id	sid	tid	token	lemma	pos
ejpam-7052	1	1	european	european	PROPN
ejpam-7052	1	2	journal	journal	PROPN
ejpam-7052	1	3	of	of	ADP
ejpam-7052	1	4	pure	pure	ADJ
ejpam-7052	1	5	and	and	CCONJ
ejpam-7052	1	6	applied	applied	ADJ
ejpam-7052	1	7	mathematics	mathematic	NOUN
ejpam-7052	1	8	2025	2025	NUM
ejpam-7052	1	9	,	,	PUNCT
ejpam-7052	1	10	vol	vol	NOUN
ejpam-7052	1	11	.	.	PROPN
ejpam-7052	1	12	18	18	NUM
ejpam-7052	1	13	,	,	PUNCT
ejpam-7052	1	14	issue	issue	NOUN
ejpam-7052	1	15	4	4	NUM
ejpam-7052	1	16	,	,	PUNCT
ejpam-7052	1	17	article	article	NOUN
ejpam-7052	1	18	number	number	NOUN
ejpam-7052	1	19	7052	7052	NUM
ejpam-7052	1	20	issn	issn	PROPN
ejpam-7052	1	21	1307	1307	NUM
ejpam-7052	1	22	-	-	SYM
ejpam-7052	1	23	5543	5543	NUM
ejpam-7052	1	24	–	–	PUNCT
ejpam-7052	1	25	ejpam.com	ejpam.com	X
ejpam-7052	1	26	published	publish	VERB
ejpam-7052	1	27	by	by	ADP
ejpam-7052	1	28	new	new	PROPN
ejpam-7052	1	29	york	york	PROPN
ejpam-7052	1	30	business	business	PROPN
ejpam-7052	1	31	global	global	ADJ
ejpam-7052	1	32	almost	almost	ADV
ejpam-7052	1	33	µ(σ1	µ(σ1	NOUN
ejpam-7052	1	34	,	,	PUNCT
ejpam-7052	1	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	1	36	multifunctions	multifunction	NOUN
ejpam-7052	1	37	prapart	prapart	VERB
ejpam-7052	1	38	pue	pue	PROPN
ejpam-7052	1	39	-	-	PUNCT
ejpam-7052	1	40	on1	on1	PROPN
ejpam-7052	1	41	,	,	PUNCT
ejpam-7052	1	42	areeyuth	areeyuth	NOUN
ejpam-7052	1	43	sama	sama	NOUN
ejpam-7052	1	44	-	-	PUNCT
ejpam-7052	1	45	ae2	ae2	PROPN
ejpam-7052	1	46	,	,	PUNCT
ejpam-7052	1	47	chawalit	chawalit	VERB
ejpam-7052	1	48	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7052	1	49	1	1	NUM
ejpam-7052	1	50	mathematics	mathematic	NOUN
ejpam-7052	1	51	and	and	CCONJ
ejpam-7052	1	52	applied	apply	VERB
ejpam-7052	1	53	mathematics	mathematics	PROPN
ejpam-7052	1	54	research	research	NOUN
ejpam-7052	1	55	unit	unit	NOUN
ejpam-7052	1	56	,	,	PUNCT
ejpam-7052	1	57	department	department	NOUN
ejpam-7052	1	58	of	of	ADP
ejpam-7052	1	59	mathematics	mathematic	NOUN
ejpam-7052	1	60	,	,	PUNCT
ejpam-7052	1	61	faculty	faculty	NOUN
ejpam-7052	1	62	of	of	ADP
ejpam-7052	1	63	science	science	NOUN
ejpam-7052	1	64	,	,	PUNCT
ejpam-7052	1	65	mahasarakham	mahasarakham	PROPN
ejpam-7052	1	66	university	university	PROPN
ejpam-7052	1	67	,	,	PUNCT
ejpam-7052	1	68	maha	maha	PROPN
ejpam-7052	1	69	sarakham	sarakham	PROPN
ejpam-7052	1	70	,	,	PUNCT
ejpam-7052	1	71	44150	44150	NUM
ejpam-7052	1	72	,	,	PUNCT
ejpam-7052	1	73	thailand	thailand	PROPN
ejpam-7052	1	74	2	2	NUM
ejpam-7052	1	75	department	department	NOUN
ejpam-7052	1	76	of	of	ADP
ejpam-7052	1	77	mathematics	mathematic	NOUN
ejpam-7052	1	78	and	and	CCONJ
ejpam-7052	1	79	computer	computer	NOUN
ejpam-7052	1	80	science	science	NOUN
ejpam-7052	1	81	,	,	PUNCT
ejpam-7052	1	82	faculty	faculty	NOUN
ejpam-7052	1	83	of	of	ADP
ejpam-7052	1	84	science	science	NOUN
ejpam-7052	1	85	and	and	CCONJ
ejpam-7052	1	86	technology	technology	NOUN
ejpam-7052	1	87	,	,	PUNCT
ejpam-7052	1	88	prince	prince	NOUN
ejpam-7052	1	89	of	of	ADP
ejpam-7052	1	90	songkla	songkla	PROPN
ejpam-7052	1	91	university	university	PROPN
ejpam-7052	1	92	,	,	PUNCT
ejpam-7052	1	93	pattani	pattani	NOUN
ejpam-7052	1	94	campus	campus	NOUN
ejpam-7052	1	95	,	,	PUNCT
ejpam-7052	1	96	pattani	pattani	NOUN
ejpam-7052	1	97	,	,	PUNCT
ejpam-7052	1	98	94000	94000	NUM
ejpam-7052	1	99	,	,	PUNCT
ejpam-7052	1	100	thailand	thailand	PROPN
ejpam-7052	1	101	abstract	abstract	PROPN
ejpam-7052	1	102	.	.	PUNCT
ejpam-7052	2	1	this	this	DET
ejpam-7052	2	2	paper	paper	NOUN
ejpam-7052	2	3	introduces	introduce	VERB
ejpam-7052	2	4	new	new	ADJ
ejpam-7052	2	5	concepts	concept	NOUN
ejpam-7052	2	6	of	of	ADP
ejpam-7052	2	7	continuous	continuous	ADJ
ejpam-7052	2	8	multifunctions	multifunction	NOUN
ejpam-7052	2	9	defined	define	VERB
ejpam-7052	2	10	between	between	ADP
ejpam-7052	2	11	a	a	DET
ejpam-7052	2	12	generalized	generalized	ADJ
ejpam-7052	2	13	topological	topological	ADJ
ejpam-7052	2	14	space	space	NOUN
ejpam-7052	2	15	and	and	CCONJ
ejpam-7052	2	16	a	a	DET
ejpam-7052	2	17	bitopological	bitopological	ADJ
ejpam-7052	2	18	space	space	NOUN
ejpam-7052	2	19	,	,	PUNCT
ejpam-7052	2	20	namely	namely	ADV
ejpam-7052	2	21	upper	upper	ADJ
ejpam-7052	2	22	almost	almost	ADV
ejpam-7052	2	23	µ(σ1	µ(σ1	NOUN
ejpam-7052	2	24	,	,	PUNCT
ejpam-7052	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	2	26	multifunctions	multifunction	NOUN
ejpam-7052	2	27	and	and	CCONJ
ejpam-7052	2	28	lower	low	ADJ
ejpam-7052	2	29	almost	almost	ADV
ejpam-7052	2	30	µ(σ1	µ(σ1	NOUN
ejpam-7052	2	31	,	,	PUNCT
ejpam-7052	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	2	33	multifunctions	multifunction	NOUN
ejpam-7052	2	34	.	.	PUNCT
ejpam-7052	3	1	moreover	moreover	ADV
ejpam-7052	3	2	,	,	PUNCT
ejpam-7052	3	3	several	several	ADJ
ejpam-7052	3	4	characterizations	characterization	NOUN
ejpam-7052	3	5	and	and	CCONJ
ejpam-7052	3	6	some	some	DET
ejpam-7052	3	7	properties	property	NOUN
ejpam-7052	3	8	concerning	concern	VERB
ejpam-7052	3	9	upper	upper	ADJ
ejpam-7052	3	10	almost	almost	ADV
ejpam-7052	3	11	µ(σ1	µ(σ1	NOUN
ejpam-7052	3	12	,	,	PUNCT
ejpam-7052	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	3	14	multifunctions	multifunction	NOUN
ejpam-7052	3	15	and	and	CCONJ
ejpam-7052	3	16	lower	low	ADJ
ejpam-7052	3	17	almost	almost	ADV
ejpam-7052	3	18	µ(σ1	µ(σ1	NOUN
ejpam-7052	3	19	,	,	PUNCT
ejpam-7052	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	3	21	multifunctions	multifunction	NOUN
ejpam-7052	3	22	are	be	AUX
ejpam-7052	3	23	investigated	investigate	VERB
ejpam-7052	3	24	.	.	PUNCT
ejpam-7052	4	1	furthermore	furthermore	ADV
ejpam-7052	4	2	,	,	PUNCT
ejpam-7052	4	3	the	the	DET
ejpam-7052	4	4	relationships	relationship	NOUN
ejpam-7052	4	5	between	between	ADP
ejpam-7052	4	6	µ(σ1	µ(σ1	NOUN
ejpam-7052	4	7	,	,	PUNCT
ejpam-7052	4	8	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7052	4	9	and	and	CCONJ
ejpam-7052	4	10	almost	almost	ADV
ejpam-7052	4	11	µ(σ1	µ(σ1	NOUN
ejpam-7052	4	12	,	,	PUNCT
ejpam-7052	4	13	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7052	4	14	are	be	AUX
ejpam-7052	4	15	considered	consider	VERB
ejpam-7052	4	16	.	.	PUNCT
ejpam-7052	5	1	2020	2020	NUM
ejpam-7052	5	2	mathematics	mathematic	NOUN
ejpam-7052	5	3	subject	subject	NOUN
ejpam-7052	5	4	classifications	classification	NOUN
ejpam-7052	5	5	:	:	PUNCT
ejpam-7052	5	6	54c08	54c08	NUM
ejpam-7052	5	7	,	,	PUNCT
ejpam-7052	5	8	54c60	54c60	NUM
ejpam-7052	5	9	key	key	ADJ
ejpam-7052	5	10	words	word	NOUN
ejpam-7052	5	11	and	and	CCONJ
ejpam-7052	5	12	phrases	phrase	NOUN
ejpam-7052	5	13	:	:	PUNCT
ejpam-7052	5	14	upper	upper	ADJ
ejpam-7052	5	15	almost	almost	ADV
ejpam-7052	5	16	µ(σ1	µ(σ1	NOUN
ejpam-7052	5	17	,	,	PUNCT
ejpam-7052	5	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	5	19	multifunction	multifunction	NOUN
ejpam-7052	5	20	,	,	PUNCT
ejpam-7052	5	21	lower	low	ADJ
ejpam-7052	5	22	almost	almost	ADV
ejpam-7052	5	23	µ(σ1	µ(σ1	NOUN
ejpam-7052	5	24	,	,	PUNCT
ejpam-7052	5	25	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	5	26	multifunction	multifunction	NOUN
ejpam-7052	5	27	1	1	NUM
ejpam-7052	5	28	.	.	PUNCT
ejpam-7052	5	29	introduction	introduction	NOUN
ejpam-7052	5	30	the	the	DET
ejpam-7052	5	31	concept	concept	NOUN
ejpam-7052	5	32	of	of	ADP
ejpam-7052	5	33	almost	almost	ADV
ejpam-7052	5	34	continuous	continuous	ADJ
ejpam-7052	5	35	functions	function	NOUN
ejpam-7052	5	36	was	be	AUX
ejpam-7052	5	37	introduced	introduce	VERB
ejpam-7052	5	38	by	by	ADP
ejpam-7052	5	39	singal	singal	NOUN
ejpam-7052	5	40	and	and	CCONJ
ejpam-7052	5	41	singal	singal	ADJ
ejpam-7052	6	1	[	[	X
ejpam-7052	6	2	1	1	NUM
ejpam-7052	6	3	]	]	PUNCT
ejpam-7052	6	4	.	.	PUNCT
ejpam-7052	7	1	munshi	munshi	PROPN
ejpam-7052	7	2	and	and	CCONJ
ejpam-7052	7	3	bassan	bassan	NOUN
ejpam-7052	7	4	[	[	X
ejpam-7052	7	5	2	2	NUM
ejpam-7052	7	6	]	]	PUNCT
ejpam-7052	7	7	studied	study	VERB
ejpam-7052	7	8	the	the	DET
ejpam-7052	7	9	notion	notion	NOUN
ejpam-7052	7	10	of	of	ADP
ejpam-7052	7	11	almost	almost	ADV
ejpam-7052	7	12	semi	semi	ADJ
ejpam-7052	7	13	-	-	ADJ
ejpam-7052	7	14	continuous	continuous	ADJ
ejpam-7052	7	15	functions	function	NOUN
ejpam-7052	7	16	.	.	PUNCT
ejpam-7052	8	1	noiri	noiri	ADV
ejpam-7052	9	1	[	[	X
ejpam-7052	9	2	3	3	X
ejpam-7052	9	3	]	]	PUNCT
ejpam-7052	9	4	introduced	introduce	VERB
ejpam-7052	9	5	and	and	CCONJ
ejpam-7052	9	6	investigated	investigate	VERB
ejpam-7052	9	7	the	the	DET
ejpam-7052	9	8	concept	concept	NOUN
ejpam-7052	9	9	of	of	ADP
ejpam-7052	9	10	almost	almost	ADV
ejpam-7052	9	11	α	α	NUM
ejpam-7052	9	12	-	-	ADJ
ejpam-7052	9	13	continuous	continuous	ADJ
ejpam-7052	9	14	functions	function	NOUN
ejpam-7052	9	15	.	.	PUNCT
ejpam-7052	10	1	nasef	nasef	NOUN
ejpam-7052	10	2	and	and	CCONJ
ejpam-7052	10	3	noiri	noiri	ADV
ejpam-7052	11	1	[	[	X
ejpam-7052	11	2	4	4	X
ejpam-7052	11	3	]	]	PUNCT
ejpam-7052	11	4	introduced	introduce	VERB
ejpam-7052	11	5	two	two	NUM
ejpam-7052	11	6	classes	class	NOUN
ejpam-7052	11	7	of	of	ADP
ejpam-7052	11	8	functions	function	NOUN
ejpam-7052	11	9	,	,	PUNCT
ejpam-7052	11	10	namely	namely	ADV
ejpam-7052	11	11	almost	almost	ADV
ejpam-7052	11	12	precontinuous	precontinuous	ADJ
ejpam-7052	11	13	functions	function	NOUN
ejpam-7052	11	14	and	and	CCONJ
ejpam-7052	11	15	almost	almost	ADV
ejpam-7052	11	16	β	β	ADJ
ejpam-7052	11	17	-	-	ADJ
ejpam-7052	11	18	continuous	continuous	ADJ
ejpam-7052	11	19	functions	function	NOUN
ejpam-7052	11	20	.	.	PUNCT
ejpam-7052	12	1	the	the	DET
ejpam-7052	12	2	class	class	NOUN
ejpam-7052	12	3	of	of	ADP
ejpam-7052	12	4	almost	almost	ADV
ejpam-7052	12	5	precontinuity	precontinuity	NOUN
ejpam-7052	12	6	is	be	AUX
ejpam-7052	12	7	a	a	DET
ejpam-7052	12	8	generalization	generalization	NOUN
ejpam-7052	12	9	of	of	ADP
ejpam-7052	12	10	almost	almost	ADV
ejpam-7052	12	11	α	α	NOUN
ejpam-7052	12	12	-	-	NOUN
ejpam-7052	12	13	continuity	continuity	NOUN
ejpam-7052	12	14	.	.	PUNCT
ejpam-7052	13	1	the	the	DET
ejpam-7052	13	2	class	class	NOUN
ejpam-7052	13	3	of	of	ADP
ejpam-7052	13	4	almost	almost	ADV
ejpam-7052	13	5	β	β	NOUN
ejpam-7052	13	6	-	-	NOUN
ejpam-7052	13	7	continuity	continuity	NOUN
ejpam-7052	13	8	is	be	AUX
ejpam-7052	13	9	a	a	DET
ejpam-7052	13	10	generalization	generalization	NOUN
ejpam-7052	13	11	of	of	ADP
ejpam-7052	13	12	almost	almost	ADV
ejpam-7052	13	13	semicontinuity	semicontinuity	NOUN
ejpam-7052	13	14	.	.	PUNCT
ejpam-7052	14	1	the	the	DET
ejpam-7052	14	2	concepts	concept	NOUN
ejpam-7052	14	3	of	of	ADP
ejpam-7052	14	4	generalized	generalized	ADJ
ejpam-7052	14	5	topological	topological	ADJ
ejpam-7052	14	6	spaces	space	NOUN
ejpam-7052	14	7	and	and	CCONJ
ejpam-7052	14	8	generalized	generalized	ADJ
ejpam-7052	14	9	neighborhood	neighborhood	NOUN
ejpam-7052	14	10	systems	system	NOUN
ejpam-7052	14	11	were	be	AUX
ejpam-7052	14	12	introduced	introduce	VERB
ejpam-7052	14	13	by	by	ADP
ejpam-7052	14	14	császár	császár	NOUN
ejpam-7052	14	15	[	[	X
ejpam-7052	14	16	5	5	NUM
ejpam-7052	14	17	]	]	PUNCT
ejpam-7052	14	18	.	.	PUNCT
ejpam-7052	15	1	the	the	DET
ejpam-7052	15	2	classes	class	NOUN
ejpam-7052	15	3	of	of	ADP
ejpam-7052	15	4	topological	topological	ADJ
ejpam-7052	15	5	spaces	space	NOUN
ejpam-7052	15	6	and	and	CCONJ
ejpam-7052	15	7	neighborhood	neighborhood	NOUN
ejpam-7052	15	8	systems	system	NOUN
ejpam-7052	15	9	are	be	AUX
ejpam-7052	15	10	contained	contain	VERB
ejpam-7052	15	11	in	in	ADP
ejpam-7052	15	12	the	the	DET
ejpam-7052	15	13	classes	class	NOUN
ejpam-7052	15	14	of	of	ADP
ejpam-7052	15	15	generalized	generalized	ADJ
ejpam-7052	15	16	topological	topological	ADJ
ejpam-7052	15	17	spaces	space	NOUN
ejpam-7052	15	18	and	and	CCONJ
ejpam-7052	15	19	generalized	generalized	ADJ
ejpam-7052	15	20	neighborhood	neighborhood	NOUN
ejpam-7052	15	21	systems	system	NOUN
ejpam-7052	15	22	,	,	PUNCT
ejpam-7052	15	23	respectively	respectively	ADV
ejpam-7052	15	24	.	.	PUNCT
ejpam-7052	16	1	moreover	moreover	ADV
ejpam-7052	16	2	,	,	PUNCT
ejpam-7052	16	3	császár	császár	PROPN
ejpam-7052	16	4	[	[	X
ejpam-7052	16	5	5	5	NUM
ejpam-7052	16	6	]	]	PUNCT
ejpam-7052	16	7	introduced	introduce	VERB
ejpam-7052	16	8	two	two	NUM
ejpam-7052	16	9	kinds	kind	NOUN
ejpam-7052	16	10	of	of	ADP
ejpam-7052	16	11	generalized	generalized	ADJ
ejpam-7052	16	12	continuous	continuous	ADJ
ejpam-7052	16	13	functions	function	NOUN
ejpam-7052	16	14	by	by	ADP
ejpam-7052	16	15	utilizing	utilize	VERB
ejpam-7052	16	16	the	the	DET
ejpam-7052	16	17	concepts	concept	NOUN
ejpam-7052	16	18	of	of	ADP
ejpam-7052	16	19	generalized	generalized	ADJ
ejpam-7052	16	20	topological	topological	ADJ
ejpam-7052	16	21	spaces	space	NOUN
ejpam-7052	16	22	and	and	CCONJ
ejpam-7052	16	23	generalized	generalized	ADJ
ejpam-7052	16	24	neighborhood	neighborhood	NOUN
ejpam-7052	16	25	systems	system	NOUN
ejpam-7052	16	26	.	.	PUNCT
ejpam-7052	17	1	kanibir	kanibir	NOUN
ejpam-7052	17	2	and	and	CCONJ
ejpam-7052	17	3	reilly	reilly	ADV
ejpam-7052	18	1	[	[	X
ejpam-7052	18	2	6	6	NUM
ejpam-7052	18	3	]	]	PUNCT
ejpam-7052	18	4	extended	extend	VERB
ejpam-7052	18	5	the	the	DET
ejpam-7052	18	6	concept	concept	NOUN
ejpam-7052	18	7	of	of	ADP
ejpam-7052	18	8	generalized	generalized	ADJ
ejpam-7052	18	9	continuous	continuous	ADJ
ejpam-7052	18	10	functions	function	NOUN
ejpam-7052	18	11	to	to	ADP
ejpam-7052	18	12	multifunctions	multifunction	NOUN
ejpam-7052	18	13	and	and	CCONJ
ejpam-7052	18	14	introduced	introduce	VERB
ejpam-7052	18	15	generalized	generalized	ADJ
ejpam-7052	18	16	continuous	continuous	ADJ
ejpam-7052	18	17	∗corresponding	∗corresponding	NOUN
ejpam-7052	18	18	author	author	NOUN
ejpam-7052	18	19	.	.	PUNCT
ejpam-7052	19	1	doi	doi	NOUN
ejpam-7052	19	2	:	:	PUNCT
ejpam-7052	19	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7052	https://doi.org/10.29020/nybg.ejpam.v18i4.7052	PROPN
ejpam-7052	19	4	email	email	NOUN
ejpam-7052	19	5	addresses	address	VERB
ejpam-7052	19	6	:	:	PUNCT
ejpam-7052	19	7	prapart.p@msu.ac.th	prapart.p@msu.ac.th	PROPN
ejpam-7052	19	8	(	(	PUNCT
ejpam-7052	19	9	p.	p.	NOUN
ejpam-7052	19	10	pue	pue	NOUN
ejpam-7052	19	11	-	-	PUNCT
ejpam-7052	19	12	on	on	ADP
ejpam-7052	19	13	)	)	PUNCT
ejpam-7052	19	14	,	,	PUNCT
ejpam-7052	19	15	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7052	19	16	(	(	PUNCT
ejpam-7052	19	17	a.	a.	PROPN
ejpam-7052	19	18	sama	sama	PROPN
ejpam-7052	19	19	-	-	PUNCT
ejpam-7052	19	20	ae	ae	PROPN
ejpam-7052	19	21	)	)	PUNCT
ejpam-7052	19	22	,	,	PUNCT
ejpam-7052	19	23	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7052	19	24	(	(	PUNCT
ejpam-7052	19	25	c.	c.	PROPN
ejpam-7052	19	26	boonpok	boonpok	PROPN
ejpam-7052	19	27	)	)	PUNCT
ejpam-7052	19	28	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7052	19	29	1	1	NUM
ejpam-7052	19	30	copyright	copyright	NOUN
ejpam-7052	19	31	:	:	PUNCT
ejpam-7052	19	32	©	©	PROPN
ejpam-7052	19	33	2025	2025	NUM
ejpam-7052	19	34	the	the	DET
ejpam-7052	19	35	author(s	author(s	NOUN
ejpam-7052	19	36	)	)	PUNCT
ejpam-7052	19	37	.	.	PUNCT
ejpam-7052	20	1	(	(	PUNCT
ejpam-7052	20	2	cc	cc	NOUN
ejpam-7052	20	3	by	by	ADP
ejpam-7052	20	4	-	-	PUNCT
ejpam-7052	20	5	nc	nc	PROPN
ejpam-7052	20	6	4.0	4.0	NUM
ejpam-7052	20	7	)	)	PUNCT
ejpam-7052	20	8	p.	p.	NOUN
ejpam-7052	20	9	pue	pue	NOUN
ejpam-7052	20	10	-	-	PUNCT
ejpam-7052	20	11	on	on	ADP
ejpam-7052	20	12	,	,	PUNCT
ejpam-7052	20	13	a.	a.	PROPN
ejpam-7052	20	14	sama	sama	PROPN
ejpam-7052	20	15	-	-	PUNCT
ejpam-7052	20	16	ae	ae	PROPN
ejpam-7052	20	17	,	,	PUNCT
ejpam-7052	20	18	c.	c.	PROPN
ejpam-7052	20	19	boonpok	boonpok	PROPN
ejpam-7052	20	20	/	/	SYM
ejpam-7052	20	21	eur	eur	PROPN
ejpam-7052	20	22	.	.	PUNCT
ejpam-7052	21	1	j.	j.	PROPN
ejpam-7052	21	2	pure	pure	PROPN
ejpam-7052	21	3	appl	appl	PROPN
ejpam-7052	21	4	.	.	PROPN
ejpam-7052	21	5	math	math	PROPN
ejpam-7052	21	6	,	,	PUNCT
ejpam-7052	21	7	18	18	NUM
ejpam-7052	21	8	(	(	PUNCT
ejpam-7052	21	9	4	4	NUM
ejpam-7052	21	10	)	)	PUNCT
ejpam-7052	21	11	(	(	PUNCT
ejpam-7052	21	12	2025	2025	NUM
ejpam-7052	21	13	)	)	PUNCT
ejpam-7052	21	14	,	,	PUNCT
ejpam-7052	21	15	7052	7052	NUM
ejpam-7052	21	16	2	2	NUM
ejpam-7052	21	17	of	of	ADP
ejpam-7052	21	18	14	14	NUM
ejpam-7052	21	19	multifunctions	multifunction	NOUN
ejpam-7052	21	20	between	between	ADP
ejpam-7052	21	21	generalized	generalized	ADJ
ejpam-7052	21	22	topological	topological	ADJ
ejpam-7052	21	23	spaces	space	NOUN
ejpam-7052	21	24	.	.	PUNCT
ejpam-7052	22	1	on	on	ADP
ejpam-7052	22	2	the	the	DET
ejpam-7052	22	3	other	other	ADJ
ejpam-7052	22	4	hand	hand	NOUN
ejpam-7052	22	5	,	,	PUNCT
ejpam-7052	22	6	the	the	DET
ejpam-7052	22	7	present	present	ADJ
ejpam-7052	22	8	authors	author	NOUN
ejpam-7052	22	9	introduced	introduce	VERB
ejpam-7052	22	10	and	and	CCONJ
ejpam-7052	22	11	investigated	investigate	VERB
ejpam-7052	22	12	four	four	NUM
ejpam-7052	22	13	classes	class	NOUN
ejpam-7052	22	14	of	of	ADP
ejpam-7052	22	15	multifunctions	multifunction	NOUN
ejpam-7052	22	16	defined	define	VERB
ejpam-7052	22	17	from	from	ADP
ejpam-7052	22	18	a	a	DET
ejpam-7052	22	19	generalized	generalized	ADJ
ejpam-7052	22	20	topological	topological	ADJ
ejpam-7052	22	21	space	space	NOUN
ejpam-7052	22	22	into	into	ADP
ejpam-7052	22	23	a	a	DET
ejpam-7052	22	24	generalized	generalized	ADJ
ejpam-7052	22	25	topological	topological	ADJ
ejpam-7052	22	26	space	space	NOUN
ejpam-7052	22	27	,	,	PUNCT
ejpam-7052	22	28	namely	namely	ADV
ejpam-7052	22	29	upper	upper	ADJ
ejpam-7052	22	30	almost	almost	ADV
ejpam-7052	22	31	β(µx	β(µx	PROPN
ejpam-7052	22	32	,	,	PUNCT
ejpam-7052	22	33	µy	µy	CCONJ
ejpam-7052	22	34	)	)	PUNCT
ejpam-7052	22	35	continuous	continuous	ADJ
ejpam-7052	22	36	multifunctions	multifunction	NOUN
ejpam-7052	23	1	[	[	X
ejpam-7052	23	2	7	7	NUM
ejpam-7052	23	3	]	]	PUNCT
ejpam-7052	23	4	,	,	PUNCT
ejpam-7052	23	5	lower	low	ADJ
ejpam-7052	23	6	almost	almost	ADV
ejpam-7052	23	7	β(µx	β(µx	PROPN
ejpam-7052	23	8	,	,	PUNCT
ejpam-7052	23	9	µy	µy	CCONJ
ejpam-7052	23	10	)	)	PUNCT
ejpam-7052	23	11	-continuous	-continuous	ADJ
ejpam-7052	23	12	multifunctions	multifunction	NOUN
ejpam-7052	24	1	[	[	X
ejpam-7052	24	2	7	7	NUM
ejpam-7052	24	3	]	]	PUNCT
ejpam-7052	24	4	,	,	PUNCT
ejpam-7052	24	5	upper	upper	ADJ
ejpam-7052	24	6	α(µx	α(µx	PROPN
ejpam-7052	24	7	,	,	PUNCT
ejpam-7052	24	8	µy	µy	CCONJ
ejpam-7052	24	9	)	)	PUNCT
ejpam-7052	24	10	-continuous	-continuous	ADJ
ejpam-7052	24	11	multifunctions	multifunction	NOUN
ejpam-7052	25	1	[	[	X
ejpam-7052	25	2	8	8	NUM
ejpam-7052	25	3	]	]	PUNCT
ejpam-7052	25	4	and	and	CCONJ
ejpam-7052	25	5	lower	low	ADJ
ejpam-7052	25	6	α(µx	α(µx	NUM
ejpam-7052	25	7	,	,	PUNCT
ejpam-7052	25	8	µy	µy	CCONJ
ejpam-7052	25	9	)	)	PUNCT
ejpam-7052	25	10	-continuous	-continuous	ADJ
ejpam-7052	25	11	multifunctions	multifunction	NOUN
ejpam-7052	26	1	[	[	X
ejpam-7052	26	2	8	8	NUM
ejpam-7052	26	3	]	]	PUNCT
ejpam-7052	26	4	.	.	PUNCT
ejpam-7052	27	1	pue	pue	NOUN
ejpam-7052	27	2	-	-	PUNCT
ejpam-7052	27	3	on	on	NOUN
ejpam-7052	27	4	et	et	PROPN
ejpam-7052	27	5	al	al	PROPN
ejpam-7052	27	6	.	.	PUNCT
ejpam-7052	28	1	[	[	X
ejpam-7052	28	2	9	9	NUM
ejpam-7052	28	3	]	]	PUNCT
ejpam-7052	28	4	introduced	introduce	VERB
ejpam-7052	28	5	and	and	CCONJ
ejpam-7052	28	6	studied	study	VERB
ejpam-7052	28	7	the	the	DET
ejpam-7052	28	8	concepts	concept	NOUN
ejpam-7052	28	9	of	of	ADP
ejpam-7052	28	10	upper	upper	ADJ
ejpam-7052	28	11	(	(	PUNCT
ejpam-7052	28	12	τ1	τ1	NOUN
ejpam-7052	28	13	,	,	PUNCT
ejpam-7052	28	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	28	15	multifunctions	multifunction	NOUN
ejpam-7052	28	16	and	and	CCONJ
ejpam-7052	28	17	lower	low	ADJ
ejpam-7052	28	18	(	(	PUNCT
ejpam-7052	28	19	τ1	τ1	NOUN
ejpam-7052	28	20	,	,	PUNCT
ejpam-7052	28	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	28	22	multifunctions	multifunction	NOUN
ejpam-7052	28	23	.	.	PUNCT
ejpam-7052	29	1	klanarong	klanarong	NOUN
ejpam-7052	29	2	et	et	PROPN
ejpam-7052	29	3	al	al	PROPN
ejpam-7052	29	4	.	.	PUNCT
ejpam-7052	30	1	[	[	X
ejpam-7052	30	2	10	10	NUM
ejpam-7052	30	3	]	]	PUNCT
ejpam-7052	30	4	introduced	introduce	VERB
ejpam-7052	30	5	and	and	CCONJ
ejpam-7052	30	6	investigated	investigate	VERB
ejpam-7052	30	7	the	the	DET
ejpam-7052	30	8	notions	notion	NOUN
ejpam-7052	30	9	of	of	ADP
ejpam-7052	30	10	upper	upper	ADJ
ejpam-7052	30	11	almost	almost	ADV
ejpam-7052	30	12	(	(	PUNCT
ejpam-7052	30	13	τ1	τ1	NOUN
ejpam-7052	30	14	,	,	PUNCT
ejpam-7052	30	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	30	16	multifunctions	multifunction	NOUN
ejpam-7052	30	17	and	and	CCONJ
ejpam-7052	30	18	lower	low	ADJ
ejpam-7052	30	19	almost	almost	ADV
ejpam-7052	30	20	(	(	PUNCT
ejpam-7052	30	21	τ1	τ1	NOUN
ejpam-7052	30	22	,	,	PUNCT
ejpam-7052	30	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	30	24	multifunctions	multifunction	NOUN
ejpam-7052	30	25	.	.	PUNCT
ejpam-7052	31	1	quite	quite	ADV
ejpam-7052	31	2	recently	recently	ADV
ejpam-7052	31	3	,	,	PUNCT
ejpam-7052	31	4	viriyapong	viriyapong	PROPN
ejpam-7052	31	5	et	et	PROPN
ejpam-7052	31	6	al	al	PROPN
ejpam-7052	31	7	.	.	PUNCT
ejpam-7052	32	1	[	[	X
ejpam-7052	32	2	11	11	NUM
ejpam-7052	32	3	]	]	PUNCT
ejpam-7052	32	4	presented	present	VERB
ejpam-7052	32	5	new	new	ADJ
ejpam-7052	32	6	classes	class	NOUN
ejpam-7052	32	7	of	of	ADP
ejpam-7052	32	8	continuous	continuous	ADJ
ejpam-7052	32	9	multifunctions	multifunction	NOUN
ejpam-7052	32	10	between	between	ADP
ejpam-7052	32	11	an	an	DET
ejpam-7052	32	12	ideal	ideal	ADJ
ejpam-7052	32	13	topological	topological	ADJ
ejpam-7052	32	14	space	space	NOUN
ejpam-7052	32	15	and	and	CCONJ
ejpam-7052	32	16	a	a	DET
ejpam-7052	32	17	bitopological	bitopological	ADJ
ejpam-7052	32	18	space	space	NOUN
ejpam-7052	32	19	,	,	PUNCT
ejpam-7052	32	20	namely	namely	ADV
ejpam-7052	32	21	upper	upper	ADJ
ejpam-7052	32	22	almost	almost	ADV
ejpam-7052	32	23	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7052	32	24	,	,	PUNCT
ejpam-7052	32	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	32	26	multifunctions	multifunction	NOUN
ejpam-7052	32	27	and	and	CCONJ
ejpam-7052	32	28	lower	low	ADJ
ejpam-7052	32	29	almost	almost	ADV
ejpam-7052	32	30	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7052	32	31	,	,	PUNCT
ejpam-7052	32	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	32	33	multifunctions	multifunction	NOUN
ejpam-7052	32	34	.	.	PUNCT
ejpam-7052	33	1	furthermore	furthermore	ADV
ejpam-7052	33	2	,	,	PUNCT
ejpam-7052	33	3	several	several	ADJ
ejpam-7052	33	4	characterizations	characterization	NOUN
ejpam-7052	33	5	and	and	CCONJ
ejpam-7052	33	6	some	some	DET
ejpam-7052	33	7	properties	property	NOUN
ejpam-7052	33	8	of	of	ADP
ejpam-7052	33	9	upper	upper	ADJ
ejpam-7052	33	10	almost	almost	ADV
ejpam-7052	33	11	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7052	33	12	,	,	PUNCT
ejpam-7052	33	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	33	14	multifunctions	multifunction	NOUN
ejpam-7052	33	15	and	and	CCONJ
ejpam-7052	33	16	lower	low	ADJ
ejpam-7052	33	17	almost	almost	ADV
ejpam-7052	33	18	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-7052	33	19	,	,	PUNCT
ejpam-7052	33	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	33	21	multifunctions	multifunction	NOUN
ejpam-7052	33	22	were	be	AUX
ejpam-7052	33	23	discussed	discuss	VERB
ejpam-7052	33	24	in	in	ADP
ejpam-7052	33	25	[	[	X
ejpam-7052	33	26	11	11	NUM
ejpam-7052	33	27	]	]	PUNCT
ejpam-7052	33	28	.	.	PUNCT
ejpam-7052	34	1	in	in	ADP
ejpam-7052	34	2	this	this	DET
ejpam-7052	34	3	paper	paper	NOUN
ejpam-7052	34	4	,	,	PUNCT
ejpam-7052	34	5	we	we	PRON
ejpam-7052	34	6	introduce	introduce	VERB
ejpam-7052	34	7	new	new	ADJ
ejpam-7052	34	8	classes	class	NOUN
ejpam-7052	34	9	of	of	ADP
ejpam-7052	34	10	multifunctions	multifunction	NOUN
ejpam-7052	34	11	between	between	ADP
ejpam-7052	34	12	a	a	DET
ejpam-7052	34	13	generalized	generalized	ADJ
ejpam-7052	34	14	topological	topological	ADJ
ejpam-7052	34	15	space	space	NOUN
ejpam-7052	34	16	and	and	CCONJ
ejpam-7052	34	17	a	a	DET
ejpam-7052	34	18	bitopological	bitopological	ADJ
ejpam-7052	34	19	space	space	NOUN
ejpam-7052	34	20	,	,	PUNCT
ejpam-7052	34	21	namely	namely	ADV
ejpam-7052	34	22	upper	upper	ADJ
ejpam-7052	34	23	almost	almost	ADV
ejpam-7052	34	24	µ(σ1	µ(σ1	NOUN
ejpam-7052	34	25	,	,	PUNCT
ejpam-7052	34	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	34	27	multifunctions	multifunction	NOUN
ejpam-7052	34	28	and	and	CCONJ
ejpam-7052	34	29	lower	low	ADJ
ejpam-7052	34	30	almost	almost	ADV
ejpam-7052	34	31	µ(σ1	µ(σ1	NOUN
ejpam-7052	34	32	,	,	PUNCT
ejpam-7052	34	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	34	34	multifunctions	multifunction	NOUN
ejpam-7052	34	35	.	.	PUNCT
ejpam-7052	35	1	we	we	PRON
ejpam-7052	35	2	also	also	ADV
ejpam-7052	35	3	investigate	investigate	VERB
ejpam-7052	35	4	several	several	ADJ
ejpam-7052	35	5	characterizations	characterization	NOUN
ejpam-7052	35	6	of	of	ADP
ejpam-7052	35	7	upper	upper	ADJ
ejpam-7052	35	8	almost	almost	ADV
ejpam-7052	35	9	µ(σ1	µ(σ1	NOUN
ejpam-7052	35	10	,	,	PUNCT
ejpam-7052	35	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	35	12	multifunctions	multifunction	NOUN
ejpam-7052	35	13	and	and	CCONJ
ejpam-7052	35	14	lower	low	ADJ
ejpam-7052	35	15	almost	almost	ADV
ejpam-7052	35	16	µ(σ1	µ(σ1	NOUN
ejpam-7052	35	17	,	,	PUNCT
ejpam-7052	35	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	35	19	multifunctions	multifunction	NOUN
ejpam-7052	35	20	.	.	PUNCT
ejpam-7052	36	1	2	2	X
ejpam-7052	36	2	.	.	X
ejpam-7052	36	3	preliminaries	preliminary	NOUN
ejpam-7052	36	4	throughout	throughout	ADP
ejpam-7052	36	5	the	the	DET
ejpam-7052	36	6	present	present	ADJ
ejpam-7052	36	7	paper	paper	NOUN
ejpam-7052	36	8	,	,	PUNCT
ejpam-7052	36	9	spaces	space	NOUN
ejpam-7052	36	10	(	(	PUNCT
ejpam-7052	36	11	x	x	NOUN
ejpam-7052	36	12	,	,	PUNCT
ejpam-7052	36	13	τ1	τ1	NOUN
ejpam-7052	36	14	,	,	PUNCT
ejpam-7052	36	15	τ2	τ2	NOUN
ejpam-7052	36	16	)	)	PUNCT
ejpam-7052	36	17	and	and	CCONJ
ejpam-7052	36	18	(	(	PUNCT
ejpam-7052	36	19	y	y	PROPN
ejpam-7052	36	20	,	,	PUNCT
ejpam-7052	36	21	σ1	σ1	PROPN
ejpam-7052	36	22	,	,	PUNCT
ejpam-7052	36	23	σ2	σ2	NOUN
ejpam-7052	36	24	)	)	PUNCT
ejpam-7052	36	25	(	(	PUNCT
ejpam-7052	36	26	or	or	CCONJ
ejpam-7052	36	27	simply	simply	ADV
ejpam-7052	36	28	x	x	X
ejpam-7052	36	29	and	and	CCONJ
ejpam-7052	36	30	y	y	PROPN
ejpam-7052	36	31	)	)	PUNCT
ejpam-7052	36	32	always	always	ADV
ejpam-7052	36	33	mean	mean	VERB
ejpam-7052	36	34	bitopological	bitopological	ADJ
ejpam-7052	36	35	spaces	space	NOUN
ejpam-7052	36	36	on	on	ADP
ejpam-7052	36	37	which	which	PRON
ejpam-7052	36	38	no	no	DET
ejpam-7052	36	39	separation	separation	NOUN
ejpam-7052	36	40	axioms	axiom	NOUN
ejpam-7052	36	41	are	be	AUX
ejpam-7052	36	42	assumed	assume	VERB
ejpam-7052	36	43	unless	unless	SCONJ
ejpam-7052	36	44	explicitly	explicitly	ADV
ejpam-7052	36	45	stated	state	VERB
ejpam-7052	36	46	.	.	PUNCT
ejpam-7052	37	1	let	let	VERB
ejpam-7052	37	2	a	a	DET
ejpam-7052	37	3	be	be	AUX
ejpam-7052	37	4	a	a	DET
ejpam-7052	37	5	subset	subset	NOUN
ejpam-7052	37	6	of	of	ADP
ejpam-7052	37	7	a	a	DET
ejpam-7052	37	8	bitopological	bitopological	ADJ
ejpam-7052	37	9	space	space	NOUN
ejpam-7052	37	10	(	(	PUNCT
ejpam-7052	37	11	x	x	NOUN
ejpam-7052	37	12	,	,	PUNCT
ejpam-7052	37	13	τ1	τ1	NOUN
ejpam-7052	37	14	,	,	PUNCT
ejpam-7052	37	15	τ2	τ2	NOUN
ejpam-7052	37	16	)	)	PUNCT
ejpam-7052	37	17	.	.	PUNCT
ejpam-7052	38	1	the	the	DET
ejpam-7052	38	2	closure	closure	NOUN
ejpam-7052	38	3	of	of	ADP
ejpam-7052	38	4	a	a	PRON
ejpam-7052	38	5	and	and	CCONJ
ejpam-7052	38	6	the	the	DET
ejpam-7052	38	7	interior	interior	NOUN
ejpam-7052	38	8	of	of	ADP
ejpam-7052	38	9	a	a	PRON
ejpam-7052	38	10	with	with	ADP
ejpam-7052	38	11	respect	respect	NOUN
ejpam-7052	38	12	to	to	ADP
ejpam-7052	38	13	τi	τi	PROPN
ejpam-7052	38	14	are	be	AUX
ejpam-7052	38	15	denoted	denote	VERB
ejpam-7052	38	16	by	by	ADP
ejpam-7052	38	17	τi	τi	NOUN
ejpam-7052	38	18	-	-	PUNCT
ejpam-7052	38	19	cl(a	cl(a	NUM
ejpam-7052	38	20	)	)	PUNCT
ejpam-7052	38	21	and	and	CCONJ
ejpam-7052	38	22	τi	τi	NOUN
ejpam-7052	38	23	-	-	PUNCT
ejpam-7052	38	24	int(a	int(a	NOUN
ejpam-7052	38	25	)	)	PUNCT
ejpam-7052	38	26	,	,	PUNCT
ejpam-7052	38	27	respectively	respectively	ADV
ejpam-7052	38	28	,	,	PUNCT
ejpam-7052	38	29	for	for	ADP
ejpam-7052	38	30	i	i	PROPN
ejpam-7052	38	31	=	=	SYM
ejpam-7052	38	32	1	1	NUM
ejpam-7052	38	33	,	,	PUNCT
ejpam-7052	38	34	2	2	NUM
ejpam-7052	38	35	.	.	X
ejpam-7052	38	36	a	a	DET
ejpam-7052	38	37	subset	subset	NOUN
ejpam-7052	38	38	a	a	PRON
ejpam-7052	38	39	of	of	ADP
ejpam-7052	38	40	a	a	DET
ejpam-7052	38	41	bitopological	bitopological	ADJ
ejpam-7052	38	42	space	space	NOUN
ejpam-7052	38	43	(	(	PUNCT
ejpam-7052	38	44	x	x	NOUN
ejpam-7052	38	45	,	,	PUNCT
ejpam-7052	38	46	τ1	τ1	NOUN
ejpam-7052	38	47	,	,	PUNCT
ejpam-7052	38	48	τ2	τ2	NOUN
ejpam-7052	38	49	)	)	PUNCT
ejpam-7052	38	50	is	be	AUX
ejpam-7052	38	51	called	call	VERB
ejpam-7052	38	52	τ1τ2	τ1τ2	VERB
ejpam-7052	38	53	-	-	ADJ
ejpam-7052	38	54	closed	closed	ADJ
ejpam-7052	38	55	[	[	X
ejpam-7052	38	56	12	12	NUM
ejpam-7052	38	57	]	]	X
ejpam-7052	38	58	if	if	SCONJ
ejpam-7052	38	59	a	a	DET
ejpam-7052	38	60	=	=	NOUN
ejpam-7052	38	61	τ1	τ1	NOUN
ejpam-7052	38	62	-	-	PUNCT
ejpam-7052	38	63	cl(τ2	cl(τ2	NOUN
ejpam-7052	38	64	-	-	PUNCT
ejpam-7052	38	65	cl(a	cl(a	NUM
ejpam-7052	38	66	)	)	PUNCT
ejpam-7052	38	67	)	)	PUNCT
ejpam-7052	38	68	.	.	PUNCT
ejpam-7052	39	1	the	the	DET
ejpam-7052	39	2	complement	complement	NOUN
ejpam-7052	39	3	of	of	ADP
ejpam-7052	39	4	a	a	DET
ejpam-7052	39	5	τ1τ2	τ1τ2	ADJ
ejpam-7052	39	6	-	-	ADJ
ejpam-7052	39	7	closed	closed	ADJ
ejpam-7052	39	8	set	set	NOUN
ejpam-7052	39	9	is	be	AUX
ejpam-7052	39	10	called	call	VERB
ejpam-7052	39	11	τ1τ2	τ1τ2	NOUN
ejpam-7052	39	12	-	-	ADJ
ejpam-7052	39	13	open	open	ADJ
ejpam-7052	39	14	.	.	PUNCT
ejpam-7052	40	1	the	the	DET
ejpam-7052	40	2	intersection	intersection	NOUN
ejpam-7052	40	3	of	of	ADP
ejpam-7052	40	4	all	all	DET
ejpam-7052	40	5	τ1τ2	τ1τ2	ADJ
ejpam-7052	40	6	-	-	ADJ
ejpam-7052	40	7	closed	closed	ADJ
ejpam-7052	40	8	sets	set	NOUN
ejpam-7052	40	9	of	of	ADP
ejpam-7052	40	10	x	x	PUNCT
ejpam-7052	40	11	containing	contain	VERB
ejpam-7052	40	12	a	a	PRON
ejpam-7052	40	13	is	be	AUX
ejpam-7052	40	14	called	call	VERB
ejpam-7052	40	15	the	the	DET
ejpam-7052	40	16	τ1τ2	τ1τ2	NOUN
ejpam-7052	40	17	-	-	NOUN
ejpam-7052	40	18	closure	closure	NOUN
ejpam-7052	40	19	[	[	X
ejpam-7052	40	20	12	12	NUM
ejpam-7052	40	21	]	]	PUNCT
ejpam-7052	40	22	of	of	ADP
ejpam-7052	40	23	a	a	PRON
ejpam-7052	40	24	and	and	CCONJ
ejpam-7052	40	25	is	be	AUX
ejpam-7052	40	26	denoted	denote	VERB
ejpam-7052	40	27	by	by	ADP
ejpam-7052	40	28	τ1τ2	τ1τ2	NOUN
ejpam-7052	40	29	-	-	NUM
ejpam-7052	40	30	cl(a	cl(a	NUM
ejpam-7052	40	31	)	)	PUNCT
ejpam-7052	40	32	.	.	PUNCT
ejpam-7052	41	1	the	the	DET
ejpam-7052	41	2	union	union	NOUN
ejpam-7052	41	3	of	of	ADP
ejpam-7052	41	4	all	all	DET
ejpam-7052	41	5	τ1τ2	τ1τ2	ADJ
ejpam-7052	41	6	-	-	ADJ
ejpam-7052	41	7	open	open	ADJ
ejpam-7052	41	8	sets	set	NOUN
ejpam-7052	41	9	of	of	ADP
ejpam-7052	41	10	x	x	PUNCT
ejpam-7052	41	11	contained	contain	VERB
ejpam-7052	41	12	in	in	ADP
ejpam-7052	41	13	a	a	PRON
ejpam-7052	41	14	is	be	AUX
ejpam-7052	41	15	called	call	VERB
ejpam-7052	41	16	the	the	DET
ejpam-7052	41	17	τ1τ2	τ1τ2	NOUN
ejpam-7052	41	18	-	-	ADJ
ejpam-7052	41	19	interior	interior	ADJ
ejpam-7052	41	20	[	[	X
ejpam-7052	41	21	12	12	NUM
ejpam-7052	41	22	]	]	PUNCT
ejpam-7052	41	23	of	of	ADP
ejpam-7052	41	24	a	a	PRON
ejpam-7052	41	25	and	and	CCONJ
ejpam-7052	41	26	is	be	AUX
ejpam-7052	41	27	denoted	denote	VERB
ejpam-7052	41	28	by	by	ADP
ejpam-7052	41	29	τ1τ2	τ1τ2	NOUN
ejpam-7052	41	30	-	-	ADJ
ejpam-7052	41	31	int(a	int(a	NOUN
ejpam-7052	41	32	)	)	PUNCT
ejpam-7052	41	33	.	.	PUNCT
ejpam-7052	42	1	lemma	lemma	PROPN
ejpam-7052	42	2	1	1	NUM
ejpam-7052	42	3	.	.	PUNCT
ejpam-7052	43	1	[	[	X
ejpam-7052	43	2	12	12	NUM
ejpam-7052	43	3	]	]	PUNCT
ejpam-7052	43	4	let	let	VERB
ejpam-7052	43	5	a	a	PRON
ejpam-7052	43	6	and	and	CCONJ
ejpam-7052	43	7	b	b	NOUN
ejpam-7052	43	8	be	be	AUX
ejpam-7052	43	9	subsets	subset	NOUN
ejpam-7052	43	10	of	of	ADP
ejpam-7052	43	11	a	a	DET
ejpam-7052	43	12	bitopological	bitopological	ADJ
ejpam-7052	43	13	space	space	NOUN
ejpam-7052	43	14	(	(	PUNCT
ejpam-7052	43	15	x	x	NOUN
ejpam-7052	43	16	,	,	PUNCT
ejpam-7052	43	17	τ1	τ1	NOUN
ejpam-7052	43	18	,	,	PUNCT
ejpam-7052	43	19	τ2	τ2	NOUN
ejpam-7052	43	20	)	)	PUNCT
ejpam-7052	43	21	.	.	PUNCT
ejpam-7052	44	1	for	for	ADP
ejpam-7052	44	2	the	the	DET
ejpam-7052	44	3	τ1τ2	τ1τ2	NOUN
ejpam-7052	44	4	-	-	NOUN
ejpam-7052	44	5	closure	closure	NOUN
ejpam-7052	44	6	,	,	PUNCT
ejpam-7052	44	7	the	the	DET
ejpam-7052	44	8	following	follow	VERB
ejpam-7052	44	9	properties	property	NOUN
ejpam-7052	44	10	hold	hold	VERB
ejpam-7052	44	11	:	:	PUNCT
ejpam-7052	44	12	(	(	PUNCT
ejpam-7052	44	13	1	1	X
ejpam-7052	44	14	)	)	PUNCT
ejpam-7052	44	15	a	a	DET
ejpam-7052	44	16	⊆	⊆	NUM
ejpam-7052	44	17	τ1τ2	τ1τ2	NOUN
ejpam-7052	44	18	-	-	NUM
ejpam-7052	44	19	cl(a	cl(a	NUM
ejpam-7052	44	20	)	)	PUNCT
ejpam-7052	44	21	and	and	CCONJ
ejpam-7052	44	22	τ1τ2	τ1τ2	NOUN
ejpam-7052	44	23	-	-	ADJ
ejpam-7052	44	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7052	44	25	-	-	PUNCT
ejpam-7052	44	26	cl(a	cl(a	NUM
ejpam-7052	44	27	)	)	PUNCT
ejpam-7052	44	28	)	)	PUNCT
ejpam-7052	45	1	=	=	PUNCT
ejpam-7052	45	2	τ1τ2	τ1τ2	NOUN
ejpam-7052	45	3	-	-	NUM
ejpam-7052	45	4	cl(a	cl(a	NUM
ejpam-7052	45	5	)	)	PUNCT
ejpam-7052	45	6	.	.	PUNCT
ejpam-7052	46	1	(	(	PUNCT
ejpam-7052	46	2	2	2	X
ejpam-7052	46	3	)	)	PUNCT
ejpam-7052	46	4	if	if	SCONJ
ejpam-7052	46	5	a	a	DET
ejpam-7052	46	6	⊆	⊆	NUM
ejpam-7052	46	7	b	b	NOUN
ejpam-7052	46	8	,	,	PUNCT
ejpam-7052	46	9	then	then	ADV
ejpam-7052	46	10	τ1τ2	τ1τ2	NOUN
ejpam-7052	46	11	-	-	NUM
ejpam-7052	46	12	cl(a	cl(a	NUM
ejpam-7052	46	13	)	)	PUNCT
ejpam-7052	46	14	⊆	⊆	NUM
ejpam-7052	46	15	τ1τ2	τ1τ2	NOUN
ejpam-7052	46	16	-	-	NOUN
ejpam-7052	46	17	cl(b	cl(b	NOUN
ejpam-7052	46	18	)	)	PUNCT
ejpam-7052	46	19	.	.	PUNCT
ejpam-7052	47	1	(	(	PUNCT
ejpam-7052	47	2	3	3	X
ejpam-7052	47	3	)	)	PUNCT
ejpam-7052	47	4	τ1τ2	τ1τ2	NOUN
ejpam-7052	47	5	-	-	NUM
ejpam-7052	47	6	cl(a	cl(a	NUM
ejpam-7052	47	7	)	)	PUNCT
ejpam-7052	47	8	is	be	AUX
ejpam-7052	47	9	τ1τ2	τ1τ2	NOUN
ejpam-7052	47	10	-	-	ADJ
ejpam-7052	47	11	closed	closed	ADJ
ejpam-7052	47	12	.	.	PUNCT
ejpam-7052	48	1	(	(	PUNCT
ejpam-7052	48	2	4	4	X
ejpam-7052	48	3	)	)	PUNCT
ejpam-7052	48	4	a	a	PRON
ejpam-7052	48	5	is	be	AUX
ejpam-7052	48	6	τ1τ2	τ1τ2	NOUN
ejpam-7052	48	7	-	-	ADJ
ejpam-7052	48	8	closed	closed	ADJ
ejpam-7052	48	9	if	if	SCONJ
ejpam-7052	48	10	and	and	CCONJ
ejpam-7052	48	11	only	only	ADV
ejpam-7052	48	12	if	if	SCONJ
ejpam-7052	48	13	a	a	DET
ejpam-7052	48	14	=	=	PUNCT
ejpam-7052	48	15	τ1τ2	τ1τ2	NOUN
ejpam-7052	48	16	-	-	NUM
ejpam-7052	48	17	cl(a	cl(a	NUM
ejpam-7052	48	18	)	)	PUNCT
ejpam-7052	48	19	.	.	PUNCT
ejpam-7052	49	1	(	(	PUNCT
ejpam-7052	49	2	5	5	X
ejpam-7052	49	3	)	)	PUNCT
ejpam-7052	49	4	τ1τ2	τ1τ2	NOUN
ejpam-7052	49	5	-	-	NOUN
ejpam-7052	49	6	cl(x	cl(x	X
ejpam-7052	49	7	−a	−a	NOUN
ejpam-7052	49	8	)	)	PUNCT
ejpam-7052	50	1	=	=	PUNCT
ejpam-7052	50	2	x	x	X
ejpam-7052	51	1	−	−	ADP
ejpam-7052	51	2	τ1τ2	τ1τ2	NOUN
ejpam-7052	51	3	-	-	PUNCT
ejpam-7052	51	4	int(a	int(a	NOUN
ejpam-7052	51	5	)	)	PUNCT
ejpam-7052	51	6	.	.	PUNCT
ejpam-7052	52	1	p.	p.	NOUN
ejpam-7052	52	2	pue	pue	NOUN
ejpam-7052	52	3	-	-	PUNCT
ejpam-7052	52	4	on	on	ADP
ejpam-7052	52	5	,	,	PUNCT
ejpam-7052	52	6	a.	a.	PROPN
ejpam-7052	52	7	sama	sama	PROPN
ejpam-7052	52	8	-	-	PUNCT
ejpam-7052	52	9	ae	ae	PROPN
ejpam-7052	52	10	,	,	PUNCT
ejpam-7052	52	11	c.	c.	PROPN
ejpam-7052	52	12	boonpok	boonpok	PROPN
ejpam-7052	52	13	/	/	SYM
ejpam-7052	52	14	eur	eur	PROPN
ejpam-7052	52	15	.	.	PUNCT
ejpam-7052	53	1	j.	j.	PROPN
ejpam-7052	53	2	pure	pure	PROPN
ejpam-7052	53	3	appl	appl	PROPN
ejpam-7052	53	4	.	.	PROPN
ejpam-7052	53	5	math	math	PROPN
ejpam-7052	53	6	,	,	PUNCT
ejpam-7052	53	7	18	18	NUM
ejpam-7052	53	8	(	(	PUNCT
ejpam-7052	53	9	4	4	NUM
ejpam-7052	53	10	)	)	PUNCT
ejpam-7052	53	11	(	(	PUNCT
ejpam-7052	53	12	2025	2025	NUM
ejpam-7052	53	13	)	)	PUNCT
ejpam-7052	53	14	,	,	PUNCT
ejpam-7052	53	15	7052	7052	NUM
ejpam-7052	53	16	3	3	NUM
ejpam-7052	53	17	of	of	ADP
ejpam-7052	53	18	14	14	NUM
ejpam-7052	53	19	a	a	DET
ejpam-7052	53	20	subset	subset	NOUN
ejpam-7052	53	21	a	a	PRON
ejpam-7052	53	22	of	of	ADP
ejpam-7052	53	23	a	a	DET
ejpam-7052	53	24	bitopological	bitopological	ADJ
ejpam-7052	53	25	space	space	NOUN
ejpam-7052	53	26	(	(	PUNCT
ejpam-7052	53	27	x	x	NOUN
ejpam-7052	53	28	,	,	PUNCT
ejpam-7052	53	29	τ1	τ1	NOUN
ejpam-7052	53	30	,	,	PUNCT
ejpam-7052	53	31	τ2	τ2	NOUN
ejpam-7052	53	32	)	)	PUNCT
ejpam-7052	53	33	is	be	AUX
ejpam-7052	53	34	said	say	VERB
ejpam-7052	53	35	to	to	PART
ejpam-7052	53	36	be	be	AUX
ejpam-7052	53	37	(	(	PUNCT
ejpam-7052	53	38	τ1	τ1	NOUN
ejpam-7052	53	39	,	,	PUNCT
ejpam-7052	53	40	τ2)r	τ2)r	NOUN
ejpam-7052	53	41	-	-	PUNCT
ejpam-7052	53	42	open	open	NOUN
ejpam-7052	53	43	[	[	X
ejpam-7052	53	44	13	13	NUM
ejpam-7052	53	45	]	]	PUNCT
ejpam-7052	53	46	(	(	PUNCT
ejpam-7052	53	47	resp	resp	NOUN
ejpam-7052	53	48	.	.	PUNCT
ejpam-7052	54	1	(	(	PUNCT
ejpam-7052	54	2	τ1	τ1	NOUN
ejpam-7052	54	3	,	,	PUNCT
ejpam-7052	54	4	τ2)s	τ2)s	NOUN
ejpam-7052	54	5	-	-	PUNCT
ejpam-7052	54	6	open	open	ADJ
ejpam-7052	54	7	[	[	X
ejpam-7052	54	8	14	14	NUM
ejpam-7052	54	9	]	]	NUM
ejpam-7052	54	10	,	,	PUNCT
ejpam-7052	54	11	(	(	PUNCT
ejpam-7052	54	12	τ1	τ1	NOUN
ejpam-7052	54	13	,	,	PUNCT
ejpam-7052	54	14	τ2)p	τ2)p	NOUN
ejpam-7052	54	15	-	-	ADJ
ejpam-7052	54	16	open	open	ADJ
ejpam-7052	54	17	[	[	X
ejpam-7052	54	18	14	14	NUM
ejpam-7052	54	19	]	]	NUM
ejpam-7052	54	20	,	,	PUNCT
ejpam-7052	54	21	(	(	PUNCT
ejpam-7052	54	22	τ1	τ1	NOUN
ejpam-7052	54	23	,	,	PUNCT
ejpam-7052	54	24	τ2)β	τ2)β	ADJ
ejpam-7052	54	25	-	-	PUNCT
ejpam-7052	54	26	open	open	NOUN
ejpam-7052	54	27	[	[	X
ejpam-7052	54	28	14	14	NUM
ejpam-7052	54	29	]	]	SYM
ejpam-7052	54	30	)	)	PUNCT
ejpam-7052	54	31	if	if	SCONJ
ejpam-7052	54	32	a	a	DET
ejpam-7052	54	33	=	=	PUNCT
ejpam-7052	54	34	τ1τ2	τ1τ2	NOUN
ejpam-7052	54	35	-	-	NOUN
ejpam-7052	54	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7052	54	37	-	-	PUNCT
ejpam-7052	54	38	cl(a	cl(a	NUM
ejpam-7052	54	39	)	)	PUNCT
ejpam-7052	54	40	)	)	PUNCT
ejpam-7052	54	41	(	(	PUNCT
ejpam-7052	54	42	resp	resp	NOUN
ejpam-7052	54	43	.	.	PUNCT
ejpam-7052	55	1	a	a	DET
ejpam-7052	55	2	⊆	⊆	NUM
ejpam-7052	55	3	τ1τ2	τ1τ2	NOUN
ejpam-7052	55	4	-	-	ADJ
ejpam-7052	55	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7052	55	6	-	-	PUNCT
ejpam-7052	55	7	int(a	int(a	NOUN
ejpam-7052	55	8	)	)	PUNCT
ejpam-7052	55	9	)	)	PUNCT
ejpam-7052	55	10	,	,	PUNCT
ejpam-7052	55	11	a	a	DET
ejpam-7052	55	12	⊆	⊆	NUM
ejpam-7052	55	13	τ1τ2	τ1τ2	NOUN
ejpam-7052	55	14	-	-	NOUN
ejpam-7052	55	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7052	55	16	-	-	PUNCT
ejpam-7052	55	17	cl(a	cl(a	NUM
ejpam-7052	55	18	)	)	PUNCT
ejpam-7052	55	19	)	)	PUNCT
ejpam-7052	55	20	,	,	PUNCT
ejpam-7052	55	21	a	a	DET
ejpam-7052	55	22	⊆	⊆	NUM
ejpam-7052	55	23	τ1τ2	τ1τ2	NOUN
ejpam-7052	55	24	-	-	PUNCT
ejpam-7052	55	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7052	55	26	-	-	PUNCT
ejpam-7052	55	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7052	55	28	-	-	PUNCT
ejpam-7052	55	29	cl(a	cl(a	NUM
ejpam-7052	55	30	)	)	PUNCT
ejpam-7052	55	31	)	)	PUNCT
ejpam-7052	55	32	)	)	PUNCT
ejpam-7052	55	33	)	)	PUNCT
ejpam-7052	55	34	.	.	PUNCT
ejpam-7052	56	1	the	the	DET
ejpam-7052	56	2	complement	complement	NOUN
ejpam-7052	56	3	of	of	ADP
ejpam-7052	56	4	a	a	DET
ejpam-7052	56	5	(	(	PUNCT
ejpam-7052	56	6	τ1	τ1	NOUN
ejpam-7052	56	7	,	,	PUNCT
ejpam-7052	56	8	τ2)r	τ2)r	NOUN
ejpam-7052	56	9	-	-	PUNCT
ejpam-7052	56	10	open	open	ADJ
ejpam-7052	56	11	(	(	PUNCT
ejpam-7052	56	12	resp	resp	NOUN
ejpam-7052	56	13	.	.	PUNCT
ejpam-7052	57	1	(	(	PUNCT
ejpam-7052	57	2	τ1	τ1	NOUN
ejpam-7052	57	3	,	,	PUNCT
ejpam-7052	57	4	τ2)s	τ2)s	NOUN
ejpam-7052	57	5	-	-	PUNCT
ejpam-7052	57	6	open	open	ADJ
ejpam-7052	57	7	,	,	PUNCT
ejpam-7052	57	8	(	(	PUNCT
ejpam-7052	57	9	τ1	τ1	NOUN
ejpam-7052	57	10	,	,	PUNCT
ejpam-7052	57	11	τ2)p	τ2)p	NOUN
ejpam-7052	57	12	-	-	ADJ
ejpam-7052	57	13	open	open	ADJ
ejpam-7052	57	14	,	,	PUNCT
ejpam-7052	57	15	(	(	PUNCT
ejpam-7052	57	16	τ1	τ1	NOUN
ejpam-7052	57	17	,	,	PUNCT
ejpam-7052	57	18	τ2)β	τ2)β	ADJ
ejpam-7052	57	19	-	-	PUNCT
ejpam-7052	57	20	open	open	ADJ
ejpam-7052	57	21	)	)	PUNCT
ejpam-7052	57	22	set	set	NOUN
ejpam-7052	57	23	is	be	AUX
ejpam-7052	57	24	said	say	VERB
ejpam-7052	57	25	to	to	PART
ejpam-7052	57	26	be	be	AUX
ejpam-7052	57	27	(	(	PUNCT
ejpam-7052	57	28	τ1	τ1	NOUN
ejpam-7052	57	29	,	,	PUNCT
ejpam-7052	57	30	τ2)r	τ2)r	NOUN
ejpam-7052	57	31	-	-	PUNCT
ejpam-7052	57	32	closed	closed	ADJ
ejpam-7052	57	33	(	(	PUNCT
ejpam-7052	57	34	resp	resp	NOUN
ejpam-7052	57	35	.	.	PUNCT
ejpam-7052	58	1	(	(	PUNCT
ejpam-7052	58	2	τ1	τ1	NOUN
ejpam-7052	58	3	,	,	PUNCT
ejpam-7052	58	4	τ2)s	τ2)s	NOUN
ejpam-7052	58	5	-	-	PUNCT
ejpam-7052	58	6	closed	closed	ADJ
ejpam-7052	58	7	,	,	PUNCT
ejpam-7052	58	8	(	(	PUNCT
ejpam-7052	58	9	τ1	τ1	NOUN
ejpam-7052	58	10	,	,	PUNCT
ejpam-7052	58	11	τ2)p	τ2)p	NOUN
ejpam-7052	58	12	-	-	PUNCT
ejpam-7052	58	13	closed	closed	ADJ
ejpam-7052	58	14	,	,	PUNCT
ejpam-7052	58	15	(	(	PUNCT
ejpam-7052	58	16	τ1	τ1	NOUN
ejpam-7052	58	17	,	,	PUNCT
ejpam-7052	58	18	τ2)β	τ2)β	ADJ
ejpam-7052	58	19	-	-	PUNCT
ejpam-7052	58	20	closed	closed	ADJ
ejpam-7052	58	21	)	)	PUNCT
ejpam-7052	58	22	.	.	PUNCT
ejpam-7052	59	1	a	a	DET
ejpam-7052	59	2	subset	subset	NOUN
ejpam-7052	59	3	a	a	PRON
ejpam-7052	59	4	of	of	ADP
ejpam-7052	59	5	a	a	DET
ejpam-7052	59	6	bitopological	bitopological	ADJ
ejpam-7052	59	7	space	space	NOUN
ejpam-7052	59	8	(	(	PUNCT
ejpam-7052	59	9	x	x	NOUN
ejpam-7052	59	10	,	,	PUNCT
ejpam-7052	59	11	τ1	τ1	NOUN
ejpam-7052	59	12	,	,	PUNCT
ejpam-7052	59	13	τ2	τ2	NOUN
ejpam-7052	59	14	)	)	PUNCT
ejpam-7052	59	15	is	be	AUX
ejpam-7052	59	16	said	say	VERB
ejpam-7052	59	17	to	to	PART
ejpam-7052	59	18	be	be	AUX
ejpam-7052	59	19	α(τ1	α(τ1	NOUN
ejpam-7052	59	20	,	,	PUNCT
ejpam-7052	59	21	τ2)-open	τ2)-open	ADJ
ejpam-7052	59	22	[	[	X
ejpam-7052	59	23	15	15	NUM
ejpam-7052	59	24	]	]	X
ejpam-7052	59	25	if	if	SCONJ
ejpam-7052	59	26	a	a	DET
ejpam-7052	59	27	⊆	⊆	NUM
ejpam-7052	59	28	τ1τ2	τ1τ2	NOUN
ejpam-7052	59	29	-	-	PUNCT
ejpam-7052	59	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7052	59	31	-	-	PUNCT
ejpam-7052	59	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7052	59	33	-	-	PUNCT
ejpam-7052	59	34	int(a	int(a	NOUN
ejpam-7052	59	35	)	)	PUNCT
ejpam-7052	59	36	)	)	PUNCT
ejpam-7052	59	37	)	)	PUNCT
ejpam-7052	59	38	.	.	PUNCT
ejpam-7052	60	1	the	the	DET
ejpam-7052	60	2	complement	complement	NOUN
ejpam-7052	60	3	of	of	ADP
ejpam-7052	60	4	an	an	DET
ejpam-7052	60	5	α(τ1	α(τ1	NOUN
ejpam-7052	60	6	,	,	PUNCT
ejpam-7052	60	7	τ2)-open	τ2)-open	ADJ
ejpam-7052	60	8	set	set	NOUN
ejpam-7052	60	9	is	be	AUX
ejpam-7052	60	10	said	say	VERB
ejpam-7052	60	11	to	to	PART
ejpam-7052	60	12	be	be	AUX
ejpam-7052	60	13	α(τ1	α(τ1	NOUN
ejpam-7052	60	14	,	,	PUNCT
ejpam-7052	60	15	τ2)-closed	τ2)-close	VERB
ejpam-7052	60	16	.	.	PUNCT
ejpam-7052	61	1	let	let	VERB
ejpam-7052	61	2	a	a	DET
ejpam-7052	61	3	be	be	AUX
ejpam-7052	61	4	a	a	DET
ejpam-7052	61	5	subset	subset	NOUN
ejpam-7052	61	6	of	of	ADP
ejpam-7052	61	7	a	a	DET
ejpam-7052	61	8	bitopological	bitopological	ADJ
ejpam-7052	61	9	space	space	NOUN
ejpam-7052	61	10	(	(	PUNCT
ejpam-7052	61	11	x	x	NOUN
ejpam-7052	61	12	,	,	PUNCT
ejpam-7052	61	13	τ1	τ1	NOUN
ejpam-7052	61	14	,	,	PUNCT
ejpam-7052	61	15	τ2	τ2	NOUN
ejpam-7052	61	16	)	)	PUNCT
ejpam-7052	61	17	.	.	PUNCT
ejpam-7052	62	1	the	the	DET
ejpam-7052	62	2	intersection	intersection	NOUN
ejpam-7052	62	3	of	of	ADP
ejpam-7052	62	4	all	all	DET
ejpam-7052	62	5	(	(	PUNCT
ejpam-7052	62	6	τ1	τ1	NOUN
ejpam-7052	62	7	,	,	PUNCT
ejpam-7052	62	8	τ2)p	τ2)p	NOUN
ejpam-7052	62	9	-	-	PUNCT
ejpam-7052	62	10	closed	closed	ADJ
ejpam-7052	62	11	(	(	PUNCT
ejpam-7052	62	12	resp	resp	NOUN
ejpam-7052	62	13	.	.	PUNCT
ejpam-7052	63	1	(	(	PUNCT
ejpam-7052	63	2	τ1	τ1	NOUN
ejpam-7052	63	3	,	,	PUNCT
ejpam-7052	63	4	τ2)s	τ2)s	NOUN
ejpam-7052	63	5	-	-	PUNCT
ejpam-7052	63	6	closed	closed	ADJ
ejpam-7052	63	7	,	,	PUNCT
ejpam-7052	63	8	α(τ1	α(τ1	NOUN
ejpam-7052	63	9	,	,	PUNCT
ejpam-7052	63	10	τ2)-closed	τ2)-closed	ADJ
ejpam-7052	63	11	)	)	PUNCT
ejpam-7052	63	12	sets	set	NOUN
ejpam-7052	63	13	of	of	ADP
ejpam-7052	63	14	x	x	PUNCT
ejpam-7052	63	15	containing	contain	VERB
ejpam-7052	63	16	a	a	PRON
ejpam-7052	63	17	is	be	AUX
ejpam-7052	63	18	called	call	VERB
ejpam-7052	63	19	the	the	DET
ejpam-7052	63	20	(	(	PUNCT
ejpam-7052	63	21	τ1	τ1	NOUN
ejpam-7052	63	22	,	,	PUNCT
ejpam-7052	63	23	τ2)p	τ2)p	NOUN
ejpam-7052	63	24	-	-	NOUN
ejpam-7052	63	25	closure	closure	NOUN
ejpam-7052	63	26	[	[	X
ejpam-7052	63	27	16	16	NUM
ejpam-7052	63	28	]	]	X
ejpam-7052	63	29	(	(	PUNCT
ejpam-7052	63	30	resp	resp	NOUN
ejpam-7052	63	31	.	.	PUNCT
ejpam-7052	64	1	(	(	PUNCT
ejpam-7052	64	2	τ1	τ1	NOUN
ejpam-7052	64	3	,	,	PUNCT
ejpam-7052	64	4	τ2)s	τ2)s	NOUN
ejpam-7052	64	5	-	-	PUNCT
ejpam-7052	64	6	closure	closure	NOUN
ejpam-7052	64	7	[	[	X
ejpam-7052	64	8	14	14	NUM
ejpam-7052	64	9	]	]	PUNCT
ejpam-7052	64	10	,	,	PUNCT
ejpam-7052	64	11	α(τ1	α(τ1	NOUN
ejpam-7052	64	12	,	,	PUNCT
ejpam-7052	64	13	τ2)-closure	τ2)-closure	NOUN
ejpam-7052	64	14	[	[	X
ejpam-7052	64	15	17	17	NUM
ejpam-7052	64	16	]	]	PUNCT
ejpam-7052	64	17	)	)	PUNCT
ejpam-7052	64	18	of	of	ADP
ejpam-7052	64	19	a	a	PRON
ejpam-7052	64	20	and	and	CCONJ
ejpam-7052	64	21	is	be	AUX
ejpam-7052	64	22	denoted	denote	VERB
ejpam-7052	64	23	by	by	ADP
ejpam-7052	64	24	(	(	PUNCT
ejpam-7052	64	25	τ1	τ1	NOUN
ejpam-7052	64	26	,	,	PUNCT
ejpam-7052	64	27	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-7052	64	28	)	)	PUNCT
ejpam-7052	64	29	(	(	PUNCT
ejpam-7052	64	30	resp	resp	NOUN
ejpam-7052	64	31	.	.	PUNCT
ejpam-7052	65	1	(	(	PUNCT
ejpam-7052	65	2	τ1	τ1	NOUN
ejpam-7052	65	3	,	,	PUNCT
ejpam-7052	65	4	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-7052	65	5	)	)	PUNCT
ejpam-7052	65	6	,	,	PUNCT
ejpam-7052	65	7	α(τ1	α(τ1	NOUN
ejpam-7052	65	8	,	,	PUNCT
ejpam-7052	65	9	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-7052	65	10	)	)	PUNCT
ejpam-7052	65	11	)	)	PUNCT
ejpam-7052	65	12	.	.	PUNCT
ejpam-7052	66	1	the	the	DET
ejpam-7052	66	2	union	union	NOUN
ejpam-7052	66	3	of	of	ADP
ejpam-7052	66	4	all	all	DET
ejpam-7052	66	5	(	(	PUNCT
ejpam-7052	66	6	τ1	τ1	NOUN
ejpam-7052	66	7	,	,	PUNCT
ejpam-7052	66	8	τ2)p	τ2)p	NOUN
ejpam-7052	66	9	-	-	ADJ
ejpam-7052	66	10	open	open	ADJ
ejpam-7052	66	11	(	(	PUNCT
ejpam-7052	66	12	resp	resp	NOUN
ejpam-7052	66	13	.	.	PUNCT
ejpam-7052	67	1	(	(	PUNCT
ejpam-7052	67	2	τ1	τ1	NOUN
ejpam-7052	67	3	,	,	PUNCT
ejpam-7052	67	4	τ2)s	τ2)s	NOUN
ejpam-7052	67	5	-	-	PUNCT
ejpam-7052	67	6	open	open	ADJ
ejpam-7052	67	7	,	,	PUNCT
ejpam-7052	67	8	α(τ1	α(τ1	NOUN
ejpam-7052	67	9	,	,	PUNCT
ejpam-7052	67	10	τ2)-open	τ2)-open	ADJ
ejpam-7052	67	11	)	)	PUNCT
ejpam-7052	67	12	sets	set	NOUN
ejpam-7052	67	13	of	of	ADP
ejpam-7052	67	14	x	x	PUNCT
ejpam-7052	67	15	contained	contain	VERB
ejpam-7052	67	16	in	in	ADP
ejpam-7052	67	17	a	a	PRON
ejpam-7052	67	18	is	be	AUX
ejpam-7052	67	19	called	call	VERB
ejpam-7052	67	20	the	the	DET
ejpam-7052	67	21	(	(	PUNCT
ejpam-7052	67	22	τ1	τ1	NOUN
ejpam-7052	67	23	,	,	PUNCT
ejpam-7052	67	24	τ2)p	τ2)p	ADJ
ejpam-7052	67	25	-	-	NOUN
ejpam-7052	67	26	interior	interior	ADJ
ejpam-7052	67	27	[	[	X
ejpam-7052	67	28	16	16	NUM
ejpam-7052	67	29	]	]	X
ejpam-7052	67	30	(	(	PUNCT
ejpam-7052	67	31	resp	resp	NOUN
ejpam-7052	67	32	.	.	PUNCT
ejpam-7052	68	1	(	(	PUNCT
ejpam-7052	68	2	τ1	τ1	NOUN
ejpam-7052	68	3	,	,	PUNCT
ejpam-7052	68	4	τ2)s	τ2)s	NOUN
ejpam-7052	68	5	-	-	ADJ
ejpam-7052	68	6	interior	interior	NOUN
ejpam-7052	68	7	[	[	X
ejpam-7052	68	8	14	14	NUM
ejpam-7052	68	9	]	]	PUNCT
ejpam-7052	68	10	,	,	PUNCT
ejpam-7052	68	11	α(τ1	α(τ1	NOUN
ejpam-7052	68	12	,	,	PUNCT
ejpam-7052	68	13	τ2)-interior	τ2)-interior	PROPN
ejpam-7052	68	14	[	[	X
ejpam-7052	68	15	17	17	NUM
ejpam-7052	68	16	]	]	PUNCT
ejpam-7052	68	17	)	)	PUNCT
ejpam-7052	68	18	of	of	ADP
ejpam-7052	68	19	a	a	PRON
ejpam-7052	68	20	and	and	CCONJ
ejpam-7052	68	21	is	be	AUX
ejpam-7052	68	22	denoted	denote	VERB
ejpam-7052	68	23	by	by	ADP
ejpam-7052	68	24	(	(	PUNCT
ejpam-7052	68	25	τ1	τ1	NOUN
ejpam-7052	68	26	,	,	PUNCT
ejpam-7052	68	27	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-7052	68	28	)	)	PUNCT
ejpam-7052	68	29	(	(	PUNCT
ejpam-7052	68	30	resp	resp	NOUN
ejpam-7052	68	31	.	.	PUNCT
ejpam-7052	69	1	(	(	PUNCT
ejpam-7052	69	2	τ1	τ1	NOUN
ejpam-7052	69	3	,	,	PUNCT
ejpam-7052	69	4	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-7052	69	5	)	)	PUNCT
ejpam-7052	69	6	,	,	PUNCT
ejpam-7052	69	7	α(τ1	α(τ1	NOUN
ejpam-7052	69	8	,	,	PUNCT
ejpam-7052	69	9	τ2)-int(a	τ2)-int(a	NOUN
ejpam-7052	69	10	)	)	PUNCT
ejpam-7052	69	11	)	)	PUNCT
ejpam-7052	69	12	.	.	PUNCT
ejpam-7052	70	1	lemma	lemma	PROPN
ejpam-7052	70	2	2	2	NUM
ejpam-7052	70	3	.	.	PUNCT
ejpam-7052	71	1	[	[	X
ejpam-7052	71	2	10	10	NUM
ejpam-7052	71	3	]	]	PUNCT
ejpam-7052	71	4	let	let	VERB
ejpam-7052	71	5	a	a	PRON
ejpam-7052	71	6	be	be	AUX
ejpam-7052	71	7	a	a	DET
ejpam-7052	71	8	subset	subset	NOUN
ejpam-7052	71	9	of	of	ADP
ejpam-7052	71	10	a	a	DET
ejpam-7052	71	11	bitopological	bitopological	ADJ
ejpam-7052	71	12	space	space	NOUN
ejpam-7052	71	13	(	(	PUNCT
ejpam-7052	71	14	x	x	NOUN
ejpam-7052	71	15	,	,	PUNCT
ejpam-7052	71	16	τ1	τ1	NOUN
ejpam-7052	71	17	,	,	PUNCT
ejpam-7052	71	18	τ2	τ2	NOUN
ejpam-7052	71	19	)	)	PUNCT
ejpam-7052	71	20	.	.	PUNCT
ejpam-7052	72	1	if	if	SCONJ
ejpam-7052	72	2	a	a	PRON
ejpam-7052	72	3	is	be	AUX
ejpam-7052	72	4	τ1τ2	τ1τ2	NOUN
ejpam-7052	72	5	-	-	ADJ
ejpam-7052	72	6	open	open	ADJ
ejpam-7052	72	7	in	in	ADP
ejpam-7052	72	8	x	x	NOUN
ejpam-7052	72	9	,	,	PUNCT
ejpam-7052	72	10	then	then	ADV
ejpam-7052	72	11	(	(	PUNCT
ejpam-7052	72	12	τ1	τ1	NOUN
ejpam-7052	72	13	,	,	PUNCT
ejpam-7052	72	14	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-7052	72	15	)	)	PUNCT
ejpam-7052	72	16	=	=	PUNCT
ejpam-7052	73	1	τ1τ2	τ1τ2	NOUN
ejpam-7052	73	2	-	-	NOUN
ejpam-7052	73	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7052	73	4	-	-	PUNCT
ejpam-7052	73	5	cl(a	cl(a	NUM
ejpam-7052	73	6	)	)	PUNCT
ejpam-7052	73	7	)	)	PUNCT
ejpam-7052	73	8	.	.	PUNCT
ejpam-7052	74	1	a	a	DET
ejpam-7052	74	2	subset	subset	NOUN
ejpam-7052	74	3	a	a	PRON
ejpam-7052	74	4	of	of	ADP
ejpam-7052	74	5	a	a	DET
ejpam-7052	74	6	bitopological	bitopological	ADJ
ejpam-7052	74	7	space	space	NOUN
ejpam-7052	74	8	(	(	PUNCT
ejpam-7052	74	9	x	x	NOUN
ejpam-7052	74	10	,	,	PUNCT
ejpam-7052	74	11	τ1	τ1	NOUN
ejpam-7052	74	12	,	,	PUNCT
ejpam-7052	74	13	τ2	τ2	NOUN
ejpam-7052	74	14	)	)	PUNCT
ejpam-7052	74	15	is	be	AUX
ejpam-7052	74	16	said	say	VERB
ejpam-7052	74	17	to	to	PART
ejpam-7052	74	18	be	be	AUX
ejpam-7052	74	19	τ1τ2	τ1τ2	NOUN
ejpam-7052	74	20	-	-	ADJ
ejpam-7052	74	21	δ	δ	NOUN
ejpam-7052	74	22	-	-	NOUN
ejpam-7052	74	23	open	open	ADJ
ejpam-7052	74	24	[	[	X
ejpam-7052	74	25	18	18	NUM
ejpam-7052	74	26	]	]	X
ejpam-7052	74	27	if	if	SCONJ
ejpam-7052	74	28	a	a	PRON
ejpam-7052	74	29	is	be	AUX
ejpam-7052	74	30	the	the	DET
ejpam-7052	74	31	union	union	NOUN
ejpam-7052	74	32	of	of	ADP
ejpam-7052	74	33	(	(	PUNCT
ejpam-7052	74	34	τ1	τ1	NOUN
ejpam-7052	74	35	,	,	PUNCT
ejpam-7052	74	36	τ2)r	τ2)r	ADJ
ejpam-7052	74	37	-	-	PUNCT
ejpam-7052	74	38	open	open	ADJ
ejpam-7052	74	39	sets	set	NOUN
ejpam-7052	74	40	of	of	ADP
ejpam-7052	74	41	x.	x.	NOUN
ejpam-7052	74	42	the	the	DET
ejpam-7052	74	43	complement	complement	NOUN
ejpam-7052	74	44	of	of	ADP
ejpam-7052	74	45	a	a	DET
ejpam-7052	74	46	τ1τ2	τ1τ2	ADJ
ejpam-7052	74	47	-	-	ADJ
ejpam-7052	74	48	δ	δ	NOUN
ejpam-7052	74	49	-	-	ADJ
ejpam-7052	74	50	open	open	ADJ
ejpam-7052	74	51	set	set	NOUN
ejpam-7052	74	52	is	be	AUX
ejpam-7052	74	53	called	call	VERB
ejpam-7052	74	54	τ1τ2	τ1τ2	NOUN
ejpam-7052	74	55	-	-	ADJ
ejpam-7052	74	56	δ	δ	NOUN
ejpam-7052	74	57	-	-	PUNCT
ejpam-7052	74	58	closed	closed	ADJ
ejpam-7052	74	59	[	[	X
ejpam-7052	74	60	18	18	NUM
ejpam-7052	74	61	]	]	PUNCT
ejpam-7052	74	62	.	.	PUNCT
ejpam-7052	75	1	the	the	DET
ejpam-7052	75	2	union	union	NOUN
ejpam-7052	75	3	of	of	ADP
ejpam-7052	75	4	all	all	DET
ejpam-7052	75	5	τ1τ2	τ1τ2	NOUN
ejpam-7052	75	6	-	-	ADJ
ejpam-7052	75	7	δ	δ	NOUN
ejpam-7052	75	8	-	-	ADJ
ejpam-7052	75	9	open	open	ADJ
ejpam-7052	75	10	sets	set	NOUN
ejpam-7052	75	11	of	of	ADP
ejpam-7052	75	12	x	x	PUNCT
ejpam-7052	75	13	contained	contain	VERB
ejpam-7052	75	14	in	in	ADP
ejpam-7052	75	15	a	a	PRON
ejpam-7052	75	16	is	be	AUX
ejpam-7052	75	17	called	call	VERB
ejpam-7052	75	18	the	the	DET
ejpam-7052	75	19	τ1τ2	τ1τ2	ADJ
ejpam-7052	75	20	-	-	ADJ
ejpam-7052	75	21	δ	δ	NOUN
ejpam-7052	75	22	-	-	NOUN
ejpam-7052	75	23	interior	interior	NOUN
ejpam-7052	75	24	[	[	X
ejpam-7052	75	25	18	18	NUM
ejpam-7052	75	26	]	]	PUNCT
ejpam-7052	75	27	of	of	ADP
ejpam-7052	75	28	a	a	PRON
ejpam-7052	75	29	and	and	CCONJ
ejpam-7052	75	30	is	be	AUX
ejpam-7052	75	31	denoted	denote	VERB
ejpam-7052	75	32	by	by	ADP
ejpam-7052	75	33	τ1τ2	τ1τ2	ADJ
ejpam-7052	75	34	-	-	ADJ
ejpam-7052	75	35	δ	δ	NOUN
ejpam-7052	75	36	-	-	PUNCT
ejpam-7052	75	37	int(a	int(a	PROPN
ejpam-7052	75	38	)	)	PUNCT
ejpam-7052	75	39	.	.	PUNCT
ejpam-7052	76	1	the	the	DET
ejpam-7052	76	2	intersection	intersection	NOUN
ejpam-7052	76	3	of	of	ADP
ejpam-7052	76	4	all	all	DET
ejpam-7052	76	5	τ1τ2δ	τ1τ2δ	NUM
ejpam-7052	76	6	-	-	PUNCT
ejpam-7052	76	7	closed	close	VERB
ejpam-7052	76	8	sets	set	NOUN
ejpam-7052	76	9	of	of	ADP
ejpam-7052	76	10	x	x	PUNCT
ejpam-7052	76	11	containing	contain	VERB
ejpam-7052	76	12	a	a	PRON
ejpam-7052	76	13	is	be	AUX
ejpam-7052	76	14	called	call	VERB
ejpam-7052	76	15	the	the	DET
ejpam-7052	76	16	τ1τ2	τ1τ2	ADJ
ejpam-7052	76	17	-	-	ADJ
ejpam-7052	76	18	δ	δ	NOUN
ejpam-7052	76	19	-	-	NOUN
ejpam-7052	76	20	closure	closure	NOUN
ejpam-7052	76	21	[	[	X
ejpam-7052	76	22	18	18	NUM
ejpam-7052	76	23	]	]	PUNCT
ejpam-7052	76	24	of	of	ADP
ejpam-7052	76	25	a	a	PRON
ejpam-7052	76	26	and	and	CCONJ
ejpam-7052	76	27	is	be	AUX
ejpam-7052	76	28	denoted	denote	VERB
ejpam-7052	76	29	by	by	ADP
ejpam-7052	76	30	τ1τ2	τ1τ2	ADJ
ejpam-7052	76	31	-	-	ADJ
ejpam-7052	76	32	δ	δ	NOUN
ejpam-7052	76	33	-	-	PUNCT
ejpam-7052	76	34	cl(a	cl(a	NUM
ejpam-7052	76	35	)	)	PUNCT
ejpam-7052	76	36	.	.	PUNCT
ejpam-7052	77	1	let	let	VERB
ejpam-7052	77	2	a	a	DET
ejpam-7052	77	3	be	be	AUX
ejpam-7052	77	4	a	a	DET
ejpam-7052	77	5	subset	subset	NOUN
ejpam-7052	77	6	of	of	ADP
ejpam-7052	77	7	a	a	DET
ejpam-7052	77	8	bitopological	bitopological	ADJ
ejpam-7052	77	9	space	space	NOUN
ejpam-7052	77	10	(	(	PUNCT
ejpam-7052	77	11	x	x	NOUN
ejpam-7052	77	12	,	,	PUNCT
ejpam-7052	77	13	τ1	τ1	NOUN
ejpam-7052	77	14	,	,	PUNCT
ejpam-7052	77	15	τ2	τ2	NOUN
ejpam-7052	77	16	)	)	PUNCT
ejpam-7052	77	17	.	.	PUNCT
ejpam-7052	78	1	a	a	DET
ejpam-7052	78	2	point	point	NOUN
ejpam-7052	78	3	x	x	X
ejpam-7052	78	4	∈	∈	NOUN
ejpam-7052	78	5	x	x	PUNCT
ejpam-7052	78	6	is	be	AUX
ejpam-7052	78	7	called	call	VERB
ejpam-7052	78	8	a	a	DET
ejpam-7052	78	9	(	(	PUNCT
ejpam-7052	78	10	τ1	τ1	NOUN
ejpam-7052	78	11	,	,	PUNCT
ejpam-7052	78	12	τ2)θ	τ2)θ	ADJ
ejpam-7052	78	13	-	-	PUNCT
ejpam-7052	78	14	cluster	cluster	NOUN
ejpam-7052	78	15	point	point	NOUN
ejpam-7052	78	16	[	[	X
ejpam-7052	78	17	13	13	NUM
ejpam-7052	78	18	]	]	PUNCT
ejpam-7052	78	19	of	of	ADP
ejpam-7052	78	20	a	a	DET
ejpam-7052	78	21	if	if	SCONJ
ejpam-7052	78	22	τ1τ2	τ1τ2	NOUN
ejpam-7052	78	23	-	-	NOUN
ejpam-7052	78	24	cl(u	cl(u	NOUN
ejpam-7052	78	25	)	)	PUNCT
ejpam-7052	78	26	∩	∩	NOUN
ejpam-7052	78	27	a	a	DET
ejpam-7052	78	28	̸=	̸=	PROPN
ejpam-7052	78	29	∅	∅	NOUN
ejpam-7052	78	30	for	for	ADP
ejpam-7052	78	31	every	every	DET
ejpam-7052	78	32	τ1τ2	τ1τ2	ADJ
ejpam-7052	78	33	-	-	ADJ
ejpam-7052	78	34	open	open	ADJ
ejpam-7052	78	35	set	set	NOUN
ejpam-7052	78	36	u	u	NOUN
ejpam-7052	78	37	containing	contain	VERB
ejpam-7052	78	38	x.	x.	NOUN
ejpam-7052	78	39	the	the	DET
ejpam-7052	78	40	set	set	NOUN
ejpam-7052	78	41	of	of	ADP
ejpam-7052	78	42	all	all	DET
ejpam-7052	78	43	(	(	PUNCT
ejpam-7052	78	44	τ1	τ1	NOUN
ejpam-7052	78	45	,	,	PUNCT
ejpam-7052	78	46	τ2)θ	τ2)θ	ADJ
ejpam-7052	78	47	-	-	PUNCT
ejpam-7052	78	48	cluster	cluster	NOUN
ejpam-7052	78	49	points	point	NOUN
ejpam-7052	78	50	of	of	ADP
ejpam-7052	78	51	a	a	PRON
ejpam-7052	78	52	is	be	AUX
ejpam-7052	78	53	called	call	VERB
ejpam-7052	78	54	the	the	DET
ejpam-7052	78	55	(	(	PUNCT
ejpam-7052	78	56	τ1	τ1	NOUN
ejpam-7052	78	57	,	,	PUNCT
ejpam-7052	78	58	τ2)θ	τ2)θ	ADJ
ejpam-7052	78	59	-	-	PUNCT
ejpam-7052	78	60	closure	closure	NOUN
ejpam-7052	78	61	[	[	X
ejpam-7052	78	62	13	13	NUM
ejpam-7052	78	63	]	]	PUNCT
ejpam-7052	78	64	of	of	ADP
ejpam-7052	78	65	a	a	PRON
ejpam-7052	78	66	and	and	CCONJ
ejpam-7052	78	67	is	be	AUX
ejpam-7052	78	68	denoted	denote	VERB
ejpam-7052	78	69	by	by	ADP
ejpam-7052	78	70	(	(	PUNCT
ejpam-7052	78	71	τ1	τ1	NOUN
ejpam-7052	78	72	,	,	PUNCT
ejpam-7052	78	73	τ2)θ	τ2)θ	NOUN
ejpam-7052	78	74	-	-	PUNCT
ejpam-7052	78	75	cl(a	cl(a	NUM
ejpam-7052	78	76	)	)	PUNCT
ejpam-7052	78	77	.	.	PUNCT
ejpam-7052	79	1	a	a	DET
ejpam-7052	79	2	subset	subset	NOUN
ejpam-7052	79	3	a	a	PRON
ejpam-7052	79	4	of	of	ADP
ejpam-7052	79	5	a	a	DET
ejpam-7052	79	6	bitopological	bitopological	ADJ
ejpam-7052	79	7	space	space	NOUN
ejpam-7052	79	8	(	(	PUNCT
ejpam-7052	79	9	x	x	NOUN
ejpam-7052	79	10	,	,	PUNCT
ejpam-7052	79	11	τ1	τ1	NOUN
ejpam-7052	79	12	,	,	PUNCT
ejpam-7052	79	13	τ2	τ2	NOUN
ejpam-7052	79	14	)	)	PUNCT
ejpam-7052	79	15	is	be	AUX
ejpam-7052	79	16	said	say	VERB
ejpam-7052	79	17	to	to	PART
ejpam-7052	79	18	be	be	AUX
ejpam-7052	79	19	(	(	PUNCT
ejpam-7052	79	20	τ1	τ1	NOUN
ejpam-7052	79	21	,	,	PUNCT
ejpam-7052	79	22	τ2)θ	τ2)θ	NOUN
ejpam-7052	79	23	-	-	PUNCT
ejpam-7052	79	24	closed	closed	ADJ
ejpam-7052	79	25	[	[	X
ejpam-7052	79	26	13	13	NUM
ejpam-7052	79	27	]	]	X
ejpam-7052	79	28	if	if	SCONJ
ejpam-7052	79	29	(	(	PUNCT
ejpam-7052	79	30	τ1	τ1	NOUN
ejpam-7052	79	31	,	,	PUNCT
ejpam-7052	79	32	τ2)θ	τ2)θ	NOUN
ejpam-7052	79	33	-	-	PUNCT
ejpam-7052	79	34	cl(a	cl(a	NUM
ejpam-7052	79	35	)	)	PUNCT
ejpam-7052	80	1	=	=	PUNCT
ejpam-7052	80	2	a.	a.	NOUN
ejpam-7052	80	3	the	the	DET
ejpam-7052	80	4	complement	complement	NOUN
ejpam-7052	80	5	of	of	ADP
ejpam-7052	80	6	a	a	DET
ejpam-7052	80	7	(	(	PUNCT
ejpam-7052	80	8	τ1	τ1	NOUN
ejpam-7052	80	9	,	,	PUNCT
ejpam-7052	80	10	τ2)θ	τ2)θ	ADJ
ejpam-7052	80	11	-	-	PUNCT
ejpam-7052	80	12	closed	close	VERB
ejpam-7052	80	13	set	set	NOUN
ejpam-7052	80	14	is	be	AUX
ejpam-7052	80	15	said	say	VERB
ejpam-7052	80	16	to	to	PART
ejpam-7052	80	17	be	be	AUX
ejpam-7052	80	18	(	(	PUNCT
ejpam-7052	80	19	τ1	τ1	NOUN
ejpam-7052	80	20	,	,	PUNCT
ejpam-7052	80	21	τ2)θ	τ2)θ	NOUN
ejpam-7052	80	22	-	-	PUNCT
ejpam-7052	80	23	open	open	ADJ
ejpam-7052	80	24	.	.	PUNCT
ejpam-7052	81	1	the	the	DET
ejpam-7052	81	2	union	union	NOUN
ejpam-7052	81	3	of	of	ADP
ejpam-7052	81	4	all	all	DET
ejpam-7052	81	5	(	(	PUNCT
ejpam-7052	81	6	τ1	τ1	NOUN
ejpam-7052	81	7	,	,	PUNCT
ejpam-7052	81	8	τ2)θ	τ2)θ	ADJ
ejpam-7052	81	9	-	-	PUNCT
ejpam-7052	81	10	open	open	ADJ
ejpam-7052	81	11	sets	set	NOUN
ejpam-7052	81	12	of	of	ADP
ejpam-7052	81	13	x	x	PUNCT
ejpam-7052	81	14	contained	contain	VERB
ejpam-7052	81	15	in	in	ADP
ejpam-7052	81	16	a	a	PRON
ejpam-7052	81	17	is	be	AUX
ejpam-7052	81	18	called	call	VERB
ejpam-7052	81	19	the	the	DET
ejpam-7052	81	20	(	(	PUNCT
ejpam-7052	81	21	τ1	τ1	NOUN
ejpam-7052	81	22	,	,	PUNCT
ejpam-7052	81	23	τ2)θ	τ2)θ	ADJ
ejpam-7052	81	24	-	-	PUNCT
ejpam-7052	81	25	interior	interior	NOUN
ejpam-7052	81	26	[	[	X
ejpam-7052	81	27	13	13	NUM
ejpam-7052	81	28	]	]	PUNCT
ejpam-7052	81	29	of	of	ADP
ejpam-7052	81	30	a	a	PRON
ejpam-7052	81	31	and	and	CCONJ
ejpam-7052	81	32	is	be	AUX
ejpam-7052	81	33	denoted	denote	VERB
ejpam-7052	81	34	by	by	ADP
ejpam-7052	81	35	(	(	PUNCT
ejpam-7052	81	36	τ1	τ1	NOUN
ejpam-7052	81	37	,	,	PUNCT
ejpam-7052	81	38	τ2)θ	τ2)θ	NOUN
ejpam-7052	81	39	-	-	PUNCT
ejpam-7052	81	40	int(a	int(a	NOUN
ejpam-7052	81	41	)	)	PUNCT
ejpam-7052	81	42	.	.	PUNCT
ejpam-7052	82	1	lemma	lemma	PROPN
ejpam-7052	82	2	3	3	X
ejpam-7052	82	3	.	.	PUNCT
ejpam-7052	83	1	[	[	X
ejpam-7052	83	2	13	13	NUM
ejpam-7052	83	3	]	]	PUNCT
ejpam-7052	83	4	for	for	ADP
ejpam-7052	83	5	a	a	DET
ejpam-7052	83	6	subset	subset	NOUN
ejpam-7052	83	7	a	a	PRON
ejpam-7052	83	8	of	of	ADP
ejpam-7052	83	9	a	a	DET
ejpam-7052	83	10	bitopological	bitopological	ADJ
ejpam-7052	83	11	space	space	NOUN
ejpam-7052	83	12	(	(	PUNCT
ejpam-7052	83	13	x	x	NOUN
ejpam-7052	83	14	,	,	PUNCT
ejpam-7052	83	15	τ1	τ1	NOUN
ejpam-7052	83	16	,	,	PUNCT
ejpam-7052	83	17	τ2	τ2	NOUN
ejpam-7052	83	18	)	)	PUNCT
ejpam-7052	83	19	,	,	PUNCT
ejpam-7052	83	20	the	the	DET
ejpam-7052	83	21	following	follow	VERB
ejpam-7052	83	22	properties	property	NOUN
ejpam-7052	83	23	hold	hold	VERB
ejpam-7052	83	24	:	:	PUNCT
ejpam-7052	83	25	(	(	PUNCT
ejpam-7052	83	26	1	1	X
ejpam-7052	83	27	)	)	PUNCT
ejpam-7052	83	28	if	if	SCONJ
ejpam-7052	83	29	a	a	PRON
ejpam-7052	83	30	is	be	AUX
ejpam-7052	83	31	τ1τ2	τ1τ2	NOUN
ejpam-7052	83	32	-	-	ADJ
ejpam-7052	83	33	open	open	ADJ
ejpam-7052	83	34	in	in	ADP
ejpam-7052	83	35	x	x	NOUN
ejpam-7052	83	36	,	,	PUNCT
ejpam-7052	83	37	then	then	ADV
ejpam-7052	83	38	τ1τ2	τ1τ2	NOUN
ejpam-7052	83	39	-	-	NUM
ejpam-7052	83	40	cl(a	cl(a	NUM
ejpam-7052	83	41	)	)	PUNCT
ejpam-7052	83	42	=	=	PUNCT
ejpam-7052	83	43	(	(	PUNCT
ejpam-7052	83	44	τ1	τ1	NOUN
ejpam-7052	83	45	,	,	PUNCT
ejpam-7052	83	46	τ2)θ	τ2)θ	NOUN
ejpam-7052	83	47	-	-	PUNCT
ejpam-7052	83	48	cl(a	cl(a	NUM
ejpam-7052	83	49	)	)	PUNCT
ejpam-7052	83	50	.	.	PUNCT
ejpam-7052	84	1	(	(	PUNCT
ejpam-7052	84	2	2	2	X
ejpam-7052	84	3	)	)	PUNCT
ejpam-7052	84	4	(	(	PUNCT
ejpam-7052	84	5	τ1	τ1	NOUN
ejpam-7052	84	6	,	,	PUNCT
ejpam-7052	84	7	τ2)θ	τ2)θ	NOUN
ejpam-7052	84	8	-	-	PUNCT
ejpam-7052	84	9	cl(a	cl(a	NUM
ejpam-7052	84	10	)	)	PUNCT
ejpam-7052	84	11	is	be	AUX
ejpam-7052	84	12	τ1τ2	τ1τ2	NOUN
ejpam-7052	84	13	-	-	ADJ
ejpam-7052	84	14	closed	closed	ADJ
ejpam-7052	84	15	in	in	ADP
ejpam-7052	84	16	x.	x.	NOUN
ejpam-7052	84	17	let	let	VERB
ejpam-7052	84	18	x	x	PRON
ejpam-7052	84	19	be	be	AUX
ejpam-7052	84	20	a	a	DET
ejpam-7052	84	21	nonempty	nonempty	ADJ
ejpam-7052	84	22	set	set	NOUN
ejpam-7052	84	23	,	,	PUNCT
ejpam-7052	84	24	and	and	CCONJ
ejpam-7052	84	25	denote	denote	VERB
ejpam-7052	84	26	p(x	p(x	PROPN
ejpam-7052	84	27	)	)	PUNCT
ejpam-7052	84	28	the	the	DET
ejpam-7052	84	29	power	power	NOUN
ejpam-7052	84	30	set	set	NOUN
ejpam-7052	84	31	of	of	ADP
ejpam-7052	84	32	x.	x.	NOUN
ejpam-7052	84	33	we	we	PRON
ejpam-7052	84	34	call	call	VERB
ejpam-7052	84	35	a	a	DET
ejpam-7052	84	36	class	class	NOUN
ejpam-7052	84	37	µ	µ	PRON
ejpam-7052	84	38	⊆	⊆	NUM
ejpam-7052	84	39	p(x	p(x	NOUN
ejpam-7052	84	40	)	)	PUNCT
ejpam-7052	84	41	a	a	DET
ejpam-7052	84	42	generalized	generalized	ADJ
ejpam-7052	84	43	topology	topology	NOUN
ejpam-7052	84	44	(	(	PUNCT
ejpam-7052	84	45	briefly	briefly	ADV
ejpam-7052	84	46	,	,	PUNCT
ejpam-7052	84	47	gt	gt	PROPN
ejpam-7052	84	48	)	)	PUNCT
ejpam-7052	84	49	if	if	SCONJ
ejpam-7052	84	50	∅	∅	NUM
ejpam-7052	84	51	∈	∈	PROPN
ejpam-7052	84	52	µ	µ	NOUN
ejpam-7052	84	53	,	,	PUNCT
ejpam-7052	84	54	and	and	CCONJ
ejpam-7052	84	55	an	an	DET
ejpam-7052	84	56	arbitrary	arbitrary	ADJ
ejpam-7052	84	57	union	union	NOUN
ejpam-7052	84	58	of	of	ADP
ejpam-7052	84	59	elements	element	NOUN
ejpam-7052	84	60	of	of	ADP
ejpam-7052	84	61	µ	µ	X
ejpam-7052	84	62	belongs	belong	VERB
ejpam-7052	84	63	to	to	ADP
ejpam-7052	84	64	µ	µ	PROPN
ejpam-7052	84	65	[	[	X
ejpam-7052	84	66	5	5	NUM
ejpam-7052	84	67	]	]	PUNCT
ejpam-7052	84	68	.	.	PUNCT
ejpam-7052	85	1	a	a	DET
ejpam-7052	85	2	set	set	NOUN
ejpam-7052	85	3	x	x	PUNCT
ejpam-7052	85	4	with	with	ADP
ejpam-7052	85	5	a	a	DET
ejpam-7052	85	6	gt	gt	PROPN
ejpam-7052	85	7	µ	µ	NOUN
ejpam-7052	85	8	on	on	ADP
ejpam-7052	85	9	it	it	PRON
ejpam-7052	85	10	is	be	AUX
ejpam-7052	85	11	said	say	VERB
ejpam-7052	85	12	to	to	PART
ejpam-7052	85	13	be	be	AUX
ejpam-7052	85	14	a	a	DET
ejpam-7052	85	15	generalized	generalized	ADJ
ejpam-7052	85	16	topological	topological	ADJ
ejpam-7052	85	17	space	space	NOUN
ejpam-7052	85	18	(	(	PUNCT
ejpam-7052	85	19	briefly	briefly	ADV
ejpam-7052	85	20	,	,	PUNCT
ejpam-7052	85	21	gts	gts	NOUN
ejpam-7052	85	22	)	)	PUNCT
ejpam-7052	85	23	and	and	CCONJ
ejpam-7052	85	24	is	be	AUX
ejpam-7052	85	25	denoted	denote	VERB
ejpam-7052	85	26	by	by	ADP
ejpam-7052	85	27	(	(	PUNCT
ejpam-7052	85	28	x,µ	x,µ	NOUN
ejpam-7052	85	29	)	)	PUNCT
ejpam-7052	85	30	.	.	PUNCT
ejpam-7052	86	1	for	for	ADP
ejpam-7052	86	2	a	a	DET
ejpam-7052	86	3	gts	gts	NOUN
ejpam-7052	86	4	(	(	PUNCT
ejpam-7052	86	5	x,µ	x,µ	NOUN
ejpam-7052	86	6	)	)	PUNCT
ejpam-7052	86	7	,	,	PUNCT
ejpam-7052	86	8	the	the	DET
ejpam-7052	86	9	elements	element	NOUN
ejpam-7052	86	10	of	of	ADP
ejpam-7052	86	11	µ	µ	NOUN
ejpam-7052	86	12	are	be	AUX
ejpam-7052	86	13	called	call	VERB
ejpam-7052	86	14	µ-open	µ-open	NOUN
ejpam-7052	86	15	sets	set	NOUN
ejpam-7052	86	16	and	and	CCONJ
ejpam-7052	86	17	the	the	DET
ejpam-7052	86	18	complements	complement	NOUN
ejpam-7052	86	19	of	of	ADP
ejpam-7052	86	20	µ-open	µ-open	NOUN
ejpam-7052	86	21	sets	set	NOUN
ejpam-7052	86	22	are	be	AUX
ejpam-7052	86	23	called	call	VERB
ejpam-7052	86	24	µ-closed	µ-close	VERB
ejpam-7052	86	25	sets	set	NOUN
ejpam-7052	86	26	.	.	PUNCT
ejpam-7052	87	1	for	for	ADP
ejpam-7052	87	2	a	a	DET
ejpam-7052	87	3	⊆	⊆	NUM
ejpam-7052	87	4	x	x	SYM
ejpam-7052	87	5	,	,	PUNCT
ejpam-7052	87	6	we	we	PRON
ejpam-7052	87	7	denote	denote	VERB
ejpam-7052	87	8	by	by	ADP
ejpam-7052	87	9	cµ(a	cµ(a	PROPN
ejpam-7052	87	10	)	)	PUNCT
ejpam-7052	87	11	the	the	DET
ejpam-7052	87	12	intersection	intersection	NOUN
ejpam-7052	87	13	of	of	ADP
ejpam-7052	87	14	all	all	DET
ejpam-7052	87	15	µ-closed	µ-close	VERB
ejpam-7052	87	16	sets	set	NOUN
ejpam-7052	87	17	containing	contain	VERB
ejpam-7052	87	18	a	a	PRON
ejpam-7052	87	19	and	and	CCONJ
ejpam-7052	87	20	by	by	ADP
ejpam-7052	87	21	iµ(a	iµ(a	PROPN
ejpam-7052	87	22	)	)	PUNCT
ejpam-7052	87	23	the	the	DET
ejpam-7052	87	24	union	union	NOUN
ejpam-7052	87	25	of	of	ADP
ejpam-7052	87	26	all	all	DET
ejpam-7052	87	27	µ-open	µ-open	NOUN
ejpam-7052	87	28	sets	set	NOUN
ejpam-7052	87	29	contained	contain	VERB
ejpam-7052	87	30	in	in	ADP
ejpam-7052	87	31	a.	a.	NOUN
ejpam-7052	87	32	then	then	ADV
ejpam-7052	87	33	,	,	PUNCT
ejpam-7052	87	34	we	we	PRON
ejpam-7052	87	35	have	have	VERB
ejpam-7052	87	36	iµ(iµ(a	iµ(iµ(a	ADJ
ejpam-7052	87	37	)	)	PUNCT
ejpam-7052	87	38	)	)	PUNCT
ejpam-7052	88	1	=	=	SYM
ejpam-7052	88	2	iµ(a	iµ(a	ADJ
ejpam-7052	88	3	)	)	PUNCT
ejpam-7052	88	4	,	,	PUNCT
ejpam-7052	88	5	cµ(cµ(a	cµ(cµ(a	PROPN
ejpam-7052	88	6	)	)	PUNCT
ejpam-7052	88	7	)	)	PUNCT
ejpam-7052	89	1	=	=	SYM
ejpam-7052	89	2	cµ(a	cµ(a	PROPN
ejpam-7052	89	3	)	)	PUNCT
ejpam-7052	89	4	,	,	PUNCT
ejpam-7052	89	5	p.	p.	NOUN
ejpam-7052	89	6	pue	pue	PROPN
ejpam-7052	89	7	-	-	PUNCT
ejpam-7052	89	8	on	on	ADP
ejpam-7052	89	9	,	,	PUNCT
ejpam-7052	89	10	a.	a.	PROPN
ejpam-7052	89	11	sama	sama	PROPN
ejpam-7052	89	12	-	-	PUNCT
ejpam-7052	89	13	ae	ae	PROPN
ejpam-7052	89	14	,	,	PUNCT
ejpam-7052	89	15	c.	c.	PROPN
ejpam-7052	89	16	boonpok	boonpok	PROPN
ejpam-7052	89	17	/	/	SYM
ejpam-7052	89	18	eur	eur	PROPN
ejpam-7052	89	19	.	.	PUNCT
ejpam-7052	90	1	j.	j.	PROPN
ejpam-7052	90	2	pure	pure	PROPN
ejpam-7052	90	3	appl	appl	PROPN
ejpam-7052	90	4	.	.	PROPN
ejpam-7052	90	5	math	math	PROPN
ejpam-7052	90	6	,	,	PUNCT
ejpam-7052	90	7	18	18	NUM
ejpam-7052	90	8	(	(	PUNCT
ejpam-7052	90	9	4	4	NUM
ejpam-7052	90	10	)	)	PUNCT
ejpam-7052	90	11	(	(	PUNCT
ejpam-7052	90	12	2025	2025	NUM
ejpam-7052	90	13	)	)	PUNCT
ejpam-7052	90	14	,	,	PUNCT
ejpam-7052	90	15	7052	7052	NUM
ejpam-7052	90	16	4	4	NUM
ejpam-7052	90	17	of	of	ADP
ejpam-7052	90	18	14	14	NUM
ejpam-7052	90	19	and	and	CCONJ
ejpam-7052	90	20	iµ(a	iµ(a	ADJ
ejpam-7052	90	21	)	)	PUNCT
ejpam-7052	90	22	=	=	SYM
ejpam-7052	91	1	x	x	SYM
ejpam-7052	91	2	−	−	PROPN
ejpam-7052	91	3	cµ(x	cµ(x	SYM
ejpam-7052	91	4	−a	−a	NOUN
ejpam-7052	91	5	)	)	PUNCT
ejpam-7052	91	6	.	.	PUNCT
ejpam-7052	92	1	according	accord	VERB
ejpam-7052	92	2	to	to	ADP
ejpam-7052	92	3	[	[	X
ejpam-7052	92	4	19	19	NUM
ejpam-7052	92	5	]	]	PUNCT
ejpam-7052	92	6	,	,	PUNCT
ejpam-7052	92	7	for	for	ADP
ejpam-7052	92	8	a	a	DET
ejpam-7052	92	9	⊆	⊆	NUM
ejpam-7052	92	10	x	x	SYM
ejpam-7052	92	11	and	and	CCONJ
ejpam-7052	92	12	x	x	SYM
ejpam-7052	92	13	∈	∈	NOUN
ejpam-7052	92	14	x	x	X
ejpam-7052	92	15	,	,	PUNCT
ejpam-7052	92	16	we	we	PRON
ejpam-7052	92	17	have	have	VERB
ejpam-7052	92	18	x	x	X
ejpam-7052	92	19	∈	∈	PROPN
ejpam-7052	92	20	cµ(a	cµ(a	NOUN
ejpam-7052	92	21	)	)	PUNCT
ejpam-7052	93	1	if	if	SCONJ
ejpam-7052	93	2	and	and	CCONJ
ejpam-7052	93	3	only	only	ADV
ejpam-7052	93	4	if	if	SCONJ
ejpam-7052	93	5	x	x	PROPN
ejpam-7052	93	6	∈	∈	PROPN
ejpam-7052	93	7	m	m	PROPN
ejpam-7052	93	8	∈	∈	NOUN
ejpam-7052	93	9	µ	µ	NOUN
ejpam-7052	93	10	implies	imply	VERB
ejpam-7052	93	11	m	m	VERB
ejpam-7052	93	12	∩a	∩a	PROPN
ejpam-7052	93	13	̸=	̸=	PROPN
ejpam-7052	93	14	∅.	∅.	ADV
ejpam-7052	93	15	by	by	ADP
ejpam-7052	93	16	a	a	DET
ejpam-7052	93	17	multifunction	multifunction	NOUN
ejpam-7052	93	18	f	f	NOUN
ejpam-7052	93	19	:	:	PUNCT
ejpam-7052	93	20	x	x	X
ejpam-7052	93	21	→	→	SYM
ejpam-7052	93	22	y	y	PROPN
ejpam-7052	93	23	,	,	PUNCT
ejpam-7052	93	24	we	we	PRON
ejpam-7052	93	25	mean	mean	VERB
ejpam-7052	93	26	a	a	DET
ejpam-7052	93	27	point	point	NOUN
ejpam-7052	93	28	-	-	PUNCT
ejpam-7052	93	29	to	to	ADP
ejpam-7052	93	30	-	-	PUNCT
ejpam-7052	93	31	set	set	VERB
ejpam-7052	93	32	correspondence	correspondence	NOUN
ejpam-7052	93	33	from	from	ADP
ejpam-7052	93	34	x	x	PUNCT
ejpam-7052	93	35	into	into	ADP
ejpam-7052	93	36	y	y	PROPN
ejpam-7052	93	37	,	,	PUNCT
ejpam-7052	93	38	and	and	CCONJ
ejpam-7052	93	39	we	we	PRON
ejpam-7052	93	40	always	always	ADV
ejpam-7052	93	41	assume	assume	VERB
ejpam-7052	93	42	that	that	SCONJ
ejpam-7052	93	43	f	f	PROPN
ejpam-7052	93	44	(	(	PUNCT
ejpam-7052	93	45	x	x	X
ejpam-7052	93	46	)	)	PUNCT
ejpam-7052	93	47	̸=	̸=	NOUN
ejpam-7052	93	48	∅	∅	NOUN
ejpam-7052	93	49	for	for	ADP
ejpam-7052	93	50	all	all	PRON
ejpam-7052	93	51	x	x	SYM
ejpam-7052	93	52	∈	∈	ADJ
ejpam-7052	93	53	x.	x.	NOUN
ejpam-7052	93	54	for	for	ADP
ejpam-7052	93	55	a	a	DET
ejpam-7052	93	56	multifunction	multifunction	NOUN
ejpam-7052	94	1	f	f	NOUN
ejpam-7052	94	2	:	:	PUNCT
ejpam-7052	94	3	x	x	X
ejpam-7052	94	4	→	→	SYM
ejpam-7052	94	5	y	y	PROPN
ejpam-7052	94	6	,	,	PUNCT
ejpam-7052	94	7	we	we	PRON
ejpam-7052	94	8	shall	shall	AUX
ejpam-7052	94	9	denote	denote	VERB
ejpam-7052	94	10	the	the	DET
ejpam-7052	94	11	upper	upper	ADJ
ejpam-7052	94	12	and	and	CCONJ
ejpam-7052	94	13	lower	low	ADJ
ejpam-7052	94	14	inverse	inverse	NOUN
ejpam-7052	94	15	of	of	ADP
ejpam-7052	94	16	a	a	DET
ejpam-7052	94	17	set	set	NOUN
ejpam-7052	94	18	b	b	PROPN
ejpam-7052	94	19	of	of	ADP
ejpam-7052	94	20	y	y	PROPN
ejpam-7052	94	21	by	by	ADP
ejpam-7052	94	22	f+(b	f+(b	NOUN
ejpam-7052	94	23	)	)	PUNCT
ejpam-7052	94	24	and	and	CCONJ
ejpam-7052	94	25	f−(b	f−(b	NOUN
ejpam-7052	94	26	)	)	PUNCT
ejpam-7052	94	27	,	,	PUNCT
ejpam-7052	94	28	respectively	respectively	ADV
ejpam-7052	94	29	,	,	PUNCT
ejpam-7052	94	30	that	that	ADV
ejpam-7052	94	31	is	is	ADV
ejpam-7052	94	32	,	,	PUNCT
ejpam-7052	94	33	f+(b	f+(b	NOUN
ejpam-7052	94	34	)	)	PUNCT
ejpam-7052	94	35	=	=	PRON
ejpam-7052	95	1	{	{	PUNCT
ejpam-7052	95	2	x	x	PUNCT
ejpam-7052	95	3	∈	∈	PROPN
ejpam-7052	95	4	x	x	INTJ
ejpam-7052	96	1	|	|	NOUN
ejpam-7052	96	2	f	f	X
ejpam-7052	96	3	(	(	PUNCT
ejpam-7052	96	4	x	x	NOUN
ejpam-7052	96	5	)	)	PUNCT
ejpam-7052	96	6	⊆	⊆	NUM
ejpam-7052	96	7	b	b	NOUN
ejpam-7052	96	8	}	}	PUNCT
ejpam-7052	96	9	and	and	CCONJ
ejpam-7052	96	10	f−(b	f−(b	PROPN
ejpam-7052	96	11	)	)	PUNCT
ejpam-7052	96	12	=	=	PRON
ejpam-7052	97	1	{	{	PUNCT
ejpam-7052	97	2	x	x	PUNCT
ejpam-7052	97	3	∈	∈	PROPN
ejpam-7052	97	4	x	x	INTJ
ejpam-7052	98	1	|	|	NOUN
ejpam-7052	98	2	f	f	X
ejpam-7052	98	3	(	(	PUNCT
ejpam-7052	98	4	x	x	NOUN
ejpam-7052	98	5	)	)	PUNCT
ejpam-7052	98	6	∩	∩	NOUN
ejpam-7052	98	7	b	b	PROPN
ejpam-7052	98	8	̸=	̸=	PROPN
ejpam-7052	98	9	∅	∅	NOUN
ejpam-7052	98	10	}	}	PUNCT
ejpam-7052	98	11	.	.	PUNCT
ejpam-7052	99	1	in	in	ADP
ejpam-7052	99	2	particular	particular	ADJ
ejpam-7052	99	3	,	,	PUNCT
ejpam-7052	99	4	f−(y	f−(y	NOUN
ejpam-7052	99	5	)	)	PUNCT
ejpam-7052	99	6	=	=	SYM
ejpam-7052	100	1	{	{	PUNCT
ejpam-7052	100	2	x	x	PUNCT
ejpam-7052	100	3	∈	∈	PROPN
ejpam-7052	100	4	x	x	INTJ
ejpam-7052	101	1	|	|	ADV
ejpam-7052	101	2	y	y	PROPN
ejpam-7052	101	3	∈	∈	PROPN
ejpam-7052	101	4	f	f	X
ejpam-7052	101	5	(	(	PUNCT
ejpam-7052	101	6	x	x	NOUN
ejpam-7052	101	7	)	)	PUNCT
ejpam-7052	101	8	}	}	PUNCT
ejpam-7052	101	9	for	for	ADP
ejpam-7052	101	10	each	each	DET
ejpam-7052	101	11	point	point	NOUN
ejpam-7052	101	12	y	y	PROPN
ejpam-7052	101	13	∈	∈	PROPN
ejpam-7052	101	14	y	y	PROPN
ejpam-7052	101	15	.	.	PUNCT
ejpam-7052	102	1	for	for	ADP
ejpam-7052	102	2	each	each	DET
ejpam-7052	102	3	a	a	DET
ejpam-7052	102	4	⊆	⊆	NUM
ejpam-7052	102	5	x	x	SYM
ejpam-7052	102	6	,	,	PUNCT
ejpam-7052	102	7	f	f	PROPN
ejpam-7052	102	8	(	(	PUNCT
ejpam-7052	102	9	a	a	NOUN
ejpam-7052	102	10	)	)	PUNCT
ejpam-7052	102	11	=	=	SYM
ejpam-7052	102	12	∪x∈af	∪x∈af	NOUN
ejpam-7052	102	13	(	(	PUNCT
ejpam-7052	102	14	x	x	NOUN
ejpam-7052	102	15	)	)	PUNCT
ejpam-7052	102	16	.	.	PUNCT
ejpam-7052	103	1	3	3	X
ejpam-7052	103	2	.	.	X
ejpam-7052	103	3	upper	upper	ADJ
ejpam-7052	103	4	and	and	CCONJ
ejpam-7052	103	5	lower	low	ADJ
ejpam-7052	103	6	almost	almost	ADV
ejpam-7052	103	7	µ(σ1	µ(σ1	NOUN
ejpam-7052	103	8	,	,	PUNCT
ejpam-7052	103	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	103	10	multifunctions	multifunction	NOUN
ejpam-7052	103	11	in	in	ADP
ejpam-7052	103	12	this	this	DET
ejpam-7052	103	13	section	section	NOUN
ejpam-7052	103	14	,	,	PUNCT
ejpam-7052	103	15	we	we	PRON
ejpam-7052	103	16	introduce	introduce	VERB
ejpam-7052	103	17	the	the	DET
ejpam-7052	103	18	concepts	concept	NOUN
ejpam-7052	103	19	of	of	ADP
ejpam-7052	103	20	upper	upper	ADJ
ejpam-7052	103	21	almost	almost	ADV
ejpam-7052	103	22	µ(σ1	µ(σ1	NOUN
ejpam-7052	103	23	,	,	PUNCT
ejpam-7052	103	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	103	25	multifunctions	multifunction	NOUN
ejpam-7052	103	26	and	and	CCONJ
ejpam-7052	103	27	lower	low	ADJ
ejpam-7052	103	28	almost	almost	ADV
ejpam-7052	103	29	µ(σ1	µ(σ1	NOUN
ejpam-7052	103	30	,	,	PUNCT
ejpam-7052	103	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	103	32	multifunctions	multifunction	NOUN
ejpam-7052	103	33	.	.	PUNCT
ejpam-7052	104	1	furthermore	furthermore	ADV
ejpam-7052	104	2	,	,	PUNCT
ejpam-7052	104	3	several	several	ADJ
ejpam-7052	104	4	characterizations	characterization	NOUN
ejpam-7052	104	5	of	of	ADP
ejpam-7052	104	6	upper	upper	ADJ
ejpam-7052	104	7	almost	almost	ADV
ejpam-7052	104	8	µ(σ1	µ(σ1	NOUN
ejpam-7052	104	9	,	,	PUNCT
ejpam-7052	104	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	104	11	multifunctions	multifunction	NOUN
ejpam-7052	104	12	and	and	CCONJ
ejpam-7052	104	13	lower	low	ADJ
ejpam-7052	104	14	almost	almost	ADV
ejpam-7052	104	15	µ(σ1	µ(σ1	NOUN
ejpam-7052	104	16	,	,	PUNCT
ejpam-7052	104	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	104	18	multifunctions	multifunction	NOUN
ejpam-7052	104	19	are	be	AUX
ejpam-7052	104	20	discussed	discuss	VERB
ejpam-7052	104	21	.	.	PUNCT
ejpam-7052	105	1	definition	definition	NOUN
ejpam-7052	105	2	1	1	NUM
ejpam-7052	105	3	.	.	PUNCT
ejpam-7052	106	1	a	a	DET
ejpam-7052	106	2	multifunction	multifunction	NOUN
ejpam-7052	106	3	f	f	NOUN
ejpam-7052	106	4	:	:	PUNCT
ejpam-7052	106	5	(	(	PUNCT
ejpam-7052	106	6	x,µ	x,µ	NOUN
ejpam-7052	106	7	)	)	PUNCT
ejpam-7052	106	8	→	→	SYM
ejpam-7052	106	9	(	(	PUNCT
ejpam-7052	106	10	y	y	PROPN
ejpam-7052	106	11	,	,	PUNCT
ejpam-7052	106	12	σ1	σ1	PROPN
ejpam-7052	106	13	,	,	PUNCT
ejpam-7052	106	14	σ2	σ2	PROPN
ejpam-7052	106	15	)	)	PUNCT
ejpam-7052	106	16	is	be	AUX
ejpam-7052	106	17	said	say	VERB
ejpam-7052	106	18	to	to	PART
ejpam-7052	106	19	be	be	AUX
ejpam-7052	106	20	upper	upper	ADJ
ejpam-7052	106	21	almost	almost	ADV
ejpam-7052	106	22	µ(σ1	µ(σ1	NOUN
ejpam-7052	106	23	,	,	PUNCT
ejpam-7052	106	24	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	106	25	at	at	ADP
ejpam-7052	106	26	a	a	DET
ejpam-7052	106	27	point	point	NOUN
ejpam-7052	106	28	x	x	SYM
ejpam-7052	106	29	∈	∈	NOUN
ejpam-7052	106	30	x	x	PUNCT
ejpam-7052	106	31	if	if	SCONJ
ejpam-7052	106	32	for	for	ADP
ejpam-7052	106	33	each	each	DET
ejpam-7052	106	34	σ1σ2	σ1σ2	VERB
ejpam-7052	106	35	-	-	ADJ
ejpam-7052	106	36	open	open	ADJ
ejpam-7052	106	37	set	set	NOUN
ejpam-7052	106	38	v	v	NOUN
ejpam-7052	106	39	of	of	ADP
ejpam-7052	106	40	y	y	PRON
ejpam-7052	106	41	such	such	ADJ
ejpam-7052	106	42	that	that	SCONJ
ejpam-7052	106	43	f	f	PROPN
ejpam-7052	106	44	(	(	PUNCT
ejpam-7052	106	45	x	x	X
ejpam-7052	106	46	)	)	PUNCT
ejpam-7052	106	47	⊆	⊆	NUM
ejpam-7052	106	48	v	v	NOUN
ejpam-7052	106	49	,	,	PUNCT
ejpam-7052	106	50	there	there	PRON
ejpam-7052	106	51	exists	exist	VERB
ejpam-7052	106	52	a	a	DET
ejpam-7052	106	53	µ-open	µ-open	NOUN
ejpam-7052	106	54	set	set	VERB
ejpam-7052	106	55	u	u	NOUN
ejpam-7052	106	56	of	of	ADP
ejpam-7052	106	57	x	x	PUNCT
ejpam-7052	106	58	containing	contain	VERB
ejpam-7052	106	59	x	x	PUNCT
ejpam-7052	106	60	such	such	ADJ
ejpam-7052	106	61	that	that	SCONJ
ejpam-7052	106	62	f	f	PROPN
ejpam-7052	106	63	(	(	PUNCT
ejpam-7052	106	64	u	u	NOUN
ejpam-7052	106	65	)	)	PUNCT
ejpam-7052	106	66	⊆	⊆	NUM
ejpam-7052	106	67	σ1σ2	σ1σ2	X
ejpam-7052	106	68	-	-	PUNCT
ejpam-7052	106	69	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	106	70	-	-	PUNCT
ejpam-7052	106	71	cl(v	cl(v	NOUN
ejpam-7052	106	72	)	)	PUNCT
ejpam-7052	106	73	)	)	PUNCT
ejpam-7052	106	74	.	.	PUNCT
ejpam-7052	107	1	a	a	DET
ejpam-7052	107	2	multifunction	multifunction	NOUN
ejpam-7052	107	3	f	f	NOUN
ejpam-7052	107	4	:	:	PUNCT
ejpam-7052	107	5	(	(	PUNCT
ejpam-7052	107	6	x,µ	x,µ	NOUN
ejpam-7052	107	7	)	)	PUNCT
ejpam-7052	107	8	→	→	SYM
ejpam-7052	107	9	(	(	PUNCT
ejpam-7052	107	10	y	y	PROPN
ejpam-7052	107	11	,	,	PUNCT
ejpam-7052	107	12	σ1	σ1	PROPN
ejpam-7052	107	13	,	,	PUNCT
ejpam-7052	107	14	σ2	σ2	PROPN
ejpam-7052	107	15	)	)	PUNCT
ejpam-7052	107	16	is	be	AUX
ejpam-7052	107	17	said	say	VERB
ejpam-7052	107	18	to	to	PART
ejpam-7052	107	19	be	be	AUX
ejpam-7052	107	20	upper	upper	ADJ
ejpam-7052	107	21	almost	almost	ADV
ejpam-7052	107	22	µ(σ1	µ(σ1	NOUN
ejpam-7052	107	23	,	,	PUNCT
ejpam-7052	107	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	107	25	if	if	SCONJ
ejpam-7052	107	26	f	f	PROPN
ejpam-7052	107	27	is	be	AUX
ejpam-7052	107	28	upper	upper	ADJ
ejpam-7052	107	29	almost	almost	ADV
ejpam-7052	107	30	µ(σ1	µ(σ1	NOUN
ejpam-7052	107	31	,	,	PUNCT
ejpam-7052	107	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	107	33	at	at	ADP
ejpam-7052	107	34	each	each	DET
ejpam-7052	107	35	point	point	NOUN
ejpam-7052	107	36	x	x	PUNCT
ejpam-7052	107	37	of	of	ADP
ejpam-7052	107	38	x.	x.	PROPN
ejpam-7052	107	39	theorem	theorem	VERB
ejpam-7052	107	40	1	1	NUM
ejpam-7052	107	41	.	.	X
ejpam-7052	107	42	for	for	ADP
ejpam-7052	107	43	a	a	DET
ejpam-7052	107	44	multifunction	multifunction	NOUN
ejpam-7052	107	45	f	f	NOUN
ejpam-7052	107	46	:	:	PUNCT
ejpam-7052	107	47	(	(	PUNCT
ejpam-7052	107	48	x,µ	x,µ	NOUN
ejpam-7052	107	49	)	)	PUNCT
ejpam-7052	107	50	→	→	SYM
ejpam-7052	107	51	(	(	PUNCT
ejpam-7052	107	52	y	y	PROPN
ejpam-7052	107	53	,	,	PUNCT
ejpam-7052	107	54	σ1	σ1	PROPN
ejpam-7052	107	55	,	,	PUNCT
ejpam-7052	107	56	σ2	σ2	NOUN
ejpam-7052	107	57	)	)	PUNCT
ejpam-7052	107	58	,	,	PUNCT
ejpam-7052	107	59	the	the	DET
ejpam-7052	107	60	following	follow	VERB
ejpam-7052	107	61	properties	property	NOUN
ejpam-7052	107	62	are	be	AUX
ejpam-7052	107	63	equivalent	equivalent	ADJ
ejpam-7052	107	64	:	:	PUNCT
ejpam-7052	107	65	(	(	PUNCT
ejpam-7052	107	66	1	1	X
ejpam-7052	107	67	)	)	PUNCT
ejpam-7052	107	68	f	f	PROPN
ejpam-7052	107	69	is	be	AUX
ejpam-7052	107	70	upper	upper	ADJ
ejpam-7052	107	71	almost	almost	ADV
ejpam-7052	107	72	µ(σ1	µ(σ1	NOUN
ejpam-7052	107	73	,	,	PUNCT
ejpam-7052	107	74	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	107	75	at	at	ADP
ejpam-7052	107	76	x	x	X
ejpam-7052	107	77	∈	∈	PROPN
ejpam-7052	107	78	x	x	X
ejpam-7052	107	79	;	;	PUNCT
ejpam-7052	108	1	(	(	PUNCT
ejpam-7052	108	2	2	2	X
ejpam-7052	108	3	)	)	PUNCT
ejpam-7052	108	4	x	x	SYM
ejpam-7052	108	5	∈	∈	NOUN
ejpam-7052	108	6	iµ(f	iµ(f	NUM
ejpam-7052	108	7	+	+	NOUN
ejpam-7052	108	8	(	(	PUNCT
ejpam-7052	108	9	σ1σ2	σ1σ2	NUM
ejpam-7052	108	10	-	-	PUNCT
ejpam-7052	108	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	108	12	-	-	PUNCT
ejpam-7052	108	13	cl(v	cl(v	NOUN
ejpam-7052	108	14	)	)	PUNCT
ejpam-7052	108	15	)	)	PUNCT
ejpam-7052	108	16	)	)	PUNCT
ejpam-7052	108	17	)	)	PUNCT
ejpam-7052	108	18	for	for	ADP
ejpam-7052	108	19	every	every	DET
ejpam-7052	108	20	σ1σ2	σ1σ2	NOUN
ejpam-7052	108	21	-	-	ADJ
ejpam-7052	108	22	open	open	ADJ
ejpam-7052	108	23	set	set	NOUN
ejpam-7052	108	24	v	v	NOUN
ejpam-7052	108	25	of	of	ADP
ejpam-7052	108	26	y	y	PROPN
ejpam-7052	108	27	containing	contain	VERB
ejpam-7052	108	28	f	f	PROPN
ejpam-7052	108	29	(	(	PUNCT
ejpam-7052	108	30	x	x	NOUN
ejpam-7052	108	31	)	)	PUNCT
ejpam-7052	108	32	;	;	PUNCT
ejpam-7052	108	33	(	(	PUNCT
ejpam-7052	108	34	3	3	X
ejpam-7052	108	35	)	)	PUNCT
ejpam-7052	108	36	x	x	SYM
ejpam-7052	108	37	∈	∈	NOUN
ejpam-7052	108	38	iµ(f	iµ(f	NUM
ejpam-7052	108	39	+	+	NOUN
ejpam-7052	108	40	(	(	PUNCT
ejpam-7052	108	41	(	(	PUNCT
ejpam-7052	108	42	σ1	σ1	PROPN
ejpam-7052	108	43	,	,	PUNCT
ejpam-7052	108	44	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7052	108	45	)	)	PUNCT
ejpam-7052	108	46	)	)	PUNCT
ejpam-7052	108	47	)	)	PUNCT
ejpam-7052	108	48	for	for	ADP
ejpam-7052	108	49	every	every	DET
ejpam-7052	108	50	σ1σ2	σ1σ2	NOUN
ejpam-7052	108	51	-	-	ADJ
ejpam-7052	108	52	open	open	ADJ
ejpam-7052	108	53	set	set	NOUN
ejpam-7052	108	54	v	v	NOUN
ejpam-7052	108	55	of	of	ADP
ejpam-7052	108	56	y	y	PROPN
ejpam-7052	108	57	containing	contain	VERB
ejpam-7052	108	58	f	f	PROPN
ejpam-7052	108	59	(	(	PUNCT
ejpam-7052	108	60	x	x	NOUN
ejpam-7052	108	61	)	)	PUNCT
ejpam-7052	108	62	;	;	PUNCT
ejpam-7052	108	63	(	(	PUNCT
ejpam-7052	108	64	4	4	X
ejpam-7052	108	65	)	)	PUNCT
ejpam-7052	108	66	x	x	SYM
ejpam-7052	108	67	∈	∈	NOUN
ejpam-7052	108	68	iµ(f	iµ(f	NUM
ejpam-7052	109	1	+	+	NOUN
ejpam-7052	110	1	(	(	PUNCT
ejpam-7052	110	2	v	v	NOUN
ejpam-7052	110	3	)	)	PUNCT
ejpam-7052	110	4	)	)	PUNCT
ejpam-7052	111	1	for	for	SCONJ
ejpam-7052	111	2	every	every	DET
ejpam-7052	111	3	(	(	PUNCT
ejpam-7052	111	4	σ1	σ1	PROPN
ejpam-7052	111	5	,	,	PUNCT
ejpam-7052	111	6	σ2)r	σ2)r	NOUN
ejpam-7052	111	7	-	-	PUNCT
ejpam-7052	111	8	open	open	ADJ
ejpam-7052	111	9	set	set	VERB
ejpam-7052	111	10	v	v	NOUN
ejpam-7052	111	11	of	of	ADP
ejpam-7052	111	12	y	y	PROPN
ejpam-7052	111	13	containing	contain	VERB
ejpam-7052	111	14	f	f	PROPN
ejpam-7052	111	15	(	(	PUNCT
ejpam-7052	111	16	x	x	NOUN
ejpam-7052	111	17	)	)	PUNCT
ejpam-7052	111	18	;	;	PUNCT
ejpam-7052	111	19	(	(	PUNCT
ejpam-7052	111	20	5	5	X
ejpam-7052	111	21	)	)	PUNCT
ejpam-7052	111	22	for	for	ADP
ejpam-7052	111	23	each	each	DET
ejpam-7052	111	24	(	(	PUNCT
ejpam-7052	111	25	σ1	σ1	PROPN
ejpam-7052	111	26	,	,	PUNCT
ejpam-7052	111	27	σ2)r	σ2)r	NOUN
ejpam-7052	111	28	-	-	PUNCT
ejpam-7052	111	29	open	open	ADJ
ejpam-7052	111	30	set	set	VERB
ejpam-7052	111	31	v	v	NOUN
ejpam-7052	111	32	of	of	ADP
ejpam-7052	111	33	y	y	PROPN
ejpam-7052	111	34	containing	contain	VERB
ejpam-7052	111	35	f	f	PROPN
ejpam-7052	111	36	(	(	PUNCT
ejpam-7052	111	37	x	x	NOUN
ejpam-7052	111	38	)	)	PUNCT
ejpam-7052	111	39	,	,	PUNCT
ejpam-7052	111	40	there	there	PRON
ejpam-7052	111	41	exists	exist	VERB
ejpam-7052	111	42	a	a	DET
ejpam-7052	111	43	µ-open	µ-open	NOUN
ejpam-7052	111	44	set	set	VERB
ejpam-7052	111	45	u	u	NOUN
ejpam-7052	111	46	of	of	ADP
ejpam-7052	111	47	x	x	PUNCT
ejpam-7052	111	48	containing	contain	VERB
ejpam-7052	111	49	x	x	PUNCT
ejpam-7052	111	50	such	such	ADJ
ejpam-7052	111	51	that	that	SCONJ
ejpam-7052	111	52	f	f	PROPN
ejpam-7052	111	53	(	(	PUNCT
ejpam-7052	111	54	u	u	NOUN
ejpam-7052	111	55	)	)	PUNCT
ejpam-7052	111	56	⊆	⊆	NUM
ejpam-7052	111	57	v	v	NOUN
ejpam-7052	111	58	.	.	PUNCT
ejpam-7052	112	1	proof	proof	NOUN
ejpam-7052	112	2	.	.	PUNCT
ejpam-7052	113	1	(	(	PUNCT
ejpam-7052	113	2	1	1	X
ejpam-7052	113	3	)	)	PUNCT
ejpam-7052	113	4	⇒	⇒	NOUN
ejpam-7052	113	5	(	(	PUNCT
ejpam-7052	113	6	2	2	NUM
ejpam-7052	113	7	):	):	PUNCT
ejpam-7052	113	8	let	let	VERB
ejpam-7052	113	9	v	v	PART
ejpam-7052	113	10	be	be	AUX
ejpam-7052	113	11	any	any	DET
ejpam-7052	113	12	σ1σ2	σ1σ2	NOUN
ejpam-7052	113	13	-	-	ADJ
ejpam-7052	113	14	open	open	ADJ
ejpam-7052	113	15	set	set	NOUN
ejpam-7052	113	16	of	of	ADP
ejpam-7052	113	17	y	y	PROPN
ejpam-7052	113	18	containing	contain	VERB
ejpam-7052	113	19	f	f	PROPN
ejpam-7052	113	20	(	(	PUNCT
ejpam-7052	113	21	x	x	NOUN
ejpam-7052	113	22	)	)	PUNCT
ejpam-7052	113	23	.	.	PUNCT
ejpam-7052	114	1	thus	thus	ADV
ejpam-7052	114	2	by	by	ADP
ejpam-7052	114	3	(	(	PUNCT
ejpam-7052	114	4	1	1	NUM
ejpam-7052	114	5	)	)	PUNCT
ejpam-7052	114	6	,	,	PUNCT
ejpam-7052	114	7	there	there	PRON
ejpam-7052	114	8	exists	exist	VERB
ejpam-7052	114	9	a	a	DET
ejpam-7052	114	10	µ-open	µ-open	NOUN
ejpam-7052	114	11	set	set	VERB
ejpam-7052	114	12	u	u	NOUN
ejpam-7052	114	13	of	of	ADP
ejpam-7052	114	14	x	x	PUNCT
ejpam-7052	114	15	containing	contain	VERB
ejpam-7052	114	16	x	x	PUNCT
ejpam-7052	114	17	such	such	ADJ
ejpam-7052	114	18	that	that	SCONJ
ejpam-7052	114	19	f	f	PROPN
ejpam-7052	114	20	(	(	PUNCT
ejpam-7052	114	21	u	u	NOUN
ejpam-7052	114	22	)	)	PUNCT
ejpam-7052	114	23	⊆	⊆	NUM
ejpam-7052	114	24	σ1σ2	σ1σ2	X
ejpam-7052	114	25	-	-	PUNCT
ejpam-7052	114	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	114	27	-	-	PUNCT
ejpam-7052	114	28	cl(v	cl(v	NOUN
ejpam-7052	114	29	)	)	PUNCT
ejpam-7052	114	30	)	)	PUNCT
ejpam-7052	114	31	.	.	PUNCT
ejpam-7052	115	1	therefore	therefore	ADV
ejpam-7052	115	2	,	,	PUNCT
ejpam-7052	115	3	x	x	PUNCT
ejpam-7052	115	4	∈	∈	PROPN
ejpam-7052	115	5	u	u	NOUN
ejpam-7052	115	6	⊆	⊆	NUM
ejpam-7052	115	7	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	115	8	-	-	PUNCT
ejpam-7052	115	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	115	10	-	-	PUNCT
ejpam-7052	115	11	cl(v	cl(v	NOUN
ejpam-7052	115	12	)	)	PUNCT
ejpam-7052	115	13	)	)	PUNCT
ejpam-7052	115	14	)	)	PUNCT
ejpam-7052	115	15	and	and	CCONJ
ejpam-7052	115	16	so	so	ADV
ejpam-7052	115	17	x	x	SYM
ejpam-7052	115	18	∈	∈	NOUN
ejpam-7052	115	19	iµ(f	iµ(f	NUM
ejpam-7052	116	1	+	+	NOUN
ejpam-7052	116	2	(	(	PUNCT
ejpam-7052	116	3	σ1σ2	σ1σ2	NUM
ejpam-7052	116	4	-	-	PUNCT
ejpam-7052	116	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	116	6	-	-	PUNCT
ejpam-7052	116	7	cl(v	cl(v	NOUN
ejpam-7052	116	8	)	)	PUNCT
ejpam-7052	116	9	)	)	PUNCT
ejpam-7052	116	10	)	)	PUNCT
ejpam-7052	116	11	)	)	PUNCT
ejpam-7052	116	12	.	.	PUNCT
ejpam-7052	117	1	(	(	PUNCT
ejpam-7052	117	2	2	2	X
ejpam-7052	117	3	)	)	PUNCT
ejpam-7052	117	4	⇒	⇒	NOUN
ejpam-7052	117	5	(	(	PUNCT
ejpam-7052	117	6	3	3	NUM
ejpam-7052	117	7	):	):	PUNCT
ejpam-7052	117	8	this	this	PRON
ejpam-7052	117	9	follows	follow	VERB
ejpam-7052	117	10	from	from	ADP
ejpam-7052	117	11	lemma	lemma	PROPN
ejpam-7052	117	12	2	2	NUM
ejpam-7052	117	13	.	.	PUNCT
ejpam-7052	117	14	(	(	PUNCT
ejpam-7052	117	15	3	3	X
ejpam-7052	117	16	)	)	PUNCT
ejpam-7052	117	17	⇒	⇒	NOUN
ejpam-7052	117	18	(	(	PUNCT
ejpam-7052	117	19	4	4	NUM
ejpam-7052	117	20	):	):	PUNCT
ejpam-7052	117	21	let	let	VERB
ejpam-7052	117	22	v	v	PART
ejpam-7052	117	23	be	be	AUX
ejpam-7052	117	24	any	any	DET
ejpam-7052	117	25	σ1σ2	σ1σ2	NOUN
ejpam-7052	117	26	-	-	ADJ
ejpam-7052	117	27	open	open	ADJ
ejpam-7052	117	28	set	set	NOUN
ejpam-7052	117	29	of	of	ADP
ejpam-7052	117	30	y	y	PROPN
ejpam-7052	117	31	containing	contain	VERB
ejpam-7052	117	32	f	f	PROPN
ejpam-7052	117	33	(	(	PUNCT
ejpam-7052	117	34	x	x	NOUN
ejpam-7052	117	35	)	)	PUNCT
ejpam-7052	117	36	.	.	PUNCT
ejpam-7052	118	1	it	it	PRON
ejpam-7052	118	2	follows	follow	VERB
ejpam-7052	118	3	from	from	ADP
ejpam-7052	118	4	lemma	lemma	PROPN
ejpam-7052	118	5	2	2	NUM
ejpam-7052	118	6	that	that	PRON
ejpam-7052	118	7	v	v	NOUN
ejpam-7052	118	8	=	=	SYM
ejpam-7052	118	9	σ1σ2	σ1σ2	NOUN
ejpam-7052	118	10	-	-	PUNCT
ejpam-7052	118	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	118	12	-	-	PUNCT
ejpam-7052	118	13	cl(v	cl(v	NOUN
ejpam-7052	118	14	)	)	PUNCT
ejpam-7052	118	15	)	)	PUNCT
ejpam-7052	119	1	=	=	SYM
ejpam-7052	119	2	(	(	PUNCT
ejpam-7052	119	3	σ1	σ1	PROPN
ejpam-7052	119	4	,	,	PUNCT
ejpam-7052	119	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7052	119	6	)	)	PUNCT
ejpam-7052	119	7	.	.	PUNCT
ejpam-7052	120	1	(	(	PUNCT
ejpam-7052	120	2	4	4	X
ejpam-7052	120	3	)	)	PUNCT
ejpam-7052	120	4	⇒	⇒	NOUN
ejpam-7052	120	5	(	(	PUNCT
ejpam-7052	120	6	5	5	NUM
ejpam-7052	120	7	):	):	PUNCT
ejpam-7052	120	8	let	let	VERB
ejpam-7052	120	9	v	v	PART
ejpam-7052	120	10	be	be	AUX
ejpam-7052	120	11	any	any	DET
ejpam-7052	120	12	(	(	PUNCT
ejpam-7052	120	13	σ1	σ1	NOUN
ejpam-7052	120	14	,	,	PUNCT
ejpam-7052	120	15	σ2)r	σ2)r	NOUN
ejpam-7052	120	16	-	-	PUNCT
ejpam-7052	120	17	open	open	ADJ
ejpam-7052	120	18	set	set	NOUN
ejpam-7052	120	19	of	of	ADP
ejpam-7052	120	20	y	y	PROPN
ejpam-7052	120	21	containing	contain	VERB
ejpam-7052	120	22	f	f	PROPN
ejpam-7052	120	23	(	(	PUNCT
ejpam-7052	120	24	x	x	NOUN
ejpam-7052	120	25	)	)	PUNCT
ejpam-7052	120	26	.	.	PUNCT
ejpam-7052	121	1	then	then	ADV
ejpam-7052	121	2	by	by	ADP
ejpam-7052	121	3	(	(	PUNCT
ejpam-7052	121	4	4	4	NUM
ejpam-7052	121	5	)	)	PUNCT
ejpam-7052	121	6	,	,	PUNCT
ejpam-7052	121	7	we	we	PRON
ejpam-7052	121	8	have	have	VERB
ejpam-7052	121	9	x	x	PART
ejpam-7052	121	10	∈	∈	NOUN
ejpam-7052	121	11	iµ(f	iµ(f	NUM
ejpam-7052	121	12	+	+	NOUN
ejpam-7052	121	13	(	(	PUNCT
ejpam-7052	121	14	v	v	NOUN
ejpam-7052	121	15	)	)	PUNCT
ejpam-7052	121	16	)	)	PUNCT
ejpam-7052	121	17	and	and	CCONJ
ejpam-7052	121	18	there	there	PRON
ejpam-7052	121	19	exists	exist	VERB
ejpam-7052	121	20	a	a	DET
ejpam-7052	121	21	µ-open	µ-open	NOUN
ejpam-7052	121	22	set	set	VERB
ejpam-7052	121	23	u	u	NOUN
ejpam-7052	121	24	of	of	ADP
ejpam-7052	121	25	x	x	PUNCT
ejpam-7052	121	26	containing	contain	VERB
ejpam-7052	121	27	x	x	PUNCT
ejpam-7052	121	28	such	such	ADJ
ejpam-7052	121	29	that	that	SCONJ
ejpam-7052	121	30	x	x	SYM
ejpam-7052	121	31	∈	∈	NUM
ejpam-7052	121	32	u	u	NOUN
ejpam-7052	121	33	⊆	⊆	NUM
ejpam-7052	121	34	f+(v	f+(v	NOUN
ejpam-7052	121	35	)	)	PUNCT
ejpam-7052	121	36	;	;	PUNCT
ejpam-7052	121	37	hence	hence	ADV
ejpam-7052	121	38	f	f	PROPN
ejpam-7052	121	39	(	(	PUNCT
ejpam-7052	121	40	u	u	NOUN
ejpam-7052	121	41	)	)	PUNCT
ejpam-7052	121	42	⊆	⊆	NUM
ejpam-7052	121	43	v	v	NOUN
ejpam-7052	121	44	.	.	PUNCT
ejpam-7052	122	1	p.	p.	NOUN
ejpam-7052	122	2	pue	pue	NOUN
ejpam-7052	122	3	-	-	PUNCT
ejpam-7052	122	4	on	on	ADP
ejpam-7052	122	5	,	,	PUNCT
ejpam-7052	122	6	a.	a.	PROPN
ejpam-7052	122	7	sama	sama	PROPN
ejpam-7052	122	8	-	-	PUNCT
ejpam-7052	122	9	ae	ae	PROPN
ejpam-7052	122	10	,	,	PUNCT
ejpam-7052	122	11	c.	c.	PROPN
ejpam-7052	122	12	boonpok	boonpok	PROPN
ejpam-7052	122	13	/	/	SYM
ejpam-7052	122	14	eur	eur	PROPN
ejpam-7052	122	15	.	.	PUNCT
ejpam-7052	123	1	j.	j.	PROPN
ejpam-7052	123	2	pure	pure	PROPN
ejpam-7052	123	3	appl	appl	PROPN
ejpam-7052	123	4	.	.	PROPN
ejpam-7052	123	5	math	math	PROPN
ejpam-7052	123	6	,	,	PUNCT
ejpam-7052	123	7	18	18	NUM
ejpam-7052	123	8	(	(	PUNCT
ejpam-7052	123	9	4	4	NUM
ejpam-7052	123	10	)	)	PUNCT
ejpam-7052	123	11	(	(	PUNCT
ejpam-7052	123	12	2025	2025	NUM
ejpam-7052	123	13	)	)	PUNCT
ejpam-7052	123	14	,	,	PUNCT
ejpam-7052	123	15	7052	7052	NUM
ejpam-7052	123	16	5	5	NUM
ejpam-7052	123	17	of	of	ADP
ejpam-7052	123	18	14	14	NUM
ejpam-7052	123	19	(	(	PUNCT
ejpam-7052	123	20	5	5	NUM
ejpam-7052	123	21	)	)	PUNCT
ejpam-7052	123	22	⇒	⇒	NOUN
ejpam-7052	123	23	(	(	PUNCT
ejpam-7052	123	24	1	1	NUM
ejpam-7052	123	25	):	):	PUNCT
ejpam-7052	123	26	let	let	VERB
ejpam-7052	123	27	v	v	PART
ejpam-7052	123	28	be	be	AUX
ejpam-7052	123	29	any	any	DET
ejpam-7052	123	30	σ1σ2	σ1σ2	NOUN
ejpam-7052	123	31	-	-	ADJ
ejpam-7052	123	32	open	open	ADJ
ejpam-7052	123	33	set	set	NOUN
ejpam-7052	123	34	of	of	ADP
ejpam-7052	123	35	y	y	PROPN
ejpam-7052	123	36	containing	contain	VERB
ejpam-7052	123	37	f	f	PROPN
ejpam-7052	123	38	(	(	PUNCT
ejpam-7052	123	39	x	x	NOUN
ejpam-7052	123	40	)	)	PUNCT
ejpam-7052	123	41	.	.	PUNCT
ejpam-7052	124	1	since	since	SCONJ
ejpam-7052	124	2	σ1σ2	σ1σ2	NOUN
ejpam-7052	124	3	-	-	PUNCT
ejpam-7052	124	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	124	5	-	-	PUNCT
ejpam-7052	124	6	cl(v	cl(v	NOUN
ejpam-7052	124	7	)	)	PUNCT
ejpam-7052	124	8	)	)	PUNCT
ejpam-7052	124	9	is	be	AUX
ejpam-7052	124	10	(	(	PUNCT
ejpam-7052	124	11	σ1	σ1	NOUN
ejpam-7052	124	12	,	,	PUNCT
ejpam-7052	124	13	σ2)r	σ2)r	NOUN
ejpam-7052	124	14	-	-	PUNCT
ejpam-7052	124	15	open	open	ADJ
ejpam-7052	124	16	,	,	PUNCT
ejpam-7052	124	17	there	there	PRON
ejpam-7052	124	18	exists	exist	VERB
ejpam-7052	124	19	a	a	DET
ejpam-7052	124	20	µ-open	µ-open	NOUN
ejpam-7052	124	21	set	set	VERB
ejpam-7052	124	22	u	u	NOUN
ejpam-7052	124	23	of	of	ADP
ejpam-7052	124	24	x	x	PUNCT
ejpam-7052	124	25	containing	contain	VERB
ejpam-7052	124	26	x	x	PUNCT
ejpam-7052	124	27	such	such	ADJ
ejpam-7052	124	28	that	that	SCONJ
ejpam-7052	124	29	f	f	PROPN
ejpam-7052	124	30	(	(	PUNCT
ejpam-7052	124	31	u	u	NOUN
ejpam-7052	124	32	)	)	PUNCT
ejpam-7052	124	33	⊆	⊆	NUM
ejpam-7052	124	34	σ1σ2	σ1σ2	X
ejpam-7052	124	35	-	-	PUNCT
ejpam-7052	124	36	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	124	37	-	-	PUNCT
ejpam-7052	124	38	cl(v	cl(v	NOUN
ejpam-7052	124	39	)	)	PUNCT
ejpam-7052	124	40	)	)	PUNCT
ejpam-7052	124	41	.	.	PUNCT
ejpam-7052	125	1	this	this	PRON
ejpam-7052	125	2	shows	show	VERB
ejpam-7052	125	3	that	that	SCONJ
ejpam-7052	125	4	f	f	PROPN
ejpam-7052	125	5	is	be	AUX
ejpam-7052	125	6	upper	upper	ADJ
ejpam-7052	125	7	almost	almost	ADV
ejpam-7052	125	8	µ(σ1	µ(σ1	NOUN
ejpam-7052	125	9	,	,	PUNCT
ejpam-7052	125	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	125	11	at	at	ADP
ejpam-7052	125	12	x	x	SYM
ejpam-7052	125	13	∈	∈	PROPN
ejpam-7052	125	14	x.	x.	NOUN
ejpam-7052	125	15	definition	definition	NOUN
ejpam-7052	125	16	2	2	NUM
ejpam-7052	125	17	.	.	PUNCT
ejpam-7052	125	18	a	a	DET
ejpam-7052	125	19	multifunction	multifunction	NOUN
ejpam-7052	126	1	f	f	NOUN
ejpam-7052	126	2	:	:	PUNCT
ejpam-7052	126	3	(	(	PUNCT
ejpam-7052	126	4	x,µ	x,µ	NOUN
ejpam-7052	126	5	)	)	PUNCT
ejpam-7052	126	6	→	→	SYM
ejpam-7052	126	7	(	(	PUNCT
ejpam-7052	126	8	y	y	PROPN
ejpam-7052	126	9	,	,	PUNCT
ejpam-7052	126	10	σ1	σ1	PROPN
ejpam-7052	126	11	,	,	PUNCT
ejpam-7052	126	12	σ2	σ2	PROPN
ejpam-7052	126	13	)	)	PUNCT
ejpam-7052	126	14	is	be	AUX
ejpam-7052	126	15	called	call	VERB
ejpam-7052	126	16	lower	low	ADJ
ejpam-7052	126	17	almost	almost	ADV
ejpam-7052	126	18	µ(σ1	µ(σ1	NOUN
ejpam-7052	126	19	,	,	PUNCT
ejpam-7052	126	20	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	126	21	at	at	ADP
ejpam-7052	126	22	a	a	DET
ejpam-7052	126	23	point	point	NOUN
ejpam-7052	126	24	x	x	SYM
ejpam-7052	126	25	∈	∈	NOUN
ejpam-7052	126	26	x	x	PUNCT
ejpam-7052	126	27	if	if	SCONJ
ejpam-7052	126	28	for	for	ADP
ejpam-7052	126	29	each	each	DET
ejpam-7052	126	30	σ1σ2	σ1σ2	VERB
ejpam-7052	126	31	-	-	ADJ
ejpam-7052	126	32	open	open	ADJ
ejpam-7052	126	33	set	set	NOUN
ejpam-7052	126	34	v	v	NOUN
ejpam-7052	126	35	of	of	ADP
ejpam-7052	126	36	y	y	PRON
ejpam-7052	126	37	such	such	ADJ
ejpam-7052	126	38	that	that	PRON
ejpam-7052	126	39	v	v	ADP
ejpam-7052	126	40	∩	∩	ADJ
ejpam-7052	126	41	f	f	X
ejpam-7052	126	42	(	(	PUNCT
ejpam-7052	126	43	x	x	X
ejpam-7052	126	44	)	)	PUNCT
ejpam-7052	126	45	̸=	̸=	NOUN
ejpam-7052	126	46	∅	∅	NOUN
ejpam-7052	126	47	,	,	PUNCT
ejpam-7052	126	48	there	there	PRON
ejpam-7052	126	49	exists	exist	VERB
ejpam-7052	126	50	a	a	DET
ejpam-7052	126	51	µ-open	µ-open	NOUN
ejpam-7052	126	52	set	set	VERB
ejpam-7052	126	53	u	u	NOUN
ejpam-7052	126	54	of	of	ADP
ejpam-7052	126	55	x	x	PUNCT
ejpam-7052	126	56	containing	contain	VERB
ejpam-7052	126	57	x	x	PUNCT
ejpam-7052	126	58	such	such	ADJ
ejpam-7052	126	59	that	that	SCONJ
ejpam-7052	126	60	σ1σ2	σ1σ2	ADV
ejpam-7052	126	61	-	-	PUNCT
ejpam-7052	126	62	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	126	63	-	-	PUNCT
ejpam-7052	126	64	cl(v	cl(v	NOUN
ejpam-7052	126	65	)	)	PUNCT
ejpam-7052	126	66	)	)	PUNCT
ejpam-7052	127	1	∩	∩	PROPN
ejpam-7052	127	2	f	f	X
ejpam-7052	127	3	(	(	PUNCT
ejpam-7052	127	4	z	z	NOUN
ejpam-7052	127	5	)	)	PUNCT
ejpam-7052	127	6	̸=	̸=	NOUN
ejpam-7052	127	7	∅	∅	NOUN
ejpam-7052	127	8	for	for	ADP
ejpam-7052	127	9	every	every	DET
ejpam-7052	127	10	z	z	NOUN
ejpam-7052	127	11	∈	∈	PROPN
ejpam-7052	127	12	u	u	NOUN
ejpam-7052	127	13	.	.	PUNCT
ejpam-7052	128	1	a	a	DET
ejpam-7052	128	2	multifunction	multifunction	NOUN
ejpam-7052	128	3	f	f	NOUN
ejpam-7052	128	4	:	:	PUNCT
ejpam-7052	128	5	(	(	PUNCT
ejpam-7052	128	6	x,µ	x,µ	NOUN
ejpam-7052	128	7	)	)	PUNCT
ejpam-7052	128	8	→	→	SYM
ejpam-7052	128	9	(	(	PUNCT
ejpam-7052	128	10	y	y	PROPN
ejpam-7052	128	11	,	,	PUNCT
ejpam-7052	128	12	σ1	σ1	PROPN
ejpam-7052	128	13	,	,	PUNCT
ejpam-7052	128	14	σ2	σ2	PROPN
ejpam-7052	128	15	)	)	PUNCT
ejpam-7052	128	16	is	be	AUX
ejpam-7052	128	17	called	call	VERB
ejpam-7052	128	18	lower	low	ADJ
ejpam-7052	128	19	almost	almost	ADV
ejpam-7052	128	20	µ(σ1	µ(σ1	NOUN
ejpam-7052	128	21	,	,	PUNCT
ejpam-7052	128	22	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	128	23	if	if	SCONJ
ejpam-7052	128	24	f	f	PROPN
ejpam-7052	128	25	is	be	AUX
ejpam-7052	128	26	lower	low	ADJ
ejpam-7052	128	27	almost	almost	ADV
ejpam-7052	128	28	µ(σ1	µ(σ1	NOUN
ejpam-7052	128	29	,	,	PUNCT
ejpam-7052	128	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	128	31	at	at	ADP
ejpam-7052	128	32	each	each	DET
ejpam-7052	128	33	point	point	NOUN
ejpam-7052	128	34	x	x	PUNCT
ejpam-7052	128	35	of	of	ADP
ejpam-7052	128	36	x.	x.	PROPN
ejpam-7052	128	37	theorem	theorem	VERB
ejpam-7052	128	38	2	2	NUM
ejpam-7052	128	39	.	.	X
ejpam-7052	128	40	for	for	ADP
ejpam-7052	128	41	a	a	DET
ejpam-7052	128	42	multifunction	multifunction	NOUN
ejpam-7052	128	43	f	f	NOUN
ejpam-7052	128	44	:	:	PUNCT
ejpam-7052	128	45	(	(	PUNCT
ejpam-7052	128	46	x,µ	x,µ	NOUN
ejpam-7052	128	47	)	)	PUNCT
ejpam-7052	128	48	→	→	SYM
ejpam-7052	128	49	(	(	PUNCT
ejpam-7052	128	50	y	y	PROPN
ejpam-7052	128	51	,	,	PUNCT
ejpam-7052	128	52	σ1	σ1	PROPN
ejpam-7052	128	53	,	,	PUNCT
ejpam-7052	128	54	σ2	σ2	NOUN
ejpam-7052	128	55	)	)	PUNCT
ejpam-7052	128	56	,	,	PUNCT
ejpam-7052	128	57	the	the	DET
ejpam-7052	128	58	following	follow	VERB
ejpam-7052	128	59	properties	property	NOUN
ejpam-7052	128	60	are	be	AUX
ejpam-7052	128	61	equivalent	equivalent	ADJ
ejpam-7052	128	62	:	:	PUNCT
ejpam-7052	128	63	(	(	PUNCT
ejpam-7052	128	64	1	1	X
ejpam-7052	128	65	)	)	PUNCT
ejpam-7052	128	66	f	f	PROPN
ejpam-7052	128	67	is	be	AUX
ejpam-7052	128	68	lower	low	ADJ
ejpam-7052	128	69	almost	almost	ADV
ejpam-7052	128	70	µ(σ1	µ(σ1	NOUN
ejpam-7052	128	71	,	,	PUNCT
ejpam-7052	128	72	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	128	73	at	at	ADP
ejpam-7052	128	74	x	x	X
ejpam-7052	128	75	∈	∈	PROPN
ejpam-7052	128	76	x	x	X
ejpam-7052	128	77	;	;	PUNCT
ejpam-7052	128	78	(	(	PUNCT
ejpam-7052	128	79	2	2	X
ejpam-7052	128	80	)	)	PUNCT
ejpam-7052	128	81	x	x	SYM
ejpam-7052	128	82	∈	∈	PROPN
ejpam-7052	128	83	iµ(f	iµ(f	NUM
ejpam-7052	128	84	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	128	85	-	-	PUNCT
ejpam-7052	128	86	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	128	87	-	-	PUNCT
ejpam-7052	128	88	cl(v	cl(v	NOUN
ejpam-7052	128	89	)	)	PUNCT
ejpam-7052	128	90	)	)	PUNCT
ejpam-7052	128	91	)	)	PUNCT
ejpam-7052	128	92	)	)	PUNCT
ejpam-7052	128	93	for	for	ADP
ejpam-7052	128	94	every	every	DET
ejpam-7052	128	95	σ1σ2	σ1σ2	NOUN
ejpam-7052	128	96	-	-	ADJ
ejpam-7052	128	97	open	open	ADJ
ejpam-7052	128	98	set	set	NOUN
ejpam-7052	128	99	v	v	NOUN
ejpam-7052	128	100	of	of	ADP
ejpam-7052	128	101	y	y	PRON
ejpam-7052	128	102	such	such	ADJ
ejpam-7052	128	103	that	that	PRON
ejpam-7052	128	104	v	v	ADP
ejpam-7052	128	105	∩	∩	ADJ
ejpam-7052	128	106	f	f	X
ejpam-7052	128	107	(	(	PUNCT
ejpam-7052	128	108	x	x	X
ejpam-7052	128	109	)	)	PUNCT
ejpam-7052	128	110	̸=	̸=	NOUN
ejpam-7052	128	111	∅	∅	NOUN
ejpam-7052	128	112	;	;	PUNCT
ejpam-7052	128	113	(	(	PUNCT
ejpam-7052	128	114	3	3	X
ejpam-7052	128	115	)	)	PUNCT
ejpam-7052	128	116	x	x	SYM
ejpam-7052	128	117	∈	∈	NOUN
ejpam-7052	128	118	iµ(f	iµ(f	NOUN
ejpam-7052	128	119	−((σ1	−((σ1	NOUN
ejpam-7052	128	120	,	,	PUNCT
ejpam-7052	128	121	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-7052	128	122	)	)	PUNCT
ejpam-7052	128	123	)	)	PUNCT
ejpam-7052	128	124	)	)	PUNCT
ejpam-7052	128	125	for	for	ADP
ejpam-7052	128	126	every	every	DET
ejpam-7052	128	127	σ1σ2	σ1σ2	NOUN
ejpam-7052	128	128	-	-	ADJ
ejpam-7052	128	129	open	open	ADJ
ejpam-7052	128	130	set	set	NOUN
ejpam-7052	128	131	v	v	NOUN
ejpam-7052	128	132	of	of	ADP
ejpam-7052	128	133	y	y	PRON
ejpam-7052	129	1	such	such	ADJ
ejpam-7052	129	2	that	that	PRON
ejpam-7052	129	3	v	v	NOUN
ejpam-7052	129	4	∩f	∩f	NOUN
ejpam-7052	129	5	(	(	PUNCT
ejpam-7052	129	6	x	x	X
ejpam-7052	129	7	)	)	PUNCT
ejpam-7052	129	8	̸=	̸=	NOUN
ejpam-7052	129	9	∅	∅	NOUN
ejpam-7052	129	10	;	;	PUNCT
ejpam-7052	129	11	(	(	PUNCT
ejpam-7052	129	12	4	4	X
ejpam-7052	129	13	)	)	PUNCT
ejpam-7052	129	14	x	x	SYM
ejpam-7052	129	15	∈	∈	NOUN
ejpam-7052	129	16	iµ(f	iµ(f	NOUN
ejpam-7052	129	17	−(v	−(v	NOUN
ejpam-7052	129	18	)	)	PUNCT
ejpam-7052	129	19	)	)	PUNCT
ejpam-7052	129	20	for	for	ADP
ejpam-7052	129	21	every	every	DET
ejpam-7052	129	22	(	(	PUNCT
ejpam-7052	129	23	σ1	σ1	PROPN
ejpam-7052	129	24	,	,	PUNCT
ejpam-7052	129	25	σ2)r	σ2)r	NOUN
ejpam-7052	129	26	-	-	PUNCT
ejpam-7052	129	27	open	open	ADJ
ejpam-7052	129	28	set	set	VERB
ejpam-7052	129	29	v	v	NOUN
ejpam-7052	129	30	of	of	ADP
ejpam-7052	129	31	y	y	PRON
ejpam-7052	129	32	such	such	ADJ
ejpam-7052	129	33	that	that	PRON
ejpam-7052	129	34	v	v	ADP
ejpam-7052	129	35	∩	∩	ADJ
ejpam-7052	129	36	f	f	X
ejpam-7052	129	37	(	(	PUNCT
ejpam-7052	129	38	x	x	X
ejpam-7052	129	39	)	)	PUNCT
ejpam-7052	129	40	̸=	̸=	NOUN
ejpam-7052	129	41	∅	∅	NOUN
ejpam-7052	129	42	;	;	PUNCT
ejpam-7052	129	43	(	(	PUNCT
ejpam-7052	129	44	5	5	X
ejpam-7052	129	45	)	)	PUNCT
ejpam-7052	129	46	for	for	ADP
ejpam-7052	129	47	each	each	DET
ejpam-7052	129	48	(	(	PUNCT
ejpam-7052	129	49	σ1	σ1	PROPN
ejpam-7052	129	50	,	,	PUNCT
ejpam-7052	129	51	σ2)r	σ2)r	NOUN
ejpam-7052	129	52	-	-	PUNCT
ejpam-7052	129	53	open	open	ADJ
ejpam-7052	129	54	set	set	VERB
ejpam-7052	129	55	v	v	NOUN
ejpam-7052	129	56	of	of	ADP
ejpam-7052	129	57	y	y	PRON
ejpam-7052	129	58	such	such	ADJ
ejpam-7052	129	59	that	that	PRON
ejpam-7052	129	60	v	v	NOUN
ejpam-7052	129	61	∩f	∩f	NOUN
ejpam-7052	129	62	(	(	PUNCT
ejpam-7052	129	63	x	x	X
ejpam-7052	129	64	)	)	PUNCT
ejpam-7052	129	65	̸=	̸=	NOUN
ejpam-7052	129	66	∅	∅	NOUN
ejpam-7052	129	67	,	,	PUNCT
ejpam-7052	129	68	there	there	PRON
ejpam-7052	129	69	exists	exist	VERB
ejpam-7052	129	70	a	a	DET
ejpam-7052	129	71	µ-open	µ-open	NOUN
ejpam-7052	129	72	set	set	VERB
ejpam-7052	129	73	u	u	NOUN
ejpam-7052	129	74	of	of	ADP
ejpam-7052	129	75	x	x	PUNCT
ejpam-7052	129	76	containing	contain	VERB
ejpam-7052	129	77	x	x	PUNCT
ejpam-7052	129	78	such	such	ADJ
ejpam-7052	129	79	that	that	SCONJ
ejpam-7052	129	80	u	u	NOUN
ejpam-7052	129	81	⊆	⊆	NUM
ejpam-7052	129	82	f−(v	f−(v	NOUN
ejpam-7052	129	83	)	)	PUNCT
ejpam-7052	129	84	.	.	PUNCT
ejpam-7052	130	1	proof	proof	NOUN
ejpam-7052	130	2	.	.	PUNCT
ejpam-7052	131	1	the	the	DET
ejpam-7052	131	2	proof	proof	NOUN
ejpam-7052	131	3	is	be	AUX
ejpam-7052	131	4	similar	similar	ADJ
ejpam-7052	131	5	to	to	ADP
ejpam-7052	131	6	that	that	PRON
ejpam-7052	131	7	of	of	ADP
ejpam-7052	131	8	theorem	theorem	ADJ
ejpam-7052	131	9	1	1	NUM
ejpam-7052	131	10	.	.	PUNCT
ejpam-7052	131	11	theorem	theorem	NOUN
ejpam-7052	131	12	3	3	NUM
ejpam-7052	131	13	.	.	X
ejpam-7052	131	14	for	for	ADP
ejpam-7052	131	15	a	a	DET
ejpam-7052	131	16	multifunction	multifunction	NOUN
ejpam-7052	132	1	f	f	NOUN
ejpam-7052	132	2	:	:	PUNCT
ejpam-7052	132	3	(	(	PUNCT
ejpam-7052	132	4	x,µ	x,µ	NOUN
ejpam-7052	132	5	)	)	PUNCT
ejpam-7052	132	6	→	→	SYM
ejpam-7052	132	7	(	(	PUNCT
ejpam-7052	132	8	y	y	PROPN
ejpam-7052	132	9	,	,	PUNCT
ejpam-7052	132	10	σ1	σ1	PROPN
ejpam-7052	132	11	,	,	PUNCT
ejpam-7052	132	12	σ2	σ2	NOUN
ejpam-7052	132	13	)	)	PUNCT
ejpam-7052	132	14	,	,	PUNCT
ejpam-7052	132	15	the	the	DET
ejpam-7052	132	16	following	follow	VERB
ejpam-7052	132	17	properties	property	NOUN
ejpam-7052	132	18	are	be	AUX
ejpam-7052	132	19	equivalent	equivalent	ADJ
ejpam-7052	132	20	:	:	PUNCT
ejpam-7052	132	21	(	(	PUNCT
ejpam-7052	132	22	1	1	X
ejpam-7052	132	23	)	)	PUNCT
ejpam-7052	132	24	f	f	PROPN
ejpam-7052	132	25	is	be	AUX
ejpam-7052	132	26	upper	upper	ADJ
ejpam-7052	132	27	almost	almost	ADV
ejpam-7052	132	28	µ(σ1	µ(σ1	NOUN
ejpam-7052	132	29	,	,	PUNCT
ejpam-7052	132	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	132	31	;	;	PUNCT
ejpam-7052	132	32	(	(	PUNCT
ejpam-7052	132	33	2	2	NUM
ejpam-7052	132	34	)	)	PUNCT
ejpam-7052	132	35	f+(v	f+(v	NOUN
ejpam-7052	132	36	)	)	PUNCT
ejpam-7052	133	1	⊆	⊆	NUM
ejpam-7052	133	2	iµ(f	iµ(f	NUM
ejpam-7052	133	3	+	+	NOUN
ejpam-7052	133	4	(	(	PUNCT
ejpam-7052	133	5	σ1σ2	σ1σ2	NUM
ejpam-7052	133	6	-	-	PUNCT
ejpam-7052	133	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	133	8	-	-	PUNCT
ejpam-7052	133	9	cl(v	cl(v	NOUN
ejpam-7052	133	10	)	)	PUNCT
ejpam-7052	133	11	)	)	PUNCT
ejpam-7052	133	12	)	)	PUNCT
ejpam-7052	133	13	)	)	PUNCT
ejpam-7052	134	1	for	for	ADP
ejpam-7052	134	2	every	every	DET
ejpam-7052	134	3	σ1σ2	σ1σ2	NOUN
ejpam-7052	134	4	-	-	ADJ
ejpam-7052	134	5	open	open	ADJ
ejpam-7052	134	6	set	set	NOUN
ejpam-7052	134	7	v	v	NOUN
ejpam-7052	134	8	of	of	ADP
ejpam-7052	134	9	y	y	PROPN
ejpam-7052	134	10	;	;	PUNCT
ejpam-7052	134	11	(	(	PUNCT
ejpam-7052	134	12	3	3	X
ejpam-7052	134	13	)	)	PUNCT
ejpam-7052	134	14	cµ(f	cµ(f	NOUN
ejpam-7052	134	15	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	134	16	-	-	PUNCT
ejpam-7052	134	17	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	134	18	-	-	PUNCT
ejpam-7052	134	19	int(k	int(k	NOUN
ejpam-7052	134	20	)	)	PUNCT
ejpam-7052	134	21	)	)	PUNCT
ejpam-7052	134	22	)	)	PUNCT
ejpam-7052	134	23	)	)	PUNCT
ejpam-7052	135	1	⊆	⊆	X
ejpam-7052	135	2	f−(k	f−(k	PROPN
ejpam-7052	135	3	)	)	PUNCT
ejpam-7052	135	4	for	for	ADP
ejpam-7052	135	5	every	every	DET
ejpam-7052	135	6	σ1σ2	σ1σ2	NUM
ejpam-7052	135	7	-	-	PUNCT
ejpam-7052	135	8	closed	closed	ADJ
ejpam-7052	135	9	set	set	NOUN
ejpam-7052	135	10	k	k	PROPN
ejpam-7052	135	11	of	of	ADP
ejpam-7052	135	12	y	y	PROPN
ejpam-7052	135	13	;	;	PUNCT
ejpam-7052	135	14	(	(	PUNCT
ejpam-7052	135	15	4	4	X
ejpam-7052	135	16	)	)	PUNCT
ejpam-7052	135	17	cµ(f	cµ(f	NOUN
ejpam-7052	135	18	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	135	19	-	-	PUNCT
ejpam-7052	135	20	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	135	21	-	-	PUNCT
ejpam-7052	135	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	135	23	-	-	PUNCT
ejpam-7052	135	24	cl(b	cl(b	NOUN
ejpam-7052	135	25	)	)	PUNCT
ejpam-7052	135	26	)	)	PUNCT
ejpam-7052	135	27	)	)	PUNCT
ejpam-7052	135	28	)	)	PUNCT
ejpam-7052	135	29	)	)	PUNCT
ejpam-7052	136	1	⊆	⊆	X
ejpam-7052	136	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	136	3	-	-	PUNCT
ejpam-7052	136	4	cl(b	cl(b	NOUN
ejpam-7052	136	5	)	)	PUNCT
ejpam-7052	136	6	)	)	PUNCT
ejpam-7052	137	1	for	for	ADP
ejpam-7052	137	2	every	every	DET
ejpam-7052	137	3	subset	subset	NOUN
ejpam-7052	137	4	b	b	PROPN
ejpam-7052	137	5	of	of	ADP
ejpam-7052	137	6	y	y	PROPN
ejpam-7052	137	7	;	;	PUNCT
ejpam-7052	137	8	(	(	PUNCT
ejpam-7052	137	9	5	5	X
ejpam-7052	137	10	)	)	PUNCT
ejpam-7052	137	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	137	12	-	-	PUNCT
ejpam-7052	137	13	int(b	int(b	NOUN
ejpam-7052	137	14	)	)	PUNCT
ejpam-7052	137	15	)	)	PUNCT
ejpam-7052	138	1	⊆	⊆	NUM
ejpam-7052	138	2	iµ(f	iµ(f	NUM
ejpam-7052	138	3	+	+	NOUN
ejpam-7052	138	4	(	(	PUNCT
ejpam-7052	138	5	σ1σ2	σ1σ2	NUM
ejpam-7052	138	6	-	-	PUNCT
ejpam-7052	138	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-7052	138	8	-	-	PUNCT
ejpam-7052	138	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	138	10	-	-	PUNCT
ejpam-7052	138	11	int(b	int(b	NOUN
ejpam-7052	138	12	)	)	PUNCT
ejpam-7052	138	13	)	)	PUNCT
ejpam-7052	138	14	)	)	PUNCT
ejpam-7052	138	15	)	)	PUNCT
ejpam-7052	138	16	)	)	PUNCT
ejpam-7052	139	1	for	for	ADP
ejpam-7052	139	2	every	every	DET
ejpam-7052	139	3	subset	subset	NOUN
ejpam-7052	139	4	b	b	PROPN
ejpam-7052	139	5	of	of	ADP
ejpam-7052	139	6	y	y	PROPN
ejpam-7052	139	7	;	;	PUNCT
ejpam-7052	139	8	(	(	PUNCT
ejpam-7052	139	9	6	6	NUM
ejpam-7052	139	10	)	)	PUNCT
ejpam-7052	139	11	f+(v	f+(v	NOUN
ejpam-7052	139	12	)	)	PUNCT
ejpam-7052	139	13	is	be	AUX
ejpam-7052	139	14	µ-open	µ-open	NOUN
ejpam-7052	139	15	in	in	ADP
ejpam-7052	139	16	x	x	PUNCT
ejpam-7052	139	17	for	for	ADP
ejpam-7052	139	18	every	every	DET
ejpam-7052	139	19	(	(	PUNCT
ejpam-7052	139	20	σ1	σ1	PROPN
ejpam-7052	139	21	,	,	PUNCT
ejpam-7052	139	22	σ2)r	σ2)r	NOUN
ejpam-7052	139	23	-	-	PUNCT
ejpam-7052	139	24	open	open	ADJ
ejpam-7052	139	25	set	set	VERB
ejpam-7052	139	26	v	v	NOUN
ejpam-7052	139	27	of	of	ADP
ejpam-7052	139	28	y	y	PROPN
ejpam-7052	139	29	;	;	PUNCT
ejpam-7052	139	30	(	(	PUNCT
ejpam-7052	139	31	7	7	X
ejpam-7052	139	32	)	)	PUNCT
ejpam-7052	139	33	f−(k	f−(k	PROPN
ejpam-7052	139	34	)	)	PUNCT
ejpam-7052	139	35	is	be	AUX
ejpam-7052	139	36	µ-closed	µ-close	VERB
ejpam-7052	139	37	in	in	ADP
ejpam-7052	139	38	x	x	PUNCT
ejpam-7052	139	39	for	for	ADP
ejpam-7052	139	40	every	every	DET
ejpam-7052	139	41	(	(	PUNCT
ejpam-7052	139	42	σ1	σ1	PROPN
ejpam-7052	139	43	,	,	PUNCT
ejpam-7052	139	44	σ2)r	σ2)r	NOUN
ejpam-7052	139	45	-	-	PUNCT
ejpam-7052	139	46	closed	close	VERB
ejpam-7052	139	47	set	set	ADJ
ejpam-7052	139	48	k	k	PROPN
ejpam-7052	139	49	of	of	ADP
ejpam-7052	139	50	y	y	PROPN
ejpam-7052	139	51	.	.	PUNCT
ejpam-7052	140	1	p.	p.	NOUN
ejpam-7052	140	2	pue	pue	NOUN
ejpam-7052	140	3	-	-	PUNCT
ejpam-7052	140	4	on	on	ADP
ejpam-7052	140	5	,	,	PUNCT
ejpam-7052	140	6	a.	a.	PROPN
ejpam-7052	140	7	sama	sama	PROPN
ejpam-7052	140	8	-	-	PUNCT
ejpam-7052	140	9	ae	ae	PROPN
ejpam-7052	140	10	,	,	PUNCT
ejpam-7052	140	11	c.	c.	PROPN
ejpam-7052	140	12	boonpok	boonpok	PROPN
ejpam-7052	140	13	/	/	SYM
ejpam-7052	140	14	eur	eur	PROPN
ejpam-7052	140	15	.	.	PUNCT
ejpam-7052	141	1	j.	j.	PROPN
ejpam-7052	141	2	pure	pure	PROPN
ejpam-7052	141	3	appl	appl	PROPN
ejpam-7052	141	4	.	.	PROPN
ejpam-7052	141	5	math	math	PROPN
ejpam-7052	141	6	,	,	PUNCT
ejpam-7052	141	7	18	18	NUM
ejpam-7052	141	8	(	(	PUNCT
ejpam-7052	141	9	4	4	NUM
ejpam-7052	141	10	)	)	PUNCT
ejpam-7052	141	11	(	(	PUNCT
ejpam-7052	141	12	2025	2025	NUM
ejpam-7052	141	13	)	)	PUNCT
ejpam-7052	141	14	,	,	PUNCT
ejpam-7052	141	15	7052	7052	NUM
ejpam-7052	141	16	6	6	NUM
ejpam-7052	141	17	of	of	ADP
ejpam-7052	141	18	14	14	NUM
ejpam-7052	141	19	proof	proof	NOUN
ejpam-7052	141	20	.	.	PUNCT
ejpam-7052	142	1	(	(	PUNCT
ejpam-7052	142	2	1	1	X
ejpam-7052	142	3	)	)	PUNCT
ejpam-7052	142	4	⇒	⇒	NOUN
ejpam-7052	142	5	(	(	PUNCT
ejpam-7052	142	6	2	2	NUM
ejpam-7052	142	7	):	):	PUNCT
ejpam-7052	142	8	let	let	VERB
ejpam-7052	142	9	v	v	PART
ejpam-7052	142	10	be	be	AUX
ejpam-7052	142	11	any	any	DET
ejpam-7052	142	12	σ1σ2	σ1σ2	NOUN
ejpam-7052	142	13	-	-	ADJ
ejpam-7052	142	14	open	open	ADJ
ejpam-7052	142	15	set	set	NOUN
ejpam-7052	142	16	of	of	ADP
ejpam-7052	142	17	y	y	PROPN
ejpam-7052	142	18	and	and	CCONJ
ejpam-7052	142	19	x	x	PROPN
ejpam-7052	142	20	∈	∈	PROPN
ejpam-7052	142	21	f+(v	f+(v	NOUN
ejpam-7052	142	22	)	)	PUNCT
ejpam-7052	142	23	.	.	PUNCT
ejpam-7052	143	1	then	then	ADV
ejpam-7052	143	2	,	,	PUNCT
ejpam-7052	143	3	f	f	PROPN
ejpam-7052	143	4	(	(	PUNCT
ejpam-7052	143	5	x	x	X
ejpam-7052	143	6	)	)	PUNCT
ejpam-7052	143	7	⊆	⊆	NUM
ejpam-7052	143	8	v	v	NOUN
ejpam-7052	143	9	.	.	PUNCT
ejpam-7052	144	1	thus	thus	ADV
ejpam-7052	144	2	by	by	ADP
ejpam-7052	144	3	theorem	theorem	NOUN
ejpam-7052	144	4	1	1	NUM
ejpam-7052	144	5	,	,	PUNCT
ejpam-7052	144	6	we	we	PRON
ejpam-7052	144	7	have	have	VERB
ejpam-7052	144	8	x	x	PART
ejpam-7052	144	9	∈	∈	NOUN
ejpam-7052	144	10	iµ(f	iµ(f	NUM
ejpam-7052	144	11	+	+	NOUN
ejpam-7052	144	12	(	(	PUNCT
ejpam-7052	144	13	σ1σ2	σ1σ2	NUM
ejpam-7052	144	14	-	-	PUNCT
ejpam-7052	144	15	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	144	16	-	-	PUNCT
ejpam-7052	144	17	cl(v	cl(v	NOUN
ejpam-7052	144	18	)	)	PUNCT
ejpam-7052	144	19	)	)	PUNCT
ejpam-7052	144	20	)	)	PUNCT
ejpam-7052	144	21	)	)	PUNCT
ejpam-7052	144	22	and	and	CCONJ
ejpam-7052	144	23	hence	hence	ADV
ejpam-7052	144	24	f+(v	f+(v	PROPN
ejpam-7052	144	25	)	)	PUNCT
ejpam-7052	145	1	⊆	⊆	NUM
ejpam-7052	145	2	iµ(f	iµ(f	NUM
ejpam-7052	145	3	+	+	NOUN
ejpam-7052	145	4	(	(	PUNCT
ejpam-7052	145	5	σ1σ2	σ1σ2	NUM
ejpam-7052	145	6	-	-	PUNCT
ejpam-7052	145	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	145	8	-	-	PUNCT
ejpam-7052	145	9	cl(v	cl(v	NOUN
ejpam-7052	145	10	)	)	PUNCT
ejpam-7052	145	11	)	)	PUNCT
ejpam-7052	145	12	)	)	PUNCT
ejpam-7052	145	13	)	)	PUNCT
ejpam-7052	145	14	.	.	PUNCT
ejpam-7052	146	1	(	(	PUNCT
ejpam-7052	146	2	2	2	X
ejpam-7052	146	3	)	)	PUNCT
ejpam-7052	146	4	⇒	⇒	NOUN
ejpam-7052	146	5	(	(	PUNCT
ejpam-7052	146	6	3	3	NUM
ejpam-7052	146	7	):	):	PUNCT
ejpam-7052	146	8	let	let	VERB
ejpam-7052	146	9	k	k	PRON
ejpam-7052	146	10	be	be	AUX
ejpam-7052	146	11	any	any	DET
ejpam-7052	146	12	σ1σ2	σ1σ2	NUM
ejpam-7052	146	13	-	-	PUNCT
ejpam-7052	146	14	closed	closed	ADJ
ejpam-7052	146	15	set	set	NOUN
ejpam-7052	146	16	of	of	ADP
ejpam-7052	146	17	y	y	PROPN
ejpam-7052	146	18	.	.	PUNCT
ejpam-7052	147	1	then	then	ADV
ejpam-7052	147	2	,	,	PUNCT
ejpam-7052	147	3	y	y	PROPN
ejpam-7052	147	4	−k	−k	PROPN
ejpam-7052	147	5	is	be	AUX
ejpam-7052	147	6	σ1σ2	σ1σ2	NOUN
ejpam-7052	147	7	-	-	ADJ
ejpam-7052	147	8	open	open	ADJ
ejpam-7052	147	9	in	in	ADP
ejpam-7052	147	10	y	y	PROPN
ejpam-7052	147	11	and	and	CCONJ
ejpam-7052	147	12	by	by	ADP
ejpam-7052	147	13	(	(	PUNCT
ejpam-7052	147	14	2	2	NUM
ejpam-7052	147	15	)	)	PUNCT
ejpam-7052	147	16	,	,	PUNCT
ejpam-7052	147	17	x	x	PUNCT
ejpam-7052	147	18	−	−	DET
ejpam-7052	147	19	f−(k	f−(k	PROPN
ejpam-7052	147	20	)	)	PUNCT
ejpam-7052	147	21	=	=	PUNCT
ejpam-7052	147	22	f+(y	f+(y	PROPN
ejpam-7052	147	23	−k	−k	PROPN
ejpam-7052	147	24	)	)	PUNCT
ejpam-7052	147	25	⊆	⊆	NUM
ejpam-7052	147	26	iµ(f	iµ(f	NUM
ejpam-7052	147	27	+	+	NOUN
ejpam-7052	147	28	(	(	PUNCT
ejpam-7052	147	29	σ1σ2	σ1σ2	NUM
ejpam-7052	147	30	-	-	PUNCT
ejpam-7052	147	31	int(σ1σ2	int(σ1σ2	VERB
ejpam-7052	147	32	-	-	PUNCT
ejpam-7052	147	33	cl(y	cl(y	NOUN
ejpam-7052	147	34	−k	−k	NOUN
ejpam-7052	147	35	)	)	PUNCT
ejpam-7052	147	36	)	)	PUNCT
ejpam-7052	147	37	)	)	PUNCT
ejpam-7052	147	38	)	)	PUNCT
ejpam-7052	148	1	=	=	PRON
ejpam-7052	148	2	iµ(x	iµ(x	VERB
ejpam-7052	148	3	−	−	ADP
ejpam-7052	148	4	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7052	148	5	-	-	PUNCT
ejpam-7052	148	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	148	7	-	-	PUNCT
ejpam-7052	148	8	int(k	int(k	NOUN
ejpam-7052	148	9	)	)	PUNCT
ejpam-7052	148	10	)	)	PUNCT
ejpam-7052	148	11	)	)	PUNCT
ejpam-7052	148	12	)	)	PUNCT
ejpam-7052	149	1	=	=	PUNCT
ejpam-7052	150	1	x	x	PUNCT
ejpam-7052	150	2	−	−	PROPN
ejpam-7052	150	3	cµ(f	cµ(f	X
ejpam-7052	150	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	150	5	-	-	PUNCT
ejpam-7052	150	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	150	7	-	-	PUNCT
ejpam-7052	150	8	int(k	int(k	NOUN
ejpam-7052	150	9	)	)	PUNCT
ejpam-7052	150	10	)	)	PUNCT
ejpam-7052	150	11	)	)	PUNCT
ejpam-7052	150	12	)	)	PUNCT
ejpam-7052	150	13	.	.	PUNCT
ejpam-7052	151	1	thus	thus	ADV
ejpam-7052	151	2	,	,	PUNCT
ejpam-7052	151	3	cµ(f−(σ1σ2	cµ(f−(σ1σ2	NUM
ejpam-7052	151	4	-	-	PUNCT
ejpam-7052	151	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	151	6	-	-	PUNCT
ejpam-7052	151	7	int(k	int(k	NOUN
ejpam-7052	151	8	)	)	PUNCT
ejpam-7052	151	9	)	)	PUNCT
ejpam-7052	151	10	)	)	PUNCT
ejpam-7052	151	11	)	)	PUNCT
ejpam-7052	151	12	⊆	⊆	NUM
ejpam-7052	151	13	f−(k	f−(k	PROPN
ejpam-7052	151	14	)	)	PUNCT
ejpam-7052	151	15	.	.	PUNCT
ejpam-7052	152	1	(	(	PUNCT
ejpam-7052	152	2	3	3	X
ejpam-7052	152	3	)	)	PUNCT
ejpam-7052	152	4	⇒	⇒	NOUN
ejpam-7052	152	5	(	(	PUNCT
ejpam-7052	152	6	4	4	NUM
ejpam-7052	152	7	):	):	PUNCT
ejpam-7052	152	8	let	let	VERB
ejpam-7052	152	9	b	b	X
ejpam-7052	152	10	be	be	AUX
ejpam-7052	152	11	any	any	DET
ejpam-7052	152	12	subset	subset	NOUN
ejpam-7052	152	13	of	of	ADP
ejpam-7052	152	14	y	y	PROPN
ejpam-7052	152	15	.	.	PUNCT
ejpam-7052	153	1	then	then	ADV
ejpam-7052	153	2	,	,	PUNCT
ejpam-7052	153	3	σ1σ2	σ1σ2	NOUN
ejpam-7052	153	4	-	-	NOUN
ejpam-7052	153	5	cl(b	cl(b	NOUN
ejpam-7052	153	6	)	)	PUNCT
ejpam-7052	153	7	is	be	AUX
ejpam-7052	153	8	a	a	DET
ejpam-7052	153	9	σ1σ2	σ1σ2	NUM
ejpam-7052	153	10	-	-	PUNCT
ejpam-7052	153	11	closed	closed	ADJ
ejpam-7052	153	12	set	set	NOUN
ejpam-7052	153	13	of	of	ADP
ejpam-7052	153	14	y	y	PROPN
ejpam-7052	153	15	and	and	CCONJ
ejpam-7052	153	16	by	by	ADP
ejpam-7052	153	17	(	(	PUNCT
ejpam-7052	153	18	3	3	NUM
ejpam-7052	153	19	)	)	PUNCT
ejpam-7052	153	20	,	,	PUNCT
ejpam-7052	153	21	cµ(f−(σ1σ2	cµ(f−(σ1σ2	NUM
ejpam-7052	153	22	-	-	PUNCT
ejpam-7052	153	23	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	153	24	-	-	PUNCT
ejpam-7052	153	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	153	26	-	-	PUNCT
ejpam-7052	153	27	cl(b	cl(b	NOUN
ejpam-7052	153	28	)	)	PUNCT
ejpam-7052	153	29	)	)	PUNCT
ejpam-7052	153	30	)	)	PUNCT
ejpam-7052	153	31	)	)	PUNCT
ejpam-7052	153	32	)	)	PUNCT
ejpam-7052	154	1	⊆	⊆	X
ejpam-7052	154	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	154	3	-	-	PUNCT
ejpam-7052	154	4	cl(b	cl(b	NOUN
ejpam-7052	154	5	)	)	PUNCT
ejpam-7052	154	6	)	)	PUNCT
ejpam-7052	154	7	.	.	PUNCT
ejpam-7052	155	1	(	(	PUNCT
ejpam-7052	155	2	4	4	X
ejpam-7052	155	3	)	)	PUNCT
ejpam-7052	155	4	⇒	⇒	NOUN
ejpam-7052	155	5	(	(	PUNCT
ejpam-7052	155	6	5	5	NUM
ejpam-7052	155	7	):	):	PUNCT
ejpam-7052	155	8	let	let	VERB
ejpam-7052	155	9	b	b	X
ejpam-7052	155	10	be	be	AUX
ejpam-7052	155	11	any	any	DET
ejpam-7052	155	12	subset	subset	NOUN
ejpam-7052	155	13	of	of	ADP
ejpam-7052	155	14	y	y	PROPN
ejpam-7052	155	15	.	.	PUNCT
ejpam-7052	156	1	then	then	ADV
ejpam-7052	156	2	,	,	PUNCT
ejpam-7052	156	3	we	we	PRON
ejpam-7052	156	4	have	have	VERB
ejpam-7052	156	5	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	156	6	-	-	PUNCT
ejpam-7052	156	7	int(b	int(b	NOUN
ejpam-7052	156	8	)	)	PUNCT
ejpam-7052	156	9	)	)	PUNCT
ejpam-7052	157	1	=	=	PUNCT
ejpam-7052	158	1	x	x	PUNCT
ejpam-7052	158	2	−	−	ADP
ejpam-7052	158	3	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	158	4	-	-	PUNCT
ejpam-7052	158	5	cl(y	cl(y	NOUN
ejpam-7052	158	6	−b	−b	NOUN
ejpam-7052	158	7	)	)	PUNCT
ejpam-7052	158	8	)	)	PUNCT
ejpam-7052	159	1	⊆	⊆	NUM
ejpam-7052	159	2	x	x	SYM
ejpam-7052	159	3	−	−	PROPN
ejpam-7052	159	4	cµ(f	cµ(f	X
ejpam-7052	159	5	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	159	6	-	-	PUNCT
ejpam-7052	159	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	159	8	-	-	PUNCT
ejpam-7052	159	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	159	10	-	-	PUNCT
ejpam-7052	159	11	cl(y	cl(y	NOUN
ejpam-7052	159	12	−b	−b	NOUN
ejpam-7052	159	13	)	)	PUNCT
ejpam-7052	159	14	)	)	PUNCT
ejpam-7052	159	15	)	)	PUNCT
ejpam-7052	159	16	)	)	PUNCT
ejpam-7052	159	17	)	)	PUNCT
ejpam-7052	160	1	=	=	PUNCT
ejpam-7052	161	1	x	x	PUNCT
ejpam-7052	162	1	−	−	NOUN
ejpam-7052	162	2	cµ(f	cµ(f	NOUN
ejpam-7052	163	1	−(y	−(y	NOUN
ejpam-7052	163	2	−	−	NOUN
ejpam-7052	163	3	σ1σ2	σ1σ2	SYM
ejpam-7052	163	4	-	-	PUNCT
ejpam-7052	163	5	int(σ1σ2	int(σ1σ2	ADV
ejpam-7052	163	6	-	-	PUNCT
ejpam-7052	163	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	163	8	-	-	PUNCT
ejpam-7052	163	9	int(b	int(b	NOUN
ejpam-7052	163	10	)	)	PUNCT
ejpam-7052	163	11	)	)	PUNCT
ejpam-7052	163	12	)	)	PUNCT
ejpam-7052	163	13	)	)	PUNCT
ejpam-7052	163	14	)	)	PUNCT
ejpam-7052	164	1	=	=	PRON
ejpam-7052	164	2	iµ(f	iµ(f	NUM
ejpam-7052	164	3	+	+	ADJ
ejpam-7052	164	4	(	(	PUNCT
ejpam-7052	164	5	σ1σ2	σ1σ2	NUM
ejpam-7052	164	6	-	-	PUNCT
ejpam-7052	164	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-7052	164	8	-	-	PUNCT
ejpam-7052	164	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	164	10	-	-	PUNCT
ejpam-7052	164	11	int(b	int(b	NOUN
ejpam-7052	164	12	)	)	PUNCT
ejpam-7052	164	13	)	)	PUNCT
ejpam-7052	164	14	)	)	PUNCT
ejpam-7052	164	15	)	)	PUNCT
ejpam-7052	164	16	)	)	PUNCT
ejpam-7052	164	17	.	.	PUNCT
ejpam-7052	165	1	(	(	PUNCT
ejpam-7052	165	2	5	5	X
ejpam-7052	165	3	)	)	PUNCT
ejpam-7052	165	4	⇒	⇒	NOUN
ejpam-7052	165	5	(	(	PUNCT
ejpam-7052	165	6	6	6	NUM
ejpam-7052	165	7	):	):	PUNCT
ejpam-7052	165	8	let	let	VERB
ejpam-7052	165	9	v	v	PART
ejpam-7052	165	10	be	be	AUX
ejpam-7052	165	11	any	any	DET
ejpam-7052	165	12	(	(	PUNCT
ejpam-7052	165	13	σ1	σ1	NOUN
ejpam-7052	165	14	,	,	PUNCT
ejpam-7052	165	15	σ2)r	σ2)r	NOUN
ejpam-7052	165	16	-	-	PUNCT
ejpam-7052	165	17	open	open	ADJ
ejpam-7052	165	18	set	set	NOUN
ejpam-7052	165	19	of	of	ADP
ejpam-7052	165	20	y	y	PROPN
ejpam-7052	165	21	.	.	PUNCT
ejpam-7052	166	1	by	by	ADP
ejpam-7052	166	2	(	(	PUNCT
ejpam-7052	166	3	5	5	NUM
ejpam-7052	166	4	)	)	PUNCT
ejpam-7052	166	5	,	,	PUNCT
ejpam-7052	166	6	we	we	PRON
ejpam-7052	166	7	have	have	VERB
ejpam-7052	166	8	f+(v	f+(v	NOUN
ejpam-7052	166	9	)	)	PUNCT
ejpam-7052	167	1	⊆	⊆	NUM
ejpam-7052	167	2	iµ(f	iµ(f	NUM
ejpam-7052	167	3	+	+	NOUN
ejpam-7052	167	4	(	(	PUNCT
ejpam-7052	167	5	v	v	NOUN
ejpam-7052	167	6	)	)	PUNCT
ejpam-7052	167	7	)	)	PUNCT
ejpam-7052	167	8	and	and	CCONJ
ejpam-7052	167	9	hence	hence	ADV
ejpam-7052	167	10	f+(v	f+(v	PROPN
ejpam-7052	167	11	)	)	PUNCT
ejpam-7052	167	12	is	be	AUX
ejpam-7052	167	13	µ-open	µ-open	NOUN
ejpam-7052	167	14	in	in	ADP
ejpam-7052	167	15	x.	x.	NOUN
ejpam-7052	167	16	(	(	PUNCT
ejpam-7052	167	17	6	6	NUM
ejpam-7052	167	18	)	)	PUNCT
ejpam-7052	167	19	⇒	⇒	NOUN
ejpam-7052	167	20	(	(	PUNCT
ejpam-7052	167	21	7	7	NUM
ejpam-7052	167	22	):	):	PUNCT
ejpam-7052	167	23	the	the	DET
ejpam-7052	167	24	proof	proof	NOUN
ejpam-7052	167	25	is	be	AUX
ejpam-7052	167	26	obvious	obvious	ADJ
ejpam-7052	167	27	.	.	PUNCT
ejpam-7052	168	1	(	(	PUNCT
ejpam-7052	168	2	7	7	X
ejpam-7052	168	3	)	)	PUNCT
ejpam-7052	168	4	⇒	⇒	NOUN
ejpam-7052	168	5	(	(	PUNCT
ejpam-7052	168	6	1	1	NUM
ejpam-7052	168	7	):	):	PUNCT
ejpam-7052	168	8	let	let	VERB
ejpam-7052	168	9	x	x	PUNCT
ejpam-7052	168	10	∈	∈	PROPN
ejpam-7052	168	11	x	x	X
ejpam-7052	168	12	and	and	CCONJ
ejpam-7052	168	13	v	v	AUX
ejpam-7052	168	14	be	be	AUX
ejpam-7052	168	15	any	any	DET
ejpam-7052	168	16	(	(	PUNCT
ejpam-7052	168	17	σ1	σ1	NOUN
ejpam-7052	168	18	,	,	PUNCT
ejpam-7052	168	19	σ2)r	σ2)r	NOUN
ejpam-7052	168	20	-	-	PUNCT
ejpam-7052	168	21	open	open	ADJ
ejpam-7052	168	22	set	set	NOUN
ejpam-7052	168	23	of	of	ADP
ejpam-7052	168	24	y	y	PROPN
ejpam-7052	168	25	containing	contain	VERB
ejpam-7052	168	26	f	f	PROPN
ejpam-7052	168	27	(	(	PUNCT
ejpam-7052	168	28	x	x	NOUN
ejpam-7052	168	29	)	)	PUNCT
ejpam-7052	168	30	.	.	PUNCT
ejpam-7052	169	1	since	since	SCONJ
ejpam-7052	169	2	y	y	PROPN
ejpam-7052	169	3	−	−	PROPN
ejpam-7052	169	4	v	v	NOUN
ejpam-7052	169	5	is	be	AUX
ejpam-7052	169	6	(	(	PUNCT
ejpam-7052	169	7	σ1	σ1	NOUN
ejpam-7052	169	8	,	,	PUNCT
ejpam-7052	169	9	σ2)r	σ2)r	NOUN
ejpam-7052	169	10	-	-	PUNCT
ejpam-7052	169	11	closed	closed	ADJ
ejpam-7052	169	12	and	and	CCONJ
ejpam-7052	169	13	by	by	ADP
ejpam-7052	169	14	(	(	PUNCT
ejpam-7052	169	15	7	7	NUM
ejpam-7052	169	16	)	)	PUNCT
ejpam-7052	169	17	,	,	PUNCT
ejpam-7052	169	18	x	x	PUNCT
ejpam-7052	169	19	−	−	NOUN
ejpam-7052	169	20	f+(v	f+(v	NOUN
ejpam-7052	169	21	)	)	PUNCT
ejpam-7052	170	1	=	=	PUNCT
ejpam-7052	170	2	f−(y	f−(y	NOUN
ejpam-7052	170	3	−	−	NOUN
ejpam-7052	170	4	v	v	NOUN
ejpam-7052	170	5	)	)	PUNCT
ejpam-7052	170	6	is	be	AUX
ejpam-7052	170	7	µ-closed	µ-close	VERB
ejpam-7052	170	8	in	in	ADP
ejpam-7052	170	9	x.	x.	NOUN
ejpam-7052	170	10	thus	thus	ADV
ejpam-7052	170	11	,	,	PUNCT
ejpam-7052	170	12	f+(v	f+(v	PROPN
ejpam-7052	170	13	)	)	PUNCT
ejpam-7052	171	1	is	be	AUX
ejpam-7052	171	2	µ-open	µ-open	NOUN
ejpam-7052	171	3	and	and	CCONJ
ejpam-7052	171	4	hence	hence	ADV
ejpam-7052	171	5	x	x	X
ejpam-7052	171	6	∈	∈	NOUN
ejpam-7052	171	7	iµ(f	iµ(f	NUM
ejpam-7052	172	1	+	+	NOUN
ejpam-7052	172	2	(	(	PUNCT
ejpam-7052	172	3	v	v	NOUN
ejpam-7052	172	4	)	)	PUNCT
ejpam-7052	172	5	)	)	PUNCT
ejpam-7052	172	6	.	.	PUNCT
ejpam-7052	173	1	then	then	ADV
ejpam-7052	173	2	,	,	PUNCT
ejpam-7052	173	3	there	there	PRON
ejpam-7052	173	4	exists	exist	VERB
ejpam-7052	173	5	a	a	DET
ejpam-7052	173	6	µ-open	µ-open	NOUN
ejpam-7052	173	7	set	set	VERB
ejpam-7052	173	8	u	u	NOUN
ejpam-7052	173	9	of	of	ADP
ejpam-7052	173	10	x	x	PUNCT
ejpam-7052	173	11	containing	contain	VERB
ejpam-7052	173	12	x	x	PUNCT
ejpam-7052	173	13	such	such	ADJ
ejpam-7052	173	14	that	that	SCONJ
ejpam-7052	173	15	f	f	PROPN
ejpam-7052	173	16	(	(	PUNCT
ejpam-7052	173	17	u	u	NOUN
ejpam-7052	173	18	)	)	PUNCT
ejpam-7052	173	19	⊆	⊆	NUM
ejpam-7052	173	20	v	v	NOUN
ejpam-7052	173	21	.	.	PUNCT
ejpam-7052	174	1	it	it	PRON
ejpam-7052	174	2	follows	follow	VERB
ejpam-7052	174	3	from	from	ADP
ejpam-7052	174	4	theorem	theorem	ADJ
ejpam-7052	174	5	1	1	NUM
ejpam-7052	174	6	that	that	SCONJ
ejpam-7052	174	7	f	f	PROPN
ejpam-7052	174	8	is	be	AUX
ejpam-7052	174	9	upper	upper	ADJ
ejpam-7052	174	10	almost	almost	ADV
ejpam-7052	174	11	µ(σ1	µ(σ1	NOUN
ejpam-7052	174	12	,	,	PUNCT
ejpam-7052	174	13	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7052	174	14	.	.	X
ejpam-7052	174	15	theorem	theorem	NOUN
ejpam-7052	174	16	4	4	NUM
ejpam-7052	174	17	.	.	X
ejpam-7052	174	18	for	for	ADP
ejpam-7052	174	19	a	a	DET
ejpam-7052	174	20	multifunction	multifunction	NOUN
ejpam-7052	174	21	f	f	NOUN
ejpam-7052	174	22	:	:	PUNCT
ejpam-7052	174	23	(	(	PUNCT
ejpam-7052	174	24	x,µ	x,µ	NOUN
ejpam-7052	174	25	)	)	PUNCT
ejpam-7052	174	26	→	→	SYM
ejpam-7052	174	27	(	(	PUNCT
ejpam-7052	174	28	y	y	PROPN
ejpam-7052	174	29	,	,	PUNCT
ejpam-7052	174	30	σ1	σ1	PROPN
ejpam-7052	174	31	,	,	PUNCT
ejpam-7052	174	32	σ2	σ2	NOUN
ejpam-7052	174	33	)	)	PUNCT
ejpam-7052	174	34	,	,	PUNCT
ejpam-7052	174	35	the	the	DET
ejpam-7052	174	36	following	follow	VERB
ejpam-7052	174	37	properties	property	NOUN
ejpam-7052	174	38	are	be	AUX
ejpam-7052	174	39	equivalent	equivalent	ADJ
ejpam-7052	174	40	:	:	PUNCT
ejpam-7052	174	41	(	(	PUNCT
ejpam-7052	174	42	1	1	X
ejpam-7052	174	43	)	)	PUNCT
ejpam-7052	174	44	f	f	PROPN
ejpam-7052	174	45	is	be	AUX
ejpam-7052	174	46	lower	low	ADJ
ejpam-7052	174	47	almost	almost	ADV
ejpam-7052	174	48	µ(σ1	µ(σ1	NOUN
ejpam-7052	174	49	,	,	PUNCT
ejpam-7052	174	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	174	51	;	;	PUNCT
ejpam-7052	174	52	(	(	PUNCT
ejpam-7052	174	53	2	2	X
ejpam-7052	174	54	)	)	PUNCT
ejpam-7052	174	55	f−(v	f−(v	NOUN
ejpam-7052	174	56	)	)	PUNCT
ejpam-7052	174	57	⊆	⊆	NUM
ejpam-7052	174	58	iµ(f	iµ(f	NUM
ejpam-7052	174	59	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	174	60	-	-	PUNCT
ejpam-7052	174	61	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	174	62	-	-	PUNCT
ejpam-7052	174	63	cl(v	cl(v	NOUN
ejpam-7052	174	64	)	)	PUNCT
ejpam-7052	174	65	)	)	PUNCT
ejpam-7052	174	66	)	)	PUNCT
ejpam-7052	174	67	)	)	PUNCT
ejpam-7052	175	1	for	for	ADP
ejpam-7052	175	2	every	every	DET
ejpam-7052	175	3	σ1σ2	σ1σ2	NOUN
ejpam-7052	175	4	-	-	ADJ
ejpam-7052	175	5	open	open	ADJ
ejpam-7052	175	6	set	set	NOUN
ejpam-7052	175	7	v	v	NOUN
ejpam-7052	175	8	of	of	ADP
ejpam-7052	175	9	y	y	PROPN
ejpam-7052	175	10	;	;	PUNCT
ejpam-7052	175	11	(	(	PUNCT
ejpam-7052	175	12	3	3	X
ejpam-7052	175	13	)	)	PUNCT
ejpam-7052	175	14	cµ(f	cµ(f	PUNCT
ejpam-7052	176	1	+	+	PROPN
ejpam-7052	176	2	(	(	PUNCT
ejpam-7052	176	3	σ1σ2	σ1σ2	NUM
ejpam-7052	176	4	-	-	PUNCT
ejpam-7052	176	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	176	6	-	-	PUNCT
ejpam-7052	176	7	int(k	int(k	NOUN
ejpam-7052	176	8	)	)	PUNCT
ejpam-7052	176	9	)	)	PUNCT
ejpam-7052	176	10	)	)	PUNCT
ejpam-7052	176	11	)	)	PUNCT
ejpam-7052	177	1	⊆	⊆	NUM
ejpam-7052	177	2	f+(k	f+(k	NOUN
ejpam-7052	177	3	)	)	PUNCT
ejpam-7052	177	4	for	for	ADP
ejpam-7052	177	5	every	every	DET
ejpam-7052	177	6	σ1σ2	σ1σ2	NUM
ejpam-7052	177	7	-	-	PUNCT
ejpam-7052	177	8	closed	closed	ADJ
ejpam-7052	177	9	set	set	NOUN
ejpam-7052	177	10	k	k	PROPN
ejpam-7052	177	11	of	of	ADP
ejpam-7052	177	12	y	y	PROPN
ejpam-7052	177	13	;	;	PUNCT
ejpam-7052	177	14	(	(	PUNCT
ejpam-7052	177	15	4	4	NUM
ejpam-7052	177	16	)	)	PUNCT
ejpam-7052	177	17	cµ(f	cµ(f	PUNCT
ejpam-7052	178	1	+	+	PROPN
ejpam-7052	178	2	(	(	PUNCT
ejpam-7052	178	3	σ1σ2	σ1σ2	NUM
ejpam-7052	178	4	-	-	PUNCT
ejpam-7052	178	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	178	6	-	-	PUNCT
ejpam-7052	178	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	178	8	-	-	PUNCT
ejpam-7052	178	9	cl(b	cl(b	NOUN
ejpam-7052	178	10	)	)	PUNCT
ejpam-7052	178	11	)	)	PUNCT
ejpam-7052	178	12	)	)	PUNCT
ejpam-7052	178	13	)	)	PUNCT
ejpam-7052	178	14	)	)	PUNCT
ejpam-7052	179	1	⊆	⊆	X
ejpam-7052	179	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	179	3	-	-	PUNCT
ejpam-7052	179	4	cl(b	cl(b	NOUN
ejpam-7052	179	5	)	)	PUNCT
ejpam-7052	179	6	)	)	PUNCT
ejpam-7052	179	7	for	for	ADP
ejpam-7052	179	8	every	every	DET
ejpam-7052	179	9	subset	subset	NOUN
ejpam-7052	179	10	b	b	PROPN
ejpam-7052	179	11	of	of	ADP
ejpam-7052	179	12	y	y	PROPN
ejpam-7052	179	13	;	;	PUNCT
ejpam-7052	179	14	(	(	PUNCT
ejpam-7052	179	15	5	5	X
ejpam-7052	179	16	)	)	PUNCT
ejpam-7052	179	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	179	18	-	-	PUNCT
ejpam-7052	179	19	int(b	int(b	NOUN
ejpam-7052	179	20	)	)	PUNCT
ejpam-7052	179	21	)	)	PUNCT
ejpam-7052	179	22	⊆	⊆	NUM
ejpam-7052	179	23	iµ(f	iµ(f	NUM
ejpam-7052	179	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	179	25	-	-	PUNCT
ejpam-7052	179	26	int(σ1σ2	int(σ1σ2	ADV
ejpam-7052	179	27	-	-	PUNCT
ejpam-7052	179	28	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	179	29	-	-	PUNCT
ejpam-7052	179	30	int(b	int(b	NOUN
ejpam-7052	179	31	)	)	PUNCT
ejpam-7052	179	32	)	)	PUNCT
ejpam-7052	179	33	)	)	PUNCT
ejpam-7052	179	34	)	)	PUNCT
ejpam-7052	179	35	)	)	PUNCT
ejpam-7052	179	36	for	for	ADP
ejpam-7052	179	37	every	every	DET
ejpam-7052	179	38	subset	subset	NOUN
ejpam-7052	179	39	b	b	PROPN
ejpam-7052	179	40	of	of	ADP
ejpam-7052	179	41	y	y	PROPN
ejpam-7052	179	42	;	;	PUNCT
ejpam-7052	179	43	(	(	PUNCT
ejpam-7052	179	44	6	6	X
ejpam-7052	179	45	)	)	PUNCT
ejpam-7052	179	46	f−(v	f−(v	NOUN
ejpam-7052	179	47	)	)	PUNCT
ejpam-7052	179	48	is	be	AUX
ejpam-7052	179	49	µ-open	µ-open	NOUN
ejpam-7052	179	50	in	in	ADP
ejpam-7052	179	51	x	x	PUNCT
ejpam-7052	179	52	for	for	ADP
ejpam-7052	179	53	every	every	DET
ejpam-7052	179	54	(	(	PUNCT
ejpam-7052	179	55	σ1	σ1	PROPN
ejpam-7052	179	56	,	,	PUNCT
ejpam-7052	179	57	σ2)r	σ2)r	NOUN
ejpam-7052	179	58	-	-	PUNCT
ejpam-7052	179	59	open	open	ADJ
ejpam-7052	179	60	set	set	VERB
ejpam-7052	179	61	v	v	NOUN
ejpam-7052	179	62	of	of	ADP
ejpam-7052	179	63	y	y	PROPN
ejpam-7052	179	64	;	;	PUNCT
ejpam-7052	179	65	p.	p.	NOUN
ejpam-7052	179	66	pue	pue	PROPN
ejpam-7052	179	67	-	-	PUNCT
ejpam-7052	179	68	on	on	ADP
ejpam-7052	179	69	,	,	PUNCT
ejpam-7052	179	70	a.	a.	PROPN
ejpam-7052	179	71	sama	sama	PROPN
ejpam-7052	179	72	-	-	PUNCT
ejpam-7052	179	73	ae	ae	PROPN
ejpam-7052	179	74	,	,	PUNCT
ejpam-7052	179	75	c.	c.	PROPN
ejpam-7052	179	76	boonpok	boonpok	PROPN
ejpam-7052	179	77	/	/	SYM
ejpam-7052	179	78	eur	eur	PROPN
ejpam-7052	179	79	.	.	PUNCT
ejpam-7052	180	1	j.	j.	PROPN
ejpam-7052	180	2	pure	pure	PROPN
ejpam-7052	180	3	appl	appl	PROPN
ejpam-7052	180	4	.	.	PROPN
ejpam-7052	180	5	math	math	PROPN
ejpam-7052	180	6	,	,	PUNCT
ejpam-7052	180	7	18	18	NUM
ejpam-7052	180	8	(	(	PUNCT
ejpam-7052	180	9	4	4	NUM
ejpam-7052	180	10	)	)	PUNCT
ejpam-7052	180	11	(	(	PUNCT
ejpam-7052	180	12	2025	2025	NUM
ejpam-7052	180	13	)	)	PUNCT
ejpam-7052	180	14	,	,	PUNCT
ejpam-7052	180	15	7052	7052	NUM
ejpam-7052	180	16	7	7	NUM
ejpam-7052	180	17	of	of	ADP
ejpam-7052	180	18	14	14	NUM
ejpam-7052	180	19	(	(	PUNCT
ejpam-7052	180	20	7	7	NUM
ejpam-7052	180	21	)	)	PUNCT
ejpam-7052	180	22	f+(k	f+(k	NUM
ejpam-7052	180	23	)	)	PUNCT
ejpam-7052	180	24	is	be	AUX
ejpam-7052	180	25	µ-closed	µ-close	VERB
ejpam-7052	180	26	in	in	ADP
ejpam-7052	180	27	x	x	PUNCT
ejpam-7052	180	28	for	for	ADP
ejpam-7052	180	29	every	every	DET
ejpam-7052	180	30	(	(	PUNCT
ejpam-7052	180	31	σ1	σ1	PROPN
ejpam-7052	180	32	,	,	PUNCT
ejpam-7052	180	33	σ2)r	σ2)r	NOUN
ejpam-7052	180	34	-	-	PUNCT
ejpam-7052	180	35	closed	close	VERB
ejpam-7052	180	36	set	set	ADJ
ejpam-7052	180	37	k	k	PROPN
ejpam-7052	180	38	of	of	ADP
ejpam-7052	180	39	y	y	PROPN
ejpam-7052	180	40	.	.	PUNCT
ejpam-7052	181	1	proof	proof	NOUN
ejpam-7052	181	2	.	.	PUNCT
ejpam-7052	182	1	the	the	DET
ejpam-7052	182	2	proof	proof	NOUN
ejpam-7052	182	3	is	be	AUX
ejpam-7052	182	4	similar	similar	ADJ
ejpam-7052	182	5	to	to	ADP
ejpam-7052	182	6	that	that	PRON
ejpam-7052	182	7	of	of	ADP
ejpam-7052	182	8	theorem	theorem	ADJ
ejpam-7052	182	9	3	3	NUM
ejpam-7052	182	10	.	.	PUNCT
ejpam-7052	182	11	theorem	theorem	NOUN
ejpam-7052	182	12	5	5	NUM
ejpam-7052	182	13	.	.	X
ejpam-7052	182	14	for	for	ADP
ejpam-7052	182	15	a	a	DET
ejpam-7052	182	16	multifunction	multifunction	NOUN
ejpam-7052	182	17	f	f	NOUN
ejpam-7052	182	18	:	:	PUNCT
ejpam-7052	182	19	(	(	PUNCT
ejpam-7052	182	20	x,µ	x,µ	NOUN
ejpam-7052	182	21	)	)	PUNCT
ejpam-7052	182	22	→	→	SYM
ejpam-7052	182	23	(	(	PUNCT
ejpam-7052	182	24	y	y	PROPN
ejpam-7052	182	25	,	,	PUNCT
ejpam-7052	182	26	σ1	σ1	PROPN
ejpam-7052	182	27	,	,	PUNCT
ejpam-7052	182	28	σ2	σ2	NOUN
ejpam-7052	182	29	)	)	PUNCT
ejpam-7052	182	30	,	,	PUNCT
ejpam-7052	182	31	the	the	DET
ejpam-7052	182	32	following	follow	VERB
ejpam-7052	182	33	properties	property	NOUN
ejpam-7052	182	34	are	be	AUX
ejpam-7052	182	35	equivalent	equivalent	ADJ
ejpam-7052	182	36	:	:	PUNCT
ejpam-7052	182	37	(	(	PUNCT
ejpam-7052	182	38	1	1	X
ejpam-7052	182	39	)	)	PUNCT
ejpam-7052	182	40	f	f	PROPN
ejpam-7052	182	41	is	be	AUX
ejpam-7052	182	42	upper	upper	ADJ
ejpam-7052	182	43	almost	almost	ADV
ejpam-7052	182	44	µ(σ1	µ(σ1	NOUN
ejpam-7052	182	45	,	,	PUNCT
ejpam-7052	182	46	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	182	47	;	;	PUNCT
ejpam-7052	182	48	(	(	PUNCT
ejpam-7052	182	49	2	2	NUM
ejpam-7052	182	50	)	)	PUNCT
ejpam-7052	182	51	cµ(f	cµ(f	NOUN
ejpam-7052	182	52	−(v	−(v	NOUN
ejpam-7052	182	53	)	)	PUNCT
ejpam-7052	182	54	)	)	PUNCT
ejpam-7052	183	1	⊆	⊆	X
ejpam-7052	183	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	183	3	-	-	PUNCT
ejpam-7052	183	4	cl(v	cl(v	NOUN
ejpam-7052	183	5	)	)	PUNCT
ejpam-7052	183	6	)	)	PUNCT
ejpam-7052	183	7	for	for	ADP
ejpam-7052	183	8	every	every	DET
ejpam-7052	183	9	(	(	PUNCT
ejpam-7052	183	10	σ1	σ1	PROPN
ejpam-7052	183	11	,	,	PUNCT
ejpam-7052	183	12	σ2)β	σ2)β	NOUN
ejpam-7052	183	13	-	-	PUNCT
ejpam-7052	183	14	open	open	NOUN
ejpam-7052	183	15	set	set	NOUN
ejpam-7052	183	16	v	v	NOUN
ejpam-7052	183	17	of	of	ADP
ejpam-7052	183	18	y	y	PROPN
ejpam-7052	183	19	;	;	PUNCT
ejpam-7052	183	20	(	(	PUNCT
ejpam-7052	183	21	3	3	X
ejpam-7052	183	22	)	)	PUNCT
ejpam-7052	183	23	cµ(f	cµ(f	NOUN
ejpam-7052	183	24	−(v	−(v	NOUN
ejpam-7052	183	25	)	)	PUNCT
ejpam-7052	183	26	)	)	PUNCT
ejpam-7052	184	1	⊆	⊆	X
ejpam-7052	184	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	184	3	-	-	PUNCT
ejpam-7052	184	4	cl(v	cl(v	NOUN
ejpam-7052	184	5	)	)	PUNCT
ejpam-7052	184	6	)	)	PUNCT
ejpam-7052	184	7	for	for	ADP
ejpam-7052	184	8	every	every	DET
ejpam-7052	184	9	(	(	PUNCT
ejpam-7052	184	10	σ1	σ1	PROPN
ejpam-7052	184	11	,	,	PUNCT
ejpam-7052	184	12	σ2)s	σ2)s	NOUN
ejpam-7052	184	13	-	-	PUNCT
ejpam-7052	184	14	open	open	NOUN
ejpam-7052	184	15	set	set	NOUN
ejpam-7052	184	16	v	v	NOUN
ejpam-7052	184	17	of	of	ADP
ejpam-7052	184	18	y	y	PROPN
ejpam-7052	184	19	.	.	PUNCT
ejpam-7052	185	1	proof	proof	NOUN
ejpam-7052	185	2	.	.	PUNCT
ejpam-7052	186	1	(	(	PUNCT
ejpam-7052	186	2	1	1	X
ejpam-7052	186	3	)	)	PUNCT
ejpam-7052	186	4	⇒	⇒	NOUN
ejpam-7052	186	5	(	(	PUNCT
ejpam-7052	186	6	2	2	NUM
ejpam-7052	186	7	):	):	PUNCT
ejpam-7052	186	8	let	let	VERB
ejpam-7052	186	9	v	v	PART
ejpam-7052	186	10	be	be	AUX
ejpam-7052	186	11	any	any	DET
ejpam-7052	186	12	(	(	PUNCT
ejpam-7052	186	13	σ1	σ1	PROPN
ejpam-7052	186	14	,	,	PUNCT
ejpam-7052	186	15	σ2)β	σ2)β	NOUN
ejpam-7052	186	16	-	-	PUNCT
ejpam-7052	186	17	open	open	ADJ
ejpam-7052	186	18	set	set	NOUN
ejpam-7052	186	19	of	of	ADP
ejpam-7052	186	20	y	y	PROPN
ejpam-7052	186	21	.	.	PUNCT
ejpam-7052	187	1	then	then	ADV
ejpam-7052	187	2	,	,	PUNCT
ejpam-7052	187	3	σ1σ2	σ1σ2	NOUN
ejpam-7052	187	4	-	-	NUM
ejpam-7052	187	5	cl(v	cl(v	NOUN
ejpam-7052	187	6	)	)	PUNCT
ejpam-7052	187	7	is	be	AUX
ejpam-7052	187	8	a	a	DET
ejpam-7052	187	9	(	(	PUNCT
ejpam-7052	187	10	σ1	σ1	NOUN
ejpam-7052	187	11	,	,	PUNCT
ejpam-7052	187	12	σ2)r	σ2)r	NOUN
ejpam-7052	187	13	-	-	PUNCT
ejpam-7052	187	14	closed	close	VERB
ejpam-7052	187	15	set	set	NOUN
ejpam-7052	187	16	of	of	ADP
ejpam-7052	187	17	y	y	PROPN
ejpam-7052	187	18	.	.	PUNCT
ejpam-7052	188	1	since	since	SCONJ
ejpam-7052	188	2	f	f	PROPN
ejpam-7052	188	3	is	be	AUX
ejpam-7052	188	4	upper	upper	ADJ
ejpam-7052	188	5	almost	almost	ADV
ejpam-7052	188	6	µ(σ1	µ(σ1	NOUN
ejpam-7052	188	7	,	,	PUNCT
ejpam-7052	188	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	188	9	and	and	CCONJ
ejpam-7052	188	10	by	by	ADP
ejpam-7052	188	11	theorem	theorem	ADJ
ejpam-7052	188	12	3	3	NUM
ejpam-7052	188	13	,	,	PUNCT
ejpam-7052	188	14	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	188	15	-	-	PUNCT
ejpam-7052	188	16	cl(v	cl(v	NOUN
ejpam-7052	188	17	)	)	PUNCT
ejpam-7052	188	18	)	)	PUNCT
ejpam-7052	188	19	is	be	AUX
ejpam-7052	188	20	µ-closed	µ-close	VERB
ejpam-7052	188	21	in	in	ADP
ejpam-7052	188	22	x.	x.	NOUN
ejpam-7052	188	23	thus	thus	ADV
ejpam-7052	188	24	,	,	PUNCT
ejpam-7052	188	25	cµ(f−(v	cµ(f−(v	NOUN
ejpam-7052	188	26	)	)	PUNCT
ejpam-7052	188	27	)	)	PUNCT
ejpam-7052	189	1	⊆	⊆	X
ejpam-7052	189	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	189	3	-	-	PUNCT
ejpam-7052	189	4	cl(v	cl(v	NOUN
ejpam-7052	189	5	)	)	PUNCT
ejpam-7052	189	6	)	)	PUNCT
ejpam-7052	189	7	.	.	PUNCT
ejpam-7052	190	1	(	(	PUNCT
ejpam-7052	190	2	2	2	X
ejpam-7052	190	3	)	)	PUNCT
ejpam-7052	190	4	⇒	⇒	NOUN
ejpam-7052	190	5	(	(	PUNCT
ejpam-7052	190	6	3	3	NUM
ejpam-7052	190	7	):	):	PUNCT
ejpam-7052	190	8	the	the	DET
ejpam-7052	190	9	proof	proof	NOUN
ejpam-7052	190	10	is	be	AUX
ejpam-7052	190	11	obvious	obvious	ADJ
ejpam-7052	190	12	.	.	PUNCT
ejpam-7052	191	1	(	(	PUNCT
ejpam-7052	191	2	3	3	X
ejpam-7052	191	3	)	)	PUNCT
ejpam-7052	191	4	⇒	⇒	NOUN
ejpam-7052	191	5	(	(	PUNCT
ejpam-7052	191	6	1	1	NUM
ejpam-7052	191	7	):	):	PUNCT
ejpam-7052	191	8	let	let	VERB
ejpam-7052	191	9	k	k	PRON
ejpam-7052	191	10	be	be	AUX
ejpam-7052	191	11	any	any	DET
ejpam-7052	191	12	(	(	PUNCT
ejpam-7052	191	13	σ1	σ1	NOUN
ejpam-7052	191	14	,	,	PUNCT
ejpam-7052	191	15	σ2)r	σ2)r	NOUN
ejpam-7052	191	16	-	-	PUNCT
ejpam-7052	191	17	closed	close	VERB
ejpam-7052	191	18	set	set	NOUN
ejpam-7052	191	19	of	of	ADP
ejpam-7052	191	20	y	y	PROPN
ejpam-7052	191	21	.	.	PUNCT
ejpam-7052	192	1	then	then	ADV
ejpam-7052	192	2	,	,	PUNCT
ejpam-7052	192	3	k	k	X
ejpam-7052	192	4	is	be	AUX
ejpam-7052	192	5	(	(	PUNCT
ejpam-7052	192	6	σ1	σ1	PROPN
ejpam-7052	192	7	,	,	PUNCT
ejpam-7052	192	8	σ2)s	σ2)s	NOUN
ejpam-7052	192	9	-	-	PUNCT
ejpam-7052	192	10	open	open	ADJ
ejpam-7052	192	11	in	in	ADP
ejpam-7052	192	12	y	y	PROPN
ejpam-7052	192	13	.	.	PUNCT
ejpam-7052	193	1	then	then	ADV
ejpam-7052	193	2	by	by	ADP
ejpam-7052	193	3	(	(	PUNCT
ejpam-7052	193	4	3	3	NUM
ejpam-7052	193	5	)	)	PUNCT
ejpam-7052	193	6	,	,	PUNCT
ejpam-7052	193	7	cµ(f−(k	cµ(f−(k	NOUN
ejpam-7052	193	8	)	)	PUNCT
ejpam-7052	193	9	)	)	PUNCT
ejpam-7052	194	1	⊆	⊆	X
ejpam-7052	194	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	194	3	-	-	PUNCT
ejpam-7052	194	4	cl(k	cl(k	NOUN
ejpam-7052	194	5	)	)	PUNCT
ejpam-7052	194	6	)	)	PUNCT
ejpam-7052	195	1	=	=	SYM
ejpam-7052	195	2	f−(k	f−(k	PROPN
ejpam-7052	195	3	)	)	PUNCT
ejpam-7052	195	4	and	and	CCONJ
ejpam-7052	195	5	hence	hence	ADV
ejpam-7052	195	6	f−(k	f−(k	PROPN
ejpam-7052	195	7	)	)	PUNCT
ejpam-7052	195	8	is	be	AUX
ejpam-7052	195	9	µ-closed	µ-close	VERB
ejpam-7052	195	10	in	in	ADP
ejpam-7052	195	11	x.	x.	NOUN
ejpam-7052	195	12	by	by	ADP
ejpam-7052	195	13	theorem	theorem	NOUN
ejpam-7052	195	14	3	3	NUM
ejpam-7052	195	15	,	,	PUNCT
ejpam-7052	195	16	f	f	PROPN
ejpam-7052	195	17	is	be	AUX
ejpam-7052	195	18	upper	upper	ADJ
ejpam-7052	195	19	almost	almost	ADV
ejpam-7052	195	20	µ(σ1	µ(σ1	NOUN
ejpam-7052	195	21	,	,	PUNCT
ejpam-7052	195	22	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7052	195	23	.	.	X
ejpam-7052	195	24	theorem	theorem	VERB
ejpam-7052	195	25	6	6	NUM
ejpam-7052	195	26	.	.	PUNCT
ejpam-7052	195	27	for	for	ADP
ejpam-7052	195	28	a	a	DET
ejpam-7052	195	29	multifunction	multifunction	NOUN
ejpam-7052	195	30	f	f	NOUN
ejpam-7052	195	31	:	:	PUNCT
ejpam-7052	195	32	(	(	PUNCT
ejpam-7052	195	33	x,µ	x,µ	NOUN
ejpam-7052	195	34	)	)	PUNCT
ejpam-7052	195	35	→	→	SYM
ejpam-7052	195	36	(	(	PUNCT
ejpam-7052	195	37	y	y	PROPN
ejpam-7052	195	38	,	,	PUNCT
ejpam-7052	195	39	σ1	σ1	PROPN
ejpam-7052	195	40	,	,	PUNCT
ejpam-7052	195	41	σ2	σ2	NOUN
ejpam-7052	195	42	)	)	PUNCT
ejpam-7052	195	43	,	,	PUNCT
ejpam-7052	195	44	the	the	DET
ejpam-7052	195	45	following	follow	VERB
ejpam-7052	195	46	properties	property	NOUN
ejpam-7052	195	47	are	be	AUX
ejpam-7052	195	48	equivalent	equivalent	ADJ
ejpam-7052	195	49	:	:	PUNCT
ejpam-7052	195	50	(	(	PUNCT
ejpam-7052	195	51	1	1	X
ejpam-7052	195	52	)	)	PUNCT
ejpam-7052	195	53	f	f	PROPN
ejpam-7052	195	54	is	be	AUX
ejpam-7052	195	55	lower	low	ADJ
ejpam-7052	195	56	almost	almost	ADV
ejpam-7052	195	57	µ(σ1	µ(σ1	NOUN
ejpam-7052	195	58	,	,	PUNCT
ejpam-7052	195	59	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	195	60	;	;	PUNCT
ejpam-7052	195	61	(	(	PUNCT
ejpam-7052	195	62	2	2	NUM
ejpam-7052	195	63	)	)	PUNCT
ejpam-7052	195	64	cµ(f	cµ(f	PUNCT
ejpam-7052	196	1	+	+	PROPN
ejpam-7052	196	2	(	(	PUNCT
ejpam-7052	196	3	v	v	NOUN
ejpam-7052	196	4	)	)	PUNCT
ejpam-7052	196	5	)	)	PUNCT
ejpam-7052	196	6	⊆	⊆	NUM
ejpam-7052	196	7	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	196	8	-	-	PUNCT
ejpam-7052	196	9	cl(v	cl(v	NOUN
ejpam-7052	196	10	)	)	PUNCT
ejpam-7052	196	11	)	)	PUNCT
ejpam-7052	196	12	for	for	ADP
ejpam-7052	196	13	every	every	DET
ejpam-7052	196	14	(	(	PUNCT
ejpam-7052	196	15	σ1	σ1	PROPN
ejpam-7052	196	16	,	,	PUNCT
ejpam-7052	196	17	σ2)β	σ2)β	NOUN
ejpam-7052	196	18	-	-	PUNCT
ejpam-7052	196	19	open	open	NOUN
ejpam-7052	196	20	set	set	NOUN
ejpam-7052	196	21	v	v	NOUN
ejpam-7052	196	22	of	of	ADP
ejpam-7052	196	23	y	y	PROPN
ejpam-7052	196	24	;	;	PUNCT
ejpam-7052	196	25	(	(	PUNCT
ejpam-7052	196	26	3	3	X
ejpam-7052	196	27	)	)	PUNCT
ejpam-7052	196	28	cµ(f	cµ(f	PUNCT
ejpam-7052	197	1	+	+	PROPN
ejpam-7052	197	2	(	(	PUNCT
ejpam-7052	197	3	v	v	NOUN
ejpam-7052	197	4	)	)	PUNCT
ejpam-7052	197	5	)	)	PUNCT
ejpam-7052	197	6	⊆	⊆	NUM
ejpam-7052	197	7	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	197	8	-	-	PUNCT
ejpam-7052	197	9	cl(v	cl(v	NOUN
ejpam-7052	197	10	)	)	PUNCT
ejpam-7052	197	11	)	)	PUNCT
ejpam-7052	197	12	for	for	SCONJ
ejpam-7052	197	13	every	every	DET
ejpam-7052	197	14	(	(	PUNCT
ejpam-7052	197	15	σ1	σ1	PROPN
ejpam-7052	197	16	,	,	PUNCT
ejpam-7052	197	17	σ2)s	σ2)s	NOUN
ejpam-7052	197	18	-	-	PUNCT
ejpam-7052	197	19	open	open	NOUN
ejpam-7052	197	20	set	set	NOUN
ejpam-7052	197	21	v	v	NOUN
ejpam-7052	197	22	of	of	ADP
ejpam-7052	197	23	y	y	PROPN
ejpam-7052	197	24	.	.	PUNCT
ejpam-7052	198	1	proof	proof	NOUN
ejpam-7052	198	2	.	.	PUNCT
ejpam-7052	199	1	the	the	DET
ejpam-7052	199	2	proof	proof	NOUN
ejpam-7052	199	3	is	be	AUX
ejpam-7052	199	4	similar	similar	ADJ
ejpam-7052	199	5	to	to	ADP
ejpam-7052	199	6	that	that	PRON
ejpam-7052	199	7	of	of	ADP
ejpam-7052	199	8	theorem	theorem	NOUN
ejpam-7052	199	9	5	5	NUM
ejpam-7052	199	10	.	.	PUNCT
ejpam-7052	199	11	lemma	lemma	PROPN
ejpam-7052	199	12	4	4	NUM
ejpam-7052	199	13	.	.	PUNCT
ejpam-7052	200	1	[	[	X
ejpam-7052	200	2	20	20	NUM
ejpam-7052	200	3	]	]	PUNCT
ejpam-7052	200	4	for	for	ADP
ejpam-7052	200	5	a	a	DET
ejpam-7052	200	6	bitopological	bitopological	ADJ
ejpam-7052	200	7	space	space	NOUN
ejpam-7052	200	8	(	(	PUNCT
ejpam-7052	200	9	x	x	NOUN
ejpam-7052	200	10	,	,	PUNCT
ejpam-7052	200	11	τ1	τ1	NOUN
ejpam-7052	200	12	,	,	PUNCT
ejpam-7052	200	13	τ2	τ2	NOUN
ejpam-7052	200	14	)	)	PUNCT
ejpam-7052	200	15	,	,	PUNCT
ejpam-7052	200	16	the	the	DET
ejpam-7052	200	17	following	follow	VERB
ejpam-7052	200	18	properties	property	NOUN
ejpam-7052	200	19	hold	hold	VERB
ejpam-7052	200	20	:	:	PUNCT
ejpam-7052	200	21	(	(	PUNCT
ejpam-7052	200	22	1	1	X
ejpam-7052	200	23	)	)	PUNCT
ejpam-7052	200	24	α(τ1	α(τ1	NOUN
ejpam-7052	200	25	,	,	PUNCT
ejpam-7052	200	26	τ2)-cl(v	τ2)-cl(v	NOUN
ejpam-7052	200	27	)	)	PUNCT
ejpam-7052	200	28	=	=	PUNCT
ejpam-7052	201	1	τ1τ2	τ1τ2	NOUN
ejpam-7052	201	2	-	-	NOUN
ejpam-7052	201	3	cl(v	cl(v	X
ejpam-7052	201	4	)	)	PUNCT
ejpam-7052	201	5	for	for	ADP
ejpam-7052	201	6	every	every	DET
ejpam-7052	201	7	(	(	PUNCT
ejpam-7052	201	8	τ1	τ1	NOUN
ejpam-7052	201	9	,	,	PUNCT
ejpam-7052	201	10	τ2)β	τ2)β	ADJ
ejpam-7052	201	11	-	-	PUNCT
ejpam-7052	201	12	open	open	NOUN
ejpam-7052	201	13	set	set	NOUN
ejpam-7052	201	14	v	v	NOUN
ejpam-7052	201	15	of	of	ADP
ejpam-7052	201	16	x	x	PRON
ejpam-7052	201	17	;	;	PUNCT
ejpam-7052	201	18	(	(	PUNCT
ejpam-7052	201	19	2	2	X
ejpam-7052	201	20	)	)	PUNCT
ejpam-7052	201	21	(	(	PUNCT
ejpam-7052	201	22	τ1	τ1	NOUN
ejpam-7052	201	23	,	,	PUNCT
ejpam-7052	201	24	τ2)-pcl(v	τ2)-pcl(v	NOUN
ejpam-7052	201	25	)	)	PUNCT
ejpam-7052	201	26	=	=	PUNCT
ejpam-7052	202	1	τ1τ2	τ1τ2	NOUN
ejpam-7052	202	2	-	-	NOUN
ejpam-7052	202	3	cl(v	cl(v	X
ejpam-7052	202	4	)	)	PUNCT
ejpam-7052	202	5	for	for	ADP
ejpam-7052	202	6	every	every	DET
ejpam-7052	202	7	(	(	PUNCT
ejpam-7052	202	8	τ1	τ1	NOUN
ejpam-7052	202	9	,	,	PUNCT
ejpam-7052	202	10	τ2)s	τ2)s	NOUN
ejpam-7052	202	11	-	-	PUNCT
ejpam-7052	202	12	open	open	ADJ
ejpam-7052	202	13	set	set	NOUN
ejpam-7052	202	14	v	v	NOUN
ejpam-7052	202	15	of	of	ADP
ejpam-7052	202	16	x.	x.	NOUN
ejpam-7052	202	17	corollary	corollary	NOUN
ejpam-7052	202	18	1	1	NUM
ejpam-7052	202	19	.	.	PUNCT
ejpam-7052	202	20	for	for	ADP
ejpam-7052	202	21	a	a	DET
ejpam-7052	202	22	multifunction	multifunction	NOUN
ejpam-7052	202	23	f	f	NOUN
ejpam-7052	202	24	:	:	PUNCT
ejpam-7052	202	25	(	(	PUNCT
ejpam-7052	202	26	x,µ	x,µ	NOUN
ejpam-7052	202	27	)	)	PUNCT
ejpam-7052	202	28	→	→	SYM
ejpam-7052	202	29	(	(	PUNCT
ejpam-7052	202	30	y	y	PROPN
ejpam-7052	202	31	,	,	PUNCT
ejpam-7052	202	32	σ1	σ1	PROPN
ejpam-7052	202	33	,	,	PUNCT
ejpam-7052	202	34	σ2	σ2	NOUN
ejpam-7052	202	35	)	)	PUNCT
ejpam-7052	202	36	,	,	PUNCT
ejpam-7052	202	37	the	the	DET
ejpam-7052	202	38	following	follow	VERB
ejpam-7052	202	39	properties	property	NOUN
ejpam-7052	202	40	are	be	AUX
ejpam-7052	202	41	equivalent	equivalent	ADJ
ejpam-7052	202	42	:	:	PUNCT
ejpam-7052	202	43	(	(	PUNCT
ejpam-7052	202	44	1	1	X
ejpam-7052	202	45	)	)	PUNCT
ejpam-7052	202	46	f	f	PROPN
ejpam-7052	202	47	is	be	AUX
ejpam-7052	202	48	upper	upper	ADJ
ejpam-7052	202	49	almost	almost	ADV
ejpam-7052	202	50	µ(σ1	µ(σ1	NOUN
ejpam-7052	202	51	,	,	PUNCT
ejpam-7052	202	52	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	202	53	;	;	PUNCT
ejpam-7052	202	54	(	(	PUNCT
ejpam-7052	202	55	2	2	NUM
ejpam-7052	202	56	)	)	PUNCT
ejpam-7052	202	57	cµ(f	cµ(f	NOUN
ejpam-7052	202	58	−(v	−(v	NOUN
ejpam-7052	202	59	)	)	PUNCT
ejpam-7052	202	60	)	)	PUNCT
ejpam-7052	203	1	⊆	⊆	NUM
ejpam-7052	203	2	f−(α(σ1	f−(α(σ1	NOUN
ejpam-7052	203	3	,	,	PUNCT
ejpam-7052	203	4	σ2)-cl(v	σ2)-cl(v	NOUN
ejpam-7052	203	5	)	)	PUNCT
ejpam-7052	203	6	)	)	PUNCT
ejpam-7052	203	7	for	for	ADP
ejpam-7052	203	8	every	every	DET
ejpam-7052	203	9	(	(	PUNCT
ejpam-7052	203	10	σ1	σ1	PROPN
ejpam-7052	203	11	,	,	PUNCT
ejpam-7052	203	12	σ2)β	σ2)β	NOUN
ejpam-7052	203	13	-	-	PUNCT
ejpam-7052	203	14	open	open	NOUN
ejpam-7052	203	15	set	set	NOUN
ejpam-7052	203	16	v	v	NOUN
ejpam-7052	203	17	of	of	ADP
ejpam-7052	203	18	y	y	PROPN
ejpam-7052	203	19	;	;	PUNCT
ejpam-7052	203	20	(	(	PUNCT
ejpam-7052	203	21	3	3	X
ejpam-7052	203	22	)	)	PUNCT
ejpam-7052	203	23	cµ(f	cµ(f	NOUN
ejpam-7052	203	24	−(v	−(v	NOUN
ejpam-7052	203	25	)	)	PUNCT
ejpam-7052	203	26	)	)	PUNCT
ejpam-7052	204	1	⊆	⊆	NUM
ejpam-7052	204	2	f−((σ1	f−((σ1	NOUN
ejpam-7052	204	3	,	,	PUNCT
ejpam-7052	204	4	σ2)-pcl(v	σ2)-pcl(v	NOUN
ejpam-7052	204	5	)	)	PUNCT
ejpam-7052	204	6	)	)	PUNCT
ejpam-7052	204	7	for	for	ADP
ejpam-7052	204	8	every	every	DET
ejpam-7052	204	9	(	(	PUNCT
ejpam-7052	204	10	σ1	σ1	PROPN
ejpam-7052	204	11	,	,	PUNCT
ejpam-7052	204	12	σ2)s	σ2)s	NOUN
ejpam-7052	204	13	-	-	PUNCT
ejpam-7052	204	14	open	open	NOUN
ejpam-7052	204	15	set	set	NOUN
ejpam-7052	204	16	v	v	NOUN
ejpam-7052	204	17	of	of	ADP
ejpam-7052	204	18	y	y	PROPN
ejpam-7052	204	19	.	.	PUNCT
ejpam-7052	205	1	corollary	corollary	ADJ
ejpam-7052	205	2	2	2	NUM
ejpam-7052	205	3	.	.	PUNCT
ejpam-7052	205	4	for	for	ADP
ejpam-7052	205	5	a	a	DET
ejpam-7052	205	6	multifunction	multifunction	NOUN
ejpam-7052	206	1	f	f	NOUN
ejpam-7052	206	2	:	:	PUNCT
ejpam-7052	206	3	(	(	PUNCT
ejpam-7052	206	4	x,µ	x,µ	NOUN
ejpam-7052	206	5	)	)	PUNCT
ejpam-7052	206	6	→	→	SYM
ejpam-7052	206	7	(	(	PUNCT
ejpam-7052	206	8	y	y	PROPN
ejpam-7052	206	9	,	,	PUNCT
ejpam-7052	206	10	σ1	σ1	PROPN
ejpam-7052	206	11	,	,	PUNCT
ejpam-7052	206	12	σ2	σ2	NOUN
ejpam-7052	206	13	)	)	PUNCT
ejpam-7052	206	14	,	,	PUNCT
ejpam-7052	206	15	the	the	DET
ejpam-7052	206	16	following	follow	VERB
ejpam-7052	206	17	properties	property	NOUN
ejpam-7052	206	18	are	be	AUX
ejpam-7052	206	19	equivalent	equivalent	ADJ
ejpam-7052	206	20	:	:	PUNCT
ejpam-7052	206	21	p.	p.	NOUN
ejpam-7052	206	22	pue	pue	NOUN
ejpam-7052	206	23	-	-	PUNCT
ejpam-7052	206	24	on	on	ADP
ejpam-7052	206	25	,	,	PUNCT
ejpam-7052	206	26	a.	a.	PROPN
ejpam-7052	206	27	sama	sama	PROPN
ejpam-7052	206	28	-	-	PUNCT
ejpam-7052	206	29	ae	ae	PROPN
ejpam-7052	206	30	,	,	PUNCT
ejpam-7052	206	31	c.	c.	PROPN
ejpam-7052	206	32	boonpok	boonpok	PROPN
ejpam-7052	206	33	/	/	SYM
ejpam-7052	206	34	eur	eur	PROPN
ejpam-7052	206	35	.	.	PUNCT
ejpam-7052	207	1	j.	j.	PROPN
ejpam-7052	207	2	pure	pure	PROPN
ejpam-7052	207	3	appl	appl	PROPN
ejpam-7052	207	4	.	.	PROPN
ejpam-7052	207	5	math	math	PROPN
ejpam-7052	207	6	,	,	PUNCT
ejpam-7052	207	7	18	18	NUM
ejpam-7052	207	8	(	(	PUNCT
ejpam-7052	207	9	4	4	NUM
ejpam-7052	207	10	)	)	PUNCT
ejpam-7052	207	11	(	(	PUNCT
ejpam-7052	207	12	2025	2025	NUM
ejpam-7052	207	13	)	)	PUNCT
ejpam-7052	207	14	,	,	PUNCT
ejpam-7052	207	15	7052	7052	NUM
ejpam-7052	207	16	8	8	NUM
ejpam-7052	207	17	of	of	ADP
ejpam-7052	207	18	14	14	NUM
ejpam-7052	207	19	(	(	PUNCT
ejpam-7052	207	20	1	1	NUM
ejpam-7052	207	21	)	)	PUNCT
ejpam-7052	207	22	f	f	PROPN
ejpam-7052	207	23	is	be	AUX
ejpam-7052	207	24	lower	low	ADJ
ejpam-7052	207	25	almost	almost	ADV
ejpam-7052	207	26	µ(σ1	µ(σ1	NOUN
ejpam-7052	207	27	,	,	PUNCT
ejpam-7052	207	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	207	29	;	;	PUNCT
ejpam-7052	207	30	(	(	PUNCT
ejpam-7052	207	31	2	2	NUM
ejpam-7052	207	32	)	)	PUNCT
ejpam-7052	207	33	cµ(f	cµ(f	PUNCT
ejpam-7052	208	1	+	+	PROPN
ejpam-7052	208	2	(	(	PUNCT
ejpam-7052	208	3	v	v	NOUN
ejpam-7052	208	4	)	)	PUNCT
ejpam-7052	208	5	)	)	PUNCT
ejpam-7052	208	6	⊆	⊆	NUM
ejpam-7052	208	7	f+(α(σ1	f+(α(σ1	NOUN
ejpam-7052	208	8	,	,	PUNCT
ejpam-7052	208	9	σ2)-cl(v	σ2)-cl(v	NOUN
ejpam-7052	208	10	)	)	PUNCT
ejpam-7052	208	11	)	)	PUNCT
ejpam-7052	208	12	for	for	ADP
ejpam-7052	208	13	every	every	DET
ejpam-7052	208	14	(	(	PUNCT
ejpam-7052	208	15	σ1	σ1	PROPN
ejpam-7052	208	16	,	,	PUNCT
ejpam-7052	208	17	σ2)β	σ2)β	NOUN
ejpam-7052	208	18	-	-	PUNCT
ejpam-7052	208	19	open	open	NOUN
ejpam-7052	208	20	set	set	NOUN
ejpam-7052	208	21	v	v	NOUN
ejpam-7052	208	22	of	of	ADP
ejpam-7052	208	23	y	y	PROPN
ejpam-7052	208	24	;	;	PUNCT
ejpam-7052	208	25	(	(	PUNCT
ejpam-7052	208	26	3	3	X
ejpam-7052	208	27	)	)	PUNCT
ejpam-7052	208	28	cµ(f	cµ(f	PUNCT
ejpam-7052	209	1	+	+	PROPN
ejpam-7052	209	2	(	(	PUNCT
ejpam-7052	209	3	v	v	NOUN
ejpam-7052	209	4	)	)	PUNCT
ejpam-7052	209	5	)	)	PUNCT
ejpam-7052	209	6	⊆	⊆	NUM
ejpam-7052	209	7	f+((σ1	f+((σ1	NOUN
ejpam-7052	209	8	,	,	PUNCT
ejpam-7052	209	9	σ2)-pcl(v	σ2)-pcl(v	NOUN
ejpam-7052	209	10	)	)	PUNCT
ejpam-7052	209	11	)	)	PUNCT
ejpam-7052	210	1	for	for	ADP
ejpam-7052	210	2	every	every	DET
ejpam-7052	210	3	(	(	PUNCT
ejpam-7052	210	4	σ1	σ1	PROPN
ejpam-7052	210	5	,	,	PUNCT
ejpam-7052	210	6	σ2)s	σ2)s	NOUN
ejpam-7052	210	7	-	-	PUNCT
ejpam-7052	210	8	open	open	NOUN
ejpam-7052	210	9	set	set	NOUN
ejpam-7052	210	10	v	v	NOUN
ejpam-7052	210	11	of	of	ADP
ejpam-7052	210	12	y	y	PROPN
ejpam-7052	210	13	.	.	PUNCT
ejpam-7052	210	14	theorem	theorem	VERB
ejpam-7052	210	15	7	7	NUM
ejpam-7052	210	16	.	.	X
ejpam-7052	210	17	for	for	ADP
ejpam-7052	210	18	a	a	DET
ejpam-7052	210	19	multifunction	multifunction	NOUN
ejpam-7052	211	1	f	f	NOUN
ejpam-7052	211	2	:	:	PUNCT
ejpam-7052	211	3	(	(	PUNCT
ejpam-7052	211	4	x,µ	x,µ	NOUN
ejpam-7052	211	5	)	)	PUNCT
ejpam-7052	211	6	→	→	SYM
ejpam-7052	211	7	(	(	PUNCT
ejpam-7052	211	8	y	y	PROPN
ejpam-7052	211	9	,	,	PUNCT
ejpam-7052	211	10	σ1	σ1	PROPN
ejpam-7052	211	11	,	,	PUNCT
ejpam-7052	211	12	σ2	σ2	NOUN
ejpam-7052	211	13	)	)	PUNCT
ejpam-7052	211	14	,	,	PUNCT
ejpam-7052	211	15	the	the	DET
ejpam-7052	211	16	following	follow	VERB
ejpam-7052	211	17	properties	property	NOUN
ejpam-7052	211	18	are	be	AUX
ejpam-7052	211	19	equivalent	equivalent	ADJ
ejpam-7052	211	20	:	:	PUNCT
ejpam-7052	211	21	(	(	PUNCT
ejpam-7052	211	22	1	1	X
ejpam-7052	211	23	)	)	PUNCT
ejpam-7052	211	24	f	f	PROPN
ejpam-7052	211	25	is	be	AUX
ejpam-7052	211	26	upper	upper	ADJ
ejpam-7052	211	27	almost	almost	ADV
ejpam-7052	211	28	µ(σ1	µ(σ1	NOUN
ejpam-7052	211	29	,	,	PUNCT
ejpam-7052	211	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	211	31	;	;	PUNCT
ejpam-7052	211	32	(	(	PUNCT
ejpam-7052	211	33	2	2	NUM
ejpam-7052	211	34	)	)	PUNCT
ejpam-7052	211	35	cµ(f	cµ(f	NOUN
ejpam-7052	211	36	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	211	37	-	-	PUNCT
ejpam-7052	211	38	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	211	39	-	-	PUNCT
ejpam-7052	211	40	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	211	41	-	-	PUNCT
ejpam-7052	211	42	cl(v	cl(v	NOUN
ejpam-7052	211	43	)	)	PUNCT
ejpam-7052	211	44	)	)	PUNCT
ejpam-7052	211	45	)	)	PUNCT
ejpam-7052	211	46	)	)	PUNCT
ejpam-7052	211	47	)	)	PUNCT
ejpam-7052	212	1	⊆	⊆	X
ejpam-7052	212	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	212	3	-	-	PUNCT
ejpam-7052	212	4	cl(v	cl(v	NOUN
ejpam-7052	212	5	)	)	PUNCT
ejpam-7052	212	6	)	)	PUNCT
ejpam-7052	212	7	for	for	ADP
ejpam-7052	212	8	every	every	DET
ejpam-7052	212	9	(	(	PUNCT
ejpam-7052	212	10	σ1	σ1	PROPN
ejpam-7052	212	11	,	,	PUNCT
ejpam-7052	212	12	σ2)p	σ2)p	NOUN
ejpam-7052	212	13	-	-	PUNCT
ejpam-7052	212	14	open	open	NOUN
ejpam-7052	212	15	set	set	NOUN
ejpam-7052	212	16	v	v	NOUN
ejpam-7052	212	17	of	of	ADP
ejpam-7052	212	18	y	y	PROPN
ejpam-7052	212	19	;	;	PUNCT
ejpam-7052	212	20	(	(	PUNCT
ejpam-7052	212	21	3	3	X
ejpam-7052	212	22	)	)	PUNCT
ejpam-7052	212	23	cµ(f	cµ(f	NOUN
ejpam-7052	212	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	212	25	-	-	PUNCT
ejpam-7052	212	26	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	212	27	-	-	PUNCT
ejpam-7052	212	28	int(v	int(v	NOUN
ejpam-7052	212	29	)	)	PUNCT
ejpam-7052	212	30	)	)	PUNCT
ejpam-7052	212	31	)	)	PUNCT
ejpam-7052	212	32	)	)	PUNCT
ejpam-7052	213	1	⊆	⊆	X
ejpam-7052	213	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	213	3	-	-	PUNCT
ejpam-7052	213	4	cl(v	cl(v	NOUN
ejpam-7052	213	5	)	)	PUNCT
ejpam-7052	213	6	)	)	PUNCT
ejpam-7052	213	7	for	for	ADP
ejpam-7052	213	8	every	every	DET
ejpam-7052	213	9	(	(	PUNCT
ejpam-7052	213	10	σ1	σ1	PROPN
ejpam-7052	213	11	,	,	PUNCT
ejpam-7052	213	12	σ2)p	σ2)p	NOUN
ejpam-7052	213	13	-	-	PUNCT
ejpam-7052	213	14	open	open	NOUN
ejpam-7052	213	15	set	set	NOUN
ejpam-7052	213	16	v	v	NOUN
ejpam-7052	213	17	of	of	ADP
ejpam-7052	213	18	y	y	PROPN
ejpam-7052	213	19	;	;	PUNCT
ejpam-7052	213	20	(	(	PUNCT
ejpam-7052	213	21	4	4	X
ejpam-7052	213	22	)	)	PUNCT
ejpam-7052	213	23	f+(v	f+(v	NOUN
ejpam-7052	213	24	)	)	PUNCT
ejpam-7052	214	1	⊆	⊆	NUM
ejpam-7052	214	2	iµ(f	iµ(f	NUM
ejpam-7052	214	3	+	+	NOUN
ejpam-7052	214	4	(	(	PUNCT
ejpam-7052	214	5	σ1σ2	σ1σ2	NUM
ejpam-7052	214	6	-	-	PUNCT
ejpam-7052	214	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	214	8	-	-	PUNCT
ejpam-7052	214	9	cl(v	cl(v	NOUN
ejpam-7052	214	10	)	)	PUNCT
ejpam-7052	214	11	)	)	PUNCT
ejpam-7052	214	12	)	)	PUNCT
ejpam-7052	214	13	)	)	PUNCT
ejpam-7052	215	1	for	for	ADP
ejpam-7052	215	2	every	every	DET
ejpam-7052	215	3	(	(	PUNCT
ejpam-7052	215	4	σ1	σ1	PROPN
ejpam-7052	215	5	,	,	PUNCT
ejpam-7052	215	6	σ2)p	σ2)p	NOUN
ejpam-7052	215	7	-	-	PUNCT
ejpam-7052	215	8	open	open	NOUN
ejpam-7052	215	9	set	set	NOUN
ejpam-7052	215	10	v	v	NOUN
ejpam-7052	215	11	of	of	ADP
ejpam-7052	215	12	y	y	PROPN
ejpam-7052	215	13	.	.	PUNCT
ejpam-7052	216	1	proof	proof	NOUN
ejpam-7052	216	2	.	.	PUNCT
ejpam-7052	217	1	(	(	PUNCT
ejpam-7052	217	2	1	1	X
ejpam-7052	217	3	)	)	PUNCT
ejpam-7052	217	4	⇒	⇒	NOUN
ejpam-7052	217	5	(	(	PUNCT
ejpam-7052	217	6	2	2	NUM
ejpam-7052	217	7	):	):	PUNCT
ejpam-7052	217	8	let	let	VERB
ejpam-7052	217	9	v	v	PART
ejpam-7052	217	10	be	be	AUX
ejpam-7052	217	11	any	any	DET
ejpam-7052	217	12	(	(	PUNCT
ejpam-7052	217	13	σ1	σ1	PROPN
ejpam-7052	217	14	,	,	PUNCT
ejpam-7052	217	15	σ2)p	σ2)p	NOUN
ejpam-7052	217	16	-	-	PUNCT
ejpam-7052	217	17	open	open	ADJ
ejpam-7052	217	18	set	set	NOUN
ejpam-7052	217	19	of	of	ADP
ejpam-7052	217	20	y	y	PROPN
ejpam-7052	217	21	.	.	PUNCT
ejpam-7052	218	1	then	then	ADV
ejpam-7052	218	2	,	,	PUNCT
ejpam-7052	218	3	σ1σ2	σ1σ2	NOUN
ejpam-7052	218	4	-	-	NUM
ejpam-7052	218	5	cl(v	cl(v	NOUN
ejpam-7052	218	6	)	)	PUNCT
ejpam-7052	218	7	is	be	AUX
ejpam-7052	218	8	σ1σ2closed	σ1σ2close	VERB
ejpam-7052	218	9	in	in	ADP
ejpam-7052	218	10	y	y	PROPN
ejpam-7052	218	11	and	and	CCONJ
ejpam-7052	218	12	by	by	ADP
ejpam-7052	218	13	theorem	theorem	NOUN
ejpam-7052	218	14	3	3	NUM
ejpam-7052	218	15	,	,	PUNCT
ejpam-7052	218	16	we	we	PRON
ejpam-7052	218	17	have	have	VERB
ejpam-7052	218	18	cµ(f	cµ(f	NOUN
ejpam-7052	218	19	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	218	20	-	-	PUNCT
ejpam-7052	218	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	218	22	-	-	PUNCT
ejpam-7052	218	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	218	24	-	-	PUNCT
ejpam-7052	218	25	cl(v	cl(v	NOUN
ejpam-7052	218	26	)	)	PUNCT
ejpam-7052	218	27	)	)	PUNCT
ejpam-7052	218	28	)	)	PUNCT
ejpam-7052	218	29	)	)	PUNCT
ejpam-7052	218	30	)	)	PUNCT
ejpam-7052	219	1	⊆	⊆	X
ejpam-7052	219	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	219	3	-	-	PUNCT
ejpam-7052	219	4	cl(v	cl(v	NOUN
ejpam-7052	219	5	)	)	PUNCT
ejpam-7052	219	6	)	)	PUNCT
ejpam-7052	219	7	.	.	PUNCT
ejpam-7052	220	1	(	(	PUNCT
ejpam-7052	220	2	2	2	X
ejpam-7052	220	3	)	)	PUNCT
ejpam-7052	220	4	⇒	⇒	NOUN
ejpam-7052	220	5	(	(	PUNCT
ejpam-7052	220	6	3	3	NUM
ejpam-7052	220	7	):	):	PUNCT
ejpam-7052	220	8	let	let	VERB
ejpam-7052	220	9	v	v	PART
ejpam-7052	220	10	be	be	AUX
ejpam-7052	220	11	any	any	DET
ejpam-7052	220	12	(	(	PUNCT
ejpam-7052	220	13	σ1	σ1	PROPN
ejpam-7052	220	14	,	,	PUNCT
ejpam-7052	220	15	σ2)p	σ2)p	NOUN
ejpam-7052	220	16	-	-	PUNCT
ejpam-7052	220	17	open	open	ADJ
ejpam-7052	220	18	set	set	NOUN
ejpam-7052	220	19	of	of	ADP
ejpam-7052	220	20	y	y	PROPN
ejpam-7052	220	21	.	.	PUNCT
ejpam-7052	221	1	by	by	ADP
ejpam-7052	221	2	(	(	PUNCT
ejpam-7052	221	3	2	2	NUM
ejpam-7052	221	4	)	)	PUNCT
ejpam-7052	221	5	,	,	PUNCT
ejpam-7052	221	6	cµ(f	cµ(f	NOUN
ejpam-7052	221	7	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	221	8	-	-	PUNCT
ejpam-7052	221	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	221	10	-	-	PUNCT
ejpam-7052	221	11	int(v	int(v	NOUN
ejpam-7052	221	12	)	)	PUNCT
ejpam-7052	221	13	)	)	PUNCT
ejpam-7052	221	14	)	)	PUNCT
ejpam-7052	221	15	)	)	PUNCT
ejpam-7052	222	1	⊆	⊆	NUM
ejpam-7052	222	2	cµ(f	cµ(f	NUM
ejpam-7052	222	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	222	4	-	-	PUNCT
ejpam-7052	222	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	222	6	-	-	PUNCT
ejpam-7052	222	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	222	8	-	-	PUNCT
ejpam-7052	222	9	cl(v	cl(v	NOUN
ejpam-7052	222	10	)	)	PUNCT
ejpam-7052	222	11	)	)	PUNCT
ejpam-7052	222	12	)	)	PUNCT
ejpam-7052	222	13	)	)	PUNCT
ejpam-7052	222	14	)	)	PUNCT
ejpam-7052	223	1	⊆	⊆	X
ejpam-7052	223	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	223	3	-	-	PUNCT
ejpam-7052	223	4	cl(v	cl(v	NOUN
ejpam-7052	223	5	)	)	PUNCT
ejpam-7052	223	6	)	)	PUNCT
ejpam-7052	223	7	.	.	PUNCT
ejpam-7052	224	1	(	(	PUNCT
ejpam-7052	224	2	3	3	X
ejpam-7052	224	3	)	)	PUNCT
ejpam-7052	224	4	⇒	⇒	NOUN
ejpam-7052	224	5	(	(	PUNCT
ejpam-7052	224	6	4	4	NUM
ejpam-7052	224	7	):	):	PUNCT
ejpam-7052	224	8	let	let	VERB
ejpam-7052	224	9	v	v	PART
ejpam-7052	224	10	be	be	AUX
ejpam-7052	224	11	any	any	DET
ejpam-7052	224	12	(	(	PUNCT
ejpam-7052	224	13	σ1	σ1	PROPN
ejpam-7052	224	14	,	,	PUNCT
ejpam-7052	224	15	σ2)p	σ2)p	NOUN
ejpam-7052	224	16	-	-	PUNCT
ejpam-7052	224	17	open	open	ADJ
ejpam-7052	224	18	set	set	NOUN
ejpam-7052	224	19	of	of	ADP
ejpam-7052	224	20	y	y	PROPN
ejpam-7052	224	21	.	.	PUNCT
ejpam-7052	225	1	thus	thus	ADV
ejpam-7052	225	2	by	by	ADP
ejpam-7052	225	3	(	(	PUNCT
ejpam-7052	225	4	3	3	NUM
ejpam-7052	225	5	)	)	PUNCT
ejpam-7052	225	6	,	,	PUNCT
ejpam-7052	225	7	we	we	PRON
ejpam-7052	225	8	have	have	VERB
ejpam-7052	225	9	x	x	PART
ejpam-7052	225	10	−	−	NOUN
ejpam-7052	225	11	iµ(f	iµ(f	NOUN
ejpam-7052	226	1	+	+	NOUN
ejpam-7052	226	2	(	(	PUNCT
ejpam-7052	226	3	σ1σ2	σ1σ2	NUM
ejpam-7052	226	4	-	-	PUNCT
ejpam-7052	226	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	226	6	-	-	PUNCT
ejpam-7052	226	7	cl(v	cl(v	NOUN
ejpam-7052	226	8	)	)	PUNCT
ejpam-7052	226	9	)	)	PUNCT
ejpam-7052	226	10	)	)	PUNCT
ejpam-7052	226	11	)	)	PUNCT
ejpam-7052	227	1	=	=	PRON
ejpam-7052	227	2	cµ(x	cµ(x	PUNCT
ejpam-7052	227	3	−	−	ADP
ejpam-7052	227	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	227	5	-	-	PUNCT
ejpam-7052	227	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	227	7	-	-	PUNCT
ejpam-7052	227	8	cl(v	cl(v	NOUN
ejpam-7052	227	9	)	)	PUNCT
ejpam-7052	227	10	)	)	PUNCT
ejpam-7052	227	11	)	)	PUNCT
ejpam-7052	227	12	)	)	PUNCT
ejpam-7052	228	1	=	=	SYM
ejpam-7052	228	2	cµ(f	cµ(f	NOUN
ejpam-7052	229	1	−(y	−(y	NOUN
ejpam-7052	229	2	−	−	NOUN
ejpam-7052	229	3	σ1σ2	σ1σ2	SYM
ejpam-7052	229	4	-	-	PUNCT
ejpam-7052	229	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	229	6	-	-	PUNCT
ejpam-7052	229	7	cl(v	cl(v	NOUN
ejpam-7052	229	8	)	)	PUNCT
ejpam-7052	229	9	)	)	PUNCT
ejpam-7052	229	10	)	)	PUNCT
ejpam-7052	229	11	)	)	PUNCT
ejpam-7052	230	1	=	=	SYM
ejpam-7052	230	2	cµ(f	cµ(f	NOUN
ejpam-7052	230	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	230	4	-	-	ADJ
ejpam-7052	230	5	cl(y	cl(y	NUM
ejpam-7052	230	6	−	−	NOUN
ejpam-7052	230	7	σ1σ2	σ1σ2	NOUN
ejpam-7052	230	8	-	-	NUM
ejpam-7052	230	9	cl(v	cl(v	NOUN
ejpam-7052	230	10	)	)	PUNCT
ejpam-7052	230	11	)	)	PUNCT
ejpam-7052	230	12	)	)	PUNCT
ejpam-7052	230	13	)	)	PUNCT
ejpam-7052	231	1	=	=	SYM
ejpam-7052	231	2	cµ(f	cµ(f	NOUN
ejpam-7052	231	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	231	4	-	-	PUNCT
ejpam-7052	231	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	231	6	-	-	PUNCT
ejpam-7052	231	7	int(y	int(y	PROPN
ejpam-7052	231	8	−	−	NUM
ejpam-7052	231	9	σ1σ2	σ1σ2	NOUN
ejpam-7052	231	10	-	-	NUM
ejpam-7052	231	11	cl(v	cl(v	NOUN
ejpam-7052	231	12	)	)	PUNCT
ejpam-7052	231	13	)	)	PUNCT
ejpam-7052	231	14	)	)	PUNCT
ejpam-7052	231	15	)	)	PUNCT
ejpam-7052	231	16	)	)	PUNCT
ejpam-7052	232	1	⊆	⊆	X
ejpam-7052	232	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	232	3	-	-	PUNCT
ejpam-7052	232	4	cl(y	cl(y	NOUN
ejpam-7052	232	5	−	−	NOUN
ejpam-7052	232	6	σ1σ2	σ1σ2	NOUN
ejpam-7052	232	7	-	-	NUM
ejpam-7052	232	8	cl(v	cl(v	NOUN
ejpam-7052	232	9	)	)	PUNCT
ejpam-7052	232	10	)	)	PUNCT
ejpam-7052	232	11	)	)	PUNCT
ejpam-7052	233	1	=	=	PUNCT
ejpam-7052	233	2	f−(y	f−(y	NOUN
ejpam-7052	233	3	−	−	ADP
ejpam-7052	233	4	σ1σ2	σ1σ2	NOUN
ejpam-7052	233	5	-	-	PUNCT
ejpam-7052	233	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	233	7	-	-	PUNCT
ejpam-7052	233	8	cl(v	cl(v	NOUN
ejpam-7052	233	9	)	)	PUNCT
ejpam-7052	233	10	)	)	PUNCT
ejpam-7052	233	11	)	)	PUNCT
ejpam-7052	234	1	⊆	⊆	NUM
ejpam-7052	234	2	f−(y	f−(y	NOUN
ejpam-7052	234	3	−	−	NOUN
ejpam-7052	234	4	v	v	NOUN
ejpam-7052	234	5	)	)	PUNCT
ejpam-7052	234	6	=	=	PUNCT
ejpam-7052	234	7	x	x	X
ejpam-7052	234	8	−	−	PROPN
ejpam-7052	234	9	f+(v	f+(v	NOUN
ejpam-7052	234	10	)	)	PUNCT
ejpam-7052	234	11	and	and	CCONJ
ejpam-7052	234	12	hence	hence	ADV
ejpam-7052	234	13	f+(v	f+(v	PROPN
ejpam-7052	234	14	)	)	PUNCT
ejpam-7052	235	1	⊆	⊆	NUM
ejpam-7052	235	2	iµ(f	iµ(f	NUM
ejpam-7052	235	3	+	+	NOUN
ejpam-7052	235	4	(	(	PUNCT
ejpam-7052	235	5	σ1σ2	σ1σ2	NUM
ejpam-7052	235	6	-	-	PUNCT
ejpam-7052	235	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	235	8	-	-	PUNCT
ejpam-7052	235	9	cl(v	cl(v	NOUN
ejpam-7052	235	10	)	)	PUNCT
ejpam-7052	235	11	)	)	PUNCT
ejpam-7052	235	12	)	)	PUNCT
ejpam-7052	235	13	)	)	PUNCT
ejpam-7052	235	14	.	.	PUNCT
ejpam-7052	236	1	(	(	PUNCT
ejpam-7052	236	2	4	4	X
ejpam-7052	236	3	)	)	PUNCT
ejpam-7052	236	4	⇒	⇒	NOUN
ejpam-7052	236	5	(	(	PUNCT
ejpam-7052	236	6	1	1	NUM
ejpam-7052	236	7	):	):	PUNCT
ejpam-7052	236	8	let	let	VERB
ejpam-7052	236	9	v	v	PART
ejpam-7052	236	10	be	be	AUX
ejpam-7052	236	11	any	any	DET
ejpam-7052	236	12	(	(	PUNCT
ejpam-7052	236	13	σ1	σ1	NOUN
ejpam-7052	236	14	,	,	PUNCT
ejpam-7052	236	15	σ2)r	σ2)r	NOUN
ejpam-7052	236	16	-	-	PUNCT
ejpam-7052	236	17	open	open	ADJ
ejpam-7052	236	18	set	set	NOUN
ejpam-7052	236	19	of	of	ADP
ejpam-7052	236	20	y	y	PROPN
ejpam-7052	236	21	.	.	PUNCT
ejpam-7052	237	1	then	then	ADV
ejpam-7052	237	2	,	,	PUNCT
ejpam-7052	237	3	v	v	NOUN
ejpam-7052	237	4	is	be	AUX
ejpam-7052	237	5	(	(	PUNCT
ejpam-7052	237	6	σ1	σ1	PROPN
ejpam-7052	237	7	,	,	PUNCT
ejpam-7052	237	8	σ2)p	σ2)p	NOUN
ejpam-7052	237	9	-	-	PUNCT
ejpam-7052	237	10	open	open	ADJ
ejpam-7052	237	11	in	in	ADP
ejpam-7052	237	12	y	y	PROPN
ejpam-7052	237	13	and	and	CCONJ
ejpam-7052	237	14	by	by	ADP
ejpam-7052	237	15	(	(	PUNCT
ejpam-7052	237	16	4	4	NUM
ejpam-7052	237	17	)	)	PUNCT
ejpam-7052	237	18	,	,	PUNCT
ejpam-7052	237	19	f+(v	f+(v	PROPN
ejpam-7052	237	20	)	)	PUNCT
ejpam-7052	238	1	⊆	⊆	NUM
ejpam-7052	238	2	iµ(f	iµ(f	NUM
ejpam-7052	238	3	+	+	NOUN
ejpam-7052	238	4	(	(	PUNCT
ejpam-7052	238	5	σ1σ2	σ1σ2	NUM
ejpam-7052	238	6	-	-	PUNCT
ejpam-7052	238	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	238	8	-	-	PUNCT
ejpam-7052	238	9	cl(v	cl(v	NOUN
ejpam-7052	238	10	)	)	PUNCT
ejpam-7052	238	11	)	)	PUNCT
ejpam-7052	238	12	)	)	PUNCT
ejpam-7052	238	13	)	)	PUNCT
ejpam-7052	239	1	=	=	PRON
ejpam-7052	239	2	iµ(f	iµ(f	NUM
ejpam-7052	239	3	+	+	ADJ
ejpam-7052	239	4	(	(	PUNCT
ejpam-7052	239	5	v	v	NOUN
ejpam-7052	239	6	)	)	PUNCT
ejpam-7052	239	7	)	)	PUNCT
ejpam-7052	239	8	.	.	PUNCT
ejpam-7052	240	1	thus	thus	ADV
ejpam-7052	240	2	,	,	PUNCT
ejpam-7052	240	3	f+(v	f+(v	PROPN
ejpam-7052	240	4	)	)	PUNCT
ejpam-7052	240	5	is	be	AUX
ejpam-7052	240	6	µ-open	µ-open	NOUN
ejpam-7052	240	7	in	in	ADP
ejpam-7052	240	8	x.	x.	NOUN
ejpam-7052	240	9	it	it	PRON
ejpam-7052	240	10	follows	follow	VERB
ejpam-7052	240	11	from	from	ADP
ejpam-7052	240	12	theorem	theorem	ADJ
ejpam-7052	240	13	3	3	NUM
ejpam-7052	240	14	that	that	SCONJ
ejpam-7052	240	15	f	f	PROPN
ejpam-7052	240	16	is	be	AUX
ejpam-7052	240	17	upper	upper	ADJ
ejpam-7052	240	18	almost	almost	ADV
ejpam-7052	240	19	µ(σ1	µ(σ1	NOUN
ejpam-7052	240	20	,	,	PUNCT
ejpam-7052	240	21	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	240	22	.	.	PUNCT
ejpam-7052	241	1	p.	p.	NOUN
ejpam-7052	241	2	pue	pue	NOUN
ejpam-7052	241	3	-	-	PUNCT
ejpam-7052	241	4	on	on	ADP
ejpam-7052	241	5	,	,	PUNCT
ejpam-7052	241	6	a.	a.	PROPN
ejpam-7052	241	7	sama	sama	PROPN
ejpam-7052	241	8	-	-	PUNCT
ejpam-7052	241	9	ae	ae	PROPN
ejpam-7052	241	10	,	,	PUNCT
ejpam-7052	241	11	c.	c.	PROPN
ejpam-7052	241	12	boonpok	boonpok	PROPN
ejpam-7052	241	13	/	/	SYM
ejpam-7052	241	14	eur	eur	PROPN
ejpam-7052	241	15	.	.	PUNCT
ejpam-7052	242	1	j.	j.	PROPN
ejpam-7052	242	2	pure	pure	PROPN
ejpam-7052	242	3	appl	appl	PROPN
ejpam-7052	242	4	.	.	PROPN
ejpam-7052	242	5	math	math	PROPN
ejpam-7052	242	6	,	,	PUNCT
ejpam-7052	242	7	18	18	NUM
ejpam-7052	242	8	(	(	PUNCT
ejpam-7052	242	9	4	4	NUM
ejpam-7052	242	10	)	)	PUNCT
ejpam-7052	242	11	(	(	PUNCT
ejpam-7052	242	12	2025	2025	NUM
ejpam-7052	242	13	)	)	PUNCT
ejpam-7052	242	14	,	,	PUNCT
ejpam-7052	242	15	7052	7052	NUM
ejpam-7052	242	16	9	9	NUM
ejpam-7052	242	17	of	of	ADP
ejpam-7052	242	18	14	14	NUM
ejpam-7052	242	19	theorem	theorem	NOUN
ejpam-7052	242	20	8	8	NUM
ejpam-7052	242	21	.	.	PUNCT
ejpam-7052	243	1	for	for	ADP
ejpam-7052	243	2	a	a	DET
ejpam-7052	243	3	multifunction	multifunction	NOUN
ejpam-7052	243	4	f	f	NOUN
ejpam-7052	243	5	:	:	PUNCT
ejpam-7052	243	6	(	(	PUNCT
ejpam-7052	243	7	x,µ	x,µ	NOUN
ejpam-7052	243	8	)	)	PUNCT
ejpam-7052	243	9	→	→	SYM
ejpam-7052	243	10	(	(	PUNCT
ejpam-7052	243	11	y	y	PROPN
ejpam-7052	243	12	,	,	PUNCT
ejpam-7052	243	13	σ1	σ1	PROPN
ejpam-7052	243	14	,	,	PUNCT
ejpam-7052	243	15	σ2	σ2	NOUN
ejpam-7052	243	16	)	)	PUNCT
ejpam-7052	243	17	,	,	PUNCT
ejpam-7052	243	18	the	the	DET
ejpam-7052	243	19	following	follow	VERB
ejpam-7052	243	20	properties	property	NOUN
ejpam-7052	243	21	are	be	AUX
ejpam-7052	243	22	equivalent	equivalent	ADJ
ejpam-7052	243	23	:	:	PUNCT
ejpam-7052	243	24	(	(	PUNCT
ejpam-7052	243	25	1	1	X
ejpam-7052	243	26	)	)	PUNCT
ejpam-7052	243	27	f	f	PROPN
ejpam-7052	243	28	is	be	AUX
ejpam-7052	243	29	lower	low	ADJ
ejpam-7052	243	30	almost	almost	ADV
ejpam-7052	243	31	µ(σ1	µ(σ1	NOUN
ejpam-7052	243	32	,	,	PUNCT
ejpam-7052	243	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	243	34	;	;	PUNCT
ejpam-7052	243	35	(	(	PUNCT
ejpam-7052	243	36	2	2	NUM
ejpam-7052	243	37	)	)	PUNCT
ejpam-7052	243	38	cµ(f	cµ(f	PUNCT
ejpam-7052	244	1	+	+	PROPN
ejpam-7052	244	2	(	(	PUNCT
ejpam-7052	244	3	σ1σ2	σ1σ2	NUM
ejpam-7052	244	4	-	-	PUNCT
ejpam-7052	244	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	244	6	-	-	PUNCT
ejpam-7052	244	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	244	8	-	-	PUNCT
ejpam-7052	244	9	cl(v	cl(v	NOUN
ejpam-7052	244	10	)	)	PUNCT
ejpam-7052	244	11	)	)	PUNCT
ejpam-7052	244	12	)	)	PUNCT
ejpam-7052	244	13	)	)	PUNCT
ejpam-7052	244	14	)	)	PUNCT
ejpam-7052	245	1	⊆	⊆	X
ejpam-7052	245	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	245	3	-	-	PUNCT
ejpam-7052	245	4	cl(v	cl(v	NOUN
ejpam-7052	245	5	)	)	PUNCT
ejpam-7052	245	6	)	)	PUNCT
ejpam-7052	245	7	for	for	ADP
ejpam-7052	245	8	every	every	DET
ejpam-7052	245	9	(	(	PUNCT
ejpam-7052	245	10	σ1	σ1	PROPN
ejpam-7052	245	11	,	,	PUNCT
ejpam-7052	245	12	σ2)p	σ2)p	NOUN
ejpam-7052	245	13	-	-	PUNCT
ejpam-7052	245	14	open	open	NOUN
ejpam-7052	245	15	set	set	NOUN
ejpam-7052	245	16	v	v	NOUN
ejpam-7052	245	17	of	of	ADP
ejpam-7052	245	18	y	y	PROPN
ejpam-7052	245	19	;	;	PUNCT
ejpam-7052	245	20	(	(	PUNCT
ejpam-7052	245	21	3	3	X
ejpam-7052	245	22	)	)	PUNCT
ejpam-7052	245	23	cµ(f	cµ(f	PUNCT
ejpam-7052	246	1	+	+	PROPN
ejpam-7052	246	2	(	(	PUNCT
ejpam-7052	246	3	σ1σ2	σ1σ2	NUM
ejpam-7052	246	4	-	-	PUNCT
ejpam-7052	246	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	246	6	-	-	PUNCT
ejpam-7052	246	7	int(v	int(v	NOUN
ejpam-7052	246	8	)	)	PUNCT
ejpam-7052	246	9	)	)	PUNCT
ejpam-7052	246	10	)	)	PUNCT
ejpam-7052	246	11	)	)	PUNCT
ejpam-7052	247	1	⊆	⊆	X
ejpam-7052	247	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	247	3	-	-	PUNCT
ejpam-7052	247	4	cl(v	cl(v	NOUN
ejpam-7052	247	5	)	)	PUNCT
ejpam-7052	247	6	)	)	PUNCT
ejpam-7052	247	7	for	for	ADP
ejpam-7052	247	8	every	every	DET
ejpam-7052	247	9	(	(	PUNCT
ejpam-7052	247	10	σ1	σ1	PROPN
ejpam-7052	247	11	,	,	PUNCT
ejpam-7052	247	12	σ2)p	σ2)p	NOUN
ejpam-7052	247	13	-	-	PUNCT
ejpam-7052	247	14	open	open	NOUN
ejpam-7052	247	15	set	set	NOUN
ejpam-7052	247	16	v	v	NOUN
ejpam-7052	247	17	of	of	ADP
ejpam-7052	247	18	y	y	PROPN
ejpam-7052	247	19	;	;	PUNCT
ejpam-7052	247	20	(	(	PUNCT
ejpam-7052	247	21	4	4	X
ejpam-7052	247	22	)	)	PUNCT
ejpam-7052	247	23	f−(v	f−(v	NOUN
ejpam-7052	247	24	)	)	PUNCT
ejpam-7052	247	25	⊆	⊆	NUM
ejpam-7052	247	26	iµ(f	iµ(f	NUM
ejpam-7052	247	27	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	247	28	-	-	PUNCT
ejpam-7052	247	29	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	247	30	-	-	PUNCT
ejpam-7052	247	31	cl(v	cl(v	NOUN
ejpam-7052	247	32	)	)	PUNCT
ejpam-7052	247	33	)	)	PUNCT
ejpam-7052	247	34	)	)	PUNCT
ejpam-7052	247	35	)	)	PUNCT
ejpam-7052	248	1	for	for	ADP
ejpam-7052	248	2	every	every	DET
ejpam-7052	248	3	(	(	PUNCT
ejpam-7052	248	4	σ1	σ1	PROPN
ejpam-7052	248	5	,	,	PUNCT
ejpam-7052	248	6	σ2)p	σ2)p	NOUN
ejpam-7052	248	7	-	-	PUNCT
ejpam-7052	248	8	open	open	NOUN
ejpam-7052	248	9	set	set	NOUN
ejpam-7052	248	10	v	v	NOUN
ejpam-7052	248	11	of	of	ADP
ejpam-7052	248	12	y	y	PROPN
ejpam-7052	248	13	.	.	PUNCT
ejpam-7052	249	1	proof	proof	NOUN
ejpam-7052	249	2	.	.	PUNCT
ejpam-7052	250	1	the	the	DET
ejpam-7052	250	2	proof	proof	NOUN
ejpam-7052	250	3	is	be	AUX
ejpam-7052	250	4	similar	similar	ADJ
ejpam-7052	250	5	to	to	ADP
ejpam-7052	250	6	that	that	PRON
ejpam-7052	250	7	of	of	ADP
ejpam-7052	250	8	theorem	theorem	NOUN
ejpam-7052	250	9	7	7	NUM
ejpam-7052	250	10	.	.	PUNCT
ejpam-7052	250	11	lemma	lemma	PROPN
ejpam-7052	250	12	5	5	NUM
ejpam-7052	250	13	.	.	PUNCT
ejpam-7052	251	1	[	[	X
ejpam-7052	251	2	21	21	NUM
ejpam-7052	251	3	]	]	PUNCT
ejpam-7052	251	4	let	let	VERB
ejpam-7052	251	5	a	a	PRON
ejpam-7052	251	6	be	be	AUX
ejpam-7052	251	7	a	a	DET
ejpam-7052	251	8	subset	subset	NOUN
ejpam-7052	251	9	of	of	ADP
ejpam-7052	251	10	a	a	DET
ejpam-7052	251	11	bitopological	bitopological	ADJ
ejpam-7052	251	12	space	space	NOUN
ejpam-7052	251	13	(	(	PUNCT
ejpam-7052	251	14	x	x	NOUN
ejpam-7052	251	15	,	,	PUNCT
ejpam-7052	251	16	τ1	τ1	NOUN
ejpam-7052	251	17	,	,	PUNCT
ejpam-7052	251	18	τ2	τ2	NOUN
ejpam-7052	251	19	)	)	PUNCT
ejpam-7052	251	20	.	.	PUNCT
ejpam-7052	252	1	then	then	ADV
ejpam-7052	252	2	,	,	PUNCT
ejpam-7052	252	3	the	the	DET
ejpam-7052	252	4	following	follow	VERB
ejpam-7052	252	5	properties	property	NOUN
ejpam-7052	252	6	hold	hold	VERB
ejpam-7052	252	7	:	:	PUNCT
ejpam-7052	252	8	(	(	PUNCT
ejpam-7052	252	9	1	1	X
ejpam-7052	252	10	)	)	PUNCT
ejpam-7052	252	11	if	if	SCONJ
ejpam-7052	252	12	a	a	PRON
ejpam-7052	252	13	is	be	AUX
ejpam-7052	252	14	τ1τ2	τ1τ2	NOUN
ejpam-7052	252	15	-	-	ADJ
ejpam-7052	252	16	open	open	ADJ
ejpam-7052	252	17	in	in	ADP
ejpam-7052	252	18	x	x	NOUN
ejpam-7052	252	19	,	,	PUNCT
ejpam-7052	252	20	then	then	ADV
ejpam-7052	252	21	τ1τ2	τ1τ2	NOUN
ejpam-7052	252	22	-	-	NUM
ejpam-7052	252	23	cl(a	cl(a	NUM
ejpam-7052	252	24	)	)	PUNCT
ejpam-7052	252	25	=	=	PUNCT
ejpam-7052	253	1	τ1τ2	τ1τ2	PROPN
ejpam-7052	253	2	-	-	ADJ
ejpam-7052	253	3	δ	δ	NOUN
ejpam-7052	253	4	-	-	PUNCT
ejpam-7052	253	5	cl(a	cl(a	NUM
ejpam-7052	253	6	)	)	PUNCT
ejpam-7052	253	7	.	.	PUNCT
ejpam-7052	254	1	(	(	PUNCT
ejpam-7052	254	2	2	2	X
ejpam-7052	254	3	)	)	PUNCT
ejpam-7052	254	4	τ1τ2	τ1τ2	NOUN
ejpam-7052	254	5	-	-	ADJ
ejpam-7052	254	6	δ	δ	NOUN
ejpam-7052	254	7	-	-	PUNCT
ejpam-7052	254	8	cl(a	cl(a	X
ejpam-7052	254	9	)	)	PUNCT
ejpam-7052	254	10	is	be	AUX
ejpam-7052	254	11	τ1τ2	τ1τ2	NOUN
ejpam-7052	254	12	-	-	ADJ
ejpam-7052	254	13	closed	closed	ADJ
ejpam-7052	254	14	.	.	PUNCT
ejpam-7052	255	1	theorem	theorem	VERB
ejpam-7052	255	2	9	9	NUM
ejpam-7052	255	3	.	.	X
ejpam-7052	255	4	for	for	ADP
ejpam-7052	255	5	a	a	DET
ejpam-7052	255	6	multifunction	multifunction	NOUN
ejpam-7052	256	1	f	f	NOUN
ejpam-7052	256	2	:	:	PUNCT
ejpam-7052	256	3	(	(	PUNCT
ejpam-7052	256	4	x,µ	x,µ	NOUN
ejpam-7052	256	5	)	)	PUNCT
ejpam-7052	256	6	→	→	SYM
ejpam-7052	256	7	(	(	PUNCT
ejpam-7052	256	8	y	y	PROPN
ejpam-7052	256	9	,	,	PUNCT
ejpam-7052	256	10	σ1	σ1	PROPN
ejpam-7052	256	11	,	,	PUNCT
ejpam-7052	256	12	σ2	σ2	NOUN
ejpam-7052	256	13	)	)	PUNCT
ejpam-7052	256	14	,	,	PUNCT
ejpam-7052	256	15	the	the	DET
ejpam-7052	256	16	following	follow	VERB
ejpam-7052	256	17	properties	property	NOUN
ejpam-7052	256	18	are	be	AUX
ejpam-7052	256	19	equivalent	equivalent	ADJ
ejpam-7052	256	20	:	:	PUNCT
ejpam-7052	256	21	(	(	PUNCT
ejpam-7052	256	22	1	1	X
ejpam-7052	256	23	)	)	PUNCT
ejpam-7052	256	24	f	f	PROPN
ejpam-7052	256	25	is	be	AUX
ejpam-7052	256	26	upper	upper	ADJ
ejpam-7052	256	27	almost	almost	ADV
ejpam-7052	256	28	µ(σ1	µ(σ1	NOUN
ejpam-7052	256	29	,	,	PUNCT
ejpam-7052	256	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	256	31	;	;	PUNCT
ejpam-7052	256	32	(	(	PUNCT
ejpam-7052	256	33	2	2	NUM
ejpam-7052	256	34	)	)	PUNCT
ejpam-7052	256	35	cµ(f	cµ(f	NOUN
ejpam-7052	256	36	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	256	37	-	-	PUNCT
ejpam-7052	256	38	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	256	39	-	-	PUNCT
ejpam-7052	256	40	int(σ1σ2	int(σ1σ2	VERB
ejpam-7052	256	41	-	-	PUNCT
ejpam-7052	256	42	δ	δ	NOUN
ejpam-7052	256	43	-	-	NOUN
ejpam-7052	256	44	cl(b	cl(b	NOUN
ejpam-7052	256	45	)	)	PUNCT
ejpam-7052	256	46	)	)	PUNCT
ejpam-7052	256	47	)	)	PUNCT
ejpam-7052	256	48	)	)	PUNCT
ejpam-7052	256	49	)	)	PUNCT
ejpam-7052	257	1	⊆	⊆	X
ejpam-7052	257	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	257	3	-	-	PUNCT
ejpam-7052	257	4	δ	δ	NOUN
ejpam-7052	257	5	-	-	NOUN
ejpam-7052	257	6	cl(b	cl(b	NOUN
ejpam-7052	257	7	)	)	PUNCT
ejpam-7052	257	8	)	)	PUNCT
ejpam-7052	257	9	for	for	ADP
ejpam-7052	257	10	every	every	DET
ejpam-7052	257	11	subset	subset	NOUN
ejpam-7052	257	12	b	b	PROPN
ejpam-7052	257	13	of	of	ADP
ejpam-7052	257	14	y	y	PROPN
ejpam-7052	257	15	;	;	PUNCT
ejpam-7052	257	16	(	(	PUNCT
ejpam-7052	257	17	3	3	X
ejpam-7052	257	18	)	)	PUNCT
ejpam-7052	257	19	cµ(f	cµ(f	NOUN
ejpam-7052	257	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	257	21	-	-	PUNCT
ejpam-7052	257	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	257	23	-	-	PUNCT
ejpam-7052	257	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	257	25	-	-	PUNCT
ejpam-7052	257	26	cl(b	cl(b	NOUN
ejpam-7052	257	27	)	)	PUNCT
ejpam-7052	257	28	)	)	PUNCT
ejpam-7052	257	29	)	)	PUNCT
ejpam-7052	257	30	)	)	PUNCT
ejpam-7052	257	31	)	)	PUNCT
ejpam-7052	258	1	⊆	⊆	X
ejpam-7052	258	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	258	3	-	-	PUNCT
ejpam-7052	258	4	δ	δ	NOUN
ejpam-7052	258	5	-	-	NOUN
ejpam-7052	258	6	cl(b	cl(b	NOUN
ejpam-7052	258	7	)	)	PUNCT
ejpam-7052	258	8	)	)	PUNCT
ejpam-7052	258	9	for	for	ADP
ejpam-7052	258	10	every	every	DET
ejpam-7052	258	11	subset	subset	NOUN
ejpam-7052	258	12	b	b	PROPN
ejpam-7052	258	13	of	of	ADP
ejpam-7052	258	14	y	y	PROPN
ejpam-7052	258	15	.	.	PUNCT
ejpam-7052	259	1	proof	proof	NOUN
ejpam-7052	259	2	.	.	PUNCT
ejpam-7052	260	1	(	(	PUNCT
ejpam-7052	260	2	1	1	X
ejpam-7052	260	3	)	)	PUNCT
ejpam-7052	260	4	⇒	⇒	NOUN
ejpam-7052	260	5	(	(	PUNCT
ejpam-7052	260	6	2	2	NUM
ejpam-7052	260	7	):	):	PUNCT
ejpam-7052	260	8	let	let	VERB
ejpam-7052	260	9	b	b	X
ejpam-7052	260	10	be	be	AUX
ejpam-7052	260	11	any	any	DET
ejpam-7052	260	12	subset	subset	NOUN
ejpam-7052	260	13	of	of	ADP
ejpam-7052	260	14	y	y	PROPN
ejpam-7052	260	15	.	.	PUNCT
ejpam-7052	261	1	by	by	ADP
ejpam-7052	261	2	lemma	lemma	PROPN
ejpam-7052	261	3	5	5	NUM
ejpam-7052	261	4	,	,	PUNCT
ejpam-7052	261	5	σ1σ2	σ1σ2	NOUN
ejpam-7052	261	6	-	-	PUNCT
ejpam-7052	261	7	δ	δ	NOUN
ejpam-7052	261	8	-	-	NOUN
ejpam-7052	261	9	cl(b	cl(b	NOUN
ejpam-7052	261	10	)	)	PUNCT
ejpam-7052	261	11	is	be	AUX
ejpam-7052	261	12	σ1σ2	σ1σ2	NOUN
ejpam-7052	261	13	-	-	ADJ
ejpam-7052	261	14	closed	closed	ADJ
ejpam-7052	261	15	in	in	ADP
ejpam-7052	261	16	y	y	PROPN
ejpam-7052	261	17	and	and	CCONJ
ejpam-7052	261	18	by	by	ADP
ejpam-7052	261	19	theorem	theorem	ADJ
ejpam-7052	261	20	3	3	NUM
ejpam-7052	261	21	,	,	PUNCT
ejpam-7052	261	22	cµ(f−(σ1σ2	cµ(f−(σ1σ2	NUM
ejpam-7052	261	23	-	-	PUNCT
ejpam-7052	261	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	261	25	-	-	PUNCT
ejpam-7052	261	26	int(σ1σ2	int(σ1σ2	VERB
ejpam-7052	261	27	-	-	PUNCT
ejpam-7052	261	28	δ	δ	NOUN
ejpam-7052	261	29	-	-	NOUN
ejpam-7052	261	30	cl(b	cl(b	NOUN
ejpam-7052	261	31	)	)	PUNCT
ejpam-7052	261	32	)	)	PUNCT
ejpam-7052	261	33	)	)	PUNCT
ejpam-7052	261	34	)	)	PUNCT
ejpam-7052	261	35	)	)	PUNCT
ejpam-7052	262	1	⊆	⊆	X
ejpam-7052	262	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	262	3	-	-	PUNCT
ejpam-7052	262	4	δ	δ	NOUN
ejpam-7052	262	5	-	-	NOUN
ejpam-7052	262	6	cl(b	cl(b	NOUN
ejpam-7052	262	7	)	)	PUNCT
ejpam-7052	262	8	)	)	PUNCT
ejpam-7052	262	9	.	.	PUNCT
ejpam-7052	263	1	(	(	PUNCT
ejpam-7052	263	2	2	2	X
ejpam-7052	263	3	)	)	PUNCT
ejpam-7052	263	4	⇒	⇒	NOUN
ejpam-7052	263	5	(	(	PUNCT
ejpam-7052	263	6	3	3	NUM
ejpam-7052	263	7	):	):	PUNCT
ejpam-7052	263	8	this	this	PRON
ejpam-7052	263	9	is	be	AUX
ejpam-7052	263	10	obvious	obvious	ADJ
ejpam-7052	263	11	since	since	SCONJ
ejpam-7052	263	12	σ1σ2	σ1σ2	NOUN
ejpam-7052	263	13	-	-	NOUN
ejpam-7052	263	14	cl(b	cl(b	NOUN
ejpam-7052	263	15	)	)	PUNCT
ejpam-7052	263	16	⊆	⊆	NUM
ejpam-7052	263	17	σ1σ2	σ1σ2	NUM
ejpam-7052	263	18	-	-	PUNCT
ejpam-7052	263	19	δ	δ	NOUN
ejpam-7052	263	20	-	-	NOUN
ejpam-7052	263	21	cl(b	cl(b	NOUN
ejpam-7052	263	22	)	)	PUNCT
ejpam-7052	263	23	.	.	PUNCT
ejpam-7052	264	1	(	(	PUNCT
ejpam-7052	264	2	3	3	X
ejpam-7052	264	3	)	)	PUNCT
ejpam-7052	264	4	⇒	⇒	NOUN
ejpam-7052	264	5	(	(	PUNCT
ejpam-7052	264	6	1	1	NUM
ejpam-7052	264	7	):	):	PUNCT
ejpam-7052	264	8	let	let	VERB
ejpam-7052	264	9	k	k	PRON
ejpam-7052	264	10	be	be	AUX
ejpam-7052	264	11	any	any	DET
ejpam-7052	264	12	(	(	PUNCT
ejpam-7052	264	13	σ1	σ1	NOUN
ejpam-7052	264	14	,	,	PUNCT
ejpam-7052	264	15	σ2)r	σ2)r	NOUN
ejpam-7052	264	16	-	-	PUNCT
ejpam-7052	264	17	closed	close	VERB
ejpam-7052	264	18	set	set	NOUN
ejpam-7052	264	19	of	of	ADP
ejpam-7052	264	20	y	y	PROPN
ejpam-7052	264	21	.	.	PUNCT
ejpam-7052	265	1	then	then	ADV
ejpam-7052	265	2	by	by	ADP
ejpam-7052	265	3	(	(	PUNCT
ejpam-7052	265	4	3	3	NUM
ejpam-7052	265	5	)	)	PUNCT
ejpam-7052	265	6	,	,	PUNCT
ejpam-7052	265	7	we	we	PRON
ejpam-7052	265	8	have	have	VERB
ejpam-7052	265	9	cµ(f	cµ(f	PROPN
ejpam-7052	265	10	−(k	−(k	NOUN
ejpam-7052	265	11	)	)	PUNCT
ejpam-7052	265	12	)	)	PUNCT
ejpam-7052	266	1	=	=	SYM
ejpam-7052	266	2	cµ(f	cµ(f	NOUN
ejpam-7052	266	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	266	4	-	-	PUNCT
ejpam-7052	266	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	266	6	-	-	PUNCT
ejpam-7052	266	7	int(k	int(k	NOUN
ejpam-7052	266	8	)	)	PUNCT
ejpam-7052	266	9	)	)	PUNCT
ejpam-7052	266	10	)	)	PUNCT
ejpam-7052	266	11	)	)	PUNCT
ejpam-7052	267	1	=	=	SYM
ejpam-7052	267	2	cµ(f	cµ(f	NOUN
ejpam-7052	267	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	267	4	-	-	PUNCT
ejpam-7052	267	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	267	6	-	-	PUNCT
ejpam-7052	267	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	267	8	-	-	PUNCT
ejpam-7052	267	9	cl(k	cl(k	NUM
ejpam-7052	267	10	)	)	PUNCT
ejpam-7052	267	11	)	)	PUNCT
ejpam-7052	267	12	)	)	PUNCT
ejpam-7052	267	13	)	)	PUNCT
ejpam-7052	267	14	)	)	PUNCT
ejpam-7052	268	1	⊆	⊆	X
ejpam-7052	268	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	268	3	-	-	PUNCT
ejpam-7052	268	4	δ	δ	NOUN
ejpam-7052	268	5	-	-	NOUN
ejpam-7052	268	6	cl(k	cl(k	NOUN
ejpam-7052	268	7	)	)	PUNCT
ejpam-7052	268	8	)	)	PUNCT
ejpam-7052	269	1	=	=	SYM
ejpam-7052	269	2	f−(k	f−(k	PROPN
ejpam-7052	269	3	)	)	PUNCT
ejpam-7052	269	4	and	and	CCONJ
ejpam-7052	269	5	hence	hence	ADV
ejpam-7052	269	6	f−(k	f−(k	PROPN
ejpam-7052	269	7	)	)	PUNCT
ejpam-7052	269	8	is	be	AUX
ejpam-7052	269	9	µ-closed	µ-close	VERB
ejpam-7052	269	10	in	in	ADP
ejpam-7052	269	11	x.	x.	NOUN
ejpam-7052	269	12	by	by	ADP
ejpam-7052	269	13	theorem	theorem	NOUN
ejpam-7052	269	14	3	3	NUM
ejpam-7052	269	15	,	,	PUNCT
ejpam-7052	269	16	f	f	PROPN
ejpam-7052	269	17	is	be	AUX
ejpam-7052	269	18	upper	upper	ADJ
ejpam-7052	269	19	almost	almost	ADV
ejpam-7052	269	20	µ(σ1	µ(σ1	NOUN
ejpam-7052	269	21	,	,	PUNCT
ejpam-7052	269	22	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7052	269	23	.	.	X
ejpam-7052	269	24	theorem	theorem	VERB
ejpam-7052	269	25	10	10	NUM
ejpam-7052	269	26	.	.	PUNCT
ejpam-7052	270	1	for	for	ADP
ejpam-7052	270	2	a	a	DET
ejpam-7052	270	3	multifunction	multifunction	NOUN
ejpam-7052	270	4	f	f	NOUN
ejpam-7052	270	5	:	:	PUNCT
ejpam-7052	270	6	(	(	PUNCT
ejpam-7052	270	7	x,µ	x,µ	NOUN
ejpam-7052	270	8	)	)	PUNCT
ejpam-7052	270	9	→	→	SYM
ejpam-7052	270	10	(	(	PUNCT
ejpam-7052	270	11	y	y	PROPN
ejpam-7052	270	12	,	,	PUNCT
ejpam-7052	270	13	σ1	σ1	PROPN
ejpam-7052	270	14	,	,	PUNCT
ejpam-7052	270	15	σ2	σ2	NOUN
ejpam-7052	270	16	)	)	PUNCT
ejpam-7052	270	17	,	,	PUNCT
ejpam-7052	270	18	the	the	DET
ejpam-7052	270	19	following	follow	VERB
ejpam-7052	270	20	properties	property	NOUN
ejpam-7052	270	21	are	be	AUX
ejpam-7052	270	22	equivalent	equivalent	ADJ
ejpam-7052	270	23	:	:	PUNCT
ejpam-7052	270	24	p.	p.	NOUN
ejpam-7052	270	25	pue	pue	NOUN
ejpam-7052	270	26	-	-	PUNCT
ejpam-7052	270	27	on	on	ADP
ejpam-7052	270	28	,	,	PUNCT
ejpam-7052	270	29	a.	a.	PROPN
ejpam-7052	270	30	sama	sama	PROPN
ejpam-7052	270	31	-	-	PUNCT
ejpam-7052	270	32	ae	ae	PROPN
ejpam-7052	270	33	,	,	PUNCT
ejpam-7052	270	34	c.	c.	PROPN
ejpam-7052	270	35	boonpok	boonpok	PROPN
ejpam-7052	270	36	/	/	SYM
ejpam-7052	270	37	eur	eur	PROPN
ejpam-7052	270	38	.	.	PUNCT
ejpam-7052	271	1	j.	j.	PROPN
ejpam-7052	271	2	pure	pure	PROPN
ejpam-7052	271	3	appl	appl	PROPN
ejpam-7052	271	4	.	.	PROPN
ejpam-7052	271	5	math	math	PROPN
ejpam-7052	271	6	,	,	PUNCT
ejpam-7052	271	7	18	18	NUM
ejpam-7052	271	8	(	(	PUNCT
ejpam-7052	271	9	4	4	NUM
ejpam-7052	271	10	)	)	PUNCT
ejpam-7052	271	11	(	(	PUNCT
ejpam-7052	271	12	2025	2025	NUM
ejpam-7052	271	13	)	)	PUNCT
ejpam-7052	271	14	,	,	PUNCT
ejpam-7052	271	15	7052	7052	NUM
ejpam-7052	271	16	10	10	NUM
ejpam-7052	271	17	of	of	ADP
ejpam-7052	271	18	14	14	NUM
ejpam-7052	271	19	(	(	PUNCT
ejpam-7052	271	20	1	1	NUM
ejpam-7052	271	21	)	)	PUNCT
ejpam-7052	271	22	f	f	PROPN
ejpam-7052	271	23	is	be	AUX
ejpam-7052	271	24	lower	low	ADJ
ejpam-7052	271	25	almost	almost	ADV
ejpam-7052	271	26	µ(σ1	µ(σ1	NOUN
ejpam-7052	271	27	,	,	PUNCT
ejpam-7052	271	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	271	29	;	;	PUNCT
ejpam-7052	271	30	(	(	PUNCT
ejpam-7052	271	31	2	2	NUM
ejpam-7052	271	32	)	)	PUNCT
ejpam-7052	271	33	cµ(f	cµ(f	PUNCT
ejpam-7052	272	1	+	+	PROPN
ejpam-7052	272	2	(	(	PUNCT
ejpam-7052	272	3	σ1σ2	σ1σ2	NUM
ejpam-7052	272	4	-	-	PUNCT
ejpam-7052	272	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	272	6	-	-	PUNCT
ejpam-7052	272	7	int(σ1σ2	int(σ1σ2	VERB
ejpam-7052	272	8	-	-	PUNCT
ejpam-7052	272	9	δ	δ	NOUN
ejpam-7052	272	10	-	-	NOUN
ejpam-7052	272	11	cl(b	cl(b	NOUN
ejpam-7052	272	12	)	)	PUNCT
ejpam-7052	272	13	)	)	PUNCT
ejpam-7052	272	14	)	)	PUNCT
ejpam-7052	272	15	)	)	PUNCT
ejpam-7052	272	16	)	)	PUNCT
ejpam-7052	273	1	⊆	⊆	X
ejpam-7052	273	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	273	3	-	-	PUNCT
ejpam-7052	273	4	δ	δ	NOUN
ejpam-7052	273	5	-	-	NOUN
ejpam-7052	273	6	cl(b	cl(b	NOUN
ejpam-7052	273	7	)	)	PUNCT
ejpam-7052	273	8	)	)	PUNCT
ejpam-7052	273	9	for	for	ADP
ejpam-7052	273	10	every	every	DET
ejpam-7052	273	11	subset	subset	NOUN
ejpam-7052	273	12	b	b	PROPN
ejpam-7052	273	13	of	of	ADP
ejpam-7052	273	14	y	y	PROPN
ejpam-7052	273	15	;	;	PUNCT
ejpam-7052	273	16	(	(	PUNCT
ejpam-7052	273	17	3	3	X
ejpam-7052	273	18	)	)	PUNCT
ejpam-7052	273	19	cµ(f	cµ(f	PUNCT
ejpam-7052	274	1	+	+	PROPN
ejpam-7052	274	2	(	(	PUNCT
ejpam-7052	274	3	σ1σ2	σ1σ2	NUM
ejpam-7052	274	4	-	-	PUNCT
ejpam-7052	274	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	274	6	-	-	PUNCT
ejpam-7052	274	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	274	8	-	-	PUNCT
ejpam-7052	274	9	cl(b	cl(b	NOUN
ejpam-7052	274	10	)	)	PUNCT
ejpam-7052	274	11	)	)	PUNCT
ejpam-7052	274	12	)	)	PUNCT
ejpam-7052	274	13	)	)	PUNCT
ejpam-7052	274	14	)	)	PUNCT
ejpam-7052	275	1	⊆	⊆	X
ejpam-7052	275	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	275	3	-	-	PUNCT
ejpam-7052	275	4	δ	δ	NOUN
ejpam-7052	275	5	-	-	NOUN
ejpam-7052	275	6	cl(b	cl(b	NOUN
ejpam-7052	275	7	)	)	PUNCT
ejpam-7052	275	8	)	)	PUNCT
ejpam-7052	275	9	for	for	ADP
ejpam-7052	275	10	every	every	DET
ejpam-7052	275	11	subset	subset	NOUN
ejpam-7052	275	12	b	b	PROPN
ejpam-7052	275	13	of	of	ADP
ejpam-7052	275	14	y	y	PROPN
ejpam-7052	275	15	.	.	PUNCT
ejpam-7052	276	1	proof	proof	NOUN
ejpam-7052	276	2	.	.	PUNCT
ejpam-7052	277	1	the	the	DET
ejpam-7052	277	2	proof	proof	NOUN
ejpam-7052	277	3	is	be	AUX
ejpam-7052	277	4	similar	similar	ADJ
ejpam-7052	277	5	to	to	ADP
ejpam-7052	277	6	that	that	PRON
ejpam-7052	277	7	of	of	ADP
ejpam-7052	277	8	theorem	theorem	NOUN
ejpam-7052	277	9	9	9	NUM
ejpam-7052	277	10	.	.	PUNCT
ejpam-7052	278	1	lemma	lemma	PROPN
ejpam-7052	278	2	6	6	NUM
ejpam-7052	278	3	.	.	PUNCT
ejpam-7052	279	1	if	if	SCONJ
ejpam-7052	279	2	f	f	PROPN
ejpam-7052	279	3	:	:	PUNCT
ejpam-7052	279	4	(	(	PUNCT
ejpam-7052	279	5	x,µ	x,µ	NOUN
ejpam-7052	279	6	)	)	PUNCT
ejpam-7052	279	7	→	→	SYM
ejpam-7052	279	8	(	(	PUNCT
ejpam-7052	279	9	y	y	PROPN
ejpam-7052	279	10	,	,	PUNCT
ejpam-7052	279	11	σ1	σ1	PROPN
ejpam-7052	279	12	,	,	PUNCT
ejpam-7052	279	13	σ2	σ2	NOUN
ejpam-7052	279	14	)	)	PUNCT
ejpam-7052	279	15	is	be	AUX
ejpam-7052	279	16	lower	low	ADJ
ejpam-7052	279	17	almost	almost	ADV
ejpam-7052	279	18	µ(σ1	µ(σ1	NOUN
ejpam-7052	279	19	,	,	PUNCT
ejpam-7052	279	20	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7052	279	21	,	,	PUNCT
ejpam-7052	279	22	then	then	ADV
ejpam-7052	279	23	for	for	ADP
ejpam-7052	279	24	each	each	DET
ejpam-7052	279	25	x	x	SYM
ejpam-7052	279	26	∈	∈	PROPN
ejpam-7052	279	27	x	x	X
ejpam-7052	279	28	and	and	CCONJ
ejpam-7052	279	29	each	each	DET
ejpam-7052	279	30	subset	subset	NOUN
ejpam-7052	279	31	b	b	PROPN
ejpam-7052	279	32	of	of	ADP
ejpam-7052	279	33	y	y	PROPN
ejpam-7052	279	34	with	with	ADP
ejpam-7052	279	35	σ1σ2	σ1σ2	PROPN
ejpam-7052	279	36	-	-	PUNCT
ejpam-7052	279	37	δ	δ	NOUN
ejpam-7052	279	38	-	-	NOUN
ejpam-7052	279	39	int(b)∩f	int(b)∩f	NOUN
ejpam-7052	279	40	(	(	PUNCT
ejpam-7052	279	41	x	x	X
ejpam-7052	279	42	)	)	PUNCT
ejpam-7052	279	43	̸=	̸=	NOUN
ejpam-7052	279	44	∅	∅	NOUN
ejpam-7052	279	45	,	,	PUNCT
ejpam-7052	279	46	there	there	PRON
ejpam-7052	279	47	exists	exist	VERB
ejpam-7052	279	48	a	a	DET
ejpam-7052	279	49	µ-open	µ-open	NOUN
ejpam-7052	279	50	set	set	VERB
ejpam-7052	279	51	u	u	NOUN
ejpam-7052	279	52	of	of	ADP
ejpam-7052	279	53	x	x	PUNCT
ejpam-7052	279	54	containing	contain	VERB
ejpam-7052	279	55	x	x	PUNCT
ejpam-7052	279	56	such	such	ADJ
ejpam-7052	279	57	that	that	SCONJ
ejpam-7052	279	58	u	u	PROPN
ejpam-7052	279	59	⊆	⊆	NUM
ejpam-7052	279	60	f−(b	f−(b	NOUN
ejpam-7052	279	61	)	)	PUNCT
ejpam-7052	279	62	.	.	PUNCT
ejpam-7052	280	1	proof	proof	NOUN
ejpam-7052	280	2	.	.	PUNCT
ejpam-7052	281	1	let	let	VERB
ejpam-7052	281	2	x	x	PUNCT
ejpam-7052	281	3	∈	∈	PROPN
ejpam-7052	281	4	x	x	X
ejpam-7052	281	5	and	and	CCONJ
ejpam-7052	281	6	b	b	X
ejpam-7052	281	7	be	be	AUX
ejpam-7052	281	8	a	a	DET
ejpam-7052	281	9	subset	subset	NOUN
ejpam-7052	281	10	of	of	ADP
ejpam-7052	281	11	y	y	PRON
ejpam-7052	281	12	with	with	ADP
ejpam-7052	281	13	σ1σ2	σ1σ2	PROPN
ejpam-7052	281	14	-	-	PUNCT
ejpam-7052	281	15	δ	δ	NOUN
ejpam-7052	281	16	-	-	PUNCT
ejpam-7052	281	17	int(b	int(b	NOUN
ejpam-7052	281	18	)	)	PUNCT
ejpam-7052	281	19	∩	∩	ADJ
ejpam-7052	281	20	f	f	PROPN
ejpam-7052	281	21	(	(	PUNCT
ejpam-7052	281	22	x	x	X
ejpam-7052	281	23	)	)	PUNCT
ejpam-7052	281	24	̸=	̸=	PROPN
ejpam-7052	281	25	∅.	∅.	ADV
ejpam-7052	281	26	since	since	SCONJ
ejpam-7052	281	27	σ1σ2	σ1σ2	NOUN
ejpam-7052	281	28	-	-	PUNCT
ejpam-7052	281	29	δ	δ	NOUN
ejpam-7052	281	30	-	-	PUNCT
ejpam-7052	281	31	int(b	int(b	NOUN
ejpam-7052	281	32	)	)	PUNCT
ejpam-7052	281	33	∩	∩	ADJ
ejpam-7052	281	34	f	f	PROPN
ejpam-7052	281	35	(	(	PUNCT
ejpam-7052	281	36	x	x	X
ejpam-7052	281	37	)	)	PUNCT
ejpam-7052	281	38	̸=	̸=	NOUN
ejpam-7052	281	39	∅	∅	NOUN
ejpam-7052	281	40	,	,	PUNCT
ejpam-7052	281	41	there	there	PRON
ejpam-7052	281	42	exists	exist	VERB
ejpam-7052	281	43	a	a	DET
ejpam-7052	281	44	nonempty	nonempty	ADJ
ejpam-7052	281	45	(	(	PUNCT
ejpam-7052	281	46	σ1	σ1	NOUN
ejpam-7052	281	47	,	,	PUNCT
ejpam-7052	281	48	σ2)r	σ2)r	NOUN
ejpam-7052	281	49	-	-	PUNCT
ejpam-7052	281	50	open	open	ADJ
ejpam-7052	281	51	set	set	VERB
ejpam-7052	281	52	v	v	NOUN
ejpam-7052	281	53	of	of	ADP
ejpam-7052	281	54	y	y	PRON
ejpam-7052	281	55	such	such	ADJ
ejpam-7052	281	56	that	that	PRON
ejpam-7052	281	57	v	v	ADP
ejpam-7052	281	58	⊆	⊆	NUM
ejpam-7052	281	59	b	b	NOUN
ejpam-7052	281	60	and	and	CCONJ
ejpam-7052	281	61	v	v	NOUN
ejpam-7052	281	62	∩	∩	ADJ
ejpam-7052	281	63	f	f	X
ejpam-7052	281	64	(	(	PUNCT
ejpam-7052	281	65	x	x	X
ejpam-7052	281	66	)	)	PUNCT
ejpam-7052	281	67	̸=	̸=	PROPN
ejpam-7052	281	68	∅.	∅.	ADV
ejpam-7052	281	69	since	since	SCONJ
ejpam-7052	281	70	f	f	PROPN
ejpam-7052	281	71	is	be	AUX
ejpam-7052	281	72	lower	low	ADJ
ejpam-7052	281	73	almost	almost	ADV
ejpam-7052	281	74	µ(σ1	µ(σ1	NOUN
ejpam-7052	281	75	,	,	PUNCT
ejpam-7052	281	76	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	281	77	,	,	PUNCT
ejpam-7052	281	78	there	there	PRON
ejpam-7052	281	79	exists	exist	VERB
ejpam-7052	281	80	a	a	DET
ejpam-7052	281	81	µ-open	µ-open	NOUN
ejpam-7052	281	82	set	set	VERB
ejpam-7052	281	83	u	u	NOUN
ejpam-7052	281	84	of	of	ADP
ejpam-7052	281	85	x	x	PUNCT
ejpam-7052	281	86	containing	contain	VERB
ejpam-7052	281	87	x	x	PUNCT
ejpam-7052	282	1	such	such	ADJ
ejpam-7052	282	2	that	that	DET
ejpam-7052	282	3	v	v	NOUN
ejpam-7052	282	4	∩f	∩f	NOUN
ejpam-7052	282	5	(	(	PUNCT
ejpam-7052	282	6	z	z	X
ejpam-7052	282	7	)	)	PUNCT
ejpam-7052	282	8	̸=	̸=	NOUN
ejpam-7052	282	9	∅	∅	NOUN
ejpam-7052	282	10	for	for	ADP
ejpam-7052	282	11	each	each	DET
ejpam-7052	282	12	z	z	NOUN
ejpam-7052	282	13	∈	∈	PROPN
ejpam-7052	282	14	u	u	NOUN
ejpam-7052	282	15	;	;	PUNCT
ejpam-7052	282	16	hence	hence	ADV
ejpam-7052	282	17	u	u	NOUN
ejpam-7052	282	18	⊆	⊆	NUM
ejpam-7052	282	19	f−(b	f−(b	NOUN
ejpam-7052	282	20	)	)	PUNCT
ejpam-7052	282	21	.	.	PUNCT
ejpam-7052	283	1	theorem	theorem	VERB
ejpam-7052	283	2	11	11	NUM
ejpam-7052	283	3	.	.	PUNCT
ejpam-7052	284	1	for	for	ADP
ejpam-7052	284	2	a	a	DET
ejpam-7052	284	3	multifunction	multifunction	NOUN
ejpam-7052	284	4	f	f	NOUN
ejpam-7052	284	5	:	:	PUNCT
ejpam-7052	284	6	(	(	PUNCT
ejpam-7052	284	7	x,µ	x,µ	NOUN
ejpam-7052	284	8	)	)	PUNCT
ejpam-7052	284	9	→	→	SYM
ejpam-7052	284	10	(	(	PUNCT
ejpam-7052	284	11	y	y	PROPN
ejpam-7052	284	12	,	,	PUNCT
ejpam-7052	284	13	σ1	σ1	PROPN
ejpam-7052	284	14	,	,	PUNCT
ejpam-7052	284	15	σ2	σ2	NOUN
ejpam-7052	284	16	)	)	PUNCT
ejpam-7052	284	17	,	,	PUNCT
ejpam-7052	284	18	the	the	DET
ejpam-7052	284	19	following	follow	VERB
ejpam-7052	284	20	properties	property	NOUN
ejpam-7052	284	21	are	be	AUX
ejpam-7052	284	22	equivalent	equivalent	ADJ
ejpam-7052	284	23	:	:	PUNCT
ejpam-7052	284	24	(	(	PUNCT
ejpam-7052	284	25	1	1	X
ejpam-7052	284	26	)	)	PUNCT
ejpam-7052	284	27	f	f	PROPN
ejpam-7052	284	28	is	be	AUX
ejpam-7052	284	29	lower	low	ADJ
ejpam-7052	284	30	almost	almost	ADV
ejpam-7052	284	31	µ(σ1	µ(σ1	NOUN
ejpam-7052	284	32	,	,	PUNCT
ejpam-7052	284	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	284	34	;	;	PUNCT
ejpam-7052	284	35	(	(	PUNCT
ejpam-7052	284	36	2	2	NUM
ejpam-7052	284	37	)	)	PUNCT
ejpam-7052	284	38	cµ(f	cµ(f	PUNCT
ejpam-7052	285	1	+	+	PROPN
ejpam-7052	285	2	(	(	PUNCT
ejpam-7052	285	3	b	b	NOUN
ejpam-7052	285	4	)	)	PUNCT
ejpam-7052	285	5	)	)	PUNCT
ejpam-7052	286	1	⊆	⊆	NUM
ejpam-7052	286	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	286	3	-	-	PUNCT
ejpam-7052	286	4	δ	δ	NOUN
ejpam-7052	286	5	-	-	NOUN
ejpam-7052	286	6	cl(b	cl(b	NOUN
ejpam-7052	286	7	)	)	PUNCT
ejpam-7052	286	8	)	)	PUNCT
ejpam-7052	286	9	for	for	ADP
ejpam-7052	286	10	every	every	DET
ejpam-7052	286	11	subset	subset	NOUN
ejpam-7052	286	12	b	b	PROPN
ejpam-7052	286	13	of	of	ADP
ejpam-7052	286	14	y	y	PROPN
ejpam-7052	286	15	;	;	PUNCT
ejpam-7052	286	16	(	(	PUNCT
ejpam-7052	286	17	3	3	X
ejpam-7052	286	18	)	)	PUNCT
ejpam-7052	286	19	f	f	NOUN
ejpam-7052	286	20	(	(	PUNCT
ejpam-7052	286	21	cµ(a	cµ(a	PROPN
ejpam-7052	286	22	)	)	PUNCT
ejpam-7052	286	23	)	)	PUNCT
ejpam-7052	286	24	⊆	⊆	X
ejpam-7052	286	25	σ1σ2	σ1σ2	X
ejpam-7052	286	26	-	-	PUNCT
ejpam-7052	286	27	δ	δ	NOUN
ejpam-7052	286	28	-	-	NOUN
ejpam-7052	286	29	cl(f	cl(f	PROPN
ejpam-7052	286	30	(	(	PUNCT
ejpam-7052	286	31	a	a	NOUN
ejpam-7052	286	32	)	)	PUNCT
ejpam-7052	286	33	)	)	PUNCT
ejpam-7052	286	34	for	for	ADP
ejpam-7052	286	35	every	every	DET
ejpam-7052	286	36	subset	subset	NOUN
ejpam-7052	286	37	a	a	PRON
ejpam-7052	286	38	of	of	ADP
ejpam-7052	286	39	x	x	PRON
ejpam-7052	286	40	;	;	PUNCT
ejpam-7052	286	41	(	(	PUNCT
ejpam-7052	286	42	4	4	X
ejpam-7052	286	43	)	)	PUNCT
ejpam-7052	286	44	f+(k	f+(k	NUM
ejpam-7052	286	45	)	)	PUNCT
ejpam-7052	286	46	is	be	AUX
ejpam-7052	286	47	µ-closed	µ-close	VERB
ejpam-7052	286	48	in	in	ADP
ejpam-7052	286	49	x	x	PUNCT
ejpam-7052	286	50	for	for	ADP
ejpam-7052	286	51	every	every	DET
ejpam-7052	286	52	σ1σ2	σ1σ2	NUM
ejpam-7052	286	53	-	-	PUNCT
ejpam-7052	286	54	δ	δ	NOUN
ejpam-7052	286	55	-	-	PUNCT
ejpam-7052	286	56	closed	close	VERB
ejpam-7052	286	57	set	set	ADJ
ejpam-7052	286	58	k	k	PROPN
ejpam-7052	286	59	of	of	ADP
ejpam-7052	286	60	y	y	PROPN
ejpam-7052	286	61	;	;	PUNCT
ejpam-7052	286	62	(	(	PUNCT
ejpam-7052	286	63	5	5	X
ejpam-7052	286	64	)	)	PUNCT
ejpam-7052	286	65	f−(v	f−(v	NOUN
ejpam-7052	286	66	)	)	PUNCT
ejpam-7052	286	67	is	be	AUX
ejpam-7052	286	68	µ-open	µ-open	NOUN
ejpam-7052	286	69	in	in	ADP
ejpam-7052	286	70	x	x	PUNCT
ejpam-7052	286	71	for	for	ADP
ejpam-7052	286	72	every	every	DET
ejpam-7052	286	73	σ1σ2	σ1σ2	NUM
ejpam-7052	286	74	-	-	PUNCT
ejpam-7052	286	75	δ	δ	NOUN
ejpam-7052	286	76	-	-	ADJ
ejpam-7052	286	77	open	open	ADJ
ejpam-7052	286	78	set	set	VERB
ejpam-7052	286	79	v	v	NOUN
ejpam-7052	286	80	of	of	ADP
ejpam-7052	286	81	y	y	PROPN
ejpam-7052	286	82	;	;	PUNCT
ejpam-7052	286	83	(	(	PUNCT
ejpam-7052	286	84	6	6	X
ejpam-7052	286	85	)	)	PUNCT
ejpam-7052	286	86	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7052	286	87	-	-	PUNCT
ejpam-7052	286	88	δ	δ	NOUN
ejpam-7052	286	89	-	-	PUNCT
ejpam-7052	286	90	int(b	int(b	NOUN
ejpam-7052	286	91	)	)	PUNCT
ejpam-7052	286	92	)	)	PUNCT
ejpam-7052	287	1	⊆	⊆	NUM
ejpam-7052	287	2	iµ(f	iµ(f	NOUN
ejpam-7052	287	3	−(b	−(b	NOUN
ejpam-7052	287	4	)	)	PUNCT
ejpam-7052	287	5	)	)	PUNCT
ejpam-7052	287	6	for	for	ADP
ejpam-7052	287	7	every	every	DET
ejpam-7052	287	8	subset	subset	NOUN
ejpam-7052	287	9	b	b	PROPN
ejpam-7052	287	10	of	of	ADP
ejpam-7052	287	11	y	y	PROPN
ejpam-7052	287	12	.	.	PUNCT
ejpam-7052	288	1	proof	proof	NOUN
ejpam-7052	288	2	.	.	PUNCT
ejpam-7052	289	1	(	(	PUNCT
ejpam-7052	289	2	1	1	X
ejpam-7052	289	3	)	)	PUNCT
ejpam-7052	289	4	⇒	⇒	NOUN
ejpam-7052	289	5	(	(	PUNCT
ejpam-7052	289	6	2	2	NUM
ejpam-7052	289	7	):	):	PUNCT
ejpam-7052	289	8	let	let	VERB
ejpam-7052	289	9	b	b	X
ejpam-7052	289	10	be	be	AUX
ejpam-7052	289	11	any	any	DET
ejpam-7052	289	12	subset	subset	NOUN
ejpam-7052	289	13	of	of	ADP
ejpam-7052	289	14	y	y	PROPN
ejpam-7052	289	15	.	.	PUNCT
ejpam-7052	289	16	suppose	suppose	VERB
ejpam-7052	289	17	that	that	SCONJ
ejpam-7052	289	18	x	x	PROPN
ejpam-7052	289	19	̸∈	̸∈	PROPN
ejpam-7052	289	20	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	289	21	-	-	PUNCT
ejpam-7052	289	22	δ	δ	NOUN
ejpam-7052	289	23	-	-	NOUN
ejpam-7052	289	24	cl(b	cl(b	NOUN
ejpam-7052	289	25	)	)	PUNCT
ejpam-7052	289	26	)	)	PUNCT
ejpam-7052	289	27	.	.	PUNCT
ejpam-7052	290	1	then	then	ADV
ejpam-7052	290	2	,	,	PUNCT
ejpam-7052	290	3	we	we	PRON
ejpam-7052	290	4	have	have	VERB
ejpam-7052	290	5	x	x	PART
ejpam-7052	290	6	∈	∈	NOUN
ejpam-7052	290	7	f−(y	f−(y	NOUN
ejpam-7052	290	8	−σ1σ2	−σ1σ2	PROPN
ejpam-7052	290	9	-	-	PUNCT
ejpam-7052	290	10	δ	δ	NOUN
ejpam-7052	290	11	-	-	NOUN
ejpam-7052	290	12	cl(b	cl(b	NOUN
ejpam-7052	290	13	)	)	PUNCT
ejpam-7052	290	14	)	)	PUNCT
ejpam-7052	291	1	=	=	PUNCT
ejpam-7052	291	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7052	291	3	-	-	PUNCT
ejpam-7052	291	4	δ	δ	NOUN
ejpam-7052	291	5	-	-	PUNCT
ejpam-7052	291	6	int(y	int(y	PROPN
ejpam-7052	291	7	−b	−b	NOUN
ejpam-7052	291	8	)	)	PUNCT
ejpam-7052	291	9	)	)	PUNCT
ejpam-7052	291	10	.	.	PUNCT
ejpam-7052	292	1	there	there	PRON
ejpam-7052	292	2	exists	exist	VERB
ejpam-7052	292	3	a	a	DET
ejpam-7052	292	4	µ-open	µ-open	NOUN
ejpam-7052	292	5	set	set	VERB
ejpam-7052	292	6	u	u	NOUN
ejpam-7052	292	7	of	of	ADP
ejpam-7052	292	8	x	x	PUNCT
ejpam-7052	292	9	containing	contain	VERB
ejpam-7052	292	10	x	x	PUNCT
ejpam-7052	292	11	such	such	ADJ
ejpam-7052	292	12	that	that	SCONJ
ejpam-7052	292	13	u	u	NOUN
ejpam-7052	292	14	⊆	⊆	NUM
ejpam-7052	292	15	f−(y	f−(y	NOUN
ejpam-7052	292	16	−b	−b	NOUN
ejpam-7052	292	17	)	)	PUNCT
ejpam-7052	293	1	=	=	PUNCT
ejpam-7052	293	2	x	x	X
ejpam-7052	294	1	−	−	NOUN
ejpam-7052	294	2	f+(b	f+(b	NOUN
ejpam-7052	294	3	)	)	PUNCT
ejpam-7052	294	4	.	.	PUNCT
ejpam-7052	295	1	thus	thus	ADV
ejpam-7052	295	2	,	,	PUNCT
ejpam-7052	295	3	u	u	NOUN
ejpam-7052	295	4	∩	∩	NOUN
ejpam-7052	295	5	f+(b	f+(b	NOUN
ejpam-7052	295	6	)	)	PUNCT
ejpam-7052	295	7	=	=	SYM
ejpam-7052	295	8	∅	∅	NOUN
ejpam-7052	295	9	and	and	CCONJ
ejpam-7052	295	10	hence	hence	ADV
ejpam-7052	295	11	x	x	X
ejpam-7052	295	12	∈	∈	NOUN
ejpam-7052	295	13	x	x	X
ejpam-7052	295	14	−	−	NOUN
ejpam-7052	295	15	cµ(f	cµ(f	X
ejpam-7052	296	1	+	+	NOUN
ejpam-7052	296	2	(	(	PUNCT
ejpam-7052	296	3	b	b	NOUN
ejpam-7052	296	4	)	)	PUNCT
ejpam-7052	296	5	)	)	PUNCT
ejpam-7052	296	6	.	.	PUNCT
ejpam-7052	297	1	this	this	PRON
ejpam-7052	297	2	shows	show	VERB
ejpam-7052	297	3	that	that	SCONJ
ejpam-7052	297	4	cµ(f	cµ(f	PUNCT
ejpam-7052	298	1	+	+	ADJ
ejpam-7052	298	2	(	(	PUNCT
ejpam-7052	298	3	b	b	NOUN
ejpam-7052	298	4	)	)	PUNCT
ejpam-7052	298	5	)	)	PUNCT
ejpam-7052	299	1	⊆	⊆	NUM
ejpam-7052	299	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	299	3	-	-	PUNCT
ejpam-7052	299	4	δ	δ	NOUN
ejpam-7052	299	5	-	-	NOUN
ejpam-7052	299	6	cl(b	cl(b	NOUN
ejpam-7052	299	7	)	)	PUNCT
ejpam-7052	299	8	)	)	PUNCT
ejpam-7052	299	9	.	.	PUNCT
ejpam-7052	300	1	(	(	PUNCT
ejpam-7052	300	2	2	2	X
ejpam-7052	300	3	)	)	PUNCT
ejpam-7052	300	4	⇒	⇒	NOUN
ejpam-7052	300	5	(	(	PUNCT
ejpam-7052	300	6	3	3	NUM
ejpam-7052	300	7	):	):	PUNCT
ejpam-7052	300	8	let	let	VERB
ejpam-7052	300	9	a	a	DET
ejpam-7052	300	10	be	be	AUX
ejpam-7052	300	11	any	any	DET
ejpam-7052	300	12	subset	subset	NOUN
ejpam-7052	300	13	of	of	ADP
ejpam-7052	300	14	x.	x.	NOUN
ejpam-7052	300	15	by	by	ADP
ejpam-7052	300	16	(	(	PUNCT
ejpam-7052	300	17	2	2	NUM
ejpam-7052	300	18	)	)	PUNCT
ejpam-7052	300	19	,	,	PUNCT
ejpam-7052	300	20	we	we	PRON
ejpam-7052	300	21	have	have	VERB
ejpam-7052	300	22	cµ(a	cµ(a	PROPN
ejpam-7052	300	23	)	)	PUNCT
ejpam-7052	300	24	⊆	⊆	NUM
ejpam-7052	300	25	cµ(f	cµ(f	X
ejpam-7052	301	1	+	+	PROPN
ejpam-7052	302	1	(	(	PUNCT
ejpam-7052	302	2	f	f	X
ejpam-7052	302	3	(	(	PUNCT
ejpam-7052	302	4	a	a	NOUN
ejpam-7052	302	5	)	)	PUNCT
ejpam-7052	302	6	)	)	PUNCT
ejpam-7052	302	7	)	)	PUNCT
ejpam-7052	302	8	⊆	⊆	X
ejpam-7052	302	9	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	302	10	-	-	PUNCT
ejpam-7052	302	11	δ	δ	NOUN
ejpam-7052	302	12	-	-	NOUN
ejpam-7052	302	13	cl(f	cl(f	PROPN
ejpam-7052	302	14	(	(	PUNCT
ejpam-7052	302	15	a	a	NOUN
ejpam-7052	302	16	)	)	PUNCT
ejpam-7052	302	17	)	)	PUNCT
ejpam-7052	302	18	)	)	PUNCT
ejpam-7052	302	19	and	and	CCONJ
ejpam-7052	302	20	hence	hence	ADV
ejpam-7052	302	21	f	f	PROPN
ejpam-7052	302	22	(	(	PUNCT
ejpam-7052	302	23	cµ(a	cµ(a	PROPN
ejpam-7052	302	24	)	)	PUNCT
ejpam-7052	302	25	)	)	PUNCT
ejpam-7052	302	26	⊆	⊆	X
ejpam-7052	302	27	σ1σ2	σ1σ2	X
ejpam-7052	302	28	-	-	PUNCT
ejpam-7052	302	29	δ	δ	NOUN
ejpam-7052	302	30	-	-	NOUN
ejpam-7052	302	31	cl(f	cl(f	PROPN
ejpam-7052	302	32	(	(	PUNCT
ejpam-7052	302	33	a	a	NOUN
ejpam-7052	302	34	)	)	PUNCT
ejpam-7052	302	35	)	)	PUNCT
ejpam-7052	302	36	.	.	PUNCT
ejpam-7052	303	1	(	(	PUNCT
ejpam-7052	303	2	3	3	X
ejpam-7052	303	3	)	)	PUNCT
ejpam-7052	303	4	⇒	⇒	NOUN
ejpam-7052	303	5	(	(	PUNCT
ejpam-7052	303	6	1	1	NUM
ejpam-7052	303	7	):	):	PUNCT
ejpam-7052	303	8	let	let	VERB
ejpam-7052	303	9	b	b	X
ejpam-7052	303	10	be	be	AUX
ejpam-7052	303	11	any	any	DET
ejpam-7052	303	12	subset	subset	NOUN
ejpam-7052	303	13	of	of	ADP
ejpam-7052	303	14	y	y	PROPN
ejpam-7052	303	15	.	.	PUNCT
ejpam-7052	304	1	then	then	ADV
ejpam-7052	304	2	,	,	PUNCT
ejpam-7052	304	3	by	by	ADP
ejpam-7052	304	4	the	the	DET
ejpam-7052	304	5	hypothesis	hypothesis	NOUN
ejpam-7052	304	6	and	and	CCONJ
ejpam-7052	304	7	lemma	lemma	PROPN
ejpam-7052	304	8	5	5	NUM
ejpam-7052	304	9	,	,	PUNCT
ejpam-7052	304	10	f	f	PROPN
ejpam-7052	304	11	(	(	PUNCT
ejpam-7052	304	12	cµ(f	cµ(f	X
ejpam-7052	304	13	+	+	ADJ
ejpam-7052	304	14	(	(	PUNCT
ejpam-7052	304	15	σ1σ2	σ1σ2	NUM
ejpam-7052	304	16	-	-	PUNCT
ejpam-7052	304	17	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	304	18	-	-	PUNCT
ejpam-7052	304	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	304	20	-	-	PUNCT
ejpam-7052	304	21	cl(b	cl(b	NOUN
ejpam-7052	304	22	)	)	PUNCT
ejpam-7052	304	23	)	)	PUNCT
ejpam-7052	304	24	)	)	PUNCT
ejpam-7052	304	25	)	)	PUNCT
ejpam-7052	304	26	)	)	PUNCT
ejpam-7052	304	27	)	)	PUNCT
ejpam-7052	305	1	⊆	⊆	X
ejpam-7052	305	2	τ1τ2	τ1τ2	PROPN
ejpam-7052	305	3	-	-	ADJ
ejpam-7052	305	4	δ	δ	NOUN
ejpam-7052	305	5	-	-	NOUN
ejpam-7052	305	6	cl(f	cl(f	PROPN
ejpam-7052	305	7	(	(	PUNCT
ejpam-7052	305	8	f+(σ1σ2	f+(σ1σ2	VERB
ejpam-7052	305	9	-	-	PUNCT
ejpam-7052	305	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	305	11	-	-	PUNCT
ejpam-7052	305	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	305	13	-	-	PUNCT
ejpam-7052	305	14	cl(b	cl(b	NOUN
ejpam-7052	305	15	)	)	PUNCT
ejpam-7052	305	16	)	)	PUNCT
ejpam-7052	305	17	)	)	PUNCT
ejpam-7052	305	18	)	)	PUNCT
ejpam-7052	305	19	)	)	PUNCT
ejpam-7052	305	20	)	)	PUNCT
ejpam-7052	306	1	p.	p.	NOUN
ejpam-7052	306	2	pue	pue	NOUN
ejpam-7052	306	3	-	-	PUNCT
ejpam-7052	306	4	on	on	ADP
ejpam-7052	306	5	,	,	PUNCT
ejpam-7052	306	6	a.	a.	PROPN
ejpam-7052	306	7	sama	sama	PROPN
ejpam-7052	306	8	-	-	PUNCT
ejpam-7052	306	9	ae	ae	PROPN
ejpam-7052	306	10	,	,	PUNCT
ejpam-7052	306	11	c.	c.	PROPN
ejpam-7052	306	12	boonpok	boonpok	PROPN
ejpam-7052	306	13	/	/	SYM
ejpam-7052	306	14	eur	eur	PROPN
ejpam-7052	306	15	.	.	PUNCT
ejpam-7052	307	1	j.	j.	PROPN
ejpam-7052	307	2	pure	pure	PROPN
ejpam-7052	307	3	appl	appl	PROPN
ejpam-7052	307	4	.	.	PROPN
ejpam-7052	307	5	math	math	PROPN
ejpam-7052	307	6	,	,	PUNCT
ejpam-7052	307	7	18	18	NUM
ejpam-7052	307	8	(	(	PUNCT
ejpam-7052	307	9	4	4	NUM
ejpam-7052	307	10	)	)	PUNCT
ejpam-7052	307	11	(	(	PUNCT
ejpam-7052	307	12	2025	2025	NUM
ejpam-7052	307	13	)	)	PUNCT
ejpam-7052	307	14	,	,	PUNCT
ejpam-7052	307	15	7052	7052	NUM
ejpam-7052	307	16	11	11	NUM
ejpam-7052	307	17	of	of	ADP
ejpam-7052	307	18	14	14	NUM
ejpam-7052	307	19	⊆	⊆	NUM
ejpam-7052	307	20	σ1σ2	σ1σ2	NOUN
ejpam-7052	307	21	-	-	PUNCT
ejpam-7052	307	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	307	23	-	-	PUNCT
ejpam-7052	307	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	307	25	-	-	PUNCT
ejpam-7052	307	26	cl(b	cl(b	NOUN
ejpam-7052	307	27	)	)	PUNCT
ejpam-7052	307	28	)	)	PUNCT
ejpam-7052	307	29	)	)	PUNCT
ejpam-7052	308	1	⊆	⊆	X
ejpam-7052	308	2	σ1σ2	σ1σ2	NUM
ejpam-7052	308	3	-	-	PUNCT
ejpam-7052	308	4	cl(b	cl(b	NOUN
ejpam-7052	308	5	)	)	PUNCT
ejpam-7052	308	6	and	and	CCONJ
ejpam-7052	308	7	hence	hence	ADV
ejpam-7052	308	8	cµ(f	cµ(f	PUNCT
ejpam-7052	309	1	+	+	ADJ
ejpam-7052	309	2	(	(	PUNCT
ejpam-7052	309	3	σ1σ2	σ1σ2	NUM
ejpam-7052	309	4	-	-	PUNCT
ejpam-7052	309	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7052	309	6	-	-	PUNCT
ejpam-7052	309	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7052	309	8	-	-	PUNCT
ejpam-7052	309	9	cl(b	cl(b	NOUN
ejpam-7052	309	10	)	)	PUNCT
ejpam-7052	309	11	)	)	PUNCT
ejpam-7052	309	12	)	)	PUNCT
ejpam-7052	309	13	)	)	PUNCT
ejpam-7052	309	14	)	)	PUNCT
ejpam-7052	310	1	⊆	⊆	X
ejpam-7052	310	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	310	3	-	-	PUNCT
ejpam-7052	310	4	cl(b	cl(b	NOUN
ejpam-7052	310	5	)	)	PUNCT
ejpam-7052	310	6	)	)	PUNCT
ejpam-7052	310	7	.	.	PUNCT
ejpam-7052	311	1	by	by	ADP
ejpam-7052	311	2	theorem	theorem	NOUN
ejpam-7052	311	3	4	4	NUM
ejpam-7052	311	4	,	,	PUNCT
ejpam-7052	311	5	f	f	PROPN
ejpam-7052	311	6	is	be	AUX
ejpam-7052	311	7	lower	low	ADJ
ejpam-7052	311	8	almost	almost	ADV
ejpam-7052	311	9	µ(σ1	µ(σ1	NOUN
ejpam-7052	311	10	,	,	PUNCT
ejpam-7052	311	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	311	12	.	.	PROPN
ejpam-7052	312	1	(	(	PUNCT
ejpam-7052	312	2	2	2	X
ejpam-7052	312	3	)	)	PUNCT
ejpam-7052	312	4	⇒	⇒	NOUN
ejpam-7052	312	5	(	(	PUNCT
ejpam-7052	312	6	4	4	NUM
ejpam-7052	312	7	):	):	PUNCT
ejpam-7052	312	8	let	let	VERB
ejpam-7052	312	9	k	k	PRON
ejpam-7052	312	10	be	be	AUX
ejpam-7052	312	11	any	any	DET
ejpam-7052	312	12	σ1σ2	σ1σ2	NUM
ejpam-7052	312	13	-	-	PUNCT
ejpam-7052	312	14	δ	δ	NOUN
ejpam-7052	312	15	-	-	PUNCT
ejpam-7052	312	16	closed	closed	ADJ
ejpam-7052	312	17	set	set	NOUN
ejpam-7052	312	18	of	of	ADP
ejpam-7052	312	19	y	y	PROPN
ejpam-7052	312	20	.	.	PUNCT
ejpam-7052	313	1	then	then	ADV
ejpam-7052	313	2	,	,	PUNCT
ejpam-7052	313	3	σ1σ2	σ1σ2	PROPN
ejpam-7052	313	4	-	-	PUNCT
ejpam-7052	313	5	δ	δ	NOUN
ejpam-7052	313	6	-	-	NOUN
ejpam-7052	313	7	cl(k	cl(k	NOUN
ejpam-7052	313	8	)	)	PUNCT
ejpam-7052	314	1	=	=	VERB
ejpam-7052	314	2	k.	k.	PROPN
ejpam-7052	314	3	by	by	ADP
ejpam-7052	314	4	(	(	PUNCT
ejpam-7052	314	5	2	2	NUM
ejpam-7052	314	6	)	)	PUNCT
ejpam-7052	314	7	,	,	PUNCT
ejpam-7052	314	8	we	we	PRON
ejpam-7052	314	9	have	have	VERB
ejpam-7052	314	10	cµ(f	cµ(f	X
ejpam-7052	315	1	+	+	ADJ
ejpam-7052	315	2	(	(	PUNCT
ejpam-7052	315	3	k	k	NOUN
ejpam-7052	315	4	)	)	PUNCT
ejpam-7052	315	5	)	)	PUNCT
ejpam-7052	316	1	⊆	⊆	NUM
ejpam-7052	316	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-7052	316	3	-	-	PUNCT
ejpam-7052	316	4	δ	δ	NOUN
ejpam-7052	316	5	-	-	NOUN
ejpam-7052	316	6	cl(k	cl(k	NOUN
ejpam-7052	316	7	)	)	PUNCT
ejpam-7052	316	8	)	)	PUNCT
ejpam-7052	317	1	=	=	SYM
ejpam-7052	317	2	f+(k	f+(k	X
ejpam-7052	317	3	)	)	PUNCT
ejpam-7052	317	4	and	and	CCONJ
ejpam-7052	317	5	so	so	ADV
ejpam-7052	317	6	f+(k	f+(k	PRON
ejpam-7052	317	7	)	)	PUNCT
ejpam-7052	317	8	is	be	AUX
ejpam-7052	317	9	µ-closed	µ-close	VERB
ejpam-7052	317	10	in	in	ADP
ejpam-7052	317	11	x.	x.	NOUN
ejpam-7052	317	12	(	(	PUNCT
ejpam-7052	317	13	4	4	NUM
ejpam-7052	317	14	)	)	PUNCT
ejpam-7052	317	15	⇒	⇒	NOUN
ejpam-7052	317	16	(	(	PUNCT
ejpam-7052	317	17	5	5	NUM
ejpam-7052	317	18	):	):	PUNCT
ejpam-7052	317	19	the	the	DET
ejpam-7052	317	20	proof	proof	NOUN
ejpam-7052	317	21	is	be	AUX
ejpam-7052	317	22	obvious	obvious	ADJ
ejpam-7052	317	23	.	.	PUNCT
ejpam-7052	318	1	(	(	PUNCT
ejpam-7052	318	2	5	5	X
ejpam-7052	318	3	)	)	PUNCT
ejpam-7052	318	4	⇒	⇒	NOUN
ejpam-7052	318	5	(	(	PUNCT
ejpam-7052	318	6	6	6	NUM
ejpam-7052	318	7	):	):	PUNCT
ejpam-7052	318	8	let	let	VERB
ejpam-7052	318	9	b	b	X
ejpam-7052	318	10	be	be	AUX
ejpam-7052	318	11	any	any	DET
ejpam-7052	318	12	subset	subset	NOUN
ejpam-7052	318	13	of	of	ADP
ejpam-7052	318	14	y	y	PROPN
ejpam-7052	318	15	.	.	PUNCT
ejpam-7052	319	1	then	then	ADV
ejpam-7052	319	2	by	by	ADP
ejpam-7052	319	3	(	(	PUNCT
ejpam-7052	319	4	5	5	NUM
ejpam-7052	319	5	)	)	PUNCT
ejpam-7052	319	6	,	,	PUNCT
ejpam-7052	319	7	we	we	PRON
ejpam-7052	319	8	have	have	VERB
ejpam-7052	319	9	f−(σ1σ2	f−(σ1σ2	VERB
ejpam-7052	319	10	-	-	PUNCT
ejpam-7052	319	11	δ	δ	NOUN
ejpam-7052	319	12	-	-	PUNCT
ejpam-7052	319	13	int(b	int(b	NOUN
ejpam-7052	319	14	)	)	PUNCT
ejpam-7052	319	15	)	)	PUNCT
ejpam-7052	320	1	=	=	SYM
ejpam-7052	320	2	iµ(f	iµ(f	NOUN
ejpam-7052	320	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7052	320	4	-	-	ADJ
ejpam-7052	320	5	δ	δ	NOUN
ejpam-7052	320	6	-	-	PUNCT
ejpam-7052	320	7	int(b	int(b	NOUN
ejpam-7052	320	8	)	)	PUNCT
ejpam-7052	320	9	)	)	PUNCT
ejpam-7052	320	10	)	)	PUNCT
ejpam-7052	321	1	⊆	⊆	NUM
ejpam-7052	321	2	iµ(f	iµ(f	NOUN
ejpam-7052	321	3	−(b	−(b	NOUN
ejpam-7052	321	4	)	)	PUNCT
ejpam-7052	321	5	)	)	PUNCT
ejpam-7052	321	6	.	.	PUNCT
ejpam-7052	322	1	(	(	PUNCT
ejpam-7052	322	2	6	6	X
ejpam-7052	322	3	)	)	PUNCT
ejpam-7052	322	4	⇒	⇒	NOUN
ejpam-7052	322	5	(	(	PUNCT
ejpam-7052	322	6	1	1	NUM
ejpam-7052	322	7	):	):	PUNCT
ejpam-7052	322	8	let	let	VERB
ejpam-7052	322	9	v	v	PART
ejpam-7052	322	10	be	be	AUX
ejpam-7052	322	11	any	any	DET
ejpam-7052	322	12	(	(	PUNCT
ejpam-7052	322	13	σ1	σ1	NOUN
ejpam-7052	322	14	,	,	PUNCT
ejpam-7052	322	15	σ2)r	σ2)r	NOUN
ejpam-7052	322	16	-	-	PUNCT
ejpam-7052	322	17	open	open	ADJ
ejpam-7052	322	18	set	set	NOUN
ejpam-7052	322	19	of	of	ADP
ejpam-7052	322	20	y	y	PROPN
ejpam-7052	322	21	.	.	PUNCT
ejpam-7052	323	1	then	then	ADV
ejpam-7052	323	2	,	,	PUNCT
ejpam-7052	323	3	we	we	PRON
ejpam-7052	323	4	have	have	VERB
ejpam-7052	323	5	v	v	NOUN
ejpam-7052	323	6	is	be	AUX
ejpam-7052	323	7	σ1σ2	σ1σ2	NOUN
ejpam-7052	323	8	-	-	PUNCT
ejpam-7052	323	9	δ	δ	NOUN
ejpam-7052	323	10	-	-	ADJ
ejpam-7052	323	11	open	open	ADJ
ejpam-7052	323	12	and	and	CCONJ
ejpam-7052	323	13	σ1σ2	σ1σ2	NOUN
ejpam-7052	323	14	-	-	PUNCT
ejpam-7052	323	15	δ	δ	NOUN
ejpam-7052	323	16	-	-	PUNCT
ejpam-7052	323	17	int(v	int(v	PROPN
ejpam-7052	323	18	)	)	PUNCT
ejpam-7052	323	19	=	=	SYM
ejpam-7052	323	20	v	v	NOUN
ejpam-7052	323	21	.	.	PUNCT
ejpam-7052	324	1	thus	thus	ADV
ejpam-7052	324	2	by	by	ADP
ejpam-7052	324	3	(	(	PUNCT
ejpam-7052	324	4	6	6	NUM
ejpam-7052	324	5	)	)	PUNCT
ejpam-7052	324	6	,	,	PUNCT
ejpam-7052	324	7	f−(v	f−(v	ADJ
ejpam-7052	324	8	)	)	PUNCT
ejpam-7052	324	9	⊆	⊆	NUM
ejpam-7052	324	10	iµ(f	iµ(f	NOUN
ejpam-7052	324	11	−(v	−(v	NOUN
ejpam-7052	324	12	)	)	PUNCT
ejpam-7052	324	13	)	)	PUNCT
ejpam-7052	324	14	and	and	CCONJ
ejpam-7052	324	15	hence	hence	ADV
ejpam-7052	324	16	f−(v	f−(v	ADJ
ejpam-7052	324	17	)	)	PUNCT
ejpam-7052	324	18	is	be	AUX
ejpam-7052	324	19	µ-open	µ-open	NOUN
ejpam-7052	324	20	in	in	ADP
ejpam-7052	324	21	x.	x.	NOUN
ejpam-7052	324	22	by	by	ADP
ejpam-7052	324	23	theorem	theorem	NOUN
ejpam-7052	324	24	4	4	NUM
ejpam-7052	324	25	,	,	PUNCT
ejpam-7052	324	26	f	f	PROPN
ejpam-7052	324	27	is	be	AUX
ejpam-7052	324	28	lower	low	ADJ
ejpam-7052	324	29	almost	almost	ADV
ejpam-7052	324	30	µ(σ1	µ(σ1	NOUN
ejpam-7052	324	31	,	,	PUNCT
ejpam-7052	324	32	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7052	324	33	.	.	NOUN
ejpam-7052	324	34	definition	definition	NOUN
ejpam-7052	324	35	3	3	NUM
ejpam-7052	324	36	.	.	PUNCT
ejpam-7052	325	1	[	[	X
ejpam-7052	325	2	22	22	NUM
ejpam-7052	325	3	]	]	PUNCT
ejpam-7052	325	4	a	a	DET
ejpam-7052	325	5	multifunction	multifunction	NOUN
ejpam-7052	325	6	f	f	NOUN
ejpam-7052	325	7	:	:	PUNCT
ejpam-7052	325	8	(	(	PUNCT
ejpam-7052	325	9	x,µ	x,µ	NOUN
ejpam-7052	325	10	)	)	PUNCT
ejpam-7052	325	11	→	→	SYM
ejpam-7052	325	12	(	(	PUNCT
ejpam-7052	325	13	y	y	PROPN
ejpam-7052	325	14	,	,	PUNCT
ejpam-7052	325	15	σ1	σ1	PROPN
ejpam-7052	325	16	,	,	PUNCT
ejpam-7052	325	17	σ2	σ2	PROPN
ejpam-7052	325	18	)	)	PUNCT
ejpam-7052	325	19	is	be	AUX
ejpam-7052	325	20	said	say	VERB
ejpam-7052	325	21	to	to	PART
ejpam-7052	325	22	be	be	AUX
ejpam-7052	325	23	upper	upper	ADJ
ejpam-7052	325	24	µ(σ1	µ(σ1	NOUN
ejpam-7052	325	25	,	,	PUNCT
ejpam-7052	325	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	325	27	at	at	ADP
ejpam-7052	325	28	a	a	DET
ejpam-7052	325	29	point	point	NOUN
ejpam-7052	325	30	x	x	SYM
ejpam-7052	325	31	∈	∈	NOUN
ejpam-7052	325	32	x	x	PUNCT
ejpam-7052	325	33	if	if	SCONJ
ejpam-7052	325	34	for	for	ADP
ejpam-7052	325	35	each	each	DET
ejpam-7052	325	36	σ1σ2	σ1σ2	VERB
ejpam-7052	325	37	-	-	ADJ
ejpam-7052	325	38	open	open	ADJ
ejpam-7052	325	39	set	set	NOUN
ejpam-7052	325	40	v	v	NOUN
ejpam-7052	325	41	of	of	ADP
ejpam-7052	325	42	y	y	PRON
ejpam-7052	325	43	such	such	ADJ
ejpam-7052	325	44	that	that	SCONJ
ejpam-7052	325	45	f	f	PROPN
ejpam-7052	325	46	(	(	PUNCT
ejpam-7052	325	47	x	x	X
ejpam-7052	325	48	)	)	PUNCT
ejpam-7052	325	49	⊆	⊆	NUM
ejpam-7052	325	50	v	v	NOUN
ejpam-7052	325	51	,	,	PUNCT
ejpam-7052	325	52	there	there	PRON
ejpam-7052	325	53	exists	exist	VERB
ejpam-7052	325	54	a	a	DET
ejpam-7052	325	55	µ-open	µ-open	NOUN
ejpam-7052	325	56	set	set	VERB
ejpam-7052	325	57	u	u	NOUN
ejpam-7052	325	58	of	of	ADP
ejpam-7052	325	59	x	x	PUNCT
ejpam-7052	325	60	containing	contain	VERB
ejpam-7052	325	61	x	x	PUNCT
ejpam-7052	325	62	such	such	ADJ
ejpam-7052	325	63	that	that	SCONJ
ejpam-7052	325	64	f	f	PROPN
ejpam-7052	325	65	(	(	PUNCT
ejpam-7052	325	66	u	u	NOUN
ejpam-7052	325	67	)	)	PUNCT
ejpam-7052	325	68	⊆	⊆	NUM
ejpam-7052	325	69	v	v	NOUN
ejpam-7052	325	70	.	.	PUNCT
ejpam-7052	326	1	a	a	DET
ejpam-7052	326	2	multifunction	multifunction	NOUN
ejpam-7052	326	3	f	f	NOUN
ejpam-7052	326	4	:	:	PUNCT
ejpam-7052	326	5	(	(	PUNCT
ejpam-7052	326	6	x,µ	x,µ	NOUN
ejpam-7052	326	7	)	)	PUNCT
ejpam-7052	326	8	→	→	SYM
ejpam-7052	326	9	(	(	PUNCT
ejpam-7052	326	10	y	y	PROPN
ejpam-7052	326	11	,	,	PUNCT
ejpam-7052	326	12	σ1	σ1	PROPN
ejpam-7052	326	13	,	,	PUNCT
ejpam-7052	326	14	σ2	σ2	PROPN
ejpam-7052	326	15	)	)	PUNCT
ejpam-7052	326	16	is	be	AUX
ejpam-7052	326	17	said	say	VERB
ejpam-7052	326	18	to	to	PART
ejpam-7052	326	19	be	be	AUX
ejpam-7052	326	20	upper	upper	ADJ
ejpam-7052	326	21	µ(σ1	µ(σ1	NOUN
ejpam-7052	326	22	,	,	PUNCT
ejpam-7052	326	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	326	24	if	if	SCONJ
ejpam-7052	326	25	f	f	PROPN
ejpam-7052	326	26	is	be	AUX
ejpam-7052	326	27	upper	upper	ADJ
ejpam-7052	326	28	µ(σ1	µ(σ1	NOUN
ejpam-7052	326	29	,	,	PUNCT
ejpam-7052	326	30	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	326	31	at	at	ADP
ejpam-7052	326	32	each	each	DET
ejpam-7052	326	33	point	point	NOUN
ejpam-7052	326	34	x	x	PUNCT
ejpam-7052	326	35	of	of	ADP
ejpam-7052	326	36	x.	x.	NOUN
ejpam-7052	326	37	definition	definition	NOUN
ejpam-7052	326	38	4	4	NUM
ejpam-7052	326	39	.	.	PUNCT
ejpam-7052	327	1	[	[	X
ejpam-7052	327	2	22	22	NUM
ejpam-7052	327	3	]	]	PUNCT
ejpam-7052	327	4	a	a	DET
ejpam-7052	327	5	multifunction	multifunction	NOUN
ejpam-7052	327	6	f	f	NOUN
ejpam-7052	327	7	:	:	PUNCT
ejpam-7052	327	8	(	(	PUNCT
ejpam-7052	327	9	x,µ	x,µ	NOUN
ejpam-7052	327	10	)	)	PUNCT
ejpam-7052	327	11	→	→	SYM
ejpam-7052	327	12	(	(	PUNCT
ejpam-7052	327	13	y	y	PROPN
ejpam-7052	327	14	,	,	PUNCT
ejpam-7052	327	15	σ1	σ1	PROPN
ejpam-7052	327	16	,	,	PUNCT
ejpam-7052	327	17	σ2	σ2	PROPN
ejpam-7052	327	18	)	)	PUNCT
ejpam-7052	327	19	is	be	AUX
ejpam-7052	327	20	said	say	VERB
ejpam-7052	327	21	to	to	PART
ejpam-7052	327	22	be	be	AUX
ejpam-7052	327	23	lower	low	ADJ
ejpam-7052	327	24	µ(σ1	µ(σ1	NOUN
ejpam-7052	327	25	,	,	PUNCT
ejpam-7052	327	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	327	27	at	at	ADP
ejpam-7052	327	28	a	a	DET
ejpam-7052	327	29	point	point	NOUN
ejpam-7052	327	30	x	x	SYM
ejpam-7052	327	31	∈	∈	NOUN
ejpam-7052	327	32	x	x	PUNCT
ejpam-7052	327	33	if	if	SCONJ
ejpam-7052	327	34	for	for	ADP
ejpam-7052	327	35	each	each	DET
ejpam-7052	327	36	σ1σ2	σ1σ2	VERB
ejpam-7052	327	37	-	-	ADJ
ejpam-7052	327	38	open	open	ADJ
ejpam-7052	327	39	set	set	NOUN
ejpam-7052	327	40	v	v	NOUN
ejpam-7052	327	41	of	of	ADP
ejpam-7052	327	42	y	y	PRON
ejpam-7052	327	43	such	such	ADJ
ejpam-7052	327	44	that	that	SCONJ
ejpam-7052	327	45	f	f	PROPN
ejpam-7052	327	46	(	(	PUNCT
ejpam-7052	327	47	x	x	NOUN
ejpam-7052	327	48	)	)	PUNCT
ejpam-7052	327	49	∩	∩	NOUN
ejpam-7052	327	50	v	v	ADP
ejpam-7052	327	51	̸=	̸=	PROPN
ejpam-7052	327	52	∅	∅	NOUN
ejpam-7052	327	53	,	,	PUNCT
ejpam-7052	327	54	there	there	PRON
ejpam-7052	327	55	exists	exist	VERB
ejpam-7052	327	56	a	a	DET
ejpam-7052	327	57	µ-open	µ-open	NOUN
ejpam-7052	327	58	set	set	VERB
ejpam-7052	327	59	u	u	NOUN
ejpam-7052	327	60	of	of	ADP
ejpam-7052	327	61	x	x	PUNCT
ejpam-7052	327	62	containing	contain	VERB
ejpam-7052	327	63	x	x	PUNCT
ejpam-7052	327	64	such	such	ADJ
ejpam-7052	327	65	that	that	SCONJ
ejpam-7052	327	66	f	f	PROPN
ejpam-7052	327	67	(	(	PUNCT
ejpam-7052	327	68	z	z	NOUN
ejpam-7052	327	69	)	)	PUNCT
ejpam-7052	327	70	∩	∩	NOUN
ejpam-7052	327	71	v	v	ADP
ejpam-7052	327	72	̸=	̸=	PROPN
ejpam-7052	327	73	∅	∅	NOUN
ejpam-7052	327	74	for	for	ADP
ejpam-7052	327	75	every	every	DET
ejpam-7052	327	76	z	z	NOUN
ejpam-7052	327	77	∈	∈	PROPN
ejpam-7052	327	78	u	u	NOUN
ejpam-7052	327	79	.	.	PUNCT
ejpam-7052	328	1	a	a	DET
ejpam-7052	328	2	multifunction	multifunction	NOUN
ejpam-7052	328	3	f	f	NOUN
ejpam-7052	328	4	:	:	PUNCT
ejpam-7052	328	5	(	(	PUNCT
ejpam-7052	328	6	x,µ	x,µ	NOUN
ejpam-7052	328	7	)	)	PUNCT
ejpam-7052	328	8	→	→	SYM
ejpam-7052	328	9	(	(	PUNCT
ejpam-7052	328	10	y	y	PROPN
ejpam-7052	328	11	,	,	PUNCT
ejpam-7052	328	12	σ1	σ1	PROPN
ejpam-7052	328	13	,	,	PUNCT
ejpam-7052	328	14	σ2	σ2	PROPN
ejpam-7052	328	15	)	)	PUNCT
ejpam-7052	328	16	is	be	AUX
ejpam-7052	328	17	said	say	VERB
ejpam-7052	328	18	to	to	PART
ejpam-7052	328	19	be	be	AUX
ejpam-7052	328	20	lower	low	ADJ
ejpam-7052	328	21	µ(σ1	µ(σ1	NOUN
ejpam-7052	328	22	,	,	PUNCT
ejpam-7052	328	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	328	24	if	if	SCONJ
ejpam-7052	328	25	f	f	PROPN
ejpam-7052	328	26	is	be	AUX
ejpam-7052	328	27	lower	low	ADJ
ejpam-7052	328	28	µ(σ1	µ(σ1	NOUN
ejpam-7052	328	29	,	,	PUNCT
ejpam-7052	328	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	328	31	at	at	ADP
ejpam-7052	328	32	each	each	DET
ejpam-7052	328	33	point	point	NOUN
ejpam-7052	328	34	x	x	PUNCT
ejpam-7052	328	35	of	of	ADP
ejpam-7052	328	36	x.	x.	NOUN
ejpam-7052	328	37	remark	remark	PROPN
ejpam-7052	328	38	1	1	NUM
ejpam-7052	328	39	.	.	PUNCT
ejpam-7052	328	40	for	for	ADP
ejpam-7052	328	41	a	a	DET
ejpam-7052	328	42	multifunction	multifunction	NOUN
ejpam-7052	328	43	f	f	NOUN
ejpam-7052	328	44	:	:	PUNCT
ejpam-7052	328	45	(	(	PUNCT
ejpam-7052	328	46	x,µ	x,µ	NOUN
ejpam-7052	328	47	)	)	PUNCT
ejpam-7052	328	48	→	→	SYM
ejpam-7052	328	49	(	(	PUNCT
ejpam-7052	328	50	y	y	PROPN
ejpam-7052	328	51	,	,	PUNCT
ejpam-7052	328	52	σ1	σ1	PROPN
ejpam-7052	328	53	,	,	PUNCT
ejpam-7052	328	54	σ2	σ2	NOUN
ejpam-7052	328	55	)	)	PUNCT
ejpam-7052	328	56	,	,	PUNCT
ejpam-7052	328	57	the	the	DET
ejpam-7052	328	58	following	follow	VERB
ejpam-7052	328	59	implication	implication	NOUN
ejpam-7052	328	60	holds	hold	VERB
ejpam-7052	328	61	:	:	PUNCT
ejpam-7052	328	62	upper	upper	ADJ
ejpam-7052	328	63	µ(σ1	µ(σ1	NOUN
ejpam-7052	328	64	,	,	PUNCT
ejpam-7052	328	65	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7052	328	66	⇒	⇒	NOUN
ejpam-7052	328	67	upper	upper	ADV
ejpam-7052	328	68	almost	almost	ADV
ejpam-7052	328	69	µ(σ1	µ(σ1	NOUN
ejpam-7052	328	70	,	,	PUNCT
ejpam-7052	328	71	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7052	328	72	.	.	PUNCT
ejpam-7052	329	1	the	the	DET
ejpam-7052	329	2	converse	converse	NOUN
ejpam-7052	329	3	of	of	ADP
ejpam-7052	329	4	the	the	DET
ejpam-7052	329	5	implication	implication	NOUN
ejpam-7052	329	6	is	be	AUX
ejpam-7052	329	7	not	not	PART
ejpam-7052	329	8	true	true	ADJ
ejpam-7052	329	9	in	in	ADP
ejpam-7052	329	10	general	general	ADJ
ejpam-7052	329	11	.	.	PUNCT
ejpam-7052	330	1	we	we	PRON
ejpam-7052	330	2	give	give	VERB
ejpam-7052	330	3	an	an	DET
ejpam-7052	330	4	example	example	NOUN
ejpam-7052	330	5	for	for	ADP
ejpam-7052	330	6	the	the	DET
ejpam-7052	330	7	implication	implication	NOUN
ejpam-7052	330	8	as	as	SCONJ
ejpam-7052	330	9	follows	follow	VERB
ejpam-7052	330	10	.	.	PUNCT
ejpam-7052	330	11	example	example	NOUN
ejpam-7052	331	1	1	1	NUM
ejpam-7052	331	2	.	.	PUNCT
ejpam-7052	331	3	let	let	VERB
ejpam-7052	331	4	x	x	PUNCT
ejpam-7052	331	5	=	=	PRON
ejpam-7052	331	6	{	{	PUNCT
ejpam-7052	331	7	1	1	NUM
ejpam-7052	331	8	,	,	PUNCT
ejpam-7052	331	9	2	2	NUM
ejpam-7052	331	10	,	,	PUNCT
ejpam-7052	331	11	3	3	NUM
ejpam-7052	331	12	}	}	PUNCT
ejpam-7052	331	13	with	with	ADP
ejpam-7052	331	14	a	a	DET
ejpam-7052	331	15	generalized	generalized	ADJ
ejpam-7052	331	16	topology	topology	NOUN
ejpam-7052	331	17	µ	µ	X
ejpam-7052	331	18	=	=	SYM
ejpam-7052	331	19	{	{	PUNCT
ejpam-7052	331	20	∅	∅	NOUN
ejpam-7052	331	21	,	,	PUNCT
ejpam-7052	331	22	{	{	PUNCT
ejpam-7052	331	23	1	1	NUM
ejpam-7052	331	24	}	}	PUNCT
ejpam-7052	331	25	,	,	PUNCT
ejpam-7052	331	26	{	{	PUNCT
ejpam-7052	331	27	2	2	NUM
ejpam-7052	331	28	}	}	PUNCT
ejpam-7052	331	29	,	,	PUNCT
ejpam-7052	331	30	{	{	PUNCT
ejpam-7052	331	31	1	1	NUM
ejpam-7052	331	32	,	,	PUNCT
ejpam-7052	331	33	2	2	NUM
ejpam-7052	331	34	}	}	PUNCT
ejpam-7052	331	35	,	,	PUNCT
ejpam-7052	331	36	x	x	NOUN
ejpam-7052	331	37	}	}	PUNCT
ejpam-7052	331	38	.	.	PUNCT
ejpam-7052	332	1	let	let	VERB
ejpam-7052	332	2	y	y	NOUN
ejpam-7052	332	3	=	=	PUNCT
ejpam-7052	332	4	{	{	PUNCT
ejpam-7052	332	5	p	p	X
ejpam-7052	332	6	,	,	PUNCT
ejpam-7052	332	7	q	q	ADJ
ejpam-7052	332	8	,	,	PUNCT
ejpam-7052	332	9	r	r	NOUN
ejpam-7052	332	10	}	}	PUNCT
ejpam-7052	332	11	with	with	ADP
ejpam-7052	332	12	topologies	topology	NOUN
ejpam-7052	332	13	σ1	σ1	NOUN
ejpam-7052	332	14	=	=	SYM
ejpam-7052	332	15	{	{	PUNCT
ejpam-7052	332	16	∅	∅	NOUN
ejpam-7052	332	17	,	,	PUNCT
ejpam-7052	332	18	{	{	PUNCT
ejpam-7052	332	19	p	p	X
ejpam-7052	332	20	}	}	PUNCT
ejpam-7052	332	21	,	,	PUNCT
ejpam-7052	332	22	{	{	PUNCT
ejpam-7052	332	23	p	p	X
ejpam-7052	332	24	,	,	PUNCT
ejpam-7052	332	25	q	q	ADJ
ejpam-7052	332	26	}	}	PUNCT
ejpam-7052	332	27	,	,	PUNCT
ejpam-7052	332	28	y	y	PROPN
ejpam-7052	332	29	}	}	PUNCT
ejpam-7052	332	30	and	and	CCONJ
ejpam-7052	332	31	σ2	σ2	PROPN
ejpam-7052	332	32	=	=	SYM
ejpam-7052	332	33	{	{	PUNCT
ejpam-7052	332	34	∅	∅	NOUN
ejpam-7052	332	35	,	,	PUNCT
ejpam-7052	332	36	{	{	PUNCT
ejpam-7052	332	37	p	p	X
ejpam-7052	332	38	}	}	PUNCT
ejpam-7052	332	39	,	,	PUNCT
ejpam-7052	332	40	{	{	PUNCT
ejpam-7052	332	41	q	q	X
ejpam-7052	332	42	}	}	PUNCT
ejpam-7052	332	43	,	,	PUNCT
ejpam-7052	332	44	{	{	PUNCT
ejpam-7052	332	45	p	p	X
ejpam-7052	332	46	,	,	PUNCT
ejpam-7052	332	47	q	q	ADJ
ejpam-7052	332	48	}	}	PUNCT
ejpam-7052	332	49	,	,	PUNCT
ejpam-7052	332	50	y	y	PROPN
ejpam-7052	332	51	}	}	PUNCT
ejpam-7052	332	52	.	.	PUNCT
ejpam-7052	333	1	a	a	DET
ejpam-7052	333	2	multifunction	multifunction	NOUN
ejpam-7052	333	3	f	f	NOUN
ejpam-7052	333	4	:	:	PUNCT
ejpam-7052	333	5	(	(	PUNCT
ejpam-7052	333	6	x,µ	x,µ	NOUN
ejpam-7052	333	7	)	)	PUNCT
ejpam-7052	333	8	→	→	SYM
ejpam-7052	333	9	(	(	PUNCT
ejpam-7052	333	10	y	y	PROPN
ejpam-7052	333	11	,	,	PUNCT
ejpam-7052	333	12	σ1	σ1	PROPN
ejpam-7052	333	13	,	,	PUNCT
ejpam-7052	333	14	σ2	σ2	PROPN
ejpam-7052	333	15	)	)	PUNCT
ejpam-7052	333	16	is	be	AUX
ejpam-7052	333	17	defined	define	VERB
ejpam-7052	333	18	as	as	SCONJ
ejpam-7052	333	19	follows	follow	VERB
ejpam-7052	333	20	:	:	PUNCT
ejpam-7052	333	21	f	f	X
ejpam-7052	333	22	(	(	PUNCT
ejpam-7052	333	23	1	1	X
ejpam-7052	333	24	)	)	PUNCT
ejpam-7052	333	25	=	=	PRON
ejpam-7052	333	26	{	{	PUNCT
ejpam-7052	333	27	r	r	NOUN
ejpam-7052	333	28	}	}	PUNCT
ejpam-7052	333	29	and	and	CCONJ
ejpam-7052	333	30	f	f	PROPN
ejpam-7052	333	31	(	(	PUNCT
ejpam-7052	333	32	2	2	NUM
ejpam-7052	333	33	)	)	PUNCT
ejpam-7052	333	34	=	=	SYM
ejpam-7052	333	35	f	f	PROPN
ejpam-7052	333	36	(	(	PUNCT
ejpam-7052	333	37	3	3	NUM
ejpam-7052	333	38	)	)	PUNCT
ejpam-7052	333	39	=	=	PRON
ejpam-7052	333	40	{	{	PUNCT
ejpam-7052	333	41	p	p	X
ejpam-7052	333	42	,	,	PUNCT
ejpam-7052	333	43	q	q	NOUN
ejpam-7052	333	44	}	}	PUNCT
ejpam-7052	333	45	.	.	PUNCT
ejpam-7052	334	1	then	then	ADV
ejpam-7052	334	2	,	,	PUNCT
ejpam-7052	334	3	f	f	PROPN
ejpam-7052	334	4	is	be	AUX
ejpam-7052	334	5	upper	upper	ADJ
ejpam-7052	334	6	almost	almost	ADV
ejpam-7052	334	7	µ(σ1	µ(σ1	NOUN
ejpam-7052	334	8	,	,	PUNCT
ejpam-7052	334	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	334	10	but	but	CCONJ
ejpam-7052	334	11	f	f	PROPN
ejpam-7052	334	12	is	be	AUX
ejpam-7052	334	13	not	not	PART
ejpam-7052	334	14	upper	upper	ADJ
ejpam-7052	334	15	µ(σ1	µ(σ1	NOUN
ejpam-7052	334	16	,	,	PUNCT
ejpam-7052	334	17	σ2)continuous	σ2)continuous	ADJ
ejpam-7052	334	18	.	.	PUNCT
ejpam-7052	335	1	recall	recall	VERB
ejpam-7052	335	2	that	that	SCONJ
ejpam-7052	335	3	a	a	DET
ejpam-7052	335	4	bitopological	bitopological	ADJ
ejpam-7052	335	5	space	space	NOUN
ejpam-7052	335	6	(	(	PUNCT
ejpam-7052	335	7	x	x	NOUN
ejpam-7052	335	8	,	,	PUNCT
ejpam-7052	335	9	τ1	τ1	NOUN
ejpam-7052	335	10	,	,	PUNCT
ejpam-7052	335	11	τ2	τ2	NOUN
ejpam-7052	335	12	)	)	PUNCT
ejpam-7052	335	13	is	be	AUX
ejpam-7052	335	14	said	say	VERB
ejpam-7052	335	15	to	to	PART
ejpam-7052	335	16	be	be	AUX
ejpam-7052	335	17	(	(	PUNCT
ejpam-7052	335	18	τ1	τ1	NOUN
ejpam-7052	335	19	,	,	PUNCT
ejpam-7052	335	20	τ2)s	τ2)s	NOUN
ejpam-7052	335	21	-	-	PUNCT
ejpam-7052	335	22	regular	regular	ADJ
ejpam-7052	335	23	[	[	X
ejpam-7052	335	24	23	23	NUM
ejpam-7052	335	25	]	]	X
ejpam-7052	335	26	if	if	SCONJ
ejpam-7052	335	27	for	for	ADP
ejpam-7052	335	28	each	each	DET
ejpam-7052	335	29	(	(	PUNCT
ejpam-7052	335	30	τ1	τ1	NOUN
ejpam-7052	335	31	,	,	PUNCT
ejpam-7052	335	32	τ2)s	τ2)s	NOUN
ejpam-7052	335	33	-	-	PUNCT
ejpam-7052	335	34	closed	close	VERB
ejpam-7052	335	35	set	set	VERB
ejpam-7052	335	36	f	f	NOUN
ejpam-7052	335	37	and	and	CCONJ
ejpam-7052	335	38	each	each	DET
ejpam-7052	335	39	x	x	PROPN
ejpam-7052	335	40	̸∈	̸∈	PROPN
ejpam-7052	335	41	f	f	PROPN
ejpam-7052	335	42	,	,	PUNCT
ejpam-7052	335	43	there	there	PRON
ejpam-7052	335	44	exist	exist	VERB
ejpam-7052	335	45	disjoint	disjoint	NOUN
ejpam-7052	335	46	(	(	PUNCT
ejpam-7052	335	47	τ1	τ1	NOUN
ejpam-7052	335	48	,	,	PUNCT
ejpam-7052	335	49	τ2)s	τ2)s	NOUN
ejpam-7052	335	50	-	-	PUNCT
ejpam-7052	335	51	open	open	ADJ
ejpam-7052	335	52	sets	set	VERB
ejpam-7052	335	53	u	u	NOUN
ejpam-7052	335	54	and	and	CCONJ
ejpam-7052	335	55	v	v	ADP
ejpam-7052	335	56	such	such	ADJ
ejpam-7052	335	57	that	that	SCONJ
ejpam-7052	335	58	x	x	SYM
ejpam-7052	335	59	∈	∈	PROPN
ejpam-7052	335	60	u	u	NOUN
ejpam-7052	335	61	and	and	CCONJ
ejpam-7052	335	62	f	f	PROPN
ejpam-7052	335	63	⊆	⊆	NUM
ejpam-7052	335	64	v	v	NOUN
ejpam-7052	335	65	.	.	PUNCT
ejpam-7052	336	1	lemma	lemma	PROPN
ejpam-7052	336	2	7	7	NUM
ejpam-7052	336	3	.	.	PUNCT
ejpam-7052	337	1	[	[	X
ejpam-7052	337	2	23	23	NUM
ejpam-7052	337	3	]	]	X
ejpam-7052	337	4	let	let	VERB
ejpam-7052	337	5	(	(	PUNCT
ejpam-7052	337	6	x	x	NOUN
ejpam-7052	337	7	,	,	PUNCT
ejpam-7052	337	8	τ1	τ1	NOUN
ejpam-7052	337	9	,	,	PUNCT
ejpam-7052	337	10	τ2	τ2	PROPN
ejpam-7052	337	11	)	)	PUNCT
ejpam-7052	337	12	be	be	VERB
ejpam-7052	337	13	a	a	DET
ejpam-7052	337	14	(	(	PUNCT
ejpam-7052	337	15	τ1	τ1	NOUN
ejpam-7052	337	16	,	,	PUNCT
ejpam-7052	337	17	τ2)s	τ2)s	NOUN
ejpam-7052	337	18	-	-	PUNCT
ejpam-7052	337	19	regular	regular	ADJ
ejpam-7052	337	20	space	space	NOUN
ejpam-7052	337	21	.	.	PUNCT
ejpam-7052	338	1	then	then	ADV
ejpam-7052	338	2	,	,	PUNCT
ejpam-7052	338	3	the	the	DET
ejpam-7052	338	4	following	follow	VERB
ejpam-7052	338	5	properties	property	NOUN
ejpam-7052	338	6	hold	hold	VERB
ejpam-7052	338	7	:	:	PUNCT
ejpam-7052	338	8	p.	p.	NOUN
ejpam-7052	338	9	pue	pue	NOUN
ejpam-7052	338	10	-	-	PUNCT
ejpam-7052	338	11	on	on	ADP
ejpam-7052	338	12	,	,	PUNCT
ejpam-7052	338	13	a.	a.	PROPN
ejpam-7052	338	14	sama	sama	PROPN
ejpam-7052	338	15	-	-	PUNCT
ejpam-7052	338	16	ae	ae	PROPN
ejpam-7052	338	17	,	,	PUNCT
ejpam-7052	338	18	c.	c.	PROPN
ejpam-7052	338	19	boonpok	boonpok	PROPN
ejpam-7052	338	20	/	/	SYM
ejpam-7052	338	21	eur	eur	PROPN
ejpam-7052	338	22	.	.	PUNCT
ejpam-7052	339	1	j.	j.	PROPN
ejpam-7052	339	2	pure	pure	PROPN
ejpam-7052	339	3	appl	appl	PROPN
ejpam-7052	339	4	.	.	PROPN
ejpam-7052	339	5	math	math	PROPN
ejpam-7052	339	6	,	,	PUNCT
ejpam-7052	339	7	18	18	NUM
ejpam-7052	339	8	(	(	PUNCT
ejpam-7052	339	9	4	4	NUM
ejpam-7052	339	10	)	)	PUNCT
ejpam-7052	339	11	(	(	PUNCT
ejpam-7052	339	12	2025	2025	NUM
ejpam-7052	339	13	)	)	PUNCT
ejpam-7052	339	14	,	,	PUNCT
ejpam-7052	339	15	7052	7052	NUM
ejpam-7052	339	16	12	12	NUM
ejpam-7052	339	17	of	of	ADP
ejpam-7052	339	18	14	14	NUM
ejpam-7052	339	19	(	(	PUNCT
ejpam-7052	339	20	1	1	NUM
ejpam-7052	339	21	)	)	PUNCT
ejpam-7052	339	22	τ1τ2	τ1τ2	NOUN
ejpam-7052	339	23	-	-	NUM
ejpam-7052	339	24	cl(a	cl(a	NUM
ejpam-7052	339	25	)	)	PUNCT
ejpam-7052	339	26	=	=	PUNCT
ejpam-7052	340	1	τ1τ2	τ1τ2	PROPN
ejpam-7052	340	2	-	-	ADJ
ejpam-7052	340	3	δ	δ	NOUN
ejpam-7052	340	4	-	-	PUNCT
ejpam-7052	340	5	cl(a	cl(a	NUM
ejpam-7052	340	6	)	)	PUNCT
ejpam-7052	340	7	for	for	ADP
ejpam-7052	340	8	every	every	DET
ejpam-7052	340	9	subset	subset	NOUN
ejpam-7052	340	10	a	a	PRON
ejpam-7052	340	11	of	of	ADP
ejpam-7052	340	12	x.	x.	NOUN
ejpam-7052	340	13	(	(	PUNCT
ejpam-7052	340	14	2	2	NUM
ejpam-7052	340	15	)	)	PUNCT
ejpam-7052	340	16	every	every	DET
ejpam-7052	340	17	τ1τ2	τ1τ2	NOUN
ejpam-7052	340	18	-	-	ADJ
ejpam-7052	340	19	open	open	ADJ
ejpam-7052	340	20	set	set	NOUN
ejpam-7052	340	21	is	be	AUX
ejpam-7052	340	22	τ1τ2	τ1τ2	ADJ
ejpam-7052	340	23	-	-	ADJ
ejpam-7052	340	24	δ	δ	NOUN
ejpam-7052	340	25	-	-	NOUN
ejpam-7052	340	26	open	open	ADJ
ejpam-7052	340	27	.	.	PUNCT
ejpam-7052	341	1	lemma	lemma	PROPN
ejpam-7052	341	2	8	8	NUM
ejpam-7052	341	3	.	.	PUNCT
ejpam-7052	342	1	[	[	X
ejpam-7052	342	2	22	22	NUM
ejpam-7052	342	3	]	]	PUNCT
ejpam-7052	342	4	for	for	ADP
ejpam-7052	342	5	a	a	DET
ejpam-7052	342	6	multifunction	multifunction	NOUN
ejpam-7052	342	7	f	f	NOUN
ejpam-7052	342	8	:	:	PUNCT
ejpam-7052	342	9	(	(	PUNCT
ejpam-7052	342	10	x,µ	x,µ	NOUN
ejpam-7052	342	11	)	)	PUNCT
ejpam-7052	342	12	→	→	SYM
ejpam-7052	342	13	(	(	PUNCT
ejpam-7052	342	14	y	y	PROPN
ejpam-7052	342	15	,	,	PUNCT
ejpam-7052	342	16	σ1	σ1	PROPN
ejpam-7052	342	17	,	,	PUNCT
ejpam-7052	342	18	σ2	σ2	NOUN
ejpam-7052	342	19	)	)	PUNCT
ejpam-7052	342	20	,	,	PUNCT
ejpam-7052	342	21	where	where	SCONJ
ejpam-7052	342	22	(	(	PUNCT
ejpam-7052	342	23	y	y	PROPN
ejpam-7052	342	24	,	,	PUNCT
ejpam-7052	342	25	σ1	σ1	PROPN
ejpam-7052	342	26	,	,	PUNCT
ejpam-7052	342	27	σ2	σ2	PROPN
ejpam-7052	342	28	)	)	PUNCT
ejpam-7052	342	29	is	be	AUX
ejpam-7052	342	30	a	a	DET
ejpam-7052	342	31	(	(	PUNCT
ejpam-7052	342	32	σ1	σ1	NOUN
ejpam-7052	342	33	,	,	PUNCT
ejpam-7052	342	34	σ2)s	σ2)s	NOUN
ejpam-7052	342	35	-	-	PUNCT
ejpam-7052	342	36	regular	regular	ADJ
ejpam-7052	342	37	space	space	NOUN
ejpam-7052	342	38	,	,	PUNCT
ejpam-7052	342	39	the	the	DET
ejpam-7052	342	40	following	follow	VERB
ejpam-7052	342	41	properties	property	NOUN
ejpam-7052	342	42	are	be	AUX
ejpam-7052	342	43	equivalent	equivalent	ADJ
ejpam-7052	342	44	:	:	PUNCT
ejpam-7052	342	45	(	(	PUNCT
ejpam-7052	342	46	1	1	X
ejpam-7052	342	47	)	)	PUNCT
ejpam-7052	342	48	f	f	PROPN
ejpam-7052	342	49	is	be	AUX
ejpam-7052	342	50	lower	low	ADJ
ejpam-7052	342	51	µ(σ1	µ(σ1	NOUN
ejpam-7052	342	52	,	,	PUNCT
ejpam-7052	342	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	342	54	;	;	PUNCT
ejpam-7052	342	55	(	(	PUNCT
ejpam-7052	342	56	2	2	X
ejpam-7052	342	57	)	)	PUNCT
ejpam-7052	342	58	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	342	59	-	-	PUNCT
ejpam-7052	342	60	δ	δ	NOUN
ejpam-7052	342	61	-	-	NOUN
ejpam-7052	342	62	cl(b	cl(b	NOUN
ejpam-7052	342	63	)	)	PUNCT
ejpam-7052	342	64	)	)	PUNCT
ejpam-7052	342	65	is	be	AUX
ejpam-7052	342	66	µ-closed	µ-close	VERB
ejpam-7052	342	67	in	in	ADP
ejpam-7052	342	68	x	x	PUNCT
ejpam-7052	342	69	for	for	ADP
ejpam-7052	342	70	every	every	DET
ejpam-7052	342	71	subset	subset	NOUN
ejpam-7052	342	72	b	b	PROPN
ejpam-7052	342	73	of	of	ADP
ejpam-7052	342	74	y	y	PROPN
ejpam-7052	342	75	;	;	PUNCT
ejpam-7052	342	76	(	(	PUNCT
ejpam-7052	342	77	3	3	X
ejpam-7052	342	78	)	)	PUNCT
ejpam-7052	342	79	f+(k	f+(k	NUM
ejpam-7052	342	80	)	)	PUNCT
ejpam-7052	342	81	is	be	AUX
ejpam-7052	342	82	µ-closed	µ-close	VERB
ejpam-7052	342	83	in	in	ADP
ejpam-7052	342	84	x	x	PUNCT
ejpam-7052	342	85	for	for	ADP
ejpam-7052	342	86	every	every	DET
ejpam-7052	342	87	σ1σ2	σ1σ2	NUM
ejpam-7052	342	88	-	-	PUNCT
ejpam-7052	342	89	δ	δ	NOUN
ejpam-7052	342	90	-	-	PUNCT
ejpam-7052	342	91	closed	close	VERB
ejpam-7052	342	92	set	set	ADJ
ejpam-7052	342	93	k	k	PROPN
ejpam-7052	342	94	of	of	ADP
ejpam-7052	342	95	y	y	PROPN
ejpam-7052	342	96	;	;	PUNCT
ejpam-7052	342	97	(	(	PUNCT
ejpam-7052	342	98	4	4	X
ejpam-7052	342	99	)	)	PUNCT
ejpam-7052	342	100	f−(v	f−(v	NOUN
ejpam-7052	342	101	)	)	PUNCT
ejpam-7052	342	102	is	be	AUX
ejpam-7052	342	103	µ-open	µ-open	NOUN
ejpam-7052	342	104	in	in	ADP
ejpam-7052	342	105	x	x	PUNCT
ejpam-7052	342	106	for	for	ADP
ejpam-7052	342	107	every	every	DET
ejpam-7052	342	108	σ1σ2	σ1σ2	NUM
ejpam-7052	342	109	-	-	PUNCT
ejpam-7052	342	110	δ	δ	NOUN
ejpam-7052	342	111	-	-	ADJ
ejpam-7052	342	112	open	open	ADJ
ejpam-7052	342	113	set	set	VERB
ejpam-7052	342	114	v	v	NOUN
ejpam-7052	342	115	of	of	ADP
ejpam-7052	342	116	y	y	PROPN
ejpam-7052	342	117	.	.	PUNCT
ejpam-7052	343	1	theorem	theorem	PROPN
ejpam-7052	343	2	12	12	NUM
ejpam-7052	343	3	.	.	PUNCT
ejpam-7052	344	1	for	for	ADP
ejpam-7052	344	2	a	a	DET
ejpam-7052	344	3	multifunction	multifunction	NOUN
ejpam-7052	344	4	f	f	NOUN
ejpam-7052	344	5	:	:	PUNCT
ejpam-7052	344	6	(	(	PUNCT
ejpam-7052	344	7	x,µ	x,µ	NOUN
ejpam-7052	344	8	)	)	PUNCT
ejpam-7052	344	9	→	→	SYM
ejpam-7052	344	10	(	(	PUNCT
ejpam-7052	344	11	y	y	PROPN
ejpam-7052	344	12	,	,	PUNCT
ejpam-7052	344	13	σ1	σ1	PROPN
ejpam-7052	344	14	,	,	PUNCT
ejpam-7052	344	15	σ2	σ2	NOUN
ejpam-7052	344	16	)	)	PUNCT
ejpam-7052	344	17	,	,	PUNCT
ejpam-7052	344	18	where	where	SCONJ
ejpam-7052	344	19	(	(	PUNCT
ejpam-7052	344	20	y	y	PROPN
ejpam-7052	344	21	,	,	PUNCT
ejpam-7052	344	22	σ1	σ1	PROPN
ejpam-7052	344	23	,	,	PUNCT
ejpam-7052	344	24	σ2	σ2	PROPN
ejpam-7052	344	25	)	)	PUNCT
ejpam-7052	344	26	is	be	AUX
ejpam-7052	344	27	a	a	DET
ejpam-7052	344	28	(	(	PUNCT
ejpam-7052	344	29	σ1	σ1	PROPN
ejpam-7052	344	30	,	,	PUNCT
ejpam-7052	344	31	σ2)sregular	σ2)sregular	ADJ
ejpam-7052	344	32	space	space	NOUN
ejpam-7052	344	33	,	,	PUNCT
ejpam-7052	344	34	the	the	DET
ejpam-7052	344	35	following	follow	VERB
ejpam-7052	344	36	properties	property	NOUN
ejpam-7052	344	37	are	be	AUX
ejpam-7052	344	38	equivalent	equivalent	ADJ
ejpam-7052	344	39	:	:	PUNCT
ejpam-7052	344	40	(	(	PUNCT
ejpam-7052	344	41	1	1	X
ejpam-7052	344	42	)	)	PUNCT
ejpam-7052	344	43	f	f	PROPN
ejpam-7052	344	44	is	be	AUX
ejpam-7052	344	45	lower	low	ADJ
ejpam-7052	344	46	µ(σ1	µ(σ1	NOUN
ejpam-7052	344	47	,	,	PUNCT
ejpam-7052	344	48	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	344	49	;	;	PUNCT
ejpam-7052	344	50	(	(	PUNCT
ejpam-7052	344	51	2	2	X
ejpam-7052	344	52	)	)	PUNCT
ejpam-7052	344	53	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7052	344	54	-	-	PUNCT
ejpam-7052	344	55	δ	δ	NOUN
ejpam-7052	344	56	-	-	NOUN
ejpam-7052	344	57	cl(b	cl(b	NOUN
ejpam-7052	344	58	)	)	PUNCT
ejpam-7052	344	59	)	)	PUNCT
ejpam-7052	344	60	is	be	AUX
ejpam-7052	344	61	µ-closed	µ-close	VERB
ejpam-7052	344	62	in	in	ADP
ejpam-7052	344	63	x	x	PUNCT
ejpam-7052	344	64	for	for	ADP
ejpam-7052	344	65	every	every	DET
ejpam-7052	344	66	subset	subset	NOUN
ejpam-7052	344	67	b	b	PROPN
ejpam-7052	344	68	of	of	ADP
ejpam-7052	344	69	y	y	PROPN
ejpam-7052	344	70	;	;	PUNCT
ejpam-7052	344	71	(	(	PUNCT
ejpam-7052	344	72	3	3	X
ejpam-7052	344	73	)	)	PUNCT
ejpam-7052	344	74	f+(k	f+(k	NUM
ejpam-7052	344	75	)	)	PUNCT
ejpam-7052	344	76	is	be	AUX
ejpam-7052	344	77	µ-closed	µ-close	VERB
ejpam-7052	344	78	in	in	ADP
ejpam-7052	344	79	x	x	PUNCT
ejpam-7052	344	80	for	for	ADP
ejpam-7052	344	81	every	every	DET
ejpam-7052	344	82	σ1σ2	σ1σ2	NUM
ejpam-7052	344	83	-	-	PUNCT
ejpam-7052	344	84	δ	δ	NOUN
ejpam-7052	344	85	-	-	PUNCT
ejpam-7052	344	86	closed	close	VERB
ejpam-7052	344	87	set	set	ADJ
ejpam-7052	344	88	k	k	PROPN
ejpam-7052	344	89	of	of	ADP
ejpam-7052	344	90	y	y	PROPN
ejpam-7052	344	91	;	;	PUNCT
ejpam-7052	344	92	(	(	PUNCT
ejpam-7052	344	93	4	4	X
ejpam-7052	344	94	)	)	PUNCT
ejpam-7052	344	95	f−(v	f−(v	NOUN
ejpam-7052	344	96	)	)	PUNCT
ejpam-7052	344	97	is	be	AUX
ejpam-7052	344	98	µ-open	µ-open	NOUN
ejpam-7052	344	99	in	in	ADP
ejpam-7052	344	100	x	x	PUNCT
ejpam-7052	344	101	for	for	ADP
ejpam-7052	344	102	every	every	DET
ejpam-7052	344	103	σ1σ2	σ1σ2	NUM
ejpam-7052	344	104	-	-	PUNCT
ejpam-7052	344	105	δ	δ	NOUN
ejpam-7052	344	106	-	-	ADJ
ejpam-7052	344	107	open	open	ADJ
ejpam-7052	344	108	set	set	VERB
ejpam-7052	344	109	v	v	NOUN
ejpam-7052	344	110	of	of	ADP
ejpam-7052	344	111	y	y	PROPN
ejpam-7052	344	112	;	;	PUNCT
ejpam-7052	344	113	(	(	PUNCT
ejpam-7052	344	114	5	5	X
ejpam-7052	344	115	)	)	PUNCT
ejpam-7052	344	116	f	f	PROPN
ejpam-7052	344	117	is	be	AUX
ejpam-7052	344	118	lower	low	ADJ
ejpam-7052	344	119	almost	almost	ADV
ejpam-7052	344	120	µ(σ1	µ(σ1	NOUN
ejpam-7052	344	121	,	,	PUNCT
ejpam-7052	345	1	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	345	2	.	.	NOUN
ejpam-7052	345	3	proof	proof	NOUN
ejpam-7052	345	4	.	.	PUNCT
ejpam-7052	346	1	the	the	DET
ejpam-7052	346	2	proofs	proof	NOUN
ejpam-7052	346	3	of	of	ADP
ejpam-7052	346	4	the	the	DET
ejpam-7052	346	5	implications	implication	NOUN
ejpam-7052	346	6	(	(	PUNCT
ejpam-7052	346	7	1	1	X
ejpam-7052	346	8	)	)	PUNCT
ejpam-7052	346	9	⇒	⇒	NOUN
ejpam-7052	346	10	(	(	PUNCT
ejpam-7052	346	11	2	2	NUM
ejpam-7052	346	12	)	)	PUNCT
ejpam-7052	346	13	⇒	⇒	NOUN
ejpam-7052	346	14	(	(	PUNCT
ejpam-7052	346	15	3	3	NUM
ejpam-7052	346	16	)	)	PUNCT
ejpam-7052	346	17	⇒	⇒	NOUN
ejpam-7052	346	18	(	(	PUNCT
ejpam-7052	346	19	4	4	X
ejpam-7052	346	20	)	)	PUNCT
ejpam-7052	346	21	are	be	AUX
ejpam-7052	346	22	similar	similar	ADJ
ejpam-7052	346	23	as	as	ADP
ejpam-7052	346	24	in	in	ADP
ejpam-7052	346	25	lemma	lemma	PROPN
ejpam-7052	346	26	8	8	NUM
ejpam-7052	346	27	.	.	PUNCT
ejpam-7052	347	1	(	(	PUNCT
ejpam-7052	347	2	4	4	X
ejpam-7052	347	3	)	)	PUNCT
ejpam-7052	347	4	⇒	⇒	NOUN
ejpam-7052	347	5	(	(	PUNCT
ejpam-7052	347	6	5	5	NUM
ejpam-7052	347	7	):	):	PUNCT
ejpam-7052	347	8	let	let	VERB
ejpam-7052	347	9	v	v	PART
ejpam-7052	347	10	be	be	AUX
ejpam-7052	347	11	any	any	DET
ejpam-7052	347	12	(	(	PUNCT
ejpam-7052	347	13	σ1	σ1	NOUN
ejpam-7052	347	14	,	,	PUNCT
ejpam-7052	347	15	σ2)r	σ2)r	NOUN
ejpam-7052	347	16	-	-	PUNCT
ejpam-7052	347	17	open	open	ADJ
ejpam-7052	347	18	set	set	NOUN
ejpam-7052	347	19	of	of	ADP
ejpam-7052	347	20	y	y	PROPN
ejpam-7052	347	21	.	.	PUNCT
ejpam-7052	348	1	then	then	ADV
ejpam-7052	348	2	,	,	PUNCT
ejpam-7052	348	3	v	v	NOUN
ejpam-7052	348	4	is	be	AUX
ejpam-7052	348	5	σ1σ2	σ1σ2	NOUN
ejpam-7052	348	6	-	-	ADJ
ejpam-7052	348	7	open	open	ADJ
ejpam-7052	348	8	in	in	ADP
ejpam-7052	348	9	y	y	PROPN
ejpam-7052	348	10	and	and	CCONJ
ejpam-7052	348	11	by	by	ADP
ejpam-7052	348	12	lemma	lemma	PROPN
ejpam-7052	348	13	7	7	NUM
ejpam-7052	348	14	,	,	PUNCT
ejpam-7052	348	15	v	v	NOUN
ejpam-7052	348	16	is	be	AUX
ejpam-7052	348	17	σ1σ2	σ1σ2	NOUN
ejpam-7052	348	18	-	-	PUNCT
ejpam-7052	348	19	δ	δ	NOUN
ejpam-7052	348	20	-	-	NOUN
ejpam-7052	348	21	open	open	ADJ
ejpam-7052	348	22	in	in	ADP
ejpam-7052	348	23	y	y	PROPN
ejpam-7052	348	24	.	.	PUNCT
ejpam-7052	349	1	by	by	ADP
ejpam-7052	349	2	(	(	PUNCT
ejpam-7052	349	3	4	4	NUM
ejpam-7052	349	4	)	)	PUNCT
ejpam-7052	349	5	,	,	PUNCT
ejpam-7052	349	6	we	we	PRON
ejpam-7052	349	7	have	have	VERB
ejpam-7052	349	8	f−(v	f−(v	NOUN
ejpam-7052	349	9	)	)	PUNCT
ejpam-7052	349	10	is	be	AUX
ejpam-7052	349	11	µ-open	µ-open	NOUN
ejpam-7052	349	12	in	in	ADP
ejpam-7052	349	13	x.	x.	NOUN
ejpam-7052	349	14	thus	thus	ADV
ejpam-7052	349	15	by	by	ADP
ejpam-7052	349	16	theorem	theorem	NOUN
ejpam-7052	349	17	4	4	NUM
ejpam-7052	349	18	,	,	PUNCT
ejpam-7052	349	19	f	f	PROPN
ejpam-7052	349	20	is	be	AUX
ejpam-7052	349	21	lower	low	ADJ
ejpam-7052	349	22	almost	almost	ADV
ejpam-7052	349	23	µ(σ1	µ(σ1	NOUN
ejpam-7052	349	24	,	,	PUNCT
ejpam-7052	349	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	349	26	.	.	PUNCT
ejpam-7052	350	1	(	(	PUNCT
ejpam-7052	350	2	5	5	X
ejpam-7052	350	3	)	)	PUNCT
ejpam-7052	350	4	⇒	⇒	NOUN
ejpam-7052	350	5	(	(	PUNCT
ejpam-7052	350	6	1	1	NUM
ejpam-7052	350	7	):	):	PUNCT
ejpam-7052	350	8	let	let	VERB
ejpam-7052	350	9	x	x	PUNCT
ejpam-7052	350	10	∈	∈	PROPN
ejpam-7052	350	11	x	x	X
ejpam-7052	350	12	and	and	CCONJ
ejpam-7052	350	13	v	v	X
ejpam-7052	350	14	be	be	AUX
ejpam-7052	350	15	any	any	DET
ejpam-7052	350	16	σ1σ2	σ1σ2	NOUN
ejpam-7052	350	17	-	-	ADJ
ejpam-7052	350	18	open	open	ADJ
ejpam-7052	350	19	set	set	NOUN
ejpam-7052	350	20	of	of	ADP
ejpam-7052	350	21	y	y	PRON
ejpam-7052	350	22	such	such	ADJ
ejpam-7052	350	23	that	that	PRON
ejpam-7052	350	24	v	v	NOUN
ejpam-7052	350	25	∩f	∩f	NOUN
ejpam-7052	350	26	(	(	PUNCT
ejpam-7052	350	27	x	x	X
ejpam-7052	350	28	)	)	PUNCT
ejpam-7052	350	29	̸=	̸=	PROPN
ejpam-7052	350	30	∅.	∅.	NOUN
ejpam-7052	350	31	since	since	SCONJ
ejpam-7052	350	32	(	(	PUNCT
ejpam-7052	350	33	y	y	PROPN
ejpam-7052	350	34	,	,	PUNCT
ejpam-7052	350	35	σ1	σ1	PROPN
ejpam-7052	350	36	,	,	PUNCT
ejpam-7052	350	37	σ2	σ2	PROPN
ejpam-7052	350	38	)	)	PUNCT
ejpam-7052	350	39	is	be	AUX
ejpam-7052	350	40	(	(	PUNCT
ejpam-7052	350	41	σ1	σ1	PROPN
ejpam-7052	350	42	,	,	PUNCT
ejpam-7052	350	43	σ2)s	σ2)s	NOUN
ejpam-7052	350	44	-	-	PUNCT
ejpam-7052	350	45	regular	regular	ADJ
ejpam-7052	350	46	,	,	PUNCT
ejpam-7052	350	47	there	there	PRON
ejpam-7052	350	48	exists	exist	VERB
ejpam-7052	350	49	a	a	DET
ejpam-7052	350	50	(	(	PUNCT
ejpam-7052	350	51	σ1	σ1	NOUN
ejpam-7052	350	52	,	,	PUNCT
ejpam-7052	350	53	σ2)r	σ2)r	NOUN
ejpam-7052	350	54	-	-	PUNCT
ejpam-7052	350	55	open	open	NOUN
ejpam-7052	350	56	set	set	NOUN
ejpam-7052	350	57	w	w	ADP
ejpam-7052	350	58	such	such	ADJ
ejpam-7052	350	59	that	that	DET
ejpam-7052	350	60	w	w	PROPN
ejpam-7052	350	61	∩f	∩f	NOUN
ejpam-7052	350	62	(	(	PUNCT
ejpam-7052	350	63	x	x	X
ejpam-7052	350	64	)	)	PUNCT
ejpam-7052	350	65	̸=	̸=	PROPN
ejpam-7052	350	66	∅	∅	NOUN
ejpam-7052	350	67	and	and	CCONJ
ejpam-7052	350	68	w	w	ADP
ejpam-7052	350	69	⊆	⊆	NUM
ejpam-7052	350	70	v	v	NOUN
ejpam-7052	350	71	.	.	PUNCT
ejpam-7052	351	1	since	since	SCONJ
ejpam-7052	351	2	f	f	PROPN
ejpam-7052	351	3	is	be	AUX
ejpam-7052	351	4	lower	low	ADJ
ejpam-7052	351	5	almost	almost	ADV
ejpam-7052	351	6	µ(σ1	µ(σ1	NOUN
ejpam-7052	351	7	,	,	PUNCT
ejpam-7052	351	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	351	9	,	,	PUNCT
ejpam-7052	351	10	there	there	PRON
ejpam-7052	351	11	exists	exist	VERB
ejpam-7052	351	12	a	a	DET
ejpam-7052	351	13	µ-open	µ-open	NOUN
ejpam-7052	351	14	set	set	VERB
ejpam-7052	351	15	u	u	NOUN
ejpam-7052	351	16	of	of	ADP
ejpam-7052	351	17	x	x	PUNCT
ejpam-7052	351	18	containing	contain	VERB
ejpam-7052	351	19	x	x	PUNCT
ejpam-7052	351	20	such	such	ADJ
ejpam-7052	351	21	that	that	SCONJ
ejpam-7052	351	22	w	w	PROPN
ejpam-7052	351	23	∩	∩	PROPN
ejpam-7052	351	24	f	f	X
ejpam-7052	351	25	(	(	PUNCT
ejpam-7052	351	26	z	z	NOUN
ejpam-7052	351	27	)	)	PUNCT
ejpam-7052	351	28	̸=	̸=	NOUN
ejpam-7052	351	29	∅	∅	NOUN
ejpam-7052	351	30	for	for	ADP
ejpam-7052	351	31	every	every	DET
ejpam-7052	351	32	z	z	NOUN
ejpam-7052	351	33	∈	∈	PROPN
ejpam-7052	351	34	u	u	NOUN
ejpam-7052	351	35	.	.	PUNCT
ejpam-7052	352	1	thus	thus	ADV
ejpam-7052	352	2	,	,	PUNCT
ejpam-7052	352	3	f	f	PROPN
ejpam-7052	352	4	(	(	PUNCT
ejpam-7052	352	5	z	z	NOUN
ejpam-7052	352	6	)	)	PUNCT
ejpam-7052	352	7	∩	∩	NOUN
ejpam-7052	352	8	v	v	ADP
ejpam-7052	352	9	̸=	̸=	PROPN
ejpam-7052	352	10	∅	∅	NOUN
ejpam-7052	352	11	for	for	ADP
ejpam-7052	352	12	every	every	DET
ejpam-7052	352	13	z	z	NOUN
ejpam-7052	352	14	∈	∈	PROPN
ejpam-7052	352	15	u	u	NOUN
ejpam-7052	352	16	.	.	PUNCT
ejpam-7052	353	1	this	this	PRON
ejpam-7052	353	2	shows	show	VERB
ejpam-7052	353	3	that	that	SCONJ
ejpam-7052	353	4	f	f	PROPN
ejpam-7052	353	5	is	be	AUX
ejpam-7052	353	6	lower	low	ADJ
ejpam-7052	353	7	µ(σ1	µ(σ1	NOUN
ejpam-7052	353	8	,	,	PUNCT
ejpam-7052	353	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7052	353	10	.	.	PUNCT
ejpam-7052	354	1	acknowledgements	acknowledgement	NOUN
ejpam-7052	354	2	this	this	DET
ejpam-7052	354	3	research	research	NOUN
ejpam-7052	354	4	project	project	NOUN
ejpam-7052	354	5	was	be	AUX
ejpam-7052	354	6	financially	financially	ADV
ejpam-7052	354	7	supported	support	VERB
ejpam-7052	354	8	by	by	ADP
ejpam-7052	354	9	mahasarakham	mahasarakham	PROPN
ejpam-7052	354	10	university	university	PROPN
ejpam-7052	354	11	.	.	PUNCT
ejpam-7052	355	1	p.	p.	NOUN
ejpam-7052	355	2	pue	pue	NOUN
ejpam-7052	355	3	-	-	PUNCT
ejpam-7052	355	4	on	on	ADP
ejpam-7052	355	5	,	,	PUNCT
ejpam-7052	355	6	a.	a.	PROPN
ejpam-7052	355	7	sama	sama	PROPN
ejpam-7052	355	8	-	-	PUNCT
ejpam-7052	355	9	ae	ae	PROPN
ejpam-7052	355	10	,	,	PUNCT
ejpam-7052	355	11	c.	c.	PROPN
ejpam-7052	355	12	boonpok	boonpok	PROPN
ejpam-7052	355	13	/	/	SYM
ejpam-7052	355	14	eur	eur	PROPN
ejpam-7052	355	15	.	.	PUNCT
ejpam-7052	356	1	j.	j.	PROPN
ejpam-7052	356	2	pure	pure	PROPN
ejpam-7052	356	3	appl	appl	PROPN
ejpam-7052	356	4	.	.	PROPN
ejpam-7052	356	5	math	math	PROPN
ejpam-7052	356	6	,	,	PUNCT
ejpam-7052	356	7	18	18	NUM
ejpam-7052	356	8	(	(	PUNCT
ejpam-7052	356	9	4	4	NUM
ejpam-7052	356	10	)	)	PUNCT
ejpam-7052	356	11	(	(	PUNCT
ejpam-7052	356	12	2025	2025	NUM
ejpam-7052	356	13	)	)	PUNCT
ejpam-7052	356	14	,	,	PUNCT
ejpam-7052	356	15	7052	7052	NUM
ejpam-7052	356	16	13	13	NUM
ejpam-7052	356	17	of	of	ADP
ejpam-7052	356	18	14	14	NUM
ejpam-7052	356	19	references	reference	NOUN
ejpam-7052	356	20	[	[	X
ejpam-7052	356	21	1	1	NUM
ejpam-7052	356	22	]	]	PUNCT
ejpam-7052	356	23	m.	m.	NOUN
ejpam-7052	356	24	k.	k.	PROPN
ejpam-7052	356	25	singal	singal	PROPN
ejpam-7052	356	26	and	and	CCONJ
ejpam-7052	356	27	a.	a.	PROPN
ejpam-7052	356	28	r.	r.	PROPN
ejpam-7052	356	29	singal	singal	PROPN
ejpam-7052	356	30	.	.	PUNCT
ejpam-7052	357	1	almost	almost	ADV
ejpam-7052	357	2	continuous	continuous	ADJ
ejpam-7052	357	3	mappings	mapping	NOUN
ejpam-7052	357	4	.	.	PUNCT
ejpam-7052	358	1	yokohama	yokohama	PROPN
ejpam-7052	358	2	mathematical	mathematical	PROPN
ejpam-7052	358	3	journal	journal	PROPN
ejpam-7052	358	4	,	,	PUNCT
ejpam-7052	358	5	16:63–73	16:63–73	PROPN
ejpam-7052	358	6	,	,	PUNCT
ejpam-7052	358	7	1968	1968	NUM
ejpam-7052	358	8	.	.	PUNCT
ejpam-7052	359	1	[	[	X
ejpam-7052	359	2	2	2	NUM
ejpam-7052	359	3	]	]	X
ejpam-7052	359	4	b.	b.	PROPN
ejpam-7052	359	5	m.	m.	PROPN
ejpam-7052	359	6	munshi	munshi	PROPN
ejpam-7052	359	7	and	and	CCONJ
ejpam-7052	359	8	d.	d.	PROPN
ejpam-7052	359	9	s.	s.	PROPN
ejpam-7052	359	10	bassan	bassan	PROPN
ejpam-7052	359	11	.	.	PUNCT
ejpam-7052	360	1	almost	almost	ADV
ejpam-7052	360	2	semi	semi	ADJ
ejpam-7052	360	3	-	-	ADJ
ejpam-7052	360	4	continuous	continuous	ADJ
ejpam-7052	360	5	mappings	mapping	NOUN
ejpam-7052	360	6	.	.	PUNCT
ejpam-7052	361	1	the	the	DET
ejpam-7052	361	2	mathematics	mathematics	PROPN
ejpam-7052	361	3	student	student	NOUN
ejpam-7052	361	4	,	,	PUNCT
ejpam-7052	361	5	49:239–248	49:239–248	PROPN
ejpam-7052	361	6	,	,	PUNCT
ejpam-7052	361	7	1981	1981	NUM
ejpam-7052	361	8	.	.	PUNCT
ejpam-7052	362	1	[	[	X
ejpam-7052	362	2	3	3	X
ejpam-7052	362	3	]	]	PUNCT
ejpam-7052	362	4	t.	t.	PROPN
ejpam-7052	362	5	noiri	noiri	PROPN
ejpam-7052	362	6	.	.	PUNCT
ejpam-7052	363	1	almost	almost	ADV
ejpam-7052	363	2	α	α	NUM
ejpam-7052	363	3	-	-	ADJ
ejpam-7052	363	4	continuous	continuous	ADJ
ejpam-7052	363	5	functions	function	NOUN
ejpam-7052	363	6	.	.	PUNCT
ejpam-7052	364	1	kyungpook	kyungpook	PROPN
ejpam-7052	364	2	mathematical	mathematical	PROPN
ejpam-7052	364	3	journal	journal	PROPN
ejpam-7052	364	4	,	,	PUNCT
ejpam-7052	364	5	28:71–77	28:71–77	PROPN
ejpam-7052	364	6	,	,	PUNCT
ejpam-7052	364	7	1988	1988	NUM
ejpam-7052	364	8	.	.	PUNCT
ejpam-7052	365	1	[	[	X
ejpam-7052	365	2	4	4	X
ejpam-7052	365	3	]	]	PUNCT
ejpam-7052	365	4	a.	a.	NOUN
ejpam-7052	365	5	a.	a.	NOUN
ejpam-7052	365	6	nasef	nasef	PROPN
ejpam-7052	365	7	and	and	CCONJ
ejpam-7052	365	8	t.	t.	PROPN
ejpam-7052	365	9	noiri	noiri	PROPN
ejpam-7052	365	10	.	.	PUNCT
ejpam-7052	366	1	some	some	DET
ejpam-7052	366	2	weak	weak	ADJ
ejpam-7052	366	3	forms	form	NOUN
ejpam-7052	366	4	of	of	ADP
ejpam-7052	366	5	almost	almost	ADV
ejpam-7052	366	6	continuity	continuity	NOUN
ejpam-7052	366	7	.	.	PUNCT
ejpam-7052	367	1	acta	acta	PROPN
ejpam-7052	367	2	mathematica	mathematica	PROPN
ejpam-7052	367	3	hungarica	hungarica	PROPN
ejpam-7052	367	4	,	,	PUNCT
ejpam-7052	367	5	74(3):211–219	74(3):211–219	PROPN
ejpam-7052	367	6	,	,	PUNCT
ejpam-7052	367	7	1997	1997	NUM
ejpam-7052	367	8	.	.	PUNCT
ejpam-7052	368	1	[	[	X
ejpam-7052	368	2	5	5	NUM
ejpam-7052	368	3	]	]	SYM
ejpam-7052	368	4	á	á	PROPN
ejpam-7052	368	5	.	.	PUNCT
ejpam-7052	368	6	császár	császár	PROPN
ejpam-7052	368	7	.	.	PUNCT
ejpam-7052	369	1	generalized	generalize	VERB
ejpam-7052	369	2	topology	topology	NOUN
ejpam-7052	369	3	,	,	PUNCT
ejpam-7052	369	4	generalized	generalize	VERB
ejpam-7052	369	5	continuity	continuity	NOUN
ejpam-7052	369	6	.	.	PUNCT
ejpam-7052	370	1	acta	acta	PROPN
ejpam-7052	370	2	mathematica	mathematica	PROPN
ejpam-7052	370	3	hungarica	hungarica	PROPN
ejpam-7052	370	4	,	,	PUNCT
ejpam-7052	370	5	96(4):351–357	96(4):351–357	NOUN
ejpam-7052	370	6	,	,	PUNCT
ejpam-7052	370	7	2002	2002	NUM
ejpam-7052	370	8	.	.	PUNCT
ejpam-7052	371	1	[	[	X
ejpam-7052	371	2	6	6	NUM
ejpam-7052	371	3	]	]	PUNCT
ejpam-7052	371	4	a.	a.	NOUN
ejpam-7052	371	5	kanibir	kanibir	NOUN
ejpam-7052	371	6	and	and	CCONJ
ejpam-7052	371	7	i.	i.	PROPN
ejpam-7052	371	8	l.	l.	PROPN
ejpam-7052	371	9	reilly	reilly	PROPN
ejpam-7052	371	10	.	.	PUNCT
ejpam-7052	372	1	generalized	generalize	VERB
ejpam-7052	372	2	continuity	continuity	NOUN
ejpam-7052	372	3	for	for	ADP
ejpam-7052	372	4	multifunctions	multifunction	NOUN
ejpam-7052	372	5	.	.	PUNCT
ejpam-7052	373	1	acta	acta	PROPN
ejpam-7052	373	2	mathematica	mathematica	PROPN
ejpam-7052	373	3	hungarica	hungarica	PROPN
ejpam-7052	373	4	,	,	PUNCT
ejpam-7052	373	5	122(3):283–292	122(3):283–292	NUM
ejpam-7052	373	6	,	,	PUNCT
ejpam-7052	373	7	2009	2009	NUM
ejpam-7052	373	8	.	.	PUNCT
ejpam-7052	374	1	[	[	X
ejpam-7052	374	2	7	7	X
ejpam-7052	374	3	]	]	X
ejpam-7052	374	4	c.	c.	PROPN
ejpam-7052	374	5	boonpok	boonpok	PROPN
ejpam-7052	374	6	.	.	PUNCT
ejpam-7052	375	1	on	on	ADP
ejpam-7052	375	2	upper	upper	ADJ
ejpam-7052	375	3	and	and	CCONJ
ejpam-7052	375	4	lower	low	ADJ
ejpam-7052	375	5	β(µx	β(µx	PROPN
ejpam-7052	375	6	,	,	PUNCT
ejpam-7052	375	7	µy	µy	CCONJ
ejpam-7052	375	8	)	)	PUNCT
ejpam-7052	375	9	-continuous	-continuous	ADJ
ejpam-7052	375	10	multifunctions	multifunction	NOUN
ejpam-7052	375	11	.	.	PUNCT
ejpam-7052	376	1	international	international	ADJ
ejpam-7052	376	2	journal	journal	PROPN
ejpam-7052	376	3	of	of	ADP
ejpam-7052	376	4	mathematics	mathematics	PROPN
ejpam-7052	376	5	and	and	CCONJ
ejpam-7052	376	6	mathematical	mathematical	ADJ
ejpam-7052	376	7	sciences	science	NOUN
ejpam-7052	376	8	,	,	PUNCT
ejpam-7052	376	9	2012:931656	2012:931656	NUM
ejpam-7052	376	10	,	,	PUNCT
ejpam-7052	376	11	2012	2012	NUM
ejpam-7052	376	12	.	.	PUNCT
ejpam-7052	377	1	[	[	X
ejpam-7052	377	2	8	8	NUM
ejpam-7052	377	3	]	]	X
ejpam-7052	377	4	n.	n.	NOUN
ejpam-7052	377	5	srisarakham	srisarakham	PROPN
ejpam-7052	377	6	and	and	CCONJ
ejpam-7052	377	7	c.	c.	PROPN
ejpam-7052	377	8	boonpok	boonpok	PROPN
ejpam-7052	377	9	.	.	PUNCT
ejpam-7052	378	1	characterizations	characterization	NOUN
ejpam-7052	378	2	of	of	ADP
ejpam-7052	378	3	upper	upper	ADJ
ejpam-7052	378	4	and	and	CCONJ
ejpam-7052	378	5	lower	low	ADJ
ejpam-7052	378	6	α(µx	α(µx	NUM
ejpam-7052	378	7	,	,	PUNCT
ejpam-7052	378	8	µy	µy	CCONJ
ejpam-7052	378	9	)	)	PUNCT
ejpam-7052	378	10	continuous	continuous	ADJ
ejpam-7052	378	11	multifunctions	multifunction	NOUN
ejpam-7052	378	12	.	.	PUNCT
ejpam-7052	379	1	journal	journal	PROPN
ejpam-7052	379	2	of	of	ADP
ejpam-7052	379	3	mathematics	mathematics	PROPN
ejpam-7052	379	4	and	and	CCONJ
ejpam-7052	379	5	computer	computer	NOUN
ejpam-7052	379	6	science	science	NOUN
ejpam-7052	379	7	,	,	PUNCT
ejpam-7052	379	8	17:255	17:255	NUM
ejpam-7052	379	9	–	–	PUNCT
ejpam-7052	379	10	265	265	NUM
ejpam-7052	379	11	,	,	PUNCT
ejpam-7052	379	12	2017	2017	NUM
ejpam-7052	379	13	.	.	PUNCT
ejpam-7052	380	1	[	[	X
ejpam-7052	380	2	9	9	NUM
ejpam-7052	380	3	]	]	X
ejpam-7052	380	4	p.	p.	NOUN
ejpam-7052	380	5	pue	pue	NOUN
ejpam-7052	380	6	-	-	PUNCT
ejpam-7052	380	7	on	on	ADP
ejpam-7052	380	8	,	,	PUNCT
ejpam-7052	380	9	s.	s.	PROPN
ejpam-7052	380	10	sompong	sompong	PROPN
ejpam-7052	380	11	,	,	PUNCT
ejpam-7052	380	12	and	and	CCONJ
ejpam-7052	380	13	c.	c.	PROPN
ejpam-7052	380	14	boonpok	boonpok	PROPN
ejpam-7052	380	15	.	.	PUNCT
ejpam-7052	381	1	upper	upper	ADJ
ejpam-7052	381	2	and	and	CCONJ
ejpam-7052	381	3	lower	low	ADJ
ejpam-7052	381	4	(	(	PUNCT
ejpam-7052	381	5	τ1	τ1	NOUN
ejpam-7052	381	6	,	,	PUNCT
ejpam-7052	381	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	381	8	multifunctions	multifunction	NOUN
ejpam-7052	381	9	.	.	PUNCT
ejpam-7052	382	1	international	international	ADJ
ejpam-7052	382	2	journal	journal	PROPN
ejpam-7052	382	3	of	of	ADP
ejpam-7052	382	4	mathematics	mathematic	NOUN
ejpam-7052	382	5	and	and	CCONJ
ejpam-7052	382	6	computer	computer	NOUN
ejpam-7052	382	7	science	science	NOUN
ejpam-7052	382	8	,	,	PUNCT
ejpam-7052	382	9	19(4):1305	19(4):1305	NUM
ejpam-7052	382	10	–	–	PUNCT
ejpam-7052	382	11	1310	1310	NUM
ejpam-7052	382	12	,	,	PUNCT
ejpam-7052	382	13	2024	2024	NUM
ejpam-7052	382	14	.	.	PUNCT
ejpam-7052	383	1	[	[	X
ejpam-7052	383	2	10	10	NUM
ejpam-7052	383	3	]	]	X
ejpam-7052	383	4	c.	c.	PROPN
ejpam-7052	383	5	klanarong	klanarong	PROPN
ejpam-7052	383	6	,	,	PUNCT
ejpam-7052	383	7	s.	s.	PROPN
ejpam-7052	383	8	sompong	sompong	PROPN
ejpam-7052	383	9	,	,	PUNCT
ejpam-7052	383	10	and	and	CCONJ
ejpam-7052	383	11	c.	c.	PROPN
ejpam-7052	383	12	boonpok	boonpok	PROPN
ejpam-7052	383	13	.	.	PUNCT
ejpam-7052	384	1	upper	upper	ADJ
ejpam-7052	384	2	and	and	CCONJ
ejpam-7052	384	3	lower	low	ADJ
ejpam-7052	384	4	almost	almost	ADV
ejpam-7052	384	5	(	(	PUNCT
ejpam-7052	384	6	τ1	τ1	NOUN
ejpam-7052	384	7	,	,	PUNCT
ejpam-7052	384	8	τ2)continuous	τ2)continuous	ADJ
ejpam-7052	384	9	multifunctions	multifunction	NOUN
ejpam-7052	384	10	.	.	PUNCT
ejpam-7052	385	1	european	european	ADJ
ejpam-7052	385	2	journal	journal	PROPN
ejpam-7052	385	3	of	of	ADP
ejpam-7052	385	4	pure	pure	ADJ
ejpam-7052	385	5	and	and	CCONJ
ejpam-7052	385	6	applied	applied	ADJ
ejpam-7052	385	7	mathematics	mathematic	NOUN
ejpam-7052	385	8	,	,	PUNCT
ejpam-7052	385	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-7052	385	10	,	,	PUNCT
ejpam-7052	385	11	2024	2024	NUM
ejpam-7052	385	12	.	.	PUNCT
ejpam-7052	386	1	[	[	X
ejpam-7052	386	2	11	11	NUM
ejpam-7052	386	3	]	]	X
ejpam-7052	386	4	c.	c.	PROPN
ejpam-7052	386	5	viriyapong	viriyapong	PROPN
ejpam-7052	386	6	,	,	PUNCT
ejpam-7052	386	7	a.	a.	PROPN
ejpam-7052	386	8	sama	sama	PROPN
ejpam-7052	386	9	-	-	PUNCT
ejpam-7052	386	10	ae	ae	PROPN
ejpam-7052	386	11	,	,	PUNCT
ejpam-7052	386	12	and	and	CCONJ
ejpam-7052	386	13	c.	c.	PROPN
ejpam-7052	386	14	boonpok	boonpok	PROPN
ejpam-7052	386	15	.	.	PUNCT
ejpam-7052	387	1	almost	almost	ADV
ejpam-7052	387	2	continuity	continuity	NOUN
ejpam-7052	387	3	for	for	ADP
ejpam-7052	387	4	multifunctions	multifunction	NOUN
ejpam-7052	387	5	defined	define	VERB
ejpam-7052	387	6	from	from	ADP
ejpam-7052	387	7	an	an	DET
ejpam-7052	387	8	ideal	ideal	ADJ
ejpam-7052	387	9	topological	topological	ADJ
ejpam-7052	387	10	space	space	NOUN
ejpam-7052	387	11	into	into	ADP
ejpam-7052	387	12	a	a	DET
ejpam-7052	387	13	bitopological	bitopological	ADJ
ejpam-7052	387	14	space	space	NOUN
ejpam-7052	387	15	.	.	PUNCT
ejpam-7052	388	1	european	european	ADJ
ejpam-7052	388	2	journal	journal	PROPN
ejpam-7052	388	3	of	of	ADP
ejpam-7052	388	4	pure	pure	ADJ
ejpam-7052	388	5	and	and	CCONJ
ejpam-7052	388	6	applied	applied	ADJ
ejpam-7052	388	7	mathematics	mathematic	NOUN
ejpam-7052	388	8	,	,	PUNCT
ejpam-7052	388	9	18(3):6566	18(3):6566	NUM
ejpam-7052	388	10	,	,	PUNCT
ejpam-7052	388	11	2025	2025	NUM
ejpam-7052	388	12	.	.	PUNCT
ejpam-7052	389	1	[	[	X
ejpam-7052	389	2	12	12	NUM
ejpam-7052	389	3	]	]	X
ejpam-7052	389	4	c.	c.	PROPN
ejpam-7052	389	5	boonpok	boonpok	PROPN
ejpam-7052	389	6	,	,	PUNCT
ejpam-7052	389	7	c.	c.	PROPN
ejpam-7052	389	8	viriyapong	viriyapong	PROPN
ejpam-7052	389	9	,	,	PUNCT
ejpam-7052	389	10	and	and	CCONJ
ejpam-7052	389	11	m.	m.	NOUN
ejpam-7052	389	12	thongmoon	thongmoon	NOUN
ejpam-7052	389	13	.	.	PUNCT
ejpam-7052	390	1	on	on	ADP
ejpam-7052	390	2	upper	upper	ADJ
ejpam-7052	390	3	and	and	CCONJ
ejpam-7052	390	4	lower	low	ADJ
ejpam-7052	390	5	(	(	PUNCT
ejpam-7052	390	6	τ1	τ1	NOUN
ejpam-7052	390	7	,	,	PUNCT
ejpam-7052	390	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7052	390	9	multifunctions	multifunction	NOUN
ejpam-7052	390	10	.	.	PUNCT
ejpam-7052	391	1	journal	journal	PROPN
ejpam-7052	391	2	of	of	ADP
ejpam-7052	391	3	mathematics	mathematics	PROPN
ejpam-7052	391	4	and	and	CCONJ
ejpam-7052	391	5	computer	computer	NOUN
ejpam-7052	391	6	science	science	NOUN
ejpam-7052	391	7	,	,	PUNCT
ejpam-7052	391	8	18:282	18:282	NUM
ejpam-7052	391	9	–	–	PUNCT
ejpam-7052	391	10	293	293	NUM
ejpam-7052	391	11	,	,	PUNCT
ejpam-7052	391	12	2018	2018	NUM
ejpam-7052	391	13	.	.	PUNCT
ejpam-7052	392	1	[	[	X
ejpam-7052	392	2	13	13	NUM
ejpam-7052	392	3	]	]	X
ejpam-7052	392	4	c.	c.	PROPN
ejpam-7052	392	5	viriyapong	viriyapong	PROPN
ejpam-7052	392	6	and	and	CCONJ
ejpam-7052	392	7	c.	c.	PROPN
ejpam-7052	392	8	boonpok	boonpok	PROPN
ejpam-7052	392	9	.	.	PUNCT
ejpam-7052	393	1	(	(	PUNCT
ejpam-7052	393	2	τ1	τ1	NOUN
ejpam-7052	393	3	,	,	PUNCT
ejpam-7052	393	4	τ2)α	τ2)α	NOUN
ejpam-7052	393	5	-	-	PUNCT
ejpam-7052	393	6	continuity	continuity	NOUN
ejpam-7052	393	7	for	for	ADP
ejpam-7052	393	8	multifunctions	multifunction	NOUN
ejpam-7052	393	9	.	.	PUNCT
ejpam-7052	394	1	journal	journal	PROPN
ejpam-7052	394	2	of	of	ADP
ejpam-7052	394	3	mathematics	mathematic	NOUN
ejpam-7052	394	4	,	,	PUNCT
ejpam-7052	394	5	2020:6285763	2020:6285763	NUM
ejpam-7052	394	6	,	,	PUNCT
ejpam-7052	394	7	2020	2020	NUM
ejpam-7052	394	8	.	.	PUNCT
ejpam-7052	395	1	[	[	X
ejpam-7052	395	2	14	14	NUM
ejpam-7052	395	3	]	]	X
ejpam-7052	395	4	c.	c.	PROPN
ejpam-7052	395	5	boonpok	boonpok	PROPN
ejpam-7052	395	6	.	.	PUNCT
ejpam-7052	396	1	(	(	PUNCT
ejpam-7052	396	2	τ1	τ1	NOUN
ejpam-7052	396	3	,	,	PUNCT
ejpam-7052	396	4	τ2)δ	τ2)δ	ADJ
ejpam-7052	396	5	-	-	PUNCT
ejpam-7052	396	6	semicontinuous	semicontinuous	ADJ
ejpam-7052	396	7	multifunctions	multifunction	NOUN
ejpam-7052	396	8	.	.	PUNCT
ejpam-7052	397	1	heliyon	heliyon	NOUN
ejpam-7052	397	2	,	,	PUNCT
ejpam-7052	397	3	6	6	NUM
ejpam-7052	397	4	:	:	SYM
ejpam-7052	397	5	e05367	e05367	PROPN
ejpam-7052	397	6	,	,	PUNCT
ejpam-7052	397	7	2020	2020	NUM
ejpam-7052	397	8	.	.	PUNCT
ejpam-7052	398	1	[	[	X
ejpam-7052	398	2	15	15	NUM
ejpam-7052	398	3	]	]	X
ejpam-7052	398	4	n.	n.	NOUN
ejpam-7052	398	5	viriyapong	viriyapong	PROPN
ejpam-7052	398	6	,	,	PUNCT
ejpam-7052	398	7	s.	s.	PROPN
ejpam-7052	398	8	sompong	sompong	PROPN
ejpam-7052	398	9	,	,	PUNCT
ejpam-7052	398	10	and	and	CCONJ
ejpam-7052	398	11	c.	c.	PROPN
ejpam-7052	398	12	boonpok	boonpok	PROPN
ejpam-7052	398	13	.	.	PUNCT
ejpam-7052	399	1	(	(	PUNCT
ejpam-7052	399	2	τ1	τ1	NOUN
ejpam-7052	399	3	,	,	PUNCT
ejpam-7052	399	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-7052	399	5	disconnectedness	disconnectedness	NOUN
ejpam-7052	399	6	in	in	ADP
ejpam-7052	399	7	bitopological	bitopological	ADJ
ejpam-7052	399	8	spaces	space	NOUN
ejpam-7052	399	9	.	.	PUNCT
ejpam-7052	400	1	international	international	ADJ
ejpam-7052	400	2	journal	journal	PROPN
ejpam-7052	400	3	of	of	ADP
ejpam-7052	400	4	mathematics	mathematic	NOUN
ejpam-7052	400	5	and	and	CCONJ
ejpam-7052	400	6	computer	computer	NOUN
ejpam-7052	400	7	science	science	NOUN
ejpam-7052	400	8	,	,	PUNCT
ejpam-7052	400	9	19(3):855–860	19(3):855–860	PROPN
ejpam-7052	400	10	,	,	PUNCT
ejpam-7052	400	11	2024	2024	NUM
ejpam-7052	400	12	.	.	PUNCT
ejpam-7052	401	1	[	[	X
ejpam-7052	401	2	16	16	NUM
ejpam-7052	401	3	]	]	X
ejpam-7052	401	4	n.	n.	PROPN
ejpam-7052	401	5	viriyapong	viriyapong	PROPN
ejpam-7052	401	6	,	,	PUNCT
ejpam-7052	401	7	s.	s.	PROPN
ejpam-7052	401	8	sompong	sompong	PROPN
ejpam-7052	401	9	,	,	PUNCT
ejpam-7052	401	10	and	and	CCONJ
ejpam-7052	401	11	c.	c.	PROPN
ejpam-7052	401	12	boonpok	boonpok	PROPN
ejpam-7052	401	13	.	.	PUNCT
ejpam-7052	402	1	upper	upper	ADJ
ejpam-7052	402	2	and	and	CCONJ
ejpam-7052	402	3	lower	low	ADJ
ejpam-7052	402	4	s-(τ1	s-(τ1	NOUN
ejpam-7052	402	5	,	,	PUNCT
ejpam-7052	402	6	τ2)p	τ2)p	ADJ
ejpam-7052	402	7	-	-	PUNCT
ejpam-7052	402	8	continuous	continuous	ADJ
ejpam-7052	402	9	multifunctions	multifunction	NOUN
ejpam-7052	402	10	.	.	PUNCT
ejpam-7052	403	1	european	european	ADJ
ejpam-7052	403	2	journal	journal	PROPN
ejpam-7052	403	3	of	of	ADP
ejpam-7052	403	4	pure	pure	ADJ
ejpam-7052	403	5	and	and	CCONJ
ejpam-7052	403	6	applied	applied	ADJ
ejpam-7052	403	7	mathematics	mathematic	NOUN
ejpam-7052	403	8	,	,	PUNCT
ejpam-7052	403	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-7052	403	10	,	,	PUNCT
ejpam-7052	403	11	2024	2024	NUM
ejpam-7052	403	12	.	.	PUNCT
ejpam-7052	404	1	[	[	X
ejpam-7052	404	2	17	17	NUM
ejpam-7052	404	3	]	]	X
ejpam-7052	404	4	c.	c.	PROPN
ejpam-7052	404	5	viriyapong	viriyapong	PROPN
ejpam-7052	404	6	,	,	PUNCT
ejpam-7052	404	7	s.	s.	PROPN
ejpam-7052	404	8	sompong	sompong	PROPN
ejpam-7052	404	9	,	,	PUNCT
ejpam-7052	404	10	and	and	CCONJ
ejpam-7052	404	11	c.	c.	PROPN
ejpam-7052	404	12	boonpok	boonpok	PROPN
ejpam-7052	404	13	.	.	PUNCT
ejpam-7052	405	1	upper	upper	ADJ
ejpam-7052	405	2	and	and	CCONJ
ejpam-7052	405	3	lower	low	ADJ
ejpam-7052	405	4	slight	slight	ADJ
ejpam-7052	405	5	α(τ1	α(τ1	NOUN
ejpam-7052	405	6	,	,	PUNCT
ejpam-7052	405	7	τ2)continuity	τ2)continuity	PROPN
ejpam-7052	405	8	.	.	PUNCT
ejpam-7052	406	1	european	european	PROPN
ejpam-7052	406	2	journal	journal	PROPN
ejpam-7052	406	3	of	of	ADP
ejpam-7052	406	4	pure	pure	ADJ
ejpam-7052	406	5	and	and	CCONJ
ejpam-7052	406	6	applied	applied	ADJ
ejpam-7052	406	7	mathematics	mathematic	NOUN
ejpam-7052	406	8	,	,	PUNCT
ejpam-7052	406	9	17(3):2142–2154	17(3):2142–2154	NUM
ejpam-7052	406	10	,	,	PUNCT
ejpam-7052	406	11	2024	2024	NUM
ejpam-7052	406	12	.	.	PUNCT
ejpam-7052	407	1	p.	p.	NOUN
ejpam-7052	407	2	pue	pue	NOUN
ejpam-7052	407	3	-	-	PUNCT
ejpam-7052	407	4	on	on	ADP
ejpam-7052	407	5	,	,	PUNCT
ejpam-7052	407	6	a.	a.	PROPN
ejpam-7052	407	7	sama	sama	PROPN
ejpam-7052	407	8	-	-	PUNCT
ejpam-7052	407	9	ae	ae	PROPN
ejpam-7052	407	10	,	,	PUNCT
ejpam-7052	407	11	c.	c.	PROPN
ejpam-7052	407	12	boonpok	boonpok	PROPN
ejpam-7052	407	13	/	/	SYM
ejpam-7052	407	14	eur	eur	PROPN
ejpam-7052	407	15	.	.	PUNCT
ejpam-7052	408	1	j.	j.	PROPN
ejpam-7052	408	2	pure	pure	PROPN
ejpam-7052	408	3	appl	appl	PROPN
ejpam-7052	408	4	.	.	PROPN
ejpam-7052	408	5	math	math	PROPN
ejpam-7052	408	6	,	,	PUNCT
ejpam-7052	408	7	18	18	NUM
ejpam-7052	408	8	(	(	PUNCT
ejpam-7052	408	9	4	4	NUM
ejpam-7052	408	10	)	)	PUNCT
ejpam-7052	408	11	(	(	PUNCT
ejpam-7052	408	12	2025	2025	NUM
ejpam-7052	408	13	)	)	PUNCT
ejpam-7052	408	14	,	,	PUNCT
ejpam-7052	408	15	7052	7052	NUM
ejpam-7052	408	16	14	14	NUM
ejpam-7052	408	17	of	of	ADP
ejpam-7052	408	18	14	14	NUM
ejpam-7052	408	19	[	[	SYM
ejpam-7052	408	20	18	18	NUM
ejpam-7052	408	21	]	]	PUNCT
ejpam-7052	408	22	c.	c.	PROPN
ejpam-7052	408	23	boonpok	boonpok	PROPN
ejpam-7052	408	24	and	and	CCONJ
ejpam-7052	408	25	p.	p.	NOUN
ejpam-7052	408	26	pue	pue	NOUN
ejpam-7052	408	27	-	-	PUNCT
ejpam-7052	408	28	on	on	ADP
ejpam-7052	408	29	.	.	PUNCT
ejpam-7052	409	1	characterizations	characterization	NOUN
ejpam-7052	409	2	of	of	ADP
ejpam-7052	409	3	almost	almost	ADV
ejpam-7052	409	4	(	(	PUNCT
ejpam-7052	409	5	τ1	τ1	NOUN
ejpam-7052	409	6	,	,	PUNCT
ejpam-7052	409	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	409	8	multifunctions	multifunction	NOUN
ejpam-7052	409	9	.	.	PUNCT
ejpam-7052	410	1	international	international	ADJ
ejpam-7052	410	2	journal	journal	NOUN
ejpam-7052	410	3	of	of	ADP
ejpam-7052	410	4	analysis	analysis	NOUN
ejpam-7052	410	5	and	and	CCONJ
ejpam-7052	410	6	applications	application	NOUN
ejpam-7052	410	7	,	,	PUNCT
ejpam-7052	410	8	22:33	22:33	NUM
ejpam-7052	410	9	,	,	PUNCT
ejpam-7052	410	10	2024	2024	NUM
ejpam-7052	410	11	.	.	PUNCT
ejpam-7052	411	1	[	[	X
ejpam-7052	411	2	19	19	NUM
ejpam-7052	411	3	]	]	SYM
ejpam-7052	411	4	á	á	PROPN
ejpam-7052	411	5	.	.	PUNCT
ejpam-7052	411	6	császár	császár	PROPN
ejpam-7052	411	7	.	.	PUNCT
ejpam-7052	412	1	δ	δ	PROPN
ejpam-7052	412	2	-	-	PUNCT
ejpam-7052	412	3	and	and	CCONJ
ejpam-7052	412	4	θ	θ	NOUN
ejpam-7052	412	5	-	-	PUNCT
ejpam-7052	412	6	modifications	modification	NOUN
ejpam-7052	412	7	of	of	ADP
ejpam-7052	412	8	generalized	generalized	ADJ
ejpam-7052	412	9	topologies	topology	NOUN
ejpam-7052	412	10	.	.	PUNCT
ejpam-7052	413	1	acta	acta	PROPN
ejpam-7052	413	2	mathematica	mathematica	PROPN
ejpam-7052	413	3	hungarica	hungarica	PROPN
ejpam-7052	413	4	,	,	PUNCT
ejpam-7052	413	5	120:274–279	120:274–279	NUM
ejpam-7052	413	6	,	,	PUNCT
ejpam-7052	413	7	2008	2008	NUM
ejpam-7052	413	8	.	.	PUNCT
ejpam-7052	414	1	[	[	X
ejpam-7052	414	2	20	20	NUM
ejpam-7052	414	3	]	]	PUNCT
ejpam-7052	414	4	p.	p.	NOUN
ejpam-7052	414	5	pue	pue	NOUN
ejpam-7052	414	6	-	-	PUNCT
ejpam-7052	414	7	on	on	ADP
ejpam-7052	414	8	,	,	PUNCT
ejpam-7052	414	9	s.	s.	PROPN
ejpam-7052	414	10	sompong	sompong	PROPN
ejpam-7052	414	11	,	,	PUNCT
ejpam-7052	414	12	and	and	CCONJ
ejpam-7052	414	13	c.	c.	PROPN
ejpam-7052	414	14	boonpok	boonpok	PROPN
ejpam-7052	414	15	.	.	PUNCT
ejpam-7052	415	1	almost	almost	ADV
ejpam-7052	415	2	contra-(τ1	contra-(τ1	NOUN
ejpam-7052	415	3	,	,	PUNCT
ejpam-7052	415	4	τ2)p	τ2)p	NOUN
ejpam-7052	415	5	-	-	PUNCT
ejpam-7052	415	6	continuity	continuity	NOUN
ejpam-7052	415	7	for	for	ADP
ejpam-7052	415	8	functions	function	NOUN
ejpam-7052	415	9	.	.	PUNCT
ejpam-7052	416	1	european	european	ADJ
ejpam-7052	416	2	journal	journal	PROPN
ejpam-7052	416	3	of	of	ADP
ejpam-7052	416	4	pure	pure	ADJ
ejpam-7052	416	5	and	and	CCONJ
ejpam-7052	416	6	applied	applied	ADJ
ejpam-7052	416	7	mathematics	mathematic	NOUN
ejpam-7052	416	8	,	,	PUNCT
ejpam-7052	416	9	18(2):6038	18(2):6038	NUM
ejpam-7052	416	10	,	,	PUNCT
ejpam-7052	416	11	2025	2025	NUM
ejpam-7052	416	12	.	.	PUNCT
ejpam-7052	417	1	[	[	X
ejpam-7052	417	2	21	21	NUM
ejpam-7052	417	3	]	]	PUNCT
ejpam-7052	417	4	m.	m.	NOUN
ejpam-7052	417	5	thongmoon	thongmoon	NOUN
ejpam-7052	417	6	,	,	PUNCT
ejpam-7052	417	7	s.	s.	PROPN
ejpam-7052	417	8	sompong	sompong	PROPN
ejpam-7052	417	9	,	,	PUNCT
ejpam-7052	417	10	and	and	CCONJ
ejpam-7052	417	11	c.	c.	PROPN
ejpam-7052	417	12	boonpok	boonpok	PROPN
ejpam-7052	417	13	.	.	PUNCT
ejpam-7052	418	1	almost	almost	ADV
ejpam-7052	418	2	(	(	PUNCT
ejpam-7052	418	3	τ1	τ1	NOUN
ejpam-7052	418	4	,	,	PUNCT
ejpam-7052	418	5	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7052	418	6	and	and	CCONJ
ejpam-7052	418	7	τ1τ2δ	τ1τ2δ	NUM
ejpam-7052	418	8	-	-	PUNCT
ejpam-7052	418	9	open	open	ADJ
ejpam-7052	418	10	sets	set	NOUN
ejpam-7052	418	11	.	.	PUNCT
ejpam-7052	419	1	international	international	ADJ
ejpam-7052	419	2	journal	journal	NOUN
ejpam-7052	419	3	of	of	ADP
ejpam-7052	419	4	analysis	analysis	NOUN
ejpam-7052	419	5	and	and	CCONJ
ejpam-7052	419	6	applications	application	NOUN
ejpam-7052	419	7	,	,	PUNCT
ejpam-7052	419	8	22:109	22:109	NUM
ejpam-7052	419	9	,	,	PUNCT
ejpam-7052	419	10	2024	2024	NUM
ejpam-7052	419	11	.	.	PUNCT
ejpam-7052	420	1	[	[	X
ejpam-7052	420	2	22	22	NUM
ejpam-7052	420	3	]	]	X
ejpam-7052	420	4	n.	n.	NOUN
ejpam-7052	420	5	chutiman	chutiman	NOUN
ejpam-7052	420	6	,	,	PUNCT
ejpam-7052	420	7	a.	a.	PROPN
ejpam-7052	420	8	sama	sama	PROPN
ejpam-7052	420	9	-	-	PUNCT
ejpam-7052	420	10	ae	ae	PROPN
ejpam-7052	420	11	,	,	PUNCT
ejpam-7052	420	12	and	and	CCONJ
ejpam-7052	420	13	c.	c.	PROPN
ejpam-7052	420	14	boonpok	boonpok	PROPN
ejpam-7052	420	15	.	.	PUNCT
ejpam-7052	421	1	on	on	ADP
ejpam-7052	421	2	upper	upper	ADJ
ejpam-7052	421	3	and	and	CCONJ
ejpam-7052	421	4	lower	low	ADJ
ejpam-7052	421	5	µ(σ1	µ(σ1	NOUN
ejpam-7052	421	6	,	,	PUNCT
ejpam-7052	421	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7052	421	8	multifunctions	multifunction	NOUN
ejpam-7052	421	9	.	.	PUNCT
ejpam-7052	422	1	(	(	PUNCT
ejpam-7052	422	2	accepted	accept	VERB
ejpam-7052	422	3	)	)	PUNCT
ejpam-7052	422	4	.	.	PUNCT
ejpam-7052	423	1	[	[	X
ejpam-7052	423	2	23	23	NUM
ejpam-7052	423	3	]	]	PUNCT
ejpam-7052	423	4	m.	m.	NOUN
ejpam-7052	423	5	thongmoon	thongmoon	NOUN
ejpam-7052	423	6	,	,	PUNCT
ejpam-7052	423	7	s.	s.	PROPN
ejpam-7052	423	8	sompong	sompong	PROPN
ejpam-7052	423	9	,	,	PUNCT
ejpam-7052	423	10	and	and	CCONJ
ejpam-7052	423	11	c.	c.	PROPN
ejpam-7052	423	12	boonpok	boonpok	PROPN
ejpam-7052	423	13	.	.	PUNCT
ejpam-7052	424	1	(	(	PUNCT
ejpam-7052	424	2	τ1	τ1	NOUN
ejpam-7052	424	3	,	,	PUNCT
ejpam-7052	424	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7052	424	5	multifunctions	multifunction	NOUN
ejpam-7052	424	6	and	and	CCONJ
ejpam-7052	424	7	τ1τ2	τ1τ2	NOUN
ejpam-7052	424	8	-	-	ADJ
ejpam-7052	424	9	δ	δ	NOUN
ejpam-7052	424	10	-	-	ADJ
ejpam-7052	424	11	open	open	ADJ
ejpam-7052	424	12	sets	set	NOUN
ejpam-7052	424	13	.	.	PUNCT
ejpam-7052	425	1	international	international	ADJ
ejpam-7052	425	2	journal	journal	NOUN
ejpam-7052	425	3	of	of	ADP
ejpam-7052	425	4	mathematics	mathematic	NOUN
ejpam-7052	425	5	and	and	CCONJ
ejpam-7052	425	6	computer	computer	NOUN
ejpam-7052	425	7	science	science	NOUN
ejpam-7052	425	8	,	,	PUNCT
ejpam-7052	425	9	19(4):1369–1375	19(4):1369–1375	NUM
ejpam-7052	425	10	,	,	PUNCT
ejpam-7052	425	11	2024	2024	NUM
ejpam-7052	425	12	.	.	PUNCT