id	sid	tid	token	lemma	pos
ejpam-7054	1	1	european	european	PROPN
ejpam-7054	1	2	journal	journal	PROPN
ejpam-7054	1	3	of	of	ADP
ejpam-7054	1	4	pure	pure	ADJ
ejpam-7054	1	5	and	and	CCONJ
ejpam-7054	1	6	applied	applied	ADJ
ejpam-7054	1	7	mathematics	mathematic	NOUN
ejpam-7054	1	8	2025	2025	NUM
ejpam-7054	1	9	,	,	PUNCT
ejpam-7054	1	10	vol	vol	NOUN
ejpam-7054	1	11	.	.	PROPN
ejpam-7054	1	12	18	18	NUM
ejpam-7054	1	13	,	,	PUNCT
ejpam-7054	1	14	issue	issue	NOUN
ejpam-7054	1	15	4	4	NUM
ejpam-7054	1	16	,	,	PUNCT
ejpam-7054	1	17	article	article	NOUN
ejpam-7054	1	18	number	number	NOUN
ejpam-7054	1	19	7054	7054	NUM
ejpam-7054	1	20	issn	issn	PROPN
ejpam-7054	1	21	1307	1307	NUM
ejpam-7054	1	22	-	-	SYM
ejpam-7054	1	23	5543	5543	NUM
ejpam-7054	1	24	–	–	PUNCT
ejpam-7054	1	25	ejpam.com	ejpam.com	X
ejpam-7054	1	26	published	publish	VERB
ejpam-7054	1	27	by	by	ADP
ejpam-7054	1	28	new	new	PROPN
ejpam-7054	1	29	york	york	PROPN
ejpam-7054	1	30	business	business	PROPN
ejpam-7054	1	31	global	global	ADJ
ejpam-7054	1	32	almost	almost	ADV
ejpam-7054	1	33	weak	weak	ADJ
ejpam-7054	1	34	µ(σ1	µ(σ1	NOUN
ejpam-7054	1	35	,	,	PUNCT
ejpam-7054	1	36	σ2)-continuity	σ2)-continuity	NOUN
ejpam-7054	1	37	for	for	ADP
ejpam-7054	1	38	multifunctions	multifunction	NOUN
ejpam-7054	1	39	butsakorn	butsakorn	PROPN
ejpam-7054	1	40	kong	kong	PROPN
ejpam-7054	1	41	-	-	PUNCT
ejpam-7054	1	42	ied1	ied1	PROPN
ejpam-7054	1	43	,	,	PUNCT
ejpam-7054	1	44	areeyuth	areeyuth	NOUN
ejpam-7054	1	45	sama	sama	NOUN
ejpam-7054	1	46	-	-	PUNCT
ejpam-7054	1	47	ae2	ae2	PROPN
ejpam-7054	1	48	,	,	PUNCT
ejpam-7054	1	49	chawalit	chawalit	VERB
ejpam-7054	1	50	boonpok1,∗	boonpok1,∗	NOUN
ejpam-7054	1	51	1	1	NUM
ejpam-7054	1	52	mathematics	mathematic	NOUN
ejpam-7054	1	53	and	and	CCONJ
ejpam-7054	1	54	applied	apply	VERB
ejpam-7054	1	55	mathematics	mathematics	PROPN
ejpam-7054	1	56	research	research	NOUN
ejpam-7054	1	57	unit	unit	NOUN
ejpam-7054	1	58	,	,	PUNCT
ejpam-7054	1	59	department	department	NOUN
ejpam-7054	1	60	of	of	ADP
ejpam-7054	1	61	mathematics	mathematic	NOUN
ejpam-7054	1	62	,	,	PUNCT
ejpam-7054	1	63	faculty	faculty	NOUN
ejpam-7054	1	64	of	of	ADP
ejpam-7054	1	65	science	science	NOUN
ejpam-7054	1	66	,	,	PUNCT
ejpam-7054	1	67	mahasarakham	mahasarakham	PROPN
ejpam-7054	1	68	university	university	PROPN
ejpam-7054	1	69	,	,	PUNCT
ejpam-7054	1	70	maha	maha	PROPN
ejpam-7054	1	71	sarakham	sarakham	PROPN
ejpam-7054	1	72	,	,	PUNCT
ejpam-7054	1	73	44150	44150	NUM
ejpam-7054	1	74	,	,	PUNCT
ejpam-7054	1	75	thailand	thailand	PROPN
ejpam-7054	1	76	2	2	NUM
ejpam-7054	1	77	department	department	NOUN
ejpam-7054	1	78	of	of	ADP
ejpam-7054	1	79	mathematics	mathematic	NOUN
ejpam-7054	1	80	and	and	CCONJ
ejpam-7054	1	81	computer	computer	NOUN
ejpam-7054	1	82	science	science	NOUN
ejpam-7054	1	83	,	,	PUNCT
ejpam-7054	1	84	faculty	faculty	NOUN
ejpam-7054	1	85	of	of	ADP
ejpam-7054	1	86	science	science	NOUN
ejpam-7054	1	87	and	and	CCONJ
ejpam-7054	1	88	technology	technology	NOUN
ejpam-7054	1	89	,	,	PUNCT
ejpam-7054	1	90	prince	prince	NOUN
ejpam-7054	1	91	of	of	ADP
ejpam-7054	1	92	songkla	songkla	PROPN
ejpam-7054	1	93	university	university	PROPN
ejpam-7054	1	94	,	,	PUNCT
ejpam-7054	1	95	pattani	pattani	NOUN
ejpam-7054	1	96	campus	campus	NOUN
ejpam-7054	1	97	,	,	PUNCT
ejpam-7054	1	98	pattani	pattani	NOUN
ejpam-7054	1	99	,	,	PUNCT
ejpam-7054	1	100	94000	94000	NUM
ejpam-7054	1	101	,	,	PUNCT
ejpam-7054	1	102	thailand	thailand	PROPN
ejpam-7054	1	103	abstract	abstract	PROPN
ejpam-7054	1	104	.	.	PUNCT
ejpam-7054	2	1	this	this	DET
ejpam-7054	2	2	paper	paper	NOUN
ejpam-7054	2	3	presents	present	VERB
ejpam-7054	2	4	new	new	ADJ
ejpam-7054	2	5	concepts	concept	NOUN
ejpam-7054	2	6	of	of	ADP
ejpam-7054	2	7	continuous	continuous	ADJ
ejpam-7054	2	8	multifunctions	multifunction	NOUN
ejpam-7054	2	9	,	,	PUNCT
ejpam-7054	2	10	called	call	VERB
ejpam-7054	2	11	upper	upper	ADJ
ejpam-7054	2	12	almost	almost	ADV
ejpam-7054	2	13	weakly	weakly	ADJ
ejpam-7054	2	14	µ(σ1	µ(σ1	NOUN
ejpam-7054	2	15	,	,	PUNCT
ejpam-7054	2	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	2	17	multifunctions	multifunction	NOUN
ejpam-7054	2	18	and	and	CCONJ
ejpam-7054	2	19	lower	low	ADJ
ejpam-7054	2	20	almost	almost	ADV
ejpam-7054	2	21	weakly	weakly	ADJ
ejpam-7054	2	22	µ(σ1	µ(σ1	NOUN
ejpam-7054	2	23	,	,	PUNCT
ejpam-7054	2	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	2	25	multifunctions	multifunction	NOUN
ejpam-7054	2	26	.	.	PUNCT
ejpam-7054	3	1	moreover	moreover	ADV
ejpam-7054	3	2	,	,	PUNCT
ejpam-7054	3	3	several	several	ADJ
ejpam-7054	3	4	characterizations	characterization	NOUN
ejpam-7054	3	5	and	and	CCONJ
ejpam-7054	3	6	some	some	DET
ejpam-7054	3	7	properties	property	NOUN
ejpam-7054	3	8	concerning	concern	VERB
ejpam-7054	3	9	upper	upper	ADJ
ejpam-7054	3	10	almost	almost	ADV
ejpam-7054	3	11	weakly	weakly	ADJ
ejpam-7054	3	12	µ(σ1	µ(σ1	NOUN
ejpam-7054	3	13	,	,	PUNCT
ejpam-7054	3	14	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	3	15	multifunctions	multifunction	NOUN
ejpam-7054	3	16	and	and	CCONJ
ejpam-7054	3	17	lower	low	ADJ
ejpam-7054	3	18	almost	almost	ADV
ejpam-7054	3	19	weakly	weakly	ADJ
ejpam-7054	3	20	µ(σ1	µ(σ1	NOUN
ejpam-7054	3	21	,	,	PUNCT
ejpam-7054	3	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	3	23	multifunctions	multifunction	NOUN
ejpam-7054	3	24	are	be	AUX
ejpam-7054	3	25	established	establish	VERB
ejpam-7054	3	26	.	.	PUNCT
ejpam-7054	4	1	2020	2020	NUM
ejpam-7054	4	2	mathematics	mathematics	PROPN
ejpam-7054	4	3	subject	subject	NOUN
ejpam-7054	4	4	classifications	classification	NOUN
ejpam-7054	4	5	:	:	PUNCT
ejpam-7054	4	6	54c08	54c08	NUM
ejpam-7054	4	7	,	,	PUNCT
ejpam-7054	4	8	54c60	54c60	NUM
ejpam-7054	4	9	key	key	ADJ
ejpam-7054	4	10	words	word	NOUN
ejpam-7054	4	11	and	and	CCONJ
ejpam-7054	4	12	phrases	phrase	NOUN
ejpam-7054	4	13	:	:	PUNCT
ejpam-7054	4	14	upper	upper	ADJ
ejpam-7054	4	15	almost	almost	ADV
ejpam-7054	4	16	weakly	weakly	ADJ
ejpam-7054	4	17	µ(σ1	µ(σ1	NOUN
ejpam-7054	4	18	,	,	PUNCT
ejpam-7054	4	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	4	20	multifunction	multifunction	NOUN
ejpam-7054	4	21	,	,	PUNCT
ejpam-7054	4	22	lower	low	ADJ
ejpam-7054	4	23	almost	almost	ADV
ejpam-7054	4	24	weakly	weakly	ADJ
ejpam-7054	4	25	µ(σ1	µ(σ1	NOUN
ejpam-7054	4	26	,	,	PUNCT
ejpam-7054	4	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	4	28	multifunction	multifunction	NOUN
ejpam-7054	4	29	1	1	NUM
ejpam-7054	4	30	.	.	PUNCT
ejpam-7054	4	31	introduction	introduction	NOUN
ejpam-7054	4	32	in	in	ADP
ejpam-7054	4	33	1968	1968	NUM
ejpam-7054	4	34	,	,	PUNCT
ejpam-7054	4	35	singal	singal	NOUN
ejpam-7054	4	36	and	and	CCONJ
ejpam-7054	4	37	singal	singal	ADJ
ejpam-7054	5	1	[	[	X
ejpam-7054	5	2	1	1	NUM
ejpam-7054	5	3	]	]	PUNCT
ejpam-7054	5	4	introduced	introduce	VERB
ejpam-7054	5	5	and	and	CCONJ
ejpam-7054	5	6	investigated	investigate	VERB
ejpam-7054	5	7	the	the	DET
ejpam-7054	5	8	concept	concept	NOUN
ejpam-7054	5	9	of	of	ADP
ejpam-7054	5	10	almost	almost	ADV
ejpam-7054	5	11	continuous	continuous	ADJ
ejpam-7054	5	12	functions	function	NOUN
ejpam-7054	5	13	.	.	PUNCT
ejpam-7054	6	1	munshi	munshi	PROPN
ejpam-7054	6	2	and	and	CCONJ
ejpam-7054	6	3	bassan	bassan	NOUN
ejpam-7054	6	4	[	[	X
ejpam-7054	6	5	2	2	NUM
ejpam-7054	6	6	]	]	PUNCT
ejpam-7054	6	7	studied	study	VERB
ejpam-7054	6	8	the	the	DET
ejpam-7054	6	9	notion	notion	NOUN
ejpam-7054	6	10	of	of	ADP
ejpam-7054	6	11	almost	almost	ADV
ejpam-7054	6	12	semi	semi	ADJ
ejpam-7054	6	13	-	-	ADJ
ejpam-7054	6	14	continuous	continuous	ADJ
ejpam-7054	6	15	functions	function	NOUN
ejpam-7054	6	16	.	.	PUNCT
ejpam-7054	7	1	noiri	noiri	ADV
ejpam-7054	8	1	[	[	X
ejpam-7054	8	2	3	3	X
ejpam-7054	8	3	]	]	PUNCT
ejpam-7054	8	4	introduced	introduce	VERB
ejpam-7054	8	5	and	and	CCONJ
ejpam-7054	8	6	investigated	investigate	VERB
ejpam-7054	8	7	the	the	DET
ejpam-7054	8	8	concept	concept	NOUN
ejpam-7054	8	9	of	of	ADP
ejpam-7054	8	10	almost	almost	ADV
ejpam-7054	8	11	α	α	NUM
ejpam-7054	8	12	-	-	ADJ
ejpam-7054	8	13	continuous	continuous	ADJ
ejpam-7054	8	14	functions	function	NOUN
ejpam-7054	8	15	.	.	PUNCT
ejpam-7054	9	1	nasef	nasef	NOUN
ejpam-7054	9	2	and	and	CCONJ
ejpam-7054	9	3	noiri	noiri	ADV
ejpam-7054	10	1	[	[	X
ejpam-7054	10	2	4	4	X
ejpam-7054	10	3	]	]	PUNCT
ejpam-7054	10	4	introduced	introduce	VERB
ejpam-7054	10	5	two	two	NUM
ejpam-7054	10	6	classes	class	NOUN
ejpam-7054	10	7	of	of	ADP
ejpam-7054	10	8	functions	function	NOUN
ejpam-7054	10	9	,	,	PUNCT
ejpam-7054	10	10	namely	namely	ADV
ejpam-7054	10	11	almost	almost	ADV
ejpam-7054	10	12	precontinuous	precontinuous	ADJ
ejpam-7054	10	13	functions	function	NOUN
ejpam-7054	10	14	and	and	CCONJ
ejpam-7054	10	15	almost	almost	ADV
ejpam-7054	10	16	β	β	ADJ
ejpam-7054	10	17	-	-	ADJ
ejpam-7054	10	18	continuous	continuous	ADJ
ejpam-7054	10	19	functions	function	NOUN
ejpam-7054	10	20	.	.	PUNCT
ejpam-7054	11	1	the	the	DET
ejpam-7054	11	2	class	class	NOUN
ejpam-7054	11	3	of	of	ADP
ejpam-7054	11	4	almost	almost	ADV
ejpam-7054	11	5	precontinuity	precontinuity	NOUN
ejpam-7054	11	6	is	be	AUX
ejpam-7054	11	7	a	a	DET
ejpam-7054	11	8	generalization	generalization	NOUN
ejpam-7054	11	9	of	of	ADP
ejpam-7054	11	10	almost	almost	ADV
ejpam-7054	11	11	α	α	NOUN
ejpam-7054	11	12	-	-	NOUN
ejpam-7054	11	13	continuity	continuity	NOUN
ejpam-7054	11	14	.	.	PUNCT
ejpam-7054	12	1	the	the	DET
ejpam-7054	12	2	class	class	NOUN
ejpam-7054	12	3	of	of	ADP
ejpam-7054	12	4	almost	almost	ADV
ejpam-7054	12	5	β	β	NOUN
ejpam-7054	12	6	-	-	NOUN
ejpam-7054	12	7	continuity	continuity	NOUN
ejpam-7054	12	8	is	be	AUX
ejpam-7054	12	9	a	a	DET
ejpam-7054	12	10	generalization	generalization	NOUN
ejpam-7054	12	11	of	of	ADP
ejpam-7054	12	12	almost	almost	ADV
ejpam-7054	12	13	semi	semi	NOUN
ejpam-7054	12	14	-	-	NOUN
ejpam-7054	12	15	continuity	continuity	NOUN
ejpam-7054	12	16	.	.	PUNCT
ejpam-7054	13	1	levine	levine	PROPN
ejpam-7054	14	1	[	[	X
ejpam-7054	14	2	5	5	NUM
ejpam-7054	14	3	]	]	PUNCT
ejpam-7054	14	4	introduced	introduce	VERB
ejpam-7054	14	5	and	and	CCONJ
ejpam-7054	14	6	investigated	investigate	VERB
ejpam-7054	14	7	the	the	DET
ejpam-7054	14	8	concept	concept	NOUN
ejpam-7054	14	9	of	of	ADP
ejpam-7054	14	10	weakly	weakly	ADJ
ejpam-7054	14	11	continuous	continuous	ADJ
ejpam-7054	14	12	functions	function	NOUN
ejpam-7054	14	13	.	.	PUNCT
ejpam-7054	15	1	husain	husain	NOUN
ejpam-7054	16	1	[	[	X
ejpam-7054	16	2	6	6	NUM
ejpam-7054	16	3	]	]	PUNCT
ejpam-7054	16	4	introduced	introduce	VERB
ejpam-7054	16	5	and	and	CCONJ
ejpam-7054	16	6	studied	study	VERB
ejpam-7054	16	7	the	the	DET
ejpam-7054	16	8	notion	notion	NOUN
ejpam-7054	16	9	of	of	ADP
ejpam-7054	16	10	almost	almost	ADV
ejpam-7054	16	11	continuous	continuous	ADJ
ejpam-7054	16	12	functions	function	NOUN
ejpam-7054	16	13	.	.	PUNCT
ejpam-7054	17	1	noiri	noiri	ADV
ejpam-7054	18	1	[	[	X
ejpam-7054	18	2	7	7	X
ejpam-7054	18	3	]	]	PUNCT
ejpam-7054	18	4	investigated	investigate	VERB
ejpam-7054	18	5	several	several	ADJ
ejpam-7054	18	6	characterizations	characterization	NOUN
ejpam-7054	18	7	of	of	ADP
ejpam-7054	18	8	almost	almost	ADV
ejpam-7054	18	9	weakly	weakly	ADJ
ejpam-7054	18	10	continuous	continuous	ADJ
ejpam-7054	18	11	functions	function	NOUN
ejpam-7054	18	12	.	.	PUNCT
ejpam-7054	19	1	rose	rise	VERB
ejpam-7054	19	2	[	[	X
ejpam-7054	19	3	8	8	NUM
ejpam-7054	19	4	]	]	PUNCT
ejpam-7054	19	5	introduced	introduce	VERB
ejpam-7054	19	6	the	the	DET
ejpam-7054	19	7	notion	notion	NOUN
ejpam-7054	19	8	of	of	ADP
ejpam-7054	19	9	subweakly	subweakly	ADJ
ejpam-7054	19	10	continuous	continuous	ADJ
ejpam-7054	19	11	functions	function	NOUN
ejpam-7054	19	12	and	and	CCONJ
ejpam-7054	19	13	investigated	investigate	VERB
ejpam-7054	19	14	the	the	DET
ejpam-7054	19	15	relationships	relationship	NOUN
ejpam-7054	19	16	between	between	ADP
ejpam-7054	19	17	subweak	subweak	NOUN
ejpam-7054	19	18	continuity	continuity	NOUN
ejpam-7054	19	19	and	and	CCONJ
ejpam-7054	19	20	weak	weak	ADJ
ejpam-7054	19	21	continuity	continuity	NOUN
ejpam-7054	19	22	.	.	PUNCT
ejpam-7054	20	1	in	in	ADP
ejpam-7054	20	2	1993	1993	NUM
ejpam-7054	20	3	,	,	PUNCT
ejpam-7054	20	4	noiri	noiri	PRON
ejpam-7054	20	5	and	and	CCONJ
ejpam-7054	20	6	popa	popa	NOUN
ejpam-7054	20	7	[	[	X
ejpam-7054	20	8	9	9	NUM
ejpam-7054	20	9	]	]	PUNCT
ejpam-7054	20	10	extended	extend	VERB
ejpam-7054	20	11	the	the	DET
ejpam-7054	20	12	concept	concept	NOUN
ejpam-7054	20	13	of	of	ADP
ejpam-7054	20	14	almost	almost	ADV
ejpam-7054	20	15	weakly	weakly	ADJ
ejpam-7054	20	16	continuous	continuous	ADJ
ejpam-7054	20	17	functions	function	NOUN
ejpam-7054	20	18	to	to	ADP
ejpam-7054	20	19	multifunctions	multifunction	NOUN
ejpam-7054	20	20	and	and	CCONJ
ejpam-7054	20	21	defined	define	VERB
ejpam-7054	20	22	upper	upper	ADJ
ejpam-7054	20	23	almost	almost	ADV
ejpam-7054	20	24	weakly	weakly	ADJ
ejpam-7054	20	25	continuous	continuous	ADJ
ejpam-7054	20	26	multifunctions	multifunction	NOUN
ejpam-7054	20	27	and	and	CCONJ
ejpam-7054	20	28	lower	low	ADJ
ejpam-7054	20	29	almost	almost	ADV
ejpam-7054	20	30	weakly	weakly	ADJ
ejpam-7054	20	31	continuous	continuous	ADJ
ejpam-7054	20	32	multifunctions	multifunction	NOUN
ejpam-7054	20	33	.	.	PUNCT
ejpam-7054	21	1	popa	popa	NOUN
ejpam-7054	21	2	and	and	CCONJ
ejpam-7054	21	3	noiri	noiri	ADV
ejpam-7054	22	1	[	[	X
ejpam-7054	22	2	10	10	NUM
ejpam-7054	22	3	]	]	PUNCT
ejpam-7054	22	4	investigated	investigate	VERB
ejpam-7054	22	5	some	some	DET
ejpam-7054	22	6	characterizations	characterization	NOUN
ejpam-7054	22	7	and	and	CCONJ
ejpam-7054	22	8	∗corresponding	∗corresponde	VERB
ejpam-7054	22	9	author	author	NOUN
ejpam-7054	22	10	.	.	PUNCT
ejpam-7054	23	1	doi	doi	NOUN
ejpam-7054	23	2	:	:	PUNCT
ejpam-7054	23	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7054	https://doi.org/10.29020/nybg.ejpam.v18i4.7054	PROPN
ejpam-7054	23	4	email	email	NOUN
ejpam-7054	23	5	addresses	address	NOUN
ejpam-7054	23	6	:	:	PUNCT
ejpam-7054	23	7	butsakorn.k@msu.ac.th	butsakorn.k@msu.ac.th	ADP
ejpam-7054	23	8	(	(	PUNCT
ejpam-7054	23	9	b.	b.	PROPN
ejpam-7054	23	10	kong	kong	PROPN
ejpam-7054	23	11	-	-	PUNCT
ejpam-7054	23	12	ied	ied	PROPN
ejpam-7054	23	13	)	)	PUNCT
ejpam-7054	23	14	,	,	PUNCT
ejpam-7054	23	15	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-7054	23	16	(	(	PUNCT
ejpam-7054	23	17	a.	a.	PROPN
ejpam-7054	23	18	sama	sama	PROPN
ejpam-7054	23	19	-	-	PUNCT
ejpam-7054	23	20	ae	ae	PROPN
ejpam-7054	23	21	)	)	PUNCT
ejpam-7054	23	22	,	,	PUNCT
ejpam-7054	23	23	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-7054	23	24	(	(	PUNCT
ejpam-7054	23	25	c.	c.	PROPN
ejpam-7054	23	26	boonpok	boonpok	PROPN
ejpam-7054	23	27	)	)	PUNCT
ejpam-7054	23	28	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-7054	24	1	1	1	NUM
ejpam-7054	24	2	copyright	copyright	NOUN
ejpam-7054	24	3	:	:	PUNCT
ejpam-7054	24	4	©	©	PROPN
ejpam-7054	24	5	2025	2025	NUM
ejpam-7054	24	6	the	the	DET
ejpam-7054	24	7	author(s	author(s	NOUN
ejpam-7054	24	8	)	)	PUNCT
ejpam-7054	24	9	.	.	PUNCT
ejpam-7054	25	1	(	(	PUNCT
ejpam-7054	25	2	cc	cc	NOUN
ejpam-7054	25	3	by	by	ADP
ejpam-7054	25	4	-	-	PUNCT
ejpam-7054	25	5	nc	nc	PROPN
ejpam-7054	25	6	4.0	4.0	NUM
ejpam-7054	25	7	)	)	PUNCT
ejpam-7054	25	8	b.	b.	PROPN
ejpam-7054	25	9	kong	kong	PROPN
ejpam-7054	25	10	-	-	PUNCT
ejpam-7054	25	11	ied	ied	PROPN
ejpam-7054	25	12	,	,	PUNCT
ejpam-7054	25	13	a.	a.	PROPN
ejpam-7054	25	14	sama	sama	PROPN
ejpam-7054	25	15	-	-	PUNCT
ejpam-7054	25	16	ae	ae	PROPN
ejpam-7054	25	17	,	,	PUNCT
ejpam-7054	25	18	c.	c.	PROPN
ejpam-7054	25	19	boonpok	boonpok	PROPN
ejpam-7054	25	20	/	/	SYM
ejpam-7054	25	21	eur	eur	PROPN
ejpam-7054	25	22	.	.	PUNCT
ejpam-7054	26	1	j.	j.	PROPN
ejpam-7054	26	2	pure	pure	PROPN
ejpam-7054	26	3	appl	appl	PROPN
ejpam-7054	26	4	.	.	PROPN
ejpam-7054	26	5	math	math	PROPN
ejpam-7054	26	6	,	,	PUNCT
ejpam-7054	26	7	18	18	NUM
ejpam-7054	26	8	(	(	PUNCT
ejpam-7054	26	9	4	4	NUM
ejpam-7054	26	10	)	)	PUNCT
ejpam-7054	26	11	(	(	PUNCT
ejpam-7054	26	12	2025	2025	NUM
ejpam-7054	26	13	)	)	PUNCT
ejpam-7054	26	14	,	,	PUNCT
ejpam-7054	26	15	7054	7054	NUM
ejpam-7054	26	16	2	2	NUM
ejpam-7054	26	17	of	of	ADP
ejpam-7054	26	18	12	12	NUM
ejpam-7054	26	19	several	several	ADJ
ejpam-7054	26	20	properties	property	NOUN
ejpam-7054	26	21	concerning	concern	VERB
ejpam-7054	26	22	upper	upper	ADJ
ejpam-7054	26	23	almost	almost	ADV
ejpam-7054	26	24	weakly	weakly	ADJ
ejpam-7054	26	25	continuous	continuous	ADJ
ejpam-7054	26	26	multifunctions	multifunction	NOUN
ejpam-7054	26	27	and	and	CCONJ
ejpam-7054	26	28	lower	low	ADJ
ejpam-7054	26	29	almost	almost	ADV
ejpam-7054	26	30	weakly	weakly	ADJ
ejpam-7054	26	31	continuous	continuous	ADJ
ejpam-7054	26	32	multifunctions	multifunction	NOUN
ejpam-7054	26	33	.	.	PUNCT
ejpam-7054	27	1	in	in	ADP
ejpam-7054	27	2	2002	2002	NUM
ejpam-7054	27	3	,	,	PUNCT
ejpam-7054	27	4	császár	császár	NOUN
ejpam-7054	28	1	[	[	X
ejpam-7054	28	2	11	11	NUM
ejpam-7054	28	3	]	]	PUNCT
ejpam-7054	28	4	introduced	introduce	VERB
ejpam-7054	28	5	the	the	DET
ejpam-7054	28	6	concepts	concept	NOUN
ejpam-7054	28	7	of	of	ADP
ejpam-7054	28	8	generalized	generalized	ADJ
ejpam-7054	28	9	topological	topological	ADJ
ejpam-7054	28	10	spaces	space	NOUN
ejpam-7054	28	11	and	and	CCONJ
ejpam-7054	28	12	generalized	generalized	ADJ
ejpam-7054	28	13	neighborhood	neighborhood	NOUN
ejpam-7054	28	14	systems	system	NOUN
ejpam-7054	28	15	.	.	PUNCT
ejpam-7054	29	1	the	the	DET
ejpam-7054	29	2	classes	class	NOUN
ejpam-7054	29	3	of	of	ADP
ejpam-7054	29	4	topological	topological	ADJ
ejpam-7054	29	5	spaces	space	NOUN
ejpam-7054	29	6	and	and	CCONJ
ejpam-7054	29	7	neighborhood	neighborhood	NOUN
ejpam-7054	29	8	systems	system	NOUN
ejpam-7054	29	9	are	be	AUX
ejpam-7054	29	10	contained	contain	VERB
ejpam-7054	29	11	in	in	ADP
ejpam-7054	29	12	the	the	DET
ejpam-7054	29	13	classes	class	NOUN
ejpam-7054	29	14	of	of	ADP
ejpam-7054	29	15	generalized	generalized	ADJ
ejpam-7054	29	16	topological	topological	ADJ
ejpam-7054	29	17	spaces	space	NOUN
ejpam-7054	29	18	and	and	CCONJ
ejpam-7054	29	19	generalized	generalized	ADJ
ejpam-7054	29	20	neighborhood	neighborhood	NOUN
ejpam-7054	29	21	systems	system	NOUN
ejpam-7054	29	22	,	,	PUNCT
ejpam-7054	29	23	respectively	respectively	ADV
ejpam-7054	29	24	.	.	PUNCT
ejpam-7054	30	1	furthermore	furthermore	ADV
ejpam-7054	30	2	,	,	PUNCT
ejpam-7054	30	3	császár	császár	PROPN
ejpam-7054	30	4	[	[	X
ejpam-7054	30	5	11	11	NUM
ejpam-7054	30	6	]	]	PUNCT
ejpam-7054	30	7	introduced	introduce	VERB
ejpam-7054	30	8	two	two	NUM
ejpam-7054	30	9	kinds	kind	NOUN
ejpam-7054	30	10	of	of	ADP
ejpam-7054	30	11	generalized	generalized	ADJ
ejpam-7054	30	12	continuous	continuous	ADJ
ejpam-7054	30	13	functions	function	NOUN
ejpam-7054	30	14	by	by	ADP
ejpam-7054	30	15	utilizing	utilize	VERB
ejpam-7054	30	16	the	the	DET
ejpam-7054	30	17	notions	notion	NOUN
ejpam-7054	30	18	of	of	ADP
ejpam-7054	30	19	generalized	generalized	ADJ
ejpam-7054	30	20	topological	topological	ADJ
ejpam-7054	30	21	spaces	space	NOUN
ejpam-7054	30	22	and	and	CCONJ
ejpam-7054	30	23	generalized	generalized	ADJ
ejpam-7054	30	24	neighborhood	neighborhood	NOUN
ejpam-7054	30	25	systems	system	NOUN
ejpam-7054	30	26	.	.	PUNCT
ejpam-7054	31	1	in	in	ADP
ejpam-7054	31	2	2009	2009	NUM
ejpam-7054	31	3	,	,	PUNCT
ejpam-7054	31	4	kanibir	kanibir	NOUN
ejpam-7054	31	5	and	and	CCONJ
ejpam-7054	31	6	reilly	reilly	ADV
ejpam-7054	32	1	[	[	X
ejpam-7054	32	2	12	12	NUM
ejpam-7054	32	3	]	]	PUNCT
ejpam-7054	32	4	extended	extend	VERB
ejpam-7054	32	5	the	the	DET
ejpam-7054	32	6	concept	concept	NOUN
ejpam-7054	32	7	of	of	ADP
ejpam-7054	32	8	generalized	generalized	ADJ
ejpam-7054	32	9	continuous	continuous	ADJ
ejpam-7054	32	10	functions	function	NOUN
ejpam-7054	32	11	to	to	ADP
ejpam-7054	32	12	multifunctions	multifunction	NOUN
ejpam-7054	32	13	and	and	CCONJ
ejpam-7054	32	14	defined	define	VERB
ejpam-7054	32	15	upper	upper	ADJ
ejpam-7054	32	16	semi	semi	ADJ
ejpam-7054	32	17	generalized	generalized	ADJ
ejpam-7054	32	18	continuous	continuous	ADJ
ejpam-7054	32	19	multifunctions	multifunction	NOUN
ejpam-7054	32	20	and	and	CCONJ
ejpam-7054	32	21	lower	low	ADJ
ejpam-7054	32	22	semi	semi	ADV
ejpam-7054	32	23	generalized	generalized	ADJ
ejpam-7054	32	24	continuous	continuous	ADJ
ejpam-7054	32	25	multifunctions	multifunction	NOUN
ejpam-7054	32	26	.	.	PUNCT
ejpam-7054	33	1	on	on	ADP
ejpam-7054	33	2	the	the	DET
ejpam-7054	33	3	other	other	ADJ
ejpam-7054	33	4	hand	hand	NOUN
ejpam-7054	33	5	,	,	PUNCT
ejpam-7054	33	6	the	the	DET
ejpam-7054	33	7	present	present	ADJ
ejpam-7054	33	8	authors	author	NOUN
ejpam-7054	33	9	introduced	introduce	VERB
ejpam-7054	33	10	and	and	CCONJ
ejpam-7054	33	11	investigated	investigate	VERB
ejpam-7054	33	12	four	four	NUM
ejpam-7054	33	13	classes	class	NOUN
ejpam-7054	33	14	of	of	ADP
ejpam-7054	33	15	multifunctions	multifunction	NOUN
ejpam-7054	33	16	defined	define	VERB
ejpam-7054	33	17	from	from	ADP
ejpam-7054	33	18	a	a	DET
ejpam-7054	33	19	generalized	generalized	ADJ
ejpam-7054	33	20	topological	topological	ADJ
ejpam-7054	33	21	space	space	NOUN
ejpam-7054	33	22	into	into	ADP
ejpam-7054	33	23	a	a	DET
ejpam-7054	33	24	generalized	generalized	ADJ
ejpam-7054	33	25	topological	topological	ADJ
ejpam-7054	33	26	space	space	NOUN
ejpam-7054	33	27	,	,	PUNCT
ejpam-7054	33	28	namely	namely	ADV
ejpam-7054	33	29	upper	upper	ADJ
ejpam-7054	33	30	β(µx	β(µx	PROPN
ejpam-7054	33	31	,	,	PUNCT
ejpam-7054	33	32	µy	µy	CCONJ
ejpam-7054	33	33	)	)	PUNCT
ejpam-7054	33	34	-continuous	-continuous	ADJ
ejpam-7054	33	35	multifunctions	multifunction	NOUN
ejpam-7054	34	1	[	[	X
ejpam-7054	34	2	13	13	NUM
ejpam-7054	34	3	]	]	PUNCT
ejpam-7054	34	4	,	,	PUNCT
ejpam-7054	34	5	lower	low	ADJ
ejpam-7054	34	6	β(µx	β(µx	PROPN
ejpam-7054	34	7	,	,	PUNCT
ejpam-7054	34	8	µy	µy	CCONJ
ejpam-7054	34	9	)	)	PUNCT
ejpam-7054	34	10	-continuous	-continuous	ADJ
ejpam-7054	34	11	multifunctions	multifunction	NOUN
ejpam-7054	35	1	[	[	X
ejpam-7054	35	2	13	13	NUM
ejpam-7054	35	3	]	]	PUNCT
ejpam-7054	35	4	,	,	PUNCT
ejpam-7054	35	5	upper	upper	ADJ
ejpam-7054	35	6	α(µx	α(µx	PROPN
ejpam-7054	35	7	,	,	PUNCT
ejpam-7054	35	8	µy	µy	CCONJ
ejpam-7054	35	9	)	)	PUNCT
ejpam-7054	35	10	-continuous	-continuous	ADJ
ejpam-7054	35	11	multifunctions	multifunction	NOUN
ejpam-7054	36	1	[	[	X
ejpam-7054	36	2	14	14	NUM
ejpam-7054	36	3	]	]	PUNCT
ejpam-7054	36	4	and	and	CCONJ
ejpam-7054	36	5	lower	low	ADJ
ejpam-7054	36	6	α(µx	α(µx	NUM
ejpam-7054	36	7	,	,	PUNCT
ejpam-7054	36	8	µy	µy	CCONJ
ejpam-7054	36	9	)	)	PUNCT
ejpam-7054	36	10	-continuous	-continuous	ADJ
ejpam-7054	36	11	multifunctions	multifunction	NOUN
ejpam-7054	37	1	[	[	X
ejpam-7054	37	2	14	14	NUM
ejpam-7054	37	3	]	]	PUNCT
ejpam-7054	37	4	.	.	PUNCT
ejpam-7054	38	1	pue	pue	NOUN
ejpam-7054	38	2	-	-	PUNCT
ejpam-7054	38	3	on	on	NOUN
ejpam-7054	38	4	et	et	PROPN
ejpam-7054	38	5	al	al	PROPN
ejpam-7054	38	6	.	.	PUNCT
ejpam-7054	39	1	[	[	X
ejpam-7054	39	2	15	15	NUM
ejpam-7054	39	3	]	]	PUNCT
ejpam-7054	39	4	introduced	introduce	VERB
ejpam-7054	39	5	and	and	CCONJ
ejpam-7054	39	6	studied	study	VERB
ejpam-7054	39	7	the	the	DET
ejpam-7054	39	8	concepts	concept	NOUN
ejpam-7054	39	9	of	of	ADP
ejpam-7054	39	10	upper	upper	ADJ
ejpam-7054	39	11	(	(	PUNCT
ejpam-7054	39	12	τ1	τ1	NOUN
ejpam-7054	39	13	,	,	PUNCT
ejpam-7054	39	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7054	39	15	continuous	continuous	ADJ
ejpam-7054	39	16	multifunctions	multifunction	NOUN
ejpam-7054	39	17	and	and	CCONJ
ejpam-7054	39	18	lower	low	ADJ
ejpam-7054	39	19	(	(	PUNCT
ejpam-7054	39	20	τ1	τ1	NOUN
ejpam-7054	39	21	,	,	PUNCT
ejpam-7054	39	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7054	39	23	continuous	continuous	ADJ
ejpam-7054	39	24	multifunctions	multifunction	NOUN
ejpam-7054	39	25	.	.	PUNCT
ejpam-7054	40	1	klanarong	klanarong	NOUN
ejpam-7054	40	2	et	et	PROPN
ejpam-7054	40	3	al	al	PROPN
ejpam-7054	40	4	.	.	PUNCT
ejpam-7054	41	1	[	[	X
ejpam-7054	41	2	16	16	NUM
ejpam-7054	41	3	]	]	PUNCT
ejpam-7054	41	4	introduced	introduce	VERB
ejpam-7054	41	5	and	and	CCONJ
ejpam-7054	41	6	investigated	investigate	VERB
ejpam-7054	41	7	the	the	DET
ejpam-7054	41	8	notions	notion	NOUN
ejpam-7054	41	9	of	of	ADP
ejpam-7054	41	10	upper	upper	ADJ
ejpam-7054	41	11	almost	almost	ADV
ejpam-7054	41	12	(	(	PUNCT
ejpam-7054	41	13	τ1	τ1	NOUN
ejpam-7054	41	14	,	,	PUNCT
ejpam-7054	41	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7054	41	16	multifunctions	multifunction	NOUN
ejpam-7054	41	17	and	and	CCONJ
ejpam-7054	41	18	lower	low	ADJ
ejpam-7054	41	19	almost	almost	ADV
ejpam-7054	41	20	(	(	PUNCT
ejpam-7054	41	21	τ1	τ1	NOUN
ejpam-7054	41	22	,	,	PUNCT
ejpam-7054	41	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7054	41	24	multifunctions	multifunction	NOUN
ejpam-7054	41	25	.	.	PUNCT
ejpam-7054	42	1	moreover	moreover	ADV
ejpam-7054	42	2	,	,	PUNCT
ejpam-7054	42	3	several	several	ADJ
ejpam-7054	42	4	characterizations	characterization	NOUN
ejpam-7054	42	5	and	and	CCONJ
ejpam-7054	42	6	some	some	DET
ejpam-7054	42	7	properties	property	NOUN
ejpam-7054	42	8	of	of	ADP
ejpam-7054	42	9	weakly	weakly	ADJ
ejpam-7054	42	10	(	(	PUNCT
ejpam-7054	42	11	τ1	τ1	NOUN
ejpam-7054	42	12	,	,	PUNCT
ejpam-7054	42	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7054	42	14	multifunctions	multifunction	NOUN
ejpam-7054	42	15	and	and	CCONJ
ejpam-7054	42	16	almost	almost	ADV
ejpam-7054	42	17	weakly	weakly	ADJ
ejpam-7054	42	18	(	(	PUNCT
ejpam-7054	42	19	τ1	τ1	NOUN
ejpam-7054	42	20	,	,	PUNCT
ejpam-7054	42	21	τ2)continuous	τ2)continuous	ADJ
ejpam-7054	42	22	multifunctions	multifunction	NOUN
ejpam-7054	42	23	were	be	AUX
ejpam-7054	42	24	established	establish	VERB
ejpam-7054	42	25	in	in	ADP
ejpam-7054	42	26	[	[	X
ejpam-7054	42	27	17	17	NUM
ejpam-7054	42	28	]	]	PUNCT
ejpam-7054	42	29	and	and	CCONJ
ejpam-7054	42	30	[	[	X
ejpam-7054	42	31	18	18	NUM
ejpam-7054	42	32	]	]	PUNCT
ejpam-7054	42	33	,	,	PUNCT
ejpam-7054	42	34	respectively	respectively	ADV
ejpam-7054	42	35	.	.	PUNCT
ejpam-7054	43	1	quite	quite	ADV
ejpam-7054	43	2	recently	recently	ADV
ejpam-7054	43	3	,	,	PUNCT
ejpam-7054	43	4	viriyapong	viriyapong	PROPN
ejpam-7054	43	5	et	et	PROPN
ejpam-7054	43	6	al	al	PROPN
ejpam-7054	43	7	.	.	PUNCT
ejpam-7054	44	1	[	[	X
ejpam-7054	44	2	19	19	NUM
ejpam-7054	44	3	]	]	PUNCT
ejpam-7054	44	4	presented	present	VERB
ejpam-7054	44	5	new	new	ADJ
ejpam-7054	44	6	classes	class	NOUN
ejpam-7054	44	7	of	of	ADP
ejpam-7054	44	8	continuous	continuous	ADJ
ejpam-7054	44	9	multifunctions	multifunction	NOUN
ejpam-7054	44	10	between	between	ADP
ejpam-7054	44	11	an	an	DET
ejpam-7054	44	12	ideal	ideal	ADJ
ejpam-7054	44	13	topological	topological	ADJ
ejpam-7054	44	14	space	space	NOUN
ejpam-7054	44	15	and	and	CCONJ
ejpam-7054	44	16	a	a	DET
ejpam-7054	44	17	bitopological	bitopological	ADJ
ejpam-7054	44	18	space	space	NOUN
ejpam-7054	44	19	,	,	PUNCT
ejpam-7054	44	20	namely	namely	ADV
ejpam-7054	44	21	upper	upper	ADJ
ejpam-7054	44	22	almost	almost	ADV
ejpam-7054	44	23	weakly	weakly	ADJ
ejpam-7054	44	24	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-7054	44	25	,	,	PUNCT
ejpam-7054	44	26	σ2)continuous	σ2)continuous	ADJ
ejpam-7054	44	27	multifunctions	multifunction	NOUN
ejpam-7054	44	28	and	and	CCONJ
ejpam-7054	44	29	lower	low	ADJ
ejpam-7054	44	30	almost	almost	ADV
ejpam-7054	44	31	weakly	weakly	ADJ
ejpam-7054	44	32	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-7054	44	33	,	,	PUNCT
ejpam-7054	44	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	44	35	multifunctions	multifunction	NOUN
ejpam-7054	44	36	.	.	PUNCT
ejpam-7054	45	1	in	in	ADP
ejpam-7054	45	2	this	this	DET
ejpam-7054	45	3	paper	paper	NOUN
ejpam-7054	45	4	,	,	PUNCT
ejpam-7054	45	5	we	we	PRON
ejpam-7054	45	6	introduce	introduce	VERB
ejpam-7054	45	7	the	the	DET
ejpam-7054	45	8	concepts	concept	NOUN
ejpam-7054	45	9	of	of	ADP
ejpam-7054	45	10	upper	upper	ADJ
ejpam-7054	45	11	almost	almost	ADV
ejpam-7054	45	12	weakly	weakly	ADJ
ejpam-7054	45	13	µ(σ1	µ(σ1	NOUN
ejpam-7054	45	14	,	,	PUNCT
ejpam-7054	45	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	45	16	multifunctions	multifunction	NOUN
ejpam-7054	45	17	and	and	CCONJ
ejpam-7054	45	18	lower	low	ADJ
ejpam-7054	45	19	almost	almost	ADV
ejpam-7054	45	20	weakly	weakly	ADJ
ejpam-7054	45	21	µ(σ1	µ(σ1	NOUN
ejpam-7054	45	22	,	,	PUNCT
ejpam-7054	45	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	45	24	multifunctions	multifunction	NOUN
ejpam-7054	45	25	.	.	PUNCT
ejpam-7054	46	1	we	we	PRON
ejpam-7054	46	2	also	also	ADV
ejpam-7054	46	3	investigate	investigate	VERB
ejpam-7054	46	4	several	several	ADJ
ejpam-7054	46	5	characterizations	characterization	NOUN
ejpam-7054	46	6	of	of	ADP
ejpam-7054	46	7	upper	upper	ADJ
ejpam-7054	46	8	almost	almost	ADV
ejpam-7054	46	9	weakly	weakly	ADJ
ejpam-7054	46	10	µ(σ1	µ(σ1	NOUN
ejpam-7054	46	11	,	,	PUNCT
ejpam-7054	46	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	46	13	multifunctions	multifunction	NOUN
ejpam-7054	46	14	and	and	CCONJ
ejpam-7054	46	15	lower	low	ADJ
ejpam-7054	46	16	almost	almost	ADV
ejpam-7054	46	17	weakly	weakly	ADJ
ejpam-7054	46	18	µ(σ1	µ(σ1	NOUN
ejpam-7054	46	19	,	,	PUNCT
ejpam-7054	46	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	46	21	multifunctions	multifunction	NOUN
ejpam-7054	46	22	.	.	PUNCT
ejpam-7054	47	1	2	2	X
ejpam-7054	47	2	.	.	X
ejpam-7054	47	3	preliminaries	preliminary	NOUN
ejpam-7054	47	4	throughout	throughout	ADP
ejpam-7054	47	5	the	the	DET
ejpam-7054	47	6	present	present	ADJ
ejpam-7054	47	7	paper	paper	NOUN
ejpam-7054	47	8	,	,	PUNCT
ejpam-7054	47	9	spaces	space	NOUN
ejpam-7054	47	10	(	(	PUNCT
ejpam-7054	47	11	x	x	NOUN
ejpam-7054	47	12	,	,	PUNCT
ejpam-7054	47	13	τ1	τ1	NOUN
ejpam-7054	47	14	,	,	PUNCT
ejpam-7054	47	15	τ2	τ2	NOUN
ejpam-7054	47	16	)	)	PUNCT
ejpam-7054	47	17	and	and	CCONJ
ejpam-7054	47	18	(	(	PUNCT
ejpam-7054	47	19	y	y	PROPN
ejpam-7054	47	20	,	,	PUNCT
ejpam-7054	47	21	σ1	σ1	PROPN
ejpam-7054	47	22	,	,	PUNCT
ejpam-7054	47	23	σ2	σ2	NOUN
ejpam-7054	47	24	)	)	PUNCT
ejpam-7054	47	25	(	(	PUNCT
ejpam-7054	47	26	or	or	CCONJ
ejpam-7054	47	27	simply	simply	ADV
ejpam-7054	47	28	x	x	X
ejpam-7054	47	29	and	and	CCONJ
ejpam-7054	47	30	y	y	PROPN
ejpam-7054	47	31	)	)	PUNCT
ejpam-7054	47	32	always	always	ADV
ejpam-7054	47	33	mean	mean	VERB
ejpam-7054	47	34	bitopological	bitopological	ADJ
ejpam-7054	47	35	spaces	space	NOUN
ejpam-7054	47	36	on	on	ADP
ejpam-7054	47	37	which	which	PRON
ejpam-7054	47	38	no	no	DET
ejpam-7054	47	39	separation	separation	NOUN
ejpam-7054	47	40	axioms	axiom	NOUN
ejpam-7054	47	41	are	be	AUX
ejpam-7054	47	42	assumed	assume	VERB
ejpam-7054	47	43	unless	unless	SCONJ
ejpam-7054	47	44	explicitly	explicitly	ADV
ejpam-7054	47	45	stated	state	VERB
ejpam-7054	47	46	.	.	PUNCT
ejpam-7054	48	1	let	let	VERB
ejpam-7054	48	2	a	a	DET
ejpam-7054	48	3	be	be	AUX
ejpam-7054	48	4	a	a	DET
ejpam-7054	48	5	subset	subset	NOUN
ejpam-7054	48	6	of	of	ADP
ejpam-7054	48	7	a	a	DET
ejpam-7054	48	8	bitopological	bitopological	ADJ
ejpam-7054	48	9	space	space	NOUN
ejpam-7054	48	10	(	(	PUNCT
ejpam-7054	48	11	x	x	NOUN
ejpam-7054	48	12	,	,	PUNCT
ejpam-7054	48	13	τ1	τ1	NOUN
ejpam-7054	48	14	,	,	PUNCT
ejpam-7054	48	15	τ2	τ2	NOUN
ejpam-7054	48	16	)	)	PUNCT
ejpam-7054	48	17	.	.	PUNCT
ejpam-7054	49	1	the	the	DET
ejpam-7054	49	2	closure	closure	NOUN
ejpam-7054	49	3	of	of	ADP
ejpam-7054	49	4	a	a	PRON
ejpam-7054	49	5	and	and	CCONJ
ejpam-7054	49	6	the	the	DET
ejpam-7054	49	7	interior	interior	NOUN
ejpam-7054	49	8	of	of	ADP
ejpam-7054	49	9	a	a	PRON
ejpam-7054	49	10	with	with	ADP
ejpam-7054	49	11	respect	respect	NOUN
ejpam-7054	49	12	to	to	ADP
ejpam-7054	49	13	τi	τi	PROPN
ejpam-7054	49	14	are	be	AUX
ejpam-7054	49	15	denoted	denote	VERB
ejpam-7054	49	16	by	by	ADP
ejpam-7054	49	17	τi	τi	NOUN
ejpam-7054	49	18	-	-	PUNCT
ejpam-7054	49	19	cl(a	cl(a	NUM
ejpam-7054	49	20	)	)	PUNCT
ejpam-7054	49	21	and	and	CCONJ
ejpam-7054	49	22	τi	τi	NOUN
ejpam-7054	49	23	-	-	PUNCT
ejpam-7054	49	24	int(a	int(a	NOUN
ejpam-7054	49	25	)	)	PUNCT
ejpam-7054	49	26	,	,	PUNCT
ejpam-7054	49	27	respectively	respectively	ADV
ejpam-7054	49	28	,	,	PUNCT
ejpam-7054	49	29	for	for	ADP
ejpam-7054	49	30	i	i	PROPN
ejpam-7054	49	31	=	=	SYM
ejpam-7054	49	32	1	1	NUM
ejpam-7054	49	33	,	,	PUNCT
ejpam-7054	49	34	2	2	NUM
ejpam-7054	49	35	.	.	X
ejpam-7054	49	36	a	a	DET
ejpam-7054	49	37	subset	subset	NOUN
ejpam-7054	49	38	a	a	PRON
ejpam-7054	49	39	of	of	ADP
ejpam-7054	49	40	a	a	DET
ejpam-7054	49	41	bitopological	bitopological	ADJ
ejpam-7054	49	42	space	space	NOUN
ejpam-7054	49	43	(	(	PUNCT
ejpam-7054	49	44	x	x	NOUN
ejpam-7054	49	45	,	,	PUNCT
ejpam-7054	49	46	τ1	τ1	NOUN
ejpam-7054	49	47	,	,	PUNCT
ejpam-7054	49	48	τ2	τ2	NOUN
ejpam-7054	49	49	)	)	PUNCT
ejpam-7054	49	50	is	be	AUX
ejpam-7054	49	51	called	call	VERB
ejpam-7054	49	52	τ1τ2	τ1τ2	VERB
ejpam-7054	49	53	-	-	ADJ
ejpam-7054	49	54	closed	closed	ADJ
ejpam-7054	49	55	[	[	X
ejpam-7054	49	56	20	20	NUM
ejpam-7054	49	57	]	]	PUNCT
ejpam-7054	49	58	if	if	SCONJ
ejpam-7054	49	59	a	a	DET
ejpam-7054	49	60	=	=	NOUN
ejpam-7054	49	61	τ1	τ1	NOUN
ejpam-7054	49	62	-	-	PUNCT
ejpam-7054	49	63	cl(τ2	cl(τ2	NOUN
ejpam-7054	49	64	-	-	PUNCT
ejpam-7054	49	65	cl(a	cl(a	NUM
ejpam-7054	49	66	)	)	PUNCT
ejpam-7054	49	67	)	)	PUNCT
ejpam-7054	49	68	.	.	PUNCT
ejpam-7054	50	1	the	the	DET
ejpam-7054	50	2	complement	complement	NOUN
ejpam-7054	50	3	of	of	ADP
ejpam-7054	50	4	a	a	DET
ejpam-7054	50	5	τ1τ2	τ1τ2	ADJ
ejpam-7054	50	6	-	-	ADJ
ejpam-7054	50	7	closed	closed	ADJ
ejpam-7054	50	8	set	set	NOUN
ejpam-7054	50	9	is	be	AUX
ejpam-7054	50	10	called	call	VERB
ejpam-7054	50	11	τ1τ2	τ1τ2	NOUN
ejpam-7054	50	12	-	-	ADJ
ejpam-7054	50	13	open	open	ADJ
ejpam-7054	50	14	.	.	PUNCT
ejpam-7054	51	1	let	let	VERB
ejpam-7054	51	2	a	a	DET
ejpam-7054	51	3	be	be	AUX
ejpam-7054	51	4	a	a	DET
ejpam-7054	51	5	subset	subset	NOUN
ejpam-7054	51	6	of	of	ADP
ejpam-7054	51	7	a	a	DET
ejpam-7054	51	8	bitopological	bitopological	ADJ
ejpam-7054	51	9	space	space	NOUN
ejpam-7054	51	10	(	(	PUNCT
ejpam-7054	51	11	x	x	NOUN
ejpam-7054	51	12	,	,	PUNCT
ejpam-7054	51	13	τ1	τ1	NOUN
ejpam-7054	51	14	,	,	PUNCT
ejpam-7054	51	15	τ2	τ2	NOUN
ejpam-7054	51	16	)	)	PUNCT
ejpam-7054	51	17	.	.	PUNCT
ejpam-7054	52	1	the	the	DET
ejpam-7054	52	2	intersection	intersection	NOUN
ejpam-7054	52	3	of	of	ADP
ejpam-7054	52	4	all	all	DET
ejpam-7054	52	5	τ1τ2	τ1τ2	ADJ
ejpam-7054	52	6	-	-	ADJ
ejpam-7054	52	7	closed	closed	ADJ
ejpam-7054	52	8	sets	set	NOUN
ejpam-7054	52	9	of	of	ADP
ejpam-7054	52	10	x	x	PUNCT
ejpam-7054	52	11	containing	contain	VERB
ejpam-7054	52	12	a	a	PRON
ejpam-7054	52	13	is	be	AUX
ejpam-7054	52	14	called	call	VERB
ejpam-7054	52	15	the	the	DET
ejpam-7054	52	16	τ1τ2	τ1τ2	NOUN
ejpam-7054	52	17	-	-	NOUN
ejpam-7054	52	18	closure	closure	NOUN
ejpam-7054	52	19	[	[	X
ejpam-7054	52	20	20	20	NUM
ejpam-7054	52	21	]	]	PUNCT
ejpam-7054	52	22	of	of	ADP
ejpam-7054	52	23	a	a	PRON
ejpam-7054	52	24	and	and	CCONJ
ejpam-7054	52	25	is	be	AUX
ejpam-7054	52	26	denoted	denote	VERB
ejpam-7054	52	27	by	by	ADP
ejpam-7054	52	28	τ1τ2	τ1τ2	NOUN
ejpam-7054	52	29	-	-	NUM
ejpam-7054	52	30	cl(a	cl(a	NUM
ejpam-7054	52	31	)	)	PUNCT
ejpam-7054	52	32	.	.	PUNCT
ejpam-7054	53	1	the	the	DET
ejpam-7054	53	2	union	union	NOUN
ejpam-7054	53	3	of	of	ADP
ejpam-7054	53	4	all	all	DET
ejpam-7054	53	5	τ1τ2	τ1τ2	ADJ
ejpam-7054	53	6	-	-	ADJ
ejpam-7054	53	7	open	open	ADJ
ejpam-7054	53	8	sets	set	NOUN
ejpam-7054	53	9	of	of	ADP
ejpam-7054	53	10	x	x	PUNCT
ejpam-7054	53	11	contained	contain	VERB
ejpam-7054	53	12	in	in	ADP
ejpam-7054	53	13	a	a	PRON
ejpam-7054	53	14	is	be	AUX
ejpam-7054	53	15	called	call	VERB
ejpam-7054	53	16	the	the	DET
ejpam-7054	53	17	τ1τ2	τ1τ2	NOUN
ejpam-7054	53	18	-	-	ADJ
ejpam-7054	53	19	interior	interior	ADJ
ejpam-7054	53	20	[	[	X
ejpam-7054	53	21	20	20	NUM
ejpam-7054	53	22	]	]	PUNCT
ejpam-7054	53	23	of	of	ADP
ejpam-7054	53	24	a	a	PRON
ejpam-7054	53	25	and	and	CCONJ
ejpam-7054	53	26	is	be	AUX
ejpam-7054	53	27	denoted	denote	VERB
ejpam-7054	53	28	by	by	ADP
ejpam-7054	53	29	τ1τ2	τ1τ2	NOUN
ejpam-7054	53	30	-	-	ADJ
ejpam-7054	53	31	int(a	int(a	NOUN
ejpam-7054	53	32	)	)	PUNCT
ejpam-7054	53	33	.	.	PUNCT
ejpam-7054	54	1	a	a	DET
ejpam-7054	54	2	subset	subset	NOUN
ejpam-7054	54	3	a	a	PRON
ejpam-7054	54	4	of	of	ADP
ejpam-7054	54	5	a	a	DET
ejpam-7054	54	6	bitopological	bitopological	ADJ
ejpam-7054	54	7	space	space	NOUN
ejpam-7054	54	8	(	(	PUNCT
ejpam-7054	54	9	x	x	NOUN
ejpam-7054	54	10	,	,	PUNCT
ejpam-7054	54	11	τ1	τ1	NOUN
ejpam-7054	54	12	,	,	PUNCT
ejpam-7054	54	13	τ2	τ2	NOUN
ejpam-7054	54	14	)	)	PUNCT
ejpam-7054	54	15	is	be	AUX
ejpam-7054	54	16	said	say	VERB
ejpam-7054	54	17	to	to	PART
ejpam-7054	54	18	be	be	AUX
ejpam-7054	54	19	(	(	PUNCT
ejpam-7054	54	20	τ1	τ1	NOUN
ejpam-7054	54	21	,	,	PUNCT
ejpam-7054	54	22	τ2)r	τ2)r	NOUN
ejpam-7054	54	23	-	-	PUNCT
ejpam-7054	54	24	open	open	NOUN
ejpam-7054	55	1	[	[	X
ejpam-7054	55	2	21	21	NUM
ejpam-7054	55	3	]	]	X
ejpam-7054	55	4	(	(	PUNCT
ejpam-7054	55	5	resp	resp	NOUN
ejpam-7054	55	6	.	.	PUNCT
ejpam-7054	56	1	(	(	PUNCT
ejpam-7054	56	2	τ1	τ1	NOUN
ejpam-7054	56	3	,	,	PUNCT
ejpam-7054	56	4	τ2)s	τ2)s	NOUN
ejpam-7054	56	5	-	-	PUNCT
ejpam-7054	56	6	open	open	ADJ
ejpam-7054	56	7	[	[	X
ejpam-7054	56	8	22	22	NUM
ejpam-7054	56	9	]	]	PUNCT
ejpam-7054	56	10	,	,	PUNCT
ejpam-7054	56	11	(	(	PUNCT
ejpam-7054	56	12	τ1	τ1	NOUN
ejpam-7054	56	13	,	,	PUNCT
ejpam-7054	56	14	τ2)p	τ2)p	NOUN
ejpam-7054	56	15	-	-	ADJ
ejpam-7054	56	16	open	open	ADJ
ejpam-7054	56	17	[	[	X
ejpam-7054	56	18	22	22	NUM
ejpam-7054	56	19	]	]	PUNCT
ejpam-7054	56	20	,	,	PUNCT
ejpam-7054	56	21	(	(	PUNCT
ejpam-7054	56	22	τ1	τ1	NOUN
ejpam-7054	56	23	,	,	PUNCT
ejpam-7054	56	24	τ2)β	τ2)β	ADJ
ejpam-7054	56	25	-	-	PUNCT
ejpam-7054	56	26	open	open	NOUN
ejpam-7054	57	1	[	[	X
ejpam-7054	57	2	22	22	NUM
ejpam-7054	57	3	]	]	SYM
ejpam-7054	57	4	)	)	PUNCT
ejpam-7054	57	5	if	if	SCONJ
ejpam-7054	57	6	a	a	DET
ejpam-7054	57	7	=	=	PUNCT
ejpam-7054	57	8	τ1τ2	τ1τ2	NOUN
ejpam-7054	57	9	-	-	NOUN
ejpam-7054	57	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7054	57	11	-	-	PUNCT
ejpam-7054	57	12	cl(a	cl(a	NUM
ejpam-7054	57	13	)	)	PUNCT
ejpam-7054	57	14	)	)	PUNCT
ejpam-7054	57	15	(	(	PUNCT
ejpam-7054	57	16	resp	resp	NOUN
ejpam-7054	57	17	.	.	PUNCT
ejpam-7054	58	1	a	a	DET
ejpam-7054	58	2	⊆	⊆	NUM
ejpam-7054	58	3	τ1τ2	τ1τ2	NOUN
ejpam-7054	58	4	-	-	ADJ
ejpam-7054	58	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7054	58	6	-	-	PUNCT
ejpam-7054	58	7	int(a	int(a	NOUN
ejpam-7054	58	8	)	)	PUNCT
ejpam-7054	58	9	)	)	PUNCT
ejpam-7054	58	10	,	,	PUNCT
ejpam-7054	58	11	a	a	DET
ejpam-7054	58	12	⊆	⊆	NUM
ejpam-7054	58	13	τ1τ2	τ1τ2	NOUN
ejpam-7054	58	14	-	-	NOUN
ejpam-7054	58	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7054	58	16	-	-	PUNCT
ejpam-7054	58	17	cl(a	cl(a	NUM
ejpam-7054	58	18	)	)	PUNCT
ejpam-7054	58	19	)	)	PUNCT
ejpam-7054	58	20	,	,	PUNCT
ejpam-7054	58	21	a	a	DET
ejpam-7054	58	22	⊆	⊆	NUM
ejpam-7054	58	23	τ1τ2	τ1τ2	NOUN
ejpam-7054	58	24	-	-	PUNCT
ejpam-7054	58	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-7054	58	26	-	-	PUNCT
ejpam-7054	58	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-7054	58	28	-	-	PUNCT
ejpam-7054	58	29	cl(a	cl(a	NUM
ejpam-7054	58	30	)	)	PUNCT
ejpam-7054	58	31	)	)	PUNCT
ejpam-7054	58	32	)	)	PUNCT
ejpam-7054	58	33	)	)	PUNCT
ejpam-7054	58	34	.	.	PUNCT
ejpam-7054	59	1	the	the	DET
ejpam-7054	59	2	complement	complement	NOUN
ejpam-7054	59	3	of	of	ADP
ejpam-7054	59	4	a	a	DET
ejpam-7054	59	5	(	(	PUNCT
ejpam-7054	59	6	τ1	τ1	NOUN
ejpam-7054	59	7	,	,	PUNCT
ejpam-7054	59	8	τ2)r	τ2)r	NOUN
ejpam-7054	59	9	-	-	PUNCT
ejpam-7054	59	10	open	open	ADJ
ejpam-7054	59	11	(	(	PUNCT
ejpam-7054	59	12	resp	resp	NOUN
ejpam-7054	59	13	.	.	PUNCT
ejpam-7054	60	1	(	(	PUNCT
ejpam-7054	60	2	τ1	τ1	NOUN
ejpam-7054	60	3	,	,	PUNCT
ejpam-7054	60	4	τ2)sb	τ2)sb	PROPN
ejpam-7054	60	5	.	.	PUNCT
ejpam-7054	61	1	kong	kong	PROPN
ejpam-7054	61	2	-	-	PUNCT
ejpam-7054	61	3	ied	ied	PROPN
ejpam-7054	61	4	,	,	PUNCT
ejpam-7054	61	5	a.	a.	PROPN
ejpam-7054	61	6	sama	sama	PROPN
ejpam-7054	61	7	-	-	PUNCT
ejpam-7054	61	8	ae	ae	PROPN
ejpam-7054	61	9	,	,	PUNCT
ejpam-7054	61	10	c.	c.	PROPN
ejpam-7054	61	11	boonpok	boonpok	PROPN
ejpam-7054	61	12	/	/	SYM
ejpam-7054	61	13	eur	eur	PROPN
ejpam-7054	61	14	.	.	PUNCT
ejpam-7054	62	1	j.	j.	PROPN
ejpam-7054	62	2	pure	pure	PROPN
ejpam-7054	62	3	appl	appl	PROPN
ejpam-7054	62	4	.	.	PROPN
ejpam-7054	62	5	math	math	PROPN
ejpam-7054	62	6	,	,	PUNCT
ejpam-7054	62	7	18	18	NUM
ejpam-7054	62	8	(	(	PUNCT
ejpam-7054	62	9	4	4	NUM
ejpam-7054	62	10	)	)	PUNCT
ejpam-7054	62	11	(	(	PUNCT
ejpam-7054	62	12	2025	2025	NUM
ejpam-7054	62	13	)	)	PUNCT
ejpam-7054	62	14	,	,	PUNCT
ejpam-7054	62	15	7054	7054	NUM
ejpam-7054	62	16	3	3	NUM
ejpam-7054	62	17	of	of	ADP
ejpam-7054	62	18	12	12	NUM
ejpam-7054	62	19	open	open	ADJ
ejpam-7054	62	20	,	,	PUNCT
ejpam-7054	62	21	(	(	PUNCT
ejpam-7054	62	22	τ1	τ1	NOUN
ejpam-7054	62	23	,	,	PUNCT
ejpam-7054	62	24	τ2)p	τ2)p	NOUN
ejpam-7054	62	25	-	-	ADJ
ejpam-7054	62	26	open	open	ADJ
ejpam-7054	62	27	,	,	PUNCT
ejpam-7054	62	28	(	(	PUNCT
ejpam-7054	62	29	τ1	τ1	NOUN
ejpam-7054	62	30	,	,	PUNCT
ejpam-7054	62	31	τ2)β	τ2)β	ADJ
ejpam-7054	62	32	-	-	PUNCT
ejpam-7054	62	33	open	open	ADJ
ejpam-7054	62	34	)	)	PUNCT
ejpam-7054	62	35	set	set	NOUN
ejpam-7054	62	36	is	be	AUX
ejpam-7054	62	37	called	call	VERB
ejpam-7054	62	38	(	(	PUNCT
ejpam-7054	62	39	τ1	τ1	NOUN
ejpam-7054	62	40	,	,	PUNCT
ejpam-7054	62	41	τ2)r	τ2)r	NOUN
ejpam-7054	62	42	-	-	PUNCT
ejpam-7054	62	43	closed	closed	ADJ
ejpam-7054	62	44	(	(	PUNCT
ejpam-7054	62	45	resp	resp	NOUN
ejpam-7054	62	46	.	.	PUNCT
ejpam-7054	63	1	(	(	PUNCT
ejpam-7054	63	2	τ1	τ1	NOUN
ejpam-7054	63	3	,	,	PUNCT
ejpam-7054	63	4	τ2)s	τ2)s	NOUN
ejpam-7054	63	5	-	-	PUNCT
ejpam-7054	63	6	closed	closed	ADJ
ejpam-7054	63	7	,	,	PUNCT
ejpam-7054	63	8	(	(	PUNCT
ejpam-7054	63	9	τ1	τ1	NOUN
ejpam-7054	63	10	,	,	PUNCT
ejpam-7054	63	11	τ2)p	τ2)p	NOUN
ejpam-7054	63	12	-	-	PUNCT
ejpam-7054	63	13	closed	closed	ADJ
ejpam-7054	63	14	,	,	PUNCT
ejpam-7054	63	15	(	(	PUNCT
ejpam-7054	63	16	τ1	τ1	NOUN
ejpam-7054	63	17	,	,	PUNCT
ejpam-7054	63	18	τ2)β	τ2)β	ADJ
ejpam-7054	63	19	-	-	PUNCT
ejpam-7054	63	20	closed	closed	ADJ
ejpam-7054	63	21	)	)	PUNCT
ejpam-7054	63	22	.	.	PUNCT
ejpam-7054	64	1	for	for	ADP
ejpam-7054	64	2	a	a	DET
ejpam-7054	64	3	subset	subset	NOUN
ejpam-7054	64	4	a	a	PRON
ejpam-7054	64	5	of	of	ADP
ejpam-7054	64	6	a	a	DET
ejpam-7054	64	7	bitopological	bitopological	ADJ
ejpam-7054	64	8	space	space	NOUN
ejpam-7054	64	9	(	(	PUNCT
ejpam-7054	64	10	x	x	NOUN
ejpam-7054	64	11	,	,	PUNCT
ejpam-7054	64	12	τ1	τ1	NOUN
ejpam-7054	64	13	,	,	PUNCT
ejpam-7054	64	14	τ2	τ2	PROPN
ejpam-7054	64	15	)	)	PUNCT
ejpam-7054	64	16	,	,	PUNCT
ejpam-7054	64	17	a	a	DET
ejpam-7054	64	18	point	point	NOUN
ejpam-7054	64	19	x	x	X
ejpam-7054	64	20	∈	∈	NOUN
ejpam-7054	64	21	x	x	PUNCT
ejpam-7054	64	22	is	be	AUX
ejpam-7054	64	23	called	call	VERB
ejpam-7054	64	24	a	a	DET
ejpam-7054	64	25	(	(	PUNCT
ejpam-7054	64	26	τ1	τ1	NOUN
ejpam-7054	64	27	,	,	PUNCT
ejpam-7054	64	28	τ2)θ	τ2)θ	ADJ
ejpam-7054	64	29	-	-	PUNCT
ejpam-7054	64	30	cluster	cluster	NOUN
ejpam-7054	64	31	point	point	NOUN
ejpam-7054	64	32	[	[	X
ejpam-7054	64	33	21	21	NUM
ejpam-7054	64	34	]	]	PUNCT
ejpam-7054	64	35	of	of	ADP
ejpam-7054	64	36	a	a	DET
ejpam-7054	64	37	if	if	SCONJ
ejpam-7054	64	38	τ1τ2	τ1τ2	NOUN
ejpam-7054	64	39	-	-	NOUN
ejpam-7054	64	40	cl(u	cl(u	NOUN
ejpam-7054	64	41	)	)	PUNCT
ejpam-7054	64	42	∩	∩	NOUN
ejpam-7054	64	43	a	a	DET
ejpam-7054	64	44	̸=	̸=	PROPN
ejpam-7054	64	45	∅	∅	NOUN
ejpam-7054	64	46	for	for	ADP
ejpam-7054	64	47	every	every	DET
ejpam-7054	64	48	τ1τ2	τ1τ2	ADJ
ejpam-7054	64	49	-	-	ADJ
ejpam-7054	64	50	open	open	ADJ
ejpam-7054	64	51	set	set	NOUN
ejpam-7054	64	52	u	u	NOUN
ejpam-7054	64	53	containing	contain	VERB
ejpam-7054	64	54	x.	x.	NOUN
ejpam-7054	64	55	the	the	DET
ejpam-7054	64	56	set	set	NOUN
ejpam-7054	64	57	of	of	ADP
ejpam-7054	64	58	all	all	DET
ejpam-7054	64	59	(	(	PUNCT
ejpam-7054	64	60	τ1	τ1	NOUN
ejpam-7054	64	61	,	,	PUNCT
ejpam-7054	64	62	τ2)θ	τ2)θ	ADJ
ejpam-7054	64	63	-	-	PUNCT
ejpam-7054	64	64	cluster	cluster	NOUN
ejpam-7054	64	65	points	point	NOUN
ejpam-7054	64	66	of	of	ADP
ejpam-7054	64	67	a	a	PRON
ejpam-7054	64	68	is	be	AUX
ejpam-7054	64	69	called	call	VERB
ejpam-7054	64	70	the	the	DET
ejpam-7054	64	71	(	(	PUNCT
ejpam-7054	64	72	τ1	τ1	NOUN
ejpam-7054	64	73	,	,	PUNCT
ejpam-7054	64	74	τ2)θ	τ2)θ	ADJ
ejpam-7054	64	75	-	-	PUNCT
ejpam-7054	64	76	closure	closure	NOUN
ejpam-7054	64	77	[	[	X
ejpam-7054	64	78	21	21	NUM
ejpam-7054	64	79	]	]	PUNCT
ejpam-7054	64	80	of	of	ADP
ejpam-7054	64	81	a	a	PRON
ejpam-7054	64	82	and	and	CCONJ
ejpam-7054	64	83	is	be	AUX
ejpam-7054	64	84	denoted	denote	VERB
ejpam-7054	64	85	by	by	ADP
ejpam-7054	64	86	(	(	PUNCT
ejpam-7054	64	87	τ1	τ1	NOUN
ejpam-7054	64	88	,	,	PUNCT
ejpam-7054	64	89	τ2)θ	τ2)θ	NOUN
ejpam-7054	64	90	-	-	PUNCT
ejpam-7054	64	91	cl(a	cl(a	NUM
ejpam-7054	64	92	)	)	PUNCT
ejpam-7054	64	93	.	.	PUNCT
ejpam-7054	65	1	a	a	DET
ejpam-7054	65	2	subset	subset	NOUN
ejpam-7054	65	3	a	a	PRON
ejpam-7054	65	4	of	of	ADP
ejpam-7054	65	5	a	a	DET
ejpam-7054	65	6	bitopological	bitopological	ADJ
ejpam-7054	65	7	space	space	NOUN
ejpam-7054	65	8	(	(	PUNCT
ejpam-7054	65	9	x	x	NOUN
ejpam-7054	65	10	,	,	PUNCT
ejpam-7054	65	11	τ1	τ1	NOUN
ejpam-7054	65	12	,	,	PUNCT
ejpam-7054	65	13	τ2	τ2	NOUN
ejpam-7054	65	14	)	)	PUNCT
ejpam-7054	65	15	is	be	AUX
ejpam-7054	65	16	said	say	VERB
ejpam-7054	65	17	to	to	PART
ejpam-7054	65	18	be	be	AUX
ejpam-7054	65	19	(	(	PUNCT
ejpam-7054	65	20	τ1	τ1	NOUN
ejpam-7054	65	21	,	,	PUNCT
ejpam-7054	65	22	τ2)θ	τ2)θ	NOUN
ejpam-7054	65	23	-	-	PUNCT
ejpam-7054	65	24	closed	closed	ADJ
ejpam-7054	65	25	[	[	X
ejpam-7054	65	26	21	21	NUM
ejpam-7054	65	27	]	]	X
ejpam-7054	65	28	if	if	SCONJ
ejpam-7054	65	29	(	(	PUNCT
ejpam-7054	65	30	τ1	τ1	NOUN
ejpam-7054	65	31	,	,	PUNCT
ejpam-7054	65	32	τ2)θ	τ2)θ	NOUN
ejpam-7054	65	33	-	-	PUNCT
ejpam-7054	65	34	cl(a	cl(a	NUM
ejpam-7054	65	35	)	)	PUNCT
ejpam-7054	66	1	=	=	PUNCT
ejpam-7054	66	2	a.	a.	NOUN
ejpam-7054	66	3	the	the	DET
ejpam-7054	66	4	complement	complement	NOUN
ejpam-7054	66	5	of	of	ADP
ejpam-7054	66	6	a	a	DET
ejpam-7054	66	7	(	(	PUNCT
ejpam-7054	66	8	τ1	τ1	NOUN
ejpam-7054	66	9	,	,	PUNCT
ejpam-7054	66	10	τ2)θ	τ2)θ	ADJ
ejpam-7054	66	11	-	-	PUNCT
ejpam-7054	66	12	closed	close	VERB
ejpam-7054	66	13	set	set	NOUN
ejpam-7054	66	14	is	be	AUX
ejpam-7054	66	15	said	say	VERB
ejpam-7054	66	16	to	to	PART
ejpam-7054	66	17	be	be	AUX
ejpam-7054	66	18	(	(	PUNCT
ejpam-7054	66	19	τ1	τ1	NOUN
ejpam-7054	66	20	,	,	PUNCT
ejpam-7054	66	21	τ2)θ	τ2)θ	NOUN
ejpam-7054	66	22	-	-	PUNCT
ejpam-7054	66	23	open	open	ADJ
ejpam-7054	66	24	.	.	PUNCT
ejpam-7054	67	1	the	the	DET
ejpam-7054	67	2	union	union	NOUN
ejpam-7054	67	3	of	of	ADP
ejpam-7054	67	4	all	all	DET
ejpam-7054	67	5	(	(	PUNCT
ejpam-7054	67	6	τ1	τ1	NOUN
ejpam-7054	67	7	,	,	PUNCT
ejpam-7054	67	8	τ2)θ	τ2)θ	ADJ
ejpam-7054	67	9	-	-	PUNCT
ejpam-7054	67	10	open	open	ADJ
ejpam-7054	67	11	sets	set	NOUN
ejpam-7054	67	12	of	of	ADP
ejpam-7054	67	13	x	x	PUNCT
ejpam-7054	67	14	contained	contain	VERB
ejpam-7054	67	15	in	in	ADP
ejpam-7054	67	16	a	a	PRON
ejpam-7054	67	17	is	be	AUX
ejpam-7054	67	18	called	call	VERB
ejpam-7054	67	19	the	the	DET
ejpam-7054	67	20	(	(	PUNCT
ejpam-7054	67	21	τ1	τ1	NOUN
ejpam-7054	67	22	,	,	PUNCT
ejpam-7054	67	23	τ2)θ	τ2)θ	ADJ
ejpam-7054	67	24	-	-	PUNCT
ejpam-7054	67	25	interior	interior	NOUN
ejpam-7054	67	26	[	[	X
ejpam-7054	67	27	21	21	NUM
ejpam-7054	67	28	]	]	PUNCT
ejpam-7054	67	29	of	of	ADP
ejpam-7054	67	30	a	a	PRON
ejpam-7054	67	31	and	and	CCONJ
ejpam-7054	67	32	is	be	AUX
ejpam-7054	67	33	denoted	denote	VERB
ejpam-7054	67	34	by	by	ADP
ejpam-7054	67	35	(	(	PUNCT
ejpam-7054	67	36	τ1	τ1	NOUN
ejpam-7054	67	37	,	,	PUNCT
ejpam-7054	67	38	τ2)θ	τ2)θ	NOUN
ejpam-7054	67	39	-	-	PUNCT
ejpam-7054	67	40	int(a	int(a	NOUN
ejpam-7054	67	41	)	)	PUNCT
ejpam-7054	67	42	.	.	PUNCT
ejpam-7054	68	1	lemma	lemma	PROPN
ejpam-7054	68	2	1	1	NUM
ejpam-7054	68	3	.	.	PUNCT
ejpam-7054	69	1	[	[	X
ejpam-7054	69	2	21	21	NUM
ejpam-7054	69	3	]	]	PUNCT
ejpam-7054	69	4	for	for	ADP
ejpam-7054	69	5	a	a	DET
ejpam-7054	69	6	subset	subset	NOUN
ejpam-7054	69	7	a	a	PRON
ejpam-7054	69	8	of	of	ADP
ejpam-7054	69	9	a	a	DET
ejpam-7054	69	10	bitopological	bitopological	ADJ
ejpam-7054	69	11	space	space	NOUN
ejpam-7054	69	12	(	(	PUNCT
ejpam-7054	69	13	x	x	NOUN
ejpam-7054	69	14	,	,	PUNCT
ejpam-7054	69	15	τ1	τ1	NOUN
ejpam-7054	69	16	,	,	PUNCT
ejpam-7054	69	17	τ2	τ2	NOUN
ejpam-7054	69	18	)	)	PUNCT
ejpam-7054	69	19	,	,	PUNCT
ejpam-7054	69	20	the	the	DET
ejpam-7054	69	21	following	follow	VERB
ejpam-7054	69	22	properties	property	NOUN
ejpam-7054	69	23	hold	hold	VERB
ejpam-7054	69	24	:	:	PUNCT
ejpam-7054	69	25	(	(	PUNCT
ejpam-7054	69	26	1	1	X
ejpam-7054	69	27	)	)	PUNCT
ejpam-7054	69	28	if	if	SCONJ
ejpam-7054	69	29	a	a	PRON
ejpam-7054	69	30	is	be	AUX
ejpam-7054	69	31	τ1τ2	τ1τ2	NOUN
ejpam-7054	69	32	-	-	ADJ
ejpam-7054	69	33	open	open	ADJ
ejpam-7054	69	34	in	in	ADP
ejpam-7054	69	35	x	x	NOUN
ejpam-7054	69	36	,	,	PUNCT
ejpam-7054	69	37	then	then	ADV
ejpam-7054	69	38	τ1τ2	τ1τ2	NOUN
ejpam-7054	69	39	-	-	NUM
ejpam-7054	69	40	cl(a	cl(a	NUM
ejpam-7054	69	41	)	)	PUNCT
ejpam-7054	69	42	=	=	PUNCT
ejpam-7054	69	43	(	(	PUNCT
ejpam-7054	69	44	τ1	τ1	NOUN
ejpam-7054	69	45	,	,	PUNCT
ejpam-7054	69	46	τ2)θ	τ2)θ	NOUN
ejpam-7054	69	47	-	-	PUNCT
ejpam-7054	69	48	cl(a	cl(a	NUM
ejpam-7054	69	49	)	)	PUNCT
ejpam-7054	69	50	.	.	PUNCT
ejpam-7054	70	1	(	(	PUNCT
ejpam-7054	70	2	2	2	X
ejpam-7054	70	3	)	)	PUNCT
ejpam-7054	70	4	(	(	PUNCT
ejpam-7054	70	5	τ1	τ1	NOUN
ejpam-7054	70	6	,	,	PUNCT
ejpam-7054	70	7	τ2)θ	τ2)θ	NOUN
ejpam-7054	70	8	-	-	PUNCT
ejpam-7054	70	9	cl(a	cl(a	NUM
ejpam-7054	70	10	)	)	PUNCT
ejpam-7054	70	11	is	be	AUX
ejpam-7054	70	12	τ1τ2	τ1τ2	NOUN
ejpam-7054	70	13	-	-	ADJ
ejpam-7054	70	14	closed	closed	ADJ
ejpam-7054	70	15	in	in	ADP
ejpam-7054	70	16	x.	x.	NOUN
ejpam-7054	70	17	let	let	VERB
ejpam-7054	70	18	x	x	PRON
ejpam-7054	70	19	be	be	AUX
ejpam-7054	70	20	a	a	DET
ejpam-7054	70	21	nonempty	nonempty	ADJ
ejpam-7054	70	22	set	set	NOUN
ejpam-7054	70	23	,	,	PUNCT
ejpam-7054	70	24	and	and	CCONJ
ejpam-7054	70	25	denote	denote	VERB
ejpam-7054	70	26	p(x	p(x	PROPN
ejpam-7054	70	27	)	)	PUNCT
ejpam-7054	70	28	the	the	DET
ejpam-7054	70	29	power	power	NOUN
ejpam-7054	70	30	set	set	NOUN
ejpam-7054	70	31	of	of	ADP
ejpam-7054	70	32	x.	x.	NOUN
ejpam-7054	70	33	we	we	PRON
ejpam-7054	70	34	call	call	VERB
ejpam-7054	70	35	a	a	DET
ejpam-7054	70	36	class	class	NOUN
ejpam-7054	70	37	µ	µ	PRON
ejpam-7054	70	38	⊆	⊆	NUM
ejpam-7054	70	39	p(x	p(x	NOUN
ejpam-7054	70	40	)	)	PUNCT
ejpam-7054	70	41	a	a	DET
ejpam-7054	70	42	generalized	generalized	ADJ
ejpam-7054	70	43	topology	topology	NOUN
ejpam-7054	70	44	(	(	PUNCT
ejpam-7054	70	45	briefly	briefly	ADV
ejpam-7054	70	46	,	,	PUNCT
ejpam-7054	70	47	gt	gt	PROPN
ejpam-7054	70	48	)	)	PUNCT
ejpam-7054	70	49	if	if	SCONJ
ejpam-7054	70	50	∅	∅	NUM
ejpam-7054	70	51	∈	∈	PROPN
ejpam-7054	70	52	µ	µ	NOUN
ejpam-7054	70	53	,	,	PUNCT
ejpam-7054	70	54	and	and	CCONJ
ejpam-7054	70	55	an	an	DET
ejpam-7054	70	56	arbitrary	arbitrary	ADJ
ejpam-7054	70	57	union	union	NOUN
ejpam-7054	70	58	of	of	ADP
ejpam-7054	70	59	elements	element	NOUN
ejpam-7054	70	60	of	of	ADP
ejpam-7054	70	61	µ	µ	X
ejpam-7054	70	62	belongs	belong	VERB
ejpam-7054	70	63	to	to	ADP
ejpam-7054	70	64	µ	µ	PROPN
ejpam-7054	70	65	[	[	PUNCT
ejpam-7054	70	66	11	11	NUM
ejpam-7054	70	67	]	]	PUNCT
ejpam-7054	70	68	.	.	PUNCT
ejpam-7054	71	1	a	a	DET
ejpam-7054	71	2	setx	setx	NOUN
ejpam-7054	71	3	with	with	ADP
ejpam-7054	71	4	a	a	DET
ejpam-7054	71	5	gt	gt	PROPN
ejpam-7054	71	6	µ	µ	NOUN
ejpam-7054	71	7	on	on	ADP
ejpam-7054	71	8	it	it	PRON
ejpam-7054	71	9	is	be	AUX
ejpam-7054	71	10	said	say	VERB
ejpam-7054	71	11	to	to	PART
ejpam-7054	71	12	be	be	AUX
ejpam-7054	71	13	a	a	DET
ejpam-7054	71	14	generalized	generalized	ADJ
ejpam-7054	71	15	topological	topological	ADJ
ejpam-7054	71	16	space	space	NOUN
ejpam-7054	71	17	(	(	PUNCT
ejpam-7054	71	18	briefly	briefly	ADV
ejpam-7054	71	19	,	,	PUNCT
ejpam-7054	71	20	gts	gts	NOUN
ejpam-7054	71	21	)	)	PUNCT
ejpam-7054	71	22	and	and	CCONJ
ejpam-7054	71	23	is	be	AUX
ejpam-7054	71	24	denoted	denote	VERB
ejpam-7054	71	25	by	by	ADP
ejpam-7054	71	26	(	(	PUNCT
ejpam-7054	71	27	x,µ	x,µ	NOUN
ejpam-7054	71	28	)	)	PUNCT
ejpam-7054	71	29	.	.	PUNCT
ejpam-7054	72	1	for	for	ADP
ejpam-7054	72	2	a	a	DET
ejpam-7054	72	3	gts	gts	NOUN
ejpam-7054	72	4	(	(	PUNCT
ejpam-7054	72	5	x,µ	x,µ	NOUN
ejpam-7054	72	6	)	)	PUNCT
ejpam-7054	72	7	,	,	PUNCT
ejpam-7054	72	8	the	the	DET
ejpam-7054	72	9	elements	element	NOUN
ejpam-7054	72	10	of	of	ADP
ejpam-7054	72	11	µ	µ	NOUN
ejpam-7054	72	12	are	be	AUX
ejpam-7054	72	13	called	call	VERB
ejpam-7054	72	14	µ-open	µ-open	NOUN
ejpam-7054	72	15	sets	set	NOUN
ejpam-7054	72	16	and	and	CCONJ
ejpam-7054	72	17	the	the	DET
ejpam-7054	72	18	complements	complement	NOUN
ejpam-7054	72	19	of	of	ADP
ejpam-7054	72	20	µ-open	µ-open	NOUN
ejpam-7054	72	21	sets	set	NOUN
ejpam-7054	72	22	are	be	AUX
ejpam-7054	72	23	called	call	VERB
ejpam-7054	72	24	µ-closed	µ-close	VERB
ejpam-7054	72	25	sets	set	NOUN
ejpam-7054	72	26	.	.	PUNCT
ejpam-7054	73	1	for	for	ADP
ejpam-7054	73	2	a	a	DET
ejpam-7054	73	3	⊆	⊆	NUM
ejpam-7054	73	4	x	x	SYM
ejpam-7054	73	5	,	,	PUNCT
ejpam-7054	73	6	we	we	PRON
ejpam-7054	73	7	denote	denote	VERB
ejpam-7054	73	8	by	by	ADP
ejpam-7054	73	9	cµ(a	cµ(a	PROPN
ejpam-7054	73	10	)	)	PUNCT
ejpam-7054	73	11	the	the	DET
ejpam-7054	73	12	intersection	intersection	NOUN
ejpam-7054	73	13	of	of	ADP
ejpam-7054	73	14	all	all	DET
ejpam-7054	73	15	µ-closed	µ-close	VERB
ejpam-7054	73	16	sets	set	NOUN
ejpam-7054	73	17	containing	contain	VERB
ejpam-7054	73	18	a	a	PRON
ejpam-7054	73	19	and	and	CCONJ
ejpam-7054	73	20	by	by	ADP
ejpam-7054	73	21	iµ(a	iµ(a	PROPN
ejpam-7054	73	22	)	)	PUNCT
ejpam-7054	73	23	the	the	DET
ejpam-7054	73	24	union	union	NOUN
ejpam-7054	73	25	of	of	ADP
ejpam-7054	73	26	all	all	DET
ejpam-7054	73	27	µ-open	µ-open	NOUN
ejpam-7054	73	28	sets	set	NOUN
ejpam-7054	73	29	contained	contain	VERB
ejpam-7054	73	30	in	in	ADP
ejpam-7054	73	31	a.	a.	NOUN
ejpam-7054	73	32	then	then	ADV
ejpam-7054	73	33	,	,	PUNCT
ejpam-7054	73	34	we	we	PRON
ejpam-7054	73	35	have	have	VERB
ejpam-7054	73	36	iµ(iµ(a	iµ(iµ(a	ADJ
ejpam-7054	73	37	)	)	PUNCT
ejpam-7054	73	38	)	)	PUNCT
ejpam-7054	74	1	=	=	SYM
ejpam-7054	74	2	iµ(a	iµ(a	ADJ
ejpam-7054	74	3	)	)	PUNCT
ejpam-7054	74	4	,	,	PUNCT
ejpam-7054	74	5	cµ(cµ(a	cµ(cµ(a	PROPN
ejpam-7054	74	6	)	)	PUNCT
ejpam-7054	74	7	)	)	PUNCT
ejpam-7054	75	1	=	=	SYM
ejpam-7054	75	2	cµ(a	cµ(a	ADJ
ejpam-7054	75	3	)	)	PUNCT
ejpam-7054	75	4	,	,	PUNCT
ejpam-7054	75	5	and	and	CCONJ
ejpam-7054	75	6	iµ(a	iµ(a	ADJ
ejpam-7054	75	7	)	)	PUNCT
ejpam-7054	75	8	=	=	SYM
ejpam-7054	76	1	x	x	SYM
ejpam-7054	76	2	−	−	PROPN
ejpam-7054	76	3	cµ(x	cµ(x	SYM
ejpam-7054	76	4	−a	−a	NOUN
ejpam-7054	76	5	)	)	PUNCT
ejpam-7054	76	6	.	.	PUNCT
ejpam-7054	77	1	according	accord	VERB
ejpam-7054	77	2	to	to	ADP
ejpam-7054	77	3	[	[	X
ejpam-7054	77	4	23	23	NUM
ejpam-7054	77	5	]	]	PUNCT
ejpam-7054	77	6	,	,	PUNCT
ejpam-7054	77	7	for	for	ADP
ejpam-7054	77	8	a	a	DET
ejpam-7054	77	9	⊆	⊆	NUM
ejpam-7054	77	10	x	x	SYM
ejpam-7054	77	11	and	and	CCONJ
ejpam-7054	77	12	x	x	SYM
ejpam-7054	77	13	∈	∈	NOUN
ejpam-7054	77	14	x	x	X
ejpam-7054	77	15	,	,	PUNCT
ejpam-7054	77	16	we	we	PRON
ejpam-7054	77	17	have	have	VERB
ejpam-7054	77	18	x	x	X
ejpam-7054	77	19	∈	∈	PROPN
ejpam-7054	77	20	cµ(a	cµ(a	NOUN
ejpam-7054	77	21	)	)	PUNCT
ejpam-7054	78	1	if	if	SCONJ
ejpam-7054	78	2	and	and	CCONJ
ejpam-7054	78	3	only	only	ADV
ejpam-7054	78	4	if	if	SCONJ
ejpam-7054	78	5	x	x	PROPN
ejpam-7054	78	6	∈	∈	PROPN
ejpam-7054	78	7	m	m	PROPN
ejpam-7054	78	8	∈	∈	NOUN
ejpam-7054	78	9	µ	µ	NOUN
ejpam-7054	78	10	implies	imply	VERB
ejpam-7054	78	11	m	m	VERB
ejpam-7054	78	12	∩a	∩a	PROPN
ejpam-7054	78	13	̸=	̸=	PROPN
ejpam-7054	78	14	∅.	∅.	ADP
ejpam-7054	78	15	a	a	DET
ejpam-7054	78	16	subset	subset	NOUN
ejpam-7054	78	17	a	a	PRON
ejpam-7054	78	18	of	of	ADP
ejpam-7054	78	19	a	a	DET
ejpam-7054	78	20	generalized	generalized	ADJ
ejpam-7054	78	21	topological	topological	ADJ
ejpam-7054	78	22	space	space	NOUN
ejpam-7054	78	23	(	(	PUNCT
ejpam-7054	78	24	x,µ	x,µ	NOUN
ejpam-7054	78	25	)	)	PUNCT
ejpam-7054	78	26	is	be	AUX
ejpam-7054	78	27	called	call	VERB
ejpam-7054	78	28	µ-preopen	µ-preopen	PROPN
ejpam-7054	78	29	[	[	X
ejpam-7054	78	30	24	24	NUM
ejpam-7054	78	31	]	]	X
ejpam-7054	78	32	if	if	SCONJ
ejpam-7054	78	33	a	a	DET
ejpam-7054	78	34	⊆	⊆	NUM
ejpam-7054	78	35	iµ(cµ(a	iµ(cµ(a	NOUN
ejpam-7054	78	36	)	)	PUNCT
ejpam-7054	78	37	)	)	PUNCT
ejpam-7054	78	38	.	.	PUNCT
ejpam-7054	79	1	the	the	DET
ejpam-7054	79	2	complement	complement	NOUN
ejpam-7054	79	3	of	of	ADP
ejpam-7054	79	4	a	a	DET
ejpam-7054	79	5	µ-preopen	µ-preopen	PROPN
ejpam-7054	79	6	set	set	NOUN
ejpam-7054	79	7	is	be	AUX
ejpam-7054	79	8	called	call	VERB
ejpam-7054	79	9	µ-preclosed	µ-preclose	VERB
ejpam-7054	79	10	.	.	PUNCT
ejpam-7054	80	1	for	for	ADP
ejpam-7054	80	2	a	a	DET
ejpam-7054	80	3	generalized	generalized	ADJ
ejpam-7054	80	4	topological	topological	ADJ
ejpam-7054	80	5	space	space	NOUN
ejpam-7054	80	6	(	(	PUNCT
ejpam-7054	80	7	x,µ	x,µ	NOUN
ejpam-7054	80	8	)	)	PUNCT
ejpam-7054	80	9	,	,	PUNCT
ejpam-7054	80	10	we	we	PRON
ejpam-7054	80	11	will	will	AUX
ejpam-7054	80	12	denote	denote	VERB
ejpam-7054	80	13	the	the	DET
ejpam-7054	80	14	class	class	NOUN
ejpam-7054	80	15	of	of	ADP
ejpam-7054	80	16	µ-preopen	µ-preopen	PROPN
ejpam-7054	80	17	sets	set	NOUN
ejpam-7054	80	18	by	by	ADP
ejpam-7054	80	19	µ(π	µ(π	NOUN
ejpam-7054	80	20	)	)	PUNCT
ejpam-7054	80	21	.	.	PUNCT
ejpam-7054	81	1	let	let	VERB
ejpam-7054	81	2	a	a	DET
ejpam-7054	81	3	be	be	AUX
ejpam-7054	81	4	a	a	DET
ejpam-7054	81	5	subset	subset	NOUN
ejpam-7054	81	6	of	of	ADP
ejpam-7054	81	7	a	a	DET
ejpam-7054	81	8	generalized	generalized	ADJ
ejpam-7054	81	9	topological	topological	ADJ
ejpam-7054	81	10	space	space	NOUN
ejpam-7054	81	11	(	(	PUNCT
ejpam-7054	81	12	x,µ	x,µ	NOUN
ejpam-7054	81	13	)	)	PUNCT
ejpam-7054	81	14	.	.	PUNCT
ejpam-7054	82	1	the	the	DET
ejpam-7054	82	2	intersection	intersection	NOUN
ejpam-7054	82	3	of	of	ADP
ejpam-7054	82	4	all	all	DET
ejpam-7054	82	5	µ-preclosed	µ-preclose	VERB
ejpam-7054	82	6	sets	set	NOUN
ejpam-7054	82	7	of	of	ADP
ejpam-7054	82	8	x	x	PUNCT
ejpam-7054	82	9	containing	contain	VERB
ejpam-7054	82	10	a	a	PRON
ejpam-7054	82	11	is	be	AUX
ejpam-7054	82	12	called	call	VERB
ejpam-7054	82	13	the	the	DET
ejpam-7054	82	14	µ-perclosure	µ-perclosure	NOUN
ejpam-7054	82	15	of	of	ADP
ejpam-7054	82	16	a	a	PRON
ejpam-7054	82	17	and	and	CCONJ
ejpam-7054	82	18	is	be	AUX
ejpam-7054	82	19	denoted	denote	VERB
ejpam-7054	82	20	by	by	ADP
ejpam-7054	82	21	cµ(π)(a	cµ(π)(a	PROPN
ejpam-7054	82	22	)	)	PUNCT
ejpam-7054	82	23	.	.	PUNCT
ejpam-7054	83	1	the	the	DET
ejpam-7054	83	2	union	union	NOUN
ejpam-7054	83	3	of	of	ADP
ejpam-7054	83	4	all	all	DET
ejpam-7054	83	5	µ-preopen	µ-preopen	PROPN
ejpam-7054	83	6	sets	set	NOUN
ejpam-7054	83	7	of	of	ADP
ejpam-7054	83	8	x	x	PUNCT
ejpam-7054	83	9	contained	contain	VERB
ejpam-7054	83	10	in	in	ADP
ejpam-7054	83	11	a	a	PRON
ejpam-7054	83	12	is	be	AUX
ejpam-7054	83	13	called	call	VERB
ejpam-7054	83	14	the	the	DET
ejpam-7054	83	15	µ-preinterior	µ-preinterior	NOUN
ejpam-7054	83	16	of	of	ADP
ejpam-7054	83	17	a	a	PRON
ejpam-7054	83	18	and	and	CCONJ
ejpam-7054	83	19	is	be	AUX
ejpam-7054	83	20	denoted	denote	VERB
ejpam-7054	83	21	by	by	ADP
ejpam-7054	83	22	iµ(π)(a	iµ(π)(a	NUM
ejpam-7054	83	23	)	)	PUNCT
ejpam-7054	83	24	.	.	PUNCT
ejpam-7054	84	1	lemma	lemma	PROPN
ejpam-7054	84	2	2	2	X
ejpam-7054	84	3	.	.	PUNCT
ejpam-7054	84	4	let	let	VERB
ejpam-7054	84	5	a	a	DET
ejpam-7054	84	6	be	be	AUX
ejpam-7054	84	7	a	a	DET
ejpam-7054	84	8	subset	subset	NOUN
ejpam-7054	84	9	of	of	ADP
ejpam-7054	84	10	a	a	DET
ejpam-7054	84	11	generalized	generalized	ADJ
ejpam-7054	84	12	topological	topological	ADJ
ejpam-7054	84	13	space	space	NOUN
ejpam-7054	84	14	(	(	PUNCT
ejpam-7054	84	15	x,µ	x,µ	NOUN
ejpam-7054	84	16	)	)	PUNCT
ejpam-7054	84	17	and	and	CCONJ
ejpam-7054	84	18	x	x	PUNCT
ejpam-7054	84	19	∈	∈	PROPN
ejpam-7054	84	20	x.	x.	NOUN
ejpam-7054	84	21	then	then	ADV
ejpam-7054	84	22	,	,	PUNCT
ejpam-7054	84	23	the	the	DET
ejpam-7054	84	24	following	follow	VERB
ejpam-7054	84	25	properties	property	NOUN
ejpam-7054	84	26	hold	hold	VERB
ejpam-7054	84	27	:	:	PUNCT
ejpam-7054	84	28	(	(	PUNCT
ejpam-7054	84	29	1	1	X
ejpam-7054	84	30	)	)	PUNCT
ejpam-7054	84	31	x	x	SYM
ejpam-7054	84	32	∈	∈	NOUN
ejpam-7054	84	33	cµ(π	cµ(π	NOUN
ejpam-7054	84	34	)	)	PUNCT
ejpam-7054	85	1	if	if	SCONJ
ejpam-7054	85	2	and	and	CCONJ
ejpam-7054	85	3	only	only	ADV
ejpam-7054	85	4	if	if	SCONJ
ejpam-7054	85	5	u	u	PROPN
ejpam-7054	85	6	∩a	∩a	PROPN
ejpam-7054	85	7	̸=	̸=	PROPN
ejpam-7054	85	8	∅	∅	NOUN
ejpam-7054	85	9	for	for	ADP
ejpam-7054	85	10	every	every	DET
ejpam-7054	85	11	µ-peropen	µ-peropen	NOUN
ejpam-7054	85	12	set	set	VERB
ejpam-7054	85	13	u	u	NOUN
ejpam-7054	85	14	of	of	ADP
ejpam-7054	85	15	x	x	PUNCT
ejpam-7054	85	16	containing	contain	VERB
ejpam-7054	85	17	x	x	PRON
ejpam-7054	85	18	;	;	PUNCT
ejpam-7054	85	19	(	(	PUNCT
ejpam-7054	85	20	2	2	X
ejpam-7054	85	21	)	)	PUNCT
ejpam-7054	85	22	a	a	PRON
ejpam-7054	85	23	is	be	AUX
ejpam-7054	85	24	µ-preclosed	µ-preclose	VERB
ejpam-7054	85	25	if	if	SCONJ
ejpam-7054	85	26	and	and	CCONJ
ejpam-7054	85	27	only	only	ADV
ejpam-7054	85	28	if	if	SCONJ
ejpam-7054	85	29	a	a	PRON
ejpam-7054	85	30	=	=	NOUN
ejpam-7054	85	31	cµ(π)(a	cµ(π)(a	NOUN
ejpam-7054	85	32	)	)	PUNCT
ejpam-7054	85	33	;	;	PUNCT
ejpam-7054	85	34	(	(	PUNCT
ejpam-7054	85	35	3	3	X
ejpam-7054	85	36	)	)	PUNCT
ejpam-7054	85	37	iµ(π)(x	iµ(π)(x	NOUN
ejpam-7054	85	38	−a	−a	NOUN
ejpam-7054	85	39	)	)	PUNCT
ejpam-7054	86	1	=	=	PUNCT
ejpam-7054	86	2	x	x	X
ejpam-7054	87	1	−	−	NOUN
ejpam-7054	87	2	cµ(π)(a	cµ(π)(a	NOUN
ejpam-7054	87	3	)	)	PUNCT
ejpam-7054	87	4	;	;	PUNCT
ejpam-7054	87	5	(	(	PUNCT
ejpam-7054	87	6	4	4	X
ejpam-7054	87	7	)	)	PUNCT
ejpam-7054	87	8	cµ(π)(x	cµ(π)(x	NOUN
ejpam-7054	87	9	−a	−a	NOUN
ejpam-7054	87	10	)	)	PUNCT
ejpam-7054	87	11	=	=	PUNCT
ejpam-7054	88	1	x	x	X
ejpam-7054	88	2	−	−	PROPN
ejpam-7054	88	3	iµ(π)(a	iµ(π)(a	NUM
ejpam-7054	88	4	)	)	PUNCT
ejpam-7054	88	5	.	.	PUNCT
ejpam-7054	89	1	by	by	ADP
ejpam-7054	89	2	a	a	DET
ejpam-7054	89	3	multifunction	multifunction	NOUN
ejpam-7054	89	4	f	f	NOUN
ejpam-7054	89	5	:	:	PUNCT
ejpam-7054	89	6	x	x	X
ejpam-7054	89	7	→	→	SYM
ejpam-7054	89	8	y	y	PROPN
ejpam-7054	89	9	,	,	PUNCT
ejpam-7054	89	10	we	we	PRON
ejpam-7054	89	11	mean	mean	VERB
ejpam-7054	89	12	a	a	DET
ejpam-7054	89	13	point	point	NOUN
ejpam-7054	89	14	-	-	PUNCT
ejpam-7054	89	15	to	to	ADP
ejpam-7054	89	16	-	-	PUNCT
ejpam-7054	89	17	set	set	VERB
ejpam-7054	89	18	correspondence	correspondence	NOUN
ejpam-7054	89	19	from	from	ADP
ejpam-7054	89	20	x	x	PUNCT
ejpam-7054	89	21	into	into	ADP
ejpam-7054	89	22	y	y	PROPN
ejpam-7054	89	23	,	,	PUNCT
ejpam-7054	89	24	and	and	CCONJ
ejpam-7054	89	25	always	always	ADV
ejpam-7054	89	26	assume	assume	VERB
ejpam-7054	89	27	that	that	SCONJ
ejpam-7054	90	1	f	f	PROPN
ejpam-7054	90	2	(	(	PUNCT
ejpam-7054	90	3	x	x	X
ejpam-7054	90	4	)	)	PUNCT
ejpam-7054	90	5	̸=	̸=	NOUN
ejpam-7054	90	6	∅	∅	NOUN
ejpam-7054	90	7	for	for	ADP
ejpam-7054	90	8	all	all	PRON
ejpam-7054	90	9	x	x	SYM
ejpam-7054	90	10	∈	∈	ADJ
ejpam-7054	90	11	x.	x.	NOUN
ejpam-7054	90	12	for	for	ADP
ejpam-7054	90	13	a	a	DET
ejpam-7054	90	14	multifunction	multifunction	NOUN
ejpam-7054	90	15	f	f	NOUN
ejpam-7054	90	16	:	:	PUNCT
ejpam-7054	90	17	x	x	X
ejpam-7054	90	18	→	→	SYM
ejpam-7054	90	19	y	y	PROPN
ejpam-7054	90	20	,	,	PUNCT
ejpam-7054	90	21	we	we	PRON
ejpam-7054	90	22	shall	shall	AUX
ejpam-7054	90	23	denote	denote	VERB
ejpam-7054	90	24	the	the	DET
ejpam-7054	90	25	upper	upper	ADJ
ejpam-7054	90	26	and	and	CCONJ
ejpam-7054	90	27	lower	low	ADJ
ejpam-7054	90	28	inverse	inverse	NOUN
ejpam-7054	90	29	of	of	ADP
ejpam-7054	90	30	a	a	DET
ejpam-7054	90	31	set	set	NOUN
ejpam-7054	90	32	b	b	PROPN
ejpam-7054	90	33	of	of	ADP
ejpam-7054	90	34	y	y	PROPN
ejpam-7054	90	35	by	by	ADP
ejpam-7054	90	36	f+(b	f+(b	NOUN
ejpam-7054	90	37	)	)	PUNCT
ejpam-7054	90	38	and	and	CCONJ
ejpam-7054	90	39	f−(b	f−(b	NOUN
ejpam-7054	90	40	)	)	PUNCT
ejpam-7054	90	41	,	,	PUNCT
ejpam-7054	90	42	respectively	respectively	ADV
ejpam-7054	90	43	,	,	PUNCT
ejpam-7054	90	44	that	that	ADV
ejpam-7054	90	45	is	is	ADV
ejpam-7054	90	46	,	,	PUNCT
ejpam-7054	90	47	f+(b	f+(b	NOUN
ejpam-7054	90	48	)	)	PUNCT
ejpam-7054	90	49	=	=	PRON
ejpam-7054	91	1	{	{	PUNCT
ejpam-7054	91	2	x	x	PUNCT
ejpam-7054	91	3	∈	∈	PROPN
ejpam-7054	91	4	x	x	INTJ
ejpam-7054	92	1	|	|	NOUN
ejpam-7054	92	2	f	f	X
ejpam-7054	92	3	(	(	PUNCT
ejpam-7054	92	4	x	x	NOUN
ejpam-7054	92	5	)	)	PUNCT
ejpam-7054	92	6	⊆	⊆	NUM
ejpam-7054	92	7	b	b	NOUN
ejpam-7054	92	8	}	}	PUNCT
ejpam-7054	92	9	and	and	CCONJ
ejpam-7054	92	10	f−(b	f−(b	PROPN
ejpam-7054	92	11	)	)	PUNCT
ejpam-7054	92	12	=	=	PRON
ejpam-7054	93	1	{	{	PUNCT
ejpam-7054	93	2	x	x	PUNCT
ejpam-7054	93	3	∈	∈	PROPN
ejpam-7054	93	4	x	x	INTJ
ejpam-7054	94	1	|	|	NOUN
ejpam-7054	94	2	f	f	X
ejpam-7054	94	3	(	(	PUNCT
ejpam-7054	94	4	x	x	NOUN
ejpam-7054	94	5	)	)	PUNCT
ejpam-7054	94	6	∩b	∩b	NOUN
ejpam-7054	94	7	̸=	̸=	PROPN
ejpam-7054	94	8	∅	∅	NOUN
ejpam-7054	94	9	}	}	PUNCT
ejpam-7054	94	10	.	.	PUNCT
ejpam-7054	95	1	b.	b.	PROPN
ejpam-7054	95	2	kong	kong	PROPN
ejpam-7054	95	3	-	-	PUNCT
ejpam-7054	95	4	ied	ied	PROPN
ejpam-7054	95	5	,	,	PUNCT
ejpam-7054	95	6	a.	a.	PROPN
ejpam-7054	95	7	sama	sama	PROPN
ejpam-7054	95	8	-	-	PUNCT
ejpam-7054	95	9	ae	ae	PROPN
ejpam-7054	95	10	,	,	PUNCT
ejpam-7054	95	11	c.	c.	PROPN
ejpam-7054	95	12	boonpok	boonpok	PROPN
ejpam-7054	95	13	/	/	SYM
ejpam-7054	95	14	eur	eur	PROPN
ejpam-7054	95	15	.	.	PUNCT
ejpam-7054	96	1	j.	j.	PROPN
ejpam-7054	96	2	pure	pure	PROPN
ejpam-7054	96	3	appl	appl	PROPN
ejpam-7054	96	4	.	.	PROPN
ejpam-7054	96	5	math	math	PROPN
ejpam-7054	96	6	,	,	PUNCT
ejpam-7054	96	7	18	18	NUM
ejpam-7054	96	8	(	(	PUNCT
ejpam-7054	96	9	4	4	NUM
ejpam-7054	96	10	)	)	PUNCT
ejpam-7054	96	11	(	(	PUNCT
ejpam-7054	96	12	2025	2025	NUM
ejpam-7054	96	13	)	)	PUNCT
ejpam-7054	96	14	,	,	PUNCT
ejpam-7054	96	15	7054	7054	NUM
ejpam-7054	96	16	4	4	NUM
ejpam-7054	96	17	of	of	ADP
ejpam-7054	96	18	12	12	NUM
ejpam-7054	96	19	3	3	NUM
ejpam-7054	96	20	.	.	PUNCT
ejpam-7054	96	21	upper	upper	ADJ
ejpam-7054	96	22	and	and	CCONJ
ejpam-7054	96	23	lower	low	ADJ
ejpam-7054	96	24	almost	almost	ADV
ejpam-7054	96	25	weakly	weakly	ADJ
ejpam-7054	96	26	µ(σ1	µ(σ1	NOUN
ejpam-7054	96	27	,	,	PUNCT
ejpam-7054	96	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	96	29	multifunctions	multifunction	NOUN
ejpam-7054	96	30	in	in	ADP
ejpam-7054	96	31	this	this	DET
ejpam-7054	96	32	section	section	NOUN
ejpam-7054	96	33	,	,	PUNCT
ejpam-7054	96	34	we	we	PRON
ejpam-7054	96	35	introduce	introduce	VERB
ejpam-7054	96	36	the	the	DET
ejpam-7054	96	37	notions	notion	NOUN
ejpam-7054	96	38	of	of	ADP
ejpam-7054	96	39	upper	upper	ADJ
ejpam-7054	96	40	almost	almost	ADV
ejpam-7054	96	41	weakly	weakly	ADJ
ejpam-7054	96	42	µ(σ1	µ(σ1	NOUN
ejpam-7054	96	43	,	,	PUNCT
ejpam-7054	96	44	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	96	45	multifunctions	multifunction	NOUN
ejpam-7054	96	46	and	and	CCONJ
ejpam-7054	96	47	lower	low	ADJ
ejpam-7054	96	48	almost	almost	ADV
ejpam-7054	96	49	weakly	weakly	ADJ
ejpam-7054	96	50	µ(σ1	µ(σ1	NOUN
ejpam-7054	96	51	,	,	PUNCT
ejpam-7054	96	52	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	96	53	multifunctions	multifunction	NOUN
ejpam-7054	96	54	.	.	PUNCT
ejpam-7054	97	1	moreover	moreover	ADV
ejpam-7054	97	2	,	,	PUNCT
ejpam-7054	97	3	several	several	ADJ
ejpam-7054	97	4	characterizations	characterization	NOUN
ejpam-7054	97	5	of	of	ADP
ejpam-7054	97	6	upper	upper	ADJ
ejpam-7054	97	7	almost	almost	ADV
ejpam-7054	97	8	weakly	weakly	ADJ
ejpam-7054	97	9	µ(σ1	µ(σ1	NOUN
ejpam-7054	97	10	,	,	PUNCT
ejpam-7054	97	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	97	12	multifunctions	multifunction	NOUN
ejpam-7054	97	13	and	and	CCONJ
ejpam-7054	97	14	lower	low	ADJ
ejpam-7054	97	15	almost	almost	ADV
ejpam-7054	97	16	weakly	weakly	ADJ
ejpam-7054	97	17	µ(σ1	µ(σ1	NOUN
ejpam-7054	97	18	,	,	PUNCT
ejpam-7054	97	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	97	20	multifunctions	multifunction	NOUN
ejpam-7054	97	21	are	be	AUX
ejpam-7054	97	22	discussed	discuss	VERB
ejpam-7054	97	23	.	.	PUNCT
ejpam-7054	98	1	definition	definition	NOUN
ejpam-7054	98	2	1	1	NUM
ejpam-7054	98	3	.	.	PUNCT
ejpam-7054	99	1	a	a	DET
ejpam-7054	99	2	multifunction	multifunction	NOUN
ejpam-7054	99	3	f	f	NOUN
ejpam-7054	99	4	:	:	PUNCT
ejpam-7054	99	5	(	(	PUNCT
ejpam-7054	99	6	x,µ	x,µ	NOUN
ejpam-7054	99	7	)	)	PUNCT
ejpam-7054	99	8	→	→	SYM
ejpam-7054	99	9	(	(	PUNCT
ejpam-7054	99	10	y	y	PROPN
ejpam-7054	99	11	,	,	PUNCT
ejpam-7054	99	12	σ1	σ1	PROPN
ejpam-7054	99	13	,	,	PUNCT
ejpam-7054	99	14	σ2	σ2	PROPN
ejpam-7054	99	15	)	)	PUNCT
ejpam-7054	99	16	is	be	AUX
ejpam-7054	99	17	said	say	VERB
ejpam-7054	99	18	to	to	PART
ejpam-7054	99	19	be	be	AUX
ejpam-7054	99	20	upper	upper	ADJ
ejpam-7054	99	21	almost	almost	ADV
ejpam-7054	99	22	weakly	weakly	ADJ
ejpam-7054	99	23	µ(σ1	µ(σ1	NOUN
ejpam-7054	99	24	,	,	PUNCT
ejpam-7054	99	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	99	26	if	if	SCONJ
ejpam-7054	99	27	for	for	ADP
ejpam-7054	99	28	each	each	DET
ejpam-7054	99	29	x	x	SYM
ejpam-7054	99	30	∈	∈	PROPN
ejpam-7054	99	31	x	x	X
ejpam-7054	99	32	and	and	CCONJ
ejpam-7054	99	33	each	each	DET
ejpam-7054	99	34	σ1σ2	σ1σ2	VERB
ejpam-7054	99	35	-	-	ADJ
ejpam-7054	99	36	open	open	ADJ
ejpam-7054	99	37	set	set	NOUN
ejpam-7054	99	38	v	v	NOUN
ejpam-7054	99	39	of	of	ADP
ejpam-7054	99	40	y	y	PRON
ejpam-7054	99	41	such	such	ADJ
ejpam-7054	99	42	that	that	SCONJ
ejpam-7054	99	43	f	f	PROPN
ejpam-7054	99	44	(	(	PUNCT
ejpam-7054	99	45	x	x	X
ejpam-7054	99	46	)	)	PUNCT
ejpam-7054	99	47	⊆	⊆	NUM
ejpam-7054	99	48	v	v	NOUN
ejpam-7054	99	49	,	,	PUNCT
ejpam-7054	99	50	x	x	SYM
ejpam-7054	99	51	∈	∈	NOUN
ejpam-7054	99	52	iµ(cµ(f	iµ(cµ(f	PUNCT
ejpam-7054	100	1	+	+	ADJ
ejpam-7054	100	2	(	(	PUNCT
ejpam-7054	100	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	100	4	-	-	NUM
ejpam-7054	100	5	cl(v	cl(v	NOUN
ejpam-7054	100	6	)	)	PUNCT
ejpam-7054	100	7	)	)	PUNCT
ejpam-7054	100	8	)	)	PUNCT
ejpam-7054	100	9	)	)	PUNCT
ejpam-7054	100	10	.	.	PUNCT
ejpam-7054	101	1	theorem	theorem	NOUN
ejpam-7054	101	2	1	1	NUM
ejpam-7054	101	3	.	.	X
ejpam-7054	101	4	for	for	ADP
ejpam-7054	101	5	a	a	DET
ejpam-7054	101	6	multifunction	multifunction	NOUN
ejpam-7054	102	1	f	f	NOUN
ejpam-7054	102	2	:	:	PUNCT
ejpam-7054	102	3	(	(	PUNCT
ejpam-7054	102	4	x,µ	x,µ	NOUN
ejpam-7054	102	5	)	)	PUNCT
ejpam-7054	102	6	→	→	SYM
ejpam-7054	102	7	(	(	PUNCT
ejpam-7054	102	8	y	y	PROPN
ejpam-7054	102	9	,	,	PUNCT
ejpam-7054	102	10	σ1	σ1	PROPN
ejpam-7054	102	11	,	,	PUNCT
ejpam-7054	102	12	σ2	σ2	NOUN
ejpam-7054	102	13	)	)	PUNCT
ejpam-7054	102	14	,	,	PUNCT
ejpam-7054	102	15	the	the	DET
ejpam-7054	102	16	following	follow	VERB
ejpam-7054	102	17	properties	property	NOUN
ejpam-7054	102	18	are	be	AUX
ejpam-7054	102	19	equivalent	equivalent	ADJ
ejpam-7054	102	20	:	:	PUNCT
ejpam-7054	102	21	(	(	PUNCT
ejpam-7054	102	22	1	1	X
ejpam-7054	102	23	)	)	PUNCT
ejpam-7054	102	24	f	f	PROPN
ejpam-7054	102	25	is	be	AUX
ejpam-7054	102	26	upper	upper	ADJ
ejpam-7054	102	27	almost	almost	ADV
ejpam-7054	102	28	weakly	weakly	ADJ
ejpam-7054	102	29	µ(σ1	µ(σ1	NOUN
ejpam-7054	102	30	,	,	PUNCT
ejpam-7054	102	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	102	32	;	;	PUNCT
ejpam-7054	102	33	(	(	PUNCT
ejpam-7054	102	34	2	2	NUM
ejpam-7054	102	35	)	)	PUNCT
ejpam-7054	102	36	f+(v	f+(v	NOUN
ejpam-7054	102	37	)	)	PUNCT
ejpam-7054	103	1	⊆	⊆	X
ejpam-7054	103	2	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	104	1	+	+	ADJ
ejpam-7054	104	2	(	(	PUNCT
ejpam-7054	104	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	104	4	-	-	NUM
ejpam-7054	104	5	cl(v	cl(v	NOUN
ejpam-7054	104	6	)	)	PUNCT
ejpam-7054	104	7	)	)	PUNCT
ejpam-7054	104	8	)	)	PUNCT
ejpam-7054	105	1	for	for	ADP
ejpam-7054	105	2	every	every	DET
ejpam-7054	105	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	105	4	-	-	ADJ
ejpam-7054	105	5	open	open	ADJ
ejpam-7054	105	6	set	set	NOUN
ejpam-7054	105	7	v	v	NOUN
ejpam-7054	105	8	of	of	ADP
ejpam-7054	105	9	y	y	PROPN
ejpam-7054	105	10	;	;	PUNCT
ejpam-7054	105	11	(	(	PUNCT
ejpam-7054	105	12	3	3	X
ejpam-7054	105	13	)	)	PUNCT
ejpam-7054	105	14	cµ(π)(f	cµ(π)(f	NOUN
ejpam-7054	105	15	−(v	−(v	NOUN
ejpam-7054	105	16	)	)	PUNCT
ejpam-7054	105	17	)	)	PUNCT
ejpam-7054	106	1	⊆	⊆	X
ejpam-7054	106	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	106	3	-	-	PUNCT
ejpam-7054	106	4	cl(v	cl(v	NOUN
ejpam-7054	106	5	)	)	PUNCT
ejpam-7054	106	6	)	)	PUNCT
ejpam-7054	106	7	for	for	ADP
ejpam-7054	106	8	every	every	DET
ejpam-7054	106	9	σ1σ2	σ1σ2	NOUN
ejpam-7054	106	10	-	-	ADJ
ejpam-7054	106	11	open	open	ADJ
ejpam-7054	106	12	set	set	NOUN
ejpam-7054	106	13	v	v	NOUN
ejpam-7054	106	14	of	of	ADP
ejpam-7054	106	15	y	y	PROPN
ejpam-7054	106	16	;	;	PUNCT
ejpam-7054	106	17	(	(	PUNCT
ejpam-7054	106	18	4	4	X
ejpam-7054	106	19	)	)	PUNCT
ejpam-7054	106	20	for	for	ADP
ejpam-7054	106	21	each	each	DET
ejpam-7054	106	22	x	x	SYM
ejpam-7054	106	23	∈	∈	PROPN
ejpam-7054	106	24	x	x	X
ejpam-7054	106	25	and	and	CCONJ
ejpam-7054	106	26	each	each	DET
ejpam-7054	106	27	σ1σ2	σ1σ2	VERB
ejpam-7054	106	28	-	-	ADJ
ejpam-7054	106	29	open	open	ADJ
ejpam-7054	106	30	set	set	NOUN
ejpam-7054	106	31	v	v	NOUN
ejpam-7054	106	32	of	of	ADP
ejpam-7054	106	33	y	y	PROPN
ejpam-7054	106	34	containing	contain	VERB
ejpam-7054	106	35	f	f	PROPN
ejpam-7054	106	36	(	(	PUNCT
ejpam-7054	106	37	x	x	NOUN
ejpam-7054	106	38	)	)	PUNCT
ejpam-7054	106	39	,	,	PUNCT
ejpam-7054	106	40	there	there	PRON
ejpam-7054	106	41	exists	exist	VERB
ejpam-7054	106	42	a	a	DET
ejpam-7054	106	43	µ-preopen	µ-preopen	PROPN
ejpam-7054	106	44	set	set	VERB
ejpam-7054	106	45	u	u	NOUN
ejpam-7054	106	46	of	of	ADP
ejpam-7054	106	47	x	x	PUNCT
ejpam-7054	106	48	containing	contain	VERB
ejpam-7054	106	49	x	x	PUNCT
ejpam-7054	106	50	such	such	ADJ
ejpam-7054	106	51	that	that	SCONJ
ejpam-7054	106	52	f	f	PROPN
ejpam-7054	106	53	(	(	PUNCT
ejpam-7054	106	54	u	u	NOUN
ejpam-7054	106	55	)	)	PUNCT
ejpam-7054	106	56	⊆	⊆	NUM
ejpam-7054	106	57	σ1σ2	σ1σ2	NOUN
ejpam-7054	106	58	-	-	NUM
ejpam-7054	106	59	cl(v	cl(v	NOUN
ejpam-7054	106	60	)	)	PUNCT
ejpam-7054	106	61	.	.	PUNCT
ejpam-7054	107	1	proof	proof	NOUN
ejpam-7054	107	2	.	.	PUNCT
ejpam-7054	108	1	(	(	PUNCT
ejpam-7054	108	2	1	1	X
ejpam-7054	108	3	)	)	PUNCT
ejpam-7054	108	4	⇒	⇒	NOUN
ejpam-7054	108	5	(	(	PUNCT
ejpam-7054	108	6	2	2	NUM
ejpam-7054	108	7	):	):	PUNCT
ejpam-7054	108	8	let	let	VERB
ejpam-7054	108	9	v	v	PART
ejpam-7054	108	10	be	be	AUX
ejpam-7054	108	11	any	any	DET
ejpam-7054	108	12	σ1σ2	σ1σ2	NOUN
ejpam-7054	108	13	-	-	ADJ
ejpam-7054	108	14	open	open	ADJ
ejpam-7054	108	15	set	set	NOUN
ejpam-7054	108	16	of	of	ADP
ejpam-7054	108	17	y	y	PROPN
ejpam-7054	108	18	and	and	CCONJ
ejpam-7054	108	19	x	x	PROPN
ejpam-7054	108	20	∈	∈	PROPN
ejpam-7054	108	21	f+(v	f+(v	NOUN
ejpam-7054	108	22	)	)	PUNCT
ejpam-7054	108	23	.	.	PUNCT
ejpam-7054	109	1	then	then	ADV
ejpam-7054	109	2	,	,	PUNCT
ejpam-7054	109	3	f	f	PROPN
ejpam-7054	109	4	(	(	PUNCT
ejpam-7054	109	5	x	x	X
ejpam-7054	109	6	)	)	PUNCT
ejpam-7054	109	7	⊆	⊆	NUM
ejpam-7054	109	8	v	v	NOUN
ejpam-7054	109	9	and	and	CCONJ
ejpam-7054	109	10	by	by	ADP
ejpam-7054	109	11	(	(	PUNCT
ejpam-7054	109	12	1	1	NUM
ejpam-7054	109	13	)	)	PUNCT
ejpam-7054	109	14	,	,	PUNCT
ejpam-7054	109	15	we	we	PRON
ejpam-7054	109	16	have	have	VERB
ejpam-7054	109	17	x	x	PART
ejpam-7054	109	18	∈	∈	PROPN
ejpam-7054	109	19	iµ(cµ(f	iµ(cµ(f	PROPN
ejpam-7054	110	1	+	+	ADJ
ejpam-7054	110	2	(	(	PUNCT
ejpam-7054	110	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	110	4	-	-	NUM
ejpam-7054	110	5	cl(v	cl(v	NOUN
ejpam-7054	110	6	)	)	PUNCT
ejpam-7054	110	7	)	)	PUNCT
ejpam-7054	110	8	)	)	PUNCT
ejpam-7054	110	9	)	)	PUNCT
ejpam-7054	111	1	and	and	CCONJ
ejpam-7054	111	2	so	so	ADV
ejpam-7054	111	3	x	x	SYM
ejpam-7054	111	4	∈	∈	NOUN
ejpam-7054	111	5	iµ(π)(f	iµ(π)(f	VERB
ejpam-7054	112	1	+	+	ADJ
ejpam-7054	112	2	(	(	PUNCT
ejpam-7054	112	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	112	4	-	-	NUM
ejpam-7054	112	5	cl(v	cl(v	NOUN
ejpam-7054	112	6	)	)	PUNCT
ejpam-7054	112	7	)	)	PUNCT
ejpam-7054	112	8	)	)	PUNCT
ejpam-7054	112	9	.	.	PUNCT
ejpam-7054	113	1	thus	thus	ADV
ejpam-7054	113	2	,	,	PUNCT
ejpam-7054	113	3	f+(v	f+(v	PROPN
ejpam-7054	113	4	)	)	PUNCT
ejpam-7054	113	5	⊆	⊆	X
ejpam-7054	113	6	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	114	1	+	+	ADJ
ejpam-7054	114	2	(	(	PUNCT
ejpam-7054	114	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	114	4	-	-	NUM
ejpam-7054	114	5	cl(v	cl(v	NOUN
ejpam-7054	114	6	)	)	PUNCT
ejpam-7054	114	7	)	)	PUNCT
ejpam-7054	114	8	)	)	PUNCT
ejpam-7054	114	9	.	.	PUNCT
ejpam-7054	115	1	(	(	PUNCT
ejpam-7054	115	2	2	2	X
ejpam-7054	115	3	)	)	PUNCT
ejpam-7054	115	4	⇒	⇒	NOUN
ejpam-7054	115	5	(	(	PUNCT
ejpam-7054	115	6	3	3	NUM
ejpam-7054	115	7	):	):	PUNCT
ejpam-7054	115	8	let	let	VERB
ejpam-7054	115	9	v	v	PART
ejpam-7054	115	10	be	be	AUX
ejpam-7054	115	11	any	any	DET
ejpam-7054	115	12	σ1σ2	σ1σ2	NOUN
ejpam-7054	115	13	-	-	ADJ
ejpam-7054	115	14	open	open	ADJ
ejpam-7054	115	15	set	set	NOUN
ejpam-7054	115	16	of	of	ADP
ejpam-7054	115	17	y	y	PROPN
ejpam-7054	115	18	.	.	PUNCT
ejpam-7054	116	1	since	since	SCONJ
ejpam-7054	116	2	y	y	PROPN
ejpam-7054	116	3	−	−	PROPN
ejpam-7054	116	4	σ1σ2	σ1σ2	NOUN
ejpam-7054	116	5	-	-	NUM
ejpam-7054	116	6	cl(v	cl(v	NOUN
ejpam-7054	116	7	)	)	PUNCT
ejpam-7054	116	8	is	be	AUX
ejpam-7054	116	9	σ1σ2	σ1σ2	NOUN
ejpam-7054	116	10	-	-	ADJ
ejpam-7054	116	11	open	open	ADJ
ejpam-7054	116	12	and	and	CCONJ
ejpam-7054	116	13	by	by	ADP
ejpam-7054	116	14	(	(	PUNCT
ejpam-7054	116	15	2	2	NUM
ejpam-7054	116	16	)	)	PUNCT
ejpam-7054	116	17	,	,	PUNCT
ejpam-7054	116	18	we	we	PRON
ejpam-7054	116	19	have	have	VERB
ejpam-7054	116	20	x	x	NOUN
ejpam-7054	116	21	−	−	PUNCT
ejpam-7054	116	22	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	116	23	-	-	PUNCT
ejpam-7054	116	24	cl(v	cl(v	NOUN
ejpam-7054	116	25	)	)	PUNCT
ejpam-7054	116	26	)	)	PUNCT
ejpam-7054	117	1	=	=	PUNCT
ejpam-7054	118	1	f+(y	f+(y	NOUN
ejpam-7054	118	2	−	−	NUM
ejpam-7054	118	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	118	4	-	-	NUM
ejpam-7054	118	5	cl(v	cl(v	NOUN
ejpam-7054	118	6	)	)	PUNCT
ejpam-7054	118	7	)	)	PUNCT
ejpam-7054	119	1	⊆	⊆	NUM
ejpam-7054	119	2	iµ(cµ(f	iµ(cµ(f	X
ejpam-7054	119	3	+	+	NOUN
ejpam-7054	119	4	(	(	PUNCT
ejpam-7054	119	5	σ1σ2	σ1σ2	NUM
ejpam-7054	119	6	-	-	PUNCT
ejpam-7054	119	7	cl(y	cl(y	NOUN
ejpam-7054	119	8	−	−	NOUN
ejpam-7054	119	9	σ1σ2	σ1σ2	NOUN
ejpam-7054	119	10	-	-	NUM
ejpam-7054	119	11	cl(v	cl(v	NOUN
ejpam-7054	119	12	)	)	PUNCT
ejpam-7054	119	13	)	)	PUNCT
ejpam-7054	119	14	)	)	PUNCT
ejpam-7054	119	15	)	)	PUNCT
ejpam-7054	119	16	)	)	PUNCT
ejpam-7054	120	1	⊆	⊆	NUM
ejpam-7054	120	2	iµ(cµ(f	iµ(cµ(f	X
ejpam-7054	120	3	+	+	PROPN
ejpam-7054	120	4	(	(	PUNCT
ejpam-7054	120	5	y	y	PROPN
ejpam-7054	120	6	−	−	PROPN
ejpam-7054	120	7	v	v	NOUN
ejpam-7054	120	8	)	)	PUNCT
ejpam-7054	120	9	)	)	PUNCT
ejpam-7054	120	10	)	)	PUNCT
ejpam-7054	121	1	=	=	PUNCT
ejpam-7054	121	2	iµ(cµ(x	iµ(cµ(x	VERB
ejpam-7054	121	3	−	−	PROPN
ejpam-7054	121	4	f−(v	f−(v	NOUN
ejpam-7054	121	5	)	)	PUNCT
ejpam-7054	121	6	)	)	PUNCT
ejpam-7054	121	7	)	)	PUNCT
ejpam-7054	122	1	=	=	PUNCT
ejpam-7054	122	2	x	x	PUNCT
ejpam-7054	122	3	−	−	NOUN
ejpam-7054	122	4	cµ(iµ(f	cµ(iµ(f	PROPN
ejpam-7054	122	5	−(v	−(v	NOUN
ejpam-7054	122	6	)	)	PUNCT
ejpam-7054	122	7	)	)	PUNCT
ejpam-7054	122	8	)	)	PUNCT
ejpam-7054	123	1	and	and	CCONJ
ejpam-7054	123	2	hence	hence	ADV
ejpam-7054	123	3	cµ(iµ(f	cµ(iµ(f	PROPN
ejpam-7054	123	4	−(v	−(v	NOUN
ejpam-7054	123	5	)	)	PUNCT
ejpam-7054	123	6	)	)	PUNCT
ejpam-7054	123	7	)	)	PUNCT
ejpam-7054	124	1	⊆	⊆	X
ejpam-7054	124	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	124	3	-	-	PUNCT
ejpam-7054	124	4	cl(v	cl(v	NOUN
ejpam-7054	124	5	)	)	PUNCT
ejpam-7054	124	6	)	)	PUNCT
ejpam-7054	124	7	.	.	PUNCT
ejpam-7054	125	1	(	(	PUNCT
ejpam-7054	125	2	3	3	X
ejpam-7054	125	3	)	)	PUNCT
ejpam-7054	125	4	⇒	⇒	NOUN
ejpam-7054	125	5	(	(	PUNCT
ejpam-7054	125	6	2	2	NUM
ejpam-7054	125	7	):	):	PUNCT
ejpam-7054	125	8	the	the	DET
ejpam-7054	125	9	proof	proof	NOUN
ejpam-7054	125	10	is	be	AUX
ejpam-7054	125	11	obvious	obvious	ADJ
ejpam-7054	125	12	.	.	PUNCT
ejpam-7054	126	1	(	(	PUNCT
ejpam-7054	126	2	2	2	X
ejpam-7054	126	3	)	)	PUNCT
ejpam-7054	126	4	⇒	⇒	NOUN
ejpam-7054	126	5	(	(	PUNCT
ejpam-7054	126	6	4	4	NUM
ejpam-7054	126	7	):	):	PUNCT
ejpam-7054	126	8	let	let	VERB
ejpam-7054	126	9	x	x	PUNCT
ejpam-7054	126	10	∈	∈	PROPN
ejpam-7054	126	11	x	x	X
ejpam-7054	126	12	and	and	CCONJ
ejpam-7054	126	13	v	v	X
ejpam-7054	126	14	be	be	AUX
ejpam-7054	126	15	any	any	DET
ejpam-7054	126	16	σ1σ2	σ1σ2	NOUN
ejpam-7054	126	17	-	-	ADJ
ejpam-7054	126	18	open	open	ADJ
ejpam-7054	126	19	set	set	NOUN
ejpam-7054	126	20	of	of	ADP
ejpam-7054	126	21	y	y	PROPN
ejpam-7054	126	22	containing	contain	VERB
ejpam-7054	126	23	f	f	PROPN
ejpam-7054	126	24	(	(	PUNCT
ejpam-7054	126	25	x	x	NOUN
ejpam-7054	126	26	)	)	PUNCT
ejpam-7054	126	27	.	.	PUNCT
ejpam-7054	127	1	by	by	ADP
ejpam-7054	127	2	(	(	PUNCT
ejpam-7054	127	3	2	2	NUM
ejpam-7054	127	4	)	)	PUNCT
ejpam-7054	127	5	,	,	PUNCT
ejpam-7054	127	6	x	x	PUNCT
ejpam-7054	127	7	∈	∈	PROPN
ejpam-7054	127	8	f+(v	f+(v	NOUN
ejpam-7054	127	9	)	)	PUNCT
ejpam-7054	128	1	⊆	⊆	X
ejpam-7054	128	2	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	129	1	+	+	ADJ
ejpam-7054	129	2	(	(	PUNCT
ejpam-7054	129	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	129	4	-	-	NUM
ejpam-7054	129	5	cl(v	cl(v	NOUN
ejpam-7054	129	6	)	)	PUNCT
ejpam-7054	129	7	)	)	PUNCT
ejpam-7054	129	8	)	)	PUNCT
ejpam-7054	130	1	and	and	CCONJ
ejpam-7054	130	2	there	there	PRON
ejpam-7054	130	3	exists	exist	VERB
ejpam-7054	130	4	a	a	DET
ejpam-7054	130	5	µ-preopen	µ-preopen	PROPN
ejpam-7054	130	6	set	set	VERB
ejpam-7054	130	7	u	u	NOUN
ejpam-7054	130	8	of	of	ADP
ejpam-7054	130	9	x	x	PUNCT
ejpam-7054	130	10	containing	contain	VERB
ejpam-7054	130	11	x	x	PUNCT
ejpam-7054	130	12	such	such	ADJ
ejpam-7054	130	13	that	that	SCONJ
ejpam-7054	130	14	f	f	PROPN
ejpam-7054	130	15	(	(	PUNCT
ejpam-7054	130	16	u	u	NOUN
ejpam-7054	130	17	)	)	PUNCT
ejpam-7054	130	18	⊆	⊆	NUM
ejpam-7054	130	19	σ1σ2	σ1σ2	NOUN
ejpam-7054	130	20	-	-	NUM
ejpam-7054	130	21	cl(v	cl(v	NOUN
ejpam-7054	130	22	)	)	PUNCT
ejpam-7054	130	23	.	.	PUNCT
ejpam-7054	131	1	(	(	PUNCT
ejpam-7054	131	2	4	4	X
ejpam-7054	131	3	)	)	PUNCT
ejpam-7054	131	4	⇒	⇒	NOUN
ejpam-7054	131	5	(	(	PUNCT
ejpam-7054	131	6	1	1	NUM
ejpam-7054	131	7	):	):	PUNCT
ejpam-7054	131	8	let	let	VERB
ejpam-7054	131	9	x	x	PUNCT
ejpam-7054	131	10	∈	∈	PROPN
ejpam-7054	131	11	x	x	X
ejpam-7054	131	12	and	and	CCONJ
ejpam-7054	131	13	v	v	X
ejpam-7054	131	14	be	be	AUX
ejpam-7054	131	15	any	any	DET
ejpam-7054	131	16	σ1σ2	σ1σ2	NOUN
ejpam-7054	131	17	-	-	ADJ
ejpam-7054	131	18	open	open	ADJ
ejpam-7054	131	19	set	set	NOUN
ejpam-7054	131	20	of	of	ADP
ejpam-7054	131	21	y	y	PROPN
ejpam-7054	131	22	containing	contain	VERB
ejpam-7054	131	23	f	f	PROPN
ejpam-7054	131	24	(	(	PUNCT
ejpam-7054	131	25	x	x	NOUN
ejpam-7054	131	26	)	)	PUNCT
ejpam-7054	131	27	.	.	PUNCT
ejpam-7054	132	1	by	by	ADP
ejpam-7054	132	2	(	(	PUNCT
ejpam-7054	132	3	4	4	NUM
ejpam-7054	132	4	)	)	PUNCT
ejpam-7054	132	5	,	,	PUNCT
ejpam-7054	132	6	there	there	PRON
ejpam-7054	132	7	exists	exist	VERB
ejpam-7054	132	8	a	a	DET
ejpam-7054	132	9	µ-preopen	µ-preopen	PROPN
ejpam-7054	132	10	set	set	VERB
ejpam-7054	132	11	u	u	NOUN
ejpam-7054	132	12	of	of	ADP
ejpam-7054	132	13	x	x	PUNCT
ejpam-7054	132	14	containing	contain	VERB
ejpam-7054	132	15	x	x	PUNCT
ejpam-7054	132	16	such	such	ADJ
ejpam-7054	132	17	that	that	SCONJ
ejpam-7054	132	18	f	f	PROPN
ejpam-7054	132	19	(	(	PUNCT
ejpam-7054	132	20	u	u	NOUN
ejpam-7054	132	21	)	)	PUNCT
ejpam-7054	132	22	⊆	⊆	NUM
ejpam-7054	132	23	σ1σ2	σ1σ2	NOUN
ejpam-7054	132	24	-	-	NUM
ejpam-7054	132	25	cl(v	cl(v	NOUN
ejpam-7054	132	26	)	)	PUNCT
ejpam-7054	132	27	;	;	PUNCT
ejpam-7054	132	28	hence	hence	ADV
ejpam-7054	132	29	u	u	NOUN
ejpam-7054	132	30	⊆	⊆	NUM
ejpam-7054	132	31	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	132	32	-	-	PUNCT
ejpam-7054	132	33	cl(v	cl(v	NOUN
ejpam-7054	132	34	)	)	PUNCT
ejpam-7054	132	35	)	)	PUNCT
ejpam-7054	132	36	.	.	PUNCT
ejpam-7054	133	1	thus	thus	ADV
ejpam-7054	133	2	,	,	PUNCT
ejpam-7054	133	3	x	x	PUNCT
ejpam-7054	133	4	∈	∈	PROPN
ejpam-7054	133	5	u	u	NOUN
ejpam-7054	133	6	⊆	⊆	NUM
ejpam-7054	133	7	iµ(cµ(u	iµ(cµ(u	NOUN
ejpam-7054	133	8	)	)	PUNCT
ejpam-7054	133	9	)	)	PUNCT
ejpam-7054	134	1	⊆	⊆	NUM
ejpam-7054	134	2	iµ(cµ(f	iµ(cµ(f	X
ejpam-7054	134	3	+	+	NOUN
ejpam-7054	134	4	(	(	PUNCT
ejpam-7054	134	5	σ1σ2	σ1σ2	NOUN
ejpam-7054	134	6	-	-	NUM
ejpam-7054	134	7	cl(v	cl(v	NOUN
ejpam-7054	134	8	)	)	PUNCT
ejpam-7054	134	9	)	)	PUNCT
ejpam-7054	134	10	)	)	PUNCT
ejpam-7054	134	11	)	)	PUNCT
ejpam-7054	134	12	.	.	PUNCT
ejpam-7054	135	1	this	this	PRON
ejpam-7054	135	2	shows	show	VERB
ejpam-7054	135	3	that	that	SCONJ
ejpam-7054	135	4	f	f	PROPN
ejpam-7054	135	5	is	be	AUX
ejpam-7054	135	6	upper	upper	ADJ
ejpam-7054	135	7	almost	almost	ADV
ejpam-7054	135	8	weakly	weakly	ADJ
ejpam-7054	135	9	µ(σ1	µ(σ1	NOUN
ejpam-7054	135	10	,	,	PUNCT
ejpam-7054	135	11	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7054	135	12	.	.	PROPN
ejpam-7054	135	13	b.	b.	PROPN
ejpam-7054	135	14	kong	kong	PROPN
ejpam-7054	135	15	-	-	PUNCT
ejpam-7054	135	16	ied	ied	PROPN
ejpam-7054	135	17	,	,	PUNCT
ejpam-7054	135	18	a.	a.	PROPN
ejpam-7054	135	19	sama	sama	PROPN
ejpam-7054	135	20	-	-	PUNCT
ejpam-7054	135	21	ae	ae	PROPN
ejpam-7054	135	22	,	,	PUNCT
ejpam-7054	135	23	c.	c.	PROPN
ejpam-7054	135	24	boonpok	boonpok	PROPN
ejpam-7054	135	25	/	/	SYM
ejpam-7054	135	26	eur	eur	PROPN
ejpam-7054	135	27	.	.	PUNCT
ejpam-7054	136	1	j.	j.	PROPN
ejpam-7054	136	2	pure	pure	PROPN
ejpam-7054	136	3	appl	appl	PROPN
ejpam-7054	136	4	.	.	PROPN
ejpam-7054	136	5	math	math	PROPN
ejpam-7054	136	6	,	,	PUNCT
ejpam-7054	136	7	18	18	NUM
ejpam-7054	136	8	(	(	PUNCT
ejpam-7054	136	9	4	4	NUM
ejpam-7054	136	10	)	)	PUNCT
ejpam-7054	136	11	(	(	PUNCT
ejpam-7054	136	12	2025	2025	NUM
ejpam-7054	136	13	)	)	PUNCT
ejpam-7054	136	14	,	,	PUNCT
ejpam-7054	136	15	7054	7054	NUM
ejpam-7054	136	16	5	5	NUM
ejpam-7054	136	17	of	of	ADP
ejpam-7054	136	18	12	12	NUM
ejpam-7054	136	19	definition	definition	NOUN
ejpam-7054	136	20	2	2	NUM
ejpam-7054	136	21	.	.	PUNCT
ejpam-7054	136	22	a	a	DET
ejpam-7054	136	23	multifunction	multifunction	NOUN
ejpam-7054	136	24	f	f	NOUN
ejpam-7054	136	25	:	:	PUNCT
ejpam-7054	136	26	(	(	PUNCT
ejpam-7054	136	27	x,µ	x,µ	NOUN
ejpam-7054	136	28	)	)	PUNCT
ejpam-7054	136	29	→	→	SYM
ejpam-7054	136	30	(	(	PUNCT
ejpam-7054	136	31	y	y	PROPN
ejpam-7054	136	32	,	,	PUNCT
ejpam-7054	136	33	σ1	σ1	PROPN
ejpam-7054	136	34	,	,	PUNCT
ejpam-7054	136	35	σ2	σ2	PROPN
ejpam-7054	136	36	)	)	PUNCT
ejpam-7054	136	37	is	be	AUX
ejpam-7054	136	38	said	say	VERB
ejpam-7054	136	39	to	to	PART
ejpam-7054	136	40	be	be	AUX
ejpam-7054	136	41	lower	low	ADJ
ejpam-7054	136	42	almost	almost	ADV
ejpam-7054	136	43	weakly	weakly	ADJ
ejpam-7054	136	44	µ(σ1	µ(σ1	NOUN
ejpam-7054	136	45	,	,	PUNCT
ejpam-7054	136	46	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	136	47	if	if	SCONJ
ejpam-7054	136	48	for	for	ADP
ejpam-7054	136	49	each	each	DET
ejpam-7054	136	50	x	x	SYM
ejpam-7054	136	51	∈	∈	PROPN
ejpam-7054	136	52	x	x	X
ejpam-7054	136	53	and	and	CCONJ
ejpam-7054	136	54	each	each	DET
ejpam-7054	136	55	σ1σ2	σ1σ2	VERB
ejpam-7054	136	56	-	-	ADJ
ejpam-7054	136	57	open	open	ADJ
ejpam-7054	136	58	set	set	NOUN
ejpam-7054	136	59	v	v	NOUN
ejpam-7054	136	60	of	of	ADP
ejpam-7054	136	61	y	y	PRON
ejpam-7054	136	62	such	such	ADJ
ejpam-7054	136	63	that	that	SCONJ
ejpam-7054	136	64	f	f	PROPN
ejpam-7054	136	65	(	(	PUNCT
ejpam-7054	136	66	x)∩v	x)∩v	PROPN
ejpam-7054	136	67	̸=	̸=	PROPN
ejpam-7054	136	68	∅	∅	NOUN
ejpam-7054	136	69	,	,	PUNCT
ejpam-7054	136	70	x	x	SYM
ejpam-7054	136	71	∈	∈	PROPN
ejpam-7054	136	72	iµ(cµ(f	iµ(cµ(f	NOUN
ejpam-7054	136	73	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	136	74	-	-	NOUN
ejpam-7054	136	75	cl(v	cl(v	NOUN
ejpam-7054	136	76	)	)	PUNCT
ejpam-7054	136	77	)	)	PUNCT
ejpam-7054	136	78	)	)	PUNCT
ejpam-7054	136	79	)	)	PUNCT
ejpam-7054	136	80	.	.	PUNCT
ejpam-7054	137	1	theorem	theorem	NOUN
ejpam-7054	137	2	2	2	NUM
ejpam-7054	137	3	.	.	X
ejpam-7054	137	4	for	for	ADP
ejpam-7054	137	5	a	a	DET
ejpam-7054	137	6	multifunction	multifunction	NOUN
ejpam-7054	138	1	f	f	NOUN
ejpam-7054	138	2	:	:	PUNCT
ejpam-7054	138	3	(	(	PUNCT
ejpam-7054	138	4	x,µ	x,µ	NOUN
ejpam-7054	138	5	)	)	PUNCT
ejpam-7054	138	6	→	→	SYM
ejpam-7054	138	7	(	(	PUNCT
ejpam-7054	138	8	y	y	PROPN
ejpam-7054	138	9	,	,	PUNCT
ejpam-7054	138	10	σ1	σ1	PROPN
ejpam-7054	138	11	,	,	PUNCT
ejpam-7054	138	12	σ2	σ2	NOUN
ejpam-7054	138	13	)	)	PUNCT
ejpam-7054	138	14	,	,	PUNCT
ejpam-7054	138	15	the	the	DET
ejpam-7054	138	16	following	follow	VERB
ejpam-7054	138	17	properties	property	NOUN
ejpam-7054	138	18	are	be	AUX
ejpam-7054	138	19	equivalent	equivalent	ADJ
ejpam-7054	138	20	:	:	PUNCT
ejpam-7054	138	21	(	(	PUNCT
ejpam-7054	138	22	1	1	X
ejpam-7054	138	23	)	)	PUNCT
ejpam-7054	138	24	f	f	PROPN
ejpam-7054	138	25	is	be	AUX
ejpam-7054	138	26	lower	low	ADJ
ejpam-7054	138	27	almost	almost	ADV
ejpam-7054	138	28	weakly	weakly	ADJ
ejpam-7054	138	29	µ(σ1	µ(σ1	NOUN
ejpam-7054	138	30	,	,	PUNCT
ejpam-7054	138	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	138	32	;	;	PUNCT
ejpam-7054	138	33	(	(	PUNCT
ejpam-7054	138	34	2	2	X
ejpam-7054	138	35	)	)	PUNCT
ejpam-7054	138	36	f−(v	f−(v	NOUN
ejpam-7054	138	37	)	)	PUNCT
ejpam-7054	138	38	⊆	⊆	NUM
ejpam-7054	138	39	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	138	40	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	138	41	-	-	NOUN
ejpam-7054	138	42	cl(v	cl(v	NOUN
ejpam-7054	138	43	)	)	PUNCT
ejpam-7054	138	44	)	)	PUNCT
ejpam-7054	138	45	)	)	PUNCT
ejpam-7054	138	46	for	for	ADP
ejpam-7054	138	47	every	every	DET
ejpam-7054	138	48	σ1σ2	σ1σ2	NOUN
ejpam-7054	138	49	-	-	ADJ
ejpam-7054	138	50	open	open	ADJ
ejpam-7054	138	51	set	set	NOUN
ejpam-7054	138	52	v	v	NOUN
ejpam-7054	138	53	of	of	ADP
ejpam-7054	138	54	y	y	PROPN
ejpam-7054	138	55	;	;	PUNCT
ejpam-7054	138	56	(	(	PUNCT
ejpam-7054	138	57	3	3	X
ejpam-7054	138	58	)	)	PUNCT
ejpam-7054	138	59	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	139	1	+	+	PROPN
ejpam-7054	139	2	(	(	PUNCT
ejpam-7054	139	3	v	v	NOUN
ejpam-7054	139	4	)	)	PUNCT
ejpam-7054	139	5	)	)	PUNCT
ejpam-7054	140	1	⊆	⊆	NUM
ejpam-7054	140	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	140	3	-	-	PUNCT
ejpam-7054	140	4	cl(v	cl(v	NOUN
ejpam-7054	140	5	)	)	PUNCT
ejpam-7054	140	6	)	)	PUNCT
ejpam-7054	140	7	for	for	ADP
ejpam-7054	140	8	every	every	DET
ejpam-7054	140	9	σ1σ2	σ1σ2	NOUN
ejpam-7054	140	10	-	-	ADJ
ejpam-7054	140	11	open	open	ADJ
ejpam-7054	140	12	set	set	NOUN
ejpam-7054	140	13	v	v	NOUN
ejpam-7054	140	14	of	of	ADP
ejpam-7054	140	15	y	y	PROPN
ejpam-7054	140	16	;	;	PUNCT
ejpam-7054	140	17	(	(	PUNCT
ejpam-7054	140	18	4	4	X
ejpam-7054	140	19	)	)	PUNCT
ejpam-7054	140	20	for	for	ADP
ejpam-7054	140	21	each	each	DET
ejpam-7054	140	22	x	x	SYM
ejpam-7054	140	23	∈	∈	PROPN
ejpam-7054	140	24	x	x	X
ejpam-7054	140	25	and	and	CCONJ
ejpam-7054	140	26	each	each	DET
ejpam-7054	140	27	σ1σ2	σ1σ2	VERB
ejpam-7054	140	28	-	-	ADJ
ejpam-7054	140	29	open	open	ADJ
ejpam-7054	140	30	set	set	NOUN
ejpam-7054	140	31	v	v	NOUN
ejpam-7054	140	32	of	of	ADP
ejpam-7054	140	33	y	y	PRON
ejpam-7054	140	34	such	such	ADJ
ejpam-7054	140	35	that	that	SCONJ
ejpam-7054	140	36	f	f	PROPN
ejpam-7054	140	37	(	(	PUNCT
ejpam-7054	140	38	x)∩v	x)∩v	PROPN
ejpam-7054	140	39	̸=	̸=	PROPN
ejpam-7054	140	40	∅	∅	NOUN
ejpam-7054	140	41	,	,	PUNCT
ejpam-7054	140	42	there	there	PRON
ejpam-7054	140	43	exists	exist	VERB
ejpam-7054	140	44	a	a	DET
ejpam-7054	140	45	µ-preopen	µ-preopen	PROPN
ejpam-7054	140	46	set	set	VERB
ejpam-7054	140	47	u	u	NOUN
ejpam-7054	140	48	of	of	ADP
ejpam-7054	140	49	x	x	PUNCT
ejpam-7054	140	50	containing	contain	VERB
ejpam-7054	140	51	x	x	PUNCT
ejpam-7054	140	52	such	such	ADJ
ejpam-7054	140	53	that	that	SCONJ
ejpam-7054	140	54	f	f	PROPN
ejpam-7054	140	55	(	(	PUNCT
ejpam-7054	140	56	z)∩	z)∩	X
ejpam-7054	140	57	σ1σ2	σ1σ2	NOUN
ejpam-7054	140	58	-	-	PUNCT
ejpam-7054	140	59	cl(v	cl(v	NOUN
ejpam-7054	140	60	)	)	PUNCT
ejpam-7054	140	61	̸=	̸=	NOUN
ejpam-7054	140	62	∅	∅	NOUN
ejpam-7054	140	63	for	for	ADP
ejpam-7054	140	64	each	each	DET
ejpam-7054	140	65	z	z	NOUN
ejpam-7054	140	66	∈	∈	PROPN
ejpam-7054	140	67	u	u	NOUN
ejpam-7054	140	68	.	.	PUNCT
ejpam-7054	141	1	proof	proof	NOUN
ejpam-7054	141	2	.	.	PUNCT
ejpam-7054	142	1	the	the	DET
ejpam-7054	142	2	proof	proof	NOUN
ejpam-7054	142	3	is	be	AUX
ejpam-7054	142	4	similar	similar	ADJ
ejpam-7054	142	5	to	to	ADP
ejpam-7054	142	6	that	that	PRON
ejpam-7054	142	7	of	of	ADP
ejpam-7054	142	8	theorem	theorem	ADJ
ejpam-7054	142	9	1	1	NUM
ejpam-7054	142	10	.	.	PUNCT
ejpam-7054	142	11	theorem	theorem	NOUN
ejpam-7054	142	12	3	3	NUM
ejpam-7054	142	13	.	.	X
ejpam-7054	142	14	for	for	ADP
ejpam-7054	142	15	a	a	DET
ejpam-7054	142	16	multifunction	multifunction	NOUN
ejpam-7054	143	1	f	f	NOUN
ejpam-7054	143	2	:	:	PUNCT
ejpam-7054	143	3	(	(	PUNCT
ejpam-7054	143	4	x,µ	x,µ	NOUN
ejpam-7054	143	5	)	)	PUNCT
ejpam-7054	143	6	→	→	SYM
ejpam-7054	143	7	(	(	PUNCT
ejpam-7054	143	8	y	y	PROPN
ejpam-7054	143	9	,	,	PUNCT
ejpam-7054	143	10	σ1	σ1	PROPN
ejpam-7054	143	11	,	,	PUNCT
ejpam-7054	143	12	σ2	σ2	NOUN
ejpam-7054	143	13	)	)	PUNCT
ejpam-7054	143	14	,	,	PUNCT
ejpam-7054	143	15	the	the	DET
ejpam-7054	143	16	following	follow	VERB
ejpam-7054	143	17	properties	property	NOUN
ejpam-7054	143	18	are	be	AUX
ejpam-7054	143	19	equivalent	equivalent	ADJ
ejpam-7054	143	20	:	:	PUNCT
ejpam-7054	143	21	(	(	PUNCT
ejpam-7054	143	22	1	1	X
ejpam-7054	143	23	)	)	PUNCT
ejpam-7054	143	24	f	f	PROPN
ejpam-7054	143	25	is	be	AUX
ejpam-7054	143	26	upper	upper	ADJ
ejpam-7054	143	27	almost	almost	ADV
ejpam-7054	143	28	weakly	weakly	ADJ
ejpam-7054	143	29	µ(σ1	µ(σ1	NOUN
ejpam-7054	143	30	,	,	PUNCT
ejpam-7054	143	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	143	32	;	;	PUNCT
ejpam-7054	143	33	(	(	PUNCT
ejpam-7054	143	34	2	2	X
ejpam-7054	143	35	)	)	PUNCT
ejpam-7054	143	36	cµ(π)(f	cµ(π)(f	NOUN
ejpam-7054	143	37	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	143	38	-	-	PUNCT
ejpam-7054	143	39	int(k	int(k	NUM
ejpam-7054	143	40	)	)	PUNCT
ejpam-7054	143	41	)	)	PUNCT
ejpam-7054	143	42	)	)	PUNCT
ejpam-7054	144	1	⊆	⊆	X
ejpam-7054	144	2	f−(k	f−(k	PROPN
ejpam-7054	144	3	)	)	PUNCT
ejpam-7054	144	4	for	for	ADP
ejpam-7054	144	5	every	every	DET
ejpam-7054	144	6	σ1σ2	σ1σ2	NUM
ejpam-7054	144	7	-	-	PUNCT
ejpam-7054	144	8	closed	closed	ADJ
ejpam-7054	144	9	set	set	NOUN
ejpam-7054	144	10	k	k	PROPN
ejpam-7054	144	11	of	of	ADP
ejpam-7054	144	12	y	y	PROPN
ejpam-7054	144	13	;	;	PUNCT
ejpam-7054	144	14	(	(	PUNCT
ejpam-7054	144	15	3	3	X
ejpam-7054	144	16	)	)	PUNCT
ejpam-7054	144	17	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	144	18	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	144	19	-	-	PUNCT
ejpam-7054	144	20	int(σ1σ2cl(b	int(σ1σ2cl(b	NOUN
ejpam-7054	144	21	)	)	PUNCT
ejpam-7054	144	22	)	)	PUNCT
ejpam-7054	144	23	)	)	PUNCT
ejpam-7054	144	24	)	)	PUNCT
ejpam-7054	145	1	⊆	⊆	X
ejpam-7054	145	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7054	145	3	-	-	PUNCT
ejpam-7054	145	4	cl(b	cl(b	NOUN
ejpam-7054	145	5	)	)	PUNCT
ejpam-7054	145	6	)	)	PUNCT
ejpam-7054	146	1	for	for	ADP
ejpam-7054	146	2	every	every	DET
ejpam-7054	146	3	subset	subset	NOUN
ejpam-7054	146	4	b	b	PROPN
ejpam-7054	146	5	of	of	ADP
ejpam-7054	146	6	y	y	PROPN
ejpam-7054	146	7	;	;	PUNCT
ejpam-7054	146	8	(	(	PUNCT
ejpam-7054	146	9	4	4	X
ejpam-7054	146	10	)	)	PUNCT
ejpam-7054	146	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	146	12	-	-	PUNCT
ejpam-7054	146	13	int(b	int(b	NOUN
ejpam-7054	146	14	)	)	PUNCT
ejpam-7054	146	15	)	)	PUNCT
ejpam-7054	147	1	⊆	⊆	X
ejpam-7054	147	2	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	148	1	+	+	ADJ
ejpam-7054	148	2	(	(	PUNCT
ejpam-7054	148	3	σ1σ2	σ1σ2	NUM
ejpam-7054	148	4	-	-	PUNCT
ejpam-7054	148	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	148	6	-	-	PUNCT
ejpam-7054	148	7	int(b	int(b	NOUN
ejpam-7054	148	8	)	)	PUNCT
ejpam-7054	148	9	)	)	PUNCT
ejpam-7054	148	10	)	)	PUNCT
ejpam-7054	148	11	)	)	PUNCT
ejpam-7054	149	1	for	for	ADP
ejpam-7054	149	2	every	every	DET
ejpam-7054	149	3	subset	subset	NOUN
ejpam-7054	149	4	b	b	PROPN
ejpam-7054	149	5	of	of	ADP
ejpam-7054	149	6	y	y	PROPN
ejpam-7054	149	7	.	.	PUNCT
ejpam-7054	150	1	proof	proof	NOUN
ejpam-7054	150	2	.	.	PUNCT
ejpam-7054	151	1	(	(	PUNCT
ejpam-7054	151	2	1	1	X
ejpam-7054	151	3	)	)	PUNCT
ejpam-7054	151	4	⇒	⇒	NOUN
ejpam-7054	151	5	(	(	PUNCT
ejpam-7054	151	6	2	2	NUM
ejpam-7054	151	7	):	):	PUNCT
ejpam-7054	151	8	let	let	VERB
ejpam-7054	151	9	k	k	PRON
ejpam-7054	151	10	be	be	AUX
ejpam-7054	151	11	any	any	DET
ejpam-7054	151	12	σ1σ2	σ1σ2	NUM
ejpam-7054	151	13	-	-	PUNCT
ejpam-7054	151	14	closed	closed	ADJ
ejpam-7054	151	15	set	set	NOUN
ejpam-7054	151	16	of	of	ADP
ejpam-7054	151	17	y	y	PROPN
ejpam-7054	151	18	.	.	PUNCT
ejpam-7054	152	1	then	then	ADV
ejpam-7054	152	2	,	,	PUNCT
ejpam-7054	152	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	152	4	-	-	PUNCT
ejpam-7054	152	5	int(k	int(k	NUM
ejpam-7054	152	6	)	)	PUNCT
ejpam-7054	152	7	is	be	AUX
ejpam-7054	152	8	σ1σ2	σ1σ2	NOUN
ejpam-7054	152	9	-	-	ADJ
ejpam-7054	152	10	open	open	ADJ
ejpam-7054	152	11	in	in	ADP
ejpam-7054	152	12	y	y	PROPN
ejpam-7054	152	13	,	,	PUNCT
ejpam-7054	152	14	by	by	ADP
ejpam-7054	152	15	theorem	theorem	NOUN
ejpam-7054	152	16	1	1	NUM
ejpam-7054	152	17	we	we	PRON
ejpam-7054	152	18	have	have	AUX
ejpam-7054	152	19	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	152	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	152	21	-	-	PUNCT
ejpam-7054	152	22	int(k	int(k	NUM
ejpam-7054	152	23	)	)	PUNCT
ejpam-7054	152	24	)	)	PUNCT
ejpam-7054	152	25	)	)	PUNCT
ejpam-7054	153	1	⊆	⊆	X
ejpam-7054	153	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7054	153	3	-	-	PUNCT
ejpam-7054	153	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	153	5	-	-	PUNCT
ejpam-7054	153	6	int(k	int(k	NOUN
ejpam-7054	153	7	)	)	PUNCT
ejpam-7054	153	8	)	)	PUNCT
ejpam-7054	153	9	)	)	PUNCT
ejpam-7054	154	1	⊆	⊆	X
ejpam-7054	154	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	154	3	-	-	PUNCT
ejpam-7054	154	4	cl(k	cl(k	NOUN
ejpam-7054	154	5	)	)	PUNCT
ejpam-7054	154	6	)	)	PUNCT
ejpam-7054	155	1	=	=	SYM
ejpam-7054	155	2	f−(k	f−(k	PROPN
ejpam-7054	155	3	)	)	PUNCT
ejpam-7054	155	4	.	.	PUNCT
ejpam-7054	156	1	(	(	PUNCT
ejpam-7054	156	2	2	2	X
ejpam-7054	156	3	)	)	PUNCT
ejpam-7054	156	4	⇒	⇒	NOUN
ejpam-7054	156	5	(	(	PUNCT
ejpam-7054	156	6	3	3	NUM
ejpam-7054	156	7	):	):	PUNCT
ejpam-7054	156	8	the	the	DET
ejpam-7054	156	9	proof	proof	NOUN
ejpam-7054	156	10	is	be	AUX
ejpam-7054	156	11	obvious	obvious	ADJ
ejpam-7054	156	12	.	.	PUNCT
ejpam-7054	157	1	(	(	PUNCT
ejpam-7054	157	2	3	3	X
ejpam-7054	157	3	)	)	PUNCT
ejpam-7054	157	4	⇒	⇒	NOUN
ejpam-7054	157	5	(	(	PUNCT
ejpam-7054	157	6	4	4	NUM
ejpam-7054	157	7	):	):	PUNCT
ejpam-7054	157	8	let	let	VERB
ejpam-7054	157	9	b	b	X
ejpam-7054	157	10	be	be	AUX
ejpam-7054	157	11	any	any	DET
ejpam-7054	157	12	subset	subset	NOUN
ejpam-7054	157	13	of	of	ADP
ejpam-7054	157	14	y	y	PROPN
ejpam-7054	157	15	.	.	PUNCT
ejpam-7054	158	1	by	by	ADP
ejpam-7054	158	2	(	(	PUNCT
ejpam-7054	158	3	3	3	NUM
ejpam-7054	158	4	)	)	PUNCT
ejpam-7054	158	5	,	,	PUNCT
ejpam-7054	158	6	we	we	PRON
ejpam-7054	158	7	have	have	VERB
ejpam-7054	158	8	x	x	X
ejpam-7054	158	9	−	−	NOUN
ejpam-7054	158	10	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	159	1	+	+	NOUN
ejpam-7054	159	2	(	(	PUNCT
ejpam-7054	159	3	σ1σ2	σ1σ2	NUM
ejpam-7054	159	4	-	-	PUNCT
ejpam-7054	159	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	159	6	-	-	PUNCT
ejpam-7054	159	7	int(b	int(b	NOUN
ejpam-7054	159	8	)	)	PUNCT
ejpam-7054	159	9	)	)	PUNCT
ejpam-7054	159	10	)	)	PUNCT
ejpam-7054	159	11	)	)	PUNCT
ejpam-7054	160	1	=	=	PUNCT
ejpam-7054	160	2	cµ(π)(x	cµ(π)(x	NOUN
ejpam-7054	160	3	−	−	ADP
ejpam-7054	160	4	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-7054	160	5	-	-	PUNCT
ejpam-7054	160	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	160	7	-	-	PUNCT
ejpam-7054	160	8	int(b	int(b	NOUN
ejpam-7054	160	9	)	)	PUNCT
ejpam-7054	160	10	)	)	PUNCT
ejpam-7054	160	11	)	)	PUNCT
ejpam-7054	160	12	)	)	PUNCT
ejpam-7054	161	1	=	=	PRON
ejpam-7054	161	2	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	161	3	−(y	−(y	NOUN
ejpam-7054	161	4	−	−	NUM
ejpam-7054	161	5	σ1σ2	σ1σ2	SYM
ejpam-7054	161	6	-	-	PUNCT
ejpam-7054	161	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	161	8	-	-	PUNCT
ejpam-7054	161	9	int(b	int(b	NOUN
ejpam-7054	161	10	)	)	PUNCT
ejpam-7054	161	11	)	)	PUNCT
ejpam-7054	161	12	)	)	PUNCT
ejpam-7054	161	13	)	)	PUNCT
ejpam-7054	162	1	=	=	PRON
ejpam-7054	162	2	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	162	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	162	4	-	-	PUNCT
ejpam-7054	162	5	int(σ1σ2	int(σ1σ2	VERB
ejpam-7054	162	6	-	-	PUNCT
ejpam-7054	162	7	cl(y	cl(y	NOUN
ejpam-7054	162	8	−b	−b	NOUN
ejpam-7054	162	9	)	)	PUNCT
ejpam-7054	162	10	)	)	PUNCT
ejpam-7054	162	11	)	)	PUNCT
ejpam-7054	162	12	)	)	PUNCT
ejpam-7054	163	1	⊆	⊆	X
ejpam-7054	163	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7054	163	3	-	-	PUNCT
ejpam-7054	163	4	cl(y	cl(y	NOUN
ejpam-7054	163	5	−b	−b	NOUN
ejpam-7054	163	6	)	)	PUNCT
ejpam-7054	163	7	)	)	PUNCT
ejpam-7054	164	1	=	=	PUNCT
ejpam-7054	164	2	x	x	X
ejpam-7054	165	1	−	−	ADP
ejpam-7054	165	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	165	3	-	-	PUNCT
ejpam-7054	165	4	int(b	int(b	NOUN
ejpam-7054	165	5	)	)	PUNCT
ejpam-7054	165	6	)	)	PUNCT
ejpam-7054	165	7	.	.	PUNCT
ejpam-7054	166	1	thus	thus	ADV
ejpam-7054	166	2	,	,	PUNCT
ejpam-7054	166	3	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	166	4	-	-	PUNCT
ejpam-7054	166	5	int(b	int(b	NOUN
ejpam-7054	166	6	)	)	PUNCT
ejpam-7054	166	7	)	)	PUNCT
ejpam-7054	167	1	⊆	⊆	X
ejpam-7054	167	2	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	168	1	+	+	ADJ
ejpam-7054	168	2	(	(	PUNCT
ejpam-7054	168	3	σ1σ2	σ1σ2	NUM
ejpam-7054	168	4	-	-	PUNCT
ejpam-7054	168	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	168	6	-	-	PUNCT
ejpam-7054	168	7	int(b	int(b	NOUN
ejpam-7054	168	8	)	)	PUNCT
ejpam-7054	168	9	)	)	PUNCT
ejpam-7054	168	10	)	)	PUNCT
ejpam-7054	168	11	)	)	PUNCT
ejpam-7054	168	12	.	.	PUNCT
ejpam-7054	169	1	(	(	PUNCT
ejpam-7054	169	2	4	4	X
ejpam-7054	169	3	)	)	PUNCT
ejpam-7054	169	4	⇒	⇒	NOUN
ejpam-7054	169	5	(	(	PUNCT
ejpam-7054	169	6	1	1	NUM
ejpam-7054	169	7	):	):	PUNCT
ejpam-7054	169	8	let	let	VERB
ejpam-7054	169	9	v	v	PART
ejpam-7054	169	10	be	be	AUX
ejpam-7054	169	11	any	any	DET
ejpam-7054	169	12	σ1σ2	σ1σ2	NOUN
ejpam-7054	169	13	-	-	ADJ
ejpam-7054	169	14	open	open	ADJ
ejpam-7054	169	15	set	set	NOUN
ejpam-7054	169	16	of	of	ADP
ejpam-7054	169	17	y	y	PROPN
ejpam-7054	169	18	.	.	PUNCT
ejpam-7054	170	1	then	then	ADV
ejpam-7054	170	2	by	by	ADP
ejpam-7054	170	3	(	(	PUNCT
ejpam-7054	170	4	5	5	NUM
ejpam-7054	170	5	)	)	PUNCT
ejpam-7054	170	6	,	,	PUNCT
ejpam-7054	170	7	we	we	PRON
ejpam-7054	170	8	have	have	VERB
ejpam-7054	170	9	f+(v	f+(v	NOUN
ejpam-7054	170	10	)	)	PUNCT
ejpam-7054	171	1	⊆	⊆	X
ejpam-7054	171	2	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	172	1	+	+	ADJ
ejpam-7054	172	2	(	(	PUNCT
ejpam-7054	172	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	172	4	-	-	NUM
ejpam-7054	172	5	cl(v	cl(v	NOUN
ejpam-7054	172	6	)	)	PUNCT
ejpam-7054	172	7	)	)	PUNCT
ejpam-7054	172	8	)	)	PUNCT
ejpam-7054	173	1	and	and	CCONJ
ejpam-7054	173	2	hence	hence	ADV
ejpam-7054	173	3	f	f	PROPN
ejpam-7054	173	4	is	be	AUX
ejpam-7054	173	5	upper	upper	ADJ
ejpam-7054	173	6	almost	almost	ADV
ejpam-7054	173	7	weakly	weakly	ADJ
ejpam-7054	173	8	µ(σ1	µ(σ1	NOUN
ejpam-7054	173	9	,	,	PUNCT
ejpam-7054	173	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	173	11	by	by	ADP
ejpam-7054	173	12	theorem	theorem	NOUN
ejpam-7054	173	13	1	1	NUM
ejpam-7054	173	14	.	.	PUNCT
ejpam-7054	173	15	b.	b.	PROPN
ejpam-7054	173	16	kong	kong	PROPN
ejpam-7054	173	17	-	-	PUNCT
ejpam-7054	173	18	ied	ied	PROPN
ejpam-7054	173	19	,	,	PUNCT
ejpam-7054	173	20	a.	a.	PROPN
ejpam-7054	173	21	sama	sama	PROPN
ejpam-7054	173	22	-	-	PUNCT
ejpam-7054	173	23	ae	ae	PROPN
ejpam-7054	173	24	,	,	PUNCT
ejpam-7054	173	25	c.	c.	PROPN
ejpam-7054	173	26	boonpok	boonpok	PROPN
ejpam-7054	173	27	/	/	SYM
ejpam-7054	173	28	eur	eur	PROPN
ejpam-7054	173	29	.	.	PUNCT
ejpam-7054	174	1	j.	j.	PROPN
ejpam-7054	174	2	pure	pure	PROPN
ejpam-7054	174	3	appl	appl	PROPN
ejpam-7054	174	4	.	.	PROPN
ejpam-7054	174	5	math	math	PROPN
ejpam-7054	174	6	,	,	PUNCT
ejpam-7054	174	7	18	18	NUM
ejpam-7054	174	8	(	(	PUNCT
ejpam-7054	174	9	4	4	NUM
ejpam-7054	174	10	)	)	PUNCT
ejpam-7054	174	11	(	(	PUNCT
ejpam-7054	174	12	2025	2025	NUM
ejpam-7054	174	13	)	)	PUNCT
ejpam-7054	174	14	,	,	PUNCT
ejpam-7054	174	15	7054	7054	NUM
ejpam-7054	174	16	6	6	NUM
ejpam-7054	174	17	of	of	ADP
ejpam-7054	174	18	12	12	NUM
ejpam-7054	174	19	theorem	theorem	NOUN
ejpam-7054	174	20	4	4	NUM
ejpam-7054	174	21	.	.	X
ejpam-7054	174	22	for	for	ADP
ejpam-7054	174	23	a	a	DET
ejpam-7054	174	24	multifunction	multifunction	NOUN
ejpam-7054	175	1	f	f	NOUN
ejpam-7054	175	2	:	:	PUNCT
ejpam-7054	175	3	(	(	PUNCT
ejpam-7054	175	4	x,µ	x,µ	NOUN
ejpam-7054	175	5	)	)	PUNCT
ejpam-7054	175	6	→	→	SYM
ejpam-7054	175	7	(	(	PUNCT
ejpam-7054	175	8	y	y	PROPN
ejpam-7054	175	9	,	,	PUNCT
ejpam-7054	175	10	σ1	σ1	PROPN
ejpam-7054	175	11	,	,	PUNCT
ejpam-7054	175	12	σ2	σ2	NOUN
ejpam-7054	175	13	)	)	PUNCT
ejpam-7054	175	14	,	,	PUNCT
ejpam-7054	175	15	the	the	DET
ejpam-7054	175	16	following	follow	VERB
ejpam-7054	175	17	properties	property	NOUN
ejpam-7054	175	18	are	be	AUX
ejpam-7054	175	19	equivalent	equivalent	ADJ
ejpam-7054	175	20	:	:	PUNCT
ejpam-7054	175	21	(	(	PUNCT
ejpam-7054	175	22	1	1	X
ejpam-7054	175	23	)	)	PUNCT
ejpam-7054	175	24	f	f	PROPN
ejpam-7054	175	25	is	be	AUX
ejpam-7054	175	26	lower	low	ADJ
ejpam-7054	175	27	almost	almost	ADV
ejpam-7054	175	28	weakly	weakly	ADJ
ejpam-7054	175	29	µ(σ1	µ(σ1	NOUN
ejpam-7054	175	30	,	,	PUNCT
ejpam-7054	175	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	175	32	;	;	PUNCT
ejpam-7054	175	33	(	(	PUNCT
ejpam-7054	175	34	2	2	X
ejpam-7054	175	35	)	)	PUNCT
ejpam-7054	175	36	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	176	1	+	+	PROPN
ejpam-7054	176	2	(	(	PUNCT
ejpam-7054	176	3	σ1σ2	σ1σ2	NUM
ejpam-7054	176	4	-	-	PUNCT
ejpam-7054	176	5	int(k	int(k	NUM
ejpam-7054	176	6	)	)	PUNCT
ejpam-7054	176	7	)	)	PUNCT
ejpam-7054	176	8	)	)	PUNCT
ejpam-7054	177	1	⊆	⊆	NUM
ejpam-7054	177	2	f+(k	f+(k	NOUN
ejpam-7054	177	3	)	)	PUNCT
ejpam-7054	177	4	for	for	ADP
ejpam-7054	177	5	every	every	DET
ejpam-7054	177	6	σ1σ2	σ1σ2	NUM
ejpam-7054	177	7	-	-	PUNCT
ejpam-7054	177	8	closed	closed	ADJ
ejpam-7054	177	9	set	set	NOUN
ejpam-7054	177	10	k	k	PROPN
ejpam-7054	177	11	of	of	ADP
ejpam-7054	177	12	y	y	PROPN
ejpam-7054	177	13	;	;	PUNCT
ejpam-7054	177	14	(	(	PUNCT
ejpam-7054	177	15	3	3	X
ejpam-7054	177	16	)	)	PUNCT
ejpam-7054	177	17	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	178	1	+	+	PROPN
ejpam-7054	178	2	(	(	PUNCT
ejpam-7054	178	3	σ1σ2	σ1σ2	NUM
ejpam-7054	178	4	-	-	PUNCT
ejpam-7054	178	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	178	6	-	-	PUNCT
ejpam-7054	178	7	cl(b	cl(b	NOUN
ejpam-7054	178	8	)	)	PUNCT
ejpam-7054	178	9	)	)	PUNCT
ejpam-7054	178	10	)	)	PUNCT
ejpam-7054	178	11	)	)	PUNCT
ejpam-7054	179	1	⊆	⊆	X
ejpam-7054	179	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	179	3	-	-	PUNCT
ejpam-7054	179	4	cl(b	cl(b	NOUN
ejpam-7054	179	5	)	)	PUNCT
ejpam-7054	179	6	)	)	PUNCT
ejpam-7054	179	7	for	for	ADP
ejpam-7054	179	8	every	every	DET
ejpam-7054	179	9	subset	subset	NOUN
ejpam-7054	179	10	b	b	PROPN
ejpam-7054	179	11	of	of	ADP
ejpam-7054	179	12	y	y	PROPN
ejpam-7054	179	13	;	;	PUNCT
ejpam-7054	179	14	(	(	PUNCT
ejpam-7054	179	15	4	4	X
ejpam-7054	179	16	)	)	PUNCT
ejpam-7054	179	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	179	18	-	-	PUNCT
ejpam-7054	179	19	int(b	int(b	NOUN
ejpam-7054	179	20	)	)	PUNCT
ejpam-7054	179	21	)	)	PUNCT
ejpam-7054	179	22	⊆	⊆	NUM
ejpam-7054	179	23	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	179	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	179	25	-	-	PUNCT
ejpam-7054	179	26	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	179	27	-	-	PUNCT
ejpam-7054	179	28	int(b	int(b	NOUN
ejpam-7054	179	29	)	)	PUNCT
ejpam-7054	179	30	)	)	PUNCT
ejpam-7054	179	31	)	)	PUNCT
ejpam-7054	179	32	)	)	PUNCT
ejpam-7054	179	33	for	for	ADP
ejpam-7054	179	34	every	every	DET
ejpam-7054	179	35	subset	subset	NOUN
ejpam-7054	179	36	b	b	PROPN
ejpam-7054	179	37	of	of	ADP
ejpam-7054	179	38	y	y	PROPN
ejpam-7054	179	39	.	.	PUNCT
ejpam-7054	180	1	proof	proof	NOUN
ejpam-7054	180	2	.	.	PUNCT
ejpam-7054	181	1	the	the	DET
ejpam-7054	181	2	proof	proof	NOUN
ejpam-7054	181	3	is	be	AUX
ejpam-7054	181	4	similar	similar	ADJ
ejpam-7054	181	5	to	to	ADP
ejpam-7054	181	6	that	that	PRON
ejpam-7054	181	7	of	of	ADP
ejpam-7054	181	8	theorem	theorem	ADJ
ejpam-7054	181	9	3	3	NUM
ejpam-7054	181	10	.	.	PUNCT
ejpam-7054	181	11	theorem	theorem	NOUN
ejpam-7054	181	12	5	5	NUM
ejpam-7054	181	13	.	.	X
ejpam-7054	181	14	for	for	ADP
ejpam-7054	181	15	a	a	DET
ejpam-7054	181	16	multifunction	multifunction	NOUN
ejpam-7054	181	17	f	f	NOUN
ejpam-7054	181	18	:	:	PUNCT
ejpam-7054	181	19	(	(	PUNCT
ejpam-7054	181	20	x,µ	x,µ	NOUN
ejpam-7054	181	21	)	)	PUNCT
ejpam-7054	181	22	→	→	SYM
ejpam-7054	181	23	(	(	PUNCT
ejpam-7054	181	24	y	y	PROPN
ejpam-7054	181	25	,	,	PUNCT
ejpam-7054	181	26	σ1	σ1	PROPN
ejpam-7054	181	27	,	,	PUNCT
ejpam-7054	181	28	σ2	σ2	NOUN
ejpam-7054	181	29	)	)	PUNCT
ejpam-7054	181	30	,	,	PUNCT
ejpam-7054	181	31	the	the	DET
ejpam-7054	181	32	following	follow	VERB
ejpam-7054	181	33	properties	property	NOUN
ejpam-7054	181	34	are	be	AUX
ejpam-7054	181	35	equivalent	equivalent	ADJ
ejpam-7054	181	36	:	:	PUNCT
ejpam-7054	181	37	(	(	PUNCT
ejpam-7054	181	38	1	1	X
ejpam-7054	181	39	)	)	PUNCT
ejpam-7054	181	40	f	f	PROPN
ejpam-7054	181	41	is	be	AUX
ejpam-7054	181	42	upper	upper	ADJ
ejpam-7054	181	43	almost	almost	ADV
ejpam-7054	181	44	weakly	weakly	ADJ
ejpam-7054	181	45	µ(σ1	µ(σ1	NOUN
ejpam-7054	181	46	,	,	PUNCT
ejpam-7054	181	47	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	181	48	;	;	PUNCT
ejpam-7054	181	49	(	(	PUNCT
ejpam-7054	181	50	2	2	X
ejpam-7054	181	51	)	)	PUNCT
ejpam-7054	181	52	cµ(π)(f	cµ(π)(f	NOUN
ejpam-7054	181	53	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	181	54	-	-	PUNCT
ejpam-7054	181	55	int((σ1	int((σ1	NUM
ejpam-7054	181	56	,	,	PUNCT
ejpam-7054	181	57	σ2)-θcl(b	σ2)-θcl(b	NOUN
ejpam-7054	181	58	)	)	PUNCT
ejpam-7054	181	59	)	)	PUNCT
ejpam-7054	181	60	)	)	PUNCT
ejpam-7054	181	61	)	)	PUNCT
ejpam-7054	182	1	⊆	⊆	NUM
ejpam-7054	182	2	f−((σ1	f−((σ1	NOUN
ejpam-7054	182	3	,	,	PUNCT
ejpam-7054	182	4	σ2)θ	σ2)θ	ADJ
ejpam-7054	182	5	-	-	PUNCT
ejpam-7054	182	6	cl(b	cl(b	NOUN
ejpam-7054	182	7	)	)	PUNCT
ejpam-7054	182	8	)	)	PUNCT
ejpam-7054	182	9	for	for	ADP
ejpam-7054	182	10	every	every	DET
ejpam-7054	182	11	subset	subset	NOUN
ejpam-7054	182	12	b	b	PROPN
ejpam-7054	182	13	of	of	ADP
ejpam-7054	182	14	y	y	PROPN
ejpam-7054	182	15	;	;	PUNCT
ejpam-7054	182	16	(	(	PUNCT
ejpam-7054	182	17	3	3	X
ejpam-7054	182	18	)	)	PUNCT
ejpam-7054	182	19	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	182	20	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	182	21	-	-	PUNCT
ejpam-7054	182	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	182	23	-	-	PUNCT
ejpam-7054	182	24	cl(v	cl(v	NOUN
ejpam-7054	182	25	)	)	PUNCT
ejpam-7054	182	26	)	)	PUNCT
ejpam-7054	182	27	)	)	PUNCT
ejpam-7054	182	28	)	)	PUNCT
ejpam-7054	183	1	⊆	⊆	X
ejpam-7054	183	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	183	3	-	-	PUNCT
ejpam-7054	183	4	cl(v	cl(v	NOUN
ejpam-7054	183	5	)	)	PUNCT
ejpam-7054	183	6	)	)	PUNCT
ejpam-7054	183	7	for	for	ADP
ejpam-7054	183	8	every	every	DET
ejpam-7054	183	9	σ1σ2	σ1σ2	NOUN
ejpam-7054	183	10	-	-	ADJ
ejpam-7054	183	11	open	open	ADJ
ejpam-7054	183	12	set	set	NOUN
ejpam-7054	183	13	v	v	NOUN
ejpam-7054	183	14	of	of	ADP
ejpam-7054	183	15	y	y	PROPN
ejpam-7054	183	16	;	;	PUNCT
ejpam-7054	183	17	(	(	PUNCT
ejpam-7054	183	18	4	4	X
ejpam-7054	183	19	)	)	PUNCT
ejpam-7054	183	20	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	183	21	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	183	22	-	-	PUNCT
ejpam-7054	183	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	183	24	-	-	PUNCT
ejpam-7054	183	25	cl(v	cl(v	NOUN
ejpam-7054	183	26	)	)	PUNCT
ejpam-7054	183	27	)	)	PUNCT
ejpam-7054	183	28	)	)	PUNCT
ejpam-7054	183	29	)	)	PUNCT
ejpam-7054	184	1	⊆	⊆	X
ejpam-7054	184	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	184	3	-	-	PUNCT
ejpam-7054	184	4	cl(v	cl(v	NOUN
ejpam-7054	184	5	)	)	PUNCT
ejpam-7054	184	6	)	)	PUNCT
ejpam-7054	184	7	for	for	ADP
ejpam-7054	184	8	every	every	DET
ejpam-7054	184	9	(	(	PUNCT
ejpam-7054	184	10	σ1	σ1	PROPN
ejpam-7054	184	11	,	,	PUNCT
ejpam-7054	184	12	σ2)p	σ2)p	NOUN
ejpam-7054	184	13	-	-	PUNCT
ejpam-7054	184	14	open	open	NOUN
ejpam-7054	184	15	set	set	NOUN
ejpam-7054	184	16	v	v	NOUN
ejpam-7054	184	17	of	of	ADP
ejpam-7054	184	18	y	y	PROPN
ejpam-7054	184	19	;	;	PUNCT
ejpam-7054	184	20	(	(	PUNCT
ejpam-7054	184	21	5	5	X
ejpam-7054	184	22	)	)	PUNCT
ejpam-7054	184	23	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	184	24	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	184	25	-	-	PUNCT
ejpam-7054	184	26	int(k	int(k	NUM
ejpam-7054	184	27	)	)	PUNCT
ejpam-7054	184	28	)	)	PUNCT
ejpam-7054	184	29	)	)	PUNCT
ejpam-7054	185	1	⊆	⊆	X
ejpam-7054	185	2	f−(k	f−(k	PROPN
ejpam-7054	185	3	)	)	PUNCT
ejpam-7054	185	4	for	for	ADP
ejpam-7054	185	5	every	every	DET
ejpam-7054	185	6	(	(	PUNCT
ejpam-7054	185	7	σ1	σ1	PROPN
ejpam-7054	185	8	,	,	PUNCT
ejpam-7054	185	9	σ2)r	σ2)r	NOUN
ejpam-7054	185	10	-	-	PUNCT
ejpam-7054	185	11	closed	close	VERB
ejpam-7054	185	12	set	set	ADJ
ejpam-7054	185	13	k	k	PROPN
ejpam-7054	185	14	of	of	ADP
ejpam-7054	185	15	y	y	PROPN
ejpam-7054	185	16	.	.	PUNCT
ejpam-7054	186	1	proof	proof	NOUN
ejpam-7054	186	2	.	.	PUNCT
ejpam-7054	187	1	(	(	PUNCT
ejpam-7054	187	2	1	1	X
ejpam-7054	187	3	)	)	PUNCT
ejpam-7054	187	4	⇒	⇒	NOUN
ejpam-7054	187	5	(	(	PUNCT
ejpam-7054	187	6	2	2	NUM
ejpam-7054	187	7	):	):	PUNCT
ejpam-7054	187	8	let	let	VERB
ejpam-7054	187	9	b	b	X
ejpam-7054	187	10	be	be	AUX
ejpam-7054	187	11	any	any	DET
ejpam-7054	187	12	subset	subset	NOUN
ejpam-7054	187	13	of	of	ADP
ejpam-7054	187	14	y	y	PROPN
ejpam-7054	187	15	.	.	PUNCT
ejpam-7054	188	1	let	let	VERB
ejpam-7054	188	2	x	x	PUNCT
ejpam-7054	188	3	∈	∈	PROPN
ejpam-7054	188	4	x	x	PUNCT
ejpam-7054	188	5	−f−((σ1	−f−((σ1	ADV
ejpam-7054	188	6	,	,	PUNCT
ejpam-7054	188	7	σ2)θ	σ2)θ	ADJ
ejpam-7054	188	8	-	-	PUNCT
ejpam-7054	188	9	cl(b	cl(b	NOUN
ejpam-7054	188	10	)	)	PUNCT
ejpam-7054	188	11	)	)	PUNCT
ejpam-7054	188	12	.	.	PUNCT
ejpam-7054	189	1	then	then	ADV
ejpam-7054	189	2	,	,	PUNCT
ejpam-7054	189	3	x	x	PUNCT
ejpam-7054	189	4	∈	∈	PROPN
ejpam-7054	189	5	f+(y	f+(y	X
ejpam-7054	189	6	−	−	PROPN
ejpam-7054	189	7	(	(	PUNCT
ejpam-7054	189	8	σ1	σ1	PROPN
ejpam-7054	189	9	,	,	PUNCT
ejpam-7054	189	10	σ2)θ	σ2)θ	NOUN
ejpam-7054	189	11	-	-	PUNCT
ejpam-7054	189	12	cl(b	cl(b	NOUN
ejpam-7054	189	13	)	)	PUNCT
ejpam-7054	189	14	)	)	PUNCT
ejpam-7054	189	15	and	and	CCONJ
ejpam-7054	189	16	(	(	PUNCT
ejpam-7054	189	17	σ1	σ1	PROPN
ejpam-7054	189	18	,	,	PUNCT
ejpam-7054	189	19	σ2)θ	σ2)θ	NOUN
ejpam-7054	189	20	-	-	PUNCT
ejpam-7054	189	21	cl(b	cl(b	NOUN
ejpam-7054	189	22	)	)	PUNCT
ejpam-7054	189	23	is	be	AUX
ejpam-7054	189	24	σ1σ2	σ1σ2	NOUN
ejpam-7054	189	25	-	-	ADJ
ejpam-7054	189	26	closed	closed	ADJ
ejpam-7054	189	27	in	in	ADP
ejpam-7054	189	28	y	y	PROPN
ejpam-7054	189	29	.	.	PUNCT
ejpam-7054	190	1	by	by	ADP
ejpam-7054	190	2	theorem	theorem	NOUN
ejpam-7054	190	3	1	1	NUM
ejpam-7054	190	4	,	,	PUNCT
ejpam-7054	190	5	there	there	PRON
ejpam-7054	190	6	exists	exist	VERB
ejpam-7054	190	7	a	a	DET
ejpam-7054	190	8	µ-preopen	µ-preopen	PROPN
ejpam-7054	190	9	set	set	VERB
ejpam-7054	190	10	u	u	NOUN
ejpam-7054	190	11	of	of	ADP
ejpam-7054	190	12	x	x	PUNCT
ejpam-7054	190	13	containing	contain	VERB
ejpam-7054	190	14	x	x	PUNCT
ejpam-7054	190	15	such	such	ADJ
ejpam-7054	190	16	that	that	SCONJ
ejpam-7054	190	17	u	u	NOUN
ejpam-7054	190	18	⊆	⊆	NUM
ejpam-7054	190	19	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	190	20	-	-	PUNCT
ejpam-7054	190	21	cl(y	cl(y	NOUN
ejpam-7054	190	22	−	−	PROPN
ejpam-7054	190	23	(	(	PUNCT
ejpam-7054	190	24	σ1	σ1	PROPN
ejpam-7054	190	25	,	,	PUNCT
ejpam-7054	190	26	σ2)θ	σ2)θ	NOUN
ejpam-7054	190	27	-	-	PUNCT
ejpam-7054	190	28	cl(b	cl(b	NOUN
ejpam-7054	190	29	)	)	PUNCT
ejpam-7054	190	30	)	)	PUNCT
ejpam-7054	190	31	)	)	PUNCT
ejpam-7054	191	1	=	=	PUNCT
ejpam-7054	192	1	f+(y	f+(y	NOUN
ejpam-7054	192	2	−	−	NUM
ejpam-7054	192	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	192	4	-	-	PUNCT
ejpam-7054	192	5	int((σ1	int((σ1	ADJ
ejpam-7054	192	6	,	,	PUNCT
ejpam-7054	192	7	σ2)θ	σ2)θ	ADJ
ejpam-7054	192	8	-	-	PUNCT
ejpam-7054	192	9	cl(b	cl(b	NOUN
ejpam-7054	192	10	)	)	PUNCT
ejpam-7054	192	11	)	)	PUNCT
ejpam-7054	192	12	)	)	PUNCT
ejpam-7054	193	1	=	=	PUNCT
ejpam-7054	193	2	x	x	X
ejpam-7054	193	3	−	−	NOUN
ejpam-7054	193	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	193	5	-	-	PUNCT
ejpam-7054	193	6	int((σ1	int((σ1	ADJ
ejpam-7054	193	7	,	,	PUNCT
ejpam-7054	193	8	σ2)θ	σ2)θ	ADJ
ejpam-7054	193	9	-	-	PUNCT
ejpam-7054	193	10	cl(b	cl(b	NOUN
ejpam-7054	193	11	)	)	PUNCT
ejpam-7054	193	12	)	)	PUNCT
ejpam-7054	193	13	)	)	PUNCT
ejpam-7054	193	14	.	.	PUNCT
ejpam-7054	194	1	thus	thus	ADV
ejpam-7054	194	2	,	,	PUNCT
ejpam-7054	194	3	u	u	PROPN
ejpam-7054	194	4	∩	∩	NOUN
ejpam-7054	194	5	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	194	6	-	-	PUNCT
ejpam-7054	194	7	int((σ1	int((σ1	PROPN
ejpam-7054	194	8	,	,	PUNCT
ejpam-7054	194	9	σ2)θ	σ2)θ	ADJ
ejpam-7054	194	10	-	-	PUNCT
ejpam-7054	194	11	cl(b	cl(b	NOUN
ejpam-7054	194	12	)	)	PUNCT
ejpam-7054	194	13	)	)	PUNCT
ejpam-7054	194	14	)	)	PUNCT
ejpam-7054	195	1	=	=	NOUN
ejpam-7054	195	2	∅	∅	NOUN
ejpam-7054	195	3	and	and	CCONJ
ejpam-7054	195	4	hence	hence	ADV
ejpam-7054	195	5	x	x	X
ejpam-7054	195	6	∈	∈	NOUN
ejpam-7054	195	7	x	x	PUNCT
ejpam-7054	195	8	−	−	NOUN
ejpam-7054	195	9	cµ(π)(f	cµ(π)(f	NOUN
ejpam-7054	195	10	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	195	11	-	-	PUNCT
ejpam-7054	195	12	int((σ1	int((σ1	ADJ
ejpam-7054	195	13	,	,	PUNCT
ejpam-7054	195	14	σ2)θ	σ2)θ	ADJ
ejpam-7054	195	15	-	-	PUNCT
ejpam-7054	195	16	cl(b	cl(b	NOUN
ejpam-7054	195	17	)	)	PUNCT
ejpam-7054	195	18	)	)	PUNCT
ejpam-7054	195	19	)	)	PUNCT
ejpam-7054	195	20	)	)	PUNCT
ejpam-7054	195	21	.	.	PUNCT
ejpam-7054	196	1	therefore	therefore	ADV
ejpam-7054	196	2	,	,	PUNCT
ejpam-7054	196	3	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	196	4	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	196	5	-	-	PUNCT
ejpam-7054	196	6	int((σ1	int((σ1	ADJ
ejpam-7054	196	7	,	,	PUNCT
ejpam-7054	196	8	σ2)θ	σ2)θ	ADJ
ejpam-7054	196	9	-	-	PUNCT
ejpam-7054	196	10	cl(b	cl(b	NOUN
ejpam-7054	196	11	)	)	PUNCT
ejpam-7054	196	12	)	)	PUNCT
ejpam-7054	196	13	)	)	PUNCT
ejpam-7054	196	14	)	)	PUNCT
ejpam-7054	197	1	⊆	⊆	NUM
ejpam-7054	197	2	f−((σ1	f−((σ1	NOUN
ejpam-7054	197	3	,	,	PUNCT
ejpam-7054	197	4	σ2)θ	σ2)θ	ADJ
ejpam-7054	197	5	-	-	PUNCT
ejpam-7054	197	6	cl(b	cl(b	NOUN
ejpam-7054	197	7	)	)	PUNCT
ejpam-7054	197	8	)	)	PUNCT
ejpam-7054	197	9	.	.	PUNCT
ejpam-7054	198	1	(	(	PUNCT
ejpam-7054	198	2	2	2	X
ejpam-7054	198	3	)	)	PUNCT
ejpam-7054	198	4	⇒	⇒	NOUN
ejpam-7054	198	5	(	(	PUNCT
ejpam-7054	198	6	3	3	NUM
ejpam-7054	198	7	):	):	PUNCT
ejpam-7054	198	8	the	the	DET
ejpam-7054	198	9	proof	proof	NOUN
ejpam-7054	198	10	is	be	AUX
ejpam-7054	198	11	obvious	obvious	ADJ
ejpam-7054	198	12	since	since	SCONJ
ejpam-7054	198	13	(	(	PUNCT
ejpam-7054	198	14	σ1	σ1	PROPN
ejpam-7054	198	15	,	,	PUNCT
ejpam-7054	198	16	σ2)θ	σ2)θ	NOUN
ejpam-7054	198	17	-	-	PUNCT
ejpam-7054	198	18	cl(v	cl(v	NOUN
ejpam-7054	198	19	)	)	PUNCT
ejpam-7054	198	20	=	=	SYM
ejpam-7054	198	21	σ1σ2	σ1σ2	NOUN
ejpam-7054	198	22	-	-	NUM
ejpam-7054	198	23	cl(v	cl(v	NOUN
ejpam-7054	198	24	)	)	PUNCT
ejpam-7054	198	25	for	for	ADP
ejpam-7054	198	26	every	every	DET
ejpam-7054	198	27	σ1σ2open	σ1σ2open	PUNCT
ejpam-7054	198	28	set	set	VERB
ejpam-7054	198	29	v	v	NOUN
ejpam-7054	198	30	of	of	ADP
ejpam-7054	198	31	y	y	PROPN
ejpam-7054	198	32	.	.	PUNCT
ejpam-7054	199	1	(	(	PUNCT
ejpam-7054	199	2	3	3	X
ejpam-7054	199	3	)	)	PUNCT
ejpam-7054	199	4	⇒	⇒	NOUN
ejpam-7054	199	5	(	(	PUNCT
ejpam-7054	199	6	4	4	NUM
ejpam-7054	199	7	):	):	PUNCT
ejpam-7054	199	8	let	let	VERB
ejpam-7054	199	9	v	v	PART
ejpam-7054	199	10	be	be	AUX
ejpam-7054	199	11	any	any	DET
ejpam-7054	199	12	(	(	PUNCT
ejpam-7054	199	13	σ1	σ1	PROPN
ejpam-7054	199	14	,	,	PUNCT
ejpam-7054	199	15	σ2)p	σ2)p	NOUN
ejpam-7054	199	16	-	-	PUNCT
ejpam-7054	199	17	open	open	ADJ
ejpam-7054	199	18	set	set	NOUN
ejpam-7054	199	19	of	of	ADP
ejpam-7054	199	20	y	y	PROPN
ejpam-7054	199	21	.	.	PUNCT
ejpam-7054	200	1	then	then	ADV
ejpam-7054	200	2	,	,	PUNCT
ejpam-7054	200	3	v	v	ADP
ejpam-7054	200	4	⊆	⊆	NUM
ejpam-7054	200	5	σ1σ2	σ1σ2	NOUN
ejpam-7054	200	6	-	-	PUNCT
ejpam-7054	200	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	200	8	-	-	PUNCT
ejpam-7054	200	9	cl(v	cl(v	NOUN
ejpam-7054	200	10	)	)	PUNCT
ejpam-7054	200	11	)	)	PUNCT
ejpam-7054	200	12	and	and	CCONJ
ejpam-7054	200	13	by	by	ADP
ejpam-7054	200	14	(	(	PUNCT
ejpam-7054	200	15	3	3	NUM
ejpam-7054	200	16	)	)	PUNCT
ejpam-7054	200	17	,	,	PUNCT
ejpam-7054	200	18	we	we	PRON
ejpam-7054	200	19	have	have	VERB
ejpam-7054	200	20	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	200	21	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	200	22	-	-	PUNCT
ejpam-7054	200	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	200	24	-	-	PUNCT
ejpam-7054	200	25	cl(v	cl(v	NOUN
ejpam-7054	200	26	)	)	PUNCT
ejpam-7054	200	27	)	)	PUNCT
ejpam-7054	200	28	)	)	PUNCT
ejpam-7054	200	29	)	)	PUNCT
ejpam-7054	201	1	=	=	PRON
ejpam-7054	201	2	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	201	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	201	4	-	-	PUNCT
ejpam-7054	201	5	int(σ1σ2	int(σ1σ2	ADV
ejpam-7054	201	6	-	-	PUNCT
ejpam-7054	201	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	201	8	-	-	PUNCT
ejpam-7054	201	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	201	10	-	-	PUNCT
ejpam-7054	201	11	cl(v	cl(v	NOUN
ejpam-7054	201	12	)	)	PUNCT
ejpam-7054	201	13	)	)	PUNCT
ejpam-7054	201	14	)	)	PUNCT
ejpam-7054	201	15	)	)	PUNCT
ejpam-7054	201	16	)	)	PUNCT
ejpam-7054	201	17	)	)	PUNCT
ejpam-7054	202	1	⊆	⊆	X
ejpam-7054	202	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7054	202	3	-	-	PUNCT
ejpam-7054	202	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	202	5	-	-	PUNCT
ejpam-7054	202	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	202	7	-	-	PUNCT
ejpam-7054	202	8	int(v	int(v	NOUN
ejpam-7054	202	9	)	)	PUNCT
ejpam-7054	202	10	)	)	PUNCT
ejpam-7054	202	11	)	)	PUNCT
ejpam-7054	202	12	)	)	PUNCT
ejpam-7054	203	1	b.	b.	PROPN
ejpam-7054	203	2	kong	kong	PROPN
ejpam-7054	203	3	-	-	PUNCT
ejpam-7054	203	4	ied	ied	PROPN
ejpam-7054	203	5	,	,	PUNCT
ejpam-7054	203	6	a.	a.	PROPN
ejpam-7054	203	7	sama	sama	PROPN
ejpam-7054	203	8	-	-	PUNCT
ejpam-7054	203	9	ae	ae	PROPN
ejpam-7054	203	10	,	,	PUNCT
ejpam-7054	203	11	c.	c.	PROPN
ejpam-7054	203	12	boonpok	boonpok	PROPN
ejpam-7054	203	13	/	/	SYM
ejpam-7054	203	14	eur	eur	PROPN
ejpam-7054	203	15	.	.	PUNCT
ejpam-7054	204	1	j.	j.	PROPN
ejpam-7054	204	2	pure	pure	PROPN
ejpam-7054	204	3	appl	appl	PROPN
ejpam-7054	204	4	.	.	PROPN
ejpam-7054	204	5	math	math	PROPN
ejpam-7054	204	6	,	,	PUNCT
ejpam-7054	204	7	18	18	NUM
ejpam-7054	204	8	(	(	PUNCT
ejpam-7054	204	9	4	4	NUM
ejpam-7054	204	10	)	)	PUNCT
ejpam-7054	204	11	(	(	PUNCT
ejpam-7054	204	12	2025	2025	NUM
ejpam-7054	204	13	)	)	PUNCT
ejpam-7054	204	14	,	,	PUNCT
ejpam-7054	204	15	7054	7054	NUM
ejpam-7054	204	16	7	7	NUM
ejpam-7054	204	17	of	of	ADP
ejpam-7054	204	18	12	12	NUM
ejpam-7054	204	19	=	=	NOUN
ejpam-7054	204	20	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	204	21	-	-	PUNCT
ejpam-7054	204	22	cl(v	cl(v	NOUN
ejpam-7054	204	23	)	)	PUNCT
ejpam-7054	204	24	)	)	PUNCT
ejpam-7054	204	25	.	.	PUNCT
ejpam-7054	205	1	(	(	PUNCT
ejpam-7054	205	2	4	4	X
ejpam-7054	205	3	)	)	PUNCT
ejpam-7054	205	4	⇒	⇒	NOUN
ejpam-7054	205	5	(	(	PUNCT
ejpam-7054	205	6	5	5	NUM
ejpam-7054	205	7	):	):	PUNCT
ejpam-7054	205	8	let	let	VERB
ejpam-7054	205	9	k	k	PRON
ejpam-7054	205	10	be	be	AUX
ejpam-7054	205	11	any	any	DET
ejpam-7054	205	12	(	(	PUNCT
ejpam-7054	205	13	σ1	σ1	NOUN
ejpam-7054	205	14	,	,	PUNCT
ejpam-7054	205	15	σ2)r	σ2)r	NOUN
ejpam-7054	205	16	-	-	PUNCT
ejpam-7054	205	17	closed	close	VERB
ejpam-7054	205	18	set	set	NOUN
ejpam-7054	205	19	of	of	ADP
ejpam-7054	205	20	y	y	PROPN
ejpam-7054	205	21	.	.	PUNCT
ejpam-7054	206	1	then	then	ADV
ejpam-7054	206	2	,	,	PUNCT
ejpam-7054	206	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	206	4	-	-	PUNCT
ejpam-7054	206	5	int(k	int(k	NOUN
ejpam-7054	206	6	)	)	PUNCT
ejpam-7054	206	7	is	be	AUX
ejpam-7054	206	8	(	(	PUNCT
ejpam-7054	206	9	σ1	σ1	PROPN
ejpam-7054	206	10	,	,	PUNCT
ejpam-7054	206	11	σ2)p	σ2)p	NOUN
ejpam-7054	206	12	-	-	PUNCT
ejpam-7054	206	13	open	open	ADJ
ejpam-7054	206	14	in	in	ADP
ejpam-7054	206	15	y	y	PROPN
ejpam-7054	206	16	and	and	CCONJ
ejpam-7054	206	17	by	by	ADP
ejpam-7054	206	18	(	(	PUNCT
ejpam-7054	206	19	4	4	NUM
ejpam-7054	206	20	)	)	PUNCT
ejpam-7054	206	21	,	,	PUNCT
ejpam-7054	206	22	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	206	23	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	206	24	-	-	PUNCT
ejpam-7054	206	25	int(k	int(k	NUM
ejpam-7054	206	26	)	)	PUNCT
ejpam-7054	206	27	)	)	PUNCT
ejpam-7054	206	28	)	)	PUNCT
ejpam-7054	207	1	=	=	PRON
ejpam-7054	207	2	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	207	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	207	4	-	-	PUNCT
ejpam-7054	207	5	int(σ1σ2	int(σ1σ2	ADV
ejpam-7054	207	6	-	-	PUNCT
ejpam-7054	207	7	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-7054	207	8	-	-	PUNCT
ejpam-7054	207	9	int(k	int(k	NOUN
ejpam-7054	207	10	)	)	PUNCT
ejpam-7054	207	11	)	)	PUNCT
ejpam-7054	207	12	)	)	PUNCT
ejpam-7054	207	13	)	)	PUNCT
ejpam-7054	207	14	)	)	PUNCT
ejpam-7054	208	1	⊆	⊆	X
ejpam-7054	208	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7054	208	3	-	-	PUNCT
ejpam-7054	208	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-7054	208	5	-	-	PUNCT
ejpam-7054	208	6	int(k	int(k	NOUN
ejpam-7054	208	7	)	)	PUNCT
ejpam-7054	208	8	)	)	PUNCT
ejpam-7054	208	9	)	)	PUNCT
ejpam-7054	209	1	=	=	SYM
ejpam-7054	209	2	f−(k	f−(k	PROPN
ejpam-7054	209	3	)	)	PUNCT
ejpam-7054	209	4	.	.	PUNCT
ejpam-7054	210	1	(	(	PUNCT
ejpam-7054	210	2	5	5	X
ejpam-7054	210	3	)	)	PUNCT
ejpam-7054	210	4	⇒	⇒	NOUN
ejpam-7054	210	5	(	(	PUNCT
ejpam-7054	210	6	1	1	NUM
ejpam-7054	210	7	):	):	PUNCT
ejpam-7054	210	8	let	let	VERB
ejpam-7054	210	9	v	v	PART
ejpam-7054	210	10	be	be	AUX
ejpam-7054	210	11	any	any	DET
ejpam-7054	210	12	σ1σ2	σ1σ2	NOUN
ejpam-7054	210	13	-	-	ADJ
ejpam-7054	210	14	open	open	ADJ
ejpam-7054	210	15	set	set	NOUN
ejpam-7054	210	16	of	of	ADP
ejpam-7054	210	17	y	y	PROPN
ejpam-7054	210	18	.	.	PUNCT
ejpam-7054	211	1	then	then	ADV
ejpam-7054	211	2	,	,	PUNCT
ejpam-7054	211	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	211	4	-	-	NUM
ejpam-7054	211	5	cl(v	cl(v	NOUN
ejpam-7054	211	6	)	)	PUNCT
ejpam-7054	211	7	is	be	AUX
ejpam-7054	211	8	(	(	PUNCT
ejpam-7054	211	9	σ1	σ1	NOUN
ejpam-7054	211	10	,	,	PUNCT
ejpam-7054	211	11	σ2)r	σ2)r	NOUN
ejpam-7054	211	12	-	-	PUNCT
ejpam-7054	211	13	closed	closed	ADJ
ejpam-7054	211	14	in	in	ADP
ejpam-7054	211	15	y	y	PROPN
ejpam-7054	211	16	and	and	CCONJ
ejpam-7054	211	17	by	by	ADP
ejpam-7054	211	18	(	(	PUNCT
ejpam-7054	211	19	5	5	NUM
ejpam-7054	211	20	)	)	PUNCT
ejpam-7054	211	21	,	,	PUNCT
ejpam-7054	211	22	cµ(π)(f	cµ(π)(f	PROPN
ejpam-7054	211	23	−(v	−(v	NOUN
ejpam-7054	211	24	)	)	PUNCT
ejpam-7054	211	25	)	)	PUNCT
ejpam-7054	212	1	⊆	⊆	NUM
ejpam-7054	212	2	cµ(π)(f	cµ(π)(f	NOUN
ejpam-7054	212	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	212	4	-	-	PUNCT
ejpam-7054	212	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	212	6	-	-	PUNCT
ejpam-7054	212	7	cl(v	cl(v	NOUN
ejpam-7054	212	8	)	)	PUNCT
ejpam-7054	212	9	)	)	PUNCT
ejpam-7054	212	10	)	)	PUNCT
ejpam-7054	212	11	)	)	PUNCT
ejpam-7054	213	1	⊆	⊆	X
ejpam-7054	213	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	213	3	-	-	PUNCT
ejpam-7054	213	4	cl(v	cl(v	NOUN
ejpam-7054	213	5	)	)	PUNCT
ejpam-7054	213	6	)	)	PUNCT
ejpam-7054	213	7	.	.	PUNCT
ejpam-7054	214	1	it	it	PRON
ejpam-7054	214	2	follows	follow	VERB
ejpam-7054	214	3	from	from	ADP
ejpam-7054	214	4	theorem	theorem	ADJ
ejpam-7054	214	5	1	1	NUM
ejpam-7054	214	6	that	that	SCONJ
ejpam-7054	214	7	f	f	PROPN
ejpam-7054	214	8	is	be	AUX
ejpam-7054	214	9	upper	upper	ADJ
ejpam-7054	214	10	almost	almost	ADV
ejpam-7054	214	11	weakly	weakly	ADJ
ejpam-7054	214	12	µ(σ1	µ(σ1	NOUN
ejpam-7054	214	13	,	,	PUNCT
ejpam-7054	214	14	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7054	214	15	.	.	X
ejpam-7054	215	1	theorem	theorem	VERB
ejpam-7054	215	2	6	6	NUM
ejpam-7054	215	3	.	.	PUNCT
ejpam-7054	215	4	for	for	ADP
ejpam-7054	215	5	a	a	DET
ejpam-7054	215	6	multifunction	multifunction	NOUN
ejpam-7054	216	1	f	f	NOUN
ejpam-7054	216	2	:	:	PUNCT
ejpam-7054	216	3	(	(	PUNCT
ejpam-7054	216	4	x,µ	x,µ	NOUN
ejpam-7054	216	5	)	)	PUNCT
ejpam-7054	216	6	→	→	SYM
ejpam-7054	216	7	(	(	PUNCT
ejpam-7054	216	8	y	y	PROPN
ejpam-7054	216	9	,	,	PUNCT
ejpam-7054	216	10	σ1	σ1	PROPN
ejpam-7054	216	11	,	,	PUNCT
ejpam-7054	216	12	σ2	σ2	NOUN
ejpam-7054	216	13	)	)	PUNCT
ejpam-7054	216	14	,	,	PUNCT
ejpam-7054	216	15	the	the	DET
ejpam-7054	216	16	following	follow	VERB
ejpam-7054	216	17	properties	property	NOUN
ejpam-7054	216	18	are	be	AUX
ejpam-7054	216	19	equivalent	equivalent	ADJ
ejpam-7054	216	20	:	:	PUNCT
ejpam-7054	216	21	(	(	PUNCT
ejpam-7054	216	22	1	1	X
ejpam-7054	216	23	)	)	PUNCT
ejpam-7054	216	24	f	f	PROPN
ejpam-7054	216	25	is	be	AUX
ejpam-7054	216	26	lower	low	ADJ
ejpam-7054	216	27	almost	almost	ADV
ejpam-7054	216	28	weakly	weakly	ADJ
ejpam-7054	216	29	µ(σ1	µ(σ1	NOUN
ejpam-7054	216	30	,	,	PUNCT
ejpam-7054	216	31	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	216	32	;	;	PUNCT
ejpam-7054	216	33	(	(	PUNCT
ejpam-7054	216	34	2	2	X
ejpam-7054	216	35	)	)	PUNCT
ejpam-7054	216	36	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	217	1	+	+	PROPN
ejpam-7054	217	2	(	(	PUNCT
ejpam-7054	217	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	217	4	-	-	PUNCT
ejpam-7054	217	5	int((σ1	int((σ1	ADJ
ejpam-7054	217	6	,	,	PUNCT
ejpam-7054	217	7	σ2)θ	σ2)θ	ADJ
ejpam-7054	217	8	-	-	PUNCT
ejpam-7054	217	9	cl(b	cl(b	NOUN
ejpam-7054	217	10	)	)	PUNCT
ejpam-7054	217	11	)	)	PUNCT
ejpam-7054	217	12	)	)	PUNCT
ejpam-7054	217	13	)	)	PUNCT
ejpam-7054	218	1	⊆	⊆	NUM
ejpam-7054	218	2	f+((σ1	f+((σ1	NOUN
ejpam-7054	218	3	,	,	PUNCT
ejpam-7054	218	4	σ2)θ	σ2)θ	ADJ
ejpam-7054	218	5	-	-	PUNCT
ejpam-7054	218	6	cl(b	cl(b	NOUN
ejpam-7054	218	7	)	)	PUNCT
ejpam-7054	218	8	)	)	PUNCT
ejpam-7054	218	9	for	for	ADP
ejpam-7054	218	10	every	every	DET
ejpam-7054	218	11	subset	subset	NOUN
ejpam-7054	218	12	b	b	PROPN
ejpam-7054	218	13	of	of	ADP
ejpam-7054	218	14	y	y	PROPN
ejpam-7054	218	15	;	;	PUNCT
ejpam-7054	218	16	(	(	PUNCT
ejpam-7054	218	17	3	3	X
ejpam-7054	218	18	)	)	PUNCT
ejpam-7054	218	19	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	219	1	+	+	PROPN
ejpam-7054	219	2	(	(	PUNCT
ejpam-7054	219	3	σ1σ2	σ1σ2	NUM
ejpam-7054	219	4	-	-	PUNCT
ejpam-7054	219	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	219	6	-	-	PUNCT
ejpam-7054	219	7	cl(v	cl(v	NOUN
ejpam-7054	219	8	)	)	PUNCT
ejpam-7054	219	9	)	)	PUNCT
ejpam-7054	219	10	)	)	PUNCT
ejpam-7054	219	11	)	)	PUNCT
ejpam-7054	220	1	⊆	⊆	X
ejpam-7054	220	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	220	3	-	-	PUNCT
ejpam-7054	220	4	cl(v	cl(v	NOUN
ejpam-7054	220	5	)	)	PUNCT
ejpam-7054	220	6	)	)	PUNCT
ejpam-7054	220	7	for	for	ADP
ejpam-7054	220	8	every	every	DET
ejpam-7054	220	9	σ1σ2	σ1σ2	NOUN
ejpam-7054	220	10	-	-	ADJ
ejpam-7054	220	11	open	open	ADJ
ejpam-7054	220	12	set	set	NOUN
ejpam-7054	220	13	v	v	NOUN
ejpam-7054	220	14	of	of	ADP
ejpam-7054	220	15	y	y	PROPN
ejpam-7054	220	16	;	;	PUNCT
ejpam-7054	220	17	(	(	PUNCT
ejpam-7054	220	18	4	4	X
ejpam-7054	220	19	)	)	PUNCT
ejpam-7054	220	20	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	221	1	+	+	PROPN
ejpam-7054	221	2	(	(	PUNCT
ejpam-7054	221	3	σ1σ2	σ1σ2	NUM
ejpam-7054	221	4	-	-	PUNCT
ejpam-7054	221	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-7054	221	6	-	-	PUNCT
ejpam-7054	221	7	cl(v	cl(v	NOUN
ejpam-7054	221	8	)	)	PUNCT
ejpam-7054	221	9	)	)	PUNCT
ejpam-7054	221	10	)	)	PUNCT
ejpam-7054	221	11	)	)	PUNCT
ejpam-7054	222	1	⊆	⊆	X
ejpam-7054	222	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	222	3	-	-	PUNCT
ejpam-7054	222	4	cl(v	cl(v	NOUN
ejpam-7054	222	5	)	)	PUNCT
ejpam-7054	222	6	)	)	PUNCT
ejpam-7054	222	7	for	for	ADP
ejpam-7054	222	8	every	every	DET
ejpam-7054	222	9	(	(	PUNCT
ejpam-7054	222	10	σ1	σ1	PROPN
ejpam-7054	222	11	,	,	PUNCT
ejpam-7054	222	12	σ2)p	σ2)p	NOUN
ejpam-7054	222	13	-	-	PUNCT
ejpam-7054	222	14	open	open	NOUN
ejpam-7054	222	15	set	set	NOUN
ejpam-7054	222	16	v	v	NOUN
ejpam-7054	222	17	of	of	ADP
ejpam-7054	222	18	y	y	PROPN
ejpam-7054	222	19	;	;	PUNCT
ejpam-7054	222	20	(	(	PUNCT
ejpam-7054	222	21	5	5	X
ejpam-7054	222	22	)	)	PUNCT
ejpam-7054	222	23	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	223	1	+	+	PROPN
ejpam-7054	223	2	(	(	PUNCT
ejpam-7054	223	3	σ1σ2	σ1σ2	NUM
ejpam-7054	223	4	-	-	PUNCT
ejpam-7054	223	5	int(k	int(k	NUM
ejpam-7054	223	6	)	)	PUNCT
ejpam-7054	223	7	)	)	PUNCT
ejpam-7054	223	8	)	)	PUNCT
ejpam-7054	224	1	⊆	⊆	NUM
ejpam-7054	224	2	f+(k	f+(k	NOUN
ejpam-7054	224	3	)	)	PUNCT
ejpam-7054	224	4	for	for	ADP
ejpam-7054	224	5	every	every	DET
ejpam-7054	224	6	(	(	PUNCT
ejpam-7054	224	7	σ1	σ1	PROPN
ejpam-7054	224	8	,	,	PUNCT
ejpam-7054	224	9	σ2)r	σ2)r	NOUN
ejpam-7054	224	10	-	-	PUNCT
ejpam-7054	224	11	closed	close	VERB
ejpam-7054	224	12	set	set	ADJ
ejpam-7054	224	13	k	k	PROPN
ejpam-7054	224	14	of	of	ADP
ejpam-7054	224	15	y	y	PROPN
ejpam-7054	224	16	.	.	PUNCT
ejpam-7054	225	1	proof	proof	NOUN
ejpam-7054	225	2	.	.	PUNCT
ejpam-7054	226	1	the	the	DET
ejpam-7054	226	2	proof	proof	NOUN
ejpam-7054	226	3	is	be	AUX
ejpam-7054	226	4	similar	similar	ADJ
ejpam-7054	226	5	to	to	ADP
ejpam-7054	226	6	that	that	PRON
ejpam-7054	226	7	of	of	ADP
ejpam-7054	226	8	theorem	theorem	NOUN
ejpam-7054	226	9	5	5	NUM
ejpam-7054	226	10	.	.	PUNCT
ejpam-7054	227	1	the	the	DET
ejpam-7054	227	2	µ-prefrontier	µ-prefrontier	NOUN
ejpam-7054	227	3	of	of	ADP
ejpam-7054	227	4	a	a	DET
ejpam-7054	227	5	subset	subset	NOUN
ejpam-7054	227	6	a	a	PRON
ejpam-7054	227	7	of	of	ADP
ejpam-7054	227	8	a	a	DET
ejpam-7054	227	9	generalized	generalized	ADJ
ejpam-7054	227	10	topological	topological	ADJ
ejpam-7054	227	11	space	space	NOUN
ejpam-7054	227	12	(	(	PUNCT
ejpam-7054	227	13	x,µ	x,µ	NOUN
ejpam-7054	227	14	)	)	PUNCT
ejpam-7054	227	15	,	,	PUNCT
ejpam-7054	227	16	denoted	denote	VERB
ejpam-7054	227	17	by	by	ADP
ejpam-7054	227	18	µ(π)fr(a	µ(π)fr(a	VERB
ejpam-7054	227	19	)	)	PUNCT
ejpam-7054	227	20	,	,	PUNCT
ejpam-7054	227	21	is	be	AUX
ejpam-7054	227	22	defined	define	VERB
ejpam-7054	227	23	by	by	ADP
ejpam-7054	227	24	µ(π)pf(a	µ(π)pf(a	PRON
ejpam-7054	227	25	)	)	PUNCT
ejpam-7054	227	26	=	=	PUNCT
ejpam-7054	227	27	cµ(π)(a	cµ(π)(a	NOUN
ejpam-7054	227	28	)	)	PUNCT
ejpam-7054	227	29	∩	∩	ADJ
ejpam-7054	227	30	cµ(π)(x	cµ(π)(x	NOUN
ejpam-7054	227	31	−a	−a	NOUN
ejpam-7054	227	32	)	)	PUNCT
ejpam-7054	228	1	=	=	SYM
ejpam-7054	228	2	cµ(π)(a)−	cµ(π)(a)−	NOUN
ejpam-7054	228	3	iµ(π)(a	iµ(π)(a	NUM
ejpam-7054	228	4	)	)	PUNCT
ejpam-7054	228	5	.	.	PUNCT
ejpam-7054	229	1	theorem	theorem	VERB
ejpam-7054	229	2	7	7	NUM
ejpam-7054	229	3	.	.	PUNCT
ejpam-7054	230	1	the	the	DET
ejpam-7054	230	2	set	set	NOUN
ejpam-7054	230	3	of	of	ADP
ejpam-7054	230	4	all	all	DET
ejpam-7054	230	5	points	point	NOUN
ejpam-7054	230	6	x	x	PUNCT
ejpam-7054	230	7	of	of	ADP
ejpam-7054	230	8	x	x	SYM
ejpam-7054	230	9	at	at	ADP
ejpam-7054	230	10	which	which	PRON
ejpam-7054	230	11	a	a	DET
ejpam-7054	230	12	multifunction	multifunction	NOUN
ejpam-7054	230	13	f	f	X
ejpam-7054	230	14	:	:	PUNCT
ejpam-7054	230	15	(	(	PUNCT
ejpam-7054	230	16	x,µ	x,µ	NOUN
ejpam-7054	230	17	)	)	PUNCT
ejpam-7054	230	18	→	→	SYM
ejpam-7054	230	19	(	(	PUNCT
ejpam-7054	230	20	y	y	PROPN
ejpam-7054	230	21	,	,	PUNCT
ejpam-7054	230	22	σ1	σ1	PROPN
ejpam-7054	230	23	,	,	PUNCT
ejpam-7054	230	24	σ2	σ2	PROPN
ejpam-7054	230	25	)	)	PUNCT
ejpam-7054	230	26	is	be	AUX
ejpam-7054	230	27	not	not	PART
ejpam-7054	230	28	upper	upper	ADJ
ejpam-7054	230	29	almost	almost	ADV
ejpam-7054	230	30	weakly	weakly	ADJ
ejpam-7054	230	31	µ(σ1	µ(σ1	NOUN
ejpam-7054	230	32	,	,	PUNCT
ejpam-7054	230	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	230	34	is	be	AUX
ejpam-7054	230	35	identical	identical	ADJ
ejpam-7054	230	36	with	with	ADP
ejpam-7054	230	37	the	the	DET
ejpam-7054	230	38	union	union	NOUN
ejpam-7054	230	39	of	of	ADP
ejpam-7054	230	40	the	the	DET
ejpam-7054	230	41	µprefrontier	µprefronti	ADJ
ejpam-7054	230	42	of	of	ADP
ejpam-7054	230	43	the	the	DET
ejpam-7054	230	44	upper	upper	ADJ
ejpam-7054	230	45	inverse	inverse	NOUN
ejpam-7054	230	46	images	image	NOUN
ejpam-7054	230	47	of	of	ADP
ejpam-7054	230	48	the	the	DET
ejpam-7054	230	49	σ1σ2	σ1σ2	NOUN
ejpam-7054	230	50	-	-	NOUN
ejpam-7054	230	51	closure	closure	NOUN
ejpam-7054	230	52	of	of	ADP
ejpam-7054	230	53	σ1σ2	σ1σ2	NOUN
ejpam-7054	230	54	-	-	PUNCT
ejpam-7054	230	55	open	open	ADJ
ejpam-7054	230	56	sets	set	NOUN
ejpam-7054	230	57	containing	contain	VERB
ejpam-7054	230	58	f	f	X
ejpam-7054	230	59	(	(	PUNCT
ejpam-7054	230	60	x	x	NOUN
ejpam-7054	230	61	)	)	PUNCT
ejpam-7054	230	62	.	.	PUNCT
ejpam-7054	231	1	proof	proof	NOUN
ejpam-7054	231	2	.	.	PUNCT
ejpam-7054	232	1	let	let	VERB
ejpam-7054	232	2	x	x	PUNCT
ejpam-7054	232	3	∈	∈	PROPN
ejpam-7054	232	4	x	x	PUNCT
ejpam-7054	232	5	at	at	ADP
ejpam-7054	232	6	which	which	PRON
ejpam-7054	232	7	f	f	NOUN
ejpam-7054	232	8	is	be	AUX
ejpam-7054	232	9	not	not	PART
ejpam-7054	232	10	upper	upper	ADJ
ejpam-7054	232	11	almost	almost	ADV
ejpam-7054	232	12	weakly	weakly	ADJ
ejpam-7054	232	13	µ(σ1	µ(σ1	NOUN
ejpam-7054	232	14	,	,	PUNCT
ejpam-7054	232	15	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7054	232	16	.	.	PUNCT
ejpam-7054	233	1	there	there	PRON
ejpam-7054	233	2	exists	exist	VERB
ejpam-7054	233	3	a	a	DET
ejpam-7054	233	4	σ1σ2	σ1σ2	NUM
ejpam-7054	233	5	-	-	ADJ
ejpam-7054	233	6	open	open	ADJ
ejpam-7054	233	7	set	set	NOUN
ejpam-7054	233	8	v	v	NOUN
ejpam-7054	233	9	of	of	ADP
ejpam-7054	233	10	y	y	PROPN
ejpam-7054	233	11	containing	contain	VERB
ejpam-7054	233	12	f	f	PROPN
ejpam-7054	233	13	(	(	PUNCT
ejpam-7054	233	14	x	x	X
ejpam-7054	233	15	)	)	PUNCT
ejpam-7054	233	16	such	such	ADJ
ejpam-7054	233	17	that	that	SCONJ
ejpam-7054	233	18	u	u	PROPN
ejpam-7054	233	19	∩	∩	NOUN
ejpam-7054	233	20	(	(	PUNCT
ejpam-7054	233	21	x	x	NOUN
ejpam-7054	233	22	−	−	PROPN
ejpam-7054	233	23	f+(v	f+(v	NOUN
ejpam-7054	233	24	)	)	PUNCT
ejpam-7054	233	25	)	)	PUNCT
ejpam-7054	234	1	̸=	̸=	NOUN
ejpam-7054	234	2	∅	∅	NOUN
ejpam-7054	234	3	for	for	ADP
ejpam-7054	234	4	every	every	DET
ejpam-7054	234	5	µ-preopen	µ-preopen	PROPN
ejpam-7054	234	6	set	set	VERB
ejpam-7054	234	7	u	u	NOUN
ejpam-7054	234	8	of	of	ADP
ejpam-7054	234	9	x	x	SYM
ejpam-7054	234	10	containing	contain	VERB
ejpam-7054	234	11	x.	x.	NOUN
ejpam-7054	234	12	therefore	therefore	ADV
ejpam-7054	234	13	,	,	PUNCT
ejpam-7054	234	14	we	we	PRON
ejpam-7054	234	15	have	have	VERB
ejpam-7054	234	16	x	x	X
ejpam-7054	234	17	∈	∈	PROPN
ejpam-7054	234	18	cµ(π)(x	cµ(π)(x	NOUN
ejpam-7054	234	19	−	−	ADP
ejpam-7054	234	20	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	234	21	-	-	PUNCT
ejpam-7054	234	22	cl(v	cl(v	NOUN
ejpam-7054	234	23	)	)	PUNCT
ejpam-7054	234	24	)	)	PUNCT
ejpam-7054	234	25	)	)	PUNCT
ejpam-7054	235	1	=	=	PUNCT
ejpam-7054	236	1	x	x	X
ejpam-7054	236	2	−	−	NOUN
ejpam-7054	236	3	iµ(π)(f	iµ(π)(f	VERB
ejpam-7054	237	1	+	+	NOUN
ejpam-7054	237	2	(	(	PUNCT
ejpam-7054	237	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	237	4	-	-	NUM
ejpam-7054	237	5	cl(v	cl(v	NOUN
ejpam-7054	237	6	)	)	PUNCT
ejpam-7054	237	7	)	)	PUNCT
ejpam-7054	237	8	)	)	PUNCT
ejpam-7054	237	9	.	.	PUNCT
ejpam-7054	238	1	b.	b.	PROPN
ejpam-7054	238	2	kong	kong	PROPN
ejpam-7054	238	3	-	-	PUNCT
ejpam-7054	238	4	ied	ied	PROPN
ejpam-7054	238	5	,	,	PUNCT
ejpam-7054	238	6	a.	a.	PROPN
ejpam-7054	238	7	sama	sama	PROPN
ejpam-7054	238	8	-	-	PUNCT
ejpam-7054	238	9	ae	ae	PROPN
ejpam-7054	238	10	,	,	PUNCT
ejpam-7054	238	11	c.	c.	PROPN
ejpam-7054	238	12	boonpok	boonpok	PROPN
ejpam-7054	238	13	/	/	SYM
ejpam-7054	238	14	eur	eur	PROPN
ejpam-7054	238	15	.	.	PUNCT
ejpam-7054	239	1	j.	j.	PROPN
ejpam-7054	239	2	pure	pure	PROPN
ejpam-7054	239	3	appl	appl	PROPN
ejpam-7054	239	4	.	.	PROPN
ejpam-7054	239	5	math	math	PROPN
ejpam-7054	239	6	,	,	PUNCT
ejpam-7054	239	7	18	18	NUM
ejpam-7054	239	8	(	(	PUNCT
ejpam-7054	239	9	4	4	NUM
ejpam-7054	239	10	)	)	PUNCT
ejpam-7054	239	11	(	(	PUNCT
ejpam-7054	239	12	2025	2025	NUM
ejpam-7054	239	13	)	)	PUNCT
ejpam-7054	239	14	,	,	PUNCT
ejpam-7054	239	15	7054	7054	NUM
ejpam-7054	239	16	8	8	NUM
ejpam-7054	239	17	of	of	ADP
ejpam-7054	239	18	12	12	NUM
ejpam-7054	239	19	since	since	SCONJ
ejpam-7054	239	20	x	x	PROPN
ejpam-7054	239	21	∈	∈	PROPN
ejpam-7054	239	22	f+(v	f+(v	NOUN
ejpam-7054	239	23	)	)	PUNCT
ejpam-7054	240	1	,	,	PUNCT
ejpam-7054	240	2	we	we	PRON
ejpam-7054	240	3	have	have	VERB
ejpam-7054	240	4	x	x	PART
ejpam-7054	240	5	∈	∈	PRON
ejpam-7054	240	6	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	241	1	+	+	ADJ
ejpam-7054	241	2	(	(	PUNCT
ejpam-7054	241	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	241	4	-	-	NUM
ejpam-7054	241	5	cl(v	cl(v	NOUN
ejpam-7054	241	6	)	)	PUNCT
ejpam-7054	241	7	)	)	PUNCT
ejpam-7054	241	8	)	)	PUNCT
ejpam-7054	242	1	and	and	CCONJ
ejpam-7054	242	2	so	so	ADV
ejpam-7054	242	3	x	x	SYM
ejpam-7054	242	4	∈	∈	PROPN
ejpam-7054	242	5	µ(π)fr(f+(σ1σ2	µ(π)fr(f+(σ1σ2	NOUN
ejpam-7054	242	6	-	-	PUNCT
ejpam-7054	242	7	cl(v	cl(v	NOUN
ejpam-7054	242	8	)	)	PUNCT
ejpam-7054	242	9	)	)	PUNCT
ejpam-7054	242	10	)	)	PUNCT
ejpam-7054	242	11	.	.	PUNCT
ejpam-7054	243	1	conversely	conversely	ADV
ejpam-7054	243	2	,	,	PUNCT
ejpam-7054	243	3	if	if	SCONJ
ejpam-7054	243	4	f	f	PROPN
ejpam-7054	243	5	is	be	AUX
ejpam-7054	243	6	upper	upper	ADJ
ejpam-7054	243	7	almost	almost	ADV
ejpam-7054	243	8	weakly	weakly	ADJ
ejpam-7054	243	9	µ(σ1	µ(σ1	NOUN
ejpam-7054	243	10	,	,	PUNCT
ejpam-7054	243	11	σ2)-continuous	σ2)-continuous	PROPN
ejpam-7054	243	12	,	,	PUNCT
ejpam-7054	243	13	then	then	ADV
ejpam-7054	243	14	for	for	ADP
ejpam-7054	243	15	any	any	DET
ejpam-7054	243	16	σ1σ2	σ1σ2	NOUN
ejpam-7054	243	17	-	-	ADJ
ejpam-7054	243	18	open	open	ADJ
ejpam-7054	243	19	set	set	NOUN
ejpam-7054	243	20	v	v	NOUN
ejpam-7054	243	21	of	of	ADP
ejpam-7054	243	22	y	y	PROPN
ejpam-7054	243	23	containing	contain	VERB
ejpam-7054	243	24	f	f	PROPN
ejpam-7054	243	25	(	(	PUNCT
ejpam-7054	243	26	x	x	X
ejpam-7054	243	27	)	)	PUNCT
ejpam-7054	243	28	there	there	PRON
ejpam-7054	243	29	exists	exist	VERB
ejpam-7054	243	30	a	a	DET
ejpam-7054	243	31	µ-preopen	µ-preopen	PROPN
ejpam-7054	243	32	set	set	VERB
ejpam-7054	243	33	u	u	NOUN
ejpam-7054	243	34	of	of	ADP
ejpam-7054	243	35	x	x	PUNCT
ejpam-7054	243	36	containing	contain	VERB
ejpam-7054	243	37	x	x	PUNCT
ejpam-7054	243	38	such	such	ADJ
ejpam-7054	243	39	that	that	SCONJ
ejpam-7054	243	40	f	f	PROPN
ejpam-7054	243	41	(	(	PUNCT
ejpam-7054	243	42	u	u	NOUN
ejpam-7054	243	43	)	)	PUNCT
ejpam-7054	243	44	⊆	⊆	NUM
ejpam-7054	243	45	σ1σ2	σ1σ2	NOUN
ejpam-7054	243	46	-	-	NUM
ejpam-7054	243	47	cl(v	cl(v	NOUN
ejpam-7054	243	48	)	)	PUNCT
ejpam-7054	243	49	;	;	PUNCT
ejpam-7054	243	50	hence	hence	ADV
ejpam-7054	243	51	u	u	NOUN
ejpam-7054	243	52	⊆	⊆	NUM
ejpam-7054	243	53	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	243	54	-	-	PUNCT
ejpam-7054	243	55	cl(v	cl(v	NOUN
ejpam-7054	243	56	)	)	PUNCT
ejpam-7054	243	57	)	)	PUNCT
ejpam-7054	243	58	.	.	PUNCT
ejpam-7054	244	1	therefore	therefore	ADV
ejpam-7054	244	2	,	,	PUNCT
ejpam-7054	244	3	x	x	PUNCT
ejpam-7054	244	4	∈	∈	PRON
ejpam-7054	244	5	iµ(π)(f	iµ(π)(f	VERB
ejpam-7054	245	1	+	+	ADJ
ejpam-7054	245	2	(	(	PUNCT
ejpam-7054	245	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	245	4	-	-	NUM
ejpam-7054	245	5	cl(v	cl(v	NOUN
ejpam-7054	245	6	)	)	PUNCT
ejpam-7054	245	7	)	)	PUNCT
ejpam-7054	245	8	)	)	PUNCT
ejpam-7054	245	9	.	.	PUNCT
ejpam-7054	246	1	this	this	PRON
ejpam-7054	246	2	contradicts	contradict	VERB
ejpam-7054	246	3	with	with	ADP
ejpam-7054	246	4	the	the	DET
ejpam-7054	246	5	fact	fact	NOUN
ejpam-7054	246	6	that	that	SCONJ
ejpam-7054	246	7	x	x	PUNCT
ejpam-7054	246	8	∈	∈	PROPN
ejpam-7054	246	9	µ(π)fr(f+(σ1σ2	µ(π)fr(f+(σ1σ2	NOUN
ejpam-7054	246	10	-	-	PUNCT
ejpam-7054	246	11	cl(v	cl(v	NOUN
ejpam-7054	246	12	)	)	PUNCT
ejpam-7054	246	13	)	)	PUNCT
ejpam-7054	246	14	)	)	PUNCT
ejpam-7054	246	15	.	.	PUNCT
ejpam-7054	247	1	thus	thus	ADV
ejpam-7054	247	2	,	,	PUNCT
ejpam-7054	247	3	f	f	PROPN
ejpam-7054	247	4	is	be	AUX
ejpam-7054	247	5	not	not	PART
ejpam-7054	247	6	upper	upper	ADJ
ejpam-7054	247	7	almost	almost	ADV
ejpam-7054	247	8	weakly	weakly	ADJ
ejpam-7054	247	9	µ(σ1	µ(σ1	NOUN
ejpam-7054	247	10	,	,	PUNCT
ejpam-7054	247	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	247	12	at	at	ADP
ejpam-7054	247	13	x.	x.	NOUN
ejpam-7054	247	14	theorem	theorem	VERB
ejpam-7054	247	15	8	8	NUM
ejpam-7054	247	16	.	.	PUNCT
ejpam-7054	248	1	the	the	DET
ejpam-7054	248	2	set	set	NOUN
ejpam-7054	248	3	of	of	ADP
ejpam-7054	248	4	all	all	DET
ejpam-7054	248	5	points	point	NOUN
ejpam-7054	248	6	x	x	PUNCT
ejpam-7054	248	7	of	of	ADP
ejpam-7054	248	8	x	x	SYM
ejpam-7054	248	9	at	at	ADP
ejpam-7054	248	10	which	which	PRON
ejpam-7054	248	11	a	a	DET
ejpam-7054	248	12	multifunction	multifunction	NOUN
ejpam-7054	248	13	f	f	X
ejpam-7054	248	14	:	:	PUNCT
ejpam-7054	248	15	(	(	PUNCT
ejpam-7054	248	16	x,µ	x,µ	NOUN
ejpam-7054	248	17	)	)	PUNCT
ejpam-7054	248	18	→	→	SYM
ejpam-7054	248	19	(	(	PUNCT
ejpam-7054	248	20	y	y	PROPN
ejpam-7054	248	21	,	,	PUNCT
ejpam-7054	248	22	σ1	σ1	PROPN
ejpam-7054	248	23	,	,	PUNCT
ejpam-7054	248	24	σ2	σ2	PROPN
ejpam-7054	248	25	)	)	PUNCT
ejpam-7054	248	26	is	be	AUX
ejpam-7054	248	27	not	not	PART
ejpam-7054	248	28	lower	low	ADJ
ejpam-7054	248	29	almost	almost	ADV
ejpam-7054	248	30	weakly	weakly	ADJ
ejpam-7054	248	31	µ(σ1	µ(σ1	NOUN
ejpam-7054	248	32	,	,	PUNCT
ejpam-7054	248	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	248	34	is	be	AUX
ejpam-7054	248	35	identical	identical	ADJ
ejpam-7054	248	36	with	with	ADP
ejpam-7054	248	37	the	the	DET
ejpam-7054	248	38	union	union	NOUN
ejpam-7054	248	39	of	of	ADP
ejpam-7054	248	40	the	the	DET
ejpam-7054	248	41	µprefrontier	µprefronti	ADJ
ejpam-7054	248	42	of	of	ADP
ejpam-7054	248	43	the	the	DET
ejpam-7054	248	44	lower	low	ADJ
ejpam-7054	248	45	inverse	inverse	NOUN
ejpam-7054	248	46	images	image	NOUN
ejpam-7054	248	47	of	of	ADP
ejpam-7054	248	48	σ1σ2	σ1σ2	NOUN
ejpam-7054	248	49	-	-	PUNCT
ejpam-7054	248	50	closure	closure	NOUN
ejpam-7054	248	51	of	of	ADP
ejpam-7054	248	52	σ1σ2	σ1σ2	NOUN
ejpam-7054	248	53	-	-	PUNCT
ejpam-7054	248	54	open	open	ADJ
ejpam-7054	248	55	sets	set	NOUN
ejpam-7054	248	56	meeting	meet	VERB
ejpam-7054	248	57	f	f	X
ejpam-7054	248	58	(	(	PUNCT
ejpam-7054	248	59	x	x	NOUN
ejpam-7054	248	60	)	)	PUNCT
ejpam-7054	248	61	.	.	PUNCT
ejpam-7054	249	1	proof	proof	NOUN
ejpam-7054	249	2	.	.	PUNCT
ejpam-7054	250	1	the	the	DET
ejpam-7054	250	2	proof	proof	NOUN
ejpam-7054	250	3	is	be	AUX
ejpam-7054	250	4	similar	similar	ADJ
ejpam-7054	250	5	to	to	ADP
ejpam-7054	250	6	that	that	PRON
ejpam-7054	250	7	of	of	ADP
ejpam-7054	250	8	theorem	theorem	ADJ
ejpam-7054	250	9	7	7	NUM
ejpam-7054	250	10	.	.	PUNCT
ejpam-7054	250	11	definition	definition	NOUN
ejpam-7054	250	12	3	3	NUM
ejpam-7054	250	13	.	.	PUNCT
ejpam-7054	250	14	a	a	DET
ejpam-7054	250	15	multifunction	multifunction	NOUN
ejpam-7054	251	1	f	f	NOUN
ejpam-7054	251	2	:	:	PUNCT
ejpam-7054	251	3	(	(	PUNCT
ejpam-7054	251	4	x,µ	x,µ	NOUN
ejpam-7054	251	5	)	)	PUNCT
ejpam-7054	251	6	→	→	SYM
ejpam-7054	251	7	(	(	PUNCT
ejpam-7054	251	8	y	y	PROPN
ejpam-7054	251	9	,	,	PUNCT
ejpam-7054	251	10	σ1	σ1	PROPN
ejpam-7054	251	11	,	,	PUNCT
ejpam-7054	251	12	σ2	σ2	PROPN
ejpam-7054	251	13	)	)	PUNCT
ejpam-7054	251	14	is	be	AUX
ejpam-7054	251	15	said	say	VERB
ejpam-7054	251	16	to	to	PART
ejpam-7054	251	17	be	be	AUX
ejpam-7054	251	18	upper	upper	ADJ
ejpam-7054	251	19	µ(σ1	µ(σ1	NOUN
ejpam-7054	251	20	,	,	PUNCT
ejpam-7054	251	21	σ2)precontinuous	σ2)precontinuous	ADJ
ejpam-7054	251	22	at	at	ADP
ejpam-7054	251	23	a	a	DET
ejpam-7054	251	24	point	point	NOUN
ejpam-7054	251	25	x	x	SYM
ejpam-7054	251	26	∈	∈	NOUN
ejpam-7054	251	27	x	x	PUNCT
ejpam-7054	251	28	if	if	SCONJ
ejpam-7054	251	29	for	for	ADP
ejpam-7054	251	30	each	each	DET
ejpam-7054	251	31	σ1σ2	σ1σ2	VERB
ejpam-7054	251	32	-	-	ADJ
ejpam-7054	251	33	open	open	ADJ
ejpam-7054	251	34	set	set	NOUN
ejpam-7054	251	35	v	v	NOUN
ejpam-7054	251	36	of	of	ADP
ejpam-7054	251	37	y	y	PRON
ejpam-7054	251	38	such	such	ADJ
ejpam-7054	251	39	that	that	SCONJ
ejpam-7054	251	40	f	f	PROPN
ejpam-7054	251	41	(	(	PUNCT
ejpam-7054	251	42	x	x	X
ejpam-7054	251	43	)	)	PUNCT
ejpam-7054	251	44	⊆	⊆	NUM
ejpam-7054	251	45	v	v	NOUN
ejpam-7054	251	46	,	,	PUNCT
ejpam-7054	251	47	there	there	PRON
ejpam-7054	251	48	exists	exist	VERB
ejpam-7054	251	49	a	a	DET
ejpam-7054	251	50	µ-preopen	µ-preopen	PROPN
ejpam-7054	251	51	set	set	VERB
ejpam-7054	251	52	u	u	NOUN
ejpam-7054	251	53	of	of	ADP
ejpam-7054	251	54	x	x	PUNCT
ejpam-7054	251	55	containing	contain	VERB
ejpam-7054	251	56	x	x	PUNCT
ejpam-7054	251	57	such	such	ADJ
ejpam-7054	251	58	that	that	SCONJ
ejpam-7054	251	59	f	f	PROPN
ejpam-7054	251	60	(	(	PUNCT
ejpam-7054	251	61	u	u	NOUN
ejpam-7054	251	62	)	)	PUNCT
ejpam-7054	251	63	⊆	⊆	NUM
ejpam-7054	251	64	v	v	NOUN
ejpam-7054	251	65	.	.	PUNCT
ejpam-7054	252	1	a	a	DET
ejpam-7054	252	2	multifunction	multifunction	NOUN
ejpam-7054	252	3	f	f	NOUN
ejpam-7054	252	4	:	:	PUNCT
ejpam-7054	252	5	(	(	PUNCT
ejpam-7054	252	6	x,µ	x,µ	NOUN
ejpam-7054	252	7	)	)	PUNCT
ejpam-7054	252	8	→	→	SYM
ejpam-7054	252	9	(	(	PUNCT
ejpam-7054	252	10	y	y	PROPN
ejpam-7054	252	11	,	,	PUNCT
ejpam-7054	252	12	σ1	σ1	PROPN
ejpam-7054	252	13	,	,	PUNCT
ejpam-7054	252	14	σ2	σ2	PROPN
ejpam-7054	252	15	)	)	PUNCT
ejpam-7054	252	16	is	be	AUX
ejpam-7054	252	17	said	say	VERB
ejpam-7054	252	18	to	to	PART
ejpam-7054	252	19	be	be	AUX
ejpam-7054	252	20	upper	upper	ADJ
ejpam-7054	252	21	µ(σ1	µ(σ1	NOUN
ejpam-7054	252	22	,	,	PUNCT
ejpam-7054	252	23	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	252	24	if	if	SCONJ
ejpam-7054	252	25	f	f	PROPN
ejpam-7054	252	26	is	be	AUX
ejpam-7054	252	27	upper	upper	ADJ
ejpam-7054	252	28	µ(σ1	µ(σ1	NOUN
ejpam-7054	252	29	,	,	PUNCT
ejpam-7054	252	30	σ2)precontinuous	σ2)precontinuous	ADJ
ejpam-7054	252	31	at	at	ADP
ejpam-7054	252	32	each	each	DET
ejpam-7054	252	33	point	point	NOUN
ejpam-7054	252	34	x	x	PUNCT
ejpam-7054	252	35	of	of	ADP
ejpam-7054	252	36	x.	x.	PROPN
ejpam-7054	252	37	theorem	theorem	VERB
ejpam-7054	252	38	9	9	NUM
ejpam-7054	252	39	.	.	PUNCT
ejpam-7054	252	40	for	for	ADP
ejpam-7054	252	41	a	a	DET
ejpam-7054	252	42	multifunction	multifunction	NOUN
ejpam-7054	252	43	f	f	NOUN
ejpam-7054	252	44	:	:	PUNCT
ejpam-7054	252	45	(	(	PUNCT
ejpam-7054	252	46	x,µ	x,µ	NOUN
ejpam-7054	252	47	)	)	PUNCT
ejpam-7054	252	48	→	→	SYM
ejpam-7054	252	49	(	(	PUNCT
ejpam-7054	252	50	y	y	PROPN
ejpam-7054	252	51	,	,	PUNCT
ejpam-7054	252	52	σ1	σ1	PROPN
ejpam-7054	252	53	,	,	PUNCT
ejpam-7054	252	54	σ2	σ2	NOUN
ejpam-7054	252	55	)	)	PUNCT
ejpam-7054	252	56	,	,	PUNCT
ejpam-7054	252	57	the	the	DET
ejpam-7054	252	58	following	follow	VERB
ejpam-7054	252	59	properties	property	NOUN
ejpam-7054	252	60	are	be	AUX
ejpam-7054	252	61	equivalent	equivalent	ADJ
ejpam-7054	252	62	:	:	PUNCT
ejpam-7054	252	63	(	(	PUNCT
ejpam-7054	252	64	1	1	X
ejpam-7054	252	65	)	)	PUNCT
ejpam-7054	252	66	f	f	PROPN
ejpam-7054	252	67	is	be	AUX
ejpam-7054	252	68	upper	upper	ADJ
ejpam-7054	252	69	µ(σ1	µ(σ1	NOUN
ejpam-7054	252	70	,	,	PUNCT
ejpam-7054	252	71	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	252	72	;	;	PUNCT
ejpam-7054	253	1	(	(	PUNCT
ejpam-7054	253	2	2	2	NUM
ejpam-7054	253	3	)	)	PUNCT
ejpam-7054	253	4	f+(v	f+(v	NOUN
ejpam-7054	253	5	)	)	PUNCT
ejpam-7054	254	1	is	be	AUX
ejpam-7054	254	2	µ-preopen	µ-preopen	VERB
ejpam-7054	254	3	in	in	ADP
ejpam-7054	254	4	x	x	PUNCT
ejpam-7054	254	5	for	for	ADP
ejpam-7054	254	6	every	every	DET
ejpam-7054	254	7	σ1σ2	σ1σ2	NOUN
ejpam-7054	254	8	-	-	ADJ
ejpam-7054	254	9	open	open	ADJ
ejpam-7054	254	10	set	set	NOUN
ejpam-7054	254	11	v	v	NOUN
ejpam-7054	254	12	of	of	ADP
ejpam-7054	254	13	y	y	PROPN
ejpam-7054	254	14	;	;	PUNCT
ejpam-7054	254	15	(	(	PUNCT
ejpam-7054	254	16	3	3	X
ejpam-7054	254	17	)	)	PUNCT
ejpam-7054	254	18	f−(k	f−(k	PROPN
ejpam-7054	254	19	)	)	PUNCT
ejpam-7054	254	20	is	be	AUX
ejpam-7054	254	21	µ-preclosed	µ-preclose	VERB
ejpam-7054	254	22	in	in	ADP
ejpam-7054	254	23	x	x	PUNCT
ejpam-7054	254	24	for	for	ADP
ejpam-7054	254	25	every	every	DET
ejpam-7054	254	26	σ1σ2	σ1σ2	NUM
ejpam-7054	254	27	-	-	PUNCT
ejpam-7054	254	28	closed	closed	ADJ
ejpam-7054	254	29	set	set	NOUN
ejpam-7054	254	30	k	k	PROPN
ejpam-7054	254	31	of	of	ADP
ejpam-7054	254	32	y	y	PROPN
ejpam-7054	254	33	;	;	PUNCT
ejpam-7054	254	34	(	(	PUNCT
ejpam-7054	254	35	4	4	X
ejpam-7054	254	36	)	)	PUNCT
ejpam-7054	254	37	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	254	38	−(b	−(b	NOUN
ejpam-7054	254	39	)	)	PUNCT
ejpam-7054	254	40	)	)	PUNCT
ejpam-7054	255	1	⊆	⊆	X
ejpam-7054	255	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7054	255	3	-	-	PUNCT
ejpam-7054	255	4	cl(b	cl(b	NOUN
ejpam-7054	255	5	)	)	PUNCT
ejpam-7054	255	6	)	)	PUNCT
ejpam-7054	256	1	for	for	ADP
ejpam-7054	256	2	every	every	DET
ejpam-7054	256	3	subset	subset	NOUN
ejpam-7054	256	4	b	b	PROPN
ejpam-7054	256	5	of	of	ADP
ejpam-7054	256	6	y	y	PROPN
ejpam-7054	256	7	;	;	PUNCT
ejpam-7054	256	8	(	(	PUNCT
ejpam-7054	256	9	5	5	X
ejpam-7054	256	10	)	)	PUNCT
ejpam-7054	256	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	256	12	-	-	PUNCT
ejpam-7054	256	13	int(b	int(b	NOUN
ejpam-7054	256	14	)	)	PUNCT
ejpam-7054	256	15	)	)	PUNCT
ejpam-7054	257	1	⊆	⊆	X
ejpam-7054	257	2	iµ(π)(f	iµ(π)(f	X
ejpam-7054	257	3	+	+	NOUN
ejpam-7054	257	4	(	(	PUNCT
ejpam-7054	257	5	b	b	NOUN
ejpam-7054	257	6	)	)	PUNCT
ejpam-7054	257	7	)	)	PUNCT
ejpam-7054	257	8	for	for	ADP
ejpam-7054	257	9	every	every	DET
ejpam-7054	257	10	subset	subset	NOUN
ejpam-7054	257	11	b	b	PROPN
ejpam-7054	257	12	of	of	ADP
ejpam-7054	257	13	y	y	PROPN
ejpam-7054	257	14	.	.	PUNCT
ejpam-7054	258	1	proof	proof	NOUN
ejpam-7054	258	2	.	.	PUNCT
ejpam-7054	259	1	(	(	PUNCT
ejpam-7054	259	2	1	1	X
ejpam-7054	259	3	)	)	PUNCT
ejpam-7054	259	4	⇒	⇒	NOUN
ejpam-7054	259	5	(	(	PUNCT
ejpam-7054	259	6	2	2	NUM
ejpam-7054	259	7	):	):	PUNCT
ejpam-7054	259	8	let	let	VERB
ejpam-7054	259	9	v	v	PART
ejpam-7054	259	10	be	be	AUX
ejpam-7054	259	11	any	any	DET
ejpam-7054	259	12	σ1σ2	σ1σ2	NOUN
ejpam-7054	259	13	-	-	ADJ
ejpam-7054	259	14	open	open	ADJ
ejpam-7054	259	15	set	set	NOUN
ejpam-7054	259	16	of	of	ADP
ejpam-7054	259	17	y	y	PROPN
ejpam-7054	259	18	and	and	CCONJ
ejpam-7054	259	19	x	x	PROPN
ejpam-7054	259	20	∈	∈	PROPN
ejpam-7054	259	21	f+(v	f+(v	NOUN
ejpam-7054	259	22	)	)	PUNCT
ejpam-7054	259	23	.	.	PUNCT
ejpam-7054	260	1	then	then	ADV
ejpam-7054	260	2	,	,	PUNCT
ejpam-7054	260	3	f	f	PROPN
ejpam-7054	260	4	(	(	PUNCT
ejpam-7054	260	5	x	x	X
ejpam-7054	260	6	)	)	PUNCT
ejpam-7054	260	7	⊆	⊆	NUM
ejpam-7054	260	8	v	v	NOUN
ejpam-7054	260	9	and	and	CCONJ
ejpam-7054	260	10	by	by	ADP
ejpam-7054	260	11	(	(	PUNCT
ejpam-7054	260	12	1	1	NUM
ejpam-7054	260	13	)	)	PUNCT
ejpam-7054	260	14	,	,	PUNCT
ejpam-7054	260	15	there	there	PRON
ejpam-7054	260	16	exists	exist	VERB
ejpam-7054	260	17	an	an	DET
ejpam-7054	260	18	µ-preopen	µ-preopen	PROPN
ejpam-7054	260	19	set	set	VERB
ejpam-7054	260	20	u	u	NOUN
ejpam-7054	260	21	of	of	ADP
ejpam-7054	260	22	x	x	PUNCT
ejpam-7054	260	23	containing	contain	VERB
ejpam-7054	260	24	x	x	PUNCT
ejpam-7054	260	25	such	such	ADJ
ejpam-7054	260	26	that	that	SCONJ
ejpam-7054	260	27	f	f	PROPN
ejpam-7054	260	28	(	(	PUNCT
ejpam-7054	260	29	u	u	NOUN
ejpam-7054	260	30	)	)	PUNCT
ejpam-7054	260	31	⊆	⊆	NUM
ejpam-7054	260	32	v	v	NOUN
ejpam-7054	260	33	.	.	PUNCT
ejpam-7054	261	1	thus	thus	ADV
ejpam-7054	261	2	,	,	PUNCT
ejpam-7054	261	3	x	x	PUNCT
ejpam-7054	261	4	∈	∈	PROPN
ejpam-7054	261	5	u	u	NOUN
ejpam-7054	261	6	⊆	⊆	NUM
ejpam-7054	261	7	f+(v	f+(v	NOUN
ejpam-7054	261	8	)	)	PUNCT
ejpam-7054	261	9	and	and	CCONJ
ejpam-7054	261	10	hence	hence	ADV
ejpam-7054	261	11	x	x	X
ejpam-7054	261	12	∈	∈	NOUN
ejpam-7054	261	13	iµ(π)(f	iµ(π)(f	VERB
ejpam-7054	262	1	+	+	ADJ
ejpam-7054	262	2	(	(	PUNCT
ejpam-7054	262	3	v	v	NOUN
ejpam-7054	262	4	)	)	PUNCT
ejpam-7054	262	5	)	)	PUNCT
ejpam-7054	262	6	.	.	PUNCT
ejpam-7054	263	1	therefore	therefore	ADV
ejpam-7054	263	2	,	,	PUNCT
ejpam-7054	263	3	f+(v	f+(v	PROPN
ejpam-7054	263	4	)	)	PUNCT
ejpam-7054	263	5	⊆	⊆	X
ejpam-7054	263	6	iµ(π)(f	iµ(π)(f	X
ejpam-7054	264	1	+	+	NOUN
ejpam-7054	264	2	(	(	PUNCT
ejpam-7054	264	3	v	v	NOUN
ejpam-7054	264	4	)	)	PUNCT
ejpam-7054	264	5	)	)	PUNCT
ejpam-7054	264	6	.	.	PUNCT
ejpam-7054	265	1	this	this	PRON
ejpam-7054	265	2	shows	show	VERB
ejpam-7054	265	3	that	that	SCONJ
ejpam-7054	265	4	f+(v	f+(v	PROPN
ejpam-7054	265	5	)	)	PUNCT
ejpam-7054	265	6	is	be	AUX
ejpam-7054	265	7	µ-preopen	µ-preopen	VERB
ejpam-7054	265	8	in	in	ADP
ejpam-7054	265	9	x.	x.	PROPN
ejpam-7054	265	10	(	(	PUNCT
ejpam-7054	265	11	2	2	NUM
ejpam-7054	265	12	)	)	PUNCT
ejpam-7054	265	13	⇒	⇒	NOUN
ejpam-7054	265	14	(	(	PUNCT
ejpam-7054	265	15	3	3	NUM
ejpam-7054	265	16	):	):	PUNCT
ejpam-7054	265	17	this	this	PRON
ejpam-7054	265	18	follows	follow	VERB
ejpam-7054	265	19	from	from	ADP
ejpam-7054	265	20	the	the	DET
ejpam-7054	265	21	fact	fact	NOUN
ejpam-7054	265	22	that	that	SCONJ
ejpam-7054	265	23	f+(y	f+(y	PROPN
ejpam-7054	265	24	−b	−b	ADV
ejpam-7054	265	25	)	)	PUNCT
ejpam-7054	265	26	=	=	PUNCT
ejpam-7054	266	1	x	x	X
ejpam-7054	266	2	−	−	PROPN
ejpam-7054	266	3	f−(b	f−(b	PROPN
ejpam-7054	266	4	)	)	PUNCT
ejpam-7054	266	5	for	for	ADP
ejpam-7054	266	6	every	every	DET
ejpam-7054	266	7	subset	subset	NOUN
ejpam-7054	266	8	b	b	PROPN
ejpam-7054	266	9	of	of	ADP
ejpam-7054	266	10	y	y	PROPN
ejpam-7054	266	11	.	.	PUNCT
ejpam-7054	267	1	(	(	PUNCT
ejpam-7054	267	2	3	3	X
ejpam-7054	267	3	)	)	PUNCT
ejpam-7054	267	4	⇒	⇒	NOUN
ejpam-7054	267	5	(	(	PUNCT
ejpam-7054	267	6	4	4	NUM
ejpam-7054	267	7	):	):	PUNCT
ejpam-7054	267	8	let	let	VERB
ejpam-7054	267	9	b	b	X
ejpam-7054	267	10	be	be	AUX
ejpam-7054	267	11	any	any	DET
ejpam-7054	267	12	subset	subset	NOUN
ejpam-7054	267	13	of	of	ADP
ejpam-7054	267	14	y	y	PROPN
ejpam-7054	267	15	.	.	PUNCT
ejpam-7054	268	1	then	then	ADV
ejpam-7054	268	2	,	,	PUNCT
ejpam-7054	268	3	σ1σ2	σ1σ2	NOUN
ejpam-7054	268	4	-	-	NOUN
ejpam-7054	268	5	cl(b	cl(b	NOUN
ejpam-7054	268	6	)	)	PUNCT
ejpam-7054	268	7	is	be	AUX
ejpam-7054	268	8	σ1σ2	σ1σ2	NOUN
ejpam-7054	268	9	-	-	ADJ
ejpam-7054	268	10	closed	closed	ADJ
ejpam-7054	268	11	in	in	ADP
ejpam-7054	268	12	y	y	PROPN
ejpam-7054	268	13	and	and	CCONJ
ejpam-7054	268	14	by	by	ADP
ejpam-7054	268	15	(	(	PUNCT
ejpam-7054	268	16	3	3	NUM
ejpam-7054	268	17	)	)	PUNCT
ejpam-7054	268	18	,	,	PUNCT
ejpam-7054	268	19	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	268	20	−(b	−(b	NOUN
ejpam-7054	268	21	)	)	PUNCT
ejpam-7054	268	22	)	)	PUNCT
ejpam-7054	269	1	⊆	⊆	NUM
ejpam-7054	269	2	cµ(π)(f	cµ(π)(f	NOUN
ejpam-7054	269	3	−(σ1σ2	−(σ1σ2	NOUN
ejpam-7054	269	4	-	-	NOUN
ejpam-7054	269	5	cl(b	cl(b	NOUN
ejpam-7054	269	6	)	)	PUNCT
ejpam-7054	269	7	)	)	PUNCT
ejpam-7054	269	8	)	)	PUNCT
ejpam-7054	270	1	=	=	PUNCT
ejpam-7054	270	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7054	270	3	-	-	PUNCT
ejpam-7054	270	4	cl(b	cl(b	NOUN
ejpam-7054	270	5	)	)	PUNCT
ejpam-7054	270	6	)	)	PUNCT
ejpam-7054	270	7	.	.	PUNCT
ejpam-7054	271	1	(	(	PUNCT
ejpam-7054	271	2	4	4	X
ejpam-7054	271	3	)	)	PUNCT
ejpam-7054	271	4	⇒	⇒	NOUN
ejpam-7054	271	5	(	(	PUNCT
ejpam-7054	271	6	5	5	NUM
ejpam-7054	271	7	):	):	PUNCT
ejpam-7054	271	8	let	let	VERB
ejpam-7054	271	9	b	b	X
ejpam-7054	271	10	be	be	AUX
ejpam-7054	271	11	any	any	DET
ejpam-7054	271	12	subset	subset	NOUN
ejpam-7054	271	13	of	of	ADP
ejpam-7054	271	14	y	y	PROPN
ejpam-7054	271	15	.	.	PUNCT
ejpam-7054	272	1	thus	thus	ADV
ejpam-7054	272	2	by	by	ADP
ejpam-7054	272	3	(	(	PUNCT
ejpam-7054	272	4	4	4	NUM
ejpam-7054	272	5	)	)	PUNCT
ejpam-7054	272	6	,	,	PUNCT
ejpam-7054	272	7	x	x	PUNCT
ejpam-7054	272	8	−	−	NOUN
ejpam-7054	272	9	iµ(π)(f	iµ(π)(f	VERB
ejpam-7054	273	1	+	+	NOUN
ejpam-7054	273	2	(	(	PUNCT
ejpam-7054	273	3	b	b	NOUN
ejpam-7054	273	4	)	)	PUNCT
ejpam-7054	273	5	)	)	PUNCT
ejpam-7054	274	1	=	=	PUNCT
ejpam-7054	274	2	cµ(π)(x	cµ(π)(x	NOUN
ejpam-7054	274	3	−	−	NOUN
ejpam-7054	274	4	f+(b	f+(b	PROPN
ejpam-7054	274	5	)	)	PUNCT
ejpam-7054	274	6	)	)	PUNCT
ejpam-7054	275	1	=	=	PRON
ejpam-7054	275	2	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	275	3	−(y	−(y	NOUN
ejpam-7054	275	4	−b	−b	NOUN
ejpam-7054	275	5	)	)	PUNCT
ejpam-7054	275	6	)	)	PUNCT
ejpam-7054	276	1	⊆	⊆	X
ejpam-7054	276	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-7054	276	3	-	-	PUNCT
ejpam-7054	276	4	cl(y	cl(y	NOUN
ejpam-7054	276	5	−b	−b	NOUN
ejpam-7054	276	6	)	)	PUNCT
ejpam-7054	276	7	)	)	PUNCT
ejpam-7054	277	1	b.	b.	PROPN
ejpam-7054	277	2	kong	kong	PROPN
ejpam-7054	277	3	-	-	PUNCT
ejpam-7054	277	4	ied	ied	PROPN
ejpam-7054	277	5	,	,	PUNCT
ejpam-7054	277	6	a.	a.	PROPN
ejpam-7054	277	7	sama	sama	PROPN
ejpam-7054	277	8	-	-	PUNCT
ejpam-7054	277	9	ae	ae	PROPN
ejpam-7054	277	10	,	,	PUNCT
ejpam-7054	277	11	c.	c.	PROPN
ejpam-7054	277	12	boonpok	boonpok	PROPN
ejpam-7054	277	13	/	/	SYM
ejpam-7054	277	14	eur	eur	PROPN
ejpam-7054	277	15	.	.	PUNCT
ejpam-7054	278	1	j.	j.	PROPN
ejpam-7054	278	2	pure	pure	PROPN
ejpam-7054	278	3	appl	appl	PROPN
ejpam-7054	278	4	.	.	PROPN
ejpam-7054	278	5	math	math	PROPN
ejpam-7054	278	6	,	,	PUNCT
ejpam-7054	278	7	18	18	NUM
ejpam-7054	278	8	(	(	PUNCT
ejpam-7054	278	9	4	4	NUM
ejpam-7054	278	10	)	)	PUNCT
ejpam-7054	278	11	(	(	PUNCT
ejpam-7054	278	12	2025	2025	NUM
ejpam-7054	278	13	)	)	PUNCT
ejpam-7054	278	14	,	,	PUNCT
ejpam-7054	278	15	7054	7054	NUM
ejpam-7054	278	16	9	9	NUM
ejpam-7054	278	17	of	of	ADP
ejpam-7054	278	18	12	12	NUM
ejpam-7054	278	19	=	=	SYM
ejpam-7054	278	20	f−(y	f−(y	NOUN
ejpam-7054	278	21	−	−	ADP
ejpam-7054	278	22	σ1σ2	σ1σ2	NOUN
ejpam-7054	278	23	-	-	PUNCT
ejpam-7054	278	24	int(b	int(b	NOUN
ejpam-7054	278	25	)	)	PUNCT
ejpam-7054	278	26	)	)	PUNCT
ejpam-7054	279	1	=	=	PUNCT
ejpam-7054	279	2	x	x	X
ejpam-7054	280	1	−	−	ADP
ejpam-7054	280	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	280	3	-	-	PUNCT
ejpam-7054	280	4	int(b	int(b	NOUN
ejpam-7054	280	5	)	)	PUNCT
ejpam-7054	280	6	)	)	PUNCT
ejpam-7054	280	7	and	and	CCONJ
ejpam-7054	280	8	so	so	ADV
ejpam-7054	280	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-7054	280	10	-	-	PUNCT
ejpam-7054	280	11	int(b	int(b	NOUN
ejpam-7054	280	12	)	)	PUNCT
ejpam-7054	280	13	)	)	PUNCT
ejpam-7054	280	14	⊆	⊆	X
ejpam-7054	280	15	iµ(π)(f	iµ(π)(f	X
ejpam-7054	280	16	+	+	NOUN
ejpam-7054	280	17	(	(	PUNCT
ejpam-7054	280	18	b	b	NOUN
ejpam-7054	280	19	)	)	PUNCT
ejpam-7054	280	20	)	)	PUNCT
ejpam-7054	280	21	.	.	PUNCT
ejpam-7054	281	1	(	(	PUNCT
ejpam-7054	281	2	5	5	X
ejpam-7054	281	3	)	)	PUNCT
ejpam-7054	281	4	⇒	⇒	NOUN
ejpam-7054	281	5	(	(	PUNCT
ejpam-7054	281	6	1	1	NUM
ejpam-7054	281	7	):	):	PUNCT
ejpam-7054	281	8	let	let	VERB
ejpam-7054	281	9	x	x	PUNCT
ejpam-7054	281	10	∈	∈	PROPN
ejpam-7054	281	11	x	x	X
ejpam-7054	281	12	and	and	CCONJ
ejpam-7054	281	13	v	v	X
ejpam-7054	281	14	be	be	AUX
ejpam-7054	281	15	any	any	DET
ejpam-7054	281	16	σ1σ2	σ1σ2	NOUN
ejpam-7054	281	17	-	-	ADJ
ejpam-7054	281	18	open	open	ADJ
ejpam-7054	281	19	set	set	NOUN
ejpam-7054	281	20	of	of	ADP
ejpam-7054	281	21	y	y	PRON
ejpam-7054	281	22	such	such	ADJ
ejpam-7054	281	23	that	that	SCONJ
ejpam-7054	281	24	f	f	PROPN
ejpam-7054	281	25	(	(	PUNCT
ejpam-7054	281	26	x	x	X
ejpam-7054	281	27	)	)	PUNCT
ejpam-7054	281	28	⊆	⊆	NUM
ejpam-7054	281	29	v	v	NOUN
ejpam-7054	281	30	.	.	PUNCT
ejpam-7054	282	1	then	then	ADV
ejpam-7054	282	2	,	,	PUNCT
ejpam-7054	282	3	x	x	X
ejpam-7054	282	4	∈	∈	NOUN
ejpam-7054	282	5	f+(v	f+(v	NOUN
ejpam-7054	282	6	)	)	PUNCT
ejpam-7054	283	1	=	=	PRON
ejpam-7054	283	2	iµ(π)(f	iµ(π)(f	VERB
ejpam-7054	284	1	+	+	ADJ
ejpam-7054	284	2	(	(	PUNCT
ejpam-7054	284	3	v	v	NOUN
ejpam-7054	284	4	)	)	PUNCT
ejpam-7054	284	5	)	)	PUNCT
ejpam-7054	284	6	.	.	PUNCT
ejpam-7054	285	1	there	there	PRON
ejpam-7054	285	2	exists	exist	VERB
ejpam-7054	285	3	a	a	DET
ejpam-7054	285	4	µ-preopen	µ-preopen	PROPN
ejpam-7054	285	5	set	set	VERB
ejpam-7054	285	6	u	u	NOUN
ejpam-7054	285	7	of	of	ADP
ejpam-7054	285	8	x	x	PUNCT
ejpam-7054	285	9	containing	contain	VERB
ejpam-7054	285	10	x	x	PUNCT
ejpam-7054	285	11	such	such	ADJ
ejpam-7054	285	12	that	that	SCONJ
ejpam-7054	285	13	u	u	NOUN
ejpam-7054	285	14	⊆	⊆	NUM
ejpam-7054	285	15	f+(v	f+(v	NOUN
ejpam-7054	285	16	)	)	PUNCT
ejpam-7054	285	17	;	;	PUNCT
ejpam-7054	285	18	hence	hence	ADV
ejpam-7054	285	19	f	f	PROPN
ejpam-7054	285	20	(	(	PUNCT
ejpam-7054	285	21	u	u	NOUN
ejpam-7054	285	22	)	)	PUNCT
ejpam-7054	285	23	⊆	⊆	NUM
ejpam-7054	285	24	v	v	NOUN
ejpam-7054	285	25	.	.	PUNCT
ejpam-7054	286	1	this	this	PRON
ejpam-7054	286	2	shows	show	VERB
ejpam-7054	286	3	that	that	SCONJ
ejpam-7054	286	4	f	f	PROPN
ejpam-7054	286	5	is	be	AUX
ejpam-7054	286	6	upper	upper	ADJ
ejpam-7054	286	7	µ(σ1	µ(σ1	NOUN
ejpam-7054	286	8	,	,	PUNCT
ejpam-7054	286	9	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	286	10	.	.	PUNCT
ejpam-7054	287	1	definition	definition	NOUN
ejpam-7054	287	2	4	4	NUM
ejpam-7054	287	3	.	.	PUNCT
ejpam-7054	288	1	a	a	DET
ejpam-7054	288	2	multifunction	multifunction	NOUN
ejpam-7054	288	3	f	f	NOUN
ejpam-7054	288	4	:	:	PUNCT
ejpam-7054	288	5	(	(	PUNCT
ejpam-7054	288	6	x,µ	x,µ	NOUN
ejpam-7054	288	7	)	)	PUNCT
ejpam-7054	288	8	→	→	SYM
ejpam-7054	288	9	(	(	PUNCT
ejpam-7054	288	10	y	y	PROPN
ejpam-7054	288	11	,	,	PUNCT
ejpam-7054	288	12	σ1	σ1	PROPN
ejpam-7054	288	13	,	,	PUNCT
ejpam-7054	288	14	σ2	σ2	PROPN
ejpam-7054	288	15	)	)	PUNCT
ejpam-7054	288	16	is	be	AUX
ejpam-7054	288	17	said	say	VERB
ejpam-7054	288	18	to	to	PART
ejpam-7054	288	19	be	be	AUX
ejpam-7054	288	20	lower	low	ADJ
ejpam-7054	288	21	µ(σ1	µ(σ1	NOUN
ejpam-7054	288	22	,	,	PUNCT
ejpam-7054	288	23	σ2)precontinuous	σ2)precontinuous	ADJ
ejpam-7054	288	24	at	at	ADP
ejpam-7054	288	25	a	a	DET
ejpam-7054	288	26	point	point	NOUN
ejpam-7054	288	27	x	x	SYM
ejpam-7054	288	28	∈	∈	NOUN
ejpam-7054	288	29	x	x	PUNCT
ejpam-7054	288	30	if	if	SCONJ
ejpam-7054	288	31	for	for	ADP
ejpam-7054	288	32	each	each	DET
ejpam-7054	288	33	σ1σ2	σ1σ2	VERB
ejpam-7054	288	34	-	-	ADJ
ejpam-7054	288	35	open	open	ADJ
ejpam-7054	288	36	set	set	NOUN
ejpam-7054	288	37	v	v	NOUN
ejpam-7054	288	38	of	of	ADP
ejpam-7054	288	39	y	y	PRON
ejpam-7054	288	40	such	such	ADJ
ejpam-7054	288	41	that	that	SCONJ
ejpam-7054	288	42	f	f	PROPN
ejpam-7054	288	43	(	(	PUNCT
ejpam-7054	288	44	x	x	NOUN
ejpam-7054	288	45	)	)	PUNCT
ejpam-7054	288	46	∩	∩	NOUN
ejpam-7054	288	47	v	v	ADP
ejpam-7054	288	48	̸=	̸=	PROPN
ejpam-7054	288	49	∅	∅	NOUN
ejpam-7054	288	50	,	,	PUNCT
ejpam-7054	288	51	there	there	PRON
ejpam-7054	288	52	exists	exist	VERB
ejpam-7054	288	53	an	an	DET
ejpam-7054	288	54	µ-preopen	µ-preopen	PROPN
ejpam-7054	288	55	set	set	VERB
ejpam-7054	288	56	u	u	NOUN
ejpam-7054	288	57	of	of	ADP
ejpam-7054	288	58	x	x	PUNCT
ejpam-7054	288	59	containing	contain	VERB
ejpam-7054	288	60	x	x	PUNCT
ejpam-7054	288	61	such	such	ADJ
ejpam-7054	288	62	that	that	SCONJ
ejpam-7054	288	63	f	f	PROPN
ejpam-7054	288	64	(	(	PUNCT
ejpam-7054	288	65	z)∩v	z)∩v	PROPN
ejpam-7054	288	66	̸=	̸=	PROPN
ejpam-7054	288	67	∅	∅	NOUN
ejpam-7054	288	68	for	for	ADP
ejpam-7054	288	69	every	every	DET
ejpam-7054	288	70	z	z	NOUN
ejpam-7054	288	71	∈	∈	PROPN
ejpam-7054	288	72	u	u	NOUN
ejpam-7054	288	73	.	.	PUNCT
ejpam-7054	289	1	a	a	DET
ejpam-7054	289	2	multifunction	multifunction	NOUN
ejpam-7054	289	3	f	f	NOUN
ejpam-7054	289	4	:	:	PUNCT
ejpam-7054	289	5	(	(	PUNCT
ejpam-7054	289	6	x,µ	x,µ	NOUN
ejpam-7054	289	7	)	)	PUNCT
ejpam-7054	289	8	→	→	SYM
ejpam-7054	289	9	(	(	PUNCT
ejpam-7054	289	10	y	y	PROPN
ejpam-7054	289	11	,	,	PUNCT
ejpam-7054	289	12	σ1	σ1	PROPN
ejpam-7054	289	13	,	,	PUNCT
ejpam-7054	289	14	σ2	σ2	PROPN
ejpam-7054	289	15	)	)	PUNCT
ejpam-7054	289	16	is	be	AUX
ejpam-7054	289	17	called	call	VERB
ejpam-7054	289	18	lower	low	ADJ
ejpam-7054	289	19	µ(σ1	µ(σ1	NOUN
ejpam-7054	289	20	,	,	PUNCT
ejpam-7054	289	21	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	289	22	if	if	SCONJ
ejpam-7054	289	23	f	f	PROPN
ejpam-7054	289	24	is	be	AUX
ejpam-7054	289	25	lower	low	ADJ
ejpam-7054	289	26	µ(σ1	µ(σ1	NOUN
ejpam-7054	289	27	,	,	PUNCT
ejpam-7054	289	28	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	289	29	at	at	ADP
ejpam-7054	289	30	each	each	DET
ejpam-7054	289	31	point	point	NOUN
ejpam-7054	289	32	x	x	PUNCT
ejpam-7054	289	33	of	of	ADP
ejpam-7054	289	34	x.	x.	PROPN
ejpam-7054	289	35	theorem	theorem	VERB
ejpam-7054	289	36	10	10	NUM
ejpam-7054	289	37	.	.	PUNCT
ejpam-7054	290	1	for	for	ADP
ejpam-7054	290	2	a	a	DET
ejpam-7054	290	3	multifunction	multifunction	NOUN
ejpam-7054	290	4	f	f	NOUN
ejpam-7054	290	5	:	:	PUNCT
ejpam-7054	290	6	(	(	PUNCT
ejpam-7054	290	7	x,µ	x,µ	NOUN
ejpam-7054	290	8	)	)	PUNCT
ejpam-7054	290	9	→	→	SYM
ejpam-7054	290	10	(	(	PUNCT
ejpam-7054	290	11	y	y	PROPN
ejpam-7054	290	12	,	,	PUNCT
ejpam-7054	290	13	σ1	σ1	PROPN
ejpam-7054	290	14	,	,	PUNCT
ejpam-7054	290	15	σ2	σ2	NOUN
ejpam-7054	290	16	)	)	PUNCT
ejpam-7054	290	17	,	,	PUNCT
ejpam-7054	290	18	the	the	DET
ejpam-7054	290	19	following	follow	VERB
ejpam-7054	290	20	properties	property	NOUN
ejpam-7054	290	21	are	be	AUX
ejpam-7054	290	22	equivalent	equivalent	ADJ
ejpam-7054	290	23	:	:	PUNCT
ejpam-7054	290	24	(	(	PUNCT
ejpam-7054	290	25	1	1	X
ejpam-7054	290	26	)	)	PUNCT
ejpam-7054	290	27	f	f	PROPN
ejpam-7054	290	28	is	be	AUX
ejpam-7054	290	29	lower	low	ADJ
ejpam-7054	290	30	µ(σ1	µ(σ1	NOUN
ejpam-7054	290	31	,	,	PUNCT
ejpam-7054	290	32	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	290	33	;	;	PUNCT
ejpam-7054	290	34	(	(	PUNCT
ejpam-7054	290	35	2	2	X
ejpam-7054	290	36	)	)	PUNCT
ejpam-7054	290	37	f−(v	f−(v	NOUN
ejpam-7054	290	38	)	)	PUNCT
ejpam-7054	290	39	is	be	AUX
ejpam-7054	290	40	µ-preopen	µ-preopen	VERB
ejpam-7054	290	41	in	in	ADP
ejpam-7054	290	42	x	x	PUNCT
ejpam-7054	290	43	for	for	ADP
ejpam-7054	290	44	every	every	DET
ejpam-7054	290	45	σ1σ2	σ1σ2	NOUN
ejpam-7054	290	46	-	-	ADJ
ejpam-7054	290	47	open	open	ADJ
ejpam-7054	290	48	set	set	NOUN
ejpam-7054	290	49	v	v	NOUN
ejpam-7054	290	50	of	of	ADP
ejpam-7054	290	51	y	y	PROPN
ejpam-7054	290	52	;	;	PUNCT
ejpam-7054	290	53	(	(	PUNCT
ejpam-7054	290	54	3	3	X
ejpam-7054	290	55	)	)	PUNCT
ejpam-7054	290	56	f+(k	f+(k	NUM
ejpam-7054	290	57	)	)	PUNCT
ejpam-7054	290	58	is	be	AUX
ejpam-7054	290	59	µ-preclosed	µ-preclose	VERB
ejpam-7054	290	60	in	in	ADP
ejpam-7054	290	61	x	x	PUNCT
ejpam-7054	290	62	for	for	ADP
ejpam-7054	290	63	every	every	DET
ejpam-7054	290	64	σ1σ2	σ1σ2	NUM
ejpam-7054	290	65	-	-	PUNCT
ejpam-7054	290	66	closed	closed	ADJ
ejpam-7054	290	67	set	set	NOUN
ejpam-7054	290	68	k	k	PROPN
ejpam-7054	290	69	of	of	ADP
ejpam-7054	290	70	y	y	PROPN
ejpam-7054	290	71	;	;	PUNCT
ejpam-7054	290	72	(	(	PUNCT
ejpam-7054	290	73	4	4	X
ejpam-7054	290	74	)	)	PUNCT
ejpam-7054	290	75	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	291	1	+	+	PROPN
ejpam-7054	291	2	(	(	PUNCT
ejpam-7054	291	3	b	b	NOUN
ejpam-7054	291	4	)	)	PUNCT
ejpam-7054	291	5	)	)	PUNCT
ejpam-7054	292	1	⊆	⊆	NUM
ejpam-7054	292	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	292	3	-	-	PUNCT
ejpam-7054	292	4	cl(b	cl(b	NOUN
ejpam-7054	292	5	)	)	PUNCT
ejpam-7054	292	6	)	)	PUNCT
ejpam-7054	292	7	for	for	ADP
ejpam-7054	292	8	every	every	DET
ejpam-7054	292	9	subset	subset	NOUN
ejpam-7054	292	10	b	b	PROPN
ejpam-7054	292	11	of	of	ADP
ejpam-7054	292	12	y	y	PROPN
ejpam-7054	292	13	;	;	PUNCT
ejpam-7054	292	14	(	(	PUNCT
ejpam-7054	292	15	5	5	X
ejpam-7054	292	16	)	)	PUNCT
ejpam-7054	292	17	f	f	NOUN
ejpam-7054	292	18	(	(	PUNCT
ejpam-7054	292	19	cµ(π)(a	cµ(π)(a	PROPN
ejpam-7054	292	20	)	)	PUNCT
ejpam-7054	292	21	)	)	PUNCT
ejpam-7054	293	1	⊆	⊆	X
ejpam-7054	293	2	σ1σ2	σ1σ2	X
ejpam-7054	293	3	-	-	NUM
ejpam-7054	293	4	cl(f	cl(f	NOUN
ejpam-7054	293	5	(	(	PUNCT
ejpam-7054	293	6	a	a	NOUN
ejpam-7054	293	7	)	)	PUNCT
ejpam-7054	293	8	)	)	PUNCT
ejpam-7054	293	9	for	for	ADP
ejpam-7054	293	10	every	every	DET
ejpam-7054	293	11	subset	subset	NOUN
ejpam-7054	293	12	a	a	PRON
ejpam-7054	293	13	of	of	ADP
ejpam-7054	293	14	x	x	PRON
ejpam-7054	293	15	;	;	PUNCT
ejpam-7054	293	16	(	(	PUNCT
ejpam-7054	293	17	6	6	X
ejpam-7054	293	18	)	)	PUNCT
ejpam-7054	293	19	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	293	20	-	-	PUNCT
ejpam-7054	293	21	int(b	int(b	NOUN
ejpam-7054	293	22	)	)	PUNCT
ejpam-7054	293	23	)	)	PUNCT
ejpam-7054	293	24	⊆	⊆	NUM
ejpam-7054	293	25	iµ(π)(f	iµ(π)(f	X
ejpam-7054	293	26	−(b	−(b	NOUN
ejpam-7054	293	27	)	)	PUNCT
ejpam-7054	293	28	)	)	PUNCT
ejpam-7054	293	29	for	for	ADP
ejpam-7054	293	30	every	every	DET
ejpam-7054	293	31	subset	subset	NOUN
ejpam-7054	293	32	b	b	PROPN
ejpam-7054	293	33	of	of	ADP
ejpam-7054	293	34	y	y	PROPN
ejpam-7054	293	35	.	.	PUNCT
ejpam-7054	294	1	proof	proof	NOUN
ejpam-7054	294	2	.	.	PUNCT
ejpam-7054	295	1	we	we	PRON
ejpam-7054	295	2	prove	prove	VERB
ejpam-7054	295	3	only	only	ADV
ejpam-7054	295	4	the	the	DET
ejpam-7054	295	5	implications	implication	NOUN
ejpam-7054	295	6	(	(	PUNCT
ejpam-7054	295	7	4	4	X
ejpam-7054	295	8	)	)	PUNCT
ejpam-7054	295	9	⇒	⇒	NOUN
ejpam-7054	295	10	(	(	PUNCT
ejpam-7054	295	11	5	5	NUM
ejpam-7054	295	12	)	)	PUNCT
ejpam-7054	295	13	and	and	CCONJ
ejpam-7054	295	14	(	(	PUNCT
ejpam-7054	295	15	5	5	X
ejpam-7054	295	16	)	)	PUNCT
ejpam-7054	295	17	⇒	⇒	NOUN
ejpam-7054	295	18	(	(	PUNCT
ejpam-7054	295	19	6	6	X
ejpam-7054	295	20	)	)	PUNCT
ejpam-7054	295	21	being	be	AUX
ejpam-7054	295	22	the	the	DET
ejpam-7054	295	23	proofs	proof	NOUN
ejpam-7054	295	24	of	of	ADP
ejpam-7054	295	25	the	the	DET
ejpam-7054	295	26	other	other	ADJ
ejpam-7054	295	27	similar	similar	ADJ
ejpam-7054	295	28	to	to	ADP
ejpam-7054	295	29	those	those	PRON
ejpam-7054	295	30	of	of	ADP
ejpam-7054	295	31	theorem	theorem	ADJ
ejpam-7054	295	32	9	9	NUM
ejpam-7054	295	33	.	.	PUNCT
ejpam-7054	295	34	(	(	PUNCT
ejpam-7054	295	35	4	4	X
ejpam-7054	295	36	)	)	PUNCT
ejpam-7054	295	37	⇒	⇒	NOUN
ejpam-7054	295	38	(	(	PUNCT
ejpam-7054	295	39	5	5	NUM
ejpam-7054	295	40	):	):	PUNCT
ejpam-7054	295	41	let	let	VERB
ejpam-7054	295	42	a	a	PRON
ejpam-7054	295	43	be	be	AUX
ejpam-7054	295	44	any	any	DET
ejpam-7054	295	45	subset	subset	NOUN
ejpam-7054	295	46	of	of	ADP
ejpam-7054	295	47	x.	x.	NOUN
ejpam-7054	295	48	by	by	ADP
ejpam-7054	295	49	(	(	PUNCT
ejpam-7054	295	50	4	4	NUM
ejpam-7054	295	51	)	)	PUNCT
ejpam-7054	295	52	,	,	PUNCT
ejpam-7054	295	53	we	we	PRON
ejpam-7054	295	54	have	have	AUX
ejpam-7054	295	55	cµ(π)(a	cµ(π)(a	NOUN
ejpam-7054	295	56	)	)	PUNCT
ejpam-7054	296	1	⊆	⊆	NUM
ejpam-7054	296	2	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	297	1	+	+	PROPN
ejpam-7054	297	2	(	(	PUNCT
ejpam-7054	297	3	f	f	X
ejpam-7054	297	4	(	(	PUNCT
ejpam-7054	297	5	a	a	NOUN
ejpam-7054	297	6	)	)	PUNCT
ejpam-7054	297	7	)	)	PUNCT
ejpam-7054	297	8	)	)	PUNCT
ejpam-7054	297	9	⊆	⊆	X
ejpam-7054	297	10	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-7054	297	11	-	-	SYM
ejpam-7054	297	12	cl(f	cl(f	NOUN
ejpam-7054	297	13	(	(	PUNCT
ejpam-7054	297	14	a	a	NOUN
ejpam-7054	297	15	)	)	PUNCT
ejpam-7054	297	16	)	)	PUNCT
ejpam-7054	297	17	)	)	PUNCT
ejpam-7054	298	1	and	and	CCONJ
ejpam-7054	298	2	hence	hence	ADV
ejpam-7054	298	3	f	f	PROPN
ejpam-7054	298	4	(	(	PUNCT
ejpam-7054	298	5	cµ(π)(a	cµ(π)(a	PROPN
ejpam-7054	298	6	)	)	PUNCT
ejpam-7054	298	7	)	)	PUNCT
ejpam-7054	299	1	⊆	⊆	X
ejpam-7054	299	2	σ1σ2	σ1σ2	X
ejpam-7054	299	3	-	-	NUM
ejpam-7054	299	4	cl(f	cl(f	NOUN
ejpam-7054	299	5	(	(	PUNCT
ejpam-7054	299	6	a	a	NOUN
ejpam-7054	299	7	)	)	PUNCT
ejpam-7054	299	8	)	)	PUNCT
ejpam-7054	299	9	.	.	PUNCT
ejpam-7054	300	1	(	(	PUNCT
ejpam-7054	300	2	5	5	X
ejpam-7054	300	3	)	)	PUNCT
ejpam-7054	300	4	⇒	⇒	NOUN
ejpam-7054	300	5	(	(	PUNCT
ejpam-7054	300	6	6	6	NUM
ejpam-7054	300	7	):	):	PUNCT
ejpam-7054	300	8	let	let	VERB
ejpam-7054	300	9	b	b	X
ejpam-7054	300	10	be	be	AUX
ejpam-7054	300	11	any	any	DET
ejpam-7054	300	12	subset	subset	NOUN
ejpam-7054	300	13	of	of	ADP
ejpam-7054	300	14	y	y	PROPN
ejpam-7054	300	15	.	.	PUNCT
ejpam-7054	301	1	by	by	ADP
ejpam-7054	301	2	(	(	PUNCT
ejpam-7054	301	3	5	5	NUM
ejpam-7054	301	4	)	)	PUNCT
ejpam-7054	301	5	,	,	PUNCT
ejpam-7054	301	6	f	f	PROPN
ejpam-7054	301	7	(	(	PUNCT
ejpam-7054	301	8	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	301	9	+	+	PROPN
ejpam-7054	301	10	(	(	PUNCT
ejpam-7054	301	11	y	y	NOUN
ejpam-7054	301	12	−b	−b	ADJ
ejpam-7054	301	13	)	)	PUNCT
ejpam-7054	301	14	)	)	PUNCT
ejpam-7054	301	15	)	)	PUNCT
ejpam-7054	302	1	⊆	⊆	X
ejpam-7054	302	2	σ1σ2	σ1σ2	X
ejpam-7054	302	3	-	-	NUM
ejpam-7054	302	4	cl(f	cl(f	NOUN
ejpam-7054	302	5	(	(	PUNCT
ejpam-7054	302	6	f+(y	f+(y	PROPN
ejpam-7054	302	7	−b	−b	PROPN
ejpam-7054	302	8	)	)	PUNCT
ejpam-7054	302	9	)	)	PUNCT
ejpam-7054	302	10	)	)	PUNCT
ejpam-7054	303	1	⊆	⊆	X
ejpam-7054	303	2	σ1σ2	σ1σ2	NUM
ejpam-7054	303	3	-	-	PUNCT
ejpam-7054	303	4	cl(y	cl(y	NOUN
ejpam-7054	303	5	−b	−b	NOUN
ejpam-7054	303	6	)	)	PUNCT
ejpam-7054	304	1	=	=	SYM
ejpam-7054	304	2	y	y	PROPN
ejpam-7054	304	3	−	−	ADP
ejpam-7054	304	4	σ1σ2	σ1σ2	X
ejpam-7054	304	5	-	-	PUNCT
ejpam-7054	304	6	int(b	int(b	NOUN
ejpam-7054	304	7	)	)	PUNCT
ejpam-7054	304	8	.	.	PUNCT
ejpam-7054	305	1	since	since	SCONJ
ejpam-7054	305	2	f	f	PROPN
ejpam-7054	305	3	(	(	PUNCT
ejpam-7054	305	4	cµ(π)(f	cµ(π)(f	VERB
ejpam-7054	305	5	+	+	PROPN
ejpam-7054	305	6	(	(	PUNCT
ejpam-7054	305	7	y	y	NOUN
ejpam-7054	305	8	−b	−b	ADJ
ejpam-7054	305	9	)	)	PUNCT
ejpam-7054	305	10	)	)	PUNCT
ejpam-7054	305	11	)	)	PUNCT
ejpam-7054	306	1	=	=	SYM
ejpam-7054	306	2	f	f	PROPN
ejpam-7054	306	3	(	(	PUNCT
ejpam-7054	306	4	cµ(π)(x	cµ(π)(x	PROPN
ejpam-7054	306	5	−	−	PROPN
ejpam-7054	306	6	f−(b	f−(b	PROPN
ejpam-7054	306	7	)	)	PUNCT
ejpam-7054	306	8	)	)	PUNCT
ejpam-7054	306	9	)	)	PUNCT
ejpam-7054	307	1	=	=	SYM
ejpam-7054	307	2	f	f	X
ejpam-7054	307	3	(	(	PUNCT
ejpam-7054	307	4	x	x	SYM
ejpam-7054	307	5	−	−	PROPN
ejpam-7054	307	6	iµ(π)(f	iµ(π)(f	PROPN
ejpam-7054	307	7	−(b	−(b	NOUN
ejpam-7054	307	8	)	)	PUNCT
ejpam-7054	307	9	)	)	PUNCT
ejpam-7054	307	10	)	)	PUNCT
ejpam-7054	307	11	,	,	PUNCT
ejpam-7054	307	12	we	we	PRON
ejpam-7054	307	13	have	have	VERB
ejpam-7054	307	14	x	x	INTJ
ejpam-7054	307	15	−	−	PROPN
ejpam-7054	307	16	iµ(π)(f	iµ(π)(f	PROPN
ejpam-7054	307	17	−(b	−(b	NOUN
ejpam-7054	307	18	)	)	PUNCT
ejpam-7054	307	19	)	)	PUNCT
ejpam-7054	308	1	⊆	⊆	NUM
ejpam-7054	308	2	f+(y	f+(y	ADP
ejpam-7054	308	3	−	−	NUM
ejpam-7054	308	4	σ1σ2	σ1σ2	SYM
ejpam-7054	308	5	-	-	PUNCT
ejpam-7054	308	6	int(b	int(b	NOUN
ejpam-7054	308	7	)	)	PUNCT
ejpam-7054	308	8	)	)	PUNCT
ejpam-7054	309	1	=	=	PUNCT
ejpam-7054	309	2	x	x	X
ejpam-7054	309	3	−	−	NOUN
ejpam-7054	309	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	309	5	-	-	PUNCT
ejpam-7054	309	6	int(b	int(b	NOUN
ejpam-7054	309	7	)	)	PUNCT
ejpam-7054	309	8	)	)	PUNCT
ejpam-7054	309	9	and	and	CCONJ
ejpam-7054	309	10	so	so	ADV
ejpam-7054	309	11	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-7054	309	12	-	-	PUNCT
ejpam-7054	309	13	int(b	int(b	NOUN
ejpam-7054	309	14	)	)	PUNCT
ejpam-7054	309	15	)	)	PUNCT
ejpam-7054	310	1	⊆	⊆	NUM
ejpam-7054	310	2	iµ(π)(f	iµ(π)(f	X
ejpam-7054	310	3	−(b	−(b	NOUN
ejpam-7054	310	4	)	)	PUNCT
ejpam-7054	310	5	)	)	PUNCT
ejpam-7054	310	6	.	.	PUNCT
ejpam-7054	311	1	recall	recall	VERB
ejpam-7054	311	2	that	that	SCONJ
ejpam-7054	311	3	a	a	DET
ejpam-7054	311	4	bitopological	bitopological	ADJ
ejpam-7054	311	5	space	space	NOUN
ejpam-7054	311	6	(	(	PUNCT
ejpam-7054	311	7	x	x	NOUN
ejpam-7054	311	8	,	,	PUNCT
ejpam-7054	311	9	τ1	τ1	NOUN
ejpam-7054	311	10	,	,	PUNCT
ejpam-7054	311	11	τ2	τ2	NOUN
ejpam-7054	311	12	)	)	PUNCT
ejpam-7054	311	13	is	be	AUX
ejpam-7054	311	14	said	say	VERB
ejpam-7054	311	15	to	to	PART
ejpam-7054	311	16	be	be	AUX
ejpam-7054	311	17	(	(	PUNCT
ejpam-7054	311	18	τ1	τ1	NOUN
ejpam-7054	311	19	,	,	PUNCT
ejpam-7054	311	20	τ2)-regular	τ2)-regular	ADJ
ejpam-7054	311	21	[	[	X
ejpam-7054	311	22	25	25	NUM
ejpam-7054	311	23	]	]	X
ejpam-7054	311	24	if	if	SCONJ
ejpam-7054	311	25	for	for	ADP
ejpam-7054	311	26	each	each	DET
ejpam-7054	311	27	τ1τ2	τ1τ2	ADJ
ejpam-7054	311	28	-	-	ADJ
ejpam-7054	311	29	closed	closed	ADJ
ejpam-7054	311	30	set	set	VERB
ejpam-7054	311	31	f	f	NOUN
ejpam-7054	311	32	and	and	CCONJ
ejpam-7054	311	33	each	each	DET
ejpam-7054	311	34	x	x	PROPN
ejpam-7054	311	35	̸∈	̸∈	PROPN
ejpam-7054	311	36	f	f	PROPN
ejpam-7054	311	37	,	,	PUNCT
ejpam-7054	311	38	there	there	PRON
ejpam-7054	311	39	exist	exist	VERB
ejpam-7054	311	40	disjoint	disjoint	ADJ
ejpam-7054	311	41	τ1τ2	τ1τ2	ADJ
ejpam-7054	311	42	-	-	ADJ
ejpam-7054	311	43	open	open	ADJ
ejpam-7054	311	44	sets	set	NOUN
ejpam-7054	311	45	u	u	NOUN
ejpam-7054	311	46	and	and	CCONJ
ejpam-7054	311	47	v	v	ADP
ejpam-7054	311	48	such	such	ADJ
ejpam-7054	311	49	that	that	SCONJ
ejpam-7054	311	50	x	x	SYM
ejpam-7054	311	51	∈	∈	PROPN
ejpam-7054	311	52	u	u	NOUN
ejpam-7054	311	53	and	and	CCONJ
ejpam-7054	311	54	f	f	PROPN
ejpam-7054	311	55	⊆	⊆	NUM
ejpam-7054	311	56	v	v	NOUN
ejpam-7054	311	57	.	.	PUNCT
ejpam-7054	312	1	b.	b.	PROPN
ejpam-7054	312	2	kong	kong	PROPN
ejpam-7054	312	3	-	-	PUNCT
ejpam-7054	312	4	ied	ied	PROPN
ejpam-7054	312	5	,	,	PUNCT
ejpam-7054	312	6	a.	a.	PROPN
ejpam-7054	312	7	sama	sama	PROPN
ejpam-7054	312	8	-	-	PUNCT
ejpam-7054	312	9	ae	ae	PROPN
ejpam-7054	312	10	,	,	PUNCT
ejpam-7054	312	11	c.	c.	PROPN
ejpam-7054	312	12	boonpok	boonpok	PROPN
ejpam-7054	312	13	/	/	SYM
ejpam-7054	312	14	eur	eur	PROPN
ejpam-7054	312	15	.	.	PUNCT
ejpam-7054	313	1	j.	j.	PROPN
ejpam-7054	313	2	pure	pure	PROPN
ejpam-7054	313	3	appl	appl	PROPN
ejpam-7054	313	4	.	.	PROPN
ejpam-7054	313	5	math	math	PROPN
ejpam-7054	313	6	,	,	PUNCT
ejpam-7054	313	7	18	18	NUM
ejpam-7054	313	8	(	(	PUNCT
ejpam-7054	313	9	4	4	NUM
ejpam-7054	313	10	)	)	PUNCT
ejpam-7054	313	11	(	(	PUNCT
ejpam-7054	313	12	2025	2025	NUM
ejpam-7054	313	13	)	)	PUNCT
ejpam-7054	313	14	,	,	PUNCT
ejpam-7054	313	15	7054	7054	NUM
ejpam-7054	313	16	10	10	NUM
ejpam-7054	313	17	of	of	ADP
ejpam-7054	313	18	12	12	NUM
ejpam-7054	313	19	lemma	lemma	PROPN
ejpam-7054	313	20	3	3	NUM
ejpam-7054	313	21	.	.	PUNCT
ejpam-7054	314	1	[	[	X
ejpam-7054	314	2	26	26	NUM
ejpam-7054	314	3	]	]	X
ejpam-7054	314	4	let	let	VERB
ejpam-7054	314	5	(	(	PUNCT
ejpam-7054	314	6	x	x	NOUN
ejpam-7054	314	7	,	,	PUNCT
ejpam-7054	314	8	τ1	τ1	NOUN
ejpam-7054	314	9	,	,	PUNCT
ejpam-7054	314	10	τ2	τ2	PROPN
ejpam-7054	314	11	)	)	PUNCT
ejpam-7054	314	12	be	be	VERB
ejpam-7054	314	13	a	a	DET
ejpam-7054	314	14	(	(	PUNCT
ejpam-7054	314	15	τ1	τ1	NOUN
ejpam-7054	314	16	,	,	PUNCT
ejpam-7054	314	17	τ2)-regular	τ2)-regular	ADJ
ejpam-7054	314	18	space	space	NOUN
ejpam-7054	314	19	.	.	PUNCT
ejpam-7054	315	1	then	then	ADV
ejpam-7054	315	2	,	,	PUNCT
ejpam-7054	315	3	the	the	DET
ejpam-7054	315	4	following	follow	VERB
ejpam-7054	315	5	properties	property	NOUN
ejpam-7054	315	6	hold	hold	VERB
ejpam-7054	315	7	:	:	PUNCT
ejpam-7054	315	8	(	(	PUNCT
ejpam-7054	315	9	1	1	X
ejpam-7054	315	10	)	)	PUNCT
ejpam-7054	315	11	τ1τ2	τ1τ2	NOUN
ejpam-7054	315	12	-	-	NUM
ejpam-7054	315	13	cl(a	cl(a	NUM
ejpam-7054	315	14	)	)	PUNCT
ejpam-7054	315	15	=	=	PUNCT
ejpam-7054	315	16	(	(	PUNCT
ejpam-7054	315	17	τ1	τ1	NOUN
ejpam-7054	315	18	,	,	PUNCT
ejpam-7054	315	19	τ2)θ	τ2)θ	NOUN
ejpam-7054	315	20	-	-	PUNCT
ejpam-7054	315	21	cl(a	cl(a	NUM
ejpam-7054	315	22	)	)	PUNCT
ejpam-7054	315	23	for	for	ADP
ejpam-7054	315	24	every	every	DET
ejpam-7054	315	25	subset	subset	NOUN
ejpam-7054	315	26	a	a	PRON
ejpam-7054	315	27	of	of	ADP
ejpam-7054	315	28	x.	x.	NOUN
ejpam-7054	315	29	(	(	PUNCT
ejpam-7054	315	30	2	2	NUM
ejpam-7054	315	31	)	)	PUNCT
ejpam-7054	315	32	every	every	DET
ejpam-7054	315	33	τ1τ2	τ1τ2	NOUN
ejpam-7054	315	34	-	-	ADJ
ejpam-7054	315	35	open	open	ADJ
ejpam-7054	315	36	set	set	NOUN
ejpam-7054	315	37	is	be	AUX
ejpam-7054	315	38	(	(	PUNCT
ejpam-7054	315	39	τ1	τ1	NOUN
ejpam-7054	315	40	,	,	PUNCT
ejpam-7054	315	41	τ2)θ	τ2)θ	ADJ
ejpam-7054	315	42	-	-	PUNCT
ejpam-7054	315	43	open	open	ADJ
ejpam-7054	315	44	.	.	PUNCT
ejpam-7054	316	1	theorem	theorem	VERB
ejpam-7054	316	2	11	11	NUM
ejpam-7054	316	3	.	.	PUNCT
ejpam-7054	317	1	for	for	ADP
ejpam-7054	317	2	a	a	DET
ejpam-7054	317	3	multifunction	multifunction	NOUN
ejpam-7054	317	4	f	f	NOUN
ejpam-7054	317	5	:	:	PUNCT
ejpam-7054	317	6	(	(	PUNCT
ejpam-7054	317	7	x,µ	x,µ	NOUN
ejpam-7054	317	8	)	)	PUNCT
ejpam-7054	317	9	→	→	SYM
ejpam-7054	317	10	(	(	PUNCT
ejpam-7054	317	11	y	y	PROPN
ejpam-7054	317	12	,	,	PUNCT
ejpam-7054	317	13	σ1	σ1	PROPN
ejpam-7054	317	14	,	,	PUNCT
ejpam-7054	317	15	σ2	σ2	NOUN
ejpam-7054	317	16	)	)	PUNCT
ejpam-7054	317	17	,	,	PUNCT
ejpam-7054	317	18	where	where	SCONJ
ejpam-7054	317	19	(	(	PUNCT
ejpam-7054	317	20	y	y	PROPN
ejpam-7054	317	21	,	,	PUNCT
ejpam-7054	317	22	σ1	σ1	PROPN
ejpam-7054	317	23	,	,	PUNCT
ejpam-7054	317	24	σ2	σ2	PROPN
ejpam-7054	317	25	)	)	PUNCT
ejpam-7054	317	26	is	be	AUX
ejpam-7054	317	27	(	(	PUNCT
ejpam-7054	317	28	σ1	σ1	PROPN
ejpam-7054	317	29	,	,	PUNCT
ejpam-7054	317	30	σ2)regular	σ2)regular	PROPN
ejpam-7054	317	31	,	,	PUNCT
ejpam-7054	317	32	the	the	DET
ejpam-7054	317	33	following	follow	VERB
ejpam-7054	317	34	properties	property	NOUN
ejpam-7054	317	35	are	be	AUX
ejpam-7054	317	36	equivalent	equivalent	ADJ
ejpam-7054	317	37	:	:	PUNCT
ejpam-7054	317	38	(	(	PUNCT
ejpam-7054	317	39	1	1	X
ejpam-7054	317	40	)	)	PUNCT
ejpam-7054	317	41	f	f	PROPN
ejpam-7054	317	42	is	be	AUX
ejpam-7054	317	43	upper	upper	ADJ
ejpam-7054	317	44	µ(σ1	µ(σ1	NOUN
ejpam-7054	317	45	,	,	PUNCT
ejpam-7054	317	46	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	317	47	;	;	PUNCT
ejpam-7054	317	48	(	(	PUNCT
ejpam-7054	317	49	2	2	X
ejpam-7054	317	50	)	)	PUNCT
ejpam-7054	317	51	f−((σ1	f−((σ1	NOUN
ejpam-7054	317	52	,	,	PUNCT
ejpam-7054	317	53	σ2)θ	σ2)θ	ADJ
ejpam-7054	317	54	-	-	PUNCT
ejpam-7054	317	55	cl(b	cl(b	NOUN
ejpam-7054	317	56	)	)	PUNCT
ejpam-7054	317	57	)	)	PUNCT
ejpam-7054	317	58	is	be	AUX
ejpam-7054	317	59	µ-preclosed	µ-preclose	VERB
ejpam-7054	317	60	in	in	ADP
ejpam-7054	317	61	x	x	PUNCT
ejpam-7054	317	62	for	for	ADP
ejpam-7054	317	63	every	every	DET
ejpam-7054	317	64	subset	subset	NOUN
ejpam-7054	317	65	b	b	PROPN
ejpam-7054	317	66	of	of	ADP
ejpam-7054	317	67	y	y	PROPN
ejpam-7054	317	68	;	;	PUNCT
ejpam-7054	317	69	(	(	PUNCT
ejpam-7054	317	70	3	3	X
ejpam-7054	317	71	)	)	PUNCT
ejpam-7054	317	72	f−(k	f−(k	PROPN
ejpam-7054	317	73	)	)	PUNCT
ejpam-7054	317	74	is	be	AUX
ejpam-7054	317	75	µ-preclosed	µ-preclose	VERB
ejpam-7054	317	76	in	in	ADP
ejpam-7054	317	77	x	x	PUNCT
ejpam-7054	317	78	for	for	ADP
ejpam-7054	317	79	every	every	DET
ejpam-7054	317	80	(	(	PUNCT
ejpam-7054	317	81	σ1	σ1	PROPN
ejpam-7054	317	82	,	,	PUNCT
ejpam-7054	317	83	σ2)θ	σ2)θ	NOUN
ejpam-7054	317	84	-	-	PUNCT
ejpam-7054	317	85	closed	close	VERB
ejpam-7054	317	86	set	set	NOUN
ejpam-7054	317	87	k	k	PROPN
ejpam-7054	317	88	of	of	ADP
ejpam-7054	317	89	y	y	PROPN
ejpam-7054	317	90	;	;	PUNCT
ejpam-7054	317	91	(	(	PUNCT
ejpam-7054	317	92	4	4	X
ejpam-7054	317	93	)	)	PUNCT
ejpam-7054	318	1	f+(v	f+(v	NOUN
ejpam-7054	318	2	)	)	PUNCT
ejpam-7054	319	1	is	be	AUX
ejpam-7054	319	2	µ-preopen	µ-preopen	VERB
ejpam-7054	319	3	in	in	ADP
ejpam-7054	319	4	x	x	PUNCT
ejpam-7054	319	5	for	for	ADP
ejpam-7054	319	6	every	every	DET
ejpam-7054	319	7	(	(	PUNCT
ejpam-7054	319	8	σ1	σ1	PROPN
ejpam-7054	319	9	,	,	PUNCT
ejpam-7054	319	10	σ2)θ	σ2)θ	NOUN
ejpam-7054	319	11	-	-	PUNCT
ejpam-7054	319	12	open	open	ADJ
ejpam-7054	319	13	set	set	NOUN
ejpam-7054	319	14	v	v	NOUN
ejpam-7054	319	15	of	of	ADP
ejpam-7054	319	16	y	y	PROPN
ejpam-7054	319	17	.	.	PUNCT
ejpam-7054	320	1	proof	proof	NOUN
ejpam-7054	320	2	.	.	PUNCT
ejpam-7054	321	1	(	(	PUNCT
ejpam-7054	321	2	1	1	X
ejpam-7054	321	3	)	)	PUNCT
ejpam-7054	321	4	⇒	⇒	NOUN
ejpam-7054	321	5	(	(	PUNCT
ejpam-7054	321	6	2	2	NUM
ejpam-7054	321	7	):	):	PUNCT
ejpam-7054	321	8	let	let	VERB
ejpam-7054	321	9	b	b	X
ejpam-7054	321	10	be	be	AUX
ejpam-7054	321	11	any	any	DET
ejpam-7054	321	12	subset	subset	NOUN
ejpam-7054	321	13	of	of	ADP
ejpam-7054	321	14	y	y	PROPN
ejpam-7054	321	15	.	.	PUNCT
ejpam-7054	322	1	then	then	ADV
ejpam-7054	322	2	,	,	PUNCT
ejpam-7054	322	3	(	(	PUNCT
ejpam-7054	322	4	σ1	σ1	PROPN
ejpam-7054	322	5	,	,	PUNCT
ejpam-7054	322	6	σ2)θ	σ2)θ	NOUN
ejpam-7054	322	7	-	-	PUNCT
ejpam-7054	322	8	cl(b	cl(b	NOUN
ejpam-7054	322	9	)	)	PUNCT
ejpam-7054	322	10	is	be	AUX
ejpam-7054	322	11	σ1σ2	σ1σ2	NOUN
ejpam-7054	322	12	-	-	ADJ
ejpam-7054	322	13	closed	closed	ADJ
ejpam-7054	322	14	in	in	ADP
ejpam-7054	322	15	y	y	PROPN
ejpam-7054	322	16	and	and	CCONJ
ejpam-7054	322	17	by	by	ADP
ejpam-7054	322	18	theorem	theorem	NOUN
ejpam-7054	322	19	9	9	NUM
ejpam-7054	322	20	,	,	PUNCT
ejpam-7054	322	21	f−((σ1	f−((σ1	NOUN
ejpam-7054	322	22	,	,	PUNCT
ejpam-7054	322	23	σ2)θ	σ2)θ	ADJ
ejpam-7054	322	24	-	-	PUNCT
ejpam-7054	322	25	cl(b	cl(b	NOUN
ejpam-7054	322	26	)	)	PUNCT
ejpam-7054	322	27	)	)	PUNCT
ejpam-7054	322	28	is	be	AUX
ejpam-7054	322	29	µ-preclosed	µ-preclose	VERB
ejpam-7054	322	30	in	in	ADP
ejpam-7054	322	31	x.	x.	NOUN
ejpam-7054	322	32	(	(	PUNCT
ejpam-7054	322	33	2	2	NUM
ejpam-7054	322	34	)	)	PUNCT
ejpam-7054	322	35	⇒	⇒	NOUN
ejpam-7054	322	36	(	(	PUNCT
ejpam-7054	322	37	3	3	NUM
ejpam-7054	322	38	):	):	PUNCT
ejpam-7054	322	39	the	the	DET
ejpam-7054	322	40	proof	proof	NOUN
ejpam-7054	322	41	is	be	AUX
ejpam-7054	322	42	obvious	obvious	ADJ
ejpam-7054	322	43	.	.	PUNCT
ejpam-7054	323	1	(	(	PUNCT
ejpam-7054	323	2	3	3	X
ejpam-7054	323	3	)	)	PUNCT
ejpam-7054	323	4	⇒	⇒	NOUN
ejpam-7054	323	5	(	(	PUNCT
ejpam-7054	323	6	4	4	NUM
ejpam-7054	323	7	):	):	PUNCT
ejpam-7054	323	8	let	let	VERB
ejpam-7054	323	9	v	v	PART
ejpam-7054	323	10	be	be	AUX
ejpam-7054	323	11	any	any	DET
ejpam-7054	323	12	(	(	PUNCT
ejpam-7054	323	13	σ1	σ1	PROPN
ejpam-7054	323	14	,	,	PUNCT
ejpam-7054	323	15	σ2)θ	σ2)θ	NOUN
ejpam-7054	323	16	-	-	PUNCT
ejpam-7054	323	17	open	open	ADJ
ejpam-7054	323	18	set	set	NOUN
ejpam-7054	323	19	of	of	ADP
ejpam-7054	323	20	y	y	PROPN
ejpam-7054	323	21	.	.	PUNCT
ejpam-7054	324	1	by	by	ADP
ejpam-7054	324	2	(	(	PUNCT
ejpam-7054	324	3	3	3	NUM
ejpam-7054	324	4	)	)	PUNCT
ejpam-7054	324	5	,	,	PUNCT
ejpam-7054	324	6	f−(y	f−(y	NOUN
ejpam-7054	324	7	−	−	NOUN
ejpam-7054	324	8	v	v	NOUN
ejpam-7054	324	9	)	)	PUNCT
ejpam-7054	324	10	is	be	AUX
ejpam-7054	324	11	µ-preclosed	µ-preclose	VERB
ejpam-7054	324	12	in	in	ADP
ejpam-7054	324	13	x	x	PUNCT
ejpam-7054	324	14	and	and	CCONJ
ejpam-7054	324	15	f−(y	f−(y	NOUN
ejpam-7054	324	16	−	−	NOUN
ejpam-7054	324	17	v	v	NOUN
ejpam-7054	324	18	)	)	PUNCT
ejpam-7054	324	19	=	=	PUNCT
ejpam-7054	325	1	x	x	X
ejpam-7054	325	2	−	−	PROPN
ejpam-7054	325	3	f+(v	f+(v	NOUN
ejpam-7054	325	4	)	)	PUNCT
ejpam-7054	325	5	.	.	PUNCT
ejpam-7054	326	1	thus	thus	ADV
ejpam-7054	326	2	,	,	PUNCT
ejpam-7054	326	3	f+(v	f+(v	PROPN
ejpam-7054	326	4	)	)	PUNCT
ejpam-7054	326	5	is	be	AUX
ejpam-7054	326	6	µ-preopen	µ-preopen	VERB
ejpam-7054	326	7	in	in	ADP
ejpam-7054	326	8	x.	x.	PROPN
ejpam-7054	326	9	(	(	PUNCT
ejpam-7054	326	10	4	4	NUM
ejpam-7054	326	11	)	)	PUNCT
ejpam-7054	326	12	⇒	⇒	NOUN
ejpam-7054	326	13	(	(	PUNCT
ejpam-7054	326	14	1	1	NUM
ejpam-7054	326	15	):	):	PUNCT
ejpam-7054	326	16	let	let	VERB
ejpam-7054	326	17	v	v	PART
ejpam-7054	326	18	be	be	AUX
ejpam-7054	326	19	any	any	DET
ejpam-7054	326	20	σ1σ2	σ1σ2	NOUN
ejpam-7054	326	21	-	-	ADJ
ejpam-7054	326	22	open	open	ADJ
ejpam-7054	326	23	set	set	NOUN
ejpam-7054	326	24	of	of	ADP
ejpam-7054	326	25	y	y	PROPN
ejpam-7054	326	26	.	.	PUNCT
ejpam-7054	327	1	since	since	SCONJ
ejpam-7054	327	2	(	(	PUNCT
ejpam-7054	327	3	y	y	PROPN
ejpam-7054	327	4	,	,	PUNCT
ejpam-7054	327	5	σ1	σ1	PROPN
ejpam-7054	327	6	,	,	PUNCT
ejpam-7054	327	7	σ2	σ2	PROPN
ejpam-7054	327	8	)	)	PUNCT
ejpam-7054	327	9	is	be	AUX
ejpam-7054	327	10	(	(	PUNCT
ejpam-7054	327	11	σ1	σ1	NOUN
ejpam-7054	327	12	,	,	PUNCT
ejpam-7054	327	13	σ2)-regular	σ2)-regular	ADJ
ejpam-7054	327	14	,	,	PUNCT
ejpam-7054	327	15	by	by	ADP
ejpam-7054	327	16	lemma	lemma	PROPN
ejpam-7054	327	17	3	3	NUM
ejpam-7054	327	18	we	we	PRON
ejpam-7054	327	19	have	have	VERB
ejpam-7054	327	20	v	v	NOUN
ejpam-7054	327	21	is	be	AUX
ejpam-7054	327	22	(	(	PUNCT
ejpam-7054	327	23	σ1	σ1	PROPN
ejpam-7054	327	24	,	,	PUNCT
ejpam-7054	327	25	σ2)θ	σ2)θ	NOUN
ejpam-7054	327	26	-	-	PUNCT
ejpam-7054	327	27	open	open	ADJ
ejpam-7054	327	28	in	in	ADP
ejpam-7054	327	29	y	y	PROPN
ejpam-7054	327	30	and	and	CCONJ
ejpam-7054	327	31	by	by	ADP
ejpam-7054	327	32	(	(	PUNCT
ejpam-7054	327	33	4	4	NUM
ejpam-7054	327	34	)	)	PUNCT
ejpam-7054	327	35	,	,	PUNCT
ejpam-7054	328	1	f+(v	f+(v	PROPN
ejpam-7054	328	2	)	)	PUNCT
ejpam-7054	329	1	is	be	AUX
ejpam-7054	329	2	µ-preopen	µ-preopen	VERB
ejpam-7054	329	3	in	in	ADP
ejpam-7054	329	4	x.	x.	NOUN
ejpam-7054	329	5	thus	thus	ADV
ejpam-7054	329	6	by	by	ADP
ejpam-7054	329	7	theorem	theorem	NOUN
ejpam-7054	329	8	9	9	NUM
ejpam-7054	329	9	,	,	PUNCT
ejpam-7054	329	10	f	f	PROPN
ejpam-7054	329	11	is	be	AUX
ejpam-7054	329	12	upper	upper	ADJ
ejpam-7054	329	13	µ(σ1	µ(σ1	NOUN
ejpam-7054	329	14	,	,	PUNCT
ejpam-7054	329	15	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	329	16	.	.	PUNCT
ejpam-7054	330	1	theorem	theorem	NOUN
ejpam-7054	330	2	12	12	NUM
ejpam-7054	330	3	.	.	PUNCT
ejpam-7054	331	1	for	for	ADP
ejpam-7054	331	2	a	a	DET
ejpam-7054	331	3	multifunction	multifunction	NOUN
ejpam-7054	331	4	f	f	NOUN
ejpam-7054	331	5	:	:	PUNCT
ejpam-7054	331	6	(	(	PUNCT
ejpam-7054	331	7	x,µ	x,µ	NOUN
ejpam-7054	331	8	)	)	PUNCT
ejpam-7054	331	9	→	→	SYM
ejpam-7054	331	10	(	(	PUNCT
ejpam-7054	331	11	y	y	PROPN
ejpam-7054	331	12	,	,	PUNCT
ejpam-7054	331	13	σ1	σ1	PROPN
ejpam-7054	331	14	,	,	PUNCT
ejpam-7054	331	15	σ2	σ2	NOUN
ejpam-7054	331	16	)	)	PUNCT
ejpam-7054	331	17	,	,	PUNCT
ejpam-7054	331	18	where	where	SCONJ
ejpam-7054	331	19	(	(	PUNCT
ejpam-7054	331	20	y	y	PROPN
ejpam-7054	331	21	,	,	PUNCT
ejpam-7054	331	22	σ1	σ1	PROPN
ejpam-7054	331	23	,	,	PUNCT
ejpam-7054	331	24	σ2	σ2	PROPN
ejpam-7054	331	25	)	)	PUNCT
ejpam-7054	331	26	is	be	AUX
ejpam-7054	331	27	(	(	PUNCT
ejpam-7054	331	28	σ1	σ1	PROPN
ejpam-7054	331	29	,	,	PUNCT
ejpam-7054	331	30	σ2)regular	σ2)regular	PROPN
ejpam-7054	331	31	,	,	PUNCT
ejpam-7054	331	32	the	the	DET
ejpam-7054	331	33	following	follow	VERB
ejpam-7054	331	34	properties	property	NOUN
ejpam-7054	331	35	are	be	AUX
ejpam-7054	331	36	equivalent	equivalent	ADJ
ejpam-7054	331	37	:	:	PUNCT
ejpam-7054	331	38	(	(	PUNCT
ejpam-7054	331	39	1	1	X
ejpam-7054	331	40	)	)	PUNCT
ejpam-7054	331	41	f	f	PROPN
ejpam-7054	331	42	is	be	AUX
ejpam-7054	331	43	lower	low	ADJ
ejpam-7054	331	44	µ(σ1	µ(σ1	NOUN
ejpam-7054	331	45	,	,	PUNCT
ejpam-7054	331	46	σ2)-precontinuous	σ2)-precontinuous	ADJ
ejpam-7054	331	47	;	;	PUNCT
ejpam-7054	331	48	(	(	PUNCT
ejpam-7054	331	49	2	2	X
ejpam-7054	331	50	)	)	PUNCT
ejpam-7054	331	51	f+((σ1	f+((σ1	NOUN
ejpam-7054	331	52	,	,	PUNCT
ejpam-7054	331	53	σ2)θ	σ2)θ	ADJ
ejpam-7054	331	54	-	-	PUNCT
ejpam-7054	331	55	cl(b	cl(b	NOUN
ejpam-7054	331	56	)	)	PUNCT
ejpam-7054	331	57	)	)	PUNCT
ejpam-7054	331	58	is	be	AUX
ejpam-7054	331	59	µ-preclosed	µ-preclose	VERB
ejpam-7054	331	60	in	in	ADP
ejpam-7054	331	61	x	x	PUNCT
ejpam-7054	331	62	for	for	ADP
ejpam-7054	331	63	every	every	DET
ejpam-7054	331	64	subset	subset	NOUN
ejpam-7054	331	65	b	b	PROPN
ejpam-7054	331	66	of	of	ADP
ejpam-7054	331	67	y	y	PROPN
ejpam-7054	331	68	;	;	PUNCT
ejpam-7054	331	69	(	(	PUNCT
ejpam-7054	331	70	3	3	X
ejpam-7054	331	71	)	)	PUNCT
ejpam-7054	331	72	f+(k	f+(k	NUM
ejpam-7054	331	73	)	)	PUNCT
ejpam-7054	331	74	is	be	AUX
ejpam-7054	331	75	µ-preclosed	µ-preclose	VERB
ejpam-7054	331	76	in	in	ADP
ejpam-7054	331	77	x	x	PUNCT
ejpam-7054	331	78	for	for	ADP
ejpam-7054	331	79	every	every	DET
ejpam-7054	331	80	(	(	PUNCT
ejpam-7054	331	81	σ1	σ1	PROPN
ejpam-7054	331	82	,	,	PUNCT
ejpam-7054	331	83	σ2)θ	σ2)θ	NOUN
ejpam-7054	331	84	-	-	PUNCT
ejpam-7054	331	85	closed	close	VERB
ejpam-7054	331	86	set	set	NOUN
ejpam-7054	331	87	k	k	PROPN
ejpam-7054	331	88	of	of	ADP
ejpam-7054	331	89	y	y	PROPN
ejpam-7054	331	90	;	;	PUNCT
ejpam-7054	331	91	(	(	PUNCT
ejpam-7054	331	92	4	4	X
ejpam-7054	331	93	)	)	PUNCT
ejpam-7054	331	94	f−(v	f−(v	NOUN
ejpam-7054	331	95	)	)	PUNCT
ejpam-7054	331	96	is	be	AUX
ejpam-7054	331	97	µ-preopen	µ-preopen	VERB
ejpam-7054	331	98	in	in	ADP
ejpam-7054	331	99	x	x	PUNCT
ejpam-7054	331	100	for	for	ADP
ejpam-7054	331	101	every	every	DET
ejpam-7054	331	102	(	(	PUNCT
ejpam-7054	331	103	σ1	σ1	PROPN
ejpam-7054	331	104	,	,	PUNCT
ejpam-7054	331	105	σ2)θ	σ2)θ	NOUN
ejpam-7054	331	106	-	-	PUNCT
ejpam-7054	331	107	open	open	ADJ
ejpam-7054	331	108	set	set	NOUN
ejpam-7054	331	109	v	v	NOUN
ejpam-7054	331	110	of	of	ADP
ejpam-7054	331	111	y	y	PROPN
ejpam-7054	331	112	;	;	PUNCT
ejpam-7054	331	113	(	(	PUNCT
ejpam-7054	331	114	5	5	X
ejpam-7054	331	115	)	)	PUNCT
ejpam-7054	331	116	f	f	PROPN
ejpam-7054	331	117	is	be	AUX
ejpam-7054	331	118	lower	low	ADJ
ejpam-7054	331	119	almost	almost	ADV
ejpam-7054	331	120	weakly	weakly	ADJ
ejpam-7054	331	121	µ(σ1	µ(σ1	NOUN
ejpam-7054	331	122	,	,	PUNCT
ejpam-7054	331	123	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	331	124	.	.	NOUN
ejpam-7054	331	125	proof	proof	NOUN
ejpam-7054	331	126	.	.	PUNCT
ejpam-7054	332	1	we	we	PRON
ejpam-7054	332	2	prove	prove	VERB
ejpam-7054	332	3	only	only	ADV
ejpam-7054	332	4	the	the	DET
ejpam-7054	332	5	implication	implication	NOUN
ejpam-7054	332	6	(	(	PUNCT
ejpam-7054	332	7	5	5	NUM
ejpam-7054	332	8	)	)	PUNCT
ejpam-7054	332	9	⇒	⇒	NOUN
ejpam-7054	332	10	(	(	PUNCT
ejpam-7054	332	11	1	1	NUM
ejpam-7054	332	12	)	)	PUNCT
ejpam-7054	332	13	,	,	PUNCT
ejpam-7054	332	14	the	the	DET
ejpam-7054	332	15	proof	proof	NOUN
ejpam-7054	332	16	of	of	ADP
ejpam-7054	332	17	the	the	DET
ejpam-7054	332	18	other	other	ADJ
ejpam-7054	332	19	being	be	AUX
ejpam-7054	332	20	similar	similar	ADJ
ejpam-7054	332	21	to	to	ADP
ejpam-7054	332	22	that	that	PRON
ejpam-7054	332	23	of	of	ADP
ejpam-7054	332	24	theprem	theprem	PROPN
ejpam-7054	332	25	11	11	NUM
ejpam-7054	332	26	.	.	PUNCT
ejpam-7054	333	1	the	the	DET
ejpam-7054	333	2	proof	proof	NOUN
ejpam-7054	333	3	of	of	ADP
ejpam-7054	333	4	the	the	DET
ejpam-7054	333	5	implication	implication	NOUN
ejpam-7054	333	6	(	(	PUNCT
ejpam-7054	333	7	4	4	X
ejpam-7054	333	8	)	)	PUNCT
ejpam-7054	333	9	⇒	⇒	NOUN
ejpam-7054	333	10	(	(	PUNCT
ejpam-7054	333	11	5	5	NUM
ejpam-7054	333	12	)	)	PUNCT
ejpam-7054	333	13	is	be	AUX
ejpam-7054	333	14	obvious	obvious	ADJ
ejpam-7054	333	15	.	.	PUNCT
ejpam-7054	334	1	(	(	PUNCT
ejpam-7054	334	2	5	5	X
ejpam-7054	334	3	)	)	PUNCT
ejpam-7054	334	4	⇒	⇒	NOUN
ejpam-7054	334	5	(	(	PUNCT
ejpam-7054	334	6	1	1	NUM
ejpam-7054	334	7	):	):	PUNCT
ejpam-7054	334	8	let	let	VERB
ejpam-7054	334	9	v	v	PART
ejpam-7054	334	10	be	be	AUX
ejpam-7054	334	11	any	any	DET
ejpam-7054	334	12	σ1σ2	σ1σ2	NOUN
ejpam-7054	334	13	-	-	ADJ
ejpam-7054	334	14	open	open	ADJ
ejpam-7054	334	15	set	set	NOUN
ejpam-7054	334	16	of	of	ADP
ejpam-7054	334	17	y	y	PROPN
ejpam-7054	334	18	and	and	CCONJ
ejpam-7054	334	19	x	x	PROPN
ejpam-7054	334	20	∈	∈	PROPN
ejpam-7054	334	21	f−(v	f−(v	NOUN
ejpam-7054	334	22	)	)	PUNCT
ejpam-7054	334	23	.	.	PUNCT
ejpam-7054	335	1	then	then	ADV
ejpam-7054	335	2	,	,	PUNCT
ejpam-7054	335	3	f	f	PROPN
ejpam-7054	335	4	(	(	PUNCT
ejpam-7054	335	5	x)∩v	x)∩v	PROPN
ejpam-7054	335	6	̸=	̸=	PROPN
ejpam-7054	335	7	∅.	∅.	NOUN
ejpam-7054	335	8	since	since	SCONJ
ejpam-7054	335	9	(	(	PUNCT
ejpam-7054	335	10	y	y	PROPN
ejpam-7054	335	11	,	,	PUNCT
ejpam-7054	335	12	σ1	σ1	PROPN
ejpam-7054	335	13	,	,	PUNCT
ejpam-7054	335	14	σ2	σ2	PROPN
ejpam-7054	335	15	)	)	PUNCT
ejpam-7054	335	16	is	be	AUX
ejpam-7054	335	17	(	(	PUNCT
ejpam-7054	335	18	σ1	σ1	NOUN
ejpam-7054	335	19	,	,	PUNCT
ejpam-7054	335	20	σ2)-regular	σ2)-regular	ADJ
ejpam-7054	335	21	,	,	PUNCT
ejpam-7054	335	22	there	there	PRON
ejpam-7054	335	23	exists	exist	VERB
ejpam-7054	335	24	a	a	DET
ejpam-7054	335	25	σ1σ2	σ1σ2	NUM
ejpam-7054	335	26	-	-	ADJ
ejpam-7054	335	27	open	open	ADJ
ejpam-7054	335	28	set	set	NOUN
ejpam-7054	335	29	w	w	PROPN
ejpam-7054	335	30	of	of	ADP
ejpam-7054	335	31	y	y	PRON
ejpam-7054	335	32	such	such	ADJ
ejpam-7054	335	33	that	that	SCONJ
ejpam-7054	335	34	f	f	PROPN
ejpam-7054	335	35	(	(	PUNCT
ejpam-7054	335	36	x)∩w	x)∩w	PROPN
ejpam-7054	335	37	̸=	̸=	PROPN
ejpam-7054	335	38	∅	∅	NOUN
ejpam-7054	335	39	and	and	CCONJ
ejpam-7054	335	40	σ1σ2	σ1σ2	NOUN
ejpam-7054	335	41	-	-	PUNCT
ejpam-7054	335	42	cl(w	cl(w	NOUN
ejpam-7054	335	43	)	)	PUNCT
ejpam-7054	335	44	⊆	⊆	NUM
ejpam-7054	335	45	v	v	NOUN
ejpam-7054	335	46	.	.	PUNCT
ejpam-7054	336	1	since	since	SCONJ
ejpam-7054	336	2	f	f	PROPN
ejpam-7054	336	3	is	be	AUX
ejpam-7054	336	4	lower	low	ADJ
ejpam-7054	336	5	almost	almost	ADV
ejpam-7054	336	6	weakly	weakly	ADJ
ejpam-7054	336	7	µ(σ1	µ(σ1	NOUN
ejpam-7054	336	8	,	,	PUNCT
ejpam-7054	336	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-7054	336	10	,	,	PUNCT
ejpam-7054	336	11	by	by	ADP
ejpam-7054	336	12	theorem	theorem	NOUN
ejpam-7054	336	13	2	2	NUM
ejpam-7054	336	14	there	there	ADV
ejpam-7054	336	15	exists	exist	VERB
ejpam-7054	336	16	a	a	DET
ejpam-7054	336	17	µ-preopen	µ-preopen	PROPN
ejpam-7054	336	18	set	set	VERB
ejpam-7054	336	19	u	u	NOUN
ejpam-7054	336	20	of	of	ADP
ejpam-7054	336	21	x	x	PUNCT
ejpam-7054	336	22	containing	contain	VERB
ejpam-7054	336	23	x	x	PUNCT
ejpam-7054	336	24	such	such	ADJ
ejpam-7054	336	25	that	that	SCONJ
ejpam-7054	336	26	u	u	NOUN
ejpam-7054	336	27	⊆	⊆	NUM
ejpam-7054	336	28	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-7054	336	29	-	-	PUNCT
ejpam-7054	336	30	cl(w	cl(w	NOUN
ejpam-7054	336	31	)	)	PUNCT
ejpam-7054	336	32	)	)	PUNCT
ejpam-7054	337	1	⊆	⊆	NUM
ejpam-7054	337	2	f−(v	f−(v	NOUN
ejpam-7054	337	3	)	)	PUNCT
ejpam-7054	337	4	.	.	PUNCT
ejpam-7054	338	1	thus	thus	ADV
ejpam-7054	338	2	,	,	PUNCT
ejpam-7054	338	3	x	x	SYM
ejpam-7054	338	4	∈	∈	PROPN
ejpam-7054	338	5	iµ(π)(f	iµ(π)(f	NOUN
ejpam-7054	338	6	−(v	−(v	NOUN
ejpam-7054	338	7	)	)	PUNCT
ejpam-7054	338	8	)	)	PUNCT
ejpam-7054	338	9	and	and	CCONJ
ejpam-7054	338	10	hence	hence	ADV
ejpam-7054	338	11	f−(v	f−(v	ADJ
ejpam-7054	338	12	)	)	PUNCT
ejpam-7054	338	13	⊆	⊆	NUM
ejpam-7054	338	14	iµ(π)(f	iµ(π)(f	ADP
ejpam-7054	338	15	−(v	−(v	NOUN
ejpam-7054	338	16	)	)	PUNCT
ejpam-7054	338	17	)	)	PUNCT
ejpam-7054	338	18	.	.	PUNCT
ejpam-7054	339	1	therefore	therefore	ADV
ejpam-7054	339	2	,	,	PUNCT
ejpam-7054	339	3	f−(v	f−(v	ADJ
ejpam-7054	339	4	)	)	PUNCT
ejpam-7054	339	5	is	be	AUX
ejpam-7054	339	6	µpreopen	µpreopen	ADJ
ejpam-7054	339	7	in	in	ADP
ejpam-7054	339	8	x	x	PUNCT
ejpam-7054	339	9	and	and	CCONJ
ejpam-7054	339	10	by	by	ADP
ejpam-7054	339	11	theorem	theorem	NOUN
ejpam-7054	339	12	10	10	NUM
ejpam-7054	339	13	,	,	PUNCT
ejpam-7054	339	14	f	f	PROPN
ejpam-7054	339	15	is	be	AUX
ejpam-7054	339	16	lower	low	ADJ
ejpam-7054	339	17	µ(σ1	µ(σ1	NOUN
ejpam-7054	339	18	,	,	PUNCT
ejpam-7054	339	19	σ2)-precontinuous	σ2)-precontinuous	PROPN
ejpam-7054	339	20	.	.	PUNCT
ejpam-7054	340	1	b.	b.	PROPN
ejpam-7054	340	2	kong	kong	PROPN
ejpam-7054	340	3	-	-	PUNCT
ejpam-7054	340	4	ied	ied	PROPN
ejpam-7054	340	5	,	,	PUNCT
ejpam-7054	340	6	a.	a.	PROPN
ejpam-7054	340	7	sama	sama	PROPN
ejpam-7054	340	8	-	-	PUNCT
ejpam-7054	340	9	ae	ae	PROPN
ejpam-7054	340	10	,	,	PUNCT
ejpam-7054	340	11	c.	c.	PROPN
ejpam-7054	340	12	boonpok	boonpok	PROPN
ejpam-7054	340	13	/	/	SYM
ejpam-7054	340	14	eur	eur	PROPN
ejpam-7054	340	15	.	.	PUNCT
ejpam-7054	341	1	j.	j.	PROPN
ejpam-7054	341	2	pure	pure	PROPN
ejpam-7054	341	3	appl	appl	PROPN
ejpam-7054	341	4	.	.	PROPN
ejpam-7054	341	5	math	math	PROPN
ejpam-7054	341	6	,	,	PUNCT
ejpam-7054	341	7	18	18	NUM
ejpam-7054	341	8	(	(	PUNCT
ejpam-7054	341	9	4	4	NUM
ejpam-7054	341	10	)	)	PUNCT
ejpam-7054	341	11	(	(	PUNCT
ejpam-7054	341	12	2025	2025	NUM
ejpam-7054	341	13	)	)	PUNCT
ejpam-7054	341	14	,	,	PUNCT
ejpam-7054	341	15	7054	7054	NUM
ejpam-7054	341	16	11	11	NUM
ejpam-7054	341	17	of	of	ADP
ejpam-7054	341	18	12	12	NUM
ejpam-7054	341	19	acknowledgements	acknowledgement	NOUN
ejpam-7054	341	20	this	this	DET
ejpam-7054	341	21	research	research	NOUN
ejpam-7054	341	22	project	project	NOUN
ejpam-7054	341	23	was	be	AUX
ejpam-7054	341	24	financially	financially	ADV
ejpam-7054	341	25	supported	support	VERB
ejpam-7054	341	26	by	by	ADP
ejpam-7054	341	27	mahasarakham	mahasarakham	PROPN
ejpam-7054	341	28	university	university	PROPN
ejpam-7054	341	29	.	.	PUNCT
ejpam-7054	342	1	references	reference	NOUN
ejpam-7054	342	2	[	[	X
ejpam-7054	342	3	1	1	NUM
ejpam-7054	342	4	]	]	PUNCT
ejpam-7054	342	5	m.	m.	NOUN
ejpam-7054	342	6	k.	k.	PROPN
ejpam-7054	342	7	singal	singal	PROPN
ejpam-7054	342	8	and	and	CCONJ
ejpam-7054	342	9	a.	a.	PROPN
ejpam-7054	342	10	r.	r.	PROPN
ejpam-7054	342	11	singal	singal	PROPN
ejpam-7054	342	12	.	.	PUNCT
ejpam-7054	343	1	almost	almost	ADV
ejpam-7054	343	2	continuous	continuous	ADJ
ejpam-7054	343	3	mappings	mapping	NOUN
ejpam-7054	343	4	.	.	PUNCT
ejpam-7054	344	1	yokohama	yokohama	PROPN
ejpam-7054	344	2	mathematical	mathematical	PROPN
ejpam-7054	344	3	journal	journal	PROPN
ejpam-7054	344	4	,	,	PUNCT
ejpam-7054	344	5	16:63–73	16:63–73	PROPN
ejpam-7054	344	6	,	,	PUNCT
ejpam-7054	344	7	1968	1968	NUM
ejpam-7054	344	8	.	.	PUNCT
ejpam-7054	345	1	[	[	X
ejpam-7054	345	2	2	2	NUM
ejpam-7054	345	3	]	]	X
ejpam-7054	345	4	b.	b.	PROPN
ejpam-7054	345	5	m.	m.	PROPN
ejpam-7054	345	6	munshi	munshi	PROPN
ejpam-7054	345	7	and	and	CCONJ
ejpam-7054	345	8	d.	d.	PROPN
ejpam-7054	345	9	s.	s.	PROPN
ejpam-7054	345	10	bassan	bassan	PROPN
ejpam-7054	345	11	.	.	PUNCT
ejpam-7054	346	1	almost	almost	ADV
ejpam-7054	346	2	semi	semi	ADJ
ejpam-7054	346	3	-	-	ADJ
ejpam-7054	346	4	continuous	continuous	ADJ
ejpam-7054	346	5	mappings	mapping	NOUN
ejpam-7054	346	6	.	.	PUNCT
ejpam-7054	347	1	the	the	DET
ejpam-7054	347	2	mathematics	mathematics	PROPN
ejpam-7054	347	3	student	student	NOUN
ejpam-7054	347	4	,	,	PUNCT
ejpam-7054	347	5	49:239–248	49:239–248	PROPN
ejpam-7054	347	6	,	,	PUNCT
ejpam-7054	347	7	1981	1981	NUM
ejpam-7054	347	8	.	.	PUNCT
ejpam-7054	348	1	[	[	X
ejpam-7054	348	2	3	3	X
ejpam-7054	348	3	]	]	PUNCT
ejpam-7054	348	4	t.	t.	PROPN
ejpam-7054	348	5	noiri	noiri	PROPN
ejpam-7054	348	6	.	.	PUNCT
ejpam-7054	349	1	almost	almost	ADV
ejpam-7054	349	2	α	α	NUM
ejpam-7054	349	3	-	-	ADJ
ejpam-7054	349	4	continuous	continuous	ADJ
ejpam-7054	349	5	functions	function	NOUN
ejpam-7054	349	6	.	.	PUNCT
ejpam-7054	350	1	kyungpook	kyungpook	PROPN
ejpam-7054	350	2	mathematical	mathematical	PROPN
ejpam-7054	350	3	journal	journal	PROPN
ejpam-7054	350	4	,	,	PUNCT
ejpam-7054	350	5	28:71–77	28:71–77	PROPN
ejpam-7054	350	6	,	,	PUNCT
ejpam-7054	350	7	1988	1988	NUM
ejpam-7054	350	8	.	.	PUNCT
ejpam-7054	351	1	[	[	X
ejpam-7054	351	2	4	4	X
ejpam-7054	351	3	]	]	PUNCT
ejpam-7054	351	4	a.	a.	NOUN
ejpam-7054	351	5	a.	a.	NOUN
ejpam-7054	351	6	nasef	nasef	PROPN
ejpam-7054	351	7	and	and	CCONJ
ejpam-7054	351	8	t.	t.	PROPN
ejpam-7054	351	9	noiri	noiri	PROPN
ejpam-7054	351	10	.	.	PUNCT
ejpam-7054	352	1	some	some	DET
ejpam-7054	352	2	weak	weak	ADJ
ejpam-7054	352	3	forms	form	NOUN
ejpam-7054	352	4	of	of	ADP
ejpam-7054	352	5	almost	almost	ADV
ejpam-7054	352	6	continuity	continuity	NOUN
ejpam-7054	352	7	.	.	PUNCT
ejpam-7054	353	1	acta	acta	PROPN
ejpam-7054	353	2	mathematica	mathematica	PROPN
ejpam-7054	353	3	hungarica	hungarica	PROPN
ejpam-7054	353	4	,	,	PUNCT
ejpam-7054	353	5	74(3):211–219	74(3):211–219	PROPN
ejpam-7054	353	6	,	,	PUNCT
ejpam-7054	353	7	1997	1997	NUM
ejpam-7054	353	8	.	.	PUNCT
ejpam-7054	354	1	[	[	X
ejpam-7054	354	2	5	5	NUM
ejpam-7054	354	3	]	]	X
ejpam-7054	354	4	n.	n.	PROPN
ejpam-7054	354	5	levine	levine	PROPN
ejpam-7054	354	6	.	.	PUNCT
ejpam-7054	355	1	a	a	DET
ejpam-7054	355	2	decomposition	decomposition	NOUN
ejpam-7054	355	3	of	of	ADP
ejpam-7054	355	4	continuity	continuity	NOUN
ejpam-7054	355	5	in	in	ADP
ejpam-7054	355	6	topological	topological	ADJ
ejpam-7054	355	7	spaces	space	NOUN
ejpam-7054	355	8	.	.	PUNCT
ejpam-7054	356	1	the	the	DET
ejpam-7054	356	2	american	american	PROPN
ejpam-7054	356	3	mathematical	mathematical	PROPN
ejpam-7054	356	4	monthly	monthly	ADV
ejpam-7054	356	5	,	,	PUNCT
ejpam-7054	356	6	68:44–46	68:44–46	NUM
ejpam-7054	356	7	,	,	PUNCT
ejpam-7054	356	8	1961	1961	NUM
ejpam-7054	356	9	.	.	PUNCT
ejpam-7054	357	1	[	[	X
ejpam-7054	357	2	6	6	NUM
ejpam-7054	357	3	]	]	PUNCT
ejpam-7054	357	4	t.	t.	PROPN
ejpam-7054	357	5	husain	husain	PROPN
ejpam-7054	357	6	.	.	PUNCT
ejpam-7054	358	1	almost	almost	ADV
ejpam-7054	358	2	continuous	continuous	ADJ
ejpam-7054	358	3	mappings	mapping	NOUN
ejpam-7054	358	4	.	.	PUNCT
ejpam-7054	359	1	prace	prace	PROPN
ejpam-7054	359	2	matematyczne	matematyczne	PROPN
ejpam-7054	359	3	,	,	PUNCT
ejpam-7054	359	4	10:1–7	10:1–7	NUM
ejpam-7054	359	5	,	,	PUNCT
ejpam-7054	359	6	1966	1966	NUM
ejpam-7054	359	7	.	.	PUNCT
ejpam-7054	360	1	[	[	X
ejpam-7054	360	2	7	7	X
ejpam-7054	360	3	]	]	PUNCT
ejpam-7054	360	4	t.	t.	PROPN
ejpam-7054	360	5	noiri	noiri	PROPN
ejpam-7054	360	6	.	.	PUNCT
ejpam-7054	361	1	properties	property	NOUN
ejpam-7054	361	2	of	of	ADP
ejpam-7054	361	3	some	some	DET
ejpam-7054	361	4	weak	weak	ADJ
ejpam-7054	361	5	forms	form	NOUN
ejpam-7054	361	6	of	of	ADP
ejpam-7054	361	7	continuity	continuity	NOUN
ejpam-7054	361	8	.	.	PUNCT
ejpam-7054	362	1	international	international	ADJ
ejpam-7054	362	2	journal	journal	PROPN
ejpam-7054	362	3	of	of	ADP
ejpam-7054	362	4	mathematics	mathematics	PROPN
ejpam-7054	362	5	and	and	CCONJ
ejpam-7054	362	6	mathematical	mathematical	ADJ
ejpam-7054	362	7	sciences	science	NOUN
ejpam-7054	362	8	,	,	PUNCT
ejpam-7054	362	9	10(1):97–111	10(1):97–111	NUM
ejpam-7054	362	10	,	,	PUNCT
ejpam-7054	362	11	1987	1987	NUM
ejpam-7054	362	12	.	.	PUNCT
ejpam-7054	363	1	[	[	X
ejpam-7054	363	2	8	8	NUM
ejpam-7054	363	3	]	]	X
ejpam-7054	363	4	d.	d.	PROPN
ejpam-7054	363	5	a.	a.	PROPN
ejpam-7054	363	6	rose	rise	VERB
ejpam-7054	363	7	.	.	PUNCT
ejpam-7054	364	1	weak	weak	ADJ
ejpam-7054	364	2	continuity	continuity	NOUN
ejpam-7054	364	3	and	and	CCONJ
ejpam-7054	364	4	almost	almost	ADV
ejpam-7054	364	5	continuity	continuity	NOUN
ejpam-7054	364	6	.	.	PUNCT
ejpam-7054	365	1	international	international	ADJ
ejpam-7054	365	2	journal	journal	PROPN
ejpam-7054	365	3	of	of	ADP
ejpam-7054	365	4	mathematics	mathematics	PROPN
ejpam-7054	365	5	and	and	CCONJ
ejpam-7054	365	6	mathematical	mathematical	ADJ
ejpam-7054	365	7	sciences	science	NOUN
ejpam-7054	365	8	,	,	PUNCT
ejpam-7054	365	9	7:311–318	7:311–318	PROPN
ejpam-7054	365	10	,	,	PUNCT
ejpam-7054	365	11	1984	1984	NUM
ejpam-7054	365	12	.	.	PUNCT
ejpam-7054	366	1	[	[	X
ejpam-7054	366	2	9	9	NUM
ejpam-7054	366	3	]	]	PUNCT
ejpam-7054	366	4	t.	t.	PROPN
ejpam-7054	366	5	noiri	noiri	PROPN
ejpam-7054	366	6	and	and	CCONJ
ejpam-7054	366	7	v.	v.	ADP
ejpam-7054	366	8	popa	popa	NOUN
ejpam-7054	366	9	.	.	PUNCT
ejpam-7054	367	1	almost	almost	ADV
ejpam-7054	367	2	weakly	weakly	ADJ
ejpam-7054	367	3	continuous	continuous	ADJ
ejpam-7054	367	4	multifunctions	multifunction	NOUN
ejpam-7054	367	5	.	.	PUNCT
ejpam-7054	368	1	demonstratio	demonstratio	PROPN
ejpam-7054	368	2	mathematica	mathematica	PROPN
ejpam-7054	368	3	,	,	PUNCT
ejpam-7054	368	4	26(2):363–380	26(2):363–380	PROPN
ejpam-7054	368	5	,	,	PUNCT
ejpam-7054	368	6	1993	1993	NUM
ejpam-7054	368	7	.	.	PUNCT
ejpam-7054	369	1	[	[	X
ejpam-7054	369	2	10	10	NUM
ejpam-7054	369	3	]	]	X
ejpam-7054	369	4	v.	v.	CCONJ
ejpam-7054	369	5	popa	popa	NOUN
ejpam-7054	369	6	and	and	CCONJ
ejpam-7054	369	7	t.	t.	PROPN
ejpam-7054	369	8	noiri	noiri	PROPN
ejpam-7054	369	9	.	.	PUNCT
ejpam-7054	370	1	some	some	DET
ejpam-7054	370	2	properties	property	NOUN
ejpam-7054	370	3	of	of	ADP
ejpam-7054	370	4	almost	almost	ADV
ejpam-7054	370	5	weakly	weakly	ADJ
ejpam-7054	370	6	continuous	continuous	ADJ
ejpam-7054	370	7	multifunctions	multifunction	NOUN
ejpam-7054	370	8	.	.	PUNCT
ejpam-7054	371	1	demonstratio	demonstratio	PROPN
ejpam-7054	371	2	mathematica	mathematica	PROPN
ejpam-7054	371	3	,	,	PUNCT
ejpam-7054	371	4	32(3):605–614	32(3):605–614	NUM
ejpam-7054	371	5	,	,	PUNCT
ejpam-7054	371	6	1999	1999	NUM
ejpam-7054	371	7	.	.	PUNCT
ejpam-7054	372	1	[	[	X
ejpam-7054	372	2	11	11	NUM
ejpam-7054	372	3	]	]	PUNCT
ejpam-7054	372	4	á.	á.	PROPN
ejpam-7054	372	5	császár	császár	PROPN
ejpam-7054	372	6	.	.	PUNCT
ejpam-7054	373	1	generalized	generalize	VERB
ejpam-7054	373	2	topology	topology	NOUN
ejpam-7054	373	3	,	,	PUNCT
ejpam-7054	373	4	generalized	generalize	VERB
ejpam-7054	373	5	continuity	continuity	NOUN
ejpam-7054	373	6	.	.	PUNCT
ejpam-7054	374	1	acta	acta	PROPN
ejpam-7054	374	2	mathematica	mathematica	PROPN
ejpam-7054	374	3	hungarica	hungarica	PROPN
ejpam-7054	374	4	,	,	PUNCT
ejpam-7054	374	5	96(4):351–357	96(4):351–357	NOUN
ejpam-7054	374	6	,	,	PUNCT
ejpam-7054	374	7	2002	2002	NUM
ejpam-7054	374	8	.	.	PUNCT
ejpam-7054	375	1	[	[	X
ejpam-7054	375	2	12	12	NUM
ejpam-7054	375	3	]	]	PUNCT
ejpam-7054	375	4	a.	a.	NOUN
ejpam-7054	375	5	kanibir	kanibir	NOUN
ejpam-7054	375	6	and	and	CCONJ
ejpam-7054	375	7	i.	i.	PROPN
ejpam-7054	375	8	l.	l.	PROPN
ejpam-7054	375	9	reilly	reilly	PROPN
ejpam-7054	375	10	.	.	PUNCT
ejpam-7054	376	1	generalized	generalize	VERB
ejpam-7054	376	2	continuity	continuity	NOUN
ejpam-7054	376	3	for	for	ADP
ejpam-7054	376	4	multifunctions	multifunction	NOUN
ejpam-7054	376	5	.	.	PUNCT
ejpam-7054	377	1	acta	acta	PROPN
ejpam-7054	377	2	mathematica	mathematica	PROPN
ejpam-7054	377	3	hungarica	hungarica	PROPN
ejpam-7054	377	4	,	,	PUNCT
ejpam-7054	377	5	122(3):283–292	122(3):283–292	NUM
ejpam-7054	377	6	,	,	PUNCT
ejpam-7054	377	7	2009	2009	NUM
ejpam-7054	377	8	.	.	PUNCT
ejpam-7054	378	1	[	[	X
ejpam-7054	378	2	13	13	NUM
ejpam-7054	378	3	]	]	PUNCT
ejpam-7054	378	4	c.	c.	PROPN
ejpam-7054	378	5	boonpok	boonpok	PROPN
ejpam-7054	378	6	.	.	PUNCT
ejpam-7054	379	1	on	on	ADP
ejpam-7054	379	2	upper	upper	ADJ
ejpam-7054	379	3	and	and	CCONJ
ejpam-7054	379	4	lower	low	ADJ
ejpam-7054	379	5	β(µx	β(µx	PROPN
ejpam-7054	379	6	,	,	PUNCT
ejpam-7054	379	7	µy	µy	CCONJ
ejpam-7054	379	8	)	)	PUNCT
ejpam-7054	379	9	-continuous	-continuous	ADJ
ejpam-7054	379	10	multifunctions	multifunction	NOUN
ejpam-7054	379	11	.	.	PUNCT
ejpam-7054	380	1	international	international	ADJ
ejpam-7054	380	2	journal	journal	PROPN
ejpam-7054	380	3	of	of	ADP
ejpam-7054	380	4	mathematics	mathematics	PROPN
ejpam-7054	380	5	and	and	CCONJ
ejpam-7054	380	6	mathematical	mathematical	ADJ
ejpam-7054	380	7	sciences	science	NOUN
ejpam-7054	380	8	,	,	PUNCT
ejpam-7054	380	9	2012:931656	2012:931656	NUM
ejpam-7054	380	10	,	,	PUNCT
ejpam-7054	380	11	2012	2012	NUM
ejpam-7054	380	12	.	.	PUNCT
ejpam-7054	381	1	[	[	X
ejpam-7054	381	2	14	14	NUM
ejpam-7054	381	3	]	]	X
ejpam-7054	381	4	n.	n.	PROPN
ejpam-7054	381	5	srisarakham	srisarakham	PROPN
ejpam-7054	381	6	and	and	CCONJ
ejpam-7054	381	7	c.	c.	PROPN
ejpam-7054	381	8	boonpok	boonpok	PROPN
ejpam-7054	381	9	.	.	PUNCT
ejpam-7054	382	1	characterizations	characterization	NOUN
ejpam-7054	382	2	of	of	ADP
ejpam-7054	382	3	upper	upper	ADJ
ejpam-7054	382	4	and	and	CCONJ
ejpam-7054	382	5	lower	low	ADJ
ejpam-7054	382	6	α(µx	α(µx	NUM
ejpam-7054	382	7	,	,	PUNCT
ejpam-7054	382	8	µy	µy	CCONJ
ejpam-7054	382	9	)	)	PUNCT
ejpam-7054	382	10	continuous	continuous	ADJ
ejpam-7054	382	11	multifunctions	multifunction	NOUN
ejpam-7054	382	12	.	.	PUNCT
ejpam-7054	383	1	journal	journal	PROPN
ejpam-7054	383	2	of	of	ADP
ejpam-7054	383	3	mathematics	mathematics	PROPN
ejpam-7054	383	4	and	and	CCONJ
ejpam-7054	383	5	computer	computer	NOUN
ejpam-7054	383	6	science	science	NOUN
ejpam-7054	383	7	,	,	PUNCT
ejpam-7054	383	8	17:255	17:255	NUM
ejpam-7054	383	9	–	–	PUNCT
ejpam-7054	383	10	265	265	NUM
ejpam-7054	383	11	,	,	PUNCT
ejpam-7054	383	12	2017	2017	NUM
ejpam-7054	383	13	.	.	PUNCT
ejpam-7054	384	1	[	[	X
ejpam-7054	384	2	15	15	NUM
ejpam-7054	384	3	]	]	X
ejpam-7054	384	4	p.	p.	NOUN
ejpam-7054	384	5	pue	pue	NOUN
ejpam-7054	384	6	-	-	PUNCT
ejpam-7054	384	7	on	on	ADP
ejpam-7054	384	8	,	,	PUNCT
ejpam-7054	384	9	s.	s.	PROPN
ejpam-7054	384	10	sompong	sompong	PROPN
ejpam-7054	384	11	,	,	PUNCT
ejpam-7054	384	12	and	and	CCONJ
ejpam-7054	384	13	c.	c.	PROPN
ejpam-7054	384	14	boonpok	boonpok	PROPN
ejpam-7054	384	15	.	.	PUNCT
ejpam-7054	385	1	upper	upper	ADJ
ejpam-7054	385	2	and	and	CCONJ
ejpam-7054	385	3	lower	low	ADJ
ejpam-7054	385	4	(	(	PUNCT
ejpam-7054	385	5	τ1	τ1	NOUN
ejpam-7054	385	6	,	,	PUNCT
ejpam-7054	385	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-7054	385	8	multifunctions	multifunction	NOUN
ejpam-7054	385	9	.	.	PUNCT
ejpam-7054	386	1	international	international	ADJ
ejpam-7054	386	2	journal	journal	PROPN
ejpam-7054	386	3	of	of	ADP
ejpam-7054	386	4	mathematics	mathematic	NOUN
ejpam-7054	386	5	and	and	CCONJ
ejpam-7054	386	6	computer	computer	NOUN
ejpam-7054	386	7	science	science	NOUN
ejpam-7054	386	8	,	,	PUNCT
ejpam-7054	386	9	19(4):1305	19(4):1305	NUM
ejpam-7054	386	10	–	–	PUNCT
ejpam-7054	386	11	1310	1310	NUM
ejpam-7054	386	12	,	,	PUNCT
ejpam-7054	386	13	2024	2024	NUM
ejpam-7054	386	14	.	.	PUNCT
ejpam-7054	387	1	[	[	X
ejpam-7054	387	2	16	16	NUM
ejpam-7054	387	3	]	]	X
ejpam-7054	387	4	c.	c.	PROPN
ejpam-7054	387	5	klanarong	klanarong	PROPN
ejpam-7054	387	6	,	,	PUNCT
ejpam-7054	387	7	s.	s.	PROPN
ejpam-7054	387	8	sompong	sompong	PROPN
ejpam-7054	387	9	,	,	PUNCT
ejpam-7054	387	10	and	and	CCONJ
ejpam-7054	387	11	c.	c.	PROPN
ejpam-7054	387	12	boonpok	boonpok	PROPN
ejpam-7054	387	13	.	.	PUNCT
ejpam-7054	388	1	upper	upper	ADJ
ejpam-7054	388	2	and	and	CCONJ
ejpam-7054	388	3	lower	low	ADJ
ejpam-7054	388	4	almost	almost	ADV
ejpam-7054	388	5	(	(	PUNCT
ejpam-7054	388	6	τ1	τ1	NOUN
ejpam-7054	388	7	,	,	PUNCT
ejpam-7054	388	8	τ2)continuous	τ2)continuous	ADJ
ejpam-7054	388	9	multifunctions	multifunction	NOUN
ejpam-7054	388	10	.	.	PUNCT
ejpam-7054	389	1	european	european	ADJ
ejpam-7054	389	2	journal	journal	PROPN
ejpam-7054	389	3	of	of	ADP
ejpam-7054	389	4	pure	pure	ADJ
ejpam-7054	389	5	and	and	CCONJ
ejpam-7054	389	6	applied	applied	ADJ
ejpam-7054	389	7	mathematics	mathematic	NOUN
ejpam-7054	389	8	,	,	PUNCT
ejpam-7054	389	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-7054	389	10	,	,	PUNCT
ejpam-7054	389	11	2024	2024	NUM
ejpam-7054	389	12	.	.	PUNCT
ejpam-7054	390	1	[	[	X
ejpam-7054	390	2	17	17	NUM
ejpam-7054	390	3	]	]	PUNCT
ejpam-7054	390	4	m.	m.	NOUN
ejpam-7054	390	5	thongmoon	thongmoon	NOUN
ejpam-7054	390	6	,	,	PUNCT
ejpam-7054	390	7	s.	s.	PROPN
ejpam-7054	390	8	sompong	sompong	PROPN
ejpam-7054	390	9	,	,	PUNCT
ejpam-7054	390	10	and	and	CCONJ
ejpam-7054	390	11	c.	c.	PROPN
ejpam-7054	390	12	boonpok	boonpok	PROPN
ejpam-7054	390	13	.	.	PUNCT
ejpam-7054	391	1	upper	upper	ADJ
ejpam-7054	391	2	and	and	CCONJ
ejpam-7054	391	3	lower	low	ADJ
ejpam-7054	391	4	weak	weak	ADJ
ejpam-7054	391	5	(	(	PUNCT
ejpam-7054	391	6	τ1	τ1	NOUN
ejpam-7054	391	7	,	,	PUNCT
ejpam-7054	391	8	τ2)continuity	τ2)continuity	PROPN
ejpam-7054	391	9	.	.	PUNCT
ejpam-7054	392	1	european	european	PROPN
ejpam-7054	392	2	journal	journal	PROPN
ejpam-7054	392	3	of	of	ADP
ejpam-7054	392	4	pure	pure	ADJ
ejpam-7054	392	5	and	and	CCONJ
ejpam-7054	392	6	applied	applied	ADJ
ejpam-7054	392	7	mathematics	mathematic	NOUN
ejpam-7054	392	8	,	,	PUNCT
ejpam-7054	392	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-7054	392	10	,	,	PUNCT
ejpam-7054	392	11	2024	2024	NUM
ejpam-7054	392	12	.	.	PUNCT
ejpam-7054	393	1	b.	b.	PROPN
ejpam-7054	393	2	kong	kong	PROPN
ejpam-7054	393	3	-	-	PUNCT
ejpam-7054	393	4	ied	ied	PROPN
ejpam-7054	393	5	,	,	PUNCT
ejpam-7054	393	6	a.	a.	PROPN
ejpam-7054	393	7	sama	sama	PROPN
ejpam-7054	393	8	-	-	PUNCT
ejpam-7054	393	9	ae	ae	PROPN
ejpam-7054	393	10	,	,	PUNCT
ejpam-7054	393	11	c.	c.	PROPN
ejpam-7054	393	12	boonpok	boonpok	PROPN
ejpam-7054	393	13	/	/	SYM
ejpam-7054	393	14	eur	eur	PROPN
ejpam-7054	393	15	.	.	PUNCT
ejpam-7054	394	1	j.	j.	PROPN
ejpam-7054	394	2	pure	pure	PROPN
ejpam-7054	394	3	appl	appl	PROPN
ejpam-7054	394	4	.	.	PROPN
ejpam-7054	394	5	math	math	PROPN
ejpam-7054	394	6	,	,	PUNCT
ejpam-7054	394	7	18	18	NUM
ejpam-7054	394	8	(	(	PUNCT
ejpam-7054	394	9	4	4	NUM
ejpam-7054	394	10	)	)	PUNCT
ejpam-7054	394	11	(	(	PUNCT
ejpam-7054	394	12	2025	2025	NUM
ejpam-7054	394	13	)	)	PUNCT
ejpam-7054	394	14	,	,	PUNCT
ejpam-7054	394	15	7054	7054	NUM
ejpam-7054	394	16	12	12	NUM
ejpam-7054	394	17	of	of	ADP
ejpam-7054	394	18	12	12	NUM
ejpam-7054	394	19	[	[	SYM
ejpam-7054	394	20	18	18	NUM
ejpam-7054	394	21	]	]	PUNCT
ejpam-7054	394	22	c.	c.	PROPN
ejpam-7054	394	23	boonpok	boonpok	PROPN
ejpam-7054	394	24	and	and	CCONJ
ejpam-7054	394	25	c.	c.	PROPN
ejpam-7054	394	26	viriyapong	viriyapong	PROPN
ejpam-7054	394	27	.	.	PUNCT
ejpam-7054	395	1	upper	upper	ADJ
ejpam-7054	395	2	and	and	CCONJ
ejpam-7054	395	3	lower	low	ADJ
ejpam-7054	395	4	weak	weak	ADJ
ejpam-7054	395	5	(	(	PUNCT
ejpam-7054	395	6	τ1	τ1	NOUN
ejpam-7054	395	7	,	,	PUNCT
ejpam-7054	395	8	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7054	395	9	.	.	PUNCT
ejpam-7054	395	10	european	european	PROPN
ejpam-7054	395	11	journal	journal	PROPN
ejpam-7054	395	12	of	of	ADP
ejpam-7054	395	13	pure	pure	ADJ
ejpam-7054	395	14	and	and	CCONJ
ejpam-7054	395	15	applied	applied	ADJ
ejpam-7054	395	16	mathematics	mathematic	NOUN
ejpam-7054	395	17	,	,	PUNCT
ejpam-7054	395	18	14(4):1212–1225	14(4):1212–1225	NUM
ejpam-7054	395	19	,	,	PUNCT
ejpam-7054	395	20	2021	2021	NUM
ejpam-7054	395	21	.	.	PUNCT
ejpam-7054	396	1	[	[	X
ejpam-7054	396	2	19	19	NUM
ejpam-7054	396	3	]	]	X
ejpam-7054	396	4	c.	c.	PROPN
ejpam-7054	396	5	viriyapong	viriyapong	PROPN
ejpam-7054	396	6	,	,	PUNCT
ejpam-7054	396	7	a.	a.	PROPN
ejpam-7054	396	8	sama	sama	PROPN
ejpam-7054	396	9	-	-	PUNCT
ejpam-7054	396	10	ae	ae	PROPN
ejpam-7054	396	11	,	,	PUNCT
ejpam-7054	396	12	and	and	CCONJ
ejpam-7054	396	13	c.	c.	PROPN
ejpam-7054	396	14	boonpok	boonpok	PROPN
ejpam-7054	396	15	.	.	PUNCT
ejpam-7054	397	1	almost	almost	ADV
ejpam-7054	397	2	weak	weak	ADJ
ejpam-7054	397	3	continuity	continuity	NOUN
ejpam-7054	397	4	for	for	ADP
ejpam-7054	397	5	multifunctions	multifunction	NOUN
ejpam-7054	397	6	defined	define	VERB
ejpam-7054	397	7	between	between	ADP
ejpam-7054	397	8	an	an	DET
ejpam-7054	397	9	ideal	ideal	ADJ
ejpam-7054	397	10	topological	topological	ADJ
ejpam-7054	397	11	space	space	NOUN
ejpam-7054	397	12	and	and	CCONJ
ejpam-7054	397	13	a	a	DET
ejpam-7054	397	14	bitopological	bitopological	ADJ
ejpam-7054	397	15	space	space	NOUN
ejpam-7054	397	16	.	.	PUNCT
ejpam-7054	398	1	european	european	ADJ
ejpam-7054	398	2	journal	journal	PROPN
ejpam-7054	398	3	of	of	ADP
ejpam-7054	398	4	pure	pure	ADJ
ejpam-7054	398	5	and	and	CCONJ
ejpam-7054	398	6	applied	applied	ADJ
ejpam-7054	398	7	mathematics	mathematic	NOUN
ejpam-7054	398	8	,	,	PUNCT
ejpam-7054	398	9	18(3):6570	18(3):6570	NUM
ejpam-7054	398	10	,	,	PUNCT
ejpam-7054	398	11	2025	2025	NUM
ejpam-7054	398	12	.	.	PUNCT
ejpam-7054	399	1	[	[	X
ejpam-7054	399	2	20	20	NUM
ejpam-7054	399	3	]	]	PUNCT
ejpam-7054	399	4	c.	c.	PROPN
ejpam-7054	399	5	boonpok	boonpok	PROPN
ejpam-7054	399	6	,	,	PUNCT
ejpam-7054	399	7	c.	c.	PROPN
ejpam-7054	399	8	viriyapong	viriyapong	PROPN
ejpam-7054	399	9	,	,	PUNCT
ejpam-7054	399	10	and	and	CCONJ
ejpam-7054	399	11	m.	m.	NOUN
ejpam-7054	399	12	thongmoon	thongmoon	NOUN
ejpam-7054	399	13	.	.	PUNCT
ejpam-7054	400	1	on	on	ADP
ejpam-7054	400	2	upper	upper	ADJ
ejpam-7054	400	3	and	and	CCONJ
ejpam-7054	400	4	lower	low	ADJ
ejpam-7054	400	5	(	(	PUNCT
ejpam-7054	400	6	τ1	τ1	NOUN
ejpam-7054	400	7	,	,	PUNCT
ejpam-7054	400	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-7054	400	9	multifunctions	multifunction	NOUN
ejpam-7054	400	10	.	.	PUNCT
ejpam-7054	401	1	journal	journal	PROPN
ejpam-7054	401	2	of	of	ADP
ejpam-7054	401	3	mathematics	mathematics	PROPN
ejpam-7054	401	4	and	and	CCONJ
ejpam-7054	401	5	computer	computer	NOUN
ejpam-7054	401	6	science	science	NOUN
ejpam-7054	401	7	,	,	PUNCT
ejpam-7054	401	8	18:282–293	18:282–293	NUM
ejpam-7054	401	9	,	,	PUNCT
ejpam-7054	401	10	2018	2018	NUM
ejpam-7054	401	11	.	.	PUNCT
ejpam-7054	402	1	[	[	X
ejpam-7054	402	2	21	21	NUM
ejpam-7054	402	3	]	]	X
ejpam-7054	402	4	c.	c.	PROPN
ejpam-7054	402	5	viriyapong	viriyapong	PROPN
ejpam-7054	402	6	and	and	CCONJ
ejpam-7054	402	7	c.	c.	PROPN
ejpam-7054	402	8	boonpok	boonpok	PROPN
ejpam-7054	402	9	.	.	PUNCT
ejpam-7054	403	1	(	(	PUNCT
ejpam-7054	403	2	τ1	τ1	NOUN
ejpam-7054	403	3	,	,	PUNCT
ejpam-7054	403	4	τ2)α	τ2)α	NOUN
ejpam-7054	403	5	-	-	PUNCT
ejpam-7054	403	6	continuity	continuity	NOUN
ejpam-7054	403	7	for	for	ADP
ejpam-7054	403	8	multifunctions	multifunction	NOUN
ejpam-7054	403	9	.	.	PUNCT
ejpam-7054	404	1	journal	journal	PROPN
ejpam-7054	404	2	of	of	ADP
ejpam-7054	404	3	mathematics	mathematic	NOUN
ejpam-7054	404	4	,	,	PUNCT
ejpam-7054	404	5	2020:6285763	2020:6285763	NUM
ejpam-7054	404	6	,	,	PUNCT
ejpam-7054	404	7	2020	2020	NUM
ejpam-7054	404	8	.	.	PUNCT
ejpam-7054	405	1	[	[	X
ejpam-7054	405	2	22	22	NUM
ejpam-7054	405	3	]	]	PUNCT
ejpam-7054	405	4	c.	c.	PROPN
ejpam-7054	405	5	boonpok	boonpok	PROPN
ejpam-7054	405	6	.	.	PUNCT
ejpam-7054	406	1	(	(	PUNCT
ejpam-7054	406	2	τ1	τ1	NOUN
ejpam-7054	406	3	,	,	PUNCT
ejpam-7054	406	4	τ2)δ	τ2)δ	ADJ
ejpam-7054	406	5	-	-	PUNCT
ejpam-7054	406	6	semicontinuous	semicontinuous	ADJ
ejpam-7054	406	7	multifunctions	multifunction	NOUN
ejpam-7054	406	8	.	.	PUNCT
ejpam-7054	407	1	heliyon	heliyon	NOUN
ejpam-7054	407	2	,	,	PUNCT
ejpam-7054	407	3	6	6	NUM
ejpam-7054	407	4	:	:	SYM
ejpam-7054	407	5	e05367	e05367	PROPN
ejpam-7054	407	6	,	,	PUNCT
ejpam-7054	407	7	2020	2020	NUM
ejpam-7054	407	8	.	.	PUNCT
ejpam-7054	408	1	[	[	X
ejpam-7054	408	2	23	23	NUM
ejpam-7054	408	3	]	]	PUNCT
ejpam-7054	408	4	á.	á.	PROPN
ejpam-7054	408	5	császár	császár	PROPN
ejpam-7054	408	6	.	.	PUNCT
ejpam-7054	409	1	δ	δ	PROPN
ejpam-7054	409	2	-	-	PUNCT
ejpam-7054	409	3	and	and	CCONJ
ejpam-7054	409	4	θ	θ	NOUN
ejpam-7054	409	5	-	-	PUNCT
ejpam-7054	409	6	modifications	modification	NOUN
ejpam-7054	409	7	of	of	ADP
ejpam-7054	409	8	generalized	generalized	ADJ
ejpam-7054	409	9	topologies	topology	NOUN
ejpam-7054	409	10	.	.	PUNCT
ejpam-7054	410	1	acta	acta	PROPN
ejpam-7054	410	2	mathematica	mathematica	PROPN
ejpam-7054	410	3	hungarica	hungarica	PROPN
ejpam-7054	410	4	,	,	PUNCT
ejpam-7054	410	5	120:274–279	120:274–279	NUM
ejpam-7054	410	6	,	,	PUNCT
ejpam-7054	410	7	2008	2008	NUM
ejpam-7054	410	8	.	.	PUNCT
ejpam-7054	411	1	[	[	X
ejpam-7054	411	2	24	24	NUM
ejpam-7054	411	3	]	]	PUNCT
ejpam-7054	411	4	á.	á.	PROPN
ejpam-7054	411	5	császár	császár	PROPN
ejpam-7054	411	6	.	.	PUNCT
ejpam-7054	412	1	generalized	generalize	VERB
ejpam-7054	412	2	open	open	ADJ
ejpam-7054	412	3	sets	set	NOUN
ejpam-7054	412	4	in	in	ADP
ejpam-7054	412	5	generalized	generalized	ADJ
ejpam-7054	412	6	topologies	topology	NOUN
ejpam-7054	412	7	.	.	PUNCT
ejpam-7054	413	1	acta	acta	PROPN
ejpam-7054	413	2	mathematica	mathematica	PROPN
ejpam-7054	413	3	hungarica	hungarica	PROPN
ejpam-7054	413	4	,	,	PUNCT
ejpam-7054	413	5	106(1	106(1	NUM
ejpam-7054	413	6	-	-	SYM
ejpam-7054	413	7	2):53–66	2):53–66	NUM
ejpam-7054	413	8	,	,	PUNCT
ejpam-7054	413	9	2005	2005	NUM
ejpam-7054	413	10	.	.	PUNCT
ejpam-7054	414	1	[	[	X
ejpam-7054	414	2	25	25	NUM
ejpam-7054	414	3	]	]	PUNCT
ejpam-7054	414	4	m.	m.	NOUN
ejpam-7054	414	5	chiangpradit	chiangpradit	NOUN
ejpam-7054	414	6	,	,	PUNCT
ejpam-7054	414	7	s.	s.	PROPN
ejpam-7054	414	8	sompong	sompong	PROPN
ejpam-7054	414	9	,	,	PUNCT
ejpam-7054	414	10	and	and	CCONJ
ejpam-7054	414	11	c.	c.	PROPN
ejpam-7054	414	12	boonpok	boonpok	PROPN
ejpam-7054	414	13	.	.	PUNCT
ejpam-7054	415	1	on	on	ADP
ejpam-7054	415	2	characterizations	characterization	NOUN
ejpam-7054	415	3	of	of	ADP
ejpam-7054	415	4	(	(	PUNCT
ejpam-7054	415	5	τ1	τ1	NOUN
ejpam-7054	415	6	,	,	PUNCT
ejpam-7054	415	7	τ2)regular	τ2)regular	ADJ
ejpam-7054	415	8	spaces	space	NOUN
ejpam-7054	415	9	.	.	PUNCT
ejpam-7054	416	1	international	international	ADJ
ejpam-7054	416	2	journal	journal	PROPN
ejpam-7054	416	3	of	of	ADP
ejpam-7054	416	4	mathematics	mathematic	NOUN
ejpam-7054	416	5	and	and	CCONJ
ejpam-7054	416	6	computer	computer	NOUN
ejpam-7054	416	7	science	science	NOUN
ejpam-7054	416	8	,	,	PUNCT
ejpam-7054	416	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-7054	416	10	,	,	PUNCT
ejpam-7054	416	11	2024	2024	NUM
ejpam-7054	416	12	.	.	PUNCT
ejpam-7054	417	1	[	[	X
ejpam-7054	417	2	26	26	NUM
ejpam-7054	417	3	]	]	X
ejpam-7054	417	4	c.	c.	PROPN
ejpam-7054	417	5	klanarong	klanarong	PROPN
ejpam-7054	417	6	,	,	PUNCT
ejpam-7054	417	7	s.	s.	PROPN
ejpam-7054	417	8	sompong	sompong	PROPN
ejpam-7054	417	9	,	,	PUNCT
ejpam-7054	417	10	and	and	CCONJ
ejpam-7054	417	11	c.	c.	PROPN
ejpam-7054	417	12	boonpok	boonpok	PROPN
ejpam-7054	417	13	.	.	PUNCT
ejpam-7054	418	1	(	(	PUNCT
ejpam-7054	418	2	τ1	τ1	NOUN
ejpam-7054	418	3	,	,	PUNCT
ejpam-7054	418	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-7054	418	5	and	and	CCONJ
ejpam-7054	418	6	(	(	PUNCT
ejpam-7054	418	7	τ1	τ1	NOUN
ejpam-7054	418	8	,	,	PUNCT
ejpam-7054	418	9	τ2)θ	τ2)θ	ADJ
ejpam-7054	418	10	-	-	PUNCT
ejpam-7054	418	11	closed	close	VERB
ejpam-7054	418	12	sets	set	NOUN
ejpam-7054	418	13	.	.	PUNCT
ejpam-7054	419	1	international	international	ADJ
ejpam-7054	419	2	journal	journal	NOUN
ejpam-7054	419	3	of	of	ADP
ejpam-7054	419	4	mathematics	mathematic	NOUN
ejpam-7054	419	5	and	and	CCONJ
ejpam-7054	419	6	computer	computer	NOUN
ejpam-7054	419	7	science	science	NOUN
ejpam-7054	419	8	,	,	PUNCT
ejpam-7054	419	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-7054	419	10	,	,	PUNCT
ejpam-7054	419	11	2024	2024	NUM
ejpam-7054	419	12	.	.	PUNCT