id	sid	tid	token	lemma	pos
ejpam-7053	1	1	european	european	PROPN
ejpam-7053	1	2	journal	journal	PROPN
ejpam-7053	1	3	of	of	ADP
ejpam-7053	1	4	pure	pure	ADJ
ejpam-7053	1	5	and	and	CCONJ
ejpam-7053	1	6	applied	applied	ADJ
ejpam-7053	1	7	mathematics	mathematic	NOUN
ejpam-7053	1	8	2025	2025	NUM
ejpam-7053	1	9	,	,	PUNCT
ejpam-7053	1	10	vol	vol	NOUN
ejpam-7053	1	11	.	.	PROPN
ejpam-7053	1	12	18	18	NUM
ejpam-7053	1	13	,	,	PUNCT
ejpam-7053	1	14	issue	issue	NOUN
ejpam-7053	1	15	4	4	NUM
ejpam-7053	1	16	,	,	PUNCT
ejpam-7053	1	17	article	article	NOUN
ejpam-7053	1	18	number	number	NOUN
ejpam-7053	1	19	7053	7053	NUM
ejpam-7053	1	20	issn	issn	PROPN
ejpam-7053	1	21	1307	1307	NUM
ejpam-7053	1	22	-	-	SYM
ejpam-7053	1	23	5543	5543	NUM
ejpam-7053	1	24	–	–	PUNCT
ejpam-7053	1	25	ejpam.com	ejpam.com	X
ejpam-7053	1	26	published	publish	VERB
ejpam-7053	1	27	by	by	ADP
ejpam-7053	1	28	new	new	PROPN
ejpam-7053	1	29	york	york	PROPN
ejpam-7053	1	30	business	business	PROPN
ejpam-7053	1	31	global	global	ADJ
ejpam-7053	1	32	weak	weak	ADJ
ejpam-7053	1	33	forms	form	NOUN
ejpam-7053	1	34	of	of	ADP
ejpam-7053	1	35	µ(σ1	µ(σ1	NOUN
ejpam-7053	1	36	,	,	PUNCT
ejpam-7053	1	37	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7053	1	38	for	for	ADP
ejpam-7053	1	39	multifunctions	multifunction	NOUN
ejpam-7053	1	40	monchaya	monchaya	PROPN
ejpam-7053	1	41	chiangpradit1	chiangpradit1	PROPN
ejpam-7053	1	42	,	,	PUNCT
ejpam-7053	1	43	areeyuth	areeyuth	NOUN
ejpam-7053	1	44	sama	sama	NOUN
ejpam-7053	1	45	-	-	PUNCT
ejpam-7053	1	46	ae2	ae2	PROPN
ejpam-7053	1	47	,	,	PUNCT
ejpam-7053	1	48	chawalit	chawalit	VERB
ejpam-7053	1	49	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7053	1	50	1	1	NUM
ejpam-7053	1	51	mathematics	mathematic	NOUN
ejpam-7053	1	52	and	and	CCONJ
ejpam-7053	1	53	applied	apply	VERB
ejpam-7053	1	54	mathematics	mathematics	PROPN
ejpam-7053	1	55	research	research	NOUN
ejpam-7053	1	56	unit	unit	NOUN
ejpam-7053	1	57	,	,	PUNCT
ejpam-7053	1	58	department	department	NOUN
ejpam-7053	1	59	of	of	ADP
ejpam-7053	1	60	mathematics	mathematic	NOUN
ejpam-7053	1	61	,	,	PUNCT
ejpam-7053	1	62	faculty	faculty	NOUN
ejpam-7053	1	63	of	of	ADP
ejpam-7053	1	64	science	science	NOUN
ejpam-7053	1	65	,	,	PUNCT
ejpam-7053	1	66	mahasarakham	mahasarakham	PROPN
ejpam-7053	1	67	university	university	PROPN
ejpam-7053	1	68	,	,	PUNCT
ejpam-7053	1	69	maha	maha	PROPN
ejpam-7053	1	70	sarakham	sarakham	PROPN
ejpam-7053	1	71	,	,	PUNCT
ejpam-7053	1	72	44150	44150	NUM
ejpam-7053	1	73	,	,	PUNCT
ejpam-7053	1	74	thailand	thailand	PROPN
ejpam-7053	1	75	2	2	NUM
ejpam-7053	1	76	department	department	NOUN
ejpam-7053	1	77	of	of	ADP
ejpam-7053	1	78	mathematics	mathematic	NOUN
ejpam-7053	1	79	and	and	CCONJ
ejpam-7053	1	80	computer	computer	NOUN
ejpam-7053	1	81	science	science	NOUN
ejpam-7053	1	82	,	,	PUNCT
ejpam-7053	1	83	faculty	faculty	NOUN
ejpam-7053	1	84	of	of	ADP
ejpam-7053	1	85	science	science	NOUN
ejpam-7053	1	86	and	and	CCONJ
ejpam-7053	1	87	technology	technology	NOUN
ejpam-7053	1	88	,	,	PUNCT
ejpam-7053	1	89	prince	prince	NOUN
ejpam-7053	1	90	of	of	ADP
ejpam-7053	1	91	songkla	songkla	PROPN
ejpam-7053	1	92	university	university	PROPN
ejpam-7053	1	93	,	,	PUNCT
ejpam-7053	1	94	pattani	pattani	NOUN
ejpam-7053	1	95	campus	campus	NOUN
ejpam-7053	1	96	,	,	PUNCT
ejpam-7053	1	97	pattani	pattani	NOUN
ejpam-7053	1	98	,	,	PUNCT
ejpam-7053	1	99	94000	94000	NUM
ejpam-7053	1	100	,	,	PUNCT
ejpam-7053	1	101	thailand	thailand	PROPN
ejpam-7053	1	102	abstract	abstract	PROPN
ejpam-7053	1	103	.	.	PUNCT
ejpam-7053	2	1	a	a	DET
ejpam-7053	2	2	new	new	ADJ
ejpam-7053	2	3	class	class	NOUN
ejpam-7053	2	4	of	of	ADP
ejpam-7053	2	5	continuous	continuous	ADJ
ejpam-7053	2	6	multifunctions	multifunction	NOUN
ejpam-7053	2	7	between	between	ADP
ejpam-7053	2	8	a	a	DET
ejpam-7053	2	9	generalized	generalized	ADJ
ejpam-7053	2	10	topological	topological	ADJ
ejpam-7053	2	11	space	space	NOUN
ejpam-7053	2	12	and	and	CCONJ
ejpam-7053	2	13	a	a	DET
ejpam-7053	2	14	bitopological	bitopological	ADJ
ejpam-7053	2	15	space	space	NOUN
ejpam-7053	2	16	,	,	PUNCT
ejpam-7053	2	17	namely	namely	ADV
ejpam-7053	2	18	upper	upper	ADJ
ejpam-7053	2	19	(	(	PUNCT
ejpam-7053	2	20	lower	low	ADJ
ejpam-7053	2	21	)	)	PUNCT
ejpam-7053	2	22	weakly	weakly	ADJ
ejpam-7053	2	23	µ(σ1	µ(σ1	NOUN
ejpam-7053	2	24	,	,	PUNCT
ejpam-7053	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	2	26	multifunctions	multifunction	NOUN
ejpam-7053	2	27	,	,	PUNCT
ejpam-7053	2	28	has	have	AUX
ejpam-7053	2	29	been	be	AUX
ejpam-7053	2	30	defined	define	VERB
ejpam-7053	2	31	and	and	CCONJ
ejpam-7053	2	32	studied	study	VERB
ejpam-7053	2	33	.	.	PUNCT
ejpam-7053	3	1	moreover	moreover	ADV
ejpam-7053	3	2	,	,	PUNCT
ejpam-7053	3	3	several	several	ADJ
ejpam-7053	3	4	characterizations	characterization	NOUN
ejpam-7053	3	5	and	and	CCONJ
ejpam-7053	3	6	some	some	DET
ejpam-7053	3	7	properties	property	NOUN
ejpam-7053	3	8	concerning	concern	VERB
ejpam-7053	3	9	upper	upper	ADJ
ejpam-7053	3	10	weakly	weakly	ADJ
ejpam-7053	3	11	µ(σ1	µ(σ1	NOUN
ejpam-7053	3	12	,	,	PUNCT
ejpam-7053	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	3	14	multifunctions	multifunction	NOUN
ejpam-7053	3	15	and	and	CCONJ
ejpam-7053	3	16	lower	low	ADJ
ejpam-7053	3	17	weakly	weakly	ADJ
ejpam-7053	3	18	µ(σ1	µ(σ1	NOUN
ejpam-7053	3	19	,	,	PUNCT
ejpam-7053	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	3	21	multifunctions	multifunction	NOUN
ejpam-7053	3	22	are	be	AUX
ejpam-7053	3	23	established	establish	VERB
ejpam-7053	3	24	.	.	PUNCT
ejpam-7053	4	1	furthermore	furthermore	ADV
ejpam-7053	4	2	,	,	PUNCT
ejpam-7053	4	3	the	the	DET
ejpam-7053	4	4	relationships	relationship	NOUN
ejpam-7053	4	5	between	between	ADP
ejpam-7053	4	6	almost	almost	ADV
ejpam-7053	4	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	4	8	,	,	PUNCT
ejpam-7053	4	9	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7053	4	10	and	and	CCONJ
ejpam-7053	4	11	weak	weak	ADJ
ejpam-7053	4	12	µ(σ1	µ(σ1	NOUN
ejpam-7053	4	13	,	,	PUNCT
ejpam-7053	4	14	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7053	4	15	are	be	AUX
ejpam-7053	4	16	considered	consider	VERB
ejpam-7053	4	17	.	.	PUNCT
ejpam-7053	5	1	2020	2020	NUM
ejpam-7053	5	2	mathematics	mathematic	NOUN
ejpam-7053	5	3	subject	subject	NOUN
ejpam-7053	5	4	classifications	classification	NOUN
ejpam-7053	5	5	:	:	PUNCT
ejpam-7053	5	6	54c08	54c08	NUM
ejpam-7053	5	7	,	,	PUNCT
ejpam-7053	5	8	54c60	54c60	NUM
ejpam-7053	5	9	key	key	ADJ
ejpam-7053	5	10	words	word	NOUN
ejpam-7053	5	11	and	and	CCONJ
ejpam-7053	5	12	phrases	phrase	NOUN
ejpam-7053	5	13	:	:	PUNCT
ejpam-7053	5	14	upper	upper	ADJ
ejpam-7053	5	15	weakly	weakly	ADJ
ejpam-7053	5	16	µ(σ1	µ(σ1	NOUN
ejpam-7053	5	17	,	,	PUNCT
ejpam-7053	5	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	5	19	multifunction	multifunction	NOUN
ejpam-7053	5	20	,	,	PUNCT
ejpam-7053	5	21	lower	low	ADJ
ejpam-7053	5	22	weakly	weakly	ADJ
ejpam-7053	5	23	µ(σ1	µ(σ1	NOUN
ejpam-7053	5	24	,	,	PUNCT
ejpam-7053	5	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	5	26	multifunction	multifunction	NOUN
ejpam-7053	5	27	1	1	NUM
ejpam-7053	5	28	.	.	PUNCT
ejpam-7053	5	29	introduction	introduction	NOUN
ejpam-7053	5	30	in	in	ADP
ejpam-7053	5	31	1961	1961	NUM
ejpam-7053	5	32	,	,	PUNCT
ejpam-7053	5	33	levine	levine	PROPN
ejpam-7053	6	1	[	[	X
ejpam-7053	6	2	1	1	NUM
ejpam-7053	6	3	]	]	PUNCT
ejpam-7053	6	4	introduced	introduce	VERB
ejpam-7053	6	5	and	and	CCONJ
ejpam-7053	6	6	investigated	investigate	VERB
ejpam-7053	6	7	the	the	DET
ejpam-7053	6	8	notion	notion	NOUN
ejpam-7053	6	9	of	of	ADP
ejpam-7053	6	10	weakly	weakly	ADJ
ejpam-7053	6	11	continuous	continuous	ADJ
ejpam-7053	6	12	functions	function	NOUN
ejpam-7053	6	13	.	.	PUNCT
ejpam-7053	7	1	husain	husain	NOUN
ejpam-7053	8	1	[	[	X
ejpam-7053	8	2	2	2	X
ejpam-7053	8	3	]	]	PUNCT
ejpam-7053	8	4	introduced	introduce	VERB
ejpam-7053	8	5	and	and	CCONJ
ejpam-7053	8	6	studied	study	VERB
ejpam-7053	8	7	the	the	DET
ejpam-7053	8	8	concept	concept	NOUN
ejpam-7053	8	9	of	of	ADP
ejpam-7053	8	10	almost	almost	ADV
ejpam-7053	8	11	continuous	continuous	ADJ
ejpam-7053	8	12	functions	function	NOUN
ejpam-7053	8	13	.	.	PUNCT
ejpam-7053	9	1	janković	janković	VERB
ejpam-7053	10	1	[	[	X
ejpam-7053	10	2	3	3	X
ejpam-7053	10	3	]	]	PUNCT
ejpam-7053	10	4	introduced	introduce	VERB
ejpam-7053	10	5	almost	almost	ADV
ejpam-7053	10	6	weak	weak	ADJ
ejpam-7053	10	7	continuity	continuity	NOUN
ejpam-7053	10	8	as	as	ADP
ejpam-7053	10	9	a	a	DET
ejpam-7053	10	10	generalization	generalization	NOUN
ejpam-7053	10	11	of	of	ADP
ejpam-7053	10	12	both	both	DET
ejpam-7053	10	13	weak	weak	ADJ
ejpam-7053	10	14	continuity	continuity	NOUN
ejpam-7053	10	15	and	and	CCONJ
ejpam-7053	10	16	almost	almost	ADV
ejpam-7053	10	17	continuity	continuity	NOUN
ejpam-7053	10	18	.	.	PUNCT
ejpam-7053	11	1	noiri	noiri	ADV
ejpam-7053	12	1	[	[	X
ejpam-7053	12	2	4	4	X
ejpam-7053	12	3	]	]	PUNCT
ejpam-7053	12	4	investigated	investigate	VERB
ejpam-7053	12	5	several	several	ADJ
ejpam-7053	12	6	characterizations	characterization	NOUN
ejpam-7053	12	7	of	of	ADP
ejpam-7053	12	8	almost	almost	ADV
ejpam-7053	12	9	weakly	weakly	ADJ
ejpam-7053	12	10	continuous	continuous	ADJ
ejpam-7053	12	11	functions	function	NOUN
ejpam-7053	12	12	.	.	PUNCT
ejpam-7053	13	1	rose	rise	VERB
ejpam-7053	13	2	[	[	X
ejpam-7053	13	3	5	5	NUM
ejpam-7053	13	4	]	]	PUNCT
ejpam-7053	13	5	introduced	introduce	VERB
ejpam-7053	13	6	the	the	DET
ejpam-7053	13	7	notion	notion	NOUN
ejpam-7053	13	8	of	of	ADP
ejpam-7053	13	9	subweakly	subweakly	ADJ
ejpam-7053	13	10	continuous	continuous	ADJ
ejpam-7053	13	11	functions	function	NOUN
ejpam-7053	13	12	and	and	CCONJ
ejpam-7053	13	13	investigated	investigate	VERB
ejpam-7053	13	14	the	the	DET
ejpam-7053	13	15	relationships	relationship	NOUN
ejpam-7053	13	16	between	between	ADP
ejpam-7053	13	17	subweak	subweak	NOUN
ejpam-7053	13	18	continuity	continuity	NOUN
ejpam-7053	13	19	and	and	CCONJ
ejpam-7053	13	20	weak	weak	ADJ
ejpam-7053	13	21	continuity	continuity	NOUN
ejpam-7053	13	22	.	.	PUNCT
ejpam-7053	14	1	popa	popa	NOUN
ejpam-7053	14	2	and	and	CCONJ
ejpam-7053	14	3	noiri	noiri	ADV
ejpam-7053	15	1	[	[	X
ejpam-7053	15	2	6	6	NUM
ejpam-7053	15	3	]	]	PUNCT
ejpam-7053	15	4	introduced	introduce	VERB
ejpam-7053	15	5	the	the	DET
ejpam-7053	15	6	concept	concept	NOUN
ejpam-7053	15	7	of	of	ADP
ejpam-7053	15	8	weakly	weakly	ADJ
ejpam-7053	15	9	(	(	PUNCT
ejpam-7053	15	10	τ	τ	PROPN
ejpam-7053	15	11	,	,	PUNCT
ejpam-7053	15	12	m)-continuous	m)-continuous	ADJ
ejpam-7053	15	13	functions	function	NOUN
ejpam-7053	15	14	as	as	ADP
ejpam-7053	15	15	functions	function	NOUN
ejpam-7053	15	16	from	from	ADP
ejpam-7053	15	17	a	a	DET
ejpam-7053	15	18	topological	topological	ADJ
ejpam-7053	15	19	space	space	NOUN
ejpam-7053	15	20	into	into	ADP
ejpam-7053	15	21	a	a	DET
ejpam-7053	15	22	set	set	NOUN
ejpam-7053	15	23	satisfying	satisfy	VERB
ejpam-7053	15	24	some	some	DET
ejpam-7053	15	25	minimal	minimal	ADJ
ejpam-7053	15	26	conditions	condition	NOUN
ejpam-7053	15	27	and	and	CCONJ
ejpam-7053	15	28	investigated	investigate	VERB
ejpam-7053	15	29	several	several	ADJ
ejpam-7053	15	30	characterizations	characterization	NOUN
ejpam-7053	15	31	of	of	ADP
ejpam-7053	15	32	weakly	weakly	ADJ
ejpam-7053	15	33	(	(	PUNCT
ejpam-7053	15	34	τ	τ	PROPN
ejpam-7053	15	35	,	,	PUNCT
ejpam-7053	15	36	m)-continuous	m)-continuous	ADJ
ejpam-7053	15	37	functions	function	NOUN
ejpam-7053	15	38	.	.	PUNCT
ejpam-7053	16	1	in	in	ADP
ejpam-7053	16	2	2002	2002	NUM
ejpam-7053	16	3	,	,	PUNCT
ejpam-7053	16	4	császár	császár	NOUN
ejpam-7053	16	5	[	[	X
ejpam-7053	16	6	7	7	NUM
ejpam-7053	16	7	]	]	PUNCT
ejpam-7053	16	8	introduced	introduce	VERB
ejpam-7053	16	9	the	the	DET
ejpam-7053	16	10	concepts	concept	NOUN
ejpam-7053	16	11	of	of	ADP
ejpam-7053	16	12	generalized	generalized	ADJ
ejpam-7053	16	13	topological	topological	ADJ
ejpam-7053	16	14	spaces	space	NOUN
ejpam-7053	16	15	and	and	CCONJ
ejpam-7053	16	16	generalized	generalized	ADJ
ejpam-7053	16	17	neighborhood	neighborhood	NOUN
ejpam-7053	16	18	systems	system	NOUN
ejpam-7053	16	19	.	.	PUNCT
ejpam-7053	17	1	the	the	DET
ejpam-7053	17	2	classes	class	NOUN
ejpam-7053	17	3	of	of	ADP
ejpam-7053	17	4	topological	topological	ADJ
ejpam-7053	17	5	spaces	space	NOUN
ejpam-7053	17	6	and	and	CCONJ
ejpam-7053	17	7	neighborhood	neighborhood	NOUN
ejpam-7053	17	8	systems	system	NOUN
ejpam-7053	17	9	are	be	AUX
ejpam-7053	17	10	contained	contain	VERB
ejpam-7053	17	11	in	in	ADP
ejpam-7053	17	12	the	the	DET
ejpam-7053	17	13	classes	class	NOUN
ejpam-7053	17	14	of	of	ADP
ejpam-7053	17	15	generalized	generalized	ADJ
ejpam-7053	17	16	topological	topological	ADJ
ejpam-7053	17	17	spaces	space	NOUN
ejpam-7053	17	18	and	and	CCONJ
ejpam-7053	17	19	generalized	generalized	ADJ
ejpam-7053	17	20	neighborhood	neighborhood	NOUN
ejpam-7053	17	21	systems	system	NOUN
ejpam-7053	17	22	,	,	PUNCT
ejpam-7053	17	23	respectively	respectively	ADV
ejpam-7053	17	24	.	.	PUNCT
ejpam-7053	18	1	furthermore	furthermore	ADV
ejpam-7053	18	2	,	,	PUNCT
ejpam-7053	18	3	császár	császár	PROPN
ejpam-7053	18	4	[	[	X
ejpam-7053	18	5	7	7	NUM
ejpam-7053	18	6	]	]	PUNCT
ejpam-7053	18	7	introduced	introduce	VERB
ejpam-7053	18	8	two	two	NUM
ejpam-7053	18	9	kinds	kind	NOUN
ejpam-7053	18	10	of	of	ADP
ejpam-7053	18	11	generalized	generalized	ADJ
ejpam-7053	18	12	continuous	continuous	ADJ
ejpam-7053	18	13	∗corresponding	∗corresponding	NOUN
ejpam-7053	18	14	author	author	NOUN
ejpam-7053	18	15	.	.	PUNCT
ejpam-7053	19	1	doi	doi	NOUN
ejpam-7053	19	2	:	:	PUNCT
ejpam-7053	19	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7053	https://doi.org/10.29020/nybg.ejpam.v18i4.7053	PROPN
ejpam-7053	19	4	email	email	NOUN
ejpam-7053	19	5	addresses	address	VERB
ejpam-7053	19	6	:	:	PUNCT
ejpam-7053	19	7	monchaya.c@msu.ac.th	monchaya.c@msu.ac.th	PROPN
ejpam-7053	19	8	(	(	PUNCT
ejpam-7053	19	9	m.	m.	NOUN
ejpam-7053	19	10	chiangpradit	chiangpradit	NOUN
ejpam-7053	19	11	)	)	PUNCT
ejpam-7053	19	12	,	,	PUNCT
ejpam-7053	19	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7053	19	14	(	(	PUNCT
ejpam-7053	19	15	a.	a.	PROPN
ejpam-7053	19	16	sama	sama	PROPN
ejpam-7053	19	17	-	-	PUNCT
ejpam-7053	19	18	ae	ae	PROPN
ejpam-7053	19	19	)	)	PUNCT
ejpam-7053	19	20	,	,	PUNCT
ejpam-7053	19	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7053	19	22	(	(	PUNCT
ejpam-7053	19	23	c.	c.	PROPN
ejpam-7053	19	24	boonpok	boonpok	PROPN
ejpam-7053	19	25	)	)	PUNCT
ejpam-7053	19	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7053	19	27	1	1	NUM
ejpam-7053	19	28	copyright	copyright	NOUN
ejpam-7053	19	29	:	:	PUNCT
ejpam-7053	20	1	©	©	PROPN
ejpam-7053	20	2	2025	2025	NUM
ejpam-7053	20	3	the	the	DET
ejpam-7053	20	4	author(s	author(s	NOUN
ejpam-7053	20	5	)	)	PUNCT
ejpam-7053	20	6	.	.	PUNCT
ejpam-7053	21	1	(	(	PUNCT
ejpam-7053	21	2	cc	cc	NOUN
ejpam-7053	21	3	by	by	ADP
ejpam-7053	21	4	-	-	PUNCT
ejpam-7053	21	5	nc	nc	PROPN
ejpam-7053	21	6	4.0	4.0	NUM
ejpam-7053	21	7	)	)	PUNCT
ejpam-7053	21	8	m.	m.	NOUN
ejpam-7053	21	9	chiangpradit	chiangpradit	NOUN
ejpam-7053	21	10	,	,	PUNCT
ejpam-7053	21	11	a.	a.	PROPN
ejpam-7053	21	12	sama	sama	PROPN
ejpam-7053	21	13	-	-	PUNCT
ejpam-7053	21	14	ae	ae	PROPN
ejpam-7053	21	15	,	,	PUNCT
ejpam-7053	21	16	c.	c.	PROPN
ejpam-7053	21	17	boonpok	boonpok	PROPN
ejpam-7053	21	18	/	/	SYM
ejpam-7053	21	19	eur	eur	PROPN
ejpam-7053	21	20	.	.	PUNCT
ejpam-7053	22	1	j.	j.	PROPN
ejpam-7053	22	2	pure	pure	PROPN
ejpam-7053	22	3	appl	appl	PROPN
ejpam-7053	22	4	.	.	PROPN
ejpam-7053	22	5	math	math	PROPN
ejpam-7053	22	6	,	,	PUNCT
ejpam-7053	22	7	18	18	NUM
ejpam-7053	22	8	(	(	PUNCT
ejpam-7053	22	9	4	4	NUM
ejpam-7053	22	10	)	)	PUNCT
ejpam-7053	22	11	(	(	PUNCT
ejpam-7053	22	12	2025	2025	NUM
ejpam-7053	22	13	)	)	PUNCT
ejpam-7053	22	14	,	,	PUNCT
ejpam-7053	22	15	7053	7053	NUM
ejpam-7053	22	16	2	2	NUM
ejpam-7053	22	17	of	of	ADP
ejpam-7053	22	18	13	13	NUM
ejpam-7053	22	19	functions	function	NOUN
ejpam-7053	22	20	by	by	ADP
ejpam-7053	22	21	utilizing	utilize	VERB
ejpam-7053	22	22	the	the	DET
ejpam-7053	22	23	notions	notion	NOUN
ejpam-7053	22	24	of	of	ADP
ejpam-7053	22	25	generalized	generalized	ADJ
ejpam-7053	22	26	topological	topological	ADJ
ejpam-7053	22	27	spaces	space	NOUN
ejpam-7053	22	28	and	and	CCONJ
ejpam-7053	22	29	generalized	generalized	ADJ
ejpam-7053	22	30	neighborhood	neighborhood	NOUN
ejpam-7053	22	31	systems	system	NOUN
ejpam-7053	22	32	.	.	PUNCT
ejpam-7053	23	1	in	in	ADP
ejpam-7053	23	2	2009	2009	NUM
ejpam-7053	23	3	,	,	PUNCT
ejpam-7053	23	4	kanibir	kanibir	NOUN
ejpam-7053	23	5	and	and	CCONJ
ejpam-7053	23	6	reilly	reilly	ADV
ejpam-7053	23	7	[	[	X
ejpam-7053	23	8	8	8	NUM
ejpam-7053	23	9	]	]	PUNCT
ejpam-7053	23	10	extended	extend	VERB
ejpam-7053	23	11	the	the	DET
ejpam-7053	23	12	concept	concept	NOUN
ejpam-7053	23	13	of	of	ADP
ejpam-7053	23	14	generalized	generalized	ADJ
ejpam-7053	23	15	continuous	continuous	ADJ
ejpam-7053	23	16	functions	function	NOUN
ejpam-7053	23	17	to	to	ADP
ejpam-7053	23	18	multifunctions	multifunction	NOUN
ejpam-7053	23	19	and	and	CCONJ
ejpam-7053	23	20	defined	define	VERB
ejpam-7053	23	21	upper	upper	ADJ
ejpam-7053	23	22	semi	semi	ADJ
ejpam-7053	23	23	generalized	generalized	ADJ
ejpam-7053	23	24	continuous	continuous	ADJ
ejpam-7053	23	25	multifunctions	multifunction	NOUN
ejpam-7053	23	26	and	and	CCONJ
ejpam-7053	23	27	lower	low	ADJ
ejpam-7053	23	28	semi	semi	ADV
ejpam-7053	23	29	generalized	generalized	ADJ
ejpam-7053	23	30	continuous	continuous	ADJ
ejpam-7053	23	31	multifunctions	multifunction	NOUN
ejpam-7053	23	32	.	.	PUNCT
ejpam-7053	24	1	on	on	ADP
ejpam-7053	24	2	the	the	DET
ejpam-7053	24	3	other	other	ADJ
ejpam-7053	24	4	hand	hand	NOUN
ejpam-7053	24	5	,	,	PUNCT
ejpam-7053	24	6	the	the	DET
ejpam-7053	24	7	present	present	ADJ
ejpam-7053	24	8	authors	author	NOUN
ejpam-7053	24	9	introduced	introduce	VERB
ejpam-7053	24	10	and	and	CCONJ
ejpam-7053	24	11	investigated	investigate	VERB
ejpam-7053	24	12	four	four	NUM
ejpam-7053	24	13	classes	class	NOUN
ejpam-7053	24	14	of	of	ADP
ejpam-7053	24	15	multifunctions	multifunction	NOUN
ejpam-7053	24	16	defined	define	VERB
ejpam-7053	24	17	from	from	ADP
ejpam-7053	24	18	a	a	DET
ejpam-7053	24	19	generalized	generalized	ADJ
ejpam-7053	24	20	topological	topological	ADJ
ejpam-7053	24	21	space	space	NOUN
ejpam-7053	24	22	into	into	ADP
ejpam-7053	24	23	a	a	DET
ejpam-7053	24	24	generalized	generalized	ADJ
ejpam-7053	24	25	topological	topological	ADJ
ejpam-7053	24	26	space	space	NOUN
ejpam-7053	24	27	,	,	PUNCT
ejpam-7053	24	28	namely	namely	ADV
ejpam-7053	24	29	upper	upper	ADJ
ejpam-7053	24	30	α(µx	α(µx	PROPN
ejpam-7053	24	31	,	,	PUNCT
ejpam-7053	24	32	µy	µy	CCONJ
ejpam-7053	24	33	)	)	PUNCT
ejpam-7053	24	34	-continuous	-continuous	ADJ
ejpam-7053	24	35	multifunctions	multifunction	NOUN
ejpam-7053	25	1	[	[	X
ejpam-7053	25	2	9	9	NUM
ejpam-7053	25	3	]	]	PUNCT
ejpam-7053	25	4	,	,	PUNCT
ejpam-7053	25	5	lower	low	ADJ
ejpam-7053	25	6	α(µx	α(µx	NUM
ejpam-7053	25	7	,	,	PUNCT
ejpam-7053	25	8	µy	µy	CCONJ
ejpam-7053	25	9	)	)	PUNCT
ejpam-7053	25	10	-continuous	-continuous	ADJ
ejpam-7053	25	11	multifunctions	multifunction	NOUN
ejpam-7053	26	1	[	[	X
ejpam-7053	26	2	9	9	NUM
ejpam-7053	26	3	]	]	PUNCT
ejpam-7053	26	4	,	,	PUNCT
ejpam-7053	26	5	upper	upper	ADJ
ejpam-7053	26	6	weakly	weakly	ADJ
ejpam-7053	26	7	β(µx	β(µx	PROPN
ejpam-7053	26	8	,	,	PUNCT
ejpam-7053	26	9	µy	µy	CCONJ
ejpam-7053	26	10	)	)	PUNCT
ejpam-7053	26	11	-continuous	-continuous	ADJ
ejpam-7053	26	12	multifunctions	multifunction	NOUN
ejpam-7053	26	13	[	[	X
ejpam-7053	26	14	10	10	NUM
ejpam-7053	26	15	]	]	PUNCT
ejpam-7053	26	16	and	and	CCONJ
ejpam-7053	26	17	lower	low	ADJ
ejpam-7053	26	18	weakly	weakly	ADJ
ejpam-7053	26	19	β(µx	β(µx	PROPN
ejpam-7053	26	20	,	,	PUNCT
ejpam-7053	26	21	µy	µy	CCONJ
ejpam-7053	26	22	)	)	PUNCT
ejpam-7053	26	23	continuous	continuous	ADJ
ejpam-7053	26	24	multifunctions	multifunction	NOUN
ejpam-7053	27	1	[	[	X
ejpam-7053	27	2	10	10	NUM
ejpam-7053	27	3	]	]	PUNCT
ejpam-7053	27	4	.	.	PUNCT
ejpam-7053	28	1	pue	pue	NOUN
ejpam-7053	28	2	-	-	PUNCT
ejpam-7053	28	3	on	on	NOUN
ejpam-7053	28	4	et	et	PROPN
ejpam-7053	28	5	al	al	PROPN
ejpam-7053	28	6	.	.	PUNCT
ejpam-7053	29	1	[	[	X
ejpam-7053	29	2	11	11	NUM
ejpam-7053	29	3	]	]	PUNCT
ejpam-7053	29	4	introduced	introduce	VERB
ejpam-7053	29	5	and	and	CCONJ
ejpam-7053	29	6	studied	study	VERB
ejpam-7053	29	7	new	new	ADJ
ejpam-7053	29	8	classes	class	NOUN
ejpam-7053	29	9	of	of	ADP
ejpam-7053	29	10	multifunctions	multifunction	NOUN
ejpam-7053	29	11	between	between	ADP
ejpam-7053	29	12	bitopological	bitopological	ADJ
ejpam-7053	29	13	spaces	space	NOUN
ejpam-7053	29	14	,	,	PUNCT
ejpam-7053	29	15	namely	namely	ADV
ejpam-7053	29	16	upper	upper	ADJ
ejpam-7053	29	17	(	(	PUNCT
ejpam-7053	29	18	τ1	τ1	NOUN
ejpam-7053	29	19	,	,	PUNCT
ejpam-7053	29	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7053	29	21	multifunctions	multifunction	NOUN
ejpam-7053	29	22	and	and	CCONJ
ejpam-7053	29	23	lower	low	ADJ
ejpam-7053	29	24	(	(	PUNCT
ejpam-7053	29	25	τ1	τ1	NOUN
ejpam-7053	29	26	,	,	PUNCT
ejpam-7053	29	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7053	29	28	multifunctions	multifunction	NOUN
ejpam-7053	29	29	.	.	PUNCT
ejpam-7053	30	1	klanarong	klanarong	NOUN
ejpam-7053	30	2	et	et	PROPN
ejpam-7053	30	3	al	al	PROPN
ejpam-7053	30	4	.	.	PUNCT
ejpam-7053	31	1	[	[	X
ejpam-7053	31	2	12	12	NUM
ejpam-7053	31	3	]	]	PUNCT
ejpam-7053	31	4	introduced	introduce	VERB
ejpam-7053	31	5	and	and	CCONJ
ejpam-7053	31	6	investigated	investigate	VERB
ejpam-7053	31	7	the	the	DET
ejpam-7053	31	8	notions	notion	NOUN
ejpam-7053	31	9	of	of	ADP
ejpam-7053	31	10	upper	upper	ADJ
ejpam-7053	31	11	almost	almost	ADV
ejpam-7053	31	12	(	(	PUNCT
ejpam-7053	31	13	τ1	τ1	NOUN
ejpam-7053	31	14	,	,	PUNCT
ejpam-7053	31	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7053	31	16	multifunctions	multifunction	NOUN
ejpam-7053	31	17	and	and	CCONJ
ejpam-7053	31	18	lower	low	ADJ
ejpam-7053	31	19	almost	almost	ADV
ejpam-7053	31	20	(	(	PUNCT
ejpam-7053	31	21	τ1	τ1	NOUN
ejpam-7053	31	22	,	,	PUNCT
ejpam-7053	31	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7053	31	24	multifunctions	multifunction	NOUN
ejpam-7053	31	25	.	.	PUNCT
ejpam-7053	32	1	thongmoon	thongmoon	NOUN
ejpam-7053	32	2	et	et	PROPN
ejpam-7053	32	3	al	al	PROPN
ejpam-7053	32	4	.	.	PUNCT
ejpam-7053	33	1	[	[	X
ejpam-7053	33	2	13	13	NUM
ejpam-7053	33	3	]	]	PUNCT
ejpam-7053	33	4	extended	extend	VERB
ejpam-7053	33	5	the	the	DET
ejpam-7053	33	6	concept	concept	NOUN
ejpam-7053	33	7	of	of	ADP
ejpam-7053	33	8	weakly	weakly	ADJ
ejpam-7053	33	9	continuous	continuous	ADJ
ejpam-7053	33	10	functions	function	NOUN
ejpam-7053	33	11	to	to	ADP
ejpam-7053	33	12	multifunctions	multifunction	NOUN
ejpam-7053	33	13	and	and	CCONJ
ejpam-7053	33	14	presented	present	VERB
ejpam-7053	33	15	two	two	NUM
ejpam-7053	33	16	classes	class	NOUN
ejpam-7053	33	17	of	of	ADP
ejpam-7053	33	18	multifunctions	multifunction	NOUN
ejpam-7053	33	19	defined	define	VERB
ejpam-7053	33	20	from	from	ADP
ejpam-7053	33	21	a	a	DET
ejpam-7053	33	22	bitopological	bitopological	ADJ
ejpam-7053	33	23	space	space	NOUN
ejpam-7053	33	24	into	into	ADP
ejpam-7053	33	25	a	a	DET
ejpam-7053	33	26	bitopological	bitopological	ADJ
ejpam-7053	33	27	space	space	NOUN
ejpam-7053	33	28	,	,	PUNCT
ejpam-7053	33	29	called	call	VERB
ejpam-7053	33	30	upper	upper	ADJ
ejpam-7053	33	31	weakly	weakly	ADJ
ejpam-7053	33	32	(	(	PUNCT
ejpam-7053	33	33	τ1	τ1	NOUN
ejpam-7053	33	34	,	,	PUNCT
ejpam-7053	33	35	τ2)continuous	τ2)continuous	ADJ
ejpam-7053	33	36	multifunctions	multifunction	NOUN
ejpam-7053	33	37	and	and	CCONJ
ejpam-7053	33	38	lower	low	ADJ
ejpam-7053	33	39	weakly	weakly	ADJ
ejpam-7053	33	40	(	(	PUNCT
ejpam-7053	33	41	τ1	τ1	NOUN
ejpam-7053	33	42	,	,	PUNCT
ejpam-7053	33	43	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7053	33	44	multifunctions	multifunction	NOUN
ejpam-7053	33	45	.	.	PUNCT
ejpam-7053	34	1	quite	quite	ADV
ejpam-7053	34	2	recently	recently	ADV
ejpam-7053	34	3	,	,	PUNCT
ejpam-7053	34	4	pue	pue	NOUN
ejpam-7053	34	5	-	-	PUNCT
ejpam-7053	34	6	on	on	NOUN
ejpam-7053	34	7	et	et	PROPN
ejpam-7053	34	8	al	al	PROPN
ejpam-7053	34	9	.	.	PUNCT
ejpam-7053	35	1	[	[	X
ejpam-7053	35	2	14	14	NUM
ejpam-7053	35	3	]	]	PUNCT
ejpam-7053	35	4	introduced	introduce	VERB
ejpam-7053	35	5	new	new	ADJ
ejpam-7053	35	6	classes	class	NOUN
ejpam-7053	35	7	of	of	ADP
ejpam-7053	35	8	continuous	continuous	ADJ
ejpam-7053	35	9	multifunctions	multifunction	NOUN
ejpam-7053	35	10	between	between	ADP
ejpam-7053	35	11	an	an	DET
ejpam-7053	35	12	ideal	ideal	ADJ
ejpam-7053	35	13	topological	topological	ADJ
ejpam-7053	35	14	space	space	NOUN
ejpam-7053	35	15	and	and	CCONJ
ejpam-7053	35	16	a	a	DET
ejpam-7053	35	17	bitopological	bitopological	ADJ
ejpam-7053	35	18	space	space	NOUN
ejpam-7053	35	19	,	,	PUNCT
ejpam-7053	35	20	namely	namely	ADV
ejpam-7053	35	21	upper	upper	ADJ
ejpam-7053	35	22	almost	almost	ADV
ejpam-7053	35	23	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7053	35	24	,	,	PUNCT
ejpam-7053	35	25	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	35	26	multifunctions	multifunction	NOUN
ejpam-7053	35	27	and	and	CCONJ
ejpam-7053	35	28	lower	low	ADJ
ejpam-7053	35	29	almost	almost	ADV
ejpam-7053	35	30	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7053	35	31	,	,	PUNCT
ejpam-7053	35	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	35	33	multifunctions	multifunction	NOUN
ejpam-7053	35	34	.	.	PUNCT
ejpam-7053	36	1	moreover	moreover	ADV
ejpam-7053	36	2	,	,	PUNCT
ejpam-7053	36	3	several	several	ADJ
ejpam-7053	36	4	characterizations	characterization	NOUN
ejpam-7053	36	5	and	and	CCONJ
ejpam-7053	36	6	some	some	DET
ejpam-7053	36	7	properties	property	NOUN
ejpam-7053	36	8	of	of	ADP
ejpam-7053	36	9	upper	upper	ADJ
ejpam-7053	36	10	almost	almost	ADV
ejpam-7053	36	11	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7053	36	12	,	,	PUNCT
ejpam-7053	36	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	36	14	multifunctions	multifunction	NOUN
ejpam-7053	36	15	and	and	CCONJ
ejpam-7053	36	16	lower	low	ADJ
ejpam-7053	36	17	almost	almost	ADV
ejpam-7053	36	18	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7053	36	19	,	,	PUNCT
ejpam-7053	36	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	36	21	multifunctions	multifunction	NOUN
ejpam-7053	36	22	were	be	AUX
ejpam-7053	36	23	discussed	discuss	VERB
ejpam-7053	36	24	in	in	ADP
ejpam-7053	36	25	[	[	X
ejpam-7053	36	26	14	14	NUM
ejpam-7053	36	27	]	]	PUNCT
ejpam-7053	36	28	.	.	PUNCT
ejpam-7053	37	1	in	in	ADP
ejpam-7053	37	2	this	this	DET
ejpam-7053	37	3	paper	paper	NOUN
ejpam-7053	37	4	,	,	PUNCT
ejpam-7053	37	5	we	we	PRON
ejpam-7053	37	6	introduce	introduce	VERB
ejpam-7053	37	7	new	new	ADJ
ejpam-7053	37	8	classes	class	NOUN
ejpam-7053	37	9	of	of	ADP
ejpam-7053	37	10	multifunctions	multifunction	NOUN
ejpam-7053	37	11	between	between	ADP
ejpam-7053	37	12	a	a	DET
ejpam-7053	37	13	generalized	generalized	ADJ
ejpam-7053	37	14	topological	topological	ADJ
ejpam-7053	37	15	space	space	NOUN
ejpam-7053	37	16	and	and	CCONJ
ejpam-7053	37	17	a	a	DET
ejpam-7053	37	18	bitopological	bitopological	ADJ
ejpam-7053	37	19	space	space	NOUN
ejpam-7053	37	20	,	,	PUNCT
ejpam-7053	37	21	namely	namely	ADV
ejpam-7053	37	22	upper	upper	ADJ
ejpam-7053	37	23	weakly	weakly	ADJ
ejpam-7053	37	24	µ(σ1	µ(σ1	NOUN
ejpam-7053	37	25	,	,	PUNCT
ejpam-7053	37	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	37	27	multifunctions	multifunction	NOUN
ejpam-7053	37	28	and	and	CCONJ
ejpam-7053	37	29	lower	low	ADJ
ejpam-7053	37	30	weakly	weakly	ADJ
ejpam-7053	37	31	µ(σ1	µ(σ1	NOUN
ejpam-7053	37	32	,	,	PUNCT
ejpam-7053	37	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	37	34	multifunctions	multifunction	NOUN
ejpam-7053	37	35	.	.	PUNCT
ejpam-7053	38	1	we	we	PRON
ejpam-7053	38	2	also	also	ADV
ejpam-7053	38	3	investigate	investigate	VERB
ejpam-7053	38	4	several	several	ADJ
ejpam-7053	38	5	characterizations	characterization	NOUN
ejpam-7053	38	6	of	of	ADP
ejpam-7053	38	7	upper	upper	ADJ
ejpam-7053	38	8	weakly	weakly	ADJ
ejpam-7053	38	9	µ(σ1	µ(σ1	NOUN
ejpam-7053	38	10	,	,	PUNCT
ejpam-7053	38	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	38	12	multifunctions	multifunction	NOUN
ejpam-7053	38	13	and	and	CCONJ
ejpam-7053	38	14	lower	low	ADJ
ejpam-7053	38	15	weakly	weakly	ADJ
ejpam-7053	38	16	µ(σ1	µ(σ1	NOUN
ejpam-7053	38	17	,	,	PUNCT
ejpam-7053	38	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	38	19	multifunctions	multifunction	NOUN
ejpam-7053	38	20	.	.	PUNCT
ejpam-7053	39	1	2	2	X
ejpam-7053	39	2	.	.	X
ejpam-7053	39	3	preliminaries	preliminary	NOUN
ejpam-7053	39	4	throughout	throughout	ADP
ejpam-7053	39	5	the	the	DET
ejpam-7053	39	6	present	present	ADJ
ejpam-7053	39	7	paper	paper	NOUN
ejpam-7053	39	8	,	,	PUNCT
ejpam-7053	39	9	spaces	space	NOUN
ejpam-7053	39	10	(	(	PUNCT
ejpam-7053	39	11	x	x	NOUN
ejpam-7053	39	12	,	,	PUNCT
ejpam-7053	39	13	τ1	τ1	NOUN
ejpam-7053	39	14	,	,	PUNCT
ejpam-7053	39	15	τ2	τ2	NOUN
ejpam-7053	39	16	)	)	PUNCT
ejpam-7053	39	17	and	and	CCONJ
ejpam-7053	39	18	(	(	PUNCT
ejpam-7053	39	19	y	y	PROPN
ejpam-7053	39	20	,	,	PUNCT
ejpam-7053	39	21	σ1	σ1	PROPN
ejpam-7053	39	22	,	,	PUNCT
ejpam-7053	39	23	σ2	σ2	NOUN
ejpam-7053	39	24	)	)	PUNCT
ejpam-7053	39	25	(	(	PUNCT
ejpam-7053	39	26	or	or	CCONJ
ejpam-7053	39	27	simply	simply	ADV
ejpam-7053	39	28	x	x	X
ejpam-7053	39	29	and	and	CCONJ
ejpam-7053	39	30	y	y	PROPN
ejpam-7053	39	31	)	)	PUNCT
ejpam-7053	39	32	always	always	ADV
ejpam-7053	39	33	mean	mean	VERB
ejpam-7053	39	34	bitopological	bitopological	ADJ
ejpam-7053	39	35	spaces	space	NOUN
ejpam-7053	39	36	on	on	ADP
ejpam-7053	39	37	which	which	PRON
ejpam-7053	39	38	no	no	DET
ejpam-7053	39	39	separation	separation	NOUN
ejpam-7053	39	40	axioms	axiom	NOUN
ejpam-7053	39	41	are	be	AUX
ejpam-7053	39	42	assumed	assume	VERB
ejpam-7053	39	43	unless	unless	SCONJ
ejpam-7053	39	44	explicitly	explicitly	ADV
ejpam-7053	39	45	stated	state	VERB
ejpam-7053	39	46	.	.	PUNCT
ejpam-7053	40	1	let	let	VERB
ejpam-7053	40	2	a	a	DET
ejpam-7053	40	3	be	be	AUX
ejpam-7053	40	4	a	a	DET
ejpam-7053	40	5	subset	subset	NOUN
ejpam-7053	40	6	of	of	ADP
ejpam-7053	40	7	a	a	DET
ejpam-7053	40	8	bitopological	bitopological	ADJ
ejpam-7053	40	9	space	space	NOUN
ejpam-7053	40	10	(	(	PUNCT
ejpam-7053	40	11	x	x	NOUN
ejpam-7053	40	12	,	,	PUNCT
ejpam-7053	40	13	τ1	τ1	NOUN
ejpam-7053	40	14	,	,	PUNCT
ejpam-7053	40	15	τ2	τ2	NOUN
ejpam-7053	40	16	)	)	PUNCT
ejpam-7053	40	17	.	.	PUNCT
ejpam-7053	41	1	the	the	DET
ejpam-7053	41	2	closure	closure	NOUN
ejpam-7053	41	3	of	of	ADP
ejpam-7053	41	4	a	a	PRON
ejpam-7053	41	5	and	and	CCONJ
ejpam-7053	41	6	the	the	DET
ejpam-7053	41	7	interior	interior	NOUN
ejpam-7053	41	8	of	of	ADP
ejpam-7053	41	9	a	a	PRON
ejpam-7053	41	10	with	with	ADP
ejpam-7053	41	11	respect	respect	NOUN
ejpam-7053	41	12	to	to	ADP
ejpam-7053	41	13	τi	τi	PROPN
ejpam-7053	41	14	are	be	AUX
ejpam-7053	41	15	denoted	denote	VERB
ejpam-7053	41	16	by	by	ADP
ejpam-7053	41	17	τi	τi	NOUN
ejpam-7053	41	18	-	-	PUNCT
ejpam-7053	41	19	cl(a	cl(a	NUM
ejpam-7053	41	20	)	)	PUNCT
ejpam-7053	41	21	and	and	CCONJ
ejpam-7053	41	22	τi	τi	NOUN
ejpam-7053	41	23	-	-	PUNCT
ejpam-7053	41	24	int(a	int(a	NOUN
ejpam-7053	41	25	)	)	PUNCT
ejpam-7053	41	26	,	,	PUNCT
ejpam-7053	41	27	respectively	respectively	ADV
ejpam-7053	41	28	,	,	PUNCT
ejpam-7053	41	29	for	for	ADP
ejpam-7053	41	30	i	i	PROPN
ejpam-7053	41	31	=	=	SYM
ejpam-7053	41	32	1	1	NUM
ejpam-7053	41	33	,	,	PUNCT
ejpam-7053	41	34	2	2	NUM
ejpam-7053	41	35	.	.	X
ejpam-7053	41	36	a	a	DET
ejpam-7053	41	37	subset	subset	NOUN
ejpam-7053	41	38	a	a	PRON
ejpam-7053	41	39	of	of	ADP
ejpam-7053	41	40	a	a	DET
ejpam-7053	41	41	bitopological	bitopological	ADJ
ejpam-7053	41	42	space	space	NOUN
ejpam-7053	41	43	(	(	PUNCT
ejpam-7053	41	44	x	x	NOUN
ejpam-7053	41	45	,	,	PUNCT
ejpam-7053	41	46	τ1	τ1	NOUN
ejpam-7053	41	47	,	,	PUNCT
ejpam-7053	41	48	τ2	τ2	NOUN
ejpam-7053	41	49	)	)	PUNCT
ejpam-7053	41	50	is	be	AUX
ejpam-7053	41	51	called	call	VERB
ejpam-7053	41	52	τ1τ2	τ1τ2	VERB
ejpam-7053	41	53	-	-	ADJ
ejpam-7053	41	54	closed	closed	ADJ
ejpam-7053	41	55	[	[	X
ejpam-7053	41	56	15	15	NUM
ejpam-7053	41	57	]	]	X
ejpam-7053	41	58	if	if	SCONJ
ejpam-7053	41	59	a	a	DET
ejpam-7053	41	60	=	=	NOUN
ejpam-7053	41	61	τ1	τ1	NOUN
ejpam-7053	41	62	-	-	PUNCT
ejpam-7053	41	63	cl(τ2	cl(τ2	NOUN
ejpam-7053	41	64	-	-	PUNCT
ejpam-7053	41	65	cl(a	cl(a	NUM
ejpam-7053	41	66	)	)	PUNCT
ejpam-7053	41	67	)	)	PUNCT
ejpam-7053	41	68	.	.	PUNCT
ejpam-7053	42	1	the	the	DET
ejpam-7053	42	2	complement	complement	NOUN
ejpam-7053	42	3	of	of	ADP
ejpam-7053	42	4	a	a	DET
ejpam-7053	42	5	τ1τ2	τ1τ2	ADJ
ejpam-7053	42	6	-	-	ADJ
ejpam-7053	42	7	closed	closed	ADJ
ejpam-7053	42	8	set	set	NOUN
ejpam-7053	42	9	is	be	AUX
ejpam-7053	42	10	called	call	VERB
ejpam-7053	42	11	τ1τ2	τ1τ2	NOUN
ejpam-7053	42	12	-	-	ADJ
ejpam-7053	42	13	open	open	ADJ
ejpam-7053	42	14	.	.	PUNCT
ejpam-7053	43	1	the	the	DET
ejpam-7053	43	2	intersection	intersection	NOUN
ejpam-7053	43	3	of	of	ADP
ejpam-7053	43	4	all	all	DET
ejpam-7053	43	5	τ1τ2	τ1τ2	ADJ
ejpam-7053	43	6	-	-	ADJ
ejpam-7053	43	7	closed	closed	ADJ
ejpam-7053	43	8	sets	set	NOUN
ejpam-7053	43	9	of	of	ADP
ejpam-7053	43	10	x	x	PUNCT
ejpam-7053	43	11	containing	contain	VERB
ejpam-7053	43	12	a	a	PRON
ejpam-7053	43	13	is	be	AUX
ejpam-7053	43	14	called	call	VERB
ejpam-7053	43	15	the	the	DET
ejpam-7053	43	16	τ1τ2	τ1τ2	NOUN
ejpam-7053	43	17	-	-	NOUN
ejpam-7053	43	18	closure	closure	NOUN
ejpam-7053	43	19	[	[	X
ejpam-7053	43	20	15	15	NUM
ejpam-7053	43	21	]	]	PUNCT
ejpam-7053	43	22	of	of	ADP
ejpam-7053	43	23	a	a	PRON
ejpam-7053	43	24	and	and	CCONJ
ejpam-7053	43	25	is	be	AUX
ejpam-7053	43	26	denoted	denote	VERB
ejpam-7053	43	27	by	by	ADP
ejpam-7053	43	28	τ1τ2	τ1τ2	NOUN
ejpam-7053	43	29	-	-	NUM
ejpam-7053	43	30	cl(a	cl(a	NUM
ejpam-7053	43	31	)	)	PUNCT
ejpam-7053	43	32	.	.	PUNCT
ejpam-7053	44	1	the	the	DET
ejpam-7053	44	2	union	union	NOUN
ejpam-7053	44	3	of	of	ADP
ejpam-7053	44	4	all	all	DET
ejpam-7053	44	5	τ1τ2	τ1τ2	ADJ
ejpam-7053	44	6	-	-	ADJ
ejpam-7053	44	7	open	open	ADJ
ejpam-7053	44	8	sets	set	NOUN
ejpam-7053	44	9	of	of	ADP
ejpam-7053	44	10	x	x	PUNCT
ejpam-7053	44	11	contained	contain	VERB
ejpam-7053	44	12	in	in	ADP
ejpam-7053	44	13	a	a	PRON
ejpam-7053	44	14	is	be	AUX
ejpam-7053	44	15	called	call	VERB
ejpam-7053	44	16	the	the	DET
ejpam-7053	44	17	τ1τ2	τ1τ2	NOUN
ejpam-7053	44	18	-	-	ADJ
ejpam-7053	44	19	interior	interior	ADJ
ejpam-7053	44	20	[	[	X
ejpam-7053	44	21	15	15	NUM
ejpam-7053	44	22	]	]	PUNCT
ejpam-7053	44	23	of	of	ADP
ejpam-7053	44	24	a	a	PRON
ejpam-7053	44	25	and	and	CCONJ
ejpam-7053	44	26	is	be	AUX
ejpam-7053	44	27	denoted	denote	VERB
ejpam-7053	44	28	by	by	ADP
ejpam-7053	44	29	τ1τ2	τ1τ2	NOUN
ejpam-7053	44	30	-	-	ADJ
ejpam-7053	44	31	int(a	int(a	NOUN
ejpam-7053	44	32	)	)	PUNCT
ejpam-7053	44	33	.	.	PUNCT
ejpam-7053	45	1	lemma	lemma	PROPN
ejpam-7053	45	2	1	1	NUM
ejpam-7053	45	3	.	.	PUNCT
ejpam-7053	46	1	[	[	X
ejpam-7053	46	2	15	15	NUM
ejpam-7053	46	3	]	]	PUNCT
ejpam-7053	46	4	let	let	VERB
ejpam-7053	46	5	a	a	PRON
ejpam-7053	46	6	and	and	CCONJ
ejpam-7053	46	7	b	b	NOUN
ejpam-7053	46	8	be	be	AUX
ejpam-7053	46	9	subsets	subset	NOUN
ejpam-7053	46	10	of	of	ADP
ejpam-7053	46	11	a	a	DET
ejpam-7053	46	12	bitopological	bitopological	ADJ
ejpam-7053	46	13	space	space	NOUN
ejpam-7053	46	14	(	(	PUNCT
ejpam-7053	46	15	x	x	NOUN
ejpam-7053	46	16	,	,	PUNCT
ejpam-7053	46	17	τ1	τ1	NOUN
ejpam-7053	46	18	,	,	PUNCT
ejpam-7053	46	19	τ2	τ2	NOUN
ejpam-7053	46	20	)	)	PUNCT
ejpam-7053	46	21	.	.	PUNCT
ejpam-7053	47	1	for	for	ADP
ejpam-7053	47	2	the	the	DET
ejpam-7053	47	3	τ1τ2	τ1τ2	NOUN
ejpam-7053	47	4	-	-	NOUN
ejpam-7053	47	5	closure	closure	NOUN
ejpam-7053	47	6	,	,	PUNCT
ejpam-7053	47	7	the	the	DET
ejpam-7053	47	8	following	follow	VERB
ejpam-7053	47	9	properties	property	NOUN
ejpam-7053	47	10	hold	hold	VERB
ejpam-7053	47	11	:	:	PUNCT
ejpam-7053	47	12	(	(	PUNCT
ejpam-7053	47	13	1	1	X
ejpam-7053	47	14	)	)	PUNCT
ejpam-7053	47	15	a	a	DET
ejpam-7053	47	16	⊆	⊆	NUM
ejpam-7053	47	17	τ1τ2	τ1τ2	NOUN
ejpam-7053	47	18	-	-	NUM
ejpam-7053	47	19	cl(a	cl(a	NUM
ejpam-7053	47	20	)	)	PUNCT
ejpam-7053	47	21	and	and	CCONJ
ejpam-7053	47	22	τ1τ2	τ1τ2	NOUN
ejpam-7053	47	23	-	-	ADJ
ejpam-7053	47	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7053	47	25	-	-	PUNCT
ejpam-7053	47	26	cl(a	cl(a	NUM
ejpam-7053	47	27	)	)	PUNCT
ejpam-7053	47	28	)	)	PUNCT
ejpam-7053	48	1	=	=	PUNCT
ejpam-7053	48	2	τ1τ2	τ1τ2	NOUN
ejpam-7053	48	3	-	-	NUM
ejpam-7053	48	4	cl(a	cl(a	NUM
ejpam-7053	48	5	)	)	PUNCT
ejpam-7053	48	6	.	.	PUNCT
ejpam-7053	49	1	(	(	PUNCT
ejpam-7053	49	2	2	2	X
ejpam-7053	49	3	)	)	PUNCT
ejpam-7053	49	4	if	if	SCONJ
ejpam-7053	49	5	a	a	DET
ejpam-7053	49	6	⊆	⊆	NUM
ejpam-7053	49	7	b	b	NOUN
ejpam-7053	49	8	,	,	PUNCT
ejpam-7053	49	9	then	then	ADV
ejpam-7053	49	10	τ1τ2	τ1τ2	NOUN
ejpam-7053	49	11	-	-	NUM
ejpam-7053	49	12	cl(a	cl(a	NUM
ejpam-7053	49	13	)	)	PUNCT
ejpam-7053	49	14	⊆	⊆	NUM
ejpam-7053	49	15	τ1τ2	τ1τ2	NOUN
ejpam-7053	49	16	-	-	NOUN
ejpam-7053	49	17	cl(b	cl(b	NOUN
ejpam-7053	49	18	)	)	PUNCT
ejpam-7053	49	19	.	.	PUNCT
ejpam-7053	50	1	m.	m.	NOUN
ejpam-7053	50	2	chiangpradit	chiangpradit	PROPN
ejpam-7053	50	3	,	,	PUNCT
ejpam-7053	50	4	a.	a.	PROPN
ejpam-7053	50	5	sama	sama	PROPN
ejpam-7053	50	6	-	-	PUNCT
ejpam-7053	50	7	ae	ae	PROPN
ejpam-7053	50	8	,	,	PUNCT
ejpam-7053	50	9	c.	c.	PROPN
ejpam-7053	50	10	boonpok	boonpok	PROPN
ejpam-7053	50	11	/	/	SYM
ejpam-7053	50	12	eur	eur	PROPN
ejpam-7053	50	13	.	.	PUNCT
ejpam-7053	51	1	j.	j.	PROPN
ejpam-7053	51	2	pure	pure	PROPN
ejpam-7053	51	3	appl	appl	PROPN
ejpam-7053	51	4	.	.	PROPN
ejpam-7053	51	5	math	math	PROPN
ejpam-7053	51	6	,	,	PUNCT
ejpam-7053	51	7	18	18	NUM
ejpam-7053	51	8	(	(	PUNCT
ejpam-7053	51	9	4	4	NUM
ejpam-7053	51	10	)	)	PUNCT
ejpam-7053	51	11	(	(	PUNCT
ejpam-7053	51	12	2025	2025	NUM
ejpam-7053	51	13	)	)	PUNCT
ejpam-7053	51	14	,	,	PUNCT
ejpam-7053	51	15	7053	7053	NUM
ejpam-7053	51	16	3	3	NUM
ejpam-7053	51	17	of	of	ADP
ejpam-7053	51	18	13	13	NUM
ejpam-7053	51	19	(	(	PUNCT
ejpam-7053	51	20	3	3	NUM
ejpam-7053	51	21	)	)	PUNCT
ejpam-7053	51	22	τ1τ2	τ1τ2	NOUN
ejpam-7053	51	23	-	-	NUM
ejpam-7053	51	24	cl(a	cl(a	NUM
ejpam-7053	51	25	)	)	PUNCT
ejpam-7053	51	26	is	be	AUX
ejpam-7053	51	27	τ1τ2	τ1τ2	NOUN
ejpam-7053	51	28	-	-	ADJ
ejpam-7053	51	29	closed	closed	ADJ
ejpam-7053	51	30	.	.	PUNCT
ejpam-7053	52	1	(	(	PUNCT
ejpam-7053	52	2	4	4	X
ejpam-7053	52	3	)	)	PUNCT
ejpam-7053	52	4	a	a	PRON
ejpam-7053	52	5	is	be	AUX
ejpam-7053	52	6	τ1τ2	τ1τ2	NOUN
ejpam-7053	52	7	-	-	ADJ
ejpam-7053	52	8	closed	closed	ADJ
ejpam-7053	52	9	if	if	SCONJ
ejpam-7053	52	10	and	and	CCONJ
ejpam-7053	52	11	only	only	ADV
ejpam-7053	52	12	if	if	SCONJ
ejpam-7053	52	13	a	a	DET
ejpam-7053	52	14	=	=	PUNCT
ejpam-7053	52	15	τ1τ2	τ1τ2	NOUN
ejpam-7053	52	16	-	-	NUM
ejpam-7053	52	17	cl(a	cl(a	NUM
ejpam-7053	52	18	)	)	PUNCT
ejpam-7053	52	19	.	.	PUNCT
ejpam-7053	53	1	(	(	PUNCT
ejpam-7053	53	2	5	5	X
ejpam-7053	53	3	)	)	PUNCT
ejpam-7053	53	4	τ1τ2	τ1τ2	NOUN
ejpam-7053	53	5	-	-	NOUN
ejpam-7053	53	6	cl(x	cl(x	X
ejpam-7053	53	7	−a	−a	NOUN
ejpam-7053	53	8	)	)	PUNCT
ejpam-7053	54	1	=	=	PUNCT
ejpam-7053	54	2	x	x	X
ejpam-7053	55	1	−	−	ADP
ejpam-7053	55	2	τ1τ2	τ1τ2	NOUN
ejpam-7053	55	3	-	-	PUNCT
ejpam-7053	55	4	int(a	int(a	NOUN
ejpam-7053	55	5	)	)	PUNCT
ejpam-7053	55	6	.	.	PUNCT
ejpam-7053	56	1	a	a	DET
ejpam-7053	56	2	subset	subset	NOUN
ejpam-7053	56	3	a	a	PRON
ejpam-7053	56	4	of	of	ADP
ejpam-7053	56	5	a	a	DET
ejpam-7053	56	6	bitopological	bitopological	ADJ
ejpam-7053	56	7	space	space	NOUN
ejpam-7053	56	8	(	(	PUNCT
ejpam-7053	56	9	x	x	NOUN
ejpam-7053	56	10	,	,	PUNCT
ejpam-7053	56	11	τ1	τ1	NOUN
ejpam-7053	56	12	,	,	PUNCT
ejpam-7053	56	13	τ2	τ2	NOUN
ejpam-7053	56	14	)	)	PUNCT
ejpam-7053	56	15	is	be	AUX
ejpam-7053	56	16	said	say	VERB
ejpam-7053	56	17	to	to	PART
ejpam-7053	56	18	be	be	AUX
ejpam-7053	56	19	(	(	PUNCT
ejpam-7053	56	20	τ1	τ1	NOUN
ejpam-7053	56	21	,	,	PUNCT
ejpam-7053	56	22	τ2)r	τ2)r	NOUN
ejpam-7053	56	23	-	-	PUNCT
ejpam-7053	56	24	open	open	NOUN
ejpam-7053	57	1	[	[	X
ejpam-7053	57	2	16	16	NUM
ejpam-7053	57	3	]	]	PUNCT
ejpam-7053	57	4	(	(	PUNCT
ejpam-7053	57	5	resp	resp	NOUN
ejpam-7053	57	6	.	.	PUNCT
ejpam-7053	58	1	(	(	PUNCT
ejpam-7053	58	2	τ1	τ1	NOUN
ejpam-7053	58	3	,	,	PUNCT
ejpam-7053	58	4	τ2)s	τ2)s	NOUN
ejpam-7053	58	5	-	-	PUNCT
ejpam-7053	58	6	open	open	ADJ
ejpam-7053	58	7	[	[	X
ejpam-7053	58	8	17	17	NUM
ejpam-7053	58	9	]	]	PUNCT
ejpam-7053	58	10	,	,	PUNCT
ejpam-7053	58	11	(	(	PUNCT
ejpam-7053	58	12	τ1	τ1	NOUN
ejpam-7053	58	13	,	,	PUNCT
ejpam-7053	58	14	τ2)p	τ2)p	NOUN
ejpam-7053	58	15	-	-	ADJ
ejpam-7053	58	16	open	open	ADJ
ejpam-7053	58	17	[	[	X
ejpam-7053	58	18	17	17	NUM
ejpam-7053	58	19	]	]	PUNCT
ejpam-7053	58	20	,	,	PUNCT
ejpam-7053	58	21	(	(	PUNCT
ejpam-7053	58	22	τ1	τ1	NOUN
ejpam-7053	58	23	,	,	PUNCT
ejpam-7053	58	24	τ2)β	τ2)β	ADJ
ejpam-7053	58	25	-	-	PUNCT
ejpam-7053	58	26	open	open	NOUN
ejpam-7053	59	1	[	[	X
ejpam-7053	59	2	17	17	NUM
ejpam-7053	59	3	]	]	SYM
ejpam-7053	59	4	)	)	PUNCT
ejpam-7053	59	5	if	if	SCONJ
ejpam-7053	59	6	a	a	DET
ejpam-7053	59	7	=	=	PUNCT
ejpam-7053	59	8	τ1τ2	τ1τ2	NOUN
ejpam-7053	59	9	-	-	NOUN
ejpam-7053	59	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7053	59	11	-	-	PUNCT
ejpam-7053	59	12	cl(a	cl(a	NUM
ejpam-7053	59	13	)	)	PUNCT
ejpam-7053	59	14	)	)	PUNCT
ejpam-7053	59	15	(	(	PUNCT
ejpam-7053	59	16	resp	resp	NOUN
ejpam-7053	59	17	.	.	PUNCT
ejpam-7053	60	1	a	a	DET
ejpam-7053	60	2	⊆	⊆	NUM
ejpam-7053	60	3	τ1τ2	τ1τ2	NOUN
ejpam-7053	60	4	-	-	ADJ
ejpam-7053	60	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7053	60	6	-	-	PUNCT
ejpam-7053	60	7	int(a	int(a	NOUN
ejpam-7053	60	8	)	)	PUNCT
ejpam-7053	60	9	)	)	PUNCT
ejpam-7053	60	10	,	,	PUNCT
ejpam-7053	60	11	a	a	DET
ejpam-7053	60	12	⊆	⊆	NUM
ejpam-7053	60	13	τ1τ2	τ1τ2	NOUN
ejpam-7053	60	14	-	-	NOUN
ejpam-7053	60	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7053	60	16	-	-	PUNCT
ejpam-7053	60	17	cl(a	cl(a	NUM
ejpam-7053	60	18	)	)	PUNCT
ejpam-7053	60	19	)	)	PUNCT
ejpam-7053	60	20	,	,	PUNCT
ejpam-7053	60	21	a	a	DET
ejpam-7053	60	22	⊆	⊆	NUM
ejpam-7053	60	23	τ1τ2	τ1τ2	NOUN
ejpam-7053	60	24	-	-	PUNCT
ejpam-7053	60	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7053	60	26	-	-	PUNCT
ejpam-7053	60	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7053	60	28	-	-	PUNCT
ejpam-7053	60	29	cl(a	cl(a	NUM
ejpam-7053	60	30	)	)	PUNCT
ejpam-7053	60	31	)	)	PUNCT
ejpam-7053	60	32	)	)	PUNCT
ejpam-7053	60	33	)	)	PUNCT
ejpam-7053	60	34	.	.	PUNCT
ejpam-7053	61	1	the	the	DET
ejpam-7053	61	2	complement	complement	NOUN
ejpam-7053	61	3	of	of	ADP
ejpam-7053	61	4	a	a	DET
ejpam-7053	61	5	(	(	PUNCT
ejpam-7053	61	6	τ1	τ1	NOUN
ejpam-7053	61	7	,	,	PUNCT
ejpam-7053	61	8	τ2)r	τ2)r	NOUN
ejpam-7053	61	9	-	-	PUNCT
ejpam-7053	61	10	open	open	ADJ
ejpam-7053	61	11	(	(	PUNCT
ejpam-7053	61	12	resp	resp	NOUN
ejpam-7053	61	13	.	.	PUNCT
ejpam-7053	62	1	(	(	PUNCT
ejpam-7053	62	2	τ1	τ1	NOUN
ejpam-7053	62	3	,	,	PUNCT
ejpam-7053	62	4	τ2)s	τ2)s	NOUN
ejpam-7053	62	5	-	-	PUNCT
ejpam-7053	62	6	open	open	ADJ
ejpam-7053	62	7	,	,	PUNCT
ejpam-7053	62	8	(	(	PUNCT
ejpam-7053	62	9	τ1	τ1	NOUN
ejpam-7053	62	10	,	,	PUNCT
ejpam-7053	62	11	τ2)p	τ2)p	NOUN
ejpam-7053	62	12	-	-	ADJ
ejpam-7053	62	13	open	open	ADJ
ejpam-7053	62	14	,	,	PUNCT
ejpam-7053	62	15	(	(	PUNCT
ejpam-7053	62	16	τ1	τ1	NOUN
ejpam-7053	62	17	,	,	PUNCT
ejpam-7053	62	18	τ2)β	τ2)β	ADJ
ejpam-7053	62	19	-	-	PUNCT
ejpam-7053	62	20	open	open	ADJ
ejpam-7053	62	21	)	)	PUNCT
ejpam-7053	62	22	set	set	NOUN
ejpam-7053	62	23	is	be	AUX
ejpam-7053	62	24	said	say	VERB
ejpam-7053	62	25	to	to	PART
ejpam-7053	62	26	be	be	AUX
ejpam-7053	62	27	(	(	PUNCT
ejpam-7053	62	28	τ1	τ1	NOUN
ejpam-7053	62	29	,	,	PUNCT
ejpam-7053	62	30	τ2)r	τ2)r	NOUN
ejpam-7053	62	31	-	-	PUNCT
ejpam-7053	62	32	closed	closed	ADJ
ejpam-7053	62	33	(	(	PUNCT
ejpam-7053	62	34	resp	resp	NOUN
ejpam-7053	62	35	.	.	PUNCT
ejpam-7053	63	1	(	(	PUNCT
ejpam-7053	63	2	τ1	τ1	NOUN
ejpam-7053	63	3	,	,	PUNCT
ejpam-7053	63	4	τ2)s	τ2)s	NOUN
ejpam-7053	63	5	-	-	PUNCT
ejpam-7053	63	6	closed	closed	ADJ
ejpam-7053	63	7	,	,	PUNCT
ejpam-7053	63	8	(	(	PUNCT
ejpam-7053	63	9	τ1	τ1	NOUN
ejpam-7053	63	10	,	,	PUNCT
ejpam-7053	63	11	τ2)p	τ2)p	NOUN
ejpam-7053	63	12	-	-	PUNCT
ejpam-7053	63	13	closed	closed	ADJ
ejpam-7053	63	14	,	,	PUNCT
ejpam-7053	63	15	(	(	PUNCT
ejpam-7053	63	16	τ1	τ1	NOUN
ejpam-7053	63	17	,	,	PUNCT
ejpam-7053	63	18	τ2)p	τ2)p	NOUN
ejpam-7053	63	19	-	-	PUNCT
ejpam-7053	63	20	closed	closed	ADJ
ejpam-7053	63	21	)	)	PUNCT
ejpam-7053	63	22	.	.	PUNCT
ejpam-7053	64	1	for	for	ADP
ejpam-7053	64	2	a	a	DET
ejpam-7053	64	3	subset	subset	NOUN
ejpam-7053	64	4	a	a	PRON
ejpam-7053	64	5	of	of	ADP
ejpam-7053	64	6	a	a	DET
ejpam-7053	64	7	bitopological	bitopological	ADJ
ejpam-7053	64	8	space	space	NOUN
ejpam-7053	64	9	(	(	PUNCT
ejpam-7053	64	10	x	x	NOUN
ejpam-7053	64	11	,	,	PUNCT
ejpam-7053	64	12	τ1	τ1	NOUN
ejpam-7053	64	13	,	,	PUNCT
ejpam-7053	64	14	τ2	τ2	PROPN
ejpam-7053	64	15	)	)	PUNCT
ejpam-7053	64	16	,	,	PUNCT
ejpam-7053	64	17	a	a	DET
ejpam-7053	64	18	point	point	NOUN
ejpam-7053	64	19	x	x	X
ejpam-7053	64	20	∈	∈	NOUN
ejpam-7053	64	21	x	x	PUNCT
ejpam-7053	64	22	is	be	AUX
ejpam-7053	64	23	called	call	VERB
ejpam-7053	64	24	a	a	DET
ejpam-7053	64	25	(	(	PUNCT
ejpam-7053	64	26	τ1	τ1	NOUN
ejpam-7053	64	27	,	,	PUNCT
ejpam-7053	64	28	τ2)θ	τ2)θ	ADJ
ejpam-7053	64	29	-	-	PUNCT
ejpam-7053	64	30	cluster	cluster	NOUN
ejpam-7053	64	31	point	point	NOUN
ejpam-7053	64	32	[	[	X
ejpam-7053	64	33	16	16	NUM
ejpam-7053	64	34	]	]	PUNCT
ejpam-7053	64	35	of	of	ADP
ejpam-7053	64	36	a	a	DET
ejpam-7053	64	37	if	if	SCONJ
ejpam-7053	64	38	τ1τ2	τ1τ2	NOUN
ejpam-7053	64	39	-	-	NOUN
ejpam-7053	64	40	cl(u	cl(u	NOUN
ejpam-7053	64	41	)	)	PUNCT
ejpam-7053	64	42	∩	∩	NOUN
ejpam-7053	64	43	a	a	DET
ejpam-7053	64	44	̸=	̸=	PROPN
ejpam-7053	64	45	∅	∅	NOUN
ejpam-7053	64	46	for	for	ADP
ejpam-7053	64	47	every	every	DET
ejpam-7053	64	48	τ1τ2	τ1τ2	ADJ
ejpam-7053	64	49	-	-	ADJ
ejpam-7053	64	50	open	open	ADJ
ejpam-7053	64	51	set	set	NOUN
ejpam-7053	64	52	u	u	NOUN
ejpam-7053	64	53	containing	contain	VERB
ejpam-7053	64	54	x.	x.	NOUN
ejpam-7053	64	55	the	the	DET
ejpam-7053	64	56	set	set	NOUN
ejpam-7053	64	57	of	of	ADP
ejpam-7053	64	58	all	all	DET
ejpam-7053	64	59	(	(	PUNCT
ejpam-7053	64	60	τ1	τ1	NOUN
ejpam-7053	64	61	,	,	PUNCT
ejpam-7053	64	62	τ2)θ	τ2)θ	ADJ
ejpam-7053	64	63	-	-	PUNCT
ejpam-7053	64	64	cluster	cluster	NOUN
ejpam-7053	64	65	points	point	NOUN
ejpam-7053	64	66	of	of	ADP
ejpam-7053	64	67	a	a	PRON
ejpam-7053	64	68	is	be	AUX
ejpam-7053	64	69	called	call	VERB
ejpam-7053	64	70	the	the	DET
ejpam-7053	64	71	(	(	PUNCT
ejpam-7053	64	72	τ1	τ1	NOUN
ejpam-7053	64	73	,	,	PUNCT
ejpam-7053	64	74	τ2)θ	τ2)θ	ADJ
ejpam-7053	64	75	-	-	PUNCT
ejpam-7053	64	76	closure	closure	NOUN
ejpam-7053	64	77	[	[	X
ejpam-7053	64	78	16	16	NUM
ejpam-7053	64	79	]	]	PUNCT
ejpam-7053	64	80	of	of	ADP
ejpam-7053	64	81	a	a	PRON
ejpam-7053	64	82	and	and	CCONJ
ejpam-7053	64	83	is	be	AUX
ejpam-7053	64	84	denoted	denote	VERB
ejpam-7053	64	85	by	by	ADP
ejpam-7053	64	86	(	(	PUNCT
ejpam-7053	64	87	τ1	τ1	NOUN
ejpam-7053	64	88	,	,	PUNCT
ejpam-7053	64	89	τ2)θ	τ2)θ	NOUN
ejpam-7053	64	90	-	-	PUNCT
ejpam-7053	64	91	cl(a	cl(a	NUM
ejpam-7053	64	92	)	)	PUNCT
ejpam-7053	64	93	.	.	PUNCT
ejpam-7053	65	1	a	a	DET
ejpam-7053	65	2	subset	subset	NOUN
ejpam-7053	65	3	a	a	PRON
ejpam-7053	65	4	of	of	ADP
ejpam-7053	65	5	a	a	DET
ejpam-7053	65	6	bitopological	bitopological	ADJ
ejpam-7053	65	7	space	space	NOUN
ejpam-7053	65	8	(	(	PUNCT
ejpam-7053	65	9	x	x	NOUN
ejpam-7053	65	10	,	,	PUNCT
ejpam-7053	65	11	τ1	τ1	NOUN
ejpam-7053	65	12	,	,	PUNCT
ejpam-7053	65	13	τ2	τ2	NOUN
ejpam-7053	65	14	)	)	PUNCT
ejpam-7053	65	15	is	be	AUX
ejpam-7053	65	16	said	say	VERB
ejpam-7053	65	17	to	to	PART
ejpam-7053	65	18	be	be	AUX
ejpam-7053	65	19	(	(	PUNCT
ejpam-7053	65	20	τ1	τ1	NOUN
ejpam-7053	65	21	,	,	PUNCT
ejpam-7053	65	22	τ2)θ	τ2)θ	NOUN
ejpam-7053	65	23	-	-	PUNCT
ejpam-7053	65	24	closed	closed	ADJ
ejpam-7053	66	1	[	[	X
ejpam-7053	66	2	16	16	NUM
ejpam-7053	66	3	]	]	X
ejpam-7053	66	4	if	if	SCONJ
ejpam-7053	66	5	(	(	PUNCT
ejpam-7053	66	6	τ1	τ1	NOUN
ejpam-7053	66	7	,	,	PUNCT
ejpam-7053	66	8	τ2)θ	τ2)θ	NOUN
ejpam-7053	66	9	-	-	PUNCT
ejpam-7053	66	10	cl(a	cl(a	NUM
ejpam-7053	66	11	)	)	PUNCT
ejpam-7053	66	12	=	=	PUNCT
ejpam-7053	66	13	a.	a.	NOUN
ejpam-7053	66	14	the	the	DET
ejpam-7053	66	15	complement	complement	NOUN
ejpam-7053	66	16	of	of	ADP
ejpam-7053	66	17	a	a	DET
ejpam-7053	66	18	(	(	PUNCT
ejpam-7053	66	19	τ1	τ1	NOUN
ejpam-7053	66	20	,	,	PUNCT
ejpam-7053	66	21	τ2)θ	τ2)θ	ADJ
ejpam-7053	66	22	-	-	PUNCT
ejpam-7053	66	23	closed	close	VERB
ejpam-7053	66	24	set	set	NOUN
ejpam-7053	66	25	is	be	AUX
ejpam-7053	66	26	said	say	VERB
ejpam-7053	66	27	to	to	PART
ejpam-7053	66	28	be	be	AUX
ejpam-7053	66	29	(	(	PUNCT
ejpam-7053	66	30	τ1	τ1	NOUN
ejpam-7053	66	31	,	,	PUNCT
ejpam-7053	66	32	τ2)θopen	τ2)θopen	PROPN
ejpam-7053	66	33	.	.	PUNCT
ejpam-7053	67	1	the	the	DET
ejpam-7053	67	2	union	union	NOUN
ejpam-7053	67	3	of	of	ADP
ejpam-7053	67	4	all	all	DET
ejpam-7053	67	5	(	(	PUNCT
ejpam-7053	67	6	τ1	τ1	NOUN
ejpam-7053	67	7	,	,	PUNCT
ejpam-7053	67	8	τ2)θ	τ2)θ	ADJ
ejpam-7053	67	9	-	-	PUNCT
ejpam-7053	67	10	open	open	ADJ
ejpam-7053	67	11	sets	set	NOUN
ejpam-7053	67	12	of	of	ADP
ejpam-7053	67	13	x	x	PUNCT
ejpam-7053	67	14	contained	contain	VERB
ejpam-7053	67	15	in	in	ADP
ejpam-7053	67	16	a	a	PRON
ejpam-7053	67	17	is	be	AUX
ejpam-7053	67	18	called	call	VERB
ejpam-7053	67	19	the	the	DET
ejpam-7053	67	20	(	(	PUNCT
ejpam-7053	67	21	τ1	τ1	NOUN
ejpam-7053	67	22	,	,	PUNCT
ejpam-7053	67	23	τ2)θ	τ2)θ	ADJ
ejpam-7053	67	24	-	-	PUNCT
ejpam-7053	67	25	interior	interior	NOUN
ejpam-7053	67	26	[	[	X
ejpam-7053	67	27	16	16	NUM
ejpam-7053	67	28	]	]	PUNCT
ejpam-7053	67	29	of	of	ADP
ejpam-7053	67	30	a	a	PRON
ejpam-7053	67	31	and	and	CCONJ
ejpam-7053	67	32	is	be	AUX
ejpam-7053	67	33	denoted	denote	VERB
ejpam-7053	67	34	by	by	ADP
ejpam-7053	67	35	(	(	PUNCT
ejpam-7053	67	36	τ1	τ1	NOUN
ejpam-7053	67	37	,	,	PUNCT
ejpam-7053	67	38	τ2)θ	τ2)θ	NOUN
ejpam-7053	67	39	-	-	PUNCT
ejpam-7053	67	40	int(a	int(a	NOUN
ejpam-7053	67	41	)	)	PUNCT
ejpam-7053	67	42	.	.	PUNCT
ejpam-7053	68	1	lemma	lemma	PROPN
ejpam-7053	68	2	2	2	NUM
ejpam-7053	68	3	.	.	PUNCT
ejpam-7053	69	1	[	[	X
ejpam-7053	69	2	16	16	NUM
ejpam-7053	69	3	]	]	PUNCT
ejpam-7053	69	4	for	for	ADP
ejpam-7053	69	5	a	a	DET
ejpam-7053	69	6	subset	subset	NOUN
ejpam-7053	69	7	a	a	PRON
ejpam-7053	69	8	of	of	ADP
ejpam-7053	69	9	a	a	DET
ejpam-7053	69	10	bitopological	bitopological	ADJ
ejpam-7053	69	11	space	space	NOUN
ejpam-7053	69	12	(	(	PUNCT
ejpam-7053	69	13	x	x	NOUN
ejpam-7053	69	14	,	,	PUNCT
ejpam-7053	69	15	τ1	τ1	NOUN
ejpam-7053	69	16	,	,	PUNCT
ejpam-7053	69	17	τ2	τ2	NOUN
ejpam-7053	69	18	)	)	PUNCT
ejpam-7053	69	19	,	,	PUNCT
ejpam-7053	69	20	the	the	DET
ejpam-7053	69	21	following	follow	VERB
ejpam-7053	69	22	properties	property	NOUN
ejpam-7053	69	23	hold	hold	VERB
ejpam-7053	69	24	:	:	PUNCT
ejpam-7053	69	25	(	(	PUNCT
ejpam-7053	69	26	1	1	X
ejpam-7053	69	27	)	)	PUNCT
ejpam-7053	69	28	if	if	SCONJ
ejpam-7053	69	29	a	a	PRON
ejpam-7053	69	30	is	be	AUX
ejpam-7053	69	31	τ1τ2	τ1τ2	NOUN
ejpam-7053	69	32	-	-	ADJ
ejpam-7053	69	33	open	open	ADJ
ejpam-7053	69	34	in	in	ADP
ejpam-7053	69	35	x	x	NOUN
ejpam-7053	69	36	,	,	PUNCT
ejpam-7053	69	37	then	then	ADV
ejpam-7053	69	38	τ1τ2	τ1τ2	NOUN
ejpam-7053	69	39	-	-	NUM
ejpam-7053	69	40	cl(a	cl(a	NUM
ejpam-7053	69	41	)	)	PUNCT
ejpam-7053	69	42	=	=	PUNCT
ejpam-7053	69	43	(	(	PUNCT
ejpam-7053	69	44	τ1	τ1	NOUN
ejpam-7053	69	45	,	,	PUNCT
ejpam-7053	69	46	τ2)θ	τ2)θ	NOUN
ejpam-7053	69	47	-	-	PUNCT
ejpam-7053	69	48	cl(a	cl(a	NUM
ejpam-7053	69	49	)	)	PUNCT
ejpam-7053	69	50	.	.	PUNCT
ejpam-7053	70	1	(	(	PUNCT
ejpam-7053	70	2	2	2	X
ejpam-7053	70	3	)	)	PUNCT
ejpam-7053	70	4	(	(	PUNCT
ejpam-7053	70	5	τ1	τ1	NOUN
ejpam-7053	70	6	,	,	PUNCT
ejpam-7053	70	7	τ2)θ	τ2)θ	NOUN
ejpam-7053	70	8	-	-	PUNCT
ejpam-7053	70	9	cl(a	cl(a	NUM
ejpam-7053	70	10	)	)	PUNCT
ejpam-7053	70	11	is	be	AUX
ejpam-7053	70	12	τ1τ2	τ1τ2	NOUN
ejpam-7053	70	13	-	-	ADJ
ejpam-7053	70	14	closed	closed	ADJ
ejpam-7053	70	15	in	in	ADP
ejpam-7053	70	16	x.	x.	NOUN
ejpam-7053	70	17	let	let	VERB
ejpam-7053	70	18	x	x	PRON
ejpam-7053	70	19	be	be	AUX
ejpam-7053	70	20	a	a	DET
ejpam-7053	70	21	nonempty	nonempty	ADJ
ejpam-7053	70	22	set	set	NOUN
ejpam-7053	70	23	,	,	PUNCT
ejpam-7053	70	24	and	and	CCONJ
ejpam-7053	70	25	denote	denote	VERB
ejpam-7053	70	26	p(x	p(x	PROPN
ejpam-7053	70	27	)	)	PUNCT
ejpam-7053	70	28	the	the	DET
ejpam-7053	70	29	power	power	NOUN
ejpam-7053	70	30	set	set	NOUN
ejpam-7053	70	31	of	of	ADP
ejpam-7053	70	32	x.	x.	NOUN
ejpam-7053	70	33	we	we	PRON
ejpam-7053	70	34	call	call	VERB
ejpam-7053	70	35	a	a	DET
ejpam-7053	70	36	class	class	NOUN
ejpam-7053	70	37	µ	µ	PRON
ejpam-7053	70	38	⊆	⊆	NUM
ejpam-7053	70	39	p(x	p(x	NOUN
ejpam-7053	70	40	)	)	PUNCT
ejpam-7053	70	41	a	a	DET
ejpam-7053	70	42	generalized	generalized	ADJ
ejpam-7053	70	43	topology	topology	NOUN
ejpam-7053	70	44	(	(	PUNCT
ejpam-7053	70	45	briefly	briefly	ADV
ejpam-7053	70	46	,	,	PUNCT
ejpam-7053	70	47	gt	gt	PROPN
ejpam-7053	70	48	)	)	PUNCT
ejpam-7053	70	49	if	if	SCONJ
ejpam-7053	70	50	∅	∅	NUM
ejpam-7053	70	51	∈	∈	PROPN
ejpam-7053	70	52	µ	µ	NOUN
ejpam-7053	70	53	,	,	PUNCT
ejpam-7053	70	54	and	and	CCONJ
ejpam-7053	70	55	an	an	DET
ejpam-7053	70	56	arbitrary	arbitrary	ADJ
ejpam-7053	70	57	union	union	NOUN
ejpam-7053	70	58	of	of	ADP
ejpam-7053	70	59	elements	element	NOUN
ejpam-7053	70	60	of	of	ADP
ejpam-7053	70	61	µ	µ	X
ejpam-7053	70	62	belongs	belong	VERB
ejpam-7053	70	63	to	to	ADP
ejpam-7053	70	64	µ	µ	PROPN
ejpam-7053	70	65	[	[	X
ejpam-7053	70	66	7	7	NUM
ejpam-7053	70	67	]	]	PUNCT
ejpam-7053	70	68	.	.	PUNCT
ejpam-7053	71	1	a	a	DET
ejpam-7053	71	2	set	set	NOUN
ejpam-7053	71	3	x	x	PUNCT
ejpam-7053	71	4	with	with	ADP
ejpam-7053	71	5	a	a	DET
ejpam-7053	71	6	gt	gt	PROPN
ejpam-7053	71	7	µ	µ	NOUN
ejpam-7053	71	8	on	on	ADP
ejpam-7053	71	9	it	it	PRON
ejpam-7053	71	10	is	be	AUX
ejpam-7053	71	11	said	say	VERB
ejpam-7053	71	12	to	to	PART
ejpam-7053	71	13	be	be	AUX
ejpam-7053	71	14	a	a	DET
ejpam-7053	71	15	generalized	generalized	ADJ
ejpam-7053	71	16	topological	topological	ADJ
ejpam-7053	71	17	space	space	NOUN
ejpam-7053	71	18	(	(	PUNCT
ejpam-7053	71	19	briefly	briefly	ADV
ejpam-7053	71	20	,	,	PUNCT
ejpam-7053	71	21	gts	gts	NOUN
ejpam-7053	71	22	)	)	PUNCT
ejpam-7053	71	23	and	and	CCONJ
ejpam-7053	71	24	is	be	AUX
ejpam-7053	71	25	denoted	denote	VERB
ejpam-7053	71	26	by	by	ADP
ejpam-7053	71	27	(	(	PUNCT
ejpam-7053	71	28	x,µ	x,µ	NOUN
ejpam-7053	71	29	)	)	PUNCT
ejpam-7053	71	30	.	.	PUNCT
ejpam-7053	72	1	for	for	ADP
ejpam-7053	72	2	a	a	DET
ejpam-7053	72	3	gts	gts	NOUN
ejpam-7053	72	4	(	(	PUNCT
ejpam-7053	72	5	x,µ	x,µ	NOUN
ejpam-7053	72	6	)	)	PUNCT
ejpam-7053	72	7	,	,	PUNCT
ejpam-7053	72	8	the	the	DET
ejpam-7053	72	9	elements	element	NOUN
ejpam-7053	72	10	of	of	ADP
ejpam-7053	72	11	µ	µ	NOUN
ejpam-7053	72	12	are	be	AUX
ejpam-7053	72	13	called	call	VERB
ejpam-7053	72	14	µ-open	µ-open	NOUN
ejpam-7053	72	15	sets	set	NOUN
ejpam-7053	72	16	and	and	CCONJ
ejpam-7053	72	17	the	the	DET
ejpam-7053	72	18	complements	complement	NOUN
ejpam-7053	72	19	of	of	ADP
ejpam-7053	72	20	µ-open	µ-open	NOUN
ejpam-7053	72	21	sets	set	NOUN
ejpam-7053	72	22	are	be	AUX
ejpam-7053	72	23	called	call	VERB
ejpam-7053	72	24	µ-closed	µ-close	VERB
ejpam-7053	72	25	sets	set	NOUN
ejpam-7053	72	26	.	.	PUNCT
ejpam-7053	73	1	for	for	ADP
ejpam-7053	73	2	a	a	DET
ejpam-7053	73	3	⊆	⊆	NUM
ejpam-7053	73	4	x	x	SYM
ejpam-7053	73	5	,	,	PUNCT
ejpam-7053	73	6	we	we	PRON
ejpam-7053	73	7	denote	denote	VERB
ejpam-7053	73	8	by	by	ADP
ejpam-7053	73	9	cµ(a	cµ(a	PROPN
ejpam-7053	73	10	)	)	PUNCT
ejpam-7053	73	11	the	the	DET
ejpam-7053	73	12	intersection	intersection	NOUN
ejpam-7053	73	13	of	of	ADP
ejpam-7053	73	14	all	all	DET
ejpam-7053	73	15	µ-closed	µ-close	VERB
ejpam-7053	73	16	sets	set	NOUN
ejpam-7053	73	17	containing	contain	VERB
ejpam-7053	73	18	a	a	PRON
ejpam-7053	73	19	and	and	CCONJ
ejpam-7053	73	20	by	by	ADP
ejpam-7053	73	21	iµ(a	iµ(a	PROPN
ejpam-7053	73	22	)	)	PUNCT
ejpam-7053	73	23	the	the	DET
ejpam-7053	73	24	union	union	NOUN
ejpam-7053	73	25	of	of	ADP
ejpam-7053	73	26	all	all	DET
ejpam-7053	73	27	µ-open	µ-open	NOUN
ejpam-7053	73	28	sets	set	NOUN
ejpam-7053	73	29	contained	contain	VERB
ejpam-7053	73	30	in	in	ADP
ejpam-7053	73	31	a.	a.	NOUN
ejpam-7053	73	32	then	then	ADV
ejpam-7053	73	33	,	,	PUNCT
ejpam-7053	73	34	we	we	PRON
ejpam-7053	73	35	have	have	VERB
ejpam-7053	73	36	iµ(iµ(a	iµ(iµ(a	ADJ
ejpam-7053	73	37	)	)	PUNCT
ejpam-7053	73	38	)	)	PUNCT
ejpam-7053	74	1	=	=	SYM
ejpam-7053	74	2	iµ(a	iµ(a	ADJ
ejpam-7053	74	3	)	)	PUNCT
ejpam-7053	74	4	,	,	PUNCT
ejpam-7053	74	5	cµ(cµ(a	cµ(cµ(a	PROPN
ejpam-7053	74	6	)	)	PUNCT
ejpam-7053	74	7	)	)	PUNCT
ejpam-7053	75	1	=	=	SYM
ejpam-7053	75	2	cµ(a	cµ(a	ADJ
ejpam-7053	75	3	)	)	PUNCT
ejpam-7053	75	4	,	,	PUNCT
ejpam-7053	75	5	and	and	CCONJ
ejpam-7053	75	6	iµ(a	iµ(a	ADJ
ejpam-7053	75	7	)	)	PUNCT
ejpam-7053	75	8	=	=	SYM
ejpam-7053	76	1	x	x	SYM
ejpam-7053	76	2	−	−	PROPN
ejpam-7053	76	3	cµ(x	cµ(x	SYM
ejpam-7053	76	4	−a	−a	NOUN
ejpam-7053	76	5	)	)	PUNCT
ejpam-7053	76	6	.	.	PUNCT
ejpam-7053	77	1	according	accord	VERB
ejpam-7053	77	2	to	to	ADP
ejpam-7053	77	3	[	[	X
ejpam-7053	77	4	18	18	NUM
ejpam-7053	77	5	]	]	PUNCT
ejpam-7053	77	6	,	,	PUNCT
ejpam-7053	77	7	for	for	ADP
ejpam-7053	77	8	a	a	DET
ejpam-7053	77	9	⊆	⊆	NUM
ejpam-7053	77	10	x	x	SYM
ejpam-7053	77	11	and	and	CCONJ
ejpam-7053	77	12	x	x	SYM
ejpam-7053	77	13	∈	∈	NOUN
ejpam-7053	77	14	x	x	X
ejpam-7053	77	15	,	,	PUNCT
ejpam-7053	77	16	we	we	PRON
ejpam-7053	77	17	have	have	VERB
ejpam-7053	77	18	x	x	X
ejpam-7053	77	19	∈	∈	PROPN
ejpam-7053	77	20	cµ(a	cµ(a	NOUN
ejpam-7053	77	21	)	)	PUNCT
ejpam-7053	78	1	if	if	SCONJ
ejpam-7053	78	2	and	and	CCONJ
ejpam-7053	78	3	only	only	ADV
ejpam-7053	78	4	if	if	SCONJ
ejpam-7053	78	5	x	x	PROPN
ejpam-7053	78	6	∈	∈	PROPN
ejpam-7053	78	7	m	m	PROPN
ejpam-7053	78	8	∈	∈	NOUN
ejpam-7053	78	9	µ	µ	NOUN
ejpam-7053	78	10	implies	imply	VERB
ejpam-7053	78	11	m	m	VERB
ejpam-7053	78	12	∩a	∩a	PROPN
ejpam-7053	78	13	̸=	̸=	PROPN
ejpam-7053	78	14	∅.	∅.	ADV
ejpam-7053	78	15	by	by	ADP
ejpam-7053	78	16	a	a	DET
ejpam-7053	78	17	multifunction	multifunction	NOUN
ejpam-7053	78	18	f	f	NOUN
ejpam-7053	78	19	:	:	PUNCT
ejpam-7053	78	20	x	x	X
ejpam-7053	78	21	→	→	SYM
ejpam-7053	78	22	y	y	PROPN
ejpam-7053	78	23	,	,	PUNCT
ejpam-7053	78	24	we	we	PRON
ejpam-7053	78	25	mean	mean	VERB
ejpam-7053	78	26	a	a	DET
ejpam-7053	78	27	point	point	NOUN
ejpam-7053	78	28	-	-	PUNCT
ejpam-7053	78	29	to	to	ADP
ejpam-7053	78	30	-	-	PUNCT
ejpam-7053	78	31	set	set	VERB
ejpam-7053	78	32	correspondence	correspondence	NOUN
ejpam-7053	78	33	from	from	ADP
ejpam-7053	78	34	x	x	PUNCT
ejpam-7053	78	35	into	into	ADP
ejpam-7053	78	36	y	y	PROPN
ejpam-7053	78	37	,	,	PUNCT
ejpam-7053	78	38	and	and	CCONJ
ejpam-7053	78	39	we	we	PRON
ejpam-7053	78	40	always	always	ADV
ejpam-7053	78	41	assume	assume	VERB
ejpam-7053	78	42	that	that	SCONJ
ejpam-7053	78	43	f	f	PROPN
ejpam-7053	78	44	(	(	PUNCT
ejpam-7053	78	45	x	x	X
ejpam-7053	78	46	)	)	PUNCT
ejpam-7053	78	47	̸=	̸=	NOUN
ejpam-7053	78	48	∅	∅	NOUN
ejpam-7053	78	49	for	for	ADP
ejpam-7053	78	50	all	all	PRON
ejpam-7053	78	51	x	x	SYM
ejpam-7053	78	52	∈	∈	ADJ
ejpam-7053	78	53	x.	x.	NOUN
ejpam-7053	78	54	for	for	ADP
ejpam-7053	78	55	a	a	DET
ejpam-7053	78	56	multifunction	multifunction	NOUN
ejpam-7053	79	1	f	f	NOUN
ejpam-7053	79	2	:	:	PUNCT
ejpam-7053	79	3	x	x	X
ejpam-7053	79	4	→	→	SYM
ejpam-7053	79	5	y	y	PROPN
ejpam-7053	79	6	,	,	PUNCT
ejpam-7053	79	7	we	we	PRON
ejpam-7053	79	8	shall	shall	AUX
ejpam-7053	79	9	denote	denote	VERB
ejpam-7053	79	10	the	the	DET
ejpam-7053	79	11	upper	upper	ADJ
ejpam-7053	79	12	and	and	CCONJ
ejpam-7053	79	13	lower	low	ADJ
ejpam-7053	79	14	inverse	inverse	NOUN
ejpam-7053	79	15	of	of	ADP
ejpam-7053	79	16	a	a	DET
ejpam-7053	79	17	set	set	NOUN
ejpam-7053	79	18	b	b	PROPN
ejpam-7053	79	19	of	of	ADP
ejpam-7053	79	20	y	y	PROPN
ejpam-7053	79	21	by	by	ADP
ejpam-7053	79	22	f+(b	f+(b	NOUN
ejpam-7053	79	23	)	)	PUNCT
ejpam-7053	79	24	and	and	CCONJ
ejpam-7053	79	25	f−(b	f−(b	NOUN
ejpam-7053	79	26	)	)	PUNCT
ejpam-7053	79	27	,	,	PUNCT
ejpam-7053	79	28	respectively	respectively	ADV
ejpam-7053	79	29	,	,	PUNCT
ejpam-7053	79	30	that	that	ADV
ejpam-7053	79	31	is	is	ADV
ejpam-7053	79	32	,	,	PUNCT
ejpam-7053	79	33	f+(b	f+(b	NOUN
ejpam-7053	79	34	)	)	PUNCT
ejpam-7053	79	35	=	=	PRON
ejpam-7053	80	1	{	{	PUNCT
ejpam-7053	80	2	x	x	PUNCT
ejpam-7053	80	3	∈	∈	PROPN
ejpam-7053	80	4	x	x	INTJ
ejpam-7053	81	1	|	|	NOUN
ejpam-7053	81	2	f	f	X
ejpam-7053	81	3	(	(	PUNCT
ejpam-7053	81	4	x	x	NOUN
ejpam-7053	81	5	)	)	PUNCT
ejpam-7053	81	6	⊆	⊆	NUM
ejpam-7053	81	7	b	b	NOUN
ejpam-7053	81	8	}	}	PUNCT
ejpam-7053	81	9	and	and	CCONJ
ejpam-7053	81	10	f−(b	f−(b	PROPN
ejpam-7053	81	11	)	)	PUNCT
ejpam-7053	81	12	=	=	PRON
ejpam-7053	82	1	{	{	PUNCT
ejpam-7053	82	2	x	x	PUNCT
ejpam-7053	82	3	∈	∈	PROPN
ejpam-7053	82	4	x	x	INTJ
ejpam-7053	83	1	|	|	NOUN
ejpam-7053	83	2	f	f	X
ejpam-7053	83	3	(	(	PUNCT
ejpam-7053	83	4	x	x	NOUN
ejpam-7053	83	5	)	)	PUNCT
ejpam-7053	83	6	∩	∩	NOUN
ejpam-7053	83	7	b	b	PROPN
ejpam-7053	83	8	̸=	̸=	PROPN
ejpam-7053	83	9	∅	∅	NOUN
ejpam-7053	83	10	}	}	PUNCT
ejpam-7053	83	11	.	.	PUNCT
ejpam-7053	84	1	in	in	ADP
ejpam-7053	84	2	particular	particular	ADJ
ejpam-7053	84	3	,	,	PUNCT
ejpam-7053	84	4	f−(y	f−(y	NOUN
ejpam-7053	84	5	)	)	PUNCT
ejpam-7053	84	6	=	=	SYM
ejpam-7053	85	1	{	{	PUNCT
ejpam-7053	85	2	x	x	PUNCT
ejpam-7053	85	3	∈	∈	PROPN
ejpam-7053	85	4	x	x	INTJ
ejpam-7053	86	1	|	|	ADV
ejpam-7053	86	2	y	y	PROPN
ejpam-7053	86	3	∈	∈	PROPN
ejpam-7053	86	4	f	f	X
ejpam-7053	86	5	(	(	PUNCT
ejpam-7053	86	6	x	x	NOUN
ejpam-7053	86	7	)	)	PUNCT
ejpam-7053	86	8	}	}	PUNCT
ejpam-7053	86	9	for	for	ADP
ejpam-7053	86	10	each	each	DET
ejpam-7053	86	11	point	point	NOUN
ejpam-7053	86	12	y	y	PROPN
ejpam-7053	86	13	∈	∈	PROPN
ejpam-7053	86	14	y	y	PROPN
ejpam-7053	86	15	.	.	PUNCT
ejpam-7053	87	1	for	for	ADP
ejpam-7053	87	2	each	each	DET
ejpam-7053	87	3	a	a	DET
ejpam-7053	87	4	⊆	⊆	NUM
ejpam-7053	87	5	x	x	SYM
ejpam-7053	87	6	,	,	PUNCT
ejpam-7053	87	7	f	f	PROPN
ejpam-7053	87	8	(	(	PUNCT
ejpam-7053	87	9	a	a	NOUN
ejpam-7053	87	10	)	)	PUNCT
ejpam-7053	87	11	=	=	SYM
ejpam-7053	87	12	∪x∈af	∪x∈af	NOUN
ejpam-7053	87	13	(	(	PUNCT
ejpam-7053	87	14	x	x	NOUN
ejpam-7053	87	15	)	)	PUNCT
ejpam-7053	87	16	.	.	PUNCT
ejpam-7053	88	1	3	3	X
ejpam-7053	88	2	.	.	X
ejpam-7053	88	3	upper	upper	ADJ
ejpam-7053	88	4	and	and	CCONJ
ejpam-7053	88	5	lower	low	ADJ
ejpam-7053	88	6	weakly	weakly	ADJ
ejpam-7053	88	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	88	8	,	,	PUNCT
ejpam-7053	88	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	88	10	multifunctions	multifunction	NOUN
ejpam-7053	88	11	in	in	ADP
ejpam-7053	88	12	this	this	DET
ejpam-7053	88	13	section	section	NOUN
ejpam-7053	88	14	,	,	PUNCT
ejpam-7053	88	15	we	we	PRON
ejpam-7053	88	16	introduce	introduce	VERB
ejpam-7053	88	17	the	the	DET
ejpam-7053	88	18	concepts	concept	NOUN
ejpam-7053	88	19	of	of	ADP
ejpam-7053	88	20	upper	upper	ADJ
ejpam-7053	88	21	weakly	weakly	ADJ
ejpam-7053	88	22	µ(σ1	µ(σ1	NOUN
ejpam-7053	88	23	,	,	PUNCT
ejpam-7053	88	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	88	25	multifunctions	multifunction	NOUN
ejpam-7053	88	26	and	and	CCONJ
ejpam-7053	88	27	lower	low	ADJ
ejpam-7053	88	28	weakly	weakly	ADJ
ejpam-7053	88	29	µ(σ1	µ(σ1	NOUN
ejpam-7053	88	30	,	,	PUNCT
ejpam-7053	88	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	88	32	multifunctions	multifunction	NOUN
ejpam-7053	88	33	.	.	PUNCT
ejpam-7053	89	1	furthermore	furthermore	ADV
ejpam-7053	89	2	,	,	PUNCT
ejpam-7053	89	3	several	several	ADJ
ejpam-7053	89	4	m.	m.	NOUN
ejpam-7053	89	5	chiangpradit	chiangpradit	NOUN
ejpam-7053	89	6	,	,	PUNCT
ejpam-7053	89	7	a.	a.	PROPN
ejpam-7053	89	8	sama	sama	PROPN
ejpam-7053	89	9	-	-	PUNCT
ejpam-7053	89	10	ae	ae	PROPN
ejpam-7053	89	11	,	,	PUNCT
ejpam-7053	89	12	c.	c.	PROPN
ejpam-7053	89	13	boonpok	boonpok	PROPN
ejpam-7053	89	14	/	/	SYM
ejpam-7053	89	15	eur	eur	PROPN
ejpam-7053	89	16	.	.	PUNCT
ejpam-7053	90	1	j.	j.	PROPN
ejpam-7053	90	2	pure	pure	PROPN
ejpam-7053	90	3	appl	appl	PROPN
ejpam-7053	90	4	.	.	PROPN
ejpam-7053	90	5	math	math	PROPN
ejpam-7053	90	6	,	,	PUNCT
ejpam-7053	90	7	18	18	NUM
ejpam-7053	90	8	(	(	PUNCT
ejpam-7053	90	9	4	4	NUM
ejpam-7053	90	10	)	)	PUNCT
ejpam-7053	90	11	(	(	PUNCT
ejpam-7053	90	12	2025	2025	NUM
ejpam-7053	90	13	)	)	PUNCT
ejpam-7053	90	14	,	,	PUNCT
ejpam-7053	90	15	7053	7053	NUM
ejpam-7053	90	16	4	4	NUM
ejpam-7053	90	17	of	of	ADP
ejpam-7053	90	18	13	13	NUM
ejpam-7053	90	19	characterizations	characterization	NOUN
ejpam-7053	90	20	of	of	ADP
ejpam-7053	90	21	upper	upper	ADJ
ejpam-7053	90	22	weakly	weakly	ADJ
ejpam-7053	90	23	µ(σ1	µ(σ1	NOUN
ejpam-7053	90	24	,	,	PUNCT
ejpam-7053	90	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	90	26	multifunctions	multifunction	NOUN
ejpam-7053	90	27	and	and	CCONJ
ejpam-7053	90	28	lower	low	ADJ
ejpam-7053	90	29	weakly	weakly	ADJ
ejpam-7053	90	30	µ(σ1	µ(σ1	NOUN
ejpam-7053	90	31	,	,	PUNCT
ejpam-7053	90	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	90	33	multifunctions	multifunction	NOUN
ejpam-7053	90	34	are	be	AUX
ejpam-7053	90	35	discussed	discuss	VERB
ejpam-7053	90	36	.	.	PUNCT
ejpam-7053	91	1	definition	definition	NOUN
ejpam-7053	91	2	1	1	NUM
ejpam-7053	91	3	.	.	PUNCT
ejpam-7053	92	1	a	a	DET
ejpam-7053	92	2	multifunction	multifunction	NOUN
ejpam-7053	92	3	f	f	NOUN
ejpam-7053	92	4	:	:	PUNCT
ejpam-7053	92	5	(	(	PUNCT
ejpam-7053	92	6	x,µ	x,µ	NOUN
ejpam-7053	92	7	)	)	PUNCT
ejpam-7053	92	8	→	→	SYM
ejpam-7053	92	9	(	(	PUNCT
ejpam-7053	92	10	y	y	PROPN
ejpam-7053	92	11	,	,	PUNCT
ejpam-7053	92	12	σ1	σ1	PROPN
ejpam-7053	92	13	,	,	PUNCT
ejpam-7053	92	14	σ2	σ2	PROPN
ejpam-7053	92	15	)	)	PUNCT
ejpam-7053	92	16	is	be	AUX
ejpam-7053	92	17	said	say	VERB
ejpam-7053	92	18	to	to	PART
ejpam-7053	92	19	be	be	AUX
ejpam-7053	92	20	upper	upper	ADJ
ejpam-7053	92	21	weakly	weakly	ADJ
ejpam-7053	92	22	µ(σ1	µ(σ1	NOUN
ejpam-7053	92	23	,	,	PUNCT
ejpam-7053	92	24	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	92	25	at	at	ADP
ejpam-7053	92	26	a	a	DET
ejpam-7053	92	27	point	point	NOUN
ejpam-7053	92	28	x	x	SYM
ejpam-7053	92	29	∈	∈	NOUN
ejpam-7053	92	30	x	x	PUNCT
ejpam-7053	92	31	if	if	SCONJ
ejpam-7053	92	32	for	for	ADP
ejpam-7053	92	33	each	each	DET
ejpam-7053	92	34	σ1σ2	σ1σ2	VERB
ejpam-7053	92	35	-	-	ADJ
ejpam-7053	92	36	open	open	ADJ
ejpam-7053	92	37	set	set	NOUN
ejpam-7053	92	38	v	v	NOUN
ejpam-7053	92	39	of	of	ADP
ejpam-7053	92	40	y	y	PRON
ejpam-7053	92	41	such	such	ADJ
ejpam-7053	92	42	that	that	SCONJ
ejpam-7053	92	43	f	f	PROPN
ejpam-7053	92	44	(	(	PUNCT
ejpam-7053	92	45	x	x	X
ejpam-7053	92	46	)	)	PUNCT
ejpam-7053	92	47	⊆	⊆	NUM
ejpam-7053	92	48	v	v	NOUN
ejpam-7053	92	49	,	,	PUNCT
ejpam-7053	92	50	there	there	PRON
ejpam-7053	92	51	exists	exist	VERB
ejpam-7053	92	52	a	a	DET
ejpam-7053	92	53	µ-open	µ-open	NOUN
ejpam-7053	92	54	set	set	VERB
ejpam-7053	92	55	u	u	NOUN
ejpam-7053	92	56	of	of	ADP
ejpam-7053	92	57	x	x	PUNCT
ejpam-7053	92	58	containing	contain	VERB
ejpam-7053	92	59	x	x	PUNCT
ejpam-7053	92	60	such	such	ADJ
ejpam-7053	92	61	that	that	SCONJ
ejpam-7053	92	62	f	f	PROPN
ejpam-7053	92	63	(	(	PUNCT
ejpam-7053	92	64	u	u	NOUN
ejpam-7053	92	65	)	)	PUNCT
ejpam-7053	92	66	⊆	⊆	NUM
ejpam-7053	92	67	σ1σ2	σ1σ2	NOUN
ejpam-7053	92	68	-	-	NUM
ejpam-7053	92	69	cl(v	cl(v	NOUN
ejpam-7053	92	70	)	)	PUNCT
ejpam-7053	92	71	.	.	PUNCT
ejpam-7053	93	1	a	a	DET
ejpam-7053	93	2	multifunction	multifunction	NOUN
ejpam-7053	93	3	f	f	NOUN
ejpam-7053	93	4	:	:	PUNCT
ejpam-7053	93	5	(	(	PUNCT
ejpam-7053	93	6	x,µ	x,µ	NOUN
ejpam-7053	93	7	)	)	PUNCT
ejpam-7053	93	8	→	→	SYM
ejpam-7053	93	9	(	(	PUNCT
ejpam-7053	93	10	y	y	PROPN
ejpam-7053	93	11	,	,	PUNCT
ejpam-7053	93	12	σ1	σ1	PROPN
ejpam-7053	93	13	,	,	PUNCT
ejpam-7053	93	14	σ2	σ2	PROPN
ejpam-7053	93	15	)	)	PUNCT
ejpam-7053	93	16	is	be	AUX
ejpam-7053	93	17	said	say	VERB
ejpam-7053	93	18	to	to	PART
ejpam-7053	93	19	be	be	AUX
ejpam-7053	93	20	upper	upper	ADJ
ejpam-7053	93	21	weakly	weakly	ADJ
ejpam-7053	93	22	µ(σ1	µ(σ1	NOUN
ejpam-7053	93	23	,	,	PUNCT
ejpam-7053	93	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	93	25	if	if	SCONJ
ejpam-7053	93	26	f	f	PROPN
ejpam-7053	93	27	is	be	AUX
ejpam-7053	93	28	upper	upper	ADJ
ejpam-7053	93	29	weakly	weakly	ADJ
ejpam-7053	93	30	µ(σ1	µ(σ1	NOUN
ejpam-7053	93	31	,	,	PUNCT
ejpam-7053	93	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	93	33	at	at	ADP
ejpam-7053	93	34	each	each	DET
ejpam-7053	93	35	point	point	NOUN
ejpam-7053	93	36	x	x	PUNCT
ejpam-7053	93	37	of	of	ADP
ejpam-7053	93	38	x.	x.	PROPN
ejpam-7053	93	39	theorem	theorem	VERB
ejpam-7053	93	40	1	1	NUM
ejpam-7053	93	41	.	.	X
ejpam-7053	93	42	for	for	ADP
ejpam-7053	93	43	a	a	DET
ejpam-7053	93	44	multifunction	multifunction	NOUN
ejpam-7053	93	45	f	f	NOUN
ejpam-7053	93	46	:	:	PUNCT
ejpam-7053	93	47	(	(	PUNCT
ejpam-7053	93	48	x,µ	x,µ	NOUN
ejpam-7053	93	49	)	)	PUNCT
ejpam-7053	93	50	→	→	SYM
ejpam-7053	93	51	(	(	PUNCT
ejpam-7053	93	52	y	y	PROPN
ejpam-7053	93	53	,	,	PUNCT
ejpam-7053	93	54	σ1	σ1	PROPN
ejpam-7053	93	55	,	,	PUNCT
ejpam-7053	93	56	σ2	σ2	NOUN
ejpam-7053	93	57	)	)	PUNCT
ejpam-7053	93	58	,	,	PUNCT
ejpam-7053	93	59	the	the	DET
ejpam-7053	93	60	following	follow	VERB
ejpam-7053	93	61	properties	property	NOUN
ejpam-7053	93	62	are	be	AUX
ejpam-7053	93	63	equivalent	equivalent	ADJ
ejpam-7053	93	64	:	:	PUNCT
ejpam-7053	93	65	(	(	PUNCT
ejpam-7053	93	66	1	1	X
ejpam-7053	93	67	)	)	PUNCT
ejpam-7053	93	68	f	f	PROPN
ejpam-7053	93	69	is	be	AUX
ejpam-7053	93	70	upper	upper	ADJ
ejpam-7053	93	71	weakly	weakly	ADJ
ejpam-7053	93	72	µ(σ1	µ(σ1	NOUN
ejpam-7053	93	73	,	,	PUNCT
ejpam-7053	93	74	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	93	75	;	;	PUNCT
ejpam-7053	93	76	(	(	PUNCT
ejpam-7053	93	77	2	2	NUM
ejpam-7053	93	78	)	)	PUNCT
ejpam-7053	93	79	f+(v	f+(v	NOUN
ejpam-7053	93	80	)	)	PUNCT
ejpam-7053	94	1	⊆	⊆	NUM
ejpam-7053	94	2	iµ(f	iµ(f	NUM
ejpam-7053	94	3	+	+	NOUN
ejpam-7053	94	4	(	(	PUNCT
ejpam-7053	94	5	σ1σ2	σ1σ2	NOUN
ejpam-7053	94	6	-	-	NUM
ejpam-7053	94	7	cl(v	cl(v	NOUN
ejpam-7053	94	8	)	)	PUNCT
ejpam-7053	94	9	)	)	PUNCT
ejpam-7053	94	10	)	)	PUNCT
ejpam-7053	95	1	for	for	ADP
ejpam-7053	95	2	every	every	DET
ejpam-7053	95	3	σ1σ2	σ1σ2	NOUN
ejpam-7053	95	4	-	-	ADJ
ejpam-7053	95	5	open	open	ADJ
ejpam-7053	95	6	set	set	NOUN
ejpam-7053	95	7	v	v	NOUN
ejpam-7053	95	8	of	of	ADP
ejpam-7053	95	9	y	y	PROPN
ejpam-7053	95	10	;	;	PUNCT
ejpam-7053	95	11	(	(	PUNCT
ejpam-7053	95	12	3	3	X
ejpam-7053	95	13	)	)	PUNCT
ejpam-7053	95	14	cµ(f	cµ(f	NOUN
ejpam-7053	95	15	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	95	16	-	-	PUNCT
ejpam-7053	95	17	int(k	int(k	NUM
ejpam-7053	95	18	)	)	PUNCT
ejpam-7053	95	19	)	)	PUNCT
ejpam-7053	95	20	)	)	PUNCT
ejpam-7053	96	1	⊆	⊆	X
ejpam-7053	96	2	f−(k	f−(k	PROPN
ejpam-7053	96	3	)	)	PUNCT
ejpam-7053	96	4	for	for	ADP
ejpam-7053	96	5	every	every	DET
ejpam-7053	96	6	σ1σ2	σ1σ2	NUM
ejpam-7053	96	7	-	-	PUNCT
ejpam-7053	96	8	closed	closed	ADJ
ejpam-7053	96	9	set	set	NOUN
ejpam-7053	96	10	k	k	PROPN
ejpam-7053	96	11	of	of	ADP
ejpam-7053	96	12	y	y	PROPN
ejpam-7053	96	13	;	;	PUNCT
ejpam-7053	96	14	(	(	PUNCT
ejpam-7053	96	15	4	4	X
ejpam-7053	96	16	)	)	PUNCT
ejpam-7053	96	17	cµ(f	cµ(f	NOUN
ejpam-7053	96	18	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	96	19	-	-	PUNCT
ejpam-7053	96	20	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	96	21	-	-	PUNCT
ejpam-7053	96	22	cl(b	cl(b	NOUN
ejpam-7053	96	23	)	)	PUNCT
ejpam-7053	96	24	)	)	PUNCT
ejpam-7053	96	25	)	)	PUNCT
ejpam-7053	96	26	)	)	PUNCT
ejpam-7053	97	1	⊆	⊆	X
ejpam-7053	97	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7053	97	3	-	-	PUNCT
ejpam-7053	97	4	cl(b	cl(b	NOUN
ejpam-7053	97	5	)	)	PUNCT
ejpam-7053	97	6	)	)	PUNCT
ejpam-7053	98	1	for	for	ADP
ejpam-7053	98	2	every	every	DET
ejpam-7053	98	3	subset	subset	NOUN
ejpam-7053	98	4	b	b	PROPN
ejpam-7053	98	5	of	of	ADP
ejpam-7053	98	6	y	y	PROPN
ejpam-7053	98	7	;	;	PUNCT
ejpam-7053	98	8	(	(	PUNCT
ejpam-7053	98	9	5	5	X
ejpam-7053	98	10	)	)	PUNCT
ejpam-7053	98	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	98	12	-	-	PUNCT
ejpam-7053	98	13	int(b	int(b	NOUN
ejpam-7053	98	14	)	)	PUNCT
ejpam-7053	98	15	)	)	PUNCT
ejpam-7053	99	1	⊆	⊆	NUM
ejpam-7053	99	2	iµ(f	iµ(f	NUM
ejpam-7053	99	3	+	+	NOUN
ejpam-7053	99	4	(	(	PUNCT
ejpam-7053	99	5	σ1σ2	σ1σ2	NUM
ejpam-7053	99	6	-	-	PUNCT
ejpam-7053	99	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	99	8	-	-	PUNCT
ejpam-7053	99	9	int(b	int(b	NOUN
ejpam-7053	99	10	)	)	PUNCT
ejpam-7053	99	11	)	)	PUNCT
ejpam-7053	99	12	)	)	PUNCT
ejpam-7053	99	13	)	)	PUNCT
ejpam-7053	100	1	for	for	ADP
ejpam-7053	100	2	every	every	DET
ejpam-7053	100	3	subset	subset	NOUN
ejpam-7053	100	4	b	b	PROPN
ejpam-7053	100	5	of	of	ADP
ejpam-7053	100	6	y	y	PROPN
ejpam-7053	100	7	;	;	PUNCT
ejpam-7053	100	8	(	(	PUNCT
ejpam-7053	100	9	6	6	NUM
ejpam-7053	100	10	)	)	PUNCT
ejpam-7053	100	11	cµ(f	cµ(f	NOUN
ejpam-7053	100	12	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	100	13	-	-	PUNCT
ejpam-7053	100	14	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	100	15	-	-	PUNCT
ejpam-7053	100	16	cl(v	cl(v	NOUN
ejpam-7053	100	17	)	)	PUNCT
ejpam-7053	100	18	)	)	PUNCT
ejpam-7053	100	19	)	)	PUNCT
ejpam-7053	100	20	)	)	PUNCT
ejpam-7053	101	1	⊆	⊆	X
ejpam-7053	101	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	101	3	-	-	PUNCT
ejpam-7053	101	4	cl(v	cl(v	NOUN
ejpam-7053	101	5	)	)	PUNCT
ejpam-7053	101	6	)	)	PUNCT
ejpam-7053	101	7	for	for	ADP
ejpam-7053	101	8	every	every	DET
ejpam-7053	101	9	σ1σ2	σ1σ2	NOUN
ejpam-7053	101	10	-	-	ADJ
ejpam-7053	101	11	open	open	ADJ
ejpam-7053	101	12	set	set	NOUN
ejpam-7053	101	13	v	v	NOUN
ejpam-7053	101	14	of	of	ADP
ejpam-7053	101	15	y	y	PROPN
ejpam-7053	101	16	;	;	PUNCT
ejpam-7053	101	17	(	(	PUNCT
ejpam-7053	101	18	7	7	X
ejpam-7053	101	19	)	)	PUNCT
ejpam-7053	101	20	cµ(f	cµ(f	NOUN
ejpam-7053	101	21	−(v	−(v	NOUN
ejpam-7053	101	22	)	)	PUNCT
ejpam-7053	101	23	)	)	PUNCT
ejpam-7053	102	1	⊆	⊆	X
ejpam-7053	102	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	102	3	-	-	PUNCT
ejpam-7053	102	4	cl(v	cl(v	NOUN
ejpam-7053	102	5	)	)	PUNCT
ejpam-7053	102	6	)	)	PUNCT
ejpam-7053	102	7	for	for	ADP
ejpam-7053	102	8	every	every	DET
ejpam-7053	102	9	σ1σ2	σ1σ2	NOUN
ejpam-7053	102	10	-	-	ADJ
ejpam-7053	102	11	open	open	ADJ
ejpam-7053	102	12	set	set	NOUN
ejpam-7053	102	13	v	v	NOUN
ejpam-7053	102	14	of	of	ADP
ejpam-7053	102	15	y	y	PROPN
ejpam-7053	102	16	;	;	PUNCT
ejpam-7053	102	17	(	(	PUNCT
ejpam-7053	102	18	8)	8)	NUM
ejpam-7053	102	19	cµ(f	cµ(f	NOUN
ejpam-7053	102	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	102	21	-	-	PUNCT
ejpam-7053	102	22	int(k	int(k	NUM
ejpam-7053	102	23	)	)	PUNCT
ejpam-7053	102	24	)	)	PUNCT
ejpam-7053	102	25	)	)	PUNCT
ejpam-7053	103	1	⊆	⊆	X
ejpam-7053	103	2	f−(k	f−(k	PROPN
ejpam-7053	103	3	)	)	PUNCT
ejpam-7053	103	4	for	for	ADP
ejpam-7053	103	5	every	every	DET
ejpam-7053	103	6	(	(	PUNCT
ejpam-7053	103	7	σ1	σ1	PROPN
ejpam-7053	103	8	,	,	PUNCT
ejpam-7053	103	9	σ2)r	σ2)r	NOUN
ejpam-7053	103	10	-	-	PUNCT
ejpam-7053	103	11	closed	close	VERB
ejpam-7053	103	12	set	set	ADJ
ejpam-7053	103	13	k	k	PROPN
ejpam-7053	103	14	of	of	ADP
ejpam-7053	103	15	y	y	PROPN
ejpam-7053	103	16	.	.	PUNCT
ejpam-7053	104	1	proof	proof	NOUN
ejpam-7053	104	2	.	.	PUNCT
ejpam-7053	105	1	(	(	PUNCT
ejpam-7053	105	2	1	1	X
ejpam-7053	105	3	)	)	PUNCT
ejpam-7053	105	4	⇒	⇒	NOUN
ejpam-7053	105	5	(	(	PUNCT
ejpam-7053	105	6	2	2	NUM
ejpam-7053	105	7	):	):	PUNCT
ejpam-7053	105	8	let	let	VERB
ejpam-7053	105	9	v	v	PART
ejpam-7053	105	10	be	be	AUX
ejpam-7053	105	11	any	any	DET
ejpam-7053	105	12	σ1σ2	σ1σ2	NOUN
ejpam-7053	105	13	-	-	ADJ
ejpam-7053	105	14	open	open	ADJ
ejpam-7053	105	15	set	set	NOUN
ejpam-7053	105	16	of	of	ADP
ejpam-7053	105	17	y	y	PRON
ejpam-7053	105	18	such	such	ADJ
ejpam-7053	105	19	that	that	SCONJ
ejpam-7053	105	20	x	x	SYM
ejpam-7053	105	21	∈	∈	PROPN
ejpam-7053	105	22	f+(v	f+(v	NOUN
ejpam-7053	105	23	)	)	PUNCT
ejpam-7053	105	24	.	.	PUNCT
ejpam-7053	106	1	then	then	ADV
ejpam-7053	106	2	,	,	PUNCT
ejpam-7053	106	3	we	we	PRON
ejpam-7053	106	4	have	have	VERB
ejpam-7053	106	5	f	f	PROPN
ejpam-7053	106	6	(	(	PUNCT
ejpam-7053	106	7	x	x	NOUN
ejpam-7053	106	8	)	)	PUNCT
ejpam-7053	106	9	⊆	⊆	NUM
ejpam-7053	106	10	v	v	NOUN
ejpam-7053	106	11	and	and	CCONJ
ejpam-7053	106	12	by	by	ADP
ejpam-7053	106	13	(	(	PUNCT
ejpam-7053	106	14	1	1	NUM
ejpam-7053	106	15	)	)	PUNCT
ejpam-7053	106	16	,	,	PUNCT
ejpam-7053	106	17	there	there	PRON
ejpam-7053	106	18	exists	exist	VERB
ejpam-7053	106	19	a	a	DET
ejpam-7053	106	20	µ-open	µ-open	NOUN
ejpam-7053	106	21	set	set	VERB
ejpam-7053	106	22	u	u	NOUN
ejpam-7053	106	23	of	of	ADP
ejpam-7053	106	24	x	x	PUNCT
ejpam-7053	106	25	containing	contain	VERB
ejpam-7053	106	26	x	x	PUNCT
ejpam-7053	106	27	such	such	ADJ
ejpam-7053	106	28	that	that	SCONJ
ejpam-7053	106	29	f	f	PROPN
ejpam-7053	106	30	(	(	PUNCT
ejpam-7053	106	31	u	u	NOUN
ejpam-7053	106	32	)	)	PUNCT
ejpam-7053	106	33	⊆	⊆	NUM
ejpam-7053	106	34	σ1σ2	σ1σ2	NOUN
ejpam-7053	106	35	-	-	NUM
ejpam-7053	106	36	cl(v	cl(v	NOUN
ejpam-7053	106	37	)	)	PUNCT
ejpam-7053	106	38	.	.	PUNCT
ejpam-7053	107	1	thus	thus	ADV
ejpam-7053	107	2	,	,	PUNCT
ejpam-7053	107	3	u	u	NOUN
ejpam-7053	107	4	⊆	⊆	NUM
ejpam-7053	107	5	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	107	6	-	-	PUNCT
ejpam-7053	107	7	cl(v	cl(v	NOUN
ejpam-7053	107	8	)	)	PUNCT
ejpam-7053	107	9	)	)	PUNCT
ejpam-7053	107	10	.	.	PUNCT
ejpam-7053	108	1	since	since	SCONJ
ejpam-7053	108	2	u	u	NOUN
ejpam-7053	108	3	is	be	AUX
ejpam-7053	108	4	µ-open	µ-open	NOUN
ejpam-7053	108	5	,	,	PUNCT
ejpam-7053	108	6	we	we	PRON
ejpam-7053	108	7	have	have	VERB
ejpam-7053	108	8	x	x	PART
ejpam-7053	108	9	∈	∈	NOUN
ejpam-7053	108	10	iµ(f	iµ(f	NUM
ejpam-7053	108	11	+	+	NOUN
ejpam-7053	108	12	(	(	PUNCT
ejpam-7053	108	13	σ1σ2	σ1σ2	NOUN
ejpam-7053	108	14	-	-	NUM
ejpam-7053	108	15	cl(v	cl(v	NOUN
ejpam-7053	108	16	)	)	PUNCT
ejpam-7053	108	17	)	)	PUNCT
ejpam-7053	108	18	)	)	PUNCT
ejpam-7053	108	19	and	and	CCONJ
ejpam-7053	108	20	so	so	ADV
ejpam-7053	108	21	f+(v	f+(v	PROPN
ejpam-7053	108	22	)	)	PUNCT
ejpam-7053	109	1	⊆	⊆	NUM
ejpam-7053	109	2	iµ(f	iµ(f	NUM
ejpam-7053	109	3	+	+	NOUN
ejpam-7053	109	4	(	(	PUNCT
ejpam-7053	109	5	σ1σ2	σ1σ2	NOUN
ejpam-7053	109	6	-	-	NUM
ejpam-7053	109	7	cl(v	cl(v	NOUN
ejpam-7053	109	8	)	)	PUNCT
ejpam-7053	109	9	)	)	PUNCT
ejpam-7053	109	10	)	)	PUNCT
ejpam-7053	109	11	.	.	PUNCT
ejpam-7053	110	1	(	(	PUNCT
ejpam-7053	110	2	2	2	X
ejpam-7053	110	3	)	)	PUNCT
ejpam-7053	110	4	⇒	⇒	NOUN
ejpam-7053	110	5	(	(	PUNCT
ejpam-7053	110	6	3	3	NUM
ejpam-7053	110	7	):	):	PUNCT
ejpam-7053	110	8	let	let	VERB
ejpam-7053	110	9	k	k	PRON
ejpam-7053	110	10	be	be	AUX
ejpam-7053	110	11	any	any	DET
ejpam-7053	110	12	σ1σ2	σ1σ2	NUM
ejpam-7053	110	13	-	-	PUNCT
ejpam-7053	110	14	closed	closed	ADJ
ejpam-7053	110	15	set	set	NOUN
ejpam-7053	110	16	of	of	ADP
ejpam-7053	110	17	y	y	PROPN
ejpam-7053	110	18	.	.	PUNCT
ejpam-7053	111	1	then	then	ADV
ejpam-7053	111	2	,	,	PUNCT
ejpam-7053	111	3	y	y	PROPN
ejpam-7053	111	4	−k	−k	PROPN
ejpam-7053	111	5	is	be	AUX
ejpam-7053	111	6	σ1σ2	σ1σ2	NOUN
ejpam-7053	111	7	-	-	ADJ
ejpam-7053	111	8	open	open	ADJ
ejpam-7053	111	9	in	in	ADP
ejpam-7053	111	10	y	y	PROPN
ejpam-7053	111	11	.	.	PUNCT
ejpam-7053	112	1	by	by	ADP
ejpam-7053	112	2	(	(	PUNCT
ejpam-7053	112	3	2	2	NUM
ejpam-7053	112	4	)	)	PUNCT
ejpam-7053	112	5	,	,	PUNCT
ejpam-7053	112	6	x	x	PUNCT
ejpam-7053	112	7	−	−	DET
ejpam-7053	112	8	f−(k	f−(k	PROPN
ejpam-7053	112	9	)	)	PUNCT
ejpam-7053	112	10	=	=	PUNCT
ejpam-7053	112	11	f+(y	f+(y	PROPN
ejpam-7053	112	12	−k	−k	PROPN
ejpam-7053	112	13	)	)	PUNCT
ejpam-7053	112	14	⊆	⊆	NUM
ejpam-7053	112	15	iµ(f	iµ(f	NUM
ejpam-7053	112	16	+	+	NOUN
ejpam-7053	112	17	(	(	PUNCT
ejpam-7053	112	18	σ1σ2	σ1σ2	NUM
ejpam-7053	112	19	-	-	PUNCT
ejpam-7053	112	20	cl(y	cl(y	NOUN
ejpam-7053	112	21	−k	−k	NOUN
ejpam-7053	112	22	)	)	PUNCT
ejpam-7053	112	23	)	)	PUNCT
ejpam-7053	112	24	)	)	PUNCT
ejpam-7053	113	1	=	=	PUNCT
ejpam-7053	114	1	x	x	PUNCT
ejpam-7053	114	2	−	−	PROPN
ejpam-7053	114	3	cµ(f	cµ(f	X
ejpam-7053	114	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	114	5	-	-	PUNCT
ejpam-7053	114	6	int(k	int(k	NUM
ejpam-7053	114	7	)	)	PUNCT
ejpam-7053	114	8	)	)	PUNCT
ejpam-7053	114	9	)	)	PUNCT
ejpam-7053	114	10	.	.	PUNCT
ejpam-7053	115	1	thus	thus	ADV
ejpam-7053	115	2	,	,	PUNCT
ejpam-7053	115	3	cµ(f−(σ1σ2	cµ(f−(σ1σ2	NUM
ejpam-7053	115	4	-	-	SYM
ejpam-7053	115	5	int(k	int(k	NUM
ejpam-7053	115	6	)	)	PUNCT
ejpam-7053	115	7	)	)	PUNCT
ejpam-7053	115	8	)	)	PUNCT
ejpam-7053	115	9	⊆	⊆	NUM
ejpam-7053	115	10	f−(k	f−(k	PROPN
ejpam-7053	115	11	)	)	PUNCT
ejpam-7053	115	12	.	.	PUNCT
ejpam-7053	116	1	(	(	PUNCT
ejpam-7053	116	2	3	3	X
ejpam-7053	116	3	)	)	PUNCT
ejpam-7053	116	4	⇒	⇒	NOUN
ejpam-7053	116	5	(	(	PUNCT
ejpam-7053	116	6	4	4	NUM
ejpam-7053	116	7	):	):	PUNCT
ejpam-7053	116	8	let	let	VERB
ejpam-7053	116	9	b	b	X
ejpam-7053	116	10	be	be	AUX
ejpam-7053	116	11	any	any	DET
ejpam-7053	116	12	subset	subset	NOUN
ejpam-7053	116	13	of	of	ADP
ejpam-7053	116	14	y	y	PROPN
ejpam-7053	116	15	.	.	PUNCT
ejpam-7053	117	1	then	then	ADV
ejpam-7053	117	2	,	,	PUNCT
ejpam-7053	117	3	σ1σ2	σ1σ2	NOUN
ejpam-7053	117	4	-	-	NOUN
ejpam-7053	117	5	cl(b	cl(b	NOUN
ejpam-7053	117	6	)	)	PUNCT
ejpam-7053	117	7	is	be	AUX
ejpam-7053	117	8	a	a	DET
ejpam-7053	117	9	σ1σ2	σ1σ2	NUM
ejpam-7053	117	10	-	-	PUNCT
ejpam-7053	117	11	closed	closed	ADJ
ejpam-7053	117	12	set	set	NOUN
ejpam-7053	117	13	of	of	ADP
ejpam-7053	117	14	y	y	PROPN
ejpam-7053	117	15	and	and	CCONJ
ejpam-7053	117	16	by	by	ADP
ejpam-7053	117	17	(	(	PUNCT
ejpam-7053	117	18	3	3	NUM
ejpam-7053	117	19	)	)	PUNCT
ejpam-7053	117	20	,	,	PUNCT
ejpam-7053	117	21	cµ(f−(σ1σ2	cµ(f−(σ1σ2	NUM
ejpam-7053	117	22	-	-	PUNCT
ejpam-7053	117	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	117	24	-	-	PUNCT
ejpam-7053	117	25	cl(b	cl(b	NOUN
ejpam-7053	117	26	)	)	PUNCT
ejpam-7053	117	27	)	)	PUNCT
ejpam-7053	117	28	)	)	PUNCT
ejpam-7053	117	29	)	)	PUNCT
ejpam-7053	118	1	⊆	⊆	X
ejpam-7053	118	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7053	118	3	-	-	PUNCT
ejpam-7053	118	4	cl(b	cl(b	NOUN
ejpam-7053	118	5	)	)	PUNCT
ejpam-7053	118	6	)	)	PUNCT
ejpam-7053	118	7	.	.	PUNCT
ejpam-7053	119	1	(	(	PUNCT
ejpam-7053	119	2	4	4	X
ejpam-7053	119	3	)	)	PUNCT
ejpam-7053	119	4	⇒	⇒	NOUN
ejpam-7053	119	5	(	(	PUNCT
ejpam-7053	119	6	5	5	NUM
ejpam-7053	119	7	):	):	PUNCT
ejpam-7053	119	8	let	let	VERB
ejpam-7053	119	9	b	b	X
ejpam-7053	119	10	be	be	AUX
ejpam-7053	119	11	any	any	DET
ejpam-7053	119	12	subset	subset	NOUN
ejpam-7053	119	13	of	of	ADP
ejpam-7053	119	14	y	y	PROPN
ejpam-7053	119	15	.	.	PUNCT
ejpam-7053	120	1	thus	thus	ADV
ejpam-7053	120	2	by	by	ADP
ejpam-7053	120	3	(	(	PUNCT
ejpam-7053	120	4	4	4	NUM
ejpam-7053	120	5	)	)	PUNCT
ejpam-7053	120	6	,	,	PUNCT
ejpam-7053	120	7	we	we	PRON
ejpam-7053	120	8	have	have	VERB
ejpam-7053	120	9	x	x	PART
ejpam-7053	120	10	−	−	NOUN
ejpam-7053	120	11	iµ(f	iµ(f	NOUN
ejpam-7053	120	12	+	+	NOUN
ejpam-7053	120	13	(	(	PUNCT
ejpam-7053	120	14	σ1σ2	σ1σ2	NUM
ejpam-7053	120	15	-	-	PUNCT
ejpam-7053	120	16	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	120	17	-	-	PUNCT
ejpam-7053	120	18	int(b	int(b	NOUN
ejpam-7053	120	19	)	)	PUNCT
ejpam-7053	120	20	)	)	PUNCT
ejpam-7053	120	21	)	)	PUNCT
ejpam-7053	120	22	)	)	PUNCT
ejpam-7053	121	1	=	=	PRON
ejpam-7053	121	2	cµ(x	cµ(x	PUNCT
ejpam-7053	121	3	−	−	ADP
ejpam-7053	121	4	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-7053	121	5	-	-	PUNCT
ejpam-7053	121	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	121	7	-	-	PUNCT
ejpam-7053	121	8	int(b	int(b	NOUN
ejpam-7053	121	9	)	)	PUNCT
ejpam-7053	121	10	)	)	PUNCT
ejpam-7053	121	11	)	)	PUNCT
ejpam-7053	121	12	)	)	PUNCT
ejpam-7053	122	1	=	=	SYM
ejpam-7053	122	2	cµ(f	cµ(f	NOUN
ejpam-7053	122	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	122	4	-	-	PUNCT
ejpam-7053	122	5	int(σ1σ2	int(σ1σ2	VERB
ejpam-7053	122	6	-	-	PUNCT
ejpam-7053	122	7	cl(y	cl(y	NOUN
ejpam-7053	122	8	−b	−b	NOUN
ejpam-7053	122	9	)	)	PUNCT
ejpam-7053	122	10	)	)	PUNCT
ejpam-7053	122	11	)	)	PUNCT
ejpam-7053	122	12	)	)	PUNCT
ejpam-7053	123	1	⊆	⊆	X
ejpam-7053	123	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7053	123	3	-	-	PUNCT
ejpam-7053	123	4	cl(y	cl(y	NOUN
ejpam-7053	123	5	−b	−b	NOUN
ejpam-7053	123	6	)	)	PUNCT
ejpam-7053	123	7	)	)	PUNCT
ejpam-7053	124	1	=	=	PUNCT
ejpam-7053	124	2	x	x	X
ejpam-7053	125	1	−	−	ADP
ejpam-7053	125	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	125	3	-	-	PUNCT
ejpam-7053	125	4	int(b	int(b	NOUN
ejpam-7053	125	5	)	)	PUNCT
ejpam-7053	125	6	)	)	PUNCT
ejpam-7053	125	7	and	and	CCONJ
ejpam-7053	125	8	hence	hence	ADV
ejpam-7053	125	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-7053	125	10	-	-	PUNCT
ejpam-7053	125	11	int(b	int(b	NOUN
ejpam-7053	125	12	)	)	PUNCT
ejpam-7053	125	13	)	)	PUNCT
ejpam-7053	126	1	⊆	⊆	NUM
ejpam-7053	126	2	iµ(f	iµ(f	NUM
ejpam-7053	126	3	+	+	NOUN
ejpam-7053	126	4	(	(	PUNCT
ejpam-7053	126	5	σ1σ2	σ1σ2	NUM
ejpam-7053	126	6	-	-	PUNCT
ejpam-7053	126	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	126	8	-	-	PUNCT
ejpam-7053	126	9	int(b	int(b	NOUN
ejpam-7053	126	10	)	)	PUNCT
ejpam-7053	126	11	)	)	PUNCT
ejpam-7053	126	12	)	)	PUNCT
ejpam-7053	126	13	)	)	PUNCT
ejpam-7053	126	14	.	.	PUNCT
ejpam-7053	127	1	(	(	PUNCT
ejpam-7053	127	2	5	5	X
ejpam-7053	127	3	)	)	PUNCT
ejpam-7053	127	4	⇒	⇒	NOUN
ejpam-7053	127	5	(	(	PUNCT
ejpam-7053	127	6	1	1	NUM
ejpam-7053	127	7	):	):	PUNCT
ejpam-7053	127	8	let	let	VERB
ejpam-7053	127	9	x	x	PUNCT
ejpam-7053	127	10	∈	∈	PROPN
ejpam-7053	127	11	x	x	X
ejpam-7053	127	12	and	and	CCONJ
ejpam-7053	127	13	v	v	X
ejpam-7053	127	14	be	be	AUX
ejpam-7053	127	15	any	any	DET
ejpam-7053	127	16	σ1σ2	σ1σ2	NOUN
ejpam-7053	127	17	-	-	ADJ
ejpam-7053	127	18	open	open	ADJ
ejpam-7053	127	19	set	set	NOUN
ejpam-7053	127	20	of	of	ADP
ejpam-7053	127	21	y	y	PRON
ejpam-7053	127	22	such	such	ADJ
ejpam-7053	127	23	that	that	SCONJ
ejpam-7053	127	24	f	f	PROPN
ejpam-7053	127	25	(	(	PUNCT
ejpam-7053	127	26	x	x	X
ejpam-7053	127	27	)	)	PUNCT
ejpam-7053	127	28	⊆	⊆	NUM
ejpam-7053	127	29	v	v	NOUN
ejpam-7053	127	30	.	.	PUNCT
ejpam-7053	128	1	by	by	ADP
ejpam-7053	128	2	(	(	PUNCT
ejpam-7053	128	3	5	5	NUM
ejpam-7053	128	4	)	)	PUNCT
ejpam-7053	128	5	,	,	PUNCT
ejpam-7053	128	6	x	x	PUNCT
ejpam-7053	128	7	∈	∈	PROPN
ejpam-7053	128	8	f+(v	f+(v	NOUN
ejpam-7053	128	9	)	)	PUNCT
ejpam-7053	128	10	⊆	⊆	NUM
ejpam-7053	128	11	iµ(f	iµ(f	NUM
ejpam-7053	128	12	+	+	NOUN
ejpam-7053	128	13	(	(	PUNCT
ejpam-7053	128	14	σ1σ2	σ1σ2	NOUN
ejpam-7053	128	15	-	-	NUM
ejpam-7053	128	16	cl(v	cl(v	NOUN
ejpam-7053	128	17	)	)	PUNCT
ejpam-7053	128	18	)	)	PUNCT
ejpam-7053	128	19	)	)	PUNCT
ejpam-7053	129	1	and	and	CCONJ
ejpam-7053	129	2	there	there	PRON
ejpam-7053	129	3	exists	exist	VERB
ejpam-7053	129	4	a	a	DET
ejpam-7053	129	5	µ-open	µ-open	NOUN
ejpam-7053	129	6	set	set	VERB
ejpam-7053	129	7	u	u	NOUN
ejpam-7053	129	8	of	of	ADP
ejpam-7053	129	9	x	x	PUNCT
ejpam-7053	129	10	containing	contain	VERB
ejpam-7053	129	11	x	x	PROPN
ejpam-7053	129	12	m.	m.	NOUN
ejpam-7053	129	13	chiangpradit	chiangpradit	NOUN
ejpam-7053	129	14	,	,	PUNCT
ejpam-7053	129	15	a.	a.	PROPN
ejpam-7053	129	16	sama	sama	PROPN
ejpam-7053	129	17	-	-	PUNCT
ejpam-7053	129	18	ae	ae	PROPN
ejpam-7053	129	19	,	,	PUNCT
ejpam-7053	129	20	c.	c.	PROPN
ejpam-7053	129	21	boonpok	boonpok	PROPN
ejpam-7053	129	22	/	/	SYM
ejpam-7053	129	23	eur	eur	PROPN
ejpam-7053	129	24	.	.	PUNCT
ejpam-7053	130	1	j.	j.	PROPN
ejpam-7053	130	2	pure	pure	PROPN
ejpam-7053	130	3	appl	appl	PROPN
ejpam-7053	130	4	.	.	PROPN
ejpam-7053	130	5	math	math	PROPN
ejpam-7053	130	6	,	,	PUNCT
ejpam-7053	130	7	18	18	NUM
ejpam-7053	130	8	(	(	PUNCT
ejpam-7053	130	9	4	4	NUM
ejpam-7053	130	10	)	)	PUNCT
ejpam-7053	130	11	(	(	PUNCT
ejpam-7053	130	12	2025	2025	NUM
ejpam-7053	130	13	)	)	PUNCT
ejpam-7053	130	14	,	,	PUNCT
ejpam-7053	130	15	7053	7053	NUM
ejpam-7053	130	16	5	5	NUM
ejpam-7053	130	17	of	of	ADP
ejpam-7053	130	18	13	13	NUM
ejpam-7053	130	19	such	such	ADJ
ejpam-7053	130	20	that	that	SCONJ
ejpam-7053	130	21	u	u	NOUN
ejpam-7053	130	22	⊆	⊆	NUM
ejpam-7053	130	23	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	130	24	-	-	PUNCT
ejpam-7053	130	25	cl(v	cl(v	NOUN
ejpam-7053	130	26	)	)	PUNCT
ejpam-7053	130	27	)	)	PUNCT
ejpam-7053	130	28	.	.	PUNCT
ejpam-7053	131	1	thus	thus	ADV
ejpam-7053	131	2	,	,	PUNCT
ejpam-7053	131	3	f	f	PROPN
ejpam-7053	131	4	(	(	PUNCT
ejpam-7053	131	5	u	u	NOUN
ejpam-7053	131	6	)	)	PUNCT
ejpam-7053	131	7	⊆	⊆	NUM
ejpam-7053	131	8	σ1σ2	σ1σ2	NOUN
ejpam-7053	131	9	-	-	NUM
ejpam-7053	131	10	cl(v	cl(v	NOUN
ejpam-7053	131	11	)	)	PUNCT
ejpam-7053	131	12	and	and	CCONJ
ejpam-7053	131	13	hence	hence	ADV
ejpam-7053	131	14	f	f	PROPN
ejpam-7053	131	15	is	be	AUX
ejpam-7053	131	16	upper	upper	ADJ
ejpam-7053	131	17	weakly	weakly	ADJ
ejpam-7053	131	18	µ(σ1	µ(σ1	NOUN
ejpam-7053	131	19	,	,	PUNCT
ejpam-7053	131	20	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	131	21	.	.	PUNCT
ejpam-7053	132	1	(	(	PUNCT
ejpam-7053	132	2	4	4	X
ejpam-7053	132	3	)	)	PUNCT
ejpam-7053	132	4	⇒	⇒	NOUN
ejpam-7053	132	5	(	(	PUNCT
ejpam-7053	132	6	6	6	NUM
ejpam-7053	132	7	)	)	PUNCT
ejpam-7053	132	8	and	and	CCONJ
ejpam-7053	132	9	(	(	PUNCT
ejpam-7053	132	10	6	6	NUM
ejpam-7053	132	11	)	)	PUNCT
ejpam-7053	132	12	⇒	⇒	NOUN
ejpam-7053	132	13	(	(	PUNCT
ejpam-7053	132	14	7	7	NUM
ejpam-7053	132	15	):	):	PUNCT
ejpam-7053	132	16	the	the	DET
ejpam-7053	132	17	proofs	proof	NOUN
ejpam-7053	132	18	are	be	AUX
ejpam-7053	132	19	obvious	obvious	ADJ
ejpam-7053	132	20	.	.	PUNCT
ejpam-7053	133	1	(	(	PUNCT
ejpam-7053	133	2	7	7	X
ejpam-7053	133	3	)	)	PUNCT
ejpam-7053	133	4	⇒	⇒	NOUN
ejpam-7053	133	5	(	(	PUNCT
ejpam-7053	133	6	8)	8)	NUM
ejpam-7053	133	7	:	:	PUNCT
ejpam-7053	133	8	let	let	VERB
ejpam-7053	133	9	k	k	X
ejpam-7053	133	10	be	be	AUX
ejpam-7053	133	11	any	any	DET
ejpam-7053	133	12	(	(	PUNCT
ejpam-7053	133	13	σ1	σ1	NOUN
ejpam-7053	133	14	,	,	PUNCT
ejpam-7053	133	15	σ2)r	σ2)r	NOUN
ejpam-7053	133	16	-	-	PUNCT
ejpam-7053	133	17	closed	close	VERB
ejpam-7053	133	18	set	set	NOUN
ejpam-7053	133	19	of	of	ADP
ejpam-7053	133	20	y	y	PROPN
ejpam-7053	133	21	.	.	PUNCT
ejpam-7053	134	1	thus	thus	ADV
ejpam-7053	134	2	by	by	ADP
ejpam-7053	134	3	(	(	PUNCT
ejpam-7053	134	4	7	7	NUM
ejpam-7053	134	5	)	)	PUNCT
ejpam-7053	134	6	,	,	PUNCT
ejpam-7053	134	7	cµ(f	cµ(f	X
ejpam-7053	134	8	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	134	9	-	-	PUNCT
ejpam-7053	134	10	int(k	int(k	NUM
ejpam-7053	134	11	)	)	PUNCT
ejpam-7053	134	12	)	)	PUNCT
ejpam-7053	134	13	)	)	PUNCT
ejpam-7053	135	1	⊆	⊆	X
ejpam-7053	135	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7053	135	3	-	-	PUNCT
ejpam-7053	135	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	135	5	-	-	PUNCT
ejpam-7053	135	6	int(k	int(k	NOUN
ejpam-7053	135	7	)	)	PUNCT
ejpam-7053	135	8	)	)	PUNCT
ejpam-7053	135	9	)	)	PUNCT
ejpam-7053	136	1	=	=	SYM
ejpam-7053	136	2	f−(k	f−(k	PROPN
ejpam-7053	136	3	)	)	PUNCT
ejpam-7053	136	4	.	.	PUNCT
ejpam-7053	137	1	(	(	PUNCT
ejpam-7053	137	2	8)	8)	NUM
ejpam-7053	137	3	⇒	⇒	NOUN
ejpam-7053	137	4	(	(	PUNCT
ejpam-7053	137	5	3	3	NUM
ejpam-7053	137	6	):	):	PUNCT
ejpam-7053	137	7	let	let	VERB
ejpam-7053	137	8	k	k	PRON
ejpam-7053	137	9	be	be	AUX
ejpam-7053	137	10	any	any	DET
ejpam-7053	137	11	σ1σ2	σ1σ2	NUM
ejpam-7053	137	12	-	-	PUNCT
ejpam-7053	137	13	closed	closed	ADJ
ejpam-7053	137	14	set	set	NOUN
ejpam-7053	137	15	of	of	ADP
ejpam-7053	137	16	y	y	PROPN
ejpam-7053	137	17	.	.	PUNCT
ejpam-7053	138	1	then	then	ADV
ejpam-7053	138	2	,	,	PUNCT
ejpam-7053	138	3	σ1σ2	σ1σ2	X
ejpam-7053	138	4	-	-	PUNCT
ejpam-7053	138	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	138	6	-	-	PUNCT
ejpam-7053	138	7	int(k	int(k	NOUN
ejpam-7053	138	8	)	)	PUNCT
ejpam-7053	138	9	)	)	PUNCT
ejpam-7053	138	10	is	be	AUX
ejpam-7053	138	11	(	(	PUNCT
ejpam-7053	138	12	σ1	σ1	PROPN
ejpam-7053	138	13	,	,	PUNCT
ejpam-7053	138	14	σ2)rclosed	σ2)rclose	VERB
ejpam-7053	138	15	in	in	ADP
ejpam-7053	138	16	y	y	PROPN
ejpam-7053	138	17	and	and	CCONJ
ejpam-7053	138	18	σ1σ2	σ1σ2	NOUN
ejpam-7053	138	19	-	-	PUNCT
ejpam-7053	138	20	int(σ1σ2	int(σ1σ2	ADV
ejpam-7053	138	21	-	-	PUNCT
ejpam-7053	138	22	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7053	138	23	-	-	PUNCT
ejpam-7053	138	24	int(k	int(k	NOUN
ejpam-7053	138	25	)	)	PUNCT
ejpam-7053	138	26	)	)	PUNCT
ejpam-7053	138	27	)	)	PUNCT
ejpam-7053	139	1	=	=	PUNCT
ejpam-7053	139	2	σ1σ2	σ1σ2	X
ejpam-7053	139	3	-	-	PUNCT
ejpam-7053	139	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	139	5	-	-	PUNCT
ejpam-7053	139	6	cl(k	cl(k	NUM
ejpam-7053	139	7	)	)	PUNCT
ejpam-7053	139	8	)	)	PUNCT
ejpam-7053	140	1	=	=	PUNCT
ejpam-7053	140	2	σ1σ2	σ1σ2	X
ejpam-7053	140	3	-	-	PUNCT
ejpam-7053	140	4	int(k	int(k	NOUN
ejpam-7053	140	5	)	)	PUNCT
ejpam-7053	140	6	.	.	PUNCT
ejpam-7053	141	1	by	by	ADP
ejpam-7053	141	2	(	(	PUNCT
ejpam-7053	141	3	8)	8)	NUM
ejpam-7053	141	4	,	,	PUNCT
ejpam-7053	141	5	cµ(f	cµ(f	NOUN
ejpam-7053	141	6	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	141	7	-	-	PUNCT
ejpam-7053	141	8	int(k	int(k	NUM
ejpam-7053	141	9	)	)	PUNCT
ejpam-7053	141	10	)	)	PUNCT
ejpam-7053	141	11	)	)	PUNCT
ejpam-7053	141	12	=	=	SYM
ejpam-7053	141	13	cµ(f	cµ(f	NOUN
ejpam-7053	141	14	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	141	15	-	-	PUNCT
ejpam-7053	141	16	int(σ1σ2	int(σ1σ2	ADV
ejpam-7053	141	17	-	-	PUNCT
ejpam-7053	141	18	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7053	141	19	-	-	PUNCT
ejpam-7053	141	20	int(k	int(k	NOUN
ejpam-7053	141	21	)	)	PUNCT
ejpam-7053	141	22	)	)	PUNCT
ejpam-7053	141	23	)	)	PUNCT
ejpam-7053	141	24	)	)	PUNCT
ejpam-7053	141	25	)	)	PUNCT
ejpam-7053	142	1	⊆	⊆	X
ejpam-7053	142	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7053	142	3	-	-	PUNCT
ejpam-7053	142	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	142	5	-	-	PUNCT
ejpam-7053	142	6	int(k	int(k	NOUN
ejpam-7053	142	7	)	)	PUNCT
ejpam-7053	142	8	)	)	PUNCT
ejpam-7053	142	9	)	)	PUNCT
ejpam-7053	143	1	⊆	⊆	NUM
ejpam-7053	143	2	f−(k	f−(k	PROPN
ejpam-7053	143	3	)	)	PUNCT
ejpam-7053	143	4	.	.	PUNCT
ejpam-7053	144	1	definition	definition	NOUN
ejpam-7053	144	2	2	2	NUM
ejpam-7053	144	3	.	.	PUNCT
ejpam-7053	144	4	a	a	DET
ejpam-7053	144	5	multifunction	multifunction	NOUN
ejpam-7053	144	6	f	f	NOUN
ejpam-7053	144	7	:	:	PUNCT
ejpam-7053	144	8	(	(	PUNCT
ejpam-7053	144	9	x,µ	x,µ	NOUN
ejpam-7053	144	10	)	)	PUNCT
ejpam-7053	144	11	→	→	SYM
ejpam-7053	144	12	(	(	PUNCT
ejpam-7053	144	13	y	y	PROPN
ejpam-7053	144	14	,	,	PUNCT
ejpam-7053	144	15	σ1	σ1	PROPN
ejpam-7053	144	16	,	,	PUNCT
ejpam-7053	144	17	σ2	σ2	PROPN
ejpam-7053	144	18	)	)	PUNCT
ejpam-7053	144	19	is	be	AUX
ejpam-7053	144	20	said	say	VERB
ejpam-7053	144	21	to	to	PART
ejpam-7053	144	22	be	be	AUX
ejpam-7053	144	23	lower	low	ADJ
ejpam-7053	144	24	weakly	weakly	ADJ
ejpam-7053	144	25	µ(σ1	µ(σ1	NOUN
ejpam-7053	144	26	,	,	PUNCT
ejpam-7053	144	27	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	144	28	at	at	ADP
ejpam-7053	144	29	a	a	DET
ejpam-7053	144	30	point	point	NOUN
ejpam-7053	144	31	x	x	SYM
ejpam-7053	144	32	∈	∈	NOUN
ejpam-7053	144	33	x	x	PUNCT
ejpam-7053	144	34	if	if	SCONJ
ejpam-7053	144	35	for	for	ADP
ejpam-7053	144	36	each	each	DET
ejpam-7053	144	37	σ1σ2	σ1σ2	VERB
ejpam-7053	144	38	-	-	ADJ
ejpam-7053	144	39	open	open	ADJ
ejpam-7053	144	40	set	set	NOUN
ejpam-7053	144	41	v	v	NOUN
ejpam-7053	144	42	of	of	ADP
ejpam-7053	144	43	y	y	PRON
ejpam-7053	144	44	such	such	ADJ
ejpam-7053	144	45	that	that	PRON
ejpam-7053	144	46	v	v	NOUN
ejpam-7053	144	47	∩f	∩f	NOUN
ejpam-7053	144	48	(	(	PUNCT
ejpam-7053	144	49	x	x	X
ejpam-7053	144	50	)	)	PUNCT
ejpam-7053	144	51	̸=	̸=	NOUN
ejpam-7053	144	52	∅	∅	NOUN
ejpam-7053	144	53	,	,	PUNCT
ejpam-7053	144	54	there	there	PRON
ejpam-7053	144	55	exists	exist	VERB
ejpam-7053	144	56	a	a	DET
ejpam-7053	144	57	µ-open	µ-open	NOUN
ejpam-7053	144	58	set	set	VERB
ejpam-7053	144	59	u	u	NOUN
ejpam-7053	144	60	of	of	ADP
ejpam-7053	144	61	x	x	PUNCT
ejpam-7053	144	62	containing	contain	VERB
ejpam-7053	144	63	x	x	PUNCT
ejpam-7053	144	64	such	such	ADJ
ejpam-7053	144	65	that	that	SCONJ
ejpam-7053	144	66	σ1σ2	σ1σ2	NOUN
ejpam-7053	144	67	-	-	NUM
ejpam-7053	144	68	cl(v	cl(v	PUNCT
ejpam-7053	144	69	)	)	PUNCT
ejpam-7053	144	70	∩f	∩f	NOUN
ejpam-7053	144	71	(	(	PUNCT
ejpam-7053	144	72	z	z	X
ejpam-7053	144	73	)	)	PUNCT
ejpam-7053	144	74	̸=	̸=	NOUN
ejpam-7053	144	75	∅	∅	NOUN
ejpam-7053	144	76	for	for	ADP
ejpam-7053	144	77	every	every	DET
ejpam-7053	144	78	z	z	NOUN
ejpam-7053	144	79	∈	∈	PROPN
ejpam-7053	144	80	u	u	NOUN
ejpam-7053	144	81	.	.	PUNCT
ejpam-7053	144	82	a	a	DET
ejpam-7053	144	83	multifunction	multifunction	NOUN
ejpam-7053	145	1	f	f	NOUN
ejpam-7053	145	2	:	:	PUNCT
ejpam-7053	145	3	(	(	PUNCT
ejpam-7053	145	4	x,µ	x,µ	NOUN
ejpam-7053	145	5	)	)	PUNCT
ejpam-7053	145	6	→	→	SYM
ejpam-7053	145	7	(	(	PUNCT
ejpam-7053	145	8	y	y	PROPN
ejpam-7053	145	9	,	,	PUNCT
ejpam-7053	145	10	σ1	σ1	PROPN
ejpam-7053	145	11	,	,	PUNCT
ejpam-7053	145	12	σ2	σ2	PROPN
ejpam-7053	145	13	)	)	PUNCT
ejpam-7053	145	14	is	be	AUX
ejpam-7053	145	15	said	say	VERB
ejpam-7053	145	16	to	to	PART
ejpam-7053	145	17	be	be	AUX
ejpam-7053	145	18	lower	low	ADJ
ejpam-7053	145	19	weakly	weakly	ADJ
ejpam-7053	145	20	µ(σ1	µ(σ1	NOUN
ejpam-7053	145	21	,	,	PUNCT
ejpam-7053	145	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	145	23	if	if	SCONJ
ejpam-7053	145	24	f	f	PROPN
ejpam-7053	145	25	is	be	AUX
ejpam-7053	145	26	lower	low	ADJ
ejpam-7053	145	27	weakly	weakly	ADJ
ejpam-7053	145	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	145	29	,	,	PUNCT
ejpam-7053	145	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	145	31	at	at	ADP
ejpam-7053	145	32	each	each	DET
ejpam-7053	145	33	point	point	NOUN
ejpam-7053	145	34	x	x	PUNCT
ejpam-7053	145	35	of	of	ADP
ejpam-7053	145	36	x.	x.	PROPN
ejpam-7053	145	37	theorem	theorem	VERB
ejpam-7053	145	38	2	2	NUM
ejpam-7053	145	39	.	.	X
ejpam-7053	145	40	for	for	ADP
ejpam-7053	145	41	a	a	DET
ejpam-7053	145	42	multifunction	multifunction	NOUN
ejpam-7053	145	43	f	f	NOUN
ejpam-7053	145	44	:	:	PUNCT
ejpam-7053	145	45	(	(	PUNCT
ejpam-7053	145	46	x,µ	x,µ	NOUN
ejpam-7053	145	47	)	)	PUNCT
ejpam-7053	145	48	→	→	SYM
ejpam-7053	145	49	(	(	PUNCT
ejpam-7053	145	50	y	y	PROPN
ejpam-7053	145	51	,	,	PUNCT
ejpam-7053	145	52	σ1	σ1	PROPN
ejpam-7053	145	53	,	,	PUNCT
ejpam-7053	145	54	σ2	σ2	NOUN
ejpam-7053	145	55	)	)	PUNCT
ejpam-7053	145	56	,	,	PUNCT
ejpam-7053	145	57	the	the	DET
ejpam-7053	145	58	following	follow	VERB
ejpam-7053	145	59	properties	property	NOUN
ejpam-7053	145	60	are	be	AUX
ejpam-7053	145	61	equivalent	equivalent	ADJ
ejpam-7053	145	62	:	:	PUNCT
ejpam-7053	145	63	(	(	PUNCT
ejpam-7053	145	64	1	1	X
ejpam-7053	145	65	)	)	PUNCT
ejpam-7053	145	66	f	f	PROPN
ejpam-7053	145	67	is	be	AUX
ejpam-7053	145	68	lower	low	ADJ
ejpam-7053	145	69	weakly	weakly	ADJ
ejpam-7053	145	70	µ(σ1	µ(σ1	NOUN
ejpam-7053	145	71	,	,	PUNCT
ejpam-7053	145	72	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	145	73	;	;	PUNCT
ejpam-7053	145	74	(	(	PUNCT
ejpam-7053	145	75	2	2	X
ejpam-7053	145	76	)	)	PUNCT
ejpam-7053	145	77	f−(v	f−(v	NOUN
ejpam-7053	145	78	)	)	PUNCT
ejpam-7053	146	1	⊆	⊆	NUM
ejpam-7053	146	2	iµ(f	iµ(f	NUM
ejpam-7053	146	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	146	4	-	-	NOUN
ejpam-7053	146	5	cl(v	cl(v	NOUN
ejpam-7053	146	6	)	)	PUNCT
ejpam-7053	146	7	)	)	PUNCT
ejpam-7053	146	8	)	)	PUNCT
ejpam-7053	147	1	for	for	ADP
ejpam-7053	147	2	every	every	DET
ejpam-7053	147	3	σ1σ2	σ1σ2	NOUN
ejpam-7053	147	4	-	-	ADJ
ejpam-7053	147	5	open	open	ADJ
ejpam-7053	147	6	set	set	NOUN
ejpam-7053	147	7	v	v	NOUN
ejpam-7053	147	8	of	of	ADP
ejpam-7053	147	9	y	y	PROPN
ejpam-7053	147	10	;	;	PUNCT
ejpam-7053	147	11	(	(	PUNCT
ejpam-7053	147	12	3	3	X
ejpam-7053	147	13	)	)	PUNCT
ejpam-7053	147	14	cµ(f	cµ(f	PUNCT
ejpam-7053	148	1	+	+	PROPN
ejpam-7053	148	2	(	(	PUNCT
ejpam-7053	148	3	σ1σ2	σ1σ2	NUM
ejpam-7053	148	4	-	-	PUNCT
ejpam-7053	148	5	int(k	int(k	NUM
ejpam-7053	148	6	)	)	PUNCT
ejpam-7053	148	7	)	)	PUNCT
ejpam-7053	148	8	)	)	PUNCT
ejpam-7053	149	1	⊆	⊆	NUM
ejpam-7053	149	2	f+(k	f+(k	NOUN
ejpam-7053	149	3	)	)	PUNCT
ejpam-7053	149	4	for	for	ADP
ejpam-7053	149	5	every	every	DET
ejpam-7053	149	6	σ1σ2	σ1σ2	NUM
ejpam-7053	149	7	-	-	PUNCT
ejpam-7053	149	8	closed	closed	ADJ
ejpam-7053	149	9	set	set	NOUN
ejpam-7053	149	10	k	k	PROPN
ejpam-7053	149	11	of	of	ADP
ejpam-7053	149	12	y	y	PROPN
ejpam-7053	149	13	;	;	PUNCT
ejpam-7053	149	14	(	(	PUNCT
ejpam-7053	149	15	4	4	NUM
ejpam-7053	149	16	)	)	PUNCT
ejpam-7053	149	17	cµ(f	cµ(f	PUNCT
ejpam-7053	150	1	+	+	PROPN
ejpam-7053	150	2	(	(	PUNCT
ejpam-7053	150	3	σ1σ2	σ1σ2	NUM
ejpam-7053	150	4	-	-	PUNCT
ejpam-7053	150	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	150	6	-	-	PUNCT
ejpam-7053	150	7	cl(b	cl(b	NOUN
ejpam-7053	150	8	)	)	PUNCT
ejpam-7053	150	9	)	)	PUNCT
ejpam-7053	150	10	)	)	PUNCT
ejpam-7053	150	11	)	)	PUNCT
ejpam-7053	151	1	⊆	⊆	X
ejpam-7053	151	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	151	3	-	-	PUNCT
ejpam-7053	151	4	cl(b	cl(b	NOUN
ejpam-7053	151	5	)	)	PUNCT
ejpam-7053	151	6	)	)	PUNCT
ejpam-7053	151	7	for	for	ADP
ejpam-7053	151	8	every	every	DET
ejpam-7053	151	9	subset	subset	NOUN
ejpam-7053	151	10	b	b	PROPN
ejpam-7053	151	11	of	of	ADP
ejpam-7053	151	12	y	y	PROPN
ejpam-7053	151	13	;	;	PUNCT
ejpam-7053	151	14	(	(	PUNCT
ejpam-7053	151	15	5	5	X
ejpam-7053	151	16	)	)	PUNCT
ejpam-7053	151	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	151	18	-	-	PUNCT
ejpam-7053	151	19	int(b	int(b	NOUN
ejpam-7053	151	20	)	)	PUNCT
ejpam-7053	151	21	)	)	PUNCT
ejpam-7053	151	22	⊆	⊆	NUM
ejpam-7053	151	23	iµ(f	iµ(f	NUM
ejpam-7053	151	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	151	25	-	-	PUNCT
ejpam-7053	151	26	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	151	27	-	-	PUNCT
ejpam-7053	151	28	int(b	int(b	NOUN
ejpam-7053	151	29	)	)	PUNCT
ejpam-7053	151	30	)	)	PUNCT
ejpam-7053	151	31	)	)	PUNCT
ejpam-7053	151	32	)	)	PUNCT
ejpam-7053	151	33	for	for	ADP
ejpam-7053	151	34	every	every	DET
ejpam-7053	151	35	subset	subset	NOUN
ejpam-7053	151	36	b	b	PROPN
ejpam-7053	151	37	of	of	ADP
ejpam-7053	151	38	y	y	PROPN
ejpam-7053	151	39	;	;	PUNCT
ejpam-7053	151	40	(	(	PUNCT
ejpam-7053	151	41	6	6	NUM
ejpam-7053	151	42	)	)	PUNCT
ejpam-7053	151	43	cµ(f	cµ(f	PUNCT
ejpam-7053	152	1	+	+	PROPN
ejpam-7053	152	2	(	(	PUNCT
ejpam-7053	152	3	σ1σ2	σ1σ2	NUM
ejpam-7053	152	4	-	-	PUNCT
ejpam-7053	152	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	152	6	-	-	PUNCT
ejpam-7053	152	7	cl(v	cl(v	NOUN
ejpam-7053	152	8	)	)	PUNCT
ejpam-7053	152	9	)	)	PUNCT
ejpam-7053	152	10	)	)	PUNCT
ejpam-7053	152	11	)	)	PUNCT
ejpam-7053	153	1	⊆	⊆	X
ejpam-7053	153	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	153	3	-	-	PUNCT
ejpam-7053	153	4	cl(v	cl(v	NOUN
ejpam-7053	153	5	)	)	PUNCT
ejpam-7053	153	6	)	)	PUNCT
ejpam-7053	153	7	for	for	ADP
ejpam-7053	153	8	every	every	DET
ejpam-7053	153	9	σ1σ2	σ1σ2	NOUN
ejpam-7053	153	10	-	-	ADJ
ejpam-7053	153	11	open	open	ADJ
ejpam-7053	153	12	set	set	NOUN
ejpam-7053	153	13	v	v	NOUN
ejpam-7053	153	14	of	of	ADP
ejpam-7053	153	15	y	y	PROPN
ejpam-7053	153	16	;	;	PUNCT
ejpam-7053	153	17	(	(	PUNCT
ejpam-7053	153	18	7	7	NUM
ejpam-7053	153	19	)	)	PUNCT
ejpam-7053	153	20	cµ(f	cµ(f	PUNCT
ejpam-7053	154	1	+	+	PROPN
ejpam-7053	154	2	(	(	PUNCT
ejpam-7053	154	3	v	v	NOUN
ejpam-7053	154	4	)	)	PUNCT
ejpam-7053	154	5	)	)	PUNCT
ejpam-7053	154	6	⊆	⊆	NUM
ejpam-7053	154	7	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	154	8	-	-	PUNCT
ejpam-7053	154	9	cl(v	cl(v	NOUN
ejpam-7053	154	10	)	)	PUNCT
ejpam-7053	154	11	)	)	PUNCT
ejpam-7053	154	12	for	for	ADP
ejpam-7053	154	13	every	every	DET
ejpam-7053	154	14	σ1σ2	σ1σ2	NOUN
ejpam-7053	154	15	-	-	ADJ
ejpam-7053	154	16	open	open	ADJ
ejpam-7053	154	17	set	set	NOUN
ejpam-7053	154	18	v	v	NOUN
ejpam-7053	154	19	of	of	ADP
ejpam-7053	154	20	y	y	PROPN
ejpam-7053	154	21	;	;	PUNCT
ejpam-7053	154	22	(	(	PUNCT
ejpam-7053	154	23	8)	8)	NUM
ejpam-7053	154	24	cµ(f	cµ(f	NOUN
ejpam-7053	154	25	+	+	PROPN
ejpam-7053	154	26	(	(	PUNCT
ejpam-7053	154	27	σ1σ2	σ1σ2	NUM
ejpam-7053	154	28	-	-	PUNCT
ejpam-7053	154	29	int(k	int(k	NUM
ejpam-7053	154	30	)	)	PUNCT
ejpam-7053	154	31	)	)	PUNCT
ejpam-7053	154	32	)	)	PUNCT
ejpam-7053	155	1	⊆	⊆	NUM
ejpam-7053	155	2	f+(k	f+(k	NOUN
ejpam-7053	155	3	)	)	PUNCT
ejpam-7053	155	4	for	for	ADP
ejpam-7053	155	5	every	every	DET
ejpam-7053	155	6	(	(	PUNCT
ejpam-7053	155	7	σ1	σ1	PROPN
ejpam-7053	155	8	,	,	PUNCT
ejpam-7053	155	9	σ2)r	σ2)r	NOUN
ejpam-7053	155	10	-	-	PUNCT
ejpam-7053	155	11	closed	close	VERB
ejpam-7053	155	12	set	set	ADJ
ejpam-7053	155	13	k	k	PROPN
ejpam-7053	155	14	of	of	ADP
ejpam-7053	155	15	y	y	PROPN
ejpam-7053	155	16	.	.	PUNCT
ejpam-7053	156	1	proof	proof	NOUN
ejpam-7053	156	2	.	.	PUNCT
ejpam-7053	157	1	the	the	DET
ejpam-7053	157	2	proof	proof	NOUN
ejpam-7053	157	3	is	be	AUX
ejpam-7053	157	4	similar	similar	ADJ
ejpam-7053	157	5	to	to	ADP
ejpam-7053	157	6	that	that	PRON
ejpam-7053	157	7	of	of	ADP
ejpam-7053	157	8	theorem	theorem	NOUN
ejpam-7053	157	9	1	1	NUM
ejpam-7053	157	10	.	.	PUNCT
ejpam-7053	157	11	definition	definition	NOUN
ejpam-7053	157	12	3	3	NUM
ejpam-7053	157	13	.	.	PUNCT
ejpam-7053	158	1	[	[	X
ejpam-7053	158	2	19	19	NUM
ejpam-7053	158	3	]	]	PUNCT
ejpam-7053	158	4	a	a	DET
ejpam-7053	158	5	multifunction	multifunction	NOUN
ejpam-7053	158	6	f	f	NOUN
ejpam-7053	158	7	:	:	PUNCT
ejpam-7053	158	8	(	(	PUNCT
ejpam-7053	158	9	x,µ	x,µ	NOUN
ejpam-7053	158	10	)	)	PUNCT
ejpam-7053	158	11	→	→	SYM
ejpam-7053	158	12	(	(	PUNCT
ejpam-7053	158	13	y	y	PROPN
ejpam-7053	158	14	,	,	PUNCT
ejpam-7053	158	15	σ1	σ1	PROPN
ejpam-7053	158	16	,	,	PUNCT
ejpam-7053	158	17	σ2	σ2	PROPN
ejpam-7053	158	18	)	)	PUNCT
ejpam-7053	158	19	is	be	AUX
ejpam-7053	158	20	said	say	VERB
ejpam-7053	158	21	to	to	PART
ejpam-7053	158	22	be	be	AUX
ejpam-7053	158	23	upper	upper	ADJ
ejpam-7053	158	24	almost	almost	ADV
ejpam-7053	158	25	µ(σ1	µ(σ1	NOUN
ejpam-7053	158	26	,	,	PUNCT
ejpam-7053	158	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	158	28	at	at	ADP
ejpam-7053	158	29	a	a	DET
ejpam-7053	158	30	point	point	NOUN
ejpam-7053	158	31	x	x	SYM
ejpam-7053	158	32	∈	∈	NOUN
ejpam-7053	158	33	x	x	PUNCT
ejpam-7053	158	34	if	if	SCONJ
ejpam-7053	158	35	for	for	ADP
ejpam-7053	158	36	each	each	DET
ejpam-7053	158	37	σ1σ2	σ1σ2	VERB
ejpam-7053	158	38	-	-	ADJ
ejpam-7053	158	39	open	open	ADJ
ejpam-7053	158	40	set	set	NOUN
ejpam-7053	158	41	v	v	NOUN
ejpam-7053	158	42	of	of	ADP
ejpam-7053	158	43	y	y	PRON
ejpam-7053	158	44	such	such	ADJ
ejpam-7053	158	45	that	that	SCONJ
ejpam-7053	158	46	f	f	PROPN
ejpam-7053	158	47	(	(	PUNCT
ejpam-7053	158	48	x	x	X
ejpam-7053	158	49	)	)	PUNCT
ejpam-7053	158	50	⊆	⊆	NUM
ejpam-7053	158	51	v	v	NOUN
ejpam-7053	158	52	,	,	PUNCT
ejpam-7053	158	53	there	there	PRON
ejpam-7053	158	54	exists	exist	VERB
ejpam-7053	158	55	a	a	DET
ejpam-7053	158	56	µ-open	µ-open	NOUN
ejpam-7053	158	57	set	set	VERB
ejpam-7053	158	58	u	u	NOUN
ejpam-7053	158	59	of	of	ADP
ejpam-7053	158	60	x	x	PUNCT
ejpam-7053	158	61	containing	contain	VERB
ejpam-7053	158	62	x	x	PUNCT
ejpam-7053	158	63	such	such	ADJ
ejpam-7053	158	64	that	that	SCONJ
ejpam-7053	158	65	f	f	PROPN
ejpam-7053	158	66	(	(	PUNCT
ejpam-7053	158	67	u	u	NOUN
ejpam-7053	158	68	)	)	PUNCT
ejpam-7053	158	69	⊆	⊆	NUM
ejpam-7053	158	70	σ1σ2	σ1σ2	X
ejpam-7053	158	71	-	-	PUNCT
ejpam-7053	158	72	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	158	73	-	-	PUNCT
ejpam-7053	158	74	cl(v	cl(v	NOUN
ejpam-7053	158	75	)	)	PUNCT
ejpam-7053	158	76	)	)	PUNCT
ejpam-7053	158	77	.	.	PUNCT
ejpam-7053	159	1	a	a	DET
ejpam-7053	159	2	multifunction	multifunction	NOUN
ejpam-7053	159	3	f	f	NOUN
ejpam-7053	159	4	:	:	PUNCT
ejpam-7053	159	5	(	(	PUNCT
ejpam-7053	159	6	x,µ	x,µ	NOUN
ejpam-7053	159	7	)	)	PUNCT
ejpam-7053	159	8	→	→	SYM
ejpam-7053	159	9	(	(	PUNCT
ejpam-7053	159	10	y	y	PROPN
ejpam-7053	159	11	,	,	PUNCT
ejpam-7053	159	12	σ1	σ1	PROPN
ejpam-7053	159	13	,	,	PUNCT
ejpam-7053	159	14	σ2	σ2	PROPN
ejpam-7053	159	15	)	)	PUNCT
ejpam-7053	159	16	is	be	AUX
ejpam-7053	159	17	said	say	VERB
ejpam-7053	159	18	to	to	PART
ejpam-7053	159	19	be	be	AUX
ejpam-7053	159	20	upper	upper	ADJ
ejpam-7053	159	21	almost	almost	ADV
ejpam-7053	159	22	µ(σ1	µ(σ1	NOUN
ejpam-7053	159	23	,	,	PUNCT
ejpam-7053	159	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	159	25	if	if	SCONJ
ejpam-7053	159	26	f	f	PROPN
ejpam-7053	159	27	is	be	AUX
ejpam-7053	159	28	upper	upper	ADJ
ejpam-7053	159	29	almost	almost	ADV
ejpam-7053	159	30	µ(σ1	µ(σ1	NOUN
ejpam-7053	159	31	,	,	PUNCT
ejpam-7053	159	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	159	33	at	at	ADP
ejpam-7053	159	34	each	each	DET
ejpam-7053	159	35	point	point	NOUN
ejpam-7053	159	36	x	x	PUNCT
ejpam-7053	159	37	of	of	ADP
ejpam-7053	159	38	x.	x.	PROPN
ejpam-7053	159	39	m.	m.	PROPN
ejpam-7053	159	40	chiangpradit	chiangpradit	PROPN
ejpam-7053	159	41	,	,	PUNCT
ejpam-7053	159	42	a.	a.	PROPN
ejpam-7053	159	43	sama	sama	PROPN
ejpam-7053	159	44	-	-	PUNCT
ejpam-7053	159	45	ae	ae	PROPN
ejpam-7053	159	46	,	,	PUNCT
ejpam-7053	159	47	c.	c.	PROPN
ejpam-7053	159	48	boonpok	boonpok	PROPN
ejpam-7053	159	49	/	/	SYM
ejpam-7053	159	50	eur	eur	PROPN
ejpam-7053	159	51	.	.	PUNCT
ejpam-7053	160	1	j.	j.	PROPN
ejpam-7053	160	2	pure	pure	PROPN
ejpam-7053	160	3	appl	appl	PROPN
ejpam-7053	160	4	.	.	PROPN
ejpam-7053	160	5	math	math	PROPN
ejpam-7053	160	6	,	,	PUNCT
ejpam-7053	160	7	18	18	NUM
ejpam-7053	160	8	(	(	PUNCT
ejpam-7053	160	9	4	4	NUM
ejpam-7053	160	10	)	)	PUNCT
ejpam-7053	160	11	(	(	PUNCT
ejpam-7053	160	12	2025	2025	NUM
ejpam-7053	160	13	)	)	PUNCT
ejpam-7053	160	14	,	,	PUNCT
ejpam-7053	160	15	7053	7053	NUM
ejpam-7053	160	16	6	6	NUM
ejpam-7053	160	17	of	of	ADP
ejpam-7053	160	18	13	13	NUM
ejpam-7053	160	19	definition	definition	NOUN
ejpam-7053	160	20	4	4	NUM
ejpam-7053	160	21	.	.	PUNCT
ejpam-7053	161	1	[	[	X
ejpam-7053	161	2	19	19	NUM
ejpam-7053	161	3	]	]	PUNCT
ejpam-7053	161	4	a	a	DET
ejpam-7053	161	5	multifunction	multifunction	NOUN
ejpam-7053	161	6	f	f	NOUN
ejpam-7053	161	7	:	:	PUNCT
ejpam-7053	161	8	(	(	PUNCT
ejpam-7053	161	9	x,µ	x,µ	NOUN
ejpam-7053	161	10	)	)	PUNCT
ejpam-7053	161	11	→	→	SYM
ejpam-7053	161	12	(	(	PUNCT
ejpam-7053	161	13	y	y	PROPN
ejpam-7053	161	14	,	,	PUNCT
ejpam-7053	161	15	σ1	σ1	PROPN
ejpam-7053	161	16	,	,	PUNCT
ejpam-7053	161	17	σ2	σ2	PROPN
ejpam-7053	161	18	)	)	PUNCT
ejpam-7053	161	19	is	be	AUX
ejpam-7053	161	20	said	say	VERB
ejpam-7053	161	21	to	to	PART
ejpam-7053	161	22	be	be	AUX
ejpam-7053	161	23	lower	low	ADJ
ejpam-7053	161	24	almost	almost	ADV
ejpam-7053	161	25	µ(σ1	µ(σ1	NOUN
ejpam-7053	161	26	,	,	PUNCT
ejpam-7053	161	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	161	28	at	at	ADP
ejpam-7053	161	29	a	a	DET
ejpam-7053	161	30	point	point	NOUN
ejpam-7053	161	31	x	x	SYM
ejpam-7053	161	32	∈	∈	NOUN
ejpam-7053	161	33	x	x	PUNCT
ejpam-7053	161	34	if	if	SCONJ
ejpam-7053	161	35	for	for	ADP
ejpam-7053	161	36	each	each	DET
ejpam-7053	161	37	σ1σ2	σ1σ2	VERB
ejpam-7053	161	38	-	-	ADJ
ejpam-7053	161	39	open	open	ADJ
ejpam-7053	161	40	set	set	NOUN
ejpam-7053	161	41	v	v	NOUN
ejpam-7053	161	42	of	of	ADP
ejpam-7053	161	43	y	y	PRON
ejpam-7053	161	44	such	such	ADJ
ejpam-7053	161	45	that	that	PRON
ejpam-7053	161	46	v	v	NOUN
ejpam-7053	161	47	∩f	∩f	NOUN
ejpam-7053	161	48	(	(	PUNCT
ejpam-7053	161	49	x	x	X
ejpam-7053	161	50	)	)	PUNCT
ejpam-7053	161	51	̸=	̸=	NOUN
ejpam-7053	161	52	∅	∅	NOUN
ejpam-7053	161	53	,	,	PUNCT
ejpam-7053	161	54	there	there	PRON
ejpam-7053	161	55	exists	exist	VERB
ejpam-7053	161	56	a	a	DET
ejpam-7053	161	57	µ-open	µ-open	NOUN
ejpam-7053	161	58	set	set	VERB
ejpam-7053	161	59	u	u	NOUN
ejpam-7053	161	60	of	of	ADP
ejpam-7053	161	61	x	x	PUNCT
ejpam-7053	161	62	containing	contain	VERB
ejpam-7053	161	63	x	x	PUNCT
ejpam-7053	161	64	such	such	ADJ
ejpam-7053	161	65	that	that	SCONJ
ejpam-7053	161	66	σ1σ2	σ1σ2	ADV
ejpam-7053	161	67	-	-	PUNCT
ejpam-7053	161	68	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	161	69	-	-	PUNCT
ejpam-7053	161	70	cl(v	cl(v	NOUN
ejpam-7053	161	71	)	)	PUNCT
ejpam-7053	161	72	)	)	PUNCT
ejpam-7053	162	1	∩f	∩f	NOUN
ejpam-7053	162	2	(	(	PUNCT
ejpam-7053	162	3	z	z	X
ejpam-7053	162	4	)	)	PUNCT
ejpam-7053	162	5	̸=	̸=	NOUN
ejpam-7053	162	6	∅	∅	NOUN
ejpam-7053	162	7	for	for	ADP
ejpam-7053	162	8	every	every	DET
ejpam-7053	162	9	z	z	NOUN
ejpam-7053	162	10	∈	∈	PROPN
ejpam-7053	162	11	u	u	NOUN
ejpam-7053	162	12	.	.	PUNCT
ejpam-7053	163	1	a	a	DET
ejpam-7053	163	2	multifunction	multifunction	NOUN
ejpam-7053	163	3	f	f	NOUN
ejpam-7053	163	4	:	:	PUNCT
ejpam-7053	163	5	(	(	PUNCT
ejpam-7053	163	6	x,µ	x,µ	NOUN
ejpam-7053	163	7	)	)	PUNCT
ejpam-7053	163	8	→	→	SYM
ejpam-7053	163	9	(	(	PUNCT
ejpam-7053	163	10	y	y	PROPN
ejpam-7053	163	11	,	,	PUNCT
ejpam-7053	163	12	σ1	σ1	PROPN
ejpam-7053	163	13	,	,	PUNCT
ejpam-7053	163	14	σ2	σ2	PROPN
ejpam-7053	163	15	)	)	PUNCT
ejpam-7053	163	16	is	be	AUX
ejpam-7053	163	17	said	say	VERB
ejpam-7053	163	18	to	to	PART
ejpam-7053	163	19	be	be	AUX
ejpam-7053	163	20	lower	low	ADJ
ejpam-7053	163	21	almost	almost	ADV
ejpam-7053	163	22	µ(σ1	µ(σ1	NOUN
ejpam-7053	163	23	,	,	PUNCT
ejpam-7053	163	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	163	25	if	if	SCONJ
ejpam-7053	163	26	f	f	PROPN
ejpam-7053	163	27	is	be	AUX
ejpam-7053	163	28	lower	low	ADJ
ejpam-7053	163	29	almost	almost	ADV
ejpam-7053	163	30	µ(σ1	µ(σ1	NOUN
ejpam-7053	163	31	,	,	PUNCT
ejpam-7053	163	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	163	33	at	at	ADP
ejpam-7053	163	34	each	each	DET
ejpam-7053	163	35	point	point	NOUN
ejpam-7053	163	36	x	x	PUNCT
ejpam-7053	163	37	of	of	ADP
ejpam-7053	163	38	x.	x.	NOUN
ejpam-7053	163	39	remark	remark	PROPN
ejpam-7053	163	40	1	1	NUM
ejpam-7053	163	41	.	.	PUNCT
ejpam-7053	163	42	for	for	ADP
ejpam-7053	163	43	a	a	DET
ejpam-7053	163	44	multifunction	multifunction	NOUN
ejpam-7053	163	45	f	f	NOUN
ejpam-7053	163	46	:	:	PUNCT
ejpam-7053	163	47	(	(	PUNCT
ejpam-7053	163	48	x,µ	x,µ	NOUN
ejpam-7053	163	49	)	)	PUNCT
ejpam-7053	163	50	→	→	SYM
ejpam-7053	163	51	(	(	PUNCT
ejpam-7053	163	52	y	y	PROPN
ejpam-7053	163	53	,	,	PUNCT
ejpam-7053	163	54	σ1	σ1	PROPN
ejpam-7053	163	55	,	,	PUNCT
ejpam-7053	163	56	σ2	σ2	NOUN
ejpam-7053	163	57	)	)	PUNCT
ejpam-7053	163	58	,	,	PUNCT
ejpam-7053	163	59	the	the	DET
ejpam-7053	163	60	following	follow	VERB
ejpam-7053	163	61	implication	implication	NOUN
ejpam-7053	163	62	holds	hold	VERB
ejpam-7053	163	63	:	:	PUNCT
ejpam-7053	163	64	upper	upper	ADJ
ejpam-7053	163	65	almost	almost	ADV
ejpam-7053	163	66	µ(σ1	µ(σ1	NOUN
ejpam-7053	163	67	,	,	PUNCT
ejpam-7053	163	68	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7053	163	69	⇒	⇒	VERB
ejpam-7053	163	70	upper	upper	ADJ
ejpam-7053	163	71	weakly	weakly	ADJ
ejpam-7053	163	72	µ(σ1	µ(σ1	NOUN
ejpam-7053	163	73	,	,	PUNCT
ejpam-7053	163	74	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7053	163	75	.	.	PUNCT
ejpam-7053	164	1	the	the	DET
ejpam-7053	164	2	converse	converse	NOUN
ejpam-7053	164	3	of	of	ADP
ejpam-7053	164	4	the	the	DET
ejpam-7053	164	5	implication	implication	NOUN
ejpam-7053	164	6	is	be	AUX
ejpam-7053	164	7	not	not	PART
ejpam-7053	164	8	true	true	ADJ
ejpam-7053	164	9	in	in	ADP
ejpam-7053	164	10	general	general	ADJ
ejpam-7053	164	11	.	.	PUNCT
ejpam-7053	165	1	we	we	PRON
ejpam-7053	165	2	give	give	VERB
ejpam-7053	165	3	an	an	DET
ejpam-7053	165	4	example	example	NOUN
ejpam-7053	165	5	for	for	ADP
ejpam-7053	165	6	the	the	DET
ejpam-7053	165	7	implication	implication	NOUN
ejpam-7053	165	8	as	as	SCONJ
ejpam-7053	165	9	follows	follow	VERB
ejpam-7053	165	10	.	.	PUNCT
ejpam-7053	165	11	example	example	NOUN
ejpam-7053	166	1	1	1	NUM
ejpam-7053	166	2	.	.	PUNCT
ejpam-7053	166	3	let	let	VERB
ejpam-7053	166	4	x	x	PUNCT
ejpam-7053	166	5	=	=	PRON
ejpam-7053	166	6	{	{	PUNCT
ejpam-7053	166	7	1	1	NUM
ejpam-7053	166	8	,	,	PUNCT
ejpam-7053	166	9	2	2	NUM
ejpam-7053	166	10	,	,	PUNCT
ejpam-7053	166	11	3	3	NUM
ejpam-7053	166	12	}	}	PUNCT
ejpam-7053	166	13	with	with	ADP
ejpam-7053	166	14	a	a	DET
ejpam-7053	166	15	generalized	generalized	ADJ
ejpam-7053	166	16	topology	topology	NOUN
ejpam-7053	166	17	µ	µ	X
ejpam-7053	166	18	=	=	SYM
ejpam-7053	166	19	{	{	PUNCT
ejpam-7053	166	20	∅	∅	NOUN
ejpam-7053	166	21	,	,	PUNCT
ejpam-7053	166	22	{	{	PUNCT
ejpam-7053	166	23	2	2	NUM
ejpam-7053	166	24	}	}	PUNCT
ejpam-7053	166	25	,	,	PUNCT
ejpam-7053	166	26	{	{	PUNCT
ejpam-7053	166	27	1	1	NUM
ejpam-7053	166	28	,	,	PUNCT
ejpam-7053	166	29	3	3	NUM
ejpam-7053	166	30	}	}	PUNCT
ejpam-7053	166	31	,	,	PUNCT
ejpam-7053	166	32	x	x	NOUN
ejpam-7053	166	33	}	}	PUNCT
ejpam-7053	166	34	.	.	PUNCT
ejpam-7053	167	1	let	let	VERB
ejpam-7053	167	2	y	y	PROPN
ejpam-7053	167	3	=	=	PUNCT
ejpam-7053	167	4	{	{	PUNCT
ejpam-7053	167	5	a	a	PRON
ejpam-7053	167	6	,	,	PUNCT
ejpam-7053	167	7	b	b	NOUN
ejpam-7053	167	8	,	,	PUNCT
ejpam-7053	167	9	c	c	NOUN
ejpam-7053	167	10	}	}	PUNCT
ejpam-7053	167	11	with	with	ADP
ejpam-7053	167	12	topologies	topology	NOUN
ejpam-7053	167	13	σ1	σ1	NOUN
ejpam-7053	167	14	=	=	SYM
ejpam-7053	167	15	{	{	PUNCT
ejpam-7053	167	16	∅	∅	NOUN
ejpam-7053	167	17	,	,	PUNCT
ejpam-7053	167	18	{	{	PUNCT
ejpam-7053	167	19	a	a	X
ejpam-7053	167	20	}	}	PUNCT
ejpam-7053	167	21	,	,	PUNCT
ejpam-7053	167	22	{	{	PUNCT
ejpam-7053	167	23	a	a	DET
ejpam-7053	167	24	,	,	PUNCT
ejpam-7053	167	25	b	b	NOUN
ejpam-7053	167	26	}	}	PUNCT
ejpam-7053	167	27	,	,	PUNCT
ejpam-7053	167	28	y	y	PROPN
ejpam-7053	167	29	}	}	PUNCT
ejpam-7053	167	30	and	and	CCONJ
ejpam-7053	167	31	σ2	σ2	PROPN
ejpam-7053	167	32	=	=	SYM
ejpam-7053	167	33	{	{	PUNCT
ejpam-7053	167	34	∅	∅	NOUN
ejpam-7053	167	35	,	,	PUNCT
ejpam-7053	167	36	{	{	PUNCT
ejpam-7053	167	37	a	a	X
ejpam-7053	167	38	}	}	PUNCT
ejpam-7053	167	39	,	,	PUNCT
ejpam-7053	167	40	{	{	PUNCT
ejpam-7053	167	41	b	b	NOUN
ejpam-7053	167	42	}	}	PUNCT
ejpam-7053	167	43	,	,	PUNCT
ejpam-7053	167	44	{	{	PUNCT
ejpam-7053	167	45	a	a	DET
ejpam-7053	167	46	,	,	PUNCT
ejpam-7053	167	47	b	b	NOUN
ejpam-7053	167	48	}	}	PUNCT
ejpam-7053	167	49	,	,	PUNCT
ejpam-7053	167	50	y	y	PROPN
ejpam-7053	167	51	}	}	PUNCT
ejpam-7053	167	52	.	.	PUNCT
ejpam-7053	168	1	a	a	DET
ejpam-7053	168	2	multifunction	multifunction	NOUN
ejpam-7053	168	3	f	f	NOUN
ejpam-7053	168	4	:	:	PUNCT
ejpam-7053	168	5	(	(	PUNCT
ejpam-7053	168	6	x,µ	x,µ	NOUN
ejpam-7053	168	7	)	)	PUNCT
ejpam-7053	168	8	→	→	SYM
ejpam-7053	168	9	(	(	PUNCT
ejpam-7053	168	10	y	y	PROPN
ejpam-7053	168	11	,	,	PUNCT
ejpam-7053	168	12	σ1	σ1	PROPN
ejpam-7053	168	13	,	,	PUNCT
ejpam-7053	168	14	σ2	σ2	PROPN
ejpam-7053	168	15	)	)	PUNCT
ejpam-7053	168	16	is	be	AUX
ejpam-7053	168	17	defined	define	VERB
ejpam-7053	168	18	as	as	SCONJ
ejpam-7053	168	19	follows	follow	VERB
ejpam-7053	168	20	:	:	PUNCT
ejpam-7053	168	21	f	f	X
ejpam-7053	168	22	(	(	PUNCT
ejpam-7053	168	23	1	1	X
ejpam-7053	168	24	)	)	PUNCT
ejpam-7053	169	1	=	=	PRON
ejpam-7053	169	2	{	{	PUNCT
ejpam-7053	169	3	a	a	NOUN
ejpam-7053	169	4	}	}	PUNCT
ejpam-7053	169	5	,	,	PUNCT
ejpam-7053	169	6	f	f	PROPN
ejpam-7053	169	7	(	(	PUNCT
ejpam-7053	169	8	2	2	NUM
ejpam-7053	169	9	)	)	PUNCT
ejpam-7053	169	10	=	=	PRON
ejpam-7053	169	11	{	{	PUNCT
ejpam-7053	169	12	b	b	NOUN
ejpam-7053	169	13	}	}	PUNCT
ejpam-7053	169	14	and	and	CCONJ
ejpam-7053	169	15	f	f	PROPN
ejpam-7053	169	16	(	(	PUNCT
ejpam-7053	169	17	3	3	X
ejpam-7053	169	18	)	)	PUNCT
ejpam-7053	169	19	=	=	PRON
ejpam-7053	169	20	{	{	PUNCT
ejpam-7053	169	21	a	a	X
ejpam-7053	169	22	,	,	PUNCT
ejpam-7053	169	23	c	c	NOUN
ejpam-7053	169	24	}	}	PUNCT
ejpam-7053	169	25	.	.	PUNCT
ejpam-7053	170	1	then	then	ADV
ejpam-7053	170	2	,	,	PUNCT
ejpam-7053	170	3	f	f	PROPN
ejpam-7053	170	4	is	be	AUX
ejpam-7053	170	5	upper	upper	ADJ
ejpam-7053	170	6	weakly	weakly	ADJ
ejpam-7053	170	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	170	8	,	,	PUNCT
ejpam-7053	170	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	170	10	but	but	CCONJ
ejpam-7053	170	11	f	f	PROPN
ejpam-7053	170	12	is	be	AUX
ejpam-7053	170	13	not	not	PART
ejpam-7053	170	14	upper	upper	ADJ
ejpam-7053	170	15	almost	almost	ADV
ejpam-7053	170	16	µ(σ1	µ(σ1	NOUN
ejpam-7053	170	17	,	,	PUNCT
ejpam-7053	170	18	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	170	19	.	.	X
ejpam-7053	170	20	theorem	theorem	NOUN
ejpam-7053	170	21	3	3	NUM
ejpam-7053	170	22	.	.	X
ejpam-7053	170	23	for	for	ADP
ejpam-7053	170	24	a	a	DET
ejpam-7053	170	25	multifunction	multifunction	NOUN
ejpam-7053	171	1	f	f	NOUN
ejpam-7053	171	2	:	:	PUNCT
ejpam-7053	171	3	(	(	PUNCT
ejpam-7053	171	4	x,µ	x,µ	NOUN
ejpam-7053	171	5	)	)	PUNCT
ejpam-7053	171	6	→	→	SYM
ejpam-7053	171	7	(	(	PUNCT
ejpam-7053	171	8	y	y	PROPN
ejpam-7053	171	9	,	,	PUNCT
ejpam-7053	171	10	σ1	σ1	PROPN
ejpam-7053	171	11	,	,	PUNCT
ejpam-7053	171	12	σ2	σ2	NOUN
ejpam-7053	171	13	)	)	PUNCT
ejpam-7053	171	14	,	,	PUNCT
ejpam-7053	171	15	the	the	DET
ejpam-7053	171	16	following	follow	VERB
ejpam-7053	171	17	properties	property	NOUN
ejpam-7053	171	18	are	be	AUX
ejpam-7053	171	19	equivalent	equivalent	ADJ
ejpam-7053	171	20	:	:	PUNCT
ejpam-7053	171	21	(	(	PUNCT
ejpam-7053	171	22	1	1	X
ejpam-7053	171	23	)	)	PUNCT
ejpam-7053	171	24	f	f	PROPN
ejpam-7053	171	25	is	be	AUX
ejpam-7053	171	26	upper	upper	ADJ
ejpam-7053	171	27	weakly	weakly	ADJ
ejpam-7053	171	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	171	29	,	,	PUNCT
ejpam-7053	171	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	171	31	;	;	PUNCT
ejpam-7053	171	32	(	(	PUNCT
ejpam-7053	171	33	2	2	NUM
ejpam-7053	171	34	)	)	PUNCT
ejpam-7053	171	35	cµ(f	cµ(f	NOUN
ejpam-7053	171	36	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	171	37	-	-	PUNCT
ejpam-7053	171	38	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	171	39	-	-	PUNCT
ejpam-7053	171	40	cl(v	cl(v	NOUN
ejpam-7053	171	41	)	)	PUNCT
ejpam-7053	171	42	)	)	PUNCT
ejpam-7053	171	43	)	)	PUNCT
ejpam-7053	171	44	)	)	PUNCT
ejpam-7053	172	1	⊆	⊆	X
ejpam-7053	172	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	172	3	-	-	PUNCT
ejpam-7053	172	4	cl(v	cl(v	NOUN
ejpam-7053	172	5	)	)	PUNCT
ejpam-7053	172	6	)	)	PUNCT
ejpam-7053	172	7	for	for	ADP
ejpam-7053	172	8	every	every	DET
ejpam-7053	172	9	(	(	PUNCT
ejpam-7053	172	10	σ1	σ1	PROPN
ejpam-7053	172	11	,	,	PUNCT
ejpam-7053	172	12	σ2)β	σ2)β	NOUN
ejpam-7053	172	13	-	-	PUNCT
ejpam-7053	172	14	open	open	NOUN
ejpam-7053	172	15	set	set	NOUN
ejpam-7053	172	16	v	v	NOUN
ejpam-7053	172	17	of	of	ADP
ejpam-7053	172	18	y	y	PROPN
ejpam-7053	172	19	;	;	PUNCT
ejpam-7053	172	20	(	(	PUNCT
ejpam-7053	172	21	3	3	X
ejpam-7053	172	22	)	)	PUNCT
ejpam-7053	172	23	cµ(f	cµ(f	NOUN
ejpam-7053	172	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	172	25	-	-	PUNCT
ejpam-7053	172	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	172	27	-	-	PUNCT
ejpam-7053	172	28	cl(v	cl(v	NOUN
ejpam-7053	172	29	)	)	PUNCT
ejpam-7053	172	30	)	)	PUNCT
ejpam-7053	172	31	)	)	PUNCT
ejpam-7053	172	32	)	)	PUNCT
ejpam-7053	173	1	⊆	⊆	X
ejpam-7053	173	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	173	3	-	-	PUNCT
ejpam-7053	173	4	cl(v	cl(v	NOUN
ejpam-7053	173	5	)	)	PUNCT
ejpam-7053	173	6	)	)	PUNCT
ejpam-7053	173	7	for	for	ADP
ejpam-7053	173	8	every	every	DET
ejpam-7053	173	9	(	(	PUNCT
ejpam-7053	173	10	σ1	σ1	PROPN
ejpam-7053	173	11	,	,	PUNCT
ejpam-7053	173	12	σ2)s	σ2)s	NOUN
ejpam-7053	173	13	-	-	PUNCT
ejpam-7053	173	14	open	open	NOUN
ejpam-7053	173	15	set	set	NOUN
ejpam-7053	173	16	v	v	NOUN
ejpam-7053	173	17	of	of	ADP
ejpam-7053	173	18	y	y	PROPN
ejpam-7053	173	19	.	.	PUNCT
ejpam-7053	174	1	proof	proof	NOUN
ejpam-7053	174	2	.	.	PUNCT
ejpam-7053	175	1	(	(	PUNCT
ejpam-7053	175	2	1	1	X
ejpam-7053	175	3	)	)	PUNCT
ejpam-7053	175	4	⇒	⇒	NOUN
ejpam-7053	175	5	(	(	PUNCT
ejpam-7053	175	6	2	2	NUM
ejpam-7053	175	7	):	):	PUNCT
ejpam-7053	175	8	this	this	PRON
ejpam-7053	175	9	follows	follow	VERB
ejpam-7053	175	10	from	from	ADP
ejpam-7053	175	11	(	(	PUNCT
ejpam-7053	175	12	4	4	NUM
ejpam-7053	175	13	)	)	PUNCT
ejpam-7053	175	14	of	of	ADP
ejpam-7053	175	15	theorem	theorem	NOUN
ejpam-7053	175	16	1	1	NUM
ejpam-7053	175	17	.	.	PUNCT
ejpam-7053	175	18	(	(	PUNCT
ejpam-7053	175	19	2	2	X
ejpam-7053	175	20	)	)	PUNCT
ejpam-7053	175	21	⇒	⇒	NOUN
ejpam-7053	175	22	(	(	PUNCT
ejpam-7053	175	23	3	3	NUM
ejpam-7053	175	24	):	):	PUNCT
ejpam-7053	175	25	the	the	DET
ejpam-7053	175	26	proof	proof	NOUN
ejpam-7053	175	27	is	be	AUX
ejpam-7053	175	28	obvious	obvious	ADJ
ejpam-7053	175	29	since	since	SCONJ
ejpam-7053	175	30	every	every	DET
ejpam-7053	175	31	(	(	PUNCT
ejpam-7053	175	32	σ1	σ1	PROPN
ejpam-7053	175	33	,	,	PUNCT
ejpam-7053	175	34	σ2)s	σ2)s	NOUN
ejpam-7053	175	35	-	-	PUNCT
ejpam-7053	175	36	open	open	ADJ
ejpam-7053	175	37	set	set	NOUN
ejpam-7053	175	38	is	be	AUX
ejpam-7053	175	39	(	(	PUNCT
ejpam-7053	175	40	σ1	σ1	PROPN
ejpam-7053	175	41	,	,	PUNCT
ejpam-7053	175	42	σ2)β	σ2)β	NOUN
ejpam-7053	175	43	-	-	PUNCT
ejpam-7053	175	44	open	open	ADJ
ejpam-7053	175	45	.	.	PUNCT
ejpam-7053	176	1	(	(	PUNCT
ejpam-7053	176	2	3	3	X
ejpam-7053	176	3	)	)	PUNCT
ejpam-7053	176	4	⇒	⇒	NOUN
ejpam-7053	176	5	(	(	PUNCT
ejpam-7053	176	6	1	1	NUM
ejpam-7053	176	7	):	):	PUNCT
ejpam-7053	176	8	since	since	SCONJ
ejpam-7053	176	9	every	every	DET
ejpam-7053	176	10	σ1σ2	σ1σ2	NUM
ejpam-7053	176	11	-	-	ADJ
ejpam-7053	176	12	open	open	ADJ
ejpam-7053	176	13	set	set	NOUN
ejpam-7053	176	14	is	be	AUX
ejpam-7053	176	15	(	(	PUNCT
ejpam-7053	176	16	σ1	σ1	PROPN
ejpam-7053	176	17	,	,	PUNCT
ejpam-7053	176	18	σ2)s	σ2)s	NOUN
ejpam-7053	176	19	-	-	PUNCT
ejpam-7053	176	20	open	open	ADJ
ejpam-7053	176	21	,	,	PUNCT
ejpam-7053	176	22	the	the	DET
ejpam-7053	176	23	proof	proof	NOUN
ejpam-7053	176	24	is	be	AUX
ejpam-7053	176	25	obvious	obvious	ADJ
ejpam-7053	176	26	by	by	ADP
ejpam-7053	176	27	(	(	PUNCT
ejpam-7053	176	28	7	7	NUM
ejpam-7053	176	29	)	)	PUNCT
ejpam-7053	176	30	of	of	ADP
ejpam-7053	176	31	theorem	theorem	ADJ
ejpam-7053	176	32	1	1	NUM
ejpam-7053	176	33	.	.	PUNCT
ejpam-7053	176	34	theorem	theorem	NOUN
ejpam-7053	176	35	4	4	NUM
ejpam-7053	176	36	.	.	X
ejpam-7053	176	37	for	for	ADP
ejpam-7053	176	38	a	a	DET
ejpam-7053	176	39	multifunction	multifunction	NOUN
ejpam-7053	176	40	f	f	NOUN
ejpam-7053	176	41	:	:	PUNCT
ejpam-7053	176	42	(	(	PUNCT
ejpam-7053	176	43	x,µ	x,µ	NOUN
ejpam-7053	176	44	)	)	PUNCT
ejpam-7053	176	45	→	→	SYM
ejpam-7053	176	46	(	(	PUNCT
ejpam-7053	176	47	y	y	PROPN
ejpam-7053	176	48	,	,	PUNCT
ejpam-7053	176	49	σ1	σ1	PROPN
ejpam-7053	176	50	,	,	PUNCT
ejpam-7053	176	51	σ2	σ2	NOUN
ejpam-7053	176	52	)	)	PUNCT
ejpam-7053	176	53	,	,	PUNCT
ejpam-7053	176	54	the	the	DET
ejpam-7053	176	55	following	follow	VERB
ejpam-7053	176	56	properties	property	NOUN
ejpam-7053	176	57	are	be	AUX
ejpam-7053	176	58	equivalent	equivalent	ADJ
ejpam-7053	176	59	:	:	PUNCT
ejpam-7053	176	60	(	(	PUNCT
ejpam-7053	176	61	1	1	X
ejpam-7053	176	62	)	)	PUNCT
ejpam-7053	176	63	f	f	PROPN
ejpam-7053	176	64	is	be	AUX
ejpam-7053	176	65	lower	low	ADJ
ejpam-7053	176	66	weakly	weakly	ADJ
ejpam-7053	176	67	µ(σ1	µ(σ1	NOUN
ejpam-7053	176	68	,	,	PUNCT
ejpam-7053	176	69	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	176	70	;	;	PUNCT
ejpam-7053	176	71	(	(	PUNCT
ejpam-7053	176	72	2	2	NUM
ejpam-7053	176	73	)	)	PUNCT
ejpam-7053	176	74	cµ(f	cµ(f	PUNCT
ejpam-7053	177	1	+	+	PROPN
ejpam-7053	177	2	(	(	PUNCT
ejpam-7053	177	3	σ1σ2	σ1σ2	NUM
ejpam-7053	177	4	-	-	PUNCT
ejpam-7053	177	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	177	6	-	-	PUNCT
ejpam-7053	177	7	cl(v	cl(v	NOUN
ejpam-7053	177	8	)	)	PUNCT
ejpam-7053	177	9	)	)	PUNCT
ejpam-7053	177	10	)	)	PUNCT
ejpam-7053	177	11	)	)	PUNCT
ejpam-7053	178	1	⊆	⊆	X
ejpam-7053	178	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	178	3	-	-	PUNCT
ejpam-7053	178	4	cl(v	cl(v	NOUN
ejpam-7053	178	5	)	)	PUNCT
ejpam-7053	178	6	)	)	PUNCT
ejpam-7053	178	7	for	for	ADP
ejpam-7053	178	8	every	every	DET
ejpam-7053	178	9	(	(	PUNCT
ejpam-7053	178	10	σ1	σ1	PROPN
ejpam-7053	178	11	,	,	PUNCT
ejpam-7053	178	12	σ2)β	σ2)β	NOUN
ejpam-7053	178	13	-	-	PUNCT
ejpam-7053	178	14	open	open	NOUN
ejpam-7053	178	15	set	set	NOUN
ejpam-7053	178	16	v	v	NOUN
ejpam-7053	178	17	of	of	ADP
ejpam-7053	178	18	y	y	PROPN
ejpam-7053	178	19	;	;	PUNCT
ejpam-7053	178	20	(	(	PUNCT
ejpam-7053	178	21	3	3	X
ejpam-7053	178	22	)	)	PUNCT
ejpam-7053	178	23	cµ(f	cµ(f	PUNCT
ejpam-7053	179	1	+	+	PROPN
ejpam-7053	179	2	(	(	PUNCT
ejpam-7053	179	3	σ1σ2	σ1σ2	NUM
ejpam-7053	179	4	-	-	PUNCT
ejpam-7053	179	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	179	6	-	-	PUNCT
ejpam-7053	179	7	cl(v	cl(v	NOUN
ejpam-7053	179	8	)	)	PUNCT
ejpam-7053	179	9	)	)	PUNCT
ejpam-7053	179	10	)	)	PUNCT
ejpam-7053	179	11	)	)	PUNCT
ejpam-7053	180	1	⊆	⊆	X
ejpam-7053	180	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	180	3	-	-	PUNCT
ejpam-7053	180	4	cl(v	cl(v	NOUN
ejpam-7053	180	5	)	)	PUNCT
ejpam-7053	180	6	)	)	PUNCT
ejpam-7053	180	7	for	for	SCONJ
ejpam-7053	180	8	every	every	DET
ejpam-7053	180	9	(	(	PUNCT
ejpam-7053	180	10	σ1	σ1	PROPN
ejpam-7053	180	11	,	,	PUNCT
ejpam-7053	180	12	σ2)s	σ2)s	NOUN
ejpam-7053	180	13	-	-	PUNCT
ejpam-7053	180	14	open	open	NOUN
ejpam-7053	180	15	set	set	NOUN
ejpam-7053	180	16	v	v	NOUN
ejpam-7053	180	17	of	of	ADP
ejpam-7053	180	18	y	y	PROPN
ejpam-7053	180	19	.	.	PUNCT
ejpam-7053	181	1	proof	proof	NOUN
ejpam-7053	181	2	.	.	PUNCT
ejpam-7053	182	1	the	the	DET
ejpam-7053	182	2	proof	proof	NOUN
ejpam-7053	182	3	is	be	AUX
ejpam-7053	182	4	similar	similar	ADJ
ejpam-7053	182	5	to	to	ADP
ejpam-7053	182	6	that	that	PRON
ejpam-7053	182	7	of	of	ADP
ejpam-7053	182	8	theorem	theorem	ADJ
ejpam-7053	182	9	3	3	NUM
ejpam-7053	182	10	.	.	PUNCT
ejpam-7053	182	11	m.	m.	NOUN
ejpam-7053	182	12	chiangpradit	chiangpradit	PROPN
ejpam-7053	182	13	,	,	PUNCT
ejpam-7053	182	14	a.	a.	PROPN
ejpam-7053	182	15	sama	sama	PROPN
ejpam-7053	182	16	-	-	PUNCT
ejpam-7053	182	17	ae	ae	PROPN
ejpam-7053	182	18	,	,	PUNCT
ejpam-7053	182	19	c.	c.	PROPN
ejpam-7053	182	20	boonpok	boonpok	PROPN
ejpam-7053	182	21	/	/	SYM
ejpam-7053	182	22	eur	eur	PROPN
ejpam-7053	182	23	.	.	PUNCT
ejpam-7053	183	1	j.	j.	PROPN
ejpam-7053	183	2	pure	pure	PROPN
ejpam-7053	183	3	appl	appl	PROPN
ejpam-7053	183	4	.	.	PROPN
ejpam-7053	183	5	math	math	PROPN
ejpam-7053	183	6	,	,	PUNCT
ejpam-7053	183	7	18	18	NUM
ejpam-7053	183	8	(	(	PUNCT
ejpam-7053	183	9	4	4	NUM
ejpam-7053	183	10	)	)	PUNCT
ejpam-7053	183	11	(	(	PUNCT
ejpam-7053	183	12	2025	2025	NUM
ejpam-7053	183	13	)	)	PUNCT
ejpam-7053	183	14	,	,	PUNCT
ejpam-7053	183	15	7053	7053	NUM
ejpam-7053	183	16	7	7	NUM
ejpam-7053	183	17	of	of	ADP
ejpam-7053	183	18	13	13	NUM
ejpam-7053	183	19	theorem	theorem	NOUN
ejpam-7053	183	20	5	5	NUM
ejpam-7053	183	21	.	.	X
ejpam-7053	183	22	for	for	ADP
ejpam-7053	183	23	a	a	DET
ejpam-7053	183	24	multifunction	multifunction	NOUN
ejpam-7053	184	1	f	f	NOUN
ejpam-7053	184	2	:	:	PUNCT
ejpam-7053	184	3	(	(	PUNCT
ejpam-7053	184	4	x,µ	x,µ	NOUN
ejpam-7053	184	5	)	)	PUNCT
ejpam-7053	184	6	→	→	SYM
ejpam-7053	184	7	(	(	PUNCT
ejpam-7053	184	8	y	y	PROPN
ejpam-7053	184	9	,	,	PUNCT
ejpam-7053	184	10	σ1	σ1	PROPN
ejpam-7053	184	11	,	,	PUNCT
ejpam-7053	184	12	σ2	σ2	NOUN
ejpam-7053	184	13	)	)	PUNCT
ejpam-7053	184	14	,	,	PUNCT
ejpam-7053	184	15	the	the	DET
ejpam-7053	184	16	following	follow	VERB
ejpam-7053	184	17	properties	property	NOUN
ejpam-7053	184	18	are	be	AUX
ejpam-7053	184	19	equivalent	equivalent	ADJ
ejpam-7053	184	20	:	:	PUNCT
ejpam-7053	184	21	(	(	PUNCT
ejpam-7053	184	22	1	1	X
ejpam-7053	184	23	)	)	PUNCT
ejpam-7053	184	24	f	f	PROPN
ejpam-7053	184	25	is	be	AUX
ejpam-7053	184	26	upper	upper	ADJ
ejpam-7053	184	27	weakly	weakly	ADJ
ejpam-7053	184	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	184	29	,	,	PUNCT
ejpam-7053	184	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	184	31	;	;	PUNCT
ejpam-7053	184	32	(	(	PUNCT
ejpam-7053	184	33	2	2	NUM
ejpam-7053	184	34	)	)	PUNCT
ejpam-7053	184	35	cµ(f	cµ(f	NOUN
ejpam-7053	184	36	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	184	37	-	-	PUNCT
ejpam-7053	184	38	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	184	39	-	-	PUNCT
ejpam-7053	184	40	cl(v	cl(v	NOUN
ejpam-7053	184	41	)	)	PUNCT
ejpam-7053	184	42	)	)	PUNCT
ejpam-7053	184	43	)	)	PUNCT
ejpam-7053	184	44	)	)	PUNCT
ejpam-7053	185	1	⊆	⊆	X
ejpam-7053	185	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	185	3	-	-	PUNCT
ejpam-7053	185	4	cl(v	cl(v	NOUN
ejpam-7053	185	5	)	)	PUNCT
ejpam-7053	185	6	)	)	PUNCT
ejpam-7053	185	7	for	for	ADP
ejpam-7053	185	8	every	every	DET
ejpam-7053	185	9	(	(	PUNCT
ejpam-7053	185	10	σ1	σ1	PROPN
ejpam-7053	185	11	,	,	PUNCT
ejpam-7053	185	12	σ2)p	σ2)p	NOUN
ejpam-7053	185	13	-	-	PUNCT
ejpam-7053	185	14	open	open	NOUN
ejpam-7053	185	15	set	set	NOUN
ejpam-7053	185	16	v	v	NOUN
ejpam-7053	185	17	of	of	ADP
ejpam-7053	185	18	y	y	PROPN
ejpam-7053	185	19	;	;	PUNCT
ejpam-7053	185	20	(	(	PUNCT
ejpam-7053	185	21	3	3	X
ejpam-7053	185	22	)	)	PUNCT
ejpam-7053	185	23	cµ(f	cµ(f	NOUN
ejpam-7053	185	24	−(v	−(v	NOUN
ejpam-7053	185	25	)	)	PUNCT
ejpam-7053	185	26	)	)	PUNCT
ejpam-7053	186	1	⊆	⊆	X
ejpam-7053	186	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	186	3	-	-	PUNCT
ejpam-7053	186	4	cl(v	cl(v	NOUN
ejpam-7053	186	5	)	)	PUNCT
ejpam-7053	186	6	)	)	PUNCT
ejpam-7053	186	7	for	for	ADP
ejpam-7053	186	8	every	every	DET
ejpam-7053	186	9	(	(	PUNCT
ejpam-7053	186	10	σ1	σ1	PROPN
ejpam-7053	186	11	,	,	PUNCT
ejpam-7053	186	12	σ2)p	σ2)p	NOUN
ejpam-7053	186	13	-	-	PUNCT
ejpam-7053	186	14	open	open	NOUN
ejpam-7053	186	15	set	set	NOUN
ejpam-7053	186	16	v	v	NOUN
ejpam-7053	186	17	of	of	ADP
ejpam-7053	186	18	y	y	PROPN
ejpam-7053	186	19	;	;	PUNCT
ejpam-7053	186	20	(	(	PUNCT
ejpam-7053	186	21	4	4	X
ejpam-7053	186	22	)	)	PUNCT
ejpam-7053	186	23	f+(v	f+(v	NOUN
ejpam-7053	186	24	)	)	PUNCT
ejpam-7053	187	1	⊆	⊆	NUM
ejpam-7053	187	2	iµ(f	iµ(f	NUM
ejpam-7053	187	3	+	+	NOUN
ejpam-7053	187	4	(	(	PUNCT
ejpam-7053	187	5	σ1σ2	σ1σ2	NOUN
ejpam-7053	187	6	-	-	NUM
ejpam-7053	187	7	cl(v	cl(v	NOUN
ejpam-7053	187	8	)	)	PUNCT
ejpam-7053	187	9	)	)	PUNCT
ejpam-7053	187	10	)	)	PUNCT
ejpam-7053	188	1	for	for	ADP
ejpam-7053	188	2	every	every	DET
ejpam-7053	188	3	(	(	PUNCT
ejpam-7053	188	4	σ1	σ1	PROPN
ejpam-7053	188	5	,	,	PUNCT
ejpam-7053	188	6	σ2)p	σ2)p	NOUN
ejpam-7053	188	7	-	-	PUNCT
ejpam-7053	188	8	open	open	NOUN
ejpam-7053	188	9	set	set	NOUN
ejpam-7053	188	10	v	v	NOUN
ejpam-7053	188	11	of	of	ADP
ejpam-7053	188	12	y	y	PROPN
ejpam-7053	188	13	.	.	PUNCT
ejpam-7053	189	1	proof	proof	NOUN
ejpam-7053	189	2	.	.	PUNCT
ejpam-7053	190	1	(	(	PUNCT
ejpam-7053	190	2	1	1	X
ejpam-7053	190	3	)	)	PUNCT
ejpam-7053	190	4	⇒	⇒	NOUN
ejpam-7053	190	5	(	(	PUNCT
ejpam-7053	190	6	2	2	NUM
ejpam-7053	190	7	):	):	PUNCT
ejpam-7053	190	8	let	let	VERB
ejpam-7053	190	9	v	v	PART
ejpam-7053	190	10	be	be	AUX
ejpam-7053	190	11	any	any	DET
ejpam-7053	190	12	(	(	PUNCT
ejpam-7053	190	13	σ1	σ1	PROPN
ejpam-7053	190	14	,	,	PUNCT
ejpam-7053	190	15	σ2)p	σ2)p	NOUN
ejpam-7053	190	16	-	-	PUNCT
ejpam-7053	190	17	open	open	ADJ
ejpam-7053	190	18	set	set	NOUN
ejpam-7053	190	19	of	of	ADP
ejpam-7053	190	20	y	y	PROPN
ejpam-7053	190	21	.	.	PUNCT
ejpam-7053	191	1	since	since	SCONJ
ejpam-7053	191	2	σ1σ2	σ1σ2	ADV
ejpam-7053	191	3	-	-	PUNCT
ejpam-7053	191	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	191	5	-	-	PUNCT
ejpam-7053	191	6	cl(v	cl(v	NOUN
ejpam-7053	191	7	)	)	PUNCT
ejpam-7053	191	8	)	)	PUNCT
ejpam-7053	191	9	is	be	AUX
ejpam-7053	191	10	σ1σ2	σ1σ2	NOUN
ejpam-7053	191	11	-	-	ADJ
ejpam-7053	191	12	open	open	ADJ
ejpam-7053	191	13	,	,	PUNCT
ejpam-7053	191	14	by	by	ADP
ejpam-7053	191	15	theorem	theorem	ADJ
ejpam-7053	191	16	1(7	1(7	PROPN
ejpam-7053	191	17	)	)	PUNCT
ejpam-7053	191	18	cµ(f	cµ(f	NOUN
ejpam-7053	191	19	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	191	20	-	-	PUNCT
ejpam-7053	191	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	191	22	-	-	PUNCT
ejpam-7053	191	23	cl(v	cl(v	NOUN
ejpam-7053	191	24	)	)	PUNCT
ejpam-7053	191	25	)	)	PUNCT
ejpam-7053	191	26	)	)	PUNCT
ejpam-7053	191	27	)	)	PUNCT
ejpam-7053	192	1	⊆	⊆	X
ejpam-7053	192	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7053	192	3	-	-	PUNCT
ejpam-7053	192	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	192	5	-	-	PUNCT
ejpam-7053	192	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	192	7	-	-	PUNCT
ejpam-7053	192	8	cl(v	cl(v	NOUN
ejpam-7053	192	9	)	)	PUNCT
ejpam-7053	192	10	)	)	PUNCT
ejpam-7053	192	11	)	)	PUNCT
ejpam-7053	192	12	)	)	PUNCT
ejpam-7053	193	1	⊆	⊆	X
ejpam-7053	193	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	193	3	-	-	PUNCT
ejpam-7053	193	4	cl(v	cl(v	NOUN
ejpam-7053	193	5	)	)	PUNCT
ejpam-7053	193	6	)	)	PUNCT
ejpam-7053	193	7	.	.	PUNCT
ejpam-7053	194	1	(	(	PUNCT
ejpam-7053	194	2	2	2	X
ejpam-7053	194	3	)	)	PUNCT
ejpam-7053	194	4	⇒	⇒	NOUN
ejpam-7053	194	5	(	(	PUNCT
ejpam-7053	194	6	3	3	NUM
ejpam-7053	194	7	):	):	PUNCT
ejpam-7053	194	8	let	let	VERB
ejpam-7053	194	9	v	v	PART
ejpam-7053	194	10	be	be	AUX
ejpam-7053	194	11	any	any	DET
ejpam-7053	194	12	(	(	PUNCT
ejpam-7053	194	13	σ1	σ1	PROPN
ejpam-7053	194	14	,	,	PUNCT
ejpam-7053	194	15	σ2)p	σ2)p	NOUN
ejpam-7053	194	16	-	-	PUNCT
ejpam-7053	194	17	open	open	ADJ
ejpam-7053	194	18	set	set	NOUN
ejpam-7053	194	19	of	of	ADP
ejpam-7053	194	20	y	y	PROPN
ejpam-7053	194	21	.	.	PUNCT
ejpam-7053	195	1	by	by	ADP
ejpam-7053	195	2	(	(	PUNCT
ejpam-7053	195	3	2	2	NUM
ejpam-7053	195	4	)	)	PUNCT
ejpam-7053	195	5	,	,	PUNCT
ejpam-7053	195	6	we	we	PRON
ejpam-7053	195	7	have	have	VERB
ejpam-7053	195	8	cµ(f	cµ(f	PROPN
ejpam-7053	195	9	−(v	−(v	NOUN
ejpam-7053	195	10	)	)	PUNCT
ejpam-7053	195	11	)	)	PUNCT
ejpam-7053	196	1	⊆	⊆	NUM
ejpam-7053	196	2	cµ(f	cµ(f	NUM
ejpam-7053	196	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	196	4	-	-	PUNCT
ejpam-7053	196	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	196	6	-	-	PUNCT
ejpam-7053	196	7	cl(v	cl(v	NOUN
ejpam-7053	196	8	)	)	PUNCT
ejpam-7053	196	9	)	)	PUNCT
ejpam-7053	196	10	)	)	PUNCT
ejpam-7053	196	11	)	)	PUNCT
ejpam-7053	197	1	⊆	⊆	X
ejpam-7053	197	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7053	197	3	-	-	PUNCT
ejpam-7053	197	4	cl(v	cl(v	NOUN
ejpam-7053	197	5	)	)	PUNCT
ejpam-7053	197	6	)	)	PUNCT
ejpam-7053	197	7	.	.	PUNCT
ejpam-7053	198	1	(	(	PUNCT
ejpam-7053	198	2	3	3	X
ejpam-7053	198	3	)	)	PUNCT
ejpam-7053	198	4	⇒	⇒	NOUN
ejpam-7053	198	5	(	(	PUNCT
ejpam-7053	198	6	4	4	NUM
ejpam-7053	198	7	):	):	PUNCT
ejpam-7053	198	8	let	let	VERB
ejpam-7053	198	9	v	v	PART
ejpam-7053	198	10	be	be	AUX
ejpam-7053	198	11	any	any	DET
ejpam-7053	198	12	(	(	PUNCT
ejpam-7053	198	13	σ1	σ1	PROPN
ejpam-7053	198	14	,	,	PUNCT
ejpam-7053	198	15	σ2)p	σ2)p	NOUN
ejpam-7053	198	16	-	-	PUNCT
ejpam-7053	198	17	open	open	ADJ
ejpam-7053	198	18	set	set	NOUN
ejpam-7053	198	19	of	of	ADP
ejpam-7053	198	20	y	y	PROPN
ejpam-7053	198	21	.	.	PUNCT
ejpam-7053	199	1	thus	thus	ADV
ejpam-7053	199	2	by	by	ADP
ejpam-7053	199	3	(	(	PUNCT
ejpam-7053	199	4	3	3	NUM
ejpam-7053	199	5	)	)	PUNCT
ejpam-7053	199	6	,	,	PUNCT
ejpam-7053	199	7	x	x	X
ejpam-7053	199	8	−	−	NOUN
ejpam-7053	199	9	iµ(f	iµ(f	X
ejpam-7053	199	10	+	+	NOUN
ejpam-7053	199	11	(	(	PUNCT
ejpam-7053	199	12	σ1σ2	σ1σ2	NOUN
ejpam-7053	199	13	-	-	NUM
ejpam-7053	199	14	cl(v	cl(v	NOUN
ejpam-7053	199	15	)	)	PUNCT
ejpam-7053	199	16	)	)	PUNCT
ejpam-7053	199	17	)	)	PUNCT
ejpam-7053	200	1	=	=	PRON
ejpam-7053	200	2	cµ(x	cµ(x	PUNCT
ejpam-7053	200	3	−	−	ADP
ejpam-7053	200	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	200	5	-	-	PUNCT
ejpam-7053	200	6	cl(v	cl(v	NOUN
ejpam-7053	200	7	)	)	PUNCT
ejpam-7053	200	8	)	)	PUNCT
ejpam-7053	200	9	)	)	PUNCT
ejpam-7053	201	1	=	=	SYM
ejpam-7053	201	2	cµ(f	cµ(f	NOUN
ejpam-7053	202	1	−(y	−(y	NOUN
ejpam-7053	202	2	−	−	NOUN
ejpam-7053	202	3	σ1σ2	σ1σ2	NOUN
ejpam-7053	202	4	-	-	NUM
ejpam-7053	202	5	cl(v	cl(v	NOUN
ejpam-7053	202	6	)	)	PUNCT
ejpam-7053	202	7	)	)	PUNCT
ejpam-7053	202	8	)	)	PUNCT
ejpam-7053	203	1	⊆	⊆	X
ejpam-7053	203	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7053	203	3	-	-	PUNCT
ejpam-7053	203	4	cl(y	cl(y	NOUN
ejpam-7053	203	5	−	−	NOUN
ejpam-7053	203	6	σ1σ2	σ1σ2	NOUN
ejpam-7053	203	7	-	-	NUM
ejpam-7053	203	8	cl(v	cl(v	NOUN
ejpam-7053	203	9	)	)	PUNCT
ejpam-7053	203	10	)	)	PUNCT
ejpam-7053	203	11	)	)	PUNCT
ejpam-7053	204	1	=	=	PUNCT
ejpam-7053	204	2	x	x	X
ejpam-7053	204	3	−	−	ADP
ejpam-7053	204	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7053	204	5	-	-	PUNCT
ejpam-7053	204	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	204	7	-	-	PUNCT
ejpam-7053	204	8	cl(v	cl(v	NOUN
ejpam-7053	204	9	)	)	PUNCT
ejpam-7053	204	10	)	)	PUNCT
ejpam-7053	204	11	)	)	PUNCT
ejpam-7053	205	1	⊆	⊆	NUM
ejpam-7053	205	2	x	x	SYM
ejpam-7053	205	3	−	−	NOUN
ejpam-7053	205	4	f+(v	f+(v	NOUN
ejpam-7053	205	5	)	)	PUNCT
ejpam-7053	205	6	and	and	CCONJ
ejpam-7053	205	7	hence	hence	ADV
ejpam-7053	205	8	f+(v	f+(v	PROPN
ejpam-7053	205	9	)	)	PUNCT
ejpam-7053	206	1	⊆	⊆	NUM
ejpam-7053	206	2	iµ(f	iµ(f	NUM
ejpam-7053	206	3	+	+	NOUN
ejpam-7053	206	4	(	(	PUNCT
ejpam-7053	206	5	σ1σ2	σ1σ2	NOUN
ejpam-7053	206	6	-	-	NUM
ejpam-7053	206	7	cl(v	cl(v	NOUN
ejpam-7053	206	8	)	)	PUNCT
ejpam-7053	206	9	)	)	PUNCT
ejpam-7053	206	10	)	)	PUNCT
ejpam-7053	206	11	.	.	PUNCT
ejpam-7053	207	1	(	(	PUNCT
ejpam-7053	207	2	4	4	X
ejpam-7053	207	3	)	)	PUNCT
ejpam-7053	207	4	⇒	⇒	NOUN
ejpam-7053	207	5	(	(	PUNCT
ejpam-7053	207	6	1	1	NUM
ejpam-7053	207	7	):	):	PUNCT
ejpam-7053	207	8	let	let	VERB
ejpam-7053	207	9	v	v	PART
ejpam-7053	207	10	be	be	AUX
ejpam-7053	207	11	any	any	DET
ejpam-7053	207	12	σ1σ2	σ1σ2	NOUN
ejpam-7053	207	13	-	-	ADJ
ejpam-7053	207	14	open	open	ADJ
ejpam-7053	207	15	set	set	NOUN
ejpam-7053	207	16	of	of	ADP
ejpam-7053	207	17	y	y	PROPN
ejpam-7053	207	18	.	.	PUNCT
ejpam-7053	208	1	then	then	ADV
ejpam-7053	208	2	,	,	PUNCT
ejpam-7053	208	3	v	v	NOUN
ejpam-7053	208	4	is	be	AUX
ejpam-7053	208	5	(	(	PUNCT
ejpam-7053	208	6	σ1	σ1	PROPN
ejpam-7053	208	7	,	,	PUNCT
ejpam-7053	208	8	σ2)p	σ2)p	NOUN
ejpam-7053	208	9	-	-	PUNCT
ejpam-7053	208	10	open	open	ADJ
ejpam-7053	208	11	in	in	ADP
ejpam-7053	208	12	y	y	PROPN
ejpam-7053	208	13	and	and	CCONJ
ejpam-7053	208	14	by	by	ADP
ejpam-7053	208	15	(	(	PUNCT
ejpam-7053	208	16	4	4	NUM
ejpam-7053	208	17	)	)	PUNCT
ejpam-7053	208	18	,	,	PUNCT
ejpam-7053	208	19	f+(v	f+(v	PROPN
ejpam-7053	208	20	)	)	PUNCT
ejpam-7053	209	1	⊆	⊆	NUM
ejpam-7053	209	2	iµ(f	iµ(f	NUM
ejpam-7053	209	3	+	+	NOUN
ejpam-7053	209	4	(	(	PUNCT
ejpam-7053	209	5	σ1σ2	σ1σ2	NOUN
ejpam-7053	209	6	-	-	NUM
ejpam-7053	209	7	cl(v	cl(v	NOUN
ejpam-7053	209	8	)	)	PUNCT
ejpam-7053	209	9	)	)	PUNCT
ejpam-7053	209	10	)	)	PUNCT
ejpam-7053	209	11	.	.	PUNCT
ejpam-7053	210	1	by	by	ADP
ejpam-7053	210	2	theorem	theorem	NOUN
ejpam-7053	210	3	1(2	1(2	NUM
ejpam-7053	210	4	)	)	PUNCT
ejpam-7053	210	5	,	,	PUNCT
ejpam-7053	210	6	f	f	PROPN
ejpam-7053	210	7	is	be	AUX
ejpam-7053	210	8	upper	upper	ADJ
ejpam-7053	210	9	weakly	weakly	ADJ
ejpam-7053	210	10	µ(σ1	µ(σ1	NOUN
ejpam-7053	210	11	,	,	PUNCT
ejpam-7053	210	12	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	210	13	.	.	X
ejpam-7053	210	14	theorem	theorem	VERB
ejpam-7053	210	15	6	6	NUM
ejpam-7053	210	16	.	.	PUNCT
ejpam-7053	210	17	for	for	ADP
ejpam-7053	210	18	a	a	DET
ejpam-7053	210	19	multifunction	multifunction	NOUN
ejpam-7053	210	20	f	f	NOUN
ejpam-7053	210	21	:	:	PUNCT
ejpam-7053	210	22	(	(	PUNCT
ejpam-7053	210	23	x,µ	x,µ	NOUN
ejpam-7053	210	24	)	)	PUNCT
ejpam-7053	210	25	→	→	SYM
ejpam-7053	210	26	(	(	PUNCT
ejpam-7053	210	27	y	y	PROPN
ejpam-7053	210	28	,	,	PUNCT
ejpam-7053	210	29	σ1	σ1	PROPN
ejpam-7053	210	30	,	,	PUNCT
ejpam-7053	210	31	σ2	σ2	NOUN
ejpam-7053	210	32	)	)	PUNCT
ejpam-7053	210	33	,	,	PUNCT
ejpam-7053	210	34	the	the	DET
ejpam-7053	210	35	following	follow	VERB
ejpam-7053	210	36	properties	property	NOUN
ejpam-7053	210	37	are	be	AUX
ejpam-7053	210	38	equivalent	equivalent	ADJ
ejpam-7053	210	39	:	:	PUNCT
ejpam-7053	210	40	(	(	PUNCT
ejpam-7053	210	41	1	1	X
ejpam-7053	210	42	)	)	PUNCT
ejpam-7053	210	43	f	f	PROPN
ejpam-7053	210	44	is	be	AUX
ejpam-7053	210	45	lower	low	ADJ
ejpam-7053	210	46	weakly	weakly	ADJ
ejpam-7053	210	47	µ(σ1	µ(σ1	NOUN
ejpam-7053	210	48	,	,	PUNCT
ejpam-7053	210	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	210	50	;	;	PUNCT
ejpam-7053	210	51	(	(	PUNCT
ejpam-7053	210	52	2	2	NUM
ejpam-7053	210	53	)	)	PUNCT
ejpam-7053	210	54	cµ(f	cµ(f	PUNCT
ejpam-7053	211	1	+	+	PROPN
ejpam-7053	211	2	(	(	PUNCT
ejpam-7053	211	3	σ1σ2	σ1σ2	NUM
ejpam-7053	211	4	-	-	PUNCT
ejpam-7053	211	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	211	6	-	-	PUNCT
ejpam-7053	211	7	cl(v	cl(v	NOUN
ejpam-7053	211	8	)	)	PUNCT
ejpam-7053	211	9	)	)	PUNCT
ejpam-7053	211	10	)	)	PUNCT
ejpam-7053	211	11	)	)	PUNCT
ejpam-7053	212	1	⊆	⊆	X
ejpam-7053	212	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	212	3	-	-	PUNCT
ejpam-7053	212	4	cl(v	cl(v	NOUN
ejpam-7053	212	5	)	)	PUNCT
ejpam-7053	212	6	)	)	PUNCT
ejpam-7053	212	7	for	for	ADP
ejpam-7053	212	8	every	every	DET
ejpam-7053	212	9	(	(	PUNCT
ejpam-7053	212	10	σ1	σ1	PROPN
ejpam-7053	212	11	,	,	PUNCT
ejpam-7053	212	12	σ2)p	σ2)p	NOUN
ejpam-7053	212	13	-	-	PUNCT
ejpam-7053	212	14	open	open	NOUN
ejpam-7053	212	15	set	set	NOUN
ejpam-7053	212	16	v	v	NOUN
ejpam-7053	212	17	of	of	ADP
ejpam-7053	212	18	y	y	PROPN
ejpam-7053	212	19	;	;	PUNCT
ejpam-7053	212	20	(	(	PUNCT
ejpam-7053	212	21	3	3	X
ejpam-7053	212	22	)	)	PUNCT
ejpam-7053	212	23	cµ(f	cµ(f	PUNCT
ejpam-7053	213	1	+	+	PROPN
ejpam-7053	213	2	(	(	PUNCT
ejpam-7053	213	3	v	v	NOUN
ejpam-7053	213	4	)	)	PUNCT
ejpam-7053	213	5	)	)	PUNCT
ejpam-7053	213	6	⊆	⊆	NUM
ejpam-7053	213	7	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	213	8	-	-	PUNCT
ejpam-7053	213	9	cl(v	cl(v	NOUN
ejpam-7053	213	10	)	)	PUNCT
ejpam-7053	213	11	)	)	PUNCT
ejpam-7053	213	12	for	for	ADP
ejpam-7053	213	13	every	every	DET
ejpam-7053	213	14	(	(	PUNCT
ejpam-7053	213	15	σ1	σ1	PROPN
ejpam-7053	213	16	,	,	PUNCT
ejpam-7053	213	17	σ2)p	σ2)p	NOUN
ejpam-7053	213	18	-	-	PUNCT
ejpam-7053	213	19	open	open	NOUN
ejpam-7053	213	20	set	set	NOUN
ejpam-7053	213	21	v	v	NOUN
ejpam-7053	213	22	of	of	ADP
ejpam-7053	213	23	y	y	PROPN
ejpam-7053	213	24	;	;	PUNCT
ejpam-7053	213	25	(	(	PUNCT
ejpam-7053	213	26	4	4	X
ejpam-7053	213	27	)	)	PUNCT
ejpam-7053	213	28	f−(v	f−(v	NOUN
ejpam-7053	213	29	)	)	PUNCT
ejpam-7053	213	30	⊆	⊆	NUM
ejpam-7053	213	31	iµ(f	iµ(f	NUM
ejpam-7053	213	32	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	213	33	-	-	NOUN
ejpam-7053	213	34	cl(v	cl(v	NOUN
ejpam-7053	213	35	)	)	PUNCT
ejpam-7053	213	36	)	)	PUNCT
ejpam-7053	213	37	)	)	PUNCT
ejpam-7053	213	38	for	for	ADP
ejpam-7053	213	39	every	every	DET
ejpam-7053	213	40	(	(	PUNCT
ejpam-7053	213	41	σ1	σ1	PROPN
ejpam-7053	213	42	,	,	PUNCT
ejpam-7053	213	43	σ2)p	σ2)p	NOUN
ejpam-7053	213	44	-	-	PUNCT
ejpam-7053	213	45	open	open	NOUN
ejpam-7053	213	46	set	set	NOUN
ejpam-7053	213	47	v	v	NOUN
ejpam-7053	213	48	of	of	ADP
ejpam-7053	213	49	y	y	PROPN
ejpam-7053	213	50	.	.	PUNCT
ejpam-7053	214	1	proof	proof	NOUN
ejpam-7053	214	2	.	.	PUNCT
ejpam-7053	215	1	the	the	DET
ejpam-7053	215	2	proof	proof	NOUN
ejpam-7053	215	3	is	be	AUX
ejpam-7053	215	4	similar	similar	ADJ
ejpam-7053	215	5	to	to	ADP
ejpam-7053	215	6	that	that	PRON
ejpam-7053	215	7	of	of	ADP
ejpam-7053	215	8	theorem	theorem	ADJ
ejpam-7053	215	9	5	5	NUM
ejpam-7053	215	10	.	.	PUNCT
ejpam-7053	215	11	m.	m.	NOUN
ejpam-7053	215	12	chiangpradit	chiangpradit	PROPN
ejpam-7053	215	13	,	,	PUNCT
ejpam-7053	215	14	a.	a.	PROPN
ejpam-7053	215	15	sama	sama	PROPN
ejpam-7053	215	16	-	-	PUNCT
ejpam-7053	215	17	ae	ae	PROPN
ejpam-7053	215	18	,	,	PUNCT
ejpam-7053	215	19	c.	c.	PROPN
ejpam-7053	215	20	boonpok	boonpok	PROPN
ejpam-7053	215	21	/	/	SYM
ejpam-7053	215	22	eur	eur	PROPN
ejpam-7053	215	23	.	.	PUNCT
ejpam-7053	216	1	j.	j.	PROPN
ejpam-7053	216	2	pure	pure	PROPN
ejpam-7053	216	3	appl	appl	PROPN
ejpam-7053	216	4	.	.	PROPN
ejpam-7053	216	5	math	math	PROPN
ejpam-7053	216	6	,	,	PUNCT
ejpam-7053	216	7	18	18	NUM
ejpam-7053	216	8	(	(	PUNCT
ejpam-7053	216	9	4	4	NUM
ejpam-7053	216	10	)	)	PUNCT
ejpam-7053	216	11	(	(	PUNCT
ejpam-7053	216	12	2025	2025	NUM
ejpam-7053	216	13	)	)	PUNCT
ejpam-7053	216	14	,	,	PUNCT
ejpam-7053	216	15	7053	7053	NUM
ejpam-7053	216	16	8	8	NUM
ejpam-7053	216	17	of	of	ADP
ejpam-7053	216	18	13	13	NUM
ejpam-7053	216	19	lemma	lemma	PROPN
ejpam-7053	216	20	3	3	NUM
ejpam-7053	216	21	.	.	PUNCT
ejpam-7053	217	1	if	if	SCONJ
ejpam-7053	217	2	f	f	PROPN
ejpam-7053	217	3	:	:	PUNCT
ejpam-7053	217	4	(	(	PUNCT
ejpam-7053	217	5	x,µ	x,µ	NOUN
ejpam-7053	217	6	)	)	PUNCT
ejpam-7053	217	7	→	→	SYM
ejpam-7053	217	8	(	(	PUNCT
ejpam-7053	217	9	y	y	PROPN
ejpam-7053	217	10	,	,	PUNCT
ejpam-7053	217	11	σ1	σ1	PROPN
ejpam-7053	217	12	,	,	PUNCT
ejpam-7053	217	13	σ2	σ2	NOUN
ejpam-7053	217	14	)	)	PUNCT
ejpam-7053	217	15	is	be	AUX
ejpam-7053	217	16	lower	low	ADJ
ejpam-7053	217	17	weakly	weakly	ADJ
ejpam-7053	217	18	µ(σ1	µ(σ1	NOUN
ejpam-7053	217	19	,	,	PUNCT
ejpam-7053	217	20	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	217	21	,	,	PUNCT
ejpam-7053	217	22	then	then	ADV
ejpam-7053	217	23	for	for	ADP
ejpam-7053	217	24	each	each	DET
ejpam-7053	217	25	x	x	SYM
ejpam-7053	217	26	∈	∈	PROPN
ejpam-7053	217	27	x	x	X
ejpam-7053	217	28	and	and	CCONJ
ejpam-7053	217	29	each	each	DET
ejpam-7053	217	30	subset	subset	NOUN
ejpam-7053	217	31	b	b	PROPN
ejpam-7053	217	32	of	of	ADP
ejpam-7053	217	33	y	y	PROPN
ejpam-7053	217	34	with	with	ADP
ejpam-7053	217	35	(	(	PUNCT
ejpam-7053	217	36	σ1	σ1	PROPN
ejpam-7053	217	37	,	,	PUNCT
ejpam-7053	217	38	σ2)θ	σ2)θ	NOUN
ejpam-7053	217	39	-	-	PUNCT
ejpam-7053	217	40	int(b)∩	int(b)∩	NOUN
ejpam-7053	217	41	f	f	NOUN
ejpam-7053	217	42	(	(	PUNCT
ejpam-7053	217	43	x	x	X
ejpam-7053	217	44	)	)	PUNCT
ejpam-7053	217	45	̸=	̸=	NOUN
ejpam-7053	217	46	∅	∅	NOUN
ejpam-7053	217	47	,	,	PUNCT
ejpam-7053	217	48	there	there	PRON
ejpam-7053	217	49	exists	exist	VERB
ejpam-7053	217	50	a	a	DET
ejpam-7053	217	51	µ-open	µ-open	NOUN
ejpam-7053	217	52	set	set	VERB
ejpam-7053	217	53	u	u	NOUN
ejpam-7053	217	54	of	of	ADP
ejpam-7053	217	55	x	x	PUNCT
ejpam-7053	217	56	containing	contain	VERB
ejpam-7053	217	57	x	x	PUNCT
ejpam-7053	217	58	such	such	ADJ
ejpam-7053	217	59	that	that	SCONJ
ejpam-7053	217	60	u	u	PROPN
ejpam-7053	217	61	⊆	⊆	NUM
ejpam-7053	217	62	f−(b	f−(b	NOUN
ejpam-7053	217	63	)	)	PUNCT
ejpam-7053	217	64	.	.	PUNCT
ejpam-7053	218	1	proof	proof	NOUN
ejpam-7053	218	2	.	.	PUNCT
ejpam-7053	219	1	since	since	SCONJ
ejpam-7053	219	2	(	(	PUNCT
ejpam-7053	219	3	σ1	σ1	PROPN
ejpam-7053	219	4	,	,	PUNCT
ejpam-7053	219	5	σ2)θ	σ2)θ	NOUN
ejpam-7053	219	6	-	-	PUNCT
ejpam-7053	219	7	int(b	int(b	NOUN
ejpam-7053	219	8	)	)	PUNCT
ejpam-7053	219	9	∩	∩	ADJ
ejpam-7053	219	10	f	f	PROPN
ejpam-7053	219	11	(	(	PUNCT
ejpam-7053	219	12	x	x	X
ejpam-7053	219	13	)	)	PUNCT
ejpam-7053	219	14	̸=	̸=	NOUN
ejpam-7053	219	15	∅	∅	NOUN
ejpam-7053	219	16	,	,	PUNCT
ejpam-7053	219	17	there	there	PRON
ejpam-7053	219	18	exists	exist	VERB
ejpam-7053	219	19	a	a	DET
ejpam-7053	219	20	nonempty	nonempty	ADJ
ejpam-7053	219	21	σ1σ2	σ1σ2	NOUN
ejpam-7053	219	22	-	-	ADJ
ejpam-7053	219	23	open	open	ADJ
ejpam-7053	219	24	set	set	NOUN
ejpam-7053	219	25	v	v	NOUN
ejpam-7053	219	26	of	of	ADP
ejpam-7053	219	27	y	y	PRON
ejpam-7053	219	28	such	such	ADJ
ejpam-7053	219	29	that	that	SCONJ
ejpam-7053	219	30	σ1σ2	σ1σ2	NOUN
ejpam-7053	219	31	-	-	NUM
ejpam-7053	219	32	cl(v	cl(v	NOUN
ejpam-7053	219	33	)	)	PUNCT
ejpam-7053	219	34	⊆	⊆	NUM
ejpam-7053	219	35	b	b	NOUN
ejpam-7053	219	36	and	and	CCONJ
ejpam-7053	219	37	v	v	NOUN
ejpam-7053	219	38	∩	∩	ADJ
ejpam-7053	219	39	f	f	X
ejpam-7053	219	40	(	(	PUNCT
ejpam-7053	219	41	x	x	X
ejpam-7053	219	42	)	)	PUNCT
ejpam-7053	219	43	̸=	̸=	PROPN
ejpam-7053	219	44	∅.	∅.	ADV
ejpam-7053	219	45	since	since	SCONJ
ejpam-7053	219	46	f	f	PROPN
ejpam-7053	219	47	is	be	AUX
ejpam-7053	219	48	lower	low	ADJ
ejpam-7053	219	49	weakly	weakly	ADJ
ejpam-7053	219	50	µ(σ1	µ(σ1	NOUN
ejpam-7053	219	51	,	,	PUNCT
ejpam-7053	219	52	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	219	53	,	,	PUNCT
ejpam-7053	219	54	there	there	PRON
ejpam-7053	219	55	exists	exist	VERB
ejpam-7053	219	56	a	a	DET
ejpam-7053	219	57	µ-open	µ-open	NOUN
ejpam-7053	219	58	set	set	VERB
ejpam-7053	219	59	u	u	NOUN
ejpam-7053	219	60	of	of	ADP
ejpam-7053	219	61	x	x	PUNCT
ejpam-7053	219	62	containing	contain	VERB
ejpam-7053	219	63	x	x	PUNCT
ejpam-7053	219	64	such	such	ADJ
ejpam-7053	219	65	that	that	SCONJ
ejpam-7053	219	66	σ1σ2	σ1σ2	NOUN
ejpam-7053	219	67	-	-	NUM
ejpam-7053	219	68	cl(v	cl(v	PUNCT
ejpam-7053	219	69	)	)	PUNCT
ejpam-7053	219	70	∩f	∩f	NOUN
ejpam-7053	219	71	(	(	PUNCT
ejpam-7053	219	72	z	z	X
ejpam-7053	219	73	)	)	PUNCT
ejpam-7053	219	74	̸=	̸=	NOUN
ejpam-7053	219	75	∅	∅	NOUN
ejpam-7053	219	76	for	for	ADP
ejpam-7053	219	77	each	each	DET
ejpam-7053	219	78	z	z	NOUN
ejpam-7053	219	79	∈	∈	PROPN
ejpam-7053	219	80	u	u	NOUN
ejpam-7053	219	81	and	and	CCONJ
ejpam-7053	219	82	hence	hence	ADV
ejpam-7053	219	83	u	u	NOUN
ejpam-7053	219	84	⊆	⊆	NUM
ejpam-7053	219	85	f−(b	f−(b	NOUN
ejpam-7053	219	86	)	)	PUNCT
ejpam-7053	219	87	.	.	PUNCT
ejpam-7053	220	1	theorem	theorem	VERB
ejpam-7053	220	2	7	7	NUM
ejpam-7053	220	3	.	.	X
ejpam-7053	220	4	for	for	ADP
ejpam-7053	220	5	a	a	DET
ejpam-7053	220	6	multifunction	multifunction	NOUN
ejpam-7053	221	1	f	f	NOUN
ejpam-7053	221	2	:	:	PUNCT
ejpam-7053	221	3	(	(	PUNCT
ejpam-7053	221	4	x,µ	x,µ	NOUN
ejpam-7053	221	5	)	)	PUNCT
ejpam-7053	221	6	→	→	SYM
ejpam-7053	221	7	(	(	PUNCT
ejpam-7053	221	8	y	y	PROPN
ejpam-7053	221	9	,	,	PUNCT
ejpam-7053	221	10	σ1	σ1	PROPN
ejpam-7053	221	11	,	,	PUNCT
ejpam-7053	221	12	σ2	σ2	NOUN
ejpam-7053	221	13	)	)	PUNCT
ejpam-7053	221	14	,	,	PUNCT
ejpam-7053	221	15	the	the	DET
ejpam-7053	221	16	following	follow	VERB
ejpam-7053	221	17	properties	property	NOUN
ejpam-7053	221	18	are	be	AUX
ejpam-7053	221	19	equivalent	equivalent	ADJ
ejpam-7053	221	20	:	:	PUNCT
ejpam-7053	221	21	(	(	PUNCT
ejpam-7053	221	22	1	1	X
ejpam-7053	221	23	)	)	PUNCT
ejpam-7053	221	24	f	f	PROPN
ejpam-7053	221	25	is	be	AUX
ejpam-7053	221	26	lower	low	ADJ
ejpam-7053	221	27	weakly	weakly	ADJ
ejpam-7053	221	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	221	29	,	,	PUNCT
ejpam-7053	221	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	221	31	;	;	PUNCT
ejpam-7053	221	32	(	(	PUNCT
ejpam-7053	221	33	2	2	NUM
ejpam-7053	221	34	)	)	PUNCT
ejpam-7053	221	35	cµ(f	cµ(f	PUNCT
ejpam-7053	222	1	+	+	PROPN
ejpam-7053	222	2	(	(	PUNCT
ejpam-7053	222	3	b	b	NOUN
ejpam-7053	222	4	)	)	PUNCT
ejpam-7053	222	5	)	)	PUNCT
ejpam-7053	222	6	⊆	⊆	NUM
ejpam-7053	222	7	f+((σ1	f+((σ1	NOUN
ejpam-7053	222	8	,	,	PUNCT
ejpam-7053	222	9	σ2)θ	σ2)θ	ADJ
ejpam-7053	222	10	-	-	PUNCT
ejpam-7053	222	11	cl(b	cl(b	NOUN
ejpam-7053	222	12	)	)	PUNCT
ejpam-7053	222	13	)	)	PUNCT
ejpam-7053	222	14	for	for	ADP
ejpam-7053	222	15	every	every	DET
ejpam-7053	222	16	subset	subset	NOUN
ejpam-7053	222	17	b	b	PROPN
ejpam-7053	222	18	of	of	ADP
ejpam-7053	222	19	y	y	PROPN
ejpam-7053	222	20	;	;	PUNCT
ejpam-7053	222	21	(	(	PUNCT
ejpam-7053	222	22	3	3	X
ejpam-7053	222	23	)	)	PUNCT
ejpam-7053	222	24	f	f	NOUN
ejpam-7053	222	25	(	(	PUNCT
ejpam-7053	222	26	cµ(a	cµ(a	PROPN
ejpam-7053	222	27	)	)	PUNCT
ejpam-7053	222	28	)	)	PUNCT
ejpam-7053	223	1	⊆	⊆	NUM
ejpam-7053	223	2	(	(	PUNCT
ejpam-7053	223	3	σ1	σ1	PROPN
ejpam-7053	223	4	,	,	PUNCT
ejpam-7053	223	5	σ2)θ	σ2)θ	NOUN
ejpam-7053	223	6	-	-	PUNCT
ejpam-7053	223	7	cl(f	cl(f	PROPN
ejpam-7053	223	8	(	(	PUNCT
ejpam-7053	223	9	a	a	NOUN
ejpam-7053	223	10	)	)	PUNCT
ejpam-7053	223	11	)	)	PUNCT
ejpam-7053	223	12	for	for	ADP
ejpam-7053	223	13	every	every	DET
ejpam-7053	223	14	subset	subset	NOUN
ejpam-7053	223	15	a	a	PRON
ejpam-7053	223	16	of	of	ADP
ejpam-7053	223	17	x.	x.	NOUN
ejpam-7053	223	18	proof	proof	NOUN
ejpam-7053	223	19	.	.	PUNCT
ejpam-7053	224	1	(	(	PUNCT
ejpam-7053	224	2	1	1	X
ejpam-7053	224	3	)	)	PUNCT
ejpam-7053	224	4	⇒	⇒	NOUN
ejpam-7053	224	5	(	(	PUNCT
ejpam-7053	224	6	2	2	NUM
ejpam-7053	224	7	):	):	PUNCT
ejpam-7053	224	8	let	let	VERB
ejpam-7053	224	9	b	b	X
ejpam-7053	224	10	be	be	AUX
ejpam-7053	224	11	any	any	DET
ejpam-7053	224	12	subset	subset	NOUN
ejpam-7053	224	13	of	of	ADP
ejpam-7053	224	14	y	y	PROPN
ejpam-7053	224	15	.	.	PUNCT
ejpam-7053	224	16	suppose	suppose	VERB
ejpam-7053	224	17	that	that	SCONJ
ejpam-7053	224	18	x	x	PROPN
ejpam-7053	224	19	̸∈	̸∈	PROPN
ejpam-7053	224	20	f+((σ1	f+((σ1	ADV
ejpam-7053	224	21	,	,	PUNCT
ejpam-7053	224	22	σ2)θ	σ2)θ	ADJ
ejpam-7053	224	23	-	-	PUNCT
ejpam-7053	224	24	cl(b	cl(b	NOUN
ejpam-7053	224	25	)	)	PUNCT
ejpam-7053	224	26	)	)	PUNCT
ejpam-7053	224	27	.	.	PUNCT
ejpam-7053	225	1	then	then	ADV
ejpam-7053	225	2	,	,	PUNCT
ejpam-7053	225	3	we	we	PRON
ejpam-7053	225	4	have	have	VERB
ejpam-7053	225	5	x	x	NOUN
ejpam-7053	225	6	∈	∈	NOUN
ejpam-7053	225	7	f−(y	f−(y	NOUN
ejpam-7053	225	8	−	−	PROPN
ejpam-7053	225	9	(	(	PUNCT
ejpam-7053	225	10	σ1	σ1	PROPN
ejpam-7053	225	11	,	,	PUNCT
ejpam-7053	225	12	σ2)θ	σ2)θ	NOUN
ejpam-7053	225	13	-	-	PUNCT
ejpam-7053	225	14	cl(b	cl(b	NOUN
ejpam-7053	225	15	)	)	PUNCT
ejpam-7053	225	16	)	)	PUNCT
ejpam-7053	226	1	=	=	SYM
ejpam-7053	226	2	f−((σ1	f−((σ1	NOUN
ejpam-7053	226	3	,	,	PUNCT
ejpam-7053	226	4	σ2)θ	σ2)θ	ADJ
ejpam-7053	226	5	-	-	PUNCT
ejpam-7053	226	6	int(y	int(y	PROPN
ejpam-7053	226	7	−	−	PROPN
ejpam-7053	226	8	b	b	NOUN
ejpam-7053	226	9	)	)	PUNCT
ejpam-7053	226	10	)	)	PUNCT
ejpam-7053	226	11	.	.	PUNCT
ejpam-7053	227	1	by	by	ADP
ejpam-7053	227	2	lemma	lemma	PROPN
ejpam-7053	227	3	3	3	NUM
ejpam-7053	227	4	,	,	PUNCT
ejpam-7053	227	5	there	there	PRON
ejpam-7053	227	6	exists	exist	VERB
ejpam-7053	227	7	a	a	DET
ejpam-7053	227	8	µ-open	µ-open	NOUN
ejpam-7053	227	9	set	set	VERB
ejpam-7053	227	10	u	u	NOUN
ejpam-7053	227	11	of	of	ADP
ejpam-7053	227	12	x	x	PUNCT
ejpam-7053	227	13	containing	contain	VERB
ejpam-7053	227	14	x	x	PUNCT
ejpam-7053	227	15	such	such	ADJ
ejpam-7053	227	16	that	that	SCONJ
ejpam-7053	227	17	u	u	NOUN
ejpam-7053	227	18	⊆	⊆	NUM
ejpam-7053	227	19	f−(y	f−(y	NOUN
ejpam-7053	227	20	−	−	NOUN
ejpam-7053	227	21	b	b	NOUN
ejpam-7053	227	22	)	)	PUNCT
ejpam-7053	227	23	=	=	PUNCT
ejpam-7053	227	24	x	x	X
ejpam-7053	228	1	−	−	NOUN
ejpam-7053	228	2	f+(b	f+(b	NOUN
ejpam-7053	228	3	)	)	PUNCT
ejpam-7053	228	4	.	.	PUNCT
ejpam-7053	229	1	therefore	therefore	ADV
ejpam-7053	229	2	,	,	PUNCT
ejpam-7053	229	3	u	u	NOUN
ejpam-7053	229	4	∩	∩	NOUN
ejpam-7053	229	5	f+(b	f+(b	NOUN
ejpam-7053	229	6	)	)	PUNCT
ejpam-7053	229	7	=	=	PUNCT
ejpam-7053	229	8	∅.	∅.	ADP
ejpam-7053	229	9	thus	thus	ADV
ejpam-7053	229	10	,	,	PUNCT
ejpam-7053	229	11	x	x	PROPN
ejpam-7053	229	12	̸∈	̸∈	PROPN
ejpam-7053	229	13	τ1τ2	τ1τ2	NOUN
ejpam-7053	229	14	-	-	NOUN
ejpam-7053	229	15	cl(f+(b	cl(f+(b	NOUN
ejpam-7053	229	16	)	)	PUNCT
ejpam-7053	229	17	)	)	PUNCT
ejpam-7053	229	18	and	and	CCONJ
ejpam-7053	229	19	hence	hence	ADV
ejpam-7053	229	20	cµ(f	cµ(f	PUNCT
ejpam-7053	230	1	+	+	PROPN
ejpam-7053	230	2	(	(	PUNCT
ejpam-7053	230	3	b	b	NOUN
ejpam-7053	230	4	)	)	PUNCT
ejpam-7053	230	5	)	)	PUNCT
ejpam-7053	230	6	⊆	⊆	NUM
ejpam-7053	230	7	f+((σ1	f+((σ1	NOUN
ejpam-7053	230	8	,	,	PUNCT
ejpam-7053	230	9	σ2)θ	σ2)θ	ADJ
ejpam-7053	230	10	-	-	PUNCT
ejpam-7053	230	11	cl(b	cl(b	NOUN
ejpam-7053	230	12	)	)	PUNCT
ejpam-7053	230	13	)	)	PUNCT
ejpam-7053	230	14	.	.	PUNCT
ejpam-7053	231	1	(	(	PUNCT
ejpam-7053	231	2	2	2	X
ejpam-7053	231	3	)	)	PUNCT
ejpam-7053	231	4	⇒	⇒	NOUN
ejpam-7053	231	5	(	(	PUNCT
ejpam-7053	231	6	3	3	NUM
ejpam-7053	231	7	):	):	PUNCT
ejpam-7053	231	8	let	let	VERB
ejpam-7053	231	9	a	a	DET
ejpam-7053	231	10	be	be	AUX
ejpam-7053	231	11	any	any	DET
ejpam-7053	231	12	subset	subset	NOUN
ejpam-7053	231	13	of	of	ADP
ejpam-7053	231	14	x.	x.	NOUN
ejpam-7053	231	15	by	by	ADP
ejpam-7053	231	16	(	(	PUNCT
ejpam-7053	231	17	2	2	NUM
ejpam-7053	231	18	)	)	PUNCT
ejpam-7053	231	19	,	,	PUNCT
ejpam-7053	231	20	we	we	PRON
ejpam-7053	231	21	have	have	VERB
ejpam-7053	231	22	cµ(a	cµ(a	PROPN
ejpam-7053	231	23	)	)	PUNCT
ejpam-7053	231	24	⊆	⊆	NUM
ejpam-7053	231	25	cµ(f	cµ(f	X
ejpam-7053	232	1	+	+	PROPN
ejpam-7053	232	2	(	(	PUNCT
ejpam-7053	232	3	f	f	X
ejpam-7053	232	4	(	(	PUNCT
ejpam-7053	232	5	a	a	NOUN
ejpam-7053	232	6	)	)	PUNCT
ejpam-7053	232	7	)	)	PUNCT
ejpam-7053	232	8	)	)	PUNCT
ejpam-7053	232	9	⊆	⊆	NUM
ejpam-7053	232	10	f+((σ1	f+((σ1	NOUN
ejpam-7053	232	11	,	,	PUNCT
ejpam-7053	232	12	σ2)θ	σ2)θ	NOUN
ejpam-7053	232	13	-	-	PUNCT
ejpam-7053	232	14	cl(f	cl(f	PROPN
ejpam-7053	232	15	(	(	PUNCT
ejpam-7053	232	16	a	a	NOUN
ejpam-7053	232	17	)	)	PUNCT
ejpam-7053	232	18	)	)	PUNCT
ejpam-7053	232	19	)	)	PUNCT
ejpam-7053	232	20	.	.	PUNCT
ejpam-7053	233	1	thus	thus	ADV
ejpam-7053	233	2	,	,	PUNCT
ejpam-7053	233	3	f	f	PROPN
ejpam-7053	233	4	(	(	PUNCT
ejpam-7053	233	5	cµ(a	cµ(a	PROPN
ejpam-7053	233	6	)	)	PUNCT
ejpam-7053	233	7	)	)	PUNCT
ejpam-7053	234	1	⊆	⊆	NUM
ejpam-7053	234	2	(	(	PUNCT
ejpam-7053	234	3	σ1	σ1	PROPN
ejpam-7053	234	4	,	,	PUNCT
ejpam-7053	234	5	σ2)θ	σ2)θ	NOUN
ejpam-7053	234	6	-	-	PUNCT
ejpam-7053	234	7	cl(f	cl(f	PROPN
ejpam-7053	234	8	(	(	PUNCT
ejpam-7053	234	9	a	a	NOUN
ejpam-7053	234	10	)	)	PUNCT
ejpam-7053	234	11	)	)	PUNCT
ejpam-7053	234	12	.	.	PUNCT
ejpam-7053	235	1	(	(	PUNCT
ejpam-7053	235	2	3	3	X
ejpam-7053	235	3	)	)	PUNCT
ejpam-7053	235	4	⇒	⇒	NOUN
ejpam-7053	235	5	(	(	PUNCT
ejpam-7053	235	6	1	1	NUM
ejpam-7053	235	7	):	):	PUNCT
ejpam-7053	235	8	let	let	VERB
ejpam-7053	235	9	v	v	PART
ejpam-7053	235	10	be	be	AUX
ejpam-7053	235	11	any	any	DET
ejpam-7053	235	12	σ1σ2	σ1σ2	NOUN
ejpam-7053	235	13	-	-	ADJ
ejpam-7053	235	14	open	open	ADJ
ejpam-7053	235	15	set	set	NOUN
ejpam-7053	235	16	of	of	ADP
ejpam-7053	235	17	y	y	PROPN
ejpam-7053	235	18	.	.	PUNCT
ejpam-7053	236	1	then	then	ADV
ejpam-7053	236	2	,	,	PUNCT
ejpam-7053	236	3	σ1σ2	σ1σ2	NOUN
ejpam-7053	236	4	-	-	NUM
ejpam-7053	236	5	cl(v	cl(v	X
ejpam-7053	236	6	)	)	PUNCT
ejpam-7053	236	7	=	=	SYM
ejpam-7053	236	8	(	(	PUNCT
ejpam-7053	236	9	σ1	σ1	PROPN
ejpam-7053	236	10	,	,	PUNCT
ejpam-7053	236	11	σ2)θ	σ2)θ	NOUN
ejpam-7053	236	12	-	-	PUNCT
ejpam-7053	236	13	cl(v	cl(v	X
ejpam-7053	236	14	)	)	PUNCT
ejpam-7053	236	15	and	and	CCONJ
ejpam-7053	236	16	by	by	ADP
ejpam-7053	236	17	(	(	PUNCT
ejpam-7053	236	18	3	3	NUM
ejpam-7053	236	19	)	)	PUNCT
ejpam-7053	236	20	,	,	PUNCT
ejpam-7053	236	21	f	f	PROPN
ejpam-7053	236	22	(	(	PUNCT
ejpam-7053	236	23	cµ(f	cµ(f	X
ejpam-7053	237	1	+	+	PROPN
ejpam-7053	237	2	(	(	PUNCT
ejpam-7053	237	3	v	v	NOUN
ejpam-7053	237	4	)	)	PUNCT
ejpam-7053	237	5	)	)	PUNCT
ejpam-7053	237	6	)	)	PUNCT
ejpam-7053	238	1	⊆	⊆	X
ejpam-7053	238	2	(	(	PUNCT
ejpam-7053	238	3	σ1	σ1	PROPN
ejpam-7053	238	4	,	,	PUNCT
ejpam-7053	238	5	σ2)θ	σ2)θ	NOUN
ejpam-7053	238	6	-	-	PUNCT
ejpam-7053	238	7	cl(f	cl(f	PROPN
ejpam-7053	238	8	(	(	PUNCT
ejpam-7053	238	9	f+(v	f+(v	PROPN
ejpam-7053	238	10	)	)	PUNCT
ejpam-7053	238	11	)	)	PUNCT
ejpam-7053	238	12	)	)	PUNCT
ejpam-7053	239	1	⊆	⊆	X
ejpam-7053	239	2	(	(	PUNCT
ejpam-7053	239	3	σ1	σ1	PROPN
ejpam-7053	239	4	,	,	PUNCT
ejpam-7053	239	5	σ2)θ	σ2)θ	NOUN
ejpam-7053	239	6	-	-	PUNCT
ejpam-7053	239	7	cl(v	cl(v	NOUN
ejpam-7053	239	8	)	)	PUNCT
ejpam-7053	239	9	=	=	SYM
ejpam-7053	239	10	σ1σ2	σ1σ2	NOUN
ejpam-7053	239	11	-	-	NUM
ejpam-7053	239	12	cl(v	cl(v	NOUN
ejpam-7053	239	13	)	)	PUNCT
ejpam-7053	239	14	.	.	PUNCT
ejpam-7053	240	1	thus	thus	ADV
ejpam-7053	240	2	,	,	PUNCT
ejpam-7053	240	3	cµ(f	cµ(f	X
ejpam-7053	241	1	+	+	ADJ
ejpam-7053	241	2	(	(	PUNCT
ejpam-7053	241	3	v	v	NOUN
ejpam-7053	241	4	)	)	PUNCT
ejpam-7053	241	5	)	)	PUNCT
ejpam-7053	241	6	⊆	⊆	NUM
ejpam-7053	241	7	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7053	241	8	-	-	PUNCT
ejpam-7053	241	9	cl(v	cl(v	NOUN
ejpam-7053	241	10	)	)	PUNCT
ejpam-7053	241	11	)	)	PUNCT
ejpam-7053	241	12	and	and	CCONJ
ejpam-7053	241	13	by	by	ADP
ejpam-7053	241	14	theorem	theorem	NOUN
ejpam-7053	241	15	1	1	NUM
ejpam-7053	241	16	,	,	PUNCT
ejpam-7053	241	17	f	f	PROPN
ejpam-7053	241	18	is	be	AUX
ejpam-7053	241	19	lower	low	ADJ
ejpam-7053	241	20	weakly	weakly	ADJ
ejpam-7053	241	21	µ(σ1	µ(σ1	NOUN
ejpam-7053	241	22	,	,	PUNCT
ejpam-7053	241	23	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	241	24	.	.	X
ejpam-7053	241	25	theorem	theorem	VERB
ejpam-7053	241	26	8	8	NUM
ejpam-7053	241	27	.	.	PUNCT
ejpam-7053	241	28	for	for	ADP
ejpam-7053	241	29	a	a	DET
ejpam-7053	241	30	multifunction	multifunction	NOUN
ejpam-7053	242	1	f	f	NOUN
ejpam-7053	242	2	:	:	PUNCT
ejpam-7053	242	3	(	(	PUNCT
ejpam-7053	242	4	x,µ	x,µ	NOUN
ejpam-7053	242	5	)	)	PUNCT
ejpam-7053	242	6	→	→	SYM
ejpam-7053	242	7	(	(	PUNCT
ejpam-7053	242	8	y	y	PROPN
ejpam-7053	242	9	,	,	PUNCT
ejpam-7053	242	10	σ1	σ1	PROPN
ejpam-7053	242	11	,	,	PUNCT
ejpam-7053	242	12	τ2	τ2	PROPN
ejpam-7053	242	13	)	)	PUNCT
ejpam-7053	242	14	,	,	PUNCT
ejpam-7053	242	15	the	the	DET
ejpam-7053	242	16	following	follow	VERB
ejpam-7053	242	17	properties	property	NOUN
ejpam-7053	242	18	are	be	AUX
ejpam-7053	242	19	equivalent	equivalent	ADJ
ejpam-7053	242	20	:	:	PUNCT
ejpam-7053	242	21	(	(	PUNCT
ejpam-7053	242	22	1	1	X
ejpam-7053	242	23	)	)	PUNCT
ejpam-7053	242	24	f	f	PROPN
ejpam-7053	242	25	is	be	AUX
ejpam-7053	242	26	upper	upper	ADJ
ejpam-7053	242	27	weakly	weakly	ADJ
ejpam-7053	242	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	242	29	,	,	PUNCT
ejpam-7053	242	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	242	31	;	;	PUNCT
ejpam-7053	242	32	(	(	PUNCT
ejpam-7053	242	33	2	2	NUM
ejpam-7053	242	34	)	)	PUNCT
ejpam-7053	242	35	cµ(f	cµ(f	NOUN
ejpam-7053	242	36	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	242	37	-	-	PUNCT
ejpam-7053	242	38	int((σ1	int((σ1	ADJ
ejpam-7053	242	39	,	,	PUNCT
ejpam-7053	242	40	σ2)θ	σ2)θ	ADJ
ejpam-7053	242	41	-	-	PUNCT
ejpam-7053	242	42	cl(b	cl(b	NOUN
ejpam-7053	242	43	)	)	PUNCT
ejpam-7053	242	44	)	)	PUNCT
ejpam-7053	242	45	)	)	PUNCT
ejpam-7053	242	46	)	)	PUNCT
ejpam-7053	243	1	⊆	⊆	NUM
ejpam-7053	243	2	f−((σ1	f−((σ1	NOUN
ejpam-7053	243	3	,	,	PUNCT
ejpam-7053	243	4	σ2)θ	σ2)θ	ADJ
ejpam-7053	243	5	-	-	PUNCT
ejpam-7053	243	6	cl(b	cl(b	NOUN
ejpam-7053	243	7	)	)	PUNCT
ejpam-7053	243	8	)	)	PUNCT
ejpam-7053	243	9	for	for	ADP
ejpam-7053	243	10	every	every	DET
ejpam-7053	243	11	subset	subset	NOUN
ejpam-7053	243	12	b	b	PROPN
ejpam-7053	243	13	of	of	ADP
ejpam-7053	243	14	y	y	PROPN
ejpam-7053	243	15	;	;	PUNCT
ejpam-7053	243	16	(	(	PUNCT
ejpam-7053	243	17	3	3	X
ejpam-7053	243	18	)	)	PUNCT
ejpam-7053	243	19	cµ(f	cµ(f	NOUN
ejpam-7053	243	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	243	21	-	-	PUNCT
ejpam-7053	243	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	243	23	-	-	PUNCT
ejpam-7053	243	24	cl(b	cl(b	NOUN
ejpam-7053	243	25	)	)	PUNCT
ejpam-7053	243	26	)	)	PUNCT
ejpam-7053	243	27	)	)	PUNCT
ejpam-7053	243	28	)	)	PUNCT
ejpam-7053	244	1	⊆	⊆	NUM
ejpam-7053	244	2	f−((σ1	f−((σ1	NOUN
ejpam-7053	244	3	,	,	PUNCT
ejpam-7053	244	4	σ2)θ	σ2)θ	ADJ
ejpam-7053	244	5	-	-	PUNCT
ejpam-7053	244	6	cl(b	cl(b	NOUN
ejpam-7053	244	7	)	)	PUNCT
ejpam-7053	244	8	)	)	PUNCT
ejpam-7053	244	9	for	for	ADP
ejpam-7053	244	10	every	every	DET
ejpam-7053	244	11	subset	subset	NOUN
ejpam-7053	244	12	b	b	PROPN
ejpam-7053	244	13	of	of	ADP
ejpam-7053	244	14	y	y	PROPN
ejpam-7053	244	15	.	.	PUNCT
ejpam-7053	245	1	proof	proof	NOUN
ejpam-7053	245	2	.	.	PUNCT
ejpam-7053	246	1	(	(	PUNCT
ejpam-7053	246	2	1	1	X
ejpam-7053	246	3	)	)	PUNCT
ejpam-7053	246	4	⇒	⇒	NOUN
ejpam-7053	246	5	(	(	PUNCT
ejpam-7053	246	6	2	2	NUM
ejpam-7053	246	7	):	):	PUNCT
ejpam-7053	246	8	let	let	VERB
ejpam-7053	246	9	b	b	X
ejpam-7053	246	10	be	be	AUX
ejpam-7053	246	11	any	any	DET
ejpam-7053	246	12	subset	subset	NOUN
ejpam-7053	246	13	of	of	ADP
ejpam-7053	246	14	y	y	PROPN
ejpam-7053	246	15	.	.	PUNCT
ejpam-7053	247	1	then	then	ADV
ejpam-7053	247	2	,	,	PUNCT
ejpam-7053	247	3	(	(	PUNCT
ejpam-7053	247	4	σ1	σ1	PROPN
ejpam-7053	247	5	,	,	PUNCT
ejpam-7053	247	6	σ2)θ	σ2)θ	NOUN
ejpam-7053	247	7	-	-	PUNCT
ejpam-7053	247	8	cl(b	cl(b	NOUN
ejpam-7053	247	9	)	)	PUNCT
ejpam-7053	247	10	is	be	AUX
ejpam-7053	247	11	σ1σ2	σ1σ2	NOUN
ejpam-7053	247	12	-	-	ADJ
ejpam-7053	247	13	closed	closed	ADJ
ejpam-7053	247	14	in	in	ADP
ejpam-7053	247	15	y	y	PROPN
ejpam-7053	247	16	and	and	CCONJ
ejpam-7053	247	17	by	by	ADP
ejpam-7053	247	18	theorem	theorem	ADJ
ejpam-7053	247	19	2	2	NUM
ejpam-7053	247	20	,	,	PUNCT
ejpam-7053	247	21	cµ(f−(σ1σ2	cµ(f−(σ1σ2	NUM
ejpam-7053	247	22	-	-	PUNCT
ejpam-7053	247	23	int((σ1	int((σ1	ADJ
ejpam-7053	247	24	,	,	PUNCT
ejpam-7053	247	25	σ2)θ	σ2)θ	ADJ
ejpam-7053	247	26	-	-	PUNCT
ejpam-7053	247	27	cl(b	cl(b	NOUN
ejpam-7053	247	28	)	)	PUNCT
ejpam-7053	247	29	)	)	PUNCT
ejpam-7053	247	30	)	)	PUNCT
ejpam-7053	247	31	)	)	PUNCT
ejpam-7053	248	1	⊆	⊆	NUM
ejpam-7053	248	2	f−((σ1	f−((σ1	NOUN
ejpam-7053	248	3	,	,	PUNCT
ejpam-7053	248	4	σ2)θ	σ2)θ	ADJ
ejpam-7053	248	5	-	-	PUNCT
ejpam-7053	248	6	cl(b	cl(b	NOUN
ejpam-7053	248	7	)	)	PUNCT
ejpam-7053	248	8	)	)	PUNCT
ejpam-7053	248	9	.	.	PUNCT
ejpam-7053	249	1	(	(	PUNCT
ejpam-7053	249	2	2	2	X
ejpam-7053	249	3	)	)	PUNCT
ejpam-7053	249	4	⇒	⇒	NOUN
ejpam-7053	249	5	(	(	PUNCT
ejpam-7053	249	6	3	3	NUM
ejpam-7053	249	7	):	):	PUNCT
ejpam-7053	249	8	the	the	DET
ejpam-7053	249	9	proof	proof	NOUN
ejpam-7053	249	10	is	be	AUX
ejpam-7053	249	11	obvious	obvious	ADJ
ejpam-7053	249	12	.	.	PUNCT
ejpam-7053	250	1	m.	m.	NOUN
ejpam-7053	250	2	chiangpradit	chiangpradit	PROPN
ejpam-7053	250	3	,	,	PUNCT
ejpam-7053	250	4	a.	a.	PROPN
ejpam-7053	250	5	sama	sama	PROPN
ejpam-7053	250	6	-	-	PUNCT
ejpam-7053	250	7	ae	ae	PROPN
ejpam-7053	250	8	,	,	PUNCT
ejpam-7053	250	9	c.	c.	PROPN
ejpam-7053	250	10	boonpok	boonpok	PROPN
ejpam-7053	250	11	/	/	SYM
ejpam-7053	250	12	eur	eur	PROPN
ejpam-7053	250	13	.	.	PUNCT
ejpam-7053	251	1	j.	j.	PROPN
ejpam-7053	251	2	pure	pure	PROPN
ejpam-7053	251	3	appl	appl	PROPN
ejpam-7053	251	4	.	.	PROPN
ejpam-7053	251	5	math	math	PROPN
ejpam-7053	251	6	,	,	PUNCT
ejpam-7053	251	7	18	18	NUM
ejpam-7053	251	8	(	(	PUNCT
ejpam-7053	251	9	4	4	NUM
ejpam-7053	251	10	)	)	PUNCT
ejpam-7053	251	11	(	(	PUNCT
ejpam-7053	251	12	2025	2025	NUM
ejpam-7053	251	13	)	)	PUNCT
ejpam-7053	251	14	,	,	PUNCT
ejpam-7053	251	15	7053	7053	NUM
ejpam-7053	251	16	9	9	NUM
ejpam-7053	251	17	of	of	ADP
ejpam-7053	251	18	13	13	NUM
ejpam-7053	251	19	(	(	PUNCT
ejpam-7053	251	20	3	3	NUM
ejpam-7053	251	21	)	)	PUNCT
ejpam-7053	251	22	⇒	⇒	NOUN
ejpam-7053	251	23	(	(	PUNCT
ejpam-7053	251	24	1	1	NUM
ejpam-7053	251	25	):	):	PUNCT
ejpam-7053	251	26	let	let	VERB
ejpam-7053	251	27	k	k	PRON
ejpam-7053	251	28	be	be	AUX
ejpam-7053	251	29	any	any	DET
ejpam-7053	251	30	(	(	PUNCT
ejpam-7053	251	31	σ1	σ1	NOUN
ejpam-7053	251	32	,	,	PUNCT
ejpam-7053	251	33	σ2)r	σ2)r	NOUN
ejpam-7053	251	34	-	-	PUNCT
ejpam-7053	251	35	closed	close	VERB
ejpam-7053	251	36	set	set	NOUN
ejpam-7053	251	37	of	of	ADP
ejpam-7053	251	38	y	y	PROPN
ejpam-7053	251	39	.	.	PUNCT
ejpam-7053	252	1	then	then	ADV
ejpam-7053	252	2	,	,	PUNCT
ejpam-7053	252	3	we	we	PRON
ejpam-7053	252	4	have	have	VERB
ejpam-7053	252	5	(	(	PUNCT
ejpam-7053	252	6	σ1	σ1	PROPN
ejpam-7053	252	7	,	,	PUNCT
ejpam-7053	252	8	σ2)θ	σ2)θ	NOUN
ejpam-7053	252	9	-	-	PUNCT
ejpam-7053	252	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	252	11	-	-	PUNCT
ejpam-7053	252	12	int(k	int(k	NOUN
ejpam-7053	252	13	)	)	PUNCT
ejpam-7053	252	14	)	)	PUNCT
ejpam-7053	253	1	=	=	SYM
ejpam-7053	253	2	σ1σ2	σ1σ2	X
ejpam-7053	253	3	-	-	PUNCT
ejpam-7053	253	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	253	5	-	-	PUNCT
ejpam-7053	253	6	int(k	int(k	NOUN
ejpam-7053	253	7	)	)	PUNCT
ejpam-7053	253	8	)	)	PUNCT
ejpam-7053	254	1	=	=	SYM
ejpam-7053	255	1	k	k	PROPN
ejpam-7053	255	2	and	and	CCONJ
ejpam-7053	255	3	hence	hence	ADV
ejpam-7053	255	4	cµ(f	cµ(f	NOUN
ejpam-7053	255	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7053	255	6	-	-	PUNCT
ejpam-7053	255	7	int(k	int(k	NUM
ejpam-7053	255	8	)	)	PUNCT
ejpam-7053	255	9	)	)	PUNCT
ejpam-7053	255	10	)	)	PUNCT
ejpam-7053	256	1	=	=	PUNCT
ejpam-7053	256	2	cµ(σ1σ2	cµ(σ1σ2	NOUN
ejpam-7053	256	3	-	-	PUNCT
ejpam-7053	256	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-7053	256	5	-	-	PUNCT
ejpam-7053	256	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7053	256	7	-	-	PUNCT
ejpam-7053	256	8	int(k	int(k	NOUN
ejpam-7053	256	9	)	)	PUNCT
ejpam-7053	256	10	)	)	PUNCT
ejpam-7053	256	11	)	)	PUNCT
ejpam-7053	256	12	)	)	PUNCT
ejpam-7053	257	1	⊆	⊆	NUM
ejpam-7053	257	2	f−((σ1	f−((σ1	NOUN
ejpam-7053	257	3	,	,	PUNCT
ejpam-7053	257	4	σ2)θ	σ2)θ	NOUN
ejpam-7053	257	5	-	-	PUNCT
ejpam-7053	257	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	257	7	-	-	PUNCT
ejpam-7053	257	8	int(k	int(k	NOUN
ejpam-7053	257	9	)	)	PUNCT
ejpam-7053	257	10	)	)	PUNCT
ejpam-7053	257	11	)	)	PUNCT
ejpam-7053	258	1	=	=	PUNCT
ejpam-7053	258	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7053	258	3	-	-	PUNCT
ejpam-7053	258	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7053	258	5	-	-	PUNCT
ejpam-7053	258	6	int(k	int(k	NOUN
ejpam-7053	258	7	)	)	PUNCT
ejpam-7053	258	8	)	)	PUNCT
ejpam-7053	258	9	)	)	PUNCT
ejpam-7053	258	10	=	=	SYM
ejpam-7053	258	11	f−(k	f−(k	PROPN
ejpam-7053	258	12	)	)	PUNCT
ejpam-7053	258	13	.	.	PUNCT
ejpam-7053	259	1	thus	thus	ADV
ejpam-7053	259	2	by	by	ADP
ejpam-7053	259	3	theorem	theorem	NOUN
ejpam-7053	259	4	2	2	NUM
ejpam-7053	259	5	,	,	PUNCT
ejpam-7053	259	6	f	f	PROPN
ejpam-7053	259	7	is	be	AUX
ejpam-7053	259	8	upper	upper	ADJ
ejpam-7053	259	9	weakly	weakly	ADJ
ejpam-7053	259	10	µ(σ1	µ(σ1	NOUN
ejpam-7053	259	11	,	,	PUNCT
ejpam-7053	259	12	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	259	13	.	.	X
ejpam-7053	259	14	theorem	theorem	VERB
ejpam-7053	259	15	9	9	NUM
ejpam-7053	259	16	.	.	X
ejpam-7053	259	17	for	for	ADP
ejpam-7053	259	18	a	a	DET
ejpam-7053	259	19	multifunction	multifunction	NOUN
ejpam-7053	259	20	f	f	NOUN
ejpam-7053	259	21	:	:	PUNCT
ejpam-7053	259	22	(	(	PUNCT
ejpam-7053	259	23	x,µ	x,µ	NOUN
ejpam-7053	259	24	)	)	PUNCT
ejpam-7053	259	25	→	→	SYM
ejpam-7053	259	26	(	(	PUNCT
ejpam-7053	259	27	y	y	PROPN
ejpam-7053	259	28	,	,	PUNCT
ejpam-7053	259	29	σ1	σ1	PROPN
ejpam-7053	259	30	,	,	PUNCT
ejpam-7053	259	31	σ2	σ2	NOUN
ejpam-7053	259	32	)	)	PUNCT
ejpam-7053	259	33	,	,	PUNCT
ejpam-7053	259	34	the	the	DET
ejpam-7053	259	35	following	follow	VERB
ejpam-7053	259	36	properties	property	NOUN
ejpam-7053	259	37	are	be	AUX
ejpam-7053	259	38	equivalent	equivalent	ADJ
ejpam-7053	259	39	:	:	PUNCT
ejpam-7053	259	40	(	(	PUNCT
ejpam-7053	259	41	1	1	X
ejpam-7053	259	42	)	)	PUNCT
ejpam-7053	259	43	f	f	PROPN
ejpam-7053	259	44	is	be	AUX
ejpam-7053	259	45	lower	low	ADJ
ejpam-7053	259	46	weakly	weakly	ADJ
ejpam-7053	259	47	µ(σ1	µ(σ1	NOUN
ejpam-7053	259	48	,	,	PUNCT
ejpam-7053	259	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	259	50	;	;	PUNCT
ejpam-7053	259	51	(	(	PUNCT
ejpam-7053	259	52	2	2	NUM
ejpam-7053	259	53	)	)	PUNCT
ejpam-7053	259	54	cµ(f	cµ(f	PUNCT
ejpam-7053	260	1	+	+	PROPN
ejpam-7053	260	2	(	(	PUNCT
ejpam-7053	260	3	σ1σ2	σ1σ2	NOUN
ejpam-7053	260	4	-	-	PUNCT
ejpam-7053	260	5	int((σ1	int((σ1	ADJ
ejpam-7053	260	6	,	,	PUNCT
ejpam-7053	260	7	σ2)θ	σ2)θ	ADJ
ejpam-7053	260	8	-	-	PUNCT
ejpam-7053	260	9	cl(b	cl(b	NOUN
ejpam-7053	260	10	)	)	PUNCT
ejpam-7053	260	11	)	)	PUNCT
ejpam-7053	260	12	)	)	PUNCT
ejpam-7053	260	13	)	)	PUNCT
ejpam-7053	261	1	⊆	⊆	NUM
ejpam-7053	261	2	f+((σ1	f+((σ1	NOUN
ejpam-7053	261	3	,	,	PUNCT
ejpam-7053	261	4	σ2)θ	σ2)θ	ADJ
ejpam-7053	261	5	-	-	PUNCT
ejpam-7053	261	6	cl(b	cl(b	NOUN
ejpam-7053	261	7	)	)	PUNCT
ejpam-7053	261	8	)	)	PUNCT
ejpam-7053	261	9	for	for	ADP
ejpam-7053	261	10	every	every	DET
ejpam-7053	261	11	subset	subset	NOUN
ejpam-7053	261	12	b	b	PROPN
ejpam-7053	261	13	of	of	ADP
ejpam-7053	261	14	y	y	PROPN
ejpam-7053	261	15	;	;	PUNCT
ejpam-7053	261	16	(	(	PUNCT
ejpam-7053	261	17	3	3	X
ejpam-7053	261	18	)	)	PUNCT
ejpam-7053	261	19	cµ(f	cµ(f	PUNCT
ejpam-7053	262	1	+	+	PROPN
ejpam-7053	262	2	(	(	PUNCT
ejpam-7053	262	3	σ1σ2	σ1σ2	NUM
ejpam-7053	262	4	-	-	PUNCT
ejpam-7053	262	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7053	262	6	-	-	PUNCT
ejpam-7053	262	7	cl(b	cl(b	NOUN
ejpam-7053	262	8	)	)	PUNCT
ejpam-7053	262	9	)	)	PUNCT
ejpam-7053	262	10	)	)	PUNCT
ejpam-7053	262	11	)	)	PUNCT
ejpam-7053	263	1	⊆	⊆	NUM
ejpam-7053	263	2	f+((σ1	f+((σ1	NOUN
ejpam-7053	263	3	,	,	PUNCT
ejpam-7053	263	4	σ2)θ	σ2)θ	ADJ
ejpam-7053	263	5	-	-	PUNCT
ejpam-7053	263	6	cl(b	cl(b	NOUN
ejpam-7053	263	7	)	)	PUNCT
ejpam-7053	263	8	)	)	PUNCT
ejpam-7053	263	9	for	for	ADP
ejpam-7053	263	10	every	every	DET
ejpam-7053	263	11	subset	subset	NOUN
ejpam-7053	263	12	b	b	PROPN
ejpam-7053	263	13	of	of	ADP
ejpam-7053	263	14	y	y	PROPN
ejpam-7053	263	15	.	.	PUNCT
ejpam-7053	264	1	proof	proof	NOUN
ejpam-7053	264	2	.	.	PUNCT
ejpam-7053	265	1	the	the	DET
ejpam-7053	265	2	proof	proof	NOUN
ejpam-7053	265	3	is	be	AUX
ejpam-7053	265	4	similar	similar	ADJ
ejpam-7053	265	5	to	to	ADP
ejpam-7053	265	6	that	that	PRON
ejpam-7053	265	7	of	of	ADP
ejpam-7053	265	8	theorem	theorem	ADJ
ejpam-7053	265	9	10	10	NUM
ejpam-7053	265	10	.	.	PUNCT
ejpam-7053	266	1	definition	definition	NOUN
ejpam-7053	266	2	5	5	NUM
ejpam-7053	266	3	.	.	PUNCT
ejpam-7053	267	1	[	[	X
ejpam-7053	267	2	20	20	NUM
ejpam-7053	267	3	]	]	PUNCT
ejpam-7053	267	4	a	a	DET
ejpam-7053	267	5	multifunction	multifunction	NOUN
ejpam-7053	267	6	f	f	NOUN
ejpam-7053	267	7	:	:	PUNCT
ejpam-7053	267	8	(	(	PUNCT
ejpam-7053	267	9	x,µ	x,µ	NOUN
ejpam-7053	267	10	)	)	PUNCT
ejpam-7053	267	11	→	→	SYM
ejpam-7053	267	12	(	(	PUNCT
ejpam-7053	267	13	y	y	PROPN
ejpam-7053	267	14	,	,	PUNCT
ejpam-7053	267	15	σ1	σ1	PROPN
ejpam-7053	267	16	,	,	PUNCT
ejpam-7053	267	17	σ2	σ2	PROPN
ejpam-7053	267	18	)	)	PUNCT
ejpam-7053	267	19	is	be	AUX
ejpam-7053	267	20	said	say	VERB
ejpam-7053	267	21	to	to	PART
ejpam-7053	267	22	be	be	AUX
ejpam-7053	267	23	upper	upper	ADJ
ejpam-7053	267	24	µ(σ1	µ(σ1	NOUN
ejpam-7053	267	25	,	,	PUNCT
ejpam-7053	267	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	267	27	at	at	ADP
ejpam-7053	267	28	a	a	DET
ejpam-7053	267	29	point	point	NOUN
ejpam-7053	267	30	x	x	SYM
ejpam-7053	267	31	∈	∈	NOUN
ejpam-7053	267	32	x	x	PUNCT
ejpam-7053	267	33	if	if	SCONJ
ejpam-7053	267	34	for	for	ADP
ejpam-7053	267	35	each	each	DET
ejpam-7053	267	36	σ1σ2	σ1σ2	VERB
ejpam-7053	267	37	-	-	ADJ
ejpam-7053	267	38	open	open	ADJ
ejpam-7053	267	39	set	set	NOUN
ejpam-7053	267	40	v	v	NOUN
ejpam-7053	267	41	of	of	ADP
ejpam-7053	267	42	y	y	PRON
ejpam-7053	267	43	such	such	ADJ
ejpam-7053	267	44	that	that	SCONJ
ejpam-7053	267	45	f	f	PROPN
ejpam-7053	267	46	(	(	PUNCT
ejpam-7053	267	47	x	x	X
ejpam-7053	267	48	)	)	PUNCT
ejpam-7053	267	49	⊆	⊆	NUM
ejpam-7053	267	50	v	v	NOUN
ejpam-7053	267	51	,	,	PUNCT
ejpam-7053	267	52	there	there	PRON
ejpam-7053	267	53	exists	exist	VERB
ejpam-7053	267	54	a	a	DET
ejpam-7053	267	55	µ-open	µ-open	NOUN
ejpam-7053	267	56	set	set	VERB
ejpam-7053	267	57	u	u	NOUN
ejpam-7053	267	58	of	of	ADP
ejpam-7053	267	59	x	x	PUNCT
ejpam-7053	267	60	containing	contain	VERB
ejpam-7053	267	61	x	x	PUNCT
ejpam-7053	267	62	such	such	ADJ
ejpam-7053	267	63	that	that	SCONJ
ejpam-7053	267	64	f	f	PROPN
ejpam-7053	267	65	(	(	PUNCT
ejpam-7053	267	66	u	u	NOUN
ejpam-7053	267	67	)	)	PUNCT
ejpam-7053	267	68	⊆	⊆	NUM
ejpam-7053	267	69	v	v	NOUN
ejpam-7053	267	70	.	.	PUNCT
ejpam-7053	268	1	a	a	DET
ejpam-7053	268	2	multifunction	multifunction	NOUN
ejpam-7053	268	3	f	f	NOUN
ejpam-7053	268	4	:	:	PUNCT
ejpam-7053	268	5	(	(	PUNCT
ejpam-7053	268	6	x,µ	x,µ	NOUN
ejpam-7053	268	7	)	)	PUNCT
ejpam-7053	268	8	→	→	SYM
ejpam-7053	268	9	(	(	PUNCT
ejpam-7053	268	10	y	y	PROPN
ejpam-7053	268	11	,	,	PUNCT
ejpam-7053	268	12	σ1	σ1	PROPN
ejpam-7053	268	13	,	,	PUNCT
ejpam-7053	268	14	σ2	σ2	PROPN
ejpam-7053	268	15	)	)	PUNCT
ejpam-7053	268	16	is	be	AUX
ejpam-7053	268	17	said	say	VERB
ejpam-7053	268	18	to	to	PART
ejpam-7053	268	19	be	be	AUX
ejpam-7053	268	20	upper	upper	ADJ
ejpam-7053	268	21	µ(σ1	µ(σ1	NOUN
ejpam-7053	268	22	,	,	PUNCT
ejpam-7053	268	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	268	24	if	if	SCONJ
ejpam-7053	268	25	f	f	PROPN
ejpam-7053	268	26	is	be	AUX
ejpam-7053	268	27	upper	upper	ADJ
ejpam-7053	268	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	268	29	,	,	PUNCT
ejpam-7053	268	30	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	268	31	at	at	ADP
ejpam-7053	268	32	each	each	DET
ejpam-7053	268	33	point	point	NOUN
ejpam-7053	268	34	x	x	PUNCT
ejpam-7053	268	35	of	of	ADP
ejpam-7053	268	36	x.	x.	NOUN
ejpam-7053	268	37	definition	definition	NOUN
ejpam-7053	268	38	6	6	NUM
ejpam-7053	268	39	.	.	PUNCT
ejpam-7053	269	1	[	[	X
ejpam-7053	269	2	20	20	NUM
ejpam-7053	269	3	]	]	PUNCT
ejpam-7053	269	4	a	a	DET
ejpam-7053	269	5	multifunction	multifunction	NOUN
ejpam-7053	269	6	f	f	NOUN
ejpam-7053	269	7	:	:	PUNCT
ejpam-7053	269	8	(	(	PUNCT
ejpam-7053	269	9	x,µ	x,µ	NOUN
ejpam-7053	269	10	)	)	PUNCT
ejpam-7053	269	11	→	→	SYM
ejpam-7053	269	12	(	(	PUNCT
ejpam-7053	269	13	y	y	PROPN
ejpam-7053	269	14	,	,	PUNCT
ejpam-7053	269	15	σ1	σ1	PROPN
ejpam-7053	269	16	,	,	PUNCT
ejpam-7053	269	17	σ2	σ2	PROPN
ejpam-7053	269	18	)	)	PUNCT
ejpam-7053	269	19	is	be	AUX
ejpam-7053	269	20	said	say	VERB
ejpam-7053	269	21	to	to	PART
ejpam-7053	269	22	be	be	AUX
ejpam-7053	269	23	lower	low	ADJ
ejpam-7053	269	24	µ(σ1	µ(σ1	NOUN
ejpam-7053	269	25	,	,	PUNCT
ejpam-7053	269	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7053	269	27	at	at	ADP
ejpam-7053	269	28	a	a	DET
ejpam-7053	269	29	point	point	NOUN
ejpam-7053	269	30	x	x	SYM
ejpam-7053	269	31	∈	∈	NOUN
ejpam-7053	269	32	x	x	PUNCT
ejpam-7053	269	33	if	if	SCONJ
ejpam-7053	269	34	for	for	ADP
ejpam-7053	269	35	each	each	DET
ejpam-7053	269	36	σ1σ2	σ1σ2	VERB
ejpam-7053	269	37	-	-	ADJ
ejpam-7053	269	38	open	open	ADJ
ejpam-7053	269	39	set	set	NOUN
ejpam-7053	269	40	v	v	NOUN
ejpam-7053	269	41	of	of	ADP
ejpam-7053	269	42	y	y	PRON
ejpam-7053	269	43	such	such	ADJ
ejpam-7053	269	44	that	that	SCONJ
ejpam-7053	269	45	f	f	PROPN
ejpam-7053	269	46	(	(	PUNCT
ejpam-7053	269	47	x	x	NOUN
ejpam-7053	269	48	)	)	PUNCT
ejpam-7053	269	49	∩	∩	NOUN
ejpam-7053	269	50	v	v	ADP
ejpam-7053	269	51	̸=	̸=	PROPN
ejpam-7053	269	52	∅	∅	NOUN
ejpam-7053	269	53	,	,	PUNCT
ejpam-7053	269	54	there	there	PRON
ejpam-7053	269	55	exists	exist	VERB
ejpam-7053	269	56	a	a	DET
ejpam-7053	269	57	µ-open	µ-open	NOUN
ejpam-7053	269	58	set	set	VERB
ejpam-7053	269	59	u	u	NOUN
ejpam-7053	269	60	of	of	ADP
ejpam-7053	269	61	x	x	PUNCT
ejpam-7053	269	62	containing	contain	VERB
ejpam-7053	269	63	x	x	PUNCT
ejpam-7053	269	64	such	such	ADJ
ejpam-7053	269	65	that	that	SCONJ
ejpam-7053	269	66	f	f	PROPN
ejpam-7053	269	67	(	(	PUNCT
ejpam-7053	269	68	z	z	NOUN
ejpam-7053	269	69	)	)	PUNCT
ejpam-7053	269	70	∩	∩	NOUN
ejpam-7053	269	71	v	v	ADP
ejpam-7053	269	72	̸=	̸=	PROPN
ejpam-7053	269	73	∅	∅	NOUN
ejpam-7053	269	74	for	for	ADP
ejpam-7053	269	75	every	every	DET
ejpam-7053	269	76	z	z	NOUN
ejpam-7053	269	77	∈	∈	PROPN
ejpam-7053	269	78	u	u	NOUN
ejpam-7053	269	79	.	.	PUNCT
ejpam-7053	270	1	a	a	DET
ejpam-7053	270	2	multifunction	multifunction	NOUN
ejpam-7053	270	3	f	f	NOUN
ejpam-7053	270	4	:	:	PUNCT
ejpam-7053	270	5	(	(	PUNCT
ejpam-7053	270	6	x,µ	x,µ	NOUN
ejpam-7053	270	7	)	)	PUNCT
ejpam-7053	270	8	→	→	SYM
ejpam-7053	270	9	(	(	PUNCT
ejpam-7053	270	10	y	y	PROPN
ejpam-7053	270	11	,	,	PUNCT
ejpam-7053	270	12	σ1	σ1	PROPN
ejpam-7053	270	13	,	,	PUNCT
ejpam-7053	270	14	σ2	σ2	PROPN
ejpam-7053	270	15	)	)	PUNCT
ejpam-7053	270	16	is	be	AUX
ejpam-7053	270	17	said	say	VERB
ejpam-7053	270	18	to	to	PART
ejpam-7053	270	19	be	be	AUX
ejpam-7053	270	20	lower	low	ADJ
ejpam-7053	270	21	µ(σ1	µ(σ1	NOUN
ejpam-7053	270	22	,	,	PUNCT
ejpam-7053	270	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	270	24	if	if	SCONJ
ejpam-7053	270	25	f	f	PROPN
ejpam-7053	270	26	is	be	AUX
ejpam-7053	270	27	lower	low	ADJ
ejpam-7053	270	28	µ(σ1	µ(σ1	NOUN
ejpam-7053	270	29	,	,	PUNCT
ejpam-7053	270	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	270	31	at	at	ADP
ejpam-7053	270	32	each	each	DET
ejpam-7053	270	33	point	point	NOUN
ejpam-7053	270	34	x	x	PUNCT
ejpam-7053	270	35	of	of	ADP
ejpam-7053	270	36	x.	x.	NOUN
ejpam-7053	270	37	recall	recall	VERB
ejpam-7053	270	38	that	that	SCONJ
ejpam-7053	270	39	a	a	DET
ejpam-7053	270	40	subset	subset	NOUN
ejpam-7053	270	41	a	a	PRON
ejpam-7053	270	42	of	of	ADP
ejpam-7053	270	43	a	a	DET
ejpam-7053	270	44	bitopological	bitopological	ADJ
ejpam-7053	270	45	space	space	NOUN
ejpam-7053	270	46	(	(	PUNCT
ejpam-7053	270	47	x	x	NOUN
ejpam-7053	270	48	,	,	PUNCT
ejpam-7053	270	49	τ1	τ1	NOUN
ejpam-7053	270	50	,	,	PUNCT
ejpam-7053	270	51	τ2	τ2	NOUN
ejpam-7053	270	52	)	)	PUNCT
ejpam-7053	270	53	is	be	AUX
ejpam-7053	270	54	said	say	VERB
ejpam-7053	270	55	to	to	PART
ejpam-7053	270	56	be	be	AUX
ejpam-7053	270	57	τ1τ2	τ1τ2	NOUN
ejpam-7053	270	58	-	-	ADJ
ejpam-7053	270	59	paracompact	paracompact	ADJ
ejpam-7053	271	1	[	[	X
ejpam-7053	271	2	15	15	NUM
ejpam-7053	271	3	]	]	X
ejpam-7053	271	4	if	if	SCONJ
ejpam-7053	271	5	every	every	DET
ejpam-7053	271	6	cover	cover	NOUN
ejpam-7053	271	7	of	of	ADP
ejpam-7053	271	8	a	a	PRON
ejpam-7053	271	9	by	by	ADP
ejpam-7053	271	10	τ1τ2	τ1τ2	ADJ
ejpam-7053	271	11	-	-	ADJ
ejpam-7053	271	12	open	open	ADJ
ejpam-7053	271	13	sets	set	NOUN
ejpam-7053	271	14	of	of	ADP
ejpam-7053	271	15	x	x	VERB
ejpam-7053	271	16	is	be	AUX
ejpam-7053	271	17	refined	refine	VERB
ejpam-7053	271	18	by	by	ADP
ejpam-7053	271	19	a	a	DET
ejpam-7053	271	20	cover	cover	NOUN
ejpam-7053	271	21	of	of	ADP
ejpam-7053	271	22	a	a	PRON
ejpam-7053	271	23	which	which	PRON
ejpam-7053	271	24	consists	consist	VERB
ejpam-7053	271	25	of	of	ADP
ejpam-7053	271	26	τ1τ2	τ1τ2	ADJ
ejpam-7053	271	27	-	-	ADJ
ejpam-7053	271	28	open	open	ADJ
ejpam-7053	271	29	sets	set	NOUN
ejpam-7053	271	30	of	of	ADP
ejpam-7053	271	31	x	x	PUNCT
ejpam-7053	271	32	and	and	CCONJ
ejpam-7053	271	33	is	be	AUX
ejpam-7053	271	34	τ1τ2	τ1τ2	NOUN
ejpam-7053	271	35	-	-	ADJ
ejpam-7053	271	36	locally	locally	ADV
ejpam-7053	271	37	finite	finite	NOUN
ejpam-7053	271	38	in	in	ADP
ejpam-7053	271	39	x.	x.	PROPN
ejpam-7053	271	40	a	a	DET
ejpam-7053	271	41	subset	subset	VERB
ejpam-7053	271	42	a	a	PRON
ejpam-7053	271	43	of	of	ADP
ejpam-7053	271	44	a	a	DET
ejpam-7053	271	45	bitopological	bitopological	ADJ
ejpam-7053	271	46	space	space	NOUN
ejpam-7053	271	47	(	(	PUNCT
ejpam-7053	271	48	x	x	NOUN
ejpam-7053	271	49	,	,	PUNCT
ejpam-7053	271	50	τ1	τ1	NOUN
ejpam-7053	271	51	,	,	PUNCT
ejpam-7053	271	52	τ2	τ2	NOUN
ejpam-7053	271	53	)	)	PUNCT
ejpam-7053	271	54	is	be	AUX
ejpam-7053	271	55	said	say	VERB
ejpam-7053	271	56	to	to	PART
ejpam-7053	271	57	be	be	AUX
ejpam-7053	271	58	τ1τ2	τ1τ2	NOUN
ejpam-7053	271	59	-	-	ADJ
ejpam-7053	271	60	regular	regular	ADJ
ejpam-7053	271	61	[	[	X
ejpam-7053	271	62	15	15	NUM
ejpam-7053	271	63	]	]	X
ejpam-7053	271	64	if	if	SCONJ
ejpam-7053	271	65	for	for	ADP
ejpam-7053	271	66	each	each	DET
ejpam-7053	271	67	x	x	SYM
ejpam-7053	271	68	∈	∈	PROPN
ejpam-7053	271	69	a	a	DET
ejpam-7053	271	70	and	and	CCONJ
ejpam-7053	271	71	each	each	PRON
ejpam-7053	271	72	τ1τ2	τ1τ2	ADJ
ejpam-7053	271	73	-	-	ADJ
ejpam-7053	271	74	open	open	ADJ
ejpam-7053	271	75	set	set	ADJ
ejpam-7053	271	76	u	u	NOUN
ejpam-7053	271	77	of	of	ADP
ejpam-7053	271	78	x	x	PUNCT
ejpam-7053	271	79	containing	contain	VERB
ejpam-7053	271	80	x	x	PRON
ejpam-7053	271	81	,	,	PUNCT
ejpam-7053	271	82	there	there	PRON
ejpam-7053	271	83	exists	exist	VERB
ejpam-7053	271	84	a	a	DET
ejpam-7053	271	85	τ1τ2	τ1τ2	NOUN
ejpam-7053	271	86	-	-	ADJ
ejpam-7053	271	87	open	open	ADJ
ejpam-7053	271	88	set	set	NOUN
ejpam-7053	271	89	v	v	NOUN
ejpam-7053	271	90	of	of	ADP
ejpam-7053	271	91	x	x	PUNCT
ejpam-7053	271	92	such	such	ADJ
ejpam-7053	271	93	that	that	SCONJ
ejpam-7053	271	94	x	x	SYM
ejpam-7053	271	95	∈	∈	NOUN
ejpam-7053	271	96	v	v	ADP
ejpam-7053	271	97	⊆	⊆	NUM
ejpam-7053	271	98	τ1τ2	τ1τ2	NOUN
ejpam-7053	271	99	-	-	NOUN
ejpam-7053	271	100	cl(v	cl(v	X
ejpam-7053	271	101	)	)	PUNCT
ejpam-7053	271	102	⊆	⊆	NUM
ejpam-7053	271	103	u	u	NOUN
ejpam-7053	271	104	.	.	PUNCT
ejpam-7053	272	1	lemma	lemma	PROPN
ejpam-7053	272	2	4	4	NUM
ejpam-7053	272	3	.	.	PUNCT
ejpam-7053	273	1	[	[	X
ejpam-7053	273	2	15	15	NUM
ejpam-7053	273	3	]	]	X
ejpam-7053	273	4	if	if	SCONJ
ejpam-7053	273	5	a	a	PRON
ejpam-7053	273	6	is	be	AUX
ejpam-7053	273	7	a	a	DET
ejpam-7053	273	8	τ1τ2	τ1τ2	ADJ
ejpam-7053	273	9	-	-	ADJ
ejpam-7053	273	10	regular	regular	ADJ
ejpam-7053	273	11	τ1τ2	τ1τ2	NOUN
ejpam-7053	273	12	-	-	ADJ
ejpam-7053	273	13	paracompact	paracompact	ADJ
ejpam-7053	273	14	set	set	NOUN
ejpam-7053	273	15	of	of	ADP
ejpam-7053	273	16	a	a	DET
ejpam-7053	273	17	bitopological	bitopological	ADJ
ejpam-7053	273	18	space	space	NOUN
ejpam-7053	273	19	(	(	PUNCT
ejpam-7053	273	20	x	x	NOUN
ejpam-7053	273	21	,	,	PUNCT
ejpam-7053	273	22	τ1	τ1	NOUN
ejpam-7053	273	23	,	,	PUNCT
ejpam-7053	273	24	τ2	τ2	NOUN
ejpam-7053	273	25	)	)	PUNCT
ejpam-7053	273	26	and	and	CCONJ
ejpam-7053	273	27	u	u	NOUN
ejpam-7053	273	28	is	be	AUX
ejpam-7053	273	29	a	a	DET
ejpam-7053	273	30	τ1τ2	τ1τ2	ADJ
ejpam-7053	273	31	-	-	ADJ
ejpam-7053	273	32	open	open	ADJ
ejpam-7053	273	33	neighbourhood	neighbourhood	NOUN
ejpam-7053	273	34	of	of	ADP
ejpam-7053	273	35	a	a	PRON
ejpam-7053	273	36	,	,	PUNCT
ejpam-7053	273	37	then	then	ADV
ejpam-7053	273	38	there	there	PRON
ejpam-7053	273	39	exists	exist	VERB
ejpam-7053	273	40	a	a	DET
ejpam-7053	273	41	τ1τ2	τ1τ2	NOUN
ejpam-7053	273	42	-	-	ADJ
ejpam-7053	273	43	open	open	ADJ
ejpam-7053	273	44	set	set	NOUN
ejpam-7053	273	45	v	v	NOUN
ejpam-7053	273	46	of	of	ADP
ejpam-7053	273	47	x	x	PUNCT
ejpam-7053	273	48	such	such	ADJ
ejpam-7053	273	49	that	that	SCONJ
ejpam-7053	273	50	a	a	DET
ejpam-7053	273	51	⊆	⊆	NUM
ejpam-7053	273	52	v	v	ADP
ejpam-7053	273	53	⊆	⊆	NUM
ejpam-7053	273	54	τ1τ2	τ1τ2	NOUN
ejpam-7053	273	55	-	-	NOUN
ejpam-7053	273	56	cl(v	cl(v	X
ejpam-7053	273	57	)	)	PUNCT
ejpam-7053	273	58	⊆	⊆	NUM
ejpam-7053	273	59	u	u	NOUN
ejpam-7053	273	60	.	.	PUNCT
ejpam-7053	273	61	m.	m.	NOUN
ejpam-7053	273	62	chiangpradit	chiangpradit	PROPN
ejpam-7053	273	63	,	,	PUNCT
ejpam-7053	273	64	a.	a.	PROPN
ejpam-7053	273	65	sama	sama	PROPN
ejpam-7053	273	66	-	-	PUNCT
ejpam-7053	273	67	ae	ae	PROPN
ejpam-7053	273	68	,	,	PUNCT
ejpam-7053	273	69	c.	c.	PROPN
ejpam-7053	273	70	boonpok	boonpok	PROPN
ejpam-7053	273	71	/	/	SYM
ejpam-7053	273	72	eur	eur	PROPN
ejpam-7053	273	73	.	.	PUNCT
ejpam-7053	274	1	j.	j.	PROPN
ejpam-7053	274	2	pure	pure	PROPN
ejpam-7053	274	3	appl	appl	PROPN
ejpam-7053	274	4	.	.	PROPN
ejpam-7053	274	5	math	math	PROPN
ejpam-7053	274	6	,	,	PUNCT
ejpam-7053	274	7	18	18	NUM
ejpam-7053	274	8	(	(	PUNCT
ejpam-7053	274	9	4	4	NUM
ejpam-7053	274	10	)	)	PUNCT
ejpam-7053	274	11	(	(	PUNCT
ejpam-7053	274	12	2025	2025	NUM
ejpam-7053	274	13	)	)	PUNCT
ejpam-7053	274	14	,	,	PUNCT
ejpam-7053	274	15	7053	7053	NUM
ejpam-7053	274	16	10	10	NUM
ejpam-7053	274	17	of	of	ADP
ejpam-7053	274	18	13	13	NUM
ejpam-7053	274	19	theorem	theorem	VERB
ejpam-7053	274	20	10	10	NUM
ejpam-7053	274	21	.	.	PUNCT
ejpam-7053	275	1	for	for	ADP
ejpam-7053	275	2	a	a	DET
ejpam-7053	275	3	multifunction	multifunction	NOUN
ejpam-7053	275	4	f	f	NOUN
ejpam-7053	275	5	:	:	PUNCT
ejpam-7053	275	6	(	(	PUNCT
ejpam-7053	275	7	x,µ	x,µ	NOUN
ejpam-7053	275	8	)	)	PUNCT
ejpam-7053	275	9	→	→	SYM
ejpam-7053	275	10	(	(	PUNCT
ejpam-7053	275	11	y	y	PROPN
ejpam-7053	275	12	,	,	PUNCT
ejpam-7053	275	13	σ1	σ1	PROPN
ejpam-7053	275	14	,	,	PUNCT
ejpam-7053	275	15	σ2	σ2	NOUN
ejpam-7053	275	16	)	)	PUNCT
ejpam-7053	275	17	such	such	ADJ
ejpam-7053	275	18	that	that	SCONJ
ejpam-7053	275	19	f	f	PROPN
ejpam-7053	275	20	(	(	PUNCT
ejpam-7053	275	21	x	x	X
ejpam-7053	275	22	)	)	PUNCT
ejpam-7053	275	23	is	be	AUX
ejpam-7053	275	24	a	a	DET
ejpam-7053	275	25	σ1σ2regular	σ1σ2regular	ADJ
ejpam-7053	275	26	σ1σ2	σ1σ2	NOUN
ejpam-7053	275	27	-	-	PUNCT
ejpam-7053	275	28	paracompact	paracompact	ADJ
ejpam-7053	275	29	set	set	NOUN
ejpam-7053	275	30	of	of	ADP
ejpam-7053	275	31	y	y	PROPN
ejpam-7053	275	32	for	for	ADP
ejpam-7053	275	33	each	each	DET
ejpam-7053	275	34	point	point	NOUN
ejpam-7053	275	35	x	x	X
ejpam-7053	275	36	∈	∈	NOUN
ejpam-7053	275	37	x	x	NOUN
ejpam-7053	275	38	,	,	PUNCT
ejpam-7053	275	39	the	the	DET
ejpam-7053	275	40	following	follow	VERB
ejpam-7053	275	41	properties	property	NOUN
ejpam-7053	275	42	are	be	AUX
ejpam-7053	275	43	equivalent	equivalent	ADJ
ejpam-7053	275	44	:	:	PUNCT
ejpam-7053	275	45	(	(	PUNCT
ejpam-7053	275	46	1	1	X
ejpam-7053	275	47	)	)	PUNCT
ejpam-7053	275	48	f	f	PROPN
ejpam-7053	275	49	is	be	AUX
ejpam-7053	275	50	upper	upper	ADJ
ejpam-7053	275	51	µ(σ1	µ(σ1	NOUN
ejpam-7053	275	52	,	,	PUNCT
ejpam-7053	275	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	275	54	;	;	PUNCT
ejpam-7053	275	55	(	(	PUNCT
ejpam-7053	275	56	2	2	X
ejpam-7053	275	57	)	)	PUNCT
ejpam-7053	275	58	f	f	PROPN
ejpam-7053	275	59	is	be	AUX
ejpam-7053	275	60	upper	upper	ADJ
ejpam-7053	275	61	almost	almost	ADV
ejpam-7053	275	62	µ(σ1	µ(σ1	NOUN
ejpam-7053	275	63	,	,	PUNCT
ejpam-7053	275	64	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	275	65	;	;	PUNCT
ejpam-7053	275	66	(	(	PUNCT
ejpam-7053	275	67	3	3	X
ejpam-7053	275	68	)	)	PUNCT
ejpam-7053	275	69	f	f	PROPN
ejpam-7053	275	70	is	be	AUX
ejpam-7053	275	71	upper	upper	ADJ
ejpam-7053	275	72	weakly	weakly	ADJ
ejpam-7053	275	73	µ(σ1	µ(σ1	NOUN
ejpam-7053	275	74	,	,	PUNCT
ejpam-7053	276	1	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	276	2	.	.	NOUN
ejpam-7053	276	3	proof	proof	NOUN
ejpam-7053	276	4	.	.	PUNCT
ejpam-7053	277	1	we	we	PRON
ejpam-7053	277	2	show	show	VERB
ejpam-7053	277	3	only	only	ADV
ejpam-7053	277	4	the	the	DET
ejpam-7053	277	5	implication	implication	NOUN
ejpam-7053	277	6	(	(	PUNCT
ejpam-7053	277	7	3	3	X
ejpam-7053	277	8	)	)	PUNCT
ejpam-7053	277	9	⇒	⇒	NOUN
ejpam-7053	277	10	(	(	PUNCT
ejpam-7053	277	11	1	1	X
ejpam-7053	277	12	)	)	PUNCT
ejpam-7053	277	13	since	since	SCONJ
ejpam-7053	277	14	the	the	DET
ejpam-7053	277	15	others	other	NOUN
ejpam-7053	277	16	are	be	AUX
ejpam-7053	277	17	obvious	obvious	ADJ
ejpam-7053	277	18	.	.	PUNCT
ejpam-7053	278	1	suppose	suppose	VERB
ejpam-7053	278	2	that	that	SCONJ
ejpam-7053	278	3	f	f	PROPN
ejpam-7053	278	4	is	be	AUX
ejpam-7053	278	5	upper	upper	ADJ
ejpam-7053	278	6	weakly	weakly	ADJ
ejpam-7053	278	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	278	8	,	,	PUNCT
ejpam-7053	278	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	279	1	.	.	PUNCT
ejpam-7053	279	2	let	let	VERB
ejpam-7053	279	3	x	x	SYM
ejpam-7053	279	4	∈	∈	PROPN
ejpam-7053	279	5	x	x	X
ejpam-7053	279	6	and	and	CCONJ
ejpam-7053	279	7	v	v	X
ejpam-7053	279	8	be	be	AUX
ejpam-7053	279	9	any	any	DET
ejpam-7053	279	10	σ1σ2	σ1σ2	NOUN
ejpam-7053	279	11	-	-	ADJ
ejpam-7053	279	12	open	open	ADJ
ejpam-7053	279	13	set	set	NOUN
ejpam-7053	279	14	of	of	ADP
ejpam-7053	279	15	y	y	PRON
ejpam-7053	279	16	such	such	ADJ
ejpam-7053	279	17	that	that	SCONJ
ejpam-7053	279	18	f	f	PROPN
ejpam-7053	279	19	(	(	PUNCT
ejpam-7053	279	20	x	x	X
ejpam-7053	279	21	)	)	PUNCT
ejpam-7053	279	22	⊆	⊆	NUM
ejpam-7053	279	23	v	v	NOUN
ejpam-7053	279	24	.	.	PUNCT
ejpam-7053	280	1	since	since	SCONJ
ejpam-7053	280	2	f	f	PROPN
ejpam-7053	280	3	(	(	PUNCT
ejpam-7053	280	4	x	x	X
ejpam-7053	280	5	)	)	PUNCT
ejpam-7053	280	6	is	be	AUX
ejpam-7053	280	7	σ1σ2	σ1σ2	NOUN
ejpam-7053	280	8	-	-	ADJ
ejpam-7053	280	9	regular	regular	ADJ
ejpam-7053	280	10	σ1σ2	σ1σ2	NOUN
ejpam-7053	280	11	-	-	PUNCT
ejpam-7053	280	12	paracompact	paracompact	ADJ
ejpam-7053	280	13	,	,	PUNCT
ejpam-7053	280	14	by	by	ADP
ejpam-7053	280	15	lemma	lemma	PROPN
ejpam-7053	280	16	4	4	NUM
ejpam-7053	280	17	there	there	ADV
ejpam-7053	280	18	exists	exist	VERB
ejpam-7053	280	19	a	a	DET
ejpam-7053	280	20	σ1σ2	σ1σ2	NUM
ejpam-7053	280	21	-	-	ADJ
ejpam-7053	280	22	open	open	ADJ
ejpam-7053	280	23	set	set	NOUN
ejpam-7053	280	24	w	w	PROPN
ejpam-7053	280	25	of	of	ADP
ejpam-7053	280	26	y	y	PRON
ejpam-7053	280	27	such	such	ADJ
ejpam-7053	280	28	that	that	SCONJ
ejpam-7053	280	29	f	f	PROPN
ejpam-7053	280	30	(	(	PUNCT
ejpam-7053	280	31	x	x	X
ejpam-7053	280	32	)	)	PUNCT
ejpam-7053	280	33	⊆	⊆	NUM
ejpam-7053	280	34	w	w	ADP
ejpam-7053	280	35	⊆	⊆	NUM
ejpam-7053	280	36	σ1σ2	σ1σ2	NOUN
ejpam-7053	280	37	-	-	PUNCT
ejpam-7053	280	38	cl(w	cl(w	NOUN
ejpam-7053	280	39	)	)	PUNCT
ejpam-7053	280	40	⊆	⊆	NUM
ejpam-7053	280	41	v	v	NOUN
ejpam-7053	280	42	.	.	PUNCT
ejpam-7053	281	1	since	since	SCONJ
ejpam-7053	281	2	f	f	PROPN
ejpam-7053	281	3	is	be	AUX
ejpam-7053	281	4	upper	upper	ADJ
ejpam-7053	281	5	weakly	weakly	ADJ
ejpam-7053	281	6	µ(σ1	µ(σ1	NOUN
ejpam-7053	281	7	,	,	PUNCT
ejpam-7053	281	8	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	281	9	,	,	PUNCT
ejpam-7053	281	10	there	there	PRON
ejpam-7053	281	11	exists	exist	VERB
ejpam-7053	281	12	a	a	DET
ejpam-7053	281	13	µ-open	µ-open	NOUN
ejpam-7053	281	14	set	set	VERB
ejpam-7053	281	15	u	u	NOUN
ejpam-7053	281	16	of	of	ADP
ejpam-7053	281	17	x	x	PUNCT
ejpam-7053	281	18	containing	contain	VERB
ejpam-7053	281	19	x	x	PUNCT
ejpam-7053	281	20	such	such	ADJ
ejpam-7053	281	21	that	that	SCONJ
ejpam-7053	281	22	f	f	PROPN
ejpam-7053	281	23	(	(	PUNCT
ejpam-7053	281	24	u	u	NOUN
ejpam-7053	281	25	)	)	PUNCT
ejpam-7053	281	26	⊆	⊆	NUM
ejpam-7053	281	27	σ1σ2	σ1σ2	NOUN
ejpam-7053	281	28	-	-	PUNCT
ejpam-7053	281	29	cl(w	cl(w	NOUN
ejpam-7053	281	30	)	)	PUNCT
ejpam-7053	281	31	;	;	PUNCT
ejpam-7053	281	32	hence	hence	ADV
ejpam-7053	281	33	f	f	PROPN
ejpam-7053	281	34	(	(	PUNCT
ejpam-7053	281	35	u	u	NOUN
ejpam-7053	281	36	)	)	PUNCT
ejpam-7053	281	37	⊆	⊆	NUM
ejpam-7053	281	38	v	v	NOUN
ejpam-7053	281	39	.	.	PUNCT
ejpam-7053	282	1	this	this	PRON
ejpam-7053	282	2	shows	show	VERB
ejpam-7053	282	3	that	that	SCONJ
ejpam-7053	282	4	f	f	PROPN
ejpam-7053	282	5	is	be	AUX
ejpam-7053	282	6	upper	upper	ADJ
ejpam-7053	282	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	282	8	,	,	PUNCT
ejpam-7053	282	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	282	10	.	.	PROPN
ejpam-7053	282	11	recall	recall	VERB
ejpam-7053	282	12	that	that	SCONJ
ejpam-7053	282	13	a	a	DET
ejpam-7053	282	14	bitopological	bitopological	ADJ
ejpam-7053	282	15	space	space	NOUN
ejpam-7053	282	16	(	(	PUNCT
ejpam-7053	282	17	x	x	NOUN
ejpam-7053	282	18	,	,	PUNCT
ejpam-7053	282	19	τ1	τ1	NOUN
ejpam-7053	282	20	,	,	PUNCT
ejpam-7053	282	21	τ2	τ2	NOUN
ejpam-7053	282	22	)	)	PUNCT
ejpam-7053	282	23	is	be	AUX
ejpam-7053	282	24	said	say	VERB
ejpam-7053	282	25	to	to	PART
ejpam-7053	282	26	be	be	AUX
ejpam-7053	282	27	τ1τ2	τ1τ2	NOUN
ejpam-7053	282	28	-	-	ADJ
ejpam-7053	282	29	compact	compact	ADJ
ejpam-7053	282	30	[	[	X
ejpam-7053	282	31	15	15	NUM
ejpam-7053	282	32	]	]	X
ejpam-7053	282	33	if	if	SCONJ
ejpam-7053	282	34	every	every	DET
ejpam-7053	282	35	cover	cover	NOUN
ejpam-7053	282	36	of	of	ADP
ejpam-7053	282	37	x	x	PUNCT
ejpam-7053	282	38	by	by	ADP
ejpam-7053	282	39	τ1τ2	τ1τ2	ADJ
ejpam-7053	282	40	-	-	ADJ
ejpam-7053	282	41	open	open	ADJ
ejpam-7053	282	42	sets	set	NOUN
ejpam-7053	282	43	of	of	ADP
ejpam-7053	282	44	x	x	PUNCT
ejpam-7053	282	45	has	have	VERB
ejpam-7053	282	46	a	a	DET
ejpam-7053	282	47	finite	finite	ADJ
ejpam-7053	282	48	subcover	subcover	PROPN
ejpam-7053	282	49	.	.	PUNCT
ejpam-7053	283	1	definition	definition	NOUN
ejpam-7053	283	2	7	7	NUM
ejpam-7053	283	3	.	.	PUNCT
ejpam-7053	284	1	[	[	X
ejpam-7053	284	2	21	21	NUM
ejpam-7053	284	3	]	]	X
ejpam-7053	284	4	a	a	DET
ejpam-7053	284	5	bitopological	bitopological	ADJ
ejpam-7053	284	6	space	space	NOUN
ejpam-7053	284	7	(	(	PUNCT
ejpam-7053	284	8	x	x	NOUN
ejpam-7053	284	9	,	,	PUNCT
ejpam-7053	284	10	τ1	τ1	NOUN
ejpam-7053	284	11	,	,	PUNCT
ejpam-7053	284	12	τ2	τ2	NOUN
ejpam-7053	284	13	)	)	PUNCT
ejpam-7053	284	14	is	be	AUX
ejpam-7053	284	15	said	say	VERB
ejpam-7053	284	16	to	to	PART
ejpam-7053	284	17	be	be	AUX
ejpam-7053	284	18	(	(	PUNCT
ejpam-7053	284	19	τ1	τ1	NOUN
ejpam-7053	284	20	,	,	PUNCT
ejpam-7053	284	21	τ2)-regular	τ2)-regular	ADJ
ejpam-7053	284	22	if	if	SCONJ
ejpam-7053	284	23	for	for	ADP
ejpam-7053	284	24	each	each	DET
ejpam-7053	284	25	τ1τ2	τ1τ2	ADJ
ejpam-7053	284	26	-	-	ADJ
ejpam-7053	284	27	closed	closed	ADJ
ejpam-7053	284	28	set	set	VERB
ejpam-7053	284	29	f	f	NOUN
ejpam-7053	284	30	and	and	CCONJ
ejpam-7053	284	31	each	each	DET
ejpam-7053	284	32	x	x	SYM
ejpam-7053	284	33	∈	∈	PROPN
ejpam-7053	284	34	x	x	X
ejpam-7053	284	35	−	−	PROPN
ejpam-7053	284	36	f	f	NOUN
ejpam-7053	284	37	,	,	PUNCT
ejpam-7053	284	38	there	there	PRON
ejpam-7053	284	39	exist	exist	VERB
ejpam-7053	284	40	disjoint	disjoint	ADJ
ejpam-7053	284	41	τ1τ2	τ1τ2	ADJ
ejpam-7053	284	42	-	-	ADJ
ejpam-7053	284	43	open	open	ADJ
ejpam-7053	284	44	sets	set	NOUN
ejpam-7053	284	45	u	u	NOUN
ejpam-7053	284	46	and	and	CCONJ
ejpam-7053	284	47	v	v	ADP
ejpam-7053	284	48	such	such	ADJ
ejpam-7053	284	49	that	that	SCONJ
ejpam-7053	284	50	x	x	SYM
ejpam-7053	284	51	∈	∈	PROPN
ejpam-7053	284	52	u	u	NOUN
ejpam-7053	284	53	and	and	CCONJ
ejpam-7053	284	54	f	f	PROPN
ejpam-7053	284	55	⊆	⊆	NUM
ejpam-7053	284	56	v	v	NOUN
ejpam-7053	284	57	.	.	PUNCT
ejpam-7053	285	1	corollary	corollary	ADJ
ejpam-7053	285	2	1	1	NUM
ejpam-7053	285	3	.	.	PUNCT
ejpam-7053	286	1	let	let	VERB
ejpam-7053	286	2	f	f	NOUN
ejpam-7053	286	3	:	:	PUNCT
ejpam-7053	286	4	(	(	PUNCT
ejpam-7053	286	5	x,µ	x,µ	NOUN
ejpam-7053	286	6	)	)	PUNCT
ejpam-7053	286	7	→	→	SYM
ejpam-7053	286	8	(	(	PUNCT
ejpam-7053	286	9	y	y	PROPN
ejpam-7053	286	10	,	,	PUNCT
ejpam-7053	286	11	σ1	σ1	PROPN
ejpam-7053	286	12	,	,	PUNCT
ejpam-7053	286	13	σ2	σ2	PROPN
ejpam-7053	286	14	)	)	PUNCT
ejpam-7053	286	15	be	be	VERB
ejpam-7053	286	16	a	a	DET
ejpam-7053	286	17	multifunction	multifunction	NOUN
ejpam-7053	286	18	such	such	ADJ
ejpam-7053	286	19	that	that	SCONJ
ejpam-7053	286	20	f	f	PROPN
ejpam-7053	286	21	(	(	PUNCT
ejpam-7053	286	22	x	x	X
ejpam-7053	286	23	)	)	PUNCT
ejpam-7053	286	24	is	be	AUX
ejpam-7053	286	25	σ1σ2compact	σ1σ2compact	VERB
ejpam-7053	286	26	for	for	ADP
ejpam-7053	286	27	each	each	DET
ejpam-7053	286	28	point	point	NOUN
ejpam-7053	286	29	x	x	X
ejpam-7053	286	30	∈	∈	NOUN
ejpam-7053	286	31	x	x	X
ejpam-7053	286	32	and	and	CCONJ
ejpam-7053	286	33	(	(	PUNCT
ejpam-7053	286	34	y	y	PROPN
ejpam-7053	286	35	,	,	PUNCT
ejpam-7053	286	36	σ1	σ1	PROPN
ejpam-7053	286	37	,	,	PUNCT
ejpam-7053	286	38	σ2	σ2	PROPN
ejpam-7053	286	39	)	)	PUNCT
ejpam-7053	286	40	is	be	AUX
ejpam-7053	286	41	(	(	PUNCT
ejpam-7053	286	42	σ1	σ1	NOUN
ejpam-7053	286	43	,	,	PUNCT
ejpam-7053	286	44	σ2)-regular	σ2)-regular	PROPN
ejpam-7053	286	45	.	.	PUNCT
ejpam-7053	287	1	then	then	ADV
ejpam-7053	287	2	,	,	PUNCT
ejpam-7053	287	3	the	the	DET
ejpam-7053	287	4	following	follow	VERB
ejpam-7053	287	5	properties	property	NOUN
ejpam-7053	287	6	are	be	AUX
ejpam-7053	287	7	equivalent	equivalent	ADJ
ejpam-7053	287	8	:	:	PUNCT
ejpam-7053	287	9	(	(	PUNCT
ejpam-7053	287	10	1	1	X
ejpam-7053	287	11	)	)	PUNCT
ejpam-7053	287	12	f	f	PROPN
ejpam-7053	287	13	is	be	AUX
ejpam-7053	287	14	upper	upper	ADJ
ejpam-7053	287	15	µ(σ1	µ(σ1	NOUN
ejpam-7053	287	16	,	,	PUNCT
ejpam-7053	287	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	287	18	;	;	PUNCT
ejpam-7053	287	19	(	(	PUNCT
ejpam-7053	287	20	2	2	X
ejpam-7053	287	21	)	)	PUNCT
ejpam-7053	287	22	f	f	PROPN
ejpam-7053	287	23	is	be	AUX
ejpam-7053	287	24	upper	upper	ADJ
ejpam-7053	287	25	almost	almost	ADV
ejpam-7053	287	26	µ(σ1	µ(σ1	NOUN
ejpam-7053	287	27	,	,	PUNCT
ejpam-7053	287	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	287	29	;	;	PUNCT
ejpam-7053	287	30	(	(	PUNCT
ejpam-7053	287	31	3	3	X
ejpam-7053	287	32	)	)	PUNCT
ejpam-7053	287	33	f	f	PROPN
ejpam-7053	287	34	is	be	AUX
ejpam-7053	287	35	upper	upper	ADJ
ejpam-7053	287	36	weakly	weakly	ADJ
ejpam-7053	287	37	µ(σ1	µ(σ1	NOUN
ejpam-7053	287	38	,	,	PUNCT
ejpam-7053	287	39	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	287	40	.	.	PUNCT
ejpam-7053	288	1	lemma	lemma	PROPN
ejpam-7053	288	2	5	5	NUM
ejpam-7053	288	3	.	.	PUNCT
ejpam-7053	289	1	[	[	X
ejpam-7053	289	2	22	22	NUM
ejpam-7053	289	3	]	]	X
ejpam-7053	289	4	if	if	SCONJ
ejpam-7053	289	5	a	a	PRON
ejpam-7053	289	6	is	be	AUX
ejpam-7053	289	7	a	a	DET
ejpam-7053	289	8	τ1τ2	τ1τ2	ADJ
ejpam-7053	289	9	-	-	ADJ
ejpam-7053	289	10	regular	regular	ADJ
ejpam-7053	289	11	set	set	NOUN
ejpam-7053	289	12	of	of	ADP
ejpam-7053	289	13	a	a	DET
ejpam-7053	289	14	bitopological	bitopological	ADJ
ejpam-7053	289	15	space	space	NOUN
ejpam-7053	289	16	(	(	PUNCT
ejpam-7053	289	17	x	x	NOUN
ejpam-7053	289	18	,	,	PUNCT
ejpam-7053	289	19	τ1	τ1	NOUN
ejpam-7053	289	20	,	,	PUNCT
ejpam-7053	289	21	τ2	τ2	NOUN
ejpam-7053	289	22	)	)	PUNCT
ejpam-7053	289	23	,	,	PUNCT
ejpam-7053	289	24	then	then	ADV
ejpam-7053	289	25	for	for	ADP
ejpam-7053	289	26	each	each	DET
ejpam-7053	289	27	τ1τ2	τ1τ2	ADJ
ejpam-7053	289	28	-	-	ADJ
ejpam-7053	289	29	open	open	ADJ
ejpam-7053	289	30	set	set	NOUN
ejpam-7053	289	31	g	g	NOUN
ejpam-7053	289	32	which	which	PRON
ejpam-7053	289	33	intersect	intersect	ADJ
ejpam-7053	289	34	a	a	X
ejpam-7053	289	35	,	,	PUNCT
ejpam-7053	289	36	there	there	PRON
ejpam-7053	289	37	exists	exist	VERB
ejpam-7053	289	38	a	a	DET
ejpam-7053	289	39	τ1τ2	τ1τ2	NOUN
ejpam-7053	289	40	-	-	ADJ
ejpam-7053	289	41	open	open	ADJ
ejpam-7053	289	42	set	set	NOUN
ejpam-7053	289	43	w	w	ADP
ejpam-7053	289	44	such	such	ADJ
ejpam-7053	289	45	that	that	SCONJ
ejpam-7053	289	46	a	a	DET
ejpam-7053	289	47	∩	∩	NOUN
ejpam-7053	289	48	w	w	PROPN
ejpam-7053	289	49	̸=	̸=	PROPN
ejpam-7053	289	50	∅	∅	NOUN
ejpam-7053	289	51	and	and	CCONJ
ejpam-7053	289	52	τ1τ2	τ1τ2	NOUN
ejpam-7053	289	53	-	-	NOUN
ejpam-7053	289	54	cl(w	cl(w	NOUN
ejpam-7053	289	55	)	)	PUNCT
ejpam-7053	289	56	⊆	⊆	NUM
ejpam-7053	289	57	g.	g.	NOUN
ejpam-7053	289	58	theorem	theorem	VERB
ejpam-7053	289	59	11	11	NUM
ejpam-7053	289	60	.	.	PUNCT
ejpam-7053	290	1	for	for	ADP
ejpam-7053	290	2	a	a	DET
ejpam-7053	290	3	multifunction	multifunction	NOUN
ejpam-7053	290	4	f	f	NOUN
ejpam-7053	290	5	:	:	PUNCT
ejpam-7053	290	6	(	(	PUNCT
ejpam-7053	290	7	x,µ	x,µ	NOUN
ejpam-7053	290	8	)	)	PUNCT
ejpam-7053	290	9	→	→	SYM
ejpam-7053	290	10	(	(	PUNCT
ejpam-7053	290	11	y	y	PROPN
ejpam-7053	290	12	,	,	PUNCT
ejpam-7053	290	13	σ1	σ1	PROPN
ejpam-7053	290	14	,	,	PUNCT
ejpam-7053	290	15	σ2	σ2	NOUN
ejpam-7053	290	16	)	)	PUNCT
ejpam-7053	290	17	such	such	ADJ
ejpam-7053	290	18	that	that	SCONJ
ejpam-7053	290	19	f	f	PROPN
ejpam-7053	290	20	(	(	PUNCT
ejpam-7053	290	21	x	x	X
ejpam-7053	290	22	)	)	PUNCT
ejpam-7053	290	23	is	be	AUX
ejpam-7053	290	24	a	a	DET
ejpam-7053	290	25	σ1σ2regular	σ1σ2regular	PROPN
ejpam-7053	290	26	set	set	NOUN
ejpam-7053	290	27	of	of	ADP
ejpam-7053	290	28	y	y	PROPN
ejpam-7053	290	29	for	for	ADP
ejpam-7053	290	30	each	each	DET
ejpam-7053	290	31	point	point	NOUN
ejpam-7053	290	32	x	x	X
ejpam-7053	290	33	∈	∈	NOUN
ejpam-7053	290	34	x	x	NOUN
ejpam-7053	290	35	,	,	PUNCT
ejpam-7053	290	36	the	the	DET
ejpam-7053	290	37	following	follow	VERB
ejpam-7053	290	38	properties	property	NOUN
ejpam-7053	290	39	are	be	AUX
ejpam-7053	290	40	equivalent	equivalent	ADJ
ejpam-7053	290	41	:	:	PUNCT
ejpam-7053	290	42	(	(	PUNCT
ejpam-7053	290	43	1	1	X
ejpam-7053	290	44	)	)	PUNCT
ejpam-7053	290	45	f	f	PROPN
ejpam-7053	290	46	is	be	AUX
ejpam-7053	290	47	lower	low	ADJ
ejpam-7053	290	48	µ(σ1	µ(σ1	NOUN
ejpam-7053	290	49	,	,	PUNCT
ejpam-7053	290	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	290	51	;	;	PUNCT
ejpam-7053	290	52	(	(	PUNCT
ejpam-7053	290	53	2	2	X
ejpam-7053	290	54	)	)	PUNCT
ejpam-7053	290	55	f	f	PROPN
ejpam-7053	290	56	is	be	AUX
ejpam-7053	290	57	lower	low	ADJ
ejpam-7053	290	58	almost	almost	ADV
ejpam-7053	290	59	µ(σ1	µ(σ1	NOUN
ejpam-7053	290	60	,	,	PUNCT
ejpam-7053	290	61	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	290	62	;	;	PUNCT
ejpam-7053	290	63	(	(	PUNCT
ejpam-7053	290	64	3	3	X
ejpam-7053	290	65	)	)	PUNCT
ejpam-7053	290	66	f	f	PROPN
ejpam-7053	290	67	is	be	AUX
ejpam-7053	290	68	lower	low	ADJ
ejpam-7053	290	69	weakly	weakly	ADJ
ejpam-7053	290	70	µ(σ1	µ(σ1	NOUN
ejpam-7053	290	71	,	,	PUNCT
ejpam-7053	290	72	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	290	73	.	.	PROPN
ejpam-7053	290	74	m.	m.	NOUN
ejpam-7053	290	75	chiangpradit	chiangpradit	PROPN
ejpam-7053	290	76	,	,	PUNCT
ejpam-7053	290	77	a.	a.	PROPN
ejpam-7053	290	78	sama	sama	PROPN
ejpam-7053	290	79	-	-	PUNCT
ejpam-7053	290	80	ae	ae	PROPN
ejpam-7053	290	81	,	,	PUNCT
ejpam-7053	290	82	c.	c.	PROPN
ejpam-7053	290	83	boonpok	boonpok	PROPN
ejpam-7053	290	84	/	/	SYM
ejpam-7053	290	85	eur	eur	PROPN
ejpam-7053	290	86	.	.	PUNCT
ejpam-7053	291	1	j.	j.	PROPN
ejpam-7053	291	2	pure	pure	PROPN
ejpam-7053	291	3	appl	appl	PROPN
ejpam-7053	291	4	.	.	PROPN
ejpam-7053	291	5	math	math	PROPN
ejpam-7053	291	6	,	,	PUNCT
ejpam-7053	291	7	18	18	NUM
ejpam-7053	291	8	(	(	PUNCT
ejpam-7053	291	9	4	4	NUM
ejpam-7053	291	10	)	)	PUNCT
ejpam-7053	291	11	(	(	PUNCT
ejpam-7053	291	12	2025	2025	NUM
ejpam-7053	291	13	)	)	PUNCT
ejpam-7053	291	14	,	,	PUNCT
ejpam-7053	291	15	7053	7053	NUM
ejpam-7053	291	16	11	11	NUM
ejpam-7053	291	17	of	of	ADP
ejpam-7053	291	18	13	13	NUM
ejpam-7053	291	19	proof	proof	NOUN
ejpam-7053	291	20	.	.	PUNCT
ejpam-7053	292	1	we	we	PRON
ejpam-7053	292	2	show	show	VERB
ejpam-7053	292	3	only	only	ADV
ejpam-7053	292	4	the	the	DET
ejpam-7053	292	5	implication	implication	NOUN
ejpam-7053	292	6	(	(	PUNCT
ejpam-7053	292	7	3	3	X
ejpam-7053	292	8	)	)	PUNCT
ejpam-7053	292	9	⇒	⇒	NOUN
ejpam-7053	292	10	(	(	PUNCT
ejpam-7053	292	11	1	1	X
ejpam-7053	292	12	)	)	PUNCT
ejpam-7053	292	13	since	since	SCONJ
ejpam-7053	292	14	the	the	DET
ejpam-7053	292	15	others	other	NOUN
ejpam-7053	292	16	are	be	AUX
ejpam-7053	292	17	obvious	obvious	ADJ
ejpam-7053	292	18	.	.	PUNCT
ejpam-7053	293	1	suppose	suppose	VERB
ejpam-7053	293	2	that	that	SCONJ
ejpam-7053	293	3	f	f	PROPN
ejpam-7053	293	4	is	be	AUX
ejpam-7053	293	5	lower	low	ADJ
ejpam-7053	293	6	weakly	weakly	ADJ
ejpam-7053	293	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	293	8	,	,	PUNCT
ejpam-7053	293	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	293	10	.	.	PUNCT
ejpam-7053	293	11	let	let	VERB
ejpam-7053	293	12	x	x	SYM
ejpam-7053	293	13	∈	∈	PROPN
ejpam-7053	293	14	x	x	X
ejpam-7053	293	15	and	and	CCONJ
ejpam-7053	293	16	v	v	X
ejpam-7053	293	17	be	be	AUX
ejpam-7053	293	18	any	any	DET
ejpam-7053	293	19	σ1σ2	σ1σ2	NOUN
ejpam-7053	293	20	-	-	ADJ
ejpam-7053	293	21	open	open	ADJ
ejpam-7053	293	22	set	set	NOUN
ejpam-7053	293	23	of	of	ADP
ejpam-7053	293	24	y	y	PRON
ejpam-7053	293	25	such	such	ADJ
ejpam-7053	293	26	that	that	PRON
ejpam-7053	293	27	v	v	ADP
ejpam-7053	293	28	∩	∩	ADJ
ejpam-7053	293	29	f	f	X
ejpam-7053	293	30	(	(	PUNCT
ejpam-7053	293	31	x	x	X
ejpam-7053	293	32	)	)	PUNCT
ejpam-7053	293	33	̸=	̸=	PROPN
ejpam-7053	293	34	∅.	∅.	ADV
ejpam-7053	293	35	since	since	SCONJ
ejpam-7053	293	36	f	f	PROPN
ejpam-7053	293	37	(	(	PUNCT
ejpam-7053	293	38	x	x	X
ejpam-7053	293	39	)	)	PUNCT
ejpam-7053	293	40	is	be	AUX
ejpam-7053	293	41	σ1σ2	σ1σ2	NOUN
ejpam-7053	293	42	-	-	ADJ
ejpam-7053	293	43	regular	regular	ADJ
ejpam-7053	293	44	,	,	PUNCT
ejpam-7053	293	45	by	by	ADP
ejpam-7053	293	46	lemma	lemma	PROPN
ejpam-7053	293	47	5	5	NUM
ejpam-7053	293	48	there	there	ADV
ejpam-7053	293	49	exists	exist	VERB
ejpam-7053	293	50	a	a	DET
ejpam-7053	293	51	σ1σ2	σ1σ2	NUM
ejpam-7053	293	52	-	-	ADJ
ejpam-7053	293	53	open	open	ADJ
ejpam-7053	293	54	set	set	NOUN
ejpam-7053	293	55	w	w	PROPN
ejpam-7053	293	56	of	of	ADP
ejpam-7053	293	57	y	y	PRON
ejpam-7053	293	58	such	such	ADJ
ejpam-7053	293	59	that	that	SCONJ
ejpam-7053	293	60	f	f	PROPN
ejpam-7053	293	61	(	(	PUNCT
ejpam-7053	293	62	x	x	NOUN
ejpam-7053	293	63	)	)	PUNCT
ejpam-7053	293	64	∩	∩	NOUN
ejpam-7053	293	65	w	w	PROPN
ejpam-7053	293	66	̸=	̸=	PROPN
ejpam-7053	293	67	∅	∅	NOUN
ejpam-7053	293	68	and	and	CCONJ
ejpam-7053	293	69	σ1σ2	σ1σ2	NOUN
ejpam-7053	293	70	-	-	PUNCT
ejpam-7053	293	71	cl(w	cl(w	NOUN
ejpam-7053	293	72	)	)	PUNCT
ejpam-7053	293	73	⊆	⊆	NUM
ejpam-7053	293	74	v	v	NOUN
ejpam-7053	293	75	.	.	PUNCT
ejpam-7053	294	1	since	since	SCONJ
ejpam-7053	294	2	f	f	PROPN
ejpam-7053	294	3	is	be	AUX
ejpam-7053	294	4	lower	low	ADJ
ejpam-7053	294	5	weakly	weakly	ADJ
ejpam-7053	294	6	µ(σ1	µ(σ1	NOUN
ejpam-7053	294	7	,	,	PUNCT
ejpam-7053	294	8	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	294	9	,	,	PUNCT
ejpam-7053	294	10	there	there	PRON
ejpam-7053	294	11	exists	exist	VERB
ejpam-7053	294	12	a	a	DET
ejpam-7053	294	13	µ-open	µ-open	NOUN
ejpam-7053	294	14	set	set	VERB
ejpam-7053	294	15	u	u	NOUN
ejpam-7053	294	16	of	of	ADP
ejpam-7053	294	17	x	x	PUNCT
ejpam-7053	294	18	containing	contain	VERB
ejpam-7053	294	19	x	x	PUNCT
ejpam-7053	294	20	such	such	ADJ
ejpam-7053	294	21	that	that	SCONJ
ejpam-7053	294	22	σ1σ2	σ1σ2	NOUN
ejpam-7053	294	23	-	-	PUNCT
ejpam-7053	294	24	cl(w	cl(w	NOUN
ejpam-7053	294	25	)	)	PUNCT
ejpam-7053	294	26	∩	∩	PROPN
ejpam-7053	294	27	f	f	X
ejpam-7053	294	28	(	(	PUNCT
ejpam-7053	294	29	z	z	NOUN
ejpam-7053	294	30	)	)	PUNCT
ejpam-7053	294	31	̸=	̸=	PROPN
ejpam-7053	294	32	∅	∅	NOUN
ejpam-7053	294	33	;	;	PUNCT
ejpam-7053	294	34	hence	hence	ADV
ejpam-7053	294	35	f	f	X
ejpam-7053	294	36	(	(	PUNCT
ejpam-7053	294	37	z	z	NOUN
ejpam-7053	294	38	)	)	PUNCT
ejpam-7053	294	39	∩	∩	NOUN
ejpam-7053	294	40	v	v	ADP
ejpam-7053	294	41	̸=	̸=	PROPN
ejpam-7053	294	42	∅	∅	NOUN
ejpam-7053	294	43	for	for	ADP
ejpam-7053	294	44	each	each	DET
ejpam-7053	294	45	z	z	NOUN
ejpam-7053	294	46	∈	∈	PROPN
ejpam-7053	294	47	u	u	NOUN
ejpam-7053	294	48	.	.	PUNCT
ejpam-7053	295	1	this	this	PRON
ejpam-7053	295	2	shows	show	VERB
ejpam-7053	295	3	that	that	SCONJ
ejpam-7053	295	4	f	f	PROPN
ejpam-7053	295	5	is	be	AUX
ejpam-7053	295	6	lower	low	ADJ
ejpam-7053	295	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	295	8	,	,	PUNCT
ejpam-7053	295	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	295	10	.	.	NOUN
ejpam-7053	295	11	definition	definition	NOUN
ejpam-7053	295	12	8	8	NUM
ejpam-7053	295	13	.	.	PUNCT
ejpam-7053	296	1	[	[	X
ejpam-7053	296	2	23	23	NUM
ejpam-7053	296	3	]	]	PUNCT
ejpam-7053	296	4	a	a	DET
ejpam-7053	296	5	bitopological	bitopological	ADJ
ejpam-7053	296	6	space	space	NOUN
ejpam-7053	296	7	(	(	PUNCT
ejpam-7053	296	8	x	x	NOUN
ejpam-7053	296	9	,	,	PUNCT
ejpam-7053	296	10	τ1	τ1	NOUN
ejpam-7053	296	11	,	,	PUNCT
ejpam-7053	296	12	τ2	τ2	NOUN
ejpam-7053	296	13	)	)	PUNCT
ejpam-7053	296	14	is	be	AUX
ejpam-7053	296	15	said	say	VERB
ejpam-7053	296	16	to	to	PART
ejpam-7053	296	17	be	be	AUX
ejpam-7053	296	18	(	(	PUNCT
ejpam-7053	296	19	τ1	τ1	NOUN
ejpam-7053	296	20	,	,	PUNCT
ejpam-7053	296	21	τ2)-normal	τ2)-normal	ADJ
ejpam-7053	296	22	if	if	SCONJ
ejpam-7053	296	23	for	for	ADP
ejpam-7053	296	24	each	each	DET
ejpam-7053	296	25	pair	pair	NOUN
ejpam-7053	296	26	of	of	ADP
ejpam-7053	296	27	disjoint	disjoint	ADJ
ejpam-7053	296	28	τ1τ2	τ1τ2	ADJ
ejpam-7053	296	29	-	-	ADJ
ejpam-7053	296	30	closed	closed	ADJ
ejpam-7053	296	31	sets	set	NOUN
ejpam-7053	296	32	f	f	PROPN
ejpam-7053	296	33	and	and	CCONJ
ejpam-7053	296	34	f	f	PROPN
ejpam-7053	296	35	′	′	NOUN
ejpam-7053	296	36	,	,	PUNCT
ejpam-7053	296	37	there	there	PRON
ejpam-7053	296	38	exist	exist	VERB
ejpam-7053	296	39	disjoint	disjoint	ADJ
ejpam-7053	296	40	τ1τ2	τ1τ2	ADJ
ejpam-7053	296	41	-	-	ADJ
ejpam-7053	296	42	open	open	ADJ
ejpam-7053	296	43	sets	set	NOUN
ejpam-7053	296	44	u	u	NOUN
ejpam-7053	296	45	and	and	CCONJ
ejpam-7053	296	46	v	v	ADP
ejpam-7053	296	47	such	such	ADJ
ejpam-7053	296	48	that	that	SCONJ
ejpam-7053	296	49	f	f	PROPN
ejpam-7053	296	50	⊆	⊆	NUM
ejpam-7053	296	51	u	u	NOUN
ejpam-7053	296	52	and	and	CCONJ
ejpam-7053	296	53	f	f	PROPN
ejpam-7053	296	54	′	′	NUM
ejpam-7053	296	55	⊆	⊆	NUM
ejpam-7053	296	56	v	v	NOUN
ejpam-7053	296	57	.	.	PUNCT
ejpam-7053	297	1	theorem	theorem	NOUN
ejpam-7053	297	2	12	12	NUM
ejpam-7053	297	3	.	.	PUNCT
ejpam-7053	298	1	if	if	SCONJ
ejpam-7053	298	2	f	f	PROPN
ejpam-7053	298	3	:	:	PUNCT
ejpam-7053	298	4	(	(	PUNCT
ejpam-7053	298	5	x,µ	x,µ	NOUN
ejpam-7053	298	6	)	)	PUNCT
ejpam-7053	298	7	→	→	SYM
ejpam-7053	298	8	(	(	PUNCT
ejpam-7053	298	9	y	y	PROPN
ejpam-7053	298	10	,	,	PUNCT
ejpam-7053	298	11	σ1	σ1	PROPN
ejpam-7053	298	12	,	,	PUNCT
ejpam-7053	298	13	σ2	σ2	PROPN
ejpam-7053	298	14	)	)	PUNCT
ejpam-7053	298	15	is	be	AUX
ejpam-7053	298	16	a	a	DET
ejpam-7053	298	17	multifunction	multifunction	NOUN
ejpam-7053	298	18	such	such	ADJ
ejpam-7053	298	19	that	that	SCONJ
ejpam-7053	298	20	f	f	PROPN
ejpam-7053	298	21	(	(	PUNCT
ejpam-7053	298	22	x	x	X
ejpam-7053	298	23	)	)	PUNCT
ejpam-7053	298	24	is	be	AUX
ejpam-7053	298	25	σ1σ2	σ1σ2	NOUN
ejpam-7053	298	26	-	-	ADJ
ejpam-7053	298	27	closed	closed	ADJ
ejpam-7053	298	28	in	in	ADP
ejpam-7053	298	29	y	y	PROPN
ejpam-7053	298	30	for	for	SCONJ
ejpam-7053	298	31	each	each	DET
ejpam-7053	298	32	x	x	SYM
ejpam-7053	298	33	∈	∈	PROPN
ejpam-7053	298	34	x	x	X
ejpam-7053	298	35	and	and	CCONJ
ejpam-7053	298	36	(	(	PUNCT
ejpam-7053	298	37	y	y	PROPN
ejpam-7053	298	38	,	,	PUNCT
ejpam-7053	298	39	σ1	σ1	PROPN
ejpam-7053	298	40	,	,	PUNCT
ejpam-7053	298	41	σ2	σ2	PROPN
ejpam-7053	298	42	)	)	PUNCT
ejpam-7053	298	43	is	be	AUX
ejpam-7053	298	44	a	a	DET
ejpam-7053	298	45	(	(	PUNCT
ejpam-7053	298	46	σ1	σ1	NOUN
ejpam-7053	298	47	,	,	PUNCT
ejpam-7053	298	48	σ2)-normal	σ2)-normal	ADJ
ejpam-7053	298	49	space	space	NOUN
ejpam-7053	298	50	,	,	PUNCT
ejpam-7053	298	51	then	then	ADV
ejpam-7053	298	52	the	the	DET
ejpam-7053	298	53	following	follow	VERB
ejpam-7053	298	54	properties	property	NOUN
ejpam-7053	298	55	are	be	AUX
ejpam-7053	298	56	equivalent	equivalent	ADJ
ejpam-7053	298	57	:	:	PUNCT
ejpam-7053	298	58	(	(	PUNCT
ejpam-7053	298	59	1	1	X
ejpam-7053	298	60	)	)	PUNCT
ejpam-7053	298	61	f	f	PROPN
ejpam-7053	298	62	is	be	AUX
ejpam-7053	298	63	upper	upper	ADJ
ejpam-7053	298	64	µ(σ1	µ(σ1	NOUN
ejpam-7053	298	65	,	,	PUNCT
ejpam-7053	298	66	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	298	67	;	;	PUNCT
ejpam-7053	298	68	(	(	PUNCT
ejpam-7053	298	69	2	2	X
ejpam-7053	298	70	)	)	PUNCT
ejpam-7053	298	71	f	f	PROPN
ejpam-7053	298	72	is	be	AUX
ejpam-7053	298	73	upper	upper	ADJ
ejpam-7053	298	74	almost	almost	ADV
ejpam-7053	298	75	µ(σ1	µ(σ1	NOUN
ejpam-7053	298	76	,	,	PUNCT
ejpam-7053	298	77	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	298	78	;	;	PUNCT
ejpam-7053	298	79	(	(	PUNCT
ejpam-7053	298	80	3	3	X
ejpam-7053	298	81	)	)	PUNCT
ejpam-7053	298	82	f	f	PROPN
ejpam-7053	298	83	is	be	AUX
ejpam-7053	298	84	upper	upper	ADJ
ejpam-7053	298	85	weakly	weakly	ADJ
ejpam-7053	298	86	µ(σ1	µ(σ1	NOUN
ejpam-7053	298	87	,	,	PUNCT
ejpam-7053	299	1	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	299	2	.	.	NOUN
ejpam-7053	299	3	proof	proof	NOUN
ejpam-7053	299	4	.	.	PUNCT
ejpam-7053	300	1	as	as	ADP
ejpam-7053	300	2	in	in	ADP
ejpam-7053	300	3	theorem	theorem	NOUN
ejpam-7053	300	4	10	10	NUM
ejpam-7053	300	5	,	,	PUNCT
ejpam-7053	300	6	we	we	PRON
ejpam-7053	300	7	prove	prove	VERB
ejpam-7053	300	8	only	only	ADV
ejpam-7053	300	9	the	the	DET
ejpam-7053	300	10	implication	implication	NOUN
ejpam-7053	300	11	(	(	PUNCT
ejpam-7053	300	12	3	3	X
ejpam-7053	300	13	)	)	PUNCT
ejpam-7053	300	14	⇒	⇒	NOUN
ejpam-7053	300	15	(	(	PUNCT
ejpam-7053	300	16	1	1	NUM
ejpam-7053	300	17	)	)	PUNCT
ejpam-7053	300	18	.	.	PUNCT
ejpam-7053	301	1	suppose	suppose	VERB
ejpam-7053	301	2	that	that	SCONJ
ejpam-7053	301	3	f	f	PROPN
ejpam-7053	301	4	is	be	AUX
ejpam-7053	301	5	upper	upper	ADJ
ejpam-7053	301	6	weakly	weakly	ADJ
ejpam-7053	301	7	µ(σ1	µ(σ1	NOUN
ejpam-7053	301	8	,	,	PUNCT
ejpam-7053	301	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	301	10	.	.	PUNCT
ejpam-7053	301	11	let	let	VERB
ejpam-7053	301	12	x	x	SYM
ejpam-7053	301	13	∈	∈	PROPN
ejpam-7053	301	14	x	x	X
ejpam-7053	301	15	and	and	CCONJ
ejpam-7053	301	16	g	g	PROPN
ejpam-7053	301	17	be	be	VERB
ejpam-7053	301	18	any	any	DET
ejpam-7053	301	19	σ1σ2	σ1σ2	NOUN
ejpam-7053	301	20	-	-	ADJ
ejpam-7053	301	21	open	open	ADJ
ejpam-7053	301	22	set	set	NOUN
ejpam-7053	301	23	of	of	ADP
ejpam-7053	301	24	y	y	PROPN
ejpam-7053	301	25	containing	contain	VERB
ejpam-7053	301	26	f	f	PROPN
ejpam-7053	301	27	(	(	PUNCT
ejpam-7053	301	28	x	x	NOUN
ejpam-7053	301	29	)	)	PUNCT
ejpam-7053	301	30	.	.	PUNCT
ejpam-7053	302	1	since	since	SCONJ
ejpam-7053	302	2	f	f	PROPN
ejpam-7053	302	3	(	(	PUNCT
ejpam-7053	302	4	x	x	X
ejpam-7053	302	5	)	)	PUNCT
ejpam-7053	302	6	is	be	AUX
ejpam-7053	302	7	σ1σ2	σ1σ2	NOUN
ejpam-7053	302	8	-	-	ADJ
ejpam-7053	302	9	closed	closed	ADJ
ejpam-7053	302	10	in	in	ADP
ejpam-7053	302	11	y	y	PROPN
ejpam-7053	302	12	,	,	PUNCT
ejpam-7053	302	13	by	by	ADP
ejpam-7053	302	14	the	the	DET
ejpam-7053	302	15	(	(	PUNCT
ejpam-7053	302	16	σ1	σ1	PROPN
ejpam-7053	302	17	,	,	PUNCT
ejpam-7053	302	18	σ2)-normality	σ2)-normality	NOUN
ejpam-7053	302	19	of	of	ADP
ejpam-7053	302	20	y	y	PRON
ejpam-7053	302	21	there	there	PRON
ejpam-7053	302	22	exists	exist	VERB
ejpam-7053	302	23	a	a	DET
ejpam-7053	302	24	σ1σ2	σ1σ2	NUM
ejpam-7053	302	25	-	-	ADJ
ejpam-7053	302	26	open	open	ADJ
ejpam-7053	302	27	set	set	NOUN
ejpam-7053	302	28	v	v	NOUN
ejpam-7053	302	29	of	of	ADP
ejpam-7053	302	30	y	y	PRON
ejpam-7053	302	31	such	such	ADJ
ejpam-7053	302	32	that	that	SCONJ
ejpam-7053	302	33	f	f	PROPN
ejpam-7053	302	34	(	(	PUNCT
ejpam-7053	302	35	x	x	X
ejpam-7053	302	36	)	)	PUNCT
ejpam-7053	302	37	⊆	⊆	NUM
ejpam-7053	302	38	v	v	ADP
ejpam-7053	302	39	⊆	⊆	NUM
ejpam-7053	302	40	σ1σ2	σ1σ2	NOUN
ejpam-7053	302	41	-	-	PUNCT
ejpam-7053	302	42	cl(v	cl(v	NOUN
ejpam-7053	302	43	)	)	PUNCT
ejpam-7053	302	44	⊆	⊆	NUM
ejpam-7053	302	45	g.	g.	NOUN
ejpam-7053	302	46	since	since	SCONJ
ejpam-7053	302	47	f	f	PROPN
ejpam-7053	302	48	is	be	AUX
ejpam-7053	302	49	upper	upper	ADJ
ejpam-7053	302	50	weakly	weakly	ADJ
ejpam-7053	302	51	µ(σ1	µ(σ1	NOUN
ejpam-7053	302	52	,	,	PUNCT
ejpam-7053	302	53	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	302	54	,	,	PUNCT
ejpam-7053	302	55	there	there	PRON
ejpam-7053	302	56	exists	exist	VERB
ejpam-7053	302	57	a	a	DET
ejpam-7053	302	58	µ-open	µ-open	NOUN
ejpam-7053	302	59	set	set	VERB
ejpam-7053	302	60	u	u	NOUN
ejpam-7053	302	61	of	of	ADP
ejpam-7053	302	62	x	x	PUNCT
ejpam-7053	302	63	containing	contain	VERB
ejpam-7053	302	64	x	x	PUNCT
ejpam-7053	302	65	such	such	ADJ
ejpam-7053	302	66	that	that	SCONJ
ejpam-7053	302	67	f	f	PROPN
ejpam-7053	302	68	(	(	PUNCT
ejpam-7053	302	69	u	u	NOUN
ejpam-7053	302	70	)	)	PUNCT
ejpam-7053	302	71	⊆	⊆	NUM
ejpam-7053	302	72	σ1σ2	σ1σ2	NOUN
ejpam-7053	302	73	-	-	NUM
ejpam-7053	302	74	cl(v	cl(v	NOUN
ejpam-7053	302	75	)	)	PUNCT
ejpam-7053	302	76	⊆	⊆	NUM
ejpam-7053	302	77	g.	g.	NOUN
ejpam-7053	302	78	this	this	PRON
ejpam-7053	302	79	shows	show	VERB
ejpam-7053	302	80	that	that	SCONJ
ejpam-7053	302	81	f	f	PROPN
ejpam-7053	302	82	is	be	AUX
ejpam-7053	302	83	upper	upper	ADJ
ejpam-7053	302	84	µ(σ1	µ(σ1	NOUN
ejpam-7053	302	85	,	,	PUNCT
ejpam-7053	302	86	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7053	302	87	.	.	PUNCT
ejpam-7053	303	1	acknowledgements	acknowledgement	NOUN
ejpam-7053	303	2	this	this	DET
ejpam-7053	303	3	research	research	NOUN
ejpam-7053	303	4	project	project	NOUN
ejpam-7053	303	5	was	be	AUX
ejpam-7053	303	6	financially	financially	ADV
ejpam-7053	303	7	supported	support	VERB
ejpam-7053	303	8	by	by	ADP
ejpam-7053	303	9	mahasarakham	mahasarakham	PROPN
ejpam-7053	303	10	university	university	PROPN
ejpam-7053	303	11	.	.	PUNCT
ejpam-7053	304	1	references	reference	NOUN
ejpam-7053	304	2	[	[	X
ejpam-7053	304	3	1	1	NUM
ejpam-7053	304	4	]	]	X
ejpam-7053	304	5	n.	n.	PROPN
ejpam-7053	304	6	levine	levine	PROPN
ejpam-7053	304	7	.	.	PUNCT
ejpam-7053	305	1	a	a	DET
ejpam-7053	305	2	decomposition	decomposition	NOUN
ejpam-7053	305	3	of	of	ADP
ejpam-7053	305	4	continuity	continuity	NOUN
ejpam-7053	305	5	in	in	ADP
ejpam-7053	305	6	topological	topological	ADJ
ejpam-7053	305	7	spaces	space	NOUN
ejpam-7053	305	8	.	.	PUNCT
ejpam-7053	306	1	the	the	DET
ejpam-7053	306	2	american	american	PROPN
ejpam-7053	306	3	mathematical	mathematical	PROPN
ejpam-7053	306	4	monthly	monthly	ADV
ejpam-7053	306	5	,	,	PUNCT
ejpam-7053	306	6	68:44–46	68:44–46	NUM
ejpam-7053	306	7	,	,	PUNCT
ejpam-7053	306	8	1961	1961	NUM
ejpam-7053	306	9	.	.	PUNCT
ejpam-7053	307	1	[	[	X
ejpam-7053	307	2	2	2	X
ejpam-7053	307	3	]	]	PUNCT
ejpam-7053	307	4	t.	t.	PROPN
ejpam-7053	307	5	husain	husain	PROPN
ejpam-7053	307	6	.	.	PUNCT
ejpam-7053	308	1	almost	almost	ADV
ejpam-7053	308	2	continuous	continuous	ADJ
ejpam-7053	308	3	mappings	mapping	NOUN
ejpam-7053	308	4	.	.	PUNCT
ejpam-7053	309	1	prace	prace	PROPN
ejpam-7053	309	2	matematyczne	matematyczne	PROPN
ejpam-7053	309	3	,	,	PUNCT
ejpam-7053	309	4	10:1–7	10:1–7	NUM
ejpam-7053	309	5	,	,	PUNCT
ejpam-7053	309	6	1966	1966	NUM
ejpam-7053	309	7	.	.	PUNCT
ejpam-7053	310	1	[	[	X
ejpam-7053	310	2	3	3	X
ejpam-7053	310	3	]	]	X
ejpam-7053	310	4	d.	d.	PROPN
ejpam-7053	310	5	s.	s.	PROPN
ejpam-7053	310	6	janković	janković	PROPN
ejpam-7053	310	7	.	.	PUNCT
ejpam-7053	311	1	θ	θ	X
ejpam-7053	311	2	-	-	ADJ
ejpam-7053	311	3	regular	regular	ADJ
ejpam-7053	311	4	spaces	space	NOUN
ejpam-7053	311	5	.	.	PUNCT
ejpam-7053	312	1	international	international	ADJ
ejpam-7053	312	2	journal	journal	PROPN
ejpam-7053	312	3	of	of	ADP
ejpam-7053	312	4	mathematics	mathematics	PROPN
ejpam-7053	312	5	and	and	CCONJ
ejpam-7053	312	6	mathematical	mathematical	ADJ
ejpam-7053	312	7	sciences	science	NOUN
ejpam-7053	312	8	,	,	PUNCT
ejpam-7053	312	9	8:615–619	8:615–619	NUM
ejpam-7053	312	10	,	,	PUNCT
ejpam-7053	312	11	1985	1985	NUM
ejpam-7053	312	12	.	.	PUNCT
ejpam-7053	313	1	[	[	X
ejpam-7053	313	2	4	4	X
ejpam-7053	313	3	]	]	PUNCT
ejpam-7053	313	4	t.	t.	PROPN
ejpam-7053	313	5	noiri	noiri	PROPN
ejpam-7053	313	6	.	.	PUNCT
ejpam-7053	314	1	properties	property	NOUN
ejpam-7053	314	2	of	of	ADP
ejpam-7053	314	3	some	some	DET
ejpam-7053	314	4	weak	weak	ADJ
ejpam-7053	314	5	forms	form	NOUN
ejpam-7053	314	6	of	of	ADP
ejpam-7053	314	7	continuity	continuity	NOUN
ejpam-7053	314	8	.	.	PUNCT
ejpam-7053	315	1	international	international	ADJ
ejpam-7053	315	2	journal	journal	PROPN
ejpam-7053	315	3	of	of	ADP
ejpam-7053	315	4	mathematics	mathematics	PROPN
ejpam-7053	315	5	and	and	CCONJ
ejpam-7053	315	6	mathematical	mathematical	ADJ
ejpam-7053	315	7	sciences	science	NOUN
ejpam-7053	315	8	,	,	PUNCT
ejpam-7053	315	9	10(1):97–111	10(1):97–111	NUM
ejpam-7053	315	10	,	,	PUNCT
ejpam-7053	315	11	1987	1987	NUM
ejpam-7053	315	12	.	.	PUNCT
ejpam-7053	316	1	[	[	X
ejpam-7053	316	2	5	5	X
ejpam-7053	316	3	]	]	PUNCT
ejpam-7053	316	4	d.	d.	PROPN
ejpam-7053	316	5	a.	a.	PROPN
ejpam-7053	316	6	rose	rise	VERB
ejpam-7053	316	7	.	.	PUNCT
ejpam-7053	317	1	weak	weak	ADJ
ejpam-7053	317	2	continuity	continuity	NOUN
ejpam-7053	317	3	and	and	CCONJ
ejpam-7053	317	4	almost	almost	ADV
ejpam-7053	317	5	continuity	continuity	NOUN
ejpam-7053	317	6	.	.	PUNCT
ejpam-7053	318	1	international	international	ADJ
ejpam-7053	318	2	journal	journal	PROPN
ejpam-7053	318	3	of	of	ADP
ejpam-7053	318	4	mathematics	mathematics	PROPN
ejpam-7053	318	5	and	and	CCONJ
ejpam-7053	318	6	mathematical	mathematical	ADJ
ejpam-7053	318	7	sciences	science	NOUN
ejpam-7053	318	8	,	,	PUNCT
ejpam-7053	318	9	7:311–318	7:311–318	PROPN
ejpam-7053	318	10	,	,	PUNCT
ejpam-7053	318	11	1984	1984	NUM
ejpam-7053	318	12	.	.	PUNCT
ejpam-7053	319	1	m.	m.	NOUN
ejpam-7053	319	2	chiangpradit	chiangpradit	PROPN
ejpam-7053	319	3	,	,	PUNCT
ejpam-7053	319	4	a.	a.	PROPN
ejpam-7053	319	5	sama	sama	PROPN
ejpam-7053	319	6	-	-	PUNCT
ejpam-7053	319	7	ae	ae	PROPN
ejpam-7053	319	8	,	,	PUNCT
ejpam-7053	319	9	c.	c.	PROPN
ejpam-7053	319	10	boonpok	boonpok	PROPN
ejpam-7053	319	11	/	/	SYM
ejpam-7053	319	12	eur	eur	PROPN
ejpam-7053	319	13	.	.	PUNCT
ejpam-7053	320	1	j.	j.	PROPN
ejpam-7053	320	2	pure	pure	PROPN
ejpam-7053	320	3	appl	appl	PROPN
ejpam-7053	320	4	.	.	PROPN
ejpam-7053	320	5	math	math	PROPN
ejpam-7053	320	6	,	,	PUNCT
ejpam-7053	320	7	18	18	NUM
ejpam-7053	320	8	(	(	PUNCT
ejpam-7053	320	9	4	4	NUM
ejpam-7053	320	10	)	)	PUNCT
ejpam-7053	320	11	(	(	PUNCT
ejpam-7053	320	12	2025	2025	NUM
ejpam-7053	320	13	)	)	PUNCT
ejpam-7053	320	14	,	,	PUNCT
ejpam-7053	320	15	7053	7053	NUM
ejpam-7053	320	16	12	12	NUM
ejpam-7053	320	17	of	of	ADP
ejpam-7053	320	18	13	13	NUM
ejpam-7053	321	1	[	[	SYM
ejpam-7053	321	2	6	6	NUM
ejpam-7053	321	3	]	]	PUNCT
ejpam-7053	321	4	v.	v.	CCONJ
ejpam-7053	321	5	popa	popa	NOUN
ejpam-7053	321	6	and	and	CCONJ
ejpam-7053	321	7	t.	t.	PROPN
ejpam-7053	321	8	noiri	noiri	PROPN
ejpam-7053	321	9	.	.	PUNCT
ejpam-7053	322	1	on	on	ADP
ejpam-7053	322	2	weakly	weakly	ADJ
ejpam-7053	322	3	(	(	PUNCT
ejpam-7053	322	4	τ	τ	PROPN
ejpam-7053	322	5	,	,	PUNCT
ejpam-7053	322	6	m)-continuous	m)-continuous	ADJ
ejpam-7053	322	7	functions	function	NOUN
ejpam-7053	322	8	.	.	PUNCT
ejpam-7053	323	1	rendiconti	rendiconti	ADJ
ejpam-7053	323	2	del	del	PROPN
ejpam-7053	323	3	circolo	circolo	PROPN
ejpam-7053	323	4	matematico	matematico	NOUN
ejpam-7053	323	5	di	di	PROPN
ejpam-7053	323	6	palermo	palermo	PROPN
ejpam-7053	323	7	series	series	PROPN
ejpam-7053	323	8	2	2	NUM
ejpam-7053	323	9	,	,	PUNCT
ejpam-7053	323	10	51:295–316	51:295–316	NUM
ejpam-7053	323	11	,	,	PUNCT
ejpam-7053	323	12	2002	2002	NUM
ejpam-7053	323	13	.	.	PUNCT
ejpam-7053	324	1	[	[	X
ejpam-7053	324	2	7	7	NUM
ejpam-7053	324	3	]	]	SYM
ejpam-7053	324	4	á	á	PROPN
ejpam-7053	324	5	.	.	PUNCT
ejpam-7053	324	6	császár	császár	PROPN
ejpam-7053	324	7	.	.	PUNCT
ejpam-7053	325	1	generalized	generalize	VERB
ejpam-7053	325	2	topology	topology	NOUN
ejpam-7053	325	3	,	,	PUNCT
ejpam-7053	325	4	generalized	generalize	VERB
ejpam-7053	325	5	continuity	continuity	NOUN
ejpam-7053	325	6	.	.	PUNCT
ejpam-7053	326	1	acta	acta	PROPN
ejpam-7053	326	2	mathematica	mathematica	PROPN
ejpam-7053	326	3	hungarica	hungarica	PROPN
ejpam-7053	326	4	,	,	PUNCT
ejpam-7053	326	5	96(4):351–357	96(4):351–357	NOUN
ejpam-7053	326	6	,	,	PUNCT
ejpam-7053	326	7	2002	2002	NUM
ejpam-7053	326	8	.	.	PUNCT
ejpam-7053	327	1	[	[	X
ejpam-7053	327	2	8	8	NUM
ejpam-7053	327	3	]	]	PUNCT
ejpam-7053	327	4	a.	a.	NOUN
ejpam-7053	327	5	kanibir	kanibir	NOUN
ejpam-7053	327	6	and	and	CCONJ
ejpam-7053	327	7	i.	i.	PROPN
ejpam-7053	327	8	l.	l.	PROPN
ejpam-7053	327	9	reilly	reilly	PROPN
ejpam-7053	327	10	.	.	PUNCT
ejpam-7053	328	1	generalized	generalize	VERB
ejpam-7053	328	2	continuity	continuity	NOUN
ejpam-7053	328	3	for	for	ADP
ejpam-7053	328	4	multifunctions	multifunction	NOUN
ejpam-7053	328	5	.	.	PUNCT
ejpam-7053	329	1	acta	acta	PROPN
ejpam-7053	329	2	mathematica	mathematica	PROPN
ejpam-7053	329	3	hungarica	hungarica	PROPN
ejpam-7053	329	4	,	,	PUNCT
ejpam-7053	329	5	122(3):283–292	122(3):283–292	NUM
ejpam-7053	329	6	,	,	PUNCT
ejpam-7053	329	7	2009	2009	NUM
ejpam-7053	329	8	.	.	PUNCT
ejpam-7053	330	1	[	[	X
ejpam-7053	330	2	9	9	NUM
ejpam-7053	330	3	]	]	X
ejpam-7053	330	4	n.	n.	NOUN
ejpam-7053	330	5	srisarakham	srisarakham	PROPN
ejpam-7053	330	6	and	and	CCONJ
ejpam-7053	330	7	c.	c.	PROPN
ejpam-7053	330	8	boonpok	boonpok	PROPN
ejpam-7053	330	9	.	.	PUNCT
ejpam-7053	331	1	characterizations	characterization	NOUN
ejpam-7053	331	2	of	of	ADP
ejpam-7053	331	3	upper	upper	ADJ
ejpam-7053	331	4	and	and	CCONJ
ejpam-7053	331	5	lower	low	ADJ
ejpam-7053	331	6	α(µx	α(µx	NUM
ejpam-7053	331	7	,	,	PUNCT
ejpam-7053	331	8	µy)continuous	µy)continuous	ADJ
ejpam-7053	331	9	multifunctions	multifunction	NOUN
ejpam-7053	331	10	.	.	PUNCT
ejpam-7053	332	1	journal	journal	PROPN
ejpam-7053	332	2	of	of	ADP
ejpam-7053	332	3	mathematics	mathematics	PROPN
ejpam-7053	332	4	and	and	CCONJ
ejpam-7053	332	5	computer	computer	NOUN
ejpam-7053	332	6	science	science	NOUN
ejpam-7053	332	7	,	,	PUNCT
ejpam-7053	332	8	17:255	17:255	NUM
ejpam-7053	332	9	–	–	PUNCT
ejpam-7053	332	10	265	265	NUM
ejpam-7053	332	11	,	,	PUNCT
ejpam-7053	332	12	2017	2017	NUM
ejpam-7053	332	13	.	.	PUNCT
ejpam-7053	333	1	[	[	X
ejpam-7053	333	2	10	10	NUM
ejpam-7053	333	3	]	]	X
ejpam-7053	333	4	c.	c.	PROPN
ejpam-7053	333	5	boonpok	boonpok	PROPN
ejpam-7053	333	6	.	.	PUNCT
ejpam-7053	334	1	on	on	ADP
ejpam-7053	334	2	upper	upper	ADJ
ejpam-7053	334	3	and	and	CCONJ
ejpam-7053	334	4	lower	low	ADJ
ejpam-7053	334	5	β(µx	β(µx	ADJ
ejpam-7053	334	6	,	,	PUNCT
ejpam-7053	334	7	µy)-continuous	µy)-continuous	ADJ
ejpam-7053	334	8	multifunctions	multifunction	NOUN
ejpam-7053	334	9	.	.	PUNCT
ejpam-7053	335	1	international	international	ADJ
ejpam-7053	335	2	journal	journal	PROPN
ejpam-7053	335	3	of	of	ADP
ejpam-7053	335	4	mathematics	mathematics	PROPN
ejpam-7053	335	5	and	and	CCONJ
ejpam-7053	335	6	mathematical	mathematical	ADJ
ejpam-7053	335	7	sciences	science	NOUN
ejpam-7053	335	8	,	,	PUNCT
ejpam-7053	335	9	2012:931656	2012:931656	NUM
ejpam-7053	335	10	,	,	PUNCT
ejpam-7053	335	11	2012	2012	NUM
ejpam-7053	335	12	.	.	PUNCT
ejpam-7053	336	1	[	[	X
ejpam-7053	336	2	11	11	NUM
ejpam-7053	336	3	]	]	X
ejpam-7053	336	4	p.	p.	NOUN
ejpam-7053	336	5	pue	pue	NOUN
ejpam-7053	336	6	-	-	PUNCT
ejpam-7053	336	7	on	on	ADP
ejpam-7053	336	8	,	,	PUNCT
ejpam-7053	336	9	s.	s.	PROPN
ejpam-7053	336	10	sompong	sompong	PROPN
ejpam-7053	336	11	,	,	PUNCT
ejpam-7053	336	12	and	and	CCONJ
ejpam-7053	336	13	c.	c.	PROPN
ejpam-7053	336	14	boonpok	boonpok	PROPN
ejpam-7053	336	15	.	.	PUNCT
ejpam-7053	337	1	upper	upper	ADJ
ejpam-7053	337	2	and	and	CCONJ
ejpam-7053	337	3	lower	low	ADJ
ejpam-7053	337	4	(	(	PUNCT
ejpam-7053	337	5	τ1	τ1	NOUN
ejpam-7053	337	6	,	,	PUNCT
ejpam-7053	337	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7053	337	8	multifunctions	multifunction	NOUN
ejpam-7053	337	9	.	.	PUNCT
ejpam-7053	338	1	international	international	ADJ
ejpam-7053	338	2	journal	journal	PROPN
ejpam-7053	338	3	of	of	ADP
ejpam-7053	338	4	mathematics	mathematic	NOUN
ejpam-7053	338	5	and	and	CCONJ
ejpam-7053	338	6	computer	computer	NOUN
ejpam-7053	338	7	science	science	NOUN
ejpam-7053	338	8	,	,	PUNCT
ejpam-7053	338	9	19(4):1305	19(4):1305	NUM
ejpam-7053	338	10	–	–	PUNCT
ejpam-7053	338	11	1310	1310	NUM
ejpam-7053	338	12	,	,	PUNCT
ejpam-7053	338	13	2024	2024	NUM
ejpam-7053	338	14	.	.	PUNCT
ejpam-7053	339	1	[	[	X
ejpam-7053	339	2	12	12	NUM
ejpam-7053	339	3	]	]	X
ejpam-7053	339	4	c.	c.	PROPN
ejpam-7053	339	5	klanarong	klanarong	PROPN
ejpam-7053	339	6	,	,	PUNCT
ejpam-7053	339	7	s.	s.	PROPN
ejpam-7053	339	8	sompong	sompong	PROPN
ejpam-7053	339	9	,	,	PUNCT
ejpam-7053	339	10	and	and	CCONJ
ejpam-7053	339	11	c.	c.	PROPN
ejpam-7053	339	12	boonpok	boonpok	PROPN
ejpam-7053	339	13	.	.	PUNCT
ejpam-7053	340	1	upper	upper	ADJ
ejpam-7053	340	2	and	and	CCONJ
ejpam-7053	340	3	lower	low	ADJ
ejpam-7053	340	4	almost	almost	ADV
ejpam-7053	340	5	(	(	PUNCT
ejpam-7053	340	6	τ1	τ1	NOUN
ejpam-7053	340	7	,	,	PUNCT
ejpam-7053	340	8	τ2)continuous	τ2)continuous	ADJ
ejpam-7053	340	9	multifunctions	multifunction	NOUN
ejpam-7053	340	10	.	.	PUNCT
ejpam-7053	341	1	european	european	ADJ
ejpam-7053	341	2	journal	journal	PROPN
ejpam-7053	341	3	of	of	ADP
ejpam-7053	341	4	pure	pure	ADJ
ejpam-7053	341	5	and	and	CCONJ
ejpam-7053	341	6	applied	applied	ADJ
ejpam-7053	341	7	mathematics	mathematic	NOUN
ejpam-7053	341	8	,	,	PUNCT
ejpam-7053	341	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-7053	341	10	,	,	PUNCT
ejpam-7053	341	11	2024	2024	NUM
ejpam-7053	341	12	.	.	PUNCT
ejpam-7053	342	1	[	[	X
ejpam-7053	342	2	13	13	NUM
ejpam-7053	342	3	]	]	PUNCT
ejpam-7053	342	4	m.	m.	NOUN
ejpam-7053	342	5	thongmoon	thongmoon	NOUN
ejpam-7053	342	6	,	,	PUNCT
ejpam-7053	342	7	s.	s.	PROPN
ejpam-7053	342	8	sompong	sompong	PROPN
ejpam-7053	342	9	,	,	PUNCT
ejpam-7053	342	10	and	and	CCONJ
ejpam-7053	342	11	c.	c.	PROPN
ejpam-7053	342	12	boonpok	boonpok	PROPN
ejpam-7053	342	13	.	.	PUNCT
ejpam-7053	343	1	upper	upper	ADJ
ejpam-7053	343	2	and	and	CCONJ
ejpam-7053	343	3	lower	low	ADJ
ejpam-7053	343	4	weak	weak	ADJ
ejpam-7053	343	5	(	(	PUNCT
ejpam-7053	343	6	τ1	τ1	NOUN
ejpam-7053	343	7	,	,	PUNCT
ejpam-7053	343	8	τ2)continuity	τ2)continuity	PROPN
ejpam-7053	343	9	.	.	PUNCT
ejpam-7053	344	1	european	european	PROPN
ejpam-7053	344	2	journal	journal	PROPN
ejpam-7053	344	3	of	of	ADP
ejpam-7053	344	4	pure	pure	ADJ
ejpam-7053	344	5	and	and	CCONJ
ejpam-7053	344	6	applied	applied	ADJ
ejpam-7053	344	7	mathematics	mathematic	NOUN
ejpam-7053	344	8	,	,	PUNCT
ejpam-7053	344	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-7053	344	10	,	,	PUNCT
ejpam-7053	344	11	2024	2024	NUM
ejpam-7053	344	12	.	.	PUNCT
ejpam-7053	345	1	[	[	X
ejpam-7053	345	2	14	14	NUM
ejpam-7053	345	3	]	]	X
ejpam-7053	345	4	p.	p.	NOUN
ejpam-7053	345	5	pue	pue	NOUN
ejpam-7053	345	6	-	-	PUNCT
ejpam-7053	345	7	on	on	ADP
ejpam-7053	345	8	,	,	PUNCT
ejpam-7053	345	9	a.	a.	PROPN
ejpam-7053	345	10	sama	sama	PROPN
ejpam-7053	345	11	-	-	PUNCT
ejpam-7053	345	12	ae	ae	PROPN
ejpam-7053	345	13	,	,	PUNCT
ejpam-7053	345	14	and	and	CCONJ
ejpam-7053	345	15	c.	c.	PROPN
ejpam-7053	345	16	boonpok	boonpok	PROPN
ejpam-7053	345	17	.	.	PUNCT
ejpam-7053	346	1	on	on	ADP
ejpam-7053	346	2	weak	weak	ADJ
ejpam-7053	346	3	forms	form	NOUN
ejpam-7053	346	4	of	of	ADP
ejpam-7053	346	5	upper	upper	ADJ
ejpam-7053	346	6	and	and	CCONJ
ejpam-7053	346	7	lower	low	ADJ
ejpam-7053	346	8	continuous	continuous	ADJ
ejpam-7053	346	9	multifunctions	multifunction	NOUN
ejpam-7053	346	10	between	between	ADP
ejpam-7053	346	11	an	an	DET
ejpam-7053	346	12	ideal	ideal	ADJ
ejpam-7053	346	13	topological	topological	ADJ
ejpam-7053	346	14	space	space	NOUN
ejpam-7053	346	15	and	and	CCONJ
ejpam-7053	346	16	a	a	DET
ejpam-7053	346	17	bitopological	bitopological	ADJ
ejpam-7053	346	18	space	space	NOUN
ejpam-7053	346	19	.	.	PUNCT
ejpam-7053	347	1	european	european	ADJ
ejpam-7053	347	2	journal	journal	PROPN
ejpam-7053	347	3	of	of	ADP
ejpam-7053	347	4	pure	pure	ADJ
ejpam-7053	347	5	and	and	CCONJ
ejpam-7053	347	6	applied	applied	ADJ
ejpam-7053	347	7	mathematics	mathematic	NOUN
ejpam-7053	347	8	,	,	PUNCT
ejpam-7053	347	9	18(3):6567	18(3):6567	NUM
ejpam-7053	347	10	,	,	PUNCT
ejpam-7053	347	11	2025	2025	NUM
ejpam-7053	347	12	.	.	PUNCT
ejpam-7053	348	1	[	[	X
ejpam-7053	348	2	15	15	NUM
ejpam-7053	348	3	]	]	X
ejpam-7053	348	4	c.	c.	PROPN
ejpam-7053	348	5	boonpok	boonpok	PROPN
ejpam-7053	348	6	,	,	PUNCT
ejpam-7053	348	7	c.	c.	PROPN
ejpam-7053	348	8	viriyapong	viriyapong	PROPN
ejpam-7053	348	9	,	,	PUNCT
ejpam-7053	348	10	and	and	CCONJ
ejpam-7053	348	11	m.	m.	NOUN
ejpam-7053	348	12	thongmoon	thongmoon	NOUN
ejpam-7053	348	13	.	.	PUNCT
ejpam-7053	349	1	on	on	ADP
ejpam-7053	349	2	upper	upper	ADJ
ejpam-7053	349	3	and	and	CCONJ
ejpam-7053	349	4	lower	low	ADJ
ejpam-7053	349	5	(	(	PUNCT
ejpam-7053	349	6	τ1	τ1	NOUN
ejpam-7053	349	7	,	,	PUNCT
ejpam-7053	349	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7053	349	9	multifunctions	multifunction	NOUN
ejpam-7053	349	10	.	.	PUNCT
ejpam-7053	350	1	journal	journal	PROPN
ejpam-7053	350	2	of	of	ADP
ejpam-7053	350	3	mathematics	mathematics	PROPN
ejpam-7053	350	4	and	and	CCONJ
ejpam-7053	350	5	computer	computer	NOUN
ejpam-7053	350	6	science	science	NOUN
ejpam-7053	350	7	,	,	PUNCT
ejpam-7053	350	8	18:282	18:282	NUM
ejpam-7053	350	9	–	–	PUNCT
ejpam-7053	350	10	293	293	NUM
ejpam-7053	350	11	,	,	PUNCT
ejpam-7053	350	12	2018	2018	NUM
ejpam-7053	350	13	.	.	PUNCT
ejpam-7053	351	1	[	[	X
ejpam-7053	351	2	16	16	NUM
ejpam-7053	351	3	]	]	X
ejpam-7053	351	4	c.	c.	PROPN
ejpam-7053	351	5	viriyapong	viriyapong	PROPN
ejpam-7053	351	6	and	and	CCONJ
ejpam-7053	351	7	c.	c.	PROPN
ejpam-7053	351	8	boonpok	boonpok	PROPN
ejpam-7053	351	9	.	.	PUNCT
ejpam-7053	352	1	(	(	PUNCT
ejpam-7053	352	2	τ1	τ1	NOUN
ejpam-7053	352	3	,	,	PUNCT
ejpam-7053	352	4	τ2)α	τ2)α	NOUN
ejpam-7053	352	5	-	-	PUNCT
ejpam-7053	352	6	continuity	continuity	NOUN
ejpam-7053	352	7	for	for	ADP
ejpam-7053	352	8	multifunctions	multifunction	NOUN
ejpam-7053	352	9	.	.	PUNCT
ejpam-7053	353	1	journal	journal	PROPN
ejpam-7053	353	2	of	of	ADP
ejpam-7053	353	3	mathematics	mathematic	NOUN
ejpam-7053	353	4	,	,	PUNCT
ejpam-7053	353	5	2020:6285763	2020:6285763	NUM
ejpam-7053	353	6	,	,	PUNCT
ejpam-7053	353	7	2020	2020	NUM
ejpam-7053	353	8	.	.	PUNCT
ejpam-7053	354	1	[	[	X
ejpam-7053	354	2	17	17	NUM
ejpam-7053	354	3	]	]	PUNCT
ejpam-7053	354	4	c.	c.	PROPN
ejpam-7053	354	5	boonpok	boonpok	PROPN
ejpam-7053	354	6	.	.	PUNCT
ejpam-7053	355	1	(	(	PUNCT
ejpam-7053	355	2	τ1	τ1	NOUN
ejpam-7053	355	3	,	,	PUNCT
ejpam-7053	355	4	τ2)δ	τ2)δ	ADJ
ejpam-7053	355	5	-	-	PUNCT
ejpam-7053	355	6	semicontinuous	semicontinuous	ADJ
ejpam-7053	355	7	multifunctions	multifunction	NOUN
ejpam-7053	355	8	.	.	PUNCT
ejpam-7053	356	1	heliyon	heliyon	NOUN
ejpam-7053	356	2	,	,	PUNCT
ejpam-7053	356	3	6	6	NUM
ejpam-7053	356	4	:	:	SYM
ejpam-7053	356	5	e05367	e05367	PROPN
ejpam-7053	356	6	,	,	PUNCT
ejpam-7053	356	7	2020	2020	NUM
ejpam-7053	356	8	.	.	PUNCT
ejpam-7053	357	1	[	[	X
ejpam-7053	357	2	18	18	NUM
ejpam-7053	357	3	]	]	SYM
ejpam-7053	357	4	á	á	PROPN
ejpam-7053	357	5	.	.	PUNCT
ejpam-7053	357	6	császár	császár	PROPN
ejpam-7053	357	7	.	.	PUNCT
ejpam-7053	358	1	δ	δ	PROPN
ejpam-7053	358	2	-	-	PUNCT
ejpam-7053	358	3	and	and	CCONJ
ejpam-7053	358	4	θ	θ	NOUN
ejpam-7053	358	5	-	-	PUNCT
ejpam-7053	358	6	modifications	modification	NOUN
ejpam-7053	358	7	of	of	ADP
ejpam-7053	358	8	generalized	generalized	ADJ
ejpam-7053	358	9	topologies	topology	NOUN
ejpam-7053	358	10	.	.	PUNCT
ejpam-7053	359	1	acta	acta	PROPN
ejpam-7053	359	2	mathematica	mathematica	PROPN
ejpam-7053	359	3	hungarica	hungarica	PROPN
ejpam-7053	359	4	,	,	PUNCT
ejpam-7053	359	5	120:274–279	120:274–279	NUM
ejpam-7053	359	6	,	,	PUNCT
ejpam-7053	359	7	2008	2008	NUM
ejpam-7053	359	8	.	.	PUNCT
ejpam-7053	360	1	[	[	X
ejpam-7053	360	2	19	19	NUM
ejpam-7053	360	3	]	]	X
ejpam-7053	360	4	p.	p.	NOUN
ejpam-7053	360	5	pue	pue	NOUN
ejpam-7053	360	6	-	-	PUNCT
ejpam-7053	360	7	on	on	ADP
ejpam-7053	360	8	,	,	PUNCT
ejpam-7053	360	9	a.	a.	PROPN
ejpam-7053	360	10	sama	sama	PROPN
ejpam-7053	360	11	-	-	PUNCT
ejpam-7053	360	12	ae	ae	PROPN
ejpam-7053	360	13	,	,	PUNCT
ejpam-7053	360	14	and	and	CCONJ
ejpam-7053	360	15	c.	c.	PROPN
ejpam-7053	360	16	boonpok	boonpok	PROPN
ejpam-7053	360	17	.	.	PUNCT
ejpam-7053	361	1	almost	almost	ADV
ejpam-7053	361	2	µ(σ1	µ(σ1	NOUN
ejpam-7053	361	3	,	,	PUNCT
ejpam-7053	361	4	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	361	5	multifunctions	multifunction	NOUN
ejpam-7053	361	6	,	,	PUNCT
ejpam-7053	361	7	submitted	submit	VERB
ejpam-7053	361	8	.	.	PUNCT
ejpam-7053	362	1	[	[	X
ejpam-7053	362	2	20	20	NUM
ejpam-7053	362	3	]	]	X
ejpam-7053	362	4	n.	n.	NOUN
ejpam-7053	362	5	chutiman	chutiman	NOUN
ejpam-7053	362	6	,	,	PUNCT
ejpam-7053	362	7	a.	a.	PROPN
ejpam-7053	362	8	sama	sama	PROPN
ejpam-7053	362	9	-	-	PUNCT
ejpam-7053	362	10	ae	ae	PROPN
ejpam-7053	362	11	,	,	PUNCT
ejpam-7053	362	12	and	and	CCONJ
ejpam-7053	362	13	c.	c.	PROPN
ejpam-7053	362	14	boonpok	boonpok	PROPN
ejpam-7053	362	15	.	.	PUNCT
ejpam-7053	363	1	on	on	ADP
ejpam-7053	363	2	upper	upper	ADJ
ejpam-7053	363	3	and	and	CCONJ
ejpam-7053	363	4	lower	low	ADJ
ejpam-7053	363	5	µ(σ1	µ(σ1	NOUN
ejpam-7053	363	6	,	,	PUNCT
ejpam-7053	363	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7053	363	8	multifunctions	multifunction	NOUN
ejpam-7053	363	9	,	,	PUNCT
ejpam-7053	363	10	submitted	submit	VERB
ejpam-7053	363	11	.	.	PUNCT
ejpam-7053	364	1	[	[	X
ejpam-7053	364	2	21	21	NUM
ejpam-7053	364	3	]	]	PUNCT
ejpam-7053	364	4	m.	m.	NOUN
ejpam-7053	364	5	chiangpradit	chiangpradit	NOUN
ejpam-7053	364	6	,	,	PUNCT
ejpam-7053	364	7	s.	s.	PROPN
ejpam-7053	364	8	sompong	sompong	PROPN
ejpam-7053	364	9	,	,	PUNCT
ejpam-7053	364	10	and	and	CCONJ
ejpam-7053	364	11	c.	c.	PROPN
ejpam-7053	364	12	boonpok	boonpok	PROPN
ejpam-7053	364	13	.	.	PUNCT
ejpam-7053	365	1	on	on	ADP
ejpam-7053	365	2	characterizations	characterization	NOUN
ejpam-7053	365	3	of	of	ADP
ejpam-7053	365	4	(	(	PUNCT
ejpam-7053	365	5	τ1	τ1	NOUN
ejpam-7053	365	6	,	,	PUNCT
ejpam-7053	365	7	τ2)regular	τ2)regular	ADJ
ejpam-7053	365	8	spaces	space	NOUN
ejpam-7053	365	9	.	.	PUNCT
ejpam-7053	366	1	international	international	ADJ
ejpam-7053	366	2	journal	journal	PROPN
ejpam-7053	366	3	of	of	ADP
ejpam-7053	366	4	mathematics	mathematic	NOUN
ejpam-7053	366	5	and	and	CCONJ
ejpam-7053	366	6	computer	computer	NOUN
ejpam-7053	366	7	science	science	NOUN
ejpam-7053	366	8	,	,	PUNCT
ejpam-7053	366	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-7053	366	10	,	,	PUNCT
ejpam-7053	366	11	2024	2024	NUM
ejpam-7053	366	12	.	.	PUNCT
ejpam-7053	367	1	[	[	X
ejpam-7053	367	2	22	22	NUM
ejpam-7053	367	3	]	]	X
ejpam-7053	367	4	b.	b.	PROPN
ejpam-7053	367	5	kong	kong	PROPN
ejpam-7053	367	6	-	-	PUNCT
ejpam-7053	367	7	ied	ied	PROPN
ejpam-7053	367	8	,	,	PUNCT
ejpam-7053	367	9	s.	s.	PROPN
ejpam-7053	367	10	sompong	sompong	PROPN
ejpam-7053	367	11	,	,	PUNCT
ejpam-7053	367	12	and	and	CCONJ
ejpam-7053	367	13	c.	c.	PROPN
ejpam-7053	367	14	boonpok	boonpok	PROPN
ejpam-7053	367	15	.	.	PUNCT
ejpam-7053	368	1	(	(	PUNCT
ejpam-7053	368	2	τ1	τ1	NOUN
ejpam-7053	368	3	,	,	PUNCT
ejpam-7053	368	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7053	368	5	and	and	CCONJ
ejpam-7053	368	6	weak	weak	ADJ
ejpam-7053	368	7	(	(	PUNCT
ejpam-7053	368	8	τ1	τ1	NOUN
ejpam-7053	368	9	,	,	PUNCT
ejpam-7053	368	10	τ2)continuity	τ2)continuity	NOUN
ejpam-7053	368	11	.	.	PUNCT
ejpam-7053	369	1	international	international	ADJ
ejpam-7053	369	2	journal	journal	PROPN
ejpam-7053	369	3	of	of	ADP
ejpam-7053	369	4	mathematics	mathematic	NOUN
ejpam-7053	369	5	and	and	CCONJ
ejpam-7053	369	6	computer	computer	NOUN
ejpam-7053	369	7	science	science	NOUN
ejpam-7053	369	8	,	,	PUNCT
ejpam-7053	369	9	19(4):1321	19(4):1321	NUM
ejpam-7053	369	10	–	–	PUNCT
ejpam-7053	369	11	1327	1327	NUM
ejpam-7053	369	12	,	,	PUNCT
ejpam-7053	369	13	2024	2024	NUM
ejpam-7053	369	14	.	.	PUNCT
ejpam-7053	370	1	[	[	X
ejpam-7053	370	2	23	23	NUM
ejpam-7053	370	3	]	]	PUNCT
ejpam-7053	370	4	m.	m.	NOUN
ejpam-7053	370	5	chiangpradit	chiangpradit	NOUN
ejpam-7053	370	6	,	,	PUNCT
ejpam-7053	370	7	s.	s.	PROPN
ejpam-7053	370	8	sompong	sompong	PROPN
ejpam-7053	370	9	,	,	PUNCT
ejpam-7053	370	10	and	and	CCONJ
ejpam-7053	370	11	c.	c.	PROPN
ejpam-7053	370	12	boonpok	boonpok	PROPN
ejpam-7053	370	13	.	.	PUNCT
ejpam-7053	371	1	on	on	ADP
ejpam-7053	371	2	characterizations	characterization	NOUN
ejpam-7053	371	3	of	of	ADP
ejpam-7053	371	4	(	(	PUNCT
ejpam-7053	371	5	τ1	τ1	NOUN
ejpam-7053	371	6	,	,	PUNCT
ejpam-7053	371	7	τ2)normal	τ2)normal	ADJ
ejpam-7053	371	8	spaces	space	NOUN
ejpam-7053	371	9	.	.	PUNCT
ejpam-7053	372	1	international	international	ADJ
ejpam-7053	372	2	journal	journal	PROPN
ejpam-7053	372	3	of	of	ADP
ejpam-7053	372	4	mathematics	mathematic	NOUN
ejpam-7053	372	5	and	and	CCONJ
ejpam-7053	372	6	computer	computer	NOUN
ejpam-7053	372	7	science	science	NOUN
ejpam-7053	372	8	,	,	PUNCT
ejpam-7053	372	9	m.	m.	NOUN
ejpam-7053	372	10	chiangpradit	chiangpradit	NOUN
ejpam-7053	372	11	,	,	PUNCT
ejpam-7053	372	12	a.	a.	PROPN
ejpam-7053	372	13	sama	sama	PROPN
ejpam-7053	372	14	-	-	PUNCT
ejpam-7053	372	15	ae	ae	PROPN
ejpam-7053	372	16	,	,	PUNCT
ejpam-7053	372	17	c.	c.	PROPN
ejpam-7053	372	18	boonpok	boonpok	PROPN
ejpam-7053	372	19	/	/	SYM
ejpam-7053	372	20	eur	eur	PROPN
ejpam-7053	372	21	.	.	PUNCT
ejpam-7053	373	1	j.	j.	PROPN
ejpam-7053	373	2	pure	pure	PROPN
ejpam-7053	373	3	appl	appl	PROPN
ejpam-7053	373	4	.	.	PROPN
ejpam-7053	373	5	math	math	PROPN
ejpam-7053	373	6	,	,	PUNCT
ejpam-7053	373	7	18	18	NUM
ejpam-7053	373	8	(	(	PUNCT
ejpam-7053	373	9	4	4	NUM
ejpam-7053	373	10	)	)	PUNCT
ejpam-7053	373	11	(	(	PUNCT
ejpam-7053	373	12	2025	2025	NUM
ejpam-7053	373	13	)	)	PUNCT
ejpam-7053	373	14	,	,	PUNCT
ejpam-7053	373	15	7053	7053	NUM
ejpam-7053	373	16	13	13	NUM
ejpam-7053	373	17	of	of	ADP
ejpam-7053	373	18	13	13	NUM
ejpam-7053	373	19	19(4):1315–1320	19(4):1315–1320	NUM
ejpam-7053	373	20	,	,	PUNCT
ejpam-7053	373	21	2024	2024	NUM
ejpam-7053	373	22	.	.	PUNCT