id	sid	tid	token	lemma	pos
ejpam-6572	1	1	european	european	PROPN
ejpam-6572	1	2	journal	journal	PROPN
ejpam-6572	1	3	of	of	ADP
ejpam-6572	1	4	pure	pure	ADJ
ejpam-6572	1	5	and	and	CCONJ
ejpam-6572	1	6	applied	applied	ADJ
ejpam-6572	1	7	mathematics	mathematic	NOUN
ejpam-6572	1	8	2025	2025	NUM
ejpam-6572	1	9	,	,	PUNCT
ejpam-6572	1	10	vol	vol	NOUN
ejpam-6572	1	11	.	.	PROPN
ejpam-6572	1	12	18	18	NUM
ejpam-6572	1	13	,	,	PUNCT
ejpam-6572	1	14	issue	issue	NOUN
ejpam-6572	1	15	3	3	NUM
ejpam-6572	1	16	,	,	PUNCT
ejpam-6572	1	17	article	article	NOUN
ejpam-6572	1	18	number	number	NOUN
ejpam-6572	1	19	6572	6572	NUM
ejpam-6572	1	20	issn	issn	PROPN
ejpam-6572	1	21	1307	1307	NUM
ejpam-6572	1	22	-	-	SYM
ejpam-6572	1	23	5543	5543	NUM
ejpam-6572	1	24	–	–	PUNCT
ejpam-6572	1	25	ejpam.com	ejpam.com	X
ejpam-6572	1	26	published	publish	VERB
ejpam-6572	1	27	by	by	ADP
ejpam-6572	1	28	new	new	PROPN
ejpam-6572	1	29	york	york	PROPN
ejpam-6572	1	30	business	business	PROPN
ejpam-6572	1	31	global	global	PROPN
ejpam-6572	1	32	almost	almost	ADV
ejpam-6572	1	33	quasi	quasi	X
ejpam-6572	1	34	τ	τ	PROPN
ejpam-6572	1	35	⋆(σ1	⋆(σ1	PROPN
ejpam-6572	1	36	,	,	PUNCT
ejpam-6572	1	37	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	1	38	and	and	CCONJ
ejpam-6572	1	39	weakly	weakly	ADJ
ejpam-6572	1	40	quasi	quasi	NOUN
ejpam-6572	1	41	τ	τ	PROPN
ejpam-6572	1	42	⋆(σ1	⋆(σ1	PROPN
ejpam-6572	1	43	,	,	PUNCT
ejpam-6572	1	44	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	1	45	functions	function	NOUN
ejpam-6572	1	46	butsakorn	butsakorn	PROPN
ejpam-6572	1	47	kong	kong	PROPN
ejpam-6572	1	48	-	-	PUNCT
ejpam-6572	1	49	ied1	ied1	PROPN
ejpam-6572	1	50	,	,	PUNCT
ejpam-6572	1	51	areeyuth	areeyuth	NOUN
ejpam-6572	1	52	sama	sama	NOUN
ejpam-6572	1	53	-	-	PUNCT
ejpam-6572	1	54	ae2	ae2	PROPN
ejpam-6572	1	55	,	,	PUNCT
ejpam-6572	1	56	chawalit	chawalit	VERB
ejpam-6572	1	57	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6572	1	58	1	1	NUM
ejpam-6572	1	59	mathematics	mathematic	NOUN
ejpam-6572	1	60	and	and	CCONJ
ejpam-6572	1	61	applied	apply	VERB
ejpam-6572	1	62	mathematics	mathematics	PROPN
ejpam-6572	1	63	research	research	NOUN
ejpam-6572	1	64	unit	unit	NOUN
ejpam-6572	1	65	,	,	PUNCT
ejpam-6572	1	66	department	department	NOUN
ejpam-6572	1	67	of	of	ADP
ejpam-6572	1	68	mathematics	mathematic	NOUN
ejpam-6572	1	69	,	,	PUNCT
ejpam-6572	1	70	faculty	faculty	NOUN
ejpam-6572	1	71	of	of	ADP
ejpam-6572	1	72	science	science	NOUN
ejpam-6572	1	73	,	,	PUNCT
ejpam-6572	1	74	mahasarakham	mahasarakham	PROPN
ejpam-6572	1	75	university	university	PROPN
ejpam-6572	1	76	,	,	PUNCT
ejpam-6572	1	77	maha	maha	PROPN
ejpam-6572	1	78	sarakham	sarakham	PROPN
ejpam-6572	1	79	,	,	PUNCT
ejpam-6572	1	80	44150	44150	NUM
ejpam-6572	1	81	,	,	PUNCT
ejpam-6572	1	82	thailand	thailand	PROPN
ejpam-6572	1	83	2	2	NUM
ejpam-6572	1	84	department	department	NOUN
ejpam-6572	1	85	of	of	ADP
ejpam-6572	1	86	mathematics	mathematic	NOUN
ejpam-6572	1	87	and	and	CCONJ
ejpam-6572	1	88	computer	computer	NOUN
ejpam-6572	1	89	science	science	NOUN
ejpam-6572	1	90	,	,	PUNCT
ejpam-6572	1	91	faculty	faculty	NOUN
ejpam-6572	1	92	of	of	ADP
ejpam-6572	1	93	science	science	NOUN
ejpam-6572	1	94	and	and	CCONJ
ejpam-6572	1	95	technology	technology	NOUN
ejpam-6572	1	96	,	,	PUNCT
ejpam-6572	1	97	prince	prince	NOUN
ejpam-6572	1	98	of	of	ADP
ejpam-6572	1	99	songkla	songkla	PROPN
ejpam-6572	1	100	university	university	PROPN
ejpam-6572	1	101	,	,	PUNCT
ejpam-6572	1	102	pattani	pattani	NOUN
ejpam-6572	1	103	campus	campus	NOUN
ejpam-6572	1	104	,	,	PUNCT
ejpam-6572	1	105	pattani	pattani	NOUN
ejpam-6572	1	106	,	,	PUNCT
ejpam-6572	1	107	94000	94000	NUM
ejpam-6572	1	108	,	,	PUNCT
ejpam-6572	1	109	thailand	thailand	PROPN
ejpam-6572	1	110	abstract	abstract	PROPN
ejpam-6572	1	111	.	.	PUNCT
ejpam-6572	2	1	this	this	DET
ejpam-6572	2	2	paper	paper	NOUN
ejpam-6572	2	3	introduces	introduce	VERB
ejpam-6572	2	4	two	two	NUM
ejpam-6572	2	5	classes	class	NOUN
ejpam-6572	2	6	of	of	ADP
ejpam-6572	2	7	continuous	continuous	ADJ
ejpam-6572	2	8	functions	function	NOUN
ejpam-6572	2	9	defined	define	VERB
ejpam-6572	2	10	between	between	ADP
ejpam-6572	2	11	an	an	DET
ejpam-6572	2	12	ideal	ideal	ADJ
ejpam-6572	2	13	topological	topological	ADJ
ejpam-6572	2	14	space	space	NOUN
ejpam-6572	2	15	and	and	CCONJ
ejpam-6572	2	16	a	a	DET
ejpam-6572	2	17	bitopological	bitopological	ADJ
ejpam-6572	2	18	space	space	NOUN
ejpam-6572	2	19	,	,	PUNCT
ejpam-6572	2	20	called	call	VERB
ejpam-6572	2	21	almost	almost	ADV
ejpam-6572	2	22	quasi	quasi	NOUN
ejpam-6572	2	23	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	2	24	,	,	PUNCT
ejpam-6572	2	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	2	26	functions	function	NOUN
ejpam-6572	2	27	and	and	CCONJ
ejpam-6572	2	28	weakly	weakly	ADJ
ejpam-6572	2	29	quasi	quasi	NOUN
ejpam-6572	2	30	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	2	31	,	,	PUNCT
ejpam-6572	2	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	2	33	functions	function	NOUN
ejpam-6572	2	34	.	.	PUNCT
ejpam-6572	3	1	furthermore	furthermore	ADV
ejpam-6572	3	2	,	,	PUNCT
ejpam-6572	3	3	several	several	ADJ
ejpam-6572	3	4	characterizations	characterization	NOUN
ejpam-6572	3	5	and	and	CCONJ
ejpam-6572	3	6	some	some	DET
ejpam-6572	3	7	properties	property	NOUN
ejpam-6572	3	8	concerning	concern	VERB
ejpam-6572	3	9	almost	almost	ADV
ejpam-6572	3	10	quasi	quasi	ADJ
ejpam-6572	3	11	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	3	12	,	,	PUNCT
ejpam-6572	3	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	3	14	functions	function	NOUN
ejpam-6572	3	15	and	and	CCONJ
ejpam-6572	3	16	weakly	weakly	ADJ
ejpam-6572	3	17	quasi	quasi	NOUN
ejpam-6572	3	18	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	3	19	,	,	PUNCT
ejpam-6572	3	20	σ2)continuous	σ2)continuous	ADJ
ejpam-6572	3	21	functions	function	NOUN
ejpam-6572	3	22	are	be	AUX
ejpam-6572	3	23	investigated	investigate	VERB
ejpam-6572	3	24	.	.	PUNCT
ejpam-6572	4	1	moreover	moreover	ADV
ejpam-6572	4	2	,	,	PUNCT
ejpam-6572	4	3	the	the	DET
ejpam-6572	4	4	relationships	relationship	NOUN
ejpam-6572	4	5	between	between	ADP
ejpam-6572	4	6	almost	almost	ADV
ejpam-6572	4	7	quasi	quasi	ADJ
ejpam-6572	4	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	4	9	,	,	PUNCT
ejpam-6572	4	10	σ2)continuity	σ2)continuity	NOUN
ejpam-6572	4	11	and	and	CCONJ
ejpam-6572	4	12	weak	weak	ADJ
ejpam-6572	4	13	quasi	quasi	NOUN
ejpam-6572	4	14	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	4	15	,	,	PUNCT
ejpam-6572	4	16	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6572	4	17	are	be	AUX
ejpam-6572	4	18	considered	consider	VERB
ejpam-6572	4	19	.	.	PUNCT
ejpam-6572	5	1	2020	2020	NUM
ejpam-6572	5	2	mathematics	mathematic	NOUN
ejpam-6572	5	3	subject	subject	NOUN
ejpam-6572	5	4	classifications	classification	NOUN
ejpam-6572	5	5	:	:	PUNCT
ejpam-6572	5	6	54c05	54c05	NUM
ejpam-6572	5	7	,	,	PUNCT
ejpam-6572	5	8	54c08	54c08	NUM
ejpam-6572	5	9	key	key	ADJ
ejpam-6572	5	10	words	word	NOUN
ejpam-6572	5	11	and	and	CCONJ
ejpam-6572	5	12	phrases	phrase	NOUN
ejpam-6572	5	13	:	:	PUNCT
ejpam-6572	5	14	almost	almost	ADV
ejpam-6572	5	15	quasi	quasi	ADJ
ejpam-6572	5	16	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	5	17	,	,	PUNCT
ejpam-6572	5	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	5	19	function	function	NOUN
ejpam-6572	5	20	,	,	PUNCT
ejpam-6572	5	21	weakly	weakly	ADJ
ejpam-6572	5	22	quasi	quasi	NOUN
ejpam-6572	5	23	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	5	24	,	,	PUNCT
ejpam-6572	5	25	σ2)continuous	σ2)continuous	ADJ
ejpam-6572	5	26	function	function	NOUN
ejpam-6572	5	27	1	1	NUM
ejpam-6572	5	28	.	.	PUNCT
ejpam-6572	5	29	introduction	introduction	NOUN
ejpam-6572	5	30	in	in	ADP
ejpam-6572	5	31	1961	1961	NUM
ejpam-6572	5	32	,	,	PUNCT
ejpam-6572	5	33	marcus	marcus	PROPN
ejpam-6572	6	1	[	[	X
ejpam-6572	6	2	1	1	X
ejpam-6572	6	3	]	]	PUNCT
ejpam-6572	6	4	introduced	introduce	VERB
ejpam-6572	6	5	the	the	DET
ejpam-6572	6	6	concept	concept	NOUN
ejpam-6572	6	7	of	of	ADP
ejpam-6572	6	8	quasi	quasi	ADJ
ejpam-6572	6	9	continuous	continuous	ADJ
ejpam-6572	6	10	functions	function	NOUN
ejpam-6572	6	11	.	.	PUNCT
ejpam-6572	7	1	popa	popa	NOUN
ejpam-6572	8	1	[	[	X
ejpam-6572	8	2	2	2	NUM
ejpam-6572	8	3	]	]	PUNCT
ejpam-6572	8	4	introduced	introduce	VERB
ejpam-6572	8	5	and	and	CCONJ
ejpam-6572	8	6	investigated	investigate	VERB
ejpam-6572	8	7	the	the	DET
ejpam-6572	8	8	notion	notion	NOUN
ejpam-6572	8	9	of	of	ADP
ejpam-6572	8	10	almost	almost	ADV
ejpam-6572	8	11	quasi	quasi	ADJ
ejpam-6572	8	12	continuous	continuous	ADJ
ejpam-6572	8	13	functions	function	NOUN
ejpam-6572	8	14	.	.	PUNCT
ejpam-6572	9	1	neubrunnovaá	neubrunnovaá	PUNCT
ejpam-6572	10	1	[	[	X
ejpam-6572	10	2	3	3	NUM
ejpam-6572	10	3	]	]	PUNCT
ejpam-6572	10	4	showed	show	VERB
ejpam-6572	10	5	that	that	SCONJ
ejpam-6572	10	6	quasi	quasi	NOUN
ejpam-6572	10	7	continuity	continuity	NOUN
ejpam-6572	10	8	is	be	AUX
ejpam-6572	10	9	equivalent	equivalent	ADJ
ejpam-6572	10	10	to	to	ADP
ejpam-6572	10	11	semi	semi	ADJ
ejpam-6572	10	12	-	-	NOUN
ejpam-6572	10	13	continuity	continuity	NOUN
ejpam-6572	10	14	due	due	ADP
ejpam-6572	10	15	to	to	ADP
ejpam-6572	10	16	levine	levine	PROPN
ejpam-6572	10	17	[	[	X
ejpam-6572	10	18	4	4	NUM
ejpam-6572	10	19	]	]	PUNCT
ejpam-6572	10	20	.	.	PUNCT
ejpam-6572	11	1	popa	popa	NOUN
ejpam-6572	11	2	and	and	CCONJ
ejpam-6572	11	3	stan	stan	PROPN
ejpam-6572	12	1	[	[	X
ejpam-6572	12	2	5	5	NUM
ejpam-6572	12	3	]	]	PUNCT
ejpam-6572	12	4	introduced	introduce	VERB
ejpam-6572	12	5	and	and	CCONJ
ejpam-6572	12	6	studied	study	VERB
ejpam-6572	12	7	the	the	DET
ejpam-6572	12	8	notion	notion	NOUN
ejpam-6572	12	9	of	of	ADP
ejpam-6572	12	10	weakly	weakly	ADJ
ejpam-6572	12	11	quasi	quasi	ADJ
ejpam-6572	12	12	continuous	continuous	ADJ
ejpam-6572	12	13	functions	function	NOUN
ejpam-6572	12	14	.	.	PUNCT
ejpam-6572	13	1	weak	weak	ADJ
ejpam-6572	13	2	quasi	quasi	NOUN
ejpam-6572	13	3	continuity	continuity	NOUN
ejpam-6572	13	4	is	be	AUX
ejpam-6572	13	5	implied	imply	VERB
ejpam-6572	13	6	by	by	ADP
ejpam-6572	13	7	quasi	quasi	NOUN
ejpam-6572	13	8	continuity	continuity	NOUN
ejpam-6572	13	9	and	and	CCONJ
ejpam-6572	13	10	weak	weak	ADJ
ejpam-6572	13	11	continuity	continuity	NOUN
ejpam-6572	13	12	[	[	X
ejpam-6572	13	13	6	6	NUM
ejpam-6572	13	14	]	]	PUNCT
ejpam-6572	13	15	which	which	PRON
ejpam-6572	13	16	are	be	AUX
ejpam-6572	13	17	independent	independent	ADJ
ejpam-6572	13	18	of	of	ADP
ejpam-6572	13	19	each	each	DET
ejpam-6572	13	20	other	other	ADJ
ejpam-6572	13	21	.	.	PUNCT
ejpam-6572	14	1	it	it	PRON
ejpam-6572	14	2	is	be	AUX
ejpam-6572	14	3	shown	show	VERB
ejpam-6572	14	4	in	in	ADP
ejpam-6572	14	5	[	[	X
ejpam-6572	14	6	7	7	X
ejpam-6572	14	7	]	]	PUNCT
ejpam-6572	14	8	that	that	SCONJ
ejpam-6572	14	9	weak	weak	ADJ
ejpam-6572	14	10	quasi	quasi	NOUN
ejpam-6572	14	11	continuity	continuity	NOUN
ejpam-6572	14	12	is	be	AUX
ejpam-6572	14	13	equivalent	equivalent	ADJ
ejpam-6572	14	14	to	to	AUX
ejpam-6572	14	15	weak	weak	ADJ
ejpam-6572	14	16	semi	semi	ADJ
ejpam-6572	14	17	-	-	NOUN
ejpam-6572	14	18	continuity	continuity	NOUN
ejpam-6572	14	19	due	due	ADP
ejpam-6572	14	20	to	to	ADP
ejpam-6572	14	21	arya	arya	PROPN
ejpam-6572	14	22	and	and	CCONJ
ejpam-6572	14	23	bhamini	bhamini	PROPN
ejpam-6572	15	1	[	[	X
ejpam-6572	15	2	8	8	NUM
ejpam-6572	15	3	]	]	PUNCT
ejpam-6572	15	4	and	and	CCONJ
ejpam-6572	15	5	kar	kar	NOUN
ejpam-6572	15	6	and	and	CCONJ
ejpam-6572	15	7	bhattacharyya	bhattacharyya	ADJ
ejpam-6572	16	1	[	[	X
ejpam-6572	16	2	9	9	NUM
ejpam-6572	16	3	]	]	PUNCT
ejpam-6572	16	4	.	.	PUNCT
ejpam-6572	17	1	in	in	ADP
ejpam-6572	17	2	1990	1990	NUM
ejpam-6572	17	3	,	,	PUNCT
ejpam-6572	17	4	janković	janković	ADJ
ejpam-6572	17	5	and	and	CCONJ
ejpam-6572	17	6	hamlett	hamlett	PROPN
ejpam-6572	18	1	[	[	X
ejpam-6572	18	2	10	10	NUM
ejpam-6572	18	3	]	]	PUNCT
ejpam-6572	18	4	introduced	introduce	VERB
ejpam-6572	18	5	the	the	DET
ejpam-6572	18	6	concept	concept	NOUN
ejpam-6572	18	7	of	of	ADP
ejpam-6572	18	8	i	i	PRON
ejpam-6572	18	9	-open	-open	PROPN
ejpam-6572	18	10	sets	set	NOUN
ejpam-6572	18	11	in	in	ADP
ejpam-6572	18	12	ideal	ideal	ADJ
ejpam-6572	18	13	topological	topological	ADJ
ejpam-6572	18	14	spaces	space	NOUN
ejpam-6572	18	15	.	.	PUNCT
ejpam-6572	19	1	abd	abd	PROPN
ejpam-6572	19	2	el	el	PROPN
ejpam-6572	19	3	-	-	PROPN
ejpam-6572	19	4	monsef	monsef	PROPN
ejpam-6572	19	5	et	et	PROPN
ejpam-6572	19	6	al	al	PROPN
ejpam-6572	19	7	.	.	PUNCT
ejpam-6572	20	1	[	[	X
ejpam-6572	20	2	11	11	NUM
ejpam-6572	20	3	]	]	PUNCT
ejpam-6572	20	4	introduced	introduce	VERB
ejpam-6572	20	5	and	and	CCONJ
ejpam-6572	20	6	studied	study	VERB
ejpam-6572	20	7	the	the	DET
ejpam-6572	20	8	concepts	concept	NOUN
ejpam-6572	20	9	of	of	ADP
ejpam-6572	20	10	i	i	PROPN
ejpam-6572	20	11	-closed	-close	VERB
ejpam-6572	20	12	sets	set	NOUN
ejpam-6572	20	13	and	and	CCONJ
ejpam-6572	20	14	i	i	PRON
ejpam-6572	20	15	-continuous	-continuous	ADJ
ejpam-6572	20	16	functions	function	NOUN
ejpam-6572	20	17	.	.	PUNCT
ejpam-6572	21	1	semi	semi	ADJ
ejpam-6572	21	2	-	-	ADJ
ejpam-6572	21	3	i	i	ADJ
ejpam-6572	21	4	-open	-open	NOUN
ejpam-6572	21	5	sets	set	NOUN
ejpam-6572	21	6	,	,	PUNCT
ejpam-6572	21	7	pre	pre	ADJ
ejpam-6572	21	8	-	-	ADJ
ejpam-6572	21	9	i	i	ADJ
ejpam-6572	21	10	-open	-open	NOUN
ejpam-6572	21	11	sets	set	NOUN
ejpam-6572	21	12	,	,	PUNCT
ejpam-6572	21	13	α	α	X
ejpam-6572	21	14	-	-	PUNCT
ejpam-6572	21	15	i	i	PRON
ejpam-6572	21	16	-open	-open	NOUN
ejpam-6572	21	17	sets	set	NOUN
ejpam-6572	21	18	,	,	PUNCT
ejpam-6572	21	19	βi	βi	X
ejpam-6572	21	20	-open	-open	NOUN
ejpam-6572	21	21	sets	set	NOUN
ejpam-6572	21	22	and	and	CCONJ
ejpam-6572	21	23	δ	δ	PROPN
ejpam-6572	21	24	-	-	PUNCT
ejpam-6572	21	25	i	i	PRON
ejpam-6572	21	26	-open	-open	NOUN
ejpam-6572	21	27	sets	set	NOUN
ejpam-6572	21	28	play	play	VERB
ejpam-6572	21	29	an	an	DET
ejpam-6572	21	30	important	important	ADJ
ejpam-6572	21	31	role	role	NOUN
ejpam-6572	21	32	in	in	ADP
ejpam-6572	21	33	the	the	DET
ejpam-6572	21	34	research	research	NOUN
ejpam-6572	21	35	of	of	ADP
ejpam-6572	21	36	generalizations	generalization	NOUN
ejpam-6572	21	37	of	of	ADP
ejpam-6572	21	38	continuity	continuity	NOUN
ejpam-6572	21	39	in	in	ADP
ejpam-6572	21	40	ideal	ideal	ADJ
ejpam-6572	21	41	topological	topological	ADJ
ejpam-6572	21	42	spaces	space	NOUN
ejpam-6572	21	43	.	.	PUNCT
ejpam-6572	22	1	using	use	VERB
ejpam-6572	22	2	these	these	DET
ejpam-6572	22	3	notions	notion	NOUN
ejpam-6572	22	4	many	many	ADJ
ejpam-6572	22	5	authors	author	NOUN
ejpam-6572	22	6	introduced	introduce	VERB
ejpam-6572	22	7	∗corresponding	∗corresponde	VERB
ejpam-6572	22	8	author	author	NOUN
ejpam-6572	22	9	.	.	PUNCT
ejpam-6572	23	1	doi	doi	NOUN
ejpam-6572	23	2	:	:	PUNCT
ejpam-6572	23	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6572	https://doi.org/10.29020/nybg.ejpam.v18i3.6572	ADJ
ejpam-6572	23	4	email	email	NOUN
ejpam-6572	23	5	addresses	address	NOUN
ejpam-6572	23	6	:	:	PUNCT
ejpam-6572	23	7	butrakorn.k@msu.ac.th	butrakorn.k@msu.ac.th	PROPN
ejpam-6572	23	8	(	(	PUNCT
ejpam-6572	23	9	b.	b.	PROPN
ejpam-6572	23	10	kong	kong	PROPN
ejpam-6572	23	11	-	-	PUNCT
ejpam-6572	23	12	ied	ied	PROPN
ejpam-6572	23	13	)	)	PUNCT
ejpam-6572	23	14	,	,	PUNCT
ejpam-6572	23	15	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6572	23	16	(	(	PUNCT
ejpam-6572	23	17	a.	a.	PROPN
ejpam-6572	23	18	sama	sama	PROPN
ejpam-6572	23	19	-	-	PUNCT
ejpam-6572	23	20	ae	ae	PROPN
ejpam-6572	23	21	)	)	PUNCT
ejpam-6572	23	22	,	,	PUNCT
ejpam-6572	23	23	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6572	23	24	(	(	PUNCT
ejpam-6572	23	25	c.	c.	PROPN
ejpam-6572	23	26	boonpok	boonpok	PROPN
ejpam-6572	23	27	)	)	PUNCT
ejpam-6572	23	28	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6572	24	1	1	1	NUM
ejpam-6572	24	2	copyright	copyright	NOUN
ejpam-6572	24	3	:	:	PUNCT
ejpam-6572	24	4	©	©	PROPN
ejpam-6572	24	5	2025	2025	NUM
ejpam-6572	24	6	the	the	DET
ejpam-6572	24	7	author(s	author(s	NOUN
ejpam-6572	24	8	)	)	PUNCT
ejpam-6572	24	9	.	.	PUNCT
ejpam-6572	25	1	(	(	PUNCT
ejpam-6572	25	2	cc	cc	NOUN
ejpam-6572	25	3	by	by	ADP
ejpam-6572	25	4	-	-	PUNCT
ejpam-6572	25	5	nc	nc	PROPN
ejpam-6572	25	6	4.0	4.0	NUM
ejpam-6572	25	7	)	)	PUNCT
ejpam-6572	25	8	b.	b.	PROPN
ejpam-6572	25	9	kong	kong	PROPN
ejpam-6572	25	10	-	-	PUNCT
ejpam-6572	25	11	ied	ied	PROPN
ejpam-6572	25	12	,	,	PUNCT
ejpam-6572	25	13	a.	a.	PROPN
ejpam-6572	25	14	sama	sama	PROPN
ejpam-6572	25	15	-	-	PUNCT
ejpam-6572	25	16	ae	ae	PROPN
ejpam-6572	25	17	,	,	PUNCT
ejpam-6572	25	18	c.	c.	PROPN
ejpam-6572	25	19	boonpok	boonpok	PROPN
ejpam-6572	25	20	/	/	SYM
ejpam-6572	25	21	eur	eur	PROPN
ejpam-6572	25	22	.	.	PUNCT
ejpam-6572	26	1	j.	j.	PROPN
ejpam-6572	26	2	pure	pure	PROPN
ejpam-6572	26	3	appl	appl	PROPN
ejpam-6572	26	4	.	.	PROPN
ejpam-6572	26	5	math	math	PROPN
ejpam-6572	26	6	,	,	PUNCT
ejpam-6572	26	7	18	18	NUM
ejpam-6572	26	8	(	(	PUNCT
ejpam-6572	26	9	3	3	NUM
ejpam-6572	26	10	)	)	PUNCT
ejpam-6572	26	11	(	(	PUNCT
ejpam-6572	26	12	2025	2025	NUM
ejpam-6572	26	13	)	)	PUNCT
ejpam-6572	26	14	,	,	PUNCT
ejpam-6572	26	15	6572	6572	NUM
ejpam-6572	26	16	2	2	NUM
ejpam-6572	26	17	of	of	ADP
ejpam-6572	26	18	12	12	NUM
ejpam-6572	26	19	and	and	CCONJ
ejpam-6572	26	20	studied	study	VERB
ejpam-6572	26	21	various	various	ADJ
ejpam-6572	26	22	types	type	NOUN
ejpam-6572	26	23	of	of	ADP
ejpam-6572	26	24	generalizations	generalization	NOUN
ejpam-6572	26	25	of	of	ADP
ejpam-6572	26	26	continuity	continuity	NOUN
ejpam-6572	26	27	for	for	ADP
ejpam-6572	26	28	functions	function	NOUN
ejpam-6572	26	29	.	.	PUNCT
ejpam-6572	27	1	hatir	hatir	PROPN
ejpam-6572	27	2	and	and	CCONJ
ejpam-6572	27	3	noiri	noiri	ADV
ejpam-6572	28	1	[	[	X
ejpam-6572	28	2	12	12	NUM
ejpam-6572	28	3	]	]	PUNCT
ejpam-6572	28	4	introduced	introduce	VERB
ejpam-6572	28	5	and	and	CCONJ
ejpam-6572	28	6	investigated	investigate	VERB
ejpam-6572	28	7	the	the	DET
ejpam-6572	28	8	notions	notion	NOUN
ejpam-6572	28	9	of	of	ADP
ejpam-6572	28	10	weakly	weakly	ADJ
ejpam-6572	28	11	pre	pre	ADJ
ejpam-6572	28	12	-	-	ADJ
ejpam-6572	28	13	i	i	PRON
ejpam-6572	28	14	-open	-open	NOUN
ejpam-6572	28	15	sets	set	NOUN
ejpam-6572	28	16	and	and	CCONJ
ejpam-6572	28	17	weakly	weakly	ADJ
ejpam-6572	28	18	prei	prei	NOUN
ejpam-6572	28	19	-continuous	-continuous	ADJ
ejpam-6572	28	20	functions	function	NOUN
ejpam-6572	28	21	.	.	PUNCT
ejpam-6572	29	1	moreover	moreover	ADV
ejpam-6572	29	2	,	,	PUNCT
ejpam-6572	29	3	hatir	hatir	PROPN
ejpam-6572	29	4	and	and	CCONJ
ejpam-6572	29	5	noiri	noiri	ADV
ejpam-6572	30	1	[	[	X
ejpam-6572	30	2	13	13	NUM
ejpam-6572	30	3	]	]	PUNCT
ejpam-6572	30	4	investigated	investigate	VERB
ejpam-6572	30	5	further	further	ADJ
ejpam-6572	30	6	properties	property	NOUN
ejpam-6572	30	7	of	of	ADP
ejpam-6572	30	8	semi	semi	ADJ
ejpam-6572	30	9	-	-	ADJ
ejpam-6572	30	10	i	i	PRON
ejpam-6572	30	11	-open	-open	NOUN
ejpam-6572	30	12	sets	set	NOUN
ejpam-6572	30	13	and	and	CCONJ
ejpam-6572	30	14	semi	semi	ADJ
ejpam-6572	30	15	-	-	ADJ
ejpam-6572	30	16	i	i	ADV
ejpam-6572	30	17	-continuous	-continuous	ADJ
ejpam-6572	30	18	functions	function	NOUN
ejpam-6572	30	19	.	.	PUNCT
ejpam-6572	31	1	on	on	ADP
ejpam-6572	31	2	the	the	DET
ejpam-6572	31	3	other	other	ADJ
ejpam-6572	31	4	hand	hand	NOUN
ejpam-6572	31	5	,	,	PUNCT
ejpam-6572	31	6	the	the	DET
ejpam-6572	31	7	present	present	ADJ
ejpam-6572	31	8	authors	author	NOUN
ejpam-6572	31	9	introduced	introduce	VERB
ejpam-6572	31	10	and	and	CCONJ
ejpam-6572	31	11	investigated	investigate	VERB
ejpam-6572	31	12	the	the	DET
ejpam-6572	31	13	concepts	concept	NOUN
ejpam-6572	31	14	of	of	ADP
ejpam-6572	31	15	pı	pı	ADJ
ejpam-6572	31	16	-	-	ADJ
ejpam-6572	31	17	continuous	continuous	ADJ
ejpam-6572	31	18	functions	function	NOUN
ejpam-6572	31	19	[	[	X
ejpam-6572	31	20	14	14	NUM
ejpam-6572	31	21	]	]	X
ejpam-6572	31	22	,	,	PUNCT
ejpam-6572	31	23	weakly	weakly	ADJ
ejpam-6572	31	24	pı	pı	ADJ
ejpam-6572	31	25	-	-	ADJ
ejpam-6572	31	26	continuous	continuous	ADJ
ejpam-6572	31	27	functions	function	NOUN
ejpam-6572	31	28	[	[	X
ejpam-6572	31	29	14	14	NUM
ejpam-6572	31	30	]	]	PUNCT
ejpam-6572	31	31	,	,	PUNCT
ejpam-6572	31	32	(	(	PUNCT
ejpam-6572	31	33	τ1	τ1	NOUN
ejpam-6572	31	34	,	,	PUNCT
ejpam-6572	31	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	31	36	functions	function	NOUN
ejpam-6572	31	37	[	[	X
ejpam-6572	31	38	15	15	NUM
ejpam-6572	31	39	]	]	PUNCT
ejpam-6572	31	40	,	,	PUNCT
ejpam-6572	31	41	almost	almost	ADV
ejpam-6572	31	42	(	(	PUNCT
ejpam-6572	31	43	τ1	τ1	NOUN
ejpam-6572	31	44	,	,	PUNCT
ejpam-6572	31	45	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	31	46	functions	function	NOUN
ejpam-6572	31	47	[	[	X
ejpam-6572	31	48	16	16	NUM
ejpam-6572	31	49	]	]	PUNCT
ejpam-6572	31	50	and	and	CCONJ
ejpam-6572	31	51	weakly	weakly	ADJ
ejpam-6572	31	52	(	(	PUNCT
ejpam-6572	31	53	τ1	τ1	NOUN
ejpam-6572	31	54	,	,	PUNCT
ejpam-6572	31	55	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	31	56	functions	function	NOUN
ejpam-6572	31	57	[	[	X
ejpam-6572	31	58	17	17	NUM
ejpam-6572	31	59	]	]	PUNCT
ejpam-6572	31	60	.	.	PUNCT
ejpam-6572	32	1	kong	kong	PROPN
ejpam-6572	32	2	-	-	PUNCT
ejpam-6572	32	3	ied	ied	PROPN
ejpam-6572	32	4	et	et	PROPN
ejpam-6572	32	5	al	al	PROPN
ejpam-6572	32	6	.	.	PUNCT
ejpam-6572	33	1	[	[	X
ejpam-6572	33	2	18	18	NUM
ejpam-6572	33	3	]	]	PUNCT
ejpam-6572	33	4	introduced	introduce	VERB
ejpam-6572	33	5	and	and	CCONJ
ejpam-6572	33	6	studied	study	VERB
ejpam-6572	33	7	the	the	DET
ejpam-6572	33	8	notion	notion	NOUN
ejpam-6572	33	9	of	of	ADP
ejpam-6572	33	10	almost	almost	ADV
ejpam-6572	33	11	quasi	quasi	X
ejpam-6572	33	12	(	(	PUNCT
ejpam-6572	33	13	τ1	τ1	NOUN
ejpam-6572	33	14	,	,	PUNCT
ejpam-6572	33	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	33	16	functions	function	NOUN
ejpam-6572	33	17	.	.	PUNCT
ejpam-6572	34	1	chiangpradit	chiangpradit	NOUN
ejpam-6572	34	2	et	et	PROPN
ejpam-6572	34	3	al	al	PROPN
ejpam-6572	34	4	.	.	PUNCT
ejpam-6572	35	1	[	[	X
ejpam-6572	35	2	19	19	NUM
ejpam-6572	35	3	]	]	PUNCT
ejpam-6572	35	4	introduced	introduce	VERB
ejpam-6572	35	5	and	and	CCONJ
ejpam-6572	35	6	investigated	investigate	VERB
ejpam-6572	35	7	the	the	DET
ejpam-6572	35	8	concept	concept	NOUN
ejpam-6572	35	9	of	of	ADP
ejpam-6572	35	10	weakly	weakly	ADJ
ejpam-6572	35	11	quasi	quasi	NOUN
ejpam-6572	35	12	(	(	PUNCT
ejpam-6572	35	13	τ1	τ1	PROPN
ejpam-6572	35	14	,	,	PUNCT
ejpam-6572	35	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	35	16	functions	function	NOUN
ejpam-6572	35	17	.	.	PUNCT
ejpam-6572	36	1	in	in	ADP
ejpam-6572	36	2	this	this	DET
ejpam-6572	36	3	paper	paper	NOUN
ejpam-6572	36	4	,	,	PUNCT
ejpam-6572	36	5	we	we	PRON
ejpam-6572	36	6	introduce	introduce	VERB
ejpam-6572	36	7	new	new	ADJ
ejpam-6572	36	8	classes	class	NOUN
ejpam-6572	36	9	of	of	ADP
ejpam-6572	36	10	functions	function	NOUN
ejpam-6572	36	11	between	between	ADP
ejpam-6572	36	12	an	an	DET
ejpam-6572	36	13	ideal	ideal	ADJ
ejpam-6572	36	14	topological	topological	ADJ
ejpam-6572	36	15	space	space	NOUN
ejpam-6572	36	16	and	and	CCONJ
ejpam-6572	36	17	a	a	DET
ejpam-6572	36	18	bitopological	bitopological	ADJ
ejpam-6572	36	19	space	space	NOUN
ejpam-6572	36	20	,	,	PUNCT
ejpam-6572	36	21	namely	namely	ADV
ejpam-6572	36	22	almost	almost	ADV
ejpam-6572	36	23	quasi	quasi	ADJ
ejpam-6572	36	24	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	36	25	,	,	PUNCT
ejpam-6572	36	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	36	27	functions	function	NOUN
ejpam-6572	36	28	and	and	CCONJ
ejpam-6572	36	29	weakly	weakly	ADJ
ejpam-6572	36	30	quasi	quasi	NOUN
ejpam-6572	36	31	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	36	32	,	,	PUNCT
ejpam-6572	36	33	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	36	34	functions	function	NOUN
ejpam-6572	36	35	.	.	PUNCT
ejpam-6572	37	1	we	we	PRON
ejpam-6572	37	2	also	also	ADV
ejpam-6572	37	3	investigate	investigate	VERB
ejpam-6572	37	4	several	several	ADJ
ejpam-6572	37	5	characterizations	characterization	NOUN
ejpam-6572	37	6	of	of	ADP
ejpam-6572	37	7	almost	almost	ADV
ejpam-6572	37	8	quasi	quasi	ADJ
ejpam-6572	37	9	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	37	10	,	,	PUNCT
ejpam-6572	37	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	37	12	functions	function	NOUN
ejpam-6572	37	13	and	and	CCONJ
ejpam-6572	37	14	weakly	weakly	ADJ
ejpam-6572	37	15	quasi	quasi	NOUN
ejpam-6572	37	16	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	37	17	,	,	PUNCT
ejpam-6572	37	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	37	19	functions	function	NOUN
ejpam-6572	37	20	.	.	PUNCT
ejpam-6572	38	1	2	2	X
ejpam-6572	38	2	.	.	X
ejpam-6572	38	3	preliminaries	preliminary	NOUN
ejpam-6572	38	4	throughout	throughout	ADP
ejpam-6572	38	5	the	the	DET
ejpam-6572	38	6	present	present	ADJ
ejpam-6572	38	7	paper	paper	NOUN
ejpam-6572	38	8	,	,	PUNCT
ejpam-6572	38	9	spaces	space	NOUN
ejpam-6572	38	10	(	(	PUNCT
ejpam-6572	38	11	x	x	NOUN
ejpam-6572	38	12	,	,	PUNCT
ejpam-6572	38	13	τ1	τ1	NOUN
ejpam-6572	38	14	,	,	PUNCT
ejpam-6572	38	15	τ2	τ2	NOUN
ejpam-6572	38	16	)	)	PUNCT
ejpam-6572	38	17	and	and	CCONJ
ejpam-6572	38	18	(	(	PUNCT
ejpam-6572	38	19	y	y	PROPN
ejpam-6572	38	20	,	,	PUNCT
ejpam-6572	38	21	σ1	σ1	PROPN
ejpam-6572	38	22	,	,	PUNCT
ejpam-6572	38	23	σ2	σ2	NOUN
ejpam-6572	38	24	)	)	PUNCT
ejpam-6572	38	25	(	(	PUNCT
ejpam-6572	38	26	or	or	CCONJ
ejpam-6572	38	27	simply	simply	ADV
ejpam-6572	38	28	x	x	X
ejpam-6572	38	29	and	and	CCONJ
ejpam-6572	38	30	y	y	PROPN
ejpam-6572	38	31	)	)	PUNCT
ejpam-6572	38	32	always	always	ADV
ejpam-6572	38	33	mean	mean	VERB
ejpam-6572	38	34	bitopological	bitopological	ADJ
ejpam-6572	38	35	spaces	space	NOUN
ejpam-6572	38	36	on	on	ADP
ejpam-6572	38	37	which	which	PRON
ejpam-6572	38	38	no	no	DET
ejpam-6572	38	39	separation	separation	NOUN
ejpam-6572	38	40	axioms	axiom	NOUN
ejpam-6572	38	41	are	be	AUX
ejpam-6572	38	42	assumed	assume	VERB
ejpam-6572	38	43	unless	unless	SCONJ
ejpam-6572	38	44	explicitly	explicitly	ADV
ejpam-6572	38	45	stated	state	VERB
ejpam-6572	38	46	.	.	PUNCT
ejpam-6572	39	1	let	let	VERB
ejpam-6572	39	2	a	a	DET
ejpam-6572	39	3	be	be	AUX
ejpam-6572	39	4	a	a	DET
ejpam-6572	39	5	subset	subset	NOUN
ejpam-6572	39	6	of	of	ADP
ejpam-6572	39	7	a	a	DET
ejpam-6572	39	8	bitopological	bitopological	ADJ
ejpam-6572	39	9	space	space	NOUN
ejpam-6572	39	10	(	(	PUNCT
ejpam-6572	39	11	x	x	NOUN
ejpam-6572	39	12	,	,	PUNCT
ejpam-6572	39	13	τ1	τ1	NOUN
ejpam-6572	39	14	,	,	PUNCT
ejpam-6572	39	15	τ2	τ2	NOUN
ejpam-6572	39	16	)	)	PUNCT
ejpam-6572	39	17	.	.	PUNCT
ejpam-6572	40	1	the	the	DET
ejpam-6572	40	2	closure	closure	NOUN
ejpam-6572	40	3	of	of	ADP
ejpam-6572	40	4	a	a	PRON
ejpam-6572	40	5	and	and	CCONJ
ejpam-6572	40	6	the	the	DET
ejpam-6572	40	7	interior	interior	NOUN
ejpam-6572	40	8	of	of	ADP
ejpam-6572	40	9	a	a	PRON
ejpam-6572	40	10	with	with	ADP
ejpam-6572	40	11	respect	respect	NOUN
ejpam-6572	40	12	to	to	ADP
ejpam-6572	40	13	τi	τi	PROPN
ejpam-6572	40	14	are	be	AUX
ejpam-6572	40	15	denoted	denote	VERB
ejpam-6572	40	16	by	by	ADP
ejpam-6572	40	17	τi	τi	NOUN
ejpam-6572	40	18	-	-	PUNCT
ejpam-6572	40	19	cl(a	cl(a	NUM
ejpam-6572	40	20	)	)	PUNCT
ejpam-6572	40	21	and	and	CCONJ
ejpam-6572	40	22	τi	τi	NOUN
ejpam-6572	40	23	-	-	PUNCT
ejpam-6572	40	24	int(a	int(a	NOUN
ejpam-6572	40	25	)	)	PUNCT
ejpam-6572	40	26	,	,	PUNCT
ejpam-6572	40	27	respectively	respectively	ADV
ejpam-6572	40	28	,	,	PUNCT
ejpam-6572	40	29	for	for	ADP
ejpam-6572	40	30	i	i	PROPN
ejpam-6572	40	31	=	=	SYM
ejpam-6572	40	32	1	1	NUM
ejpam-6572	40	33	,	,	PUNCT
ejpam-6572	40	34	2	2	NUM
ejpam-6572	40	35	.	.	X
ejpam-6572	40	36	a	a	DET
ejpam-6572	40	37	subset	subset	NOUN
ejpam-6572	40	38	a	a	PRON
ejpam-6572	40	39	of	of	ADP
ejpam-6572	40	40	a	a	DET
ejpam-6572	40	41	bitopological	bitopological	ADJ
ejpam-6572	40	42	space	space	NOUN
ejpam-6572	40	43	(	(	PUNCT
ejpam-6572	40	44	x	x	NOUN
ejpam-6572	40	45	,	,	PUNCT
ejpam-6572	40	46	τ1	τ1	NOUN
ejpam-6572	40	47	,	,	PUNCT
ejpam-6572	40	48	τ2	τ2	NOUN
ejpam-6572	40	49	)	)	PUNCT
ejpam-6572	40	50	is	be	AUX
ejpam-6572	40	51	called	call	VERB
ejpam-6572	40	52	τ1τ2	τ1τ2	VERB
ejpam-6572	40	53	-	-	ADJ
ejpam-6572	40	54	closed	closed	ADJ
ejpam-6572	40	55	[	[	X
ejpam-6572	40	56	20	20	NUM
ejpam-6572	40	57	]	]	PUNCT
ejpam-6572	40	58	if	if	SCONJ
ejpam-6572	40	59	a	a	DET
ejpam-6572	40	60	=	=	NOUN
ejpam-6572	40	61	τ1	τ1	NOUN
ejpam-6572	40	62	-	-	PUNCT
ejpam-6572	40	63	cl(τ2	cl(τ2	NOUN
ejpam-6572	40	64	-	-	PUNCT
ejpam-6572	40	65	cl(a	cl(a	NUM
ejpam-6572	40	66	)	)	PUNCT
ejpam-6572	40	67	)	)	PUNCT
ejpam-6572	40	68	.	.	PUNCT
ejpam-6572	41	1	the	the	DET
ejpam-6572	41	2	complement	complement	NOUN
ejpam-6572	41	3	of	of	ADP
ejpam-6572	41	4	a	a	DET
ejpam-6572	41	5	τ1τ2	τ1τ2	ADJ
ejpam-6572	41	6	-	-	ADJ
ejpam-6572	41	7	closed	closed	ADJ
ejpam-6572	41	8	set	set	NOUN
ejpam-6572	41	9	is	be	AUX
ejpam-6572	41	10	called	call	VERB
ejpam-6572	41	11	τ1τ2	τ1τ2	NOUN
ejpam-6572	41	12	-	-	ADJ
ejpam-6572	41	13	open	open	ADJ
ejpam-6572	41	14	.	.	PUNCT
ejpam-6572	42	1	the	the	DET
ejpam-6572	42	2	intersection	intersection	NOUN
ejpam-6572	42	3	of	of	ADP
ejpam-6572	42	4	all	all	DET
ejpam-6572	42	5	τ1τ2	τ1τ2	ADJ
ejpam-6572	42	6	-	-	ADJ
ejpam-6572	42	7	closed	closed	ADJ
ejpam-6572	42	8	sets	set	NOUN
ejpam-6572	42	9	of	of	ADP
ejpam-6572	42	10	x	x	PUNCT
ejpam-6572	42	11	containing	contain	VERB
ejpam-6572	42	12	a	a	PRON
ejpam-6572	42	13	is	be	AUX
ejpam-6572	42	14	called	call	VERB
ejpam-6572	42	15	the	the	DET
ejpam-6572	42	16	τ1τ2	τ1τ2	NOUN
ejpam-6572	42	17	-	-	NOUN
ejpam-6572	42	18	closure	closure	NOUN
ejpam-6572	42	19	[	[	X
ejpam-6572	42	20	20	20	NUM
ejpam-6572	42	21	]	]	PUNCT
ejpam-6572	42	22	of	of	ADP
ejpam-6572	42	23	a	a	PRON
ejpam-6572	42	24	and	and	CCONJ
ejpam-6572	42	25	is	be	AUX
ejpam-6572	42	26	denoted	denote	VERB
ejpam-6572	42	27	by	by	ADP
ejpam-6572	42	28	τ1τ2	τ1τ2	NOUN
ejpam-6572	42	29	-	-	NUM
ejpam-6572	42	30	cl(a	cl(a	NUM
ejpam-6572	42	31	)	)	PUNCT
ejpam-6572	42	32	.	.	PUNCT
ejpam-6572	43	1	the	the	DET
ejpam-6572	43	2	union	union	NOUN
ejpam-6572	43	3	of	of	ADP
ejpam-6572	43	4	all	all	DET
ejpam-6572	43	5	τ1τ2	τ1τ2	ADJ
ejpam-6572	43	6	-	-	ADJ
ejpam-6572	43	7	open	open	ADJ
ejpam-6572	43	8	sets	set	NOUN
ejpam-6572	43	9	of	of	ADP
ejpam-6572	43	10	x	x	PUNCT
ejpam-6572	43	11	contained	contain	VERB
ejpam-6572	43	12	in	in	ADP
ejpam-6572	43	13	a	a	PRON
ejpam-6572	43	14	is	be	AUX
ejpam-6572	43	15	called	call	VERB
ejpam-6572	43	16	the	the	DET
ejpam-6572	43	17	τ1τ2	τ1τ2	NOUN
ejpam-6572	43	18	-	-	ADJ
ejpam-6572	43	19	interior	interior	ADJ
ejpam-6572	43	20	[	[	X
ejpam-6572	43	21	20	20	NUM
ejpam-6572	43	22	]	]	PUNCT
ejpam-6572	43	23	of	of	ADP
ejpam-6572	43	24	a	a	PRON
ejpam-6572	43	25	and	and	CCONJ
ejpam-6572	43	26	is	be	AUX
ejpam-6572	43	27	denoted	denote	VERB
ejpam-6572	43	28	by	by	ADP
ejpam-6572	43	29	τ1τ2	τ1τ2	NOUN
ejpam-6572	43	30	-	-	ADJ
ejpam-6572	43	31	int(a	int(a	NOUN
ejpam-6572	43	32	)	)	PUNCT
ejpam-6572	43	33	.	.	PUNCT
ejpam-6572	44	1	lemma	lemma	PROPN
ejpam-6572	44	2	1	1	NUM
ejpam-6572	44	3	.	.	PUNCT
ejpam-6572	45	1	[	[	X
ejpam-6572	45	2	20	20	NUM
ejpam-6572	45	3	]	]	PUNCT
ejpam-6572	45	4	let	let	VERB
ejpam-6572	45	5	a	a	PRON
ejpam-6572	45	6	and	and	CCONJ
ejpam-6572	45	7	b	b	NOUN
ejpam-6572	45	8	be	be	AUX
ejpam-6572	45	9	subsets	subset	NOUN
ejpam-6572	45	10	of	of	ADP
ejpam-6572	45	11	a	a	DET
ejpam-6572	45	12	bitopological	bitopological	ADJ
ejpam-6572	45	13	space	space	NOUN
ejpam-6572	45	14	(	(	PUNCT
ejpam-6572	45	15	x	x	NOUN
ejpam-6572	45	16	,	,	PUNCT
ejpam-6572	45	17	τ1	τ1	NOUN
ejpam-6572	45	18	,	,	PUNCT
ejpam-6572	45	19	τ2	τ2	NOUN
ejpam-6572	45	20	)	)	PUNCT
ejpam-6572	45	21	.	.	PUNCT
ejpam-6572	46	1	for	for	ADP
ejpam-6572	46	2	the	the	DET
ejpam-6572	46	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6572	46	4	,	,	PUNCT
ejpam-6572	46	5	the	the	DET
ejpam-6572	46	6	following	follow	VERB
ejpam-6572	46	7	properties	property	NOUN
ejpam-6572	46	8	hold	hold	VERB
ejpam-6572	46	9	:	:	PUNCT
ejpam-6572	46	10	(	(	PUNCT
ejpam-6572	46	11	1	1	X
ejpam-6572	46	12	)	)	PUNCT
ejpam-6572	46	13	a	a	DET
ejpam-6572	46	14	⊆	⊆	NUM
ejpam-6572	46	15	τ1τ2	τ1τ2	NOUN
ejpam-6572	46	16	-	-	NUM
ejpam-6572	46	17	cl(a	cl(a	NUM
ejpam-6572	46	18	)	)	PUNCT
ejpam-6572	46	19	and	and	CCONJ
ejpam-6572	46	20	τ1τ2	τ1τ2	NOUN
ejpam-6572	46	21	-	-	ADJ
ejpam-6572	46	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6572	46	23	-	-	PUNCT
ejpam-6572	46	24	cl(a	cl(a	NUM
ejpam-6572	46	25	)	)	PUNCT
ejpam-6572	46	26	)	)	PUNCT
ejpam-6572	47	1	=	=	PUNCT
ejpam-6572	47	2	τ1τ2	τ1τ2	NOUN
ejpam-6572	47	3	-	-	NUM
ejpam-6572	47	4	cl(a	cl(a	NUM
ejpam-6572	47	5	)	)	PUNCT
ejpam-6572	47	6	.	.	PUNCT
ejpam-6572	48	1	(	(	PUNCT
ejpam-6572	48	2	2	2	X
ejpam-6572	48	3	)	)	PUNCT
ejpam-6572	48	4	if	if	SCONJ
ejpam-6572	48	5	a	a	DET
ejpam-6572	48	6	⊆	⊆	NUM
ejpam-6572	48	7	b	b	NOUN
ejpam-6572	48	8	,	,	PUNCT
ejpam-6572	48	9	then	then	ADV
ejpam-6572	48	10	τ1τ2	τ1τ2	NOUN
ejpam-6572	48	11	-	-	NUM
ejpam-6572	48	12	cl(a	cl(a	NUM
ejpam-6572	48	13	)	)	PUNCT
ejpam-6572	48	14	⊆	⊆	NUM
ejpam-6572	48	15	τ1τ2	τ1τ2	NOUN
ejpam-6572	48	16	-	-	NOUN
ejpam-6572	48	17	cl(b	cl(b	NOUN
ejpam-6572	48	18	)	)	PUNCT
ejpam-6572	48	19	.	.	PUNCT
ejpam-6572	49	1	(	(	PUNCT
ejpam-6572	49	2	3	3	X
ejpam-6572	49	3	)	)	PUNCT
ejpam-6572	49	4	τ1τ2	τ1τ2	NOUN
ejpam-6572	49	5	-	-	NUM
ejpam-6572	49	6	cl(a	cl(a	NUM
ejpam-6572	49	7	)	)	PUNCT
ejpam-6572	49	8	is	be	AUX
ejpam-6572	49	9	τ1τ2	τ1τ2	NOUN
ejpam-6572	49	10	-	-	ADJ
ejpam-6572	49	11	closed	closed	ADJ
ejpam-6572	49	12	.	.	PUNCT
ejpam-6572	50	1	(	(	PUNCT
ejpam-6572	50	2	4	4	X
ejpam-6572	50	3	)	)	PUNCT
ejpam-6572	50	4	a	a	PRON
ejpam-6572	50	5	is	be	AUX
ejpam-6572	50	6	τ1τ2	τ1τ2	NOUN
ejpam-6572	50	7	-	-	ADJ
ejpam-6572	50	8	closed	closed	ADJ
ejpam-6572	50	9	if	if	SCONJ
ejpam-6572	50	10	and	and	CCONJ
ejpam-6572	50	11	only	only	ADV
ejpam-6572	50	12	if	if	SCONJ
ejpam-6572	50	13	a	a	DET
ejpam-6572	50	14	=	=	PUNCT
ejpam-6572	50	15	τ1τ2	τ1τ2	NOUN
ejpam-6572	50	16	-	-	NUM
ejpam-6572	50	17	cl(a	cl(a	NUM
ejpam-6572	50	18	)	)	PUNCT
ejpam-6572	50	19	.	.	PUNCT
ejpam-6572	51	1	(	(	PUNCT
ejpam-6572	51	2	5	5	X
ejpam-6572	51	3	)	)	PUNCT
ejpam-6572	51	4	τ1τ2	τ1τ2	NOUN
ejpam-6572	51	5	-	-	NOUN
ejpam-6572	51	6	cl(x	cl(x	X
ejpam-6572	51	7	−a	−a	NOUN
ejpam-6572	51	8	)	)	PUNCT
ejpam-6572	52	1	=	=	PUNCT
ejpam-6572	52	2	x	x	X
ejpam-6572	53	1	−	−	ADP
ejpam-6572	53	2	τ1τ2	τ1τ2	NOUN
ejpam-6572	53	3	-	-	PUNCT
ejpam-6572	53	4	int(a	int(a	NOUN
ejpam-6572	53	5	)	)	PUNCT
ejpam-6572	53	6	.	.	PUNCT
ejpam-6572	54	1	a	a	DET
ejpam-6572	54	2	subseta	subseta	NOUN
ejpam-6572	54	3	of	of	ADP
ejpam-6572	54	4	a	a	DET
ejpam-6572	54	5	bitopological	bitopological	ADJ
ejpam-6572	54	6	space	space	NOUN
ejpam-6572	54	7	(	(	PUNCT
ejpam-6572	54	8	x	x	NOUN
ejpam-6572	54	9	,	,	PUNCT
ejpam-6572	54	10	τ1	τ1	NOUN
ejpam-6572	54	11	,	,	PUNCT
ejpam-6572	54	12	τ2	τ2	NOUN
ejpam-6572	54	13	)	)	PUNCT
ejpam-6572	54	14	is	be	AUX
ejpam-6572	54	15	called	call	VERB
ejpam-6572	54	16	(	(	PUNCT
ejpam-6572	54	17	τ1	τ1	NOUN
ejpam-6572	54	18	,	,	PUNCT
ejpam-6572	54	19	τ2)r	τ2)r	NOUN
ejpam-6572	54	20	-	-	PUNCT
ejpam-6572	54	21	open	open	NOUN
ejpam-6572	54	22	[	[	X
ejpam-6572	54	23	21	21	NUM
ejpam-6572	54	24	]	]	X
ejpam-6572	54	25	(	(	PUNCT
ejpam-6572	54	26	resp	resp	NOUN
ejpam-6572	54	27	.	.	PUNCT
ejpam-6572	55	1	(	(	PUNCT
ejpam-6572	55	2	τ1	τ1	NOUN
ejpam-6572	55	3	,	,	PUNCT
ejpam-6572	55	4	τ2)sopen	τ2)sopen	VERB
ejpam-6572	55	5	[	[	X
ejpam-6572	55	6	22	22	NUM
ejpam-6572	55	7	]	]	PUNCT
ejpam-6572	55	8	,	,	PUNCT
ejpam-6572	55	9	(	(	PUNCT
ejpam-6572	55	10	τ1	τ1	NOUN
ejpam-6572	55	11	,	,	PUNCT
ejpam-6572	55	12	τ2)p	τ2)p	NOUN
ejpam-6572	55	13	-	-	ADJ
ejpam-6572	55	14	open	open	ADJ
ejpam-6572	55	15	[	[	X
ejpam-6572	55	16	22	22	NUM
ejpam-6572	55	17	]	]	PUNCT
ejpam-6572	55	18	,	,	PUNCT
ejpam-6572	55	19	(	(	PUNCT
ejpam-6572	55	20	τ1	τ1	NOUN
ejpam-6572	55	21	,	,	PUNCT
ejpam-6572	55	22	τ2)β	τ2)β	ADJ
ejpam-6572	55	23	-	-	PUNCT
ejpam-6572	55	24	open	open	NOUN
ejpam-6572	56	1	[	[	X
ejpam-6572	56	2	22	22	NUM
ejpam-6572	56	3	]	]	SYM
ejpam-6572	56	4	)	)	PUNCT
ejpam-6572	56	5	if	if	SCONJ
ejpam-6572	56	6	a	a	DET
ejpam-6572	56	7	=	=	PUNCT
ejpam-6572	56	8	τ1τ2	τ1τ2	NOUN
ejpam-6572	56	9	-	-	NOUN
ejpam-6572	56	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6572	56	11	-	-	PUNCT
ejpam-6572	56	12	cl(a	cl(a	NUM
ejpam-6572	56	13	)	)	PUNCT
ejpam-6572	56	14	)	)	PUNCT
ejpam-6572	56	15	(	(	PUNCT
ejpam-6572	56	16	resp	resp	NOUN
ejpam-6572	56	17	.	.	PUNCT
ejpam-6572	57	1	a	a	DET
ejpam-6572	57	2	⊆	⊆	NUM
ejpam-6572	57	3	τ1τ2	τ1τ2	NOUN
ejpam-6572	57	4	-	-	ADJ
ejpam-6572	57	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6572	57	6	-	-	PUNCT
ejpam-6572	57	7	int(a	int(a	NOUN
ejpam-6572	57	8	)	)	PUNCT
ejpam-6572	57	9	)	)	PUNCT
ejpam-6572	57	10	,	,	PUNCT
ejpam-6572	57	11	a	a	DET
ejpam-6572	57	12	⊆	⊆	NUM
ejpam-6572	57	13	τ1τ2	τ1τ2	NOUN
ejpam-6572	57	14	-	-	NOUN
ejpam-6572	57	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6572	57	16	-	-	PUNCT
ejpam-6572	57	17	cl(a	cl(a	NUM
ejpam-6572	57	18	)	)	PUNCT
ejpam-6572	57	19	)	)	PUNCT
ejpam-6572	57	20	,	,	PUNCT
ejpam-6572	57	21	a	a	DET
ejpam-6572	57	22	⊆	⊆	NUM
ejpam-6572	57	23	τ1τ2	τ1τ2	NOUN
ejpam-6572	57	24	-	-	PUNCT
ejpam-6572	57	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6572	57	26	-	-	PUNCT
ejpam-6572	57	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6572	57	28	-	-	PUNCT
ejpam-6572	57	29	cl(a	cl(a	NUM
ejpam-6572	57	30	)	)	PUNCT
ejpam-6572	57	31	)	)	PUNCT
ejpam-6572	57	32	)	)	PUNCT
ejpam-6572	57	33	)	)	PUNCT
ejpam-6572	57	34	.	.	PUNCT
ejpam-6572	58	1	the	the	DET
ejpam-6572	58	2	complement	complement	NOUN
ejpam-6572	58	3	of	of	ADP
ejpam-6572	58	4	a	a	DET
ejpam-6572	58	5	(	(	PUNCT
ejpam-6572	58	6	τ1	τ1	NOUN
ejpam-6572	58	7	,	,	PUNCT
ejpam-6572	58	8	τ2)r	τ2)r	NOUN
ejpam-6572	58	9	-	-	PUNCT
ejpam-6572	58	10	open	open	ADJ
ejpam-6572	58	11	(	(	PUNCT
ejpam-6572	58	12	resp	resp	NOUN
ejpam-6572	58	13	.	.	PUNCT
ejpam-6572	59	1	(	(	PUNCT
ejpam-6572	59	2	τ1	τ1	NOUN
ejpam-6572	59	3	,	,	PUNCT
ejpam-6572	59	4	τ2)s	τ2)s	NOUN
ejpam-6572	59	5	-	-	PUNCT
ejpam-6572	59	6	open	open	ADJ
ejpam-6572	59	7	,	,	PUNCT
ejpam-6572	59	8	(	(	PUNCT
ejpam-6572	59	9	τ1	τ1	NOUN
ejpam-6572	59	10	,	,	PUNCT
ejpam-6572	59	11	τ2)p	τ2)p	NOUN
ejpam-6572	59	12	-	-	ADJ
ejpam-6572	59	13	open	open	ADJ
ejpam-6572	59	14	,	,	PUNCT
ejpam-6572	59	15	(	(	PUNCT
ejpam-6572	59	16	τ1	τ1	NOUN
ejpam-6572	59	17	,	,	PUNCT
ejpam-6572	59	18	τ2)β	τ2)β	ADJ
ejpam-6572	59	19	-	-	PUNCT
ejpam-6572	59	20	open	open	ADJ
ejpam-6572	59	21	)	)	PUNCT
ejpam-6572	59	22	set	set	NOUN
ejpam-6572	59	23	is	be	AUX
ejpam-6572	59	24	said	say	VERB
ejpam-6572	59	25	to	to	PART
ejpam-6572	59	26	be	be	AUX
ejpam-6572	59	27	(	(	PUNCT
ejpam-6572	59	28	τ1	τ1	NOUN
ejpam-6572	59	29	,	,	PUNCT
ejpam-6572	59	30	τ2)r	τ2)r	NOUN
ejpam-6572	59	31	-	-	PUNCT
ejpam-6572	59	32	closed	closed	ADJ
ejpam-6572	59	33	(	(	PUNCT
ejpam-6572	59	34	resp	resp	NOUN
ejpam-6572	59	35	.	.	PUNCT
ejpam-6572	60	1	(	(	PUNCT
ejpam-6572	60	2	τ1	τ1	NOUN
ejpam-6572	60	3	,	,	PUNCT
ejpam-6572	60	4	τ2)s	τ2)s	NOUN
ejpam-6572	60	5	-	-	PUNCT
ejpam-6572	60	6	closed	closed	ADJ
ejpam-6572	60	7	,	,	PUNCT
ejpam-6572	60	8	(	(	PUNCT
ejpam-6572	60	9	τ1	τ1	NOUN
ejpam-6572	60	10	,	,	PUNCT
ejpam-6572	60	11	τ2)p	τ2)p	NOUN
ejpam-6572	60	12	-	-	PUNCT
ejpam-6572	60	13	closed	closed	ADJ
ejpam-6572	60	14	,	,	PUNCT
ejpam-6572	60	15	(	(	PUNCT
ejpam-6572	60	16	τ1	τ1	NOUN
ejpam-6572	60	17	,	,	PUNCT
ejpam-6572	60	18	τ2)β	τ2)β	ADJ
ejpam-6572	60	19	-	-	PUNCT
ejpam-6572	60	20	closed	closed	ADJ
ejpam-6572	60	21	)	)	PUNCT
ejpam-6572	60	22	.	.	PUNCT
ejpam-6572	61	1	a	a	DET
ejpam-6572	61	2	subset	subset	NOUN
ejpam-6572	61	3	a	a	PRON
ejpam-6572	61	4	of	of	ADP
ejpam-6572	61	5	a	a	DET
ejpam-6572	61	6	bitopological	bitopological	ADJ
ejpam-6572	61	7	space	space	NOUN
ejpam-6572	61	8	(	(	PUNCT
ejpam-6572	61	9	x	x	NOUN
ejpam-6572	61	10	,	,	PUNCT
ejpam-6572	61	11	τ1	τ1	NOUN
ejpam-6572	61	12	,	,	PUNCT
ejpam-6572	61	13	τ2	τ2	NOUN
ejpam-6572	61	14	)	)	PUNCT
ejpam-6572	61	15	is	be	AUX
ejpam-6572	61	16	said	say	VERB
ejpam-6572	61	17	to	to	PART
ejpam-6572	61	18	be	be	AUX
ejpam-6572	61	19	α(τ1	α(τ1	NOUN
ejpam-6572	61	20	,	,	PUNCT
ejpam-6572	61	21	τ2)-open	τ2)-open	ADJ
ejpam-6572	61	22	[	[	X
ejpam-6572	61	23	23	23	NUM
ejpam-6572	61	24	]	]	PUNCT
ejpam-6572	61	25	if	if	SCONJ
ejpam-6572	61	26	a	a	DET
ejpam-6572	61	27	⊆	⊆	NUM
ejpam-6572	61	28	b.	b.	PROPN
ejpam-6572	61	29	kong	kong	PROPN
ejpam-6572	61	30	-	-	PUNCT
ejpam-6572	61	31	ied	ied	PROPN
ejpam-6572	61	32	,	,	PUNCT
ejpam-6572	61	33	a.	a.	PROPN
ejpam-6572	61	34	sama	sama	PROPN
ejpam-6572	61	35	-	-	PUNCT
ejpam-6572	61	36	ae	ae	PROPN
ejpam-6572	61	37	,	,	PUNCT
ejpam-6572	61	38	c.	c.	PROPN
ejpam-6572	61	39	boonpok	boonpok	PROPN
ejpam-6572	61	40	/	/	SYM
ejpam-6572	61	41	eur	eur	PROPN
ejpam-6572	61	42	.	.	PUNCT
ejpam-6572	62	1	j.	j.	PROPN
ejpam-6572	62	2	pure	pure	PROPN
ejpam-6572	62	3	appl	appl	PROPN
ejpam-6572	62	4	.	.	PROPN
ejpam-6572	62	5	math	math	PROPN
ejpam-6572	62	6	,	,	PUNCT
ejpam-6572	62	7	18	18	NUM
ejpam-6572	62	8	(	(	PUNCT
ejpam-6572	62	9	3	3	NUM
ejpam-6572	62	10	)	)	PUNCT
ejpam-6572	62	11	(	(	PUNCT
ejpam-6572	62	12	2025	2025	NUM
ejpam-6572	62	13	)	)	PUNCT
ejpam-6572	62	14	,	,	PUNCT
ejpam-6572	62	15	6572	6572	NUM
ejpam-6572	62	16	3	3	NUM
ejpam-6572	62	17	of	of	ADP
ejpam-6572	62	18	12	12	NUM
ejpam-6572	62	19	τ1τ2	τ1τ2	NOUN
ejpam-6572	62	20	-	-	NOUN
ejpam-6572	62	21	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6572	62	22	-	-	PUNCT
ejpam-6572	62	23	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6572	62	24	-	-	PUNCT
ejpam-6572	62	25	int(a	int(a	NOUN
ejpam-6572	62	26	)	)	PUNCT
ejpam-6572	62	27	)	)	PUNCT
ejpam-6572	62	28	)	)	PUNCT
ejpam-6572	62	29	.	.	PUNCT
ejpam-6572	63	1	the	the	DET
ejpam-6572	63	2	complement	complement	NOUN
ejpam-6572	63	3	of	of	ADP
ejpam-6572	63	4	an	an	DET
ejpam-6572	63	5	α(τ1	α(τ1	NOUN
ejpam-6572	63	6	,	,	PUNCT
ejpam-6572	63	7	τ2)-open	τ2)-open	ADJ
ejpam-6572	63	8	set	set	NOUN
ejpam-6572	63	9	is	be	AUX
ejpam-6572	63	10	said	say	VERB
ejpam-6572	63	11	to	to	PART
ejpam-6572	63	12	be	be	AUX
ejpam-6572	63	13	α(τ1	α(τ1	NOUN
ejpam-6572	63	14	,	,	PUNCT
ejpam-6572	63	15	τ2)-closed	τ2)-close	VERB
ejpam-6572	63	16	.	.	PUNCT
ejpam-6572	64	1	let	let	VERB
ejpam-6572	64	2	a	a	DET
ejpam-6572	64	3	be	be	AUX
ejpam-6572	64	4	a	a	DET
ejpam-6572	64	5	subset	subset	NOUN
ejpam-6572	64	6	of	of	ADP
ejpam-6572	64	7	a	a	DET
ejpam-6572	64	8	bitopological	bitopological	ADJ
ejpam-6572	64	9	space	space	NOUN
ejpam-6572	64	10	(	(	PUNCT
ejpam-6572	64	11	x	x	NOUN
ejpam-6572	64	12	,	,	PUNCT
ejpam-6572	64	13	τ1	τ1	NOUN
ejpam-6572	64	14	,	,	PUNCT
ejpam-6572	64	15	τ2	τ2	NOUN
ejpam-6572	64	16	)	)	PUNCT
ejpam-6572	64	17	.	.	PUNCT
ejpam-6572	65	1	the	the	DET
ejpam-6572	65	2	intersection	intersection	NOUN
ejpam-6572	65	3	of	of	ADP
ejpam-6572	65	4	all	all	DET
ejpam-6572	65	5	(	(	PUNCT
ejpam-6572	65	6	τ1	τ1	NOUN
ejpam-6572	65	7	,	,	PUNCT
ejpam-6572	65	8	τ2)p	τ2)p	NOUN
ejpam-6572	65	9	-	-	PUNCT
ejpam-6572	65	10	closed	closed	ADJ
ejpam-6572	65	11	(	(	PUNCT
ejpam-6572	65	12	resp	resp	NOUN
ejpam-6572	65	13	.	.	PUNCT
ejpam-6572	66	1	(	(	PUNCT
ejpam-6572	66	2	τ1	τ1	NOUN
ejpam-6572	66	3	,	,	PUNCT
ejpam-6572	66	4	τ2)s	τ2)s	NOUN
ejpam-6572	66	5	-	-	PUNCT
ejpam-6572	66	6	closed	closed	ADJ
ejpam-6572	66	7	,	,	PUNCT
ejpam-6572	66	8	α(τ1	α(τ1	NOUN
ejpam-6572	66	9	,	,	PUNCT
ejpam-6572	66	10	τ2)-closed	τ2)-closed	ADJ
ejpam-6572	66	11	)	)	PUNCT
ejpam-6572	66	12	sets	set	NOUN
ejpam-6572	66	13	of	of	ADP
ejpam-6572	66	14	x	x	PUNCT
ejpam-6572	66	15	containing	contain	VERB
ejpam-6572	66	16	a	a	PRON
ejpam-6572	66	17	is	be	AUX
ejpam-6572	66	18	called	call	VERB
ejpam-6572	66	19	the	the	DET
ejpam-6572	66	20	(	(	PUNCT
ejpam-6572	66	21	τ1	τ1	NOUN
ejpam-6572	66	22	,	,	PUNCT
ejpam-6572	66	23	τ2)p	τ2)p	NOUN
ejpam-6572	66	24	-	-	NOUN
ejpam-6572	66	25	closure	closure	NOUN
ejpam-6572	66	26	[	[	X
ejpam-6572	66	27	24	24	NUM
ejpam-6572	66	28	]	]	PUNCT
ejpam-6572	66	29	(	(	PUNCT
ejpam-6572	66	30	resp	resp	NOUN
ejpam-6572	66	31	.	.	PUNCT
ejpam-6572	67	1	(	(	PUNCT
ejpam-6572	67	2	τ1	τ1	NOUN
ejpam-6572	67	3	,	,	PUNCT
ejpam-6572	67	4	τ2)s	τ2)s	NOUN
ejpam-6572	67	5	-	-	PUNCT
ejpam-6572	67	6	closure	closure	NOUN
ejpam-6572	67	7	[	[	X
ejpam-6572	67	8	22	22	NUM
ejpam-6572	67	9	]	]	PUNCT
ejpam-6572	67	10	,	,	PUNCT
ejpam-6572	67	11	α(τ1	α(τ1	NOUN
ejpam-6572	67	12	,	,	PUNCT
ejpam-6572	67	13	τ2)-closure	τ2)-closure	NOUN
ejpam-6572	67	14	[	[	X
ejpam-6572	67	15	25	25	NUM
ejpam-6572	67	16	]	]	PUNCT
ejpam-6572	67	17	)	)	PUNCT
ejpam-6572	67	18	of	of	ADP
ejpam-6572	67	19	a	a	PRON
ejpam-6572	67	20	and	and	CCONJ
ejpam-6572	67	21	is	be	AUX
ejpam-6572	67	22	denoted	denote	VERB
ejpam-6572	67	23	by	by	ADP
ejpam-6572	67	24	(	(	PUNCT
ejpam-6572	67	25	τ1	τ1	NOUN
ejpam-6572	67	26	,	,	PUNCT
ejpam-6572	67	27	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6572	67	28	)	)	PUNCT
ejpam-6572	67	29	(	(	PUNCT
ejpam-6572	67	30	resp	resp	NOUN
ejpam-6572	67	31	.	.	PUNCT
ejpam-6572	68	1	(	(	PUNCT
ejpam-6572	68	2	τ1	τ1	NOUN
ejpam-6572	68	3	,	,	PUNCT
ejpam-6572	68	4	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6572	68	5	)	)	PUNCT
ejpam-6572	68	6	,	,	PUNCT
ejpam-6572	68	7	α(τ1	α(τ1	NOUN
ejpam-6572	68	8	,	,	PUNCT
ejpam-6572	68	9	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-6572	68	10	)	)	PUNCT
ejpam-6572	68	11	)	)	PUNCT
ejpam-6572	68	12	.	.	PUNCT
ejpam-6572	69	1	the	the	DET
ejpam-6572	69	2	union	union	NOUN
ejpam-6572	69	3	of	of	ADP
ejpam-6572	69	4	all	all	DET
ejpam-6572	69	5	(	(	PUNCT
ejpam-6572	69	6	τ1	τ1	NOUN
ejpam-6572	69	7	,	,	PUNCT
ejpam-6572	69	8	τ2)p	τ2)p	NOUN
ejpam-6572	69	9	-	-	ADJ
ejpam-6572	69	10	open	open	ADJ
ejpam-6572	69	11	(	(	PUNCT
ejpam-6572	69	12	resp	resp	NOUN
ejpam-6572	69	13	.	.	PUNCT
ejpam-6572	70	1	(	(	PUNCT
ejpam-6572	70	2	τ1	τ1	NOUN
ejpam-6572	70	3	,	,	PUNCT
ejpam-6572	70	4	τ2)s	τ2)s	NOUN
ejpam-6572	70	5	-	-	PUNCT
ejpam-6572	70	6	open	open	ADJ
ejpam-6572	70	7	,	,	PUNCT
ejpam-6572	70	8	α(τ1	α(τ1	NOUN
ejpam-6572	70	9	,	,	PUNCT
ejpam-6572	70	10	τ2)-open	τ2)-open	ADJ
ejpam-6572	70	11	)	)	PUNCT
ejpam-6572	70	12	sets	set	NOUN
ejpam-6572	70	13	of	of	ADP
ejpam-6572	70	14	x	x	PUNCT
ejpam-6572	70	15	contained	contain	VERB
ejpam-6572	70	16	in	in	ADP
ejpam-6572	70	17	a	a	PRON
ejpam-6572	70	18	is	be	AUX
ejpam-6572	70	19	called	call	VERB
ejpam-6572	70	20	the	the	DET
ejpam-6572	70	21	(	(	PUNCT
ejpam-6572	70	22	τ1	τ1	NOUN
ejpam-6572	70	23	,	,	PUNCT
ejpam-6572	70	24	τ2)p	τ2)p	ADJ
ejpam-6572	70	25	-	-	NOUN
ejpam-6572	70	26	interior	interior	ADJ
ejpam-6572	70	27	[	[	X
ejpam-6572	70	28	24	24	NUM
ejpam-6572	70	29	]	]	PUNCT
ejpam-6572	70	30	(	(	PUNCT
ejpam-6572	70	31	resp	resp	NOUN
ejpam-6572	70	32	.	.	PUNCT
ejpam-6572	71	1	(	(	PUNCT
ejpam-6572	71	2	τ1	τ1	NOUN
ejpam-6572	71	3	,	,	PUNCT
ejpam-6572	71	4	τ2)s	τ2)s	NOUN
ejpam-6572	71	5	-	-	ADJ
ejpam-6572	71	6	interior	interior	ADJ
ejpam-6572	71	7	[	[	X
ejpam-6572	71	8	22	22	NUM
ejpam-6572	71	9	]	]	PUNCT
ejpam-6572	71	10	,	,	PUNCT
ejpam-6572	71	11	α(τ1	α(τ1	NOUN
ejpam-6572	71	12	,	,	PUNCT
ejpam-6572	71	13	τ2)-interior	τ2)-interior	PROPN
ejpam-6572	71	14	[	[	X
ejpam-6572	71	15	25	25	NUM
ejpam-6572	71	16	]	]	PUNCT
ejpam-6572	71	17	)	)	PUNCT
ejpam-6572	71	18	of	of	ADP
ejpam-6572	71	19	a	a	PRON
ejpam-6572	71	20	and	and	CCONJ
ejpam-6572	71	21	is	be	AUX
ejpam-6572	71	22	denoted	denote	VERB
ejpam-6572	71	23	by	by	ADP
ejpam-6572	71	24	(	(	PUNCT
ejpam-6572	71	25	τ1	τ1	NOUN
ejpam-6572	71	26	,	,	PUNCT
ejpam-6572	71	27	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6572	71	28	)	)	PUNCT
ejpam-6572	71	29	(	(	PUNCT
ejpam-6572	71	30	resp	resp	NOUN
ejpam-6572	71	31	.	.	PUNCT
ejpam-6572	72	1	(	(	PUNCT
ejpam-6572	72	2	τ1	τ1	NOUN
ejpam-6572	72	3	,	,	PUNCT
ejpam-6572	72	4	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6572	72	5	)	)	PUNCT
ejpam-6572	72	6	,	,	PUNCT
ejpam-6572	72	7	α(τ1	α(τ1	NOUN
ejpam-6572	72	8	,	,	PUNCT
ejpam-6572	72	9	τ2)-int(a	τ2)-int(a	NOUN
ejpam-6572	72	10	)	)	PUNCT
ejpam-6572	72	11	)	)	PUNCT
ejpam-6572	72	12	.	.	PUNCT
ejpam-6572	73	1	lemma	lemma	PROPN
ejpam-6572	73	2	2	2	NUM
ejpam-6572	73	3	.	.	PUNCT
ejpam-6572	74	1	[	[	X
ejpam-6572	74	2	18	18	NUM
ejpam-6572	74	3	]	]	PUNCT
ejpam-6572	74	4	for	for	ADP
ejpam-6572	74	5	a	a	DET
ejpam-6572	74	6	subset	subset	NOUN
ejpam-6572	74	7	a	a	PRON
ejpam-6572	74	8	of	of	ADP
ejpam-6572	74	9	a	a	DET
ejpam-6572	74	10	bitopological	bitopological	ADJ
ejpam-6572	74	11	space	space	NOUN
ejpam-6572	74	12	(	(	PUNCT
ejpam-6572	74	13	x	x	NOUN
ejpam-6572	74	14	,	,	PUNCT
ejpam-6572	74	15	τ1	τ1	NOUN
ejpam-6572	74	16	,	,	PUNCT
ejpam-6572	74	17	τ2	τ2	NOUN
ejpam-6572	74	18	)	)	PUNCT
ejpam-6572	74	19	,	,	PUNCT
ejpam-6572	74	20	the	the	DET
ejpam-6572	74	21	following	follow	VERB
ejpam-6572	74	22	properties	property	NOUN
ejpam-6572	74	23	hold	hold	VERB
ejpam-6572	74	24	:	:	PUNCT
ejpam-6572	74	25	(	(	PUNCT
ejpam-6572	74	26	1	1	X
ejpam-6572	74	27	)	)	PUNCT
ejpam-6572	74	28	(	(	PUNCT
ejpam-6572	74	29	τ1	τ1	NOUN
ejpam-6572	74	30	,	,	PUNCT
ejpam-6572	74	31	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6572	74	32	)	)	PUNCT
ejpam-6572	74	33	=	=	PUNCT
ejpam-6572	75	1	τ1τ2	τ1τ2	NOUN
ejpam-6572	75	2	-	-	NOUN
ejpam-6572	75	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6572	75	4	-	-	PUNCT
ejpam-6572	75	5	cl(a	cl(a	NUM
ejpam-6572	75	6	)	)	PUNCT
ejpam-6572	75	7	)	)	PUNCT
ejpam-6572	75	8	∪a	∪a	X
ejpam-6572	76	1	[	[	X
ejpam-6572	76	2	17	17	NUM
ejpam-6572	76	3	]	]	SYM
ejpam-6572	76	4	;	;	PUNCT
ejpam-6572	76	5	(	(	PUNCT
ejpam-6572	76	6	2	2	X
ejpam-6572	76	7	)	)	PUNCT
ejpam-6572	76	8	(	(	PUNCT
ejpam-6572	76	9	τ1	τ1	NOUN
ejpam-6572	76	10	,	,	PUNCT
ejpam-6572	76	11	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6572	76	12	)	)	PUNCT
ejpam-6572	77	1	=	=	PUNCT
ejpam-6572	77	2	τ1τ2	τ1τ2	NOUN
ejpam-6572	77	3	-	-	ADJ
ejpam-6572	77	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6572	77	5	-	-	PUNCT
ejpam-6572	77	6	int(a	int(a	NOUN
ejpam-6572	77	7	)	)	PUNCT
ejpam-6572	77	8	)	)	PUNCT
ejpam-6572	78	1	∩a	∩a	PROPN
ejpam-6572	78	2	.	.	PUNCT
ejpam-6572	79	1	for	for	ADP
ejpam-6572	79	2	a	a	DET
ejpam-6572	79	3	subset	subset	NOUN
ejpam-6572	79	4	a	a	PRON
ejpam-6572	79	5	of	of	ADP
ejpam-6572	79	6	a	a	DET
ejpam-6572	79	7	bitopological	bitopological	ADJ
ejpam-6572	79	8	space	space	NOUN
ejpam-6572	79	9	(	(	PUNCT
ejpam-6572	79	10	x	x	NOUN
ejpam-6572	79	11	,	,	PUNCT
ejpam-6572	79	12	τ1	τ1	NOUN
ejpam-6572	79	13	,	,	PUNCT
ejpam-6572	79	14	τ2	τ2	PROPN
ejpam-6572	79	15	)	)	PUNCT
ejpam-6572	79	16	,	,	PUNCT
ejpam-6572	79	17	a	a	DET
ejpam-6572	79	18	point	point	NOUN
ejpam-6572	79	19	x	x	X
ejpam-6572	79	20	∈	∈	NOUN
ejpam-6572	79	21	x	x	PUNCT
ejpam-6572	79	22	is	be	AUX
ejpam-6572	79	23	called	call	VERB
ejpam-6572	79	24	a	a	DET
ejpam-6572	79	25	(	(	PUNCT
ejpam-6572	79	26	τ1	τ1	NOUN
ejpam-6572	79	27	,	,	PUNCT
ejpam-6572	79	28	τ2)θcluster	τ2)θcluster	NOUN
ejpam-6572	79	29	point	point	NOUN
ejpam-6572	79	30	of	of	ADP
ejpam-6572	79	31	a	a	DET
ejpam-6572	79	32	if	if	SCONJ
ejpam-6572	79	33	τ1τ2	τ1τ2	NOUN
ejpam-6572	79	34	-	-	NOUN
ejpam-6572	79	35	cl(u	cl(u	NOUN
ejpam-6572	79	36	)	)	PUNCT
ejpam-6572	79	37	∩	∩	NOUN
ejpam-6572	79	38	a	a	DET
ejpam-6572	79	39	̸=	̸=	PROPN
ejpam-6572	79	40	∅	∅	NOUN
ejpam-6572	79	41	for	for	ADP
ejpam-6572	79	42	every	every	DET
ejpam-6572	79	43	τ1τ2	τ1τ2	ADJ
ejpam-6572	79	44	-	-	ADJ
ejpam-6572	79	45	open	open	ADJ
ejpam-6572	79	46	set	set	NOUN
ejpam-6572	79	47	u	u	NOUN
ejpam-6572	79	48	containing	contain	VERB
ejpam-6572	79	49	x.	x.	NOUN
ejpam-6572	79	50	the	the	DET
ejpam-6572	79	51	set	set	NOUN
ejpam-6572	79	52	of	of	ADP
ejpam-6572	79	53	all	all	DET
ejpam-6572	79	54	(	(	PUNCT
ejpam-6572	79	55	τ1	τ1	NOUN
ejpam-6572	79	56	,	,	PUNCT
ejpam-6572	79	57	τ2)θ	τ2)θ	ADJ
ejpam-6572	79	58	-	-	PUNCT
ejpam-6572	79	59	cluster	cluster	NOUN
ejpam-6572	79	60	points	point	NOUN
ejpam-6572	79	61	of	of	ADP
ejpam-6572	79	62	a	a	PRON
ejpam-6572	79	63	is	be	AUX
ejpam-6572	79	64	called	call	VERB
ejpam-6572	79	65	the	the	DET
ejpam-6572	79	66	(	(	PUNCT
ejpam-6572	79	67	τ1	τ1	NOUN
ejpam-6572	79	68	,	,	PUNCT
ejpam-6572	79	69	τ2)θ	τ2)θ	NOUN
ejpam-6572	79	70	-	-	PUNCT
ejpam-6572	79	71	closure	closure	NOUN
ejpam-6572	79	72	of	of	ADP
ejpam-6572	79	73	a	a	PRON
ejpam-6572	79	74	and	and	CCONJ
ejpam-6572	79	75	is	be	AUX
ejpam-6572	79	76	denoted	denote	VERB
ejpam-6572	79	77	by	by	ADP
ejpam-6572	79	78	(	(	PUNCT
ejpam-6572	79	79	τ1	τ1	NOUN
ejpam-6572	79	80	,	,	PUNCT
ejpam-6572	79	81	τ2)θ	τ2)θ	NOUN
ejpam-6572	79	82	-	-	PUNCT
ejpam-6572	79	83	cl(a	cl(a	NUM
ejpam-6572	79	84	)	)	PUNCT
ejpam-6572	79	85	.	.	PUNCT
ejpam-6572	80	1	a	a	DET
ejpam-6572	80	2	subset	subset	NOUN
ejpam-6572	80	3	a	a	PRON
ejpam-6572	80	4	of	of	ADP
ejpam-6572	80	5	a	a	DET
ejpam-6572	80	6	bitopological	bitopological	ADJ
ejpam-6572	80	7	space	space	NOUN
ejpam-6572	80	8	(	(	PUNCT
ejpam-6572	80	9	x	x	NOUN
ejpam-6572	80	10	,	,	PUNCT
ejpam-6572	80	11	τ1	τ1	NOUN
ejpam-6572	80	12	,	,	PUNCT
ejpam-6572	80	13	τ2	τ2	NOUN
ejpam-6572	80	14	)	)	PUNCT
ejpam-6572	80	15	is	be	AUX
ejpam-6572	80	16	said	say	VERB
ejpam-6572	80	17	to	to	PART
ejpam-6572	80	18	be	be	AUX
ejpam-6572	80	19	(	(	PUNCT
ejpam-6572	80	20	τ1	τ1	NOUN
ejpam-6572	80	21	,	,	PUNCT
ejpam-6572	80	22	τ2)θ	τ2)θ	NOUN
ejpam-6572	80	23	-	-	PUNCT
ejpam-6572	80	24	closed	closed	ADJ
ejpam-6572	80	25	if	if	SCONJ
ejpam-6572	80	26	(	(	PUNCT
ejpam-6572	80	27	τ1	τ1	NOUN
ejpam-6572	80	28	,	,	PUNCT
ejpam-6572	80	29	τ2)θ	τ2)θ	NOUN
ejpam-6572	80	30	-	-	PUNCT
ejpam-6572	80	31	cl(a	cl(a	NUM
ejpam-6572	80	32	)	)	PUNCT
ejpam-6572	80	33	=	=	PUNCT
ejpam-6572	80	34	a.	a.	NOUN
ejpam-6572	80	35	the	the	DET
ejpam-6572	80	36	complement	complement	NOUN
ejpam-6572	80	37	of	of	ADP
ejpam-6572	80	38	a	a	DET
ejpam-6572	80	39	(	(	PUNCT
ejpam-6572	80	40	τ1	τ1	NOUN
ejpam-6572	80	41	,	,	PUNCT
ejpam-6572	80	42	τ2)θ	τ2)θ	ADJ
ejpam-6572	80	43	-	-	PUNCT
ejpam-6572	80	44	closed	close	VERB
ejpam-6572	80	45	set	set	NOUN
ejpam-6572	80	46	is	be	AUX
ejpam-6572	80	47	said	say	VERB
ejpam-6572	80	48	to	to	PART
ejpam-6572	80	49	be	be	AUX
ejpam-6572	80	50	(	(	PUNCT
ejpam-6572	80	51	τ1	τ1	NOUN
ejpam-6572	80	52	,	,	PUNCT
ejpam-6572	80	53	τ2)θ	τ2)θ	NOUN
ejpam-6572	80	54	-	-	PUNCT
ejpam-6572	80	55	open	open	ADJ
ejpam-6572	80	56	.	.	PUNCT
ejpam-6572	81	1	the	the	DET
ejpam-6572	81	2	union	union	NOUN
ejpam-6572	81	3	of	of	ADP
ejpam-6572	81	4	all	all	DET
ejpam-6572	81	5	(	(	PUNCT
ejpam-6572	81	6	τ1	τ1	NOUN
ejpam-6572	81	7	,	,	PUNCT
ejpam-6572	81	8	τ2)θ	τ2)θ	ADJ
ejpam-6572	81	9	-	-	PUNCT
ejpam-6572	81	10	open	open	ADJ
ejpam-6572	81	11	sets	set	NOUN
ejpam-6572	81	12	of	of	ADP
ejpam-6572	81	13	x	x	PUNCT
ejpam-6572	81	14	contained	contain	VERB
ejpam-6572	81	15	in	in	ADP
ejpam-6572	81	16	a	a	PRON
ejpam-6572	81	17	is	be	AUX
ejpam-6572	81	18	called	call	VERB
ejpam-6572	81	19	the	the	DET
ejpam-6572	81	20	(	(	PUNCT
ejpam-6572	81	21	τ1	τ1	NOUN
ejpam-6572	81	22	,	,	PUNCT
ejpam-6572	81	23	τ2)θ	τ2)θ	ADJ
ejpam-6572	81	24	-	-	PUNCT
ejpam-6572	81	25	interior	interior	NOUN
ejpam-6572	81	26	of	of	ADP
ejpam-6572	81	27	a	a	PRON
ejpam-6572	81	28	and	and	CCONJ
ejpam-6572	81	29	is	be	AUX
ejpam-6572	81	30	denoted	denote	VERB
ejpam-6572	81	31	by	by	ADP
ejpam-6572	81	32	(	(	PUNCT
ejpam-6572	81	33	τ1	τ1	NOUN
ejpam-6572	81	34	,	,	PUNCT
ejpam-6572	81	35	τ2)θ	τ2)θ	NOUN
ejpam-6572	81	36	-	-	PUNCT
ejpam-6572	81	37	int(a	int(a	NOUN
ejpam-6572	81	38	)	)	PUNCT
ejpam-6572	82	1	[	[	X
ejpam-6572	82	2	21	21	NUM
ejpam-6572	82	3	]	]	PUNCT
ejpam-6572	82	4	.	.	PUNCT
ejpam-6572	83	1	lemma	lemma	PROPN
ejpam-6572	83	2	3	3	X
ejpam-6572	83	3	.	.	PUNCT
ejpam-6572	84	1	[	[	X
ejpam-6572	84	2	21	21	NUM
ejpam-6572	84	3	]	]	PUNCT
ejpam-6572	84	4	for	for	ADP
ejpam-6572	84	5	a	a	DET
ejpam-6572	84	6	subset	subset	NOUN
ejpam-6572	84	7	a	a	PRON
ejpam-6572	84	8	of	of	ADP
ejpam-6572	84	9	a	a	DET
ejpam-6572	84	10	bitopological	bitopological	ADJ
ejpam-6572	84	11	space	space	NOUN
ejpam-6572	84	12	(	(	PUNCT
ejpam-6572	84	13	x	x	NOUN
ejpam-6572	84	14	,	,	PUNCT
ejpam-6572	84	15	τ1	τ1	NOUN
ejpam-6572	84	16	,	,	PUNCT
ejpam-6572	84	17	τ2	τ2	NOUN
ejpam-6572	84	18	)	)	PUNCT
ejpam-6572	84	19	,	,	PUNCT
ejpam-6572	84	20	the	the	DET
ejpam-6572	84	21	following	follow	VERB
ejpam-6572	84	22	properties	property	NOUN
ejpam-6572	84	23	hold	hold	VERB
ejpam-6572	84	24	:	:	PUNCT
ejpam-6572	84	25	(	(	PUNCT
ejpam-6572	84	26	1	1	X
ejpam-6572	84	27	)	)	PUNCT
ejpam-6572	84	28	if	if	SCONJ
ejpam-6572	84	29	a	a	PRON
ejpam-6572	84	30	is	be	AUX
ejpam-6572	84	31	τ1τ2	τ1τ2	NOUN
ejpam-6572	84	32	-	-	ADJ
ejpam-6572	84	33	open	open	ADJ
ejpam-6572	84	34	in	in	ADP
ejpam-6572	84	35	x	x	NOUN
ejpam-6572	84	36	,	,	PUNCT
ejpam-6572	84	37	then	then	ADV
ejpam-6572	84	38	τ1τ2	τ1τ2	NOUN
ejpam-6572	84	39	-	-	NUM
ejpam-6572	84	40	cl(a	cl(a	NUM
ejpam-6572	84	41	)	)	PUNCT
ejpam-6572	84	42	=	=	PUNCT
ejpam-6572	84	43	(	(	PUNCT
ejpam-6572	84	44	τ1	τ1	NOUN
ejpam-6572	84	45	,	,	PUNCT
ejpam-6572	84	46	τ2)θ	τ2)θ	NOUN
ejpam-6572	84	47	-	-	PUNCT
ejpam-6572	84	48	cl(a	cl(a	NUM
ejpam-6572	84	49	)	)	PUNCT
ejpam-6572	84	50	.	.	PUNCT
ejpam-6572	85	1	(	(	PUNCT
ejpam-6572	85	2	2	2	X
ejpam-6572	85	3	)	)	PUNCT
ejpam-6572	85	4	(	(	PUNCT
ejpam-6572	85	5	τ1	τ1	NOUN
ejpam-6572	85	6	,	,	PUNCT
ejpam-6572	85	7	τ2)θ	τ2)θ	NOUN
ejpam-6572	85	8	-	-	PUNCT
ejpam-6572	85	9	cl(a	cl(a	NUM
ejpam-6572	85	10	)	)	PUNCT
ejpam-6572	85	11	is	be	AUX
ejpam-6572	85	12	τ1τ2	τ1τ2	NOUN
ejpam-6572	85	13	-	-	ADJ
ejpam-6572	85	14	closed	closed	ADJ
ejpam-6572	85	15	in	in	ADP
ejpam-6572	85	16	x.	x.	NOUN
ejpam-6572	85	17	an	an	DET
ejpam-6572	85	18	ideal	ideal	NOUN
ejpam-6572	85	19	i	i	PRON
ejpam-6572	85	20	on	on	ADP
ejpam-6572	85	21	a	a	DET
ejpam-6572	85	22	topological	topological	ADJ
ejpam-6572	85	23	space	space	NOUN
ejpam-6572	85	24	(	(	PUNCT
ejpam-6572	85	25	x	x	X
ejpam-6572	85	26	,	,	PUNCT
ejpam-6572	85	27	τ	τ	X
ejpam-6572	85	28	)	)	PUNCT
ejpam-6572	85	29	is	be	AUX
ejpam-6572	85	30	a	a	DET
ejpam-6572	85	31	nonempty	nonempty	ADJ
ejpam-6572	85	32	collection	collection	NOUN
ejpam-6572	85	33	of	of	ADP
ejpam-6572	85	34	subsets	subset	NOUN
ejpam-6572	85	35	of	of	ADP
ejpam-6572	85	36	x	x	PUNCT
ejpam-6572	85	37	satisfying	satisfy	VERB
ejpam-6572	85	38	the	the	DET
ejpam-6572	85	39	following	follow	VERB
ejpam-6572	85	40	properties	property	NOUN
ejpam-6572	85	41	:	:	PUNCT
ejpam-6572	85	42	(	(	PUNCT
ejpam-6572	85	43	1	1	X
ejpam-6572	85	44	)	)	PUNCT
ejpam-6572	85	45	a	a	DET
ejpam-6572	85	46	∈	∈	NOUN
ejpam-6572	86	1	i	i	PRON
ejpam-6572	86	2	and	and	CCONJ
ejpam-6572	86	3	b	b	X
ejpam-6572	86	4	⊆	⊆	NUM
ejpam-6572	86	5	a	a	DET
ejpam-6572	86	6	imply	imply	NOUN
ejpam-6572	86	7	b	b	X
ejpam-6572	86	8	∈	∈	PROPN
ejpam-6572	86	9	i	i	PRON
ejpam-6572	86	10	;	;	PUNCT
ejpam-6572	86	11	(	(	PUNCT
ejpam-6572	86	12	2	2	X
ejpam-6572	86	13	)	)	PUNCT
ejpam-6572	87	1	a	a	DET
ejpam-6572	87	2	∈	∈	NOUN
ejpam-6572	87	3	i	i	PRON
ejpam-6572	87	4	and	and	CCONJ
ejpam-6572	87	5	b	b	X
ejpam-6572	87	6	∈	∈	NOUN
ejpam-6572	88	1	i	i	PRON
ejpam-6572	88	2	imply	imply	VERB
ejpam-6572	88	3	a	a	DET
ejpam-6572	88	4	∪	∪	X
ejpam-6572	88	5	b	b	NOUN
ejpam-6572	88	6	∈	∈	NOUN
ejpam-6572	89	1	i	i	PRON
ejpam-6572	89	2	.	.	PUNCT
ejpam-6572	90	1	a	a	DET
ejpam-6572	90	2	topological	topological	ADJ
ejpam-6572	90	3	space	space	NOUN
ejpam-6572	90	4	(	(	PUNCT
ejpam-6572	90	5	x	x	X
ejpam-6572	90	6	,	,	PUNCT
ejpam-6572	90	7	τ	τ	X
ejpam-6572	90	8	)	)	PUNCT
ejpam-6572	90	9	with	with	ADP
ejpam-6572	90	10	an	an	DET
ejpam-6572	90	11	ideal	ideal	ADJ
ejpam-6572	90	12	i	i	PRON
ejpam-6572	90	13	on	on	ADP
ejpam-6572	90	14	x	x	SYM
ejpam-6572	90	15	is	be	AUX
ejpam-6572	90	16	called	call	VERB
ejpam-6572	90	17	an	an	DET
ejpam-6572	90	18	ideal	ideal	ADJ
ejpam-6572	90	19	topological	topological	ADJ
ejpam-6572	90	20	space	space	NOUN
ejpam-6572	90	21	and	and	CCONJ
ejpam-6572	90	22	is	be	AUX
ejpam-6572	90	23	denoted	denote	VERB
ejpam-6572	90	24	by	by	ADP
ejpam-6572	90	25	(	(	PUNCT
ejpam-6572	90	26	x	x	X
ejpam-6572	90	27	,	,	PUNCT
ejpam-6572	90	28	τ	τ	PROPN
ejpam-6572	90	29	,	,	PUNCT
ejpam-6572	90	30	i	i	NOUN
ejpam-6572	90	31	)	)	PUNCT
ejpam-6572	90	32	.	.	PUNCT
ejpam-6572	91	1	for	for	ADP
ejpam-6572	91	2	an	an	DET
ejpam-6572	91	3	ideal	ideal	ADJ
ejpam-6572	91	4	topological	topological	ADJ
ejpam-6572	91	5	space	space	NOUN
ejpam-6572	91	6	(	(	PUNCT
ejpam-6572	91	7	x	x	X
ejpam-6572	91	8	,	,	PUNCT
ejpam-6572	91	9	τ	τ	PROPN
ejpam-6572	91	10	,	,	PUNCT
ejpam-6572	91	11	i	i	PROPN
ejpam-6572	91	12	)	)	PUNCT
ejpam-6572	91	13	and	and	CCONJ
ejpam-6572	91	14	a	a	DET
ejpam-6572	91	15	subset	subset	NOUN
ejpam-6572	91	16	a	a	PRON
ejpam-6572	91	17	of	of	ADP
ejpam-6572	91	18	x	x	PRON
ejpam-6572	91	19	,	,	PUNCT
ejpam-6572	91	20	a⋆(i	a⋆(i	PROPN
ejpam-6572	91	21	)	)	PUNCT
ejpam-6572	91	22	is	be	AUX
ejpam-6572	91	23	defined	define	VERB
ejpam-6572	91	24	as	as	SCONJ
ejpam-6572	91	25	follows	follow	VERB
ejpam-6572	91	26	:	:	PUNCT
ejpam-6572	91	27	a⋆(i	a⋆(i	NOUN
ejpam-6572	91	28	)	)	PUNCT
ejpam-6572	92	1	=	=	PUNCT
ejpam-6572	92	2	{	{	PUNCT
ejpam-6572	92	3	x	x	PUNCT
ejpam-6572	92	4	∈	∈	PROPN
ejpam-6572	92	5	x	x	X
ejpam-6572	92	6	:	:	PUNCT
ejpam-6572	92	7	u	u	X
ejpam-6572	92	8	∩a	∩a	PROPN
ejpam-6572	92	9	̸∈	̸∈	PROPN
ejpam-6572	92	10	i	i	PRON
ejpam-6572	92	11	for	for	ADP
ejpam-6572	92	12	every	every	DET
ejpam-6572	92	13	open	open	ADJ
ejpam-6572	92	14	neighbourhood	neighbourhood	NOUN
ejpam-6572	92	15	u	u	NOUN
ejpam-6572	92	16	of	of	ADP
ejpam-6572	92	17	x	x	NOUN
ejpam-6572	92	18	}	}	PUNCT
ejpam-6572	92	19	.	.	PUNCT
ejpam-6572	93	1	in	in	ADP
ejpam-6572	93	2	case	case	NOUN
ejpam-6572	93	3	there	there	PRON
ejpam-6572	93	4	is	be	VERB
ejpam-6572	93	5	no	no	DET
ejpam-6572	93	6	chance	chance	NOUN
ejpam-6572	93	7	for	for	ADP
ejpam-6572	93	8	confusion	confusion	NOUN
ejpam-6572	93	9	,	,	PUNCT
ejpam-6572	93	10	a⋆(i	a⋆(i	NOUN
ejpam-6572	93	11	)	)	PUNCT
ejpam-6572	93	12	is	be	AUX
ejpam-6572	93	13	simply	simply	ADV
ejpam-6572	93	14	written	write	VERB
ejpam-6572	93	15	as	as	ADP
ejpam-6572	93	16	a⋆.	a⋆.	NOUN
ejpam-6572	93	17	in	in	ADP
ejpam-6572	93	18	[	[	X
ejpam-6572	93	19	26	26	NUM
ejpam-6572	93	20	]	]	PUNCT
ejpam-6572	93	21	,	,	PUNCT
ejpam-6572	93	22	a⋆	a⋆	ADV
ejpam-6572	93	23	is	be	AUX
ejpam-6572	93	24	called	call	VERB
ejpam-6572	93	25	the	the	DET
ejpam-6572	93	26	local	local	ADJ
ejpam-6572	93	27	function	function	NOUN
ejpam-6572	93	28	of	of	ADP
ejpam-6572	93	29	a	a	PRON
ejpam-6572	93	30	with	with	ADP
ejpam-6572	93	31	respect	respect	NOUN
ejpam-6572	93	32	to	to	ADP
ejpam-6572	93	33	i	i	PRON
ejpam-6572	93	34	and	and	CCONJ
ejpam-6572	93	35	τ	τ	PROPN
ejpam-6572	93	36	and	and	CCONJ
ejpam-6572	93	37	cl⋆(a	cl⋆(a	PROPN
ejpam-6572	93	38	)	)	PUNCT
ejpam-6572	93	39	=	=	NOUN
ejpam-6572	93	40	a⋆∪a	a⋆∪a	NOUN
ejpam-6572	93	41	defines	define	VERB
ejpam-6572	93	42	a	a	DET
ejpam-6572	93	43	kuratowski	kuratowski	ADJ
ejpam-6572	93	44	closure	closure	NOUN
ejpam-6572	93	45	operator	operator	NOUN
ejpam-6572	93	46	for	for	ADP
ejpam-6572	93	47	a	a	DET
ejpam-6572	93	48	topology	topology	NOUN
ejpam-6572	93	49	τ⋆(i	τ⋆(i	NOUN
ejpam-6572	93	50	)	)	PUNCT
ejpam-6572	93	51	finer	fine	ADJ
ejpam-6572	93	52	than	than	ADP
ejpam-6572	93	53	τ	τ	PROPN
ejpam-6572	93	54	.	.	PUNCT
ejpam-6572	94	1	a	a	DET
ejpam-6572	94	2	subset	subset	NOUN
ejpam-6572	94	3	a	a	PRON
ejpam-6572	94	4	is	be	AUX
ejpam-6572	94	5	said	say	VERB
ejpam-6572	94	6	to	to	PART
ejpam-6572	94	7	be	be	AUX
ejpam-6572	94	8	⋆-closed	⋆-close	VERB
ejpam-6572	94	9	[	[	X
ejpam-6572	94	10	10	10	NUM
ejpam-6572	94	11	]	]	X
ejpam-6572	94	12	if	if	SCONJ
ejpam-6572	94	13	a⋆	a⋆	ADJ
ejpam-6572	94	14	⊆	⊆	NUM
ejpam-6572	94	15	a.	a.	NOUN
ejpam-6572	94	16	the	the	DET
ejpam-6572	94	17	interior	interior	NOUN
ejpam-6572	94	18	of	of	ADP
ejpam-6572	94	19	a	a	DET
ejpam-6572	94	20	subset	subset	NOUN
ejpam-6572	94	21	a	a	DET
ejpam-6572	94	22	in	in	ADP
ejpam-6572	94	23	(	(	PUNCT
ejpam-6572	94	24	x	x	X
ejpam-6572	94	25	,	,	PUNCT
ejpam-6572	94	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6572	94	27	)	)	PUNCT
ejpam-6572	94	28	)	)	PUNCT
ejpam-6572	94	29	is	be	AUX
ejpam-6572	94	30	denoted	denote	VERB
ejpam-6572	94	31	by	by	ADP
ejpam-6572	94	32	int⋆(a	int⋆(a	NOUN
ejpam-6572	94	33	)	)	PUNCT
ejpam-6572	94	34	.	.	PUNCT
ejpam-6572	95	1	a	a	DET
ejpam-6572	95	2	subset	subset	NOUN
ejpam-6572	95	3	a	a	PRON
ejpam-6572	95	4	of	of	ADP
ejpam-6572	95	5	an	an	DET
ejpam-6572	95	6	ideal	ideal	ADJ
ejpam-6572	95	7	topological	topological	ADJ
ejpam-6572	95	8	space	space	NOUN
ejpam-6572	95	9	(	(	PUNCT
ejpam-6572	95	10	x	x	X
ejpam-6572	95	11	,	,	PUNCT
ejpam-6572	95	12	τ	τ	PROPN
ejpam-6572	95	13	,	,	PUNCT
ejpam-6572	95	14	i	i	PROPN
ejpam-6572	95	15	)	)	PUNCT
ejpam-6572	95	16	is	be	AUX
ejpam-6572	95	17	said	say	VERB
ejpam-6572	95	18	to	to	PART
ejpam-6572	95	19	be	be	AUX
ejpam-6572	95	20	semi⋆-i	semi⋆-i	X
ejpam-6572	95	21	-open	-open	VERB
ejpam-6572	95	22	[	[	PUNCT
ejpam-6572	95	23	27	27	NUM
ejpam-6572	95	24	]	]	PUNCT
ejpam-6572	95	25	(	(	PUNCT
ejpam-6572	95	26	resp	resp	NOUN
ejpam-6572	95	27	.	.	PUNCT
ejpam-6572	96	1	semi	semi	ADJ
ejpam-6572	96	2	-	-	VERB
ejpam-6572	96	3	i	i	PRON
ejpam-6572	96	4	-open	-open	NOUN
ejpam-6572	97	1	[	[	X
ejpam-6572	97	2	13	13	NUM
ejpam-6572	97	3	]	]	SYM
ejpam-6572	97	4	)	)	PUNCT
ejpam-6572	97	5	if	if	SCONJ
ejpam-6572	97	6	a	a	DET
ejpam-6572	97	7	⊆	⊆	NUM
ejpam-6572	97	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6572	97	9	)	)	PUNCT
ejpam-6572	97	10	)	)	PUNCT
ejpam-6572	97	11	(	(	PUNCT
ejpam-6572	97	12	resp	resp	NOUN
ejpam-6572	97	13	.	.	PUNCT
ejpam-6572	98	1	a	a	DET
ejpam-6572	98	2	⊆	⊆	NUM
ejpam-6572	98	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6572	98	4	)	)	PUNCT
ejpam-6572	98	5	)	)	PUNCT
ejpam-6572	98	6	)	)	PUNCT
ejpam-6572	98	7	.	.	PUNCT
ejpam-6572	99	1	the	the	DET
ejpam-6572	99	2	complement	complement	NOUN
ejpam-6572	99	3	of	of	ADP
ejpam-6572	99	4	a	a	DET
ejpam-6572	99	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6572	99	6	-open	-open	ADJ
ejpam-6572	99	7	(	(	PUNCT
ejpam-6572	99	8	resp	resp	NOUN
ejpam-6572	99	9	.	.	PUNCT
ejpam-6572	100	1	semi	semi	ADJ
ejpam-6572	100	2	-	-	VERB
ejpam-6572	100	3	i	i	PRON
ejpam-6572	100	4	-open	-open	NOUN
ejpam-6572	100	5	)	)	PUNCT
ejpam-6572	101	1	set	set	NOUN
ejpam-6572	101	2	is	be	AUX
ejpam-6572	101	3	said	say	VERB
ejpam-6572	101	4	to	to	PART
ejpam-6572	101	5	be	be	AUX
ejpam-6572	101	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6572	101	7	-closed	-close	VERB
ejpam-6572	101	8	[	[	PUNCT
ejpam-6572	101	9	27	27	NUM
ejpam-6572	101	10	]	]	PUNCT
ejpam-6572	101	11	(	(	PUNCT
ejpam-6572	101	12	resp	resp	NOUN
ejpam-6572	101	13	.	.	PUNCT
ejpam-6572	102	1	semi	semi	ADJ
ejpam-6572	102	2	-	-	VERB
ejpam-6572	102	3	i	i	PRON
ejpam-6572	102	4	-closed	-close	VERB
ejpam-6572	103	1	[	[	PUNCT
ejpam-6572	103	2	13	13	NUM
ejpam-6572	103	3	]	]	NUM
ejpam-6572	103	4	)	)	PUNCT
ejpam-6572	103	5	.	.	PUNCT
ejpam-6572	104	1	a	a	DET
ejpam-6572	104	2	subset	subset	NOUN
ejpam-6572	104	3	a	a	DET
ejpam-6572	104	4	b.	b.	PROPN
ejpam-6572	104	5	kong	kong	PROPN
ejpam-6572	104	6	-	-	PUNCT
ejpam-6572	104	7	ied	ied	PROPN
ejpam-6572	104	8	,	,	PUNCT
ejpam-6572	104	9	a.	a.	PROPN
ejpam-6572	104	10	sama	sama	PROPN
ejpam-6572	104	11	-	-	PUNCT
ejpam-6572	104	12	ae	ae	PROPN
ejpam-6572	104	13	,	,	PUNCT
ejpam-6572	104	14	c.	c.	PROPN
ejpam-6572	104	15	boonpok	boonpok	PROPN
ejpam-6572	104	16	/	/	SYM
ejpam-6572	104	17	eur	eur	PROPN
ejpam-6572	104	18	.	.	PUNCT
ejpam-6572	105	1	j.	j.	PROPN
ejpam-6572	105	2	pure	pure	PROPN
ejpam-6572	105	3	appl	appl	PROPN
ejpam-6572	105	4	.	.	PROPN
ejpam-6572	105	5	math	math	PROPN
ejpam-6572	105	6	,	,	PUNCT
ejpam-6572	105	7	18	18	NUM
ejpam-6572	105	8	(	(	PUNCT
ejpam-6572	105	9	3	3	NUM
ejpam-6572	105	10	)	)	PUNCT
ejpam-6572	105	11	(	(	PUNCT
ejpam-6572	105	12	2025	2025	NUM
ejpam-6572	105	13	)	)	PUNCT
ejpam-6572	105	14	,	,	PUNCT
ejpam-6572	105	15	6572	6572	NUM
ejpam-6572	105	16	4	4	NUM
ejpam-6572	105	17	of	of	ADP
ejpam-6572	105	18	12	12	NUM
ejpam-6572	105	19	of	of	ADP
ejpam-6572	105	20	an	an	DET
ejpam-6572	105	21	ideal	ideal	ADJ
ejpam-6572	105	22	topological	topological	ADJ
ejpam-6572	105	23	space	space	NOUN
ejpam-6572	105	24	(	(	PUNCT
ejpam-6572	105	25	x	x	X
ejpam-6572	105	26	,	,	PUNCT
ejpam-6572	105	27	τ	τ	PROPN
ejpam-6572	105	28	,	,	PUNCT
ejpam-6572	105	29	i	i	PROPN
ejpam-6572	105	30	)	)	PUNCT
ejpam-6572	105	31	is	be	AUX
ejpam-6572	105	32	said	say	VERB
ejpam-6572	105	33	to	to	PART
ejpam-6572	105	34	be	be	AUX
ejpam-6572	105	35	semi	semi	ADJ
ejpam-6572	105	36	-	-	ADJ
ejpam-6572	105	37	i	i	PRON
ejpam-6572	105	38	⋆-open	⋆-open	VERB
ejpam-6572	106	1	[	[	X
ejpam-6572	106	2	28	28	NUM
ejpam-6572	106	3	]	]	X
ejpam-6572	106	4	if	if	SCONJ
ejpam-6572	106	5	a	a	DET
ejpam-6572	106	6	⊆	⊆	NUM
ejpam-6572	106	7	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-6572	106	8	)	)	PUNCT
ejpam-6572	106	9	)	)	PUNCT
ejpam-6572	106	10	.	.	PUNCT
ejpam-6572	107	1	the	the	DET
ejpam-6572	107	2	complement	complement	NOUN
ejpam-6572	107	3	of	of	ADP
ejpam-6572	107	4	a	a	DET
ejpam-6572	107	5	semi	semi	NOUN
ejpam-6572	107	6	-	-	ADJ
ejpam-6572	107	7	i	i	PRON
ejpam-6572	107	8	⋆-open	⋆-open	ADJ
ejpam-6572	107	9	set	set	VERB
ejpam-6572	107	10	is	be	AUX
ejpam-6572	107	11	called	call	VERB
ejpam-6572	107	12	semi	semi	ADJ
ejpam-6572	107	13	-	-	ADJ
ejpam-6572	107	14	i	i	PRON
ejpam-6572	107	15	⋆-closed	⋆-close	VERB
ejpam-6572	108	1	[	[	X
ejpam-6572	108	2	28	28	NUM
ejpam-6572	108	3	]	]	PUNCT
ejpam-6572	108	4	.	.	PUNCT
ejpam-6572	109	1	for	for	ADP
ejpam-6572	109	2	a	a	DET
ejpam-6572	109	3	subset	subset	NOUN
ejpam-6572	109	4	a	a	PRON
ejpam-6572	109	5	of	of	ADP
ejpam-6572	109	6	an	an	DET
ejpam-6572	109	7	ideal	ideal	ADJ
ejpam-6572	109	8	topological	topological	ADJ
ejpam-6572	109	9	space	space	NOUN
ejpam-6572	109	10	(	(	PUNCT
ejpam-6572	109	11	x	x	X
ejpam-6572	109	12	,	,	PUNCT
ejpam-6572	109	13	τ	τ	PROPN
ejpam-6572	109	14	,	,	PUNCT
ejpam-6572	109	15	i	i	NOUN
ejpam-6572	109	16	)	)	PUNCT
ejpam-6572	109	17	,	,	PUNCT
ejpam-6572	109	18	the	the	DET
ejpam-6572	109	19	intersection	intersection	NOUN
ejpam-6572	109	20	of	of	ADP
ejpam-6572	109	21	all	all	DET
ejpam-6572	109	22	semi	semi	ADJ
ejpam-6572	109	23	-	-	ADJ
ejpam-6572	109	24	i	i	PRON
ejpam-6572	109	25	-closed	-closed	ADJ
ejpam-6572	109	26	sets	set	NOUN
ejpam-6572	109	27	containing	contain	VERB
ejpam-6572	109	28	a	a	PRON
ejpam-6572	109	29	is	be	AUX
ejpam-6572	109	30	called	call	VERB
ejpam-6572	109	31	the	the	DET
ejpam-6572	109	32	semi	semi	NOUN
ejpam-6572	109	33	-	-	ADJ
ejpam-6572	109	34	i	i	PRON
ejpam-6572	109	35	⋆-closure	⋆-closure	NOUN
ejpam-6572	110	1	[	[	X
ejpam-6572	110	2	28	28	NUM
ejpam-6572	110	3	]	]	PUNCT
ejpam-6572	110	4	of	of	ADP
ejpam-6572	110	5	a	a	PRON
ejpam-6572	110	6	and	and	CCONJ
ejpam-6572	110	7	is	be	AUX
ejpam-6572	110	8	denoted	denote	VERB
ejpam-6572	110	9	by	by	ADP
ejpam-6572	110	10	scl⋆(a	scl⋆(a	NOUN
ejpam-6572	110	11	)	)	PUNCT
ejpam-6572	110	12	(	(	PUNCT
ejpam-6572	110	13	scli	scli	PROPN
ejpam-6572	110	14	⋆(a	⋆(a	PRON
ejpam-6572	110	15	)	)	PUNCT
ejpam-6572	111	1	[	[	X
ejpam-6572	111	2	28	28	NUM
ejpam-6572	111	3	]	]	NUM
ejpam-6572	111	4	)	)	PUNCT
ejpam-6572	111	5	.	.	PUNCT
ejpam-6572	112	1	the	the	DET
ejpam-6572	112	2	union	union	NOUN
ejpam-6572	112	3	of	of	ADP
ejpam-6572	112	4	all	all	DET
ejpam-6572	112	5	semi	semi	ADJ
ejpam-6572	112	6	-	-	ADJ
ejpam-6572	112	7	i	i	PRON
ejpam-6572	112	8	-open	-open	NOUN
ejpam-6572	112	9	sets	set	NOUN
ejpam-6572	112	10	contained	contain	VERB
ejpam-6572	112	11	in	in	ADP
ejpam-6572	112	12	a	a	PRON
ejpam-6572	112	13	is	be	AUX
ejpam-6572	112	14	called	call	VERB
ejpam-6572	112	15	the	the	DET
ejpam-6572	112	16	semi	semi	NOUN
ejpam-6572	112	17	-	-	ADJ
ejpam-6572	112	18	i	i	PRON
ejpam-6572	112	19	⋆-interior	⋆-interior	PUNCT
ejpam-6572	113	1	[	[	X
ejpam-6572	113	2	28	28	NUM
ejpam-6572	113	3	]	]	PUNCT
ejpam-6572	113	4	of	of	ADP
ejpam-6572	113	5	a	a	PRON
ejpam-6572	113	6	and	and	CCONJ
ejpam-6572	113	7	is	be	AUX
ejpam-6572	113	8	denoted	denote	VERB
ejpam-6572	113	9	by	by	ADP
ejpam-6572	113	10	sint⋆(a	sint⋆(a	PROPN
ejpam-6572	113	11	)	)	PUNCT
ejpam-6572	114	1	(	(	PUNCT
ejpam-6572	114	2	sinti	sinti	PROPN
ejpam-6572	114	3	⋆(a	⋆(a	NOUN
ejpam-6572	114	4	)	)	PUNCT
ejpam-6572	115	1	[	[	X
ejpam-6572	115	2	28	28	NUM
ejpam-6572	115	3	]	]	NUM
ejpam-6572	115	4	)	)	PUNCT
ejpam-6572	115	5	.	.	PUNCT
ejpam-6572	116	1	lemma	lemma	PROPN
ejpam-6572	116	2	4	4	NUM
ejpam-6572	116	3	.	.	PUNCT
ejpam-6572	117	1	[	[	X
ejpam-6572	117	2	28	28	NUM
ejpam-6572	117	3	]	]	PUNCT
ejpam-6572	117	4	for	for	ADP
ejpam-6572	117	5	a	a	DET
ejpam-6572	117	6	subset	subset	NOUN
ejpam-6572	117	7	a	a	PRON
ejpam-6572	117	8	of	of	ADP
ejpam-6572	117	9	a	a	DET
ejpam-6572	117	10	an	an	DET
ejpam-6572	117	11	ideal	ideal	ADJ
ejpam-6572	117	12	topological	topological	ADJ
ejpam-6572	117	13	space	space	NOUN
ejpam-6572	117	14	(	(	PUNCT
ejpam-6572	117	15	x	x	X
ejpam-6572	117	16	,	,	PUNCT
ejpam-6572	117	17	τ	τ	PROPN
ejpam-6572	117	18	,	,	PUNCT
ejpam-6572	117	19	i	i	NOUN
ejpam-6572	117	20	)	)	PUNCT
ejpam-6572	117	21	,	,	PUNCT
ejpam-6572	117	22	the	the	DET
ejpam-6572	117	23	following	follow	VERB
ejpam-6572	117	24	properties	property	NOUN
ejpam-6572	117	25	hold	hold	VERB
ejpam-6572	117	26	:	:	PUNCT
ejpam-6572	117	27	(	(	PUNCT
ejpam-6572	117	28	1	1	X
ejpam-6572	117	29	)	)	PUNCT
ejpam-6572	117	30	scl⋆(a	scl⋆(a	NUM
ejpam-6572	117	31	)	)	PUNCT
ejpam-6572	117	32	=	=	PUNCT
ejpam-6572	117	33	a	a	DET
ejpam-6572	117	34	∪	∪	ADJ
ejpam-6572	117	35	int⋆(cl⋆(a	int⋆(cl⋆(a	NOUN
ejpam-6572	117	36	)	)	PUNCT
ejpam-6572	117	37	)	)	PUNCT
ejpam-6572	117	38	;	;	PUNCT
ejpam-6572	117	39	(	(	PUNCT
ejpam-6572	117	40	2	2	X
ejpam-6572	117	41	)	)	PUNCT
ejpam-6572	117	42	sint⋆(a	sint⋆(a	PROPN
ejpam-6572	117	43	)	)	PUNCT
ejpam-6572	118	1	=	=	PUNCT
ejpam-6572	118	2	a	a	DET
ejpam-6572	118	3	∩	∩	ADJ
ejpam-6572	118	4	cl⋆(int⋆(a	cl⋆(int⋆(a	NOUN
ejpam-6572	118	5	)	)	PUNCT
ejpam-6572	118	6	)	)	PUNCT
ejpam-6572	118	7	.	.	PUNCT
ejpam-6572	119	1	3	3	X
ejpam-6572	119	2	.	.	X
ejpam-6572	119	3	almost	almost	ADV
ejpam-6572	119	4	quasi	quasi	VERB
ejpam-6572	119	5	τ	τ	PROPN
ejpam-6572	119	6	⋆(σ1	⋆(σ1	PROPN
ejpam-6572	119	7	,	,	PUNCT
ejpam-6572	119	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	119	9	functions	function	NOUN
ejpam-6572	119	10	in	in	ADP
ejpam-6572	119	11	this	this	DET
ejpam-6572	119	12	section	section	NOUN
ejpam-6572	119	13	,	,	PUNCT
ejpam-6572	119	14	we	we	PRON
ejpam-6572	119	15	introduce	introduce	VERB
ejpam-6572	119	16	the	the	DET
ejpam-6572	119	17	concept	concept	NOUN
ejpam-6572	119	18	of	of	ADP
ejpam-6572	119	19	almost	almost	ADV
ejpam-6572	119	20	quasi	quasi	ADJ
ejpam-6572	119	21	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	119	22	,	,	PUNCT
ejpam-6572	119	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	119	24	functions	function	NOUN
ejpam-6572	119	25	.	.	PUNCT
ejpam-6572	120	1	moreover	moreover	ADV
ejpam-6572	120	2	,	,	PUNCT
ejpam-6572	120	3	some	some	DET
ejpam-6572	120	4	characterizations	characterization	NOUN
ejpam-6572	120	5	of	of	ADP
ejpam-6572	120	6	almost	almost	ADV
ejpam-6572	120	7	quasi	quasi	ADJ
ejpam-6572	120	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	120	9	,	,	PUNCT
ejpam-6572	120	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	120	11	functions	function	NOUN
ejpam-6572	120	12	are	be	AUX
ejpam-6572	120	13	considered	consider	VERB
ejpam-6572	120	14	.	.	PUNCT
ejpam-6572	121	1	definition	definition	NOUN
ejpam-6572	121	2	1	1	NUM
ejpam-6572	121	3	.	.	PUNCT
ejpam-6572	122	1	a	a	DET
ejpam-6572	122	2	function	function	NOUN
ejpam-6572	122	3	f	f	NOUN
ejpam-6572	122	4	:	:	PUNCT
ejpam-6572	122	5	(	(	PUNCT
ejpam-6572	122	6	x	x	X
ejpam-6572	122	7	,	,	PUNCT
ejpam-6572	122	8	τ	τ	PROPN
ejpam-6572	122	9	,	,	PUNCT
ejpam-6572	122	10	i	i	NOUN
ejpam-6572	122	11	)	)	PUNCT
ejpam-6572	122	12	→	→	PUNCT
ejpam-6572	122	13	(	(	PUNCT
ejpam-6572	122	14	y	y	PROPN
ejpam-6572	122	15	,	,	PUNCT
ejpam-6572	122	16	σ1	σ1	PROPN
ejpam-6572	122	17	,	,	PUNCT
ejpam-6572	122	18	σ2	σ2	PROPN
ejpam-6572	122	19	)	)	PUNCT
ejpam-6572	122	20	is	be	AUX
ejpam-6572	122	21	said	say	VERB
ejpam-6572	122	22	to	to	PART
ejpam-6572	122	23	be	be	AUX
ejpam-6572	122	24	almost	almost	ADV
ejpam-6572	122	25	quasi	quasi	ADJ
ejpam-6572	122	26	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	122	27	,	,	PUNCT
ejpam-6572	122	28	σ2)continuous	σ2)continuous	ADJ
ejpam-6572	122	29	at	at	ADP
ejpam-6572	122	30	a	a	DET
ejpam-6572	122	31	point	point	NOUN
ejpam-6572	122	32	x	x	SYM
ejpam-6572	122	33	∈	∈	NOUN
ejpam-6572	122	34	x	x	INTJ
ejpam-6572	122	35	if	if	SCONJ
ejpam-6572	122	36	for	for	ADP
ejpam-6572	122	37	every	every	DET
ejpam-6572	122	38	σ1σ2	σ1σ2	NOUN
ejpam-6572	122	39	-	-	ADJ
ejpam-6572	122	40	open	open	ADJ
ejpam-6572	122	41	set	set	NOUN
ejpam-6572	122	42	v	v	NOUN
ejpam-6572	122	43	of	of	ADP
ejpam-6572	122	44	y	y	NOUN
ejpam-6572	122	45	containing	contain	VERB
ejpam-6572	122	46	f(x	f(x	PROPN
ejpam-6572	122	47	)	)	PUNCT
ejpam-6572	122	48	and	and	CCONJ
ejpam-6572	122	49	each	each	DET
ejpam-6572	122	50	⋆-open	⋆-open	ADV
ejpam-6572	122	51	set	set	VERB
ejpam-6572	122	52	u	u	NOUN
ejpam-6572	122	53	of	of	ADP
ejpam-6572	122	54	x	x	SYM
ejpam-6572	122	55	containing	contain	VERB
ejpam-6572	122	56	x	x	PRON
ejpam-6572	122	57	,	,	PUNCT
ejpam-6572	122	58	there	there	PRON
ejpam-6572	122	59	exists	exist	VERB
ejpam-6572	122	60	a	a	DET
ejpam-6572	122	61	nonempty	nonempty	ADJ
ejpam-6572	122	62	⋆-open	⋆-open	NOUN
ejpam-6572	122	63	set	set	VERB
ejpam-6572	122	64	g	g	PROPN
ejpam-6572	123	1	such	such	ADJ
ejpam-6572	123	2	that	that	SCONJ
ejpam-6572	123	3	g	g	PROPN
ejpam-6572	123	4	⊆	⊆	NUM
ejpam-6572	123	5	u	u	NOUN
ejpam-6572	123	6	and	and	CCONJ
ejpam-6572	123	7	f(g	f(g	NOUN
ejpam-6572	123	8	)	)	PUNCT
ejpam-6572	124	1	⊆	⊆	NUM
ejpam-6572	124	2	(	(	PUNCT
ejpam-6572	124	3	σ1	σ1	PROPN
ejpam-6572	124	4	,	,	PUNCT
ejpam-6572	124	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	124	6	)	)	PUNCT
ejpam-6572	124	7	.	.	PUNCT
ejpam-6572	125	1	a	a	DET
ejpam-6572	125	2	function	function	NOUN
ejpam-6572	125	3	f	f	NOUN
ejpam-6572	125	4	:	:	PUNCT
ejpam-6572	125	5	(	(	PUNCT
ejpam-6572	125	6	x	x	X
ejpam-6572	125	7	,	,	PUNCT
ejpam-6572	125	8	τ	τ	PROPN
ejpam-6572	125	9	,	,	PUNCT
ejpam-6572	125	10	i	i	NOUN
ejpam-6572	125	11	)	)	PUNCT
ejpam-6572	125	12	→	→	PUNCT
ejpam-6572	125	13	(	(	PUNCT
ejpam-6572	125	14	y	y	PROPN
ejpam-6572	125	15	,	,	PUNCT
ejpam-6572	125	16	σ1	σ1	PROPN
ejpam-6572	125	17	,	,	PUNCT
ejpam-6572	125	18	σ2	σ2	PROPN
ejpam-6572	125	19	)	)	PUNCT
ejpam-6572	125	20	is	be	AUX
ejpam-6572	125	21	said	say	VERB
ejpam-6572	125	22	to	to	PART
ejpam-6572	125	23	be	be	AUX
ejpam-6572	125	24	almost	almost	ADV
ejpam-6572	125	25	quasi	quasi	ADJ
ejpam-6572	125	26	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	125	27	,	,	PUNCT
ejpam-6572	125	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	125	29	if	if	SCONJ
ejpam-6572	125	30	f	f	PROPN
ejpam-6572	125	31	is	be	AUX
ejpam-6572	125	32	almost	almost	ADV
ejpam-6572	125	33	quasi	quasi	ADJ
ejpam-6572	125	34	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	125	35	,	,	PUNCT
ejpam-6572	125	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	125	37	at	at	ADP
ejpam-6572	125	38	each	each	DET
ejpam-6572	125	39	point	point	NOUN
ejpam-6572	125	40	of	of	ADP
ejpam-6572	125	41	x.	x.	NOUN
ejpam-6572	125	42	theorem	theorem	VERB
ejpam-6572	125	43	1	1	NUM
ejpam-6572	125	44	.	.	PUNCT
ejpam-6572	125	45	for	for	ADP
ejpam-6572	125	46	a	a	DET
ejpam-6572	125	47	function	function	NOUN
ejpam-6572	125	48	f	f	NOUN
ejpam-6572	125	49	:	:	PUNCT
ejpam-6572	125	50	(	(	PUNCT
ejpam-6572	125	51	x	x	X
ejpam-6572	125	52	,	,	PUNCT
ejpam-6572	125	53	τ	τ	PROPN
ejpam-6572	125	54	,	,	PUNCT
ejpam-6572	125	55	i	i	NOUN
ejpam-6572	125	56	)	)	PUNCT
ejpam-6572	125	57	→	→	PUNCT
ejpam-6572	125	58	(	(	PUNCT
ejpam-6572	125	59	y	y	PROPN
ejpam-6572	125	60	,	,	PUNCT
ejpam-6572	125	61	σ1	σ1	PROPN
ejpam-6572	125	62	,	,	PUNCT
ejpam-6572	125	63	σ2	σ2	NOUN
ejpam-6572	125	64	)	)	PUNCT
ejpam-6572	125	65	,	,	PUNCT
ejpam-6572	125	66	the	the	DET
ejpam-6572	125	67	following	follow	VERB
ejpam-6572	125	68	properties	property	NOUN
ejpam-6572	125	69	are	be	AUX
ejpam-6572	125	70	equivalent	equivalent	ADJ
ejpam-6572	125	71	:	:	PUNCT
ejpam-6572	125	72	(	(	PUNCT
ejpam-6572	125	73	1	1	X
ejpam-6572	125	74	)	)	PUNCT
ejpam-6572	125	75	f	f	NOUN
ejpam-6572	125	76	is	be	AUX
ejpam-6572	125	77	almost	almost	ADV
ejpam-6572	125	78	quasi	quasi	ADJ
ejpam-6572	125	79	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	125	80	,	,	PUNCT
ejpam-6572	125	81	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	125	82	at	at	ADP
ejpam-6572	125	83	x	x	X
ejpam-6572	125	84	∈	∈	PROPN
ejpam-6572	125	85	x	x	X
ejpam-6572	125	86	;	;	PUNCT
ejpam-6572	125	87	(	(	PUNCT
ejpam-6572	125	88	2	2	X
ejpam-6572	125	89	)	)	PUNCT
ejpam-6572	125	90	for	for	ADP
ejpam-6572	125	91	every	every	DET
ejpam-6572	125	92	σ1σ2	σ1σ2	NOUN
ejpam-6572	125	93	-	-	ADJ
ejpam-6572	125	94	open	open	ADJ
ejpam-6572	125	95	set	set	NOUN
ejpam-6572	125	96	v	v	NOUN
ejpam-6572	125	97	of	of	ADP
ejpam-6572	125	98	y	y	NOUN
ejpam-6572	125	99	containing	contain	VERB
ejpam-6572	125	100	f(x	f(x	PROPN
ejpam-6572	125	101	)	)	PUNCT
ejpam-6572	125	102	,	,	PUNCT
ejpam-6572	125	103	there	there	PRON
ejpam-6572	125	104	exists	exist	VERB
ejpam-6572	125	105	a	a	DET
ejpam-6572	125	106	semi	semi	NOUN
ejpam-6572	125	107	-	-	ADJ
ejpam-6572	125	108	i	i	PRON
ejpam-6572	125	109	⋆-open	⋆-open	VERB
ejpam-6572	125	110	set	set	VERB
ejpam-6572	125	111	u	u	NOUN
ejpam-6572	125	112	of	of	ADP
ejpam-6572	125	113	x	x	PUNCT
ejpam-6572	125	114	containing	contain	VERB
ejpam-6572	125	115	x	x	PUNCT
ejpam-6572	125	116	such	such	ADJ
ejpam-6572	125	117	that	that	DET
ejpam-6572	125	118	f(u	f(u	PROPN
ejpam-6572	125	119	)	)	PUNCT
ejpam-6572	125	120	⊆	⊆	NUM
ejpam-6572	125	121	(	(	PUNCT
ejpam-6572	125	122	σ1	σ1	PROPN
ejpam-6572	125	123	,	,	PUNCT
ejpam-6572	125	124	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	125	125	)	)	PUNCT
ejpam-6572	125	126	;	;	PUNCT
ejpam-6572	125	127	(	(	PUNCT
ejpam-6572	125	128	3	3	X
ejpam-6572	125	129	)	)	PUNCT
ejpam-6572	125	130	x	x	SYM
ejpam-6572	125	131	∈	∈	NOUN
ejpam-6572	125	132	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	125	133	,	,	PUNCT
ejpam-6572	125	134	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	125	135	)	)	PUNCT
ejpam-6572	125	136	)	)	PUNCT
ejpam-6572	125	137	)	)	PUNCT
ejpam-6572	126	1	for	for	ADP
ejpam-6572	126	2	every	every	DET
ejpam-6572	126	3	σ1σ2	σ1σ2	NOUN
ejpam-6572	126	4	-	-	ADJ
ejpam-6572	126	5	open	open	ADJ
ejpam-6572	126	6	set	set	NOUN
ejpam-6572	126	7	v	v	NOUN
ejpam-6572	126	8	of	of	ADP
ejpam-6572	126	9	y	y	NOUN
ejpam-6572	126	10	containing	contain	VERB
ejpam-6572	126	11	f(x	f(x	PROPN
ejpam-6572	126	12	)	)	PUNCT
ejpam-6572	126	13	;	;	PUNCT
ejpam-6572	126	14	(	(	PUNCT
ejpam-6572	126	15	4	4	X
ejpam-6572	126	16	)	)	PUNCT
ejpam-6572	126	17	x	x	SYM
ejpam-6572	126	18	∈	∈	NOUN
ejpam-6572	126	19	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	NOUN
ejpam-6572	126	20	,	,	PUNCT
ejpam-6572	126	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	126	22	)	)	PUNCT
ejpam-6572	126	23	)	)	PUNCT
ejpam-6572	126	24	)	)	PUNCT
ejpam-6572	126	25	)	)	PUNCT
ejpam-6572	126	26	for	for	ADP
ejpam-6572	126	27	every	every	DET
ejpam-6572	126	28	σ1σ2	σ1σ2	NOUN
ejpam-6572	126	29	-	-	ADJ
ejpam-6572	126	30	open	open	ADJ
ejpam-6572	126	31	set	set	NOUN
ejpam-6572	126	32	v	v	NOUN
ejpam-6572	126	33	of	of	ADP
ejpam-6572	126	34	y	y	NOUN
ejpam-6572	126	35	containing	contain	VERB
ejpam-6572	126	36	f(x	f(x	PROPN
ejpam-6572	126	37	)	)	PUNCT
ejpam-6572	126	38	.	.	PUNCT
ejpam-6572	127	1	proof	proof	NOUN
ejpam-6572	127	2	.	.	PUNCT
ejpam-6572	128	1	(	(	PUNCT
ejpam-6572	128	2	1	1	X
ejpam-6572	128	3	)	)	PUNCT
ejpam-6572	128	4	⇒	⇒	NOUN
ejpam-6572	128	5	(	(	PUNCT
ejpam-6572	128	6	2	2	NUM
ejpam-6572	128	7	):	):	PUNCT
ejpam-6572	128	8	let	let	VERB
ejpam-6572	128	9	u	u	PRON
ejpam-6572	128	10	(	(	PUNCT
ejpam-6572	128	11	x	x	NOUN
ejpam-6572	128	12	)	)	PUNCT
ejpam-6572	128	13	be	be	VERB
ejpam-6572	128	14	the	the	DET
ejpam-6572	128	15	family	family	NOUN
ejpam-6572	128	16	of	of	ADP
ejpam-6572	128	17	all	all	DET
ejpam-6572	128	18	⋆-open	⋆-open	ADJ
ejpam-6572	128	19	sets	set	NOUN
ejpam-6572	128	20	of	of	ADP
ejpam-6572	128	21	x	x	PUNCT
ejpam-6572	128	22	containing	contain	VERB
ejpam-6572	128	23	x.	x.	NOUN
ejpam-6572	128	24	let	let	VERB
ejpam-6572	128	25	v	v	PART
ejpam-6572	128	26	be	be	AUX
ejpam-6572	128	27	any	any	DET
ejpam-6572	128	28	σ1σ2	σ1σ2	NOUN
ejpam-6572	128	29	-	-	ADJ
ejpam-6572	128	30	open	open	ADJ
ejpam-6572	128	31	set	set	NOUN
ejpam-6572	128	32	of	of	ADP
ejpam-6572	128	33	y	y	PROPN
ejpam-6572	128	34	containing	contain	VERB
ejpam-6572	128	35	f(x	f(x	PROPN
ejpam-6572	128	36	)	)	PUNCT
ejpam-6572	128	37	.	.	PUNCT
ejpam-6572	129	1	for	for	ADP
ejpam-6572	129	2	eachh	eachh	PROPN
ejpam-6572	129	3	∈	∈	PROPN
ejpam-6572	129	4	u	u	PROPN
ejpam-6572	129	5	(	(	PUNCT
ejpam-6572	129	6	x	x	NOUN
ejpam-6572	129	7	)	)	PUNCT
ejpam-6572	129	8	,	,	PUNCT
ejpam-6572	129	9	there	there	PRON
ejpam-6572	129	10	exists	exist	VERB
ejpam-6572	129	11	a	a	DET
ejpam-6572	129	12	nonempty	nonempty	ADJ
ejpam-6572	129	13	⋆open	⋆open	PUNCT
ejpam-6572	129	14	set	set	NOUN
ejpam-6572	129	15	gh	gh	PROPN
ejpam-6572	129	16	such	such	ADJ
ejpam-6572	129	17	that	that	SCONJ
ejpam-6572	129	18	gh	gh	PROPN
ejpam-6572	129	19	⊆	⊆	NUM
ejpam-6572	129	20	h	h	NOUN
ejpam-6572	129	21	,	,	PUNCT
ejpam-6572	129	22	f(gh	f(gh	ADJ
ejpam-6572	129	23	)	)	PUNCT
ejpam-6572	129	24	⊆	⊆	NUM
ejpam-6572	129	25	(	(	PUNCT
ejpam-6572	129	26	σ1	σ1	PROPN
ejpam-6572	129	27	,	,	PUNCT
ejpam-6572	129	28	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	129	29	)	)	PUNCT
ejpam-6572	129	30	.	.	PUNCT
ejpam-6572	130	1	let	let	VERB
ejpam-6572	130	2	w	w	NOUN
ejpam-6572	130	3	=	=	PUNCT
ejpam-6572	130	4	∪{gh	∪{gh	NOUN
ejpam-6572	130	5	|	|	ADV
ejpam-6572	130	6	h	h	NOUN
ejpam-6572	130	7	∈	∈	PROPN
ejpam-6572	130	8	u	u	NOUN
ejpam-6572	130	9	(	(	PUNCT
ejpam-6572	130	10	x	x	NOUN
ejpam-6572	130	11	)	)	PUNCT
ejpam-6572	130	12	}	}	PUNCT
ejpam-6572	130	13	.	.	PUNCT
ejpam-6572	131	1	then	then	ADV
ejpam-6572	131	2	,	,	PUNCT
ejpam-6572	131	3	w	w	PROPN
ejpam-6572	131	4	is	be	AUX
ejpam-6572	131	5	⋆-open	⋆-open	ADJ
ejpam-6572	131	6	in	in	ADP
ejpam-6572	131	7	x	x	SYM
ejpam-6572	131	8	and	and	CCONJ
ejpam-6572	131	9	x	x	SYM
ejpam-6572	131	10	∈	∈	PROPN
ejpam-6572	131	11	cl⋆(w	cl⋆(w	PROPN
ejpam-6572	131	12	)	)	PUNCT
ejpam-6572	131	13	.	.	PUNCT
ejpam-6572	132	1	put	put	VERB
ejpam-6572	132	2	u	u	NOUN
ejpam-6572	132	3	=	=	NOUN
ejpam-6572	132	4	w	w	NOUN
ejpam-6572	132	5	∪	∪	X
ejpam-6572	132	6	{	{	PUNCT
ejpam-6572	132	7	x	x	NOUN
ejpam-6572	132	8	}	}	PUNCT
ejpam-6572	132	9	.	.	PUNCT
ejpam-6572	133	1	then	then	ADV
ejpam-6572	133	2	,	,	PUNCT
ejpam-6572	133	3	u	u	NOUN
ejpam-6572	133	4	is	be	AUX
ejpam-6572	133	5	a	a	DET
ejpam-6572	133	6	semi	semi	ADJ
ejpam-6572	133	7	-	-	ADJ
ejpam-6572	133	8	i	i	PRON
ejpam-6572	133	9	⋆-open	⋆-open	VERB
ejpam-6572	133	10	set	set	VERB
ejpam-6572	133	11	of	of	ADP
ejpam-6572	133	12	x	x	PUNCT
ejpam-6572	133	13	containing	contain	VERB
ejpam-6572	133	14	x	x	PROPN
ejpam-6572	133	15	and	and	CCONJ
ejpam-6572	133	16	f(u	f(u	PROPN
ejpam-6572	133	17	)	)	PUNCT
ejpam-6572	134	1	⊆	⊆	NUM
ejpam-6572	134	2	(	(	PUNCT
ejpam-6572	134	3	σ1	σ1	PROPN
ejpam-6572	134	4	,	,	PUNCT
ejpam-6572	134	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	134	6	)	)	PUNCT
ejpam-6572	134	7	.	.	PUNCT
ejpam-6572	135	1	(	(	PUNCT
ejpam-6572	135	2	2	2	X
ejpam-6572	135	3	)	)	PUNCT
ejpam-6572	135	4	⇒	⇒	NOUN
ejpam-6572	135	5	(	(	PUNCT
ejpam-6572	135	6	3	3	NUM
ejpam-6572	135	7	):	):	PUNCT
ejpam-6572	135	8	let	let	VERB
ejpam-6572	135	9	v	v	PART
ejpam-6572	135	10	be	be	AUX
ejpam-6572	135	11	any	any	DET
ejpam-6572	135	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	135	13	-	-	ADJ
ejpam-6572	135	14	open	open	ADJ
ejpam-6572	135	15	set	set	NOUN
ejpam-6572	135	16	of	of	ADP
ejpam-6572	135	17	y	y	PROPN
ejpam-6572	135	18	containing	contain	VERB
ejpam-6572	135	19	f(x	f(x	PROPN
ejpam-6572	135	20	)	)	PUNCT
ejpam-6572	135	21	.	.	PUNCT
ejpam-6572	136	1	then	then	ADV
ejpam-6572	136	2	,	,	PUNCT
ejpam-6572	136	3	there	there	PRON
ejpam-6572	136	4	exists	exist	VERB
ejpam-6572	136	5	a	a	DET
ejpam-6572	136	6	semi	semi	NOUN
ejpam-6572	136	7	-	-	ADJ
ejpam-6572	136	8	i	i	PRON
ejpam-6572	136	9	⋆-open	⋆-open	VERB
ejpam-6572	136	10	set	set	VERB
ejpam-6572	136	11	u	u	NOUN
ejpam-6572	136	12	of	of	ADP
ejpam-6572	136	13	x	x	PUNCT
ejpam-6572	136	14	containing	contain	VERB
ejpam-6572	136	15	x	x	PUNCT
ejpam-6572	136	16	such	such	ADJ
ejpam-6572	136	17	that	that	DET
ejpam-6572	136	18	f(u	f(u	PROPN
ejpam-6572	136	19	)	)	PUNCT
ejpam-6572	136	20	⊆	⊆	NUM
ejpam-6572	136	21	(	(	PUNCT
ejpam-6572	136	22	σ1	σ1	PROPN
ejpam-6572	136	23	,	,	PUNCT
ejpam-6572	136	24	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	136	25	)	)	PUNCT
ejpam-6572	136	26	.	.	PUNCT
ejpam-6572	137	1	thus	thus	ADV
ejpam-6572	137	2	,	,	PUNCT
ejpam-6572	137	3	x	x	PUNCT
ejpam-6572	137	4	∈	∈	PROPN
ejpam-6572	137	5	u	u	NOUN
ejpam-6572	137	6	⊆	⊆	NUM
ejpam-6572	137	7	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	137	8	,	,	PUNCT
ejpam-6572	137	9	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	137	10	)	)	PUNCT
ejpam-6572	137	11	)	)	PUNCT
ejpam-6572	138	1	b.	b.	PROPN
ejpam-6572	138	2	kong	kong	PROPN
ejpam-6572	138	3	-	-	PUNCT
ejpam-6572	138	4	ied	ied	PROPN
ejpam-6572	138	5	,	,	PUNCT
ejpam-6572	138	6	a.	a.	PROPN
ejpam-6572	138	7	sama	sama	PROPN
ejpam-6572	138	8	-	-	PUNCT
ejpam-6572	138	9	ae	ae	PROPN
ejpam-6572	138	10	,	,	PUNCT
ejpam-6572	138	11	c.	c.	PROPN
ejpam-6572	138	12	boonpok	boonpok	PROPN
ejpam-6572	138	13	/	/	SYM
ejpam-6572	138	14	eur	eur	PROPN
ejpam-6572	138	15	.	.	PUNCT
ejpam-6572	139	1	j.	j.	PROPN
ejpam-6572	139	2	pure	pure	PROPN
ejpam-6572	139	3	appl	appl	PROPN
ejpam-6572	139	4	.	.	PROPN
ejpam-6572	139	5	math	math	PROPN
ejpam-6572	139	6	,	,	PUNCT
ejpam-6572	139	7	18	18	NUM
ejpam-6572	139	8	(	(	PUNCT
ejpam-6572	139	9	3	3	NUM
ejpam-6572	139	10	)	)	PUNCT
ejpam-6572	139	11	(	(	PUNCT
ejpam-6572	139	12	2025	2025	NUM
ejpam-6572	139	13	)	)	PUNCT
ejpam-6572	139	14	,	,	PUNCT
ejpam-6572	139	15	6572	6572	NUM
ejpam-6572	139	16	5	5	NUM
ejpam-6572	139	17	of	of	ADP
ejpam-6572	139	18	12	12	NUM
ejpam-6572	139	19	and	and	CCONJ
ejpam-6572	139	20	hence	hence	ADV
ejpam-6572	139	21	x	x	PART
ejpam-6572	139	22	∈	∈	PROPN
ejpam-6572	139	23	u	u	NOUN
ejpam-6572	139	24	⊆	⊆	NUM
ejpam-6572	139	25	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	139	26	,	,	PUNCT
ejpam-6572	139	27	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	139	28	)	)	PUNCT
ejpam-6572	139	29	)	)	PUNCT
ejpam-6572	139	30	)	)	PUNCT
ejpam-6572	139	31	.	.	PUNCT
ejpam-6572	140	1	(	(	PUNCT
ejpam-6572	140	2	3	3	X
ejpam-6572	140	3	)	)	PUNCT
ejpam-6572	140	4	⇒	⇒	NOUN
ejpam-6572	140	5	(	(	PUNCT
ejpam-6572	140	6	4	4	NUM
ejpam-6572	140	7	):	):	PUNCT
ejpam-6572	140	8	let	let	VERB
ejpam-6572	140	9	v	v	PART
ejpam-6572	140	10	be	be	AUX
ejpam-6572	140	11	any	any	DET
ejpam-6572	140	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	140	13	-	-	ADJ
ejpam-6572	140	14	open	open	ADJ
ejpam-6572	140	15	set	set	NOUN
ejpam-6572	140	16	of	of	ADP
ejpam-6572	140	17	y	y	PROPN
ejpam-6572	140	18	containing	contain	VERB
ejpam-6572	140	19	f(x	f(x	PROPN
ejpam-6572	140	20	)	)	PUNCT
ejpam-6572	140	21	.	.	PUNCT
ejpam-6572	141	1	thus	thus	ADV
ejpam-6572	141	2	by	by	ADP
ejpam-6572	141	3	(	(	PUNCT
ejpam-6572	141	4	3	3	NUM
ejpam-6572	141	5	)	)	PUNCT
ejpam-6572	141	6	,	,	PUNCT
ejpam-6572	141	7	x	x	PUNCT
ejpam-6572	141	8	∈	∈	NOUN
ejpam-6572	141	9	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	141	10	,	,	PUNCT
ejpam-6572	141	11	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	141	12	)	)	PUNCT
ejpam-6572	141	13	)	)	PUNCT
ejpam-6572	141	14	)	)	PUNCT
ejpam-6572	141	15	.	.	PUNCT
ejpam-6572	142	1	now	now	ADV
ejpam-6572	142	2	,	,	PUNCT
ejpam-6572	142	3	put	put	VERB
ejpam-6572	142	4	u	u	NOUN
ejpam-6572	142	5	=	=	NOUN
ejpam-6572	142	6	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	142	7	,	,	PUNCT
ejpam-6572	142	8	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	142	9	)	)	PUNCT
ejpam-6572	142	10	)	)	PUNCT
ejpam-6572	142	11	)	)	PUNCT
ejpam-6572	142	12	.	.	PUNCT
ejpam-6572	143	1	then	then	ADV
ejpam-6572	143	2	,	,	PUNCT
ejpam-6572	143	3	we	we	PRON
ejpam-6572	143	4	have	have	VERB
ejpam-6572	143	5	u	u	NOUN
ejpam-6572	143	6	is	be	AUX
ejpam-6572	143	7	semi	semi	ADJ
ejpam-6572	143	8	-	-	ADJ
ejpam-6572	143	9	i	i	PRON
ejpam-6572	143	10	⋆-open	⋆-open	VERB
ejpam-6572	143	11	in	in	ADP
ejpam-6572	143	12	x	x	X
ejpam-6572	143	13	and	and	CCONJ
ejpam-6572	143	14	by	by	ADP
ejpam-6572	143	15	lemma	lemma	PROPN
ejpam-6572	143	16	4	4	NUM
ejpam-6572	143	17	,	,	PUNCT
ejpam-6572	143	18	x	x	SYM
ejpam-6572	143	19	∈	∈	PROPN
ejpam-6572	143	20	u	u	NOUN
ejpam-6572	143	21	⊆	⊆	NUM
ejpam-6572	143	22	cl⋆(int⋆(u	cl⋆(int⋆(u	NOUN
ejpam-6572	143	23	)	)	PUNCT
ejpam-6572	143	24	)	)	PUNCT
ejpam-6572	144	1	⊆	⊆	X
ejpam-6572	144	2	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	ADJ
ejpam-6572	144	3	,	,	PUNCT
ejpam-6572	144	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	144	5	)	)	PUNCT
ejpam-6572	144	6	)	)	PUNCT
ejpam-6572	144	7	)	)	PUNCT
ejpam-6572	144	8	)	)	PUNCT
ejpam-6572	144	9	.	.	PUNCT
ejpam-6572	145	1	(	(	PUNCT
ejpam-6572	145	2	4	4	X
ejpam-6572	145	3	)	)	PUNCT
ejpam-6572	145	4	⇒	⇒	NOUN
ejpam-6572	145	5	(	(	PUNCT
ejpam-6572	145	6	1	1	NUM
ejpam-6572	145	7	):	):	PUNCT
ejpam-6572	145	8	let	let	VERB
ejpam-6572	145	9	u	u	PRON
ejpam-6572	145	10	be	be	AUX
ejpam-6572	145	11	any	any	DET
ejpam-6572	145	12	⋆-open	⋆-open	ADJ
ejpam-6572	145	13	set	set	NOUN
ejpam-6572	145	14	of	of	ADP
ejpam-6572	145	15	x	x	PUNCT
ejpam-6572	145	16	containing	contain	VERB
ejpam-6572	145	17	x	x	X
ejpam-6572	145	18	and	and	CCONJ
ejpam-6572	145	19	v	v	AUX
ejpam-6572	145	20	be	be	AUX
ejpam-6572	145	21	any	any	DET
ejpam-6572	145	22	σ1σ2	σ1σ2	NOUN
ejpam-6572	145	23	-	-	ADJ
ejpam-6572	145	24	open	open	ADJ
ejpam-6572	145	25	set	set	NOUN
ejpam-6572	145	26	of	of	ADP
ejpam-6572	145	27	y	y	PROPN
ejpam-6572	145	28	containing	contain	VERB
ejpam-6572	145	29	f(x	f(x	PROPN
ejpam-6572	145	30	)	)	PUNCT
ejpam-6572	145	31	.	.	PUNCT
ejpam-6572	146	1	then	then	ADV
ejpam-6572	146	2	,	,	PUNCT
ejpam-6572	146	3	we	we	PRON
ejpam-6572	146	4	have	have	VERB
ejpam-6572	146	5	x	x	PART
ejpam-6572	146	6	∈	∈	NOUN
ejpam-6572	146	7	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	NOUN
ejpam-6572	146	8	,	,	PUNCT
ejpam-6572	146	9	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	146	10	)	)	PUNCT
ejpam-6572	146	11	)	)	PUNCT
ejpam-6572	146	12	)	)	PUNCT
ejpam-6572	146	13	)	)	PUNCT
ejpam-6572	147	1	and	and	CCONJ
ejpam-6572	147	2	hence	hence	ADV
ejpam-6572	147	3	u	u	NOUN
ejpam-6572	147	4	∩	∩	NOUN
ejpam-6572	147	5	int⋆(f−1((σ1	int⋆(f−1((σ1	NOUN
ejpam-6572	147	6	,	,	PUNCT
ejpam-6572	147	7	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	147	8	)	)	PUNCT
ejpam-6572	147	9	)	)	PUNCT
ejpam-6572	147	10	)	)	PUNCT
ejpam-6572	148	1	̸=	̸=	PROPN
ejpam-6572	148	2	∅.	∅.	ADV
ejpam-6572	148	3	put	put	VERB
ejpam-6572	148	4	w	w	NOUN
ejpam-6572	148	5	=	=	SYM
ejpam-6572	148	6	u	u	NOUN
ejpam-6572	148	7	∩	∩	NOUN
ejpam-6572	148	8	int⋆(f−1((σ1	int⋆(f−1((σ1	NOUN
ejpam-6572	148	9	,	,	PUNCT
ejpam-6572	148	10	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	148	11	)	)	PUNCT
ejpam-6572	148	12	)	)	PUNCT
ejpam-6572	148	13	)	)	PUNCT
ejpam-6572	148	14	.	.	PUNCT
ejpam-6572	149	1	then	then	ADV
ejpam-6572	149	2	,	,	PUNCT
ejpam-6572	149	3	w	w	PROPN
ejpam-6572	149	4	is	be	AUX
ejpam-6572	149	5	a	a	DET
ejpam-6572	149	6	nonempty	nonempty	ADJ
ejpam-6572	149	7	⋆-open	⋆-open	NOUN
ejpam-6572	149	8	set	set	NOUN
ejpam-6572	149	9	of	of	ADP
ejpam-6572	149	10	x	x	PUNCT
ejpam-6572	149	11	such	such	ADJ
ejpam-6572	149	12	that	that	SCONJ
ejpam-6572	149	13	w	w	ADP
ejpam-6572	149	14	⊆	⊆	NUM
ejpam-6572	149	15	u	u	NOUN
ejpam-6572	149	16	,	,	PUNCT
ejpam-6572	149	17	f(w	f(w	PROPN
ejpam-6572	149	18	)	)	PUNCT
ejpam-6572	150	1	⊆	⊆	NUM
ejpam-6572	150	2	(	(	PUNCT
ejpam-6572	150	3	σ1	σ1	PROPN
ejpam-6572	150	4	,	,	PUNCT
ejpam-6572	150	5	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	150	6	)	)	PUNCT
ejpam-6572	150	7	.	.	PUNCT
ejpam-6572	151	1	this	this	PRON
ejpam-6572	151	2	shows	show	VERB
ejpam-6572	151	3	that	that	SCONJ
ejpam-6572	151	4	f	f	PROPN
ejpam-6572	151	5	is	be	AUX
ejpam-6572	151	6	almost	almost	ADV
ejpam-6572	151	7	quasi	quasi	ADJ
ejpam-6572	151	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	151	9	,	,	PUNCT
ejpam-6572	151	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	151	11	at	at	ADP
ejpam-6572	151	12	x.	x.	NOUN
ejpam-6572	151	13	theorem	theorem	NOUN
ejpam-6572	151	14	2	2	NUM
ejpam-6572	151	15	.	.	X
ejpam-6572	151	16	for	for	ADP
ejpam-6572	151	17	a	a	DET
ejpam-6572	151	18	function	function	NOUN
ejpam-6572	151	19	f	f	NOUN
ejpam-6572	151	20	:	:	PUNCT
ejpam-6572	151	21	(	(	PUNCT
ejpam-6572	151	22	x	x	X
ejpam-6572	151	23	,	,	PUNCT
ejpam-6572	151	24	τ	τ	PROPN
ejpam-6572	151	25	,	,	PUNCT
ejpam-6572	151	26	i	i	NOUN
ejpam-6572	151	27	)	)	PUNCT
ejpam-6572	151	28	→	→	PUNCT
ejpam-6572	151	29	(	(	PUNCT
ejpam-6572	151	30	y	y	PROPN
ejpam-6572	151	31	,	,	PUNCT
ejpam-6572	151	32	σ1	σ1	PROPN
ejpam-6572	151	33	,	,	PUNCT
ejpam-6572	151	34	σ2	σ2	NOUN
ejpam-6572	151	35	)	)	PUNCT
ejpam-6572	151	36	,	,	PUNCT
ejpam-6572	151	37	the	the	DET
ejpam-6572	151	38	following	follow	VERB
ejpam-6572	151	39	properties	property	NOUN
ejpam-6572	151	40	are	be	AUX
ejpam-6572	151	41	equivalent	equivalent	ADJ
ejpam-6572	151	42	:	:	PUNCT
ejpam-6572	151	43	(	(	PUNCT
ejpam-6572	151	44	1	1	X
ejpam-6572	151	45	)	)	PUNCT
ejpam-6572	151	46	f	f	NOUN
ejpam-6572	151	47	is	be	AUX
ejpam-6572	151	48	almost	almost	ADV
ejpam-6572	151	49	quasi	quasi	ADJ
ejpam-6572	151	50	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	151	51	,	,	PUNCT
ejpam-6572	151	52	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	151	53	;	;	PUNCT
ejpam-6572	151	54	(	(	PUNCT
ejpam-6572	151	55	2	2	X
ejpam-6572	151	56	)	)	PUNCT
ejpam-6572	151	57	for	for	ADP
ejpam-6572	151	58	each	each	DET
ejpam-6572	151	59	x	x	SYM
ejpam-6572	151	60	∈	∈	PROPN
ejpam-6572	151	61	x	x	X
ejpam-6572	151	62	and	and	CCONJ
ejpam-6572	151	63	every	every	DET
ejpam-6572	151	64	σ1σ2	σ1σ2	NUM
ejpam-6572	151	65	-	-	ADJ
ejpam-6572	151	66	open	open	ADJ
ejpam-6572	151	67	set	set	NOUN
ejpam-6572	151	68	v	v	NOUN
ejpam-6572	151	69	of	of	ADP
ejpam-6572	151	70	y	y	NOUN
ejpam-6572	151	71	containing	contain	VERB
ejpam-6572	151	72	f(x	f(x	PROPN
ejpam-6572	151	73	)	)	PUNCT
ejpam-6572	151	74	,	,	PUNCT
ejpam-6572	151	75	there	there	PRON
ejpam-6572	151	76	exists	exist	VERB
ejpam-6572	151	77	a	a	DET
ejpam-6572	151	78	semi	semi	NOUN
ejpam-6572	151	79	-	-	ADJ
ejpam-6572	151	80	i	i	PRON
ejpam-6572	151	81	⋆-open	⋆-open	VERB
ejpam-6572	151	82	set	set	VERB
ejpam-6572	151	83	u	u	NOUN
ejpam-6572	151	84	of	of	ADP
ejpam-6572	151	85	x	x	PUNCT
ejpam-6572	151	86	containing	contain	VERB
ejpam-6572	151	87	x	x	PUNCT
ejpam-6572	151	88	such	such	ADJ
ejpam-6572	151	89	that	that	DET
ejpam-6572	151	90	f(u	f(u	PROPN
ejpam-6572	151	91	)	)	PUNCT
ejpam-6572	151	92	⊆	⊆	NUM
ejpam-6572	151	93	(	(	PUNCT
ejpam-6572	151	94	σ1	σ1	PROPN
ejpam-6572	151	95	,	,	PUNCT
ejpam-6572	151	96	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	151	97	)	)	PUNCT
ejpam-6572	151	98	;	;	PUNCT
ejpam-6572	151	99	(	(	PUNCT
ejpam-6572	151	100	3	3	X
ejpam-6572	151	101	)	)	PUNCT
ejpam-6572	151	102	f−1(v	f−1(v	NOUN
ejpam-6572	151	103	)	)	PUNCT
ejpam-6572	151	104	is	be	AUX
ejpam-6572	151	105	semi	semi	ADJ
ejpam-6572	151	106	-	-	ADJ
ejpam-6572	151	107	i	i	PRON
ejpam-6572	151	108	⋆-open	⋆-open	VERB
ejpam-6572	151	109	in	in	ADP
ejpam-6572	151	110	x	x	PUNCT
ejpam-6572	151	111	for	for	ADP
ejpam-6572	151	112	every	every	DET
ejpam-6572	151	113	(	(	PUNCT
ejpam-6572	151	114	σ1	σ1	PROPN
ejpam-6572	151	115	,	,	PUNCT
ejpam-6572	151	116	σ2)r	σ2)r	NOUN
ejpam-6572	151	117	-	-	PUNCT
ejpam-6572	151	118	open	open	ADJ
ejpam-6572	151	119	set	set	VERB
ejpam-6572	151	120	v	v	NOUN
ejpam-6572	151	121	of	of	ADP
ejpam-6572	151	122	y	y	PROPN
ejpam-6572	151	123	;	;	PUNCT
ejpam-6572	151	124	(	(	PUNCT
ejpam-6572	151	125	4	4	X
ejpam-6572	151	126	)	)	PUNCT
ejpam-6572	151	127	f−1(v	f−1(v	NOUN
ejpam-6572	151	128	)	)	PUNCT
ejpam-6572	152	1	⊆	⊆	NUM
ejpam-6572	152	2	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	152	3	,	,	PUNCT
ejpam-6572	152	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	152	5	)	)	PUNCT
ejpam-6572	152	6	)	)	PUNCT
ejpam-6572	152	7	)	)	PUNCT
ejpam-6572	152	8	for	for	ADP
ejpam-6572	152	9	every	every	DET
ejpam-6572	152	10	σ1σ2	σ1σ2	NOUN
ejpam-6572	152	11	-	-	ADJ
ejpam-6572	152	12	open	open	ADJ
ejpam-6572	152	13	set	set	NOUN
ejpam-6572	152	14	v	v	NOUN
ejpam-6572	152	15	of	of	ADP
ejpam-6572	152	16	y	y	PROPN
ejpam-6572	152	17	;	;	PUNCT
ejpam-6572	152	18	(	(	PUNCT
ejpam-6572	152	19	5	5	X
ejpam-6572	152	20	)	)	PUNCT
ejpam-6572	152	21	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	152	22	-	-	PUNCT
ejpam-6572	152	23	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	152	24	-	-	PUNCT
ejpam-6572	152	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	152	26	-	-	PUNCT
ejpam-6572	152	27	cl(b	cl(b	NOUN
ejpam-6572	152	28	)	)	PUNCT
ejpam-6572	152	29	)	)	PUNCT
ejpam-6572	152	30	)	)	PUNCT
ejpam-6572	152	31	)	)	PUNCT
ejpam-6572	152	32	)	)	PUNCT
ejpam-6572	153	1	⊆	⊆	NUM
ejpam-6572	153	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	153	3	-	-	PUNCT
ejpam-6572	153	4	cl(b	cl(b	NOUN
ejpam-6572	153	5	)	)	PUNCT
ejpam-6572	153	6	)	)	PUNCT
ejpam-6572	153	7	for	for	ADP
ejpam-6572	153	8	every	every	DET
ejpam-6572	153	9	subset	subset	NOUN
ejpam-6572	153	10	b	b	PROPN
ejpam-6572	153	11	of	of	ADP
ejpam-6572	153	12	y	y	PROPN
ejpam-6572	153	13	;	;	PUNCT
ejpam-6572	153	14	(	(	PUNCT
ejpam-6572	153	15	6	6	X
ejpam-6572	153	16	)	)	PUNCT
ejpam-6572	153	17	f−1(v	f−1(v	NOUN
ejpam-6572	153	18	)	)	PUNCT
ejpam-6572	153	19	⊆	⊆	NUM
ejpam-6572	153	20	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	ADJ
ejpam-6572	153	21	,	,	PUNCT
ejpam-6572	153	22	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	153	23	)	)	PUNCT
ejpam-6572	153	24	)	)	PUNCT
ejpam-6572	153	25	)	)	PUNCT
ejpam-6572	153	26	)	)	PUNCT
ejpam-6572	153	27	for	for	ADP
ejpam-6572	153	28	every	every	DET
ejpam-6572	153	29	σ1σ2	σ1σ2	NOUN
ejpam-6572	153	30	-	-	ADJ
ejpam-6572	153	31	open	open	ADJ
ejpam-6572	153	32	set	set	NOUN
ejpam-6572	153	33	v	v	NOUN
ejpam-6572	153	34	of	of	ADP
ejpam-6572	153	35	y	y	PROPN
ejpam-6572	153	36	.	.	PUNCT
ejpam-6572	154	1	proof	proof	NOUN
ejpam-6572	154	2	.	.	PUNCT
ejpam-6572	155	1	(	(	PUNCT
ejpam-6572	155	2	1	1	X
ejpam-6572	155	3	)	)	PUNCT
ejpam-6572	155	4	⇒	⇒	NOUN
ejpam-6572	155	5	(	(	PUNCT
ejpam-6572	155	6	2	2	NUM
ejpam-6572	155	7	):	):	PUNCT
ejpam-6572	155	8	the	the	DET
ejpam-6572	155	9	proof	proof	NOUN
ejpam-6572	155	10	follows	follow	VERB
ejpam-6572	155	11	from	from	ADP
ejpam-6572	155	12	theorem	theorem	ADJ
ejpam-6572	155	13	1	1	NUM
ejpam-6572	155	14	.	.	PUNCT
ejpam-6572	155	15	(	(	PUNCT
ejpam-6572	155	16	2	2	X
ejpam-6572	155	17	)	)	PUNCT
ejpam-6572	155	18	⇒	⇒	NOUN
ejpam-6572	155	19	(	(	PUNCT
ejpam-6572	155	20	3	3	NUM
ejpam-6572	155	21	):	):	PUNCT
ejpam-6572	155	22	let	let	VERB
ejpam-6572	155	23	v	v	PART
ejpam-6572	155	24	be	be	AUX
ejpam-6572	155	25	any	any	DET
ejpam-6572	155	26	(	(	PUNCT
ejpam-6572	155	27	σ1	σ1	NOUN
ejpam-6572	155	28	,	,	PUNCT
ejpam-6572	155	29	σ2)r	σ2)r	NOUN
ejpam-6572	155	30	-	-	PUNCT
ejpam-6572	155	31	open	open	ADJ
ejpam-6572	155	32	set	set	NOUN
ejpam-6572	155	33	of	of	ADP
ejpam-6572	155	34	y	y	PROPN
ejpam-6572	155	35	and	and	CCONJ
ejpam-6572	155	36	x	x	PROPN
ejpam-6572	155	37	∈	∈	PROPN
ejpam-6572	155	38	f−1(v	f−1(v	NOUN
ejpam-6572	155	39	)	)	PUNCT
ejpam-6572	155	40	.	.	PUNCT
ejpam-6572	156	1	then	then	ADV
ejpam-6572	156	2	,	,	PUNCT
ejpam-6572	156	3	f(x	f(x	PROPN
ejpam-6572	156	4	)	)	PUNCT
ejpam-6572	156	5	∈	∈	PROPN
ejpam-6572	156	6	v	v	NOUN
ejpam-6572	156	7	and	and	CCONJ
ejpam-6572	156	8	there	there	PRON
ejpam-6572	156	9	exists	exist	VERB
ejpam-6572	156	10	a	a	DET
ejpam-6572	156	11	semi	semi	NOUN
ejpam-6572	156	12	-	-	ADJ
ejpam-6572	156	13	i	i	PRON
ejpam-6572	156	14	⋆-open	⋆-open	VERB
ejpam-6572	156	15	set	set	VERB
ejpam-6572	156	16	u	u	NOUN
ejpam-6572	156	17	of	of	ADP
ejpam-6572	156	18	x	x	PUNCT
ejpam-6572	156	19	containing	contain	VERB
ejpam-6572	156	20	x	x	PUNCT
ejpam-6572	156	21	such	such	ADJ
ejpam-6572	156	22	that	that	DET
ejpam-6572	156	23	f(u	f(u	PROPN
ejpam-6572	156	24	)	)	PUNCT
ejpam-6572	156	25	⊆	⊆	NUM
ejpam-6572	156	26	v	v	NOUN
ejpam-6572	156	27	.	.	PUNCT
ejpam-6572	157	1	thus	thus	ADV
ejpam-6572	157	2	,	,	PUNCT
ejpam-6572	157	3	x	x	PUNCT
ejpam-6572	157	4	∈	∈	PROPN
ejpam-6572	157	5	u	u	NOUN
ejpam-6572	157	6	⊆	⊆	NUM
ejpam-6572	157	7	f−1(v	f−1(v	NOUN
ejpam-6572	157	8	)	)	PUNCT
ejpam-6572	157	9	and	and	CCONJ
ejpam-6572	157	10	hence	hence	ADV
ejpam-6572	157	11	x	x	X
ejpam-6572	157	12	∈	∈	PROPN
ejpam-6572	157	13	sint⋆(f−1(v	sint⋆(f−1(v	NOUN
ejpam-6572	157	14	)	)	PUNCT
ejpam-6572	157	15	)	)	PUNCT
ejpam-6572	157	16	.	.	PUNCT
ejpam-6572	158	1	therefore	therefore	ADV
ejpam-6572	158	2	,	,	PUNCT
ejpam-6572	158	3	f−1(v	f−1(v	PROPN
ejpam-6572	158	4	)	)	PUNCT
ejpam-6572	159	1	⊆	⊆	NUM
ejpam-6572	159	2	sint⋆(f−1(v	sint⋆(f−1(v	NOUN
ejpam-6572	159	3	)	)	PUNCT
ejpam-6572	159	4	)	)	PUNCT
ejpam-6572	159	5	.	.	PUNCT
ejpam-6572	160	1	this	this	PRON
ejpam-6572	160	2	shows	show	VERB
ejpam-6572	160	3	that	that	DET
ejpam-6572	160	4	f−1(v	f−1(v	PROPN
ejpam-6572	160	5	)	)	PUNCT
ejpam-6572	160	6	is	be	AUX
ejpam-6572	160	7	semi	semi	ADJ
ejpam-6572	160	8	-	-	ADJ
ejpam-6572	160	9	i	i	PRON
ejpam-6572	160	10	⋆-open	⋆-open	VERB
ejpam-6572	160	11	in	in	ADP
ejpam-6572	160	12	x.	x.	PROPN
ejpam-6572	160	13	(	(	PUNCT
ejpam-6572	160	14	3	3	NUM
ejpam-6572	160	15	)	)	PUNCT
ejpam-6572	160	16	⇒	⇒	NOUN
ejpam-6572	160	17	(	(	PUNCT
ejpam-6572	160	18	4	4	NUM
ejpam-6572	160	19	):	):	PUNCT
ejpam-6572	160	20	let	let	VERB
ejpam-6572	160	21	v	v	PART
ejpam-6572	160	22	be	be	AUX
ejpam-6572	160	23	any	any	DET
ejpam-6572	160	24	σ1σ2	σ1σ2	NOUN
ejpam-6572	160	25	-	-	ADJ
ejpam-6572	160	26	open	open	ADJ
ejpam-6572	160	27	set	set	NOUN
ejpam-6572	160	28	of	of	ADP
ejpam-6572	160	29	y	y	PROPN
ejpam-6572	160	30	and	and	CCONJ
ejpam-6572	160	31	x	x	PROPN
ejpam-6572	160	32	∈	∈	PROPN
ejpam-6572	160	33	f−1(v	f−1(v	NOUN
ejpam-6572	160	34	)	)	PUNCT
ejpam-6572	160	35	.	.	PUNCT
ejpam-6572	161	1	then	then	ADV
ejpam-6572	161	2	,	,	PUNCT
ejpam-6572	161	3	we	we	PRON
ejpam-6572	161	4	have	have	VERB
ejpam-6572	161	5	f(x	f(x	PROPN
ejpam-6572	161	6	)	)	PUNCT
ejpam-6572	161	7	∈	∈	PROPN
ejpam-6572	161	8	v	v	ADP
ejpam-6572	161	9	⊆	⊆	NUM
ejpam-6572	161	10	(	(	PUNCT
ejpam-6572	161	11	σ1	σ1	PROPN
ejpam-6572	161	12	,	,	PUNCT
ejpam-6572	161	13	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	161	14	)	)	PUNCT
ejpam-6572	161	15	.	.	PUNCT
ejpam-6572	162	1	thus	thus	ADV
ejpam-6572	162	2	,	,	PUNCT
ejpam-6572	162	3	x	x	SYM
ejpam-6572	162	4	∈	∈	NOUN
ejpam-6572	162	5	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	162	6	,	,	PUNCT
ejpam-6572	162	7	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	162	8	)	)	PUNCT
ejpam-6572	162	9	)	)	PUNCT
ejpam-6572	162	10	.	.	PUNCT
ejpam-6572	163	1	by	by	ADP
ejpam-6572	163	2	lemma	lemma	PROPN
ejpam-6572	163	3	2	2	NUM
ejpam-6572	163	4	,	,	PUNCT
ejpam-6572	163	5	(	(	PUNCT
ejpam-6572	163	6	σ1	σ1	PROPN
ejpam-6572	163	7	,	,	PUNCT
ejpam-6572	163	8	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	163	9	)	)	PUNCT
ejpam-6572	163	10	is	be	AUX
ejpam-6572	163	11	(	(	PUNCT
ejpam-6572	163	12	σ1	σ1	NOUN
ejpam-6572	163	13	,	,	PUNCT
ejpam-6572	163	14	σ2)r	σ2)r	NOUN
ejpam-6572	163	15	-	-	PUNCT
ejpam-6572	163	16	open	open	ADJ
ejpam-6572	163	17	in	in	ADP
ejpam-6572	163	18	y	y	PROPN
ejpam-6572	163	19	.	.	PUNCT
ejpam-6572	164	1	by	by	ADP
ejpam-6572	164	2	(	(	PUNCT
ejpam-6572	164	3	3	3	NUM
ejpam-6572	164	4	)	)	PUNCT
ejpam-6572	164	5	,	,	PUNCT
ejpam-6572	164	6	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	164	7	,	,	PUNCT
ejpam-6572	164	8	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	164	9	)	)	PUNCT
ejpam-6572	164	10	)	)	PUNCT
ejpam-6572	164	11	is	be	AUX
ejpam-6572	164	12	semi	semi	ADJ
ejpam-6572	164	13	-	-	ADJ
ejpam-6572	164	14	i	i	PRON
ejpam-6572	164	15	⋆-open	⋆-open	VERB
ejpam-6572	164	16	in	in	ADP
ejpam-6572	164	17	x	x	X
ejpam-6572	164	18	and	and	CCONJ
ejpam-6572	164	19	so	so	ADV
ejpam-6572	164	20	x	x	SYM
ejpam-6572	164	21	∈	∈	NOUN
ejpam-6572	164	22	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	164	23	,	,	PUNCT
ejpam-6572	164	24	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	164	25	)	)	PUNCT
ejpam-6572	164	26	)	)	PUNCT
ejpam-6572	164	27	)	)	PUNCT
ejpam-6572	164	28	.	.	PUNCT
ejpam-6572	165	1	thus	thus	ADV
ejpam-6572	165	2	,	,	PUNCT
ejpam-6572	165	3	f−1(v	f−1(v	PROPN
ejpam-6572	165	4	)	)	PUNCT
ejpam-6572	166	1	⊆	⊆	NUM
ejpam-6572	166	2	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	166	3	,	,	PUNCT
ejpam-6572	166	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	166	5	)	)	PUNCT
ejpam-6572	166	6	)	)	PUNCT
ejpam-6572	166	7	)	)	PUNCT
ejpam-6572	166	8	.	.	PUNCT
ejpam-6572	167	1	(	(	PUNCT
ejpam-6572	167	2	4	4	X
ejpam-6572	167	3	)	)	PUNCT
ejpam-6572	167	4	⇒	⇒	NOUN
ejpam-6572	167	5	(	(	PUNCT
ejpam-6572	167	6	5	5	NUM
ejpam-6572	167	7	):	):	PUNCT
ejpam-6572	167	8	let	let	VERB
ejpam-6572	167	9	b	b	X
ejpam-6572	167	10	be	be	AUX
ejpam-6572	167	11	any	any	DET
ejpam-6572	167	12	subset	subset	NOUN
ejpam-6572	167	13	of	of	ADP
ejpam-6572	167	14	y	y	PROPN
ejpam-6572	167	15	.	.	PUNCT
ejpam-6572	168	1	then	then	ADV
ejpam-6572	168	2	,	,	PUNCT
ejpam-6572	168	3	we	we	PRON
ejpam-6572	168	4	have	have	AUX
ejpam-6572	168	5	y	y	NOUN
ejpam-6572	168	6	−	−	VERB
ejpam-6572	168	7	σ1σ2	σ1σ2	NOUN
ejpam-6572	168	8	-	-	PUNCT
ejpam-6572	168	9	cl(b	cl(b	NOUN
ejpam-6572	168	10	)	)	PUNCT
ejpam-6572	168	11	is	be	AUX
ejpam-6572	168	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	168	13	-	-	ADJ
ejpam-6572	168	14	open	open	ADJ
ejpam-6572	168	15	in	in	ADP
ejpam-6572	168	16	y	y	PROPN
ejpam-6572	168	17	.	.	PUNCT
ejpam-6572	169	1	thus	thus	ADV
ejpam-6572	169	2	by	by	ADP
ejpam-6572	169	3	(	(	PUNCT
ejpam-6572	169	4	4	4	NUM
ejpam-6572	169	5	)	)	PUNCT
ejpam-6572	169	6	and	and	CCONJ
ejpam-6572	169	7	lemma	lemma	PROPN
ejpam-6572	169	8	2	2	NUM
ejpam-6572	169	9	,	,	PUNCT
ejpam-6572	169	10	x	x	PUNCT
ejpam-6572	169	11	−	−	NOUN
ejpam-6572	169	12	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	169	13	-	-	PUNCT
ejpam-6572	169	14	cl(b	cl(b	NOUN
ejpam-6572	169	15	)	)	PUNCT
ejpam-6572	169	16	)	)	PUNCT
ejpam-6572	170	1	=	=	SYM
ejpam-6572	170	2	f−1(y	f−1(y	PROPN
ejpam-6572	170	3	−	−	NUM
ejpam-6572	170	4	σ1σ2	σ1σ2	NOUN
ejpam-6572	170	5	-	-	PUNCT
ejpam-6572	170	6	cl(b	cl(b	NOUN
ejpam-6572	170	7	)	)	PUNCT
ejpam-6572	170	8	)	)	PUNCT
ejpam-6572	171	1	b.	b.	PROPN
ejpam-6572	171	2	kong	kong	PROPN
ejpam-6572	171	3	-	-	PUNCT
ejpam-6572	171	4	ied	ied	PROPN
ejpam-6572	171	5	,	,	PUNCT
ejpam-6572	171	6	a.	a.	PROPN
ejpam-6572	171	7	sama	sama	PROPN
ejpam-6572	171	8	-	-	PUNCT
ejpam-6572	171	9	ae	ae	PROPN
ejpam-6572	171	10	,	,	PUNCT
ejpam-6572	171	11	c.	c.	PROPN
ejpam-6572	171	12	boonpok	boonpok	PROPN
ejpam-6572	171	13	/	/	SYM
ejpam-6572	171	14	eur	eur	PROPN
ejpam-6572	171	15	.	.	PUNCT
ejpam-6572	172	1	j.	j.	PROPN
ejpam-6572	172	2	pure	pure	PROPN
ejpam-6572	172	3	appl	appl	PROPN
ejpam-6572	172	4	.	.	PROPN
ejpam-6572	172	5	math	math	PROPN
ejpam-6572	172	6	,	,	PUNCT
ejpam-6572	172	7	18	18	NUM
ejpam-6572	172	8	(	(	PUNCT
ejpam-6572	172	9	3	3	NUM
ejpam-6572	172	10	)	)	PUNCT
ejpam-6572	172	11	(	(	PUNCT
ejpam-6572	172	12	2025	2025	NUM
ejpam-6572	172	13	)	)	PUNCT
ejpam-6572	172	14	,	,	PUNCT
ejpam-6572	172	15	6572	6572	NUM
ejpam-6572	172	16	6	6	NUM
ejpam-6572	172	17	of	of	ADP
ejpam-6572	172	18	12	12	NUM
ejpam-6572	172	19	⊆	⊆	NUM
ejpam-6572	172	20	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	172	21	,	,	PUNCT
ejpam-6572	172	22	σ2)-scl(y	σ2)-scl(y	NOUN
ejpam-6572	172	23	−	−	NOUN
ejpam-6572	172	24	σ1σ2	σ1σ2	NOUN
ejpam-6572	172	25	-	-	NOUN
ejpam-6572	172	26	cl(b	cl(b	NOUN
ejpam-6572	172	27	)	)	PUNCT
ejpam-6572	172	28	)	)	PUNCT
ejpam-6572	172	29	)	)	PUNCT
ejpam-6572	172	30	)	)	PUNCT
ejpam-6572	173	1	=	=	PUNCT
ejpam-6572	173	2	sint⋆(x	sint⋆(x	NOUN
ejpam-6572	173	3	−	−	NOUN
ejpam-6572	173	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	173	5	-	-	PUNCT
ejpam-6572	173	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	173	7	-	-	PUNCT
ejpam-6572	173	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	173	9	-	-	PUNCT
ejpam-6572	173	10	cl(b	cl(b	NOUN
ejpam-6572	173	11	)	)	PUNCT
ejpam-6572	173	12	)	)	PUNCT
ejpam-6572	173	13	)	)	PUNCT
ejpam-6572	173	14	)	)	PUNCT
ejpam-6572	173	15	)	)	PUNCT
ejpam-6572	174	1	=	=	PUNCT
ejpam-6572	174	2	x	x	X
ejpam-6572	174	3	−	−	NOUN
ejpam-6572	174	4	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NUM
ejpam-6572	174	5	-	-	PUNCT
ejpam-6572	174	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	174	7	-	-	PUNCT
ejpam-6572	174	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	174	9	-	-	PUNCT
ejpam-6572	174	10	cl(b	cl(b	NOUN
ejpam-6572	174	11	)	)	PUNCT
ejpam-6572	174	12	)	)	PUNCT
ejpam-6572	174	13	)	)	PUNCT
ejpam-6572	174	14	)	)	PUNCT
ejpam-6572	174	15	)	)	PUNCT
ejpam-6572	175	1	and	and	CCONJ
ejpam-6572	175	2	hence	hence	ADV
ejpam-6572	175	3	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	ADV
ejpam-6572	175	4	-	-	PUNCT
ejpam-6572	175	5	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	175	6	-	-	PUNCT
ejpam-6572	175	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	175	8	-	-	PUNCT
ejpam-6572	175	9	cl(b	cl(b	NOUN
ejpam-6572	175	10	)	)	PUNCT
ejpam-6572	175	11	)	)	PUNCT
ejpam-6572	175	12	)	)	PUNCT
ejpam-6572	175	13	)	)	PUNCT
ejpam-6572	175	14	)	)	PUNCT
ejpam-6572	176	1	⊆	⊆	NUM
ejpam-6572	176	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	176	3	-	-	PUNCT
ejpam-6572	176	4	cl(b	cl(b	NOUN
ejpam-6572	176	5	)	)	PUNCT
ejpam-6572	176	6	)	)	PUNCT
ejpam-6572	176	7	.	.	PUNCT
ejpam-6572	177	1	(	(	PUNCT
ejpam-6572	177	2	5	5	X
ejpam-6572	177	3	)	)	PUNCT
ejpam-6572	177	4	⇒	⇒	NOUN
ejpam-6572	177	5	(	(	PUNCT
ejpam-6572	177	6	6	6	NUM
ejpam-6572	177	7	):	):	PUNCT
ejpam-6572	177	8	let	let	VERB
ejpam-6572	177	9	v	v	PART
ejpam-6572	177	10	be	be	AUX
ejpam-6572	177	11	any	any	DET
ejpam-6572	177	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	177	13	-	-	ADJ
ejpam-6572	177	14	open	open	ADJ
ejpam-6572	177	15	set	set	NOUN
ejpam-6572	177	16	of	of	ADP
ejpam-6572	177	17	y	y	PROPN
ejpam-6572	177	18	.	.	PUNCT
ejpam-6572	178	1	then	then	ADV
ejpam-6572	178	2	,	,	PUNCT
ejpam-6572	178	3	y	y	PROPN
ejpam-6572	178	4	−v	−v	NOUN
ejpam-6572	178	5	is	be	AUX
ejpam-6572	178	6	σ1σ2	σ1σ2	NOUN
ejpam-6572	178	7	-	-	ADJ
ejpam-6572	178	8	closed	closed	ADJ
ejpam-6572	178	9	in	in	ADP
ejpam-6572	178	10	y	y	PROPN
ejpam-6572	178	11	.	.	PUNCT
ejpam-6572	179	1	by	by	ADP
ejpam-6572	179	2	(	(	PUNCT
ejpam-6572	179	3	5	5	NUM
ejpam-6572	179	4	)	)	PUNCT
ejpam-6572	179	5	and	and	CCONJ
ejpam-6572	179	6	lemma	lemma	PROPN
ejpam-6572	179	7	4	4	NUM
ejpam-6572	179	8	,	,	PUNCT
ejpam-6572	179	9	int⋆(cl⋆(f−1(σ1σ2	int⋆(cl⋆(f−1(σ1σ2	NOUN
ejpam-6572	179	10	-	-	PUNCT
ejpam-6572	179	11	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	179	12	-	-	PUNCT
ejpam-6572	179	13	int(y	int(y	PROPN
ejpam-6572	179	14	−	−	PROPN
ejpam-6572	179	15	v	v	NOUN
ejpam-6572	179	16	)	)	PUNCT
ejpam-6572	179	17	)	)	PUNCT
ejpam-6572	179	18	)	)	PUNCT
ejpam-6572	179	19	)	)	PUNCT
ejpam-6572	179	20	)	)	PUNCT
ejpam-6572	180	1	⊆	⊆	X
ejpam-6572	180	2	f−1(y	f−1(y	NOUN
ejpam-6572	180	3	−	−	PROPN
ejpam-6572	180	4	v	v	NOUN
ejpam-6572	180	5	)	)	PUNCT
ejpam-6572	180	6	=	=	PUNCT
ejpam-6572	180	7	x	x	PUNCT
ejpam-6572	180	8	−	−	PROPN
ejpam-6572	180	9	f−1(v	f−1(v	PROPN
ejpam-6572	180	10	)	)	PUNCT
ejpam-6572	180	11	.	.	PUNCT
ejpam-6572	181	1	moreover	moreover	ADV
ejpam-6572	181	2	,	,	PUNCT
ejpam-6572	181	3	we	we	PRON
ejpam-6572	181	4	have	have	VERB
ejpam-6572	181	5	int⋆(cl⋆(f−1(σ1σ2	int⋆(cl⋆(f−1(σ1σ2	VERB
ejpam-6572	181	6	-	-	PUNCT
ejpam-6572	181	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	181	8	-	-	PUNCT
ejpam-6572	181	9	int(y	int(y	PROPN
ejpam-6572	181	10	−	−	PROPN
ejpam-6572	181	11	v	v	NOUN
ejpam-6572	181	12	)	)	PUNCT
ejpam-6572	181	13	)	)	PUNCT
ejpam-6572	181	14	)	)	PUNCT
ejpam-6572	181	15	)	)	PUNCT
ejpam-6572	181	16	)	)	PUNCT
ejpam-6572	182	1	=	=	PUNCT
ejpam-6572	183	1	int⋆(cl⋆(f−1(y	int⋆(cl⋆(f−1(y	ADJ
ejpam-6572	183	2	−	−	PUNCT
ejpam-6572	183	3	σ1σ2	σ1σ2	X
ejpam-6572	183	4	-	-	PUNCT
ejpam-6572	183	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	183	6	-	-	PUNCT
ejpam-6572	183	7	cl(v	cl(v	NOUN
ejpam-6572	183	8	)	)	PUNCT
ejpam-6572	183	9	)	)	PUNCT
ejpam-6572	183	10	)	)	PUNCT
ejpam-6572	183	11	)	)	PUNCT
ejpam-6572	183	12	)	)	PUNCT
ejpam-6572	184	1	=	=	PRON
ejpam-6572	184	2	int⋆(cl⋆(x	int⋆(cl⋆(x	VERB
ejpam-6572	184	3	−	−	NOUN
ejpam-6572	184	4	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	184	5	,	,	PUNCT
ejpam-6572	184	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	184	7	)	)	PUNCT
ejpam-6572	184	8	)	)	PUNCT
ejpam-6572	184	9	)	)	PUNCT
ejpam-6572	184	10	)	)	PUNCT
ejpam-6572	185	1	=	=	PUNCT
ejpam-6572	185	2	x	x	X
ejpam-6572	186	1	−	−	NOUN
ejpam-6572	186	2	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	NOUN
ejpam-6572	186	3	,	,	PUNCT
ejpam-6572	186	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	186	5	)	)	PUNCT
ejpam-6572	186	6	)	)	PUNCT
ejpam-6572	186	7	)	)	PUNCT
ejpam-6572	186	8	)	)	PUNCT
ejpam-6572	186	9	.	.	PUNCT
ejpam-6572	187	1	thus	thus	ADV
ejpam-6572	187	2	,	,	PUNCT
ejpam-6572	187	3	f−1(v	f−1(v	PROPN
ejpam-6572	187	4	)	)	PUNCT
ejpam-6572	188	1	⊆	⊆	NUM
ejpam-6572	188	2	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	ADJ
ejpam-6572	188	3	,	,	PUNCT
ejpam-6572	188	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	188	5	)	)	PUNCT
ejpam-6572	188	6	)	)	PUNCT
ejpam-6572	188	7	)	)	PUNCT
ejpam-6572	188	8	)	)	PUNCT
ejpam-6572	188	9	.	.	PUNCT
ejpam-6572	189	1	(	(	PUNCT
ejpam-6572	189	2	6	6	X
ejpam-6572	189	3	)	)	PUNCT
ejpam-6572	189	4	⇒	⇒	NOUN
ejpam-6572	189	5	(	(	PUNCT
ejpam-6572	189	6	1	1	NUM
ejpam-6572	189	7	):	):	PUNCT
ejpam-6572	189	8	let	let	VERB
ejpam-6572	189	9	x	x	PUNCT
ejpam-6572	189	10	∈	∈	PROPN
ejpam-6572	189	11	x	x	X
ejpam-6572	189	12	and	and	CCONJ
ejpam-6572	189	13	v	v	X
ejpam-6572	189	14	be	be	AUX
ejpam-6572	189	15	any	any	DET
ejpam-6572	189	16	σ1σ2	σ1σ2	NOUN
ejpam-6572	189	17	-	-	ADJ
ejpam-6572	189	18	open	open	ADJ
ejpam-6572	189	19	set	set	NOUN
ejpam-6572	189	20	of	of	ADP
ejpam-6572	189	21	y	y	PROPN
ejpam-6572	189	22	containing	contain	VERB
ejpam-6572	189	23	f(x	f(x	PROPN
ejpam-6572	189	24	)	)	PUNCT
ejpam-6572	189	25	.	.	PUNCT
ejpam-6572	190	1	by	by	ADP
ejpam-6572	190	2	(	(	PUNCT
ejpam-6572	190	3	6	6	NUM
ejpam-6572	190	4	)	)	PUNCT
ejpam-6572	190	5	,	,	PUNCT
ejpam-6572	190	6	we	we	PRON
ejpam-6572	190	7	have	have	VERB
ejpam-6572	190	8	x	x	X
ejpam-6572	190	9	∈	∈	PROPN
ejpam-6572	190	10	f−1(v	f−1(v	NOUN
ejpam-6572	190	11	)	)	PUNCT
ejpam-6572	191	1	⊆	⊆	NUM
ejpam-6572	191	2	cl⋆(int⋆(f−1((σ1	cl⋆(int⋆(f−1((σ1	ADJ
ejpam-6572	191	3	,	,	PUNCT
ejpam-6572	191	4	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	191	5	)	)	PUNCT
ejpam-6572	191	6	)	)	PUNCT
ejpam-6572	191	7	)	)	PUNCT
ejpam-6572	191	8	)	)	PUNCT
ejpam-6572	191	9	and	and	CCONJ
ejpam-6572	191	10	by	by	ADP
ejpam-6572	191	11	lemma	lemma	PROPN
ejpam-6572	191	12	4	4	NUM
ejpam-6572	191	13	,	,	PUNCT
ejpam-6572	191	14	x	x	SYM
ejpam-6572	191	15	∈	∈	PROPN
ejpam-6572	191	16	f−1(v	f−1(v	NOUN
ejpam-6572	191	17	)	)	PUNCT
ejpam-6572	191	18	⊆	⊆	NUM
ejpam-6572	191	19	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	191	20	,	,	PUNCT
ejpam-6572	191	21	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	191	22	)	)	PUNCT
ejpam-6572	191	23	)	)	PUNCT
ejpam-6572	191	24	)	)	PUNCT
ejpam-6572	191	25	.	.	PUNCT
ejpam-6572	192	1	put	put	VERB
ejpam-6572	192	2	u	u	NOUN
ejpam-6572	192	3	=	=	NOUN
ejpam-6572	192	4	sint⋆(f−1((σ1	sint⋆(f−1((σ1	NOUN
ejpam-6572	192	5	,	,	PUNCT
ejpam-6572	192	6	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	192	7	)	)	PUNCT
ejpam-6572	192	8	)	)	PUNCT
ejpam-6572	192	9	)	)	PUNCT
ejpam-6572	192	10	.	.	PUNCT
ejpam-6572	193	1	then	then	ADV
ejpam-6572	193	2	,	,	PUNCT
ejpam-6572	193	3	u	u	NOUN
ejpam-6572	193	4	is	be	AUX
ejpam-6572	193	5	semi	semi	ADJ
ejpam-6572	193	6	-	-	ADJ
ejpam-6572	193	7	i	i	PRON
ejpam-6572	193	8	⋆-open	⋆-open	VERB
ejpam-6572	193	9	set	set	VERB
ejpam-6572	193	10	of	of	ADP
ejpam-6572	193	11	x	x	PUNCT
ejpam-6572	193	12	containing	contain	VERB
ejpam-6572	193	13	x	x	PUNCT
ejpam-6572	193	14	such	such	ADJ
ejpam-6572	193	15	that	that	DET
ejpam-6572	193	16	f(u	f(u	PROPN
ejpam-6572	193	17	)	)	PUNCT
ejpam-6572	193	18	⊆	⊆	NUM
ejpam-6572	193	19	(	(	PUNCT
ejpam-6572	193	20	σ1	σ1	PROPN
ejpam-6572	193	21	,	,	PUNCT
ejpam-6572	193	22	σ2)-scl(v	σ2)-scl(v	NOUN
ejpam-6572	193	23	)	)	PUNCT
ejpam-6572	193	24	.	.	PUNCT
ejpam-6572	194	1	this	this	PRON
ejpam-6572	194	2	shows	show	VERB
ejpam-6572	194	3	that	that	SCONJ
ejpam-6572	194	4	f	f	PROPN
ejpam-6572	194	5	is	be	AUX
ejpam-6572	194	6	almost	almost	ADV
ejpam-6572	194	7	quasi	quasi	ADJ
ejpam-6572	194	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	194	9	,	,	PUNCT
ejpam-6572	194	10	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6572	194	11	.	.	X
ejpam-6572	194	12	theorem	theorem	NOUN
ejpam-6572	194	13	3	3	NUM
ejpam-6572	194	14	.	.	X
ejpam-6572	194	15	for	for	ADP
ejpam-6572	194	16	a	a	DET
ejpam-6572	194	17	function	function	NOUN
ejpam-6572	194	18	f	f	NOUN
ejpam-6572	194	19	:	:	PUNCT
ejpam-6572	194	20	(	(	PUNCT
ejpam-6572	194	21	x	x	X
ejpam-6572	194	22	,	,	PUNCT
ejpam-6572	194	23	τ	τ	PROPN
ejpam-6572	194	24	,	,	PUNCT
ejpam-6572	194	25	i	i	NOUN
ejpam-6572	194	26	)	)	PUNCT
ejpam-6572	194	27	→	→	PUNCT
ejpam-6572	194	28	(	(	PUNCT
ejpam-6572	194	29	y	y	PROPN
ejpam-6572	194	30	,	,	PUNCT
ejpam-6572	194	31	σ1	σ1	PROPN
ejpam-6572	194	32	,	,	PUNCT
ejpam-6572	194	33	σ2	σ2	NOUN
ejpam-6572	194	34	)	)	PUNCT
ejpam-6572	194	35	,	,	PUNCT
ejpam-6572	194	36	the	the	DET
ejpam-6572	194	37	following	follow	VERB
ejpam-6572	194	38	properties	property	NOUN
ejpam-6572	194	39	are	be	AUX
ejpam-6572	194	40	equivalent	equivalent	ADJ
ejpam-6572	194	41	:	:	PUNCT
ejpam-6572	194	42	(	(	PUNCT
ejpam-6572	194	43	1	1	X
ejpam-6572	194	44	)	)	PUNCT
ejpam-6572	194	45	f	f	NOUN
ejpam-6572	194	46	is	be	AUX
ejpam-6572	194	47	almost	almost	ADV
ejpam-6572	194	48	quasi	quasi	ADJ
ejpam-6572	194	49	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	194	50	,	,	PUNCT
ejpam-6572	194	51	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	194	52	;	;	PUNCT
ejpam-6572	194	53	(	(	PUNCT
ejpam-6572	194	54	2	2	X
ejpam-6572	194	55	)	)	PUNCT
ejpam-6572	194	56	scl⋆(f−1(v	scl⋆(f−1(v	NOUN
ejpam-6572	194	57	)	)	PUNCT
ejpam-6572	194	58	)	)	PUNCT
ejpam-6572	195	1	⊆	⊆	NUM
ejpam-6572	195	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	195	3	-	-	PUNCT
ejpam-6572	195	4	cl(v	cl(v	NOUN
ejpam-6572	195	5	)	)	PUNCT
ejpam-6572	195	6	)	)	PUNCT
ejpam-6572	195	7	for	for	ADP
ejpam-6572	195	8	every	every	DET
ejpam-6572	195	9	(	(	PUNCT
ejpam-6572	195	10	σ1	σ1	PROPN
ejpam-6572	195	11	,	,	PUNCT
ejpam-6572	195	12	σ2)β	σ2)β	NOUN
ejpam-6572	195	13	-	-	PUNCT
ejpam-6572	195	14	open	open	NOUN
ejpam-6572	195	15	set	set	NOUN
ejpam-6572	195	16	v	v	NOUN
ejpam-6572	195	17	of	of	ADP
ejpam-6572	195	18	y	y	PROPN
ejpam-6572	195	19	;	;	PUNCT
ejpam-6572	195	20	(	(	PUNCT
ejpam-6572	195	21	3	3	X
ejpam-6572	195	22	)	)	PUNCT
ejpam-6572	195	23	scl⋆(f−1(v	scl⋆(f−1(v	NOUN
ejpam-6572	195	24	)	)	PUNCT
ejpam-6572	195	25	)	)	PUNCT
ejpam-6572	196	1	⊆	⊆	NUM
ejpam-6572	196	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	196	3	-	-	PUNCT
ejpam-6572	196	4	cl(v	cl(v	NOUN
ejpam-6572	196	5	)	)	PUNCT
ejpam-6572	196	6	)	)	PUNCT
ejpam-6572	196	7	for	for	ADP
ejpam-6572	196	8	every	every	DET
ejpam-6572	196	9	(	(	PUNCT
ejpam-6572	196	10	σ1	σ1	PROPN
ejpam-6572	196	11	,	,	PUNCT
ejpam-6572	196	12	σ2)s	σ2)s	NOUN
ejpam-6572	196	13	-	-	PUNCT
ejpam-6572	196	14	open	open	NOUN
ejpam-6572	196	15	set	set	NOUN
ejpam-6572	196	16	v	v	NOUN
ejpam-6572	196	17	of	of	ADP
ejpam-6572	196	18	y	y	PROPN
ejpam-6572	196	19	;	;	PUNCT
ejpam-6572	196	20	(	(	PUNCT
ejpam-6572	196	21	4	4	X
ejpam-6572	196	22	)	)	PUNCT
ejpam-6572	196	23	f−1(v	f−1(v	NOUN
ejpam-6572	196	24	)	)	PUNCT
ejpam-6572	196	25	⊆	⊆	X
ejpam-6572	196	26	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	VERB
ejpam-6572	196	27	-	-	PUNCT
ejpam-6572	196	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	196	29	-	-	PUNCT
ejpam-6572	196	30	cl(v	cl(v	NOUN
ejpam-6572	196	31	)	)	PUNCT
ejpam-6572	196	32	)	)	PUNCT
ejpam-6572	196	33	)	)	PUNCT
ejpam-6572	196	34	)	)	PUNCT
ejpam-6572	197	1	for	for	ADP
ejpam-6572	197	2	every	every	DET
ejpam-6572	197	3	(	(	PUNCT
ejpam-6572	197	4	σ1	σ1	PROPN
ejpam-6572	197	5	,	,	PUNCT
ejpam-6572	197	6	σ2)p	σ2)p	NOUN
ejpam-6572	197	7	-	-	PUNCT
ejpam-6572	197	8	open	open	NOUN
ejpam-6572	197	9	set	set	NOUN
ejpam-6572	197	10	v	v	NOUN
ejpam-6572	197	11	of	of	ADP
ejpam-6572	197	12	y	y	PROPN
ejpam-6572	197	13	.	.	PUNCT
ejpam-6572	198	1	proof	proof	NOUN
ejpam-6572	198	2	.	.	PUNCT
ejpam-6572	199	1	the	the	DET
ejpam-6572	199	2	proof	proof	NOUN
ejpam-6572	199	3	follows	follow	VERB
ejpam-6572	199	4	from	from	ADP
ejpam-6572	199	5	theorem	theorem	ADJ
ejpam-6572	199	6	2	2	NUM
ejpam-6572	199	7	.	.	PUNCT
ejpam-6572	199	8	b.	b.	PROPN
ejpam-6572	199	9	kong	kong	PROPN
ejpam-6572	199	10	-	-	PUNCT
ejpam-6572	199	11	ied	ied	PROPN
ejpam-6572	199	12	,	,	PUNCT
ejpam-6572	199	13	a.	a.	PROPN
ejpam-6572	199	14	sama	sama	PROPN
ejpam-6572	199	15	-	-	PUNCT
ejpam-6572	199	16	ae	ae	PROPN
ejpam-6572	199	17	,	,	PUNCT
ejpam-6572	199	18	c.	c.	PROPN
ejpam-6572	199	19	boonpok	boonpok	PROPN
ejpam-6572	199	20	/	/	SYM
ejpam-6572	199	21	eur	eur	PROPN
ejpam-6572	199	22	.	.	PUNCT
ejpam-6572	200	1	j.	j.	PROPN
ejpam-6572	200	2	pure	pure	PROPN
ejpam-6572	200	3	appl	appl	PROPN
ejpam-6572	200	4	.	.	PROPN
ejpam-6572	200	5	math	math	PROPN
ejpam-6572	200	6	,	,	PUNCT
ejpam-6572	200	7	18	18	NUM
ejpam-6572	200	8	(	(	PUNCT
ejpam-6572	200	9	3	3	NUM
ejpam-6572	200	10	)	)	PUNCT
ejpam-6572	200	11	(	(	PUNCT
ejpam-6572	200	12	2025	2025	NUM
ejpam-6572	200	13	)	)	PUNCT
ejpam-6572	200	14	,	,	PUNCT
ejpam-6572	200	15	6572	6572	NUM
ejpam-6572	200	16	7	7	NUM
ejpam-6572	200	17	of	of	ADP
ejpam-6572	200	18	12	12	NUM
ejpam-6572	200	19	4	4	NUM
ejpam-6572	200	20	.	.	PUNCT
ejpam-6572	201	1	weakly	weakly	ADJ
ejpam-6572	201	2	quasi	quasi	NOUN
ejpam-6572	201	3	τ	τ	PROPN
ejpam-6572	201	4	⋆(σ1	⋆(σ1	PROPN
ejpam-6572	201	5	,	,	PUNCT
ejpam-6572	201	6	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	201	7	functions	function	NOUN
ejpam-6572	201	8	in	in	ADP
ejpam-6572	201	9	this	this	DET
ejpam-6572	201	10	section	section	NOUN
ejpam-6572	201	11	,	,	PUNCT
ejpam-6572	201	12	we	we	PRON
ejpam-6572	201	13	introduce	introduce	VERB
ejpam-6572	201	14	the	the	DET
ejpam-6572	201	15	notion	notion	NOUN
ejpam-6572	201	16	of	of	ADP
ejpam-6572	201	17	weakly	weakly	ADJ
ejpam-6572	201	18	quasi	quasi	NOUN
ejpam-6572	201	19	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6572	201	20	,	,	PUNCT
ejpam-6572	201	21	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	201	22	functions	function	NOUN
ejpam-6572	201	23	.	.	PUNCT
ejpam-6572	202	1	furthermore	furthermore	ADV
ejpam-6572	202	2	,	,	PUNCT
ejpam-6572	202	3	several	several	ADJ
ejpam-6572	202	4	characterizations	characterization	NOUN
ejpam-6572	202	5	of	of	ADP
ejpam-6572	202	6	weakly	weakly	ADJ
ejpam-6572	202	7	quasi	quasi	NOUN
ejpam-6572	202	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	202	9	,	,	PUNCT
ejpam-6572	202	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	202	11	functions	function	NOUN
ejpam-6572	202	12	are	be	AUX
ejpam-6572	202	13	discussed	discuss	VERB
ejpam-6572	202	14	.	.	PUNCT
ejpam-6572	203	1	definition	definition	NOUN
ejpam-6572	203	2	2	2	NUM
ejpam-6572	203	3	.	.	PUNCT
ejpam-6572	204	1	a	a	DET
ejpam-6572	204	2	function	function	NOUN
ejpam-6572	204	3	f	f	NOUN
ejpam-6572	204	4	:	:	PUNCT
ejpam-6572	204	5	(	(	PUNCT
ejpam-6572	204	6	x	x	X
ejpam-6572	204	7	,	,	PUNCT
ejpam-6572	204	8	τ	τ	PROPN
ejpam-6572	204	9	,	,	PUNCT
ejpam-6572	204	10	i	i	NOUN
ejpam-6572	204	11	)	)	PUNCT
ejpam-6572	204	12	→	→	PUNCT
ejpam-6572	204	13	(	(	PUNCT
ejpam-6572	204	14	y	y	PROPN
ejpam-6572	204	15	,	,	PUNCT
ejpam-6572	204	16	σ1	σ1	PROPN
ejpam-6572	204	17	,	,	PUNCT
ejpam-6572	204	18	σ2	σ2	PROPN
ejpam-6572	204	19	)	)	PUNCT
ejpam-6572	204	20	is	be	AUX
ejpam-6572	204	21	said	say	VERB
ejpam-6572	204	22	to	to	PART
ejpam-6572	204	23	be	be	AUX
ejpam-6572	204	24	weakly	weakly	ADJ
ejpam-6572	204	25	quasi	quasi	ADJ
ejpam-6572	204	26	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	204	27	,	,	PUNCT
ejpam-6572	204	28	σ2)continuous	σ2)continuous	ADJ
ejpam-6572	204	29	at	at	ADP
ejpam-6572	204	30	a	a	DET
ejpam-6572	204	31	point	point	NOUN
ejpam-6572	204	32	x	x	SYM
ejpam-6572	204	33	∈	∈	NOUN
ejpam-6572	204	34	x	x	PUNCT
ejpam-6572	204	35	if	if	SCONJ
ejpam-6572	204	36	for	for	ADP
ejpam-6572	204	37	each	each	DET
ejpam-6572	204	38	σ1σ2	σ1σ2	VERB
ejpam-6572	204	39	-	-	ADJ
ejpam-6572	204	40	open	open	ADJ
ejpam-6572	204	41	set	set	NOUN
ejpam-6572	204	42	v	v	NOUN
ejpam-6572	204	43	of	of	ADP
ejpam-6572	204	44	y	y	NOUN
ejpam-6572	204	45	containing	contain	VERB
ejpam-6572	204	46	f(x	f(x	PROPN
ejpam-6572	204	47	)	)	PUNCT
ejpam-6572	204	48	and	and	CCONJ
ejpam-6572	204	49	each	each	PRON
ejpam-6572	204	50	⋆-open	⋆-open	ADV
ejpam-6572	204	51	set	set	VERB
ejpam-6572	204	52	u	u	NOUN
ejpam-6572	204	53	of	of	ADP
ejpam-6572	204	54	x	x	SYM
ejpam-6572	204	55	containing	contain	VERB
ejpam-6572	204	56	x	x	PRON
ejpam-6572	204	57	,	,	PUNCT
ejpam-6572	204	58	there	there	PRON
ejpam-6572	204	59	exists	exist	VERB
ejpam-6572	205	1	a	a	DET
ejpam-6572	205	2	nonempty	nonempty	ADJ
ejpam-6572	205	3	⋆-open	⋆-open	NOUN
ejpam-6572	205	4	set	set	VERB
ejpam-6572	205	5	g	g	PROPN
ejpam-6572	205	6	such	such	ADJ
ejpam-6572	205	7	that	that	SCONJ
ejpam-6572	205	8	g	g	PROPN
ejpam-6572	205	9	⊆	⊆	NUM
ejpam-6572	205	10	u	u	NOUN
ejpam-6572	205	11	,	,	PUNCT
ejpam-6572	205	12	f(g	f(g	PROPN
ejpam-6572	205	13	)	)	PUNCT
ejpam-6572	205	14	⊆	⊆	NUM
ejpam-6572	205	15	σ1σ2	σ1σ2	NOUN
ejpam-6572	205	16	-	-	NUM
ejpam-6572	205	17	cl(v	cl(v	NOUN
ejpam-6572	205	18	)	)	PUNCT
ejpam-6572	205	19	.	.	PUNCT
ejpam-6572	206	1	a	a	DET
ejpam-6572	206	2	function	function	NOUN
ejpam-6572	206	3	f	f	NOUN
ejpam-6572	206	4	:	:	PUNCT
ejpam-6572	206	5	(	(	PUNCT
ejpam-6572	206	6	x	x	X
ejpam-6572	206	7	,	,	PUNCT
ejpam-6572	206	8	τ	τ	PROPN
ejpam-6572	206	9	,	,	PUNCT
ejpam-6572	206	10	i	i	NOUN
ejpam-6572	206	11	)	)	PUNCT
ejpam-6572	206	12	→	→	PUNCT
ejpam-6572	206	13	(	(	PUNCT
ejpam-6572	206	14	y	y	PROPN
ejpam-6572	206	15	,	,	PUNCT
ejpam-6572	206	16	σ1	σ1	PROPN
ejpam-6572	206	17	,	,	PUNCT
ejpam-6572	206	18	σ2	σ2	PROPN
ejpam-6572	206	19	)	)	PUNCT
ejpam-6572	206	20	is	be	AUX
ejpam-6572	206	21	said	say	VERB
ejpam-6572	206	22	to	to	PART
ejpam-6572	206	23	be	be	AUX
ejpam-6572	206	24	weakly	weakly	ADJ
ejpam-6572	206	25	quasi	quasi	ADJ
ejpam-6572	206	26	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	206	27	,	,	PUNCT
ejpam-6572	206	28	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	206	29	if	if	SCONJ
ejpam-6572	206	30	f	f	PROPN
ejpam-6572	206	31	is	be	AUX
ejpam-6572	206	32	weakly	weakly	ADJ
ejpam-6572	206	33	quasi	quasi	ADJ
ejpam-6572	206	34	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	206	35	,	,	PUNCT
ejpam-6572	206	36	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	206	37	at	at	ADP
ejpam-6572	206	38	each	each	DET
ejpam-6572	206	39	point	point	NOUN
ejpam-6572	206	40	of	of	ADP
ejpam-6572	206	41	x.	x.	NOUN
ejpam-6572	206	42	remark	remark	PROPN
ejpam-6572	206	43	1	1	NUM
ejpam-6572	206	44	.	.	PUNCT
ejpam-6572	207	1	for	for	ADP
ejpam-6572	207	2	a	a	DET
ejpam-6572	207	3	function	function	NOUN
ejpam-6572	207	4	f	f	NOUN
ejpam-6572	207	5	:	:	PUNCT
ejpam-6572	207	6	(	(	PUNCT
ejpam-6572	207	7	x	x	X
ejpam-6572	207	8	,	,	PUNCT
ejpam-6572	207	9	τ	τ	PROPN
ejpam-6572	207	10	,	,	PUNCT
ejpam-6572	207	11	i	i	NOUN
ejpam-6572	207	12	)	)	PUNCT
ejpam-6572	207	13	→	→	PUNCT
ejpam-6572	207	14	(	(	PUNCT
ejpam-6572	207	15	y	y	PROPN
ejpam-6572	207	16	,	,	PUNCT
ejpam-6572	207	17	σ1	σ1	PROPN
ejpam-6572	207	18	,	,	PUNCT
ejpam-6572	207	19	σ2	σ2	NOUN
ejpam-6572	207	20	)	)	PUNCT
ejpam-6572	207	21	,	,	PUNCT
ejpam-6572	207	22	the	the	DET
ejpam-6572	207	23	following	follow	VERB
ejpam-6572	207	24	implication	implication	NOUN
ejpam-6572	207	25	holds	hold	VERB
ejpam-6572	207	26	:	:	PUNCT
ejpam-6572	207	27	almost	almost	ADV
ejpam-6572	207	28	quasi	quasi	ADJ
ejpam-6572	207	29	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	207	30	,	,	PUNCT
ejpam-6572	207	31	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6572	207	32	⇒	⇒	NOUN
ejpam-6572	207	33	weakly	weakly	ADJ
ejpam-6572	207	34	quasi	quasi	NOUN
ejpam-6572	207	35	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	207	36	,	,	PUNCT
ejpam-6572	207	37	σ2)-continuity	σ2)-continuity	NOUN
ejpam-6572	207	38	.	.	PUNCT
ejpam-6572	208	1	the	the	DET
ejpam-6572	208	2	converse	converse	NOUN
ejpam-6572	208	3	of	of	ADP
ejpam-6572	208	4	the	the	DET
ejpam-6572	208	5	implication	implication	NOUN
ejpam-6572	208	6	is	be	AUX
ejpam-6572	208	7	not	not	PART
ejpam-6572	208	8	true	true	ADJ
ejpam-6572	208	9	in	in	ADP
ejpam-6572	208	10	general	general	ADJ
ejpam-6572	208	11	.	.	PUNCT
ejpam-6572	209	1	we	we	PRON
ejpam-6572	209	2	give	give	VERB
ejpam-6572	209	3	an	an	DET
ejpam-6572	209	4	example	example	NOUN
ejpam-6572	209	5	for	for	ADP
ejpam-6572	209	6	the	the	DET
ejpam-6572	209	7	implication	implication	NOUN
ejpam-6572	209	8	as	as	SCONJ
ejpam-6572	209	9	follows	follow	VERB
ejpam-6572	209	10	.	.	PUNCT
ejpam-6572	209	11	example	example	NOUN
ejpam-6572	210	1	1	1	NUM
ejpam-6572	210	2	.	.	PUNCT
ejpam-6572	210	3	let	let	VERB
ejpam-6572	210	4	x	x	PUNCT
ejpam-6572	210	5	=	=	PRON
ejpam-6572	210	6	{	{	PUNCT
ejpam-6572	210	7	1	1	NUM
ejpam-6572	210	8	,	,	PUNCT
ejpam-6572	210	9	2	2	NUM
ejpam-6572	210	10	,	,	PUNCT
ejpam-6572	210	11	3	3	NUM
ejpam-6572	210	12	}	}	PUNCT
ejpam-6572	210	13	with	with	ADP
ejpam-6572	210	14	a	a	DET
ejpam-6572	210	15	topology	topology	NOUN
ejpam-6572	210	16	τ	τ	X
ejpam-6572	210	17	=	=	SYM
ejpam-6572	210	18	{	{	PUNCT
ejpam-6572	210	19	∅	∅	NOUN
ejpam-6572	210	20	,	,	PUNCT
ejpam-6572	210	21	{	{	PUNCT
ejpam-6572	210	22	1	1	NUM
ejpam-6572	210	23	}	}	PUNCT
ejpam-6572	210	24	,	,	PUNCT
ejpam-6572	210	25	{	{	PUNCT
ejpam-6572	210	26	2	2	NUM
ejpam-6572	210	27	,	,	PUNCT
ejpam-6572	210	28	3	3	NUM
ejpam-6572	210	29	}	}	PUNCT
ejpam-6572	210	30	,	,	PUNCT
ejpam-6572	210	31	x	x	NOUN
ejpam-6572	210	32	}	}	PUNCT
ejpam-6572	210	33	and	and	CCONJ
ejpam-6572	210	34	an	an	DET
ejpam-6572	210	35	ideal	ideal	NOUN
ejpam-6572	210	36	i	i	X
ejpam-6572	210	37	=	=	SYM
ejpam-6572	210	38	{	{	PUNCT
ejpam-6572	210	39	∅	∅	NOUN
ejpam-6572	210	40	,	,	PUNCT
ejpam-6572	210	41	{	{	PUNCT
ejpam-6572	210	42	1	1	NUM
ejpam-6572	210	43	}	}	PUNCT
ejpam-6572	210	44	}	}	PUNCT
ejpam-6572	210	45	.	.	PUNCT
ejpam-6572	211	1	let	let	VERB
ejpam-6572	211	2	y	y	PROPN
ejpam-6572	211	3	=	=	PUNCT
ejpam-6572	211	4	{	{	PUNCT
ejpam-6572	211	5	a	a	PRON
ejpam-6572	211	6	,	,	PUNCT
ejpam-6572	211	7	b	b	NOUN
ejpam-6572	211	8	,	,	PUNCT
ejpam-6572	211	9	c	c	NOUN
ejpam-6572	211	10	}	}	PUNCT
ejpam-6572	211	11	with	with	ADP
ejpam-6572	211	12	topologies	topology	NOUN
ejpam-6572	211	13	σ1	σ1	NOUN
ejpam-6572	211	14	=	=	SYM
ejpam-6572	211	15	{	{	PUNCT
ejpam-6572	211	16	∅	∅	NOUN
ejpam-6572	211	17	,	,	PUNCT
ejpam-6572	211	18	{	{	PUNCT
ejpam-6572	211	19	a	a	X
ejpam-6572	211	20	}	}	PUNCT
ejpam-6572	211	21	,	,	PUNCT
ejpam-6572	211	22	{	{	PUNCT
ejpam-6572	211	23	b	b	NOUN
ejpam-6572	211	24	}	}	PUNCT
ejpam-6572	211	25	,	,	PUNCT
ejpam-6572	211	26	{	{	PUNCT
ejpam-6572	211	27	a	a	DET
ejpam-6572	211	28	,	,	PUNCT
ejpam-6572	211	29	b	b	NOUN
ejpam-6572	211	30	}	}	PUNCT
ejpam-6572	211	31	,	,	PUNCT
ejpam-6572	211	32	{	{	PUNCT
ejpam-6572	211	33	a	a	X
ejpam-6572	211	34	,	,	PUNCT
ejpam-6572	211	35	c	c	NOUN
ejpam-6572	211	36	}	}	PUNCT
ejpam-6572	211	37	,	,	PUNCT
ejpam-6572	211	38	y	y	PROPN
ejpam-6572	211	39	}	}	PUNCT
ejpam-6572	211	40	and	and	CCONJ
ejpam-6572	211	41	σ2	σ2	PROPN
ejpam-6572	211	42	=	=	SYM
ejpam-6572	211	43	{	{	PUNCT
ejpam-6572	211	44	∅	∅	NOUN
ejpam-6572	211	45	,	,	PUNCT
ejpam-6572	211	46	{	{	PUNCT
ejpam-6572	211	47	a	a	X
ejpam-6572	211	48	}	}	PUNCT
ejpam-6572	211	49	,	,	PUNCT
ejpam-6572	211	50	{	{	PUNCT
ejpam-6572	211	51	b	b	NOUN
ejpam-6572	211	52	}	}	PUNCT
ejpam-6572	211	53	,	,	PUNCT
ejpam-6572	211	54	{	{	PUNCT
ejpam-6572	211	55	a	a	DET
ejpam-6572	211	56	,	,	PUNCT
ejpam-6572	211	57	b	b	NOUN
ejpam-6572	211	58	}	}	PUNCT
ejpam-6572	211	59	,	,	PUNCT
ejpam-6572	211	60	y	y	PROPN
ejpam-6572	211	61	}	}	PUNCT
ejpam-6572	211	62	.	.	PUNCT
ejpam-6572	212	1	a	a	DET
ejpam-6572	212	2	function	function	NOUN
ejpam-6572	212	3	f	f	NOUN
ejpam-6572	212	4	:	:	PUNCT
ejpam-6572	212	5	(	(	PUNCT
ejpam-6572	212	6	x	x	X
ejpam-6572	212	7	,	,	PUNCT
ejpam-6572	212	8	τ	τ	PROPN
ejpam-6572	212	9	,	,	PUNCT
ejpam-6572	212	10	i	i	NOUN
ejpam-6572	212	11	)	)	PUNCT
ejpam-6572	212	12	→	→	PUNCT
ejpam-6572	212	13	(	(	PUNCT
ejpam-6572	212	14	y	y	PROPN
ejpam-6572	212	15	,	,	PUNCT
ejpam-6572	212	16	σ1	σ1	PROPN
ejpam-6572	212	17	,	,	PUNCT
ejpam-6572	212	18	σ2	σ2	PROPN
ejpam-6572	212	19	)	)	PUNCT
ejpam-6572	212	20	is	be	AUX
ejpam-6572	212	21	defined	define	VERB
ejpam-6572	212	22	as	as	SCONJ
ejpam-6572	212	23	follows	follow	VERB
ejpam-6572	212	24	:	:	PUNCT
ejpam-6572	212	25	f(1	f(1	X
ejpam-6572	212	26	)	)	PUNCT
ejpam-6572	212	27	=	=	SYM
ejpam-6572	213	1	a	a	PRON
ejpam-6572	213	2	,	,	PUNCT
ejpam-6572	213	3	f(2	f(2	PROPN
ejpam-6572	213	4	)	)	PUNCT
ejpam-6572	213	5	=	=	SYM
ejpam-6572	213	6	b	b	PROPN
ejpam-6572	213	7	and	and	CCONJ
ejpam-6572	213	8	f(3	f(3	PROPN
ejpam-6572	213	9	)	)	PUNCT
ejpam-6572	214	1	=	=	SYM
ejpam-6572	214	2	c.	c.	NOUN
ejpam-6572	214	3	then	then	ADV
ejpam-6572	214	4	,	,	PUNCT
ejpam-6572	214	5	f	f	PROPN
ejpam-6572	214	6	is	be	AUX
ejpam-6572	214	7	weakly	weakly	ADJ
ejpam-6572	214	8	quasi	quasi	ADJ
ejpam-6572	214	9	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	214	10	,	,	PUNCT
ejpam-6572	214	11	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	214	12	but	but	CCONJ
ejpam-6572	214	13	f	f	PROPN
ejpam-6572	214	14	is	be	AUX
ejpam-6572	214	15	not	not	PART
ejpam-6572	214	16	almost	almost	ADV
ejpam-6572	214	17	quasi	quasi	ADJ
ejpam-6572	214	18	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	214	19	,	,	PUNCT
ejpam-6572	214	20	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6572	214	21	.	.	X
ejpam-6572	215	1	theorem	theorem	NOUN
ejpam-6572	215	2	4	4	NUM
ejpam-6572	215	3	.	.	X
ejpam-6572	215	4	for	for	ADP
ejpam-6572	215	5	a	a	DET
ejpam-6572	215	6	function	function	NOUN
ejpam-6572	215	7	f	f	NOUN
ejpam-6572	215	8	:	:	PUNCT
ejpam-6572	215	9	(	(	PUNCT
ejpam-6572	215	10	x	x	X
ejpam-6572	215	11	,	,	PUNCT
ejpam-6572	215	12	τ	τ	PROPN
ejpam-6572	215	13	,	,	PUNCT
ejpam-6572	215	14	i	i	NOUN
ejpam-6572	215	15	)	)	PUNCT
ejpam-6572	215	16	→	→	PUNCT
ejpam-6572	215	17	(	(	PUNCT
ejpam-6572	215	18	y	y	PROPN
ejpam-6572	215	19	,	,	PUNCT
ejpam-6572	215	20	σ1	σ1	PROPN
ejpam-6572	215	21	,	,	PUNCT
ejpam-6572	215	22	σ2	σ2	NOUN
ejpam-6572	215	23	)	)	PUNCT
ejpam-6572	215	24	,	,	PUNCT
ejpam-6572	215	25	the	the	DET
ejpam-6572	215	26	following	follow	VERB
ejpam-6572	215	27	properties	property	NOUN
ejpam-6572	215	28	are	be	AUX
ejpam-6572	215	29	equivalent	equivalent	ADJ
ejpam-6572	215	30	:	:	PUNCT
ejpam-6572	215	31	(	(	PUNCT
ejpam-6572	215	32	1	1	X
ejpam-6572	215	33	)	)	PUNCT
ejpam-6572	215	34	f	f	PROPN
ejpam-6572	215	35	is	be	AUX
ejpam-6572	215	36	weakly	weakly	ADJ
ejpam-6572	215	37	quasi	quasi	ADJ
ejpam-6572	215	38	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	215	39	,	,	PUNCT
ejpam-6572	215	40	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	215	41	;	;	PUNCT
ejpam-6572	215	42	(	(	PUNCT
ejpam-6572	215	43	2	2	X
ejpam-6572	215	44	)	)	PUNCT
ejpam-6572	215	45	for	for	ADP
ejpam-6572	215	46	each	each	DET
ejpam-6572	215	47	x	x	SYM
ejpam-6572	215	48	∈	∈	PROPN
ejpam-6572	215	49	x	x	X
ejpam-6572	215	50	and	and	CCONJ
ejpam-6572	215	51	every	every	DET
ejpam-6572	215	52	σ1σ2	σ1σ2	NUM
ejpam-6572	215	53	-	-	ADJ
ejpam-6572	215	54	open	open	ADJ
ejpam-6572	215	55	set	set	NOUN
ejpam-6572	215	56	v	v	NOUN
ejpam-6572	215	57	of	of	ADP
ejpam-6572	215	58	y	y	NOUN
ejpam-6572	215	59	containing	contain	VERB
ejpam-6572	215	60	f(x	f(x	PROPN
ejpam-6572	215	61	)	)	PUNCT
ejpam-6572	215	62	,	,	PUNCT
ejpam-6572	215	63	there	there	PRON
ejpam-6572	215	64	exists	exist	VERB
ejpam-6572	215	65	a	a	DET
ejpam-6572	215	66	semi	semi	NOUN
ejpam-6572	215	67	-	-	ADJ
ejpam-6572	215	68	i	i	PRON
ejpam-6572	215	69	⋆-open	⋆-open	VERB
ejpam-6572	215	70	set	set	VERB
ejpam-6572	215	71	of	of	ADP
ejpam-6572	215	72	x	x	PUNCT
ejpam-6572	215	73	containing	contain	VERB
ejpam-6572	215	74	x	x	PUNCT
ejpam-6572	215	75	such	such	ADJ
ejpam-6572	215	76	that	that	DET
ejpam-6572	215	77	f(u	f(u	PROPN
ejpam-6572	215	78	)	)	PUNCT
ejpam-6572	216	1	⊆	⊆	NUM
ejpam-6572	216	2	σ1σ2	σ1σ2	NOUN
ejpam-6572	216	3	-	-	NUM
ejpam-6572	216	4	cl(v	cl(v	NOUN
ejpam-6572	216	5	)	)	PUNCT
ejpam-6572	216	6	;	;	PUNCT
ejpam-6572	216	7	(	(	PUNCT
ejpam-6572	216	8	3	3	X
ejpam-6572	216	9	)	)	PUNCT
ejpam-6572	216	10	int⋆(cl⋆(f−1(σ1σ2	int⋆(cl⋆(f−1(σ1σ2	NOUN
ejpam-6572	216	11	-	-	PUNCT
ejpam-6572	216	12	int(k	int(k	NOUN
ejpam-6572	216	13	)	)	PUNCT
ejpam-6572	216	14	)	)	PUNCT
ejpam-6572	216	15	)	)	PUNCT
ejpam-6572	216	16	)	)	PUNCT
ejpam-6572	217	1	⊆	⊆	NUM
ejpam-6572	217	2	f−1(k	f−1(k	PROPN
ejpam-6572	217	3	)	)	PUNCT
ejpam-6572	217	4	for	for	ADP
ejpam-6572	217	5	every	every	DET
ejpam-6572	217	6	σ1σ2	σ1σ2	NUM
ejpam-6572	217	7	-	-	PUNCT
ejpam-6572	217	8	closed	closed	ADJ
ejpam-6572	217	9	set	set	NOUN
ejpam-6572	217	10	k	k	PROPN
ejpam-6572	217	11	of	of	ADP
ejpam-6572	217	12	y	y	PROPN
ejpam-6572	217	13	;	;	PUNCT
ejpam-6572	217	14	(	(	PUNCT
ejpam-6572	217	15	4	4	X
ejpam-6572	217	16	)	)	PUNCT
ejpam-6572	217	17	f−1(v	f−1(v	NOUN
ejpam-6572	217	18	)	)	PUNCT
ejpam-6572	217	19	⊆	⊆	NUM
ejpam-6572	217	20	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	217	21	-	-	PUNCT
ejpam-6572	217	22	cl(v	cl(v	NOUN
ejpam-6572	217	23	)	)	PUNCT
ejpam-6572	217	24	)	)	PUNCT
ejpam-6572	217	25	)	)	PUNCT
ejpam-6572	218	1	for	for	ADP
ejpam-6572	218	2	every	every	DET
ejpam-6572	218	3	σ1σ2	σ1σ2	NOUN
ejpam-6572	218	4	-	-	ADJ
ejpam-6572	218	5	open	open	ADJ
ejpam-6572	218	6	set	set	NOUN
ejpam-6572	218	7	v	v	NOUN
ejpam-6572	218	8	of	of	ADP
ejpam-6572	218	9	y	y	PROPN
ejpam-6572	218	10	;	;	PUNCT
ejpam-6572	218	11	(	(	PUNCT
ejpam-6572	218	12	5	5	X
ejpam-6572	218	13	)	)	PUNCT
ejpam-6572	218	14	scl⋆(f−1(v	scl⋆(f−1(v	NOUN
ejpam-6572	218	15	)	)	PUNCT
ejpam-6572	218	16	)	)	PUNCT
ejpam-6572	218	17	⊆	⊆	NUM
ejpam-6572	218	18	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	218	19	-	-	PUNCT
ejpam-6572	218	20	cl(v	cl(v	NOUN
ejpam-6572	218	21	)	)	PUNCT
ejpam-6572	218	22	)	)	PUNCT
ejpam-6572	218	23	for	for	ADP
ejpam-6572	218	24	every	every	DET
ejpam-6572	218	25	σ1σ2	σ1σ2	NUM
ejpam-6572	218	26	-	-	ADJ
ejpam-6572	218	27	open	open	ADJ
ejpam-6572	218	28	sets	set	NOUN
ejpam-6572	218	29	v	v	ADP
ejpam-6572	218	30	of	of	ADP
ejpam-6572	218	31	y	y	PROPN
ejpam-6572	218	32	.	.	PUNCT
ejpam-6572	219	1	proof	proof	NOUN
ejpam-6572	219	2	.	.	PUNCT
ejpam-6572	220	1	(	(	PUNCT
ejpam-6572	220	2	1	1	X
ejpam-6572	220	3	)	)	PUNCT
ejpam-6572	220	4	⇒	⇒	NOUN
ejpam-6572	220	5	(	(	PUNCT
ejpam-6572	220	6	2	2	NUM
ejpam-6572	220	7	):	):	PUNCT
ejpam-6572	220	8	let	let	VERB
ejpam-6572	220	9	u	u	PRON
ejpam-6572	220	10	(	(	PUNCT
ejpam-6572	220	11	x	x	NOUN
ejpam-6572	220	12	)	)	PUNCT
ejpam-6572	220	13	be	be	VERB
ejpam-6572	220	14	the	the	DET
ejpam-6572	220	15	family	family	NOUN
ejpam-6572	220	16	of	of	ADP
ejpam-6572	220	17	all	all	DET
ejpam-6572	220	18	⋆-open	⋆-open	ADJ
ejpam-6572	220	19	sets	set	NOUN
ejpam-6572	220	20	of	of	ADP
ejpam-6572	220	21	x	x	PUNCT
ejpam-6572	220	22	containing	contain	VERB
ejpam-6572	220	23	x.	x.	NOUN
ejpam-6572	220	24	let	let	VERB
ejpam-6572	220	25	v	v	PART
ejpam-6572	220	26	be	be	AUX
ejpam-6572	220	27	any	any	DET
ejpam-6572	220	28	σ1σ2	σ1σ2	NOUN
ejpam-6572	220	29	-	-	ADJ
ejpam-6572	220	30	open	open	ADJ
ejpam-6572	220	31	set	set	NOUN
ejpam-6572	220	32	of	of	ADP
ejpam-6572	220	33	y	y	PROPN
ejpam-6572	220	34	containing	contain	VERB
ejpam-6572	220	35	f(x	f(x	PROPN
ejpam-6572	220	36	)	)	PUNCT
ejpam-6572	220	37	.	.	PUNCT
ejpam-6572	221	1	for	for	ADP
ejpam-6572	221	2	eachh	eachh	PROPN
ejpam-6572	221	3	∈	∈	PROPN
ejpam-6572	221	4	u	u	PROPN
ejpam-6572	221	5	(	(	PUNCT
ejpam-6572	221	6	x	x	NOUN
ejpam-6572	221	7	)	)	PUNCT
ejpam-6572	221	8	,	,	PUNCT
ejpam-6572	221	9	there	there	PRON
ejpam-6572	221	10	exists	exist	VERB
ejpam-6572	221	11	a	a	DET
ejpam-6572	221	12	nonempty	nonempty	ADJ
ejpam-6572	221	13	⋆open	⋆open	PUNCT
ejpam-6572	221	14	set	set	NOUN
ejpam-6572	221	15	gh	gh	PROPN
ejpam-6572	221	16	such	such	ADJ
ejpam-6572	221	17	that	that	SCONJ
ejpam-6572	221	18	gh	gh	PROPN
ejpam-6572	221	19	⊆	⊆	NUM
ejpam-6572	221	20	h	h	NOUN
ejpam-6572	221	21	and	and	CCONJ
ejpam-6572	221	22	f(gh	f(gh	ADJ
ejpam-6572	221	23	)	)	PUNCT
ejpam-6572	221	24	⊆	⊆	NUM
ejpam-6572	221	25	σ1σ2	σ1σ2	NOUN
ejpam-6572	221	26	-	-	NUM
ejpam-6572	221	27	cl(v	cl(v	NOUN
ejpam-6572	221	28	)	)	PUNCT
ejpam-6572	221	29	.	.	PUNCT
ejpam-6572	222	1	put	put	VERB
ejpam-6572	222	2	w	w	NOUN
ejpam-6572	222	3	=	=	PUNCT
ejpam-6572	222	4	∪{gh	∪{gh	ADP
ejpam-6572	223	1	|	|	ADV
ejpam-6572	223	2	h	h	NOUN
ejpam-6572	223	3	∈	∈	PROPN
ejpam-6572	223	4	u	u	NOUN
ejpam-6572	223	5	(	(	PUNCT
ejpam-6572	223	6	x	x	NOUN
ejpam-6572	223	7	)	)	PUNCT
ejpam-6572	223	8	}	}	PUNCT
ejpam-6572	223	9	.	.	PUNCT
ejpam-6572	224	1	then	then	ADV
ejpam-6572	224	2	,	,	PUNCT
ejpam-6572	224	3	w	w	PROPN
ejpam-6572	224	4	is	be	AUX
ejpam-6572	224	5	⋆-open	⋆-open	ADJ
ejpam-6572	224	6	in	in	ADP
ejpam-6572	224	7	x	x	SYM
ejpam-6572	224	8	and	and	CCONJ
ejpam-6572	224	9	x	x	SYM
ejpam-6572	224	10	∈	∈	PROPN
ejpam-6572	224	11	cl⋆(w	cl⋆(w	PROPN
ejpam-6572	224	12	)	)	PUNCT
ejpam-6572	224	13	.	.	PUNCT
ejpam-6572	225	1	let	let	VERB
ejpam-6572	225	2	u	u	NOUN
ejpam-6572	226	1	=	=	NOUN
ejpam-6572	226	2	w	w	NOUN
ejpam-6572	226	3	∪	∪	X
ejpam-6572	226	4	{	{	PUNCT
ejpam-6572	226	5	x	x	NOUN
ejpam-6572	226	6	}	}	PUNCT
ejpam-6572	226	7	.	.	PUNCT
ejpam-6572	227	1	then	then	ADV
ejpam-6572	227	2	,	,	PUNCT
ejpam-6572	227	3	u	u	NOUN
ejpam-6572	227	4	is	be	AUX
ejpam-6572	227	5	a	a	DET
ejpam-6572	227	6	semi	semi	ADJ
ejpam-6572	227	7	-	-	ADJ
ejpam-6572	227	8	i	i	PRON
ejpam-6572	227	9	⋆-open	⋆-open	VERB
ejpam-6572	227	10	set	set	VERB
ejpam-6572	227	11	of	of	ADP
ejpam-6572	227	12	x	x	PUNCT
ejpam-6572	227	13	containing	contain	VERB
ejpam-6572	227	14	x	x	PROPN
ejpam-6572	227	15	and	and	CCONJ
ejpam-6572	227	16	f(u	f(u	PROPN
ejpam-6572	227	17	)	)	PUNCT
ejpam-6572	227	18	⊆	⊆	NUM
ejpam-6572	227	19	σ1σ2	σ1σ2	NOUN
ejpam-6572	227	20	-	-	NUM
ejpam-6572	227	21	cl(v	cl(v	NOUN
ejpam-6572	227	22	)	)	PUNCT
ejpam-6572	227	23	.	.	PUNCT
ejpam-6572	228	1	(	(	PUNCT
ejpam-6572	228	2	2	2	X
ejpam-6572	228	3	)	)	PUNCT
ejpam-6572	228	4	⇒	⇒	NOUN
ejpam-6572	228	5	(	(	PUNCT
ejpam-6572	228	6	4	4	NUM
ejpam-6572	228	7	):	):	PUNCT
ejpam-6572	228	8	let	let	VERB
ejpam-6572	228	9	v	v	PART
ejpam-6572	228	10	be	be	AUX
ejpam-6572	228	11	any	any	DET
ejpam-6572	228	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	228	13	-	-	ADJ
ejpam-6572	228	14	open	open	ADJ
ejpam-6572	228	15	set	set	NOUN
ejpam-6572	228	16	of	of	ADP
ejpam-6572	228	17	y	y	PROPN
ejpam-6572	228	18	and	and	CCONJ
ejpam-6572	228	19	x	x	PROPN
ejpam-6572	228	20	∈	∈	PROPN
ejpam-6572	228	21	f−1(v	f−1(v	NOUN
ejpam-6572	228	22	)	)	PUNCT
ejpam-6572	228	23	.	.	PUNCT
ejpam-6572	229	1	then	then	ADV
ejpam-6572	229	2	,	,	PUNCT
ejpam-6572	229	3	f(x	f(x	PROPN
ejpam-6572	229	4	)	)	PUNCT
ejpam-6572	229	5	∈	∈	PROPN
ejpam-6572	229	6	v	v	NOUN
ejpam-6572	229	7	and	and	CCONJ
ejpam-6572	229	8	there	there	PRON
ejpam-6572	229	9	exists	exist	VERB
ejpam-6572	229	10	a	a	DET
ejpam-6572	229	11	semi	semi	NOUN
ejpam-6572	229	12	-	-	ADJ
ejpam-6572	229	13	i	i	PRON
ejpam-6572	229	14	⋆-open	⋆-open	VERB
ejpam-6572	229	15	set	set	VERB
ejpam-6572	229	16	u	u	NOUN
ejpam-6572	229	17	of	of	ADP
ejpam-6572	229	18	x	x	PUNCT
ejpam-6572	229	19	containing	contain	VERB
ejpam-6572	229	20	x	x	PUNCT
ejpam-6572	229	21	such	such	ADJ
ejpam-6572	229	22	that	that	DET
ejpam-6572	229	23	f(u	f(u	PROPN
ejpam-6572	229	24	)	)	PUNCT
ejpam-6572	230	1	⊆	⊆	NUM
ejpam-6572	230	2	σ1σ2	σ1σ2	NOUN
ejpam-6572	230	3	-	-	NUM
ejpam-6572	230	4	cl(v	cl(v	NOUN
ejpam-6572	230	5	)	)	PUNCT
ejpam-6572	230	6	.	.	PUNCT
ejpam-6572	231	1	thus	thus	ADV
ejpam-6572	231	2	,	,	PUNCT
ejpam-6572	231	3	x	x	PUNCT
ejpam-6572	231	4	∈	∈	PROPN
ejpam-6572	231	5	u	u	NOUN
ejpam-6572	231	6	⊆	⊆	NUM
ejpam-6572	231	7	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	231	8	-	-	PUNCT
ejpam-6572	231	9	cl(v	cl(v	NOUN
ejpam-6572	231	10	)	)	PUNCT
ejpam-6572	231	11	)	)	PUNCT
ejpam-6572	231	12	)	)	PUNCT
ejpam-6572	231	13	b.	b.	PROPN
ejpam-6572	231	14	kong	kong	PROPN
ejpam-6572	231	15	-	-	PUNCT
ejpam-6572	231	16	ied	ied	PROPN
ejpam-6572	231	17	,	,	PUNCT
ejpam-6572	231	18	a.	a.	PROPN
ejpam-6572	231	19	sama	sama	PROPN
ejpam-6572	231	20	-	-	PUNCT
ejpam-6572	231	21	ae	ae	PROPN
ejpam-6572	231	22	,	,	PUNCT
ejpam-6572	231	23	c.	c.	PROPN
ejpam-6572	231	24	boonpok	boonpok	PROPN
ejpam-6572	231	25	/	/	SYM
ejpam-6572	231	26	eur	eur	PROPN
ejpam-6572	231	27	.	.	PUNCT
ejpam-6572	232	1	j.	j.	PROPN
ejpam-6572	232	2	pure	pure	PROPN
ejpam-6572	232	3	appl	appl	PROPN
ejpam-6572	232	4	.	.	PROPN
ejpam-6572	232	5	math	math	PROPN
ejpam-6572	232	6	,	,	PUNCT
ejpam-6572	232	7	18	18	NUM
ejpam-6572	232	8	(	(	PUNCT
ejpam-6572	232	9	3	3	NUM
ejpam-6572	232	10	)	)	PUNCT
ejpam-6572	232	11	(	(	PUNCT
ejpam-6572	232	12	2025	2025	NUM
ejpam-6572	232	13	)	)	PUNCT
ejpam-6572	232	14	,	,	PUNCT
ejpam-6572	232	15	6572	6572	NUM
ejpam-6572	232	16	8	8	NUM
ejpam-6572	232	17	of	of	ADP
ejpam-6572	232	18	12	12	NUM
ejpam-6572	232	19	and	and	CCONJ
ejpam-6572	232	20	hence	hence	ADV
ejpam-6572	232	21	f−1(v	f−1(v	NOUN
ejpam-6572	232	22	)	)	PUNCT
ejpam-6572	232	23	⊆	⊆	NUM
ejpam-6572	232	24	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	232	25	-	-	PUNCT
ejpam-6572	232	26	cl(v	cl(v	NOUN
ejpam-6572	232	27	)	)	PUNCT
ejpam-6572	232	28	)	)	PUNCT
ejpam-6572	232	29	)	)	PUNCT
ejpam-6572	232	30	.	.	PUNCT
ejpam-6572	233	1	(	(	PUNCT
ejpam-6572	233	2	4	4	X
ejpam-6572	233	3	)	)	PUNCT
ejpam-6572	233	4	⇒	⇒	NOUN
ejpam-6572	233	5	(	(	PUNCT
ejpam-6572	233	6	5	5	NUM
ejpam-6572	233	7	):	):	PUNCT
ejpam-6572	233	8	let	let	VERB
ejpam-6572	233	9	v	v	PART
ejpam-6572	233	10	be	be	AUX
ejpam-6572	233	11	any	any	DET
ejpam-6572	233	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	233	13	-	-	ADJ
ejpam-6572	233	14	open	open	ADJ
ejpam-6572	233	15	set	set	NOUN
ejpam-6572	233	16	of	of	ADP
ejpam-6572	233	17	y	y	PROPN
ejpam-6572	233	18	.	.	PUNCT
ejpam-6572	234	1	then	then	ADV
ejpam-6572	234	2	by	by	ADP
ejpam-6572	234	3	(	(	PUNCT
ejpam-6572	234	4	4	4	NUM
ejpam-6572	234	5	)	)	PUNCT
ejpam-6572	234	6	,	,	PUNCT
ejpam-6572	234	7	we	we	PRON
ejpam-6572	234	8	have	have	VERB
ejpam-6572	234	9	x	x	PART
ejpam-6572	234	10	−	−	PRON
ejpam-6572	234	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	234	12	-	-	PUNCT
ejpam-6572	234	13	cl(v	cl(v	NOUN
ejpam-6572	234	14	)	)	PUNCT
ejpam-6572	234	15	)	)	PUNCT
ejpam-6572	235	1	=	=	PUNCT
ejpam-6572	235	2	f−1(y	f−1(y	PROPN
ejpam-6572	235	3	−	−	NOUN
ejpam-6572	235	4	σ1σ2	σ1σ2	NOUN
ejpam-6572	235	5	-	-	NUM
ejpam-6572	235	6	cl(v	cl(v	NOUN
ejpam-6572	235	7	)	)	PUNCT
ejpam-6572	235	8	)	)	PUNCT
ejpam-6572	236	1	⊆	⊆	X
ejpam-6572	236	2	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	236	3	-	-	PUNCT
ejpam-6572	236	4	cl(y	cl(y	NOUN
ejpam-6572	236	5	−	−	NOUN
ejpam-6572	236	6	σ1σ2	σ1σ2	NOUN
ejpam-6572	236	7	-	-	NUM
ejpam-6572	236	8	cl(v	cl(v	NOUN
ejpam-6572	236	9	)	)	PUNCT
ejpam-6572	236	10	)	)	PUNCT
ejpam-6572	236	11	)	)	PUNCT
ejpam-6572	236	12	)	)	PUNCT
ejpam-6572	237	1	=	=	PUNCT
ejpam-6572	238	1	sint⋆(f−1(y	sint⋆(f−1(y	NUM
ejpam-6572	239	1	−	−	NOUN
ejpam-6572	239	2	σ1σ2	σ1σ2	NUM
ejpam-6572	239	3	-	-	PUNCT
ejpam-6572	239	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	239	5	-	-	PUNCT
ejpam-6572	239	6	cl(v	cl(v	NOUN
ejpam-6572	239	7	)	)	PUNCT
ejpam-6572	239	8	)	)	PUNCT
ejpam-6572	239	9	)	)	PUNCT
ejpam-6572	239	10	)	)	PUNCT
ejpam-6572	240	1	⊆	⊆	NUM
ejpam-6572	240	2	sint⋆(f−1(y	sint⋆(f−1(y	NUM
ejpam-6572	240	3	−	−	PROPN
ejpam-6572	240	4	v	v	NOUN
ejpam-6572	240	5	)	)	PUNCT
ejpam-6572	240	6	)	)	PUNCT
ejpam-6572	241	1	=	=	PUNCT
ejpam-6572	241	2	sint⋆(x	sint⋆(x	PROPN
ejpam-6572	241	3	−	−	PROPN
ejpam-6572	241	4	f−1(v	f−1(v	NOUN
ejpam-6572	241	5	)	)	PUNCT
ejpam-6572	241	6	)	)	PUNCT
ejpam-6572	242	1	=	=	PUNCT
ejpam-6572	242	2	x	x	PUNCT
ejpam-6572	243	1	−	−	PROPN
ejpam-6572	243	2	scl⋆(f−1(v	scl⋆(f−1(v	PROPN
ejpam-6572	243	3	)	)	PUNCT
ejpam-6572	243	4	)	)	PUNCT
ejpam-6572	243	5	and	and	CCONJ
ejpam-6572	243	6	so	so	ADV
ejpam-6572	243	7	scl⋆(f−1(v	scl⋆(f−1(v	PROPN
ejpam-6572	243	8	)	)	PUNCT
ejpam-6572	243	9	)	)	PUNCT
ejpam-6572	244	1	⊆	⊆	NUM
ejpam-6572	244	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	244	3	-	-	PUNCT
ejpam-6572	244	4	cl(v	cl(v	NOUN
ejpam-6572	244	5	)	)	PUNCT
ejpam-6572	244	6	)	)	PUNCT
ejpam-6572	244	7	.	.	PUNCT
ejpam-6572	245	1	(	(	PUNCT
ejpam-6572	245	2	5	5	X
ejpam-6572	245	3	)	)	PUNCT
ejpam-6572	245	4	⇒	⇒	NOUN
ejpam-6572	245	5	(	(	PUNCT
ejpam-6572	245	6	3	3	NUM
ejpam-6572	245	7	):	):	PUNCT
ejpam-6572	245	8	let	let	VERB
ejpam-6572	245	9	k	k	PRON
ejpam-6572	245	10	be	be	AUX
ejpam-6572	245	11	any	any	DET
ejpam-6572	245	12	σ1σ2	σ1σ2	NUM
ejpam-6572	245	13	-	-	PUNCT
ejpam-6572	245	14	closed	closed	ADJ
ejpam-6572	245	15	set	set	NOUN
ejpam-6572	245	16	of	of	ADP
ejpam-6572	245	17	y	y	PROPN
ejpam-6572	245	18	.	.	PUNCT
ejpam-6572	246	1	thus	thus	ADV
ejpam-6572	246	2	by	by	ADP
ejpam-6572	246	3	(	(	PUNCT
ejpam-6572	246	4	5	5	NUM
ejpam-6572	246	5	)	)	PUNCT
ejpam-6572	246	6	and	and	CCONJ
ejpam-6572	246	7	lemma	lemma	PROPN
ejpam-6572	246	8	4	4	NUM
ejpam-6572	246	9	,	,	PUNCT
ejpam-6572	246	10	int⋆(cl⋆(f−1(σ1σ2	int⋆(cl⋆(f−1(σ1σ2	NOUN
ejpam-6572	246	11	-	-	SYM
ejpam-6572	246	12	int(k	int(k	NUM
ejpam-6572	246	13	)	)	PUNCT
ejpam-6572	246	14	)	)	PUNCT
ejpam-6572	246	15	)	)	PUNCT
ejpam-6572	246	16	)	)	PUNCT
ejpam-6572	247	1	⊆	⊆	X
ejpam-6572	247	2	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NUM
ejpam-6572	247	3	-	-	SYM
ejpam-6572	247	4	int(k	int(k	NOUN
ejpam-6572	247	5	)	)	PUNCT
ejpam-6572	247	6	)	)	PUNCT
ejpam-6572	247	7	)	)	PUNCT
ejpam-6572	248	1	⊆	⊆	NUM
ejpam-6572	248	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	248	3	-	-	PUNCT
ejpam-6572	248	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6572	248	5	-	-	PUNCT
ejpam-6572	248	6	int(k	int(k	NOUN
ejpam-6572	248	7	)	)	PUNCT
ejpam-6572	248	8	)	)	PUNCT
ejpam-6572	248	9	)	)	PUNCT
ejpam-6572	249	1	⊆	⊆	NUM
ejpam-6572	249	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	249	3	-	-	PUNCT
ejpam-6572	249	4	cl(k	cl(k	NOUN
ejpam-6572	249	5	)	)	PUNCT
ejpam-6572	249	6	)	)	PUNCT
ejpam-6572	250	1	=	=	PUNCT
ejpam-6572	250	2	f−1(k	f−1(k	PROPN
ejpam-6572	250	3	)	)	PUNCT
ejpam-6572	250	4	.	.	PUNCT
ejpam-6572	251	1	(	(	PUNCT
ejpam-6572	251	2	3	3	X
ejpam-6572	251	3	)	)	PUNCT
ejpam-6572	251	4	⇒	⇒	NOUN
ejpam-6572	251	5	(	(	PUNCT
ejpam-6572	251	6	4	4	NUM
ejpam-6572	251	7	):	):	PUNCT
ejpam-6572	251	8	let	let	VERB
ejpam-6572	251	9	v	v	PART
ejpam-6572	251	10	be	be	AUX
ejpam-6572	251	11	any	any	DET
ejpam-6572	251	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	251	13	-	-	ADJ
ejpam-6572	251	14	open	open	ADJ
ejpam-6572	251	15	set	set	NOUN
ejpam-6572	251	16	of	of	ADP
ejpam-6572	251	17	y	y	PROPN
ejpam-6572	251	18	.	.	PUNCT
ejpam-6572	252	1	by	by	ADP
ejpam-6572	252	2	(	(	PUNCT
ejpam-6572	252	3	3	3	X
ejpam-6572	252	4	)	)	PUNCT
ejpam-6572	252	5	and	and	CCONJ
ejpam-6572	252	6	lemma	lemma	PROPN
ejpam-6572	252	7	4	4	NUM
ejpam-6572	252	8	,	,	PUNCT
ejpam-6572	252	9	x	x	PUNCT
ejpam-6572	252	10	−	−	NOUN
ejpam-6572	252	11	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	252	12	-	-	PUNCT
ejpam-6572	252	13	cl(v	cl(v	NOUN
ejpam-6572	252	14	)	)	PUNCT
ejpam-6572	252	15	)	)	PUNCT
ejpam-6572	252	16	)	)	PUNCT
ejpam-6572	253	1	=	=	SYM
ejpam-6572	253	2	scl⋆(f−1(y	scl⋆(f−1(y	PROPN
ejpam-6572	253	3	−	−	NUM
ejpam-6572	253	4	σ1σ2	σ1σ2	NOUN
ejpam-6572	253	5	-	-	NUM
ejpam-6572	253	6	cl(v	cl(v	NOUN
ejpam-6572	253	7	)	)	PUNCT
ejpam-6572	253	8	)	)	PUNCT
ejpam-6572	253	9	)	)	PUNCT
ejpam-6572	254	1	⊆	⊆	NUM
ejpam-6572	254	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	254	3	-	-	PUNCT
ejpam-6572	254	4	cl(y	cl(y	NOUN
ejpam-6572	254	5	−	−	NOUN
ejpam-6572	254	6	σ1σ2	σ1σ2	NOUN
ejpam-6572	254	7	-	-	NUM
ejpam-6572	254	8	cl(v	cl(v	NOUN
ejpam-6572	254	9	)	)	PUNCT
ejpam-6572	254	10	)	)	PUNCT
ejpam-6572	254	11	)	)	PUNCT
ejpam-6572	255	1	=	=	PUNCT
ejpam-6572	255	2	f−1(y	f−1(y	PROPN
ejpam-6572	255	3	−	−	NOUN
ejpam-6572	255	4	σ1σ2	σ1σ2	X
ejpam-6572	255	5	-	-	PUNCT
ejpam-6572	255	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	255	7	-	-	PUNCT
ejpam-6572	255	8	cl(v	cl(v	NOUN
ejpam-6572	255	9	)	)	PUNCT
ejpam-6572	255	10	)	)	PUNCT
ejpam-6572	255	11	)	)	PUNCT
ejpam-6572	256	1	⊆	⊆	X
ejpam-6572	256	2	f−1(y	f−1(y	NOUN
ejpam-6572	256	3	−	−	PROPN
ejpam-6572	256	4	v	v	NOUN
ejpam-6572	256	5	)	)	PUNCT
ejpam-6572	256	6	=	=	PUNCT
ejpam-6572	256	7	x	x	PUNCT
ejpam-6572	256	8	−	−	PROPN
ejpam-6572	256	9	f−1(v	f−1(v	PROPN
ejpam-6572	256	10	)	)	PUNCT
ejpam-6572	256	11	and	and	CCONJ
ejpam-6572	256	12	hence	hence	ADV
ejpam-6572	256	13	f−1(v	f−1(v	NOUN
ejpam-6572	256	14	)	)	PUNCT
ejpam-6572	256	15	⊆	⊆	NUM
ejpam-6572	256	16	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	256	17	-	-	PUNCT
ejpam-6572	256	18	cl(v	cl(v	NOUN
ejpam-6572	256	19	)	)	PUNCT
ejpam-6572	256	20	)	)	PUNCT
ejpam-6572	256	21	)	)	PUNCT
ejpam-6572	256	22	.	.	PUNCT
ejpam-6572	257	1	(	(	PUNCT
ejpam-6572	257	2	4	4	X
ejpam-6572	257	3	)	)	PUNCT
ejpam-6572	257	4	⇒	⇒	NOUN
ejpam-6572	257	5	(	(	PUNCT
ejpam-6572	257	6	1	1	NUM
ejpam-6572	257	7	):	):	PUNCT
ejpam-6572	257	8	let	let	VERB
ejpam-6572	257	9	x	x	PUNCT
ejpam-6572	257	10	∈	∈	PROPN
ejpam-6572	257	11	x	x	X
ejpam-6572	257	12	and	and	CCONJ
ejpam-6572	257	13	v	v	X
ejpam-6572	257	14	be	be	AUX
ejpam-6572	257	15	any	any	DET
ejpam-6572	257	16	σ1σ2	σ1σ2	NOUN
ejpam-6572	257	17	-	-	ADJ
ejpam-6572	257	18	open	open	ADJ
ejpam-6572	257	19	set	set	NOUN
ejpam-6572	257	20	of	of	ADP
ejpam-6572	257	21	y	y	PROPN
ejpam-6572	257	22	containing	contain	VERB
ejpam-6572	257	23	f(x	f(x	PROPN
ejpam-6572	257	24	)	)	PUNCT
ejpam-6572	257	25	.	.	PUNCT
ejpam-6572	258	1	by	by	ADP
ejpam-6572	258	2	(	(	PUNCT
ejpam-6572	258	3	4	4	NUM
ejpam-6572	258	4	)	)	PUNCT
ejpam-6572	258	5	,	,	PUNCT
ejpam-6572	258	6	we	we	PRON
ejpam-6572	258	7	have	have	VERB
ejpam-6572	258	8	f−1(v	f−1(v	NOUN
ejpam-6572	258	9	)	)	PUNCT
ejpam-6572	259	1	⊆	⊆	NUM
ejpam-6572	259	2	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	259	3	-	-	PUNCT
ejpam-6572	259	4	cl(v	cl(v	NOUN
ejpam-6572	259	5	)	)	PUNCT
ejpam-6572	259	6	)	)	PUNCT
ejpam-6572	259	7	)	)	PUNCT
ejpam-6572	259	8	.	.	PUNCT
ejpam-6572	260	1	put	put	VERB
ejpam-6572	260	2	u	u	NOUN
ejpam-6572	260	3	=	=	NOUN
ejpam-6572	260	4	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	260	5	-	-	NOUN
ejpam-6572	260	6	cl(v	cl(v	NOUN
ejpam-6572	260	7	)	)	PUNCT
ejpam-6572	260	8	)	)	PUNCT
ejpam-6572	260	9	)	)	PUNCT
ejpam-6572	260	10	.	.	PUNCT
ejpam-6572	261	1	then	then	ADV
ejpam-6572	261	2	,	,	PUNCT
ejpam-6572	261	3	u	u	NOUN
ejpam-6572	261	4	is	be	AUX
ejpam-6572	261	5	a	a	DET
ejpam-6572	261	6	semi	semi	ADJ
ejpam-6572	261	7	-	-	ADJ
ejpam-6572	261	8	i	i	PRON
ejpam-6572	261	9	⋆-open	⋆-open	VERB
ejpam-6572	261	10	set	set	VERB
ejpam-6572	261	11	of	of	ADP
ejpam-6572	261	12	x	x	PUNCT
ejpam-6572	261	13	containing	contain	VERB
ejpam-6572	261	14	x	x	PUNCT
ejpam-6572	261	15	such	such	ADJ
ejpam-6572	261	16	that	that	DET
ejpam-6572	261	17	f(u	f(u	PROPN
ejpam-6572	261	18	)	)	PUNCT
ejpam-6572	262	1	⊆	⊆	NUM
ejpam-6572	262	2	σ1σ2	σ1σ2	NOUN
ejpam-6572	262	3	-	-	NUM
ejpam-6572	262	4	cl(v	cl(v	NOUN
ejpam-6572	262	5	)	)	PUNCT
ejpam-6572	262	6	.	.	PUNCT
ejpam-6572	263	1	this	this	PRON
ejpam-6572	263	2	shows	show	VERB
ejpam-6572	263	3	that	that	SCONJ
ejpam-6572	263	4	f	f	PROPN
ejpam-6572	263	5	is	be	AUX
ejpam-6572	263	6	weakly	weakly	ADJ
ejpam-6572	263	7	quasi	quasi	ADJ
ejpam-6572	263	8	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	263	9	,	,	PUNCT
ejpam-6572	263	10	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6572	263	11	.	.	X
ejpam-6572	263	12	theorem	theorem	NOUN
ejpam-6572	263	13	5	5	NUM
ejpam-6572	263	14	.	.	X
ejpam-6572	263	15	for	for	ADP
ejpam-6572	263	16	a	a	DET
ejpam-6572	263	17	function	function	NOUN
ejpam-6572	263	18	f	f	NOUN
ejpam-6572	263	19	:	:	PUNCT
ejpam-6572	263	20	(	(	PUNCT
ejpam-6572	263	21	x	x	X
ejpam-6572	263	22	,	,	PUNCT
ejpam-6572	263	23	τ	τ	PROPN
ejpam-6572	263	24	,	,	PUNCT
ejpam-6572	263	25	i	i	NOUN
ejpam-6572	263	26	)	)	PUNCT
ejpam-6572	263	27	→	→	PUNCT
ejpam-6572	263	28	(	(	PUNCT
ejpam-6572	263	29	y	y	PROPN
ejpam-6572	263	30	,	,	PUNCT
ejpam-6572	263	31	σ1	σ1	PROPN
ejpam-6572	263	32	,	,	PUNCT
ejpam-6572	263	33	σ2	σ2	NOUN
ejpam-6572	263	34	)	)	PUNCT
ejpam-6572	263	35	,	,	PUNCT
ejpam-6572	263	36	the	the	DET
ejpam-6572	263	37	following	follow	VERB
ejpam-6572	263	38	properties	property	NOUN
ejpam-6572	263	39	are	be	AUX
ejpam-6572	263	40	equivalent	equivalent	ADJ
ejpam-6572	263	41	:	:	PUNCT
ejpam-6572	263	42	(	(	PUNCT
ejpam-6572	263	43	1	1	X
ejpam-6572	263	44	)	)	PUNCT
ejpam-6572	263	45	f	f	PROPN
ejpam-6572	263	46	is	be	AUX
ejpam-6572	263	47	weakly	weakly	ADJ
ejpam-6572	263	48	quasi	quasi	ADJ
ejpam-6572	263	49	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	263	50	,	,	PUNCT
ejpam-6572	263	51	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	263	52	;	;	PUNCT
ejpam-6572	263	53	(	(	PUNCT
ejpam-6572	263	54	2	2	X
ejpam-6572	263	55	)	)	PUNCT
ejpam-6572	263	56	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	263	57	-	-	PUNCT
ejpam-6572	263	58	int((σ1	int((σ1	PROPN
ejpam-6572	263	59	,	,	PUNCT
ejpam-6572	263	60	σ2)θ	σ2)θ	ADJ
ejpam-6572	263	61	-	-	PUNCT
ejpam-6572	263	62	cl(b	cl(b	NOUN
ejpam-6572	263	63	)	)	PUNCT
ejpam-6572	263	64	)	)	PUNCT
ejpam-6572	263	65	)	)	PUNCT
ejpam-6572	263	66	)	)	PUNCT
ejpam-6572	264	1	⊆	⊆	NUM
ejpam-6572	264	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	264	3	,	,	PUNCT
ejpam-6572	264	4	σ2)θ	σ2)θ	NOUN
ejpam-6572	264	5	-	-	PUNCT
ejpam-6572	264	6	cl(b	cl(b	NOUN
ejpam-6572	264	7	)	)	PUNCT
ejpam-6572	264	8	)	)	PUNCT
ejpam-6572	264	9	for	for	ADP
ejpam-6572	264	10	every	every	DET
ejpam-6572	264	11	subset	subset	NOUN
ejpam-6572	264	12	b	b	PROPN
ejpam-6572	264	13	of	of	ADP
ejpam-6572	264	14	y	y	PROPN
ejpam-6572	264	15	;	;	PUNCT
ejpam-6572	264	16	(	(	PUNCT
ejpam-6572	264	17	3	3	X
ejpam-6572	264	18	)	)	PUNCT
ejpam-6572	264	19	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	264	20	-	-	PUNCT
ejpam-6572	264	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	264	22	-	-	PUNCT
ejpam-6572	264	23	cl(b	cl(b	NOUN
ejpam-6572	264	24	)	)	PUNCT
ejpam-6572	264	25	)	)	PUNCT
ejpam-6572	264	26	)	)	PUNCT
ejpam-6572	264	27	)	)	PUNCT
ejpam-6572	265	1	⊆	⊆	NUM
ejpam-6572	265	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	265	3	,	,	PUNCT
ejpam-6572	265	4	σ2)θ	σ2)θ	NOUN
ejpam-6572	265	5	-	-	PUNCT
ejpam-6572	265	6	cl(b	cl(b	NOUN
ejpam-6572	265	7	)	)	PUNCT
ejpam-6572	265	8	)	)	PUNCT
ejpam-6572	265	9	for	for	ADP
ejpam-6572	265	10	every	every	DET
ejpam-6572	265	11	subset	subset	NOUN
ejpam-6572	265	12	b	b	PROPN
ejpam-6572	265	13	of	of	ADP
ejpam-6572	265	14	y	y	PROPN
ejpam-6572	265	15	;	;	PUNCT
ejpam-6572	265	16	(	(	PUNCT
ejpam-6572	265	17	4	4	X
ejpam-6572	265	18	)	)	PUNCT
ejpam-6572	265	19	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	265	20	-	-	PUNCT
ejpam-6572	265	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	265	22	-	-	PUNCT
ejpam-6572	265	23	cl(v	cl(v	NOUN
ejpam-6572	265	24	)	)	PUNCT
ejpam-6572	265	25	)	)	PUNCT
ejpam-6572	265	26	)	)	PUNCT
ejpam-6572	265	27	)	)	PUNCT
ejpam-6572	266	1	⊆	⊆	NUM
ejpam-6572	266	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	266	3	-	-	PUNCT
ejpam-6572	266	4	cl(v	cl(v	NOUN
ejpam-6572	266	5	)	)	PUNCT
ejpam-6572	266	6	)	)	PUNCT
ejpam-6572	266	7	for	for	ADP
ejpam-6572	266	8	every	every	DET
ejpam-6572	266	9	σ1σ2	σ1σ2	NOUN
ejpam-6572	266	10	-	-	ADJ
ejpam-6572	266	11	open	open	ADJ
ejpam-6572	266	12	set	set	NOUN
ejpam-6572	266	13	v	v	NOUN
ejpam-6572	266	14	of	of	ADP
ejpam-6572	266	15	y	y	PROPN
ejpam-6572	266	16	;	;	PUNCT
ejpam-6572	266	17	b.	b.	PROPN
ejpam-6572	266	18	kong	kong	PROPN
ejpam-6572	266	19	-	-	PUNCT
ejpam-6572	266	20	ied	ied	PROPN
ejpam-6572	266	21	,	,	PUNCT
ejpam-6572	266	22	a.	a.	PROPN
ejpam-6572	266	23	sama	sama	PROPN
ejpam-6572	266	24	-	-	PUNCT
ejpam-6572	266	25	ae	ae	PROPN
ejpam-6572	266	26	,	,	PUNCT
ejpam-6572	266	27	c.	c.	PROPN
ejpam-6572	266	28	boonpok	boonpok	PROPN
ejpam-6572	266	29	/	/	SYM
ejpam-6572	266	30	eur	eur	PROPN
ejpam-6572	266	31	.	.	PUNCT
ejpam-6572	267	1	j.	j.	PROPN
ejpam-6572	267	2	pure	pure	PROPN
ejpam-6572	267	3	appl	appl	PROPN
ejpam-6572	267	4	.	.	PROPN
ejpam-6572	267	5	math	math	PROPN
ejpam-6572	267	6	,	,	PUNCT
ejpam-6572	267	7	18	18	NUM
ejpam-6572	267	8	(	(	PUNCT
ejpam-6572	267	9	3	3	NUM
ejpam-6572	267	10	)	)	PUNCT
ejpam-6572	267	11	(	(	PUNCT
ejpam-6572	267	12	2025	2025	NUM
ejpam-6572	267	13	)	)	PUNCT
ejpam-6572	267	14	,	,	PUNCT
ejpam-6572	267	15	6572	6572	NUM
ejpam-6572	267	16	9	9	NUM
ejpam-6572	267	17	of	of	ADP
ejpam-6572	267	18	12	12	NUM
ejpam-6572	267	19	(	(	PUNCT
ejpam-6572	267	20	5	5	NUM
ejpam-6572	267	21	)	)	PUNCT
ejpam-6572	267	22	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	267	23	-	-	PUNCT
ejpam-6572	267	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	267	25	-	-	PUNCT
ejpam-6572	267	26	cl(v	cl(v	NOUN
ejpam-6572	267	27	)	)	PUNCT
ejpam-6572	267	28	)	)	PUNCT
ejpam-6572	267	29	)	)	PUNCT
ejpam-6572	267	30	)	)	PUNCT
ejpam-6572	268	1	⊆	⊆	NUM
ejpam-6572	268	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	268	3	-	-	PUNCT
ejpam-6572	268	4	cl(v	cl(v	NOUN
ejpam-6572	268	5	)	)	PUNCT
ejpam-6572	268	6	)	)	PUNCT
ejpam-6572	268	7	for	for	ADP
ejpam-6572	268	8	every	every	DET
ejpam-6572	268	9	(	(	PUNCT
ejpam-6572	268	10	σ1	σ1	PROPN
ejpam-6572	268	11	,	,	PUNCT
ejpam-6572	268	12	σ2)p	σ2)p	NOUN
ejpam-6572	268	13	-	-	PUNCT
ejpam-6572	268	14	open	open	NOUN
ejpam-6572	268	15	set	set	NOUN
ejpam-6572	268	16	v	v	NOUN
ejpam-6572	268	17	of	of	ADP
ejpam-6572	268	18	y	y	PROPN
ejpam-6572	268	19	;	;	PUNCT
ejpam-6572	268	20	(	(	PUNCT
ejpam-6572	268	21	6	6	X
ejpam-6572	268	22	)	)	PUNCT
ejpam-6572	268	23	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	268	24	-	-	SYM
ejpam-6572	268	25	int(k	int(k	NOUN
ejpam-6572	268	26	)	)	PUNCT
ejpam-6572	268	27	)	)	PUNCT
ejpam-6572	268	28	)	)	PUNCT
ejpam-6572	269	1	⊆	⊆	NUM
ejpam-6572	269	2	f−1(k	f−1(k	PROPN
ejpam-6572	269	3	)	)	PUNCT
ejpam-6572	269	4	for	for	ADP
ejpam-6572	269	5	every	every	DET
ejpam-6572	269	6	(	(	PUNCT
ejpam-6572	269	7	σ1	σ1	PROPN
ejpam-6572	269	8	,	,	PUNCT
ejpam-6572	269	9	σ2)r	σ2)r	NOUN
ejpam-6572	269	10	-	-	PUNCT
ejpam-6572	269	11	closed	close	VERB
ejpam-6572	269	12	set	set	ADJ
ejpam-6572	269	13	k	k	PROPN
ejpam-6572	269	14	of	of	ADP
ejpam-6572	269	15	y	y	PROPN
ejpam-6572	269	16	.	.	PUNCT
ejpam-6572	270	1	proof	proof	NOUN
ejpam-6572	270	2	.	.	PUNCT
ejpam-6572	271	1	(	(	PUNCT
ejpam-6572	271	2	1	1	X
ejpam-6572	271	3	)	)	PUNCT
ejpam-6572	271	4	⇒	⇒	NOUN
ejpam-6572	271	5	(	(	PUNCT
ejpam-6572	271	6	2	2	NUM
ejpam-6572	271	7	):	):	PUNCT
ejpam-6572	271	8	let	let	VERB
ejpam-6572	271	9	b	b	X
ejpam-6572	271	10	be	be	AUX
ejpam-6572	271	11	any	any	DET
ejpam-6572	271	12	subset	subset	NOUN
ejpam-6572	271	13	of	of	ADP
ejpam-6572	271	14	y	y	PROPN
ejpam-6572	271	15	.	.	PUNCT
ejpam-6572	272	1	since	since	SCONJ
ejpam-6572	272	2	(	(	PUNCT
ejpam-6572	272	3	σ1	σ1	PROPN
ejpam-6572	272	4	,	,	PUNCT
ejpam-6572	272	5	σ2)θ	σ2)θ	NOUN
ejpam-6572	272	6	-	-	PUNCT
ejpam-6572	272	7	cl(b	cl(b	NOUN
ejpam-6572	272	8	)	)	PUNCT
ejpam-6572	272	9	is	be	AUX
ejpam-6572	272	10	σ1σ2	σ1σ2	NOUN
ejpam-6572	272	11	-	-	ADJ
ejpam-6572	272	12	closed	closed	ADJ
ejpam-6572	272	13	in	in	ADP
ejpam-6572	272	14	y	y	PROPN
ejpam-6572	272	15	,	,	PUNCT
ejpam-6572	272	16	by	by	ADP
ejpam-6572	272	17	theorem	theorem	NOUN
ejpam-6572	272	18	4	4	NUM
ejpam-6572	272	19	we	we	PRON
ejpam-6572	272	20	have	have	VERB
ejpam-6572	272	21	int⋆(cl⋆(f−1(σ1σ2	int⋆(cl⋆(f−1(σ1σ2	NOUN
ejpam-6572	272	22	-	-	PUNCT
ejpam-6572	272	23	int((σ1	int((σ1	ADJ
ejpam-6572	272	24	,	,	PUNCT
ejpam-6572	272	25	σ2)θ	σ2)θ	ADJ
ejpam-6572	272	26	-	-	PUNCT
ejpam-6572	272	27	cl(b	cl(b	NOUN
ejpam-6572	272	28	)	)	PUNCT
ejpam-6572	272	29	)	)	PUNCT
ejpam-6572	272	30	)	)	PUNCT
ejpam-6572	272	31	)	)	PUNCT
ejpam-6572	272	32	)	)	PUNCT
ejpam-6572	273	1	⊆	⊆	NUM
ejpam-6572	273	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	273	3	,	,	PUNCT
ejpam-6572	273	4	σ2)θ	σ2)θ	NOUN
ejpam-6572	273	5	-	-	PUNCT
ejpam-6572	273	6	cl(b	cl(b	NOUN
ejpam-6572	273	7	)	)	PUNCT
ejpam-6572	273	8	)	)	PUNCT
ejpam-6572	273	9	and	and	CCONJ
ejpam-6572	273	10	by	by	ADP
ejpam-6572	273	11	lemma	lemma	PROPN
ejpam-6572	273	12	4	4	NUM
ejpam-6572	273	13	,	,	PUNCT
ejpam-6572	273	14	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	273	15	-	-	PUNCT
ejpam-6572	273	16	int((σ1	int((σ1	PROPN
ejpam-6572	273	17	,	,	PUNCT
ejpam-6572	273	18	σ2)θ	σ2)θ	ADJ
ejpam-6572	273	19	-	-	PUNCT
ejpam-6572	273	20	cl(b	cl(b	NOUN
ejpam-6572	273	21	)	)	PUNCT
ejpam-6572	273	22	)	)	PUNCT
ejpam-6572	273	23	)	)	PUNCT
ejpam-6572	273	24	)	)	PUNCT
ejpam-6572	274	1	⊆	⊆	NUM
ejpam-6572	274	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6572	274	3	,	,	PUNCT
ejpam-6572	274	4	σ2)θ	σ2)θ	NOUN
ejpam-6572	274	5	-	-	PUNCT
ejpam-6572	274	6	cl(b	cl(b	NOUN
ejpam-6572	274	7	)	)	PUNCT
ejpam-6572	274	8	)	)	PUNCT
ejpam-6572	274	9	.	.	PUNCT
ejpam-6572	275	1	(	(	PUNCT
ejpam-6572	275	2	2	2	X
ejpam-6572	275	3	)	)	PUNCT
ejpam-6572	275	4	⇒	⇒	NOUN
ejpam-6572	275	5	(	(	PUNCT
ejpam-6572	275	6	3	3	NUM
ejpam-6572	275	7	):	):	PUNCT
ejpam-6572	275	8	this	this	PRON
ejpam-6572	275	9	is	be	AUX
ejpam-6572	275	10	obvious	obvious	ADJ
ejpam-6572	275	11	since	since	SCONJ
ejpam-6572	275	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	275	13	-	-	NOUN
ejpam-6572	275	14	cl(b	cl(b	NOUN
ejpam-6572	275	15	)	)	PUNCT
ejpam-6572	275	16	⊆	⊆	NUM
ejpam-6572	275	17	(	(	PUNCT
ejpam-6572	275	18	σ1	σ1	PROPN
ejpam-6572	275	19	,	,	PUNCT
ejpam-6572	275	20	σ2)θ	σ2)θ	NOUN
ejpam-6572	275	21	-	-	PUNCT
ejpam-6572	275	22	cl(b	cl(b	NOUN
ejpam-6572	275	23	)	)	PUNCT
ejpam-6572	275	24	for	for	ADP
ejpam-6572	275	25	every	every	DET
ejpam-6572	275	26	subset	subset	NOUN
ejpam-6572	275	27	b	b	PROPN
ejpam-6572	275	28	of	of	ADP
ejpam-6572	275	29	y	y	PROPN
ejpam-6572	275	30	.	.	PUNCT
ejpam-6572	276	1	(	(	PUNCT
ejpam-6572	276	2	3	3	X
ejpam-6572	276	3	)	)	PUNCT
ejpam-6572	276	4	⇒	⇒	NOUN
ejpam-6572	276	5	(	(	PUNCT
ejpam-6572	276	6	4	4	NUM
ejpam-6572	276	7	):	):	PUNCT
ejpam-6572	276	8	this	this	PRON
ejpam-6572	276	9	is	be	AUX
ejpam-6572	276	10	obvious	obvious	ADJ
ejpam-6572	276	11	since	since	SCONJ
ejpam-6572	276	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	276	13	-	-	NOUN
ejpam-6572	276	14	cl(v	cl(v	X
ejpam-6572	276	15	)	)	PUNCT
ejpam-6572	277	1	=	=	SYM
ejpam-6572	277	2	(	(	PUNCT
ejpam-6572	277	3	σ1	σ1	PROPN
ejpam-6572	277	4	,	,	PUNCT
ejpam-6572	277	5	σ2)θ	σ2)θ	NOUN
ejpam-6572	277	6	-	-	PUNCT
ejpam-6572	277	7	cl(v	cl(v	NOUN
ejpam-6572	277	8	)	)	PUNCT
ejpam-6572	277	9	for	for	ADP
ejpam-6572	277	10	every	every	DET
ejpam-6572	277	11	σ1σ2	σ1σ2	NOUN
ejpam-6572	277	12	-	-	ADJ
ejpam-6572	277	13	open	open	ADJ
ejpam-6572	277	14	set	set	NOUN
ejpam-6572	277	15	v	v	NOUN
ejpam-6572	277	16	of	of	ADP
ejpam-6572	277	17	y	y	PROPN
ejpam-6572	277	18	.	.	PUNCT
ejpam-6572	278	1	(	(	PUNCT
ejpam-6572	278	2	4	4	X
ejpam-6572	278	3	)	)	PUNCT
ejpam-6572	278	4	⇒	⇒	NOUN
ejpam-6572	278	5	(	(	PUNCT
ejpam-6572	278	6	5	5	NUM
ejpam-6572	278	7	):	):	PUNCT
ejpam-6572	278	8	let	let	VERB
ejpam-6572	278	9	v	v	PART
ejpam-6572	278	10	be	be	AUX
ejpam-6572	278	11	any	any	DET
ejpam-6572	278	12	(	(	PUNCT
ejpam-6572	278	13	σ1	σ1	PROPN
ejpam-6572	278	14	,	,	PUNCT
ejpam-6572	278	15	σ2)p	σ2)p	NOUN
ejpam-6572	278	16	-	-	PUNCT
ejpam-6572	278	17	open	open	ADJ
ejpam-6572	278	18	set	set	NOUN
ejpam-6572	278	19	of	of	ADP
ejpam-6572	278	20	y	y	PROPN
ejpam-6572	278	21	.	.	PUNCT
ejpam-6572	279	1	then	then	ADV
ejpam-6572	279	2	,	,	PUNCT
ejpam-6572	279	3	v	v	ADP
ejpam-6572	279	4	⊆	⊆	NUM
ejpam-6572	279	5	σ1σ2	σ1σ2	NOUN
ejpam-6572	279	6	-	-	PUNCT
ejpam-6572	279	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	279	8	-	-	PUNCT
ejpam-6572	279	9	cl(v	cl(v	NOUN
ejpam-6572	279	10	)	)	PUNCT
ejpam-6572	279	11	)	)	PUNCT
ejpam-6572	279	12	and	and	CCONJ
ejpam-6572	279	13	σ1σ2	σ1σ2	NOUN
ejpam-6572	279	14	-	-	NUM
ejpam-6572	279	15	cl(v	cl(v	X
ejpam-6572	279	16	)	)	PUNCT
ejpam-6572	280	1	=	=	SYM
ejpam-6572	280	2	σ1σ2	σ1σ2	X
ejpam-6572	280	3	-	-	PUNCT
ejpam-6572	280	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	280	5	-	-	PUNCT
ejpam-6572	280	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	280	7	-	-	PUNCT
ejpam-6572	280	8	cl(v	cl(v	NOUN
ejpam-6572	280	9	)	)	PUNCT
ejpam-6572	280	10	)	)	PUNCT
ejpam-6572	280	11	)	)	PUNCT
ejpam-6572	280	12	.	.	PUNCT
ejpam-6572	281	1	now	now	ADV
ejpam-6572	281	2	,	,	PUNCT
ejpam-6572	281	3	put	put	VERB
ejpam-6572	281	4	g	g	NOUN
ejpam-6572	281	5	=	=	SYM
ejpam-6572	281	6	σ1σ2	σ1σ2	NOUN
ejpam-6572	281	7	-	-	PUNCT
ejpam-6572	281	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	281	9	-	-	PUNCT
ejpam-6572	281	10	cl(v	cl(v	NOUN
ejpam-6572	281	11	)	)	PUNCT
ejpam-6572	281	12	)	)	PUNCT
ejpam-6572	281	13	,	,	PUNCT
ejpam-6572	281	14	then	then	ADV
ejpam-6572	281	15	g	g	PROPN
ejpam-6572	281	16	is	be	AUX
ejpam-6572	281	17	σ1σ2	σ1σ2	NOUN
ejpam-6572	281	18	-	-	ADJ
ejpam-6572	281	19	open	open	ADJ
ejpam-6572	281	20	in	in	ADP
ejpam-6572	281	21	y	y	PROPN
ejpam-6572	281	22	and	and	CCONJ
ejpam-6572	281	23	σ1σ2	σ1σ2	NOUN
ejpam-6572	281	24	-	-	NUM
ejpam-6572	281	25	cl(g	cl(g	ADJ
ejpam-6572	281	26	)	)	PUNCT
ejpam-6572	281	27	=	=	PUNCT
ejpam-6572	281	28	σ1σ2	σ1σ2	NOUN
ejpam-6572	281	29	-	-	NUM
ejpam-6572	281	30	cl(v	cl(v	NOUN
ejpam-6572	281	31	)	)	PUNCT
ejpam-6572	281	32	.	.	PUNCT
ejpam-6572	282	1	thus	thus	ADV
ejpam-6572	282	2	by	by	ADP
ejpam-6572	282	3	(	(	PUNCT
ejpam-6572	282	4	4	4	NUM
ejpam-6572	282	5	)	)	PUNCT
ejpam-6572	282	6	,	,	PUNCT
ejpam-6572	282	7	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	X
ejpam-6572	282	8	-	-	PUNCT
ejpam-6572	282	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	282	10	-	-	PUNCT
ejpam-6572	282	11	cl(v	cl(v	NOUN
ejpam-6572	282	12	)	)	PUNCT
ejpam-6572	282	13	)	)	PUNCT
ejpam-6572	282	14	)	)	PUNCT
ejpam-6572	282	15	)	)	PUNCT
ejpam-6572	283	1	⊆	⊆	NUM
ejpam-6572	283	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	283	3	-	-	PUNCT
ejpam-6572	283	4	cl(v	cl(v	NOUN
ejpam-6572	283	5	)	)	PUNCT
ejpam-6572	283	6	)	)	PUNCT
ejpam-6572	283	7	.	.	PUNCT
ejpam-6572	284	1	(	(	PUNCT
ejpam-6572	284	2	5	5	X
ejpam-6572	284	3	)	)	PUNCT
ejpam-6572	284	4	⇒	⇒	NOUN
ejpam-6572	284	5	(	(	PUNCT
ejpam-6572	284	6	6	6	NUM
ejpam-6572	284	7	):	):	PUNCT
ejpam-6572	284	8	let	let	VERB
ejpam-6572	284	9	k	k	PRON
ejpam-6572	284	10	be	be	AUX
ejpam-6572	284	11	any	any	DET
ejpam-6572	284	12	(	(	PUNCT
ejpam-6572	284	13	σ1	σ1	NOUN
ejpam-6572	284	14	,	,	PUNCT
ejpam-6572	284	15	σ2)r	σ2)r	NOUN
ejpam-6572	284	16	-	-	PUNCT
ejpam-6572	284	17	closed	close	VERB
ejpam-6572	284	18	set	set	NOUN
ejpam-6572	284	19	of	of	ADP
ejpam-6572	284	20	y	y	PROPN
ejpam-6572	284	21	.	.	PUNCT
ejpam-6572	285	1	since	since	SCONJ
ejpam-6572	285	2	σ1σ2	σ1σ2	NOUN
ejpam-6572	285	3	-	-	SYM
ejpam-6572	285	4	int(k	int(k	NOUN
ejpam-6572	285	5	)	)	PUNCT
ejpam-6572	285	6	is	be	AUX
ejpam-6572	285	7	(	(	PUNCT
ejpam-6572	285	8	σ1	σ1	PROPN
ejpam-6572	285	9	,	,	PUNCT
ejpam-6572	285	10	σ2)p	σ2)p	NOUN
ejpam-6572	285	11	-	-	PUNCT
ejpam-6572	285	12	open	open	ADJ
ejpam-6572	285	13	in	in	ADP
ejpam-6572	285	14	y	y	PROPN
ejpam-6572	285	15	and	and	CCONJ
ejpam-6572	285	16	by	by	ADP
ejpam-6572	285	17	(	(	PUNCT
ejpam-6572	285	18	5	5	NUM
ejpam-6572	285	19	)	)	PUNCT
ejpam-6572	285	20	,	,	PUNCT
ejpam-6572	285	21	we	we	PRON
ejpam-6572	285	22	have	have	VERB
ejpam-6572	285	23	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	285	24	-	-	SYM
ejpam-6572	285	25	int(k	int(k	NOUN
ejpam-6572	285	26	)	)	PUNCT
ejpam-6572	285	27	)	)	PUNCT
ejpam-6572	285	28	)	)	PUNCT
ejpam-6572	286	1	=	=	SYM
ejpam-6572	286	2	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	X
ejpam-6572	286	3	-	-	PUNCT
ejpam-6572	286	4	int(σ1σ2	int(σ1σ2	ADV
ejpam-6572	286	5	-	-	PUNCT
ejpam-6572	286	6	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6572	286	7	-	-	PUNCT
ejpam-6572	286	8	int(k	int(k	NOUN
ejpam-6572	286	9	)	)	PUNCT
ejpam-6572	286	10	)	)	PUNCT
ejpam-6572	286	11	)	)	PUNCT
ejpam-6572	286	12	)	)	PUNCT
ejpam-6572	286	13	)	)	PUNCT
ejpam-6572	287	1	⊆	⊆	NUM
ejpam-6572	287	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	287	3	-	-	PUNCT
ejpam-6572	287	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6572	287	5	-	-	PUNCT
ejpam-6572	287	6	int(k	int(k	NOUN
ejpam-6572	287	7	)	)	PUNCT
ejpam-6572	287	8	)	)	PUNCT
ejpam-6572	287	9	)	)	PUNCT
ejpam-6572	288	1	=	=	PUNCT
ejpam-6572	288	2	f−1(k	f−1(k	PROPN
ejpam-6572	288	3	)	)	PUNCT
ejpam-6572	288	4	.	.	PUNCT
ejpam-6572	289	1	(	(	PUNCT
ejpam-6572	289	2	6	6	X
ejpam-6572	289	3	)	)	PUNCT
ejpam-6572	289	4	⇒	⇒	NOUN
ejpam-6572	289	5	(	(	PUNCT
ejpam-6572	289	6	1	1	NUM
ejpam-6572	289	7	):	):	PUNCT
ejpam-6572	289	8	let	let	VERB
ejpam-6572	289	9	v	v	PART
ejpam-6572	289	10	be	be	AUX
ejpam-6572	289	11	any	any	DET
ejpam-6572	289	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	289	13	-	-	ADJ
ejpam-6572	289	14	open	open	ADJ
ejpam-6572	289	15	set	set	NOUN
ejpam-6572	289	16	of	of	ADP
ejpam-6572	289	17	y	y	PROPN
ejpam-6572	289	18	.	.	PUNCT
ejpam-6572	290	1	then	then	ADV
ejpam-6572	290	2	,	,	PUNCT
ejpam-6572	290	3	σ1σ2	σ1σ2	NOUN
ejpam-6572	290	4	-	-	NUM
ejpam-6572	290	5	cl(v	cl(v	NOUN
ejpam-6572	290	6	)	)	PUNCT
ejpam-6572	290	7	is	be	AUX
ejpam-6572	290	8	(	(	PUNCT
ejpam-6572	290	9	σ1	σ1	NOUN
ejpam-6572	290	10	,	,	PUNCT
ejpam-6572	290	11	σ2)r	σ2)r	NOUN
ejpam-6572	290	12	-	-	PUNCT
ejpam-6572	290	13	closed	closed	ADJ
ejpam-6572	290	14	in	in	ADP
ejpam-6572	290	15	y	y	PROPN
ejpam-6572	290	16	and	and	CCONJ
ejpam-6572	290	17	by	by	ADP
ejpam-6572	290	18	(	(	PUNCT
ejpam-6572	290	19	6	6	NUM
ejpam-6572	290	20	)	)	PUNCT
ejpam-6572	290	21	,	,	PUNCT
ejpam-6572	290	22	scl⋆(f−1(v	scl⋆(f−1(v	PROPN
ejpam-6572	290	23	)	)	PUNCT
ejpam-6572	290	24	)	)	PUNCT
ejpam-6572	291	1	⊆	⊆	X
ejpam-6572	291	2	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NUM
ejpam-6572	291	3	-	-	PUNCT
ejpam-6572	291	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	291	5	-	-	PUNCT
ejpam-6572	291	6	cl(v	cl(v	NOUN
ejpam-6572	291	7	)	)	PUNCT
ejpam-6572	291	8	)	)	PUNCT
ejpam-6572	291	9	)	)	PUNCT
ejpam-6572	291	10	)	)	PUNCT
ejpam-6572	292	1	⊆	⊆	NUM
ejpam-6572	292	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	292	3	-	-	PUNCT
ejpam-6572	292	4	cl(v	cl(v	NOUN
ejpam-6572	292	5	)	)	PUNCT
ejpam-6572	292	6	)	)	PUNCT
ejpam-6572	292	7	.	.	PUNCT
ejpam-6572	293	1	it	it	PRON
ejpam-6572	293	2	follows	follow	VERB
ejpam-6572	293	3	from	from	ADP
ejpam-6572	293	4	theorem	theorem	ADJ
ejpam-6572	293	5	4	4	NUM
ejpam-6572	293	6	that	that	PRON
ejpam-6572	293	7	f	f	PROPN
ejpam-6572	293	8	is	be	AUX
ejpam-6572	293	9	weakly	weakly	ADJ
ejpam-6572	293	10	quasi	quasi	ADJ
ejpam-6572	293	11	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	293	12	,	,	PUNCT
ejpam-6572	293	13	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6572	293	14	.	.	X
ejpam-6572	293	15	theorem	theorem	VERB
ejpam-6572	293	16	6	6	NUM
ejpam-6572	293	17	.	.	PUNCT
ejpam-6572	293	18	for	for	ADP
ejpam-6572	293	19	a	a	DET
ejpam-6572	293	20	function	function	NOUN
ejpam-6572	293	21	f	f	NOUN
ejpam-6572	293	22	:	:	PUNCT
ejpam-6572	293	23	(	(	PUNCT
ejpam-6572	293	24	x	x	X
ejpam-6572	293	25	,	,	PUNCT
ejpam-6572	293	26	τ	τ	PROPN
ejpam-6572	293	27	,	,	PUNCT
ejpam-6572	293	28	i	i	NOUN
ejpam-6572	293	29	)	)	PUNCT
ejpam-6572	293	30	→	→	PUNCT
ejpam-6572	293	31	(	(	PUNCT
ejpam-6572	293	32	y	y	PROPN
ejpam-6572	293	33	,	,	PUNCT
ejpam-6572	293	34	σ1	σ1	PROPN
ejpam-6572	293	35	,	,	PUNCT
ejpam-6572	293	36	σ2	σ2	NOUN
ejpam-6572	293	37	)	)	PUNCT
ejpam-6572	293	38	,	,	PUNCT
ejpam-6572	293	39	the	the	DET
ejpam-6572	293	40	following	follow	VERB
ejpam-6572	293	41	properties	property	NOUN
ejpam-6572	293	42	are	be	AUX
ejpam-6572	293	43	equivalent	equivalent	ADJ
ejpam-6572	293	44	:	:	PUNCT
ejpam-6572	293	45	(	(	PUNCT
ejpam-6572	293	46	1	1	X
ejpam-6572	293	47	)	)	PUNCT
ejpam-6572	293	48	f	f	PROPN
ejpam-6572	293	49	is	be	AUX
ejpam-6572	293	50	weakly	weakly	ADJ
ejpam-6572	293	51	quasi	quasi	ADJ
ejpam-6572	293	52	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	293	53	,	,	PUNCT
ejpam-6572	293	54	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	293	55	;	;	PUNCT
ejpam-6572	293	56	(	(	PUNCT
ejpam-6572	293	57	2	2	X
ejpam-6572	293	58	)	)	PUNCT
ejpam-6572	293	59	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	293	60	-	-	PUNCT
ejpam-6572	293	61	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	293	62	-	-	PUNCT
ejpam-6572	293	63	cl(v	cl(v	NOUN
ejpam-6572	293	64	)	)	PUNCT
ejpam-6572	293	65	)	)	PUNCT
ejpam-6572	293	66	)	)	PUNCT
ejpam-6572	293	67	)	)	PUNCT
ejpam-6572	294	1	⊆	⊆	NUM
ejpam-6572	294	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	294	3	-	-	PUNCT
ejpam-6572	294	4	cl(v	cl(v	NOUN
ejpam-6572	294	5	)	)	PUNCT
ejpam-6572	294	6	)	)	PUNCT
ejpam-6572	294	7	for	for	ADP
ejpam-6572	294	8	every	every	DET
ejpam-6572	294	9	(	(	PUNCT
ejpam-6572	294	10	σ1	σ1	PROPN
ejpam-6572	294	11	,	,	PUNCT
ejpam-6572	294	12	σ2)β	σ2)β	NOUN
ejpam-6572	294	13	-	-	PUNCT
ejpam-6572	294	14	open	open	NOUN
ejpam-6572	294	15	set	set	NOUN
ejpam-6572	294	16	v	v	NOUN
ejpam-6572	294	17	of	of	ADP
ejpam-6572	294	18	y	y	PROPN
ejpam-6572	294	19	;	;	PUNCT
ejpam-6572	294	20	(	(	PUNCT
ejpam-6572	294	21	3	3	X
ejpam-6572	294	22	)	)	PUNCT
ejpam-6572	294	23	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	294	24	-	-	PUNCT
ejpam-6572	294	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	294	26	-	-	PUNCT
ejpam-6572	294	27	cl(v	cl(v	NOUN
ejpam-6572	294	28	)	)	PUNCT
ejpam-6572	294	29	)	)	PUNCT
ejpam-6572	294	30	)	)	PUNCT
ejpam-6572	294	31	)	)	PUNCT
ejpam-6572	295	1	⊆	⊆	NUM
ejpam-6572	295	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	295	3	-	-	PUNCT
ejpam-6572	295	4	cl(v	cl(v	NOUN
ejpam-6572	295	5	)	)	PUNCT
ejpam-6572	295	6	)	)	PUNCT
ejpam-6572	295	7	for	for	ADP
ejpam-6572	295	8	every	every	DET
ejpam-6572	295	9	(	(	PUNCT
ejpam-6572	295	10	σ1	σ1	PROPN
ejpam-6572	295	11	,	,	PUNCT
ejpam-6572	295	12	σ2)s	σ2)s	NOUN
ejpam-6572	295	13	-	-	PUNCT
ejpam-6572	295	14	open	open	NOUN
ejpam-6572	295	15	set	set	NOUN
ejpam-6572	295	16	v	v	NOUN
ejpam-6572	295	17	of	of	ADP
ejpam-6572	295	18	y	y	PROPN
ejpam-6572	295	19	;	;	PUNCT
ejpam-6572	295	20	(	(	PUNCT
ejpam-6572	295	21	4	4	X
ejpam-6572	295	22	)	)	PUNCT
ejpam-6572	295	23	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	NOUN
ejpam-6572	295	24	-	-	PUNCT
ejpam-6572	295	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	295	26	-	-	PUNCT
ejpam-6572	295	27	cl(v	cl(v	NOUN
ejpam-6572	295	28	)	)	PUNCT
ejpam-6572	295	29	)	)	PUNCT
ejpam-6572	295	30	)	)	PUNCT
ejpam-6572	295	31	)	)	PUNCT
ejpam-6572	296	1	⊆	⊆	NUM
ejpam-6572	296	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	296	3	-	-	PUNCT
ejpam-6572	296	4	cl(v	cl(v	NOUN
ejpam-6572	296	5	)	)	PUNCT
ejpam-6572	296	6	)	)	PUNCT
ejpam-6572	296	7	for	for	ADP
ejpam-6572	296	8	every	every	DET
ejpam-6572	296	9	(	(	PUNCT
ejpam-6572	296	10	σ1	σ1	PROPN
ejpam-6572	296	11	,	,	PUNCT
ejpam-6572	296	12	σ2)p	σ2)p	NOUN
ejpam-6572	296	13	-	-	PUNCT
ejpam-6572	296	14	open	open	NOUN
ejpam-6572	296	15	set	set	NOUN
ejpam-6572	296	16	v	v	NOUN
ejpam-6572	296	17	of	of	ADP
ejpam-6572	296	18	y	y	PROPN
ejpam-6572	296	19	.	.	PUNCT
ejpam-6572	297	1	proof	proof	NOUN
ejpam-6572	297	2	.	.	PUNCT
ejpam-6572	298	1	(	(	PUNCT
ejpam-6572	298	2	1	1	X
ejpam-6572	298	3	)	)	PUNCT
ejpam-6572	298	4	⇒	⇒	NOUN
ejpam-6572	298	5	(	(	PUNCT
ejpam-6572	298	6	2	2	NUM
ejpam-6572	298	7	):	):	PUNCT
ejpam-6572	298	8	let	let	VERB
ejpam-6572	298	9	v	v	PART
ejpam-6572	298	10	be	be	AUX
ejpam-6572	298	11	any	any	DET
ejpam-6572	298	12	(	(	PUNCT
ejpam-6572	298	13	σ1	σ1	PROPN
ejpam-6572	298	14	,	,	PUNCT
ejpam-6572	298	15	σ2)β	σ2)β	NOUN
ejpam-6572	298	16	-	-	PUNCT
ejpam-6572	298	17	open	open	ADJ
ejpam-6572	298	18	set	set	NOUN
ejpam-6572	298	19	of	of	ADP
ejpam-6572	298	20	y	y	PROPN
ejpam-6572	298	21	.	.	PUNCT
ejpam-6572	299	1	then	then	ADV
ejpam-6572	299	2	,	,	PUNCT
ejpam-6572	299	3	we	we	PRON
ejpam-6572	299	4	have	have	VERB
ejpam-6572	299	5	v	v	ADP
ejpam-6572	299	6	⊆	⊆	NUM
ejpam-6572	299	7	σ1σ2	σ1σ2	NOUN
ejpam-6572	299	8	-	-	PUNCT
ejpam-6572	299	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	299	10	-	-	PUNCT
ejpam-6572	299	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	299	12	-	-	PUNCT
ejpam-6572	299	13	cl(v	cl(v	NOUN
ejpam-6572	299	14	)	)	PUNCT
ejpam-6572	299	15	)	)	PUNCT
ejpam-6572	299	16	)	)	PUNCT
ejpam-6572	300	1	b.	b.	PROPN
ejpam-6572	300	2	kong	kong	PROPN
ejpam-6572	300	3	-	-	PUNCT
ejpam-6572	300	4	ied	ied	PROPN
ejpam-6572	300	5	,	,	PUNCT
ejpam-6572	300	6	a.	a.	PROPN
ejpam-6572	300	7	sama	sama	PROPN
ejpam-6572	300	8	-	-	PUNCT
ejpam-6572	300	9	ae	ae	PROPN
ejpam-6572	300	10	,	,	PUNCT
ejpam-6572	300	11	c.	c.	PROPN
ejpam-6572	300	12	boonpok	boonpok	PROPN
ejpam-6572	300	13	/	/	SYM
ejpam-6572	300	14	eur	eur	PROPN
ejpam-6572	300	15	.	.	PUNCT
ejpam-6572	301	1	j.	j.	PROPN
ejpam-6572	301	2	pure	pure	PROPN
ejpam-6572	301	3	appl	appl	PROPN
ejpam-6572	301	4	.	.	PROPN
ejpam-6572	301	5	math	math	PROPN
ejpam-6572	301	6	,	,	PUNCT
ejpam-6572	301	7	18	18	NUM
ejpam-6572	301	8	(	(	PUNCT
ejpam-6572	301	9	3	3	NUM
ejpam-6572	301	10	)	)	PUNCT
ejpam-6572	301	11	(	(	PUNCT
ejpam-6572	301	12	2025	2025	NUM
ejpam-6572	301	13	)	)	PUNCT
ejpam-6572	301	14	,	,	PUNCT
ejpam-6572	301	15	6572	6572	NUM
ejpam-6572	301	16	10	10	NUM
ejpam-6572	301	17	of	of	ADP
ejpam-6572	301	18	12	12	NUM
ejpam-6572	301	19	and	and	CCONJ
ejpam-6572	301	20	hence	hence	ADV
ejpam-6572	301	21	σ1σ2	σ1σ2	NOUN
ejpam-6572	301	22	-	-	NOUN
ejpam-6572	301	23	cl(v	cl(v	NOUN
ejpam-6572	301	24	)	)	PUNCT
ejpam-6572	302	1	=	=	SYM
ejpam-6572	302	2	σ1σ2	σ1σ2	X
ejpam-6572	302	3	-	-	PUNCT
ejpam-6572	302	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6572	302	5	-	-	PUNCT
ejpam-6572	302	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	302	7	-	-	PUNCT
ejpam-6572	302	8	cl(v	cl(v	NOUN
ejpam-6572	302	9	)	)	PUNCT
ejpam-6572	302	10	)	)	PUNCT
ejpam-6572	302	11	)	)	PUNCT
ejpam-6572	302	12	.	.	PUNCT
ejpam-6572	303	1	since	since	SCONJ
ejpam-6572	303	2	σ1σ2	σ1σ2	NOUN
ejpam-6572	303	3	-	-	NOUN
ejpam-6572	303	4	cl(v	cl(v	NOUN
ejpam-6572	303	5	)	)	PUNCT
ejpam-6572	303	6	is	be	AUX
ejpam-6572	303	7	(	(	PUNCT
ejpam-6572	303	8	σ1	σ1	PROPN
ejpam-6572	303	9	,	,	PUNCT
ejpam-6572	303	10	σ2)rclosed	σ2)rclose	VERB
ejpam-6572	303	11	in	in	ADP
ejpam-6572	303	12	y	y	PROPN
ejpam-6572	303	13	and	and	CCONJ
ejpam-6572	303	14	by	by	ADP
ejpam-6572	303	15	theorem	theorem	NOUN
ejpam-6572	303	16	5	5	NUM
ejpam-6572	303	17	,	,	PUNCT
ejpam-6572	303	18	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	X
ejpam-6572	303	19	-	-	PUNCT
ejpam-6572	303	20	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	303	21	-	-	PUNCT
ejpam-6572	303	22	cl(v	cl(v	NOUN
ejpam-6572	303	23	)	)	PUNCT
ejpam-6572	303	24	)	)	PUNCT
ejpam-6572	303	25	)	)	PUNCT
ejpam-6572	303	26	)	)	PUNCT
ejpam-6572	304	1	⊆	⊆	NUM
ejpam-6572	304	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	304	3	-	-	PUNCT
ejpam-6572	304	4	cl(v	cl(v	NOUN
ejpam-6572	304	5	)	)	PUNCT
ejpam-6572	304	6	)	)	PUNCT
ejpam-6572	304	7	.	.	PUNCT
ejpam-6572	305	1	(	(	PUNCT
ejpam-6572	305	2	2	2	X
ejpam-6572	305	3	)	)	PUNCT
ejpam-6572	305	4	⇒	⇒	NOUN
ejpam-6572	305	5	(	(	PUNCT
ejpam-6572	305	6	3	3	NUM
ejpam-6572	305	7	):	):	PUNCT
ejpam-6572	305	8	this	this	PRON
ejpam-6572	305	9	is	be	AUX
ejpam-6572	305	10	obvious	obvious	ADJ
ejpam-6572	305	11	since	since	SCONJ
ejpam-6572	305	12	every	every	DET
ejpam-6572	305	13	(	(	PUNCT
ejpam-6572	305	14	σ1	σ1	PROPN
ejpam-6572	305	15	,	,	PUNCT
ejpam-6572	305	16	σ2)s	σ2)s	NOUN
ejpam-6572	305	17	-	-	PUNCT
ejpam-6572	305	18	open	open	ADJ
ejpam-6572	305	19	set	set	NOUN
ejpam-6572	305	20	is	be	AUX
ejpam-6572	305	21	(	(	PUNCT
ejpam-6572	305	22	σ1	σ1	PROPN
ejpam-6572	305	23	,	,	PUNCT
ejpam-6572	305	24	σ2)β	σ2)β	NOUN
ejpam-6572	305	25	-	-	PUNCT
ejpam-6572	305	26	open	open	ADJ
ejpam-6572	305	27	.	.	PUNCT
ejpam-6572	306	1	(	(	PUNCT
ejpam-6572	306	2	3	3	X
ejpam-6572	306	3	)	)	PUNCT
ejpam-6572	306	4	⇒	⇒	NOUN
ejpam-6572	306	5	(	(	PUNCT
ejpam-6572	306	6	4	4	NUM
ejpam-6572	306	7	):	):	PUNCT
ejpam-6572	306	8	let	let	VERB
ejpam-6572	306	9	v	v	PART
ejpam-6572	306	10	be	be	AUX
ejpam-6572	306	11	any	any	DET
ejpam-6572	306	12	(	(	PUNCT
ejpam-6572	306	13	σ1	σ1	PROPN
ejpam-6572	306	14	,	,	PUNCT
ejpam-6572	306	15	σ2)p	σ2)p	NOUN
ejpam-6572	306	16	-	-	PUNCT
ejpam-6572	306	17	open	open	ADJ
ejpam-6572	306	18	set	set	NOUN
ejpam-6572	306	19	of	of	ADP
ejpam-6572	306	20	y	y	PROPN
ejpam-6572	306	21	.	.	PUNCT
ejpam-6572	307	1	since	since	SCONJ
ejpam-6572	307	2	σ1σ2	σ1σ2	NOUN
ejpam-6572	307	3	-	-	NOUN
ejpam-6572	307	4	cl(v	cl(v	NOUN
ejpam-6572	307	5	)	)	PUNCT
ejpam-6572	307	6	is	be	AUX
ejpam-6572	307	7	(	(	PUNCT
ejpam-6572	307	8	σ1	σ1	NOUN
ejpam-6572	307	9	,	,	PUNCT
ejpam-6572	307	10	σ2)r	σ2)r	NOUN
ejpam-6572	307	11	-	-	PUNCT
ejpam-6572	307	12	closed	closed	ADJ
ejpam-6572	307	13	,	,	PUNCT
ejpam-6572	307	14	we	we	PRON
ejpam-6572	307	15	have	have	VERB
ejpam-6572	307	16	σ1σ2	σ1σ2	NOUN
ejpam-6572	307	17	-	-	NUM
ejpam-6572	307	18	cl(v	cl(v	NOUN
ejpam-6572	307	19	)	)	PUNCT
ejpam-6572	307	20	is	be	AUX
ejpam-6572	307	21	(	(	PUNCT
ejpam-6572	307	22	σ1	σ1	PROPN
ejpam-6572	307	23	,	,	PUNCT
ejpam-6572	307	24	σ2)s	σ2)s	NOUN
ejpam-6572	307	25	-	-	PUNCT
ejpam-6572	307	26	open	open	ADJ
ejpam-6572	307	27	in	in	ADP
ejpam-6572	307	28	y	y	PROPN
ejpam-6572	307	29	.	.	PUNCT
ejpam-6572	308	1	thus	thus	ADV
ejpam-6572	308	2	by	by	ADP
ejpam-6572	308	3	(	(	PUNCT
ejpam-6572	308	4	3	3	NUM
ejpam-6572	308	5	)	)	PUNCT
ejpam-6572	308	6	,	,	PUNCT
ejpam-6572	308	7	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	X
ejpam-6572	308	8	-	-	PUNCT
ejpam-6572	308	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	308	10	-	-	PUNCT
ejpam-6572	308	11	cl(v	cl(v	NOUN
ejpam-6572	308	12	)	)	PUNCT
ejpam-6572	308	13	)	)	PUNCT
ejpam-6572	308	14	)	)	PUNCT
ejpam-6572	308	15	)	)	PUNCT
ejpam-6572	309	1	⊆	⊆	NUM
ejpam-6572	309	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	309	3	-	-	PUNCT
ejpam-6572	309	4	cl(v	cl(v	NOUN
ejpam-6572	309	5	)	)	PUNCT
ejpam-6572	309	6	)	)	PUNCT
ejpam-6572	309	7	.	.	PUNCT
ejpam-6572	310	1	(	(	PUNCT
ejpam-6572	310	2	4	4	X
ejpam-6572	310	3	)	)	PUNCT
ejpam-6572	310	4	⇒	⇒	NOUN
ejpam-6572	310	5	(	(	PUNCT
ejpam-6572	310	6	1	1	NUM
ejpam-6572	310	7	):	):	PUNCT
ejpam-6572	310	8	let	let	VERB
ejpam-6572	310	9	v	v	PART
ejpam-6572	310	10	be	be	AUX
ejpam-6572	310	11	any	any	DET
ejpam-6572	310	12	σ1σ2	σ1σ2	NOUN
ejpam-6572	310	13	-	-	ADJ
ejpam-6572	310	14	open	open	ADJ
ejpam-6572	310	15	set	set	NOUN
ejpam-6572	310	16	of	of	ADP
ejpam-6572	310	17	y	y	PROPN
ejpam-6572	310	18	.	.	PUNCT
ejpam-6572	311	1	then	then	ADV
ejpam-6572	311	2	,	,	PUNCT
ejpam-6572	311	3	v	v	NOUN
ejpam-6572	311	4	is	be	AUX
ejpam-6572	311	5	(	(	PUNCT
ejpam-6572	311	6	σ1	σ1	PROPN
ejpam-6572	311	7	,	,	PUNCT
ejpam-6572	311	8	σ2)p	σ2)p	NOUN
ejpam-6572	311	9	-	-	PUNCT
ejpam-6572	311	10	open	open	ADJ
ejpam-6572	311	11	in	in	ADP
ejpam-6572	311	12	y	y	PROPN
ejpam-6572	311	13	.	.	PUNCT
ejpam-6572	312	1	by	by	ADP
ejpam-6572	312	2	(	(	PUNCT
ejpam-6572	312	3	4	4	NUM
ejpam-6572	312	4	)	)	PUNCT
ejpam-6572	312	5	,	,	PUNCT
ejpam-6572	312	6	scl⋆(f−1(σ1σ2	scl⋆(f−1(σ1σ2	X
ejpam-6572	312	7	-	-	PUNCT
ejpam-6572	312	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	312	9	-	-	PUNCT
ejpam-6572	312	10	cl(v	cl(v	NOUN
ejpam-6572	312	11	)	)	PUNCT
ejpam-6572	312	12	)	)	PUNCT
ejpam-6572	312	13	)	)	PUNCT
ejpam-6572	312	14	)	)	PUNCT
ejpam-6572	313	1	⊆	⊆	NUM
ejpam-6572	313	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	313	3	-	-	PUNCT
ejpam-6572	313	4	cl(v	cl(v	NOUN
ejpam-6572	313	5	)	)	PUNCT
ejpam-6572	313	6	)	)	PUNCT
ejpam-6572	313	7	.	.	PUNCT
ejpam-6572	314	1	it	it	PRON
ejpam-6572	314	2	follows	follow	VERB
ejpam-6572	314	3	from	from	ADP
ejpam-6572	314	4	theorem	theorem	ADJ
ejpam-6572	314	5	5	5	NUM
ejpam-6572	314	6	that	that	SCONJ
ejpam-6572	314	7	f	f	PROPN
ejpam-6572	314	8	is	be	AUX
ejpam-6572	314	9	weakly	weakly	ADJ
ejpam-6572	314	10	quasi	quasi	ADJ
ejpam-6572	314	11	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	314	12	,	,	PUNCT
ejpam-6572	314	13	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6572	314	14	.	.	X
ejpam-6572	314	15	theorem	theorem	VERB
ejpam-6572	314	16	7	7	NUM
ejpam-6572	314	17	.	.	X
ejpam-6572	314	18	for	for	ADP
ejpam-6572	314	19	a	a	DET
ejpam-6572	314	20	function	function	NOUN
ejpam-6572	314	21	f	f	NOUN
ejpam-6572	314	22	:	:	PUNCT
ejpam-6572	314	23	(	(	PUNCT
ejpam-6572	314	24	x	x	X
ejpam-6572	314	25	,	,	PUNCT
ejpam-6572	314	26	τ	τ	PROPN
ejpam-6572	314	27	,	,	PUNCT
ejpam-6572	314	28	i	i	NOUN
ejpam-6572	314	29	)	)	PUNCT
ejpam-6572	314	30	→	→	PUNCT
ejpam-6572	314	31	(	(	PUNCT
ejpam-6572	314	32	y	y	PROPN
ejpam-6572	314	33	,	,	PUNCT
ejpam-6572	314	34	σ1	σ1	PROPN
ejpam-6572	314	35	,	,	PUNCT
ejpam-6572	314	36	σ2	σ2	NOUN
ejpam-6572	314	37	)	)	PUNCT
ejpam-6572	314	38	,	,	PUNCT
ejpam-6572	314	39	the	the	DET
ejpam-6572	314	40	following	follow	VERB
ejpam-6572	314	41	properties	property	NOUN
ejpam-6572	314	42	are	be	AUX
ejpam-6572	314	43	equivalent	equivalent	ADJ
ejpam-6572	314	44	:	:	PUNCT
ejpam-6572	314	45	(	(	PUNCT
ejpam-6572	314	46	1	1	X
ejpam-6572	314	47	)	)	PUNCT
ejpam-6572	314	48	f	f	PROPN
ejpam-6572	314	49	is	be	AUX
ejpam-6572	314	50	weakly	weakly	ADJ
ejpam-6572	314	51	quasi	quasi	ADJ
ejpam-6572	314	52	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	314	53	,	,	PUNCT
ejpam-6572	314	54	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	314	55	;	;	PUNCT
ejpam-6572	314	56	(	(	PUNCT
ejpam-6572	314	57	2	2	X
ejpam-6572	314	58	)	)	PUNCT
ejpam-6572	314	59	int⋆(cl⋆(f−1(v	int⋆(cl⋆(f−1(v	NOUN
ejpam-6572	314	60	)	)	PUNCT
ejpam-6572	314	61	)	)	PUNCT
ejpam-6572	314	62	)	)	PUNCT
ejpam-6572	315	1	⊆	⊆	NUM
ejpam-6572	315	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	315	3	-	-	PUNCT
ejpam-6572	315	4	cl(v	cl(v	NOUN
ejpam-6572	315	5	)	)	PUNCT
ejpam-6572	315	6	)	)	PUNCT
ejpam-6572	315	7	for	for	ADP
ejpam-6572	315	8	every	every	DET
ejpam-6572	315	9	(	(	PUNCT
ejpam-6572	315	10	σ1	σ1	PROPN
ejpam-6572	315	11	,	,	PUNCT
ejpam-6572	315	12	σ2)p	σ2)p	NOUN
ejpam-6572	315	13	-	-	PUNCT
ejpam-6572	315	14	open	open	NOUN
ejpam-6572	315	15	set	set	NOUN
ejpam-6572	315	16	v	v	NOUN
ejpam-6572	315	17	of	of	ADP
ejpam-6572	315	18	y	y	PROPN
ejpam-6572	315	19	;	;	PUNCT
ejpam-6572	315	20	(	(	PUNCT
ejpam-6572	315	21	3	3	X
ejpam-6572	315	22	)	)	PUNCT
ejpam-6572	315	23	scl⋆(f−1(v	scl⋆(f−1(v	NOUN
ejpam-6572	315	24	)	)	PUNCT
ejpam-6572	315	25	)	)	PUNCT
ejpam-6572	316	1	⊆	⊆	NUM
ejpam-6572	316	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	316	3	-	-	PUNCT
ejpam-6572	316	4	cl(v	cl(v	NOUN
ejpam-6572	316	5	)	)	PUNCT
ejpam-6572	316	6	)	)	PUNCT
ejpam-6572	316	7	for	for	ADP
ejpam-6572	316	8	every	every	DET
ejpam-6572	316	9	(	(	PUNCT
ejpam-6572	316	10	σ1	σ1	PROPN
ejpam-6572	316	11	,	,	PUNCT
ejpam-6572	316	12	σ2)p	σ2)p	NOUN
ejpam-6572	316	13	-	-	PUNCT
ejpam-6572	316	14	open	open	NOUN
ejpam-6572	316	15	set	set	NOUN
ejpam-6572	316	16	v	v	NOUN
ejpam-6572	316	17	of	of	ADP
ejpam-6572	316	18	y	y	PROPN
ejpam-6572	316	19	;	;	PUNCT
ejpam-6572	316	20	(	(	PUNCT
ejpam-6572	316	21	4	4	X
ejpam-6572	316	22	)	)	PUNCT
ejpam-6572	316	23	f−1(v	f−1(v	NOUN
ejpam-6572	316	24	)	)	PUNCT
ejpam-6572	317	1	⊆	⊆	NUM
ejpam-6572	317	2	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	317	3	-	-	PUNCT
ejpam-6572	317	4	cl(v	cl(v	NOUN
ejpam-6572	317	5	)	)	PUNCT
ejpam-6572	317	6	)	)	PUNCT
ejpam-6572	317	7	)	)	PUNCT
ejpam-6572	317	8	for	for	ADP
ejpam-6572	317	9	every	every	DET
ejpam-6572	317	10	(	(	PUNCT
ejpam-6572	317	11	σ1	σ1	PROPN
ejpam-6572	317	12	,	,	PUNCT
ejpam-6572	317	13	σ2)p	σ2)p	NOUN
ejpam-6572	317	14	-	-	PUNCT
ejpam-6572	317	15	open	open	NOUN
ejpam-6572	317	16	set	set	NOUN
ejpam-6572	317	17	v	v	NOUN
ejpam-6572	317	18	of	of	ADP
ejpam-6572	317	19	y	y	PROPN
ejpam-6572	317	20	.	.	PUNCT
ejpam-6572	318	1	proof	proof	NOUN
ejpam-6572	318	2	.	.	PUNCT
ejpam-6572	319	1	(	(	PUNCT
ejpam-6572	319	2	1	1	X
ejpam-6572	319	3	)	)	PUNCT
ejpam-6572	319	4	⇒	⇒	NOUN
ejpam-6572	319	5	(	(	PUNCT
ejpam-6572	319	6	2	2	NUM
ejpam-6572	319	7	):	):	PUNCT
ejpam-6572	319	8	let	let	VERB
ejpam-6572	319	9	v	v	PART
ejpam-6572	319	10	be	be	AUX
ejpam-6572	319	11	any	any	DET
ejpam-6572	319	12	(	(	PUNCT
ejpam-6572	319	13	σ1	σ1	PROPN
ejpam-6572	319	14	,	,	PUNCT
ejpam-6572	319	15	σ2)p	σ2)p	NOUN
ejpam-6572	319	16	-	-	PUNCT
ejpam-6572	319	17	open	open	ADJ
ejpam-6572	319	18	set	set	NOUN
ejpam-6572	319	19	of	of	ADP
ejpam-6572	319	20	y	y	PROPN
ejpam-6572	319	21	.	.	PUNCT
ejpam-6572	320	1	since	since	SCONJ
ejpam-6572	320	2	f	f	PROPN
ejpam-6572	320	3	is	be	AUX
ejpam-6572	320	4	weakly	weakly	ADJ
ejpam-6572	320	5	quasi	quasi	ADJ
ejpam-6572	320	6	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6572	320	7	,	,	PUNCT
ejpam-6572	320	8	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6572	320	9	,	,	PUNCT
ejpam-6572	320	10	by	by	ADP
ejpam-6572	320	11	theorem	theorem	NOUN
ejpam-6572	320	12	5	5	NUM
ejpam-6572	320	13	and	and	CCONJ
ejpam-6572	320	14	lemma	lemma	PROPN
ejpam-6572	320	15	4	4	NUM
ejpam-6572	320	16	int⋆(cl⋆(f−1(v	int⋆(cl⋆(f−1(v	NOUN
ejpam-6572	320	17	)	)	PUNCT
ejpam-6572	320	18	)	)	PUNCT
ejpam-6572	320	19	)	)	PUNCT
ejpam-6572	320	20	⊆	⊆	X
ejpam-6572	320	21	int⋆(cl⋆(f−1(σ1σ2	int⋆(cl⋆(f−1(σ1σ2	VERB
ejpam-6572	320	22	-	-	PUNCT
ejpam-6572	320	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	320	24	-	-	PUNCT
ejpam-6572	320	25	cl(v	cl(v	NOUN
ejpam-6572	320	26	)	)	PUNCT
ejpam-6572	320	27	)	)	PUNCT
ejpam-6572	320	28	)	)	PUNCT
ejpam-6572	320	29	)	)	PUNCT
ejpam-6572	320	30	)	)	PUNCT
ejpam-6572	320	31	⊆	⊆	NUM
ejpam-6572	320	32	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	320	33	-	-	PUNCT
ejpam-6572	320	34	cl(v	cl(v	NOUN
ejpam-6572	320	35	)	)	PUNCT
ejpam-6572	320	36	)	)	PUNCT
ejpam-6572	320	37	.	.	PUNCT
ejpam-6572	321	1	(	(	PUNCT
ejpam-6572	321	2	2	2	X
ejpam-6572	321	3	)	)	PUNCT
ejpam-6572	321	4	⇒	⇒	NOUN
ejpam-6572	321	5	(	(	PUNCT
ejpam-6572	321	6	3	3	NUM
ejpam-6572	321	7	):	):	PUNCT
ejpam-6572	321	8	let	let	VERB
ejpam-6572	321	9	v	v	PART
ejpam-6572	321	10	be	be	AUX
ejpam-6572	321	11	any	any	DET
ejpam-6572	321	12	(	(	PUNCT
ejpam-6572	321	13	σ1	σ1	PROPN
ejpam-6572	321	14	,	,	PUNCT
ejpam-6572	321	15	σ2)p	σ2)p	NOUN
ejpam-6572	321	16	-	-	PUNCT
ejpam-6572	321	17	open	open	ADJ
ejpam-6572	321	18	set	set	NOUN
ejpam-6572	321	19	of	of	ADP
ejpam-6572	321	20	y	y	PROPN
ejpam-6572	321	21	.	.	PUNCT
ejpam-6572	322	1	by	by	ADP
ejpam-6572	322	2	(	(	PUNCT
ejpam-6572	322	3	2	2	NUM
ejpam-6572	322	4	)	)	PUNCT
ejpam-6572	322	5	and	and	CCONJ
ejpam-6572	322	6	lemma	lemma	PROPN
ejpam-6572	322	7	4	4	NUM
ejpam-6572	322	8	,	,	PUNCT
ejpam-6572	322	9	we	we	PRON
ejpam-6572	322	10	have	have	VERB
ejpam-6572	322	11	scl⋆(f−1(v	scl⋆(f−1(v	PROPN
ejpam-6572	322	12	)	)	PUNCT
ejpam-6572	322	13	)	)	PUNCT
ejpam-6572	323	1	=	=	SYM
ejpam-6572	323	2	f−1(v	f−1(v	PROPN
ejpam-6572	323	3	)	)	PUNCT
ejpam-6572	323	4	∪	∪	VERB
ejpam-6572	323	5	int⋆(cl⋆(f−1(v	int⋆(cl⋆(f−1(v	PROPN
ejpam-6572	323	6	)	)	PUNCT
ejpam-6572	323	7	)	)	PUNCT
ejpam-6572	323	8	)	)	PUNCT
ejpam-6572	324	1	⊆	⊆	NUM
ejpam-6572	324	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	324	3	-	-	PUNCT
ejpam-6572	324	4	cl(v	cl(v	NOUN
ejpam-6572	324	5	)	)	PUNCT
ejpam-6572	324	6	)	)	PUNCT
ejpam-6572	324	7	.	.	PUNCT
ejpam-6572	325	1	(	(	PUNCT
ejpam-6572	325	2	3	3	X
ejpam-6572	325	3	)	)	PUNCT
ejpam-6572	325	4	⇒	⇒	NOUN
ejpam-6572	325	5	(	(	PUNCT
ejpam-6572	325	6	4	4	NUM
ejpam-6572	325	7	):	):	PUNCT
ejpam-6572	325	8	let	let	VERB
ejpam-6572	325	9	v	v	PART
ejpam-6572	325	10	be	be	AUX
ejpam-6572	325	11	any	any	DET
ejpam-6572	325	12	(	(	PUNCT
ejpam-6572	325	13	σ1	σ1	PROPN
ejpam-6572	325	14	,	,	PUNCT
ejpam-6572	325	15	σ2)p	σ2)p	NOUN
ejpam-6572	325	16	-	-	PUNCT
ejpam-6572	325	17	open	open	ADJ
ejpam-6572	325	18	set	set	NOUN
ejpam-6572	325	19	of	of	ADP
ejpam-6572	325	20	y	y	PROPN
ejpam-6572	325	21	.	.	PUNCT
ejpam-6572	326	1	thus	thus	ADV
ejpam-6572	326	2	by	by	ADP
ejpam-6572	326	3	(	(	PUNCT
ejpam-6572	326	4	3	3	NUM
ejpam-6572	326	5	)	)	PUNCT
ejpam-6572	326	6	,	,	PUNCT
ejpam-6572	326	7	x	x	PUNCT
ejpam-6572	326	8	−	−	NOUN
ejpam-6572	326	9	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	326	10	-	-	PUNCT
ejpam-6572	326	11	cl(v	cl(v	NOUN
ejpam-6572	326	12	)	)	PUNCT
ejpam-6572	326	13	)	)	PUNCT
ejpam-6572	326	14	)	)	PUNCT
ejpam-6572	327	1	=	=	PRON
ejpam-6572	327	2	scl⋆(x	scl⋆(x	NOUN
ejpam-6572	327	3	−	−	PROPN
ejpam-6572	327	4	(	(	PUNCT
ejpam-6572	327	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	327	6	-	-	PUNCT
ejpam-6572	327	7	cl(v	cl(v	NOUN
ejpam-6572	327	8	)	)	PUNCT
ejpam-6572	327	9	)	)	PUNCT
ejpam-6572	327	10	)	)	PUNCT
ejpam-6572	327	11	)	)	PUNCT
ejpam-6572	328	1	=	=	PRON
ejpam-6572	328	2	scl⋆(x	scl⋆(x	NOUN
ejpam-6572	328	3	−	−	NOUN
ejpam-6572	328	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	328	5	-	-	PUNCT
ejpam-6572	328	6	cl(v	cl(v	NOUN
ejpam-6572	328	7	)	)	PUNCT
ejpam-6572	328	8	)	)	PUNCT
ejpam-6572	328	9	)	)	PUNCT
ejpam-6572	329	1	=	=	SYM
ejpam-6572	329	2	scl⋆(f−1(y	scl⋆(f−1(y	PROPN
ejpam-6572	329	3	−	−	NUM
ejpam-6572	329	4	σ1σ2	σ1σ2	NOUN
ejpam-6572	329	5	-	-	NUM
ejpam-6572	329	6	cl(v	cl(v	NOUN
ejpam-6572	329	7	)	)	PUNCT
ejpam-6572	329	8	)	)	PUNCT
ejpam-6572	329	9	)	)	PUNCT
ejpam-6572	330	1	⊆	⊆	NUM
ejpam-6572	330	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6572	330	3	-	-	PUNCT
ejpam-6572	330	4	cl(y	cl(y	NOUN
ejpam-6572	330	5	−	−	NOUN
ejpam-6572	330	6	σ1σ2	σ1σ2	NOUN
ejpam-6572	330	7	-	-	NUM
ejpam-6572	330	8	cl(v	cl(v	NOUN
ejpam-6572	330	9	)	)	PUNCT
ejpam-6572	330	10	)	)	PUNCT
ejpam-6572	330	11	)	)	PUNCT
ejpam-6572	331	1	=	=	PUNCT
ejpam-6572	331	2	x	x	X
ejpam-6572	331	3	−	−	PRON
ejpam-6572	331	4	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6572	331	5	-	-	PUNCT
ejpam-6572	331	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6572	331	7	-	-	PUNCT
ejpam-6572	331	8	cl(v	cl(v	NOUN
ejpam-6572	331	9	)	)	PUNCT
ejpam-6572	331	10	)	)	PUNCT
ejpam-6572	331	11	)	)	PUNCT
ejpam-6572	332	1	⊆	⊆	NUM
ejpam-6572	332	2	x	x	SYM
ejpam-6572	332	3	−	−	PROPN
ejpam-6572	332	4	f−1(v	f−1(v	PROPN
ejpam-6572	332	5	)	)	PUNCT
ejpam-6572	332	6	and	and	CCONJ
ejpam-6572	332	7	hence	hence	ADV
ejpam-6572	332	8	f−1(v	f−1(v	NOUN
ejpam-6572	332	9	)	)	PUNCT
ejpam-6572	332	10	⊆	⊆	NUM
ejpam-6572	332	11	sint⋆(f−1(σ1σ2	sint⋆(f−1(σ1σ2	NOUN
ejpam-6572	332	12	-	-	PUNCT
ejpam-6572	332	13	cl(v	cl(v	NOUN
ejpam-6572	332	14	)	)	PUNCT
ejpam-6572	332	15	)	)	PUNCT
ejpam-6572	332	16	)	)	PUNCT
ejpam-6572	332	17	.	.	PUNCT
ejpam-6572	333	1	(	(	PUNCT
ejpam-6572	333	2	4	4	X
ejpam-6572	333	3	)	)	PUNCT
ejpam-6572	333	4	⇒	⇒	NOUN
ejpam-6572	333	5	(	(	PUNCT
ejpam-6572	333	6	1	1	NUM
ejpam-6572	333	7	):	):	PUNCT
ejpam-6572	333	8	since	since	SCONJ
ejpam-6572	333	9	every	every	DET
ejpam-6572	333	10	σ1σ2	σ1σ2	NUM
ejpam-6572	333	11	-	-	ADJ
ejpam-6572	333	12	open	open	ADJ
ejpam-6572	333	13	set	set	NOUN
ejpam-6572	333	14	is	be	AUX
ejpam-6572	333	15	(	(	PUNCT
ejpam-6572	333	16	σ1	σ1	PROPN
ejpam-6572	333	17	,	,	PUNCT
ejpam-6572	333	18	σ2)p	σ2)p	NOUN
ejpam-6572	333	19	-	-	PUNCT
ejpam-6572	333	20	open	open	ADJ
ejpam-6572	333	21	,	,	PUNCT
ejpam-6572	333	22	this	this	PRON
ejpam-6572	333	23	follows	follow	VERB
ejpam-6572	333	24	from	from	ADP
ejpam-6572	333	25	theorem	theorem	ADJ
ejpam-6572	333	26	4	4	NUM
ejpam-6572	333	27	.	.	PUNCT
ejpam-6572	333	28	b.	b.	PROPN
ejpam-6572	333	29	kong	kong	PROPN
ejpam-6572	333	30	-	-	PUNCT
ejpam-6572	333	31	ied	ied	PROPN
ejpam-6572	333	32	,	,	PUNCT
ejpam-6572	333	33	a.	a.	PROPN
ejpam-6572	333	34	sama	sama	PROPN
ejpam-6572	333	35	-	-	PUNCT
ejpam-6572	333	36	ae	ae	PROPN
ejpam-6572	333	37	,	,	PUNCT
ejpam-6572	333	38	c.	c.	PROPN
ejpam-6572	333	39	boonpok	boonpok	PROPN
ejpam-6572	333	40	/	/	SYM
ejpam-6572	333	41	eur	eur	PROPN
ejpam-6572	333	42	.	.	PUNCT
ejpam-6572	334	1	j.	j.	PROPN
ejpam-6572	334	2	pure	pure	PROPN
ejpam-6572	334	3	appl	appl	PROPN
ejpam-6572	334	4	.	.	PROPN
ejpam-6572	334	5	math	math	PROPN
ejpam-6572	334	6	,	,	PUNCT
ejpam-6572	334	7	18	18	NUM
ejpam-6572	334	8	(	(	PUNCT
ejpam-6572	334	9	3	3	NUM
ejpam-6572	334	10	)	)	PUNCT
ejpam-6572	334	11	(	(	PUNCT
ejpam-6572	334	12	2025	2025	NUM
ejpam-6572	334	13	)	)	PUNCT
ejpam-6572	334	14	,	,	PUNCT
ejpam-6572	334	15	6572	6572	NUM
ejpam-6572	334	16	11	11	NUM
ejpam-6572	334	17	of	of	ADP
ejpam-6572	334	18	12	12	NUM
ejpam-6572	334	19	acknowledgements	acknowledgement	NOUN
ejpam-6572	334	20	this	this	DET
ejpam-6572	334	21	research	research	NOUN
ejpam-6572	334	22	project	project	NOUN
ejpam-6572	334	23	was	be	AUX
ejpam-6572	334	24	financially	financially	ADV
ejpam-6572	334	25	supported	support	VERB
ejpam-6572	334	26	by	by	ADP
ejpam-6572	334	27	mahasarakham	mahasarakham	PROPN
ejpam-6572	334	28	university	university	PROPN
ejpam-6572	334	29	.	.	PUNCT
ejpam-6572	335	1	references	reference	NOUN
ejpam-6572	335	2	[	[	X
ejpam-6572	335	3	1	1	X
ejpam-6572	335	4	]	]	PUNCT
ejpam-6572	335	5	s.	s.	PROPN
ejpam-6572	335	6	marcus	marcus	PROPN
ejpam-6572	335	7	.	.	PUNCT
ejpam-6572	336	1	sur	sur	PROPN
ejpam-6572	336	2	les	les	PROPN
ejpam-6572	336	3	fonctions	fonctions	PROPN
ejpam-6572	336	4	quasicontinues	quasicontinue	NOUN
ejpam-6572	336	5	au	au	ADP
ejpam-6572	336	6	sense	sense	NOUN
ejpam-6572	336	7	de	de	X
ejpam-6572	336	8	s.	s.	PROPN
ejpam-6572	336	9	kempisty	kempisty	PROPN
ejpam-6572	336	10	.	.	PUNCT
ejpam-6572	337	1	colloquium	colloquium	NOUN
ejpam-6572	337	2	mathematicum	mathematicum	PROPN
ejpam-6572	337	3	,	,	PUNCT
ejpam-6572	337	4	8:47–53	8:47–53	NUM
ejpam-6572	337	5	,	,	PUNCT
ejpam-6572	337	6	1961	1961	NUM
ejpam-6572	337	7	.	.	PUNCT
ejpam-6572	338	1	[	[	X
ejpam-6572	338	2	2	2	X
ejpam-6572	338	3	]	]	PUNCT
ejpam-6572	338	4	v.	v.	CCONJ
ejpam-6572	338	5	popa	popa	NOUN
ejpam-6572	338	6	.	.	PUNCT
ejpam-6572	339	1	on	on	ADP
ejpam-6572	339	2	the	the	DET
ejpam-6572	339	3	decomposition	decomposition	NOUN
ejpam-6572	339	4	of	of	ADP
ejpam-6572	339	5	the	the	DET
ejpam-6572	339	6	quasi	quasi	NOUN
ejpam-6572	339	7	-	-	NOUN
ejpam-6572	339	8	continuity	continuity	NOUN
ejpam-6572	339	9	in	in	ADP
ejpam-6572	339	10	topological	topological	ADJ
ejpam-6572	339	11	spaces	space	NOUN
ejpam-6572	339	12	(	(	PUNCT
ejpam-6572	339	13	romanian	romanian	ADJ
ejpam-6572	339	14	)	)	PUNCT
ejpam-6572	339	15	.	.	PUNCT
ejpam-6572	340	1	studii	studii	PROPN
ejpam-6572	340	2	şi	şi	PROPN
ejpam-6572	340	3	cercetǎri	cercetǎri	NOUN
ejpam-6572	340	4	de	de	X
ejpam-6572	340	5	matematicǎ	matematicǎ	NOUN
ejpam-6572	340	6	,	,	PUNCT
ejpam-6572	340	7	30:31–35	30:31–35	NUM
ejpam-6572	340	8	,	,	PUNCT
ejpam-6572	340	9	1978	1978	NUM
ejpam-6572	340	10	.	.	PUNCT
ejpam-6572	341	1	[	[	X
ejpam-6572	341	2	3	3	NUM
ejpam-6572	341	3	]	]	PUNCT
ejpam-6572	341	4	a.	a.	NOUN
ejpam-6572	341	5	neubrunnová.	neubrunnová.	PROPN
ejpam-6572	341	6	on	on	ADP
ejpam-6572	341	7	certain	certain	ADJ
ejpam-6572	341	8	generalizations	generalization	NOUN
ejpam-6572	341	9	of	of	ADP
ejpam-6572	341	10	the	the	DET
ejpam-6572	341	11	notion	notion	NOUN
ejpam-6572	341	12	of	of	ADP
ejpam-6572	341	13	continuity	continuity	NOUN
ejpam-6572	341	14	.	.	PUNCT
ejpam-6572	342	1	matematický	matematický	ADJ
ejpam-6572	342	2	c̆asopis	c̆asopis	PROPN
ejpam-6572	342	3	,	,	PUNCT
ejpam-6572	342	4	23:374–380	23:374–380	NUM
ejpam-6572	342	5	,	,	PUNCT
ejpam-6572	342	6	1973	1973	NUM
ejpam-6572	342	7	.	.	PUNCT
ejpam-6572	343	1	[	[	X
ejpam-6572	343	2	4	4	NUM
ejpam-6572	343	3	]	]	X
ejpam-6572	343	4	n.	n.	PROPN
ejpam-6572	343	5	levine	levine	PROPN
ejpam-6572	343	6	.	.	PUNCT
ejpam-6572	344	1	semi	semi	ADJ
ejpam-6572	344	2	-	-	ADJ
ejpam-6572	344	3	open	open	ADJ
ejpam-6572	344	4	sets	set	NOUN
ejpam-6572	344	5	and	and	CCONJ
ejpam-6572	344	6	semi	semi	ADJ
ejpam-6572	344	7	-	-	NOUN
ejpam-6572	344	8	continuity	continuity	NOUN
ejpam-6572	344	9	in	in	ADP
ejpam-6572	344	10	topological	topological	ADJ
ejpam-6572	344	11	spaces	space	NOUN
ejpam-6572	344	12	.	.	PUNCT
ejpam-6572	345	1	the	the	DET
ejpam-6572	345	2	american	american	PROPN
ejpam-6572	345	3	mathematical	mathematical	PROPN
ejpam-6572	345	4	monthly	monthly	ADV
ejpam-6572	345	5	,	,	PUNCT
ejpam-6572	345	6	70:36–41	70:36–41	NUM
ejpam-6572	345	7	,	,	PUNCT
ejpam-6572	345	8	1963	1963	NUM
ejpam-6572	345	9	.	.	PUNCT
ejpam-6572	346	1	[	[	X
ejpam-6572	346	2	5	5	X
ejpam-6572	346	3	]	]	PUNCT
ejpam-6572	346	4	v.	v.	CCONJ
ejpam-6572	346	5	popa	popa	NOUN
ejpam-6572	346	6	and	and	CCONJ
ejpam-6572	346	7	c.	c.	PROPN
ejpam-6572	346	8	stan	stan	PROPN
ejpam-6572	346	9	.	.	PUNCT
ejpam-6572	347	1	on	on	ADP
ejpam-6572	347	2	a	a	DET
ejpam-6572	347	3	decomposition	decomposition	NOUN
ejpam-6572	347	4	of	of	ADP
ejpam-6572	347	5	quasicontinuity	quasicontinuity	NOUN
ejpam-6572	347	6	in	in	ADP
ejpam-6572	347	7	topological	topological	ADJ
ejpam-6572	347	8	spaces	space	NOUN
ejpam-6572	347	9	.	.	PUNCT
ejpam-6572	348	1	studii	studii	PROPN
ejpam-6572	348	2	şi	şi	PROPN
ejpam-6572	348	3	cercetǎri	cercetǎri	PROPN
ejpam-6572	348	4	de	de	X
ejpam-6572	348	5	matematicǎ	matematicǎ	NOUN
ejpam-6572	348	6	,	,	PUNCT
ejpam-6572	348	7	25:41–43	25:41–43	NUM
ejpam-6572	348	8	,	,	PUNCT
ejpam-6572	348	9	1973	1973	NUM
ejpam-6572	348	10	.	.	PUNCT
ejpam-6572	349	1	[	[	X
ejpam-6572	349	2	6	6	NUM
ejpam-6572	349	3	]	]	X
ejpam-6572	349	4	n.	n.	PROPN
ejpam-6572	349	5	levine	levine	PROPN
ejpam-6572	349	6	.	.	PUNCT
ejpam-6572	350	1	a	a	DET
ejpam-6572	350	2	decomposition	decomposition	NOUN
ejpam-6572	350	3	of	of	ADP
ejpam-6572	350	4	continuity	continuity	NOUN
ejpam-6572	350	5	in	in	ADP
ejpam-6572	350	6	topological	topological	ADJ
ejpam-6572	350	7	spaces	space	NOUN
ejpam-6572	350	8	.	.	PUNCT
ejpam-6572	351	1	the	the	DET
ejpam-6572	351	2	american	american	PROPN
ejpam-6572	351	3	mathematical	mathematical	PROPN
ejpam-6572	351	4	monthly	monthly	ADV
ejpam-6572	351	5	,	,	PUNCT
ejpam-6572	351	6	68:44–46	68:44–46	NUM
ejpam-6572	351	7	,	,	PUNCT
ejpam-6572	351	8	1961	1961	NUM
ejpam-6572	351	9	.	.	PUNCT
ejpam-6572	352	1	[	[	X
ejpam-6572	352	2	7	7	X
ejpam-6572	352	3	]	]	PUNCT
ejpam-6572	352	4	t.	t.	PROPN
ejpam-6572	352	5	noiri	noiri	PROPN
ejpam-6572	352	6	.	.	PUNCT
ejpam-6572	353	1	properties	property	NOUN
ejpam-6572	353	2	of	of	ADP
ejpam-6572	353	3	some	some	DET
ejpam-6572	353	4	weak	weak	ADJ
ejpam-6572	353	5	forms	form	NOUN
ejpam-6572	353	6	of	of	ADP
ejpam-6572	353	7	continuity	continuity	NOUN
ejpam-6572	353	8	.	.	PUNCT
ejpam-6572	354	1	international	international	ADJ
ejpam-6572	354	2	journal	journal	PROPN
ejpam-6572	354	3	of	of	ADP
ejpam-6572	354	4	mathematics	mathematics	PROPN
ejpam-6572	354	5	and	and	CCONJ
ejpam-6572	354	6	mathematical	mathematical	ADJ
ejpam-6572	354	7	sciences	science	NOUN
ejpam-6572	354	8	,	,	PUNCT
ejpam-6572	354	9	10:97–111	10:97–111	NUM
ejpam-6572	354	10	,	,	PUNCT
ejpam-6572	354	11	1987	1987	NUM
ejpam-6572	354	12	.	.	PUNCT
ejpam-6572	355	1	[	[	X
ejpam-6572	355	2	8	8	NUM
ejpam-6572	355	3	]	]	PUNCT
ejpam-6572	355	4	s.	s.	PROPN
ejpam-6572	355	5	p.	p.	PROPN
ejpam-6572	355	6	arya	arya	PROPN
ejpam-6572	355	7	and	and	CCONJ
ejpam-6572	355	8	m.	m.	PROPN
ejpam-6572	355	9	p.	p.	PROPN
ejpam-6572	355	10	bhamini	bhamini	PROPN
ejpam-6572	355	11	.	.	PUNCT
ejpam-6572	356	1	some	some	DET
ejpam-6572	356	2	weaker	weak	ADJ
ejpam-6572	356	3	forms	form	NOUN
ejpam-6572	356	4	of	of	ADP
ejpam-6572	356	5	semi	semi	ADJ
ejpam-6572	356	6	-	-	ADJ
ejpam-6572	356	7	continuous	continuous	ADJ
ejpam-6572	356	8	functions	function	NOUN
ejpam-6572	356	9	.	.	PUNCT
ejpam-6572	357	1	ganita	ganita	NOUN
ejpam-6572	357	2	,	,	PUNCT
ejpam-6572	357	3	33:124–134	33:124–134	NUM
ejpam-6572	357	4	,	,	PUNCT
ejpam-6572	357	5	1982	1982	NUM
ejpam-6572	357	6	.	.	PUNCT
ejpam-6572	358	1	[	[	X
ejpam-6572	358	2	9	9	NUM
ejpam-6572	358	3	]	]	PUNCT
ejpam-6572	358	4	a.	a.	NOUN
ejpam-6572	358	5	kar	kar	PROPN
ejpam-6572	358	6	and	and	CCONJ
ejpam-6572	358	7	p.	p.	PROPN
ejpam-6572	358	8	bhattacharyya	bhattacharyya	PROPN
ejpam-6572	358	9	.	.	PUNCT
ejpam-6572	359	1	weakly	weakly	ADJ
ejpam-6572	359	2	semi	semi	ADJ
ejpam-6572	359	3	-	-	ADJ
ejpam-6572	359	4	continuous	continuous	ADJ
ejpam-6572	359	5	functions	function	NOUN
ejpam-6572	359	6	.	.	PUNCT
ejpam-6572	360	1	the	the	DET
ejpam-6572	360	2	journal	journal	NOUN
ejpam-6572	360	3	of	of	ADP
ejpam-6572	360	4	the	the	DET
ejpam-6572	360	5	indian	indian	PROPN
ejpam-6572	360	6	academy	academy	PROPN
ejpam-6572	360	7	of	of	ADP
ejpam-6572	360	8	mathematics	mathematics	PROPN
ejpam-6572	360	9	,	,	PUNCT
ejpam-6572	360	10	8:83–93	8:83–93	NUM
ejpam-6572	360	11	,	,	PUNCT
ejpam-6572	360	12	1986	1986	NUM
ejpam-6572	360	13	.	.	PUNCT
ejpam-6572	361	1	[	[	X
ejpam-6572	361	2	10	10	NUM
ejpam-6572	361	3	]	]	X
ejpam-6572	361	4	d.	d.	PROPN
ejpam-6572	361	5	janković	janković	PROPN
ejpam-6572	361	6	and	and	CCONJ
ejpam-6572	361	7	t.	t.	PROPN
ejpam-6572	361	8	r.	r.	PROPN
ejpam-6572	361	9	hamlett	hamlett	PROPN
ejpam-6572	361	10	.	.	PUNCT
ejpam-6572	362	1	new	new	ADJ
ejpam-6572	362	2	topologies	topology	NOUN
ejpam-6572	362	3	from	from	ADP
ejpam-6572	362	4	old	old	ADJ
ejpam-6572	362	5	via	via	ADP
ejpam-6572	362	6	ideals	ideal	NOUN
ejpam-6572	362	7	.	.	PUNCT
ejpam-6572	363	1	the	the	DET
ejpam-6572	363	2	american	american	PROPN
ejpam-6572	363	3	mathematical	mathematical	PROPN
ejpam-6572	363	4	monthly	monthly	ADV
ejpam-6572	363	5	,	,	PUNCT
ejpam-6572	363	6	97:295–310	97:295–310	PROPN
ejpam-6572	363	7	,	,	PUNCT
ejpam-6572	363	8	1990	1990	NUM
ejpam-6572	363	9	.	.	PUNCT
ejpam-6572	364	1	[	[	X
ejpam-6572	364	2	11	11	NUM
ejpam-6572	364	3	]	]	PUNCT
ejpam-6572	364	4	m.	m.	NOUN
ejpam-6572	364	5	e.	e.	PROPN
ejpam-6572	364	6	abd	abd	PROPN
ejpam-6572	364	7	el	el	PROPN
ejpam-6572	364	8	-	-	PROPN
ejpam-6572	364	9	monsef	monsef	PROPN
ejpam-6572	364	10	,	,	PUNCT
ejpam-6572	364	11	e.	e.	PROPN
ejpam-6572	364	12	f.	f.	PROPN
ejpam-6572	364	13	lashien	lashien	PROPN
ejpam-6572	364	14	,	,	PUNCT
ejpam-6572	364	15	and	and	CCONJ
ejpam-6572	364	16	a.	a.	NOUN
ejpam-6572	364	17	a.	a.	NOUN
ejpam-6572	364	18	nasef	nasef	PROPN
ejpam-6572	364	19	.	.	PUNCT
ejpam-6572	365	1	on	on	ADP
ejpam-6572	365	2	i	i	PRON
ejpam-6572	365	3	-open	-open	PROPN
ejpam-6572	365	4	sets	set	NOUN
ejpam-6572	365	5	and	and	CCONJ
ejpam-6572	365	6	i	i	PRON
ejpam-6572	365	7	continuous	continuous	ADJ
ejpam-6572	365	8	functions	function	NOUN
ejpam-6572	365	9	.	.	PUNCT
ejpam-6572	366	1	kyungpook	kyungpook	PROPN
ejpam-6572	366	2	mathematical	mathematical	PROPN
ejpam-6572	366	3	journal	journal	PROPN
ejpam-6572	366	4	,	,	PUNCT
ejpam-6572	366	5	32:21–30	32:21–30	NUM
ejpam-6572	366	6	,	,	PUNCT
ejpam-6572	366	7	1992	1992	NUM
ejpam-6572	366	8	.	.	PUNCT
ejpam-6572	367	1	[	[	X
ejpam-6572	367	2	12	12	NUM
ejpam-6572	367	3	]	]	X
ejpam-6572	367	4	e.	e.	PROPN
ejpam-6572	367	5	hatir	hatir	PROPN
ejpam-6572	367	6	and	and	CCONJ
ejpam-6572	367	7	t.	t.	PROPN
ejpam-6572	367	8	noiri	noiri	PROPN
ejpam-6572	367	9	.	.	PUNCT
ejpam-6572	368	1	weakly	weakly	ADJ
ejpam-6572	368	2	pre	pre	ADJ
ejpam-6572	368	3	-	-	ADJ
ejpam-6572	368	4	i	i	PRON
ejpam-6572	368	5	-	-	PUNCT
ejpam-6572	368	6	open	open	ADJ
ejpam-6572	368	7	sets	set	NOUN
ejpam-6572	368	8	and	and	CCONJ
ejpam-6572	368	9	decomposition	decomposition	NOUN
ejpam-6572	368	10	of	of	ADP
ejpam-6572	368	11	continuity	continuity	NOUN
ejpam-6572	368	12	.	.	PUNCT
ejpam-6572	369	1	acta	acta	PROPN
ejpam-6572	369	2	mathematica	mathematica	PROPN
ejpam-6572	369	3	hungarica	hungarica	PROPN
ejpam-6572	369	4	,	,	PUNCT
ejpam-6572	369	5	106(3):227–238	106(3):227–238	NUM
ejpam-6572	369	6	,	,	PUNCT
ejpam-6572	369	7	2005	2005	NUM
ejpam-6572	369	8	.	.	PUNCT
ejpam-6572	370	1	[	[	X
ejpam-6572	370	2	13	13	NUM
ejpam-6572	370	3	]	]	X
ejpam-6572	370	4	e.	e.	PROPN
ejpam-6572	370	5	hatir	hatir	PROPN
ejpam-6572	370	6	and	and	CCONJ
ejpam-6572	370	7	t.	t.	PROPN
ejpam-6572	370	8	noiri	noiri	PROPN
ejpam-6572	370	9	.	.	PUNCT
ejpam-6572	371	1	on	on	ADP
ejpam-6572	371	2	decompositions	decomposition	NOUN
ejpam-6572	371	3	of	of	ADP
ejpam-6572	371	4	continuity	continuity	NOUN
ejpam-6572	371	5	via	via	ADP
ejpam-6572	371	6	idealization	idealization	NOUN
ejpam-6572	371	7	.	.	PUNCT
ejpam-6572	372	1	acta	acta	PROPN
ejpam-6572	372	2	mathematica	mathematica	PROPN
ejpam-6572	372	3	hungarica	hungarica	PROPN
ejpam-6572	372	4	,	,	PUNCT
ejpam-6572	372	5	96:341–349	96:341–349	PROPN
ejpam-6572	372	6	,	,	PUNCT
ejpam-6572	372	7	2002	2002	NUM
ejpam-6572	372	8	.	.	PUNCT
ejpam-6572	373	1	[	[	X
ejpam-6572	373	2	14	14	NUM
ejpam-6572	373	3	]	]	X
ejpam-6572	373	4	c.	c.	PROPN
ejpam-6572	373	5	boonpok	boonpok	PROPN
ejpam-6572	373	6	.	.	PUNCT
ejpam-6572	374	1	pı	pı	NOUN
ejpam-6572	374	2	-	-	NOUN
ejpam-6572	374	3	continuity	continuity	NOUN
ejpam-6572	374	4	and	and	CCONJ
ejpam-6572	374	5	weak	weak	ADJ
ejpam-6572	374	6	pı	pı	NOUN
ejpam-6572	374	7	-	-	NOUN
ejpam-6572	374	8	continuity	continuity	NOUN
ejpam-6572	374	9	.	.	PUNCT
ejpam-6572	375	1	carpathian	carpathian	ADJ
ejpam-6572	375	2	mathematical	mathematical	ADJ
ejpam-6572	375	3	publications	publication	NOUN
ejpam-6572	375	4	,	,	PUNCT
ejpam-6572	375	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6572	375	6	,	,	PUNCT
ejpam-6572	375	7	2025	2025	NUM
ejpam-6572	375	8	.	.	PUNCT
ejpam-6572	376	1	[	[	X
ejpam-6572	376	2	15	15	NUM
ejpam-6572	376	3	]	]	X
ejpam-6572	376	4	c.	c.	PROPN
ejpam-6572	376	5	boonpok	boonpok	PROPN
ejpam-6572	376	6	and	and	CCONJ
ejpam-6572	376	7	n.	n.	PROPN
ejpam-6572	376	8	srisarakham	srisarakham	PROPN
ejpam-6572	376	9	.	.	PUNCT
ejpam-6572	377	1	(	(	PUNCT
ejpam-6572	377	2	τ1	τ1	NOUN
ejpam-6572	377	3	,	,	PUNCT
ejpam-6572	377	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6572	377	5	for	for	ADP
ejpam-6572	377	6	functions	function	NOUN
ejpam-6572	377	7	.	.	PUNCT
ejpam-6572	378	1	asia	asia	PROPN
ejpam-6572	378	2	pacific	pacific	PROPN
ejpam-6572	378	3	journal	journal	PROPN
ejpam-6572	378	4	of	of	ADP
ejpam-6572	378	5	mathematics	mathematic	NOUN
ejpam-6572	378	6	,	,	PUNCT
ejpam-6572	378	7	11:21	11:21	NUM
ejpam-6572	378	8	,	,	PUNCT
ejpam-6572	378	9	2024	2024	NUM
ejpam-6572	378	10	.	.	PUNCT
ejpam-6572	379	1	[	[	X
ejpam-6572	379	2	16	16	NUM
ejpam-6572	379	3	]	]	X
ejpam-6572	379	4	c.	c.	PROPN
ejpam-6572	379	5	boonpok	boonpok	PROPN
ejpam-6572	379	6	and	and	CCONJ
ejpam-6572	379	7	p.	p.	NOUN
ejpam-6572	379	8	pue	pue	NOUN
ejpam-6572	379	9	-	-	PUNCT
ejpam-6572	379	10	on	on	ADP
ejpam-6572	379	11	.	.	PUNCT
ejpam-6572	380	1	characterizations	characterization	NOUN
ejpam-6572	380	2	of	of	ADP
ejpam-6572	380	3	almost	almost	ADV
ejpam-6572	380	4	(	(	PUNCT
ejpam-6572	380	5	τ1	τ1	NOUN
ejpam-6572	380	6	,	,	PUNCT
ejpam-6572	380	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	380	8	multifunctions	multifunction	NOUN
ejpam-6572	380	9	.	.	PUNCT
ejpam-6572	381	1	international	international	ADJ
ejpam-6572	381	2	journal	journal	NOUN
ejpam-6572	381	3	of	of	ADP
ejpam-6572	381	4	analysis	analysis	NOUN
ejpam-6572	381	5	and	and	CCONJ
ejpam-6572	381	6	applications	application	NOUN
ejpam-6572	381	7	,	,	PUNCT
ejpam-6572	381	8	22:33	22:33	NUM
ejpam-6572	381	9	,	,	PUNCT
ejpam-6572	381	10	2024	2024	NUM
ejpam-6572	381	11	.	.	PUNCT
ejpam-6572	382	1	[	[	X
ejpam-6572	382	2	17	17	NUM
ejpam-6572	382	3	]	]	X
ejpam-6572	382	4	c.	c.	PROPN
ejpam-6572	382	5	boonpok	boonpok	PROPN
ejpam-6572	382	6	and	and	CCONJ
ejpam-6572	382	7	c.	c.	PROPN
ejpam-6572	382	8	khanarong	khanarong	PROPN
ejpam-6572	382	9	.	.	PUNCT
ejpam-6572	383	1	on	on	ADP
ejpam-6572	383	2	weakly	weakly	ADJ
ejpam-6572	383	3	(	(	PUNCT
ejpam-6572	383	4	τ1	τ1	NOUN
ejpam-6572	383	5	,	,	PUNCT
ejpam-6572	383	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	383	7	functions	function	NOUN
ejpam-6572	383	8	.	.	PUNCT
ejpam-6572	384	1	european	european	ADJ
ejpam-6572	384	2	journal	journal	PROPN
ejpam-6572	384	3	of	of	ADP
ejpam-6572	384	4	pure	pure	ADJ
ejpam-6572	384	5	and	and	CCONJ
ejpam-6572	384	6	applied	applied	ADJ
ejpam-6572	384	7	mathematics	mathematic	NOUN
ejpam-6572	384	8	,	,	PUNCT
ejpam-6572	384	9	17(1):416–425	17(1):416–425	NUM
ejpam-6572	384	10	,	,	PUNCT
ejpam-6572	384	11	2024	2024	NUM
ejpam-6572	384	12	.	.	PUNCT
ejpam-6572	385	1	[	[	X
ejpam-6572	385	2	18	18	NUM
ejpam-6572	385	3	]	]	X
ejpam-6572	385	4	b.	b.	PROPN
ejpam-6572	385	5	kong	kong	PROPN
ejpam-6572	385	6	-	-	PUNCT
ejpam-6572	385	7	ied	ied	PROPN
ejpam-6572	385	8	,	,	PUNCT
ejpam-6572	385	9	s.	s.	PROPN
ejpam-6572	385	10	sompong	sompong	PROPN
ejpam-6572	385	11	,	,	PUNCT
ejpam-6572	385	12	and	and	CCONJ
ejpam-6572	385	13	c.	c.	PROPN
ejpam-6572	385	14	boonpok	boonpok	PROPN
ejpam-6572	385	15	.	.	PUNCT
ejpam-6572	386	1	almost	almost	ADV
ejpam-6572	386	2	quasi	quasi	X
ejpam-6572	386	3	(	(	PUNCT
ejpam-6572	386	4	τ1	τ1	NOUN
ejpam-6572	386	5	,	,	PUNCT
ejpam-6572	386	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	386	7	functions	function	NOUN
ejpam-6572	386	8	.	.	PUNCT
ejpam-6572	387	1	asia	asia	PROPN
ejpam-6572	387	2	pacific	pacific	PROPN
ejpam-6572	387	3	journal	journal	PROPN
ejpam-6572	387	4	of	of	ADP
ejpam-6572	387	5	mathematics	mathematic	NOUN
ejpam-6572	387	6	,	,	PUNCT
ejpam-6572	387	7	11:64	11:64	NUM
ejpam-6572	387	8	,	,	PUNCT
ejpam-6572	387	9	2024	2024	NUM
ejpam-6572	387	10	.	.	PUNCT
ejpam-6572	388	1	b.	b.	PROPN
ejpam-6572	388	2	kong	kong	PROPN
ejpam-6572	388	3	-	-	PUNCT
ejpam-6572	388	4	ied	ied	PROPN
ejpam-6572	388	5	,	,	PUNCT
ejpam-6572	388	6	a.	a.	PROPN
ejpam-6572	388	7	sama	sama	PROPN
ejpam-6572	388	8	-	-	PUNCT
ejpam-6572	388	9	ae	ae	PROPN
ejpam-6572	388	10	,	,	PUNCT
ejpam-6572	388	11	c.	c.	PROPN
ejpam-6572	388	12	boonpok	boonpok	PROPN
ejpam-6572	388	13	/	/	SYM
ejpam-6572	388	14	eur	eur	PROPN
ejpam-6572	388	15	.	.	PUNCT
ejpam-6572	389	1	j.	j.	PROPN
ejpam-6572	389	2	pure	pure	PROPN
ejpam-6572	389	3	appl	appl	PROPN
ejpam-6572	389	4	.	.	PROPN
ejpam-6572	389	5	math	math	PROPN
ejpam-6572	389	6	,	,	PUNCT
ejpam-6572	389	7	18	18	NUM
ejpam-6572	389	8	(	(	PUNCT
ejpam-6572	389	9	3	3	NUM
ejpam-6572	389	10	)	)	PUNCT
ejpam-6572	389	11	(	(	PUNCT
ejpam-6572	389	12	2025	2025	NUM
ejpam-6572	389	13	)	)	PUNCT
ejpam-6572	389	14	,	,	PUNCT
ejpam-6572	389	15	6572	6572	NUM
ejpam-6572	389	16	12	12	NUM
ejpam-6572	389	17	of	of	ADP
ejpam-6572	389	18	12	12	NUM
ejpam-6572	389	19	[	[	SYM
ejpam-6572	389	20	19	19	NUM
ejpam-6572	389	21	]	]	PUNCT
ejpam-6572	389	22	m.	m.	NOUN
ejpam-6572	389	23	chiangpradit	chiangpradit	NOUN
ejpam-6572	389	24	,	,	PUNCT
ejpam-6572	389	25	s.	s.	PROPN
ejpam-6572	389	26	sompong	sompong	PROPN
ejpam-6572	389	27	,	,	PUNCT
ejpam-6572	389	28	and	and	CCONJ
ejpam-6572	389	29	c.	c.	PROPN
ejpam-6572	389	30	boonpok	boonpok	PROPN
ejpam-6572	389	31	.	.	PUNCT
ejpam-6572	390	1	weakly	weakly	ADJ
ejpam-6572	390	2	quasi	quasi	NOUN
ejpam-6572	390	3	(	(	PUNCT
ejpam-6572	390	4	τ1	τ1	PROPN
ejpam-6572	390	5	,	,	PUNCT
ejpam-6572	390	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6572	390	7	functions	function	NOUN
ejpam-6572	390	8	.	.	PUNCT
ejpam-6572	391	1	international	international	ADJ
ejpam-6572	391	2	journal	journal	NOUN
ejpam-6572	391	3	of	of	ADP
ejpam-6572	391	4	analysis	analysis	NOUN
ejpam-6572	391	5	and	and	CCONJ
ejpam-6572	391	6	applications	application	NOUN
ejpam-6572	391	7	,	,	PUNCT
ejpam-6572	391	8	22:125	22:125	NUM
ejpam-6572	391	9	,	,	PUNCT
ejpam-6572	391	10	2024	2024	NUM
ejpam-6572	391	11	.	.	PUNCT
ejpam-6572	392	1	[	[	X
ejpam-6572	392	2	20	20	NUM
ejpam-6572	392	3	]	]	PUNCT
ejpam-6572	392	4	c.	c.	PROPN
ejpam-6572	392	5	boonpok	boonpok	PROPN
ejpam-6572	392	6	,	,	PUNCT
ejpam-6572	392	7	c.	c.	PROPN
ejpam-6572	392	8	viriyapong	viriyapong	PROPN
ejpam-6572	392	9	,	,	PUNCT
ejpam-6572	392	10	and	and	CCONJ
ejpam-6572	392	11	m.	m.	NOUN
ejpam-6572	392	12	thongmoon	thongmoon	NOUN
ejpam-6572	392	13	.	.	PUNCT
ejpam-6572	393	1	on	on	ADP
ejpam-6572	393	2	upper	upper	ADJ
ejpam-6572	393	3	and	and	CCONJ
ejpam-6572	393	4	lower	low	ADJ
ejpam-6572	393	5	(	(	PUNCT
ejpam-6572	393	6	τ1	τ1	NOUN
ejpam-6572	393	7	,	,	PUNCT
ejpam-6572	393	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6572	393	9	multifunctions	multifunction	NOUN
ejpam-6572	393	10	.	.	PUNCT
ejpam-6572	394	1	journal	journal	PROPN
ejpam-6572	394	2	of	of	ADP
ejpam-6572	394	3	mathematics	mathematics	PROPN
ejpam-6572	394	4	and	and	CCONJ
ejpam-6572	394	5	computer	computer	NOUN
ejpam-6572	394	6	science	science	NOUN
ejpam-6572	394	7	,	,	PUNCT
ejpam-6572	394	8	18:282–293	18:282–293	NUM
ejpam-6572	394	9	,	,	PUNCT
ejpam-6572	394	10	2018	2018	NUM
ejpam-6572	394	11	.	.	PUNCT
ejpam-6572	395	1	[	[	X
ejpam-6572	395	2	21	21	NUM
ejpam-6572	395	3	]	]	X
ejpam-6572	395	4	c.	c.	PROPN
ejpam-6572	395	5	viriyapong	viriyapong	PROPN
ejpam-6572	395	6	and	and	CCONJ
ejpam-6572	395	7	c.	c.	PROPN
ejpam-6572	395	8	boonpok	boonpok	PROPN
ejpam-6572	395	9	.	.	PUNCT
ejpam-6572	396	1	(	(	PUNCT
ejpam-6572	396	2	τ1	τ1	NOUN
ejpam-6572	396	3	,	,	PUNCT
ejpam-6572	396	4	τ2)α	τ2)α	NOUN
ejpam-6572	396	5	-	-	PUNCT
ejpam-6572	396	6	continuity	continuity	NOUN
ejpam-6572	396	7	for	for	ADP
ejpam-6572	396	8	multifunctions	multifunction	NOUN
ejpam-6572	396	9	.	.	PUNCT
ejpam-6572	397	1	journal	journal	PROPN
ejpam-6572	397	2	of	of	ADP
ejpam-6572	397	3	mathematics	mathematic	NOUN
ejpam-6572	397	4	,	,	PUNCT
ejpam-6572	397	5	2020:6285763	2020:6285763	NUM
ejpam-6572	397	6	,	,	PUNCT
ejpam-6572	397	7	2020	2020	NUM
ejpam-6572	397	8	.	.	PUNCT
ejpam-6572	398	1	[	[	X
ejpam-6572	398	2	22	22	NUM
ejpam-6572	398	3	]	]	PUNCT
ejpam-6572	398	4	c.	c.	PROPN
ejpam-6572	398	5	boonpok	boonpok	PROPN
ejpam-6572	398	6	.	.	PUNCT
ejpam-6572	399	1	(	(	PUNCT
ejpam-6572	399	2	τ1	τ1	NOUN
ejpam-6572	399	3	,	,	PUNCT
ejpam-6572	399	4	τ2)δ	τ2)δ	ADJ
ejpam-6572	399	5	-	-	PUNCT
ejpam-6572	399	6	semicontinuous	semicontinuous	ADJ
ejpam-6572	399	7	multifunctions	multifunction	NOUN
ejpam-6572	399	8	.	.	PUNCT
ejpam-6572	400	1	heliyon	heliyon	NOUN
ejpam-6572	400	2	,	,	PUNCT
ejpam-6572	400	3	6	6	NUM
ejpam-6572	400	4	:	:	SYM
ejpam-6572	400	5	e05367	e05367	PROPN
ejpam-6572	400	6	,	,	PUNCT
ejpam-6572	400	7	2020	2020	NUM
ejpam-6572	400	8	.	.	PUNCT
ejpam-6572	401	1	[	[	X
ejpam-6572	401	2	23	23	NUM
ejpam-6572	401	3	]	]	X
ejpam-6572	401	4	n.	n.	PROPN
ejpam-6572	401	5	viriyapong	viriyapong	PROPN
ejpam-6572	401	6	,	,	PUNCT
ejpam-6572	401	7	s.	s.	PROPN
ejpam-6572	401	8	sompong	sompong	PROPN
ejpam-6572	401	9	,	,	PUNCT
ejpam-6572	401	10	and	and	CCONJ
ejpam-6572	401	11	c.	c.	PROPN
ejpam-6572	401	12	boonpok	boonpok	PROPN
ejpam-6572	401	13	.	.	PUNCT
ejpam-6572	402	1	(	(	PUNCT
ejpam-6572	402	2	τ1	τ1	NOUN
ejpam-6572	402	3	,	,	PUNCT
ejpam-6572	402	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6572	402	5	disconnectedness	disconnectedness	NOUN
ejpam-6572	402	6	in	in	ADP
ejpam-6572	402	7	bitopological	bitopological	ADJ
ejpam-6572	402	8	spaces	space	NOUN
ejpam-6572	402	9	.	.	PUNCT
ejpam-6572	403	1	international	international	ADJ
ejpam-6572	403	2	journal	journal	PROPN
ejpam-6572	403	3	of	of	ADP
ejpam-6572	403	4	mathematics	mathematic	NOUN
ejpam-6572	403	5	and	and	CCONJ
ejpam-6572	403	6	computer	computer	NOUN
ejpam-6572	403	7	science	science	NOUN
ejpam-6572	403	8	,	,	PUNCT
ejpam-6572	403	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6572	403	10	,	,	PUNCT
ejpam-6572	403	11	2024	2024	NUM
ejpam-6572	403	12	.	.	PUNCT
ejpam-6572	404	1	[	[	X
ejpam-6572	404	2	24	24	NUM
ejpam-6572	404	3	]	]	X
ejpam-6572	404	4	n.	n.	PROPN
ejpam-6572	404	5	viriyapong	viriyapong	PROPN
ejpam-6572	404	6	,	,	PUNCT
ejpam-6572	404	7	s.	s.	PROPN
ejpam-6572	404	8	sompong	sompong	PROPN
ejpam-6572	404	9	,	,	PUNCT
ejpam-6572	404	10	and	and	CCONJ
ejpam-6572	404	11	c.	c.	PROPN
ejpam-6572	404	12	boonpok	boonpok	PROPN
ejpam-6572	404	13	.	.	PUNCT
ejpam-6572	405	1	upper	upper	ADJ
ejpam-6572	405	2	and	and	CCONJ
ejpam-6572	405	3	lower	low	ADJ
ejpam-6572	405	4	s-(τ1	s-(τ1	NOUN
ejpam-6572	405	5	,	,	PUNCT
ejpam-6572	405	6	τ2)p	τ2)p	ADJ
ejpam-6572	405	7	-	-	PUNCT
ejpam-6572	405	8	continuous	continuous	ADJ
ejpam-6572	405	9	multifunctions	multifunction	NOUN
ejpam-6572	405	10	.	.	PUNCT
ejpam-6572	406	1	european	european	ADJ
ejpam-6572	406	2	journal	journal	PROPN
ejpam-6572	406	3	of	of	ADP
ejpam-6572	406	4	pure	pure	ADJ
ejpam-6572	406	5	and	and	CCONJ
ejpam-6572	406	6	applied	applied	ADJ
ejpam-6572	406	7	mathematics	mathematic	NOUN
ejpam-6572	406	8	,	,	PUNCT
ejpam-6572	406	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-6572	406	10	,	,	PUNCT
ejpam-6572	406	11	2024	2024	NUM
ejpam-6572	406	12	.	.	PUNCT
ejpam-6572	407	1	[	[	X
ejpam-6572	407	2	25	25	NUM
ejpam-6572	407	3	]	]	X
ejpam-6572	407	4	c.	c.	PROPN
ejpam-6572	407	5	viriyapong	viriyapong	PROPN
ejpam-6572	407	6	,	,	PUNCT
ejpam-6572	407	7	s.	s.	PROPN
ejpam-6572	407	8	sompong	sompong	PROPN
ejpam-6572	407	9	,	,	PUNCT
ejpam-6572	407	10	and	and	CCONJ
ejpam-6572	407	11	c.	c.	PROPN
ejpam-6572	407	12	boonpok	boonpok	PROPN
ejpam-6572	407	13	.	.	PUNCT
ejpam-6572	408	1	upper	upper	ADJ
ejpam-6572	408	2	and	and	CCONJ
ejpam-6572	408	3	lower	low	ADJ
ejpam-6572	408	4	slight	slight	ADJ
ejpam-6572	408	5	α(τ1	α(τ1	NOUN
ejpam-6572	408	6	,	,	PUNCT
ejpam-6572	408	7	τ2)continuity	τ2)continuity	PROPN
ejpam-6572	408	8	.	.	PUNCT
ejpam-6572	409	1	european	european	PROPN
ejpam-6572	409	2	journal	journal	PROPN
ejpam-6572	409	3	of	of	ADP
ejpam-6572	409	4	pure	pure	ADJ
ejpam-6572	409	5	and	and	CCONJ
ejpam-6572	409	6	applied	applied	ADJ
ejpam-6572	409	7	mathematics	mathematic	NOUN
ejpam-6572	409	8	,	,	PUNCT
ejpam-6572	409	9	17(3):2142–2154	17(3):2142–2154	NUM
ejpam-6572	409	10	,	,	PUNCT
ejpam-6572	409	11	2024	2024	NUM
ejpam-6572	409	12	.	.	PUNCT
ejpam-6572	410	1	[	[	X
ejpam-6572	410	2	26	26	NUM
ejpam-6572	410	3	]	]	PUNCT
ejpam-6572	410	4	k.	k.	PROPN
ejpam-6572	410	5	kuratowski	kuratowski	PROPN
ejpam-6572	410	6	.	.	PUNCT
ejpam-6572	411	1	topology	topology	PROPN
ejpam-6572	411	2	,	,	PUNCT
ejpam-6572	411	3	vol	vol	NOUN
ejpam-6572	411	4	.	.	PUNCT
ejpam-6572	411	5	i.	i.	PROPN
ejpam-6572	411	6	academic	academic	PROPN
ejpam-6572	411	7	press	press	PROPN
ejpam-6572	411	8	,	,	PUNCT
ejpam-6572	411	9	new	new	PROPN
ejpam-6572	411	10	york	york	PROPN
ejpam-6572	411	11	,	,	PUNCT
ejpam-6572	411	12	1966	1966	NUM
ejpam-6572	411	13	.	.	PUNCT
ejpam-6572	412	1	[	[	X
ejpam-6572	412	2	27	27	NUM
ejpam-6572	412	3	]	]	X
ejpam-6572	412	4	e.	e.	PROPN
ejpam-6572	412	5	ekici	ekici	PROPN
ejpam-6572	412	6	and	and	CCONJ
ejpam-6572	412	7	t.	t.	PROPN
ejpam-6572	412	8	noiri	noiri	PROPN
ejpam-6572	412	9	.	.	PUNCT
ejpam-6572	413	1	⋆-extremally	⋆-extremally	ADV
ejpam-6572	413	2	disconnected	disconnect	VERB
ejpam-6572	413	3	ideal	ideal	ADJ
ejpam-6572	413	4	topological	topological	ADJ
ejpam-6572	413	5	spaces	space	NOUN
ejpam-6572	413	6	.	.	PUNCT
ejpam-6572	414	1	acta	acta	PROPN
ejpam-6572	414	2	mathematica	mathematica	PROPN
ejpam-6572	414	3	hungarica	hungarica	PROPN
ejpam-6572	414	4	,	,	PUNCT
ejpam-6572	414	5	122:81–90	122:81–90	NUM
ejpam-6572	414	6	,	,	PUNCT
ejpam-6572	414	7	2009	2009	NUM
ejpam-6572	414	8	.	.	PUNCT
ejpam-6572	415	1	[	[	X
ejpam-6572	415	2	28	28	NUM
ejpam-6572	415	3	]	]	X
ejpam-6572	415	4	c.	c.	PROPN
ejpam-6572	415	5	boonpok	boonpok	PROPN
ejpam-6572	415	6	.	.	PUNCT
ejpam-6572	416	1	weak	weak	ADJ
ejpam-6572	416	2	quasi	quasi	ADJ
ejpam-6572	416	3	continuity	continuity	NOUN
ejpam-6572	416	4	for	for	ADP
ejpam-6572	416	5	multifunctions	multifunction	NOUN
ejpam-6572	416	6	in	in	ADP
ejpam-6572	416	7	ideal	ideal	ADJ
ejpam-6572	416	8	topological	topological	ADJ
ejpam-6572	416	9	spaces	space	NOUN
ejpam-6572	416	10	.	.	PUNCT
ejpam-6572	417	1	advances	advance	NOUN
ejpam-6572	417	2	in	in	ADP
ejpam-6572	417	3	mathematics	mathematic	NOUN
ejpam-6572	417	4	:	:	PUNCT
ejpam-6572	417	5	scientific	scientific	ADJ
ejpam-6572	417	6	journal	journal	NOUN
ejpam-6572	417	7	,	,	PUNCT
ejpam-6572	417	8	9(1):339–355	9(1):339–355	NUM
ejpam-6572	417	9	,	,	PUNCT
ejpam-6572	417	10	2020	2020	NUM
ejpam-6572	417	11	.	.	PUNCT