id	sid	tid	token	lemma	pos
ejpam-6566	1	1	european	european	PROPN
ejpam-6566	1	2	journal	journal	PROPN
ejpam-6566	1	3	of	of	ADP
ejpam-6566	1	4	pure	pure	ADJ
ejpam-6566	1	5	and	and	CCONJ
ejpam-6566	1	6	applied	applied	ADJ
ejpam-6566	1	7	mathematics	mathematic	NOUN
ejpam-6566	1	8	2025	2025	NUM
ejpam-6566	1	9	,	,	PUNCT
ejpam-6566	1	10	vol	vol	NOUN
ejpam-6566	1	11	.	.	PROPN
ejpam-6566	1	12	18	18	NUM
ejpam-6566	1	13	,	,	PUNCT
ejpam-6566	1	14	issue	issue	NOUN
ejpam-6566	1	15	3	3	NUM
ejpam-6566	1	16	,	,	PUNCT
ejpam-6566	1	17	article	article	NOUN
ejpam-6566	1	18	number	number	NOUN
ejpam-6566	1	19	6566	6566	NUM
ejpam-6566	1	20	issn	issn	PROPN
ejpam-6566	1	21	1307	1307	NUM
ejpam-6566	1	22	-	-	SYM
ejpam-6566	1	23	5543	5543	NUM
ejpam-6566	1	24	–	–	PUNCT
ejpam-6566	1	25	ejpam.com	ejpam.com	X
ejpam-6566	1	26	published	publish	VERB
ejpam-6566	1	27	by	by	ADP
ejpam-6566	1	28	new	new	PROPN
ejpam-6566	1	29	york	york	PROPN
ejpam-6566	1	30	business	business	NOUN
ejpam-6566	1	31	global	global	ADJ
ejpam-6566	1	32	almost	almost	ADV
ejpam-6566	1	33	continuity	continuity	NOUN
ejpam-6566	1	34	for	for	ADP
ejpam-6566	1	35	multifunctions	multifunction	NOUN
ejpam-6566	1	36	defined	define	VERB
ejpam-6566	1	37	from	from	ADP
ejpam-6566	1	38	an	an	DET
ejpam-6566	1	39	ideal	ideal	ADJ
ejpam-6566	1	40	topological	topological	ADJ
ejpam-6566	1	41	space	space	NOUN
ejpam-6566	1	42	into	into	ADP
ejpam-6566	1	43	a	a	DET
ejpam-6566	1	44	bitopological	bitopological	ADJ
ejpam-6566	1	45	space	space	NOUN
ejpam-6566	1	46	chokchai	chokchai	ADJ
ejpam-6566	1	47	viriyapong1	viriyapong1	PROPN
ejpam-6566	1	48	,	,	PUNCT
ejpam-6566	1	49	areeyuth	areeyuth	NOUN
ejpam-6566	1	50	sama	sama	NOUN
ejpam-6566	1	51	-	-	PUNCT
ejpam-6566	1	52	ae2	ae2	PROPN
ejpam-6566	1	53	,	,	PUNCT
ejpam-6566	1	54	chawalit	chawalit	VERB
ejpam-6566	1	55	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6566	1	56	1	1	NUM
ejpam-6566	1	57	mathematics	mathematic	NOUN
ejpam-6566	1	58	and	and	CCONJ
ejpam-6566	1	59	applied	apply	VERB
ejpam-6566	1	60	mathematics	mathematics	PROPN
ejpam-6566	1	61	research	research	NOUN
ejpam-6566	1	62	unit	unit	NOUN
ejpam-6566	1	63	,	,	PUNCT
ejpam-6566	1	64	department	department	NOUN
ejpam-6566	1	65	of	of	ADP
ejpam-6566	1	66	mathematics	mathematic	NOUN
ejpam-6566	1	67	,	,	PUNCT
ejpam-6566	1	68	faculty	faculty	NOUN
ejpam-6566	1	69	of	of	ADP
ejpam-6566	1	70	science	science	NOUN
ejpam-6566	1	71	,	,	PUNCT
ejpam-6566	1	72	mahasarakham	mahasarakham	PROPN
ejpam-6566	1	73	university	university	PROPN
ejpam-6566	1	74	,	,	PUNCT
ejpam-6566	1	75	maha	maha	PROPN
ejpam-6566	1	76	sarakham	sarakham	PROPN
ejpam-6566	1	77	,	,	PUNCT
ejpam-6566	1	78	44150	44150	NUM
ejpam-6566	1	79	,	,	PUNCT
ejpam-6566	1	80	thailand	thailand	PROPN
ejpam-6566	1	81	2	2	NUM
ejpam-6566	1	82	department	department	NOUN
ejpam-6566	1	83	of	of	ADP
ejpam-6566	1	84	mathematics	mathematic	NOUN
ejpam-6566	1	85	and	and	CCONJ
ejpam-6566	1	86	computer	computer	NOUN
ejpam-6566	1	87	science	science	NOUN
ejpam-6566	1	88	,	,	PUNCT
ejpam-6566	1	89	faculty	faculty	NOUN
ejpam-6566	1	90	of	of	ADP
ejpam-6566	1	91	science	science	NOUN
ejpam-6566	1	92	and	and	CCONJ
ejpam-6566	1	93	technology	technology	NOUN
ejpam-6566	1	94	,	,	PUNCT
ejpam-6566	1	95	prince	prince	NOUN
ejpam-6566	1	96	of	of	ADP
ejpam-6566	1	97	songkla	songkla	PROPN
ejpam-6566	1	98	university	university	PROPN
ejpam-6566	1	99	,	,	PUNCT
ejpam-6566	1	100	pattani	pattani	NOUN
ejpam-6566	1	101	campus	campus	NOUN
ejpam-6566	1	102	,	,	PUNCT
ejpam-6566	1	103	pattani	pattani	NOUN
ejpam-6566	1	104	,	,	PUNCT
ejpam-6566	1	105	94000	94000	NUM
ejpam-6566	1	106	,	,	PUNCT
ejpam-6566	1	107	thailand	thailand	PROPN
ejpam-6566	1	108	abstract	abstract	PROPN
ejpam-6566	1	109	.	.	PUNCT
ejpam-6566	2	1	this	this	DET
ejpam-6566	2	2	paper	paper	NOUN
ejpam-6566	2	3	presents	present	VERB
ejpam-6566	2	4	new	new	ADJ
ejpam-6566	2	5	concepts	concept	NOUN
ejpam-6566	2	6	of	of	ADP
ejpam-6566	2	7	continuous	continuous	ADJ
ejpam-6566	2	8	multifunctions	multifunction	NOUN
ejpam-6566	2	9	defined	define	VERB
ejpam-6566	2	10	between	between	ADP
ejpam-6566	2	11	an	an	DET
ejpam-6566	2	12	ideal	ideal	ADJ
ejpam-6566	2	13	topological	topological	ADJ
ejpam-6566	2	14	space	space	NOUN
ejpam-6566	2	15	and	and	CCONJ
ejpam-6566	2	16	a	a	DET
ejpam-6566	2	17	bitopological	bitopological	ADJ
ejpam-6566	2	18	space	space	NOUN
ejpam-6566	2	19	,	,	PUNCT
ejpam-6566	2	20	namely	namely	ADV
ejpam-6566	2	21	upper	upper	ADJ
ejpam-6566	2	22	almost	almost	ADV
ejpam-6566	2	23	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	2	24	,	,	PUNCT
ejpam-6566	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	2	26	multifunctions	multifunction	NOUN
ejpam-6566	2	27	and	and	CCONJ
ejpam-6566	2	28	lower	low	ADJ
ejpam-6566	2	29	almost	almost	ADV
ejpam-6566	2	30	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	2	31	,	,	PUNCT
ejpam-6566	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	2	33	multifunctions	multifunction	NOUN
ejpam-6566	2	34	.	.	PUNCT
ejpam-6566	3	1	moreover	moreover	ADV
ejpam-6566	3	2	,	,	PUNCT
ejpam-6566	3	3	several	several	ADJ
ejpam-6566	3	4	characterizations	characterization	NOUN
ejpam-6566	3	5	and	and	CCONJ
ejpam-6566	3	6	some	some	DET
ejpam-6566	3	7	properties	property	NOUN
ejpam-6566	3	8	concerning	concern	VERB
ejpam-6566	3	9	upper	upper	ADJ
ejpam-6566	3	10	almost	almost	ADV
ejpam-6566	3	11	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	3	12	,	,	PUNCT
ejpam-6566	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	3	14	multifunctions	multifunction	NOUN
ejpam-6566	3	15	and	and	CCONJ
ejpam-6566	3	16	lower	low	ADJ
ejpam-6566	3	17	almost	almost	ADV
ejpam-6566	3	18	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	3	19	,	,	PUNCT
ejpam-6566	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	3	21	multifunctions	multifunction	NOUN
ejpam-6566	3	22	are	be	AUX
ejpam-6566	3	23	considered	consider	VERB
ejpam-6566	3	24	.	.	PUNCT
ejpam-6566	4	1	furthermore	furthermore	ADV
ejpam-6566	4	2	,	,	PUNCT
ejpam-6566	4	3	the	the	DET
ejpam-6566	4	4	relationships	relationship	NOUN
ejpam-6566	4	5	between	between	ADP
ejpam-6566	4	6	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6566	4	7	,	,	PUNCT
ejpam-6566	4	8	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6566	4	9	and	and	CCONJ
ejpam-6566	4	10	almost	almost	ADV
ejpam-6566	4	11	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	4	12	,	,	PUNCT
ejpam-6566	4	13	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6566	4	14	are	be	AUX
ejpam-6566	4	15	discussed	discuss	VERB
ejpam-6566	4	16	.	.	PUNCT
ejpam-6566	5	1	2020	2020	NUM
ejpam-6566	5	2	mathematics	mathematic	NOUN
ejpam-6566	5	3	subject	subject	NOUN
ejpam-6566	5	4	classifications	classification	NOUN
ejpam-6566	5	5	:	:	PUNCT
ejpam-6566	5	6	54c08	54c08	NUM
ejpam-6566	5	7	,	,	PUNCT
ejpam-6566	5	8	54c60	54c60	NUM
ejpam-6566	5	9	key	key	ADJ
ejpam-6566	5	10	words	word	NOUN
ejpam-6566	5	11	and	and	CCONJ
ejpam-6566	5	12	phrases	phrase	NOUN
ejpam-6566	5	13	:	:	PUNCT
ejpam-6566	5	14	upper	upper	ADJ
ejpam-6566	5	15	almost	almost	ADV
ejpam-6566	5	16	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	5	17	,	,	PUNCT
ejpam-6566	5	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	5	19	multifunction	multifunction	NOUN
ejpam-6566	5	20	,	,	PUNCT
ejpam-6566	5	21	lower	low	ADJ
ejpam-6566	5	22	almost	almost	ADV
ejpam-6566	5	23	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	5	24	,	,	PUNCT
ejpam-6566	5	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	5	26	multifunction	multifunction	NOUN
ejpam-6566	5	27	1	1	NUM
ejpam-6566	5	28	.	.	PUNCT
ejpam-6566	6	1	introduction	introduction	NOUN
ejpam-6566	6	2	it	it	PRON
ejpam-6566	6	3	is	be	AUX
ejpam-6566	6	4	well	well	ADV
ejpam-6566	6	5	-	-	PUNCT
ejpam-6566	6	6	known	know	VERB
ejpam-6566	6	7	that	that	SCONJ
ejpam-6566	6	8	the	the	DET
ejpam-6566	6	9	branch	branch	NOUN
ejpam-6566	6	10	of	of	ADP
ejpam-6566	6	11	mathematics	mathematic	NOUN
ejpam-6566	6	12	called	call	VERB
ejpam-6566	6	13	topology	topology	NOUN
ejpam-6566	6	14	is	be	AUX
ejpam-6566	6	15	related	relate	VERB
ejpam-6566	6	16	to	to	ADP
ejpam-6566	6	17	all	all	DET
ejpam-6566	6	18	questions	question	NOUN
ejpam-6566	6	19	directly	directly	ADV
ejpam-6566	6	20	or	or	CCONJ
ejpam-6566	6	21	indirectly	indirectly	ADV
ejpam-6566	6	22	concerned	concerned	ADJ
ejpam-6566	6	23	with	with	ADP
ejpam-6566	6	24	continuity	continuity	NOUN
ejpam-6566	6	25	.	.	PUNCT
ejpam-6566	7	1	singal	singal	NOUN
ejpam-6566	7	2	and	and	CCONJ
ejpam-6566	7	3	singal	singal	ADJ
ejpam-6566	7	4	[	[	X
ejpam-6566	7	5	1	1	NUM
ejpam-6566	7	6	]	]	PUNCT
ejpam-6566	7	7	introduced	introduce	VERB
ejpam-6566	7	8	the	the	DET
ejpam-6566	7	9	concept	concept	NOUN
ejpam-6566	7	10	of	of	ADP
ejpam-6566	7	11	almost	almost	ADV
ejpam-6566	7	12	continuous	continuous	ADJ
ejpam-6566	7	13	functions	function	NOUN
ejpam-6566	7	14	as	as	ADP
ejpam-6566	7	15	a	a	DET
ejpam-6566	7	16	generalization	generalization	NOUN
ejpam-6566	7	17	of	of	ADP
ejpam-6566	7	18	continuity	continuity	NOUN
ejpam-6566	7	19	.	.	PUNCT
ejpam-6566	8	1	munshi	munshi	PROPN
ejpam-6566	8	2	and	and	CCONJ
ejpam-6566	8	3	bassan	bassan	NOUN
ejpam-6566	8	4	[	[	X
ejpam-6566	8	5	2	2	NUM
ejpam-6566	8	6	]	]	PUNCT
ejpam-6566	8	7	studied	study	VERB
ejpam-6566	8	8	the	the	DET
ejpam-6566	8	9	notion	notion	NOUN
ejpam-6566	8	10	of	of	ADP
ejpam-6566	8	11	almost	almost	ADV
ejpam-6566	8	12	semi	semi	ADJ
ejpam-6566	8	13	-	-	ADJ
ejpam-6566	8	14	continuous	continuous	ADJ
ejpam-6566	8	15	functions	function	NOUN
ejpam-6566	8	16	.	.	PUNCT
ejpam-6566	9	1	noiri	noiri	ADV
ejpam-6566	10	1	[	[	X
ejpam-6566	10	2	3	3	X
ejpam-6566	10	3	]	]	PUNCT
ejpam-6566	10	4	introduced	introduce	VERB
ejpam-6566	10	5	and	and	CCONJ
ejpam-6566	10	6	investigated	investigate	VERB
ejpam-6566	10	7	the	the	DET
ejpam-6566	10	8	concept	concept	NOUN
ejpam-6566	10	9	of	of	ADP
ejpam-6566	10	10	almost	almost	ADV
ejpam-6566	10	11	α	α	NUM
ejpam-6566	10	12	-	-	ADJ
ejpam-6566	10	13	continuous	continuous	ADJ
ejpam-6566	10	14	functions	function	NOUN
ejpam-6566	10	15	.	.	PUNCT
ejpam-6566	11	1	nasef	nasef	NOUN
ejpam-6566	11	2	and	and	CCONJ
ejpam-6566	11	3	noiri	noiri	ADV
ejpam-6566	12	1	[	[	X
ejpam-6566	12	2	4	4	X
ejpam-6566	12	3	]	]	PUNCT
ejpam-6566	12	4	introduced	introduce	VERB
ejpam-6566	12	5	two	two	NUM
ejpam-6566	12	6	classes	class	NOUN
ejpam-6566	12	7	of	of	ADP
ejpam-6566	12	8	functions	function	NOUN
ejpam-6566	12	9	,	,	PUNCT
ejpam-6566	12	10	namely	namely	ADV
ejpam-6566	12	11	almost	almost	ADV
ejpam-6566	12	12	precontinuous	precontinuous	ADJ
ejpam-6566	12	13	functions	function	NOUN
ejpam-6566	12	14	and	and	CCONJ
ejpam-6566	12	15	almost	almost	ADV
ejpam-6566	12	16	β	β	ADJ
ejpam-6566	12	17	-	-	ADJ
ejpam-6566	12	18	continuous	continuous	ADJ
ejpam-6566	12	19	functions	function	NOUN
ejpam-6566	12	20	.	.	PUNCT
ejpam-6566	13	1	the	the	DET
ejpam-6566	13	2	class	class	NOUN
ejpam-6566	13	3	of	of	ADP
ejpam-6566	13	4	almost	almost	ADV
ejpam-6566	13	5	precontinuity	precontinuity	NOUN
ejpam-6566	13	6	is	be	AUX
ejpam-6566	13	7	a	a	DET
ejpam-6566	13	8	generalization	generalization	NOUN
ejpam-6566	13	9	of	of	ADP
ejpam-6566	13	10	almost	almost	ADV
ejpam-6566	13	11	α	α	NOUN
ejpam-6566	13	12	-	-	NOUN
ejpam-6566	13	13	continuity	continuity	NOUN
ejpam-6566	13	14	.	.	PUNCT
ejpam-6566	14	1	the	the	DET
ejpam-6566	14	2	class	class	NOUN
ejpam-6566	14	3	of	of	ADP
ejpam-6566	14	4	almost	almost	ADV
ejpam-6566	14	5	β	β	NOUN
ejpam-6566	14	6	-	-	NOUN
ejpam-6566	14	7	continuity	continuity	NOUN
ejpam-6566	14	8	is	be	AUX
ejpam-6566	14	9	a	a	DET
ejpam-6566	14	10	generalization	generalization	NOUN
ejpam-6566	14	11	of	of	ADP
ejpam-6566	14	12	almost	almost	ADV
ejpam-6566	14	13	semi	semi	NOUN
ejpam-6566	14	14	-	-	NOUN
ejpam-6566	14	15	continuity	continuity	NOUN
ejpam-6566	14	16	.	.	PUNCT
ejpam-6566	15	1	popa	popa	NOUN
ejpam-6566	16	1	[	[	X
ejpam-6566	16	2	5	5	NUM
ejpam-6566	16	3	]	]	PUNCT
ejpam-6566	16	4	introduced	introduce	VERB
ejpam-6566	16	5	and	and	CCONJ
ejpam-6566	16	6	studied	study	VERB
ejpam-6566	16	7	the	the	DET
ejpam-6566	16	8	concepts	concept	NOUN
ejpam-6566	16	9	of	of	ADP
ejpam-6566	16	10	upper	upper	ADJ
ejpam-6566	16	11	almost	almost	ADV
ejpam-6566	16	12	continuous	continuous	ADJ
ejpam-6566	16	13	multifunctions	multifunction	NOUN
ejpam-6566	16	14	and	and	CCONJ
ejpam-6566	16	15	lower	low	ADJ
ejpam-6566	16	16	almost	almost	ADV
ejpam-6566	16	17	continuous	continuous	ADJ
ejpam-6566	16	18	multifunctions	multifunction	NOUN
ejpam-6566	16	19	.	.	PUNCT
ejpam-6566	17	1	furthermore	furthermore	ADV
ejpam-6566	17	2	,	,	PUNCT
ejpam-6566	17	3	popa	popa	NOUN
ejpam-6566	17	4	and	and	CCONJ
ejpam-6566	17	5	noiri	noiri	ADV
ejpam-6566	18	1	[	[	X
ejpam-6566	18	2	6	6	NUM
ejpam-6566	18	3	]	]	PUNCT
ejpam-6566	18	4	introduced	introduce	VERB
ejpam-6566	18	5	and	and	CCONJ
ejpam-6566	18	6	investigated	investigate	VERB
ejpam-6566	18	7	the	the	DET
ejpam-6566	18	8	notions	notion	NOUN
ejpam-6566	18	9	of	of	ADP
ejpam-6566	18	10	upper	upper	ADJ
ejpam-6566	18	11	almost	almost	ADV
ejpam-6566	18	12	quasi	quasi	ADJ
ejpam-6566	18	13	-	-	ADJ
ejpam-6566	18	14	continuous	continuous	ADJ
ejpam-6566	18	15	multifunctions	multifunction	NOUN
ejpam-6566	18	16	and	and	CCONJ
ejpam-6566	18	17	lower	low	ADJ
ejpam-6566	18	18	almost	almost	ADV
ejpam-6566	18	19	quasi	quasi	ADJ
ejpam-6566	18	20	-	-	ADJ
ejpam-6566	18	21	continuous	continuous	ADJ
ejpam-6566	18	22	multifunctions	multifunction	NOUN
ejpam-6566	18	23	.	.	PUNCT
ejpam-6566	19	1	popa	popa	NOUN
ejpam-6566	19	2	et	et	PROPN
ejpam-6566	19	3	al	al	PROPN
ejpam-6566	19	4	.	.	PUNCT
ejpam-6566	20	1	[	[	X
ejpam-6566	20	2	7	7	X
ejpam-6566	20	3	]	]	PUNCT
ejpam-6566	20	4	introduced	introduce	VERB
ejpam-6566	20	5	the	the	DET
ejpam-6566	20	6	concepts	concept	NOUN
ejpam-6566	20	7	of	of	ADP
ejpam-6566	20	8	upper	upper	ADJ
ejpam-6566	20	9	almost	almost	ADV
ejpam-6566	20	10	∗corresponding	∗corresponde	VERB
ejpam-6566	20	11	author	author	NOUN
ejpam-6566	20	12	.	.	PUNCT
ejpam-6566	21	1	doi	doi	NOUN
ejpam-6566	21	2	:	:	PUNCT
ejpam-6566	21	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6566	https://doi.org/10.29020/nybg.ejpam.v18i3.6566	DET
ejpam-6566	21	4	email	email	NOUN
ejpam-6566	21	5	addresses	address	VERB
ejpam-6566	21	6	:	:	PUNCT
ejpam-6566	21	7	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	PROPN
ejpam-6566	21	8	(	(	PUNCT
ejpam-6566	21	9	c.	c.	PROPN
ejpam-6566	21	10	viriyapong	viriyapong	PROPN
ejpam-6566	21	11	)	)	PUNCT
ejpam-6566	21	12	,	,	PUNCT
ejpam-6566	21	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6566	21	14	(	(	PUNCT
ejpam-6566	21	15	a.	a.	PROPN
ejpam-6566	21	16	sama	sama	PROPN
ejpam-6566	21	17	-	-	PUNCT
ejpam-6566	21	18	ae	ae	PROPN
ejpam-6566	21	19	)	)	PUNCT
ejpam-6566	21	20	,	,	PUNCT
ejpam-6566	21	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6566	21	22	(	(	PUNCT
ejpam-6566	21	23	c.	c.	PROPN
ejpam-6566	21	24	boonpok	boonpok	PROPN
ejpam-6566	21	25	)	)	PUNCT
ejpam-6566	21	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6566	22	1	1	1	NUM
ejpam-6566	22	2	copyright	copyright	NOUN
ejpam-6566	22	3	:	:	PUNCT
ejpam-6566	22	4	©	©	PROPN
ejpam-6566	22	5	2025	2025	NUM
ejpam-6566	22	6	the	the	DET
ejpam-6566	22	7	author(s	author(s	NOUN
ejpam-6566	22	8	)	)	PUNCT
ejpam-6566	22	9	.	.	PUNCT
ejpam-6566	23	1	(	(	PUNCT
ejpam-6566	23	2	cc	cc	NOUN
ejpam-6566	23	3	by	by	ADP
ejpam-6566	23	4	-	-	PUNCT
ejpam-6566	23	5	nc	nc	PROPN
ejpam-6566	23	6	4.0	4.0	NUM
ejpam-6566	23	7	)	)	PUNCT
ejpam-6566	23	8	c.	c.	PROPN
ejpam-6566	23	9	viriyapong	viriyapong	PROPN
ejpam-6566	23	10	,	,	PUNCT
ejpam-6566	23	11	a.	a.	PROPN
ejpam-6566	23	12	sama	sama	PROPN
ejpam-6566	23	13	-	-	PUNCT
ejpam-6566	23	14	ae	ae	PROPN
ejpam-6566	23	15	,	,	PUNCT
ejpam-6566	23	16	c.	c.	PROPN
ejpam-6566	23	17	boonpok	boonpok	PROPN
ejpam-6566	23	18	/	/	SYM
ejpam-6566	23	19	eur	eur	PROPN
ejpam-6566	23	20	.	.	PUNCT
ejpam-6566	24	1	j.	j.	PROPN
ejpam-6566	24	2	pure	pure	PROPN
ejpam-6566	24	3	appl	appl	PROPN
ejpam-6566	24	4	.	.	PROPN
ejpam-6566	24	5	math	math	PROPN
ejpam-6566	24	6	,	,	PUNCT
ejpam-6566	24	7	18	18	NUM
ejpam-6566	24	8	(	(	PUNCT
ejpam-6566	24	9	3	3	NUM
ejpam-6566	24	10	)	)	PUNCT
ejpam-6566	24	11	(	(	PUNCT
ejpam-6566	24	12	2025	2025	NUM
ejpam-6566	24	13	)	)	PUNCT
ejpam-6566	24	14	,	,	PUNCT
ejpam-6566	24	15	6566	6566	NUM
ejpam-6566	24	16	2	2	NUM
ejpam-6566	24	17	of	of	ADP
ejpam-6566	24	18	14	14	NUM
ejpam-6566	24	19	precontinuous	precontinuous	ADJ
ejpam-6566	24	20	multifunctions	multifunction	NOUN
ejpam-6566	24	21	and	and	CCONJ
ejpam-6566	24	22	lower	low	ADJ
ejpam-6566	24	23	almost	almost	ADV
ejpam-6566	24	24	precontinuous	precontinuous	ADJ
ejpam-6566	24	25	multifunctions	multifunction	NOUN
ejpam-6566	24	26	.	.	PUNCT
ejpam-6566	25	1	noiri	noiri	PROPN
ejpam-6566	25	2	and	and	CCONJ
ejpam-6566	25	3	popa	popa	NOUN
ejpam-6566	25	4	[	[	X
ejpam-6566	25	5	8	8	NUM
ejpam-6566	25	6	]	]	PUNCT
ejpam-6566	25	7	introduced	introduce	VERB
ejpam-6566	25	8	the	the	DET
ejpam-6566	25	9	concepts	concept	NOUN
ejpam-6566	25	10	of	of	ADP
ejpam-6566	25	11	upper	upper	ADJ
ejpam-6566	25	12	almost	almost	ADV
ejpam-6566	25	13	β	β	ADJ
ejpam-6566	25	14	-	-	ADJ
ejpam-6566	25	15	continuous	continuous	ADJ
ejpam-6566	25	16	multifunctions	multifunction	NOUN
ejpam-6566	25	17	and	and	CCONJ
ejpam-6566	25	18	lower	low	ADJ
ejpam-6566	25	19	almost	almost	ADV
ejpam-6566	25	20	β	β	ADJ
ejpam-6566	25	21	-	-	ADJ
ejpam-6566	25	22	continuous	continuous	ADJ
ejpam-6566	25	23	multifunctions	multifunction	NOUN
ejpam-6566	25	24	.	.	PUNCT
ejpam-6566	26	1	moreover	moreover	ADV
ejpam-6566	26	2	,	,	PUNCT
ejpam-6566	26	3	some	some	DET
ejpam-6566	26	4	characterizations	characterization	NOUN
ejpam-6566	26	5	of	of	ADP
ejpam-6566	26	6	upper	upper	ADJ
ejpam-6566	26	7	almost	almost	ADV
ejpam-6566	26	8	β	β	ADJ
ejpam-6566	26	9	-	-	ADJ
ejpam-6566	26	10	continuous	continuous	ADJ
ejpam-6566	26	11	multifunctions	multifunction	NOUN
ejpam-6566	26	12	and	and	CCONJ
ejpam-6566	26	13	lower	low	ADJ
ejpam-6566	26	14	almost	almost	ADV
ejpam-6566	26	15	β	β	ADJ
ejpam-6566	26	16	-	-	ADJ
ejpam-6566	26	17	continuous	continuous	ADJ
ejpam-6566	26	18	multifunctions	multifunction	NOUN
ejpam-6566	26	19	were	be	AUX
ejpam-6566	26	20	presented	present	VERB
ejpam-6566	26	21	in	in	ADP
ejpam-6566	26	22	[	[	PUNCT
ejpam-6566	26	23	9	9	NUM
ejpam-6566	26	24	]	]	PUNCT
ejpam-6566	26	25	.	.	PUNCT
ejpam-6566	27	1	popa	popa	NOUN
ejpam-6566	27	2	and	and	CCONJ
ejpam-6566	27	3	noiri	noiri	ADV
ejpam-6566	28	1	[	[	X
ejpam-6566	28	2	10	10	NUM
ejpam-6566	28	3	]	]	PUNCT
ejpam-6566	28	4	introduced	introduce	VERB
ejpam-6566	28	5	and	and	CCONJ
ejpam-6566	28	6	investigated	investigate	VERB
ejpam-6566	28	7	the	the	DET
ejpam-6566	28	8	notions	notion	NOUN
ejpam-6566	28	9	of	of	ADP
ejpam-6566	28	10	upper	upper	ADJ
ejpam-6566	28	11	almost	almost	ADV
ejpam-6566	28	12	αcontinuous	αcontinuous	ADJ
ejpam-6566	28	13	multifunctions	multifunction	NOUN
ejpam-6566	28	14	and	and	CCONJ
ejpam-6566	28	15	lower	low	ADJ
ejpam-6566	28	16	almost	almost	ADV
ejpam-6566	28	17	α	α	ADJ
ejpam-6566	28	18	-	-	ADJ
ejpam-6566	28	19	continuous	continuous	ADJ
ejpam-6566	28	20	multifunctions	multifunction	NOUN
ejpam-6566	28	21	.	.	PUNCT
ejpam-6566	29	1	pue	pue	NOUN
ejpam-6566	29	2	-	-	PUNCT
ejpam-6566	29	3	on	on	NOUN
ejpam-6566	29	4	et	et	PROPN
ejpam-6566	29	5	al	al	PROPN
ejpam-6566	29	6	.	.	PUNCT
ejpam-6566	30	1	[	[	X
ejpam-6566	30	2	11	11	NUM
ejpam-6566	30	3	]	]	PUNCT
ejpam-6566	30	4	introduced	introduce	VERB
ejpam-6566	30	5	and	and	CCONJ
ejpam-6566	30	6	studied	study	VERB
ejpam-6566	30	7	the	the	DET
ejpam-6566	30	8	concepts	concept	NOUN
ejpam-6566	30	9	of	of	ADP
ejpam-6566	30	10	upper	upper	ADJ
ejpam-6566	30	11	(	(	PUNCT
ejpam-6566	30	12	τ1	τ1	NOUN
ejpam-6566	30	13	,	,	PUNCT
ejpam-6566	30	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6566	30	15	multifunctions	multifunction	NOUN
ejpam-6566	30	16	and	and	CCONJ
ejpam-6566	30	17	lower	low	ADJ
ejpam-6566	30	18	(	(	PUNCT
ejpam-6566	30	19	τ1	τ1	NOUN
ejpam-6566	30	20	,	,	PUNCT
ejpam-6566	30	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6566	30	22	multifunctions	multifunction	NOUN
ejpam-6566	30	23	.	.	PUNCT
ejpam-6566	31	1	klanarong	klanarong	NOUN
ejpam-6566	31	2	et	et	PROPN
ejpam-6566	31	3	al	al	PROPN
ejpam-6566	31	4	.	.	PUNCT
ejpam-6566	32	1	[	[	X
ejpam-6566	32	2	12	12	NUM
ejpam-6566	32	3	]	]	PUNCT
ejpam-6566	32	4	introduced	introduce	VERB
ejpam-6566	32	5	and	and	CCONJ
ejpam-6566	32	6	investigated	investigate	VERB
ejpam-6566	32	7	the	the	DET
ejpam-6566	32	8	notions	notion	NOUN
ejpam-6566	32	9	of	of	ADP
ejpam-6566	32	10	upper	upper	ADJ
ejpam-6566	32	11	almost	almost	ADV
ejpam-6566	32	12	(	(	PUNCT
ejpam-6566	32	13	τ1	τ1	NOUN
ejpam-6566	32	14	,	,	PUNCT
ejpam-6566	32	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6566	32	16	multifunctions	multifunction	NOUN
ejpam-6566	32	17	and	and	CCONJ
ejpam-6566	32	18	lower	low	ADJ
ejpam-6566	32	19	almost	almost	ADV
ejpam-6566	32	20	(	(	PUNCT
ejpam-6566	32	21	τ1	τ1	NOUN
ejpam-6566	32	22	,	,	PUNCT
ejpam-6566	32	23	τ2)continuous	τ2)continuous	ADJ
ejpam-6566	32	24	multifunctions	multifunction	NOUN
ejpam-6566	32	25	.	.	PUNCT
ejpam-6566	33	1	the	the	DET
ejpam-6566	33	2	notion	notion	NOUN
ejpam-6566	33	3	of	of	ADP
ejpam-6566	33	4	ideal	ideal	ADJ
ejpam-6566	33	5	topological	topological	ADJ
ejpam-6566	33	6	spaces	space	NOUN
ejpam-6566	33	7	was	be	AUX
ejpam-6566	33	8	introduced	introduce	VERB
ejpam-6566	33	9	and	and	CCONJ
ejpam-6566	33	10	studied	study	VERB
ejpam-6566	33	11	by	by	ADP
ejpam-6566	33	12	kuratowski	kuratowski	NOUN
ejpam-6566	33	13	[	[	X
ejpam-6566	33	14	13	13	NUM
ejpam-6566	33	15	]	]	PUNCT
ejpam-6566	33	16	and	and	CCONJ
ejpam-6566	33	17	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-6566	33	18	[	[	X
ejpam-6566	33	19	14	14	NUM
ejpam-6566	33	20	]	]	SYM
ejpam-6566	33	21	.	.	PUNCT
ejpam-6566	34	1	stronger	strong	ADJ
ejpam-6566	34	2	and	and	CCONJ
ejpam-6566	34	3	weaker	weak	ADJ
ejpam-6566	34	4	forms	form	NOUN
ejpam-6566	34	5	of	of	ADP
ejpam-6566	34	6	open	open	ADJ
ejpam-6566	34	7	sets	set	NOUN
ejpam-6566	34	8	in	in	ADP
ejpam-6566	34	9	ideal	ideal	ADJ
ejpam-6566	34	10	topological	topological	ADJ
ejpam-6566	34	11	spaces	space	NOUN
ejpam-6566	34	12	such	such	ADJ
ejpam-6566	34	13	as	as	ADP
ejpam-6566	34	14	semi	semi	ADJ
ejpam-6566	34	15	-	-	ADJ
ejpam-6566	34	16	i	i	ADJ
ejpam-6566	34	17	-open	-open	NOUN
ejpam-6566	34	18	sets	set	NOUN
ejpam-6566	34	19	,	,	PUNCT
ejpam-6566	34	20	pre	pre	ADJ
ejpam-6566	34	21	-	-	ADJ
ejpam-6566	34	22	i	i	ADJ
ejpam-6566	34	23	-open	-open	NOUN
ejpam-6566	34	24	sets	set	NOUN
ejpam-6566	34	25	,	,	PUNCT
ejpam-6566	34	26	α	α	X
ejpam-6566	34	27	-	-	PUNCT
ejpam-6566	34	28	i	i	PRON
ejpam-6566	34	29	-open	-open	NOUN
ejpam-6566	34	30	sets	set	NOUN
ejpam-6566	34	31	,	,	PUNCT
ejpam-6566	34	32	β	β	X
ejpam-6566	34	33	-	-	ADJ
ejpam-6566	34	34	i	i	PRON
ejpam-6566	34	35	-open	-open	NOUN
ejpam-6566	34	36	sets	set	NOUN
ejpam-6566	34	37	and	and	CCONJ
ejpam-6566	34	38	δ	δ	PROPN
ejpam-6566	34	39	-	-	PUNCT
ejpam-6566	34	40	i	i	PRON
ejpam-6566	34	41	-open	-open	NOUN
ejpam-6566	34	42	sets	set	NOUN
ejpam-6566	34	43	play	play	VERB
ejpam-6566	34	44	an	an	DET
ejpam-6566	34	45	important	important	ADJ
ejpam-6566	34	46	role	role	NOUN
ejpam-6566	34	47	in	in	ADP
ejpam-6566	34	48	the	the	DET
ejpam-6566	34	49	research	research	NOUN
ejpam-6566	34	50	of	of	ADP
ejpam-6566	34	51	generalizations	generalization	NOUN
ejpam-6566	34	52	of	of	ADP
ejpam-6566	34	53	continuity	continuity	NOUN
ejpam-6566	34	54	.	.	PUNCT
ejpam-6566	35	1	using	use	VERB
ejpam-6566	35	2	these	these	DET
ejpam-6566	35	3	notions	notion	NOUN
ejpam-6566	35	4	many	many	ADJ
ejpam-6566	35	5	authors	author	NOUN
ejpam-6566	35	6	introduced	introduce	VERB
ejpam-6566	35	7	and	and	CCONJ
ejpam-6566	35	8	studied	study	VERB
ejpam-6566	35	9	various	various	ADJ
ejpam-6566	35	10	types	type	NOUN
ejpam-6566	35	11	of	of	ADP
ejpam-6566	35	12	generalizations	generalization	NOUN
ejpam-6566	35	13	of	of	ADP
ejpam-6566	35	14	continuity	continuity	NOUN
ejpam-6566	35	15	for	for	ADP
ejpam-6566	35	16	functions	function	NOUN
ejpam-6566	35	17	and	and	CCONJ
ejpam-6566	35	18	multifunctions	multifunction	NOUN
ejpam-6566	35	19	.	.	PUNCT
ejpam-6566	36	1	hatir	hatir	PROPN
ejpam-6566	36	2	and	and	CCONJ
ejpam-6566	36	3	noiri	noiri	ADV
ejpam-6566	37	1	[	[	X
ejpam-6566	37	2	15	15	NUM
ejpam-6566	37	3	]	]	PUNCT
ejpam-6566	37	4	introduced	introduce	VERB
ejpam-6566	37	5	and	and	CCONJ
ejpam-6566	37	6	investigated	investigate	VERB
ejpam-6566	37	7	the	the	DET
ejpam-6566	37	8	notions	notion	NOUN
ejpam-6566	37	9	of	of	ADP
ejpam-6566	37	10	weakly	weakly	ADJ
ejpam-6566	37	11	pre	pre	ADJ
ejpam-6566	37	12	-	-	ADJ
ejpam-6566	37	13	i	i	PRON
ejpam-6566	37	14	-open	-open	NOUN
ejpam-6566	37	15	sets	set	NOUN
ejpam-6566	37	16	and	and	CCONJ
ejpam-6566	37	17	weakly	weakly	ADJ
ejpam-6566	37	18	prei	prei	NOUN
ejpam-6566	37	19	-continuous	-continuous	ADJ
ejpam-6566	37	20	functions	function	NOUN
ejpam-6566	37	21	.	.	PUNCT
ejpam-6566	38	1	furthermore	furthermore	ADV
ejpam-6566	38	2	,	,	PUNCT
ejpam-6566	38	3	hatir	hatir	PROPN
ejpam-6566	38	4	and	and	CCONJ
ejpam-6566	38	5	noiri	noiri	ADV
ejpam-6566	39	1	[	[	X
ejpam-6566	39	2	16	16	NUM
ejpam-6566	39	3	]	]	PUNCT
ejpam-6566	39	4	investigated	investigate	VERB
ejpam-6566	39	5	further	further	ADJ
ejpam-6566	39	6	properties	property	NOUN
ejpam-6566	39	7	of	of	ADP
ejpam-6566	39	8	semi	semi	ADJ
ejpam-6566	39	9	-	-	ADJ
ejpam-6566	39	10	i	i	PRON
ejpam-6566	39	11	-open	-open	NOUN
ejpam-6566	39	12	sets	set	NOUN
ejpam-6566	39	13	and	and	CCONJ
ejpam-6566	39	14	semi	semi	ADJ
ejpam-6566	39	15	-	-	ADJ
ejpam-6566	39	16	i	i	ADV
ejpam-6566	39	17	-continuous	-continuous	ADJ
ejpam-6566	39	18	functions	function	NOUN
ejpam-6566	39	19	.	.	PUNCT
ejpam-6566	40	1	on	on	ADP
ejpam-6566	40	2	the	the	DET
ejpam-6566	40	3	other	other	ADJ
ejpam-6566	40	4	hand	hand	NOUN
ejpam-6566	40	5	,	,	PUNCT
ejpam-6566	40	6	the	the	DET
ejpam-6566	40	7	present	present	ADJ
ejpam-6566	40	8	author	author	NOUN
ejpam-6566	40	9	[	[	X
ejpam-6566	40	10	17	17	NUM
ejpam-6566	40	11	]	]	PUNCT
ejpam-6566	40	12	introduced	introduce	VERB
ejpam-6566	40	13	the	the	DET
ejpam-6566	40	14	concepts	concept	NOUN
ejpam-6566	40	15	of	of	ADP
ejpam-6566	40	16	upper	upper	ADJ
ejpam-6566	40	17	⋆-continuous	⋆-continuous	ADJ
ejpam-6566	40	18	multifunctions	multifunction	NOUN
ejpam-6566	40	19	and	and	CCONJ
ejpam-6566	40	20	lower	low	ADJ
ejpam-6566	40	21	⋆continuous	⋆continuous	ADJ
ejpam-6566	40	22	multifunctions	multifunction	NOUN
ejpam-6566	40	23	.	.	PUNCT
ejpam-6566	41	1	moreover	moreover	ADV
ejpam-6566	41	2	,	,	PUNCT
ejpam-6566	41	3	several	several	ADJ
ejpam-6566	41	4	characterizations	characterization	NOUN
ejpam-6566	41	5	of	of	ADP
ejpam-6566	41	6	upper	upper	ADJ
ejpam-6566	41	7	⋆-continuous	⋆-continuous	ADJ
ejpam-6566	41	8	multifunctions	multifunction	NOUN
ejpam-6566	41	9	,	,	PUNCT
ejpam-6566	41	10	lower	low	ADJ
ejpam-6566	41	11	⋆-continuous	⋆-continuous	ADJ
ejpam-6566	41	12	multifunctions	multifunction	NOUN
ejpam-6566	41	13	,	,	PUNCT
ejpam-6566	41	14	upper	upper	ADJ
ejpam-6566	41	15	almost	almost	ADV
ejpam-6566	41	16	⋆-continuous	⋆-continuous	ADJ
ejpam-6566	41	17	multifunctions	multifunction	NOUN
ejpam-6566	41	18	and	and	CCONJ
ejpam-6566	41	19	lower	low	ADJ
ejpam-6566	41	20	almost	almost	ADV
ejpam-6566	41	21	⋆-continuous	⋆-continuous	ADJ
ejpam-6566	41	22	multifunctions	multifunction	NOUN
ejpam-6566	41	23	were	be	AUX
ejpam-6566	41	24	established	establish	VERB
ejpam-6566	41	25	in	in	ADP
ejpam-6566	41	26	[	[	X
ejpam-6566	41	27	17	17	NUM
ejpam-6566	41	28	]	]	X
ejpam-6566	41	29	.	.	PUNCT
ejpam-6566	42	1	quite	quite	ADV
ejpam-6566	42	2	recently	recently	ADV
ejpam-6566	42	3	,	,	PUNCT
ejpam-6566	42	4	the	the	DET
ejpam-6566	42	5	present	present	ADJ
ejpam-6566	42	6	author	author	NOUN
ejpam-6566	42	7	[	[	X
ejpam-6566	42	8	18	18	NUM
ejpam-6566	42	9	]	]	PUNCT
ejpam-6566	42	10	introduced	introduce	VERB
ejpam-6566	42	11	and	and	CCONJ
ejpam-6566	42	12	studied	study	VERB
ejpam-6566	42	13	the	the	DET
ejpam-6566	42	14	notions	notion	NOUN
ejpam-6566	42	15	of	of	ADP
ejpam-6566	42	16	pı	pı	ADJ
ejpam-6566	42	17	-	-	ADJ
ejpam-6566	42	18	continuous	continuous	ADJ
ejpam-6566	42	19	multifunctions	multifunction	NOUN
ejpam-6566	42	20	and	and	CCONJ
ejpam-6566	43	1	weakly	weakly	ADJ
ejpam-6566	43	2	pı	pı	ADJ
ejpam-6566	43	3	-	-	ADJ
ejpam-6566	43	4	continuous	continuous	ADJ
ejpam-6566	43	5	multifunctions	multifunction	NOUN
ejpam-6566	43	6	.	.	PUNCT
ejpam-6566	44	1	in	in	ADP
ejpam-6566	44	2	this	this	DET
ejpam-6566	44	3	paper	paper	NOUN
ejpam-6566	44	4	,	,	PUNCT
ejpam-6566	44	5	we	we	PRON
ejpam-6566	44	6	introduce	introduce	VERB
ejpam-6566	44	7	new	new	ADJ
ejpam-6566	44	8	classes	class	NOUN
ejpam-6566	44	9	of	of	ADP
ejpam-6566	44	10	multifunctions	multifunction	NOUN
ejpam-6566	44	11	between	between	ADP
ejpam-6566	44	12	an	an	DET
ejpam-6566	44	13	ideal	ideal	ADJ
ejpam-6566	44	14	topological	topological	ADJ
ejpam-6566	44	15	space	space	NOUN
ejpam-6566	44	16	and	and	CCONJ
ejpam-6566	44	17	a	a	DET
ejpam-6566	44	18	bitopological	bitopological	ADJ
ejpam-6566	44	19	space	space	NOUN
ejpam-6566	44	20	,	,	PUNCT
ejpam-6566	44	21	namely	namely	ADV
ejpam-6566	44	22	upper	upper	ADJ
ejpam-6566	44	23	almost	almost	ADV
ejpam-6566	44	24	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	44	25	,	,	PUNCT
ejpam-6566	44	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	44	27	multifunctions	multifunction	NOUN
ejpam-6566	44	28	and	and	CCONJ
ejpam-6566	44	29	lower	low	ADJ
ejpam-6566	44	30	almost	almost	ADV
ejpam-6566	44	31	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	44	32	,	,	PUNCT
ejpam-6566	44	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	44	34	multifunctions	multifunction	NOUN
ejpam-6566	44	35	.	.	PUNCT
ejpam-6566	45	1	we	we	PRON
ejpam-6566	45	2	also	also	ADV
ejpam-6566	45	3	investigate	investigate	VERB
ejpam-6566	45	4	several	several	ADJ
ejpam-6566	45	5	characterizations	characterization	NOUN
ejpam-6566	45	6	of	of	ADP
ejpam-6566	45	7	upper	upper	ADJ
ejpam-6566	45	8	almost	almost	ADV
ejpam-6566	45	9	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	45	10	,	,	PUNCT
ejpam-6566	45	11	σ2)continuous	σ2)continuous	ADJ
ejpam-6566	45	12	multifunctions	multifunction	NOUN
ejpam-6566	45	13	and	and	CCONJ
ejpam-6566	45	14	lower	low	ADJ
ejpam-6566	45	15	almost	almost	ADV
ejpam-6566	45	16	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	45	17	,	,	PUNCT
ejpam-6566	45	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	45	19	multifunctions	multifunction	NOUN
ejpam-6566	45	20	.	.	PUNCT
ejpam-6566	46	1	2	2	X
ejpam-6566	46	2	.	.	X
ejpam-6566	46	3	preliminaries	preliminary	NOUN
ejpam-6566	46	4	throughout	throughout	ADP
ejpam-6566	46	5	the	the	DET
ejpam-6566	46	6	present	present	ADJ
ejpam-6566	46	7	paper	paper	NOUN
ejpam-6566	46	8	,	,	PUNCT
ejpam-6566	46	9	spaces	space	NOUN
ejpam-6566	46	10	(	(	PUNCT
ejpam-6566	46	11	x	x	NOUN
ejpam-6566	46	12	,	,	PUNCT
ejpam-6566	46	13	τ1	τ1	NOUN
ejpam-6566	46	14	,	,	PUNCT
ejpam-6566	46	15	τ2	τ2	NOUN
ejpam-6566	46	16	)	)	PUNCT
ejpam-6566	46	17	and	and	CCONJ
ejpam-6566	46	18	(	(	PUNCT
ejpam-6566	46	19	y	y	PROPN
ejpam-6566	46	20	,	,	PUNCT
ejpam-6566	46	21	σ1	σ1	PROPN
ejpam-6566	46	22	,	,	PUNCT
ejpam-6566	46	23	σ2	σ2	NOUN
ejpam-6566	46	24	)	)	PUNCT
ejpam-6566	46	25	(	(	PUNCT
ejpam-6566	46	26	or	or	CCONJ
ejpam-6566	46	27	simply	simply	ADV
ejpam-6566	46	28	x	x	X
ejpam-6566	46	29	and	and	CCONJ
ejpam-6566	46	30	y	y	PROPN
ejpam-6566	46	31	)	)	PUNCT
ejpam-6566	46	32	always	always	ADV
ejpam-6566	46	33	mean	mean	VERB
ejpam-6566	46	34	bitopological	bitopological	ADJ
ejpam-6566	46	35	spaces	space	NOUN
ejpam-6566	46	36	on	on	ADP
ejpam-6566	46	37	which	which	PRON
ejpam-6566	46	38	no	no	DET
ejpam-6566	46	39	separation	separation	NOUN
ejpam-6566	46	40	axioms	axiom	NOUN
ejpam-6566	46	41	are	be	AUX
ejpam-6566	46	42	assumed	assume	VERB
ejpam-6566	46	43	unless	unless	SCONJ
ejpam-6566	46	44	explicitly	explicitly	ADV
ejpam-6566	46	45	stated	state	VERB
ejpam-6566	46	46	.	.	PUNCT
ejpam-6566	47	1	let	let	VERB
ejpam-6566	47	2	a	a	DET
ejpam-6566	47	3	be	be	AUX
ejpam-6566	47	4	a	a	DET
ejpam-6566	47	5	subset	subset	NOUN
ejpam-6566	47	6	of	of	ADP
ejpam-6566	47	7	a	a	DET
ejpam-6566	47	8	bitopological	bitopological	ADJ
ejpam-6566	47	9	space	space	NOUN
ejpam-6566	47	10	(	(	PUNCT
ejpam-6566	47	11	x	x	NOUN
ejpam-6566	47	12	,	,	PUNCT
ejpam-6566	47	13	τ1	τ1	NOUN
ejpam-6566	47	14	,	,	PUNCT
ejpam-6566	47	15	τ2	τ2	NOUN
ejpam-6566	47	16	)	)	PUNCT
ejpam-6566	47	17	.	.	PUNCT
ejpam-6566	48	1	the	the	DET
ejpam-6566	48	2	closure	closure	NOUN
ejpam-6566	48	3	of	of	ADP
ejpam-6566	48	4	a	a	PRON
ejpam-6566	48	5	and	and	CCONJ
ejpam-6566	48	6	the	the	DET
ejpam-6566	48	7	interior	interior	NOUN
ejpam-6566	48	8	of	of	ADP
ejpam-6566	48	9	a	a	PRON
ejpam-6566	48	10	with	with	ADP
ejpam-6566	48	11	respect	respect	NOUN
ejpam-6566	48	12	to	to	ADP
ejpam-6566	48	13	τi	τi	PROPN
ejpam-6566	48	14	are	be	AUX
ejpam-6566	48	15	denoted	denote	VERB
ejpam-6566	48	16	by	by	ADP
ejpam-6566	48	17	τi	τi	NOUN
ejpam-6566	48	18	-	-	PUNCT
ejpam-6566	48	19	cl(a	cl(a	NUM
ejpam-6566	48	20	)	)	PUNCT
ejpam-6566	48	21	and	and	CCONJ
ejpam-6566	48	22	τi	τi	NOUN
ejpam-6566	48	23	-	-	PUNCT
ejpam-6566	48	24	int(a	int(a	NOUN
ejpam-6566	48	25	)	)	PUNCT
ejpam-6566	48	26	,	,	PUNCT
ejpam-6566	48	27	respectively	respectively	ADV
ejpam-6566	48	28	,	,	PUNCT
ejpam-6566	48	29	for	for	ADP
ejpam-6566	48	30	i	i	PROPN
ejpam-6566	48	31	=	=	SYM
ejpam-6566	48	32	1	1	NUM
ejpam-6566	48	33	,	,	PUNCT
ejpam-6566	48	34	2	2	NUM
ejpam-6566	48	35	.	.	X
ejpam-6566	48	36	a	a	DET
ejpam-6566	48	37	subset	subset	NOUN
ejpam-6566	48	38	a	a	PRON
ejpam-6566	48	39	of	of	ADP
ejpam-6566	48	40	a	a	DET
ejpam-6566	48	41	bitopological	bitopological	ADJ
ejpam-6566	48	42	space	space	NOUN
ejpam-6566	48	43	(	(	PUNCT
ejpam-6566	48	44	x	x	NOUN
ejpam-6566	48	45	,	,	PUNCT
ejpam-6566	48	46	τ1	τ1	NOUN
ejpam-6566	48	47	,	,	PUNCT
ejpam-6566	48	48	τ2	τ2	NOUN
ejpam-6566	48	49	)	)	PUNCT
ejpam-6566	48	50	is	be	AUX
ejpam-6566	48	51	called	call	VERB
ejpam-6566	48	52	τ1τ2	τ1τ2	VERB
ejpam-6566	48	53	-	-	ADJ
ejpam-6566	48	54	closed	closed	ADJ
ejpam-6566	48	55	[	[	X
ejpam-6566	48	56	19	19	NUM
ejpam-6566	48	57	]	]	X
ejpam-6566	48	58	if	if	SCONJ
ejpam-6566	48	59	a	a	DET
ejpam-6566	48	60	=	=	NOUN
ejpam-6566	48	61	τ1	τ1	NOUN
ejpam-6566	48	62	-	-	PUNCT
ejpam-6566	48	63	cl(τ2	cl(τ2	NOUN
ejpam-6566	48	64	-	-	PUNCT
ejpam-6566	48	65	cl(a	cl(a	NUM
ejpam-6566	48	66	)	)	PUNCT
ejpam-6566	48	67	)	)	PUNCT
ejpam-6566	48	68	.	.	PUNCT
ejpam-6566	49	1	the	the	DET
ejpam-6566	49	2	complement	complement	NOUN
ejpam-6566	49	3	of	of	ADP
ejpam-6566	49	4	a	a	DET
ejpam-6566	49	5	τ1τ2	τ1τ2	ADJ
ejpam-6566	49	6	-	-	ADJ
ejpam-6566	49	7	closed	closed	ADJ
ejpam-6566	49	8	set	set	NOUN
ejpam-6566	49	9	is	be	AUX
ejpam-6566	49	10	called	call	VERB
ejpam-6566	49	11	τ1τ2	τ1τ2	NOUN
ejpam-6566	49	12	-	-	ADJ
ejpam-6566	49	13	open	open	ADJ
ejpam-6566	49	14	.	.	PUNCT
ejpam-6566	50	1	the	the	DET
ejpam-6566	50	2	intersection	intersection	NOUN
ejpam-6566	50	3	of	of	ADP
ejpam-6566	50	4	all	all	DET
ejpam-6566	50	5	τ1τ2	τ1τ2	ADJ
ejpam-6566	50	6	-	-	ADJ
ejpam-6566	50	7	closed	closed	ADJ
ejpam-6566	50	8	sets	set	NOUN
ejpam-6566	50	9	of	of	ADP
ejpam-6566	50	10	x	x	PUNCT
ejpam-6566	50	11	containing	contain	VERB
ejpam-6566	50	12	a	a	PRON
ejpam-6566	50	13	is	be	AUX
ejpam-6566	50	14	called	call	VERB
ejpam-6566	50	15	the	the	DET
ejpam-6566	50	16	τ1τ2	τ1τ2	NOUN
ejpam-6566	50	17	-	-	NOUN
ejpam-6566	50	18	closure	closure	NOUN
ejpam-6566	50	19	[	[	X
ejpam-6566	50	20	19	19	NUM
ejpam-6566	50	21	]	]	PUNCT
ejpam-6566	50	22	of	of	ADP
ejpam-6566	50	23	a	a	PRON
ejpam-6566	50	24	and	and	CCONJ
ejpam-6566	50	25	is	be	AUX
ejpam-6566	50	26	denoted	denote	VERB
ejpam-6566	50	27	by	by	ADP
ejpam-6566	50	28	τ1τ2	τ1τ2	NOUN
ejpam-6566	50	29	-	-	NUM
ejpam-6566	50	30	cl(a	cl(a	NUM
ejpam-6566	50	31	)	)	PUNCT
ejpam-6566	50	32	.	.	PUNCT
ejpam-6566	51	1	the	the	DET
ejpam-6566	51	2	union	union	NOUN
ejpam-6566	51	3	of	of	ADP
ejpam-6566	51	4	all	all	DET
ejpam-6566	51	5	τ1τ2	τ1τ2	ADJ
ejpam-6566	51	6	-	-	ADJ
ejpam-6566	51	7	open	open	ADJ
ejpam-6566	51	8	sets	set	NOUN
ejpam-6566	51	9	of	of	ADP
ejpam-6566	51	10	x	x	PUNCT
ejpam-6566	51	11	contained	contain	VERB
ejpam-6566	51	12	in	in	ADP
ejpam-6566	51	13	a	a	PRON
ejpam-6566	51	14	is	be	AUX
ejpam-6566	51	15	called	call	VERB
ejpam-6566	51	16	the	the	DET
ejpam-6566	51	17	τ1τ2	τ1τ2	NOUN
ejpam-6566	51	18	-	-	ADJ
ejpam-6566	51	19	interior	interior	ADJ
ejpam-6566	51	20	[	[	X
ejpam-6566	51	21	19	19	NUM
ejpam-6566	51	22	]	]	PUNCT
ejpam-6566	51	23	of	of	ADP
ejpam-6566	51	24	a	a	PRON
ejpam-6566	51	25	and	and	CCONJ
ejpam-6566	51	26	is	be	AUX
ejpam-6566	51	27	denoted	denote	VERB
ejpam-6566	51	28	by	by	ADP
ejpam-6566	51	29	τ1τ2	τ1τ2	NOUN
ejpam-6566	51	30	-	-	ADJ
ejpam-6566	51	31	int(a	int(a	NOUN
ejpam-6566	51	32	)	)	PUNCT
ejpam-6566	51	33	.	.	PUNCT
ejpam-6566	52	1	lemma	lemma	PROPN
ejpam-6566	52	2	1	1	NUM
ejpam-6566	52	3	.	.	PUNCT
ejpam-6566	53	1	[	[	X
ejpam-6566	53	2	19	19	NUM
ejpam-6566	53	3	]	]	PUNCT
ejpam-6566	53	4	let	let	VERB
ejpam-6566	53	5	a	a	PRON
ejpam-6566	53	6	and	and	CCONJ
ejpam-6566	53	7	b	b	NOUN
ejpam-6566	53	8	be	be	AUX
ejpam-6566	53	9	subsets	subset	NOUN
ejpam-6566	53	10	of	of	ADP
ejpam-6566	53	11	a	a	DET
ejpam-6566	53	12	bitopological	bitopological	ADJ
ejpam-6566	53	13	space	space	NOUN
ejpam-6566	53	14	(	(	PUNCT
ejpam-6566	53	15	x	x	NOUN
ejpam-6566	53	16	,	,	PUNCT
ejpam-6566	53	17	τ1	τ1	NOUN
ejpam-6566	53	18	,	,	PUNCT
ejpam-6566	53	19	τ2	τ2	NOUN
ejpam-6566	53	20	)	)	PUNCT
ejpam-6566	53	21	.	.	PUNCT
ejpam-6566	54	1	for	for	ADP
ejpam-6566	54	2	the	the	DET
ejpam-6566	54	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6566	54	4	,	,	PUNCT
ejpam-6566	54	5	the	the	DET
ejpam-6566	54	6	following	follow	VERB
ejpam-6566	54	7	properties	property	NOUN
ejpam-6566	54	8	hold	hold	VERB
ejpam-6566	54	9	:	:	PUNCT
ejpam-6566	54	10	c.	c.	PROPN
ejpam-6566	54	11	viriyapong	viriyapong	PROPN
ejpam-6566	54	12	,	,	PUNCT
ejpam-6566	54	13	a.	a.	PROPN
ejpam-6566	54	14	sama	sama	PROPN
ejpam-6566	54	15	-	-	PUNCT
ejpam-6566	54	16	ae	ae	PROPN
ejpam-6566	54	17	,	,	PUNCT
ejpam-6566	54	18	c.	c.	PROPN
ejpam-6566	54	19	boonpok	boonpok	PROPN
ejpam-6566	54	20	/	/	SYM
ejpam-6566	54	21	eur	eur	PROPN
ejpam-6566	54	22	.	.	PUNCT
ejpam-6566	55	1	j.	j.	PROPN
ejpam-6566	55	2	pure	pure	PROPN
ejpam-6566	55	3	appl	appl	PROPN
ejpam-6566	55	4	.	.	PROPN
ejpam-6566	55	5	math	math	PROPN
ejpam-6566	55	6	,	,	PUNCT
ejpam-6566	55	7	18	18	NUM
ejpam-6566	55	8	(	(	PUNCT
ejpam-6566	55	9	3	3	NUM
ejpam-6566	55	10	)	)	PUNCT
ejpam-6566	55	11	(	(	PUNCT
ejpam-6566	55	12	2025	2025	NUM
ejpam-6566	55	13	)	)	PUNCT
ejpam-6566	55	14	,	,	PUNCT
ejpam-6566	55	15	6566	6566	NUM
ejpam-6566	55	16	3	3	NUM
ejpam-6566	55	17	of	of	ADP
ejpam-6566	55	18	14	14	NUM
ejpam-6566	55	19	(	(	PUNCT
ejpam-6566	55	20	1	1	NUM
ejpam-6566	55	21	)	)	PUNCT
ejpam-6566	55	22	a	a	DET
ejpam-6566	55	23	⊆	⊆	NUM
ejpam-6566	55	24	τ1τ2	τ1τ2	NOUN
ejpam-6566	55	25	-	-	NUM
ejpam-6566	55	26	cl(a	cl(a	NUM
ejpam-6566	55	27	)	)	PUNCT
ejpam-6566	55	28	and	and	CCONJ
ejpam-6566	55	29	τ1τ2	τ1τ2	NOUN
ejpam-6566	55	30	-	-	ADJ
ejpam-6566	55	31	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6566	55	32	-	-	PUNCT
ejpam-6566	55	33	cl(a	cl(a	NUM
ejpam-6566	55	34	)	)	PUNCT
ejpam-6566	55	35	)	)	PUNCT
ejpam-6566	56	1	=	=	PUNCT
ejpam-6566	56	2	τ1τ2	τ1τ2	NOUN
ejpam-6566	56	3	-	-	NUM
ejpam-6566	56	4	cl(a	cl(a	NUM
ejpam-6566	56	5	)	)	PUNCT
ejpam-6566	56	6	.	.	PUNCT
ejpam-6566	57	1	(	(	PUNCT
ejpam-6566	57	2	2	2	X
ejpam-6566	57	3	)	)	PUNCT
ejpam-6566	57	4	if	if	SCONJ
ejpam-6566	57	5	a	a	DET
ejpam-6566	57	6	⊆	⊆	NUM
ejpam-6566	57	7	b	b	NOUN
ejpam-6566	57	8	,	,	PUNCT
ejpam-6566	57	9	then	then	ADV
ejpam-6566	57	10	τ1τ2	τ1τ2	NOUN
ejpam-6566	57	11	-	-	NUM
ejpam-6566	57	12	cl(a	cl(a	NUM
ejpam-6566	57	13	)	)	PUNCT
ejpam-6566	57	14	⊆	⊆	NUM
ejpam-6566	57	15	τ1τ2	τ1τ2	NOUN
ejpam-6566	57	16	-	-	NOUN
ejpam-6566	57	17	cl(b	cl(b	NOUN
ejpam-6566	57	18	)	)	PUNCT
ejpam-6566	57	19	.	.	PUNCT
ejpam-6566	58	1	(	(	PUNCT
ejpam-6566	58	2	3	3	X
ejpam-6566	58	3	)	)	PUNCT
ejpam-6566	58	4	τ1τ2	τ1τ2	NOUN
ejpam-6566	58	5	-	-	NUM
ejpam-6566	58	6	cl(a	cl(a	NUM
ejpam-6566	58	7	)	)	PUNCT
ejpam-6566	58	8	is	be	AUX
ejpam-6566	58	9	τ1τ2	τ1τ2	NOUN
ejpam-6566	58	10	-	-	ADJ
ejpam-6566	58	11	closed	closed	ADJ
ejpam-6566	58	12	.	.	PUNCT
ejpam-6566	59	1	(	(	PUNCT
ejpam-6566	59	2	4	4	X
ejpam-6566	59	3	)	)	PUNCT
ejpam-6566	59	4	a	a	PRON
ejpam-6566	59	5	is	be	AUX
ejpam-6566	59	6	τ1τ2	τ1τ2	NOUN
ejpam-6566	59	7	-	-	ADJ
ejpam-6566	59	8	closed	closed	ADJ
ejpam-6566	59	9	if	if	SCONJ
ejpam-6566	59	10	and	and	CCONJ
ejpam-6566	59	11	only	only	ADV
ejpam-6566	59	12	if	if	SCONJ
ejpam-6566	59	13	a	a	DET
ejpam-6566	59	14	=	=	PUNCT
ejpam-6566	59	15	τ1τ2	τ1τ2	NOUN
ejpam-6566	59	16	-	-	NUM
ejpam-6566	59	17	cl(a	cl(a	NUM
ejpam-6566	59	18	)	)	PUNCT
ejpam-6566	59	19	.	.	PUNCT
ejpam-6566	60	1	(	(	PUNCT
ejpam-6566	60	2	5	5	X
ejpam-6566	60	3	)	)	PUNCT
ejpam-6566	60	4	τ1τ2	τ1τ2	NOUN
ejpam-6566	60	5	-	-	NOUN
ejpam-6566	60	6	cl(x	cl(x	X
ejpam-6566	60	7	−a	−a	NOUN
ejpam-6566	60	8	)	)	PUNCT
ejpam-6566	61	1	=	=	PUNCT
ejpam-6566	61	2	x	x	X
ejpam-6566	62	1	−	−	ADP
ejpam-6566	62	2	τ1τ2	τ1τ2	NOUN
ejpam-6566	62	3	-	-	PUNCT
ejpam-6566	62	4	int(a	int(a	NOUN
ejpam-6566	62	5	)	)	PUNCT
ejpam-6566	62	6	.	.	PUNCT
ejpam-6566	63	1	a	a	DET
ejpam-6566	63	2	subset	subset	NOUN
ejpam-6566	63	3	a	a	PRON
ejpam-6566	63	4	of	of	ADP
ejpam-6566	63	5	a	a	DET
ejpam-6566	63	6	bitopological	bitopological	ADJ
ejpam-6566	63	7	space	space	NOUN
ejpam-6566	63	8	(	(	PUNCT
ejpam-6566	63	9	x	x	NOUN
ejpam-6566	63	10	,	,	PUNCT
ejpam-6566	63	11	τ1	τ1	NOUN
ejpam-6566	63	12	,	,	PUNCT
ejpam-6566	63	13	τ2	τ2	NOUN
ejpam-6566	63	14	)	)	PUNCT
ejpam-6566	63	15	is	be	AUX
ejpam-6566	63	16	said	say	VERB
ejpam-6566	63	17	to	to	PART
ejpam-6566	63	18	be	be	AUX
ejpam-6566	63	19	(	(	PUNCT
ejpam-6566	63	20	τ1	τ1	NOUN
ejpam-6566	63	21	,	,	PUNCT
ejpam-6566	63	22	τ2)r	τ2)r	NOUN
ejpam-6566	63	23	-	-	PUNCT
ejpam-6566	63	24	open	open	NOUN
ejpam-6566	64	1	[	[	X
ejpam-6566	64	2	20	20	NUM
ejpam-6566	64	3	]	]	PUNCT
ejpam-6566	64	4	(	(	PUNCT
ejpam-6566	64	5	resp	resp	NOUN
ejpam-6566	64	6	.	.	PUNCT
ejpam-6566	65	1	(	(	PUNCT
ejpam-6566	65	2	τ1	τ1	NOUN
ejpam-6566	65	3	,	,	PUNCT
ejpam-6566	65	4	τ2)s	τ2)s	NOUN
ejpam-6566	65	5	-	-	PUNCT
ejpam-6566	65	6	open	open	ADJ
ejpam-6566	65	7	[	[	X
ejpam-6566	65	8	21	21	NUM
ejpam-6566	65	9	]	]	PUNCT
ejpam-6566	65	10	,	,	PUNCT
ejpam-6566	65	11	(	(	PUNCT
ejpam-6566	65	12	τ1	τ1	NOUN
ejpam-6566	65	13	,	,	PUNCT
ejpam-6566	65	14	τ2)p	τ2)p	NOUN
ejpam-6566	65	15	-	-	ADJ
ejpam-6566	65	16	open	open	ADJ
ejpam-6566	65	17	[	[	X
ejpam-6566	65	18	21	21	NUM
ejpam-6566	65	19	]	]	PUNCT
ejpam-6566	65	20	,	,	PUNCT
ejpam-6566	65	21	(	(	PUNCT
ejpam-6566	65	22	τ1	τ1	NOUN
ejpam-6566	65	23	,	,	PUNCT
ejpam-6566	65	24	τ2)β	τ2)β	ADJ
ejpam-6566	65	25	-	-	PUNCT
ejpam-6566	65	26	open	open	NOUN
ejpam-6566	66	1	[	[	X
ejpam-6566	66	2	21	21	NUM
ejpam-6566	66	3	]	]	PUNCT
ejpam-6566	66	4	)	)	PUNCT
ejpam-6566	66	5	if	if	SCONJ
ejpam-6566	66	6	a	a	DET
ejpam-6566	66	7	=	=	PUNCT
ejpam-6566	66	8	τ1τ2	τ1τ2	NOUN
ejpam-6566	66	9	-	-	NOUN
ejpam-6566	66	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6566	66	11	-	-	PUNCT
ejpam-6566	66	12	cl(a	cl(a	NUM
ejpam-6566	66	13	)	)	PUNCT
ejpam-6566	66	14	)	)	PUNCT
ejpam-6566	66	15	(	(	PUNCT
ejpam-6566	66	16	resp	resp	NOUN
ejpam-6566	66	17	.	.	PUNCT
ejpam-6566	67	1	a	a	DET
ejpam-6566	67	2	⊆	⊆	NUM
ejpam-6566	67	3	τ1τ2	τ1τ2	NOUN
ejpam-6566	67	4	-	-	ADJ
ejpam-6566	67	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6566	67	6	-	-	PUNCT
ejpam-6566	67	7	int(a	int(a	NOUN
ejpam-6566	67	8	)	)	PUNCT
ejpam-6566	67	9	)	)	PUNCT
ejpam-6566	67	10	,	,	PUNCT
ejpam-6566	67	11	a	a	DET
ejpam-6566	67	12	⊆	⊆	NUM
ejpam-6566	67	13	τ1τ2	τ1τ2	NOUN
ejpam-6566	67	14	-	-	NOUN
ejpam-6566	67	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6566	67	16	-	-	PUNCT
ejpam-6566	67	17	cl(a	cl(a	NUM
ejpam-6566	67	18	)	)	PUNCT
ejpam-6566	67	19	)	)	PUNCT
ejpam-6566	67	20	,	,	PUNCT
ejpam-6566	67	21	a	a	DET
ejpam-6566	67	22	⊆	⊆	NUM
ejpam-6566	67	23	τ1τ2	τ1τ2	NOUN
ejpam-6566	67	24	-	-	PUNCT
ejpam-6566	67	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6566	67	26	-	-	PUNCT
ejpam-6566	67	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6566	67	28	-	-	PUNCT
ejpam-6566	67	29	cl(a	cl(a	NUM
ejpam-6566	67	30	)	)	PUNCT
ejpam-6566	67	31	)	)	PUNCT
ejpam-6566	67	32	)	)	PUNCT
ejpam-6566	67	33	)	)	PUNCT
ejpam-6566	67	34	.	.	PUNCT
ejpam-6566	68	1	the	the	DET
ejpam-6566	68	2	complement	complement	NOUN
ejpam-6566	68	3	of	of	ADP
ejpam-6566	68	4	a	a	DET
ejpam-6566	68	5	(	(	PUNCT
ejpam-6566	68	6	τ1	τ1	NOUN
ejpam-6566	68	7	,	,	PUNCT
ejpam-6566	68	8	τ2)r	τ2)r	NOUN
ejpam-6566	68	9	-	-	PUNCT
ejpam-6566	68	10	open	open	ADJ
ejpam-6566	68	11	(	(	PUNCT
ejpam-6566	68	12	resp	resp	NOUN
ejpam-6566	68	13	.	.	PUNCT
ejpam-6566	69	1	(	(	PUNCT
ejpam-6566	69	2	τ1	τ1	NOUN
ejpam-6566	69	3	,	,	PUNCT
ejpam-6566	69	4	τ2)s	τ2)s	NOUN
ejpam-6566	69	5	-	-	PUNCT
ejpam-6566	69	6	open	open	ADJ
ejpam-6566	69	7	,	,	PUNCT
ejpam-6566	69	8	(	(	PUNCT
ejpam-6566	69	9	τ1	τ1	NOUN
ejpam-6566	69	10	,	,	PUNCT
ejpam-6566	69	11	τ2)p	τ2)p	NOUN
ejpam-6566	69	12	-	-	ADJ
ejpam-6566	69	13	open	open	ADJ
ejpam-6566	69	14	,	,	PUNCT
ejpam-6566	69	15	(	(	PUNCT
ejpam-6566	69	16	τ1	τ1	NOUN
ejpam-6566	69	17	,	,	PUNCT
ejpam-6566	69	18	τ2)β	τ2)β	ADJ
ejpam-6566	69	19	-	-	PUNCT
ejpam-6566	69	20	open	open	ADJ
ejpam-6566	69	21	)	)	PUNCT
ejpam-6566	69	22	set	set	NOUN
ejpam-6566	69	23	is	be	AUX
ejpam-6566	69	24	said	say	VERB
ejpam-6566	69	25	to	to	PART
ejpam-6566	69	26	be	be	AUX
ejpam-6566	69	27	(	(	PUNCT
ejpam-6566	69	28	τ1	τ1	NOUN
ejpam-6566	69	29	,	,	PUNCT
ejpam-6566	69	30	τ2)r	τ2)r	NOUN
ejpam-6566	69	31	-	-	PUNCT
ejpam-6566	69	32	closed	closed	ADJ
ejpam-6566	69	33	(	(	PUNCT
ejpam-6566	69	34	resp	resp	NOUN
ejpam-6566	69	35	.	.	PUNCT
ejpam-6566	70	1	(	(	PUNCT
ejpam-6566	70	2	τ1	τ1	NOUN
ejpam-6566	70	3	,	,	PUNCT
ejpam-6566	70	4	τ2)s	τ2)s	NOUN
ejpam-6566	70	5	-	-	PUNCT
ejpam-6566	70	6	closed	closed	ADJ
ejpam-6566	70	7	,	,	PUNCT
ejpam-6566	70	8	(	(	PUNCT
ejpam-6566	70	9	τ1	τ1	NOUN
ejpam-6566	70	10	,	,	PUNCT
ejpam-6566	70	11	τ2)p	τ2)p	NOUN
ejpam-6566	70	12	-	-	PUNCT
ejpam-6566	70	13	closed	closed	ADJ
ejpam-6566	70	14	,	,	PUNCT
ejpam-6566	70	15	(	(	PUNCT
ejpam-6566	70	16	τ1	τ1	NOUN
ejpam-6566	70	17	,	,	PUNCT
ejpam-6566	70	18	τ2)β	τ2)β	ADJ
ejpam-6566	70	19	-	-	PUNCT
ejpam-6566	70	20	closed	closed	ADJ
ejpam-6566	70	21	)	)	PUNCT
ejpam-6566	70	22	.	.	PUNCT
ejpam-6566	71	1	a	a	DET
ejpam-6566	71	2	subset	subset	NOUN
ejpam-6566	71	3	a	a	PRON
ejpam-6566	71	4	of	of	ADP
ejpam-6566	71	5	a	a	DET
ejpam-6566	71	6	bitopological	bitopological	ADJ
ejpam-6566	71	7	space	space	NOUN
ejpam-6566	71	8	(	(	PUNCT
ejpam-6566	71	9	x	x	NOUN
ejpam-6566	71	10	,	,	PUNCT
ejpam-6566	71	11	τ1	τ1	NOUN
ejpam-6566	71	12	,	,	PUNCT
ejpam-6566	71	13	τ2	τ2	NOUN
ejpam-6566	71	14	)	)	PUNCT
ejpam-6566	71	15	is	be	AUX
ejpam-6566	71	16	said	say	VERB
ejpam-6566	71	17	to	to	PART
ejpam-6566	71	18	be	be	AUX
ejpam-6566	71	19	α(τ1	α(τ1	NOUN
ejpam-6566	71	20	,	,	PUNCT
ejpam-6566	71	21	τ2)-open	τ2)-open	ADJ
ejpam-6566	72	1	[	[	X
ejpam-6566	72	2	22	22	NUM
ejpam-6566	72	3	]	]	PUNCT
ejpam-6566	72	4	if	if	SCONJ
ejpam-6566	72	5	a	a	DET
ejpam-6566	72	6	⊆	⊆	NUM
ejpam-6566	72	7	τ1τ2	τ1τ2	NOUN
ejpam-6566	72	8	-	-	PUNCT
ejpam-6566	72	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6566	72	10	-	-	PUNCT
ejpam-6566	72	11	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6566	72	12	-	-	PUNCT
ejpam-6566	72	13	int(a	int(a	NOUN
ejpam-6566	72	14	)	)	PUNCT
ejpam-6566	72	15	)	)	PUNCT
ejpam-6566	72	16	)	)	PUNCT
ejpam-6566	72	17	.	.	PUNCT
ejpam-6566	73	1	the	the	DET
ejpam-6566	73	2	complement	complement	NOUN
ejpam-6566	73	3	of	of	ADP
ejpam-6566	73	4	an	an	DET
ejpam-6566	73	5	α(τ1	α(τ1	NOUN
ejpam-6566	73	6	,	,	PUNCT
ejpam-6566	73	7	τ2)-open	τ2)-open	ADJ
ejpam-6566	73	8	set	set	NOUN
ejpam-6566	73	9	is	be	AUX
ejpam-6566	73	10	said	say	VERB
ejpam-6566	73	11	to	to	PART
ejpam-6566	73	12	be	be	AUX
ejpam-6566	73	13	α(τ1	α(τ1	NOUN
ejpam-6566	73	14	,	,	PUNCT
ejpam-6566	73	15	τ2)-closed	τ2)-close	VERB
ejpam-6566	73	16	.	.	PUNCT
ejpam-6566	74	1	let	let	VERB
ejpam-6566	74	2	a	a	DET
ejpam-6566	74	3	be	be	AUX
ejpam-6566	74	4	a	a	DET
ejpam-6566	74	5	subset	subset	NOUN
ejpam-6566	74	6	of	of	ADP
ejpam-6566	74	7	a	a	DET
ejpam-6566	74	8	bitopological	bitopological	ADJ
ejpam-6566	74	9	space	space	NOUN
ejpam-6566	74	10	(	(	PUNCT
ejpam-6566	74	11	x	x	NOUN
ejpam-6566	74	12	,	,	PUNCT
ejpam-6566	74	13	τ1	τ1	NOUN
ejpam-6566	74	14	,	,	PUNCT
ejpam-6566	74	15	τ2	τ2	NOUN
ejpam-6566	74	16	)	)	PUNCT
ejpam-6566	74	17	.	.	PUNCT
ejpam-6566	75	1	the	the	DET
ejpam-6566	75	2	intersection	intersection	NOUN
ejpam-6566	75	3	of	of	ADP
ejpam-6566	75	4	all	all	DET
ejpam-6566	75	5	(	(	PUNCT
ejpam-6566	75	6	τ1	τ1	NOUN
ejpam-6566	75	7	,	,	PUNCT
ejpam-6566	75	8	τ2)p	τ2)p	NOUN
ejpam-6566	75	9	-	-	PUNCT
ejpam-6566	75	10	closed	closed	ADJ
ejpam-6566	75	11	(	(	PUNCT
ejpam-6566	75	12	resp	resp	NOUN
ejpam-6566	75	13	.	.	PUNCT
ejpam-6566	76	1	(	(	PUNCT
ejpam-6566	76	2	τ1	τ1	NOUN
ejpam-6566	76	3	,	,	PUNCT
ejpam-6566	76	4	τ2)s	τ2)s	NOUN
ejpam-6566	76	5	-	-	PUNCT
ejpam-6566	76	6	closed	closed	ADJ
ejpam-6566	76	7	,	,	PUNCT
ejpam-6566	76	8	α(τ1	α(τ1	NOUN
ejpam-6566	76	9	,	,	PUNCT
ejpam-6566	76	10	τ2)-closed	τ2)-closed	ADJ
ejpam-6566	76	11	)	)	PUNCT
ejpam-6566	76	12	sets	set	NOUN
ejpam-6566	76	13	of	of	ADP
ejpam-6566	76	14	x	x	PUNCT
ejpam-6566	76	15	containing	contain	VERB
ejpam-6566	76	16	a	a	PRON
ejpam-6566	76	17	is	be	AUX
ejpam-6566	76	18	called	call	VERB
ejpam-6566	76	19	the	the	DET
ejpam-6566	76	20	(	(	PUNCT
ejpam-6566	76	21	τ1	τ1	NOUN
ejpam-6566	76	22	,	,	PUNCT
ejpam-6566	76	23	τ2)p	τ2)p	NOUN
ejpam-6566	76	24	-	-	NOUN
ejpam-6566	76	25	closure	closure	NOUN
ejpam-6566	76	26	[	[	X
ejpam-6566	76	27	23	23	NUM
ejpam-6566	76	28	]	]	PUNCT
ejpam-6566	76	29	(	(	PUNCT
ejpam-6566	76	30	resp	resp	NOUN
ejpam-6566	76	31	.	.	PUNCT
ejpam-6566	77	1	(	(	PUNCT
ejpam-6566	77	2	τ1	τ1	NOUN
ejpam-6566	77	3	,	,	PUNCT
ejpam-6566	77	4	τ2)s	τ2)s	NOUN
ejpam-6566	77	5	-	-	PUNCT
ejpam-6566	77	6	closure	closure	NOUN
ejpam-6566	77	7	[	[	X
ejpam-6566	77	8	21	21	NUM
ejpam-6566	77	9	]	]	PUNCT
ejpam-6566	77	10	,	,	PUNCT
ejpam-6566	77	11	α(τ1	α(τ1	NOUN
ejpam-6566	77	12	,	,	PUNCT
ejpam-6566	77	13	τ2)-closure	τ2)-closure	NOUN
ejpam-6566	77	14	[	[	X
ejpam-6566	77	15	24	24	NUM
ejpam-6566	77	16	]	]	PUNCT
ejpam-6566	77	17	)	)	PUNCT
ejpam-6566	77	18	of	of	ADP
ejpam-6566	77	19	a	a	PRON
ejpam-6566	77	20	and	and	CCONJ
ejpam-6566	77	21	is	be	AUX
ejpam-6566	77	22	denoted	denote	VERB
ejpam-6566	77	23	by	by	ADP
ejpam-6566	77	24	(	(	PUNCT
ejpam-6566	77	25	τ1	τ1	NOUN
ejpam-6566	77	26	,	,	PUNCT
ejpam-6566	77	27	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6566	77	28	)	)	PUNCT
ejpam-6566	77	29	(	(	PUNCT
ejpam-6566	77	30	resp	resp	NOUN
ejpam-6566	77	31	.	.	PUNCT
ejpam-6566	78	1	(	(	PUNCT
ejpam-6566	78	2	τ1	τ1	NOUN
ejpam-6566	78	3	,	,	PUNCT
ejpam-6566	78	4	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6566	78	5	)	)	PUNCT
ejpam-6566	78	6	,	,	PUNCT
ejpam-6566	78	7	α(τ1	α(τ1	NOUN
ejpam-6566	78	8	,	,	PUNCT
ejpam-6566	78	9	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-6566	78	10	)	)	PUNCT
ejpam-6566	78	11	)	)	PUNCT
ejpam-6566	78	12	.	.	PUNCT
ejpam-6566	79	1	the	the	DET
ejpam-6566	79	2	union	union	NOUN
ejpam-6566	79	3	of	of	ADP
ejpam-6566	79	4	all	all	DET
ejpam-6566	79	5	(	(	PUNCT
ejpam-6566	79	6	τ1	τ1	NOUN
ejpam-6566	79	7	,	,	PUNCT
ejpam-6566	79	8	τ2)p	τ2)p	NOUN
ejpam-6566	79	9	-	-	ADJ
ejpam-6566	79	10	open	open	ADJ
ejpam-6566	79	11	(	(	PUNCT
ejpam-6566	79	12	resp	resp	NOUN
ejpam-6566	79	13	.	.	PUNCT
ejpam-6566	80	1	(	(	PUNCT
ejpam-6566	80	2	τ1	τ1	NOUN
ejpam-6566	80	3	,	,	PUNCT
ejpam-6566	80	4	τ2)s	τ2)s	NOUN
ejpam-6566	80	5	-	-	PUNCT
ejpam-6566	80	6	open	open	ADJ
ejpam-6566	80	7	,	,	PUNCT
ejpam-6566	80	8	α(τ1	α(τ1	NOUN
ejpam-6566	80	9	,	,	PUNCT
ejpam-6566	80	10	τ2)-open	τ2)-open	ADJ
ejpam-6566	80	11	)	)	PUNCT
ejpam-6566	80	12	sets	set	NOUN
ejpam-6566	80	13	of	of	ADP
ejpam-6566	80	14	x	x	PUNCT
ejpam-6566	80	15	contained	contain	VERB
ejpam-6566	80	16	in	in	ADP
ejpam-6566	80	17	a	a	PRON
ejpam-6566	80	18	is	be	AUX
ejpam-6566	80	19	called	call	VERB
ejpam-6566	80	20	the	the	DET
ejpam-6566	80	21	(	(	PUNCT
ejpam-6566	80	22	τ1	τ1	NOUN
ejpam-6566	80	23	,	,	PUNCT
ejpam-6566	80	24	τ2)p	τ2)p	ADJ
ejpam-6566	80	25	-	-	NOUN
ejpam-6566	80	26	interior	interior	ADJ
ejpam-6566	80	27	[	[	X
ejpam-6566	80	28	23	23	NUM
ejpam-6566	80	29	]	]	PUNCT
ejpam-6566	80	30	(	(	PUNCT
ejpam-6566	80	31	resp	resp	NOUN
ejpam-6566	80	32	.	.	PUNCT
ejpam-6566	81	1	(	(	PUNCT
ejpam-6566	81	2	τ1	τ1	NOUN
ejpam-6566	81	3	,	,	PUNCT
ejpam-6566	81	4	τ2)s	τ2)s	NOUN
ejpam-6566	81	5	-	-	ADJ
ejpam-6566	81	6	interior	interior	NOUN
ejpam-6566	81	7	[	[	X
ejpam-6566	81	8	21	21	NUM
ejpam-6566	81	9	]	]	PUNCT
ejpam-6566	81	10	,	,	PUNCT
ejpam-6566	81	11	α(τ1	α(τ1	NOUN
ejpam-6566	81	12	,	,	PUNCT
ejpam-6566	81	13	τ2)-interior	τ2)-interior	PROPN
ejpam-6566	81	14	[	[	X
ejpam-6566	81	15	24	24	NUM
ejpam-6566	81	16	]	]	PUNCT
ejpam-6566	81	17	)	)	PUNCT
ejpam-6566	81	18	of	of	ADP
ejpam-6566	81	19	a	a	PRON
ejpam-6566	81	20	and	and	CCONJ
ejpam-6566	81	21	is	be	AUX
ejpam-6566	81	22	denoted	denote	VERB
ejpam-6566	81	23	by	by	ADP
ejpam-6566	81	24	(	(	PUNCT
ejpam-6566	81	25	τ1	τ1	NOUN
ejpam-6566	81	26	,	,	PUNCT
ejpam-6566	81	27	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6566	81	28	)	)	PUNCT
ejpam-6566	81	29	(	(	PUNCT
ejpam-6566	81	30	resp	resp	NOUN
ejpam-6566	81	31	.	.	PUNCT
ejpam-6566	82	1	(	(	PUNCT
ejpam-6566	82	2	τ1	τ1	NOUN
ejpam-6566	82	3	,	,	PUNCT
ejpam-6566	82	4	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6566	82	5	)	)	PUNCT
ejpam-6566	82	6	,	,	PUNCT
ejpam-6566	82	7	α(τ1	α(τ1	NOUN
ejpam-6566	82	8	,	,	PUNCT
ejpam-6566	82	9	τ2)-int(a	τ2)-int(a	NOUN
ejpam-6566	82	10	)	)	PUNCT
ejpam-6566	82	11	)	)	PUNCT
ejpam-6566	82	12	.	.	PUNCT
ejpam-6566	83	1	a	a	DET
ejpam-6566	83	2	subset	subset	NOUN
ejpam-6566	83	3	a	a	PRON
ejpam-6566	83	4	of	of	ADP
ejpam-6566	83	5	a	a	DET
ejpam-6566	83	6	bitopological	bitopological	ADJ
ejpam-6566	83	7	space	space	NOUN
ejpam-6566	83	8	(	(	PUNCT
ejpam-6566	83	9	x	x	NOUN
ejpam-6566	83	10	,	,	PUNCT
ejpam-6566	83	11	τ1	τ1	NOUN
ejpam-6566	83	12	,	,	PUNCT
ejpam-6566	83	13	τ2	τ2	NOUN
ejpam-6566	83	14	)	)	PUNCT
ejpam-6566	83	15	is	be	AUX
ejpam-6566	83	16	said	say	VERB
ejpam-6566	83	17	to	to	PART
ejpam-6566	83	18	be	be	AUX
ejpam-6566	83	19	τ1τ2	τ1τ2	NOUN
ejpam-6566	83	20	-	-	ADJ
ejpam-6566	83	21	δ	δ	NOUN
ejpam-6566	83	22	-	-	NOUN
ejpam-6566	83	23	open	open	ADJ
ejpam-6566	83	24	if	if	SCONJ
ejpam-6566	83	25	a	a	PRON
ejpam-6566	83	26	is	be	AUX
ejpam-6566	83	27	the	the	DET
ejpam-6566	83	28	union	union	NOUN
ejpam-6566	83	29	of	of	ADP
ejpam-6566	83	30	(	(	PUNCT
ejpam-6566	83	31	τ1	τ1	NOUN
ejpam-6566	83	32	,	,	PUNCT
ejpam-6566	83	33	τ2)r	τ2)r	ADJ
ejpam-6566	83	34	-	-	PUNCT
ejpam-6566	83	35	open	open	ADJ
ejpam-6566	83	36	sets	set	NOUN
ejpam-6566	83	37	of	of	ADP
ejpam-6566	83	38	x.	x.	NOUN
ejpam-6566	83	39	the	the	DET
ejpam-6566	83	40	complement	complement	NOUN
ejpam-6566	83	41	of	of	ADP
ejpam-6566	83	42	a	a	DET
ejpam-6566	83	43	τ1τ2	τ1τ2	ADJ
ejpam-6566	83	44	-	-	ADJ
ejpam-6566	83	45	δ	δ	NOUN
ejpam-6566	83	46	-	-	ADJ
ejpam-6566	83	47	open	open	ADJ
ejpam-6566	83	48	set	set	NOUN
ejpam-6566	83	49	is	be	AUX
ejpam-6566	83	50	called	call	VERB
ejpam-6566	83	51	τ1τ2	τ1τ2	NOUN
ejpam-6566	83	52	-	-	ADJ
ejpam-6566	83	53	δ	δ	NOUN
ejpam-6566	83	54	-	-	PUNCT
ejpam-6566	83	55	closed	closed	ADJ
ejpam-6566	83	56	.	.	PUNCT
ejpam-6566	84	1	the	the	DET
ejpam-6566	84	2	union	union	NOUN
ejpam-6566	84	3	of	of	ADP
ejpam-6566	84	4	all	all	DET
ejpam-6566	84	5	τ1τ2	τ1τ2	NOUN
ejpam-6566	84	6	-	-	ADJ
ejpam-6566	84	7	δ	δ	NOUN
ejpam-6566	84	8	-	-	ADJ
ejpam-6566	84	9	open	open	ADJ
ejpam-6566	84	10	sets	set	NOUN
ejpam-6566	84	11	of	of	ADP
ejpam-6566	84	12	x	x	PUNCT
ejpam-6566	84	13	contained	contain	VERB
ejpam-6566	84	14	in	in	ADP
ejpam-6566	84	15	a	a	PRON
ejpam-6566	84	16	is	be	AUX
ejpam-6566	84	17	called	call	VERB
ejpam-6566	84	18	the	the	DET
ejpam-6566	84	19	τ1τ2	τ1τ2	ADJ
ejpam-6566	84	20	-	-	ADJ
ejpam-6566	84	21	δ	δ	NOUN
ejpam-6566	84	22	-	-	NOUN
ejpam-6566	84	23	interior	interior	NOUN
ejpam-6566	84	24	of	of	ADP
ejpam-6566	84	25	a	a	PRON
ejpam-6566	84	26	and	and	CCONJ
ejpam-6566	84	27	is	be	AUX
ejpam-6566	84	28	denoted	denote	VERB
ejpam-6566	84	29	by	by	ADP
ejpam-6566	84	30	τ1τ2	τ1τ2	ADJ
ejpam-6566	84	31	-	-	ADJ
ejpam-6566	84	32	δ	δ	NOUN
ejpam-6566	84	33	-	-	PUNCT
ejpam-6566	84	34	int(a	int(a	PROPN
ejpam-6566	84	35	)	)	PUNCT
ejpam-6566	84	36	.	.	PUNCT
ejpam-6566	85	1	the	the	DET
ejpam-6566	85	2	intersection	intersection	NOUN
ejpam-6566	85	3	of	of	ADP
ejpam-6566	85	4	all	all	DET
ejpam-6566	85	5	τ1τ2	τ1τ2	NOUN
ejpam-6566	85	6	-	-	ADJ
ejpam-6566	85	7	δ	δ	NOUN
ejpam-6566	85	8	-	-	PUNCT
ejpam-6566	85	9	closed	close	VERB
ejpam-6566	85	10	sets	set	NOUN
ejpam-6566	85	11	of	of	ADP
ejpam-6566	85	12	x	x	PUNCT
ejpam-6566	85	13	containing	contain	VERB
ejpam-6566	85	14	a	a	PRON
ejpam-6566	85	15	is	be	AUX
ejpam-6566	85	16	called	call	VERB
ejpam-6566	85	17	the	the	DET
ejpam-6566	85	18	τ1τ2	τ1τ2	ADJ
ejpam-6566	85	19	-	-	ADJ
ejpam-6566	85	20	δ	δ	NOUN
ejpam-6566	85	21	-	-	NOUN
ejpam-6566	85	22	closure	closure	NOUN
ejpam-6566	85	23	of	of	ADP
ejpam-6566	85	24	a	a	PRON
ejpam-6566	85	25	and	and	CCONJ
ejpam-6566	85	26	is	be	AUX
ejpam-6566	85	27	denoted	denote	VERB
ejpam-6566	85	28	by	by	ADP
ejpam-6566	85	29	τ1τ2	τ1τ2	ADJ
ejpam-6566	85	30	-	-	ADJ
ejpam-6566	85	31	δ	δ	NOUN
ejpam-6566	85	32	-	-	PUNCT
ejpam-6566	85	33	cl(a	cl(a	X
ejpam-6566	85	34	)	)	PUNCT
ejpam-6566	86	1	[	[	X
ejpam-6566	86	2	25	25	NUM
ejpam-6566	86	3	]	]	PUNCT
ejpam-6566	86	4	.	.	PUNCT
ejpam-6566	87	1	for	for	ADP
ejpam-6566	87	2	a	a	DET
ejpam-6566	87	3	subset	subset	NOUN
ejpam-6566	87	4	a	a	PRON
ejpam-6566	87	5	of	of	ADP
ejpam-6566	87	6	a	a	DET
ejpam-6566	87	7	bitopological	bitopological	ADJ
ejpam-6566	87	8	space	space	NOUN
ejpam-6566	87	9	(	(	PUNCT
ejpam-6566	87	10	x	x	NOUN
ejpam-6566	87	11	,	,	PUNCT
ejpam-6566	87	12	τ1	τ1	NOUN
ejpam-6566	87	13	,	,	PUNCT
ejpam-6566	87	14	τ2	τ2	PROPN
ejpam-6566	87	15	)	)	PUNCT
ejpam-6566	87	16	,	,	PUNCT
ejpam-6566	87	17	a	a	DET
ejpam-6566	87	18	point	point	NOUN
ejpam-6566	87	19	x	x	X
ejpam-6566	87	20	∈	∈	NOUN
ejpam-6566	87	21	x	x	PUNCT
ejpam-6566	87	22	is	be	AUX
ejpam-6566	87	23	called	call	VERB
ejpam-6566	87	24	a	a	DET
ejpam-6566	87	25	(	(	PUNCT
ejpam-6566	87	26	τ1	τ1	NOUN
ejpam-6566	87	27	,	,	PUNCT
ejpam-6566	87	28	τ2)θ	τ2)θ	ADJ
ejpam-6566	87	29	-	-	PUNCT
ejpam-6566	87	30	cluster	cluster	NOUN
ejpam-6566	87	31	point	point	NOUN
ejpam-6566	87	32	of	of	ADP
ejpam-6566	87	33	a	a	DET
ejpam-6566	87	34	if	if	SCONJ
ejpam-6566	87	35	τ1τ2	τ1τ2	NOUN
ejpam-6566	87	36	-	-	NOUN
ejpam-6566	87	37	cl(u	cl(u	NOUN
ejpam-6566	87	38	)	)	PUNCT
ejpam-6566	87	39	∩a	∩a	PROPN
ejpam-6566	87	40	̸=	̸=	PROPN
ejpam-6566	87	41	∅	∅	NOUN
ejpam-6566	87	42	for	for	ADP
ejpam-6566	87	43	every	every	DET
ejpam-6566	87	44	τ1τ2	τ1τ2	ADJ
ejpam-6566	87	45	-	-	ADJ
ejpam-6566	87	46	open	open	ADJ
ejpam-6566	87	47	set	set	NOUN
ejpam-6566	87	48	u	u	NOUN
ejpam-6566	87	49	containing	contain	VERB
ejpam-6566	87	50	x.	x.	NOUN
ejpam-6566	87	51	the	the	DET
ejpam-6566	87	52	set	set	NOUN
ejpam-6566	87	53	of	of	ADP
ejpam-6566	87	54	all	all	DET
ejpam-6566	87	55	(	(	PUNCT
ejpam-6566	87	56	τ1	τ1	NOUN
ejpam-6566	87	57	,	,	PUNCT
ejpam-6566	87	58	τ2)θ	τ2)θ	ADJ
ejpam-6566	87	59	-	-	PUNCT
ejpam-6566	87	60	cluster	cluster	NOUN
ejpam-6566	87	61	points	point	NOUN
ejpam-6566	87	62	of	of	ADP
ejpam-6566	87	63	a	a	PRON
ejpam-6566	87	64	is	be	AUX
ejpam-6566	87	65	called	call	VERB
ejpam-6566	87	66	the	the	DET
ejpam-6566	87	67	(	(	PUNCT
ejpam-6566	87	68	τ1	τ1	NOUN
ejpam-6566	87	69	,	,	PUNCT
ejpam-6566	87	70	τ2)θ	τ2)θ	NOUN
ejpam-6566	87	71	-	-	PUNCT
ejpam-6566	87	72	closure	closure	NOUN
ejpam-6566	87	73	of	of	ADP
ejpam-6566	87	74	a	a	PRON
ejpam-6566	87	75	and	and	CCONJ
ejpam-6566	87	76	is	be	AUX
ejpam-6566	87	77	denoted	denote	VERB
ejpam-6566	87	78	by	by	ADP
ejpam-6566	87	79	(	(	PUNCT
ejpam-6566	87	80	τ1	τ1	NOUN
ejpam-6566	87	81	,	,	PUNCT
ejpam-6566	87	82	τ2)θ	τ2)θ	NOUN
ejpam-6566	87	83	-	-	PUNCT
ejpam-6566	87	84	cl(a	cl(a	NUM
ejpam-6566	87	85	)	)	PUNCT
ejpam-6566	87	86	.	.	PUNCT
ejpam-6566	88	1	a	a	DET
ejpam-6566	88	2	subset	subset	NOUN
ejpam-6566	88	3	a	a	PRON
ejpam-6566	88	4	of	of	ADP
ejpam-6566	88	5	a	a	DET
ejpam-6566	88	6	bitopological	bitopological	ADJ
ejpam-6566	88	7	space	space	NOUN
ejpam-6566	88	8	(	(	PUNCT
ejpam-6566	88	9	x	x	NOUN
ejpam-6566	88	10	,	,	PUNCT
ejpam-6566	88	11	τ1	τ1	NOUN
ejpam-6566	88	12	,	,	PUNCT
ejpam-6566	88	13	τ2	τ2	NOUN
ejpam-6566	88	14	)	)	PUNCT
ejpam-6566	88	15	is	be	AUX
ejpam-6566	88	16	said	say	VERB
ejpam-6566	88	17	to	to	PART
ejpam-6566	88	18	be	be	AUX
ejpam-6566	88	19	(	(	PUNCT
ejpam-6566	88	20	τ1	τ1	NOUN
ejpam-6566	88	21	,	,	PUNCT
ejpam-6566	88	22	τ2)θ	τ2)θ	NOUN
ejpam-6566	88	23	-	-	PUNCT
ejpam-6566	88	24	closed	closed	ADJ
ejpam-6566	88	25	if	if	SCONJ
ejpam-6566	88	26	(	(	PUNCT
ejpam-6566	88	27	τ1	τ1	NOUN
ejpam-6566	88	28	,	,	PUNCT
ejpam-6566	88	29	τ2)θ	τ2)θ	NOUN
ejpam-6566	88	30	-	-	PUNCT
ejpam-6566	88	31	cl(a	cl(a	NUM
ejpam-6566	88	32	)	)	PUNCT
ejpam-6566	88	33	=	=	PUNCT
ejpam-6566	88	34	a.	a.	NOUN
ejpam-6566	88	35	the	the	DET
ejpam-6566	88	36	complement	complement	NOUN
ejpam-6566	88	37	of	of	ADP
ejpam-6566	88	38	a	a	DET
ejpam-6566	88	39	(	(	PUNCT
ejpam-6566	88	40	τ1	τ1	NOUN
ejpam-6566	88	41	,	,	PUNCT
ejpam-6566	88	42	τ2)θ	τ2)θ	ADJ
ejpam-6566	88	43	-	-	PUNCT
ejpam-6566	88	44	closed	close	VERB
ejpam-6566	88	45	set	set	NOUN
ejpam-6566	88	46	is	be	AUX
ejpam-6566	88	47	said	say	VERB
ejpam-6566	88	48	to	to	PART
ejpam-6566	88	49	be	be	AUX
ejpam-6566	88	50	(	(	PUNCT
ejpam-6566	88	51	τ1	τ1	NOUN
ejpam-6566	88	52	,	,	PUNCT
ejpam-6566	88	53	τ2)θ	τ2)θ	NOUN
ejpam-6566	88	54	-	-	PUNCT
ejpam-6566	88	55	open	open	ADJ
ejpam-6566	88	56	.	.	PUNCT
ejpam-6566	89	1	the	the	DET
ejpam-6566	89	2	union	union	NOUN
ejpam-6566	89	3	of	of	ADP
ejpam-6566	89	4	all	all	DET
ejpam-6566	89	5	(	(	PUNCT
ejpam-6566	89	6	τ1	τ1	NOUN
ejpam-6566	89	7	,	,	PUNCT
ejpam-6566	89	8	τ2)θ	τ2)θ	ADJ
ejpam-6566	89	9	-	-	PUNCT
ejpam-6566	89	10	open	open	ADJ
ejpam-6566	89	11	sets	set	NOUN
ejpam-6566	89	12	of	of	ADP
ejpam-6566	89	13	x	x	PUNCT
ejpam-6566	89	14	contained	contain	VERB
ejpam-6566	89	15	in	in	ADP
ejpam-6566	89	16	a	a	PRON
ejpam-6566	89	17	is	be	AUX
ejpam-6566	89	18	called	call	VERB
ejpam-6566	89	19	the	the	DET
ejpam-6566	89	20	(	(	PUNCT
ejpam-6566	89	21	τ1	τ1	NOUN
ejpam-6566	89	22	,	,	PUNCT
ejpam-6566	89	23	τ2)θ	τ2)θ	ADJ
ejpam-6566	89	24	-	-	PUNCT
ejpam-6566	89	25	interior	interior	NOUN
ejpam-6566	89	26	of	of	ADP
ejpam-6566	89	27	a	a	PRON
ejpam-6566	89	28	and	and	CCONJ
ejpam-6566	89	29	is	be	AUX
ejpam-6566	89	30	denoted	denote	VERB
ejpam-6566	89	31	by	by	ADP
ejpam-6566	89	32	(	(	PUNCT
ejpam-6566	89	33	τ1	τ1	NOUN
ejpam-6566	89	34	,	,	PUNCT
ejpam-6566	89	35	τ2)θ	τ2)θ	NOUN
ejpam-6566	89	36	-	-	PUNCT
ejpam-6566	89	37	int(a	int(a	NOUN
ejpam-6566	89	38	)	)	PUNCT
ejpam-6566	90	1	[	[	X
ejpam-6566	90	2	20	20	NUM
ejpam-6566	90	3	]	]	PUNCT
ejpam-6566	90	4	.	.	PUNCT
ejpam-6566	91	1	an	an	DET
ejpam-6566	91	2	ideal	ideal	NOUN
ejpam-6566	91	3	i	i	PRON
ejpam-6566	91	4	on	on	ADP
ejpam-6566	91	5	a	a	DET
ejpam-6566	91	6	topological	topological	ADJ
ejpam-6566	91	7	space	space	NOUN
ejpam-6566	91	8	(	(	PUNCT
ejpam-6566	91	9	x	x	X
ejpam-6566	91	10	,	,	PUNCT
ejpam-6566	91	11	τ	τ	X
ejpam-6566	91	12	)	)	PUNCT
ejpam-6566	91	13	is	be	AUX
ejpam-6566	91	14	a	a	DET
ejpam-6566	91	15	nonempty	nonempty	ADJ
ejpam-6566	91	16	collection	collection	NOUN
ejpam-6566	91	17	of	of	ADP
ejpam-6566	91	18	subsets	subset	NOUN
ejpam-6566	91	19	of	of	ADP
ejpam-6566	91	20	x	x	PUNCT
ejpam-6566	91	21	satisfying	satisfy	VERB
ejpam-6566	91	22	the	the	DET
ejpam-6566	91	23	following	follow	VERB
ejpam-6566	91	24	properties	property	NOUN
ejpam-6566	91	25	:	:	PUNCT
ejpam-6566	91	26	(	(	PUNCT
ejpam-6566	91	27	1	1	X
ejpam-6566	91	28	)	)	PUNCT
ejpam-6566	91	29	a	a	DET
ejpam-6566	91	30	∈	∈	NOUN
ejpam-6566	91	31	i	i	PRON
ejpam-6566	91	32	and	and	CCONJ
ejpam-6566	91	33	b	b	X
ejpam-6566	91	34	⊆	⊆	NUM
ejpam-6566	91	35	a	a	DET
ejpam-6566	91	36	imply	imply	NOUN
ejpam-6566	91	37	b	b	X
ejpam-6566	91	38	∈	∈	PROPN
ejpam-6566	91	39	i	i	PRON
ejpam-6566	91	40	;	;	PUNCT
ejpam-6566	91	41	(	(	PUNCT
ejpam-6566	91	42	2	2	X
ejpam-6566	91	43	)	)	PUNCT
ejpam-6566	92	1	a	a	PRON
ejpam-6566	92	2	∈	∈	NOUN
ejpam-6566	93	1	i	i	PRON
ejpam-6566	93	2	and	and	CCONJ
ejpam-6566	93	3	b	b	X
ejpam-6566	93	4	∈	∈	NOUN
ejpam-6566	94	1	i	i	PRON
ejpam-6566	94	2	imply	imply	VERB
ejpam-6566	94	3	a	a	DET
ejpam-6566	94	4	∪	∪	X
ejpam-6566	94	5	b	b	NOUN
ejpam-6566	94	6	∈	∈	NOUN
ejpam-6566	95	1	i	i	PRON
ejpam-6566	95	2	.	.	PUNCT
ejpam-6566	96	1	a	a	DET
ejpam-6566	96	2	topological	topological	ADJ
ejpam-6566	96	3	space	space	NOUN
ejpam-6566	96	4	(	(	PUNCT
ejpam-6566	96	5	x	x	X
ejpam-6566	96	6	,	,	PUNCT
ejpam-6566	96	7	τ	τ	X
ejpam-6566	96	8	)	)	PUNCT
ejpam-6566	96	9	with	with	ADP
ejpam-6566	96	10	an	an	DET
ejpam-6566	96	11	ideal	ideal	ADJ
ejpam-6566	96	12	i	i	PRON
ejpam-6566	96	13	on	on	ADP
ejpam-6566	96	14	x	x	SYM
ejpam-6566	96	15	is	be	AUX
ejpam-6566	96	16	called	call	VERB
ejpam-6566	96	17	an	an	DET
ejpam-6566	96	18	ideal	ideal	ADJ
ejpam-6566	96	19	topological	topological	ADJ
ejpam-6566	96	20	space	space	NOUN
ejpam-6566	96	21	and	and	CCONJ
ejpam-6566	96	22	is	be	AUX
ejpam-6566	96	23	denoted	denote	VERB
ejpam-6566	96	24	by	by	ADP
ejpam-6566	96	25	(	(	PUNCT
ejpam-6566	96	26	x	x	X
ejpam-6566	96	27	,	,	PUNCT
ejpam-6566	96	28	τ	τ	PROPN
ejpam-6566	96	29	,	,	PUNCT
ejpam-6566	96	30	i	i	NOUN
ejpam-6566	96	31	)	)	PUNCT
ejpam-6566	96	32	.	.	PUNCT
ejpam-6566	97	1	for	for	ADP
ejpam-6566	97	2	an	an	DET
ejpam-6566	97	3	ideal	ideal	ADJ
ejpam-6566	97	4	topological	topological	ADJ
ejpam-6566	97	5	space	space	NOUN
ejpam-6566	97	6	(	(	PUNCT
ejpam-6566	97	7	x	x	X
ejpam-6566	97	8	,	,	PUNCT
ejpam-6566	97	9	τ	τ	PROPN
ejpam-6566	97	10	,	,	PUNCT
ejpam-6566	97	11	i	i	PROPN
ejpam-6566	97	12	)	)	PUNCT
ejpam-6566	97	13	and	and	CCONJ
ejpam-6566	97	14	a	a	DET
ejpam-6566	97	15	subset	subset	NOUN
ejpam-6566	97	16	a	a	PRON
ejpam-6566	97	17	of	of	ADP
ejpam-6566	97	18	x	x	PRON
ejpam-6566	97	19	,	,	PUNCT
ejpam-6566	97	20	a⋆(i	a⋆(i	PROPN
ejpam-6566	97	21	)	)	PUNCT
ejpam-6566	97	22	is	be	AUX
ejpam-6566	97	23	defined	define	VERB
ejpam-6566	97	24	as	as	SCONJ
ejpam-6566	97	25	follows	follow	VERB
ejpam-6566	97	26	:	:	PUNCT
ejpam-6566	97	27	a⋆(i	a⋆(i	NOUN
ejpam-6566	97	28	)	)	PUNCT
ejpam-6566	98	1	=	=	PUNCT
ejpam-6566	98	2	{	{	PUNCT
ejpam-6566	98	3	x	x	PUNCT
ejpam-6566	98	4	∈	∈	PROPN
ejpam-6566	98	5	x	x	X
ejpam-6566	98	6	:	:	PUNCT
ejpam-6566	98	7	u	u	X
ejpam-6566	98	8	∩a	∩a	PROPN
ejpam-6566	98	9	̸∈	̸∈	PROPN
ejpam-6566	98	10	i	i	PRON
ejpam-6566	98	11	for	for	ADP
ejpam-6566	98	12	every	every	DET
ejpam-6566	98	13	open	open	ADJ
ejpam-6566	98	14	neighbourhood	neighbourhood	NOUN
ejpam-6566	98	15	u	u	NOUN
ejpam-6566	98	16	of	of	ADP
ejpam-6566	98	17	x	x	NOUN
ejpam-6566	98	18	}	}	PUNCT
ejpam-6566	98	19	.	.	PUNCT
ejpam-6566	99	1	in	in	ADP
ejpam-6566	99	2	case	case	NOUN
ejpam-6566	99	3	there	there	PRON
ejpam-6566	99	4	is	be	VERB
ejpam-6566	99	5	no	no	DET
ejpam-6566	99	6	chance	chance	NOUN
ejpam-6566	99	7	for	for	ADP
ejpam-6566	99	8	confusion	confusion	NOUN
ejpam-6566	99	9	,	,	PUNCT
ejpam-6566	99	10	a⋆(i	a⋆(i	NOUN
ejpam-6566	99	11	)	)	PUNCT
ejpam-6566	99	12	is	be	AUX
ejpam-6566	99	13	simply	simply	ADV
ejpam-6566	99	14	written	write	VERB
ejpam-6566	99	15	as	as	ADP
ejpam-6566	99	16	a⋆.	a⋆.	NOUN
ejpam-6566	99	17	in	in	ADP
ejpam-6566	99	18	[	[	X
ejpam-6566	99	19	13	13	NUM
ejpam-6566	99	20	]	]	PUNCT
ejpam-6566	99	21	,	,	PUNCT
ejpam-6566	99	22	a⋆	a⋆	ADV
ejpam-6566	99	23	is	be	AUX
ejpam-6566	99	24	called	call	VERB
ejpam-6566	99	25	the	the	DET
ejpam-6566	99	26	local	local	ADJ
ejpam-6566	99	27	function	function	NOUN
ejpam-6566	99	28	of	of	ADP
ejpam-6566	99	29	a	a	PRON
ejpam-6566	99	30	with	with	ADP
ejpam-6566	99	31	respect	respect	NOUN
ejpam-6566	99	32	to	to	ADP
ejpam-6566	99	33	i	i	PRON
ejpam-6566	99	34	and	and	CCONJ
ejpam-6566	99	35	τ	τ	PROPN
ejpam-6566	99	36	and	and	CCONJ
ejpam-6566	99	37	cl⋆(a	cl⋆(a	PROPN
ejpam-6566	99	38	)	)	PUNCT
ejpam-6566	99	39	=	=	NOUN
ejpam-6566	99	40	a⋆∪a	a⋆∪a	NOUN
ejpam-6566	99	41	defines	define	VERB
ejpam-6566	99	42	a	a	DET
ejpam-6566	99	43	kuratowski	kuratowski	ADJ
ejpam-6566	99	44	c.	c.	PROPN
ejpam-6566	99	45	viriyapong	viriyapong	PROPN
ejpam-6566	99	46	,	,	PUNCT
ejpam-6566	99	47	a.	a.	PROPN
ejpam-6566	99	48	sama	sama	PROPN
ejpam-6566	99	49	-	-	PUNCT
ejpam-6566	99	50	ae	ae	PROPN
ejpam-6566	99	51	,	,	PUNCT
ejpam-6566	99	52	c.	c.	PROPN
ejpam-6566	99	53	boonpok	boonpok	PROPN
ejpam-6566	99	54	/	/	SYM
ejpam-6566	99	55	eur	eur	PROPN
ejpam-6566	99	56	.	.	PUNCT
ejpam-6566	100	1	j.	j.	PROPN
ejpam-6566	100	2	pure	pure	PROPN
ejpam-6566	100	3	appl	appl	PROPN
ejpam-6566	100	4	.	.	PROPN
ejpam-6566	100	5	math	math	PROPN
ejpam-6566	100	6	,	,	PUNCT
ejpam-6566	100	7	18	18	NUM
ejpam-6566	100	8	(	(	PUNCT
ejpam-6566	100	9	3	3	NUM
ejpam-6566	100	10	)	)	PUNCT
ejpam-6566	100	11	(	(	PUNCT
ejpam-6566	100	12	2025	2025	NUM
ejpam-6566	100	13	)	)	PUNCT
ejpam-6566	100	14	,	,	PUNCT
ejpam-6566	100	15	6566	6566	NUM
ejpam-6566	100	16	4	4	NUM
ejpam-6566	100	17	of	of	ADP
ejpam-6566	100	18	14	14	NUM
ejpam-6566	100	19	closure	closure	NOUN
ejpam-6566	100	20	operator	operator	NOUN
ejpam-6566	100	21	for	for	ADP
ejpam-6566	100	22	a	a	DET
ejpam-6566	100	23	topology	topology	NOUN
ejpam-6566	100	24	τ⋆(i	τ⋆(i	NOUN
ejpam-6566	100	25	)	)	PUNCT
ejpam-6566	100	26	finer	fine	ADJ
ejpam-6566	100	27	than	than	ADP
ejpam-6566	100	28	τ	τ	PROPN
ejpam-6566	100	29	.	.	PUNCT
ejpam-6566	101	1	a	a	DET
ejpam-6566	101	2	subset	subset	NOUN
ejpam-6566	101	3	a	a	PRON
ejpam-6566	101	4	is	be	AUX
ejpam-6566	101	5	said	say	VERB
ejpam-6566	101	6	to	to	PART
ejpam-6566	101	7	be	be	AUX
ejpam-6566	101	8	⋆-closed	⋆-close	VERB
ejpam-6566	101	9	[	[	X
ejpam-6566	101	10	26	26	NUM
ejpam-6566	101	11	]	]	X
ejpam-6566	101	12	if	if	SCONJ
ejpam-6566	101	13	a⋆	a⋆	ADJ
ejpam-6566	101	14	⊆	⊆	NUM
ejpam-6566	101	15	a.	a.	NOUN
ejpam-6566	101	16	the	the	DET
ejpam-6566	101	17	interior	interior	NOUN
ejpam-6566	101	18	of	of	ADP
ejpam-6566	101	19	a	a	DET
ejpam-6566	101	20	subset	subset	NOUN
ejpam-6566	101	21	a	a	DET
ejpam-6566	101	22	in	in	ADP
ejpam-6566	101	23	(	(	PUNCT
ejpam-6566	101	24	x	x	X
ejpam-6566	101	25	,	,	PUNCT
ejpam-6566	101	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6566	101	27	)	)	PUNCT
ejpam-6566	101	28	)	)	PUNCT
ejpam-6566	101	29	is	be	AUX
ejpam-6566	101	30	denoted	denote	VERB
ejpam-6566	101	31	by	by	ADP
ejpam-6566	101	32	int⋆(a	int⋆(a	NOUN
ejpam-6566	101	33	)	)	PUNCT
ejpam-6566	101	34	.	.	PUNCT
ejpam-6566	102	1	a	a	DET
ejpam-6566	102	2	subset	subset	NOUN
ejpam-6566	102	3	a	a	PRON
ejpam-6566	102	4	of	of	ADP
ejpam-6566	102	5	an	an	DET
ejpam-6566	102	6	ideal	ideal	ADJ
ejpam-6566	102	7	topological	topological	ADJ
ejpam-6566	102	8	space	space	NOUN
ejpam-6566	102	9	(	(	PUNCT
ejpam-6566	102	10	x	x	X
ejpam-6566	102	11	,	,	PUNCT
ejpam-6566	102	12	τ	τ	PROPN
ejpam-6566	102	13	,	,	PUNCT
ejpam-6566	102	14	i	i	PROPN
ejpam-6566	102	15	)	)	PUNCT
ejpam-6566	102	16	is	be	AUX
ejpam-6566	102	17	said	say	VERB
ejpam-6566	102	18	to	to	PART
ejpam-6566	102	19	be	be	AUX
ejpam-6566	102	20	semi⋆-i	semi⋆-i	X
ejpam-6566	102	21	-open	-open	VERB
ejpam-6566	102	22	[	[	PUNCT
ejpam-6566	102	23	27	27	NUM
ejpam-6566	102	24	]	]	PUNCT
ejpam-6566	102	25	(	(	PUNCT
ejpam-6566	102	26	resp	resp	NOUN
ejpam-6566	102	27	.	.	PUNCT
ejpam-6566	103	1	semi	semi	ADJ
ejpam-6566	103	2	-	-	VERB
ejpam-6566	103	3	i	i	PRON
ejpam-6566	103	4	-open	-open	NOUN
ejpam-6566	104	1	[	[	X
ejpam-6566	104	2	16	16	NUM
ejpam-6566	104	3	]	]	SYM
ejpam-6566	104	4	)	)	PUNCT
ejpam-6566	104	5	if	if	SCONJ
ejpam-6566	104	6	a	a	DET
ejpam-6566	104	7	⊆	⊆	NUM
ejpam-6566	104	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6566	104	9	)	)	PUNCT
ejpam-6566	104	10	)	)	PUNCT
ejpam-6566	104	11	(	(	PUNCT
ejpam-6566	104	12	resp	resp	NOUN
ejpam-6566	104	13	.	.	PUNCT
ejpam-6566	105	1	a	a	DET
ejpam-6566	105	2	⊆	⊆	NUM
ejpam-6566	105	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6566	105	4	)	)	PUNCT
ejpam-6566	105	5	)	)	PUNCT
ejpam-6566	105	6	)	)	PUNCT
ejpam-6566	105	7	.	.	PUNCT
ejpam-6566	106	1	the	the	DET
ejpam-6566	106	2	complement	complement	NOUN
ejpam-6566	106	3	of	of	ADP
ejpam-6566	106	4	a	a	DET
ejpam-6566	106	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6566	106	6	-open	-open	ADJ
ejpam-6566	106	7	(	(	PUNCT
ejpam-6566	106	8	resp	resp	NOUN
ejpam-6566	106	9	.	.	PUNCT
ejpam-6566	107	1	semi	semi	ADJ
ejpam-6566	107	2	-	-	VERB
ejpam-6566	107	3	i	i	PRON
ejpam-6566	107	4	-open	-open	NOUN
ejpam-6566	107	5	)	)	PUNCT
ejpam-6566	108	1	set	set	NOUN
ejpam-6566	108	2	is	be	AUX
ejpam-6566	108	3	said	say	VERB
ejpam-6566	108	4	to	to	PART
ejpam-6566	108	5	be	be	AUX
ejpam-6566	108	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6566	108	7	-closed	-close	VERB
ejpam-6566	108	8	[	[	PUNCT
ejpam-6566	108	9	27	27	NUM
ejpam-6566	108	10	]	]	PUNCT
ejpam-6566	108	11	(	(	PUNCT
ejpam-6566	108	12	resp	resp	NOUN
ejpam-6566	108	13	.	.	PUNCT
ejpam-6566	109	1	semi	semi	ADJ
ejpam-6566	109	2	-	-	VERB
ejpam-6566	109	3	i	i	PRON
ejpam-6566	109	4	-closed	-close	VERB
ejpam-6566	109	5	[	[	X
ejpam-6566	109	6	16	16	NUM
ejpam-6566	109	7	]	]	PUNCT
ejpam-6566	109	8	)	)	PUNCT
ejpam-6566	109	9	.	.	PUNCT
ejpam-6566	110	1	by	by	ADP
ejpam-6566	110	2	a	a	DET
ejpam-6566	110	3	multifunction	multifunction	NOUN
ejpam-6566	110	4	f	f	NOUN
ejpam-6566	110	5	:	:	PUNCT
ejpam-6566	110	6	x	x	X
ejpam-6566	110	7	→	→	SYM
ejpam-6566	110	8	y	y	PROPN
ejpam-6566	110	9	,	,	PUNCT
ejpam-6566	110	10	we	we	PRON
ejpam-6566	110	11	mean	mean	VERB
ejpam-6566	110	12	a	a	DET
ejpam-6566	110	13	point	point	NOUN
ejpam-6566	110	14	-	-	PUNCT
ejpam-6566	110	15	to	to	ADP
ejpam-6566	110	16	-	-	PUNCT
ejpam-6566	110	17	set	set	VERB
ejpam-6566	110	18	correspondence	correspondence	NOUN
ejpam-6566	110	19	from	from	ADP
ejpam-6566	110	20	x	x	PUNCT
ejpam-6566	110	21	into	into	ADP
ejpam-6566	110	22	y	y	PROPN
ejpam-6566	110	23	,	,	PUNCT
ejpam-6566	110	24	and	and	CCONJ
ejpam-6566	110	25	we	we	PRON
ejpam-6566	110	26	always	always	ADV
ejpam-6566	110	27	assume	assume	VERB
ejpam-6566	110	28	that	that	SCONJ
ejpam-6566	110	29	f	f	PROPN
ejpam-6566	110	30	(	(	PUNCT
ejpam-6566	110	31	x	x	X
ejpam-6566	110	32	)	)	PUNCT
ejpam-6566	110	33	̸=	̸=	NOUN
ejpam-6566	110	34	∅	∅	NOUN
ejpam-6566	110	35	for	for	ADP
ejpam-6566	110	36	all	all	PRON
ejpam-6566	110	37	x	x	SYM
ejpam-6566	110	38	∈	∈	ADJ
ejpam-6566	110	39	x.	x.	NOUN
ejpam-6566	110	40	for	for	ADP
ejpam-6566	110	41	a	a	DET
ejpam-6566	110	42	multifunction	multifunction	NOUN
ejpam-6566	110	43	f	f	NOUN
ejpam-6566	110	44	:	:	PUNCT
ejpam-6566	110	45	x	x	X
ejpam-6566	110	46	→	→	SYM
ejpam-6566	110	47	y	y	PROPN
ejpam-6566	110	48	,	,	PUNCT
ejpam-6566	110	49	we	we	PRON
ejpam-6566	110	50	shall	shall	AUX
ejpam-6566	110	51	denote	denote	VERB
ejpam-6566	110	52	the	the	DET
ejpam-6566	110	53	upper	upper	ADJ
ejpam-6566	110	54	and	and	CCONJ
ejpam-6566	110	55	lower	low	ADJ
ejpam-6566	110	56	inverse	inverse	NOUN
ejpam-6566	110	57	of	of	ADP
ejpam-6566	110	58	a	a	DET
ejpam-6566	110	59	set	set	NOUN
ejpam-6566	110	60	b	b	PROPN
ejpam-6566	110	61	of	of	ADP
ejpam-6566	110	62	y	y	PROPN
ejpam-6566	110	63	by	by	ADP
ejpam-6566	110	64	f+(b	f+(b	NOUN
ejpam-6566	110	65	)	)	PUNCT
ejpam-6566	110	66	and	and	CCONJ
ejpam-6566	110	67	f−(b	f−(b	NOUN
ejpam-6566	110	68	)	)	PUNCT
ejpam-6566	110	69	,	,	PUNCT
ejpam-6566	110	70	respectively	respectively	ADV
ejpam-6566	110	71	,	,	PUNCT
ejpam-6566	110	72	that	that	ADV
ejpam-6566	110	73	is	is	ADV
ejpam-6566	110	74	,	,	PUNCT
ejpam-6566	110	75	f+(b	f+(b	NOUN
ejpam-6566	110	76	)	)	PUNCT
ejpam-6566	110	77	=	=	PRON
ejpam-6566	111	1	{	{	PUNCT
ejpam-6566	111	2	x	x	PUNCT
ejpam-6566	111	3	∈	∈	PROPN
ejpam-6566	111	4	x	x	INTJ
ejpam-6566	112	1	|	|	NOUN
ejpam-6566	112	2	f	f	X
ejpam-6566	112	3	(	(	PUNCT
ejpam-6566	112	4	x	x	NOUN
ejpam-6566	112	5	)	)	PUNCT
ejpam-6566	112	6	⊆	⊆	NUM
ejpam-6566	112	7	b	b	NOUN
ejpam-6566	112	8	}	}	PUNCT
ejpam-6566	112	9	and	and	CCONJ
ejpam-6566	112	10	f−(b	f−(b	PROPN
ejpam-6566	112	11	)	)	PUNCT
ejpam-6566	112	12	=	=	PRON
ejpam-6566	113	1	{	{	PUNCT
ejpam-6566	113	2	x	x	PUNCT
ejpam-6566	113	3	∈	∈	PROPN
ejpam-6566	113	4	x	x	INTJ
ejpam-6566	114	1	|	|	NOUN
ejpam-6566	114	2	f	f	X
ejpam-6566	114	3	(	(	PUNCT
ejpam-6566	114	4	x	x	NOUN
ejpam-6566	114	5	)	)	PUNCT
ejpam-6566	114	6	∩	∩	NOUN
ejpam-6566	114	7	b	b	PROPN
ejpam-6566	114	8	̸=	̸=	PROPN
ejpam-6566	114	9	∅	∅	NOUN
ejpam-6566	114	10	}	}	PUNCT
ejpam-6566	114	11	.	.	PUNCT
ejpam-6566	115	1	in	in	ADP
ejpam-6566	115	2	particular	particular	ADJ
ejpam-6566	115	3	,	,	PUNCT
ejpam-6566	115	4	f−(y	f−(y	NOUN
ejpam-6566	115	5	)	)	PUNCT
ejpam-6566	115	6	=	=	SYM
ejpam-6566	116	1	{	{	PUNCT
ejpam-6566	116	2	x	x	PUNCT
ejpam-6566	116	3	∈	∈	PROPN
ejpam-6566	116	4	x	x	INTJ
ejpam-6566	117	1	|	|	ADV
ejpam-6566	117	2	y	y	PROPN
ejpam-6566	117	3	∈	∈	PROPN
ejpam-6566	117	4	f	f	X
ejpam-6566	117	5	(	(	PUNCT
ejpam-6566	117	6	x	x	NOUN
ejpam-6566	117	7	)	)	PUNCT
ejpam-6566	117	8	}	}	PUNCT
ejpam-6566	117	9	for	for	ADP
ejpam-6566	117	10	each	each	DET
ejpam-6566	117	11	point	point	NOUN
ejpam-6566	117	12	y	y	PROPN
ejpam-6566	117	13	∈	∈	PROPN
ejpam-6566	117	14	y	y	PROPN
ejpam-6566	117	15	.	.	PUNCT
ejpam-6566	118	1	for	for	ADP
ejpam-6566	118	2	each	each	DET
ejpam-6566	118	3	a	a	DET
ejpam-6566	118	4	⊆	⊆	NUM
ejpam-6566	118	5	x	x	SYM
ejpam-6566	118	6	,	,	PUNCT
ejpam-6566	118	7	f	f	PROPN
ejpam-6566	118	8	(	(	PUNCT
ejpam-6566	118	9	a	a	NOUN
ejpam-6566	118	10	)	)	PUNCT
ejpam-6566	118	11	=	=	SYM
ejpam-6566	118	12	∪x∈af	∪x∈af	NOUN
ejpam-6566	118	13	(	(	PUNCT
ejpam-6566	118	14	x	x	NOUN
ejpam-6566	118	15	)	)	PUNCT
ejpam-6566	118	16	.	.	PUNCT
ejpam-6566	119	1	3	3	X
ejpam-6566	119	2	.	.	X
ejpam-6566	119	3	upper	upper	ADJ
ejpam-6566	119	4	and	and	CCONJ
ejpam-6566	119	5	lower	low	ADJ
ejpam-6566	119	6	almost	almost	ADV
ejpam-6566	119	7	τ	τ	NOUN
ejpam-6566	119	8	⋆(σ1	⋆(σ1	NOUN
ejpam-6566	119	9	,	,	PUNCT
ejpam-6566	119	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	119	11	multifunctions	multifunction	NOUN
ejpam-6566	119	12	in	in	ADP
ejpam-6566	119	13	this	this	DET
ejpam-6566	119	14	section	section	NOUN
ejpam-6566	119	15	,	,	PUNCT
ejpam-6566	119	16	we	we	PRON
ejpam-6566	119	17	introduce	introduce	VERB
ejpam-6566	119	18	the	the	DET
ejpam-6566	119	19	concepts	concept	NOUN
ejpam-6566	119	20	of	of	ADP
ejpam-6566	119	21	upper	upper	ADJ
ejpam-6566	119	22	almost	almost	ADV
ejpam-6566	119	23	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	119	24	,	,	PUNCT
ejpam-6566	119	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	119	26	multifunctions	multifunction	NOUN
ejpam-6566	119	27	and	and	CCONJ
ejpam-6566	119	28	lower	low	ADJ
ejpam-6566	119	29	almost	almost	ADV
ejpam-6566	119	30	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	119	31	,	,	PUNCT
ejpam-6566	119	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	119	33	multifunctions	multifunction	NOUN
ejpam-6566	119	34	.	.	PUNCT
ejpam-6566	120	1	furthermore	furthermore	ADV
ejpam-6566	120	2	,	,	PUNCT
ejpam-6566	120	3	several	several	ADJ
ejpam-6566	120	4	characterizations	characterization	NOUN
ejpam-6566	120	5	of	of	ADP
ejpam-6566	120	6	upper	upper	ADJ
ejpam-6566	120	7	almost	almost	ADV
ejpam-6566	120	8	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	120	9	,	,	PUNCT
ejpam-6566	120	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	120	11	multifunctions	multifunction	NOUN
ejpam-6566	120	12	and	and	CCONJ
ejpam-6566	120	13	lower	low	ADJ
ejpam-6566	120	14	almost	almost	ADV
ejpam-6566	120	15	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	120	16	,	,	PUNCT
ejpam-6566	120	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	120	18	multifunctions	multifunction	NOUN
ejpam-6566	120	19	are	be	AUX
ejpam-6566	120	20	discussed	discuss	VERB
ejpam-6566	120	21	.	.	PUNCT
ejpam-6566	121	1	definition	definition	NOUN
ejpam-6566	121	2	1	1	NUM
ejpam-6566	121	3	.	.	PUNCT
ejpam-6566	122	1	a	a	DET
ejpam-6566	122	2	multifunction	multifunction	NOUN
ejpam-6566	122	3	f	f	NOUN
ejpam-6566	122	4	:	:	PUNCT
ejpam-6566	122	5	(	(	PUNCT
ejpam-6566	122	6	x	x	X
ejpam-6566	122	7	,	,	PUNCT
ejpam-6566	122	8	τ	τ	PROPN
ejpam-6566	122	9	,	,	PUNCT
ejpam-6566	122	10	i	i	NOUN
ejpam-6566	122	11	)	)	PUNCT
ejpam-6566	122	12	→	→	PUNCT
ejpam-6566	122	13	(	(	PUNCT
ejpam-6566	122	14	y	y	PROPN
ejpam-6566	122	15	,	,	PUNCT
ejpam-6566	122	16	σ1	σ1	PROPN
ejpam-6566	122	17	,	,	PUNCT
ejpam-6566	122	18	σ2	σ2	PROPN
ejpam-6566	122	19	)	)	PUNCT
ejpam-6566	122	20	is	be	AUX
ejpam-6566	122	21	said	say	VERB
ejpam-6566	122	22	to	to	PART
ejpam-6566	122	23	be	be	AUX
ejpam-6566	122	24	upper	upper	ADJ
ejpam-6566	122	25	almost	almost	ADV
ejpam-6566	122	26	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	122	27	,	,	PUNCT
ejpam-6566	122	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	122	29	at	at	ADP
ejpam-6566	122	30	a	a	DET
ejpam-6566	122	31	point	point	NOUN
ejpam-6566	122	32	x	x	SYM
ejpam-6566	122	33	∈	∈	NOUN
ejpam-6566	122	34	x	x	PUNCT
ejpam-6566	122	35	if	if	SCONJ
ejpam-6566	122	36	for	for	ADP
ejpam-6566	122	37	each	each	DET
ejpam-6566	122	38	σ1σ2	σ1σ2	VERB
ejpam-6566	122	39	-	-	ADJ
ejpam-6566	122	40	open	open	ADJ
ejpam-6566	122	41	set	set	NOUN
ejpam-6566	122	42	v	v	NOUN
ejpam-6566	122	43	of	of	ADP
ejpam-6566	122	44	y	y	PRON
ejpam-6566	122	45	such	such	ADJ
ejpam-6566	122	46	that	that	SCONJ
ejpam-6566	122	47	f	f	PROPN
ejpam-6566	122	48	(	(	PUNCT
ejpam-6566	122	49	x	x	X
ejpam-6566	122	50	)	)	PUNCT
ejpam-6566	122	51	⊆	⊆	NUM
ejpam-6566	122	52	v	v	NOUN
ejpam-6566	122	53	,	,	PUNCT
ejpam-6566	122	54	there	there	PRON
ejpam-6566	122	55	exists	exist	VERB
ejpam-6566	122	56	a	a	DET
ejpam-6566	122	57	⋆-open	⋆-open	ADJ
ejpam-6566	122	58	set	set	NOUN
ejpam-6566	122	59	u	u	NOUN
ejpam-6566	122	60	of	of	ADP
ejpam-6566	122	61	x	x	PUNCT
ejpam-6566	122	62	containing	contain	VERB
ejpam-6566	122	63	x	x	PUNCT
ejpam-6566	122	64	such	such	ADJ
ejpam-6566	122	65	that	that	SCONJ
ejpam-6566	122	66	f	f	PROPN
ejpam-6566	122	67	(	(	PUNCT
ejpam-6566	122	68	u	u	NOUN
ejpam-6566	122	69	)	)	PUNCT
ejpam-6566	122	70	⊆	⊆	NUM
ejpam-6566	122	71	σ1σ2	σ1σ2	X
ejpam-6566	122	72	-	-	PUNCT
ejpam-6566	122	73	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	122	74	-	-	PUNCT
ejpam-6566	122	75	cl(v	cl(v	NOUN
ejpam-6566	122	76	)	)	PUNCT
ejpam-6566	122	77	)	)	PUNCT
ejpam-6566	122	78	.	.	PUNCT
ejpam-6566	123	1	a	a	DET
ejpam-6566	123	2	multifunction	multifunction	NOUN
ejpam-6566	123	3	f	f	NOUN
ejpam-6566	123	4	:	:	PUNCT
ejpam-6566	123	5	(	(	PUNCT
ejpam-6566	123	6	x	x	X
ejpam-6566	123	7	,	,	PUNCT
ejpam-6566	123	8	τ	τ	PROPN
ejpam-6566	123	9	,	,	PUNCT
ejpam-6566	123	10	i	i	NOUN
ejpam-6566	123	11	)	)	PUNCT
ejpam-6566	123	12	→	→	PUNCT
ejpam-6566	123	13	(	(	PUNCT
ejpam-6566	123	14	y	y	PROPN
ejpam-6566	123	15	,	,	PUNCT
ejpam-6566	123	16	σ1	σ1	PROPN
ejpam-6566	123	17	,	,	PUNCT
ejpam-6566	123	18	σ2	σ2	PROPN
ejpam-6566	123	19	)	)	PUNCT
ejpam-6566	123	20	is	be	AUX
ejpam-6566	123	21	said	say	VERB
ejpam-6566	123	22	to	to	PART
ejpam-6566	123	23	be	be	AUX
ejpam-6566	123	24	upper	upper	ADJ
ejpam-6566	123	25	almost	almost	ADV
ejpam-6566	123	26	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	123	27	,	,	PUNCT
ejpam-6566	123	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	123	29	if	if	SCONJ
ejpam-6566	123	30	f	f	PROPN
ejpam-6566	123	31	is	be	AUX
ejpam-6566	123	32	upper	upper	ADJ
ejpam-6566	123	33	almost	almost	ADV
ejpam-6566	123	34	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	123	35	,	,	PUNCT
ejpam-6566	123	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	123	37	at	at	ADP
ejpam-6566	123	38	each	each	DET
ejpam-6566	123	39	point	point	NOUN
ejpam-6566	123	40	x	x	PUNCT
ejpam-6566	123	41	of	of	ADP
ejpam-6566	123	42	x.	x.	PROPN
ejpam-6566	123	43	lemma	lemma	PROPN
ejpam-6566	123	44	2	2	X
ejpam-6566	123	45	.	.	PUNCT
ejpam-6566	124	1	[	[	X
ejpam-6566	124	2	12	12	NUM
ejpam-6566	124	3	]	]	PUNCT
ejpam-6566	124	4	let	let	VERB
ejpam-6566	124	5	a	a	PRON
ejpam-6566	124	6	be	be	AUX
ejpam-6566	124	7	a	a	DET
ejpam-6566	124	8	subset	subset	NOUN
ejpam-6566	124	9	of	of	ADP
ejpam-6566	124	10	a	a	DET
ejpam-6566	124	11	bitopological	bitopological	ADJ
ejpam-6566	124	12	space	space	NOUN
ejpam-6566	124	13	(	(	PUNCT
ejpam-6566	124	14	x	x	NOUN
ejpam-6566	124	15	,	,	PUNCT
ejpam-6566	124	16	τ1	τ1	NOUN
ejpam-6566	124	17	,	,	PUNCT
ejpam-6566	124	18	τ2	τ2	NOUN
ejpam-6566	124	19	)	)	PUNCT
ejpam-6566	124	20	.	.	PUNCT
ejpam-6566	125	1	if	if	SCONJ
ejpam-6566	125	2	a	a	PRON
ejpam-6566	125	3	is	be	AUX
ejpam-6566	125	4	τ1τ2	τ1τ2	NOUN
ejpam-6566	125	5	-	-	ADJ
ejpam-6566	125	6	open	open	ADJ
ejpam-6566	125	7	in	in	ADP
ejpam-6566	125	8	x	x	NOUN
ejpam-6566	125	9	,	,	PUNCT
ejpam-6566	125	10	then	then	ADV
ejpam-6566	125	11	(	(	PUNCT
ejpam-6566	125	12	τ1	τ1	NOUN
ejpam-6566	125	13	,	,	PUNCT
ejpam-6566	125	14	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6566	125	15	)	)	PUNCT
ejpam-6566	125	16	=	=	PUNCT
ejpam-6566	126	1	τ1τ2	τ1τ2	NOUN
ejpam-6566	126	2	-	-	NOUN
ejpam-6566	126	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6566	126	4	-	-	PUNCT
ejpam-6566	126	5	cl(a	cl(a	NUM
ejpam-6566	126	6	)	)	PUNCT
ejpam-6566	126	7	)	)	PUNCT
ejpam-6566	126	8	.	.	PUNCT
ejpam-6566	127	1	theorem	theorem	NOUN
ejpam-6566	127	2	1	1	NUM
ejpam-6566	127	3	.	.	X
ejpam-6566	127	4	for	for	ADP
ejpam-6566	127	5	a	a	DET
ejpam-6566	127	6	multifunction	multifunction	NOUN
ejpam-6566	128	1	f	f	NOUN
ejpam-6566	128	2	:	:	PUNCT
ejpam-6566	128	3	(	(	PUNCT
ejpam-6566	128	4	x	x	X
ejpam-6566	128	5	,	,	PUNCT
ejpam-6566	128	6	τ	τ	PROPN
ejpam-6566	128	7	,	,	PUNCT
ejpam-6566	128	8	i	i	NOUN
ejpam-6566	128	9	)	)	PUNCT
ejpam-6566	128	10	→	→	PUNCT
ejpam-6566	128	11	(	(	PUNCT
ejpam-6566	128	12	y	y	PROPN
ejpam-6566	128	13	,	,	PUNCT
ejpam-6566	128	14	σ1	σ1	PROPN
ejpam-6566	128	15	,	,	PUNCT
ejpam-6566	128	16	σ2	σ2	NOUN
ejpam-6566	128	17	)	)	PUNCT
ejpam-6566	128	18	,	,	PUNCT
ejpam-6566	128	19	the	the	DET
ejpam-6566	128	20	following	follow	VERB
ejpam-6566	128	21	properties	property	NOUN
ejpam-6566	128	22	are	be	AUX
ejpam-6566	128	23	equivalent	equivalent	ADJ
ejpam-6566	128	24	:	:	PUNCT
ejpam-6566	128	25	(	(	PUNCT
ejpam-6566	128	26	1	1	X
ejpam-6566	128	27	)	)	PUNCT
ejpam-6566	128	28	f	f	PROPN
ejpam-6566	128	29	is	be	AUX
ejpam-6566	128	30	upper	upper	ADJ
ejpam-6566	128	31	almost	almost	ADV
ejpam-6566	128	32	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	128	33	,	,	PUNCT
ejpam-6566	128	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	128	35	at	at	ADP
ejpam-6566	128	36	x	x	X
ejpam-6566	128	37	∈	∈	PROPN
ejpam-6566	128	38	x	x	X
ejpam-6566	128	39	;	;	PUNCT
ejpam-6566	128	40	(	(	PUNCT
ejpam-6566	128	41	2	2	X
ejpam-6566	128	42	)	)	PUNCT
ejpam-6566	128	43	x	x	SYM
ejpam-6566	128	44	∈	∈	NOUN
ejpam-6566	128	45	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	128	46	-	-	PUNCT
ejpam-6566	128	47	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	128	48	-	-	PUNCT
ejpam-6566	128	49	cl(v	cl(v	NOUN
ejpam-6566	128	50	)	)	PUNCT
ejpam-6566	128	51	)	)	PUNCT
ejpam-6566	128	52	)	)	PUNCT
ejpam-6566	128	53	)	)	PUNCT
ejpam-6566	128	54	for	for	ADP
ejpam-6566	128	55	every	every	DET
ejpam-6566	128	56	σ1σ2	σ1σ2	NOUN
ejpam-6566	128	57	-	-	ADJ
ejpam-6566	128	58	open	open	ADJ
ejpam-6566	128	59	set	set	NOUN
ejpam-6566	128	60	v	v	NOUN
ejpam-6566	128	61	of	of	ADP
ejpam-6566	128	62	y	y	PROPN
ejpam-6566	128	63	containing	contain	VERB
ejpam-6566	128	64	f	f	PROPN
ejpam-6566	128	65	(	(	PUNCT
ejpam-6566	128	66	x	x	NOUN
ejpam-6566	128	67	)	)	PUNCT
ejpam-6566	128	68	;	;	PUNCT
ejpam-6566	128	69	(	(	PUNCT
ejpam-6566	128	70	3	3	X
ejpam-6566	128	71	)	)	PUNCT
ejpam-6566	128	72	x	x	SYM
ejpam-6566	128	73	∈	∈	NOUN
ejpam-6566	128	74	int⋆(f+((σ1	int⋆(f+((σ1	NOUN
ejpam-6566	128	75	,	,	PUNCT
ejpam-6566	128	76	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6566	128	77	)	)	PUNCT
ejpam-6566	128	78	)	)	PUNCT
ejpam-6566	128	79	)	)	PUNCT
ejpam-6566	128	80	for	for	ADP
ejpam-6566	128	81	every	every	DET
ejpam-6566	128	82	σ1σ2	σ1σ2	NOUN
ejpam-6566	128	83	-	-	ADJ
ejpam-6566	128	84	open	open	ADJ
ejpam-6566	128	85	set	set	NOUN
ejpam-6566	128	86	v	v	NOUN
ejpam-6566	128	87	of	of	ADP
ejpam-6566	128	88	y	y	PROPN
ejpam-6566	128	89	containing	contain	VERB
ejpam-6566	128	90	f	f	PROPN
ejpam-6566	128	91	(	(	PUNCT
ejpam-6566	128	92	x	x	NOUN
ejpam-6566	128	93	)	)	PUNCT
ejpam-6566	128	94	;	;	PUNCT
ejpam-6566	128	95	(	(	PUNCT
ejpam-6566	128	96	4	4	X
ejpam-6566	128	97	)	)	PUNCT
ejpam-6566	128	98	x	x	SYM
ejpam-6566	128	99	∈	∈	PROPN
ejpam-6566	128	100	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6566	128	101	)	)	PUNCT
ejpam-6566	128	102	)	)	PUNCT
ejpam-6566	128	103	for	for	ADP
ejpam-6566	128	104	every	every	DET
ejpam-6566	128	105	(	(	PUNCT
ejpam-6566	128	106	σ1	σ1	PROPN
ejpam-6566	128	107	,	,	PUNCT
ejpam-6566	128	108	σ2)r	σ2)r	NOUN
ejpam-6566	128	109	-	-	PUNCT
ejpam-6566	128	110	open	open	ADJ
ejpam-6566	128	111	set	set	VERB
ejpam-6566	128	112	v	v	NOUN
ejpam-6566	128	113	of	of	ADP
ejpam-6566	128	114	y	y	PROPN
ejpam-6566	128	115	containing	contain	VERB
ejpam-6566	128	116	f	f	PROPN
ejpam-6566	128	117	(	(	PUNCT
ejpam-6566	128	118	x	x	NOUN
ejpam-6566	128	119	)	)	PUNCT
ejpam-6566	128	120	;	;	PUNCT
ejpam-6566	128	121	(	(	PUNCT
ejpam-6566	128	122	5	5	X
ejpam-6566	128	123	)	)	PUNCT
ejpam-6566	128	124	for	for	ADP
ejpam-6566	128	125	each	each	DET
ejpam-6566	128	126	(	(	PUNCT
ejpam-6566	128	127	σ1	σ1	PROPN
ejpam-6566	128	128	,	,	PUNCT
ejpam-6566	128	129	σ2)r	σ2)r	NOUN
ejpam-6566	128	130	-	-	PUNCT
ejpam-6566	128	131	open	open	ADJ
ejpam-6566	128	132	set	set	VERB
ejpam-6566	128	133	v	v	NOUN
ejpam-6566	128	134	of	of	ADP
ejpam-6566	128	135	y	y	PROPN
ejpam-6566	128	136	containing	contain	VERB
ejpam-6566	128	137	f	f	PROPN
ejpam-6566	128	138	(	(	PUNCT
ejpam-6566	128	139	x	x	NOUN
ejpam-6566	128	140	)	)	PUNCT
ejpam-6566	128	141	,	,	PUNCT
ejpam-6566	128	142	there	there	PRON
ejpam-6566	128	143	exists	exist	VERB
ejpam-6566	128	144	a	a	DET
ejpam-6566	128	145	⋆-open	⋆-open	ADJ
ejpam-6566	128	146	set	set	NOUN
ejpam-6566	128	147	u	u	NOUN
ejpam-6566	128	148	of	of	ADP
ejpam-6566	128	149	x	x	PUNCT
ejpam-6566	128	150	containing	contain	VERB
ejpam-6566	128	151	x	x	PUNCT
ejpam-6566	128	152	such	such	ADJ
ejpam-6566	128	153	that	that	SCONJ
ejpam-6566	128	154	f	f	PROPN
ejpam-6566	128	155	(	(	PUNCT
ejpam-6566	128	156	u	u	NOUN
ejpam-6566	128	157	)	)	PUNCT
ejpam-6566	128	158	⊆	⊆	NUM
ejpam-6566	128	159	v	v	NOUN
ejpam-6566	128	160	.	.	PUNCT
ejpam-6566	129	1	proof	proof	NOUN
ejpam-6566	129	2	.	.	PUNCT
ejpam-6566	130	1	(	(	PUNCT
ejpam-6566	130	2	1	1	X
ejpam-6566	130	3	)	)	PUNCT
ejpam-6566	130	4	⇒	⇒	NOUN
ejpam-6566	130	5	(	(	PUNCT
ejpam-6566	130	6	2	2	NUM
ejpam-6566	130	7	):	):	PUNCT
ejpam-6566	130	8	let	let	VERB
ejpam-6566	130	9	v	v	PART
ejpam-6566	130	10	be	be	AUX
ejpam-6566	130	11	any	any	DET
ejpam-6566	130	12	σ1σ2	σ1σ2	NOUN
ejpam-6566	130	13	-	-	ADJ
ejpam-6566	130	14	open	open	ADJ
ejpam-6566	130	15	set	set	NOUN
ejpam-6566	130	16	of	of	ADP
ejpam-6566	130	17	y	y	PROPN
ejpam-6566	130	18	containing	contain	VERB
ejpam-6566	130	19	f	f	PROPN
ejpam-6566	130	20	(	(	PUNCT
ejpam-6566	130	21	x	x	NOUN
ejpam-6566	130	22	)	)	PUNCT
ejpam-6566	130	23	.	.	PUNCT
ejpam-6566	131	1	thus	thus	ADV
ejpam-6566	131	2	by	by	ADP
ejpam-6566	131	3	(	(	PUNCT
ejpam-6566	131	4	1	1	NUM
ejpam-6566	131	5	)	)	PUNCT
ejpam-6566	131	6	,	,	PUNCT
ejpam-6566	131	7	there	there	PRON
ejpam-6566	131	8	exists	exist	VERB
ejpam-6566	131	9	a	a	DET
ejpam-6566	131	10	⋆-open	⋆-open	ADJ
ejpam-6566	131	11	set	set	NOUN
ejpam-6566	131	12	u	u	NOUN
ejpam-6566	131	13	of	of	ADP
ejpam-6566	131	14	x	x	PUNCT
ejpam-6566	131	15	containing	contain	VERB
ejpam-6566	131	16	x	x	PUNCT
ejpam-6566	131	17	such	such	ADJ
ejpam-6566	131	18	that	that	SCONJ
ejpam-6566	131	19	f	f	PROPN
ejpam-6566	131	20	(	(	PUNCT
ejpam-6566	131	21	u	u	NOUN
ejpam-6566	131	22	)	)	PUNCT
ejpam-6566	131	23	⊆	⊆	NUM
ejpam-6566	131	24	σ1σ2	σ1σ2	X
ejpam-6566	131	25	-	-	PUNCT
ejpam-6566	131	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	131	27	-	-	PUNCT
ejpam-6566	131	28	cl(v	cl(v	NOUN
ejpam-6566	131	29	)	)	PUNCT
ejpam-6566	131	30	)	)	PUNCT
ejpam-6566	131	31	.	.	PUNCT
ejpam-6566	132	1	therefore	therefore	ADV
ejpam-6566	132	2	,	,	PUNCT
ejpam-6566	132	3	x	x	PUNCT
ejpam-6566	132	4	∈	∈	PROPN
ejpam-6566	132	5	u	u	NOUN
ejpam-6566	132	6	⊆	⊆	NUM
ejpam-6566	132	7	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	132	8	-	-	PUNCT
ejpam-6566	132	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	132	10	-	-	PUNCT
ejpam-6566	132	11	cl(v	cl(v	NOUN
ejpam-6566	132	12	)	)	PUNCT
ejpam-6566	132	13	)	)	PUNCT
ejpam-6566	132	14	)	)	PUNCT
ejpam-6566	133	1	and	and	CCONJ
ejpam-6566	133	2	so	so	ADV
ejpam-6566	133	3	x	x	SYM
ejpam-6566	133	4	∈	∈	ADJ
ejpam-6566	133	5	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	133	6	-	-	PUNCT
ejpam-6566	133	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	133	8	-	-	PUNCT
ejpam-6566	133	9	cl(v	cl(v	NOUN
ejpam-6566	133	10	)	)	PUNCT
ejpam-6566	133	11	)	)	PUNCT
ejpam-6566	133	12	)	)	PUNCT
ejpam-6566	133	13	)	)	PUNCT
ejpam-6566	133	14	.	.	PUNCT
ejpam-6566	134	1	c.	c.	PROPN
ejpam-6566	134	2	viriyapong	viriyapong	PROPN
ejpam-6566	134	3	,	,	PUNCT
ejpam-6566	134	4	a.	a.	PROPN
ejpam-6566	134	5	sama	sama	PROPN
ejpam-6566	134	6	-	-	PUNCT
ejpam-6566	134	7	ae	ae	PROPN
ejpam-6566	134	8	,	,	PUNCT
ejpam-6566	134	9	c.	c.	PROPN
ejpam-6566	134	10	boonpok	boonpok	PROPN
ejpam-6566	134	11	/	/	SYM
ejpam-6566	134	12	eur	eur	PROPN
ejpam-6566	134	13	.	.	PUNCT
ejpam-6566	135	1	j.	j.	PROPN
ejpam-6566	135	2	pure	pure	PROPN
ejpam-6566	135	3	appl	appl	PROPN
ejpam-6566	135	4	.	.	PROPN
ejpam-6566	135	5	math	math	PROPN
ejpam-6566	135	6	,	,	PUNCT
ejpam-6566	135	7	18	18	NUM
ejpam-6566	135	8	(	(	PUNCT
ejpam-6566	135	9	3	3	NUM
ejpam-6566	135	10	)	)	PUNCT
ejpam-6566	135	11	(	(	PUNCT
ejpam-6566	135	12	2025	2025	NUM
ejpam-6566	135	13	)	)	PUNCT
ejpam-6566	135	14	,	,	PUNCT
ejpam-6566	135	15	6566	6566	NUM
ejpam-6566	135	16	5	5	NUM
ejpam-6566	135	17	of	of	ADP
ejpam-6566	135	18	14	14	NUM
ejpam-6566	135	19	(	(	PUNCT
ejpam-6566	135	20	2	2	NUM
ejpam-6566	135	21	)	)	PUNCT
ejpam-6566	135	22	⇒	⇒	NOUN
ejpam-6566	135	23	(	(	PUNCT
ejpam-6566	135	24	3	3	NUM
ejpam-6566	135	25	):	):	PUNCT
ejpam-6566	135	26	this	this	PRON
ejpam-6566	135	27	follows	follow	VERB
ejpam-6566	135	28	from	from	ADP
ejpam-6566	135	29	lemma	lemma	PROPN
ejpam-6566	135	30	2	2	NUM
ejpam-6566	135	31	.	.	PUNCT
ejpam-6566	136	1	(	(	PUNCT
ejpam-6566	136	2	3	3	X
ejpam-6566	136	3	)	)	PUNCT
ejpam-6566	136	4	⇒	⇒	NOUN
ejpam-6566	136	5	(	(	PUNCT
ejpam-6566	136	6	4	4	NUM
ejpam-6566	136	7	):	):	PUNCT
ejpam-6566	136	8	let	let	VERB
ejpam-6566	136	9	v	v	PART
ejpam-6566	136	10	be	be	AUX
ejpam-6566	136	11	any	any	DET
ejpam-6566	136	12	σ1σ2	σ1σ2	NOUN
ejpam-6566	136	13	-	-	ADJ
ejpam-6566	136	14	open	open	ADJ
ejpam-6566	136	15	set	set	NOUN
ejpam-6566	136	16	of	of	ADP
ejpam-6566	136	17	y	y	PROPN
ejpam-6566	136	18	containing	contain	VERB
ejpam-6566	136	19	f	f	PROPN
ejpam-6566	136	20	(	(	PUNCT
ejpam-6566	136	21	x	x	NOUN
ejpam-6566	136	22	)	)	PUNCT
ejpam-6566	136	23	.	.	PUNCT
ejpam-6566	137	1	it	it	PRON
ejpam-6566	137	2	follows	follow	VERB
ejpam-6566	137	3	from	from	ADP
ejpam-6566	137	4	lemma	lemma	PROPN
ejpam-6566	137	5	2	2	NUM
ejpam-6566	137	6	that	that	PRON
ejpam-6566	137	7	v	v	NOUN
ejpam-6566	137	8	=	=	SYM
ejpam-6566	137	9	σ1σ2	σ1σ2	NOUN
ejpam-6566	137	10	-	-	PUNCT
ejpam-6566	137	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	137	12	-	-	PUNCT
ejpam-6566	137	13	cl(v	cl(v	NOUN
ejpam-6566	137	14	)	)	PUNCT
ejpam-6566	137	15	)	)	PUNCT
ejpam-6566	138	1	=	=	SYM
ejpam-6566	138	2	(	(	PUNCT
ejpam-6566	138	3	σ1	σ1	PROPN
ejpam-6566	138	4	,	,	PUNCT
ejpam-6566	138	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6566	138	6	)	)	PUNCT
ejpam-6566	138	7	.	.	PUNCT
ejpam-6566	139	1	(	(	PUNCT
ejpam-6566	139	2	4	4	X
ejpam-6566	139	3	)	)	PUNCT
ejpam-6566	139	4	⇒	⇒	NOUN
ejpam-6566	139	5	(	(	PUNCT
ejpam-6566	139	6	5	5	NUM
ejpam-6566	139	7	):	):	PUNCT
ejpam-6566	139	8	let	let	VERB
ejpam-6566	139	9	v	v	PART
ejpam-6566	139	10	be	be	AUX
ejpam-6566	139	11	any	any	DET
ejpam-6566	139	12	(	(	PUNCT
ejpam-6566	139	13	σ1	σ1	NOUN
ejpam-6566	139	14	,	,	PUNCT
ejpam-6566	139	15	σ2)r	σ2)r	NOUN
ejpam-6566	139	16	-	-	PUNCT
ejpam-6566	139	17	open	open	ADJ
ejpam-6566	139	18	set	set	NOUN
ejpam-6566	139	19	of	of	ADP
ejpam-6566	139	20	y	y	PROPN
ejpam-6566	139	21	containing	contain	VERB
ejpam-6566	139	22	f	f	PROPN
ejpam-6566	139	23	(	(	PUNCT
ejpam-6566	139	24	x	x	NOUN
ejpam-6566	139	25	)	)	PUNCT
ejpam-6566	139	26	.	.	PUNCT
ejpam-6566	140	1	then	then	ADV
ejpam-6566	140	2	by	by	ADP
ejpam-6566	140	3	(	(	PUNCT
ejpam-6566	140	4	4	4	NUM
ejpam-6566	140	5	)	)	PUNCT
ejpam-6566	140	6	,	,	PUNCT
ejpam-6566	140	7	we	we	PRON
ejpam-6566	140	8	have	have	VERB
ejpam-6566	140	9	x	x	X
ejpam-6566	140	10	∈	∈	PROPN
ejpam-6566	140	11	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6566	140	12	)	)	PUNCT
ejpam-6566	140	13	)	)	PUNCT
ejpam-6566	140	14	and	and	CCONJ
ejpam-6566	140	15	there	there	PRON
ejpam-6566	140	16	exists	exist	VERB
ejpam-6566	140	17	a	a	DET
ejpam-6566	140	18	⋆-open	⋆-open	ADJ
ejpam-6566	140	19	set	set	NOUN
ejpam-6566	140	20	u	u	NOUN
ejpam-6566	140	21	of	of	ADP
ejpam-6566	140	22	x	x	PUNCT
ejpam-6566	140	23	containing	contain	VERB
ejpam-6566	140	24	x	x	PUNCT
ejpam-6566	140	25	such	such	ADJ
ejpam-6566	140	26	that	that	SCONJ
ejpam-6566	140	27	x	x	SYM
ejpam-6566	140	28	∈	∈	NUM
ejpam-6566	140	29	u	u	NOUN
ejpam-6566	140	30	⊆	⊆	NUM
ejpam-6566	140	31	f+(v	f+(v	NOUN
ejpam-6566	140	32	)	)	PUNCT
ejpam-6566	140	33	;	;	PUNCT
ejpam-6566	140	34	hence	hence	ADV
ejpam-6566	140	35	f	f	PROPN
ejpam-6566	140	36	(	(	PUNCT
ejpam-6566	140	37	u	u	NOUN
ejpam-6566	140	38	)	)	PUNCT
ejpam-6566	140	39	⊆	⊆	NUM
ejpam-6566	140	40	v	v	NOUN
ejpam-6566	140	41	.	.	PUNCT
ejpam-6566	141	1	(	(	PUNCT
ejpam-6566	141	2	5	5	X
ejpam-6566	141	3	)	)	PUNCT
ejpam-6566	141	4	⇒	⇒	NOUN
ejpam-6566	141	5	(	(	PUNCT
ejpam-6566	141	6	1	1	NUM
ejpam-6566	141	7	):	):	PUNCT
ejpam-6566	141	8	let	let	VERB
ejpam-6566	141	9	v	v	PART
ejpam-6566	141	10	be	be	AUX
ejpam-6566	141	11	any	any	DET
ejpam-6566	141	12	σ1σ2	σ1σ2	NOUN
ejpam-6566	141	13	-	-	ADJ
ejpam-6566	141	14	open	open	ADJ
ejpam-6566	141	15	set	set	NOUN
ejpam-6566	141	16	of	of	ADP
ejpam-6566	141	17	y	y	PROPN
ejpam-6566	141	18	containing	contain	VERB
ejpam-6566	141	19	f	f	PROPN
ejpam-6566	141	20	(	(	PUNCT
ejpam-6566	141	21	x	x	NOUN
ejpam-6566	141	22	)	)	PUNCT
ejpam-6566	141	23	.	.	PUNCT
ejpam-6566	142	1	since	since	SCONJ
ejpam-6566	142	2	σ1σ2	σ1σ2	NOUN
ejpam-6566	142	3	-	-	PUNCT
ejpam-6566	142	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	142	5	-	-	PUNCT
ejpam-6566	142	6	cl(v	cl(v	NOUN
ejpam-6566	142	7	)	)	PUNCT
ejpam-6566	142	8	)	)	PUNCT
ejpam-6566	142	9	is	be	AUX
ejpam-6566	142	10	(	(	PUNCT
ejpam-6566	142	11	σ1	σ1	NOUN
ejpam-6566	142	12	,	,	PUNCT
ejpam-6566	142	13	σ2)r	σ2)r	NOUN
ejpam-6566	142	14	-	-	PUNCT
ejpam-6566	142	15	open	open	ADJ
ejpam-6566	142	16	in	in	ADP
ejpam-6566	142	17	y	y	PROPN
ejpam-6566	142	18	and	and	CCONJ
ejpam-6566	142	19	by	by	ADP
ejpam-6566	142	20	(	(	PUNCT
ejpam-6566	142	21	5	5	NUM
ejpam-6566	142	22	)	)	PUNCT
ejpam-6566	142	23	,	,	PUNCT
ejpam-6566	142	24	there	there	PRON
ejpam-6566	142	25	exists	exist	VERB
ejpam-6566	142	26	a	a	DET
ejpam-6566	142	27	⋆-open	⋆-open	ADJ
ejpam-6566	142	28	set	set	NOUN
ejpam-6566	142	29	u	u	NOUN
ejpam-6566	142	30	of	of	ADP
ejpam-6566	142	31	x	x	PUNCT
ejpam-6566	142	32	containing	contain	VERB
ejpam-6566	142	33	x	x	PUNCT
ejpam-6566	142	34	such	such	ADJ
ejpam-6566	142	35	that	that	SCONJ
ejpam-6566	142	36	f	f	PROPN
ejpam-6566	142	37	(	(	PUNCT
ejpam-6566	142	38	u	u	NOUN
ejpam-6566	142	39	)	)	PUNCT
ejpam-6566	142	40	⊆	⊆	NUM
ejpam-6566	142	41	σ1σ2	σ1σ2	X
ejpam-6566	142	42	-	-	PUNCT
ejpam-6566	142	43	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	142	44	-	-	PUNCT
ejpam-6566	142	45	cl(v	cl(v	NOUN
ejpam-6566	142	46	)	)	PUNCT
ejpam-6566	142	47	)	)	PUNCT
ejpam-6566	142	48	.	.	PUNCT
ejpam-6566	143	1	this	this	PRON
ejpam-6566	143	2	shows	show	VERB
ejpam-6566	143	3	that	that	SCONJ
ejpam-6566	143	4	f	f	PROPN
ejpam-6566	143	5	is	be	AUX
ejpam-6566	143	6	upper	upper	ADJ
ejpam-6566	143	7	almost	almost	ADV
ejpam-6566	143	8	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	143	9	,	,	PUNCT
ejpam-6566	143	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	143	11	at	at	ADP
ejpam-6566	143	12	x	x	SYM
ejpam-6566	143	13	∈	∈	PROPN
ejpam-6566	143	14	x.	x.	NOUN
ejpam-6566	143	15	definition	definition	NOUN
ejpam-6566	143	16	2	2	NUM
ejpam-6566	143	17	.	.	PUNCT
ejpam-6566	143	18	a	a	DET
ejpam-6566	143	19	multifunction	multifunction	NOUN
ejpam-6566	144	1	f	f	NOUN
ejpam-6566	144	2	:	:	PUNCT
ejpam-6566	144	3	(	(	PUNCT
ejpam-6566	144	4	x	x	X
ejpam-6566	144	5	,	,	PUNCT
ejpam-6566	144	6	τ	τ	PROPN
ejpam-6566	144	7	,	,	PUNCT
ejpam-6566	144	8	i	i	NOUN
ejpam-6566	144	9	)	)	PUNCT
ejpam-6566	144	10	→	→	PUNCT
ejpam-6566	144	11	(	(	PUNCT
ejpam-6566	144	12	y	y	PROPN
ejpam-6566	144	13	,	,	PUNCT
ejpam-6566	144	14	σ1	σ1	PROPN
ejpam-6566	144	15	,	,	PUNCT
ejpam-6566	144	16	σ2	σ2	PROPN
ejpam-6566	144	17	)	)	PUNCT
ejpam-6566	144	18	is	be	AUX
ejpam-6566	144	19	called	call	VERB
ejpam-6566	144	20	lower	low	ADJ
ejpam-6566	144	21	almost	almost	ADV
ejpam-6566	144	22	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	144	23	,	,	PUNCT
ejpam-6566	144	24	σ2)continuous	σ2)continuous	ADJ
ejpam-6566	144	25	at	at	ADP
ejpam-6566	144	26	a	a	DET
ejpam-6566	144	27	point	point	NOUN
ejpam-6566	144	28	x	x	SYM
ejpam-6566	144	29	∈	∈	NOUN
ejpam-6566	144	30	x	x	PUNCT
ejpam-6566	144	31	if	if	SCONJ
ejpam-6566	144	32	for	for	ADP
ejpam-6566	144	33	each	each	DET
ejpam-6566	144	34	σ1σ2	σ1σ2	VERB
ejpam-6566	144	35	-	-	ADJ
ejpam-6566	144	36	open	open	ADJ
ejpam-6566	144	37	set	set	NOUN
ejpam-6566	144	38	v	v	NOUN
ejpam-6566	144	39	of	of	ADP
ejpam-6566	144	40	y	y	PRON
ejpam-6566	144	41	such	such	ADJ
ejpam-6566	144	42	that	that	PRON
ejpam-6566	144	43	v	v	ADP
ejpam-6566	144	44	∩	∩	ADJ
ejpam-6566	144	45	f	f	X
ejpam-6566	144	46	(	(	PUNCT
ejpam-6566	144	47	x	x	X
ejpam-6566	144	48	)	)	PUNCT
ejpam-6566	144	49	̸=	̸=	NOUN
ejpam-6566	144	50	∅	∅	NOUN
ejpam-6566	144	51	,	,	PUNCT
ejpam-6566	144	52	there	there	PRON
ejpam-6566	144	53	exists	exist	VERB
ejpam-6566	144	54	a	a	DET
ejpam-6566	144	55	⋆-open	⋆-open	ADJ
ejpam-6566	144	56	set	set	NOUN
ejpam-6566	144	57	u	u	NOUN
ejpam-6566	144	58	of	of	ADP
ejpam-6566	144	59	x	x	PUNCT
ejpam-6566	144	60	containing	contain	VERB
ejpam-6566	144	61	x	x	PUNCT
ejpam-6566	144	62	such	such	ADJ
ejpam-6566	144	63	that	that	SCONJ
ejpam-6566	144	64	σ1σ2	σ1σ2	ADV
ejpam-6566	144	65	-	-	PUNCT
ejpam-6566	144	66	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	144	67	-	-	PUNCT
ejpam-6566	144	68	cl(v	cl(v	NOUN
ejpam-6566	144	69	)	)	PUNCT
ejpam-6566	144	70	)	)	PUNCT
ejpam-6566	145	1	∩	∩	PROPN
ejpam-6566	145	2	f	f	X
ejpam-6566	145	3	(	(	PUNCT
ejpam-6566	145	4	z	z	NOUN
ejpam-6566	145	5	)	)	PUNCT
ejpam-6566	145	6	̸=	̸=	NOUN
ejpam-6566	145	7	∅	∅	NOUN
ejpam-6566	145	8	for	for	ADP
ejpam-6566	145	9	every	every	DET
ejpam-6566	145	10	z	z	NOUN
ejpam-6566	145	11	∈	∈	PROPN
ejpam-6566	145	12	u	u	NOUN
ejpam-6566	145	13	.	.	PUNCT
ejpam-6566	146	1	a	a	DET
ejpam-6566	146	2	multifunction	multifunction	NOUN
ejpam-6566	146	3	f	f	NOUN
ejpam-6566	146	4	:	:	PUNCT
ejpam-6566	146	5	(	(	PUNCT
ejpam-6566	146	6	x	x	X
ejpam-6566	146	7	,	,	PUNCT
ejpam-6566	146	8	τ	τ	PROPN
ejpam-6566	146	9	,	,	PUNCT
ejpam-6566	146	10	i	i	NOUN
ejpam-6566	146	11	)	)	PUNCT
ejpam-6566	146	12	→	→	PUNCT
ejpam-6566	146	13	(	(	PUNCT
ejpam-6566	146	14	y	y	PROPN
ejpam-6566	146	15	,	,	PUNCT
ejpam-6566	146	16	σ1	σ1	PROPN
ejpam-6566	146	17	,	,	PUNCT
ejpam-6566	146	18	σ2	σ2	PROPN
ejpam-6566	146	19	)	)	PUNCT
ejpam-6566	146	20	is	be	AUX
ejpam-6566	146	21	called	call	VERB
ejpam-6566	146	22	lower	low	ADJ
ejpam-6566	146	23	almost	almost	ADV
ejpam-6566	146	24	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	146	25	,	,	PUNCT
ejpam-6566	146	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	146	27	if	if	SCONJ
ejpam-6566	146	28	f	f	PROPN
ejpam-6566	146	29	is	be	AUX
ejpam-6566	146	30	lower	low	ADJ
ejpam-6566	146	31	almost	almost	ADV
ejpam-6566	146	32	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	146	33	,	,	PUNCT
ejpam-6566	146	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	146	35	at	at	ADP
ejpam-6566	146	36	each	each	DET
ejpam-6566	146	37	point	point	NOUN
ejpam-6566	146	38	x	x	PUNCT
ejpam-6566	146	39	of	of	ADP
ejpam-6566	146	40	x.	x.	PROPN
ejpam-6566	146	41	theorem	theorem	VERB
ejpam-6566	146	42	2	2	NUM
ejpam-6566	146	43	.	.	X
ejpam-6566	146	44	for	for	ADP
ejpam-6566	146	45	a	a	DET
ejpam-6566	146	46	multifunction	multifunction	NOUN
ejpam-6566	146	47	f	f	NOUN
ejpam-6566	146	48	:	:	PUNCT
ejpam-6566	146	49	(	(	PUNCT
ejpam-6566	146	50	x	x	X
ejpam-6566	146	51	,	,	PUNCT
ejpam-6566	146	52	τ	τ	PROPN
ejpam-6566	146	53	,	,	PUNCT
ejpam-6566	146	54	i	i	NOUN
ejpam-6566	146	55	)	)	PUNCT
ejpam-6566	146	56	→	→	PUNCT
ejpam-6566	146	57	(	(	PUNCT
ejpam-6566	146	58	y	y	PROPN
ejpam-6566	146	59	,	,	PUNCT
ejpam-6566	146	60	σ1	σ1	PROPN
ejpam-6566	146	61	,	,	PUNCT
ejpam-6566	146	62	σ2	σ2	NOUN
ejpam-6566	146	63	)	)	PUNCT
ejpam-6566	146	64	,	,	PUNCT
ejpam-6566	146	65	the	the	DET
ejpam-6566	146	66	following	follow	VERB
ejpam-6566	146	67	properties	property	NOUN
ejpam-6566	146	68	are	be	AUX
ejpam-6566	146	69	equivalent	equivalent	ADJ
ejpam-6566	146	70	:	:	PUNCT
ejpam-6566	146	71	(	(	PUNCT
ejpam-6566	146	72	1	1	X
ejpam-6566	146	73	)	)	PUNCT
ejpam-6566	146	74	f	f	PROPN
ejpam-6566	146	75	is	be	AUX
ejpam-6566	146	76	lower	low	ADJ
ejpam-6566	146	77	almost	almost	ADV
ejpam-6566	146	78	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	146	79	,	,	PUNCT
ejpam-6566	146	80	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	146	81	at	at	ADP
ejpam-6566	146	82	x	x	X
ejpam-6566	146	83	∈	∈	PROPN
ejpam-6566	146	84	x	x	X
ejpam-6566	146	85	;	;	PUNCT
ejpam-6566	146	86	(	(	PUNCT
ejpam-6566	146	87	2	2	X
ejpam-6566	146	88	)	)	PUNCT
ejpam-6566	146	89	x	x	SYM
ejpam-6566	146	90	∈	∈	PROPN
ejpam-6566	146	91	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	VERB
ejpam-6566	146	92	-	-	PUNCT
ejpam-6566	146	93	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	146	94	-	-	PUNCT
ejpam-6566	146	95	cl(v	cl(v	NOUN
ejpam-6566	146	96	)	)	PUNCT
ejpam-6566	146	97	)	)	PUNCT
ejpam-6566	146	98	)	)	PUNCT
ejpam-6566	146	99	)	)	PUNCT
ejpam-6566	147	1	for	for	ADP
ejpam-6566	147	2	every	every	DET
ejpam-6566	147	3	σ1σ2	σ1σ2	NOUN
ejpam-6566	147	4	-	-	ADJ
ejpam-6566	147	5	open	open	ADJ
ejpam-6566	147	6	set	set	NOUN
ejpam-6566	147	7	v	v	NOUN
ejpam-6566	147	8	of	of	ADP
ejpam-6566	147	9	y	y	PRON
ejpam-6566	148	1	such	such	ADJ
ejpam-6566	148	2	that	that	PRON
ejpam-6566	148	3	v	v	ADP
ejpam-6566	148	4	∩	∩	ADJ
ejpam-6566	148	5	f	f	X
ejpam-6566	148	6	(	(	PUNCT
ejpam-6566	148	7	x	x	X
ejpam-6566	148	8	)	)	PUNCT
ejpam-6566	148	9	̸=	̸=	NOUN
ejpam-6566	148	10	∅	∅	NOUN
ejpam-6566	148	11	;	;	PUNCT
ejpam-6566	148	12	(	(	PUNCT
ejpam-6566	148	13	3	3	X
ejpam-6566	148	14	)	)	PUNCT
ejpam-6566	148	15	x	x	SYM
ejpam-6566	148	16	∈	∈	NOUN
ejpam-6566	148	17	int⋆(f−((σ1	int⋆(f−((σ1	NOUN
ejpam-6566	148	18	,	,	PUNCT
ejpam-6566	148	19	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6566	148	20	)	)	PUNCT
ejpam-6566	148	21	)	)	PUNCT
ejpam-6566	148	22	)	)	PUNCT
ejpam-6566	149	1	for	for	ADP
ejpam-6566	149	2	every	every	DET
ejpam-6566	149	3	σ1σ2	σ1σ2	NOUN
ejpam-6566	149	4	-	-	ADJ
ejpam-6566	149	5	open	open	ADJ
ejpam-6566	149	6	set	set	NOUN
ejpam-6566	149	7	v	v	NOUN
ejpam-6566	149	8	of	of	ADP
ejpam-6566	149	9	y	y	PRON
ejpam-6566	149	10	such	such	ADJ
ejpam-6566	149	11	that	that	PRON
ejpam-6566	149	12	v	v	ADP
ejpam-6566	149	13	∩	∩	ADJ
ejpam-6566	149	14	f	f	X
ejpam-6566	149	15	(	(	PUNCT
ejpam-6566	149	16	x	x	X
ejpam-6566	149	17	)	)	PUNCT
ejpam-6566	149	18	̸=	̸=	PROPN
ejpam-6566	149	19	∅	∅	NOUN
ejpam-6566	149	20	;	;	PUNCT
ejpam-6566	149	21	(	(	PUNCT
ejpam-6566	149	22	4	4	X
ejpam-6566	149	23	)	)	PUNCT
ejpam-6566	149	24	x	x	SYM
ejpam-6566	149	25	∈	∈	PROPN
ejpam-6566	149	26	int⋆(f−(v	int⋆(f−(v	PUNCT
ejpam-6566	149	27	)	)	PUNCT
ejpam-6566	149	28	)	)	PUNCT
ejpam-6566	150	1	for	for	SCONJ
ejpam-6566	150	2	every	every	DET
ejpam-6566	150	3	(	(	PUNCT
ejpam-6566	150	4	σ1	σ1	PROPN
ejpam-6566	150	5	,	,	PUNCT
ejpam-6566	150	6	σ2)r	σ2)r	NOUN
ejpam-6566	150	7	-	-	PUNCT
ejpam-6566	150	8	open	open	ADJ
ejpam-6566	150	9	set	set	VERB
ejpam-6566	150	10	v	v	NOUN
ejpam-6566	150	11	of	of	ADP
ejpam-6566	150	12	y	y	PRON
ejpam-6566	150	13	such	such	ADJ
ejpam-6566	150	14	that	that	PRON
ejpam-6566	150	15	v	v	ADP
ejpam-6566	150	16	∩	∩	ADJ
ejpam-6566	150	17	f	f	X
ejpam-6566	150	18	(	(	PUNCT
ejpam-6566	150	19	x	x	X
ejpam-6566	150	20	)	)	PUNCT
ejpam-6566	150	21	̸=	̸=	NOUN
ejpam-6566	150	22	∅	∅	NOUN
ejpam-6566	150	23	;	;	PUNCT
ejpam-6566	150	24	(	(	PUNCT
ejpam-6566	150	25	5	5	X
ejpam-6566	150	26	)	)	PUNCT
ejpam-6566	150	27	for	for	ADP
ejpam-6566	150	28	each	each	DET
ejpam-6566	150	29	(	(	PUNCT
ejpam-6566	150	30	σ1	σ1	PROPN
ejpam-6566	150	31	,	,	PUNCT
ejpam-6566	150	32	σ2)r	σ2)r	NOUN
ejpam-6566	150	33	-	-	PUNCT
ejpam-6566	150	34	open	open	ADJ
ejpam-6566	150	35	set	set	VERB
ejpam-6566	150	36	v	v	NOUN
ejpam-6566	150	37	of	of	ADP
ejpam-6566	150	38	y	y	PRON
ejpam-6566	150	39	such	such	ADJ
ejpam-6566	150	40	that	that	PRON
ejpam-6566	150	41	v	v	ADP
ejpam-6566	150	42	∩	∩	ADJ
ejpam-6566	150	43	f	f	X
ejpam-6566	150	44	(	(	PUNCT
ejpam-6566	150	45	x	x	X
ejpam-6566	150	46	)	)	PUNCT
ejpam-6566	150	47	̸=	̸=	NOUN
ejpam-6566	150	48	∅	∅	NOUN
ejpam-6566	150	49	,	,	PUNCT
ejpam-6566	150	50	there	there	PRON
ejpam-6566	150	51	exists	exist	VERB
ejpam-6566	150	52	a	a	DET
ejpam-6566	150	53	⋆-open	⋆-open	ADJ
ejpam-6566	150	54	set	set	NOUN
ejpam-6566	150	55	u	u	NOUN
ejpam-6566	150	56	of	of	ADP
ejpam-6566	150	57	x	x	PUNCT
ejpam-6566	150	58	containing	contain	VERB
ejpam-6566	150	59	x	x	PUNCT
ejpam-6566	150	60	such	such	ADJ
ejpam-6566	150	61	that	that	SCONJ
ejpam-6566	150	62	u	u	NOUN
ejpam-6566	150	63	⊆	⊆	NUM
ejpam-6566	150	64	f−(v	f−(v	NOUN
ejpam-6566	150	65	)	)	PUNCT
ejpam-6566	150	66	.	.	PUNCT
ejpam-6566	151	1	proof	proof	NOUN
ejpam-6566	151	2	.	.	PUNCT
ejpam-6566	152	1	the	the	DET
ejpam-6566	152	2	proof	proof	NOUN
ejpam-6566	152	3	is	be	AUX
ejpam-6566	152	4	similar	similar	ADJ
ejpam-6566	152	5	to	to	ADP
ejpam-6566	152	6	that	that	PRON
ejpam-6566	152	7	of	of	ADP
ejpam-6566	152	8	theorem	theorem	ADJ
ejpam-6566	152	9	1	1	NUM
ejpam-6566	152	10	.	.	PUNCT
ejpam-6566	152	11	theorem	theorem	NOUN
ejpam-6566	152	12	3	3	NUM
ejpam-6566	152	13	.	.	X
ejpam-6566	152	14	for	for	ADP
ejpam-6566	152	15	a	a	DET
ejpam-6566	152	16	multifunction	multifunction	NOUN
ejpam-6566	152	17	f	f	NOUN
ejpam-6566	152	18	:	:	PUNCT
ejpam-6566	152	19	(	(	PUNCT
ejpam-6566	152	20	x	x	X
ejpam-6566	152	21	,	,	PUNCT
ejpam-6566	152	22	τ	τ	PROPN
ejpam-6566	152	23	,	,	PUNCT
ejpam-6566	152	24	i	i	NOUN
ejpam-6566	152	25	)	)	PUNCT
ejpam-6566	152	26	→	→	PUNCT
ejpam-6566	152	27	(	(	PUNCT
ejpam-6566	152	28	y	y	PROPN
ejpam-6566	152	29	,	,	PUNCT
ejpam-6566	152	30	σ1	σ1	PROPN
ejpam-6566	152	31	,	,	PUNCT
ejpam-6566	152	32	σ2	σ2	NOUN
ejpam-6566	152	33	)	)	PUNCT
ejpam-6566	152	34	,	,	PUNCT
ejpam-6566	152	35	the	the	DET
ejpam-6566	152	36	following	follow	VERB
ejpam-6566	152	37	properties	property	NOUN
ejpam-6566	152	38	are	be	AUX
ejpam-6566	152	39	equivalent	equivalent	ADJ
ejpam-6566	152	40	:	:	PUNCT
ejpam-6566	152	41	(	(	PUNCT
ejpam-6566	152	42	1	1	X
ejpam-6566	152	43	)	)	PUNCT
ejpam-6566	152	44	f	f	PROPN
ejpam-6566	152	45	is	be	AUX
ejpam-6566	152	46	upper	upper	ADJ
ejpam-6566	152	47	almost	almost	ADV
ejpam-6566	152	48	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	152	49	,	,	PUNCT
ejpam-6566	152	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	152	51	;	;	PUNCT
ejpam-6566	152	52	(	(	PUNCT
ejpam-6566	152	53	2	2	NUM
ejpam-6566	152	54	)	)	PUNCT
ejpam-6566	152	55	f+(v	f+(v	NOUN
ejpam-6566	152	56	)	)	PUNCT
ejpam-6566	153	1	⊆	⊆	X
ejpam-6566	153	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	153	3	-	-	PUNCT
ejpam-6566	153	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	153	5	-	-	PUNCT
ejpam-6566	153	6	cl(v	cl(v	NOUN
ejpam-6566	153	7	)	)	PUNCT
ejpam-6566	153	8	)	)	PUNCT
ejpam-6566	153	9	)	)	PUNCT
ejpam-6566	153	10	)	)	PUNCT
ejpam-6566	153	11	for	for	ADP
ejpam-6566	153	12	every	every	DET
ejpam-6566	153	13	σ1σ2	σ1σ2	NOUN
ejpam-6566	153	14	-	-	ADJ
ejpam-6566	153	15	open	open	ADJ
ejpam-6566	153	16	set	set	NOUN
ejpam-6566	153	17	v	v	NOUN
ejpam-6566	153	18	of	of	ADP
ejpam-6566	153	19	y	y	PROPN
ejpam-6566	153	20	;	;	PUNCT
ejpam-6566	153	21	(	(	PUNCT
ejpam-6566	153	22	3	3	X
ejpam-6566	153	23	)	)	PUNCT
ejpam-6566	153	24	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	153	25	-	-	PUNCT
ejpam-6566	153	26	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	153	27	-	-	PUNCT
ejpam-6566	153	28	int(k	int(k	NOUN
ejpam-6566	153	29	)	)	PUNCT
ejpam-6566	153	30	)	)	PUNCT
ejpam-6566	153	31	)	)	PUNCT
ejpam-6566	153	32	)	)	PUNCT
ejpam-6566	154	1	⊆	⊆	X
ejpam-6566	154	2	f−(k	f−(k	PROPN
ejpam-6566	154	3	)	)	PUNCT
ejpam-6566	154	4	for	for	ADP
ejpam-6566	154	5	every	every	DET
ejpam-6566	154	6	σ1σ2	σ1σ2	NUM
ejpam-6566	154	7	-	-	PUNCT
ejpam-6566	154	8	closed	closed	ADJ
ejpam-6566	154	9	set	set	NOUN
ejpam-6566	154	10	k	k	PROPN
ejpam-6566	154	11	of	of	ADP
ejpam-6566	154	12	y	y	PROPN
ejpam-6566	154	13	;	;	PUNCT
ejpam-6566	154	14	c.	c.	PROPN
ejpam-6566	154	15	viriyapong	viriyapong	PROPN
ejpam-6566	154	16	,	,	PUNCT
ejpam-6566	154	17	a.	a.	PROPN
ejpam-6566	154	18	sama	sama	PROPN
ejpam-6566	154	19	-	-	PUNCT
ejpam-6566	154	20	ae	ae	PROPN
ejpam-6566	154	21	,	,	PUNCT
ejpam-6566	154	22	c.	c.	PROPN
ejpam-6566	154	23	boonpok	boonpok	PROPN
ejpam-6566	154	24	/	/	SYM
ejpam-6566	154	25	eur	eur	PROPN
ejpam-6566	154	26	.	.	PUNCT
ejpam-6566	155	1	j.	j.	PROPN
ejpam-6566	155	2	pure	pure	PROPN
ejpam-6566	155	3	appl	appl	PROPN
ejpam-6566	155	4	.	.	PROPN
ejpam-6566	155	5	math	math	PROPN
ejpam-6566	155	6	,	,	PUNCT
ejpam-6566	155	7	18	18	NUM
ejpam-6566	155	8	(	(	PUNCT
ejpam-6566	155	9	3	3	NUM
ejpam-6566	155	10	)	)	PUNCT
ejpam-6566	155	11	(	(	PUNCT
ejpam-6566	155	12	2025	2025	NUM
ejpam-6566	155	13	)	)	PUNCT
ejpam-6566	155	14	,	,	PUNCT
ejpam-6566	155	15	6566	6566	NUM
ejpam-6566	155	16	6	6	NUM
ejpam-6566	155	17	of	of	ADP
ejpam-6566	155	18	14	14	NUM
ejpam-6566	155	19	(	(	PUNCT
ejpam-6566	155	20	4	4	NUM
ejpam-6566	155	21	)	)	PUNCT
ejpam-6566	155	22	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	155	23	-	-	PUNCT
ejpam-6566	155	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	155	25	-	-	PUNCT
ejpam-6566	155	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	155	27	-	-	PUNCT
ejpam-6566	155	28	cl(b	cl(b	NOUN
ejpam-6566	155	29	)	)	PUNCT
ejpam-6566	155	30	)	)	PUNCT
ejpam-6566	155	31	)	)	PUNCT
ejpam-6566	155	32	)	)	PUNCT
ejpam-6566	155	33	)	)	PUNCT
ejpam-6566	156	1	⊆	⊆	X
ejpam-6566	156	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	156	3	-	-	PUNCT
ejpam-6566	156	4	cl(b	cl(b	NOUN
ejpam-6566	156	5	)	)	PUNCT
ejpam-6566	156	6	)	)	PUNCT
ejpam-6566	157	1	for	for	ADP
ejpam-6566	157	2	every	every	DET
ejpam-6566	157	3	subset	subset	NOUN
ejpam-6566	157	4	b	b	PROPN
ejpam-6566	157	5	of	of	ADP
ejpam-6566	157	6	y	y	PROPN
ejpam-6566	157	7	;	;	PUNCT
ejpam-6566	157	8	(	(	PUNCT
ejpam-6566	157	9	5	5	X
ejpam-6566	157	10	)	)	PUNCT
ejpam-6566	157	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	157	12	-	-	PUNCT
ejpam-6566	157	13	int(b	int(b	NOUN
ejpam-6566	157	14	)	)	PUNCT
ejpam-6566	157	15	)	)	PUNCT
ejpam-6566	157	16	⊆	⊆	X
ejpam-6566	157	17	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	157	18	-	-	PUNCT
ejpam-6566	157	19	int(σ1σ2	int(σ1σ2	ADV
ejpam-6566	157	20	-	-	PUNCT
ejpam-6566	157	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	157	22	-	-	PUNCT
ejpam-6566	157	23	int(b	int(b	NOUN
ejpam-6566	157	24	)	)	PUNCT
ejpam-6566	157	25	)	)	PUNCT
ejpam-6566	157	26	)	)	PUNCT
ejpam-6566	157	27	)	)	PUNCT
ejpam-6566	157	28	)	)	PUNCT
ejpam-6566	157	29	for	for	ADP
ejpam-6566	157	30	every	every	DET
ejpam-6566	157	31	subset	subset	NOUN
ejpam-6566	157	32	b	b	PROPN
ejpam-6566	157	33	of	of	ADP
ejpam-6566	157	34	y	y	PROPN
ejpam-6566	157	35	;	;	PUNCT
ejpam-6566	157	36	(	(	PUNCT
ejpam-6566	157	37	6	6	NUM
ejpam-6566	157	38	)	)	PUNCT
ejpam-6566	157	39	f+(v	f+(v	NOUN
ejpam-6566	157	40	)	)	PUNCT
ejpam-6566	157	41	is	be	AUX
ejpam-6566	157	42	⋆-open	⋆-open	ADJ
ejpam-6566	157	43	in	in	ADP
ejpam-6566	157	44	x	x	PUNCT
ejpam-6566	157	45	for	for	ADP
ejpam-6566	157	46	every	every	DET
ejpam-6566	157	47	(	(	PUNCT
ejpam-6566	157	48	σ1	σ1	PROPN
ejpam-6566	157	49	,	,	PUNCT
ejpam-6566	157	50	σ2)r	σ2)r	NOUN
ejpam-6566	157	51	-	-	PUNCT
ejpam-6566	157	52	open	open	ADJ
ejpam-6566	157	53	set	set	VERB
ejpam-6566	157	54	v	v	NOUN
ejpam-6566	157	55	of	of	ADP
ejpam-6566	157	56	y	y	PROPN
ejpam-6566	157	57	;	;	PUNCT
ejpam-6566	157	58	(	(	PUNCT
ejpam-6566	157	59	7	7	X
ejpam-6566	157	60	)	)	PUNCT
ejpam-6566	157	61	f−(k	f−(k	PROPN
ejpam-6566	157	62	)	)	PUNCT
ejpam-6566	157	63	is	be	AUX
ejpam-6566	157	64	⋆-closed	⋆-close	VERB
ejpam-6566	157	65	in	in	ADP
ejpam-6566	157	66	x	x	PUNCT
ejpam-6566	157	67	for	for	ADP
ejpam-6566	157	68	every	every	DET
ejpam-6566	157	69	(	(	PUNCT
ejpam-6566	157	70	σ1	σ1	PROPN
ejpam-6566	157	71	,	,	PUNCT
ejpam-6566	157	72	σ2)r	σ2)r	NOUN
ejpam-6566	157	73	-	-	PUNCT
ejpam-6566	157	74	closed	close	VERB
ejpam-6566	157	75	set	set	ADJ
ejpam-6566	157	76	k	k	PROPN
ejpam-6566	157	77	of	of	ADP
ejpam-6566	157	78	y	y	PROPN
ejpam-6566	157	79	.	.	PUNCT
ejpam-6566	158	1	proof	proof	NOUN
ejpam-6566	158	2	.	.	PUNCT
ejpam-6566	159	1	(	(	PUNCT
ejpam-6566	159	2	1	1	X
ejpam-6566	159	3	)	)	PUNCT
ejpam-6566	159	4	⇒	⇒	NOUN
ejpam-6566	159	5	(	(	PUNCT
ejpam-6566	159	6	2	2	NUM
ejpam-6566	159	7	):	):	PUNCT
ejpam-6566	159	8	let	let	VERB
ejpam-6566	159	9	v	v	PART
ejpam-6566	159	10	be	be	AUX
ejpam-6566	159	11	any	any	DET
ejpam-6566	159	12	σ1σ2	σ1σ2	NOUN
ejpam-6566	159	13	-	-	ADJ
ejpam-6566	159	14	open	open	ADJ
ejpam-6566	159	15	set	set	NOUN
ejpam-6566	159	16	of	of	ADP
ejpam-6566	159	17	y	y	PROPN
ejpam-6566	159	18	and	and	CCONJ
ejpam-6566	159	19	x	x	PROPN
ejpam-6566	159	20	∈	∈	PROPN
ejpam-6566	159	21	f+(v	f+(v	NOUN
ejpam-6566	159	22	)	)	PUNCT
ejpam-6566	159	23	.	.	PUNCT
ejpam-6566	160	1	then	then	ADV
ejpam-6566	160	2	,	,	PUNCT
ejpam-6566	160	3	f	f	PROPN
ejpam-6566	160	4	(	(	PUNCT
ejpam-6566	160	5	x	x	X
ejpam-6566	160	6	)	)	PUNCT
ejpam-6566	160	7	⊆	⊆	NUM
ejpam-6566	160	8	v	v	NOUN
ejpam-6566	160	9	.	.	PUNCT
ejpam-6566	161	1	thus	thus	ADV
ejpam-6566	161	2	by	by	ADP
ejpam-6566	161	3	theorem	theorem	NOUN
ejpam-6566	161	4	1	1	NUM
ejpam-6566	161	5	,	,	PUNCT
ejpam-6566	161	6	we	we	PRON
ejpam-6566	161	7	have	have	VERB
ejpam-6566	161	8	x	x	PART
ejpam-6566	161	9	∈	∈	VERB
ejpam-6566	161	10	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	161	11	-	-	PUNCT
ejpam-6566	161	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	161	13	-	-	PUNCT
ejpam-6566	161	14	cl(v	cl(v	NOUN
ejpam-6566	161	15	)	)	PUNCT
ejpam-6566	161	16	)	)	PUNCT
ejpam-6566	161	17	)	)	PUNCT
ejpam-6566	161	18	)	)	PUNCT
ejpam-6566	161	19	and	and	CCONJ
ejpam-6566	161	20	hence	hence	ADV
ejpam-6566	161	21	f+(v	f+(v	NOUN
ejpam-6566	161	22	)	)	PUNCT
ejpam-6566	162	1	⊆	⊆	X
ejpam-6566	162	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	162	3	-	-	PUNCT
ejpam-6566	162	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	162	5	-	-	PUNCT
ejpam-6566	162	6	cl(v	cl(v	NOUN
ejpam-6566	162	7	)	)	PUNCT
ejpam-6566	162	8	)	)	PUNCT
ejpam-6566	162	9	)	)	PUNCT
ejpam-6566	162	10	)	)	PUNCT
ejpam-6566	162	11	.	.	PUNCT
ejpam-6566	163	1	(	(	PUNCT
ejpam-6566	163	2	2	2	X
ejpam-6566	163	3	)	)	PUNCT
ejpam-6566	163	4	⇒	⇒	NOUN
ejpam-6566	163	5	(	(	PUNCT
ejpam-6566	163	6	3	3	NUM
ejpam-6566	163	7	):	):	PUNCT
ejpam-6566	163	8	let	let	VERB
ejpam-6566	163	9	k	k	PRON
ejpam-6566	163	10	be	be	AUX
ejpam-6566	163	11	any	any	DET
ejpam-6566	163	12	σ1σ2	σ1σ2	NUM
ejpam-6566	163	13	-	-	PUNCT
ejpam-6566	163	14	closed	closed	ADJ
ejpam-6566	163	15	set	set	NOUN
ejpam-6566	163	16	of	of	ADP
ejpam-6566	163	17	y	y	PROPN
ejpam-6566	163	18	.	.	PUNCT
ejpam-6566	164	1	then	then	ADV
ejpam-6566	164	2	,	,	PUNCT
ejpam-6566	164	3	y	y	PROPN
ejpam-6566	164	4	−k	−k	PROPN
ejpam-6566	164	5	is	be	AUX
ejpam-6566	164	6	σ1σ2	σ1σ2	NOUN
ejpam-6566	164	7	-	-	ADJ
ejpam-6566	164	8	open	open	ADJ
ejpam-6566	164	9	in	in	ADP
ejpam-6566	164	10	y	y	PROPN
ejpam-6566	164	11	and	and	CCONJ
ejpam-6566	164	12	by	by	ADP
ejpam-6566	164	13	(	(	PUNCT
ejpam-6566	164	14	2	2	NUM
ejpam-6566	164	15	)	)	PUNCT
ejpam-6566	164	16	,	,	PUNCT
ejpam-6566	164	17	x	x	PUNCT
ejpam-6566	164	18	−	−	DET
ejpam-6566	164	19	f−(k	f−(k	PROPN
ejpam-6566	164	20	)	)	PUNCT
ejpam-6566	164	21	=	=	PUNCT
ejpam-6566	164	22	f+(y	f+(y	PROPN
ejpam-6566	164	23	−k	−k	PROPN
ejpam-6566	164	24	)	)	PUNCT
ejpam-6566	164	25	⊆	⊆	NUM
ejpam-6566	164	26	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	164	27	-	-	PUNCT
ejpam-6566	164	28	int(σ1σ2	int(σ1σ2	VERB
ejpam-6566	164	29	-	-	PUNCT
ejpam-6566	164	30	cl(y	cl(y	NOUN
ejpam-6566	164	31	−k	−k	NOUN
ejpam-6566	164	32	)	)	PUNCT
ejpam-6566	164	33	)	)	PUNCT
ejpam-6566	164	34	)	)	PUNCT
ejpam-6566	164	35	)	)	PUNCT
ejpam-6566	165	1	=	=	PRON
ejpam-6566	165	2	int⋆(x	int⋆(x	NOUN
ejpam-6566	165	3	−	−	NOUN
ejpam-6566	165	4	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6566	165	5	-	-	PUNCT
ejpam-6566	165	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	165	7	-	-	PUNCT
ejpam-6566	165	8	int(k	int(k	NOUN
ejpam-6566	165	9	)	)	PUNCT
ejpam-6566	165	10	)	)	PUNCT
ejpam-6566	165	11	)	)	PUNCT
ejpam-6566	165	12	)	)	PUNCT
ejpam-6566	166	1	=	=	PUNCT
ejpam-6566	166	2	x	x	X
ejpam-6566	166	3	−	−	ADP
ejpam-6566	166	4	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	166	5	-	-	PUNCT
ejpam-6566	166	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	166	7	-	-	PUNCT
ejpam-6566	166	8	int(k	int(k	NOUN
ejpam-6566	166	9	)	)	PUNCT
ejpam-6566	166	10	)	)	PUNCT
ejpam-6566	166	11	)	)	PUNCT
ejpam-6566	166	12	)	)	PUNCT
ejpam-6566	166	13	.	.	PUNCT
ejpam-6566	167	1	thus	thus	ADV
ejpam-6566	167	2	,	,	PUNCT
ejpam-6566	167	3	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	167	4	-	-	PUNCT
ejpam-6566	167	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	167	6	-	-	PUNCT
ejpam-6566	167	7	int(k	int(k	NOUN
ejpam-6566	167	8	)	)	PUNCT
ejpam-6566	167	9	)	)	PUNCT
ejpam-6566	167	10	)	)	PUNCT
ejpam-6566	167	11	)	)	PUNCT
ejpam-6566	168	1	⊆	⊆	NUM
ejpam-6566	168	2	f−(k	f−(k	PROPN
ejpam-6566	168	3	)	)	PUNCT
ejpam-6566	168	4	.	.	PUNCT
ejpam-6566	169	1	(	(	PUNCT
ejpam-6566	169	2	3	3	X
ejpam-6566	169	3	)	)	PUNCT
ejpam-6566	169	4	⇒	⇒	NOUN
ejpam-6566	169	5	(	(	PUNCT
ejpam-6566	169	6	4	4	NUM
ejpam-6566	169	7	):	):	PUNCT
ejpam-6566	169	8	let	let	VERB
ejpam-6566	169	9	b	b	X
ejpam-6566	169	10	be	be	AUX
ejpam-6566	169	11	any	any	DET
ejpam-6566	169	12	subset	subset	NOUN
ejpam-6566	169	13	of	of	ADP
ejpam-6566	169	14	y	y	PROPN
ejpam-6566	169	15	.	.	PUNCT
ejpam-6566	170	1	then	then	ADV
ejpam-6566	170	2	,	,	PUNCT
ejpam-6566	170	3	σ1σ2	σ1σ2	NOUN
ejpam-6566	170	4	-	-	NOUN
ejpam-6566	170	5	cl(b	cl(b	NOUN
ejpam-6566	170	6	)	)	PUNCT
ejpam-6566	170	7	is	be	AUX
ejpam-6566	170	8	a	a	DET
ejpam-6566	170	9	σ1σ2	σ1σ2	NUM
ejpam-6566	170	10	-	-	PUNCT
ejpam-6566	170	11	closed	closed	ADJ
ejpam-6566	170	12	set	set	NOUN
ejpam-6566	170	13	of	of	ADP
ejpam-6566	170	14	y	y	PROPN
ejpam-6566	170	15	and	and	CCONJ
ejpam-6566	170	16	by	by	ADP
ejpam-6566	170	17	(	(	PUNCT
ejpam-6566	170	18	3	3	NUM
ejpam-6566	170	19	)	)	PUNCT
ejpam-6566	170	20	,	,	PUNCT
ejpam-6566	170	21	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	170	22	-	-	PUNCT
ejpam-6566	170	23	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	170	24	-	-	PUNCT
ejpam-6566	170	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	170	26	-	-	PUNCT
ejpam-6566	170	27	cl(b	cl(b	NOUN
ejpam-6566	170	28	)	)	PUNCT
ejpam-6566	170	29	)	)	PUNCT
ejpam-6566	170	30	)	)	PUNCT
ejpam-6566	170	31	)	)	PUNCT
ejpam-6566	170	32	)	)	PUNCT
ejpam-6566	171	1	⊆	⊆	X
ejpam-6566	171	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	171	3	-	-	PUNCT
ejpam-6566	171	4	cl(b	cl(b	NOUN
ejpam-6566	171	5	)	)	PUNCT
ejpam-6566	171	6	)	)	PUNCT
ejpam-6566	171	7	.	.	PUNCT
ejpam-6566	172	1	(	(	PUNCT
ejpam-6566	172	2	4	4	X
ejpam-6566	172	3	)	)	PUNCT
ejpam-6566	172	4	⇒	⇒	NOUN
ejpam-6566	172	5	(	(	PUNCT
ejpam-6566	172	6	5	5	NUM
ejpam-6566	172	7	):	):	PUNCT
ejpam-6566	172	8	let	let	VERB
ejpam-6566	172	9	b	b	X
ejpam-6566	172	10	be	be	AUX
ejpam-6566	172	11	any	any	DET
ejpam-6566	172	12	subset	subset	NOUN
ejpam-6566	172	13	of	of	ADP
ejpam-6566	172	14	y	y	PROPN
ejpam-6566	172	15	.	.	PUNCT
ejpam-6566	173	1	thus	thus	ADV
ejpam-6566	173	2	by	by	ADP
ejpam-6566	173	3	(	(	PUNCT
ejpam-6566	173	4	4	4	NUM
ejpam-6566	173	5	)	)	PUNCT
ejpam-6566	173	6	,	,	PUNCT
ejpam-6566	173	7	we	we	PRON
ejpam-6566	173	8	have	have	VERB
ejpam-6566	173	9	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	173	10	-	-	PUNCT
ejpam-6566	173	11	int(b	int(b	NOUN
ejpam-6566	173	12	)	)	PUNCT
ejpam-6566	173	13	)	)	PUNCT
ejpam-6566	174	1	=	=	PUNCT
ejpam-6566	175	1	x	x	PUNCT
ejpam-6566	175	2	−	−	ADP
ejpam-6566	175	3	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	175	4	-	-	PUNCT
ejpam-6566	175	5	cl(y	cl(y	NOUN
ejpam-6566	175	6	−b	−b	NOUN
ejpam-6566	175	7	)	)	PUNCT
ejpam-6566	175	8	)	)	PUNCT
ejpam-6566	176	1	⊆	⊆	NUM
ejpam-6566	176	2	x	x	SYM
ejpam-6566	176	3	−	−	NOUN
ejpam-6566	176	4	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	176	5	-	-	PUNCT
ejpam-6566	176	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	176	7	-	-	PUNCT
ejpam-6566	176	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	176	9	-	-	PUNCT
ejpam-6566	176	10	cl(y	cl(y	NOUN
ejpam-6566	176	11	−b	−b	NOUN
ejpam-6566	176	12	)	)	PUNCT
ejpam-6566	176	13	)	)	PUNCT
ejpam-6566	176	14	)	)	PUNCT
ejpam-6566	176	15	)	)	PUNCT
ejpam-6566	176	16	)	)	PUNCT
ejpam-6566	177	1	=	=	PUNCT
ejpam-6566	177	2	x	x	SYM
ejpam-6566	177	3	−	−	ADP
ejpam-6566	177	4	cl⋆(f−(y	cl⋆(f−(y	NUM
ejpam-6566	177	5	−	−	NUM
ejpam-6566	177	6	σ1σ2	σ1σ2	SYM
ejpam-6566	177	7	-	-	PUNCT
ejpam-6566	177	8	int(σ1σ2	int(σ1σ2	ADV
ejpam-6566	177	9	-	-	PUNCT
ejpam-6566	177	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	177	11	-	-	PUNCT
ejpam-6566	177	12	int(b	int(b	NOUN
ejpam-6566	177	13	)	)	PUNCT
ejpam-6566	177	14	)	)	PUNCT
ejpam-6566	177	15	)	)	PUNCT
ejpam-6566	177	16	)	)	PUNCT
ejpam-6566	177	17	)	)	PUNCT
ejpam-6566	178	1	=	=	SYM
ejpam-6566	178	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	178	3	-	-	PUNCT
ejpam-6566	178	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-6566	178	5	-	-	PUNCT
ejpam-6566	178	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	178	7	-	-	PUNCT
ejpam-6566	178	8	int(b	int(b	NOUN
ejpam-6566	178	9	)	)	PUNCT
ejpam-6566	178	10	)	)	PUNCT
ejpam-6566	178	11	)	)	PUNCT
ejpam-6566	178	12	)	)	PUNCT
ejpam-6566	178	13	)	)	PUNCT
ejpam-6566	178	14	.	.	PUNCT
ejpam-6566	179	1	(	(	PUNCT
ejpam-6566	179	2	5	5	X
ejpam-6566	179	3	)	)	PUNCT
ejpam-6566	179	4	⇒	⇒	NOUN
ejpam-6566	179	5	(	(	PUNCT
ejpam-6566	179	6	6	6	NUM
ejpam-6566	179	7	):	):	PUNCT
ejpam-6566	179	8	let	let	VERB
ejpam-6566	179	9	v	v	PART
ejpam-6566	179	10	be	be	AUX
ejpam-6566	179	11	any	any	DET
ejpam-6566	179	12	(	(	PUNCT
ejpam-6566	179	13	σ1	σ1	NOUN
ejpam-6566	179	14	,	,	PUNCT
ejpam-6566	179	15	σ2)r	σ2)r	NOUN
ejpam-6566	179	16	-	-	PUNCT
ejpam-6566	179	17	open	open	ADJ
ejpam-6566	179	18	set	set	NOUN
ejpam-6566	179	19	of	of	ADP
ejpam-6566	179	20	y	y	PROPN
ejpam-6566	179	21	.	.	PUNCT
ejpam-6566	180	1	by	by	ADP
ejpam-6566	180	2	(	(	PUNCT
ejpam-6566	180	3	5	5	NUM
ejpam-6566	180	4	)	)	PUNCT
ejpam-6566	180	5	,	,	PUNCT
ejpam-6566	180	6	we	we	PRON
ejpam-6566	180	7	have	have	VERB
ejpam-6566	180	8	f+(v	f+(v	NOUN
ejpam-6566	180	9	)	)	PUNCT
ejpam-6566	181	1	⊆	⊆	NUM
ejpam-6566	181	2	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6566	181	3	)	)	PUNCT
ejpam-6566	181	4	)	)	PUNCT
ejpam-6566	181	5	and	and	CCONJ
ejpam-6566	181	6	hence	hence	ADV
ejpam-6566	181	7	f+(v	f+(v	PROPN
ejpam-6566	181	8	)	)	PUNCT
ejpam-6566	181	9	is	be	AUX
ejpam-6566	181	10	⋆-open	⋆-open	ADJ
ejpam-6566	181	11	in	in	ADP
ejpam-6566	181	12	x.	x.	PROPN
ejpam-6566	181	13	(	(	PUNCT
ejpam-6566	181	14	6	6	NUM
ejpam-6566	181	15	)	)	PUNCT
ejpam-6566	181	16	⇒	⇒	NOUN
ejpam-6566	181	17	(	(	PUNCT
ejpam-6566	181	18	7	7	NUM
ejpam-6566	181	19	):	):	PUNCT
ejpam-6566	181	20	the	the	DET
ejpam-6566	181	21	proof	proof	NOUN
ejpam-6566	181	22	is	be	AUX
ejpam-6566	181	23	obvious	obvious	ADJ
ejpam-6566	181	24	.	.	PUNCT
ejpam-6566	182	1	(	(	PUNCT
ejpam-6566	182	2	7	7	X
ejpam-6566	182	3	)	)	PUNCT
ejpam-6566	182	4	⇒	⇒	NOUN
ejpam-6566	182	5	(	(	PUNCT
ejpam-6566	182	6	1	1	NUM
ejpam-6566	182	7	):	):	PUNCT
ejpam-6566	182	8	let	let	VERB
ejpam-6566	182	9	x	x	PUNCT
ejpam-6566	182	10	∈	∈	PROPN
ejpam-6566	182	11	x	x	X
ejpam-6566	182	12	and	and	CCONJ
ejpam-6566	182	13	v	v	AUX
ejpam-6566	182	14	be	be	AUX
ejpam-6566	182	15	any	any	DET
ejpam-6566	182	16	(	(	PUNCT
ejpam-6566	182	17	σ1	σ1	NOUN
ejpam-6566	182	18	,	,	PUNCT
ejpam-6566	182	19	σ2)r	σ2)r	NOUN
ejpam-6566	182	20	-	-	PUNCT
ejpam-6566	182	21	open	open	ADJ
ejpam-6566	182	22	set	set	NOUN
ejpam-6566	182	23	of	of	ADP
ejpam-6566	182	24	y	y	PROPN
ejpam-6566	182	25	containing	contain	VERB
ejpam-6566	182	26	f	f	PROPN
ejpam-6566	182	27	(	(	PUNCT
ejpam-6566	182	28	x	x	NOUN
ejpam-6566	182	29	)	)	PUNCT
ejpam-6566	182	30	.	.	PUNCT
ejpam-6566	183	1	since	since	SCONJ
ejpam-6566	183	2	y	y	PROPN
ejpam-6566	183	3	−	−	PROPN
ejpam-6566	183	4	v	v	NOUN
ejpam-6566	183	5	is	be	AUX
ejpam-6566	183	6	(	(	PUNCT
ejpam-6566	183	7	σ1	σ1	NOUN
ejpam-6566	183	8	,	,	PUNCT
ejpam-6566	183	9	σ2)r	σ2)r	NOUN
ejpam-6566	183	10	-	-	PUNCT
ejpam-6566	183	11	closed	closed	ADJ
ejpam-6566	183	12	and	and	CCONJ
ejpam-6566	183	13	by	by	ADP
ejpam-6566	183	14	(	(	PUNCT
ejpam-6566	183	15	7	7	NUM
ejpam-6566	183	16	)	)	PUNCT
ejpam-6566	183	17	,	,	PUNCT
ejpam-6566	183	18	x	x	PUNCT
ejpam-6566	183	19	−	−	NOUN
ejpam-6566	183	20	f+(v	f+(v	NOUN
ejpam-6566	183	21	)	)	PUNCT
ejpam-6566	184	1	=	=	PUNCT
ejpam-6566	184	2	f−(y	f−(y	NOUN
ejpam-6566	184	3	−	−	NOUN
ejpam-6566	184	4	v	v	NOUN
ejpam-6566	184	5	)	)	PUNCT
ejpam-6566	184	6	is	be	AUX
ejpam-6566	184	7	⋆-closed	⋆-close	VERB
ejpam-6566	184	8	in	in	ADP
ejpam-6566	184	9	x.	x.	PROPN
ejpam-6566	184	10	thus	thus	ADV
ejpam-6566	184	11	,	,	PUNCT
ejpam-6566	184	12	f+(v	f+(v	PROPN
ejpam-6566	184	13	)	)	PUNCT
ejpam-6566	185	1	is	be	AUX
ejpam-6566	185	2	⋆-open	⋆-open	ADJ
ejpam-6566	185	3	and	and	CCONJ
ejpam-6566	185	4	hence	hence	ADV
ejpam-6566	185	5	x	x	X
ejpam-6566	185	6	∈	∈	PROPN
ejpam-6566	185	7	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6566	185	8	)	)	PUNCT
ejpam-6566	185	9	)	)	PUNCT
ejpam-6566	185	10	.	.	PUNCT
ejpam-6566	186	1	then	then	ADV
ejpam-6566	186	2	,	,	PUNCT
ejpam-6566	186	3	there	there	PRON
ejpam-6566	186	4	exists	exist	VERB
ejpam-6566	186	5	a	a	DET
ejpam-6566	186	6	⋆-open	⋆-open	ADJ
ejpam-6566	186	7	set	set	NOUN
ejpam-6566	186	8	u	u	NOUN
ejpam-6566	186	9	of	of	ADP
ejpam-6566	186	10	x	x	PUNCT
ejpam-6566	186	11	containing	contain	VERB
ejpam-6566	186	12	x	x	PUNCT
ejpam-6566	186	13	such	such	ADJ
ejpam-6566	186	14	that	that	SCONJ
ejpam-6566	186	15	f	f	PROPN
ejpam-6566	186	16	(	(	PUNCT
ejpam-6566	186	17	u	u	NOUN
ejpam-6566	186	18	)	)	PUNCT
ejpam-6566	186	19	⊆	⊆	NUM
ejpam-6566	186	20	v	v	NOUN
ejpam-6566	186	21	.	.	PUNCT
ejpam-6566	187	1	it	it	PRON
ejpam-6566	187	2	follows	follow	VERB
ejpam-6566	187	3	from	from	ADP
ejpam-6566	187	4	theorem	theorem	ADJ
ejpam-6566	187	5	1	1	NUM
ejpam-6566	187	6	that	that	SCONJ
ejpam-6566	187	7	f	f	PROPN
ejpam-6566	187	8	is	be	AUX
ejpam-6566	187	9	upper	upper	ADJ
ejpam-6566	187	10	almost	almost	ADV
ejpam-6566	187	11	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	187	12	,	,	PUNCT
ejpam-6566	187	13	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	187	14	.	.	X
ejpam-6566	187	15	theorem	theorem	NOUN
ejpam-6566	187	16	4	4	NUM
ejpam-6566	187	17	.	.	X
ejpam-6566	187	18	for	for	ADP
ejpam-6566	187	19	a	a	DET
ejpam-6566	187	20	multifunction	multifunction	NOUN
ejpam-6566	187	21	f	f	NOUN
ejpam-6566	187	22	:	:	PUNCT
ejpam-6566	187	23	(	(	PUNCT
ejpam-6566	187	24	x	x	X
ejpam-6566	187	25	,	,	PUNCT
ejpam-6566	187	26	τ	τ	PROPN
ejpam-6566	187	27	,	,	PUNCT
ejpam-6566	187	28	i	i	NOUN
ejpam-6566	187	29	)	)	PUNCT
ejpam-6566	187	30	→	→	PUNCT
ejpam-6566	187	31	(	(	PUNCT
ejpam-6566	187	32	y	y	PROPN
ejpam-6566	187	33	,	,	PUNCT
ejpam-6566	187	34	σ1	σ1	PROPN
ejpam-6566	187	35	,	,	PUNCT
ejpam-6566	187	36	σ2	σ2	NOUN
ejpam-6566	187	37	)	)	PUNCT
ejpam-6566	187	38	,	,	PUNCT
ejpam-6566	187	39	the	the	DET
ejpam-6566	187	40	following	follow	VERB
ejpam-6566	187	41	properties	property	NOUN
ejpam-6566	187	42	are	be	AUX
ejpam-6566	187	43	equivalent	equivalent	ADJ
ejpam-6566	187	44	:	:	PUNCT
ejpam-6566	187	45	(	(	PUNCT
ejpam-6566	187	46	1	1	X
ejpam-6566	187	47	)	)	PUNCT
ejpam-6566	187	48	f	f	PROPN
ejpam-6566	187	49	is	be	AUX
ejpam-6566	187	50	lower	low	ADJ
ejpam-6566	187	51	almost	almost	ADV
ejpam-6566	187	52	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	187	53	,	,	PUNCT
ejpam-6566	187	54	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	187	55	;	;	PUNCT
ejpam-6566	187	56	(	(	PUNCT
ejpam-6566	187	57	2	2	X
ejpam-6566	187	58	)	)	PUNCT
ejpam-6566	187	59	f−(v	f−(v	NOUN
ejpam-6566	187	60	)	)	PUNCT
ejpam-6566	188	1	⊆	⊆	NUM
ejpam-6566	188	2	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	NOUN
ejpam-6566	188	3	-	-	PUNCT
ejpam-6566	188	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	188	5	-	-	PUNCT
ejpam-6566	188	6	cl(v	cl(v	NOUN
ejpam-6566	188	7	)	)	PUNCT
ejpam-6566	188	8	)	)	PUNCT
ejpam-6566	188	9	)	)	PUNCT
ejpam-6566	188	10	)	)	PUNCT
ejpam-6566	188	11	for	for	ADP
ejpam-6566	188	12	every	every	DET
ejpam-6566	188	13	σ1σ2	σ1σ2	NOUN
ejpam-6566	188	14	-	-	ADJ
ejpam-6566	188	15	open	open	ADJ
ejpam-6566	188	16	set	set	NOUN
ejpam-6566	188	17	v	v	NOUN
ejpam-6566	188	18	of	of	ADP
ejpam-6566	188	19	y	y	PROPN
ejpam-6566	188	20	;	;	PUNCT
ejpam-6566	188	21	c.	c.	PROPN
ejpam-6566	188	22	viriyapong	viriyapong	PROPN
ejpam-6566	188	23	,	,	PUNCT
ejpam-6566	188	24	a.	a.	PROPN
ejpam-6566	188	25	sama	sama	PROPN
ejpam-6566	188	26	-	-	PUNCT
ejpam-6566	188	27	ae	ae	PROPN
ejpam-6566	188	28	,	,	PUNCT
ejpam-6566	188	29	c.	c.	PROPN
ejpam-6566	188	30	boonpok	boonpok	PROPN
ejpam-6566	188	31	/	/	SYM
ejpam-6566	188	32	eur	eur	PROPN
ejpam-6566	188	33	.	.	PUNCT
ejpam-6566	189	1	j.	j.	PROPN
ejpam-6566	189	2	pure	pure	PROPN
ejpam-6566	189	3	appl	appl	PROPN
ejpam-6566	189	4	.	.	PROPN
ejpam-6566	189	5	math	math	PROPN
ejpam-6566	189	6	,	,	PUNCT
ejpam-6566	189	7	18	18	NUM
ejpam-6566	189	8	(	(	PUNCT
ejpam-6566	189	9	3	3	NUM
ejpam-6566	189	10	)	)	PUNCT
ejpam-6566	189	11	(	(	PUNCT
ejpam-6566	189	12	2025	2025	NUM
ejpam-6566	189	13	)	)	PUNCT
ejpam-6566	189	14	,	,	PUNCT
ejpam-6566	189	15	6566	6566	NUM
ejpam-6566	189	16	7	7	NUM
ejpam-6566	189	17	of	of	ADP
ejpam-6566	189	18	14	14	NUM
ejpam-6566	189	19	(	(	PUNCT
ejpam-6566	189	20	3	3	NUM
ejpam-6566	189	21	)	)	PUNCT
ejpam-6566	189	22	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	189	23	-	-	PUNCT
ejpam-6566	189	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	189	25	-	-	PUNCT
ejpam-6566	189	26	int(k	int(k	NOUN
ejpam-6566	189	27	)	)	PUNCT
ejpam-6566	189	28	)	)	PUNCT
ejpam-6566	189	29	)	)	PUNCT
ejpam-6566	189	30	)	)	PUNCT
ejpam-6566	190	1	⊆	⊆	NUM
ejpam-6566	190	2	f+(k	f+(k	NOUN
ejpam-6566	190	3	)	)	PUNCT
ejpam-6566	190	4	for	for	ADP
ejpam-6566	190	5	every	every	DET
ejpam-6566	190	6	σ1σ2	σ1σ2	NUM
ejpam-6566	190	7	-	-	PUNCT
ejpam-6566	190	8	closed	closed	ADJ
ejpam-6566	190	9	set	set	NOUN
ejpam-6566	190	10	k	k	PROPN
ejpam-6566	190	11	of	of	ADP
ejpam-6566	190	12	y	y	PROPN
ejpam-6566	190	13	;	;	PUNCT
ejpam-6566	190	14	(	(	PUNCT
ejpam-6566	190	15	4	4	X
ejpam-6566	190	16	)	)	PUNCT
ejpam-6566	190	17	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	190	18	-	-	PUNCT
ejpam-6566	190	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	190	20	-	-	PUNCT
ejpam-6566	190	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	190	22	-	-	PUNCT
ejpam-6566	190	23	cl(b	cl(b	NOUN
ejpam-6566	190	24	)	)	PUNCT
ejpam-6566	190	25	)	)	PUNCT
ejpam-6566	190	26	)	)	PUNCT
ejpam-6566	190	27	)	)	PUNCT
ejpam-6566	190	28	)	)	PUNCT
ejpam-6566	191	1	⊆	⊆	X
ejpam-6566	191	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	191	3	-	-	PUNCT
ejpam-6566	191	4	cl(b	cl(b	NOUN
ejpam-6566	191	5	)	)	PUNCT
ejpam-6566	191	6	)	)	PUNCT
ejpam-6566	191	7	for	for	ADP
ejpam-6566	191	8	every	every	DET
ejpam-6566	191	9	subset	subset	NOUN
ejpam-6566	191	10	b	b	PROPN
ejpam-6566	191	11	of	of	ADP
ejpam-6566	191	12	y	y	PROPN
ejpam-6566	191	13	;	;	PUNCT
ejpam-6566	191	14	(	(	PUNCT
ejpam-6566	191	15	5	5	X
ejpam-6566	191	16	)	)	PUNCT
ejpam-6566	191	17	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	191	18	-	-	PUNCT
ejpam-6566	191	19	int(b	int(b	NOUN
ejpam-6566	191	20	)	)	PUNCT
ejpam-6566	191	21	)	)	PUNCT
ejpam-6566	192	1	⊆	⊆	NUM
ejpam-6566	192	2	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	NOUN
ejpam-6566	192	3	-	-	PUNCT
ejpam-6566	192	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-6566	192	5	-	-	PUNCT
ejpam-6566	192	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	192	7	-	-	PUNCT
ejpam-6566	192	8	int(b	int(b	NOUN
ejpam-6566	192	9	)	)	PUNCT
ejpam-6566	192	10	)	)	PUNCT
ejpam-6566	192	11	)	)	PUNCT
ejpam-6566	192	12	)	)	PUNCT
ejpam-6566	192	13	)	)	PUNCT
ejpam-6566	192	14	for	for	ADP
ejpam-6566	192	15	every	every	DET
ejpam-6566	192	16	subset	subset	NOUN
ejpam-6566	192	17	b	b	PROPN
ejpam-6566	192	18	of	of	ADP
ejpam-6566	192	19	y	y	PROPN
ejpam-6566	192	20	;	;	PUNCT
ejpam-6566	192	21	(	(	PUNCT
ejpam-6566	192	22	6	6	X
ejpam-6566	192	23	)	)	PUNCT
ejpam-6566	192	24	f−(v	f−(v	NOUN
ejpam-6566	192	25	)	)	PUNCT
ejpam-6566	192	26	is	be	AUX
ejpam-6566	192	27	⋆-open	⋆-open	ADJ
ejpam-6566	192	28	in	in	ADP
ejpam-6566	192	29	x	x	PUNCT
ejpam-6566	192	30	for	for	ADP
ejpam-6566	192	31	every	every	DET
ejpam-6566	192	32	(	(	PUNCT
ejpam-6566	192	33	σ1	σ1	PROPN
ejpam-6566	192	34	,	,	PUNCT
ejpam-6566	192	35	σ2)r	σ2)r	NOUN
ejpam-6566	192	36	-	-	PUNCT
ejpam-6566	192	37	open	open	ADJ
ejpam-6566	192	38	set	set	VERB
ejpam-6566	192	39	v	v	NOUN
ejpam-6566	192	40	of	of	ADP
ejpam-6566	192	41	y	y	PROPN
ejpam-6566	192	42	;	;	PUNCT
ejpam-6566	192	43	(	(	PUNCT
ejpam-6566	192	44	7	7	X
ejpam-6566	192	45	)	)	PUNCT
ejpam-6566	192	46	f+(k	f+(k	NUM
ejpam-6566	192	47	)	)	PUNCT
ejpam-6566	192	48	is	be	AUX
ejpam-6566	192	49	⋆-closed	⋆-close	VERB
ejpam-6566	192	50	in	in	ADP
ejpam-6566	192	51	x	x	PUNCT
ejpam-6566	192	52	for	for	ADP
ejpam-6566	192	53	every	every	DET
ejpam-6566	192	54	(	(	PUNCT
ejpam-6566	192	55	σ1	σ1	PROPN
ejpam-6566	192	56	,	,	PUNCT
ejpam-6566	192	57	σ2)r	σ2)r	NOUN
ejpam-6566	192	58	-	-	PUNCT
ejpam-6566	192	59	closed	close	VERB
ejpam-6566	192	60	set	set	ADJ
ejpam-6566	192	61	k	k	PROPN
ejpam-6566	192	62	of	of	ADP
ejpam-6566	192	63	y	y	PROPN
ejpam-6566	192	64	.	.	PUNCT
ejpam-6566	193	1	proof	proof	NOUN
ejpam-6566	193	2	.	.	PUNCT
ejpam-6566	194	1	the	the	DET
ejpam-6566	194	2	proof	proof	NOUN
ejpam-6566	194	3	is	be	AUX
ejpam-6566	194	4	similar	similar	ADJ
ejpam-6566	194	5	to	to	ADP
ejpam-6566	194	6	that	that	PRON
ejpam-6566	194	7	of	of	ADP
ejpam-6566	194	8	theorem	theorem	ADJ
ejpam-6566	194	9	3	3	NUM
ejpam-6566	194	10	.	.	PUNCT
ejpam-6566	194	11	theorem	theorem	NOUN
ejpam-6566	194	12	5	5	NUM
ejpam-6566	194	13	.	.	X
ejpam-6566	194	14	for	for	ADP
ejpam-6566	194	15	a	a	DET
ejpam-6566	194	16	multifunction	multifunction	NOUN
ejpam-6566	194	17	f	f	NOUN
ejpam-6566	194	18	:	:	PUNCT
ejpam-6566	194	19	(	(	PUNCT
ejpam-6566	194	20	x	x	X
ejpam-6566	194	21	,	,	PUNCT
ejpam-6566	194	22	τ	τ	PROPN
ejpam-6566	194	23	,	,	PUNCT
ejpam-6566	194	24	i	i	NOUN
ejpam-6566	194	25	)	)	PUNCT
ejpam-6566	194	26	→	→	PUNCT
ejpam-6566	194	27	(	(	PUNCT
ejpam-6566	194	28	y	y	PROPN
ejpam-6566	194	29	,	,	PUNCT
ejpam-6566	194	30	σ1	σ1	PROPN
ejpam-6566	194	31	,	,	PUNCT
ejpam-6566	194	32	σ2	σ2	NOUN
ejpam-6566	194	33	)	)	PUNCT
ejpam-6566	194	34	,	,	PUNCT
ejpam-6566	194	35	the	the	DET
ejpam-6566	194	36	following	follow	VERB
ejpam-6566	194	37	properties	property	NOUN
ejpam-6566	194	38	are	be	AUX
ejpam-6566	194	39	equivalent	equivalent	ADJ
ejpam-6566	194	40	:	:	PUNCT
ejpam-6566	194	41	(	(	PUNCT
ejpam-6566	194	42	1	1	X
ejpam-6566	194	43	)	)	PUNCT
ejpam-6566	194	44	f	f	PROPN
ejpam-6566	194	45	is	be	AUX
ejpam-6566	194	46	upper	upper	ADJ
ejpam-6566	194	47	almost	almost	ADV
ejpam-6566	194	48	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	194	49	,	,	PUNCT
ejpam-6566	194	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	194	51	;	;	PUNCT
ejpam-6566	194	52	(	(	PUNCT
ejpam-6566	194	53	2	2	X
ejpam-6566	194	54	)	)	SYM
ejpam-6566	194	55	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6566	194	56	)	)	PUNCT
ejpam-6566	194	57	)	)	PUNCT
ejpam-6566	195	1	⊆	⊆	X
ejpam-6566	195	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	195	3	-	-	PUNCT
ejpam-6566	195	4	cl(v	cl(v	NOUN
ejpam-6566	195	5	)	)	PUNCT
ejpam-6566	195	6	)	)	PUNCT
ejpam-6566	195	7	for	for	ADP
ejpam-6566	195	8	every	every	DET
ejpam-6566	195	9	(	(	PUNCT
ejpam-6566	195	10	σ1	σ1	PROPN
ejpam-6566	195	11	,	,	PUNCT
ejpam-6566	195	12	σ2)β	σ2)β	NOUN
ejpam-6566	195	13	-	-	PUNCT
ejpam-6566	195	14	open	open	NOUN
ejpam-6566	195	15	set	set	NOUN
ejpam-6566	195	16	v	v	NOUN
ejpam-6566	195	17	of	of	ADP
ejpam-6566	195	18	y	y	PROPN
ejpam-6566	195	19	;	;	PUNCT
ejpam-6566	195	20	(	(	PUNCT
ejpam-6566	195	21	3	3	X
ejpam-6566	195	22	)	)	SYM
ejpam-6566	195	23	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6566	195	24	)	)	PUNCT
ejpam-6566	195	25	)	)	PUNCT
ejpam-6566	196	1	⊆	⊆	X
ejpam-6566	196	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	196	3	-	-	PUNCT
ejpam-6566	196	4	cl(v	cl(v	NOUN
ejpam-6566	196	5	)	)	PUNCT
ejpam-6566	196	6	)	)	PUNCT
ejpam-6566	196	7	for	for	ADP
ejpam-6566	196	8	every	every	DET
ejpam-6566	196	9	(	(	PUNCT
ejpam-6566	196	10	σ1	σ1	PROPN
ejpam-6566	196	11	,	,	PUNCT
ejpam-6566	196	12	σ2)s	σ2)s	NOUN
ejpam-6566	196	13	-	-	PUNCT
ejpam-6566	196	14	open	open	NOUN
ejpam-6566	196	15	set	set	NOUN
ejpam-6566	196	16	v	v	NOUN
ejpam-6566	196	17	of	of	ADP
ejpam-6566	196	18	y	y	PROPN
ejpam-6566	196	19	.	.	PUNCT
ejpam-6566	197	1	proof	proof	NOUN
ejpam-6566	197	2	.	.	PUNCT
ejpam-6566	198	1	(	(	PUNCT
ejpam-6566	198	2	1	1	X
ejpam-6566	198	3	)	)	PUNCT
ejpam-6566	198	4	⇒	⇒	NOUN
ejpam-6566	198	5	(	(	PUNCT
ejpam-6566	198	6	2	2	NUM
ejpam-6566	198	7	):	):	PUNCT
ejpam-6566	198	8	let	let	VERB
ejpam-6566	198	9	v	v	PART
ejpam-6566	198	10	be	be	AUX
ejpam-6566	198	11	any	any	DET
ejpam-6566	198	12	(	(	PUNCT
ejpam-6566	198	13	σ1	σ1	PROPN
ejpam-6566	198	14	,	,	PUNCT
ejpam-6566	198	15	σ2)β	σ2)β	NOUN
ejpam-6566	198	16	-	-	PUNCT
ejpam-6566	198	17	open	open	ADJ
ejpam-6566	198	18	set	set	NOUN
ejpam-6566	198	19	of	of	ADP
ejpam-6566	198	20	y	y	PROPN
ejpam-6566	198	21	.	.	PUNCT
ejpam-6566	199	1	then	then	ADV
ejpam-6566	199	2	,	,	PUNCT
ejpam-6566	199	3	σ1σ2	σ1σ2	NOUN
ejpam-6566	199	4	-	-	NUM
ejpam-6566	199	5	cl(v	cl(v	NOUN
ejpam-6566	199	6	)	)	PUNCT
ejpam-6566	199	7	is	be	AUX
ejpam-6566	199	8	a	a	DET
ejpam-6566	199	9	(	(	PUNCT
ejpam-6566	199	10	σ1	σ1	NOUN
ejpam-6566	199	11	,	,	PUNCT
ejpam-6566	199	12	σ2)r	σ2)r	NOUN
ejpam-6566	199	13	-	-	PUNCT
ejpam-6566	199	14	closed	close	VERB
ejpam-6566	199	15	set	set	NOUN
ejpam-6566	199	16	of	of	ADP
ejpam-6566	199	17	y	y	PROPN
ejpam-6566	199	18	.	.	PUNCT
ejpam-6566	200	1	since	since	SCONJ
ejpam-6566	200	2	f	f	PROPN
ejpam-6566	200	3	is	be	AUX
ejpam-6566	200	4	upper	upper	ADJ
ejpam-6566	200	5	almost	almost	ADV
ejpam-6566	200	6	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	200	7	,	,	PUNCT
ejpam-6566	200	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	200	9	and	and	CCONJ
ejpam-6566	200	10	by	by	ADP
ejpam-6566	200	11	theorem	theorem	ADJ
ejpam-6566	200	12	3	3	NUM
ejpam-6566	200	13	,	,	PUNCT
ejpam-6566	200	14	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	200	15	-	-	PUNCT
ejpam-6566	200	16	cl(v	cl(v	NOUN
ejpam-6566	200	17	)	)	PUNCT
ejpam-6566	200	18	)	)	PUNCT
ejpam-6566	200	19	is	be	AUX
ejpam-6566	200	20	⋆-closed	⋆-close	VERB
ejpam-6566	200	21	in	in	ADP
ejpam-6566	200	22	x.	x.	NOUN
ejpam-6566	200	23	thus	thus	ADV
ejpam-6566	200	24	,	,	PUNCT
ejpam-6566	200	25	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6566	200	26	)	)	PUNCT
ejpam-6566	200	27	)	)	PUNCT
ejpam-6566	201	1	⊆	⊆	NUM
ejpam-6566	201	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	201	3	-	-	PUNCT
ejpam-6566	201	4	cl(v	cl(v	NOUN
ejpam-6566	201	5	)	)	PUNCT
ejpam-6566	201	6	)	)	PUNCT
ejpam-6566	201	7	.	.	PUNCT
ejpam-6566	202	1	(	(	PUNCT
ejpam-6566	202	2	2	2	X
ejpam-6566	202	3	)	)	PUNCT
ejpam-6566	202	4	⇒	⇒	NOUN
ejpam-6566	202	5	(	(	PUNCT
ejpam-6566	202	6	3	3	NUM
ejpam-6566	202	7	):	):	PUNCT
ejpam-6566	202	8	the	the	DET
ejpam-6566	202	9	proof	proof	NOUN
ejpam-6566	202	10	is	be	AUX
ejpam-6566	202	11	obvious	obvious	ADJ
ejpam-6566	202	12	.	.	PUNCT
ejpam-6566	203	1	(	(	PUNCT
ejpam-6566	203	2	3	3	X
ejpam-6566	203	3	)	)	PUNCT
ejpam-6566	203	4	⇒	⇒	NOUN
ejpam-6566	203	5	(	(	PUNCT
ejpam-6566	203	6	1	1	NUM
ejpam-6566	203	7	):	):	PUNCT
ejpam-6566	203	8	let	let	VERB
ejpam-6566	203	9	k	k	PRON
ejpam-6566	203	10	be	be	AUX
ejpam-6566	203	11	any	any	DET
ejpam-6566	203	12	(	(	PUNCT
ejpam-6566	203	13	σ1	σ1	NOUN
ejpam-6566	203	14	,	,	PUNCT
ejpam-6566	203	15	σ2)r	σ2)r	NOUN
ejpam-6566	203	16	-	-	PUNCT
ejpam-6566	203	17	closed	close	VERB
ejpam-6566	203	18	set	set	NOUN
ejpam-6566	203	19	of	of	ADP
ejpam-6566	203	20	y	y	PROPN
ejpam-6566	203	21	.	.	PUNCT
ejpam-6566	204	1	then	then	ADV
ejpam-6566	204	2	,	,	PUNCT
ejpam-6566	204	3	k	k	X
ejpam-6566	204	4	is	be	AUX
ejpam-6566	204	5	(	(	PUNCT
ejpam-6566	204	6	σ1	σ1	PROPN
ejpam-6566	204	7	,	,	PUNCT
ejpam-6566	204	8	σ2)s	σ2)s	NOUN
ejpam-6566	204	9	-	-	PUNCT
ejpam-6566	204	10	open	open	ADJ
ejpam-6566	204	11	in	in	ADP
ejpam-6566	204	12	y	y	PROPN
ejpam-6566	204	13	.	.	PUNCT
ejpam-6566	205	1	then	then	ADV
ejpam-6566	205	2	by	by	ADP
ejpam-6566	205	3	(	(	PUNCT
ejpam-6566	205	4	3	3	NUM
ejpam-6566	205	5	)	)	PUNCT
ejpam-6566	205	6	,	,	PUNCT
ejpam-6566	205	7	cl⋆(f−(k	cl⋆(f−(k	NUM
ejpam-6566	205	8	)	)	PUNCT
ejpam-6566	205	9	)	)	PUNCT
ejpam-6566	206	1	⊆	⊆	X
ejpam-6566	206	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	206	3	-	-	PUNCT
ejpam-6566	206	4	cl(k	cl(k	NOUN
ejpam-6566	206	5	)	)	PUNCT
ejpam-6566	206	6	)	)	PUNCT
ejpam-6566	207	1	=	=	SYM
ejpam-6566	207	2	f−(k	f−(k	PROPN
ejpam-6566	207	3	)	)	PUNCT
ejpam-6566	207	4	and	and	CCONJ
ejpam-6566	207	5	hence	hence	ADV
ejpam-6566	207	6	f−(k	f−(k	PROPN
ejpam-6566	207	7	)	)	PUNCT
ejpam-6566	207	8	is	be	AUX
ejpam-6566	207	9	⋆-closed	⋆-close	VERB
ejpam-6566	207	10	in	in	ADP
ejpam-6566	207	11	x.	x.	NOUN
ejpam-6566	207	12	by	by	ADP
ejpam-6566	207	13	theorem	theorem	NOUN
ejpam-6566	207	14	3	3	NUM
ejpam-6566	207	15	,	,	PUNCT
ejpam-6566	207	16	f	f	PROPN
ejpam-6566	207	17	is	be	AUX
ejpam-6566	207	18	upper	upper	ADJ
ejpam-6566	207	19	almost	almost	ADV
ejpam-6566	207	20	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	207	21	,	,	PUNCT
ejpam-6566	207	22	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	207	23	.	.	X
ejpam-6566	207	24	theorem	theorem	VERB
ejpam-6566	207	25	6	6	NUM
ejpam-6566	207	26	.	.	PUNCT
ejpam-6566	207	27	for	for	ADP
ejpam-6566	207	28	a	a	DET
ejpam-6566	207	29	multifunction	multifunction	NOUN
ejpam-6566	207	30	f	f	NOUN
ejpam-6566	207	31	:	:	PUNCT
ejpam-6566	207	32	(	(	PUNCT
ejpam-6566	207	33	x	x	X
ejpam-6566	207	34	,	,	PUNCT
ejpam-6566	207	35	τ	τ	PROPN
ejpam-6566	207	36	,	,	PUNCT
ejpam-6566	207	37	i	i	NOUN
ejpam-6566	207	38	)	)	PUNCT
ejpam-6566	207	39	→	→	PUNCT
ejpam-6566	207	40	(	(	PUNCT
ejpam-6566	207	41	y	y	PROPN
ejpam-6566	207	42	,	,	PUNCT
ejpam-6566	207	43	σ1	σ1	PROPN
ejpam-6566	207	44	,	,	PUNCT
ejpam-6566	207	45	σ2	σ2	NOUN
ejpam-6566	207	46	)	)	PUNCT
ejpam-6566	207	47	,	,	PUNCT
ejpam-6566	207	48	the	the	DET
ejpam-6566	207	49	following	follow	VERB
ejpam-6566	207	50	properties	property	NOUN
ejpam-6566	207	51	are	be	AUX
ejpam-6566	207	52	equivalent	equivalent	ADJ
ejpam-6566	207	53	:	:	PUNCT
ejpam-6566	207	54	(	(	PUNCT
ejpam-6566	207	55	1	1	X
ejpam-6566	207	56	)	)	PUNCT
ejpam-6566	207	57	f	f	PROPN
ejpam-6566	207	58	is	be	AUX
ejpam-6566	207	59	lower	low	ADJ
ejpam-6566	207	60	almost	almost	ADV
ejpam-6566	207	61	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	207	62	,	,	PUNCT
ejpam-6566	207	63	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	207	64	;	;	PUNCT
ejpam-6566	207	65	(	(	PUNCT
ejpam-6566	207	66	2	2	X
ejpam-6566	207	67	)	)	PUNCT
ejpam-6566	207	68	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-6566	207	69	)	)	PUNCT
ejpam-6566	207	70	)	)	PUNCT
ejpam-6566	208	1	⊆	⊆	X
ejpam-6566	208	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	208	3	-	-	PUNCT
ejpam-6566	208	4	cl(v	cl(v	NOUN
ejpam-6566	208	5	)	)	PUNCT
ejpam-6566	208	6	)	)	PUNCT
ejpam-6566	208	7	for	for	ADP
ejpam-6566	208	8	every	every	DET
ejpam-6566	208	9	(	(	PUNCT
ejpam-6566	208	10	σ1	σ1	PROPN
ejpam-6566	208	11	,	,	PUNCT
ejpam-6566	208	12	σ2)β	σ2)β	NOUN
ejpam-6566	208	13	-	-	PUNCT
ejpam-6566	208	14	open	open	NOUN
ejpam-6566	208	15	set	set	NOUN
ejpam-6566	208	16	v	v	NOUN
ejpam-6566	208	17	of	of	ADP
ejpam-6566	208	18	y	y	PROPN
ejpam-6566	208	19	;	;	PUNCT
ejpam-6566	208	20	(	(	PUNCT
ejpam-6566	208	21	3	3	X
ejpam-6566	208	22	)	)	PUNCT
ejpam-6566	208	23	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-6566	208	24	)	)	PUNCT
ejpam-6566	208	25	)	)	PUNCT
ejpam-6566	209	1	⊆	⊆	X
ejpam-6566	209	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	209	3	-	-	PUNCT
ejpam-6566	209	4	cl(v	cl(v	NOUN
ejpam-6566	209	5	)	)	PUNCT
ejpam-6566	209	6	)	)	PUNCT
ejpam-6566	209	7	for	for	SCONJ
ejpam-6566	209	8	every	every	DET
ejpam-6566	209	9	(	(	PUNCT
ejpam-6566	209	10	σ1	σ1	PROPN
ejpam-6566	209	11	,	,	PUNCT
ejpam-6566	209	12	σ2)s	σ2)s	NOUN
ejpam-6566	209	13	-	-	PUNCT
ejpam-6566	209	14	open	open	NOUN
ejpam-6566	209	15	set	set	NOUN
ejpam-6566	209	16	v	v	NOUN
ejpam-6566	209	17	of	of	ADP
ejpam-6566	209	18	y	y	PROPN
ejpam-6566	209	19	.	.	PUNCT
ejpam-6566	210	1	proof	proof	NOUN
ejpam-6566	210	2	.	.	PUNCT
ejpam-6566	211	1	the	the	DET
ejpam-6566	211	2	proof	proof	NOUN
ejpam-6566	211	3	is	be	AUX
ejpam-6566	211	4	similar	similar	ADJ
ejpam-6566	211	5	to	to	ADP
ejpam-6566	211	6	that	that	PRON
ejpam-6566	211	7	of	of	ADP
ejpam-6566	211	8	theorem	theorem	NOUN
ejpam-6566	211	9	5	5	NUM
ejpam-6566	211	10	.	.	PUNCT
ejpam-6566	211	11	lemma	lemma	PROPN
ejpam-6566	211	12	3	3	X
ejpam-6566	211	13	.	.	PUNCT
ejpam-6566	212	1	[	[	X
ejpam-6566	212	2	28	28	NUM
ejpam-6566	212	3	]	]	PUNCT
ejpam-6566	212	4	for	for	ADP
ejpam-6566	212	5	a	a	DET
ejpam-6566	212	6	bitopological	bitopological	ADJ
ejpam-6566	212	7	space	space	NOUN
ejpam-6566	212	8	(	(	PUNCT
ejpam-6566	212	9	x	x	NOUN
ejpam-6566	212	10	,	,	PUNCT
ejpam-6566	212	11	τ1	τ1	NOUN
ejpam-6566	212	12	,	,	PUNCT
ejpam-6566	212	13	τ2	τ2	NOUN
ejpam-6566	212	14	)	)	PUNCT
ejpam-6566	212	15	,	,	PUNCT
ejpam-6566	212	16	the	the	DET
ejpam-6566	212	17	following	follow	VERB
ejpam-6566	212	18	properties	property	NOUN
ejpam-6566	212	19	hold	hold	VERB
ejpam-6566	212	20	:	:	PUNCT
ejpam-6566	212	21	(	(	PUNCT
ejpam-6566	212	22	1	1	X
ejpam-6566	212	23	)	)	PUNCT
ejpam-6566	212	24	α(τ1	α(τ1	NOUN
ejpam-6566	212	25	,	,	PUNCT
ejpam-6566	212	26	τ2)-cl(v	τ2)-cl(v	NOUN
ejpam-6566	212	27	)	)	PUNCT
ejpam-6566	212	28	=	=	PUNCT
ejpam-6566	213	1	τ1τ2	τ1τ2	NOUN
ejpam-6566	213	2	-	-	NOUN
ejpam-6566	213	3	cl(v	cl(v	X
ejpam-6566	213	4	)	)	PUNCT
ejpam-6566	213	5	for	for	ADP
ejpam-6566	213	6	every	every	DET
ejpam-6566	213	7	(	(	PUNCT
ejpam-6566	213	8	τ1	τ1	NOUN
ejpam-6566	213	9	,	,	PUNCT
ejpam-6566	213	10	τ2)β	τ2)β	ADJ
ejpam-6566	213	11	-	-	PUNCT
ejpam-6566	213	12	open	open	NOUN
ejpam-6566	213	13	set	set	NOUN
ejpam-6566	213	14	v	v	NOUN
ejpam-6566	213	15	of	of	ADP
ejpam-6566	213	16	x	x	PRON
ejpam-6566	213	17	;	;	PUNCT
ejpam-6566	213	18	(	(	PUNCT
ejpam-6566	213	19	2	2	X
ejpam-6566	213	20	)	)	PUNCT
ejpam-6566	213	21	(	(	PUNCT
ejpam-6566	213	22	τ1	τ1	NOUN
ejpam-6566	213	23	,	,	PUNCT
ejpam-6566	213	24	τ2)-pcl(v	τ2)-pcl(v	NOUN
ejpam-6566	213	25	)	)	PUNCT
ejpam-6566	213	26	=	=	PUNCT
ejpam-6566	214	1	τ1τ2	τ1τ2	NOUN
ejpam-6566	214	2	-	-	NOUN
ejpam-6566	214	3	cl(v	cl(v	X
ejpam-6566	214	4	)	)	PUNCT
ejpam-6566	214	5	for	for	ADP
ejpam-6566	214	6	every	every	DET
ejpam-6566	214	7	(	(	PUNCT
ejpam-6566	214	8	τ1	τ1	NOUN
ejpam-6566	214	9	,	,	PUNCT
ejpam-6566	214	10	τ2)s	τ2)s	NOUN
ejpam-6566	214	11	-	-	PUNCT
ejpam-6566	214	12	open	open	ADJ
ejpam-6566	214	13	set	set	NOUN
ejpam-6566	214	14	v	v	NOUN
ejpam-6566	214	15	of	of	ADP
ejpam-6566	214	16	x.	x.	PROPN
ejpam-6566	214	17	c.	c.	PROPN
ejpam-6566	214	18	viriyapong	viriyapong	PROPN
ejpam-6566	214	19	,	,	PUNCT
ejpam-6566	214	20	a.	a.	PROPN
ejpam-6566	214	21	sama	sama	PROPN
ejpam-6566	214	22	-	-	PUNCT
ejpam-6566	214	23	ae	ae	PROPN
ejpam-6566	214	24	,	,	PUNCT
ejpam-6566	214	25	c.	c.	PROPN
ejpam-6566	214	26	boonpok	boonpok	PROPN
ejpam-6566	214	27	/	/	SYM
ejpam-6566	214	28	eur	eur	PROPN
ejpam-6566	214	29	.	.	PUNCT
ejpam-6566	215	1	j.	j.	PROPN
ejpam-6566	215	2	pure	pure	PROPN
ejpam-6566	215	3	appl	appl	PROPN
ejpam-6566	215	4	.	.	PROPN
ejpam-6566	215	5	math	math	PROPN
ejpam-6566	215	6	,	,	PUNCT
ejpam-6566	215	7	18	18	NUM
ejpam-6566	215	8	(	(	PUNCT
ejpam-6566	215	9	3	3	NUM
ejpam-6566	215	10	)	)	PUNCT
ejpam-6566	215	11	(	(	PUNCT
ejpam-6566	215	12	2025	2025	NUM
ejpam-6566	215	13	)	)	PUNCT
ejpam-6566	215	14	,	,	PUNCT
ejpam-6566	215	15	6566	6566	NUM
ejpam-6566	215	16	8	8	NUM
ejpam-6566	215	17	of	of	ADP
ejpam-6566	215	18	14	14	NUM
ejpam-6566	215	19	corollary	corollary	ADJ
ejpam-6566	215	20	1	1	NUM
ejpam-6566	215	21	.	.	PUNCT
ejpam-6566	216	1	for	for	ADP
ejpam-6566	216	2	a	a	DET
ejpam-6566	216	3	multifunction	multifunction	NOUN
ejpam-6566	216	4	f	f	NOUN
ejpam-6566	216	5	:	:	PUNCT
ejpam-6566	216	6	(	(	PUNCT
ejpam-6566	216	7	x	x	X
ejpam-6566	216	8	,	,	PUNCT
ejpam-6566	216	9	τ	τ	PROPN
ejpam-6566	216	10	,	,	PUNCT
ejpam-6566	216	11	i	i	NOUN
ejpam-6566	216	12	)	)	PUNCT
ejpam-6566	216	13	→	→	PUNCT
ejpam-6566	216	14	(	(	PUNCT
ejpam-6566	216	15	y	y	PROPN
ejpam-6566	216	16	,	,	PUNCT
ejpam-6566	216	17	σ1	σ1	PROPN
ejpam-6566	216	18	,	,	PUNCT
ejpam-6566	216	19	σ2	σ2	NOUN
ejpam-6566	216	20	)	)	PUNCT
ejpam-6566	216	21	,	,	PUNCT
ejpam-6566	216	22	the	the	DET
ejpam-6566	216	23	following	follow	VERB
ejpam-6566	216	24	properties	property	NOUN
ejpam-6566	216	25	are	be	AUX
ejpam-6566	216	26	equivalent	equivalent	ADJ
ejpam-6566	216	27	:	:	PUNCT
ejpam-6566	216	28	(	(	PUNCT
ejpam-6566	216	29	1	1	X
ejpam-6566	216	30	)	)	PUNCT
ejpam-6566	216	31	f	f	PROPN
ejpam-6566	216	32	is	be	AUX
ejpam-6566	216	33	upper	upper	ADJ
ejpam-6566	216	34	almost	almost	ADV
ejpam-6566	216	35	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	216	36	,	,	PUNCT
ejpam-6566	216	37	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	216	38	;	;	PUNCT
ejpam-6566	216	39	(	(	PUNCT
ejpam-6566	216	40	2	2	X
ejpam-6566	216	41	)	)	SYM
ejpam-6566	216	42	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6566	216	43	)	)	PUNCT
ejpam-6566	216	44	)	)	PUNCT
ejpam-6566	217	1	⊆	⊆	NUM
ejpam-6566	217	2	f−(α(σ1	f−(α(σ1	NOUN
ejpam-6566	217	3	,	,	PUNCT
ejpam-6566	217	4	σ2)-cl(v	σ2)-cl(v	NOUN
ejpam-6566	217	5	)	)	PUNCT
ejpam-6566	217	6	)	)	PUNCT
ejpam-6566	217	7	for	for	ADP
ejpam-6566	217	8	every	every	DET
ejpam-6566	217	9	(	(	PUNCT
ejpam-6566	217	10	σ1	σ1	PROPN
ejpam-6566	217	11	,	,	PUNCT
ejpam-6566	217	12	σ2)β	σ2)β	NOUN
ejpam-6566	217	13	-	-	PUNCT
ejpam-6566	217	14	open	open	NOUN
ejpam-6566	217	15	set	set	NOUN
ejpam-6566	217	16	v	v	NOUN
ejpam-6566	217	17	of	of	ADP
ejpam-6566	217	18	y	y	PROPN
ejpam-6566	217	19	;	;	PUNCT
ejpam-6566	217	20	(	(	PUNCT
ejpam-6566	217	21	3	3	X
ejpam-6566	217	22	)	)	SYM
ejpam-6566	217	23	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6566	217	24	)	)	PUNCT
ejpam-6566	217	25	)	)	PUNCT
ejpam-6566	217	26	⊆	⊆	NUM
ejpam-6566	217	27	f−((σ1	f−((σ1	NOUN
ejpam-6566	217	28	,	,	PUNCT
ejpam-6566	217	29	σ2)-pcl(v	σ2)-pcl(v	NOUN
ejpam-6566	217	30	)	)	PUNCT
ejpam-6566	217	31	)	)	PUNCT
ejpam-6566	217	32	for	for	ADP
ejpam-6566	217	33	every	every	DET
ejpam-6566	217	34	(	(	PUNCT
ejpam-6566	217	35	σ1	σ1	PROPN
ejpam-6566	217	36	,	,	PUNCT
ejpam-6566	217	37	σ2)s	σ2)s	NOUN
ejpam-6566	217	38	-	-	PUNCT
ejpam-6566	217	39	open	open	NOUN
ejpam-6566	217	40	set	set	NOUN
ejpam-6566	217	41	v	v	NOUN
ejpam-6566	217	42	of	of	ADP
ejpam-6566	217	43	y	y	PROPN
ejpam-6566	217	44	.	.	PUNCT
ejpam-6566	218	1	corollary	corollary	ADJ
ejpam-6566	218	2	2	2	NUM
ejpam-6566	218	3	.	.	PUNCT
ejpam-6566	218	4	for	for	ADP
ejpam-6566	218	5	a	a	DET
ejpam-6566	218	6	multifunction	multifunction	NOUN
ejpam-6566	219	1	f	f	NOUN
ejpam-6566	219	2	:	:	PUNCT
ejpam-6566	219	3	(	(	PUNCT
ejpam-6566	219	4	x	x	X
ejpam-6566	219	5	,	,	PUNCT
ejpam-6566	219	6	τ	τ	PROPN
ejpam-6566	219	7	,	,	PUNCT
ejpam-6566	219	8	i	i	NOUN
ejpam-6566	219	9	)	)	PUNCT
ejpam-6566	219	10	→	→	PUNCT
ejpam-6566	219	11	(	(	PUNCT
ejpam-6566	219	12	y	y	PROPN
ejpam-6566	219	13	,	,	PUNCT
ejpam-6566	219	14	σ1	σ1	PROPN
ejpam-6566	219	15	,	,	PUNCT
ejpam-6566	219	16	σ2	σ2	NOUN
ejpam-6566	219	17	)	)	PUNCT
ejpam-6566	219	18	,	,	PUNCT
ejpam-6566	219	19	the	the	DET
ejpam-6566	219	20	following	follow	VERB
ejpam-6566	219	21	properties	property	NOUN
ejpam-6566	219	22	are	be	AUX
ejpam-6566	219	23	equivalent	equivalent	ADJ
ejpam-6566	219	24	:	:	PUNCT
ejpam-6566	219	25	(	(	PUNCT
ejpam-6566	219	26	1	1	X
ejpam-6566	219	27	)	)	PUNCT
ejpam-6566	219	28	f	f	PROPN
ejpam-6566	219	29	is	be	AUX
ejpam-6566	219	30	lower	low	ADJ
ejpam-6566	219	31	almost	almost	ADV
ejpam-6566	219	32	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	219	33	,	,	PUNCT
ejpam-6566	219	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	219	35	;	;	PUNCT
ejpam-6566	219	36	(	(	PUNCT
ejpam-6566	219	37	2	2	X
ejpam-6566	219	38	)	)	PUNCT
ejpam-6566	219	39	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-6566	219	40	)	)	PUNCT
ejpam-6566	219	41	)	)	PUNCT
ejpam-6566	220	1	⊆	⊆	NUM
ejpam-6566	220	2	f+(α(σ1	f+(α(σ1	NOUN
ejpam-6566	220	3	,	,	PUNCT
ejpam-6566	220	4	σ2)-cl(v	σ2)-cl(v	NOUN
ejpam-6566	220	5	)	)	PUNCT
ejpam-6566	220	6	)	)	PUNCT
ejpam-6566	220	7	for	for	ADP
ejpam-6566	220	8	every	every	DET
ejpam-6566	220	9	(	(	PUNCT
ejpam-6566	220	10	σ1	σ1	PROPN
ejpam-6566	220	11	,	,	PUNCT
ejpam-6566	220	12	σ2)β	σ2)β	NOUN
ejpam-6566	220	13	-	-	PUNCT
ejpam-6566	220	14	open	open	NOUN
ejpam-6566	220	15	set	set	NOUN
ejpam-6566	220	16	v	v	NOUN
ejpam-6566	220	17	of	of	ADP
ejpam-6566	220	18	y	y	PROPN
ejpam-6566	220	19	;	;	PUNCT
ejpam-6566	220	20	(	(	PUNCT
ejpam-6566	220	21	3	3	X
ejpam-6566	220	22	)	)	PUNCT
ejpam-6566	220	23	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-6566	220	24	)	)	PUNCT
ejpam-6566	220	25	)	)	PUNCT
ejpam-6566	221	1	⊆	⊆	NUM
ejpam-6566	221	2	f+((σ1	f+((σ1	NOUN
ejpam-6566	221	3	,	,	PUNCT
ejpam-6566	221	4	σ2)-pcl(v	σ2)-pcl(v	NOUN
ejpam-6566	221	5	)	)	PUNCT
ejpam-6566	221	6	)	)	PUNCT
ejpam-6566	222	1	for	for	ADP
ejpam-6566	222	2	every	every	DET
ejpam-6566	222	3	(	(	PUNCT
ejpam-6566	222	4	σ1	σ1	PROPN
ejpam-6566	222	5	,	,	PUNCT
ejpam-6566	222	6	σ2)s	σ2)s	NOUN
ejpam-6566	222	7	-	-	PUNCT
ejpam-6566	222	8	open	open	NOUN
ejpam-6566	222	9	set	set	NOUN
ejpam-6566	222	10	v	v	NOUN
ejpam-6566	222	11	of	of	ADP
ejpam-6566	222	12	y	y	PROPN
ejpam-6566	222	13	.	.	PUNCT
ejpam-6566	222	14	theorem	theorem	VERB
ejpam-6566	222	15	7	7	NUM
ejpam-6566	222	16	.	.	X
ejpam-6566	222	17	for	for	ADP
ejpam-6566	222	18	a	a	DET
ejpam-6566	222	19	multifunction	multifunction	NOUN
ejpam-6566	222	20	f	f	NOUN
ejpam-6566	222	21	:	:	PUNCT
ejpam-6566	222	22	(	(	PUNCT
ejpam-6566	222	23	x	x	X
ejpam-6566	222	24	,	,	PUNCT
ejpam-6566	222	25	τ	τ	PROPN
ejpam-6566	222	26	,	,	PUNCT
ejpam-6566	222	27	i	i	NOUN
ejpam-6566	222	28	)	)	PUNCT
ejpam-6566	222	29	→	→	PUNCT
ejpam-6566	222	30	(	(	PUNCT
ejpam-6566	222	31	y	y	PROPN
ejpam-6566	222	32	,	,	PUNCT
ejpam-6566	222	33	σ1	σ1	PROPN
ejpam-6566	222	34	,	,	PUNCT
ejpam-6566	222	35	σ2	σ2	NOUN
ejpam-6566	222	36	)	)	PUNCT
ejpam-6566	222	37	,	,	PUNCT
ejpam-6566	222	38	the	the	DET
ejpam-6566	222	39	following	follow	VERB
ejpam-6566	222	40	properties	property	NOUN
ejpam-6566	222	41	are	be	AUX
ejpam-6566	222	42	equivalent	equivalent	ADJ
ejpam-6566	222	43	:	:	PUNCT
ejpam-6566	222	44	(	(	PUNCT
ejpam-6566	222	45	1	1	X
ejpam-6566	222	46	)	)	PUNCT
ejpam-6566	222	47	f	f	PROPN
ejpam-6566	222	48	is	be	AUX
ejpam-6566	222	49	upper	upper	ADJ
ejpam-6566	222	50	almost	almost	ADV
ejpam-6566	222	51	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	222	52	,	,	PUNCT
ejpam-6566	222	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	222	54	;	;	PUNCT
ejpam-6566	222	55	(	(	PUNCT
ejpam-6566	222	56	2	2	X
ejpam-6566	222	57	)	)	PUNCT
ejpam-6566	222	58	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	222	59	-	-	PUNCT
ejpam-6566	222	60	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	222	61	-	-	PUNCT
ejpam-6566	222	62	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	222	63	-	-	PUNCT
ejpam-6566	222	64	cl(v	cl(v	NOUN
ejpam-6566	222	65	)	)	PUNCT
ejpam-6566	222	66	)	)	PUNCT
ejpam-6566	222	67	)	)	PUNCT
ejpam-6566	222	68	)	)	PUNCT
ejpam-6566	222	69	)	)	PUNCT
ejpam-6566	223	1	⊆	⊆	X
ejpam-6566	223	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	223	3	-	-	PUNCT
ejpam-6566	223	4	cl(v	cl(v	NOUN
ejpam-6566	223	5	)	)	PUNCT
ejpam-6566	223	6	)	)	PUNCT
ejpam-6566	223	7	for	for	ADP
ejpam-6566	223	8	every	every	DET
ejpam-6566	223	9	(	(	PUNCT
ejpam-6566	223	10	σ1	σ1	PROPN
ejpam-6566	223	11	,	,	PUNCT
ejpam-6566	223	12	σ2)p	σ2)p	NOUN
ejpam-6566	223	13	-	-	PUNCT
ejpam-6566	223	14	open	open	NOUN
ejpam-6566	223	15	set	set	NOUN
ejpam-6566	223	16	v	v	NOUN
ejpam-6566	223	17	of	of	ADP
ejpam-6566	223	18	y	y	PROPN
ejpam-6566	223	19	;	;	PUNCT
ejpam-6566	223	20	(	(	PUNCT
ejpam-6566	223	21	3	3	X
ejpam-6566	223	22	)	)	PUNCT
ejpam-6566	223	23	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	223	24	-	-	PUNCT
ejpam-6566	223	25	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	223	26	-	-	PUNCT
ejpam-6566	223	27	int(v	int(v	NOUN
ejpam-6566	223	28	)	)	PUNCT
ejpam-6566	223	29	)	)	PUNCT
ejpam-6566	223	30	)	)	PUNCT
ejpam-6566	223	31	)	)	PUNCT
ejpam-6566	224	1	⊆	⊆	X
ejpam-6566	224	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	224	3	-	-	PUNCT
ejpam-6566	224	4	cl(v	cl(v	NOUN
ejpam-6566	224	5	)	)	PUNCT
ejpam-6566	224	6	)	)	PUNCT
ejpam-6566	224	7	for	for	ADP
ejpam-6566	224	8	every	every	DET
ejpam-6566	224	9	(	(	PUNCT
ejpam-6566	224	10	σ1	σ1	PROPN
ejpam-6566	224	11	,	,	PUNCT
ejpam-6566	224	12	σ2)p	σ2)p	NOUN
ejpam-6566	224	13	-	-	PUNCT
ejpam-6566	224	14	open	open	NOUN
ejpam-6566	224	15	set	set	NOUN
ejpam-6566	224	16	v	v	NOUN
ejpam-6566	224	17	of	of	ADP
ejpam-6566	224	18	y	y	PROPN
ejpam-6566	224	19	;	;	PUNCT
ejpam-6566	224	20	(	(	PUNCT
ejpam-6566	224	21	4	4	X
ejpam-6566	224	22	)	)	PUNCT
ejpam-6566	224	23	f+(v	f+(v	NOUN
ejpam-6566	224	24	)	)	PUNCT
ejpam-6566	225	1	⊆	⊆	X
ejpam-6566	225	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	225	3	-	-	PUNCT
ejpam-6566	225	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	225	5	-	-	PUNCT
ejpam-6566	225	6	cl(v	cl(v	NOUN
ejpam-6566	225	7	)	)	PUNCT
ejpam-6566	225	8	)	)	PUNCT
ejpam-6566	225	9	)	)	PUNCT
ejpam-6566	225	10	)	)	PUNCT
ejpam-6566	225	11	for	for	ADP
ejpam-6566	225	12	every	every	DET
ejpam-6566	225	13	(	(	PUNCT
ejpam-6566	225	14	σ1	σ1	PROPN
ejpam-6566	225	15	,	,	PUNCT
ejpam-6566	225	16	σ2)p	σ2)p	NOUN
ejpam-6566	225	17	-	-	PUNCT
ejpam-6566	225	18	open	open	NOUN
ejpam-6566	225	19	set	set	NOUN
ejpam-6566	225	20	v	v	NOUN
ejpam-6566	225	21	of	of	ADP
ejpam-6566	225	22	y	y	PROPN
ejpam-6566	225	23	.	.	PUNCT
ejpam-6566	226	1	proof	proof	NOUN
ejpam-6566	226	2	.	.	PUNCT
ejpam-6566	227	1	(	(	PUNCT
ejpam-6566	227	2	1	1	X
ejpam-6566	227	3	)	)	PUNCT
ejpam-6566	227	4	⇒	⇒	NOUN
ejpam-6566	227	5	(	(	PUNCT
ejpam-6566	227	6	2	2	NUM
ejpam-6566	227	7	):	):	PUNCT
ejpam-6566	227	8	let	let	VERB
ejpam-6566	227	9	v	v	PART
ejpam-6566	227	10	be	be	AUX
ejpam-6566	227	11	any	any	DET
ejpam-6566	227	12	(	(	PUNCT
ejpam-6566	227	13	σ1	σ1	PROPN
ejpam-6566	227	14	,	,	PUNCT
ejpam-6566	227	15	σ2)p	σ2)p	NOUN
ejpam-6566	227	16	-	-	PUNCT
ejpam-6566	227	17	open	open	ADJ
ejpam-6566	227	18	set	set	NOUN
ejpam-6566	227	19	of	of	ADP
ejpam-6566	227	20	y	y	PROPN
ejpam-6566	227	21	.	.	PUNCT
ejpam-6566	228	1	then	then	ADV
ejpam-6566	228	2	,	,	PUNCT
ejpam-6566	228	3	σ1σ2	σ1σ2	NOUN
ejpam-6566	228	4	-	-	NUM
ejpam-6566	228	5	cl(v	cl(v	NOUN
ejpam-6566	228	6	)	)	PUNCT
ejpam-6566	228	7	is	be	AUX
ejpam-6566	228	8	σ1σ2closed	σ1σ2close	VERB
ejpam-6566	228	9	in	in	ADP
ejpam-6566	228	10	y	y	PROPN
ejpam-6566	228	11	and	and	CCONJ
ejpam-6566	228	12	by	by	ADP
ejpam-6566	228	13	theorem	theorem	NOUN
ejpam-6566	228	14	3	3	NUM
ejpam-6566	228	15	,	,	PUNCT
ejpam-6566	228	16	we	we	PRON
ejpam-6566	228	17	have	have	VERB
ejpam-6566	228	18	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6566	228	19	-	-	PUNCT
ejpam-6566	228	20	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	228	21	-	-	PUNCT
ejpam-6566	228	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	228	23	-	-	PUNCT
ejpam-6566	228	24	cl(v	cl(v	NOUN
ejpam-6566	228	25	)	)	PUNCT
ejpam-6566	228	26	)	)	PUNCT
ejpam-6566	228	27	)	)	PUNCT
ejpam-6566	228	28	)	)	PUNCT
ejpam-6566	228	29	)	)	PUNCT
ejpam-6566	229	1	⊆	⊆	X
ejpam-6566	229	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	229	3	-	-	PUNCT
ejpam-6566	229	4	cl(v	cl(v	NOUN
ejpam-6566	229	5	)	)	PUNCT
ejpam-6566	229	6	)	)	PUNCT
ejpam-6566	229	7	.	.	PUNCT
ejpam-6566	230	1	(	(	PUNCT
ejpam-6566	230	2	2	2	X
ejpam-6566	230	3	)	)	PUNCT
ejpam-6566	230	4	⇒	⇒	NOUN
ejpam-6566	230	5	(	(	PUNCT
ejpam-6566	230	6	3	3	NUM
ejpam-6566	230	7	):	):	PUNCT
ejpam-6566	230	8	let	let	VERB
ejpam-6566	230	9	v	v	PART
ejpam-6566	230	10	be	be	AUX
ejpam-6566	230	11	any	any	DET
ejpam-6566	230	12	(	(	PUNCT
ejpam-6566	230	13	σ1	σ1	PROPN
ejpam-6566	230	14	,	,	PUNCT
ejpam-6566	230	15	σ2)p	σ2)p	NOUN
ejpam-6566	230	16	-	-	PUNCT
ejpam-6566	230	17	open	open	ADJ
ejpam-6566	230	18	set	set	NOUN
ejpam-6566	230	19	of	of	ADP
ejpam-6566	230	20	y	y	PROPN
ejpam-6566	230	21	.	.	PUNCT
ejpam-6566	231	1	by	by	ADP
ejpam-6566	231	2	(	(	PUNCT
ejpam-6566	231	3	2	2	NUM
ejpam-6566	231	4	)	)	PUNCT
ejpam-6566	231	5	,	,	PUNCT
ejpam-6566	231	6	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	231	7	-	-	PUNCT
ejpam-6566	231	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	231	9	-	-	PUNCT
ejpam-6566	231	10	int(v	int(v	NOUN
ejpam-6566	231	11	)	)	PUNCT
ejpam-6566	231	12	)	)	PUNCT
ejpam-6566	231	13	)	)	PUNCT
ejpam-6566	231	14	)	)	PUNCT
ejpam-6566	232	1	⊆	⊆	NUM
ejpam-6566	232	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	232	3	-	-	PUNCT
ejpam-6566	232	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	232	5	-	-	PUNCT
ejpam-6566	232	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	232	7	-	-	PUNCT
ejpam-6566	232	8	cl(v	cl(v	NOUN
ejpam-6566	232	9	)	)	PUNCT
ejpam-6566	232	10	)	)	PUNCT
ejpam-6566	232	11	)	)	PUNCT
ejpam-6566	232	12	)	)	PUNCT
ejpam-6566	232	13	)	)	PUNCT
ejpam-6566	233	1	⊆	⊆	X
ejpam-6566	233	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	233	3	-	-	PUNCT
ejpam-6566	233	4	cl(v	cl(v	NOUN
ejpam-6566	233	5	)	)	PUNCT
ejpam-6566	233	6	)	)	PUNCT
ejpam-6566	233	7	.	.	PUNCT
ejpam-6566	234	1	(	(	PUNCT
ejpam-6566	234	2	3	3	X
ejpam-6566	234	3	)	)	PUNCT
ejpam-6566	234	4	⇒	⇒	NOUN
ejpam-6566	234	5	(	(	PUNCT
ejpam-6566	234	6	4	4	NUM
ejpam-6566	234	7	):	):	PUNCT
ejpam-6566	234	8	let	let	VERB
ejpam-6566	234	9	v	v	PART
ejpam-6566	234	10	be	be	AUX
ejpam-6566	234	11	any	any	DET
ejpam-6566	234	12	(	(	PUNCT
ejpam-6566	234	13	σ1	σ1	PROPN
ejpam-6566	234	14	,	,	PUNCT
ejpam-6566	234	15	σ2)p	σ2)p	NOUN
ejpam-6566	234	16	-	-	PUNCT
ejpam-6566	234	17	open	open	ADJ
ejpam-6566	234	18	set	set	NOUN
ejpam-6566	234	19	of	of	ADP
ejpam-6566	234	20	y	y	PROPN
ejpam-6566	234	21	.	.	PUNCT
ejpam-6566	235	1	thus	thus	ADV
ejpam-6566	235	2	by	by	ADP
ejpam-6566	235	3	(	(	PUNCT
ejpam-6566	235	4	3	3	NUM
ejpam-6566	235	5	)	)	PUNCT
ejpam-6566	235	6	,	,	PUNCT
ejpam-6566	235	7	we	we	PRON
ejpam-6566	235	8	have	have	VERB
ejpam-6566	235	9	x	x	PRON
ejpam-6566	235	10	−	−	PUNCT
ejpam-6566	235	11	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	235	12	-	-	PUNCT
ejpam-6566	235	13	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	235	14	-	-	PUNCT
ejpam-6566	235	15	cl(v	cl(v	NOUN
ejpam-6566	235	16	)	)	PUNCT
ejpam-6566	235	17	)	)	PUNCT
ejpam-6566	235	18	)	)	PUNCT
ejpam-6566	235	19	)	)	PUNCT
ejpam-6566	236	1	=	=	PUNCT
ejpam-6566	236	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6566	236	3	−	−	NOUN
ejpam-6566	236	4	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	236	5	-	-	PUNCT
ejpam-6566	236	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	236	7	-	-	PUNCT
ejpam-6566	236	8	cl(v	cl(v	NOUN
ejpam-6566	236	9	)	)	PUNCT
ejpam-6566	236	10	)	)	PUNCT
ejpam-6566	236	11	)	)	PUNCT
ejpam-6566	236	12	)	)	PUNCT
ejpam-6566	237	1	=	=	NOUN
ejpam-6566	237	2	cl⋆(f−(y	cl⋆(f−(y	NUM
ejpam-6566	237	3	−	−	NUM
ejpam-6566	237	4	σ1σ2	σ1σ2	X
ejpam-6566	237	5	-	-	PUNCT
ejpam-6566	237	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	237	7	-	-	PUNCT
ejpam-6566	237	8	cl(v	cl(v	NOUN
ejpam-6566	237	9	)	)	PUNCT
ejpam-6566	237	10	)	)	PUNCT
ejpam-6566	237	11	)	)	PUNCT
ejpam-6566	237	12	)	)	PUNCT
ejpam-6566	238	1	=	=	SYM
ejpam-6566	238	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6566	238	3	-	-	PUNCT
ejpam-6566	238	4	cl(y	cl(y	NOUN
ejpam-6566	238	5	−	−	NOUN
ejpam-6566	238	6	σ1σ2	σ1σ2	NOUN
ejpam-6566	238	7	-	-	NUM
ejpam-6566	238	8	cl(v	cl(v	NOUN
ejpam-6566	238	9	)	)	PUNCT
ejpam-6566	238	10	)	)	PUNCT
ejpam-6566	238	11	)	)	PUNCT
ejpam-6566	238	12	)	)	PUNCT
ejpam-6566	239	1	=	=	SYM
ejpam-6566	239	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6566	239	3	-	-	PUNCT
ejpam-6566	239	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	239	5	-	-	PUNCT
ejpam-6566	239	6	int(y	int(y	PROPN
ejpam-6566	239	7	−	−	NUM
ejpam-6566	239	8	σ1σ2	σ1σ2	NOUN
ejpam-6566	239	9	-	-	NUM
ejpam-6566	239	10	cl(v	cl(v	NOUN
ejpam-6566	239	11	)	)	PUNCT
ejpam-6566	239	12	)	)	PUNCT
ejpam-6566	239	13	)	)	PUNCT
ejpam-6566	239	14	)	)	PUNCT
ejpam-6566	239	15	)	)	PUNCT
ejpam-6566	240	1	⊆	⊆	X
ejpam-6566	240	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	240	3	-	-	PUNCT
ejpam-6566	240	4	cl(y	cl(y	NOUN
ejpam-6566	240	5	−	−	NOUN
ejpam-6566	240	6	σ1σ2	σ1σ2	NOUN
ejpam-6566	240	7	-	-	NUM
ejpam-6566	240	8	cl(v	cl(v	NOUN
ejpam-6566	240	9	)	)	PUNCT
ejpam-6566	240	10	)	)	PUNCT
ejpam-6566	240	11	)	)	PUNCT
ejpam-6566	241	1	c.	c.	PROPN
ejpam-6566	241	2	viriyapong	viriyapong	PROPN
ejpam-6566	241	3	,	,	PUNCT
ejpam-6566	241	4	a.	a.	PROPN
ejpam-6566	241	5	sama	sama	PROPN
ejpam-6566	241	6	-	-	PUNCT
ejpam-6566	241	7	ae	ae	PROPN
ejpam-6566	241	8	,	,	PUNCT
ejpam-6566	241	9	c.	c.	PROPN
ejpam-6566	241	10	boonpok	boonpok	PROPN
ejpam-6566	241	11	/	/	SYM
ejpam-6566	241	12	eur	eur	PROPN
ejpam-6566	241	13	.	.	PUNCT
ejpam-6566	242	1	j.	j.	PROPN
ejpam-6566	242	2	pure	pure	PROPN
ejpam-6566	242	3	appl	appl	PROPN
ejpam-6566	242	4	.	.	PROPN
ejpam-6566	242	5	math	math	PROPN
ejpam-6566	242	6	,	,	PUNCT
ejpam-6566	242	7	18	18	NUM
ejpam-6566	242	8	(	(	PUNCT
ejpam-6566	242	9	3	3	NUM
ejpam-6566	242	10	)	)	PUNCT
ejpam-6566	242	11	(	(	PUNCT
ejpam-6566	242	12	2025	2025	NUM
ejpam-6566	242	13	)	)	PUNCT
ejpam-6566	242	14	,	,	PUNCT
ejpam-6566	242	15	6566	6566	NUM
ejpam-6566	242	16	9	9	NUM
ejpam-6566	242	17	of	of	ADP
ejpam-6566	242	18	14	14	NUM
ejpam-6566	242	19	=	=	SYM
ejpam-6566	242	20	f−(y	f−(y	NOUN
ejpam-6566	242	21	−	−	NOUN
ejpam-6566	242	22	σ1σ2	σ1σ2	NOUN
ejpam-6566	242	23	-	-	PUNCT
ejpam-6566	242	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	242	25	-	-	PUNCT
ejpam-6566	242	26	cl(v	cl(v	NOUN
ejpam-6566	242	27	)	)	PUNCT
ejpam-6566	242	28	)	)	PUNCT
ejpam-6566	242	29	)	)	PUNCT
ejpam-6566	243	1	⊆	⊆	NUM
ejpam-6566	243	2	f−(y	f−(y	NOUN
ejpam-6566	243	3	−	−	NOUN
ejpam-6566	243	4	v	v	NOUN
ejpam-6566	243	5	)	)	PUNCT
ejpam-6566	243	6	=	=	PUNCT
ejpam-6566	243	7	x	x	X
ejpam-6566	243	8	−	−	PROPN
ejpam-6566	243	9	f+(v	f+(v	NOUN
ejpam-6566	243	10	)	)	PUNCT
ejpam-6566	243	11	and	and	CCONJ
ejpam-6566	243	12	hence	hence	ADV
ejpam-6566	243	13	f+(v	f+(v	NOUN
ejpam-6566	243	14	)	)	PUNCT
ejpam-6566	244	1	⊆	⊆	X
ejpam-6566	244	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	244	3	-	-	PUNCT
ejpam-6566	244	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	244	5	-	-	PUNCT
ejpam-6566	244	6	cl(v	cl(v	NOUN
ejpam-6566	244	7	)	)	PUNCT
ejpam-6566	244	8	)	)	PUNCT
ejpam-6566	244	9	)	)	PUNCT
ejpam-6566	244	10	)	)	PUNCT
ejpam-6566	244	11	.	.	PUNCT
ejpam-6566	245	1	(	(	PUNCT
ejpam-6566	245	2	4	4	X
ejpam-6566	245	3	)	)	PUNCT
ejpam-6566	245	4	⇒	⇒	NOUN
ejpam-6566	245	5	(	(	PUNCT
ejpam-6566	245	6	1	1	NUM
ejpam-6566	245	7	):	):	PUNCT
ejpam-6566	245	8	let	let	VERB
ejpam-6566	245	9	v	v	PART
ejpam-6566	245	10	be	be	AUX
ejpam-6566	245	11	any	any	DET
ejpam-6566	245	12	(	(	PUNCT
ejpam-6566	245	13	σ1	σ1	NOUN
ejpam-6566	245	14	,	,	PUNCT
ejpam-6566	245	15	σ2)r	σ2)r	NOUN
ejpam-6566	245	16	-	-	PUNCT
ejpam-6566	245	17	open	open	ADJ
ejpam-6566	245	18	set	set	NOUN
ejpam-6566	245	19	of	of	ADP
ejpam-6566	245	20	y	y	PROPN
ejpam-6566	245	21	.	.	PUNCT
ejpam-6566	246	1	then	then	ADV
ejpam-6566	246	2	,	,	PUNCT
ejpam-6566	246	3	v	v	NOUN
ejpam-6566	246	4	is	be	AUX
ejpam-6566	246	5	(	(	PUNCT
ejpam-6566	246	6	σ1	σ1	PROPN
ejpam-6566	246	7	,	,	PUNCT
ejpam-6566	246	8	σ2)p	σ2)p	NOUN
ejpam-6566	246	9	-	-	PUNCT
ejpam-6566	246	10	open	open	ADJ
ejpam-6566	246	11	in	in	ADP
ejpam-6566	246	12	y	y	PROPN
ejpam-6566	246	13	and	and	CCONJ
ejpam-6566	246	14	by	by	ADP
ejpam-6566	246	15	(	(	PUNCT
ejpam-6566	246	16	4	4	NUM
ejpam-6566	246	17	)	)	PUNCT
ejpam-6566	246	18	,	,	PUNCT
ejpam-6566	246	19	f+(v	f+(v	PROPN
ejpam-6566	246	20	)	)	PUNCT
ejpam-6566	247	1	⊆	⊆	X
ejpam-6566	247	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6566	247	3	-	-	PUNCT
ejpam-6566	247	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	247	5	-	-	PUNCT
ejpam-6566	247	6	cl(v	cl(v	NOUN
ejpam-6566	247	7	)	)	PUNCT
ejpam-6566	247	8	)	)	PUNCT
ejpam-6566	247	9	)	)	PUNCT
ejpam-6566	247	10	)	)	PUNCT
ejpam-6566	248	1	=	=	SYM
ejpam-6566	248	2	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6566	248	3	)	)	PUNCT
ejpam-6566	248	4	)	)	PUNCT
ejpam-6566	248	5	.	.	PUNCT
ejpam-6566	249	1	thus	thus	ADV
ejpam-6566	249	2	,	,	PUNCT
ejpam-6566	249	3	f+(v	f+(v	PROPN
ejpam-6566	249	4	)	)	PUNCT
ejpam-6566	249	5	is	be	AUX
ejpam-6566	249	6	⋆-open	⋆-open	ADJ
ejpam-6566	249	7	in	in	ADP
ejpam-6566	249	8	x.	x.	NOUN
ejpam-6566	249	9	it	it	PRON
ejpam-6566	249	10	follows	follow	VERB
ejpam-6566	249	11	from	from	ADP
ejpam-6566	249	12	theorem	theorem	ADJ
ejpam-6566	249	13	3	3	NUM
ejpam-6566	249	14	that	that	SCONJ
ejpam-6566	249	15	f	f	PROPN
ejpam-6566	249	16	is	be	AUX
ejpam-6566	249	17	upper	upper	ADJ
ejpam-6566	249	18	almost	almost	ADV
ejpam-6566	249	19	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	249	20	,	,	PUNCT
ejpam-6566	249	21	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	249	22	.	.	X
ejpam-6566	249	23	theorem	theorem	VERB
ejpam-6566	249	24	8	8	NUM
ejpam-6566	249	25	.	.	PUNCT
ejpam-6566	249	26	for	for	ADP
ejpam-6566	249	27	a	a	DET
ejpam-6566	249	28	multifunction	multifunction	NOUN
ejpam-6566	249	29	f	f	NOUN
ejpam-6566	249	30	:	:	PUNCT
ejpam-6566	249	31	(	(	PUNCT
ejpam-6566	249	32	x	x	X
ejpam-6566	249	33	,	,	PUNCT
ejpam-6566	249	34	τ	τ	PROPN
ejpam-6566	249	35	,	,	PUNCT
ejpam-6566	249	36	i	i	NOUN
ejpam-6566	249	37	)	)	PUNCT
ejpam-6566	249	38	→	→	PUNCT
ejpam-6566	249	39	(	(	PUNCT
ejpam-6566	249	40	y	y	PROPN
ejpam-6566	249	41	,	,	PUNCT
ejpam-6566	249	42	σ1	σ1	PROPN
ejpam-6566	249	43	,	,	PUNCT
ejpam-6566	249	44	σ2	σ2	NOUN
ejpam-6566	249	45	)	)	PUNCT
ejpam-6566	249	46	,	,	PUNCT
ejpam-6566	249	47	the	the	DET
ejpam-6566	249	48	following	follow	VERB
ejpam-6566	249	49	properties	property	NOUN
ejpam-6566	249	50	are	be	AUX
ejpam-6566	249	51	equivalent	equivalent	ADJ
ejpam-6566	249	52	:	:	PUNCT
ejpam-6566	249	53	(	(	PUNCT
ejpam-6566	249	54	1	1	X
ejpam-6566	249	55	)	)	PUNCT
ejpam-6566	249	56	f	f	PROPN
ejpam-6566	249	57	is	be	AUX
ejpam-6566	249	58	lower	low	ADJ
ejpam-6566	249	59	almost	almost	ADV
ejpam-6566	249	60	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	249	61	,	,	PUNCT
ejpam-6566	249	62	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	249	63	;	;	PUNCT
ejpam-6566	249	64	(	(	PUNCT
ejpam-6566	249	65	2	2	X
ejpam-6566	249	66	)	)	PUNCT
ejpam-6566	249	67	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	249	68	-	-	PUNCT
ejpam-6566	249	69	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	249	70	-	-	PUNCT
ejpam-6566	249	71	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	249	72	-	-	PUNCT
ejpam-6566	249	73	cl(v	cl(v	NOUN
ejpam-6566	249	74	)	)	PUNCT
ejpam-6566	249	75	)	)	PUNCT
ejpam-6566	249	76	)	)	PUNCT
ejpam-6566	249	77	)	)	PUNCT
ejpam-6566	249	78	)	)	PUNCT
ejpam-6566	250	1	⊆	⊆	X
ejpam-6566	250	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	250	3	-	-	PUNCT
ejpam-6566	250	4	cl(v	cl(v	NOUN
ejpam-6566	250	5	)	)	PUNCT
ejpam-6566	250	6	)	)	PUNCT
ejpam-6566	250	7	for	for	ADP
ejpam-6566	250	8	every	every	DET
ejpam-6566	250	9	(	(	PUNCT
ejpam-6566	250	10	σ1	σ1	PROPN
ejpam-6566	250	11	,	,	PUNCT
ejpam-6566	250	12	σ2)p	σ2)p	NOUN
ejpam-6566	250	13	-	-	PUNCT
ejpam-6566	250	14	open	open	NOUN
ejpam-6566	250	15	set	set	NOUN
ejpam-6566	250	16	v	v	NOUN
ejpam-6566	250	17	of	of	ADP
ejpam-6566	250	18	y	y	PROPN
ejpam-6566	250	19	;	;	PUNCT
ejpam-6566	250	20	(	(	PUNCT
ejpam-6566	250	21	3	3	X
ejpam-6566	250	22	)	)	PUNCT
ejpam-6566	250	23	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	250	24	-	-	PUNCT
ejpam-6566	250	25	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	250	26	-	-	PUNCT
ejpam-6566	250	27	int(v	int(v	NOUN
ejpam-6566	250	28	)	)	PUNCT
ejpam-6566	250	29	)	)	PUNCT
ejpam-6566	250	30	)	)	PUNCT
ejpam-6566	250	31	)	)	PUNCT
ejpam-6566	251	1	⊆	⊆	X
ejpam-6566	251	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	251	3	-	-	PUNCT
ejpam-6566	251	4	cl(v	cl(v	NOUN
ejpam-6566	251	5	)	)	PUNCT
ejpam-6566	251	6	)	)	PUNCT
ejpam-6566	251	7	for	for	ADP
ejpam-6566	251	8	every	every	DET
ejpam-6566	251	9	(	(	PUNCT
ejpam-6566	251	10	σ1	σ1	PROPN
ejpam-6566	251	11	,	,	PUNCT
ejpam-6566	251	12	σ2)p	σ2)p	NOUN
ejpam-6566	251	13	-	-	PUNCT
ejpam-6566	251	14	open	open	NOUN
ejpam-6566	251	15	set	set	NOUN
ejpam-6566	251	16	v	v	NOUN
ejpam-6566	251	17	of	of	ADP
ejpam-6566	251	18	y	y	PROPN
ejpam-6566	251	19	;	;	PUNCT
ejpam-6566	251	20	(	(	PUNCT
ejpam-6566	251	21	4	4	X
ejpam-6566	251	22	)	)	PUNCT
ejpam-6566	251	23	f−(v	f−(v	NOUN
ejpam-6566	251	24	)	)	PUNCT
ejpam-6566	251	25	⊆	⊆	NUM
ejpam-6566	251	26	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	NOUN
ejpam-6566	251	27	-	-	PUNCT
ejpam-6566	251	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	251	29	-	-	PUNCT
ejpam-6566	251	30	cl(v	cl(v	NOUN
ejpam-6566	251	31	)	)	PUNCT
ejpam-6566	251	32	)	)	PUNCT
ejpam-6566	251	33	)	)	PUNCT
ejpam-6566	251	34	)	)	PUNCT
ejpam-6566	252	1	for	for	ADP
ejpam-6566	252	2	every	every	DET
ejpam-6566	252	3	(	(	PUNCT
ejpam-6566	252	4	σ1	σ1	PROPN
ejpam-6566	252	5	,	,	PUNCT
ejpam-6566	252	6	σ2)p	σ2)p	NOUN
ejpam-6566	252	7	-	-	PUNCT
ejpam-6566	252	8	open	open	NOUN
ejpam-6566	252	9	set	set	NOUN
ejpam-6566	252	10	v	v	NOUN
ejpam-6566	252	11	of	of	ADP
ejpam-6566	252	12	y	y	PROPN
ejpam-6566	252	13	.	.	PUNCT
ejpam-6566	253	1	proof	proof	NOUN
ejpam-6566	253	2	.	.	PUNCT
ejpam-6566	254	1	the	the	DET
ejpam-6566	254	2	proof	proof	NOUN
ejpam-6566	254	3	is	be	AUX
ejpam-6566	254	4	similar	similar	ADJ
ejpam-6566	254	5	to	to	ADP
ejpam-6566	254	6	that	that	PRON
ejpam-6566	254	7	of	of	ADP
ejpam-6566	254	8	theorem	theorem	NOUN
ejpam-6566	254	9	7	7	NUM
ejpam-6566	254	10	.	.	PUNCT
ejpam-6566	254	11	lemma	lemma	PROPN
ejpam-6566	254	12	4	4	NUM
ejpam-6566	254	13	.	.	PUNCT
ejpam-6566	255	1	[	[	X
ejpam-6566	255	2	29	29	NUM
ejpam-6566	255	3	]	]	PUNCT
ejpam-6566	255	4	let	let	VERB
ejpam-6566	255	5	a	a	PRON
ejpam-6566	255	6	be	be	AUX
ejpam-6566	255	7	a	a	DET
ejpam-6566	255	8	subset	subset	NOUN
ejpam-6566	255	9	of	of	ADP
ejpam-6566	255	10	a	a	DET
ejpam-6566	255	11	bitopological	bitopological	ADJ
ejpam-6566	255	12	space	space	NOUN
ejpam-6566	255	13	(	(	PUNCT
ejpam-6566	255	14	x	x	NOUN
ejpam-6566	255	15	,	,	PUNCT
ejpam-6566	255	16	τ1	τ1	NOUN
ejpam-6566	255	17	,	,	PUNCT
ejpam-6566	255	18	τ2	τ2	NOUN
ejpam-6566	255	19	)	)	PUNCT
ejpam-6566	255	20	.	.	PUNCT
ejpam-6566	256	1	then	then	ADV
ejpam-6566	256	2	,	,	PUNCT
ejpam-6566	256	3	the	the	DET
ejpam-6566	256	4	following	follow	VERB
ejpam-6566	256	5	properties	property	NOUN
ejpam-6566	256	6	hold	hold	VERB
ejpam-6566	256	7	:	:	PUNCT
ejpam-6566	256	8	(	(	PUNCT
ejpam-6566	256	9	1	1	X
ejpam-6566	256	10	)	)	PUNCT
ejpam-6566	256	11	if	if	SCONJ
ejpam-6566	256	12	a	a	PRON
ejpam-6566	256	13	is	be	AUX
ejpam-6566	256	14	τ1τ2	τ1τ2	NOUN
ejpam-6566	256	15	-	-	ADJ
ejpam-6566	256	16	open	open	ADJ
ejpam-6566	256	17	in	in	ADP
ejpam-6566	256	18	x	x	NOUN
ejpam-6566	256	19	,	,	PUNCT
ejpam-6566	256	20	then	then	ADV
ejpam-6566	256	21	τ1τ2	τ1τ2	NOUN
ejpam-6566	256	22	-	-	NUM
ejpam-6566	256	23	cl(a	cl(a	NUM
ejpam-6566	256	24	)	)	PUNCT
ejpam-6566	256	25	=	=	PUNCT
ejpam-6566	257	1	τ1τ2	τ1τ2	PROPN
ejpam-6566	257	2	-	-	ADJ
ejpam-6566	257	3	δ	δ	NOUN
ejpam-6566	257	4	-	-	PUNCT
ejpam-6566	257	5	cl(a	cl(a	NUM
ejpam-6566	257	6	)	)	PUNCT
ejpam-6566	257	7	.	.	PUNCT
ejpam-6566	258	1	(	(	PUNCT
ejpam-6566	258	2	2	2	X
ejpam-6566	258	3	)	)	PUNCT
ejpam-6566	258	4	τ1τ2	τ1τ2	NOUN
ejpam-6566	258	5	-	-	ADJ
ejpam-6566	258	6	δ	δ	NOUN
ejpam-6566	258	7	-	-	PUNCT
ejpam-6566	258	8	cl(a	cl(a	X
ejpam-6566	258	9	)	)	PUNCT
ejpam-6566	258	10	is	be	AUX
ejpam-6566	258	11	τ1τ2	τ1τ2	NOUN
ejpam-6566	258	12	-	-	ADJ
ejpam-6566	258	13	closed	closed	ADJ
ejpam-6566	258	14	.	.	PUNCT
ejpam-6566	259	1	theorem	theorem	VERB
ejpam-6566	259	2	9	9	NUM
ejpam-6566	259	3	.	.	X
ejpam-6566	259	4	for	for	ADP
ejpam-6566	259	5	a	a	DET
ejpam-6566	259	6	multifunction	multifunction	NOUN
ejpam-6566	260	1	f	f	NOUN
ejpam-6566	260	2	:	:	PUNCT
ejpam-6566	260	3	(	(	PUNCT
ejpam-6566	260	4	x	x	X
ejpam-6566	260	5	,	,	PUNCT
ejpam-6566	260	6	τ	τ	PROPN
ejpam-6566	260	7	,	,	PUNCT
ejpam-6566	260	8	i	i	NOUN
ejpam-6566	260	9	)	)	PUNCT
ejpam-6566	260	10	→	→	PUNCT
ejpam-6566	260	11	(	(	PUNCT
ejpam-6566	260	12	y	y	PROPN
ejpam-6566	260	13	,	,	PUNCT
ejpam-6566	260	14	σ1	σ1	PROPN
ejpam-6566	260	15	,	,	PUNCT
ejpam-6566	260	16	σ2	σ2	NOUN
ejpam-6566	260	17	)	)	PUNCT
ejpam-6566	260	18	,	,	PUNCT
ejpam-6566	260	19	the	the	DET
ejpam-6566	260	20	following	follow	VERB
ejpam-6566	260	21	properties	property	NOUN
ejpam-6566	260	22	are	be	AUX
ejpam-6566	260	23	equivalent	equivalent	ADJ
ejpam-6566	260	24	:	:	PUNCT
ejpam-6566	260	25	(	(	PUNCT
ejpam-6566	260	26	1	1	X
ejpam-6566	260	27	)	)	PUNCT
ejpam-6566	260	28	f	f	PROPN
ejpam-6566	260	29	is	be	AUX
ejpam-6566	260	30	upper	upper	ADJ
ejpam-6566	260	31	almost	almost	ADV
ejpam-6566	260	32	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	260	33	,	,	PUNCT
ejpam-6566	260	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	260	35	;	;	PUNCT
ejpam-6566	260	36	(	(	PUNCT
ejpam-6566	260	37	2	2	X
ejpam-6566	260	38	)	)	PUNCT
ejpam-6566	260	39	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	260	40	-	-	PUNCT
ejpam-6566	260	41	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	260	42	-	-	PUNCT
ejpam-6566	260	43	int(σ1σ2	int(σ1σ2	VERB
ejpam-6566	260	44	-	-	PUNCT
ejpam-6566	260	45	δ	δ	NOUN
ejpam-6566	260	46	-	-	NOUN
ejpam-6566	260	47	cl(b	cl(b	NOUN
ejpam-6566	260	48	)	)	PUNCT
ejpam-6566	260	49	)	)	PUNCT
ejpam-6566	260	50	)	)	PUNCT
ejpam-6566	260	51	)	)	PUNCT
ejpam-6566	260	52	)	)	PUNCT
ejpam-6566	261	1	⊆	⊆	X
ejpam-6566	261	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	261	3	-	-	PUNCT
ejpam-6566	261	4	δ	δ	NOUN
ejpam-6566	261	5	-	-	NOUN
ejpam-6566	261	6	cl(b	cl(b	NOUN
ejpam-6566	261	7	)	)	PUNCT
ejpam-6566	261	8	)	)	PUNCT
ejpam-6566	261	9	for	for	ADP
ejpam-6566	261	10	every	every	DET
ejpam-6566	261	11	subset	subset	NOUN
ejpam-6566	261	12	b	b	PROPN
ejpam-6566	261	13	of	of	ADP
ejpam-6566	261	14	y	y	PROPN
ejpam-6566	261	15	;	;	PUNCT
ejpam-6566	261	16	(	(	PUNCT
ejpam-6566	261	17	3	3	X
ejpam-6566	261	18	)	)	PUNCT
ejpam-6566	261	19	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	261	20	-	-	PUNCT
ejpam-6566	261	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	261	22	-	-	PUNCT
ejpam-6566	261	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	261	24	-	-	PUNCT
ejpam-6566	261	25	cl(b	cl(b	NOUN
ejpam-6566	261	26	)	)	PUNCT
ejpam-6566	261	27	)	)	PUNCT
ejpam-6566	261	28	)	)	PUNCT
ejpam-6566	261	29	)	)	PUNCT
ejpam-6566	261	30	)	)	PUNCT
ejpam-6566	262	1	⊆	⊆	X
ejpam-6566	262	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	262	3	-	-	PUNCT
ejpam-6566	262	4	δ	δ	NOUN
ejpam-6566	262	5	-	-	NOUN
ejpam-6566	262	6	cl(b	cl(b	NOUN
ejpam-6566	262	7	)	)	PUNCT
ejpam-6566	262	8	)	)	PUNCT
ejpam-6566	262	9	for	for	ADP
ejpam-6566	262	10	every	every	DET
ejpam-6566	262	11	subset	subset	NOUN
ejpam-6566	262	12	b	b	PROPN
ejpam-6566	262	13	of	of	ADP
ejpam-6566	262	14	y	y	PROPN
ejpam-6566	262	15	.	.	PUNCT
ejpam-6566	263	1	proof	proof	NOUN
ejpam-6566	263	2	.	.	PUNCT
ejpam-6566	264	1	(	(	PUNCT
ejpam-6566	264	2	1	1	X
ejpam-6566	264	3	)	)	PUNCT
ejpam-6566	264	4	⇒	⇒	NOUN
ejpam-6566	264	5	(	(	PUNCT
ejpam-6566	264	6	2	2	NUM
ejpam-6566	264	7	):	):	PUNCT
ejpam-6566	264	8	let	let	VERB
ejpam-6566	264	9	b	b	X
ejpam-6566	264	10	be	be	AUX
ejpam-6566	264	11	any	any	DET
ejpam-6566	264	12	subset	subset	NOUN
ejpam-6566	264	13	of	of	ADP
ejpam-6566	264	14	y	y	PROPN
ejpam-6566	264	15	.	.	PUNCT
ejpam-6566	265	1	by	by	ADP
ejpam-6566	265	2	lemma	lemma	PROPN
ejpam-6566	265	3	4	4	NUM
ejpam-6566	265	4	,	,	PUNCT
ejpam-6566	265	5	σ1σ2	σ1σ2	NOUN
ejpam-6566	265	6	-	-	PUNCT
ejpam-6566	265	7	δ	δ	NOUN
ejpam-6566	265	8	-	-	NOUN
ejpam-6566	265	9	cl(b	cl(b	NOUN
ejpam-6566	265	10	)	)	PUNCT
ejpam-6566	265	11	is	be	AUX
ejpam-6566	265	12	σ1σ2	σ1σ2	NOUN
ejpam-6566	265	13	-	-	ADJ
ejpam-6566	265	14	closed	closed	ADJ
ejpam-6566	265	15	in	in	ADP
ejpam-6566	265	16	y	y	PROPN
ejpam-6566	265	17	and	and	CCONJ
ejpam-6566	265	18	by	by	ADP
ejpam-6566	265	19	theorem	theorem	ADJ
ejpam-6566	265	20	3	3	NUM
ejpam-6566	265	21	,	,	PUNCT
ejpam-6566	265	22	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	265	23	-	-	PUNCT
ejpam-6566	265	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	265	25	-	-	PUNCT
ejpam-6566	265	26	int(σ1σ2	int(σ1σ2	VERB
ejpam-6566	265	27	-	-	PUNCT
ejpam-6566	265	28	δ	δ	NOUN
ejpam-6566	265	29	-	-	NOUN
ejpam-6566	265	30	cl(b	cl(b	NOUN
ejpam-6566	265	31	)	)	PUNCT
ejpam-6566	265	32	)	)	PUNCT
ejpam-6566	265	33	)	)	PUNCT
ejpam-6566	265	34	)	)	PUNCT
ejpam-6566	265	35	)	)	PUNCT
ejpam-6566	266	1	⊆	⊆	X
ejpam-6566	266	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	266	3	-	-	PUNCT
ejpam-6566	266	4	δ	δ	NOUN
ejpam-6566	266	5	-	-	NOUN
ejpam-6566	266	6	cl(b	cl(b	NOUN
ejpam-6566	266	7	)	)	PUNCT
ejpam-6566	266	8	)	)	PUNCT
ejpam-6566	266	9	.	.	PUNCT
ejpam-6566	267	1	(	(	PUNCT
ejpam-6566	267	2	2	2	X
ejpam-6566	267	3	)	)	PUNCT
ejpam-6566	267	4	⇒	⇒	NOUN
ejpam-6566	267	5	(	(	PUNCT
ejpam-6566	267	6	3	3	NUM
ejpam-6566	267	7	):	):	PUNCT
ejpam-6566	267	8	this	this	PRON
ejpam-6566	267	9	is	be	AUX
ejpam-6566	267	10	obvious	obvious	ADJ
ejpam-6566	267	11	since	since	SCONJ
ejpam-6566	267	12	σ1σ2	σ1σ2	NOUN
ejpam-6566	267	13	-	-	NOUN
ejpam-6566	267	14	cl(b	cl(b	NOUN
ejpam-6566	267	15	)	)	PUNCT
ejpam-6566	267	16	⊆	⊆	NUM
ejpam-6566	267	17	σ1σ2	σ1σ2	NUM
ejpam-6566	267	18	-	-	PUNCT
ejpam-6566	267	19	δ	δ	NOUN
ejpam-6566	267	20	-	-	NOUN
ejpam-6566	267	21	cl(b	cl(b	NOUN
ejpam-6566	267	22	)	)	PUNCT
ejpam-6566	267	23	for	for	ADP
ejpam-6566	267	24	every	every	DET
ejpam-6566	267	25	subset	subset	NOUN
ejpam-6566	267	26	b	b	PROPN
ejpam-6566	267	27	of	of	ADP
ejpam-6566	267	28	y	y	PROPN
ejpam-6566	267	29	.	.	PUNCT
ejpam-6566	268	1	(	(	PUNCT
ejpam-6566	268	2	3	3	X
ejpam-6566	268	3	)	)	PUNCT
ejpam-6566	268	4	⇒	⇒	NOUN
ejpam-6566	268	5	(	(	PUNCT
ejpam-6566	268	6	1	1	NUM
ejpam-6566	268	7	):	):	PUNCT
ejpam-6566	268	8	let	let	VERB
ejpam-6566	268	9	k	k	PRON
ejpam-6566	268	10	be	be	AUX
ejpam-6566	268	11	any	any	DET
ejpam-6566	268	12	(	(	PUNCT
ejpam-6566	268	13	σ1	σ1	NOUN
ejpam-6566	268	14	,	,	PUNCT
ejpam-6566	268	15	σ2)r	σ2)r	NOUN
ejpam-6566	268	16	-	-	PUNCT
ejpam-6566	268	17	closed	close	VERB
ejpam-6566	268	18	set	set	NOUN
ejpam-6566	268	19	of	of	ADP
ejpam-6566	268	20	y	y	PROPN
ejpam-6566	268	21	.	.	PUNCT
ejpam-6566	269	1	then	then	ADV
ejpam-6566	269	2	by	by	ADP
ejpam-6566	269	3	(	(	PUNCT
ejpam-6566	269	4	3	3	NUM
ejpam-6566	269	5	)	)	PUNCT
ejpam-6566	269	6	,	,	PUNCT
ejpam-6566	269	7	we	we	PRON
ejpam-6566	269	8	have	have	VERB
ejpam-6566	269	9	cl⋆(f−(k	cl⋆(f−(k	NUM
ejpam-6566	269	10	)	)	PUNCT
ejpam-6566	269	11	)	)	PUNCT
ejpam-6566	270	1	=	=	SYM
ejpam-6566	270	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6566	270	3	-	-	PUNCT
ejpam-6566	270	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	270	5	-	-	PUNCT
ejpam-6566	270	6	int(k	int(k	NOUN
ejpam-6566	270	7	)	)	PUNCT
ejpam-6566	270	8	)	)	PUNCT
ejpam-6566	270	9	)	)	PUNCT
ejpam-6566	270	10	)	)	PUNCT
ejpam-6566	271	1	c.	c.	PROPN
ejpam-6566	271	2	viriyapong	viriyapong	PROPN
ejpam-6566	271	3	,	,	PUNCT
ejpam-6566	271	4	a.	a.	PROPN
ejpam-6566	271	5	sama	sama	PROPN
ejpam-6566	271	6	-	-	PUNCT
ejpam-6566	271	7	ae	ae	PROPN
ejpam-6566	271	8	,	,	PUNCT
ejpam-6566	271	9	c.	c.	PROPN
ejpam-6566	271	10	boonpok	boonpok	PROPN
ejpam-6566	271	11	/	/	SYM
ejpam-6566	271	12	eur	eur	PROPN
ejpam-6566	271	13	.	.	PUNCT
ejpam-6566	272	1	j.	j.	PROPN
ejpam-6566	272	2	pure	pure	PROPN
ejpam-6566	272	3	appl	appl	PROPN
ejpam-6566	272	4	.	.	PROPN
ejpam-6566	272	5	math	math	PROPN
ejpam-6566	272	6	,	,	PUNCT
ejpam-6566	272	7	18	18	NUM
ejpam-6566	272	8	(	(	PUNCT
ejpam-6566	272	9	3	3	NUM
ejpam-6566	272	10	)	)	PUNCT
ejpam-6566	272	11	(	(	PUNCT
ejpam-6566	272	12	2025	2025	NUM
ejpam-6566	272	13	)	)	PUNCT
ejpam-6566	272	14	,	,	PUNCT
ejpam-6566	272	15	6566	6566	NUM
ejpam-6566	272	16	10	10	NUM
ejpam-6566	272	17	of	of	ADP
ejpam-6566	272	18	14	14	NUM
ejpam-6566	272	19	=	=	SYM
ejpam-6566	272	20	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6566	272	21	-	-	PUNCT
ejpam-6566	272	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	272	23	-	-	PUNCT
ejpam-6566	272	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	272	25	-	-	PUNCT
ejpam-6566	272	26	cl(k	cl(k	NUM
ejpam-6566	272	27	)	)	PUNCT
ejpam-6566	272	28	)	)	PUNCT
ejpam-6566	272	29	)	)	PUNCT
ejpam-6566	272	30	)	)	PUNCT
ejpam-6566	272	31	)	)	PUNCT
ejpam-6566	273	1	⊆	⊆	X
ejpam-6566	273	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	273	3	-	-	PUNCT
ejpam-6566	273	4	δ	δ	NOUN
ejpam-6566	273	5	-	-	NOUN
ejpam-6566	273	6	cl(k	cl(k	NOUN
ejpam-6566	273	7	)	)	PUNCT
ejpam-6566	273	8	)	)	PUNCT
ejpam-6566	274	1	=	=	SYM
ejpam-6566	274	2	f−(k	f−(k	PROPN
ejpam-6566	274	3	)	)	PUNCT
ejpam-6566	274	4	and	and	CCONJ
ejpam-6566	274	5	hence	hence	ADV
ejpam-6566	274	6	f−(k	f−(k	PROPN
ejpam-6566	274	7	)	)	PUNCT
ejpam-6566	274	8	is	be	AUX
ejpam-6566	274	9	⋆-closed	⋆-close	VERB
ejpam-6566	274	10	inx	inx	PROPN
ejpam-6566	274	11	.	.	PROPN
ejpam-6566	275	1	by	by	ADP
ejpam-6566	275	2	theorem	theorem	NOUN
ejpam-6566	275	3	3	3	NUM
ejpam-6566	275	4	,	,	PUNCT
ejpam-6566	275	5	f	f	PROPN
ejpam-6566	275	6	is	be	AUX
ejpam-6566	275	7	upper	upper	ADJ
ejpam-6566	275	8	almost	almost	ADV
ejpam-6566	275	9	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	275	10	,	,	PUNCT
ejpam-6566	275	11	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	275	12	.	.	X
ejpam-6566	275	13	theorem	theorem	NOUN
ejpam-6566	275	14	10	10	NUM
ejpam-6566	275	15	.	.	PUNCT
ejpam-6566	276	1	for	for	ADP
ejpam-6566	276	2	a	a	DET
ejpam-6566	276	3	multifunction	multifunction	NOUN
ejpam-6566	276	4	f	f	NOUN
ejpam-6566	276	5	:	:	PUNCT
ejpam-6566	276	6	(	(	PUNCT
ejpam-6566	276	7	x	x	X
ejpam-6566	276	8	,	,	PUNCT
ejpam-6566	276	9	τ	τ	PROPN
ejpam-6566	276	10	,	,	PUNCT
ejpam-6566	276	11	i	i	NOUN
ejpam-6566	276	12	)	)	PUNCT
ejpam-6566	276	13	→	→	PUNCT
ejpam-6566	276	14	(	(	PUNCT
ejpam-6566	276	15	y	y	PROPN
ejpam-6566	276	16	,	,	PUNCT
ejpam-6566	276	17	σ1	σ1	PROPN
ejpam-6566	276	18	,	,	PUNCT
ejpam-6566	276	19	σ2	σ2	NOUN
ejpam-6566	276	20	)	)	PUNCT
ejpam-6566	276	21	,	,	PUNCT
ejpam-6566	276	22	the	the	DET
ejpam-6566	276	23	following	follow	VERB
ejpam-6566	276	24	properties	property	NOUN
ejpam-6566	276	25	are	be	AUX
ejpam-6566	276	26	equivalent	equivalent	ADJ
ejpam-6566	276	27	:	:	PUNCT
ejpam-6566	276	28	(	(	PUNCT
ejpam-6566	276	29	1	1	X
ejpam-6566	276	30	)	)	PUNCT
ejpam-6566	276	31	f	f	PROPN
ejpam-6566	276	32	is	be	AUX
ejpam-6566	276	33	lower	low	ADJ
ejpam-6566	276	34	almost	almost	ADV
ejpam-6566	276	35	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	276	36	,	,	PUNCT
ejpam-6566	276	37	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	276	38	;	;	PUNCT
ejpam-6566	276	39	(	(	PUNCT
ejpam-6566	276	40	2	2	X
ejpam-6566	276	41	)	)	PUNCT
ejpam-6566	276	42	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	276	43	-	-	PUNCT
ejpam-6566	276	44	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	276	45	-	-	PUNCT
ejpam-6566	276	46	int(σ1σ2	int(σ1σ2	VERB
ejpam-6566	276	47	-	-	PUNCT
ejpam-6566	276	48	δ	δ	NOUN
ejpam-6566	276	49	-	-	NOUN
ejpam-6566	276	50	cl(b	cl(b	NOUN
ejpam-6566	276	51	)	)	PUNCT
ejpam-6566	276	52	)	)	PUNCT
ejpam-6566	276	53	)	)	PUNCT
ejpam-6566	276	54	)	)	PUNCT
ejpam-6566	276	55	)	)	PUNCT
ejpam-6566	277	1	⊆	⊆	X
ejpam-6566	277	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	277	3	-	-	PUNCT
ejpam-6566	277	4	δ	δ	NOUN
ejpam-6566	277	5	-	-	NOUN
ejpam-6566	277	6	cl(b	cl(b	NOUN
ejpam-6566	277	7	)	)	PUNCT
ejpam-6566	277	8	)	)	PUNCT
ejpam-6566	277	9	for	for	ADP
ejpam-6566	277	10	every	every	DET
ejpam-6566	277	11	subset	subset	NOUN
ejpam-6566	277	12	b	b	PROPN
ejpam-6566	277	13	of	of	ADP
ejpam-6566	277	14	y	y	PROPN
ejpam-6566	277	15	;	;	PUNCT
ejpam-6566	277	16	(	(	PUNCT
ejpam-6566	277	17	3	3	X
ejpam-6566	277	18	)	)	PUNCT
ejpam-6566	277	19	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	277	20	-	-	PUNCT
ejpam-6566	277	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	277	22	-	-	PUNCT
ejpam-6566	277	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	277	24	-	-	PUNCT
ejpam-6566	277	25	cl(b	cl(b	NOUN
ejpam-6566	277	26	)	)	PUNCT
ejpam-6566	277	27	)	)	PUNCT
ejpam-6566	277	28	)	)	PUNCT
ejpam-6566	277	29	)	)	PUNCT
ejpam-6566	277	30	)	)	PUNCT
ejpam-6566	278	1	⊆	⊆	X
ejpam-6566	278	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	278	3	-	-	PUNCT
ejpam-6566	278	4	δ	δ	NOUN
ejpam-6566	278	5	-	-	NOUN
ejpam-6566	278	6	cl(b	cl(b	NOUN
ejpam-6566	278	7	)	)	PUNCT
ejpam-6566	278	8	)	)	PUNCT
ejpam-6566	278	9	for	for	ADP
ejpam-6566	278	10	every	every	DET
ejpam-6566	278	11	subset	subset	NOUN
ejpam-6566	278	12	b	b	PROPN
ejpam-6566	278	13	of	of	ADP
ejpam-6566	278	14	y	y	PROPN
ejpam-6566	278	15	.	.	PUNCT
ejpam-6566	279	1	proof	proof	NOUN
ejpam-6566	279	2	.	.	PUNCT
ejpam-6566	280	1	the	the	DET
ejpam-6566	280	2	proof	proof	NOUN
ejpam-6566	280	3	is	be	AUX
ejpam-6566	280	4	similar	similar	ADJ
ejpam-6566	280	5	to	to	ADP
ejpam-6566	280	6	that	that	PRON
ejpam-6566	280	7	of	of	ADP
ejpam-6566	280	8	theorem	theorem	NOUN
ejpam-6566	280	9	9	9	NUM
ejpam-6566	280	10	.	.	PUNCT
ejpam-6566	281	1	lemma	lemma	PROPN
ejpam-6566	281	2	5	5	NUM
ejpam-6566	281	3	.	.	PUNCT
ejpam-6566	282	1	if	if	SCONJ
ejpam-6566	282	2	f	f	PROPN
ejpam-6566	282	3	:	:	PUNCT
ejpam-6566	282	4	(	(	PUNCT
ejpam-6566	282	5	x	x	X
ejpam-6566	282	6	,	,	PUNCT
ejpam-6566	282	7	τ	τ	PROPN
ejpam-6566	282	8	,	,	PUNCT
ejpam-6566	282	9	i	i	NOUN
ejpam-6566	282	10	)	)	PUNCT
ejpam-6566	282	11	→	→	PUNCT
ejpam-6566	282	12	(	(	PUNCT
ejpam-6566	282	13	y	y	PROPN
ejpam-6566	282	14	,	,	PUNCT
ejpam-6566	282	15	σ1	σ1	PROPN
ejpam-6566	282	16	,	,	PUNCT
ejpam-6566	282	17	σ2	σ2	NOUN
ejpam-6566	282	18	)	)	PUNCT
ejpam-6566	282	19	is	be	AUX
ejpam-6566	282	20	lower	low	ADJ
ejpam-6566	282	21	almost	almost	ADV
ejpam-6566	282	22	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	282	23	,	,	PUNCT
ejpam-6566	282	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	282	25	,	,	PUNCT
ejpam-6566	282	26	then	then	ADV
ejpam-6566	282	27	for	for	ADP
ejpam-6566	282	28	each	each	DET
ejpam-6566	282	29	x	x	SYM
ejpam-6566	282	30	∈	∈	PROPN
ejpam-6566	282	31	x	x	X
ejpam-6566	282	32	and	and	CCONJ
ejpam-6566	282	33	each	each	DET
ejpam-6566	282	34	subset	subset	NOUN
ejpam-6566	282	35	b	b	PROPN
ejpam-6566	282	36	of	of	ADP
ejpam-6566	282	37	y	y	PROPN
ejpam-6566	282	38	with	with	ADP
ejpam-6566	282	39	σ1σ2	σ1σ2	PROPN
ejpam-6566	282	40	-	-	PUNCT
ejpam-6566	282	41	δ	δ	NOUN
ejpam-6566	282	42	-	-	PUNCT
ejpam-6566	282	43	int(b	int(b	NOUN
ejpam-6566	282	44	)	)	PUNCT
ejpam-6566	282	45	∩	∩	ADJ
ejpam-6566	282	46	f	f	PROPN
ejpam-6566	282	47	(	(	PUNCT
ejpam-6566	282	48	x	x	X
ejpam-6566	282	49	)	)	PUNCT
ejpam-6566	282	50	̸=	̸=	NOUN
ejpam-6566	282	51	∅	∅	NOUN
ejpam-6566	282	52	,	,	PUNCT
ejpam-6566	282	53	there	there	PRON
ejpam-6566	282	54	exists	exist	VERB
ejpam-6566	282	55	a	a	DET
ejpam-6566	282	56	⋆-open	⋆-open	ADJ
ejpam-6566	282	57	set	set	NOUN
ejpam-6566	282	58	u	u	NOUN
ejpam-6566	282	59	of	of	ADP
ejpam-6566	282	60	x	x	PUNCT
ejpam-6566	282	61	containing	contain	VERB
ejpam-6566	282	62	x	x	PUNCT
ejpam-6566	282	63	such	such	ADJ
ejpam-6566	282	64	that	that	SCONJ
ejpam-6566	282	65	u	u	PROPN
ejpam-6566	282	66	⊆	⊆	NUM
ejpam-6566	282	67	f−(b	f−(b	NOUN
ejpam-6566	282	68	)	)	PUNCT
ejpam-6566	282	69	.	.	PUNCT
ejpam-6566	283	1	proof	proof	NOUN
ejpam-6566	283	2	.	.	PUNCT
ejpam-6566	284	1	let	let	VERB
ejpam-6566	284	2	x	x	PUNCT
ejpam-6566	284	3	∈	∈	PROPN
ejpam-6566	284	4	x	x	X
ejpam-6566	284	5	and	and	CCONJ
ejpam-6566	284	6	b	b	X
ejpam-6566	284	7	be	be	AUX
ejpam-6566	284	8	a	a	DET
ejpam-6566	284	9	subset	subset	NOUN
ejpam-6566	284	10	of	of	ADP
ejpam-6566	284	11	y	y	PRON
ejpam-6566	284	12	with	with	ADP
ejpam-6566	284	13	σ1σ2	σ1σ2	PROPN
ejpam-6566	284	14	-	-	PUNCT
ejpam-6566	284	15	δ	δ	NOUN
ejpam-6566	284	16	-	-	PUNCT
ejpam-6566	284	17	int(b	int(b	NOUN
ejpam-6566	284	18	)	)	PUNCT
ejpam-6566	284	19	∩	∩	ADJ
ejpam-6566	284	20	f	f	PROPN
ejpam-6566	284	21	(	(	PUNCT
ejpam-6566	284	22	x	x	X
ejpam-6566	284	23	)	)	PUNCT
ejpam-6566	284	24	̸=	̸=	PROPN
ejpam-6566	284	25	∅.	∅.	ADV
ejpam-6566	284	26	since	since	SCONJ
ejpam-6566	284	27	σ1σ2	σ1σ2	NOUN
ejpam-6566	284	28	-	-	PUNCT
ejpam-6566	284	29	δ	δ	NOUN
ejpam-6566	284	30	-	-	PUNCT
ejpam-6566	284	31	int(b	int(b	NOUN
ejpam-6566	284	32	)	)	PUNCT
ejpam-6566	284	33	∩	∩	ADJ
ejpam-6566	284	34	f	f	PROPN
ejpam-6566	284	35	(	(	PUNCT
ejpam-6566	284	36	x	x	X
ejpam-6566	284	37	)	)	PUNCT
ejpam-6566	284	38	̸=	̸=	NOUN
ejpam-6566	284	39	∅	∅	NOUN
ejpam-6566	284	40	,	,	PUNCT
ejpam-6566	284	41	there	there	PRON
ejpam-6566	284	42	exists	exist	VERB
ejpam-6566	284	43	a	a	DET
ejpam-6566	284	44	nonempty	nonempty	ADJ
ejpam-6566	284	45	(	(	PUNCT
ejpam-6566	284	46	σ1	σ1	NOUN
ejpam-6566	284	47	,	,	PUNCT
ejpam-6566	284	48	σ2)r	σ2)r	NOUN
ejpam-6566	284	49	-	-	PUNCT
ejpam-6566	284	50	open	open	ADJ
ejpam-6566	284	51	set	set	VERB
ejpam-6566	284	52	v	v	NOUN
ejpam-6566	284	53	of	of	ADP
ejpam-6566	284	54	y	y	PRON
ejpam-6566	284	55	such	such	ADJ
ejpam-6566	284	56	that	that	PRON
ejpam-6566	284	57	v	v	ADP
ejpam-6566	284	58	⊆	⊆	NUM
ejpam-6566	284	59	b	b	NOUN
ejpam-6566	284	60	and	and	CCONJ
ejpam-6566	284	61	v	v	NOUN
ejpam-6566	284	62	∩	∩	ADJ
ejpam-6566	284	63	f	f	X
ejpam-6566	284	64	(	(	PUNCT
ejpam-6566	284	65	x	x	X
ejpam-6566	284	66	)	)	PUNCT
ejpam-6566	284	67	̸=	̸=	PROPN
ejpam-6566	284	68	∅.	∅.	ADV
ejpam-6566	284	69	since	since	SCONJ
ejpam-6566	284	70	f	f	PROPN
ejpam-6566	284	71	is	be	AUX
ejpam-6566	284	72	lower	low	ADJ
ejpam-6566	284	73	almost	almost	ADV
ejpam-6566	284	74	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	284	75	,	,	PUNCT
ejpam-6566	284	76	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	284	77	,	,	PUNCT
ejpam-6566	284	78	there	there	PRON
ejpam-6566	284	79	exists	exist	VERB
ejpam-6566	284	80	a	a	DET
ejpam-6566	284	81	⋆-open	⋆-open	ADJ
ejpam-6566	284	82	set	set	NOUN
ejpam-6566	284	83	u	u	NOUN
ejpam-6566	284	84	of	of	ADP
ejpam-6566	284	85	x	x	PUNCT
ejpam-6566	284	86	containing	contain	VERB
ejpam-6566	284	87	x	x	PUNCT
ejpam-6566	284	88	such	such	ADJ
ejpam-6566	284	89	that	that	DET
ejpam-6566	284	90	v	v	NOUN
ejpam-6566	284	91	∩f	∩f	NOUN
ejpam-6566	284	92	(	(	PUNCT
ejpam-6566	284	93	z	z	X
ejpam-6566	284	94	)	)	PUNCT
ejpam-6566	284	95	̸=	̸=	NOUN
ejpam-6566	284	96	∅	∅	NOUN
ejpam-6566	284	97	for	for	ADP
ejpam-6566	284	98	each	each	DET
ejpam-6566	284	99	z	z	NOUN
ejpam-6566	284	100	∈	∈	PROPN
ejpam-6566	284	101	u	u	NOUN
ejpam-6566	284	102	;	;	PUNCT
ejpam-6566	284	103	hence	hence	ADV
ejpam-6566	284	104	u	u	NOUN
ejpam-6566	284	105	⊆	⊆	NUM
ejpam-6566	284	106	f−(b	f−(b	NOUN
ejpam-6566	284	107	)	)	PUNCT
ejpam-6566	284	108	.	.	PUNCT
ejpam-6566	285	1	theorem	theorem	VERB
ejpam-6566	285	2	11	11	NUM
ejpam-6566	285	3	.	.	PUNCT
ejpam-6566	286	1	for	for	ADP
ejpam-6566	286	2	a	a	DET
ejpam-6566	286	3	multifunction	multifunction	NOUN
ejpam-6566	286	4	f	f	NOUN
ejpam-6566	286	5	:	:	PUNCT
ejpam-6566	286	6	(	(	PUNCT
ejpam-6566	286	7	x	x	X
ejpam-6566	286	8	,	,	PUNCT
ejpam-6566	286	9	τ	τ	PROPN
ejpam-6566	286	10	,	,	PUNCT
ejpam-6566	286	11	i	i	NOUN
ejpam-6566	286	12	)	)	PUNCT
ejpam-6566	286	13	→	→	PUNCT
ejpam-6566	286	14	(	(	PUNCT
ejpam-6566	286	15	y	y	PROPN
ejpam-6566	286	16	,	,	PUNCT
ejpam-6566	286	17	σ1	σ1	PROPN
ejpam-6566	286	18	,	,	PUNCT
ejpam-6566	286	19	σ2	σ2	NOUN
ejpam-6566	286	20	)	)	PUNCT
ejpam-6566	286	21	,	,	PUNCT
ejpam-6566	286	22	the	the	DET
ejpam-6566	286	23	following	follow	VERB
ejpam-6566	286	24	properties	property	NOUN
ejpam-6566	286	25	are	be	AUX
ejpam-6566	286	26	equivalent	equivalent	ADJ
ejpam-6566	286	27	:	:	PUNCT
ejpam-6566	286	28	(	(	PUNCT
ejpam-6566	286	29	1	1	X
ejpam-6566	286	30	)	)	PUNCT
ejpam-6566	286	31	f	f	PROPN
ejpam-6566	286	32	is	be	AUX
ejpam-6566	286	33	lower	low	ADJ
ejpam-6566	286	34	almost	almost	ADV
ejpam-6566	286	35	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	286	36	,	,	PUNCT
ejpam-6566	286	37	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	286	38	;	;	PUNCT
ejpam-6566	286	39	(	(	PUNCT
ejpam-6566	286	40	2	2	X
ejpam-6566	286	41	)	)	PUNCT
ejpam-6566	286	42	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6566	286	43	)	)	PUNCT
ejpam-6566	286	44	)	)	PUNCT
ejpam-6566	287	1	⊆	⊆	X
ejpam-6566	287	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	287	3	-	-	PUNCT
ejpam-6566	287	4	δ	δ	NOUN
ejpam-6566	287	5	-	-	NOUN
ejpam-6566	287	6	cl(b	cl(b	NOUN
ejpam-6566	287	7	)	)	PUNCT
ejpam-6566	287	8	)	)	PUNCT
ejpam-6566	287	9	for	for	ADP
ejpam-6566	287	10	every	every	DET
ejpam-6566	287	11	subset	subset	NOUN
ejpam-6566	287	12	b	b	PROPN
ejpam-6566	287	13	of	of	ADP
ejpam-6566	287	14	y	y	PROPN
ejpam-6566	287	15	;	;	PUNCT
ejpam-6566	287	16	(	(	PUNCT
ejpam-6566	287	17	3	3	X
ejpam-6566	287	18	)	)	PUNCT
ejpam-6566	287	19	f	f	NOUN
ejpam-6566	287	20	(	(	PUNCT
ejpam-6566	287	21	cl⋆(a	cl⋆(a	PROPN
ejpam-6566	287	22	)	)	PUNCT
ejpam-6566	287	23	)	)	PUNCT
ejpam-6566	287	24	⊆	⊆	X
ejpam-6566	287	25	σ1σ2	σ1σ2	PROPN
ejpam-6566	287	26	-	-	PUNCT
ejpam-6566	287	27	δ	δ	NOUN
ejpam-6566	287	28	-	-	NOUN
ejpam-6566	287	29	cl(f	cl(f	PROPN
ejpam-6566	287	30	(	(	PUNCT
ejpam-6566	287	31	a	a	NOUN
ejpam-6566	287	32	)	)	PUNCT
ejpam-6566	287	33	)	)	PUNCT
ejpam-6566	287	34	for	for	ADP
ejpam-6566	287	35	every	every	DET
ejpam-6566	287	36	subset	subset	NOUN
ejpam-6566	287	37	a	a	PRON
ejpam-6566	287	38	of	of	ADP
ejpam-6566	287	39	x	x	PRON
ejpam-6566	287	40	;	;	PUNCT
ejpam-6566	287	41	(	(	PUNCT
ejpam-6566	287	42	4	4	X
ejpam-6566	287	43	)	)	PUNCT
ejpam-6566	287	44	f+(k	f+(k	NUM
ejpam-6566	287	45	)	)	PUNCT
ejpam-6566	287	46	is	be	AUX
ejpam-6566	287	47	⋆-closed	⋆-close	VERB
ejpam-6566	287	48	in	in	ADP
ejpam-6566	287	49	x	x	PUNCT
ejpam-6566	287	50	for	for	ADP
ejpam-6566	287	51	every	every	DET
ejpam-6566	287	52	σ1σ2	σ1σ2	NUM
ejpam-6566	287	53	-	-	PUNCT
ejpam-6566	287	54	δ	δ	NOUN
ejpam-6566	287	55	-	-	PUNCT
ejpam-6566	287	56	closed	close	VERB
ejpam-6566	287	57	set	set	ADJ
ejpam-6566	287	58	k	k	PROPN
ejpam-6566	287	59	of	of	ADP
ejpam-6566	287	60	y	y	PROPN
ejpam-6566	287	61	;	;	PUNCT
ejpam-6566	287	62	(	(	PUNCT
ejpam-6566	287	63	5	5	X
ejpam-6566	287	64	)	)	PUNCT
ejpam-6566	287	65	f−(v	f−(v	NOUN
ejpam-6566	287	66	)	)	PUNCT
ejpam-6566	287	67	is	be	AUX
ejpam-6566	287	68	⋆-open	⋆-open	ADJ
ejpam-6566	287	69	in	in	ADP
ejpam-6566	287	70	x	x	PUNCT
ejpam-6566	287	71	for	for	SCONJ
ejpam-6566	287	72	every	every	DET
ejpam-6566	287	73	σ1σ2	σ1σ2	NUM
ejpam-6566	287	74	-	-	PUNCT
ejpam-6566	287	75	δ	δ	NOUN
ejpam-6566	287	76	-	-	ADJ
ejpam-6566	287	77	open	open	ADJ
ejpam-6566	287	78	set	set	VERB
ejpam-6566	287	79	v	v	NOUN
ejpam-6566	287	80	of	of	ADP
ejpam-6566	287	81	y	y	PROPN
ejpam-6566	287	82	;	;	PUNCT
ejpam-6566	287	83	(	(	PUNCT
ejpam-6566	287	84	6	6	X
ejpam-6566	287	85	)	)	PUNCT
ejpam-6566	287	86	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6566	287	87	-	-	PUNCT
ejpam-6566	287	88	δ	δ	NOUN
ejpam-6566	287	89	-	-	PUNCT
ejpam-6566	287	90	int(b	int(b	NOUN
ejpam-6566	287	91	)	)	PUNCT
ejpam-6566	287	92	)	)	PUNCT
ejpam-6566	288	1	⊆	⊆	NUM
ejpam-6566	288	2	int⋆(f−(b	int⋆(f−(b	NOUN
ejpam-6566	288	3	)	)	PUNCT
ejpam-6566	288	4	)	)	PUNCT
ejpam-6566	288	5	for	for	ADP
ejpam-6566	288	6	every	every	DET
ejpam-6566	288	7	subset	subset	NOUN
ejpam-6566	288	8	b	b	PROPN
ejpam-6566	288	9	of	of	ADP
ejpam-6566	288	10	y	y	PROPN
ejpam-6566	288	11	.	.	PUNCT
ejpam-6566	289	1	proof	proof	NOUN
ejpam-6566	289	2	.	.	PUNCT
ejpam-6566	290	1	(	(	PUNCT
ejpam-6566	290	2	1	1	X
ejpam-6566	290	3	)	)	PUNCT
ejpam-6566	290	4	⇒	⇒	NOUN
ejpam-6566	290	5	(	(	PUNCT
ejpam-6566	290	6	2	2	NUM
ejpam-6566	290	7	):	):	PUNCT
ejpam-6566	290	8	let	let	VERB
ejpam-6566	290	9	b	b	X
ejpam-6566	290	10	be	be	AUX
ejpam-6566	290	11	any	any	DET
ejpam-6566	290	12	subset	subset	NOUN
ejpam-6566	290	13	of	of	ADP
ejpam-6566	290	14	y	y	PROPN
ejpam-6566	290	15	.	.	PUNCT
ejpam-6566	290	16	suppose	suppose	VERB
ejpam-6566	290	17	that	that	SCONJ
ejpam-6566	290	18	x	x	PROPN
ejpam-6566	290	19	̸∈	̸∈	PROPN
ejpam-6566	290	20	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	290	21	-	-	PUNCT
ejpam-6566	290	22	δ	δ	NOUN
ejpam-6566	290	23	-	-	NOUN
ejpam-6566	290	24	cl(b	cl(b	NOUN
ejpam-6566	290	25	)	)	PUNCT
ejpam-6566	290	26	)	)	PUNCT
ejpam-6566	290	27	.	.	PUNCT
ejpam-6566	291	1	then	then	ADV
ejpam-6566	291	2	,	,	PUNCT
ejpam-6566	291	3	we	we	PRON
ejpam-6566	291	4	have	have	VERB
ejpam-6566	291	5	x	x	PART
ejpam-6566	291	6	∈	∈	NOUN
ejpam-6566	291	7	f−(y	f−(y	NOUN
ejpam-6566	291	8	−σ1σ2	−σ1σ2	PROPN
ejpam-6566	291	9	-	-	PUNCT
ejpam-6566	291	10	δ	δ	NOUN
ejpam-6566	291	11	-	-	NOUN
ejpam-6566	291	12	cl(b	cl(b	NOUN
ejpam-6566	291	13	)	)	PUNCT
ejpam-6566	291	14	)	)	PUNCT
ejpam-6566	292	1	=	=	PUNCT
ejpam-6566	292	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6566	292	3	-	-	PUNCT
ejpam-6566	292	4	δ	δ	NOUN
ejpam-6566	292	5	-	-	PUNCT
ejpam-6566	292	6	int(y	int(y	PROPN
ejpam-6566	292	7	−b	−b	NOUN
ejpam-6566	292	8	)	)	PUNCT
ejpam-6566	292	9	)	)	PUNCT
ejpam-6566	292	10	.	.	PUNCT
ejpam-6566	293	1	there	there	PRON
ejpam-6566	293	2	exists	exist	VERB
ejpam-6566	293	3	a	a	DET
ejpam-6566	293	4	⋆-open	⋆-open	ADJ
ejpam-6566	293	5	set	set	NOUN
ejpam-6566	293	6	u	u	NOUN
ejpam-6566	293	7	of	of	ADP
ejpam-6566	293	8	x	x	PUNCT
ejpam-6566	293	9	containing	contain	VERB
ejpam-6566	293	10	x	x	PUNCT
ejpam-6566	293	11	such	such	ADJ
ejpam-6566	293	12	that	that	SCONJ
ejpam-6566	293	13	u	u	NOUN
ejpam-6566	293	14	⊆	⊆	NUM
ejpam-6566	293	15	f−(y	f−(y	NOUN
ejpam-6566	293	16	−b	−b	NOUN
ejpam-6566	293	17	)	)	PUNCT
ejpam-6566	294	1	=	=	PUNCT
ejpam-6566	294	2	x	x	X
ejpam-6566	295	1	−	−	NOUN
ejpam-6566	295	2	f+(b	f+(b	NOUN
ejpam-6566	295	3	)	)	PUNCT
ejpam-6566	295	4	.	.	PUNCT
ejpam-6566	296	1	thus	thus	ADV
ejpam-6566	296	2	,	,	PUNCT
ejpam-6566	296	3	u	u	NOUN
ejpam-6566	296	4	∩	∩	NOUN
ejpam-6566	296	5	f+(b	f+(b	NOUN
ejpam-6566	296	6	)	)	PUNCT
ejpam-6566	296	7	=	=	SYM
ejpam-6566	296	8	∅	∅	NOUN
ejpam-6566	296	9	and	and	CCONJ
ejpam-6566	296	10	hence	hence	ADV
ejpam-6566	297	1	x	x	X
ejpam-6566	297	2	∈	∈	ADJ
ejpam-6566	297	3	x	x	PUNCT
ejpam-6566	297	4	−	−	NOUN
ejpam-6566	297	5	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6566	297	6	)	)	PUNCT
ejpam-6566	297	7	)	)	PUNCT
ejpam-6566	297	8	.	.	PUNCT
ejpam-6566	298	1	this	this	PRON
ejpam-6566	298	2	shows	show	VERB
ejpam-6566	298	3	that	that	SCONJ
ejpam-6566	298	4	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6566	298	5	)	)	PUNCT
ejpam-6566	298	6	)	)	PUNCT
ejpam-6566	299	1	⊆	⊆	X
ejpam-6566	299	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	299	3	-	-	PUNCT
ejpam-6566	299	4	δ	δ	NOUN
ejpam-6566	299	5	-	-	NOUN
ejpam-6566	299	6	cl(b	cl(b	NOUN
ejpam-6566	299	7	)	)	PUNCT
ejpam-6566	299	8	)	)	PUNCT
ejpam-6566	299	9	.	.	PUNCT
ejpam-6566	300	1	c.	c.	PROPN
ejpam-6566	300	2	viriyapong	viriyapong	PROPN
ejpam-6566	300	3	,	,	PUNCT
ejpam-6566	300	4	a.	a.	PROPN
ejpam-6566	300	5	sama	sama	PROPN
ejpam-6566	300	6	-	-	PUNCT
ejpam-6566	300	7	ae	ae	PROPN
ejpam-6566	300	8	,	,	PUNCT
ejpam-6566	300	9	c.	c.	PROPN
ejpam-6566	300	10	boonpok	boonpok	PROPN
ejpam-6566	300	11	/	/	SYM
ejpam-6566	300	12	eur	eur	PROPN
ejpam-6566	300	13	.	.	PUNCT
ejpam-6566	301	1	j.	j.	PROPN
ejpam-6566	301	2	pure	pure	PROPN
ejpam-6566	301	3	appl	appl	PROPN
ejpam-6566	301	4	.	.	PROPN
ejpam-6566	301	5	math	math	PROPN
ejpam-6566	301	6	,	,	PUNCT
ejpam-6566	301	7	18	18	NUM
ejpam-6566	301	8	(	(	PUNCT
ejpam-6566	301	9	3	3	NUM
ejpam-6566	301	10	)	)	PUNCT
ejpam-6566	301	11	(	(	PUNCT
ejpam-6566	301	12	2025	2025	NUM
ejpam-6566	301	13	)	)	PUNCT
ejpam-6566	301	14	,	,	PUNCT
ejpam-6566	301	15	6566	6566	NUM
ejpam-6566	301	16	11	11	NUM
ejpam-6566	301	17	of	of	ADP
ejpam-6566	301	18	14	14	NUM
ejpam-6566	301	19	(	(	PUNCT
ejpam-6566	301	20	2	2	NUM
ejpam-6566	301	21	)	)	PUNCT
ejpam-6566	301	22	⇒	⇒	NOUN
ejpam-6566	301	23	(	(	PUNCT
ejpam-6566	301	24	3	3	NUM
ejpam-6566	301	25	):	):	PUNCT
ejpam-6566	301	26	let	let	VERB
ejpam-6566	301	27	a	a	DET
ejpam-6566	301	28	be	be	AUX
ejpam-6566	301	29	any	any	DET
ejpam-6566	301	30	subset	subset	NOUN
ejpam-6566	301	31	of	of	ADP
ejpam-6566	301	32	x.	x.	NOUN
ejpam-6566	301	33	by	by	ADP
ejpam-6566	301	34	(	(	PUNCT
ejpam-6566	301	35	2	2	NUM
ejpam-6566	301	36	)	)	PUNCT
ejpam-6566	301	37	,	,	PUNCT
ejpam-6566	301	38	we	we	PRON
ejpam-6566	301	39	have	have	VERB
ejpam-6566	301	40	cl⋆(a	cl⋆(a	NOUN
ejpam-6566	301	41	)	)	PUNCT
ejpam-6566	301	42	⊆	⊆	NUM
ejpam-6566	301	43	τ1τ2	τ1τ2	NOUN
ejpam-6566	301	44	-	-	NOUN
ejpam-6566	301	45	cl(f	cl(f	NOUN
ejpam-6566	302	1	+	+	PROPN
ejpam-6566	303	1	(	(	PUNCT
ejpam-6566	303	2	f	f	X
ejpam-6566	303	3	(	(	PUNCT
ejpam-6566	303	4	a	a	NOUN
ejpam-6566	303	5	)	)	PUNCT
ejpam-6566	303	6	)	)	PUNCT
ejpam-6566	303	7	)	)	PUNCT
ejpam-6566	303	8	⊆	⊆	X
ejpam-6566	303	9	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	303	10	-	-	PUNCT
ejpam-6566	303	11	δ	δ	NOUN
ejpam-6566	303	12	-	-	NOUN
ejpam-6566	303	13	cl(f	cl(f	PROPN
ejpam-6566	303	14	(	(	PUNCT
ejpam-6566	303	15	a	a	NOUN
ejpam-6566	303	16	)	)	PUNCT
ejpam-6566	303	17	)	)	PUNCT
ejpam-6566	303	18	)	)	PUNCT
ejpam-6566	303	19	and	and	CCONJ
ejpam-6566	303	20	hence	hence	ADV
ejpam-6566	303	21	f	f	PROPN
ejpam-6566	303	22	(	(	PUNCT
ejpam-6566	303	23	cl⋆(a	cl⋆(a	PROPN
ejpam-6566	303	24	)	)	PUNCT
ejpam-6566	303	25	)	)	PUNCT
ejpam-6566	303	26	⊆	⊆	X
ejpam-6566	303	27	σ1σ2	σ1σ2	PROPN
ejpam-6566	303	28	-	-	PUNCT
ejpam-6566	303	29	δ	δ	NOUN
ejpam-6566	303	30	-	-	NOUN
ejpam-6566	303	31	cl(f	cl(f	PROPN
ejpam-6566	303	32	(	(	PUNCT
ejpam-6566	303	33	a	a	NOUN
ejpam-6566	303	34	)	)	PUNCT
ejpam-6566	303	35	)	)	PUNCT
ejpam-6566	303	36	.	.	PUNCT
ejpam-6566	304	1	(	(	PUNCT
ejpam-6566	304	2	3	3	X
ejpam-6566	304	3	)	)	PUNCT
ejpam-6566	304	4	⇒	⇒	NOUN
ejpam-6566	304	5	(	(	PUNCT
ejpam-6566	304	6	1	1	NUM
ejpam-6566	304	7	):	):	PUNCT
ejpam-6566	304	8	let	let	VERB
ejpam-6566	304	9	b	b	X
ejpam-6566	304	10	be	be	AUX
ejpam-6566	304	11	any	any	DET
ejpam-6566	304	12	subset	subset	NOUN
ejpam-6566	304	13	of	of	ADP
ejpam-6566	304	14	y	y	PROPN
ejpam-6566	304	15	.	.	PUNCT
ejpam-6566	305	1	then	then	ADV
ejpam-6566	305	2	,	,	PUNCT
ejpam-6566	305	3	by	by	ADP
ejpam-6566	305	4	the	the	DET
ejpam-6566	305	5	hypothesis	hypothesis	NOUN
ejpam-6566	305	6	and	and	CCONJ
ejpam-6566	305	7	lemma	lemma	PROPN
ejpam-6566	305	8	4	4	NUM
ejpam-6566	305	9	,	,	PUNCT
ejpam-6566	305	10	f	f	X
ejpam-6566	305	11	(	(	PUNCT
ejpam-6566	305	12	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6566	305	13	-	-	PUNCT
ejpam-6566	305	14	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	305	15	-	-	PUNCT
ejpam-6566	305	16	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	305	17	-	-	PUNCT
ejpam-6566	305	18	cl(b	cl(b	NOUN
ejpam-6566	305	19	)	)	PUNCT
ejpam-6566	305	20	)	)	PUNCT
ejpam-6566	305	21	)	)	PUNCT
ejpam-6566	305	22	)	)	PUNCT
ejpam-6566	305	23	)	)	PUNCT
ejpam-6566	305	24	)	)	PUNCT
ejpam-6566	306	1	⊆	⊆	X
ejpam-6566	306	2	τ1τ2	τ1τ2	PROPN
ejpam-6566	306	3	-	-	ADJ
ejpam-6566	306	4	δ	δ	NOUN
ejpam-6566	306	5	-	-	NOUN
ejpam-6566	306	6	cl(f	cl(f	PROPN
ejpam-6566	306	7	(	(	PUNCT
ejpam-6566	306	8	f+(σ1σ2	f+(σ1σ2	VERB
ejpam-6566	306	9	-	-	PUNCT
ejpam-6566	306	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	306	11	-	-	PUNCT
ejpam-6566	306	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	306	13	-	-	PUNCT
ejpam-6566	306	14	cl(b	cl(b	NOUN
ejpam-6566	306	15	)	)	PUNCT
ejpam-6566	306	16	)	)	PUNCT
ejpam-6566	306	17	)	)	PUNCT
ejpam-6566	306	18	)	)	PUNCT
ejpam-6566	306	19	)	)	PUNCT
ejpam-6566	306	20	)	)	PUNCT
ejpam-6566	307	1	⊆	⊆	X
ejpam-6566	307	2	σ1σ2	σ1σ2	X
ejpam-6566	307	3	-	-	PUNCT
ejpam-6566	307	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	307	5	-	-	PUNCT
ejpam-6566	307	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	307	7	-	-	PUNCT
ejpam-6566	307	8	cl(b	cl(b	NOUN
ejpam-6566	307	9	)	)	PUNCT
ejpam-6566	307	10	)	)	PUNCT
ejpam-6566	307	11	)	)	PUNCT
ejpam-6566	308	1	⊆	⊆	X
ejpam-6566	308	2	σ1σ2	σ1σ2	NUM
ejpam-6566	308	3	-	-	PUNCT
ejpam-6566	308	4	cl(b	cl(b	NOUN
ejpam-6566	308	5	)	)	PUNCT
ejpam-6566	308	6	and	and	CCONJ
ejpam-6566	308	7	hence	hence	ADV
ejpam-6566	308	8	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	ADV
ejpam-6566	308	9	-	-	PUNCT
ejpam-6566	308	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6566	308	11	-	-	PUNCT
ejpam-6566	308	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6566	308	13	-	-	PUNCT
ejpam-6566	308	14	cl(b	cl(b	NOUN
ejpam-6566	308	15	)	)	PUNCT
ejpam-6566	308	16	)	)	PUNCT
ejpam-6566	308	17	)	)	PUNCT
ejpam-6566	308	18	)	)	PUNCT
ejpam-6566	308	19	)	)	PUNCT
ejpam-6566	309	1	⊆	⊆	X
ejpam-6566	309	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	309	3	-	-	PUNCT
ejpam-6566	309	4	cl(b	cl(b	NOUN
ejpam-6566	309	5	)	)	PUNCT
ejpam-6566	309	6	)	)	PUNCT
ejpam-6566	309	7	.	.	PUNCT
ejpam-6566	310	1	by	by	ADP
ejpam-6566	310	2	theorem	theorem	NOUN
ejpam-6566	310	3	4	4	NUM
ejpam-6566	310	4	,	,	PUNCT
ejpam-6566	310	5	f	f	PROPN
ejpam-6566	310	6	is	be	AUX
ejpam-6566	310	7	lower	low	ADJ
ejpam-6566	310	8	almost	almost	ADV
ejpam-6566	310	9	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	310	10	,	,	PUNCT
ejpam-6566	310	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	310	12	.	.	PUNCT
ejpam-6566	311	1	(	(	PUNCT
ejpam-6566	311	2	2	2	X
ejpam-6566	311	3	)	)	PUNCT
ejpam-6566	311	4	⇒	⇒	NOUN
ejpam-6566	311	5	(	(	PUNCT
ejpam-6566	311	6	4	4	NUM
ejpam-6566	311	7	):	):	PUNCT
ejpam-6566	311	8	let	let	VERB
ejpam-6566	311	9	k	k	PRON
ejpam-6566	311	10	be	be	AUX
ejpam-6566	311	11	any	any	DET
ejpam-6566	311	12	σ1σ2	σ1σ2	NUM
ejpam-6566	311	13	-	-	PUNCT
ejpam-6566	311	14	δ	δ	NOUN
ejpam-6566	311	15	-	-	PUNCT
ejpam-6566	311	16	closed	closed	ADJ
ejpam-6566	311	17	set	set	NOUN
ejpam-6566	311	18	of	of	ADP
ejpam-6566	311	19	y	y	PROPN
ejpam-6566	311	20	.	.	PUNCT
ejpam-6566	312	1	then	then	ADV
ejpam-6566	312	2	,	,	PUNCT
ejpam-6566	312	3	σ1σ2	σ1σ2	PROPN
ejpam-6566	312	4	-	-	PUNCT
ejpam-6566	312	5	δ	δ	NOUN
ejpam-6566	312	6	-	-	NOUN
ejpam-6566	312	7	cl(k	cl(k	NOUN
ejpam-6566	312	8	)	)	PUNCT
ejpam-6566	313	1	=	=	VERB
ejpam-6566	313	2	k.	k.	PROPN
ejpam-6566	313	3	by	by	ADP
ejpam-6566	313	4	(	(	PUNCT
ejpam-6566	313	5	2	2	NUM
ejpam-6566	313	6	)	)	PUNCT
ejpam-6566	313	7	,	,	PUNCT
ejpam-6566	313	8	we	we	PRON
ejpam-6566	313	9	have	have	VERB
ejpam-6566	313	10	cl⋆(f+(k	cl⋆(f+(k	ADJ
ejpam-6566	313	11	)	)	PUNCT
ejpam-6566	313	12	)	)	PUNCT
ejpam-6566	314	1	⊆	⊆	X
ejpam-6566	314	2	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6566	314	3	-	-	PUNCT
ejpam-6566	314	4	δ	δ	NOUN
ejpam-6566	314	5	-	-	NOUN
ejpam-6566	314	6	cl(k	cl(k	NOUN
ejpam-6566	314	7	)	)	PUNCT
ejpam-6566	314	8	)	)	PUNCT
ejpam-6566	315	1	=	=	SYM
ejpam-6566	315	2	f+(k	f+(k	X
ejpam-6566	315	3	)	)	PUNCT
ejpam-6566	315	4	and	and	CCONJ
ejpam-6566	315	5	hence	hence	ADV
ejpam-6566	315	6	f+(k	f+(k	NUM
ejpam-6566	315	7	)	)	PUNCT
ejpam-6566	315	8	is	be	AUX
ejpam-6566	315	9	⋆-closed	⋆-close	VERB
ejpam-6566	315	10	in	in	ADP
ejpam-6566	315	11	x.	x.	PROPN
ejpam-6566	315	12	(	(	PUNCT
ejpam-6566	315	13	4	4	NUM
ejpam-6566	315	14	)	)	PUNCT
ejpam-6566	315	15	⇒	⇒	NOUN
ejpam-6566	315	16	(	(	PUNCT
ejpam-6566	315	17	5	5	NUM
ejpam-6566	315	18	):	):	PUNCT
ejpam-6566	315	19	the	the	DET
ejpam-6566	315	20	proof	proof	NOUN
ejpam-6566	315	21	is	be	AUX
ejpam-6566	315	22	obvious	obvious	ADJ
ejpam-6566	315	23	.	.	PUNCT
ejpam-6566	316	1	(	(	PUNCT
ejpam-6566	316	2	5	5	X
ejpam-6566	316	3	)	)	PUNCT
ejpam-6566	316	4	⇒	⇒	NOUN
ejpam-6566	316	5	(	(	PUNCT
ejpam-6566	316	6	6	6	NUM
ejpam-6566	316	7	):	):	PUNCT
ejpam-6566	316	8	let	let	VERB
ejpam-6566	316	9	b	b	X
ejpam-6566	316	10	be	be	AUX
ejpam-6566	316	11	any	any	DET
ejpam-6566	316	12	subset	subset	NOUN
ejpam-6566	316	13	of	of	ADP
ejpam-6566	316	14	y	y	PROPN
ejpam-6566	316	15	.	.	PUNCT
ejpam-6566	317	1	then	then	ADV
ejpam-6566	317	2	by	by	ADP
ejpam-6566	317	3	(	(	PUNCT
ejpam-6566	317	4	5	5	NUM
ejpam-6566	317	5	)	)	PUNCT
ejpam-6566	317	6	,	,	PUNCT
ejpam-6566	317	7	we	we	PRON
ejpam-6566	317	8	have	have	VERB
ejpam-6566	317	9	f−(σ1σ2	f−(σ1σ2	VERB
ejpam-6566	317	10	-	-	PUNCT
ejpam-6566	317	11	δ	δ	NOUN
ejpam-6566	317	12	-	-	PUNCT
ejpam-6566	317	13	int(b	int(b	NOUN
ejpam-6566	317	14	)	)	PUNCT
ejpam-6566	317	15	)	)	PUNCT
ejpam-6566	318	1	=	=	SYM
ejpam-6566	318	2	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	VERB
ejpam-6566	318	3	-	-	PUNCT
ejpam-6566	318	4	δ	δ	PROPN
ejpam-6566	318	5	-	-	PUNCT
ejpam-6566	318	6	int(b	int(b	NOUN
ejpam-6566	318	7	)	)	PUNCT
ejpam-6566	318	8	)	)	PUNCT
ejpam-6566	318	9	)	)	PUNCT
ejpam-6566	319	1	⊆	⊆	NUM
ejpam-6566	319	2	int⋆(f−(b	int⋆(f−(b	NOUN
ejpam-6566	319	3	)	)	PUNCT
ejpam-6566	319	4	)	)	PUNCT
ejpam-6566	319	5	.	.	PUNCT
ejpam-6566	320	1	(	(	PUNCT
ejpam-6566	320	2	6	6	X
ejpam-6566	320	3	)	)	PUNCT
ejpam-6566	320	4	⇒	⇒	NOUN
ejpam-6566	320	5	(	(	PUNCT
ejpam-6566	320	6	1	1	NUM
ejpam-6566	320	7	):	):	PUNCT
ejpam-6566	320	8	let	let	VERB
ejpam-6566	320	9	v	v	PART
ejpam-6566	320	10	be	be	AUX
ejpam-6566	320	11	any	any	DET
ejpam-6566	320	12	(	(	PUNCT
ejpam-6566	320	13	σ1	σ1	NOUN
ejpam-6566	320	14	,	,	PUNCT
ejpam-6566	320	15	σ2)r	σ2)r	NOUN
ejpam-6566	320	16	-	-	PUNCT
ejpam-6566	320	17	open	open	ADJ
ejpam-6566	320	18	set	set	NOUN
ejpam-6566	320	19	of	of	ADP
ejpam-6566	320	20	y	y	PROPN
ejpam-6566	320	21	.	.	PUNCT
ejpam-6566	321	1	then	then	ADV
ejpam-6566	321	2	,	,	PUNCT
ejpam-6566	321	3	we	we	PRON
ejpam-6566	321	4	have	have	VERB
ejpam-6566	321	5	v	v	NOUN
ejpam-6566	321	6	is	be	AUX
ejpam-6566	321	7	σ1σ2	σ1σ2	NOUN
ejpam-6566	321	8	-	-	PUNCT
ejpam-6566	321	9	δ	δ	NOUN
ejpam-6566	321	10	-	-	ADJ
ejpam-6566	321	11	open	open	ADJ
ejpam-6566	321	12	and	and	CCONJ
ejpam-6566	321	13	σ1σ2	σ1σ2	NOUN
ejpam-6566	321	14	-	-	PUNCT
ejpam-6566	321	15	δ	δ	NOUN
ejpam-6566	321	16	-	-	PUNCT
ejpam-6566	321	17	int(v	int(v	PROPN
ejpam-6566	321	18	)	)	PUNCT
ejpam-6566	321	19	=	=	SYM
ejpam-6566	321	20	v	v	NOUN
ejpam-6566	321	21	.	.	PUNCT
ejpam-6566	322	1	thus	thus	ADV
ejpam-6566	322	2	by	by	ADP
ejpam-6566	322	3	(	(	PUNCT
ejpam-6566	322	4	6	6	NUM
ejpam-6566	322	5	)	)	PUNCT
ejpam-6566	322	6	,	,	PUNCT
ejpam-6566	322	7	f−(v	f−(v	ADJ
ejpam-6566	322	8	)	)	PUNCT
ejpam-6566	322	9	⊆	⊆	NUM
ejpam-6566	322	10	int⋆(f−(v	int⋆(f−(v	NOUN
ejpam-6566	322	11	)	)	PUNCT
ejpam-6566	322	12	)	)	PUNCT
ejpam-6566	322	13	and	and	CCONJ
ejpam-6566	322	14	hence	hence	ADV
ejpam-6566	322	15	f−(v	f−(v	ADJ
ejpam-6566	322	16	)	)	PUNCT
ejpam-6566	322	17	is	be	AUX
ejpam-6566	322	18	⋆-open	⋆-open	ADJ
ejpam-6566	322	19	in	in	ADP
ejpam-6566	322	20	x.	x.	NOUN
ejpam-6566	322	21	by	by	ADP
ejpam-6566	322	22	theorem	theorem	NOUN
ejpam-6566	322	23	4	4	NUM
ejpam-6566	322	24	,	,	PUNCT
ejpam-6566	322	25	f	f	PROPN
ejpam-6566	322	26	is	be	AUX
ejpam-6566	322	27	lower	low	ADJ
ejpam-6566	322	28	almost	almost	ADV
ejpam-6566	322	29	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	322	30	,	,	PUNCT
ejpam-6566	322	31	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	322	32	.	.	NOUN
ejpam-6566	322	33	definition	definition	NOUN
ejpam-6566	322	34	3	3	NUM
ejpam-6566	322	35	.	.	PUNCT
ejpam-6566	323	1	[	[	X
ejpam-6566	323	2	30	30	NUM
ejpam-6566	323	3	]	]	X
ejpam-6566	323	4	a	a	DET
ejpam-6566	323	5	multifunction	multifunction	NOUN
ejpam-6566	323	6	f	f	NOUN
ejpam-6566	323	7	:	:	PUNCT
ejpam-6566	323	8	(	(	PUNCT
ejpam-6566	323	9	x	x	X
ejpam-6566	323	10	,	,	PUNCT
ejpam-6566	323	11	τ	τ	PROPN
ejpam-6566	323	12	,	,	PUNCT
ejpam-6566	323	13	i	i	NOUN
ejpam-6566	323	14	)	)	PUNCT
ejpam-6566	323	15	→	→	PUNCT
ejpam-6566	323	16	(	(	PUNCT
ejpam-6566	323	17	y	y	PROPN
ejpam-6566	323	18	,	,	PUNCT
ejpam-6566	323	19	σ1	σ1	PROPN
ejpam-6566	323	20	,	,	PUNCT
ejpam-6566	323	21	σ2	σ2	PROPN
ejpam-6566	323	22	)	)	PUNCT
ejpam-6566	323	23	is	be	AUX
ejpam-6566	323	24	said	say	VERB
ejpam-6566	323	25	to	to	PART
ejpam-6566	323	26	be	be	AUX
ejpam-6566	323	27	upper	upper	ADJ
ejpam-6566	323	28	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	323	29	,	,	PUNCT
ejpam-6566	323	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	323	31	at	at	ADP
ejpam-6566	323	32	a	a	DET
ejpam-6566	323	33	point	point	NOUN
ejpam-6566	323	34	x	x	SYM
ejpam-6566	323	35	∈	∈	NOUN
ejpam-6566	323	36	x	x	PUNCT
ejpam-6566	323	37	if	if	SCONJ
ejpam-6566	323	38	for	for	ADP
ejpam-6566	323	39	each	each	DET
ejpam-6566	323	40	σ1σ2	σ1σ2	VERB
ejpam-6566	323	41	-	-	ADJ
ejpam-6566	323	42	open	open	ADJ
ejpam-6566	323	43	set	set	NOUN
ejpam-6566	323	44	v	v	NOUN
ejpam-6566	323	45	of	of	ADP
ejpam-6566	323	46	y	y	PRON
ejpam-6566	323	47	such	such	ADJ
ejpam-6566	323	48	that	that	SCONJ
ejpam-6566	323	49	f	f	PROPN
ejpam-6566	323	50	(	(	PUNCT
ejpam-6566	323	51	x	x	X
ejpam-6566	323	52	)	)	PUNCT
ejpam-6566	323	53	⊆	⊆	NUM
ejpam-6566	323	54	v	v	NOUN
ejpam-6566	323	55	,	,	PUNCT
ejpam-6566	323	56	there	there	PRON
ejpam-6566	323	57	exists	exist	VERB
ejpam-6566	323	58	a	a	DET
ejpam-6566	323	59	⋆-open	⋆-open	ADJ
ejpam-6566	323	60	set	set	NOUN
ejpam-6566	323	61	u	u	NOUN
ejpam-6566	323	62	of	of	ADP
ejpam-6566	323	63	x	x	PUNCT
ejpam-6566	323	64	containing	contain	VERB
ejpam-6566	323	65	x	x	PUNCT
ejpam-6566	323	66	such	such	ADJ
ejpam-6566	323	67	that	that	SCONJ
ejpam-6566	323	68	f	f	PROPN
ejpam-6566	323	69	(	(	PUNCT
ejpam-6566	323	70	u	u	NOUN
ejpam-6566	323	71	)	)	PUNCT
ejpam-6566	323	72	⊆	⊆	NUM
ejpam-6566	323	73	v	v	NOUN
ejpam-6566	323	74	.	.	PUNCT
ejpam-6566	324	1	a	a	DET
ejpam-6566	324	2	multifunction	multifunction	NOUN
ejpam-6566	324	3	f	f	NOUN
ejpam-6566	324	4	:	:	PUNCT
ejpam-6566	324	5	(	(	PUNCT
ejpam-6566	324	6	x	x	X
ejpam-6566	324	7	,	,	PUNCT
ejpam-6566	324	8	τ	τ	PROPN
ejpam-6566	324	9	,	,	PUNCT
ejpam-6566	324	10	i	i	NOUN
ejpam-6566	324	11	)	)	PUNCT
ejpam-6566	324	12	→	→	PUNCT
ejpam-6566	324	13	(	(	PUNCT
ejpam-6566	324	14	y	y	PROPN
ejpam-6566	324	15	,	,	PUNCT
ejpam-6566	324	16	σ1	σ1	PROPN
ejpam-6566	324	17	,	,	PUNCT
ejpam-6566	324	18	σ2	σ2	PROPN
ejpam-6566	324	19	)	)	PUNCT
ejpam-6566	324	20	is	be	AUX
ejpam-6566	324	21	said	say	VERB
ejpam-6566	324	22	to	to	PART
ejpam-6566	324	23	be	be	AUX
ejpam-6566	324	24	upper	upper	ADJ
ejpam-6566	324	25	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6566	324	26	,	,	PUNCT
ejpam-6566	324	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	324	28	if	if	SCONJ
ejpam-6566	324	29	f	f	PROPN
ejpam-6566	324	30	is	be	AUX
ejpam-6566	324	31	upper	upper	ADJ
ejpam-6566	324	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6566	324	33	,	,	PUNCT
ejpam-6566	324	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	324	35	at	at	ADP
ejpam-6566	324	36	each	each	DET
ejpam-6566	324	37	point	point	NOUN
ejpam-6566	324	38	x	x	PUNCT
ejpam-6566	324	39	of	of	ADP
ejpam-6566	324	40	x.	x.	NOUN
ejpam-6566	324	41	definition	definition	NOUN
ejpam-6566	324	42	4	4	NUM
ejpam-6566	324	43	.	.	PUNCT
ejpam-6566	325	1	[	[	X
ejpam-6566	325	2	30	30	NUM
ejpam-6566	325	3	]	]	X
ejpam-6566	325	4	a	a	DET
ejpam-6566	325	5	multifunction	multifunction	NOUN
ejpam-6566	325	6	f	f	NOUN
ejpam-6566	325	7	:	:	PUNCT
ejpam-6566	325	8	(	(	PUNCT
ejpam-6566	325	9	x	x	X
ejpam-6566	325	10	,	,	PUNCT
ejpam-6566	325	11	τ	τ	PROPN
ejpam-6566	325	12	,	,	PUNCT
ejpam-6566	325	13	i	i	NOUN
ejpam-6566	325	14	)	)	PUNCT
ejpam-6566	325	15	→	→	PUNCT
ejpam-6566	325	16	(	(	PUNCT
ejpam-6566	325	17	y	y	PROPN
ejpam-6566	325	18	,	,	PUNCT
ejpam-6566	325	19	σ1	σ1	PROPN
ejpam-6566	325	20	,	,	PUNCT
ejpam-6566	325	21	σ2	σ2	PROPN
ejpam-6566	325	22	)	)	PUNCT
ejpam-6566	325	23	is	be	AUX
ejpam-6566	325	24	called	call	VERB
ejpam-6566	325	25	lower	low	ADJ
ejpam-6566	325	26	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	325	27	,	,	PUNCT
ejpam-6566	325	28	σ2)continuous	σ2)continuous	ADJ
ejpam-6566	325	29	at	at	ADP
ejpam-6566	325	30	a	a	DET
ejpam-6566	325	31	point	point	NOUN
ejpam-6566	325	32	x	x	SYM
ejpam-6566	325	33	∈	∈	NOUN
ejpam-6566	325	34	x	x	PUNCT
ejpam-6566	325	35	if	if	SCONJ
ejpam-6566	325	36	for	for	ADP
ejpam-6566	325	37	each	each	DET
ejpam-6566	325	38	σ1σ2	σ1σ2	VERB
ejpam-6566	325	39	-	-	ADJ
ejpam-6566	325	40	open	open	ADJ
ejpam-6566	325	41	set	set	NOUN
ejpam-6566	325	42	v	v	NOUN
ejpam-6566	325	43	of	of	ADP
ejpam-6566	325	44	y	y	PRON
ejpam-6566	325	45	such	such	ADJ
ejpam-6566	325	46	that	that	SCONJ
ejpam-6566	325	47	f	f	PROPN
ejpam-6566	325	48	(	(	PUNCT
ejpam-6566	325	49	x	x	NOUN
ejpam-6566	325	50	)	)	PUNCT
ejpam-6566	325	51	∩	∩	NOUN
ejpam-6566	325	52	v	v	ADP
ejpam-6566	325	53	̸=	̸=	PROPN
ejpam-6566	325	54	∅	∅	NOUN
ejpam-6566	325	55	,	,	PUNCT
ejpam-6566	325	56	there	there	PRON
ejpam-6566	325	57	exists	exist	VERB
ejpam-6566	325	58	a	a	DET
ejpam-6566	325	59	⋆-open	⋆-open	ADJ
ejpam-6566	325	60	set	set	NOUN
ejpam-6566	325	61	u	u	NOUN
ejpam-6566	325	62	of	of	ADP
ejpam-6566	325	63	x	x	PUNCT
ejpam-6566	325	64	containing	contain	VERB
ejpam-6566	325	65	x	x	PUNCT
ejpam-6566	325	66	such	such	ADJ
ejpam-6566	325	67	that	that	SCONJ
ejpam-6566	325	68	f	f	PROPN
ejpam-6566	325	69	(	(	PUNCT
ejpam-6566	325	70	z	z	NOUN
ejpam-6566	325	71	)	)	PUNCT
ejpam-6566	325	72	∩	∩	NOUN
ejpam-6566	325	73	v	v	ADP
ejpam-6566	325	74	̸=	̸=	PROPN
ejpam-6566	325	75	∅	∅	NOUN
ejpam-6566	325	76	for	for	ADP
ejpam-6566	325	77	every	every	DET
ejpam-6566	325	78	z	z	NOUN
ejpam-6566	325	79	∈	∈	PROPN
ejpam-6566	325	80	u	u	NOUN
ejpam-6566	325	81	.	.	PUNCT
ejpam-6566	326	1	a	a	DET
ejpam-6566	326	2	multifunction	multifunction	NOUN
ejpam-6566	326	3	f	f	NOUN
ejpam-6566	326	4	:	:	PUNCT
ejpam-6566	326	5	(	(	PUNCT
ejpam-6566	326	6	x	x	X
ejpam-6566	326	7	,	,	PUNCT
ejpam-6566	326	8	τ	τ	PROPN
ejpam-6566	326	9	,	,	PUNCT
ejpam-6566	326	10	i	i	NOUN
ejpam-6566	326	11	)	)	PUNCT
ejpam-6566	326	12	→	→	PUNCT
ejpam-6566	326	13	(	(	PUNCT
ejpam-6566	326	14	y	y	PROPN
ejpam-6566	326	15	,	,	PUNCT
ejpam-6566	326	16	σ1	σ1	PROPN
ejpam-6566	326	17	,	,	PUNCT
ejpam-6566	326	18	σ2	σ2	PROPN
ejpam-6566	326	19	)	)	PUNCT
ejpam-6566	326	20	is	be	AUX
ejpam-6566	326	21	called	call	VERB
ejpam-6566	326	22	lower	low	ADJ
ejpam-6566	326	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	326	24	,	,	PUNCT
ejpam-6566	326	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	326	26	if	if	SCONJ
ejpam-6566	326	27	f	f	PROPN
ejpam-6566	326	28	is	be	AUX
ejpam-6566	326	29	lower	low	ADJ
ejpam-6566	326	30	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	326	31	,	,	PUNCT
ejpam-6566	326	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	326	33	at	at	ADP
ejpam-6566	326	34	each	each	DET
ejpam-6566	326	35	point	point	NOUN
ejpam-6566	326	36	x	x	PUNCT
ejpam-6566	326	37	of	of	ADP
ejpam-6566	326	38	x.	x.	NOUN
ejpam-6566	326	39	remark	remark	PROPN
ejpam-6566	326	40	1	1	NUM
ejpam-6566	326	41	.	.	PUNCT
ejpam-6566	326	42	for	for	ADP
ejpam-6566	326	43	a	a	DET
ejpam-6566	326	44	multifunction	multifunction	NOUN
ejpam-6566	326	45	f	f	NOUN
ejpam-6566	326	46	:	:	PUNCT
ejpam-6566	326	47	(	(	PUNCT
ejpam-6566	326	48	x	x	X
ejpam-6566	326	49	,	,	PUNCT
ejpam-6566	326	50	τ	τ	PROPN
ejpam-6566	326	51	,	,	PUNCT
ejpam-6566	326	52	i	i	NOUN
ejpam-6566	326	53	)	)	PUNCT
ejpam-6566	326	54	→	→	PUNCT
ejpam-6566	326	55	(	(	PUNCT
ejpam-6566	326	56	y	y	PROPN
ejpam-6566	326	57	,	,	PUNCT
ejpam-6566	326	58	σ1	σ1	PROPN
ejpam-6566	326	59	,	,	PUNCT
ejpam-6566	326	60	σ2	σ2	NOUN
ejpam-6566	326	61	)	)	PUNCT
ejpam-6566	326	62	,	,	PUNCT
ejpam-6566	326	63	the	the	DET
ejpam-6566	326	64	following	follow	VERB
ejpam-6566	326	65	implication	implication	NOUN
ejpam-6566	326	66	holds	hold	VERB
ejpam-6566	326	67	:	:	PUNCT
ejpam-6566	326	68	upper	upper	ADJ
ejpam-6566	326	69	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6566	326	70	,	,	PUNCT
ejpam-6566	326	71	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6566	326	72	⇒	⇒	NOUN
ejpam-6566	326	73	upper	upper	ADJ
ejpam-6566	326	74	almost	almost	ADV
ejpam-6566	326	75	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	326	76	,	,	PUNCT
ejpam-6566	326	77	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6566	326	78	.	.	PUNCT
ejpam-6566	327	1	the	the	DET
ejpam-6566	327	2	converse	converse	NOUN
ejpam-6566	327	3	of	of	ADP
ejpam-6566	327	4	the	the	DET
ejpam-6566	327	5	implication	implication	NOUN
ejpam-6566	327	6	is	be	AUX
ejpam-6566	327	7	not	not	PART
ejpam-6566	327	8	true	true	ADJ
ejpam-6566	327	9	in	in	ADP
ejpam-6566	327	10	general	general	ADJ
ejpam-6566	327	11	.	.	PUNCT
ejpam-6566	328	1	we	we	PRON
ejpam-6566	328	2	give	give	VERB
ejpam-6566	328	3	an	an	DET
ejpam-6566	328	4	example	example	NOUN
ejpam-6566	328	5	for	for	ADP
ejpam-6566	328	6	the	the	DET
ejpam-6566	328	7	implication	implication	NOUN
ejpam-6566	328	8	as	as	SCONJ
ejpam-6566	328	9	follows	follow	VERB
ejpam-6566	328	10	.	.	PUNCT
ejpam-6566	329	1	c.	c.	PROPN
ejpam-6566	329	2	viriyapong	viriyapong	PROPN
ejpam-6566	329	3	,	,	PUNCT
ejpam-6566	329	4	a.	a.	PROPN
ejpam-6566	329	5	sama	sama	PROPN
ejpam-6566	329	6	-	-	PUNCT
ejpam-6566	329	7	ae	ae	PROPN
ejpam-6566	329	8	,	,	PUNCT
ejpam-6566	329	9	c.	c.	PROPN
ejpam-6566	329	10	boonpok	boonpok	PROPN
ejpam-6566	329	11	/	/	SYM
ejpam-6566	329	12	eur	eur	PROPN
ejpam-6566	329	13	.	.	PUNCT
ejpam-6566	330	1	j.	j.	PROPN
ejpam-6566	330	2	pure	pure	PROPN
ejpam-6566	330	3	appl	appl	PROPN
ejpam-6566	330	4	.	.	PROPN
ejpam-6566	330	5	math	math	PROPN
ejpam-6566	330	6	,	,	PUNCT
ejpam-6566	330	7	18	18	NUM
ejpam-6566	330	8	(	(	PUNCT
ejpam-6566	330	9	3	3	NUM
ejpam-6566	330	10	)	)	PUNCT
ejpam-6566	330	11	(	(	PUNCT
ejpam-6566	330	12	2025	2025	NUM
ejpam-6566	330	13	)	)	PUNCT
ejpam-6566	330	14	,	,	PUNCT
ejpam-6566	330	15	6566	6566	NUM
ejpam-6566	330	16	12	12	NUM
ejpam-6566	330	17	of	of	ADP
ejpam-6566	330	18	14	14	NUM
ejpam-6566	330	19	example	example	NOUN
ejpam-6566	331	1	1	1	NUM
ejpam-6566	331	2	.	.	PUNCT
ejpam-6566	332	1	let	let	VERB
ejpam-6566	332	2	x	x	PUNCT
ejpam-6566	332	3	=	=	PRON
ejpam-6566	332	4	{	{	PUNCT
ejpam-6566	332	5	1	1	NUM
ejpam-6566	332	6	,	,	PUNCT
ejpam-6566	332	7	2	2	NUM
ejpam-6566	332	8	,	,	PUNCT
ejpam-6566	332	9	3	3	NUM
ejpam-6566	332	10	}	}	PUNCT
ejpam-6566	332	11	with	with	ADP
ejpam-6566	332	12	a	a	DET
ejpam-6566	332	13	topology	topology	NOUN
ejpam-6566	332	14	τ	τ	X
ejpam-6566	332	15	=	=	SYM
ejpam-6566	332	16	{	{	PUNCT
ejpam-6566	332	17	∅	∅	NOUN
ejpam-6566	332	18	,	,	PUNCT
ejpam-6566	332	19	{	{	PUNCT
ejpam-6566	332	20	1	1	NUM
ejpam-6566	332	21	}	}	PUNCT
ejpam-6566	332	22	,	,	PUNCT
ejpam-6566	332	23	{	{	PUNCT
ejpam-6566	332	24	2	2	NUM
ejpam-6566	332	25	}	}	PUNCT
ejpam-6566	332	26	,	,	PUNCT
ejpam-6566	332	27	{	{	PUNCT
ejpam-6566	332	28	1	1	NUM
ejpam-6566	332	29	,	,	PUNCT
ejpam-6566	332	30	2	2	NUM
ejpam-6566	332	31	}	}	PUNCT
ejpam-6566	332	32	,	,	PUNCT
ejpam-6566	332	33	x	x	NOUN
ejpam-6566	332	34	}	}	PUNCT
ejpam-6566	332	35	and	and	CCONJ
ejpam-6566	332	36	an	an	DET
ejpam-6566	332	37	ideal	ideal	NOUN
ejpam-6566	332	38	i	i	X
ejpam-6566	332	39	=	=	SYM
ejpam-6566	332	40	{	{	PUNCT
ejpam-6566	332	41	∅	∅	NOUN
ejpam-6566	332	42	,	,	PUNCT
ejpam-6566	332	43	{	{	PUNCT
ejpam-6566	332	44	1	1	NUM
ejpam-6566	332	45	}	}	PUNCT
ejpam-6566	332	46	}	}	PUNCT
ejpam-6566	332	47	.	.	PUNCT
ejpam-6566	333	1	let	let	VERB
ejpam-6566	333	2	y	y	PROPN
ejpam-6566	333	3	=	=	PUNCT
ejpam-6566	333	4	{	{	PUNCT
ejpam-6566	333	5	a	a	PRON
ejpam-6566	333	6	,	,	PUNCT
ejpam-6566	333	7	b	b	NOUN
ejpam-6566	333	8	,	,	PUNCT
ejpam-6566	333	9	c	c	NOUN
ejpam-6566	333	10	}	}	PUNCT
ejpam-6566	333	11	with	with	ADP
ejpam-6566	333	12	topologies	topology	NOUN
ejpam-6566	333	13	σ1	σ1	NOUN
ejpam-6566	333	14	=	=	SYM
ejpam-6566	333	15	{	{	PUNCT
ejpam-6566	333	16	∅	∅	NOUN
ejpam-6566	333	17	,	,	PUNCT
ejpam-6566	333	18	{	{	PUNCT
ejpam-6566	333	19	a	a	X
ejpam-6566	333	20	}	}	PUNCT
ejpam-6566	333	21	,	,	PUNCT
ejpam-6566	333	22	{	{	PUNCT
ejpam-6566	333	23	a	a	DET
ejpam-6566	333	24	,	,	PUNCT
ejpam-6566	333	25	b	b	NOUN
ejpam-6566	333	26	}	}	PUNCT
ejpam-6566	333	27	,	,	PUNCT
ejpam-6566	333	28	y	y	PROPN
ejpam-6566	333	29	}	}	PUNCT
ejpam-6566	333	30	and	and	CCONJ
ejpam-6566	333	31	σ2	σ2	PROPN
ejpam-6566	333	32	=	=	SYM
ejpam-6566	333	33	{	{	PUNCT
ejpam-6566	333	34	∅	∅	NOUN
ejpam-6566	333	35	,	,	PUNCT
ejpam-6566	333	36	{	{	PUNCT
ejpam-6566	333	37	a	a	X
ejpam-6566	333	38	}	}	PUNCT
ejpam-6566	333	39	,	,	PUNCT
ejpam-6566	333	40	{	{	PUNCT
ejpam-6566	333	41	b	b	NOUN
ejpam-6566	333	42	}	}	PUNCT
ejpam-6566	333	43	,	,	PUNCT
ejpam-6566	333	44	{	{	PUNCT
ejpam-6566	333	45	a	a	DET
ejpam-6566	333	46	,	,	PUNCT
ejpam-6566	333	47	b	b	NOUN
ejpam-6566	333	48	}	}	PUNCT
ejpam-6566	333	49	,	,	PUNCT
ejpam-6566	333	50	y	y	PROPN
ejpam-6566	333	51	}	}	PUNCT
ejpam-6566	333	52	.	.	PUNCT
ejpam-6566	334	1	a	a	DET
ejpam-6566	334	2	multifunction	multifunction	NOUN
ejpam-6566	334	3	f	f	NOUN
ejpam-6566	334	4	:	:	PUNCT
ejpam-6566	334	5	(	(	PUNCT
ejpam-6566	334	6	x	x	X
ejpam-6566	334	7	,	,	PUNCT
ejpam-6566	334	8	τ	τ	PROPN
ejpam-6566	334	9	,	,	PUNCT
ejpam-6566	334	10	i	i	NOUN
ejpam-6566	334	11	)	)	PUNCT
ejpam-6566	334	12	→	→	PUNCT
ejpam-6566	334	13	(	(	PUNCT
ejpam-6566	334	14	y	y	PROPN
ejpam-6566	334	15	,	,	PUNCT
ejpam-6566	334	16	σ1	σ1	PROPN
ejpam-6566	334	17	,	,	PUNCT
ejpam-6566	334	18	σ2	σ2	PROPN
ejpam-6566	334	19	)	)	PUNCT
ejpam-6566	334	20	is	be	AUX
ejpam-6566	334	21	defined	define	VERB
ejpam-6566	334	22	as	as	SCONJ
ejpam-6566	334	23	follows	follow	VERB
ejpam-6566	334	24	:	:	PUNCT
ejpam-6566	334	25	f	f	X
ejpam-6566	334	26	(	(	PUNCT
ejpam-6566	334	27	1	1	X
ejpam-6566	334	28	)	)	PUNCT
ejpam-6566	334	29	=	=	PRON
ejpam-6566	335	1	{	{	PUNCT
ejpam-6566	335	2	c	c	NOUN
ejpam-6566	335	3	}	}	PUNCT
ejpam-6566	335	4	and	and	CCONJ
ejpam-6566	335	5	f	f	X
ejpam-6566	335	6	(	(	PUNCT
ejpam-6566	335	7	2	2	NUM
ejpam-6566	335	8	)	)	PUNCT
ejpam-6566	335	9	=	=	SYM
ejpam-6566	335	10	f	f	PROPN
ejpam-6566	335	11	(	(	PUNCT
ejpam-6566	335	12	3	3	NUM
ejpam-6566	335	13	)	)	PUNCT
ejpam-6566	335	14	=	=	PRON
ejpam-6566	335	15	{	{	PUNCT
ejpam-6566	335	16	a	a	PRON
ejpam-6566	335	17	,	,	PUNCT
ejpam-6566	335	18	b	b	NOUN
ejpam-6566	335	19	}	}	PUNCT
ejpam-6566	335	20	.	.	PUNCT
ejpam-6566	336	1	then	then	ADV
ejpam-6566	336	2	,	,	PUNCT
ejpam-6566	336	3	f	f	PROPN
ejpam-6566	336	4	is	be	AUX
ejpam-6566	336	5	upper	upper	ADJ
ejpam-6566	336	6	almost	almost	ADV
ejpam-6566	336	7	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	336	8	,	,	PUNCT
ejpam-6566	336	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	336	10	but	but	CCONJ
ejpam-6566	336	11	f	f	PROPN
ejpam-6566	336	12	is	be	AUX
ejpam-6566	336	13	not	not	PART
ejpam-6566	336	14	upper	upper	ADJ
ejpam-6566	336	15	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6566	336	16	,	,	PUNCT
ejpam-6566	336	17	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	336	18	.	.	NOUN
ejpam-6566	336	19	definition	definition	NOUN
ejpam-6566	336	20	5	5	NUM
ejpam-6566	336	21	.	.	PUNCT
ejpam-6566	337	1	[	[	X
ejpam-6566	337	2	31	31	NUM
ejpam-6566	337	3	]	]	PUNCT
ejpam-6566	337	4	a	a	DET
ejpam-6566	337	5	bitopological	bitopological	ADJ
ejpam-6566	337	6	space	space	NOUN
ejpam-6566	337	7	(	(	PUNCT
ejpam-6566	337	8	x	x	NOUN
ejpam-6566	337	9	,	,	PUNCT
ejpam-6566	337	10	τ1	τ1	NOUN
ejpam-6566	337	11	,	,	PUNCT
ejpam-6566	337	12	τ2	τ2	NOUN
ejpam-6566	337	13	)	)	PUNCT
ejpam-6566	337	14	is	be	AUX
ejpam-6566	337	15	said	say	VERB
ejpam-6566	337	16	to	to	PART
ejpam-6566	337	17	be	be	AUX
ejpam-6566	337	18	(	(	PUNCT
ejpam-6566	337	19	τ1	τ1	NOUN
ejpam-6566	337	20	,	,	PUNCT
ejpam-6566	337	21	τ2)s	τ2)s	NOUN
ejpam-6566	337	22	-	-	PUNCT
ejpam-6566	337	23	regular	regular	ADJ
ejpam-6566	337	24	if	if	SCONJ
ejpam-6566	337	25	for	for	ADP
ejpam-6566	337	26	each	each	DET
ejpam-6566	337	27	(	(	PUNCT
ejpam-6566	337	28	τ1	τ1	NOUN
ejpam-6566	337	29	,	,	PUNCT
ejpam-6566	337	30	τ2)s	τ2)s	NOUN
ejpam-6566	337	31	-	-	PUNCT
ejpam-6566	337	32	closed	close	VERB
ejpam-6566	337	33	set	set	VERB
ejpam-6566	337	34	f	f	NOUN
ejpam-6566	337	35	and	and	CCONJ
ejpam-6566	337	36	each	each	DET
ejpam-6566	337	37	x	x	PROPN
ejpam-6566	337	38	̸∈	̸∈	PROPN
ejpam-6566	337	39	f	f	PROPN
ejpam-6566	337	40	,	,	PUNCT
ejpam-6566	337	41	there	there	PRON
ejpam-6566	337	42	exist	exist	VERB
ejpam-6566	337	43	disjoint	disjoint	NOUN
ejpam-6566	337	44	(	(	PUNCT
ejpam-6566	337	45	τ1	τ1	NOUN
ejpam-6566	337	46	,	,	PUNCT
ejpam-6566	337	47	τ2)s	τ2)s	NOUN
ejpam-6566	337	48	-	-	PUNCT
ejpam-6566	337	49	open	open	ADJ
ejpam-6566	337	50	sets	set	VERB
ejpam-6566	337	51	u	u	NOUN
ejpam-6566	337	52	and	and	CCONJ
ejpam-6566	337	53	v	v	ADP
ejpam-6566	337	54	such	such	ADJ
ejpam-6566	337	55	that	that	SCONJ
ejpam-6566	337	56	x	x	SYM
ejpam-6566	337	57	∈	∈	PROPN
ejpam-6566	337	58	u	u	NOUN
ejpam-6566	337	59	and	and	CCONJ
ejpam-6566	337	60	f	f	PROPN
ejpam-6566	337	61	⊆	⊆	NUM
ejpam-6566	337	62	v	v	NOUN
ejpam-6566	337	63	.	.	PUNCT
ejpam-6566	338	1	lemma	lemma	PROPN
ejpam-6566	338	2	6	6	NUM
ejpam-6566	338	3	.	.	PUNCT
ejpam-6566	339	1	[	[	X
ejpam-6566	339	2	31	31	NUM
ejpam-6566	339	3	]	]	X
ejpam-6566	339	4	let	let	VERB
ejpam-6566	339	5	(	(	PUNCT
ejpam-6566	339	6	x	x	NOUN
ejpam-6566	339	7	,	,	PUNCT
ejpam-6566	339	8	τ1	τ1	NOUN
ejpam-6566	339	9	,	,	PUNCT
ejpam-6566	339	10	τ2	τ2	PROPN
ejpam-6566	339	11	)	)	PUNCT
ejpam-6566	339	12	be	be	VERB
ejpam-6566	339	13	a	a	DET
ejpam-6566	339	14	(	(	PUNCT
ejpam-6566	339	15	τ1	τ1	NOUN
ejpam-6566	339	16	,	,	PUNCT
ejpam-6566	339	17	τ2)s	τ2)s	NOUN
ejpam-6566	339	18	-	-	PUNCT
ejpam-6566	339	19	regular	regular	ADJ
ejpam-6566	339	20	space	space	NOUN
ejpam-6566	339	21	.	.	PUNCT
ejpam-6566	340	1	then	then	ADV
ejpam-6566	340	2	,	,	PUNCT
ejpam-6566	340	3	the	the	DET
ejpam-6566	340	4	following	follow	VERB
ejpam-6566	340	5	properties	property	NOUN
ejpam-6566	340	6	hold	hold	VERB
ejpam-6566	340	7	:	:	PUNCT
ejpam-6566	340	8	(	(	PUNCT
ejpam-6566	340	9	1	1	X
ejpam-6566	340	10	)	)	PUNCT
ejpam-6566	340	11	τ1τ2	τ1τ2	NOUN
ejpam-6566	340	12	-	-	NUM
ejpam-6566	340	13	cl(a	cl(a	NUM
ejpam-6566	340	14	)	)	PUNCT
ejpam-6566	340	15	=	=	PUNCT
ejpam-6566	341	1	τ1τ2	τ1τ2	PROPN
ejpam-6566	341	2	-	-	ADJ
ejpam-6566	341	3	δ	δ	NOUN
ejpam-6566	341	4	-	-	PUNCT
ejpam-6566	341	5	cl(a	cl(a	NUM
ejpam-6566	341	6	)	)	PUNCT
ejpam-6566	341	7	for	for	ADP
ejpam-6566	341	8	every	every	DET
ejpam-6566	341	9	subset	subset	NOUN
ejpam-6566	341	10	a	a	PRON
ejpam-6566	341	11	of	of	ADP
ejpam-6566	341	12	x.	x.	NOUN
ejpam-6566	341	13	(	(	PUNCT
ejpam-6566	341	14	2	2	NUM
ejpam-6566	341	15	)	)	PUNCT
ejpam-6566	341	16	every	every	DET
ejpam-6566	341	17	τ1τ2	τ1τ2	NOUN
ejpam-6566	341	18	-	-	ADJ
ejpam-6566	341	19	open	open	ADJ
ejpam-6566	341	20	set	set	NOUN
ejpam-6566	341	21	is	be	AUX
ejpam-6566	341	22	τ1τ2	τ1τ2	ADJ
ejpam-6566	341	23	-	-	ADJ
ejpam-6566	341	24	δ	δ	NOUN
ejpam-6566	341	25	-	-	NOUN
ejpam-6566	341	26	open	open	ADJ
ejpam-6566	341	27	.	.	PUNCT
ejpam-6566	342	1	lemma	lemma	PROPN
ejpam-6566	342	2	7	7	NUM
ejpam-6566	342	3	.	.	PUNCT
ejpam-6566	343	1	[	[	X
ejpam-6566	343	2	31	31	NUM
ejpam-6566	343	3	]	]	PUNCT
ejpam-6566	343	4	for	for	ADP
ejpam-6566	343	5	a	a	DET
ejpam-6566	343	6	multifunction	multifunction	NOUN
ejpam-6566	343	7	f	f	NOUN
ejpam-6566	343	8	:	:	PUNCT
ejpam-6566	343	9	(	(	PUNCT
ejpam-6566	343	10	x	x	X
ejpam-6566	343	11	,	,	PUNCT
ejpam-6566	343	12	τ	τ	PROPN
ejpam-6566	343	13	,	,	PUNCT
ejpam-6566	343	14	i	i	NOUN
ejpam-6566	343	15	)	)	PUNCT
ejpam-6566	343	16	→	→	PUNCT
ejpam-6566	343	17	(	(	PUNCT
ejpam-6566	343	18	y	y	PROPN
ejpam-6566	343	19	,	,	PUNCT
ejpam-6566	343	20	σ1	σ1	PROPN
ejpam-6566	343	21	,	,	PUNCT
ejpam-6566	343	22	σ2	σ2	NOUN
ejpam-6566	343	23	)	)	PUNCT
ejpam-6566	343	24	,	,	PUNCT
ejpam-6566	343	25	where	where	SCONJ
ejpam-6566	343	26	(	(	PUNCT
ejpam-6566	343	27	y	y	PROPN
ejpam-6566	343	28	,	,	PUNCT
ejpam-6566	343	29	σ1	σ1	PROPN
ejpam-6566	343	30	,	,	PUNCT
ejpam-6566	343	31	σ2	σ2	PROPN
ejpam-6566	343	32	)	)	PUNCT
ejpam-6566	343	33	is	be	AUX
ejpam-6566	343	34	a	a	DET
ejpam-6566	343	35	(	(	PUNCT
ejpam-6566	343	36	σ1	σ1	NOUN
ejpam-6566	343	37	,	,	PUNCT
ejpam-6566	343	38	σ2)s	σ2)s	NOUN
ejpam-6566	343	39	-	-	PUNCT
ejpam-6566	343	40	regular	regular	ADJ
ejpam-6566	343	41	space	space	NOUN
ejpam-6566	343	42	,	,	PUNCT
ejpam-6566	343	43	the	the	DET
ejpam-6566	343	44	following	follow	VERB
ejpam-6566	343	45	properties	property	NOUN
ejpam-6566	343	46	are	be	AUX
ejpam-6566	343	47	equivalent	equivalent	ADJ
ejpam-6566	343	48	:	:	PUNCT
ejpam-6566	343	49	(	(	PUNCT
ejpam-6566	343	50	1	1	X
ejpam-6566	343	51	)	)	PUNCT
ejpam-6566	343	52	f	f	PROPN
ejpam-6566	343	53	is	be	AUX
ejpam-6566	343	54	lower	low	ADJ
ejpam-6566	343	55	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	343	56	,	,	PUNCT
ejpam-6566	343	57	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	343	58	;	;	PUNCT
ejpam-6566	343	59	(	(	PUNCT
ejpam-6566	343	60	2	2	X
ejpam-6566	343	61	)	)	PUNCT
ejpam-6566	343	62	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	343	63	-	-	PUNCT
ejpam-6566	343	64	δ	δ	NOUN
ejpam-6566	343	65	-	-	NOUN
ejpam-6566	343	66	cl(b	cl(b	NOUN
ejpam-6566	343	67	)	)	PUNCT
ejpam-6566	343	68	)	)	PUNCT
ejpam-6566	343	69	is	be	AUX
ejpam-6566	343	70	⋆-closed	⋆-close	VERB
ejpam-6566	343	71	in	in	ADP
ejpam-6566	343	72	x	x	PUNCT
ejpam-6566	343	73	for	for	ADP
ejpam-6566	343	74	every	every	DET
ejpam-6566	343	75	subset	subset	NOUN
ejpam-6566	343	76	b	b	PROPN
ejpam-6566	343	77	of	of	ADP
ejpam-6566	343	78	y	y	PROPN
ejpam-6566	343	79	;	;	PUNCT
ejpam-6566	343	80	(	(	PUNCT
ejpam-6566	343	81	3	3	X
ejpam-6566	343	82	)	)	PUNCT
ejpam-6566	343	83	f+(k	f+(k	NUM
ejpam-6566	343	84	)	)	PUNCT
ejpam-6566	343	85	is	be	AUX
ejpam-6566	343	86	⋆-closed	⋆-close	VERB
ejpam-6566	343	87	in	in	ADP
ejpam-6566	343	88	x	x	PUNCT
ejpam-6566	343	89	for	for	ADP
ejpam-6566	343	90	every	every	DET
ejpam-6566	343	91	σ1σ2	σ1σ2	NUM
ejpam-6566	343	92	-	-	PUNCT
ejpam-6566	343	93	δ	δ	NOUN
ejpam-6566	343	94	-	-	PUNCT
ejpam-6566	343	95	closed	close	VERB
ejpam-6566	343	96	set	set	ADJ
ejpam-6566	343	97	k	k	PROPN
ejpam-6566	343	98	of	of	ADP
ejpam-6566	343	99	y	y	PROPN
ejpam-6566	343	100	;	;	PUNCT
ejpam-6566	343	101	(	(	PUNCT
ejpam-6566	343	102	4	4	X
ejpam-6566	343	103	)	)	PUNCT
ejpam-6566	343	104	f−(v	f−(v	NOUN
ejpam-6566	343	105	)	)	PUNCT
ejpam-6566	343	106	is	be	AUX
ejpam-6566	343	107	⋆-open	⋆-open	ADJ
ejpam-6566	343	108	in	in	ADP
ejpam-6566	343	109	x	x	PUNCT
ejpam-6566	343	110	for	for	SCONJ
ejpam-6566	343	111	every	every	DET
ejpam-6566	343	112	σ1σ2	σ1σ2	NUM
ejpam-6566	343	113	-	-	PUNCT
ejpam-6566	343	114	δ	δ	NOUN
ejpam-6566	343	115	-	-	ADJ
ejpam-6566	343	116	open	open	ADJ
ejpam-6566	343	117	set	set	VERB
ejpam-6566	343	118	v	v	NOUN
ejpam-6566	343	119	of	of	ADP
ejpam-6566	343	120	y	y	PROPN
ejpam-6566	343	121	.	.	PUNCT
ejpam-6566	344	1	theorem	theorem	PROPN
ejpam-6566	344	2	12	12	NUM
ejpam-6566	344	3	.	.	PUNCT
ejpam-6566	345	1	for	for	ADP
ejpam-6566	345	2	a	a	DET
ejpam-6566	345	3	multifunction	multifunction	NOUN
ejpam-6566	345	4	f	f	NOUN
ejpam-6566	345	5	:	:	PUNCT
ejpam-6566	345	6	(	(	PUNCT
ejpam-6566	345	7	x	x	X
ejpam-6566	345	8	,	,	PUNCT
ejpam-6566	345	9	τ	τ	PROPN
ejpam-6566	345	10	,	,	PUNCT
ejpam-6566	345	11	i	i	NOUN
ejpam-6566	345	12	)	)	PUNCT
ejpam-6566	345	13	→	→	PUNCT
ejpam-6566	345	14	(	(	PUNCT
ejpam-6566	345	15	y	y	PROPN
ejpam-6566	345	16	,	,	PUNCT
ejpam-6566	345	17	σ1	σ1	PROPN
ejpam-6566	345	18	,	,	PUNCT
ejpam-6566	345	19	σ2	σ2	NOUN
ejpam-6566	345	20	)	)	PUNCT
ejpam-6566	345	21	,	,	PUNCT
ejpam-6566	345	22	where	where	SCONJ
ejpam-6566	345	23	(	(	PUNCT
ejpam-6566	345	24	y	y	PROPN
ejpam-6566	345	25	,	,	PUNCT
ejpam-6566	345	26	σ1	σ1	PROPN
ejpam-6566	345	27	,	,	PUNCT
ejpam-6566	345	28	σ2	σ2	PROPN
ejpam-6566	345	29	)	)	PUNCT
ejpam-6566	345	30	is	be	AUX
ejpam-6566	345	31	a	a	DET
ejpam-6566	345	32	(	(	PUNCT
ejpam-6566	345	33	σ1	σ1	NOUN
ejpam-6566	345	34	,	,	PUNCT
ejpam-6566	345	35	σ2)s	σ2)s	NOUN
ejpam-6566	345	36	-	-	PUNCT
ejpam-6566	345	37	regular	regular	ADJ
ejpam-6566	345	38	space	space	NOUN
ejpam-6566	345	39	,	,	PUNCT
ejpam-6566	345	40	the	the	DET
ejpam-6566	345	41	following	follow	VERB
ejpam-6566	345	42	properties	property	NOUN
ejpam-6566	345	43	are	be	AUX
ejpam-6566	345	44	equivalent	equivalent	ADJ
ejpam-6566	345	45	:	:	PUNCT
ejpam-6566	345	46	(	(	PUNCT
ejpam-6566	345	47	1	1	X
ejpam-6566	345	48	)	)	PUNCT
ejpam-6566	345	49	f	f	PROPN
ejpam-6566	345	50	is	be	AUX
ejpam-6566	345	51	lower	low	ADJ
ejpam-6566	345	52	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	345	53	,	,	PUNCT
ejpam-6566	345	54	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	345	55	;	;	PUNCT
ejpam-6566	345	56	(	(	PUNCT
ejpam-6566	345	57	2	2	X
ejpam-6566	345	58	)	)	PUNCT
ejpam-6566	345	59	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6566	345	60	-	-	PUNCT
ejpam-6566	345	61	δ	δ	NOUN
ejpam-6566	345	62	-	-	NOUN
ejpam-6566	345	63	cl(b	cl(b	NOUN
ejpam-6566	345	64	)	)	PUNCT
ejpam-6566	345	65	)	)	PUNCT
ejpam-6566	345	66	is	be	AUX
ejpam-6566	345	67	⋆-closed	⋆-close	VERB
ejpam-6566	345	68	in	in	ADP
ejpam-6566	345	69	x	x	PUNCT
ejpam-6566	345	70	for	for	ADP
ejpam-6566	345	71	every	every	DET
ejpam-6566	345	72	subset	subset	NOUN
ejpam-6566	345	73	b	b	PROPN
ejpam-6566	345	74	of	of	ADP
ejpam-6566	345	75	y	y	PROPN
ejpam-6566	345	76	;	;	PUNCT
ejpam-6566	345	77	(	(	PUNCT
ejpam-6566	345	78	3	3	X
ejpam-6566	345	79	)	)	PUNCT
ejpam-6566	345	80	f+(k	f+(k	NUM
ejpam-6566	345	81	)	)	PUNCT
ejpam-6566	345	82	is	be	AUX
ejpam-6566	345	83	⋆-closed	⋆-close	VERB
ejpam-6566	345	84	in	in	ADP
ejpam-6566	345	85	x	x	PUNCT
ejpam-6566	345	86	for	for	ADP
ejpam-6566	345	87	every	every	DET
ejpam-6566	345	88	σ1σ2	σ1σ2	NUM
ejpam-6566	345	89	-	-	PUNCT
ejpam-6566	345	90	δ	δ	NOUN
ejpam-6566	345	91	-	-	PUNCT
ejpam-6566	345	92	closed	close	VERB
ejpam-6566	345	93	set	set	ADJ
ejpam-6566	345	94	k	k	PROPN
ejpam-6566	345	95	of	of	ADP
ejpam-6566	345	96	y	y	PROPN
ejpam-6566	345	97	;	;	PUNCT
ejpam-6566	345	98	(	(	PUNCT
ejpam-6566	345	99	4	4	X
ejpam-6566	345	100	)	)	PUNCT
ejpam-6566	345	101	f−(v	f−(v	NOUN
ejpam-6566	345	102	)	)	PUNCT
ejpam-6566	345	103	is	be	AUX
ejpam-6566	345	104	⋆-open	⋆-open	ADJ
ejpam-6566	345	105	in	in	ADP
ejpam-6566	345	106	x	x	PUNCT
ejpam-6566	345	107	for	for	SCONJ
ejpam-6566	345	108	every	every	DET
ejpam-6566	345	109	σ1σ2	σ1σ2	NUM
ejpam-6566	345	110	-	-	PUNCT
ejpam-6566	345	111	δ	δ	NOUN
ejpam-6566	345	112	-	-	ADJ
ejpam-6566	345	113	open	open	ADJ
ejpam-6566	345	114	set	set	VERB
ejpam-6566	345	115	v	v	NOUN
ejpam-6566	345	116	of	of	ADP
ejpam-6566	345	117	y	y	PROPN
ejpam-6566	345	118	;	;	PUNCT
ejpam-6566	345	119	(	(	PUNCT
ejpam-6566	345	120	5	5	X
ejpam-6566	345	121	)	)	PUNCT
ejpam-6566	345	122	f	f	PROPN
ejpam-6566	345	123	is	be	AUX
ejpam-6566	345	124	lower	low	ADJ
ejpam-6566	345	125	almost	almost	ADV
ejpam-6566	345	126	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	345	127	,	,	PUNCT
ejpam-6566	345	128	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	345	129	.	.	NOUN
ejpam-6566	345	130	proof	proof	NOUN
ejpam-6566	345	131	.	.	PUNCT
ejpam-6566	346	1	the	the	DET
ejpam-6566	346	2	proofs	proof	NOUN
ejpam-6566	346	3	of	of	ADP
ejpam-6566	346	4	the	the	DET
ejpam-6566	346	5	implications	implication	NOUN
ejpam-6566	346	6	(	(	PUNCT
ejpam-6566	346	7	1	1	X
ejpam-6566	346	8	)	)	PUNCT
ejpam-6566	346	9	⇒	⇒	NOUN
ejpam-6566	346	10	(	(	PUNCT
ejpam-6566	346	11	2	2	NUM
ejpam-6566	346	12	)	)	PUNCT
ejpam-6566	346	13	⇒	⇒	NOUN
ejpam-6566	346	14	(	(	PUNCT
ejpam-6566	346	15	3	3	NUM
ejpam-6566	346	16	)	)	PUNCT
ejpam-6566	346	17	⇒	⇒	NOUN
ejpam-6566	346	18	(	(	PUNCT
ejpam-6566	346	19	4	4	X
ejpam-6566	346	20	)	)	PUNCT
ejpam-6566	346	21	are	be	AUX
ejpam-6566	346	22	similar	similar	ADJ
ejpam-6566	346	23	as	as	ADP
ejpam-6566	346	24	in	in	ADP
ejpam-6566	346	25	lemma	lemma	PROPN
ejpam-6566	346	26	7	7	NUM
ejpam-6566	346	27	.	.	PUNCT
ejpam-6566	346	28	(	(	PUNCT
ejpam-6566	346	29	4	4	X
ejpam-6566	346	30	)	)	PUNCT
ejpam-6566	346	31	⇒	⇒	NOUN
ejpam-6566	346	32	(	(	PUNCT
ejpam-6566	346	33	5	5	NUM
ejpam-6566	346	34	):	):	PUNCT
ejpam-6566	346	35	let	let	VERB
ejpam-6566	346	36	v	v	PART
ejpam-6566	346	37	be	be	AUX
ejpam-6566	346	38	any	any	DET
ejpam-6566	346	39	(	(	PUNCT
ejpam-6566	346	40	σ1	σ1	NOUN
ejpam-6566	346	41	,	,	PUNCT
ejpam-6566	346	42	σ2)r	σ2)r	NOUN
ejpam-6566	346	43	-	-	PUNCT
ejpam-6566	346	44	open	open	ADJ
ejpam-6566	346	45	set	set	NOUN
ejpam-6566	346	46	of	of	ADP
ejpam-6566	346	47	y	y	PROPN
ejpam-6566	346	48	.	.	PUNCT
ejpam-6566	347	1	then	then	ADV
ejpam-6566	347	2	,	,	PUNCT
ejpam-6566	347	3	v	v	NOUN
ejpam-6566	347	4	is	be	AUX
ejpam-6566	347	5	σ1σ2	σ1σ2	NOUN
ejpam-6566	347	6	-	-	ADJ
ejpam-6566	347	7	open	open	ADJ
ejpam-6566	347	8	in	in	ADP
ejpam-6566	347	9	y	y	PROPN
ejpam-6566	347	10	and	and	CCONJ
ejpam-6566	347	11	by	by	ADP
ejpam-6566	347	12	lemma	lemma	PROPN
ejpam-6566	347	13	6	6	NUM
ejpam-6566	347	14	,	,	PUNCT
ejpam-6566	347	15	v	v	NOUN
ejpam-6566	347	16	is	be	AUX
ejpam-6566	347	17	σ1σ2	σ1σ2	NOUN
ejpam-6566	347	18	-	-	PUNCT
ejpam-6566	347	19	δ	δ	NOUN
ejpam-6566	347	20	-	-	NOUN
ejpam-6566	347	21	open	open	ADJ
ejpam-6566	347	22	in	in	ADP
ejpam-6566	347	23	y	y	PROPN
ejpam-6566	347	24	.	.	PUNCT
ejpam-6566	348	1	by	by	ADP
ejpam-6566	348	2	(	(	PUNCT
ejpam-6566	348	3	4	4	NUM
ejpam-6566	348	4	)	)	PUNCT
ejpam-6566	348	5	,	,	PUNCT
ejpam-6566	348	6	we	we	PRON
ejpam-6566	348	7	have	have	VERB
ejpam-6566	348	8	f−(v	f−(v	NOUN
ejpam-6566	348	9	)	)	PUNCT
ejpam-6566	348	10	is	be	AUX
ejpam-6566	348	11	⋆-open	⋆-open	ADJ
ejpam-6566	348	12	in	in	ADP
ejpam-6566	348	13	x.	x.	NOUN
ejpam-6566	348	14	thus	thus	ADV
ejpam-6566	348	15	by	by	ADP
ejpam-6566	348	16	theorem	theorem	NOUN
ejpam-6566	348	17	4	4	NUM
ejpam-6566	348	18	,	,	PUNCT
ejpam-6566	348	19	f	f	PROPN
ejpam-6566	348	20	is	be	AUX
ejpam-6566	348	21	lower	low	ADJ
ejpam-6566	348	22	almost	almost	ADV
ejpam-6566	348	23	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	348	24	,	,	PUNCT
ejpam-6566	348	25	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	348	26	.	.	PUNCT
ejpam-6566	349	1	c.	c.	PROPN
ejpam-6566	349	2	viriyapong	viriyapong	PROPN
ejpam-6566	349	3	,	,	PUNCT
ejpam-6566	349	4	a.	a.	PROPN
ejpam-6566	349	5	sama	sama	PROPN
ejpam-6566	349	6	-	-	PUNCT
ejpam-6566	349	7	ae	ae	PROPN
ejpam-6566	349	8	,	,	PUNCT
ejpam-6566	349	9	c.	c.	PROPN
ejpam-6566	349	10	boonpok	boonpok	PROPN
ejpam-6566	349	11	/	/	SYM
ejpam-6566	349	12	eur	eur	PROPN
ejpam-6566	349	13	.	.	PUNCT
ejpam-6566	350	1	j.	j.	PROPN
ejpam-6566	350	2	pure	pure	PROPN
ejpam-6566	350	3	appl	appl	PROPN
ejpam-6566	350	4	.	.	PROPN
ejpam-6566	350	5	math	math	PROPN
ejpam-6566	350	6	,	,	PUNCT
ejpam-6566	350	7	18	18	NUM
ejpam-6566	350	8	(	(	PUNCT
ejpam-6566	350	9	3	3	NUM
ejpam-6566	350	10	)	)	PUNCT
ejpam-6566	350	11	(	(	PUNCT
ejpam-6566	350	12	2025	2025	NUM
ejpam-6566	350	13	)	)	PUNCT
ejpam-6566	350	14	,	,	PUNCT
ejpam-6566	350	15	6566	6566	NUM
ejpam-6566	350	16	13	13	NUM
ejpam-6566	350	17	of	of	ADP
ejpam-6566	350	18	14	14	NUM
ejpam-6566	350	19	(	(	PUNCT
ejpam-6566	350	20	5	5	NUM
ejpam-6566	350	21	)	)	PUNCT
ejpam-6566	350	22	⇒	⇒	NOUN
ejpam-6566	350	23	(	(	PUNCT
ejpam-6566	350	24	1	1	NUM
ejpam-6566	350	25	):	):	PUNCT
ejpam-6566	350	26	let	let	VERB
ejpam-6566	350	27	x	x	PUNCT
ejpam-6566	350	28	∈	∈	PROPN
ejpam-6566	350	29	x	x	X
ejpam-6566	350	30	and	and	CCONJ
ejpam-6566	350	31	v	v	X
ejpam-6566	350	32	be	be	AUX
ejpam-6566	350	33	any	any	DET
ejpam-6566	350	34	σ1σ2	σ1σ2	NOUN
ejpam-6566	350	35	-	-	ADJ
ejpam-6566	350	36	open	open	ADJ
ejpam-6566	350	37	set	set	NOUN
ejpam-6566	350	38	of	of	ADP
ejpam-6566	350	39	y	y	PRON
ejpam-6566	350	40	such	such	ADJ
ejpam-6566	350	41	that	that	PRON
ejpam-6566	350	42	v	v	NOUN
ejpam-6566	350	43	∩f	∩f	NOUN
ejpam-6566	350	44	(	(	PUNCT
ejpam-6566	350	45	x	x	X
ejpam-6566	350	46	)	)	PUNCT
ejpam-6566	350	47	̸=	̸=	PROPN
ejpam-6566	350	48	∅.	∅.	NOUN
ejpam-6566	350	49	since	since	SCONJ
ejpam-6566	350	50	(	(	PUNCT
ejpam-6566	350	51	y	y	PROPN
ejpam-6566	350	52	,	,	PUNCT
ejpam-6566	350	53	σ1	σ1	PROPN
ejpam-6566	350	54	,	,	PUNCT
ejpam-6566	350	55	σ2	σ2	PROPN
ejpam-6566	350	56	)	)	PUNCT
ejpam-6566	350	57	is	be	AUX
ejpam-6566	350	58	(	(	PUNCT
ejpam-6566	350	59	σ1	σ1	PROPN
ejpam-6566	350	60	,	,	PUNCT
ejpam-6566	350	61	σ2)s	σ2)s	NOUN
ejpam-6566	350	62	-	-	PUNCT
ejpam-6566	350	63	regular	regular	ADJ
ejpam-6566	350	64	,	,	PUNCT
ejpam-6566	350	65	there	there	PRON
ejpam-6566	350	66	exists	exist	VERB
ejpam-6566	350	67	a	a	DET
ejpam-6566	350	68	(	(	PUNCT
ejpam-6566	350	69	σ1	σ1	NOUN
ejpam-6566	350	70	,	,	PUNCT
ejpam-6566	350	71	σ2)r	σ2)r	NOUN
ejpam-6566	350	72	-	-	PUNCT
ejpam-6566	350	73	open	open	NOUN
ejpam-6566	350	74	set	set	NOUN
ejpam-6566	350	75	w	w	ADP
ejpam-6566	350	76	such	such	ADJ
ejpam-6566	350	77	that	that	DET
ejpam-6566	350	78	w	w	PROPN
ejpam-6566	350	79	∩f	∩f	NOUN
ejpam-6566	350	80	(	(	PUNCT
ejpam-6566	350	81	x	x	X
ejpam-6566	350	82	)	)	PUNCT
ejpam-6566	350	83	̸=	̸=	PROPN
ejpam-6566	350	84	∅	∅	NOUN
ejpam-6566	350	85	and	and	CCONJ
ejpam-6566	350	86	w	w	ADP
ejpam-6566	350	87	⊆	⊆	NUM
ejpam-6566	350	88	v	v	NOUN
ejpam-6566	350	89	.	.	PUNCT
ejpam-6566	351	1	since	since	SCONJ
ejpam-6566	351	2	f	f	PROPN
ejpam-6566	351	3	is	be	AUX
ejpam-6566	351	4	lower	low	ADJ
ejpam-6566	351	5	almost	almost	ADV
ejpam-6566	351	6	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6566	351	7	,	,	PUNCT
ejpam-6566	351	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6566	351	9	,	,	PUNCT
ejpam-6566	351	10	there	there	PRON
ejpam-6566	351	11	exists	exist	VERB
ejpam-6566	351	12	a	a	DET
ejpam-6566	351	13	⋆-open	⋆-open	ADJ
ejpam-6566	351	14	set	set	NOUN
ejpam-6566	351	15	u	u	NOUN
ejpam-6566	351	16	of	of	ADP
ejpam-6566	351	17	x	x	PUNCT
ejpam-6566	351	18	containing	contain	VERB
ejpam-6566	351	19	x	x	PUNCT
ejpam-6566	351	20	such	such	ADJ
ejpam-6566	351	21	that	that	SCONJ
ejpam-6566	351	22	w	w	PROPN
ejpam-6566	351	23	∩	∩	PROPN
ejpam-6566	351	24	f	f	X
ejpam-6566	351	25	(	(	PUNCT
ejpam-6566	351	26	z	z	NOUN
ejpam-6566	351	27	)	)	PUNCT
ejpam-6566	351	28	̸=	̸=	NOUN
ejpam-6566	351	29	∅	∅	NOUN
ejpam-6566	351	30	for	for	ADP
ejpam-6566	351	31	every	every	DET
ejpam-6566	351	32	z	z	NOUN
ejpam-6566	351	33	∈	∈	PROPN
ejpam-6566	351	34	u	u	NOUN
ejpam-6566	351	35	.	.	PUNCT
ejpam-6566	352	1	thus	thus	ADV
ejpam-6566	352	2	,	,	PUNCT
ejpam-6566	352	3	f	f	PROPN
ejpam-6566	352	4	(	(	PUNCT
ejpam-6566	352	5	z	z	NOUN
ejpam-6566	352	6	)	)	PUNCT
ejpam-6566	352	7	∩	∩	NOUN
ejpam-6566	352	8	v	v	ADP
ejpam-6566	352	9	̸=	̸=	PROPN
ejpam-6566	352	10	∅	∅	NOUN
ejpam-6566	352	11	for	for	ADP
ejpam-6566	352	12	every	every	DET
ejpam-6566	352	13	z	z	NOUN
ejpam-6566	352	14	∈	∈	PROPN
ejpam-6566	352	15	u	u	NOUN
ejpam-6566	352	16	.	.	PUNCT
ejpam-6566	353	1	this	this	PRON
ejpam-6566	353	2	shows	show	VERB
ejpam-6566	353	3	that	that	SCONJ
ejpam-6566	353	4	f	f	PROPN
ejpam-6566	353	5	is	be	AUX
ejpam-6566	353	6	lower	low	ADJ
ejpam-6566	353	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6566	353	8	,	,	PUNCT
ejpam-6566	353	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6566	353	10	.	.	PUNCT
ejpam-6566	354	1	acknowledgements	acknowledgement	NOUN
ejpam-6566	354	2	this	this	DET
ejpam-6566	354	3	research	research	NOUN
ejpam-6566	354	4	project	project	NOUN
ejpam-6566	354	5	was	be	AUX
ejpam-6566	354	6	financially	financially	ADV
ejpam-6566	354	7	supported	support	VERB
ejpam-6566	354	8	by	by	ADP
ejpam-6566	354	9	mahasarakham	mahasarakham	PROPN
ejpam-6566	354	10	university	university	PROPN
ejpam-6566	354	11	.	.	PUNCT
ejpam-6566	355	1	references	reference	NOUN
ejpam-6566	355	2	[	[	X
ejpam-6566	355	3	1	1	NUM
ejpam-6566	355	4	]	]	PUNCT
ejpam-6566	355	5	m.	m.	NOUN
ejpam-6566	355	6	k.	k.	PROPN
ejpam-6566	355	7	singal	singal	PROPN
ejpam-6566	355	8	and	and	CCONJ
ejpam-6566	355	9	a.	a.	PROPN
ejpam-6566	355	10	r.	r.	PROPN
ejpam-6566	355	11	singal	singal	PROPN
ejpam-6566	355	12	.	.	PUNCT
ejpam-6566	356	1	almost	almost	ADV
ejpam-6566	356	2	continuous	continuous	ADJ
ejpam-6566	356	3	mappings	mapping	NOUN
ejpam-6566	356	4	.	.	PUNCT
ejpam-6566	357	1	yokohama	yokohama	PROPN
ejpam-6566	357	2	mathematical	mathematical	PROPN
ejpam-6566	357	3	journal	journal	PROPN
ejpam-6566	357	4	,	,	PUNCT
ejpam-6566	357	5	16:63–73	16:63–73	PROPN
ejpam-6566	357	6	,	,	PUNCT
ejpam-6566	357	7	1968	1968	NUM
ejpam-6566	357	8	.	.	PUNCT
ejpam-6566	358	1	[	[	X
ejpam-6566	358	2	2	2	NUM
ejpam-6566	358	3	]	]	X
ejpam-6566	358	4	b.	b.	PROPN
ejpam-6566	358	5	m.	m.	PROPN
ejpam-6566	358	6	munshi	munshi	PROPN
ejpam-6566	358	7	and	and	CCONJ
ejpam-6566	358	8	d.	d.	PROPN
ejpam-6566	358	9	s.	s.	PROPN
ejpam-6566	358	10	bassan	bassan	PROPN
ejpam-6566	358	11	.	.	PUNCT
ejpam-6566	359	1	almost	almost	ADV
ejpam-6566	359	2	semi	semi	ADJ
ejpam-6566	359	3	-	-	ADJ
ejpam-6566	359	4	continuous	continuous	ADJ
ejpam-6566	359	5	mappings	mapping	NOUN
ejpam-6566	359	6	.	.	PUNCT
ejpam-6566	360	1	the	the	DET
ejpam-6566	360	2	mathematics	mathematics	PROPN
ejpam-6566	360	3	student	student	NOUN
ejpam-6566	360	4	,	,	PUNCT
ejpam-6566	360	5	49:239–248	49:239–248	PROPN
ejpam-6566	360	6	,	,	PUNCT
ejpam-6566	360	7	1981	1981	NUM
ejpam-6566	360	8	.	.	PUNCT
ejpam-6566	361	1	[	[	X
ejpam-6566	361	2	3	3	X
ejpam-6566	361	3	]	]	PUNCT
ejpam-6566	361	4	t.	t.	PROPN
ejpam-6566	361	5	noiri	noiri	PROPN
ejpam-6566	361	6	.	.	PUNCT
ejpam-6566	362	1	almost	almost	ADV
ejpam-6566	362	2	α	α	NUM
ejpam-6566	362	3	-	-	ADJ
ejpam-6566	362	4	continuous	continuous	ADJ
ejpam-6566	362	5	functions	function	NOUN
ejpam-6566	362	6	.	.	PUNCT
ejpam-6566	363	1	kyungpook	kyungpook	PROPN
ejpam-6566	363	2	mathematical	mathematical	PROPN
ejpam-6566	363	3	journal	journal	PROPN
ejpam-6566	363	4	,	,	PUNCT
ejpam-6566	363	5	28:71–77	28:71–77	PROPN
ejpam-6566	363	6	,	,	PUNCT
ejpam-6566	363	7	1988	1988	NUM
ejpam-6566	363	8	.	.	PUNCT
ejpam-6566	364	1	[	[	X
ejpam-6566	364	2	4	4	X
ejpam-6566	364	3	]	]	PUNCT
ejpam-6566	364	4	a.	a.	NOUN
ejpam-6566	364	5	a.	a.	NOUN
ejpam-6566	364	6	nasef	nasef	PROPN
ejpam-6566	364	7	and	and	CCONJ
ejpam-6566	364	8	t.	t.	PROPN
ejpam-6566	364	9	noiri	noiri	PROPN
ejpam-6566	364	10	.	.	PUNCT
ejpam-6566	365	1	some	some	DET
ejpam-6566	365	2	weak	weak	ADJ
ejpam-6566	365	3	forms	form	NOUN
ejpam-6566	365	4	of	of	ADP
ejpam-6566	365	5	almost	almost	ADV
ejpam-6566	365	6	continuity	continuity	NOUN
ejpam-6566	365	7	.	.	PUNCT
ejpam-6566	366	1	acta	acta	PROPN
ejpam-6566	366	2	mathematica	mathematica	PROPN
ejpam-6566	366	3	hungarica	hungarica	PROPN
ejpam-6566	366	4	,	,	PUNCT
ejpam-6566	366	5	74(3):211–219	74(3):211–219	PROPN
ejpam-6566	366	6	,	,	PUNCT
ejpam-6566	366	7	1997	1997	NUM
ejpam-6566	366	8	.	.	PUNCT
ejpam-6566	367	1	[	[	X
ejpam-6566	367	2	5	5	X
ejpam-6566	367	3	]	]	PUNCT
ejpam-6566	367	4	v.	v.	CCONJ
ejpam-6566	367	5	popa	popa	NOUN
ejpam-6566	367	6	.	.	PUNCT
ejpam-6566	368	1	almost	almost	ADV
ejpam-6566	368	2	continuous	continuous	ADJ
ejpam-6566	368	3	multifunctions	multifunction	NOUN
ejpam-6566	368	4	.	.	PUNCT
ejpam-6566	369	1	matematički	matematički	PROPN
ejpam-6566	369	2	vesnik	vesnik	PROPN
ejpam-6566	369	3	,	,	PUNCT
ejpam-6566	369	4	6(9)(34):75–84	6(9)(34):75–84	NOUN
ejpam-6566	369	5	,	,	PUNCT
ejpam-6566	369	6	1982	1982	NUM
ejpam-6566	369	7	.	.	PUNCT
ejpam-6566	370	1	[	[	X
ejpam-6566	370	2	6	6	NUM
ejpam-6566	370	3	]	]	PUNCT
ejpam-6566	370	4	v.	v.	CCONJ
ejpam-6566	370	5	popa	popa	NOUN
ejpam-6566	370	6	and	and	CCONJ
ejpam-6566	370	7	t.	t.	PROPN
ejpam-6566	370	8	noiri	noiri	PROPN
ejpam-6566	370	9	.	.	PUNCT
ejpam-6566	371	1	on	on	ADP
ejpam-6566	371	2	upper	upper	ADJ
ejpam-6566	371	3	and	and	CCONJ
ejpam-6566	371	4	lower	low	ADJ
ejpam-6566	371	5	almost	almost	ADV
ejpam-6566	371	6	quasi	quasi	ADJ
ejpam-6566	371	7	-	-	ADJ
ejpam-6566	371	8	continuous	continuous	ADJ
ejpam-6566	371	9	multifunctions	multifunction	NOUN
ejpam-6566	371	10	.	.	PUNCT
ejpam-6566	372	1	bulletin	bulletin	NOUN
ejpam-6566	372	2	of	of	ADP
ejpam-6566	372	3	the	the	DET
ejpam-6566	372	4	institute	institute	NOUN
ejpam-6566	372	5	of	of	ADP
ejpam-6566	372	6	mathematics	mathematics	PROPN
ejpam-6566	372	7	,	,	PUNCT
ejpam-6566	372	8	academia	academia	PROPN
ejpam-6566	372	9	sinica	sinica	PROPN
ejpam-6566	372	10	,	,	PUNCT
ejpam-6566	372	11	21:337–349	21:337–349	NUM
ejpam-6566	372	12	,	,	PUNCT
ejpam-6566	372	13	1993	1993	NUM
ejpam-6566	372	14	.	.	PUNCT
ejpam-6566	373	1	[	[	X
ejpam-6566	373	2	7	7	X
ejpam-6566	373	3	]	]	PUNCT
ejpam-6566	373	4	v.	v.	CCONJ
ejpam-6566	373	5	popa	popa	NOUN
ejpam-6566	373	6	,	,	PUNCT
ejpam-6566	373	7	t.	t.	PROPN
ejpam-6566	373	8	noiri	noiri	PROPN
ejpam-6566	373	9	,	,	PUNCT
ejpam-6566	373	10	and	and	CCONJ
ejpam-6566	373	11	m.	m.	NOUN
ejpam-6566	373	12	ganster	ganster	NOUN
ejpam-6566	373	13	.	.	PUNCT
ejpam-6566	374	1	on	on	ADP
ejpam-6566	374	2	upper	upper	ADJ
ejpam-6566	374	3	and	and	CCONJ
ejpam-6566	374	4	lower	low	ADJ
ejpam-6566	374	5	almost	almost	ADV
ejpam-6566	374	6	precontinuous	precontinuous	ADJ
ejpam-6566	374	7	multifunctions	multifunction	NOUN
ejpam-6566	374	8	.	.	PUNCT
ejpam-6566	375	1	far	far	PROPN
ejpam-6566	375	2	east	east	PROPN
ejpam-6566	375	3	journal	journal	PROPN
ejpam-6566	375	4	of	of	ADP
ejpam-6566	375	5	mathematical	mathematical	ADJ
ejpam-6566	375	6	sciences	science	NOUN
ejpam-6566	375	7	,	,	PUNCT
ejpam-6566	375	8	special	special	ADJ
ejpam-6566	375	9	volume(part	volume(part	NOUN
ejpam-6566	375	10	i):49–68	i):49–68	NOUN
ejpam-6566	375	11	,	,	PUNCT
ejpam-6566	375	12	1997	1997	NUM
ejpam-6566	375	13	.	.	PUNCT
ejpam-6566	376	1	[	[	X
ejpam-6566	376	2	8	8	X
ejpam-6566	376	3	]	]	PUNCT
ejpam-6566	376	4	t.	t.	PROPN
ejpam-6566	376	5	noiri	noiri	PROPN
ejpam-6566	376	6	and	and	CCONJ
ejpam-6566	376	7	v.	v.	ADP
ejpam-6566	376	8	popa	popa	NOUN
ejpam-6566	376	9	.	.	PUNCT
ejpam-6566	377	1	on	on	ADP
ejpam-6566	377	2	upper	upper	ADJ
ejpam-6566	377	3	and	and	CCONJ
ejpam-6566	377	4	lower	low	ADJ
ejpam-6566	377	5	almost	almost	ADV
ejpam-6566	377	6	β	β	ADJ
ejpam-6566	377	7	-	-	ADJ
ejpam-6566	377	8	continuous	continuous	ADJ
ejpam-6566	377	9	multifunctions	multifunction	NOUN
ejpam-6566	377	10	.	.	PUNCT
ejpam-6566	378	1	acta	acta	PROPN
ejpam-6566	378	2	mathematica	mathematica	PROPN
ejpam-6566	378	3	hungarica	hungarica	PROPN
ejpam-6566	378	4	,	,	PUNCT
ejpam-6566	378	5	82:57–73	82:57–73	PROPN
ejpam-6566	378	6	,	,	PUNCT
ejpam-6566	378	7	1999	1999	NUM
ejpam-6566	378	8	.	.	PUNCT
ejpam-6566	379	1	[	[	X
ejpam-6566	379	2	9	9	NUM
ejpam-6566	379	3	]	]	PUNCT
ejpam-6566	379	4	v.	v.	CCONJ
ejpam-6566	379	5	popa	popa	NOUN
ejpam-6566	379	6	and	and	CCONJ
ejpam-6566	379	7	t.	t.	PROPN
ejpam-6566	379	8	noiri	noiri	PROPN
ejpam-6566	379	9	.	.	PUNCT
ejpam-6566	380	1	on	on	ADP
ejpam-6566	380	2	upper	upper	ADJ
ejpam-6566	380	3	and	and	CCONJ
ejpam-6566	380	4	lower	low	ADJ
ejpam-6566	380	5	weakly	weakly	ADJ
ejpam-6566	380	6	β	β	ADJ
ejpam-6566	380	7	-	-	ADJ
ejpam-6566	380	8	continuous	continuous	ADJ
ejpam-6566	380	9	multifunctions	multifunction	NOUN
ejpam-6566	380	10	.	.	PUNCT
ejpam-6566	381	1	annales	annales	PROPN
ejpam-6566	381	2	universitatis	universitatis	PROPN
ejpam-6566	381	3	scientiarum	scientiarum	PROPN
ejpam-6566	381	4	budapestinensis	budapestinensis	PROPN
ejpam-6566	381	5	,	,	PUNCT
ejpam-6566	381	6	43:25–48	43:25–48	PROPN
ejpam-6566	381	7	,	,	PUNCT
ejpam-6566	381	8	2000	2000	NUM
ejpam-6566	381	9	.	.	PUNCT
ejpam-6566	382	1	[	[	X
ejpam-6566	382	2	10	10	NUM
ejpam-6566	382	3	]	]	X
ejpam-6566	382	4	v.	v.	CCONJ
ejpam-6566	382	5	popa	popa	NOUN
ejpam-6566	382	6	and	and	CCONJ
ejpam-6566	382	7	t.	t.	PROPN
ejpam-6566	382	8	noiri	noiri	PROPN
ejpam-6566	382	9	.	.	PUNCT
ejpam-6566	383	1	on	on	ADP
ejpam-6566	383	2	upper	upper	ADJ
ejpam-6566	383	3	and	and	CCONJ
ejpam-6566	383	4	lower	low	ADJ
ejpam-6566	383	5	almost	almost	ADV
ejpam-6566	383	6	α	α	ADJ
ejpam-6566	383	7	-	-	ADJ
ejpam-6566	383	8	continuous	continuous	ADJ
ejpam-6566	383	9	multifunctions	multifunction	NOUN
ejpam-6566	383	10	.	.	PUNCT
ejpam-6566	384	1	demonstratio	demonstratio	PROPN
ejpam-6566	384	2	mathematica	mathematica	PROPN
ejpam-6566	384	3	,	,	PUNCT
ejpam-6566	384	4	29:381–396	29:381–396	PROPN
ejpam-6566	384	5	,	,	PUNCT
ejpam-6566	384	6	1996	1996	NUM
ejpam-6566	384	7	.	.	PUNCT
ejpam-6566	385	1	[	[	X
ejpam-6566	385	2	11	11	NUM
ejpam-6566	385	3	]	]	X
ejpam-6566	385	4	p.	p.	NOUN
ejpam-6566	385	5	pue	pue	NOUN
ejpam-6566	385	6	-	-	PUNCT
ejpam-6566	385	7	on	on	ADP
ejpam-6566	385	8	,	,	PUNCT
ejpam-6566	385	9	s.	s.	PROPN
ejpam-6566	385	10	sompong	sompong	PROPN
ejpam-6566	385	11	,	,	PUNCT
ejpam-6566	385	12	and	and	CCONJ
ejpam-6566	385	13	c.	c.	PROPN
ejpam-6566	385	14	boonpok	boonpok	PROPN
ejpam-6566	385	15	.	.	PUNCT
ejpam-6566	386	1	upper	upper	ADJ
ejpam-6566	386	2	and	and	CCONJ
ejpam-6566	386	3	lower	low	ADJ
ejpam-6566	386	4	(	(	PUNCT
ejpam-6566	386	5	τ1	τ1	NOUN
ejpam-6566	386	6	,	,	PUNCT
ejpam-6566	386	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6566	386	8	multifunctions	multifunction	NOUN
ejpam-6566	386	9	.	.	PUNCT
ejpam-6566	387	1	international	international	ADJ
ejpam-6566	387	2	journal	journal	PROPN
ejpam-6566	387	3	of	of	ADP
ejpam-6566	387	4	mathematics	mathematic	NOUN
ejpam-6566	387	5	and	and	CCONJ
ejpam-6566	387	6	computer	computer	NOUN
ejpam-6566	387	7	science	science	NOUN
ejpam-6566	387	8	,	,	PUNCT
ejpam-6566	387	9	19(4):1305	19(4):1305	NUM
ejpam-6566	387	10	–	–	PUNCT
ejpam-6566	387	11	1310	1310	NUM
ejpam-6566	387	12	,	,	PUNCT
ejpam-6566	387	13	2024	2024	NUM
ejpam-6566	387	14	.	.	PUNCT
ejpam-6566	388	1	[	[	X
ejpam-6566	388	2	12	12	NUM
ejpam-6566	388	3	]	]	X
ejpam-6566	388	4	c.	c.	PROPN
ejpam-6566	388	5	klanarong	klanarong	PROPN
ejpam-6566	388	6	,	,	PUNCT
ejpam-6566	388	7	s.	s.	PROPN
ejpam-6566	388	8	sompong	sompong	PROPN
ejpam-6566	388	9	,	,	PUNCT
ejpam-6566	388	10	and	and	CCONJ
ejpam-6566	388	11	c.	c.	PROPN
ejpam-6566	388	12	boonpok	boonpok	PROPN
ejpam-6566	388	13	.	.	PUNCT
ejpam-6566	389	1	upper	upper	ADJ
ejpam-6566	389	2	and	and	CCONJ
ejpam-6566	389	3	lower	low	ADJ
ejpam-6566	389	4	almost	almost	ADV
ejpam-6566	389	5	(	(	PUNCT
ejpam-6566	389	6	τ1	τ1	NOUN
ejpam-6566	389	7	,	,	PUNCT
ejpam-6566	389	8	τ2)continuous	τ2)continuous	ADJ
ejpam-6566	389	9	multifunctions	multifunction	NOUN
ejpam-6566	389	10	.	.	PUNCT
ejpam-6566	390	1	european	european	ADJ
ejpam-6566	390	2	journal	journal	PROPN
ejpam-6566	390	3	of	of	ADP
ejpam-6566	390	4	pure	pure	ADJ
ejpam-6566	390	5	and	and	CCONJ
ejpam-6566	390	6	applied	applied	ADJ
ejpam-6566	390	7	mathematics	mathematic	NOUN
ejpam-6566	390	8	,	,	PUNCT
ejpam-6566	390	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-6566	390	10	,	,	PUNCT
ejpam-6566	390	11	2024	2024	NUM
ejpam-6566	390	12	.	.	PUNCT
ejpam-6566	391	1	[	[	X
ejpam-6566	391	2	13	13	NUM
ejpam-6566	391	3	]	]	PUNCT
ejpam-6566	391	4	k.	k.	PROPN
ejpam-6566	391	5	kuratowski	kuratowski	PROPN
ejpam-6566	391	6	.	.	PUNCT
ejpam-6566	392	1	topology	topology	PROPN
ejpam-6566	392	2	,	,	PUNCT
ejpam-6566	392	3	vol	vol	NOUN
ejpam-6566	392	4	.	.	PUNCT
ejpam-6566	392	5	i.	i.	PROPN
ejpam-6566	392	6	academic	academic	PROPN
ejpam-6566	392	7	press	press	PROPN
ejpam-6566	392	8	,	,	PUNCT
ejpam-6566	392	9	new	new	PROPN
ejpam-6566	392	10	york	york	PROPN
ejpam-6566	392	11	,	,	PUNCT
ejpam-6566	392	12	1966	1966	NUM
ejpam-6566	392	13	.	.	PUNCT
ejpam-6566	393	1	[	[	X
ejpam-6566	393	2	14	14	NUM
ejpam-6566	393	3	]	]	X
ejpam-6566	393	4	r.	r.	PROPN
ejpam-6566	393	5	vaidyanathaswamy	vaidyanathaswamy	PROPN
ejpam-6566	393	6	.	.	PUNCT
ejpam-6566	394	1	the	the	DET
ejpam-6566	394	2	localisation	localisation	NOUN
ejpam-6566	394	3	theory	theory	NOUN
ejpam-6566	394	4	in	in	ADP
ejpam-6566	394	5	set	set	NOUN
ejpam-6566	394	6	-	-	PUNCT
ejpam-6566	394	7	topology	topology	NOUN
ejpam-6566	394	8	.	.	PUNCT
ejpam-6566	395	1	proceedings	proceeding	NOUN
ejpam-6566	395	2	of	of	ADP
ejpam-6566	395	3	the	the	DET
ejpam-6566	395	4	indian	indian	PROPN
ejpam-6566	395	5	academy	academy	PROPN
ejpam-6566	395	6	of	of	ADP
ejpam-6566	395	7	sciences	science	NOUN
ejpam-6566	395	8	-	-	PUNCT
ejpam-6566	395	9	section	section	NOUN
ejpam-6566	395	10	a	a	PRON
ejpam-6566	395	11	,	,	PUNCT
ejpam-6566	395	12	20:51–61	20:51–61	NUM
ejpam-6566	395	13	,	,	PUNCT
ejpam-6566	395	14	1944	1944	NUM
ejpam-6566	395	15	.	.	PUNCT
ejpam-6566	396	1	c.	c.	PROPN
ejpam-6566	396	2	viriyapong	viriyapong	PROPN
ejpam-6566	396	3	,	,	PUNCT
ejpam-6566	396	4	a.	a.	PROPN
ejpam-6566	396	5	sama	sama	PROPN
ejpam-6566	396	6	-	-	PUNCT
ejpam-6566	396	7	ae	ae	PROPN
ejpam-6566	396	8	,	,	PUNCT
ejpam-6566	396	9	c.	c.	PROPN
ejpam-6566	396	10	boonpok	boonpok	PROPN
ejpam-6566	396	11	/	/	SYM
ejpam-6566	396	12	eur	eur	PROPN
ejpam-6566	396	13	.	.	PUNCT
ejpam-6566	397	1	j.	j.	PROPN
ejpam-6566	397	2	pure	pure	PROPN
ejpam-6566	397	3	appl	appl	PROPN
ejpam-6566	397	4	.	.	PROPN
ejpam-6566	397	5	math	math	PROPN
ejpam-6566	397	6	,	,	PUNCT
ejpam-6566	397	7	18	18	NUM
ejpam-6566	397	8	(	(	PUNCT
ejpam-6566	397	9	3	3	NUM
ejpam-6566	397	10	)	)	PUNCT
ejpam-6566	397	11	(	(	PUNCT
ejpam-6566	397	12	2025	2025	NUM
ejpam-6566	397	13	)	)	PUNCT
ejpam-6566	397	14	,	,	PUNCT
ejpam-6566	397	15	6566	6566	NUM
ejpam-6566	397	16	14	14	NUM
ejpam-6566	397	17	of	of	ADP
ejpam-6566	397	18	14	14	NUM
ejpam-6566	398	1	[	[	SYM
ejpam-6566	398	2	15	15	NUM
ejpam-6566	398	3	]	]	X
ejpam-6566	398	4	e.	e.	PROPN
ejpam-6566	398	5	hatir	hatir	PROPN
ejpam-6566	398	6	and	and	CCONJ
ejpam-6566	398	7	t.	t.	PROPN
ejpam-6566	398	8	noiri	noiri	PROPN
ejpam-6566	398	9	.	.	PUNCT
ejpam-6566	399	1	weakly	weakly	ADJ
ejpam-6566	399	2	pre	pre	ADJ
ejpam-6566	399	3	-	-	ADJ
ejpam-6566	399	4	i	i	PRON
ejpam-6566	399	5	-	-	PUNCT
ejpam-6566	399	6	open	open	ADJ
ejpam-6566	399	7	sets	set	NOUN
ejpam-6566	399	8	and	and	CCONJ
ejpam-6566	399	9	decomposition	decomposition	NOUN
ejpam-6566	399	10	of	of	ADP
ejpam-6566	399	11	continuity	continuity	NOUN
ejpam-6566	399	12	.	.	PUNCT
ejpam-6566	400	1	acta	acta	PROPN
ejpam-6566	400	2	mathematica	mathematica	PROPN
ejpam-6566	400	3	hungarica	hungarica	PROPN
ejpam-6566	400	4	,	,	PUNCT
ejpam-6566	400	5	106(3):227–238	106(3):227–238	NUM
ejpam-6566	400	6	,	,	PUNCT
ejpam-6566	400	7	2005	2005	NUM
ejpam-6566	400	8	.	.	PUNCT
ejpam-6566	401	1	[	[	X
ejpam-6566	401	2	16	16	NUM
ejpam-6566	401	3	]	]	X
ejpam-6566	401	4	e.	e.	PROPN
ejpam-6566	401	5	hatir	hatir	PROPN
ejpam-6566	401	6	and	and	CCONJ
ejpam-6566	401	7	t.	t.	PROPN
ejpam-6566	401	8	noiri	noiri	PROPN
ejpam-6566	401	9	.	.	PUNCT
ejpam-6566	402	1	on	on	ADP
ejpam-6566	402	2	decompositions	decomposition	NOUN
ejpam-6566	402	3	of	of	ADP
ejpam-6566	402	4	continuity	continuity	NOUN
ejpam-6566	402	5	via	via	ADP
ejpam-6566	402	6	idealization	idealization	NOUN
ejpam-6566	402	7	.	.	PUNCT
ejpam-6566	403	1	acta	acta	PROPN
ejpam-6566	403	2	mathematica	mathematica	PROPN
ejpam-6566	403	3	hungarica	hungarica	PROPN
ejpam-6566	403	4	,	,	PUNCT
ejpam-6566	403	5	96:341–349	96:341–349	PROPN
ejpam-6566	403	6	,	,	PUNCT
ejpam-6566	403	7	2002	2002	NUM
ejpam-6566	403	8	.	.	PUNCT
ejpam-6566	404	1	[	[	X
ejpam-6566	404	2	17	17	NUM
ejpam-6566	404	3	]	]	PUNCT
ejpam-6566	404	4	c.	c.	PROPN
ejpam-6566	404	5	boonpok	boonpok	PROPN
ejpam-6566	404	6	.	.	PUNCT
ejpam-6566	405	1	on	on	ADP
ejpam-6566	405	2	continuous	continuous	ADJ
ejpam-6566	405	3	multifunctions	multifunction	NOUN
ejpam-6566	405	4	in	in	ADP
ejpam-6566	405	5	ideal	ideal	ADJ
ejpam-6566	405	6	topological	topological	ADJ
ejpam-6566	405	7	spaces	space	NOUN
ejpam-6566	405	8	.	.	PUNCT
ejpam-6566	406	1	lobachevskii	lobachevskii	PROPN
ejpam-6566	406	2	journal	journal	PROPN
ejpam-6566	406	3	of	of	ADP
ejpam-6566	406	4	mathematics	mathematic	NOUN
ejpam-6566	406	5	,	,	PUNCT
ejpam-6566	406	6	40(1):24–35	40(1):24–35	NUM
ejpam-6566	406	7	,	,	PUNCT
ejpam-6566	406	8	2019	2019	NUM
ejpam-6566	406	9	.	.	PUNCT
ejpam-6566	407	1	[	[	X
ejpam-6566	407	2	18	18	NUM
ejpam-6566	407	3	]	]	PUNCT
ejpam-6566	407	4	c.	c.	PROPN
ejpam-6566	407	5	boonpok	boonpok	PROPN
ejpam-6566	407	6	.	.	PUNCT
ejpam-6566	408	1	pı	pı	NOUN
ejpam-6566	408	2	-	-	NOUN
ejpam-6566	408	3	continuity	continuity	NOUN
ejpam-6566	408	4	and	and	CCONJ
ejpam-6566	408	5	weak	weak	ADJ
ejpam-6566	408	6	pı	pı	NOUN
ejpam-6566	408	7	-	-	NOUN
ejpam-6566	408	8	continuity	continuity	NOUN
ejpam-6566	408	9	.	.	PUNCT
ejpam-6566	409	1	carpathian	carpathian	ADJ
ejpam-6566	409	2	mathematical	mathematical	ADJ
ejpam-6566	409	3	publications	publication	NOUN
ejpam-6566	409	4	,	,	PUNCT
ejpam-6566	409	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6566	409	6	,	,	PUNCT
ejpam-6566	409	7	2025	2025	NUM
ejpam-6566	409	8	.	.	PUNCT
ejpam-6566	410	1	[	[	X
ejpam-6566	410	2	19	19	NUM
ejpam-6566	410	3	]	]	PUNCT
ejpam-6566	410	4	c.	c.	PROPN
ejpam-6566	410	5	boonpok	boonpok	PROPN
ejpam-6566	410	6	,	,	PUNCT
ejpam-6566	410	7	c.	c.	PROPN
ejpam-6566	410	8	viriyapong	viriyapong	PROPN
ejpam-6566	410	9	,	,	PUNCT
ejpam-6566	410	10	and	and	CCONJ
ejpam-6566	410	11	m.	m.	NOUN
ejpam-6566	410	12	thongmoon	thongmoon	NOUN
ejpam-6566	410	13	.	.	PUNCT
ejpam-6566	411	1	on	on	ADP
ejpam-6566	411	2	upper	upper	ADJ
ejpam-6566	411	3	and	and	CCONJ
ejpam-6566	411	4	lower	low	ADJ
ejpam-6566	411	5	(	(	PUNCT
ejpam-6566	411	6	τ1	τ1	NOUN
ejpam-6566	411	7	,	,	PUNCT
ejpam-6566	411	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6566	411	9	multifunctions	multifunction	NOUN
ejpam-6566	411	10	.	.	PUNCT
ejpam-6566	412	1	journal	journal	PROPN
ejpam-6566	412	2	of	of	ADP
ejpam-6566	412	3	mathematics	mathematics	PROPN
ejpam-6566	412	4	and	and	CCONJ
ejpam-6566	412	5	computer	computer	NOUN
ejpam-6566	412	6	science	science	NOUN
ejpam-6566	412	7	,	,	PUNCT
ejpam-6566	412	8	18:282–293	18:282–293	NUM
ejpam-6566	412	9	,	,	PUNCT
ejpam-6566	412	10	2018	2018	NUM
ejpam-6566	412	11	.	.	PUNCT
ejpam-6566	413	1	[	[	X
ejpam-6566	413	2	20	20	NUM
ejpam-6566	413	3	]	]	X
ejpam-6566	413	4	c.	c.	PROPN
ejpam-6566	413	5	viriyapong	viriyapong	PROPN
ejpam-6566	413	6	and	and	CCONJ
ejpam-6566	413	7	c.	c.	PROPN
ejpam-6566	413	8	boonpok	boonpok	PROPN
ejpam-6566	413	9	.	.	PUNCT
ejpam-6566	414	1	(	(	PUNCT
ejpam-6566	414	2	τ1	τ1	NOUN
ejpam-6566	414	3	,	,	PUNCT
ejpam-6566	414	4	τ2)α	τ2)α	NOUN
ejpam-6566	414	5	-	-	PUNCT
ejpam-6566	414	6	continuity	continuity	NOUN
ejpam-6566	414	7	for	for	ADP
ejpam-6566	414	8	multifunctions	multifunction	NOUN
ejpam-6566	414	9	.	.	PUNCT
ejpam-6566	415	1	journal	journal	PROPN
ejpam-6566	415	2	of	of	ADP
ejpam-6566	415	3	mathematics	mathematic	NOUN
ejpam-6566	415	4	,	,	PUNCT
ejpam-6566	415	5	2020:6285763	2020:6285763	NUM
ejpam-6566	415	6	,	,	PUNCT
ejpam-6566	415	7	2020	2020	NUM
ejpam-6566	415	8	.	.	PUNCT
ejpam-6566	416	1	[	[	X
ejpam-6566	416	2	21	21	NUM
ejpam-6566	416	3	]	]	X
ejpam-6566	416	4	c.	c.	PROPN
ejpam-6566	416	5	boonpok	boonpok	PROPN
ejpam-6566	416	6	.	.	PUNCT
ejpam-6566	417	1	(	(	PUNCT
ejpam-6566	417	2	τ1	τ1	NOUN
ejpam-6566	417	3	,	,	PUNCT
ejpam-6566	417	4	τ2)δ	τ2)δ	ADJ
ejpam-6566	417	5	-	-	PUNCT
ejpam-6566	417	6	semicontinuous	semicontinuous	ADJ
ejpam-6566	417	7	multifunctions	multifunction	NOUN
ejpam-6566	417	8	.	.	PUNCT
ejpam-6566	418	1	heliyon	heliyon	NOUN
ejpam-6566	418	2	,	,	PUNCT
ejpam-6566	418	3	6	6	NUM
ejpam-6566	418	4	:	:	SYM
ejpam-6566	418	5	e05367	e05367	PROPN
ejpam-6566	418	6	,	,	PUNCT
ejpam-6566	418	7	2020	2020	NUM
ejpam-6566	418	8	.	.	PUNCT
ejpam-6566	419	1	[	[	X
ejpam-6566	419	2	22	22	NUM
ejpam-6566	419	3	]	]	X
ejpam-6566	419	4	n.	n.	PROPN
ejpam-6566	419	5	viriyapong	viriyapong	PROPN
ejpam-6566	419	6	,	,	PUNCT
ejpam-6566	419	7	s.	s.	PROPN
ejpam-6566	419	8	sompong	sompong	PROPN
ejpam-6566	419	9	,	,	PUNCT
ejpam-6566	419	10	and	and	CCONJ
ejpam-6566	419	11	c.	c.	PROPN
ejpam-6566	419	12	boonpok	boonpok	PROPN
ejpam-6566	419	13	.	.	PUNCT
ejpam-6566	420	1	(	(	PUNCT
ejpam-6566	420	2	τ1	τ1	NOUN
ejpam-6566	420	3	,	,	PUNCT
ejpam-6566	420	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6566	420	5	disconnectedness	disconnectedness	NOUN
ejpam-6566	420	6	in	in	ADP
ejpam-6566	420	7	bitopological	bitopological	ADJ
ejpam-6566	420	8	spaces	space	NOUN
ejpam-6566	420	9	.	.	PUNCT
ejpam-6566	421	1	international	international	ADJ
ejpam-6566	421	2	journal	journal	PROPN
ejpam-6566	421	3	of	of	ADP
ejpam-6566	421	4	mathematics	mathematic	NOUN
ejpam-6566	421	5	and	and	CCONJ
ejpam-6566	421	6	computer	computer	NOUN
ejpam-6566	421	7	science	science	NOUN
ejpam-6566	421	8	,	,	PUNCT
ejpam-6566	421	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6566	421	10	,	,	PUNCT
ejpam-6566	421	11	2024	2024	NUM
ejpam-6566	421	12	.	.	PUNCT
ejpam-6566	422	1	[	[	X
ejpam-6566	422	2	23	23	NUM
ejpam-6566	422	3	]	]	X
ejpam-6566	422	4	n.	n.	PROPN
ejpam-6566	422	5	viriyapong	viriyapong	PROPN
ejpam-6566	422	6	,	,	PUNCT
ejpam-6566	422	7	s.	s.	PROPN
ejpam-6566	422	8	sompong	sompong	PROPN
ejpam-6566	422	9	,	,	PUNCT
ejpam-6566	422	10	and	and	CCONJ
ejpam-6566	422	11	c.	c.	PROPN
ejpam-6566	422	12	boonpok	boonpok	PROPN
ejpam-6566	422	13	.	.	PUNCT
ejpam-6566	423	1	upper	upper	ADJ
ejpam-6566	423	2	and	and	CCONJ
ejpam-6566	423	3	lower	low	ADJ
ejpam-6566	423	4	s-(τ1	s-(τ1	NOUN
ejpam-6566	423	5	,	,	PUNCT
ejpam-6566	423	6	τ2)p	τ2)p	ADJ
ejpam-6566	423	7	-	-	PUNCT
ejpam-6566	423	8	continuous	continuous	ADJ
ejpam-6566	423	9	multifunctions	multifunction	NOUN
ejpam-6566	423	10	.	.	PUNCT
ejpam-6566	424	1	european	european	ADJ
ejpam-6566	424	2	journal	journal	PROPN
ejpam-6566	424	3	of	of	ADP
ejpam-6566	424	4	pure	pure	ADJ
ejpam-6566	424	5	and	and	CCONJ
ejpam-6566	424	6	applied	applied	ADJ
ejpam-6566	424	7	mathematics	mathematic	NOUN
ejpam-6566	424	8	,	,	PUNCT
ejpam-6566	424	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-6566	424	10	,	,	PUNCT
ejpam-6566	424	11	2024	2024	NUM
ejpam-6566	424	12	.	.	PUNCT
ejpam-6566	425	1	[	[	X
ejpam-6566	425	2	24	24	NUM
ejpam-6566	425	3	]	]	PUNCT
ejpam-6566	425	4	c.	c.	PROPN
ejpam-6566	425	5	viriyapong	viriyapong	PROPN
ejpam-6566	425	6	,	,	PUNCT
ejpam-6566	425	7	s.	s.	PROPN
ejpam-6566	425	8	sompong	sompong	PROPN
ejpam-6566	425	9	,	,	PUNCT
ejpam-6566	425	10	and	and	CCONJ
ejpam-6566	425	11	c.	c.	PROPN
ejpam-6566	425	12	boonpok	boonpok	PROPN
ejpam-6566	425	13	.	.	PUNCT
ejpam-6566	426	1	upper	upper	ADJ
ejpam-6566	426	2	and	and	CCONJ
ejpam-6566	426	3	lower	low	ADJ
ejpam-6566	426	4	slight	slight	ADJ
ejpam-6566	426	5	α(τ1	α(τ1	NOUN
ejpam-6566	426	6	,	,	PUNCT
ejpam-6566	426	7	τ2)continuity	τ2)continuity	PROPN
ejpam-6566	426	8	.	.	PUNCT
ejpam-6566	427	1	european	european	PROPN
ejpam-6566	427	2	journal	journal	PROPN
ejpam-6566	427	3	of	of	ADP
ejpam-6566	427	4	pure	pure	ADJ
ejpam-6566	427	5	and	and	CCONJ
ejpam-6566	427	6	applied	applied	ADJ
ejpam-6566	427	7	mathematics	mathematic	NOUN
ejpam-6566	427	8	,	,	PUNCT
ejpam-6566	427	9	17(3):2142–2154	17(3):2142–2154	NUM
ejpam-6566	427	10	,	,	PUNCT
ejpam-6566	427	11	2024	2024	NUM
ejpam-6566	427	12	.	.	PUNCT
ejpam-6566	428	1	[	[	X
ejpam-6566	428	2	25	25	NUM
ejpam-6566	428	3	]	]	PUNCT
ejpam-6566	428	4	c.	c.	PROPN
ejpam-6566	428	5	boonpok	boonpok	PROPN
ejpam-6566	428	6	and	and	CCONJ
ejpam-6566	428	7	p.	p.	NOUN
ejpam-6566	428	8	pue	pue	NOUN
ejpam-6566	428	9	-	-	PUNCT
ejpam-6566	428	10	on	on	ADP
ejpam-6566	428	11	.	.	PUNCT
ejpam-6566	429	1	characterizations	characterization	NOUN
ejpam-6566	429	2	of	of	ADP
ejpam-6566	429	3	almost	almost	ADV
ejpam-6566	429	4	(	(	PUNCT
ejpam-6566	429	5	τ1	τ1	NOUN
ejpam-6566	429	6	,	,	PUNCT
ejpam-6566	429	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6566	429	8	multifunctions	multifunction	NOUN
ejpam-6566	429	9	.	.	PUNCT
ejpam-6566	430	1	international	international	ADJ
ejpam-6566	430	2	journal	journal	NOUN
ejpam-6566	430	3	of	of	ADP
ejpam-6566	430	4	analysis	analysis	NOUN
ejpam-6566	430	5	and	and	CCONJ
ejpam-6566	430	6	applications	application	NOUN
ejpam-6566	430	7	,	,	PUNCT
ejpam-6566	430	8	22:33	22:33	NUM
ejpam-6566	430	9	,	,	PUNCT
ejpam-6566	430	10	2024	2024	NUM
ejpam-6566	430	11	.	.	PUNCT
ejpam-6566	431	1	[	[	X
ejpam-6566	431	2	26	26	NUM
ejpam-6566	431	3	]	]	X
ejpam-6566	431	4	d.	d.	PROPN
ejpam-6566	431	5	janković	janković	PROPN
ejpam-6566	431	6	and	and	CCONJ
ejpam-6566	431	7	t.	t.	PROPN
ejpam-6566	431	8	r.	r.	PROPN
ejpam-6566	431	9	hamlett	hamlett	PROPN
ejpam-6566	431	10	.	.	PUNCT
ejpam-6566	432	1	new	new	ADJ
ejpam-6566	432	2	topologies	topology	NOUN
ejpam-6566	432	3	from	from	ADP
ejpam-6566	432	4	old	old	ADJ
ejpam-6566	432	5	via	via	ADP
ejpam-6566	432	6	ideals	ideal	NOUN
ejpam-6566	432	7	.	.	PUNCT
ejpam-6566	433	1	the	the	DET
ejpam-6566	433	2	american	american	PROPN
ejpam-6566	433	3	mathematical	mathematical	PROPN
ejpam-6566	433	4	monthly	monthly	ADV
ejpam-6566	433	5	,	,	PUNCT
ejpam-6566	433	6	97:295–310	97:295–310	PROPN
ejpam-6566	433	7	,	,	PUNCT
ejpam-6566	433	8	1990	1990	NUM
ejpam-6566	433	9	.	.	PUNCT
ejpam-6566	434	1	[	[	X
ejpam-6566	434	2	27	27	NUM
ejpam-6566	434	3	]	]	X
ejpam-6566	434	4	e.	e.	PROPN
ejpam-6566	434	5	ekici	ekici	PROPN
ejpam-6566	434	6	and	and	CCONJ
ejpam-6566	434	7	t.	t.	PROPN
ejpam-6566	434	8	noiri	noiri	PROPN
ejpam-6566	434	9	.	.	PUNCT
ejpam-6566	435	1	⋆-extremally	⋆-extremally	ADV
ejpam-6566	435	2	disconnected	disconnect	VERB
ejpam-6566	435	3	ideal	ideal	ADJ
ejpam-6566	435	4	topological	topological	ADJ
ejpam-6566	435	5	spaces	space	NOUN
ejpam-6566	435	6	.	.	PUNCT
ejpam-6566	436	1	acta	acta	PROPN
ejpam-6566	436	2	mathematica	mathematica	PROPN
ejpam-6566	436	3	hungarica	hungarica	PROPN
ejpam-6566	436	4	,	,	PUNCT
ejpam-6566	436	5	122:81–90	122:81–90	NUM
ejpam-6566	436	6	,	,	PUNCT
ejpam-6566	436	7	2009	2009	NUM
ejpam-6566	436	8	.	.	PUNCT
ejpam-6566	437	1	[	[	X
ejpam-6566	437	2	28	28	NUM
ejpam-6566	437	3	]	]	X
ejpam-6566	437	4	p.	p.	NOUN
ejpam-6566	437	5	pue	pue	NOUN
ejpam-6566	437	6	-	-	PUNCT
ejpam-6566	437	7	on	on	ADP
ejpam-6566	437	8	,	,	PUNCT
ejpam-6566	437	9	s.	s.	PROPN
ejpam-6566	437	10	sompong	sompong	PROPN
ejpam-6566	437	11	,	,	PUNCT
ejpam-6566	437	12	and	and	CCONJ
ejpam-6566	437	13	c.	c.	PROPN
ejpam-6566	437	14	boonpok	boonpok	PROPN
ejpam-6566	437	15	.	.	PUNCT
ejpam-6566	438	1	almost	almost	ADV
ejpam-6566	438	2	contra-(τ1	contra-(τ1	NOUN
ejpam-6566	438	3	,	,	PUNCT
ejpam-6566	438	4	τ2)p	τ2)p	NOUN
ejpam-6566	438	5	-	-	PUNCT
ejpam-6566	438	6	continuity	continuity	NOUN
ejpam-6566	438	7	for	for	ADP
ejpam-6566	438	8	functions	function	NOUN
ejpam-6566	438	9	.	.	PUNCT
ejpam-6566	439	1	european	european	ADJ
ejpam-6566	439	2	journal	journal	PROPN
ejpam-6566	439	3	of	of	ADP
ejpam-6566	439	4	pure	pure	ADJ
ejpam-6566	439	5	and	and	CCONJ
ejpam-6566	439	6	applied	applied	ADJ
ejpam-6566	439	7	mathematics	mathematic	NOUN
ejpam-6566	439	8	,	,	PUNCT
ejpam-6566	439	9	18(2):6038	18(2):6038	NUM
ejpam-6566	439	10	,	,	PUNCT
ejpam-6566	439	11	2025	2025	NUM
ejpam-6566	439	12	.	.	PUNCT
ejpam-6566	440	1	[	[	X
ejpam-6566	440	2	29	29	NUM
ejpam-6566	440	3	]	]	PUNCT
ejpam-6566	440	4	m.	m.	NOUN
ejpam-6566	440	5	thongmoon	thongmoon	NOUN
ejpam-6566	440	6	,	,	PUNCT
ejpam-6566	440	7	s.	s.	PROPN
ejpam-6566	440	8	sompong	sompong	PROPN
ejpam-6566	440	9	,	,	PUNCT
ejpam-6566	440	10	and	and	CCONJ
ejpam-6566	440	11	c.	c.	PROPN
ejpam-6566	440	12	boonpok	boonpok	PROPN
ejpam-6566	440	13	.	.	PUNCT
ejpam-6566	441	1	almost	almost	ADV
ejpam-6566	441	2	(	(	PUNCT
ejpam-6566	441	3	τ1	τ1	NOUN
ejpam-6566	441	4	,	,	PUNCT
ejpam-6566	441	5	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6566	441	6	and	and	CCONJ
ejpam-6566	441	7	τ1τ2δ	τ1τ2δ	NUM
ejpam-6566	441	8	-	-	PUNCT
ejpam-6566	441	9	open	open	ADJ
ejpam-6566	441	10	sets	set	NOUN
ejpam-6566	441	11	.	.	PUNCT
ejpam-6566	442	1	international	international	ADJ
ejpam-6566	442	2	journal	journal	NOUN
ejpam-6566	442	3	of	of	ADP
ejpam-6566	442	4	analysis	analysis	NOUN
ejpam-6566	442	5	and	and	CCONJ
ejpam-6566	442	6	applications	application	NOUN
ejpam-6566	442	7	,	,	PUNCT
ejpam-6566	442	8	22:109	22:109	NUM
ejpam-6566	442	9	,	,	PUNCT
ejpam-6566	442	10	2024	2024	NUM
ejpam-6566	442	11	.	.	PUNCT
ejpam-6566	443	1	[	[	X
ejpam-6566	443	2	30	30	NUM
ejpam-6566	443	3	]	]	X
ejpam-6566	443	4	j.	j.	PROPN
ejpam-6566	443	5	khampakdee	khampakdee	PROPN
ejpam-6566	443	6	,	,	PUNCT
ejpam-6566	443	7	a.	a.	PROPN
ejpam-6566	443	8	sama	sama	PROPN
ejpam-6566	443	9	-	-	PUNCT
ejpam-6566	443	10	ae	ae	PROPN
ejpam-6566	443	11	,	,	PUNCT
ejpam-6566	443	12	and	and	CCONJ
ejpam-6566	443	13	c.	c.	PROPN
ejpam-6566	443	14	boonpok	boonpok	PROPN
ejpam-6566	443	15	.	.	PUNCT
ejpam-6566	444	1	upper	upper	ADJ
ejpam-6566	444	2	and	and	CCONJ
ejpam-6566	444	3	lower	low	ADJ
ejpam-6566	444	4	continuous	continuous	ADJ
ejpam-6566	444	5	multifunctions	multifunction	NOUN
ejpam-6566	444	6	defined	define	VERB
ejpam-6566	444	7	between	between	ADP
ejpam-6566	444	8	an	an	DET
ejpam-6566	444	9	ideal	ideal	ADJ
ejpam-6566	444	10	topological	topological	ADJ
ejpam-6566	444	11	space	space	NOUN
ejpam-6566	444	12	and	and	CCONJ
ejpam-6566	444	13	a	a	DET
ejpam-6566	444	14	bitopological	bitopological	ADJ
ejpam-6566	444	15	space	space	NOUN
ejpam-6566	444	16	.	.	PUNCT
ejpam-6566	445	1	(	(	PUNCT
ejpam-6566	445	2	submitted	submit	VERB
ejpam-6566	445	3	)	)	PUNCT
ejpam-6566	445	4	.	.	PUNCT
ejpam-6566	446	1	[	[	X
ejpam-6566	446	2	31	31	NUM
ejpam-6566	446	3	]	]	PUNCT
ejpam-6566	446	4	m.	m.	NOUN
ejpam-6566	446	5	thongmoon	thongmoon	NOUN
ejpam-6566	446	6	,	,	PUNCT
ejpam-6566	446	7	s.	s.	PROPN
ejpam-6566	446	8	sompong	sompong	PROPN
ejpam-6566	446	9	,	,	PUNCT
ejpam-6566	446	10	and	and	CCONJ
ejpam-6566	446	11	c.	c.	PROPN
ejpam-6566	446	12	boonpok	boonpok	PROPN
ejpam-6566	446	13	.	.	PUNCT
ejpam-6566	447	1	(	(	PUNCT
ejpam-6566	447	2	τ1	τ1	NOUN
ejpam-6566	447	3	,	,	PUNCT
ejpam-6566	447	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6566	447	5	multifunctions	multifunction	NOUN
ejpam-6566	447	6	and	and	CCONJ
ejpam-6566	447	7	τ1τ2	τ1τ2	NOUN
ejpam-6566	447	8	-	-	ADJ
ejpam-6566	447	9	δ	δ	NOUN
ejpam-6566	447	10	-	-	ADJ
ejpam-6566	447	11	open	open	ADJ
ejpam-6566	447	12	sets	set	NOUN
ejpam-6566	447	13	.	.	PUNCT
ejpam-6566	448	1	international	international	ADJ
ejpam-6566	448	2	journal	journal	NOUN
ejpam-6566	448	3	of	of	ADP
ejpam-6566	448	4	mathematics	mathematic	NOUN
ejpam-6566	448	5	and	and	CCONJ
ejpam-6566	448	6	computer	computer	NOUN
ejpam-6566	448	7	science	science	NOUN
ejpam-6566	448	8	,	,	PUNCT
ejpam-6566	448	9	19(4):1369–1375	19(4):1369–1375	NUM
ejpam-6566	448	10	,	,	PUNCT
ejpam-6566	448	11	2024	2024	NUM
ejpam-6566	448	12	.	.	PUNCT