id	sid	tid	token	lemma	pos
ejpam-6568	1	1	european	european	PROPN
ejpam-6568	1	2	journal	journal	PROPN
ejpam-6568	1	3	of	of	ADP
ejpam-6568	1	4	pure	pure	ADJ
ejpam-6568	1	5	and	and	CCONJ
ejpam-6568	1	6	applied	applied	ADJ
ejpam-6568	1	7	mathematics	mathematic	NOUN
ejpam-6568	1	8	2025	2025	NUM
ejpam-6568	1	9	,	,	PUNCT
ejpam-6568	1	10	vol	vol	NOUN
ejpam-6568	1	11	.	.	PROPN
ejpam-6568	1	12	18	18	NUM
ejpam-6568	1	13	,	,	PUNCT
ejpam-6568	1	14	issue	issue	NOUN
ejpam-6568	1	15	3	3	NUM
ejpam-6568	1	16	,	,	PUNCT
ejpam-6568	1	17	article	article	NOUN
ejpam-6568	1	18	number	number	NOUN
ejpam-6568	1	19	6568	6568	NUM
ejpam-6568	1	20	issn	issn	PROPN
ejpam-6568	1	21	1307	1307	NUM
ejpam-6568	1	22	-	-	SYM
ejpam-6568	1	23	5543	5543	NUM
ejpam-6568	1	24	–	–	PUNCT
ejpam-6568	1	25	ejpam.com	ejpam.com	X
ejpam-6568	1	26	published	publish	VERB
ejpam-6568	1	27	by	by	ADP
ejpam-6568	1	28	new	new	PROPN
ejpam-6568	1	29	york	york	PROPN
ejpam-6568	1	30	business	business	PROPN
ejpam-6568	1	31	global	global	PROPN
ejpam-6568	1	32	on	on	ADP
ejpam-6568	1	33	almost	almost	ADV
ejpam-6568	1	34	τ	τ	NOUN
ejpam-6568	1	35	⋆(σ1	⋆(σ1	NOUN
ejpam-6568	1	36	,	,	PUNCT
ejpam-6568	1	37	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6568	1	38	and	and	CCONJ
ejpam-6568	1	39	weak	weak	ADJ
ejpam-6568	1	40	τ	τ	NOUN
ejpam-6568	1	41	⋆(σ1	⋆(σ1	NOUN
ejpam-6568	1	42	,	,	PUNCT
ejpam-6568	1	43	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6568	1	44	nongluk	nongluk	NOUN
ejpam-6568	1	45	viriyapong1	viriyapong1	PROPN
ejpam-6568	1	46	,	,	PUNCT
ejpam-6568	1	47	areeyuth	areeyuth	NOUN
ejpam-6568	1	48	sama	sama	NOUN
ejpam-6568	1	49	-	-	PUNCT
ejpam-6568	1	50	ae2	ae2	PROPN
ejpam-6568	1	51	,	,	PUNCT
ejpam-6568	1	52	chawalit	chawalit	VERB
ejpam-6568	1	53	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6568	1	54	1	1	NUM
ejpam-6568	1	55	mathematics	mathematic	NOUN
ejpam-6568	1	56	and	and	CCONJ
ejpam-6568	1	57	applied	apply	VERB
ejpam-6568	1	58	mathematics	mathematics	PROPN
ejpam-6568	1	59	research	research	NOUN
ejpam-6568	1	60	unit	unit	NOUN
ejpam-6568	1	61	,	,	PUNCT
ejpam-6568	1	62	department	department	NOUN
ejpam-6568	1	63	of	of	ADP
ejpam-6568	1	64	mathematics	mathematic	NOUN
ejpam-6568	1	65	,	,	PUNCT
ejpam-6568	1	66	faculty	faculty	NOUN
ejpam-6568	1	67	of	of	ADP
ejpam-6568	1	68	science	science	NOUN
ejpam-6568	1	69	,	,	PUNCT
ejpam-6568	1	70	mahasarakham	mahasarakham	PROPN
ejpam-6568	1	71	university	university	PROPN
ejpam-6568	1	72	,	,	PUNCT
ejpam-6568	1	73	maha	maha	PROPN
ejpam-6568	1	74	sarakham	sarakham	PROPN
ejpam-6568	1	75	,	,	PUNCT
ejpam-6568	1	76	44150	44150	NUM
ejpam-6568	1	77	,	,	PUNCT
ejpam-6568	1	78	thailand	thailand	PROPN
ejpam-6568	1	79	2	2	NUM
ejpam-6568	1	80	department	department	NOUN
ejpam-6568	1	81	of	of	ADP
ejpam-6568	1	82	mathematics	mathematic	NOUN
ejpam-6568	1	83	and	and	CCONJ
ejpam-6568	1	84	computer	computer	NOUN
ejpam-6568	1	85	science	science	NOUN
ejpam-6568	1	86	,	,	PUNCT
ejpam-6568	1	87	faculty	faculty	NOUN
ejpam-6568	1	88	of	of	ADP
ejpam-6568	1	89	science	science	NOUN
ejpam-6568	1	90	and	and	CCONJ
ejpam-6568	1	91	technology	technology	NOUN
ejpam-6568	1	92	,	,	PUNCT
ejpam-6568	1	93	prince	prince	NOUN
ejpam-6568	1	94	of	of	ADP
ejpam-6568	1	95	songkla	songkla	PROPN
ejpam-6568	1	96	university	university	PROPN
ejpam-6568	1	97	,	,	PUNCT
ejpam-6568	1	98	pattani	pattani	NOUN
ejpam-6568	1	99	campus	campus	NOUN
ejpam-6568	1	100	,	,	PUNCT
ejpam-6568	1	101	pattani	pattani	NOUN
ejpam-6568	1	102	,	,	PUNCT
ejpam-6568	1	103	94000	94000	NUM
ejpam-6568	1	104	,	,	PUNCT
ejpam-6568	1	105	thailand	thailand	PROPN
ejpam-6568	1	106	abstract	abstract	PROPN
ejpam-6568	1	107	.	.	PUNCT
ejpam-6568	2	1	this	this	DET
ejpam-6568	2	2	paper	paper	NOUN
ejpam-6568	2	3	is	be	AUX
ejpam-6568	2	4	concerned	concern	VERB
ejpam-6568	2	5	with	with	ADP
ejpam-6568	2	6	the	the	DET
ejpam-6568	2	7	concepts	concept	NOUN
ejpam-6568	2	8	of	of	ADP
ejpam-6568	2	9	almost	almost	ADV
ejpam-6568	2	10	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	2	11	,	,	PUNCT
ejpam-6568	2	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	2	13	functions	function	NOUN
ejpam-6568	2	14	and	and	CCONJ
ejpam-6568	2	15	weakly	weakly	ADJ
ejpam-6568	2	16	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	2	17	,	,	PUNCT
ejpam-6568	2	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	2	19	functions	function	NOUN
ejpam-6568	2	20	.	.	PUNCT
ejpam-6568	3	1	moreover	moreover	ADV
ejpam-6568	3	2	,	,	PUNCT
ejpam-6568	3	3	some	some	DET
ejpam-6568	3	4	characterizations	characterization	NOUN
ejpam-6568	3	5	of	of	ADP
ejpam-6568	3	6	almost	almost	ADV
ejpam-6568	3	7	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	3	8	,	,	PUNCT
ejpam-6568	3	9	σ2)continuous	σ2)continuous	ADJ
ejpam-6568	3	10	functions	function	NOUN
ejpam-6568	3	11	and	and	CCONJ
ejpam-6568	3	12	weakly	weakly	ADJ
ejpam-6568	3	13	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	3	14	,	,	PUNCT
ejpam-6568	3	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	3	16	functions	function	NOUN
ejpam-6568	3	17	are	be	AUX
ejpam-6568	3	18	investigated	investigate	VERB
ejpam-6568	3	19	.	.	PUNCT
ejpam-6568	4	1	furthermore	furthermore	ADV
ejpam-6568	4	2	,	,	PUNCT
ejpam-6568	4	3	the	the	DET
ejpam-6568	4	4	relationships	relationship	NOUN
ejpam-6568	4	5	between	between	ADP
ejpam-6568	4	6	almost	almost	ADV
ejpam-6568	4	7	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	4	8	,	,	PUNCT
ejpam-6568	4	9	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6568	4	10	and	and	CCONJ
ejpam-6568	4	11	weak	weak	ADJ
ejpam-6568	4	12	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	4	13	,	,	PUNCT
ejpam-6568	4	14	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6568	4	15	are	be	AUX
ejpam-6568	4	16	considered	consider	VERB
ejpam-6568	4	17	.	.	PUNCT
ejpam-6568	5	1	2020	2020	NUM
ejpam-6568	5	2	mathematics	mathematic	NOUN
ejpam-6568	5	3	subject	subject	NOUN
ejpam-6568	5	4	classifications	classification	NOUN
ejpam-6568	5	5	:	:	PUNCT
ejpam-6568	5	6	54c05	54c05	NUM
ejpam-6568	5	7	,	,	PUNCT
ejpam-6568	5	8	54c08	54c08	NUM
ejpam-6568	5	9	key	key	ADJ
ejpam-6568	5	10	words	word	NOUN
ejpam-6568	5	11	and	and	CCONJ
ejpam-6568	5	12	phrases	phrase	NOUN
ejpam-6568	5	13	:	:	PUNCT
ejpam-6568	5	14	almost	almost	ADV
ejpam-6568	5	15	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	5	16	,	,	PUNCT
ejpam-6568	5	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	5	18	function	function	NOUN
ejpam-6568	5	19	,	,	PUNCT
ejpam-6568	5	20	weakly	weakly	ADJ
ejpam-6568	5	21	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	5	22	,	,	PUNCT
ejpam-6568	5	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	5	24	function	function	NOUN
ejpam-6568	5	25	1	1	NUM
ejpam-6568	5	26	.	.	PUNCT
ejpam-6568	5	27	introduction	introduction	NOUN
ejpam-6568	5	28	in	in	ADP
ejpam-6568	5	29	1968	1968	NUM
ejpam-6568	5	30	,	,	PUNCT
ejpam-6568	5	31	singal	singal	NOUN
ejpam-6568	5	32	and	and	CCONJ
ejpam-6568	5	33	singal	singal	ADJ
ejpam-6568	6	1	[	[	X
ejpam-6568	6	2	1	1	NUM
ejpam-6568	6	3	]	]	PUNCT
ejpam-6568	6	4	introduced	introduce	VERB
ejpam-6568	6	5	the	the	DET
ejpam-6568	6	6	concept	concept	NOUN
ejpam-6568	6	7	of	of	ADP
ejpam-6568	6	8	almost	almost	ADV
ejpam-6568	6	9	continuous	continuous	ADJ
ejpam-6568	6	10	functions	function	NOUN
ejpam-6568	6	11	as	as	ADP
ejpam-6568	6	12	a	a	DET
ejpam-6568	6	13	generalization	generalization	NOUN
ejpam-6568	6	14	of	of	ADP
ejpam-6568	6	15	continuity	continuity	NOUN
ejpam-6568	6	16	.	.	PUNCT
ejpam-6568	7	1	munshi	munshi	PROPN
ejpam-6568	7	2	and	and	CCONJ
ejpam-6568	7	3	bassan	bassan	NOUN
ejpam-6568	7	4	[	[	X
ejpam-6568	7	5	2	2	NUM
ejpam-6568	7	6	]	]	PUNCT
ejpam-6568	7	7	studied	study	VERB
ejpam-6568	7	8	the	the	DET
ejpam-6568	7	9	notion	notion	NOUN
ejpam-6568	7	10	of	of	ADP
ejpam-6568	7	11	almost	almost	ADV
ejpam-6568	7	12	semi	semi	ADJ
ejpam-6568	7	13	-	-	ADJ
ejpam-6568	7	14	continuous	continuous	ADJ
ejpam-6568	7	15	functions	function	NOUN
ejpam-6568	7	16	.	.	PUNCT
ejpam-6568	8	1	noiri	noiri	ADV
ejpam-6568	9	1	[	[	X
ejpam-6568	9	2	3	3	X
ejpam-6568	9	3	]	]	PUNCT
ejpam-6568	9	4	introduced	introduce	VERB
ejpam-6568	9	5	and	and	CCONJ
ejpam-6568	9	6	investigated	investigate	VERB
ejpam-6568	9	7	the	the	DET
ejpam-6568	9	8	concept	concept	NOUN
ejpam-6568	9	9	of	of	ADP
ejpam-6568	9	10	almost	almost	ADV
ejpam-6568	9	11	α	α	NUM
ejpam-6568	9	12	-	-	ADJ
ejpam-6568	9	13	continuous	continuous	ADJ
ejpam-6568	9	14	functions	function	NOUN
ejpam-6568	9	15	.	.	PUNCT
ejpam-6568	10	1	nasef	nasef	NOUN
ejpam-6568	10	2	and	and	CCONJ
ejpam-6568	10	3	noiri	noiri	ADV
ejpam-6568	11	1	[	[	X
ejpam-6568	11	2	4	4	X
ejpam-6568	11	3	]	]	PUNCT
ejpam-6568	11	4	introduced	introduce	VERB
ejpam-6568	11	5	two	two	NUM
ejpam-6568	11	6	classes	class	NOUN
ejpam-6568	11	7	of	of	ADP
ejpam-6568	11	8	functions	function	NOUN
ejpam-6568	11	9	,	,	PUNCT
ejpam-6568	11	10	namely	namely	ADV
ejpam-6568	11	11	almost	almost	ADV
ejpam-6568	11	12	precontinuous	precontinuous	ADJ
ejpam-6568	11	13	functions	function	NOUN
ejpam-6568	11	14	and	and	CCONJ
ejpam-6568	11	15	almost	almost	ADV
ejpam-6568	11	16	β	β	ADJ
ejpam-6568	11	17	-	-	ADJ
ejpam-6568	11	18	continuous	continuous	ADJ
ejpam-6568	11	19	functions	function	NOUN
ejpam-6568	11	20	.	.	PUNCT
ejpam-6568	12	1	the	the	DET
ejpam-6568	12	2	class	class	NOUN
ejpam-6568	12	3	of	of	ADP
ejpam-6568	12	4	almost	almost	ADV
ejpam-6568	12	5	precontinuity	precontinuity	NOUN
ejpam-6568	12	6	is	be	AUX
ejpam-6568	12	7	a	a	DET
ejpam-6568	12	8	generalization	generalization	NOUN
ejpam-6568	12	9	of	of	ADP
ejpam-6568	12	10	almost	almost	ADV
ejpam-6568	12	11	α	α	NOUN
ejpam-6568	12	12	-	-	NOUN
ejpam-6568	12	13	continuity	continuity	NOUN
ejpam-6568	12	14	.	.	PUNCT
ejpam-6568	13	1	the	the	DET
ejpam-6568	13	2	class	class	NOUN
ejpam-6568	13	3	of	of	ADP
ejpam-6568	13	4	almost	almost	ADV
ejpam-6568	13	5	β	β	NOUN
ejpam-6568	13	6	-	-	NOUN
ejpam-6568	13	7	continuity	continuity	NOUN
ejpam-6568	13	8	is	be	AUX
ejpam-6568	13	9	a	a	DET
ejpam-6568	13	10	generalization	generalization	NOUN
ejpam-6568	13	11	of	of	ADP
ejpam-6568	13	12	almost	almost	ADV
ejpam-6568	13	13	semi	semi	NOUN
ejpam-6568	13	14	-	-	NOUN
ejpam-6568	13	15	continuity	continuity	NOUN
ejpam-6568	13	16	.	.	PUNCT
ejpam-6568	14	1	levine	levine	PROPN
ejpam-6568	15	1	[	[	X
ejpam-6568	15	2	5	5	NUM
ejpam-6568	15	3	]	]	PUNCT
ejpam-6568	15	4	introduced	introduce	VERB
ejpam-6568	15	5	and	and	CCONJ
ejpam-6568	15	6	investigated	investigate	VERB
ejpam-6568	15	7	the	the	DET
ejpam-6568	15	8	concept	concept	NOUN
ejpam-6568	15	9	of	of	ADP
ejpam-6568	15	10	weakly	weakly	ADJ
ejpam-6568	15	11	continuous	continuous	ADJ
ejpam-6568	15	12	functions	function	NOUN
ejpam-6568	15	13	.	.	PUNCT
ejpam-6568	16	1	husain	husain	NOUN
ejpam-6568	17	1	[	[	X
ejpam-6568	17	2	6	6	NUM
ejpam-6568	17	3	]	]	PUNCT
ejpam-6568	17	4	introduced	introduce	VERB
ejpam-6568	17	5	and	and	CCONJ
ejpam-6568	17	6	studied	study	VERB
ejpam-6568	17	7	the	the	DET
ejpam-6568	17	8	notion	notion	NOUN
ejpam-6568	17	9	of	of	ADP
ejpam-6568	17	10	almost	almost	ADV
ejpam-6568	17	11	continuous	continuous	ADJ
ejpam-6568	17	12	functions	function	NOUN
ejpam-6568	17	13	.	.	PUNCT
ejpam-6568	18	1	janković	janković	PUNCT
ejpam-6568	19	1	[	[	X
ejpam-6568	19	2	7	7	X
ejpam-6568	19	3	]	]	PUNCT
ejpam-6568	19	4	introduced	introduce	VERB
ejpam-6568	19	5	almost	almost	ADV
ejpam-6568	19	6	weak	weak	ADJ
ejpam-6568	19	7	continuity	continuity	NOUN
ejpam-6568	19	8	as	as	ADP
ejpam-6568	19	9	a	a	DET
ejpam-6568	19	10	generalization	generalization	NOUN
ejpam-6568	19	11	of	of	ADP
ejpam-6568	19	12	both	both	DET
ejpam-6568	19	13	weak	weak	ADJ
ejpam-6568	19	14	continuity	continuity	NOUN
ejpam-6568	19	15	and	and	CCONJ
ejpam-6568	19	16	almost	almost	ADV
ejpam-6568	19	17	continuity	continuity	NOUN
ejpam-6568	19	18	.	.	PUNCT
ejpam-6568	20	1	noiri	noiri	ADV
ejpam-6568	21	1	[	[	X
ejpam-6568	21	2	8	8	NUM
ejpam-6568	21	3	]	]	PUNCT
ejpam-6568	21	4	investigated	investigate	VERB
ejpam-6568	21	5	several	several	ADJ
ejpam-6568	21	6	characterizations	characterization	NOUN
ejpam-6568	21	7	of	of	ADP
ejpam-6568	21	8	almost	almost	ADV
ejpam-6568	21	9	weakly	weakly	ADJ
ejpam-6568	21	10	continuous	continuous	ADJ
ejpam-6568	21	11	functions	function	NOUN
ejpam-6568	21	12	.	.	PUNCT
ejpam-6568	22	1	rose	rise	VERB
ejpam-6568	23	1	[	[	X
ejpam-6568	23	2	9	9	NUM
ejpam-6568	23	3	]	]	PUNCT
ejpam-6568	23	4	introduced	introduce	VERB
ejpam-6568	23	5	the	the	DET
ejpam-6568	23	6	notion	notion	NOUN
ejpam-6568	23	7	of	of	ADP
ejpam-6568	23	8	subweakly	subweakly	ADJ
ejpam-6568	23	9	continuous	continuous	ADJ
ejpam-6568	23	10	functions	function	NOUN
ejpam-6568	23	11	and	and	CCONJ
ejpam-6568	23	12	investigated	investigate	VERB
ejpam-6568	23	13	the	the	DET
ejpam-6568	23	14	relationships	relationship	NOUN
ejpam-6568	23	15	between	between	ADP
ejpam-6568	23	16	subweak	subweak	NOUN
ejpam-6568	23	17	continuity	continuity	NOUN
ejpam-6568	23	18	and	and	CCONJ
ejpam-6568	23	19	weak	weak	ADJ
ejpam-6568	23	20	continuity	continuity	NOUN
ejpam-6568	23	21	.	.	PUNCT
ejpam-6568	24	1	popa	popa	NOUN
ejpam-6568	24	2	and	and	CCONJ
ejpam-6568	24	3	noiri	noiri	ADV
ejpam-6568	25	1	[	[	X
ejpam-6568	25	2	10	10	NUM
ejpam-6568	25	3	]	]	PUNCT
ejpam-6568	25	4	introduced	introduce	VERB
ejpam-6568	25	5	the	the	DET
ejpam-6568	25	6	concept	concept	NOUN
ejpam-6568	25	7	of	of	ADP
ejpam-6568	25	8	weakly	weakly	ADJ
ejpam-6568	25	9	∗corresponding	∗corresponde	VERB
ejpam-6568	25	10	author	author	NOUN
ejpam-6568	25	11	.	.	PUNCT
ejpam-6568	26	1	doi	doi	NOUN
ejpam-6568	26	2	:	:	PUNCT
ejpam-6568	26	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6568	https://doi.org/10.29020/nybg.ejpam.v18i3.6568	PROPN
ejpam-6568	26	4	email	email	NOUN
ejpam-6568	26	5	addresses	address	NOUN
ejpam-6568	26	6	:	:	PUNCT
ejpam-6568	26	7	nongluk.h@msu.ac.th	nongluk.h@msu.ac.th	PROPN
ejpam-6568	26	8	(	(	PUNCT
ejpam-6568	26	9	n.	n.	NOUN
ejpam-6568	26	10	viriyapong	viriyapong	PROPN
ejpam-6568	26	11	)	)	PUNCT
ejpam-6568	26	12	,	,	PUNCT
ejpam-6568	26	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6568	26	14	(	(	PUNCT
ejpam-6568	26	15	a.	a.	PROPN
ejpam-6568	26	16	sama	sama	PROPN
ejpam-6568	26	17	-	-	PUNCT
ejpam-6568	26	18	ae	ae	PROPN
ejpam-6568	26	19	)	)	PUNCT
ejpam-6568	26	20	,	,	PUNCT
ejpam-6568	26	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6568	26	22	(	(	PUNCT
ejpam-6568	26	23	c.	c.	PROPN
ejpam-6568	26	24	boonpok	boonpok	PROPN
ejpam-6568	26	25	)	)	PUNCT
ejpam-6568	26	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6568	27	1	1	1	NUM
ejpam-6568	27	2	copyright	copyright	NOUN
ejpam-6568	27	3	:	:	PUNCT
ejpam-6568	27	4	©	©	PROPN
ejpam-6568	27	5	2025	2025	NUM
ejpam-6568	27	6	the	the	DET
ejpam-6568	27	7	author(s	author(s	NOUN
ejpam-6568	27	8	)	)	PUNCT
ejpam-6568	27	9	.	.	PUNCT
ejpam-6568	28	1	(	(	PUNCT
ejpam-6568	28	2	cc	cc	NOUN
ejpam-6568	28	3	by	by	ADP
ejpam-6568	28	4	-	-	PUNCT
ejpam-6568	28	5	nc	nc	PROPN
ejpam-6568	28	6	4.0	4.0	NUM
ejpam-6568	28	7	)	)	PUNCT
ejpam-6568	28	8	n.	n.	NOUN
ejpam-6568	28	9	viriyapong	viriyapong	PROPN
ejpam-6568	28	10	,	,	PUNCT
ejpam-6568	28	11	a.	a.	PROPN
ejpam-6568	28	12	sama	sama	PROPN
ejpam-6568	28	13	-	-	PUNCT
ejpam-6568	28	14	ae	ae	PROPN
ejpam-6568	28	15	,	,	PUNCT
ejpam-6568	28	16	c.	c.	PROPN
ejpam-6568	28	17	boonpok	boonpok	PROPN
ejpam-6568	28	18	/	/	SYM
ejpam-6568	28	19	eur	eur	PROPN
ejpam-6568	28	20	.	.	PUNCT
ejpam-6568	29	1	j.	j.	PROPN
ejpam-6568	29	2	pure	pure	PROPN
ejpam-6568	29	3	appl	appl	PROPN
ejpam-6568	29	4	.	.	PROPN
ejpam-6568	29	5	math	math	PROPN
ejpam-6568	29	6	,	,	PUNCT
ejpam-6568	29	7	18	18	NUM
ejpam-6568	29	8	(	(	PUNCT
ejpam-6568	29	9	3	3	NUM
ejpam-6568	29	10	)	)	PUNCT
ejpam-6568	29	11	(	(	PUNCT
ejpam-6568	29	12	2025	2025	NUM
ejpam-6568	29	13	)	)	PUNCT
ejpam-6568	29	14	,	,	PUNCT
ejpam-6568	29	15	6568	6568	NUM
ejpam-6568	29	16	2	2	NUM
ejpam-6568	29	17	of	of	ADP
ejpam-6568	29	18	18	18	NUM
ejpam-6568	29	19	(	(	PUNCT
ejpam-6568	29	20	τ	τ	PROPN
ejpam-6568	29	21	,	,	PUNCT
ejpam-6568	29	22	m)-continuous	m)-continuous	ADJ
ejpam-6568	29	23	functions	function	NOUN
ejpam-6568	29	24	as	as	ADP
ejpam-6568	29	25	functions	function	NOUN
ejpam-6568	29	26	from	from	ADP
ejpam-6568	29	27	a	a	DET
ejpam-6568	29	28	topological	topological	ADJ
ejpam-6568	29	29	space	space	NOUN
ejpam-6568	29	30	into	into	ADP
ejpam-6568	29	31	a	a	DET
ejpam-6568	29	32	set	set	NOUN
ejpam-6568	29	33	satisfying	satisfy	VERB
ejpam-6568	29	34	some	some	DET
ejpam-6568	29	35	minimal	minimal	ADJ
ejpam-6568	29	36	conditions	condition	NOUN
ejpam-6568	29	37	and	and	CCONJ
ejpam-6568	29	38	investigated	investigate	VERB
ejpam-6568	29	39	several	several	ADJ
ejpam-6568	29	40	characterizations	characterization	NOUN
ejpam-6568	29	41	of	of	ADP
ejpam-6568	29	42	weakly	weakly	ADJ
ejpam-6568	29	43	(	(	PUNCT
ejpam-6568	29	44	τ	τ	PROPN
ejpam-6568	29	45	,	,	PUNCT
ejpam-6568	29	46	m)-continuous	m)-continuous	ADJ
ejpam-6568	29	47	functions	function	NOUN
ejpam-6568	29	48	.	.	PUNCT
ejpam-6568	30	1	ekici	ekici	NOUN
ejpam-6568	30	2	et	et	PROPN
ejpam-6568	30	3	al	al	PROPN
ejpam-6568	30	4	.	.	PUNCT
ejpam-6568	31	1	[	[	X
ejpam-6568	31	2	11	11	NUM
ejpam-6568	31	3	]	]	PUNCT
ejpam-6568	31	4	introduced	introduce	VERB
ejpam-6568	31	5	and	and	CCONJ
ejpam-6568	31	6	studied	study	VERB
ejpam-6568	31	7	the	the	DET
ejpam-6568	31	8	concept	concept	NOUN
ejpam-6568	31	9	of	of	ADP
ejpam-6568	31	10	weakly	weakly	ADJ
ejpam-6568	31	11	λ	λ	ADJ
ejpam-6568	31	12	-	-	ADJ
ejpam-6568	31	13	continuous	continuous	ADJ
ejpam-6568	31	14	functions	function	NOUN
ejpam-6568	31	15	.	.	PUNCT
ejpam-6568	32	1	in	in	ADP
ejpam-6568	32	2	1992	1992	NUM
ejpam-6568	32	3	,	,	PUNCT
ejpam-6568	32	4	abd	abd	PROPN
ejpam-6568	32	5	el	el	PROPN
ejpam-6568	32	6	-	-	PROPN
ejpam-6568	32	7	monsef	monsef	PROPN
ejpam-6568	32	8	et	et	PROPN
ejpam-6568	32	9	al	al	PROPN
ejpam-6568	32	10	.	.	PUNCT
ejpam-6568	33	1	[	[	X
ejpam-6568	33	2	12	12	NUM
ejpam-6568	33	3	]	]	PUNCT
ejpam-6568	33	4	introduced	introduce	VERB
ejpam-6568	33	5	and	and	CCONJ
ejpam-6568	33	6	studied	study	VERB
ejpam-6568	33	7	the	the	DET
ejpam-6568	33	8	notions	notion	NOUN
ejpam-6568	33	9	of	of	ADP
ejpam-6568	33	10	i	i	PRON
ejpam-6568	33	11	-closed	-close	VERB
ejpam-6568	33	12	sets	set	NOUN
ejpam-6568	33	13	and	and	CCONJ
ejpam-6568	33	14	i	i	PRON
ejpam-6568	33	15	-continuous	-continuous	ADJ
ejpam-6568	33	16	functions	function	NOUN
ejpam-6568	33	17	.	.	PUNCT
ejpam-6568	34	1	semi	semi	ADJ
ejpam-6568	34	2	-	-	ADJ
ejpam-6568	34	3	i	i	ADJ
ejpam-6568	34	4	-open	-open	NOUN
ejpam-6568	34	5	sets	set	NOUN
ejpam-6568	34	6	,	,	PUNCT
ejpam-6568	34	7	pre	pre	ADJ
ejpam-6568	34	8	-	-	ADJ
ejpam-6568	34	9	i	i	ADJ
ejpam-6568	34	10	-open	-open	NOUN
ejpam-6568	34	11	sets	set	NOUN
ejpam-6568	34	12	,	,	PUNCT
ejpam-6568	34	13	α	α	X
ejpam-6568	34	14	-	-	PUNCT
ejpam-6568	34	15	i	i	PRON
ejpam-6568	34	16	open	open	ADJ
ejpam-6568	34	17	sets	set	NOUN
ejpam-6568	34	18	,	,	PUNCT
ejpam-6568	34	19	β	β	X
ejpam-6568	34	20	-	-	ADJ
ejpam-6568	34	21	i	i	PRON
ejpam-6568	34	22	-open	-open	NOUN
ejpam-6568	34	23	sets	set	NOUN
ejpam-6568	34	24	and	and	CCONJ
ejpam-6568	34	25	δ	δ	PROPN
ejpam-6568	34	26	-	-	PUNCT
ejpam-6568	34	27	i	i	PRON
ejpam-6568	34	28	-open	-open	NOUN
ejpam-6568	34	29	sets	set	NOUN
ejpam-6568	34	30	play	play	VERB
ejpam-6568	34	31	an	an	DET
ejpam-6568	34	32	important	important	ADJ
ejpam-6568	34	33	role	role	NOUN
ejpam-6568	34	34	in	in	ADP
ejpam-6568	34	35	the	the	DET
ejpam-6568	34	36	research	research	NOUN
ejpam-6568	34	37	of	of	ADP
ejpam-6568	34	38	generalizations	generalization	NOUN
ejpam-6568	34	39	of	of	ADP
ejpam-6568	34	40	continuity	continuity	NOUN
ejpam-6568	34	41	.	.	PUNCT
ejpam-6568	35	1	using	use	VERB
ejpam-6568	35	2	these	these	DET
ejpam-6568	35	3	notions	notion	NOUN
ejpam-6568	35	4	many	many	ADJ
ejpam-6568	35	5	authors	author	NOUN
ejpam-6568	35	6	introduced	introduce	VERB
ejpam-6568	35	7	and	and	CCONJ
ejpam-6568	35	8	studied	study	VERB
ejpam-6568	35	9	various	various	ADJ
ejpam-6568	35	10	types	type	NOUN
ejpam-6568	35	11	of	of	ADP
ejpam-6568	35	12	generalizations	generalization	NOUN
ejpam-6568	35	13	of	of	ADP
ejpam-6568	35	14	continuity	continuity	NOUN
ejpam-6568	35	15	for	for	ADP
ejpam-6568	35	16	functions	function	NOUN
ejpam-6568	35	17	and	and	CCONJ
ejpam-6568	35	18	multifunctions	multifunction	NOUN
ejpam-6568	35	19	.	.	PUNCT
ejpam-6568	36	1	hatir	hatir	PROPN
ejpam-6568	36	2	and	and	CCONJ
ejpam-6568	36	3	noiri	noiri	ADV
ejpam-6568	37	1	[	[	X
ejpam-6568	37	2	13	13	NUM
ejpam-6568	37	3	]	]	PUNCT
ejpam-6568	37	4	introduced	introduce	VERB
ejpam-6568	37	5	and	and	CCONJ
ejpam-6568	37	6	investigated	investigate	VERB
ejpam-6568	37	7	the	the	DET
ejpam-6568	37	8	notions	notion	NOUN
ejpam-6568	37	9	of	of	ADP
ejpam-6568	37	10	weakly	weakly	ADJ
ejpam-6568	37	11	pre	pre	ADJ
ejpam-6568	37	12	-	-	ADJ
ejpam-6568	37	13	i	i	PRON
ejpam-6568	37	14	-open	-open	NOUN
ejpam-6568	37	15	sets	set	NOUN
ejpam-6568	37	16	and	and	CCONJ
ejpam-6568	37	17	weakly	weakly	ADJ
ejpam-6568	37	18	pre	pre	ADJ
ejpam-6568	37	19	-	-	ADJ
ejpam-6568	37	20	i	i	ADJ
ejpam-6568	37	21	-continuous	-continuous	ADJ
ejpam-6568	37	22	functions	function	NOUN
ejpam-6568	37	23	.	.	PUNCT
ejpam-6568	38	1	moreover	moreover	ADV
ejpam-6568	38	2	,	,	PUNCT
ejpam-6568	38	3	hatir	hatir	PROPN
ejpam-6568	38	4	and	and	CCONJ
ejpam-6568	38	5	noiri	noiri	ADV
ejpam-6568	39	1	[	[	X
ejpam-6568	39	2	14	14	NUM
ejpam-6568	39	3	]	]	PUNCT
ejpam-6568	39	4	investigated	investigate	VERB
ejpam-6568	39	5	further	further	ADJ
ejpam-6568	39	6	properties	property	NOUN
ejpam-6568	39	7	of	of	ADP
ejpam-6568	39	8	semi	semi	ADJ
ejpam-6568	39	9	-	-	ADJ
ejpam-6568	39	10	i	i	PRON
ejpam-6568	39	11	-open	-open	NOUN
ejpam-6568	39	12	sets	set	NOUN
ejpam-6568	39	13	and	and	CCONJ
ejpam-6568	39	14	semi	semi	ADJ
ejpam-6568	39	15	-	-	ADJ
ejpam-6568	39	16	i	i	ADJ
ejpam-6568	39	17	-continuous	-continuous	ADJ
ejpam-6568	39	18	functions	function	NOUN
ejpam-6568	39	19	[	[	X
ejpam-6568	39	20	15	15	NUM
ejpam-6568	39	21	]	]	PUNCT
ejpam-6568	39	22	.	.	PUNCT
ejpam-6568	40	1	on	on	ADP
ejpam-6568	40	2	the	the	DET
ejpam-6568	40	3	other	other	ADJ
ejpam-6568	40	4	hand	hand	NOUN
ejpam-6568	40	5	,	,	PUNCT
ejpam-6568	40	6	the	the	DET
ejpam-6568	40	7	present	present	ADJ
ejpam-6568	40	8	author	author	NOUN
ejpam-6568	40	9	introduced	introduce	VERB
ejpam-6568	40	10	and	and	CCONJ
ejpam-6568	40	11	studied	study	VERB
ejpam-6568	40	12	the	the	DET
ejpam-6568	40	13	concepts	concept	NOUN
ejpam-6568	40	14	of	of	ADP
ejpam-6568	40	15	⋆-continuous	⋆-continuous	ADJ
ejpam-6568	40	16	functions	function	NOUN
ejpam-6568	41	1	[	[	X
ejpam-6568	41	2	16	16	NUM
ejpam-6568	41	3	]	]	PUNCT
ejpam-6568	41	4	,	,	PUNCT
ejpam-6568	41	5	θ	θ	PROPN
ejpam-6568	41	6	-	-	ADJ
ejpam-6568	41	7	i	i	NOUN
ejpam-6568	41	8	-continuous	-continuous	ADJ
ejpam-6568	41	9	functions	function	NOUN
ejpam-6568	41	10	[	[	X
ejpam-6568	41	11	17	17	NUM
ejpam-6568	41	12	]	]	PUNCT
ejpam-6568	41	13	,	,	PUNCT
ejpam-6568	41	14	weakly	weakly	ADJ
ejpam-6568	41	15	⋆-continuous	⋆-continuous	ADJ
ejpam-6568	41	16	functions	function	NOUN
ejpam-6568	41	17	[	[	X
ejpam-6568	41	18	18	18	NUM
ejpam-6568	41	19	]	]	X
ejpam-6568	41	20	,	,	PUNCT
ejpam-6568	41	21	θ(⋆)-continuous	θ(⋆)-continuous	ADJ
ejpam-6568	41	22	functions	function	NOUN
ejpam-6568	41	23	[	[	X
ejpam-6568	41	24	18	18	NUM
ejpam-6568	41	25	]	]	PUNCT
ejpam-6568	41	26	,	,	PUNCT
ejpam-6568	41	27	almost	almost	ADV
ejpam-6568	41	28	⋆-precontinuous	⋆-precontinuous	ADJ
ejpam-6568	41	29	functions	function	NOUN
ejpam-6568	41	30	[	[	X
ejpam-6568	41	31	19	19	NUM
ejpam-6568	41	32	]	]	PUNCT
ejpam-6568	41	33	,	,	PUNCT
ejpam-6568	41	34	weakly	weakly	ADJ
ejpam-6568	41	35	⋆-precontinuous	⋆-precontinuous	ADJ
ejpam-6568	41	36	functions	function	NOUN
ejpam-6568	41	37	[	[	X
ejpam-6568	41	38	19	19	NUM
ejpam-6568	41	39	]	]	PUNCT
ejpam-6568	41	40	,	,	PUNCT
ejpam-6568	41	41	pı	pı	ADJ
ejpam-6568	41	42	-	-	ADJ
ejpam-6568	41	43	continuous	continuous	ADJ
ejpam-6568	41	44	functions	function	NOUN
ejpam-6568	41	45	[	[	X
ejpam-6568	41	46	20	20	NUM
ejpam-6568	41	47	]	]	PUNCT
ejpam-6568	41	48	and	and	CCONJ
ejpam-6568	41	49	weakly	weakly	ADJ
ejpam-6568	41	50	pı	pı	ADJ
ejpam-6568	41	51	-	-	ADJ
ejpam-6568	41	52	continuous	continuous	ADJ
ejpam-6568	41	53	functions	function	NOUN
ejpam-6568	41	54	[	[	X
ejpam-6568	41	55	20	20	NUM
ejpam-6568	41	56	]	]	PUNCT
ejpam-6568	41	57	.	.	PUNCT
ejpam-6568	42	1	recently	recently	ADV
ejpam-6568	42	2	,	,	PUNCT
ejpam-6568	42	3	boonpok	boonpok	PROPN
ejpam-6568	42	4	and	and	CCONJ
ejpam-6568	42	5	srisarakham	srisarakham	PROPN
ejpam-6568	42	6	[	[	X
ejpam-6568	42	7	21	21	NUM
ejpam-6568	42	8	]	]	PUNCT
ejpam-6568	42	9	introduced	introduce	VERB
ejpam-6568	42	10	and	and	CCONJ
ejpam-6568	42	11	investigated	investigate	VERB
ejpam-6568	42	12	the	the	DET
ejpam-6568	42	13	notion	notion	NOUN
ejpam-6568	42	14	of	of	ADP
ejpam-6568	42	15	(	(	PUNCT
ejpam-6568	42	16	τ1	τ1	PROPN
ejpam-6568	42	17	,	,	PUNCT
ejpam-6568	42	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6568	42	19	functions	function	NOUN
ejpam-6568	42	20	.	.	PUNCT
ejpam-6568	43	1	furthermore	furthermore	ADV
ejpam-6568	43	2	,	,	PUNCT
ejpam-6568	43	3	some	some	DET
ejpam-6568	43	4	characterizations	characterization	NOUN
ejpam-6568	43	5	of	of	ADP
ejpam-6568	43	6	almost	almost	ADV
ejpam-6568	43	7	(	(	PUNCT
ejpam-6568	43	8	τ1	τ1	NOUN
ejpam-6568	43	9	,	,	PUNCT
ejpam-6568	43	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6568	43	11	functions	function	NOUN
ejpam-6568	43	12	and	and	CCONJ
ejpam-6568	43	13	weakly	weakly	ADJ
ejpam-6568	43	14	(	(	PUNCT
ejpam-6568	43	15	τ1	τ1	NOUN
ejpam-6568	43	16	,	,	PUNCT
ejpam-6568	43	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6568	43	18	functions	function	NOUN
ejpam-6568	43	19	were	be	AUX
ejpam-6568	43	20	established	establish	VERB
ejpam-6568	43	21	in	in	ADP
ejpam-6568	43	22	[	[	X
ejpam-6568	43	23	22	22	NUM
ejpam-6568	43	24	]	]	PUNCT
ejpam-6568	43	25	and	and	CCONJ
ejpam-6568	43	26	[	[	X
ejpam-6568	43	27	23	23	NUM
ejpam-6568	43	28	]	]	X
ejpam-6568	43	29	,	,	PUNCT
ejpam-6568	43	30	respectively	respectively	ADV
ejpam-6568	43	31	.	.	PUNCT
ejpam-6568	44	1	in	in	ADP
ejpam-6568	44	2	this	this	DET
ejpam-6568	44	3	paper	paper	NOUN
ejpam-6568	44	4	,	,	PUNCT
ejpam-6568	44	5	we	we	PRON
ejpam-6568	44	6	introduce	introduce	VERB
ejpam-6568	44	7	the	the	DET
ejpam-6568	44	8	concepts	concept	NOUN
ejpam-6568	44	9	of	of	ADP
ejpam-6568	44	10	almost	almost	ADV
ejpam-6568	44	11	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	44	12	,	,	PUNCT
ejpam-6568	44	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	44	14	functions	function	NOUN
ejpam-6568	44	15	and	and	CCONJ
ejpam-6568	44	16	weakly	weakly	ADJ
ejpam-6568	44	17	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	44	18	,	,	PUNCT
ejpam-6568	44	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	44	20	functions	function	NOUN
ejpam-6568	44	21	.	.	PUNCT
ejpam-6568	45	1	we	we	PRON
ejpam-6568	45	2	also	also	ADV
ejpam-6568	45	3	investigate	investigate	VERB
ejpam-6568	45	4	several	several	ADJ
ejpam-6568	45	5	characterizations	characterization	NOUN
ejpam-6568	45	6	of	of	ADP
ejpam-6568	45	7	almost	almost	ADV
ejpam-6568	45	8	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	45	9	,	,	PUNCT
ejpam-6568	45	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	45	11	functions	function	NOUN
ejpam-6568	45	12	and	and	CCONJ
ejpam-6568	45	13	weakly	weakly	ADJ
ejpam-6568	45	14	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	45	15	,	,	PUNCT
ejpam-6568	45	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	45	17	functions	function	NOUN
ejpam-6568	45	18	.	.	PUNCT
ejpam-6568	46	1	2	2	X
ejpam-6568	46	2	.	.	X
ejpam-6568	46	3	preliminaries	preliminary	NOUN
ejpam-6568	46	4	throughout	throughout	ADP
ejpam-6568	46	5	the	the	DET
ejpam-6568	46	6	present	present	ADJ
ejpam-6568	46	7	paper	paper	NOUN
ejpam-6568	46	8	,	,	PUNCT
ejpam-6568	46	9	spaces	space	NOUN
ejpam-6568	46	10	(	(	PUNCT
ejpam-6568	46	11	x	x	NOUN
ejpam-6568	46	12	,	,	PUNCT
ejpam-6568	46	13	τ1	τ1	NOUN
ejpam-6568	46	14	,	,	PUNCT
ejpam-6568	46	15	τ2	τ2	NOUN
ejpam-6568	46	16	)	)	PUNCT
ejpam-6568	46	17	and	and	CCONJ
ejpam-6568	46	18	(	(	PUNCT
ejpam-6568	46	19	y	y	PROPN
ejpam-6568	46	20	,	,	PUNCT
ejpam-6568	46	21	σ1	σ1	PROPN
ejpam-6568	46	22	,	,	PUNCT
ejpam-6568	46	23	σ2	σ2	NOUN
ejpam-6568	46	24	)	)	PUNCT
ejpam-6568	46	25	(	(	PUNCT
ejpam-6568	46	26	or	or	CCONJ
ejpam-6568	46	27	simply	simply	ADV
ejpam-6568	46	28	x	x	X
ejpam-6568	46	29	and	and	CCONJ
ejpam-6568	46	30	y	y	PROPN
ejpam-6568	46	31	)	)	PUNCT
ejpam-6568	46	32	always	always	ADV
ejpam-6568	46	33	mean	mean	VERB
ejpam-6568	46	34	bitopological	bitopological	ADJ
ejpam-6568	46	35	spaces	space	NOUN
ejpam-6568	46	36	on	on	ADP
ejpam-6568	46	37	which	which	PRON
ejpam-6568	46	38	no	no	DET
ejpam-6568	46	39	separation	separation	NOUN
ejpam-6568	46	40	axioms	axiom	NOUN
ejpam-6568	46	41	are	be	AUX
ejpam-6568	46	42	assumed	assume	VERB
ejpam-6568	46	43	unless	unless	SCONJ
ejpam-6568	46	44	explicitly	explicitly	ADV
ejpam-6568	46	45	stated	state	VERB
ejpam-6568	46	46	.	.	PUNCT
ejpam-6568	47	1	let	let	VERB
ejpam-6568	47	2	a	a	DET
ejpam-6568	47	3	be	be	AUX
ejpam-6568	47	4	a	a	DET
ejpam-6568	47	5	subset	subset	NOUN
ejpam-6568	47	6	of	of	ADP
ejpam-6568	47	7	a	a	DET
ejpam-6568	47	8	bitopological	bitopological	ADJ
ejpam-6568	47	9	space	space	NOUN
ejpam-6568	47	10	(	(	PUNCT
ejpam-6568	47	11	x	x	NOUN
ejpam-6568	47	12	,	,	PUNCT
ejpam-6568	47	13	τ1	τ1	NOUN
ejpam-6568	47	14	,	,	PUNCT
ejpam-6568	47	15	τ2	τ2	NOUN
ejpam-6568	47	16	)	)	PUNCT
ejpam-6568	47	17	.	.	PUNCT
ejpam-6568	48	1	the	the	DET
ejpam-6568	48	2	closure	closure	NOUN
ejpam-6568	48	3	of	of	ADP
ejpam-6568	48	4	a	a	PRON
ejpam-6568	48	5	and	and	CCONJ
ejpam-6568	48	6	the	the	DET
ejpam-6568	48	7	interior	interior	NOUN
ejpam-6568	48	8	of	of	ADP
ejpam-6568	48	9	a	a	PRON
ejpam-6568	48	10	with	with	ADP
ejpam-6568	48	11	respect	respect	NOUN
ejpam-6568	48	12	to	to	ADP
ejpam-6568	48	13	τi	τi	PROPN
ejpam-6568	48	14	are	be	AUX
ejpam-6568	48	15	denoted	denote	VERB
ejpam-6568	48	16	by	by	ADP
ejpam-6568	48	17	τi	τi	NOUN
ejpam-6568	48	18	-	-	PUNCT
ejpam-6568	48	19	cl(a	cl(a	NUM
ejpam-6568	48	20	)	)	PUNCT
ejpam-6568	48	21	and	and	CCONJ
ejpam-6568	48	22	τi	τi	NOUN
ejpam-6568	48	23	-	-	PUNCT
ejpam-6568	48	24	int(a	int(a	NOUN
ejpam-6568	48	25	)	)	PUNCT
ejpam-6568	48	26	,	,	PUNCT
ejpam-6568	48	27	respectively	respectively	ADV
ejpam-6568	48	28	,	,	PUNCT
ejpam-6568	48	29	for	for	ADP
ejpam-6568	48	30	i	i	PROPN
ejpam-6568	48	31	=	=	SYM
ejpam-6568	48	32	1	1	NUM
ejpam-6568	48	33	,	,	PUNCT
ejpam-6568	48	34	2	2	NUM
ejpam-6568	48	35	.	.	X
ejpam-6568	48	36	a	a	DET
ejpam-6568	48	37	subset	subset	NOUN
ejpam-6568	48	38	a	a	PRON
ejpam-6568	48	39	of	of	ADP
ejpam-6568	48	40	a	a	DET
ejpam-6568	48	41	bitopological	bitopological	ADJ
ejpam-6568	48	42	space	space	NOUN
ejpam-6568	48	43	(	(	PUNCT
ejpam-6568	48	44	x	x	NOUN
ejpam-6568	48	45	,	,	PUNCT
ejpam-6568	48	46	τ1	τ1	NOUN
ejpam-6568	48	47	,	,	PUNCT
ejpam-6568	48	48	τ2	τ2	NOUN
ejpam-6568	48	49	)	)	PUNCT
ejpam-6568	48	50	is	be	AUX
ejpam-6568	48	51	called	call	VERB
ejpam-6568	48	52	τ1τ2	τ1τ2	VERB
ejpam-6568	48	53	-	-	ADJ
ejpam-6568	48	54	closed	closed	ADJ
ejpam-6568	48	55	[	[	X
ejpam-6568	48	56	24	24	NUM
ejpam-6568	48	57	]	]	X
ejpam-6568	48	58	if	if	SCONJ
ejpam-6568	48	59	a	a	DET
ejpam-6568	48	60	=	=	NOUN
ejpam-6568	48	61	τ1	τ1	NOUN
ejpam-6568	48	62	-	-	PUNCT
ejpam-6568	48	63	cl(τ2	cl(τ2	NOUN
ejpam-6568	48	64	-	-	PUNCT
ejpam-6568	48	65	cl(a	cl(a	NUM
ejpam-6568	48	66	)	)	PUNCT
ejpam-6568	48	67	)	)	PUNCT
ejpam-6568	48	68	.	.	PUNCT
ejpam-6568	49	1	the	the	DET
ejpam-6568	49	2	complement	complement	NOUN
ejpam-6568	49	3	of	of	ADP
ejpam-6568	49	4	a	a	DET
ejpam-6568	49	5	τ1τ2	τ1τ2	ADJ
ejpam-6568	49	6	-	-	ADJ
ejpam-6568	49	7	closed	closed	ADJ
ejpam-6568	49	8	set	set	NOUN
ejpam-6568	49	9	is	be	AUX
ejpam-6568	49	10	called	call	VERB
ejpam-6568	49	11	τ1τ2	τ1τ2	NOUN
ejpam-6568	49	12	-	-	ADJ
ejpam-6568	49	13	open	open	ADJ
ejpam-6568	49	14	.	.	PUNCT
ejpam-6568	50	1	the	the	DET
ejpam-6568	50	2	intersection	intersection	NOUN
ejpam-6568	50	3	of	of	ADP
ejpam-6568	50	4	all	all	DET
ejpam-6568	50	5	τ1τ2	τ1τ2	ADJ
ejpam-6568	50	6	-	-	ADJ
ejpam-6568	50	7	closed	closed	ADJ
ejpam-6568	50	8	sets	set	NOUN
ejpam-6568	50	9	of	of	ADP
ejpam-6568	50	10	x	x	PUNCT
ejpam-6568	50	11	containing	contain	VERB
ejpam-6568	50	12	a	a	PRON
ejpam-6568	50	13	is	be	AUX
ejpam-6568	50	14	called	call	VERB
ejpam-6568	50	15	the	the	DET
ejpam-6568	50	16	τ1τ2	τ1τ2	NOUN
ejpam-6568	50	17	-	-	NOUN
ejpam-6568	50	18	closure	closure	NOUN
ejpam-6568	50	19	[	[	X
ejpam-6568	50	20	24	24	NUM
ejpam-6568	50	21	]	]	PUNCT
ejpam-6568	50	22	of	of	ADP
ejpam-6568	50	23	a	a	PRON
ejpam-6568	50	24	and	and	CCONJ
ejpam-6568	50	25	is	be	AUX
ejpam-6568	50	26	denoted	denote	VERB
ejpam-6568	50	27	by	by	ADP
ejpam-6568	50	28	τ1τ2	τ1τ2	NOUN
ejpam-6568	50	29	-	-	NUM
ejpam-6568	50	30	cl(a	cl(a	NUM
ejpam-6568	50	31	)	)	PUNCT
ejpam-6568	50	32	.	.	PUNCT
ejpam-6568	51	1	the	the	DET
ejpam-6568	51	2	union	union	NOUN
ejpam-6568	51	3	of	of	ADP
ejpam-6568	51	4	all	all	DET
ejpam-6568	51	5	τ1τ2	τ1τ2	ADJ
ejpam-6568	51	6	-	-	ADJ
ejpam-6568	51	7	open	open	ADJ
ejpam-6568	51	8	sets	set	NOUN
ejpam-6568	51	9	of	of	ADP
ejpam-6568	51	10	x	x	PUNCT
ejpam-6568	51	11	contained	contain	VERB
ejpam-6568	51	12	in	in	ADP
ejpam-6568	51	13	a	a	PRON
ejpam-6568	51	14	is	be	AUX
ejpam-6568	51	15	called	call	VERB
ejpam-6568	51	16	the	the	DET
ejpam-6568	51	17	τ1τ2	τ1τ2	NOUN
ejpam-6568	51	18	-	-	ADJ
ejpam-6568	51	19	interior	interior	ADJ
ejpam-6568	51	20	[	[	X
ejpam-6568	51	21	24	24	NUM
ejpam-6568	51	22	]	]	PUNCT
ejpam-6568	51	23	of	of	ADP
ejpam-6568	51	24	a	a	PRON
ejpam-6568	51	25	and	and	CCONJ
ejpam-6568	51	26	is	be	AUX
ejpam-6568	51	27	denoted	denote	VERB
ejpam-6568	51	28	by	by	ADP
ejpam-6568	51	29	τ1τ2	τ1τ2	NOUN
ejpam-6568	51	30	-	-	ADJ
ejpam-6568	51	31	int(a	int(a	NOUN
ejpam-6568	51	32	)	)	PUNCT
ejpam-6568	51	33	.	.	PUNCT
ejpam-6568	52	1	lemma	lemma	PROPN
ejpam-6568	52	2	1	1	NUM
ejpam-6568	52	3	.	.	PUNCT
ejpam-6568	53	1	[	[	X
ejpam-6568	53	2	24	24	NUM
ejpam-6568	53	3	]	]	PUNCT
ejpam-6568	53	4	let	let	VERB
ejpam-6568	53	5	a	a	PRON
ejpam-6568	53	6	and	and	CCONJ
ejpam-6568	53	7	b	b	NOUN
ejpam-6568	53	8	be	be	AUX
ejpam-6568	53	9	subsets	subset	NOUN
ejpam-6568	53	10	of	of	ADP
ejpam-6568	53	11	a	a	DET
ejpam-6568	53	12	bitopological	bitopological	ADJ
ejpam-6568	53	13	space	space	NOUN
ejpam-6568	53	14	(	(	PUNCT
ejpam-6568	53	15	x	x	NOUN
ejpam-6568	53	16	,	,	PUNCT
ejpam-6568	53	17	τ1	τ1	NOUN
ejpam-6568	53	18	,	,	PUNCT
ejpam-6568	53	19	τ2	τ2	NOUN
ejpam-6568	53	20	)	)	PUNCT
ejpam-6568	53	21	.	.	PUNCT
ejpam-6568	54	1	for	for	ADP
ejpam-6568	54	2	the	the	DET
ejpam-6568	54	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6568	54	4	,	,	PUNCT
ejpam-6568	54	5	the	the	DET
ejpam-6568	54	6	following	follow	VERB
ejpam-6568	54	7	properties	property	NOUN
ejpam-6568	54	8	hold	hold	VERB
ejpam-6568	54	9	:	:	PUNCT
ejpam-6568	54	10	(	(	PUNCT
ejpam-6568	54	11	1	1	X
ejpam-6568	54	12	)	)	PUNCT
ejpam-6568	54	13	a	a	DET
ejpam-6568	54	14	⊆	⊆	NUM
ejpam-6568	54	15	τ1τ2	τ1τ2	NOUN
ejpam-6568	54	16	-	-	NUM
ejpam-6568	54	17	cl(a	cl(a	NUM
ejpam-6568	54	18	)	)	PUNCT
ejpam-6568	54	19	and	and	CCONJ
ejpam-6568	54	20	τ1τ2	τ1τ2	NOUN
ejpam-6568	54	21	-	-	ADJ
ejpam-6568	54	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6568	54	23	-	-	PUNCT
ejpam-6568	54	24	cl(a	cl(a	NUM
ejpam-6568	54	25	)	)	PUNCT
ejpam-6568	54	26	)	)	PUNCT
ejpam-6568	55	1	=	=	PUNCT
ejpam-6568	55	2	τ1τ2	τ1τ2	NOUN
ejpam-6568	55	3	-	-	NUM
ejpam-6568	55	4	cl(a	cl(a	NUM
ejpam-6568	55	5	)	)	PUNCT
ejpam-6568	55	6	.	.	PUNCT
ejpam-6568	56	1	(	(	PUNCT
ejpam-6568	56	2	2	2	X
ejpam-6568	56	3	)	)	PUNCT
ejpam-6568	56	4	if	if	SCONJ
ejpam-6568	56	5	a	a	DET
ejpam-6568	56	6	⊆	⊆	NUM
ejpam-6568	56	7	b	b	NOUN
ejpam-6568	56	8	,	,	PUNCT
ejpam-6568	56	9	then	then	ADV
ejpam-6568	56	10	τ1τ2	τ1τ2	NOUN
ejpam-6568	56	11	-	-	NUM
ejpam-6568	56	12	cl(a	cl(a	NUM
ejpam-6568	56	13	)	)	PUNCT
ejpam-6568	56	14	⊆	⊆	NUM
ejpam-6568	56	15	τ1τ2	τ1τ2	NOUN
ejpam-6568	56	16	-	-	NOUN
ejpam-6568	56	17	cl(b	cl(b	NOUN
ejpam-6568	56	18	)	)	PUNCT
ejpam-6568	56	19	.	.	PUNCT
ejpam-6568	57	1	(	(	PUNCT
ejpam-6568	57	2	3	3	X
ejpam-6568	57	3	)	)	PUNCT
ejpam-6568	57	4	τ1τ2	τ1τ2	NOUN
ejpam-6568	57	5	-	-	NUM
ejpam-6568	57	6	cl(a	cl(a	NUM
ejpam-6568	57	7	)	)	PUNCT
ejpam-6568	57	8	is	be	AUX
ejpam-6568	57	9	τ1τ2	τ1τ2	NOUN
ejpam-6568	57	10	-	-	ADJ
ejpam-6568	57	11	closed	closed	ADJ
ejpam-6568	57	12	.	.	PUNCT
ejpam-6568	58	1	(	(	PUNCT
ejpam-6568	58	2	4	4	X
ejpam-6568	58	3	)	)	PUNCT
ejpam-6568	58	4	a	a	PRON
ejpam-6568	58	5	is	be	AUX
ejpam-6568	58	6	τ1τ2	τ1τ2	NOUN
ejpam-6568	58	7	-	-	ADJ
ejpam-6568	58	8	closed	closed	ADJ
ejpam-6568	58	9	if	if	SCONJ
ejpam-6568	58	10	and	and	CCONJ
ejpam-6568	58	11	only	only	ADV
ejpam-6568	58	12	if	if	SCONJ
ejpam-6568	58	13	a	a	DET
ejpam-6568	58	14	=	=	PUNCT
ejpam-6568	58	15	τ1τ2	τ1τ2	NOUN
ejpam-6568	58	16	-	-	NUM
ejpam-6568	58	17	cl(a	cl(a	NUM
ejpam-6568	58	18	)	)	PUNCT
ejpam-6568	58	19	.	.	PUNCT
ejpam-6568	59	1	n.	n.	PROPN
ejpam-6568	59	2	viriyapong	viriyapong	PROPN
ejpam-6568	59	3	,	,	PUNCT
ejpam-6568	59	4	a.	a.	PROPN
ejpam-6568	59	5	sama	sama	PROPN
ejpam-6568	59	6	-	-	PUNCT
ejpam-6568	59	7	ae	ae	PROPN
ejpam-6568	59	8	,	,	PUNCT
ejpam-6568	59	9	c.	c.	PROPN
ejpam-6568	59	10	boonpok	boonpok	PROPN
ejpam-6568	59	11	/	/	SYM
ejpam-6568	59	12	eur	eur	PROPN
ejpam-6568	59	13	.	.	PUNCT
ejpam-6568	60	1	j.	j.	PROPN
ejpam-6568	60	2	pure	pure	PROPN
ejpam-6568	60	3	appl	appl	PROPN
ejpam-6568	60	4	.	.	PROPN
ejpam-6568	60	5	math	math	PROPN
ejpam-6568	60	6	,	,	PUNCT
ejpam-6568	60	7	18	18	NUM
ejpam-6568	60	8	(	(	PUNCT
ejpam-6568	60	9	3	3	NUM
ejpam-6568	60	10	)	)	PUNCT
ejpam-6568	60	11	(	(	PUNCT
ejpam-6568	60	12	2025	2025	NUM
ejpam-6568	60	13	)	)	PUNCT
ejpam-6568	60	14	,	,	PUNCT
ejpam-6568	60	15	6568	6568	NUM
ejpam-6568	60	16	3	3	NUM
ejpam-6568	60	17	of	of	ADP
ejpam-6568	60	18	18	18	NUM
ejpam-6568	60	19	(	(	PUNCT
ejpam-6568	60	20	5	5	NUM
ejpam-6568	60	21	)	)	PUNCT
ejpam-6568	60	22	τ1τ2	τ1τ2	NOUN
ejpam-6568	60	23	-	-	NOUN
ejpam-6568	60	24	cl(x	cl(x	X
ejpam-6568	60	25	−a	−a	NOUN
ejpam-6568	60	26	)	)	PUNCT
ejpam-6568	60	27	=	=	PUNCT
ejpam-6568	61	1	x	x	X
ejpam-6568	61	2	−	−	ADP
ejpam-6568	61	3	τ1τ2	τ1τ2	NOUN
ejpam-6568	61	4	-	-	PUNCT
ejpam-6568	61	5	int(a	int(a	NOUN
ejpam-6568	61	6	)	)	PUNCT
ejpam-6568	61	7	.	.	PUNCT
ejpam-6568	62	1	a	a	DET
ejpam-6568	62	2	subset	subset	NOUN
ejpam-6568	62	3	a	a	PRON
ejpam-6568	62	4	of	of	ADP
ejpam-6568	62	5	a	a	DET
ejpam-6568	62	6	bitopological	bitopological	ADJ
ejpam-6568	62	7	space	space	NOUN
ejpam-6568	62	8	(	(	PUNCT
ejpam-6568	62	9	x	x	NOUN
ejpam-6568	62	10	,	,	PUNCT
ejpam-6568	62	11	τ1	τ1	NOUN
ejpam-6568	62	12	,	,	PUNCT
ejpam-6568	62	13	τ2	τ2	NOUN
ejpam-6568	62	14	)	)	PUNCT
ejpam-6568	62	15	is	be	AUX
ejpam-6568	62	16	said	say	VERB
ejpam-6568	62	17	to	to	PART
ejpam-6568	62	18	be	be	AUX
ejpam-6568	62	19	(	(	PUNCT
ejpam-6568	62	20	τ1	τ1	NOUN
ejpam-6568	62	21	,	,	PUNCT
ejpam-6568	62	22	τ2)r	τ2)r	NOUN
ejpam-6568	62	23	-	-	PUNCT
ejpam-6568	62	24	open	open	NOUN
ejpam-6568	63	1	[	[	X
ejpam-6568	63	2	25	25	NUM
ejpam-6568	63	3	]	]	PUNCT
ejpam-6568	63	4	(	(	PUNCT
ejpam-6568	63	5	resp	resp	NOUN
ejpam-6568	63	6	.	.	PUNCT
ejpam-6568	64	1	(	(	PUNCT
ejpam-6568	64	2	τ1	τ1	NOUN
ejpam-6568	64	3	,	,	PUNCT
ejpam-6568	64	4	τ2)s	τ2)s	NOUN
ejpam-6568	64	5	-	-	PUNCT
ejpam-6568	64	6	open	open	ADJ
ejpam-6568	64	7	[	[	X
ejpam-6568	64	8	26	26	NUM
ejpam-6568	64	9	]	]	PUNCT
ejpam-6568	64	10	,	,	PUNCT
ejpam-6568	64	11	(	(	PUNCT
ejpam-6568	64	12	τ1	τ1	NOUN
ejpam-6568	64	13	,	,	PUNCT
ejpam-6568	64	14	τ2)p	τ2)p	NOUN
ejpam-6568	64	15	-	-	ADJ
ejpam-6568	64	16	open	open	ADJ
ejpam-6568	65	1	[	[	X
ejpam-6568	65	2	26	26	NUM
ejpam-6568	65	3	]	]	PUNCT
ejpam-6568	65	4	,	,	PUNCT
ejpam-6568	65	5	(	(	PUNCT
ejpam-6568	65	6	τ1	τ1	NOUN
ejpam-6568	65	7	,	,	PUNCT
ejpam-6568	65	8	τ2)β	τ2)β	ADJ
ejpam-6568	65	9	-	-	PUNCT
ejpam-6568	65	10	open	open	NOUN
ejpam-6568	66	1	[	[	X
ejpam-6568	66	2	26	26	NUM
ejpam-6568	66	3	]	]	SYM
ejpam-6568	66	4	)	)	PUNCT
ejpam-6568	66	5	if	if	SCONJ
ejpam-6568	66	6	a	a	DET
ejpam-6568	66	7	=	=	PUNCT
ejpam-6568	66	8	τ1τ2	τ1τ2	NOUN
ejpam-6568	66	9	-	-	NOUN
ejpam-6568	66	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6568	66	11	-	-	PUNCT
ejpam-6568	66	12	cl(a	cl(a	NUM
ejpam-6568	66	13	)	)	PUNCT
ejpam-6568	66	14	)	)	PUNCT
ejpam-6568	66	15	(	(	PUNCT
ejpam-6568	66	16	resp	resp	NOUN
ejpam-6568	66	17	.	.	PUNCT
ejpam-6568	67	1	a	a	DET
ejpam-6568	67	2	⊆	⊆	NUM
ejpam-6568	67	3	τ1τ2	τ1τ2	NOUN
ejpam-6568	67	4	-	-	ADJ
ejpam-6568	67	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6568	67	6	-	-	PUNCT
ejpam-6568	67	7	int(a	int(a	NOUN
ejpam-6568	67	8	)	)	PUNCT
ejpam-6568	67	9	)	)	PUNCT
ejpam-6568	67	10	,	,	PUNCT
ejpam-6568	67	11	a	a	DET
ejpam-6568	67	12	⊆	⊆	NUM
ejpam-6568	67	13	τ1τ2	τ1τ2	NOUN
ejpam-6568	67	14	-	-	NOUN
ejpam-6568	67	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6568	67	16	-	-	PUNCT
ejpam-6568	67	17	cl(a	cl(a	NUM
ejpam-6568	67	18	)	)	PUNCT
ejpam-6568	67	19	)	)	PUNCT
ejpam-6568	67	20	,	,	PUNCT
ejpam-6568	67	21	a	a	DET
ejpam-6568	67	22	⊆	⊆	NUM
ejpam-6568	67	23	τ1τ2	τ1τ2	NOUN
ejpam-6568	67	24	-	-	PUNCT
ejpam-6568	67	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6568	67	26	-	-	PUNCT
ejpam-6568	67	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6568	67	28	-	-	PUNCT
ejpam-6568	67	29	cl(a	cl(a	NUM
ejpam-6568	67	30	)	)	PUNCT
ejpam-6568	67	31	)	)	PUNCT
ejpam-6568	67	32	)	)	PUNCT
ejpam-6568	67	33	)	)	PUNCT
ejpam-6568	67	34	.	.	PUNCT
ejpam-6568	68	1	the	the	DET
ejpam-6568	68	2	complement	complement	NOUN
ejpam-6568	68	3	of	of	ADP
ejpam-6568	68	4	a	a	DET
ejpam-6568	68	5	(	(	PUNCT
ejpam-6568	68	6	τ1	τ1	NOUN
ejpam-6568	68	7	,	,	PUNCT
ejpam-6568	68	8	τ2)r	τ2)r	NOUN
ejpam-6568	68	9	-	-	PUNCT
ejpam-6568	68	10	open	open	ADJ
ejpam-6568	68	11	(	(	PUNCT
ejpam-6568	68	12	resp	resp	NOUN
ejpam-6568	68	13	.	.	PUNCT
ejpam-6568	69	1	(	(	PUNCT
ejpam-6568	69	2	τ1	τ1	NOUN
ejpam-6568	69	3	,	,	PUNCT
ejpam-6568	69	4	τ2)s	τ2)s	NOUN
ejpam-6568	69	5	-	-	PUNCT
ejpam-6568	69	6	open	open	ADJ
ejpam-6568	69	7	,	,	PUNCT
ejpam-6568	69	8	(	(	PUNCT
ejpam-6568	69	9	τ1	τ1	NOUN
ejpam-6568	69	10	,	,	PUNCT
ejpam-6568	69	11	τ2)p	τ2)p	NOUN
ejpam-6568	69	12	-	-	ADJ
ejpam-6568	69	13	open	open	ADJ
ejpam-6568	69	14	,	,	PUNCT
ejpam-6568	69	15	(	(	PUNCT
ejpam-6568	69	16	τ1	τ1	NOUN
ejpam-6568	69	17	,	,	PUNCT
ejpam-6568	69	18	τ2)β	τ2)β	ADJ
ejpam-6568	69	19	-	-	PUNCT
ejpam-6568	69	20	open	open	ADJ
ejpam-6568	69	21	)	)	PUNCT
ejpam-6568	69	22	set	set	NOUN
ejpam-6568	69	23	is	be	AUX
ejpam-6568	69	24	said	say	VERB
ejpam-6568	69	25	to	to	PART
ejpam-6568	69	26	be	be	AUX
ejpam-6568	69	27	(	(	PUNCT
ejpam-6568	69	28	τ1	τ1	NOUN
ejpam-6568	69	29	,	,	PUNCT
ejpam-6568	69	30	τ2)r	τ2)r	NOUN
ejpam-6568	69	31	-	-	PUNCT
ejpam-6568	69	32	closed	closed	ADJ
ejpam-6568	69	33	(	(	PUNCT
ejpam-6568	69	34	resp	resp	NOUN
ejpam-6568	69	35	.	.	PUNCT
ejpam-6568	70	1	(	(	PUNCT
ejpam-6568	70	2	τ1	τ1	NOUN
ejpam-6568	70	3	,	,	PUNCT
ejpam-6568	70	4	τ2)s	τ2)s	NOUN
ejpam-6568	70	5	-	-	PUNCT
ejpam-6568	70	6	closed	closed	ADJ
ejpam-6568	70	7	,	,	PUNCT
ejpam-6568	70	8	(	(	PUNCT
ejpam-6568	70	9	τ1	τ1	NOUN
ejpam-6568	70	10	,	,	PUNCT
ejpam-6568	70	11	τ2)p	τ2)p	NOUN
ejpam-6568	70	12	-	-	PUNCT
ejpam-6568	70	13	closed	closed	ADJ
ejpam-6568	70	14	,	,	PUNCT
ejpam-6568	70	15	(	(	PUNCT
ejpam-6568	70	16	τ1	τ1	NOUN
ejpam-6568	70	17	,	,	PUNCT
ejpam-6568	70	18	τ2)β	τ2)β	ADJ
ejpam-6568	70	19	-	-	PUNCT
ejpam-6568	70	20	closed	closed	ADJ
ejpam-6568	70	21	)	)	PUNCT
ejpam-6568	70	22	.	.	PUNCT
ejpam-6568	71	1	a	a	DET
ejpam-6568	71	2	subset	subset	NOUN
ejpam-6568	71	3	a	a	PRON
ejpam-6568	71	4	of	of	ADP
ejpam-6568	71	5	a	a	DET
ejpam-6568	71	6	bitopological	bitopological	ADJ
ejpam-6568	71	7	space	space	NOUN
ejpam-6568	71	8	(	(	PUNCT
ejpam-6568	71	9	x	x	NOUN
ejpam-6568	71	10	,	,	PUNCT
ejpam-6568	71	11	τ1	τ1	NOUN
ejpam-6568	71	12	,	,	PUNCT
ejpam-6568	71	13	τ2	τ2	NOUN
ejpam-6568	71	14	)	)	PUNCT
ejpam-6568	71	15	is	be	AUX
ejpam-6568	71	16	said	say	VERB
ejpam-6568	71	17	to	to	PART
ejpam-6568	71	18	be	be	AUX
ejpam-6568	71	19	α(τ1	α(τ1	NOUN
ejpam-6568	71	20	,	,	PUNCT
ejpam-6568	71	21	τ2)-open	τ2)-open	ADJ
ejpam-6568	71	22	[	[	X
ejpam-6568	71	23	27	27	NUM
ejpam-6568	71	24	]	]	PUNCT
ejpam-6568	71	25	if	if	SCONJ
ejpam-6568	71	26	a	a	DET
ejpam-6568	71	27	⊆	⊆	NUM
ejpam-6568	71	28	τ1τ2	τ1τ2	NOUN
ejpam-6568	71	29	-	-	PUNCT
ejpam-6568	71	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6568	71	31	-	-	PUNCT
ejpam-6568	71	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6568	71	33	-	-	PUNCT
ejpam-6568	71	34	int(a	int(a	NOUN
ejpam-6568	71	35	)	)	PUNCT
ejpam-6568	71	36	)	)	PUNCT
ejpam-6568	71	37	)	)	PUNCT
ejpam-6568	71	38	.	.	PUNCT
ejpam-6568	72	1	the	the	DET
ejpam-6568	72	2	complement	complement	NOUN
ejpam-6568	72	3	of	of	ADP
ejpam-6568	72	4	an	an	DET
ejpam-6568	72	5	α(τ1	α(τ1	NOUN
ejpam-6568	72	6	,	,	PUNCT
ejpam-6568	72	7	τ2)-open	τ2)-open	ADJ
ejpam-6568	72	8	set	set	NOUN
ejpam-6568	72	9	is	be	AUX
ejpam-6568	72	10	said	say	VERB
ejpam-6568	72	11	to	to	PART
ejpam-6568	72	12	be	be	AUX
ejpam-6568	72	13	α(τ1	α(τ1	NOUN
ejpam-6568	72	14	,	,	PUNCT
ejpam-6568	72	15	τ2)-closed	τ2)-closed	ADJ
ejpam-6568	72	16	.	.	PUNCT
ejpam-6568	73	1	a	a	DET
ejpam-6568	73	2	subset	subset	NOUN
ejpam-6568	73	3	a	a	PRON
ejpam-6568	73	4	of	of	ADP
ejpam-6568	73	5	a	a	DET
ejpam-6568	73	6	bitopological	bitopological	ADJ
ejpam-6568	73	7	space	space	NOUN
ejpam-6568	73	8	(	(	PUNCT
ejpam-6568	73	9	x	x	NOUN
ejpam-6568	73	10	,	,	PUNCT
ejpam-6568	73	11	τ1	τ1	NOUN
ejpam-6568	73	12	,	,	PUNCT
ejpam-6568	73	13	τ2	τ2	NOUN
ejpam-6568	73	14	)	)	PUNCT
ejpam-6568	73	15	is	be	AUX
ejpam-6568	73	16	said	say	VERB
ejpam-6568	73	17	to	to	PART
ejpam-6568	73	18	be	be	AUX
ejpam-6568	73	19	τ1τ2	τ1τ2	NOUN
ejpam-6568	73	20	-	-	ADJ
ejpam-6568	73	21	δ	δ	NOUN
ejpam-6568	73	22	-	-	NOUN
ejpam-6568	73	23	open	open	ADJ
ejpam-6568	73	24	[	[	X
ejpam-6568	73	25	16	16	NUM
ejpam-6568	73	26	]	]	X
ejpam-6568	73	27	if	if	SCONJ
ejpam-6568	73	28	a	a	PRON
ejpam-6568	73	29	is	be	AUX
ejpam-6568	73	30	the	the	DET
ejpam-6568	73	31	union	union	NOUN
ejpam-6568	73	32	of	of	ADP
ejpam-6568	73	33	(	(	PUNCT
ejpam-6568	73	34	τ1	τ1	NOUN
ejpam-6568	73	35	,	,	PUNCT
ejpam-6568	73	36	τ2)r	τ2)r	ADJ
ejpam-6568	73	37	-	-	PUNCT
ejpam-6568	73	38	open	open	ADJ
ejpam-6568	73	39	sets	set	NOUN
ejpam-6568	73	40	of	of	ADP
ejpam-6568	73	41	x.	x.	NOUN
ejpam-6568	73	42	the	the	DET
ejpam-6568	73	43	complement	complement	NOUN
ejpam-6568	73	44	of	of	ADP
ejpam-6568	73	45	a	a	DET
ejpam-6568	73	46	τ1τ2	τ1τ2	ADJ
ejpam-6568	73	47	-	-	ADJ
ejpam-6568	73	48	δ	δ	NOUN
ejpam-6568	73	49	-	-	ADJ
ejpam-6568	73	50	open	open	ADJ
ejpam-6568	73	51	set	set	NOUN
ejpam-6568	73	52	is	be	AUX
ejpam-6568	73	53	called	call	VERB
ejpam-6568	73	54	τ1τ2	τ1τ2	NOUN
ejpam-6568	73	55	-	-	ADJ
ejpam-6568	73	56	δ	δ	NOUN
ejpam-6568	73	57	-	-	PUNCT
ejpam-6568	73	58	closed	closed	ADJ
ejpam-6568	73	59	[	[	X
ejpam-6568	73	60	16	16	NUM
ejpam-6568	73	61	]	]	PUNCT
ejpam-6568	73	62	.	.	PUNCT
ejpam-6568	74	1	the	the	DET
ejpam-6568	74	2	union	union	NOUN
ejpam-6568	74	3	of	of	ADP
ejpam-6568	74	4	all	all	DET
ejpam-6568	74	5	τ1τ2	τ1τ2	NOUN
ejpam-6568	74	6	-	-	ADJ
ejpam-6568	74	7	δ	δ	NOUN
ejpam-6568	74	8	-	-	ADJ
ejpam-6568	74	9	open	open	ADJ
ejpam-6568	74	10	sets	set	NOUN
ejpam-6568	74	11	of	of	ADP
ejpam-6568	74	12	x	x	PUNCT
ejpam-6568	74	13	contained	contain	VERB
ejpam-6568	74	14	in	in	ADP
ejpam-6568	74	15	a	a	PRON
ejpam-6568	74	16	is	be	AUX
ejpam-6568	74	17	called	call	VERB
ejpam-6568	74	18	the	the	DET
ejpam-6568	74	19	τ1τ2	τ1τ2	ADJ
ejpam-6568	74	20	-	-	ADJ
ejpam-6568	74	21	δ	δ	NOUN
ejpam-6568	74	22	-	-	NOUN
ejpam-6568	74	23	interior	interior	NOUN
ejpam-6568	74	24	[	[	X
ejpam-6568	74	25	16	16	NUM
ejpam-6568	74	26	]	]	PUNCT
ejpam-6568	74	27	of	of	ADP
ejpam-6568	74	28	a	a	PRON
ejpam-6568	74	29	and	and	CCONJ
ejpam-6568	74	30	is	be	AUX
ejpam-6568	74	31	denoted	denote	VERB
ejpam-6568	74	32	by	by	ADP
ejpam-6568	74	33	τ1τ2	τ1τ2	ADJ
ejpam-6568	74	34	-	-	ADJ
ejpam-6568	74	35	δ	δ	NOUN
ejpam-6568	74	36	-	-	PUNCT
ejpam-6568	74	37	int(a	int(a	PROPN
ejpam-6568	74	38	)	)	PUNCT
ejpam-6568	74	39	.	.	PUNCT
ejpam-6568	75	1	the	the	DET
ejpam-6568	75	2	intersection	intersection	NOUN
ejpam-6568	75	3	of	of	ADP
ejpam-6568	75	4	all	all	DET
ejpam-6568	75	5	τ1τ2δ	τ1τ2δ	NUM
ejpam-6568	75	6	-	-	PUNCT
ejpam-6568	75	7	closed	close	VERB
ejpam-6568	75	8	sets	set	NOUN
ejpam-6568	75	9	of	of	ADP
ejpam-6568	75	10	x	x	PUNCT
ejpam-6568	75	11	containing	contain	VERB
ejpam-6568	75	12	a	a	PRON
ejpam-6568	75	13	is	be	AUX
ejpam-6568	75	14	called	call	VERB
ejpam-6568	75	15	the	the	DET
ejpam-6568	75	16	τ1τ2	τ1τ2	ADJ
ejpam-6568	75	17	-	-	ADJ
ejpam-6568	75	18	δ	δ	NOUN
ejpam-6568	75	19	-	-	NOUN
ejpam-6568	75	20	closure	closure	NOUN
ejpam-6568	75	21	[	[	X
ejpam-6568	75	22	16	16	NUM
ejpam-6568	75	23	]	]	PUNCT
ejpam-6568	75	24	of	of	ADP
ejpam-6568	75	25	a	a	PRON
ejpam-6568	75	26	and	and	CCONJ
ejpam-6568	75	27	is	be	AUX
ejpam-6568	75	28	denoted	denote	VERB
ejpam-6568	75	29	by	by	ADP
ejpam-6568	75	30	τ1τ2	τ1τ2	ADJ
ejpam-6568	75	31	-	-	ADJ
ejpam-6568	75	32	δ	δ	NOUN
ejpam-6568	75	33	-	-	PUNCT
ejpam-6568	75	34	cl(a	cl(a	NUM
ejpam-6568	75	35	)	)	PUNCT
ejpam-6568	75	36	.	.	PUNCT
ejpam-6568	76	1	let	let	VERB
ejpam-6568	76	2	a	a	DET
ejpam-6568	76	3	be	be	AUX
ejpam-6568	76	4	a	a	DET
ejpam-6568	76	5	subset	subset	NOUN
ejpam-6568	76	6	of	of	ADP
ejpam-6568	76	7	a	a	DET
ejpam-6568	76	8	bitopological	bitopological	ADJ
ejpam-6568	76	9	space	space	NOUN
ejpam-6568	76	10	(	(	PUNCT
ejpam-6568	76	11	x	x	NOUN
ejpam-6568	76	12	,	,	PUNCT
ejpam-6568	76	13	τ1	τ1	NOUN
ejpam-6568	76	14	,	,	PUNCT
ejpam-6568	76	15	τ2	τ2	NOUN
ejpam-6568	76	16	)	)	PUNCT
ejpam-6568	76	17	.	.	PUNCT
ejpam-6568	77	1	a	a	DET
ejpam-6568	77	2	point	point	NOUN
ejpam-6568	77	3	x	x	X
ejpam-6568	77	4	∈	∈	NOUN
ejpam-6568	77	5	x	x	PUNCT
ejpam-6568	77	6	is	be	AUX
ejpam-6568	77	7	called	call	VERB
ejpam-6568	77	8	a	a	DET
ejpam-6568	77	9	(	(	PUNCT
ejpam-6568	77	10	τ1	τ1	NOUN
ejpam-6568	77	11	,	,	PUNCT
ejpam-6568	77	12	τ2)θ	τ2)θ	ADJ
ejpam-6568	77	13	-	-	PUNCT
ejpam-6568	77	14	cluster	cluster	NOUN
ejpam-6568	77	15	point	point	NOUN
ejpam-6568	77	16	[	[	X
ejpam-6568	77	17	25	25	NUM
ejpam-6568	77	18	]	]	PUNCT
ejpam-6568	77	19	of	of	ADP
ejpam-6568	77	20	a	a	DET
ejpam-6568	77	21	if	if	SCONJ
ejpam-6568	77	22	τ1τ2	τ1τ2	NOUN
ejpam-6568	77	23	-	-	NOUN
ejpam-6568	77	24	cl(u	cl(u	NOUN
ejpam-6568	77	25	)	)	PUNCT
ejpam-6568	77	26	∩	∩	NOUN
ejpam-6568	77	27	a	a	DET
ejpam-6568	77	28	̸=	̸=	PROPN
ejpam-6568	77	29	∅	∅	NOUN
ejpam-6568	77	30	for	for	ADP
ejpam-6568	77	31	every	every	DET
ejpam-6568	77	32	τ1τ2	τ1τ2	ADJ
ejpam-6568	77	33	-	-	ADJ
ejpam-6568	77	34	open	open	ADJ
ejpam-6568	77	35	set	set	NOUN
ejpam-6568	77	36	u	u	NOUN
ejpam-6568	77	37	containing	contain	VERB
ejpam-6568	77	38	x.	x.	NOUN
ejpam-6568	77	39	the	the	DET
ejpam-6568	77	40	set	set	NOUN
ejpam-6568	77	41	of	of	ADP
ejpam-6568	77	42	all	all	DET
ejpam-6568	77	43	(	(	PUNCT
ejpam-6568	77	44	τ1	τ1	NOUN
ejpam-6568	77	45	,	,	PUNCT
ejpam-6568	77	46	τ2)θ	τ2)θ	ADJ
ejpam-6568	77	47	-	-	PUNCT
ejpam-6568	77	48	cluster	cluster	NOUN
ejpam-6568	77	49	points	point	NOUN
ejpam-6568	77	50	of	of	ADP
ejpam-6568	77	51	a	a	PRON
ejpam-6568	77	52	is	be	AUX
ejpam-6568	77	53	called	call	VERB
ejpam-6568	77	54	the	the	DET
ejpam-6568	77	55	(	(	PUNCT
ejpam-6568	77	56	τ1	τ1	NOUN
ejpam-6568	77	57	,	,	PUNCT
ejpam-6568	77	58	τ2)θ	τ2)θ	ADJ
ejpam-6568	77	59	-	-	PUNCT
ejpam-6568	77	60	closure	closure	NOUN
ejpam-6568	77	61	[	[	X
ejpam-6568	77	62	25	25	NUM
ejpam-6568	77	63	]	]	PUNCT
ejpam-6568	77	64	of	of	ADP
ejpam-6568	77	65	a	a	PRON
ejpam-6568	77	66	and	and	CCONJ
ejpam-6568	77	67	is	be	AUX
ejpam-6568	77	68	denoted	denote	VERB
ejpam-6568	77	69	by	by	ADP
ejpam-6568	77	70	(	(	PUNCT
ejpam-6568	77	71	τ1	τ1	NOUN
ejpam-6568	77	72	,	,	PUNCT
ejpam-6568	77	73	τ2)θ	τ2)θ	NOUN
ejpam-6568	77	74	-	-	PUNCT
ejpam-6568	77	75	cl(a	cl(a	NUM
ejpam-6568	77	76	)	)	PUNCT
ejpam-6568	77	77	.	.	PUNCT
ejpam-6568	78	1	a	a	DET
ejpam-6568	78	2	subset	subset	NOUN
ejpam-6568	78	3	a	a	PRON
ejpam-6568	78	4	of	of	ADP
ejpam-6568	78	5	a	a	DET
ejpam-6568	78	6	bitopological	bitopological	ADJ
ejpam-6568	78	7	space	space	NOUN
ejpam-6568	78	8	(	(	PUNCT
ejpam-6568	78	9	x	x	NOUN
ejpam-6568	78	10	,	,	PUNCT
ejpam-6568	78	11	τ1	τ1	NOUN
ejpam-6568	78	12	,	,	PUNCT
ejpam-6568	78	13	τ2	τ2	NOUN
ejpam-6568	78	14	)	)	PUNCT
ejpam-6568	78	15	is	be	AUX
ejpam-6568	78	16	said	say	VERB
ejpam-6568	78	17	to	to	PART
ejpam-6568	78	18	be	be	AUX
ejpam-6568	78	19	(	(	PUNCT
ejpam-6568	78	20	τ1	τ1	NOUN
ejpam-6568	78	21	,	,	PUNCT
ejpam-6568	78	22	τ2)θ	τ2)θ	NOUN
ejpam-6568	78	23	-	-	PUNCT
ejpam-6568	78	24	closed	closed	ADJ
ejpam-6568	78	25	[	[	X
ejpam-6568	78	26	25	25	NUM
ejpam-6568	78	27	]	]	X
ejpam-6568	78	28	if	if	SCONJ
ejpam-6568	78	29	(	(	PUNCT
ejpam-6568	78	30	τ1	τ1	NOUN
ejpam-6568	78	31	,	,	PUNCT
ejpam-6568	78	32	τ2)θ	τ2)θ	NOUN
ejpam-6568	78	33	-	-	PUNCT
ejpam-6568	78	34	cl(a	cl(a	NUM
ejpam-6568	78	35	)	)	PUNCT
ejpam-6568	79	1	=	=	PUNCT
ejpam-6568	79	2	a.	a.	NOUN
ejpam-6568	79	3	the	the	DET
ejpam-6568	79	4	complement	complement	NOUN
ejpam-6568	79	5	of	of	ADP
ejpam-6568	79	6	a	a	DET
ejpam-6568	79	7	(	(	PUNCT
ejpam-6568	79	8	τ1	τ1	NOUN
ejpam-6568	79	9	,	,	PUNCT
ejpam-6568	79	10	τ2)θ	τ2)θ	ADJ
ejpam-6568	79	11	-	-	PUNCT
ejpam-6568	79	12	closed	close	VERB
ejpam-6568	79	13	set	set	NOUN
ejpam-6568	79	14	is	be	AUX
ejpam-6568	79	15	said	say	VERB
ejpam-6568	79	16	to	to	PART
ejpam-6568	79	17	be	be	AUX
ejpam-6568	79	18	(	(	PUNCT
ejpam-6568	79	19	τ1	τ1	NOUN
ejpam-6568	79	20	,	,	PUNCT
ejpam-6568	79	21	τ2)θ	τ2)θ	NOUN
ejpam-6568	79	22	-	-	PUNCT
ejpam-6568	79	23	open	open	ADJ
ejpam-6568	79	24	.	.	PUNCT
ejpam-6568	80	1	the	the	DET
ejpam-6568	80	2	union	union	NOUN
ejpam-6568	80	3	of	of	ADP
ejpam-6568	80	4	all	all	DET
ejpam-6568	80	5	(	(	PUNCT
ejpam-6568	80	6	τ1	τ1	NOUN
ejpam-6568	80	7	,	,	PUNCT
ejpam-6568	80	8	τ2)θ	τ2)θ	ADJ
ejpam-6568	80	9	-	-	PUNCT
ejpam-6568	80	10	open	open	ADJ
ejpam-6568	80	11	sets	set	NOUN
ejpam-6568	80	12	of	of	ADP
ejpam-6568	80	13	x	x	PUNCT
ejpam-6568	80	14	contained	contain	VERB
ejpam-6568	80	15	in	in	ADP
ejpam-6568	80	16	a	a	PRON
ejpam-6568	80	17	is	be	AUX
ejpam-6568	80	18	called	call	VERB
ejpam-6568	80	19	the	the	DET
ejpam-6568	80	20	(	(	PUNCT
ejpam-6568	80	21	τ1	τ1	NOUN
ejpam-6568	80	22	,	,	PUNCT
ejpam-6568	80	23	τ2)θ	τ2)θ	ADJ
ejpam-6568	80	24	-	-	PUNCT
ejpam-6568	80	25	interior	interior	NOUN
ejpam-6568	80	26	[	[	X
ejpam-6568	80	27	25	25	NUM
ejpam-6568	80	28	]	]	PUNCT
ejpam-6568	80	29	of	of	ADP
ejpam-6568	80	30	a	a	PRON
ejpam-6568	80	31	and	and	CCONJ
ejpam-6568	80	32	is	be	AUX
ejpam-6568	80	33	denoted	denote	VERB
ejpam-6568	80	34	by	by	ADP
ejpam-6568	80	35	(	(	PUNCT
ejpam-6568	80	36	τ1	τ1	NOUN
ejpam-6568	80	37	,	,	PUNCT
ejpam-6568	80	38	τ2)θ	τ2)θ	NOUN
ejpam-6568	80	39	-	-	PUNCT
ejpam-6568	80	40	int(a	int(a	NOUN
ejpam-6568	80	41	)	)	PUNCT
ejpam-6568	80	42	.	.	PUNCT
ejpam-6568	81	1	an	an	DET
ejpam-6568	81	2	ideal	ideal	NOUN
ejpam-6568	81	3	i	i	PRON
ejpam-6568	81	4	on	on	ADP
ejpam-6568	81	5	a	a	DET
ejpam-6568	81	6	topological	topological	ADJ
ejpam-6568	81	7	space	space	NOUN
ejpam-6568	81	8	(	(	PUNCT
ejpam-6568	81	9	x	x	X
ejpam-6568	81	10	,	,	PUNCT
ejpam-6568	81	11	τ	τ	X
ejpam-6568	81	12	)	)	PUNCT
ejpam-6568	81	13	is	be	AUX
ejpam-6568	81	14	a	a	DET
ejpam-6568	81	15	nonempty	nonempty	ADJ
ejpam-6568	81	16	collection	collection	NOUN
ejpam-6568	81	17	of	of	ADP
ejpam-6568	81	18	subsets	subset	NOUN
ejpam-6568	81	19	of	of	ADP
ejpam-6568	81	20	x	x	PUNCT
ejpam-6568	81	21	satisfying	satisfy	VERB
ejpam-6568	81	22	the	the	DET
ejpam-6568	81	23	following	follow	VERB
ejpam-6568	81	24	properties	property	NOUN
ejpam-6568	81	25	:	:	PUNCT
ejpam-6568	81	26	(	(	PUNCT
ejpam-6568	81	27	1	1	X
ejpam-6568	81	28	)	)	PUNCT
ejpam-6568	81	29	a	a	DET
ejpam-6568	81	30	∈	∈	NOUN
ejpam-6568	81	31	i	i	PRON
ejpam-6568	81	32	and	and	CCONJ
ejpam-6568	81	33	b	b	X
ejpam-6568	81	34	⊆	⊆	NUM
ejpam-6568	81	35	a	a	DET
ejpam-6568	81	36	imply	imply	NOUN
ejpam-6568	81	37	b	b	X
ejpam-6568	81	38	∈	∈	PROPN
ejpam-6568	81	39	i	i	PRON
ejpam-6568	81	40	;	;	PUNCT
ejpam-6568	81	41	(	(	PUNCT
ejpam-6568	81	42	2	2	X
ejpam-6568	81	43	)	)	PUNCT
ejpam-6568	82	1	a	a	PRON
ejpam-6568	82	2	∈	∈	NOUN
ejpam-6568	83	1	i	i	PRON
ejpam-6568	83	2	and	and	CCONJ
ejpam-6568	83	3	b	b	X
ejpam-6568	83	4	∈	∈	NOUN
ejpam-6568	84	1	i	i	PRON
ejpam-6568	84	2	imply	imply	VERB
ejpam-6568	84	3	a	a	DET
ejpam-6568	84	4	∪	∪	X
ejpam-6568	84	5	b	b	NOUN
ejpam-6568	84	6	∈	∈	NOUN
ejpam-6568	85	1	i	i	PRON
ejpam-6568	85	2	.	.	PUNCT
ejpam-6568	86	1	a	a	DET
ejpam-6568	86	2	topological	topological	ADJ
ejpam-6568	86	3	space	space	NOUN
ejpam-6568	86	4	(	(	PUNCT
ejpam-6568	86	5	x	x	X
ejpam-6568	86	6	,	,	PUNCT
ejpam-6568	86	7	τ	τ	X
ejpam-6568	86	8	)	)	PUNCT
ejpam-6568	86	9	with	with	ADP
ejpam-6568	86	10	an	an	DET
ejpam-6568	86	11	ideal	ideal	ADJ
ejpam-6568	86	12	i	i	PRON
ejpam-6568	86	13	on	on	ADP
ejpam-6568	86	14	x	x	SYM
ejpam-6568	86	15	is	be	AUX
ejpam-6568	86	16	called	call	VERB
ejpam-6568	86	17	an	an	DET
ejpam-6568	86	18	ideal	ideal	ADJ
ejpam-6568	86	19	topological	topological	ADJ
ejpam-6568	86	20	space	space	NOUN
ejpam-6568	86	21	and	and	CCONJ
ejpam-6568	86	22	is	be	AUX
ejpam-6568	86	23	denoted	denote	VERB
ejpam-6568	86	24	by	by	ADP
ejpam-6568	86	25	(	(	PUNCT
ejpam-6568	86	26	x	x	X
ejpam-6568	86	27	,	,	PUNCT
ejpam-6568	86	28	τ	τ	PROPN
ejpam-6568	86	29	,	,	PUNCT
ejpam-6568	86	30	i	i	NOUN
ejpam-6568	86	31	)	)	PUNCT
ejpam-6568	86	32	.	.	PUNCT
ejpam-6568	87	1	for	for	ADP
ejpam-6568	87	2	an	an	DET
ejpam-6568	87	3	ideal	ideal	ADJ
ejpam-6568	87	4	topological	topological	ADJ
ejpam-6568	87	5	space	space	NOUN
ejpam-6568	87	6	(	(	PUNCT
ejpam-6568	87	7	x	x	X
ejpam-6568	87	8	,	,	PUNCT
ejpam-6568	87	9	τ	τ	PROPN
ejpam-6568	87	10	,	,	PUNCT
ejpam-6568	87	11	i	i	PROPN
ejpam-6568	87	12	)	)	PUNCT
ejpam-6568	87	13	and	and	CCONJ
ejpam-6568	87	14	a	a	DET
ejpam-6568	87	15	subset	subset	NOUN
ejpam-6568	87	16	a	a	PRON
ejpam-6568	87	17	of	of	ADP
ejpam-6568	87	18	x	x	PRON
ejpam-6568	87	19	,	,	PUNCT
ejpam-6568	87	20	a⋆(i	a⋆(i	PROPN
ejpam-6568	87	21	)	)	PUNCT
ejpam-6568	87	22	is	be	AUX
ejpam-6568	87	23	defined	define	VERB
ejpam-6568	87	24	as	as	SCONJ
ejpam-6568	87	25	follows	follow	VERB
ejpam-6568	87	26	:	:	PUNCT
ejpam-6568	87	27	a⋆(i	a⋆(i	NOUN
ejpam-6568	87	28	)	)	PUNCT
ejpam-6568	88	1	=	=	PUNCT
ejpam-6568	88	2	{	{	PUNCT
ejpam-6568	88	3	x	x	PUNCT
ejpam-6568	88	4	∈	∈	PROPN
ejpam-6568	88	5	x	x	X
ejpam-6568	88	6	:	:	PUNCT
ejpam-6568	88	7	u	u	X
ejpam-6568	88	8	∩a	∩a	PROPN
ejpam-6568	88	9	̸∈	̸∈	PROPN
ejpam-6568	88	10	i	i	PRON
ejpam-6568	88	11	for	for	ADP
ejpam-6568	88	12	every	every	DET
ejpam-6568	88	13	open	open	ADJ
ejpam-6568	88	14	neighbourhood	neighbourhood	NOUN
ejpam-6568	88	15	u	u	NOUN
ejpam-6568	88	16	of	of	ADP
ejpam-6568	88	17	x	x	NOUN
ejpam-6568	88	18	}	}	PUNCT
ejpam-6568	88	19	.	.	PUNCT
ejpam-6568	89	1	in	in	ADP
ejpam-6568	89	2	case	case	NOUN
ejpam-6568	89	3	there	there	PRON
ejpam-6568	89	4	is	be	VERB
ejpam-6568	89	5	no	no	DET
ejpam-6568	89	6	chance	chance	NOUN
ejpam-6568	89	7	for	for	ADP
ejpam-6568	89	8	confusion	confusion	NOUN
ejpam-6568	89	9	,	,	PUNCT
ejpam-6568	89	10	a⋆(i	a⋆(i	NOUN
ejpam-6568	89	11	)	)	PUNCT
ejpam-6568	89	12	is	be	AUX
ejpam-6568	89	13	simply	simply	ADV
ejpam-6568	89	14	written	write	VERB
ejpam-6568	89	15	as	as	ADP
ejpam-6568	89	16	a⋆.	a⋆.	NOUN
ejpam-6568	89	17	in	in	ADP
ejpam-6568	89	18	[	[	X
ejpam-6568	89	19	28	28	NUM
ejpam-6568	89	20	]	]	PUNCT
ejpam-6568	89	21	,	,	PUNCT
ejpam-6568	89	22	a⋆	a⋆	ADV
ejpam-6568	89	23	is	be	AUX
ejpam-6568	89	24	called	call	VERB
ejpam-6568	89	25	the	the	DET
ejpam-6568	89	26	local	local	ADJ
ejpam-6568	89	27	function	function	NOUN
ejpam-6568	89	28	of	of	ADP
ejpam-6568	89	29	a	a	PRON
ejpam-6568	89	30	with	with	ADP
ejpam-6568	89	31	respect	respect	NOUN
ejpam-6568	89	32	to	to	ADP
ejpam-6568	89	33	i	i	PRON
ejpam-6568	89	34	and	and	CCONJ
ejpam-6568	89	35	τ	τ	PROPN
ejpam-6568	89	36	and	and	CCONJ
ejpam-6568	89	37	cl⋆(a	cl⋆(a	PROPN
ejpam-6568	89	38	)	)	PUNCT
ejpam-6568	89	39	=	=	NOUN
ejpam-6568	89	40	a⋆∪a	a⋆∪a	NOUN
ejpam-6568	89	41	defines	define	VERB
ejpam-6568	89	42	a	a	DET
ejpam-6568	89	43	kuratowski	kuratowski	ADJ
ejpam-6568	89	44	closure	closure	NOUN
ejpam-6568	89	45	operator	operator	NOUN
ejpam-6568	89	46	for	for	ADP
ejpam-6568	89	47	a	a	DET
ejpam-6568	89	48	topology	topology	NOUN
ejpam-6568	89	49	τ⋆(i	τ⋆(i	NOUN
ejpam-6568	89	50	)	)	PUNCT
ejpam-6568	89	51	finer	fine	ADJ
ejpam-6568	89	52	than	than	ADP
ejpam-6568	89	53	τ	τ	PROPN
ejpam-6568	89	54	.	.	PUNCT
ejpam-6568	90	1	a	a	DET
ejpam-6568	90	2	subset	subset	NOUN
ejpam-6568	90	3	a	a	PRON
ejpam-6568	90	4	is	be	AUX
ejpam-6568	90	5	said	say	VERB
ejpam-6568	90	6	to	to	PART
ejpam-6568	90	7	be	be	AUX
ejpam-6568	90	8	⋆-closed	⋆-close	VERB
ejpam-6568	90	9	[	[	X
ejpam-6568	90	10	29	29	NUM
ejpam-6568	90	11	]	]	X
ejpam-6568	90	12	if	if	SCONJ
ejpam-6568	90	13	a⋆	a⋆	ADJ
ejpam-6568	90	14	⊆	⊆	NUM
ejpam-6568	90	15	a.	a.	NOUN
ejpam-6568	90	16	the	the	DET
ejpam-6568	90	17	interior	interior	NOUN
ejpam-6568	90	18	of	of	ADP
ejpam-6568	90	19	a	a	DET
ejpam-6568	90	20	subset	subset	NOUN
ejpam-6568	90	21	a	a	DET
ejpam-6568	90	22	in	in	ADP
ejpam-6568	90	23	(	(	PUNCT
ejpam-6568	90	24	x	x	X
ejpam-6568	90	25	,	,	PUNCT
ejpam-6568	90	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6568	90	27	)	)	PUNCT
ejpam-6568	90	28	)	)	PUNCT
ejpam-6568	90	29	is	be	AUX
ejpam-6568	90	30	denoted	denote	VERB
ejpam-6568	90	31	by	by	ADP
ejpam-6568	90	32	int⋆(a	int⋆(a	NOUN
ejpam-6568	90	33	)	)	PUNCT
ejpam-6568	90	34	.	.	PUNCT
ejpam-6568	91	1	a	a	DET
ejpam-6568	91	2	subset	subset	NOUN
ejpam-6568	91	3	a	a	PRON
ejpam-6568	91	4	of	of	ADP
ejpam-6568	91	5	an	an	DET
ejpam-6568	91	6	ideal	ideal	ADJ
ejpam-6568	91	7	topological	topological	ADJ
ejpam-6568	91	8	space	space	NOUN
ejpam-6568	91	9	(	(	PUNCT
ejpam-6568	91	10	x	x	X
ejpam-6568	91	11	,	,	PUNCT
ejpam-6568	91	12	τ	τ	PROPN
ejpam-6568	91	13	,	,	PUNCT
ejpam-6568	91	14	i	i	PROPN
ejpam-6568	91	15	)	)	PUNCT
ejpam-6568	91	16	is	be	AUX
ejpam-6568	91	17	said	say	VERB
ejpam-6568	91	18	to	to	PART
ejpam-6568	91	19	be	be	AUX
ejpam-6568	91	20	semi⋆-i	semi⋆-i	PUNCT
ejpam-6568	91	21	-open	-open	VERB
ejpam-6568	91	22	[	[	PUNCT
ejpam-6568	91	23	30	30	NUM
ejpam-6568	91	24	]	]	PUNCT
ejpam-6568	91	25	(	(	PUNCT
ejpam-6568	91	26	resp	resp	NOUN
ejpam-6568	91	27	.	.	PUNCT
ejpam-6568	92	1	semi	semi	ADJ
ejpam-6568	92	2	-	-	VERB
ejpam-6568	92	3	i	i	PRON
ejpam-6568	92	4	-open	-open	NOUN
ejpam-6568	93	1	[	[	X
ejpam-6568	93	2	15	15	NUM
ejpam-6568	93	3	]	]	PUNCT
ejpam-6568	93	4	)	)	PUNCT
ejpam-6568	93	5	if	if	SCONJ
ejpam-6568	93	6	a	a	DET
ejpam-6568	93	7	⊆	⊆	NUM
ejpam-6568	93	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6568	93	9	)	)	PUNCT
ejpam-6568	93	10	)	)	PUNCT
ejpam-6568	93	11	(	(	PUNCT
ejpam-6568	93	12	resp	resp	NOUN
ejpam-6568	93	13	.	.	PUNCT
ejpam-6568	94	1	a	a	DET
ejpam-6568	94	2	⊆	⊆	NUM
ejpam-6568	94	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6568	94	4	)	)	PUNCT
ejpam-6568	94	5	)	)	PUNCT
ejpam-6568	94	6	)	)	PUNCT
ejpam-6568	94	7	.	.	PUNCT
ejpam-6568	95	1	the	the	DET
ejpam-6568	95	2	complement	complement	NOUN
ejpam-6568	95	3	of	of	ADP
ejpam-6568	95	4	a	a	DET
ejpam-6568	95	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6568	95	6	-open	-open	ADJ
ejpam-6568	95	7	(	(	PUNCT
ejpam-6568	95	8	resp	resp	NOUN
ejpam-6568	95	9	.	.	PUNCT
ejpam-6568	96	1	semi	semi	ADJ
ejpam-6568	96	2	-	-	VERB
ejpam-6568	96	3	i	i	PRON
ejpam-6568	96	4	-open	-open	NOUN
ejpam-6568	96	5	)	)	PUNCT
ejpam-6568	97	1	set	set	NOUN
ejpam-6568	97	2	is	be	AUX
ejpam-6568	97	3	said	say	VERB
ejpam-6568	97	4	to	to	PART
ejpam-6568	97	5	be	be	AUX
ejpam-6568	97	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6568	97	7	-closed	-close	VERB
ejpam-6568	97	8	[	[	PUNCT
ejpam-6568	97	9	30	30	NUM
ejpam-6568	97	10	]	]	PUNCT
ejpam-6568	97	11	(	(	PUNCT
ejpam-6568	97	12	resp	resp	NOUN
ejpam-6568	97	13	.	.	PUNCT
ejpam-6568	98	1	semi	semi	ADJ
ejpam-6568	98	2	-	-	VERB
ejpam-6568	98	3	i	i	PRON
ejpam-6568	98	4	-closed	-close	VERB
ejpam-6568	98	5	[	[	X
ejpam-6568	98	6	15	15	NUM
ejpam-6568	98	7	]	]	NUM
ejpam-6568	98	8	)	)	PUNCT
ejpam-6568	98	9	.	.	PUNCT
ejpam-6568	99	1	3	3	X
ejpam-6568	99	2	.	.	X
ejpam-6568	99	3	on	on	ADP
ejpam-6568	99	4	almost	almost	ADV
ejpam-6568	99	5	τ	τ	NOUN
ejpam-6568	99	6	⋆(σ1	⋆(σ1	NOUN
ejpam-6568	99	7	,	,	PUNCT
ejpam-6568	99	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	99	9	functions	function	NOUN
ejpam-6568	99	10	in	in	ADP
ejpam-6568	99	11	this	this	DET
ejpam-6568	99	12	section	section	NOUN
ejpam-6568	99	13	,	,	PUNCT
ejpam-6568	99	14	we	we	PRON
ejpam-6568	99	15	introduce	introduce	VERB
ejpam-6568	99	16	the	the	DET
ejpam-6568	99	17	concept	concept	NOUN
ejpam-6568	99	18	of	of	ADP
ejpam-6568	99	19	almost	almost	ADV
ejpam-6568	99	20	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	99	21	,	,	PUNCT
ejpam-6568	99	22	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	99	23	functions	function	NOUN
ejpam-6568	99	24	.	.	PUNCT
ejpam-6568	100	1	moreover	moreover	ADV
ejpam-6568	100	2	,	,	PUNCT
ejpam-6568	100	3	some	some	DET
ejpam-6568	100	4	characterizations	characterization	NOUN
ejpam-6568	100	5	of	of	ADP
ejpam-6568	100	6	almost	almost	ADV
ejpam-6568	100	7	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	100	8	,	,	PUNCT
ejpam-6568	100	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	100	10	functions	function	NOUN
ejpam-6568	100	11	are	be	AUX
ejpam-6568	100	12	discussed	discuss	VERB
ejpam-6568	100	13	.	.	PUNCT
ejpam-6568	101	1	definition	definition	NOUN
ejpam-6568	101	2	1	1	NUM
ejpam-6568	101	3	.	.	PUNCT
ejpam-6568	102	1	a	a	DET
ejpam-6568	102	2	function	function	NOUN
ejpam-6568	102	3	f	f	NOUN
ejpam-6568	102	4	:	:	PUNCT
ejpam-6568	102	5	(	(	PUNCT
ejpam-6568	102	6	x	x	X
ejpam-6568	102	7	,	,	PUNCT
ejpam-6568	102	8	τ	τ	PROPN
ejpam-6568	102	9	,	,	PUNCT
ejpam-6568	102	10	i	i	NOUN
ejpam-6568	102	11	)	)	PUNCT
ejpam-6568	102	12	→	→	PUNCT
ejpam-6568	102	13	(	(	PUNCT
ejpam-6568	102	14	y	y	PROPN
ejpam-6568	102	15	,	,	PUNCT
ejpam-6568	102	16	σ1	σ1	PROPN
ejpam-6568	102	17	,	,	PUNCT
ejpam-6568	102	18	σ2	σ2	PROPN
ejpam-6568	102	19	)	)	PUNCT
ejpam-6568	102	20	is	be	AUX
ejpam-6568	102	21	said	say	VERB
ejpam-6568	102	22	to	to	PART
ejpam-6568	102	23	be	be	AUX
ejpam-6568	102	24	almost	almost	ADV
ejpam-6568	102	25	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	102	26	,	,	PUNCT
ejpam-6568	102	27	σ2)continuous	σ2)continuous	ADJ
ejpam-6568	102	28	at	at	ADP
ejpam-6568	102	29	a	a	DET
ejpam-6568	102	30	point	point	NOUN
ejpam-6568	102	31	x	x	SYM
ejpam-6568	102	32	∈	∈	NOUN
ejpam-6568	102	33	x	x	PUNCT
ejpam-6568	102	34	if	if	SCONJ
ejpam-6568	102	35	for	for	ADP
ejpam-6568	102	36	each	each	DET
ejpam-6568	102	37	σ1σ2	σ1σ2	VERB
ejpam-6568	102	38	-	-	ADJ
ejpam-6568	102	39	open	open	ADJ
ejpam-6568	102	40	set	set	NOUN
ejpam-6568	102	41	v	v	NOUN
ejpam-6568	102	42	of	of	ADP
ejpam-6568	102	43	y	y	NOUN
ejpam-6568	102	44	containing	contain	VERB
ejpam-6568	102	45	f(x	f(x	PROPN
ejpam-6568	102	46	)	)	PUNCT
ejpam-6568	102	47	,	,	PUNCT
ejpam-6568	102	48	there	there	PRON
ejpam-6568	102	49	n.	n.	PROPN
ejpam-6568	102	50	viriyapong	viriyapong	PROPN
ejpam-6568	102	51	,	,	PUNCT
ejpam-6568	102	52	a.	a.	PROPN
ejpam-6568	102	53	sama	sama	PROPN
ejpam-6568	102	54	-	-	PUNCT
ejpam-6568	102	55	ae	ae	PROPN
ejpam-6568	102	56	,	,	PUNCT
ejpam-6568	102	57	c.	c.	PROPN
ejpam-6568	102	58	boonpok	boonpok	PROPN
ejpam-6568	102	59	/	/	SYM
ejpam-6568	102	60	eur	eur	PROPN
ejpam-6568	102	61	.	.	PUNCT
ejpam-6568	103	1	j.	j.	PROPN
ejpam-6568	103	2	pure	pure	PROPN
ejpam-6568	103	3	appl	appl	PROPN
ejpam-6568	103	4	.	.	PROPN
ejpam-6568	103	5	math	math	PROPN
ejpam-6568	103	6	,	,	PUNCT
ejpam-6568	103	7	18	18	NUM
ejpam-6568	103	8	(	(	PUNCT
ejpam-6568	103	9	3	3	NUM
ejpam-6568	103	10	)	)	PUNCT
ejpam-6568	103	11	(	(	PUNCT
ejpam-6568	103	12	2025	2025	NUM
ejpam-6568	103	13	)	)	PUNCT
ejpam-6568	103	14	,	,	PUNCT
ejpam-6568	103	15	6568	6568	NUM
ejpam-6568	103	16	4	4	NUM
ejpam-6568	103	17	of	of	ADP
ejpam-6568	103	18	18	18	NUM
ejpam-6568	103	19	exists	exist	VERB
ejpam-6568	103	20	a	a	DET
ejpam-6568	103	21	⋆-open	⋆-open	ADJ
ejpam-6568	103	22	set	set	NOUN
ejpam-6568	103	23	u	u	NOUN
ejpam-6568	103	24	of	of	ADP
ejpam-6568	103	25	x	x	PUNCT
ejpam-6568	103	26	containing	contain	VERB
ejpam-6568	103	27	x	x	PUNCT
ejpam-6568	103	28	such	such	ADJ
ejpam-6568	103	29	that	that	DET
ejpam-6568	103	30	f(u	f(u	PROPN
ejpam-6568	103	31	)	)	PUNCT
ejpam-6568	103	32	⊆	⊆	NUM
ejpam-6568	103	33	σ1σ2	σ1σ2	X
ejpam-6568	103	34	-	-	PUNCT
ejpam-6568	103	35	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	103	36	-	-	PUNCT
ejpam-6568	103	37	cl(v	cl(v	NOUN
ejpam-6568	103	38	)	)	PUNCT
ejpam-6568	103	39	)	)	PUNCT
ejpam-6568	103	40	.	.	PUNCT
ejpam-6568	104	1	a	a	DET
ejpam-6568	104	2	function	function	NOUN
ejpam-6568	104	3	f	f	NOUN
ejpam-6568	104	4	:	:	PUNCT
ejpam-6568	104	5	(	(	PUNCT
ejpam-6568	104	6	x	x	X
ejpam-6568	104	7	,	,	PUNCT
ejpam-6568	104	8	τ	τ	PROPN
ejpam-6568	104	9	,	,	PUNCT
ejpam-6568	104	10	i	i	NOUN
ejpam-6568	104	11	)	)	PUNCT
ejpam-6568	104	12	→	→	PUNCT
ejpam-6568	104	13	(	(	PUNCT
ejpam-6568	104	14	y	y	PROPN
ejpam-6568	104	15	,	,	PUNCT
ejpam-6568	104	16	σ1	σ1	PROPN
ejpam-6568	104	17	,	,	PUNCT
ejpam-6568	104	18	σ2	σ2	PROPN
ejpam-6568	104	19	)	)	PUNCT
ejpam-6568	104	20	is	be	AUX
ejpam-6568	104	21	said	say	VERB
ejpam-6568	104	22	to	to	PART
ejpam-6568	104	23	be	be	AUX
ejpam-6568	104	24	almost	almost	ADV
ejpam-6568	104	25	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	104	26	,	,	PUNCT
ejpam-6568	104	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	104	28	if	if	SCONJ
ejpam-6568	104	29	f	f	PROPN
ejpam-6568	104	30	is	be	AUX
ejpam-6568	104	31	almost	almost	ADV
ejpam-6568	104	32	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	104	33	,	,	PUNCT
ejpam-6568	104	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	104	35	at	at	ADP
ejpam-6568	104	36	each	each	DET
ejpam-6568	104	37	point	point	NOUN
ejpam-6568	104	38	x	x	PUNCT
ejpam-6568	104	39	of	of	ADP
ejpam-6568	104	40	x.	x.	PROPN
ejpam-6568	104	41	lemma	lemma	PROPN
ejpam-6568	105	1	2	2	X
ejpam-6568	105	2	.	.	PUNCT
ejpam-6568	106	1	[	[	X
ejpam-6568	106	2	31	31	NUM
ejpam-6568	106	3	]	]	PUNCT
ejpam-6568	106	4	let	let	VERB
ejpam-6568	106	5	a	a	PRON
ejpam-6568	106	6	be	be	AUX
ejpam-6568	106	7	a	a	DET
ejpam-6568	106	8	subset	subset	NOUN
ejpam-6568	106	9	of	of	ADP
ejpam-6568	106	10	a	a	DET
ejpam-6568	106	11	bitopological	bitopological	ADJ
ejpam-6568	106	12	space	space	NOUN
ejpam-6568	106	13	(	(	PUNCT
ejpam-6568	106	14	x	x	NOUN
ejpam-6568	106	15	,	,	PUNCT
ejpam-6568	106	16	τ1	τ1	NOUN
ejpam-6568	106	17	,	,	PUNCT
ejpam-6568	106	18	τ2	τ2	NOUN
ejpam-6568	106	19	)	)	PUNCT
ejpam-6568	106	20	.	.	PUNCT
ejpam-6568	107	1	if	if	SCONJ
ejpam-6568	107	2	a	a	PRON
ejpam-6568	107	3	is	be	AUX
ejpam-6568	107	4	τ1τ2	τ1τ2	NOUN
ejpam-6568	107	5	-	-	ADJ
ejpam-6568	107	6	open	open	ADJ
ejpam-6568	107	7	in	in	ADP
ejpam-6568	107	8	x	x	NOUN
ejpam-6568	107	9	,	,	PUNCT
ejpam-6568	107	10	then	then	ADV
ejpam-6568	107	11	(	(	PUNCT
ejpam-6568	107	12	τ1	τ1	NOUN
ejpam-6568	107	13	,	,	PUNCT
ejpam-6568	107	14	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6568	107	15	)	)	PUNCT
ejpam-6568	107	16	=	=	PUNCT
ejpam-6568	108	1	τ1τ2	τ1τ2	NOUN
ejpam-6568	108	2	-	-	NOUN
ejpam-6568	108	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6568	108	4	-	-	PUNCT
ejpam-6568	108	5	cl(a	cl(a	NUM
ejpam-6568	108	6	)	)	PUNCT
ejpam-6568	108	7	)	)	PUNCT
ejpam-6568	108	8	.	.	PUNCT
ejpam-6568	109	1	theorem	theorem	NOUN
ejpam-6568	109	2	1	1	NUM
ejpam-6568	109	3	.	.	X
ejpam-6568	109	4	for	for	ADP
ejpam-6568	109	5	a	a	DET
ejpam-6568	109	6	function	function	NOUN
ejpam-6568	109	7	f	f	NOUN
ejpam-6568	109	8	:	:	PUNCT
ejpam-6568	109	9	(	(	PUNCT
ejpam-6568	109	10	x	x	X
ejpam-6568	109	11	,	,	PUNCT
ejpam-6568	109	12	τ	τ	PROPN
ejpam-6568	109	13	,	,	PUNCT
ejpam-6568	109	14	i	i	NOUN
ejpam-6568	109	15	)	)	PUNCT
ejpam-6568	109	16	→	→	PUNCT
ejpam-6568	109	17	(	(	PUNCT
ejpam-6568	109	18	y	y	PROPN
ejpam-6568	109	19	,	,	PUNCT
ejpam-6568	109	20	σ1	σ1	PROPN
ejpam-6568	109	21	,	,	PUNCT
ejpam-6568	109	22	σ2	σ2	NOUN
ejpam-6568	109	23	)	)	PUNCT
ejpam-6568	109	24	,	,	PUNCT
ejpam-6568	109	25	the	the	DET
ejpam-6568	109	26	following	follow	VERB
ejpam-6568	109	27	properties	property	NOUN
ejpam-6568	109	28	are	be	AUX
ejpam-6568	109	29	equivalent	equivalent	ADJ
ejpam-6568	109	30	:	:	PUNCT
ejpam-6568	109	31	(	(	PUNCT
ejpam-6568	109	32	1	1	X
ejpam-6568	109	33	)	)	PUNCT
ejpam-6568	109	34	f	f	NOUN
ejpam-6568	109	35	is	be	AUX
ejpam-6568	109	36	almost	almost	ADV
ejpam-6568	109	37	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	109	38	,	,	PUNCT
ejpam-6568	109	39	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	109	40	at	at	ADP
ejpam-6568	109	41	x	x	X
ejpam-6568	109	42	∈	∈	PROPN
ejpam-6568	109	43	x	x	X
ejpam-6568	109	44	;	;	PUNCT
ejpam-6568	109	45	(	(	PUNCT
ejpam-6568	109	46	2	2	X
ejpam-6568	109	47	)	)	PUNCT
ejpam-6568	109	48	x	x	SYM
ejpam-6568	109	49	∈	∈	NOUN
ejpam-6568	109	50	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	ADV
ejpam-6568	109	51	-	-	PUNCT
ejpam-6568	109	52	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	109	53	-	-	PUNCT
ejpam-6568	109	54	cl(v	cl(v	NOUN
ejpam-6568	109	55	)	)	PUNCT
ejpam-6568	109	56	)	)	PUNCT
ejpam-6568	109	57	)	)	PUNCT
ejpam-6568	109	58	)	)	PUNCT
ejpam-6568	109	59	for	for	ADP
ejpam-6568	109	60	every	every	DET
ejpam-6568	109	61	σ1σ2	σ1σ2	NOUN
ejpam-6568	109	62	-	-	ADJ
ejpam-6568	109	63	open	open	ADJ
ejpam-6568	109	64	set	set	NOUN
ejpam-6568	109	65	v	v	NOUN
ejpam-6568	109	66	of	of	ADP
ejpam-6568	109	67	y	y	NOUN
ejpam-6568	109	68	containing	contain	VERB
ejpam-6568	109	69	f(x	f(x	PROPN
ejpam-6568	109	70	)	)	PUNCT
ejpam-6568	109	71	;	;	PUNCT
ejpam-6568	109	72	(	(	PUNCT
ejpam-6568	109	73	3	3	X
ejpam-6568	109	74	)	)	PUNCT
ejpam-6568	109	75	x	x	SYM
ejpam-6568	109	76	∈	∈	PROPN
ejpam-6568	109	77	int⋆(f−1((σ1	int⋆(f−1((σ1	NOUN
ejpam-6568	109	78	,	,	PUNCT
ejpam-6568	109	79	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6568	109	80	)	)	PUNCT
ejpam-6568	109	81	)	)	PUNCT
ejpam-6568	109	82	)	)	PUNCT
ejpam-6568	109	83	for	for	ADP
ejpam-6568	109	84	every	every	DET
ejpam-6568	109	85	σ1σ2	σ1σ2	NOUN
ejpam-6568	109	86	-	-	ADJ
ejpam-6568	109	87	open	open	ADJ
ejpam-6568	109	88	set	set	NOUN
ejpam-6568	109	89	v	v	NOUN
ejpam-6568	109	90	of	of	ADP
ejpam-6568	109	91	y	y	NOUN
ejpam-6568	109	92	containing	contain	VERB
ejpam-6568	109	93	f(x	f(x	PROPN
ejpam-6568	109	94	)	)	PUNCT
ejpam-6568	109	95	;	;	PUNCT
ejpam-6568	109	96	(	(	PUNCT
ejpam-6568	109	97	4	4	X
ejpam-6568	109	98	)	)	PUNCT
ejpam-6568	109	99	x	x	SYM
ejpam-6568	109	100	∈	∈	PROPN
ejpam-6568	109	101	int⋆(f−1(v	int⋆(f−1(v	NUM
ejpam-6568	109	102	)	)	PUNCT
ejpam-6568	109	103	)	)	PUNCT
ejpam-6568	109	104	for	for	ADP
ejpam-6568	109	105	every	every	DET
ejpam-6568	109	106	(	(	PUNCT
ejpam-6568	109	107	σ1	σ1	PROPN
ejpam-6568	109	108	,	,	PUNCT
ejpam-6568	109	109	σ2)r	σ2)r	NOUN
ejpam-6568	109	110	-	-	PUNCT
ejpam-6568	109	111	open	open	ADJ
ejpam-6568	109	112	set	set	VERB
ejpam-6568	109	113	v	v	NOUN
ejpam-6568	109	114	of	of	ADP
ejpam-6568	109	115	y	y	NOUN
ejpam-6568	109	116	containing	contain	VERB
ejpam-6568	109	117	f(x	f(x	PROPN
ejpam-6568	109	118	)	)	PUNCT
ejpam-6568	109	119	;	;	PUNCT
ejpam-6568	109	120	(	(	PUNCT
ejpam-6568	109	121	5	5	X
ejpam-6568	109	122	)	)	PUNCT
ejpam-6568	109	123	for	for	ADP
ejpam-6568	109	124	each	each	DET
ejpam-6568	109	125	(	(	PUNCT
ejpam-6568	109	126	σ1	σ1	PROPN
ejpam-6568	109	127	,	,	PUNCT
ejpam-6568	109	128	σ2)r	σ2)r	NOUN
ejpam-6568	109	129	-	-	PUNCT
ejpam-6568	109	130	open	open	ADJ
ejpam-6568	109	131	set	set	VERB
ejpam-6568	109	132	v	v	NOUN
ejpam-6568	109	133	of	of	ADP
ejpam-6568	109	134	y	y	NOUN
ejpam-6568	109	135	containing	contain	VERB
ejpam-6568	109	136	f(x	f(x	PROPN
ejpam-6568	109	137	)	)	PUNCT
ejpam-6568	109	138	,	,	PUNCT
ejpam-6568	109	139	there	there	PRON
ejpam-6568	109	140	exists	exist	VERB
ejpam-6568	109	141	a	a	DET
ejpam-6568	109	142	⋆-open	⋆-open	ADJ
ejpam-6568	109	143	set	set	NOUN
ejpam-6568	109	144	u	u	NOUN
ejpam-6568	109	145	of	of	ADP
ejpam-6568	109	146	x	x	PUNCT
ejpam-6568	109	147	containing	contain	VERB
ejpam-6568	109	148	x	x	PUNCT
ejpam-6568	109	149	such	such	ADJ
ejpam-6568	109	150	that	that	DET
ejpam-6568	109	151	f(u	f(u	PROPN
ejpam-6568	109	152	)	)	PUNCT
ejpam-6568	109	153	⊆	⊆	NUM
ejpam-6568	109	154	v	v	NOUN
ejpam-6568	109	155	.	.	PUNCT
ejpam-6568	110	1	proof	proof	NOUN
ejpam-6568	110	2	.	.	PUNCT
ejpam-6568	111	1	(	(	PUNCT
ejpam-6568	111	2	1	1	X
ejpam-6568	111	3	)	)	PUNCT
ejpam-6568	111	4	⇒	⇒	NOUN
ejpam-6568	111	5	(	(	PUNCT
ejpam-6568	111	6	2	2	NUM
ejpam-6568	111	7	):	):	PUNCT
ejpam-6568	111	8	let	let	VERB
ejpam-6568	111	9	v	v	PART
ejpam-6568	111	10	be	be	AUX
ejpam-6568	111	11	any	any	DET
ejpam-6568	111	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	111	13	-	-	ADJ
ejpam-6568	111	14	open	open	ADJ
ejpam-6568	111	15	set	set	NOUN
ejpam-6568	111	16	of	of	ADP
ejpam-6568	111	17	y	y	PROPN
ejpam-6568	111	18	containing	contain	VERB
ejpam-6568	111	19	f(x	f(x	PROPN
ejpam-6568	111	20	)	)	PUNCT
ejpam-6568	111	21	.	.	PUNCT
ejpam-6568	112	1	thus	thus	ADV
ejpam-6568	112	2	by	by	ADP
ejpam-6568	112	3	(	(	PUNCT
ejpam-6568	112	4	1	1	NUM
ejpam-6568	112	5	)	)	PUNCT
ejpam-6568	112	6	,	,	PUNCT
ejpam-6568	112	7	there	there	PRON
ejpam-6568	112	8	exists	exist	VERB
ejpam-6568	112	9	a	a	DET
ejpam-6568	112	10	⋆-open	⋆-open	ADJ
ejpam-6568	112	11	set	set	VERB
ejpam-6568	112	12	u	u	PROPN
ejpam-6568	112	13	ofx	ofx	NOUN
ejpam-6568	112	14	containing	contain	VERB
ejpam-6568	112	15	x	x	PUNCT
ejpam-6568	112	16	such	such	ADJ
ejpam-6568	112	17	that	that	DET
ejpam-6568	112	18	f(u	f(u	PROPN
ejpam-6568	112	19	)	)	PUNCT
ejpam-6568	112	20	⊆	⊆	NUM
ejpam-6568	112	21	σ1σ2	σ1σ2	X
ejpam-6568	112	22	-	-	PUNCT
ejpam-6568	112	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	112	24	-	-	PUNCT
ejpam-6568	112	25	cl(v	cl(v	NOUN
ejpam-6568	112	26	)	)	PUNCT
ejpam-6568	112	27	)	)	PUNCT
ejpam-6568	112	28	.	.	PUNCT
ejpam-6568	113	1	therefore	therefore	ADV
ejpam-6568	113	2	,	,	PUNCT
ejpam-6568	113	3	x	x	PUNCT
ejpam-6568	113	4	∈	∈	PROPN
ejpam-6568	113	5	u	u	NOUN
ejpam-6568	113	6	⊆	⊆	NUM
ejpam-6568	113	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	113	8	-	-	PUNCT
ejpam-6568	113	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	113	10	-	-	PUNCT
ejpam-6568	113	11	cl(v	cl(v	NOUN
ejpam-6568	113	12	)	)	PUNCT
ejpam-6568	113	13	)	)	PUNCT
ejpam-6568	113	14	)	)	PUNCT
ejpam-6568	113	15	and	and	CCONJ
ejpam-6568	113	16	hence	hence	ADV
ejpam-6568	113	17	x	x	X
ejpam-6568	113	18	∈	∈	NOUN
ejpam-6568	113	19	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	ADV
ejpam-6568	113	20	-	-	PUNCT
ejpam-6568	113	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	113	22	-	-	PUNCT
ejpam-6568	113	23	cl(v	cl(v	NOUN
ejpam-6568	113	24	)	)	PUNCT
ejpam-6568	113	25	)	)	PUNCT
ejpam-6568	113	26	)	)	PUNCT
ejpam-6568	113	27	)	)	PUNCT
ejpam-6568	113	28	.	.	PUNCT
ejpam-6568	114	1	(	(	PUNCT
ejpam-6568	114	2	2	2	X
ejpam-6568	114	3	)	)	PUNCT
ejpam-6568	114	4	⇒	⇒	NOUN
ejpam-6568	114	5	(	(	PUNCT
ejpam-6568	114	6	3	3	NUM
ejpam-6568	114	7	):	):	PUNCT
ejpam-6568	114	8	this	this	PRON
ejpam-6568	114	9	follows	follow	VERB
ejpam-6568	114	10	from	from	ADP
ejpam-6568	114	11	lemma	lemma	PROPN
ejpam-6568	114	12	2	2	NUM
ejpam-6568	114	13	.	.	PUNCT
ejpam-6568	114	14	(	(	PUNCT
ejpam-6568	114	15	3	3	X
ejpam-6568	114	16	)	)	PUNCT
ejpam-6568	114	17	⇒	⇒	NOUN
ejpam-6568	114	18	(	(	PUNCT
ejpam-6568	114	19	4	4	NUM
ejpam-6568	114	20	):	):	PUNCT
ejpam-6568	114	21	let	let	VERB
ejpam-6568	114	22	v	v	PART
ejpam-6568	114	23	be	be	AUX
ejpam-6568	114	24	any	any	DET
ejpam-6568	114	25	(	(	PUNCT
ejpam-6568	114	26	σ1	σ1	NOUN
ejpam-6568	114	27	,	,	PUNCT
ejpam-6568	114	28	σ2)r	σ2)r	NOUN
ejpam-6568	114	29	-	-	PUNCT
ejpam-6568	114	30	open	open	ADJ
ejpam-6568	114	31	set	set	NOUN
ejpam-6568	114	32	of	of	ADP
ejpam-6568	114	33	y	y	PROPN
ejpam-6568	114	34	containing	contain	VERB
ejpam-6568	114	35	f(x	f(x	PROPN
ejpam-6568	114	36	)	)	PUNCT
ejpam-6568	114	37	.	.	PUNCT
ejpam-6568	115	1	it	it	PRON
ejpam-6568	115	2	follows	follow	VERB
ejpam-6568	115	3	from	from	ADP
ejpam-6568	115	4	lemma	lemma	PROPN
ejpam-6568	115	5	2	2	NUM
ejpam-6568	115	6	that	that	PRON
ejpam-6568	115	7	v	v	NOUN
ejpam-6568	115	8	=	=	SYM
ejpam-6568	115	9	σ1σ2	σ1σ2	NOUN
ejpam-6568	115	10	-	-	PUNCT
ejpam-6568	115	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	115	12	-	-	PUNCT
ejpam-6568	115	13	cl(v	cl(v	NOUN
ejpam-6568	115	14	)	)	PUNCT
ejpam-6568	115	15	)	)	PUNCT
ejpam-6568	116	1	=	=	SYM
ejpam-6568	116	2	(	(	PUNCT
ejpam-6568	116	3	σ1	σ1	PROPN
ejpam-6568	116	4	,	,	PUNCT
ejpam-6568	116	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6568	116	6	)	)	PUNCT
ejpam-6568	116	7	.	.	PUNCT
ejpam-6568	117	1	(	(	PUNCT
ejpam-6568	117	2	4	4	X
ejpam-6568	117	3	)	)	PUNCT
ejpam-6568	117	4	⇒	⇒	NOUN
ejpam-6568	117	5	(	(	PUNCT
ejpam-6568	117	6	5	5	NUM
ejpam-6568	117	7	):	):	PUNCT
ejpam-6568	117	8	let	let	VERB
ejpam-6568	117	9	v	v	PART
ejpam-6568	117	10	be	be	AUX
ejpam-6568	117	11	any	any	DET
ejpam-6568	117	12	(	(	PUNCT
ejpam-6568	117	13	σ1	σ1	NOUN
ejpam-6568	117	14	,	,	PUNCT
ejpam-6568	117	15	σ2)r	σ2)r	NOUN
ejpam-6568	117	16	-	-	PUNCT
ejpam-6568	117	17	open	open	ADJ
ejpam-6568	117	18	set	set	NOUN
ejpam-6568	117	19	of	of	ADP
ejpam-6568	117	20	y	y	PROPN
ejpam-6568	117	21	containing	contain	VERB
ejpam-6568	117	22	f(x	f(x	PROPN
ejpam-6568	117	23	)	)	PUNCT
ejpam-6568	117	24	.	.	PUNCT
ejpam-6568	118	1	then	then	ADV
ejpam-6568	118	2	by	by	ADP
ejpam-6568	118	3	(	(	PUNCT
ejpam-6568	118	4	4	4	NUM
ejpam-6568	118	5	)	)	PUNCT
ejpam-6568	118	6	,	,	PUNCT
ejpam-6568	118	7	we	we	PRON
ejpam-6568	118	8	have	have	VERB
ejpam-6568	118	9	x	x	X
ejpam-6568	118	10	∈	∈	PROPN
ejpam-6568	118	11	int⋆(f−1(v	int⋆(f−1(v	NUM
ejpam-6568	118	12	)	)	PUNCT
ejpam-6568	118	13	)	)	PUNCT
ejpam-6568	118	14	and	and	CCONJ
ejpam-6568	118	15	there	there	PRON
ejpam-6568	118	16	exists	exist	VERB
ejpam-6568	118	17	a	a	DET
ejpam-6568	118	18	⋆-open	⋆-open	ADJ
ejpam-6568	118	19	set	set	NOUN
ejpam-6568	118	20	u	u	NOUN
ejpam-6568	118	21	of	of	ADP
ejpam-6568	118	22	x	x	PUNCT
ejpam-6568	118	23	containing	contain	VERB
ejpam-6568	118	24	x	x	PUNCT
ejpam-6568	118	25	such	such	ADJ
ejpam-6568	118	26	that	that	SCONJ
ejpam-6568	118	27	u	u	PROPN
ejpam-6568	118	28	⊆	⊆	NUM
ejpam-6568	118	29	f−1(v	f−1(v	NOUN
ejpam-6568	118	30	)	)	PUNCT
ejpam-6568	118	31	;	;	PUNCT
ejpam-6568	118	32	hence	hence	ADV
ejpam-6568	118	33	f(u	f(u	PROPN
ejpam-6568	118	34	)	)	PUNCT
ejpam-6568	118	35	⊆	⊆	NUM
ejpam-6568	118	36	v	v	NOUN
ejpam-6568	118	37	.	.	PUNCT
ejpam-6568	119	1	(	(	PUNCT
ejpam-6568	119	2	5	5	X
ejpam-6568	119	3	)	)	PUNCT
ejpam-6568	119	4	⇒	⇒	NOUN
ejpam-6568	119	5	(	(	PUNCT
ejpam-6568	119	6	1	1	NUM
ejpam-6568	119	7	):	):	PUNCT
ejpam-6568	119	8	let	let	VERB
ejpam-6568	119	9	v	v	PART
ejpam-6568	119	10	be	be	AUX
ejpam-6568	119	11	any	any	DET
ejpam-6568	119	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	119	13	-	-	ADJ
ejpam-6568	119	14	open	open	ADJ
ejpam-6568	119	15	set	set	NOUN
ejpam-6568	119	16	of	of	ADP
ejpam-6568	119	17	y	y	PROPN
ejpam-6568	119	18	containing	contain	VERB
ejpam-6568	119	19	f(x	f(x	PROPN
ejpam-6568	119	20	)	)	PUNCT
ejpam-6568	119	21	.	.	PUNCT
ejpam-6568	120	1	since	since	SCONJ
ejpam-6568	120	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	120	3	-	-	PUNCT
ejpam-6568	120	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	120	5	-	-	PUNCT
ejpam-6568	120	6	cl(v	cl(v	NOUN
ejpam-6568	120	7	)	)	PUNCT
ejpam-6568	120	8	)	)	PUNCT
ejpam-6568	120	9	is	be	AUX
ejpam-6568	120	10	(	(	PUNCT
ejpam-6568	120	11	σ1	σ1	NOUN
ejpam-6568	120	12	,	,	PUNCT
ejpam-6568	120	13	σ2)r	σ2)r	NOUN
ejpam-6568	120	14	-	-	PUNCT
ejpam-6568	120	15	open	open	ADJ
ejpam-6568	120	16	,	,	PUNCT
ejpam-6568	120	17	there	there	PRON
ejpam-6568	120	18	exists	exist	VERB
ejpam-6568	120	19	a	a	DET
ejpam-6568	120	20	⋆-open	⋆-open	ADJ
ejpam-6568	120	21	set	set	NOUN
ejpam-6568	120	22	u	u	NOUN
ejpam-6568	120	23	of	of	ADP
ejpam-6568	120	24	x	x	PUNCT
ejpam-6568	120	25	containing	contain	VERB
ejpam-6568	120	26	x	x	PUNCT
ejpam-6568	120	27	such	such	ADJ
ejpam-6568	120	28	that	that	DET
ejpam-6568	120	29	f(u	f(u	PROPN
ejpam-6568	120	30	)	)	PUNCT
ejpam-6568	120	31	⊆	⊆	NUM
ejpam-6568	120	32	σ1σ2	σ1σ2	X
ejpam-6568	120	33	-	-	PUNCT
ejpam-6568	120	34	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	120	35	-	-	PUNCT
ejpam-6568	120	36	cl(v	cl(v	NOUN
ejpam-6568	120	37	)	)	PUNCT
ejpam-6568	120	38	)	)	PUNCT
ejpam-6568	120	39	.	.	PUNCT
ejpam-6568	121	1	this	this	PRON
ejpam-6568	121	2	shows	show	VERB
ejpam-6568	121	3	that	that	SCONJ
ejpam-6568	121	4	f	f	PROPN
ejpam-6568	121	5	is	be	AUX
ejpam-6568	121	6	almost	almost	ADV
ejpam-6568	121	7	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	121	8	,	,	PUNCT
ejpam-6568	121	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	121	10	at	at	ADP
ejpam-6568	121	11	x.	x.	NOUN
ejpam-6568	121	12	theorem	theorem	NOUN
ejpam-6568	121	13	2	2	NUM
ejpam-6568	121	14	.	.	X
ejpam-6568	121	15	for	for	ADP
ejpam-6568	121	16	a	a	DET
ejpam-6568	121	17	function	function	NOUN
ejpam-6568	121	18	f	f	NOUN
ejpam-6568	121	19	:	:	PUNCT
ejpam-6568	121	20	(	(	PUNCT
ejpam-6568	121	21	x	x	X
ejpam-6568	121	22	,	,	PUNCT
ejpam-6568	121	23	τ	τ	PROPN
ejpam-6568	121	24	,	,	PUNCT
ejpam-6568	121	25	i	i	NOUN
ejpam-6568	121	26	)	)	PUNCT
ejpam-6568	121	27	→	→	PUNCT
ejpam-6568	121	28	(	(	PUNCT
ejpam-6568	121	29	y	y	PROPN
ejpam-6568	121	30	,	,	PUNCT
ejpam-6568	121	31	σ1	σ1	PROPN
ejpam-6568	121	32	,	,	PUNCT
ejpam-6568	121	33	σ2	σ2	NOUN
ejpam-6568	121	34	)	)	PUNCT
ejpam-6568	121	35	,	,	PUNCT
ejpam-6568	121	36	the	the	DET
ejpam-6568	121	37	following	follow	VERB
ejpam-6568	121	38	properties	property	NOUN
ejpam-6568	121	39	are	be	AUX
ejpam-6568	121	40	equivalent	equivalent	ADJ
ejpam-6568	121	41	:	:	PUNCT
ejpam-6568	121	42	(	(	PUNCT
ejpam-6568	121	43	1	1	X
ejpam-6568	121	44	)	)	PUNCT
ejpam-6568	121	45	f	f	NOUN
ejpam-6568	121	46	is	be	AUX
ejpam-6568	121	47	almost	almost	ADV
ejpam-6568	121	48	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	121	49	,	,	PUNCT
ejpam-6568	121	50	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	121	51	;	;	PUNCT
ejpam-6568	121	52	(	(	PUNCT
ejpam-6568	121	53	2	2	X
ejpam-6568	121	54	)	)	PUNCT
ejpam-6568	121	55	f−1(v	f−1(v	NOUN
ejpam-6568	121	56	)	)	PUNCT
ejpam-6568	122	1	⊆	⊆	X
ejpam-6568	122	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	122	3	-	-	PUNCT
ejpam-6568	122	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	122	5	-	-	PUNCT
ejpam-6568	122	6	cl(v	cl(v	NOUN
ejpam-6568	122	7	)	)	PUNCT
ejpam-6568	122	8	)	)	PUNCT
ejpam-6568	122	9	)	)	PUNCT
ejpam-6568	122	10	)	)	PUNCT
ejpam-6568	122	11	for	for	ADP
ejpam-6568	122	12	every	every	DET
ejpam-6568	122	13	σ1σ2	σ1σ2	NOUN
ejpam-6568	122	14	-	-	ADJ
ejpam-6568	122	15	open	open	ADJ
ejpam-6568	122	16	set	set	NOUN
ejpam-6568	122	17	v	v	NOUN
ejpam-6568	122	18	of	of	ADP
ejpam-6568	122	19	y	y	PROPN
ejpam-6568	122	20	;	;	PUNCT
ejpam-6568	122	21	(	(	PUNCT
ejpam-6568	122	22	3	3	X
ejpam-6568	122	23	)	)	PUNCT
ejpam-6568	122	24	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	122	25	-	-	PUNCT
ejpam-6568	122	26	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	122	27	-	-	PUNCT
ejpam-6568	122	28	int(k	int(k	NOUN
ejpam-6568	122	29	)	)	PUNCT
ejpam-6568	122	30	)	)	PUNCT
ejpam-6568	122	31	)	)	PUNCT
ejpam-6568	122	32	)	)	PUNCT
ejpam-6568	122	33	⊆	⊆	NUM
ejpam-6568	122	34	f−1(k	f−1(k	PROPN
ejpam-6568	122	35	)	)	PUNCT
ejpam-6568	122	36	for	for	ADP
ejpam-6568	122	37	every	every	DET
ejpam-6568	122	38	σ1σ2	σ1σ2	NUM
ejpam-6568	122	39	-	-	PUNCT
ejpam-6568	122	40	closed	closed	ADJ
ejpam-6568	122	41	set	set	NOUN
ejpam-6568	122	42	k	k	PROPN
ejpam-6568	122	43	of	of	ADP
ejpam-6568	122	44	y	y	PROPN
ejpam-6568	122	45	;	;	PUNCT
ejpam-6568	122	46	n.	n.	PROPN
ejpam-6568	122	47	viriyapong	viriyapong	PROPN
ejpam-6568	122	48	,	,	PUNCT
ejpam-6568	122	49	a.	a.	PROPN
ejpam-6568	122	50	sama	sama	PROPN
ejpam-6568	122	51	-	-	PUNCT
ejpam-6568	122	52	ae	ae	PROPN
ejpam-6568	122	53	,	,	PUNCT
ejpam-6568	122	54	c.	c.	PROPN
ejpam-6568	122	55	boonpok	boonpok	PROPN
ejpam-6568	122	56	/	/	SYM
ejpam-6568	122	57	eur	eur	PROPN
ejpam-6568	122	58	.	.	PUNCT
ejpam-6568	123	1	j.	j.	PROPN
ejpam-6568	123	2	pure	pure	PROPN
ejpam-6568	123	3	appl	appl	PROPN
ejpam-6568	123	4	.	.	PROPN
ejpam-6568	123	5	math	math	PROPN
ejpam-6568	123	6	,	,	PUNCT
ejpam-6568	123	7	18	18	NUM
ejpam-6568	123	8	(	(	PUNCT
ejpam-6568	123	9	3	3	NUM
ejpam-6568	123	10	)	)	PUNCT
ejpam-6568	123	11	(	(	PUNCT
ejpam-6568	123	12	2025	2025	NUM
ejpam-6568	123	13	)	)	PUNCT
ejpam-6568	123	14	,	,	PUNCT
ejpam-6568	123	15	6568	6568	NUM
ejpam-6568	123	16	5	5	NUM
ejpam-6568	123	17	of	of	ADP
ejpam-6568	123	18	18	18	NUM
ejpam-6568	123	19	(	(	PUNCT
ejpam-6568	123	20	4	4	NUM
ejpam-6568	123	21	)	)	PUNCT
ejpam-6568	123	22	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	123	23	-	-	PUNCT
ejpam-6568	123	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	123	25	-	-	PUNCT
ejpam-6568	123	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	123	27	-	-	PUNCT
ejpam-6568	123	28	cl(b	cl(b	NOUN
ejpam-6568	123	29	)	)	PUNCT
ejpam-6568	123	30	)	)	PUNCT
ejpam-6568	123	31	)	)	PUNCT
ejpam-6568	123	32	)	)	PUNCT
ejpam-6568	123	33	)	)	PUNCT
ejpam-6568	124	1	⊆	⊆	NUM
ejpam-6568	124	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	124	3	-	-	PUNCT
ejpam-6568	124	4	cl(b	cl(b	NOUN
ejpam-6568	124	5	)	)	PUNCT
ejpam-6568	124	6	)	)	PUNCT
ejpam-6568	124	7	for	for	ADP
ejpam-6568	124	8	every	every	DET
ejpam-6568	124	9	subset	subset	NOUN
ejpam-6568	124	10	b	b	PROPN
ejpam-6568	124	11	of	of	ADP
ejpam-6568	124	12	y	y	PROPN
ejpam-6568	124	13	;	;	PUNCT
ejpam-6568	124	14	(	(	PUNCT
ejpam-6568	124	15	5	5	X
ejpam-6568	124	16	)	)	PUNCT
ejpam-6568	124	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	124	18	-	-	PUNCT
ejpam-6568	124	19	int(b	int(b	NOUN
ejpam-6568	124	20	)	)	PUNCT
ejpam-6568	124	21	)	)	PUNCT
ejpam-6568	124	22	⊆	⊆	X
ejpam-6568	124	23	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	124	24	-	-	PUNCT
ejpam-6568	124	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	124	26	-	-	PUNCT
ejpam-6568	124	27	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	124	28	-	-	PUNCT
ejpam-6568	124	29	int(b	int(b	NOUN
ejpam-6568	124	30	)	)	PUNCT
ejpam-6568	124	31	)	)	PUNCT
ejpam-6568	124	32	)	)	PUNCT
ejpam-6568	124	33	)	)	PUNCT
ejpam-6568	124	34	)	)	PUNCT
ejpam-6568	124	35	for	for	ADP
ejpam-6568	124	36	every	every	DET
ejpam-6568	124	37	subset	subset	NOUN
ejpam-6568	124	38	b	b	PROPN
ejpam-6568	124	39	of	of	ADP
ejpam-6568	124	40	y	y	PROPN
ejpam-6568	124	41	;	;	PUNCT
ejpam-6568	124	42	(	(	PUNCT
ejpam-6568	124	43	6	6	X
ejpam-6568	124	44	)	)	PUNCT
ejpam-6568	124	45	f−1(v	f−1(v	NOUN
ejpam-6568	124	46	)	)	PUNCT
ejpam-6568	124	47	is	be	AUX
ejpam-6568	124	48	⋆-open	⋆-open	ADJ
ejpam-6568	124	49	in	in	ADP
ejpam-6568	124	50	x	x	PUNCT
ejpam-6568	124	51	for	for	ADP
ejpam-6568	124	52	every	every	DET
ejpam-6568	124	53	(	(	PUNCT
ejpam-6568	124	54	σ1	σ1	PROPN
ejpam-6568	124	55	,	,	PUNCT
ejpam-6568	124	56	σ2)r	σ2)r	NOUN
ejpam-6568	124	57	-	-	PUNCT
ejpam-6568	124	58	open	open	ADJ
ejpam-6568	124	59	set	set	VERB
ejpam-6568	124	60	v	v	NOUN
ejpam-6568	124	61	of	of	ADP
ejpam-6568	124	62	y	y	PROPN
ejpam-6568	124	63	;	;	PUNCT
ejpam-6568	124	64	(	(	PUNCT
ejpam-6568	124	65	7	7	X
ejpam-6568	124	66	)	)	PUNCT
ejpam-6568	124	67	f−1(k	f−1(k	PROPN
ejpam-6568	124	68	)	)	PUNCT
ejpam-6568	124	69	is	be	AUX
ejpam-6568	124	70	⋆-closed	⋆-close	VERB
ejpam-6568	124	71	in	in	ADP
ejpam-6568	124	72	x	x	PUNCT
ejpam-6568	124	73	for	for	ADP
ejpam-6568	124	74	every	every	DET
ejpam-6568	124	75	(	(	PUNCT
ejpam-6568	124	76	σ1	σ1	PROPN
ejpam-6568	124	77	,	,	PUNCT
ejpam-6568	124	78	σ2)r	σ2)r	NOUN
ejpam-6568	124	79	-	-	PUNCT
ejpam-6568	124	80	closed	close	VERB
ejpam-6568	124	81	set	set	ADJ
ejpam-6568	124	82	k	k	PROPN
ejpam-6568	124	83	of	of	ADP
ejpam-6568	124	84	y	y	PROPN
ejpam-6568	124	85	.	.	PUNCT
ejpam-6568	125	1	proof	proof	NOUN
ejpam-6568	125	2	.	.	PUNCT
ejpam-6568	126	1	(	(	PUNCT
ejpam-6568	126	2	1	1	X
ejpam-6568	126	3	)	)	PUNCT
ejpam-6568	126	4	⇒	⇒	NOUN
ejpam-6568	126	5	(	(	PUNCT
ejpam-6568	126	6	2	2	NUM
ejpam-6568	126	7	):	):	PUNCT
ejpam-6568	126	8	let	let	VERB
ejpam-6568	126	9	v	v	PART
ejpam-6568	126	10	be	be	AUX
ejpam-6568	126	11	any	any	DET
ejpam-6568	126	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	126	13	-	-	ADJ
ejpam-6568	126	14	open	open	ADJ
ejpam-6568	126	15	set	set	NOUN
ejpam-6568	126	16	of	of	ADP
ejpam-6568	126	17	y	y	PROPN
ejpam-6568	126	18	and	and	CCONJ
ejpam-6568	126	19	x	x	PROPN
ejpam-6568	126	20	∈	∈	PROPN
ejpam-6568	126	21	f−1(v	f−1(v	NOUN
ejpam-6568	126	22	)	)	PUNCT
ejpam-6568	126	23	.	.	PUNCT
ejpam-6568	127	1	then	then	ADV
ejpam-6568	127	2	,	,	PUNCT
ejpam-6568	127	3	f(x	f(x	PROPN
ejpam-6568	127	4	)	)	PUNCT
ejpam-6568	127	5	∈	∈	PROPN
ejpam-6568	127	6	v	v	NOUN
ejpam-6568	127	7	.	.	PUNCT
ejpam-6568	128	1	thus	thus	ADV
ejpam-6568	128	2	by	by	ADP
ejpam-6568	128	3	theorem	theorem	NOUN
ejpam-6568	128	4	1	1	NUM
ejpam-6568	128	5	,	,	PUNCT
ejpam-6568	128	6	we	we	PRON
ejpam-6568	128	7	have	have	VERB
ejpam-6568	128	8	x	x	PART
ejpam-6568	128	9	∈	∈	PROPN
ejpam-6568	128	10	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	128	11	-	-	PUNCT
ejpam-6568	128	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	128	13	-	-	PUNCT
ejpam-6568	128	14	cl(v	cl(v	NOUN
ejpam-6568	128	15	)	)	PUNCT
ejpam-6568	128	16	)	)	PUNCT
ejpam-6568	128	17	)	)	PUNCT
ejpam-6568	128	18	)	)	PUNCT
ejpam-6568	128	19	and	and	CCONJ
ejpam-6568	128	20	hence	hence	ADV
ejpam-6568	128	21	f−1(v	f−1(v	NOUN
ejpam-6568	128	22	)	)	PUNCT
ejpam-6568	129	1	⊆	⊆	X
ejpam-6568	129	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	129	3	-	-	PUNCT
ejpam-6568	129	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	129	5	-	-	PUNCT
ejpam-6568	129	6	cl(v	cl(v	NOUN
ejpam-6568	129	7	)	)	PUNCT
ejpam-6568	129	8	)	)	PUNCT
ejpam-6568	129	9	)	)	PUNCT
ejpam-6568	129	10	)	)	PUNCT
ejpam-6568	129	11	.	.	PUNCT
ejpam-6568	130	1	(	(	PUNCT
ejpam-6568	130	2	2	2	X
ejpam-6568	130	3	)	)	PUNCT
ejpam-6568	130	4	⇒	⇒	NOUN
ejpam-6568	130	5	(	(	PUNCT
ejpam-6568	130	6	3	3	NUM
ejpam-6568	130	7	):	):	PUNCT
ejpam-6568	130	8	let	let	VERB
ejpam-6568	130	9	k	k	PRON
ejpam-6568	130	10	be	be	AUX
ejpam-6568	130	11	any	any	DET
ejpam-6568	130	12	σ1σ2	σ1σ2	NUM
ejpam-6568	130	13	-	-	PUNCT
ejpam-6568	130	14	closed	closed	ADJ
ejpam-6568	130	15	set	set	NOUN
ejpam-6568	130	16	of	of	ADP
ejpam-6568	130	17	y	y	PROPN
ejpam-6568	130	18	.	.	PUNCT
ejpam-6568	131	1	then	then	ADV
ejpam-6568	131	2	,	,	PUNCT
ejpam-6568	131	3	y	y	PROPN
ejpam-6568	131	4	−	−	PROPN
ejpam-6568	131	5	k	k	PROPN
ejpam-6568	131	6	is	be	AUX
ejpam-6568	131	7	σ1σ2	σ1σ2	NOUN
ejpam-6568	131	8	-	-	ADJ
ejpam-6568	131	9	open	open	ADJ
ejpam-6568	131	10	in	in	ADP
ejpam-6568	131	11	y	y	PROPN
ejpam-6568	131	12	and	and	CCONJ
ejpam-6568	131	13	by	by	ADP
ejpam-6568	131	14	(	(	PUNCT
ejpam-6568	131	15	2	2	NUM
ejpam-6568	131	16	)	)	PUNCT
ejpam-6568	131	17	,	,	PUNCT
ejpam-6568	131	18	x	x	PUNCT
ejpam-6568	131	19	−	−	PROPN
ejpam-6568	131	20	f−1(k	f−1(k	PROPN
ejpam-6568	131	21	)	)	PUNCT
ejpam-6568	132	1	=	=	SYM
ejpam-6568	132	2	f−1(y	f−1(y	PROPN
ejpam-6568	132	3	−	−	PROPN
ejpam-6568	132	4	k	k	NOUN
ejpam-6568	132	5	)	)	PUNCT
ejpam-6568	132	6	⊆	⊆	NUM
ejpam-6568	132	7	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	132	8	-	-	PUNCT
ejpam-6568	132	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	132	10	-	-	PUNCT
ejpam-6568	132	11	cl(y	cl(y	NOUN
ejpam-6568	132	12	−	−	PROPN
ejpam-6568	132	13	k	k	NOUN
ejpam-6568	132	14	)	)	PUNCT
ejpam-6568	132	15	)	)	PUNCT
ejpam-6568	132	16	)	)	PUNCT
ejpam-6568	132	17	)	)	PUNCT
ejpam-6568	133	1	=	=	PRON
ejpam-6568	133	2	int⋆(x	int⋆(x	NOUN
ejpam-6568	133	3	−	−	NUM
ejpam-6568	133	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	133	5	-	-	PUNCT
ejpam-6568	133	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	133	7	-	-	PUNCT
ejpam-6568	133	8	int(k	int(k	NOUN
ejpam-6568	133	9	)	)	PUNCT
ejpam-6568	133	10	)	)	PUNCT
ejpam-6568	133	11	)	)	PUNCT
ejpam-6568	133	12	)	)	PUNCT
ejpam-6568	134	1	=	=	PUNCT
ejpam-6568	134	2	x	x	X
ejpam-6568	135	1	−	−	PRON
ejpam-6568	135	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	135	3	-	-	PUNCT
ejpam-6568	135	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	135	5	-	-	PUNCT
ejpam-6568	135	6	int(k	int(k	NOUN
ejpam-6568	135	7	)	)	PUNCT
ejpam-6568	135	8	)	)	PUNCT
ejpam-6568	135	9	)	)	PUNCT
ejpam-6568	135	10	)	)	PUNCT
ejpam-6568	135	11	.	.	PUNCT
ejpam-6568	136	1	thus	thus	ADV
ejpam-6568	136	2	,	,	PUNCT
ejpam-6568	136	3	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	136	4	-	-	PUNCT
ejpam-6568	136	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	136	6	-	-	PUNCT
ejpam-6568	136	7	int(k	int(k	NOUN
ejpam-6568	136	8	)	)	PUNCT
ejpam-6568	136	9	)	)	PUNCT
ejpam-6568	136	10	)	)	PUNCT
ejpam-6568	136	11	)	)	PUNCT
ejpam-6568	136	12	⊆	⊆	NUM
ejpam-6568	136	13	f−1(k	f−1(k	NOUN
ejpam-6568	136	14	)	)	PUNCT
ejpam-6568	136	15	.	.	PUNCT
ejpam-6568	137	1	(	(	PUNCT
ejpam-6568	137	2	3	3	X
ejpam-6568	137	3	)	)	PUNCT
ejpam-6568	137	4	⇒	⇒	NOUN
ejpam-6568	137	5	(	(	PUNCT
ejpam-6568	137	6	4	4	NUM
ejpam-6568	137	7	):	):	PUNCT
ejpam-6568	137	8	let	let	VERB
ejpam-6568	137	9	b	b	X
ejpam-6568	137	10	be	be	AUX
ejpam-6568	137	11	any	any	DET
ejpam-6568	137	12	subset	subset	NOUN
ejpam-6568	137	13	of	of	ADP
ejpam-6568	137	14	y	y	PROPN
ejpam-6568	137	15	.	.	PUNCT
ejpam-6568	138	1	then	then	ADV
ejpam-6568	138	2	,	,	PUNCT
ejpam-6568	138	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	138	4	-	-	NOUN
ejpam-6568	138	5	cl(b	cl(b	NOUN
ejpam-6568	138	6	)	)	PUNCT
ejpam-6568	138	7	is	be	AUX
ejpam-6568	138	8	a	a	DET
ejpam-6568	138	9	σ1σ2	σ1σ2	NUM
ejpam-6568	138	10	-	-	PUNCT
ejpam-6568	138	11	closed	closed	ADJ
ejpam-6568	138	12	set	set	NOUN
ejpam-6568	138	13	of	of	ADP
ejpam-6568	138	14	y	y	PROPN
ejpam-6568	138	15	and	and	CCONJ
ejpam-6568	138	16	by	by	ADP
ejpam-6568	138	17	(	(	PUNCT
ejpam-6568	138	18	3	3	NUM
ejpam-6568	138	19	)	)	PUNCT
ejpam-6568	138	20	,	,	PUNCT
ejpam-6568	138	21	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	138	22	-	-	PUNCT
ejpam-6568	138	23	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	138	24	-	-	PUNCT
ejpam-6568	138	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	138	26	-	-	PUNCT
ejpam-6568	138	27	cl(b	cl(b	NOUN
ejpam-6568	138	28	)	)	PUNCT
ejpam-6568	138	29	)	)	PUNCT
ejpam-6568	138	30	)	)	PUNCT
ejpam-6568	138	31	)	)	PUNCT
ejpam-6568	138	32	)	)	PUNCT
ejpam-6568	139	1	⊆	⊆	NUM
ejpam-6568	139	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	139	3	-	-	PUNCT
ejpam-6568	139	4	cl(b	cl(b	NOUN
ejpam-6568	139	5	)	)	PUNCT
ejpam-6568	139	6	)	)	PUNCT
ejpam-6568	139	7	.	.	PUNCT
ejpam-6568	140	1	(	(	PUNCT
ejpam-6568	140	2	4	4	X
ejpam-6568	140	3	)	)	PUNCT
ejpam-6568	140	4	⇒	⇒	NOUN
ejpam-6568	140	5	(	(	PUNCT
ejpam-6568	140	6	5	5	NUM
ejpam-6568	140	7	):	):	PUNCT
ejpam-6568	140	8	let	let	VERB
ejpam-6568	140	9	b	b	X
ejpam-6568	140	10	be	be	AUX
ejpam-6568	140	11	any	any	DET
ejpam-6568	140	12	subset	subset	NOUN
ejpam-6568	140	13	of	of	ADP
ejpam-6568	140	14	y	y	PROPN
ejpam-6568	140	15	.	.	PUNCT
ejpam-6568	141	1	then	then	ADV
ejpam-6568	141	2	by	by	ADP
ejpam-6568	141	3	(	(	PUNCT
ejpam-6568	141	4	4	4	NUM
ejpam-6568	141	5	)	)	PUNCT
ejpam-6568	141	6	,	,	PUNCT
ejpam-6568	141	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	141	8	-	-	PUNCT
ejpam-6568	141	9	int(b	int(b	NOUN
ejpam-6568	141	10	)	)	PUNCT
ejpam-6568	141	11	)	)	PUNCT
ejpam-6568	142	1	=	=	PUNCT
ejpam-6568	143	1	x	x	X
ejpam-6568	143	2	−	−	PRON
ejpam-6568	143	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	143	4	-	-	PUNCT
ejpam-6568	143	5	cl(y	cl(y	NOUN
ejpam-6568	143	6	−b	−b	NOUN
ejpam-6568	143	7	)	)	PUNCT
ejpam-6568	143	8	)	)	PUNCT
ejpam-6568	144	1	⊆	⊆	NUM
ejpam-6568	144	2	x	x	X
ejpam-6568	144	3	−	−	PRON
ejpam-6568	144	4	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	144	5	-	-	PUNCT
ejpam-6568	144	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	144	7	-	-	PUNCT
ejpam-6568	144	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	144	9	-	-	PUNCT
ejpam-6568	144	10	cl(y	cl(y	NOUN
ejpam-6568	144	11	−b	−b	NOUN
ejpam-6568	144	12	)	)	PUNCT
ejpam-6568	144	13	)	)	PUNCT
ejpam-6568	144	14	)	)	PUNCT
ejpam-6568	144	15	)	)	PUNCT
ejpam-6568	144	16	)	)	PUNCT
ejpam-6568	145	1	=	=	PUNCT
ejpam-6568	146	1	x	x	X
ejpam-6568	146	2	−	−	PROPN
ejpam-6568	146	3	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	146	4	−	−	VERB
ejpam-6568	146	5	σ1σ2	σ1σ2	NUM
ejpam-6568	146	6	-	-	PUNCT
ejpam-6568	146	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-6568	146	8	-	-	PUNCT
ejpam-6568	146	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	146	10	-	-	PUNCT
ejpam-6568	146	11	int(b	int(b	NOUN
ejpam-6568	146	12	)	)	PUNCT
ejpam-6568	146	13	)	)	PUNCT
ejpam-6568	146	14	)	)	PUNCT
ejpam-6568	146	15	)	)	PUNCT
ejpam-6568	146	16	)	)	PUNCT
ejpam-6568	147	1	=	=	SYM
ejpam-6568	147	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	X
ejpam-6568	147	3	-	-	PUNCT
ejpam-6568	147	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-6568	147	5	-	-	PUNCT
ejpam-6568	147	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	147	7	-	-	PUNCT
ejpam-6568	147	8	int(b	int(b	NOUN
ejpam-6568	147	9	)	)	PUNCT
ejpam-6568	147	10	)	)	PUNCT
ejpam-6568	147	11	)	)	PUNCT
ejpam-6568	147	12	)	)	PUNCT
ejpam-6568	147	13	)	)	PUNCT
ejpam-6568	147	14	.	.	PUNCT
ejpam-6568	148	1	(	(	PUNCT
ejpam-6568	148	2	5	5	X
ejpam-6568	148	3	)	)	PUNCT
ejpam-6568	148	4	⇒	⇒	NOUN
ejpam-6568	148	5	(	(	PUNCT
ejpam-6568	148	6	6	6	NUM
ejpam-6568	148	7	):	):	PUNCT
ejpam-6568	148	8	let	let	VERB
ejpam-6568	148	9	v	v	PART
ejpam-6568	148	10	be	be	AUX
ejpam-6568	148	11	any	any	DET
ejpam-6568	148	12	(	(	PUNCT
ejpam-6568	148	13	σ1	σ1	NOUN
ejpam-6568	148	14	,	,	PUNCT
ejpam-6568	148	15	σ2)r	σ2)r	NOUN
ejpam-6568	148	16	-	-	PUNCT
ejpam-6568	148	17	open	open	ADJ
ejpam-6568	148	18	set	set	NOUN
ejpam-6568	148	19	of	of	ADP
ejpam-6568	148	20	y	y	PROPN
ejpam-6568	148	21	.	.	PUNCT
ejpam-6568	149	1	by	by	ADP
ejpam-6568	149	2	(	(	PUNCT
ejpam-6568	149	3	5	5	NUM
ejpam-6568	149	4	)	)	PUNCT
ejpam-6568	149	5	,	,	PUNCT
ejpam-6568	149	6	we	we	PRON
ejpam-6568	149	7	have	have	VERB
ejpam-6568	149	8	f−1(v	f−1(v	NOUN
ejpam-6568	149	9	)	)	PUNCT
ejpam-6568	150	1	⊆	⊆	NUM
ejpam-6568	150	2	int⋆(f−1(v	int⋆(f−1(v	NUM
ejpam-6568	150	3	)	)	PUNCT
ejpam-6568	150	4	)	)	PUNCT
ejpam-6568	150	5	and	and	CCONJ
ejpam-6568	150	6	hence	hence	ADV
ejpam-6568	150	7	f−1(v	f−1(v	PROPN
ejpam-6568	150	8	)	)	PUNCT
ejpam-6568	150	9	is	be	AUX
ejpam-6568	150	10	⋆-open	⋆-open	ADJ
ejpam-6568	150	11	in	in	ADP
ejpam-6568	150	12	x.	x.	PROPN
ejpam-6568	150	13	(	(	PUNCT
ejpam-6568	150	14	6	6	NUM
ejpam-6568	150	15	)	)	PUNCT
ejpam-6568	150	16	⇒	⇒	NOUN
ejpam-6568	150	17	(	(	PUNCT
ejpam-6568	150	18	7	7	NUM
ejpam-6568	150	19	):	):	PUNCT
ejpam-6568	150	20	the	the	DET
ejpam-6568	150	21	proof	proof	NOUN
ejpam-6568	150	22	is	be	AUX
ejpam-6568	150	23	obvious	obvious	ADJ
ejpam-6568	150	24	.	.	PUNCT
ejpam-6568	151	1	(	(	PUNCT
ejpam-6568	151	2	7	7	X
ejpam-6568	151	3	)	)	PUNCT
ejpam-6568	151	4	⇒	⇒	NOUN
ejpam-6568	151	5	(	(	PUNCT
ejpam-6568	151	6	1	1	NUM
ejpam-6568	151	7	):	):	PUNCT
ejpam-6568	151	8	let	let	VERB
ejpam-6568	151	9	x	x	PUNCT
ejpam-6568	151	10	∈	∈	PROPN
ejpam-6568	151	11	x	x	X
ejpam-6568	151	12	and	and	CCONJ
ejpam-6568	151	13	v	v	AUX
ejpam-6568	151	14	be	be	AUX
ejpam-6568	151	15	any	any	DET
ejpam-6568	151	16	(	(	PUNCT
ejpam-6568	151	17	σ1	σ1	NOUN
ejpam-6568	151	18	,	,	PUNCT
ejpam-6568	151	19	σ2)r	σ2)r	NOUN
ejpam-6568	151	20	-	-	PUNCT
ejpam-6568	151	21	open	open	ADJ
ejpam-6568	151	22	set	set	NOUN
ejpam-6568	151	23	of	of	ADP
ejpam-6568	151	24	y	y	PROPN
ejpam-6568	151	25	containing	contain	VERB
ejpam-6568	151	26	f(x	f(x	PROPN
ejpam-6568	151	27	)	)	PUNCT
ejpam-6568	151	28	.	.	PUNCT
ejpam-6568	152	1	since	since	SCONJ
ejpam-6568	152	2	y	y	PROPN
ejpam-6568	152	3	−	−	PROPN
ejpam-6568	152	4	v	v	NOUN
ejpam-6568	152	5	is	be	AUX
ejpam-6568	152	6	(	(	PUNCT
ejpam-6568	152	7	σ1	σ1	NOUN
ejpam-6568	152	8	,	,	PUNCT
ejpam-6568	152	9	σ2)r	σ2)r	NOUN
ejpam-6568	152	10	-	-	PUNCT
ejpam-6568	152	11	closed	closed	ADJ
ejpam-6568	152	12	and	and	CCONJ
ejpam-6568	152	13	by	by	ADP
ejpam-6568	152	14	(	(	PUNCT
ejpam-6568	152	15	7	7	NUM
ejpam-6568	152	16	)	)	PUNCT
ejpam-6568	152	17	,	,	PUNCT
ejpam-6568	152	18	x	x	PUNCT
ejpam-6568	152	19	−	−	PROPN
ejpam-6568	152	20	f−1(v	f−1(v	NOUN
ejpam-6568	152	21	)	)	PUNCT
ejpam-6568	153	1	=	=	PUNCT
ejpam-6568	153	2	f−1(y	f−1(y	PROPN
ejpam-6568	153	3	−	−	PROPN
ejpam-6568	153	4	v	v	NOUN
ejpam-6568	153	5	)	)	PUNCT
ejpam-6568	153	6	is	be	AUX
ejpam-6568	153	7	⋆-closed	⋆-close	VERB
ejpam-6568	153	8	in	in	ADP
ejpam-6568	153	9	x.	x.	PROPN
ejpam-6568	153	10	thus	thus	ADV
ejpam-6568	153	11	,	,	PUNCT
ejpam-6568	153	12	f−1(v	f−1(v	PROPN
ejpam-6568	153	13	)	)	PUNCT
ejpam-6568	153	14	is	be	AUX
ejpam-6568	153	15	⋆-open	⋆-open	ADJ
ejpam-6568	153	16	and	and	CCONJ
ejpam-6568	153	17	hence	hence	ADV
ejpam-6568	153	18	x	x	X
ejpam-6568	153	19	∈	∈	PROPN
ejpam-6568	153	20	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6568	153	21	)	)	PUNCT
ejpam-6568	153	22	)	)	PUNCT
ejpam-6568	153	23	.	.	PUNCT
ejpam-6568	154	1	then	then	ADV
ejpam-6568	154	2	,	,	PUNCT
ejpam-6568	154	3	there	there	PRON
ejpam-6568	154	4	exists	exist	VERB
ejpam-6568	154	5	a	a	DET
ejpam-6568	154	6	⋆-open	⋆-open	ADJ
ejpam-6568	154	7	set	set	NOUN
ejpam-6568	154	8	u	u	NOUN
ejpam-6568	154	9	of	of	ADP
ejpam-6568	154	10	x	x	PUNCT
ejpam-6568	154	11	containing	contain	VERB
ejpam-6568	154	12	x	x	PUNCT
ejpam-6568	154	13	such	such	ADJ
ejpam-6568	154	14	that	that	DET
ejpam-6568	154	15	f(u	f(u	PROPN
ejpam-6568	154	16	)	)	PUNCT
ejpam-6568	154	17	⊆	⊆	NUM
ejpam-6568	154	18	v	v	NOUN
ejpam-6568	154	19	.	.	PUNCT
ejpam-6568	155	1	it	it	PRON
ejpam-6568	155	2	follows	follow	VERB
ejpam-6568	155	3	from	from	ADP
ejpam-6568	155	4	theorem	theorem	ADJ
ejpam-6568	155	5	1	1	NUM
ejpam-6568	155	6	that	that	SCONJ
ejpam-6568	155	7	f	f	PROPN
ejpam-6568	155	8	is	be	AUX
ejpam-6568	155	9	almost	almost	ADV
ejpam-6568	155	10	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	155	11	,	,	PUNCT
ejpam-6568	155	12	σ2)continuous	σ2)continuous	ADJ
ejpam-6568	155	13	.	.	PUNCT
ejpam-6568	156	1	theorem	theorem	NOUN
ejpam-6568	156	2	3	3	NUM
ejpam-6568	156	3	.	.	X
ejpam-6568	156	4	for	for	ADP
ejpam-6568	156	5	a	a	DET
ejpam-6568	156	6	function	function	NOUN
ejpam-6568	156	7	f	f	NOUN
ejpam-6568	156	8	:	:	PUNCT
ejpam-6568	156	9	(	(	PUNCT
ejpam-6568	156	10	x	x	X
ejpam-6568	156	11	,	,	PUNCT
ejpam-6568	156	12	τ	τ	PROPN
ejpam-6568	156	13	,	,	PUNCT
ejpam-6568	156	14	i	i	NOUN
ejpam-6568	156	15	)	)	PUNCT
ejpam-6568	156	16	→	→	PUNCT
ejpam-6568	156	17	(	(	PUNCT
ejpam-6568	156	18	y	y	PROPN
ejpam-6568	156	19	,	,	PUNCT
ejpam-6568	156	20	σ1	σ1	PROPN
ejpam-6568	156	21	,	,	PUNCT
ejpam-6568	156	22	σ2	σ2	NOUN
ejpam-6568	156	23	)	)	PUNCT
ejpam-6568	156	24	,	,	PUNCT
ejpam-6568	156	25	the	the	DET
ejpam-6568	156	26	following	follow	VERB
ejpam-6568	156	27	properties	property	NOUN
ejpam-6568	156	28	are	be	AUX
ejpam-6568	156	29	equivalent	equivalent	ADJ
ejpam-6568	156	30	:	:	PUNCT
ejpam-6568	156	31	(	(	PUNCT
ejpam-6568	156	32	1	1	X
ejpam-6568	156	33	)	)	PUNCT
ejpam-6568	156	34	f	f	NOUN
ejpam-6568	156	35	is	be	AUX
ejpam-6568	156	36	almost	almost	ADV
ejpam-6568	156	37	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	156	38	,	,	PUNCT
ejpam-6568	156	39	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	156	40	;	;	PUNCT
ejpam-6568	156	41	(	(	PUNCT
ejpam-6568	156	42	2	2	X
ejpam-6568	156	43	)	)	PUNCT
ejpam-6568	156	44	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	156	45	)	)	PUNCT
ejpam-6568	156	46	)	)	PUNCT
ejpam-6568	157	1	⊆	⊆	NUM
ejpam-6568	157	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	157	3	-	-	PUNCT
ejpam-6568	157	4	cl(v	cl(v	NOUN
ejpam-6568	157	5	)	)	PUNCT
ejpam-6568	157	6	)	)	PUNCT
ejpam-6568	157	7	for	for	ADP
ejpam-6568	157	8	every	every	DET
ejpam-6568	157	9	(	(	PUNCT
ejpam-6568	157	10	σ1	σ1	PROPN
ejpam-6568	157	11	,	,	PUNCT
ejpam-6568	157	12	σ2)β	σ2)β	NOUN
ejpam-6568	157	13	-	-	PUNCT
ejpam-6568	157	14	open	open	NOUN
ejpam-6568	157	15	set	set	NOUN
ejpam-6568	157	16	v	v	NOUN
ejpam-6568	157	17	of	of	ADP
ejpam-6568	157	18	y	y	PROPN
ejpam-6568	157	19	;	;	PUNCT
ejpam-6568	157	20	(	(	PUNCT
ejpam-6568	157	21	3	3	X
ejpam-6568	157	22	)	)	PUNCT
ejpam-6568	157	23	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	157	24	)	)	PUNCT
ejpam-6568	157	25	)	)	PUNCT
ejpam-6568	158	1	⊆	⊆	NUM
ejpam-6568	158	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	158	3	-	-	PUNCT
ejpam-6568	158	4	cl(v	cl(v	NOUN
ejpam-6568	158	5	)	)	PUNCT
ejpam-6568	158	6	)	)	PUNCT
ejpam-6568	158	7	for	for	ADP
ejpam-6568	158	8	every	every	DET
ejpam-6568	158	9	(	(	PUNCT
ejpam-6568	158	10	σ1	σ1	PROPN
ejpam-6568	158	11	,	,	PUNCT
ejpam-6568	158	12	σ2)s	σ2)s	NOUN
ejpam-6568	158	13	-	-	PUNCT
ejpam-6568	158	14	open	open	NOUN
ejpam-6568	158	15	set	set	NOUN
ejpam-6568	158	16	v	v	NOUN
ejpam-6568	158	17	of	of	ADP
ejpam-6568	158	18	y	y	PROPN
ejpam-6568	158	19	.	.	PUNCT
ejpam-6568	159	1	n.	n.	PROPN
ejpam-6568	159	2	viriyapong	viriyapong	PROPN
ejpam-6568	159	3	,	,	PUNCT
ejpam-6568	159	4	a.	a.	PROPN
ejpam-6568	159	5	sama	sama	PROPN
ejpam-6568	159	6	-	-	PUNCT
ejpam-6568	159	7	ae	ae	PROPN
ejpam-6568	159	8	,	,	PUNCT
ejpam-6568	159	9	c.	c.	PROPN
ejpam-6568	159	10	boonpok	boonpok	PROPN
ejpam-6568	159	11	/	/	SYM
ejpam-6568	159	12	eur	eur	PROPN
ejpam-6568	159	13	.	.	PUNCT
ejpam-6568	160	1	j.	j.	PROPN
ejpam-6568	160	2	pure	pure	PROPN
ejpam-6568	160	3	appl	appl	PROPN
ejpam-6568	160	4	.	.	PROPN
ejpam-6568	160	5	math	math	PROPN
ejpam-6568	160	6	,	,	PUNCT
ejpam-6568	160	7	18	18	NUM
ejpam-6568	160	8	(	(	PUNCT
ejpam-6568	160	9	3	3	NUM
ejpam-6568	160	10	)	)	PUNCT
ejpam-6568	160	11	(	(	PUNCT
ejpam-6568	160	12	2025	2025	NUM
ejpam-6568	160	13	)	)	PUNCT
ejpam-6568	160	14	,	,	PUNCT
ejpam-6568	160	15	6568	6568	NUM
ejpam-6568	160	16	6	6	NUM
ejpam-6568	160	17	of	of	ADP
ejpam-6568	160	18	18	18	NUM
ejpam-6568	160	19	proof	proof	NOUN
ejpam-6568	160	20	.	.	PUNCT
ejpam-6568	161	1	(	(	PUNCT
ejpam-6568	161	2	1	1	X
ejpam-6568	161	3	)	)	PUNCT
ejpam-6568	161	4	⇒	⇒	NOUN
ejpam-6568	161	5	(	(	PUNCT
ejpam-6568	161	6	2	2	NUM
ejpam-6568	161	7	):	):	PUNCT
ejpam-6568	161	8	let	let	VERB
ejpam-6568	161	9	v	v	PART
ejpam-6568	161	10	be	be	AUX
ejpam-6568	161	11	any	any	DET
ejpam-6568	161	12	(	(	PUNCT
ejpam-6568	161	13	σ1	σ1	PROPN
ejpam-6568	161	14	,	,	PUNCT
ejpam-6568	161	15	σ2)β	σ2)β	NOUN
ejpam-6568	161	16	-	-	PUNCT
ejpam-6568	161	17	open	open	ADJ
ejpam-6568	161	18	set	set	NOUN
ejpam-6568	161	19	of	of	ADP
ejpam-6568	161	20	y	y	PROPN
ejpam-6568	161	21	.	.	PUNCT
ejpam-6568	162	1	then	then	ADV
ejpam-6568	162	2	,	,	PUNCT
ejpam-6568	162	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	162	4	-	-	NUM
ejpam-6568	162	5	cl(v	cl(v	NOUN
ejpam-6568	162	6	)	)	PUNCT
ejpam-6568	162	7	is	be	AUX
ejpam-6568	162	8	a	a	DET
ejpam-6568	162	9	(	(	PUNCT
ejpam-6568	162	10	σ1	σ1	NOUN
ejpam-6568	162	11	,	,	PUNCT
ejpam-6568	162	12	σ2)r	σ2)r	NOUN
ejpam-6568	162	13	-	-	PUNCT
ejpam-6568	162	14	closed	close	VERB
ejpam-6568	162	15	set	set	NOUN
ejpam-6568	162	16	of	of	ADP
ejpam-6568	162	17	y	y	PROPN
ejpam-6568	162	18	.	.	PUNCT
ejpam-6568	163	1	since	since	SCONJ
ejpam-6568	163	2	f	f	PROPN
ejpam-6568	163	3	is	be	AUX
ejpam-6568	163	4	almost	almost	ADV
ejpam-6568	163	5	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	163	6	,	,	PUNCT
ejpam-6568	163	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	163	8	,	,	PUNCT
ejpam-6568	163	9	by	by	ADP
ejpam-6568	163	10	theorem	theorem	NOUN
ejpam-6568	163	11	2	2	NUM
ejpam-6568	163	12	we	we	PRON
ejpam-6568	163	13	have	have	AUX
ejpam-6568	163	14	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	163	15	-	-	PUNCT
ejpam-6568	163	16	cl(v	cl(v	NOUN
ejpam-6568	163	17	)	)	PUNCT
ejpam-6568	163	18	)	)	PUNCT
ejpam-6568	163	19	is	be	AUX
ejpam-6568	163	20	⋆-closed	⋆-close	VERB
ejpam-6568	163	21	in	in	ADP
ejpam-6568	163	22	x.	x.	PROPN
ejpam-6568	163	23	thus	thus	ADV
ejpam-6568	163	24	,	,	PUNCT
ejpam-6568	163	25	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	163	26	)	)	PUNCT
ejpam-6568	163	27	)	)	PUNCT
ejpam-6568	164	1	⊆	⊆	NUM
ejpam-6568	164	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	164	3	-	-	PUNCT
ejpam-6568	164	4	cl(v	cl(v	NOUN
ejpam-6568	164	5	)	)	PUNCT
ejpam-6568	164	6	)	)	PUNCT
ejpam-6568	164	7	.	.	PUNCT
ejpam-6568	165	1	(	(	PUNCT
ejpam-6568	165	2	2	2	X
ejpam-6568	165	3	)	)	PUNCT
ejpam-6568	165	4	⇒	⇒	NOUN
ejpam-6568	165	5	(	(	PUNCT
ejpam-6568	165	6	3	3	NUM
ejpam-6568	165	7	):	):	PUNCT
ejpam-6568	165	8	the	the	DET
ejpam-6568	165	9	proof	proof	NOUN
ejpam-6568	165	10	is	be	AUX
ejpam-6568	165	11	obvious	obvious	ADJ
ejpam-6568	165	12	.	.	PUNCT
ejpam-6568	166	1	(	(	PUNCT
ejpam-6568	166	2	3	3	X
ejpam-6568	166	3	)	)	PUNCT
ejpam-6568	166	4	⇒	⇒	NOUN
ejpam-6568	166	5	(	(	PUNCT
ejpam-6568	166	6	1	1	NUM
ejpam-6568	166	7	):	):	PUNCT
ejpam-6568	166	8	let	let	VERB
ejpam-6568	166	9	k	k	PRON
ejpam-6568	166	10	be	be	AUX
ejpam-6568	166	11	any	any	DET
ejpam-6568	166	12	(	(	PUNCT
ejpam-6568	166	13	σ1	σ1	NOUN
ejpam-6568	166	14	,	,	PUNCT
ejpam-6568	166	15	σ2)r	σ2)r	NOUN
ejpam-6568	166	16	-	-	PUNCT
ejpam-6568	166	17	closed	close	VERB
ejpam-6568	166	18	set	set	NOUN
ejpam-6568	166	19	of	of	ADP
ejpam-6568	166	20	y	y	PROPN
ejpam-6568	166	21	.	.	PUNCT
ejpam-6568	167	1	then	then	ADV
ejpam-6568	167	2	,	,	PUNCT
ejpam-6568	167	3	k	k	X
ejpam-6568	167	4	is	be	AUX
ejpam-6568	167	5	(	(	PUNCT
ejpam-6568	167	6	σ1	σ1	PROPN
ejpam-6568	167	7	,	,	PUNCT
ejpam-6568	167	8	σ2)s	σ2)s	NOUN
ejpam-6568	167	9	-	-	PUNCT
ejpam-6568	167	10	open	open	ADJ
ejpam-6568	167	11	in	in	ADP
ejpam-6568	167	12	y	y	PROPN
ejpam-6568	167	13	.	.	PUNCT
ejpam-6568	168	1	then	then	ADV
ejpam-6568	168	2	by	by	ADP
ejpam-6568	168	3	(	(	PUNCT
ejpam-6568	168	4	3	3	NUM
ejpam-6568	168	5	)	)	PUNCT
ejpam-6568	168	6	,	,	PUNCT
ejpam-6568	168	7	cl⋆(f−1(k	cl⋆(f−1(k	PROPN
ejpam-6568	168	8	)	)	PUNCT
ejpam-6568	168	9	)	)	PUNCT
ejpam-6568	168	10	⊆	⊆	NUM
ejpam-6568	168	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	168	12	-	-	PUNCT
ejpam-6568	168	13	cl(k	cl(k	NOUN
ejpam-6568	168	14	)	)	PUNCT
ejpam-6568	168	15	)	)	PUNCT
ejpam-6568	169	1	=	=	PUNCT
ejpam-6568	169	2	f−1(k	f−1(k	PROPN
ejpam-6568	169	3	)	)	PUNCT
ejpam-6568	169	4	and	and	CCONJ
ejpam-6568	169	5	hence	hence	ADV
ejpam-6568	169	6	f−1(k	f−1(k	PROPN
ejpam-6568	169	7	)	)	PUNCT
ejpam-6568	169	8	is	be	AUX
ejpam-6568	169	9	⋆-closed	⋆-close	VERB
ejpam-6568	169	10	in	in	ADP
ejpam-6568	169	11	x.	x.	NOUN
ejpam-6568	169	12	by	by	ADP
ejpam-6568	169	13	theorem	theorem	NOUN
ejpam-6568	169	14	2	2	NUM
ejpam-6568	169	15	,	,	PUNCT
ejpam-6568	169	16	f	f	PROPN
ejpam-6568	169	17	is	be	AUX
ejpam-6568	169	18	almost	almost	ADV
ejpam-6568	169	19	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	169	20	,	,	PUNCT
ejpam-6568	169	21	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	169	22	.	.	PUNCT
ejpam-6568	170	1	lemma	lemma	PROPN
ejpam-6568	170	2	3	3	NUM
ejpam-6568	170	3	.	.	PUNCT
ejpam-6568	171	1	[	[	X
ejpam-6568	171	2	32	32	NUM
ejpam-6568	171	3	]	]	PUNCT
ejpam-6568	171	4	for	for	ADP
ejpam-6568	171	5	a	a	DET
ejpam-6568	171	6	bitopological	bitopological	ADJ
ejpam-6568	171	7	space	space	NOUN
ejpam-6568	171	8	(	(	PUNCT
ejpam-6568	171	9	x	x	NOUN
ejpam-6568	171	10	,	,	PUNCT
ejpam-6568	171	11	τ1	τ1	NOUN
ejpam-6568	171	12	,	,	PUNCT
ejpam-6568	171	13	τ2	τ2	NOUN
ejpam-6568	171	14	)	)	PUNCT
ejpam-6568	171	15	,	,	PUNCT
ejpam-6568	171	16	the	the	DET
ejpam-6568	171	17	following	follow	VERB
ejpam-6568	171	18	properties	property	NOUN
ejpam-6568	171	19	hold	hold	VERB
ejpam-6568	171	20	:	:	PUNCT
ejpam-6568	171	21	(	(	PUNCT
ejpam-6568	171	22	1	1	X
ejpam-6568	171	23	)	)	PUNCT
ejpam-6568	171	24	α(τ1	α(τ1	NOUN
ejpam-6568	171	25	,	,	PUNCT
ejpam-6568	171	26	τ2)-cl(v	τ2)-cl(v	NOUN
ejpam-6568	171	27	)	)	PUNCT
ejpam-6568	171	28	=	=	PUNCT
ejpam-6568	172	1	τ1τ2	τ1τ2	NOUN
ejpam-6568	172	2	-	-	NOUN
ejpam-6568	172	3	cl(v	cl(v	X
ejpam-6568	172	4	)	)	PUNCT
ejpam-6568	172	5	for	for	ADP
ejpam-6568	172	6	every	every	DET
ejpam-6568	172	7	(	(	PUNCT
ejpam-6568	172	8	τ1	τ1	NOUN
ejpam-6568	172	9	,	,	PUNCT
ejpam-6568	172	10	τ2)β	τ2)β	ADJ
ejpam-6568	172	11	-	-	PUNCT
ejpam-6568	172	12	open	open	NOUN
ejpam-6568	172	13	set	set	NOUN
ejpam-6568	172	14	v	v	NOUN
ejpam-6568	172	15	of	of	ADP
ejpam-6568	172	16	x	x	PRON
ejpam-6568	172	17	;	;	PUNCT
ejpam-6568	172	18	(	(	PUNCT
ejpam-6568	172	19	2	2	X
ejpam-6568	172	20	)	)	PUNCT
ejpam-6568	172	21	(	(	PUNCT
ejpam-6568	172	22	τ1	τ1	NOUN
ejpam-6568	172	23	,	,	PUNCT
ejpam-6568	172	24	τ2)-pcl(v	τ2)-pcl(v	NOUN
ejpam-6568	172	25	)	)	PUNCT
ejpam-6568	172	26	=	=	PUNCT
ejpam-6568	173	1	τ1τ2	τ1τ2	NOUN
ejpam-6568	173	2	-	-	NOUN
ejpam-6568	173	3	cl(v	cl(v	X
ejpam-6568	173	4	)	)	PUNCT
ejpam-6568	173	5	for	for	ADP
ejpam-6568	173	6	every	every	DET
ejpam-6568	173	7	(	(	PUNCT
ejpam-6568	173	8	τ1	τ1	NOUN
ejpam-6568	173	9	,	,	PUNCT
ejpam-6568	173	10	τ2)s	τ2)s	NOUN
ejpam-6568	173	11	-	-	PUNCT
ejpam-6568	173	12	open	open	ADJ
ejpam-6568	173	13	set	set	NOUN
ejpam-6568	173	14	v	v	NOUN
ejpam-6568	173	15	of	of	ADP
ejpam-6568	173	16	x.	x.	NOUN
ejpam-6568	173	17	corollary	corollary	NOUN
ejpam-6568	173	18	1	1	NUM
ejpam-6568	173	19	.	.	PUNCT
ejpam-6568	174	1	for	for	ADP
ejpam-6568	174	2	a	a	DET
ejpam-6568	174	3	function	function	NOUN
ejpam-6568	174	4	f	f	NOUN
ejpam-6568	174	5	:	:	PUNCT
ejpam-6568	174	6	(	(	PUNCT
ejpam-6568	174	7	x	x	X
ejpam-6568	174	8	,	,	PUNCT
ejpam-6568	174	9	τ	τ	PROPN
ejpam-6568	174	10	,	,	PUNCT
ejpam-6568	174	11	i	i	NOUN
ejpam-6568	174	12	)	)	PUNCT
ejpam-6568	174	13	→	→	PUNCT
ejpam-6568	174	14	(	(	PUNCT
ejpam-6568	174	15	y	y	PROPN
ejpam-6568	174	16	,	,	PUNCT
ejpam-6568	174	17	σ1	σ1	PROPN
ejpam-6568	174	18	,	,	PUNCT
ejpam-6568	174	19	σ2	σ2	NOUN
ejpam-6568	174	20	)	)	PUNCT
ejpam-6568	174	21	,	,	PUNCT
ejpam-6568	174	22	the	the	DET
ejpam-6568	174	23	following	follow	VERB
ejpam-6568	174	24	properties	property	NOUN
ejpam-6568	174	25	are	be	AUX
ejpam-6568	174	26	equivalent	equivalent	ADJ
ejpam-6568	174	27	:	:	PUNCT
ejpam-6568	174	28	(	(	PUNCT
ejpam-6568	174	29	1	1	X
ejpam-6568	174	30	)	)	PUNCT
ejpam-6568	174	31	f	f	NOUN
ejpam-6568	174	32	is	be	AUX
ejpam-6568	174	33	almost	almost	ADV
ejpam-6568	174	34	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	174	35	,	,	PUNCT
ejpam-6568	174	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	174	37	;	;	PUNCT
ejpam-6568	174	38	(	(	PUNCT
ejpam-6568	174	39	2	2	X
ejpam-6568	174	40	)	)	PUNCT
ejpam-6568	174	41	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	174	42	)	)	PUNCT
ejpam-6568	174	43	)	)	PUNCT
ejpam-6568	175	1	⊆	⊆	X
ejpam-6568	175	2	f−1(α(σ1	f−1(α(σ1	NOUN
ejpam-6568	175	3	,	,	PUNCT
ejpam-6568	175	4	σ2)-cl(v	σ2)-cl(v	NOUN
ejpam-6568	175	5	)	)	PUNCT
ejpam-6568	175	6	)	)	PUNCT
ejpam-6568	175	7	for	for	ADP
ejpam-6568	175	8	every	every	DET
ejpam-6568	175	9	(	(	PUNCT
ejpam-6568	175	10	σ1	σ1	PROPN
ejpam-6568	175	11	,	,	PUNCT
ejpam-6568	175	12	σ2)β	σ2)β	NOUN
ejpam-6568	175	13	-	-	PUNCT
ejpam-6568	175	14	open	open	NOUN
ejpam-6568	175	15	set	set	NOUN
ejpam-6568	175	16	v	v	NOUN
ejpam-6568	175	17	of	of	ADP
ejpam-6568	175	18	y	y	PROPN
ejpam-6568	175	19	;	;	PUNCT
ejpam-6568	175	20	(	(	PUNCT
ejpam-6568	175	21	3	3	X
ejpam-6568	175	22	)	)	PUNCT
ejpam-6568	175	23	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	175	24	)	)	PUNCT
ejpam-6568	175	25	)	)	PUNCT
ejpam-6568	176	1	⊆	⊆	NUM
ejpam-6568	176	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	176	3	,	,	PUNCT
ejpam-6568	176	4	σ2)-pcl(v	σ2)-pcl(v	NOUN
ejpam-6568	176	5	)	)	PUNCT
ejpam-6568	176	6	)	)	PUNCT
ejpam-6568	176	7	for	for	ADP
ejpam-6568	176	8	every	every	DET
ejpam-6568	176	9	(	(	PUNCT
ejpam-6568	176	10	σ1	σ1	PROPN
ejpam-6568	176	11	,	,	PUNCT
ejpam-6568	176	12	σ2)s	σ2)s	NOUN
ejpam-6568	176	13	-	-	PUNCT
ejpam-6568	176	14	open	open	NOUN
ejpam-6568	176	15	set	set	NOUN
ejpam-6568	176	16	v	v	NOUN
ejpam-6568	176	17	of	of	ADP
ejpam-6568	176	18	y	y	PROPN
ejpam-6568	176	19	.	.	PUNCT
ejpam-6568	177	1	theorem	theorem	ADJ
ejpam-6568	177	2	4	4	NUM
ejpam-6568	177	3	.	.	X
ejpam-6568	177	4	for	for	ADP
ejpam-6568	177	5	a	a	DET
ejpam-6568	177	6	function	function	NOUN
ejpam-6568	177	7	f	f	NOUN
ejpam-6568	177	8	:	:	PUNCT
ejpam-6568	177	9	(	(	PUNCT
ejpam-6568	177	10	x	x	X
ejpam-6568	177	11	,	,	PUNCT
ejpam-6568	177	12	τ	τ	PROPN
ejpam-6568	177	13	,	,	PUNCT
ejpam-6568	177	14	i	i	NOUN
ejpam-6568	177	15	)	)	PUNCT
ejpam-6568	177	16	→	→	PUNCT
ejpam-6568	177	17	(	(	PUNCT
ejpam-6568	177	18	y	y	PROPN
ejpam-6568	177	19	,	,	PUNCT
ejpam-6568	177	20	σ1	σ1	PROPN
ejpam-6568	177	21	,	,	PUNCT
ejpam-6568	177	22	σ2	σ2	NOUN
ejpam-6568	177	23	)	)	PUNCT
ejpam-6568	177	24	,	,	PUNCT
ejpam-6568	177	25	the	the	DET
ejpam-6568	177	26	following	follow	VERB
ejpam-6568	177	27	properties	property	NOUN
ejpam-6568	177	28	are	be	AUX
ejpam-6568	177	29	equivalent	equivalent	ADJ
ejpam-6568	177	30	:	:	PUNCT
ejpam-6568	177	31	(	(	PUNCT
ejpam-6568	177	32	1	1	X
ejpam-6568	177	33	)	)	PUNCT
ejpam-6568	177	34	f	f	NOUN
ejpam-6568	177	35	is	be	AUX
ejpam-6568	177	36	almost	almost	ADV
ejpam-6568	177	37	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	177	38	,	,	PUNCT
ejpam-6568	177	39	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	177	40	;	;	PUNCT
ejpam-6568	177	41	(	(	PUNCT
ejpam-6568	177	42	2	2	X
ejpam-6568	177	43	)	)	PUNCT
ejpam-6568	177	44	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	177	45	-	-	PUNCT
ejpam-6568	177	46	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	177	47	-	-	PUNCT
ejpam-6568	177	48	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	177	49	-	-	PUNCT
ejpam-6568	177	50	cl(v	cl(v	NOUN
ejpam-6568	177	51	)	)	PUNCT
ejpam-6568	177	52	)	)	PUNCT
ejpam-6568	177	53	)	)	PUNCT
ejpam-6568	177	54	)	)	PUNCT
ejpam-6568	177	55	)	)	PUNCT
ejpam-6568	178	1	⊆	⊆	NUM
ejpam-6568	178	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	178	3	-	-	PUNCT
ejpam-6568	178	4	cl(v	cl(v	NOUN
ejpam-6568	178	5	)	)	PUNCT
ejpam-6568	178	6	)	)	PUNCT
ejpam-6568	178	7	for	for	SCONJ
ejpam-6568	178	8	every	every	DET
ejpam-6568	178	9	(	(	PUNCT
ejpam-6568	178	10	σ1	σ1	PROPN
ejpam-6568	178	11	,	,	PUNCT
ejpam-6568	178	12	σ2)popen	σ2)popen	VERB
ejpam-6568	178	13	set	set	VERB
ejpam-6568	178	14	v	v	NOUN
ejpam-6568	178	15	of	of	ADP
ejpam-6568	178	16	y	y	PROPN
ejpam-6568	178	17	;	;	PUNCT
ejpam-6568	178	18	(	(	PUNCT
ejpam-6568	178	19	3	3	X
ejpam-6568	178	20	)	)	PUNCT
ejpam-6568	178	21	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	178	22	-	-	PUNCT
ejpam-6568	178	23	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	178	24	-	-	PUNCT
ejpam-6568	178	25	int(v	int(v	NOUN
ejpam-6568	178	26	)	)	PUNCT
ejpam-6568	178	27	)	)	PUNCT
ejpam-6568	178	28	)	)	PUNCT
ejpam-6568	178	29	)	)	PUNCT
ejpam-6568	179	1	⊆	⊆	NUM
ejpam-6568	179	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	179	3	-	-	PUNCT
ejpam-6568	179	4	cl(v	cl(v	NOUN
ejpam-6568	179	5	)	)	PUNCT
ejpam-6568	179	6	)	)	PUNCT
ejpam-6568	179	7	for	for	ADP
ejpam-6568	179	8	every	every	DET
ejpam-6568	179	9	(	(	PUNCT
ejpam-6568	179	10	σ1	σ1	PROPN
ejpam-6568	179	11	,	,	PUNCT
ejpam-6568	179	12	σ2)p	σ2)p	NOUN
ejpam-6568	179	13	-	-	PUNCT
ejpam-6568	179	14	open	open	NOUN
ejpam-6568	179	15	set	set	NOUN
ejpam-6568	179	16	v	v	NOUN
ejpam-6568	179	17	of	of	ADP
ejpam-6568	179	18	y	y	PROPN
ejpam-6568	179	19	;	;	PUNCT
ejpam-6568	179	20	(	(	PUNCT
ejpam-6568	179	21	4	4	X
ejpam-6568	179	22	)	)	PUNCT
ejpam-6568	179	23	f−1(v	f−1(v	NOUN
ejpam-6568	179	24	)	)	PUNCT
ejpam-6568	180	1	⊆	⊆	X
ejpam-6568	180	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	180	3	-	-	PUNCT
ejpam-6568	180	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	180	5	-	-	PUNCT
ejpam-6568	180	6	cl(v	cl(v	NOUN
ejpam-6568	180	7	)	)	PUNCT
ejpam-6568	180	8	)	)	PUNCT
ejpam-6568	180	9	)	)	PUNCT
ejpam-6568	180	10	)	)	PUNCT
ejpam-6568	180	11	for	for	ADP
ejpam-6568	180	12	every	every	DET
ejpam-6568	180	13	(	(	PUNCT
ejpam-6568	180	14	σ1	σ1	PROPN
ejpam-6568	180	15	,	,	PUNCT
ejpam-6568	180	16	σ2)p	σ2)p	NOUN
ejpam-6568	180	17	-	-	PUNCT
ejpam-6568	180	18	open	open	NOUN
ejpam-6568	180	19	set	set	NOUN
ejpam-6568	180	20	v	v	NOUN
ejpam-6568	180	21	of	of	ADP
ejpam-6568	180	22	y	y	PROPN
ejpam-6568	180	23	.	.	PUNCT
ejpam-6568	181	1	proof	proof	NOUN
ejpam-6568	181	2	.	.	PUNCT
ejpam-6568	182	1	(	(	PUNCT
ejpam-6568	182	2	1	1	X
ejpam-6568	182	3	)	)	PUNCT
ejpam-6568	182	4	⇒	⇒	NOUN
ejpam-6568	182	5	(	(	PUNCT
ejpam-6568	182	6	2	2	NUM
ejpam-6568	182	7	):	):	PUNCT
ejpam-6568	182	8	let	let	VERB
ejpam-6568	182	9	v	v	PART
ejpam-6568	182	10	be	be	AUX
ejpam-6568	182	11	any	any	DET
ejpam-6568	182	12	(	(	PUNCT
ejpam-6568	182	13	σ1	σ1	PROPN
ejpam-6568	182	14	,	,	PUNCT
ejpam-6568	182	15	σ2)p	σ2)p	NOUN
ejpam-6568	182	16	-	-	PUNCT
ejpam-6568	182	17	open	open	ADJ
ejpam-6568	182	18	set	set	NOUN
ejpam-6568	182	19	of	of	ADP
ejpam-6568	182	20	y	y	PROPN
ejpam-6568	182	21	.	.	PUNCT
ejpam-6568	183	1	then	then	ADV
ejpam-6568	183	2	,	,	PUNCT
ejpam-6568	183	3	we	we	PRON
ejpam-6568	183	4	have	have	VERB
ejpam-6568	183	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	183	6	-	-	NUM
ejpam-6568	183	7	cl(v	cl(v	NOUN
ejpam-6568	183	8	)	)	PUNCT
ejpam-6568	183	9	is	be	AUX
ejpam-6568	183	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	183	11	-	-	ADJ
ejpam-6568	183	12	closed	closed	ADJ
ejpam-6568	183	13	in	in	ADP
ejpam-6568	183	14	y	y	PROPN
ejpam-6568	183	15	and	and	CCONJ
ejpam-6568	183	16	by	by	ADP
ejpam-6568	183	17	theorem	theorem	NOUN
ejpam-6568	183	18	2	2	NUM
ejpam-6568	183	19	,	,	PUNCT
ejpam-6568	183	20	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	183	21	-	-	PUNCT
ejpam-6568	183	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	183	23	-	-	PUNCT
ejpam-6568	183	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	183	25	-	-	PUNCT
ejpam-6568	183	26	cl(v	cl(v	NOUN
ejpam-6568	183	27	)	)	PUNCT
ejpam-6568	183	28	)	)	PUNCT
ejpam-6568	183	29	)	)	PUNCT
ejpam-6568	183	30	)	)	PUNCT
ejpam-6568	183	31	)	)	PUNCT
ejpam-6568	184	1	⊆	⊆	NUM
ejpam-6568	184	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	184	3	-	-	PUNCT
ejpam-6568	184	4	cl(v	cl(v	NOUN
ejpam-6568	184	5	)	)	PUNCT
ejpam-6568	184	6	)	)	PUNCT
ejpam-6568	184	7	.	.	PUNCT
ejpam-6568	185	1	(	(	PUNCT
ejpam-6568	185	2	2	2	X
ejpam-6568	185	3	)	)	PUNCT
ejpam-6568	185	4	⇒	⇒	NOUN
ejpam-6568	185	5	(	(	PUNCT
ejpam-6568	185	6	3	3	NUM
ejpam-6568	185	7	):	):	PUNCT
ejpam-6568	185	8	let	let	VERB
ejpam-6568	185	9	v	v	PART
ejpam-6568	185	10	be	be	AUX
ejpam-6568	185	11	any	any	DET
ejpam-6568	185	12	(	(	PUNCT
ejpam-6568	185	13	σ1	σ1	PROPN
ejpam-6568	185	14	,	,	PUNCT
ejpam-6568	185	15	σ2)p	σ2)p	NOUN
ejpam-6568	185	16	-	-	PUNCT
ejpam-6568	185	17	open	open	ADJ
ejpam-6568	185	18	set	set	NOUN
ejpam-6568	185	19	of	of	ADP
ejpam-6568	185	20	y	y	PROPN
ejpam-6568	185	21	.	.	PUNCT
ejpam-6568	186	1	by	by	ADP
ejpam-6568	186	2	(	(	PUNCT
ejpam-6568	186	3	2	2	NUM
ejpam-6568	186	4	)	)	PUNCT
ejpam-6568	186	5	,	,	PUNCT
ejpam-6568	186	6	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	186	7	-	-	PUNCT
ejpam-6568	186	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	186	9	-	-	PUNCT
ejpam-6568	186	10	int(v	int(v	NOUN
ejpam-6568	186	11	)	)	PUNCT
ejpam-6568	186	12	)	)	PUNCT
ejpam-6568	186	13	)	)	PUNCT
ejpam-6568	186	14	)	)	PUNCT
ejpam-6568	187	1	⊆	⊆	X
ejpam-6568	187	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	187	3	-	-	PUNCT
ejpam-6568	187	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	187	5	-	-	PUNCT
ejpam-6568	187	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	187	7	-	-	PUNCT
ejpam-6568	187	8	cl(v	cl(v	NOUN
ejpam-6568	187	9	)	)	PUNCT
ejpam-6568	187	10	)	)	PUNCT
ejpam-6568	187	11	)	)	PUNCT
ejpam-6568	187	12	)	)	PUNCT
ejpam-6568	187	13	)	)	PUNCT
ejpam-6568	188	1	⊆	⊆	NUM
ejpam-6568	188	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	188	3	-	-	PUNCT
ejpam-6568	188	4	cl(v	cl(v	NOUN
ejpam-6568	188	5	)	)	PUNCT
ejpam-6568	188	6	)	)	PUNCT
ejpam-6568	188	7	.	.	PUNCT
ejpam-6568	189	1	(	(	PUNCT
ejpam-6568	189	2	3	3	X
ejpam-6568	189	3	)	)	PUNCT
ejpam-6568	189	4	⇒	⇒	NOUN
ejpam-6568	189	5	(	(	PUNCT
ejpam-6568	189	6	4	4	NUM
ejpam-6568	189	7	):	):	PUNCT
ejpam-6568	189	8	let	let	VERB
ejpam-6568	189	9	v	v	PART
ejpam-6568	189	10	be	be	AUX
ejpam-6568	189	11	any	any	DET
ejpam-6568	189	12	(	(	PUNCT
ejpam-6568	189	13	σ1	σ1	PROPN
ejpam-6568	189	14	,	,	PUNCT
ejpam-6568	189	15	σ2)p	σ2)p	NOUN
ejpam-6568	189	16	-	-	PUNCT
ejpam-6568	189	17	open	open	ADJ
ejpam-6568	189	18	set	set	NOUN
ejpam-6568	189	19	of	of	ADP
ejpam-6568	189	20	y	y	PROPN
ejpam-6568	189	21	.	.	PUNCT
ejpam-6568	190	1	thus	thus	ADV
ejpam-6568	190	2	by	by	ADP
ejpam-6568	190	3	(	(	PUNCT
ejpam-6568	190	4	3	3	NUM
ejpam-6568	190	5	)	)	PUNCT
ejpam-6568	190	6	,	,	PUNCT
ejpam-6568	190	7	we	we	PRON
ejpam-6568	190	8	have	have	VERB
ejpam-6568	190	9	x	x	X
ejpam-6568	190	10	−	−	VERB
ejpam-6568	190	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	ADV
ejpam-6568	190	12	-	-	PUNCT
ejpam-6568	190	13	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	190	14	-	-	PUNCT
ejpam-6568	190	15	cl(v	cl(v	NOUN
ejpam-6568	190	16	)	)	PUNCT
ejpam-6568	190	17	)	)	PUNCT
ejpam-6568	190	18	)	)	PUNCT
ejpam-6568	190	19	)	)	PUNCT
ejpam-6568	191	1	=	=	PUNCT
ejpam-6568	191	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6568	191	3	−	−	NOUN
ejpam-6568	191	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	191	5	-	-	PUNCT
ejpam-6568	191	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	191	7	-	-	PUNCT
ejpam-6568	191	8	cl(v	cl(v	NOUN
ejpam-6568	191	9	)	)	PUNCT
ejpam-6568	191	10	)	)	PUNCT
ejpam-6568	191	11	)	)	PUNCT
ejpam-6568	191	12	)	)	PUNCT
ejpam-6568	192	1	n.	n.	PROPN
ejpam-6568	192	2	viriyapong	viriyapong	PROPN
ejpam-6568	192	3	,	,	PUNCT
ejpam-6568	192	4	a.	a.	PROPN
ejpam-6568	192	5	sama	sama	PROPN
ejpam-6568	192	6	-	-	PUNCT
ejpam-6568	192	7	ae	ae	PROPN
ejpam-6568	192	8	,	,	PUNCT
ejpam-6568	192	9	c.	c.	PROPN
ejpam-6568	192	10	boonpok	boonpok	PROPN
ejpam-6568	192	11	/	/	SYM
ejpam-6568	192	12	eur	eur	PROPN
ejpam-6568	192	13	.	.	PUNCT
ejpam-6568	193	1	j.	j.	PROPN
ejpam-6568	193	2	pure	pure	PROPN
ejpam-6568	193	3	appl	appl	PROPN
ejpam-6568	193	4	.	.	PROPN
ejpam-6568	193	5	math	math	PROPN
ejpam-6568	193	6	,	,	PUNCT
ejpam-6568	193	7	18	18	NUM
ejpam-6568	193	8	(	(	PUNCT
ejpam-6568	193	9	3	3	NUM
ejpam-6568	193	10	)	)	PUNCT
ejpam-6568	193	11	(	(	PUNCT
ejpam-6568	193	12	2025	2025	NUM
ejpam-6568	193	13	)	)	PUNCT
ejpam-6568	193	14	,	,	PUNCT
ejpam-6568	193	15	6568	6568	NUM
ejpam-6568	193	16	7	7	NUM
ejpam-6568	193	17	of	of	ADP
ejpam-6568	193	18	18	18	NUM
ejpam-6568	193	19	=	=	SYM
ejpam-6568	193	20	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	193	21	−	−	VERB
ejpam-6568	193	22	σ1σ2	σ1σ2	SYM
ejpam-6568	193	23	-	-	PUNCT
ejpam-6568	193	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	193	25	-	-	PUNCT
ejpam-6568	193	26	cl(v	cl(v	NOUN
ejpam-6568	193	27	)	)	PUNCT
ejpam-6568	193	28	)	)	PUNCT
ejpam-6568	193	29	)	)	PUNCT
ejpam-6568	193	30	)	)	PUNCT
ejpam-6568	194	1	=	=	PUNCT
ejpam-6568	194	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	194	3	-	-	PUNCT
ejpam-6568	194	4	cl(y	cl(y	NOUN
ejpam-6568	194	5	−	−	NOUN
ejpam-6568	194	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	194	7	-	-	NUM
ejpam-6568	194	8	cl(v	cl(v	NOUN
ejpam-6568	194	9	)	)	PUNCT
ejpam-6568	194	10	)	)	PUNCT
ejpam-6568	194	11	)	)	PUNCT
ejpam-6568	194	12	)	)	PUNCT
ejpam-6568	195	1	=	=	PUNCT
ejpam-6568	196	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	196	2	-	-	PUNCT
ejpam-6568	196	3	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	196	4	-	-	PUNCT
ejpam-6568	196	5	int(y	int(y	PROPN
ejpam-6568	196	6	−	−	NUM
ejpam-6568	196	7	σ1σ2	σ1σ2	NOUN
ejpam-6568	196	8	-	-	NUM
ejpam-6568	196	9	cl(v	cl(v	NOUN
ejpam-6568	196	10	)	)	PUNCT
ejpam-6568	196	11	)	)	PUNCT
ejpam-6568	196	12	)	)	PUNCT
ejpam-6568	196	13	)	)	PUNCT
ejpam-6568	196	14	)	)	PUNCT
ejpam-6568	197	1	⊆	⊆	NUM
ejpam-6568	197	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	197	3	-	-	PUNCT
ejpam-6568	197	4	cl(y	cl(y	NOUN
ejpam-6568	197	5	−	−	NOUN
ejpam-6568	197	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	197	7	-	-	NUM
ejpam-6568	197	8	cl(v	cl(v	NOUN
ejpam-6568	197	9	)	)	PUNCT
ejpam-6568	197	10	)	)	PUNCT
ejpam-6568	197	11	)	)	PUNCT
ejpam-6568	198	1	=	=	PUNCT
ejpam-6568	198	2	f−1(y	f−1(y	PROPN
ejpam-6568	198	3	−	−	NOUN
ejpam-6568	198	4	σ1σ2	σ1σ2	X
ejpam-6568	198	5	-	-	PUNCT
ejpam-6568	198	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	198	7	-	-	PUNCT
ejpam-6568	198	8	cl(v	cl(v	NOUN
ejpam-6568	198	9	)	)	PUNCT
ejpam-6568	198	10	)	)	PUNCT
ejpam-6568	198	11	)	)	PUNCT
ejpam-6568	199	1	⊆	⊆	X
ejpam-6568	199	2	f−1(y	f−1(y	NOUN
ejpam-6568	199	3	−	−	PROPN
ejpam-6568	199	4	v	v	NOUN
ejpam-6568	199	5	)	)	PUNCT
ejpam-6568	199	6	=	=	PUNCT
ejpam-6568	199	7	x	x	PUNCT
ejpam-6568	199	8	−	−	PROPN
ejpam-6568	199	9	f−1(v	f−1(v	PROPN
ejpam-6568	199	10	)	)	PUNCT
ejpam-6568	199	11	and	and	CCONJ
ejpam-6568	199	12	hence	hence	ADV
ejpam-6568	199	13	f−1(v	f−1(v	NOUN
ejpam-6568	199	14	)	)	PUNCT
ejpam-6568	199	15	⊆	⊆	X
ejpam-6568	199	16	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	199	17	-	-	PUNCT
ejpam-6568	199	18	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	199	19	-	-	PUNCT
ejpam-6568	199	20	cl(v	cl(v	NOUN
ejpam-6568	199	21	)	)	PUNCT
ejpam-6568	199	22	)	)	PUNCT
ejpam-6568	199	23	)	)	PUNCT
ejpam-6568	199	24	)	)	PUNCT
ejpam-6568	199	25	.	.	PUNCT
ejpam-6568	200	1	(	(	PUNCT
ejpam-6568	200	2	4	4	X
ejpam-6568	200	3	)	)	PUNCT
ejpam-6568	200	4	⇒	⇒	NOUN
ejpam-6568	200	5	(	(	PUNCT
ejpam-6568	200	6	1	1	NUM
ejpam-6568	200	7	):	):	PUNCT
ejpam-6568	200	8	let	let	VERB
ejpam-6568	200	9	v	v	PART
ejpam-6568	200	10	be	be	AUX
ejpam-6568	200	11	any	any	DET
ejpam-6568	200	12	(	(	PUNCT
ejpam-6568	200	13	σ1	σ1	NOUN
ejpam-6568	200	14	,	,	PUNCT
ejpam-6568	200	15	σ2)r	σ2)r	NOUN
ejpam-6568	200	16	-	-	PUNCT
ejpam-6568	200	17	open	open	ADJ
ejpam-6568	200	18	set	set	NOUN
ejpam-6568	200	19	of	of	ADP
ejpam-6568	200	20	y	y	PROPN
ejpam-6568	200	21	.	.	PUNCT
ejpam-6568	201	1	then	then	ADV
ejpam-6568	201	2	,	,	PUNCT
ejpam-6568	201	3	v	v	NOUN
ejpam-6568	201	4	is	be	AUX
ejpam-6568	201	5	(	(	PUNCT
ejpam-6568	201	6	σ1	σ1	PROPN
ejpam-6568	201	7	,	,	PUNCT
ejpam-6568	201	8	σ2)p	σ2)p	NOUN
ejpam-6568	201	9	-	-	PUNCT
ejpam-6568	201	10	open	open	ADJ
ejpam-6568	201	11	in	in	ADP
ejpam-6568	201	12	y	y	PROPN
ejpam-6568	201	13	and	and	CCONJ
ejpam-6568	201	14	by	by	ADP
ejpam-6568	201	15	(	(	PUNCT
ejpam-6568	201	16	4	4	NUM
ejpam-6568	201	17	)	)	PUNCT
ejpam-6568	201	18	,	,	PUNCT
ejpam-6568	201	19	f−1(v	f−1(v	PROPN
ejpam-6568	201	20	)	)	PUNCT
ejpam-6568	201	21	⊆	⊆	X
ejpam-6568	201	22	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	201	23	-	-	PUNCT
ejpam-6568	201	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	201	25	-	-	PUNCT
ejpam-6568	201	26	cl(v	cl(v	NOUN
ejpam-6568	201	27	)	)	PUNCT
ejpam-6568	201	28	)	)	PUNCT
ejpam-6568	201	29	)	)	PUNCT
ejpam-6568	201	30	)	)	PUNCT
ejpam-6568	202	1	=	=	SYM
ejpam-6568	202	2	int⋆(f−1(v	int⋆(f−1(v	NOUN
ejpam-6568	202	3	)	)	PUNCT
ejpam-6568	202	4	)	)	PUNCT
ejpam-6568	202	5	.	.	PUNCT
ejpam-6568	203	1	thus	thus	ADV
ejpam-6568	203	2	,	,	PUNCT
ejpam-6568	203	3	f−1(v	f−1(v	PROPN
ejpam-6568	203	4	)	)	PUNCT
ejpam-6568	203	5	is	be	AUX
ejpam-6568	203	6	⋆-open	⋆-open	ADJ
ejpam-6568	203	7	in	in	ADP
ejpam-6568	203	8	x.	x.	NOUN
ejpam-6568	203	9	it	it	PRON
ejpam-6568	203	10	follows	follow	VERB
ejpam-6568	203	11	from	from	ADP
ejpam-6568	203	12	theorem	theorem	ADJ
ejpam-6568	203	13	2	2	NUM
ejpam-6568	203	14	that	that	PRON
ejpam-6568	203	15	f	f	PROPN
ejpam-6568	203	16	is	be	AUX
ejpam-6568	203	17	almost	almost	ADV
ejpam-6568	203	18	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	203	19	,	,	PUNCT
ejpam-6568	203	20	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	203	21	.	.	PUNCT
ejpam-6568	204	1	lemma	lemma	PROPN
ejpam-6568	204	2	4	4	NUM
ejpam-6568	204	3	.	.	PUNCT
ejpam-6568	205	1	[	[	X
ejpam-6568	205	2	33	33	NUM
ejpam-6568	205	3	]	]	PUNCT
ejpam-6568	205	4	let	let	VERB
ejpam-6568	205	5	a	a	PRON
ejpam-6568	205	6	be	be	AUX
ejpam-6568	205	7	a	a	DET
ejpam-6568	205	8	subset	subset	NOUN
ejpam-6568	205	9	of	of	ADP
ejpam-6568	205	10	a	a	DET
ejpam-6568	205	11	bitopological	bitopological	ADJ
ejpam-6568	205	12	space	space	NOUN
ejpam-6568	205	13	(	(	PUNCT
ejpam-6568	205	14	x	x	NOUN
ejpam-6568	205	15	,	,	PUNCT
ejpam-6568	205	16	τ1	τ1	NOUN
ejpam-6568	205	17	,	,	PUNCT
ejpam-6568	205	18	τ2	τ2	NOUN
ejpam-6568	205	19	)	)	PUNCT
ejpam-6568	205	20	.	.	PUNCT
ejpam-6568	206	1	then	then	ADV
ejpam-6568	206	2	,	,	PUNCT
ejpam-6568	206	3	the	the	DET
ejpam-6568	206	4	following	follow	VERB
ejpam-6568	206	5	properties	property	NOUN
ejpam-6568	206	6	hold	hold	VERB
ejpam-6568	206	7	:	:	PUNCT
ejpam-6568	206	8	(	(	PUNCT
ejpam-6568	206	9	1	1	X
ejpam-6568	206	10	)	)	PUNCT
ejpam-6568	206	11	if	if	SCONJ
ejpam-6568	206	12	a	a	PRON
ejpam-6568	206	13	is	be	AUX
ejpam-6568	206	14	τ1τ2	τ1τ2	NOUN
ejpam-6568	206	15	-	-	ADJ
ejpam-6568	206	16	open	open	ADJ
ejpam-6568	206	17	in	in	ADP
ejpam-6568	206	18	x	x	NOUN
ejpam-6568	206	19	,	,	PUNCT
ejpam-6568	206	20	then	then	ADV
ejpam-6568	206	21	τ1τ2	τ1τ2	NOUN
ejpam-6568	206	22	-	-	NUM
ejpam-6568	206	23	cl(a	cl(a	NUM
ejpam-6568	206	24	)	)	PUNCT
ejpam-6568	206	25	=	=	PUNCT
ejpam-6568	207	1	τ1τ2	τ1τ2	PROPN
ejpam-6568	207	2	-	-	ADJ
ejpam-6568	207	3	δ	δ	NOUN
ejpam-6568	207	4	-	-	PUNCT
ejpam-6568	207	5	cl(a	cl(a	NUM
ejpam-6568	207	6	)	)	PUNCT
ejpam-6568	207	7	.	.	PUNCT
ejpam-6568	208	1	(	(	PUNCT
ejpam-6568	208	2	2	2	X
ejpam-6568	208	3	)	)	PUNCT
ejpam-6568	208	4	τ1τ2	τ1τ2	NOUN
ejpam-6568	208	5	-	-	ADJ
ejpam-6568	208	6	δ	δ	NOUN
ejpam-6568	208	7	-	-	PUNCT
ejpam-6568	208	8	cl(a	cl(a	X
ejpam-6568	208	9	)	)	PUNCT
ejpam-6568	208	10	is	be	AUX
ejpam-6568	208	11	τ1τ2	τ1τ2	NOUN
ejpam-6568	208	12	-	-	ADJ
ejpam-6568	208	13	closed	closed	ADJ
ejpam-6568	208	14	.	.	PUNCT
ejpam-6568	209	1	theorem	theorem	NOUN
ejpam-6568	209	2	5	5	NUM
ejpam-6568	209	3	.	.	X
ejpam-6568	209	4	for	for	ADP
ejpam-6568	209	5	a	a	DET
ejpam-6568	209	6	function	function	NOUN
ejpam-6568	209	7	f	f	NOUN
ejpam-6568	209	8	:	:	PUNCT
ejpam-6568	209	9	(	(	PUNCT
ejpam-6568	209	10	x	x	X
ejpam-6568	209	11	,	,	PUNCT
ejpam-6568	209	12	τ	τ	PROPN
ejpam-6568	209	13	,	,	PUNCT
ejpam-6568	209	14	i	i	NOUN
ejpam-6568	209	15	)	)	PUNCT
ejpam-6568	209	16	→	→	PUNCT
ejpam-6568	209	17	(	(	PUNCT
ejpam-6568	209	18	y	y	PROPN
ejpam-6568	209	19	,	,	PUNCT
ejpam-6568	209	20	σ1	σ1	PROPN
ejpam-6568	209	21	,	,	PUNCT
ejpam-6568	209	22	σ2	σ2	NOUN
ejpam-6568	209	23	)	)	PUNCT
ejpam-6568	209	24	,	,	PUNCT
ejpam-6568	209	25	the	the	DET
ejpam-6568	209	26	following	follow	VERB
ejpam-6568	209	27	properties	property	NOUN
ejpam-6568	209	28	are	be	AUX
ejpam-6568	209	29	equivalent	equivalent	ADJ
ejpam-6568	209	30	:	:	PUNCT
ejpam-6568	209	31	(	(	PUNCT
ejpam-6568	209	32	1	1	X
ejpam-6568	209	33	)	)	PUNCT
ejpam-6568	209	34	f	f	NOUN
ejpam-6568	209	35	is	be	AUX
ejpam-6568	209	36	almost	almost	ADV
ejpam-6568	209	37	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	209	38	,	,	PUNCT
ejpam-6568	209	39	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	209	40	;	;	PUNCT
ejpam-6568	209	41	(	(	PUNCT
ejpam-6568	209	42	2	2	X
ejpam-6568	209	43	)	)	PUNCT
ejpam-6568	209	44	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	209	45	-	-	PUNCT
ejpam-6568	209	46	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	209	47	-	-	PUNCT
ejpam-6568	209	48	int(σ1σ2	int(σ1σ2	VERB
ejpam-6568	209	49	-	-	PUNCT
ejpam-6568	209	50	δ	δ	NOUN
ejpam-6568	209	51	-	-	NOUN
ejpam-6568	209	52	cl(b	cl(b	NOUN
ejpam-6568	209	53	)	)	PUNCT
ejpam-6568	209	54	)	)	PUNCT
ejpam-6568	209	55	)	)	PUNCT
ejpam-6568	209	56	)	)	PUNCT
ejpam-6568	209	57	)	)	PUNCT
ejpam-6568	210	1	⊆	⊆	NUM
ejpam-6568	210	2	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	210	3	-	-	PUNCT
ejpam-6568	210	4	δ	δ	NOUN
ejpam-6568	210	5	-	-	NOUN
ejpam-6568	210	6	cl(b	cl(b	NOUN
ejpam-6568	210	7	)	)	PUNCT
ejpam-6568	210	8	)	)	PUNCT
ejpam-6568	210	9	for	for	ADP
ejpam-6568	210	10	every	every	DET
ejpam-6568	210	11	subset	subset	NOUN
ejpam-6568	210	12	b	b	PROPN
ejpam-6568	210	13	of	of	ADP
ejpam-6568	210	14	y	y	PROPN
ejpam-6568	210	15	;	;	PUNCT
ejpam-6568	210	16	(	(	PUNCT
ejpam-6568	210	17	3	3	X
ejpam-6568	210	18	)	)	PUNCT
ejpam-6568	210	19	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	210	20	-	-	PUNCT
ejpam-6568	210	21	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	210	22	-	-	PUNCT
ejpam-6568	210	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	210	24	-	-	PUNCT
ejpam-6568	210	25	cl(b	cl(b	NOUN
ejpam-6568	210	26	)	)	PUNCT
ejpam-6568	210	27	)	)	PUNCT
ejpam-6568	210	28	)	)	PUNCT
ejpam-6568	210	29	)	)	PUNCT
ejpam-6568	210	30	)	)	PUNCT
ejpam-6568	211	1	⊆	⊆	NUM
ejpam-6568	211	2	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	211	3	-	-	PUNCT
ejpam-6568	211	4	δ	δ	NOUN
ejpam-6568	211	5	-	-	NOUN
ejpam-6568	211	6	cl(b	cl(b	NOUN
ejpam-6568	211	7	)	)	PUNCT
ejpam-6568	211	8	)	)	PUNCT
ejpam-6568	211	9	for	for	ADP
ejpam-6568	211	10	every	every	DET
ejpam-6568	211	11	subset	subset	NOUN
ejpam-6568	211	12	b	b	PROPN
ejpam-6568	211	13	of	of	ADP
ejpam-6568	211	14	y	y	PROPN
ejpam-6568	211	15	.	.	PUNCT
ejpam-6568	212	1	proof	proof	NOUN
ejpam-6568	212	2	.	.	PUNCT
ejpam-6568	213	1	(	(	PUNCT
ejpam-6568	213	2	1	1	X
ejpam-6568	213	3	)	)	PUNCT
ejpam-6568	213	4	⇒	⇒	NOUN
ejpam-6568	213	5	(	(	PUNCT
ejpam-6568	213	6	2	2	NUM
ejpam-6568	213	7	):	):	PUNCT
ejpam-6568	213	8	let	let	VERB
ejpam-6568	213	9	b	b	X
ejpam-6568	213	10	be	be	AUX
ejpam-6568	213	11	any	any	DET
ejpam-6568	213	12	subset	subset	NOUN
ejpam-6568	213	13	of	of	ADP
ejpam-6568	213	14	y	y	PROPN
ejpam-6568	213	15	.	.	PUNCT
ejpam-6568	214	1	by	by	ADP
ejpam-6568	214	2	lemma	lemma	PROPN
ejpam-6568	214	3	4	4	NUM
ejpam-6568	214	4	,	,	PUNCT
ejpam-6568	214	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	214	6	-	-	PUNCT
ejpam-6568	214	7	δ	δ	NOUN
ejpam-6568	214	8	-	-	NOUN
ejpam-6568	214	9	cl(b	cl(b	NOUN
ejpam-6568	214	10	)	)	PUNCT
ejpam-6568	214	11	is	be	AUX
ejpam-6568	214	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	214	13	-	-	ADJ
ejpam-6568	214	14	closed	closed	ADJ
ejpam-6568	214	15	in	in	ADP
ejpam-6568	214	16	y	y	PROPN
ejpam-6568	214	17	and	and	CCONJ
ejpam-6568	214	18	by	by	ADP
ejpam-6568	214	19	theorem	theorem	NOUN
ejpam-6568	214	20	2	2	NUM
ejpam-6568	214	21	,	,	PUNCT
ejpam-6568	214	22	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	214	23	-	-	PUNCT
ejpam-6568	214	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	214	25	-	-	PUNCT
ejpam-6568	214	26	int(σ1σ2	int(σ1σ2	VERB
ejpam-6568	214	27	-	-	PUNCT
ejpam-6568	214	28	δ	δ	NOUN
ejpam-6568	214	29	-	-	NOUN
ejpam-6568	214	30	cl(b	cl(b	NOUN
ejpam-6568	214	31	)	)	PUNCT
ejpam-6568	214	32	)	)	PUNCT
ejpam-6568	214	33	)	)	PUNCT
ejpam-6568	214	34	)	)	PUNCT
ejpam-6568	214	35	)	)	PUNCT
ejpam-6568	215	1	⊆	⊆	NUM
ejpam-6568	215	2	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	215	3	-	-	PUNCT
ejpam-6568	215	4	δ	δ	NOUN
ejpam-6568	215	5	-	-	NOUN
ejpam-6568	215	6	cl(b	cl(b	NOUN
ejpam-6568	215	7	)	)	PUNCT
ejpam-6568	215	8	)	)	PUNCT
ejpam-6568	215	9	.	.	PUNCT
ejpam-6568	216	1	(	(	PUNCT
ejpam-6568	216	2	2	2	X
ejpam-6568	216	3	)	)	PUNCT
ejpam-6568	216	4	⇒	⇒	NOUN
ejpam-6568	216	5	(	(	PUNCT
ejpam-6568	216	6	3	3	NUM
ejpam-6568	216	7	):	):	PUNCT
ejpam-6568	216	8	this	this	PRON
ejpam-6568	216	9	is	be	AUX
ejpam-6568	216	10	obvious	obvious	ADJ
ejpam-6568	216	11	since	since	SCONJ
ejpam-6568	216	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	216	13	-	-	NOUN
ejpam-6568	216	14	cl(b	cl(b	NOUN
ejpam-6568	216	15	)	)	PUNCT
ejpam-6568	216	16	⊆	⊆	NUM
ejpam-6568	216	17	σ1σ2	σ1σ2	NUM
ejpam-6568	216	18	-	-	PUNCT
ejpam-6568	216	19	δ	δ	NOUN
ejpam-6568	216	20	-	-	NOUN
ejpam-6568	216	21	cl(b	cl(b	NOUN
ejpam-6568	216	22	)	)	PUNCT
ejpam-6568	216	23	for	for	ADP
ejpam-6568	216	24	every	every	DET
ejpam-6568	216	25	subset	subset	NOUN
ejpam-6568	216	26	b	b	PROPN
ejpam-6568	216	27	of	of	ADP
ejpam-6568	216	28	y	y	PROPN
ejpam-6568	216	29	.	.	PUNCT
ejpam-6568	217	1	(	(	PUNCT
ejpam-6568	217	2	3	3	X
ejpam-6568	217	3	)	)	PUNCT
ejpam-6568	217	4	⇒	⇒	NOUN
ejpam-6568	217	5	(	(	PUNCT
ejpam-6568	217	6	1	1	NUM
ejpam-6568	217	7	):	):	PUNCT
ejpam-6568	217	8	let	let	VERB
ejpam-6568	217	9	k	k	PRON
ejpam-6568	217	10	be	be	AUX
ejpam-6568	217	11	any	any	DET
ejpam-6568	217	12	(	(	PUNCT
ejpam-6568	217	13	σ1	σ1	NOUN
ejpam-6568	217	14	,	,	PUNCT
ejpam-6568	217	15	σ2)r	σ2)r	NOUN
ejpam-6568	217	16	-	-	PUNCT
ejpam-6568	217	17	closed	close	VERB
ejpam-6568	217	18	set	set	NOUN
ejpam-6568	217	19	of	of	ADP
ejpam-6568	217	20	y	y	PROPN
ejpam-6568	217	21	.	.	PUNCT
ejpam-6568	218	1	then	then	ADV
ejpam-6568	218	2	by	by	ADP
ejpam-6568	218	3	(	(	PUNCT
ejpam-6568	218	4	3	3	NUM
ejpam-6568	218	5	)	)	PUNCT
ejpam-6568	218	6	,	,	PUNCT
ejpam-6568	218	7	we	we	PRON
ejpam-6568	218	8	have	have	VERB
ejpam-6568	218	9	cl⋆(f−1(k	cl⋆(f−1(k	NOUN
ejpam-6568	218	10	)	)	PUNCT
ejpam-6568	218	11	)	)	PUNCT
ejpam-6568	219	1	=	=	PUNCT
ejpam-6568	220	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	220	2	-	-	PUNCT
ejpam-6568	220	3	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	220	4	-	-	PUNCT
ejpam-6568	220	5	int(k	int(k	NOUN
ejpam-6568	220	6	)	)	PUNCT
ejpam-6568	220	7	)	)	PUNCT
ejpam-6568	220	8	)	)	PUNCT
ejpam-6568	220	9	)	)	PUNCT
ejpam-6568	221	1	=	=	PUNCT
ejpam-6568	222	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	222	2	-	-	PUNCT
ejpam-6568	222	3	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	222	4	-	-	PUNCT
ejpam-6568	222	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	222	6	-	-	PUNCT
ejpam-6568	222	7	cl(k	cl(k	NUM
ejpam-6568	222	8	)	)	PUNCT
ejpam-6568	222	9	)	)	PUNCT
ejpam-6568	222	10	)	)	PUNCT
ejpam-6568	222	11	)	)	PUNCT
ejpam-6568	222	12	)	)	PUNCT
ejpam-6568	223	1	⊆	⊆	NUM
ejpam-6568	223	2	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	223	3	-	-	PUNCT
ejpam-6568	223	4	δ	δ	NOUN
ejpam-6568	223	5	-	-	NOUN
ejpam-6568	223	6	cl(k	cl(k	NOUN
ejpam-6568	223	7	)	)	PUNCT
ejpam-6568	223	8	)	)	PUNCT
ejpam-6568	224	1	=	=	PUNCT
ejpam-6568	224	2	f−1(k	f−1(k	PROPN
ejpam-6568	224	3	)	)	PUNCT
ejpam-6568	224	4	and	and	CCONJ
ejpam-6568	224	5	hence	hence	ADV
ejpam-6568	224	6	f−1(k	f−1(k	PROPN
ejpam-6568	224	7	)	)	PUNCT
ejpam-6568	224	8	is	be	AUX
ejpam-6568	224	9	⋆-closed	⋆-close	VERB
ejpam-6568	224	10	in	in	ADP
ejpam-6568	224	11	x.	x.	NOUN
ejpam-6568	224	12	by	by	ADP
ejpam-6568	224	13	theorem	theorem	NOUN
ejpam-6568	224	14	2	2	NUM
ejpam-6568	224	15	,	,	PUNCT
ejpam-6568	224	16	f	f	PROPN
ejpam-6568	224	17	is	be	AUX
ejpam-6568	224	18	almost	almost	ADV
ejpam-6568	224	19	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	224	20	,	,	PUNCT
ejpam-6568	224	21	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	224	22	.	.	PUNCT
ejpam-6568	225	1	n.	n.	PROPN
ejpam-6568	225	2	viriyapong	viriyapong	PROPN
ejpam-6568	225	3	,	,	PUNCT
ejpam-6568	225	4	a.	a.	PROPN
ejpam-6568	225	5	sama	sama	PROPN
ejpam-6568	225	6	-	-	PUNCT
ejpam-6568	225	7	ae	ae	PROPN
ejpam-6568	225	8	,	,	PUNCT
ejpam-6568	225	9	c.	c.	PROPN
ejpam-6568	225	10	boonpok	boonpok	PROPN
ejpam-6568	225	11	/	/	SYM
ejpam-6568	225	12	eur	eur	PROPN
ejpam-6568	225	13	.	.	PUNCT
ejpam-6568	226	1	j.	j.	PROPN
ejpam-6568	226	2	pure	pure	PROPN
ejpam-6568	226	3	appl	appl	PROPN
ejpam-6568	226	4	.	.	PROPN
ejpam-6568	226	5	math	math	PROPN
ejpam-6568	226	6	,	,	PUNCT
ejpam-6568	226	7	18	18	NUM
ejpam-6568	226	8	(	(	PUNCT
ejpam-6568	226	9	3	3	NUM
ejpam-6568	226	10	)	)	PUNCT
ejpam-6568	226	11	(	(	PUNCT
ejpam-6568	226	12	2025	2025	NUM
ejpam-6568	226	13	)	)	PUNCT
ejpam-6568	226	14	,	,	PUNCT
ejpam-6568	226	15	6568	6568	NUM
ejpam-6568	226	16	8	8	NUM
ejpam-6568	226	17	of	of	ADP
ejpam-6568	226	18	18	18	NUM
ejpam-6568	226	19	4	4	NUM
ejpam-6568	226	20	.	.	PUNCT
ejpam-6568	227	1	on	on	ADP
ejpam-6568	227	2	weakly	weakly	ADJ
ejpam-6568	227	3	τ	τ	PROPN
ejpam-6568	227	4	⋆(σ1	⋆(σ1	NOUN
ejpam-6568	227	5	,	,	PUNCT
ejpam-6568	227	6	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	227	7	functions	function	NOUN
ejpam-6568	227	8	in	in	ADP
ejpam-6568	227	9	this	this	DET
ejpam-6568	227	10	section	section	NOUN
ejpam-6568	227	11	,	,	PUNCT
ejpam-6568	227	12	we	we	PRON
ejpam-6568	227	13	introduce	introduce	VERB
ejpam-6568	227	14	the	the	DET
ejpam-6568	227	15	notion	notion	NOUN
ejpam-6568	227	16	of	of	ADP
ejpam-6568	227	17	weakly	weakly	ADJ
ejpam-6568	227	18	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	227	19	,	,	PUNCT
ejpam-6568	227	20	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	227	21	functions	function	NOUN
ejpam-6568	227	22	.	.	PUNCT
ejpam-6568	228	1	moreover	moreover	ADV
ejpam-6568	228	2	,	,	PUNCT
ejpam-6568	228	3	some	some	DET
ejpam-6568	228	4	characterizations	characterization	NOUN
ejpam-6568	228	5	of	of	ADP
ejpam-6568	228	6	weakly	weakly	ADJ
ejpam-6568	228	7	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6568	228	8	,	,	PUNCT
ejpam-6568	228	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	228	10	functions	function	NOUN
ejpam-6568	228	11	are	be	AUX
ejpam-6568	228	12	discussed	discuss	VERB
ejpam-6568	228	13	.	.	PUNCT
ejpam-6568	229	1	definition	definition	NOUN
ejpam-6568	229	2	2	2	NUM
ejpam-6568	229	3	.	.	PUNCT
ejpam-6568	230	1	a	a	DET
ejpam-6568	230	2	function	function	NOUN
ejpam-6568	230	3	f	f	NOUN
ejpam-6568	230	4	:	:	PUNCT
ejpam-6568	230	5	(	(	PUNCT
ejpam-6568	230	6	x	x	X
ejpam-6568	230	7	,	,	PUNCT
ejpam-6568	230	8	τ	τ	PROPN
ejpam-6568	230	9	,	,	PUNCT
ejpam-6568	230	10	i	i	NOUN
ejpam-6568	230	11	)	)	PUNCT
ejpam-6568	230	12	→	→	PUNCT
ejpam-6568	230	13	(	(	PUNCT
ejpam-6568	230	14	y	y	PROPN
ejpam-6568	230	15	,	,	PUNCT
ejpam-6568	230	16	σ1	σ1	PROPN
ejpam-6568	230	17	,	,	PUNCT
ejpam-6568	230	18	σ2	σ2	PROPN
ejpam-6568	230	19	)	)	PUNCT
ejpam-6568	230	20	is	be	AUX
ejpam-6568	230	21	said	say	VERB
ejpam-6568	230	22	to	to	PART
ejpam-6568	230	23	be	be	AUX
ejpam-6568	230	24	weakly	weakly	ADJ
ejpam-6568	230	25	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	230	26	,	,	PUNCT
ejpam-6568	230	27	σ2)continuous	σ2)continuous	ADJ
ejpam-6568	230	28	at	at	ADP
ejpam-6568	230	29	a	a	DET
ejpam-6568	230	30	point	point	NOUN
ejpam-6568	230	31	x	x	SYM
ejpam-6568	230	32	∈	∈	NOUN
ejpam-6568	230	33	x	x	PUNCT
ejpam-6568	230	34	if	if	SCONJ
ejpam-6568	230	35	for	for	ADP
ejpam-6568	230	36	each	each	DET
ejpam-6568	230	37	σ1σ2	σ1σ2	VERB
ejpam-6568	230	38	-	-	ADJ
ejpam-6568	230	39	open	open	ADJ
ejpam-6568	230	40	set	set	NOUN
ejpam-6568	230	41	v	v	NOUN
ejpam-6568	230	42	of	of	ADP
ejpam-6568	230	43	y	y	NOUN
ejpam-6568	230	44	containing	contain	VERB
ejpam-6568	230	45	f(x	f(x	PROPN
ejpam-6568	230	46	)	)	PUNCT
ejpam-6568	230	47	,	,	PUNCT
ejpam-6568	230	48	there	there	PRON
ejpam-6568	230	49	exists	exist	VERB
ejpam-6568	230	50	a	a	DET
ejpam-6568	230	51	⋆-open	⋆-open	ADJ
ejpam-6568	230	52	set	set	NOUN
ejpam-6568	230	53	u	u	NOUN
ejpam-6568	230	54	of	of	ADP
ejpam-6568	230	55	x	x	PUNCT
ejpam-6568	230	56	containing	contain	VERB
ejpam-6568	230	57	x	x	PUNCT
ejpam-6568	230	58	such	such	ADJ
ejpam-6568	230	59	that	that	DET
ejpam-6568	230	60	f(u	f(u	PROPN
ejpam-6568	230	61	)	)	PUNCT
ejpam-6568	230	62	⊆	⊆	NUM
ejpam-6568	230	63	σ1σ2	σ1σ2	NOUN
ejpam-6568	230	64	-	-	NUM
ejpam-6568	230	65	cl(v	cl(v	NOUN
ejpam-6568	230	66	)	)	PUNCT
ejpam-6568	230	67	.	.	PUNCT
ejpam-6568	231	1	a	a	DET
ejpam-6568	231	2	function	function	NOUN
ejpam-6568	231	3	f	f	NOUN
ejpam-6568	231	4	:	:	PUNCT
ejpam-6568	231	5	(	(	PUNCT
ejpam-6568	231	6	x	x	X
ejpam-6568	231	7	,	,	PUNCT
ejpam-6568	231	8	τ	τ	PROPN
ejpam-6568	231	9	,	,	PUNCT
ejpam-6568	231	10	i	i	NOUN
ejpam-6568	231	11	)	)	PUNCT
ejpam-6568	231	12	→	→	PUNCT
ejpam-6568	231	13	(	(	PUNCT
ejpam-6568	231	14	y	y	PROPN
ejpam-6568	231	15	,	,	PUNCT
ejpam-6568	231	16	σ1	σ1	PROPN
ejpam-6568	231	17	,	,	PUNCT
ejpam-6568	231	18	σ2	σ2	PROPN
ejpam-6568	231	19	)	)	PUNCT
ejpam-6568	231	20	is	be	AUX
ejpam-6568	231	21	said	say	VERB
ejpam-6568	231	22	to	to	PART
ejpam-6568	231	23	be	be	AUX
ejpam-6568	231	24	weakly	weakly	ADJ
ejpam-6568	231	25	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	231	26	,	,	PUNCT
ejpam-6568	231	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	231	28	if	if	SCONJ
ejpam-6568	231	29	f	f	PROPN
ejpam-6568	231	30	is	be	AUX
ejpam-6568	231	31	weakly	weakly	ADJ
ejpam-6568	231	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	231	33	,	,	PUNCT
ejpam-6568	231	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	231	35	at	at	ADP
ejpam-6568	231	36	each	each	DET
ejpam-6568	231	37	point	point	NOUN
ejpam-6568	231	38	x	x	PUNCT
ejpam-6568	231	39	of	of	ADP
ejpam-6568	231	40	x.	x.	NOUN
ejpam-6568	231	41	remark	remark	PROPN
ejpam-6568	231	42	1	1	NUM
ejpam-6568	231	43	.	.	PUNCT
ejpam-6568	232	1	for	for	ADP
ejpam-6568	232	2	a	a	DET
ejpam-6568	232	3	function	function	NOUN
ejpam-6568	232	4	f	f	NOUN
ejpam-6568	232	5	:	:	PUNCT
ejpam-6568	232	6	(	(	PUNCT
ejpam-6568	232	7	x	x	X
ejpam-6568	232	8	,	,	PUNCT
ejpam-6568	232	9	τ	τ	PROPN
ejpam-6568	232	10	,	,	PUNCT
ejpam-6568	232	11	i	i	NOUN
ejpam-6568	232	12	)	)	PUNCT
ejpam-6568	232	13	→	→	PUNCT
ejpam-6568	232	14	(	(	PUNCT
ejpam-6568	232	15	y	y	PROPN
ejpam-6568	232	16	,	,	PUNCT
ejpam-6568	232	17	σ1	σ1	PROPN
ejpam-6568	232	18	,	,	PUNCT
ejpam-6568	232	19	σ2	σ2	NOUN
ejpam-6568	232	20	)	)	PUNCT
ejpam-6568	232	21	,	,	PUNCT
ejpam-6568	232	22	the	the	DET
ejpam-6568	232	23	following	follow	VERB
ejpam-6568	232	24	implication	implication	NOUN
ejpam-6568	232	25	holds	hold	VERB
ejpam-6568	232	26	:	:	PUNCT
ejpam-6568	232	27	almost	almost	ADV
ejpam-6568	232	28	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6568	232	29	,	,	PUNCT
ejpam-6568	232	30	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6568	232	31	⇒	⇒	NOUN
ejpam-6568	232	32	weakly	weakly	ADJ
ejpam-6568	232	33	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	232	34	,	,	PUNCT
ejpam-6568	232	35	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6568	232	36	.	.	PUNCT
ejpam-6568	233	1	the	the	DET
ejpam-6568	233	2	converse	converse	NOUN
ejpam-6568	233	3	of	of	ADP
ejpam-6568	233	4	the	the	DET
ejpam-6568	233	5	implication	implication	NOUN
ejpam-6568	233	6	is	be	AUX
ejpam-6568	233	7	not	not	PART
ejpam-6568	233	8	true	true	ADJ
ejpam-6568	233	9	in	in	ADP
ejpam-6568	233	10	general	general	ADJ
ejpam-6568	233	11	.	.	PUNCT
ejpam-6568	234	1	we	we	PRON
ejpam-6568	234	2	give	give	VERB
ejpam-6568	234	3	an	an	DET
ejpam-6568	234	4	example	example	NOUN
ejpam-6568	234	5	for	for	ADP
ejpam-6568	234	6	the	the	DET
ejpam-6568	234	7	implication	implication	NOUN
ejpam-6568	234	8	as	as	SCONJ
ejpam-6568	234	9	follows	follow	VERB
ejpam-6568	234	10	.	.	PUNCT
ejpam-6568	234	11	example	example	NOUN
ejpam-6568	235	1	1	1	NUM
ejpam-6568	235	2	.	.	PUNCT
ejpam-6568	235	3	let	let	VERB
ejpam-6568	235	4	x	x	PUNCT
ejpam-6568	235	5	=	=	PRON
ejpam-6568	235	6	{	{	PUNCT
ejpam-6568	235	7	1	1	NUM
ejpam-6568	235	8	,	,	PUNCT
ejpam-6568	235	9	2	2	NUM
ejpam-6568	235	10	,	,	PUNCT
ejpam-6568	235	11	3	3	NUM
ejpam-6568	235	12	}	}	PUNCT
ejpam-6568	235	13	with	with	ADP
ejpam-6568	235	14	a	a	DET
ejpam-6568	235	15	topology	topology	NOUN
ejpam-6568	235	16	τ	τ	X
ejpam-6568	235	17	=	=	SYM
ejpam-6568	235	18	{	{	PUNCT
ejpam-6568	235	19	∅	∅	NOUN
ejpam-6568	235	20	,	,	PUNCT
ejpam-6568	235	21	{	{	PUNCT
ejpam-6568	235	22	2	2	NUM
ejpam-6568	235	23	}	}	PUNCT
ejpam-6568	235	24	,	,	PUNCT
ejpam-6568	235	25	{	{	PUNCT
ejpam-6568	235	26	1	1	NUM
ejpam-6568	235	27	,	,	PUNCT
ejpam-6568	235	28	3	3	NUM
ejpam-6568	235	29	}	}	PUNCT
ejpam-6568	235	30	,	,	PUNCT
ejpam-6568	235	31	x	x	NOUN
ejpam-6568	235	32	}	}	PUNCT
ejpam-6568	235	33	and	and	CCONJ
ejpam-6568	235	34	an	an	DET
ejpam-6568	235	35	ideal	ideal	NOUN
ejpam-6568	235	36	i	i	X
ejpam-6568	235	37	=	=	SYM
ejpam-6568	235	38	{	{	PUNCT
ejpam-6568	235	39	∅	∅	NOUN
ejpam-6568	235	40	}	}	PUNCT
ejpam-6568	235	41	.	.	PUNCT
ejpam-6568	236	1	let	let	VERB
ejpam-6568	236	2	y	y	PROPN
ejpam-6568	236	3	=	=	PUNCT
ejpam-6568	236	4	{	{	PUNCT
ejpam-6568	236	5	a	a	PRON
ejpam-6568	236	6	,	,	PUNCT
ejpam-6568	236	7	b	b	NOUN
ejpam-6568	236	8	,	,	PUNCT
ejpam-6568	236	9	c	c	NOUN
ejpam-6568	236	10	}	}	PUNCT
ejpam-6568	236	11	with	with	ADP
ejpam-6568	236	12	topologies	topology	NOUN
ejpam-6568	236	13	σ1	σ1	NOUN
ejpam-6568	236	14	=	=	SYM
ejpam-6568	236	15	{	{	PUNCT
ejpam-6568	236	16	∅	∅	NOUN
ejpam-6568	236	17	,	,	PUNCT
ejpam-6568	236	18	{	{	PUNCT
ejpam-6568	236	19	a	a	X
ejpam-6568	236	20	}	}	PUNCT
ejpam-6568	236	21	,	,	PUNCT
ejpam-6568	236	22	{	{	PUNCT
ejpam-6568	236	23	a	a	DET
ejpam-6568	236	24	,	,	PUNCT
ejpam-6568	236	25	b	b	NOUN
ejpam-6568	236	26	}	}	PUNCT
ejpam-6568	236	27	,	,	PUNCT
ejpam-6568	236	28	y	y	PROPN
ejpam-6568	236	29	}	}	PUNCT
ejpam-6568	236	30	and	and	CCONJ
ejpam-6568	236	31	σ2	σ2	PROPN
ejpam-6568	236	32	=	=	SYM
ejpam-6568	236	33	{	{	PUNCT
ejpam-6568	236	34	∅	∅	NOUN
ejpam-6568	236	35	,	,	PUNCT
ejpam-6568	236	36	{	{	PUNCT
ejpam-6568	236	37	a	a	X
ejpam-6568	236	38	}	}	PUNCT
ejpam-6568	236	39	,	,	PUNCT
ejpam-6568	236	40	{	{	PUNCT
ejpam-6568	236	41	b	b	NOUN
ejpam-6568	236	42	}	}	PUNCT
ejpam-6568	236	43	,	,	PUNCT
ejpam-6568	236	44	{	{	PUNCT
ejpam-6568	236	45	a	a	DET
ejpam-6568	236	46	,	,	PUNCT
ejpam-6568	236	47	b	b	NOUN
ejpam-6568	236	48	}	}	PUNCT
ejpam-6568	236	49	,	,	PUNCT
ejpam-6568	236	50	y	y	PROPN
ejpam-6568	236	51	}	}	PUNCT
ejpam-6568	236	52	.	.	PUNCT
ejpam-6568	237	1	a	a	DET
ejpam-6568	237	2	function	function	NOUN
ejpam-6568	237	3	f	f	NOUN
ejpam-6568	237	4	:	:	PUNCT
ejpam-6568	237	5	(	(	PUNCT
ejpam-6568	237	6	x	x	X
ejpam-6568	237	7	,	,	PUNCT
ejpam-6568	237	8	τ	τ	PROPN
ejpam-6568	237	9	,	,	PUNCT
ejpam-6568	237	10	i	i	NOUN
ejpam-6568	237	11	)	)	PUNCT
ejpam-6568	237	12	→	→	PUNCT
ejpam-6568	237	13	(	(	PUNCT
ejpam-6568	237	14	y	y	PROPN
ejpam-6568	237	15	,	,	PUNCT
ejpam-6568	237	16	σ1	σ1	PROPN
ejpam-6568	237	17	,	,	PUNCT
ejpam-6568	237	18	σ2	σ2	PROPN
ejpam-6568	237	19	)	)	PUNCT
ejpam-6568	237	20	is	be	AUX
ejpam-6568	237	21	defined	define	VERB
ejpam-6568	237	22	as	as	SCONJ
ejpam-6568	237	23	follows	follow	VERB
ejpam-6568	237	24	:	:	PUNCT
ejpam-6568	237	25	f(1	f(1	X
ejpam-6568	237	26	)	)	PUNCT
ejpam-6568	237	27	=	=	SYM
ejpam-6568	238	1	a	a	PRON
ejpam-6568	238	2	,	,	PUNCT
ejpam-6568	238	3	f(2	f(2	PROPN
ejpam-6568	238	4	)	)	PUNCT
ejpam-6568	238	5	=	=	SYM
ejpam-6568	238	6	b	b	PROPN
ejpam-6568	238	7	and	and	CCONJ
ejpam-6568	238	8	f(3	f(3	PROPN
ejpam-6568	238	9	)	)	PUNCT
ejpam-6568	239	1	=	=	SYM
ejpam-6568	239	2	c.	c.	NOUN
ejpam-6568	239	3	then	then	ADV
ejpam-6568	239	4	,	,	PUNCT
ejpam-6568	239	5	f	f	PROPN
ejpam-6568	239	6	is	be	AUX
ejpam-6568	239	7	weakly	weakly	ADJ
ejpam-6568	239	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	239	9	,	,	PUNCT
ejpam-6568	239	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	239	11	but	but	CCONJ
ejpam-6568	239	12	f	f	PROPN
ejpam-6568	239	13	is	be	AUX
ejpam-6568	239	14	not	not	PART
ejpam-6568	239	15	almost	almost	ADV
ejpam-6568	239	16	τ⋆(σ1	τ⋆(σ1	ADJ
ejpam-6568	239	17	,	,	PUNCT
ejpam-6568	239	18	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	239	19	.	.	X
ejpam-6568	239	20	theorem	theorem	VERB
ejpam-6568	239	21	6	6	NUM
ejpam-6568	239	22	.	.	PUNCT
ejpam-6568	239	23	for	for	ADP
ejpam-6568	239	24	a	a	DET
ejpam-6568	239	25	function	function	NOUN
ejpam-6568	239	26	f	f	NOUN
ejpam-6568	239	27	:	:	PUNCT
ejpam-6568	239	28	(	(	PUNCT
ejpam-6568	239	29	x	x	X
ejpam-6568	239	30	,	,	PUNCT
ejpam-6568	239	31	τ	τ	PROPN
ejpam-6568	239	32	,	,	PUNCT
ejpam-6568	239	33	i	i	NOUN
ejpam-6568	239	34	)	)	PUNCT
ejpam-6568	239	35	→	→	PUNCT
ejpam-6568	239	36	(	(	PUNCT
ejpam-6568	239	37	y	y	PROPN
ejpam-6568	239	38	,	,	PUNCT
ejpam-6568	239	39	σ1	σ1	PROPN
ejpam-6568	239	40	,	,	PUNCT
ejpam-6568	239	41	σ2	σ2	NOUN
ejpam-6568	239	42	)	)	PUNCT
ejpam-6568	239	43	,	,	PUNCT
ejpam-6568	239	44	the	the	DET
ejpam-6568	239	45	following	follow	VERB
ejpam-6568	239	46	properties	property	NOUN
ejpam-6568	239	47	are	be	AUX
ejpam-6568	239	48	equivalent	equivalent	ADJ
ejpam-6568	239	49	:	:	PUNCT
ejpam-6568	239	50	(	(	PUNCT
ejpam-6568	239	51	1	1	X
ejpam-6568	239	52	)	)	PUNCT
ejpam-6568	239	53	f	f	PROPN
ejpam-6568	239	54	is	be	AUX
ejpam-6568	239	55	weakly	weakly	ADJ
ejpam-6568	239	56	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	239	57	,	,	PUNCT
ejpam-6568	239	58	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	239	59	;	;	PUNCT
ejpam-6568	239	60	(	(	PUNCT
ejpam-6568	239	61	2	2	X
ejpam-6568	239	62	)	)	PUNCT
ejpam-6568	239	63	f−1(v	f−1(v	NOUN
ejpam-6568	239	64	)	)	PUNCT
ejpam-6568	240	1	⊆	⊆	X
ejpam-6568	240	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	240	3	-	-	PUNCT
ejpam-6568	240	4	cl(v	cl(v	NOUN
ejpam-6568	240	5	)	)	PUNCT
ejpam-6568	240	6	)	)	PUNCT
ejpam-6568	240	7	)	)	PUNCT
ejpam-6568	240	8	for	for	ADP
ejpam-6568	240	9	every	every	DET
ejpam-6568	240	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	240	11	-	-	ADJ
ejpam-6568	240	12	open	open	ADJ
ejpam-6568	240	13	set	set	NOUN
ejpam-6568	240	14	v	v	NOUN
ejpam-6568	240	15	of	of	ADP
ejpam-6568	240	16	y	y	PROPN
ejpam-6568	240	17	;	;	PUNCT
ejpam-6568	240	18	(	(	PUNCT
ejpam-6568	240	19	3	3	X
ejpam-6568	240	20	)	)	PUNCT
ejpam-6568	240	21	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	240	22	-	-	PUNCT
ejpam-6568	240	23	int(k	int(k	PROPN
ejpam-6568	240	24	)	)	PUNCT
ejpam-6568	240	25	)	)	PUNCT
ejpam-6568	240	26	)	)	PUNCT
ejpam-6568	240	27	⊆	⊆	NUM
ejpam-6568	240	28	f−1(k	f−1(k	PROPN
ejpam-6568	240	29	)	)	PUNCT
ejpam-6568	240	30	for	for	ADP
ejpam-6568	240	31	every	every	DET
ejpam-6568	240	32	σ1σ2	σ1σ2	NUM
ejpam-6568	240	33	-	-	PUNCT
ejpam-6568	240	34	closed	closed	ADJ
ejpam-6568	240	35	set	set	NOUN
ejpam-6568	240	36	k	k	PROPN
ejpam-6568	240	37	of	of	ADP
ejpam-6568	240	38	y	y	PROPN
ejpam-6568	240	39	;	;	PUNCT
ejpam-6568	240	40	(	(	PUNCT
ejpam-6568	240	41	4	4	X
ejpam-6568	240	42	)	)	PUNCT
ejpam-6568	240	43	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	240	44	-	-	PUNCT
ejpam-6568	240	45	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	240	46	-	-	PUNCT
ejpam-6568	240	47	cl(b	cl(b	NOUN
ejpam-6568	240	48	)	)	PUNCT
ejpam-6568	240	49	)	)	PUNCT
ejpam-6568	240	50	)	)	PUNCT
ejpam-6568	240	51	)	)	PUNCT
ejpam-6568	240	52	⊆	⊆	NUM
ejpam-6568	240	53	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	240	54	-	-	PUNCT
ejpam-6568	240	55	cl(b	cl(b	NOUN
ejpam-6568	240	56	)	)	PUNCT
ejpam-6568	240	57	)	)	PUNCT
ejpam-6568	240	58	for	for	ADP
ejpam-6568	240	59	every	every	DET
ejpam-6568	240	60	subset	subset	NOUN
ejpam-6568	240	61	b	b	PROPN
ejpam-6568	240	62	of	of	ADP
ejpam-6568	240	63	y	y	PROPN
ejpam-6568	240	64	;	;	PUNCT
ejpam-6568	240	65	(	(	PUNCT
ejpam-6568	240	66	5	5	X
ejpam-6568	240	67	)	)	PUNCT
ejpam-6568	240	68	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	240	69	-	-	PUNCT
ejpam-6568	240	70	int(b	int(b	NOUN
ejpam-6568	240	71	)	)	PUNCT
ejpam-6568	240	72	)	)	PUNCT
ejpam-6568	241	1	⊆	⊆	X
ejpam-6568	241	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	241	3	-	-	PUNCT
ejpam-6568	241	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	241	5	-	-	PUNCT
ejpam-6568	241	6	int(b	int(b	NOUN
ejpam-6568	241	7	)	)	PUNCT
ejpam-6568	241	8	)	)	PUNCT
ejpam-6568	241	9	)	)	PUNCT
ejpam-6568	241	10	)	)	PUNCT
ejpam-6568	241	11	for	for	ADP
ejpam-6568	241	12	every	every	DET
ejpam-6568	241	13	subset	subset	NOUN
ejpam-6568	241	14	b	b	PROPN
ejpam-6568	241	15	of	of	ADP
ejpam-6568	241	16	y	y	PROPN
ejpam-6568	241	17	;	;	PUNCT
ejpam-6568	241	18	(	(	PUNCT
ejpam-6568	241	19	6	6	X
ejpam-6568	241	20	)	)	PUNCT
ejpam-6568	241	21	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	241	22	-	-	PUNCT
ejpam-6568	241	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	241	24	-	-	PUNCT
ejpam-6568	241	25	cl(v	cl(v	NOUN
ejpam-6568	241	26	)	)	PUNCT
ejpam-6568	241	27	)	)	PUNCT
ejpam-6568	241	28	)	)	PUNCT
ejpam-6568	241	29	)	)	PUNCT
ejpam-6568	241	30	⊆	⊆	NUM
ejpam-6568	241	31	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	241	32	-	-	PUNCT
ejpam-6568	241	33	cl(v	cl(v	NOUN
ejpam-6568	241	34	)	)	PUNCT
ejpam-6568	241	35	)	)	PUNCT
ejpam-6568	241	36	for	for	ADP
ejpam-6568	241	37	every	every	DET
ejpam-6568	241	38	σ1σ2	σ1σ2	NOUN
ejpam-6568	241	39	-	-	ADJ
ejpam-6568	241	40	open	open	ADJ
ejpam-6568	241	41	set	set	NOUN
ejpam-6568	241	42	v	v	NOUN
ejpam-6568	241	43	of	of	ADP
ejpam-6568	241	44	y	y	PROPN
ejpam-6568	241	45	;	;	PUNCT
ejpam-6568	241	46	(	(	PUNCT
ejpam-6568	241	47	7	7	X
ejpam-6568	241	48	)	)	PUNCT
ejpam-6568	241	49	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	241	50	)	)	PUNCT
ejpam-6568	241	51	)	)	PUNCT
ejpam-6568	241	52	⊆	⊆	NUM
ejpam-6568	241	53	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	241	54	-	-	PUNCT
ejpam-6568	241	55	cl(v	cl(v	NOUN
ejpam-6568	241	56	)	)	PUNCT
ejpam-6568	241	57	)	)	PUNCT
ejpam-6568	241	58	for	for	ADP
ejpam-6568	241	59	every	every	DET
ejpam-6568	241	60	σ1σ2	σ1σ2	NOUN
ejpam-6568	241	61	-	-	ADJ
ejpam-6568	241	62	open	open	ADJ
ejpam-6568	241	63	set	set	NOUN
ejpam-6568	241	64	v	v	NOUN
ejpam-6568	241	65	of	of	ADP
ejpam-6568	241	66	y	y	PROPN
ejpam-6568	241	67	;	;	PUNCT
ejpam-6568	241	68	(	(	PUNCT
ejpam-6568	241	69	8)	8)	NUM
ejpam-6568	241	70	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	241	71	-	-	PUNCT
ejpam-6568	241	72	int(k	int(k	PROPN
ejpam-6568	241	73	)	)	PUNCT
ejpam-6568	241	74	)	)	PUNCT
ejpam-6568	241	75	)	)	PUNCT
ejpam-6568	241	76	⊆	⊆	NUM
ejpam-6568	241	77	f−1(k	f−1(k	PROPN
ejpam-6568	241	78	)	)	PUNCT
ejpam-6568	241	79	for	for	ADP
ejpam-6568	241	80	every	every	DET
ejpam-6568	241	81	(	(	PUNCT
ejpam-6568	241	82	σ1	σ1	PROPN
ejpam-6568	241	83	,	,	PUNCT
ejpam-6568	241	84	σ2)r	σ2)r	NOUN
ejpam-6568	241	85	-	-	PUNCT
ejpam-6568	241	86	closed	close	VERB
ejpam-6568	241	87	set	set	ADJ
ejpam-6568	241	88	k	k	PROPN
ejpam-6568	241	89	of	of	ADP
ejpam-6568	241	90	y	y	PROPN
ejpam-6568	241	91	.	.	PUNCT
ejpam-6568	242	1	n.	n.	PROPN
ejpam-6568	242	2	viriyapong	viriyapong	PROPN
ejpam-6568	242	3	,	,	PUNCT
ejpam-6568	242	4	a.	a.	PROPN
ejpam-6568	242	5	sama	sama	PROPN
ejpam-6568	242	6	-	-	PUNCT
ejpam-6568	242	7	ae	ae	PROPN
ejpam-6568	242	8	,	,	PUNCT
ejpam-6568	242	9	c.	c.	PROPN
ejpam-6568	242	10	boonpok	boonpok	PROPN
ejpam-6568	242	11	/	/	SYM
ejpam-6568	242	12	eur	eur	PROPN
ejpam-6568	242	13	.	.	PUNCT
ejpam-6568	243	1	j.	j.	PROPN
ejpam-6568	243	2	pure	pure	PROPN
ejpam-6568	243	3	appl	appl	PROPN
ejpam-6568	243	4	.	.	PROPN
ejpam-6568	243	5	math	math	PROPN
ejpam-6568	243	6	,	,	PUNCT
ejpam-6568	243	7	18	18	NUM
ejpam-6568	243	8	(	(	PUNCT
ejpam-6568	243	9	3	3	NUM
ejpam-6568	243	10	)	)	PUNCT
ejpam-6568	243	11	(	(	PUNCT
ejpam-6568	243	12	2025	2025	NUM
ejpam-6568	243	13	)	)	PUNCT
ejpam-6568	243	14	,	,	PUNCT
ejpam-6568	243	15	6568	6568	NUM
ejpam-6568	243	16	9	9	NUM
ejpam-6568	243	17	of	of	ADP
ejpam-6568	243	18	18	18	NUM
ejpam-6568	243	19	proof	proof	NOUN
ejpam-6568	243	20	.	.	PUNCT
ejpam-6568	244	1	(	(	PUNCT
ejpam-6568	244	2	1	1	X
ejpam-6568	244	3	)	)	PUNCT
ejpam-6568	244	4	⇒	⇒	NOUN
ejpam-6568	244	5	(	(	PUNCT
ejpam-6568	244	6	2	2	NUM
ejpam-6568	244	7	):	):	PUNCT
ejpam-6568	244	8	let	let	VERB
ejpam-6568	244	9	v	v	PART
ejpam-6568	244	10	be	be	AUX
ejpam-6568	244	11	any	any	DET
ejpam-6568	244	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	244	13	-	-	ADJ
ejpam-6568	244	14	open	open	ADJ
ejpam-6568	244	15	set	set	NOUN
ejpam-6568	244	16	of	of	ADP
ejpam-6568	244	17	y	y	PRON
ejpam-6568	244	18	such	such	ADJ
ejpam-6568	244	19	that	that	SCONJ
ejpam-6568	244	20	x	x	SYM
ejpam-6568	244	21	∈	∈	PROPN
ejpam-6568	244	22	f−1(v	f−1(v	NOUN
ejpam-6568	244	23	)	)	PUNCT
ejpam-6568	244	24	.	.	PUNCT
ejpam-6568	245	1	then	then	ADV
ejpam-6568	245	2	,	,	PUNCT
ejpam-6568	245	3	we	we	PRON
ejpam-6568	245	4	have	have	VERB
ejpam-6568	245	5	f(x	f(x	PROPN
ejpam-6568	245	6	)	)	PUNCT
ejpam-6568	245	7	∈	∈	PROPN
ejpam-6568	245	8	v	v	NOUN
ejpam-6568	245	9	.	.	PUNCT
ejpam-6568	246	1	by	by	ADP
ejpam-6568	246	2	(	(	PUNCT
ejpam-6568	246	3	1	1	NUM
ejpam-6568	246	4	)	)	PUNCT
ejpam-6568	246	5	,	,	PUNCT
ejpam-6568	246	6	there	there	PRON
ejpam-6568	246	7	exists	exist	VERB
ejpam-6568	246	8	a	a	DET
ejpam-6568	246	9	⋆-open	⋆-open	ADJ
ejpam-6568	246	10	set	set	NOUN
ejpam-6568	246	11	u	u	NOUN
ejpam-6568	246	12	of	of	ADP
ejpam-6568	246	13	x	x	PUNCT
ejpam-6568	246	14	containing	contain	VERB
ejpam-6568	246	15	x	x	PUNCT
ejpam-6568	246	16	such	such	ADJ
ejpam-6568	246	17	that	that	DET
ejpam-6568	246	18	f(u	f(u	PROPN
ejpam-6568	246	19	)	)	PUNCT
ejpam-6568	246	20	⊆	⊆	NUM
ejpam-6568	246	21	σ1σ2	σ1σ2	NOUN
ejpam-6568	246	22	-	-	NUM
ejpam-6568	246	23	cl(v	cl(v	NOUN
ejpam-6568	246	24	)	)	PUNCT
ejpam-6568	246	25	.	.	PUNCT
ejpam-6568	247	1	therefore	therefore	ADV
ejpam-6568	247	2	,	,	PUNCT
ejpam-6568	247	3	u	u	NOUN
ejpam-6568	247	4	⊆	⊆	NUM
ejpam-6568	247	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	247	6	-	-	PUNCT
ejpam-6568	247	7	cl(v	cl(v	NOUN
ejpam-6568	247	8	)	)	PUNCT
ejpam-6568	247	9	)	)	PUNCT
ejpam-6568	247	10	.	.	PUNCT
ejpam-6568	248	1	since	since	SCONJ
ejpam-6568	248	2	u	u	NOUN
ejpam-6568	248	3	is	be	AUX
ejpam-6568	248	4	⋆-open	⋆-open	ADJ
ejpam-6568	248	5	,	,	PUNCT
ejpam-6568	248	6	we	we	PRON
ejpam-6568	248	7	have	have	VERB
ejpam-6568	248	8	x	x	PART
ejpam-6568	248	9	∈	∈	PROPN
ejpam-6568	248	10	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NOUN
ejpam-6568	248	11	-	-	PUNCT
ejpam-6568	248	12	cl(v	cl(v	NOUN
ejpam-6568	248	13	)	)	PUNCT
ejpam-6568	248	14	)	)	PUNCT
ejpam-6568	248	15	)	)	PUNCT
ejpam-6568	248	16	and	and	CCONJ
ejpam-6568	248	17	hence	hence	ADV
ejpam-6568	248	18	f−1(v	f−1(v	NOUN
ejpam-6568	248	19	)	)	PUNCT
ejpam-6568	248	20	⊆	⊆	X
ejpam-6568	248	21	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	248	22	-	-	PUNCT
ejpam-6568	248	23	cl(v	cl(v	NOUN
ejpam-6568	248	24	)	)	PUNCT
ejpam-6568	248	25	)	)	PUNCT
ejpam-6568	248	26	)	)	PUNCT
ejpam-6568	248	27	.	.	PUNCT
ejpam-6568	249	1	(	(	PUNCT
ejpam-6568	249	2	2	2	X
ejpam-6568	249	3	)	)	PUNCT
ejpam-6568	249	4	⇒	⇒	NOUN
ejpam-6568	249	5	(	(	PUNCT
ejpam-6568	249	6	3	3	NUM
ejpam-6568	249	7	):	):	PUNCT
ejpam-6568	249	8	let	let	VERB
ejpam-6568	249	9	k	k	PRON
ejpam-6568	249	10	be	be	AUX
ejpam-6568	249	11	any	any	DET
ejpam-6568	249	12	σ1σ2	σ1σ2	NUM
ejpam-6568	249	13	-	-	PUNCT
ejpam-6568	249	14	closed	closed	ADJ
ejpam-6568	249	15	set	set	NOUN
ejpam-6568	249	16	of	of	ADP
ejpam-6568	249	17	y	y	PROPN
ejpam-6568	249	18	.	.	PUNCT
ejpam-6568	250	1	then	then	ADV
ejpam-6568	250	2	,	,	PUNCT
ejpam-6568	250	3	y	y	PROPN
ejpam-6568	250	4	−k	−k	PROPN
ejpam-6568	250	5	is	be	AUX
ejpam-6568	250	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	250	7	-	-	ADJ
ejpam-6568	250	8	open	open	ADJ
ejpam-6568	250	9	in	in	ADP
ejpam-6568	250	10	y	y	PROPN
ejpam-6568	250	11	.	.	PUNCT
ejpam-6568	251	1	by	by	ADP
ejpam-6568	251	2	(	(	PUNCT
ejpam-6568	251	3	2	2	NUM
ejpam-6568	251	4	)	)	PUNCT
ejpam-6568	251	5	,	,	PUNCT
ejpam-6568	251	6	x	x	PUNCT
ejpam-6568	251	7	−	−	PROPN
ejpam-6568	251	8	f−1(k	f−1(k	PROPN
ejpam-6568	251	9	)	)	PUNCT
ejpam-6568	251	10	=	=	SYM
ejpam-6568	251	11	f−1(y	f−1(y	PROPN
ejpam-6568	251	12	−k	−k	PROPN
ejpam-6568	251	13	)	)	PUNCT
ejpam-6568	251	14	⊆	⊆	NUM
ejpam-6568	251	15	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	251	16	-	-	PUNCT
ejpam-6568	251	17	cl(y	cl(y	NOUN
ejpam-6568	251	18	−k	−k	NOUN
ejpam-6568	251	19	)	)	PUNCT
ejpam-6568	251	20	)	)	PUNCT
ejpam-6568	251	21	)	)	PUNCT
ejpam-6568	252	1	=	=	PUNCT
ejpam-6568	252	2	x	x	SYM
ejpam-6568	252	3	−cl⋆(f−1(σ1σ2	−cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	252	4	-	-	PUNCT
ejpam-6568	252	5	int(k	int(k	NOUN
ejpam-6568	252	6	)	)	PUNCT
ejpam-6568	252	7	)	)	PUNCT
ejpam-6568	252	8	)	)	PUNCT
ejpam-6568	252	9	.	.	PUNCT
ejpam-6568	253	1	thus	thus	ADV
ejpam-6568	253	2	,	,	PUNCT
ejpam-6568	253	3	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	253	4	-	-	PUNCT
ejpam-6568	253	5	int(k	int(k	PROPN
ejpam-6568	253	6	)	)	PUNCT
ejpam-6568	253	7	)	)	PUNCT
ejpam-6568	253	8	)	)	PUNCT
ejpam-6568	253	9	⊆	⊆	NUM
ejpam-6568	253	10	f−1(k	f−1(k	NOUN
ejpam-6568	253	11	)	)	PUNCT
ejpam-6568	253	12	.	.	PUNCT
ejpam-6568	254	1	(	(	PUNCT
ejpam-6568	254	2	3	3	X
ejpam-6568	254	3	)	)	PUNCT
ejpam-6568	254	4	⇒	⇒	NOUN
ejpam-6568	254	5	(	(	PUNCT
ejpam-6568	254	6	4	4	NUM
ejpam-6568	254	7	):	):	PUNCT
ejpam-6568	254	8	let	let	VERB
ejpam-6568	254	9	b	b	X
ejpam-6568	254	10	be	be	AUX
ejpam-6568	254	11	any	any	DET
ejpam-6568	254	12	subset	subset	NOUN
ejpam-6568	254	13	of	of	ADP
ejpam-6568	254	14	y	y	PROPN
ejpam-6568	254	15	.	.	PUNCT
ejpam-6568	255	1	then	then	ADV
ejpam-6568	255	2	,	,	PUNCT
ejpam-6568	255	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	255	4	-	-	NOUN
ejpam-6568	255	5	cl(b	cl(b	NOUN
ejpam-6568	255	6	)	)	PUNCT
ejpam-6568	255	7	is	be	AUX
ejpam-6568	255	8	a	a	DET
ejpam-6568	255	9	σ1σ2	σ1σ2	NUM
ejpam-6568	255	10	-	-	PUNCT
ejpam-6568	255	11	closed	closed	ADJ
ejpam-6568	255	12	set	set	NOUN
ejpam-6568	255	13	of	of	ADP
ejpam-6568	255	14	y	y	PROPN
ejpam-6568	255	15	and	and	CCONJ
ejpam-6568	255	16	by	by	ADP
ejpam-6568	255	17	(	(	PUNCT
ejpam-6568	255	18	3	3	NUM
ejpam-6568	255	19	)	)	PUNCT
ejpam-6568	255	20	,	,	PUNCT
ejpam-6568	255	21	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	255	22	-	-	PUNCT
ejpam-6568	255	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	255	24	-	-	PUNCT
ejpam-6568	255	25	cl(b	cl(b	NOUN
ejpam-6568	255	26	)	)	PUNCT
ejpam-6568	255	27	)	)	PUNCT
ejpam-6568	255	28	)	)	PUNCT
ejpam-6568	255	29	)	)	PUNCT
ejpam-6568	256	1	⊆	⊆	NUM
ejpam-6568	256	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	256	3	-	-	PUNCT
ejpam-6568	256	4	cl(b	cl(b	NOUN
ejpam-6568	256	5	)	)	PUNCT
ejpam-6568	256	6	)	)	PUNCT
ejpam-6568	256	7	.	.	PUNCT
ejpam-6568	257	1	(	(	PUNCT
ejpam-6568	257	2	4	4	X
ejpam-6568	257	3	)	)	PUNCT
ejpam-6568	257	4	⇒	⇒	NOUN
ejpam-6568	257	5	(	(	PUNCT
ejpam-6568	257	6	5	5	NUM
ejpam-6568	257	7	):	):	PUNCT
ejpam-6568	257	8	let	let	VERB
ejpam-6568	257	9	b	b	X
ejpam-6568	257	10	be	be	AUX
ejpam-6568	257	11	any	any	DET
ejpam-6568	257	12	subset	subset	NOUN
ejpam-6568	257	13	of	of	ADP
ejpam-6568	257	14	y	y	PROPN
ejpam-6568	257	15	.	.	PUNCT
ejpam-6568	258	1	thus	thus	ADV
ejpam-6568	258	2	by	by	ADP
ejpam-6568	258	3	(	(	PUNCT
ejpam-6568	258	4	4	4	NUM
ejpam-6568	258	5	)	)	PUNCT
ejpam-6568	258	6	,	,	PUNCT
ejpam-6568	258	7	we	we	PRON
ejpam-6568	258	8	have	have	VERB
ejpam-6568	258	9	x	x	INTJ
ejpam-6568	258	10	−	−	VERB
ejpam-6568	258	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	ADV
ejpam-6568	258	12	-	-	PUNCT
ejpam-6568	258	13	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	258	14	-	-	PUNCT
ejpam-6568	258	15	int(b	int(b	NOUN
ejpam-6568	258	16	)	)	PUNCT
ejpam-6568	258	17	)	)	PUNCT
ejpam-6568	258	18	)	)	PUNCT
ejpam-6568	258	19	)	)	PUNCT
ejpam-6568	259	1	=	=	PUNCT
ejpam-6568	259	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6568	259	3	−	−	NOUN
ejpam-6568	259	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	259	5	-	-	PUNCT
ejpam-6568	259	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	259	7	-	-	PUNCT
ejpam-6568	259	8	int(b	int(b	NOUN
ejpam-6568	259	9	)	)	PUNCT
ejpam-6568	259	10	)	)	PUNCT
ejpam-6568	259	11	)	)	PUNCT
ejpam-6568	259	12	)	)	PUNCT
ejpam-6568	260	1	=	=	SYM
ejpam-6568	261	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	261	2	-	-	PUNCT
ejpam-6568	261	3	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	261	4	-	-	PUNCT
ejpam-6568	261	5	cl(y	cl(y	NOUN
ejpam-6568	261	6	−b	−b	NOUN
ejpam-6568	261	7	)	)	PUNCT
ejpam-6568	261	8	)	)	PUNCT
ejpam-6568	261	9	)	)	PUNCT
ejpam-6568	261	10	)	)	PUNCT
ejpam-6568	262	1	⊆	⊆	NUM
ejpam-6568	262	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	262	3	-	-	PUNCT
ejpam-6568	262	4	cl(y	cl(y	NOUN
ejpam-6568	262	5	−b	−b	NOUN
ejpam-6568	262	6	)	)	PUNCT
ejpam-6568	262	7	)	)	PUNCT
ejpam-6568	263	1	=	=	PUNCT
ejpam-6568	263	2	x	x	X
ejpam-6568	263	3	−	−	PRON
ejpam-6568	263	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	263	5	-	-	PUNCT
ejpam-6568	263	6	int(b	int(b	NOUN
ejpam-6568	263	7	)	)	PUNCT
ejpam-6568	263	8	)	)	PUNCT
ejpam-6568	263	9	and	and	CCONJ
ejpam-6568	263	10	hence	hence	ADV
ejpam-6568	263	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	263	12	-	-	PUNCT
ejpam-6568	263	13	int(b	int(b	NOUN
ejpam-6568	263	14	)	)	PUNCT
ejpam-6568	263	15	)	)	PUNCT
ejpam-6568	264	1	⊆	⊆	X
ejpam-6568	264	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	264	3	-	-	PUNCT
ejpam-6568	264	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	264	5	-	-	PUNCT
ejpam-6568	264	6	int(b	int(b	NOUN
ejpam-6568	264	7	)	)	PUNCT
ejpam-6568	264	8	)	)	PUNCT
ejpam-6568	264	9	)	)	PUNCT
ejpam-6568	264	10	)	)	PUNCT
ejpam-6568	264	11	.	.	PUNCT
ejpam-6568	265	1	(	(	PUNCT
ejpam-6568	265	2	5	5	X
ejpam-6568	265	3	)	)	PUNCT
ejpam-6568	265	4	⇒	⇒	NOUN
ejpam-6568	265	5	(	(	PUNCT
ejpam-6568	265	6	1	1	NUM
ejpam-6568	265	7	):	):	PUNCT
ejpam-6568	265	8	let	let	VERB
ejpam-6568	265	9	x	x	PUNCT
ejpam-6568	265	10	∈	∈	PROPN
ejpam-6568	265	11	x	x	X
ejpam-6568	265	12	and	and	CCONJ
ejpam-6568	265	13	v	v	X
ejpam-6568	265	14	be	be	AUX
ejpam-6568	265	15	any	any	DET
ejpam-6568	265	16	σ1σ2	σ1σ2	NOUN
ejpam-6568	265	17	-	-	ADJ
ejpam-6568	265	18	open	open	ADJ
ejpam-6568	265	19	set	set	NOUN
ejpam-6568	265	20	of	of	ADP
ejpam-6568	265	21	y	y	PROPN
ejpam-6568	265	22	containing	contain	VERB
ejpam-6568	265	23	f(x	f(x	PROPN
ejpam-6568	265	24	)	)	PUNCT
ejpam-6568	265	25	.	.	PUNCT
ejpam-6568	266	1	by	by	ADP
ejpam-6568	266	2	(	(	PUNCT
ejpam-6568	266	3	5	5	NUM
ejpam-6568	266	4	)	)	PUNCT
ejpam-6568	266	5	,	,	PUNCT
ejpam-6568	266	6	x	x	PUNCT
ejpam-6568	266	7	∈	∈	PROPN
ejpam-6568	266	8	f−1(v	f−1(v	NOUN
ejpam-6568	266	9	)	)	PUNCT
ejpam-6568	266	10	⊆	⊆	X
ejpam-6568	266	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	266	12	-	-	PUNCT
ejpam-6568	266	13	cl(v	cl(v	NOUN
ejpam-6568	266	14	)	)	PUNCT
ejpam-6568	266	15	)	)	PUNCT
ejpam-6568	266	16	)	)	PUNCT
ejpam-6568	266	17	and	and	CCONJ
ejpam-6568	266	18	there	there	PRON
ejpam-6568	266	19	exists	exist	VERB
ejpam-6568	266	20	a	a	DET
ejpam-6568	266	21	⋆-open	⋆-open	ADJ
ejpam-6568	266	22	set	set	NOUN
ejpam-6568	266	23	u	u	NOUN
ejpam-6568	266	24	of	of	ADP
ejpam-6568	266	25	x	x	PUNCT
ejpam-6568	266	26	containing	contain	VERB
ejpam-6568	266	27	x	x	PUNCT
ejpam-6568	266	28	such	such	ADJ
ejpam-6568	266	29	that	that	SCONJ
ejpam-6568	266	30	u	u	PROPN
ejpam-6568	266	31	⊆	⊆	NUM
ejpam-6568	266	32	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	266	33	-	-	PUNCT
ejpam-6568	266	34	cl(v	cl(v	NOUN
ejpam-6568	266	35	)	)	PUNCT
ejpam-6568	266	36	)	)	PUNCT
ejpam-6568	266	37	.	.	PUNCT
ejpam-6568	267	1	thus	thus	ADV
ejpam-6568	267	2	,	,	PUNCT
ejpam-6568	267	3	f(u	f(u	PROPN
ejpam-6568	267	4	)	)	PUNCT
ejpam-6568	267	5	⊆	⊆	NUM
ejpam-6568	267	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	267	7	-	-	NUM
ejpam-6568	267	8	cl(v	cl(v	NOUN
ejpam-6568	267	9	)	)	PUNCT
ejpam-6568	267	10	and	and	CCONJ
ejpam-6568	267	11	hence	hence	ADV
ejpam-6568	267	12	f	f	PROPN
ejpam-6568	267	13	is	be	AUX
ejpam-6568	267	14	weakly	weakly	ADJ
ejpam-6568	267	15	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	267	16	,	,	PUNCT
ejpam-6568	267	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	267	18	.	.	PUNCT
ejpam-6568	267	19	(	(	PUNCT
ejpam-6568	267	20	4	4	X
ejpam-6568	267	21	)	)	PUNCT
ejpam-6568	267	22	⇒	⇒	NOUN
ejpam-6568	267	23	(	(	PUNCT
ejpam-6568	267	24	6	6	NUM
ejpam-6568	267	25	)	)	PUNCT
ejpam-6568	267	26	and	and	CCONJ
ejpam-6568	267	27	(	(	PUNCT
ejpam-6568	267	28	6	6	NUM
ejpam-6568	267	29	)	)	PUNCT
ejpam-6568	267	30	⇒	⇒	NOUN
ejpam-6568	267	31	(	(	PUNCT
ejpam-6568	267	32	7	7	NUM
ejpam-6568	267	33	):	):	PUNCT
ejpam-6568	267	34	the	the	DET
ejpam-6568	267	35	proofs	proof	NOUN
ejpam-6568	267	36	are	be	AUX
ejpam-6568	267	37	obvious	obvious	ADJ
ejpam-6568	267	38	.	.	PUNCT
ejpam-6568	268	1	(	(	PUNCT
ejpam-6568	268	2	7	7	X
ejpam-6568	268	3	)	)	PUNCT
ejpam-6568	268	4	⇒	⇒	NOUN
ejpam-6568	268	5	(	(	PUNCT
ejpam-6568	268	6	8)	8)	NUM
ejpam-6568	268	7	:	:	PUNCT
ejpam-6568	268	8	let	let	VERB
ejpam-6568	268	9	k	k	X
ejpam-6568	268	10	be	be	AUX
ejpam-6568	268	11	any	any	DET
ejpam-6568	268	12	(	(	PUNCT
ejpam-6568	268	13	σ1	σ1	NOUN
ejpam-6568	268	14	,	,	PUNCT
ejpam-6568	268	15	σ2)r	σ2)r	NOUN
ejpam-6568	268	16	-	-	PUNCT
ejpam-6568	268	17	closed	close	VERB
ejpam-6568	268	18	set	set	NOUN
ejpam-6568	268	19	of	of	ADP
ejpam-6568	268	20	y	y	PROPN
ejpam-6568	268	21	.	.	PUNCT
ejpam-6568	269	1	thus	thus	ADV
ejpam-6568	269	2	by	by	ADP
ejpam-6568	269	3	(	(	PUNCT
ejpam-6568	269	4	7	7	NUM
ejpam-6568	269	5	)	)	PUNCT
ejpam-6568	269	6	,	,	PUNCT
ejpam-6568	269	7	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	269	8	-	-	PUNCT
ejpam-6568	269	9	int(k	int(k	PROPN
ejpam-6568	269	10	)	)	PUNCT
ejpam-6568	269	11	)	)	PUNCT
ejpam-6568	269	12	)	)	PUNCT
ejpam-6568	270	1	⊆	⊆	NUM
ejpam-6568	270	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	270	3	-	-	PUNCT
ejpam-6568	270	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	270	5	-	-	PUNCT
ejpam-6568	270	6	int(k	int(k	NOUN
ejpam-6568	270	7	)	)	PUNCT
ejpam-6568	270	8	)	)	PUNCT
ejpam-6568	270	9	)	)	PUNCT
ejpam-6568	271	1	=	=	PUNCT
ejpam-6568	271	2	f−1(k	f−1(k	PROPN
ejpam-6568	271	3	)	)	PUNCT
ejpam-6568	271	4	.	.	PUNCT
ejpam-6568	272	1	(	(	PUNCT
ejpam-6568	272	2	8)	8)	NUM
ejpam-6568	272	3	⇒	⇒	NOUN
ejpam-6568	272	4	(	(	PUNCT
ejpam-6568	272	5	3	3	NUM
ejpam-6568	272	6	):	):	PUNCT
ejpam-6568	272	7	letk	letk	ADJ
ejpam-6568	272	8	be	be	AUX
ejpam-6568	272	9	any	any	DET
ejpam-6568	272	10	σ1σ2	σ1σ2	NUM
ejpam-6568	272	11	-	-	PUNCT
ejpam-6568	272	12	closed	closed	ADJ
ejpam-6568	272	13	set	set	NOUN
ejpam-6568	272	14	of	of	ADP
ejpam-6568	272	15	y	y	PROPN
ejpam-6568	272	16	.	.	PUNCT
ejpam-6568	273	1	then	then	ADV
ejpam-6568	273	2	,	,	PUNCT
ejpam-6568	273	3	σ1σ2	σ1σ2	X
ejpam-6568	273	4	-	-	PUNCT
ejpam-6568	273	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	273	6	-	-	PUNCT
ejpam-6568	273	7	int(k	int(k	NOUN
ejpam-6568	273	8	)	)	PUNCT
ejpam-6568	273	9	)	)	PUNCT
ejpam-6568	273	10	is	be	AUX
ejpam-6568	273	11	(	(	PUNCT
ejpam-6568	273	12	σ1	σ1	PROPN
ejpam-6568	273	13	,	,	PUNCT
ejpam-6568	273	14	σ2)rclosed	σ2)rclose	VERB
ejpam-6568	273	15	in	in	ADP
ejpam-6568	273	16	y	y	PROPN
ejpam-6568	273	17	and	and	CCONJ
ejpam-6568	273	18	σ1σ2	σ1σ2	NOUN
ejpam-6568	273	19	-	-	PUNCT
ejpam-6568	273	20	int(σ1σ2	int(σ1σ2	ADV
ejpam-6568	273	21	-	-	PUNCT
ejpam-6568	273	22	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	273	23	-	-	PUNCT
ejpam-6568	273	24	int(k	int(k	NOUN
ejpam-6568	273	25	)	)	PUNCT
ejpam-6568	273	26	)	)	PUNCT
ejpam-6568	273	27	)	)	PUNCT
ejpam-6568	274	1	=	=	PUNCT
ejpam-6568	274	2	σ1σ2	σ1σ2	X
ejpam-6568	274	3	-	-	PUNCT
ejpam-6568	274	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	274	5	-	-	PUNCT
ejpam-6568	274	6	cl(k	cl(k	NUM
ejpam-6568	274	7	)	)	PUNCT
ejpam-6568	274	8	)	)	PUNCT
ejpam-6568	275	1	=	=	PUNCT
ejpam-6568	275	2	σ1σ2	σ1σ2	X
ejpam-6568	275	3	-	-	PUNCT
ejpam-6568	275	4	int(k	int(k	NOUN
ejpam-6568	275	5	)	)	PUNCT
ejpam-6568	275	6	.	.	PUNCT
ejpam-6568	276	1	by	by	ADP
ejpam-6568	276	2	(	(	PUNCT
ejpam-6568	276	3	8)	8)	NUM
ejpam-6568	276	4	,	,	PUNCT
ejpam-6568	276	5	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	276	6	-	-	PUNCT
ejpam-6568	276	7	int(k	int(k	PROPN
ejpam-6568	276	8	)	)	PUNCT
ejpam-6568	276	9	)	)	PUNCT
ejpam-6568	276	10	)	)	PUNCT
ejpam-6568	277	1	=	=	SYM
ejpam-6568	278	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	278	2	-	-	PUNCT
ejpam-6568	278	3	int(σ1σ2	int(σ1σ2	ADV
ejpam-6568	278	4	-	-	PUNCT
ejpam-6568	278	5	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	278	6	-	-	PUNCT
ejpam-6568	278	7	int(k	int(k	NOUN
ejpam-6568	278	8	)	)	PUNCT
ejpam-6568	278	9	)	)	PUNCT
ejpam-6568	278	10	)	)	PUNCT
ejpam-6568	278	11	)	)	PUNCT
ejpam-6568	278	12	)	)	PUNCT
ejpam-6568	279	1	⊆	⊆	NUM
ejpam-6568	279	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	279	3	-	-	PUNCT
ejpam-6568	279	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	279	5	-	-	PUNCT
ejpam-6568	279	6	int(k	int(k	NOUN
ejpam-6568	279	7	)	)	PUNCT
ejpam-6568	279	8	)	)	PUNCT
ejpam-6568	279	9	)	)	PUNCT
ejpam-6568	280	1	⊆	⊆	NUM
ejpam-6568	280	2	f−1(k	f−1(k	NOUN
ejpam-6568	280	3	)	)	PUNCT
ejpam-6568	280	4	.	.	PUNCT
ejpam-6568	281	1	theorem	theorem	VERB
ejpam-6568	281	2	7	7	NUM
ejpam-6568	281	3	.	.	X
ejpam-6568	281	4	for	for	ADP
ejpam-6568	281	5	a	a	DET
ejpam-6568	281	6	function	function	NOUN
ejpam-6568	281	7	f	f	NOUN
ejpam-6568	281	8	:	:	PUNCT
ejpam-6568	281	9	(	(	PUNCT
ejpam-6568	281	10	x	x	X
ejpam-6568	281	11	,	,	PUNCT
ejpam-6568	281	12	τ	τ	PROPN
ejpam-6568	281	13	,	,	PUNCT
ejpam-6568	281	14	i	i	NOUN
ejpam-6568	281	15	)	)	PUNCT
ejpam-6568	281	16	→	→	PUNCT
ejpam-6568	281	17	(	(	PUNCT
ejpam-6568	281	18	y	y	PROPN
ejpam-6568	281	19	,	,	PUNCT
ejpam-6568	281	20	σ1	σ1	PROPN
ejpam-6568	281	21	,	,	PUNCT
ejpam-6568	281	22	σ2	σ2	NOUN
ejpam-6568	281	23	)	)	PUNCT
ejpam-6568	281	24	,	,	PUNCT
ejpam-6568	281	25	the	the	DET
ejpam-6568	281	26	following	follow	VERB
ejpam-6568	281	27	properties	property	NOUN
ejpam-6568	281	28	are	be	AUX
ejpam-6568	281	29	equivalent	equivalent	ADJ
ejpam-6568	281	30	:	:	PUNCT
ejpam-6568	281	31	(	(	PUNCT
ejpam-6568	281	32	1	1	X
ejpam-6568	281	33	)	)	PUNCT
ejpam-6568	281	34	f	f	PROPN
ejpam-6568	281	35	is	be	AUX
ejpam-6568	281	36	weakly	weakly	ADJ
ejpam-6568	281	37	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	281	38	,	,	PUNCT
ejpam-6568	281	39	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	281	40	;	;	PUNCT
ejpam-6568	281	41	(	(	PUNCT
ejpam-6568	281	42	2	2	X
ejpam-6568	281	43	)	)	PUNCT
ejpam-6568	281	44	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	281	45	-	-	PUNCT
ejpam-6568	281	46	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	281	47	-	-	PUNCT
ejpam-6568	281	48	cl(v	cl(v	NOUN
ejpam-6568	281	49	)	)	PUNCT
ejpam-6568	281	50	)	)	PUNCT
ejpam-6568	281	51	)	)	PUNCT
ejpam-6568	281	52	)	)	PUNCT
ejpam-6568	282	1	⊆	⊆	NUM
ejpam-6568	282	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	282	3	-	-	PUNCT
ejpam-6568	282	4	cl(v	cl(v	NOUN
ejpam-6568	282	5	)	)	PUNCT
ejpam-6568	282	6	)	)	PUNCT
ejpam-6568	282	7	for	for	ADP
ejpam-6568	282	8	every	every	DET
ejpam-6568	282	9	(	(	PUNCT
ejpam-6568	282	10	σ1	σ1	PROPN
ejpam-6568	282	11	,	,	PUNCT
ejpam-6568	282	12	σ2)β	σ2)β	NOUN
ejpam-6568	282	13	-	-	PUNCT
ejpam-6568	282	14	open	open	NOUN
ejpam-6568	282	15	set	set	NOUN
ejpam-6568	282	16	v	v	NOUN
ejpam-6568	282	17	of	of	ADP
ejpam-6568	282	18	y	y	PROPN
ejpam-6568	282	19	;	;	PUNCT
ejpam-6568	282	20	n.	n.	PROPN
ejpam-6568	282	21	viriyapong	viriyapong	PROPN
ejpam-6568	282	22	,	,	PUNCT
ejpam-6568	282	23	a.	a.	PROPN
ejpam-6568	282	24	sama	sama	PROPN
ejpam-6568	282	25	-	-	PUNCT
ejpam-6568	282	26	ae	ae	PROPN
ejpam-6568	282	27	,	,	PUNCT
ejpam-6568	282	28	c.	c.	PROPN
ejpam-6568	282	29	boonpok	boonpok	PROPN
ejpam-6568	282	30	/	/	SYM
ejpam-6568	282	31	eur	eur	PROPN
ejpam-6568	282	32	.	.	PUNCT
ejpam-6568	283	1	j.	j.	PROPN
ejpam-6568	283	2	pure	pure	PROPN
ejpam-6568	283	3	appl	appl	PROPN
ejpam-6568	283	4	.	.	PROPN
ejpam-6568	283	5	math	math	PROPN
ejpam-6568	283	6	,	,	PUNCT
ejpam-6568	283	7	18	18	NUM
ejpam-6568	283	8	(	(	PUNCT
ejpam-6568	283	9	3	3	NUM
ejpam-6568	283	10	)	)	PUNCT
ejpam-6568	283	11	(	(	PUNCT
ejpam-6568	283	12	2025	2025	NUM
ejpam-6568	283	13	)	)	PUNCT
ejpam-6568	283	14	,	,	PUNCT
ejpam-6568	283	15	6568	6568	NUM
ejpam-6568	283	16	10	10	NUM
ejpam-6568	283	17	of	of	ADP
ejpam-6568	283	18	18	18	NUM
ejpam-6568	283	19	(	(	PUNCT
ejpam-6568	283	20	3	3	NUM
ejpam-6568	283	21	)	)	PUNCT
ejpam-6568	283	22	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	283	23	-	-	PUNCT
ejpam-6568	283	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	283	25	-	-	PUNCT
ejpam-6568	283	26	cl(v	cl(v	NOUN
ejpam-6568	283	27	)	)	PUNCT
ejpam-6568	283	28	)	)	PUNCT
ejpam-6568	283	29	)	)	PUNCT
ejpam-6568	283	30	)	)	PUNCT
ejpam-6568	284	1	⊆	⊆	NUM
ejpam-6568	284	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	284	3	-	-	PUNCT
ejpam-6568	284	4	cl(v	cl(v	NOUN
ejpam-6568	284	5	)	)	PUNCT
ejpam-6568	284	6	)	)	PUNCT
ejpam-6568	284	7	for	for	ADP
ejpam-6568	284	8	every	every	DET
ejpam-6568	284	9	(	(	PUNCT
ejpam-6568	284	10	σ1	σ1	PROPN
ejpam-6568	284	11	,	,	PUNCT
ejpam-6568	284	12	σ2)s	σ2)s	NOUN
ejpam-6568	284	13	-	-	PUNCT
ejpam-6568	284	14	open	open	NOUN
ejpam-6568	284	15	set	set	NOUN
ejpam-6568	284	16	v	v	NOUN
ejpam-6568	284	17	of	of	ADP
ejpam-6568	284	18	y	y	PROPN
ejpam-6568	284	19	.	.	PUNCT
ejpam-6568	285	1	proof	proof	NOUN
ejpam-6568	285	2	.	.	PUNCT
ejpam-6568	286	1	(	(	PUNCT
ejpam-6568	286	2	1	1	X
ejpam-6568	286	3	)	)	PUNCT
ejpam-6568	286	4	⇒	⇒	NOUN
ejpam-6568	286	5	(	(	PUNCT
ejpam-6568	286	6	2	2	NUM
ejpam-6568	286	7	):	):	PUNCT
ejpam-6568	286	8	this	this	PRON
ejpam-6568	286	9	follows	follow	VERB
ejpam-6568	286	10	from	from	ADP
ejpam-6568	286	11	(	(	PUNCT
ejpam-6568	286	12	4	4	NUM
ejpam-6568	286	13	)	)	PUNCT
ejpam-6568	286	14	of	of	ADP
ejpam-6568	286	15	theorem	theorem	NOUN
ejpam-6568	286	16	6	6	NUM
ejpam-6568	286	17	.	.	PUNCT
ejpam-6568	286	18	(	(	PUNCT
ejpam-6568	286	19	2	2	X
ejpam-6568	286	20	)	)	PUNCT
ejpam-6568	286	21	⇒	⇒	NOUN
ejpam-6568	286	22	(	(	PUNCT
ejpam-6568	286	23	3	3	NUM
ejpam-6568	286	24	):	):	PUNCT
ejpam-6568	286	25	the	the	DET
ejpam-6568	286	26	proof	proof	NOUN
ejpam-6568	286	27	is	be	AUX
ejpam-6568	286	28	obvious	obvious	ADJ
ejpam-6568	286	29	since	since	SCONJ
ejpam-6568	286	30	every	every	DET
ejpam-6568	286	31	(	(	PUNCT
ejpam-6568	286	32	σ1	σ1	PROPN
ejpam-6568	286	33	,	,	PUNCT
ejpam-6568	286	34	σ2)s	σ2)s	NOUN
ejpam-6568	286	35	-	-	PUNCT
ejpam-6568	286	36	open	open	ADJ
ejpam-6568	286	37	set	set	NOUN
ejpam-6568	286	38	is	be	AUX
ejpam-6568	286	39	(	(	PUNCT
ejpam-6568	286	40	σ1	σ1	PROPN
ejpam-6568	286	41	,	,	PUNCT
ejpam-6568	286	42	σ2)β	σ2)β	NOUN
ejpam-6568	286	43	-	-	PUNCT
ejpam-6568	286	44	open	open	ADJ
ejpam-6568	286	45	.	.	PUNCT
ejpam-6568	287	1	(	(	PUNCT
ejpam-6568	287	2	3	3	X
ejpam-6568	287	3	)	)	PUNCT
ejpam-6568	287	4	⇒	⇒	NOUN
ejpam-6568	287	5	(	(	PUNCT
ejpam-6568	287	6	1	1	NUM
ejpam-6568	287	7	):	):	PUNCT
ejpam-6568	287	8	since	since	SCONJ
ejpam-6568	287	9	every	every	DET
ejpam-6568	287	10	σ1σ2	σ1σ2	NUM
ejpam-6568	287	11	-	-	ADJ
ejpam-6568	287	12	open	open	ADJ
ejpam-6568	287	13	set	set	NOUN
ejpam-6568	287	14	is	be	AUX
ejpam-6568	287	15	(	(	PUNCT
ejpam-6568	287	16	σ1	σ1	PROPN
ejpam-6568	287	17	,	,	PUNCT
ejpam-6568	287	18	σ2)s	σ2)s	NOUN
ejpam-6568	287	19	-	-	PUNCT
ejpam-6568	287	20	open	open	ADJ
ejpam-6568	287	21	,	,	PUNCT
ejpam-6568	287	22	the	the	DET
ejpam-6568	287	23	proof	proof	NOUN
ejpam-6568	287	24	is	be	AUX
ejpam-6568	287	25	obvious	obvious	ADJ
ejpam-6568	287	26	by	by	ADP
ejpam-6568	287	27	(	(	PUNCT
ejpam-6568	287	28	7	7	NUM
ejpam-6568	287	29	)	)	PUNCT
ejpam-6568	287	30	of	of	ADP
ejpam-6568	287	31	theorem	theorem	ADJ
ejpam-6568	287	32	6	6	NUM
ejpam-6568	287	33	.	.	PUNCT
ejpam-6568	287	34	theorem	theorem	ADJ
ejpam-6568	287	35	8	8	NUM
ejpam-6568	287	36	.	.	PUNCT
ejpam-6568	287	37	for	for	ADP
ejpam-6568	287	38	a	a	DET
ejpam-6568	287	39	function	function	NOUN
ejpam-6568	287	40	f	f	NOUN
ejpam-6568	287	41	:	:	PUNCT
ejpam-6568	287	42	(	(	PUNCT
ejpam-6568	287	43	x	x	X
ejpam-6568	287	44	,	,	PUNCT
ejpam-6568	287	45	τ	τ	PROPN
ejpam-6568	287	46	,	,	PUNCT
ejpam-6568	287	47	i	i	NOUN
ejpam-6568	287	48	)	)	PUNCT
ejpam-6568	287	49	→	→	PUNCT
ejpam-6568	287	50	(	(	PUNCT
ejpam-6568	287	51	y	y	PROPN
ejpam-6568	287	52	,	,	PUNCT
ejpam-6568	287	53	σ1	σ1	PROPN
ejpam-6568	287	54	,	,	PUNCT
ejpam-6568	287	55	σ2	σ2	NOUN
ejpam-6568	287	56	)	)	PUNCT
ejpam-6568	287	57	,	,	PUNCT
ejpam-6568	287	58	the	the	DET
ejpam-6568	287	59	following	follow	VERB
ejpam-6568	287	60	properties	property	NOUN
ejpam-6568	287	61	are	be	AUX
ejpam-6568	287	62	equivalent	equivalent	ADJ
ejpam-6568	287	63	:	:	PUNCT
ejpam-6568	287	64	(	(	PUNCT
ejpam-6568	287	65	1	1	X
ejpam-6568	287	66	)	)	PUNCT
ejpam-6568	287	67	f	f	PROPN
ejpam-6568	287	68	is	be	AUX
ejpam-6568	287	69	weakly	weakly	ADJ
ejpam-6568	287	70	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	287	71	,	,	PUNCT
ejpam-6568	287	72	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	287	73	;	;	PUNCT
ejpam-6568	287	74	(	(	PUNCT
ejpam-6568	287	75	2	2	X
ejpam-6568	287	76	)	)	PUNCT
ejpam-6568	287	77	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	287	78	-	-	PUNCT
ejpam-6568	287	79	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	287	80	-	-	PUNCT
ejpam-6568	287	81	cl(v	cl(v	NOUN
ejpam-6568	287	82	)	)	PUNCT
ejpam-6568	287	83	)	)	PUNCT
ejpam-6568	287	84	)	)	PUNCT
ejpam-6568	287	85	)	)	PUNCT
ejpam-6568	288	1	⊆	⊆	NUM
ejpam-6568	288	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	288	3	-	-	PUNCT
ejpam-6568	288	4	cl(v	cl(v	NOUN
ejpam-6568	288	5	)	)	PUNCT
ejpam-6568	288	6	)	)	PUNCT
ejpam-6568	288	7	for	for	ADP
ejpam-6568	288	8	every	every	DET
ejpam-6568	288	9	(	(	PUNCT
ejpam-6568	288	10	σ1	σ1	PROPN
ejpam-6568	288	11	,	,	PUNCT
ejpam-6568	288	12	σ2)p	σ2)p	NOUN
ejpam-6568	288	13	-	-	PUNCT
ejpam-6568	288	14	open	open	NOUN
ejpam-6568	288	15	set	set	NOUN
ejpam-6568	288	16	v	v	NOUN
ejpam-6568	288	17	of	of	ADP
ejpam-6568	288	18	y	y	PROPN
ejpam-6568	288	19	;	;	PUNCT
ejpam-6568	288	20	(	(	PUNCT
ejpam-6568	288	21	3	3	X
ejpam-6568	288	22	)	)	PUNCT
ejpam-6568	288	23	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	288	24	)	)	PUNCT
ejpam-6568	288	25	)	)	PUNCT
ejpam-6568	289	1	⊆	⊆	NUM
ejpam-6568	289	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	289	3	-	-	PUNCT
ejpam-6568	289	4	cl(v	cl(v	NOUN
ejpam-6568	289	5	)	)	PUNCT
ejpam-6568	289	6	)	)	PUNCT
ejpam-6568	289	7	for	for	ADP
ejpam-6568	289	8	every	every	DET
ejpam-6568	289	9	(	(	PUNCT
ejpam-6568	289	10	σ1	σ1	PROPN
ejpam-6568	289	11	,	,	PUNCT
ejpam-6568	289	12	σ2)p	σ2)p	NOUN
ejpam-6568	289	13	-	-	PUNCT
ejpam-6568	289	14	open	open	NOUN
ejpam-6568	289	15	set	set	NOUN
ejpam-6568	289	16	v	v	NOUN
ejpam-6568	289	17	of	of	ADP
ejpam-6568	289	18	y	y	PROPN
ejpam-6568	289	19	;	;	PUNCT
ejpam-6568	289	20	(	(	PUNCT
ejpam-6568	289	21	4	4	X
ejpam-6568	289	22	)	)	PUNCT
ejpam-6568	289	23	f−1(v	f−1(v	NOUN
ejpam-6568	289	24	)	)	PUNCT
ejpam-6568	290	1	⊆	⊆	X
ejpam-6568	290	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	290	3	-	-	PUNCT
ejpam-6568	290	4	cl(v	cl(v	NOUN
ejpam-6568	290	5	)	)	PUNCT
ejpam-6568	290	6	)	)	PUNCT
ejpam-6568	290	7	)	)	PUNCT
ejpam-6568	290	8	for	for	ADP
ejpam-6568	290	9	every	every	DET
ejpam-6568	290	10	(	(	PUNCT
ejpam-6568	290	11	σ1	σ1	PROPN
ejpam-6568	290	12	,	,	PUNCT
ejpam-6568	290	13	σ2)p	σ2)p	NOUN
ejpam-6568	290	14	-	-	PUNCT
ejpam-6568	290	15	open	open	NOUN
ejpam-6568	290	16	set	set	NOUN
ejpam-6568	290	17	v	v	NOUN
ejpam-6568	290	18	of	of	ADP
ejpam-6568	290	19	y	y	PROPN
ejpam-6568	290	20	.	.	PUNCT
ejpam-6568	291	1	proof	proof	NOUN
ejpam-6568	291	2	.	.	PUNCT
ejpam-6568	292	1	(	(	PUNCT
ejpam-6568	292	2	1	1	X
ejpam-6568	292	3	)	)	PUNCT
ejpam-6568	292	4	⇒	⇒	NOUN
ejpam-6568	292	5	(	(	PUNCT
ejpam-6568	292	6	2	2	NUM
ejpam-6568	292	7	):	):	PUNCT
ejpam-6568	292	8	let	let	VERB
ejpam-6568	292	9	v	v	PART
ejpam-6568	292	10	be	be	AUX
ejpam-6568	292	11	any	any	DET
ejpam-6568	292	12	(	(	PUNCT
ejpam-6568	292	13	σ1	σ1	PROPN
ejpam-6568	292	14	,	,	PUNCT
ejpam-6568	292	15	σ2)p	σ2)p	NOUN
ejpam-6568	292	16	-	-	PUNCT
ejpam-6568	292	17	open	open	ADJ
ejpam-6568	292	18	set	set	NOUN
ejpam-6568	292	19	of	of	ADP
ejpam-6568	292	20	y	y	PROPN
ejpam-6568	292	21	.	.	PUNCT
ejpam-6568	293	1	since	since	SCONJ
ejpam-6568	293	2	σ1σ2	σ1σ2	ADV
ejpam-6568	293	3	-	-	PUNCT
ejpam-6568	293	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	293	5	-	-	PUNCT
ejpam-6568	293	6	cl(v	cl(v	NOUN
ejpam-6568	293	7	)	)	PUNCT
ejpam-6568	293	8	)	)	PUNCT
ejpam-6568	293	9	is	be	AUX
ejpam-6568	293	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	293	11	-	-	ADJ
ejpam-6568	293	12	open	open	ADJ
ejpam-6568	293	13	,	,	PUNCT
ejpam-6568	293	14	by	by	ADP
ejpam-6568	293	15	theorem	theorem	NOUN
ejpam-6568	293	16	6	6	NUM
ejpam-6568	293	17	(	(	PUNCT
ejpam-6568	293	18	7	7	NUM
ejpam-6568	293	19	)	)	PUNCT
ejpam-6568	293	20	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	293	21	-	-	PUNCT
ejpam-6568	293	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	293	23	-	-	PUNCT
ejpam-6568	293	24	cl(v	cl(v	NOUN
ejpam-6568	293	25	)	)	PUNCT
ejpam-6568	293	26	)	)	PUNCT
ejpam-6568	293	27	)	)	PUNCT
ejpam-6568	293	28	)	)	PUNCT
ejpam-6568	294	1	⊆	⊆	NUM
ejpam-6568	294	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	294	3	-	-	PUNCT
ejpam-6568	294	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	294	5	-	-	PUNCT
ejpam-6568	294	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	294	7	-	-	PUNCT
ejpam-6568	294	8	cl(v	cl(v	NOUN
ejpam-6568	294	9	)	)	PUNCT
ejpam-6568	294	10	)	)	PUNCT
ejpam-6568	294	11	)	)	PUNCT
ejpam-6568	294	12	)	)	PUNCT
ejpam-6568	295	1	⊆	⊆	NUM
ejpam-6568	295	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	295	3	-	-	PUNCT
ejpam-6568	295	4	cl(v	cl(v	NOUN
ejpam-6568	295	5	)	)	PUNCT
ejpam-6568	295	6	)	)	PUNCT
ejpam-6568	295	7	.	.	PUNCT
ejpam-6568	296	1	(	(	PUNCT
ejpam-6568	296	2	2	2	X
ejpam-6568	296	3	)	)	PUNCT
ejpam-6568	296	4	⇒	⇒	NOUN
ejpam-6568	296	5	(	(	PUNCT
ejpam-6568	296	6	3	3	NUM
ejpam-6568	296	7	):	):	PUNCT
ejpam-6568	296	8	let	let	VERB
ejpam-6568	296	9	v	v	PART
ejpam-6568	296	10	be	be	AUX
ejpam-6568	296	11	any	any	DET
ejpam-6568	296	12	(	(	PUNCT
ejpam-6568	296	13	σ1	σ1	PROPN
ejpam-6568	296	14	,	,	PUNCT
ejpam-6568	296	15	σ2)p	σ2)p	NOUN
ejpam-6568	296	16	-	-	PUNCT
ejpam-6568	296	17	open	open	ADJ
ejpam-6568	296	18	set	set	NOUN
ejpam-6568	296	19	of	of	ADP
ejpam-6568	296	20	y	y	PROPN
ejpam-6568	296	21	.	.	PUNCT
ejpam-6568	297	1	by	by	ADP
ejpam-6568	297	2	(	(	PUNCT
ejpam-6568	297	3	2	2	NUM
ejpam-6568	297	4	)	)	PUNCT
ejpam-6568	297	5	,	,	PUNCT
ejpam-6568	297	6	we	we	PRON
ejpam-6568	297	7	have	have	VERB
ejpam-6568	297	8	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	297	9	)	)	PUNCT
ejpam-6568	297	10	)	)	PUNCT
ejpam-6568	298	1	⊆	⊆	X
ejpam-6568	298	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	298	3	-	-	PUNCT
ejpam-6568	298	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	298	5	-	-	PUNCT
ejpam-6568	298	6	cl(v	cl(v	NOUN
ejpam-6568	298	7	)	)	PUNCT
ejpam-6568	298	8	)	)	PUNCT
ejpam-6568	298	9	)	)	PUNCT
ejpam-6568	298	10	)	)	PUNCT
ejpam-6568	298	11	⊆	⊆	NUM
ejpam-6568	298	12	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	298	13	-	-	PUNCT
ejpam-6568	298	14	cl(v	cl(v	NOUN
ejpam-6568	298	15	)	)	PUNCT
ejpam-6568	298	16	)	)	PUNCT
ejpam-6568	298	17	.	.	PUNCT
ejpam-6568	299	1	(	(	PUNCT
ejpam-6568	299	2	3	3	X
ejpam-6568	299	3	)	)	PUNCT
ejpam-6568	299	4	⇒	⇒	NOUN
ejpam-6568	299	5	(	(	PUNCT
ejpam-6568	299	6	4	4	NUM
ejpam-6568	299	7	):	):	PUNCT
ejpam-6568	299	8	let	let	VERB
ejpam-6568	299	9	v	v	PART
ejpam-6568	299	10	be	be	AUX
ejpam-6568	299	11	any	any	DET
ejpam-6568	299	12	(	(	PUNCT
ejpam-6568	299	13	σ1	σ1	PROPN
ejpam-6568	299	14	,	,	PUNCT
ejpam-6568	299	15	σ2)p	σ2)p	NOUN
ejpam-6568	299	16	-	-	PUNCT
ejpam-6568	299	17	open	open	ADJ
ejpam-6568	299	18	set	set	NOUN
ejpam-6568	299	19	of	of	ADP
ejpam-6568	299	20	y	y	PROPN
ejpam-6568	299	21	.	.	PUNCT
ejpam-6568	300	1	thus	thus	ADV
ejpam-6568	300	2	by	by	ADP
ejpam-6568	300	3	(	(	PUNCT
ejpam-6568	300	4	3	3	NUM
ejpam-6568	300	5	)	)	PUNCT
ejpam-6568	300	6	,	,	PUNCT
ejpam-6568	300	7	x	x	X
ejpam-6568	300	8	−	−	PROPN
ejpam-6568	300	9	int⋆(f−1(cl⋆(v	int⋆(f−1(cl⋆(v	NOUN
ejpam-6568	300	10	)	)	PUNCT
ejpam-6568	300	11	)	)	PUNCT
ejpam-6568	300	12	)	)	PUNCT
ejpam-6568	301	1	=	=	PUNCT
ejpam-6568	301	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6568	301	3	−	−	PROPN
ejpam-6568	301	4	f−1(cl⋆(v	f−1(cl⋆(v	NOUN
ejpam-6568	301	5	)	)	PUNCT
ejpam-6568	301	6	)	)	PUNCT
ejpam-6568	301	7	)	)	PUNCT
ejpam-6568	302	1	=	=	PUNCT
ejpam-6568	302	2	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	302	3	−	−	PROPN
ejpam-6568	302	4	cl⋆(v	cl⋆(v	PROPN
ejpam-6568	302	5	)	)	PUNCT
ejpam-6568	302	6	)	)	PUNCT
ejpam-6568	302	7	)	)	PUNCT
ejpam-6568	303	1	⊆	⊆	NUM
ejpam-6568	303	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	303	3	-	-	PUNCT
ejpam-6568	303	4	cl(y	cl(y	NOUN
ejpam-6568	303	5	−	−	NOUN
ejpam-6568	303	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	303	7	-	-	NUM
ejpam-6568	303	8	cl(v	cl(v	NOUN
ejpam-6568	303	9	)	)	PUNCT
ejpam-6568	303	10	)	)	PUNCT
ejpam-6568	303	11	)	)	PUNCT
ejpam-6568	304	1	=	=	PUNCT
ejpam-6568	304	2	x	x	X
ejpam-6568	304	3	−	−	PRON
ejpam-6568	304	4	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	304	5	-	-	PUNCT
ejpam-6568	304	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	304	7	-	-	PUNCT
ejpam-6568	304	8	cl(v	cl(v	NOUN
ejpam-6568	304	9	)	)	PUNCT
ejpam-6568	304	10	)	)	PUNCT
ejpam-6568	304	11	)	)	PUNCT
ejpam-6568	305	1	⊆	⊆	NUM
ejpam-6568	305	2	x	x	SYM
ejpam-6568	305	3	−	−	PROPN
ejpam-6568	305	4	f−1(v	f−1(v	PROPN
ejpam-6568	305	5	)	)	PUNCT
ejpam-6568	305	6	and	and	CCONJ
ejpam-6568	305	7	so	so	ADV
ejpam-6568	305	8	f−1(v	f−1(v	PROPN
ejpam-6568	305	9	)	)	PUNCT
ejpam-6568	305	10	⊆	⊆	X
ejpam-6568	305	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	305	12	-	-	PUNCT
ejpam-6568	305	13	cl(v	cl(v	NOUN
ejpam-6568	305	14	)	)	PUNCT
ejpam-6568	305	15	)	)	PUNCT
ejpam-6568	305	16	)	)	PUNCT
ejpam-6568	305	17	.	.	PUNCT
ejpam-6568	306	1	(	(	PUNCT
ejpam-6568	306	2	4	4	X
ejpam-6568	306	3	)	)	PUNCT
ejpam-6568	306	4	⇒	⇒	NOUN
ejpam-6568	306	5	(	(	PUNCT
ejpam-6568	306	6	1	1	NUM
ejpam-6568	306	7	):	):	PUNCT
ejpam-6568	306	8	let	let	VERB
ejpam-6568	306	9	v	v	PART
ejpam-6568	306	10	be	be	AUX
ejpam-6568	306	11	any	any	DET
ejpam-6568	306	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	306	13	-	-	ADJ
ejpam-6568	306	14	open	open	ADJ
ejpam-6568	306	15	set	set	NOUN
ejpam-6568	306	16	of	of	ADP
ejpam-6568	306	17	y	y	PROPN
ejpam-6568	306	18	.	.	PUNCT
ejpam-6568	307	1	then	then	ADV
ejpam-6568	307	2	,	,	PUNCT
ejpam-6568	307	3	v	v	NOUN
ejpam-6568	307	4	is	be	AUX
ejpam-6568	307	5	(	(	PUNCT
ejpam-6568	307	6	σ1	σ1	PROPN
ejpam-6568	307	7	,	,	PUNCT
ejpam-6568	307	8	σ2)p	σ2)p	NOUN
ejpam-6568	307	9	-	-	PUNCT
ejpam-6568	307	10	open	open	ADJ
ejpam-6568	307	11	in	in	ADP
ejpam-6568	307	12	y	y	PROPN
ejpam-6568	307	13	and	and	CCONJ
ejpam-6568	307	14	by	by	ADP
ejpam-6568	307	15	(	(	PUNCT
ejpam-6568	307	16	4	4	NUM
ejpam-6568	307	17	)	)	PUNCT
ejpam-6568	307	18	,	,	PUNCT
ejpam-6568	307	19	f−1(v	f−1(v	PROPN
ejpam-6568	307	20	)	)	PUNCT
ejpam-6568	308	1	⊆	⊆	X
ejpam-6568	308	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	308	3	-	-	PUNCT
ejpam-6568	308	4	cl(v	cl(v	NOUN
ejpam-6568	308	5	)	)	PUNCT
ejpam-6568	308	6	)	)	PUNCT
ejpam-6568	308	7	)	)	PUNCT
ejpam-6568	308	8	.	.	PUNCT
ejpam-6568	309	1	by	by	ADP
ejpam-6568	309	2	theorem	theorem	NOUN
ejpam-6568	309	3	6	6	NUM
ejpam-6568	309	4	(	(	PUNCT
ejpam-6568	309	5	2	2	NUM
ejpam-6568	309	6	)	)	PUNCT
ejpam-6568	309	7	,	,	PUNCT
ejpam-6568	309	8	f	f	PROPN
ejpam-6568	309	9	is	be	AUX
ejpam-6568	309	10	weakly	weakly	ADJ
ejpam-6568	309	11	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	309	12	,	,	PUNCT
ejpam-6568	309	13	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	309	14	.	.	PUNCT
ejpam-6568	310	1	lemma	lemma	PROPN
ejpam-6568	310	2	5	5	NUM
ejpam-6568	310	3	.	.	PUNCT
ejpam-6568	311	1	[	[	X
ejpam-6568	311	2	25	25	NUM
ejpam-6568	311	3	]	]	PUNCT
ejpam-6568	311	4	for	for	ADP
ejpam-6568	311	5	a	a	DET
ejpam-6568	311	6	subset	subset	NOUN
ejpam-6568	311	7	a	a	PRON
ejpam-6568	311	8	of	of	ADP
ejpam-6568	311	9	a	a	DET
ejpam-6568	311	10	bitopological	bitopological	ADJ
ejpam-6568	311	11	space	space	NOUN
ejpam-6568	311	12	(	(	PUNCT
ejpam-6568	311	13	x	x	NOUN
ejpam-6568	311	14	,	,	PUNCT
ejpam-6568	311	15	τ1	τ1	NOUN
ejpam-6568	311	16	,	,	PUNCT
ejpam-6568	311	17	τ2	τ2	NOUN
ejpam-6568	311	18	)	)	PUNCT
ejpam-6568	311	19	,	,	PUNCT
ejpam-6568	311	20	the	the	DET
ejpam-6568	311	21	following	follow	VERB
ejpam-6568	311	22	properties	property	NOUN
ejpam-6568	311	23	hold	hold	VERB
ejpam-6568	311	24	:	:	PUNCT
ejpam-6568	311	25	n.	n.	PROPN
ejpam-6568	311	26	viriyapong	viriyapong	PROPN
ejpam-6568	311	27	,	,	PUNCT
ejpam-6568	311	28	a.	a.	PROPN
ejpam-6568	311	29	sama	sama	PROPN
ejpam-6568	311	30	-	-	PUNCT
ejpam-6568	311	31	ae	ae	PROPN
ejpam-6568	311	32	,	,	PUNCT
ejpam-6568	311	33	c.	c.	PROPN
ejpam-6568	311	34	boonpok	boonpok	PROPN
ejpam-6568	311	35	/	/	SYM
ejpam-6568	311	36	eur	eur	PROPN
ejpam-6568	311	37	.	.	PUNCT
ejpam-6568	312	1	j.	j.	PROPN
ejpam-6568	312	2	pure	pure	PROPN
ejpam-6568	312	3	appl	appl	PROPN
ejpam-6568	312	4	.	.	PROPN
ejpam-6568	312	5	math	math	PROPN
ejpam-6568	312	6	,	,	PUNCT
ejpam-6568	312	7	18	18	NUM
ejpam-6568	312	8	(	(	PUNCT
ejpam-6568	312	9	3	3	NUM
ejpam-6568	312	10	)	)	PUNCT
ejpam-6568	312	11	(	(	PUNCT
ejpam-6568	312	12	2025	2025	NUM
ejpam-6568	312	13	)	)	PUNCT
ejpam-6568	312	14	,	,	PUNCT
ejpam-6568	312	15	6568	6568	NUM
ejpam-6568	312	16	11	11	NUM
ejpam-6568	312	17	of	of	ADP
ejpam-6568	312	18	18	18	NUM
ejpam-6568	312	19	(	(	PUNCT
ejpam-6568	312	20	1	1	NUM
ejpam-6568	312	21	)	)	PUNCT
ejpam-6568	312	22	if	if	SCONJ
ejpam-6568	312	23	a	a	PRON
ejpam-6568	312	24	is	be	AUX
ejpam-6568	312	25	τ2τ2	τ2τ2	VERB
ejpam-6568	312	26	-	-	VERB
ejpam-6568	312	27	open	open	ADJ
ejpam-6568	312	28	in	in	ADP
ejpam-6568	312	29	x	x	NOUN
ejpam-6568	312	30	,	,	PUNCT
ejpam-6568	312	31	then	then	ADV
ejpam-6568	312	32	τ1τ2	τ1τ2	NOUN
ejpam-6568	312	33	-	-	NUM
ejpam-6568	312	34	cl(a	cl(a	NUM
ejpam-6568	312	35	)	)	PUNCT
ejpam-6568	312	36	=	=	PUNCT
ejpam-6568	312	37	(	(	PUNCT
ejpam-6568	312	38	τ1	τ1	NOUN
ejpam-6568	312	39	,	,	PUNCT
ejpam-6568	312	40	τ2)θ	τ2)θ	NOUN
ejpam-6568	312	41	-	-	PUNCT
ejpam-6568	312	42	cl(a	cl(a	NUM
ejpam-6568	312	43	)	)	PUNCT
ejpam-6568	312	44	.	.	PUNCT
ejpam-6568	313	1	(	(	PUNCT
ejpam-6568	313	2	2	2	X
ejpam-6568	313	3	)	)	PUNCT
ejpam-6568	313	4	(	(	PUNCT
ejpam-6568	313	5	τ1	τ1	NOUN
ejpam-6568	313	6	,	,	PUNCT
ejpam-6568	313	7	τ2)θ	τ2)θ	NOUN
ejpam-6568	313	8	-	-	PUNCT
ejpam-6568	313	9	cl(a	cl(a	NUM
ejpam-6568	313	10	)	)	PUNCT
ejpam-6568	313	11	is	be	AUX
ejpam-6568	313	12	τ1τ2	τ1τ2	NOUN
ejpam-6568	313	13	-	-	ADJ
ejpam-6568	313	14	closed	closed	ADJ
ejpam-6568	313	15	in	in	ADP
ejpam-6568	313	16	x.	x.	NOUN
ejpam-6568	313	17	theorem	theorem	VERB
ejpam-6568	313	18	9	9	NUM
ejpam-6568	313	19	.	.	PUNCT
ejpam-6568	314	1	for	for	ADP
ejpam-6568	314	2	a	a	DET
ejpam-6568	314	3	function	function	NOUN
ejpam-6568	314	4	f	f	NOUN
ejpam-6568	314	5	:	:	PUNCT
ejpam-6568	314	6	(	(	PUNCT
ejpam-6568	314	7	x	x	X
ejpam-6568	314	8	,	,	PUNCT
ejpam-6568	314	9	τ	τ	PROPN
ejpam-6568	314	10	,	,	PUNCT
ejpam-6568	314	11	i	i	NOUN
ejpam-6568	314	12	)	)	PUNCT
ejpam-6568	314	13	→	→	PUNCT
ejpam-6568	314	14	(	(	PUNCT
ejpam-6568	314	15	y	y	PROPN
ejpam-6568	314	16	,	,	PUNCT
ejpam-6568	314	17	σ1	σ1	PROPN
ejpam-6568	314	18	,	,	PUNCT
ejpam-6568	314	19	σ2	σ2	NOUN
ejpam-6568	314	20	)	)	PUNCT
ejpam-6568	314	21	,	,	PUNCT
ejpam-6568	314	22	the	the	DET
ejpam-6568	314	23	following	follow	VERB
ejpam-6568	314	24	properties	property	NOUN
ejpam-6568	314	25	are	be	AUX
ejpam-6568	314	26	equivalent	equivalent	ADJ
ejpam-6568	314	27	:	:	PUNCT
ejpam-6568	314	28	(	(	PUNCT
ejpam-6568	314	29	1	1	X
ejpam-6568	314	30	)	)	PUNCT
ejpam-6568	314	31	f	f	PROPN
ejpam-6568	314	32	is	be	AUX
ejpam-6568	314	33	weakly	weakly	ADJ
ejpam-6568	314	34	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	314	35	,	,	PUNCT
ejpam-6568	314	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	314	37	;	;	PUNCT
ejpam-6568	314	38	(	(	PUNCT
ejpam-6568	314	39	2	2	X
ejpam-6568	314	40	)	)	PUNCT
ejpam-6568	314	41	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	314	42	-	-	PUNCT
ejpam-6568	314	43	int((σ1	int((σ1	PROPN
ejpam-6568	314	44	,	,	PUNCT
ejpam-6568	314	45	σ2)θ	σ2)θ	ADJ
ejpam-6568	314	46	-	-	PUNCT
ejpam-6568	314	47	cl(b	cl(b	NOUN
ejpam-6568	314	48	)	)	PUNCT
ejpam-6568	314	49	)	)	PUNCT
ejpam-6568	314	50	)	)	PUNCT
ejpam-6568	314	51	)	)	PUNCT
ejpam-6568	315	1	⊆	⊆	NUM
ejpam-6568	315	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	315	3	,	,	PUNCT
ejpam-6568	315	4	σ2)θ	σ2)θ	NOUN
ejpam-6568	315	5	-	-	PUNCT
ejpam-6568	315	6	cl(b	cl(b	NOUN
ejpam-6568	315	7	)	)	PUNCT
ejpam-6568	315	8	)	)	PUNCT
ejpam-6568	315	9	for	for	ADP
ejpam-6568	315	10	every	every	DET
ejpam-6568	315	11	subset	subset	NOUN
ejpam-6568	315	12	b	b	PROPN
ejpam-6568	315	13	of	of	ADP
ejpam-6568	315	14	y	y	PROPN
ejpam-6568	315	15	;	;	PUNCT
ejpam-6568	315	16	(	(	PUNCT
ejpam-6568	315	17	3	3	X
ejpam-6568	315	18	)	)	PUNCT
ejpam-6568	315	19	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	315	20	-	-	PUNCT
ejpam-6568	315	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	315	22	-	-	PUNCT
ejpam-6568	315	23	cl(b	cl(b	NOUN
ejpam-6568	315	24	)	)	PUNCT
ejpam-6568	315	25	)	)	PUNCT
ejpam-6568	315	26	)	)	PUNCT
ejpam-6568	315	27	)	)	PUNCT
ejpam-6568	316	1	⊆	⊆	NUM
ejpam-6568	316	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	316	3	,	,	PUNCT
ejpam-6568	316	4	σ2)θ	σ2)θ	NOUN
ejpam-6568	316	5	-	-	PUNCT
ejpam-6568	316	6	cl(b	cl(b	NOUN
ejpam-6568	316	7	)	)	PUNCT
ejpam-6568	316	8	)	)	PUNCT
ejpam-6568	316	9	for	for	ADP
ejpam-6568	316	10	every	every	DET
ejpam-6568	316	11	subset	subset	NOUN
ejpam-6568	316	12	b	b	PROPN
ejpam-6568	316	13	of	of	ADP
ejpam-6568	316	14	y	y	PROPN
ejpam-6568	316	15	.	.	PUNCT
ejpam-6568	317	1	proof	proof	NOUN
ejpam-6568	317	2	.	.	PUNCT
ejpam-6568	318	1	(	(	PUNCT
ejpam-6568	318	2	1	1	X
ejpam-6568	318	3	)	)	PUNCT
ejpam-6568	318	4	⇒	⇒	NOUN
ejpam-6568	318	5	(	(	PUNCT
ejpam-6568	318	6	2	2	NUM
ejpam-6568	318	7	):	):	PUNCT
ejpam-6568	318	8	let	let	VERB
ejpam-6568	318	9	b	b	X
ejpam-6568	318	10	be	be	AUX
ejpam-6568	318	11	any	any	DET
ejpam-6568	318	12	subset	subset	NOUN
ejpam-6568	318	13	of	of	ADP
ejpam-6568	318	14	y	y	PROPN
ejpam-6568	318	15	.	.	PUNCT
ejpam-6568	319	1	by	by	ADP
ejpam-6568	319	2	lemma	lemma	PROPN
ejpam-6568	319	3	5	5	NUM
ejpam-6568	319	4	,	,	PUNCT
ejpam-6568	319	5	(	(	PUNCT
ejpam-6568	319	6	σ1	σ1	PROPN
ejpam-6568	319	7	,	,	PUNCT
ejpam-6568	319	8	σ2)θ	σ2)θ	NOUN
ejpam-6568	319	9	-	-	PUNCT
ejpam-6568	319	10	cl(b	cl(b	NOUN
ejpam-6568	319	11	)	)	PUNCT
ejpam-6568	319	12	is	be	AUX
ejpam-6568	319	13	σ1σ2closed	σ1σ2close	VERB
ejpam-6568	319	14	in	in	ADP
ejpam-6568	319	15	y	y	PROPN
ejpam-6568	319	16	and	and	CCONJ
ejpam-6568	319	17	by	by	ADP
ejpam-6568	319	18	theorem	theorem	NOUN
ejpam-6568	319	19	6	6	NUM
ejpam-6568	319	20	,	,	PUNCT
ejpam-6568	319	21	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	319	22	-	-	PUNCT
ejpam-6568	319	23	int((σ1	int((σ1	PROPN
ejpam-6568	319	24	,	,	PUNCT
ejpam-6568	319	25	σ2)θ	σ2)θ	ADJ
ejpam-6568	319	26	-	-	PUNCT
ejpam-6568	319	27	cl(b	cl(b	NOUN
ejpam-6568	319	28	)	)	PUNCT
ejpam-6568	319	29	)	)	PUNCT
ejpam-6568	319	30	)	)	PUNCT
ejpam-6568	319	31	)	)	PUNCT
ejpam-6568	320	1	⊆	⊆	NUM
ejpam-6568	320	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	320	3	,	,	PUNCT
ejpam-6568	320	4	σ2)θ	σ2)θ	NOUN
ejpam-6568	320	5	-	-	PUNCT
ejpam-6568	320	6	cl(b	cl(b	NOUN
ejpam-6568	320	7	)	)	PUNCT
ejpam-6568	320	8	)	)	PUNCT
ejpam-6568	320	9	.	.	PUNCT
ejpam-6568	321	1	(	(	PUNCT
ejpam-6568	321	2	2	2	X
ejpam-6568	321	3	)	)	PUNCT
ejpam-6568	321	4	⇒	⇒	NOUN
ejpam-6568	321	5	(	(	PUNCT
ejpam-6568	321	6	3	3	NUM
ejpam-6568	321	7	):	):	PUNCT
ejpam-6568	321	8	the	the	DET
ejpam-6568	321	9	proof	proof	NOUN
ejpam-6568	321	10	is	be	AUX
ejpam-6568	321	11	obvious	obvious	ADJ
ejpam-6568	321	12	.	.	PUNCT
ejpam-6568	322	1	(	(	PUNCT
ejpam-6568	322	2	3	3	X
ejpam-6568	322	3	)	)	PUNCT
ejpam-6568	322	4	⇒	⇒	NOUN
ejpam-6568	322	5	(	(	PUNCT
ejpam-6568	322	6	1	1	NUM
ejpam-6568	322	7	):	):	PUNCT
ejpam-6568	322	8	let	let	VERB
ejpam-6568	322	9	k	k	PRON
ejpam-6568	322	10	be	be	AUX
ejpam-6568	322	11	any	any	DET
ejpam-6568	322	12	(	(	PUNCT
ejpam-6568	322	13	σ1	σ1	NOUN
ejpam-6568	322	14	,	,	PUNCT
ejpam-6568	322	15	σ2)r	σ2)r	NOUN
ejpam-6568	322	16	-	-	PUNCT
ejpam-6568	322	17	closed	close	VERB
ejpam-6568	322	18	set	set	NOUN
ejpam-6568	322	19	of	of	ADP
ejpam-6568	322	20	y	y	PROPN
ejpam-6568	322	21	.	.	PUNCT
ejpam-6568	323	1	then	then	ADV
ejpam-6568	323	2	,	,	PUNCT
ejpam-6568	323	3	we	we	PRON
ejpam-6568	323	4	have	have	VERB
ejpam-6568	323	5	(	(	PUNCT
ejpam-6568	323	6	σ1	σ1	PROPN
ejpam-6568	323	7	,	,	PUNCT
ejpam-6568	323	8	σ2)θ	σ2)θ	NOUN
ejpam-6568	323	9	-	-	PUNCT
ejpam-6568	323	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	323	11	-	-	PUNCT
ejpam-6568	323	12	int(k	int(k	NOUN
ejpam-6568	323	13	)	)	PUNCT
ejpam-6568	323	14	)	)	PUNCT
ejpam-6568	324	1	=	=	SYM
ejpam-6568	324	2	σ1σ2	σ1σ2	X
ejpam-6568	324	3	-	-	PUNCT
ejpam-6568	324	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	324	5	-	-	PUNCT
ejpam-6568	324	6	int(k	int(k	NOUN
ejpam-6568	324	7	)	)	PUNCT
ejpam-6568	324	8	)	)	PUNCT
ejpam-6568	325	1	=	=	SYM
ejpam-6568	326	1	k	k	PROPN
ejpam-6568	326	2	and	and	CCONJ
ejpam-6568	326	3	hence	hence	ADV
ejpam-6568	326	4	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	326	5	-	-	PUNCT
ejpam-6568	326	6	int(k	int(k	PROPN
ejpam-6568	326	7	)	)	PUNCT
ejpam-6568	326	8	)	)	PUNCT
ejpam-6568	326	9	)	)	PUNCT
ejpam-6568	327	1	=	=	SYM
ejpam-6568	328	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	328	2	-	-	PUNCT
ejpam-6568	328	3	int(σ1σ2	int(σ1σ2	ADV
ejpam-6568	328	4	-	-	PUNCT
ejpam-6568	328	5	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	328	6	-	-	PUNCT
ejpam-6568	328	7	int(k	int(k	NOUN
ejpam-6568	328	8	)	)	PUNCT
ejpam-6568	328	9	)	)	PUNCT
ejpam-6568	328	10	)	)	PUNCT
ejpam-6568	328	11	)	)	PUNCT
ejpam-6568	328	12	)	)	PUNCT
ejpam-6568	329	1	⊆	⊆	NUM
ejpam-6568	329	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	329	3	,	,	PUNCT
ejpam-6568	329	4	σ2)θ	σ2)θ	NOUN
ejpam-6568	329	5	-	-	PUNCT
ejpam-6568	329	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	329	7	-	-	PUNCT
ejpam-6568	329	8	int(k	int(k	NOUN
ejpam-6568	329	9	)	)	PUNCT
ejpam-6568	329	10	)	)	PUNCT
ejpam-6568	329	11	)	)	PUNCT
ejpam-6568	330	1	=	=	PRON
ejpam-6568	330	2	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	330	3	-	-	PUNCT
ejpam-6568	330	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	330	5	-	-	PUNCT
ejpam-6568	330	6	int(k	int(k	NOUN
ejpam-6568	330	7	)	)	PUNCT
ejpam-6568	330	8	)	)	PUNCT
ejpam-6568	330	9	)	)	PUNCT
ejpam-6568	331	1	=	=	PUNCT
ejpam-6568	331	2	f−1(k	f−1(k	PROPN
ejpam-6568	331	3	)	)	PUNCT
ejpam-6568	331	4	.	.	PUNCT
ejpam-6568	332	1	thus	thus	ADV
ejpam-6568	332	2	,	,	PUNCT
ejpam-6568	332	3	by	by	ADP
ejpam-6568	332	4	theorem	theorem	NOUN
ejpam-6568	332	5	6	6	NUM
ejpam-6568	332	6	f	f	NOUN
ejpam-6568	332	7	is	be	AUX
ejpam-6568	332	8	weakly	weakly	ADJ
ejpam-6568	332	9	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	332	10	,	,	PUNCT
ejpam-6568	332	11	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	332	12	.	.	X
ejpam-6568	332	13	theorem	theorem	VERB
ejpam-6568	332	14	10	10	NUM
ejpam-6568	332	15	.	.	PUNCT
ejpam-6568	333	1	a	a	DET
ejpam-6568	333	2	function	function	NOUN
ejpam-6568	333	3	f	f	NOUN
ejpam-6568	333	4	:	:	PUNCT
ejpam-6568	333	5	(	(	PUNCT
ejpam-6568	333	6	x	x	X
ejpam-6568	333	7	,	,	PUNCT
ejpam-6568	333	8	τ	τ	PROPN
ejpam-6568	333	9	,	,	PUNCT
ejpam-6568	333	10	i	i	NOUN
ejpam-6568	333	11	)	)	PUNCT
ejpam-6568	333	12	→	→	PUNCT
ejpam-6568	333	13	(	(	PUNCT
ejpam-6568	333	14	y	y	PROPN
ejpam-6568	333	15	,	,	PUNCT
ejpam-6568	333	16	σ1	σ1	PROPN
ejpam-6568	333	17	,	,	PUNCT
ejpam-6568	333	18	σ2	σ2	NOUN
ejpam-6568	333	19	)	)	PUNCT
ejpam-6568	333	20	is	be	AUX
ejpam-6568	333	21	weakly	weakly	ADJ
ejpam-6568	333	22	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	333	23	,	,	PUNCT
ejpam-6568	333	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	333	25	at	at	ADP
ejpam-6568	333	26	x	x	X
ejpam-6568	333	27	∈	∈	PROPN
ejpam-6568	333	28	x	x	SYM
ejpam-6568	333	29	if	if	SCONJ
ejpam-6568	333	30	and	and	CCONJ
ejpam-6568	333	31	only	only	ADV
ejpam-6568	333	32	if	if	SCONJ
ejpam-6568	333	33	x	x	SYM
ejpam-6568	333	34	∈	∈	VERB
ejpam-6568	333	35	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	333	36	-	-	PUNCT
ejpam-6568	333	37	cl(v	cl(v	NOUN
ejpam-6568	333	38	)	)	PUNCT
ejpam-6568	333	39	)	)	PUNCT
ejpam-6568	333	40	)	)	PUNCT
ejpam-6568	334	1	for	for	ADP
ejpam-6568	334	2	every	every	DET
ejpam-6568	334	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	334	4	-	-	ADJ
ejpam-6568	334	5	open	open	ADJ
ejpam-6568	334	6	set	set	NOUN
ejpam-6568	334	7	v	v	NOUN
ejpam-6568	334	8	of	of	ADP
ejpam-6568	334	9	y	y	NOUN
ejpam-6568	334	10	containing	contain	VERB
ejpam-6568	334	11	f(x	f(x	PROPN
ejpam-6568	334	12	)	)	PUNCT
ejpam-6568	334	13	.	.	PUNCT
ejpam-6568	335	1	proof	proof	NOUN
ejpam-6568	335	2	.	.	PUNCT
ejpam-6568	336	1	let	let	VERB
ejpam-6568	336	2	x	x	PUNCT
ejpam-6568	336	3	∈	∈	PROPN
ejpam-6568	336	4	x	x	X
ejpam-6568	336	5	and	and	CCONJ
ejpam-6568	336	6	v	v	X
ejpam-6568	336	7	be	be	AUX
ejpam-6568	336	8	any	any	DET
ejpam-6568	336	9	σ1σ2	σ1σ2	NOUN
ejpam-6568	336	10	-	-	ADJ
ejpam-6568	336	11	open	open	ADJ
ejpam-6568	336	12	set	set	NOUN
ejpam-6568	336	13	of	of	ADP
ejpam-6568	336	14	y	y	PROPN
ejpam-6568	336	15	containing	contain	VERB
ejpam-6568	336	16	f(x	f(x	PROPN
ejpam-6568	336	17	)	)	PUNCT
ejpam-6568	336	18	.	.	PUNCT
ejpam-6568	337	1	then	then	ADV
ejpam-6568	337	2	,	,	PUNCT
ejpam-6568	337	3	there	there	PRON
ejpam-6568	337	4	exists	exist	VERB
ejpam-6568	337	5	a	a	DET
ejpam-6568	337	6	⋆-open	⋆-open	ADJ
ejpam-6568	337	7	set	set	NOUN
ejpam-6568	337	8	u	u	NOUN
ejpam-6568	337	9	of	of	ADP
ejpam-6568	337	10	x	x	PUNCT
ejpam-6568	337	11	containing	contain	VERB
ejpam-6568	337	12	x	x	PUNCT
ejpam-6568	337	13	such	such	ADJ
ejpam-6568	337	14	that	that	DET
ejpam-6568	337	15	f(u	f(u	PROPN
ejpam-6568	337	16	)	)	PUNCT
ejpam-6568	337	17	⊆	⊆	NUM
ejpam-6568	337	18	σ1σ2	σ1σ2	NOUN
ejpam-6568	337	19	-	-	NUM
ejpam-6568	337	20	cl(v	cl(v	NOUN
ejpam-6568	337	21	)	)	PUNCT
ejpam-6568	337	22	.	.	PUNCT
ejpam-6568	338	1	thus	thus	ADV
ejpam-6568	338	2	,	,	PUNCT
ejpam-6568	338	3	u	u	PROPN
ejpam-6568	338	4	⊆	⊆	NUM
ejpam-6568	338	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	338	6	-	-	PUNCT
ejpam-6568	338	7	cl(v	cl(v	NOUN
ejpam-6568	338	8	)	)	PUNCT
ejpam-6568	338	9	)	)	PUNCT
ejpam-6568	338	10	and	and	CCONJ
ejpam-6568	338	11	hence	hence	ADV
ejpam-6568	338	12	x	x	X
ejpam-6568	338	13	∈	∈	NOUN
ejpam-6568	338	14	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	338	15	-	-	PUNCT
ejpam-6568	338	16	cl(v	cl(v	NOUN
ejpam-6568	338	17	)	)	PUNCT
ejpam-6568	338	18	)	)	PUNCT
ejpam-6568	338	19	)	)	PUNCT
ejpam-6568	338	20	.	.	PUNCT
ejpam-6568	339	1	conversely	conversely	ADV
ejpam-6568	339	2	,	,	PUNCT
ejpam-6568	339	3	let	let	VERB
ejpam-6568	339	4	v	v	PART
ejpam-6568	339	5	be	be	AUX
ejpam-6568	339	6	any	any	DET
ejpam-6568	339	7	σ1σ2	σ1σ2	NOUN
ejpam-6568	339	8	-	-	ADJ
ejpam-6568	339	9	open	open	ADJ
ejpam-6568	339	10	set	set	NOUN
ejpam-6568	339	11	of	of	ADP
ejpam-6568	339	12	y	y	PROPN
ejpam-6568	339	13	containing	contain	VERB
ejpam-6568	339	14	f(x	f(x	PROPN
ejpam-6568	339	15	)	)	PUNCT
ejpam-6568	339	16	.	.	PUNCT
ejpam-6568	340	1	by	by	ADP
ejpam-6568	340	2	the	the	DET
ejpam-6568	340	3	hypothesis	hypothesis	NOUN
ejpam-6568	340	4	,	,	PUNCT
ejpam-6568	340	5	x	x	X
ejpam-6568	340	6	∈	∈	NOUN
ejpam-6568	340	7	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	340	8	-	-	PUNCT
ejpam-6568	340	9	cl(v	cl(v	NOUN
ejpam-6568	340	10	)	)	PUNCT
ejpam-6568	340	11	)	)	PUNCT
ejpam-6568	340	12	)	)	PUNCT
ejpam-6568	340	13	.	.	PUNCT
ejpam-6568	341	1	then	then	ADV
ejpam-6568	341	2	,	,	PUNCT
ejpam-6568	341	3	there	there	PRON
ejpam-6568	341	4	exists	exist	VERB
ejpam-6568	341	5	a	a	DET
ejpam-6568	341	6	⋆-open	⋆-open	ADJ
ejpam-6568	341	7	set	set	NOUN
ejpam-6568	341	8	u	u	NOUN
ejpam-6568	341	9	of	of	ADP
ejpam-6568	341	10	x	x	SYM
ejpam-6568	341	11	such	such	ADJ
ejpam-6568	341	12	that	that	SCONJ
ejpam-6568	341	13	x	x	SYM
ejpam-6568	341	14	∈	∈	PROPN
ejpam-6568	341	15	u	u	NOUN
ejpam-6568	341	16	⊆	⊆	NUM
ejpam-6568	341	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	341	18	-	-	PUNCT
ejpam-6568	341	19	cl(v	cl(v	NOUN
ejpam-6568	341	20	)	)	PUNCT
ejpam-6568	341	21	)	)	PUNCT
ejpam-6568	341	22	.	.	PUNCT
ejpam-6568	342	1	thus	thus	ADV
ejpam-6568	342	2	,	,	PUNCT
ejpam-6568	342	3	f(u	f(u	PROPN
ejpam-6568	342	4	)	)	PUNCT
ejpam-6568	342	5	⊆	⊆	NUM
ejpam-6568	342	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	342	7	-	-	NUM
ejpam-6568	342	8	cl(v	cl(v	NOUN
ejpam-6568	342	9	)	)	PUNCT
ejpam-6568	342	10	and	and	CCONJ
ejpam-6568	342	11	so	so	ADV
ejpam-6568	342	12	f	f	PROPN
ejpam-6568	342	13	is	be	AUX
ejpam-6568	342	14	weakly	weakly	ADJ
ejpam-6568	342	15	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	342	16	,	,	PUNCT
ejpam-6568	342	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	342	18	at	at	ADP
ejpam-6568	342	19	x.	x.	NOUN
ejpam-6568	342	20	theorem	theorem	VERB
ejpam-6568	342	21	11	11	NUM
ejpam-6568	342	22	.	.	PUNCT
ejpam-6568	343	1	a	a	DET
ejpam-6568	343	2	function	function	NOUN
ejpam-6568	343	3	f	f	NOUN
ejpam-6568	343	4	:	:	PUNCT
ejpam-6568	343	5	(	(	PUNCT
ejpam-6568	343	6	x	x	X
ejpam-6568	343	7	,	,	PUNCT
ejpam-6568	343	8	τ	τ	PROPN
ejpam-6568	343	9	,	,	PUNCT
ejpam-6568	343	10	i	i	NOUN
ejpam-6568	343	11	)	)	PUNCT
ejpam-6568	343	12	→	→	PUNCT
ejpam-6568	343	13	(	(	PUNCT
ejpam-6568	343	14	y	y	PROPN
ejpam-6568	343	15	,	,	PUNCT
ejpam-6568	343	16	σ1	σ1	PROPN
ejpam-6568	343	17	,	,	PUNCT
ejpam-6568	343	18	σ2	σ2	NOUN
ejpam-6568	343	19	)	)	PUNCT
ejpam-6568	343	20	is	be	AUX
ejpam-6568	343	21	weakly	weakly	ADJ
ejpam-6568	343	22	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	343	23	,	,	PUNCT
ejpam-6568	343	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	343	25	if	if	SCONJ
ejpam-6568	343	26	and	and	CCONJ
ejpam-6568	343	27	only	only	ADV
ejpam-6568	343	28	if	if	SCONJ
ejpam-6568	343	29	f−1(v	f−1(v	PROPN
ejpam-6568	343	30	)	)	PUNCT
ejpam-6568	343	31	⊆	⊆	X
ejpam-6568	343	32	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	343	33	-	-	PUNCT
ejpam-6568	343	34	cl(v	cl(v	NOUN
ejpam-6568	343	35	)	)	PUNCT
ejpam-6568	343	36	)	)	PUNCT
ejpam-6568	343	37	)	)	PUNCT
ejpam-6568	344	1	for	for	ADP
ejpam-6568	344	2	every	every	DET
ejpam-6568	344	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	344	4	-	-	ADJ
ejpam-6568	344	5	open	open	ADJ
ejpam-6568	344	6	set	set	NOUN
ejpam-6568	344	7	v	v	NOUN
ejpam-6568	344	8	of	of	ADP
ejpam-6568	344	9	y	y	PROPN
ejpam-6568	344	10	.	.	PUNCT
ejpam-6568	345	1	n.	n.	PROPN
ejpam-6568	345	2	viriyapong	viriyapong	PROPN
ejpam-6568	345	3	,	,	PUNCT
ejpam-6568	345	4	a.	a.	PROPN
ejpam-6568	345	5	sama	sama	PROPN
ejpam-6568	345	6	-	-	PUNCT
ejpam-6568	345	7	ae	ae	PROPN
ejpam-6568	345	8	,	,	PUNCT
ejpam-6568	345	9	c.	c.	PROPN
ejpam-6568	345	10	boonpok	boonpok	PROPN
ejpam-6568	345	11	/	/	SYM
ejpam-6568	345	12	eur	eur	PROPN
ejpam-6568	345	13	.	.	PUNCT
ejpam-6568	346	1	j.	j.	PROPN
ejpam-6568	346	2	pure	pure	PROPN
ejpam-6568	346	3	appl	appl	PROPN
ejpam-6568	346	4	.	.	PROPN
ejpam-6568	346	5	math	math	PROPN
ejpam-6568	346	6	,	,	PUNCT
ejpam-6568	346	7	18	18	NUM
ejpam-6568	346	8	(	(	PUNCT
ejpam-6568	346	9	3	3	NUM
ejpam-6568	346	10	)	)	PUNCT
ejpam-6568	346	11	(	(	PUNCT
ejpam-6568	346	12	2025	2025	NUM
ejpam-6568	346	13	)	)	PUNCT
ejpam-6568	346	14	,	,	PUNCT
ejpam-6568	346	15	6568	6568	NUM
ejpam-6568	346	16	12	12	NUM
ejpam-6568	346	17	of	of	ADP
ejpam-6568	346	18	18	18	NUM
ejpam-6568	346	19	proof	proof	NOUN
ejpam-6568	346	20	.	.	PUNCT
ejpam-6568	347	1	let	let	VERB
ejpam-6568	347	2	v	v	PART
ejpam-6568	347	3	be	be	AUX
ejpam-6568	347	4	any	any	DET
ejpam-6568	347	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	347	6	-	-	ADJ
ejpam-6568	347	7	open	open	ADJ
ejpam-6568	347	8	set	set	NOUN
ejpam-6568	347	9	of	of	ADP
ejpam-6568	347	10	y	y	PROPN
ejpam-6568	347	11	and	and	CCONJ
ejpam-6568	347	12	x	x	PROPN
ejpam-6568	347	13	∈	∈	PROPN
ejpam-6568	347	14	f−1(v	f−1(v	NOUN
ejpam-6568	347	15	)	)	PUNCT
ejpam-6568	347	16	.	.	PUNCT
ejpam-6568	348	1	then	then	ADV
ejpam-6568	348	2	,	,	PUNCT
ejpam-6568	348	3	f(x	f(x	PROPN
ejpam-6568	348	4	)	)	PUNCT
ejpam-6568	348	5	∈	∈	PROPN
ejpam-6568	348	6	v	v	NOUN
ejpam-6568	348	7	.	.	PUNCT
ejpam-6568	349	1	since	since	SCONJ
ejpam-6568	349	2	f	f	PROPN
ejpam-6568	349	3	is	be	AUX
ejpam-6568	349	4	weakly	weakly	ADJ
ejpam-6568	349	5	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	349	6	,	,	PUNCT
ejpam-6568	349	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	349	8	at	at	ADP
ejpam-6568	349	9	x	x	X
ejpam-6568	349	10	,	,	PUNCT
ejpam-6568	349	11	by	by	ADP
ejpam-6568	349	12	theorem	theorem	NOUN
ejpam-6568	349	13	10	10	NUM
ejpam-6568	349	14	we	we	PRON
ejpam-6568	349	15	have	have	VERB
ejpam-6568	349	16	x	x	PART
ejpam-6568	349	17	∈	∈	PROPN
ejpam-6568	349	18	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NOUN
ejpam-6568	349	19	-	-	PUNCT
ejpam-6568	349	20	cl(v	cl(v	NOUN
ejpam-6568	349	21	)	)	PUNCT
ejpam-6568	349	22	)	)	PUNCT
ejpam-6568	349	23	)	)	PUNCT
ejpam-6568	349	24	and	and	CCONJ
ejpam-6568	349	25	hence	hence	ADV
ejpam-6568	349	26	f−1(v	f−1(v	NOUN
ejpam-6568	349	27	)	)	PUNCT
ejpam-6568	349	28	⊆	⊆	X
ejpam-6568	349	29	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	349	30	-	-	PUNCT
ejpam-6568	349	31	cl(v	cl(v	NOUN
ejpam-6568	349	32	)	)	PUNCT
ejpam-6568	349	33	)	)	PUNCT
ejpam-6568	349	34	)	)	PUNCT
ejpam-6568	349	35	.	.	PUNCT
ejpam-6568	350	1	conversely	conversely	ADV
ejpam-6568	350	2	,	,	PUNCT
ejpam-6568	350	3	let	let	VERB
ejpam-6568	350	4	x	x	X
ejpam-6568	350	5	∈	∈	PROPN
ejpam-6568	350	6	x	x	X
ejpam-6568	350	7	and	and	CCONJ
ejpam-6568	350	8	v	v	AUX
ejpam-6568	350	9	be	be	AUX
ejpam-6568	350	10	any	any	DET
ejpam-6568	350	11	σ1σ2	σ1σ2	NOUN
ejpam-6568	350	12	-	-	ADJ
ejpam-6568	350	13	open	open	ADJ
ejpam-6568	350	14	set	set	NOUN
ejpam-6568	350	15	of	of	ADP
ejpam-6568	350	16	y	y	PROPN
ejpam-6568	350	17	containing	contain	VERB
ejpam-6568	350	18	f(x	f(x	PROPN
ejpam-6568	350	19	)	)	PUNCT
ejpam-6568	350	20	.	.	PUNCT
ejpam-6568	351	1	then	then	ADV
ejpam-6568	351	2	,	,	PUNCT
ejpam-6568	351	3	we	we	PRON
ejpam-6568	351	4	have	have	VERB
ejpam-6568	351	5	x	x	X
ejpam-6568	351	6	∈	∈	PROPN
ejpam-6568	351	7	f−1(v	f−1(v	NOUN
ejpam-6568	351	8	)	)	PUNCT
ejpam-6568	352	1	⊆	⊆	X
ejpam-6568	352	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	352	3	-	-	PUNCT
ejpam-6568	352	4	cl(v	cl(v	NOUN
ejpam-6568	352	5	)	)	PUNCT
ejpam-6568	352	6	)	)	PUNCT
ejpam-6568	352	7	)	)	PUNCT
ejpam-6568	352	8	.	.	PUNCT
ejpam-6568	353	1	by	by	ADP
ejpam-6568	353	2	theorem	theorem	NOUN
ejpam-6568	353	3	10	10	NUM
ejpam-6568	353	4	,	,	PUNCT
ejpam-6568	353	5	f	f	PROPN
ejpam-6568	353	6	is	be	AUX
ejpam-6568	353	7	weakly	weakly	ADJ
ejpam-6568	353	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	353	9	,	,	PUNCT
ejpam-6568	353	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	353	11	at	at	ADP
ejpam-6568	353	12	x.	x.	NOUN
ejpam-6568	353	13	this	this	PRON
ejpam-6568	353	14	shows	show	VERB
ejpam-6568	353	15	that	that	SCONJ
ejpam-6568	353	16	f	f	PROPN
ejpam-6568	353	17	is	be	AUX
ejpam-6568	353	18	weakly	weakly	ADJ
ejpam-6568	353	19	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	353	20	,	,	PUNCT
ejpam-6568	353	21	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	353	22	.	.	X
ejpam-6568	353	23	theorem	theorem	NOUN
ejpam-6568	353	24	12	12	NUM
ejpam-6568	353	25	.	.	PUNCT
ejpam-6568	354	1	a	a	DET
ejpam-6568	354	2	function	function	NOUN
ejpam-6568	354	3	f	f	NOUN
ejpam-6568	354	4	:	:	PUNCT
ejpam-6568	354	5	(	(	PUNCT
ejpam-6568	354	6	x	x	X
ejpam-6568	354	7	,	,	PUNCT
ejpam-6568	354	8	τ	τ	PROPN
ejpam-6568	354	9	,	,	PUNCT
ejpam-6568	354	10	i	i	NOUN
ejpam-6568	354	11	)	)	PUNCT
ejpam-6568	354	12	→	→	PUNCT
ejpam-6568	354	13	(	(	PUNCT
ejpam-6568	354	14	y	y	PROPN
ejpam-6568	354	15	,	,	PUNCT
ejpam-6568	354	16	σ1	σ1	PROPN
ejpam-6568	354	17	,	,	PUNCT
ejpam-6568	354	18	σ2	σ2	NOUN
ejpam-6568	354	19	)	)	PUNCT
ejpam-6568	354	20	is	be	AUX
ejpam-6568	354	21	weakly	weakly	ADJ
ejpam-6568	354	22	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	354	23	,	,	PUNCT
ejpam-6568	354	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	354	25	if	if	SCONJ
ejpam-6568	354	26	and	and	CCONJ
ejpam-6568	354	27	only	only	ADV
ejpam-6568	354	28	if	if	SCONJ
ejpam-6568	354	29	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	354	30	)	)	PUNCT
ejpam-6568	354	31	)	)	PUNCT
ejpam-6568	355	1	⊆	⊆	NUM
ejpam-6568	355	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	355	3	-	-	PUNCT
ejpam-6568	355	4	cl(v	cl(v	NOUN
ejpam-6568	355	5	)	)	PUNCT
ejpam-6568	355	6	)	)	PUNCT
ejpam-6568	355	7	for	for	ADP
ejpam-6568	355	8	every	every	DET
ejpam-6568	355	9	σ1σ2	σ1σ2	NOUN
ejpam-6568	355	10	-	-	ADJ
ejpam-6568	355	11	open	open	ADJ
ejpam-6568	355	12	set	set	NOUN
ejpam-6568	355	13	v	v	NOUN
ejpam-6568	355	14	of	of	ADP
ejpam-6568	355	15	y	y	PROPN
ejpam-6568	355	16	.	.	PUNCT
ejpam-6568	356	1	proof	proof	NOUN
ejpam-6568	356	2	.	.	PUNCT
ejpam-6568	357	1	let	let	VERB
ejpam-6568	357	2	v	v	PART
ejpam-6568	357	3	be	be	AUX
ejpam-6568	357	4	any	any	DET
ejpam-6568	357	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	357	6	-	-	ADJ
ejpam-6568	357	7	open	open	ADJ
ejpam-6568	357	8	set	set	NOUN
ejpam-6568	357	9	of	of	ADP
ejpam-6568	357	10	y	y	PROPN
ejpam-6568	357	11	.	.	PUNCT
ejpam-6568	358	1	suppose	suppose	VERB
ejpam-6568	358	2	that	that	SCONJ
ejpam-6568	358	3	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	358	4	)	)	PUNCT
ejpam-6568	358	5	)	)	PUNCT
ejpam-6568	359	1	⊈	⊈	PROPN
ejpam-6568	359	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	359	3	-	-	PUNCT
ejpam-6568	359	4	cl(v	cl(v	NOUN
ejpam-6568	359	5	)	)	PUNCT
ejpam-6568	359	6	)	)	PUNCT
ejpam-6568	359	7	.	.	PUNCT
ejpam-6568	360	1	there	there	PRON
ejpam-6568	360	2	exists	exist	VERB
ejpam-6568	360	3	x	x	X
ejpam-6568	360	4	∈	∈	PROPN
ejpam-6568	360	5	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	360	6	)	)	PUNCT
ejpam-6568	360	7	)	)	PUNCT
ejpam-6568	360	8	,	,	PUNCT
ejpam-6568	360	9	but	but	CCONJ
ejpam-6568	360	10	x	x	X
ejpam-6568	360	11	̸∈	̸∈	PROPN
ejpam-6568	360	12	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	360	13	-	-	PUNCT
ejpam-6568	360	14	cl(v	cl(v	NOUN
ejpam-6568	360	15	)	)	PUNCT
ejpam-6568	360	16	)	)	PUNCT
ejpam-6568	360	17	.	.	PUNCT
ejpam-6568	361	1	then	then	ADV
ejpam-6568	361	2	,	,	PUNCT
ejpam-6568	361	3	f(x	f(x	PROPN
ejpam-6568	361	4	)	)	PUNCT
ejpam-6568	361	5	̸∈	̸∈	PROPN
ejpam-6568	361	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	361	7	-	-	NUM
ejpam-6568	361	8	cl(v	cl(v	NOUN
ejpam-6568	361	9	)	)	PUNCT
ejpam-6568	361	10	and	and	CCONJ
ejpam-6568	361	11	there	there	PRON
ejpam-6568	361	12	exists	exist	VERB
ejpam-6568	361	13	a	a	DET
ejpam-6568	361	14	⋆-open	⋆-open	ADJ
ejpam-6568	361	15	set	set	NOUN
ejpam-6568	361	16	w	w	PROPN
ejpam-6568	361	17	of	of	ADP
ejpam-6568	361	18	y	y	PROPN
ejpam-6568	361	19	containing	contain	VERB
ejpam-6568	361	20	f(x	f(x	PROPN
ejpam-6568	361	21	)	)	PUNCT
ejpam-6568	361	22	such	such	ADJ
ejpam-6568	361	23	that	that	SCONJ
ejpam-6568	361	24	w	w	PROPN
ejpam-6568	361	25	∩	∩	NOUN
ejpam-6568	361	26	v	v	NOUN
ejpam-6568	361	27	=	=	PUNCT
ejpam-6568	361	28	∅.	∅.	ADP
ejpam-6568	361	29	thus	thus	ADV
ejpam-6568	361	30	,	,	PUNCT
ejpam-6568	361	31	σ1σ2	σ1σ2	NOUN
ejpam-6568	361	32	-	-	PUNCT
ejpam-6568	361	33	cl(w	cl(w	NOUN
ejpam-6568	361	34	)	)	PUNCT
ejpam-6568	361	35	∩	∩	NOUN
ejpam-6568	361	36	v	v	NOUN
ejpam-6568	361	37	=	=	PUNCT
ejpam-6568	361	38	∅.	∅.	NOUN
ejpam-6568	361	39	since	since	SCONJ
ejpam-6568	361	40	f	f	PROPN
ejpam-6568	361	41	is	be	AUX
ejpam-6568	361	42	weakly	weakly	ADJ
ejpam-6568	361	43	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	361	44	,	,	PUNCT
ejpam-6568	361	45	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	361	46	at	at	ADP
ejpam-6568	361	47	x	x	X
ejpam-6568	361	48	,	,	PUNCT
ejpam-6568	361	49	there	there	PRON
ejpam-6568	361	50	exists	exist	VERB
ejpam-6568	361	51	a	a	DET
ejpam-6568	361	52	⋆-open	⋆-open	ADJ
ejpam-6568	361	53	set	set	NOUN
ejpam-6568	361	54	u	u	NOUN
ejpam-6568	361	55	of	of	ADP
ejpam-6568	361	56	x	x	PUNCT
ejpam-6568	361	57	containing	contain	VERB
ejpam-6568	361	58	x	x	PUNCT
ejpam-6568	361	59	such	such	ADJ
ejpam-6568	361	60	that	that	DET
ejpam-6568	361	61	f(u	f(u	PROPN
ejpam-6568	361	62	)	)	PUNCT
ejpam-6568	361	63	⊆	⊆	NUM
ejpam-6568	361	64	σ1σ2	σ1σ2	NOUN
ejpam-6568	361	65	-	-	PUNCT
ejpam-6568	361	66	cl(w	cl(w	NOUN
ejpam-6568	361	67	)	)	PUNCT
ejpam-6568	361	68	.	.	PUNCT
ejpam-6568	362	1	therefore	therefore	ADV
ejpam-6568	362	2	,	,	PUNCT
ejpam-6568	362	3	f(u	f(u	PROPN
ejpam-6568	362	4	)	)	PUNCT
ejpam-6568	362	5	∩	∩	NOUN
ejpam-6568	362	6	v	v	NOUN
ejpam-6568	362	7	=	=	PUNCT
ejpam-6568	362	8	∅.	∅.	NOUN
ejpam-6568	362	9	since	since	SCONJ
ejpam-6568	362	10	x	x	PROPN
ejpam-6568	362	11	∈	∈	PROPN
ejpam-6568	362	12	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	362	13	)	)	PUNCT
ejpam-6568	362	14	)	)	PUNCT
ejpam-6568	362	15	,	,	PUNCT
ejpam-6568	362	16	u	u	NOUN
ejpam-6568	362	17	∩f−1(v	∩f−1(v	NOUN
ejpam-6568	362	18	)	)	PUNCT
ejpam-6568	362	19	̸=	̸=	NOUN
ejpam-6568	362	20	∅	∅	NOUN
ejpam-6568	362	21	and	and	CCONJ
ejpam-6568	362	22	f(u)∩v	f(u)∩v	PROPN
ejpam-6568	362	23	̸=	̸=	PROPN
ejpam-6568	362	24	∅	∅	NOUN
ejpam-6568	362	25	,	,	PUNCT
ejpam-6568	362	26	which	which	PRON
ejpam-6568	362	27	is	be	AUX
ejpam-6568	362	28	a	a	DET
ejpam-6568	362	29	contradiction	contradiction	NOUN
ejpam-6568	362	30	.	.	PUNCT
ejpam-6568	363	1	this	this	PRON
ejpam-6568	363	2	shows	show	VERB
ejpam-6568	363	3	that	that	SCONJ
ejpam-6568	363	4	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	363	5	)	)	PUNCT
ejpam-6568	363	6	)	)	PUNCT
ejpam-6568	364	1	⊆	⊆	NUM
ejpam-6568	364	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	364	3	-	-	PUNCT
ejpam-6568	364	4	cl(v	cl(v	NOUN
ejpam-6568	364	5	)	)	PUNCT
ejpam-6568	364	6	)	)	PUNCT
ejpam-6568	364	7	.	.	PUNCT
ejpam-6568	365	1	conversely	conversely	ADV
ejpam-6568	365	2	,	,	PUNCT
ejpam-6568	365	3	let	let	VERB
ejpam-6568	365	4	v	v	PART
ejpam-6568	365	5	be	be	AUX
ejpam-6568	365	6	any	any	DET
ejpam-6568	365	7	σ1σ2	σ1σ2	NOUN
ejpam-6568	365	8	-	-	ADJ
ejpam-6568	365	9	open	open	ADJ
ejpam-6568	365	10	set	set	NOUN
ejpam-6568	365	11	of	of	ADP
ejpam-6568	365	12	y	y	PROPN
ejpam-6568	365	13	.	.	PUNCT
ejpam-6568	366	1	then	then	ADV
ejpam-6568	366	2	,	,	PUNCT
ejpam-6568	366	3	y	y	PROPN
ejpam-6568	366	4	−	−	NUM
ejpam-6568	366	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	366	6	-	-	NUM
ejpam-6568	366	7	cl(v	cl(v	NOUN
ejpam-6568	366	8	)	)	PUNCT
ejpam-6568	366	9	is	be	AUX
ejpam-6568	366	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	366	11	-	-	ADJ
ejpam-6568	366	12	open	open	ADJ
ejpam-6568	366	13	in	in	ADP
ejpam-6568	366	14	y	y	PROPN
ejpam-6568	366	15	.	.	PUNCT
ejpam-6568	367	1	by	by	ADP
ejpam-6568	367	2	the	the	DET
ejpam-6568	367	3	hypothesis	hypothesis	NOUN
ejpam-6568	367	4	,	,	PUNCT
ejpam-6568	367	5	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	367	6	−σ1σ2	−σ1σ2	PROPN
ejpam-6568	367	7	-	-	PUNCT
ejpam-6568	367	8	cl(v	cl(v	NOUN
ejpam-6568	367	9	)	)	PUNCT
ejpam-6568	367	10	)	)	PUNCT
ejpam-6568	367	11	)	)	PUNCT
ejpam-6568	368	1	⊆	⊆	NUM
ejpam-6568	368	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	368	3	-	-	PUNCT
ejpam-6568	368	4	cl(y	cl(y	NOUN
ejpam-6568	368	5	−σ1σ2	−σ1σ2	NOUN
ejpam-6568	368	6	-	-	NOUN
ejpam-6568	368	7	cl(v	cl(v	NOUN
ejpam-6568	368	8	)	)	PUNCT
ejpam-6568	368	9	)	)	PUNCT
ejpam-6568	368	10	)	)	PUNCT
ejpam-6568	368	11	.	.	PUNCT
ejpam-6568	369	1	thus	thus	ADV
ejpam-6568	369	2	,	,	PUNCT
ejpam-6568	369	3	x	x	PUNCT
ejpam-6568	369	4	−	−	ADP
ejpam-6568	369	5	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	369	6	-	-	PUNCT
ejpam-6568	369	7	cl(v	cl(v	NOUN
ejpam-6568	369	8	)	)	PUNCT
ejpam-6568	369	9	)	)	PUNCT
ejpam-6568	369	10	)	)	PUNCT
ejpam-6568	370	1	⊆	⊆	NUM
ejpam-6568	370	2	x	x	X
ejpam-6568	370	3	−	−	PRON
ejpam-6568	370	4	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	370	5	-	-	PUNCT
ejpam-6568	370	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	370	7	-	-	PUNCT
ejpam-6568	370	8	cl(v	cl(v	NOUN
ejpam-6568	370	9	)	)	PUNCT
ejpam-6568	370	10	)	)	PUNCT
ejpam-6568	370	11	)	)	PUNCT
ejpam-6568	371	1	⊆	⊆	NUM
ejpam-6568	371	2	x	x	SYM
ejpam-6568	371	3	−	−	PROPN
ejpam-6568	371	4	f−1(v	f−1(v	PROPN
ejpam-6568	371	5	)	)	PUNCT
ejpam-6568	371	6	and	and	CCONJ
ejpam-6568	371	7	hence	hence	ADV
ejpam-6568	371	8	f−1(v	f−1(v	NOUN
ejpam-6568	371	9	)	)	PUNCT
ejpam-6568	371	10	⊆	⊆	X
ejpam-6568	371	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	371	12	-	-	PUNCT
ejpam-6568	371	13	cl(v	cl(v	NOUN
ejpam-6568	371	14	)	)	PUNCT
ejpam-6568	371	15	)	)	PUNCT
ejpam-6568	371	16	)	)	PUNCT
ejpam-6568	371	17	.	.	PUNCT
ejpam-6568	372	1	by	by	ADP
ejpam-6568	372	2	theorem	theorem	NOUN
ejpam-6568	372	3	11	11	NUM
ejpam-6568	372	4	,	,	PUNCT
ejpam-6568	372	5	f	f	PROPN
ejpam-6568	372	6	is	be	AUX
ejpam-6568	372	7	weakly	weakly	ADJ
ejpam-6568	372	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	372	9	,	,	PUNCT
ejpam-6568	372	10	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	372	11	.	.	X
ejpam-6568	372	12	theorem	theorem	VERB
ejpam-6568	372	13	13	13	NUM
ejpam-6568	372	14	.	.	PUNCT
ejpam-6568	373	1	for	for	ADP
ejpam-6568	373	2	a	a	DET
ejpam-6568	373	3	function	function	NOUN
ejpam-6568	373	4	(	(	PUNCT
ejpam-6568	373	5	x	x	X
ejpam-6568	373	6	,	,	PUNCT
ejpam-6568	373	7	τ	τ	PROPN
ejpam-6568	373	8	,	,	PUNCT
ejpam-6568	373	9	i	i	NOUN
ejpam-6568	373	10	)	)	PUNCT
ejpam-6568	373	11	→	→	PUNCT
ejpam-6568	373	12	(	(	PUNCT
ejpam-6568	373	13	y	y	PROPN
ejpam-6568	373	14	,	,	PUNCT
ejpam-6568	373	15	σ1	σ1	PROPN
ejpam-6568	373	16	,	,	PUNCT
ejpam-6568	373	17	σ2	σ2	NOUN
ejpam-6568	373	18	)	)	PUNCT
ejpam-6568	373	19	,	,	PUNCT
ejpam-6568	373	20	the	the	DET
ejpam-6568	373	21	following	follow	VERB
ejpam-6568	373	22	properties	property	NOUN
ejpam-6568	373	23	are	be	AUX
ejpam-6568	373	24	equivalent	equivalent	ADJ
ejpam-6568	373	25	:	:	PUNCT
ejpam-6568	373	26	(	(	PUNCT
ejpam-6568	373	27	1	1	X
ejpam-6568	373	28	)	)	PUNCT
ejpam-6568	373	29	f	f	PROPN
ejpam-6568	373	30	is	be	AUX
ejpam-6568	373	31	weakly	weakly	ADJ
ejpam-6568	373	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	373	33	,	,	PUNCT
ejpam-6568	373	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	373	35	;	;	PUNCT
ejpam-6568	373	36	(	(	PUNCT
ejpam-6568	373	37	2	2	X
ejpam-6568	373	38	)	)	PUNCT
ejpam-6568	373	39	f−1(v	f−1(v	NOUN
ejpam-6568	373	40	)	)	PUNCT
ejpam-6568	373	41	⊆	⊆	X
ejpam-6568	373	42	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	373	43	-	-	PUNCT
ejpam-6568	373	44	cl(v	cl(v	NOUN
ejpam-6568	373	45	)	)	PUNCT
ejpam-6568	373	46	)	)	PUNCT
ejpam-6568	373	47	)	)	PUNCT
ejpam-6568	374	1	for	for	ADP
ejpam-6568	374	2	every	every	DET
ejpam-6568	374	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	374	4	-	-	ADJ
ejpam-6568	374	5	open	open	ADJ
ejpam-6568	374	6	set	set	NOUN
ejpam-6568	374	7	v	v	NOUN
ejpam-6568	374	8	of	of	ADP
ejpam-6568	374	9	y	y	PROPN
ejpam-6568	374	10	;	;	PUNCT
ejpam-6568	374	11	(	(	PUNCT
ejpam-6568	374	12	3	3	X
ejpam-6568	374	13	)	)	PUNCT
ejpam-6568	374	14	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	374	15	-	-	PUNCT
ejpam-6568	374	16	int(k	int(k	PROPN
ejpam-6568	374	17	)	)	PUNCT
ejpam-6568	374	18	)	)	PUNCT
ejpam-6568	374	19	)	)	PUNCT
ejpam-6568	374	20	⊆	⊆	NUM
ejpam-6568	374	21	f−1(k	f−1(k	PROPN
ejpam-6568	374	22	)	)	PUNCT
ejpam-6568	374	23	for	for	ADP
ejpam-6568	374	24	every	every	DET
ejpam-6568	374	25	σ1σ2	σ1σ2	NUM
ejpam-6568	374	26	-	-	PUNCT
ejpam-6568	374	27	closed	closed	ADJ
ejpam-6568	374	28	set	set	NOUN
ejpam-6568	374	29	k	k	PROPN
ejpam-6568	374	30	of	of	ADP
ejpam-6568	374	31	y	y	PROPN
ejpam-6568	374	32	;	;	PUNCT
ejpam-6568	374	33	(	(	PUNCT
ejpam-6568	374	34	4	4	X
ejpam-6568	374	35	)	)	PUNCT
ejpam-6568	374	36	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	374	37	-	-	PUNCT
ejpam-6568	374	38	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	374	39	-	-	PUNCT
ejpam-6568	374	40	cl(b	cl(b	NOUN
ejpam-6568	374	41	)	)	PUNCT
ejpam-6568	374	42	)	)	PUNCT
ejpam-6568	374	43	)	)	PUNCT
ejpam-6568	374	44	)	)	PUNCT
ejpam-6568	375	1	⊆	⊆	NUM
ejpam-6568	375	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	375	3	-	-	PUNCT
ejpam-6568	375	4	cl(b	cl(b	NOUN
ejpam-6568	375	5	)	)	PUNCT
ejpam-6568	375	6	)	)	PUNCT
ejpam-6568	375	7	for	for	ADP
ejpam-6568	375	8	every	every	DET
ejpam-6568	375	9	subset	subset	NOUN
ejpam-6568	375	10	b	b	PROPN
ejpam-6568	375	11	of	of	ADP
ejpam-6568	375	12	y	y	PROPN
ejpam-6568	375	13	;	;	PUNCT
ejpam-6568	375	14	(	(	PUNCT
ejpam-6568	375	15	5	5	X
ejpam-6568	375	16	)	)	PUNCT
ejpam-6568	375	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	375	18	-	-	PUNCT
ejpam-6568	375	19	int(b	int(b	NOUN
ejpam-6568	375	20	)	)	PUNCT
ejpam-6568	375	21	)	)	PUNCT
ejpam-6568	376	1	⊆	⊆	X
ejpam-6568	376	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	376	3	-	-	PUNCT
ejpam-6568	376	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	376	5	-	-	PUNCT
ejpam-6568	376	6	int(b	int(b	NOUN
ejpam-6568	376	7	)	)	PUNCT
ejpam-6568	376	8	)	)	PUNCT
ejpam-6568	376	9	)	)	PUNCT
ejpam-6568	376	10	)	)	PUNCT
ejpam-6568	376	11	for	for	ADP
ejpam-6568	376	12	every	every	DET
ejpam-6568	376	13	subset	subset	NOUN
ejpam-6568	376	14	b	b	PROPN
ejpam-6568	376	15	of	of	ADP
ejpam-6568	376	16	y	y	PROPN
ejpam-6568	376	17	;	;	PUNCT
ejpam-6568	376	18	(	(	PUNCT
ejpam-6568	376	19	6	6	X
ejpam-6568	376	20	)	)	PUNCT
ejpam-6568	376	21	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	376	22	)	)	PUNCT
ejpam-6568	376	23	)	)	PUNCT
ejpam-6568	376	24	⊆	⊆	NUM
ejpam-6568	376	25	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	376	26	-	-	PUNCT
ejpam-6568	376	27	cl(v	cl(v	NOUN
ejpam-6568	376	28	)	)	PUNCT
ejpam-6568	376	29	)	)	PUNCT
ejpam-6568	376	30	for	for	ADP
ejpam-6568	376	31	every	every	DET
ejpam-6568	376	32	σ1σ2	σ1σ2	NOUN
ejpam-6568	376	33	-	-	ADJ
ejpam-6568	376	34	open	open	ADJ
ejpam-6568	376	35	set	set	NOUN
ejpam-6568	376	36	v	v	NOUN
ejpam-6568	376	37	of	of	ADP
ejpam-6568	376	38	y	y	PROPN
ejpam-6568	376	39	.	.	PUNCT
ejpam-6568	377	1	proof	proof	NOUN
ejpam-6568	377	2	.	.	PUNCT
ejpam-6568	378	1	(	(	PUNCT
ejpam-6568	378	2	1	1	X
ejpam-6568	378	3	)	)	PUNCT
ejpam-6568	378	4	⇒	⇒	NOUN
ejpam-6568	378	5	(	(	PUNCT
ejpam-6568	378	6	2	2	NUM
ejpam-6568	378	7	):	):	PUNCT
ejpam-6568	378	8	it	it	PRON
ejpam-6568	378	9	follows	follow	VERB
ejpam-6568	378	10	from	from	ADP
ejpam-6568	378	11	theorem	theorem	ADJ
ejpam-6568	378	12	11	11	NUM
ejpam-6568	378	13	.	.	PUNCT
ejpam-6568	379	1	(	(	PUNCT
ejpam-6568	379	2	2	2	X
ejpam-6568	379	3	)	)	PUNCT
ejpam-6568	379	4	⇒	⇒	NOUN
ejpam-6568	379	5	(	(	PUNCT
ejpam-6568	379	6	3	3	NUM
ejpam-6568	379	7	):	):	PUNCT
ejpam-6568	379	8	let	let	VERB
ejpam-6568	379	9	k	k	PRON
ejpam-6568	379	10	be	be	AUX
ejpam-6568	379	11	any	any	DET
ejpam-6568	379	12	σ1σ2	σ1σ2	NUM
ejpam-6568	379	13	-	-	PUNCT
ejpam-6568	379	14	closed	closed	ADJ
ejpam-6568	379	15	set	set	NOUN
ejpam-6568	379	16	of	of	ADP
ejpam-6568	379	17	y	y	PROPN
ejpam-6568	379	18	.	.	PUNCT
ejpam-6568	380	1	then	then	ADV
ejpam-6568	380	2	,	,	PUNCT
ejpam-6568	380	3	y	y	PROPN
ejpam-6568	380	4	−k	−k	PROPN
ejpam-6568	380	5	is	be	AUX
ejpam-6568	380	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	380	7	-	-	ADJ
ejpam-6568	380	8	open	open	ADJ
ejpam-6568	380	9	in	in	ADP
ejpam-6568	380	10	y	y	PROPN
ejpam-6568	380	11	and	and	CCONJ
ejpam-6568	380	12	by	by	ADP
ejpam-6568	380	13	(	(	PUNCT
ejpam-6568	380	14	2	2	NUM
ejpam-6568	380	15	)	)	PUNCT
ejpam-6568	380	16	,	,	PUNCT
ejpam-6568	380	17	x	x	PUNCT
ejpam-6568	380	18	−	−	PROPN
ejpam-6568	380	19	f−1(k	f−1(k	PROPN
ejpam-6568	380	20	)	)	PUNCT
ejpam-6568	380	21	=	=	SYM
ejpam-6568	380	22	f−1(y	f−1(y	PROPN
ejpam-6568	380	23	−k	−k	PROPN
ejpam-6568	380	24	)	)	PUNCT
ejpam-6568	380	25	⊆	⊆	NUM
ejpam-6568	380	26	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	380	27	-	-	PUNCT
ejpam-6568	380	28	cl(y	cl(y	NOUN
ejpam-6568	380	29	−k	−k	NOUN
ejpam-6568	380	30	)	)	PUNCT
ejpam-6568	380	31	)	)	PUNCT
ejpam-6568	380	32	)	)	PUNCT
ejpam-6568	381	1	=	=	SYM
ejpam-6568	381	2	int⋆(f−1(y	int⋆(f−1(y	PRON
ejpam-6568	381	3	−	−	VERB
ejpam-6568	381	4	σ1σ2	σ1σ2	NUM
ejpam-6568	381	5	-	-	PUNCT
ejpam-6568	381	6	int(k	int(k	NOUN
ejpam-6568	381	7	)	)	PUNCT
ejpam-6568	381	8	)	)	PUNCT
ejpam-6568	381	9	)	)	PUNCT
ejpam-6568	382	1	=	=	PUNCT
ejpam-6568	382	2	x	x	X
ejpam-6568	383	1	−	−	PRON
ejpam-6568	383	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	383	3	-	-	PUNCT
ejpam-6568	383	4	int(k	int(k	PROPN
ejpam-6568	383	5	)	)	PUNCT
ejpam-6568	383	6	)	)	PUNCT
ejpam-6568	383	7	)	)	PUNCT
ejpam-6568	383	8	.	.	PUNCT
ejpam-6568	384	1	thus	thus	ADV
ejpam-6568	384	2	,	,	PUNCT
ejpam-6568	384	3	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	384	4	-	-	PUNCT
ejpam-6568	384	5	int(k	int(k	PROPN
ejpam-6568	384	6	)	)	PUNCT
ejpam-6568	384	7	)	)	PUNCT
ejpam-6568	384	8	)	)	PUNCT
ejpam-6568	384	9	⊆	⊆	NUM
ejpam-6568	384	10	f−1(k	f−1(k	NOUN
ejpam-6568	384	11	)	)	PUNCT
ejpam-6568	384	12	.	.	PUNCT
ejpam-6568	385	1	n.	n.	PROPN
ejpam-6568	385	2	viriyapong	viriyapong	PROPN
ejpam-6568	385	3	,	,	PUNCT
ejpam-6568	385	4	a.	a.	PROPN
ejpam-6568	385	5	sama	sama	PROPN
ejpam-6568	385	6	-	-	PUNCT
ejpam-6568	385	7	ae	ae	PROPN
ejpam-6568	385	8	,	,	PUNCT
ejpam-6568	385	9	c.	c.	PROPN
ejpam-6568	385	10	boonpok	boonpok	PROPN
ejpam-6568	385	11	/	/	SYM
ejpam-6568	385	12	eur	eur	PROPN
ejpam-6568	385	13	.	.	PUNCT
ejpam-6568	386	1	j.	j.	PROPN
ejpam-6568	386	2	pure	pure	PROPN
ejpam-6568	386	3	appl	appl	PROPN
ejpam-6568	386	4	.	.	PROPN
ejpam-6568	386	5	math	math	PROPN
ejpam-6568	386	6	,	,	PUNCT
ejpam-6568	386	7	18	18	NUM
ejpam-6568	386	8	(	(	PUNCT
ejpam-6568	386	9	3	3	NUM
ejpam-6568	386	10	)	)	PUNCT
ejpam-6568	386	11	(	(	PUNCT
ejpam-6568	386	12	2025	2025	NUM
ejpam-6568	386	13	)	)	PUNCT
ejpam-6568	386	14	,	,	PUNCT
ejpam-6568	386	15	6568	6568	NUM
ejpam-6568	386	16	13	13	NUM
ejpam-6568	386	17	of	of	ADP
ejpam-6568	386	18	18	18	NUM
ejpam-6568	386	19	(	(	PUNCT
ejpam-6568	386	20	3	3	NUM
ejpam-6568	386	21	)	)	PUNCT
ejpam-6568	386	22	⇒	⇒	NOUN
ejpam-6568	386	23	(	(	PUNCT
ejpam-6568	386	24	4	4	NUM
ejpam-6568	386	25	):	):	PUNCT
ejpam-6568	386	26	let	let	VERB
ejpam-6568	386	27	b	b	X
ejpam-6568	386	28	be	be	AUX
ejpam-6568	386	29	any	any	DET
ejpam-6568	386	30	subset	subset	NOUN
ejpam-6568	386	31	of	of	ADP
ejpam-6568	386	32	y	y	PROPN
ejpam-6568	386	33	.	.	PUNCT
ejpam-6568	387	1	then	then	ADV
ejpam-6568	387	2	,	,	PUNCT
ejpam-6568	387	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	387	4	-	-	NOUN
ejpam-6568	387	5	cl(b	cl(b	NOUN
ejpam-6568	387	6	)	)	PUNCT
ejpam-6568	387	7	is	be	AUX
ejpam-6568	387	8	σ1σ2	σ1σ2	NOUN
ejpam-6568	387	9	-	-	ADJ
ejpam-6568	387	10	closed	closed	ADJ
ejpam-6568	387	11	in	in	ADP
ejpam-6568	387	12	y	y	PROPN
ejpam-6568	387	13	.	.	PUNCT
ejpam-6568	388	1	by	by	ADP
ejpam-6568	388	2	(	(	PUNCT
ejpam-6568	388	3	3	3	NUM
ejpam-6568	388	4	)	)	PUNCT
ejpam-6568	388	5	,	,	PUNCT
ejpam-6568	388	6	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	388	7	-	-	PUNCT
ejpam-6568	388	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	388	9	-	-	PUNCT
ejpam-6568	388	10	cl(b	cl(b	NOUN
ejpam-6568	388	11	)	)	PUNCT
ejpam-6568	388	12	)	)	PUNCT
ejpam-6568	388	13	)	)	PUNCT
ejpam-6568	388	14	)	)	PUNCT
ejpam-6568	388	15	⊆	⊆	NUM
ejpam-6568	388	16	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	388	17	-	-	PUNCT
ejpam-6568	388	18	cl(b	cl(b	NOUN
ejpam-6568	388	19	)	)	PUNCT
ejpam-6568	388	20	)	)	PUNCT
ejpam-6568	388	21	.	.	PUNCT
ejpam-6568	389	1	(	(	PUNCT
ejpam-6568	389	2	4	4	X
ejpam-6568	389	3	)	)	PUNCT
ejpam-6568	389	4	⇒	⇒	NOUN
ejpam-6568	389	5	(	(	PUNCT
ejpam-6568	389	6	5	5	NUM
ejpam-6568	389	7	):	):	PUNCT
ejpam-6568	389	8	let	let	VERB
ejpam-6568	389	9	b	b	X
ejpam-6568	389	10	be	be	AUX
ejpam-6568	389	11	any	any	DET
ejpam-6568	389	12	subset	subset	NOUN
ejpam-6568	389	13	of	of	ADP
ejpam-6568	389	14	y	y	PROPN
ejpam-6568	389	15	.	.	PUNCT
ejpam-6568	390	1	thus	thus	ADV
ejpam-6568	390	2	by	by	ADP
ejpam-6568	390	3	(	(	PUNCT
ejpam-6568	390	4	4	4	NUM
ejpam-6568	390	5	)	)	PUNCT
ejpam-6568	390	6	,	,	PUNCT
ejpam-6568	390	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	390	8	-	-	PUNCT
ejpam-6568	390	9	int(b	int(b	NOUN
ejpam-6568	390	10	)	)	PUNCT
ejpam-6568	390	11	)	)	PUNCT
ejpam-6568	391	1	=	=	PUNCT
ejpam-6568	392	1	x	x	X
ejpam-6568	392	2	−	−	PRON
ejpam-6568	392	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	392	4	-	-	PUNCT
ejpam-6568	392	5	cl(y	cl(y	NOUN
ejpam-6568	392	6	−b	−b	NOUN
ejpam-6568	392	7	)	)	PUNCT
ejpam-6568	392	8	)	)	PUNCT
ejpam-6568	393	1	⊆	⊆	NUM
ejpam-6568	393	2	x	x	X
ejpam-6568	393	3	−	−	PRON
ejpam-6568	393	4	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	393	5	-	-	PUNCT
ejpam-6568	393	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	393	7	-	-	PUNCT
ejpam-6568	393	8	cl(y	cl(y	NOUN
ejpam-6568	393	9	−b	−b	NOUN
ejpam-6568	393	10	)	)	PUNCT
ejpam-6568	393	11	)	)	PUNCT
ejpam-6568	393	12	)	)	PUNCT
ejpam-6568	393	13	)	)	PUNCT
ejpam-6568	394	1	=	=	PUNCT
ejpam-6568	394	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	394	3	-	-	PUNCT
ejpam-6568	394	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	394	5	-	-	PUNCT
ejpam-6568	394	6	int(b	int(b	NOUN
ejpam-6568	394	7	)	)	PUNCT
ejpam-6568	394	8	)	)	PUNCT
ejpam-6568	394	9	)	)	PUNCT
ejpam-6568	394	10	)	)	PUNCT
ejpam-6568	394	11	.	.	PUNCT
ejpam-6568	395	1	(	(	PUNCT
ejpam-6568	395	2	5	5	X
ejpam-6568	395	3	)	)	PUNCT
ejpam-6568	395	4	⇒	⇒	NOUN
ejpam-6568	395	5	(	(	PUNCT
ejpam-6568	395	6	6	6	NUM
ejpam-6568	395	7	):	):	PUNCT
ejpam-6568	395	8	let	let	VERB
ejpam-6568	395	9	v	v	PART
ejpam-6568	395	10	be	be	AUX
ejpam-6568	395	11	any	any	DET
ejpam-6568	395	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	395	13	-	-	ADJ
ejpam-6568	395	14	open	open	ADJ
ejpam-6568	395	15	set	set	NOUN
ejpam-6568	395	16	of	of	ADP
ejpam-6568	395	17	y	y	PROPN
ejpam-6568	395	18	and	and	CCONJ
ejpam-6568	395	19	x	x	PROPN
ejpam-6568	395	20	̸∈	̸∈	PROPN
ejpam-6568	395	21	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	395	22	-	-	PUNCT
ejpam-6568	395	23	cl(v	cl(v	NOUN
ejpam-6568	395	24	)	)	PUNCT
ejpam-6568	395	25	)	)	PUNCT
ejpam-6568	395	26	.	.	PUNCT
ejpam-6568	396	1	then	then	ADV
ejpam-6568	396	2	,	,	PUNCT
ejpam-6568	396	3	there	there	PRON
ejpam-6568	396	4	exists	exist	VERB
ejpam-6568	396	5	a	a	DET
ejpam-6568	396	6	σ1σ2	σ1σ2	NUM
ejpam-6568	396	7	-	-	ADJ
ejpam-6568	396	8	open	open	ADJ
ejpam-6568	396	9	set	set	NOUN
ejpam-6568	396	10	u	u	NOUN
ejpam-6568	396	11	of	of	ADP
ejpam-6568	396	12	y	y	PROPN
ejpam-6568	396	13	containing	contain	VERB
ejpam-6568	396	14	f(x	f(x	PROPN
ejpam-6568	396	15	)	)	PUNCT
ejpam-6568	396	16	such	such	ADJ
ejpam-6568	396	17	that	that	SCONJ
ejpam-6568	396	18	u	u	PROPN
ejpam-6568	396	19	∩	∩	NOUN
ejpam-6568	396	20	v	v	ADJ
ejpam-6568	396	21	=	=	PUNCT
ejpam-6568	396	22	∅.	∅.	X
ejpam-6568	396	23	by	by	ADP
ejpam-6568	396	24	(	(	PUNCT
ejpam-6568	396	25	5	5	NUM
ejpam-6568	396	26	)	)	PUNCT
ejpam-6568	396	27	,	,	PUNCT
ejpam-6568	396	28	x	x	PUNCT
ejpam-6568	396	29	∈	∈	PROPN
ejpam-6568	396	30	f−1(u	f−1(u	PROPN
ejpam-6568	396	31	)	)	PUNCT
ejpam-6568	396	32	⊆	⊆	NUM
ejpam-6568	396	33	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	396	34	-	-	PUNCT
ejpam-6568	396	35	cl(u	cl(u	NOUN
ejpam-6568	396	36	)	)	PUNCT
ejpam-6568	396	37	)	)	PUNCT
ejpam-6568	396	38	)	)	PUNCT
ejpam-6568	396	39	and	and	CCONJ
ejpam-6568	396	40	there	there	PRON
ejpam-6568	396	41	exists	exist	VERB
ejpam-6568	396	42	a	a	DET
ejpam-6568	396	43	⋆-open	⋆-open	ADJ
ejpam-6568	396	44	set	set	NOUN
ejpam-6568	396	45	g	g	NOUN
ejpam-6568	396	46	of	of	ADP
ejpam-6568	396	47	x	x	PUNCT
ejpam-6568	396	48	containing	contain	VERB
ejpam-6568	396	49	x	x	PUNCT
ejpam-6568	396	50	such	such	ADJ
ejpam-6568	396	51	that	that	DET
ejpam-6568	396	52	f(g	f(g	NOUN
ejpam-6568	396	53	)	)	PUNCT
ejpam-6568	396	54	⊆	⊆	NUM
ejpam-6568	396	55	σ1σ2	σ1σ2	NOUN
ejpam-6568	396	56	-	-	PUNCT
ejpam-6568	396	57	cl(u	cl(u	NUM
ejpam-6568	396	58	)	)	PUNCT
ejpam-6568	396	59	.	.	PUNCT
ejpam-6568	397	1	thus	thus	ADV
ejpam-6568	397	2	,	,	PUNCT
ejpam-6568	397	3	g∩f−1(v	g∩f−1(v	NOUN
ejpam-6568	397	4	)	)	PUNCT
ejpam-6568	398	1	=	=	NOUN
ejpam-6568	398	2	∅	∅	NOUN
ejpam-6568	399	1	and	and	CCONJ
ejpam-6568	399	2	so	so	ADV
ejpam-6568	399	3	x	x	PUNCT
ejpam-6568	399	4	̸∈	̸∈	PROPN
ejpam-6568	399	5	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	399	6	)	)	PUNCT
ejpam-6568	399	7	)	)	PUNCT
ejpam-6568	399	8	.	.	PUNCT
ejpam-6568	400	1	this	this	PRON
ejpam-6568	400	2	shows	show	VERB
ejpam-6568	400	3	that	that	SCONJ
ejpam-6568	400	4	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	400	5	)	)	PUNCT
ejpam-6568	400	6	)	)	PUNCT
ejpam-6568	401	1	⊆	⊆	NUM
ejpam-6568	401	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	401	3	-	-	PUNCT
ejpam-6568	401	4	cl(v	cl(v	NOUN
ejpam-6568	401	5	)	)	PUNCT
ejpam-6568	401	6	)	)	PUNCT
ejpam-6568	401	7	.	.	PUNCT
ejpam-6568	402	1	(	(	PUNCT
ejpam-6568	402	2	6	6	X
ejpam-6568	402	3	)	)	PUNCT
ejpam-6568	402	4	⇒	⇒	NOUN
ejpam-6568	402	5	(	(	PUNCT
ejpam-6568	402	6	1	1	NUM
ejpam-6568	402	7	):	):	PUNCT
ejpam-6568	402	8	let	let	VERB
ejpam-6568	402	9	x	x	PUNCT
ejpam-6568	402	10	∈	∈	PROPN
ejpam-6568	402	11	x	x	X
ejpam-6568	402	12	and	and	CCONJ
ejpam-6568	402	13	v	v	X
ejpam-6568	402	14	be	be	AUX
ejpam-6568	402	15	any	any	DET
ejpam-6568	402	16	be	be	AUX
ejpam-6568	402	17	any	any	DET
ejpam-6568	402	18	σ1σ2	σ1σ2	NOUN
ejpam-6568	402	19	-	-	ADJ
ejpam-6568	402	20	open	open	ADJ
ejpam-6568	402	21	set	set	NOUN
ejpam-6568	402	22	of	of	ADP
ejpam-6568	402	23	y	y	PROPN
ejpam-6568	402	24	containing	contain	VERB
ejpam-6568	402	25	f(x	f(x	PROPN
ejpam-6568	402	26	)	)	PUNCT
ejpam-6568	402	27	.	.	PUNCT
ejpam-6568	403	1	since	since	SCONJ
ejpam-6568	403	2	v	v	NUM
ejpam-6568	403	3	=	=	SYM
ejpam-6568	403	4	σ1σ2	σ1σ2	NUM
ejpam-6568	403	5	-	-	PUNCT
ejpam-6568	403	6	int(v	int(v	NOUN
ejpam-6568	403	7	)	)	PUNCT
ejpam-6568	403	8	⊆	⊆	NUM
ejpam-6568	403	9	σ1σ2	σ1σ2	X
ejpam-6568	403	10	-	-	PUNCT
ejpam-6568	403	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	403	12	-	-	PUNCT
ejpam-6568	403	13	cl(v	cl(v	NOUN
ejpam-6568	403	14	)	)	PUNCT
ejpam-6568	403	15	)	)	PUNCT
ejpam-6568	403	16	,	,	PUNCT
ejpam-6568	403	17	by	by	ADP
ejpam-6568	403	18	(	(	PUNCT
ejpam-6568	403	19	6	6	X
ejpam-6568	403	20	)	)	PUNCT
ejpam-6568	403	21	we	we	PRON
ejpam-6568	403	22	have	have	VERB
ejpam-6568	403	23	x	x	X
ejpam-6568	403	24	∈	∈	PROPN
ejpam-6568	403	25	f−1(v	f−1(v	NOUN
ejpam-6568	403	26	)	)	PUNCT
ejpam-6568	403	27	⊆	⊆	NUM
ejpam-6568	403	28	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	403	29	-	-	PUNCT
ejpam-6568	403	30	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	403	31	-	-	PUNCT
ejpam-6568	403	32	cl(v	cl(v	NOUN
ejpam-6568	403	33	)	)	PUNCT
ejpam-6568	403	34	)	)	PUNCT
ejpam-6568	403	35	)	)	PUNCT
ejpam-6568	404	1	=	=	PUNCT
ejpam-6568	405	1	x	x	X
ejpam-6568	405	2	−	−	PRON
ejpam-6568	405	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	405	4	-	-	PUNCT
ejpam-6568	405	5	cl(y	cl(y	NOUN
ejpam-6568	405	6	−	−	NOUN
ejpam-6568	405	7	σ1σ2	σ1σ2	NOUN
ejpam-6568	405	8	-	-	NUM
ejpam-6568	405	9	cl(v	cl(v	NOUN
ejpam-6568	405	10	)	)	PUNCT
ejpam-6568	405	11	)	)	PUNCT
ejpam-6568	405	12	)	)	PUNCT
ejpam-6568	406	1	⊆	⊆	NUM
ejpam-6568	406	2	x	x	PUNCT
ejpam-6568	406	3	−	−	PROPN
ejpam-6568	406	4	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	406	5	−	−	PROPN
ejpam-6568	406	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	406	7	-	-	NUM
ejpam-6568	406	8	cl(v	cl(v	NOUN
ejpam-6568	406	9	)	)	PUNCT
ejpam-6568	406	10	)	)	PUNCT
ejpam-6568	406	11	)	)	PUNCT
ejpam-6568	407	1	=	=	SYM
ejpam-6568	407	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	407	3	-	-	PUNCT
ejpam-6568	407	4	cl(v	cl(v	NOUN
ejpam-6568	407	5	)	)	PUNCT
ejpam-6568	407	6	)	)	PUNCT
ejpam-6568	407	7	)	)	PUNCT
ejpam-6568	407	8	.	.	PUNCT
ejpam-6568	408	1	there	there	PRON
ejpam-6568	408	2	exists	exist	VERB
ejpam-6568	408	3	a	a	DET
ejpam-6568	408	4	⋆-open	⋆-open	ADJ
ejpam-6568	408	5	set	set	NOUN
ejpam-6568	408	6	u	u	NOUN
ejpam-6568	408	7	of	of	ADP
ejpam-6568	408	8	x	x	PUNCT
ejpam-6568	408	9	containing	contain	VERB
ejpam-6568	408	10	x	x	PUNCT
ejpam-6568	408	11	such	such	ADJ
ejpam-6568	408	12	that	that	SCONJ
ejpam-6568	408	13	u	u	PROPN
ejpam-6568	408	14	⊆	⊆	NUM
ejpam-6568	408	15	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	408	16	-	-	PUNCT
ejpam-6568	408	17	cl(v	cl(v	NOUN
ejpam-6568	408	18	)	)	PUNCT
ejpam-6568	408	19	)	)	PUNCT
ejpam-6568	408	20	;	;	PUNCT
ejpam-6568	408	21	hence	hence	ADV
ejpam-6568	408	22	f(u	f(u	PROPN
ejpam-6568	408	23	)	)	PUNCT
ejpam-6568	409	1	⊆	⊆	NUM
ejpam-6568	409	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	409	3	-	-	NUM
ejpam-6568	409	4	cl(v	cl(v	NOUN
ejpam-6568	409	5	)	)	PUNCT
ejpam-6568	409	6	.	.	PUNCT
ejpam-6568	410	1	thus	thus	ADV
ejpam-6568	410	2	,	,	PUNCT
ejpam-6568	410	3	f	f	PROPN
ejpam-6568	410	4	is	be	AUX
ejpam-6568	410	5	weakly	weakly	ADJ
ejpam-6568	410	6	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	410	7	,	,	PUNCT
ejpam-6568	410	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	410	9	at	at	ADP
ejpam-6568	410	10	x.	x.	NOUN
ejpam-6568	410	11	this	this	PRON
ejpam-6568	410	12	shows	show	VERB
ejpam-6568	410	13	that	that	SCONJ
ejpam-6568	410	14	f	f	PROPN
ejpam-6568	410	15	is	be	AUX
ejpam-6568	410	16	weakly	weakly	ADJ
ejpam-6568	410	17	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	410	18	,	,	PUNCT
ejpam-6568	410	19	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	410	20	.	.	X
ejpam-6568	410	21	theorem	theorem	VERB
ejpam-6568	410	22	14	14	NUM
ejpam-6568	410	23	.	.	PUNCT
ejpam-6568	411	1	for	for	ADP
ejpam-6568	411	2	a	a	DET
ejpam-6568	411	3	function	function	NOUN
ejpam-6568	411	4	(	(	PUNCT
ejpam-6568	411	5	x	x	X
ejpam-6568	411	6	,	,	PUNCT
ejpam-6568	411	7	τ	τ	PROPN
ejpam-6568	411	8	,	,	PUNCT
ejpam-6568	411	9	i	i	NOUN
ejpam-6568	411	10	)	)	PUNCT
ejpam-6568	411	11	→	→	PUNCT
ejpam-6568	411	12	(	(	PUNCT
ejpam-6568	411	13	y	y	PROPN
ejpam-6568	411	14	,	,	PUNCT
ejpam-6568	411	15	σ1	σ1	PROPN
ejpam-6568	411	16	,	,	PUNCT
ejpam-6568	411	17	σ2	σ2	NOUN
ejpam-6568	411	18	)	)	PUNCT
ejpam-6568	411	19	,	,	PUNCT
ejpam-6568	411	20	the	the	DET
ejpam-6568	411	21	following	follow	VERB
ejpam-6568	411	22	properties	property	NOUN
ejpam-6568	411	23	are	be	AUX
ejpam-6568	411	24	equivalent	equivalent	ADJ
ejpam-6568	411	25	:	:	PUNCT
ejpam-6568	411	26	(	(	PUNCT
ejpam-6568	411	27	1	1	X
ejpam-6568	411	28	)	)	PUNCT
ejpam-6568	411	29	f	f	PROPN
ejpam-6568	411	30	is	be	AUX
ejpam-6568	411	31	weakly	weakly	ADJ
ejpam-6568	411	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	411	33	,	,	PUNCT
ejpam-6568	411	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	411	35	;	;	PUNCT
ejpam-6568	411	36	(	(	PUNCT
ejpam-6568	411	37	2	2	X
ejpam-6568	411	38	)	)	PUNCT
ejpam-6568	411	39	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	411	40	-	-	PUNCT
ejpam-6568	411	41	int(k	int(k	PROPN
ejpam-6568	411	42	)	)	PUNCT
ejpam-6568	411	43	)	)	PUNCT
ejpam-6568	411	44	)	)	PUNCT
ejpam-6568	412	1	⊆	⊆	NUM
ejpam-6568	412	2	f−1(k	f−1(k	PROPN
ejpam-6568	412	3	)	)	PUNCT
ejpam-6568	412	4	for	for	ADP
ejpam-6568	412	5	every	every	DET
ejpam-6568	412	6	(	(	PUNCT
ejpam-6568	412	7	σ1	σ1	PROPN
ejpam-6568	412	8	,	,	PUNCT
ejpam-6568	412	9	σ2)r	σ2)r	NOUN
ejpam-6568	412	10	-	-	PUNCT
ejpam-6568	412	11	closed	close	VERB
ejpam-6568	412	12	set	set	ADJ
ejpam-6568	412	13	k	k	PROPN
ejpam-6568	412	14	of	of	ADP
ejpam-6568	412	15	y	y	PROPN
ejpam-6568	412	16	;	;	PUNCT
ejpam-6568	412	17	(	(	PUNCT
ejpam-6568	412	18	3	3	X
ejpam-6568	412	19	)	)	PUNCT
ejpam-6568	412	20	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	412	21	-	-	PUNCT
ejpam-6568	412	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	412	23	-	-	PUNCT
ejpam-6568	412	24	cl(v	cl(v	NOUN
ejpam-6568	412	25	)	)	PUNCT
ejpam-6568	412	26	)	)	PUNCT
ejpam-6568	412	27	)	)	PUNCT
ejpam-6568	412	28	)	)	PUNCT
ejpam-6568	412	29	⊆	⊆	NUM
ejpam-6568	412	30	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	412	31	-	-	PUNCT
ejpam-6568	412	32	cl(v	cl(v	NOUN
ejpam-6568	412	33	)	)	PUNCT
ejpam-6568	412	34	)	)	PUNCT
ejpam-6568	412	35	for	for	ADP
ejpam-6568	412	36	every	every	DET
ejpam-6568	412	37	(	(	PUNCT
ejpam-6568	412	38	σ1	σ1	PROPN
ejpam-6568	412	39	,	,	PUNCT
ejpam-6568	412	40	σ2)β	σ2)β	NOUN
ejpam-6568	412	41	-	-	PUNCT
ejpam-6568	412	42	open	open	NOUN
ejpam-6568	412	43	set	set	NOUN
ejpam-6568	412	44	v	v	NOUN
ejpam-6568	412	45	of	of	ADP
ejpam-6568	412	46	y	y	PROPN
ejpam-6568	412	47	;	;	PUNCT
ejpam-6568	412	48	(	(	PUNCT
ejpam-6568	412	49	4	4	X
ejpam-6568	412	50	)	)	PUNCT
ejpam-6568	412	51	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	412	52	-	-	PUNCT
ejpam-6568	412	53	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	412	54	-	-	PUNCT
ejpam-6568	412	55	cl(v	cl(v	NOUN
ejpam-6568	412	56	)	)	PUNCT
ejpam-6568	412	57	)	)	PUNCT
ejpam-6568	412	58	)	)	PUNCT
ejpam-6568	412	59	)	)	PUNCT
ejpam-6568	412	60	⊆	⊆	NUM
ejpam-6568	412	61	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	412	62	-	-	PUNCT
ejpam-6568	412	63	cl(v	cl(v	NOUN
ejpam-6568	412	64	)	)	PUNCT
ejpam-6568	412	65	)	)	PUNCT
ejpam-6568	412	66	for	for	ADP
ejpam-6568	412	67	every	every	DET
ejpam-6568	412	68	(	(	PUNCT
ejpam-6568	412	69	σ1	σ1	PROPN
ejpam-6568	412	70	,	,	PUNCT
ejpam-6568	412	71	σ2)s	σ2)s	NOUN
ejpam-6568	412	72	-	-	PUNCT
ejpam-6568	412	73	open	open	NOUN
ejpam-6568	412	74	set	set	NOUN
ejpam-6568	412	75	v	v	NOUN
ejpam-6568	412	76	of	of	ADP
ejpam-6568	412	77	y	y	PROPN
ejpam-6568	412	78	.	.	PUNCT
ejpam-6568	413	1	proof	proof	NOUN
ejpam-6568	413	2	.	.	PUNCT
ejpam-6568	414	1	(	(	PUNCT
ejpam-6568	414	2	1	1	X
ejpam-6568	414	3	)	)	PUNCT
ejpam-6568	414	4	⇒	⇒	NOUN
ejpam-6568	414	5	(	(	PUNCT
ejpam-6568	414	6	2	2	NUM
ejpam-6568	414	7	):	):	PUNCT
ejpam-6568	414	8	let	let	VERB
ejpam-6568	414	9	k	k	PRON
ejpam-6568	414	10	be	be	AUX
ejpam-6568	414	11	any	any	DET
ejpam-6568	414	12	(	(	PUNCT
ejpam-6568	414	13	σ1	σ1	NOUN
ejpam-6568	414	14	,	,	PUNCT
ejpam-6568	414	15	σ2)r	σ2)r	NOUN
ejpam-6568	414	16	-	-	PUNCT
ejpam-6568	414	17	closed	close	VERB
ejpam-6568	414	18	set	set	NOUN
ejpam-6568	414	19	of	of	ADP
ejpam-6568	414	20	y	y	PROPN
ejpam-6568	414	21	.	.	PUNCT
ejpam-6568	415	1	then	then	ADV
ejpam-6568	415	2	,	,	PUNCT
ejpam-6568	415	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	415	4	-	-	PUNCT
ejpam-6568	415	5	int(k	int(k	NUM
ejpam-6568	415	6	)	)	PUNCT
ejpam-6568	415	7	is	be	AUX
ejpam-6568	415	8	σ1σ2	σ1σ2	NOUN
ejpam-6568	415	9	-	-	ADJ
ejpam-6568	415	10	open	open	ADJ
ejpam-6568	415	11	in	in	ADP
ejpam-6568	415	12	y	y	PROPN
ejpam-6568	415	13	,	,	PUNCT
ejpam-6568	415	14	by	by	ADP
ejpam-6568	415	15	theorem	theorem	NOUN
ejpam-6568	415	16	13	13	NUM
ejpam-6568	415	17	(	(	PUNCT
ejpam-6568	415	18	6	6	NUM
ejpam-6568	415	19	)	)	PUNCT
ejpam-6568	415	20	we	we	PRON
ejpam-6568	415	21	have	have	VERB
ejpam-6568	415	22	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	415	23	-	-	PUNCT
ejpam-6568	415	24	int(k	int(k	PROPN
ejpam-6568	415	25	)	)	PUNCT
ejpam-6568	415	26	)	)	PUNCT
ejpam-6568	415	27	)	)	PUNCT
ejpam-6568	416	1	⊆	⊆	NUM
ejpam-6568	416	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	416	3	-	-	PUNCT
ejpam-6568	416	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	416	5	-	-	PUNCT
ejpam-6568	416	6	int(k	int(k	NOUN
ejpam-6568	416	7	)	)	PUNCT
ejpam-6568	416	8	)	)	PUNCT
ejpam-6568	416	9	)	)	PUNCT
ejpam-6568	417	1	=	=	PUNCT
ejpam-6568	417	2	f−1(k	f−1(k	PROPN
ejpam-6568	417	3	)	)	PUNCT
ejpam-6568	417	4	.	.	PUNCT
ejpam-6568	418	1	(	(	PUNCT
ejpam-6568	418	2	2	2	X
ejpam-6568	418	3	)	)	PUNCT
ejpam-6568	418	4	⇒	⇒	NOUN
ejpam-6568	418	5	(	(	PUNCT
ejpam-6568	418	6	3	3	NUM
ejpam-6568	418	7	):	):	PUNCT
ejpam-6568	418	8	let	let	VERB
ejpam-6568	418	9	v	v	PART
ejpam-6568	418	10	be	be	AUX
ejpam-6568	418	11	any	any	DET
ejpam-6568	418	12	(	(	PUNCT
ejpam-6568	418	13	σ1	σ1	PROPN
ejpam-6568	418	14	,	,	PUNCT
ejpam-6568	418	15	σ2)β	σ2)β	NOUN
ejpam-6568	418	16	-	-	PUNCT
ejpam-6568	418	17	open	open	ADJ
ejpam-6568	418	18	set	set	NOUN
ejpam-6568	418	19	of	of	ADP
ejpam-6568	418	20	y	y	PROPN
ejpam-6568	418	21	.	.	PUNCT
ejpam-6568	419	1	then	then	ADV
ejpam-6568	419	2	,	,	PUNCT
ejpam-6568	419	3	we	we	PRON
ejpam-6568	419	4	have	have	VERB
ejpam-6568	419	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	419	6	-	-	NUM
ejpam-6568	419	7	cl(v	cl(v	NOUN
ejpam-6568	419	8	)	)	PUNCT
ejpam-6568	420	1	⊆	⊆	NUM
ejpam-6568	420	2	σ1σ2	σ1σ2	X
ejpam-6568	420	3	-	-	PUNCT
ejpam-6568	420	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	420	5	-	-	PUNCT
ejpam-6568	420	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	420	7	-	-	PUNCT
ejpam-6568	420	8	cl(v	cl(v	NOUN
ejpam-6568	420	9	)	)	PUNCT
ejpam-6568	420	10	)	)	PUNCT
ejpam-6568	420	11	)	)	PUNCT
ejpam-6568	421	1	⊆	⊆	X
ejpam-6568	421	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	421	3	-	-	PUNCT
ejpam-6568	421	4	cl(v	cl(v	X
ejpam-6568	421	5	)	)	PUNCT
ejpam-6568	421	6	n.	n.	PROPN
ejpam-6568	421	7	viriyapong	viriyapong	PROPN
ejpam-6568	421	8	,	,	PUNCT
ejpam-6568	421	9	a.	a.	PROPN
ejpam-6568	421	10	sama	sama	PROPN
ejpam-6568	421	11	-	-	PUNCT
ejpam-6568	421	12	ae	ae	PROPN
ejpam-6568	421	13	,	,	PUNCT
ejpam-6568	421	14	c.	c.	PROPN
ejpam-6568	421	15	boonpok	boonpok	PROPN
ejpam-6568	421	16	/	/	SYM
ejpam-6568	421	17	eur	eur	PROPN
ejpam-6568	421	18	.	.	PUNCT
ejpam-6568	422	1	j.	j.	PROPN
ejpam-6568	422	2	pure	pure	PROPN
ejpam-6568	422	3	appl	appl	PROPN
ejpam-6568	422	4	.	.	PROPN
ejpam-6568	422	5	math	math	PROPN
ejpam-6568	422	6	,	,	PUNCT
ejpam-6568	422	7	18	18	NUM
ejpam-6568	422	8	(	(	PUNCT
ejpam-6568	422	9	3	3	NUM
ejpam-6568	422	10	)	)	PUNCT
ejpam-6568	422	11	(	(	PUNCT
ejpam-6568	422	12	2025	2025	NUM
ejpam-6568	422	13	)	)	PUNCT
ejpam-6568	422	14	,	,	PUNCT
ejpam-6568	422	15	6568	6568	NUM
ejpam-6568	422	16	14	14	NUM
ejpam-6568	422	17	of	of	ADP
ejpam-6568	422	18	18	18	NUM
ejpam-6568	422	19	and	and	CCONJ
ejpam-6568	422	20	hence	hence	ADV
ejpam-6568	422	21	σ1σ2	σ1σ2	NOUN
ejpam-6568	422	22	-	-	NUM
ejpam-6568	422	23	cl(v	cl(v	NOUN
ejpam-6568	422	24	)	)	PUNCT
ejpam-6568	422	25	is	be	AUX
ejpam-6568	422	26	(	(	PUNCT
ejpam-6568	422	27	σ1	σ1	NOUN
ejpam-6568	422	28	,	,	PUNCT
ejpam-6568	422	29	σ2)r	σ2)r	NOUN
ejpam-6568	422	30	-	-	PUNCT
ejpam-6568	422	31	closed	closed	ADJ
ejpam-6568	422	32	.	.	PUNCT
ejpam-6568	423	1	by	by	ADP
ejpam-6568	423	2	(	(	PUNCT
ejpam-6568	423	3	2	2	NUM
ejpam-6568	423	4	)	)	PUNCT
ejpam-6568	423	5	,	,	PUNCT
ejpam-6568	423	6	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	423	7	-	-	PUNCT
ejpam-6568	423	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	423	9	-	-	PUNCT
ejpam-6568	423	10	cl(v	cl(v	NOUN
ejpam-6568	423	11	)	)	PUNCT
ejpam-6568	423	12	)	)	PUNCT
ejpam-6568	423	13	)	)	PUNCT
ejpam-6568	423	14	)	)	PUNCT
ejpam-6568	423	15	⊆	⊆	NUM
ejpam-6568	423	16	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	423	17	-	-	PUNCT
ejpam-6568	423	18	cl(v	cl(v	NOUN
ejpam-6568	423	19	)	)	PUNCT
ejpam-6568	423	20	)	)	PUNCT
ejpam-6568	423	21	.	.	PUNCT
ejpam-6568	424	1	(	(	PUNCT
ejpam-6568	424	2	3	3	X
ejpam-6568	424	3	)	)	PUNCT
ejpam-6568	424	4	⇒	⇒	NOUN
ejpam-6568	424	5	(	(	PUNCT
ejpam-6568	424	6	4	4	NUM
ejpam-6568	424	7	):	):	PUNCT
ejpam-6568	424	8	this	this	PRON
ejpam-6568	424	9	is	be	AUX
ejpam-6568	424	10	obvious	obvious	ADJ
ejpam-6568	424	11	.	.	PUNCT
ejpam-6568	425	1	(	(	PUNCT
ejpam-6568	425	2	4	4	X
ejpam-6568	425	3	)	)	PUNCT
ejpam-6568	425	4	⇒	⇒	NOUN
ejpam-6568	425	5	(	(	PUNCT
ejpam-6568	425	6	1	1	NUM
ejpam-6568	425	7	):	):	PUNCT
ejpam-6568	425	8	let	let	VERB
ejpam-6568	425	9	v	v	PART
ejpam-6568	425	10	be	be	AUX
ejpam-6568	425	11	any	any	DET
ejpam-6568	425	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	425	13	-	-	ADJ
ejpam-6568	425	14	open	open	ADJ
ejpam-6568	425	15	set	set	NOUN
ejpam-6568	425	16	of	of	ADP
ejpam-6568	425	17	y	y	PROPN
ejpam-6568	425	18	.	.	PUNCT
ejpam-6568	426	1	then	then	ADV
ejpam-6568	426	2	,	,	PUNCT
ejpam-6568	426	3	we	we	PRON
ejpam-6568	426	4	have	have	VERB
ejpam-6568	426	5	v	v	NOUN
ejpam-6568	426	6	is	be	AUX
ejpam-6568	426	7	(	(	PUNCT
ejpam-6568	426	8	σ1	σ1	PROPN
ejpam-6568	426	9	,	,	PUNCT
ejpam-6568	426	10	σ2)s	σ2)s	NOUN
ejpam-6568	426	11	-	-	PUNCT
ejpam-6568	426	12	open	open	ADJ
ejpam-6568	426	13	in	in	ADP
ejpam-6568	426	14	y	y	PROPN
ejpam-6568	426	15	.	.	PUNCT
ejpam-6568	427	1	by	by	ADP
ejpam-6568	427	2	(	(	PUNCT
ejpam-6568	427	3	4	4	NUM
ejpam-6568	427	4	)	)	PUNCT
ejpam-6568	427	5	,	,	PUNCT
ejpam-6568	427	6	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	427	7	)	)	PUNCT
ejpam-6568	427	8	)	)	PUNCT
ejpam-6568	428	1	⊆	⊆	X
ejpam-6568	428	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	428	3	-	-	PUNCT
ejpam-6568	428	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	428	5	-	-	PUNCT
ejpam-6568	428	6	cl(v	cl(v	NOUN
ejpam-6568	428	7	)	)	PUNCT
ejpam-6568	428	8	)	)	PUNCT
ejpam-6568	428	9	)	)	PUNCT
ejpam-6568	428	10	)	)	PUNCT
ejpam-6568	428	11	⊆	⊆	NUM
ejpam-6568	428	12	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	428	13	-	-	PUNCT
ejpam-6568	428	14	cl(v	cl(v	NOUN
ejpam-6568	428	15	)	)	PUNCT
ejpam-6568	428	16	)	)	PUNCT
ejpam-6568	428	17	and	and	CCONJ
ejpam-6568	428	18	by	by	ADP
ejpam-6568	428	19	theorem	theorem	NOUN
ejpam-6568	428	20	13	13	NUM
ejpam-6568	428	21	(	(	PUNCT
ejpam-6568	428	22	6	6	NUM
ejpam-6568	428	23	)	)	PUNCT
ejpam-6568	428	24	,	,	PUNCT
ejpam-6568	428	25	f	f	PROPN
ejpam-6568	428	26	is	be	AUX
ejpam-6568	428	27	weakly	weakly	ADJ
ejpam-6568	428	28	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	428	29	,	,	PUNCT
ejpam-6568	428	30	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	428	31	.	.	X
ejpam-6568	428	32	theorem	theorem	VERB
ejpam-6568	428	33	15	15	NUM
ejpam-6568	428	34	.	.	PUNCT
ejpam-6568	429	1	for	for	ADP
ejpam-6568	429	2	a	a	DET
ejpam-6568	429	3	function	function	NOUN
ejpam-6568	429	4	(	(	PUNCT
ejpam-6568	429	5	x	x	X
ejpam-6568	429	6	,	,	PUNCT
ejpam-6568	429	7	τ	τ	PROPN
ejpam-6568	429	8	,	,	PUNCT
ejpam-6568	429	9	i	i	NOUN
ejpam-6568	429	10	)	)	PUNCT
ejpam-6568	429	11	→	→	PUNCT
ejpam-6568	429	12	(	(	PUNCT
ejpam-6568	429	13	y	y	PROPN
ejpam-6568	429	14	,	,	PUNCT
ejpam-6568	429	15	σ1	σ1	PROPN
ejpam-6568	429	16	,	,	PUNCT
ejpam-6568	429	17	σ2	σ2	NOUN
ejpam-6568	429	18	)	)	PUNCT
ejpam-6568	429	19	,	,	PUNCT
ejpam-6568	429	20	the	the	DET
ejpam-6568	429	21	following	follow	VERB
ejpam-6568	429	22	properties	property	NOUN
ejpam-6568	429	23	are	be	AUX
ejpam-6568	429	24	equivalent	equivalent	ADJ
ejpam-6568	429	25	:	:	PUNCT
ejpam-6568	429	26	(	(	PUNCT
ejpam-6568	429	27	1	1	X
ejpam-6568	429	28	)	)	PUNCT
ejpam-6568	429	29	f	f	PROPN
ejpam-6568	429	30	is	be	AUX
ejpam-6568	429	31	weakly	weakly	ADJ
ejpam-6568	429	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	429	33	,	,	PUNCT
ejpam-6568	429	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	429	35	;	;	PUNCT
ejpam-6568	429	36	(	(	PUNCT
ejpam-6568	429	37	2	2	X
ejpam-6568	429	38	)	)	PUNCT
ejpam-6568	429	39	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	429	40	-	-	PUNCT
ejpam-6568	429	41	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	429	42	-	-	PUNCT
ejpam-6568	429	43	cl(v	cl(v	NOUN
ejpam-6568	429	44	)	)	PUNCT
ejpam-6568	429	45	)	)	PUNCT
ejpam-6568	429	46	)	)	PUNCT
ejpam-6568	429	47	)	)	PUNCT
ejpam-6568	430	1	⊆	⊆	NUM
ejpam-6568	430	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	430	3	-	-	PUNCT
ejpam-6568	430	4	cl(v	cl(v	NOUN
ejpam-6568	430	5	)	)	PUNCT
ejpam-6568	430	6	)	)	PUNCT
ejpam-6568	430	7	for	for	ADP
ejpam-6568	430	8	every	every	DET
ejpam-6568	430	9	(	(	PUNCT
ejpam-6568	430	10	σ1	σ1	PROPN
ejpam-6568	430	11	,	,	PUNCT
ejpam-6568	430	12	σ2)p	σ2)p	NOUN
ejpam-6568	430	13	-	-	PUNCT
ejpam-6568	430	14	open	open	NOUN
ejpam-6568	430	15	set	set	NOUN
ejpam-6568	430	16	v	v	NOUN
ejpam-6568	430	17	of	of	ADP
ejpam-6568	430	18	y	y	PROPN
ejpam-6568	430	19	;	;	PUNCT
ejpam-6568	430	20	(	(	PUNCT
ejpam-6568	430	21	3	3	X
ejpam-6568	430	22	)	)	PUNCT
ejpam-6568	430	23	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	430	24	)	)	PUNCT
ejpam-6568	430	25	)	)	PUNCT
ejpam-6568	431	1	⊆	⊆	NUM
ejpam-6568	431	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	431	3	-	-	PUNCT
ejpam-6568	431	4	cl(v	cl(v	NOUN
ejpam-6568	431	5	)	)	PUNCT
ejpam-6568	431	6	)	)	PUNCT
ejpam-6568	431	7	for	for	ADP
ejpam-6568	431	8	every	every	DET
ejpam-6568	431	9	(	(	PUNCT
ejpam-6568	431	10	σ1	σ1	PROPN
ejpam-6568	431	11	,	,	PUNCT
ejpam-6568	431	12	σ2)p	σ2)p	NOUN
ejpam-6568	431	13	-	-	PUNCT
ejpam-6568	431	14	open	open	NOUN
ejpam-6568	431	15	set	set	NOUN
ejpam-6568	431	16	v	v	NOUN
ejpam-6568	431	17	of	of	ADP
ejpam-6568	431	18	y	y	PROPN
ejpam-6568	431	19	;	;	PUNCT
ejpam-6568	431	20	(	(	PUNCT
ejpam-6568	431	21	4	4	X
ejpam-6568	431	22	)	)	PUNCT
ejpam-6568	431	23	f−1(v	f−1(v	NOUN
ejpam-6568	431	24	)	)	PUNCT
ejpam-6568	432	1	⊆	⊆	X
ejpam-6568	432	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	432	3	-	-	PUNCT
ejpam-6568	432	4	cl(v	cl(v	NOUN
ejpam-6568	432	5	)	)	PUNCT
ejpam-6568	432	6	)	)	PUNCT
ejpam-6568	432	7	)	)	PUNCT
ejpam-6568	432	8	for	for	ADP
ejpam-6568	432	9	every	every	DET
ejpam-6568	432	10	(	(	PUNCT
ejpam-6568	432	11	σ1	σ1	PROPN
ejpam-6568	432	12	,	,	PUNCT
ejpam-6568	432	13	σ2)p	σ2)p	NOUN
ejpam-6568	432	14	-	-	PUNCT
ejpam-6568	432	15	open	open	NOUN
ejpam-6568	432	16	set	set	NOUN
ejpam-6568	432	17	v	v	NOUN
ejpam-6568	432	18	of	of	ADP
ejpam-6568	432	19	y	y	PROPN
ejpam-6568	432	20	.	.	PUNCT
ejpam-6568	433	1	proof	proof	NOUN
ejpam-6568	433	2	.	.	PUNCT
ejpam-6568	434	1	(	(	PUNCT
ejpam-6568	434	2	1	1	X
ejpam-6568	434	3	)	)	PUNCT
ejpam-6568	434	4	⇒	⇒	NOUN
ejpam-6568	434	5	(	(	PUNCT
ejpam-6568	434	6	2	2	NUM
ejpam-6568	434	7	):	):	PUNCT
ejpam-6568	434	8	let	let	VERB
ejpam-6568	434	9	v	v	PART
ejpam-6568	434	10	be	be	AUX
ejpam-6568	434	11	any	any	DET
ejpam-6568	434	12	(	(	PUNCT
ejpam-6568	434	13	σ1	σ1	PROPN
ejpam-6568	434	14	,	,	PUNCT
ejpam-6568	434	15	σ2)p	σ2)p	NOUN
ejpam-6568	434	16	-	-	PUNCT
ejpam-6568	434	17	open	open	ADJ
ejpam-6568	434	18	set	set	NOUN
ejpam-6568	434	19	of	of	ADP
ejpam-6568	434	20	y	y	PROPN
ejpam-6568	434	21	.	.	PUNCT
ejpam-6568	435	1	then	then	ADV
ejpam-6568	435	2	,	,	PUNCT
ejpam-6568	435	3	we	we	PRON
ejpam-6568	435	4	have	have	VERB
ejpam-6568	435	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	435	6	-	-	NUM
ejpam-6568	435	7	cl(v	cl(v	NOUN
ejpam-6568	435	8	)	)	PUNCT
ejpam-6568	436	1	⊆	⊆	NUM
ejpam-6568	436	2	σ1σ2	σ1σ2	X
ejpam-6568	436	3	-	-	PUNCT
ejpam-6568	436	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	436	5	-	-	PUNCT
ejpam-6568	436	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	436	7	-	-	PUNCT
ejpam-6568	436	8	cl(v	cl(v	NOUN
ejpam-6568	436	9	)	)	PUNCT
ejpam-6568	436	10	)	)	PUNCT
ejpam-6568	436	11	)	)	PUNCT
ejpam-6568	436	12	and	and	CCONJ
ejpam-6568	436	13	hence	hence	ADV
ejpam-6568	436	14	σ1σ2	σ1σ2	NOUN
ejpam-6568	436	15	-	-	NOUN
ejpam-6568	436	16	cl(v	cl(v	NOUN
ejpam-6568	436	17	)	)	PUNCT
ejpam-6568	436	18	is	be	AUX
ejpam-6568	436	19	(	(	PUNCT
ejpam-6568	436	20	σ1	σ1	NOUN
ejpam-6568	436	21	,	,	PUNCT
ejpam-6568	436	22	σ2)r	σ2)r	NOUN
ejpam-6568	436	23	-	-	PUNCT
ejpam-6568	436	24	closed	closed	ADJ
ejpam-6568	436	25	in	in	ADP
ejpam-6568	436	26	y	y	PROPN
ejpam-6568	436	27	.	.	PUNCT
ejpam-6568	437	1	thus	thus	ADV
ejpam-6568	437	2	,	,	PUNCT
ejpam-6568	437	3	by	by	ADP
ejpam-6568	437	4	theorem	theorem	NOUN
ejpam-6568	437	5	14	14	NUM
ejpam-6568	437	6	(	(	PUNCT
ejpam-6568	437	7	2	2	NUM
ejpam-6568	437	8	)	)	PUNCT
ejpam-6568	437	9	we	we	PRON
ejpam-6568	437	10	have	have	VERB
ejpam-6568	437	11	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	437	12	-	-	PUNCT
ejpam-6568	437	13	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	437	14	-	-	PUNCT
ejpam-6568	437	15	cl(v	cl(v	NOUN
ejpam-6568	437	16	)	)	PUNCT
ejpam-6568	437	17	)	)	PUNCT
ejpam-6568	437	18	)	)	PUNCT
ejpam-6568	437	19	)	)	PUNCT
ejpam-6568	438	1	⊆	⊆	NUM
ejpam-6568	438	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	438	3	-	-	PUNCT
ejpam-6568	438	4	cl(v	cl(v	NOUN
ejpam-6568	438	5	)	)	PUNCT
ejpam-6568	438	6	)	)	PUNCT
ejpam-6568	438	7	.	.	PUNCT
ejpam-6568	439	1	(	(	PUNCT
ejpam-6568	439	2	2	2	X
ejpam-6568	439	3	)	)	PUNCT
ejpam-6568	439	4	⇒	⇒	NOUN
ejpam-6568	439	5	(	(	PUNCT
ejpam-6568	439	6	3	3	NUM
ejpam-6568	439	7	):	):	PUNCT
ejpam-6568	439	8	the	the	DET
ejpam-6568	439	9	proof	proof	NOUN
ejpam-6568	439	10	is	be	AUX
ejpam-6568	439	11	obvious	obvious	ADJ
ejpam-6568	439	12	.	.	PUNCT
ejpam-6568	440	1	(	(	PUNCT
ejpam-6568	440	2	3	3	X
ejpam-6568	440	3	)	)	PUNCT
ejpam-6568	440	4	⇒	⇒	NOUN
ejpam-6568	440	5	(	(	PUNCT
ejpam-6568	440	6	4	4	NUM
ejpam-6568	440	7	):	):	PUNCT
ejpam-6568	440	8	let	let	VERB
ejpam-6568	440	9	v	v	PART
ejpam-6568	440	10	be	be	AUX
ejpam-6568	440	11	any	any	DET
ejpam-6568	440	12	(	(	PUNCT
ejpam-6568	440	13	σ1	σ1	PROPN
ejpam-6568	440	14	,	,	PUNCT
ejpam-6568	440	15	σ2)p	σ2)p	NOUN
ejpam-6568	440	16	-	-	PUNCT
ejpam-6568	440	17	open	open	ADJ
ejpam-6568	440	18	set	set	NOUN
ejpam-6568	440	19	of	of	ADP
ejpam-6568	440	20	y	y	PROPN
ejpam-6568	440	21	.	.	PUNCT
ejpam-6568	441	1	by	by	ADP
ejpam-6568	441	2	(	(	PUNCT
ejpam-6568	441	3	3	3	NUM
ejpam-6568	441	4	)	)	PUNCT
ejpam-6568	441	5	,	,	PUNCT
ejpam-6568	441	6	f−1(v	f−1(v	PROPN
ejpam-6568	441	7	)	)	PUNCT
ejpam-6568	441	8	⊆	⊆	NUM
ejpam-6568	441	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	441	10	-	-	PUNCT
ejpam-6568	441	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	441	12	-	-	PUNCT
ejpam-6568	441	13	cl(v	cl(v	NOUN
ejpam-6568	441	14	)	)	PUNCT
ejpam-6568	441	15	)	)	PUNCT
ejpam-6568	441	16	)	)	PUNCT
ejpam-6568	442	1	=	=	PUNCT
ejpam-6568	443	1	x	x	X
ejpam-6568	443	2	−	−	PRON
ejpam-6568	443	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	443	4	-	-	PUNCT
ejpam-6568	443	5	cl(y	cl(y	NOUN
ejpam-6568	443	6	−	−	NOUN
ejpam-6568	443	7	σ1σ2	σ1σ2	NOUN
ejpam-6568	443	8	-	-	NUM
ejpam-6568	443	9	cl(v	cl(v	NOUN
ejpam-6568	443	10	)	)	PUNCT
ejpam-6568	443	11	)	)	PUNCT
ejpam-6568	443	12	)	)	PUNCT
ejpam-6568	444	1	⊆	⊆	NUM
ejpam-6568	444	2	x	x	PUNCT
ejpam-6568	444	3	−	−	PROPN
ejpam-6568	444	4	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	444	5	−	−	PROPN
ejpam-6568	444	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	444	7	-	-	NUM
ejpam-6568	444	8	cl(v	cl(v	NOUN
ejpam-6568	444	9	)	)	PUNCT
ejpam-6568	444	10	)	)	PUNCT
ejpam-6568	444	11	)	)	PUNCT
ejpam-6568	445	1	=	=	SYM
ejpam-6568	445	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	445	3	-	-	PUNCT
ejpam-6568	445	4	cl(v	cl(v	NOUN
ejpam-6568	445	5	)	)	PUNCT
ejpam-6568	445	6	)	)	PUNCT
ejpam-6568	445	7	)	)	PUNCT
ejpam-6568	445	8	.	.	PUNCT
ejpam-6568	446	1	(	(	PUNCT
ejpam-6568	446	2	4	4	X
ejpam-6568	446	3	)	)	PUNCT
ejpam-6568	446	4	⇒	⇒	NOUN
ejpam-6568	446	5	(	(	PUNCT
ejpam-6568	446	6	1	1	NUM
ejpam-6568	446	7	):	):	PUNCT
ejpam-6568	446	8	let	let	VERB
ejpam-6568	446	9	v	v	PART
ejpam-6568	446	10	be	be	AUX
ejpam-6568	446	11	any	any	DET
ejpam-6568	446	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	446	13	-	-	ADJ
ejpam-6568	446	14	open	open	ADJ
ejpam-6568	446	15	set	set	NOUN
ejpam-6568	446	16	of	of	ADP
ejpam-6568	446	17	y	y	PROPN
ejpam-6568	446	18	.	.	PUNCT
ejpam-6568	447	1	then	then	ADV
ejpam-6568	447	2	,	,	PUNCT
ejpam-6568	447	3	v	v	NOUN
ejpam-6568	447	4	is	be	AUX
ejpam-6568	447	5	(	(	PUNCT
ejpam-6568	447	6	σ1	σ1	PROPN
ejpam-6568	447	7	,	,	PUNCT
ejpam-6568	447	8	σ2)p	σ2)p	NOUN
ejpam-6568	447	9	-	-	PUNCT
ejpam-6568	447	10	open	open	ADJ
ejpam-6568	447	11	in	in	ADP
ejpam-6568	447	12	y	y	PROPN
ejpam-6568	447	13	.	.	PUNCT
ejpam-6568	448	1	thus	thus	ADV
ejpam-6568	448	2	by	by	ADP
ejpam-6568	448	3	(	(	PUNCT
ejpam-6568	448	4	4	4	NUM
ejpam-6568	448	5	)	)	PUNCT
ejpam-6568	448	6	and	and	CCONJ
ejpam-6568	448	7	theorem	theorem	VERB
ejpam-6568	448	8	13	13	NUM
ejpam-6568	448	9	(	(	PUNCT
ejpam-6568	448	10	2	2	NUM
ejpam-6568	448	11	)	)	PUNCT
ejpam-6568	448	12	,	,	PUNCT
ejpam-6568	448	13	f	f	PROPN
ejpam-6568	448	14	is	be	AUX
ejpam-6568	448	15	weakly	weakly	ADJ
ejpam-6568	448	16	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	448	17	,	,	PUNCT
ejpam-6568	448	18	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	448	19	.	.	X
ejpam-6568	448	20	theorem	theorem	VERB
ejpam-6568	448	21	16	16	NUM
ejpam-6568	448	22	.	.	PUNCT
ejpam-6568	449	1	for	for	ADP
ejpam-6568	449	2	a	a	DET
ejpam-6568	449	3	function	function	NOUN
ejpam-6568	449	4	(	(	PUNCT
ejpam-6568	449	5	x	x	X
ejpam-6568	449	6	,	,	PUNCT
ejpam-6568	449	7	τ	τ	PROPN
ejpam-6568	449	8	,	,	PUNCT
ejpam-6568	449	9	i	i	NOUN
ejpam-6568	449	10	)	)	PUNCT
ejpam-6568	449	11	→	→	PUNCT
ejpam-6568	449	12	(	(	PUNCT
ejpam-6568	449	13	y	y	PROPN
ejpam-6568	449	14	,	,	PUNCT
ejpam-6568	449	15	σ1	σ1	PROPN
ejpam-6568	449	16	,	,	PUNCT
ejpam-6568	449	17	σ2	σ2	NOUN
ejpam-6568	449	18	)	)	PUNCT
ejpam-6568	449	19	,	,	PUNCT
ejpam-6568	449	20	the	the	DET
ejpam-6568	449	21	following	follow	VERB
ejpam-6568	449	22	properties	property	NOUN
ejpam-6568	449	23	are	be	AUX
ejpam-6568	449	24	equivalent	equivalent	ADJ
ejpam-6568	449	25	:	:	PUNCT
ejpam-6568	449	26	(	(	PUNCT
ejpam-6568	449	27	1	1	X
ejpam-6568	449	28	)	)	PUNCT
ejpam-6568	449	29	f	f	PROPN
ejpam-6568	449	30	is	be	AUX
ejpam-6568	449	31	weakly	weakly	ADJ
ejpam-6568	449	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	449	33	,	,	PUNCT
ejpam-6568	449	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	449	35	;	;	PUNCT
ejpam-6568	449	36	(	(	PUNCT
ejpam-6568	449	37	2	2	X
ejpam-6568	449	38	)	)	PUNCT
ejpam-6568	449	39	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	449	40	-	-	PUNCT
ejpam-6568	449	41	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	449	42	-	-	PUNCT
ejpam-6568	449	43	cl(b	cl(b	NOUN
ejpam-6568	449	44	)	)	PUNCT
ejpam-6568	449	45	)	)	PUNCT
ejpam-6568	449	46	)	)	PUNCT
ejpam-6568	449	47	)	)	PUNCT
ejpam-6568	450	1	⊆	⊆	NUM
ejpam-6568	450	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	450	3	-	-	PUNCT
ejpam-6568	450	4	cl(b	cl(b	NOUN
ejpam-6568	450	5	)	)	PUNCT
ejpam-6568	450	6	)	)	PUNCT
ejpam-6568	450	7	for	for	ADP
ejpam-6568	450	8	every	every	DET
ejpam-6568	450	9	subset	subset	NOUN
ejpam-6568	450	10	b	b	PROPN
ejpam-6568	450	11	of	of	ADP
ejpam-6568	450	12	y	y	PROPN
ejpam-6568	450	13	;	;	PUNCT
ejpam-6568	450	14	n.	n.	PROPN
ejpam-6568	450	15	viriyapong	viriyapong	PROPN
ejpam-6568	450	16	,	,	PUNCT
ejpam-6568	450	17	a.	a.	PROPN
ejpam-6568	450	18	sama	sama	PROPN
ejpam-6568	450	19	-	-	PUNCT
ejpam-6568	450	20	ae	ae	PROPN
ejpam-6568	450	21	,	,	PUNCT
ejpam-6568	450	22	c.	c.	PROPN
ejpam-6568	450	23	boonpok	boonpok	PROPN
ejpam-6568	450	24	/	/	SYM
ejpam-6568	450	25	eur	eur	PROPN
ejpam-6568	450	26	.	.	PUNCT
ejpam-6568	451	1	j.	j.	PROPN
ejpam-6568	451	2	pure	pure	PROPN
ejpam-6568	451	3	appl	appl	PROPN
ejpam-6568	451	4	.	.	PROPN
ejpam-6568	451	5	math	math	PROPN
ejpam-6568	451	6	,	,	PUNCT
ejpam-6568	451	7	18	18	NUM
ejpam-6568	451	8	(	(	PUNCT
ejpam-6568	451	9	3	3	NUM
ejpam-6568	451	10	)	)	PUNCT
ejpam-6568	451	11	(	(	PUNCT
ejpam-6568	451	12	2025	2025	NUM
ejpam-6568	451	13	)	)	PUNCT
ejpam-6568	451	14	,	,	PUNCT
ejpam-6568	451	15	6568	6568	NUM
ejpam-6568	451	16	15	15	NUM
ejpam-6568	451	17	of	of	ADP
ejpam-6568	451	18	18	18	NUM
ejpam-6568	451	19	(	(	PUNCT
ejpam-6568	451	20	3	3	NUM
ejpam-6568	451	21	)	)	PUNCT
ejpam-6568	451	22	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	451	23	-	-	PUNCT
ejpam-6568	451	24	int(k	int(k	PROPN
ejpam-6568	451	25	)	)	PUNCT
ejpam-6568	451	26	)	)	PUNCT
ejpam-6568	451	27	)	)	PUNCT
ejpam-6568	452	1	⊆	⊆	NUM
ejpam-6568	452	2	f−1(k	f−1(k	PROPN
ejpam-6568	452	3	)	)	PUNCT
ejpam-6568	452	4	for	for	ADP
ejpam-6568	452	5	every	every	DET
ejpam-6568	452	6	(	(	PUNCT
ejpam-6568	452	7	σ1	σ1	PROPN
ejpam-6568	452	8	,	,	PUNCT
ejpam-6568	452	9	σ2)r	σ2)r	NOUN
ejpam-6568	452	10	-	-	PUNCT
ejpam-6568	452	11	closed	close	VERB
ejpam-6568	452	12	set	set	ADJ
ejpam-6568	452	13	k	k	PROPN
ejpam-6568	452	14	of	of	ADP
ejpam-6568	452	15	y	y	PROPN
ejpam-6568	452	16	;	;	PUNCT
ejpam-6568	452	17	(	(	PUNCT
ejpam-6568	452	18	4	4	X
ejpam-6568	452	19	)	)	PUNCT
ejpam-6568	452	20	cl(f−1(v	cl(f−1(v	NOUN
ejpam-6568	452	21	)	)	PUNCT
ejpam-6568	452	22	)	)	PUNCT
ejpam-6568	453	1	⊆	⊆	NUM
ejpam-6568	453	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	453	3	-	-	PUNCT
ejpam-6568	453	4	cl(v	cl(v	NOUN
ejpam-6568	453	5	)	)	PUNCT
ejpam-6568	453	6	)	)	PUNCT
ejpam-6568	453	7	for	for	ADP
ejpam-6568	453	8	every	every	DET
ejpam-6568	453	9	σ1σ2	σ1σ2	NOUN
ejpam-6568	453	10	-	-	ADJ
ejpam-6568	453	11	open	open	ADJ
ejpam-6568	453	12	set	set	NOUN
ejpam-6568	453	13	v	v	NOUN
ejpam-6568	453	14	of	of	ADP
ejpam-6568	453	15	y	y	PROPN
ejpam-6568	453	16	;	;	PUNCT
ejpam-6568	453	17	(	(	PUNCT
ejpam-6568	453	18	5	5	X
ejpam-6568	453	19	)	)	PUNCT
ejpam-6568	453	20	f−1(v	f−1(v	NOUN
ejpam-6568	453	21	)	)	PUNCT
ejpam-6568	453	22	⊆	⊆	X
ejpam-6568	453	23	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	453	24	-	-	PUNCT
ejpam-6568	453	25	cl(v	cl(v	NOUN
ejpam-6568	453	26	)	)	PUNCT
ejpam-6568	453	27	)	)	PUNCT
ejpam-6568	453	28	)	)	PUNCT
ejpam-6568	453	29	for	for	ADP
ejpam-6568	453	30	every	every	DET
ejpam-6568	453	31	σ1σ2	σ1σ2	NOUN
ejpam-6568	453	32	-	-	ADJ
ejpam-6568	453	33	open	open	ADJ
ejpam-6568	453	34	set	set	NOUN
ejpam-6568	453	35	v	v	NOUN
ejpam-6568	453	36	of	of	ADP
ejpam-6568	453	37	y	y	PROPN
ejpam-6568	453	38	;	;	PUNCT
ejpam-6568	453	39	(	(	PUNCT
ejpam-6568	453	40	6	6	X
ejpam-6568	453	41	)	)	PUNCT
ejpam-6568	453	42	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	453	43	)	)	PUNCT
ejpam-6568	453	44	)	)	PUNCT
ejpam-6568	454	1	⊆	⊆	NUM
ejpam-6568	454	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	454	3	-	-	PUNCT
ejpam-6568	454	4	cl(v	cl(v	NOUN
ejpam-6568	454	5	)	)	PUNCT
ejpam-6568	454	6	)	)	PUNCT
ejpam-6568	454	7	for	for	ADP
ejpam-6568	454	8	every	every	DET
ejpam-6568	454	9	(	(	PUNCT
ejpam-6568	454	10	σ1	σ1	PROPN
ejpam-6568	454	11	,	,	PUNCT
ejpam-6568	454	12	σ2)p	σ2)p	NOUN
ejpam-6568	454	13	-	-	PUNCT
ejpam-6568	454	14	open	open	NOUN
ejpam-6568	454	15	set	set	NOUN
ejpam-6568	454	16	v	v	NOUN
ejpam-6568	454	17	of	of	ADP
ejpam-6568	454	18	y	y	PROPN
ejpam-6568	454	19	;	;	PUNCT
ejpam-6568	454	20	(	(	PUNCT
ejpam-6568	454	21	7	7	X
ejpam-6568	454	22	)	)	PUNCT
ejpam-6568	454	23	f−1(v	f−1(v	NOUN
ejpam-6568	454	24	)	)	PUNCT
ejpam-6568	455	1	⊆	⊆	X
ejpam-6568	455	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	455	3	-	-	PUNCT
ejpam-6568	455	4	cl(v	cl(v	NOUN
ejpam-6568	455	5	)	)	PUNCT
ejpam-6568	455	6	)	)	PUNCT
ejpam-6568	455	7	)	)	PUNCT
ejpam-6568	455	8	for	for	ADP
ejpam-6568	455	9	every	every	DET
ejpam-6568	455	10	(	(	PUNCT
ejpam-6568	455	11	σ1	σ1	PROPN
ejpam-6568	455	12	,	,	PUNCT
ejpam-6568	455	13	σ2)p	σ2)p	NOUN
ejpam-6568	455	14	-	-	PUNCT
ejpam-6568	455	15	open	open	NOUN
ejpam-6568	455	16	set	set	NOUN
ejpam-6568	455	17	v	v	NOUN
ejpam-6568	455	18	of	of	ADP
ejpam-6568	455	19	y	y	PROPN
ejpam-6568	455	20	.	.	PUNCT
ejpam-6568	456	1	proof	proof	NOUN
ejpam-6568	456	2	.	.	PUNCT
ejpam-6568	457	1	(	(	PUNCT
ejpam-6568	457	2	1	1	X
ejpam-6568	457	3	)	)	PUNCT
ejpam-6568	457	4	⇒	⇒	NOUN
ejpam-6568	457	5	(	(	PUNCT
ejpam-6568	457	6	2	2	NUM
ejpam-6568	457	7	):	):	PUNCT
ejpam-6568	457	8	let	let	VERB
ejpam-6568	457	9	b	b	X
ejpam-6568	457	10	be	be	AUX
ejpam-6568	457	11	any	any	DET
ejpam-6568	457	12	subset	subset	NOUN
ejpam-6568	457	13	of	of	ADP
ejpam-6568	457	14	y	y	PROPN
ejpam-6568	457	15	and	and	CCONJ
ejpam-6568	457	16	x	x	PROPN
ejpam-6568	457	17	̸∈	̸∈	PROPN
ejpam-6568	457	18	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	457	19	-	-	PUNCT
ejpam-6568	457	20	cl(b	cl(b	NOUN
ejpam-6568	457	21	)	)	PUNCT
ejpam-6568	457	22	)	)	PUNCT
ejpam-6568	457	23	.	.	PUNCT
ejpam-6568	458	1	then	then	ADV
ejpam-6568	458	2	,	,	PUNCT
ejpam-6568	458	3	we	we	PRON
ejpam-6568	458	4	have	have	VERB
ejpam-6568	458	5	f(x	f(x	PROPN
ejpam-6568	458	6	)	)	PUNCT
ejpam-6568	458	7	̸∈	̸∈	PROPN
ejpam-6568	458	8	σ1σ2	σ1σ2	NOUN
ejpam-6568	458	9	-	-	NUM
ejpam-6568	458	10	cl(b	cl(b	NOUN
ejpam-6568	458	11	)	)	PUNCT
ejpam-6568	458	12	and	and	CCONJ
ejpam-6568	458	13	there	there	PRON
ejpam-6568	458	14	exists	exist	VERB
ejpam-6568	458	15	a	a	DET
ejpam-6568	458	16	σ1σ2	σ1σ2	NUM
ejpam-6568	458	17	-	-	ADJ
ejpam-6568	458	18	open	open	ADJ
ejpam-6568	458	19	set	set	NOUN
ejpam-6568	458	20	u	u	NOUN
ejpam-6568	458	21	of	of	ADP
ejpam-6568	458	22	y	y	PROPN
ejpam-6568	458	23	containing	contain	VERB
ejpam-6568	458	24	f(x	f(x	PROPN
ejpam-6568	458	25	)	)	PUNCT
ejpam-6568	459	1	such	such	ADJ
ejpam-6568	459	2	that	that	SCONJ
ejpam-6568	459	3	u	u	PROPN
ejpam-6568	459	4	∩	∩	NOUN
ejpam-6568	459	5	b	b	NOUN
ejpam-6568	459	6	=	=	X
ejpam-6568	459	7	∅.	∅.	VERB
ejpam-6568	459	8	therefore	therefore	ADV
ejpam-6568	459	9	,	,	PUNCT
ejpam-6568	459	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	459	11	-	-	PUNCT
ejpam-6568	459	12	cl(u	cl(u	NOUN
ejpam-6568	459	13	)	)	PUNCT
ejpam-6568	459	14	∩	∩	NOUN
ejpam-6568	459	15	σ1σ2	σ1σ2	X
ejpam-6568	459	16	-	-	PUNCT
ejpam-6568	459	17	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	459	18	-	-	PUNCT
ejpam-6568	459	19	cl(b	cl(b	NOUN
ejpam-6568	459	20	)	)	PUNCT
ejpam-6568	459	21	)	)	PUNCT
ejpam-6568	460	1	=	=	PUNCT
ejpam-6568	460	2	∅.	∅.	NOUN
ejpam-6568	460	3	since	since	SCONJ
ejpam-6568	460	4	f	f	PROPN
ejpam-6568	460	5	is	be	AUX
ejpam-6568	460	6	weakly	weakly	ADJ
ejpam-6568	460	7	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	460	8	,	,	PUNCT
ejpam-6568	460	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	460	10	at	at	ADP
ejpam-6568	460	11	x	x	X
ejpam-6568	460	12	,	,	PUNCT
ejpam-6568	460	13	there	there	PRON
ejpam-6568	460	14	exists	exist	VERB
ejpam-6568	460	15	a	a	DET
ejpam-6568	460	16	⋆-open	⋆-open	ADJ
ejpam-6568	460	17	set	set	NOUN
ejpam-6568	460	18	w	w	PROPN
ejpam-6568	460	19	of	of	ADP
ejpam-6568	460	20	x	x	PUNCT
ejpam-6568	460	21	containing	contain	VERB
ejpam-6568	460	22	x	x	PUNCT
ejpam-6568	460	23	such	such	ADJ
ejpam-6568	460	24	that	that	SCONJ
ejpam-6568	460	25	f(w	f(w	PROPN
ejpam-6568	460	26	)	)	PUNCT
ejpam-6568	461	1	⊆	⊆	NUM
ejpam-6568	461	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	461	3	-	-	PUNCT
ejpam-6568	461	4	cl(u	cl(u	NUM
ejpam-6568	461	5	)	)	PUNCT
ejpam-6568	461	6	.	.	PUNCT
ejpam-6568	462	1	thus	thus	ADV
ejpam-6568	462	2	,	,	PUNCT
ejpam-6568	462	3	w	w	PROPN
ejpam-6568	462	4	∩	∩	NOUN
ejpam-6568	462	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	462	6	-	-	PUNCT
ejpam-6568	462	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	462	8	-	-	PUNCT
ejpam-6568	462	9	cl(b	cl(b	NOUN
ejpam-6568	462	10	)	)	PUNCT
ejpam-6568	462	11	)	)	PUNCT
ejpam-6568	462	12	)	)	PUNCT
ejpam-6568	463	1	=	=	NOUN
ejpam-6568	463	2	∅	∅	NOUN
ejpam-6568	463	3	and	and	CCONJ
ejpam-6568	463	4	hence	hence	ADV
ejpam-6568	463	5	x	x	X
ejpam-6568	463	6	̸∈	̸∈	PROPN
ejpam-6568	463	7	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	463	8	-	-	PUNCT
ejpam-6568	463	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	463	10	-	-	PUNCT
ejpam-6568	463	11	cl(b	cl(b	NOUN
ejpam-6568	463	12	)	)	PUNCT
ejpam-6568	463	13	)	)	PUNCT
ejpam-6568	463	14	)	)	PUNCT
ejpam-6568	463	15	)	)	PUNCT
ejpam-6568	463	16	.	.	PUNCT
ejpam-6568	464	1	this	this	PRON
ejpam-6568	464	2	shows	show	VERB
ejpam-6568	464	3	that	that	SCONJ
ejpam-6568	464	4	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	NOUN
ejpam-6568	464	5	-	-	PUNCT
ejpam-6568	464	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	464	7	-	-	PUNCT
ejpam-6568	464	8	cl(b	cl(b	NOUN
ejpam-6568	464	9	)	)	PUNCT
ejpam-6568	464	10	)	)	PUNCT
ejpam-6568	464	11	)	)	PUNCT
ejpam-6568	464	12	)	)	PUNCT
ejpam-6568	464	13	⊆	⊆	NUM
ejpam-6568	464	14	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	464	15	-	-	PUNCT
ejpam-6568	464	16	cl(b	cl(b	NOUN
ejpam-6568	464	17	)	)	PUNCT
ejpam-6568	464	18	)	)	PUNCT
ejpam-6568	464	19	.	.	PUNCT
ejpam-6568	465	1	(	(	PUNCT
ejpam-6568	465	2	2	2	X
ejpam-6568	465	3	)	)	PUNCT
ejpam-6568	465	4	⇒	⇒	NOUN
ejpam-6568	465	5	(	(	PUNCT
ejpam-6568	465	6	3	3	NUM
ejpam-6568	465	7	):	):	PUNCT
ejpam-6568	465	8	let	let	VERB
ejpam-6568	465	9	k	k	PRON
ejpam-6568	465	10	be	be	AUX
ejpam-6568	465	11	any	any	DET
ejpam-6568	465	12	(	(	PUNCT
ejpam-6568	465	13	σ1	σ1	NOUN
ejpam-6568	465	14	,	,	PUNCT
ejpam-6568	465	15	σ2)r	σ2)r	NOUN
ejpam-6568	465	16	-	-	PUNCT
ejpam-6568	465	17	closed	close	VERB
ejpam-6568	465	18	set	set	NOUN
ejpam-6568	465	19	of	of	ADP
ejpam-6568	465	20	y	y	PROPN
ejpam-6568	465	21	.	.	PUNCT
ejpam-6568	466	1	then	then	ADV
ejpam-6568	466	2	by	by	ADP
ejpam-6568	466	3	(	(	PUNCT
ejpam-6568	466	4	2	2	NUM
ejpam-6568	466	5	)	)	PUNCT
ejpam-6568	466	6	,	,	PUNCT
ejpam-6568	466	7	we	we	PRON
ejpam-6568	466	8	have	have	VERB
ejpam-6568	466	9	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	PROPN
ejpam-6568	466	10	-	-	PUNCT
ejpam-6568	466	11	int(k	int(k	PROPN
ejpam-6568	466	12	)	)	PUNCT
ejpam-6568	466	13	)	)	PUNCT
ejpam-6568	466	14	)	)	PUNCT
ejpam-6568	467	1	=	=	SYM
ejpam-6568	468	1	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	468	2	-	-	PUNCT
ejpam-6568	468	3	int(σ1σ2	int(σ1σ2	ADV
ejpam-6568	468	4	-	-	PUNCT
ejpam-6568	468	5	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	468	6	-	-	PUNCT
ejpam-6568	468	7	int(k	int(k	NOUN
ejpam-6568	468	8	)	)	PUNCT
ejpam-6568	468	9	)	)	PUNCT
ejpam-6568	468	10	)	)	PUNCT
ejpam-6568	468	11	)	)	PUNCT
ejpam-6568	468	12	)	)	PUNCT
ejpam-6568	469	1	⊆	⊆	NUM
ejpam-6568	469	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	469	3	-	-	PUNCT
ejpam-6568	469	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6568	469	5	-	-	PUNCT
ejpam-6568	469	6	int(k	int(k	NOUN
ejpam-6568	469	7	)	)	PUNCT
ejpam-6568	469	8	)	)	PUNCT
ejpam-6568	469	9	)	)	PUNCT
ejpam-6568	470	1	=	=	PUNCT
ejpam-6568	470	2	f−1(k	f−1(k	PROPN
ejpam-6568	470	3	)	)	PUNCT
ejpam-6568	470	4	.	.	PUNCT
ejpam-6568	471	1	(	(	PUNCT
ejpam-6568	471	2	3	3	X
ejpam-6568	471	3	)	)	PUNCT
ejpam-6568	471	4	⇒	⇒	NOUN
ejpam-6568	471	5	(	(	PUNCT
ejpam-6568	471	6	4	4	NUM
ejpam-6568	471	7	):	):	PUNCT
ejpam-6568	471	8	let	let	VERB
ejpam-6568	471	9	v	v	PART
ejpam-6568	471	10	be	be	AUX
ejpam-6568	471	11	any	any	DET
ejpam-6568	471	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	471	13	-	-	ADJ
ejpam-6568	471	14	open	open	ADJ
ejpam-6568	471	15	set	set	NOUN
ejpam-6568	471	16	of	of	ADP
ejpam-6568	471	17	y	y	PROPN
ejpam-6568	471	18	.	.	PUNCT
ejpam-6568	472	1	then	then	ADV
ejpam-6568	472	2	,	,	PUNCT
ejpam-6568	472	3	σ1σ2	σ1σ2	NOUN
ejpam-6568	472	4	-	-	NUM
ejpam-6568	472	5	cl(v	cl(v	NOUN
ejpam-6568	472	6	)	)	PUNCT
ejpam-6568	472	7	is	be	AUX
ejpam-6568	472	8	(	(	PUNCT
ejpam-6568	472	9	σ1	σ1	NOUN
ejpam-6568	472	10	,	,	PUNCT
ejpam-6568	472	11	σ2)r	σ2)r	NOUN
ejpam-6568	472	12	-	-	PUNCT
ejpam-6568	472	13	closed	closed	ADJ
ejpam-6568	472	14	in	in	ADP
ejpam-6568	472	15	y	y	PROPN
ejpam-6568	472	16	.	.	PUNCT
ejpam-6568	473	1	by	by	ADP
ejpam-6568	473	2	(	(	PUNCT
ejpam-6568	473	3	3	3	NUM
ejpam-6568	473	4	)	)	PUNCT
ejpam-6568	473	5	,	,	PUNCT
ejpam-6568	473	6	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	473	7	)	)	PUNCT
ejpam-6568	473	8	)	)	PUNCT
ejpam-6568	474	1	⊆	⊆	X
ejpam-6568	474	2	cl⋆(f−1(σ1σ2	cl⋆(f−1(σ1σ2	ADJ
ejpam-6568	474	3	-	-	PUNCT
ejpam-6568	474	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	474	5	-	-	PUNCT
ejpam-6568	474	6	cl(v	cl(v	NOUN
ejpam-6568	474	7	)	)	PUNCT
ejpam-6568	474	8	)	)	PUNCT
ejpam-6568	474	9	)	)	PUNCT
ejpam-6568	474	10	)	)	PUNCT
ejpam-6568	474	11	⊆	⊆	NUM
ejpam-6568	474	12	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	474	13	-	-	PUNCT
ejpam-6568	474	14	cl(v	cl(v	NOUN
ejpam-6568	474	15	)	)	PUNCT
ejpam-6568	474	16	)	)	PUNCT
ejpam-6568	474	17	.	.	PUNCT
ejpam-6568	475	1	(	(	PUNCT
ejpam-6568	475	2	4	4	X
ejpam-6568	475	3	)	)	PUNCT
ejpam-6568	475	4	⇒	⇒	NOUN
ejpam-6568	475	5	(	(	PUNCT
ejpam-6568	475	6	5	5	NUM
ejpam-6568	475	7	):	):	PUNCT
ejpam-6568	475	8	let	let	VERB
ejpam-6568	475	9	v	v	PART
ejpam-6568	475	10	be	be	AUX
ejpam-6568	475	11	any	any	DET
ejpam-6568	475	12	σ1σ2	σ1σ2	NOUN
ejpam-6568	475	13	-	-	ADJ
ejpam-6568	475	14	open	open	ADJ
ejpam-6568	475	15	set	set	NOUN
ejpam-6568	475	16	of	of	ADP
ejpam-6568	475	17	y	y	PROPN
ejpam-6568	475	18	.	.	PUNCT
ejpam-6568	476	1	since	since	SCONJ
ejpam-6568	476	2	y	y	PROPN
ejpam-6568	476	3	−σ1σ2	−σ1σ2	PROPN
ejpam-6568	476	4	-	-	PUNCT
ejpam-6568	476	5	cl(v	cl(v	NOUN
ejpam-6568	476	6	)	)	PUNCT
ejpam-6568	476	7	is	be	AUX
ejpam-6568	476	8	σ1σ2	σ1σ2	NOUN
ejpam-6568	476	9	-	-	ADJ
ejpam-6568	476	10	open	open	ADJ
ejpam-6568	476	11	in	in	ADP
ejpam-6568	476	12	y	y	PROPN
ejpam-6568	476	13	,	,	PUNCT
ejpam-6568	476	14	by	by	ADP
ejpam-6568	476	15	(	(	PUNCT
ejpam-6568	476	16	4	4	X
ejpam-6568	476	17	)	)	PUNCT
ejpam-6568	476	18	we	we	PRON
ejpam-6568	476	19	have	have	VERB
ejpam-6568	476	20	x	x	X
ejpam-6568	476	21	−	−	VERB
ejpam-6568	476	22	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	476	23	-	-	PUNCT
ejpam-6568	476	24	cl(v	cl(v	NOUN
ejpam-6568	476	25	)	)	PUNCT
ejpam-6568	476	26	)	)	PUNCT
ejpam-6568	476	27	)	)	PUNCT
ejpam-6568	477	1	=	=	PUNCT
ejpam-6568	477	2	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	477	3	−	−	VERB
ejpam-6568	477	4	σ1σ2	σ1σ2	NOUN
ejpam-6568	477	5	-	-	NUM
ejpam-6568	477	6	cl(v	cl(v	NOUN
ejpam-6568	477	7	)	)	PUNCT
ejpam-6568	477	8	)	)	PUNCT
ejpam-6568	477	9	)	)	PUNCT
ejpam-6568	478	1	⊆	⊆	NUM
ejpam-6568	478	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	478	3	-	-	PUNCT
ejpam-6568	478	4	cl(y	cl(y	NOUN
ejpam-6568	478	5	−	−	NOUN
ejpam-6568	478	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	478	7	-	-	NUM
ejpam-6568	478	8	cl(v	cl(v	NOUN
ejpam-6568	478	9	)	)	PUNCT
ejpam-6568	478	10	)	)	PUNCT
ejpam-6568	478	11	)	)	PUNCT
ejpam-6568	479	1	⊆	⊆	NUM
ejpam-6568	479	2	x	x	SYM
ejpam-6568	479	3	−	−	PROPN
ejpam-6568	479	4	f−1(v	f−1(v	PROPN
ejpam-6568	479	5	)	)	PUNCT
ejpam-6568	479	6	and	and	CCONJ
ejpam-6568	479	7	hence	hence	ADV
ejpam-6568	479	8	f−1(v	f−1(v	NOUN
ejpam-6568	479	9	)	)	PUNCT
ejpam-6568	479	10	⊆	⊆	X
ejpam-6568	479	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	479	12	-	-	PUNCT
ejpam-6568	479	13	cl(v	cl(v	NOUN
ejpam-6568	479	14	)	)	PUNCT
ejpam-6568	479	15	)	)	PUNCT
ejpam-6568	479	16	)	)	PUNCT
ejpam-6568	479	17	.	.	PUNCT
ejpam-6568	480	1	(	(	PUNCT
ejpam-6568	480	2	5	5	X
ejpam-6568	480	3	)	)	PUNCT
ejpam-6568	480	4	⇒	⇒	NOUN
ejpam-6568	480	5	(	(	PUNCT
ejpam-6568	480	6	1	1	NUM
ejpam-6568	480	7	):	):	PUNCT
ejpam-6568	480	8	let	let	VERB
ejpam-6568	480	9	x	x	PUNCT
ejpam-6568	480	10	∈	∈	PROPN
ejpam-6568	480	11	x	x	X
ejpam-6568	480	12	and	and	CCONJ
ejpam-6568	480	13	v	v	X
ejpam-6568	480	14	be	be	AUX
ejpam-6568	480	15	any	any	DET
ejpam-6568	480	16	σ1σ2	σ1σ2	NOUN
ejpam-6568	480	17	-	-	ADJ
ejpam-6568	480	18	open	open	ADJ
ejpam-6568	480	19	set	set	NOUN
ejpam-6568	480	20	of	of	ADP
ejpam-6568	480	21	y	y	PROPN
ejpam-6568	480	22	containing	contain	VERB
ejpam-6568	480	23	f(x	f(x	PROPN
ejpam-6568	480	24	)	)	PUNCT
ejpam-6568	480	25	.	.	PUNCT
ejpam-6568	481	1	by	by	ADP
ejpam-6568	481	2	(	(	PUNCT
ejpam-6568	481	3	5	5	NUM
ejpam-6568	481	4	)	)	PUNCT
ejpam-6568	481	5	,	,	PUNCT
ejpam-6568	481	6	x	x	PUNCT
ejpam-6568	481	7	∈	∈	PROPN
ejpam-6568	481	8	f−1(v	f−1(v	NOUN
ejpam-6568	481	9	)	)	PUNCT
ejpam-6568	481	10	⊆	⊆	X
ejpam-6568	481	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	481	12	-	-	PUNCT
ejpam-6568	481	13	cl(v	cl(v	NOUN
ejpam-6568	481	14	)	)	PUNCT
ejpam-6568	481	15	)	)	PUNCT
ejpam-6568	481	16	)	)	PUNCT
ejpam-6568	481	17	.	.	PUNCT
ejpam-6568	482	1	put	put	VERB
ejpam-6568	482	2	w	w	NOUN
ejpam-6568	482	3	=	=	PUNCT
ejpam-6568	482	4	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	482	5	-	-	PUNCT
ejpam-6568	482	6	cl(v	cl(v	NOUN
ejpam-6568	482	7	)	)	PUNCT
ejpam-6568	482	8	)	)	PUNCT
ejpam-6568	482	9	)	)	PUNCT
ejpam-6568	482	10	.	.	PUNCT
ejpam-6568	483	1	then	then	ADV
ejpam-6568	483	2	,	,	PUNCT
ejpam-6568	483	3	w	w	PROPN
ejpam-6568	483	4	is	be	AUX
ejpam-6568	483	5	⋆-open	⋆-open	ADJ
ejpam-6568	483	6	set	set	VERB
ejpam-6568	483	7	of	of	ADP
ejpam-6568	483	8	x	x	PUNCT
ejpam-6568	483	9	containing	contain	VERB
ejpam-6568	483	10	x	x	PUNCT
ejpam-6568	484	1	such	such	ADJ
ejpam-6568	484	2	that	that	SCONJ
ejpam-6568	484	3	f(w	f(w	PROPN
ejpam-6568	484	4	)	)	PUNCT
ejpam-6568	484	5	⊆	⊆	X
ejpam-6568	484	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	484	7	-	-	NUM
ejpam-6568	484	8	cl(v	cl(v	NOUN
ejpam-6568	484	9	)	)	PUNCT
ejpam-6568	484	10	.	.	PUNCT
ejpam-6568	485	1	thus	thus	ADV
ejpam-6568	485	2	,	,	PUNCT
ejpam-6568	485	3	f	f	PROPN
ejpam-6568	485	4	is	be	AUX
ejpam-6568	485	5	weakly	weakly	ADJ
ejpam-6568	485	6	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	485	7	,	,	PUNCT
ejpam-6568	485	8	σ2)continuous	σ2)continuous	ADJ
ejpam-6568	485	9	at	at	ADP
ejpam-6568	485	10	x.	x.	NOUN
ejpam-6568	485	11	this	this	PRON
ejpam-6568	485	12	shows	show	VERB
ejpam-6568	485	13	that	that	SCONJ
ejpam-6568	485	14	f	f	PROPN
ejpam-6568	485	15	is	be	AUX
ejpam-6568	485	16	weakly	weakly	ADJ
ejpam-6568	485	17	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	485	18	,	,	PUNCT
ejpam-6568	485	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	485	20	.	.	PUNCT
ejpam-6568	486	1	(	(	PUNCT
ejpam-6568	486	2	1	1	X
ejpam-6568	486	3	)	)	PUNCT
ejpam-6568	486	4	⇒	⇒	NOUN
ejpam-6568	486	5	(	(	PUNCT
ejpam-6568	486	6	6	6	NUM
ejpam-6568	486	7	):	):	PUNCT
ejpam-6568	486	8	let	let	VERB
ejpam-6568	486	9	v	v	PART
ejpam-6568	486	10	be	be	AUX
ejpam-6568	486	11	any	any	DET
ejpam-6568	486	12	(	(	PUNCT
ejpam-6568	486	13	σ1	σ1	PROPN
ejpam-6568	486	14	,	,	PUNCT
ejpam-6568	486	15	σ2)p	σ2)p	NOUN
ejpam-6568	486	16	-	-	PUNCT
ejpam-6568	486	17	open	open	ADJ
ejpam-6568	486	18	set	set	NOUN
ejpam-6568	486	19	of	of	ADP
ejpam-6568	486	20	y	y	PROPN
ejpam-6568	486	21	and	and	CCONJ
ejpam-6568	486	22	x	x	PROPN
ejpam-6568	486	23	̸∈	̸∈	PROPN
ejpam-6568	486	24	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	486	25	-	-	PUNCT
ejpam-6568	486	26	cl(v	cl(v	NOUN
ejpam-6568	486	27	)	)	PUNCT
ejpam-6568	486	28	)	)	PUNCT
ejpam-6568	486	29	.	.	PUNCT
ejpam-6568	487	1	then	then	ADV
ejpam-6568	487	2	,	,	PUNCT
ejpam-6568	487	3	f(x	f(x	PROPN
ejpam-6568	487	4	)	)	PUNCT
ejpam-6568	487	5	̸∈	̸∈	PROPN
ejpam-6568	487	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	487	7	-	-	NUM
ejpam-6568	487	8	cl(v	cl(v	NOUN
ejpam-6568	487	9	)	)	PUNCT
ejpam-6568	487	10	and	and	CCONJ
ejpam-6568	487	11	there	there	PRON
ejpam-6568	487	12	exists	exist	VERB
ejpam-6568	487	13	a	a	DET
ejpam-6568	487	14	σ1σ2	σ1σ2	NUM
ejpam-6568	487	15	-	-	ADJ
ejpam-6568	487	16	open	open	ADJ
ejpam-6568	487	17	set	set	NOUN
ejpam-6568	487	18	g	g	NOUN
ejpam-6568	487	19	of	of	ADP
ejpam-6568	487	20	y	y	PROPN
ejpam-6568	487	21	containing	contain	VERB
ejpam-6568	487	22	f(x	f(x	PROPN
ejpam-6568	487	23	)	)	PUNCT
ejpam-6568	487	24	such	such	ADJ
ejpam-6568	487	25	that	that	SCONJ
ejpam-6568	487	26	g	g	PROPN
ejpam-6568	487	27	∩	∩	NOUN
ejpam-6568	487	28	v	v	X
ejpam-6568	487	29	=	=	PUNCT
ejpam-6568	487	30	∅.	∅.	NOUN
ejpam-6568	487	31	since	since	SCONJ
ejpam-6568	487	32	v	v	NOUN
ejpam-6568	487	33	is	be	AUX
ejpam-6568	487	34	(	(	PUNCT
ejpam-6568	487	35	σ1	σ1	PROPN
ejpam-6568	487	36	,	,	PUNCT
ejpam-6568	487	37	σ2)p	σ2)p	NOUN
ejpam-6568	487	38	-	-	PUNCT
ejpam-6568	487	39	open	open	ADJ
ejpam-6568	487	40	,	,	PUNCT
ejpam-6568	487	41	we	we	PRON
ejpam-6568	487	42	have	have	VERB
ejpam-6568	487	43	v	v	NUM
ejpam-6568	487	44	∩	∩	ADJ
ejpam-6568	487	45	σ1σ2	σ1σ2	NOUN
ejpam-6568	487	46	-	-	PUNCT
ejpam-6568	487	47	cl(g	cl(g	ADJ
ejpam-6568	487	48	)	)	PUNCT
ejpam-6568	487	49	⊆	⊆	NUM
ejpam-6568	487	50	σ1σ2	σ1σ2	X
ejpam-6568	487	51	-	-	PUNCT
ejpam-6568	487	52	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	487	53	-	-	PUNCT
ejpam-6568	487	54	cl(v	cl(v	NOUN
ejpam-6568	487	55	)	)	PUNCT
ejpam-6568	487	56	)	)	PUNCT
ejpam-6568	488	1	∩	∩	NOUN
ejpam-6568	488	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	488	3	-	-	PUNCT
ejpam-6568	488	4	cl(g	cl(g	ADJ
ejpam-6568	488	5	)	)	PUNCT
ejpam-6568	488	6	⊆	⊆	NUM
ejpam-6568	488	7	σ1σ2	σ1σ2	X
ejpam-6568	488	8	-	-	PUNCT
ejpam-6568	488	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	488	10	-	-	PUNCT
ejpam-6568	488	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	488	12	-	-	PUNCT
ejpam-6568	488	13	cl(v	cl(v	NOUN
ejpam-6568	488	14	)	)	PUNCT
ejpam-6568	488	15	)	)	PUNCT
ejpam-6568	488	16	∩g	∩g	PROPN
ejpam-6568	488	17	)	)	PUNCT
ejpam-6568	488	18	n.	n.	PROPN
ejpam-6568	488	19	viriyapong	viriyapong	PROPN
ejpam-6568	488	20	,	,	PUNCT
ejpam-6568	488	21	a.	a.	PROPN
ejpam-6568	488	22	sama	sama	PROPN
ejpam-6568	488	23	-	-	PUNCT
ejpam-6568	488	24	ae	ae	PROPN
ejpam-6568	488	25	,	,	PUNCT
ejpam-6568	488	26	c.	c.	PROPN
ejpam-6568	488	27	boonpok	boonpok	PROPN
ejpam-6568	488	28	/	/	SYM
ejpam-6568	488	29	eur	eur	PROPN
ejpam-6568	488	30	.	.	PUNCT
ejpam-6568	489	1	j.	j.	PROPN
ejpam-6568	489	2	pure	pure	PROPN
ejpam-6568	489	3	appl	appl	PROPN
ejpam-6568	489	4	.	.	PROPN
ejpam-6568	489	5	math	math	PROPN
ejpam-6568	489	6	,	,	PUNCT
ejpam-6568	489	7	18	18	NUM
ejpam-6568	489	8	(	(	PUNCT
ejpam-6568	489	9	3	3	NUM
ejpam-6568	489	10	)	)	PUNCT
ejpam-6568	489	11	(	(	PUNCT
ejpam-6568	489	12	2025	2025	NUM
ejpam-6568	489	13	)	)	PUNCT
ejpam-6568	489	14	,	,	PUNCT
ejpam-6568	489	15	6568	6568	NUM
ejpam-6568	489	16	16	16	NUM
ejpam-6568	489	17	of	of	ADP
ejpam-6568	489	18	18	18	NUM
ejpam-6568	489	19	⊆	⊆	NUM
ejpam-6568	489	20	σ1σ2	σ1σ2	NUM
ejpam-6568	489	21	-	-	PUNCT
ejpam-6568	489	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	489	23	-	-	PUNCT
ejpam-6568	489	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	489	25	-	-	PUNCT
ejpam-6568	489	26	cl(v	cl(v	NOUN
ejpam-6568	489	27	)	)	PUNCT
ejpam-6568	489	28	∩g	∩g	PROPN
ejpam-6568	489	29	)	)	PUNCT
ejpam-6568	489	30	)	)	PUNCT
ejpam-6568	490	1	⊆	⊆	X
ejpam-6568	490	2	σ1σ2	σ1σ2	X
ejpam-6568	490	3	-	-	PUNCT
ejpam-6568	490	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6568	490	5	-	-	PUNCT
ejpam-6568	490	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	490	7	-	-	PUNCT
ejpam-6568	490	8	cl(v	cl(v	NOUN
ejpam-6568	490	9	∩g	∩g	NOUN
ejpam-6568	490	10	)	)	PUNCT
ejpam-6568	490	11	)	)	PUNCT
ejpam-6568	490	12	)	)	PUNCT
ejpam-6568	490	13	⊆	⊆	X
ejpam-6568	490	14	σ1σ2	σ1σ2	NUM
ejpam-6568	490	15	-	-	PUNCT
ejpam-6568	490	16	cl(v	cl(v	NOUN
ejpam-6568	490	17	∩g	∩g	NOUN
ejpam-6568	490	18	)	)	PUNCT
ejpam-6568	490	19	=	=	PUNCT
ejpam-6568	490	20	∅.	∅.	NOUN
ejpam-6568	490	21	since	since	SCONJ
ejpam-6568	490	22	f	f	PROPN
ejpam-6568	490	23	is	be	AUX
ejpam-6568	490	24	weakly	weakly	ADJ
ejpam-6568	490	25	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	490	26	,	,	PUNCT
ejpam-6568	490	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	490	28	at	at	ADP
ejpam-6568	490	29	x	x	X
ejpam-6568	490	30	,	,	PUNCT
ejpam-6568	490	31	there	there	PRON
ejpam-6568	490	32	exists	exist	VERB
ejpam-6568	490	33	a	a	DET
ejpam-6568	490	34	⋆-open	⋆-open	ADJ
ejpam-6568	490	35	set	set	NOUN
ejpam-6568	490	36	w	w	PROPN
ejpam-6568	490	37	of	of	ADP
ejpam-6568	490	38	x	x	PUNCT
ejpam-6568	490	39	containing	contain	VERB
ejpam-6568	490	40	x	x	PUNCT
ejpam-6568	490	41	such	such	ADJ
ejpam-6568	490	42	that	that	SCONJ
ejpam-6568	490	43	f(w	f(w	PROPN
ejpam-6568	490	44	)	)	PUNCT
ejpam-6568	491	1	⊆	⊆	X
ejpam-6568	491	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	491	3	-	-	PUNCT
ejpam-6568	491	4	cl(g	cl(g	NUM
ejpam-6568	491	5	)	)	PUNCT
ejpam-6568	491	6	.	.	PUNCT
ejpam-6568	492	1	thus	thus	ADV
ejpam-6568	492	2	,	,	PUNCT
ejpam-6568	492	3	f(w	f(w	PROPN
ejpam-6568	492	4	)	)	PUNCT
ejpam-6568	492	5	∩	∩	NOUN
ejpam-6568	492	6	v	v	NOUN
ejpam-6568	492	7	=	=	NOUN
ejpam-6568	492	8	∅	∅	NOUN
ejpam-6568	492	9	and	and	CCONJ
ejpam-6568	492	10	hence	hence	ADV
ejpam-6568	492	11	w	w	ADP
ejpam-6568	492	12	∩	∩	ADJ
ejpam-6568	492	13	f−1(v	f−1(v	NOUN
ejpam-6568	492	14	)	)	PUNCT
ejpam-6568	493	1	=	=	PUNCT
ejpam-6568	493	2	∅.	∅.	VERB
ejpam-6568	493	3	therefore	therefore	ADV
ejpam-6568	493	4	,	,	PUNCT
ejpam-6568	493	5	x	x	PROPN
ejpam-6568	493	6	̸∈	̸∈	PROPN
ejpam-6568	493	7	cl⋆(f−1(v	cl⋆(f−1(v	PROPN
ejpam-6568	493	8	)	)	PUNCT
ejpam-6568	493	9	.	.	PUNCT
ejpam-6568	494	1	this	this	PRON
ejpam-6568	494	2	shows	show	VERB
ejpam-6568	494	3	that	that	SCONJ
ejpam-6568	494	4	cl⋆(f−1(v	cl⋆(f−1(v	NOUN
ejpam-6568	494	5	)	)	PUNCT
ejpam-6568	494	6	)	)	PUNCT
ejpam-6568	495	1	⊆	⊆	NUM
ejpam-6568	495	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	495	3	-	-	PUNCT
ejpam-6568	495	4	cl(v	cl(v	NOUN
ejpam-6568	495	5	)	)	PUNCT
ejpam-6568	495	6	)	)	PUNCT
ejpam-6568	495	7	.	.	PUNCT
ejpam-6568	496	1	(	(	PUNCT
ejpam-6568	496	2	6	6	X
ejpam-6568	496	3	)	)	PUNCT
ejpam-6568	496	4	⇒	⇒	NOUN
ejpam-6568	496	5	(	(	PUNCT
ejpam-6568	496	6	7	7	NUM
ejpam-6568	496	7	):	):	PUNCT
ejpam-6568	496	8	let	let	VERB
ejpam-6568	496	9	v	v	PART
ejpam-6568	496	10	be	be	AUX
ejpam-6568	496	11	any	any	DET
ejpam-6568	496	12	(	(	PUNCT
ejpam-6568	496	13	σ1	σ1	PROPN
ejpam-6568	496	14	,	,	PUNCT
ejpam-6568	496	15	σ2)p	σ2)p	NOUN
ejpam-6568	496	16	-	-	PUNCT
ejpam-6568	496	17	open	open	ADJ
ejpam-6568	496	18	set	set	NOUN
ejpam-6568	496	19	of	of	ADP
ejpam-6568	496	20	y	y	PROPN
ejpam-6568	496	21	.	.	PUNCT
ejpam-6568	497	1	then	then	ADV
ejpam-6568	497	2	,	,	PUNCT
ejpam-6568	497	3	y	y	PROPN
ejpam-6568	497	4	−	−	NUM
ejpam-6568	497	5	σ1σ2	σ1σ2	NOUN
ejpam-6568	497	6	-	-	NUM
ejpam-6568	497	7	cl(v	cl(v	NOUN
ejpam-6568	497	8	)	)	PUNCT
ejpam-6568	497	9	is	be	AUX
ejpam-6568	497	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	497	11	-	-	ADJ
ejpam-6568	497	12	open	open	ADJ
ejpam-6568	497	13	and	and	CCONJ
ejpam-6568	497	14	hence	hence	ADV
ejpam-6568	497	15	y	y	NOUN
ejpam-6568	497	16	−	−	NUM
ejpam-6568	497	17	σ1σ2	σ1σ2	NOUN
ejpam-6568	497	18	-	-	NUM
ejpam-6568	497	19	cl(v	cl(v	NOUN
ejpam-6568	497	20	)	)	PUNCT
ejpam-6568	497	21	is	be	AUX
ejpam-6568	497	22	(	(	PUNCT
ejpam-6568	497	23	σ1	σ1	PROPN
ejpam-6568	497	24	,	,	PUNCT
ejpam-6568	497	25	σ2)p	σ2)p	NOUN
ejpam-6568	497	26	-	-	PUNCT
ejpam-6568	497	27	open	open	ADJ
ejpam-6568	497	28	in	in	ADP
ejpam-6568	497	29	y	y	PROPN
ejpam-6568	497	30	.	.	PUNCT
ejpam-6568	498	1	then	then	ADV
ejpam-6568	498	2	by	by	ADP
ejpam-6568	498	3	(	(	PUNCT
ejpam-6568	498	4	6	6	NUM
ejpam-6568	498	5	)	)	PUNCT
ejpam-6568	498	6	,	,	PUNCT
ejpam-6568	498	7	we	we	PRON
ejpam-6568	498	8	have	have	VERB
ejpam-6568	498	9	x	x	X
ejpam-6568	498	10	−	−	VERB
ejpam-6568	498	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	498	12	-	-	PUNCT
ejpam-6568	498	13	cl(v	cl(v	NOUN
ejpam-6568	498	14	)	)	PUNCT
ejpam-6568	498	15	)	)	PUNCT
ejpam-6568	498	16	)	)	PUNCT
ejpam-6568	499	1	=	=	PUNCT
ejpam-6568	499	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6568	499	3	−	−	NOUN
ejpam-6568	499	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	499	5	-	-	PUNCT
ejpam-6568	499	6	cl(v	cl(v	NOUN
ejpam-6568	499	7	)	)	PUNCT
ejpam-6568	499	8	)	)	PUNCT
ejpam-6568	499	9	)	)	PUNCT
ejpam-6568	500	1	=	=	PUNCT
ejpam-6568	500	2	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	500	3	−	−	VERB
ejpam-6568	500	4	σ1σ2	σ1σ2	NOUN
ejpam-6568	500	5	-	-	NUM
ejpam-6568	500	6	cl(v	cl(v	NOUN
ejpam-6568	500	7	)	)	PUNCT
ejpam-6568	500	8	)	)	PUNCT
ejpam-6568	500	9	)	)	PUNCT
ejpam-6568	501	1	⊆	⊆	NUM
ejpam-6568	501	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6568	501	3	-	-	PUNCT
ejpam-6568	501	4	cl(y	cl(y	NOUN
ejpam-6568	501	5	−	−	NOUN
ejpam-6568	501	6	σ1σ2	σ1σ2	NOUN
ejpam-6568	501	7	-	-	NUM
ejpam-6568	501	8	cl(v	cl(v	NOUN
ejpam-6568	501	9	)	)	PUNCT
ejpam-6568	501	10	)	)	PUNCT
ejpam-6568	501	11	)	)	PUNCT
ejpam-6568	502	1	=	=	PUNCT
ejpam-6568	502	2	f−1(y	f−1(y	PROPN
ejpam-6568	502	3	−	−	NOUN
ejpam-6568	502	4	σ1σ2	σ1σ2	X
ejpam-6568	502	5	-	-	PUNCT
ejpam-6568	502	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	502	7	-	-	PUNCT
ejpam-6568	502	8	cl(v	cl(v	NOUN
ejpam-6568	502	9	)	)	PUNCT
ejpam-6568	502	10	)	)	PUNCT
ejpam-6568	502	11	)	)	PUNCT
ejpam-6568	503	1	=	=	PUNCT
ejpam-6568	503	2	x	x	X
ejpam-6568	503	3	−	−	PRON
ejpam-6568	503	4	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6568	503	5	-	-	PUNCT
ejpam-6568	503	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6568	503	7	-	-	PUNCT
ejpam-6568	503	8	cl(v	cl(v	NOUN
ejpam-6568	503	9	)	)	PUNCT
ejpam-6568	503	10	)	)	PUNCT
ejpam-6568	503	11	)	)	PUNCT
ejpam-6568	504	1	⊆	⊆	NUM
ejpam-6568	504	2	x	x	SYM
ejpam-6568	504	3	−	−	PROPN
ejpam-6568	504	4	f−1(v	f−1(v	PROPN
ejpam-6568	504	5	)	)	PUNCT
ejpam-6568	504	6	and	and	CCONJ
ejpam-6568	504	7	hence	hence	ADV
ejpam-6568	504	8	f−1(v	f−1(v	NOUN
ejpam-6568	504	9	)	)	PUNCT
ejpam-6568	504	10	⊆	⊆	X
ejpam-6568	504	11	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	504	12	-	-	PUNCT
ejpam-6568	504	13	cl(v	cl(v	NOUN
ejpam-6568	504	14	)	)	PUNCT
ejpam-6568	504	15	)	)	PUNCT
ejpam-6568	504	16	)	)	PUNCT
ejpam-6568	504	17	.	.	PUNCT
ejpam-6568	505	1	(	(	PUNCT
ejpam-6568	505	2	7	7	X
ejpam-6568	505	3	)	)	PUNCT
ejpam-6568	505	4	⇒	⇒	NOUN
ejpam-6568	505	5	(	(	PUNCT
ejpam-6568	505	6	1	1	NUM
ejpam-6568	505	7	):	):	PUNCT
ejpam-6568	505	8	let	let	VERB
ejpam-6568	505	9	x	x	PUNCT
ejpam-6568	505	10	∈	∈	PROPN
ejpam-6568	505	11	x	x	X
ejpam-6568	505	12	and	and	CCONJ
ejpam-6568	505	13	v	v	X
ejpam-6568	505	14	be	be	AUX
ejpam-6568	505	15	any	any	DET
ejpam-6568	505	16	σ1σ2	σ1σ2	NOUN
ejpam-6568	505	17	-	-	ADJ
ejpam-6568	505	18	open	open	ADJ
ejpam-6568	505	19	set	set	NOUN
ejpam-6568	505	20	of	of	ADP
ejpam-6568	505	21	y	y	PROPN
ejpam-6568	505	22	containing	contain	VERB
ejpam-6568	505	23	f(x	f(x	PROPN
ejpam-6568	505	24	)	)	PUNCT
ejpam-6568	505	25	.	.	PUNCT
ejpam-6568	506	1	since	since	SCONJ
ejpam-6568	506	2	v	v	NOUN
ejpam-6568	506	3	is	be	AUX
ejpam-6568	506	4	(	(	PUNCT
ejpam-6568	506	5	σ1	σ1	PROPN
ejpam-6568	506	6	,	,	PUNCT
ejpam-6568	506	7	σ2)p	σ2)p	NOUN
ejpam-6568	506	8	-	-	PUNCT
ejpam-6568	506	9	open	open	ADJ
ejpam-6568	506	10	in	in	ADP
ejpam-6568	506	11	y	y	PROPN
ejpam-6568	506	12	and	and	CCONJ
ejpam-6568	506	13	by	by	ADP
ejpam-6568	506	14	(	(	PUNCT
ejpam-6568	506	15	7	7	NUM
ejpam-6568	506	16	)	)	PUNCT
ejpam-6568	506	17	,	,	PUNCT
ejpam-6568	506	18	x	x	PUNCT
ejpam-6568	506	19	∈	∈	PROPN
ejpam-6568	506	20	f−1(v	f−1(v	NOUN
ejpam-6568	506	21	)	)	PUNCT
ejpam-6568	507	1	⊆	⊆	X
ejpam-6568	507	2	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	NUM
ejpam-6568	507	3	-	-	PUNCT
ejpam-6568	507	4	cl(v	cl(v	NOUN
ejpam-6568	507	5	)	)	PUNCT
ejpam-6568	507	6	)	)	PUNCT
ejpam-6568	507	7	)	)	PUNCT
ejpam-6568	507	8	.	.	PUNCT
ejpam-6568	508	1	put	put	VERB
ejpam-6568	508	2	u	u	NOUN
ejpam-6568	508	3	=	=	NOUN
ejpam-6568	508	4	int⋆(f−1(σ1σ2	int⋆(f−1(σ1σ2	X
ejpam-6568	508	5	-	-	PUNCT
ejpam-6568	508	6	cl(v	cl(v	NOUN
ejpam-6568	508	7	)	)	PUNCT
ejpam-6568	508	8	)	)	PUNCT
ejpam-6568	508	9	)	)	PUNCT
ejpam-6568	508	10	.	.	PUNCT
ejpam-6568	509	1	then	then	ADV
ejpam-6568	509	2	,	,	PUNCT
ejpam-6568	509	3	u	u	NOUN
ejpam-6568	509	4	is	be	AUX
ejpam-6568	509	5	a	a	DET
ejpam-6568	509	6	⋆-open	⋆-open	ADJ
ejpam-6568	509	7	set	set	NOUN
ejpam-6568	509	8	of	of	ADP
ejpam-6568	509	9	x	x	PUNCT
ejpam-6568	509	10	containing	contain	VERB
ejpam-6568	509	11	x	x	PUNCT
ejpam-6568	509	12	such	such	ADJ
ejpam-6568	509	13	that	that	DET
ejpam-6568	509	14	f(u	f(u	PROPN
ejpam-6568	509	15	)	)	PUNCT
ejpam-6568	509	16	⊆	⊆	NUM
ejpam-6568	509	17	σ1σ2	σ1σ2	NOUN
ejpam-6568	509	18	-	-	NUM
ejpam-6568	509	19	cl(v	cl(v	NOUN
ejpam-6568	509	20	)	)	PUNCT
ejpam-6568	509	21	.	.	PUNCT
ejpam-6568	510	1	thus	thus	ADV
ejpam-6568	510	2	,	,	PUNCT
ejpam-6568	510	3	f	f	PROPN
ejpam-6568	510	4	is	be	AUX
ejpam-6568	510	5	weakly	weakly	ADJ
ejpam-6568	510	6	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	510	7	,	,	PUNCT
ejpam-6568	510	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	510	9	at	at	ADP
ejpam-6568	510	10	x.	x.	NOUN
ejpam-6568	510	11	this	this	PRON
ejpam-6568	510	12	shows	show	VERB
ejpam-6568	510	13	that	that	SCONJ
ejpam-6568	510	14	f	f	PROPN
ejpam-6568	510	15	is	be	AUX
ejpam-6568	510	16	weakly	weakly	ADJ
ejpam-6568	510	17	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	510	18	,	,	PUNCT
ejpam-6568	510	19	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	510	20	.	.	X
ejpam-6568	510	21	theorem	theorem	VERB
ejpam-6568	510	22	17	17	NUM
ejpam-6568	510	23	.	.	PUNCT
ejpam-6568	511	1	for	for	ADP
ejpam-6568	511	2	a	a	DET
ejpam-6568	511	3	function	function	NOUN
ejpam-6568	511	4	(	(	PUNCT
ejpam-6568	511	5	x	x	X
ejpam-6568	511	6	,	,	PUNCT
ejpam-6568	511	7	τ	τ	PROPN
ejpam-6568	511	8	,	,	PUNCT
ejpam-6568	511	9	i	i	NOUN
ejpam-6568	511	10	)	)	PUNCT
ejpam-6568	511	11	→	→	PUNCT
ejpam-6568	511	12	(	(	PUNCT
ejpam-6568	511	13	y	y	PROPN
ejpam-6568	511	14	,	,	PUNCT
ejpam-6568	511	15	σ1	σ1	PROPN
ejpam-6568	511	16	,	,	PUNCT
ejpam-6568	511	17	σ2	σ2	NOUN
ejpam-6568	511	18	)	)	PUNCT
ejpam-6568	511	19	,	,	PUNCT
ejpam-6568	511	20	the	the	DET
ejpam-6568	511	21	following	follow	VERB
ejpam-6568	511	22	properties	property	NOUN
ejpam-6568	511	23	are	be	AUX
ejpam-6568	511	24	equivalent	equivalent	ADJ
ejpam-6568	511	25	:	:	PUNCT
ejpam-6568	511	26	(	(	PUNCT
ejpam-6568	511	27	1	1	X
ejpam-6568	511	28	)	)	PUNCT
ejpam-6568	511	29	f	f	PROPN
ejpam-6568	511	30	is	be	AUX
ejpam-6568	511	31	weakly	weakly	ADJ
ejpam-6568	511	32	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	511	33	,	,	PUNCT
ejpam-6568	511	34	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	511	35	;	;	PUNCT
ejpam-6568	511	36	(	(	PUNCT
ejpam-6568	511	37	2	2	X
ejpam-6568	511	38	)	)	PUNCT
ejpam-6568	511	39	f(cl⋆(a	f(cl⋆(a	NOUN
ejpam-6568	511	40	)	)	PUNCT
ejpam-6568	511	41	)	)	PUNCT
ejpam-6568	512	1	⊆	⊆	NUM
ejpam-6568	512	2	(	(	PUNCT
ejpam-6568	512	3	σ1	σ1	PROPN
ejpam-6568	512	4	,	,	PUNCT
ejpam-6568	512	5	σ2)θ	σ2)θ	NOUN
ejpam-6568	512	6	-	-	PUNCT
ejpam-6568	512	7	cl(f(a	cl(f(a	NOUN
ejpam-6568	512	8	)	)	PUNCT
ejpam-6568	512	9	)	)	PUNCT
ejpam-6568	512	10	for	for	ADP
ejpam-6568	512	11	every	every	DET
ejpam-6568	512	12	subset	subset	NOUN
ejpam-6568	512	13	a	a	PRON
ejpam-6568	512	14	of	of	ADP
ejpam-6568	512	15	x	x	PRON
ejpam-6568	512	16	;	;	PUNCT
ejpam-6568	512	17	(	(	PUNCT
ejpam-6568	512	18	3	3	X
ejpam-6568	512	19	)	)	PUNCT
ejpam-6568	512	20	cl⋆(f−1(b	cl⋆(f−1(b	NOUN
ejpam-6568	512	21	)	)	PUNCT
ejpam-6568	512	22	)	)	PUNCT
ejpam-6568	513	1	⊆	⊆	NUM
ejpam-6568	513	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	513	3	,	,	PUNCT
ejpam-6568	513	4	σ2)θ	σ2)θ	NOUN
ejpam-6568	513	5	-	-	PUNCT
ejpam-6568	513	6	cl(b	cl(b	NOUN
ejpam-6568	513	7	)	)	PUNCT
ejpam-6568	513	8	)	)	PUNCT
ejpam-6568	513	9	for	for	ADP
ejpam-6568	513	10	every	every	DET
ejpam-6568	513	11	subset	subset	NOUN
ejpam-6568	513	12	b	b	PROPN
ejpam-6568	513	13	of	of	ADP
ejpam-6568	513	14	y	y	PROPN
ejpam-6568	513	15	.	.	PUNCT
ejpam-6568	514	1	proof	proof	NOUN
ejpam-6568	514	2	.	.	PUNCT
ejpam-6568	515	1	(	(	PUNCT
ejpam-6568	515	2	1	1	X
ejpam-6568	515	3	)	)	PUNCT
ejpam-6568	515	4	⇒	⇒	NOUN
ejpam-6568	515	5	(	(	PUNCT
ejpam-6568	515	6	2	2	NUM
ejpam-6568	515	7	):	):	PUNCT
ejpam-6568	515	8	let	let	VERB
ejpam-6568	515	9	a	a	DET
ejpam-6568	515	10	be	be	AUX
ejpam-6568	515	11	any	any	DET
ejpam-6568	515	12	subset	subset	NOUN
ejpam-6568	515	13	of	of	ADP
ejpam-6568	515	14	x.	x.	PROPN
ejpam-6568	515	15	suppose	suppose	VERB
ejpam-6568	515	16	that	that	SCONJ
ejpam-6568	515	17	x	x	PUNCT
ejpam-6568	515	18	∈	∈	PROPN
ejpam-6568	515	19	cl⋆(a	cl⋆(a	PROPN
ejpam-6568	515	20	)	)	PUNCT
ejpam-6568	515	21	and	and	CCONJ
ejpam-6568	515	22	g	g	NOUN
ejpam-6568	515	23	is	be	AUX
ejpam-6568	515	24	any	any	DET
ejpam-6568	515	25	σ1σ2	σ1σ2	NOUN
ejpam-6568	515	26	-	-	ADJ
ejpam-6568	515	27	open	open	ADJ
ejpam-6568	515	28	set	set	NOUN
ejpam-6568	515	29	of	of	ADP
ejpam-6568	515	30	y	y	PROPN
ejpam-6568	515	31	containing	contain	VERB
ejpam-6568	515	32	f(x	f(x	PROPN
ejpam-6568	515	33	)	)	PUNCT
ejpam-6568	515	34	.	.	PUNCT
ejpam-6568	516	1	since	since	SCONJ
ejpam-6568	516	2	f	f	PROPN
ejpam-6568	516	3	is	be	AUX
ejpam-6568	516	4	weakly	weakly	ADJ
ejpam-6568	516	5	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	516	6	,	,	PUNCT
ejpam-6568	516	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	516	8	,	,	PUNCT
ejpam-6568	516	9	there	there	PRON
ejpam-6568	516	10	exists	exist	VERB
ejpam-6568	516	11	a	a	DET
ejpam-6568	516	12	⋆-open	⋆-open	ADJ
ejpam-6568	516	13	set	set	NOUN
ejpam-6568	516	14	u	u	NOUN
ejpam-6568	516	15	of	of	ADP
ejpam-6568	516	16	x	x	PUNCT
ejpam-6568	516	17	containing	contain	VERB
ejpam-6568	516	18	x	x	PUNCT
ejpam-6568	516	19	such	such	ADJ
ejpam-6568	516	20	that	that	DET
ejpam-6568	516	21	f(u	f(u	PROPN
ejpam-6568	516	22	)	)	PUNCT
ejpam-6568	516	23	⊆	⊆	NUM
ejpam-6568	516	24	σ1σ2	σ1σ2	NOUN
ejpam-6568	516	25	-	-	PUNCT
ejpam-6568	516	26	cl(g	cl(g	NUM
ejpam-6568	516	27	)	)	PUNCT
ejpam-6568	516	28	.	.	PUNCT
ejpam-6568	517	1	since	since	SCONJ
ejpam-6568	517	2	x	x	PROPN
ejpam-6568	517	3	∈	∈	PROPN
ejpam-6568	517	4	cl⋆(a	cl⋆(a	PROPN
ejpam-6568	517	5	)	)	PUNCT
ejpam-6568	517	6	,	,	PUNCT
ejpam-6568	517	7	we	we	PRON
ejpam-6568	517	8	have	have	VERB
ejpam-6568	517	9	u	u	NOUN
ejpam-6568	517	10	∩	∩	NOUN
ejpam-6568	517	11	a	a	DET
ejpam-6568	517	12	̸=	̸=	PROPN
ejpam-6568	517	13	∅.	∅.	NOUN
ejpam-6568	517	14	it	it	PRON
ejpam-6568	517	15	follows	follow	VERB
ejpam-6568	517	16	that	that	SCONJ
ejpam-6568	517	17	∅	∅	NOUN
ejpam-6568	517	18	̸=	̸=	PROPN
ejpam-6568	517	19	f(u	f(u	PROPN
ejpam-6568	517	20	)	)	PUNCT
ejpam-6568	517	21	∩	∩	NOUN
ejpam-6568	517	22	f(a	f(a	NOUN
ejpam-6568	517	23	)	)	PUNCT
ejpam-6568	517	24	⊆	⊆	NUM
ejpam-6568	517	25	σ1σ2	σ1σ2	NOUN
ejpam-6568	517	26	-	-	PUNCT
ejpam-6568	517	27	cl(g	cl(g	ADJ
ejpam-6568	517	28	)	)	PUNCT
ejpam-6568	517	29	∩	∩	ADJ
ejpam-6568	517	30	f(a	f(a	NOUN
ejpam-6568	517	31	)	)	PUNCT
ejpam-6568	517	32	.	.	PUNCT
ejpam-6568	518	1	thus	thus	ADV
ejpam-6568	518	2	,	,	PUNCT
ejpam-6568	518	3	f(x	f(x	PROPN
ejpam-6568	518	4	)	)	PUNCT
ejpam-6568	518	5	∈	∈	PROPN
ejpam-6568	518	6	(	(	PUNCT
ejpam-6568	518	7	σ1	σ1	PROPN
ejpam-6568	518	8	,	,	PUNCT
ejpam-6568	518	9	σ2)θ	σ2)θ	NOUN
ejpam-6568	518	10	-	-	PUNCT
ejpam-6568	518	11	cl(f(a	cl(f(a	NOUN
ejpam-6568	518	12	)	)	PUNCT
ejpam-6568	518	13	)	)	PUNCT
ejpam-6568	518	14	and	and	CCONJ
ejpam-6568	518	15	hence	hence	ADV
ejpam-6568	518	16	f(cl⋆(a	f(cl⋆(a	NUM
ejpam-6568	518	17	)	)	PUNCT
ejpam-6568	518	18	)	)	PUNCT
ejpam-6568	519	1	⊆	⊆	NUM
ejpam-6568	519	2	(	(	PUNCT
ejpam-6568	519	3	σ1	σ1	PROPN
ejpam-6568	519	4	,	,	PUNCT
ejpam-6568	519	5	σ2)θ	σ2)θ	NOUN
ejpam-6568	519	6	-	-	PUNCT
ejpam-6568	519	7	cl(f(a	cl(f(a	NOUN
ejpam-6568	519	8	)	)	PUNCT
ejpam-6568	519	9	)	)	PUNCT
ejpam-6568	519	10	.	.	PUNCT
ejpam-6568	520	1	(	(	PUNCT
ejpam-6568	520	2	2	2	X
ejpam-6568	520	3	)	)	PUNCT
ejpam-6568	520	4	⇒	⇒	NOUN
ejpam-6568	520	5	(	(	PUNCT
ejpam-6568	520	6	3	3	NUM
ejpam-6568	520	7	):	):	PUNCT
ejpam-6568	520	8	let	let	VERB
ejpam-6568	520	9	b	b	X
ejpam-6568	520	10	be	be	AUX
ejpam-6568	520	11	any	any	DET
ejpam-6568	520	12	subset	subset	NOUN
ejpam-6568	520	13	of	of	ADP
ejpam-6568	520	14	y	y	PROPN
ejpam-6568	520	15	.	.	PUNCT
ejpam-6568	521	1	then	then	ADV
ejpam-6568	521	2	,	,	PUNCT
ejpam-6568	521	3	we	we	PRON
ejpam-6568	521	4	have	have	VERB
ejpam-6568	521	5	f(cl⋆(f−1(b	f(cl⋆(f−1(b	VERB
ejpam-6568	521	6	)	)	PUNCT
ejpam-6568	521	7	)	)	PUNCT
ejpam-6568	521	8	)	)	PUNCT
ejpam-6568	522	1	⊆	⊆	X
ejpam-6568	522	2	(	(	PUNCT
ejpam-6568	522	3	σ1	σ1	PROPN
ejpam-6568	522	4	,	,	PUNCT
ejpam-6568	522	5	σ2)θ	σ2)θ	ADJ
ejpam-6568	522	6	-	-	PUNCT
ejpam-6568	522	7	cl(f(f	cl(f(f	ADJ
ejpam-6568	522	8	−1(b	−1(b	NOUN
ejpam-6568	522	9	)	)	PUNCT
ejpam-6568	522	10	)	)	PUNCT
ejpam-6568	522	11	)	)	PUNCT
ejpam-6568	522	12	⊆	⊆	X
ejpam-6568	522	13	(	(	PUNCT
ejpam-6568	522	14	σ1	σ1	PROPN
ejpam-6568	522	15	,	,	PUNCT
ejpam-6568	522	16	σ2)θ	σ2)θ	NOUN
ejpam-6568	522	17	-	-	PUNCT
ejpam-6568	522	18	cl(b	cl(b	NOUN
ejpam-6568	522	19	)	)	PUNCT
ejpam-6568	522	20	n.	n.	PROPN
ejpam-6568	522	21	viriyapong	viriyapong	PROPN
ejpam-6568	522	22	,	,	PUNCT
ejpam-6568	522	23	a.	a.	PROPN
ejpam-6568	522	24	sama	sama	PROPN
ejpam-6568	522	25	-	-	PUNCT
ejpam-6568	522	26	ae	ae	PROPN
ejpam-6568	522	27	,	,	PUNCT
ejpam-6568	522	28	c.	c.	PROPN
ejpam-6568	522	29	boonpok	boonpok	PROPN
ejpam-6568	522	30	/	/	SYM
ejpam-6568	522	31	eur	eur	PROPN
ejpam-6568	522	32	.	.	PUNCT
ejpam-6568	523	1	j.	j.	PROPN
ejpam-6568	523	2	pure	pure	PROPN
ejpam-6568	523	3	appl	appl	PROPN
ejpam-6568	523	4	.	.	PROPN
ejpam-6568	523	5	math	math	PROPN
ejpam-6568	523	6	,	,	PUNCT
ejpam-6568	523	7	18	18	NUM
ejpam-6568	523	8	(	(	PUNCT
ejpam-6568	523	9	3	3	NUM
ejpam-6568	523	10	)	)	PUNCT
ejpam-6568	523	11	(	(	PUNCT
ejpam-6568	523	12	2025	2025	NUM
ejpam-6568	523	13	)	)	PUNCT
ejpam-6568	523	14	,	,	PUNCT
ejpam-6568	523	15	6568	6568	NUM
ejpam-6568	523	16	17	17	NUM
ejpam-6568	523	17	of	of	ADP
ejpam-6568	523	18	18	18	NUM
ejpam-6568	523	19	and	and	CCONJ
ejpam-6568	523	20	hence	hence	ADV
ejpam-6568	523	21	cl⋆(f−1(b	cl⋆(f−1(b	PROPN
ejpam-6568	523	22	)	)	PUNCT
ejpam-6568	523	23	)	)	PUNCT
ejpam-6568	524	1	⊆	⊆	NUM
ejpam-6568	524	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	524	3	,	,	PUNCT
ejpam-6568	524	4	σ2)θ	σ2)θ	NOUN
ejpam-6568	524	5	-	-	PUNCT
ejpam-6568	524	6	cl(b	cl(b	NOUN
ejpam-6568	524	7	)	)	PUNCT
ejpam-6568	524	8	)	)	PUNCT
ejpam-6568	524	9	.	.	PUNCT
ejpam-6568	525	1	(	(	PUNCT
ejpam-6568	525	2	3	3	X
ejpam-6568	525	3	)	)	PUNCT
ejpam-6568	525	4	⇒	⇒	NOUN
ejpam-6568	525	5	(	(	PUNCT
ejpam-6568	525	6	1	1	NUM
ejpam-6568	525	7	):	):	PUNCT
ejpam-6568	525	8	let	let	VERB
ejpam-6568	525	9	x	x	PUNCT
ejpam-6568	525	10	∈	∈	PROPN
ejpam-6568	525	11	x	x	X
ejpam-6568	525	12	and	and	CCONJ
ejpam-6568	525	13	v	v	X
ejpam-6568	525	14	be	be	AUX
ejpam-6568	525	15	any	any	DET
ejpam-6568	525	16	σ1σ2	σ1σ2	NOUN
ejpam-6568	525	17	-	-	ADJ
ejpam-6568	525	18	open	open	ADJ
ejpam-6568	525	19	set	set	NOUN
ejpam-6568	525	20	of	of	ADP
ejpam-6568	525	21	y	y	PROPN
ejpam-6568	525	22	containing	contain	VERB
ejpam-6568	525	23	f(x	f(x	PROPN
ejpam-6568	525	24	)	)	PUNCT
ejpam-6568	525	25	.	.	PUNCT
ejpam-6568	526	1	since	since	SCONJ
ejpam-6568	526	2	σ1σ2	σ1σ2	NOUN
ejpam-6568	526	3	-	-	PUNCT
ejpam-6568	526	4	cl(v	cl(v	NOUN
ejpam-6568	526	5	)	)	PUNCT
ejpam-6568	526	6	∩	∩	NOUN
ejpam-6568	526	7	(	(	PUNCT
ejpam-6568	526	8	y	y	PROPN
ejpam-6568	526	9	−	−	PROPN
ejpam-6568	526	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	526	11	-	-	NUM
ejpam-6568	526	12	cl(v	cl(v	NOUN
ejpam-6568	526	13	)	)	PUNCT
ejpam-6568	526	14	)	)	PUNCT
ejpam-6568	526	15	=	=	NOUN
ejpam-6568	526	16	∅	∅	NOUN
ejpam-6568	526	17	,	,	PUNCT
ejpam-6568	526	18	f(x	f(x	PROPN
ejpam-6568	526	19	)	)	PUNCT
ejpam-6568	526	20	̸∈	̸∈	PROPN
ejpam-6568	526	21	(	(	PUNCT
ejpam-6568	526	22	σ1	σ1	PROPN
ejpam-6568	526	23	,	,	PUNCT
ejpam-6568	526	24	σ2)θ	σ2)θ	NOUN
ejpam-6568	526	25	-	-	PUNCT
ejpam-6568	526	26	cl(y	cl(y	NOUN
ejpam-6568	526	27	−	−	NOUN
ejpam-6568	526	28	σ1σ2	σ1σ2	NOUN
ejpam-6568	526	29	-	-	NUM
ejpam-6568	526	30	cl(v	cl(v	NOUN
ejpam-6568	526	31	)	)	PUNCT
ejpam-6568	526	32	)	)	PUNCT
ejpam-6568	526	33	and	and	CCONJ
ejpam-6568	526	34	hence	hence	ADV
ejpam-6568	526	35	x	x	X
ejpam-6568	526	36	̸∈	̸∈	PROPN
ejpam-6568	526	37	f−1((σ1	f−1((σ1	NOUN
ejpam-6568	526	38	,	,	PUNCT
ejpam-6568	526	39	σ2)θ	σ2)θ	ADJ
ejpam-6568	526	40	-	-	PUNCT
ejpam-6568	526	41	cl(y	cl(y	NOUN
ejpam-6568	526	42	−	−	NOUN
ejpam-6568	526	43	σ1σ2	σ1σ2	NOUN
ejpam-6568	526	44	-	-	NUM
ejpam-6568	526	45	cl(v	cl(v	NOUN
ejpam-6568	526	46	)	)	PUNCT
ejpam-6568	526	47	)	)	PUNCT
ejpam-6568	526	48	)	)	PUNCT
ejpam-6568	526	49	.	.	PUNCT
ejpam-6568	527	1	by	by	ADP
ejpam-6568	527	2	(	(	PUNCT
ejpam-6568	527	3	3	3	NUM
ejpam-6568	527	4	)	)	PUNCT
ejpam-6568	527	5	,	,	PUNCT
ejpam-6568	527	6	x	x	PROPN
ejpam-6568	527	7	̸∈	̸∈	PROPN
ejpam-6568	527	8	cl⋆(f−1(y	cl⋆(f−1(y	PROPN
ejpam-6568	527	9	−	−	VERB
ejpam-6568	527	10	σ1σ2	σ1σ2	NOUN
ejpam-6568	527	11	-	-	NUM
ejpam-6568	527	12	cl(v	cl(v	NOUN
ejpam-6568	527	13	)	)	PUNCT
ejpam-6568	527	14	)	)	PUNCT
ejpam-6568	527	15	)	)	PUNCT
ejpam-6568	528	1	and	and	CCONJ
ejpam-6568	528	2	there	there	PRON
ejpam-6568	528	3	exists	exist	VERB
ejpam-6568	528	4	a	a	DET
ejpam-6568	528	5	⋆-open	⋆-open	ADJ
ejpam-6568	528	6	set	set	NOUN
ejpam-6568	528	7	u	u	NOUN
ejpam-6568	528	8	of	of	ADP
ejpam-6568	528	9	x	x	PUNCT
ejpam-6568	528	10	containing	contain	VERB
ejpam-6568	528	11	x	x	PUNCT
ejpam-6568	528	12	such	such	ADJ
ejpam-6568	528	13	that	that	SCONJ
ejpam-6568	528	14	u	u	PROPN
ejpam-6568	528	15	∩	∩	NOUN
ejpam-6568	528	16	f−1(y	f−1(y	NOUN
ejpam-6568	528	17	−	−	PUNCT
ejpam-6568	528	18	σ1σ2	σ1σ2	NOUN
ejpam-6568	528	19	-	-	NUM
ejpam-6568	528	20	cl(v	cl(v	NOUN
ejpam-6568	528	21	)	)	PUNCT
ejpam-6568	528	22	)	)	PUNCT
ejpam-6568	529	1	=	=	NOUN
ejpam-6568	529	2	∅	∅	NOUN
ejpam-6568	529	3	;	;	PUNCT
ejpam-6568	529	4	hence	hence	ADV
ejpam-6568	529	5	f(u	f(u	ADJ
ejpam-6568	529	6	)	)	PUNCT
ejpam-6568	529	7	∩	∩	NOUN
ejpam-6568	529	8	(	(	PUNCT
ejpam-6568	529	9	y	y	PROPN
ejpam-6568	529	10	−	−	PROPN
ejpam-6568	529	11	σ1σ2	σ1σ2	NOUN
ejpam-6568	529	12	-	-	NUM
ejpam-6568	529	13	cl(v	cl(v	NOUN
ejpam-6568	529	14	)	)	PUNCT
ejpam-6568	529	15	)	)	PUNCT
ejpam-6568	530	1	=	=	PUNCT
ejpam-6568	530	2	∅.	∅.	ADP
ejpam-6568	530	3	thus	thus	ADV
ejpam-6568	530	4	,	,	PUNCT
ejpam-6568	530	5	f(u	f(u	PROPN
ejpam-6568	530	6	)	)	PUNCT
ejpam-6568	530	7	⊆	⊆	NUM
ejpam-6568	530	8	σ1σ2	σ1σ2	NOUN
ejpam-6568	530	9	-	-	NUM
ejpam-6568	530	10	cl(v	cl(v	NOUN
ejpam-6568	530	11	)	)	PUNCT
ejpam-6568	530	12	and	and	CCONJ
ejpam-6568	530	13	so	so	ADV
ejpam-6568	530	14	f	f	PROPN
ejpam-6568	530	15	is	be	AUX
ejpam-6568	530	16	weakly	weakly	ADJ
ejpam-6568	530	17	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	530	18	,	,	PUNCT
ejpam-6568	530	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6568	530	20	at	at	ADP
ejpam-6568	530	21	x.	x.	NOUN
ejpam-6568	530	22	this	this	PRON
ejpam-6568	530	23	shows	show	VERB
ejpam-6568	530	24	that	that	SCONJ
ejpam-6568	530	25	f	f	PROPN
ejpam-6568	530	26	is	be	AUX
ejpam-6568	530	27	weakly	weakly	ADJ
ejpam-6568	530	28	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6568	530	29	,	,	PUNCT
ejpam-6568	530	30	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6568	530	31	.	.	PUNCT
ejpam-6568	530	32	acknowledgements	acknowledgement	NOUN
ejpam-6568	530	33	this	this	DET
ejpam-6568	530	34	research	research	NOUN
ejpam-6568	530	35	project	project	NOUN
ejpam-6568	530	36	was	be	AUX
ejpam-6568	530	37	financially	financially	ADV
ejpam-6568	530	38	supported	support	VERB
ejpam-6568	530	39	by	by	ADP
ejpam-6568	530	40	mahasarakham	mahasarakham	PROPN
ejpam-6568	530	41	university	university	PROPN
ejpam-6568	530	42	.	.	PUNCT
ejpam-6568	531	1	references	reference	NOUN
ejpam-6568	531	2	[	[	X
ejpam-6568	531	3	1	1	NUM
ejpam-6568	531	4	]	]	PUNCT
ejpam-6568	531	5	m.	m.	NOUN
ejpam-6568	531	6	k.	k.	PROPN
ejpam-6568	531	7	singal	singal	PROPN
ejpam-6568	531	8	and	and	CCONJ
ejpam-6568	531	9	a.	a.	PROPN
ejpam-6568	531	10	r.	r.	PROPN
ejpam-6568	531	11	singal	singal	PROPN
ejpam-6568	531	12	.	.	PUNCT
ejpam-6568	532	1	almost	almost	ADV
ejpam-6568	532	2	continuous	continuous	ADJ
ejpam-6568	532	3	mappings	mapping	NOUN
ejpam-6568	532	4	.	.	PUNCT
ejpam-6568	533	1	yokohama	yokohama	PROPN
ejpam-6568	533	2	mathematical	mathematical	PROPN
ejpam-6568	533	3	journal	journal	PROPN
ejpam-6568	533	4	,	,	PUNCT
ejpam-6568	533	5	16:63–73	16:63–73	PROPN
ejpam-6568	533	6	,	,	PUNCT
ejpam-6568	533	7	1968	1968	NUM
ejpam-6568	533	8	.	.	PUNCT
ejpam-6568	534	1	[	[	X
ejpam-6568	534	2	2	2	NUM
ejpam-6568	534	3	]	]	X
ejpam-6568	534	4	b.	b.	PROPN
ejpam-6568	534	5	m.	m.	PROPN
ejpam-6568	534	6	munshi	munshi	PROPN
ejpam-6568	534	7	and	and	CCONJ
ejpam-6568	534	8	d.	d.	PROPN
ejpam-6568	534	9	s.	s.	PROPN
ejpam-6568	534	10	bassan	bassan	PROPN
ejpam-6568	534	11	.	.	PUNCT
ejpam-6568	535	1	almost	almost	ADV
ejpam-6568	535	2	semi	semi	ADJ
ejpam-6568	535	3	-	-	ADJ
ejpam-6568	535	4	continuous	continuous	ADJ
ejpam-6568	535	5	mappings	mapping	NOUN
ejpam-6568	535	6	.	.	PUNCT
ejpam-6568	536	1	the	the	DET
ejpam-6568	536	2	mathematics	mathematics	PROPN
ejpam-6568	536	3	student	student	NOUN
ejpam-6568	536	4	,	,	PUNCT
ejpam-6568	536	5	49:239–248	49:239–248	PROPN
ejpam-6568	536	6	,	,	PUNCT
ejpam-6568	536	7	1981	1981	NUM
ejpam-6568	536	8	.	.	PUNCT
ejpam-6568	537	1	[	[	X
ejpam-6568	537	2	3	3	X
ejpam-6568	537	3	]	]	PUNCT
ejpam-6568	537	4	t.	t.	PROPN
ejpam-6568	537	5	noiri	noiri	PROPN
ejpam-6568	537	6	.	.	PUNCT
ejpam-6568	538	1	almost	almost	ADV
ejpam-6568	538	2	α	α	NUM
ejpam-6568	538	3	-	-	ADJ
ejpam-6568	538	4	continuous	continuous	ADJ
ejpam-6568	538	5	functions	function	NOUN
ejpam-6568	538	6	.	.	PUNCT
ejpam-6568	539	1	kyungpook	kyungpook	PROPN
ejpam-6568	539	2	mathematical	mathematical	PROPN
ejpam-6568	539	3	journal	journal	PROPN
ejpam-6568	539	4	,	,	PUNCT
ejpam-6568	539	5	28:71–77	28:71–77	PROPN
ejpam-6568	539	6	,	,	PUNCT
ejpam-6568	539	7	1988	1988	NUM
ejpam-6568	539	8	.	.	PUNCT
ejpam-6568	540	1	[	[	X
ejpam-6568	540	2	4	4	X
ejpam-6568	540	3	]	]	PUNCT
ejpam-6568	540	4	a.	a.	NOUN
ejpam-6568	540	5	a.	a.	NOUN
ejpam-6568	540	6	nasef	nasef	PROPN
ejpam-6568	540	7	and	and	CCONJ
ejpam-6568	540	8	t.	t.	PROPN
ejpam-6568	540	9	noiri	noiri	PROPN
ejpam-6568	540	10	.	.	PUNCT
ejpam-6568	541	1	some	some	DET
ejpam-6568	541	2	weak	weak	ADJ
ejpam-6568	541	3	forms	form	NOUN
ejpam-6568	541	4	of	of	ADP
ejpam-6568	541	5	almost	almost	ADV
ejpam-6568	541	6	continuity	continuity	NOUN
ejpam-6568	541	7	.	.	PUNCT
ejpam-6568	542	1	acta	acta	PROPN
ejpam-6568	542	2	mathematica	mathematica	PROPN
ejpam-6568	542	3	hungarica	hungarica	PROPN
ejpam-6568	542	4	,	,	PUNCT
ejpam-6568	542	5	74(3):211–219	74(3):211–219	PROPN
ejpam-6568	542	6	,	,	PUNCT
ejpam-6568	542	7	1997	1997	NUM
ejpam-6568	542	8	.	.	PUNCT
ejpam-6568	543	1	[	[	X
ejpam-6568	543	2	5	5	NUM
ejpam-6568	543	3	]	]	X
ejpam-6568	543	4	n.	n.	PROPN
ejpam-6568	543	5	levine	levine	PROPN
ejpam-6568	543	6	.	.	PUNCT
ejpam-6568	544	1	a	a	DET
ejpam-6568	544	2	decomposition	decomposition	NOUN
ejpam-6568	544	3	of	of	ADP
ejpam-6568	544	4	continuity	continuity	NOUN
ejpam-6568	544	5	in	in	ADP
ejpam-6568	544	6	topological	topological	ADJ
ejpam-6568	544	7	spaces	space	NOUN
ejpam-6568	544	8	.	.	PUNCT
ejpam-6568	545	1	the	the	DET
ejpam-6568	545	2	american	american	PROPN
ejpam-6568	545	3	mathematical	mathematical	PROPN
ejpam-6568	545	4	monthly	monthly	ADV
ejpam-6568	545	5	,	,	PUNCT
ejpam-6568	545	6	68:44–46	68:44–46	NUM
ejpam-6568	545	7	,	,	PUNCT
ejpam-6568	545	8	1961	1961	NUM
ejpam-6568	545	9	.	.	PUNCT
ejpam-6568	546	1	[	[	X
ejpam-6568	546	2	6	6	NUM
ejpam-6568	546	3	]	]	PUNCT
ejpam-6568	546	4	t.	t.	PROPN
ejpam-6568	546	5	husain	husain	PROPN
ejpam-6568	546	6	.	.	PUNCT
ejpam-6568	547	1	almost	almost	ADV
ejpam-6568	547	2	continuous	continuous	ADJ
ejpam-6568	547	3	mappings	mapping	NOUN
ejpam-6568	547	4	.	.	PUNCT
ejpam-6568	548	1	prace	prace	PROPN
ejpam-6568	548	2	matematyczne	matematyczne	PROPN
ejpam-6568	548	3	,	,	PUNCT
ejpam-6568	548	4	10:1–7	10:1–7	NUM
ejpam-6568	548	5	,	,	PUNCT
ejpam-6568	548	6	1966	1966	NUM
ejpam-6568	548	7	.	.	PUNCT
ejpam-6568	549	1	[	[	X
ejpam-6568	549	2	7	7	X
ejpam-6568	549	3	]	]	X
ejpam-6568	550	1	d.	d.	PROPN
ejpam-6568	550	2	s.	s.	PROPN
ejpam-6568	550	3	janković.	janković.	PROPN
ejpam-6568	550	4	θ	θ	PROPN
ejpam-6568	550	5	-	-	ADJ
ejpam-6568	550	6	regular	regular	ADJ
ejpam-6568	550	7	spaces	space	NOUN
ejpam-6568	550	8	.	.	PUNCT
ejpam-6568	551	1	international	international	ADJ
ejpam-6568	551	2	journal	journal	PROPN
ejpam-6568	551	3	of	of	ADP
ejpam-6568	551	4	mathematics	mathematics	PROPN
ejpam-6568	551	5	and	and	CCONJ
ejpam-6568	551	6	mathematical	mathematical	ADJ
ejpam-6568	551	7	sciences	science	NOUN
ejpam-6568	551	8	,	,	PUNCT
ejpam-6568	551	9	8:615–619	8:615–619	NUM
ejpam-6568	551	10	,	,	PUNCT
ejpam-6568	551	11	1985	1985	NUM
ejpam-6568	551	12	.	.	PUNCT
ejpam-6568	552	1	[	[	X
ejpam-6568	552	2	8	8	NUM
ejpam-6568	552	3	]	]	PUNCT
ejpam-6568	552	4	t.	t.	PROPN
ejpam-6568	552	5	noiri	noiri	PROPN
ejpam-6568	552	6	.	.	PUNCT
ejpam-6568	553	1	properties	property	NOUN
ejpam-6568	553	2	of	of	ADP
ejpam-6568	553	3	some	some	DET
ejpam-6568	553	4	weak	weak	ADJ
ejpam-6568	553	5	forms	form	NOUN
ejpam-6568	553	6	of	of	ADP
ejpam-6568	553	7	continuity	continuity	NOUN
ejpam-6568	553	8	.	.	PUNCT
ejpam-6568	554	1	international	international	ADJ
ejpam-6568	554	2	journal	journal	PROPN
ejpam-6568	554	3	of	of	ADP
ejpam-6568	554	4	mathematics	mathematics	PROPN
ejpam-6568	554	5	and	and	CCONJ
ejpam-6568	554	6	mathematical	mathematical	ADJ
ejpam-6568	554	7	sciences	science	NOUN
ejpam-6568	554	8	,	,	PUNCT
ejpam-6568	554	9	10(1):97–111	10(1):97–111	NUM
ejpam-6568	554	10	,	,	PUNCT
ejpam-6568	554	11	1987	1987	NUM
ejpam-6568	554	12	.	.	PUNCT
ejpam-6568	555	1	[	[	X
ejpam-6568	555	2	9	9	NUM
ejpam-6568	555	3	]	]	X
ejpam-6568	555	4	d.	d.	PROPN
ejpam-6568	555	5	a.	a.	PROPN
ejpam-6568	555	6	rose	rise	VERB
ejpam-6568	555	7	.	.	PUNCT
ejpam-6568	556	1	weak	weak	ADJ
ejpam-6568	556	2	continuity	continuity	NOUN
ejpam-6568	556	3	and	and	CCONJ
ejpam-6568	556	4	almost	almost	ADV
ejpam-6568	556	5	continuity	continuity	NOUN
ejpam-6568	556	6	.	.	PUNCT
ejpam-6568	557	1	international	international	ADJ
ejpam-6568	557	2	journal	journal	PROPN
ejpam-6568	557	3	of	of	ADP
ejpam-6568	557	4	mathematics	mathematics	PROPN
ejpam-6568	557	5	and	and	CCONJ
ejpam-6568	557	6	mathematical	mathematical	ADJ
ejpam-6568	557	7	sciences	science	NOUN
ejpam-6568	557	8	,	,	PUNCT
ejpam-6568	557	9	7:311–318	7:311–318	PROPN
ejpam-6568	557	10	,	,	PUNCT
ejpam-6568	557	11	1984	1984	NUM
ejpam-6568	557	12	.	.	PUNCT
ejpam-6568	558	1	[	[	X
ejpam-6568	558	2	10	10	NUM
ejpam-6568	558	3	]	]	X
ejpam-6568	558	4	v.	v.	CCONJ
ejpam-6568	558	5	popa	popa	NOUN
ejpam-6568	558	6	and	and	CCONJ
ejpam-6568	558	7	t.	t.	PROPN
ejpam-6568	558	8	noiri	noiri	PROPN
ejpam-6568	558	9	.	.	PUNCT
ejpam-6568	559	1	on	on	ADP
ejpam-6568	559	2	weakly	weakly	ADJ
ejpam-6568	559	3	(	(	PUNCT
ejpam-6568	559	4	τ	τ	PROPN
ejpam-6568	559	5	,	,	PUNCT
ejpam-6568	559	6	m)-continuous	m)-continuous	ADJ
ejpam-6568	559	7	functions	function	NOUN
ejpam-6568	559	8	.	.	PUNCT
ejpam-6568	560	1	rendiconti	rendiconti	ADJ
ejpam-6568	560	2	del	del	PROPN
ejpam-6568	560	3	circolo	circolo	PROPN
ejpam-6568	560	4	matematico	matematico	NOUN
ejpam-6568	560	5	di	di	PROPN
ejpam-6568	560	6	palermo	palermo	PROPN
ejpam-6568	560	7	series	series	PROPN
ejpam-6568	560	8	2	2	NUM
ejpam-6568	560	9	,	,	PUNCT
ejpam-6568	560	10	51:295–316	51:295–316	NUM
ejpam-6568	560	11	,	,	PUNCT
ejpam-6568	560	12	2002	2002	NUM
ejpam-6568	560	13	.	.	PUNCT
ejpam-6568	561	1	[	[	X
ejpam-6568	561	2	11	11	NUM
ejpam-6568	561	3	]	]	X
ejpam-6568	561	4	e.	e.	PROPN
ejpam-6568	561	5	ekici	ekici	PROPN
ejpam-6568	561	6	,	,	PUNCT
ejpam-6568	561	7	s.	s.	PROPN
ejpam-6568	561	8	jafari	jafari	PROPN
ejpam-6568	561	9	,	,	PUNCT
ejpam-6568	561	10	m.	m.	PROPN
ejpam-6568	561	11	caldas	caldas	PROPN
ejpam-6568	561	12	,	,	PUNCT
ejpam-6568	561	13	and	and	CCONJ
ejpam-6568	561	14	t.	t.	PROPN
ejpam-6568	561	15	noiri	noiri	PROPN
ejpam-6568	561	16	.	.	PUNCT
ejpam-6568	562	1	weakly	weakly	ADJ
ejpam-6568	562	2	λ	λ	ADJ
ejpam-6568	562	3	-	-	ADJ
ejpam-6568	562	4	continuous	continuous	ADJ
ejpam-6568	562	5	functions	function	NOUN
ejpam-6568	562	6	.	.	PUNCT
ejpam-6568	563	1	novi	novi	PROPN
ejpam-6568	563	2	sad	sad	PROPN
ejpam-6568	563	3	journal	journal	PROPN
ejpam-6568	563	4	of	of	ADP
ejpam-6568	563	5	mathematics	mathematic	NOUN
ejpam-6568	563	6	,	,	PUNCT
ejpam-6568	563	7	38:47–56	38:47–56	NUM
ejpam-6568	563	8	,	,	PUNCT
ejpam-6568	563	9	2008	2008	NUM
ejpam-6568	563	10	.	.	PUNCT
ejpam-6568	564	1	[	[	X
ejpam-6568	564	2	12	12	NUM
ejpam-6568	564	3	]	]	PUNCT
ejpam-6568	564	4	m.	m.	NOUN
ejpam-6568	564	5	e.	e.	PROPN
ejpam-6568	564	6	abd	abd	PROPN
ejpam-6568	565	1	el	el	PROPN
ejpam-6568	565	2	-	-	PROPN
ejpam-6568	565	3	monsef	monsef	PROPN
ejpam-6568	565	4	,	,	PUNCT
ejpam-6568	565	5	e.	e.	PROPN
ejpam-6568	565	6	f.	f.	PROPN
ejpam-6568	565	7	lashien	lashien	PROPN
ejpam-6568	565	8	,	,	PUNCT
ejpam-6568	565	9	and	and	CCONJ
ejpam-6568	565	10	a.	a.	NOUN
ejpam-6568	565	11	a.	a.	NOUN
ejpam-6568	565	12	nasef	nasef	PROPN
ejpam-6568	565	13	.	.	PUNCT
ejpam-6568	566	1	on	on	ADP
ejpam-6568	566	2	i	i	PRON
ejpam-6568	566	3	-open	-open	PROPN
ejpam-6568	566	4	sets	set	NOUN
ejpam-6568	566	5	and	and	CCONJ
ejpam-6568	566	6	i	i	PRON
ejpam-6568	566	7	continuous	continuous	ADJ
ejpam-6568	566	8	functions	function	NOUN
ejpam-6568	566	9	.	.	PUNCT
ejpam-6568	567	1	kyungpook	kyungpook	PROPN
ejpam-6568	567	2	mathematical	mathematical	PROPN
ejpam-6568	567	3	journal	journal	PROPN
ejpam-6568	567	4	,	,	PUNCT
ejpam-6568	567	5	32:21–30	32:21–30	NUM
ejpam-6568	567	6	,	,	PUNCT
ejpam-6568	567	7	1992	1992	NUM
ejpam-6568	567	8	.	.	PUNCT
ejpam-6568	568	1	[	[	X
ejpam-6568	568	2	13	13	NUM
ejpam-6568	568	3	]	]	X
ejpam-6568	568	4	e.	e.	PROPN
ejpam-6568	568	5	hatir	hatir	PROPN
ejpam-6568	568	6	and	and	CCONJ
ejpam-6568	568	7	t.	t.	PROPN
ejpam-6568	568	8	noiri	noiri	PROPN
ejpam-6568	568	9	.	.	PUNCT
ejpam-6568	569	1	weakly	weakly	ADJ
ejpam-6568	569	2	pre	pre	ADJ
ejpam-6568	569	3	-	-	ADJ
ejpam-6568	569	4	i	i	PRON
ejpam-6568	569	5	-	-	PUNCT
ejpam-6568	569	6	open	open	ADJ
ejpam-6568	569	7	sets	set	NOUN
ejpam-6568	569	8	and	and	CCONJ
ejpam-6568	569	9	decomposition	decomposition	NOUN
ejpam-6568	569	10	of	of	ADP
ejpam-6568	569	11	continuity	continuity	NOUN
ejpam-6568	569	12	.	.	PUNCT
ejpam-6568	570	1	acta	acta	PROPN
ejpam-6568	570	2	mathematica	mathematica	PROPN
ejpam-6568	570	3	hungarica	hungarica	PROPN
ejpam-6568	570	4	,	,	PUNCT
ejpam-6568	570	5	106(3):227–238	106(3):227–238	NUM
ejpam-6568	570	6	,	,	PUNCT
ejpam-6568	570	7	2005	2005	NUM
ejpam-6568	570	8	.	.	PUNCT
ejpam-6568	571	1	n.	n.	PROPN
ejpam-6568	571	2	viriyapong	viriyapong	PROPN
ejpam-6568	571	3	,	,	PUNCT
ejpam-6568	571	4	a.	a.	PROPN
ejpam-6568	571	5	sama	sama	PROPN
ejpam-6568	571	6	-	-	PUNCT
ejpam-6568	571	7	ae	ae	PROPN
ejpam-6568	571	8	,	,	PUNCT
ejpam-6568	571	9	c.	c.	PROPN
ejpam-6568	571	10	boonpok	boonpok	PROPN
ejpam-6568	571	11	/	/	SYM
ejpam-6568	571	12	eur	eur	PROPN
ejpam-6568	571	13	.	.	PUNCT
ejpam-6568	572	1	j.	j.	PROPN
ejpam-6568	572	2	pure	pure	PROPN
ejpam-6568	572	3	appl	appl	PROPN
ejpam-6568	572	4	.	.	PROPN
ejpam-6568	572	5	math	math	PROPN
ejpam-6568	572	6	,	,	PUNCT
ejpam-6568	572	7	18	18	NUM
ejpam-6568	572	8	(	(	PUNCT
ejpam-6568	572	9	3	3	NUM
ejpam-6568	572	10	)	)	PUNCT
ejpam-6568	572	11	(	(	PUNCT
ejpam-6568	572	12	2025	2025	NUM
ejpam-6568	572	13	)	)	PUNCT
ejpam-6568	572	14	,	,	PUNCT
ejpam-6568	572	15	6568	6568	NUM
ejpam-6568	572	16	18	18	NUM
ejpam-6568	572	17	of	of	ADP
ejpam-6568	572	18	18	18	NUM
ejpam-6568	572	19	[	[	SYM
ejpam-6568	572	20	14	14	NUM
ejpam-6568	572	21	]	]	X
ejpam-6568	572	22	e.	e.	PROPN
ejpam-6568	572	23	hatir	hatir	PROPN
ejpam-6568	572	24	and	and	CCONJ
ejpam-6568	572	25	t.	t.	PROPN
ejpam-6568	572	26	noiri	noiri	PROPN
ejpam-6568	572	27	.	.	PUNCT
ejpam-6568	573	1	on	on	ADP
ejpam-6568	573	2	semi	semi	ADJ
ejpam-6568	573	3	-	-	ADJ
ejpam-6568	573	4	i	i	PRON
ejpam-6568	573	5	-	-	PUNCT
ejpam-6568	573	6	open	open	ADJ
ejpam-6568	573	7	sets	set	NOUN
ejpam-6568	573	8	and	and	CCONJ
ejpam-6568	573	9	semi	semi	ADJ
ejpam-6568	573	10	-	-	ADJ
ejpam-6568	573	11	i	i	ADV
ejpam-6568	573	12	-	-	PUNCT
ejpam-6568	573	13	continuous	continuous	ADJ
ejpam-6568	573	14	functions	function	NOUN
ejpam-6568	573	15	.	.	PUNCT
ejpam-6568	574	1	acta	acta	PROPN
ejpam-6568	574	2	mathematica	mathematica	PROPN
ejpam-6568	574	3	hungarica	hungarica	PROPN
ejpam-6568	574	4	,	,	PUNCT
ejpam-6568	574	5	107(4):345–353	107(4):345–353	NUM
ejpam-6568	574	6	,	,	PUNCT
ejpam-6568	574	7	2005	2005	NUM
ejpam-6568	574	8	.	.	PUNCT
ejpam-6568	575	1	[	[	X
ejpam-6568	575	2	15	15	NUM
ejpam-6568	575	3	]	]	X
ejpam-6568	575	4	e.	e.	PROPN
ejpam-6568	575	5	hatir	hatir	PROPN
ejpam-6568	575	6	and	and	CCONJ
ejpam-6568	575	7	t.	t.	PROPN
ejpam-6568	575	8	noiri	noiri	PROPN
ejpam-6568	575	9	.	.	PUNCT
ejpam-6568	576	1	on	on	ADP
ejpam-6568	576	2	decompositions	decomposition	NOUN
ejpam-6568	576	3	of	of	ADP
ejpam-6568	576	4	continuity	continuity	NOUN
ejpam-6568	576	5	via	via	ADP
ejpam-6568	576	6	idealization	idealization	NOUN
ejpam-6568	576	7	.	.	PUNCT
ejpam-6568	577	1	acta	acta	PROPN
ejpam-6568	577	2	mathematica	mathematica	PROPN
ejpam-6568	577	3	hungarica	hungarica	PROPN
ejpam-6568	577	4	,	,	PUNCT
ejpam-6568	577	5	96:341–349	96:341–349	PROPN
ejpam-6568	577	6	,	,	PUNCT
ejpam-6568	577	7	2002	2002	NUM
ejpam-6568	577	8	.	.	PUNCT
ejpam-6568	578	1	[	[	X
ejpam-6568	578	2	16	16	NUM
ejpam-6568	578	3	]	]	X
ejpam-6568	578	4	c.	c.	PROPN
ejpam-6568	578	5	boonpok	boonpok	PROPN
ejpam-6568	578	6	and	and	CCONJ
ejpam-6568	578	7	p.	p.	NOUN
ejpam-6568	578	8	pue	pue	NOUN
ejpam-6568	578	9	-	-	PUNCT
ejpam-6568	578	10	on	on	ADP
ejpam-6568	578	11	.	.	PUNCT
ejpam-6568	579	1	characterizations	characterization	NOUN
ejpam-6568	579	2	of	of	ADP
ejpam-6568	579	3	almost	almost	ADV
ejpam-6568	579	4	(	(	PUNCT
ejpam-6568	579	5	τ1	τ1	NOUN
ejpam-6568	579	6	,	,	PUNCT
ejpam-6568	579	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6568	579	8	multifunctions	multifunction	NOUN
ejpam-6568	579	9	.	.	PUNCT
ejpam-6568	580	1	international	international	ADJ
ejpam-6568	580	2	journal	journal	NOUN
ejpam-6568	580	3	of	of	ADP
ejpam-6568	580	4	analysis	analysis	NOUN
ejpam-6568	580	5	and	and	CCONJ
ejpam-6568	580	6	applications	application	NOUN
ejpam-6568	580	7	,	,	PUNCT
ejpam-6568	580	8	22:33	22:33	NUM
ejpam-6568	580	9	,	,	PUNCT
ejpam-6568	580	10	2024	2024	NUM
ejpam-6568	580	11	.	.	PUNCT
ejpam-6568	581	1	[	[	X
ejpam-6568	581	2	17	17	NUM
ejpam-6568	581	3	]	]	PUNCT
ejpam-6568	581	4	c.	c.	PROPN
ejpam-6568	581	5	boonpok	boonpok	PROPN
ejpam-6568	581	6	.	.	PUNCT
ejpam-6568	582	1	on	on	ADP
ejpam-6568	582	2	characterizations	characterization	NOUN
ejpam-6568	582	3	of	of	ADP
ejpam-6568	582	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-6568	582	5	ideal	ideal	ADJ
ejpam-6568	582	6	topological	topological	ADJ
ejpam-6568	582	7	spaces	space	NOUN
ejpam-6568	582	8	.	.	PUNCT
ejpam-6568	583	1	journal	journal	NOUN
ejpam-6568	583	2	of	of	ADP
ejpam-6568	583	3	mathematics	mathematic	NOUN
ejpam-6568	583	4	,	,	PUNCT
ejpam-6568	583	5	2020:9387601	2020:9387601	NUM
ejpam-6568	583	6	,	,	PUNCT
ejpam-6568	583	7	2020	2020	NUM
ejpam-6568	583	8	.	.	PUNCT
ejpam-6568	584	1	[	[	X
ejpam-6568	584	2	18	18	NUM
ejpam-6568	584	3	]	]	PUNCT
ejpam-6568	584	4	c.	c.	PROPN
ejpam-6568	584	5	boonpok	boonpok	PROPN
ejpam-6568	584	6	.	.	PUNCT
ejpam-6568	585	1	weak	weak	ADJ
ejpam-6568	585	2	openness	openness	NOUN
ejpam-6568	585	3	and	and	CCONJ
ejpam-6568	585	4	weak	weak	ADJ
ejpam-6568	585	5	continuity	continuity	NOUN
ejpam-6568	585	6	in	in	ADP
ejpam-6568	585	7	ideal	ideal	ADJ
ejpam-6568	585	8	topological	topological	ADJ
ejpam-6568	585	9	spaces	space	NOUN
ejpam-6568	585	10	.	.	PUNCT
ejpam-6568	586	1	mathematica	mathematica	PROPN
ejpam-6568	586	2	,	,	PUNCT
ejpam-6568	586	3	64(2):173–185	64(2):173–185	NOUN
ejpam-6568	586	4	,	,	PUNCT
ejpam-6568	586	5	2022	2022	NUM
ejpam-6568	586	6	.	.	PUNCT
ejpam-6568	587	1	[	[	X
ejpam-6568	587	2	19	19	NUM
ejpam-6568	587	3	]	]	PUNCT
ejpam-6568	587	4	c.	c.	PROPN
ejpam-6568	587	5	boonpok	boonpok	PROPN
ejpam-6568	587	6	.	.	PUNCT
ejpam-6568	588	1	θ(⋆)-continuity	θ(⋆)-continuity	NOUN
ejpam-6568	588	2	.	.	PUNCT
ejpam-6568	589	1	mathematica	mathematica	PROPN
ejpam-6568	589	2	,	,	PUNCT
ejpam-6568	589	3	65(1):31–42	65(1):31–42	NUM
ejpam-6568	589	4	,	,	PUNCT
ejpam-6568	589	5	2023	2023	NUM
ejpam-6568	589	6	.	.	PUNCT
ejpam-6568	590	1	[	[	X
ejpam-6568	590	2	20	20	NUM
ejpam-6568	590	3	]	]	PUNCT
ejpam-6568	590	4	c.	c.	PROPN
ejpam-6568	590	5	boonpok	boonpok	PROPN
ejpam-6568	590	6	.	.	PUNCT
ejpam-6568	591	1	pı	pı	NOUN
ejpam-6568	591	2	-	-	NOUN
ejpam-6568	591	3	continuity	continuity	NOUN
ejpam-6568	591	4	and	and	CCONJ
ejpam-6568	591	5	weak	weak	ADJ
ejpam-6568	591	6	pı	pı	NOUN
ejpam-6568	591	7	-	-	NOUN
ejpam-6568	591	8	continuity	continuity	NOUN
ejpam-6568	591	9	.	.	PUNCT
ejpam-6568	592	1	carpathian	carpathian	ADJ
ejpam-6568	592	2	mathematical	mathematical	ADJ
ejpam-6568	592	3	publications	publication	NOUN
ejpam-6568	592	4	,	,	PUNCT
ejpam-6568	592	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6568	592	6	,	,	PUNCT
ejpam-6568	592	7	2025	2025	NUM
ejpam-6568	592	8	.	.	PUNCT
ejpam-6568	593	1	[	[	X
ejpam-6568	593	2	21	21	NUM
ejpam-6568	593	3	]	]	X
ejpam-6568	593	4	c.	c.	PROPN
ejpam-6568	593	5	boonpok	boonpok	PROPN
ejpam-6568	593	6	and	and	CCONJ
ejpam-6568	593	7	n.	n.	PROPN
ejpam-6568	593	8	srisarakham	srisarakham	PROPN
ejpam-6568	593	9	.	.	PUNCT
ejpam-6568	594	1	(	(	PUNCT
ejpam-6568	594	2	τ1	τ1	NOUN
ejpam-6568	594	3	,	,	PUNCT
ejpam-6568	594	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6568	594	5	for	for	ADP
ejpam-6568	594	6	functions	function	NOUN
ejpam-6568	594	7	.	.	PUNCT
ejpam-6568	595	1	asia	asia	PROPN
ejpam-6568	595	2	pacific	pacific	PROPN
ejpam-6568	595	3	journal	journal	PROPN
ejpam-6568	595	4	of	of	ADP
ejpam-6568	595	5	mathematics	mathematic	NOUN
ejpam-6568	595	6	,	,	PUNCT
ejpam-6568	595	7	11:21	11:21	NUM
ejpam-6568	595	8	,	,	PUNCT
ejpam-6568	595	9	2024	2024	NUM
ejpam-6568	595	10	.	.	PUNCT
ejpam-6568	596	1	[	[	X
ejpam-6568	596	2	22	22	NUM
ejpam-6568	596	3	]	]	PUNCT
ejpam-6568	596	4	c.	c.	PROPN
ejpam-6568	596	5	boonpok	boonpok	PROPN
ejpam-6568	596	6	and	and	CCONJ
ejpam-6568	596	7	p.	p.	NOUN
ejpam-6568	596	8	pue	pue	NOUN
ejpam-6568	596	9	-	-	PUNCT
ejpam-6568	596	10	on	on	ADP
ejpam-6568	596	11	.	.	PUNCT
ejpam-6568	597	1	characterizations	characterization	NOUN
ejpam-6568	597	2	of	of	ADP
ejpam-6568	597	3	almost	almost	ADV
ejpam-6568	597	4	(	(	PUNCT
ejpam-6568	597	5	τ1	τ1	NOUN
ejpam-6568	597	6	,	,	PUNCT
ejpam-6568	597	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6568	597	8	multifunctions	multifunction	NOUN
ejpam-6568	597	9	.	.	PUNCT
ejpam-6568	598	1	international	international	ADJ
ejpam-6568	598	2	journal	journal	NOUN
ejpam-6568	598	3	of	of	ADP
ejpam-6568	598	4	analysis	analysis	NOUN
ejpam-6568	598	5	and	and	CCONJ
ejpam-6568	598	6	applications	application	NOUN
ejpam-6568	598	7	,	,	PUNCT
ejpam-6568	598	8	22:33	22:33	NUM
ejpam-6568	598	9	,	,	PUNCT
ejpam-6568	598	10	2024	2024	NUM
ejpam-6568	598	11	.	.	PUNCT
ejpam-6568	599	1	[	[	X
ejpam-6568	599	2	23	23	NUM
ejpam-6568	599	3	]	]	X
ejpam-6568	599	4	c.	c.	PROPN
ejpam-6568	599	5	boonpok	boonpok	PROPN
ejpam-6568	599	6	and	and	CCONJ
ejpam-6568	599	7	c.	c.	PROPN
ejpam-6568	599	8	khanarong	khanarong	PROPN
ejpam-6568	599	9	.	.	PUNCT
ejpam-6568	600	1	on	on	ADP
ejpam-6568	600	2	weakly	weakly	ADJ
ejpam-6568	600	3	(	(	PUNCT
ejpam-6568	600	4	τ1	τ1	NOUN
ejpam-6568	600	5	,	,	PUNCT
ejpam-6568	600	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6568	600	7	functions	function	NOUN
ejpam-6568	600	8	.	.	PUNCT
ejpam-6568	601	1	european	european	ADJ
ejpam-6568	601	2	journal	journal	PROPN
ejpam-6568	601	3	of	of	ADP
ejpam-6568	601	4	pure	pure	ADJ
ejpam-6568	601	5	and	and	CCONJ
ejpam-6568	601	6	applied	applied	ADJ
ejpam-6568	601	7	mathematics	mathematic	NOUN
ejpam-6568	601	8	,	,	PUNCT
ejpam-6568	601	9	17(1):416–425	17(1):416–425	NUM
ejpam-6568	601	10	,	,	PUNCT
ejpam-6568	601	11	2024	2024	NUM
ejpam-6568	601	12	.	.	PUNCT
ejpam-6568	602	1	[	[	X
ejpam-6568	602	2	24	24	NUM
ejpam-6568	602	3	]	]	PUNCT
ejpam-6568	602	4	c.	c.	PROPN
ejpam-6568	602	5	boonpok	boonpok	PROPN
ejpam-6568	602	6	,	,	PUNCT
ejpam-6568	602	7	c.	c.	PROPN
ejpam-6568	602	8	viriyapong	viriyapong	PROPN
ejpam-6568	602	9	,	,	PUNCT
ejpam-6568	602	10	and	and	CCONJ
ejpam-6568	602	11	m.	m.	NOUN
ejpam-6568	602	12	thongmoon	thongmoon	NOUN
ejpam-6568	602	13	.	.	PUNCT
ejpam-6568	603	1	on	on	ADP
ejpam-6568	603	2	upper	upper	ADJ
ejpam-6568	603	3	and	and	CCONJ
ejpam-6568	603	4	lower	low	ADJ
ejpam-6568	603	5	(	(	PUNCT
ejpam-6568	603	6	τ1	τ1	NOUN
ejpam-6568	603	7	,	,	PUNCT
ejpam-6568	603	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6568	603	9	multifunctions	multifunction	NOUN
ejpam-6568	603	10	.	.	PUNCT
ejpam-6568	604	1	journal	journal	PROPN
ejpam-6568	604	2	of	of	ADP
ejpam-6568	604	3	mathematics	mathematics	PROPN
ejpam-6568	604	4	and	and	CCONJ
ejpam-6568	604	5	computer	computer	NOUN
ejpam-6568	604	6	science	science	NOUN
ejpam-6568	604	7	,	,	PUNCT
ejpam-6568	604	8	18:282–293	18:282–293	NUM
ejpam-6568	604	9	,	,	PUNCT
ejpam-6568	604	10	2018	2018	NUM
ejpam-6568	604	11	.	.	PUNCT
ejpam-6568	605	1	[	[	X
ejpam-6568	605	2	25	25	NUM
ejpam-6568	605	3	]	]	X
ejpam-6568	605	4	c.	c.	PROPN
ejpam-6568	605	5	viriyapong	viriyapong	PROPN
ejpam-6568	605	6	and	and	CCONJ
ejpam-6568	605	7	c.	c.	PROPN
ejpam-6568	605	8	boonpok	boonpok	PROPN
ejpam-6568	605	9	.	.	PUNCT
ejpam-6568	606	1	(	(	PUNCT
ejpam-6568	606	2	τ1	τ1	NOUN
ejpam-6568	606	3	,	,	PUNCT
ejpam-6568	606	4	τ2)α	τ2)α	NOUN
ejpam-6568	606	5	-	-	PUNCT
ejpam-6568	606	6	continuity	continuity	NOUN
ejpam-6568	606	7	for	for	ADP
ejpam-6568	606	8	multifunctions	multifunction	NOUN
ejpam-6568	606	9	.	.	PUNCT
ejpam-6568	607	1	journal	journal	PROPN
ejpam-6568	607	2	of	of	ADP
ejpam-6568	607	3	mathematics	mathematic	NOUN
ejpam-6568	607	4	,	,	PUNCT
ejpam-6568	607	5	2020:6285763	2020:6285763	NUM
ejpam-6568	607	6	,	,	PUNCT
ejpam-6568	607	7	2020	2020	NUM
ejpam-6568	607	8	.	.	PUNCT
ejpam-6568	608	1	[	[	X
ejpam-6568	608	2	26	26	NUM
ejpam-6568	608	3	]	]	PUNCT
ejpam-6568	608	4	c.	c.	PROPN
ejpam-6568	608	5	boonpok	boonpok	PROPN
ejpam-6568	608	6	.	.	PUNCT
ejpam-6568	609	1	(	(	PUNCT
ejpam-6568	609	2	τ1	τ1	NOUN
ejpam-6568	609	3	,	,	PUNCT
ejpam-6568	609	4	τ2)δ	τ2)δ	ADJ
ejpam-6568	609	5	-	-	PUNCT
ejpam-6568	609	6	semicontinuous	semicontinuous	ADJ
ejpam-6568	609	7	multifunctions	multifunction	NOUN
ejpam-6568	609	8	.	.	PUNCT
ejpam-6568	610	1	heliyon	heliyon	NOUN
ejpam-6568	610	2	,	,	PUNCT
ejpam-6568	610	3	6	6	NUM
ejpam-6568	610	4	:	:	SYM
ejpam-6568	610	5	e05367	e05367	PROPN
ejpam-6568	610	6	,	,	PUNCT
ejpam-6568	610	7	2020	2020	NUM
ejpam-6568	610	8	.	.	PUNCT
ejpam-6568	611	1	[	[	X
ejpam-6568	611	2	27	27	NUM
ejpam-6568	611	3	]	]	X
ejpam-6568	611	4	n.	n.	PROPN
ejpam-6568	611	5	viriyapong	viriyapong	PROPN
ejpam-6568	611	6	,	,	PUNCT
ejpam-6568	611	7	s.	s.	PROPN
ejpam-6568	611	8	sompong	sompong	PROPN
ejpam-6568	611	9	,	,	PUNCT
ejpam-6568	611	10	and	and	CCONJ
ejpam-6568	611	11	c.	c.	PROPN
ejpam-6568	611	12	boonpok	boonpok	PROPN
ejpam-6568	611	13	.	.	PUNCT
ejpam-6568	612	1	(	(	PUNCT
ejpam-6568	612	2	τ1	τ1	NOUN
ejpam-6568	612	3	,	,	PUNCT
ejpam-6568	612	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6568	612	5	disconnectedness	disconnectedness	NOUN
ejpam-6568	612	6	in	in	ADP
ejpam-6568	612	7	bitopological	bitopological	ADJ
ejpam-6568	612	8	spaces	space	NOUN
ejpam-6568	612	9	.	.	PUNCT
ejpam-6568	613	1	international	international	ADJ
ejpam-6568	613	2	journal	journal	PROPN
ejpam-6568	613	3	of	of	ADP
ejpam-6568	613	4	mathematics	mathematic	NOUN
ejpam-6568	613	5	and	and	CCONJ
ejpam-6568	613	6	computer	computer	NOUN
ejpam-6568	613	7	science	science	NOUN
ejpam-6568	613	8	,	,	PUNCT
ejpam-6568	613	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6568	613	10	,	,	PUNCT
ejpam-6568	613	11	2024	2024	NUM
ejpam-6568	613	12	.	.	PUNCT
ejpam-6568	614	1	[	[	X
ejpam-6568	614	2	28	28	NUM
ejpam-6568	614	3	]	]	PUNCT
ejpam-6568	614	4	k.	k.	PROPN
ejpam-6568	614	5	kuratowski	kuratowski	PROPN
ejpam-6568	614	6	.	.	PUNCT
ejpam-6568	615	1	topology	topology	PROPN
ejpam-6568	615	2	,	,	PUNCT
ejpam-6568	615	3	vol	vol	NOUN
ejpam-6568	615	4	.	.	PUNCT
ejpam-6568	615	5	i.	i.	PROPN
ejpam-6568	615	6	academic	academic	PROPN
ejpam-6568	615	7	press	press	PROPN
ejpam-6568	615	8	,	,	PUNCT
ejpam-6568	615	9	new	new	PROPN
ejpam-6568	615	10	york	york	PROPN
ejpam-6568	615	11	,	,	PUNCT
ejpam-6568	615	12	1966	1966	NUM
ejpam-6568	615	13	.	.	PUNCT
ejpam-6568	616	1	[	[	X
ejpam-6568	616	2	29	29	NUM
ejpam-6568	616	3	]	]	X
ejpam-6568	616	4	d.	d.	PROPN
ejpam-6568	616	5	janković	janković	PROPN
ejpam-6568	616	6	and	and	CCONJ
ejpam-6568	616	7	t.	t.	PROPN
ejpam-6568	616	8	r.	r.	PROPN
ejpam-6568	616	9	hamlett	hamlett	PROPN
ejpam-6568	616	10	.	.	PUNCT
ejpam-6568	617	1	new	new	ADJ
ejpam-6568	617	2	topologies	topology	NOUN
ejpam-6568	617	3	from	from	ADP
ejpam-6568	617	4	old	old	ADJ
ejpam-6568	617	5	via	via	ADP
ejpam-6568	617	6	ideals	ideal	NOUN
ejpam-6568	617	7	.	.	PUNCT
ejpam-6568	618	1	the	the	DET
ejpam-6568	618	2	american	american	PROPN
ejpam-6568	618	3	mathematical	mathematical	PROPN
ejpam-6568	618	4	monthly	monthly	ADV
ejpam-6568	618	5	,	,	PUNCT
ejpam-6568	618	6	97:295–310	97:295–310	PROPN
ejpam-6568	618	7	,	,	PUNCT
ejpam-6568	618	8	1990	1990	NUM
ejpam-6568	618	9	.	.	PUNCT
ejpam-6568	619	1	[	[	X
ejpam-6568	619	2	30	30	NUM
ejpam-6568	619	3	]	]	X
ejpam-6568	619	4	e.	e.	PROPN
ejpam-6568	619	5	ekici	ekici	PROPN
ejpam-6568	619	6	and	and	CCONJ
ejpam-6568	619	7	t.	t.	PROPN
ejpam-6568	619	8	noiri	noiri	PROPN
ejpam-6568	619	9	.	.	PUNCT
ejpam-6568	620	1	⋆-extremally	⋆-extremally	ADV
ejpam-6568	620	2	disconnected	disconnect	VERB
ejpam-6568	620	3	ideal	ideal	ADJ
ejpam-6568	620	4	topological	topological	ADJ
ejpam-6568	620	5	spaces	space	NOUN
ejpam-6568	620	6	.	.	PUNCT
ejpam-6568	621	1	acta	acta	PROPN
ejpam-6568	621	2	mathematica	mathematica	PROPN
ejpam-6568	621	3	hungarica	hungarica	PROPN
ejpam-6568	621	4	,	,	PUNCT
ejpam-6568	621	5	122:81–90	122:81–90	NUM
ejpam-6568	621	6	,	,	PUNCT
ejpam-6568	621	7	2009	2009	NUM
ejpam-6568	621	8	.	.	PUNCT
ejpam-6568	622	1	[	[	X
ejpam-6568	622	2	31	31	NUM
ejpam-6568	622	3	]	]	X
ejpam-6568	622	4	c.	c.	PROPN
ejpam-6568	622	5	klanarong	klanarong	PROPN
ejpam-6568	622	6	,	,	PUNCT
ejpam-6568	622	7	s.	s.	PROPN
ejpam-6568	622	8	sompong	sompong	PROPN
ejpam-6568	622	9	,	,	PUNCT
ejpam-6568	622	10	and	and	CCONJ
ejpam-6568	622	11	c.	c.	PROPN
ejpam-6568	622	12	boonpok	boonpok	PROPN
ejpam-6568	622	13	.	.	PUNCT
ejpam-6568	623	1	upper	upper	ADJ
ejpam-6568	623	2	and	and	CCONJ
ejpam-6568	623	3	lower	low	ADJ
ejpam-6568	623	4	almost	almost	ADV
ejpam-6568	623	5	(	(	PUNCT
ejpam-6568	623	6	τ1	τ1	NOUN
ejpam-6568	623	7	,	,	PUNCT
ejpam-6568	623	8	τ2)continuous	τ2)continuous	ADJ
ejpam-6568	623	9	multifunctions	multifunction	NOUN
ejpam-6568	623	10	.	.	PUNCT
ejpam-6568	624	1	european	european	ADJ
ejpam-6568	624	2	journal	journal	PROPN
ejpam-6568	624	3	of	of	ADP
ejpam-6568	624	4	pure	pure	ADJ
ejpam-6568	624	5	and	and	CCONJ
ejpam-6568	624	6	applied	applied	ADJ
ejpam-6568	624	7	mathematics	mathematic	NOUN
ejpam-6568	624	8	,	,	PUNCT
ejpam-6568	624	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-6568	624	10	,	,	PUNCT
ejpam-6568	624	11	2024	2024	NUM
ejpam-6568	624	12	.	.	PUNCT
ejpam-6568	625	1	[	[	X
ejpam-6568	625	2	32	32	NUM
ejpam-6568	625	3	]	]	PUNCT
ejpam-6568	625	4	p.	p.	NOUN
ejpam-6568	625	5	pue	pue	NOUN
ejpam-6568	625	6	-	-	PUNCT
ejpam-6568	625	7	on	on	ADP
ejpam-6568	625	8	,	,	PUNCT
ejpam-6568	625	9	s.	s.	PROPN
ejpam-6568	625	10	sompong	sompong	PROPN
ejpam-6568	625	11	,	,	PUNCT
ejpam-6568	625	12	and	and	CCONJ
ejpam-6568	625	13	c.	c.	PROPN
ejpam-6568	625	14	boonpok	boonpok	PROPN
ejpam-6568	625	15	.	.	PUNCT
ejpam-6568	626	1	almost	almost	ADV
ejpam-6568	626	2	contra-(τ1	contra-(τ1	NOUN
ejpam-6568	626	3	,	,	PUNCT
ejpam-6568	626	4	τ2)p	τ2)p	NOUN
ejpam-6568	626	5	-	-	PUNCT
ejpam-6568	626	6	continuity	continuity	NOUN
ejpam-6568	626	7	for	for	ADP
ejpam-6568	626	8	functions	function	NOUN
ejpam-6568	626	9	.	.	PUNCT
ejpam-6568	627	1	european	european	ADJ
ejpam-6568	627	2	journal	journal	PROPN
ejpam-6568	627	3	of	of	ADP
ejpam-6568	627	4	pure	pure	ADJ
ejpam-6568	627	5	and	and	CCONJ
ejpam-6568	627	6	applied	applied	ADJ
ejpam-6568	627	7	mathematics	mathematic	NOUN
ejpam-6568	627	8	,	,	PUNCT
ejpam-6568	627	9	18(2):6038	18(2):6038	NUM
ejpam-6568	627	10	,	,	PUNCT
ejpam-6568	627	11	2025	2025	NUM
ejpam-6568	627	12	.	.	PUNCT
ejpam-6568	628	1	[	[	X
ejpam-6568	628	2	33	33	NUM
ejpam-6568	628	3	]	]	PUNCT
ejpam-6568	628	4	m.	m.	NOUN
ejpam-6568	628	5	thongmoon	thongmoon	NOUN
ejpam-6568	628	6	,	,	PUNCT
ejpam-6568	628	7	s.	s.	PROPN
ejpam-6568	628	8	sompong	sompong	PROPN
ejpam-6568	628	9	,	,	PUNCT
ejpam-6568	628	10	and	and	CCONJ
ejpam-6568	628	11	c.	c.	PROPN
ejpam-6568	628	12	boonpok	boonpok	PROPN
ejpam-6568	628	13	.	.	PUNCT
ejpam-6568	629	1	almost	almost	ADV
ejpam-6568	629	2	(	(	PUNCT
ejpam-6568	629	3	τ1	τ1	NOUN
ejpam-6568	629	4	,	,	PUNCT
ejpam-6568	629	5	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6568	629	6	and	and	CCONJ
ejpam-6568	629	7	τ1τ2δ	τ1τ2δ	NUM
ejpam-6568	629	8	-	-	PUNCT
ejpam-6568	629	9	open	open	ADJ
ejpam-6568	629	10	sets	set	NOUN
ejpam-6568	629	11	.	.	PUNCT
ejpam-6568	630	1	international	international	ADJ
ejpam-6568	630	2	journal	journal	NOUN
ejpam-6568	630	3	of	of	ADP
ejpam-6568	630	4	analysis	analysis	NOUN
ejpam-6568	630	5	and	and	CCONJ
ejpam-6568	630	6	applications	application	NOUN
ejpam-6568	630	7	,	,	PUNCT
ejpam-6568	630	8	22:109	22:109	NUM
ejpam-6568	630	9	,	,	PUNCT
ejpam-6568	630	10	2024	2024	NUM
ejpam-6568	630	11	.	.	PUNCT