id	sid	tid	token	lemma	pos
ejpam-6567	1	1	european	european	PROPN
ejpam-6567	1	2	journal	journal	PROPN
ejpam-6567	1	3	of	of	ADP
ejpam-6567	1	4	pure	pure	ADJ
ejpam-6567	1	5	and	and	CCONJ
ejpam-6567	1	6	applied	applied	ADJ
ejpam-6567	1	7	mathematics	mathematic	NOUN
ejpam-6567	1	8	2025	2025	NUM
ejpam-6567	1	9	,	,	PUNCT
ejpam-6567	1	10	vol	vol	NOUN
ejpam-6567	1	11	.	.	PROPN
ejpam-6567	1	12	18	18	NUM
ejpam-6567	1	13	,	,	PUNCT
ejpam-6567	1	14	issue	issue	NOUN
ejpam-6567	1	15	3	3	NUM
ejpam-6567	1	16	,	,	PUNCT
ejpam-6567	1	17	article	article	NOUN
ejpam-6567	1	18	number	number	NOUN
ejpam-6567	1	19	6567	6567	NUM
ejpam-6567	1	20	issn	issn	PROPN
ejpam-6567	1	21	1307	1307	NUM
ejpam-6567	1	22	-	-	SYM
ejpam-6567	1	23	5543	5543	NUM
ejpam-6567	1	24	–	–	PUNCT
ejpam-6567	1	25	ejpam.com	ejpam.com	X
ejpam-6567	1	26	published	publish	VERB
ejpam-6567	1	27	by	by	ADP
ejpam-6567	1	28	new	new	PROPN
ejpam-6567	1	29	york	york	PROPN
ejpam-6567	1	30	business	business	PROPN
ejpam-6567	1	31	global	global	ADJ
ejpam-6567	1	32	on	on	ADP
ejpam-6567	1	33	weak	weak	ADJ
ejpam-6567	1	34	forms	form	NOUN
ejpam-6567	1	35	of	of	ADP
ejpam-6567	1	36	upper	upper	ADJ
ejpam-6567	1	37	and	and	CCONJ
ejpam-6567	1	38	lower	low	ADJ
ejpam-6567	1	39	continuous	continuous	ADJ
ejpam-6567	1	40	multifunctions	multifunction	NOUN
ejpam-6567	1	41	between	between	ADP
ejpam-6567	1	42	an	an	DET
ejpam-6567	1	43	ideal	ideal	ADJ
ejpam-6567	1	44	topological	topological	ADJ
ejpam-6567	1	45	space	space	NOUN
ejpam-6567	1	46	and	and	CCONJ
ejpam-6567	1	47	a	a	DET
ejpam-6567	1	48	bitopological	bitopological	ADJ
ejpam-6567	1	49	space	space	NOUN
ejpam-6567	1	50	prapart	prapart	NOUN
ejpam-6567	1	51	pue	pue	PROPN
ejpam-6567	1	52	-	-	PUNCT
ejpam-6567	1	53	on1	on1	PROPN
ejpam-6567	1	54	,	,	PUNCT
ejpam-6567	1	55	areeyuth	areeyuth	NOUN
ejpam-6567	1	56	sama	sama	NOUN
ejpam-6567	1	57	-	-	PUNCT
ejpam-6567	1	58	ae2	ae2	PROPN
ejpam-6567	1	59	,	,	PUNCT
ejpam-6567	1	60	chawalit	chawalit	VERB
ejpam-6567	1	61	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6567	1	62	1	1	NUM
ejpam-6567	1	63	mathematics	mathematic	NOUN
ejpam-6567	1	64	and	and	CCONJ
ejpam-6567	1	65	applied	apply	VERB
ejpam-6567	1	66	mathematics	mathematics	PROPN
ejpam-6567	1	67	research	research	NOUN
ejpam-6567	1	68	unit	unit	NOUN
ejpam-6567	1	69	,	,	PUNCT
ejpam-6567	1	70	department	department	NOUN
ejpam-6567	1	71	of	of	ADP
ejpam-6567	1	72	mathematics	mathematic	NOUN
ejpam-6567	1	73	,	,	PUNCT
ejpam-6567	1	74	faculty	faculty	NOUN
ejpam-6567	1	75	of	of	ADP
ejpam-6567	1	76	science	science	NOUN
ejpam-6567	1	77	,	,	PUNCT
ejpam-6567	1	78	mahasarakham	mahasarakham	PROPN
ejpam-6567	1	79	university	university	PROPN
ejpam-6567	1	80	,	,	PUNCT
ejpam-6567	1	81	maha	maha	PROPN
ejpam-6567	1	82	sarakham	sarakham	PROPN
ejpam-6567	1	83	,	,	PUNCT
ejpam-6567	1	84	44150	44150	NUM
ejpam-6567	1	85	,	,	PUNCT
ejpam-6567	1	86	thailand	thailand	PROPN
ejpam-6567	1	87	2	2	NUM
ejpam-6567	1	88	department	department	NOUN
ejpam-6567	1	89	of	of	ADP
ejpam-6567	1	90	mathematics	mathematic	NOUN
ejpam-6567	1	91	and	and	CCONJ
ejpam-6567	1	92	computer	computer	NOUN
ejpam-6567	1	93	science	science	NOUN
ejpam-6567	1	94	,	,	PUNCT
ejpam-6567	1	95	faculty	faculty	NOUN
ejpam-6567	1	96	of	of	ADP
ejpam-6567	1	97	science	science	NOUN
ejpam-6567	1	98	and	and	CCONJ
ejpam-6567	1	99	technology	technology	NOUN
ejpam-6567	1	100	,	,	PUNCT
ejpam-6567	1	101	prince	prince	NOUN
ejpam-6567	1	102	of	of	ADP
ejpam-6567	1	103	songkla	songkla	PROPN
ejpam-6567	1	104	university	university	PROPN
ejpam-6567	1	105	,	,	PUNCT
ejpam-6567	1	106	pattani	pattani	NOUN
ejpam-6567	1	107	campus	campus	NOUN
ejpam-6567	1	108	,	,	PUNCT
ejpam-6567	1	109	pattani	pattani	NOUN
ejpam-6567	1	110	,	,	PUNCT
ejpam-6567	1	111	94000	94000	NUM
ejpam-6567	1	112	,	,	PUNCT
ejpam-6567	1	113	thailand	thailand	PROPN
ejpam-6567	1	114	abstract	abstract	PROPN
ejpam-6567	1	115	.	.	PUNCT
ejpam-6567	2	1	this	this	DET
ejpam-6567	2	2	paper	paper	NOUN
ejpam-6567	2	3	presents	present	VERB
ejpam-6567	2	4	new	new	ADJ
ejpam-6567	2	5	concepts	concept	NOUN
ejpam-6567	2	6	of	of	ADP
ejpam-6567	2	7	continuous	continuous	ADJ
ejpam-6567	2	8	multifunctions	multifunction	NOUN
ejpam-6567	2	9	defined	define	VERB
ejpam-6567	2	10	from	from	ADP
ejpam-6567	2	11	an	an	DET
ejpam-6567	2	12	ideal	ideal	ADJ
ejpam-6567	2	13	topological	topological	ADJ
ejpam-6567	2	14	space	space	NOUN
ejpam-6567	2	15	into	into	ADP
ejpam-6567	2	16	a	a	DET
ejpam-6567	2	17	bitopological	bitopological	ADJ
ejpam-6567	2	18	space	space	NOUN
ejpam-6567	2	19	,	,	PUNCT
ejpam-6567	2	20	called	call	VERB
ejpam-6567	2	21	upper	upper	ADJ
ejpam-6567	2	22	weakly	weakly	ADJ
ejpam-6567	2	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	2	24	,	,	PUNCT
ejpam-6567	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	2	26	multifunctions	multifunction	NOUN
ejpam-6567	2	27	and	and	CCONJ
ejpam-6567	2	28	lower	low	ADJ
ejpam-6567	2	29	weakly	weakly	ADJ
ejpam-6567	2	30	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	2	31	,	,	PUNCT
ejpam-6567	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	2	33	multifunctions	multifunction	NOUN
ejpam-6567	2	34	.	.	PUNCT
ejpam-6567	3	1	furthermore	furthermore	ADV
ejpam-6567	3	2	,	,	PUNCT
ejpam-6567	3	3	several	several	ADJ
ejpam-6567	3	4	characterizations	characterization	NOUN
ejpam-6567	3	5	and	and	CCONJ
ejpam-6567	3	6	some	some	DET
ejpam-6567	3	7	properties	property	NOUN
ejpam-6567	3	8	concerning	concern	VERB
ejpam-6567	3	9	upper	upper	ADJ
ejpam-6567	3	10	weakly	weakly	ADJ
ejpam-6567	3	11	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	3	12	,	,	PUNCT
ejpam-6567	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	3	14	multifunctions	multifunction	NOUN
ejpam-6567	3	15	and	and	CCONJ
ejpam-6567	3	16	lower	low	ADJ
ejpam-6567	3	17	weakly	weakly	ADJ
ejpam-6567	3	18	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	3	19	,	,	PUNCT
ejpam-6567	3	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	3	21	multifunctions	multifunction	NOUN
ejpam-6567	3	22	are	be	AUX
ejpam-6567	3	23	investigated	investigate	VERB
ejpam-6567	3	24	.	.	PUNCT
ejpam-6567	4	1	moreover	moreover	ADV
ejpam-6567	4	2	,	,	PUNCT
ejpam-6567	4	3	the	the	DET
ejpam-6567	4	4	relationships	relationship	NOUN
ejpam-6567	4	5	between	between	ADP
ejpam-6567	4	6	almost	almost	ADV
ejpam-6567	4	7	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	4	8	,	,	PUNCT
ejpam-6567	4	9	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6567	4	10	and	and	CCONJ
ejpam-6567	4	11	weak	weak	ADJ
ejpam-6567	4	12	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	4	13	,	,	PUNCT
ejpam-6567	4	14	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6567	4	15	are	be	AUX
ejpam-6567	4	16	established	establish	VERB
ejpam-6567	4	17	.	.	PUNCT
ejpam-6567	5	1	2020	2020	NUM
ejpam-6567	5	2	mathematics	mathematics	PROPN
ejpam-6567	5	3	subject	subject	NOUN
ejpam-6567	5	4	classifications	classification	NOUN
ejpam-6567	5	5	:	:	PUNCT
ejpam-6567	5	6	54c08	54c08	NUM
ejpam-6567	5	7	,	,	PUNCT
ejpam-6567	5	8	54c60	54c60	NUM
ejpam-6567	5	9	key	key	ADJ
ejpam-6567	5	10	words	word	NOUN
ejpam-6567	5	11	and	and	CCONJ
ejpam-6567	5	12	phrases	phrase	NOUN
ejpam-6567	5	13	:	:	PUNCT
ejpam-6567	5	14	upper	upper	ADJ
ejpam-6567	5	15	weakly	weakly	ADJ
ejpam-6567	5	16	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	5	17	,	,	PUNCT
ejpam-6567	5	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	5	19	multifunction	multifunction	NOUN
ejpam-6567	5	20	,	,	PUNCT
ejpam-6567	5	21	lower	low	ADJ
ejpam-6567	5	22	weakly	weakly	ADJ
ejpam-6567	5	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	5	24	,	,	PUNCT
ejpam-6567	5	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	5	26	multifunction	multifunction	NOUN
ejpam-6567	5	27	1	1	NUM
ejpam-6567	5	28	.	.	PUNCT
ejpam-6567	5	29	introduction	introduction	NOUN
ejpam-6567	5	30	the	the	DET
ejpam-6567	5	31	concept	concept	NOUN
ejpam-6567	5	32	of	of	ADP
ejpam-6567	5	33	weakly	weakly	ADJ
ejpam-6567	5	34	continuous	continuous	ADJ
ejpam-6567	5	35	functions	function	NOUN
ejpam-6567	5	36	was	be	AUX
ejpam-6567	5	37	introduced	introduce	VERB
ejpam-6567	5	38	by	by	ADP
ejpam-6567	5	39	levine	levine	PROPN
ejpam-6567	6	1	[	[	X
ejpam-6567	6	2	1	1	NUM
ejpam-6567	6	3	]	]	PUNCT
ejpam-6567	6	4	.	.	PUNCT
ejpam-6567	7	1	husain	husain	PROPN
ejpam-6567	8	1	[	[	X
ejpam-6567	8	2	2	2	X
ejpam-6567	8	3	]	]	PUNCT
ejpam-6567	8	4	introduced	introduce	VERB
ejpam-6567	8	5	and	and	CCONJ
ejpam-6567	8	6	studied	study	VERB
ejpam-6567	8	7	the	the	DET
ejpam-6567	8	8	notion	notion	NOUN
ejpam-6567	8	9	of	of	ADP
ejpam-6567	8	10	almost	almost	ADV
ejpam-6567	8	11	continuous	continuous	ADJ
ejpam-6567	8	12	functions	function	NOUN
ejpam-6567	8	13	.	.	PUNCT
ejpam-6567	9	1	janković	janković	PUNCT
ejpam-6567	10	1	[	[	X
ejpam-6567	10	2	3	3	X
ejpam-6567	10	3	]	]	PUNCT
ejpam-6567	10	4	introduced	introduce	VERB
ejpam-6567	10	5	almost	almost	ADV
ejpam-6567	10	6	weak	weak	ADJ
ejpam-6567	10	7	continuity	continuity	NOUN
ejpam-6567	10	8	as	as	ADP
ejpam-6567	10	9	a	a	DET
ejpam-6567	10	10	generalization	generalization	NOUN
ejpam-6567	10	11	of	of	ADP
ejpam-6567	10	12	both	both	DET
ejpam-6567	10	13	weak	weak	ADJ
ejpam-6567	10	14	continuity	continuity	NOUN
ejpam-6567	10	15	and	and	CCONJ
ejpam-6567	10	16	almost	almost	ADV
ejpam-6567	10	17	continuity	continuity	NOUN
ejpam-6567	10	18	.	.	PUNCT
ejpam-6567	11	1	noiri	noiri	ADV
ejpam-6567	12	1	[	[	X
ejpam-6567	12	2	4	4	X
ejpam-6567	12	3	]	]	PUNCT
ejpam-6567	12	4	investigated	investigate	VERB
ejpam-6567	12	5	several	several	ADJ
ejpam-6567	12	6	characterizations	characterization	NOUN
ejpam-6567	12	7	of	of	ADP
ejpam-6567	12	8	almost	almost	ADV
ejpam-6567	12	9	weakly	weakly	ADJ
ejpam-6567	12	10	continuous	continuous	ADJ
ejpam-6567	12	11	functions	function	NOUN
ejpam-6567	12	12	.	.	PUNCT
ejpam-6567	13	1	rose	rise	VERB
ejpam-6567	13	2	[	[	X
ejpam-6567	13	3	5	5	NUM
ejpam-6567	13	4	]	]	PUNCT
ejpam-6567	13	5	introduced	introduce	VERB
ejpam-6567	13	6	the	the	DET
ejpam-6567	13	7	notion	notion	NOUN
ejpam-6567	13	8	of	of	ADP
ejpam-6567	13	9	subweakly	subweakly	ADJ
ejpam-6567	13	10	continuous	continuous	ADJ
ejpam-6567	13	11	functions	function	NOUN
ejpam-6567	13	12	and	and	CCONJ
ejpam-6567	13	13	investigated	investigate	VERB
ejpam-6567	13	14	the	the	DET
ejpam-6567	13	15	relationships	relationship	NOUN
ejpam-6567	13	16	between	between	ADP
ejpam-6567	13	17	subweak	subweak	NOUN
ejpam-6567	13	18	continuity	continuity	NOUN
ejpam-6567	13	19	and	and	CCONJ
ejpam-6567	13	20	weak	weak	ADJ
ejpam-6567	13	21	continuity	continuity	NOUN
ejpam-6567	13	22	.	.	PUNCT
ejpam-6567	14	1	popa	popa	NOUN
ejpam-6567	14	2	and	and	CCONJ
ejpam-6567	14	3	noiri	noiri	ADV
ejpam-6567	15	1	[	[	X
ejpam-6567	15	2	6	6	NUM
ejpam-6567	15	3	]	]	PUNCT
ejpam-6567	15	4	introduced	introduce	VERB
ejpam-6567	15	5	the	the	DET
ejpam-6567	15	6	concept	concept	NOUN
ejpam-6567	15	7	of	of	ADP
ejpam-6567	15	8	weakly	weakly	ADJ
ejpam-6567	15	9	(	(	PUNCT
ejpam-6567	15	10	τ	τ	PROPN
ejpam-6567	15	11	,	,	PUNCT
ejpam-6567	15	12	m)-continuous	m)-continuous	ADJ
ejpam-6567	15	13	functions	function	NOUN
ejpam-6567	15	14	as	as	ADP
ejpam-6567	15	15	functions	function	NOUN
ejpam-6567	15	16	from	from	ADP
ejpam-6567	15	17	a	a	DET
ejpam-6567	15	18	topological	topological	ADJ
ejpam-6567	15	19	space	space	NOUN
ejpam-6567	15	20	into	into	ADP
ejpam-6567	15	21	a	a	DET
ejpam-6567	15	22	set	set	NOUN
ejpam-6567	15	23	satisfying	satisfy	VERB
ejpam-6567	15	24	some	some	DET
ejpam-6567	15	25	minimal	minimal	ADJ
ejpam-6567	15	26	conditions	condition	NOUN
ejpam-6567	15	27	and	and	CCONJ
ejpam-6567	15	28	investigated	investigate	VERB
ejpam-6567	15	29	several	several	ADJ
ejpam-6567	15	30	characterizations	characterization	NOUN
ejpam-6567	15	31	of	of	ADP
ejpam-6567	15	32	weakly	weakly	ADJ
ejpam-6567	15	33	(	(	PUNCT
ejpam-6567	15	34	τ	τ	PROPN
ejpam-6567	15	35	,	,	PUNCT
ejpam-6567	15	36	m)-continuous	m)-continuous	ADJ
ejpam-6567	15	37	functions	function	NOUN
ejpam-6567	15	38	.	.	PUNCT
ejpam-6567	16	1	ekici	ekici	NOUN
ejpam-6567	16	2	et	et	PROPN
ejpam-6567	16	3	al	al	PROPN
ejpam-6567	16	4	.	.	PUNCT
ejpam-6567	17	1	[	[	X
ejpam-6567	17	2	7	7	X
ejpam-6567	17	3	]	]	PUNCT
ejpam-6567	17	4	introduced	introduce	VERB
ejpam-6567	17	5	and	and	CCONJ
ejpam-6567	17	6	studied	study	VERB
ejpam-6567	17	7	the	the	DET
ejpam-6567	17	8	concept	concept	NOUN
ejpam-6567	17	9	of	of	ADP
ejpam-6567	17	10	weakly	weakly	ADJ
ejpam-6567	17	11	λ	λ	ADJ
ejpam-6567	17	12	-	-	ADJ
ejpam-6567	17	13	continuous	continuous	ADJ
ejpam-6567	17	14	functions	function	NOUN
ejpam-6567	17	15	.	.	PUNCT
ejpam-6567	18	1	popa	popa	NOUN
ejpam-6567	19	1	[	[	X
ejpam-6567	19	2	8	8	NUM
ejpam-6567	19	3	]	]	PUNCT
ejpam-6567	19	4	and	and	CCONJ
ejpam-6567	19	5	smithson	smithson	PROPN
ejpam-6567	19	6	[	[	X
ejpam-6567	19	7	9	9	NUM
ejpam-6567	19	8	]	]	PUNCT
ejpam-6567	19	9	independently	independently	ADV
ejpam-6567	19	10	introduced	introduce	VERB
ejpam-6567	19	11	the	the	DET
ejpam-6567	19	12	notion	notion	NOUN
ejpam-6567	19	13	of	of	ADP
ejpam-6567	19	14	∗corresponding	∗corresponde	VERB
ejpam-6567	19	15	author	author	NOUN
ejpam-6567	19	16	.	.	PUNCT
ejpam-6567	20	1	doi	doi	NOUN
ejpam-6567	20	2	:	:	PUNCT
ejpam-6567	20	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6567	https://doi.org/10.29020/nybg.ejpam.v18i3.6567	PRON
ejpam-6567	20	4	email	email	NOUN
ejpam-6567	20	5	addresses	address	NOUN
ejpam-6567	20	6	:	:	PUNCT
ejpam-6567	20	7	prapart.p@msu.ac.th	prapart.p@msu.ac.th	PROPN
ejpam-6567	20	8	(	(	PUNCT
ejpam-6567	20	9	p.	p.	NOUN
ejpam-6567	20	10	pue	pue	NOUN
ejpam-6567	20	11	-	-	PUNCT
ejpam-6567	20	12	on	on	ADP
ejpam-6567	20	13	)	)	PUNCT
ejpam-6567	20	14	,	,	PUNCT
ejpam-6567	20	15	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6567	20	16	(	(	PUNCT
ejpam-6567	20	17	a.	a.	PROPN
ejpam-6567	20	18	sama	sama	PROPN
ejpam-6567	20	19	-	-	PUNCT
ejpam-6567	20	20	ae	ae	PROPN
ejpam-6567	20	21	)	)	PUNCT
ejpam-6567	20	22	,	,	PUNCT
ejpam-6567	20	23	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6567	20	24	(	(	PUNCT
ejpam-6567	20	25	c.	c.	PROPN
ejpam-6567	20	26	boonpok	boonpok	PROPN
ejpam-6567	20	27	)	)	PUNCT
ejpam-6567	20	28	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6567	21	1	1	1	NUM
ejpam-6567	21	2	copyright	copyright	NOUN
ejpam-6567	21	3	:	:	PUNCT
ejpam-6567	21	4	©	©	PROPN
ejpam-6567	21	5	2025	2025	NUM
ejpam-6567	21	6	the	the	DET
ejpam-6567	21	7	author(s	author(s	NOUN
ejpam-6567	21	8	)	)	PUNCT
ejpam-6567	21	9	.	.	PUNCT
ejpam-6567	22	1	(	(	PUNCT
ejpam-6567	22	2	cc	cc	NOUN
ejpam-6567	22	3	by	by	ADP
ejpam-6567	22	4	-	-	PUNCT
ejpam-6567	22	5	nc	nc	PROPN
ejpam-6567	22	6	4.0	4.0	NUM
ejpam-6567	22	7	)	)	PUNCT
ejpam-6567	22	8	p.	p.	NOUN
ejpam-6567	22	9	pue	pue	NOUN
ejpam-6567	22	10	-	-	PUNCT
ejpam-6567	22	11	on	on	ADP
ejpam-6567	22	12	,	,	PUNCT
ejpam-6567	22	13	a.	a.	PROPN
ejpam-6567	22	14	sama	sama	PROPN
ejpam-6567	22	15	-	-	PUNCT
ejpam-6567	22	16	ae	ae	PROPN
ejpam-6567	22	17	,	,	PUNCT
ejpam-6567	22	18	c.	c.	PROPN
ejpam-6567	22	19	boonpok	boonpok	PROPN
ejpam-6567	22	20	/	/	SYM
ejpam-6567	22	21	eur	eur	PROPN
ejpam-6567	22	22	.	.	PUNCT
ejpam-6567	23	1	j.	j.	PROPN
ejpam-6567	23	2	pure	pure	PROPN
ejpam-6567	23	3	appl	appl	PROPN
ejpam-6567	23	4	.	.	PROPN
ejpam-6567	23	5	math	math	PROPN
ejpam-6567	23	6	,	,	PUNCT
ejpam-6567	23	7	18	18	NUM
ejpam-6567	23	8	(	(	PUNCT
ejpam-6567	23	9	3	3	NUM
ejpam-6567	23	10	)	)	PUNCT
ejpam-6567	23	11	(	(	PUNCT
ejpam-6567	23	12	2025	2025	NUM
ejpam-6567	23	13	)	)	PUNCT
ejpam-6567	23	14	,	,	PUNCT
ejpam-6567	23	15	6567	6567	NUM
ejpam-6567	23	16	2	2	NUM
ejpam-6567	23	17	of	of	ADP
ejpam-6567	23	18	13	13	NUM
ejpam-6567	23	19	weakly	weakly	ADJ
ejpam-6567	23	20	continuous	continuous	ADJ
ejpam-6567	23	21	multifunctions	multifunction	NOUN
ejpam-6567	23	22	.	.	PUNCT
ejpam-6567	24	1	popa	popa	NOUN
ejpam-6567	24	2	and	and	CCONJ
ejpam-6567	24	3	noiri	noiri	ADV
ejpam-6567	25	1	[	[	X
ejpam-6567	25	2	10	10	NUM
ejpam-6567	25	3	]	]	PUNCT
ejpam-6567	25	4	introduced	introduce	VERB
ejpam-6567	25	5	a	a	DET
ejpam-6567	25	6	class	class	NOUN
ejpam-6567	25	7	of	of	ADP
ejpam-6567	25	8	multifunctions	multifunction	NOUN
ejpam-6567	25	9	called	call	VERB
ejpam-6567	25	10	weakly	weakly	ADJ
ejpam-6567	25	11	α	α	ADJ
ejpam-6567	25	12	-	-	ADJ
ejpam-6567	25	13	continuous	continuous	ADJ
ejpam-6567	25	14	multifunctions	multifunction	NOUN
ejpam-6567	25	15	.	.	PUNCT
ejpam-6567	26	1	furthermore	furthermore	ADV
ejpam-6567	26	2	,	,	PUNCT
ejpam-6567	26	3	popa	popa	NOUN
ejpam-6567	26	4	and	and	CCONJ
ejpam-6567	26	5	noiri	noiri	ADV
ejpam-6567	27	1	[	[	X
ejpam-6567	27	2	11	11	NUM
ejpam-6567	27	3	]	]	PUNCT
ejpam-6567	27	4	investigated	investigate	VERB
ejpam-6567	27	5	some	some	DET
ejpam-6567	27	6	characterizations	characterization	NOUN
ejpam-6567	27	7	of	of	ADP
ejpam-6567	27	8	upper	upper	ADJ
ejpam-6567	27	9	and	and	CCONJ
ejpam-6567	27	10	lower	low	ADJ
ejpam-6567	27	11	weakly	weakly	ADJ
ejpam-6567	27	12	β	β	ADJ
ejpam-6567	27	13	-	-	ADJ
ejpam-6567	27	14	continuous	continuous	ADJ
ejpam-6567	27	15	multifunctions	multifunction	NOUN
ejpam-6567	27	16	.	.	PUNCT
ejpam-6567	28	1	noiri	noiri	PROPN
ejpam-6567	28	2	and	and	CCONJ
ejpam-6567	28	3	popa	popa	NOUN
ejpam-6567	29	1	[	[	X
ejpam-6567	29	2	12	12	NUM
ejpam-6567	29	3	]	]	PUNCT
ejpam-6567	29	4	introduced	introduce	VERB
ejpam-6567	29	5	and	and	CCONJ
ejpam-6567	29	6	investigated	investigate	VERB
ejpam-6567	29	7	the	the	DET
ejpam-6567	29	8	notion	notion	NOUN
ejpam-6567	29	9	of	of	ADP
ejpam-6567	29	10	weakly	weakly	ADJ
ejpam-6567	29	11	m	m	ADJ
ejpam-6567	29	12	-	-	ADJ
ejpam-6567	29	13	continuous	continuous	ADJ
ejpam-6567	29	14	multifunctions	multifunction	NOUN
ejpam-6567	29	15	as	as	ADP
ejpam-6567	29	16	a	a	DET
ejpam-6567	29	17	multifunction	multifunction	NOUN
ejpam-6567	29	18	from	from	ADP
ejpam-6567	29	19	a	a	DET
ejpam-6567	29	20	set	set	NOUN
ejpam-6567	29	21	satisfying	satisfy	VERB
ejpam-6567	29	22	certain	certain	ADJ
ejpam-6567	29	23	minimal	minimal	ADJ
ejpam-6567	29	24	condition	condition	NOUN
ejpam-6567	29	25	into	into	ADP
ejpam-6567	29	26	a	a	DET
ejpam-6567	29	27	topological	topological	ADJ
ejpam-6567	29	28	space	space	NOUN
ejpam-6567	29	29	.	.	PUNCT
ejpam-6567	30	1	pue	pue	NOUN
ejpam-6567	30	2	-	-	PUNCT
ejpam-6567	30	3	on	on	NOUN
ejpam-6567	30	4	et	et	PROPN
ejpam-6567	30	5	al	al	PROPN
ejpam-6567	30	6	.	.	PUNCT
ejpam-6567	31	1	[	[	X
ejpam-6567	31	2	13	13	NUM
ejpam-6567	31	3	]	]	PUNCT
ejpam-6567	31	4	introduced	introduce	VERB
ejpam-6567	31	5	and	and	CCONJ
ejpam-6567	31	6	studied	study	VERB
ejpam-6567	31	7	the	the	DET
ejpam-6567	31	8	concepts	concept	NOUN
ejpam-6567	31	9	of	of	ADP
ejpam-6567	31	10	upper	upper	ADJ
ejpam-6567	31	11	(	(	PUNCT
ejpam-6567	31	12	τ1	τ1	NOUN
ejpam-6567	31	13	,	,	PUNCT
ejpam-6567	31	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6567	31	15	multifunctions	multifunction	NOUN
ejpam-6567	31	16	and	and	CCONJ
ejpam-6567	31	17	lower	low	ADJ
ejpam-6567	31	18	(	(	PUNCT
ejpam-6567	31	19	τ1	τ1	NOUN
ejpam-6567	31	20	,	,	PUNCT
ejpam-6567	31	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6567	31	22	multifunctions	multifunction	NOUN
ejpam-6567	31	23	.	.	PUNCT
ejpam-6567	32	1	klanarong	klanarong	NOUN
ejpam-6567	32	2	et	et	PROPN
ejpam-6567	32	3	al	al	PROPN
ejpam-6567	32	4	.	.	PUNCT
ejpam-6567	33	1	[	[	X
ejpam-6567	33	2	14	14	NUM
ejpam-6567	33	3	]	]	PUNCT
ejpam-6567	33	4	introduced	introduce	VERB
ejpam-6567	33	5	and	and	CCONJ
ejpam-6567	33	6	investigated	investigate	VERB
ejpam-6567	33	7	the	the	DET
ejpam-6567	33	8	notions	notion	NOUN
ejpam-6567	33	9	of	of	ADP
ejpam-6567	33	10	upper	upper	ADJ
ejpam-6567	33	11	almost	almost	ADV
ejpam-6567	33	12	(	(	PUNCT
ejpam-6567	33	13	τ1	τ1	NOUN
ejpam-6567	33	14	,	,	PUNCT
ejpam-6567	33	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6567	33	16	multifunctions	multifunction	NOUN
ejpam-6567	33	17	and	and	CCONJ
ejpam-6567	33	18	lower	low	ADJ
ejpam-6567	33	19	almost	almost	ADV
ejpam-6567	33	20	(	(	PUNCT
ejpam-6567	33	21	τ1	τ1	NOUN
ejpam-6567	33	22	,	,	PUNCT
ejpam-6567	33	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6567	33	24	multifunctions	multifunction	NOUN
ejpam-6567	33	25	.	.	PUNCT
ejpam-6567	34	1	thongmoon	thongmoon	NOUN
ejpam-6567	34	2	et	et	PROPN
ejpam-6567	34	3	al	al	PROPN
ejpam-6567	34	4	.	.	PUNCT
ejpam-6567	35	1	[	[	X
ejpam-6567	35	2	15	15	NUM
ejpam-6567	35	3	]	]	PUNCT
ejpam-6567	35	4	introduced	introduce	VERB
ejpam-6567	35	5	and	and	CCONJ
ejpam-6567	35	6	studied	study	VERB
ejpam-6567	35	7	the	the	DET
ejpam-6567	35	8	concepts	concept	NOUN
ejpam-6567	35	9	of	of	ADP
ejpam-6567	35	10	upper	upper	ADJ
ejpam-6567	35	11	weakly	weakly	ADJ
ejpam-6567	35	12	(	(	PUNCT
ejpam-6567	35	13	τ1	τ1	NOUN
ejpam-6567	35	14	,	,	PUNCT
ejpam-6567	35	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6567	35	16	multifunctions	multifunction	NOUN
ejpam-6567	35	17	and	and	CCONJ
ejpam-6567	35	18	lower	low	ADJ
ejpam-6567	35	19	weakly	weakly	ADJ
ejpam-6567	35	20	(	(	PUNCT
ejpam-6567	35	21	τ1	τ1	NOUN
ejpam-6567	35	22	,	,	PUNCT
ejpam-6567	35	23	τ2)continuous	τ2)continuous	ADJ
ejpam-6567	35	24	multifunctions	multifunction	NOUN
ejpam-6567	35	25	.	.	PUNCT
ejpam-6567	36	1	the	the	DET
ejpam-6567	36	2	concept	concept	NOUN
ejpam-6567	36	3	of	of	ADP
ejpam-6567	36	4	ideal	ideal	ADJ
ejpam-6567	36	5	topological	topological	ADJ
ejpam-6567	36	6	spaces	space	NOUN
ejpam-6567	36	7	was	be	AUX
ejpam-6567	36	8	introduced	introduce	VERB
ejpam-6567	36	9	and	and	CCONJ
ejpam-6567	36	10	studied	study	VERB
ejpam-6567	36	11	by	by	ADP
ejpam-6567	36	12	kuratowski	kuratowski	PROPN
ejpam-6567	36	13	[	[	X
ejpam-6567	36	14	16	16	NUM
ejpam-6567	36	15	]	]	PUNCT
ejpam-6567	36	16	and	and	CCONJ
ejpam-6567	36	17	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-6567	36	18	[	[	X
ejpam-6567	36	19	17	17	NUM
ejpam-6567	36	20	]	]	PUNCT
ejpam-6567	36	21	.	.	PUNCT
ejpam-6567	37	1	weaker	weak	ADJ
ejpam-6567	37	2	and	and	CCONJ
ejpam-6567	37	3	stronger	strong	ADJ
ejpam-6567	37	4	forms	form	NOUN
ejpam-6567	37	5	of	of	ADP
ejpam-6567	37	6	open	open	ADJ
ejpam-6567	37	7	sets	set	NOUN
ejpam-6567	37	8	in	in	ADP
ejpam-6567	37	9	ideal	ideal	ADJ
ejpam-6567	37	10	topological	topological	ADJ
ejpam-6567	37	11	spaces	space	NOUN
ejpam-6567	37	12	such	such	ADJ
ejpam-6567	37	13	as	as	ADP
ejpam-6567	37	14	semi	semi	ADJ
ejpam-6567	37	15	-	-	ADJ
ejpam-6567	37	16	i	i	ADJ
ejpam-6567	37	17	-open	-open	NOUN
ejpam-6567	37	18	sets	set	NOUN
ejpam-6567	37	19	,	,	PUNCT
ejpam-6567	37	20	pre	pre	ADJ
ejpam-6567	37	21	-	-	ADJ
ejpam-6567	37	22	i	i	ADJ
ejpam-6567	37	23	-open	-open	NOUN
ejpam-6567	37	24	sets	set	NOUN
ejpam-6567	37	25	,	,	PUNCT
ejpam-6567	37	26	α	α	X
ejpam-6567	37	27	-	-	PUNCT
ejpam-6567	37	28	i	i	PRON
ejpam-6567	37	29	open	open	ADJ
ejpam-6567	37	30	sets	set	NOUN
ejpam-6567	37	31	,	,	PUNCT
ejpam-6567	37	32	β	β	X
ejpam-6567	37	33	-	-	ADJ
ejpam-6567	37	34	i	i	PRON
ejpam-6567	37	35	-open	-open	NOUN
ejpam-6567	37	36	sets	set	NOUN
ejpam-6567	37	37	and	and	CCONJ
ejpam-6567	37	38	δ	δ	PROPN
ejpam-6567	37	39	-	-	PUNCT
ejpam-6567	37	40	i	i	PRON
ejpam-6567	37	41	-open	-open	NOUN
ejpam-6567	37	42	sets	set	NOUN
ejpam-6567	37	43	play	play	VERB
ejpam-6567	37	44	an	an	DET
ejpam-6567	37	45	important	important	ADJ
ejpam-6567	37	46	role	role	NOUN
ejpam-6567	37	47	in	in	ADP
ejpam-6567	37	48	the	the	DET
ejpam-6567	37	49	research	research	NOUN
ejpam-6567	37	50	of	of	ADP
ejpam-6567	37	51	generalizations	generalization	NOUN
ejpam-6567	37	52	of	of	ADP
ejpam-6567	37	53	continuity	continuity	NOUN
ejpam-6567	37	54	.	.	PUNCT
ejpam-6567	38	1	using	use	VERB
ejpam-6567	38	2	these	these	DET
ejpam-6567	38	3	notions	notion	NOUN
ejpam-6567	38	4	many	many	ADJ
ejpam-6567	38	5	authors	author	NOUN
ejpam-6567	38	6	introduced	introduce	VERB
ejpam-6567	38	7	and	and	CCONJ
ejpam-6567	38	8	studied	study	VERB
ejpam-6567	38	9	various	various	ADJ
ejpam-6567	38	10	types	type	NOUN
ejpam-6567	38	11	of	of	ADP
ejpam-6567	38	12	generalizations	generalization	NOUN
ejpam-6567	38	13	of	of	ADP
ejpam-6567	38	14	continuity	continuity	NOUN
ejpam-6567	38	15	for	for	ADP
ejpam-6567	38	16	functions	function	NOUN
ejpam-6567	38	17	and	and	CCONJ
ejpam-6567	38	18	multifunctions	multifunction	NOUN
ejpam-6567	38	19	.	.	PUNCT
ejpam-6567	39	1	hatir	hatir	PROPN
ejpam-6567	39	2	and	and	CCONJ
ejpam-6567	39	3	noiri	noiri	ADV
ejpam-6567	40	1	[	[	X
ejpam-6567	40	2	18	18	NUM
ejpam-6567	40	3	]	]	PUNCT
ejpam-6567	40	4	introduced	introduce	VERB
ejpam-6567	40	5	and	and	CCONJ
ejpam-6567	40	6	investigated	investigate	VERB
ejpam-6567	40	7	the	the	DET
ejpam-6567	40	8	notions	notion	NOUN
ejpam-6567	40	9	of	of	ADP
ejpam-6567	40	10	weakly	weakly	ADJ
ejpam-6567	40	11	pre	pre	ADJ
ejpam-6567	40	12	-	-	ADJ
ejpam-6567	40	13	i	i	PRON
ejpam-6567	40	14	-open	-open	NOUN
ejpam-6567	40	15	sets	set	NOUN
ejpam-6567	40	16	and	and	CCONJ
ejpam-6567	40	17	weakly	weakly	ADJ
ejpam-6567	40	18	pre	pre	ADJ
ejpam-6567	40	19	-	-	ADJ
ejpam-6567	40	20	i	i	ADJ
ejpam-6567	40	21	-continuous	-continuous	ADJ
ejpam-6567	40	22	functions	function	NOUN
ejpam-6567	40	23	.	.	PUNCT
ejpam-6567	41	1	moreover	moreover	ADV
ejpam-6567	41	2	,	,	PUNCT
ejpam-6567	41	3	hatir	hatir	PROPN
ejpam-6567	41	4	and	and	CCONJ
ejpam-6567	41	5	noiri	noiri	ADV
ejpam-6567	42	1	[	[	X
ejpam-6567	42	2	19	19	NUM
ejpam-6567	42	3	]	]	PUNCT
ejpam-6567	42	4	investigated	investigate	VERB
ejpam-6567	42	5	further	further	ADJ
ejpam-6567	42	6	properties	property	NOUN
ejpam-6567	42	7	of	of	ADP
ejpam-6567	42	8	semi	semi	ADJ
ejpam-6567	42	9	-	-	ADJ
ejpam-6567	42	10	i	i	PRON
ejpam-6567	42	11	-open	-open	NOUN
ejpam-6567	42	12	sets	set	NOUN
ejpam-6567	42	13	and	and	CCONJ
ejpam-6567	42	14	semi	semi	ADJ
ejpam-6567	42	15	-	-	ADJ
ejpam-6567	42	16	i	i	ADV
ejpam-6567	42	17	-continuous	-continuous	ADJ
ejpam-6567	42	18	functions	function	NOUN
ejpam-6567	42	19	.	.	PUNCT
ejpam-6567	43	1	on	on	ADP
ejpam-6567	43	2	the	the	DET
ejpam-6567	43	3	other	other	ADJ
ejpam-6567	43	4	hand	hand	NOUN
ejpam-6567	43	5	,	,	PUNCT
ejpam-6567	43	6	the	the	DET
ejpam-6567	43	7	present	present	ADJ
ejpam-6567	43	8	author	author	NOUN
ejpam-6567	43	9	[	[	X
ejpam-6567	43	10	20	20	NUM
ejpam-6567	43	11	]	]	PUNCT
ejpam-6567	43	12	introduced	introduce	VERB
ejpam-6567	43	13	the	the	DET
ejpam-6567	43	14	notions	notion	NOUN
ejpam-6567	43	15	of	of	ADP
ejpam-6567	43	16	upper	upper	ADJ
ejpam-6567	43	17	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	43	18	multifunctions	multifunction	NOUN
ejpam-6567	43	19	and	and	CCONJ
ejpam-6567	43	20	lower	low	ADJ
ejpam-6567	43	21	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	43	22	multifunctions	multifunction	NOUN
ejpam-6567	43	23	.	.	PUNCT
ejpam-6567	44	1	furthermore	furthermore	ADV
ejpam-6567	44	2	,	,	PUNCT
ejpam-6567	44	3	several	several	ADJ
ejpam-6567	44	4	characterizations	characterization	NOUN
ejpam-6567	44	5	of	of	ADP
ejpam-6567	44	6	upper	upper	ADJ
ejpam-6567	44	7	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	44	8	multifunctions	multifunction	NOUN
ejpam-6567	44	9	,	,	PUNCT
ejpam-6567	44	10	lower	low	ADJ
ejpam-6567	44	11	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	44	12	multifunctions	multifunction	NOUN
ejpam-6567	44	13	,	,	PUNCT
ejpam-6567	44	14	upper	upper	ADJ
ejpam-6567	44	15	almost	almost	ADV
ejpam-6567	44	16	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	44	17	multifunctions	multifunction	NOUN
ejpam-6567	44	18	,	,	PUNCT
ejpam-6567	44	19	lower	low	ADJ
ejpam-6567	44	20	almost	almost	ADV
ejpam-6567	44	21	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	44	22	multifunctions	multifunction	NOUN
ejpam-6567	44	23	,	,	PUNCT
ejpam-6567	44	24	upper	upper	ADJ
ejpam-6567	44	25	weakly	weakly	ADJ
ejpam-6567	44	26	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	44	27	multifunctions	multifunction	NOUN
ejpam-6567	44	28	and	and	CCONJ
ejpam-6567	44	29	lower	low	ADJ
ejpam-6567	44	30	weakly	weakly	ADJ
ejpam-6567	44	31	⋆-continuous	⋆-continuous	ADJ
ejpam-6567	44	32	multifunctions	multifunction	NOUN
ejpam-6567	44	33	were	be	AUX
ejpam-6567	44	34	considered	consider	VERB
ejpam-6567	44	35	in	in	ADP
ejpam-6567	44	36	[	[	X
ejpam-6567	44	37	20	20	NUM
ejpam-6567	44	38	]	]	PUNCT
ejpam-6567	44	39	.	.	PUNCT
ejpam-6567	45	1	quite	quite	ADV
ejpam-6567	45	2	recently	recently	ADV
ejpam-6567	45	3	,	,	PUNCT
ejpam-6567	45	4	the	the	DET
ejpam-6567	45	5	present	present	ADJ
ejpam-6567	45	6	author	author	NOUN
ejpam-6567	45	7	[	[	X
ejpam-6567	45	8	21	21	NUM
ejpam-6567	45	9	]	]	PUNCT
ejpam-6567	45	10	introduced	introduce	VERB
ejpam-6567	45	11	and	and	CCONJ
ejpam-6567	45	12	studied	study	VERB
ejpam-6567	45	13	the	the	DET
ejpam-6567	45	14	notions	notion	NOUN
ejpam-6567	45	15	of	of	ADP
ejpam-6567	45	16	pı	pı	ADJ
ejpam-6567	45	17	-	-	ADJ
ejpam-6567	45	18	continuous	continuous	ADJ
ejpam-6567	45	19	multifunctions	multifunction	NOUN
ejpam-6567	45	20	and	and	CCONJ
ejpam-6567	46	1	weakly	weakly	ADJ
ejpam-6567	46	2	pı	pı	ADJ
ejpam-6567	46	3	-	-	ADJ
ejpam-6567	46	4	continuous	continuous	ADJ
ejpam-6567	46	5	multifunctions	multifunction	NOUN
ejpam-6567	46	6	.	.	PUNCT
ejpam-6567	47	1	in	in	ADP
ejpam-6567	47	2	this	this	DET
ejpam-6567	47	3	paper	paper	NOUN
ejpam-6567	47	4	,	,	PUNCT
ejpam-6567	47	5	we	we	PRON
ejpam-6567	47	6	introduce	introduce	VERB
ejpam-6567	47	7	new	new	ADJ
ejpam-6567	47	8	classes	class	NOUN
ejpam-6567	47	9	of	of	ADP
ejpam-6567	47	10	multifunctions	multifunction	NOUN
ejpam-6567	47	11	between	between	ADP
ejpam-6567	47	12	an	an	DET
ejpam-6567	47	13	ideal	ideal	ADJ
ejpam-6567	47	14	topological	topological	ADJ
ejpam-6567	47	15	space	space	NOUN
ejpam-6567	47	16	and	and	CCONJ
ejpam-6567	47	17	a	a	DET
ejpam-6567	47	18	bitopological	bitopological	ADJ
ejpam-6567	47	19	space	space	NOUN
ejpam-6567	47	20	,	,	PUNCT
ejpam-6567	47	21	namely	namely	ADV
ejpam-6567	47	22	upper	upper	ADJ
ejpam-6567	47	23	weakly	weakly	ADJ
ejpam-6567	47	24	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	47	25	,	,	PUNCT
ejpam-6567	47	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	47	27	multifunctions	multifunction	NOUN
ejpam-6567	47	28	and	and	CCONJ
ejpam-6567	47	29	lower	low	ADJ
ejpam-6567	47	30	weakly	weakly	ADJ
ejpam-6567	47	31	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	47	32	,	,	PUNCT
ejpam-6567	47	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	47	34	multifunctions	multifunction	NOUN
ejpam-6567	47	35	.	.	PUNCT
ejpam-6567	48	1	we	we	PRON
ejpam-6567	48	2	also	also	ADV
ejpam-6567	48	3	investigate	investigate	VERB
ejpam-6567	48	4	several	several	ADJ
ejpam-6567	48	5	characterizations	characterization	NOUN
ejpam-6567	48	6	of	of	ADP
ejpam-6567	48	7	upper	upper	ADJ
ejpam-6567	48	8	weakly	weakly	ADJ
ejpam-6567	48	9	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	48	10	,	,	PUNCT
ejpam-6567	48	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	48	12	multifunctions	multifunction	NOUN
ejpam-6567	48	13	and	and	CCONJ
ejpam-6567	48	14	lower	low	ADJ
ejpam-6567	48	15	weakly	weakly	ADJ
ejpam-6567	48	16	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	48	17	,	,	PUNCT
ejpam-6567	48	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	48	19	multifunctions	multifunction	NOUN
ejpam-6567	48	20	.	.	PUNCT
ejpam-6567	49	1	2	2	X
ejpam-6567	49	2	.	.	X
ejpam-6567	49	3	preliminaries	preliminary	NOUN
ejpam-6567	49	4	throughout	throughout	ADP
ejpam-6567	49	5	the	the	DET
ejpam-6567	49	6	present	present	ADJ
ejpam-6567	49	7	paper	paper	NOUN
ejpam-6567	49	8	,	,	PUNCT
ejpam-6567	49	9	spaces	space	NOUN
ejpam-6567	49	10	(	(	PUNCT
ejpam-6567	49	11	x	x	NOUN
ejpam-6567	49	12	,	,	PUNCT
ejpam-6567	49	13	τ1	τ1	NOUN
ejpam-6567	49	14	,	,	PUNCT
ejpam-6567	49	15	τ2	τ2	NOUN
ejpam-6567	49	16	)	)	PUNCT
ejpam-6567	49	17	and	and	CCONJ
ejpam-6567	49	18	(	(	PUNCT
ejpam-6567	49	19	y	y	PROPN
ejpam-6567	49	20	,	,	PUNCT
ejpam-6567	49	21	σ1	σ1	PROPN
ejpam-6567	49	22	,	,	PUNCT
ejpam-6567	49	23	σ2	σ2	NOUN
ejpam-6567	49	24	)	)	PUNCT
ejpam-6567	49	25	(	(	PUNCT
ejpam-6567	49	26	or	or	CCONJ
ejpam-6567	49	27	simply	simply	ADV
ejpam-6567	49	28	x	x	X
ejpam-6567	49	29	and	and	CCONJ
ejpam-6567	49	30	y	y	PROPN
ejpam-6567	49	31	)	)	PUNCT
ejpam-6567	49	32	always	always	ADV
ejpam-6567	49	33	mean	mean	VERB
ejpam-6567	49	34	bitopological	bitopological	ADJ
ejpam-6567	49	35	spaces	space	NOUN
ejpam-6567	49	36	on	on	ADP
ejpam-6567	49	37	which	which	PRON
ejpam-6567	49	38	no	no	DET
ejpam-6567	49	39	separation	separation	NOUN
ejpam-6567	49	40	axioms	axiom	NOUN
ejpam-6567	49	41	are	be	AUX
ejpam-6567	49	42	assumed	assume	VERB
ejpam-6567	49	43	unless	unless	SCONJ
ejpam-6567	49	44	explicitly	explicitly	ADV
ejpam-6567	49	45	stated	state	VERB
ejpam-6567	49	46	.	.	PUNCT
ejpam-6567	50	1	let	let	VERB
ejpam-6567	50	2	a	a	DET
ejpam-6567	50	3	be	be	AUX
ejpam-6567	50	4	a	a	DET
ejpam-6567	50	5	subset	subset	NOUN
ejpam-6567	50	6	of	of	ADP
ejpam-6567	50	7	a	a	DET
ejpam-6567	50	8	bitopological	bitopological	ADJ
ejpam-6567	50	9	space	space	NOUN
ejpam-6567	50	10	(	(	PUNCT
ejpam-6567	50	11	x	x	NOUN
ejpam-6567	50	12	,	,	PUNCT
ejpam-6567	50	13	τ1	τ1	NOUN
ejpam-6567	50	14	,	,	PUNCT
ejpam-6567	50	15	τ2	τ2	NOUN
ejpam-6567	50	16	)	)	PUNCT
ejpam-6567	50	17	.	.	PUNCT
ejpam-6567	51	1	the	the	DET
ejpam-6567	51	2	closure	closure	NOUN
ejpam-6567	51	3	of	of	ADP
ejpam-6567	51	4	a	a	PRON
ejpam-6567	51	5	and	and	CCONJ
ejpam-6567	51	6	the	the	DET
ejpam-6567	51	7	interior	interior	NOUN
ejpam-6567	51	8	of	of	ADP
ejpam-6567	51	9	a	a	PRON
ejpam-6567	51	10	with	with	ADP
ejpam-6567	51	11	respect	respect	NOUN
ejpam-6567	51	12	to	to	ADP
ejpam-6567	51	13	τi	τi	PROPN
ejpam-6567	51	14	are	be	AUX
ejpam-6567	51	15	denoted	denote	VERB
ejpam-6567	51	16	by	by	ADP
ejpam-6567	51	17	τi	τi	NOUN
ejpam-6567	51	18	-	-	PUNCT
ejpam-6567	51	19	cl(a	cl(a	NUM
ejpam-6567	51	20	)	)	PUNCT
ejpam-6567	51	21	and	and	CCONJ
ejpam-6567	51	22	τi	τi	NOUN
ejpam-6567	51	23	-	-	PUNCT
ejpam-6567	51	24	int(a	int(a	NOUN
ejpam-6567	51	25	)	)	PUNCT
ejpam-6567	51	26	,	,	PUNCT
ejpam-6567	51	27	respectively	respectively	ADV
ejpam-6567	51	28	,	,	PUNCT
ejpam-6567	51	29	for	for	ADP
ejpam-6567	51	30	i	i	PROPN
ejpam-6567	51	31	=	=	SYM
ejpam-6567	51	32	1	1	NUM
ejpam-6567	51	33	,	,	PUNCT
ejpam-6567	51	34	2	2	NUM
ejpam-6567	51	35	.	.	X
ejpam-6567	51	36	a	a	DET
ejpam-6567	51	37	subset	subset	NOUN
ejpam-6567	51	38	a	a	PRON
ejpam-6567	51	39	of	of	ADP
ejpam-6567	51	40	a	a	DET
ejpam-6567	51	41	bitopological	bitopological	ADJ
ejpam-6567	51	42	space	space	NOUN
ejpam-6567	51	43	(	(	PUNCT
ejpam-6567	51	44	x	x	NOUN
ejpam-6567	51	45	,	,	PUNCT
ejpam-6567	51	46	τ1	τ1	NOUN
ejpam-6567	51	47	,	,	PUNCT
ejpam-6567	51	48	τ2	τ2	NOUN
ejpam-6567	51	49	)	)	PUNCT
ejpam-6567	51	50	is	be	AUX
ejpam-6567	51	51	called	call	VERB
ejpam-6567	51	52	τ1τ2	τ1τ2	VERB
ejpam-6567	51	53	-	-	ADJ
ejpam-6567	51	54	closed	closed	ADJ
ejpam-6567	51	55	[	[	X
ejpam-6567	51	56	22	22	NUM
ejpam-6567	51	57	]	]	PUNCT
ejpam-6567	51	58	if	if	SCONJ
ejpam-6567	51	59	a	a	DET
ejpam-6567	51	60	=	=	NOUN
ejpam-6567	51	61	τ1	τ1	NOUN
ejpam-6567	51	62	-	-	PUNCT
ejpam-6567	51	63	cl(τ2	cl(τ2	NOUN
ejpam-6567	51	64	-	-	PUNCT
ejpam-6567	51	65	cl(a	cl(a	NUM
ejpam-6567	51	66	)	)	PUNCT
ejpam-6567	51	67	)	)	PUNCT
ejpam-6567	51	68	.	.	PUNCT
ejpam-6567	52	1	the	the	DET
ejpam-6567	52	2	complement	complement	NOUN
ejpam-6567	52	3	of	of	ADP
ejpam-6567	52	4	a	a	DET
ejpam-6567	52	5	τ1τ2	τ1τ2	ADJ
ejpam-6567	52	6	-	-	ADJ
ejpam-6567	52	7	closed	closed	ADJ
ejpam-6567	52	8	set	set	NOUN
ejpam-6567	52	9	is	be	AUX
ejpam-6567	52	10	called	call	VERB
ejpam-6567	52	11	τ1τ2	τ1τ2	NOUN
ejpam-6567	52	12	-	-	ADJ
ejpam-6567	52	13	open	open	ADJ
ejpam-6567	52	14	.	.	PUNCT
ejpam-6567	53	1	the	the	DET
ejpam-6567	53	2	intersection	intersection	NOUN
ejpam-6567	53	3	of	of	ADP
ejpam-6567	53	4	all	all	DET
ejpam-6567	53	5	τ1τ2	τ1τ2	ADJ
ejpam-6567	53	6	-	-	ADJ
ejpam-6567	53	7	closed	closed	ADJ
ejpam-6567	53	8	sets	set	NOUN
ejpam-6567	53	9	of	of	ADP
ejpam-6567	53	10	x	x	PUNCT
ejpam-6567	53	11	containing	contain	VERB
ejpam-6567	53	12	a	a	PRON
ejpam-6567	53	13	is	be	AUX
ejpam-6567	53	14	called	call	VERB
ejpam-6567	53	15	the	the	DET
ejpam-6567	53	16	τ1τ2	τ1τ2	NOUN
ejpam-6567	53	17	-	-	NOUN
ejpam-6567	53	18	closure	closure	NOUN
ejpam-6567	53	19	[	[	X
ejpam-6567	53	20	22	22	NUM
ejpam-6567	53	21	]	]	PUNCT
ejpam-6567	53	22	of	of	ADP
ejpam-6567	53	23	a	a	PRON
ejpam-6567	53	24	and	and	CCONJ
ejpam-6567	53	25	is	be	AUX
ejpam-6567	53	26	denoted	denote	VERB
ejpam-6567	53	27	by	by	ADP
ejpam-6567	53	28	τ1τ2	τ1τ2	NOUN
ejpam-6567	53	29	-	-	NUM
ejpam-6567	53	30	cl(a	cl(a	NUM
ejpam-6567	53	31	)	)	PUNCT
ejpam-6567	53	32	.	.	PUNCT
ejpam-6567	54	1	the	the	DET
ejpam-6567	54	2	union	union	NOUN
ejpam-6567	54	3	of	of	ADP
ejpam-6567	54	4	all	all	DET
ejpam-6567	54	5	τ1τ2	τ1τ2	ADJ
ejpam-6567	54	6	-	-	ADJ
ejpam-6567	54	7	open	open	ADJ
ejpam-6567	54	8	sets	set	NOUN
ejpam-6567	54	9	of	of	ADP
ejpam-6567	54	10	x	x	PUNCT
ejpam-6567	54	11	contained	contain	VERB
ejpam-6567	54	12	in	in	ADP
ejpam-6567	54	13	a	a	PRON
ejpam-6567	54	14	is	be	AUX
ejpam-6567	54	15	called	call	VERB
ejpam-6567	54	16	the	the	DET
ejpam-6567	54	17	τ1τ2	τ1τ2	NOUN
ejpam-6567	54	18	-	-	ADJ
ejpam-6567	54	19	interior	interior	ADJ
ejpam-6567	54	20	[	[	X
ejpam-6567	54	21	22	22	NUM
ejpam-6567	54	22	]	]	PUNCT
ejpam-6567	54	23	of	of	ADP
ejpam-6567	54	24	a	a	PRON
ejpam-6567	54	25	and	and	CCONJ
ejpam-6567	54	26	is	be	AUX
ejpam-6567	54	27	denoted	denote	VERB
ejpam-6567	54	28	by	by	ADP
ejpam-6567	54	29	τ1τ2	τ1τ2	NOUN
ejpam-6567	54	30	-	-	ADJ
ejpam-6567	54	31	int(a	int(a	NOUN
ejpam-6567	54	32	)	)	PUNCT
ejpam-6567	54	33	.	.	PUNCT
ejpam-6567	55	1	lemma	lemma	PROPN
ejpam-6567	55	2	1	1	NUM
ejpam-6567	55	3	.	.	PUNCT
ejpam-6567	56	1	[	[	X
ejpam-6567	56	2	22	22	NUM
ejpam-6567	56	3	]	]	PUNCT
ejpam-6567	56	4	let	let	VERB
ejpam-6567	56	5	a	a	PRON
ejpam-6567	56	6	and	and	CCONJ
ejpam-6567	56	7	b	b	NOUN
ejpam-6567	56	8	be	be	AUX
ejpam-6567	56	9	subsets	subset	NOUN
ejpam-6567	56	10	of	of	ADP
ejpam-6567	56	11	a	a	DET
ejpam-6567	56	12	bitopological	bitopological	ADJ
ejpam-6567	56	13	space	space	NOUN
ejpam-6567	56	14	(	(	PUNCT
ejpam-6567	56	15	x	x	NOUN
ejpam-6567	56	16	,	,	PUNCT
ejpam-6567	56	17	τ1	τ1	NOUN
ejpam-6567	56	18	,	,	PUNCT
ejpam-6567	56	19	τ2	τ2	NOUN
ejpam-6567	56	20	)	)	PUNCT
ejpam-6567	56	21	.	.	PUNCT
ejpam-6567	57	1	for	for	ADP
ejpam-6567	57	2	the	the	DET
ejpam-6567	57	3	τ1τ2p	τ1τ2p	NUM
ejpam-6567	57	4	.	.	PUNCT
ejpam-6567	58	1	pue	pue	NOUN
ejpam-6567	58	2	-	-	PUNCT
ejpam-6567	58	3	on	on	ADP
ejpam-6567	58	4	,	,	PUNCT
ejpam-6567	58	5	a.	a.	PROPN
ejpam-6567	58	6	sama	sama	PROPN
ejpam-6567	58	7	-	-	PUNCT
ejpam-6567	58	8	ae	ae	PROPN
ejpam-6567	58	9	,	,	PUNCT
ejpam-6567	58	10	c.	c.	PROPN
ejpam-6567	58	11	boonpok	boonpok	PROPN
ejpam-6567	58	12	/	/	SYM
ejpam-6567	58	13	eur	eur	PROPN
ejpam-6567	58	14	.	.	PUNCT
ejpam-6567	59	1	j.	j.	PROPN
ejpam-6567	59	2	pure	pure	PROPN
ejpam-6567	59	3	appl	appl	PROPN
ejpam-6567	59	4	.	.	PROPN
ejpam-6567	59	5	math	math	PROPN
ejpam-6567	59	6	,	,	PUNCT
ejpam-6567	59	7	18	18	NUM
ejpam-6567	59	8	(	(	PUNCT
ejpam-6567	59	9	3	3	NUM
ejpam-6567	59	10	)	)	PUNCT
ejpam-6567	59	11	(	(	PUNCT
ejpam-6567	59	12	2025	2025	NUM
ejpam-6567	59	13	)	)	PUNCT
ejpam-6567	59	14	,	,	PUNCT
ejpam-6567	59	15	6567	6567	NUM
ejpam-6567	59	16	3	3	NUM
ejpam-6567	59	17	of	of	ADP
ejpam-6567	59	18	13	13	NUM
ejpam-6567	59	19	closure	closure	NOUN
ejpam-6567	59	20	,	,	PUNCT
ejpam-6567	59	21	the	the	DET
ejpam-6567	59	22	following	follow	VERB
ejpam-6567	59	23	properties	property	NOUN
ejpam-6567	59	24	hold	hold	VERB
ejpam-6567	59	25	:	:	PUNCT
ejpam-6567	59	26	(	(	PUNCT
ejpam-6567	59	27	1	1	X
ejpam-6567	59	28	)	)	PUNCT
ejpam-6567	59	29	a	a	DET
ejpam-6567	59	30	⊆	⊆	NUM
ejpam-6567	59	31	τ1τ2	τ1τ2	NOUN
ejpam-6567	59	32	-	-	NUM
ejpam-6567	59	33	cl(a	cl(a	NUM
ejpam-6567	59	34	)	)	PUNCT
ejpam-6567	59	35	and	and	CCONJ
ejpam-6567	59	36	τ1τ2	τ1τ2	NOUN
ejpam-6567	59	37	-	-	ADJ
ejpam-6567	59	38	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6567	59	39	-	-	PUNCT
ejpam-6567	59	40	cl(a	cl(a	NUM
ejpam-6567	59	41	)	)	PUNCT
ejpam-6567	59	42	)	)	PUNCT
ejpam-6567	60	1	=	=	PUNCT
ejpam-6567	60	2	τ1τ2	τ1τ2	NOUN
ejpam-6567	60	3	-	-	NUM
ejpam-6567	60	4	cl(a	cl(a	NUM
ejpam-6567	60	5	)	)	PUNCT
ejpam-6567	60	6	.	.	PUNCT
ejpam-6567	61	1	(	(	PUNCT
ejpam-6567	61	2	2	2	X
ejpam-6567	61	3	)	)	PUNCT
ejpam-6567	61	4	if	if	SCONJ
ejpam-6567	61	5	a	a	DET
ejpam-6567	61	6	⊆	⊆	NUM
ejpam-6567	61	7	b	b	NOUN
ejpam-6567	61	8	,	,	PUNCT
ejpam-6567	61	9	then	then	ADV
ejpam-6567	61	10	τ1τ2	τ1τ2	NOUN
ejpam-6567	61	11	-	-	NUM
ejpam-6567	61	12	cl(a	cl(a	NUM
ejpam-6567	61	13	)	)	PUNCT
ejpam-6567	61	14	⊆	⊆	NUM
ejpam-6567	61	15	τ1τ2	τ1τ2	NOUN
ejpam-6567	61	16	-	-	NOUN
ejpam-6567	61	17	cl(b	cl(b	NOUN
ejpam-6567	61	18	)	)	PUNCT
ejpam-6567	61	19	.	.	PUNCT
ejpam-6567	62	1	(	(	PUNCT
ejpam-6567	62	2	3	3	X
ejpam-6567	62	3	)	)	PUNCT
ejpam-6567	62	4	τ1τ2	τ1τ2	NOUN
ejpam-6567	62	5	-	-	NUM
ejpam-6567	62	6	cl(a	cl(a	NUM
ejpam-6567	62	7	)	)	PUNCT
ejpam-6567	62	8	is	be	AUX
ejpam-6567	62	9	τ1τ2	τ1τ2	NOUN
ejpam-6567	62	10	-	-	ADJ
ejpam-6567	62	11	closed	closed	ADJ
ejpam-6567	62	12	.	.	PUNCT
ejpam-6567	63	1	(	(	PUNCT
ejpam-6567	63	2	4	4	X
ejpam-6567	63	3	)	)	PUNCT
ejpam-6567	63	4	a	a	PRON
ejpam-6567	63	5	is	be	AUX
ejpam-6567	63	6	τ1τ2	τ1τ2	NOUN
ejpam-6567	63	7	-	-	ADJ
ejpam-6567	63	8	closed	closed	ADJ
ejpam-6567	63	9	if	if	SCONJ
ejpam-6567	63	10	and	and	CCONJ
ejpam-6567	63	11	only	only	ADV
ejpam-6567	63	12	if	if	SCONJ
ejpam-6567	63	13	a	a	DET
ejpam-6567	63	14	=	=	PUNCT
ejpam-6567	63	15	τ1τ2	τ1τ2	NOUN
ejpam-6567	63	16	-	-	NUM
ejpam-6567	63	17	cl(a	cl(a	NUM
ejpam-6567	63	18	)	)	PUNCT
ejpam-6567	63	19	.	.	PUNCT
ejpam-6567	64	1	(	(	PUNCT
ejpam-6567	64	2	5	5	X
ejpam-6567	64	3	)	)	PUNCT
ejpam-6567	64	4	τ1τ2	τ1τ2	NOUN
ejpam-6567	64	5	-	-	NOUN
ejpam-6567	64	6	cl(x	cl(x	X
ejpam-6567	64	7	−a	−a	NOUN
ejpam-6567	64	8	)	)	PUNCT
ejpam-6567	65	1	=	=	PUNCT
ejpam-6567	65	2	x	x	X
ejpam-6567	66	1	−	−	ADP
ejpam-6567	66	2	τ1τ2	τ1τ2	NOUN
ejpam-6567	66	3	-	-	PUNCT
ejpam-6567	66	4	int(a	int(a	NOUN
ejpam-6567	66	5	)	)	PUNCT
ejpam-6567	66	6	.	.	PUNCT
ejpam-6567	67	1	a	a	DET
ejpam-6567	67	2	subset	subset	NOUN
ejpam-6567	67	3	a	a	PRON
ejpam-6567	67	4	of	of	ADP
ejpam-6567	67	5	a	a	DET
ejpam-6567	67	6	bitopological	bitopological	ADJ
ejpam-6567	67	7	space	space	NOUN
ejpam-6567	67	8	(	(	PUNCT
ejpam-6567	67	9	x	x	NOUN
ejpam-6567	67	10	,	,	PUNCT
ejpam-6567	67	11	τ1	τ1	NOUN
ejpam-6567	67	12	,	,	PUNCT
ejpam-6567	67	13	τ2	τ2	NOUN
ejpam-6567	67	14	)	)	PUNCT
ejpam-6567	67	15	is	be	AUX
ejpam-6567	67	16	said	say	VERB
ejpam-6567	67	17	to	to	PART
ejpam-6567	67	18	be	be	AUX
ejpam-6567	67	19	(	(	PUNCT
ejpam-6567	67	20	τ1	τ1	NOUN
ejpam-6567	67	21	,	,	PUNCT
ejpam-6567	67	22	τ2)r	τ2)r	NOUN
ejpam-6567	67	23	-	-	PUNCT
ejpam-6567	67	24	open	open	NOUN
ejpam-6567	68	1	[	[	X
ejpam-6567	68	2	23	23	NUM
ejpam-6567	68	3	]	]	PUNCT
ejpam-6567	68	4	(	(	PUNCT
ejpam-6567	68	5	resp	resp	NOUN
ejpam-6567	68	6	.	.	PUNCT
ejpam-6567	69	1	(	(	PUNCT
ejpam-6567	69	2	τ1	τ1	NOUN
ejpam-6567	69	3	,	,	PUNCT
ejpam-6567	69	4	τ2)s	τ2)s	NOUN
ejpam-6567	69	5	-	-	PUNCT
ejpam-6567	69	6	open	open	ADJ
ejpam-6567	69	7	[	[	X
ejpam-6567	69	8	24	24	NUM
ejpam-6567	69	9	]	]	PUNCT
ejpam-6567	69	10	,	,	PUNCT
ejpam-6567	69	11	(	(	PUNCT
ejpam-6567	69	12	τ1	τ1	NOUN
ejpam-6567	69	13	,	,	PUNCT
ejpam-6567	69	14	τ2)p	τ2)p	NOUN
ejpam-6567	69	15	-	-	ADJ
ejpam-6567	69	16	open	open	ADJ
ejpam-6567	69	17	[	[	X
ejpam-6567	69	18	24	24	NUM
ejpam-6567	69	19	]	]	PUNCT
ejpam-6567	69	20	,	,	PUNCT
ejpam-6567	69	21	(	(	PUNCT
ejpam-6567	69	22	τ1	τ1	NOUN
ejpam-6567	69	23	,	,	PUNCT
ejpam-6567	69	24	τ2)β	τ2)β	ADJ
ejpam-6567	69	25	-	-	PUNCT
ejpam-6567	69	26	open	open	NOUN
ejpam-6567	69	27	[	[	X
ejpam-6567	69	28	24	24	NUM
ejpam-6567	69	29	]	]	PUNCT
ejpam-6567	69	30	)	)	PUNCT
ejpam-6567	69	31	if	if	SCONJ
ejpam-6567	69	32	a	a	DET
ejpam-6567	69	33	=	=	PUNCT
ejpam-6567	69	34	τ1τ2	τ1τ2	NOUN
ejpam-6567	69	35	-	-	NOUN
ejpam-6567	69	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6567	69	37	-	-	PUNCT
ejpam-6567	69	38	cl(a	cl(a	NUM
ejpam-6567	69	39	)	)	PUNCT
ejpam-6567	69	40	)	)	PUNCT
ejpam-6567	69	41	(	(	PUNCT
ejpam-6567	69	42	resp	resp	NOUN
ejpam-6567	69	43	.	.	PUNCT
ejpam-6567	70	1	a	a	DET
ejpam-6567	70	2	⊆	⊆	NUM
ejpam-6567	70	3	τ1τ2	τ1τ2	NOUN
ejpam-6567	70	4	-	-	ADJ
ejpam-6567	70	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6567	70	6	-	-	PUNCT
ejpam-6567	70	7	int(a	int(a	NOUN
ejpam-6567	70	8	)	)	PUNCT
ejpam-6567	70	9	)	)	PUNCT
ejpam-6567	70	10	,	,	PUNCT
ejpam-6567	70	11	a	a	DET
ejpam-6567	70	12	⊆	⊆	NUM
ejpam-6567	70	13	τ1τ2	τ1τ2	NOUN
ejpam-6567	70	14	-	-	NOUN
ejpam-6567	70	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6567	70	16	-	-	PUNCT
ejpam-6567	70	17	cl(a	cl(a	NUM
ejpam-6567	70	18	)	)	PUNCT
ejpam-6567	70	19	)	)	PUNCT
ejpam-6567	70	20	,	,	PUNCT
ejpam-6567	70	21	a	a	DET
ejpam-6567	70	22	⊆	⊆	NUM
ejpam-6567	70	23	τ1τ2	τ1τ2	NOUN
ejpam-6567	70	24	-	-	PUNCT
ejpam-6567	70	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6567	70	26	-	-	PUNCT
ejpam-6567	70	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6567	70	28	-	-	PUNCT
ejpam-6567	70	29	cl(a	cl(a	NUM
ejpam-6567	70	30	)	)	PUNCT
ejpam-6567	70	31	)	)	PUNCT
ejpam-6567	70	32	)	)	PUNCT
ejpam-6567	70	33	)	)	PUNCT
ejpam-6567	70	34	.	.	PUNCT
ejpam-6567	71	1	the	the	DET
ejpam-6567	71	2	complement	complement	NOUN
ejpam-6567	71	3	of	of	ADP
ejpam-6567	71	4	a	a	DET
ejpam-6567	71	5	(	(	PUNCT
ejpam-6567	71	6	τ1	τ1	NOUN
ejpam-6567	71	7	,	,	PUNCT
ejpam-6567	71	8	τ2)r	τ2)r	NOUN
ejpam-6567	71	9	-	-	PUNCT
ejpam-6567	71	10	open	open	ADJ
ejpam-6567	71	11	(	(	PUNCT
ejpam-6567	71	12	resp	resp	NOUN
ejpam-6567	71	13	.	.	PUNCT
ejpam-6567	72	1	(	(	PUNCT
ejpam-6567	72	2	τ1	τ1	NOUN
ejpam-6567	72	3	,	,	PUNCT
ejpam-6567	72	4	τ2)s	τ2)s	NOUN
ejpam-6567	72	5	-	-	PUNCT
ejpam-6567	72	6	open	open	ADJ
ejpam-6567	72	7	,	,	PUNCT
ejpam-6567	72	8	(	(	PUNCT
ejpam-6567	72	9	τ1	τ1	NOUN
ejpam-6567	72	10	,	,	PUNCT
ejpam-6567	72	11	τ2)p	τ2)p	NOUN
ejpam-6567	72	12	-	-	ADJ
ejpam-6567	72	13	open	open	ADJ
ejpam-6567	72	14	,	,	PUNCT
ejpam-6567	72	15	(	(	PUNCT
ejpam-6567	72	16	τ1	τ1	NOUN
ejpam-6567	72	17	,	,	PUNCT
ejpam-6567	72	18	τ2)β	τ2)β	ADJ
ejpam-6567	72	19	-	-	PUNCT
ejpam-6567	72	20	open	open	ADJ
ejpam-6567	72	21	)	)	PUNCT
ejpam-6567	72	22	set	set	NOUN
ejpam-6567	72	23	is	be	AUX
ejpam-6567	72	24	said	say	VERB
ejpam-6567	72	25	to	to	PART
ejpam-6567	72	26	be	be	AUX
ejpam-6567	72	27	(	(	PUNCT
ejpam-6567	72	28	τ1	τ1	NOUN
ejpam-6567	72	29	,	,	PUNCT
ejpam-6567	72	30	τ2)r	τ2)r	NOUN
ejpam-6567	72	31	-	-	PUNCT
ejpam-6567	72	32	closed	closed	ADJ
ejpam-6567	72	33	(	(	PUNCT
ejpam-6567	72	34	resp	resp	NOUN
ejpam-6567	72	35	.	.	PUNCT
ejpam-6567	73	1	(	(	PUNCT
ejpam-6567	73	2	τ1	τ1	NOUN
ejpam-6567	73	3	,	,	PUNCT
ejpam-6567	73	4	τ2)s	τ2)s	NOUN
ejpam-6567	73	5	-	-	PUNCT
ejpam-6567	73	6	closed	closed	ADJ
ejpam-6567	73	7	,	,	PUNCT
ejpam-6567	73	8	(	(	PUNCT
ejpam-6567	73	9	τ1	τ1	NOUN
ejpam-6567	73	10	,	,	PUNCT
ejpam-6567	73	11	τ2)p	τ2)p	NOUN
ejpam-6567	73	12	-	-	PUNCT
ejpam-6567	73	13	closed	closed	ADJ
ejpam-6567	73	14	,	,	PUNCT
ejpam-6567	73	15	(	(	PUNCT
ejpam-6567	73	16	τ1	τ1	NOUN
ejpam-6567	73	17	,	,	PUNCT
ejpam-6567	73	18	τ2)β	τ2)β	ADJ
ejpam-6567	73	19	-	-	PUNCT
ejpam-6567	73	20	closed	closed	ADJ
ejpam-6567	73	21	)	)	PUNCT
ejpam-6567	73	22	.	.	PUNCT
ejpam-6567	74	1	a	a	DET
ejpam-6567	74	2	subset	subset	NOUN
ejpam-6567	74	3	a	a	PRON
ejpam-6567	74	4	of	of	ADP
ejpam-6567	74	5	a	a	DET
ejpam-6567	74	6	bitopological	bitopological	ADJ
ejpam-6567	74	7	space	space	NOUN
ejpam-6567	74	8	(	(	PUNCT
ejpam-6567	74	9	x	x	NOUN
ejpam-6567	74	10	,	,	PUNCT
ejpam-6567	74	11	τ1	τ1	NOUN
ejpam-6567	74	12	,	,	PUNCT
ejpam-6567	74	13	τ2	τ2	NOUN
ejpam-6567	74	14	)	)	PUNCT
ejpam-6567	74	15	is	be	AUX
ejpam-6567	74	16	said	say	VERB
ejpam-6567	74	17	to	to	PART
ejpam-6567	74	18	be	be	AUX
ejpam-6567	74	19	α(τ1	α(τ1	NOUN
ejpam-6567	74	20	,	,	PUNCT
ejpam-6567	74	21	τ2)-open	τ2)-open	ADJ
ejpam-6567	74	22	[	[	X
ejpam-6567	74	23	25	25	NUM
ejpam-6567	74	24	]	]	PUNCT
ejpam-6567	74	25	if	if	SCONJ
ejpam-6567	74	26	a	a	DET
ejpam-6567	74	27	⊆	⊆	NUM
ejpam-6567	74	28	τ1τ2	τ1τ2	NOUN
ejpam-6567	74	29	-	-	PUNCT
ejpam-6567	74	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6567	74	31	-	-	PUNCT
ejpam-6567	74	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6567	74	33	-	-	PUNCT
ejpam-6567	74	34	int(a	int(a	NOUN
ejpam-6567	74	35	)	)	PUNCT
ejpam-6567	74	36	)	)	PUNCT
ejpam-6567	74	37	)	)	PUNCT
ejpam-6567	74	38	.	.	PUNCT
ejpam-6567	75	1	the	the	DET
ejpam-6567	75	2	complement	complement	NOUN
ejpam-6567	75	3	of	of	ADP
ejpam-6567	75	4	an	an	DET
ejpam-6567	75	5	α(τ1	α(τ1	NOUN
ejpam-6567	75	6	,	,	PUNCT
ejpam-6567	75	7	τ2)-open	τ2)-open	ADJ
ejpam-6567	75	8	set	set	NOUN
ejpam-6567	75	9	is	be	AUX
ejpam-6567	75	10	said	say	VERB
ejpam-6567	75	11	to	to	PART
ejpam-6567	75	12	be	be	AUX
ejpam-6567	75	13	α(τ1	α(τ1	NOUN
ejpam-6567	75	14	,	,	PUNCT
ejpam-6567	75	15	τ2)-closed	τ2)-close	VERB
ejpam-6567	75	16	.	.	PUNCT
ejpam-6567	76	1	let	let	VERB
ejpam-6567	76	2	a	a	DET
ejpam-6567	76	3	be	be	AUX
ejpam-6567	76	4	a	a	DET
ejpam-6567	76	5	subset	subset	NOUN
ejpam-6567	76	6	of	of	ADP
ejpam-6567	76	7	a	a	DET
ejpam-6567	76	8	bitopological	bitopological	ADJ
ejpam-6567	76	9	space	space	NOUN
ejpam-6567	76	10	(	(	PUNCT
ejpam-6567	76	11	x	x	NOUN
ejpam-6567	76	12	,	,	PUNCT
ejpam-6567	76	13	τ1	τ1	NOUN
ejpam-6567	76	14	,	,	PUNCT
ejpam-6567	76	15	τ2	τ2	NOUN
ejpam-6567	76	16	)	)	PUNCT
ejpam-6567	76	17	.	.	PUNCT
ejpam-6567	77	1	the	the	DET
ejpam-6567	77	2	intersection	intersection	NOUN
ejpam-6567	77	3	of	of	ADP
ejpam-6567	77	4	all	all	DET
ejpam-6567	77	5	(	(	PUNCT
ejpam-6567	77	6	τ1	τ1	NOUN
ejpam-6567	77	7	,	,	PUNCT
ejpam-6567	77	8	τ2)p	τ2)p	NOUN
ejpam-6567	77	9	-	-	PUNCT
ejpam-6567	77	10	closed	closed	ADJ
ejpam-6567	77	11	(	(	PUNCT
ejpam-6567	77	12	resp	resp	NOUN
ejpam-6567	77	13	.	.	PUNCT
ejpam-6567	78	1	(	(	PUNCT
ejpam-6567	78	2	τ1	τ1	NOUN
ejpam-6567	78	3	,	,	PUNCT
ejpam-6567	78	4	τ2)s	τ2)s	NOUN
ejpam-6567	78	5	-	-	PUNCT
ejpam-6567	78	6	closed	closed	ADJ
ejpam-6567	78	7	,	,	PUNCT
ejpam-6567	78	8	α(τ1	α(τ1	NOUN
ejpam-6567	78	9	,	,	PUNCT
ejpam-6567	78	10	τ2)-closed	τ2)-closed	ADJ
ejpam-6567	78	11	)	)	PUNCT
ejpam-6567	78	12	sets	set	NOUN
ejpam-6567	78	13	of	of	ADP
ejpam-6567	78	14	x	x	PUNCT
ejpam-6567	78	15	containing	contain	VERB
ejpam-6567	78	16	a	a	PRON
ejpam-6567	78	17	is	be	AUX
ejpam-6567	78	18	called	call	VERB
ejpam-6567	78	19	the	the	DET
ejpam-6567	78	20	(	(	PUNCT
ejpam-6567	78	21	τ1	τ1	NOUN
ejpam-6567	78	22	,	,	PUNCT
ejpam-6567	78	23	τ2)p	τ2)p	NOUN
ejpam-6567	78	24	-	-	NOUN
ejpam-6567	78	25	closure	closure	NOUN
ejpam-6567	78	26	[	[	X
ejpam-6567	78	27	26	26	NUM
ejpam-6567	78	28	]	]	PUNCT
ejpam-6567	78	29	(	(	PUNCT
ejpam-6567	78	30	resp	resp	NOUN
ejpam-6567	78	31	.	.	PUNCT
ejpam-6567	79	1	(	(	PUNCT
ejpam-6567	79	2	τ1	τ1	NOUN
ejpam-6567	79	3	,	,	PUNCT
ejpam-6567	79	4	τ2)s	τ2)s	NOUN
ejpam-6567	79	5	-	-	PUNCT
ejpam-6567	79	6	closure	closure	NOUN
ejpam-6567	79	7	[	[	X
ejpam-6567	79	8	24	24	NUM
ejpam-6567	79	9	]	]	PUNCT
ejpam-6567	79	10	,	,	PUNCT
ejpam-6567	79	11	α(τ1	α(τ1	NOUN
ejpam-6567	79	12	,	,	PUNCT
ejpam-6567	79	13	τ2)-closure	τ2)-closure	NOUN
ejpam-6567	79	14	[	[	X
ejpam-6567	79	15	27	27	NUM
ejpam-6567	79	16	]	]	PUNCT
ejpam-6567	79	17	)	)	PUNCT
ejpam-6567	79	18	of	of	ADP
ejpam-6567	79	19	a	a	PRON
ejpam-6567	79	20	and	and	CCONJ
ejpam-6567	79	21	is	be	AUX
ejpam-6567	79	22	denoted	denote	VERB
ejpam-6567	79	23	by	by	ADP
ejpam-6567	79	24	(	(	PUNCT
ejpam-6567	79	25	τ1	τ1	NOUN
ejpam-6567	79	26	,	,	PUNCT
ejpam-6567	79	27	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6567	79	28	)	)	PUNCT
ejpam-6567	79	29	(	(	PUNCT
ejpam-6567	79	30	resp	resp	NOUN
ejpam-6567	79	31	.	.	PUNCT
ejpam-6567	80	1	(	(	PUNCT
ejpam-6567	80	2	τ1	τ1	NOUN
ejpam-6567	80	3	,	,	PUNCT
ejpam-6567	80	4	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6567	80	5	)	)	PUNCT
ejpam-6567	80	6	,	,	PUNCT
ejpam-6567	80	7	α(τ1	α(τ1	NOUN
ejpam-6567	80	8	,	,	PUNCT
ejpam-6567	80	9	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-6567	80	10	)	)	PUNCT
ejpam-6567	80	11	)	)	PUNCT
ejpam-6567	80	12	.	.	PUNCT
ejpam-6567	81	1	the	the	DET
ejpam-6567	81	2	union	union	NOUN
ejpam-6567	81	3	of	of	ADP
ejpam-6567	81	4	all	all	DET
ejpam-6567	81	5	(	(	PUNCT
ejpam-6567	81	6	τ1	τ1	NOUN
ejpam-6567	81	7	,	,	PUNCT
ejpam-6567	81	8	τ2)p	τ2)p	NOUN
ejpam-6567	81	9	-	-	ADJ
ejpam-6567	81	10	open	open	ADJ
ejpam-6567	81	11	(	(	PUNCT
ejpam-6567	81	12	resp	resp	NOUN
ejpam-6567	81	13	.	.	PUNCT
ejpam-6567	82	1	(	(	PUNCT
ejpam-6567	82	2	τ1	τ1	NOUN
ejpam-6567	82	3	,	,	PUNCT
ejpam-6567	82	4	τ2)s	τ2)s	NOUN
ejpam-6567	82	5	-	-	PUNCT
ejpam-6567	82	6	open	open	ADJ
ejpam-6567	82	7	,	,	PUNCT
ejpam-6567	82	8	α(τ1	α(τ1	NOUN
ejpam-6567	82	9	,	,	PUNCT
ejpam-6567	82	10	τ2)-open	τ2)-open	ADJ
ejpam-6567	82	11	)	)	PUNCT
ejpam-6567	82	12	sets	set	NOUN
ejpam-6567	82	13	of	of	ADP
ejpam-6567	82	14	x	x	PUNCT
ejpam-6567	82	15	contained	contain	VERB
ejpam-6567	82	16	in	in	ADP
ejpam-6567	82	17	a	a	PRON
ejpam-6567	82	18	is	be	AUX
ejpam-6567	82	19	called	call	VERB
ejpam-6567	82	20	the	the	DET
ejpam-6567	82	21	(	(	PUNCT
ejpam-6567	82	22	τ1	τ1	NOUN
ejpam-6567	82	23	,	,	PUNCT
ejpam-6567	82	24	τ2)p	τ2)p	ADJ
ejpam-6567	82	25	-	-	NOUN
ejpam-6567	82	26	interior	interior	ADJ
ejpam-6567	82	27	[	[	X
ejpam-6567	82	28	26	26	NUM
ejpam-6567	82	29	]	]	PUNCT
ejpam-6567	82	30	(	(	PUNCT
ejpam-6567	82	31	resp	resp	NOUN
ejpam-6567	82	32	.	.	PUNCT
ejpam-6567	83	1	(	(	PUNCT
ejpam-6567	83	2	τ1	τ1	NOUN
ejpam-6567	83	3	,	,	PUNCT
ejpam-6567	83	4	τ2)s	τ2)s	NOUN
ejpam-6567	83	5	-	-	ADJ
ejpam-6567	83	6	interior	interior	NOUN
ejpam-6567	83	7	[	[	X
ejpam-6567	83	8	24	24	NUM
ejpam-6567	83	9	]	]	PUNCT
ejpam-6567	83	10	,	,	PUNCT
ejpam-6567	83	11	α(τ1	α(τ1	NOUN
ejpam-6567	83	12	,	,	PUNCT
ejpam-6567	83	13	τ2)-interior	τ2)-interior	PROPN
ejpam-6567	83	14	[	[	X
ejpam-6567	83	15	27	27	NUM
ejpam-6567	83	16	]	]	PUNCT
ejpam-6567	83	17	)	)	PUNCT
ejpam-6567	83	18	of	of	ADP
ejpam-6567	83	19	a	a	PRON
ejpam-6567	83	20	and	and	CCONJ
ejpam-6567	83	21	is	be	AUX
ejpam-6567	83	22	denoted	denote	VERB
ejpam-6567	83	23	by	by	ADP
ejpam-6567	83	24	(	(	PUNCT
ejpam-6567	83	25	τ1	τ1	NOUN
ejpam-6567	83	26	,	,	PUNCT
ejpam-6567	83	27	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6567	83	28	)	)	PUNCT
ejpam-6567	83	29	(	(	PUNCT
ejpam-6567	83	30	resp	resp	NOUN
ejpam-6567	83	31	.	.	PUNCT
ejpam-6567	84	1	(	(	PUNCT
ejpam-6567	84	2	τ1	τ1	NOUN
ejpam-6567	84	3	,	,	PUNCT
ejpam-6567	84	4	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6567	84	5	)	)	PUNCT
ejpam-6567	84	6	,	,	PUNCT
ejpam-6567	84	7	α(τ1	α(τ1	NOUN
ejpam-6567	84	8	,	,	PUNCT
ejpam-6567	84	9	τ2)-int(a	τ2)-int(a	NOUN
ejpam-6567	84	10	)	)	PUNCT
ejpam-6567	84	11	)	)	PUNCT
ejpam-6567	84	12	.	.	PUNCT
ejpam-6567	85	1	for	for	ADP
ejpam-6567	85	2	a	a	DET
ejpam-6567	85	3	subset	subset	NOUN
ejpam-6567	85	4	a	a	PRON
ejpam-6567	85	5	of	of	ADP
ejpam-6567	85	6	a	a	DET
ejpam-6567	85	7	bitopological	bitopological	ADJ
ejpam-6567	85	8	space	space	NOUN
ejpam-6567	85	9	(	(	PUNCT
ejpam-6567	85	10	x	x	NOUN
ejpam-6567	85	11	,	,	PUNCT
ejpam-6567	85	12	τ1	τ1	NOUN
ejpam-6567	85	13	,	,	PUNCT
ejpam-6567	85	14	τ2	τ2	PROPN
ejpam-6567	85	15	)	)	PUNCT
ejpam-6567	85	16	,	,	PUNCT
ejpam-6567	85	17	a	a	DET
ejpam-6567	85	18	point	point	NOUN
ejpam-6567	85	19	x	x	X
ejpam-6567	85	20	∈	∈	NOUN
ejpam-6567	85	21	x	x	PUNCT
ejpam-6567	85	22	is	be	AUX
ejpam-6567	85	23	called	call	VERB
ejpam-6567	85	24	a	a	DET
ejpam-6567	85	25	(	(	PUNCT
ejpam-6567	85	26	τ1	τ1	NOUN
ejpam-6567	85	27	,	,	PUNCT
ejpam-6567	85	28	τ2)θ	τ2)θ	ADJ
ejpam-6567	85	29	-	-	PUNCT
ejpam-6567	85	30	cluster	cluster	NOUN
ejpam-6567	85	31	point	point	NOUN
ejpam-6567	85	32	of	of	ADP
ejpam-6567	85	33	a	a	DET
ejpam-6567	85	34	if	if	SCONJ
ejpam-6567	85	35	τ1τ2	τ1τ2	NOUN
ejpam-6567	85	36	-	-	NOUN
ejpam-6567	85	37	cl(u	cl(u	NOUN
ejpam-6567	85	38	)	)	PUNCT
ejpam-6567	85	39	∩a	∩a	PROPN
ejpam-6567	85	40	̸=	̸=	PROPN
ejpam-6567	85	41	∅	∅	NOUN
ejpam-6567	85	42	for	for	ADP
ejpam-6567	85	43	every	every	DET
ejpam-6567	85	44	τ1τ2	τ1τ2	ADJ
ejpam-6567	85	45	-	-	ADJ
ejpam-6567	85	46	open	open	ADJ
ejpam-6567	85	47	set	set	NOUN
ejpam-6567	85	48	u	u	NOUN
ejpam-6567	85	49	containing	contain	VERB
ejpam-6567	85	50	x.	x.	NOUN
ejpam-6567	85	51	the	the	DET
ejpam-6567	85	52	set	set	NOUN
ejpam-6567	85	53	of	of	ADP
ejpam-6567	85	54	all	all	DET
ejpam-6567	85	55	(	(	PUNCT
ejpam-6567	85	56	τ1	τ1	NOUN
ejpam-6567	85	57	,	,	PUNCT
ejpam-6567	85	58	τ2)θ	τ2)θ	ADJ
ejpam-6567	85	59	-	-	PUNCT
ejpam-6567	85	60	cluster	cluster	NOUN
ejpam-6567	85	61	points	point	NOUN
ejpam-6567	85	62	of	of	ADP
ejpam-6567	85	63	a	a	PRON
ejpam-6567	85	64	is	be	AUX
ejpam-6567	85	65	called	call	VERB
ejpam-6567	85	66	the	the	DET
ejpam-6567	85	67	(	(	PUNCT
ejpam-6567	85	68	τ1	τ1	NOUN
ejpam-6567	85	69	,	,	PUNCT
ejpam-6567	85	70	τ2)θ	τ2)θ	NOUN
ejpam-6567	85	71	-	-	PUNCT
ejpam-6567	85	72	closure	closure	NOUN
ejpam-6567	85	73	of	of	ADP
ejpam-6567	85	74	a	a	PRON
ejpam-6567	85	75	and	and	CCONJ
ejpam-6567	85	76	is	be	AUX
ejpam-6567	85	77	denoted	denote	VERB
ejpam-6567	85	78	by	by	ADP
ejpam-6567	85	79	(	(	PUNCT
ejpam-6567	85	80	τ1	τ1	NOUN
ejpam-6567	85	81	,	,	PUNCT
ejpam-6567	85	82	τ2)θ	τ2)θ	NOUN
ejpam-6567	85	83	-	-	PUNCT
ejpam-6567	85	84	cl(a	cl(a	NUM
ejpam-6567	85	85	)	)	PUNCT
ejpam-6567	85	86	.	.	PUNCT
ejpam-6567	86	1	a	a	DET
ejpam-6567	86	2	subset	subset	NOUN
ejpam-6567	86	3	a	a	PRON
ejpam-6567	86	4	of	of	ADP
ejpam-6567	86	5	a	a	DET
ejpam-6567	86	6	bitopological	bitopological	ADJ
ejpam-6567	86	7	space	space	NOUN
ejpam-6567	86	8	(	(	PUNCT
ejpam-6567	86	9	x	x	NOUN
ejpam-6567	86	10	,	,	PUNCT
ejpam-6567	86	11	τ1	τ1	NOUN
ejpam-6567	86	12	,	,	PUNCT
ejpam-6567	86	13	τ2	τ2	NOUN
ejpam-6567	86	14	)	)	PUNCT
ejpam-6567	86	15	is	be	AUX
ejpam-6567	86	16	said	say	VERB
ejpam-6567	86	17	to	to	PART
ejpam-6567	86	18	be	be	AUX
ejpam-6567	86	19	(	(	PUNCT
ejpam-6567	86	20	τ1	τ1	NOUN
ejpam-6567	86	21	,	,	PUNCT
ejpam-6567	86	22	τ2)θ	τ2)θ	NOUN
ejpam-6567	86	23	-	-	PUNCT
ejpam-6567	86	24	closed	closed	ADJ
ejpam-6567	86	25	if	if	SCONJ
ejpam-6567	86	26	(	(	PUNCT
ejpam-6567	86	27	τ1	τ1	NOUN
ejpam-6567	86	28	,	,	PUNCT
ejpam-6567	86	29	τ2)θ	τ2)θ	NOUN
ejpam-6567	86	30	-	-	PUNCT
ejpam-6567	86	31	cl(a	cl(a	NUM
ejpam-6567	86	32	)	)	PUNCT
ejpam-6567	86	33	=	=	PUNCT
ejpam-6567	86	34	a.	a.	NOUN
ejpam-6567	86	35	the	the	DET
ejpam-6567	86	36	complement	complement	NOUN
ejpam-6567	86	37	of	of	ADP
ejpam-6567	86	38	a	a	DET
ejpam-6567	86	39	(	(	PUNCT
ejpam-6567	86	40	τ1	τ1	NOUN
ejpam-6567	86	41	,	,	PUNCT
ejpam-6567	86	42	τ2)θ	τ2)θ	ADJ
ejpam-6567	86	43	-	-	PUNCT
ejpam-6567	86	44	closed	close	VERB
ejpam-6567	86	45	set	set	NOUN
ejpam-6567	86	46	is	be	AUX
ejpam-6567	86	47	said	say	VERB
ejpam-6567	86	48	to	to	PART
ejpam-6567	86	49	be	be	AUX
ejpam-6567	86	50	(	(	PUNCT
ejpam-6567	86	51	τ1	τ1	NOUN
ejpam-6567	86	52	,	,	PUNCT
ejpam-6567	86	53	τ2)θ	τ2)θ	NOUN
ejpam-6567	86	54	-	-	PUNCT
ejpam-6567	86	55	open	open	ADJ
ejpam-6567	86	56	.	.	PUNCT
ejpam-6567	87	1	the	the	DET
ejpam-6567	87	2	union	union	NOUN
ejpam-6567	87	3	of	of	ADP
ejpam-6567	87	4	all	all	DET
ejpam-6567	87	5	(	(	PUNCT
ejpam-6567	87	6	τ1	τ1	NOUN
ejpam-6567	87	7	,	,	PUNCT
ejpam-6567	87	8	τ2)θ	τ2)θ	ADJ
ejpam-6567	87	9	-	-	PUNCT
ejpam-6567	87	10	open	open	ADJ
ejpam-6567	87	11	sets	set	NOUN
ejpam-6567	87	12	of	of	ADP
ejpam-6567	87	13	x	x	PUNCT
ejpam-6567	87	14	contained	contain	VERB
ejpam-6567	87	15	in	in	ADP
ejpam-6567	87	16	a	a	PRON
ejpam-6567	87	17	is	be	AUX
ejpam-6567	87	18	called	call	VERB
ejpam-6567	87	19	the	the	DET
ejpam-6567	87	20	(	(	PUNCT
ejpam-6567	87	21	τ1	τ1	NOUN
ejpam-6567	87	22	,	,	PUNCT
ejpam-6567	87	23	τ2)θ	τ2)θ	ADJ
ejpam-6567	87	24	-	-	PUNCT
ejpam-6567	87	25	interior	interior	NOUN
ejpam-6567	87	26	of	of	ADP
ejpam-6567	87	27	a	a	PRON
ejpam-6567	87	28	and	and	CCONJ
ejpam-6567	87	29	is	be	AUX
ejpam-6567	87	30	denoted	denote	VERB
ejpam-6567	87	31	by	by	ADP
ejpam-6567	87	32	(	(	PUNCT
ejpam-6567	87	33	τ1	τ1	NOUN
ejpam-6567	87	34	,	,	PUNCT
ejpam-6567	87	35	τ2)θ	τ2)θ	NOUN
ejpam-6567	87	36	-	-	PUNCT
ejpam-6567	87	37	int(a	int(a	NOUN
ejpam-6567	87	38	)	)	PUNCT
ejpam-6567	88	1	[	[	X
ejpam-6567	88	2	23	23	NUM
ejpam-6567	88	3	]	]	PUNCT
ejpam-6567	88	4	.	.	PUNCT
ejpam-6567	89	1	an	an	DET
ejpam-6567	89	2	ideal	ideal	NOUN
ejpam-6567	89	3	i	i	PRON
ejpam-6567	89	4	on	on	ADP
ejpam-6567	89	5	a	a	DET
ejpam-6567	89	6	topological	topological	ADJ
ejpam-6567	89	7	space	space	NOUN
ejpam-6567	89	8	(	(	PUNCT
ejpam-6567	89	9	x	x	X
ejpam-6567	89	10	,	,	PUNCT
ejpam-6567	89	11	τ	τ	X
ejpam-6567	89	12	)	)	PUNCT
ejpam-6567	89	13	is	be	AUX
ejpam-6567	89	14	a	a	DET
ejpam-6567	89	15	nonempty	nonempty	ADJ
ejpam-6567	89	16	collection	collection	NOUN
ejpam-6567	89	17	of	of	ADP
ejpam-6567	89	18	subsets	subset	NOUN
ejpam-6567	89	19	of	of	ADP
ejpam-6567	89	20	x	x	PUNCT
ejpam-6567	89	21	satisfying	satisfy	VERB
ejpam-6567	89	22	the	the	DET
ejpam-6567	89	23	following	follow	VERB
ejpam-6567	89	24	properties	property	NOUN
ejpam-6567	89	25	:	:	PUNCT
ejpam-6567	89	26	(	(	PUNCT
ejpam-6567	89	27	1	1	X
ejpam-6567	89	28	)	)	PUNCT
ejpam-6567	89	29	a	a	DET
ejpam-6567	89	30	∈	∈	NOUN
ejpam-6567	89	31	i	i	PRON
ejpam-6567	89	32	and	and	CCONJ
ejpam-6567	89	33	b	b	X
ejpam-6567	89	34	⊆	⊆	NUM
ejpam-6567	89	35	a	a	DET
ejpam-6567	89	36	imply	imply	NOUN
ejpam-6567	89	37	b	b	X
ejpam-6567	89	38	∈	∈	PROPN
ejpam-6567	89	39	i	i	PRON
ejpam-6567	89	40	;	;	PUNCT
ejpam-6567	89	41	(	(	PUNCT
ejpam-6567	89	42	2	2	X
ejpam-6567	89	43	)	)	PUNCT
ejpam-6567	90	1	a	a	PRON
ejpam-6567	90	2	∈	∈	NOUN
ejpam-6567	91	1	i	i	PRON
ejpam-6567	91	2	and	and	CCONJ
ejpam-6567	91	3	b	b	X
ejpam-6567	91	4	∈	∈	NOUN
ejpam-6567	92	1	i	i	PRON
ejpam-6567	92	2	imply	imply	VERB
ejpam-6567	92	3	a	a	DET
ejpam-6567	92	4	∪	∪	X
ejpam-6567	92	5	b	b	NOUN
ejpam-6567	92	6	∈	∈	NOUN
ejpam-6567	93	1	i	i	PRON
ejpam-6567	93	2	.	.	PUNCT
ejpam-6567	94	1	a	a	DET
ejpam-6567	94	2	topological	topological	ADJ
ejpam-6567	94	3	space	space	NOUN
ejpam-6567	94	4	(	(	PUNCT
ejpam-6567	94	5	x	x	X
ejpam-6567	94	6	,	,	PUNCT
ejpam-6567	94	7	τ	τ	X
ejpam-6567	94	8	)	)	PUNCT
ejpam-6567	94	9	with	with	ADP
ejpam-6567	94	10	an	an	DET
ejpam-6567	94	11	ideal	ideal	ADJ
ejpam-6567	94	12	i	i	PRON
ejpam-6567	94	13	on	on	ADP
ejpam-6567	94	14	x	x	SYM
ejpam-6567	94	15	is	be	AUX
ejpam-6567	94	16	called	call	VERB
ejpam-6567	94	17	an	an	DET
ejpam-6567	94	18	ideal	ideal	ADJ
ejpam-6567	94	19	topological	topological	ADJ
ejpam-6567	94	20	space	space	NOUN
ejpam-6567	94	21	and	and	CCONJ
ejpam-6567	94	22	is	be	AUX
ejpam-6567	94	23	denoted	denote	VERB
ejpam-6567	94	24	by	by	ADP
ejpam-6567	94	25	(	(	PUNCT
ejpam-6567	94	26	x	x	X
ejpam-6567	94	27	,	,	PUNCT
ejpam-6567	94	28	τ	τ	PROPN
ejpam-6567	94	29	,	,	PUNCT
ejpam-6567	94	30	i	i	NOUN
ejpam-6567	94	31	)	)	PUNCT
ejpam-6567	94	32	.	.	PUNCT
ejpam-6567	95	1	for	for	ADP
ejpam-6567	95	2	an	an	DET
ejpam-6567	95	3	ideal	ideal	ADJ
ejpam-6567	95	4	topological	topological	ADJ
ejpam-6567	95	5	space	space	NOUN
ejpam-6567	95	6	(	(	PUNCT
ejpam-6567	95	7	x	x	X
ejpam-6567	95	8	,	,	PUNCT
ejpam-6567	95	9	τ	τ	PROPN
ejpam-6567	95	10	,	,	PUNCT
ejpam-6567	95	11	i	i	PROPN
ejpam-6567	95	12	)	)	PUNCT
ejpam-6567	95	13	and	and	CCONJ
ejpam-6567	95	14	a	a	DET
ejpam-6567	95	15	subset	subset	NOUN
ejpam-6567	95	16	a	a	PRON
ejpam-6567	95	17	of	of	ADP
ejpam-6567	95	18	x	x	PRON
ejpam-6567	95	19	,	,	PUNCT
ejpam-6567	95	20	a⋆(i	a⋆(i	PROPN
ejpam-6567	95	21	)	)	PUNCT
ejpam-6567	95	22	is	be	AUX
ejpam-6567	95	23	defined	define	VERB
ejpam-6567	95	24	as	as	SCONJ
ejpam-6567	95	25	follows	follow	VERB
ejpam-6567	95	26	:	:	PUNCT
ejpam-6567	95	27	a⋆(i	a⋆(i	NOUN
ejpam-6567	95	28	)	)	PUNCT
ejpam-6567	96	1	=	=	PUNCT
ejpam-6567	96	2	{	{	PUNCT
ejpam-6567	96	3	x	x	PUNCT
ejpam-6567	96	4	∈	∈	PROPN
ejpam-6567	96	5	x	x	X
ejpam-6567	96	6	:	:	PUNCT
ejpam-6567	96	7	u	u	X
ejpam-6567	96	8	∩a	∩a	PROPN
ejpam-6567	96	9	̸∈	̸∈	PROPN
ejpam-6567	96	10	i	i	PRON
ejpam-6567	96	11	for	for	ADP
ejpam-6567	96	12	every	every	DET
ejpam-6567	96	13	open	open	ADJ
ejpam-6567	96	14	neighbourhood	neighbourhood	NOUN
ejpam-6567	96	15	u	u	NOUN
ejpam-6567	96	16	of	of	ADP
ejpam-6567	96	17	x	x	NOUN
ejpam-6567	96	18	}	}	PUNCT
ejpam-6567	96	19	.	.	PUNCT
ejpam-6567	97	1	in	in	ADP
ejpam-6567	97	2	case	case	NOUN
ejpam-6567	97	3	there	there	PRON
ejpam-6567	97	4	is	be	VERB
ejpam-6567	97	5	no	no	DET
ejpam-6567	97	6	chance	chance	NOUN
ejpam-6567	97	7	for	for	ADP
ejpam-6567	97	8	confusion	confusion	NOUN
ejpam-6567	97	9	,	,	PUNCT
ejpam-6567	97	10	a⋆(i	a⋆(i	NOUN
ejpam-6567	97	11	)	)	PUNCT
ejpam-6567	97	12	is	be	AUX
ejpam-6567	97	13	simply	simply	ADV
ejpam-6567	97	14	written	write	VERB
ejpam-6567	97	15	as	as	ADP
ejpam-6567	97	16	a⋆.	a⋆.	NOUN
ejpam-6567	97	17	in	in	ADP
ejpam-6567	97	18	[	[	X
ejpam-6567	97	19	16	16	NUM
ejpam-6567	97	20	]	]	PUNCT
ejpam-6567	97	21	,	,	PUNCT
ejpam-6567	97	22	a⋆	a⋆	ADV
ejpam-6567	97	23	is	be	AUX
ejpam-6567	97	24	called	call	VERB
ejpam-6567	97	25	the	the	DET
ejpam-6567	97	26	local	local	ADJ
ejpam-6567	97	27	function	function	NOUN
ejpam-6567	97	28	of	of	ADP
ejpam-6567	97	29	a	a	PRON
ejpam-6567	97	30	with	with	ADP
ejpam-6567	97	31	respect	respect	NOUN
ejpam-6567	97	32	to	to	ADP
ejpam-6567	97	33	i	i	PRON
ejpam-6567	97	34	and	and	CCONJ
ejpam-6567	97	35	τ	τ	PROPN
ejpam-6567	97	36	and	and	CCONJ
ejpam-6567	97	37	cl⋆(a	cl⋆(a	PROPN
ejpam-6567	97	38	)	)	PUNCT
ejpam-6567	97	39	=	=	NOUN
ejpam-6567	97	40	a⋆∪a	a⋆∪a	NOUN
ejpam-6567	97	41	defines	define	VERB
ejpam-6567	97	42	a	a	DET
ejpam-6567	97	43	kuratowski	kuratowski	ADJ
ejpam-6567	97	44	closure	closure	NOUN
ejpam-6567	97	45	operator	operator	NOUN
ejpam-6567	97	46	for	for	ADP
ejpam-6567	97	47	a	a	DET
ejpam-6567	97	48	topology	topology	NOUN
ejpam-6567	97	49	τ⋆(i	τ⋆(i	NOUN
ejpam-6567	97	50	)	)	PUNCT
ejpam-6567	97	51	finer	fine	ADJ
ejpam-6567	97	52	than	than	ADP
ejpam-6567	97	53	τ	τ	PROPN
ejpam-6567	97	54	.	.	PUNCT
ejpam-6567	98	1	a	a	DET
ejpam-6567	98	2	subset	subset	NOUN
ejpam-6567	98	3	a	a	PRON
ejpam-6567	98	4	is	be	AUX
ejpam-6567	98	5	said	say	VERB
ejpam-6567	98	6	to	to	PART
ejpam-6567	98	7	be	be	AUX
ejpam-6567	98	8	⋆-closed	⋆-close	VERB
ejpam-6567	98	9	[	[	X
ejpam-6567	98	10	3	3	X
ejpam-6567	98	11	]	]	X
ejpam-6567	98	12	if	if	SCONJ
ejpam-6567	98	13	a⋆	a⋆	ADJ
ejpam-6567	98	14	⊆	⊆	NUM
ejpam-6567	98	15	a.	a.	NOUN
ejpam-6567	98	16	the	the	DET
ejpam-6567	98	17	interior	interior	NOUN
ejpam-6567	98	18	of	of	ADP
ejpam-6567	98	19	a	a	DET
ejpam-6567	98	20	subset	subset	NOUN
ejpam-6567	98	21	a	a	DET
ejpam-6567	98	22	in	in	ADP
ejpam-6567	98	23	(	(	PUNCT
ejpam-6567	98	24	x	x	X
ejpam-6567	98	25	,	,	PUNCT
ejpam-6567	98	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6567	98	27	)	)	PUNCT
ejpam-6567	98	28	)	)	PUNCT
ejpam-6567	98	29	is	be	AUX
ejpam-6567	98	30	denoted	denote	VERB
ejpam-6567	98	31	by	by	ADP
ejpam-6567	98	32	int⋆(a	int⋆(a	NOUN
ejpam-6567	98	33	)	)	PUNCT
ejpam-6567	98	34	.	.	PUNCT
ejpam-6567	99	1	a	a	DET
ejpam-6567	99	2	subset	subset	NOUN
ejpam-6567	99	3	a	a	PRON
ejpam-6567	99	4	of	of	ADP
ejpam-6567	99	5	an	an	DET
ejpam-6567	99	6	ideal	ideal	ADJ
ejpam-6567	99	7	topological	topological	ADJ
ejpam-6567	99	8	space	space	NOUN
ejpam-6567	99	9	(	(	PUNCT
ejpam-6567	99	10	x	x	X
ejpam-6567	99	11	,	,	PUNCT
ejpam-6567	99	12	τ	τ	PROPN
ejpam-6567	99	13	,	,	PUNCT
ejpam-6567	99	14	i	i	PROPN
ejpam-6567	99	15	)	)	PUNCT
ejpam-6567	99	16	is	be	AUX
ejpam-6567	99	17	said	say	VERB
ejpam-6567	99	18	to	to	PART
ejpam-6567	99	19	be	be	AUX
ejpam-6567	99	20	semi⋆-i	semi⋆-i	X
ejpam-6567	99	21	-open	-open	VERB
ejpam-6567	99	22	[	[	X
ejpam-6567	99	23	28	28	NUM
ejpam-6567	99	24	]	]	X
ejpam-6567	99	25	(	(	PUNCT
ejpam-6567	99	26	resp	resp	NOUN
ejpam-6567	99	27	.	.	PUNCT
ejpam-6567	100	1	semi	semi	ADJ
ejpam-6567	100	2	-	-	ADJ
ejpam-6567	100	3	i	i	PRON
ejpam-6567	100	4	-open	-open	ADJ
ejpam-6567	100	5	p.	p.	NOUN
ejpam-6567	100	6	pue	pue	NOUN
ejpam-6567	100	7	-	-	PUNCT
ejpam-6567	100	8	on	on	ADP
ejpam-6567	100	9	,	,	PUNCT
ejpam-6567	100	10	a.	a.	PROPN
ejpam-6567	100	11	sama	sama	PROPN
ejpam-6567	100	12	-	-	PUNCT
ejpam-6567	100	13	ae	ae	PROPN
ejpam-6567	100	14	,	,	PUNCT
ejpam-6567	100	15	c.	c.	PROPN
ejpam-6567	100	16	boonpok	boonpok	PROPN
ejpam-6567	100	17	/	/	SYM
ejpam-6567	100	18	eur	eur	PROPN
ejpam-6567	100	19	.	.	PUNCT
ejpam-6567	101	1	j.	j.	PROPN
ejpam-6567	101	2	pure	pure	PROPN
ejpam-6567	101	3	appl	appl	PROPN
ejpam-6567	101	4	.	.	PROPN
ejpam-6567	101	5	math	math	PROPN
ejpam-6567	101	6	,	,	PUNCT
ejpam-6567	101	7	18	18	NUM
ejpam-6567	101	8	(	(	PUNCT
ejpam-6567	101	9	3	3	NUM
ejpam-6567	101	10	)	)	PUNCT
ejpam-6567	101	11	(	(	PUNCT
ejpam-6567	101	12	2025	2025	NUM
ejpam-6567	101	13	)	)	PUNCT
ejpam-6567	101	14	,	,	PUNCT
ejpam-6567	101	15	6567	6567	NUM
ejpam-6567	101	16	4	4	NUM
ejpam-6567	101	17	of	of	ADP
ejpam-6567	101	18	13	13	NUM
ejpam-6567	102	1	[	[	SYM
ejpam-6567	102	2	19	19	NUM
ejpam-6567	102	3	]	]	SYM
ejpam-6567	102	4	)	)	PUNCT
ejpam-6567	102	5	if	if	SCONJ
ejpam-6567	102	6	a	a	DET
ejpam-6567	102	7	⊆	⊆	NUM
ejpam-6567	102	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6567	102	9	)	)	PUNCT
ejpam-6567	102	10	)	)	PUNCT
ejpam-6567	102	11	(	(	PUNCT
ejpam-6567	102	12	resp	resp	NOUN
ejpam-6567	102	13	.	.	PUNCT
ejpam-6567	103	1	a	a	DET
ejpam-6567	103	2	⊆	⊆	NUM
ejpam-6567	103	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6567	103	4	)	)	PUNCT
ejpam-6567	103	5	)	)	PUNCT
ejpam-6567	103	6	)	)	PUNCT
ejpam-6567	103	7	.	.	PUNCT
ejpam-6567	104	1	the	the	DET
ejpam-6567	104	2	complement	complement	NOUN
ejpam-6567	104	3	of	of	ADP
ejpam-6567	104	4	a	a	DET
ejpam-6567	104	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6567	104	6	-open	-open	ADJ
ejpam-6567	104	7	(	(	PUNCT
ejpam-6567	104	8	resp	resp	NOUN
ejpam-6567	104	9	.	.	PUNCT
ejpam-6567	105	1	semi	semi	ADJ
ejpam-6567	105	2	-	-	VERB
ejpam-6567	105	3	i	i	PRON
ejpam-6567	105	4	-open	-open	NOUN
ejpam-6567	105	5	)	)	PUNCT
ejpam-6567	106	1	set	set	NOUN
ejpam-6567	106	2	is	be	AUX
ejpam-6567	106	3	said	say	VERB
ejpam-6567	106	4	to	to	PART
ejpam-6567	106	5	be	be	AUX
ejpam-6567	106	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6567	106	7	-closed	-close	VERB
ejpam-6567	106	8	[	[	X
ejpam-6567	106	9	28	28	NUM
ejpam-6567	106	10	]	]	X
ejpam-6567	106	11	(	(	PUNCT
ejpam-6567	106	12	resp	resp	NOUN
ejpam-6567	106	13	.	.	PUNCT
ejpam-6567	107	1	semi	semi	ADJ
ejpam-6567	107	2	-	-	VERB
ejpam-6567	107	3	i	i	PRON
ejpam-6567	107	4	-closed	-close	VERB
ejpam-6567	107	5	[	[	X
ejpam-6567	107	6	19	19	NUM
ejpam-6567	107	7	]	]	PUNCT
ejpam-6567	107	8	)	)	PUNCT
ejpam-6567	107	9	.	.	PUNCT
ejpam-6567	108	1	by	by	ADP
ejpam-6567	108	2	a	a	DET
ejpam-6567	108	3	multifunction	multifunction	NOUN
ejpam-6567	108	4	f	f	NOUN
ejpam-6567	108	5	:	:	PUNCT
ejpam-6567	108	6	x	x	X
ejpam-6567	108	7	→	→	SYM
ejpam-6567	108	8	y	y	PROPN
ejpam-6567	108	9	,	,	PUNCT
ejpam-6567	108	10	we	we	PRON
ejpam-6567	108	11	mean	mean	VERB
ejpam-6567	108	12	a	a	DET
ejpam-6567	108	13	point	point	NOUN
ejpam-6567	108	14	-	-	PUNCT
ejpam-6567	108	15	to	to	ADP
ejpam-6567	108	16	-	-	PUNCT
ejpam-6567	108	17	set	set	VERB
ejpam-6567	108	18	correspondence	correspondence	NOUN
ejpam-6567	108	19	from	from	ADP
ejpam-6567	108	20	x	x	PUNCT
ejpam-6567	108	21	into	into	ADP
ejpam-6567	108	22	y	y	PROPN
ejpam-6567	108	23	,	,	PUNCT
ejpam-6567	108	24	and	and	CCONJ
ejpam-6567	108	25	we	we	PRON
ejpam-6567	108	26	always	always	ADV
ejpam-6567	108	27	assume	assume	VERB
ejpam-6567	108	28	that	that	SCONJ
ejpam-6567	108	29	f	f	PROPN
ejpam-6567	108	30	(	(	PUNCT
ejpam-6567	108	31	x	x	X
ejpam-6567	108	32	)	)	PUNCT
ejpam-6567	108	33	̸=	̸=	NOUN
ejpam-6567	108	34	∅	∅	NOUN
ejpam-6567	108	35	for	for	ADP
ejpam-6567	108	36	all	all	PRON
ejpam-6567	108	37	x	x	SYM
ejpam-6567	108	38	∈	∈	ADJ
ejpam-6567	108	39	x.	x.	NOUN
ejpam-6567	108	40	for	for	ADP
ejpam-6567	108	41	a	a	DET
ejpam-6567	108	42	multifunction	multifunction	NOUN
ejpam-6567	108	43	f	f	NOUN
ejpam-6567	108	44	:	:	PUNCT
ejpam-6567	108	45	x	x	X
ejpam-6567	108	46	→	→	SYM
ejpam-6567	108	47	y	y	PROPN
ejpam-6567	108	48	,	,	PUNCT
ejpam-6567	108	49	we	we	PRON
ejpam-6567	108	50	shall	shall	AUX
ejpam-6567	108	51	denote	denote	VERB
ejpam-6567	108	52	the	the	DET
ejpam-6567	108	53	upper	upper	ADJ
ejpam-6567	108	54	and	and	CCONJ
ejpam-6567	108	55	lower	low	ADJ
ejpam-6567	108	56	inverse	inverse	NOUN
ejpam-6567	108	57	of	of	ADP
ejpam-6567	108	58	a	a	DET
ejpam-6567	108	59	set	set	NOUN
ejpam-6567	108	60	b	b	PROPN
ejpam-6567	108	61	of	of	ADP
ejpam-6567	108	62	y	y	PROPN
ejpam-6567	108	63	by	by	ADP
ejpam-6567	108	64	f+(b	f+(b	NOUN
ejpam-6567	108	65	)	)	PUNCT
ejpam-6567	108	66	and	and	CCONJ
ejpam-6567	108	67	f−(b	f−(b	NOUN
ejpam-6567	108	68	)	)	PUNCT
ejpam-6567	108	69	,	,	PUNCT
ejpam-6567	108	70	respectively	respectively	ADV
ejpam-6567	108	71	,	,	PUNCT
ejpam-6567	108	72	that	that	ADV
ejpam-6567	108	73	is	is	ADV
ejpam-6567	108	74	,	,	PUNCT
ejpam-6567	108	75	f+(b	f+(b	NOUN
ejpam-6567	108	76	)	)	PUNCT
ejpam-6567	108	77	=	=	PRON
ejpam-6567	109	1	{	{	PUNCT
ejpam-6567	109	2	x	x	PUNCT
ejpam-6567	109	3	∈	∈	PROPN
ejpam-6567	109	4	x	x	INTJ
ejpam-6567	110	1	|	|	NOUN
ejpam-6567	110	2	f	f	X
ejpam-6567	110	3	(	(	PUNCT
ejpam-6567	110	4	x	x	NOUN
ejpam-6567	110	5	)	)	PUNCT
ejpam-6567	110	6	⊆	⊆	NUM
ejpam-6567	110	7	b	b	NOUN
ejpam-6567	110	8	}	}	PUNCT
ejpam-6567	110	9	and	and	CCONJ
ejpam-6567	110	10	f−(b	f−(b	PROPN
ejpam-6567	110	11	)	)	PUNCT
ejpam-6567	110	12	=	=	PRON
ejpam-6567	111	1	{	{	PUNCT
ejpam-6567	111	2	x	x	PUNCT
ejpam-6567	111	3	∈	∈	PROPN
ejpam-6567	111	4	x	x	INTJ
ejpam-6567	112	1	|	|	NOUN
ejpam-6567	112	2	f	f	X
ejpam-6567	112	3	(	(	PUNCT
ejpam-6567	112	4	x	x	NOUN
ejpam-6567	112	5	)	)	PUNCT
ejpam-6567	112	6	∩	∩	NOUN
ejpam-6567	112	7	b	b	PROPN
ejpam-6567	112	8	̸=	̸=	PROPN
ejpam-6567	112	9	∅	∅	NOUN
ejpam-6567	112	10	}	}	PUNCT
ejpam-6567	112	11	.	.	PUNCT
ejpam-6567	113	1	in	in	ADP
ejpam-6567	113	2	particular	particular	ADJ
ejpam-6567	113	3	,	,	PUNCT
ejpam-6567	113	4	f−(y	f−(y	NOUN
ejpam-6567	113	5	)	)	PUNCT
ejpam-6567	113	6	=	=	SYM
ejpam-6567	114	1	{	{	PUNCT
ejpam-6567	114	2	x	x	PUNCT
ejpam-6567	114	3	∈	∈	PROPN
ejpam-6567	114	4	x	x	INTJ
ejpam-6567	115	1	|	|	ADV
ejpam-6567	115	2	y	y	PROPN
ejpam-6567	115	3	∈	∈	PROPN
ejpam-6567	115	4	f	f	X
ejpam-6567	115	5	(	(	PUNCT
ejpam-6567	115	6	x	x	NOUN
ejpam-6567	115	7	)	)	PUNCT
ejpam-6567	115	8	}	}	PUNCT
ejpam-6567	115	9	for	for	ADP
ejpam-6567	115	10	each	each	DET
ejpam-6567	115	11	point	point	NOUN
ejpam-6567	115	12	y	y	PROPN
ejpam-6567	115	13	∈	∈	PROPN
ejpam-6567	115	14	y	y	PROPN
ejpam-6567	115	15	.	.	PUNCT
ejpam-6567	116	1	for	for	ADP
ejpam-6567	116	2	each	each	DET
ejpam-6567	116	3	a	a	DET
ejpam-6567	116	4	⊆	⊆	NUM
ejpam-6567	116	5	x	x	SYM
ejpam-6567	116	6	,	,	PUNCT
ejpam-6567	116	7	f	f	PROPN
ejpam-6567	116	8	(	(	PUNCT
ejpam-6567	116	9	a	a	NOUN
ejpam-6567	116	10	)	)	PUNCT
ejpam-6567	116	11	=	=	SYM
ejpam-6567	116	12	∪x∈af	∪x∈af	NOUN
ejpam-6567	116	13	(	(	PUNCT
ejpam-6567	116	14	x	x	NOUN
ejpam-6567	116	15	)	)	PUNCT
ejpam-6567	116	16	.	.	PUNCT
ejpam-6567	117	1	3	3	X
ejpam-6567	117	2	.	.	X
ejpam-6567	117	3	upper	upper	ADJ
ejpam-6567	117	4	and	and	CCONJ
ejpam-6567	117	5	lower	low	ADJ
ejpam-6567	117	6	weakly	weakly	ADJ
ejpam-6567	117	7	τ	τ	PROPN
ejpam-6567	117	8	⋆(σ1	⋆(σ1	NOUN
ejpam-6567	117	9	,	,	PUNCT
ejpam-6567	117	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	117	11	multifunctions	multifunction	NOUN
ejpam-6567	117	12	in	in	ADP
ejpam-6567	117	13	this	this	DET
ejpam-6567	117	14	section	section	NOUN
ejpam-6567	117	15	,	,	PUNCT
ejpam-6567	117	16	we	we	PRON
ejpam-6567	117	17	introduce	introduce	VERB
ejpam-6567	117	18	the	the	DET
ejpam-6567	117	19	concepts	concept	NOUN
ejpam-6567	117	20	of	of	ADP
ejpam-6567	117	21	upper	upper	ADJ
ejpam-6567	117	22	weakly	weakly	ADJ
ejpam-6567	117	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	117	24	,	,	PUNCT
ejpam-6567	117	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	117	26	multifunctions	multifunction	NOUN
ejpam-6567	117	27	and	and	CCONJ
ejpam-6567	117	28	lower	low	ADJ
ejpam-6567	117	29	weakly	weakly	ADJ
ejpam-6567	117	30	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	117	31	,	,	PUNCT
ejpam-6567	117	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	117	33	multifunctions	multifunction	NOUN
ejpam-6567	117	34	.	.	PUNCT
ejpam-6567	118	1	furthermore	furthermore	ADV
ejpam-6567	118	2	,	,	PUNCT
ejpam-6567	118	3	several	several	ADJ
ejpam-6567	118	4	characterizations	characterization	NOUN
ejpam-6567	118	5	of	of	ADP
ejpam-6567	118	6	upper	upper	ADJ
ejpam-6567	118	7	weakly	weakly	ADJ
ejpam-6567	118	8	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	118	9	,	,	PUNCT
ejpam-6567	118	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	118	11	multifunctions	multifunction	NOUN
ejpam-6567	118	12	and	and	CCONJ
ejpam-6567	118	13	lower	low	ADJ
ejpam-6567	118	14	weakly	weakly	ADJ
ejpam-6567	118	15	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	118	16	,	,	PUNCT
ejpam-6567	118	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	118	18	multifunctions	multifunction	NOUN
ejpam-6567	118	19	are	be	AUX
ejpam-6567	118	20	discussed	discuss	VERB
ejpam-6567	118	21	.	.	PUNCT
ejpam-6567	119	1	definition	definition	NOUN
ejpam-6567	119	2	1	1	NUM
ejpam-6567	119	3	.	.	PUNCT
ejpam-6567	120	1	a	a	DET
ejpam-6567	120	2	multifunction	multifunction	NOUN
ejpam-6567	120	3	f	f	NOUN
ejpam-6567	120	4	:	:	PUNCT
ejpam-6567	120	5	(	(	PUNCT
ejpam-6567	120	6	x	x	X
ejpam-6567	120	7	,	,	PUNCT
ejpam-6567	120	8	τ	τ	PROPN
ejpam-6567	120	9	,	,	PUNCT
ejpam-6567	120	10	i	i	NOUN
ejpam-6567	120	11	)	)	PUNCT
ejpam-6567	120	12	→	→	PUNCT
ejpam-6567	120	13	(	(	PUNCT
ejpam-6567	120	14	y	y	PROPN
ejpam-6567	120	15	,	,	PUNCT
ejpam-6567	120	16	σ1	σ1	PROPN
ejpam-6567	120	17	,	,	PUNCT
ejpam-6567	120	18	σ2	σ2	PROPN
ejpam-6567	120	19	)	)	PUNCT
ejpam-6567	120	20	is	be	AUX
ejpam-6567	120	21	said	say	VERB
ejpam-6567	120	22	to	to	PART
ejpam-6567	120	23	be	be	AUX
ejpam-6567	120	24	upper	upper	ADJ
ejpam-6567	120	25	weakly	weakly	ADJ
ejpam-6567	120	26	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	120	27	,	,	PUNCT
ejpam-6567	120	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	120	29	at	at	ADP
ejpam-6567	120	30	a	a	DET
ejpam-6567	120	31	point	point	NOUN
ejpam-6567	120	32	x	x	SYM
ejpam-6567	120	33	∈	∈	NOUN
ejpam-6567	120	34	x	x	PUNCT
ejpam-6567	120	35	if	if	SCONJ
ejpam-6567	120	36	for	for	ADP
ejpam-6567	120	37	each	each	DET
ejpam-6567	120	38	σ1σ2	σ1σ2	VERB
ejpam-6567	120	39	-	-	ADJ
ejpam-6567	120	40	open	open	ADJ
ejpam-6567	120	41	set	set	NOUN
ejpam-6567	120	42	v	v	NOUN
ejpam-6567	120	43	of	of	ADP
ejpam-6567	120	44	y	y	PRON
ejpam-6567	120	45	such	such	ADJ
ejpam-6567	120	46	that	that	SCONJ
ejpam-6567	120	47	f	f	PROPN
ejpam-6567	120	48	(	(	PUNCT
ejpam-6567	120	49	x	x	X
ejpam-6567	120	50	)	)	PUNCT
ejpam-6567	120	51	⊆	⊆	NUM
ejpam-6567	120	52	v	v	NOUN
ejpam-6567	120	53	,	,	PUNCT
ejpam-6567	120	54	there	there	PRON
ejpam-6567	120	55	exists	exist	VERB
ejpam-6567	120	56	a	a	DET
ejpam-6567	120	57	⋆-open	⋆-open	ADJ
ejpam-6567	120	58	set	set	NOUN
ejpam-6567	120	59	u	u	NOUN
ejpam-6567	120	60	of	of	ADP
ejpam-6567	120	61	x	x	PUNCT
ejpam-6567	120	62	containing	contain	VERB
ejpam-6567	120	63	x	x	PUNCT
ejpam-6567	120	64	such	such	ADJ
ejpam-6567	120	65	that	that	SCONJ
ejpam-6567	120	66	f	f	PROPN
ejpam-6567	120	67	(	(	PUNCT
ejpam-6567	120	68	u	u	NOUN
ejpam-6567	120	69	)	)	PUNCT
ejpam-6567	120	70	⊆	⊆	NUM
ejpam-6567	120	71	σ1σ2	σ1σ2	NOUN
ejpam-6567	120	72	-	-	NUM
ejpam-6567	120	73	cl(v	cl(v	NOUN
ejpam-6567	120	74	)	)	PUNCT
ejpam-6567	120	75	.	.	PUNCT
ejpam-6567	121	1	a	a	DET
ejpam-6567	121	2	multifunction	multifunction	NOUN
ejpam-6567	121	3	f	f	NOUN
ejpam-6567	121	4	:	:	PUNCT
ejpam-6567	121	5	(	(	PUNCT
ejpam-6567	121	6	x	x	X
ejpam-6567	121	7	,	,	PUNCT
ejpam-6567	121	8	τ	τ	PROPN
ejpam-6567	121	9	,	,	PUNCT
ejpam-6567	121	10	i	i	NOUN
ejpam-6567	121	11	)	)	PUNCT
ejpam-6567	121	12	→	→	PUNCT
ejpam-6567	121	13	(	(	PUNCT
ejpam-6567	121	14	y	y	PROPN
ejpam-6567	121	15	,	,	PUNCT
ejpam-6567	121	16	σ1	σ1	PROPN
ejpam-6567	121	17	,	,	PUNCT
ejpam-6567	121	18	σ2	σ2	PROPN
ejpam-6567	121	19	)	)	PUNCT
ejpam-6567	121	20	is	be	AUX
ejpam-6567	121	21	said	say	VERB
ejpam-6567	121	22	to	to	PART
ejpam-6567	121	23	be	be	AUX
ejpam-6567	121	24	upper	upper	ADJ
ejpam-6567	121	25	weakly	weakly	ADJ
ejpam-6567	121	26	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	121	27	,	,	PUNCT
ejpam-6567	121	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	121	29	if	if	SCONJ
ejpam-6567	121	30	f	f	PROPN
ejpam-6567	121	31	is	be	AUX
ejpam-6567	121	32	upper	upper	ADJ
ejpam-6567	121	33	weakly	weakly	ADJ
ejpam-6567	121	34	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	121	35	,	,	PUNCT
ejpam-6567	121	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	121	37	at	at	ADP
ejpam-6567	121	38	each	each	DET
ejpam-6567	121	39	point	point	NOUN
ejpam-6567	121	40	x	x	PUNCT
ejpam-6567	121	41	of	of	ADP
ejpam-6567	121	42	x.	x.	NOUN
ejpam-6567	121	43	definition	definition	NOUN
ejpam-6567	121	44	2	2	NUM
ejpam-6567	121	45	.	.	PUNCT
ejpam-6567	122	1	[	[	X
ejpam-6567	122	2	29	29	NUM
ejpam-6567	122	3	]	]	PUNCT
ejpam-6567	122	4	a	a	DET
ejpam-6567	122	5	multifunction	multifunction	NOUN
ejpam-6567	122	6	f	f	NOUN
ejpam-6567	122	7	:	:	PUNCT
ejpam-6567	122	8	(	(	PUNCT
ejpam-6567	122	9	x	x	X
ejpam-6567	122	10	,	,	PUNCT
ejpam-6567	122	11	τ	τ	PROPN
ejpam-6567	122	12	,	,	PUNCT
ejpam-6567	122	13	i	i	NOUN
ejpam-6567	122	14	)	)	PUNCT
ejpam-6567	122	15	→	→	PUNCT
ejpam-6567	122	16	(	(	PUNCT
ejpam-6567	122	17	y	y	PROPN
ejpam-6567	122	18	,	,	PUNCT
ejpam-6567	122	19	σ1	σ1	PROPN
ejpam-6567	122	20	,	,	PUNCT
ejpam-6567	122	21	σ2	σ2	PROPN
ejpam-6567	122	22	)	)	PUNCT
ejpam-6567	122	23	is	be	AUX
ejpam-6567	122	24	said	say	VERB
ejpam-6567	122	25	to	to	PART
ejpam-6567	122	26	be	be	AUX
ejpam-6567	122	27	:	:	PUNCT
ejpam-6567	122	28	(	(	PUNCT
ejpam-6567	122	29	i	i	NOUN
ejpam-6567	122	30	)	)	PUNCT
ejpam-6567	122	31	upper	upper	ADJ
ejpam-6567	122	32	almost	almost	ADV
ejpam-6567	122	33	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	122	34	,	,	PUNCT
ejpam-6567	122	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	122	36	if	if	SCONJ
ejpam-6567	122	37	for	for	ADP
ejpam-6567	122	38	each	each	DET
ejpam-6567	122	39	point	point	NOUN
ejpam-6567	122	40	x	x	X
ejpam-6567	122	41	∈	∈	NOUN
ejpam-6567	122	42	x	x	X
ejpam-6567	122	43	and	and	CCONJ
ejpam-6567	122	44	each	each	DET
ejpam-6567	122	45	σ1σ2	σ1σ2	VERB
ejpam-6567	122	46	-	-	ADJ
ejpam-6567	122	47	open	open	ADJ
ejpam-6567	122	48	set	set	NOUN
ejpam-6567	122	49	v	v	NOUN
ejpam-6567	122	50	of	of	ADP
ejpam-6567	122	51	y	y	PRON
ejpam-6567	122	52	such	such	ADJ
ejpam-6567	122	53	that	that	SCONJ
ejpam-6567	122	54	f	f	PROPN
ejpam-6567	122	55	(	(	PUNCT
ejpam-6567	122	56	x	x	X
ejpam-6567	122	57	)	)	PUNCT
ejpam-6567	122	58	⊆	⊆	NUM
ejpam-6567	122	59	v	v	NOUN
ejpam-6567	122	60	,	,	PUNCT
ejpam-6567	122	61	there	there	PRON
ejpam-6567	122	62	exists	exist	VERB
ejpam-6567	122	63	a	a	DET
ejpam-6567	122	64	⋆-open	⋆-open	ADJ
ejpam-6567	122	65	set	set	NOUN
ejpam-6567	122	66	u	u	NOUN
ejpam-6567	122	67	of	of	ADP
ejpam-6567	122	68	x	x	PUNCT
ejpam-6567	122	69	containing	contain	VERB
ejpam-6567	122	70	x	x	PUNCT
ejpam-6567	122	71	such	such	ADJ
ejpam-6567	122	72	that	that	SCONJ
ejpam-6567	122	73	f	f	PROPN
ejpam-6567	122	74	(	(	PUNCT
ejpam-6567	122	75	u	u	NOUN
ejpam-6567	122	76	)	)	PUNCT
ejpam-6567	122	77	⊆	⊆	NUM
ejpam-6567	122	78	σ1σ2	σ1σ2	X
ejpam-6567	122	79	-	-	PUNCT
ejpam-6567	122	80	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	122	81	-	-	PUNCT
ejpam-6567	122	82	cl(v	cl(v	NOUN
ejpam-6567	122	83	)	)	PUNCT
ejpam-6567	122	84	)	)	PUNCT
ejpam-6567	122	85	;	;	PUNCT
ejpam-6567	122	86	(	(	PUNCT
ejpam-6567	122	87	ii	ii	NOUN
ejpam-6567	122	88	)	)	PUNCT
ejpam-6567	122	89	lower	low	ADJ
ejpam-6567	122	90	almost	almost	ADV
ejpam-6567	122	91	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	122	92	,	,	PUNCT
ejpam-6567	122	93	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	122	94	if	if	SCONJ
ejpam-6567	122	95	for	for	ADP
ejpam-6567	122	96	each	each	DET
ejpam-6567	122	97	point	point	NOUN
ejpam-6567	122	98	x	x	X
ejpam-6567	122	99	∈	∈	NOUN
ejpam-6567	122	100	x	x	X
ejpam-6567	122	101	and	and	CCONJ
ejpam-6567	122	102	each	each	DET
ejpam-6567	122	103	σ1σ2	σ1σ2	VERB
ejpam-6567	122	104	-	-	ADJ
ejpam-6567	122	105	open	open	ADJ
ejpam-6567	122	106	set	set	NOUN
ejpam-6567	122	107	v	v	NOUN
ejpam-6567	122	108	of	of	ADP
ejpam-6567	122	109	y	y	PRON
ejpam-6567	122	110	such	such	ADJ
ejpam-6567	122	111	that	that	PRON
ejpam-6567	122	112	v	v	ADP
ejpam-6567	122	113	∩	∩	ADJ
ejpam-6567	122	114	f	f	X
ejpam-6567	122	115	(	(	PUNCT
ejpam-6567	122	116	x	x	X
ejpam-6567	122	117	)	)	PUNCT
ejpam-6567	122	118	̸=	̸=	NOUN
ejpam-6567	122	119	∅	∅	NOUN
ejpam-6567	122	120	,	,	PUNCT
ejpam-6567	122	121	there	there	PRON
ejpam-6567	122	122	exists	exist	VERB
ejpam-6567	122	123	a	a	DET
ejpam-6567	122	124	⋆-open	⋆-open	ADJ
ejpam-6567	122	125	set	set	NOUN
ejpam-6567	122	126	u	u	NOUN
ejpam-6567	122	127	of	of	ADP
ejpam-6567	122	128	x	x	PUNCT
ejpam-6567	122	129	containing	contain	VERB
ejpam-6567	122	130	x	x	PUNCT
ejpam-6567	122	131	such	such	ADJ
ejpam-6567	122	132	that	that	SCONJ
ejpam-6567	122	133	σ1σ2	σ1σ2	ADV
ejpam-6567	122	134	-	-	PUNCT
ejpam-6567	122	135	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	122	136	-	-	PUNCT
ejpam-6567	122	137	cl(v	cl(v	NOUN
ejpam-6567	122	138	)	)	PUNCT
ejpam-6567	122	139	)	)	PUNCT
ejpam-6567	122	140	∩	∩	PROPN
ejpam-6567	122	141	f	f	X
ejpam-6567	122	142	(	(	PUNCT
ejpam-6567	122	143	z	z	NOUN
ejpam-6567	122	144	)	)	PUNCT
ejpam-6567	122	145	̸=	̸=	NOUN
ejpam-6567	122	146	∅	∅	NOUN
ejpam-6567	122	147	for	for	ADP
ejpam-6567	122	148	every	every	DET
ejpam-6567	122	149	z	z	NOUN
ejpam-6567	122	150	∈	∈	PROPN
ejpam-6567	122	151	u	u	PROPN
ejpam-6567	122	152	.	.	PUNCT
ejpam-6567	122	153	remark	remark	PROPN
ejpam-6567	122	154	1	1	NUM
ejpam-6567	122	155	.	.	PUNCT
ejpam-6567	122	156	for	for	ADP
ejpam-6567	122	157	a	a	DET
ejpam-6567	122	158	multifunction	multifunction	NOUN
ejpam-6567	122	159	f	f	NOUN
ejpam-6567	122	160	:	:	PUNCT
ejpam-6567	122	161	(	(	PUNCT
ejpam-6567	122	162	x	x	X
ejpam-6567	122	163	,	,	PUNCT
ejpam-6567	122	164	τ	τ	PROPN
ejpam-6567	122	165	,	,	PUNCT
ejpam-6567	122	166	i	i	NOUN
ejpam-6567	122	167	)	)	PUNCT
ejpam-6567	122	168	→	→	PUNCT
ejpam-6567	122	169	(	(	PUNCT
ejpam-6567	122	170	y	y	PROPN
ejpam-6567	122	171	,	,	PUNCT
ejpam-6567	122	172	σ1	σ1	PROPN
ejpam-6567	122	173	,	,	PUNCT
ejpam-6567	122	174	σ2	σ2	NOUN
ejpam-6567	122	175	)	)	PUNCT
ejpam-6567	122	176	,	,	PUNCT
ejpam-6567	122	177	the	the	DET
ejpam-6567	122	178	following	follow	VERB
ejpam-6567	122	179	implication	implication	NOUN
ejpam-6567	122	180	holds	hold	VERB
ejpam-6567	122	181	:	:	PUNCT
ejpam-6567	122	182	upper	upper	ADJ
ejpam-6567	122	183	almost	almost	ADV
ejpam-6567	122	184	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	122	185	,	,	PUNCT
ejpam-6567	122	186	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6567	122	187	⇒	⇒	VERB
ejpam-6567	122	188	upper	upper	ADJ
ejpam-6567	122	189	weakly	weakly	ADJ
ejpam-6567	122	190	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	122	191	,	,	PUNCT
ejpam-6567	122	192	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6567	122	193	.	.	PUNCT
ejpam-6567	123	1	the	the	DET
ejpam-6567	123	2	converse	converse	NOUN
ejpam-6567	123	3	of	of	ADP
ejpam-6567	123	4	the	the	DET
ejpam-6567	123	5	implication	implication	NOUN
ejpam-6567	123	6	is	be	AUX
ejpam-6567	123	7	not	not	PART
ejpam-6567	123	8	true	true	ADJ
ejpam-6567	123	9	in	in	ADP
ejpam-6567	123	10	general	general	ADJ
ejpam-6567	123	11	.	.	PUNCT
ejpam-6567	124	1	we	we	PRON
ejpam-6567	124	2	give	give	VERB
ejpam-6567	124	3	an	an	DET
ejpam-6567	124	4	example	example	NOUN
ejpam-6567	124	5	for	for	ADP
ejpam-6567	124	6	the	the	DET
ejpam-6567	124	7	implication	implication	NOUN
ejpam-6567	124	8	as	as	SCONJ
ejpam-6567	124	9	follows	follow	VERB
ejpam-6567	124	10	.	.	PUNCT
ejpam-6567	124	11	example	example	NOUN
ejpam-6567	125	1	1	1	NUM
ejpam-6567	125	2	.	.	PUNCT
ejpam-6567	125	3	let	let	VERB
ejpam-6567	125	4	x	x	PUNCT
ejpam-6567	125	5	=	=	PRON
ejpam-6567	125	6	{	{	PUNCT
ejpam-6567	125	7	1	1	NUM
ejpam-6567	125	8	,	,	PUNCT
ejpam-6567	125	9	2	2	NUM
ejpam-6567	125	10	,	,	PUNCT
ejpam-6567	125	11	3	3	NUM
ejpam-6567	125	12	}	}	PUNCT
ejpam-6567	125	13	with	with	ADP
ejpam-6567	125	14	a	a	DET
ejpam-6567	125	15	topology	topology	NOUN
ejpam-6567	125	16	τ	τ	X
ejpam-6567	125	17	=	=	SYM
ejpam-6567	125	18	{	{	PUNCT
ejpam-6567	125	19	∅	∅	NOUN
ejpam-6567	125	20	,	,	PUNCT
ejpam-6567	125	21	{	{	PUNCT
ejpam-6567	125	22	2	2	NUM
ejpam-6567	125	23	}	}	PUNCT
ejpam-6567	125	24	,	,	PUNCT
ejpam-6567	125	25	{	{	PUNCT
ejpam-6567	125	26	1	1	NUM
ejpam-6567	125	27	,	,	PUNCT
ejpam-6567	125	28	3	3	NUM
ejpam-6567	125	29	}	}	PUNCT
ejpam-6567	125	30	,	,	PUNCT
ejpam-6567	125	31	x	x	NOUN
ejpam-6567	125	32	}	}	PUNCT
ejpam-6567	125	33	and	and	CCONJ
ejpam-6567	125	34	an	an	DET
ejpam-6567	125	35	ideal	ideal	NOUN
ejpam-6567	125	36	i	i	X
ejpam-6567	125	37	=	=	SYM
ejpam-6567	125	38	{	{	PUNCT
ejpam-6567	125	39	∅	∅	NOUN
ejpam-6567	125	40	,	,	PUNCT
ejpam-6567	125	41	{	{	PUNCT
ejpam-6567	125	42	2	2	NUM
ejpam-6567	125	43	}	}	PUNCT
ejpam-6567	125	44	}	}	PUNCT
ejpam-6567	125	45	.	.	PUNCT
ejpam-6567	126	1	let	let	VERB
ejpam-6567	126	2	y	y	PROPN
ejpam-6567	126	3	=	=	PUNCT
ejpam-6567	126	4	{	{	PUNCT
ejpam-6567	126	5	a	a	PRON
ejpam-6567	126	6	,	,	PUNCT
ejpam-6567	126	7	b	b	NOUN
ejpam-6567	126	8	,	,	PUNCT
ejpam-6567	126	9	c	c	NOUN
ejpam-6567	126	10	}	}	PUNCT
ejpam-6567	126	11	with	with	ADP
ejpam-6567	126	12	topologies	topology	NOUN
ejpam-6567	126	13	σ1	σ1	NOUN
ejpam-6567	126	14	=	=	SYM
ejpam-6567	126	15	{	{	PUNCT
ejpam-6567	126	16	∅	∅	NOUN
ejpam-6567	126	17	,	,	PUNCT
ejpam-6567	126	18	{	{	PUNCT
ejpam-6567	126	19	a	a	X
ejpam-6567	126	20	}	}	PUNCT
ejpam-6567	126	21	,	,	PUNCT
ejpam-6567	126	22	{	{	PUNCT
ejpam-6567	126	23	a	a	DET
ejpam-6567	126	24	,	,	PUNCT
ejpam-6567	126	25	b	b	NOUN
ejpam-6567	126	26	}	}	PUNCT
ejpam-6567	126	27	,	,	PUNCT
ejpam-6567	126	28	y	y	PROPN
ejpam-6567	126	29	}	}	PUNCT
ejpam-6567	126	30	and	and	CCONJ
ejpam-6567	126	31	σ2	σ2	PROPN
ejpam-6567	126	32	=	=	SYM
ejpam-6567	126	33	{	{	PUNCT
ejpam-6567	126	34	∅	∅	NOUN
ejpam-6567	126	35	,	,	PUNCT
ejpam-6567	126	36	{	{	PUNCT
ejpam-6567	126	37	a	a	X
ejpam-6567	126	38	}	}	PUNCT
ejpam-6567	126	39	,	,	PUNCT
ejpam-6567	126	40	{	{	PUNCT
ejpam-6567	126	41	b	b	NOUN
ejpam-6567	126	42	}	}	PUNCT
ejpam-6567	126	43	,	,	PUNCT
ejpam-6567	126	44	{	{	PUNCT
ejpam-6567	126	45	a	a	DET
ejpam-6567	126	46	,	,	PUNCT
ejpam-6567	126	47	b	b	NOUN
ejpam-6567	126	48	}	}	PUNCT
ejpam-6567	126	49	,	,	PUNCT
ejpam-6567	126	50	y	y	PROPN
ejpam-6567	126	51	}	}	PUNCT
ejpam-6567	126	52	.	.	PUNCT
ejpam-6567	127	1	a	a	DET
ejpam-6567	127	2	multifunction	multifunction	NOUN
ejpam-6567	127	3	f	f	NOUN
ejpam-6567	127	4	:	:	PUNCT
ejpam-6567	127	5	(	(	PUNCT
ejpam-6567	127	6	x	x	X
ejpam-6567	127	7	,	,	PUNCT
ejpam-6567	127	8	τ	τ	PROPN
ejpam-6567	127	9	,	,	PUNCT
ejpam-6567	127	10	i	i	NOUN
ejpam-6567	127	11	)	)	PUNCT
ejpam-6567	127	12	→	→	PUNCT
ejpam-6567	127	13	(	(	PUNCT
ejpam-6567	127	14	y	y	PROPN
ejpam-6567	127	15	,	,	PUNCT
ejpam-6567	127	16	σ1	σ1	PROPN
ejpam-6567	127	17	,	,	PUNCT
ejpam-6567	127	18	σ2	σ2	PROPN
ejpam-6567	127	19	)	)	PUNCT
ejpam-6567	127	20	is	be	AUX
ejpam-6567	127	21	defined	define	VERB
ejpam-6567	127	22	as	as	SCONJ
ejpam-6567	127	23	follows	follow	VERB
ejpam-6567	127	24	:	:	PUNCT
ejpam-6567	127	25	f	f	X
ejpam-6567	127	26	(	(	PUNCT
ejpam-6567	127	27	1	1	X
ejpam-6567	127	28	)	)	PUNCT
ejpam-6567	128	1	=	=	PRON
ejpam-6567	128	2	{	{	PUNCT
ejpam-6567	128	3	a	a	NOUN
ejpam-6567	128	4	}	}	PUNCT
ejpam-6567	128	5	,	,	PUNCT
ejpam-6567	128	6	f	f	PROPN
ejpam-6567	128	7	(	(	PUNCT
ejpam-6567	128	8	2	2	NUM
ejpam-6567	128	9	)	)	PUNCT
ejpam-6567	128	10	=	=	PRON
ejpam-6567	128	11	{	{	PUNCT
ejpam-6567	128	12	b	b	NOUN
ejpam-6567	128	13	}	}	PUNCT
ejpam-6567	128	14	and	and	CCONJ
ejpam-6567	128	15	f	f	PROPN
ejpam-6567	128	16	(	(	PUNCT
ejpam-6567	128	17	3	3	X
ejpam-6567	128	18	)	)	PUNCT
ejpam-6567	128	19	=	=	PRON
ejpam-6567	128	20	{	{	PUNCT
ejpam-6567	128	21	a	a	X
ejpam-6567	128	22	,	,	PUNCT
ejpam-6567	128	23	c	c	NOUN
ejpam-6567	128	24	}	}	PUNCT
ejpam-6567	128	25	.	.	PUNCT
ejpam-6567	129	1	then	then	ADV
ejpam-6567	129	2	,	,	PUNCT
ejpam-6567	129	3	f	f	PROPN
ejpam-6567	129	4	is	be	AUX
ejpam-6567	129	5	upper	upper	ADJ
ejpam-6567	129	6	weakly	weakly	ADJ
ejpam-6567	129	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	129	8	,	,	PUNCT
ejpam-6567	129	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	129	10	but	but	CCONJ
ejpam-6567	129	11	f	f	PROPN
ejpam-6567	129	12	is	be	AUX
ejpam-6567	129	13	not	not	PART
ejpam-6567	129	14	upper	upper	ADJ
ejpam-6567	129	15	almost	almost	ADV
ejpam-6567	129	16	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	129	17	,	,	PUNCT
ejpam-6567	129	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	129	19	.	.	PUNCT
ejpam-6567	130	1	p.	p.	NOUN
ejpam-6567	130	2	pue	pue	NOUN
ejpam-6567	130	3	-	-	PUNCT
ejpam-6567	130	4	on	on	ADP
ejpam-6567	130	5	,	,	PUNCT
ejpam-6567	130	6	a.	a.	PROPN
ejpam-6567	130	7	sama	sama	PROPN
ejpam-6567	130	8	-	-	PUNCT
ejpam-6567	130	9	ae	ae	PROPN
ejpam-6567	130	10	,	,	PUNCT
ejpam-6567	130	11	c.	c.	PROPN
ejpam-6567	130	12	boonpok	boonpok	PROPN
ejpam-6567	130	13	/	/	SYM
ejpam-6567	130	14	eur	eur	PROPN
ejpam-6567	130	15	.	.	PUNCT
ejpam-6567	131	1	j.	j.	PROPN
ejpam-6567	131	2	pure	pure	PROPN
ejpam-6567	131	3	appl	appl	PROPN
ejpam-6567	131	4	.	.	PROPN
ejpam-6567	131	5	math	math	PROPN
ejpam-6567	131	6	,	,	PUNCT
ejpam-6567	131	7	18	18	NUM
ejpam-6567	131	8	(	(	PUNCT
ejpam-6567	131	9	3	3	NUM
ejpam-6567	131	10	)	)	PUNCT
ejpam-6567	131	11	(	(	PUNCT
ejpam-6567	131	12	2025	2025	NUM
ejpam-6567	131	13	)	)	PUNCT
ejpam-6567	131	14	,	,	PUNCT
ejpam-6567	131	15	6567	6567	NUM
ejpam-6567	131	16	5	5	NUM
ejpam-6567	131	17	of	of	ADP
ejpam-6567	131	18	13	13	NUM
ejpam-6567	131	19	theorem	theorem	NOUN
ejpam-6567	131	20	1	1	NUM
ejpam-6567	131	21	.	.	PUNCT
ejpam-6567	131	22	for	for	ADP
ejpam-6567	131	23	a	a	DET
ejpam-6567	131	24	multifunction	multifunction	NOUN
ejpam-6567	131	25	f	f	NOUN
ejpam-6567	131	26	:	:	PUNCT
ejpam-6567	131	27	(	(	PUNCT
ejpam-6567	131	28	x	x	X
ejpam-6567	131	29	,	,	PUNCT
ejpam-6567	131	30	τ	τ	PROPN
ejpam-6567	131	31	,	,	PUNCT
ejpam-6567	131	32	i	i	NOUN
ejpam-6567	131	33	)	)	PUNCT
ejpam-6567	131	34	→	→	PUNCT
ejpam-6567	131	35	(	(	PUNCT
ejpam-6567	131	36	y	y	PROPN
ejpam-6567	131	37	,	,	PUNCT
ejpam-6567	131	38	σ1	σ1	PROPN
ejpam-6567	131	39	,	,	PUNCT
ejpam-6567	131	40	σ2	σ2	NOUN
ejpam-6567	131	41	)	)	PUNCT
ejpam-6567	131	42	,	,	PUNCT
ejpam-6567	131	43	the	the	DET
ejpam-6567	131	44	following	follow	VERB
ejpam-6567	131	45	properties	property	NOUN
ejpam-6567	131	46	are	be	AUX
ejpam-6567	131	47	equivalent	equivalent	ADJ
ejpam-6567	131	48	:	:	PUNCT
ejpam-6567	131	49	(	(	PUNCT
ejpam-6567	131	50	1	1	X
ejpam-6567	131	51	)	)	PUNCT
ejpam-6567	131	52	f	f	PROPN
ejpam-6567	131	53	is	be	AUX
ejpam-6567	131	54	upper	upper	ADJ
ejpam-6567	131	55	weakly	weakly	ADJ
ejpam-6567	131	56	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	131	57	,	,	PUNCT
ejpam-6567	131	58	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	131	59	;	;	PUNCT
ejpam-6567	131	60	(	(	PUNCT
ejpam-6567	131	61	2	2	NUM
ejpam-6567	131	62	)	)	PUNCT
ejpam-6567	131	63	f+(v	f+(v	NOUN
ejpam-6567	131	64	)	)	PUNCT
ejpam-6567	132	1	⊆	⊆	X
ejpam-6567	132	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	132	3	-	-	PUNCT
ejpam-6567	132	4	cl(v	cl(v	NOUN
ejpam-6567	132	5	)	)	PUNCT
ejpam-6567	132	6	)	)	PUNCT
ejpam-6567	132	7	)	)	PUNCT
ejpam-6567	132	8	for	for	ADP
ejpam-6567	132	9	every	every	DET
ejpam-6567	132	10	σ1σ2	σ1σ2	NOUN
ejpam-6567	132	11	-	-	ADJ
ejpam-6567	132	12	open	open	ADJ
ejpam-6567	132	13	set	set	NOUN
ejpam-6567	132	14	v	v	NOUN
ejpam-6567	132	15	of	of	ADP
ejpam-6567	132	16	y	y	PROPN
ejpam-6567	132	17	;	;	PUNCT
ejpam-6567	132	18	(	(	PUNCT
ejpam-6567	132	19	3	3	X
ejpam-6567	132	20	)	)	PUNCT
ejpam-6567	132	21	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	PROPN
ejpam-6567	132	22	-	-	PUNCT
ejpam-6567	132	23	int(k	int(k	NOUN
ejpam-6567	132	24	)	)	PUNCT
ejpam-6567	132	25	)	)	PUNCT
ejpam-6567	132	26	)	)	PUNCT
ejpam-6567	133	1	⊆	⊆	X
ejpam-6567	133	2	f−(k	f−(k	PROPN
ejpam-6567	133	3	)	)	PUNCT
ejpam-6567	133	4	for	for	ADP
ejpam-6567	133	5	every	every	DET
ejpam-6567	133	6	σ1σ2	σ1σ2	NUM
ejpam-6567	133	7	-	-	PUNCT
ejpam-6567	133	8	closed	closed	ADJ
ejpam-6567	133	9	set	set	NOUN
ejpam-6567	133	10	k	k	PROPN
ejpam-6567	133	11	of	of	ADP
ejpam-6567	133	12	y	y	PROPN
ejpam-6567	133	13	;	;	PUNCT
ejpam-6567	133	14	(	(	PUNCT
ejpam-6567	133	15	4	4	X
ejpam-6567	133	16	)	)	PUNCT
ejpam-6567	133	17	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	133	18	-	-	PUNCT
ejpam-6567	133	19	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	133	20	-	-	PUNCT
ejpam-6567	133	21	cl(b	cl(b	NOUN
ejpam-6567	133	22	)	)	PUNCT
ejpam-6567	133	23	)	)	PUNCT
ejpam-6567	133	24	)	)	PUNCT
ejpam-6567	133	25	)	)	PUNCT
ejpam-6567	133	26	⊆	⊆	X
ejpam-6567	133	27	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6567	133	28	-	-	PUNCT
ejpam-6567	133	29	cl(b	cl(b	NOUN
ejpam-6567	133	30	)	)	PUNCT
ejpam-6567	133	31	)	)	PUNCT
ejpam-6567	133	32	for	for	ADP
ejpam-6567	133	33	every	every	DET
ejpam-6567	133	34	subset	subset	NOUN
ejpam-6567	133	35	b	b	PROPN
ejpam-6567	133	36	of	of	ADP
ejpam-6567	133	37	y	y	PROPN
ejpam-6567	133	38	;	;	PUNCT
ejpam-6567	133	39	(	(	PUNCT
ejpam-6567	133	40	5	5	X
ejpam-6567	133	41	)	)	PUNCT
ejpam-6567	133	42	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	133	43	-	-	PUNCT
ejpam-6567	133	44	int(b	int(b	NOUN
ejpam-6567	133	45	)	)	PUNCT
ejpam-6567	133	46	)	)	PUNCT
ejpam-6567	133	47	⊆	⊆	X
ejpam-6567	133	48	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	133	49	-	-	PUNCT
ejpam-6567	133	50	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6567	133	51	-	-	PUNCT
ejpam-6567	133	52	int(b	int(b	NOUN
ejpam-6567	133	53	)	)	PUNCT
ejpam-6567	133	54	)	)	PUNCT
ejpam-6567	133	55	)	)	PUNCT
ejpam-6567	133	56	)	)	PUNCT
ejpam-6567	133	57	for	for	ADP
ejpam-6567	133	58	every	every	DET
ejpam-6567	133	59	subset	subset	NOUN
ejpam-6567	133	60	b	b	PROPN
ejpam-6567	133	61	of	of	ADP
ejpam-6567	133	62	y	y	PROPN
ejpam-6567	133	63	;	;	PUNCT
ejpam-6567	133	64	(	(	PUNCT
ejpam-6567	133	65	6	6	X
ejpam-6567	133	66	)	)	PUNCT
ejpam-6567	133	67	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	133	68	-	-	PUNCT
ejpam-6567	133	69	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	133	70	-	-	PUNCT
ejpam-6567	133	71	cl(v	cl(v	NOUN
ejpam-6567	133	72	)	)	PUNCT
ejpam-6567	133	73	)	)	PUNCT
ejpam-6567	133	74	)	)	PUNCT
ejpam-6567	133	75	)	)	PUNCT
ejpam-6567	133	76	⊆	⊆	X
ejpam-6567	133	77	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	133	78	-	-	PUNCT
ejpam-6567	133	79	cl(v	cl(v	NOUN
ejpam-6567	133	80	)	)	PUNCT
ejpam-6567	133	81	)	)	PUNCT
ejpam-6567	133	82	for	for	ADP
ejpam-6567	133	83	every	every	DET
ejpam-6567	133	84	σ1σ2	σ1σ2	NOUN
ejpam-6567	133	85	-	-	ADJ
ejpam-6567	133	86	open	open	ADJ
ejpam-6567	133	87	set	set	NOUN
ejpam-6567	133	88	v	v	NOUN
ejpam-6567	133	89	of	of	ADP
ejpam-6567	133	90	y	y	PROPN
ejpam-6567	133	91	;	;	PUNCT
ejpam-6567	133	92	(	(	PUNCT
ejpam-6567	133	93	7	7	X
ejpam-6567	133	94	)	)	SYM
ejpam-6567	133	95	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6567	133	96	)	)	PUNCT
ejpam-6567	133	97	)	)	PUNCT
ejpam-6567	134	1	⊆	⊆	X
ejpam-6567	134	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	134	3	-	-	PUNCT
ejpam-6567	134	4	cl(v	cl(v	NOUN
ejpam-6567	134	5	)	)	PUNCT
ejpam-6567	134	6	)	)	PUNCT
ejpam-6567	134	7	for	for	ADP
ejpam-6567	134	8	every	every	DET
ejpam-6567	134	9	σ1σ2	σ1σ2	NOUN
ejpam-6567	134	10	-	-	ADJ
ejpam-6567	134	11	open	open	ADJ
ejpam-6567	134	12	set	set	NOUN
ejpam-6567	134	13	v	v	NOUN
ejpam-6567	134	14	of	of	ADP
ejpam-6567	134	15	y	y	PROPN
ejpam-6567	134	16	;	;	PUNCT
ejpam-6567	134	17	(	(	PUNCT
ejpam-6567	134	18	8)	8)	NUM
ejpam-6567	134	19	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	134	20	-	-	PUNCT
ejpam-6567	134	21	int(k	int(k	NOUN
ejpam-6567	134	22	)	)	PUNCT
ejpam-6567	134	23	)	)	PUNCT
ejpam-6567	134	24	)	)	PUNCT
ejpam-6567	135	1	⊆	⊆	X
ejpam-6567	135	2	f−(k	f−(k	PROPN
ejpam-6567	135	3	)	)	PUNCT
ejpam-6567	135	4	for	for	ADP
ejpam-6567	135	5	every	every	DET
ejpam-6567	135	6	(	(	PUNCT
ejpam-6567	135	7	σ1	σ1	PROPN
ejpam-6567	135	8	,	,	PUNCT
ejpam-6567	135	9	σ2)r	σ2)r	NOUN
ejpam-6567	135	10	-	-	PUNCT
ejpam-6567	135	11	closed	close	VERB
ejpam-6567	135	12	set	set	ADJ
ejpam-6567	135	13	k	k	PROPN
ejpam-6567	135	14	of	of	ADP
ejpam-6567	135	15	y	y	PROPN
ejpam-6567	135	16	.	.	PUNCT
ejpam-6567	136	1	proof	proof	NOUN
ejpam-6567	136	2	.	.	PUNCT
ejpam-6567	137	1	(	(	PUNCT
ejpam-6567	137	2	1	1	X
ejpam-6567	137	3	)	)	PUNCT
ejpam-6567	137	4	⇒	⇒	NOUN
ejpam-6567	137	5	(	(	PUNCT
ejpam-6567	137	6	2	2	NUM
ejpam-6567	137	7	):	):	PUNCT
ejpam-6567	137	8	let	let	VERB
ejpam-6567	137	9	v	v	PART
ejpam-6567	137	10	be	be	AUX
ejpam-6567	137	11	any	any	DET
ejpam-6567	137	12	σ1σ2	σ1σ2	NOUN
ejpam-6567	137	13	-	-	ADJ
ejpam-6567	137	14	open	open	ADJ
ejpam-6567	137	15	set	set	NOUN
ejpam-6567	137	16	of	of	ADP
ejpam-6567	137	17	y	y	PRON
ejpam-6567	137	18	such	such	ADJ
ejpam-6567	137	19	that	that	SCONJ
ejpam-6567	137	20	x	x	SYM
ejpam-6567	137	21	∈	∈	PROPN
ejpam-6567	137	22	f+(v	f+(v	NOUN
ejpam-6567	137	23	)	)	PUNCT
ejpam-6567	137	24	.	.	PUNCT
ejpam-6567	138	1	then	then	ADV
ejpam-6567	138	2	,	,	PUNCT
ejpam-6567	138	3	we	we	PRON
ejpam-6567	138	4	have	have	VERB
ejpam-6567	138	5	f	f	PROPN
ejpam-6567	138	6	(	(	PUNCT
ejpam-6567	138	7	x	x	NOUN
ejpam-6567	138	8	)	)	PUNCT
ejpam-6567	138	9	⊆	⊆	NUM
ejpam-6567	138	10	v	v	NOUN
ejpam-6567	138	11	and	and	CCONJ
ejpam-6567	138	12	by	by	ADP
ejpam-6567	138	13	(	(	PUNCT
ejpam-6567	138	14	1	1	NUM
ejpam-6567	138	15	)	)	PUNCT
ejpam-6567	138	16	,	,	PUNCT
ejpam-6567	138	17	there	there	PRON
ejpam-6567	138	18	exists	exist	VERB
ejpam-6567	138	19	a	a	DET
ejpam-6567	138	20	⋆-open	⋆-open	ADJ
ejpam-6567	138	21	set	set	NOUN
ejpam-6567	138	22	u	u	NOUN
ejpam-6567	138	23	of	of	ADP
ejpam-6567	138	24	x	x	PUNCT
ejpam-6567	138	25	containing	contain	VERB
ejpam-6567	138	26	x	x	PUNCT
ejpam-6567	138	27	such	such	ADJ
ejpam-6567	138	28	that	that	SCONJ
ejpam-6567	138	29	f	f	PROPN
ejpam-6567	138	30	(	(	PUNCT
ejpam-6567	138	31	u	u	NOUN
ejpam-6567	138	32	)	)	PUNCT
ejpam-6567	138	33	⊆	⊆	NUM
ejpam-6567	138	34	σ1σ2	σ1σ2	NOUN
ejpam-6567	138	35	-	-	NUM
ejpam-6567	138	36	cl(v	cl(v	NOUN
ejpam-6567	138	37	)	)	PUNCT
ejpam-6567	138	38	.	.	PUNCT
ejpam-6567	139	1	thus	thus	ADV
ejpam-6567	139	2	,	,	PUNCT
ejpam-6567	139	3	u	u	NOUN
ejpam-6567	139	4	⊆	⊆	NUM
ejpam-6567	139	5	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	139	6	-	-	PUNCT
ejpam-6567	139	7	cl(v	cl(v	NOUN
ejpam-6567	139	8	)	)	PUNCT
ejpam-6567	139	9	)	)	PUNCT
ejpam-6567	139	10	.	.	PUNCT
ejpam-6567	140	1	since	since	SCONJ
ejpam-6567	140	2	u	u	NOUN
ejpam-6567	140	3	is	be	AUX
ejpam-6567	140	4	⋆-open	⋆-open	ADJ
ejpam-6567	140	5	,	,	PUNCT
ejpam-6567	140	6	we	we	PRON
ejpam-6567	140	7	have	have	VERB
ejpam-6567	140	8	x	x	X
ejpam-6567	140	9	∈	∈	VERB
ejpam-6567	140	10	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	140	11	-	-	PUNCT
ejpam-6567	140	12	cl(v	cl(v	NOUN
ejpam-6567	140	13	)	)	PUNCT
ejpam-6567	140	14	)	)	PUNCT
ejpam-6567	140	15	)	)	PUNCT
ejpam-6567	141	1	and	and	CCONJ
ejpam-6567	141	2	so	so	ADV
ejpam-6567	141	3	f+(v	f+(v	PROPN
ejpam-6567	141	4	)	)	PUNCT
ejpam-6567	142	1	⊆	⊆	X
ejpam-6567	142	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	142	3	-	-	PUNCT
ejpam-6567	142	4	cl(v	cl(v	NOUN
ejpam-6567	142	5	)	)	PUNCT
ejpam-6567	142	6	)	)	PUNCT
ejpam-6567	142	7	)	)	PUNCT
ejpam-6567	142	8	.	.	PUNCT
ejpam-6567	143	1	(	(	PUNCT
ejpam-6567	143	2	2	2	X
ejpam-6567	143	3	)	)	PUNCT
ejpam-6567	143	4	⇒	⇒	NOUN
ejpam-6567	143	5	(	(	PUNCT
ejpam-6567	143	6	3	3	NUM
ejpam-6567	143	7	):	):	PUNCT
ejpam-6567	143	8	let	let	VERB
ejpam-6567	143	9	k	k	PRON
ejpam-6567	143	10	be	be	AUX
ejpam-6567	143	11	any	any	DET
ejpam-6567	143	12	σ1σ2	σ1σ2	NUM
ejpam-6567	143	13	-	-	PUNCT
ejpam-6567	143	14	closed	closed	ADJ
ejpam-6567	143	15	set	set	NOUN
ejpam-6567	143	16	of	of	ADP
ejpam-6567	143	17	y	y	PROPN
ejpam-6567	143	18	.	.	PUNCT
ejpam-6567	144	1	then	then	ADV
ejpam-6567	144	2	,	,	PUNCT
ejpam-6567	144	3	y	y	PROPN
ejpam-6567	144	4	−k	−k	PROPN
ejpam-6567	144	5	is	be	AUX
ejpam-6567	144	6	σ1σ2	σ1σ2	NOUN
ejpam-6567	144	7	-	-	ADJ
ejpam-6567	144	8	open	open	ADJ
ejpam-6567	144	9	in	in	ADP
ejpam-6567	144	10	y	y	PROPN
ejpam-6567	144	11	.	.	PUNCT
ejpam-6567	145	1	by	by	ADP
ejpam-6567	145	2	(	(	PUNCT
ejpam-6567	145	3	2	2	NUM
ejpam-6567	145	4	)	)	PUNCT
ejpam-6567	145	5	,	,	PUNCT
ejpam-6567	145	6	x	x	PUNCT
ejpam-6567	145	7	−	−	DET
ejpam-6567	145	8	f−(k	f−(k	PROPN
ejpam-6567	145	9	)	)	PUNCT
ejpam-6567	145	10	=	=	PUNCT
ejpam-6567	146	1	f+(y	f+(y	PROPN
ejpam-6567	146	2	−	−	PROPN
ejpam-6567	146	3	k	k	NOUN
ejpam-6567	146	4	)	)	PUNCT
ejpam-6567	146	5	⊆	⊆	NUM
ejpam-6567	146	6	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	146	7	-	-	PUNCT
ejpam-6567	146	8	cl(y	cl(y	NOUN
ejpam-6567	146	9	−	−	PROPN
ejpam-6567	146	10	k	k	NOUN
ejpam-6567	146	11	)	)	PUNCT
ejpam-6567	146	12	)	)	PUNCT
ejpam-6567	146	13	)	)	PUNCT
ejpam-6567	147	1	=	=	PUNCT
ejpam-6567	147	2	x	x	X
ejpam-6567	147	3	−	−	NOUN
ejpam-6567	147	4	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	PROPN
ejpam-6567	147	5	-	-	PUNCT
ejpam-6567	147	6	int(k	int(k	NOUN
ejpam-6567	147	7	)	)	PUNCT
ejpam-6567	147	8	)	)	PUNCT
ejpam-6567	147	9	)	)	PUNCT
ejpam-6567	147	10	.	.	PUNCT
ejpam-6567	148	1	thus	thus	ADV
ejpam-6567	148	2	,	,	PUNCT
ejpam-6567	148	3	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	148	4	-	-	PUNCT
ejpam-6567	148	5	int(k	int(k	NOUN
ejpam-6567	148	6	)	)	PUNCT
ejpam-6567	148	7	)	)	PUNCT
ejpam-6567	148	8	)	)	PUNCT
ejpam-6567	149	1	⊆	⊆	NUM
ejpam-6567	149	2	f−(k	f−(k	PROPN
ejpam-6567	149	3	)	)	PUNCT
ejpam-6567	149	4	.	.	PUNCT
ejpam-6567	150	1	(	(	PUNCT
ejpam-6567	150	2	3	3	X
ejpam-6567	150	3	)	)	PUNCT
ejpam-6567	150	4	⇒	⇒	NOUN
ejpam-6567	150	5	(	(	PUNCT
ejpam-6567	150	6	4	4	NUM
ejpam-6567	150	7	):	):	PUNCT
ejpam-6567	150	8	let	let	VERB
ejpam-6567	150	9	b	b	X
ejpam-6567	150	10	be	be	AUX
ejpam-6567	150	11	any	any	DET
ejpam-6567	150	12	subset	subset	NOUN
ejpam-6567	150	13	of	of	ADP
ejpam-6567	150	14	y	y	PROPN
ejpam-6567	150	15	.	.	PUNCT
ejpam-6567	151	1	then	then	ADV
ejpam-6567	151	2	,	,	PUNCT
ejpam-6567	151	3	σ1σ2	σ1σ2	NOUN
ejpam-6567	151	4	-	-	NOUN
ejpam-6567	151	5	cl(b	cl(b	NOUN
ejpam-6567	151	6	)	)	PUNCT
ejpam-6567	151	7	is	be	AUX
ejpam-6567	151	8	a	a	DET
ejpam-6567	151	9	σ1σ2	σ1σ2	NUM
ejpam-6567	151	10	-	-	PUNCT
ejpam-6567	151	11	closed	closed	ADJ
ejpam-6567	151	12	set	set	NOUN
ejpam-6567	151	13	of	of	ADP
ejpam-6567	151	14	y	y	PROPN
ejpam-6567	151	15	and	and	CCONJ
ejpam-6567	151	16	by	by	ADP
ejpam-6567	151	17	(	(	PUNCT
ejpam-6567	151	18	3	3	NUM
ejpam-6567	151	19	)	)	PUNCT
ejpam-6567	151	20	,	,	PUNCT
ejpam-6567	151	21	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6567	151	22	-	-	PUNCT
ejpam-6567	151	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	151	24	-	-	PUNCT
ejpam-6567	151	25	cl(b	cl(b	NOUN
ejpam-6567	151	26	)	)	PUNCT
ejpam-6567	151	27	)	)	PUNCT
ejpam-6567	151	28	)	)	PUNCT
ejpam-6567	151	29	)	)	PUNCT
ejpam-6567	152	1	⊆	⊆	X
ejpam-6567	152	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6567	152	3	-	-	PUNCT
ejpam-6567	152	4	cl(b	cl(b	NOUN
ejpam-6567	152	5	)	)	PUNCT
ejpam-6567	152	6	)	)	PUNCT
ejpam-6567	152	7	.	.	PUNCT
ejpam-6567	153	1	(	(	PUNCT
ejpam-6567	153	2	4	4	X
ejpam-6567	153	3	)	)	PUNCT
ejpam-6567	153	4	⇒	⇒	NOUN
ejpam-6567	153	5	(	(	PUNCT
ejpam-6567	153	6	5	5	NUM
ejpam-6567	153	7	):	):	PUNCT
ejpam-6567	153	8	let	let	VERB
ejpam-6567	153	9	b	b	X
ejpam-6567	153	10	be	be	AUX
ejpam-6567	153	11	any	any	DET
ejpam-6567	153	12	subset	subset	NOUN
ejpam-6567	153	13	of	of	ADP
ejpam-6567	153	14	y	y	PROPN
ejpam-6567	153	15	.	.	PUNCT
ejpam-6567	154	1	thus	thus	ADV
ejpam-6567	154	2	by	by	ADP
ejpam-6567	154	3	(	(	PUNCT
ejpam-6567	154	4	4	4	NUM
ejpam-6567	154	5	)	)	PUNCT
ejpam-6567	154	6	,	,	PUNCT
ejpam-6567	154	7	we	we	PRON
ejpam-6567	154	8	have	have	VERB
ejpam-6567	154	9	x	x	PRON
ejpam-6567	154	10	−	−	PUNCT
ejpam-6567	154	11	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	154	12	-	-	PUNCT
ejpam-6567	154	13	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	154	14	-	-	PUNCT
ejpam-6567	154	15	int(b	int(b	NOUN
ejpam-6567	154	16	)	)	PUNCT
ejpam-6567	154	17	)	)	PUNCT
ejpam-6567	154	18	)	)	PUNCT
ejpam-6567	154	19	)	)	PUNCT
ejpam-6567	155	1	=	=	PUNCT
ejpam-6567	155	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6567	155	3	−	−	NOUN
ejpam-6567	155	4	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-6567	155	5	-	-	PUNCT
ejpam-6567	155	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	155	7	-	-	PUNCT
ejpam-6567	155	8	int(b	int(b	NOUN
ejpam-6567	155	9	)	)	PUNCT
ejpam-6567	155	10	)	)	PUNCT
ejpam-6567	155	11	)	)	PUNCT
ejpam-6567	155	12	)	)	PUNCT
ejpam-6567	156	1	=	=	SYM
ejpam-6567	156	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6567	156	3	-	-	PUNCT
ejpam-6567	156	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	156	5	-	-	PUNCT
ejpam-6567	156	6	cl(y	cl(y	NOUN
ejpam-6567	156	7	−b	−b	NOUN
ejpam-6567	156	8	)	)	PUNCT
ejpam-6567	156	9	)	)	PUNCT
ejpam-6567	156	10	)	)	PUNCT
ejpam-6567	156	11	)	)	PUNCT
ejpam-6567	157	1	⊆	⊆	X
ejpam-6567	157	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6567	157	3	-	-	PUNCT
ejpam-6567	157	4	cl(y	cl(y	NOUN
ejpam-6567	157	5	−b	−b	NOUN
ejpam-6567	157	6	)	)	PUNCT
ejpam-6567	157	7	)	)	PUNCT
ejpam-6567	158	1	=	=	PUNCT
ejpam-6567	158	2	x	x	X
ejpam-6567	159	1	−	−	ADP
ejpam-6567	159	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	159	3	-	-	PUNCT
ejpam-6567	159	4	int(b	int(b	NOUN
ejpam-6567	159	5	)	)	PUNCT
ejpam-6567	159	6	)	)	PUNCT
ejpam-6567	159	7	and	and	CCONJ
ejpam-6567	159	8	hence	hence	ADV
ejpam-6567	159	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-6567	159	10	-	-	PUNCT
ejpam-6567	159	11	int(b	int(b	NOUN
ejpam-6567	159	12	)	)	PUNCT
ejpam-6567	159	13	)	)	PUNCT
ejpam-6567	159	14	⊆	⊆	X
ejpam-6567	159	15	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	159	16	-	-	PUNCT
ejpam-6567	159	17	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6567	159	18	-	-	PUNCT
ejpam-6567	159	19	int(b	int(b	NOUN
ejpam-6567	159	20	)	)	PUNCT
ejpam-6567	159	21	)	)	PUNCT
ejpam-6567	159	22	)	)	PUNCT
ejpam-6567	159	23	)	)	PUNCT
ejpam-6567	159	24	.	.	PUNCT
ejpam-6567	160	1	(	(	PUNCT
ejpam-6567	160	2	5	5	X
ejpam-6567	160	3	)	)	PUNCT
ejpam-6567	160	4	⇒	⇒	NOUN
ejpam-6567	160	5	(	(	PUNCT
ejpam-6567	160	6	1	1	NUM
ejpam-6567	160	7	):	):	PUNCT
ejpam-6567	160	8	let	let	VERB
ejpam-6567	160	9	x	x	PUNCT
ejpam-6567	160	10	∈	∈	PROPN
ejpam-6567	160	11	x	x	X
ejpam-6567	160	12	and	and	CCONJ
ejpam-6567	160	13	v	v	X
ejpam-6567	160	14	be	be	AUX
ejpam-6567	160	15	any	any	DET
ejpam-6567	160	16	σ1σ2	σ1σ2	NOUN
ejpam-6567	160	17	-	-	ADJ
ejpam-6567	160	18	open	open	ADJ
ejpam-6567	160	19	set	set	NOUN
ejpam-6567	160	20	of	of	ADP
ejpam-6567	160	21	y	y	PRON
ejpam-6567	160	22	such	such	ADJ
ejpam-6567	160	23	that	that	SCONJ
ejpam-6567	160	24	f	f	PROPN
ejpam-6567	160	25	(	(	PUNCT
ejpam-6567	160	26	x	x	X
ejpam-6567	160	27	)	)	PUNCT
ejpam-6567	160	28	⊆	⊆	NUM
ejpam-6567	160	29	v	v	NOUN
ejpam-6567	160	30	.	.	PUNCT
ejpam-6567	161	1	by	by	ADP
ejpam-6567	161	2	(	(	PUNCT
ejpam-6567	161	3	5	5	NUM
ejpam-6567	161	4	)	)	PUNCT
ejpam-6567	161	5	,	,	PUNCT
ejpam-6567	161	6	x	x	PUNCT
ejpam-6567	161	7	∈	∈	PROPN
ejpam-6567	161	8	f+(v	f+(v	NOUN
ejpam-6567	161	9	)	)	PUNCT
ejpam-6567	161	10	⊆	⊆	X
ejpam-6567	161	11	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	161	12	-	-	PUNCT
ejpam-6567	161	13	cl(v	cl(v	NOUN
ejpam-6567	161	14	)	)	PUNCT
ejpam-6567	161	15	)	)	PUNCT
ejpam-6567	161	16	)	)	PUNCT
ejpam-6567	162	1	and	and	CCONJ
ejpam-6567	162	2	there	there	PRON
ejpam-6567	162	3	exists	exist	VERB
ejpam-6567	162	4	a	a	DET
ejpam-6567	162	5	⋆-open	⋆-open	ADJ
ejpam-6567	162	6	set	set	NOUN
ejpam-6567	162	7	u	u	NOUN
ejpam-6567	162	8	of	of	ADP
ejpam-6567	162	9	x	x	PUNCT
ejpam-6567	162	10	containing	contain	VERB
ejpam-6567	162	11	x	x	PUNCT
ejpam-6567	162	12	such	such	ADJ
ejpam-6567	162	13	that	that	SCONJ
ejpam-6567	162	14	u	u	NOUN
ejpam-6567	162	15	⊆	⊆	NUM
ejpam-6567	162	16	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	162	17	-	-	PUNCT
ejpam-6567	162	18	cl(v	cl(v	NOUN
ejpam-6567	162	19	)	)	PUNCT
ejpam-6567	162	20	)	)	PUNCT
ejpam-6567	162	21	.	.	PUNCT
ejpam-6567	163	1	thus	thus	ADV
ejpam-6567	163	2	,	,	PUNCT
ejpam-6567	163	3	f	f	PROPN
ejpam-6567	163	4	(	(	PUNCT
ejpam-6567	163	5	u	u	NOUN
ejpam-6567	163	6	)	)	PUNCT
ejpam-6567	163	7	⊆	⊆	NUM
ejpam-6567	163	8	σ1σ2	σ1σ2	NOUN
ejpam-6567	163	9	-	-	NUM
ejpam-6567	163	10	cl(v	cl(v	NOUN
ejpam-6567	163	11	)	)	PUNCT
ejpam-6567	163	12	and	and	CCONJ
ejpam-6567	163	13	hence	hence	ADV
ejpam-6567	163	14	f	f	PROPN
ejpam-6567	163	15	is	be	AUX
ejpam-6567	163	16	upper	upper	ADJ
ejpam-6567	163	17	weakly	weakly	ADJ
ejpam-6567	163	18	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	163	19	,	,	PUNCT
ejpam-6567	163	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	163	21	.	.	PUNCT
ejpam-6567	164	1	(	(	PUNCT
ejpam-6567	164	2	4	4	X
ejpam-6567	164	3	)	)	PUNCT
ejpam-6567	164	4	⇒	⇒	NOUN
ejpam-6567	164	5	(	(	PUNCT
ejpam-6567	164	6	6	6	NUM
ejpam-6567	164	7	)	)	PUNCT
ejpam-6567	164	8	and	and	CCONJ
ejpam-6567	164	9	(	(	PUNCT
ejpam-6567	164	10	6	6	NUM
ejpam-6567	164	11	)	)	PUNCT
ejpam-6567	164	12	⇒	⇒	NOUN
ejpam-6567	164	13	(	(	PUNCT
ejpam-6567	164	14	7	7	NUM
ejpam-6567	164	15	):	):	PUNCT
ejpam-6567	164	16	the	the	DET
ejpam-6567	164	17	proofs	proof	NOUN
ejpam-6567	164	18	are	be	AUX
ejpam-6567	164	19	obvious	obvious	ADJ
ejpam-6567	164	20	.	.	PUNCT
ejpam-6567	165	1	(	(	PUNCT
ejpam-6567	165	2	7	7	X
ejpam-6567	165	3	)	)	PUNCT
ejpam-6567	165	4	⇒	⇒	NOUN
ejpam-6567	165	5	(	(	PUNCT
ejpam-6567	165	6	8)	8)	NUM
ejpam-6567	165	7	:	:	PUNCT
ejpam-6567	165	8	let	let	VERB
ejpam-6567	165	9	k	k	X
ejpam-6567	165	10	be	be	AUX
ejpam-6567	165	11	any	any	DET
ejpam-6567	165	12	(	(	PUNCT
ejpam-6567	165	13	σ1	σ1	NOUN
ejpam-6567	165	14	,	,	PUNCT
ejpam-6567	165	15	σ2)r	σ2)r	NOUN
ejpam-6567	165	16	-	-	PUNCT
ejpam-6567	165	17	closed	close	VERB
ejpam-6567	165	18	set	set	NOUN
ejpam-6567	165	19	of	of	ADP
ejpam-6567	165	20	y	y	PROPN
ejpam-6567	165	21	.	.	PUNCT
ejpam-6567	166	1	thus	thus	ADV
ejpam-6567	166	2	by	by	ADP
ejpam-6567	166	3	(	(	PUNCT
ejpam-6567	166	4	7	7	NUM
ejpam-6567	166	5	)	)	PUNCT
ejpam-6567	166	6	,	,	PUNCT
ejpam-6567	166	7	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	PROPN
ejpam-6567	166	8	-	-	PUNCT
ejpam-6567	166	9	int(k	int(k	NOUN
ejpam-6567	166	10	)	)	PUNCT
ejpam-6567	166	11	)	)	PUNCT
ejpam-6567	166	12	)	)	PUNCT
ejpam-6567	167	1	⊆	⊆	X
ejpam-6567	167	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6567	167	3	-	-	PUNCT
ejpam-6567	167	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	167	5	-	-	PUNCT
ejpam-6567	167	6	int(k	int(k	NOUN
ejpam-6567	167	7	)	)	PUNCT
ejpam-6567	167	8	)	)	PUNCT
ejpam-6567	167	9	)	)	PUNCT
ejpam-6567	168	1	=	=	SYM
ejpam-6567	168	2	f−(k	f−(k	PROPN
ejpam-6567	168	3	)	)	PUNCT
ejpam-6567	168	4	.	.	PUNCT
ejpam-6567	169	1	p.	p.	NOUN
ejpam-6567	169	2	pue	pue	NOUN
ejpam-6567	169	3	-	-	PUNCT
ejpam-6567	169	4	on	on	ADP
ejpam-6567	169	5	,	,	PUNCT
ejpam-6567	169	6	a.	a.	PROPN
ejpam-6567	169	7	sama	sama	PROPN
ejpam-6567	169	8	-	-	PUNCT
ejpam-6567	169	9	ae	ae	PROPN
ejpam-6567	169	10	,	,	PUNCT
ejpam-6567	169	11	c.	c.	PROPN
ejpam-6567	169	12	boonpok	boonpok	PROPN
ejpam-6567	169	13	/	/	SYM
ejpam-6567	169	14	eur	eur	PROPN
ejpam-6567	169	15	.	.	PUNCT
ejpam-6567	170	1	j.	j.	PROPN
ejpam-6567	170	2	pure	pure	PROPN
ejpam-6567	170	3	appl	appl	PROPN
ejpam-6567	170	4	.	.	PROPN
ejpam-6567	170	5	math	math	PROPN
ejpam-6567	170	6	,	,	PUNCT
ejpam-6567	170	7	18	18	NUM
ejpam-6567	170	8	(	(	PUNCT
ejpam-6567	170	9	3	3	NUM
ejpam-6567	170	10	)	)	PUNCT
ejpam-6567	170	11	(	(	PUNCT
ejpam-6567	170	12	2025	2025	NUM
ejpam-6567	170	13	)	)	PUNCT
ejpam-6567	170	14	,	,	PUNCT
ejpam-6567	170	15	6567	6567	NUM
ejpam-6567	170	16	6	6	NUM
ejpam-6567	170	17	of	of	ADP
ejpam-6567	170	18	13	13	NUM
ejpam-6567	170	19	(	(	PUNCT
ejpam-6567	170	20	8)	8)	NUM
ejpam-6567	170	21	⇒	⇒	NOUN
ejpam-6567	170	22	(	(	PUNCT
ejpam-6567	170	23	3	3	NUM
ejpam-6567	170	24	):	):	PUNCT
ejpam-6567	170	25	letk	letk	ADJ
ejpam-6567	170	26	be	be	AUX
ejpam-6567	170	27	any	any	DET
ejpam-6567	170	28	σ1σ2	σ1σ2	NUM
ejpam-6567	170	29	-	-	PUNCT
ejpam-6567	170	30	closed	closed	ADJ
ejpam-6567	170	31	set	set	NOUN
ejpam-6567	170	32	of	of	ADP
ejpam-6567	170	33	y	y	PROPN
ejpam-6567	170	34	.	.	PUNCT
ejpam-6567	171	1	then	then	ADV
ejpam-6567	171	2	,	,	PUNCT
ejpam-6567	171	3	σ1σ2	σ1σ2	X
ejpam-6567	171	4	-	-	PUNCT
ejpam-6567	171	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	171	6	-	-	PUNCT
ejpam-6567	171	7	int(k	int(k	NOUN
ejpam-6567	171	8	)	)	PUNCT
ejpam-6567	171	9	)	)	PUNCT
ejpam-6567	171	10	is	be	AUX
ejpam-6567	171	11	(	(	PUNCT
ejpam-6567	171	12	σ1	σ1	PROPN
ejpam-6567	171	13	,	,	PUNCT
ejpam-6567	171	14	σ2)rclosed	σ2)rclose	VERB
ejpam-6567	171	15	in	in	ADP
ejpam-6567	171	16	y	y	PROPN
ejpam-6567	171	17	and	and	CCONJ
ejpam-6567	171	18	σ1σ2	σ1σ2	NOUN
ejpam-6567	171	19	-	-	PUNCT
ejpam-6567	171	20	int(σ1σ2	int(σ1σ2	ADV
ejpam-6567	171	21	-	-	PUNCT
ejpam-6567	171	22	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6567	171	23	-	-	PUNCT
ejpam-6567	171	24	int(k	int(k	NOUN
ejpam-6567	171	25	)	)	PUNCT
ejpam-6567	171	26	)	)	PUNCT
ejpam-6567	171	27	)	)	PUNCT
ejpam-6567	172	1	=	=	PUNCT
ejpam-6567	172	2	σ1σ2	σ1σ2	X
ejpam-6567	172	3	-	-	PUNCT
ejpam-6567	172	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	172	5	-	-	PUNCT
ejpam-6567	172	6	cl(k	cl(k	NUM
ejpam-6567	172	7	)	)	PUNCT
ejpam-6567	172	8	)	)	PUNCT
ejpam-6567	173	1	=	=	PUNCT
ejpam-6567	173	2	σ1σ2	σ1σ2	X
ejpam-6567	173	3	-	-	PUNCT
ejpam-6567	173	4	int(k	int(k	NOUN
ejpam-6567	173	5	)	)	PUNCT
ejpam-6567	173	6	.	.	PUNCT
ejpam-6567	174	1	by	by	ADP
ejpam-6567	174	2	(	(	PUNCT
ejpam-6567	174	3	8)	8)	NUM
ejpam-6567	174	4	,	,	PUNCT
ejpam-6567	174	5	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	174	6	-	-	PUNCT
ejpam-6567	174	7	int(k	int(k	NOUN
ejpam-6567	174	8	)	)	PUNCT
ejpam-6567	174	9	)	)	PUNCT
ejpam-6567	174	10	)	)	PUNCT
ejpam-6567	175	1	=	=	SYM
ejpam-6567	175	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6567	175	3	-	-	PUNCT
ejpam-6567	175	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	175	5	-	-	PUNCT
ejpam-6567	175	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6567	175	7	-	-	PUNCT
ejpam-6567	175	8	int(k	int(k	NOUN
ejpam-6567	175	9	)	)	PUNCT
ejpam-6567	175	10	)	)	PUNCT
ejpam-6567	175	11	)	)	PUNCT
ejpam-6567	175	12	)	)	PUNCT
ejpam-6567	175	13	)	)	PUNCT
ejpam-6567	176	1	⊆	⊆	X
ejpam-6567	176	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6567	176	3	-	-	PUNCT
ejpam-6567	176	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	176	5	-	-	PUNCT
ejpam-6567	176	6	int(k	int(k	NOUN
ejpam-6567	176	7	)	)	PUNCT
ejpam-6567	176	8	)	)	PUNCT
ejpam-6567	176	9	)	)	PUNCT
ejpam-6567	177	1	⊆	⊆	NUM
ejpam-6567	177	2	f−(k	f−(k	PROPN
ejpam-6567	177	3	)	)	PUNCT
ejpam-6567	177	4	.	.	PUNCT
ejpam-6567	178	1	definition	definition	NOUN
ejpam-6567	178	2	3	3	NUM
ejpam-6567	178	3	.	.	PUNCT
ejpam-6567	179	1	a	a	DET
ejpam-6567	179	2	multifunction	multifunction	NOUN
ejpam-6567	179	3	f	f	NOUN
ejpam-6567	179	4	:	:	PUNCT
ejpam-6567	179	5	(	(	PUNCT
ejpam-6567	179	6	x	x	X
ejpam-6567	179	7	,	,	PUNCT
ejpam-6567	179	8	τ	τ	PROPN
ejpam-6567	179	9	,	,	PUNCT
ejpam-6567	179	10	i	i	NOUN
ejpam-6567	179	11	)	)	PUNCT
ejpam-6567	179	12	→	→	PUNCT
ejpam-6567	179	13	(	(	PUNCT
ejpam-6567	179	14	y	y	PROPN
ejpam-6567	179	15	,	,	PUNCT
ejpam-6567	179	16	σ1	σ1	PROPN
ejpam-6567	179	17	,	,	PUNCT
ejpam-6567	179	18	σ2	σ2	PROPN
ejpam-6567	179	19	)	)	PUNCT
ejpam-6567	179	20	is	be	AUX
ejpam-6567	179	21	said	say	VERB
ejpam-6567	179	22	to	to	PART
ejpam-6567	179	23	be	be	AUX
ejpam-6567	179	24	lower	lower	ADV
ejpam-6567	179	25	weakly	weakly	ADJ
ejpam-6567	179	26	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	179	27	,	,	PUNCT
ejpam-6567	179	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	179	29	at	at	ADP
ejpam-6567	179	30	a	a	DET
ejpam-6567	179	31	point	point	NOUN
ejpam-6567	179	32	x	x	SYM
ejpam-6567	179	33	∈	∈	NOUN
ejpam-6567	179	34	x	x	PUNCT
ejpam-6567	179	35	if	if	SCONJ
ejpam-6567	179	36	for	for	ADP
ejpam-6567	179	37	each	each	DET
ejpam-6567	179	38	σ1σ2	σ1σ2	VERB
ejpam-6567	179	39	-	-	ADJ
ejpam-6567	179	40	open	open	ADJ
ejpam-6567	179	41	set	set	NOUN
ejpam-6567	179	42	v	v	NOUN
ejpam-6567	179	43	of	of	ADP
ejpam-6567	179	44	y	y	PRON
ejpam-6567	179	45	such	such	ADJ
ejpam-6567	179	46	that	that	PRON
ejpam-6567	179	47	v	v	ADP
ejpam-6567	179	48	∩	∩	ADJ
ejpam-6567	179	49	f	f	X
ejpam-6567	179	50	(	(	PUNCT
ejpam-6567	179	51	x	x	X
ejpam-6567	179	52	)	)	PUNCT
ejpam-6567	179	53	̸=	̸=	NOUN
ejpam-6567	179	54	∅	∅	NOUN
ejpam-6567	179	55	,	,	PUNCT
ejpam-6567	179	56	there	there	PRON
ejpam-6567	179	57	exists	exist	VERB
ejpam-6567	179	58	a	a	DET
ejpam-6567	179	59	⋆-open	⋆-open	ADJ
ejpam-6567	179	60	set	set	NOUN
ejpam-6567	179	61	u	u	NOUN
ejpam-6567	179	62	of	of	ADP
ejpam-6567	179	63	x	x	PUNCT
ejpam-6567	179	64	containing	contain	VERB
ejpam-6567	179	65	x	x	PUNCT
ejpam-6567	179	66	such	such	ADJ
ejpam-6567	179	67	that	that	SCONJ
ejpam-6567	179	68	σ1σ2	σ1σ2	NOUN
ejpam-6567	179	69	-	-	PUNCT
ejpam-6567	179	70	cl(v	cl(v	NOUN
ejpam-6567	179	71	)	)	PUNCT
ejpam-6567	179	72	∩	∩	PROPN
ejpam-6567	179	73	f	f	X
ejpam-6567	179	74	(	(	PUNCT
ejpam-6567	179	75	z	z	NOUN
ejpam-6567	179	76	)	)	PUNCT
ejpam-6567	179	77	̸=	̸=	NOUN
ejpam-6567	179	78	∅	∅	NOUN
ejpam-6567	179	79	for	for	ADP
ejpam-6567	179	80	every	every	DET
ejpam-6567	179	81	z	z	NOUN
ejpam-6567	179	82	∈	∈	PROPN
ejpam-6567	179	83	u	u	NOUN
ejpam-6567	179	84	.	.	PUNCT
ejpam-6567	180	1	a	a	DET
ejpam-6567	180	2	multifunction	multifunction	NOUN
ejpam-6567	180	3	f	f	NOUN
ejpam-6567	180	4	:	:	PUNCT
ejpam-6567	180	5	(	(	PUNCT
ejpam-6567	180	6	x	x	X
ejpam-6567	180	7	,	,	PUNCT
ejpam-6567	180	8	τ	τ	PROPN
ejpam-6567	180	9	,	,	PUNCT
ejpam-6567	180	10	i	i	NOUN
ejpam-6567	180	11	)	)	PUNCT
ejpam-6567	180	12	→	→	PUNCT
ejpam-6567	180	13	(	(	PUNCT
ejpam-6567	180	14	y	y	PROPN
ejpam-6567	180	15	,	,	PUNCT
ejpam-6567	180	16	σ1	σ1	PROPN
ejpam-6567	180	17	,	,	PUNCT
ejpam-6567	180	18	σ2	σ2	PROPN
ejpam-6567	180	19	)	)	PUNCT
ejpam-6567	180	20	is	be	AUX
ejpam-6567	180	21	said	say	VERB
ejpam-6567	180	22	to	to	PART
ejpam-6567	180	23	be	be	AUX
ejpam-6567	180	24	lower	lower	ADV
ejpam-6567	180	25	weakly	weakly	ADJ
ejpam-6567	180	26	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	180	27	,	,	PUNCT
ejpam-6567	180	28	σ2)continuous	σ2)continuous	ADJ
ejpam-6567	180	29	if	if	SCONJ
ejpam-6567	180	30	f	f	PROPN
ejpam-6567	180	31	is	be	AUX
ejpam-6567	180	32	lower	lower	ADV
ejpam-6567	180	33	weakly	weakly	ADJ
ejpam-6567	180	34	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	180	35	,	,	PUNCT
ejpam-6567	180	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	180	37	at	at	ADP
ejpam-6567	180	38	each	each	DET
ejpam-6567	180	39	point	point	NOUN
ejpam-6567	180	40	x	x	PUNCT
ejpam-6567	180	41	of	of	ADP
ejpam-6567	180	42	x.	x.	PROPN
ejpam-6567	180	43	theorem	theorem	VERB
ejpam-6567	180	44	2	2	NUM
ejpam-6567	180	45	.	.	X
ejpam-6567	180	46	for	for	ADP
ejpam-6567	180	47	a	a	DET
ejpam-6567	180	48	multifunction	multifunction	NOUN
ejpam-6567	180	49	f	f	NOUN
ejpam-6567	180	50	:	:	PUNCT
ejpam-6567	180	51	(	(	PUNCT
ejpam-6567	180	52	x	x	X
ejpam-6567	180	53	,	,	PUNCT
ejpam-6567	180	54	τ	τ	PROPN
ejpam-6567	180	55	,	,	PUNCT
ejpam-6567	180	56	i	i	NOUN
ejpam-6567	180	57	)	)	PUNCT
ejpam-6567	180	58	→	→	PUNCT
ejpam-6567	180	59	(	(	PUNCT
ejpam-6567	180	60	y	y	PROPN
ejpam-6567	180	61	,	,	PUNCT
ejpam-6567	180	62	σ1	σ1	PROPN
ejpam-6567	180	63	,	,	PUNCT
ejpam-6567	180	64	σ2	σ2	NOUN
ejpam-6567	180	65	)	)	PUNCT
ejpam-6567	180	66	,	,	PUNCT
ejpam-6567	180	67	the	the	DET
ejpam-6567	180	68	following	follow	VERB
ejpam-6567	180	69	properties	property	NOUN
ejpam-6567	180	70	are	be	AUX
ejpam-6567	180	71	equivalent	equivalent	ADJ
ejpam-6567	180	72	:	:	PUNCT
ejpam-6567	180	73	(	(	PUNCT
ejpam-6567	180	74	1	1	X
ejpam-6567	180	75	)	)	PUNCT
ejpam-6567	180	76	f	f	PROPN
ejpam-6567	180	77	is	be	AUX
ejpam-6567	180	78	lower	low	ADJ
ejpam-6567	180	79	weakly	weakly	ADJ
ejpam-6567	180	80	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	180	81	,	,	PUNCT
ejpam-6567	180	82	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	180	83	;	;	PUNCT
ejpam-6567	180	84	(	(	PUNCT
ejpam-6567	180	85	2	2	X
ejpam-6567	180	86	)	)	PUNCT
ejpam-6567	180	87	f−(v	f−(v	NOUN
ejpam-6567	180	88	)	)	PUNCT
ejpam-6567	181	1	⊆	⊆	NUM
ejpam-6567	181	2	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	NOUN
ejpam-6567	181	3	-	-	PUNCT
ejpam-6567	181	4	cl(v	cl(v	NOUN
ejpam-6567	181	5	)	)	PUNCT
ejpam-6567	181	6	)	)	PUNCT
ejpam-6567	181	7	)	)	PUNCT
ejpam-6567	182	1	for	for	ADP
ejpam-6567	182	2	every	every	DET
ejpam-6567	182	3	σ1σ2	σ1σ2	NOUN
ejpam-6567	182	4	-	-	ADJ
ejpam-6567	182	5	open	open	ADJ
ejpam-6567	182	6	set	set	NOUN
ejpam-6567	182	7	v	v	NOUN
ejpam-6567	182	8	of	of	ADP
ejpam-6567	182	9	y	y	PROPN
ejpam-6567	182	10	;	;	PUNCT
ejpam-6567	182	11	(	(	PUNCT
ejpam-6567	182	12	3	3	X
ejpam-6567	182	13	)	)	PUNCT
ejpam-6567	182	14	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	182	15	-	-	PUNCT
ejpam-6567	182	16	int(k	int(k	NOUN
ejpam-6567	182	17	)	)	PUNCT
ejpam-6567	182	18	)	)	PUNCT
ejpam-6567	182	19	)	)	PUNCT
ejpam-6567	182	20	⊆	⊆	NUM
ejpam-6567	182	21	f+(k	f+(k	NOUN
ejpam-6567	182	22	)	)	PUNCT
ejpam-6567	182	23	for	for	ADP
ejpam-6567	182	24	every	every	DET
ejpam-6567	182	25	σ1σ2	σ1σ2	NUM
ejpam-6567	182	26	-	-	PUNCT
ejpam-6567	182	27	closed	closed	ADJ
ejpam-6567	182	28	set	set	NOUN
ejpam-6567	182	29	k	k	PROPN
ejpam-6567	182	30	of	of	ADP
ejpam-6567	182	31	y	y	PROPN
ejpam-6567	182	32	;	;	PUNCT
ejpam-6567	182	33	(	(	PUNCT
ejpam-6567	182	34	4	4	X
ejpam-6567	182	35	)	)	PUNCT
ejpam-6567	182	36	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	182	37	-	-	PUNCT
ejpam-6567	182	38	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	182	39	-	-	PUNCT
ejpam-6567	182	40	cl(b	cl(b	NOUN
ejpam-6567	182	41	)	)	PUNCT
ejpam-6567	182	42	)	)	PUNCT
ejpam-6567	182	43	)	)	PUNCT
ejpam-6567	182	44	)	)	PUNCT
ejpam-6567	182	45	⊆	⊆	X
ejpam-6567	182	46	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	182	47	-	-	PUNCT
ejpam-6567	182	48	cl(b	cl(b	NOUN
ejpam-6567	182	49	)	)	PUNCT
ejpam-6567	182	50	)	)	PUNCT
ejpam-6567	182	51	for	for	ADP
ejpam-6567	182	52	every	every	DET
ejpam-6567	182	53	subset	subset	NOUN
ejpam-6567	182	54	b	b	PROPN
ejpam-6567	182	55	of	of	ADP
ejpam-6567	182	56	y	y	PROPN
ejpam-6567	182	57	;	;	PUNCT
ejpam-6567	182	58	(	(	PUNCT
ejpam-6567	182	59	5	5	X
ejpam-6567	182	60	)	)	PUNCT
ejpam-6567	182	61	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	182	62	-	-	PUNCT
ejpam-6567	182	63	int(b	int(b	NOUN
ejpam-6567	182	64	)	)	PUNCT
ejpam-6567	182	65	)	)	PUNCT
ejpam-6567	182	66	⊆	⊆	NUM
ejpam-6567	182	67	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	VERB
ejpam-6567	182	68	-	-	PUNCT
ejpam-6567	182	69	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	182	70	-	-	PUNCT
ejpam-6567	182	71	int(b	int(b	NOUN
ejpam-6567	182	72	)	)	PUNCT
ejpam-6567	182	73	)	)	PUNCT
ejpam-6567	182	74	)	)	PUNCT
ejpam-6567	182	75	)	)	PUNCT
ejpam-6567	182	76	for	for	ADP
ejpam-6567	182	77	every	every	DET
ejpam-6567	182	78	subset	subset	NOUN
ejpam-6567	182	79	b	b	PROPN
ejpam-6567	182	80	of	of	ADP
ejpam-6567	182	81	y	y	PROPN
ejpam-6567	182	82	;	;	PUNCT
ejpam-6567	182	83	(	(	PUNCT
ejpam-6567	182	84	6	6	X
ejpam-6567	182	85	)	)	PUNCT
ejpam-6567	182	86	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	182	87	-	-	PUNCT
ejpam-6567	182	88	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	182	89	-	-	PUNCT
ejpam-6567	182	90	cl(v	cl(v	NOUN
ejpam-6567	182	91	)	)	PUNCT
ejpam-6567	182	92	)	)	PUNCT
ejpam-6567	182	93	)	)	PUNCT
ejpam-6567	182	94	)	)	PUNCT
ejpam-6567	182	95	⊆	⊆	X
ejpam-6567	182	96	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	182	97	-	-	PUNCT
ejpam-6567	182	98	cl(v	cl(v	NOUN
ejpam-6567	182	99	)	)	PUNCT
ejpam-6567	182	100	)	)	PUNCT
ejpam-6567	182	101	for	for	ADP
ejpam-6567	182	102	every	every	DET
ejpam-6567	182	103	σ1σ2	σ1σ2	NOUN
ejpam-6567	182	104	-	-	ADJ
ejpam-6567	182	105	open	open	ADJ
ejpam-6567	182	106	set	set	NOUN
ejpam-6567	182	107	v	v	NOUN
ejpam-6567	182	108	of	of	ADP
ejpam-6567	182	109	y	y	PROPN
ejpam-6567	182	110	;	;	PUNCT
ejpam-6567	182	111	(	(	PUNCT
ejpam-6567	182	112	7	7	X
ejpam-6567	182	113	)	)	PUNCT
ejpam-6567	182	114	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-6567	182	115	)	)	PUNCT
ejpam-6567	182	116	)	)	PUNCT
ejpam-6567	182	117	⊆	⊆	X
ejpam-6567	182	118	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	182	119	-	-	PUNCT
ejpam-6567	182	120	cl(v	cl(v	NOUN
ejpam-6567	182	121	)	)	PUNCT
ejpam-6567	182	122	)	)	PUNCT
ejpam-6567	182	123	for	for	ADP
ejpam-6567	182	124	every	every	DET
ejpam-6567	182	125	σ1σ2	σ1σ2	NOUN
ejpam-6567	182	126	-	-	ADJ
ejpam-6567	182	127	open	open	ADJ
ejpam-6567	182	128	set	set	NOUN
ejpam-6567	182	129	v	v	NOUN
ejpam-6567	182	130	of	of	ADP
ejpam-6567	182	131	y	y	PROPN
ejpam-6567	182	132	;	;	PUNCT
ejpam-6567	182	133	(	(	PUNCT
ejpam-6567	182	134	8)	8)	NUM
ejpam-6567	182	135	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	182	136	-	-	PUNCT
ejpam-6567	182	137	int(k	int(k	NOUN
ejpam-6567	182	138	)	)	PUNCT
ejpam-6567	182	139	)	)	PUNCT
ejpam-6567	182	140	)	)	PUNCT
ejpam-6567	183	1	⊆	⊆	NUM
ejpam-6567	183	2	f+(k	f+(k	NOUN
ejpam-6567	183	3	)	)	PUNCT
ejpam-6567	183	4	for	for	ADP
ejpam-6567	183	5	every	every	DET
ejpam-6567	183	6	(	(	PUNCT
ejpam-6567	183	7	σ1	σ1	PROPN
ejpam-6567	183	8	,	,	PUNCT
ejpam-6567	183	9	σ2)r	σ2)r	NOUN
ejpam-6567	183	10	-	-	PUNCT
ejpam-6567	183	11	closed	close	VERB
ejpam-6567	183	12	set	set	ADJ
ejpam-6567	183	13	k	k	PROPN
ejpam-6567	183	14	of	of	ADP
ejpam-6567	183	15	y	y	PROPN
ejpam-6567	183	16	.	.	PUNCT
ejpam-6567	184	1	proof	proof	NOUN
ejpam-6567	184	2	.	.	PUNCT
ejpam-6567	185	1	the	the	DET
ejpam-6567	185	2	proof	proof	NOUN
ejpam-6567	185	3	is	be	AUX
ejpam-6567	185	4	similar	similar	ADJ
ejpam-6567	185	5	to	to	ADP
ejpam-6567	185	6	that	that	PRON
ejpam-6567	185	7	of	of	ADP
ejpam-6567	185	8	theorem	theorem	ADJ
ejpam-6567	185	9	1	1	NUM
ejpam-6567	185	10	.	.	PUNCT
ejpam-6567	185	11	theorem	theorem	NOUN
ejpam-6567	185	12	3	3	NUM
ejpam-6567	185	13	.	.	X
ejpam-6567	185	14	for	for	ADP
ejpam-6567	185	15	a	a	DET
ejpam-6567	185	16	multifunction	multifunction	NOUN
ejpam-6567	185	17	f	f	NOUN
ejpam-6567	185	18	:	:	PUNCT
ejpam-6567	185	19	(	(	PUNCT
ejpam-6567	185	20	x	x	X
ejpam-6567	185	21	,	,	PUNCT
ejpam-6567	185	22	τ	τ	PROPN
ejpam-6567	185	23	,	,	PUNCT
ejpam-6567	185	24	i	i	NOUN
ejpam-6567	185	25	)	)	PUNCT
ejpam-6567	185	26	→	→	PUNCT
ejpam-6567	185	27	(	(	PUNCT
ejpam-6567	185	28	y	y	PROPN
ejpam-6567	185	29	,	,	PUNCT
ejpam-6567	185	30	σ1	σ1	PROPN
ejpam-6567	185	31	,	,	PUNCT
ejpam-6567	185	32	σ2	σ2	NOUN
ejpam-6567	185	33	)	)	PUNCT
ejpam-6567	185	34	,	,	PUNCT
ejpam-6567	185	35	the	the	DET
ejpam-6567	185	36	following	follow	VERB
ejpam-6567	185	37	properties	property	NOUN
ejpam-6567	185	38	are	be	AUX
ejpam-6567	185	39	equivalent	equivalent	ADJ
ejpam-6567	185	40	:	:	PUNCT
ejpam-6567	185	41	(	(	PUNCT
ejpam-6567	185	42	1	1	X
ejpam-6567	185	43	)	)	PUNCT
ejpam-6567	185	44	f	f	PROPN
ejpam-6567	185	45	is	be	AUX
ejpam-6567	185	46	upper	upper	ADJ
ejpam-6567	185	47	weakly	weakly	ADJ
ejpam-6567	185	48	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	185	49	,	,	PUNCT
ejpam-6567	185	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	185	51	;	;	PUNCT
ejpam-6567	185	52	(	(	PUNCT
ejpam-6567	185	53	2	2	X
ejpam-6567	185	54	)	)	PUNCT
ejpam-6567	185	55	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	185	56	-	-	PUNCT
ejpam-6567	185	57	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	185	58	-	-	PUNCT
ejpam-6567	185	59	cl(v	cl(v	NOUN
ejpam-6567	185	60	)	)	PUNCT
ejpam-6567	185	61	)	)	PUNCT
ejpam-6567	185	62	)	)	PUNCT
ejpam-6567	185	63	)	)	PUNCT
ejpam-6567	186	1	⊆	⊆	X
ejpam-6567	186	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	186	3	-	-	PUNCT
ejpam-6567	186	4	cl(v	cl(v	NOUN
ejpam-6567	186	5	)	)	PUNCT
ejpam-6567	186	6	)	)	PUNCT
ejpam-6567	186	7	for	for	ADP
ejpam-6567	186	8	every	every	DET
ejpam-6567	186	9	(	(	PUNCT
ejpam-6567	186	10	σ1	σ1	PROPN
ejpam-6567	186	11	,	,	PUNCT
ejpam-6567	186	12	σ2)β	σ2)β	NOUN
ejpam-6567	186	13	-	-	PUNCT
ejpam-6567	186	14	open	open	NOUN
ejpam-6567	186	15	set	set	NOUN
ejpam-6567	186	16	v	v	NOUN
ejpam-6567	186	17	of	of	ADP
ejpam-6567	186	18	y	y	PROPN
ejpam-6567	186	19	;	;	PUNCT
ejpam-6567	186	20	(	(	PUNCT
ejpam-6567	186	21	3	3	X
ejpam-6567	186	22	)	)	PUNCT
ejpam-6567	186	23	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	186	24	-	-	PUNCT
ejpam-6567	186	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	186	26	-	-	PUNCT
ejpam-6567	186	27	cl(v	cl(v	NOUN
ejpam-6567	186	28	)	)	PUNCT
ejpam-6567	186	29	)	)	PUNCT
ejpam-6567	186	30	)	)	PUNCT
ejpam-6567	186	31	)	)	PUNCT
ejpam-6567	187	1	⊆	⊆	X
ejpam-6567	187	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	187	3	-	-	PUNCT
ejpam-6567	187	4	cl(v	cl(v	NOUN
ejpam-6567	187	5	)	)	PUNCT
ejpam-6567	187	6	)	)	PUNCT
ejpam-6567	187	7	for	for	ADP
ejpam-6567	187	8	every	every	DET
ejpam-6567	187	9	(	(	PUNCT
ejpam-6567	187	10	σ1	σ1	PROPN
ejpam-6567	187	11	,	,	PUNCT
ejpam-6567	187	12	σ2)s	σ2)s	NOUN
ejpam-6567	187	13	-	-	PUNCT
ejpam-6567	187	14	open	open	NOUN
ejpam-6567	187	15	set	set	NOUN
ejpam-6567	187	16	v	v	NOUN
ejpam-6567	187	17	of	of	ADP
ejpam-6567	187	18	y	y	PROPN
ejpam-6567	187	19	.	.	PUNCT
ejpam-6567	188	1	p.	p.	NOUN
ejpam-6567	188	2	pue	pue	NOUN
ejpam-6567	188	3	-	-	PUNCT
ejpam-6567	188	4	on	on	ADP
ejpam-6567	188	5	,	,	PUNCT
ejpam-6567	188	6	a.	a.	PROPN
ejpam-6567	188	7	sama	sama	PROPN
ejpam-6567	188	8	-	-	PUNCT
ejpam-6567	188	9	ae	ae	PROPN
ejpam-6567	188	10	,	,	PUNCT
ejpam-6567	188	11	c.	c.	PROPN
ejpam-6567	188	12	boonpok	boonpok	PROPN
ejpam-6567	188	13	/	/	SYM
ejpam-6567	188	14	eur	eur	PROPN
ejpam-6567	188	15	.	.	PUNCT
ejpam-6567	189	1	j.	j.	PROPN
ejpam-6567	189	2	pure	pure	PROPN
ejpam-6567	189	3	appl	appl	PROPN
ejpam-6567	189	4	.	.	PROPN
ejpam-6567	189	5	math	math	PROPN
ejpam-6567	189	6	,	,	PUNCT
ejpam-6567	189	7	18	18	NUM
ejpam-6567	189	8	(	(	PUNCT
ejpam-6567	189	9	3	3	NUM
ejpam-6567	189	10	)	)	PUNCT
ejpam-6567	189	11	(	(	PUNCT
ejpam-6567	189	12	2025	2025	NUM
ejpam-6567	189	13	)	)	PUNCT
ejpam-6567	189	14	,	,	PUNCT
ejpam-6567	189	15	6567	6567	NUM
ejpam-6567	189	16	7	7	NUM
ejpam-6567	189	17	of	of	ADP
ejpam-6567	189	18	13	13	NUM
ejpam-6567	189	19	proof	proof	NOUN
ejpam-6567	189	20	.	.	PUNCT
ejpam-6567	190	1	(	(	PUNCT
ejpam-6567	190	2	1	1	X
ejpam-6567	190	3	)	)	PUNCT
ejpam-6567	190	4	⇒	⇒	NOUN
ejpam-6567	190	5	(	(	PUNCT
ejpam-6567	190	6	2	2	NUM
ejpam-6567	190	7	):	):	PUNCT
ejpam-6567	190	8	this	this	PRON
ejpam-6567	190	9	follows	follow	VERB
ejpam-6567	190	10	from	from	ADP
ejpam-6567	190	11	(	(	PUNCT
ejpam-6567	190	12	4	4	NUM
ejpam-6567	190	13	)	)	PUNCT
ejpam-6567	190	14	of	of	ADP
ejpam-6567	190	15	theorem	theorem	NOUN
ejpam-6567	190	16	1	1	NUM
ejpam-6567	190	17	.	.	PUNCT
ejpam-6567	190	18	(	(	PUNCT
ejpam-6567	190	19	2	2	X
ejpam-6567	190	20	)	)	PUNCT
ejpam-6567	190	21	⇒	⇒	NOUN
ejpam-6567	190	22	(	(	PUNCT
ejpam-6567	190	23	3	3	NUM
ejpam-6567	190	24	):	):	PUNCT
ejpam-6567	190	25	the	the	DET
ejpam-6567	190	26	proof	proof	NOUN
ejpam-6567	190	27	is	be	AUX
ejpam-6567	190	28	obvious	obvious	ADJ
ejpam-6567	190	29	since	since	SCONJ
ejpam-6567	190	30	every	every	DET
ejpam-6567	190	31	(	(	PUNCT
ejpam-6567	190	32	σ1	σ1	PROPN
ejpam-6567	190	33	,	,	PUNCT
ejpam-6567	190	34	σ2)s	σ2)s	NOUN
ejpam-6567	190	35	-	-	PUNCT
ejpam-6567	190	36	open	open	ADJ
ejpam-6567	190	37	set	set	NOUN
ejpam-6567	190	38	is	be	AUX
ejpam-6567	190	39	(	(	PUNCT
ejpam-6567	190	40	σ1	σ1	PROPN
ejpam-6567	190	41	,	,	PUNCT
ejpam-6567	190	42	σ2)β	σ2)β	NOUN
ejpam-6567	190	43	-	-	PUNCT
ejpam-6567	190	44	open	open	ADJ
ejpam-6567	190	45	.	.	PUNCT
ejpam-6567	191	1	(	(	PUNCT
ejpam-6567	191	2	3	3	X
ejpam-6567	191	3	)	)	PUNCT
ejpam-6567	191	4	⇒	⇒	NOUN
ejpam-6567	191	5	(	(	PUNCT
ejpam-6567	191	6	1	1	NUM
ejpam-6567	191	7	):	):	PUNCT
ejpam-6567	191	8	since	since	SCONJ
ejpam-6567	191	9	every	every	DET
ejpam-6567	191	10	σ1σ2	σ1σ2	NUM
ejpam-6567	191	11	-	-	ADJ
ejpam-6567	191	12	open	open	ADJ
ejpam-6567	191	13	set	set	NOUN
ejpam-6567	191	14	is	be	AUX
ejpam-6567	191	15	(	(	PUNCT
ejpam-6567	191	16	σ1	σ1	PROPN
ejpam-6567	191	17	,	,	PUNCT
ejpam-6567	191	18	σ2)s	σ2)s	NOUN
ejpam-6567	191	19	-	-	PUNCT
ejpam-6567	191	20	open	open	ADJ
ejpam-6567	191	21	,	,	PUNCT
ejpam-6567	191	22	the	the	DET
ejpam-6567	191	23	proof	proof	NOUN
ejpam-6567	191	24	is	be	AUX
ejpam-6567	191	25	obvious	obvious	ADJ
ejpam-6567	191	26	by	by	ADP
ejpam-6567	191	27	(	(	PUNCT
ejpam-6567	191	28	7	7	NUM
ejpam-6567	191	29	)	)	PUNCT
ejpam-6567	191	30	of	of	ADP
ejpam-6567	191	31	theorem	theorem	ADJ
ejpam-6567	191	32	1	1	NUM
ejpam-6567	191	33	.	.	PUNCT
ejpam-6567	191	34	theorem	theorem	NOUN
ejpam-6567	191	35	4	4	NUM
ejpam-6567	191	36	.	.	X
ejpam-6567	191	37	for	for	ADP
ejpam-6567	191	38	a	a	DET
ejpam-6567	191	39	multifunction	multifunction	NOUN
ejpam-6567	191	40	f	f	NOUN
ejpam-6567	191	41	:	:	PUNCT
ejpam-6567	191	42	(	(	PUNCT
ejpam-6567	191	43	x	x	X
ejpam-6567	191	44	,	,	PUNCT
ejpam-6567	191	45	τ	τ	PROPN
ejpam-6567	191	46	,	,	PUNCT
ejpam-6567	191	47	i	i	NOUN
ejpam-6567	191	48	)	)	PUNCT
ejpam-6567	191	49	→	→	PUNCT
ejpam-6567	191	50	(	(	PUNCT
ejpam-6567	191	51	y	y	PROPN
ejpam-6567	191	52	,	,	PUNCT
ejpam-6567	191	53	σ1	σ1	PROPN
ejpam-6567	191	54	,	,	PUNCT
ejpam-6567	191	55	σ2	σ2	NOUN
ejpam-6567	191	56	)	)	PUNCT
ejpam-6567	191	57	,	,	PUNCT
ejpam-6567	191	58	the	the	DET
ejpam-6567	191	59	following	follow	VERB
ejpam-6567	191	60	properties	property	NOUN
ejpam-6567	191	61	are	be	AUX
ejpam-6567	191	62	equivalent	equivalent	ADJ
ejpam-6567	191	63	:	:	PUNCT
ejpam-6567	191	64	(	(	PUNCT
ejpam-6567	191	65	1	1	X
ejpam-6567	191	66	)	)	PUNCT
ejpam-6567	191	67	f	f	PROPN
ejpam-6567	191	68	is	be	AUX
ejpam-6567	191	69	lower	low	ADJ
ejpam-6567	191	70	weakly	weakly	ADJ
ejpam-6567	191	71	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	191	72	,	,	PUNCT
ejpam-6567	191	73	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	191	74	;	;	PUNCT
ejpam-6567	191	75	(	(	PUNCT
ejpam-6567	191	76	2	2	X
ejpam-6567	191	77	)	)	PUNCT
ejpam-6567	191	78	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	191	79	-	-	PUNCT
ejpam-6567	191	80	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	191	81	-	-	PUNCT
ejpam-6567	191	82	cl(v	cl(v	NOUN
ejpam-6567	191	83	)	)	PUNCT
ejpam-6567	191	84	)	)	PUNCT
ejpam-6567	191	85	)	)	PUNCT
ejpam-6567	191	86	)	)	PUNCT
ejpam-6567	192	1	⊆	⊆	X
ejpam-6567	192	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	192	3	-	-	PUNCT
ejpam-6567	192	4	cl(v	cl(v	NOUN
ejpam-6567	192	5	)	)	PUNCT
ejpam-6567	192	6	)	)	PUNCT
ejpam-6567	192	7	for	for	ADP
ejpam-6567	192	8	every	every	DET
ejpam-6567	192	9	(	(	PUNCT
ejpam-6567	192	10	σ1	σ1	PROPN
ejpam-6567	192	11	,	,	PUNCT
ejpam-6567	192	12	σ2)β	σ2)β	NOUN
ejpam-6567	192	13	-	-	PUNCT
ejpam-6567	192	14	open	open	NOUN
ejpam-6567	192	15	set	set	NOUN
ejpam-6567	192	16	v	v	NOUN
ejpam-6567	192	17	of	of	ADP
ejpam-6567	192	18	y	y	PROPN
ejpam-6567	192	19	;	;	PUNCT
ejpam-6567	192	20	(	(	PUNCT
ejpam-6567	192	21	3	3	X
ejpam-6567	192	22	)	)	PUNCT
ejpam-6567	192	23	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	192	24	-	-	PUNCT
ejpam-6567	192	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	192	26	-	-	PUNCT
ejpam-6567	192	27	cl(v	cl(v	NOUN
ejpam-6567	192	28	)	)	PUNCT
ejpam-6567	192	29	)	)	PUNCT
ejpam-6567	192	30	)	)	PUNCT
ejpam-6567	192	31	)	)	PUNCT
ejpam-6567	192	32	⊆	⊆	X
ejpam-6567	192	33	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	192	34	-	-	PUNCT
ejpam-6567	192	35	cl(v	cl(v	NOUN
ejpam-6567	192	36	)	)	PUNCT
ejpam-6567	192	37	)	)	PUNCT
ejpam-6567	192	38	for	for	SCONJ
ejpam-6567	192	39	every	every	DET
ejpam-6567	192	40	(	(	PUNCT
ejpam-6567	192	41	σ1	σ1	PROPN
ejpam-6567	192	42	,	,	PUNCT
ejpam-6567	192	43	σ2)s	σ2)s	NOUN
ejpam-6567	192	44	-	-	PUNCT
ejpam-6567	192	45	open	open	NOUN
ejpam-6567	192	46	set	set	NOUN
ejpam-6567	192	47	v	v	NOUN
ejpam-6567	192	48	of	of	ADP
ejpam-6567	192	49	y	y	PROPN
ejpam-6567	192	50	.	.	PUNCT
ejpam-6567	193	1	proof	proof	NOUN
ejpam-6567	193	2	.	.	PUNCT
ejpam-6567	194	1	the	the	DET
ejpam-6567	194	2	proof	proof	NOUN
ejpam-6567	194	3	is	be	AUX
ejpam-6567	194	4	similar	similar	ADJ
ejpam-6567	194	5	to	to	ADP
ejpam-6567	194	6	that	that	PRON
ejpam-6567	194	7	of	of	ADP
ejpam-6567	194	8	theorem	theorem	ADJ
ejpam-6567	194	9	3	3	NUM
ejpam-6567	194	10	.	.	PUNCT
ejpam-6567	194	11	theorem	theorem	NOUN
ejpam-6567	194	12	5	5	NUM
ejpam-6567	194	13	.	.	X
ejpam-6567	194	14	for	for	ADP
ejpam-6567	194	15	a	a	DET
ejpam-6567	194	16	multifunction	multifunction	NOUN
ejpam-6567	194	17	f	f	NOUN
ejpam-6567	194	18	:	:	PUNCT
ejpam-6567	194	19	(	(	PUNCT
ejpam-6567	194	20	x	x	X
ejpam-6567	194	21	,	,	PUNCT
ejpam-6567	194	22	τ	τ	PROPN
ejpam-6567	194	23	,	,	PUNCT
ejpam-6567	194	24	i	i	NOUN
ejpam-6567	194	25	)	)	PUNCT
ejpam-6567	194	26	→	→	PUNCT
ejpam-6567	194	27	(	(	PUNCT
ejpam-6567	194	28	y	y	PROPN
ejpam-6567	194	29	,	,	PUNCT
ejpam-6567	194	30	σ1	σ1	PROPN
ejpam-6567	194	31	,	,	PUNCT
ejpam-6567	194	32	σ2	σ2	NOUN
ejpam-6567	194	33	)	)	PUNCT
ejpam-6567	194	34	,	,	PUNCT
ejpam-6567	194	35	the	the	DET
ejpam-6567	194	36	following	follow	VERB
ejpam-6567	194	37	properties	property	NOUN
ejpam-6567	194	38	are	be	AUX
ejpam-6567	194	39	equivalent	equivalent	ADJ
ejpam-6567	194	40	:	:	PUNCT
ejpam-6567	194	41	(	(	PUNCT
ejpam-6567	194	42	1	1	X
ejpam-6567	194	43	)	)	PUNCT
ejpam-6567	194	44	f	f	PROPN
ejpam-6567	194	45	is	be	AUX
ejpam-6567	194	46	upper	upper	ADJ
ejpam-6567	194	47	weakly	weakly	ADJ
ejpam-6567	194	48	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	194	49	,	,	PUNCT
ejpam-6567	194	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	194	51	;	;	PUNCT
ejpam-6567	194	52	(	(	PUNCT
ejpam-6567	194	53	2	2	X
ejpam-6567	194	54	)	)	PUNCT
ejpam-6567	194	55	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	194	56	-	-	PUNCT
ejpam-6567	194	57	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	194	58	-	-	PUNCT
ejpam-6567	194	59	cl(v	cl(v	NOUN
ejpam-6567	194	60	)	)	PUNCT
ejpam-6567	194	61	)	)	PUNCT
ejpam-6567	194	62	)	)	PUNCT
ejpam-6567	194	63	)	)	PUNCT
ejpam-6567	195	1	⊆	⊆	X
ejpam-6567	195	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	195	3	-	-	PUNCT
ejpam-6567	195	4	cl(v	cl(v	NOUN
ejpam-6567	195	5	)	)	PUNCT
ejpam-6567	195	6	)	)	PUNCT
ejpam-6567	195	7	for	for	ADP
ejpam-6567	195	8	every	every	DET
ejpam-6567	195	9	(	(	PUNCT
ejpam-6567	195	10	σ1	σ1	PROPN
ejpam-6567	195	11	,	,	PUNCT
ejpam-6567	195	12	σ2)p	σ2)p	NOUN
ejpam-6567	195	13	-	-	PUNCT
ejpam-6567	195	14	open	open	NOUN
ejpam-6567	195	15	set	set	NOUN
ejpam-6567	195	16	v	v	NOUN
ejpam-6567	195	17	of	of	ADP
ejpam-6567	195	18	y	y	PROPN
ejpam-6567	195	19	;	;	PUNCT
ejpam-6567	195	20	(	(	PUNCT
ejpam-6567	195	21	3	3	X
ejpam-6567	195	22	)	)	SYM
ejpam-6567	195	23	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6567	195	24	)	)	PUNCT
ejpam-6567	195	25	)	)	PUNCT
ejpam-6567	195	26	⊆	⊆	X
ejpam-6567	195	27	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	195	28	-	-	PUNCT
ejpam-6567	195	29	cl(v	cl(v	NOUN
ejpam-6567	195	30	)	)	PUNCT
ejpam-6567	195	31	)	)	PUNCT
ejpam-6567	195	32	for	for	ADP
ejpam-6567	195	33	every	every	DET
ejpam-6567	195	34	(	(	PUNCT
ejpam-6567	195	35	σ1	σ1	PROPN
ejpam-6567	195	36	,	,	PUNCT
ejpam-6567	195	37	σ2)p	σ2)p	NOUN
ejpam-6567	195	38	-	-	PUNCT
ejpam-6567	195	39	open	open	NOUN
ejpam-6567	195	40	set	set	NOUN
ejpam-6567	195	41	v	v	NOUN
ejpam-6567	195	42	of	of	ADP
ejpam-6567	195	43	y	y	PROPN
ejpam-6567	195	44	;	;	PUNCT
ejpam-6567	195	45	(	(	PUNCT
ejpam-6567	195	46	4	4	X
ejpam-6567	195	47	)	)	PUNCT
ejpam-6567	195	48	f+(v	f+(v	NOUN
ejpam-6567	195	49	)	)	PUNCT
ejpam-6567	196	1	⊆	⊆	X
ejpam-6567	196	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	196	3	-	-	PUNCT
ejpam-6567	196	4	cl(v	cl(v	NOUN
ejpam-6567	196	5	)	)	PUNCT
ejpam-6567	196	6	)	)	PUNCT
ejpam-6567	196	7	)	)	PUNCT
ejpam-6567	196	8	for	for	ADP
ejpam-6567	196	9	every	every	DET
ejpam-6567	196	10	(	(	PUNCT
ejpam-6567	196	11	σ1	σ1	PROPN
ejpam-6567	196	12	,	,	PUNCT
ejpam-6567	196	13	σ2)p	σ2)p	NOUN
ejpam-6567	196	14	-	-	PUNCT
ejpam-6567	196	15	open	open	NOUN
ejpam-6567	196	16	set	set	NOUN
ejpam-6567	196	17	v	v	NOUN
ejpam-6567	196	18	of	of	ADP
ejpam-6567	196	19	y	y	PROPN
ejpam-6567	196	20	.	.	PUNCT
ejpam-6567	197	1	proof	proof	NOUN
ejpam-6567	197	2	.	.	PUNCT
ejpam-6567	198	1	(	(	PUNCT
ejpam-6567	198	2	1	1	X
ejpam-6567	198	3	)	)	PUNCT
ejpam-6567	198	4	⇒	⇒	NOUN
ejpam-6567	198	5	(	(	PUNCT
ejpam-6567	198	6	2	2	NUM
ejpam-6567	198	7	):	):	PUNCT
ejpam-6567	198	8	let	let	VERB
ejpam-6567	198	9	v	v	PART
ejpam-6567	198	10	be	be	AUX
ejpam-6567	198	11	any	any	DET
ejpam-6567	198	12	(	(	PUNCT
ejpam-6567	198	13	σ1	σ1	PROPN
ejpam-6567	198	14	,	,	PUNCT
ejpam-6567	198	15	σ2)p	σ2)p	NOUN
ejpam-6567	198	16	-	-	PUNCT
ejpam-6567	198	17	open	open	ADJ
ejpam-6567	198	18	set	set	NOUN
ejpam-6567	198	19	of	of	ADP
ejpam-6567	198	20	y	y	PROPN
ejpam-6567	198	21	.	.	PUNCT
ejpam-6567	199	1	since	since	SCONJ
ejpam-6567	199	2	σ1σ2	σ1σ2	ADV
ejpam-6567	199	3	-	-	PUNCT
ejpam-6567	199	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	199	5	-	-	PUNCT
ejpam-6567	199	6	cl(v	cl(v	NOUN
ejpam-6567	199	7	)	)	PUNCT
ejpam-6567	199	8	)	)	PUNCT
ejpam-6567	199	9	is	be	AUX
ejpam-6567	199	10	σ1σ2	σ1σ2	NOUN
ejpam-6567	199	11	-	-	ADJ
ejpam-6567	199	12	open	open	ADJ
ejpam-6567	199	13	,	,	PUNCT
ejpam-6567	199	14	by	by	ADP
ejpam-6567	199	15	theorem	theorem	ADJ
ejpam-6567	199	16	1(7	1(7	PROPN
ejpam-6567	199	17	)	)	PUNCT
ejpam-6567	199	18	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	199	19	-	-	PUNCT
ejpam-6567	199	20	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	199	21	-	-	PUNCT
ejpam-6567	199	22	cl(v	cl(v	NOUN
ejpam-6567	199	23	)	)	PUNCT
ejpam-6567	199	24	)	)	PUNCT
ejpam-6567	199	25	)	)	PUNCT
ejpam-6567	199	26	)	)	PUNCT
ejpam-6567	200	1	⊆	⊆	X
ejpam-6567	200	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6567	200	3	-	-	PUNCT
ejpam-6567	200	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	200	5	-	-	PUNCT
ejpam-6567	200	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	200	7	-	-	PUNCT
ejpam-6567	200	8	cl(v	cl(v	NOUN
ejpam-6567	200	9	)	)	PUNCT
ejpam-6567	200	10	)	)	PUNCT
ejpam-6567	200	11	)	)	PUNCT
ejpam-6567	200	12	)	)	PUNCT
ejpam-6567	201	1	⊆	⊆	X
ejpam-6567	201	2	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	201	3	-	-	PUNCT
ejpam-6567	201	4	cl(v	cl(v	NOUN
ejpam-6567	201	5	)	)	PUNCT
ejpam-6567	201	6	)	)	PUNCT
ejpam-6567	201	7	.	.	PUNCT
ejpam-6567	202	1	(	(	PUNCT
ejpam-6567	202	2	2	2	X
ejpam-6567	202	3	)	)	PUNCT
ejpam-6567	202	4	⇒	⇒	NOUN
ejpam-6567	202	5	(	(	PUNCT
ejpam-6567	202	6	3	3	NUM
ejpam-6567	202	7	):	):	PUNCT
ejpam-6567	202	8	let	let	VERB
ejpam-6567	202	9	v	v	PART
ejpam-6567	202	10	be	be	AUX
ejpam-6567	202	11	any	any	DET
ejpam-6567	202	12	(	(	PUNCT
ejpam-6567	202	13	σ1	σ1	PROPN
ejpam-6567	202	14	,	,	PUNCT
ejpam-6567	202	15	σ2)p	σ2)p	NOUN
ejpam-6567	202	16	-	-	PUNCT
ejpam-6567	202	17	open	open	ADJ
ejpam-6567	202	18	set	set	NOUN
ejpam-6567	202	19	of	of	ADP
ejpam-6567	202	20	y	y	PROPN
ejpam-6567	202	21	.	.	PUNCT
ejpam-6567	203	1	by	by	ADP
ejpam-6567	203	2	(	(	PUNCT
ejpam-6567	203	3	2	2	NUM
ejpam-6567	203	4	)	)	PUNCT
ejpam-6567	203	5	,	,	PUNCT
ejpam-6567	203	6	we	we	PRON
ejpam-6567	203	7	have	have	VERB
ejpam-6567	203	8	cl⋆(f−(v	cl⋆(f−(v	NUM
ejpam-6567	203	9	)	)	PUNCT
ejpam-6567	203	10	)	)	PUNCT
ejpam-6567	204	1	⊆	⊆	NUM
ejpam-6567	204	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6567	204	3	-	-	PUNCT
ejpam-6567	204	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	204	5	-	-	PUNCT
ejpam-6567	204	6	cl(v	cl(v	NOUN
ejpam-6567	204	7	)	)	PUNCT
ejpam-6567	204	8	)	)	PUNCT
ejpam-6567	204	9	)	)	PUNCT
ejpam-6567	204	10	)	)	PUNCT
ejpam-6567	204	11	⊆	⊆	X
ejpam-6567	204	12	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6567	204	13	-	-	PUNCT
ejpam-6567	204	14	cl(v	cl(v	NOUN
ejpam-6567	204	15	)	)	PUNCT
ejpam-6567	204	16	)	)	PUNCT
ejpam-6567	204	17	.	.	PUNCT
ejpam-6567	205	1	(	(	PUNCT
ejpam-6567	205	2	3	3	X
ejpam-6567	205	3	)	)	PUNCT
ejpam-6567	205	4	⇒	⇒	NOUN
ejpam-6567	205	5	(	(	PUNCT
ejpam-6567	205	6	4	4	NUM
ejpam-6567	205	7	):	):	PUNCT
ejpam-6567	205	8	let	let	VERB
ejpam-6567	205	9	v	v	PART
ejpam-6567	205	10	be	be	AUX
ejpam-6567	205	11	any	any	DET
ejpam-6567	205	12	(	(	PUNCT
ejpam-6567	205	13	σ1	σ1	PROPN
ejpam-6567	205	14	,	,	PUNCT
ejpam-6567	205	15	σ2)p	σ2)p	NOUN
ejpam-6567	205	16	-	-	PUNCT
ejpam-6567	205	17	open	open	ADJ
ejpam-6567	205	18	set	set	NOUN
ejpam-6567	205	19	of	of	ADP
ejpam-6567	205	20	y	y	PROPN
ejpam-6567	205	21	.	.	PUNCT
ejpam-6567	206	1	thus	thus	ADV
ejpam-6567	206	2	by	by	ADP
ejpam-6567	206	3	(	(	PUNCT
ejpam-6567	206	4	3	3	NUM
ejpam-6567	206	5	)	)	PUNCT
ejpam-6567	206	6	,	,	PUNCT
ejpam-6567	206	7	x	x	PUNCT
ejpam-6567	206	8	−	−	NOUN
ejpam-6567	206	9	int⋆(f+(cl⋆(v	int⋆(f+(cl⋆(v	NOUN
ejpam-6567	206	10	)	)	PUNCT
ejpam-6567	206	11	)	)	PUNCT
ejpam-6567	206	12	)	)	PUNCT
ejpam-6567	207	1	=	=	PUNCT
ejpam-6567	207	2	cl⋆(x	cl⋆(x	PROPN
ejpam-6567	207	3	−	−	PROPN
ejpam-6567	207	4	f+(cl⋆(v	f+(cl⋆(v	PROPN
ejpam-6567	207	5	)	)	PUNCT
ejpam-6567	207	6	)	)	PUNCT
ejpam-6567	207	7	)	)	PUNCT
ejpam-6567	208	1	=	=	NOUN
ejpam-6567	208	2	cl⋆(f−(y	cl⋆(f−(y	NUM
ejpam-6567	208	3	−	−	ADP
ejpam-6567	208	4	cl⋆(v	cl⋆(v	NOUN
ejpam-6567	208	5	)	)	PUNCT
ejpam-6567	208	6	)	)	PUNCT
ejpam-6567	208	7	)	)	PUNCT
ejpam-6567	208	8	⊆	⊆	X
ejpam-6567	208	9	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6567	208	10	-	-	PUNCT
ejpam-6567	208	11	cl(y	cl(y	NOUN
ejpam-6567	208	12	−	−	NOUN
ejpam-6567	208	13	σ1σ2	σ1σ2	NOUN
ejpam-6567	208	14	-	-	NUM
ejpam-6567	208	15	cl(v	cl(v	NOUN
ejpam-6567	208	16	)	)	PUNCT
ejpam-6567	208	17	)	)	PUNCT
ejpam-6567	208	18	)	)	PUNCT
ejpam-6567	209	1	=	=	PUNCT
ejpam-6567	209	2	x	x	X
ejpam-6567	209	3	−	−	ADP
ejpam-6567	209	4	f+(σ1σ2	f+(σ1σ2	ADJ
ejpam-6567	209	5	-	-	PUNCT
ejpam-6567	209	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	209	7	-	-	PUNCT
ejpam-6567	209	8	cl(v	cl(v	NOUN
ejpam-6567	209	9	)	)	PUNCT
ejpam-6567	209	10	)	)	PUNCT
ejpam-6567	209	11	)	)	PUNCT
ejpam-6567	210	1	p.	p.	NOUN
ejpam-6567	210	2	pue	pue	NOUN
ejpam-6567	210	3	-	-	PUNCT
ejpam-6567	210	4	on	on	ADP
ejpam-6567	210	5	,	,	PUNCT
ejpam-6567	210	6	a.	a.	PROPN
ejpam-6567	210	7	sama	sama	PROPN
ejpam-6567	210	8	-	-	PUNCT
ejpam-6567	210	9	ae	ae	PROPN
ejpam-6567	210	10	,	,	PUNCT
ejpam-6567	210	11	c.	c.	PROPN
ejpam-6567	210	12	boonpok	boonpok	PROPN
ejpam-6567	210	13	/	/	SYM
ejpam-6567	210	14	eur	eur	PROPN
ejpam-6567	210	15	.	.	PUNCT
ejpam-6567	211	1	j.	j.	PROPN
ejpam-6567	211	2	pure	pure	PROPN
ejpam-6567	211	3	appl	appl	PROPN
ejpam-6567	211	4	.	.	PROPN
ejpam-6567	211	5	math	math	PROPN
ejpam-6567	211	6	,	,	PUNCT
ejpam-6567	211	7	18	18	NUM
ejpam-6567	211	8	(	(	PUNCT
ejpam-6567	211	9	3	3	NUM
ejpam-6567	211	10	)	)	PUNCT
ejpam-6567	211	11	(	(	PUNCT
ejpam-6567	211	12	2025	2025	NUM
ejpam-6567	211	13	)	)	PUNCT
ejpam-6567	211	14	,	,	PUNCT
ejpam-6567	211	15	6567	6567	NUM
ejpam-6567	211	16	8	8	NUM
ejpam-6567	211	17	of	of	ADP
ejpam-6567	211	18	13	13	NUM
ejpam-6567	211	19	⊆	⊆	NUM
ejpam-6567	211	20	x	x	SYM
ejpam-6567	211	21	−	−	PROPN
ejpam-6567	212	1	f+(v	f+(v	NOUN
ejpam-6567	212	2	)	)	PUNCT
ejpam-6567	213	1	and	and	CCONJ
ejpam-6567	213	2	hence	hence	ADV
ejpam-6567	213	3	f+(v	f+(v	NOUN
ejpam-6567	213	4	)	)	PUNCT
ejpam-6567	214	1	⊆	⊆	X
ejpam-6567	214	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	214	3	-	-	PUNCT
ejpam-6567	214	4	cl(v	cl(v	NOUN
ejpam-6567	214	5	)	)	PUNCT
ejpam-6567	214	6	)	)	PUNCT
ejpam-6567	214	7	)	)	PUNCT
ejpam-6567	214	8	.	.	PUNCT
ejpam-6567	215	1	(	(	PUNCT
ejpam-6567	215	2	4	4	X
ejpam-6567	215	3	)	)	PUNCT
ejpam-6567	215	4	⇒	⇒	NOUN
ejpam-6567	215	5	(	(	PUNCT
ejpam-6567	215	6	1	1	NUM
ejpam-6567	215	7	):	):	PUNCT
ejpam-6567	215	8	let	let	VERB
ejpam-6567	215	9	v	v	PART
ejpam-6567	215	10	be	be	AUX
ejpam-6567	215	11	any	any	DET
ejpam-6567	215	12	σ1σ2	σ1σ2	NOUN
ejpam-6567	215	13	-	-	ADJ
ejpam-6567	215	14	open	open	ADJ
ejpam-6567	215	15	set	set	NOUN
ejpam-6567	215	16	of	of	ADP
ejpam-6567	215	17	y	y	PROPN
ejpam-6567	215	18	.	.	PUNCT
ejpam-6567	216	1	then	then	ADV
ejpam-6567	216	2	,	,	PUNCT
ejpam-6567	216	3	v	v	NOUN
ejpam-6567	216	4	is	be	AUX
ejpam-6567	216	5	(	(	PUNCT
ejpam-6567	216	6	σ1	σ1	PROPN
ejpam-6567	216	7	,	,	PUNCT
ejpam-6567	216	8	σ2)p	σ2)p	NOUN
ejpam-6567	216	9	-	-	PUNCT
ejpam-6567	216	10	open	open	ADJ
ejpam-6567	216	11	in	in	ADP
ejpam-6567	216	12	y	y	PROPN
ejpam-6567	216	13	and	and	CCONJ
ejpam-6567	216	14	by	by	ADP
ejpam-6567	216	15	(	(	PUNCT
ejpam-6567	216	16	4	4	NUM
ejpam-6567	216	17	)	)	PUNCT
ejpam-6567	216	18	,	,	PUNCT
ejpam-6567	216	19	f+(v	f+(v	PROPN
ejpam-6567	216	20	)	)	PUNCT
ejpam-6567	217	1	⊆	⊆	X
ejpam-6567	217	2	int⋆(f+(σ1σ2	int⋆(f+(σ1σ2	NOUN
ejpam-6567	217	3	-	-	PUNCT
ejpam-6567	217	4	cl(v	cl(v	NOUN
ejpam-6567	217	5	)	)	PUNCT
ejpam-6567	217	6	)	)	PUNCT
ejpam-6567	217	7	)	)	PUNCT
ejpam-6567	217	8	.	.	PUNCT
ejpam-6567	218	1	by	by	ADP
ejpam-6567	218	2	theorem	theorem	NOUN
ejpam-6567	218	3	1(2	1(2	NUM
ejpam-6567	218	4	)	)	PUNCT
ejpam-6567	218	5	,	,	PUNCT
ejpam-6567	218	6	f	f	PROPN
ejpam-6567	218	7	is	be	AUX
ejpam-6567	218	8	upper	upper	ADJ
ejpam-6567	218	9	weakly	weakly	ADJ
ejpam-6567	218	10	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	218	11	,	,	PUNCT
ejpam-6567	218	12	σ2)continuous	σ2)continuous	ADJ
ejpam-6567	218	13	.	.	PUNCT
ejpam-6567	219	1	theorem	theorem	VERB
ejpam-6567	219	2	6	6	NUM
ejpam-6567	219	3	.	.	PUNCT
ejpam-6567	219	4	for	for	ADP
ejpam-6567	219	5	a	a	DET
ejpam-6567	219	6	multifunction	multifunction	NOUN
ejpam-6567	220	1	f	f	NOUN
ejpam-6567	220	2	:	:	PUNCT
ejpam-6567	220	3	(	(	PUNCT
ejpam-6567	220	4	x	x	X
ejpam-6567	220	5	,	,	PUNCT
ejpam-6567	220	6	τ	τ	PROPN
ejpam-6567	220	7	,	,	PUNCT
ejpam-6567	220	8	i	i	NOUN
ejpam-6567	220	9	)	)	PUNCT
ejpam-6567	220	10	→	→	PUNCT
ejpam-6567	220	11	(	(	PUNCT
ejpam-6567	220	12	y	y	PROPN
ejpam-6567	220	13	,	,	PUNCT
ejpam-6567	220	14	σ1	σ1	PROPN
ejpam-6567	220	15	,	,	PUNCT
ejpam-6567	220	16	σ2	σ2	NOUN
ejpam-6567	220	17	)	)	PUNCT
ejpam-6567	220	18	,	,	PUNCT
ejpam-6567	220	19	the	the	DET
ejpam-6567	220	20	following	follow	VERB
ejpam-6567	220	21	properties	property	NOUN
ejpam-6567	220	22	are	be	AUX
ejpam-6567	220	23	equivalent	equivalent	ADJ
ejpam-6567	220	24	:	:	PUNCT
ejpam-6567	220	25	(	(	PUNCT
ejpam-6567	220	26	1	1	X
ejpam-6567	220	27	)	)	PUNCT
ejpam-6567	220	28	f	f	PROPN
ejpam-6567	220	29	is	be	AUX
ejpam-6567	220	30	lower	low	ADJ
ejpam-6567	220	31	weakly	weakly	ADJ
ejpam-6567	220	32	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	220	33	,	,	PUNCT
ejpam-6567	220	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	220	35	;	;	PUNCT
ejpam-6567	220	36	(	(	PUNCT
ejpam-6567	220	37	2	2	X
ejpam-6567	220	38	)	)	PUNCT
ejpam-6567	220	39	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	220	40	-	-	PUNCT
ejpam-6567	220	41	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	220	42	-	-	PUNCT
ejpam-6567	220	43	cl(v	cl(v	NOUN
ejpam-6567	220	44	)	)	PUNCT
ejpam-6567	220	45	)	)	PUNCT
ejpam-6567	220	46	)	)	PUNCT
ejpam-6567	220	47	)	)	PUNCT
ejpam-6567	221	1	⊆	⊆	X
ejpam-6567	221	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	221	3	-	-	PUNCT
ejpam-6567	221	4	cl(v	cl(v	NOUN
ejpam-6567	221	5	)	)	PUNCT
ejpam-6567	221	6	)	)	PUNCT
ejpam-6567	221	7	for	for	ADP
ejpam-6567	221	8	every	every	DET
ejpam-6567	221	9	(	(	PUNCT
ejpam-6567	221	10	σ1	σ1	PROPN
ejpam-6567	221	11	,	,	PUNCT
ejpam-6567	221	12	σ2)p	σ2)p	NOUN
ejpam-6567	221	13	-	-	PUNCT
ejpam-6567	221	14	open	open	NOUN
ejpam-6567	221	15	set	set	NOUN
ejpam-6567	221	16	v	v	NOUN
ejpam-6567	221	17	of	of	ADP
ejpam-6567	221	18	y	y	PROPN
ejpam-6567	221	19	;	;	PUNCT
ejpam-6567	221	20	(	(	PUNCT
ejpam-6567	221	21	3	3	X
ejpam-6567	221	22	)	)	PUNCT
ejpam-6567	221	23	cl⋆(f+(v	cl⋆(f+(v	NOUN
ejpam-6567	221	24	)	)	PUNCT
ejpam-6567	221	25	)	)	PUNCT
ejpam-6567	222	1	⊆	⊆	X
ejpam-6567	222	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	222	3	-	-	PUNCT
ejpam-6567	222	4	cl(v	cl(v	NOUN
ejpam-6567	222	5	)	)	PUNCT
ejpam-6567	222	6	)	)	PUNCT
ejpam-6567	222	7	for	for	ADP
ejpam-6567	222	8	every	every	DET
ejpam-6567	222	9	(	(	PUNCT
ejpam-6567	222	10	σ1	σ1	PROPN
ejpam-6567	222	11	,	,	PUNCT
ejpam-6567	222	12	σ2)p	σ2)p	NOUN
ejpam-6567	222	13	-	-	PUNCT
ejpam-6567	222	14	open	open	NOUN
ejpam-6567	222	15	set	set	NOUN
ejpam-6567	222	16	v	v	NOUN
ejpam-6567	222	17	of	of	ADP
ejpam-6567	222	18	y	y	PROPN
ejpam-6567	222	19	;	;	PUNCT
ejpam-6567	222	20	(	(	PUNCT
ejpam-6567	222	21	4	4	X
ejpam-6567	222	22	)	)	PUNCT
ejpam-6567	222	23	f−(v	f−(v	NOUN
ejpam-6567	222	24	)	)	PUNCT
ejpam-6567	223	1	⊆	⊆	NUM
ejpam-6567	223	2	int⋆(f−(σ1σ2	int⋆(f−(σ1σ2	NOUN
ejpam-6567	223	3	-	-	PUNCT
ejpam-6567	223	4	cl(v	cl(v	NOUN
ejpam-6567	223	5	)	)	PUNCT
ejpam-6567	223	6	)	)	PUNCT
ejpam-6567	223	7	)	)	PUNCT
ejpam-6567	224	1	for	for	ADP
ejpam-6567	224	2	every	every	DET
ejpam-6567	224	3	(	(	PUNCT
ejpam-6567	224	4	σ1	σ1	PROPN
ejpam-6567	224	5	,	,	PUNCT
ejpam-6567	224	6	σ2)p	σ2)p	NOUN
ejpam-6567	224	7	-	-	PUNCT
ejpam-6567	224	8	open	open	NOUN
ejpam-6567	224	9	set	set	NOUN
ejpam-6567	224	10	v	v	NOUN
ejpam-6567	224	11	of	of	ADP
ejpam-6567	224	12	y	y	PROPN
ejpam-6567	224	13	.	.	PUNCT
ejpam-6567	225	1	proof	proof	NOUN
ejpam-6567	225	2	.	.	PUNCT
ejpam-6567	226	1	the	the	DET
ejpam-6567	226	2	proof	proof	NOUN
ejpam-6567	226	3	is	be	AUX
ejpam-6567	226	4	similar	similar	ADJ
ejpam-6567	226	5	to	to	ADP
ejpam-6567	226	6	that	that	PRON
ejpam-6567	226	7	of	of	ADP
ejpam-6567	226	8	theorem	theorem	NOUN
ejpam-6567	226	9	5	5	NUM
ejpam-6567	226	10	.	.	PUNCT
ejpam-6567	226	11	lemma	lemma	PROPN
ejpam-6567	226	12	2	2	NUM
ejpam-6567	226	13	.	.	PUNCT
ejpam-6567	227	1	if	if	SCONJ
ejpam-6567	227	2	f	f	PROPN
ejpam-6567	227	3	:	:	PUNCT
ejpam-6567	227	4	(	(	PUNCT
ejpam-6567	227	5	x	x	X
ejpam-6567	227	6	,	,	PUNCT
ejpam-6567	227	7	τ	τ	PROPN
ejpam-6567	227	8	,	,	PUNCT
ejpam-6567	227	9	i	i	NOUN
ejpam-6567	227	10	)	)	PUNCT
ejpam-6567	227	11	→	→	PUNCT
ejpam-6567	227	12	(	(	PUNCT
ejpam-6567	227	13	y	y	PROPN
ejpam-6567	227	14	,	,	PUNCT
ejpam-6567	227	15	σ1	σ1	PROPN
ejpam-6567	227	16	,	,	PUNCT
ejpam-6567	227	17	σ2	σ2	NOUN
ejpam-6567	227	18	)	)	PUNCT
ejpam-6567	227	19	is	be	AUX
ejpam-6567	227	20	lower	low	ADJ
ejpam-6567	227	21	weakly	weakly	ADJ
ejpam-6567	227	22	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	227	23	,	,	PUNCT
ejpam-6567	227	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	227	25	,	,	PUNCT
ejpam-6567	227	26	then	then	ADV
ejpam-6567	227	27	for	for	ADP
ejpam-6567	227	28	each	each	DET
ejpam-6567	227	29	x	x	SYM
ejpam-6567	227	30	∈	∈	PROPN
ejpam-6567	227	31	x	x	X
ejpam-6567	227	32	and	and	CCONJ
ejpam-6567	227	33	each	each	DET
ejpam-6567	227	34	subset	subset	NOUN
ejpam-6567	227	35	b	b	PROPN
ejpam-6567	227	36	of	of	ADP
ejpam-6567	227	37	y	y	PROPN
ejpam-6567	227	38	with	with	ADP
ejpam-6567	227	39	(	(	PUNCT
ejpam-6567	227	40	σ1	σ1	PROPN
ejpam-6567	227	41	,	,	PUNCT
ejpam-6567	227	42	σ2)θ	σ2)θ	NOUN
ejpam-6567	227	43	-	-	PUNCT
ejpam-6567	227	44	int(b)∩f	int(b)∩f	NOUN
ejpam-6567	227	45	(	(	PUNCT
ejpam-6567	227	46	x	x	X
ejpam-6567	227	47	)	)	PUNCT
ejpam-6567	227	48	̸=	̸=	NOUN
ejpam-6567	227	49	∅	∅	NOUN
ejpam-6567	227	50	,	,	PUNCT
ejpam-6567	227	51	there	there	PRON
ejpam-6567	227	52	exists	exist	VERB
ejpam-6567	227	53	a	a	DET
ejpam-6567	227	54	⋆-open	⋆-open	ADJ
ejpam-6567	227	55	set	set	NOUN
ejpam-6567	227	56	u	u	NOUN
ejpam-6567	227	57	of	of	ADP
ejpam-6567	227	58	x	x	PUNCT
ejpam-6567	227	59	containing	contain	VERB
ejpam-6567	227	60	x	x	PUNCT
ejpam-6567	227	61	such	such	ADJ
ejpam-6567	227	62	that	that	SCONJ
ejpam-6567	227	63	u	u	PROPN
ejpam-6567	227	64	⊆	⊆	NUM
ejpam-6567	227	65	f−(b	f−(b	NOUN
ejpam-6567	227	66	)	)	PUNCT
ejpam-6567	227	67	.	.	PUNCT
ejpam-6567	228	1	proof	proof	NOUN
ejpam-6567	228	2	.	.	PUNCT
ejpam-6567	229	1	since	since	SCONJ
ejpam-6567	229	2	(	(	PUNCT
ejpam-6567	229	3	σ1	σ1	PROPN
ejpam-6567	229	4	,	,	PUNCT
ejpam-6567	229	5	σ2)θ	σ2)θ	NOUN
ejpam-6567	229	6	-	-	PUNCT
ejpam-6567	229	7	int(b	int(b	NOUN
ejpam-6567	229	8	)	)	PUNCT
ejpam-6567	229	9	∩	∩	ADJ
ejpam-6567	229	10	f	f	PROPN
ejpam-6567	229	11	(	(	PUNCT
ejpam-6567	229	12	x	x	X
ejpam-6567	229	13	)	)	PUNCT
ejpam-6567	229	14	̸=	̸=	NOUN
ejpam-6567	229	15	∅	∅	NOUN
ejpam-6567	229	16	,	,	PUNCT
ejpam-6567	229	17	there	there	PRON
ejpam-6567	229	18	exists	exist	VERB
ejpam-6567	229	19	a	a	DET
ejpam-6567	229	20	nonempty	nonempty	ADJ
ejpam-6567	229	21	σ1σ2	σ1σ2	NOUN
ejpam-6567	229	22	-	-	ADJ
ejpam-6567	229	23	open	open	ADJ
ejpam-6567	229	24	set	set	NOUN
ejpam-6567	229	25	v	v	NOUN
ejpam-6567	229	26	of	of	ADP
ejpam-6567	229	27	y	y	PRON
ejpam-6567	229	28	such	such	ADJ
ejpam-6567	229	29	that	that	SCONJ
ejpam-6567	229	30	σ1σ2	σ1σ2	NOUN
ejpam-6567	229	31	-	-	NUM
ejpam-6567	229	32	cl(v	cl(v	NOUN
ejpam-6567	229	33	)	)	PUNCT
ejpam-6567	229	34	⊆	⊆	NUM
ejpam-6567	229	35	b	b	NOUN
ejpam-6567	229	36	and	and	CCONJ
ejpam-6567	229	37	v	v	NOUN
ejpam-6567	229	38	∩	∩	ADJ
ejpam-6567	229	39	f	f	X
ejpam-6567	229	40	(	(	PUNCT
ejpam-6567	229	41	x	x	X
ejpam-6567	229	42	)	)	PUNCT
ejpam-6567	229	43	̸=	̸=	PROPN
ejpam-6567	229	44	∅.	∅.	ADV
ejpam-6567	229	45	since	since	SCONJ
ejpam-6567	229	46	f	f	PROPN
ejpam-6567	229	47	is	be	AUX
ejpam-6567	229	48	lower	low	ADJ
ejpam-6567	229	49	weakly	weakly	ADJ
ejpam-6567	229	50	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	229	51	,	,	PUNCT
ejpam-6567	229	52	σ2)continuous	σ2)continuous	ADJ
ejpam-6567	229	53	,	,	PUNCT
ejpam-6567	229	54	there	there	PRON
ejpam-6567	229	55	exists	exist	VERB
ejpam-6567	229	56	a	a	DET
ejpam-6567	229	57	⋆-open	⋆-open	ADJ
ejpam-6567	229	58	set	set	NOUN
ejpam-6567	229	59	u	u	NOUN
ejpam-6567	229	60	of	of	ADP
ejpam-6567	229	61	x	x	PUNCT
ejpam-6567	229	62	containing	contain	VERB
ejpam-6567	229	63	x	x	PUNCT
ejpam-6567	229	64	such	such	ADJ
ejpam-6567	229	65	that	that	SCONJ
ejpam-6567	229	66	σ1σ2	σ1σ2	NOUN
ejpam-6567	229	67	-	-	NUM
ejpam-6567	229	68	cl(v	cl(v	PUNCT
ejpam-6567	229	69	)	)	PUNCT
ejpam-6567	229	70	∩f	∩f	NOUN
ejpam-6567	229	71	(	(	PUNCT
ejpam-6567	229	72	z	z	X
ejpam-6567	229	73	)	)	PUNCT
ejpam-6567	229	74	̸=	̸=	NOUN
ejpam-6567	229	75	∅	∅	NOUN
ejpam-6567	229	76	for	for	ADP
ejpam-6567	229	77	each	each	DET
ejpam-6567	229	78	z	z	NOUN
ejpam-6567	229	79	∈	∈	PROPN
ejpam-6567	229	80	u	u	NOUN
ejpam-6567	229	81	and	and	CCONJ
ejpam-6567	229	82	hence	hence	ADV
ejpam-6567	229	83	u	u	NOUN
ejpam-6567	229	84	⊆	⊆	NUM
ejpam-6567	229	85	f−(b	f−(b	NOUN
ejpam-6567	229	86	)	)	PUNCT
ejpam-6567	229	87	.	.	PUNCT
ejpam-6567	230	1	theorem	theorem	VERB
ejpam-6567	230	2	7	7	NUM
ejpam-6567	230	3	.	.	X
ejpam-6567	230	4	for	for	ADP
ejpam-6567	230	5	a	a	DET
ejpam-6567	230	6	multifunction	multifunction	NOUN
ejpam-6567	231	1	f	f	NOUN
ejpam-6567	231	2	:	:	PUNCT
ejpam-6567	231	3	(	(	PUNCT
ejpam-6567	231	4	x	x	X
ejpam-6567	231	5	,	,	PUNCT
ejpam-6567	231	6	τ	τ	PROPN
ejpam-6567	231	7	,	,	PUNCT
ejpam-6567	231	8	i	i	NOUN
ejpam-6567	231	9	)	)	PUNCT
ejpam-6567	231	10	→	→	PUNCT
ejpam-6567	231	11	(	(	PUNCT
ejpam-6567	231	12	y	y	PROPN
ejpam-6567	231	13	,	,	PUNCT
ejpam-6567	231	14	σ1	σ1	PROPN
ejpam-6567	231	15	,	,	PUNCT
ejpam-6567	231	16	σ2	σ2	NOUN
ejpam-6567	231	17	)	)	PUNCT
ejpam-6567	231	18	,	,	PUNCT
ejpam-6567	231	19	the	the	DET
ejpam-6567	231	20	following	follow	VERB
ejpam-6567	231	21	properties	property	NOUN
ejpam-6567	231	22	are	be	AUX
ejpam-6567	231	23	equivalent	equivalent	ADJ
ejpam-6567	231	24	:	:	PUNCT
ejpam-6567	231	25	(	(	PUNCT
ejpam-6567	231	26	1	1	X
ejpam-6567	231	27	)	)	PUNCT
ejpam-6567	231	28	f	f	PROPN
ejpam-6567	231	29	is	be	AUX
ejpam-6567	231	30	lower	low	ADJ
ejpam-6567	231	31	weakly	weakly	ADJ
ejpam-6567	231	32	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	231	33	,	,	PUNCT
ejpam-6567	231	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	231	35	;	;	PUNCT
ejpam-6567	231	36	(	(	PUNCT
ejpam-6567	231	37	2	2	X
ejpam-6567	231	38	)	)	PUNCT
ejpam-6567	231	39	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6567	231	40	)	)	PUNCT
ejpam-6567	231	41	)	)	PUNCT
ejpam-6567	231	42	⊆	⊆	NUM
ejpam-6567	231	43	f+((σ1	f+((σ1	NOUN
ejpam-6567	231	44	,	,	PUNCT
ejpam-6567	231	45	σ2)θ	σ2)θ	ADJ
ejpam-6567	231	46	-	-	PUNCT
ejpam-6567	231	47	cl(b	cl(b	NOUN
ejpam-6567	231	48	)	)	PUNCT
ejpam-6567	231	49	)	)	PUNCT
ejpam-6567	231	50	for	for	ADP
ejpam-6567	231	51	every	every	DET
ejpam-6567	231	52	subset	subset	NOUN
ejpam-6567	231	53	b	b	PROPN
ejpam-6567	231	54	of	of	ADP
ejpam-6567	231	55	y	y	PROPN
ejpam-6567	231	56	;	;	PUNCT
ejpam-6567	231	57	(	(	PUNCT
ejpam-6567	231	58	3	3	X
ejpam-6567	231	59	)	)	PUNCT
ejpam-6567	231	60	f	f	NOUN
ejpam-6567	231	61	(	(	PUNCT
ejpam-6567	231	62	cl⋆(a	cl⋆(a	PROPN
ejpam-6567	231	63	)	)	PUNCT
ejpam-6567	231	64	)	)	PUNCT
ejpam-6567	232	1	⊆	⊆	NUM
ejpam-6567	232	2	(	(	PUNCT
ejpam-6567	232	3	σ1	σ1	PROPN
ejpam-6567	232	4	,	,	PUNCT
ejpam-6567	232	5	σ2)θ	σ2)θ	NOUN
ejpam-6567	232	6	-	-	PUNCT
ejpam-6567	232	7	cl(f	cl(f	PROPN
ejpam-6567	232	8	(	(	PUNCT
ejpam-6567	232	9	a	a	NOUN
ejpam-6567	232	10	)	)	PUNCT
ejpam-6567	232	11	)	)	PUNCT
ejpam-6567	232	12	for	for	ADP
ejpam-6567	232	13	every	every	DET
ejpam-6567	232	14	subset	subset	NOUN
ejpam-6567	232	15	a	a	PRON
ejpam-6567	232	16	of	of	ADP
ejpam-6567	232	17	x.	x.	NOUN
ejpam-6567	232	18	proof	proof	NOUN
ejpam-6567	232	19	.	.	PUNCT
ejpam-6567	233	1	(	(	PUNCT
ejpam-6567	233	2	1	1	X
ejpam-6567	233	3	)	)	PUNCT
ejpam-6567	233	4	⇒	⇒	NOUN
ejpam-6567	233	5	(	(	PUNCT
ejpam-6567	233	6	2	2	NUM
ejpam-6567	233	7	):	):	PUNCT
ejpam-6567	233	8	let	let	VERB
ejpam-6567	233	9	b	b	X
ejpam-6567	233	10	be	be	AUX
ejpam-6567	233	11	any	any	DET
ejpam-6567	233	12	subset	subset	NOUN
ejpam-6567	233	13	of	of	ADP
ejpam-6567	233	14	y	y	PROPN
ejpam-6567	233	15	.	.	PUNCT
ejpam-6567	233	16	suppose	suppose	VERB
ejpam-6567	233	17	that	that	SCONJ
ejpam-6567	233	18	x	x	PROPN
ejpam-6567	233	19	̸∈	̸∈	PROPN
ejpam-6567	233	20	f+((σ1	f+((σ1	ADV
ejpam-6567	233	21	,	,	PUNCT
ejpam-6567	233	22	σ2)θ	σ2)θ	ADJ
ejpam-6567	233	23	-	-	PUNCT
ejpam-6567	233	24	cl(b	cl(b	NOUN
ejpam-6567	233	25	)	)	PUNCT
ejpam-6567	233	26	)	)	PUNCT
ejpam-6567	233	27	.	.	PUNCT
ejpam-6567	234	1	then	then	ADV
ejpam-6567	234	2	,	,	PUNCT
ejpam-6567	234	3	we	we	PRON
ejpam-6567	234	4	have	have	VERB
ejpam-6567	234	5	x	x	NOUN
ejpam-6567	234	6	∈	∈	NOUN
ejpam-6567	234	7	f−(y	f−(y	NOUN
ejpam-6567	234	8	−	−	PROPN
ejpam-6567	234	9	(	(	PUNCT
ejpam-6567	234	10	σ1	σ1	PROPN
ejpam-6567	234	11	,	,	PUNCT
ejpam-6567	234	12	σ2)θ	σ2)θ	NOUN
ejpam-6567	234	13	-	-	PUNCT
ejpam-6567	234	14	cl(b	cl(b	NOUN
ejpam-6567	234	15	)	)	PUNCT
ejpam-6567	234	16	)	)	PUNCT
ejpam-6567	235	1	=	=	SYM
ejpam-6567	235	2	f−((σ1	f−((σ1	NOUN
ejpam-6567	235	3	,	,	PUNCT
ejpam-6567	235	4	σ2)θ	σ2)θ	ADJ
ejpam-6567	235	5	-	-	PUNCT
ejpam-6567	235	6	int(y	int(y	PROPN
ejpam-6567	235	7	−	−	PROPN
ejpam-6567	235	8	b	b	NOUN
ejpam-6567	235	9	)	)	PUNCT
ejpam-6567	235	10	)	)	PUNCT
ejpam-6567	235	11	.	.	PUNCT
ejpam-6567	236	1	by	by	ADP
ejpam-6567	236	2	lemma	lemma	PROPN
ejpam-6567	236	3	3	3	NUM
ejpam-6567	236	4	,	,	PUNCT
ejpam-6567	236	5	there	there	PRON
ejpam-6567	236	6	exists	exist	VERB
ejpam-6567	236	7	a	a	DET
ejpam-6567	236	8	⋆-open	⋆-open	ADJ
ejpam-6567	236	9	set	set	NOUN
ejpam-6567	236	10	u	u	NOUN
ejpam-6567	236	11	of	of	ADP
ejpam-6567	236	12	x	x	PUNCT
ejpam-6567	236	13	containing	contain	VERB
ejpam-6567	236	14	x	x	PUNCT
ejpam-6567	236	15	such	such	ADJ
ejpam-6567	236	16	that	that	SCONJ
ejpam-6567	236	17	u	u	NOUN
ejpam-6567	236	18	⊆	⊆	NUM
ejpam-6567	236	19	f−(y	f−(y	NOUN
ejpam-6567	236	20	−	−	NOUN
ejpam-6567	236	21	b	b	NOUN
ejpam-6567	236	22	)	)	PUNCT
ejpam-6567	236	23	=	=	PUNCT
ejpam-6567	236	24	x	x	X
ejpam-6567	237	1	−	−	NOUN
ejpam-6567	237	2	f+(b	f+(b	NOUN
ejpam-6567	237	3	)	)	PUNCT
ejpam-6567	237	4	.	.	PUNCT
ejpam-6567	238	1	thus	thus	ADV
ejpam-6567	238	2	,	,	PUNCT
ejpam-6567	238	3	u	u	NOUN
ejpam-6567	238	4	∩	∩	NOUN
ejpam-6567	238	5	f+(b	f+(b	NOUN
ejpam-6567	238	6	)	)	PUNCT
ejpam-6567	238	7	=	=	SYM
ejpam-6567	238	8	∅	∅	NOUN
ejpam-6567	238	9	and	and	CCONJ
ejpam-6567	238	10	hence	hence	ADV
ejpam-6567	238	11	x	x	X
ejpam-6567	238	12	̸∈	̸∈	PROPN
ejpam-6567	238	13	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6567	238	14	)	)	PUNCT
ejpam-6567	238	15	)	)	PUNCT
ejpam-6567	238	16	.	.	PUNCT
ejpam-6567	239	1	this	this	PRON
ejpam-6567	239	2	shows	show	VERB
ejpam-6567	239	3	that	that	SCONJ
ejpam-6567	239	4	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6567	239	5	)	)	PUNCT
ejpam-6567	239	6	)	)	PUNCT
ejpam-6567	240	1	⊆	⊆	NUM
ejpam-6567	240	2	f+((σ1	f+((σ1	NOUN
ejpam-6567	240	3	,	,	PUNCT
ejpam-6567	240	4	σ2)θ	σ2)θ	ADJ
ejpam-6567	240	5	-	-	PUNCT
ejpam-6567	240	6	cl(b	cl(b	NOUN
ejpam-6567	240	7	)	)	PUNCT
ejpam-6567	240	8	)	)	PUNCT
ejpam-6567	240	9	.	.	PUNCT
ejpam-6567	241	1	p.	p.	NOUN
ejpam-6567	241	2	pue	pue	NOUN
ejpam-6567	241	3	-	-	PUNCT
ejpam-6567	241	4	on	on	ADP
ejpam-6567	241	5	,	,	PUNCT
ejpam-6567	241	6	a.	a.	PROPN
ejpam-6567	241	7	sama	sama	PROPN
ejpam-6567	241	8	-	-	PUNCT
ejpam-6567	241	9	ae	ae	PROPN
ejpam-6567	241	10	,	,	PUNCT
ejpam-6567	241	11	c.	c.	PROPN
ejpam-6567	241	12	boonpok	boonpok	PROPN
ejpam-6567	241	13	/	/	SYM
ejpam-6567	241	14	eur	eur	PROPN
ejpam-6567	241	15	.	.	PUNCT
ejpam-6567	242	1	j.	j.	PROPN
ejpam-6567	242	2	pure	pure	PROPN
ejpam-6567	242	3	appl	appl	PROPN
ejpam-6567	242	4	.	.	PROPN
ejpam-6567	242	5	math	math	PROPN
ejpam-6567	242	6	,	,	PUNCT
ejpam-6567	242	7	18	18	NUM
ejpam-6567	242	8	(	(	PUNCT
ejpam-6567	242	9	3	3	NUM
ejpam-6567	242	10	)	)	PUNCT
ejpam-6567	242	11	(	(	PUNCT
ejpam-6567	242	12	2025	2025	NUM
ejpam-6567	242	13	)	)	PUNCT
ejpam-6567	242	14	,	,	PUNCT
ejpam-6567	242	15	6567	6567	NUM
ejpam-6567	242	16	9	9	NUM
ejpam-6567	242	17	of	of	ADP
ejpam-6567	242	18	13	13	NUM
ejpam-6567	242	19	(	(	PUNCT
ejpam-6567	242	20	2	2	NUM
ejpam-6567	242	21	)	)	PUNCT
ejpam-6567	242	22	⇒	⇒	NOUN
ejpam-6567	242	23	(	(	PUNCT
ejpam-6567	242	24	3	3	NUM
ejpam-6567	242	25	):	):	PUNCT
ejpam-6567	242	26	let	let	VERB
ejpam-6567	242	27	a	a	DET
ejpam-6567	242	28	be	be	AUX
ejpam-6567	242	29	any	any	DET
ejpam-6567	242	30	subset	subset	NOUN
ejpam-6567	242	31	of	of	ADP
ejpam-6567	242	32	x.	x.	NOUN
ejpam-6567	242	33	by	by	ADP
ejpam-6567	242	34	(	(	PUNCT
ejpam-6567	242	35	2	2	NUM
ejpam-6567	242	36	)	)	PUNCT
ejpam-6567	242	37	,	,	PUNCT
ejpam-6567	242	38	we	we	PRON
ejpam-6567	242	39	have	have	VERB
ejpam-6567	242	40	cl⋆(a	cl⋆(a	NOUN
ejpam-6567	242	41	)	)	PUNCT
ejpam-6567	242	42	⊆	⊆	NUM
ejpam-6567	242	43	cl⋆(f+(f	cl⋆(f+(f	NOUN
ejpam-6567	242	44	(	(	PUNCT
ejpam-6567	242	45	a	a	NOUN
ejpam-6567	242	46	)	)	PUNCT
ejpam-6567	242	47	)	)	PUNCT
ejpam-6567	242	48	)	)	PUNCT
ejpam-6567	243	1	⊆	⊆	NUM
ejpam-6567	243	2	f+((σ1	f+((σ1	NOUN
ejpam-6567	243	3	,	,	PUNCT
ejpam-6567	243	4	σ2)θ	σ2)θ	NOUN
ejpam-6567	243	5	-	-	PUNCT
ejpam-6567	243	6	cl(f	cl(f	PROPN
ejpam-6567	243	7	(	(	PUNCT
ejpam-6567	243	8	a	a	NOUN
ejpam-6567	243	9	)	)	PUNCT
ejpam-6567	243	10	)	)	PUNCT
ejpam-6567	243	11	)	)	PUNCT
ejpam-6567	243	12	.	.	PUNCT
ejpam-6567	244	1	thus	thus	ADV
ejpam-6567	244	2	,	,	PUNCT
ejpam-6567	244	3	f	f	PROPN
ejpam-6567	244	4	(	(	PUNCT
ejpam-6567	244	5	cl⋆(a	cl⋆(a	PROPN
ejpam-6567	244	6	)	)	PUNCT
ejpam-6567	244	7	)	)	PUNCT
ejpam-6567	244	8	⊆	⊆	NUM
ejpam-6567	244	9	(	(	PUNCT
ejpam-6567	244	10	σ1	σ1	PROPN
ejpam-6567	244	11	,	,	PUNCT
ejpam-6567	244	12	σ2)θ	σ2)θ	NOUN
ejpam-6567	244	13	-	-	PUNCT
ejpam-6567	244	14	cl(f	cl(f	PROPN
ejpam-6567	244	15	(	(	PUNCT
ejpam-6567	244	16	a	a	NOUN
ejpam-6567	244	17	)	)	PUNCT
ejpam-6567	244	18	)	)	PUNCT
ejpam-6567	244	19	.	.	PUNCT
ejpam-6567	245	1	(	(	PUNCT
ejpam-6567	245	2	3	3	X
ejpam-6567	245	3	)	)	PUNCT
ejpam-6567	245	4	⇒	⇒	NOUN
ejpam-6567	245	5	(	(	PUNCT
ejpam-6567	245	6	1	1	NUM
ejpam-6567	245	7	):	):	PUNCT
ejpam-6567	245	8	let	let	VERB
ejpam-6567	245	9	v	v	PART
ejpam-6567	245	10	be	be	AUX
ejpam-6567	245	11	any	any	DET
ejpam-6567	245	12	σ1σ2	σ1σ2	NOUN
ejpam-6567	245	13	-	-	ADJ
ejpam-6567	245	14	open	open	ADJ
ejpam-6567	245	15	set	set	NOUN
ejpam-6567	245	16	of	of	ADP
ejpam-6567	245	17	y	y	PROPN
ejpam-6567	245	18	.	.	PUNCT
ejpam-6567	246	1	then	then	ADV
ejpam-6567	246	2	,	,	PUNCT
ejpam-6567	246	3	σ1σ2	σ1σ2	NOUN
ejpam-6567	246	4	-	-	NUM
ejpam-6567	246	5	cl(v	cl(v	X
ejpam-6567	246	6	)	)	PUNCT
ejpam-6567	246	7	=	=	SYM
ejpam-6567	246	8	(	(	PUNCT
ejpam-6567	246	9	σ1	σ1	PROPN
ejpam-6567	246	10	,	,	PUNCT
ejpam-6567	246	11	σ2)θ	σ2)θ	NOUN
ejpam-6567	246	12	-	-	PUNCT
ejpam-6567	246	13	cl(v	cl(v	X
ejpam-6567	246	14	)	)	PUNCT
ejpam-6567	246	15	and	and	CCONJ
ejpam-6567	246	16	by	by	ADP
ejpam-6567	246	17	(	(	PUNCT
ejpam-6567	246	18	3	3	NUM
ejpam-6567	246	19	)	)	PUNCT
ejpam-6567	246	20	,	,	PUNCT
ejpam-6567	246	21	f	f	PROPN
ejpam-6567	246	22	(	(	PUNCT
ejpam-6567	246	23	cl⋆(f+(v	cl⋆(f+(v	PROPN
ejpam-6567	246	24	)	)	PUNCT
ejpam-6567	246	25	)	)	PUNCT
ejpam-6567	246	26	)	)	PUNCT
ejpam-6567	247	1	⊆	⊆	X
ejpam-6567	247	2	(	(	PUNCT
ejpam-6567	247	3	σ1	σ1	PROPN
ejpam-6567	247	4	,	,	PUNCT
ejpam-6567	247	5	σ2)θ	σ2)θ	NOUN
ejpam-6567	247	6	-	-	PUNCT
ejpam-6567	247	7	cl(f	cl(f	PROPN
ejpam-6567	247	8	(	(	PUNCT
ejpam-6567	247	9	f+(v	f+(v	PROPN
ejpam-6567	247	10	)	)	PUNCT
ejpam-6567	247	11	)	)	PUNCT
ejpam-6567	247	12	)	)	PUNCT
ejpam-6567	248	1	⊆	⊆	X
ejpam-6567	248	2	(	(	PUNCT
ejpam-6567	248	3	σ1	σ1	PROPN
ejpam-6567	248	4	,	,	PUNCT
ejpam-6567	248	5	σ2)θ	σ2)θ	NOUN
ejpam-6567	248	6	-	-	PUNCT
ejpam-6567	248	7	cl(v	cl(v	NOUN
ejpam-6567	248	8	)	)	PUNCT
ejpam-6567	248	9	=	=	SYM
ejpam-6567	248	10	σ1σ2	σ1σ2	NOUN
ejpam-6567	248	11	-	-	NUM
ejpam-6567	248	12	cl(v	cl(v	NOUN
ejpam-6567	248	13	)	)	PUNCT
ejpam-6567	248	14	.	.	PUNCT
ejpam-6567	249	1	thus	thus	ADV
ejpam-6567	249	2	,	,	PUNCT
ejpam-6567	249	3	cl⋆(f+(v	cl⋆(f+(v	PROPN
ejpam-6567	249	4	)	)	PUNCT
ejpam-6567	249	5	)	)	PUNCT
ejpam-6567	250	1	⊆	⊆	X
ejpam-6567	250	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6567	250	3	-	-	PUNCT
ejpam-6567	250	4	cl(v	cl(v	NOUN
ejpam-6567	250	5	)	)	PUNCT
ejpam-6567	250	6	)	)	PUNCT
ejpam-6567	250	7	and	and	CCONJ
ejpam-6567	250	8	by	by	ADP
ejpam-6567	250	9	theorem	theorem	NOUN
ejpam-6567	250	10	1	1	NUM
ejpam-6567	250	11	,	,	PUNCT
ejpam-6567	250	12	f	f	PROPN
ejpam-6567	250	13	is	be	AUX
ejpam-6567	250	14	lower	low	ADJ
ejpam-6567	250	15	weakly	weakly	ADJ
ejpam-6567	250	16	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	250	17	,	,	PUNCT
ejpam-6567	250	18	σ2)continuous	σ2)continuous	ADJ
ejpam-6567	250	19	.	.	PUNCT
ejpam-6567	251	1	theorem	theorem	VERB
ejpam-6567	251	2	8	8	NUM
ejpam-6567	251	3	.	.	PUNCT
ejpam-6567	252	1	for	for	ADP
ejpam-6567	252	2	a	a	DET
ejpam-6567	252	3	multifunction	multifunction	NOUN
ejpam-6567	252	4	f	f	NOUN
ejpam-6567	252	5	:	:	PUNCT
ejpam-6567	252	6	(	(	PUNCT
ejpam-6567	252	7	x	x	X
ejpam-6567	252	8	,	,	PUNCT
ejpam-6567	252	9	τ	τ	PROPN
ejpam-6567	252	10	,	,	PUNCT
ejpam-6567	252	11	i	i	NOUN
ejpam-6567	252	12	)	)	PUNCT
ejpam-6567	252	13	→	→	PUNCT
ejpam-6567	252	14	(	(	PUNCT
ejpam-6567	252	15	y	y	PROPN
ejpam-6567	252	16	,	,	PUNCT
ejpam-6567	252	17	σ1	σ1	PROPN
ejpam-6567	252	18	,	,	PUNCT
ejpam-6567	252	19	σ2	σ2	NOUN
ejpam-6567	252	20	)	)	PUNCT
ejpam-6567	252	21	,	,	PUNCT
ejpam-6567	252	22	the	the	DET
ejpam-6567	252	23	following	follow	VERB
ejpam-6567	252	24	properties	property	NOUN
ejpam-6567	252	25	are	be	AUX
ejpam-6567	252	26	equivalent	equivalent	ADJ
ejpam-6567	252	27	:	:	PUNCT
ejpam-6567	252	28	(	(	PUNCT
ejpam-6567	252	29	1	1	X
ejpam-6567	252	30	)	)	PUNCT
ejpam-6567	252	31	f	f	PROPN
ejpam-6567	252	32	is	be	AUX
ejpam-6567	252	33	upper	upper	ADJ
ejpam-6567	252	34	weakly	weakly	ADJ
ejpam-6567	252	35	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	252	36	,	,	PUNCT
ejpam-6567	252	37	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	252	38	;	;	PUNCT
ejpam-6567	252	39	(	(	PUNCT
ejpam-6567	252	40	2	2	X
ejpam-6567	252	41	)	)	PUNCT
ejpam-6567	252	42	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	252	43	-	-	PUNCT
ejpam-6567	252	44	int((σ1	int((σ1	PROPN
ejpam-6567	252	45	,	,	PUNCT
ejpam-6567	252	46	σ2)θ	σ2)θ	ADJ
ejpam-6567	252	47	-	-	PUNCT
ejpam-6567	252	48	cl(b	cl(b	NOUN
ejpam-6567	252	49	)	)	PUNCT
ejpam-6567	252	50	)	)	PUNCT
ejpam-6567	252	51	)	)	PUNCT
ejpam-6567	252	52	)	)	PUNCT
ejpam-6567	253	1	⊆	⊆	NUM
ejpam-6567	253	2	f−((σ1	f−((σ1	NOUN
ejpam-6567	253	3	,	,	PUNCT
ejpam-6567	253	4	σ2)θ	σ2)θ	ADJ
ejpam-6567	253	5	-	-	PUNCT
ejpam-6567	253	6	cl(b	cl(b	NOUN
ejpam-6567	253	7	)	)	PUNCT
ejpam-6567	253	8	)	)	PUNCT
ejpam-6567	253	9	for	for	ADP
ejpam-6567	253	10	every	every	DET
ejpam-6567	253	11	subset	subset	NOUN
ejpam-6567	253	12	b	b	PROPN
ejpam-6567	253	13	of	of	ADP
ejpam-6567	253	14	y	y	PROPN
ejpam-6567	253	15	;	;	PUNCT
ejpam-6567	253	16	(	(	PUNCT
ejpam-6567	253	17	3	3	X
ejpam-6567	253	18	)	)	PUNCT
ejpam-6567	253	19	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	253	20	-	-	PUNCT
ejpam-6567	253	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	253	22	-	-	PUNCT
ejpam-6567	253	23	cl(b	cl(b	NOUN
ejpam-6567	253	24	)	)	PUNCT
ejpam-6567	253	25	)	)	PUNCT
ejpam-6567	253	26	)	)	PUNCT
ejpam-6567	253	27	)	)	PUNCT
ejpam-6567	253	28	⊆	⊆	NUM
ejpam-6567	253	29	f−((σ1	f−((σ1	NOUN
ejpam-6567	253	30	,	,	PUNCT
ejpam-6567	253	31	σ2)θ	σ2)θ	ADJ
ejpam-6567	253	32	-	-	PUNCT
ejpam-6567	253	33	cl(b	cl(b	NOUN
ejpam-6567	253	34	)	)	PUNCT
ejpam-6567	253	35	)	)	PUNCT
ejpam-6567	253	36	for	for	ADP
ejpam-6567	253	37	every	every	DET
ejpam-6567	253	38	subset	subset	NOUN
ejpam-6567	253	39	b	b	PROPN
ejpam-6567	253	40	of	of	ADP
ejpam-6567	253	41	y	y	PROPN
ejpam-6567	253	42	.	.	PUNCT
ejpam-6567	254	1	proof	proof	NOUN
ejpam-6567	254	2	.	.	PUNCT
ejpam-6567	255	1	(	(	PUNCT
ejpam-6567	255	2	1	1	X
ejpam-6567	255	3	)	)	PUNCT
ejpam-6567	255	4	⇒	⇒	NOUN
ejpam-6567	255	5	(	(	PUNCT
ejpam-6567	255	6	2	2	NUM
ejpam-6567	255	7	):	):	PUNCT
ejpam-6567	255	8	let	let	VERB
ejpam-6567	255	9	b	b	X
ejpam-6567	255	10	be	be	AUX
ejpam-6567	255	11	any	any	DET
ejpam-6567	255	12	subset	subset	NOUN
ejpam-6567	255	13	of	of	ADP
ejpam-6567	255	14	y	y	PROPN
ejpam-6567	255	15	.	.	PUNCT
ejpam-6567	256	1	then	then	ADV
ejpam-6567	256	2	,	,	PUNCT
ejpam-6567	256	3	(	(	PUNCT
ejpam-6567	256	4	σ1	σ1	PROPN
ejpam-6567	256	5	,	,	PUNCT
ejpam-6567	256	6	σ2)θ	σ2)θ	NOUN
ejpam-6567	256	7	-	-	PUNCT
ejpam-6567	256	8	cl(b	cl(b	NOUN
ejpam-6567	256	9	)	)	PUNCT
ejpam-6567	256	10	is	be	AUX
ejpam-6567	256	11	σ1σ2	σ1σ2	NOUN
ejpam-6567	256	12	-	-	ADJ
ejpam-6567	256	13	closed	closed	ADJ
ejpam-6567	256	14	in	in	ADP
ejpam-6567	256	15	y	y	PROPN
ejpam-6567	256	16	and	and	CCONJ
ejpam-6567	256	17	by	by	ADP
ejpam-6567	256	18	theorem	theorem	ADJ
ejpam-6567	256	19	2	2	NUM
ejpam-6567	256	20	,	,	PUNCT
ejpam-6567	256	21	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	NOUN
ejpam-6567	256	22	-	-	PUNCT
ejpam-6567	256	23	int((σ1	int((σ1	NUM
ejpam-6567	256	24	,	,	PUNCT
ejpam-6567	256	25	σ2)θ	σ2)θ	ADJ
ejpam-6567	256	26	-	-	PUNCT
ejpam-6567	256	27	cl(b	cl(b	NOUN
ejpam-6567	256	28	)	)	PUNCT
ejpam-6567	256	29	)	)	PUNCT
ejpam-6567	256	30	)	)	PUNCT
ejpam-6567	256	31	)	)	PUNCT
ejpam-6567	257	1	⊆	⊆	NUM
ejpam-6567	257	2	f−((σ1	f−((σ1	NOUN
ejpam-6567	257	3	,	,	PUNCT
ejpam-6567	257	4	σ2)θ	σ2)θ	ADJ
ejpam-6567	257	5	-	-	PUNCT
ejpam-6567	257	6	cl(b	cl(b	NOUN
ejpam-6567	257	7	)	)	PUNCT
ejpam-6567	257	8	)	)	PUNCT
ejpam-6567	257	9	.	.	PUNCT
ejpam-6567	258	1	(	(	PUNCT
ejpam-6567	258	2	2	2	X
ejpam-6567	258	3	)	)	PUNCT
ejpam-6567	258	4	⇒	⇒	NOUN
ejpam-6567	258	5	(	(	PUNCT
ejpam-6567	258	6	3	3	NUM
ejpam-6567	258	7	):	):	PUNCT
ejpam-6567	258	8	the	the	DET
ejpam-6567	258	9	proof	proof	NOUN
ejpam-6567	258	10	is	be	AUX
ejpam-6567	258	11	obvious	obvious	ADJ
ejpam-6567	258	12	.	.	PUNCT
ejpam-6567	259	1	(	(	PUNCT
ejpam-6567	259	2	3	3	X
ejpam-6567	259	3	)	)	PUNCT
ejpam-6567	259	4	⇒	⇒	NOUN
ejpam-6567	259	5	(	(	PUNCT
ejpam-6567	259	6	1	1	NUM
ejpam-6567	259	7	):	):	PUNCT
ejpam-6567	259	8	let	let	VERB
ejpam-6567	259	9	k	k	PRON
ejpam-6567	259	10	be	be	AUX
ejpam-6567	259	11	any	any	DET
ejpam-6567	259	12	(	(	PUNCT
ejpam-6567	259	13	σ1	σ1	NOUN
ejpam-6567	259	14	,	,	PUNCT
ejpam-6567	259	15	σ2)r	σ2)r	NOUN
ejpam-6567	259	16	-	-	PUNCT
ejpam-6567	259	17	closed	close	VERB
ejpam-6567	259	18	set	set	NOUN
ejpam-6567	259	19	of	of	ADP
ejpam-6567	259	20	y	y	PROPN
ejpam-6567	259	21	.	.	PUNCT
ejpam-6567	260	1	then	then	ADV
ejpam-6567	260	2	,	,	PUNCT
ejpam-6567	260	3	we	we	PRON
ejpam-6567	260	4	have	have	VERB
ejpam-6567	260	5	(	(	PUNCT
ejpam-6567	260	6	σ1	σ1	PROPN
ejpam-6567	260	7	,	,	PUNCT
ejpam-6567	260	8	σ2)θ	σ2)θ	NOUN
ejpam-6567	260	9	-	-	PUNCT
ejpam-6567	260	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	260	11	-	-	PUNCT
ejpam-6567	260	12	int(k	int(k	NOUN
ejpam-6567	260	13	)	)	PUNCT
ejpam-6567	260	14	)	)	PUNCT
ejpam-6567	261	1	=	=	SYM
ejpam-6567	261	2	σ1σ2	σ1σ2	X
ejpam-6567	261	3	-	-	PUNCT
ejpam-6567	261	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	261	5	-	-	PUNCT
ejpam-6567	261	6	int(k	int(k	NOUN
ejpam-6567	261	7	)	)	PUNCT
ejpam-6567	261	8	)	)	PUNCT
ejpam-6567	262	1	=	=	SYM
ejpam-6567	263	1	k	k	NOUN
ejpam-6567	263	2	and	and	CCONJ
ejpam-6567	263	3	hence	hence	ADV
ejpam-6567	263	4	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6567	263	5	-	-	PUNCT
ejpam-6567	263	6	int(k	int(k	NOUN
ejpam-6567	263	7	)	)	PUNCT
ejpam-6567	263	8	)	)	PUNCT
ejpam-6567	263	9	)	)	PUNCT
ejpam-6567	264	1	=	=	PUNCT
ejpam-6567	264	2	cl⋆(σ1σ2	cl⋆(σ1σ2	VERB
ejpam-6567	264	3	-	-	PUNCT
ejpam-6567	264	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	264	5	-	-	PUNCT
ejpam-6567	264	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6567	264	7	-	-	PUNCT
ejpam-6567	264	8	int(k	int(k	NOUN
ejpam-6567	264	9	)	)	PUNCT
ejpam-6567	264	10	)	)	PUNCT
ejpam-6567	264	11	)	)	PUNCT
ejpam-6567	264	12	)	)	PUNCT
ejpam-6567	265	1	⊆	⊆	NUM
ejpam-6567	265	2	f−((σ1	f−((σ1	NOUN
ejpam-6567	265	3	,	,	PUNCT
ejpam-6567	265	4	σ2)θ	σ2)θ	NOUN
ejpam-6567	265	5	-	-	PUNCT
ejpam-6567	265	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	265	7	-	-	PUNCT
ejpam-6567	265	8	int(k	int(k	NOUN
ejpam-6567	265	9	)	)	PUNCT
ejpam-6567	265	10	)	)	PUNCT
ejpam-6567	265	11	)	)	PUNCT
ejpam-6567	266	1	=	=	PUNCT
ejpam-6567	266	2	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6567	266	3	-	-	PUNCT
ejpam-6567	266	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6567	266	5	-	-	PUNCT
ejpam-6567	266	6	int(k	int(k	NOUN
ejpam-6567	266	7	)	)	PUNCT
ejpam-6567	266	8	)	)	PUNCT
ejpam-6567	266	9	)	)	PUNCT
ejpam-6567	266	10	=	=	SYM
ejpam-6567	266	11	f−(k	f−(k	PROPN
ejpam-6567	266	12	)	)	PUNCT
ejpam-6567	266	13	.	.	PUNCT
ejpam-6567	267	1	thus	thus	ADV
ejpam-6567	267	2	by	by	ADP
ejpam-6567	267	3	theorem	theorem	NOUN
ejpam-6567	267	4	2	2	NUM
ejpam-6567	267	5	,	,	PUNCT
ejpam-6567	267	6	f	f	PROPN
ejpam-6567	267	7	is	be	AUX
ejpam-6567	267	8	upper	upper	ADJ
ejpam-6567	267	9	weakly	weakly	ADJ
ejpam-6567	267	10	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	267	11	,	,	PUNCT
ejpam-6567	267	12	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6567	267	13	.	.	X
ejpam-6567	267	14	theorem	theorem	NOUN
ejpam-6567	267	15	9	9	NUM
ejpam-6567	267	16	.	.	X
ejpam-6567	267	17	for	for	ADP
ejpam-6567	267	18	a	a	DET
ejpam-6567	267	19	multifunction	multifunction	NOUN
ejpam-6567	267	20	f	f	NOUN
ejpam-6567	267	21	:	:	PUNCT
ejpam-6567	267	22	(	(	PUNCT
ejpam-6567	267	23	x	x	X
ejpam-6567	267	24	,	,	PUNCT
ejpam-6567	267	25	τ	τ	PROPN
ejpam-6567	267	26	,	,	PUNCT
ejpam-6567	267	27	i	i	NOUN
ejpam-6567	267	28	)	)	PUNCT
ejpam-6567	267	29	→	→	PUNCT
ejpam-6567	267	30	(	(	PUNCT
ejpam-6567	267	31	y	y	PROPN
ejpam-6567	267	32	,	,	PUNCT
ejpam-6567	267	33	σ1	σ1	PROPN
ejpam-6567	267	34	,	,	PUNCT
ejpam-6567	267	35	σ2	σ2	NOUN
ejpam-6567	267	36	)	)	PUNCT
ejpam-6567	267	37	,	,	PUNCT
ejpam-6567	267	38	the	the	DET
ejpam-6567	267	39	following	follow	VERB
ejpam-6567	267	40	properties	property	NOUN
ejpam-6567	267	41	are	be	AUX
ejpam-6567	267	42	equivalent	equivalent	ADJ
ejpam-6567	267	43	:	:	PUNCT
ejpam-6567	267	44	(	(	PUNCT
ejpam-6567	267	45	1	1	X
ejpam-6567	267	46	)	)	PUNCT
ejpam-6567	267	47	f	f	PROPN
ejpam-6567	267	48	is	be	AUX
ejpam-6567	267	49	lower	low	ADJ
ejpam-6567	267	50	weakly	weakly	ADJ
ejpam-6567	267	51	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	267	52	,	,	PUNCT
ejpam-6567	267	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	267	54	;	;	PUNCT
ejpam-6567	267	55	(	(	PUNCT
ejpam-6567	267	56	2	2	X
ejpam-6567	267	57	)	)	PUNCT
ejpam-6567	267	58	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	267	59	-	-	PUNCT
ejpam-6567	267	60	int((σ1	int((σ1	NUM
ejpam-6567	267	61	,	,	PUNCT
ejpam-6567	267	62	σ2)θ	σ2)θ	ADJ
ejpam-6567	267	63	-	-	PUNCT
ejpam-6567	267	64	cl(b	cl(b	NOUN
ejpam-6567	267	65	)	)	PUNCT
ejpam-6567	267	66	)	)	PUNCT
ejpam-6567	267	67	)	)	PUNCT
ejpam-6567	267	68	)	)	PUNCT
ejpam-6567	268	1	⊆	⊆	NUM
ejpam-6567	268	2	f+((σ1	f+((σ1	NOUN
ejpam-6567	268	3	,	,	PUNCT
ejpam-6567	268	4	σ2)θ	σ2)θ	ADJ
ejpam-6567	268	5	-	-	PUNCT
ejpam-6567	268	6	cl(b	cl(b	NOUN
ejpam-6567	268	7	)	)	PUNCT
ejpam-6567	268	8	)	)	PUNCT
ejpam-6567	268	9	for	for	ADP
ejpam-6567	268	10	every	every	DET
ejpam-6567	268	11	subset	subset	NOUN
ejpam-6567	268	12	b	b	PROPN
ejpam-6567	268	13	of	of	ADP
ejpam-6567	268	14	y	y	PROPN
ejpam-6567	268	15	;	;	PUNCT
ejpam-6567	268	16	(	(	PUNCT
ejpam-6567	268	17	3	3	X
ejpam-6567	268	18	)	)	PUNCT
ejpam-6567	268	19	cl⋆(f+(σ1σ2	cl⋆(f+(σ1σ2	NOUN
ejpam-6567	268	20	-	-	PUNCT
ejpam-6567	268	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6567	268	22	-	-	PUNCT
ejpam-6567	268	23	cl(b	cl(b	NOUN
ejpam-6567	268	24	)	)	PUNCT
ejpam-6567	268	25	)	)	PUNCT
ejpam-6567	268	26	)	)	PUNCT
ejpam-6567	268	27	)	)	PUNCT
ejpam-6567	269	1	⊆	⊆	NUM
ejpam-6567	269	2	f+((σ1	f+((σ1	NOUN
ejpam-6567	269	3	,	,	PUNCT
ejpam-6567	269	4	σ2)θ	σ2)θ	ADJ
ejpam-6567	269	5	-	-	PUNCT
ejpam-6567	269	6	cl(b	cl(b	NOUN
ejpam-6567	269	7	)	)	PUNCT
ejpam-6567	269	8	)	)	PUNCT
ejpam-6567	269	9	for	for	ADP
ejpam-6567	269	10	every	every	DET
ejpam-6567	269	11	subset	subset	NOUN
ejpam-6567	269	12	b	b	PROPN
ejpam-6567	269	13	of	of	ADP
ejpam-6567	269	14	y	y	PROPN
ejpam-6567	269	15	.	.	PUNCT
ejpam-6567	270	1	p.	p.	NOUN
ejpam-6567	270	2	pue	pue	NOUN
ejpam-6567	270	3	-	-	PUNCT
ejpam-6567	270	4	on	on	ADP
ejpam-6567	270	5	,	,	PUNCT
ejpam-6567	270	6	a.	a.	PROPN
ejpam-6567	270	7	sama	sama	PROPN
ejpam-6567	270	8	-	-	PUNCT
ejpam-6567	270	9	ae	ae	PROPN
ejpam-6567	270	10	,	,	PUNCT
ejpam-6567	270	11	c.	c.	PROPN
ejpam-6567	270	12	boonpok	boonpok	PROPN
ejpam-6567	270	13	/	/	SYM
ejpam-6567	270	14	eur	eur	PROPN
ejpam-6567	270	15	.	.	PUNCT
ejpam-6567	271	1	j.	j.	PROPN
ejpam-6567	271	2	pure	pure	PROPN
ejpam-6567	271	3	appl	appl	PROPN
ejpam-6567	271	4	.	.	PROPN
ejpam-6567	271	5	math	math	PROPN
ejpam-6567	271	6	,	,	PUNCT
ejpam-6567	271	7	18	18	NUM
ejpam-6567	271	8	(	(	PUNCT
ejpam-6567	271	9	3	3	NUM
ejpam-6567	271	10	)	)	PUNCT
ejpam-6567	271	11	(	(	PUNCT
ejpam-6567	271	12	2025	2025	NUM
ejpam-6567	271	13	)	)	PUNCT
ejpam-6567	271	14	,	,	PUNCT
ejpam-6567	271	15	6567	6567	NUM
ejpam-6567	271	16	10	10	NUM
ejpam-6567	271	17	of	of	ADP
ejpam-6567	271	18	13	13	NUM
ejpam-6567	271	19	proof	proof	NOUN
ejpam-6567	271	20	.	.	PUNCT
ejpam-6567	272	1	the	the	DET
ejpam-6567	272	2	proof	proof	NOUN
ejpam-6567	272	3	is	be	AUX
ejpam-6567	272	4	similar	similar	ADJ
ejpam-6567	272	5	to	to	ADP
ejpam-6567	272	6	that	that	PRON
ejpam-6567	272	7	of	of	ADP
ejpam-6567	272	8	theorem	theorem	ADJ
ejpam-6567	272	9	10	10	NUM
ejpam-6567	272	10	.	.	PUNCT
ejpam-6567	273	1	recall	recall	VERB
ejpam-6567	273	2	that	that	SCONJ
ejpam-6567	273	3	a	a	DET
ejpam-6567	273	4	subset	subset	NOUN
ejpam-6567	273	5	a	a	PRON
ejpam-6567	273	6	of	of	ADP
ejpam-6567	273	7	a	a	DET
ejpam-6567	273	8	bitopological	bitopological	ADJ
ejpam-6567	273	9	space	space	NOUN
ejpam-6567	273	10	(	(	PUNCT
ejpam-6567	273	11	x	x	NOUN
ejpam-6567	273	12	,	,	PUNCT
ejpam-6567	273	13	τ1	τ1	NOUN
ejpam-6567	273	14	,	,	PUNCT
ejpam-6567	273	15	τ2	τ2	NOUN
ejpam-6567	273	16	)	)	PUNCT
ejpam-6567	273	17	is	be	AUX
ejpam-6567	273	18	said	say	VERB
ejpam-6567	273	19	to	to	PART
ejpam-6567	273	20	be	be	AUX
ejpam-6567	273	21	τ1τ2	τ1τ2	NOUN
ejpam-6567	273	22	-	-	ADJ
ejpam-6567	273	23	paracompact	paracompact	ADJ
ejpam-6567	274	1	[	[	X
ejpam-6567	274	2	22	22	NUM
ejpam-6567	274	3	]	]	PUNCT
ejpam-6567	274	4	if	if	SCONJ
ejpam-6567	274	5	every	every	DET
ejpam-6567	274	6	cover	cover	NOUN
ejpam-6567	274	7	of	of	ADP
ejpam-6567	274	8	a	a	PRON
ejpam-6567	274	9	by	by	ADP
ejpam-6567	274	10	τ1τ2	τ1τ2	ADJ
ejpam-6567	274	11	-	-	ADJ
ejpam-6567	274	12	open	open	ADJ
ejpam-6567	274	13	sets	set	NOUN
ejpam-6567	274	14	of	of	ADP
ejpam-6567	274	15	x	x	VERB
ejpam-6567	274	16	is	be	AUX
ejpam-6567	274	17	refined	refine	VERB
ejpam-6567	274	18	by	by	ADP
ejpam-6567	274	19	a	a	DET
ejpam-6567	274	20	cover	cover	NOUN
ejpam-6567	274	21	of	of	ADP
ejpam-6567	274	22	a	a	PRON
ejpam-6567	274	23	which	which	PRON
ejpam-6567	274	24	consists	consist	VERB
ejpam-6567	274	25	of	of	ADP
ejpam-6567	274	26	τ1τ2	τ1τ2	ADJ
ejpam-6567	274	27	-	-	ADJ
ejpam-6567	274	28	open	open	ADJ
ejpam-6567	274	29	sets	set	NOUN
ejpam-6567	274	30	of	of	ADP
ejpam-6567	274	31	x	x	PUNCT
ejpam-6567	274	32	and	and	CCONJ
ejpam-6567	274	33	is	be	AUX
ejpam-6567	274	34	τ1τ2	τ1τ2	NOUN
ejpam-6567	274	35	-	-	ADJ
ejpam-6567	274	36	locally	locally	ADV
ejpam-6567	274	37	finite	finite	NOUN
ejpam-6567	274	38	in	in	ADP
ejpam-6567	274	39	x.	x.	PROPN
ejpam-6567	274	40	a	a	DET
ejpam-6567	274	41	subset	subset	VERB
ejpam-6567	274	42	a	a	PRON
ejpam-6567	274	43	of	of	ADP
ejpam-6567	274	44	a	a	DET
ejpam-6567	274	45	bitopological	bitopological	ADJ
ejpam-6567	274	46	space	space	NOUN
ejpam-6567	274	47	(	(	PUNCT
ejpam-6567	274	48	x	x	NOUN
ejpam-6567	274	49	,	,	PUNCT
ejpam-6567	274	50	τ1	τ1	NOUN
ejpam-6567	274	51	,	,	PUNCT
ejpam-6567	274	52	τ2	τ2	NOUN
ejpam-6567	274	53	)	)	PUNCT
ejpam-6567	274	54	is	be	AUX
ejpam-6567	274	55	said	say	VERB
ejpam-6567	274	56	to	to	PART
ejpam-6567	274	57	be	be	AUX
ejpam-6567	274	58	τ1τ2	τ1τ2	NOUN
ejpam-6567	274	59	-	-	ADJ
ejpam-6567	274	60	regular	regular	ADJ
ejpam-6567	274	61	[	[	X
ejpam-6567	274	62	22	22	NUM
ejpam-6567	274	63	]	]	X
ejpam-6567	274	64	if	if	SCONJ
ejpam-6567	274	65	for	for	ADP
ejpam-6567	274	66	each	each	DET
ejpam-6567	274	67	x	x	SYM
ejpam-6567	274	68	∈	∈	PROPN
ejpam-6567	274	69	a	a	DET
ejpam-6567	274	70	and	and	CCONJ
ejpam-6567	274	71	each	each	PRON
ejpam-6567	274	72	τ1τ2	τ1τ2	ADJ
ejpam-6567	274	73	-	-	ADJ
ejpam-6567	274	74	open	open	ADJ
ejpam-6567	274	75	set	set	ADJ
ejpam-6567	274	76	u	u	NOUN
ejpam-6567	274	77	of	of	ADP
ejpam-6567	274	78	x	x	PUNCT
ejpam-6567	274	79	containing	contain	VERB
ejpam-6567	274	80	x	x	PRON
ejpam-6567	274	81	,	,	PUNCT
ejpam-6567	274	82	there	there	PRON
ejpam-6567	274	83	exists	exist	VERB
ejpam-6567	274	84	a	a	DET
ejpam-6567	274	85	τ1τ2	τ1τ2	NOUN
ejpam-6567	274	86	-	-	ADJ
ejpam-6567	274	87	open	open	ADJ
ejpam-6567	274	88	set	set	NOUN
ejpam-6567	274	89	v	v	NOUN
ejpam-6567	274	90	of	of	ADP
ejpam-6567	274	91	x	x	PUNCT
ejpam-6567	274	92	such	such	ADJ
ejpam-6567	274	93	that	that	SCONJ
ejpam-6567	274	94	x	x	SYM
ejpam-6567	274	95	∈	∈	NOUN
ejpam-6567	274	96	v	v	ADP
ejpam-6567	274	97	⊆	⊆	NUM
ejpam-6567	274	98	τ1τ2	τ1τ2	NOUN
ejpam-6567	274	99	-	-	NOUN
ejpam-6567	274	100	cl(v	cl(v	X
ejpam-6567	274	101	)	)	PUNCT
ejpam-6567	274	102	⊆	⊆	NUM
ejpam-6567	274	103	u	u	NOUN
ejpam-6567	274	104	.	.	PUNCT
ejpam-6567	275	1	lemma	lemma	PROPN
ejpam-6567	275	2	3	3	X
ejpam-6567	275	3	.	.	PUNCT
ejpam-6567	276	1	[	[	X
ejpam-6567	276	2	22	22	NUM
ejpam-6567	276	3	]	]	X
ejpam-6567	276	4	if	if	SCONJ
ejpam-6567	276	5	a	a	PRON
ejpam-6567	276	6	is	be	AUX
ejpam-6567	276	7	a	a	DET
ejpam-6567	276	8	τ1τ2	τ1τ2	ADJ
ejpam-6567	276	9	-	-	ADJ
ejpam-6567	276	10	regular	regular	ADJ
ejpam-6567	276	11	τ1τ2	τ1τ2	NOUN
ejpam-6567	276	12	-	-	ADJ
ejpam-6567	276	13	paracompact	paracompact	ADJ
ejpam-6567	276	14	set	set	NOUN
ejpam-6567	276	15	of	of	ADP
ejpam-6567	276	16	a	a	DET
ejpam-6567	276	17	bitopological	bitopological	ADJ
ejpam-6567	276	18	space	space	NOUN
ejpam-6567	276	19	(	(	PUNCT
ejpam-6567	276	20	x	x	NOUN
ejpam-6567	276	21	,	,	PUNCT
ejpam-6567	276	22	τ1	τ1	NOUN
ejpam-6567	276	23	,	,	PUNCT
ejpam-6567	276	24	τ2	τ2	NOUN
ejpam-6567	276	25	)	)	PUNCT
ejpam-6567	276	26	and	and	CCONJ
ejpam-6567	276	27	u	u	NOUN
ejpam-6567	276	28	is	be	AUX
ejpam-6567	276	29	a	a	DET
ejpam-6567	276	30	τ1τ2	τ1τ2	ADJ
ejpam-6567	276	31	-	-	ADJ
ejpam-6567	276	32	open	open	ADJ
ejpam-6567	276	33	neighbourhood	neighbourhood	NOUN
ejpam-6567	276	34	of	of	ADP
ejpam-6567	276	35	a	a	PRON
ejpam-6567	276	36	,	,	PUNCT
ejpam-6567	276	37	then	then	ADV
ejpam-6567	276	38	there	there	PRON
ejpam-6567	276	39	exists	exist	VERB
ejpam-6567	276	40	a	a	DET
ejpam-6567	276	41	τ1τ2	τ1τ2	NOUN
ejpam-6567	276	42	-	-	ADJ
ejpam-6567	276	43	open	open	ADJ
ejpam-6567	276	44	set	set	NOUN
ejpam-6567	276	45	v	v	NOUN
ejpam-6567	276	46	of	of	ADP
ejpam-6567	276	47	x	x	PUNCT
ejpam-6567	276	48	such	such	ADJ
ejpam-6567	276	49	that	that	SCONJ
ejpam-6567	276	50	a	a	DET
ejpam-6567	276	51	⊆	⊆	NUM
ejpam-6567	276	52	v	v	ADP
ejpam-6567	276	53	⊆	⊆	NUM
ejpam-6567	276	54	τ1τ2	τ1τ2	NOUN
ejpam-6567	276	55	-	-	NOUN
ejpam-6567	276	56	cl(v	cl(v	X
ejpam-6567	276	57	)	)	PUNCT
ejpam-6567	276	58	⊆	⊆	NUM
ejpam-6567	276	59	u	u	NOUN
ejpam-6567	276	60	.	.	PUNCT
ejpam-6567	277	1	theorem	theorem	VERB
ejpam-6567	277	2	10	10	NUM
ejpam-6567	277	3	.	.	PUNCT
ejpam-6567	278	1	for	for	ADP
ejpam-6567	278	2	a	a	DET
ejpam-6567	278	3	multifunction	multifunction	NOUN
ejpam-6567	278	4	f	f	NOUN
ejpam-6567	278	5	:	:	PUNCT
ejpam-6567	278	6	(	(	PUNCT
ejpam-6567	278	7	x	x	X
ejpam-6567	278	8	,	,	PUNCT
ejpam-6567	278	9	τ	τ	PROPN
ejpam-6567	278	10	,	,	PUNCT
ejpam-6567	278	11	i	i	NOUN
ejpam-6567	278	12	)	)	PUNCT
ejpam-6567	278	13	→	→	PUNCT
ejpam-6567	278	14	(	(	PUNCT
ejpam-6567	278	15	y	y	PROPN
ejpam-6567	278	16	,	,	PUNCT
ejpam-6567	278	17	σ1	σ1	PROPN
ejpam-6567	278	18	,	,	PUNCT
ejpam-6567	278	19	σ2	σ2	NOUN
ejpam-6567	278	20	)	)	PUNCT
ejpam-6567	278	21	such	such	ADJ
ejpam-6567	278	22	that	that	SCONJ
ejpam-6567	278	23	f	f	PROPN
ejpam-6567	278	24	(	(	PUNCT
ejpam-6567	278	25	x	x	X
ejpam-6567	278	26	)	)	PUNCT
ejpam-6567	278	27	is	be	AUX
ejpam-6567	278	28	a	a	DET
ejpam-6567	278	29	σ1σ2regular	σ1σ2regular	ADJ
ejpam-6567	278	30	σ1σ2	σ1σ2	NOUN
ejpam-6567	278	31	-	-	PUNCT
ejpam-6567	278	32	paracompact	paracompact	ADJ
ejpam-6567	278	33	set	set	NOUN
ejpam-6567	278	34	of	of	ADP
ejpam-6567	278	35	y	y	PROPN
ejpam-6567	278	36	for	for	ADP
ejpam-6567	278	37	each	each	DET
ejpam-6567	278	38	point	point	NOUN
ejpam-6567	278	39	x	x	X
ejpam-6567	278	40	∈	∈	NOUN
ejpam-6567	278	41	x	x	NOUN
ejpam-6567	278	42	,	,	PUNCT
ejpam-6567	278	43	the	the	DET
ejpam-6567	278	44	following	follow	VERB
ejpam-6567	278	45	properties	property	NOUN
ejpam-6567	278	46	are	be	AUX
ejpam-6567	278	47	equivalent	equivalent	ADJ
ejpam-6567	278	48	:	:	PUNCT
ejpam-6567	278	49	(	(	PUNCT
ejpam-6567	278	50	1	1	X
ejpam-6567	278	51	)	)	PUNCT
ejpam-6567	278	52	f	f	PROPN
ejpam-6567	278	53	is	be	AUX
ejpam-6567	278	54	upper	upper	ADJ
ejpam-6567	278	55	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6567	278	56	,	,	PUNCT
ejpam-6567	278	57	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	278	58	;	;	PUNCT
ejpam-6567	278	59	(	(	PUNCT
ejpam-6567	278	60	2	2	X
ejpam-6567	278	61	)	)	PUNCT
ejpam-6567	278	62	f	f	PROPN
ejpam-6567	278	63	is	be	AUX
ejpam-6567	278	64	upper	upper	ADJ
ejpam-6567	278	65	almost	almost	ADV
ejpam-6567	278	66	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	278	67	,	,	PUNCT
ejpam-6567	278	68	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	278	69	;	;	PUNCT
ejpam-6567	278	70	(	(	PUNCT
ejpam-6567	278	71	3	3	X
ejpam-6567	278	72	)	)	PUNCT
ejpam-6567	278	73	f	f	PROPN
ejpam-6567	278	74	is	be	AUX
ejpam-6567	278	75	upper	upper	ADJ
ejpam-6567	278	76	weakly	weakly	ADJ
ejpam-6567	278	77	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	278	78	,	,	PUNCT
ejpam-6567	278	79	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	278	80	.	.	NOUN
ejpam-6567	278	81	proof	proof	NOUN
ejpam-6567	278	82	.	.	PUNCT
ejpam-6567	279	1	we	we	PRON
ejpam-6567	279	2	show	show	VERB
ejpam-6567	279	3	only	only	ADV
ejpam-6567	279	4	the	the	DET
ejpam-6567	279	5	implication	implication	NOUN
ejpam-6567	279	6	(	(	PUNCT
ejpam-6567	279	7	3	3	X
ejpam-6567	279	8	)	)	PUNCT
ejpam-6567	279	9	⇒	⇒	NOUN
ejpam-6567	279	10	(	(	PUNCT
ejpam-6567	279	11	1	1	X
ejpam-6567	279	12	)	)	PUNCT
ejpam-6567	279	13	since	since	SCONJ
ejpam-6567	279	14	the	the	DET
ejpam-6567	279	15	others	other	NOUN
ejpam-6567	279	16	are	be	AUX
ejpam-6567	279	17	obvious	obvious	ADJ
ejpam-6567	279	18	.	.	PUNCT
ejpam-6567	280	1	suppose	suppose	VERB
ejpam-6567	280	2	that	that	SCONJ
ejpam-6567	280	3	f	f	PROPN
ejpam-6567	280	4	is	be	AUX
ejpam-6567	280	5	upper	upper	ADJ
ejpam-6567	280	6	weakly	weakly	ADJ
ejpam-6567	280	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	280	8	,	,	PUNCT
ejpam-6567	280	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	280	10	.	.	PUNCT
ejpam-6567	280	11	let	let	VERB
ejpam-6567	280	12	x	x	SYM
ejpam-6567	280	13	∈	∈	PROPN
ejpam-6567	280	14	x	x	X
ejpam-6567	280	15	and	and	CCONJ
ejpam-6567	280	16	v	v	X
ejpam-6567	280	17	be	be	AUX
ejpam-6567	280	18	any	any	DET
ejpam-6567	280	19	σ1σ2	σ1σ2	NOUN
ejpam-6567	280	20	-	-	ADJ
ejpam-6567	280	21	open	open	ADJ
ejpam-6567	280	22	set	set	NOUN
ejpam-6567	280	23	of	of	ADP
ejpam-6567	280	24	y	y	PRON
ejpam-6567	280	25	such	such	ADJ
ejpam-6567	280	26	that	that	SCONJ
ejpam-6567	280	27	f	f	PROPN
ejpam-6567	280	28	(	(	PUNCT
ejpam-6567	280	29	x	x	X
ejpam-6567	280	30	)	)	PUNCT
ejpam-6567	280	31	⊆	⊆	NUM
ejpam-6567	280	32	v	v	NOUN
ejpam-6567	280	33	.	.	PUNCT
ejpam-6567	281	1	since	since	SCONJ
ejpam-6567	281	2	f	f	PROPN
ejpam-6567	281	3	(	(	PUNCT
ejpam-6567	281	4	x	x	X
ejpam-6567	281	5	)	)	PUNCT
ejpam-6567	281	6	is	be	AUX
ejpam-6567	281	7	σ1σ2	σ1σ2	NOUN
ejpam-6567	281	8	-	-	ADJ
ejpam-6567	281	9	regular	regular	ADJ
ejpam-6567	281	10	σ1σ2	σ1σ2	NOUN
ejpam-6567	281	11	-	-	PUNCT
ejpam-6567	281	12	paracompact	paracompact	ADJ
ejpam-6567	281	13	,	,	PUNCT
ejpam-6567	281	14	by	by	ADP
ejpam-6567	281	15	lemma	lemma	PROPN
ejpam-6567	281	16	3	3	NUM
ejpam-6567	281	17	there	there	ADV
ejpam-6567	281	18	exists	exist	VERB
ejpam-6567	281	19	a	a	DET
ejpam-6567	281	20	σ1σ2	σ1σ2	NUM
ejpam-6567	281	21	-	-	ADJ
ejpam-6567	281	22	open	open	ADJ
ejpam-6567	281	23	set	set	NOUN
ejpam-6567	281	24	w	w	PROPN
ejpam-6567	281	25	of	of	ADP
ejpam-6567	281	26	y	y	PRON
ejpam-6567	281	27	such	such	ADJ
ejpam-6567	281	28	that	that	SCONJ
ejpam-6567	281	29	f	f	PROPN
ejpam-6567	281	30	(	(	PUNCT
ejpam-6567	281	31	x	x	X
ejpam-6567	281	32	)	)	PUNCT
ejpam-6567	281	33	⊆	⊆	NUM
ejpam-6567	281	34	w	w	ADP
ejpam-6567	281	35	⊆	⊆	NUM
ejpam-6567	281	36	σ1σ2	σ1σ2	NOUN
ejpam-6567	281	37	-	-	PUNCT
ejpam-6567	281	38	cl(w	cl(w	NOUN
ejpam-6567	281	39	)	)	PUNCT
ejpam-6567	281	40	⊆	⊆	NUM
ejpam-6567	281	41	v	v	NOUN
ejpam-6567	281	42	.	.	PUNCT
ejpam-6567	282	1	since	since	SCONJ
ejpam-6567	282	2	f	f	PROPN
ejpam-6567	282	3	is	be	AUX
ejpam-6567	282	4	upper	upper	ADJ
ejpam-6567	282	5	weakly	weakly	ADJ
ejpam-6567	282	6	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	282	7	,	,	PUNCT
ejpam-6567	282	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	282	9	,	,	PUNCT
ejpam-6567	282	10	there	there	PRON
ejpam-6567	282	11	exists	exist	VERB
ejpam-6567	282	12	a	a	DET
ejpam-6567	282	13	⋆-open	⋆-open	ADJ
ejpam-6567	282	14	set	set	NOUN
ejpam-6567	282	15	u	u	NOUN
ejpam-6567	282	16	of	of	ADP
ejpam-6567	282	17	x	x	PUNCT
ejpam-6567	282	18	containing	contain	VERB
ejpam-6567	282	19	x	x	PUNCT
ejpam-6567	282	20	such	such	ADJ
ejpam-6567	282	21	that	that	SCONJ
ejpam-6567	282	22	f	f	PROPN
ejpam-6567	282	23	(	(	PUNCT
ejpam-6567	282	24	u	u	NOUN
ejpam-6567	282	25	)	)	PUNCT
ejpam-6567	282	26	⊆	⊆	NUM
ejpam-6567	282	27	σ1σ2	σ1σ2	NOUN
ejpam-6567	282	28	-	-	PUNCT
ejpam-6567	282	29	cl(w	cl(w	NOUN
ejpam-6567	282	30	)	)	PUNCT
ejpam-6567	282	31	;	;	PUNCT
ejpam-6567	282	32	hence	hence	ADV
ejpam-6567	282	33	f	f	PROPN
ejpam-6567	282	34	(	(	PUNCT
ejpam-6567	282	35	u	u	NOUN
ejpam-6567	282	36	)	)	PUNCT
ejpam-6567	282	37	⊆	⊆	NUM
ejpam-6567	282	38	v	v	NOUN
ejpam-6567	282	39	.	.	PUNCT
ejpam-6567	283	1	this	this	PRON
ejpam-6567	283	2	shows	show	VERB
ejpam-6567	283	3	that	that	SCONJ
ejpam-6567	283	4	f	f	PROPN
ejpam-6567	283	5	is	be	AUX
ejpam-6567	283	6	upper	upper	ADJ
ejpam-6567	283	7	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6567	283	8	,	,	PUNCT
ejpam-6567	283	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6567	283	10	.	.	PROPN
ejpam-6567	283	11	recall	recall	VERB
ejpam-6567	283	12	that	that	SCONJ
ejpam-6567	283	13	a	a	DET
ejpam-6567	283	14	bitopological	bitopological	ADJ
ejpam-6567	283	15	space	space	NOUN
ejpam-6567	283	16	(	(	PUNCT
ejpam-6567	283	17	x	x	NOUN
ejpam-6567	283	18	,	,	PUNCT
ejpam-6567	283	19	τ1	τ1	NOUN
ejpam-6567	283	20	,	,	PUNCT
ejpam-6567	283	21	τ2	τ2	NOUN
ejpam-6567	283	22	)	)	PUNCT
ejpam-6567	283	23	is	be	AUX
ejpam-6567	283	24	said	say	VERB
ejpam-6567	283	25	to	to	PART
ejpam-6567	283	26	be	be	AUX
ejpam-6567	283	27	τ1τ2	τ1τ2	NOUN
ejpam-6567	283	28	-	-	ADJ
ejpam-6567	283	29	compact	compact	ADJ
ejpam-6567	283	30	[	[	X
ejpam-6567	283	31	22	22	NUM
ejpam-6567	283	32	]	]	PUNCT
ejpam-6567	283	33	if	if	SCONJ
ejpam-6567	283	34	every	every	DET
ejpam-6567	283	35	cover	cover	NOUN
ejpam-6567	283	36	of	of	ADP
ejpam-6567	283	37	x	x	PUNCT
ejpam-6567	283	38	by	by	ADP
ejpam-6567	283	39	τ1τ2	τ1τ2	ADJ
ejpam-6567	283	40	-	-	ADJ
ejpam-6567	283	41	open	open	ADJ
ejpam-6567	283	42	sets	set	NOUN
ejpam-6567	283	43	of	of	ADP
ejpam-6567	283	44	x	x	PUNCT
ejpam-6567	283	45	has	have	VERB
ejpam-6567	283	46	a	a	DET
ejpam-6567	283	47	finite	finite	ADJ
ejpam-6567	283	48	subcover	subcover	PROPN
ejpam-6567	283	49	.	.	PUNCT
ejpam-6567	284	1	definition	definition	NOUN
ejpam-6567	284	2	4	4	NUM
ejpam-6567	284	3	.	.	PUNCT
ejpam-6567	285	1	[	[	X
ejpam-6567	285	2	30	30	NUM
ejpam-6567	285	3	]	]	X
ejpam-6567	285	4	a	a	DET
ejpam-6567	285	5	bitopological	bitopological	ADJ
ejpam-6567	285	6	space	space	NOUN
ejpam-6567	285	7	(	(	PUNCT
ejpam-6567	285	8	x	x	NOUN
ejpam-6567	285	9	,	,	PUNCT
ejpam-6567	285	10	τ1	τ1	NOUN
ejpam-6567	285	11	,	,	PUNCT
ejpam-6567	285	12	τ2	τ2	NOUN
ejpam-6567	285	13	)	)	PUNCT
ejpam-6567	285	14	is	be	AUX
ejpam-6567	285	15	said	say	VERB
ejpam-6567	285	16	to	to	PART
ejpam-6567	285	17	be	be	AUX
ejpam-6567	285	18	(	(	PUNCT
ejpam-6567	285	19	τ1	τ1	NOUN
ejpam-6567	285	20	,	,	PUNCT
ejpam-6567	285	21	τ2)-regular	τ2)-regular	ADJ
ejpam-6567	285	22	if	if	SCONJ
ejpam-6567	285	23	for	for	ADP
ejpam-6567	285	24	each	each	DET
ejpam-6567	285	25	τ1τ2	τ1τ2	ADJ
ejpam-6567	285	26	-	-	ADJ
ejpam-6567	285	27	closed	closed	ADJ
ejpam-6567	285	28	set	set	VERB
ejpam-6567	285	29	f	f	NOUN
ejpam-6567	285	30	and	and	CCONJ
ejpam-6567	285	31	each	each	DET
ejpam-6567	285	32	x	x	SYM
ejpam-6567	285	33	∈	∈	PROPN
ejpam-6567	285	34	x	x	X
ejpam-6567	285	35	−	−	PROPN
ejpam-6567	285	36	f	f	NOUN
ejpam-6567	285	37	,	,	PUNCT
ejpam-6567	285	38	there	there	PRON
ejpam-6567	285	39	exist	exist	VERB
ejpam-6567	285	40	disjoint	disjoint	ADJ
ejpam-6567	285	41	τ1τ2	τ1τ2	ADJ
ejpam-6567	285	42	-	-	ADJ
ejpam-6567	285	43	open	open	ADJ
ejpam-6567	285	44	sets	set	NOUN
ejpam-6567	285	45	u	u	NOUN
ejpam-6567	285	46	and	and	CCONJ
ejpam-6567	285	47	v	v	ADP
ejpam-6567	285	48	such	such	ADJ
ejpam-6567	285	49	that	that	SCONJ
ejpam-6567	285	50	x	x	SYM
ejpam-6567	285	51	∈	∈	PROPN
ejpam-6567	285	52	u	u	NOUN
ejpam-6567	285	53	and	and	CCONJ
ejpam-6567	285	54	f	f	PROPN
ejpam-6567	285	55	⊆	⊆	NUM
ejpam-6567	285	56	v	v	NOUN
ejpam-6567	285	57	.	.	PUNCT
ejpam-6567	286	1	corollary	corollary	ADJ
ejpam-6567	286	2	1	1	NUM
ejpam-6567	286	3	.	.	PUNCT
ejpam-6567	287	1	let	let	VERB
ejpam-6567	287	2	f	f	NOUN
ejpam-6567	287	3	:	:	PUNCT
ejpam-6567	287	4	(	(	PUNCT
ejpam-6567	287	5	x	x	X
ejpam-6567	287	6	,	,	PUNCT
ejpam-6567	287	7	τ	τ	PROPN
ejpam-6567	287	8	,	,	PUNCT
ejpam-6567	287	9	i	i	NOUN
ejpam-6567	287	10	)	)	PUNCT
ejpam-6567	287	11	→	→	PUNCT
ejpam-6567	287	12	(	(	PUNCT
ejpam-6567	287	13	y	y	PROPN
ejpam-6567	287	14	,	,	PUNCT
ejpam-6567	287	15	σ1	σ1	PROPN
ejpam-6567	287	16	,	,	PUNCT
ejpam-6567	287	17	σ2	σ2	PROPN
ejpam-6567	287	18	)	)	PUNCT
ejpam-6567	287	19	be	be	VERB
ejpam-6567	287	20	a	a	DET
ejpam-6567	287	21	multifunction	multifunction	NOUN
ejpam-6567	287	22	such	such	ADJ
ejpam-6567	287	23	that	that	SCONJ
ejpam-6567	287	24	f	f	PROPN
ejpam-6567	287	25	(	(	PUNCT
ejpam-6567	287	26	x	x	X
ejpam-6567	287	27	)	)	PUNCT
ejpam-6567	287	28	is	be	AUX
ejpam-6567	287	29	σ1σ2compact	σ1σ2compact	VERB
ejpam-6567	287	30	for	for	ADP
ejpam-6567	287	31	each	each	DET
ejpam-6567	287	32	point	point	NOUN
ejpam-6567	287	33	x	x	X
ejpam-6567	287	34	∈	∈	NOUN
ejpam-6567	287	35	x	x	X
ejpam-6567	287	36	and	and	CCONJ
ejpam-6567	287	37	(	(	PUNCT
ejpam-6567	287	38	y	y	PROPN
ejpam-6567	287	39	,	,	PUNCT
ejpam-6567	287	40	σ1	σ1	PROPN
ejpam-6567	287	41	,	,	PUNCT
ejpam-6567	287	42	σ2	σ2	PROPN
ejpam-6567	287	43	)	)	PUNCT
ejpam-6567	287	44	is	be	AUX
ejpam-6567	287	45	(	(	PUNCT
ejpam-6567	287	46	σ1	σ1	NOUN
ejpam-6567	287	47	,	,	PUNCT
ejpam-6567	287	48	σ2)-regular	σ2)-regular	PROPN
ejpam-6567	287	49	.	.	PUNCT
ejpam-6567	288	1	then	then	ADV
ejpam-6567	288	2	,	,	PUNCT
ejpam-6567	288	3	the	the	DET
ejpam-6567	288	4	following	follow	VERB
ejpam-6567	288	5	properties	property	NOUN
ejpam-6567	288	6	are	be	AUX
ejpam-6567	288	7	equivalent	equivalent	ADJ
ejpam-6567	288	8	:	:	PUNCT
ejpam-6567	288	9	(	(	PUNCT
ejpam-6567	288	10	1	1	X
ejpam-6567	288	11	)	)	PUNCT
ejpam-6567	288	12	f	f	PROPN
ejpam-6567	288	13	is	be	AUX
ejpam-6567	288	14	upper	upper	ADJ
ejpam-6567	288	15	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6567	288	16	,	,	PUNCT
ejpam-6567	288	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	288	18	;	;	PUNCT
ejpam-6567	288	19	(	(	PUNCT
ejpam-6567	288	20	2	2	X
ejpam-6567	288	21	)	)	PUNCT
ejpam-6567	288	22	f	f	PROPN
ejpam-6567	288	23	is	be	AUX
ejpam-6567	288	24	upper	upper	ADJ
ejpam-6567	288	25	almost	almost	ADV
ejpam-6567	288	26	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	288	27	,	,	PUNCT
ejpam-6567	288	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	288	29	;	;	PUNCT
ejpam-6567	288	30	(	(	PUNCT
ejpam-6567	288	31	3	3	X
ejpam-6567	288	32	)	)	PUNCT
ejpam-6567	288	33	f	f	PROPN
ejpam-6567	288	34	is	be	AUX
ejpam-6567	288	35	upper	upper	ADJ
ejpam-6567	289	1	weakly	weakly	ADJ
ejpam-6567	289	2	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	289	3	,	,	PUNCT
ejpam-6567	289	4	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6567	289	5	.	.	PUNCT
ejpam-6567	290	1	lemma	lemma	PROPN
ejpam-6567	290	2	4	4	NUM
ejpam-6567	290	3	.	.	PUNCT
ejpam-6567	291	1	[	[	X
ejpam-6567	291	2	31	31	NUM
ejpam-6567	291	3	]	]	PUNCT
ejpam-6567	291	4	if	if	SCONJ
ejpam-6567	291	5	a	a	PRON
ejpam-6567	291	6	is	be	AUX
ejpam-6567	291	7	a	a	DET
ejpam-6567	291	8	τ1τ2	τ1τ2	ADJ
ejpam-6567	291	9	-	-	ADJ
ejpam-6567	291	10	regular	regular	ADJ
ejpam-6567	291	11	set	set	NOUN
ejpam-6567	291	12	of	of	ADP
ejpam-6567	291	13	a	a	DET
ejpam-6567	291	14	bitopological	bitopological	ADJ
ejpam-6567	291	15	space	space	NOUN
ejpam-6567	291	16	(	(	PUNCT
ejpam-6567	291	17	x	x	NOUN
ejpam-6567	291	18	,	,	PUNCT
ejpam-6567	291	19	τ1	τ1	NOUN
ejpam-6567	291	20	,	,	PUNCT
ejpam-6567	291	21	τ2	τ2	NOUN
ejpam-6567	291	22	)	)	PUNCT
ejpam-6567	291	23	,	,	PUNCT
ejpam-6567	291	24	then	then	ADV
ejpam-6567	291	25	for	for	ADP
ejpam-6567	291	26	each	each	DET
ejpam-6567	291	27	τ1τ2	τ1τ2	ADJ
ejpam-6567	291	28	-	-	ADJ
ejpam-6567	291	29	open	open	ADJ
ejpam-6567	291	30	set	set	NOUN
ejpam-6567	291	31	g	g	NOUN
ejpam-6567	291	32	which	which	PRON
ejpam-6567	291	33	intersect	intersect	ADJ
ejpam-6567	291	34	a	a	X
ejpam-6567	291	35	,	,	PUNCT
ejpam-6567	291	36	there	there	PRON
ejpam-6567	291	37	exists	exist	VERB
ejpam-6567	291	38	a	a	DET
ejpam-6567	291	39	τ1τ2	τ1τ2	NOUN
ejpam-6567	291	40	-	-	ADJ
ejpam-6567	291	41	open	open	ADJ
ejpam-6567	291	42	set	set	NOUN
ejpam-6567	291	43	w	w	ADP
ejpam-6567	291	44	such	such	ADJ
ejpam-6567	291	45	that	that	SCONJ
ejpam-6567	291	46	a	a	DET
ejpam-6567	291	47	∩	∩	NOUN
ejpam-6567	291	48	w	w	PROPN
ejpam-6567	291	49	̸=	̸=	PROPN
ejpam-6567	291	50	∅	∅	NOUN
ejpam-6567	291	51	and	and	CCONJ
ejpam-6567	291	52	τ1τ2	τ1τ2	NOUN
ejpam-6567	291	53	-	-	NOUN
ejpam-6567	291	54	cl(w	cl(w	NOUN
ejpam-6567	291	55	)	)	PUNCT
ejpam-6567	291	56	⊆	⊆	NUM
ejpam-6567	291	57	g.	g.	PROPN
ejpam-6567	291	58	p.	p.	NOUN
ejpam-6567	291	59	pue	pue	PROPN
ejpam-6567	291	60	-	-	PUNCT
ejpam-6567	291	61	on	on	ADP
ejpam-6567	291	62	,	,	PUNCT
ejpam-6567	291	63	a.	a.	PROPN
ejpam-6567	291	64	sama	sama	PROPN
ejpam-6567	291	65	-	-	PUNCT
ejpam-6567	291	66	ae	ae	PROPN
ejpam-6567	291	67	,	,	PUNCT
ejpam-6567	291	68	c.	c.	PROPN
ejpam-6567	291	69	boonpok	boonpok	PROPN
ejpam-6567	291	70	/	/	SYM
ejpam-6567	291	71	eur	eur	PROPN
ejpam-6567	291	72	.	.	PUNCT
ejpam-6567	292	1	j.	j.	PROPN
ejpam-6567	292	2	pure	pure	PROPN
ejpam-6567	292	3	appl	appl	PROPN
ejpam-6567	292	4	.	.	PROPN
ejpam-6567	292	5	math	math	PROPN
ejpam-6567	292	6	,	,	PUNCT
ejpam-6567	292	7	18	18	NUM
ejpam-6567	292	8	(	(	PUNCT
ejpam-6567	292	9	3	3	NUM
ejpam-6567	292	10	)	)	PUNCT
ejpam-6567	292	11	(	(	PUNCT
ejpam-6567	292	12	2025	2025	NUM
ejpam-6567	292	13	)	)	PUNCT
ejpam-6567	292	14	,	,	PUNCT
ejpam-6567	292	15	6567	6567	NUM
ejpam-6567	292	16	11	11	NUM
ejpam-6567	292	17	of	of	ADP
ejpam-6567	292	18	13	13	NUM
ejpam-6567	292	19	theorem	theorem	VERB
ejpam-6567	292	20	11	11	NUM
ejpam-6567	292	21	.	.	PUNCT
ejpam-6567	293	1	for	for	ADP
ejpam-6567	293	2	a	a	DET
ejpam-6567	293	3	multifunction	multifunction	NOUN
ejpam-6567	293	4	f	f	NOUN
ejpam-6567	293	5	:	:	PUNCT
ejpam-6567	293	6	(	(	PUNCT
ejpam-6567	293	7	x	x	X
ejpam-6567	293	8	,	,	PUNCT
ejpam-6567	293	9	τ	τ	PROPN
ejpam-6567	293	10	,	,	PUNCT
ejpam-6567	293	11	i	i	NOUN
ejpam-6567	293	12	)	)	PUNCT
ejpam-6567	293	13	→	→	PUNCT
ejpam-6567	293	14	(	(	PUNCT
ejpam-6567	293	15	y	y	PROPN
ejpam-6567	293	16	,	,	PUNCT
ejpam-6567	293	17	σ1	σ1	PROPN
ejpam-6567	293	18	,	,	PUNCT
ejpam-6567	293	19	σ2	σ2	NOUN
ejpam-6567	293	20	)	)	PUNCT
ejpam-6567	293	21	such	such	ADJ
ejpam-6567	293	22	that	that	SCONJ
ejpam-6567	293	23	f	f	PROPN
ejpam-6567	293	24	(	(	PUNCT
ejpam-6567	293	25	x	x	X
ejpam-6567	293	26	)	)	PUNCT
ejpam-6567	293	27	is	be	AUX
ejpam-6567	293	28	a	a	DET
ejpam-6567	293	29	σ1σ2regular	σ1σ2regular	PROPN
ejpam-6567	293	30	set	set	NOUN
ejpam-6567	293	31	of	of	ADP
ejpam-6567	293	32	y	y	PROPN
ejpam-6567	293	33	for	for	ADP
ejpam-6567	293	34	each	each	DET
ejpam-6567	293	35	x	x	SYM
ejpam-6567	293	36	∈	∈	PROPN
ejpam-6567	293	37	x	x	NOUN
ejpam-6567	293	38	,	,	PUNCT
ejpam-6567	293	39	the	the	DET
ejpam-6567	293	40	following	follow	VERB
ejpam-6567	293	41	properties	property	NOUN
ejpam-6567	293	42	are	be	AUX
ejpam-6567	293	43	equivalent	equivalent	ADJ
ejpam-6567	293	44	:	:	PUNCT
ejpam-6567	293	45	(	(	PUNCT
ejpam-6567	293	46	1	1	X
ejpam-6567	293	47	)	)	PUNCT
ejpam-6567	293	48	f	f	PROPN
ejpam-6567	293	49	is	be	AUX
ejpam-6567	293	50	lower	low	ADJ
ejpam-6567	293	51	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	293	52	,	,	PUNCT
ejpam-6567	293	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	293	54	;	;	PUNCT
ejpam-6567	293	55	(	(	PUNCT
ejpam-6567	293	56	2	2	X
ejpam-6567	293	57	)	)	PUNCT
ejpam-6567	293	58	f	f	PROPN
ejpam-6567	293	59	is	be	AUX
ejpam-6567	293	60	lower	low	ADJ
ejpam-6567	293	61	almost	almost	ADV
ejpam-6567	293	62	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	293	63	,	,	PUNCT
ejpam-6567	293	64	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	293	65	;	;	PUNCT
ejpam-6567	293	66	(	(	PUNCT
ejpam-6567	293	67	3	3	X
ejpam-6567	293	68	)	)	PUNCT
ejpam-6567	293	69	f	f	PROPN
ejpam-6567	293	70	is	be	AUX
ejpam-6567	293	71	lower	low	ADJ
ejpam-6567	293	72	weakly	weakly	ADJ
ejpam-6567	293	73	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	293	74	,	,	PUNCT
ejpam-6567	293	75	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	293	76	.	.	NOUN
ejpam-6567	293	77	proof	proof	NOUN
ejpam-6567	293	78	.	.	PUNCT
ejpam-6567	294	1	we	we	PRON
ejpam-6567	294	2	show	show	VERB
ejpam-6567	294	3	only	only	ADV
ejpam-6567	294	4	the	the	DET
ejpam-6567	294	5	implication	implication	NOUN
ejpam-6567	294	6	(	(	PUNCT
ejpam-6567	294	7	3	3	X
ejpam-6567	294	8	)	)	PUNCT
ejpam-6567	294	9	⇒	⇒	NOUN
ejpam-6567	294	10	(	(	PUNCT
ejpam-6567	294	11	1	1	X
ejpam-6567	294	12	)	)	PUNCT
ejpam-6567	294	13	since	since	SCONJ
ejpam-6567	294	14	the	the	DET
ejpam-6567	294	15	others	other	NOUN
ejpam-6567	294	16	are	be	AUX
ejpam-6567	294	17	obvious	obvious	ADJ
ejpam-6567	294	18	.	.	PUNCT
ejpam-6567	295	1	suppose	suppose	VERB
ejpam-6567	295	2	that	that	SCONJ
ejpam-6567	295	3	f	f	PROPN
ejpam-6567	295	4	is	be	AUX
ejpam-6567	295	5	lower	low	ADJ
ejpam-6567	295	6	weakly	weakly	ADJ
ejpam-6567	295	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	295	8	,	,	PUNCT
ejpam-6567	295	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	295	10	.	.	PUNCT
ejpam-6567	295	11	let	let	VERB
ejpam-6567	295	12	x	x	SYM
ejpam-6567	295	13	∈	∈	PROPN
ejpam-6567	295	14	x	x	X
ejpam-6567	295	15	and	and	CCONJ
ejpam-6567	295	16	v	v	X
ejpam-6567	295	17	be	be	AUX
ejpam-6567	295	18	any	any	DET
ejpam-6567	295	19	σ1σ2	σ1σ2	NOUN
ejpam-6567	295	20	-	-	ADJ
ejpam-6567	295	21	open	open	ADJ
ejpam-6567	295	22	set	set	NOUN
ejpam-6567	295	23	of	of	ADP
ejpam-6567	295	24	y	y	PRON
ejpam-6567	295	25	such	such	ADJ
ejpam-6567	295	26	that	that	PRON
ejpam-6567	295	27	v	v	ADP
ejpam-6567	295	28	∩	∩	ADJ
ejpam-6567	295	29	f	f	X
ejpam-6567	295	30	(	(	PUNCT
ejpam-6567	295	31	x	x	X
ejpam-6567	295	32	)	)	PUNCT
ejpam-6567	295	33	̸=	̸=	PROPN
ejpam-6567	295	34	∅.	∅.	ADV
ejpam-6567	295	35	since	since	SCONJ
ejpam-6567	295	36	f	f	PROPN
ejpam-6567	295	37	(	(	PUNCT
ejpam-6567	295	38	x	x	X
ejpam-6567	295	39	)	)	PUNCT
ejpam-6567	295	40	is	be	AUX
ejpam-6567	295	41	σ1σ2	σ1σ2	NOUN
ejpam-6567	295	42	-	-	ADJ
ejpam-6567	295	43	regular	regular	ADJ
ejpam-6567	295	44	,	,	PUNCT
ejpam-6567	295	45	by	by	ADP
ejpam-6567	295	46	lemma	lemma	PROPN
ejpam-6567	295	47	4	4	NUM
ejpam-6567	295	48	there	there	ADV
ejpam-6567	295	49	exists	exist	VERB
ejpam-6567	295	50	a	a	DET
ejpam-6567	295	51	σ1σ2	σ1σ2	NUM
ejpam-6567	295	52	-	-	ADJ
ejpam-6567	295	53	open	open	ADJ
ejpam-6567	295	54	set	set	NOUN
ejpam-6567	295	55	w	w	PROPN
ejpam-6567	295	56	of	of	ADP
ejpam-6567	295	57	y	y	PRON
ejpam-6567	295	58	such	such	ADJ
ejpam-6567	295	59	that	that	SCONJ
ejpam-6567	295	60	f	f	PROPN
ejpam-6567	295	61	(	(	PUNCT
ejpam-6567	295	62	x	x	NOUN
ejpam-6567	295	63	)	)	PUNCT
ejpam-6567	295	64	∩	∩	NOUN
ejpam-6567	295	65	w	w	PROPN
ejpam-6567	295	66	̸=	̸=	PROPN
ejpam-6567	295	67	∅	∅	NOUN
ejpam-6567	295	68	and	and	CCONJ
ejpam-6567	295	69	σ1σ2	σ1σ2	NOUN
ejpam-6567	295	70	-	-	PUNCT
ejpam-6567	295	71	cl(w	cl(w	NOUN
ejpam-6567	295	72	)	)	PUNCT
ejpam-6567	295	73	⊆	⊆	NUM
ejpam-6567	295	74	v	v	NOUN
ejpam-6567	295	75	.	.	PUNCT
ejpam-6567	296	1	since	since	SCONJ
ejpam-6567	296	2	f	f	PROPN
ejpam-6567	296	3	is	be	AUX
ejpam-6567	296	4	lower	low	ADJ
ejpam-6567	296	5	weakly	weakly	ADJ
ejpam-6567	296	6	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	296	7	,	,	PUNCT
ejpam-6567	296	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	296	9	,	,	PUNCT
ejpam-6567	296	10	there	there	PRON
ejpam-6567	296	11	exists	exist	VERB
ejpam-6567	296	12	a	a	DET
ejpam-6567	296	13	⋆-open	⋆-open	ADJ
ejpam-6567	296	14	set	set	NOUN
ejpam-6567	296	15	u	u	NOUN
ejpam-6567	296	16	of	of	ADP
ejpam-6567	296	17	x	x	PUNCT
ejpam-6567	296	18	containing	contain	VERB
ejpam-6567	296	19	x	x	PUNCT
ejpam-6567	296	20	such	such	ADJ
ejpam-6567	296	21	that	that	SCONJ
ejpam-6567	296	22	σ1σ2	σ1σ2	NOUN
ejpam-6567	296	23	-	-	PUNCT
ejpam-6567	296	24	cl(w	cl(w	NOUN
ejpam-6567	296	25	)	)	PUNCT
ejpam-6567	296	26	∩	∩	PROPN
ejpam-6567	296	27	f	f	X
ejpam-6567	296	28	(	(	PUNCT
ejpam-6567	296	29	z	z	NOUN
ejpam-6567	296	30	)	)	PUNCT
ejpam-6567	296	31	̸=	̸=	PROPN
ejpam-6567	296	32	∅	∅	NOUN
ejpam-6567	296	33	;	;	PUNCT
ejpam-6567	296	34	hence	hence	ADV
ejpam-6567	296	35	f	f	X
ejpam-6567	296	36	(	(	PUNCT
ejpam-6567	296	37	z	z	NOUN
ejpam-6567	296	38	)	)	PUNCT
ejpam-6567	296	39	∩	∩	NOUN
ejpam-6567	296	40	v	v	ADP
ejpam-6567	296	41	̸=	̸=	PROPN
ejpam-6567	296	42	∅	∅	NOUN
ejpam-6567	296	43	for	for	ADP
ejpam-6567	296	44	each	each	DET
ejpam-6567	296	45	z	z	NOUN
ejpam-6567	296	46	∈	∈	PROPN
ejpam-6567	296	47	u	u	NOUN
ejpam-6567	296	48	.	.	PUNCT
ejpam-6567	297	1	this	this	PRON
ejpam-6567	297	2	shows	show	VERB
ejpam-6567	297	3	that	that	SCONJ
ejpam-6567	297	4	f	f	PROPN
ejpam-6567	297	5	is	be	AUX
ejpam-6567	297	6	lower	low	ADJ
ejpam-6567	297	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	297	8	,	,	PUNCT
ejpam-6567	297	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	297	10	.	.	NOUN
ejpam-6567	297	11	definition	definition	NOUN
ejpam-6567	297	12	5	5	NUM
ejpam-6567	297	13	.	.	PUNCT
ejpam-6567	298	1	[	[	X
ejpam-6567	298	2	32	32	NUM
ejpam-6567	298	3	]	]	PUNCT
ejpam-6567	298	4	a	a	DET
ejpam-6567	298	5	bitopological	bitopological	ADJ
ejpam-6567	298	6	space	space	NOUN
ejpam-6567	298	7	(	(	PUNCT
ejpam-6567	298	8	x	x	NOUN
ejpam-6567	298	9	,	,	PUNCT
ejpam-6567	298	10	τ1	τ1	NOUN
ejpam-6567	298	11	,	,	PUNCT
ejpam-6567	298	12	τ2	τ2	NOUN
ejpam-6567	298	13	)	)	PUNCT
ejpam-6567	298	14	is	be	AUX
ejpam-6567	298	15	said	say	VERB
ejpam-6567	298	16	to	to	PART
ejpam-6567	298	17	be	be	AUX
ejpam-6567	298	18	(	(	PUNCT
ejpam-6567	298	19	τ1	τ1	NOUN
ejpam-6567	298	20	,	,	PUNCT
ejpam-6567	298	21	τ2)-normal	τ2)-normal	ADJ
ejpam-6567	298	22	if	if	SCONJ
ejpam-6567	298	23	for	for	ADP
ejpam-6567	298	24	each	each	DET
ejpam-6567	298	25	pair	pair	NOUN
ejpam-6567	298	26	of	of	ADP
ejpam-6567	298	27	disjoint	disjoint	ADJ
ejpam-6567	298	28	τ1τ2	τ1τ2	ADJ
ejpam-6567	298	29	-	-	ADJ
ejpam-6567	298	30	closed	closed	ADJ
ejpam-6567	298	31	sets	set	NOUN
ejpam-6567	298	32	f	f	PROPN
ejpam-6567	298	33	and	and	CCONJ
ejpam-6567	298	34	f	f	PROPN
ejpam-6567	298	35	′	′	NOUN
ejpam-6567	298	36	,	,	PUNCT
ejpam-6567	298	37	there	there	PRON
ejpam-6567	298	38	exist	exist	VERB
ejpam-6567	298	39	disjoint	disjoint	ADJ
ejpam-6567	298	40	τ1τ2	τ1τ2	ADJ
ejpam-6567	298	41	-	-	ADJ
ejpam-6567	298	42	open	open	ADJ
ejpam-6567	298	43	sets	set	NOUN
ejpam-6567	298	44	u	u	NOUN
ejpam-6567	298	45	and	and	CCONJ
ejpam-6567	298	46	v	v	ADP
ejpam-6567	298	47	such	such	ADJ
ejpam-6567	298	48	that	that	SCONJ
ejpam-6567	298	49	f	f	PROPN
ejpam-6567	298	50	⊆	⊆	NUM
ejpam-6567	298	51	u	u	NOUN
ejpam-6567	298	52	and	and	CCONJ
ejpam-6567	298	53	f	f	PROPN
ejpam-6567	298	54	′	′	NUM
ejpam-6567	298	55	⊆	⊆	NUM
ejpam-6567	298	56	v	v	NOUN
ejpam-6567	298	57	.	.	PUNCT
ejpam-6567	299	1	theorem	theorem	NOUN
ejpam-6567	299	2	12	12	NUM
ejpam-6567	299	3	.	.	PUNCT
ejpam-6567	300	1	if	if	SCONJ
ejpam-6567	300	2	f	f	PROPN
ejpam-6567	300	3	:	:	PUNCT
ejpam-6567	300	4	(	(	PUNCT
ejpam-6567	300	5	x	x	X
ejpam-6567	300	6	,	,	PUNCT
ejpam-6567	300	7	τ	τ	PROPN
ejpam-6567	300	8	,	,	PUNCT
ejpam-6567	300	9	i	i	NOUN
ejpam-6567	300	10	)	)	PUNCT
ejpam-6567	300	11	→	→	PUNCT
ejpam-6567	300	12	(	(	PUNCT
ejpam-6567	300	13	y	y	PROPN
ejpam-6567	300	14	,	,	PUNCT
ejpam-6567	300	15	σ1	σ1	PROPN
ejpam-6567	300	16	,	,	PUNCT
ejpam-6567	300	17	σ2	σ2	PROPN
ejpam-6567	300	18	)	)	PUNCT
ejpam-6567	300	19	is	be	AUX
ejpam-6567	300	20	a	a	DET
ejpam-6567	300	21	multifunction	multifunction	NOUN
ejpam-6567	300	22	such	such	ADJ
ejpam-6567	300	23	that	that	SCONJ
ejpam-6567	300	24	f	f	PROPN
ejpam-6567	300	25	(	(	PUNCT
ejpam-6567	300	26	x	x	X
ejpam-6567	300	27	)	)	PUNCT
ejpam-6567	300	28	is	be	AUX
ejpam-6567	300	29	σ1σ2closed	σ1σ2close	VERB
ejpam-6567	300	30	in	in	ADP
ejpam-6567	300	31	y	y	PROPN
ejpam-6567	300	32	for	for	ADP
ejpam-6567	300	33	each	each	DET
ejpam-6567	300	34	x	x	SYM
ejpam-6567	300	35	∈	∈	PROPN
ejpam-6567	300	36	x	x	X
ejpam-6567	300	37	and	and	CCONJ
ejpam-6567	300	38	(	(	PUNCT
ejpam-6567	300	39	y	y	PROPN
ejpam-6567	300	40	,	,	PUNCT
ejpam-6567	300	41	σ1	σ1	PROPN
ejpam-6567	300	42	,	,	PUNCT
ejpam-6567	300	43	σ2	σ2	PROPN
ejpam-6567	300	44	)	)	PUNCT
ejpam-6567	300	45	is	be	AUX
ejpam-6567	300	46	a	a	DET
ejpam-6567	300	47	(	(	PUNCT
ejpam-6567	300	48	σ1	σ1	NOUN
ejpam-6567	300	49	,	,	PUNCT
ejpam-6567	300	50	σ2)-normal	σ2)-normal	ADJ
ejpam-6567	300	51	space	space	NOUN
ejpam-6567	300	52	,	,	PUNCT
ejpam-6567	300	53	then	then	ADV
ejpam-6567	300	54	the	the	DET
ejpam-6567	300	55	following	follow	VERB
ejpam-6567	300	56	properties	property	NOUN
ejpam-6567	300	57	are	be	AUX
ejpam-6567	300	58	equivalent	equivalent	ADJ
ejpam-6567	300	59	:	:	PUNCT
ejpam-6567	300	60	(	(	PUNCT
ejpam-6567	300	61	1	1	X
ejpam-6567	300	62	)	)	PUNCT
ejpam-6567	300	63	f	f	PROPN
ejpam-6567	300	64	is	be	AUX
ejpam-6567	300	65	upper	upper	ADJ
ejpam-6567	300	66	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6567	300	67	,	,	PUNCT
ejpam-6567	300	68	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	300	69	;	;	PUNCT
ejpam-6567	300	70	(	(	PUNCT
ejpam-6567	300	71	2	2	X
ejpam-6567	300	72	)	)	PUNCT
ejpam-6567	300	73	f	f	PROPN
ejpam-6567	300	74	is	be	AUX
ejpam-6567	300	75	upper	upper	ADJ
ejpam-6567	300	76	almost	almost	ADV
ejpam-6567	300	77	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6567	300	78	,	,	PUNCT
ejpam-6567	300	79	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	300	80	;	;	PUNCT
ejpam-6567	300	81	(	(	PUNCT
ejpam-6567	300	82	3	3	X
ejpam-6567	300	83	)	)	PUNCT
ejpam-6567	300	84	f	f	PROPN
ejpam-6567	300	85	is	be	AUX
ejpam-6567	300	86	upper	upper	ADJ
ejpam-6567	300	87	weakly	weakly	ADJ
ejpam-6567	300	88	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	300	89	,	,	PUNCT
ejpam-6567	300	90	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	300	91	.	.	NOUN
ejpam-6567	300	92	proof	proof	NOUN
ejpam-6567	300	93	.	.	PUNCT
ejpam-6567	301	1	as	as	ADP
ejpam-6567	301	2	in	in	ADP
ejpam-6567	301	3	theorem	theorem	NOUN
ejpam-6567	301	4	10	10	NUM
ejpam-6567	301	5	,	,	PUNCT
ejpam-6567	301	6	we	we	PRON
ejpam-6567	301	7	prove	prove	VERB
ejpam-6567	301	8	only	only	ADV
ejpam-6567	301	9	the	the	DET
ejpam-6567	301	10	implication	implication	NOUN
ejpam-6567	301	11	(	(	PUNCT
ejpam-6567	301	12	3	3	X
ejpam-6567	301	13	)	)	PUNCT
ejpam-6567	301	14	⇒	⇒	NOUN
ejpam-6567	301	15	(	(	PUNCT
ejpam-6567	301	16	1	1	NUM
ejpam-6567	301	17	)	)	PUNCT
ejpam-6567	301	18	.	.	PUNCT
ejpam-6567	302	1	suppose	suppose	VERB
ejpam-6567	302	2	that	that	SCONJ
ejpam-6567	302	3	f	f	PROPN
ejpam-6567	302	4	is	be	AUX
ejpam-6567	302	5	upper	upper	ADJ
ejpam-6567	302	6	weakly	weakly	ADJ
ejpam-6567	302	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	302	8	,	,	PUNCT
ejpam-6567	302	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	302	10	.	.	PUNCT
ejpam-6567	302	11	let	let	VERB
ejpam-6567	302	12	x	x	SYM
ejpam-6567	302	13	∈	∈	PROPN
ejpam-6567	302	14	x	x	X
ejpam-6567	302	15	and	and	CCONJ
ejpam-6567	302	16	g	g	PROPN
ejpam-6567	302	17	be	be	VERB
ejpam-6567	302	18	any	any	DET
ejpam-6567	302	19	σ1σ2	σ1σ2	NOUN
ejpam-6567	302	20	-	-	ADJ
ejpam-6567	302	21	open	open	ADJ
ejpam-6567	302	22	set	set	NOUN
ejpam-6567	302	23	of	of	ADP
ejpam-6567	302	24	y	y	PROPN
ejpam-6567	302	25	containing	contain	VERB
ejpam-6567	302	26	f	f	PROPN
ejpam-6567	302	27	(	(	PUNCT
ejpam-6567	302	28	x	x	NOUN
ejpam-6567	302	29	)	)	PUNCT
ejpam-6567	302	30	.	.	PUNCT
ejpam-6567	303	1	since	since	SCONJ
ejpam-6567	303	2	f	f	PROPN
ejpam-6567	303	3	(	(	PUNCT
ejpam-6567	303	4	x	x	X
ejpam-6567	303	5	)	)	PUNCT
ejpam-6567	303	6	is	be	AUX
ejpam-6567	303	7	σ1σ2	σ1σ2	NOUN
ejpam-6567	303	8	-	-	ADJ
ejpam-6567	303	9	closed	closed	ADJ
ejpam-6567	303	10	in	in	ADP
ejpam-6567	303	11	y	y	PROPN
ejpam-6567	303	12	,	,	PUNCT
ejpam-6567	303	13	by	by	ADP
ejpam-6567	303	14	the	the	DET
ejpam-6567	303	15	(	(	PUNCT
ejpam-6567	303	16	σ1	σ1	PROPN
ejpam-6567	303	17	,	,	PUNCT
ejpam-6567	303	18	σ2)-normality	σ2)-normality	NOUN
ejpam-6567	303	19	of	of	ADP
ejpam-6567	303	20	y	y	PRON
ejpam-6567	303	21	there	there	PRON
ejpam-6567	303	22	exists	exist	VERB
ejpam-6567	303	23	a	a	DET
ejpam-6567	303	24	σ1σ2	σ1σ2	NUM
ejpam-6567	303	25	-	-	ADJ
ejpam-6567	303	26	open	open	ADJ
ejpam-6567	303	27	set	set	NOUN
ejpam-6567	303	28	v	v	NOUN
ejpam-6567	303	29	of	of	ADP
ejpam-6567	303	30	y	y	PRON
ejpam-6567	303	31	such	such	ADJ
ejpam-6567	303	32	that	that	SCONJ
ejpam-6567	303	33	f	f	PROPN
ejpam-6567	303	34	(	(	PUNCT
ejpam-6567	303	35	x	x	X
ejpam-6567	303	36	)	)	PUNCT
ejpam-6567	303	37	⊆	⊆	NUM
ejpam-6567	303	38	v	v	ADP
ejpam-6567	303	39	⊆	⊆	NUM
ejpam-6567	303	40	σ1σ2	σ1σ2	NOUN
ejpam-6567	303	41	-	-	PUNCT
ejpam-6567	303	42	cl(v	cl(v	NOUN
ejpam-6567	303	43	)	)	PUNCT
ejpam-6567	303	44	⊆	⊆	NUM
ejpam-6567	303	45	g.	g.	NOUN
ejpam-6567	303	46	since	since	SCONJ
ejpam-6567	303	47	f	f	PROPN
ejpam-6567	303	48	is	be	AUX
ejpam-6567	303	49	upper	upper	ADJ
ejpam-6567	303	50	weakly	weakly	ADJ
ejpam-6567	303	51	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6567	303	52	,	,	PUNCT
ejpam-6567	303	53	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6567	303	54	,	,	PUNCT
ejpam-6567	303	55	there	there	PRON
ejpam-6567	303	56	exists	exist	VERB
ejpam-6567	303	57	a	a	DET
ejpam-6567	303	58	⋆-open	⋆-open	ADJ
ejpam-6567	303	59	set	set	NOUN
ejpam-6567	303	60	u	u	NOUN
ejpam-6567	303	61	of	of	ADP
ejpam-6567	303	62	x	x	PUNCT
ejpam-6567	303	63	containing	contain	VERB
ejpam-6567	303	64	x	x	PUNCT
ejpam-6567	303	65	such	such	ADJ
ejpam-6567	303	66	that	that	SCONJ
ejpam-6567	303	67	f	f	PROPN
ejpam-6567	303	68	(	(	PUNCT
ejpam-6567	303	69	u	u	NOUN
ejpam-6567	303	70	)	)	PUNCT
ejpam-6567	303	71	⊆	⊆	NUM
ejpam-6567	303	72	σ1σ2	σ1σ2	NOUN
ejpam-6567	303	73	-	-	NUM
ejpam-6567	303	74	cl(v	cl(v	NOUN
ejpam-6567	303	75	)	)	PUNCT
ejpam-6567	303	76	⊆	⊆	NUM
ejpam-6567	303	77	g.	g.	NOUN
ejpam-6567	303	78	this	this	PRON
ejpam-6567	303	79	shows	show	VERB
ejpam-6567	303	80	that	that	SCONJ
ejpam-6567	303	81	f	f	PROPN
ejpam-6567	303	82	is	be	AUX
ejpam-6567	303	83	upper	upper	ADJ
ejpam-6567	303	84	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6567	303	85	,	,	PUNCT
ejpam-6567	303	86	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6567	303	87	.	.	PUNCT
ejpam-6567	304	1	acknowledgements	acknowledgement	NOUN
ejpam-6567	304	2	this	this	DET
ejpam-6567	304	3	research	research	NOUN
ejpam-6567	304	4	project	project	NOUN
ejpam-6567	304	5	was	be	AUX
ejpam-6567	304	6	financially	financially	ADV
ejpam-6567	304	7	supported	support	VERB
ejpam-6567	304	8	by	by	ADP
ejpam-6567	304	9	mahasarakham	mahasarakham	PROPN
ejpam-6567	304	10	university	university	PROPN
ejpam-6567	304	11	.	.	PUNCT
ejpam-6567	305	1	p.	p.	NOUN
ejpam-6567	305	2	pue	pue	NOUN
ejpam-6567	305	3	-	-	PUNCT
ejpam-6567	305	4	on	on	ADP
ejpam-6567	305	5	,	,	PUNCT
ejpam-6567	305	6	a.	a.	PROPN
ejpam-6567	305	7	sama	sama	PROPN
ejpam-6567	305	8	-	-	PUNCT
ejpam-6567	305	9	ae	ae	PROPN
ejpam-6567	305	10	,	,	PUNCT
ejpam-6567	305	11	c.	c.	PROPN
ejpam-6567	305	12	boonpok	boonpok	PROPN
ejpam-6567	305	13	/	/	SYM
ejpam-6567	305	14	eur	eur	PROPN
ejpam-6567	305	15	.	.	PUNCT
ejpam-6567	306	1	j.	j.	PROPN
ejpam-6567	306	2	pure	pure	PROPN
ejpam-6567	306	3	appl	appl	PROPN
ejpam-6567	306	4	.	.	PROPN
ejpam-6567	306	5	math	math	PROPN
ejpam-6567	306	6	,	,	PUNCT
ejpam-6567	306	7	18	18	NUM
ejpam-6567	306	8	(	(	PUNCT
ejpam-6567	306	9	3	3	NUM
ejpam-6567	306	10	)	)	PUNCT
ejpam-6567	306	11	(	(	PUNCT
ejpam-6567	306	12	2025	2025	NUM
ejpam-6567	306	13	)	)	PUNCT
ejpam-6567	306	14	,	,	PUNCT
ejpam-6567	306	15	6567	6567	NUM
ejpam-6567	306	16	12	12	NUM
ejpam-6567	306	17	of	of	ADP
ejpam-6567	306	18	13	13	NUM
ejpam-6567	306	19	references	reference	NOUN
ejpam-6567	306	20	[	[	X
ejpam-6567	306	21	1	1	NUM
ejpam-6567	306	22	]	]	X
ejpam-6567	306	23	n.	n.	PROPN
ejpam-6567	306	24	levine	levine	PROPN
ejpam-6567	306	25	.	.	PUNCT
ejpam-6567	307	1	a	a	DET
ejpam-6567	307	2	decomposition	decomposition	NOUN
ejpam-6567	307	3	of	of	ADP
ejpam-6567	307	4	continuity	continuity	NOUN
ejpam-6567	307	5	in	in	ADP
ejpam-6567	307	6	topological	topological	ADJ
ejpam-6567	307	7	spaces	space	NOUN
ejpam-6567	307	8	.	.	PUNCT
ejpam-6567	308	1	the	the	DET
ejpam-6567	308	2	american	american	PROPN
ejpam-6567	308	3	mathematical	mathematical	PROPN
ejpam-6567	308	4	monthly	monthly	ADV
ejpam-6567	308	5	,	,	PUNCT
ejpam-6567	308	6	68:44–46	68:44–46	NUM
ejpam-6567	308	7	,	,	PUNCT
ejpam-6567	308	8	1961	1961	NUM
ejpam-6567	308	9	.	.	PUNCT
ejpam-6567	309	1	[	[	X
ejpam-6567	309	2	2	2	X
ejpam-6567	309	3	]	]	PUNCT
ejpam-6567	309	4	t.	t.	PROPN
ejpam-6567	309	5	husain	husain	PROPN
ejpam-6567	309	6	.	.	PUNCT
ejpam-6567	310	1	almost	almost	ADV
ejpam-6567	310	2	continuous	continuous	ADJ
ejpam-6567	310	3	mappings	mapping	NOUN
ejpam-6567	310	4	.	.	PUNCT
ejpam-6567	311	1	prace	prace	PROPN
ejpam-6567	311	2	matematyczne	matematyczne	PROPN
ejpam-6567	311	3	,	,	PUNCT
ejpam-6567	311	4	10:1–7	10:1–7	NUM
ejpam-6567	311	5	,	,	PUNCT
ejpam-6567	311	6	1966	1966	NUM
ejpam-6567	311	7	.	.	PUNCT
ejpam-6567	312	1	[	[	X
ejpam-6567	312	2	3	3	X
ejpam-6567	312	3	]	]	X
ejpam-6567	312	4	d.	d.	PROPN
ejpam-6567	312	5	s.	s.	PROPN
ejpam-6567	312	6	janković.	janković.	PROPN
ejpam-6567	312	7	θ	θ	PROPN
ejpam-6567	312	8	-	-	ADJ
ejpam-6567	312	9	regular	regular	ADJ
ejpam-6567	312	10	spaces	space	NOUN
ejpam-6567	312	11	.	.	PUNCT
ejpam-6567	313	1	international	international	ADJ
ejpam-6567	313	2	journal	journal	PROPN
ejpam-6567	313	3	of	of	ADP
ejpam-6567	313	4	mathematics	mathematics	PROPN
ejpam-6567	313	5	and	and	CCONJ
ejpam-6567	313	6	mathematical	mathematical	ADJ
ejpam-6567	313	7	sciences	science	NOUN
ejpam-6567	313	8	,	,	PUNCT
ejpam-6567	313	9	8:615–619	8:615–619	NUM
ejpam-6567	313	10	,	,	PUNCT
ejpam-6567	313	11	1985	1985	NUM
ejpam-6567	313	12	.	.	PUNCT
ejpam-6567	314	1	[	[	X
ejpam-6567	314	2	4	4	X
ejpam-6567	314	3	]	]	PUNCT
ejpam-6567	314	4	t.	t.	PROPN
ejpam-6567	314	5	noiri	noiri	PROPN
ejpam-6567	314	6	.	.	PUNCT
ejpam-6567	315	1	properties	property	NOUN
ejpam-6567	315	2	of	of	ADP
ejpam-6567	315	3	some	some	DET
ejpam-6567	315	4	weak	weak	ADJ
ejpam-6567	315	5	forms	form	NOUN
ejpam-6567	315	6	of	of	ADP
ejpam-6567	315	7	continuity	continuity	NOUN
ejpam-6567	315	8	.	.	PUNCT
ejpam-6567	316	1	international	international	ADJ
ejpam-6567	316	2	journal	journal	PROPN
ejpam-6567	316	3	of	of	ADP
ejpam-6567	316	4	mathematics	mathematics	PROPN
ejpam-6567	316	5	and	and	CCONJ
ejpam-6567	316	6	mathematical	mathematical	ADJ
ejpam-6567	316	7	sciences	science	NOUN
ejpam-6567	316	8	,	,	PUNCT
ejpam-6567	316	9	10(1):97–111	10(1):97–111	NUM
ejpam-6567	316	10	,	,	PUNCT
ejpam-6567	316	11	1987	1987	NUM
ejpam-6567	316	12	.	.	PUNCT
ejpam-6567	317	1	[	[	X
ejpam-6567	317	2	5	5	X
ejpam-6567	317	3	]	]	PUNCT
ejpam-6567	317	4	d.	d.	PROPN
ejpam-6567	317	5	a.	a.	PROPN
ejpam-6567	317	6	rose	rise	VERB
ejpam-6567	317	7	.	.	PUNCT
ejpam-6567	318	1	weak	weak	ADJ
ejpam-6567	318	2	continuity	continuity	NOUN
ejpam-6567	318	3	and	and	CCONJ
ejpam-6567	318	4	almost	almost	ADV
ejpam-6567	318	5	continuity	continuity	NOUN
ejpam-6567	318	6	.	.	PUNCT
ejpam-6567	319	1	international	international	ADJ
ejpam-6567	319	2	journal	journal	PROPN
ejpam-6567	319	3	of	of	ADP
ejpam-6567	319	4	mathematics	mathematics	PROPN
ejpam-6567	319	5	and	and	CCONJ
ejpam-6567	319	6	mathematical	mathematical	ADJ
ejpam-6567	319	7	sciences	science	NOUN
ejpam-6567	319	8	,	,	PUNCT
ejpam-6567	319	9	7:311–318	7:311–318	PROPN
ejpam-6567	319	10	,	,	PUNCT
ejpam-6567	319	11	1984	1984	NUM
ejpam-6567	319	12	.	.	PUNCT
ejpam-6567	320	1	[	[	X
ejpam-6567	320	2	6	6	NUM
ejpam-6567	320	3	]	]	PUNCT
ejpam-6567	320	4	v.	v.	CCONJ
ejpam-6567	320	5	popa	popa	NOUN
ejpam-6567	320	6	and	and	CCONJ
ejpam-6567	320	7	t.	t.	PROPN
ejpam-6567	320	8	noiri	noiri	PROPN
ejpam-6567	320	9	.	.	PUNCT
ejpam-6567	321	1	on	on	ADP
ejpam-6567	321	2	weakly	weakly	ADJ
ejpam-6567	321	3	(	(	PUNCT
ejpam-6567	321	4	τ	τ	PROPN
ejpam-6567	321	5	,	,	PUNCT
ejpam-6567	321	6	m)-continuous	m)-continuous	ADJ
ejpam-6567	321	7	functions	function	NOUN
ejpam-6567	321	8	.	.	PUNCT
ejpam-6567	322	1	rendiconti	rendiconti	ADJ
ejpam-6567	322	2	del	del	PROPN
ejpam-6567	322	3	circolo	circolo	PROPN
ejpam-6567	322	4	matematico	matematico	NOUN
ejpam-6567	322	5	di	di	PROPN
ejpam-6567	322	6	palermo	palermo	PROPN
ejpam-6567	322	7	series	series	PROPN
ejpam-6567	322	8	2	2	NUM
ejpam-6567	322	9	,	,	PUNCT
ejpam-6567	322	10	51:295–316	51:295–316	NUM
ejpam-6567	322	11	,	,	PUNCT
ejpam-6567	322	12	2002	2002	NUM
ejpam-6567	322	13	.	.	PUNCT
ejpam-6567	323	1	[	[	X
ejpam-6567	323	2	7	7	X
ejpam-6567	323	3	]	]	X
ejpam-6567	323	4	e.	e.	PROPN
ejpam-6567	323	5	ekici	ekici	PROPN
ejpam-6567	323	6	,	,	PUNCT
ejpam-6567	323	7	s.	s.	PROPN
ejpam-6567	323	8	jafari	jafari	PROPN
ejpam-6567	323	9	,	,	PUNCT
ejpam-6567	323	10	m.	m.	PROPN
ejpam-6567	323	11	caldas	caldas	PROPN
ejpam-6567	323	12	,	,	PUNCT
ejpam-6567	323	13	and	and	CCONJ
ejpam-6567	323	14	t.	t.	PROPN
ejpam-6567	323	15	noiri	noiri	PROPN
ejpam-6567	323	16	.	.	PUNCT
ejpam-6567	324	1	weakly	weakly	ADJ
ejpam-6567	324	2	λ	λ	ADJ
ejpam-6567	324	3	-	-	ADJ
ejpam-6567	324	4	continuous	continuous	ADJ
ejpam-6567	324	5	functions	function	NOUN
ejpam-6567	324	6	.	.	PUNCT
ejpam-6567	325	1	novi	novi	PROPN
ejpam-6567	325	2	sad	sad	PROPN
ejpam-6567	325	3	journal	journal	PROPN
ejpam-6567	325	4	of	of	ADP
ejpam-6567	325	5	mathematics	mathematic	NOUN
ejpam-6567	325	6	,	,	PUNCT
ejpam-6567	325	7	38:47–56	38:47–56	NUM
ejpam-6567	325	8	,	,	PUNCT
ejpam-6567	325	9	2008	2008	NUM
ejpam-6567	325	10	.	.	PUNCT
ejpam-6567	326	1	[	[	X
ejpam-6567	326	2	8	8	NUM
ejpam-6567	326	3	]	]	PUNCT
ejpam-6567	326	4	v.	v.	CCONJ
ejpam-6567	326	5	popa	popa	NOUN
ejpam-6567	326	6	.	.	PUNCT
ejpam-6567	327	1	weakly	weakly	ADJ
ejpam-6567	327	2	continuous	continuous	ADJ
ejpam-6567	327	3	multifunctions	multifunction	NOUN
ejpam-6567	327	4	.	.	PUNCT
ejpam-6567	328	1	bollettino	bollettino	PROPN
ejpam-6567	328	2	dell	dell	PROPN
ejpam-6567	328	3	’	'	PUNCT
ejpam-6567	328	4	unione	unione	PROPN
ejpam-6567	328	5	matematica	matematica	PROPN
ejpam-6567	328	6	italiana	italiana	PROPN
ejpam-6567	328	7	(	(	PUNCT
ejpam-6567	328	8	5	5	NUM
ejpam-6567	328	9	)	)	PUNCT
ejpam-6567	328	10	,	,	PUNCT
ejpam-6567	328	11	15(a):379–388	15(a):379–388	NUM
ejpam-6567	328	12	,	,	PUNCT
ejpam-6567	328	13	1978	1978	NUM
ejpam-6567	328	14	.	.	PUNCT
ejpam-6567	329	1	[	[	X
ejpam-6567	329	2	9	9	NUM
ejpam-6567	329	3	]	]	PUNCT
ejpam-6567	329	4	r.	r.	PROPN
ejpam-6567	329	5	e.	e.	PROPN
ejpam-6567	329	6	smithson	smithson	PROPN
ejpam-6567	329	7	.	.	PUNCT
ejpam-6567	330	1	almost	almost	ADV
ejpam-6567	330	2	and	and	CCONJ
ejpam-6567	330	3	weak	weak	ADJ
ejpam-6567	330	4	continuity	continuity	NOUN
ejpam-6567	330	5	for	for	ADP
ejpam-6567	330	6	multifunctions	multifunction	NOUN
ejpam-6567	330	7	.	.	PUNCT
ejpam-6567	331	1	bulletin	bulletin	NOUN
ejpam-6567	331	2	of	of	ADP
ejpam-6567	331	3	the	the	DET
ejpam-6567	331	4	calcutta	calcutta	PROPN
ejpam-6567	331	5	mathematical	mathematical	ADJ
ejpam-6567	331	6	society	society	NOUN
ejpam-6567	331	7	,	,	PUNCT
ejpam-6567	331	8	70:383–390	70:383–390	NUM
ejpam-6567	331	9	,	,	PUNCT
ejpam-6567	331	10	1978	1978	NUM
ejpam-6567	331	11	.	.	PUNCT
ejpam-6567	332	1	[	[	X
ejpam-6567	332	2	10	10	NUM
ejpam-6567	332	3	]	]	X
ejpam-6567	332	4	v.	v.	CCONJ
ejpam-6567	332	5	popa	popa	NOUN
ejpam-6567	332	6	and	and	CCONJ
ejpam-6567	332	7	t.	t.	PROPN
ejpam-6567	332	8	noiri	noiri	PROPN
ejpam-6567	332	9	.	.	PUNCT
ejpam-6567	333	1	on	on	ADP
ejpam-6567	333	2	upper	upper	ADJ
ejpam-6567	333	3	and	and	CCONJ
ejpam-6567	333	4	lower	low	ADJ
ejpam-6567	333	5	weakly	weakly	ADJ
ejpam-6567	333	6	α	α	ADJ
ejpam-6567	333	7	-	-	ADJ
ejpam-6567	333	8	continuous	continuous	ADJ
ejpam-6567	333	9	multifunctions	multifunction	NOUN
ejpam-6567	333	10	.	.	PUNCT
ejpam-6567	334	1	novi	novi	PROPN
ejpam-6567	334	2	sad	sad	PROPN
ejpam-6567	334	3	journal	journal	PROPN
ejpam-6567	334	4	of	of	ADP
ejpam-6567	334	5	mathematics	mathematic	NOUN
ejpam-6567	334	6	,	,	PUNCT
ejpam-6567	334	7	32(1):7–24	32(1):7–24	NUM
ejpam-6567	334	8	,	,	PUNCT
ejpam-6567	334	9	2002	2002	NUM
ejpam-6567	334	10	.	.	PUNCT
ejpam-6567	335	1	[	[	X
ejpam-6567	335	2	11	11	NUM
ejpam-6567	335	3	]	]	PUNCT
ejpam-6567	335	4	v.	v.	CCONJ
ejpam-6567	335	5	popa	popa	NOUN
ejpam-6567	335	6	and	and	CCONJ
ejpam-6567	335	7	t.	t.	PROPN
ejpam-6567	335	8	noiri	noiri	PROPN
ejpam-6567	335	9	.	.	PUNCT
ejpam-6567	336	1	on	on	ADP
ejpam-6567	336	2	upper	upper	ADJ
ejpam-6567	336	3	and	and	CCONJ
ejpam-6567	336	4	lower	low	ADJ
ejpam-6567	336	5	weakly	weakly	ADJ
ejpam-6567	336	6	β	β	ADJ
ejpam-6567	336	7	-	-	ADJ
ejpam-6567	336	8	continuous	continuous	ADJ
ejpam-6567	336	9	multifunctions	multifunction	NOUN
ejpam-6567	336	10	.	.	PUNCT
ejpam-6567	337	1	annales	annales	PROPN
ejpam-6567	337	2	universitatis	universitatis	PROPN
ejpam-6567	337	3	scientiarum	scientiarum	PROPN
ejpam-6567	337	4	budapestinensis	budapestinensis	PROPN
ejpam-6567	337	5	de	de	PROPN
ejpam-6567	337	6	rolando	rolando	PROPN
ejpam-6567	337	7	eötvös	eötvös	PROPN
ejpam-6567	337	8	nominatae	nominatae	NOUN
ejpam-6567	337	9	sectio	sectio	NOUN
ejpam-6567	337	10	mathematica	mathematica	PROPN
ejpam-6567	337	11	,	,	PUNCT
ejpam-6567	337	12	43:25–48	43:25–48	PROPN
ejpam-6567	337	13	,	,	PUNCT
ejpam-6567	337	14	2000	2000	NUM
ejpam-6567	337	15	.	.	PUNCT
ejpam-6567	338	1	[	[	X
ejpam-6567	338	2	12	12	NUM
ejpam-6567	338	3	]	]	PUNCT
ejpam-6567	338	4	t.	t.	PROPN
ejpam-6567	338	5	noiri	noiri	PROPN
ejpam-6567	338	6	and	and	CCONJ
ejpam-6567	338	7	v.	v.	ADP
ejpam-6567	338	8	popa	popa	NOUN
ejpam-6567	338	9	.	.	PUNCT
ejpam-6567	339	1	a	a	DET
ejpam-6567	339	2	unified	unified	ADJ
ejpam-6567	339	3	theory	theory	NOUN
ejpam-6567	339	4	of	of	ADP
ejpam-6567	339	5	weak	weak	ADJ
ejpam-6567	339	6	continuity	continuity	NOUN
ejpam-6567	339	7	for	for	ADP
ejpam-6567	339	8	multifunctions	multifunction	NOUN
ejpam-6567	339	9	.	.	PUNCT
ejpam-6567	340	1	studii	studii	PROPN
ejpam-6567	340	2	şi	şi	PROPN
ejpam-6567	340	3	cercetǎri	cercetǎri	VERB
ejpam-6567	340	4	ştiinţific	ştiinţific	NOUN
ejpam-6567	340	5	.	.	PUNCT
ejpam-6567	341	1	seria	seria	PROPN
ejpam-6567	341	2	:	:	PUNCT
ejpam-6567	341	3	matematicǎ.	matematicǎ.	VERB
ejpam-6567	341	4	universitatea	universitatea	ADV
ejpam-6567	341	5	din	din	PROPN
ejpam-6567	341	6	bacǎu	bacǎu	PROPN
ejpam-6567	341	7	,	,	PUNCT
ejpam-6567	341	8	16:167–200	16:167–200	NUM
ejpam-6567	341	9	,	,	PUNCT
ejpam-6567	341	10	2006	2006	NUM
ejpam-6567	341	11	.	.	PUNCT
ejpam-6567	342	1	[	[	X
ejpam-6567	342	2	13	13	NUM
ejpam-6567	342	3	]	]	X
ejpam-6567	342	4	p.	p.	NOUN
ejpam-6567	342	5	pue	pue	NOUN
ejpam-6567	342	6	-	-	PUNCT
ejpam-6567	342	7	on	on	ADP
ejpam-6567	342	8	,	,	PUNCT
ejpam-6567	342	9	s.	s.	PROPN
ejpam-6567	342	10	sompong	sompong	PROPN
ejpam-6567	342	11	,	,	PUNCT
ejpam-6567	342	12	and	and	CCONJ
ejpam-6567	342	13	c.	c.	PROPN
ejpam-6567	342	14	boonpok	boonpok	PROPN
ejpam-6567	342	15	.	.	PUNCT
ejpam-6567	343	1	upper	upper	ADJ
ejpam-6567	343	2	and	and	CCONJ
ejpam-6567	343	3	lower	low	ADJ
ejpam-6567	343	4	(	(	PUNCT
ejpam-6567	343	5	τ1	τ1	NOUN
ejpam-6567	343	6	,	,	PUNCT
ejpam-6567	343	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6567	343	8	multifunctions	multifunction	NOUN
ejpam-6567	343	9	.	.	PUNCT
ejpam-6567	344	1	international	international	ADJ
ejpam-6567	344	2	journal	journal	PROPN
ejpam-6567	344	3	of	of	ADP
ejpam-6567	344	4	mathematics	mathematic	NOUN
ejpam-6567	344	5	and	and	CCONJ
ejpam-6567	344	6	computer	computer	NOUN
ejpam-6567	344	7	science	science	NOUN
ejpam-6567	344	8	,	,	PUNCT
ejpam-6567	344	9	19(4):1305	19(4):1305	NUM
ejpam-6567	344	10	–	–	PUNCT
ejpam-6567	344	11	1310	1310	NUM
ejpam-6567	344	12	,	,	PUNCT
ejpam-6567	344	13	2024	2024	NUM
ejpam-6567	344	14	.	.	PUNCT
ejpam-6567	345	1	[	[	X
ejpam-6567	345	2	14	14	NUM
ejpam-6567	345	3	]	]	X
ejpam-6567	345	4	c.	c.	PROPN
ejpam-6567	345	5	klanarong	klanarong	PROPN
ejpam-6567	345	6	,	,	PUNCT
ejpam-6567	345	7	s.	s.	PROPN
ejpam-6567	345	8	sompong	sompong	PROPN
ejpam-6567	345	9	,	,	PUNCT
ejpam-6567	345	10	and	and	CCONJ
ejpam-6567	345	11	c.	c.	PROPN
ejpam-6567	345	12	boonpok	boonpok	PROPN
ejpam-6567	345	13	.	.	PUNCT
ejpam-6567	346	1	upper	upper	ADJ
ejpam-6567	346	2	and	and	CCONJ
ejpam-6567	346	3	lower	low	ADJ
ejpam-6567	346	4	almost	almost	ADV
ejpam-6567	346	5	(	(	PUNCT
ejpam-6567	346	6	τ1	τ1	NOUN
ejpam-6567	346	7	,	,	PUNCT
ejpam-6567	346	8	τ2)continuous	τ2)continuous	ADJ
ejpam-6567	346	9	multifunctions	multifunction	NOUN
ejpam-6567	346	10	.	.	PUNCT
ejpam-6567	347	1	european	european	ADJ
ejpam-6567	347	2	journal	journal	PROPN
ejpam-6567	347	3	of	of	ADP
ejpam-6567	347	4	pure	pure	ADJ
ejpam-6567	347	5	and	and	CCONJ
ejpam-6567	347	6	applied	applied	ADJ
ejpam-6567	347	7	mathematics	mathematic	NOUN
ejpam-6567	347	8	,	,	PUNCT
ejpam-6567	347	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-6567	347	10	,	,	PUNCT
ejpam-6567	347	11	2024	2024	NUM
ejpam-6567	347	12	.	.	PUNCT
ejpam-6567	348	1	[	[	X
ejpam-6567	348	2	15	15	NUM
ejpam-6567	348	3	]	]	X
ejpam-6567	348	4	m.	m.	NOUN
ejpam-6567	348	5	thongmoon	thongmoon	NOUN
ejpam-6567	348	6	,	,	PUNCT
ejpam-6567	348	7	s.	s.	PROPN
ejpam-6567	348	8	sompong	sompong	PROPN
ejpam-6567	348	9	,	,	PUNCT
ejpam-6567	348	10	and	and	CCONJ
ejpam-6567	348	11	c.	c.	PROPN
ejpam-6567	348	12	boonpok	boonpok	PROPN
ejpam-6567	348	13	.	.	PUNCT
ejpam-6567	349	1	upper	upper	ADJ
ejpam-6567	349	2	and	and	CCONJ
ejpam-6567	349	3	lower	low	ADJ
ejpam-6567	349	4	weak	weak	ADJ
ejpam-6567	349	5	(	(	PUNCT
ejpam-6567	349	6	τ1	τ1	NOUN
ejpam-6567	349	7	,	,	PUNCT
ejpam-6567	349	8	τ2)continuity	τ2)continuity	PROPN
ejpam-6567	349	9	.	.	PUNCT
ejpam-6567	350	1	european	european	PROPN
ejpam-6567	350	2	journal	journal	PROPN
ejpam-6567	350	3	of	of	ADP
ejpam-6567	350	4	pure	pure	ADJ
ejpam-6567	350	5	and	and	CCONJ
ejpam-6567	350	6	applied	applied	ADJ
ejpam-6567	350	7	mathematics	mathematic	NOUN
ejpam-6567	350	8	,	,	PUNCT
ejpam-6567	350	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-6567	350	10	,	,	PUNCT
ejpam-6567	350	11	2024	2024	NUM
ejpam-6567	350	12	.	.	PUNCT
ejpam-6567	351	1	[	[	X
ejpam-6567	351	2	16	16	NUM
ejpam-6567	351	3	]	]	PUNCT
ejpam-6567	351	4	k.	k.	PROPN
ejpam-6567	351	5	kuratowski	kuratowski	PROPN
ejpam-6567	351	6	.	.	PUNCT
ejpam-6567	352	1	topology	topology	PROPN
ejpam-6567	352	2	,	,	PUNCT
ejpam-6567	352	3	vol	vol	NOUN
ejpam-6567	352	4	.	.	PUNCT
ejpam-6567	352	5	i.	i.	PROPN
ejpam-6567	352	6	academic	academic	PROPN
ejpam-6567	352	7	press	press	PROPN
ejpam-6567	352	8	,	,	PUNCT
ejpam-6567	352	9	new	new	PROPN
ejpam-6567	352	10	york	york	PROPN
ejpam-6567	352	11	,	,	PUNCT
ejpam-6567	352	12	1966	1966	NUM
ejpam-6567	352	13	.	.	PUNCT
ejpam-6567	353	1	[	[	X
ejpam-6567	353	2	17	17	NUM
ejpam-6567	353	3	]	]	X
ejpam-6567	353	4	r.	r.	PROPN
ejpam-6567	353	5	vaidyanathaswamy	vaidyanathaswamy	PROPN
ejpam-6567	353	6	.	.	PUNCT
ejpam-6567	354	1	the	the	DET
ejpam-6567	354	2	localisation	localisation	NOUN
ejpam-6567	354	3	theory	theory	NOUN
ejpam-6567	354	4	in	in	ADP
ejpam-6567	354	5	set	set	NOUN
ejpam-6567	354	6	-	-	PUNCT
ejpam-6567	354	7	topology	topology	NOUN
ejpam-6567	354	8	.	.	PUNCT
ejpam-6567	355	1	proceedings	proceeding	NOUN
ejpam-6567	355	2	of	of	ADP
ejpam-6567	355	3	the	the	DET
ejpam-6567	355	4	indian	indian	PROPN
ejpam-6567	355	5	academy	academy	PROPN
ejpam-6567	355	6	of	of	ADP
ejpam-6567	355	7	sciences	science	NOUN
ejpam-6567	355	8	-	-	PUNCT
ejpam-6567	355	9	section	section	NOUN
ejpam-6567	355	10	a	a	PRON
ejpam-6567	355	11	,	,	PUNCT
ejpam-6567	355	12	20:51–61	20:51–61	NUM
ejpam-6567	355	13	,	,	PUNCT
ejpam-6567	355	14	1944	1944	NUM
ejpam-6567	355	15	.	.	PUNCT
ejpam-6567	356	1	[	[	X
ejpam-6567	356	2	18	18	NUM
ejpam-6567	356	3	]	]	X
ejpam-6567	356	4	e.	e.	PROPN
ejpam-6567	356	5	hatir	hatir	PROPN
ejpam-6567	356	6	and	and	CCONJ
ejpam-6567	356	7	t.	t.	PROPN
ejpam-6567	356	8	noiri	noiri	PROPN
ejpam-6567	356	9	.	.	PUNCT
ejpam-6567	357	1	weakly	weakly	ADJ
ejpam-6567	357	2	pre	pre	ADJ
ejpam-6567	357	3	-	-	ADJ
ejpam-6567	357	4	i	i	PRON
ejpam-6567	357	5	-	-	PUNCT
ejpam-6567	357	6	open	open	ADJ
ejpam-6567	357	7	sets	set	NOUN
ejpam-6567	357	8	and	and	CCONJ
ejpam-6567	357	9	decomposition	decomposition	NOUN
ejpam-6567	357	10	of	of	ADP
ejpam-6567	357	11	continuity	continuity	NOUN
ejpam-6567	357	12	.	.	PUNCT
ejpam-6567	358	1	acta	acta	PROPN
ejpam-6567	358	2	mathematica	mathematica	PROPN
ejpam-6567	358	3	hungarica	hungarica	PROPN
ejpam-6567	358	4	,	,	PUNCT
ejpam-6567	358	5	106(3):227–238	106(3):227–238	NUM
ejpam-6567	358	6	,	,	PUNCT
ejpam-6567	358	7	2005	2005	NUM
ejpam-6567	358	8	.	.	PUNCT
ejpam-6567	359	1	[	[	X
ejpam-6567	359	2	19	19	NUM
ejpam-6567	359	3	]	]	X
ejpam-6567	359	4	e.	e.	PROPN
ejpam-6567	359	5	hatir	hatir	PROPN
ejpam-6567	359	6	and	and	CCONJ
ejpam-6567	359	7	t.	t.	PROPN
ejpam-6567	359	8	noiri	noiri	PROPN
ejpam-6567	359	9	.	.	PUNCT
ejpam-6567	360	1	on	on	ADP
ejpam-6567	360	2	decompositions	decomposition	NOUN
ejpam-6567	360	3	of	of	ADP
ejpam-6567	360	4	continuity	continuity	NOUN
ejpam-6567	360	5	via	via	ADP
ejpam-6567	360	6	idealization	idealization	NOUN
ejpam-6567	360	7	.	.	PUNCT
ejpam-6567	361	1	acta	acta	PROPN
ejpam-6567	361	2	mathematica	mathematica	PROPN
ejpam-6567	361	3	hungarica	hungarica	PROPN
ejpam-6567	361	4	,	,	PUNCT
ejpam-6567	361	5	96:341–349	96:341–349	PROPN
ejpam-6567	361	6	,	,	PUNCT
ejpam-6567	361	7	2002	2002	NUM
ejpam-6567	361	8	.	.	PUNCT
ejpam-6567	362	1	[	[	X
ejpam-6567	362	2	20	20	NUM
ejpam-6567	362	3	]	]	PUNCT
ejpam-6567	362	4	c.	c.	PROPN
ejpam-6567	362	5	boonpok	boonpok	PROPN
ejpam-6567	362	6	.	.	PUNCT
ejpam-6567	363	1	on	on	ADP
ejpam-6567	363	2	continuous	continuous	ADJ
ejpam-6567	363	3	multifunctions	multifunction	NOUN
ejpam-6567	363	4	in	in	ADP
ejpam-6567	363	5	ideal	ideal	ADJ
ejpam-6567	363	6	topological	topological	ADJ
ejpam-6567	363	7	spaces	space	NOUN
ejpam-6567	363	8	.	.	PUNCT
ejpam-6567	364	1	lobachevskii	lobachevskii	PROPN
ejpam-6567	364	2	p.	p.	PROPN
ejpam-6567	364	3	pue	pue	PROPN
ejpam-6567	364	4	-	-	PUNCT
ejpam-6567	364	5	on	on	ADP
ejpam-6567	364	6	,	,	PUNCT
ejpam-6567	364	7	a.	a.	PROPN
ejpam-6567	364	8	sama	sama	PROPN
ejpam-6567	364	9	-	-	PUNCT
ejpam-6567	364	10	ae	ae	PROPN
ejpam-6567	364	11	,	,	PUNCT
ejpam-6567	364	12	c.	c.	PROPN
ejpam-6567	364	13	boonpok	boonpok	PROPN
ejpam-6567	364	14	/	/	SYM
ejpam-6567	364	15	eur	eur	PROPN
ejpam-6567	364	16	.	.	PUNCT
ejpam-6567	365	1	j.	j.	PROPN
ejpam-6567	365	2	pure	pure	PROPN
ejpam-6567	365	3	appl	appl	PROPN
ejpam-6567	365	4	.	.	PROPN
ejpam-6567	365	5	math	math	PROPN
ejpam-6567	365	6	,	,	PUNCT
ejpam-6567	365	7	18	18	NUM
ejpam-6567	365	8	(	(	PUNCT
ejpam-6567	365	9	3	3	NUM
ejpam-6567	365	10	)	)	PUNCT
ejpam-6567	365	11	(	(	PUNCT
ejpam-6567	365	12	2025	2025	NUM
ejpam-6567	365	13	)	)	PUNCT
ejpam-6567	365	14	,	,	PUNCT
ejpam-6567	365	15	6567	6567	NUM
ejpam-6567	365	16	13	13	NUM
ejpam-6567	365	17	of	of	ADP
ejpam-6567	365	18	13	13	NUM
ejpam-6567	365	19	journal	journal	NOUN
ejpam-6567	365	20	of	of	ADP
ejpam-6567	365	21	mathematics	mathematic	NOUN
ejpam-6567	365	22	,	,	PUNCT
ejpam-6567	365	23	40(1):24–35	40(1):24–35	NUM
ejpam-6567	365	24	,	,	PUNCT
ejpam-6567	365	25	2019	2019	NUM
ejpam-6567	365	26	.	.	PUNCT
ejpam-6567	366	1	[	[	X
ejpam-6567	366	2	21	21	NUM
ejpam-6567	366	3	]	]	X
ejpam-6567	366	4	c.	c.	PROPN
ejpam-6567	366	5	boonpok	boonpok	PROPN
ejpam-6567	366	6	.	.	PUNCT
ejpam-6567	367	1	pı	pı	NOUN
ejpam-6567	367	2	-	-	NOUN
ejpam-6567	367	3	continuity	continuity	NOUN
ejpam-6567	367	4	and	and	CCONJ
ejpam-6567	367	5	weak	weak	ADJ
ejpam-6567	367	6	pı	pı	NOUN
ejpam-6567	367	7	-	-	NOUN
ejpam-6567	367	8	continuity	continuity	NOUN
ejpam-6567	367	9	.	.	PUNCT
ejpam-6567	368	1	carpathian	carpathian	ADJ
ejpam-6567	368	2	mathematical	mathematical	ADJ
ejpam-6567	368	3	publications	publication	NOUN
ejpam-6567	368	4	,	,	PUNCT
ejpam-6567	368	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6567	368	6	,	,	PUNCT
ejpam-6567	368	7	2025	2025	NUM
ejpam-6567	368	8	.	.	PUNCT
ejpam-6567	369	1	[	[	X
ejpam-6567	369	2	22	22	NUM
ejpam-6567	369	3	]	]	PUNCT
ejpam-6567	369	4	c.	c.	PROPN
ejpam-6567	369	5	boonpok	boonpok	PROPN
ejpam-6567	369	6	,	,	PUNCT
ejpam-6567	369	7	c.	c.	PROPN
ejpam-6567	369	8	viriyapong	viriyapong	PROPN
ejpam-6567	369	9	,	,	PUNCT
ejpam-6567	369	10	and	and	CCONJ
ejpam-6567	369	11	m.	m.	NOUN
ejpam-6567	369	12	thongmoon	thongmoon	NOUN
ejpam-6567	369	13	.	.	PUNCT
ejpam-6567	370	1	on	on	ADP
ejpam-6567	370	2	upper	upper	ADJ
ejpam-6567	370	3	and	and	CCONJ
ejpam-6567	370	4	lower	low	ADJ
ejpam-6567	370	5	(	(	PUNCT
ejpam-6567	370	6	τ1	τ1	NOUN
ejpam-6567	370	7	,	,	PUNCT
ejpam-6567	370	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6567	370	9	multifunctions	multifunction	NOUN
ejpam-6567	370	10	.	.	PUNCT
ejpam-6567	371	1	journal	journal	PROPN
ejpam-6567	371	2	of	of	ADP
ejpam-6567	371	3	mathematics	mathematics	PROPN
ejpam-6567	371	4	and	and	CCONJ
ejpam-6567	371	5	computer	computer	NOUN
ejpam-6567	371	6	science	science	NOUN
ejpam-6567	371	7	,	,	PUNCT
ejpam-6567	371	8	18:282–293	18:282–293	NUM
ejpam-6567	371	9	,	,	PUNCT
ejpam-6567	371	10	2018	2018	NUM
ejpam-6567	371	11	.	.	PUNCT
ejpam-6567	372	1	[	[	X
ejpam-6567	372	2	23	23	NUM
ejpam-6567	372	3	]	]	X
ejpam-6567	372	4	c.	c.	PROPN
ejpam-6567	372	5	viriyapong	viriyapong	PROPN
ejpam-6567	372	6	and	and	CCONJ
ejpam-6567	372	7	c.	c.	PROPN
ejpam-6567	372	8	boonpok	boonpok	PROPN
ejpam-6567	372	9	.	.	PUNCT
ejpam-6567	373	1	(	(	PUNCT
ejpam-6567	373	2	τ1	τ1	NOUN
ejpam-6567	373	3	,	,	PUNCT
ejpam-6567	373	4	τ2)α	τ2)α	NOUN
ejpam-6567	373	5	-	-	PUNCT
ejpam-6567	373	6	continuity	continuity	NOUN
ejpam-6567	373	7	for	for	ADP
ejpam-6567	373	8	multifunctions	multifunction	NOUN
ejpam-6567	373	9	.	.	PUNCT
ejpam-6567	374	1	journal	journal	PROPN
ejpam-6567	374	2	of	of	ADP
ejpam-6567	374	3	mathematics	mathematic	NOUN
ejpam-6567	374	4	,	,	PUNCT
ejpam-6567	374	5	2020:6285763	2020:6285763	NUM
ejpam-6567	374	6	,	,	PUNCT
ejpam-6567	374	7	2020	2020	NUM
ejpam-6567	374	8	.	.	PUNCT
ejpam-6567	375	1	[	[	X
ejpam-6567	375	2	24	24	NUM
ejpam-6567	375	3	]	]	PUNCT
ejpam-6567	375	4	c.	c.	PROPN
ejpam-6567	375	5	boonpok	boonpok	PROPN
ejpam-6567	375	6	.	.	PUNCT
ejpam-6567	376	1	(	(	PUNCT
ejpam-6567	376	2	τ1	τ1	NOUN
ejpam-6567	376	3	,	,	PUNCT
ejpam-6567	376	4	τ2)δ	τ2)δ	ADJ
ejpam-6567	376	5	-	-	PUNCT
ejpam-6567	376	6	semicontinuous	semicontinuous	ADJ
ejpam-6567	376	7	multifunctions	multifunction	NOUN
ejpam-6567	376	8	.	.	PUNCT
ejpam-6567	377	1	heliyon	heliyon	NOUN
ejpam-6567	377	2	,	,	PUNCT
ejpam-6567	377	3	6	6	NUM
ejpam-6567	377	4	:	:	SYM
ejpam-6567	377	5	e05367	e05367	PROPN
ejpam-6567	377	6	,	,	PUNCT
ejpam-6567	377	7	2020	2020	NUM
ejpam-6567	377	8	.	.	PUNCT
ejpam-6567	378	1	[	[	X
ejpam-6567	378	2	25	25	NUM
ejpam-6567	378	3	]	]	X
ejpam-6567	378	4	n.	n.	PROPN
ejpam-6567	378	5	viriyapong	viriyapong	PROPN
ejpam-6567	378	6	,	,	PUNCT
ejpam-6567	378	7	s.	s.	PROPN
ejpam-6567	378	8	sompong	sompong	PROPN
ejpam-6567	378	9	,	,	PUNCT
ejpam-6567	378	10	and	and	CCONJ
ejpam-6567	378	11	c.	c.	PROPN
ejpam-6567	378	12	boonpok	boonpok	PROPN
ejpam-6567	378	13	.	.	PUNCT
ejpam-6567	379	1	(	(	PUNCT
ejpam-6567	379	2	τ1	τ1	NOUN
ejpam-6567	379	3	,	,	PUNCT
ejpam-6567	379	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6567	379	5	disconnectedness	disconnectedness	NOUN
ejpam-6567	379	6	in	in	ADP
ejpam-6567	379	7	bitopological	bitopological	ADJ
ejpam-6567	379	8	spaces	space	NOUN
ejpam-6567	379	9	.	.	PUNCT
ejpam-6567	380	1	international	international	ADJ
ejpam-6567	380	2	journal	journal	PROPN
ejpam-6567	380	3	of	of	ADP
ejpam-6567	380	4	mathematics	mathematic	NOUN
ejpam-6567	380	5	and	and	CCONJ
ejpam-6567	380	6	computer	computer	NOUN
ejpam-6567	380	7	science	science	NOUN
ejpam-6567	380	8	,	,	PUNCT
ejpam-6567	380	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6567	380	10	,	,	PUNCT
ejpam-6567	380	11	2024	2024	NUM
ejpam-6567	380	12	.	.	PUNCT
ejpam-6567	381	1	[	[	X
ejpam-6567	381	2	26	26	NUM
ejpam-6567	381	3	]	]	X
ejpam-6567	381	4	n.	n.	PROPN
ejpam-6567	381	5	viriyapong	viriyapong	PROPN
ejpam-6567	381	6	,	,	PUNCT
ejpam-6567	381	7	s.	s.	PROPN
ejpam-6567	381	8	sompong	sompong	PROPN
ejpam-6567	381	9	,	,	PUNCT
ejpam-6567	381	10	and	and	CCONJ
ejpam-6567	381	11	c.	c.	PROPN
ejpam-6567	381	12	boonpok	boonpok	PROPN
ejpam-6567	381	13	.	.	PUNCT
ejpam-6567	382	1	upper	upper	ADJ
ejpam-6567	382	2	and	and	CCONJ
ejpam-6567	382	3	lower	low	ADJ
ejpam-6567	382	4	s-(τ1	s-(τ1	NOUN
ejpam-6567	382	5	,	,	PUNCT
ejpam-6567	382	6	τ2)p	τ2)p	ADJ
ejpam-6567	382	7	-	-	PUNCT
ejpam-6567	382	8	continuous	continuous	ADJ
ejpam-6567	382	9	multifunctions	multifunction	NOUN
ejpam-6567	382	10	.	.	PUNCT
ejpam-6567	383	1	european	european	ADJ
ejpam-6567	383	2	journal	journal	PROPN
ejpam-6567	383	3	of	of	ADP
ejpam-6567	383	4	pure	pure	ADJ
ejpam-6567	383	5	and	and	CCONJ
ejpam-6567	383	6	applied	applied	ADJ
ejpam-6567	383	7	mathematics	mathematic	NOUN
ejpam-6567	383	8	,	,	PUNCT
ejpam-6567	383	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-6567	383	10	,	,	PUNCT
ejpam-6567	383	11	2024	2024	NUM
ejpam-6567	383	12	.	.	PUNCT
ejpam-6567	384	1	[	[	X
ejpam-6567	384	2	27	27	NUM
ejpam-6567	384	3	]	]	X
ejpam-6567	384	4	c.	c.	PROPN
ejpam-6567	384	5	viriyapong	viriyapong	PROPN
ejpam-6567	384	6	,	,	PUNCT
ejpam-6567	384	7	s.	s.	PROPN
ejpam-6567	384	8	sompong	sompong	PROPN
ejpam-6567	384	9	,	,	PUNCT
ejpam-6567	384	10	and	and	CCONJ
ejpam-6567	384	11	c.	c.	PROPN
ejpam-6567	384	12	boonpok	boonpok	PROPN
ejpam-6567	384	13	.	.	PUNCT
ejpam-6567	385	1	upper	upper	ADJ
ejpam-6567	385	2	and	and	CCONJ
ejpam-6567	385	3	lower	low	ADJ
ejpam-6567	385	4	slight	slight	ADJ
ejpam-6567	385	5	α(τ1	α(τ1	NOUN
ejpam-6567	385	6	,	,	PUNCT
ejpam-6567	385	7	τ2)continuity	τ2)continuity	PROPN
ejpam-6567	385	8	.	.	PUNCT
ejpam-6567	386	1	european	european	PROPN
ejpam-6567	386	2	journal	journal	PROPN
ejpam-6567	386	3	of	of	ADP
ejpam-6567	386	4	pure	pure	ADJ
ejpam-6567	386	5	and	and	CCONJ
ejpam-6567	386	6	applied	applied	ADJ
ejpam-6567	386	7	mathematics	mathematic	NOUN
ejpam-6567	386	8	,	,	PUNCT
ejpam-6567	386	9	17(3):2142–2154	17(3):2142–2154	NUM
ejpam-6567	386	10	,	,	PUNCT
ejpam-6567	386	11	2024	2024	NUM
ejpam-6567	386	12	.	.	PUNCT
ejpam-6567	387	1	[	[	X
ejpam-6567	387	2	28	28	NUM
ejpam-6567	387	3	]	]	X
ejpam-6567	387	4	e.	e.	PROPN
ejpam-6567	387	5	ekici	ekici	PROPN
ejpam-6567	387	6	,	,	PUNCT
ejpam-6567	387	7	,	,	PUNCT
ejpam-6567	387	8	and	and	CCONJ
ejpam-6567	387	9	t.	t.	PROPN
ejpam-6567	387	10	noiri	noiri	PROPN
ejpam-6567	387	11	.	.	PUNCT
ejpam-6567	388	1	⋆-extremally	⋆-extremally	ADV
ejpam-6567	388	2	disconnected	disconnect	VERB
ejpam-6567	388	3	ideal	ideal	ADJ
ejpam-6567	388	4	topological	topological	ADJ
ejpam-6567	388	5	spaces	space	NOUN
ejpam-6567	388	6	.	.	PUNCT
ejpam-6567	389	1	acta	acta	PROPN
ejpam-6567	389	2	mathematica	mathematica	PROPN
ejpam-6567	389	3	hungarica	hungarica	PROPN
ejpam-6567	389	4	,	,	PUNCT
ejpam-6567	389	5	122:81–90	122:81–90	NUM
ejpam-6567	389	6	,	,	PUNCT
ejpam-6567	389	7	2009	2009	NUM
ejpam-6567	389	8	.	.	PUNCT
ejpam-6567	390	1	[	[	X
ejpam-6567	390	2	29	29	NUM
ejpam-6567	390	3	]	]	X
ejpam-6567	390	4	c.	c.	PROPN
ejpam-6567	390	5	viriyapong	viriyapong	PROPN
ejpam-6567	390	6	,	,	PUNCT
ejpam-6567	390	7	a.	a.	PROPN
ejpam-6567	390	8	sama	sama	PROPN
ejpam-6567	390	9	-	-	PUNCT
ejpam-6567	390	10	ae	ae	PROPN
ejpam-6567	390	11	,	,	PUNCT
ejpam-6567	390	12	and	and	CCONJ
ejpam-6567	390	13	c.	c.	PROPN
ejpam-6567	390	14	boonpok	boonpok	PROPN
ejpam-6567	390	15	.	.	PUNCT
ejpam-6567	391	1	almost	almost	ADV
ejpam-6567	391	2	continuity	continuity	NOUN
ejpam-6567	391	3	for	for	ADP
ejpam-6567	391	4	multifunctions	multifunction	NOUN
ejpam-6567	391	5	defined	define	VERB
ejpam-6567	391	6	from	from	ADP
ejpam-6567	391	7	an	an	DET
ejpam-6567	391	8	ideal	ideal	ADJ
ejpam-6567	391	9	topological	topological	ADJ
ejpam-6567	391	10	space	space	NOUN
ejpam-6567	391	11	into	into	ADP
ejpam-6567	391	12	a	a	DET
ejpam-6567	391	13	bitopological	bitopological	ADJ
ejpam-6567	391	14	space	space	NOUN
ejpam-6567	391	15	.	.	PUNCT
ejpam-6567	392	1	(	(	PUNCT
ejpam-6567	392	2	submitted	submit	VERB
ejpam-6567	392	3	)	)	PUNCT
ejpam-6567	392	4	.	.	PUNCT
ejpam-6567	393	1	[	[	X
ejpam-6567	393	2	30	30	NUM
ejpam-6567	393	3	]	]	X
ejpam-6567	393	4	m.	m.	NOUN
ejpam-6567	393	5	chiangpradit	chiangpradit	NOUN
ejpam-6567	393	6	,	,	PUNCT
ejpam-6567	393	7	s.	s.	PROPN
ejpam-6567	393	8	sompong	sompong	PROPN
ejpam-6567	393	9	,	,	PUNCT
ejpam-6567	393	10	and	and	CCONJ
ejpam-6567	393	11	c.	c.	PROPN
ejpam-6567	393	12	boonpok	boonpok	PROPN
ejpam-6567	393	13	.	.	PUNCT
ejpam-6567	394	1	on	on	ADP
ejpam-6567	394	2	characterizations	characterization	NOUN
ejpam-6567	394	3	of	of	ADP
ejpam-6567	394	4	(	(	PUNCT
ejpam-6567	394	5	τ1	τ1	NOUN
ejpam-6567	394	6	,	,	PUNCT
ejpam-6567	394	7	τ2)regular	τ2)regular	ADJ
ejpam-6567	394	8	spaces	space	NOUN
ejpam-6567	394	9	.	.	PUNCT
ejpam-6567	395	1	international	international	ADJ
ejpam-6567	395	2	journal	journal	PROPN
ejpam-6567	395	3	of	of	ADP
ejpam-6567	395	4	mathematics	mathematic	NOUN
ejpam-6567	395	5	and	and	CCONJ
ejpam-6567	395	6	computer	computer	NOUN
ejpam-6567	395	7	science	science	NOUN
ejpam-6567	395	8	,	,	PUNCT
ejpam-6567	395	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-6567	395	10	,	,	PUNCT
ejpam-6567	395	11	2024	2024	NUM
ejpam-6567	395	12	.	.	PUNCT
ejpam-6567	396	1	[	[	X
ejpam-6567	396	2	31	31	NUM
ejpam-6567	396	3	]	]	PUNCT
ejpam-6567	396	4	b.	b.	PROPN
ejpam-6567	396	5	kong	kong	PROPN
ejpam-6567	396	6	-	-	PUNCT
ejpam-6567	396	7	ied	ied	PROPN
ejpam-6567	396	8	,	,	PUNCT
ejpam-6567	396	9	s.	s.	PROPN
ejpam-6567	396	10	sompong	sompong	PROPN
ejpam-6567	396	11	,	,	PUNCT
ejpam-6567	396	12	and	and	CCONJ
ejpam-6567	396	13	c.	c.	PROPN
ejpam-6567	396	14	boonpok	boonpok	PROPN
ejpam-6567	396	15	.	.	PUNCT
ejpam-6567	397	1	(	(	PUNCT
ejpam-6567	397	2	τ1	τ1	NOUN
ejpam-6567	397	3	,	,	PUNCT
ejpam-6567	397	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6567	397	5	and	and	CCONJ
ejpam-6567	397	6	weak	weak	ADJ
ejpam-6567	397	7	(	(	PUNCT
ejpam-6567	397	8	τ1	τ1	NOUN
ejpam-6567	397	9	,	,	PUNCT
ejpam-6567	397	10	τ2)continuity	τ2)continuity	NOUN
ejpam-6567	397	11	.	.	PUNCT
ejpam-6567	398	1	international	international	ADJ
ejpam-6567	398	2	journal	journal	PROPN
ejpam-6567	398	3	of	of	ADP
ejpam-6567	398	4	mathematics	mathematic	NOUN
ejpam-6567	398	5	and	and	CCONJ
ejpam-6567	398	6	computer	computer	NOUN
ejpam-6567	398	7	science	science	NOUN
ejpam-6567	398	8	,	,	PUNCT
ejpam-6567	398	9	19(4):1321	19(4):1321	NUM
ejpam-6567	398	10	–	–	PUNCT
ejpam-6567	398	11	1327	1327	NUM
ejpam-6567	398	12	,	,	PUNCT
ejpam-6567	398	13	2024	2024	NUM
ejpam-6567	398	14	.	.	PUNCT
ejpam-6567	399	1	[	[	X
ejpam-6567	399	2	32	32	NUM
ejpam-6567	399	3	]	]	PUNCT
ejpam-6567	399	4	m.	m.	NOUN
ejpam-6567	399	5	chiangpradit	chiangpradit	NOUN
ejpam-6567	399	6	,	,	PUNCT
ejpam-6567	399	7	s.	s.	PROPN
ejpam-6567	399	8	sompong	sompong	PROPN
ejpam-6567	399	9	,	,	PUNCT
ejpam-6567	399	10	and	and	CCONJ
ejpam-6567	399	11	c.	c.	PROPN
ejpam-6567	399	12	boonpok	boonpok	PROPN
ejpam-6567	399	13	.	.	PUNCT
ejpam-6567	400	1	on	on	ADP
ejpam-6567	400	2	characterizations	characterization	NOUN
ejpam-6567	400	3	of	of	ADP
ejpam-6567	400	4	(	(	PUNCT
ejpam-6567	400	5	τ1	τ1	NOUN
ejpam-6567	400	6	,	,	PUNCT
ejpam-6567	400	7	τ2)normal	τ2)normal	ADJ
ejpam-6567	400	8	spaces	space	NOUN
ejpam-6567	400	9	.	.	PUNCT
ejpam-6567	401	1	international	international	ADJ
ejpam-6567	401	2	journal	journal	PROPN
ejpam-6567	401	3	of	of	ADP
ejpam-6567	401	4	mathematics	mathematic	NOUN
ejpam-6567	401	5	and	and	CCONJ
ejpam-6567	401	6	computer	computer	NOUN
ejpam-6567	401	7	science	science	NOUN
ejpam-6567	401	8	,	,	PUNCT
ejpam-6567	401	9	19(4):1315–1320	19(4):1315–1320	NUM
ejpam-6567	401	10	,	,	PUNCT
ejpam-6567	401	11	2024	2024	NUM
ejpam-6567	401	12	.	.	PUNCT