejpam-5238	146	24	τ1τ2	τ1τ2	NOUN
ejpam-5238	146	25	-	-	NOUN
ejpam-5238	146	26	cl(f	cl(f	NOUN
ejpam-5238	146	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	146	28	-	-	PUNCT
ejpam-5238	146	29	int(k	int(k	NUM
ejpam-5238	146	30	)	)	PUNCT
ejpam-5238	146	31	)	)	PUNCT
ejpam-5238	146	32	)	)	PUNCT
ejpam-5238	147	1	⊆	⊆	NUM
ejpam-5238	147	2	f−1(k	f−1(k	PROPN
ejpam-5238	147	3	)	)	PUNCT
ejpam-5238	147	4	for	for	ADP
ejpam-5238	147	5	every	every	DET
ejpam-5238	147	6	σ1σ2	σ1σ2	NUM
ejpam-5238	147	7	-	-	PUNCT
ejpam-5238	147	8	closed	closed	ADJ
ejpam-5238	147	9	set	set	NOUN
ejpam-5238	147	10	k	k	PROPN
ejpam-5238	147	11	of	of	ADP
ejpam-5238	147	12	y	y	PROPN
ejpam-5238	147	13	;	;	PUNCT
ejpam-5238	147	14	(	(	PUNCT
ejpam-5238	147	15	4	4	X
ejpam-5238	147	16	)	)	PUNCT
ejpam-5238	147	17	τ1τ2	τ1τ2	NOUN
ejpam-5238	147	18	-	-	NOUN
ejpam-5238	147	19	cl(f	cl(f	NOUN
ejpam-5238	147	20	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	147	21	-	-	PUNCT
ejpam-5238	147	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	147	23	-	-	PUNCT
ejpam-5238	147	24	cl(b	cl(b	NOUN
ejpam-5238	147	25	)	)	PUNCT
ejpam-5238	147	26	)	)	PUNCT
ejpam-5238	147	27	)	)	PUNCT
ejpam-5238	147	28	)	)	PUNCT
ejpam-5238	148	1	⊆	⊆	NUM
ejpam-5238	148	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	148	3	-	-	PUNCT
ejpam-5238	148	4	cl(b	cl(b	NOUN
ejpam-5238	148	5	)	)	PUNCT
ejpam-5238	148	6	)	)	PUNCT
ejpam-5238	148	7	for	for	ADP
ejpam-5238	148	8	every	every	DET
ejpam-5238	148	9	subset	subset	NOUN
ejpam-5238	148	10	b	b	PROPN
ejpam-5238	148	11	of	of	ADP
ejpam-5238	148	12	y	y	PROPN
ejpam-5238	148	13	;	;	PUNCT
ejpam-5238	148	14	(	(	PUNCT
ejpam-5238	148	15	5	5	X
ejpam-5238	148	16	)	)	PUNCT
ejpam-5238	148	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	148	18	-	-	PUNCT
ejpam-5238	148	19	int(b	int(b	NOUN
ejpam-5238	148	20	)	)	PUNCT
ejpam-5238	148	21	)	)	PUNCT
ejpam-5238	149	1	⊆	⊆	X
ejpam-5238	149	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	149	3	-	-	NUM
ejpam-5238	149	4	int(f	int(f	VERB
ejpam-5238	149	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	149	6	-	-	PUNCT
ejpam-5238	149	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	149	8	-	-	PUNCT
ejpam-5238	149	9	int(b	int(b	NOUN
ejpam-5238	149	10	)	)	PUNCT
ejpam-5238	149	11	)	)	PUNCT
ejpam-5238	149	12	)	)	PUNCT
ejpam-5238	149	13	)	)	PUNCT
ejpam-5238	149	14	for	for	ADP
ejpam-5238	149	15	every	every	DET
ejpam-5238	149	16	subset	subset	NOUN
ejpam-5238	149	17	b	b	PROPN
ejpam-5238	149	18	of	of	ADP
ejpam-5238	149	19	y	y	PROPN
ejpam-5238	149	20	;	;	PUNCT
ejpam-5238	149	21	(	(	PUNCT
ejpam-5238	149	22	6	6	X
ejpam-5238	149	23	)	)	PUNCT
ejpam-5238	149	24	τ1τ2	τ1τ2	NOUN
ejpam-5238	149	25	-	-	NOUN
ejpam-5238	149	26	cl(f	cl(f	NOUN
ejpam-5238	149	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	149	28	-	-	PUNCT
ejpam-5238	149	29	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	149	30	-	-	PUNCT
ejpam-5238	149	31	cl(v	cl(v	NOUN
ejpam-5238	149	32	)	)	PUNCT
ejpam-5238	149	33	)	)	PUNCT
ejpam-5238	149	34	)	)	PUNCT
ejpam-5238	149	35	)	)	PUNCT
ejpam-5238	150	1	⊆	⊆	NUM
ejpam-5238	150	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	150	3	-	-	PUNCT
ejpam-5238	150	4	cl(v	cl(v	NOUN
ejpam-5238	150	5	)	)	PUNCT
ejpam-5238	150	6	)	)	PUNCT
ejpam-5238	150	7	for	for	ADP
ejpam-5238	150	8	every	every	DET
ejpam-5238	150	9	σ1σ2	σ1σ2	NOUN
ejpam-5238	150	10	-	-	ADJ
ejpam-5238	150	11	open	open	ADJ
ejpam-5238	150	12	set	set	NOUN
ejpam-5238	150	13	v	v	NOUN
ejpam-5238	150	14	of	of	ADP
ejpam-5238	150	15	y	y	PROPN
ejpam-5238	150	16	;	;	PUNCT
ejpam-5238	150	17	(	(	PUNCT
ejpam-5238	150	18	7	7	X
ejpam-5238	150	19	)	)	PUNCT
ejpam-5238	150	20	τ1τ2	τ1τ2	NOUN
ejpam-5238	150	21	-	-	NOUN
ejpam-5238	150	22	cl(f	cl(f	PRON
ejpam-5238	150	23	−1(v	−1(v	NOUN
ejpam-5238	150	24	)	)	PUNCT
ejpam-5238	150	25	)	)	PUNCT
ejpam-5238	151	1	⊆	⊆	NUM
ejpam-5238	151	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	151	3	-	-	PUNCT
ejpam-5238	151	4	cl(v	cl(v	NOUN
ejpam-5238	151	5	)	)	PUNCT
ejpam-5238	151	6	)	)	PUNCT
ejpam-5238	151	7	for	for	ADP
ejpam-5238	151	8	every	every	DET
ejpam-5238	151	9	σ1σ2	σ1σ2	NOUN
ejpam-5238	151	10	-	-	ADJ
ejpam-5238	151	11	open	open	ADJ
ejpam-5238	151	12	set	set	NOUN
ejpam-5238	151	13	v	v	NOUN
ejpam-5238	151	14	of	of	ADP
ejpam-5238	151	15	y	y	PROPN
ejpam-5238	151	16	;	;	PUNCT
ejpam-5238	151	17	(	(	PUNCT
ejpam-5238	151	18	8)	8)	NUM
ejpam-5238	151	19	τ1τ2	τ1τ2	NOUN
ejpam-5238	151	20	-	-	NOUN
ejpam-5238	151	21	cl(f	cl(f	NOUN
ejpam-5238	151	22	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	151	23	-	-	PUNCT
ejpam-5238	151	24	int(k	int(k	NUM
ejpam-5238	151	25	)	)	PUNCT
ejpam-5238	151	26	)	)	PUNCT
ejpam-5238	151	27	)	)	PUNCT
ejpam-5238	152	1	⊆	⊆	NUM
ejpam-5238	152	2	f−1(k	f−1(k	PROPN
ejpam-5238	152	3	)	)	PUNCT
ejpam-5238	152	4	for	for	ADP
ejpam-5238	152	5	every	every	DET
ejpam-5238	152	6	(	(	PUNCT
ejpam-5238	152	7	σ1	σ1	PROPN
ejpam-5238	152	8	,	,	PUNCT
ejpam-5238	152	9	σ2)r	σ2)r	NOUN
ejpam-5238	152	10	-	-	PUNCT
ejpam-5238	152	11	closed	close	VERB
ejpam-5238	152	12	set	set	ADJ
ejpam-5238	152	13	k	k	PROPN
ejpam-5238	152	14	of	of	ADP
ejpam-5238	152	15	y	y	PROPN
ejpam-5238	152	16	.	.	PUNCT
ejpam-5238	153	1	theorem	theorem	VERB
ejpam-5238	153	2	3	3	NUM
ejpam-5238	153	3	.	.	X
ejpam-5238	153	4	for	for	ADP
ejpam-5238	153	5	a	a	DET
ejpam-5238	153	6	multifunction	multifunction	NOUN
ejpam-5238	154	1	f	f	NOUN
ejpam-5238	154	2	:	:	PUNCT
ejpam-5238	154	3	(	(	PUNCT
ejpam-5238	154	4	x	x	NOUN
ejpam-5238	154	5	,	,	PUNCT
ejpam-5238	154	6	τ1	τ1	NOUN
ejpam-5238	154	7	,	,	PUNCT
ejpam-5238	154	8	τ2	τ2	NOUN
ejpam-5238	154	9	)	)	PUNCT
ejpam-5238	154	10	→	→	SYM
ejpam-5238	154	11	(	(	PUNCT
ejpam-5238	154	12	y	y	PROPN
ejpam-5238	154	13	,	,	PUNCT
ejpam-5238	154	14	σ1	σ1	PROPN
ejpam-5238	154	15	,	,	PUNCT
ejpam-5238	154	16	σ2	σ2	NOUN
ejpam-5238	154	17	)	)	PUNCT
ejpam-5238	154	18	,	,	PUNCT
ejpam-5238	154	19	the	the	DET
ejpam-5238	154	20	following	follow	VERB
ejpam-5238	154	21	properties	property	NOUN
ejpam-5238	154	22	are	be	AUX
ejpam-5238	154	23	equivalent	equivalent	ADJ
ejpam-5238	154	24	:	:	PUNCT
ejpam-5238	154	25	m.	m.	NOUN
ejpam-5238	154	26	thongmoon	thongmoon	NOUN
ejpam-5238	154	27	,	,	PUNCT
ejpam-5238	154	28	s.	s.	PROPN
ejpam-5238	154	29	sompong	sompong	PROPN
ejpam-5238	154	30	,	,	PUNCT
ejpam-5238	154	31	c.	c.	PROPN
ejpam-5238	154	32	boonpok	boonpok	PROPN
ejpam-5238	154	33	/	/	SYM
ejpam-5238	154	34	eur	eur	PROPN
ejpam-5238	154	35	.	.	PUNCT
ejpam-5238	155	1	j.	j.	PROPN
ejpam-5238	155	2	pure	pure	PROPN
ejpam-5238	155	3	appl	appl	PROPN
ejpam-5238	155	4	.	.	PROPN
ejpam-5238	155	5	math	math	PROPN
ejpam-5238	155	6	,	,	PUNCT
ejpam-5238	155	7	17	17	NUM
ejpam-5238	155	8	(	(	PUNCT
ejpam-5238	155	9	3	3	NUM
ejpam-5238	155	10	)	)	PUNCT
ejpam-5238	155	11	(	(	PUNCT
ejpam-5238	155	12	2024	2024	NUM
ejpam-5238	155	13	)	)	PUNCT
ejpam-5238	155	14	,	,	PUNCT
ejpam-5238	155	15	1705	1705	NUM
ejpam-5238	155	16	-	-	SYM
ejpam-5238	155	17	1716	1716	NUM
ejpam-5238	155	18	1710	1710	NUM
ejpam-5238	155	19	(	(	PUNCT
ejpam-5238	155	20	1	1	X
ejpam-5238	155	21	)	)	PUNCT
ejpam-5238	155	22	f	f	PROPN
ejpam-5238	155	23	is	be	AUX
ejpam-5238	155	24	upper	upper	ADJ
ejpam-5238	155	25	weakly	weakly	ADJ
ejpam-5238	155	26	(	(	PUNCT
ejpam-5238	155	27	τ1	τ1	NOUN
ejpam-5238	155	28	,	,	PUNCT
ejpam-5238	155	29	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	155	30	;	;	PUNCT
ejpam-5238	155	31	(	(	PUNCT
ejpam-5238	155	32	2	2	X
ejpam-5238	155	33	)	)	PUNCT
ejpam-5238	155	34	τ1τ2	τ1τ2	NOUN
ejpam-5238	155	35	-	-	NOUN
ejpam-5238	155	36	cl(f	cl(f	NOUN
ejpam-5238	155	37	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	155	38	-	-	PUNCT
ejpam-5238	155	39	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	155	40	-	-	PUNCT
ejpam-5238	155	41	cl(v	cl(v	NOUN
ejpam-5238	155	42	)	)	PUNCT
ejpam-5238	155	43	)	)	PUNCT
ejpam-5238	155	44	)	)	PUNCT
ejpam-5238	155	45	)	)	PUNCT
ejpam-5238	156	1	⊆	⊆	X
ejpam-5238	156	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	156	3	-	-	PUNCT
ejpam-5238	156	4	cl(v	cl(v	NOUN
ejpam-5238	156	5	)	)	PUNCT
ejpam-5238	156	6	)	)	PUNCT
ejpam-5238	156	7	for	for	ADP
ejpam-5238	156	8	every	every	DET
ejpam-5238	156	9	(	(	PUNCT
ejpam-5238	156	10	σ1	σ1	PROPN
ejpam-5238	156	11	,	,	PUNCT
ejpam-5238	156	12	σ2)β	σ2)β	NOUN
ejpam-5238	156	13	-	-	PUNCT
ejpam-5238	156	14	open	open	NOUN
ejpam-5238	156	15	set	set	NOUN
ejpam-5238	156	16	v	v	NOUN
ejpam-5238	156	17	of	of	ADP
ejpam-5238	156	18	y	y	PROPN
ejpam-5238	156	19	;	;	PUNCT
ejpam-5238	156	20	(	(	PUNCT
ejpam-5238	156	21	3	3	X
ejpam-5238	156	22	)	)	PUNCT
ejpam-5238	156	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	156	24	-	-	NOUN
ejpam-5238	156	25	cl(f	cl(f	NOUN
ejpam-5238	156	26	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	156	27	-	-	PUNCT
ejpam-5238	156	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	156	29	-	-	PUNCT
ejpam-5238	156	30	cl(v	cl(v	NOUN
ejpam-5238	156	31	)	)	PUNCT
ejpam-5238	156	32	)	)	PUNCT
ejpam-5238	156	33	)	)	PUNCT
ejpam-5238	156	34	)	)	PUNCT
ejpam-5238	157	1	⊆	⊆	X
ejpam-5238	157	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	157	3	-	-	PUNCT
ejpam-5238	157	4	cl(v	cl(v	NOUN
ejpam-5238	157	5	)	)	PUNCT
ejpam-5238	157	6	)	)	PUNCT
ejpam-5238	157	7	for	for	ADP
ejpam-5238	157	8	every	every	DET
ejpam-5238	157	9	(	(	PUNCT
ejpam-5238	157	10	σ1	σ1	PROPN
ejpam-5238	157	11	,	,	PUNCT
ejpam-5238	157	12	σ2)s	σ2)s	NOUN
ejpam-5238	157	13	-	-	PUNCT
ejpam-5238	157	14	open	open	NOUN
ejpam-5238	157	15	set	set	NOUN
ejpam-5238	157	16	v	v	NOUN
ejpam-5238	157	17	of	of	ADP
ejpam-5238	157	18	y	y	PROPN
ejpam-5238	157	19	.	.	PUNCT
ejpam-5238	158	1	proof	proof	NOUN
ejpam-5238	158	2	.	.	PUNCT
ejpam-5238	159	1	(	(	PUNCT
ejpam-5238	159	2	1	1	X
ejpam-5238	159	3	)	)	PUNCT
ejpam-5238	159	4	⇒	⇒	NOUN
ejpam-5238	159	5	(	(	PUNCT
ejpam-5238	159	6	2	2	NUM
ejpam-5238	159	7	):	):	PUNCT
ejpam-5238	159	8	this	this	PRON
ejpam-5238	159	9	follows	follow	VERB
ejpam-5238	159	10	from	from	ADP
ejpam-5238	159	11	(	(	PUNCT
ejpam-5238	159	12	4	4	NUM
ejpam-5238	159	13	)	)	PUNCT
ejpam-5238	159	14	of	of	ADP
ejpam-5238	159	15	theorem	theorem	NOUN
ejpam-5238	159	16	1	1	NUM
ejpam-5238	159	17	.	.	PUNCT
ejpam-5238	159	18	(	(	PUNCT
ejpam-5238	159	19	2	2	X
ejpam-5238	159	20	)	)	PUNCT
ejpam-5238	159	21	⇒	⇒	NOUN
ejpam-5238	159	22	(	(	PUNCT
ejpam-5238	159	23	3	3	NUM
ejpam-5238	159	24	):	):	PUNCT
ejpam-5238	159	25	the	the	DET
ejpam-5238	159	26	proof	proof	NOUN
ejpam-5238	159	27	is	be	AUX
ejpam-5238	159	28	obvious	obvious	ADJ
ejpam-5238	159	29	since	since	SCONJ
ejpam-5238	159	30	every	every	DET
ejpam-5238	159	31	(	(	PUNCT
ejpam-5238	159	32	σ1	σ1	PROPN
ejpam-5238	159	33	,	,	PUNCT
ejpam-5238	159	34	σ2)s	σ2)s	NOUN
ejpam-5238	159	35	-	-	PUNCT
ejpam-5238	159	36	open	open	ADJ
ejpam-5238	159	37	set	set	NOUN
ejpam-5238	159	38	is	be	AUX
ejpam-5238	159	39	(	(	PUNCT
ejpam-5238	159	40	σ1	σ1	PROPN
ejpam-5238	159	41	,	,	PUNCT
ejpam-5238	159	42	σ2)β	σ2)β	NOUN
ejpam-5238	159	43	-	-	PUNCT
ejpam-5238	159	44	open	open	ADJ
ejpam-5238	159	45	.	.	PUNCT
ejpam-5238	160	1	(	(	PUNCT
ejpam-5238	160	2	3	3	X
ejpam-5238	160	3	)	)	PUNCT
ejpam-5238	160	4	⇒	⇒	NOUN
ejpam-5238	160	5	(	(	PUNCT
ejpam-5238	160	6	1	1	NUM
ejpam-5238	160	7	):	):	PUNCT
ejpam-5238	160	8	since	since	SCONJ
ejpam-5238	160	9	every	every	DET
ejpam-5238	160	10	σ1σ2	σ1σ2	NUM
ejpam-5238	160	11	-	-	ADJ
ejpam-5238	160	12	open	open	ADJ
ejpam-5238	160	13	set	set	NOUN
ejpam-5238	160	14	is	be	AUX
ejpam-5238	160	15	(	(	PUNCT
ejpam-5238	160	16	σ1	σ1	PROPN
ejpam-5238	160	17	,	,	PUNCT
ejpam-5238	160	18	σ2)s	σ2)s	NOUN
ejpam-5238	160	19	-	-	PUNCT
ejpam-5238	160	20	open	open	ADJ
ejpam-5238	160	21	,	,	PUNCT
ejpam-5238	160	22	the	the	DET
ejpam-5238	160	23	proof	proof	NOUN
ejpam-5238	160	24	is	be	AUX
ejpam-5238	160	25	obvious	obvious	ADJ
ejpam-5238	160	26	by	by	ADP
ejpam-5238	160	27	(	(	PUNCT
ejpam-5238	160	28	7	7	NUM
ejpam-5238	160	29	)	)	PUNCT
ejpam-5238	160	30	of	of	ADP
ejpam-5238	160	31	theorem	theorem	ADJ
ejpam-5238	160	32	1	1	NUM
ejpam-5238	160	33	.	.	PUNCT
ejpam-5238	160	34	theorem	theorem	NOUN
ejpam-5238	160	35	4	4	NUM
ejpam-5238	160	36	.	.	X
ejpam-5238	160	37	for	for	ADP
ejpam-5238	160	38	a	a	DET
ejpam-5238	160	39	multifunction	multifunction	NOUN
ejpam-5238	160	40	f	f	NOUN
ejpam-5238	160	41	:	:	PUNCT
ejpam-5238	160	42	(	(	PUNCT
ejpam-5238	160	43	x	x	NOUN
ejpam-5238	160	44	,	,	PUNCT
ejpam-5238	160	45	τ1	τ1	NOUN
ejpam-5238	160	46	,	,	PUNCT
ejpam-5238	160	47	τ2	τ2	NOUN
ejpam-5238	160	48	)	)	PUNCT
ejpam-5238	160	49	→	→	SYM
ejpam-5238	160	50	(	(	PUNCT
ejpam-5238	160	51	y	y	PROPN
ejpam-5238	160	52	,	,	PUNCT
ejpam-5238	160	53	σ1	σ1	PROPN
ejpam-5238	160	54	,	,	PUNCT
ejpam-5238	160	55	σ2	σ2	NOUN
ejpam-5238	160	56	)	)	PUNCT
ejpam-5238	160	57	,	,	PUNCT
ejpam-5238	160	58	the	the	DET
ejpam-5238	160	59	following	follow	VERB
ejpam-5238	160	60	properties	property	NOUN
ejpam-5238	160	61	are	be	AUX
ejpam-5238	160	62	equivalent	equivalent	ADJ
ejpam-5238	160	63	:	:	PUNCT
ejpam-5238	160	64	(	(	PUNCT
ejpam-5238	160	65	1	1	X
ejpam-5238	160	66	)	)	PUNCT
ejpam-5238	160	67	f	f	PROPN
ejpam-5238	160	68	is	be	AUX
ejpam-5238	160	69	lower	low	ADJ
ejpam-5238	160	70	weakly	weakly	ADJ
ejpam-5238	160	71	(	(	PUNCT
ejpam-5238	160	72	τ1	τ1	NOUN
ejpam-5238	160	73	,	,	PUNCT
ejpam-5238	160	74	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	160	75	;	;	PUNCT
ejpam-5238	160	76	(	(	PUNCT
ejpam-5238	160	77	2	2	X
ejpam-5238	160	78	)	)	PUNCT
ejpam-5238	160	79	τ1τ2	τ1τ2	NOUN
ejpam-5238	160	80	-	-	NOUN
ejpam-5238	160	81	cl(f	cl(f	NOUN
ejpam-5238	160	82	+	+	NOUN
ejpam-5238	160	83	(	(	PUNCT
ejpam-5238	160	84	σ1σ2	σ1σ2	NUM
ejpam-5238	160	85	-	-	PUNCT
ejpam-5238	160	86	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	160	87	-	-	PUNCT
ejpam-5238	160	88	cl(v	cl(v	NOUN
ejpam-5238	160	89	)	)	PUNCT
ejpam-5238	160	90	)	)	PUNCT
ejpam-5238	160	91	)	)	PUNCT
ejpam-5238	160	92	)	)	PUNCT
ejpam-5238	161	1	⊆	⊆	X
ejpam-5238	161	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	161	3	-	-	PUNCT
ejpam-5238	161	4	cl(v	cl(v	NOUN
ejpam-5238	161	5	)	)	PUNCT
ejpam-5238	161	6	)	)	PUNCT
ejpam-5238	161	7	for	for	ADP
ejpam-5238	161	8	every	every	DET
ejpam-5238	161	9	(	(	PUNCT
ejpam-5238	161	10	σ1	σ1	PROPN
ejpam-5238	161	11	,	,	PUNCT
ejpam-5238	161	12	σ2)β	σ2)β	NOUN
ejpam-5238	161	13	-	-	PUNCT
ejpam-5238	161	14	open	open	NOUN
ejpam-5238	161	15	set	set	NOUN
ejpam-5238	161	16	v	v	NOUN
ejpam-5238	161	17	of	of	ADP
ejpam-5238	161	18	y	y	PROPN
ejpam-5238	161	19	;	;	PUNCT
ejpam-5238	161	20	(	(	PUNCT
ejpam-5238	161	21	3	3	X
ejpam-5238	161	22	)	)	PUNCT
ejpam-5238	161	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	161	24	-	-	NOUN
ejpam-5238	161	25	cl(f	cl(f	NOUN
ejpam-5238	161	26	+	+	NOUN
ejpam-5238	161	27	(	(	PUNCT
ejpam-5238	161	28	σ1σ2	σ1σ2	NUM
ejpam-5238	161	29	-	-	PUNCT
ejpam-5238	161	30	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	161	31	-	-	PUNCT
ejpam-5238	161	32	cl(v	cl(v	NOUN
ejpam-5238	161	33	)	)	PUNCT
ejpam-5238	161	34	)	)	PUNCT
ejpam-5238	161	35	)	)	PUNCT
ejpam-5238	161	36	)	)	PUNCT
ejpam-5238	162	1	⊆	⊆	X
ejpam-5238	162	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	162	3	-	-	PUNCT
ejpam-5238	162	4	cl(v	cl(v	NOUN
ejpam-5238	162	5	)	)	PUNCT
ejpam-5238	162	6	)	)	PUNCT
ejpam-5238	162	7	for	for	SCONJ
ejpam-5238	162	8	every	every	DET
ejpam-5238	162	9	(	(	PUNCT
ejpam-5238	162	10	σ1	σ1	PROPN
ejpam-5238	162	11	,	,	PUNCT
ejpam-5238	162	12	σ2)s	σ2)s	NOUN
ejpam-5238	162	13	-	-	PUNCT
ejpam-5238	162	14	open	open	NOUN
ejpam-5238	162	15	set	set	NOUN
ejpam-5238	162	16	v	v	NOUN
ejpam-5238	162	17	of	of	ADP
ejpam-5238	162	18	y	y	PROPN
ejpam-5238	162	19	.	.	PUNCT
ejpam-5238	163	1	proof	proof	NOUN
ejpam-5238	163	2	.	.	PUNCT
ejpam-5238	164	1	the	the	DET
ejpam-5238	164	2	proof	proof	NOUN
ejpam-5238	164	3	is	be	AUX
ejpam-5238	164	4	similar	similar	ADJ
ejpam-5238	164	5	to	to	ADP
ejpam-5238	164	6	that	that	PRON
ejpam-5238	164	7	of	of	ADP
ejpam-5238	164	8	theorem	theorem	ADJ
ejpam-5238	164	9	3	3	NUM
ejpam-5238	164	10	.	.	PUNCT
ejpam-5238	164	11	corollary	corollary	ADJ
ejpam-5238	164	12	2	2	NUM
ejpam-5238	164	13	.	.	PUNCT
ejpam-5238	164	14	for	for	ADP
ejpam-5238	164	15	a	a	DET
ejpam-5238	164	16	function	function	NOUN
ejpam-5238	164	17	f	f	NOUN
ejpam-5238	164	18	:	:	PUNCT
ejpam-5238	164	19	(	(	PUNCT
ejpam-5238	164	20	x	x	NOUN
ejpam-5238	164	21	,	,	PUNCT
ejpam-5238	164	22	τ1	τ1	NOUN
ejpam-5238	164	23	,	,	PUNCT
ejpam-5238	164	24	τ2	τ2	NOUN
ejpam-5238	164	25	)	)	PUNCT
ejpam-5238	164	26	→	→	SYM
ejpam-5238	164	27	(	(	PUNCT
ejpam-5238	164	28	y	y	PROPN
ejpam-5238	164	29	,	,	PUNCT
ejpam-5238	164	30	σ1	σ1	PROPN
ejpam-5238	164	31	,	,	PUNCT
ejpam-5238	164	32	σ2	σ2	NOUN
ejpam-5238	164	33	)	)	PUNCT
ejpam-5238	164	34	,	,	PUNCT
ejpam-5238	164	35	the	the	DET
ejpam-5238	164	36	following	follow	VERB
ejpam-5238	164	37	properties	property	NOUN
ejpam-5238	164	38	are	be	AUX
ejpam-5238	164	39	equivalent	equivalent	ADJ
ejpam-5238	164	40	:	:	PUNCT
ejpam-5238	164	41	(	(	PUNCT
ejpam-5238	164	42	1	1	X
ejpam-5238	164	43	)	)	PUNCT
ejpam-5238	164	44	f	f	PROPN
ejpam-5238	164	45	is	be	AUX
ejpam-5238	164	46	weakly	weakly	ADJ
ejpam-5238	164	47	(	(	PUNCT
ejpam-5238	164	48	τ1	τ1	NOUN
ejpam-5238	164	49	,	,	PUNCT
ejpam-5238	164	50	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	164	51	;	;	PUNCT
ejpam-5238	164	52	(	(	PUNCT
ejpam-5238	164	53	2	2	X
ejpam-5238	164	54	)	)	PUNCT
ejpam-5238	164	55	τ1τ2	τ1τ2	NOUN
ejpam-5238	164	56	-	-	NOUN
ejpam-5238	164	57	cl(f	cl(f	NOUN
ejpam-5238	164	58	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	164	59	-	-	PUNCT
ejpam-5238	164	60	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	164	61	-	-	PUNCT
ejpam-5238	164	62	cl(v	cl(v	NOUN
ejpam-5238	164	63	)	)	PUNCT
ejpam-5238	164	64	)	)	PUNCT
ejpam-5238	164	65	)	)	PUNCT
ejpam-5238	164	66	)	)	PUNCT
ejpam-5238	165	1	⊆	⊆	NUM
ejpam-5238	165	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	165	3	-	-	PUNCT
ejpam-5238	165	4	cl(v	cl(v	NOUN
ejpam-5238	165	5	)	)	PUNCT
ejpam-5238	165	6	)	)	PUNCT
ejpam-5238	165	7	for	for	ADP
ejpam-5238	165	8	every	every	DET
ejpam-5238	165	9	(	(	PUNCT
ejpam-5238	165	10	σ1	σ1	PROPN
ejpam-5238	165	11	,	,	PUNCT
ejpam-5238	165	12	σ2)β	σ2)β	NOUN
ejpam-5238	165	13	-	-	PUNCT
ejpam-5238	165	14	open	open	NOUN
ejpam-5238	165	15	set	set	NOUN
ejpam-5238	165	16	v	v	NOUN
ejpam-5238	165	17	of	of	ADP
ejpam-5238	165	18	y	y	PROPN
ejpam-5238	165	19	;	;	PUNCT
ejpam-5238	165	20	(	(	PUNCT
ejpam-5238	165	21	3	3	X
ejpam-5238	165	22	)	)	PUNCT
ejpam-5238	165	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	165	24	-	-	NOUN
ejpam-5238	165	25	cl(f	cl(f	NOUN
ejpam-5238	165	26	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	165	27	-	-	PUNCT
ejpam-5238	165	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	165	29	-	-	PUNCT
ejpam-5238	165	30	cl(v	cl(v	NOUN
ejpam-5238	165	31	)	)	PUNCT
ejpam-5238	165	32	)	)	PUNCT
ejpam-5238	165	33	)	)	PUNCT
ejpam-5238	165	34	)	)	PUNCT
ejpam-5238	166	1	⊆	⊆	NUM
ejpam-5238	166	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	166	3	-	-	PUNCT
ejpam-5238	166	4	cl(v	cl(v	NOUN
ejpam-5238	166	5	)	)	PUNCT
ejpam-5238	166	6	)	)	PUNCT
ejpam-5238	166	7	for	for	ADP
ejpam-5238	166	8	every	every	DET
ejpam-5238	166	9	(	(	PUNCT
ejpam-5238	166	10	σ1	σ1	PROPN
ejpam-5238	166	11	,	,	PUNCT
ejpam-5238	166	12	σ2)s	σ2)s	NOUN
ejpam-5238	166	13	-	-	PUNCT
ejpam-5238	166	14	open	open	NOUN
ejpam-5238	166	15	set	set	NOUN
ejpam-5238	166	16	v	v	NOUN
ejpam-5238	166	17	of	of	ADP
ejpam-5238	166	18	y	y	PROPN
ejpam-5238	166	19	.	.	PUNCT
ejpam-5238	167	1	theorem	theorem	ADJ
ejpam-5238	167	2	5	5	NUM
ejpam-5238	167	3	.	.	X
ejpam-5238	167	4	for	for	ADP
ejpam-5238	167	5	a	a	DET
ejpam-5238	167	6	multifunction	multifunction	NOUN
ejpam-5238	168	1	f	f	NOUN
ejpam-5238	168	2	:	:	PUNCT
ejpam-5238	168	3	(	(	PUNCT
ejpam-5238	168	4	x	x	NOUN
ejpam-5238	168	5	,	,	PUNCT
ejpam-5238	168	6	τ1	τ1	NOUN
ejpam-5238	168	7	,	,	PUNCT
ejpam-5238	168	8	τ2	τ2	NOUN
ejpam-5238	168	9	)	)	PUNCT
ejpam-5238	168	10	→	→	SYM
ejpam-5238	168	11	(	(	PUNCT
ejpam-5238	168	12	y	y	PROPN
ejpam-5238	168	13	,	,	PUNCT
ejpam-5238	168	14	σ1	σ1	PROPN
ejpam-5238	168	15	,	,	PUNCT
ejpam-5238	168	16	σ2	σ2	NOUN
ejpam-5238	168	17	)	)	PUNCT
ejpam-5238	168	18	,	,	PUNCT
ejpam-5238	168	19	the	the	DET
ejpam-5238	168	20	following	follow	VERB
ejpam-5238	168	21	properties	property	NOUN
ejpam-5238	168	22	are	be	AUX
ejpam-5238	168	23	equivalent	equivalent	ADJ
ejpam-5238	168	24	:	:	PUNCT
ejpam-5238	168	25	(	(	PUNCT
ejpam-5238	168	26	1	1	X
ejpam-5238	168	27	)	)	PUNCT
ejpam-5238	168	28	f	f	PROPN
ejpam-5238	168	29	is	be	AUX
ejpam-5238	168	30	upper	upper	ADJ
ejpam-5238	168	31	weakly	weakly	ADJ
ejpam-5238	168	32	(	(	PUNCT
ejpam-5238	168	33	τ1	τ1	NOUN
ejpam-5238	168	34	,	,	PUNCT
ejpam-5238	168	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	168	36	;	;	PUNCT
ejpam-5238	168	37	(	(	PUNCT
ejpam-5238	168	38	2	2	X
ejpam-5238	168	39	)	)	PUNCT
ejpam-5238	168	40	τ1τ2	τ1τ2	NOUN
ejpam-5238	168	41	-	-	NOUN
ejpam-5238	168	42	cl(f	cl(f	NOUN
ejpam-5238	168	43	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	168	44	-	-	PUNCT
ejpam-5238	168	45	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	168	46	-	-	PUNCT
ejpam-5238	168	47	cl(v	cl(v	NOUN
ejpam-5238	168	48	)	)	PUNCT
ejpam-5238	168	49	)	)	PUNCT
ejpam-5238	168	50	)	)	PUNCT
ejpam-5238	168	51	)	)	PUNCT
ejpam-5238	169	1	⊆	⊆	X
ejpam-5238	169	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	169	3	-	-	PUNCT
ejpam-5238	169	4	cl(v	cl(v	NOUN
ejpam-5238	169	5	)	)	PUNCT
ejpam-5238	169	6	)	)	PUNCT
ejpam-5238	169	7	for	for	ADP
ejpam-5238	169	8	every	every	DET
ejpam-5238	169	9	(	(	PUNCT
ejpam-5238	169	10	σ1	σ1	PROPN
ejpam-5238	169	11	,	,	PUNCT
ejpam-5238	169	12	σ2)p	σ2)p	NOUN
ejpam-5238	169	13	-	-	PUNCT
ejpam-5238	169	14	open	open	NOUN
ejpam-5238	169	15	set	set	NOUN
ejpam-5238	169	16	v	v	NOUN
ejpam-5238	169	17	of	of	ADP
ejpam-5238	169	18	y	y	PROPN
ejpam-5238	169	19	;	;	PUNCT
ejpam-5238	169	20	(	(	PUNCT
ejpam-5238	169	21	3	3	X
ejpam-5238	169	22	)	)	PUNCT
ejpam-5238	169	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	169	24	-	-	NOUN
ejpam-5238	169	25	cl(f	cl(f	NUM
ejpam-5238	169	26	−(v	−(v	NOUN
ejpam-5238	169	27	)	)	PUNCT
ejpam-5238	169	28	)	)	PUNCT
ejpam-5238	170	1	⊆	⊆	X
ejpam-5238	170	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	170	3	-	-	PUNCT
ejpam-5238	170	4	cl(v	cl(v	NOUN
ejpam-5238	170	5	)	)	PUNCT
ejpam-5238	170	6	)	)	PUNCT
ejpam-5238	170	7	for	for	ADP
ejpam-5238	170	8	every	every	DET
ejpam-5238	170	9	(	(	PUNCT
ejpam-5238	170	10	σ1	σ1	PROPN
ejpam-5238	170	11	,	,	PUNCT
ejpam-5238	170	12	σ2)p	σ2)p	NOUN
ejpam-5238	170	13	-	-	PUNCT
ejpam-5238	170	14	open	open	NOUN
ejpam-5238	170	15	set	set	NOUN
ejpam-5238	170	16	v	v	NOUN
ejpam-5238	170	17	of	of	ADP
ejpam-5238	170	18	y	y	PROPN
ejpam-5238	170	19	;	;	PUNCT
ejpam-5238	170	20	(	(	PUNCT
ejpam-5238	170	21	4	4	X
ejpam-5238	170	22	)	)	PUNCT
ejpam-5238	170	23	f+(v	f+(v	NOUN
ejpam-5238	170	24	)	)	PUNCT
ejpam-5238	171	1	⊆	⊆	X
ejpam-5238	171	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	171	3	-	-	NUM
ejpam-5238	171	4	int(f	int(f	VERB
ejpam-5238	171	5	+	+	ADJ
ejpam-5238	171	6	(	(	PUNCT
ejpam-5238	171	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	171	8	-	-	NUM
ejpam-5238	171	9	cl(v	cl(v	NOUN
ejpam-5238	171	10	)	)	PUNCT
ejpam-5238	171	11	)	)	PUNCT
ejpam-5238	171	12	)	)	PUNCT
ejpam-5238	171	13	for	for	ADP
ejpam-5238	171	14	every	every	DET
ejpam-5238	171	15	(	(	PUNCT
ejpam-5238	171	16	σ1	σ1	PROPN
ejpam-5238	171	17	,	,	PUNCT
ejpam-5238	171	18	σ2)p	σ2)p	NOUN
ejpam-5238	171	19	-	-	PUNCT
ejpam-5238	171	20	open	open	NOUN
ejpam-5238	171	21	set	set	NOUN
ejpam-5238	171	22	v	v	NOUN
ejpam-5238	171	23	of	of	ADP
ejpam-5238	171	24	y	y	PROPN
ejpam-5238	171	25	.	.	PUNCT
ejpam-5238	172	1	m.	m.	NOUN
ejpam-5238	172	2	thongmoon	thongmoon	PROPN
ejpam-5238	172	3	,	,	PUNCT
ejpam-5238	172	4	s.	s.	PROPN
ejpam-5238	172	5	sompong	sompong	PROPN
ejpam-5238	172	6	,	,	PUNCT
ejpam-5238	172	7	c.	c.	PROPN
ejpam-5238	172	8	boonpok	boonpok	PROPN
ejpam-5238	172	9	/	/	SYM
ejpam-5238	172	10	eur	eur	PROPN
ejpam-5238	172	11	.	.	PUNCT
ejpam-5238	173	1	j.	j.	PROPN
ejpam-5238	173	2	pure	pure	PROPN
ejpam-5238	173	3	appl	appl	PROPN
ejpam-5238	173	4	.	.	PROPN
ejpam-5238	173	5	math	math	PROPN
ejpam-5238	173	6	,	,	PUNCT
ejpam-5238	173	7	17	17	NUM
ejpam-5238	173	8	(	(	PUNCT
ejpam-5238	173	9	3	3	NUM
ejpam-5238	173	10	)	)	PUNCT
ejpam-5238	173	11	(	(	PUNCT
ejpam-5238	173	12	2024	2024	NUM
ejpam-5238	173	13	)	)	PUNCT
ejpam-5238	173	14	,	,	PUNCT
ejpam-5238	173	15	1705	1705	NUM
ejpam-5238	173	16	-	-	SYM
ejpam-5238	173	17	1716	1716	NUM
ejpam-5238	173	18	1711	1711	NUM
ejpam-5238	173	19	proof	proof	NOUN
ejpam-5238	173	20	.	.	PUNCT
ejpam-5238	174	1	(	(	PUNCT
ejpam-5238	174	2	1	1	X
ejpam-5238	174	3	)	)	PUNCT
ejpam-5238	174	4	⇒	⇒	NOUN
ejpam-5238	174	5	(	(	PUNCT
ejpam-5238	174	6	2	2	NUM
ejpam-5238	174	7	):	):	PUNCT
ejpam-5238	174	8	let	let	VERB
ejpam-5238	174	9	v	v	PART
ejpam-5238	174	10	be	be	AUX
ejpam-5238	174	11	any	any	DET
ejpam-5238	174	12	(	(	PUNCT
ejpam-5238	174	13	σ1	σ1	PROPN
ejpam-5238	174	14	,	,	PUNCT
ejpam-5238	174	15	σ2)p	σ2)p	NOUN
ejpam-5238	174	16	-	-	PUNCT
ejpam-5238	174	17	open	open	ADJ
ejpam-5238	174	18	set	set	NOUN
ejpam-5238	174	19	of	of	ADP
ejpam-5238	174	20	y	y	PROPN
ejpam-5238	174	21	.	.	PUNCT
ejpam-5238	175	1	since	since	SCONJ
ejpam-5238	175	2	σ1σ2	σ1σ2	ADV
ejpam-5238	175	3	-	-	PUNCT
ejpam-5238	175	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	175	5	-	-	PUNCT
ejpam-5238	175	6	cl(v	cl(v	NOUN
ejpam-5238	175	7	)	)	PUNCT
ejpam-5238	175	8	)	)	PUNCT
ejpam-5238	175	9	is	be	AUX
ejpam-5238	175	10	σ1σ2	σ1σ2	NOUN
ejpam-5238	175	11	-	-	ADJ
ejpam-5238	175	12	open	open	ADJ
ejpam-5238	175	13	,	,	PUNCT
ejpam-5238	175	14	by	by	ADP
ejpam-5238	175	15	theorem	theorem	ADJ
ejpam-5238	175	16	1(7	1(7	NUM
ejpam-5238	175	17	)	)	PUNCT
ejpam-5238	175	18	τ1τ2	τ1τ2	NOUN
ejpam-5238	175	19	-	-	NOUN
ejpam-5238	175	20	cl(f	cl(f	NOUN
ejpam-5238	175	21	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	175	22	-	-	PUNCT
ejpam-5238	175	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	175	24	-	-	PUNCT
ejpam-5238	175	25	cl(v	cl(v	NOUN
ejpam-5238	175	26	)	)	PUNCT
ejpam-5238	175	27	)	)	PUNCT
ejpam-5238	175	28	)	)	PUNCT
ejpam-5238	175	29	)	)	PUNCT
ejpam-5238	176	1	⊆	⊆	X
ejpam-5238	176	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-5238	176	3	-	-	PUNCT
ejpam-5238	176	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5238	176	5	-	-	PUNCT
ejpam-5238	176	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	176	7	-	-	PUNCT
ejpam-5238	176	8	cl(v	cl(v	NOUN
ejpam-5238	176	9	)	)	PUNCT
ejpam-5238	176	10	)	)	PUNCT
ejpam-5238	176	11	)	)	PUNCT
ejpam-5238	176	12	)	)	PUNCT
ejpam-5238	177	1	⊆	⊆	X
ejpam-5238	177	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	177	3	-	-	PUNCT
ejpam-5238	177	4	cl(v	cl(v	NOUN
ejpam-5238	177	5	)	)	PUNCT
ejpam-5238	177	6	)	)	PUNCT
ejpam-5238	177	7	.	.	PUNCT
ejpam-5238	178	1	(	(	PUNCT
ejpam-5238	178	2	2	2	X
ejpam-5238	178	3	)	)	PUNCT
ejpam-5238	178	4	⇒	⇒	NOUN
ejpam-5238	178	5	(	(	PUNCT
ejpam-5238	178	6	3	3	NUM
ejpam-5238	178	7	):	):	PUNCT
ejpam-5238	178	8	let	let	VERB
ejpam-5238	178	9	v	v	PART
ejpam-5238	178	10	be	be	AUX
ejpam-5238	178	11	any	any	DET
ejpam-5238	178	12	(	(	PUNCT
ejpam-5238	178	13	σ1	σ1	PROPN
ejpam-5238	178	14	,	,	PUNCT
ejpam-5238	178	15	σ2)p	σ2)p	NOUN
ejpam-5238	178	16	-	-	PUNCT
ejpam-5238	178	17	open	open	ADJ
ejpam-5238	178	18	set	set	NOUN
ejpam-5238	178	19	of	of	ADP
ejpam-5238	178	20	y	y	PROPN
ejpam-5238	178	21	.	.	PUNCT
ejpam-5238	179	1	by	by	ADP
ejpam-5238	179	2	(	(	PUNCT
ejpam-5238	179	3	2	2	NUM
ejpam-5238	179	4	)	)	PUNCT
ejpam-5238	179	5	,	,	PUNCT
ejpam-5238	179	6	we	we	PRON
ejpam-5238	179	7	have	have	VERB
ejpam-5238	179	8	τ1τ2	τ1τ2	NOUN
ejpam-5238	179	9	-	-	ADJ
ejpam-5238	179	10	cl(f	cl(f	NUM
ejpam-5238	179	11	−(v	−(v	NOUN
ejpam-5238	179	12	)	)	PUNCT
ejpam-5238	179	13	)	)	PUNCT
ejpam-5238	180	1	⊆	⊆	X
ejpam-5238	180	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	180	3	-	-	ADJ
ejpam-5238	180	4	cl(f	cl(f	NOUN
ejpam-5238	180	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	180	6	-	-	PUNCT
ejpam-5238	180	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	180	8	-	-	PUNCT
ejpam-5238	180	9	cl(v	cl(v	NOUN
ejpam-5238	180	10	)	)	PUNCT
ejpam-5238	180	11	)	)	PUNCT
ejpam-5238	180	12	)	)	PUNCT
ejpam-5238	180	13	)	)	PUNCT
ejpam-5238	181	1	⊆	⊆	X
ejpam-5238	181	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-5238	181	3	-	-	PUNCT
ejpam-5238	181	4	cl(v	cl(v	NOUN
ejpam-5238	181	5	)	)	PUNCT
ejpam-5238	181	6	)	)	PUNCT
ejpam-5238	181	7	.	.	PUNCT
ejpam-5238	182	1	(	(	PUNCT
ejpam-5238	182	2	3	3	X
ejpam-5238	182	3	)	)	PUNCT
ejpam-5238	182	4	⇒	⇒	NOUN
ejpam-5238	182	5	(	(	PUNCT
ejpam-5238	182	6	4	4	NUM
ejpam-5238	182	7	):	):	PUNCT
ejpam-5238	182	8	let	let	VERB
ejpam-5238	182	9	v	v	PART
ejpam-5238	182	10	be	be	AUX
ejpam-5238	182	11	any	any	DET
ejpam-5238	182	12	(	(	PUNCT
ejpam-5238	182	13	σ1	σ1	PROPN
ejpam-5238	182	14	,	,	PUNCT
ejpam-5238	182	15	σ2)p	σ2)p	NOUN
ejpam-5238	182	16	-	-	PUNCT
ejpam-5238	182	17	open	open	ADJ
ejpam-5238	182	18	set	set	NOUN
ejpam-5238	182	19	of	of	ADP
ejpam-5238	182	20	y	y	PROPN
ejpam-5238	182	21	.	.	PUNCT
ejpam-5238	183	1	thus	thus	ADV
ejpam-5238	183	2	by	by	ADP
ejpam-5238	183	3	(	(	PUNCT
ejpam-5238	183	4	3	3	NUM
ejpam-5238	183	5	)	)	PUNCT
ejpam-5238	183	6	,	,	PUNCT
ejpam-5238	183	7	x	x	X
ejpam-5238	183	8	−	−	ADP
ejpam-5238	183	9	τ1τ2	τ1τ2	NOUN
ejpam-5238	183	10	-	-	NUM
ejpam-5238	183	11	int(f	int(f	VERB
ejpam-5238	183	12	+	+	ADJ
ejpam-5238	183	13	(	(	PUNCT
ejpam-5238	183	14	σ1σ2	σ1σ2	NOUN
ejpam-5238	183	15	-	-	NUM
ejpam-5238	183	16	cl(v	cl(v	NOUN
ejpam-5238	183	17	)	)	PUNCT
ejpam-5238	183	18	)	)	PUNCT
ejpam-5238	184	1	=	=	PUNCT
ejpam-5238	184	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	184	3	-	-	NOUN
ejpam-5238	184	4	cl(x	cl(x	SYM
ejpam-5238	184	5	−	−	ADP
ejpam-5238	184	6	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	184	7	-	-	PUNCT
ejpam-5238	184	8	cl(v	cl(v	NOUN
ejpam-5238	184	9	)	)	PUNCT
ejpam-5238	184	10	)	)	PUNCT
ejpam-5238	184	11	)	)	PUNCT
ejpam-5238	185	1	=	=	PUNCT
ejpam-5238	185	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	185	3	-	-	PROPN
ejpam-5238	185	4	cl(f	cl(f	NOUN
ejpam-5238	185	5	−(y	−(y	NOUN
ejpam-5238	185	6	−	−	NOUN
ejpam-5238	185	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	185	8	-	-	NUM
ejpam-5238	185	9	cl(v	cl(v	NOUN
ejpam-5238	185	10	)	)	PUNCT
ejpam-5238	185	11	)	)	PUNCT
ejpam-5238	185	12	)	)	PUNCT
ejpam-5238	186	1	⊆	⊆	X
ejpam-5238	186	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-5238	186	3	-	-	PUNCT
ejpam-5238	186	4	cl(y	cl(y	NOUN
ejpam-5238	186	5	−	−	NOUN
ejpam-5238	186	6	σ1σ2	σ1σ2	NOUN
ejpam-5238	186	7	-	-	NUM
ejpam-5238	186	8	cl(v	cl(v	NOUN
ejpam-5238	186	9	)	)	PUNCT
ejpam-5238	186	10	)	)	PUNCT
ejpam-5238	186	11	)	)	PUNCT
ejpam-5238	187	1	=	=	PUNCT
ejpam-5238	187	2	x	x	X
ejpam-5238	187	3	−	−	ADP
ejpam-5238	187	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-5238	187	5	-	-	PUNCT
ejpam-5238	187	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	187	7	-	-	PUNCT
ejpam-5238	187	8	cl(v	cl(v	NOUN
ejpam-5238	187	9	)	)	PUNCT
ejpam-5238	187	10	)	)	PUNCT
ejpam-5238	187	11	)	)	PUNCT
ejpam-5238	188	1	⊆	⊆	NUM
ejpam-5238	188	2	x	x	SYM
ejpam-5238	188	3	−	−	NOUN
ejpam-5238	188	4	f+(v	f+(v	NOUN
ejpam-5238	188	5	)	)	PUNCT
ejpam-5238	188	6	and	and	CCONJ
ejpam-5238	188	7	hence	hence	ADV
ejpam-5238	188	8	f+(v	f+(v	NOUN
ejpam-5238	188	9	)	)	PUNCT
ejpam-5238	189	1	⊆	⊆	X
ejpam-5238	189	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	189	3	-	-	NUM
ejpam-5238	189	4	int(f	int(f	VERB
ejpam-5238	189	5	+	+	ADJ
ejpam-5238	189	6	(	(	PUNCT
ejpam-5238	189	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	189	8	-	-	NUM
ejpam-5238	189	9	cl(v	cl(v	NOUN
ejpam-5238	189	10	)	)	PUNCT
ejpam-5238	189	11	)	)	PUNCT
ejpam-5238	189	12	)	)	PUNCT
ejpam-5238	189	13	.	.	PUNCT
ejpam-5238	190	1	(	(	PUNCT
ejpam-5238	190	2	4	4	X
ejpam-5238	190	3	)	)	PUNCT
ejpam-5238	190	4	⇒	⇒	NOUN
ejpam-5238	190	5	(	(	PUNCT
ejpam-5238	190	6	1	1	NUM
ejpam-5238	190	7	):	):	PUNCT
ejpam-5238	190	8	let	let	VERB
ejpam-5238	190	9	v	v	PART
ejpam-5238	190	10	be	be	AUX
ejpam-5238	190	11	any	any	DET
ejpam-5238	190	12	σ1σ2	σ1σ2	NOUN
ejpam-5238	190	13	-	-	ADJ
ejpam-5238	190	14	open	open	ADJ
ejpam-5238	190	15	set	set	NOUN
ejpam-5238	190	16	of	of	ADP
ejpam-5238	190	17	y	y	PROPN
ejpam-5238	190	18	.	.	PUNCT
ejpam-5238	191	1	then	then	ADV
ejpam-5238	191	2	,	,	PUNCT
ejpam-5238	191	3	v	v	NOUN
ejpam-5238	191	4	is	be	AUX
ejpam-5238	191	5	(	(	PUNCT
ejpam-5238	191	6	σ1	σ1	PROPN
ejpam-5238	191	7	,	,	PUNCT
ejpam-5238	191	8	σ2)p	σ2)p	NOUN
ejpam-5238	191	9	-	-	PUNCT
ejpam-5238	191	10	open	open	ADJ
ejpam-5238	191	11	and	and	CCONJ
ejpam-5238	191	12	by	by	ADP
ejpam-5238	191	13	(	(	PUNCT
ejpam-5238	191	14	4	4	NUM
ejpam-5238	191	15	)	)	PUNCT
ejpam-5238	191	16	,	,	PUNCT
ejpam-5238	191	17	f+(v	f+(v	PROPN
ejpam-5238	191	18	)	)	PUNCT
ejpam-5238	192	1	⊆	⊆	X
ejpam-5238	192	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	192	3	-	-	NUM
ejpam-5238	192	4	int(f	int(f	VERB
ejpam-5238	192	5	+	+	ADJ
ejpam-5238	192	6	(	(	PUNCT
ejpam-5238	192	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	192	8	-	-	NUM
ejpam-5238	192	9	cl(v	cl(v	NOUN
ejpam-5238	192	10	)	)	PUNCT
ejpam-5238	192	11	)	)	PUNCT
ejpam-5238	192	12	)	)	PUNCT
ejpam-5238	192	13	.	.	PUNCT
ejpam-5238	193	1	by	by	ADP
ejpam-5238	193	2	theorem	theorem	NOUN
ejpam-5238	193	3	1(2	1(2	NUM
ejpam-5238	193	4	)	)	PUNCT
ejpam-5238	193	5	,	,	PUNCT
ejpam-5238	193	6	f	f	PROPN
ejpam-5238	193	7	is	be	AUX
ejpam-5238	193	8	upper	upper	ADJ
ejpam-5238	193	9	weakly	weakly	ADJ
ejpam-5238	193	10	(	(	PUNCT
ejpam-5238	193	11	τ1	τ1	NOUN
ejpam-5238	193	12	,	,	PUNCT
ejpam-5238	193	13	τ2)continuous	τ2)continuous	ADJ
ejpam-5238	193	14	.	.	PUNCT
ejpam-5238	194	1	theorem	theorem	VERB
ejpam-5238	194	2	6	6	NUM
ejpam-5238	194	3	.	.	PUNCT
ejpam-5238	194	4	for	for	ADP
ejpam-5238	194	5	a	a	DET
ejpam-5238	194	6	multifunction	multifunction	NOUN
ejpam-5238	195	1	f	f	NOUN
ejpam-5238	195	2	:	:	PUNCT
ejpam-5238	195	3	(	(	PUNCT
ejpam-5238	195	4	x	x	NOUN
ejpam-5238	195	5	,	,	PUNCT
ejpam-5238	195	6	τ1	τ1	NOUN
ejpam-5238	195	7	,	,	PUNCT
ejpam-5238	195	8	τ2	τ2	NOUN
ejpam-5238	195	9	)	)	PUNCT
ejpam-5238	195	10	→	→	SYM
ejpam-5238	195	11	(	(	PUNCT
ejpam-5238	195	12	y	y	PROPN
ejpam-5238	195	13	,	,	PUNCT
ejpam-5238	195	14	σ1	σ1	PROPN
ejpam-5238	195	15	,	,	PUNCT
ejpam-5238	195	16	σ2	σ2	NOUN
ejpam-5238	195	17	)	)	PUNCT
ejpam-5238	195	18	,	,	PUNCT
ejpam-5238	195	19	the	the	DET
ejpam-5238	195	20	following	follow	VERB
ejpam-5238	195	21	properties	property	NOUN
ejpam-5238	195	22	are	be	AUX
ejpam-5238	195	23	equivalent	equivalent	ADJ
ejpam-5238	195	24	:	:	PUNCT
ejpam-5238	195	25	(	(	PUNCT
ejpam-5238	195	26	1	1	X
ejpam-5238	195	27	)	)	PUNCT
ejpam-5238	195	28	f	f	PROPN
ejpam-5238	195	29	is	be	AUX
ejpam-5238	195	30	lower	low	ADJ
ejpam-5238	195	31	weakly	weakly	ADJ
ejpam-5238	195	32	(	(	PUNCT
ejpam-5238	195	33	τ1	τ1	NOUN
ejpam-5238	195	34	,	,	PUNCT
ejpam-5238	195	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	195	36	;	;	PUNCT
ejpam-5238	195	37	(	(	PUNCT
ejpam-5238	195	38	2	2	X
ejpam-5238	195	39	)	)	PUNCT
ejpam-5238	195	40	τ1τ2	τ1τ2	NOUN
ejpam-5238	195	41	-	-	NOUN
ejpam-5238	195	42	cl(f	cl(f	NOUN
ejpam-5238	195	43	+	+	NOUN
ejpam-5238	195	44	(	(	PUNCT
ejpam-5238	195	45	σ1σ2	σ1σ2	NUM
ejpam-5238	195	46	-	-	PUNCT
ejpam-5238	195	47	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	195	48	-	-	PUNCT
ejpam-5238	195	49	cl(v	cl(v	NOUN
ejpam-5238	195	50	)	)	PUNCT
ejpam-5238	195	51	)	)	PUNCT
ejpam-5238	195	52	)	)	PUNCT
ejpam-5238	195	53	)	)	PUNCT
ejpam-5238	196	1	⊆	⊆	X
ejpam-5238	196	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	196	3	-	-	PUNCT
ejpam-5238	196	4	cl(v	cl(v	NOUN
ejpam-5238	196	5	)	)	PUNCT
ejpam-5238	196	6	)	)	PUNCT
ejpam-5238	196	7	for	for	ADP
ejpam-5238	196	8	every	every	DET
ejpam-5238	196	9	(	(	PUNCT
ejpam-5238	196	10	σ1	σ1	PROPN
ejpam-5238	196	11	,	,	PUNCT
ejpam-5238	196	12	σ2)p	σ2)p	NOUN
ejpam-5238	196	13	-	-	PUNCT
ejpam-5238	196	14	open	open	NOUN
ejpam-5238	196	15	set	set	NOUN
ejpam-5238	196	16	v	v	NOUN
ejpam-5238	196	17	of	of	ADP
ejpam-5238	196	18	y	y	PROPN
ejpam-5238	196	19	;	;	PUNCT
ejpam-5238	196	20	(	(	PUNCT
ejpam-5238	196	21	3	3	X
ejpam-5238	196	22	)	)	PUNCT
ejpam-5238	196	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	196	24	-	-	NOUN
ejpam-5238	196	25	cl(f	cl(f	NOUN
ejpam-5238	196	26	+	+	NOUN
ejpam-5238	196	27	(	(	PUNCT
ejpam-5238	196	28	v	v	NOUN
ejpam-5238	196	29	)	)	PUNCT
ejpam-5238	196	30	)	)	PUNCT
ejpam-5238	196	31	⊆	⊆	NUM
ejpam-5238	196	32	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	196	33	-	-	PUNCT
ejpam-5238	196	34	cl(v	cl(v	NOUN
ejpam-5238	196	35	)	)	PUNCT
ejpam-5238	196	36	)	)	PUNCT
ejpam-5238	196	37	for	for	ADP
ejpam-5238	196	38	every	every	DET
ejpam-5238	196	39	(	(	PUNCT
ejpam-5238	196	40	σ1	σ1	PROPN
ejpam-5238	196	41	,	,	PUNCT
ejpam-5238	196	42	σ2)p	σ2)p	NOUN
ejpam-5238	196	43	-	-	PUNCT
ejpam-5238	196	44	open	open	NOUN
ejpam-5238	196	45	set	set	NOUN
ejpam-5238	196	46	v	v	NOUN
ejpam-5238	196	47	of	of	ADP
ejpam-5238	196	48	y	y	PROPN
ejpam-5238	196	49	;	;	PUNCT
ejpam-5238	196	50	(	(	PUNCT
ejpam-5238	196	51	4	4	X
ejpam-5238	196	52	)	)	PUNCT
ejpam-5238	196	53	f−(v	f−(v	NOUN
ejpam-5238	196	54	)	)	PUNCT
ejpam-5238	196	55	⊆	⊆	NUM
ejpam-5238	196	56	τ1τ2	τ1τ2	NOUN
ejpam-5238	196	57	-	-	NUM
ejpam-5238	196	58	int(f	int(f	PRON
ejpam-5238	196	59	−(σ1σ2	−(σ1σ2	NOUN
ejpam-5238	196	60	-	-	NOUN
ejpam-5238	196	61	cl(v	cl(v	NOUN
ejpam-5238	196	62	)	)	PUNCT
ejpam-5238	196	63	)	)	PUNCT
ejpam-5238	196	64	)	)	PUNCT
ejpam-5238	196	65	for	for	ADP
ejpam-5238	196	66	every	every	DET
ejpam-5238	196	67	(	(	PUNCT
ejpam-5238	196	68	σ1	σ1	PROPN
ejpam-5238	196	69	,	,	PUNCT
ejpam-5238	196	70	σ2)p	σ2)p	NOUN
ejpam-5238	196	71	-	-	PUNCT
ejpam-5238	196	72	open	open	NOUN
ejpam-5238	196	73	set	set	NOUN
ejpam-5238	196	74	v	v	NOUN
ejpam-5238	196	75	of	of	ADP
ejpam-5238	196	76	y	y	PROPN
ejpam-5238	196	77	.	.	PUNCT
ejpam-5238	197	1	proof	proof	NOUN
ejpam-5238	197	2	.	.	PUNCT
ejpam-5238	198	1	the	the	DET
ejpam-5238	198	2	proof	proof	NOUN
ejpam-5238	198	3	is	be	AUX
ejpam-5238	198	4	similar	similar	ADJ
ejpam-5238	198	5	to	to	ADP
ejpam-5238	198	6	that	that	PRON
ejpam-5238	198	7	of	of	ADP
ejpam-5238	198	8	theorem	theorem	ADJ
ejpam-5238	198	9	5	5	NUM
ejpam-5238	198	10	.	.	PUNCT
ejpam-5238	198	11	corollary	corollary	ADJ
ejpam-5238	198	12	3	3	NUM
ejpam-5238	198	13	.	.	PUNCT
ejpam-5238	199	1	for	for	ADP
ejpam-5238	199	2	a	a	DET
ejpam-5238	199	3	function	function	NOUN
ejpam-5238	199	4	f	f	NOUN
ejpam-5238	199	5	:	:	PUNCT
ejpam-5238	199	6	(	(	PUNCT
ejpam-5238	199	7	x	x	NOUN
ejpam-5238	199	8	,	,	PUNCT
ejpam-5238	199	9	τ1	τ1	NOUN
ejpam-5238	199	10	,	,	PUNCT
ejpam-5238	199	11	τ2	τ2	NOUN
ejpam-5238	199	12	)	)	PUNCT
ejpam-5238	199	13	→	→	SYM
ejpam-5238	199	14	(	(	PUNCT
ejpam-5238	199	15	y	y	PROPN
ejpam-5238	199	16	,	,	PUNCT
ejpam-5238	199	17	σ1	σ1	PROPN
ejpam-5238	199	18	,	,	PUNCT
ejpam-5238	199	19	σ2	σ2	NOUN
ejpam-5238	199	20	)	)	PUNCT
ejpam-5238	199	21	,	,	PUNCT
ejpam-5238	199	22	the	the	DET
ejpam-5238	199	23	following	follow	VERB
ejpam-5238	199	24	properties	property	NOUN
ejpam-5238	199	25	are	be	AUX
ejpam-5238	199	26	equivalent	equivalent	ADJ
ejpam-5238	199	27	:	:	PUNCT
ejpam-5238	199	28	(	(	PUNCT
ejpam-5238	199	29	1	1	X
ejpam-5238	199	30	)	)	PUNCT
ejpam-5238	199	31	f	f	PROPN
ejpam-5238	199	32	is	be	AUX
ejpam-5238	199	33	weakly	weakly	ADJ
ejpam-5238	199	34	(	(	PUNCT
ejpam-5238	199	35	τ1	τ1	NOUN
ejpam-5238	199	36	,	,	PUNCT
ejpam-5238	199	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	199	38	;	;	PUNCT
ejpam-5238	199	39	(	(	PUNCT
ejpam-5238	199	40	2	2	X
ejpam-5238	199	41	)	)	PUNCT
ejpam-5238	199	42	τ1τ2	τ1τ2	NOUN
ejpam-5238	199	43	-	-	NOUN
ejpam-5238	199	44	cl(f	cl(f	NOUN
ejpam-5238	199	45	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	199	46	-	-	PUNCT
ejpam-5238	199	47	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5238	199	48	-	-	PUNCT
ejpam-5238	199	49	cl(v	cl(v	NOUN
ejpam-5238	199	50	)	)	PUNCT
ejpam-5238	199	51	)	)	PUNCT
ejpam-5238	199	52	)	)	PUNCT
ejpam-5238	199	53	)	)	PUNCT
ejpam-5238	200	1	⊆	⊆	NUM
ejpam-5238	200	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	200	3	-	-	PUNCT
ejpam-5238	200	4	cl(v	cl(v	NOUN
ejpam-5238	200	5	)	)	PUNCT
ejpam-5238	200	6	)	)	PUNCT
ejpam-5238	200	7	for	for	ADP
ejpam-5238	200	8	every	every	DET
ejpam-5238	200	9	(	(	PUNCT
ejpam-5238	200	10	σ1	σ1	PROPN
ejpam-5238	200	11	,	,	PUNCT
ejpam-5238	200	12	σ2)p	σ2)p	NOUN
ejpam-5238	200	13	-	-	PUNCT
ejpam-5238	200	14	open	open	NOUN
ejpam-5238	200	15	set	set	NOUN
ejpam-5238	200	16	v	v	NOUN
ejpam-5238	200	17	of	of	ADP
ejpam-5238	200	18	y	y	PROPN
ejpam-5238	200	19	;	;	PUNCT
ejpam-5238	200	20	(	(	PUNCT
ejpam-5238	200	21	3	3	X
ejpam-5238	200	22	)	)	PUNCT
ejpam-5238	200	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	200	24	-	-	NOUN
ejpam-5238	200	25	cl(f	cl(f	PRON
ejpam-5238	200	26	−1(v	−1(v	NOUN
ejpam-5238	200	27	)	)	PUNCT
ejpam-5238	200	28	)	)	PUNCT
ejpam-5238	201	1	⊆	⊆	NUM
ejpam-5238	201	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5238	201	3	-	-	PUNCT
ejpam-5238	201	4	cl(v	cl(v	NOUN
ejpam-5238	201	5	)	)	PUNCT
ejpam-5238	201	6	)	)	PUNCT
ejpam-5238	201	7	for	for	ADP
ejpam-5238	201	8	every	every	DET
ejpam-5238	201	9	(	(	PUNCT
ejpam-5238	201	10	σ1	σ1	PROPN
ejpam-5238	201	11	,	,	PUNCT
ejpam-5238	201	12	σ2)p	σ2)p	NOUN
ejpam-5238	201	13	-	-	PUNCT
ejpam-5238	201	14	open	open	NOUN
ejpam-5238	201	15	set	set	NOUN
ejpam-5238	201	16	v	v	NOUN
ejpam-5238	201	17	of	of	ADP
ejpam-5238	201	18	y	y	PROPN
ejpam-5238	201	19	;	;	PUNCT
ejpam-5238	201	20	(	(	PUNCT
ejpam-5238	201	21	4	4	X
ejpam-5238	201	22	)	)	PUNCT
ejpam-5238	201	23	f−1(v	f−1(v	NOUN
ejpam-5238	201	24	)	)	PUNCT
ejpam-5238	202	1	⊆	⊆	X
ejpam-5238	202	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	202	3	-	-	NUM
ejpam-5238	202	4	int(f	int(f	PRON
ejpam-5238	202	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5238	202	6	-	-	PUNCT
ejpam-5238	202	7	cl(v	cl(v	NOUN
ejpam-5238	202	8	)	)	PUNCT
ejpam-5238	202	9	)	)	PUNCT
ejpam-5238	202	10	)	)	PUNCT
ejpam-5238	202	11	for	for	ADP
ejpam-5238	202	12	every	every	DET
ejpam-5238	202	13	(	(	PUNCT
ejpam-5238	202	14	σ1	σ1	PROPN
ejpam-5238	202	15	,	,	PUNCT
ejpam-5238	202	16	σ2)p	σ2)p	NOUN
ejpam-5238	202	17	-	-	PUNCT
ejpam-5238	202	18	open	open	NOUN
ejpam-5238	202	19	set	set	NOUN
ejpam-5238	202	20	v	v	NOUN
ejpam-5238	202	21	of	of	ADP
ejpam-5238	202	22	y	y	PROPN
ejpam-5238	202	23	.	.	PUNCT
ejpam-5238	203	1	m.	m.	NOUN
ejpam-5238	203	2	thongmoon	thongmoon	PROPN
ejpam-5238	203	3	,	,	PUNCT
ejpam-5238	203	4	s.	s.	PROPN
ejpam-5238	203	5	sompong	sompong	PROPN
ejpam-5238	203	6	,	,	PUNCT
ejpam-5238	203	7	c.	c.	PROPN
ejpam-5238	203	8	boonpok	boonpok	PROPN
ejpam-5238	203	9	/	/	SYM
ejpam-5238	203	10	eur	eur	PROPN
ejpam-5238	203	11	.	.	PUNCT
ejpam-5238	204	1	j.	j.	PROPN
ejpam-5238	204	2	pure	pure	PROPN
ejpam-5238	204	3	appl	appl	PROPN
ejpam-5238	204	4	.	.	PROPN
ejpam-5238	204	5	math	math	PROPN
ejpam-5238	204	6	,	,	PUNCT
ejpam-5238	204	7	17	17	NUM
ejpam-5238	204	8	(	(	PUNCT
ejpam-5238	204	9	3	3	NUM
ejpam-5238	204	10	)	)	PUNCT
ejpam-5238	204	11	(	(	PUNCT
ejpam-5238	204	12	2024	2024	NUM
ejpam-5238	204	13	)	)	PUNCT
ejpam-5238	204	14	,	,	PUNCT
ejpam-5238	204	15	1705	1705	NUM
ejpam-5238	204	16	-	-	SYM
ejpam-5238	204	17	1716	1716	NUM
ejpam-5238	204	18	1712	1712	NUM
ejpam-5238	204	19	4	4	NUM
ejpam-5238	204	20	.	.	PUNCT
ejpam-5238	205	1	several	several	ADJ
ejpam-5238	205	2	characterizations	characterization	NOUN
ejpam-5238	205	3	recall	recall	VERB
ejpam-5238	205	4	that	that	SCONJ
ejpam-5238	205	5	a	a	DET
ejpam-5238	205	6	bitopological	bitopological	ADJ
ejpam-5238	205	7	space	space	NOUN
ejpam-5238	205	8	(	(	PUNCT
ejpam-5238	205	9	x	x	NOUN
ejpam-5238	205	10	,	,	PUNCT
ejpam-5238	205	11	τ1	τ1	NOUN
ejpam-5238	205	12	,	,	PUNCT
ejpam-5238	205	13	τ2	τ2	NOUN
ejpam-5238	205	14	)	)	PUNCT
ejpam-5238	205	15	is	be	AUX
ejpam-5238	205	16	said	say	VERB
ejpam-5238	205	17	to	to	PART
ejpam-5238	205	18	be	be	AUX
ejpam-5238	205	19	τ1τ2	τ1τ2	NOUN
ejpam-5238	205	20	-	-	ADJ
ejpam-5238	205	21	compact	compact	ADJ
ejpam-5238	205	22	[	[	X
ejpam-5238	205	23	20	20	NUM
ejpam-5238	205	24	]	]	PUNCT
ejpam-5238	205	25	if	if	SCONJ
ejpam-5238	205	26	every	every	DET
ejpam-5238	205	27	cover	cover	NOUN
ejpam-5238	205	28	of	of	ADP
ejpam-5238	205	29	x	x	PUNCT
ejpam-5238	205	30	by	by	ADP
ejpam-5238	205	31	τ1τ2	τ1τ2	ADJ
ejpam-5238	205	32	-	-	ADJ
ejpam-5238	205	33	open	open	ADJ
ejpam-5238	205	34	sets	set	NOUN
ejpam-5238	205	35	of	of	ADP
ejpam-5238	205	36	x	x	PUNCT
ejpam-5238	205	37	has	have	VERB
ejpam-5238	205	38	a	a	DET
ejpam-5238	205	39	finite	finite	ADJ
ejpam-5238	205	40	subcover	subcover	PROPN
ejpam-5238	205	41	.	.	PUNCT
ejpam-5238	206	1	definition	definition	NOUN
ejpam-5238	206	2	4	4	NUM
ejpam-5238	206	3	.	.	PUNCT
ejpam-5238	207	1	a	a	DET
ejpam-5238	207	2	bitopological	bitopological	ADJ
ejpam-5238	207	3	space	space	NOUN
ejpam-5238	207	4	(	(	PUNCT
ejpam-5238	207	5	x	x	NOUN
ejpam-5238	207	6	,	,	PUNCT
ejpam-5238	207	7	τ1	τ1	NOUN
ejpam-5238	207	8	,	,	PUNCT
ejpam-5238	207	9	τ2	τ2	NOUN
ejpam-5238	207	10	)	)	PUNCT
ejpam-5238	207	11	is	be	AUX
ejpam-5238	207	12	said	say	VERB
ejpam-5238	207	13	to	to	PART
ejpam-5238	207	14	be	be	AUX
ejpam-5238	207	15	quasi	quasi	X
ejpam-5238	207	16	(	(	PUNCT
ejpam-5238	207	17	τ1	τ1	NOUN
ejpam-5238	207	18	,	,	PUNCT
ejpam-5238	207	19	τ2)-h	τ2)-h	PUNCT
ejpam-5238	207	20	-closed	-close	VERB
ejpam-5238	207	21	if	if	SCONJ
ejpam-5238	207	22	every	every	DET
ejpam-5238	207	23	τ1τ2	τ1τ2	ADJ
ejpam-5238	207	24	-	-	ADJ
ejpam-5238	207	25	open	open	ADJ
ejpam-5238	207	26	cover	cover	NOUN
ejpam-5238	207	27	{	{	PUNCT
ejpam-5238	207	28	uγ	uγ	ADV
ejpam-5238	207	29	|	|	ADV
ejpam-5238	207	30	γ	γ	X
ejpam-5238	207	31	∈	∈	PROPN
ejpam-5238	207	32	γ	γ	X
ejpam-5238	207	33	}	}	PUNCT
ejpam-5238	207	34	,	,	PUNCT
ejpam-5238	207	35	there	there	PRON
ejpam-5238	207	36	exists	exist	VERB
ejpam-5238	207	37	a	a	DET
ejpam-5238	207	38	finite	finite	NOUN
ejpam-5238	207	39	subset	subset	NOUN
ejpam-5238	207	40	γ0	γ0	NOUN
ejpam-5238	207	41	of	of	ADP
ejpam-5238	207	42	γ	γ	NOUN
ejpam-5238	208	1	such	such	ADJ
ejpam-5238	208	2	that	that	SCONJ
ejpam-5238	208	3	x	x	X
ejpam-5238	208	4	=	=	PUNCT
ejpam-5238	208	5	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-5238	208	6	-	-	NOUN
ejpam-5238	208	7	cl(uγ	cl(uγ	NOUN
ejpam-5238	208	8	)	)	PUNCT
ejpam-5238	208	9	|	|	ADV
ejpam-5238	208	10	γ	γ	PROPN
ejpam-5238	208	11	∈	∈	PROPN
ejpam-5238	208	12	γ0	γ0	PROPN
ejpam-5238	208	13	}	}	PUNCT
ejpam-5238	208	14	.	.	PUNCT
ejpam-5238	209	1	theorem	theorem	VERB
ejpam-5238	209	2	7	7	NUM
ejpam-5238	209	3	.	.	PUNCT
ejpam-5238	210	1	let	let	VERB
ejpam-5238	210	2	f	f	NOUN
ejpam-5238	210	3	:	:	PUNCT
ejpam-5238	210	4	(	(	PUNCT
ejpam-5238	210	5	x	x	NOUN
ejpam-5238	210	6	,	,	PUNCT
ejpam-5238	210	7	τ1	τ1	NOUN
ejpam-5238	210	8	,	,	PUNCT
ejpam-5238	210	9	τ2	τ2	NOUN
ejpam-5238	210	10	)	)	PUNCT
ejpam-5238	210	11	→	→	SYM
ejpam-5238	210	12	(	(	PUNCT
ejpam-5238	210	13	y	y	PROPN
ejpam-5238	210	14	,	,	PUNCT
ejpam-5238	210	15	σ1	σ1	PROPN
ejpam-5238	210	16	,	,	PUNCT
ejpam-5238	210	17	σ2	σ2	PROPN
ejpam-5238	210	18	)	)	PUNCT
ejpam-5238	210	19	be	be	VERB
ejpam-5238	210	20	an	an	DET
ejpam-5238	210	21	upper	upper	ADJ
ejpam-5238	210	22	weakly	weakly	ADJ
ejpam-5238	210	23	(	(	PUNCT
ejpam-5238	210	24	τ1	τ1	NOUN
ejpam-5238	210	25	,	,	PUNCT
ejpam-5238	210	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	210	27	surjective	surjective	ADJ
ejpam-5238	210	28	multifunction	multifunction	NOUN
ejpam-5238	210	29	such	such	ADJ
ejpam-5238	210	30	that	that	SCONJ
ejpam-5238	210	31	f	f	PROPN
ejpam-5238	210	32	(	(	PUNCT
ejpam-5238	210	33	x	x	X
ejpam-5238	210	34	)	)	PUNCT
ejpam-5238	210	35	is	be	AUX
ejpam-5238	210	36	σ1σ2	σ1σ2	NOUN
ejpam-5238	210	37	-	-	ADJ
ejpam-5238	210	38	compact	compact	ADJ
ejpam-5238	210	39	for	for	ADP
ejpam-5238	210	40	each	each	DET
ejpam-5238	210	41	x	x	SYM
ejpam-5238	210	42	∈	∈	PROPN
ejpam-5238	210	43	x.	x.	NOUN
ejpam-5238	211	1	if	if	SCONJ
ejpam-5238	211	2	(	(	PUNCT
ejpam-5238	211	3	x	x	NOUN
ejpam-5238	211	4	,	,	PUNCT
ejpam-5238	211	5	τ1	τ1	NOUN
ejpam-5238	211	6	,	,	PUNCT
ejpam-5238	211	7	τ2	τ2	NOUN
ejpam-5238	211	8	)	)	PUNCT
ejpam-5238	211	9	is	be	AUX
ejpam-5238	211	10	τ1τ2	τ1τ2	NOUN
ejpam-5238	211	11	-	-	ADJ
ejpam-5238	211	12	compact	compact	ADJ
ejpam-5238	211	13	,	,	PUNCT
ejpam-5238	211	14	then	then	ADV
ejpam-5238	211	15	(	(	PUNCT
ejpam-5238	211	16	y	y	PROPN
ejpam-5238	211	17	,	,	PUNCT
ejpam-5238	211	18	σ1	σ1	PROPN
ejpam-5238	211	19	,	,	PUNCT
ejpam-5238	211	20	σ2	σ2	PROPN
ejpam-5238	211	21	)	)	PUNCT
ejpam-5238	211	22	is	be	AUX
ejpam-5238	211	23	quasi	quasi	X
ejpam-5238	211	24	(	(	PUNCT
ejpam-5238	211	25	σ1	σ1	PROPN
ejpam-5238	211	26	,	,	PUNCT
ejpam-5238	211	27	σ2)-h	σ2)-h	PROPN
ejpam-5238	211	28	-closed	-closed	ADJ
ejpam-5238	211	29	.	.	PUNCT
ejpam-5238	212	1	proof	proof	NOUN
ejpam-5238	212	2	.	.	PUNCT
ejpam-5238	213	1	let	let	VERB
ejpam-5238	213	2	{	{	PUNCT
ejpam-5238	213	3	vγ	vγ	VERB
ejpam-5238	213	4	|	|	ADV
ejpam-5238	213	5	γ	γ	PROPN
ejpam-5238	213	6	∈	∈	PROPN
ejpam-5238	213	7	γ	γ	AUX
ejpam-5238	213	8	}	}	PUNCT
ejpam-5238	213	9	be	be	VERB
ejpam-5238	213	10	any	any	DET
ejpam-5238	213	11	σ1σ2	σ1σ2	NOUN
ejpam-5238	213	12	-	-	PUNCT
ejpam-5238	213	13	open	open	ADJ
ejpam-5238	213	14	cover	cover	NOUN
ejpam-5238	213	15	of	of	ADP
ejpam-5238	213	16	y	y	PROPN
ejpam-5238	213	17	.	.	PUNCT
ejpam-5238	214	1	for	for	ADP
ejpam-5238	214	2	each	each	DET
ejpam-5238	214	3	x	x	SYM
ejpam-5238	214	4	∈	∈	PROPN
ejpam-5238	214	5	x	x	X
ejpam-5238	214	6	,	,	PUNCT
ejpam-5238	214	7	f	f	PROPN
ejpam-5238	214	8	(	(	PUNCT
ejpam-5238	214	9	x	x	X
ejpam-5238	214	10	)	)	PUNCT
ejpam-5238	214	11	is	be	AUX
ejpam-5238	214	12	σ1σ2compact	σ1σ2compact	PUNCT
ejpam-5238	214	13	and	and	CCONJ
ejpam-5238	214	14	there	there	PRON
ejpam-5238	214	15	exists	exist	VERB
ejpam-5238	214	16	a	a	DET
ejpam-5238	214	17	finite	finite	NOUN
ejpam-5238	214	18	subset	subset	NOUN
ejpam-5238	214	19	γ(x	γ(x	NOUN
ejpam-5238	214	20	)	)	PUNCT
ejpam-5238	214	21	of	of	ADP
ejpam-5238	214	22	γ	γ	PRON
ejpam-5238	214	23	such	such	ADJ
ejpam-5238	214	24	that	that	SCONJ
ejpam-5238	214	25	f	f	PROPN
ejpam-5238	214	26	(	(	PUNCT
ejpam-5238	214	27	x	x	X
ejpam-5238	214	28	)	)	PUNCT
ejpam-5238	214	29	⊆	⊆	NUM
ejpam-5238	214	30	∪{vγ	∪{vγ	PROPN
ejpam-5238	214	31	|	|	ADV
ejpam-5238	214	32	γ	γ	X
ejpam-5238	214	33	∈	∈	PROPN
ejpam-5238	214	34	γ(x	γ(x	PROPN
ejpam-5238	214	35	)	)	PUNCT
ejpam-5238	214	36	}	}	PUNCT
ejpam-5238	214	37	.	.	PUNCT
ejpam-5238	215	1	now	now	ADV
ejpam-5238	215	2	,	,	PUNCT
ejpam-5238	215	3	set	set	VERB
ejpam-5238	215	4	v	v	NOUN
ejpam-5238	215	5	(	(	PUNCT
ejpam-5238	215	6	x	x	NOUN
ejpam-5238	215	7	)	)	PUNCT
ejpam-5238	215	8	=	=	SYM
ejpam-5238	215	9	∪{vγ	∪{vγ	PROPN
ejpam-5238	215	10	|	|	ADV
ejpam-5238	215	11	γ	γ	X
ejpam-5238	215	12	∈	∈	PROPN
ejpam-5238	215	13	γ(x	γ(x	PROPN
ejpam-5238	215	14	)	)	PUNCT
ejpam-5238	215	15	}	}	PUNCT
ejpam-5238	215	16	.	.	PUNCT
ejpam-5238	216	1	since	since	SCONJ
ejpam-5238	216	2	f	f	PROPN
ejpam-5238	216	3	is	be	AUX
ejpam-5238	216	4	upper	upper	ADJ
ejpam-5238	216	5	weakly	weakly	ADJ
ejpam-5238	216	6	(	(	PUNCT
ejpam-5238	216	7	τ1	τ1	NOUN
ejpam-5238	216	8	,	,	PUNCT
ejpam-5238	216	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	216	10	,	,	PUNCT
ejpam-5238	216	11	there	there	PRON
ejpam-5238	216	12	exists	exist	VERB
ejpam-5238	216	13	a	a	DET
ejpam-5238	216	14	τ1τ2	τ1τ2	NOUN
ejpam-5238	216	15	-	-	ADJ
ejpam-5238	216	16	open	open	ADJ
ejpam-5238	216	17	set	set	ADJ
ejpam-5238	216	18	u(x	u(x	NOUN
ejpam-5238	216	19	)	)	PUNCT
ejpam-5238	216	20	of	of	ADP
ejpam-5238	216	21	x	x	SYM
ejpam-5238	216	22	containing	contain	VERB
ejpam-5238	216	23	x	x	PUNCT
ejpam-5238	216	24	such	such	ADJ
ejpam-5238	216	25	that	that	SCONJ
ejpam-5238	216	26	f	f	PROPN
ejpam-5238	216	27	(	(	PUNCT
ejpam-5238	216	28	u(x	u(x	PROPN
ejpam-5238	216	29	)	)	PUNCT
ejpam-5238	216	30	)	)	PUNCT
ejpam-5238	216	31	⊆	⊆	X
ejpam-5238	216	32	σ1σ2	σ1σ2	NOUN
ejpam-5238	216	33	-	-	PUNCT
ejpam-5238	216	34	cl(v	cl(v	PRON
ejpam-5238	216	35	(	(	PUNCT
ejpam-5238	216	36	x	x	NOUN
ejpam-5238	216	37	)	)	PUNCT
ejpam-5238	216	38	)	)	PUNCT
ejpam-5238	216	39	.	.	PUNCT
ejpam-5238	217	1	the	the	DET
ejpam-5238	217	2	family	family	NOUN
ejpam-5238	217	3	{	{	PUNCT
ejpam-5238	217	4	u(x	u(x	PROPN
ejpam-5238	217	5	)	)	PUNCT
ejpam-5238	217	6	|	|	ADV
ejpam-5238	217	7	x	x	SYM
ejpam-5238	217	8	∈	∈	NOUN
ejpam-5238	217	9	x	x	X
ejpam-5238	217	10	}	}	PUNCT
ejpam-5238	217	11	is	be	AUX
ejpam-5238	217	12	a	a	DET
ejpam-5238	217	13	τ1τ2	τ1τ2	ADJ
ejpam-5238	217	14	-	-	ADJ
ejpam-5238	217	15	open	open	ADJ
ejpam-5238	217	16	cover	cover	NOUN
ejpam-5238	217	17	of	of	ADP
ejpam-5238	217	18	x	x	PUNCT
ejpam-5238	217	19	by	by	ADP
ejpam-5238	217	20	τ1τ2	τ1τ2	ADJ
ejpam-5238	217	21	-	-	ADJ
ejpam-5238	217	22	open	open	ADJ
ejpam-5238	217	23	sets	set	NOUN
ejpam-5238	217	24	.	.	PUNCT
ejpam-5238	218	1	since	since	SCONJ
ejpam-5238	218	2	(	(	PUNCT
ejpam-5238	218	3	x	x	NOUN
ejpam-5238	218	4	,	,	PUNCT
ejpam-5238	218	5	τ1	τ1	NOUN
ejpam-5238	218	6	,	,	PUNCT
ejpam-5238	218	7	τ2	τ2	NOUN
ejpam-5238	218	8	)	)	PUNCT
ejpam-5238	218	9	is	be	AUX
ejpam-5238	218	10	τ1τ2	τ1τ2	NOUN
ejpam-5238	218	11	-	-	ADJ
ejpam-5238	218	12	compact	compact	ADJ
ejpam-5238	218	13	,	,	PUNCT
ejpam-5238	218	14	there	there	PRON
ejpam-5238	218	15	exists	exist	VERB
ejpam-5238	218	16	a	a	DET
ejpam-5238	218	17	finite	finite	ADJ
ejpam-5238	218	18	number	number	NOUN
ejpam-5238	218	19	of	of	ADP
ejpam-5238	218	20	points	point	NOUN
ejpam-5238	218	21	,	,	PUNCT
ejpam-5238	218	22	say	say	INTJ
ejpam-5238	218	23	,	,	PUNCT
ejpam-5238	218	24	x1	x1	PROPN
ejpam-5238	218	25	,	,	PUNCT
ejpam-5238	218	26	x2	x2	PROPN
ejpam-5238	218	27	,	,	PUNCT
ejpam-5238	218	28	...	...	PUNCT
ejpam-5238	218	29	,	,	PUNCT
ejpam-5238	218	30	xn	xn	PROPN
ejpam-5238	219	1	in	in	ADP
ejpam-5238	219	2	x	x	X
ejpam-5238	219	3	such	such	ADJ
ejpam-5238	219	4	that	that	SCONJ
ejpam-5238	219	5	x	x	X
ejpam-5238	219	6	=	=	PUNCT
ejpam-5238	219	7	∪{u(xi	∪{u(xi	X
ejpam-5238	219	8	)	)	PUNCT
ejpam-5238	219	9	|	|	ADV
ejpam-5238	219	10	1	1	NUM
ejpam-5238	219	11	≤	≤	NUM
ejpam-5238	219	12	i	i	PRON
ejpam-5238	219	13	≤	≤	NOUN
ejpam-5238	219	14	n	n	CCONJ
ejpam-5238	219	15	}	}	PUNCT
ejpam-5238	219	16	.	.	PUNCT
ejpam-5238	220	1	thus	thus	ADV
ejpam-5238	220	2	,	,	PUNCT
ejpam-5238	220	3	y	y	PROPN
ejpam-5238	220	4	=	=	SYM
ejpam-5238	220	5	f	f	PROPN
ejpam-5238	220	6	(	(	PUNCT
ejpam-5238	220	7	x	x	NOUN
ejpam-5238	220	8	)	)	PUNCT
ejpam-5238	220	9	=	=	SYM
ejpam-5238	220	10	∪{f	∪{f	PROPN
ejpam-5238	220	11	(	(	PUNCT
ejpam-5238	220	12	u(xi	u(xi	PROPN
ejpam-5238	220	13	)	)	PUNCT
ejpam-5238	220	14	)	)	PUNCT
ejpam-5238	221	1	|	|	ADV
ejpam-5238	221	2	1	1	NUM
ejpam-5238	221	3	≤	≤	NUM
ejpam-5238	221	4	i	i	PRON
ejpam-5238	221	5	≤	≤	NOUN
ejpam-5238	221	6	n	n	CCONJ
ejpam-5238	221	7	}	}	PUNCT
ejpam-5238	221	8	⊆	⊆	NUM
ejpam-5238	221	9	∪{σ1σ2	∪{σ1σ2	NOUN
ejpam-5238	221	10	-	-	PUNCT
ejpam-5238	221	11	cl(v	cl(v	PRON
ejpam-5238	221	12	(	(	PUNCT
ejpam-5238	221	13	xi	xi	NOUN
ejpam-5238	221	14	)	)	PUNCT
ejpam-5238	221	15	)	)	PUNCT
ejpam-5238	222	1	|	|	ADV
ejpam-5238	222	2	1	1	NUM
ejpam-5238	222	3	≤	≤	NUM
ejpam-5238	222	4	i	i	PRON
ejpam-5238	222	5	≤	≤	NOUN
ejpam-5238	222	6	n	n	CCONJ
ejpam-5238	222	7	}	}	PUNCT
ejpam-5238	222	8	⊆	⊆	NUM
ejpam-5238	222	9	∪{σ1σ2	∪{σ1σ2	NOUN
ejpam-5238	222	10	-	-	PUNCT
ejpam-5238	222	11	cl(vγ	cl(vγ	ADJ
ejpam-5238	222	12	)	)	PUNCT
ejpam-5238	223	1	|	|	ADV
ejpam-5238	223	2	γ	γ	PROPN
ejpam-5238	223	3	∈	∈	PROPN
ejpam-5238	223	4	γ(xi	γ(xi	PROPN
ejpam-5238	223	5	)	)	PUNCT
ejpam-5238	223	6	,	,	PUNCT
ejpam-5238	223	7	1	1	NUM
ejpam-5238	223	8	≤	≤	NUM
ejpam-5238	223	9	i	i	PRON
ejpam-5238	223	10	≤	≤	NOUN
ejpam-5238	223	11	n	n	CCONJ
ejpam-5238	223	12	}	}	PUNCT
ejpam-5238	223	13	.	.	PUNCT
ejpam-5238	224	1	this	this	PRON
ejpam-5238	224	2	shows	show	VERB
ejpam-5238	224	3	that	that	SCONJ
ejpam-5238	224	4	(	(	PUNCT
ejpam-5238	224	5	y	y	PROPN
ejpam-5238	224	6	,	,	PUNCT
ejpam-5238	224	7	σ1	σ1	PROPN
ejpam-5238	224	8	,	,	PUNCT
ejpam-5238	224	9	σ2	σ2	PROPN
ejpam-5238	224	10	)	)	PUNCT
ejpam-5238	224	11	is	be	AUX
ejpam-5238	224	12	quasi	quasi	X
ejpam-5238	224	13	(	(	PUNCT
ejpam-5238	224	14	σ1	σ1	PROPN
ejpam-5238	224	15	,	,	PUNCT
ejpam-5238	224	16	σ2)-h	σ2)-h	PROPN
ejpam-5238	224	17	-closed	-closed	ADJ
ejpam-5238	224	18	.	.	PUNCT
ejpam-5238	225	1	the	the	DET
ejpam-5238	225	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	225	3	-	-	NOUN
ejpam-5238	225	4	frontier	frontier	NOUN
ejpam-5238	225	5	[	[	X
ejpam-5238	225	6	17	17	NUM
ejpam-5238	225	7	]	]	PUNCT
ejpam-5238	225	8	of	of	ADP
ejpam-5238	225	9	a	a	DET
ejpam-5238	225	10	subset	subset	NOUN
ejpam-5238	225	11	a	a	PRON
ejpam-5238	225	12	of	of	ADP
ejpam-5238	225	13	a	a	DET
ejpam-5238	225	14	bitopological	bitopological	ADJ
ejpam-5238	225	15	space	space	NOUN
ejpam-5238	225	16	(	(	PUNCT
ejpam-5238	225	17	x	x	NOUN
ejpam-5238	225	18	,	,	PUNCT
ejpam-5238	225	19	τ1	τ1	NOUN
ejpam-5238	225	20	,	,	PUNCT
ejpam-5238	225	21	τ2	τ2	PROPN
ejpam-5238	225	22	)	)	PUNCT
ejpam-5238	225	23	,	,	PUNCT
ejpam-5238	225	24	denoted	denote	VERB
ejpam-5238	225	25	by	by	ADP
ejpam-5238	225	26	τ1τ2	τ1τ2	NOUN
ejpam-5238	225	27	-	-	ADJ
ejpam-5238	225	28	fr(a	fr(a	NUM
ejpam-5238	225	29	)	)	PUNCT
ejpam-5238	225	30	,	,	PUNCT
ejpam-5238	225	31	is	be	AUX
ejpam-5238	225	32	defined	define	VERB
ejpam-5238	225	33	by	by	ADP
ejpam-5238	225	34	τ1τ2	τ1τ2	NOUN
ejpam-5238	225	35	-	-	ADJ
ejpam-5238	225	36	fr(a	fr(a	ADJ
ejpam-5238	225	37	)	)	PUNCT
ejpam-5238	226	1	=	=	PUNCT
ejpam-5238	226	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	226	3	-	-	NUM
ejpam-5238	226	4	cl(a	cl(a	NUM
ejpam-5238	226	5	)	)	PUNCT
ejpam-5238	226	6	∩	∩	NOUN
ejpam-5238	226	7	τ1τ2	τ1τ2	NOUN
ejpam-5238	226	8	-	-	ADJ
ejpam-5238	226	9	cl(x	cl(x	SYM
ejpam-5238	226	10	−a	−a	NOUN
ejpam-5238	226	11	)	)	PUNCT
ejpam-5238	226	12	=	=	PUNCT
ejpam-5238	227	1	τ1τ2	τ1τ2	ADJ
ejpam-5238	227	2	-	-	ADJ
ejpam-5238	227	3	cl(a)−	cl(a)−	ADJ
ejpam-5238	227	4	τ1τ2	τ1τ2	NOUN
ejpam-5238	227	5	-	-	ADJ
ejpam-5238	227	6	int(a	int(a	NOUN
ejpam-5238	227	7	)	)	PUNCT
ejpam-5238	227	8	.	.	PUNCT
ejpam-5238	228	1	theorem	theorem	ADJ
ejpam-5238	228	2	8	8	NUM
ejpam-5238	228	3	.	.	PUNCT
ejpam-5238	229	1	the	the	DET
ejpam-5238	229	2	set	set	NOUN
ejpam-5238	229	3	of	of	ADP
ejpam-5238	229	4	all	all	DET
ejpam-5238	229	5	points	point	NOUN
ejpam-5238	229	6	x	x	PUNCT
ejpam-5238	229	7	of	of	ADP
ejpam-5238	229	8	x	x	SYM
ejpam-5238	229	9	at	at	ADP
ejpam-5238	229	10	which	which	PRON
ejpam-5238	229	11	a	a	DET
ejpam-5238	229	12	multifunction	multifunction	NOUN
ejpam-5238	230	1	f	f	NOUN
ejpam-5238	230	2	:	:	PUNCT
ejpam-5238	230	3	(	(	PUNCT
ejpam-5238	230	4	x	x	NOUN
ejpam-5238	230	5	,	,	PUNCT
ejpam-5238	230	6	τ1	τ1	NOUN
ejpam-5238	230	7	,	,	PUNCT
ejpam-5238	230	8	τ2	τ2	NOUN
ejpam-5238	230	9	)	)	PUNCT
ejpam-5238	230	10	→	→	SYM
ejpam-5238	230	11	(	(	PUNCT
ejpam-5238	230	12	y	y	PROPN
ejpam-5238	230	13	,	,	PUNCT
ejpam-5238	230	14	σ1	σ1	PROPN
ejpam-5238	230	15	,	,	PUNCT
ejpam-5238	230	16	σ2	σ2	PROPN
ejpam-5238	230	17	)	)	PUNCT
ejpam-5238	230	18	is	be	AUX
ejpam-5238	230	19	not	not	PART
ejpam-5238	230	20	upper	upper	ADJ
ejpam-5238	230	21	weakly	weakly	ADJ
ejpam-5238	230	22	(	(	PUNCT
ejpam-5238	230	23	τ1	τ1	NOUN
ejpam-5238	230	24	,	,	PUNCT
ejpam-5238	230	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	230	26	is	be	AUX
ejpam-5238	230	27	identical	identical	ADJ
ejpam-5238	230	28	with	with	ADP
ejpam-5238	230	29	the	the	DET
ejpam-5238	230	30	union	union	NOUN
ejpam-5238	230	31	of	of	ADP
ejpam-5238	230	32	the	the	DET
ejpam-5238	230	33	τ1τ2	τ1τ2	NOUN
ejpam-5238	230	34	-	-	NOUN
ejpam-5238	230	35	frontier	frontier	NOUN
ejpam-5238	230	36	of	of	ADP
ejpam-5238	230	37	the	the	DET
ejpam-5238	230	38	upper	upper	ADJ
ejpam-5238	230	39	inverse	inverse	NOUN
ejpam-5238	230	40	images	image	NOUN
ejpam-5238	230	41	of	of	ADP
ejpam-5238	230	42	the	the	DET
ejpam-5238	230	43	σ1σ2	σ1σ2	NOUN
ejpam-5238	230	44	-	-	PUNCT
ejpam-5238	230	45	closures	closure	NOUN
ejpam-5238	230	46	of	of	ADP
ejpam-5238	230	47	σ1σ2	σ1σ2	NOUN
ejpam-5238	230	48	-	-	PUNCT
ejpam-5238	230	49	open	open	ADJ
ejpam-5238	230	50	sets	set	NOUN
ejpam-5238	230	51	containing	contain	VERB
ejpam-5238	230	52	f	f	X
ejpam-5238	230	53	(	(	PUNCT
ejpam-5238	230	54	x	x	NOUN
ejpam-5238	230	55	)	)	PUNCT
ejpam-5238	230	56	.	.	PUNCT
ejpam-5238	231	1	proof	proof	NOUN
ejpam-5238	231	2	.	.	PUNCT
ejpam-5238	232	1	let	let	VERB
ejpam-5238	232	2	x	x	PRON
ejpam-5238	232	3	be	be	AUX
ejpam-5238	232	4	a	a	DET
ejpam-5238	232	5	point	point	NOUN
ejpam-5238	232	6	of	of	ADP
ejpam-5238	232	7	x	x	PUNCT
ejpam-5238	232	8	at	at	ADP
ejpam-5238	232	9	which	which	PRON
ejpam-5238	232	10	f	f	NOUN
ejpam-5238	232	11	is	be	AUX
ejpam-5238	232	12	not	not	PART
ejpam-5238	232	13	upper	upper	ADJ
ejpam-5238	232	14	weakly	weakly	ADJ
ejpam-5238	232	15	(	(	PUNCT
ejpam-5238	232	16	τ1	τ1	NOUN
ejpam-5238	232	17	,	,	PUNCT
ejpam-5238	232	18	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5238	232	19	.	.	PUNCT
ejpam-5238	233	1	then	then	ADV
ejpam-5238	233	2	,	,	PUNCT
ejpam-5238	233	3	there	there	PRON
ejpam-5238	233	4	exists	exist	VERB
ejpam-5238	233	5	a	a	DET
ejpam-5238	233	6	σ1σ2	σ1σ2	NUM
ejpam-5238	233	7	-	-	ADJ
ejpam-5238	233	8	open	open	ADJ
ejpam-5238	233	9	set	set	NOUN
ejpam-5238	233	10	v	v	NOUN
ejpam-5238	233	11	containing	contain	VERB
ejpam-5238	233	12	f	f	X
ejpam-5238	233	13	(	(	PUNCT
ejpam-5238	233	14	x	x	X
ejpam-5238	233	15	)	)	PUNCT
ejpam-5238	233	16	such	such	ADJ
ejpam-5238	233	17	that	that	SCONJ
ejpam-5238	233	18	u	u	PROPN
ejpam-5238	233	19	∩	∩	NOUN
ejpam-5238	233	20	(	(	PUNCT
ejpam-5238	233	21	x	x	SYM
ejpam-5238	233	22	−	−	PRON
ejpam-5238	233	23	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	233	24	-	-	PUNCT
ejpam-5238	233	25	cl(v	cl(v	NOUN
ejpam-5238	233	26	)	)	PUNCT
ejpam-5238	233	27	)	)	PUNCT
ejpam-5238	233	28	)	)	PUNCT
ejpam-5238	234	1	̸=	̸=	NOUN
ejpam-5238	234	2	∅	∅	NOUN
ejpam-5238	234	3	for	for	ADP
ejpam-5238	234	4	every	every	DET
ejpam-5238	234	5	τ1τ2	τ1τ2	ADJ
ejpam-5238	234	6	-	-	ADJ
ejpam-5238	234	7	open	open	ADJ
ejpam-5238	234	8	set	set	NOUN
ejpam-5238	234	9	u	u	NOUN
ejpam-5238	234	10	containing	contain	VERB
ejpam-5238	234	11	x.	x.	NOUN
ejpam-5238	234	12	then	then	ADV
ejpam-5238	234	13	,	,	PUNCT
ejpam-5238	234	14	we	we	PRON
ejpam-5238	234	15	have	have	VERB
ejpam-5238	234	16	x	x	PART
ejpam-5238	234	17	∈	∈	PROPN
ejpam-5238	234	18	τ1τ2	τ1τ2	NOUN
ejpam-5238	234	19	-	-	ADJ
ejpam-5238	234	20	cl(x	cl(x	SYM
ejpam-5238	234	21	−f+(σ1σ2	−f+(σ1σ2	NOUN
ejpam-5238	234	22	-	-	PUNCT
ejpam-5238	234	23	cl(v	cl(v	NOUN
ejpam-5238	234	24	)	)	PUNCT
ejpam-5238	234	25	)	)	PUNCT
ejpam-5238	234	26	)	)	PUNCT
ejpam-5238	234	27	.	.	PUNCT
ejpam-5238	235	1	since	since	SCONJ
ejpam-5238	235	2	x	x	PROPN
ejpam-5238	235	3	∈	∈	PROPN
ejpam-5238	235	4	f+(v	f+(v	NOUN
ejpam-5238	235	5	)	)	PUNCT
ejpam-5238	235	6	,	,	PUNCT
ejpam-5238	235	7	x	x	PUNCT
ejpam-5238	235	8	∈	∈	PROPN
ejpam-5238	235	9	τ1τ2	τ1τ2	NOUN
ejpam-5238	235	10	-	-	NOUN
ejpam-5238	235	11	cl(f	cl(f	NOUN
ejpam-5238	235	12	+	+	NOUN
ejpam-5238	235	13	(	(	PUNCT
ejpam-5238	235	14	σ1σ2	σ1σ2	NOUN
ejpam-5238	235	15	-	-	NUM
ejpam-5238	235	16	cl(v	cl(v	NOUN
ejpam-5238	235	17	)	)	PUNCT
ejpam-5238	235	18	)	)	PUNCT
ejpam-5238	235	19	)	)	PUNCT
ejpam-5238	235	20	and	and	CCONJ
ejpam-5238	235	21	hence	hence	ADV
ejpam-5238	235	22	x	x	X
ejpam-5238	235	23	∈	∈	PRON
ejpam-5238	235	24	τ1τ2	τ1τ2	NOUN
ejpam-5238	235	25	-	-	ADJ
ejpam-5238	235	26	fr(f	fr(f	PUNCT
ejpam-5238	235	27	+	+	ADJ
ejpam-5238	235	28	(	(	PUNCT
ejpam-5238	235	29	σ1σ2	σ1σ2	NOUN
ejpam-5238	235	30	-	-	NUM
ejpam-5238	235	31	cl(v	cl(v	NOUN
ejpam-5238	235	32	)	)	PUNCT
ejpam-5238	235	33	)	)	PUNCT
ejpam-5238	235	34	)	)	PUNCT
ejpam-5238	235	35	.	.	PUNCT
ejpam-5238	236	1	conversely	conversely	ADV
ejpam-5238	236	2	,	,	PUNCT
ejpam-5238	236	3	suppose	suppose	VERB
ejpam-5238	236	4	that	that	SCONJ
ejpam-5238	236	5	v	v	NOUN
ejpam-5238	236	6	is	be	AUX
ejpam-5238	236	7	a	a	DET
ejpam-5238	236	8	σ1σ2	σ1σ2	NOUN
ejpam-5238	236	9	-	-	ADJ
ejpam-5238	236	10	open	open	ADJ
ejpam-5238	236	11	set	set	NOUN
ejpam-5238	236	12	of	of	ADP
ejpam-5238	236	13	y	y	PROPN
ejpam-5238	236	14	containing	contain	VERB
ejpam-5238	236	15	f	f	PROPN
ejpam-5238	236	16	(	(	PUNCT
ejpam-5238	236	17	x	x	X
ejpam-5238	236	18	)	)	PUNCT
ejpam-5238	236	19	such	such	ADJ
ejpam-5238	236	20	that	that	SCONJ
ejpam-5238	236	21	x	x	PUNCT
ejpam-5238	236	22	∈	∈	PRON
ejpam-5238	236	23	τ1τ2	τ1τ2	NOUN
ejpam-5238	236	24	-	-	ADJ
ejpam-5238	236	25	fr(f	fr(f	PUNCT
ejpam-5238	236	26	+	+	ADJ
ejpam-5238	236	27	(	(	PUNCT
ejpam-5238	236	28	σ1σ2	σ1σ2	NOUN
ejpam-5238	236	29	-	-	NUM
ejpam-5238	236	30	cl(v	cl(v	NOUN
ejpam-5238	236	31	)	)	PUNCT
ejpam-5238	236	32	)	)	PUNCT
ejpam-5238	236	33	)	)	PUNCT
ejpam-5238	236	34	.	.	PUNCT
ejpam-5238	237	1	m.	m.	NOUN
ejpam-5238	237	2	thongmoon	thongmoon	PROPN
ejpam-5238	237	3	,	,	PUNCT
ejpam-5238	237	4	s.	s.	PROPN
ejpam-5238	237	5	sompong	sompong	PROPN
ejpam-5238	237	6	,	,	PUNCT
ejpam-5238	237	7	c.	c.	PROPN
ejpam-5238	237	8	boonpok	boonpok	PROPN
ejpam-5238	237	9	/	/	SYM
ejpam-5238	237	10	eur	eur	PROPN
ejpam-5238	237	11	.	.	PUNCT
ejpam-5238	238	1	j.	j.	PROPN
ejpam-5238	238	2	pure	pure	PROPN
ejpam-5238	238	3	appl	appl	PROPN
ejpam-5238	238	4	.	.	PROPN
ejpam-5238	238	5	math	math	PROPN
ejpam-5238	238	6	,	,	PUNCT
ejpam-5238	238	7	17	17	NUM
ejpam-5238	238	8	(	(	PUNCT
ejpam-5238	238	9	3	3	NUM
ejpam-5238	238	10	)	)	PUNCT
ejpam-5238	238	11	(	(	PUNCT
ejpam-5238	238	12	2024	2024	NUM
ejpam-5238	238	13	)	)	PUNCT
ejpam-5238	238	14	,	,	PUNCT
ejpam-5238	238	15	1705	1705	NUM
ejpam-5238	238	16	-	-	SYM
ejpam-5238	238	17	1716	1716	NUM
ejpam-5238	238	18	1713	1713	NUM
ejpam-5238	238	19	if	if	SCONJ
ejpam-5238	238	20	f	f	PROPN
ejpam-5238	238	21	is	be	AUX
ejpam-5238	238	22	upper	upper	ADJ
ejpam-5238	238	23	weakly	weakly	ADJ
ejpam-5238	238	24	(	(	PUNCT
ejpam-5238	238	25	τ1	τ1	NOUN
ejpam-5238	238	26	,	,	PUNCT
ejpam-5238	238	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	238	28	at	at	ADP
ejpam-5238	238	29	x	x	X
ejpam-5238	238	30	,	,	PUNCT
ejpam-5238	238	31	there	there	PRON
ejpam-5238	238	32	exists	exist	VERB
ejpam-5238	238	33	a	a	DET
ejpam-5238	238	34	τ1τ2	τ1τ2	NOUN
ejpam-5238	238	35	-	-	ADJ
ejpam-5238	238	36	open	open	ADJ
ejpam-5238	238	37	set	set	ADJ
ejpam-5238	238	38	u	u	NOUN
ejpam-5238	238	39	of	of	ADP
ejpam-5238	238	40	x	x	PUNCT
ejpam-5238	238	41	containing	contain	VERB
ejpam-5238	238	42	x	x	PUNCT
ejpam-5238	238	43	such	such	ADJ
ejpam-5238	238	44	that	that	SCONJ
ejpam-5238	238	45	u	u	NOUN
ejpam-5238	238	46	⊆	⊆	NUM
ejpam-5238	238	47	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	238	48	-	-	PUNCT
ejpam-5238	238	49	cl(v	cl(v	NOUN
ejpam-5238	238	50	)	)	PUNCT
ejpam-5238	238	51	)	)	PUNCT
ejpam-5238	238	52	;	;	PUNCT
ejpam-5238	238	53	hence	hence	ADV
ejpam-5238	238	54	x	x	X
ejpam-5238	238	55	∈	∈	PRON
ejpam-5238	238	56	τ1τ2	τ1τ2	NOUN
ejpam-5238	238	57	-	-	NUM
ejpam-5238	238	58	int(f	int(f	VERB
ejpam-5238	238	59	+	+	ADJ
ejpam-5238	238	60	(	(	PUNCT
ejpam-5238	238	61	σ1σ2	σ1σ2	NOUN
ejpam-5238	238	62	-	-	NUM
ejpam-5238	238	63	cl(v	cl(v	NOUN
ejpam-5238	238	64	)	)	PUNCT
ejpam-5238	238	65	)	)	PUNCT
ejpam-5238	238	66	)	)	PUNCT
ejpam-5238	238	67	.	.	PUNCT
ejpam-5238	239	1	this	this	PRON
ejpam-5238	239	2	is	be	AUX
ejpam-5238	239	3	a	a	DET
ejpam-5238	239	4	contradiction	contradiction	NOUN
ejpam-5238	239	5	and	and	CCONJ
ejpam-5238	239	6	hence	hence	ADV
ejpam-5238	239	7	f	f	PROPN
ejpam-5238	239	8	is	be	AUX
ejpam-5238	239	9	not	not	PART
ejpam-5238	239	10	upper	upper	ADJ
ejpam-5238	239	11	weakly	weakly	ADJ
ejpam-5238	239	12	(	(	PUNCT
ejpam-5238	239	13	τ1	τ1	NOUN
ejpam-5238	239	14	,	,	PUNCT
ejpam-5238	239	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	239	16	at	at	ADP
ejpam-5238	239	17	x.	x.	NOUN
ejpam-5238	239	18	theorem	theorem	VERB
ejpam-5238	239	19	9	9	NUM
ejpam-5238	239	20	.	.	PUNCT
ejpam-5238	240	1	the	the	DET
ejpam-5238	240	2	set	set	NOUN
ejpam-5238	240	3	of	of	ADP
ejpam-5238	240	4	all	all	DET
ejpam-5238	240	5	points	point	NOUN
ejpam-5238	240	6	of	of	ADP
ejpam-5238	240	7	x	x	PUNCT
ejpam-5238	240	8	at	at	ADP
ejpam-5238	240	9	which	which	PRON
ejpam-5238	240	10	a	a	DET
ejpam-5238	240	11	multifunction	multifunction	NOUN
ejpam-5238	240	12	f	f	NOUN
ejpam-5238	240	13	:	:	PUNCT
ejpam-5238	240	14	(	(	PUNCT
ejpam-5238	240	15	x	x	NOUN
ejpam-5238	240	16	,	,	PUNCT
ejpam-5238	240	17	τ1	τ1	NOUN
ejpam-5238	240	18	,	,	PUNCT
ejpam-5238	240	19	τ2	τ2	NOUN
ejpam-5238	240	20	)	)	PUNCT
ejpam-5238	240	21	→	→	SYM
ejpam-5238	240	22	(	(	PUNCT
ejpam-5238	240	23	y	y	PROPN
ejpam-5238	240	24	,	,	PUNCT
ejpam-5238	240	25	σ1	σ1	PROPN
ejpam-5238	240	26	,	,	PUNCT
ejpam-5238	240	27	σ2	σ2	PROPN
ejpam-5238	240	28	)	)	PUNCT
ejpam-5238	240	29	is	be	AUX
ejpam-5238	240	30	not	not	PART
ejpam-5238	240	31	lower	low	ADJ
ejpam-5238	240	32	weakly	weakly	ADJ
ejpam-5238	240	33	(	(	PUNCT
ejpam-5238	240	34	τ1	τ1	NOUN
ejpam-5238	240	35	,	,	PUNCT
ejpam-5238	240	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	240	37	is	be	AUX
ejpam-5238	240	38	identical	identical	ADJ
ejpam-5238	240	39	with	with	ADP
ejpam-5238	240	40	the	the	DET
ejpam-5238	240	41	union	union	NOUN
ejpam-5238	240	42	of	of	ADP
ejpam-5238	240	43	the	the	DET
ejpam-5238	240	44	τ1τ2	τ1τ2	NOUN
ejpam-5238	240	45	-	-	NOUN
ejpam-5238	240	46	frontier	frontier	NOUN
ejpam-5238	240	47	of	of	ADP
ejpam-5238	240	48	the	the	DET
ejpam-5238	240	49	lower	low	ADJ
ejpam-5238	240	50	inverse	inverse	NOUN
ejpam-5238	240	51	images	image	NOUN
ejpam-5238	240	52	of	of	ADP
ejpam-5238	240	53	the	the	DET
ejpam-5238	240	54	σ1σ2	σ1σ2	NOUN
ejpam-5238	240	55	-	-	PUNCT
ejpam-5238	240	56	closures	closure	NOUN
ejpam-5238	240	57	of	of	ADP
ejpam-5238	240	58	σ1σ2	σ1σ2	NOUN
ejpam-5238	240	59	-	-	PUNCT
ejpam-5238	240	60	open	open	ADJ
ejpam-5238	240	61	sets	set	NOUN
ejpam-5238	240	62	meeting	meet	VERB
ejpam-5238	240	63	f	f	X
ejpam-5238	240	64	(	(	PUNCT
ejpam-5238	240	65	x	x	NOUN
ejpam-5238	240	66	)	)	PUNCT
ejpam-5238	240	67	.	.	PUNCT
ejpam-5238	241	1	proof	proof	NOUN
ejpam-5238	241	2	.	.	PUNCT
ejpam-5238	242	1	the	the	DET
ejpam-5238	242	2	proof	proof	NOUN
ejpam-5238	242	3	is	be	AUX
ejpam-5238	242	4	similar	similar	ADJ
ejpam-5238	242	5	to	to	ADP
ejpam-5238	242	6	that	that	PRON
ejpam-5238	242	7	of	of	ADP
ejpam-5238	242	8	theorem	theorem	ADJ
ejpam-5238	242	9	8	8	NUM
ejpam-5238	242	10	.	.	PUNCT
ejpam-5238	242	11	definition	definition	NOUN
ejpam-5238	242	12	5	5	NUM
ejpam-5238	242	13	.	.	PUNCT
ejpam-5238	243	1	[	[	X
ejpam-5238	243	2	20	20	NUM
ejpam-5238	243	3	]	]	PUNCT
ejpam-5238	243	4	a	a	DET
ejpam-5238	243	5	bitopological	bitopological	ADJ
ejpam-5238	243	6	space	space	NOUN
ejpam-5238	243	7	(	(	PUNCT
ejpam-5238	243	8	x	x	NOUN
ejpam-5238	243	9	,	,	PUNCT
ejpam-5238	243	10	τ1	τ1	NOUN
ejpam-5238	243	11	,	,	PUNCT
ejpam-5238	243	12	τ2	τ2	NOUN
ejpam-5238	243	13	)	)	PUNCT
ejpam-5238	243	14	is	be	AUX
ejpam-5238	243	15	said	say	VERB
ejpam-5238	243	16	to	to	PART
ejpam-5238	243	17	be	be	AUX
ejpam-5238	243	18	τ1τ2	τ1τ2	NOUN
ejpam-5238	243	19	-	-	ADJ
ejpam-5238	243	20	connected	connected	ADJ
ejpam-5238	243	21	if	if	SCONJ
ejpam-5238	243	22	x	x	PRON
ejpam-5238	243	23	can	can	AUX
ejpam-5238	243	24	not	not	PART
ejpam-5238	243	25	be	be	AUX
ejpam-5238	243	26	written	write	VERB
ejpam-5238	243	27	as	as	ADP
ejpam-5238	243	28	the	the	DET
ejpam-5238	243	29	union	union	NOUN
ejpam-5238	243	30	of	of	ADP
ejpam-5238	243	31	two	two	NUM
ejpam-5238	243	32	nonempty	nonempty	ADV
ejpam-5238	243	33	disjoint	disjoint	NOUN
ejpam-5238	243	34	τ1τ2	τ1τ2	ADJ
ejpam-5238	243	35	-	-	ADJ
ejpam-5238	243	36	open	open	ADJ
ejpam-5238	243	37	sets	set	NOUN
ejpam-5238	243	38	.	.	PUNCT
ejpam-5238	244	1	recall	recall	VERB
ejpam-5238	244	2	that	that	SCONJ
ejpam-5238	244	3	a	a	DET
ejpam-5238	244	4	subset	subset	NOUN
ejpam-5238	244	5	a	a	PRON
ejpam-5238	244	6	of	of	ADP
ejpam-5238	244	7	a	a	DET
ejpam-5238	244	8	bitopological	bitopological	ADJ
ejpam-5238	244	9	space	space	NOUN
ejpam-5238	244	10	(	(	PUNCT
ejpam-5238	244	11	x	x	NOUN
ejpam-5238	244	12	,	,	PUNCT
ejpam-5238	244	13	τ1	τ1	NOUN
ejpam-5238	244	14	,	,	PUNCT
ejpam-5238	244	15	τ2	τ2	NOUN
ejpam-5238	244	16	)	)	PUNCT
ejpam-5238	244	17	is	be	AUX
ejpam-5238	244	18	said	say	VERB
ejpam-5238	244	19	to	to	PART
ejpam-5238	244	20	be	be	AUX
ejpam-5238	244	21	τ1τ2	τ1τ2	NOUN
ejpam-5238	244	22	-	-	ADJ
ejpam-5238	244	23	clopen	clopen	ADJ
ejpam-5238	245	1	[	[	X
ejpam-5238	245	2	20	20	NUM
ejpam-5238	245	3	]	]	X
ejpam-5238	245	4	if	if	SCONJ
ejpam-5238	245	5	a	a	PRON
ejpam-5238	245	6	is	be	AUX
ejpam-5238	245	7	both	both	PRON
ejpam-5238	245	8	τ1τ2	τ1τ2	ADJ
ejpam-5238	245	9	-	-	ADJ
ejpam-5238	245	10	open	open	ADJ
ejpam-5238	245	11	and	and	CCONJ
ejpam-5238	245	12	τ1τ2	τ1τ2	NOUN
ejpam-5238	245	13	-	-	ADJ
ejpam-5238	245	14	closed	closed	ADJ
ejpam-5238	245	15	.	.	PUNCT
ejpam-5238	246	1	theorem	theorem	VERB
ejpam-5238	246	2	10	10	NUM
ejpam-5238	246	3	.	.	PUNCT
ejpam-5238	247	1	if	if	SCONJ
ejpam-5238	247	2	f	f	PROPN
ejpam-5238	247	3	:	:	PUNCT
ejpam-5238	247	4	(	(	PUNCT
ejpam-5238	247	5	x	x	NOUN
ejpam-5238	247	6	,	,	PUNCT
ejpam-5238	247	7	τ1	τ1	NOUN
ejpam-5238	247	8	,	,	PUNCT
ejpam-5238	247	9	τ2	τ2	NOUN
ejpam-5238	247	10	)	)	PUNCT
ejpam-5238	247	11	→	→	SYM
ejpam-5238	247	12	(	(	PUNCT
ejpam-5238	247	13	y	y	PROPN
ejpam-5238	247	14	,	,	PUNCT
ejpam-5238	247	15	σ1	σ1	PROPN
ejpam-5238	247	16	,	,	PUNCT
ejpam-5238	247	17	σ2	σ2	PROPN
ejpam-5238	247	18	)	)	PUNCT
ejpam-5238	247	19	is	be	AUX
ejpam-5238	247	20	an	an	DET
ejpam-5238	247	21	upper	upper	ADJ
ejpam-5238	247	22	or	or	CCONJ
ejpam-5238	247	23	lower	low	ADJ
ejpam-5238	247	24	weakly	weakly	ADJ
ejpam-5238	247	25	(	(	PUNCT
ejpam-5238	247	26	τ1	τ1	NOUN
ejpam-5238	247	27	,	,	PUNCT
ejpam-5238	247	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	247	29	surjective	surjective	ADJ
ejpam-5238	247	30	multifunction	multifunction	NOUN
ejpam-5238	247	31	such	such	ADJ
ejpam-5238	247	32	that	that	SCONJ
ejpam-5238	247	33	f	f	PROPN
ejpam-5238	247	34	(	(	PUNCT
ejpam-5238	247	35	x	x	X
ejpam-5238	247	36	)	)	PUNCT
ejpam-5238	247	37	is	be	AUX
ejpam-5238	247	38	σ1σ2	σ1σ2	NOUN
ejpam-5238	247	39	-	-	PUNCT
ejpam-5238	247	40	connected	connected	ADJ
ejpam-5238	247	41	for	for	ADP
ejpam-5238	247	42	each	each	DET
ejpam-5238	247	43	x	x	SYM
ejpam-5238	247	44	∈	∈	PROPN
ejpam-5238	247	45	x	x	X
ejpam-5238	247	46	and	and	CCONJ
ejpam-5238	247	47	(	(	PUNCT
ejpam-5238	247	48	x	x	NOUN
ejpam-5238	247	49	,	,	PUNCT
ejpam-5238	247	50	τ1	τ1	NOUN
ejpam-5238	247	51	,	,	PUNCT
ejpam-5238	247	52	τ2	τ2	NOUN
ejpam-5238	247	53	)	)	PUNCT
ejpam-5238	247	54	is	be	AUX
ejpam-5238	247	55	τ1τ2	τ1τ2	NOUN
ejpam-5238	247	56	-	-	ADJ
ejpam-5238	247	57	connected	connected	ADJ
ejpam-5238	247	58	,	,	PUNCT
ejpam-5238	247	59	then	then	ADV
ejpam-5238	247	60	(	(	PUNCT
ejpam-5238	247	61	y	y	PROPN
ejpam-5238	247	62	,	,	PUNCT
ejpam-5238	247	63	σ1	σ1	PROPN
ejpam-5238	247	64	,	,	PUNCT
ejpam-5238	247	65	σ2	σ2	PROPN
ejpam-5238	247	66	)	)	PUNCT
ejpam-5238	247	67	is	be	AUX
ejpam-5238	247	68	σ1σ2	σ1σ2	NOUN
ejpam-5238	247	69	-	-	PUNCT
ejpam-5238	247	70	connected	connected	ADJ
ejpam-5238	247	71	.	.	PUNCT
ejpam-5238	248	1	proof	proof	NOUN
ejpam-5238	248	2	.	.	PUNCT
ejpam-5238	249	1	suppose	suppose	VERB
ejpam-5238	249	2	that	that	SCONJ
ejpam-5238	249	3	(	(	PUNCT
ejpam-5238	249	4	y	y	PROPN
ejpam-5238	249	5	,	,	PUNCT
ejpam-5238	249	6	σ1	σ1	PROPN
ejpam-5238	249	7	,	,	PUNCT
ejpam-5238	249	8	σ2	σ2	PROPN
ejpam-5238	249	9	)	)	PUNCT
ejpam-5238	249	10	is	be	AUX
ejpam-5238	249	11	not	not	PART
ejpam-5238	249	12	σ1σ2	σ1σ2	VERB
ejpam-5238	249	13	-	-	PUNCT
ejpam-5238	249	14	connected	connect	VERB
ejpam-5238	249	15	.	.	PUNCT
ejpam-5238	250	1	there	there	PRON
ejpam-5238	250	2	exist	exist	VERB
ejpam-5238	250	3	non	non	ADJ
ejpam-5238	250	4	-	-	ADJ
ejpam-5238	250	5	empty	empty	ADJ
ejpam-5238	250	6	σ1σ2open	σ1σ2open	PUNCT
ejpam-5238	250	7	sets	set	VERB
ejpam-5238	250	8	u	u	NOUN
ejpam-5238	250	9	and	and	CCONJ
ejpam-5238	250	10	v	v	NOUN
ejpam-5238	250	11	of	of	ADP
ejpam-5238	250	12	y	y	PRON
ejpam-5238	250	13	such	such	ADJ
ejpam-5238	250	14	that	that	SCONJ
ejpam-5238	250	15	u	u	NOUN
ejpam-5238	250	16	∩v	∩v	NOUN
ejpam-5238	250	17	=	=	SYM
ejpam-5238	250	18	∅	∅	NOUN
ejpam-5238	250	19	and	and	CCONJ
ejpam-5238	250	20	u	u	NOUN
ejpam-5238	250	21	∪v	∪v	ADP
ejpam-5238	250	22	=	=	SYM
ejpam-5238	250	23	y	y	PROPN
ejpam-5238	250	24	.	.	PUNCT
ejpam-5238	251	1	since	since	SCONJ
ejpam-5238	251	2	f	f	PROPN
ejpam-5238	251	3	(	(	PUNCT
ejpam-5238	251	4	x	x	X
ejpam-5238	251	5	)	)	PUNCT
ejpam-5238	251	6	is	be	AUX
ejpam-5238	251	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	251	8	-	-	PUNCT
ejpam-5238	251	9	connected	connected	ADJ
ejpam-5238	251	10	for	for	ADP
ejpam-5238	251	11	each	each	DET
ejpam-5238	251	12	x	x	SYM
ejpam-5238	251	13	∈	∈	PROPN
ejpam-5238	251	14	x	x	NOUN
ejpam-5238	251	15	,	,	PUNCT
ejpam-5238	251	16	either	either	CCONJ
ejpam-5238	251	17	f	f	PROPN
ejpam-5238	251	18	(	(	PUNCT
ejpam-5238	251	19	x	x	X
ejpam-5238	251	20	)	)	PUNCT
ejpam-5238	251	21	⊆	⊆	NUM
ejpam-5238	251	22	u	u	NOUN
ejpam-5238	251	23	or	or	CCONJ
ejpam-5238	251	24	f	f	PROPN
ejpam-5238	251	25	(	(	PUNCT
ejpam-5238	251	26	x	x	NOUN
ejpam-5238	251	27	)	)	PUNCT
ejpam-5238	251	28	⊆	⊆	NUM
ejpam-5238	251	29	v	v	NOUN
ejpam-5238	251	30	.	.	PUNCT
ejpam-5238	252	1	if	if	SCONJ
ejpam-5238	252	2	x	x	SYM
ejpam-5238	252	3	∈	∈	NOUN
ejpam-5238	252	4	f+(u	f+(u	PUNCT
ejpam-5238	252	5	∪v	∪v	NUM
ejpam-5238	252	6	)	)	PUNCT
ejpam-5238	252	7	,	,	PUNCT
ejpam-5238	252	8	then	then	ADV
ejpam-5238	252	9	f	f	X
ejpam-5238	252	10	(	(	PUNCT
ejpam-5238	252	11	x	x	NOUN
ejpam-5238	252	12	)	)	PUNCT
ejpam-5238	252	13	⊆	⊆	NUM
ejpam-5238	252	14	u	u	NOUN
ejpam-5238	252	15	∪v	∪v	PUNCT
ejpam-5238	252	16	and	and	CCONJ
ejpam-5238	252	17	hence	hence	ADV
ejpam-5238	252	18	x	x	PART
ejpam-5238	252	19	∈	∈	NOUN
ejpam-5238	252	20	f+(u)∪f+(v	f+(u)∪f+(v	NOUN
ejpam-5238	252	21	)	)	PUNCT
ejpam-5238	252	22	.	.	PUNCT
ejpam-5238	253	1	moreover	moreover	ADV
ejpam-5238	253	2	,	,	PUNCT
ejpam-5238	253	3	since	since	SCONJ
ejpam-5238	253	4	f	f	PROPN
ejpam-5238	253	5	is	be	AUX
ejpam-5238	253	6	surjective	surjective	ADJ
ejpam-5238	253	7	,	,	PUNCT
ejpam-5238	253	8	there	there	PRON
ejpam-5238	253	9	exist	exist	VERB
ejpam-5238	253	10	x	x	PUNCT
ejpam-5238	253	11	and	and	CCONJ
ejpam-5238	253	12	y	y	PROPN
ejpam-5238	253	13	in	in	ADP
ejpam-5238	253	14	x	x	PUNCT
ejpam-5238	253	15	such	such	ADJ
ejpam-5238	253	16	that	that	SCONJ
ejpam-5238	253	17	f	f	PROPN
ejpam-5238	253	18	(	(	PUNCT
ejpam-5238	253	19	x	x	X
ejpam-5238	253	20	)	)	PUNCT
ejpam-5238	253	21	⊆	⊆	NUM
ejpam-5238	253	22	u	u	NOUN
ejpam-5238	253	23	and	and	CCONJ
ejpam-5238	253	24	f	f	PROPN
ejpam-5238	253	25	(	(	PUNCT
ejpam-5238	253	26	y	y	PROPN
ejpam-5238	253	27	)	)	PUNCT
ejpam-5238	253	28	⊆	⊆	NUM
ejpam-5238	253	29	v	v	NOUN
ejpam-5238	253	30	;	;	PUNCT
ejpam-5238	253	31	hence	hence	ADV
ejpam-5238	253	32	x	x	SYM
ejpam-5238	253	33	∈	∈	PROPN
ejpam-5238	253	34	f+(u	f+(u	NUM
ejpam-5238	253	35	)	)	PUNCT
ejpam-5238	253	36	and	and	CCONJ
ejpam-5238	253	37	y	y	PROPN
ejpam-5238	253	38	∈	∈	PROPN
ejpam-5238	253	39	f+(v	f+(v	PROPN
ejpam-5238	253	40	)	)	PUNCT
ejpam-5238	253	41	.	.	PUNCT
ejpam-5238	254	1	therefore	therefore	ADV
ejpam-5238	254	2	,	,	PUNCT
ejpam-5238	254	3	we	we	PRON
ejpam-5238	254	4	obtain	obtain	VERB
ejpam-5238	254	5	the	the	DET
ejpam-5238	254	6	following	following	NOUN
ejpam-5238	254	7	:	:	PUNCT
ejpam-5238	254	8	(	(	PUNCT
ejpam-5238	254	9	1	1	X
ejpam-5238	254	10	)	)	PUNCT
ejpam-5238	254	11	f+(u	f+(u	NUM
ejpam-5238	254	12	)	)	PUNCT
ejpam-5238	254	13	∪	∪	ADP
ejpam-5238	254	14	f+(v	f+(v	NOUN
ejpam-5238	254	15	)	)	PUNCT
ejpam-5238	255	1	=	=	PUNCT
ejpam-5238	256	1	f+(u	f+(u	PUNCT
ejpam-5238	256	2	∪	∪	ADP
ejpam-5238	256	3	v	v	NOUN
ejpam-5238	256	4	)	)	PUNCT
ejpam-5238	256	5	=	=	SYM
ejpam-5238	257	1	x	x	X
ejpam-5238	257	2	;	;	PUNCT
ejpam-5238	257	3	(	(	PUNCT
ejpam-5238	257	4	2	2	X
ejpam-5238	257	5	)	)	PUNCT
ejpam-5238	257	6	f+(u	f+(u	NUM
ejpam-5238	257	7	)	)	PUNCT
ejpam-5238	257	8	∩	∩	NOUN
ejpam-5238	257	9	f+(v	f+(v	NOUN
ejpam-5238	257	10	)	)	PUNCT
ejpam-5238	257	11	=	=	SYM
ejpam-5238	258	1	f+(u	f+(u	NUM
ejpam-5238	258	2	∩	∩	NOUN
ejpam-5238	258	3	v	v	NOUN
ejpam-5238	258	4	)	)	PUNCT
ejpam-5238	258	5	=	=	NOUN
ejpam-5238	258	6	∅	∅	NOUN
ejpam-5238	258	7	;	;	PUNCT
ejpam-5238	258	8	(	(	PUNCT
ejpam-5238	258	9	3	3	X
ejpam-5238	258	10	)	)	PUNCT
ejpam-5238	258	11	f+(u	f+(u	NUM
ejpam-5238	258	12	)	)	PUNCT
ejpam-5238	258	13	̸=	̸=	PROPN
ejpam-5238	258	14	∅	∅	NOUN
ejpam-5238	258	15	and	and	CCONJ
ejpam-5238	258	16	f+(v	f+(v	NUM
ejpam-5238	258	17	)	)	PUNCT
ejpam-5238	259	1	̸=	̸=	PROPN
ejpam-5238	259	2	∅.	∅.	ADP
ejpam-5238	259	3	next	next	ADV
ejpam-5238	259	4	,	,	PUNCT
ejpam-5238	259	5	we	we	PRON
ejpam-5238	259	6	show	show	VERB
ejpam-5238	259	7	that	that	PRON
ejpam-5238	259	8	f+(u	f+(u	NUM
ejpam-5238	259	9	)	)	PUNCT
ejpam-5238	259	10	and	and	CCONJ
ejpam-5238	259	11	f+(v	f+(v	NUM
ejpam-5238	259	12	)	)	PUNCT
ejpam-5238	259	13	are	be	AUX
ejpam-5238	259	14	τ1τ2	τ1τ2	NOUN
ejpam-5238	259	15	-	-	ADJ
ejpam-5238	259	16	open	open	ADJ
ejpam-5238	259	17	in	in	ADP
ejpam-5238	259	18	x.	x.	PROPN
ejpam-5238	259	19	(	(	PUNCT
ejpam-5238	259	20	i	i	NOUN
ejpam-5238	259	21	)	)	PUNCT
ejpam-5238	259	22	let	let	VERB
ejpam-5238	259	23	f	f	PRON
ejpam-5238	259	24	be	be	AUX
ejpam-5238	259	25	upper	upper	ADJ
ejpam-5238	259	26	weakly	weakly	ADJ
ejpam-5238	259	27	(	(	PUNCT
ejpam-5238	259	28	τ1	τ1	NOUN
ejpam-5238	259	29	,	,	PUNCT
ejpam-5238	259	30	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	259	31	.	.	PUNCT
ejpam-5238	260	1	by	by	ADP
ejpam-5238	260	2	theorem	theorem	NOUN
ejpam-5238	260	3	1	1	NUM
ejpam-5238	260	4	,	,	PUNCT
ejpam-5238	260	5	f+(v	f+(v	PROPN
ejpam-5238	260	6	)	)	PUNCT
ejpam-5238	260	7	⊆	⊆	X
ejpam-5238	260	8	τ1τ2	τ1τ2	NOUN
ejpam-5238	260	9	-	-	NUM
ejpam-5238	260	10	int(f	int(f	VERB
ejpam-5238	260	11	+	+	ADJ
ejpam-5238	260	12	(	(	PUNCT
ejpam-5238	260	13	σ1σ2	σ1σ2	NOUN
ejpam-5238	260	14	-	-	NUM
ejpam-5238	260	15	cl(v	cl(v	NOUN
ejpam-5238	260	16	)	)	PUNCT
ejpam-5238	260	17	)	)	PUNCT
ejpam-5238	260	18	)	)	PUNCT
ejpam-5238	261	1	=	=	PUNCT
ejpam-5238	261	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	261	3	-	-	NUM
ejpam-5238	261	4	int(f	int(f	VERB
ejpam-5238	261	5	+	+	ADJ
ejpam-5238	261	6	(	(	PUNCT
ejpam-5238	261	7	v	v	NOUN
ejpam-5238	261	8	)	)	PUNCT
ejpam-5238	261	9	)	)	PUNCT
ejpam-5238	261	10	since	since	SCONJ
ejpam-5238	261	11	v	v	NUM
ejpam-5238	261	12	is	be	AUX
ejpam-5238	261	13	σ1σ2	σ1σ2	NOUN
ejpam-5238	261	14	-	-	PUNCT
ejpam-5238	261	15	clopen	clopen	ADJ
ejpam-5238	261	16	.	.	PUNCT
ejpam-5238	262	1	thus	thus	ADV
ejpam-5238	262	2	,	,	PUNCT
ejpam-5238	262	3	f+(v	f+(v	PROPN
ejpam-5238	262	4	)	)	PUNCT
ejpam-5238	263	1	=	=	PUNCT
ejpam-5238	263	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	263	3	-	-	NUM
ejpam-5238	263	4	int(f	int(f	VERB
ejpam-5238	263	5	+	+	ADJ
ejpam-5238	263	6	(	(	PUNCT
ejpam-5238	263	7	v	v	NOUN
ejpam-5238	263	8	)	)	PUNCT
ejpam-5238	263	9	)	)	PUNCT
ejpam-5238	263	10	and	and	CCONJ
ejpam-5238	263	11	hence	hence	ADV
ejpam-5238	263	12	f+(v	f+(v	PROPN
ejpam-5238	263	13	)	)	PUNCT
ejpam-5238	263	14	is	be	AUX
ejpam-5238	263	15	τ1τ2	τ1τ2	NOUN
ejpam-5238	263	16	-	-	ADJ
ejpam-5238	263	17	open	open	ADJ
ejpam-5238	263	18	in	in	ADP
ejpam-5238	263	19	x.	x.	NOUN
ejpam-5238	263	20	similarly	similarly	ADV
ejpam-5238	263	21	,	,	PUNCT
ejpam-5238	263	22	we	we	PRON
ejpam-5238	263	23	obtain	obtain	VERB
ejpam-5238	263	24	f+(u	f+(u	PUNCT
ejpam-5238	263	25	)	)	PUNCT
ejpam-5238	263	26	is	be	AUX
ejpam-5238	263	27	τ1τ2	τ1τ2	NOUN
ejpam-5238	263	28	-	-	ADJ
ejpam-5238	263	29	open	open	ADJ
ejpam-5238	263	30	in	in	ADP
ejpam-5238	263	31	x.	x.	NOUN
ejpam-5238	263	32	consequently	consequently	ADV
ejpam-5238	263	33	,	,	PUNCT
ejpam-5238	263	34	this	this	PRON
ejpam-5238	263	35	shows	show	VERB
ejpam-5238	263	36	that	that	SCONJ
ejpam-5238	263	37	(	(	PUNCT
ejpam-5238	263	38	x	x	NOUN
ejpam-5238	263	39	,	,	PUNCT
ejpam-5238	263	40	τ1	τ1	NOUN
ejpam-5238	263	41	,	,	PUNCT
ejpam-5238	263	42	τ2	τ2	NOUN
ejpam-5238	263	43	)	)	PUNCT
ejpam-5238	263	44	is	be	AUX
ejpam-5238	263	45	not	not	PART
ejpam-5238	263	46	τ1τ2	τ1τ2	ADJ
ejpam-5238	263	47	-	-	VERB
ejpam-5238	263	48	connected	connected	ADJ
ejpam-5238	263	49	.	.	PUNCT
ejpam-5238	264	1	(	(	PUNCT
ejpam-5238	264	2	ii	ii	NOUN
ejpam-5238	264	3	)	)	PUNCT
ejpam-5238	264	4	let	let	VERB
ejpam-5238	264	5	f	f	PRON
ejpam-5238	264	6	be	be	AUX
ejpam-5238	264	7	lower	lower	ADV
ejpam-5238	264	8	weakly	weakly	ADJ
ejpam-5238	264	9	(	(	PUNCT
ejpam-5238	264	10	τ1	τ1	NOUN
ejpam-5238	264	11	,	,	PUNCT
ejpam-5238	264	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	264	13	.	.	PUNCT
ejpam-5238	265	1	by	by	ADP
ejpam-5238	265	2	theorem	theorem	NOUN
ejpam-5238	265	3	2	2	NUM
ejpam-5238	265	4	,	,	PUNCT
ejpam-5238	265	5	τ1τ2	τ1τ2	NOUN
ejpam-5238	265	6	-	-	NOUN
ejpam-5238	265	7	cl(f	cl(f	NOUN
ejpam-5238	265	8	+	+	NOUN
ejpam-5238	265	9	(	(	PUNCT
ejpam-5238	265	10	v	v	NOUN
ejpam-5238	265	11	)	)	PUNCT
ejpam-5238	265	12	)	)	PUNCT
ejpam-5238	266	1	⊆	⊆	NUM
ejpam-5238	266	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-5238	266	3	-	-	PUNCT
ejpam-5238	266	4	cl(v	cl(v	NOUN
ejpam-5238	266	5	)	)	PUNCT
ejpam-5238	266	6	)	)	PUNCT
ejpam-5238	267	1	=	=	PUNCT
ejpam-5238	267	2	f+(v	f+(v	NOUN
ejpam-5238	267	3	)	)	PUNCT
ejpam-5238	267	4	since	since	SCONJ
ejpam-5238	267	5	v	v	NUM
ejpam-5238	267	6	is	be	AUX
ejpam-5238	267	7	σ1σ2	σ1σ2	NOUN
ejpam-5238	267	8	-	-	PUNCT
ejpam-5238	267	9	clopen	clopen	ADJ
ejpam-5238	267	10	.	.	PUNCT
ejpam-5238	268	1	therefore	therefore	ADV
ejpam-5238	268	2	,	,	PUNCT
ejpam-5238	268	3	f+(v	f+(v	PROPN
ejpam-5238	268	4	)	)	PUNCT
ejpam-5238	269	1	=	=	PUNCT
ejpam-5238	269	2	τ1τ2	τ1τ2	NOUN
ejpam-5238	269	3	-	-	NOUN
ejpam-5238	269	4	cl(f	cl(f	NOUN
ejpam-5238	269	5	+	+	NOUN
ejpam-5238	269	6	(	(	PUNCT
ejpam-5238	269	7	v	v	NOUN
ejpam-5238	269	8	)	)	PUNCT
ejpam-5238	269	9	)	)	PUNCT
ejpam-5238	269	10	and	and	CCONJ
ejpam-5238	269	11	so	so	ADV
ejpam-5238	269	12	f+(v	f+(v	PROPN
ejpam-5238	269	13	)	)	PUNCT
ejpam-5238	269	14	is	be	AUX
ejpam-5238	269	15	τ1τ2	τ1τ2	NOUN
ejpam-5238	269	16	-	-	ADJ
ejpam-5238	269	17	closed	closed	ADJ
ejpam-5238	269	18	in	in	ADP
ejpam-5238	269	19	x.	x.	NOUN
ejpam-5238	269	20	thus	thus	ADV
ejpam-5238	269	21	,	,	PUNCT
ejpam-5238	269	22	we	we	PRON
ejpam-5238	269	23	have	have	AUX
ejpam-5238	269	24	f+(u	f+(u	PUNCT
ejpam-5238	269	25	)	)	PUNCT
ejpam-5238	269	26	is	be	AUX
ejpam-5238	269	27	τ1τ2	τ1τ2	NOUN
ejpam-5238	269	28	-	-	ADJ
ejpam-5238	269	29	open	open	ADJ
ejpam-5238	269	30	in	in	ADP
ejpam-5238	269	31	x.	x.	NOUN
ejpam-5238	269	32	similarly	similarly	ADV
ejpam-5238	269	33	,	,	PUNCT
ejpam-5238	269	34	we	we	PRON
ejpam-5238	269	35	obtain	obtain	VERB
ejpam-5238	269	36	f+(v	f+(v	NOUN
ejpam-5238	269	37	)	)	PUNCT
ejpam-5238	269	38	is	be	AUX
ejpam-5238	269	39	τ1τ2	τ1τ2	NOUN
ejpam-5238	269	40	-	-	ADJ
ejpam-5238	269	41	open	open	ADJ
ejpam-5238	269	42	in	in	ADP
ejpam-5238	269	43	x.	x.	NOUN
ejpam-5238	269	44	consequently	consequently	ADV
ejpam-5238	269	45	,	,	PUNCT
ejpam-5238	269	46	this	this	PRON
ejpam-5238	269	47	shows	show	VERB
ejpam-5238	269	48	that	that	SCONJ
ejpam-5238	269	49	(	(	PUNCT
ejpam-5238	269	50	x	x	NOUN
ejpam-5238	269	51	,	,	PUNCT
ejpam-5238	269	52	τ1	τ1	NOUN
ejpam-5238	269	53	,	,	PUNCT
ejpam-5238	269	54	τ2	τ2	NOUN
ejpam-5238	269	55	)	)	PUNCT
ejpam-5238	269	56	is	be	AUX
ejpam-5238	269	57	not	not	PART
ejpam-5238	269	58	τ1τ2	τ1τ2	ADJ
ejpam-5238	269	59	-	-	VERB
ejpam-5238	269	60	connected	connected	ADJ
ejpam-5238	269	61	.	.	PUNCT
ejpam-5238	270	1	this	this	PRON
ejpam-5238	270	2	completes	complete	VERB
ejpam-5238	270	3	the	the	DET
ejpam-5238	270	4	proof	proof	NOUN
ejpam-5238	270	5	.	.	PUNCT
ejpam-5238	271	1	references	reference	NOUN
ejpam-5238	271	2	1714	1714	NUM
ejpam-5238	271	3	acknowledgements	acknowledgement	NOUN
ejpam-5238	271	4	this	this	DET
ejpam-5238	271	5	research	research	NOUN
ejpam-5238	271	6	project	project	NOUN
ejpam-5238	271	7	was	be	AUX
ejpam-5238	271	8	financially	financially	ADV
ejpam-5238	271	9	supported	support	VERB
ejpam-5238	271	10	by	by	ADP
ejpam-5238	271	11	mahasarakham	mahasarakham	PROPN
ejpam-5238	271	12	university	university	PROPN
ejpam-5238	271	13	.	.	PUNCT
ejpam-5238	272	1	references	reference	NOUN
ejpam-5238	272	2	[	[	X
ejpam-5238	272	3	1	1	NUM
ejpam-5238	272	4	]	]	PUNCT
ejpam-5238	272	5	c.	c.	PROPN
ejpam-5238	272	6	berge	berge	PROPN
ejpam-5238	272	7	.	.	PUNCT
ejpam-5238	273	1	espaces	espace	VERB
ejpam-5238	273	2	topologiques	topologique	NOUN
ejpam-5238	273	3	fonctions	fonction	NOUN
ejpam-5238	273	4	multivoques	multivoque	NOUN
ejpam-5238	273	5	.	.	PUNCT
ejpam-5238	274	1	dunod	dunod	PROPN
ejpam-5238	274	2	,	,	PUNCT
ejpam-5238	274	3	paris	paris	PROPN
ejpam-5238	274	4	,	,	PUNCT
ejpam-5238	274	5	1959	1959	NUM
ejpam-5238	274	6	.	.	PUNCT
ejpam-5238	275	1	[	[	X
ejpam-5238	275	2	2	2	NUM
ejpam-5238	275	3	]	]	PUNCT
ejpam-5238	275	4	c.	c.	PROPN
ejpam-5238	275	5	boonpok	boonpok	PROPN
ejpam-5238	275	6	.	.	PUNCT
ejpam-5238	276	1	almost	almost	ADV
ejpam-5238	276	2	(	(	PUNCT
ejpam-5238	276	3	g	g	NOUN
ejpam-5238	276	4	,	,	PUNCT
ejpam-5238	276	5	m)-continuous	m)-continuous	ADJ
ejpam-5238	276	6	functions	function	NOUN
ejpam-5238	276	7	.	.	PUNCT
ejpam-5238	277	1	international	international	ADJ
ejpam-5238	277	2	journal	journal	PROPN
ejpam-5238	277	3	of	of	ADP
ejpam-5238	277	4	mathematical	mathematical	ADJ
ejpam-5238	277	5	analysis	analysis	NOUN
ejpam-5238	277	6	,	,	PUNCT
ejpam-5238	277	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-5238	277	8	,	,	PUNCT
ejpam-5238	277	9	2010	2010	NUM
ejpam-5238	277	10	.	.	PUNCT
ejpam-5238	278	1	[	[	X
ejpam-5238	278	2	3	3	X
ejpam-5238	278	3	]	]	PUNCT
ejpam-5238	278	4	c.	c.	PROPN
ejpam-5238	278	5	boonpok	boonpok	PROPN
ejpam-5238	278	6	.	.	PUNCT
ejpam-5238	279	1	m	m	VERB
ejpam-5238	279	2	-continuous	-continuous	ADJ
ejpam-5238	279	3	functions	function	NOUN
ejpam-5238	279	4	in	in	ADP
ejpam-5238	279	5	biminimal	biminimal	NOUN
ejpam-5238	279	6	structure	structure	NOUN
ejpam-5238	279	7	spaces	space	NOUN
ejpam-5238	279	8	.	.	PUNCT
ejpam-5238	280	1	far	far	PROPN
ejpam-5238	280	2	east	east	PROPN
ejpam-5238	280	3	journal	journal	PROPN
ejpam-5238	280	4	of	of	ADP
ejpam-5238	280	5	mathematical	mathematical	ADJ
ejpam-5238	280	6	sciences	science	NOUN
ejpam-5238	280	7	,	,	PUNCT
ejpam-5238	280	8	43(1):41–58	43(1):41–58	NUM
ejpam-5238	280	9	,	,	PUNCT
ejpam-5238	280	10	2010	2010	NUM
ejpam-5238	280	11	.	.	PUNCT
ejpam-5238	281	1	[	[	X
ejpam-5238	281	2	4	4	NUM
ejpam-5238	281	3	]	]	PUNCT
ejpam-5238	281	4	c.	c.	PROPN
ejpam-5238	281	5	boonpok	boonpok	PROPN
ejpam-5238	281	6	.	.	PUNCT
ejpam-5238	282	1	on	on	ADP
ejpam-5238	282	2	continuous	continuous	ADJ
ejpam-5238	282	3	multifunctions	multifunction	NOUN
ejpam-5238	282	4	in	in	ADP
ejpam-5238	282	5	ideal	ideal	ADJ
ejpam-5238	282	6	topological	topological	ADJ
ejpam-5238	282	7	spaces	space	NOUN
ejpam-5238	282	8	.	.	PUNCT
ejpam-5238	283	1	lobachevskii	lobachevskii	PROPN
ejpam-5238	283	2	journal	journal	PROPN
ejpam-5238	283	3	of	of	ADP
ejpam-5238	283	4	mathematics	mathematic	NOUN
ejpam-5238	283	5	,	,	PUNCT
ejpam-5238	283	6	40(1):24–35	40(1):24–35	NUM
ejpam-5238	283	7	,	,	PUNCT
ejpam-5238	283	8	2019	2019	NUM
ejpam-5238	283	9	.	.	PUNCT
ejpam-5238	284	1	[	[	X
ejpam-5238	284	2	5	5	X
ejpam-5238	284	3	]	]	PUNCT
ejpam-5238	284	4	c.	c.	PROPN
ejpam-5238	284	5	boonpok	boonpok	PROPN
ejpam-5238	284	6	.	.	PUNCT
ejpam-5238	285	1	on	on	ADP
ejpam-5238	285	2	characterizations	characterization	NOUN
ejpam-5238	285	3	of	of	ADP
ejpam-5238	285	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5238	285	5	ideal	ideal	ADJ
ejpam-5238	285	6	topological	topological	ADJ
ejpam-5238	285	7	spaces	space	NOUN
ejpam-5238	285	8	.	.	PUNCT
ejpam-5238	286	1	journal	journal	NOUN
ejpam-5238	286	2	of	of	ADP
ejpam-5238	286	3	mathematics	mathematic	NOUN
ejpam-5238	286	4	,	,	PUNCT
ejpam-5238	286	5	2020:9387601	2020:9387601	NUM
ejpam-5238	286	6	,	,	PUNCT
ejpam-5238	286	7	2020	2020	NUM
ejpam-5238	286	8	.	.	PUNCT
ejpam-5238	287	1	[	[	X
ejpam-5238	287	2	6	6	NUM
ejpam-5238	287	3	]	]	PUNCT
ejpam-5238	287	4	c.	c.	PROPN
ejpam-5238	287	5	boonpok	boonpok	PROPN
ejpam-5238	287	6	.	.	PUNCT
ejpam-5238	288	1	(	(	PUNCT
ejpam-5238	288	2	τ1	τ1	NOUN
ejpam-5238	288	3	,	,	PUNCT
ejpam-5238	288	4	τ2)δ	τ2)δ	ADJ
ejpam-5238	288	5	-	-	PUNCT
ejpam-5238	288	6	semicontinuous	semicontinuous	ADJ
ejpam-5238	288	7	multifunctions	multifunction	NOUN
ejpam-5238	288	8	.	.	PUNCT
ejpam-5238	289	1	heliyon	heliyon	NOUN
ejpam-5238	289	2	,	,	PUNCT
ejpam-5238	289	3	6	6	NUM
ejpam-5238	289	4	:	:	SYM
ejpam-5238	289	5	e05367	e05367	PROPN
ejpam-5238	289	6	,	,	PUNCT
ejpam-5238	289	7	2020	2020	NUM
ejpam-5238	289	8	.	.	PUNCT
ejpam-5238	290	1	[	[	X
ejpam-5238	290	2	7	7	X
ejpam-5238	290	3	]	]	X
ejpam-5238	290	4	c.	c.	PROPN
ejpam-5238	290	5	boonpok	boonpok	PROPN
ejpam-5238	290	6	.	.	PUNCT
ejpam-5238	291	1	on	on	ADP
ejpam-5238	291	2	some	some	DET
ejpam-5238	291	3	closed	closed	ADJ
ejpam-5238	291	4	sets	set	NOUN
ejpam-5238	291	5	and	and	CCONJ
ejpam-5238	291	6	low	low	ADJ
ejpam-5238	291	7	separation	separation	NOUN
ejpam-5238	291	8	axioms	axiom	NOUN
ejpam-5238	291	9	via	via	ADP
ejpam-5238	291	10	topological	topological	ADJ
ejpam-5238	291	11	ideals	ideal	NOUN
ejpam-5238	291	12	.	.	PUNCT
ejpam-5238	292	1	european	european	ADJ
ejpam-5238	292	2	journal	journal	PROPN
ejpam-5238	292	3	of	of	ADP
ejpam-5238	292	4	pure	pure	ADJ
ejpam-5238	292	5	and	and	CCONJ
ejpam-5238	292	6	applied	applied	ADJ
ejpam-5238	292	7	mathematics	mathematic	NOUN
ejpam-5238	292	8	,	,	PUNCT
ejpam-5238	292	9	15(3):300–309	15(3):300–309	NOUN
ejpam-5238	292	10	,	,	PUNCT
ejpam-5238	292	11	2022	2022	NUM
ejpam-5238	292	12	.	.	PUNCT
ejpam-5238	293	1	[	[	X
ejpam-5238	293	2	8	8	NUM
ejpam-5238	293	3	]	]	X
ejpam-5238	293	4	c.	c.	PROPN
ejpam-5238	293	5	boonpok	boonpok	PROPN
ejpam-5238	293	6	.	.	PUNCT
ejpam-5238	294	1	on	on	ADP
ejpam-5238	294	2	some	some	DET
ejpam-5238	294	3	spaces	space	NOUN
ejpam-5238	294	4	via	via	ADP
ejpam-5238	294	5	topological	topological	ADJ
ejpam-5238	294	6	ideals	ideal	NOUN
ejpam-5238	294	7	.	.	PUNCT
ejpam-5238	295	1	open	open	ADJ
ejpam-5238	295	2	mathematics	mathematic	NOUN
ejpam-5238	295	3	,	,	PUNCT
ejpam-5238	295	4	21:20230118	21:20230118	NUM
ejpam-5238	295	5	,	,	PUNCT
ejpam-5238	295	6	2023	2023	NUM
ejpam-5238	295	7	.	.	PUNCT
ejpam-5238	296	1	[	[	X
ejpam-5238	296	2	9	9	NUM
ejpam-5238	296	3	]	]	PUNCT
ejpam-5238	296	4	c.	c.	PROPN
ejpam-5238	296	5	boonpok	boonpok	PROPN
ejpam-5238	296	6	.	.	PUNCT
ejpam-5238	297	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5238	297	2	.	.	PUNCT
ejpam-5238	298	1	mathematica	mathematica	PROPN
ejpam-5238	298	2	,	,	PUNCT
ejpam-5238	298	3	65(1):31–42	65(1):31–42	NUM
ejpam-5238	298	4	,	,	PUNCT
ejpam-5238	298	5	2023	2023	NUM
ejpam-5238	298	6	.	.	PUNCT
ejpam-5238	299	1	[	[	X
ejpam-5238	299	2	10	10	NUM
ejpam-5238	299	3	]	]	X
ejpam-5238	299	4	c.	c.	PROPN
ejpam-5238	299	5	boonpok	boonpok	PROPN
ejpam-5238	299	6	and	and	CCONJ
ejpam-5238	299	7	j.	j.	PROPN
ejpam-5238	299	8	khampakdee	khampakdee	PROPN
ejpam-5238	299	9	.	.	PUNCT
ejpam-5238	300	1	almost	almost	ADV
ejpam-5238	300	2	strong	strong	ADJ
ejpam-5238	300	3	θ(λ	θ(λ	PROPN
ejpam-5238	300	4	,	,	PUNCT
ejpam-5238	300	5	p)-continuity	p)-continuity	NOUN
ejpam-5238	300	6	for	for	ADP
ejpam-5238	300	7	functions	function	NOUN
ejpam-5238	300	8	.	.	PUNCT
ejpam-5238	301	1	european	european	ADJ
ejpam-5238	301	2	journal	journal	PROPN
ejpam-5238	301	3	of	of	ADP
ejpam-5238	301	4	pure	pure	ADJ
ejpam-5238	301	5	and	and	CCONJ
ejpam-5238	301	6	applied	applied	ADJ
ejpam-5238	301	7	mathematics	mathematic	NOUN
ejpam-5238	301	8	,	,	PUNCT
ejpam-5238	301	9	17(1):300–309	17(1):300–309	PROPN
ejpam-5238	301	10	,	,	PUNCT
ejpam-5238	301	11	2024	2024	NUM
ejpam-5238	301	12	.	.	PUNCT
ejpam-5238	302	1	[	[	X
ejpam-5238	302	2	11	11	NUM
ejpam-5238	302	3	]	]	X
ejpam-5238	302	4	c.	c.	PROPN
ejpam-5238	302	5	boonpok	boonpok	PROPN
ejpam-5238	302	6	and	and	CCONJ
ejpam-5238	302	7	c.	c.	PROPN
ejpam-5238	302	8	klanarong	klanarong	PROPN
ejpam-5238	302	9	.	.	PUNCT
ejpam-5238	303	1	on	on	ADP
ejpam-5238	303	2	weakly	weakly	ADJ
ejpam-5238	303	3	(	(	PUNCT
ejpam-5238	303	4	τ1	τ1	NOUN
ejpam-5238	303	5	,	,	PUNCT
ejpam-5238	303	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	303	7	functions	function	NOUN
ejpam-5238	303	8	.	.	PUNCT
ejpam-5238	304	1	european	european	ADJ
ejpam-5238	304	2	journal	journal	PROPN
ejpam-5238	304	3	of	of	ADP
ejpam-5238	304	4	pure	pure	ADJ
ejpam-5238	304	5	and	and	CCONJ
ejpam-5238	304	6	applied	applied	ADJ
ejpam-5238	304	7	mathematics	mathematic	NOUN
ejpam-5238	304	8	,	,	PUNCT
ejpam-5238	304	9	17(1):416–425	17(1):416–425	NUM
ejpam-5238	304	10	,	,	PUNCT
ejpam-5238	304	11	2024	2024	NUM
ejpam-5238	304	12	.	.	PUNCT
ejpam-5238	305	1	[	[	X
ejpam-5238	305	2	12	12	NUM
ejpam-5238	305	3	]	]	X
ejpam-5238	305	4	c.	c.	PROPN
ejpam-5238	305	5	boonpok	boonpok	PROPN
ejpam-5238	305	6	and	and	CCONJ
ejpam-5238	305	7	p.	p.	NOUN
ejpam-5238	305	8	pue	pue	NOUN
ejpam-5238	305	9	-	-	PUNCT
ejpam-5238	305	10	on	on	ADP
ejpam-5238	305	11	.	.	PUNCT
ejpam-5238	306	1	continuity	continuity	NOUN
ejpam-5238	306	2	for	for	ADP
ejpam-5238	306	3	multifunctions	multifunction	NOUN
ejpam-5238	306	4	in	in	ADP
ejpam-5238	306	5	ideal	ideal	ADJ
ejpam-5238	306	6	topological	topological	ADJ
ejpam-5238	306	7	spaces	space	NOUN
ejpam-5238	306	8	.	.	PUNCT
ejpam-5238	307	1	wseas	wseas	VERB
ejpam-5238	307	2	transactions	transaction	NOUN
ejpam-5238	307	3	on	on	ADP
ejpam-5238	307	4	mathematics	mathematic	NOUN
ejpam-5238	307	5	,	,	PUNCT
ejpam-5238	307	6	19:624–631	19:624–631	NUM
ejpam-5238	307	7	,	,	PUNCT
ejpam-5238	307	8	2020	2020	NUM
ejpam-5238	307	9	.	.	PUNCT
ejpam-5238	308	1	[	[	X
ejpam-5238	308	2	13	13	NUM
ejpam-5238	308	3	]	]	PUNCT
ejpam-5238	308	4	c.	c.	PROPN
ejpam-5238	308	5	boonpok	boonpok	PROPN
ejpam-5238	308	6	and	and	CCONJ
ejpam-5238	308	7	p.	p.	NOUN
ejpam-5238	308	8	pue	pue	NOUN
ejpam-5238	308	9	-	-	PUNCT
ejpam-5238	308	10	on	on	ADP
ejpam-5238	308	11	.	.	PUNCT
ejpam-5238	309	1	upper	upper	ADJ
ejpam-5238	309	2	and	and	CCONJ
ejpam-5238	309	3	lower	low	ADJ
ejpam-5238	309	4	weakly	weakly	ADJ
ejpam-5238	309	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5238	309	6	multifunctions	multifunction	NOUN
ejpam-5238	309	7	.	.	PUNCT
ejpam-5238	310	1	international	international	ADJ
ejpam-5238	310	2	journal	journal	NOUN
ejpam-5238	310	3	of	of	ADP
ejpam-5238	310	4	analysis	analysis	NOUN
ejpam-5238	310	5	and	and	CCONJ
ejpam-5238	310	6	applications	application	NOUN
ejpam-5238	310	7	,	,	PUNCT
ejpam-5238	310	8	21:90	21:90	NUM
ejpam-5238	310	9	,	,	PUNCT
ejpam-5238	310	10	2023	2023	NUM
ejpam-5238	310	11	.	.	PUNCT
ejpam-5238	311	1	[	[	X
ejpam-5238	311	2	14	14	NUM
ejpam-5238	311	3	]	]	X
ejpam-5238	311	4	c.	c.	PROPN
ejpam-5238	311	5	boonpok	boonpok	PROPN
ejpam-5238	311	6	and	and	CCONJ
ejpam-5238	311	7	p.	p.	NOUN
ejpam-5238	311	8	pue	pue	NOUN
ejpam-5238	311	9	-	-	PUNCT
ejpam-5238	311	10	on	on	ADP
ejpam-5238	311	11	.	.	PUNCT
ejpam-5238	312	1	upper	upper	ADJ
ejpam-5238	312	2	and	and	CCONJ
ejpam-5238	312	3	lower	low	ADJ
ejpam-5238	312	4	weakly	weakly	ADJ
ejpam-5238	312	5	(	(	PUNCT
ejpam-5238	312	6	λ	λ	NOUN
ejpam-5238	312	7	,	,	PUNCT
ejpam-5238	312	8	sp)-continuous	sp)-continuous	ADJ
ejpam-5238	312	9	multifunctions	multifunction	NOUN
ejpam-5238	312	10	.	.	PUNCT
ejpam-5238	313	1	european	european	PROPN
ejpam-5238	313	2	journal	journal	PROPN
ejpam-5238	313	3	of	of	ADP
ejpam-5238	313	4	pure	pure	ADJ
ejpam-5238	313	5	and	and	CCONJ
ejpam-5238	313	6	applied	applied	ADJ
ejpam-5238	313	7	mathematics	mathematic	NOUN
ejpam-5238	313	8	,	,	PUNCT
ejpam-5238	313	9	16(2):1047–1058	16(2):1047–1058	NUM
ejpam-5238	313	10	,	,	PUNCT
ejpam-5238	313	11	2023	2023	NUM
ejpam-5238	313	12	.	.	PUNCT
ejpam-5238	314	1	[	[	X
ejpam-5238	314	2	15	15	NUM
ejpam-5238	314	3	]	]	X
ejpam-5238	314	4	c.	c.	PROPN
ejpam-5238	314	5	boonpok	boonpok	PROPN
ejpam-5238	314	6	and	and	CCONJ
ejpam-5238	314	7	p.	p.	NOUN
ejpam-5238	314	8	pue	pue	NOUN
ejpam-5238	314	9	-	-	PUNCT
ejpam-5238	314	10	on	on	ADP
ejpam-5238	314	11	.	.	PUNCT
ejpam-5238	315	1	characterizations	characterization	NOUN
ejpam-5238	315	2	of	of	ADP
ejpam-5238	315	3	almost	almost	ADV
ejpam-5238	315	4	(	(	PUNCT
ejpam-5238	315	5	τ1	τ1	NOUN
ejpam-5238	315	6	,	,	PUNCT
ejpam-5238	315	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	315	8	functions	function	NOUN
ejpam-5238	315	9	.	.	PUNCT
ejpam-5238	316	1	international	international	ADJ
ejpam-5238	316	2	journal	journal	NOUN
ejpam-5238	316	3	of	of	ADP
ejpam-5238	316	4	analysis	analysis	NOUN
ejpam-5238	316	5	and	and	CCONJ
ejpam-5238	316	6	applications	application	NOUN
ejpam-5238	316	7	,	,	PUNCT
ejpam-5238	316	8	22:33	22:33	NUM
ejpam-5238	316	9	,	,	PUNCT
ejpam-5238	316	10	2024	2024	NUM
ejpam-5238	316	11	.	.	PUNCT
ejpam-5238	317	1	references	reference	NOUN
ejpam-5238	317	2	1715	1715	NUM
ejpam-5238	317	3	[	[	X
ejpam-5238	317	4	16	16	NUM
ejpam-5238	317	5	]	]	PUNCT
ejpam-5238	317	6	c.	c.	PROPN
ejpam-5238	317	7	boonpok	boonpok	PROPN
ejpam-5238	317	8	and	and	CCONJ
ejpam-5238	317	9	n.	n.	PROPN
ejpam-5238	317	10	srisarakham	srisarakham	PROPN
ejpam-5238	317	11	.	.	PUNCT
ejpam-5238	318	1	weak	weak	ADJ
ejpam-5238	318	2	forms	form	NOUN
ejpam-5238	318	3	of	of	ADP
ejpam-5238	318	4	(	(	PUNCT
ejpam-5238	318	5	λ	λ	PROPN
ejpam-5238	318	6	,	,	PUNCT
ejpam-5238	318	7	b)-open	b)-open	VERB
ejpam-5238	318	8	sets	set	NOUN
ejpam-5238	318	9	and	and	CCONJ
ejpam-5238	318	10	weak	weak	ADJ
ejpam-5238	318	11	(	(	PUNCT
ejpam-5238	318	12	λ	λ	NOUN
ejpam-5238	318	13	,	,	PUNCT
ejpam-5238	318	14	b)continuity	b)continuity	NOUN
ejpam-5238	318	15	.	.	PUNCT
ejpam-5238	319	1	european	european	PROPN
ejpam-5238	319	2	journal	journal	PROPN
ejpam-5238	319	3	of	of	ADP
ejpam-5238	319	4	pure	pure	ADJ
ejpam-5238	319	5	and	and	CCONJ
ejpam-5238	319	6	applied	applied	ADJ
ejpam-5238	319	7	mathematics	mathematic	NOUN
ejpam-5238	319	8	,	,	PUNCT
ejpam-5238	319	9	16(1):29–43	16(1):29–43	NUM
ejpam-5238	319	10	,	,	PUNCT
ejpam-5238	319	11	2023	2023	NUM
ejpam-5238	319	12	.	.	PUNCT
ejpam-5238	320	1	[	[	X
ejpam-5238	320	2	17	17	NUM
ejpam-5238	320	3	]	]	X
ejpam-5238	320	4	c.	c.	PROPN
ejpam-5238	320	5	boonpok	boonpok	PROPN
ejpam-5238	320	6	and	and	CCONJ
ejpam-5238	320	7	n.	n.	PROPN
ejpam-5238	320	8	srisarakham	srisarakham	PROPN
ejpam-5238	320	9	.	.	PUNCT
ejpam-5238	321	1	(	(	PUNCT
ejpam-5238	321	2	τ1	τ1	NOUN
ejpam-5238	321	3	,	,	PUNCT
ejpam-5238	321	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5238	321	5	for	for	ADP
ejpam-5238	321	6	functions	function	NOUN
ejpam-5238	321	7	.	.	PUNCT
ejpam-5238	322	1	asia	asia	PROPN
ejpam-5238	322	2	pacific	pacific	PROPN
ejpam-5238	322	3	journal	journal	PROPN
ejpam-5238	322	4	of	of	ADP
ejpam-5238	322	5	mathematics	mathematic	NOUN
ejpam-5238	322	6	,	,	PUNCT
ejpam-5238	322	7	11:21	11:21	NUM
ejpam-5238	322	8	,	,	PUNCT
ejpam-5238	322	9	2024	2024	NUM
ejpam-5238	322	10	.	.	PUNCT
ejpam-5238	323	1	[	[	X
ejpam-5238	323	2	18	18	NUM
ejpam-5238	323	3	]	]	PUNCT
ejpam-5238	323	4	c.	c.	PROPN
ejpam-5238	323	5	boonpok	boonpok	PROPN
ejpam-5238	323	6	and	and	CCONJ
ejpam-5238	323	7	c.	c.	PROPN
ejpam-5238	323	8	viriyapong	viriyapong	PROPN
ejpam-5238	323	9	.	.	PUNCT
ejpam-5238	324	1	almost	almost	ADV
ejpam-5238	324	2	weak	weak	ADJ
ejpam-5238	324	3	continuity	continuity	NOUN
ejpam-5238	324	4	for	for	ADP
ejpam-5238	324	5	multifunctions	multifunction	NOUN
ejpam-5238	324	6	in	in	ADP
ejpam-5238	324	7	ideal	ideal	ADJ
ejpam-5238	324	8	topological	topological	ADJ
ejpam-5238	324	9	spaces	space	NOUN
ejpam-5238	324	10	.	.	PUNCT
ejpam-5238	325	1	wseas	wseas	VERB
ejpam-5238	325	2	transactions	transaction	NOUN
ejpam-5238	325	3	on	on	ADP
ejpam-5238	325	4	mathematics	mathematic	NOUN
ejpam-5238	325	5	,	,	PUNCT
ejpam-5238	325	6	19:367–372	19:367–372	PROPN
ejpam-5238	325	7	,	,	PUNCT
ejpam-5238	325	8	2020	2020	NUM
ejpam-5238	325	9	.	.	PUNCT
ejpam-5238	326	1	[	[	X
ejpam-5238	326	2	19	19	NUM
ejpam-5238	326	3	]	]	X
ejpam-5238	326	4	c.	c.	PROPN
ejpam-5238	326	5	boonpok	boonpok	PROPN
ejpam-5238	326	6	and	and	CCONJ
ejpam-5238	326	7	c.	c.	PROPN
ejpam-5238	326	8	viriyapong	viriyapong	PROPN
ejpam-5238	326	9	.	.	PUNCT
ejpam-5238	327	1	upper	upper	ADJ
ejpam-5238	327	2	and	and	CCONJ
ejpam-5238	327	3	lower	low	ADJ
ejpam-5238	327	4	almost	almost	ADV
ejpam-5238	327	5	weak	weak	ADJ
ejpam-5238	327	6	(	(	PUNCT
ejpam-5238	327	7	τ1	τ1	NOUN
ejpam-5238	327	8	,	,	PUNCT
ejpam-5238	327	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5238	327	10	.	.	PUNCT
ejpam-5238	328	1	european	european	PROPN
ejpam-5238	328	2	journal	journal	PROPN
ejpam-5238	328	3	of	of	ADP
ejpam-5238	328	4	pure	pure	ADJ
ejpam-5238	328	5	and	and	CCONJ
ejpam-5238	328	6	applied	applied	ADJ
ejpam-5238	328	7	mathematics	mathematic	NOUN
ejpam-5238	328	8	,	,	PUNCT
ejpam-5238	328	9	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-5238	328	10	,	,	PUNCT
ejpam-5238	328	11	2021	2021	NUM
ejpam-5238	328	12	.	.	PUNCT
ejpam-5238	329	1	[	[	X
ejpam-5238	329	2	20	20	NUM
ejpam-5238	329	3	]	]	PUNCT
ejpam-5238	329	4	c.	c.	PROPN
ejpam-5238	329	5	boonpok	boonpok	PROPN
ejpam-5238	329	6	,	,	PUNCT
ejpam-5238	329	7	c.	c.	PROPN
ejpam-5238	329	8	viriyapong	viriyapong	PROPN
ejpam-5238	329	9	,	,	PUNCT
ejpam-5238	329	10	and	and	CCONJ
ejpam-5238	329	11	m.	m.	NOUN
ejpam-5238	329	12	thongmoon	thongmoon	NOUN
ejpam-5238	329	13	.	.	PUNCT
ejpam-5238	330	1	on	on	ADP
ejpam-5238	330	2	upper	upper	ADJ
ejpam-5238	330	3	and	and	CCONJ
ejpam-5238	330	4	lower	low	ADJ
ejpam-5238	330	5	(	(	PUNCT
ejpam-5238	330	6	τ1	τ1	NOUN
ejpam-5238	330	7	,	,	PUNCT
ejpam-5238	330	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-5238	330	9	multifunctions	multifunction	NOUN
ejpam-5238	330	10	.	.	PUNCT
ejpam-5238	331	1	journal	journal	PROPN
ejpam-5238	331	2	of	of	ADP
ejpam-5238	331	3	mathematics	mathematics	PROPN
ejpam-5238	331	4	and	and	CCONJ
ejpam-5238	331	5	computer	computer	NOUN
ejpam-5238	331	6	science	science	NOUN
ejpam-5238	331	7	,	,	PUNCT
ejpam-5238	331	8	18:282–293	18:282–293	NUM
ejpam-5238	331	9	,	,	PUNCT
ejpam-5238	331	10	2018	2018	NUM
ejpam-5238	331	11	.	.	PUNCT
ejpam-5238	332	1	[	[	X
ejpam-5238	332	2	21	21	NUM
ejpam-5238	332	3	]	]	X
ejpam-5238	332	4	t.	t.	PROPN
ejpam-5238	332	5	duangphui	duangphui	PROPN
ejpam-5238	332	6	,	,	PUNCT
ejpam-5238	332	7	c.	c.	PROPN
ejpam-5238	332	8	boonpok	boonpok	PROPN
ejpam-5238	332	9	,	,	PUNCT
ejpam-5238	332	10	and	and	CCONJ
ejpam-5238	332	11	c.	c.	PROPN
ejpam-5238	332	12	viriyapong	viriyapong	PROPN
ejpam-5238	332	13	.	.	PUNCT
ejpam-5238	333	1	continuous	continuous	ADJ
ejpam-5238	333	2	functions	function	NOUN
ejpam-5238	333	3	on	on	ADP
ejpam-5238	333	4	bigeneralized	bigeneralize	VERB
ejpam-5238	333	5	topological	topological	ADJ
ejpam-5238	333	6	spaces	space	NOUN
ejpam-5238	333	7	.	.	PUNCT
ejpam-5238	334	1	international	international	ADJ
ejpam-5238	334	2	journal	journal	PROPN
ejpam-5238	334	3	of	of	ADP
ejpam-5238	334	4	mathematical	mathematical	ADJ
ejpam-5238	334	5	analysis	analysis	NOUN
ejpam-5238	334	6	,	,	PUNCT
ejpam-5238	334	7	5(24):1165	5(24):1165	NUM
ejpam-5238	334	8	–	–	PUNCT
ejpam-5238	334	9	1174	1174	NUM
ejpam-5238	334	10	,	,	PUNCT
ejpam-5238	334	11	2011	2011	NUM
ejpam-5238	334	12	.	.	PUNCT
ejpam-5238	335	1	[	[	X
ejpam-5238	335	2	22	22	NUM
ejpam-5238	335	3	]	]	X
ejpam-5238	335	4	e.	e.	PROPN
ejpam-5238	335	5	ekici	ekici	PROPN
ejpam-5238	335	6	,	,	PUNCT
ejpam-5238	335	7	s.	s.	PROPN
ejpam-5238	335	8	jafari	jafari	PROPN
ejpam-5238	335	9	,	,	PUNCT
ejpam-5238	335	10	m.	m.	PROPN
ejpam-5238	335	11	caldas	caldas	PROPN
ejpam-5238	335	12	,	,	PUNCT
ejpam-5238	335	13	and	and	CCONJ
ejpam-5238	335	14	t.	t.	PROPN
ejpam-5238	335	15	noiri	noiri	PROPN
ejpam-5238	335	16	.	.	PUNCT
ejpam-5238	336	1	weakly	weakly	ADJ
ejpam-5238	336	2	λ	λ	ADJ
ejpam-5238	336	3	-	-	ADJ
ejpam-5238	336	4	continuous	continuous	ADJ
ejpam-5238	336	5	functions	function	NOUN
ejpam-5238	336	6	.	.	PUNCT
ejpam-5238	337	1	novi	novi	PROPN
ejpam-5238	337	2	sad	sad	PROPN
ejpam-5238	337	3	journal	journal	PROPN
ejpam-5238	337	4	of	of	ADP
ejpam-5238	337	5	mathematics	mathematic	NOUN
ejpam-5238	337	6	,	,	PUNCT
ejpam-5238	337	7	38:47–56	38:47–56	NUM
ejpam-5238	337	8	,	,	PUNCT
ejpam-5238	337	9	2008	2008	NUM
ejpam-5238	337	10	.	.	PUNCT
ejpam-5238	338	1	[	[	X
ejpam-5238	338	2	23	23	NUM
ejpam-5238	338	3	]	]	PUNCT
ejpam-5238	338	4	t.	t.	PROPN
ejpam-5238	338	5	husain	husain	PROPN
ejpam-5238	338	6	.	.	PUNCT
ejpam-5238	339	1	almost	almost	ADV
ejpam-5238	339	2	continuous	continuous	ADJ
ejpam-5238	339	3	mappings	mapping	NOUN
ejpam-5238	339	4	.	.	PUNCT
ejpam-5238	340	1	prace	prace	PROPN
ejpam-5238	340	2	matematyczne	matematyczne	PROPN
ejpam-5238	340	3	,	,	PUNCT
ejpam-5238	340	4	10:1–7	10:1–7	NUM
ejpam-5238	340	5	,	,	PUNCT
ejpam-5238	340	6	1966	1966	NUM
ejpam-5238	340	7	.	.	PUNCT
ejpam-5238	341	1	[	[	X
ejpam-5238	341	2	24	24	NUM
ejpam-5238	341	3	]	]	X
ejpam-5238	341	4	d.	d.	PROPN
ejpam-5238	341	5	s.	s.	PROPN
ejpam-5238	341	6	janković.	janković.	PROPN
ejpam-5238	341	7	θ	θ	PROPN
ejpam-5238	341	8	-	-	ADJ
ejpam-5238	341	9	regular	regular	ADJ
ejpam-5238	341	10	spaces	space	NOUN
ejpam-5238	341	11	.	.	PUNCT
ejpam-5238	342	1	international	international	ADJ
ejpam-5238	342	2	journal	journal	PROPN
ejpam-5238	342	3	of	of	ADP
ejpam-5238	342	4	mathematics	mathematics	PROPN
ejpam-5238	342	5	and	and	CCONJ
ejpam-5238	342	6	mathematical	mathematical	ADJ
ejpam-5238	342	7	sciences	science	NOUN
ejpam-5238	342	8	,	,	PUNCT
ejpam-5238	342	9	8:615–619	8:615–619	NUM
ejpam-5238	342	10	,	,	PUNCT
ejpam-5238	342	11	1985	1985	NUM
ejpam-5238	342	12	.	.	PUNCT
ejpam-5238	343	1	[	[	X
ejpam-5238	343	2	25	25	NUM
ejpam-5238	343	3	]	]	PUNCT
ejpam-5238	343	4	k.	k.	PROPN
ejpam-5238	343	5	laprom	laprom	PROPN
ejpam-5238	343	6	,	,	PUNCT
ejpam-5238	343	7	c.	c.	PROPN
ejpam-5238	343	8	boonpok	boonpok	PROPN
ejpam-5238	343	9	,	,	PUNCT
ejpam-5238	343	10	and	and	CCONJ
ejpam-5238	343	11	c.	c.	PROPN
ejpam-5238	343	12	viriyapong	viriyapong	PROPN
ejpam-5238	343	13	.	.	PUNCT
ejpam-5238	344	1	β(τ1	β(τ1	PROPN
ejpam-5238	344	2	,	,	PUNCT
ejpam-5238	344	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5238	344	4	multifunctions	multifunction	NOUN
ejpam-5238	344	5	on	on	ADP
ejpam-5238	344	6	bitopological	bitopological	ADJ
ejpam-5238	344	7	spaces	space	NOUN
ejpam-5238	344	8	.	.	PUNCT
ejpam-5238	345	1	journal	journal	NOUN
ejpam-5238	345	2	of	of	ADP
ejpam-5238	345	3	mathematics	mathematic	NOUN
ejpam-5238	345	4	,	,	PUNCT
ejpam-5238	345	5	2020:4020971	2020:4020971	NUM
ejpam-5238	345	6	,	,	PUNCT
ejpam-5238	345	7	2020	2020	NUM
ejpam-5238	345	8	.	.	PUNCT
ejpam-5238	346	1	[	[	X
ejpam-5238	346	2	26	26	NUM
ejpam-5238	346	3	]	]	X
ejpam-5238	346	4	n.	n.	PROPN
ejpam-5238	346	5	levine	levine	PROPN
ejpam-5238	346	6	.	.	PUNCT
ejpam-5238	347	1	a	a	DET
ejpam-5238	347	2	decomposition	decomposition	NOUN
ejpam-5238	347	3	of	of	ADP
ejpam-5238	347	4	continuity	continuity	NOUN
ejpam-5238	347	5	in	in	ADP
ejpam-5238	347	6	topological	topological	ADJ
ejpam-5238	347	7	spaces	space	NOUN
ejpam-5238	347	8	.	.	PUNCT
ejpam-5238	348	1	the	the	DET
ejpam-5238	348	2	american	american	PROPN
ejpam-5238	348	3	mathematical	mathematical	PROPN
ejpam-5238	348	4	monthly	monthly	ADV
ejpam-5238	348	5	,	,	PUNCT
ejpam-5238	348	6	68:44–46	68:44–46	NUM
ejpam-5238	348	7	,	,	PUNCT
ejpam-5238	348	8	1961	1961	NUM
ejpam-5238	348	9	.	.	PUNCT
ejpam-5238	349	1	[	[	X
ejpam-5238	349	2	27	27	NUM
ejpam-5238	349	3	]	]	PUNCT
ejpam-5238	349	4	t.	t.	PROPN
ejpam-5238	349	5	noiri	noiri	PROPN
ejpam-5238	349	6	.	.	PUNCT
ejpam-5238	350	1	properties	property	NOUN
ejpam-5238	350	2	of	of	ADP
ejpam-5238	350	3	some	some	DET
ejpam-5238	350	4	weak	weak	ADJ
ejpam-5238	350	5	forms	form	NOUN
ejpam-5238	350	6	of	of	ADP
ejpam-5238	350	7	continuity	continuity	NOUN
ejpam-5238	350	8	.	.	PUNCT
ejpam-5238	351	1	international	international	ADJ
ejpam-5238	351	2	journal	journal	PROPN
ejpam-5238	351	3	of	of	ADP
ejpam-5238	351	4	mathematics	mathematics	PROPN
ejpam-5238	351	5	and	and	CCONJ
ejpam-5238	351	6	mathematical	mathematical	ADJ
ejpam-5238	351	7	sciences	science	NOUN
ejpam-5238	351	8	,	,	PUNCT
ejpam-5238	351	9	10(1):97–111	10(1):97–111	NUM
ejpam-5238	351	10	,	,	PUNCT
ejpam-5238	351	11	1987	1987	NUM
ejpam-5238	351	12	.	.	PUNCT
ejpam-5238	352	1	[	[	X
ejpam-5238	352	2	28	28	NUM
ejpam-5238	352	3	]	]	X
ejpam-5238	352	4	t.	t.	PROPN
ejpam-5238	352	5	noiri	noiri	PROPN
ejpam-5238	352	6	and	and	CCONJ
ejpam-5238	352	7	v.	v.	ADP
ejpam-5238	352	8	popa	popa	NOUN
ejpam-5238	352	9	.	.	PUNCT
ejpam-5238	353	1	a	a	DET
ejpam-5238	353	2	unified	unified	ADJ
ejpam-5238	353	3	theory	theory	NOUN
ejpam-5238	353	4	of	of	ADP
ejpam-5238	353	5	weak	weak	ADJ
ejpam-5238	353	6	continuity	continuity	NOUN
ejpam-5238	353	7	for	for	ADP
ejpam-5238	353	8	multifunctions	multifunction	NOUN
ejpam-5238	353	9	.	.	PUNCT
ejpam-5238	354	1	studii	studii	PROPN
ejpam-5238	354	2	şi	şi	PROPN
ejpam-5238	354	3	cercetǎri	cercetǎri	VERB
ejpam-5238	354	4	ştiinţifice	ştiinţifice	NOUN
ejpam-5238	354	5	,	,	PUNCT
ejpam-5238	354	6	ser	ser	NOUN
ejpam-5238	354	7	.	.	PUNCT
ejpam-5238	355	1	matematicǎ-universitatea	matematicǎ-universitatea	PROPN
ejpam-5238	356	1	din	din	VERB
ejpam-5238	356	2	bacǎu	bacǎu	PROPN
ejpam-5238	356	3	,	,	PUNCT
ejpam-5238	356	4	16:167–200	16:167–200	NUM
ejpam-5238	356	5	,	,	PUNCT
ejpam-5238	356	6	2006	2006	NUM
ejpam-5238	356	7	.	.	PUNCT
ejpam-5238	357	1	[	[	X
ejpam-5238	357	2	29	29	NUM
ejpam-5238	357	3	]	]	PUNCT
ejpam-5238	357	4	v.	v.	CCONJ
ejpam-5238	357	5	popa	popa	NOUN
ejpam-5238	357	6	.	.	PUNCT
ejpam-5238	358	1	weakly	weakly	ADJ
ejpam-5238	358	2	continuous	continuous	ADJ
ejpam-5238	358	3	multifunctions	multifunction	NOUN
ejpam-5238	358	4	.	.	PUNCT
ejpam-5238	359	1	bollettino	bollettino	PROPN
ejpam-5238	359	2	dell	dell	PROPN
ejpam-5238	359	3	’	'	PUNCT
ejpam-5238	359	4	unione	unione	PROPN
ejpam-5238	359	5	matematica	matematica	PROPN
ejpam-5238	359	6	italiana	italiana	PROPN
ejpam-5238	359	7	,	,	PUNCT
ejpam-5238	359	8	15(a)(5):379–388	15(a)(5):379–388	NUM
ejpam-5238	359	9	,	,	PUNCT
ejpam-5238	359	10	1978	1978	NUM
ejpam-5238	359	11	.	.	PUNCT
ejpam-5238	360	1	[	[	X
ejpam-5238	360	2	30	30	NUM
ejpam-5238	360	3	]	]	X
ejpam-5238	360	4	v.	v.	CCONJ
ejpam-5238	360	5	popa	popa	NOUN
ejpam-5238	360	6	and	and	CCONJ
ejpam-5238	360	7	t.	t.	PROPN
ejpam-5238	360	8	noiri	noiri	PROPN
ejpam-5238	360	9	.	.	PUNCT
ejpam-5238	361	1	on	on	ADP
ejpam-5238	361	2	upper	upper	ADJ
ejpam-5238	361	3	and	and	CCONJ
ejpam-5238	361	4	lower	low	ADJ
ejpam-5238	361	5	weakly	weakly	ADJ
ejpam-5238	361	6	β	β	ADJ
ejpam-5238	361	7	-	-	ADJ
ejpam-5238	361	8	continuous	continuous	ADJ
ejpam-5238	361	9	multifunctions	multifunction	NOUN
ejpam-5238	361	10	.	.	PUNCT
ejpam-5238	362	1	annales	annales	PROPN
ejpam-5238	362	2	universitatis	universitatis	PROPN
ejpam-5238	362	3	scientiarium	scientiarium	PROPN
ejpam-5238	362	4	budapestinensis	budapestinensis	NOUN
ejpam-5238	362	5	de	de	PROPN
ejpam-5238	362	6	rolando	rolando	PROPN
ejpam-5238	362	7	eötvös	eötvös	PROPN
ejpam-5238	362	8	nominatae	nominatae	PROPN
ejpam-5238	362	9	.	.	PUNCT
ejpam-5238	363	1	sectio	sectio	PROPN
ejpam-5238	363	2	mathematica	mathematica	PROPN
ejpam-5238	363	3	,	,	PUNCT
ejpam-5238	363	4	43:25–48	43:25–48	PROPN
ejpam-5238	363	5	,	,	PUNCT
ejpam-5238	363	6	2000	2000	NUM
ejpam-5238	363	7	.	.	PUNCT
ejpam-5238	364	1	[	[	X
ejpam-5238	364	2	31	31	NUM
ejpam-5238	364	3	]	]	PUNCT
ejpam-5238	364	4	v.	v.	CCONJ
ejpam-5238	364	5	popa	popa	NOUN
ejpam-5238	364	6	and	and	CCONJ
ejpam-5238	364	7	t.	t.	PROPN
ejpam-5238	364	8	noiri	noiri	PROPN
ejpam-5238	364	9	.	.	PUNCT
ejpam-5238	365	1	on	on	ADP
ejpam-5238	365	2	upper	upper	ADJ
ejpam-5238	365	3	and	and	CCONJ
ejpam-5238	365	4	lower	low	ADJ
ejpam-5238	365	5	weakly	weakly	ADJ
ejpam-5238	365	6	α	α	ADJ
ejpam-5238	365	7	-	-	ADJ
ejpam-5238	365	8	continuous	continuous	ADJ
ejpam-5238	365	9	multifunctions	multifunction	NOUN
ejpam-5238	365	10	.	.	PUNCT
ejpam-5238	366	1	novi	novi	PROPN
ejpam-5238	366	2	sad	sad	PROPN
ejpam-5238	366	3	journal	journal	PROPN
ejpam-5238	366	4	of	of	ADP
ejpam-5238	366	5	mathematics	mathematic	NOUN
ejpam-5238	366	6	,	,	PUNCT
ejpam-5238	366	7	32(1):7–24	32(1):7–24	NUM
ejpam-5238	366	8	,	,	PUNCT
ejpam-5238	366	9	2002	2002	NUM
ejpam-5238	366	10	.	.	PUNCT
ejpam-5238	367	1	references	reference	NOUN
ejpam-5238	367	2	1716	1716	NUM
ejpam-5238	367	3	[	[	X
ejpam-5238	367	4	32	32	NUM
ejpam-5238	367	5	]	]	PUNCT
ejpam-5238	367	6	v.	v.	CCONJ
ejpam-5238	367	7	popa	popa	NOUN
ejpam-5238	367	8	and	and	CCONJ
ejpam-5238	367	9	t.	t.	PROPN
ejpam-5238	367	10	noiri	noiri	PROPN
ejpam-5238	367	11	.	.	PUNCT
ejpam-5238	368	1	on	on	ADP
ejpam-5238	368	2	weakly	weakly	ADJ
ejpam-5238	368	3	(	(	PUNCT
ejpam-5238	368	4	τ	τ	PROPN
ejpam-5238	368	5	,	,	PUNCT
ejpam-5238	368	6	m)-continuous	m)-continuous	ADJ
ejpam-5238	368	7	functions	function	NOUN
ejpam-5238	368	8	.	.	PUNCT
ejpam-5238	369	1	rendiconti	rendiconti	ADJ
ejpam-5238	369	2	del	del	PROPN
ejpam-5238	369	3	circolo	circolo	PROPN
ejpam-5238	369	4	matematico	matematico	NOUN
ejpam-5238	369	5	di	di	X
ejpam-5238	369	6	palermo	palermo	NOUN
ejpam-5238	369	7	(	(	PUNCT
ejpam-5238	369	8	2	2	NUM
ejpam-5238	369	9	)	)	PUNCT
ejpam-5238	369	10	,	,	PUNCT
ejpam-5238	369	11	51:295–316	51:295–316	PROPN
ejpam-5238	369	12	,	,	PUNCT
ejpam-5238	369	13	2002	2002	NUM
ejpam-5238	369	14	.	.	PUNCT
ejpam-5238	370	1	[	[	X
ejpam-5238	370	2	33	33	NUM
ejpam-5238	370	3	]	]	PUNCT
ejpam-5238	370	4	p.	p.	NOUN
ejpam-5238	370	5	pue	pue	NOUN
ejpam-5238	370	6	-	-	PUNCT
ejpam-5238	370	7	on	on	ADP
ejpam-5238	370	8	and	and	CCONJ
ejpam-5238	370	9	c.	c.	PROPN
ejpam-5238	370	10	boonpok	boonpok	PROPN
ejpam-5238	370	11	.	.	PUNCT
ejpam-5238	371	1	θ(λ	θ(λ	PROPN
ejpam-5238	371	2	,	,	PUNCT
ejpam-5238	371	3	p)-continuity	p)-continuity	NOUN
ejpam-5238	371	4	for	for	ADP
ejpam-5238	371	5	functions	function	NOUN
ejpam-5238	371	6	.	.	PUNCT
ejpam-5238	372	1	international	international	ADJ
ejpam-5238	372	2	journal	journal	NOUN
ejpam-5238	372	3	of	of	ADP
ejpam-5238	372	4	mathematics	mathematic	NOUN
ejpam-5238	372	5	and	and	CCONJ
ejpam-5238	372	6	computer	computer	NOUN
ejpam-5238	372	7	science	science	NOUN
ejpam-5238	372	8	,	,	PUNCT
ejpam-5238	372	9	19(2):491–495	19(2):491–495	NUM
ejpam-5238	372	10	,	,	PUNCT
ejpam-5238	372	11	2024	2024	NUM
ejpam-5238	372	12	.	.	PUNCT
ejpam-5238	373	1	[	[	X
ejpam-5238	373	2	34	34	NUM
ejpam-5238	373	3	]	]	X
ejpam-5238	373	4	d.	d.	PROPN
ejpam-5238	373	5	a.	a.	PROPN
ejpam-5238	373	6	rose	rise	VERB
ejpam-5238	373	7	.	.	PUNCT
ejpam-5238	374	1	weak	weak	ADJ
ejpam-5238	374	2	continuity	continuity	NOUN
ejpam-5238	374	3	and	and	CCONJ
ejpam-5238	374	4	almost	almost	ADV
ejpam-5238	374	5	continuity	continuity	NOUN
ejpam-5238	374	6	.	.	PUNCT
ejpam-5238	375	1	international	international	ADJ
ejpam-5238	375	2	journal	journal	PROPN
ejpam-5238	375	3	of	of	ADP
ejpam-5238	375	4	mathematics	mathematics	PROPN
ejpam-5238	375	5	and	and	CCONJ
ejpam-5238	375	6	mathematical	mathematical	ADJ
ejpam-5238	375	7	sciences	science	NOUN
ejpam-5238	375	8	,	,	PUNCT
ejpam-5238	375	9	7:311–318	7:311–318	PROPN
ejpam-5238	375	10	,	,	PUNCT
ejpam-5238	375	11	1984	1984	NUM
ejpam-5238	375	12	.	.	PUNCT
ejpam-5238	376	1	[	[	X
ejpam-5238	376	2	35	35	NUM
ejpam-5238	376	3	]	]	X
ejpam-5238	376	4	r.	r.	PROPN
ejpam-5238	376	5	e.	e.	PROPN
ejpam-5238	376	6	smithson	smithson	PROPN
ejpam-5238	376	7	.	.	PUNCT
ejpam-5238	377	1	almost	almost	ADV
ejpam-5238	377	2	and	and	CCONJ
ejpam-5238	377	3	weak	weak	ADJ
ejpam-5238	377	4	continuity	continuity	NOUN
ejpam-5238	377	5	for	for	ADP
ejpam-5238	377	6	multifunctions	multifunction	NOUN
ejpam-5238	377	7	.	.	PUNCT
ejpam-5238	378	1	bulletin	bulletin	NOUN
ejpam-5238	378	2	of	of	ADP
ejpam-5238	378	3	the	the	DET
ejpam-5238	378	4	calcutta	calcutta	PROPN
ejpam-5238	378	5	mathematical	mathematical	ADJ
ejpam-5238	378	6	society	society	NOUN
ejpam-5238	378	7	,	,	PUNCT
ejpam-5238	378	8	70:383–390	70:383–390	NUM
ejpam-5238	378	9	,	,	PUNCT
ejpam-5238	378	10	1978	1978	NUM
ejpam-5238	378	11	.	.	PUNCT
ejpam-5238	379	1	[	[	X
ejpam-5238	379	2	36	36	NUM
ejpam-5238	379	3	]	]	X
ejpam-5238	379	4	n.	n.	PROPN
ejpam-5238	379	5	srisarakham	srisarakham	PROPN
ejpam-5238	379	6	and	and	CCONJ
ejpam-5238	379	7	c.	c.	PROPN
ejpam-5238	379	8	boonpok	boonpok	PROPN
ejpam-5238	379	9	.	.	PUNCT
ejpam-5238	380	1	almost	almost	ADV
ejpam-5238	380	2	(	(	PUNCT
ejpam-5238	380	3	λ	λ	NOUN
ejpam-5238	380	4	,	,	PUNCT
ejpam-5238	380	5	p)-continuous	p)-continuous	ADJ
ejpam-5238	380	6	functions	function	NOUN
ejpam-5238	380	7	.	.	PUNCT
ejpam-5238	381	1	international	international	ADJ
ejpam-5238	381	2	journal	journal	PROPN
ejpam-5238	381	3	of	of	ADP
ejpam-5238	381	4	mathematics	mathematic	NOUN
ejpam-5238	381	5	and	and	CCONJ
ejpam-5238	381	6	computer	computer	NOUN
ejpam-5238	381	7	science	science	NOUN
ejpam-5238	381	8	,	,	PUNCT
ejpam-5238	381	9	18(2):255–259	18(2):255–259	NUM
ejpam-5238	381	10	,	,	PUNCT
ejpam-5238	381	11	2023	2023	NUM
ejpam-5238	381	12	.	.	PUNCT
ejpam-5238	382	1	[	[	X
ejpam-5238	382	2	37	37	NUM
ejpam-5238	382	3	]	]	X
ejpam-5238	382	4	n.	n.	NOUN
ejpam-5238	382	5	srisarakham	srisarakham	PROPN
ejpam-5238	382	6	and	and	CCONJ
ejpam-5238	382	7	c.	c.	PROPN
ejpam-5238	382	8	boonpok	boonpok	PROPN
ejpam-5238	382	9	.	.	PUNCT
ejpam-5238	383	1	on	on	ADP
ejpam-5238	383	2	characterizations	characterization	NOUN
ejpam-5238	383	3	of	of	ADP
ejpam-5238	383	4	δp(λ	δp(λ	NOUN
ejpam-5238	383	5	,	,	PUNCT
ejpam-5238	383	6	s)-d1	s)-d1	NOUN
ejpam-5238	383	7	spaces	space	NOUN
ejpam-5238	383	8	.	.	PUNCT
ejpam-5238	384	1	international	international	ADJ
ejpam-5238	384	2	journal	journal	PROPN
ejpam-5238	384	3	of	of	ADP
ejpam-5238	384	4	mathematics	mathematic	NOUN
ejpam-5238	384	5	and	and	CCONJ
ejpam-5238	384	6	computer	computer	NOUN
ejpam-5238	384	7	science	science	NOUN
ejpam-5238	384	8	,	,	PUNCT
ejpam-5238	384	9	18(4):743–747	18(4):743–747	PROPN
ejpam-5238	384	10	,	,	PUNCT
ejpam-5238	384	11	2023	2023	NUM
ejpam-5238	384	12	.	.	PUNCT
ejpam-5238	385	1	[	[	X
ejpam-5238	385	2	38	38	NUM
ejpam-5238	385	3	]	]	PUNCT
ejpam-5238	385	4	m.	m.	NOUN
ejpam-5238	385	5	thongmoon	thongmoon	NOUN
ejpam-5238	385	6	and	and	CCONJ
ejpam-5238	385	7	c.	c.	PROPN
ejpam-5238	385	8	boonpok	boonpok	PROPN
ejpam-5238	385	9	.	.	PUNCT
ejpam-5238	386	1	strongly	strongly	ADV
ejpam-5238	386	2	θ(λ	θ(λ	PROPN
ejpam-5238	386	3	,	,	PUNCT
ejpam-5238	386	4	p)-continuous	p)-continuous	ADJ
ejpam-5238	386	5	functions	function	NOUN
ejpam-5238	386	6	.	.	PUNCT
ejpam-5238	387	1	international	international	ADJ
ejpam-5238	387	2	journal	journal	PROPN
ejpam-5238	387	3	of	of	ADP
ejpam-5238	387	4	mathematics	mathematic	NOUN
ejpam-5238	387	5	and	and	CCONJ
ejpam-5238	387	6	computer	computer	NOUN
ejpam-5238	387	7	science	science	NOUN
ejpam-5238	387	8	,	,	PUNCT
ejpam-5238	387	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5238	387	10	,	,	PUNCT
ejpam-5238	387	11	2024	2024	NUM
ejpam-5238	387	12	.	.	PUNCT
ejpam-5238	388	1	[	[	X
ejpam-5238	388	2	39	39	NUM
ejpam-5238	388	3	]	]	PUNCT
ejpam-5238	388	4	c.	c.	PROPN
ejpam-5238	388	5	viriyapong	viriyapong	PROPN
ejpam-5238	388	6	and	and	CCONJ
ejpam-5238	388	7	c.	c.	PROPN
ejpam-5238	388	8	boonpok	boonpok	PROPN
ejpam-5238	388	9	.	.	PUNCT
ejpam-5238	389	1	(	(	PUNCT
ejpam-5238	389	2	τ1	τ1	NOUN
ejpam-5238	389	3	,	,	PUNCT
ejpam-5238	389	4	τ2)α	τ2)α	NOUN
ejpam-5238	389	5	-	-	PUNCT
ejpam-5238	389	6	continuity	continuity	NOUN
ejpam-5238	389	7	for	for	ADP
ejpam-5238	389	8	multifunctions	multifunction	NOUN
ejpam-5238	389	9	.	.	PUNCT
ejpam-5238	390	1	journal	journal	PROPN
ejpam-5238	390	2	of	of	ADP
ejpam-5238	390	3	mathematics	mathematic	NOUN
ejpam-5238	390	4	,	,	PUNCT
ejpam-5238	390	5	2020:6285763	2020:6285763	NUM
ejpam-5238	390	6	,	,	PUNCT
ejpam-5238	390	7	2020	2020	NUM
ejpam-5238	390	8	.	.	PUNCT
ejpam-5238	391	1	[	[	X
ejpam-5238	391	2	40	40	NUM
ejpam-5238	391	3	]	]	PUNCT
ejpam-5238	391	4	c.	c.	PROPN
ejpam-5238	391	5	viriyapong	viriyapong	PROPN
ejpam-5238	391	6	and	and	CCONJ
ejpam-5238	391	7	c.	c.	PROPN
ejpam-5238	391	8	boonpok	boonpok	PROPN
ejpam-5238	391	9	.	.	PUNCT
ejpam-5238	392	1	(	(	PUNCT
ejpam-5238	392	2	λ	λ	X
ejpam-5238	392	3	,	,	PUNCT
ejpam-5238	392	4	sp)-continuous	sp)-continuous	ADJ
ejpam-5238	392	5	functions	function	NOUN
ejpam-5238	392	6	.	.	PUNCT
ejpam-5238	393	1	wseas	wseas	VERB
ejpam-5238	393	2	transactions	transaction	NOUN
ejpam-5238	393	3	on	on	ADP
ejpam-5238	393	4	mathematics	mathematic	NOUN
ejpam-5238	393	5	,	,	PUNCT
ejpam-5238	393	6	21:380–385	21:380–385	NUM
ejpam-5238	393	7	,	,	PUNCT
ejpam-5238	393	8	2022	2022	NUM
ejpam-5238	393	9	.	.	PUNCT
ejpam-5238	394	1	[	[	X
ejpam-5238	394	2	41	41	NUM
ejpam-5238	394	3	]	]	X
ejpam-5238	394	4	c.	c.	PROPN
ejpam-5238	394	5	viriyapong	viriyapong	PROPN
ejpam-5238	394	6	and	and	CCONJ
ejpam-5238	394	7	c.	c.	PROPN
ejpam-5238	394	8	boonpok	boonpok	PROPN
ejpam-5238	394	9	.	.	PUNCT
ejpam-5238	395	1	weak	weak	ADJ
ejpam-5238	395	2	quasi	quasi	NOUN
ejpam-5238	395	3	(	(	PUNCT
ejpam-5238	395	4	λ	λ	PROPN
ejpam-5238	395	5	,	,	PUNCT
ejpam-5238	395	6	sp)-continuity	sp)-continuity	NOUN
ejpam-5238	395	7	for	for	ADP
ejpam-5238	395	8	multifunctions	multifunction	NOUN
ejpam-5238	395	9	.	.	PUNCT
ejpam-5238	396	1	international	international	ADJ
ejpam-5238	396	2	journal	journal	PROPN
ejpam-5238	396	3	of	of	ADP
ejpam-5238	396	4	mathematics	mathematic	NOUN
ejpam-5238	396	5	and	and	CCONJ
ejpam-5238	396	6	computer	computer	NOUN
ejpam-5238	396	7	science	science	NOUN
ejpam-5238	396	8	,	,	PUNCT
ejpam-5238	396	9	17(3):1201–1209	17(3):1201–1209	NUM
ejpam-5238	396	10	,	,	PUNCT
ejpam-5238	396	11	2022	2022	NUM
ejpam-5238	396	12	.	.	PUNCT
ejpam-5238	397	1	[	[	X
ejpam-5238	397	2	42	42	NUM
ejpam-5238	397	3	]	]	X
ejpam-5238	397	4	n.	n.	PROPN
ejpam-5238	397	5	viriyapong	viriyapong	PROPN
ejpam-5238	397	6	,	,	PUNCT
ejpam-5238	397	7	s.	s.	PROPN
ejpam-5238	397	8	sompong	sompong	PROPN
ejpam-5238	397	9	,	,	PUNCT
ejpam-5238	397	10	and	and	CCONJ
ejpam-5238	397	11	c.	c.	PROPN
ejpam-5238	397	12	boonpok	boonpok	PROPN
ejpam-5238	397	13	.	.	PUNCT
ejpam-5238	398	1	(	(	PUNCT
ejpam-5238	398	2	τ1	τ1	NOUN
ejpam-5238	398	3	,	,	PUNCT
ejpam-5238	398	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5238	398	5	disconnectedness	disconnectedness	NOUN
ejpam-5238	398	6	in	in	ADP
ejpam-5238	398	7	bitopological	bitopological	ADJ
ejpam-5238	398	8	spaces	space	NOUN
ejpam-5238	398	9	.	.	PUNCT
ejpam-5238	399	1	international	international	ADJ
ejpam-5238	399	2	journal	journal	PROPN
ejpam-5238	399	3	of	of	ADP
ejpam-5238	399	4	mathematics	mathematic	NOUN
ejpam-5238	399	5	and	and	CCONJ
ejpam-5238	399	6	computer	computer	NOUN
ejpam-5238	399	7	science	science	NOUN
ejpam-5238	399	8	,	,	PUNCT
ejpam-5238	399	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5238	399	10	,	,	PUNCT
ejpam-5238	399	11	2024	2024	NUM
ejpam-5238	399	12	.	.	PUNCT