id	sid	tid	token	lemma	pos
ejpam-6573	1	1	european	european	PROPN
ejpam-6573	1	2	journal	journal	PROPN
ejpam-6573	1	3	of	of	ADP
ejpam-6573	1	4	pure	pure	ADJ
ejpam-6573	1	5	and	and	CCONJ
ejpam-6573	1	6	applied	applied	ADJ
ejpam-6573	1	7	mathematics	mathematic	NOUN
ejpam-6573	1	8	2025	2025	NUM
ejpam-6573	1	9	,	,	PUNCT
ejpam-6573	1	10	vol	vol	NOUN
ejpam-6573	1	11	.	.	PROPN
ejpam-6573	1	12	18	18	NUM
ejpam-6573	1	13	,	,	PUNCT
ejpam-6573	1	14	issue	issue	NOUN
ejpam-6573	1	15	3	3	NUM
ejpam-6573	1	16	,	,	PUNCT
ejpam-6573	1	17	article	article	NOUN
ejpam-6573	1	18	number	number	NOUN
ejpam-6573	1	19	6573	6573	NUM
ejpam-6573	1	20	issn	issn	VERB
ejpam-6573	1	21	1307	1307	NUM
ejpam-6573	1	22	-	-	SYM
ejpam-6573	1	23	5543	5543	NUM
ejpam-6573	1	24	–	–	PUNCT
ejpam-6573	1	25	ejpam.com	ejpam.com	X
ejpam-6573	1	26	published	publish	VERB
ejpam-6573	1	27	by	by	ADP
ejpam-6573	1	28	new	new	PROPN
ejpam-6573	1	29	york	york	PROPN
ejpam-6573	1	30	business	business	PROPN
ejpam-6573	1	31	global	global	ADJ
ejpam-6573	1	32	on	on	ADP
ejpam-6573	1	33	weak	weak	ADJ
ejpam-6573	1	34	forms	form	NOUN
ejpam-6573	1	35	of	of	ADP
ejpam-6573	1	36	open	open	ADJ
ejpam-6573	1	37	and	and	CCONJ
ejpam-6573	1	38	closed	closed	ADJ
ejpam-6573	1	39	functions	function	NOUN
ejpam-6573	1	40	via	via	ADP
ejpam-6573	1	41	(	(	PUNCT
ejpam-6573	1	42	τ1	τ1	NOUN
ejpam-6573	1	43	,	,	PUNCT
ejpam-6573	1	44	τ2)β	τ2)β	ADJ
ejpam-6573	1	45	-	-	PUNCT
ejpam-6573	1	46	open	open	ADJ
ejpam-6573	1	47	sets	set	NOUN
ejpam-6573	1	48	monchaya	monchaya	NOUN
ejpam-6573	1	49	chiangpradit1	chiangpradit1	NOUN
ejpam-6573	1	50	,	,	PUNCT
ejpam-6573	1	51	areeyuth	areeyuth	NOUN
ejpam-6573	1	52	sama	sama	NOUN
ejpam-6573	1	53	-	-	PUNCT
ejpam-6573	1	54	ae2	ae2	PROPN
ejpam-6573	1	55	,	,	PUNCT
ejpam-6573	1	56	chawalit	chawalit	VERB
ejpam-6573	1	57	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6573	1	58	1	1	NUM
ejpam-6573	1	59	mathematics	mathematic	NOUN
ejpam-6573	1	60	and	and	CCONJ
ejpam-6573	1	61	applied	apply	VERB
ejpam-6573	1	62	mathematics	mathematics	PROPN
ejpam-6573	1	63	research	research	NOUN
ejpam-6573	1	64	unit	unit	NOUN
ejpam-6573	1	65	,	,	PUNCT
ejpam-6573	1	66	department	department	NOUN
ejpam-6573	1	67	of	of	ADP
ejpam-6573	1	68	mathematics	mathematic	NOUN
ejpam-6573	1	69	,	,	PUNCT
ejpam-6573	1	70	faculty	faculty	NOUN
ejpam-6573	1	71	of	of	ADP
ejpam-6573	1	72	science	science	NOUN
ejpam-6573	1	73	,	,	PUNCT
ejpam-6573	1	74	mahasarakham	mahasarakham	PROPN
ejpam-6573	1	75	university	university	PROPN
ejpam-6573	1	76	,	,	PUNCT
ejpam-6573	1	77	maha	maha	PROPN
ejpam-6573	1	78	sarakham	sarakham	PROPN
ejpam-6573	1	79	,	,	PUNCT
ejpam-6573	1	80	44150	44150	NUM
ejpam-6573	1	81	,	,	PUNCT
ejpam-6573	1	82	thailand	thailand	PROPN
ejpam-6573	1	83	2	2	NUM
ejpam-6573	1	84	department	department	NOUN
ejpam-6573	1	85	of	of	ADP
ejpam-6573	1	86	mathematics	mathematic	NOUN
ejpam-6573	1	87	and	and	CCONJ
ejpam-6573	1	88	computer	computer	NOUN
ejpam-6573	1	89	science	science	NOUN
ejpam-6573	1	90	,	,	PUNCT
ejpam-6573	1	91	faculty	faculty	NOUN
ejpam-6573	1	92	of	of	ADP
ejpam-6573	1	93	science	science	NOUN
ejpam-6573	1	94	and	and	CCONJ
ejpam-6573	1	95	technology	technology	NOUN
ejpam-6573	1	96	,	,	PUNCT
ejpam-6573	1	97	prince	prince	NOUN
ejpam-6573	1	98	of	of	ADP
ejpam-6573	1	99	songkla	songkla	PROPN
ejpam-6573	1	100	university	university	PROPN
ejpam-6573	1	101	,	,	PUNCT
ejpam-6573	1	102	pattani	pattani	NOUN
ejpam-6573	1	103	campus	campus	NOUN
ejpam-6573	1	104	,	,	PUNCT
ejpam-6573	1	105	pattani	pattani	NOUN
ejpam-6573	1	106	,	,	PUNCT
ejpam-6573	1	107	94000	94000	NUM
ejpam-6573	1	108	,	,	PUNCT
ejpam-6573	1	109	thailand	thailand	PROPN
ejpam-6573	1	110	abstract	abstract	PROPN
ejpam-6573	1	111	.	.	PUNCT
ejpam-6573	2	1	this	this	DET
ejpam-6573	2	2	paper	paper	NOUN
ejpam-6573	2	3	is	be	AUX
ejpam-6573	2	4	concerned	concern	VERB
ejpam-6573	2	5	with	with	ADP
ejpam-6573	2	6	the	the	DET
ejpam-6573	2	7	concepts	concept	NOUN
ejpam-6573	2	8	of	of	ADP
ejpam-6573	2	9	weakly	weakly	ADJ
ejpam-6573	2	10	(	(	PUNCT
ejpam-6573	2	11	τ1	τ1	NOUN
ejpam-6573	2	12	,	,	PUNCT
ejpam-6573	2	13	τ2)β	τ2)β	ADJ
ejpam-6573	2	14	-	-	PUNCT
ejpam-6573	2	15	open	open	ADJ
ejpam-6573	2	16	functions	function	NOUN
ejpam-6573	2	17	and	and	CCONJ
ejpam-6573	2	18	weakly	weakly	ADJ
ejpam-6573	2	19	(	(	PUNCT
ejpam-6573	2	20	τ1	τ1	NOUN
ejpam-6573	2	21	,	,	PUNCT
ejpam-6573	2	22	τ2)β	τ2)β	ADJ
ejpam-6573	2	23	-	-	PUNCT
ejpam-6573	2	24	closed	close	VERB
ejpam-6573	2	25	functions	function	NOUN
ejpam-6573	2	26	.	.	PUNCT
ejpam-6573	3	1	furthermore	furthermore	ADV
ejpam-6573	3	2	,	,	PUNCT
ejpam-6573	3	3	several	several	ADJ
ejpam-6573	3	4	characterizations	characterization	NOUN
ejpam-6573	3	5	of	of	ADP
ejpam-6573	3	6	weakly	weakly	ADJ
ejpam-6573	3	7	(	(	PUNCT
ejpam-6573	3	8	τ1	τ1	NOUN
ejpam-6573	3	9	,	,	PUNCT
ejpam-6573	3	10	τ2)β	τ2)β	ADJ
ejpam-6573	3	11	-	-	PUNCT
ejpam-6573	3	12	open	open	ADJ
ejpam-6573	3	13	functions	function	NOUN
ejpam-6573	3	14	and	and	CCONJ
ejpam-6573	3	15	weakly	weakly	ADJ
ejpam-6573	3	16	(	(	PUNCT
ejpam-6573	3	17	τ1	τ1	NOUN
ejpam-6573	3	18	,	,	PUNCT
ejpam-6573	3	19	τ2)β	τ2)β	ADJ
ejpam-6573	3	20	-	-	PUNCT
ejpam-6573	3	21	closed	close	VERB
ejpam-6573	3	22	functions	function	NOUN
ejpam-6573	3	23	are	be	AUX
ejpam-6573	3	24	investigated	investigate	VERB
ejpam-6573	3	25	.	.	PUNCT
ejpam-6573	4	1	2020	2020	NUM
ejpam-6573	4	2	mathematics	mathematic	NOUN
ejpam-6573	4	3	subject	subject	NOUN
ejpam-6573	4	4	classifications	classification	NOUN
ejpam-6573	4	5	:	:	PUNCT
ejpam-6573	4	6	54c10	54c10	NUM
ejpam-6573	4	7	,	,	PUNCT
ejpam-6573	4	8	54e55	54e55	NUM
ejpam-6573	4	9	key	key	ADJ
ejpam-6573	4	10	words	word	NOUN
ejpam-6573	4	11	and	and	CCONJ
ejpam-6573	4	12	phrases	phrase	NOUN
ejpam-6573	4	13	:	:	PUNCT
ejpam-6573	4	14	weakly	weakly	ADJ
ejpam-6573	4	15	(	(	PUNCT
ejpam-6573	4	16	τ1	τ1	NOUN
ejpam-6573	4	17	,	,	PUNCT
ejpam-6573	4	18	τ2)β	τ2)β	ADJ
ejpam-6573	4	19	-	-	PUNCT
ejpam-6573	4	20	open	open	ADJ
ejpam-6573	4	21	function	function	NOUN
ejpam-6573	4	22	,	,	PUNCT
ejpam-6573	4	23	weakly	weakly	ADJ
ejpam-6573	4	24	(	(	PUNCT
ejpam-6573	4	25	τ1	τ1	NOUN
ejpam-6573	4	26	,	,	PUNCT
ejpam-6573	4	27	τ2)β	τ2)β	ADJ
ejpam-6573	4	28	-	-	PUNCT
ejpam-6573	4	29	closed	close	VERB
ejpam-6573	4	30	function	function	NOUN
ejpam-6573	4	31	1	1	NUM
ejpam-6573	4	32	.	.	PUNCT
ejpam-6573	5	1	introduction	introduction	NOUN
ejpam-6573	5	2	it	it	PRON
ejpam-6573	5	3	is	be	AUX
ejpam-6573	5	4	well	well	ADV
ejpam-6573	5	5	-	-	PUNCT
ejpam-6573	5	6	known	know	VERB
ejpam-6573	5	7	that	that	SCONJ
ejpam-6573	5	8	the	the	DET
ejpam-6573	5	9	branch	branch	NOUN
ejpam-6573	5	10	of	of	ADP
ejpam-6573	5	11	mathematics	mathematic	NOUN
ejpam-6573	5	12	called	call	VERB
ejpam-6573	5	13	topology	topology	NOUN
ejpam-6573	5	14	is	be	AUX
ejpam-6573	5	15	related	relate	VERB
ejpam-6573	5	16	to	to	ADP
ejpam-6573	5	17	all	all	DET
ejpam-6573	5	18	questions	question	NOUN
ejpam-6573	5	19	directly	directly	ADV
ejpam-6573	5	20	or	or	CCONJ
ejpam-6573	5	21	indirectly	indirectly	ADV
ejpam-6573	5	22	concerned	concerned	ADJ
ejpam-6573	5	23	with	with	ADP
ejpam-6573	5	24	openness	openness	NOUN
ejpam-6573	5	25	and	and	CCONJ
ejpam-6573	5	26	closedness	closedness	NOUN
ejpam-6573	5	27	.	.	PUNCT
ejpam-6573	6	1	semi	semi	ADJ
ejpam-6573	6	2	-	-	ADJ
ejpam-6573	6	3	open	open	ADJ
ejpam-6573	6	4	sets	set	NOUN
ejpam-6573	6	5	,	,	PUNCT
ejpam-6573	6	6	preopen	preopen	ADJ
ejpam-6573	6	7	sets	set	NOUN
ejpam-6573	6	8	,	,	PUNCT
ejpam-6573	6	9	α	α	NOUN
ejpam-6573	6	10	-	-	ADJ
ejpam-6573	6	11	open	open	ADJ
ejpam-6573	6	12	sets	set	NOUN
ejpam-6573	6	13	,	,	PUNCT
ejpam-6573	6	14	β	β	ADJ
ejpam-6573	6	15	-	-	ADJ
ejpam-6573	6	16	open	open	ADJ
ejpam-6573	6	17	sets	set	NOUN
ejpam-6573	6	18	,	,	PUNCT
ejpam-6573	6	19	b	b	X
ejpam-6573	6	20	-	-	PUNCT
ejpam-6573	6	21	open	open	ADJ
ejpam-6573	6	22	sets	set	NOUN
ejpam-6573	6	23	,	,	PUNCT
ejpam-6573	6	24	δ	δ	NOUN
ejpam-6573	6	25	-	-	ADJ
ejpam-6573	6	26	open	open	ADJ
ejpam-6573	6	27	sets	set	NOUN
ejpam-6573	6	28	and	and	CCONJ
ejpam-6573	6	29	θ	θ	ADJ
ejpam-6573	6	30	-	-	ADJ
ejpam-6573	6	31	open	open	ADJ
ejpam-6573	6	32	sets	set	NOUN
ejpam-6573	6	33	play	play	VERB
ejpam-6573	6	34	an	an	DET
ejpam-6573	6	35	important	important	ADJ
ejpam-6573	6	36	role	role	NOUN
ejpam-6573	6	37	in	in	ADP
ejpam-6573	6	38	the	the	DET
ejpam-6573	6	39	researches	research	NOUN
ejpam-6573	6	40	of	of	ADP
ejpam-6573	6	41	generalizations	generalization	NOUN
ejpam-6573	6	42	of	of	ADP
ejpam-6573	6	43	open	open	ADJ
ejpam-6573	6	44	functions	function	NOUN
ejpam-6573	6	45	and	and	CCONJ
ejpam-6573	6	46	closed	closed	ADJ
ejpam-6573	6	47	functions	function	NOUN
ejpam-6573	6	48	.	.	PUNCT
ejpam-6573	7	1	by	by	ADP
ejpam-6573	7	2	using	use	VERB
ejpam-6573	7	3	these	these	DET
ejpam-6573	7	4	sets	set	NOUN
ejpam-6573	7	5	,	,	PUNCT
ejpam-6573	7	6	many	many	ADJ
ejpam-6573	7	7	authors	author	NOUN
ejpam-6573	7	8	introduced	introduce	VERB
ejpam-6573	7	9	and	and	CCONJ
ejpam-6573	7	10	studied	study	VERB
ejpam-6573	7	11	various	various	ADJ
ejpam-6573	7	12	types	type	NOUN
ejpam-6573	7	13	of	of	ADP
ejpam-6573	7	14	open	open	ADJ
ejpam-6573	7	15	functions	function	NOUN
ejpam-6573	7	16	and	and	CCONJ
ejpam-6573	7	17	closed	closed	ADJ
ejpam-6573	7	18	functions	function	NOUN
ejpam-6573	7	19	.	.	PUNCT
ejpam-6573	8	1	the	the	DET
ejpam-6573	8	2	concept	concept	NOUN
ejpam-6573	8	3	of	of	ADP
ejpam-6573	8	4	weakly	weakly	ADJ
ejpam-6573	8	5	open	open	ADJ
ejpam-6573	8	6	functions	function	NOUN
ejpam-6573	8	7	was	be	AUX
ejpam-6573	8	8	first	first	ADV
ejpam-6573	8	9	introduced	introduce	VERB
ejpam-6573	8	10	by	by	ADP
ejpam-6573	8	11	rose	rose	NOUN
ejpam-6573	8	12	[	[	X
ejpam-6573	8	13	1	1	NUM
ejpam-6573	8	14	]	]	PUNCT
ejpam-6573	8	15	.	.	PUNCT
ejpam-6573	9	1	rose	rise	VERB
ejpam-6573	9	2	and	and	CCONJ
ejpam-6573	9	3	janković	janković	ADJ
ejpam-6573	10	1	[	[	X
ejpam-6573	10	2	2	2	X
ejpam-6573	10	3	]	]	PUNCT
ejpam-6573	10	4	investigated	investigate	VERB
ejpam-6573	10	5	some	some	PRON
ejpam-6573	10	6	of	of	ADP
ejpam-6573	10	7	the	the	DET
ejpam-6573	10	8	fundamental	fundamental	ADJ
ejpam-6573	10	9	properties	property	NOUN
ejpam-6573	10	10	of	of	ADP
ejpam-6573	10	11	weakly	weakly	ADJ
ejpam-6573	10	12	closed	closed	ADJ
ejpam-6573	10	13	functions	function	NOUN
ejpam-6573	10	14	.	.	PUNCT
ejpam-6573	11	1	caldas	calda	NOUN
ejpam-6573	11	2	and	and	CCONJ
ejpam-6573	11	3	navalagi	navalagi	ADJ
ejpam-6573	11	4	[	[	X
ejpam-6573	11	5	3	3	NUM
ejpam-6573	11	6	]	]	PUNCT
ejpam-6573	11	7	introduced	introduce	VERB
ejpam-6573	11	8	two	two	NUM
ejpam-6573	11	9	new	new	ADJ
ejpam-6573	11	10	classes	class	NOUN
ejpam-6573	11	11	of	of	ADP
ejpam-6573	11	12	functions	function	NOUN
ejpam-6573	11	13	called	call	VERB
ejpam-6573	11	14	weakly	weakly	ADJ
ejpam-6573	11	15	preopen	preopen	ADJ
ejpam-6573	11	16	functions	function	NOUN
ejpam-6573	11	17	and	and	CCONJ
ejpam-6573	11	18	weakly	weakly	ADJ
ejpam-6573	11	19	preclosed	preclose	VERB
ejpam-6573	11	20	functions	function	NOUN
ejpam-6573	11	21	as	as	ADP
ejpam-6573	11	22	generalization	generalization	NOUN
ejpam-6573	11	23	of	of	ADP
ejpam-6573	11	24	weak	weak	ADJ
ejpam-6573	11	25	openness	openness	NOUN
ejpam-6573	11	26	and	and	CCONJ
ejpam-6573	11	27	weak	weak	ADJ
ejpam-6573	11	28	closedness	closedness	NOUN
ejpam-6573	11	29	due	due	ADP
ejpam-6573	11	30	to	to	ADP
ejpam-6573	11	31	[	[	X
ejpam-6573	11	32	1	1	NUM
ejpam-6573	11	33	]	]	PUNCT
ejpam-6573	11	34	and	and	CCONJ
ejpam-6573	11	35	[	[	X
ejpam-6573	11	36	2	2	NUM
ejpam-6573	11	37	]	]	PUNCT
ejpam-6573	11	38	,	,	PUNCT
ejpam-6573	11	39	respectively	respectively	ADV
ejpam-6573	11	40	.	.	PUNCT
ejpam-6573	12	1	moreover	moreover	ADV
ejpam-6573	12	2	,	,	PUNCT
ejpam-6573	12	3	caldas	calda	NOUN
ejpam-6573	12	4	and	and	CCONJ
ejpam-6573	12	5	navalagi	navalagi	ADJ
ejpam-6573	12	6	[	[	X
ejpam-6573	12	7	4	4	NUM
ejpam-6573	12	8	]	]	PUNCT
ejpam-6573	12	9	introduced	introduce	VERB
ejpam-6573	12	10	and	and	CCONJ
ejpam-6573	12	11	investigated	investigate	VERB
ejpam-6573	12	12	the	the	DET
ejpam-6573	12	13	concepts	concept	NOUN
ejpam-6573	12	14	of	of	ADP
ejpam-6573	12	15	weakly	weakly	ADJ
ejpam-6573	12	16	semi	semi	ADJ
ejpam-6573	12	17	-	-	ADJ
ejpam-6573	12	18	open	open	ADJ
ejpam-6573	12	19	functions	function	NOUN
ejpam-6573	12	20	and	and	CCONJ
ejpam-6573	12	21	weakly	weakly	ADJ
ejpam-6573	12	22	semi	semi	ADJ
ejpam-6573	12	23	-	-	ADJ
ejpam-6573	12	24	closed	closed	ADJ
ejpam-6573	12	25	functions	function	NOUN
ejpam-6573	12	26	as	as	ADP
ejpam-6573	12	27	a	a	DET
ejpam-6573	12	28	new	new	ADJ
ejpam-6573	12	29	generalization	generalization	NOUN
ejpam-6573	12	30	of	of	ADP
ejpam-6573	12	31	weakly	weakly	ADJ
ejpam-6573	12	32	open	open	ADJ
ejpam-6573	12	33	functions	function	NOUN
ejpam-6573	12	34	and	and	CCONJ
ejpam-6573	12	35	weakly	weakly	ADJ
ejpam-6573	12	36	closed	closed	ADJ
ejpam-6573	12	37	functions	function	NOUN
ejpam-6573	12	38	,	,	PUNCT
ejpam-6573	12	39	respectively	respectively	ADV
ejpam-6573	12	40	.	.	PUNCT
ejpam-6573	13	1	noiri	noiri	PROPN
ejpam-6573	13	2	et	et	PROPN
ejpam-6573	13	3	al	al	PROPN
ejpam-6573	13	4	.	.	PUNCT
ejpam-6573	14	1	[	[	X
ejpam-6573	14	2	5	5	NUM
ejpam-6573	14	3	]	]	PUNCT
ejpam-6573	14	4	introduced	introduce	VERB
ejpam-6573	14	5	and	and	CCONJ
ejpam-6573	14	6	studied	study	VERB
ejpam-6573	14	7	two	two	NUM
ejpam-6573	14	8	new	new	ADJ
ejpam-6573	14	9	classes	class	NOUN
ejpam-6573	14	10	of	of	ADP
ejpam-6573	14	11	functions	function	NOUN
ejpam-6573	14	12	called	call	VERB
ejpam-6573	14	13	weakly	weakly	ADJ
ejpam-6573	14	14	b	b	NOUN
ejpam-6573	14	15	-	-	PUNCT
ejpam-6573	14	16	θ	θ	ADJ
ejpam-6573	14	17	-	-	PUNCT
ejpam-6573	14	18	open	open	ADJ
ejpam-6573	14	19	functions	function	NOUN
ejpam-6573	14	20	and	and	CCONJ
ejpam-6573	14	21	weakly	weakly	ADJ
ejpam-6573	14	22	b	b	NOUN
ejpam-6573	14	23	-	-	PUNCT
ejpam-6573	14	24	θ	θ	ADJ
ejpam-6573	14	25	-	-	PUNCT
ejpam-6573	14	26	open	open	ADJ
ejpam-6573	14	27	functions	function	NOUN
ejpam-6573	14	28	by	by	ADP
ejpam-6573	14	29	utilizing	utilize	VERB
ejpam-6573	14	30	the	the	DET
ejpam-6573	14	31	notions	notion	NOUN
ejpam-6573	14	32	of	of	ADP
ejpam-6573	14	33	b	b	NOUN
ejpam-6573	14	34	-	-	PUNCT
ejpam-6573	14	35	θ	θ	ADJ
ejpam-6573	14	36	-	-	ADJ
ejpam-6573	14	37	open	open	ADJ
ejpam-6573	14	38	sets	set	NOUN
ejpam-6573	14	39	and	and	CCONJ
ejpam-6573	14	40	the	the	DET
ejpam-6573	14	41	b	b	PROPN
ejpam-6573	14	42	-	-	PUNCT
ejpam-6573	14	43	θ	θ	NOUN
ejpam-6573	14	44	-	-	PUNCT
ejpam-6573	14	45	closure	closure	NOUN
ejpam-6573	14	46	operator	operator	NOUN
ejpam-6573	14	47	.	.	PUNCT
ejpam-6573	15	1	weak	weak	ADJ
ejpam-6573	15	2	b	b	X
ejpam-6573	15	3	-	-	PUNCT
ejpam-6573	15	4	θ	θ	NOUN
ejpam-6573	15	5	-	-	PUNCT
ejpam-6573	15	6	openness	openness	NOUN
ejpam-6573	15	7	(	(	PUNCT
ejpam-6573	15	8	resp	resp	NOUN
ejpam-6573	15	9	.	.	PUNCT
ejpam-6573	16	1	b	b	X
ejpam-6573	16	2	-	-	PUNCT
ejpam-6573	16	3	θ	θ	NOUN
ejpam-6573	16	4	-	-	PUNCT
ejpam-6573	16	5	closedness	closedness	NOUN
ejpam-6573	16	6	)	)	PUNCT
ejpam-6573	17	1	is	be	AUX
ejpam-6573	17	2	∗corresponding	∗corresponde	VERB
ejpam-6573	17	3	author	author	NOUN
ejpam-6573	17	4	.	.	PUNCT
ejpam-6573	18	1	doi	doi	NOUN
ejpam-6573	18	2	:	:	PUNCT
ejpam-6573	18	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6573	https://doi.org/10.29020/nybg.ejpam.v18i3.6573	NOUN
ejpam-6573	18	4	email	email	NOUN
ejpam-6573	18	5	addresses	address	NOUN
ejpam-6573	18	6	:	:	PUNCT
ejpam-6573	18	7	monchaya.c@msu.ac.th	monchaya.c@msu.ac.th	PROPN
ejpam-6573	18	8	(	(	PUNCT
ejpam-6573	18	9	m.	m.	NOUN
ejpam-6573	18	10	chiangpradit	chiangpradit	NOUN
ejpam-6573	18	11	)	)	PUNCT
ejpam-6573	18	12	,	,	PUNCT
ejpam-6573	18	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6573	18	14	(	(	PUNCT
ejpam-6573	18	15	a.	a.	PROPN
ejpam-6573	18	16	sama	sama	PROPN
ejpam-6573	18	17	-	-	PUNCT
ejpam-6573	18	18	ae	ae	PROPN
ejpam-6573	18	19	)	)	PUNCT
ejpam-6573	18	20	,	,	PUNCT
ejpam-6573	18	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6573	18	22	(	(	PUNCT
ejpam-6573	18	23	c.	c.	PROPN
ejpam-6573	18	24	boonpok	boonpok	PROPN
ejpam-6573	18	25	)	)	PUNCT
ejpam-6573	18	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6573	18	27	1	1	NUM
ejpam-6573	18	28	copyright	copyright	NOUN
ejpam-6573	18	29	:	:	PUNCT
ejpam-6573	19	1	©	©	PROPN
ejpam-6573	19	2	2025	2025	NUM
ejpam-6573	19	3	the	the	DET
ejpam-6573	19	4	author(s	author(s	NOUN
ejpam-6573	19	5	)	)	PUNCT
ejpam-6573	19	6	.	.	PUNCT
ejpam-6573	20	1	(	(	PUNCT
ejpam-6573	20	2	cc	cc	NOUN
ejpam-6573	20	3	by	by	ADP
ejpam-6573	20	4	-	-	PUNCT
ejpam-6573	20	5	nc	nc	PROPN
ejpam-6573	20	6	4.0	4.0	NUM
ejpam-6573	20	7	)	)	PUNCT
ejpam-6573	20	8	m.	m.	NOUN
ejpam-6573	20	9	chiangpradit	chiangpradit	NOUN
ejpam-6573	20	10	,	,	PUNCT
ejpam-6573	20	11	a.	a.	PROPN
ejpam-6573	20	12	sama	sama	PROPN
ejpam-6573	20	13	-	-	PUNCT
ejpam-6573	20	14	ae	ae	PROPN
ejpam-6573	20	15	,	,	PUNCT
ejpam-6573	20	16	c.	c.	PROPN
ejpam-6573	20	17	boonpok	boonpok	PROPN
ejpam-6573	20	18	/	/	SYM
ejpam-6573	20	19	eur	eur	PROPN
ejpam-6573	20	20	.	.	PUNCT
ejpam-6573	21	1	j.	j.	PROPN
ejpam-6573	21	2	pure	pure	PROPN
ejpam-6573	21	3	appl	appl	PROPN
ejpam-6573	21	4	.	.	PROPN
ejpam-6573	21	5	math	math	PROPN
ejpam-6573	21	6	,	,	PUNCT
ejpam-6573	21	7	18	18	NUM
ejpam-6573	21	8	(	(	PUNCT
ejpam-6573	21	9	3	3	NUM
ejpam-6573	21	10	)	)	PUNCT
ejpam-6573	21	11	(	(	PUNCT
ejpam-6573	21	12	2025	2025	NUM
ejpam-6573	21	13	)	)	PUNCT
ejpam-6573	21	14	,	,	PUNCT
ejpam-6573	21	15	6573	6573	NUM
ejpam-6573	21	16	2	2	NUM
ejpam-6573	21	17	of	of	ADP
ejpam-6573	21	18	8	8	NUM
ejpam-6573	21	19	a	a	DET
ejpam-6573	21	20	generalization	generalization	NOUN
ejpam-6573	21	21	of	of	ADP
ejpam-6573	21	22	both	both	DET
ejpam-6573	21	23	θ	θ	NOUN
ejpam-6573	21	24	-	-	NOUN
ejpam-6573	21	25	preopenness	preopenness	NOUN
ejpam-6573	21	26	and	and	CCONJ
ejpam-6573	21	27	weak	weak	ADJ
ejpam-6573	21	28	semi	semi	ADJ
ejpam-6573	21	29	-	-	ADJ
ejpam-6573	21	30	θ	θ	ADJ
ejpam-6573	21	31	-	-	PUNCT
ejpam-6573	21	32	openness	openness	NOUN
ejpam-6573	21	33	(	(	PUNCT
ejpam-6573	21	34	resp	resp	NOUN
ejpam-6573	21	35	.	.	PUNCT
ejpam-6573	22	1	θ	θ	NOUN
ejpam-6573	22	2	-	-	PUNCT
ejpam-6573	22	3	preclosedness	preclosedness	NOUN
ejpam-6573	22	4	and	and	CCONJ
ejpam-6573	22	5	weak	weak	ADJ
ejpam-6573	22	6	semi	semi	ADJ
ejpam-6573	22	7	-	-	ADJ
ejpam-6573	22	8	θ	θ	ADJ
ejpam-6573	22	9	-	-	PUNCT
ejpam-6573	22	10	closedness	closedness	ADJ
ejpam-6573	22	11	)	)	PUNCT
ejpam-6573	22	12	.	.	PUNCT
ejpam-6573	23	1	caldas	caldas	PROPN
ejpam-6573	23	2	and	and	CCONJ
ejpam-6573	23	3	navalagi	navalagi	ADJ
ejpam-6573	23	4	[	[	X
ejpam-6573	23	5	6	6	NUM
ejpam-6573	23	6	]	]	PUNCT
ejpam-6573	23	7	introduced	introduce	VERB
ejpam-6573	23	8	and	and	CCONJ
ejpam-6573	23	9	investigated	investigate	VERB
ejpam-6573	23	10	the	the	DET
ejpam-6573	23	11	notions	notion	NOUN
ejpam-6573	23	12	of	of	ADP
ejpam-6573	23	13	weakly	weakly	ADJ
ejpam-6573	23	14	β	β	X
ejpam-6573	23	15	-	-	ADJ
ejpam-6573	23	16	open	open	ADJ
ejpam-6573	23	17	functions	function	NOUN
ejpam-6573	23	18	and	and	CCONJ
ejpam-6573	23	19	weakly	weakly	ADJ
ejpam-6573	23	20	β	β	NOUN
ejpam-6573	23	21	-	-	ADJ
ejpam-6573	23	22	closed	closed	ADJ
ejpam-6573	23	23	functions	function	NOUN
ejpam-6573	23	24	.	.	PUNCT
ejpam-6573	24	1	quite	quite	ADV
ejpam-6573	24	2	recently	recently	ADV
ejpam-6573	24	3	,	,	PUNCT
ejpam-6573	24	4	chutiman	chutiman	NOUN
ejpam-6573	24	5	and	and	CCONJ
ejpam-6573	24	6	boonpok	boonpok	NOUN
ejpam-6573	25	1	[	[	X
ejpam-6573	25	2	7	7	NUM
ejpam-6573	25	3	]	]	PUNCT
ejpam-6573	25	4	studied	study	VERB
ejpam-6573	25	5	some	some	DET
ejpam-6573	25	6	properties	property	NOUN
ejpam-6573	25	7	of	of	ADP
ejpam-6573	25	8	weakly	weakly	ADJ
ejpam-6573	25	9	b(λ	b(λ	NOUN
ejpam-6573	25	10	,	,	PUNCT
ejpam-6573	25	11	p)-open	p)-open	NOUN
ejpam-6573	25	12	functions	function	NOUN
ejpam-6573	25	13	.	.	PUNCT
ejpam-6573	26	1	on	on	ADP
ejpam-6573	26	2	the	the	DET
ejpam-6573	26	3	other	other	ADJ
ejpam-6573	26	4	hand	hand	NOUN
ejpam-6573	26	5	,	,	PUNCT
ejpam-6573	26	6	the	the	DET
ejpam-6573	26	7	present	present	ADJ
ejpam-6573	26	8	authors	author	NOUN
ejpam-6573	26	9	introduced	introduce	VERB
ejpam-6573	26	10	and	and	CCONJ
ejpam-6573	26	11	studied	study	VERB
ejpam-6573	26	12	the	the	DET
ejpam-6573	26	13	notions	notion	NOUN
ejpam-6573	26	14	of	of	ADP
ejpam-6573	26	15	semi-(i	semi-(i	PROPN
ejpam-6573	26	16	,	,	PUNCT
ejpam-6573	26	17	j	j	NOUN
ejpam-6573	26	18	)	)	PUNCT
ejpam-6573	26	19	-open	-open	NOUN
ejpam-6573	26	20	functions	function	NOUN
ejpam-6573	26	21	[	[	X
ejpam-6573	26	22	8	8	NUM
ejpam-6573	26	23	]	]	PUNCT
ejpam-6573	26	24	,	,	PUNCT
ejpam-6573	26	25	semi-(i	semi-(i	PROPN
ejpam-6573	26	26	,	,	PUNCT
ejpam-6573	26	27	j	j	PROPN
ejpam-6573	26	28	)	)	PUNCT
ejpam-6573	26	29	-closed	-close	VERB
ejpam-6573	26	30	functions	function	NOUN
ejpam-6573	26	31	[	[	X
ejpam-6573	26	32	8	8	NUM
ejpam-6573	26	33	]	]	PUNCT
ejpam-6573	26	34	,	,	PUNCT
ejpam-6573	26	35	weakly	weakly	ADJ
ejpam-6573	26	36	s(λ	s(λ	NOUN
ejpam-6573	26	37	,	,	PUNCT
ejpam-6573	26	38	p)-open	p)-open	VERB
ejpam-6573	26	39	functions	function	NOUN
ejpam-6573	26	40	[	[	X
ejpam-6573	26	41	9	9	NUM
ejpam-6573	26	42	]	]	PUNCT
ejpam-6573	26	43	,	,	PUNCT
ejpam-6573	26	44	weakly	weakly	ADJ
ejpam-6573	26	45	s(λ	s(λ	PROPN
ejpam-6573	26	46	,	,	PUNCT
ejpam-6573	26	47	p)-closed	p)-close	VERB
ejpam-6573	26	48	functions	function	NOUN
ejpam-6573	26	49	[	[	X
ejpam-6573	26	50	9	9	NUM
ejpam-6573	26	51	]	]	PUNCT
ejpam-6573	26	52	,	,	PUNCT
ejpam-6573	26	53	weakly	weakly	ADJ
ejpam-6573	26	54	δ(λ	δ(λ	PROPN
ejpam-6573	26	55	,	,	PUNCT
ejpam-6573	26	56	p)-open	p)-open	VERB
ejpam-6573	26	57	functions	function	NOUN
ejpam-6573	26	58	[	[	X
ejpam-6573	26	59	10	10	NUM
ejpam-6573	26	60	]	]	PUNCT
ejpam-6573	26	61	,	,	PUNCT
ejpam-6573	26	62	weakly	weakly	ADJ
ejpam-6573	26	63	δ(λ	δ(λ	PROPN
ejpam-6573	26	64	,	,	PUNCT
ejpam-6573	26	65	p)-closed	p)-close	VERB
ejpam-6573	26	66	functions	function	NOUN
ejpam-6573	26	67	[	[	X
ejpam-6573	26	68	11	11	NUM
ejpam-6573	26	69	]	]	PUNCT
ejpam-6573	26	70	,	,	PUNCT
ejpam-6573	26	71	weakly	weakly	ADJ
ejpam-6573	26	72	β(λ	β(λ	NOUN
ejpam-6573	26	73	,	,	PUNCT
ejpam-6573	26	74	p)-open	p)-open	VERB
ejpam-6573	26	75	functions	function	NOUN
ejpam-6573	26	76	[	[	X
ejpam-6573	26	77	12	12	NUM
ejpam-6573	26	78	]	]	PUNCT
ejpam-6573	26	79	,	,	PUNCT
ejpam-6573	26	80	weakly	weakly	ADJ
ejpam-6573	26	81	β(λ	β(λ	NOUN
ejpam-6573	26	82	,	,	PUNCT
ejpam-6573	26	83	p)-closed	p)-close	VERB
ejpam-6573	26	84	functions	function	NOUN
ejpam-6573	26	85	[	[	X
ejpam-6573	26	86	12	12	NUM
ejpam-6573	26	87	]	]	PUNCT
ejpam-6573	26	88	,	,	PUNCT
ejpam-6573	26	89	weakly	weakly	ADJ
ejpam-6573	26	90	p(λ	p(λ	NOUN
ejpam-6573	26	91	,	,	PUNCT
ejpam-6573	26	92	p)-open	p)-open	NOUN
ejpam-6573	26	93	functions	function	NOUN
ejpam-6573	26	94	[	[	PUNCT
ejpam-6573	26	95	13	13	NUM
ejpam-6573	26	96	]	]	PUNCT
ejpam-6573	26	97	,	,	PUNCT
ejpam-6573	26	98	weakly	weakly	ADJ
ejpam-6573	26	99	p(λ	p(λ	NOUN
ejpam-6573	26	100	,	,	PUNCT
ejpam-6573	26	101	p)-closed	p)-close	VERB
ejpam-6573	26	102	functions	function	NOUN
ejpam-6573	26	103	[	[	PUNCT
ejpam-6573	26	104	13	13	NUM
ejpam-6573	26	105	]	]	PUNCT
ejpam-6573	26	106	,	,	PUNCT
ejpam-6573	26	107	weakly	weakly	ADJ
ejpam-6573	26	108	θs(λ	θs(λ	NOUN
ejpam-6573	26	109	,	,	PUNCT
ejpam-6573	26	110	p)-open	p)-open	VERB
ejpam-6573	26	111	functions	function	NOUN
ejpam-6573	26	112	[	[	X
ejpam-6573	26	113	14	14	NUM
ejpam-6573	26	114	]	]	PUNCT
ejpam-6573	26	115	and	and	CCONJ
ejpam-6573	26	116	weakly	weakly	ADJ
ejpam-6573	26	117	θs(λ	θs(λ	NOUN
ejpam-6573	26	118	,	,	PUNCT
ejpam-6573	26	119	p)-closed	p)-close	VERB
ejpam-6573	26	120	functions	function	NOUN
ejpam-6573	26	121	[	[	X
ejpam-6573	26	122	14	14	NUM
ejpam-6573	26	123	]	]	PUNCT
ejpam-6573	26	124	.	.	PUNCT
ejpam-6573	27	1	in	in	ADP
ejpam-6573	27	2	this	this	DET
ejpam-6573	27	3	paper	paper	NOUN
ejpam-6573	27	4	,	,	PUNCT
ejpam-6573	27	5	we	we	PRON
ejpam-6573	27	6	introduce	introduce	VERB
ejpam-6573	27	7	the	the	DET
ejpam-6573	27	8	notions	notion	NOUN
ejpam-6573	27	9	of	of	ADP
ejpam-6573	27	10	weakly	weakly	ADJ
ejpam-6573	27	11	(	(	PUNCT
ejpam-6573	27	12	τ1	τ1	NOUN
ejpam-6573	27	13	,	,	PUNCT
ejpam-6573	27	14	τ2)β	τ2)β	ADJ
ejpam-6573	27	15	-	-	PUNCT
ejpam-6573	27	16	open	open	ADJ
ejpam-6573	27	17	functions	function	NOUN
ejpam-6573	27	18	and	and	CCONJ
ejpam-6573	27	19	weakly	weakly	ADJ
ejpam-6573	27	20	(	(	PUNCT
ejpam-6573	27	21	τ1	τ1	NOUN
ejpam-6573	27	22	,	,	PUNCT
ejpam-6573	27	23	τ2)β	τ2)β	ADJ
ejpam-6573	27	24	-	-	PUNCT
ejpam-6573	27	25	closed	close	VERB
ejpam-6573	27	26	functions	function	NOUN
ejpam-6573	27	27	.	.	PUNCT
ejpam-6573	28	1	furthermore	furthermore	ADV
ejpam-6573	28	2	,	,	PUNCT
ejpam-6573	28	3	several	several	ADJ
ejpam-6573	28	4	characterizations	characterization	NOUN
ejpam-6573	28	5	of	of	ADP
ejpam-6573	28	6	weakly	weakly	ADJ
ejpam-6573	28	7	(	(	PUNCT
ejpam-6573	28	8	τ1	τ1	NOUN
ejpam-6573	28	9	,	,	PUNCT
ejpam-6573	28	10	τ2)β	τ2)β	ADJ
ejpam-6573	28	11	-	-	PUNCT
ejpam-6573	28	12	open	open	ADJ
ejpam-6573	28	13	functions	function	NOUN
ejpam-6573	28	14	and	and	CCONJ
ejpam-6573	28	15	weakly	weakly	ADJ
ejpam-6573	28	16	(	(	PUNCT
ejpam-6573	28	17	τ1	τ1	NOUN
ejpam-6573	28	18	,	,	PUNCT
ejpam-6573	28	19	τ2)β	τ2)β	ADJ
ejpam-6573	28	20	-	-	PUNCT
ejpam-6573	28	21	closed	close	VERB
ejpam-6573	28	22	functions	function	NOUN
ejpam-6573	28	23	are	be	AUX
ejpam-6573	28	24	investigated	investigate	VERB
ejpam-6573	28	25	.	.	PUNCT
ejpam-6573	29	1	2	2	X
ejpam-6573	29	2	.	.	X
ejpam-6573	29	3	preliminaries	preliminary	NOUN
ejpam-6573	29	4	throughout	throughout	ADP
ejpam-6573	29	5	the	the	DET
ejpam-6573	29	6	present	present	ADJ
ejpam-6573	29	7	paper	paper	NOUN
ejpam-6573	29	8	,	,	PUNCT
ejpam-6573	29	9	spaces	space	NOUN
ejpam-6573	29	10	(	(	PUNCT
ejpam-6573	29	11	x	x	NOUN
ejpam-6573	29	12	,	,	PUNCT
ejpam-6573	29	13	τ1	τ1	NOUN
ejpam-6573	29	14	,	,	PUNCT
ejpam-6573	29	15	τ2	τ2	NOUN
ejpam-6573	29	16	)	)	PUNCT
ejpam-6573	29	17	and	and	CCONJ
ejpam-6573	29	18	(	(	PUNCT
ejpam-6573	29	19	y	y	PROPN
ejpam-6573	29	20	,	,	PUNCT
ejpam-6573	29	21	σ1	σ1	PROPN
ejpam-6573	29	22	,	,	PUNCT
ejpam-6573	29	23	σ2	σ2	NOUN
ejpam-6573	29	24	)	)	PUNCT
ejpam-6573	29	25	(	(	PUNCT
ejpam-6573	29	26	or	or	CCONJ
ejpam-6573	29	27	simply	simply	ADV
ejpam-6573	29	28	x	x	X
ejpam-6573	29	29	and	and	CCONJ
ejpam-6573	29	30	y	y	PROPN
ejpam-6573	29	31	)	)	PUNCT
ejpam-6573	29	32	always	always	ADV
ejpam-6573	29	33	mean	mean	VERB
ejpam-6573	29	34	bitopological	bitopological	ADJ
ejpam-6573	29	35	spaces	space	NOUN
ejpam-6573	29	36	on	on	ADP
ejpam-6573	29	37	which	which	PRON
ejpam-6573	29	38	no	no	DET
ejpam-6573	29	39	separation	separation	NOUN
ejpam-6573	29	40	axioms	axiom	NOUN
ejpam-6573	29	41	are	be	AUX
ejpam-6573	29	42	assumed	assume	VERB
ejpam-6573	29	43	unless	unless	SCONJ
ejpam-6573	29	44	explicitly	explicitly	ADV
ejpam-6573	29	45	stated	state	VERB
ejpam-6573	29	46	.	.	PUNCT
ejpam-6573	30	1	let	let	VERB
ejpam-6573	30	2	a	a	DET
ejpam-6573	30	3	be	be	AUX
ejpam-6573	30	4	a	a	DET
ejpam-6573	30	5	subset	subset	NOUN
ejpam-6573	30	6	of	of	ADP
ejpam-6573	30	7	a	a	DET
ejpam-6573	30	8	bitopological	bitopological	ADJ
ejpam-6573	30	9	space	space	NOUN
ejpam-6573	30	10	(	(	PUNCT
ejpam-6573	30	11	x	x	NOUN
ejpam-6573	30	12	,	,	PUNCT
ejpam-6573	30	13	τ1	τ1	NOUN
ejpam-6573	30	14	,	,	PUNCT
ejpam-6573	30	15	τ2	τ2	NOUN
ejpam-6573	30	16	)	)	PUNCT
ejpam-6573	30	17	.	.	PUNCT
ejpam-6573	31	1	the	the	DET
ejpam-6573	31	2	closure	closure	NOUN
ejpam-6573	31	3	of	of	ADP
ejpam-6573	31	4	a	a	PRON
ejpam-6573	31	5	and	and	CCONJ
ejpam-6573	31	6	the	the	DET
ejpam-6573	31	7	interior	interior	NOUN
ejpam-6573	31	8	of	of	ADP
ejpam-6573	31	9	a	a	PRON
ejpam-6573	31	10	with	with	ADP
ejpam-6573	31	11	respect	respect	NOUN
ejpam-6573	31	12	to	to	ADP
ejpam-6573	31	13	τi	τi	PROPN
ejpam-6573	31	14	are	be	AUX
ejpam-6573	31	15	denoted	denote	VERB
ejpam-6573	31	16	by	by	ADP
ejpam-6573	31	17	τi	τi	NOUN
ejpam-6573	31	18	-	-	PUNCT
ejpam-6573	31	19	cl(a	cl(a	NUM
ejpam-6573	31	20	)	)	PUNCT
ejpam-6573	31	21	and	and	CCONJ
ejpam-6573	31	22	τi	τi	NOUN
ejpam-6573	31	23	-	-	PUNCT
ejpam-6573	31	24	int(a	int(a	NOUN
ejpam-6573	31	25	)	)	PUNCT
ejpam-6573	31	26	,	,	PUNCT
ejpam-6573	31	27	respectively	respectively	ADV
ejpam-6573	31	28	,	,	PUNCT
ejpam-6573	31	29	for	for	ADP
ejpam-6573	31	30	i	i	PROPN
ejpam-6573	31	31	=	=	SYM
ejpam-6573	31	32	1	1	NUM
ejpam-6573	31	33	,	,	PUNCT
ejpam-6573	31	34	2	2	NUM
ejpam-6573	31	35	.	.	X
ejpam-6573	31	36	a	a	DET
ejpam-6573	31	37	subset	subset	NOUN
ejpam-6573	31	38	a	a	PRON
ejpam-6573	31	39	of	of	ADP
ejpam-6573	31	40	a	a	DET
ejpam-6573	31	41	bitopological	bitopological	ADJ
ejpam-6573	31	42	space	space	NOUN
ejpam-6573	31	43	(	(	PUNCT
ejpam-6573	31	44	x	x	NOUN
ejpam-6573	31	45	,	,	PUNCT
ejpam-6573	31	46	τ1	τ1	NOUN
ejpam-6573	31	47	,	,	PUNCT
ejpam-6573	31	48	τ2	τ2	NOUN
ejpam-6573	31	49	)	)	PUNCT
ejpam-6573	31	50	is	be	AUX
ejpam-6573	31	51	called	call	VERB
ejpam-6573	31	52	τ1τ2	τ1τ2	VERB
ejpam-6573	31	53	-	-	ADJ
ejpam-6573	31	54	closed	closed	ADJ
ejpam-6573	31	55	[	[	X
ejpam-6573	31	56	15	15	NUM
ejpam-6573	31	57	]	]	X
ejpam-6573	31	58	if	if	SCONJ
ejpam-6573	31	59	a	a	DET
ejpam-6573	31	60	=	=	NOUN
ejpam-6573	31	61	τ1	τ1	NOUN
ejpam-6573	31	62	-	-	PUNCT
ejpam-6573	31	63	cl(τ2	cl(τ2	NOUN
ejpam-6573	31	64	-	-	PUNCT
ejpam-6573	31	65	cl(a	cl(a	NUM
ejpam-6573	31	66	)	)	PUNCT
ejpam-6573	31	67	)	)	PUNCT
ejpam-6573	31	68	.	.	PUNCT
ejpam-6573	32	1	the	the	DET
ejpam-6573	32	2	complement	complement	NOUN
ejpam-6573	32	3	of	of	ADP
ejpam-6573	32	4	a	a	DET
ejpam-6573	32	5	τ1τ2	τ1τ2	ADJ
ejpam-6573	32	6	-	-	ADJ
ejpam-6573	32	7	closed	closed	ADJ
ejpam-6573	32	8	set	set	NOUN
ejpam-6573	32	9	is	be	AUX
ejpam-6573	32	10	called	call	VERB
ejpam-6573	32	11	τ1τ2	τ1τ2	NOUN
ejpam-6573	32	12	-	-	ADJ
ejpam-6573	32	13	open	open	ADJ
ejpam-6573	32	14	.	.	PUNCT
ejpam-6573	33	1	let	let	VERB
ejpam-6573	33	2	a	a	DET
ejpam-6573	33	3	be	be	AUX
ejpam-6573	33	4	a	a	DET
ejpam-6573	33	5	subset	subset	NOUN
ejpam-6573	33	6	of	of	ADP
ejpam-6573	33	7	a	a	DET
ejpam-6573	33	8	bitopological	bitopological	ADJ
ejpam-6573	33	9	space	space	NOUN
ejpam-6573	33	10	(	(	PUNCT
ejpam-6573	33	11	x	x	NOUN
ejpam-6573	33	12	,	,	PUNCT
ejpam-6573	33	13	τ1	τ1	NOUN
ejpam-6573	33	14	,	,	PUNCT
ejpam-6573	33	15	τ2	τ2	NOUN
ejpam-6573	33	16	)	)	PUNCT
ejpam-6573	33	17	.	.	PUNCT
ejpam-6573	34	1	the	the	DET
ejpam-6573	34	2	intersection	intersection	NOUN
ejpam-6573	34	3	of	of	ADP
ejpam-6573	34	4	all	all	DET
ejpam-6573	34	5	τ1τ2	τ1τ2	ADJ
ejpam-6573	34	6	-	-	ADJ
ejpam-6573	34	7	closed	closed	ADJ
ejpam-6573	34	8	sets	set	NOUN
ejpam-6573	34	9	of	of	ADP
ejpam-6573	34	10	x	x	PUNCT
ejpam-6573	34	11	containing	contain	VERB
ejpam-6573	34	12	a	a	PRON
ejpam-6573	34	13	is	be	AUX
ejpam-6573	34	14	called	call	VERB
ejpam-6573	34	15	the	the	DET
ejpam-6573	34	16	τ1τ2	τ1τ2	NOUN
ejpam-6573	34	17	-	-	NOUN
ejpam-6573	34	18	closure	closure	NOUN
ejpam-6573	34	19	[	[	X
ejpam-6573	34	20	15	15	NUM
ejpam-6573	34	21	]	]	PUNCT
ejpam-6573	34	22	of	of	ADP
ejpam-6573	34	23	a	a	PRON
ejpam-6573	34	24	and	and	CCONJ
ejpam-6573	34	25	is	be	AUX
ejpam-6573	34	26	denoted	denote	VERB
ejpam-6573	34	27	by	by	ADP
ejpam-6573	34	28	τ1τ2	τ1τ2	NOUN
ejpam-6573	34	29	-	-	NUM
ejpam-6573	34	30	cl(a	cl(a	NUM
ejpam-6573	34	31	)	)	PUNCT
ejpam-6573	34	32	.	.	PUNCT
ejpam-6573	35	1	the	the	DET
ejpam-6573	35	2	union	union	NOUN
ejpam-6573	35	3	of	of	ADP
ejpam-6573	35	4	all	all	DET
ejpam-6573	35	5	τ1τ2	τ1τ2	ADJ
ejpam-6573	35	6	-	-	ADJ
ejpam-6573	35	7	open	open	ADJ
ejpam-6573	35	8	sets	set	NOUN
ejpam-6573	35	9	of	of	ADP
ejpam-6573	35	10	x	x	PUNCT
ejpam-6573	35	11	contained	contain	VERB
ejpam-6573	35	12	in	in	ADP
ejpam-6573	35	13	a	a	PRON
ejpam-6573	35	14	is	be	AUX
ejpam-6573	35	15	called	call	VERB
ejpam-6573	35	16	the	the	DET
ejpam-6573	35	17	τ1τ2	τ1τ2	NOUN
ejpam-6573	35	18	-	-	ADJ
ejpam-6573	35	19	interior	interior	ADJ
ejpam-6573	35	20	[	[	X
ejpam-6573	35	21	15	15	NUM
ejpam-6573	35	22	]	]	PUNCT
ejpam-6573	35	23	of	of	ADP
ejpam-6573	35	24	a	a	PRON
ejpam-6573	35	25	and	and	CCONJ
ejpam-6573	35	26	is	be	AUX
ejpam-6573	35	27	denoted	denote	VERB
ejpam-6573	35	28	by	by	ADP
ejpam-6573	35	29	τ1τ2	τ1τ2	NOUN
ejpam-6573	35	30	-	-	ADJ
ejpam-6573	35	31	int(a	int(a	NOUN
ejpam-6573	35	32	)	)	PUNCT
ejpam-6573	35	33	.	.	PUNCT
ejpam-6573	36	1	lemma	lemma	PROPN
ejpam-6573	36	2	1	1	NUM
ejpam-6573	36	3	.	.	PUNCT
ejpam-6573	37	1	[	[	X
ejpam-6573	37	2	15	15	NUM
ejpam-6573	37	3	]	]	PUNCT
ejpam-6573	37	4	let	let	VERB
ejpam-6573	37	5	a	a	PRON
ejpam-6573	37	6	and	and	CCONJ
ejpam-6573	37	7	b	b	NOUN
ejpam-6573	37	8	be	be	AUX
ejpam-6573	37	9	subsets	subset	NOUN
ejpam-6573	37	10	of	of	ADP
ejpam-6573	37	11	a	a	DET
ejpam-6573	37	12	bitopological	bitopological	ADJ
ejpam-6573	37	13	space	space	NOUN
ejpam-6573	37	14	(	(	PUNCT
ejpam-6573	37	15	x	x	NOUN
ejpam-6573	37	16	,	,	PUNCT
ejpam-6573	37	17	τ1	τ1	NOUN
ejpam-6573	37	18	,	,	PUNCT
ejpam-6573	37	19	τ2	τ2	NOUN
ejpam-6573	37	20	)	)	PUNCT
ejpam-6573	37	21	.	.	PUNCT
ejpam-6573	38	1	for	for	ADP
ejpam-6573	38	2	the	the	DET
ejpam-6573	38	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6573	38	4	,	,	PUNCT
ejpam-6573	38	5	the	the	DET
ejpam-6573	38	6	following	follow	VERB
ejpam-6573	38	7	properties	property	NOUN
ejpam-6573	38	8	hold	hold	VERB
ejpam-6573	38	9	:	:	PUNCT
ejpam-6573	38	10	(	(	PUNCT
ejpam-6573	38	11	1	1	X
ejpam-6573	38	12	)	)	PUNCT
ejpam-6573	38	13	a	a	DET
ejpam-6573	38	14	⊆	⊆	NUM
ejpam-6573	38	15	τ1τ2	τ1τ2	NOUN
ejpam-6573	38	16	-	-	NUM
ejpam-6573	38	17	cl(a	cl(a	NUM
ejpam-6573	38	18	)	)	PUNCT
ejpam-6573	38	19	and	and	CCONJ
ejpam-6573	38	20	τ1τ2	τ1τ2	NOUN
ejpam-6573	38	21	-	-	ADJ
ejpam-6573	38	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6573	38	23	-	-	PUNCT
ejpam-6573	38	24	cl(a	cl(a	NUM
ejpam-6573	38	25	)	)	PUNCT
ejpam-6573	38	26	)	)	PUNCT
ejpam-6573	39	1	=	=	PUNCT
ejpam-6573	39	2	τ1τ2	τ1τ2	NOUN
ejpam-6573	39	3	-	-	NUM
ejpam-6573	39	4	cl(a	cl(a	NUM
ejpam-6573	39	5	)	)	PUNCT
ejpam-6573	39	6	.	.	PUNCT
ejpam-6573	40	1	(	(	PUNCT
ejpam-6573	40	2	2	2	X
ejpam-6573	40	3	)	)	PUNCT
ejpam-6573	40	4	if	if	SCONJ
ejpam-6573	40	5	a	a	DET
ejpam-6573	40	6	⊆	⊆	NUM
ejpam-6573	40	7	b	b	NOUN
ejpam-6573	40	8	,	,	PUNCT
ejpam-6573	40	9	then	then	ADV
ejpam-6573	40	10	τ1τ2	τ1τ2	NOUN
ejpam-6573	40	11	-	-	NUM
ejpam-6573	40	12	cl(a	cl(a	NUM
ejpam-6573	40	13	)	)	PUNCT
ejpam-6573	40	14	⊆	⊆	NUM
ejpam-6573	40	15	τ1τ2	τ1τ2	NOUN
ejpam-6573	40	16	-	-	NOUN
ejpam-6573	40	17	cl(b	cl(b	NOUN
ejpam-6573	40	18	)	)	PUNCT
ejpam-6573	40	19	.	.	PUNCT
ejpam-6573	41	1	(	(	PUNCT
ejpam-6573	41	2	3	3	X
ejpam-6573	41	3	)	)	PUNCT
ejpam-6573	41	4	τ1τ2	τ1τ2	NOUN
ejpam-6573	41	5	-	-	NUM
ejpam-6573	41	6	cl(a	cl(a	NUM
ejpam-6573	41	7	)	)	PUNCT
ejpam-6573	41	8	is	be	AUX
ejpam-6573	41	9	τ1τ2	τ1τ2	NOUN
ejpam-6573	41	10	-	-	ADJ
ejpam-6573	41	11	closed	closed	ADJ
ejpam-6573	41	12	.	.	PUNCT
ejpam-6573	42	1	(	(	PUNCT
ejpam-6573	42	2	4	4	X
ejpam-6573	42	3	)	)	PUNCT
ejpam-6573	42	4	a	a	PRON
ejpam-6573	42	5	is	be	AUX
ejpam-6573	42	6	τ1τ2	τ1τ2	NOUN
ejpam-6573	42	7	-	-	ADJ
ejpam-6573	42	8	closed	closed	ADJ
ejpam-6573	42	9	if	if	SCONJ
ejpam-6573	42	10	and	and	CCONJ
ejpam-6573	42	11	only	only	ADV
ejpam-6573	42	12	if	if	SCONJ
ejpam-6573	42	13	a	a	DET
ejpam-6573	42	14	=	=	PUNCT
ejpam-6573	42	15	τ1τ2	τ1τ2	NOUN
ejpam-6573	42	16	-	-	NUM
ejpam-6573	42	17	cl(a	cl(a	NUM
ejpam-6573	42	18	)	)	PUNCT
ejpam-6573	42	19	.	.	PUNCT
ejpam-6573	43	1	(	(	PUNCT
ejpam-6573	43	2	5	5	X
ejpam-6573	43	3	)	)	PUNCT
ejpam-6573	43	4	τ1τ2	τ1τ2	NOUN
ejpam-6573	43	5	-	-	NOUN
ejpam-6573	43	6	cl(x	cl(x	X
ejpam-6573	43	7	−a	−a	NOUN
ejpam-6573	43	8	)	)	PUNCT
ejpam-6573	44	1	=	=	PUNCT
ejpam-6573	44	2	x	x	X
ejpam-6573	45	1	−	−	ADP
ejpam-6573	45	2	τ1τ2	τ1τ2	NOUN
ejpam-6573	45	3	-	-	PUNCT
ejpam-6573	45	4	int(a	int(a	NOUN
ejpam-6573	45	5	)	)	PUNCT
ejpam-6573	45	6	.	.	PUNCT
ejpam-6573	46	1	a	a	DET
ejpam-6573	46	2	subset	subset	NOUN
ejpam-6573	46	3	a	a	PRON
ejpam-6573	46	4	of	of	ADP
ejpam-6573	46	5	a	a	DET
ejpam-6573	46	6	bitopological	bitopological	ADJ
ejpam-6573	46	7	space	space	NOUN
ejpam-6573	46	8	(	(	PUNCT
ejpam-6573	46	9	x	x	NOUN
ejpam-6573	46	10	,	,	PUNCT
ejpam-6573	46	11	τ1	τ1	NOUN
ejpam-6573	46	12	,	,	PUNCT
ejpam-6573	46	13	τ2	τ2	NOUN
ejpam-6573	46	14	)	)	PUNCT
ejpam-6573	46	15	is	be	AUX
ejpam-6573	46	16	said	say	VERB
ejpam-6573	46	17	to	to	PART
ejpam-6573	46	18	be	be	AUX
ejpam-6573	46	19	(	(	PUNCT
ejpam-6573	46	20	τ1	τ1	NOUN
ejpam-6573	46	21	,	,	PUNCT
ejpam-6573	46	22	τ2)r	τ2)r	NOUN
ejpam-6573	46	23	-	-	PUNCT
ejpam-6573	46	24	open	open	NOUN
ejpam-6573	47	1	[	[	X
ejpam-6573	47	2	16	16	NUM
ejpam-6573	47	3	]	]	PUNCT
ejpam-6573	47	4	(	(	PUNCT
ejpam-6573	47	5	resp	resp	NOUN
ejpam-6573	47	6	.	.	PUNCT
ejpam-6573	48	1	(	(	PUNCT
ejpam-6573	48	2	τ1	τ1	NOUN
ejpam-6573	48	3	,	,	PUNCT
ejpam-6573	48	4	τ2)s	τ2)s	NOUN
ejpam-6573	48	5	-	-	PUNCT
ejpam-6573	48	6	open	open	ADJ
ejpam-6573	48	7	[	[	X
ejpam-6573	48	8	17	17	NUM
ejpam-6573	48	9	]	]	PUNCT
ejpam-6573	48	10	,	,	PUNCT
ejpam-6573	48	11	(	(	PUNCT
ejpam-6573	48	12	τ1	τ1	NOUN
ejpam-6573	48	13	,	,	PUNCT
ejpam-6573	48	14	τ2)p	τ2)p	NOUN
ejpam-6573	48	15	-	-	ADJ
ejpam-6573	48	16	open	open	ADJ
ejpam-6573	48	17	[	[	X
ejpam-6573	48	18	17	17	NUM
ejpam-6573	48	19	]	]	PUNCT
ejpam-6573	48	20	,	,	PUNCT
ejpam-6573	48	21	(	(	PUNCT
ejpam-6573	48	22	τ1	τ1	NOUN
ejpam-6573	48	23	,	,	PUNCT
ejpam-6573	48	24	τ2)β	τ2)β	ADJ
ejpam-6573	48	25	-	-	PUNCT
ejpam-6573	48	26	open	open	NOUN
ejpam-6573	49	1	[	[	X
ejpam-6573	49	2	17	17	NUM
ejpam-6573	49	3	]	]	SYM
ejpam-6573	49	4	)	)	PUNCT
ejpam-6573	49	5	if	if	SCONJ
ejpam-6573	49	6	a	a	DET
ejpam-6573	49	7	=	=	PUNCT
ejpam-6573	49	8	τ1τ2	τ1τ2	NOUN
ejpam-6573	49	9	-	-	NOUN
ejpam-6573	49	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	49	11	-	-	PUNCT
ejpam-6573	49	12	cl(a	cl(a	NUM
ejpam-6573	49	13	)	)	PUNCT
ejpam-6573	49	14	)	)	PUNCT
ejpam-6573	49	15	(	(	PUNCT
ejpam-6573	49	16	resp	resp	NOUN
ejpam-6573	49	17	.	.	PUNCT
ejpam-6573	50	1	a	a	DET
ejpam-6573	50	2	⊆	⊆	NUM
ejpam-6573	50	3	τ1τ2	τ1τ2	NOUN
ejpam-6573	50	4	-	-	ADJ
ejpam-6573	50	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6573	50	6	-	-	PUNCT
ejpam-6573	50	7	int(a	int(a	NOUN
ejpam-6573	50	8	)	)	PUNCT
ejpam-6573	50	9	)	)	PUNCT
ejpam-6573	50	10	,	,	PUNCT
ejpam-6573	50	11	a	a	DET
ejpam-6573	50	12	⊆	⊆	NUM
ejpam-6573	50	13	τ1τ2	τ1τ2	NOUN
ejpam-6573	50	14	-	-	NOUN
ejpam-6573	50	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	50	16	-	-	PUNCT
ejpam-6573	50	17	cl(a	cl(a	NUM
ejpam-6573	50	18	)	)	PUNCT
ejpam-6573	50	19	)	)	PUNCT
ejpam-6573	50	20	,	,	PUNCT
ejpam-6573	50	21	a	a	DET
ejpam-6573	50	22	⊆	⊆	NUM
ejpam-6573	50	23	τ1τ2	τ1τ2	NOUN
ejpam-6573	50	24	-	-	PUNCT
ejpam-6573	50	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6573	50	26	-	-	PUNCT
ejpam-6573	50	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	50	28	-	-	PUNCT
ejpam-6573	50	29	cl(a	cl(a	NUM
ejpam-6573	50	30	)	)	PUNCT
ejpam-6573	50	31	)	)	PUNCT
ejpam-6573	50	32	)	)	PUNCT
ejpam-6573	50	33	)	)	PUNCT
ejpam-6573	50	34	.	.	PUNCT
ejpam-6573	51	1	the	the	DET
ejpam-6573	51	2	complement	complement	NOUN
ejpam-6573	51	3	of	of	ADP
ejpam-6573	51	4	a	a	DET
ejpam-6573	51	5	(	(	PUNCT
ejpam-6573	51	6	τ1	τ1	NOUN
ejpam-6573	51	7	,	,	PUNCT
ejpam-6573	51	8	τ2)r	τ2)r	NOUN
ejpam-6573	51	9	-	-	PUNCT
ejpam-6573	51	10	open	open	ADJ
ejpam-6573	51	11	(	(	PUNCT
ejpam-6573	51	12	resp	resp	NOUN
ejpam-6573	51	13	.	.	PUNCT
ejpam-6573	52	1	(	(	PUNCT
ejpam-6573	52	2	τ1	τ1	NOUN
ejpam-6573	52	3	,	,	PUNCT
ejpam-6573	52	4	τ2)s	τ2)s	NOUN
ejpam-6573	52	5	-	-	PUNCT
ejpam-6573	52	6	open	open	ADJ
ejpam-6573	52	7	,	,	PUNCT
ejpam-6573	52	8	(	(	PUNCT
ejpam-6573	52	9	τ1	τ1	NOUN
ejpam-6573	52	10	,	,	PUNCT
ejpam-6573	52	11	τ2)p	τ2)p	NOUN
ejpam-6573	52	12	-	-	ADJ
ejpam-6573	52	13	open	open	ADJ
ejpam-6573	52	14	,	,	PUNCT
ejpam-6573	52	15	(	(	PUNCT
ejpam-6573	52	16	τ1	τ1	NOUN
ejpam-6573	52	17	,	,	PUNCT
ejpam-6573	52	18	τ2)β	τ2)β	ADJ
ejpam-6573	52	19	-	-	PUNCT
ejpam-6573	52	20	open	open	ADJ
ejpam-6573	52	21	)	)	PUNCT
ejpam-6573	52	22	set	set	NOUN
ejpam-6573	52	23	is	be	AUX
ejpam-6573	52	24	called	call	VERB
ejpam-6573	52	25	(	(	PUNCT
ejpam-6573	52	26	τ1	τ1	NOUN
ejpam-6573	52	27	,	,	PUNCT
ejpam-6573	52	28	τ2)r	τ2)r	NOUN
ejpam-6573	52	29	-	-	PUNCT
ejpam-6573	52	30	closed	closed	ADJ
ejpam-6573	52	31	(	(	PUNCT
ejpam-6573	52	32	resp	resp	NOUN
ejpam-6573	52	33	.	.	PUNCT
ejpam-6573	53	1	(	(	PUNCT
ejpam-6573	53	2	τ1	τ1	NOUN
ejpam-6573	53	3	,	,	PUNCT
ejpam-6573	53	4	τ2)s	τ2)s	NOUN
ejpam-6573	53	5	-	-	PUNCT
ejpam-6573	53	6	closed	closed	ADJ
ejpam-6573	53	7	,	,	PUNCT
ejpam-6573	53	8	(	(	PUNCT
ejpam-6573	53	9	τ1	τ1	NOUN
ejpam-6573	53	10	,	,	PUNCT
ejpam-6573	53	11	τ2)p	τ2)p	NOUN
ejpam-6573	53	12	-	-	PUNCT
ejpam-6573	53	13	closed	closed	ADJ
ejpam-6573	53	14	,	,	PUNCT
ejpam-6573	53	15	(	(	PUNCT
ejpam-6573	53	16	τ1	τ1	NOUN
ejpam-6573	53	17	,	,	PUNCT
ejpam-6573	53	18	τ2)β	τ2)β	ADJ
ejpam-6573	53	19	-	-	PUNCT
ejpam-6573	53	20	closed	closed	ADJ
ejpam-6573	53	21	)	)	PUNCT
ejpam-6573	53	22	.	.	PUNCT
ejpam-6573	54	1	a	a	DET
ejpam-6573	54	2	subset	subset	NOUN
ejpam-6573	54	3	a	a	PRON
ejpam-6573	54	4	of	of	ADP
ejpam-6573	54	5	a	a	DET
ejpam-6573	54	6	bitopological	bitopological	ADJ
ejpam-6573	54	7	space	space	NOUN
ejpam-6573	54	8	(	(	PUNCT
ejpam-6573	54	9	x	x	NOUN
ejpam-6573	54	10	,	,	PUNCT
ejpam-6573	54	11	τ1	τ1	NOUN
ejpam-6573	54	12	,	,	PUNCT
ejpam-6573	54	13	τ2	τ2	NOUN
ejpam-6573	54	14	)	)	PUNCT
ejpam-6573	54	15	is	be	AUX
ejpam-6573	54	16	said	say	VERB
ejpam-6573	54	17	to	to	PART
ejpam-6573	54	18	be	be	AUX
ejpam-6573	54	19	α(τ1	α(τ1	NOUN
ejpam-6573	54	20	,	,	PUNCT
ejpam-6573	54	21	τ2)-open	τ2)-open	ADJ
ejpam-6573	54	22	[	[	X
ejpam-6573	54	23	18	18	NUM
ejpam-6573	54	24	]	]	PUNCT
ejpam-6573	54	25	if	if	SCONJ
ejpam-6573	54	26	a	a	DET
ejpam-6573	54	27	⊆	⊆	NUM
ejpam-6573	54	28	m.	m.	NOUN
ejpam-6573	54	29	chiangpradit	chiangpradit	NOUN
ejpam-6573	54	30	,	,	PUNCT
ejpam-6573	54	31	a.	a.	PROPN
ejpam-6573	54	32	sama	sama	PROPN
ejpam-6573	54	33	-	-	PUNCT
ejpam-6573	54	34	ae	ae	PROPN
ejpam-6573	54	35	,	,	PUNCT
ejpam-6573	54	36	c.	c.	PROPN
ejpam-6573	54	37	boonpok	boonpok	PROPN
ejpam-6573	54	38	/	/	SYM
ejpam-6573	54	39	eur	eur	PROPN
ejpam-6573	54	40	.	.	PUNCT
ejpam-6573	55	1	j.	j.	PROPN
ejpam-6573	55	2	pure	pure	PROPN
ejpam-6573	55	3	appl	appl	PROPN
ejpam-6573	55	4	.	.	PROPN
ejpam-6573	55	5	math	math	PROPN
ejpam-6573	55	6	,	,	PUNCT
ejpam-6573	55	7	18	18	NUM
ejpam-6573	55	8	(	(	PUNCT
ejpam-6573	55	9	3	3	NUM
ejpam-6573	55	10	)	)	PUNCT
ejpam-6573	55	11	(	(	PUNCT
ejpam-6573	55	12	2025	2025	NUM
ejpam-6573	55	13	)	)	PUNCT
ejpam-6573	55	14	,	,	PUNCT
ejpam-6573	55	15	6573	6573	NUM
ejpam-6573	55	16	3	3	NUM
ejpam-6573	55	17	of	of	ADP
ejpam-6573	55	18	8	8	NUM
ejpam-6573	55	19	τ1τ2	τ1τ2	NOUN
ejpam-6573	55	20	-	-	NOUN
ejpam-6573	55	21	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	55	22	-	-	PUNCT
ejpam-6573	55	23	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6573	55	24	-	-	PUNCT
ejpam-6573	55	25	int(a	int(a	NOUN
ejpam-6573	55	26	)	)	PUNCT
ejpam-6573	55	27	)	)	PUNCT
ejpam-6573	55	28	)	)	PUNCT
ejpam-6573	55	29	.	.	PUNCT
ejpam-6573	56	1	the	the	DET
ejpam-6573	56	2	complement	complement	NOUN
ejpam-6573	56	3	of	of	ADP
ejpam-6573	56	4	an	an	DET
ejpam-6573	56	5	α(τ1	α(τ1	NOUN
ejpam-6573	56	6	,	,	PUNCT
ejpam-6573	56	7	τ2)-open	τ2)-open	ADJ
ejpam-6573	56	8	set	set	NOUN
ejpam-6573	56	9	is	be	AUX
ejpam-6573	56	10	said	say	VERB
ejpam-6573	56	11	to	to	PART
ejpam-6573	56	12	be	be	AUX
ejpam-6573	56	13	α(τ1	α(τ1	NOUN
ejpam-6573	56	14	,	,	PUNCT
ejpam-6573	56	15	τ2)-closed	τ2)-close	VERB
ejpam-6573	56	16	.	.	PUNCT
ejpam-6573	57	1	let	let	VERB
ejpam-6573	57	2	a	a	DET
ejpam-6573	57	3	be	be	AUX
ejpam-6573	57	4	a	a	DET
ejpam-6573	57	5	subset	subset	NOUN
ejpam-6573	57	6	of	of	ADP
ejpam-6573	57	7	a	a	DET
ejpam-6573	57	8	bitopological	bitopological	ADJ
ejpam-6573	57	9	space	space	NOUN
ejpam-6573	57	10	(	(	PUNCT
ejpam-6573	57	11	x	x	NOUN
ejpam-6573	57	12	,	,	PUNCT
ejpam-6573	57	13	τ1	τ1	NOUN
ejpam-6573	57	14	,	,	PUNCT
ejpam-6573	57	15	τ2	τ2	NOUN
ejpam-6573	57	16	)	)	PUNCT
ejpam-6573	57	17	.	.	PUNCT
ejpam-6573	58	1	the	the	DET
ejpam-6573	58	2	intersection	intersection	NOUN
ejpam-6573	58	3	of	of	ADP
ejpam-6573	58	4	all	all	DET
ejpam-6573	58	5	(	(	PUNCT
ejpam-6573	58	6	τ1	τ1	NOUN
ejpam-6573	58	7	,	,	PUNCT
ejpam-6573	58	8	τ2)β	τ2)β	ADJ
ejpam-6573	58	9	-	-	PUNCT
ejpam-6573	58	10	closed	close	VERB
ejpam-6573	58	11	sets	set	NOUN
ejpam-6573	58	12	of	of	ADP
ejpam-6573	58	13	x	x	PUNCT
ejpam-6573	58	14	containing	contain	VERB
ejpam-6573	58	15	a	a	PRON
ejpam-6573	58	16	is	be	AUX
ejpam-6573	58	17	called	call	VERB
ejpam-6573	58	18	the	the	DET
ejpam-6573	58	19	(	(	PUNCT
ejpam-6573	58	20	τ1	τ1	NOUN
ejpam-6573	58	21	,	,	PUNCT
ejpam-6573	58	22	τ2)β	τ2)β	ADJ
ejpam-6573	58	23	-	-	PUNCT
ejpam-6573	58	24	closure	closure	NOUN
ejpam-6573	58	25	[	[	X
ejpam-6573	58	26	19	19	NUM
ejpam-6573	58	27	]	]	PUNCT
ejpam-6573	58	28	of	of	ADP
ejpam-6573	58	29	a	a	PRON
ejpam-6573	58	30	and	and	CCONJ
ejpam-6573	58	31	is	be	AUX
ejpam-6573	58	32	denoted	denote	VERB
ejpam-6573	58	33	by	by	ADP
ejpam-6573	58	34	(	(	PUNCT
ejpam-6573	58	35	τ1	τ1	NOUN
ejpam-6573	58	36	,	,	PUNCT
ejpam-6573	58	37	τ2)β	τ2)β	NOUN
ejpam-6573	58	38	-	-	PUNCT
ejpam-6573	58	39	cl(a	cl(a	NUM
ejpam-6573	58	40	)	)	PUNCT
ejpam-6573	58	41	.	.	PUNCT
ejpam-6573	59	1	the	the	DET
ejpam-6573	59	2	union	union	NOUN
ejpam-6573	59	3	of	of	ADP
ejpam-6573	59	4	all	all	DET
ejpam-6573	59	5	(	(	PUNCT
ejpam-6573	59	6	τ1	τ1	NOUN
ejpam-6573	59	7	,	,	PUNCT
ejpam-6573	59	8	τ2)β	τ2)β	ADJ
ejpam-6573	59	9	-	-	PUNCT
ejpam-6573	59	10	open	open	ADJ
ejpam-6573	59	11	sets	set	NOUN
ejpam-6573	59	12	of	of	ADP
ejpam-6573	59	13	x	x	PUNCT
ejpam-6573	59	14	contained	contain	VERB
ejpam-6573	59	15	in	in	ADP
ejpam-6573	59	16	a	a	PRON
ejpam-6573	59	17	is	be	AUX
ejpam-6573	59	18	called	call	VERB
ejpam-6573	59	19	the	the	DET
ejpam-6573	59	20	(	(	PUNCT
ejpam-6573	59	21	τ1	τ1	NOUN
ejpam-6573	59	22	,	,	PUNCT
ejpam-6573	59	23	τ2)β	τ2)β	ADJ
ejpam-6573	59	24	-	-	PUNCT
ejpam-6573	59	25	interior	interior	NOUN
ejpam-6573	59	26	[	[	X
ejpam-6573	59	27	19	19	NUM
ejpam-6573	59	28	]	]	PUNCT
ejpam-6573	59	29	of	of	ADP
ejpam-6573	59	30	a	a	PRON
ejpam-6573	59	31	and	and	CCONJ
ejpam-6573	59	32	is	be	AUX
ejpam-6573	59	33	denoted	denote	VERB
ejpam-6573	59	34	by	by	ADP
ejpam-6573	59	35	(	(	PUNCT
ejpam-6573	59	36	τ1	τ1	NOUN
ejpam-6573	59	37	,	,	PUNCT
ejpam-6573	59	38	τ2)β	τ2)β	NOUN
ejpam-6573	59	39	-	-	PUNCT
ejpam-6573	59	40	int(a	int(a	NOUN
ejpam-6573	59	41	)	)	PUNCT
ejpam-6573	59	42	.	.	PUNCT
ejpam-6573	60	1	lemma	lemma	PROPN
ejpam-6573	60	2	2	2	NUM
ejpam-6573	60	3	.	.	PUNCT
ejpam-6573	61	1	[	[	X
ejpam-6573	61	2	19	19	NUM
ejpam-6573	61	3	]	]	PUNCT
ejpam-6573	61	4	for	for	ADP
ejpam-6573	61	5	subsets	subset	NOUN
ejpam-6573	61	6	a	a	PRON
ejpam-6573	61	7	and	and	CCONJ
ejpam-6573	61	8	b	b	NOUN
ejpam-6573	61	9	of	of	ADP
ejpam-6573	61	10	a	a	DET
ejpam-6573	61	11	bitopological	bitopological	ADJ
ejpam-6573	61	12	space	space	NOUN
ejpam-6573	61	13	(	(	PUNCT
ejpam-6573	61	14	x	x	NOUN
ejpam-6573	61	15	,	,	PUNCT
ejpam-6573	61	16	τ1	τ1	NOUN
ejpam-6573	61	17	,	,	PUNCT
ejpam-6573	61	18	τ2	τ2	NOUN
ejpam-6573	61	19	)	)	PUNCT
ejpam-6573	61	20	,	,	PUNCT
ejpam-6573	61	21	the	the	DET
ejpam-6573	61	22	following	follow	VERB
ejpam-6573	61	23	properties	property	NOUN
ejpam-6573	61	24	hold	hold	VERB
ejpam-6573	61	25	:	:	PUNCT
ejpam-6573	61	26	(	(	PUNCT
ejpam-6573	61	27	1	1	X
ejpam-6573	61	28	)	)	PUNCT
ejpam-6573	61	29	a	a	DET
ejpam-6573	61	30	⊆	⊆	NUM
ejpam-6573	61	31	(	(	PUNCT
ejpam-6573	61	32	τ1	τ1	NOUN
ejpam-6573	61	33	,	,	PUNCT
ejpam-6573	61	34	τ2)β	τ2)β	NOUN
ejpam-6573	61	35	-	-	PUNCT
ejpam-6573	61	36	cl(a	cl(a	NUM
ejpam-6573	61	37	)	)	PUNCT
ejpam-6573	61	38	and	and	CCONJ
ejpam-6573	61	39	(	(	PUNCT
ejpam-6573	61	40	τ1	τ1	NOUN
ejpam-6573	61	41	,	,	PUNCT
ejpam-6573	61	42	τ2)β	τ2)β	NOUN
ejpam-6573	61	43	-	-	PUNCT
ejpam-6573	61	44	cl((τ1	cl((τ1	PROPN
ejpam-6573	61	45	,	,	PUNCT
ejpam-6573	61	46	τ2)β	τ2)β	NOUN
ejpam-6573	61	47	-	-	PUNCT
ejpam-6573	61	48	cl(a	cl(a	NUM
ejpam-6573	61	49	)	)	PUNCT
ejpam-6573	61	50	)	)	PUNCT
ejpam-6573	62	1	=	=	PRON
ejpam-6573	62	2	(	(	PUNCT
ejpam-6573	62	3	τ1	τ1	NOUN
ejpam-6573	62	4	,	,	PUNCT
ejpam-6573	62	5	τ2)β	τ2)β	NOUN
ejpam-6573	62	6	-	-	PUNCT
ejpam-6573	62	7	cl(a	cl(a	NUM
ejpam-6573	62	8	)	)	PUNCT
ejpam-6573	62	9	.	.	PUNCT
ejpam-6573	63	1	(	(	PUNCT
ejpam-6573	63	2	2	2	X
ejpam-6573	63	3	)	)	PUNCT
ejpam-6573	63	4	if	if	SCONJ
ejpam-6573	63	5	a	a	DET
ejpam-6573	63	6	⊆	⊆	NUM
ejpam-6573	63	7	b	b	NOUN
ejpam-6573	63	8	,	,	PUNCT
ejpam-6573	63	9	then	then	ADV
ejpam-6573	63	10	(	(	PUNCT
ejpam-6573	63	11	τ1	τ1	NOUN
ejpam-6573	63	12	,	,	PUNCT
ejpam-6573	63	13	τ2)β	τ2)β	NOUN
ejpam-6573	63	14	-	-	PUNCT
ejpam-6573	63	15	cl(a	cl(a	NUM
ejpam-6573	63	16	)	)	PUNCT
ejpam-6573	63	17	⊆	⊆	NUM
ejpam-6573	63	18	(	(	PUNCT
ejpam-6573	63	19	τ1	τ1	NOUN
ejpam-6573	63	20	,	,	PUNCT
ejpam-6573	63	21	τ2)β	τ2)β	ADJ
ejpam-6573	63	22	-	-	PUNCT
ejpam-6573	63	23	cl(b	cl(b	NOUN
ejpam-6573	63	24	)	)	PUNCT
ejpam-6573	63	25	.	.	PUNCT
ejpam-6573	64	1	(	(	PUNCT
ejpam-6573	64	2	3	3	X
ejpam-6573	64	3	)	)	PUNCT
ejpam-6573	64	4	(	(	PUNCT
ejpam-6573	64	5	τ1	τ1	NOUN
ejpam-6573	64	6	,	,	PUNCT
ejpam-6573	64	7	τ2)β	τ2)β	NOUN
ejpam-6573	64	8	-	-	PUNCT
ejpam-6573	64	9	cl(a	cl(a	NUM
ejpam-6573	64	10	)	)	PUNCT
ejpam-6573	64	11	is	be	AUX
ejpam-6573	64	12	(	(	PUNCT
ejpam-6573	64	13	τ1	τ1	NOUN
ejpam-6573	64	14	,	,	PUNCT
ejpam-6573	64	15	τ2)β	τ2)β	NOUN
ejpam-6573	64	16	-	-	PUNCT
ejpam-6573	64	17	closed	closed	ADJ
ejpam-6573	64	18	.	.	PUNCT
ejpam-6573	65	1	(	(	PUNCT
ejpam-6573	65	2	4	4	X
ejpam-6573	65	3	)	)	PUNCT
ejpam-6573	65	4	a	a	DET
ejpam-6573	65	5	is	is	NOUN
ejpam-6573	65	6	(	(	PUNCT
ejpam-6573	65	7	τ1	τ1	NOUN
ejpam-6573	65	8	,	,	PUNCT
ejpam-6573	65	9	τ2)β	τ2)β	NOUN
ejpam-6573	65	10	-	-	PUNCT
ejpam-6573	65	11	closed	close	VERB
ejpam-6573	65	12	if	if	SCONJ
ejpam-6573	65	13	and	and	CCONJ
ejpam-6573	65	14	only	only	ADV
ejpam-6573	65	15	if	if	SCONJ
ejpam-6573	65	16	a	a	DET
ejpam-6573	65	17	=	=	X
ejpam-6573	65	18	(	(	PUNCT
ejpam-6573	65	19	τ1	τ1	PROPN
ejpam-6573	65	20	,	,	PUNCT
ejpam-6573	65	21	τ2)β	τ2)β	NOUN
ejpam-6573	65	22	-	-	PUNCT
ejpam-6573	65	23	cl(a	cl(a	NUM
ejpam-6573	65	24	)	)	PUNCT
ejpam-6573	65	25	.	.	PUNCT
ejpam-6573	66	1	(	(	PUNCT
ejpam-6573	66	2	5	5	NUM
ejpam-6573	66	3	)	)	PUNCT
ejpam-6573	66	4	(	(	PUNCT
ejpam-6573	66	5	τ1	τ1	NOUN
ejpam-6573	66	6	,	,	PUNCT
ejpam-6573	66	7	τ2)β	τ2)β	ADJ
ejpam-6573	66	8	-	-	PUNCT
ejpam-6573	66	9	cl(x	cl(x	NOUN
ejpam-6573	66	10	−a	−a	NOUN
ejpam-6573	66	11	)	)	PUNCT
ejpam-6573	67	1	=	=	PUNCT
ejpam-6573	67	2	x	x	X
ejpam-6573	67	3	−	−	PROPN
ejpam-6573	67	4	(	(	PUNCT
ejpam-6573	67	5	τ1	τ1	NOUN
ejpam-6573	67	6	,	,	PUNCT
ejpam-6573	67	7	τ2)β	τ2)β	NOUN
ejpam-6573	67	8	-	-	PUNCT
ejpam-6573	67	9	int(a	int(a	NOUN
ejpam-6573	67	10	)	)	PUNCT
ejpam-6573	67	11	.	.	PUNCT
ejpam-6573	68	1	for	for	ADP
ejpam-6573	68	2	a	a	DET
ejpam-6573	68	3	subset	subset	NOUN
ejpam-6573	68	4	a	a	PRON
ejpam-6573	68	5	of	of	ADP
ejpam-6573	68	6	a	a	DET
ejpam-6573	68	7	bitopological	bitopological	ADJ
ejpam-6573	68	8	space	space	NOUN
ejpam-6573	68	9	(	(	PUNCT
ejpam-6573	68	10	x	x	NOUN
ejpam-6573	68	11	,	,	PUNCT
ejpam-6573	68	12	τ1	τ1	NOUN
ejpam-6573	68	13	,	,	PUNCT
ejpam-6573	68	14	τ2	τ2	PROPN
ejpam-6573	68	15	)	)	PUNCT
ejpam-6573	68	16	,	,	PUNCT
ejpam-6573	68	17	a	a	DET
ejpam-6573	68	18	point	point	NOUN
ejpam-6573	68	19	x	x	X
ejpam-6573	68	20	∈	∈	NOUN
ejpam-6573	68	21	x	x	PUNCT
ejpam-6573	68	22	is	be	AUX
ejpam-6573	68	23	called	call	VERB
ejpam-6573	68	24	(	(	PUNCT
ejpam-6573	68	25	τ1	τ1	NOUN
ejpam-6573	68	26	,	,	PUNCT
ejpam-6573	68	27	τ2)θcluster	τ2)θcluster	NOUN
ejpam-6573	68	28	point	point	NOUN
ejpam-6573	68	29	of	of	ADP
ejpam-6573	68	30	a	a	DET
ejpam-6573	68	31	if	if	SCONJ
ejpam-6573	68	32	τ1τ2	τ1τ2	NOUN
ejpam-6573	68	33	-	-	NOUN
ejpam-6573	68	34	cl(u	cl(u	NOUN
ejpam-6573	68	35	)	)	PUNCT
ejpam-6573	68	36	∩	∩	NOUN
ejpam-6573	68	37	a	a	DET
ejpam-6573	68	38	̸=	̸=	PROPN
ejpam-6573	68	39	∅	∅	NOUN
ejpam-6573	68	40	for	for	ADP
ejpam-6573	68	41	every	every	DET
ejpam-6573	68	42	τ1τ2	τ1τ2	ADJ
ejpam-6573	68	43	-	-	ADJ
ejpam-6573	68	44	open	open	ADJ
ejpam-6573	68	45	set	set	NOUN
ejpam-6573	68	46	u	u	NOUN
ejpam-6573	68	47	containing	contain	VERB
ejpam-6573	68	48	x.	x.	NOUN
ejpam-6573	68	49	the	the	DET
ejpam-6573	68	50	set	set	NOUN
ejpam-6573	68	51	of	of	ADP
ejpam-6573	68	52	all	all	DET
ejpam-6573	68	53	(	(	PUNCT
ejpam-6573	68	54	τ1	τ1	NOUN
ejpam-6573	68	55	,	,	PUNCT
ejpam-6573	68	56	τ2)θ	τ2)θ	ADJ
ejpam-6573	68	57	-	-	PUNCT
ejpam-6573	68	58	cluster	cluster	NOUN
ejpam-6573	68	59	points	point	NOUN
ejpam-6573	68	60	of	of	ADP
ejpam-6573	68	61	a	a	PRON
ejpam-6573	68	62	is	be	AUX
ejpam-6573	68	63	called	call	VERB
ejpam-6573	68	64	the	the	DET
ejpam-6573	68	65	(	(	PUNCT
ejpam-6573	68	66	τ1	τ1	NOUN
ejpam-6573	68	67	,	,	PUNCT
ejpam-6573	68	68	τ2)θ	τ2)θ	NOUN
ejpam-6573	68	69	-	-	PUNCT
ejpam-6573	68	70	closure	closure	NOUN
ejpam-6573	68	71	of	of	ADP
ejpam-6573	68	72	a	a	PRON
ejpam-6573	68	73	and	and	CCONJ
ejpam-6573	68	74	is	be	AUX
ejpam-6573	68	75	denoted	denote	VERB
ejpam-6573	68	76	by	by	ADP
ejpam-6573	68	77	(	(	PUNCT
ejpam-6573	68	78	τ1	τ1	NOUN
ejpam-6573	68	79	,	,	PUNCT
ejpam-6573	68	80	τ2)θ	τ2)θ	NOUN
ejpam-6573	68	81	-	-	PUNCT
ejpam-6573	68	82	cl(a	cl(a	NUM
ejpam-6573	68	83	)	)	PUNCT
ejpam-6573	68	84	.	.	PUNCT
ejpam-6573	69	1	a	a	DET
ejpam-6573	69	2	subset	subset	NOUN
ejpam-6573	69	3	a	a	PRON
ejpam-6573	69	4	of	of	ADP
ejpam-6573	69	5	a	a	DET
ejpam-6573	69	6	bitopological	bitopological	ADJ
ejpam-6573	69	7	space	space	NOUN
ejpam-6573	69	8	(	(	PUNCT
ejpam-6573	69	9	x	x	NOUN
ejpam-6573	69	10	,	,	PUNCT
ejpam-6573	69	11	τ1	τ1	NOUN
ejpam-6573	69	12	,	,	PUNCT
ejpam-6573	69	13	τ2	τ2	NOUN
ejpam-6573	69	14	)	)	PUNCT
ejpam-6573	69	15	is	be	AUX
ejpam-6573	69	16	said	say	VERB
ejpam-6573	69	17	to	to	PART
ejpam-6573	69	18	be	be	AUX
ejpam-6573	69	19	(	(	PUNCT
ejpam-6573	69	20	τ1	τ1	NOUN
ejpam-6573	69	21	,	,	PUNCT
ejpam-6573	69	22	τ2)θ	τ2)θ	NOUN
ejpam-6573	69	23	-	-	PUNCT
ejpam-6573	69	24	closed	closed	ADJ
ejpam-6573	69	25	if	if	SCONJ
ejpam-6573	69	26	a	a	DET
ejpam-6573	69	27	=	=	X
ejpam-6573	69	28	(	(	PUNCT
ejpam-6573	69	29	τ1	τ1	NOUN
ejpam-6573	69	30	,	,	PUNCT
ejpam-6573	69	31	τ2)θ	τ2)θ	NOUN
ejpam-6573	69	32	-	-	PUNCT
ejpam-6573	69	33	cl(a	cl(a	NUM
ejpam-6573	69	34	)	)	PUNCT
ejpam-6573	69	35	.	.	PUNCT
ejpam-6573	70	1	the	the	DET
ejpam-6573	70	2	complement	complement	NOUN
ejpam-6573	70	3	of	of	ADP
ejpam-6573	70	4	a	a	DET
ejpam-6573	70	5	(	(	PUNCT
ejpam-6573	70	6	τ1	τ1	NOUN
ejpam-6573	70	7	,	,	PUNCT
ejpam-6573	70	8	τ2)θ	τ2)θ	ADJ
ejpam-6573	70	9	-	-	PUNCT
ejpam-6573	70	10	closed	close	VERB
ejpam-6573	70	11	set	set	NOUN
ejpam-6573	70	12	is	be	AUX
ejpam-6573	70	13	said	say	VERB
ejpam-6573	70	14	to	to	PART
ejpam-6573	70	15	be	be	AUX
ejpam-6573	70	16	(	(	PUNCT
ejpam-6573	70	17	τ1	τ1	NOUN
ejpam-6573	70	18	,	,	PUNCT
ejpam-6573	70	19	τ2)θ	τ2)θ	NOUN
ejpam-6573	70	20	-	-	PUNCT
ejpam-6573	70	21	open	open	ADJ
ejpam-6573	70	22	.	.	PUNCT
ejpam-6573	71	1	the	the	DET
ejpam-6573	71	2	union	union	NOUN
ejpam-6573	71	3	of	of	ADP
ejpam-6573	71	4	all	all	DET
ejpam-6573	71	5	(	(	PUNCT
ejpam-6573	71	6	τ1	τ1	NOUN
ejpam-6573	71	7	,	,	PUNCT
ejpam-6573	71	8	τ2)θ	τ2)θ	ADJ
ejpam-6573	71	9	-	-	PUNCT
ejpam-6573	71	10	open	open	ADJ
ejpam-6573	71	11	sets	set	NOUN
ejpam-6573	71	12	contained	contain	VERB
ejpam-6573	71	13	in	in	ADP
ejpam-6573	71	14	a	a	PRON
ejpam-6573	71	15	is	be	AUX
ejpam-6573	71	16	called	call	VERB
ejpam-6573	71	17	the	the	DET
ejpam-6573	71	18	(	(	PUNCT
ejpam-6573	71	19	τ1	τ1	NOUN
ejpam-6573	71	20	,	,	PUNCT
ejpam-6573	71	21	τ2)θ	τ2)θ	ADJ
ejpam-6573	71	22	-	-	PUNCT
ejpam-6573	71	23	interior	interior	NOUN
ejpam-6573	71	24	of	of	ADP
ejpam-6573	71	25	a	a	PRON
ejpam-6573	71	26	and	and	CCONJ
ejpam-6573	71	27	is	be	AUX
ejpam-6573	71	28	denoted	denote	VERB
ejpam-6573	71	29	by	by	ADP
ejpam-6573	71	30	(	(	PUNCT
ejpam-6573	71	31	τ1	τ1	NOUN
ejpam-6573	71	32	,	,	PUNCT
ejpam-6573	71	33	τ2)θ	τ2)θ	NOUN
ejpam-6573	71	34	-	-	PUNCT
ejpam-6573	71	35	int(a	int(a	NOUN
ejpam-6573	71	36	)	)	PUNCT
ejpam-6573	72	1	[	[	X
ejpam-6573	72	2	16	16	NUM
ejpam-6573	72	3	]	]	PUNCT
ejpam-6573	72	4	.	.	PUNCT
ejpam-6573	73	1	lemma	lemma	PROPN
ejpam-6573	73	2	3	3	X
ejpam-6573	73	3	.	.	PUNCT
ejpam-6573	74	1	[	[	X
ejpam-6573	74	2	16	16	NUM
ejpam-6573	74	3	]	]	PUNCT
ejpam-6573	74	4	for	for	ADP
ejpam-6573	74	5	a	a	DET
ejpam-6573	74	6	subset	subset	NOUN
ejpam-6573	74	7	a	a	PRON
ejpam-6573	74	8	of	of	ADP
ejpam-6573	74	9	a	a	DET
ejpam-6573	74	10	bitopological	bitopological	ADJ
ejpam-6573	74	11	space	space	NOUN
ejpam-6573	74	12	(	(	PUNCT
ejpam-6573	74	13	x	x	NOUN
ejpam-6573	74	14	,	,	PUNCT
ejpam-6573	74	15	τ1	τ1	NOUN
ejpam-6573	74	16	,	,	PUNCT
ejpam-6573	74	17	τ2	τ2	NOUN
ejpam-6573	74	18	)	)	PUNCT
ejpam-6573	74	19	,	,	PUNCT
ejpam-6573	74	20	the	the	DET
ejpam-6573	74	21	following	follow	VERB
ejpam-6573	74	22	properties	property	NOUN
ejpam-6573	74	23	hold	hold	VERB
ejpam-6573	74	24	:	:	PUNCT
ejpam-6573	74	25	(	(	PUNCT
ejpam-6573	74	26	1	1	X
ejpam-6573	74	27	)	)	PUNCT
ejpam-6573	74	28	if	if	SCONJ
ejpam-6573	74	29	a	a	PRON
ejpam-6573	74	30	is	be	AUX
ejpam-6573	74	31	τ2τ2	τ2τ2	VERB
ejpam-6573	74	32	-	-	VERB
ejpam-6573	74	33	open	open	ADJ
ejpam-6573	74	34	in	in	ADP
ejpam-6573	74	35	x	x	NOUN
ejpam-6573	74	36	,	,	PUNCT
ejpam-6573	74	37	then	then	ADV
ejpam-6573	74	38	τ1τ2	τ1τ2	NOUN
ejpam-6573	74	39	-	-	NUM
ejpam-6573	74	40	cl(a	cl(a	NUM
ejpam-6573	74	41	)	)	PUNCT
ejpam-6573	74	42	=	=	PUNCT
ejpam-6573	74	43	(	(	PUNCT
ejpam-6573	74	44	τ1	τ1	NOUN
ejpam-6573	74	45	,	,	PUNCT
ejpam-6573	74	46	τ2)θ	τ2)θ	NOUN
ejpam-6573	74	47	-	-	PUNCT
ejpam-6573	74	48	cl(a	cl(a	NUM
ejpam-6573	74	49	)	)	PUNCT
ejpam-6573	74	50	.	.	PUNCT
ejpam-6573	75	1	(	(	PUNCT
ejpam-6573	75	2	2	2	X
ejpam-6573	75	3	)	)	PUNCT
ejpam-6573	75	4	(	(	PUNCT
ejpam-6573	75	5	τ1	τ1	NOUN
ejpam-6573	75	6	,	,	PUNCT
ejpam-6573	75	7	τ2)θ	τ2)θ	NOUN
ejpam-6573	75	8	-	-	PUNCT
ejpam-6573	75	9	cl(a	cl(a	NUM
ejpam-6573	75	10	)	)	PUNCT
ejpam-6573	75	11	is	be	AUX
ejpam-6573	75	12	τ1τ2	τ1τ2	NOUN
ejpam-6573	75	13	-	-	ADJ
ejpam-6573	75	14	closed	closed	ADJ
ejpam-6573	75	15	in	in	ADP
ejpam-6573	75	16	x.	x.	NOUN
ejpam-6573	75	17	3	3	NUM
ejpam-6573	75	18	.	.	PUNCT
ejpam-6573	75	19	characterizations	characterization	NOUN
ejpam-6573	75	20	of	of	ADP
ejpam-6573	75	21	weakly	weakly	ADJ
ejpam-6573	75	22	(	(	PUNCT
ejpam-6573	75	23	τ1	τ1	NOUN
ejpam-6573	75	24	,	,	PUNCT
ejpam-6573	75	25	τ2)β	τ2)β	ADJ
ejpam-6573	75	26	-	-	PUNCT
ejpam-6573	75	27	open	open	ADJ
ejpam-6573	75	28	functions	function	NOUN
ejpam-6573	75	29	in	in	ADP
ejpam-6573	75	30	this	this	DET
ejpam-6573	75	31	section	section	NOUN
ejpam-6573	75	32	,	,	PUNCT
ejpam-6573	75	33	we	we	PRON
ejpam-6573	75	34	introduce	introduce	VERB
ejpam-6573	75	35	the	the	DET
ejpam-6573	75	36	concept	concept	NOUN
ejpam-6573	75	37	of	of	ADP
ejpam-6573	75	38	weakly	weakly	ADJ
ejpam-6573	75	39	(	(	PUNCT
ejpam-6573	75	40	τ1	τ1	NOUN
ejpam-6573	75	41	,	,	PUNCT
ejpam-6573	75	42	τ2)β	τ2)β	ADJ
ejpam-6573	75	43	-	-	PUNCT
ejpam-6573	75	44	open	open	ADJ
ejpam-6573	75	45	functions	function	NOUN
ejpam-6573	75	46	.	.	PUNCT
ejpam-6573	76	1	moreover	moreover	ADV
ejpam-6573	76	2	,	,	PUNCT
ejpam-6573	76	3	some	some	DET
ejpam-6573	76	4	characterizations	characterization	NOUN
ejpam-6573	76	5	of	of	ADP
ejpam-6573	76	6	weakly	weakly	ADJ
ejpam-6573	76	7	(	(	PUNCT
ejpam-6573	76	8	τ1	τ1	NOUN
ejpam-6573	76	9	,	,	PUNCT
ejpam-6573	76	10	τ2)β	τ2)β	ADJ
ejpam-6573	76	11	-	-	PUNCT
ejpam-6573	76	12	open	open	ADJ
ejpam-6573	76	13	functions	function	NOUN
ejpam-6573	76	14	are	be	AUX
ejpam-6573	76	15	discussed	discuss	VERB
ejpam-6573	76	16	.	.	PUNCT
ejpam-6573	77	1	definition	definition	NOUN
ejpam-6573	77	2	1	1	NUM
ejpam-6573	77	3	.	.	PUNCT
ejpam-6573	78	1	a	a	DET
ejpam-6573	78	2	functions	function	NOUN
ejpam-6573	78	3	f	f	X
ejpam-6573	78	4	:	:	PUNCT
ejpam-6573	78	5	(	(	PUNCT
ejpam-6573	78	6	x	x	NOUN
ejpam-6573	78	7	,	,	PUNCT
ejpam-6573	78	8	τ1	τ1	NOUN
ejpam-6573	78	9	,	,	PUNCT
ejpam-6573	78	10	τ2	τ2	NOUN
ejpam-6573	78	11	)	)	PUNCT
ejpam-6573	78	12	→	→	SYM
ejpam-6573	78	13	(	(	PUNCT
ejpam-6573	78	14	y	y	PROPN
ejpam-6573	78	15	,	,	PUNCT
ejpam-6573	78	16	σ1	σ1	PROPN
ejpam-6573	78	17	,	,	PUNCT
ejpam-6573	78	18	σ2	σ2	PROPN
ejpam-6573	78	19	)	)	PUNCT
ejpam-6573	78	20	is	be	AUX
ejpam-6573	78	21	said	say	VERB
ejpam-6573	78	22	to	to	PART
ejpam-6573	78	23	be	be	AUX
ejpam-6573	78	24	weakly	weakly	ADJ
ejpam-6573	78	25	(	(	PUNCT
ejpam-6573	78	26	τ1	τ1	NOUN
ejpam-6573	78	27	,	,	PUNCT
ejpam-6573	78	28	τ2)β	τ2)β	ADJ
ejpam-6573	78	29	-	-	PUNCT
ejpam-6573	78	30	open	open	ADJ
ejpam-6573	78	31	if	if	SCONJ
ejpam-6573	78	32	f(u	f(u	PROPN
ejpam-6573	78	33	)	)	PUNCT
ejpam-6573	78	34	⊆	⊆	NUM
ejpam-6573	78	35	(	(	PUNCT
ejpam-6573	78	36	σ1	σ1	PROPN
ejpam-6573	78	37	,	,	PUNCT
ejpam-6573	78	38	σ2)β	σ2)β	NOUN
ejpam-6573	78	39	-	-	PUNCT
ejpam-6573	78	40	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	78	41	-	-	NOUN
ejpam-6573	78	42	cl(u	cl(u	NOUN
ejpam-6573	78	43	)	)	PUNCT
ejpam-6573	78	44	)	)	PUNCT
ejpam-6573	78	45	)	)	PUNCT
ejpam-6573	78	46	for	for	ADP
ejpam-6573	78	47	every	every	DET
ejpam-6573	78	48	τ1τ2	τ1τ2	ADJ
ejpam-6573	78	49	-	-	ADJ
ejpam-6573	78	50	open	open	ADJ
ejpam-6573	78	51	set	set	ADJ
ejpam-6573	78	52	u	u	NOUN
ejpam-6573	78	53	of	of	ADP
ejpam-6573	78	54	x.	x.	PROPN
ejpam-6573	78	55	theorem	theorem	VERB
ejpam-6573	78	56	1	1	NUM
ejpam-6573	78	57	.	.	PUNCT
ejpam-6573	78	58	for	for	ADP
ejpam-6573	78	59	a	a	DET
ejpam-6573	78	60	function	function	NOUN
ejpam-6573	78	61	f	f	NOUN
ejpam-6573	78	62	:	:	PUNCT
ejpam-6573	78	63	(	(	PUNCT
ejpam-6573	78	64	x	x	NOUN
ejpam-6573	78	65	,	,	PUNCT
ejpam-6573	78	66	τ1	τ1	NOUN
ejpam-6573	78	67	,	,	PUNCT
ejpam-6573	78	68	τ2	τ2	NOUN
ejpam-6573	78	69	)	)	PUNCT
ejpam-6573	78	70	→	→	SYM
ejpam-6573	78	71	(	(	PUNCT
ejpam-6573	78	72	y	y	NOUN
ejpam-6573	78	73	,	,	PUNCT
ejpam-6573	78	74	σ1σ2	σ1σ2	NOUN
ejpam-6573	78	75	)	)	PUNCT
ejpam-6573	78	76	,	,	PUNCT
ejpam-6573	78	77	the	the	DET
ejpam-6573	78	78	following	follow	VERB
ejpam-6573	78	79	properties	property	NOUN
ejpam-6573	78	80	are	be	AUX
ejpam-6573	78	81	equivalent	equivalent	ADJ
ejpam-6573	78	82	:	:	PUNCT
ejpam-6573	78	83	(	(	PUNCT
ejpam-6573	78	84	1	1	X
ejpam-6573	78	85	)	)	PUNCT
ejpam-6573	78	86	f	f	PROPN
ejpam-6573	78	87	is	be	AUX
ejpam-6573	78	88	weakly	weakly	ADJ
ejpam-6573	78	89	(	(	PUNCT
ejpam-6573	78	90	τ1	τ1	NOUN
ejpam-6573	78	91	,	,	PUNCT
ejpam-6573	78	92	τ2)β	τ2)β	ADJ
ejpam-6573	78	93	-	-	PUNCT
ejpam-6573	78	94	open	open	ADJ
ejpam-6573	78	95	;	;	PUNCT
ejpam-6573	78	96	(	(	PUNCT
ejpam-6573	78	97	2	2	X
ejpam-6573	78	98	)	)	PUNCT
ejpam-6573	78	99	f((τ1	f((τ1	PROPN
ejpam-6573	78	100	,	,	PUNCT
ejpam-6573	78	101	τ2)θ	τ2)θ	NOUN
ejpam-6573	78	102	-	-	PUNCT
ejpam-6573	78	103	int(a	int(a	NOUN
ejpam-6573	78	104	)	)	PUNCT
ejpam-6573	78	105	)	)	PUNCT
ejpam-6573	79	1	⊆	⊆	NUM
ejpam-6573	79	2	(	(	PUNCT
ejpam-6573	79	3	σ1	σ1	PROPN
ejpam-6573	79	4	,	,	PUNCT
ejpam-6573	79	5	σ2)β	σ2)β	NOUN
ejpam-6573	79	6	-	-	PUNCT
ejpam-6573	79	7	int(f(a	int(f(a	NOUN
ejpam-6573	79	8	)	)	PUNCT
ejpam-6573	79	9	)	)	PUNCT
ejpam-6573	79	10	for	for	ADP
ejpam-6573	79	11	every	every	DET
ejpam-6573	79	12	subset	subset	NOUN
ejpam-6573	79	13	a	a	PRON
ejpam-6573	79	14	of	of	ADP
ejpam-6573	79	15	x	x	NOUN
ejpam-6573	79	16	;	;	PUNCT
ejpam-6573	79	17	m.	m.	NOUN
ejpam-6573	79	18	chiangpradit	chiangpradit	NOUN
ejpam-6573	79	19	,	,	PUNCT
ejpam-6573	79	20	a.	a.	PROPN
ejpam-6573	79	21	sama	sama	PROPN
ejpam-6573	79	22	-	-	PUNCT
ejpam-6573	79	23	ae	ae	PROPN
ejpam-6573	79	24	,	,	PUNCT
ejpam-6573	79	25	c.	c.	PROPN
ejpam-6573	79	26	boonpok	boonpok	PROPN
ejpam-6573	79	27	/	/	SYM
ejpam-6573	79	28	eur	eur	PROPN
ejpam-6573	79	29	.	.	PUNCT
ejpam-6573	80	1	j.	j.	PROPN
ejpam-6573	80	2	pure	pure	PROPN
ejpam-6573	80	3	appl	appl	PROPN
ejpam-6573	80	4	.	.	PROPN
ejpam-6573	80	5	math	math	PROPN
ejpam-6573	80	6	,	,	PUNCT
ejpam-6573	80	7	18	18	NUM
ejpam-6573	80	8	(	(	PUNCT
ejpam-6573	80	9	3	3	NUM
ejpam-6573	80	10	)	)	PUNCT
ejpam-6573	80	11	(	(	PUNCT
ejpam-6573	80	12	2025	2025	NUM
ejpam-6573	80	13	)	)	PUNCT
ejpam-6573	80	14	,	,	PUNCT
ejpam-6573	80	15	6573	6573	NUM
ejpam-6573	80	16	4	4	NUM
ejpam-6573	80	17	of	of	ADP
ejpam-6573	80	18	8	8	NUM
ejpam-6573	80	19	(	(	PUNCT
ejpam-6573	80	20	3	3	NUM
ejpam-6573	80	21	)	)	PUNCT
ejpam-6573	80	22	(	(	PUNCT
ejpam-6573	80	23	τ1	τ1	NOUN
ejpam-6573	80	24	,	,	PUNCT
ejpam-6573	80	25	τ2)θ	τ2)θ	NOUN
ejpam-6573	80	26	-	-	PUNCT
ejpam-6573	80	27	int(f	int(f	PROPN
ejpam-6573	80	28	−1(b	−1(b	NOUN
ejpam-6573	80	29	)	)	PUNCT
ejpam-6573	80	30	)	)	PUNCT
ejpam-6573	81	1	⊆	⊆	NUM
ejpam-6573	81	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	81	3	,	,	PUNCT
ejpam-6573	81	4	σ2)β	σ2)β	NOUN
ejpam-6573	81	5	-	-	PUNCT
ejpam-6573	81	6	int(b	int(b	NOUN
ejpam-6573	81	7	)	)	PUNCT
ejpam-6573	81	8	)	)	PUNCT
ejpam-6573	81	9	for	for	ADP
ejpam-6573	81	10	every	every	DET
ejpam-6573	81	11	subset	subset	NOUN
ejpam-6573	81	12	b	b	PROPN
ejpam-6573	81	13	of	of	ADP
ejpam-6573	81	14	y	y	PROPN
ejpam-6573	81	15	;	;	PUNCT
ejpam-6573	81	16	(	(	PUNCT
ejpam-6573	81	17	4	4	X
ejpam-6573	81	18	)	)	PUNCT
ejpam-6573	81	19	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	81	20	,	,	PUNCT
ejpam-6573	81	21	σ2)β	σ2)β	NOUN
ejpam-6573	81	22	-	-	PUNCT
ejpam-6573	81	23	cl(b	cl(b	NOUN
ejpam-6573	81	24	)	)	PUNCT
ejpam-6573	81	25	)	)	PUNCT
ejpam-6573	82	1	⊆	⊆	NUM
ejpam-6573	82	2	(	(	PUNCT
ejpam-6573	82	3	τ1	τ1	NOUN
ejpam-6573	82	4	,	,	PUNCT
ejpam-6573	82	5	τ2)θ	τ2)θ	PROPN
ejpam-6573	82	6	-	-	PUNCT
ejpam-6573	82	7	cl(f	cl(f	NOUN
ejpam-6573	82	8	−1(b	−1(b	NOUN
ejpam-6573	82	9	)	)	PUNCT
ejpam-6573	82	10	)	)	PUNCT
ejpam-6573	82	11	for	for	ADP
ejpam-6573	82	12	every	every	DET
ejpam-6573	82	13	subset	subset	NOUN
ejpam-6573	82	14	b	b	PROPN
ejpam-6573	82	15	of	of	ADP
ejpam-6573	82	16	y	y	PROPN
ejpam-6573	82	17	;	;	PUNCT
ejpam-6573	82	18	(	(	PUNCT
ejpam-6573	82	19	5	5	X
ejpam-6573	82	20	)	)	PUNCT
ejpam-6573	82	21	for	for	ADP
ejpam-6573	82	22	each	each	DET
ejpam-6573	82	23	x	x	SYM
ejpam-6573	82	24	∈	∈	PROPN
ejpam-6573	82	25	x	x	X
ejpam-6573	82	26	and	and	CCONJ
ejpam-6573	82	27	each	each	DET
ejpam-6573	82	28	τ1τ2	τ1τ2	ADJ
ejpam-6573	82	29	-	-	ADJ
ejpam-6573	82	30	open	open	ADJ
ejpam-6573	82	31	set	set	ADJ
ejpam-6573	82	32	u	u	NOUN
ejpam-6573	82	33	of	of	ADP
ejpam-6573	82	34	x	x	PUNCT
ejpam-6573	82	35	containing	contain	VERB
ejpam-6573	82	36	x	x	PRON
ejpam-6573	82	37	,	,	PUNCT
ejpam-6573	82	38	there	there	PRON
ejpam-6573	82	39	exists	exist	VERB
ejpam-6573	82	40	a	a	DET
ejpam-6573	82	41	(	(	PUNCT
ejpam-6573	82	42	σ1	σ1	PROPN
ejpam-6573	82	43	,	,	PUNCT
ejpam-6573	82	44	σ2)βopen	σ2)βopen	PROPN
ejpam-6573	82	45	set	set	VERB
ejpam-6573	82	46	v	v	NOUN
ejpam-6573	82	47	of	of	ADP
ejpam-6573	82	48	y	y	NOUN
ejpam-6573	82	49	containing	contain	VERB
ejpam-6573	82	50	f(x	f(x	PROPN
ejpam-6573	82	51	)	)	PUNCT
ejpam-6573	82	52	such	such	ADJ
ejpam-6573	82	53	that	that	SCONJ
ejpam-6573	82	54	v	v	ADP
ejpam-6573	82	55	⊆	⊆	NUM
ejpam-6573	82	56	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	82	57	-	-	PUNCT
ejpam-6573	82	58	cl(u	cl(u	NOUN
ejpam-6573	82	59	)	)	PUNCT
ejpam-6573	82	60	)	)	PUNCT
ejpam-6573	82	61	;	;	PUNCT
ejpam-6573	82	62	(	(	PUNCT
ejpam-6573	82	63	6	6	X
ejpam-6573	82	64	)	)	PUNCT
ejpam-6573	82	65	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	82	66	-	-	PUNCT
ejpam-6573	82	67	int(k	int(k	NOUN
ejpam-6573	82	68	)	)	PUNCT
ejpam-6573	82	69	)	)	PUNCT
ejpam-6573	83	1	⊆	⊆	NUM
ejpam-6573	83	2	(	(	PUNCT
ejpam-6573	83	3	σ1	σ1	PROPN
ejpam-6573	83	4	,	,	PUNCT
ejpam-6573	83	5	σ2)β	σ2)β	NOUN
ejpam-6573	83	6	-	-	PUNCT
ejpam-6573	83	7	int(f(k	int(f(k	NOUN
ejpam-6573	83	8	)	)	PUNCT
ejpam-6573	83	9	)	)	PUNCT
ejpam-6573	83	10	for	for	ADP
ejpam-6573	83	11	every	every	DET
ejpam-6573	83	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	83	13	-	-	ADJ
ejpam-6573	83	14	closed	closed	ADJ
ejpam-6573	83	15	set	set	ADJ
ejpam-6573	83	16	k	k	PROPN
ejpam-6573	83	17	of	of	ADP
ejpam-6573	83	18	x	x	PROPN
ejpam-6573	83	19	;	;	PUNCT
ejpam-6573	83	20	(	(	PUNCT
ejpam-6573	83	21	7	7	X
ejpam-6573	83	22	)	)	PUNCT
ejpam-6573	83	23	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	83	24	-	-	PUNCT
ejpam-6573	83	25	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	83	26	-	-	PUNCT
ejpam-6573	83	27	cl(u	cl(u	NOUN
ejpam-6573	83	28	)	)	PUNCT
ejpam-6573	83	29	)	)	PUNCT
ejpam-6573	83	30	)	)	PUNCT
ejpam-6573	84	1	⊆	⊆	X
ejpam-6573	84	2	(	(	PUNCT
ejpam-6573	84	3	σ1	σ1	PROPN
ejpam-6573	84	4	,	,	PUNCT
ejpam-6573	84	5	σ2)β	σ2)β	NOUN
ejpam-6573	84	6	-	-	PUNCT
ejpam-6573	84	7	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	84	8	-	-	NOUN
ejpam-6573	84	9	cl(u	cl(u	NOUN
ejpam-6573	84	10	)	)	PUNCT
ejpam-6573	84	11	)	)	PUNCT
ejpam-6573	84	12	)	)	PUNCT
ejpam-6573	84	13	for	for	ADP
ejpam-6573	84	14	every	every	DET
ejpam-6573	84	15	τ1τ2	τ1τ2	ADJ
ejpam-6573	84	16	-	-	ADJ
ejpam-6573	84	17	open	open	ADJ
ejpam-6573	84	18	set	set	ADJ
ejpam-6573	84	19	u	u	NOUN
ejpam-6573	84	20	of	of	ADP
ejpam-6573	84	21	x	x	PRON
ejpam-6573	84	22	;	;	PUNCT
ejpam-6573	84	23	(	(	PUNCT
ejpam-6573	84	24	8)	8)	NUM
ejpam-6573	84	25	f(u	f(u	NOUN
ejpam-6573	84	26	)	)	PUNCT
ejpam-6573	85	1	⊆	⊆	NUM
ejpam-6573	85	2	(	(	PUNCT
ejpam-6573	85	3	σ1	σ1	PROPN
ejpam-6573	85	4	,	,	PUNCT
ejpam-6573	85	5	σ2)β	σ2)β	NOUN
ejpam-6573	85	6	-	-	PUNCT
ejpam-6573	85	7	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	85	8	-	-	NOUN
ejpam-6573	85	9	cl(u	cl(u	NOUN
ejpam-6573	85	10	)	)	PUNCT
ejpam-6573	85	11	)	)	PUNCT
ejpam-6573	85	12	)	)	PUNCT
ejpam-6573	86	1	for	for	ADP
ejpam-6573	86	2	every	every	DET
ejpam-6573	86	3	(	(	PUNCT
ejpam-6573	86	4	τ1	τ1	NOUN
ejpam-6573	86	5	,	,	PUNCT
ejpam-6573	86	6	τ2)p	τ2)p	ADJ
ejpam-6573	86	7	-	-	PUNCT
ejpam-6573	86	8	open	open	ADJ
ejpam-6573	86	9	set	set	NOUN
ejpam-6573	86	10	u	u	NOUN
ejpam-6573	86	11	of	of	ADP
ejpam-6573	86	12	x	x	PRON
ejpam-6573	86	13	;	;	PUNCT
ejpam-6573	86	14	(	(	PUNCT
ejpam-6573	86	15	9	9	X
ejpam-6573	86	16	)	)	PUNCT
ejpam-6573	86	17	f(u	f(u	PROPN
ejpam-6573	86	18	)	)	PUNCT
ejpam-6573	86	19	⊆	⊆	NUM
ejpam-6573	86	20	(	(	PUNCT
ejpam-6573	86	21	σ1	σ1	PROPN
ejpam-6573	86	22	,	,	PUNCT
ejpam-6573	86	23	σ2)β	σ2)β	NOUN
ejpam-6573	86	24	-	-	PUNCT
ejpam-6573	86	25	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	86	26	-	-	NOUN
ejpam-6573	86	27	cl(u	cl(u	NOUN
ejpam-6573	86	28	)	)	PUNCT
ejpam-6573	86	29	)	)	PUNCT
ejpam-6573	86	30	)	)	PUNCT
ejpam-6573	86	31	for	for	ADP
ejpam-6573	86	32	every	every	DET
ejpam-6573	86	33	α(τ1	α(τ1	NOUN
ejpam-6573	86	34	,	,	PUNCT
ejpam-6573	86	35	τ2)-open	τ2)-open	ADJ
ejpam-6573	86	36	set	set	ADJ
ejpam-6573	86	37	u	u	NOUN
ejpam-6573	86	38	of	of	ADP
ejpam-6573	86	39	x.	x.	NOUN
ejpam-6573	86	40	proof	proof	NOUN
ejpam-6573	86	41	.	.	PUNCT
ejpam-6573	87	1	(	(	PUNCT
ejpam-6573	87	2	1	1	X
ejpam-6573	87	3	)	)	PUNCT
ejpam-6573	87	4	⇒	⇒	NOUN
ejpam-6573	87	5	(	(	PUNCT
ejpam-6573	87	6	2	2	NUM
ejpam-6573	87	7	):	):	PUNCT
ejpam-6573	87	8	let	let	VERB
ejpam-6573	87	9	a	a	PRON
ejpam-6573	87	10	be	be	AUX
ejpam-6573	87	11	any	any	DET
ejpam-6573	87	12	subset	subset	NOUN
ejpam-6573	87	13	of	of	ADP
ejpam-6573	87	14	x	x	PUNCT
ejpam-6573	87	15	and	and	CCONJ
ejpam-6573	87	16	x	x	PROPN
ejpam-6573	87	17	∈	∈	PROPN
ejpam-6573	87	18	(	(	PUNCT
ejpam-6573	87	19	τ1	τ1	NOUN
ejpam-6573	87	20	,	,	PUNCT
ejpam-6573	87	21	τ2)θ	τ2)θ	NOUN
ejpam-6573	87	22	-	-	PUNCT
ejpam-6573	87	23	int(a	int(a	NOUN
ejpam-6573	87	24	)	)	PUNCT
ejpam-6573	87	25	.	.	PUNCT
ejpam-6573	88	1	then	then	ADV
ejpam-6573	88	2	,	,	PUNCT
ejpam-6573	88	3	there	there	PRON
ejpam-6573	88	4	exists	exist	VERB
ejpam-6573	88	5	a	a	DET
ejpam-6573	88	6	τ1τ2	τ1τ2	NOUN
ejpam-6573	88	7	-	-	ADJ
ejpam-6573	88	8	open	open	ADJ
ejpam-6573	88	9	set	set	ADJ
ejpam-6573	88	10	u	u	NOUN
ejpam-6573	88	11	of	of	ADP
ejpam-6573	88	12	x	x	SYM
ejpam-6573	88	13	such	such	ADJ
ejpam-6573	88	14	that	that	SCONJ
ejpam-6573	88	15	x	x	SYM
ejpam-6573	88	16	∈	∈	PROPN
ejpam-6573	88	17	u	u	NOUN
ejpam-6573	88	18	⊆	⊆	NUM
ejpam-6573	88	19	τ1τ2	τ1τ2	NOUN
ejpam-6573	88	20	-	-	NOUN
ejpam-6573	88	21	cl(u	cl(u	ADJ
ejpam-6573	88	22	)	)	PUNCT
ejpam-6573	88	23	⊆	⊆	NUM
ejpam-6573	88	24	a.	a.	NOUN
ejpam-6573	88	25	therefore	therefore	ADV
ejpam-6573	88	26	,	,	PUNCT
ejpam-6573	88	27	f(x	f(x	PROPN
ejpam-6573	88	28	)	)	PUNCT
ejpam-6573	88	29	∈	∈	PROPN
ejpam-6573	88	30	f(u	f(u	PROPN
ejpam-6573	88	31	)	)	PUNCT
ejpam-6573	88	32	⊆	⊆	NUM
ejpam-6573	88	33	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	88	34	-	-	PUNCT
ejpam-6573	88	35	cl(u	cl(u	NOUN
ejpam-6573	88	36	)	)	PUNCT
ejpam-6573	88	37	)	)	PUNCT
ejpam-6573	89	1	⊆	⊆	NUM
ejpam-6573	89	2	f(a	f(a	NOUN
ejpam-6573	89	3	)	)	PUNCT
ejpam-6573	89	4	.	.	PUNCT
ejpam-6573	90	1	since	since	SCONJ
ejpam-6573	90	2	f	f	PROPN
ejpam-6573	90	3	is	be	AUX
ejpam-6573	90	4	weakly	weakly	ADJ
ejpam-6573	90	5	(	(	PUNCT
ejpam-6573	90	6	τ1	τ1	NOUN
ejpam-6573	90	7	,	,	PUNCT
ejpam-6573	90	8	τ2)β	τ2)β	ADJ
ejpam-6573	90	9	-	-	PUNCT
ejpam-6573	90	10	open	open	ADJ
ejpam-6573	90	11	,	,	PUNCT
ejpam-6573	90	12	f(u	f(u	PROPN
ejpam-6573	90	13	)	)	PUNCT
ejpam-6573	90	14	⊆	⊆	NUM
ejpam-6573	90	15	(	(	PUNCT
ejpam-6573	90	16	σ1	σ1	PROPN
ejpam-6573	90	17	,	,	PUNCT
ejpam-6573	90	18	σ2)β	σ2)β	NOUN
ejpam-6573	90	19	-	-	PUNCT
ejpam-6573	90	20	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	90	21	-	-	NOUN
ejpam-6573	90	22	cl(u	cl(u	NOUN
ejpam-6573	90	23	)	)	PUNCT
ejpam-6573	90	24	)	)	PUNCT
ejpam-6573	90	25	)	)	PUNCT
ejpam-6573	91	1	⊆	⊆	X
ejpam-6573	91	2	(	(	PUNCT
ejpam-6573	91	3	σ1	σ1	PROPN
ejpam-6573	91	4	,	,	PUNCT
ejpam-6573	91	5	σ2)β	σ2)β	NOUN
ejpam-6573	91	6	-	-	PUNCT
ejpam-6573	91	7	int(f(a	int(f(a	NOUN
ejpam-6573	91	8	)	)	PUNCT
ejpam-6573	91	9	)	)	PUNCT
ejpam-6573	91	10	.	.	PUNCT
ejpam-6573	92	1	it	it	PRON
ejpam-6573	92	2	implies	imply	VERB
ejpam-6573	92	3	that	that	SCONJ
ejpam-6573	92	4	f(x	f(x	PROPN
ejpam-6573	92	5	)	)	PUNCT
ejpam-6573	92	6	∈	∈	PROPN
ejpam-6573	92	7	(	(	PUNCT
ejpam-6573	92	8	σ1	σ1	PROPN
ejpam-6573	92	9	,	,	PUNCT
ejpam-6573	92	10	σ2)β	σ2)β	NOUN
ejpam-6573	92	11	-	-	PUNCT
ejpam-6573	92	12	int(f(a	int(f(a	NOUN
ejpam-6573	92	13	)	)	PUNCT
ejpam-6573	92	14	)	)	PUNCT
ejpam-6573	92	15	.	.	PUNCT
ejpam-6573	93	1	thus	thus	ADV
ejpam-6573	93	2	,	,	PUNCT
ejpam-6573	93	3	x	x	SYM
ejpam-6573	93	4	∈	∈	NOUN
ejpam-6573	93	5	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	93	6	,	,	PUNCT
ejpam-6573	93	7	σ2)β	σ2)β	NOUN
ejpam-6573	93	8	-	-	PUNCT
ejpam-6573	93	9	int(f(a	int(f(a	NOUN
ejpam-6573	93	10	)	)	PUNCT
ejpam-6573	93	11	)	)	PUNCT
ejpam-6573	93	12	)	)	PUNCT
ejpam-6573	93	13	and	and	CCONJ
ejpam-6573	93	14	hence	hence	ADV
ejpam-6573	93	15	(	(	PUNCT
ejpam-6573	93	16	τ1	τ1	NOUN
ejpam-6573	93	17	,	,	PUNCT
ejpam-6573	93	18	τ2)θ	τ2)θ	ADJ
ejpam-6573	93	19	-	-	PUNCT
ejpam-6573	93	20	int(a	int(a	PROPN
ejpam-6573	93	21	)	)	PUNCT
ejpam-6573	93	22	⊆	⊆	NUM
ejpam-6573	93	23	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	93	24	,	,	PUNCT
ejpam-6573	93	25	σ2)β	σ2)β	NOUN
ejpam-6573	93	26	-	-	PUNCT
ejpam-6573	93	27	int(f(a	int(f(a	NOUN
ejpam-6573	93	28	)	)	PUNCT
ejpam-6573	93	29	)	)	PUNCT
ejpam-6573	93	30	)	)	PUNCT
ejpam-6573	93	31	.	.	PUNCT
ejpam-6573	94	1	this	this	PRON
ejpam-6573	94	2	shows	show	VERB
ejpam-6573	94	3	that	that	SCONJ
ejpam-6573	94	4	f((τ1	f((τ1	PROPN
ejpam-6573	94	5	,	,	PUNCT
ejpam-6573	94	6	τ2)θ	τ2)θ	NOUN
ejpam-6573	94	7	-	-	PUNCT
ejpam-6573	94	8	int(a	int(a	NOUN
ejpam-6573	94	9	)	)	PUNCT
ejpam-6573	94	10	)	)	PUNCT
ejpam-6573	95	1	⊆	⊆	NUM
ejpam-6573	95	2	(	(	PUNCT
ejpam-6573	95	3	σ1	σ1	PROPN
ejpam-6573	95	4	,	,	PUNCT
ejpam-6573	95	5	σ2)β	σ2)β	NOUN
ejpam-6573	95	6	-	-	PUNCT
ejpam-6573	95	7	int(f(a	int(f(a	NOUN
ejpam-6573	95	8	)	)	PUNCT
ejpam-6573	95	9	)	)	PUNCT
ejpam-6573	95	10	.	.	PUNCT
ejpam-6573	96	1	(	(	PUNCT
ejpam-6573	96	2	2	2	X
ejpam-6573	96	3	)	)	PUNCT
ejpam-6573	96	4	⇒	⇒	NOUN
ejpam-6573	96	5	(	(	PUNCT
ejpam-6573	96	6	1	1	NUM
ejpam-6573	96	7	):	):	PUNCT
ejpam-6573	96	8	let	let	VERB
ejpam-6573	96	9	u	u	PRON
ejpam-6573	96	10	be	be	AUX
ejpam-6573	96	11	any	any	DET
ejpam-6573	96	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	96	13	-	-	ADJ
ejpam-6573	96	14	open	open	ADJ
ejpam-6573	96	15	set	set	NOUN
ejpam-6573	96	16	of	of	ADP
ejpam-6573	96	17	x.	x.	NOUN
ejpam-6573	96	18	as	as	ADP
ejpam-6573	96	19	u	u	NOUN
ejpam-6573	96	20	⊆	⊆	NUM
ejpam-6573	96	21	(	(	PUNCT
ejpam-6573	96	22	τ1	τ1	NOUN
ejpam-6573	96	23	,	,	PUNCT
ejpam-6573	96	24	τ2)θ	τ2)θ	ADJ
ejpam-6573	96	25	-	-	PUNCT
ejpam-6573	96	26	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	96	27	-	-	PUNCT
ejpam-6573	96	28	cl(u	cl(u	NOUN
ejpam-6573	96	29	)	)	PUNCT
ejpam-6573	96	30	)	)	PUNCT
ejpam-6573	96	31	implies	imply	VERB
ejpam-6573	96	32	f(u	f(u	PROPN
ejpam-6573	96	33	)	)	PUNCT
ejpam-6573	96	34	⊆	⊆	NUM
ejpam-6573	96	35	f((τ1	f((τ1	NOUN
ejpam-6573	96	36	,	,	PUNCT
ejpam-6573	96	37	τ2)θ	τ2)θ	ADJ
ejpam-6573	96	38	-	-	PUNCT
ejpam-6573	96	39	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	96	40	-	-	PUNCT
ejpam-6573	96	41	cl(u	cl(u	NOUN
ejpam-6573	96	42	)	)	PUNCT
ejpam-6573	96	43	)	)	PUNCT
ejpam-6573	96	44	)	)	PUNCT
ejpam-6573	97	1	⊆	⊆	X
ejpam-6573	97	2	(	(	PUNCT
ejpam-6573	97	3	σ1	σ1	PROPN
ejpam-6573	97	4	,	,	PUNCT
ejpam-6573	97	5	σ2)β	σ2)β	NOUN
ejpam-6573	97	6	-	-	PUNCT
ejpam-6573	97	7	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	97	8	-	-	NOUN
ejpam-6573	97	9	cl(u	cl(u	NOUN
ejpam-6573	97	10	)	)	PUNCT
ejpam-6573	97	11	)	)	PUNCT
ejpam-6573	97	12	)	)	PUNCT
ejpam-6573	97	13	.	.	PUNCT
ejpam-6573	98	1	thus	thus	ADV
ejpam-6573	98	2	,	,	PUNCT
ejpam-6573	98	3	f	f	PROPN
ejpam-6573	98	4	is	be	AUX
ejpam-6573	98	5	weakly	weakly	ADJ
ejpam-6573	98	6	(	(	PUNCT
ejpam-6573	98	7	τ1	τ1	NOUN
ejpam-6573	98	8	,	,	PUNCT
ejpam-6573	98	9	τ2)β	τ2)β	ADJ
ejpam-6573	98	10	-	-	PUNCT
ejpam-6573	98	11	open	open	ADJ
ejpam-6573	98	12	.	.	PUNCT
ejpam-6573	99	1	(	(	PUNCT
ejpam-6573	99	2	2	2	X
ejpam-6573	99	3	)	)	PUNCT
ejpam-6573	99	4	⇒	⇒	NOUN
ejpam-6573	99	5	(	(	PUNCT
ejpam-6573	99	6	3	3	NUM
ejpam-6573	99	7	):	):	PUNCT
ejpam-6573	99	8	let	let	VERB
ejpam-6573	99	9	b	b	X
ejpam-6573	99	10	be	be	AUX
ejpam-6573	99	11	any	any	DET
ejpam-6573	99	12	subset	subset	NOUN
ejpam-6573	99	13	of	of	ADP
ejpam-6573	99	14	y	y	PROPN
ejpam-6573	99	15	.	.	PUNCT
ejpam-6573	100	1	thus	thus	ADV
ejpam-6573	100	2	by	by	ADP
ejpam-6573	100	3	(	(	PUNCT
ejpam-6573	100	4	2	2	NUM
ejpam-6573	100	5	)	)	PUNCT
ejpam-6573	100	6	,	,	PUNCT
ejpam-6573	100	7	we	we	PRON
ejpam-6573	100	8	have	have	AUX
ejpam-6573	100	9	f((τ1	f((τ1	NOUN
ejpam-6573	100	10	,	,	PUNCT
ejpam-6573	100	11	τ2)θ	τ2)θ	ADJ
ejpam-6573	100	12	-	-	PUNCT
ejpam-6573	100	13	int(f	int(f	PROPN
ejpam-6573	100	14	−1(b	−1(b	NOUN
ejpam-6573	100	15	)	)	PUNCT
ejpam-6573	100	16	)	)	PUNCT
ejpam-6573	101	1	⊆	⊆	NUM
ejpam-6573	101	2	(	(	PUNCT
ejpam-6573	101	3	σ1	σ1	PROPN
ejpam-6573	101	4	,	,	PUNCT
ejpam-6573	101	5	σ2)β	σ2)β	NOUN
ejpam-6573	101	6	-	-	PUNCT
ejpam-6573	101	7	int(b	int(b	NOUN
ejpam-6573	101	8	)	)	PUNCT
ejpam-6573	101	9	and	and	CCONJ
ejpam-6573	101	10	hence	hence	ADV
ejpam-6573	101	11	(	(	PUNCT
ejpam-6573	101	12	τ1	τ1	NOUN
ejpam-6573	101	13	,	,	PUNCT
ejpam-6573	101	14	τ2)θ	τ2)θ	ADJ
ejpam-6573	101	15	-	-	PUNCT
ejpam-6573	101	16	int(f	int(f	PROPN
ejpam-6573	101	17	−1(b	−1(b	NOUN
ejpam-6573	101	18	)	)	PUNCT
ejpam-6573	101	19	)	)	PUNCT
ejpam-6573	101	20	⊆	⊆	NUM
ejpam-6573	101	21	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	101	22	,	,	PUNCT
ejpam-6573	101	23	σ2)β	σ2)β	NOUN
ejpam-6573	101	24	-	-	PUNCT
ejpam-6573	101	25	int(b	int(b	NOUN
ejpam-6573	101	26	)	)	PUNCT
ejpam-6573	101	27	)	)	PUNCT
ejpam-6573	101	28	.	.	PUNCT
ejpam-6573	102	1	(	(	PUNCT
ejpam-6573	102	2	3	3	X
ejpam-6573	102	3	)	)	PUNCT
ejpam-6573	102	4	⇒	⇒	NOUN
ejpam-6573	102	5	(	(	PUNCT
ejpam-6573	102	6	2	2	NUM
ejpam-6573	102	7	):	):	PUNCT
ejpam-6573	102	8	the	the	DET
ejpam-6573	102	9	proof	proof	NOUN
ejpam-6573	102	10	is	be	AUX
ejpam-6573	102	11	obvious	obvious	ADJ
ejpam-6573	102	12	.	.	PUNCT
ejpam-6573	103	1	(	(	PUNCT
ejpam-6573	103	2	3	3	X
ejpam-6573	103	3	)	)	PUNCT
ejpam-6573	103	4	⇒	⇒	NOUN
ejpam-6573	103	5	(	(	PUNCT
ejpam-6573	103	6	4	4	NUM
ejpam-6573	103	7	):	):	PUNCT
ejpam-6573	103	8	let	let	VERB
ejpam-6573	103	9	b	b	X
ejpam-6573	103	10	be	be	AUX
ejpam-6573	103	11	any	any	DET
ejpam-6573	103	12	subset	subset	NOUN
ejpam-6573	103	13	of	of	ADP
ejpam-6573	103	14	y	y	PROPN
ejpam-6573	103	15	.	.	PUNCT
ejpam-6573	104	1	using	use	VERB
ejpam-6573	104	2	(	(	PUNCT
ejpam-6573	104	3	3	3	NUM
ejpam-6573	104	4	)	)	PUNCT
ejpam-6573	104	5	,	,	PUNCT
ejpam-6573	104	6	we	we	PRON
ejpam-6573	104	7	have	have	VERB
ejpam-6573	104	8	x	x	X
ejpam-6573	104	9	−	−	PROPN
ejpam-6573	104	10	(	(	PUNCT
ejpam-6573	104	11	τ1	τ1	NOUN
ejpam-6573	104	12	,	,	PUNCT
ejpam-6573	104	13	τ2)θ	τ2)θ	PROPN
ejpam-6573	104	14	-	-	PUNCT
ejpam-6573	104	15	cl(f	cl(f	NOUN
ejpam-6573	104	16	−1(b	−1(b	NOUN
ejpam-6573	104	17	)	)	PUNCT
ejpam-6573	104	18	)	)	PUNCT
ejpam-6573	105	1	=	=	PRON
ejpam-6573	105	2	(	(	PUNCT
ejpam-6573	105	3	τ1	τ1	NOUN
ejpam-6573	105	4	,	,	PUNCT
ejpam-6573	105	5	τ2)θ	τ2)θ	ADJ
ejpam-6573	105	6	-	-	PUNCT
ejpam-6573	105	7	int(x	int(x	PROPN
ejpam-6573	105	8	−	−	NOUN
ejpam-6573	105	9	f−1(b	f−1(b	PROPN
ejpam-6573	105	10	)	)	PUNCT
ejpam-6573	105	11	)	)	PUNCT
ejpam-6573	106	1	=	=	PRON
ejpam-6573	106	2	(	(	PUNCT
ejpam-6573	106	3	τ1	τ1	NOUN
ejpam-6573	106	4	,	,	PUNCT
ejpam-6573	106	5	τ2)θ	τ2)θ	NOUN
ejpam-6573	106	6	-	-	PUNCT
ejpam-6573	106	7	int(f	int(f	PROPN
ejpam-6573	106	8	−1(y	−1(y	VERB
ejpam-6573	106	9	−b	−b	ADJ
ejpam-6573	106	10	)	)	PUNCT
ejpam-6573	106	11	)	)	PUNCT
ejpam-6573	107	1	⊆	⊆	NUM
ejpam-6573	107	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	107	3	,	,	PUNCT
ejpam-6573	107	4	σ2)β	σ2)β	NOUN
ejpam-6573	107	5	-	-	PUNCT
ejpam-6573	107	6	int(y	int(y	PROPN
ejpam-6573	107	7	−b	−b	NOUN
ejpam-6573	107	8	)	)	PUNCT
ejpam-6573	107	9	)	)	PUNCT
ejpam-6573	108	1	=	=	SYM
ejpam-6573	108	2	f−1(y	f−1(y	PROPN
ejpam-6573	109	1	−	−	PROPN
ejpam-6573	110	1	(	(	PUNCT
ejpam-6573	111	1	σ1	σ1	PROPN
ejpam-6573	111	2	,	,	PUNCT
ejpam-6573	111	3	σ2)β	σ2)β	NOUN
ejpam-6573	111	4	-	-	PUNCT
ejpam-6573	111	5	cl(b	cl(b	NOUN
ejpam-6573	111	6	)	)	PUNCT
ejpam-6573	111	7	)	)	PUNCT
ejpam-6573	112	1	=	=	PUNCT
ejpam-6573	112	2	x	x	X
ejpam-6573	113	1	−	−	NOUN
ejpam-6573	113	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	113	3	,	,	PUNCT
ejpam-6573	113	4	σ2)β	σ2)β	NOUN
ejpam-6573	113	5	-	-	PUNCT
ejpam-6573	113	6	cl(b	cl(b	NOUN
ejpam-6573	113	7	)	)	PUNCT
ejpam-6573	113	8	)	)	PUNCT
ejpam-6573	113	9	and	and	CCONJ
ejpam-6573	113	10	so	so	ADV
ejpam-6573	113	11	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	113	12	,	,	PUNCT
ejpam-6573	113	13	σ2)β	σ2)β	NOUN
ejpam-6573	113	14	-	-	PUNCT
ejpam-6573	113	15	cl(b	cl(b	NOUN
ejpam-6573	113	16	)	)	PUNCT
ejpam-6573	113	17	)	)	PUNCT
ejpam-6573	114	1	⊆	⊆	NUM
ejpam-6573	114	2	(	(	PUNCT
ejpam-6573	114	3	τ1	τ1	NOUN
ejpam-6573	114	4	,	,	PUNCT
ejpam-6573	114	5	τ2)θ	τ2)θ	PROPN
ejpam-6573	114	6	-	-	PUNCT
ejpam-6573	114	7	cl(f	cl(f	NOUN
ejpam-6573	114	8	−1(b	−1(b	NOUN
ejpam-6573	114	9	)	)	PUNCT
ejpam-6573	114	10	)	)	PUNCT
ejpam-6573	114	11	.	.	PUNCT
ejpam-6573	115	1	m.	m.	NOUN
ejpam-6573	115	2	chiangpradit	chiangpradit	PROPN
ejpam-6573	115	3	,	,	PUNCT
ejpam-6573	115	4	a.	a.	PROPN
ejpam-6573	115	5	sama	sama	PROPN
ejpam-6573	115	6	-	-	PUNCT
ejpam-6573	115	7	ae	ae	PROPN
ejpam-6573	115	8	,	,	PUNCT
ejpam-6573	115	9	c.	c.	PROPN
ejpam-6573	115	10	boonpok	boonpok	PROPN
ejpam-6573	115	11	/	/	SYM
ejpam-6573	115	12	eur	eur	PROPN
ejpam-6573	115	13	.	.	PUNCT
ejpam-6573	116	1	j.	j.	PROPN
ejpam-6573	116	2	pure	pure	PROPN
ejpam-6573	116	3	appl	appl	PROPN
ejpam-6573	116	4	.	.	PROPN
ejpam-6573	116	5	math	math	PROPN
ejpam-6573	116	6	,	,	PUNCT
ejpam-6573	116	7	18	18	NUM
ejpam-6573	116	8	(	(	PUNCT
ejpam-6573	116	9	3	3	NUM
ejpam-6573	116	10	)	)	PUNCT
ejpam-6573	116	11	(	(	PUNCT
ejpam-6573	116	12	2025	2025	NUM
ejpam-6573	116	13	)	)	PUNCT
ejpam-6573	116	14	,	,	PUNCT
ejpam-6573	116	15	6573	6573	NUM
ejpam-6573	116	16	5	5	NUM
ejpam-6573	116	17	of	of	ADP
ejpam-6573	116	18	8	8	NUM
ejpam-6573	116	19	(	(	PUNCT
ejpam-6573	116	20	4	4	NUM
ejpam-6573	116	21	)	)	PUNCT
ejpam-6573	116	22	⇒	⇒	NOUN
ejpam-6573	116	23	(	(	PUNCT
ejpam-6573	116	24	3	3	NUM
ejpam-6573	116	25	):	):	PUNCT
ejpam-6573	116	26	let	let	VERB
ejpam-6573	116	27	b	b	X
ejpam-6573	116	28	be	be	AUX
ejpam-6573	116	29	any	any	DET
ejpam-6573	116	30	subset	subset	NOUN
ejpam-6573	116	31	of	of	ADP
ejpam-6573	116	32	y	y	PROPN
ejpam-6573	116	33	.	.	PUNCT
ejpam-6573	117	1	then	then	ADV
ejpam-6573	117	2	by	by	ADP
ejpam-6573	117	3	(	(	PUNCT
ejpam-6573	117	4	4	4	NUM
ejpam-6573	117	5	)	)	PUNCT
ejpam-6573	117	6	,	,	PUNCT
ejpam-6573	117	7	x	x	PUNCT
ejpam-6573	117	8	−	−	NOUN
ejpam-6573	117	9	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	117	10	,	,	PUNCT
ejpam-6573	117	11	σ2)β	σ2)β	NOUN
ejpam-6573	117	12	-	-	PUNCT
ejpam-6573	117	13	int(b	int(b	NOUN
ejpam-6573	117	14	)	)	PUNCT
ejpam-6573	117	15	)	)	PUNCT
ejpam-6573	118	1	⊆	⊆	NUM
ejpam-6573	118	2	x	x	SYM
ejpam-6573	118	3	−	−	PROPN
ejpam-6573	118	4	(	(	PUNCT
ejpam-6573	118	5	τ1	τ1	NOUN
ejpam-6573	118	6	,	,	PUNCT
ejpam-6573	118	7	τ2)θ	τ2)θ	NOUN
ejpam-6573	118	8	-	-	PUNCT
ejpam-6573	118	9	int(f	int(f	PROPN
ejpam-6573	118	10	−1(b	−1(b	NOUN
ejpam-6573	118	11	)	)	PUNCT
ejpam-6573	118	12	)	)	PUNCT
ejpam-6573	118	13	.	.	PUNCT
ejpam-6573	119	1	thus	thus	ADV
ejpam-6573	119	2	,	,	PUNCT
ejpam-6573	119	3	(	(	PUNCT
ejpam-6573	119	4	τ1	τ1	NOUN
ejpam-6573	119	5	,	,	PUNCT
ejpam-6573	119	6	τ2)θ	τ2)θ	ADJ
ejpam-6573	119	7	-	-	PUNCT
ejpam-6573	119	8	int(f	int(f	PROPN
ejpam-6573	119	9	−1(b	−1(b	NOUN
ejpam-6573	119	10	)	)	PUNCT
ejpam-6573	119	11	)	)	PUNCT
ejpam-6573	120	1	⊆	⊆	NUM
ejpam-6573	120	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	120	3	,	,	PUNCT
ejpam-6573	120	4	σ2)β	σ2)β	NOUN
ejpam-6573	120	5	-	-	PUNCT
ejpam-6573	120	6	int(b	int(b	NOUN
ejpam-6573	120	7	)	)	PUNCT
ejpam-6573	120	8	)	)	PUNCT
ejpam-6573	120	9	.	.	PUNCT
ejpam-6573	121	1	(	(	PUNCT
ejpam-6573	121	2	1	1	X
ejpam-6573	121	3	)	)	PUNCT
ejpam-6573	121	4	⇒	⇒	NOUN
ejpam-6573	121	5	(	(	PUNCT
ejpam-6573	121	6	5	5	NUM
ejpam-6573	121	7	):	):	PUNCT
ejpam-6573	121	8	let	let	VERB
ejpam-6573	121	9	x	x	PUNCT
ejpam-6573	121	10	∈	∈	PROPN
ejpam-6573	121	11	x	x	X
ejpam-6573	121	12	and	and	CCONJ
ejpam-6573	121	13	u	u	PRON
ejpam-6573	121	14	be	be	VERB
ejpam-6573	121	15	any	any	DET
ejpam-6573	121	16	τ1τ2	τ1τ2	ADJ
ejpam-6573	121	17	-	-	ADJ
ejpam-6573	121	18	open	open	ADJ
ejpam-6573	121	19	set	set	NOUN
ejpam-6573	121	20	of	of	ADP
ejpam-6573	121	21	x	x	PUNCT
ejpam-6573	121	22	containing	contain	VERB
ejpam-6573	121	23	x.	x.	NOUN
ejpam-6573	121	24	since	since	SCONJ
ejpam-6573	121	25	f	f	PROPN
ejpam-6573	121	26	is	be	AUX
ejpam-6573	121	27	weakly	weakly	ADJ
ejpam-6573	121	28	(	(	PUNCT
ejpam-6573	121	29	τ1	τ1	NOUN
ejpam-6573	121	30	,	,	PUNCT
ejpam-6573	121	31	τ2)β	τ2)β	ADJ
ejpam-6573	121	32	-	-	PUNCT
ejpam-6573	121	33	open	open	ADJ
ejpam-6573	121	34	,	,	PUNCT
ejpam-6573	121	35	f(x	f(x	PROPN
ejpam-6573	121	36	)	)	PUNCT
ejpam-6573	121	37	∈	∈	PROPN
ejpam-6573	121	38	f(u	f(u	PROPN
ejpam-6573	121	39	)	)	PUNCT
ejpam-6573	121	40	⊆	⊆	NUM
ejpam-6573	121	41	(	(	PUNCT
ejpam-6573	121	42	σ1	σ1	PROPN
ejpam-6573	121	43	,	,	PUNCT
ejpam-6573	121	44	σ2)β	σ2)β	NOUN
ejpam-6573	121	45	-	-	PUNCT
ejpam-6573	121	46	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	121	47	-	-	NOUN
ejpam-6573	121	48	cl(u	cl(u	NOUN
ejpam-6573	121	49	)	)	PUNCT
ejpam-6573	121	50	)	)	PUNCT
ejpam-6573	121	51	)	)	PUNCT
ejpam-6573	121	52	.	.	PUNCT
ejpam-6573	122	1	put	put	VERB
ejpam-6573	122	2	v	v	NOUN
ejpam-6573	122	3	=	=	SYM
ejpam-6573	122	4	(	(	PUNCT
ejpam-6573	122	5	σ1	σ1	PROPN
ejpam-6573	122	6	,	,	PUNCT
ejpam-6573	122	7	σ2)β	σ2)β	NOUN
ejpam-6573	122	8	-	-	PUNCT
ejpam-6573	122	9	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	122	10	-	-	NOUN
ejpam-6573	122	11	cl(u	cl(u	NOUN
ejpam-6573	122	12	)	)	PUNCT
ejpam-6573	122	13	)	)	PUNCT
ejpam-6573	122	14	)	)	PUNCT
ejpam-6573	122	15	.	.	PUNCT
ejpam-6573	123	1	then	then	ADV
ejpam-6573	123	2	,	,	PUNCT
ejpam-6573	123	3	v	v	NOUN
ejpam-6573	123	4	is	be	AUX
ejpam-6573	123	5	a	a	DET
ejpam-6573	123	6	(	(	PUNCT
ejpam-6573	123	7	σ1	σ1	PROPN
ejpam-6573	123	8	,	,	PUNCT
ejpam-6573	123	9	σ2)β	σ2)β	NOUN
ejpam-6573	123	10	-	-	PUNCT
ejpam-6573	123	11	open	open	ADJ
ejpam-6573	123	12	set	set	NOUN
ejpam-6573	123	13	of	of	ADP
ejpam-6573	123	14	y	y	PROPN
ejpam-6573	123	15	containing	contain	VERB
ejpam-6573	123	16	f(x	f(x	PROPN
ejpam-6573	123	17	)	)	PUNCT
ejpam-6573	123	18	such	such	ADJ
ejpam-6573	123	19	that	that	SCONJ
ejpam-6573	123	20	v	v	ADP
ejpam-6573	123	21	⊆	⊆	NUM
ejpam-6573	123	22	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	123	23	-	-	PUNCT
ejpam-6573	123	24	cl(u	cl(u	NOUN
ejpam-6573	123	25	)	)	PUNCT
ejpam-6573	123	26	)	)	PUNCT
ejpam-6573	123	27	.	.	PUNCT
ejpam-6573	124	1	(	(	PUNCT
ejpam-6573	124	2	5	5	X
ejpam-6573	124	3	)	)	PUNCT
ejpam-6573	124	4	⇒	⇒	NOUN
ejpam-6573	124	5	(	(	PUNCT
ejpam-6573	124	6	1	1	NUM
ejpam-6573	124	7	):	):	PUNCT
ejpam-6573	124	8	let	let	VERB
ejpam-6573	124	9	u	u	PRON
ejpam-6573	124	10	be	be	AUX
ejpam-6573	124	11	any	any	DET
ejpam-6573	124	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	124	13	-	-	ADJ
ejpam-6573	124	14	open	open	ADJ
ejpam-6573	124	15	set	set	NOUN
ejpam-6573	124	16	of	of	ADP
ejpam-6573	124	17	x	x	PUNCT
ejpam-6573	124	18	and	and	CCONJ
ejpam-6573	124	19	y	y	PROPN
ejpam-6573	124	20	∈	∈	PROPN
ejpam-6573	124	21	f(u	f(u	PROPN
ejpam-6573	124	22	)	)	PUNCT
ejpam-6573	124	23	.	.	PUNCT
ejpam-6573	125	1	it	it	PRON
ejpam-6573	125	2	follows	follow	VERB
ejpam-6573	125	3	from	from	ADP
ejpam-6573	125	4	(	(	PUNCT
ejpam-6573	125	5	5	5	NUM
ejpam-6573	125	6	)	)	PUNCT
ejpam-6573	126	1	that	that	PRON
ejpam-6573	126	2	v	v	ADP
ejpam-6573	126	3	⊆	⊆	NUM
ejpam-6573	126	4	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	126	5	-	-	PUNCT
ejpam-6573	126	6	cl(u	cl(u	NOUN
ejpam-6573	126	7	)	)	PUNCT
ejpam-6573	126	8	)	)	PUNCT
ejpam-6573	126	9	for	for	ADP
ejpam-6573	126	10	some	some	DET
ejpam-6573	126	11	(	(	PUNCT
ejpam-6573	126	12	σ1	σ1	PROPN
ejpam-6573	126	13	,	,	PUNCT
ejpam-6573	126	14	σ2)β	σ2)β	NOUN
ejpam-6573	126	15	-	-	PUNCT
ejpam-6573	126	16	open	open	NOUN
ejpam-6573	126	17	set	set	NOUN
ejpam-6573	126	18	v	v	NOUN
ejpam-6573	126	19	of	of	ADP
ejpam-6573	126	20	y	y	PROPN
ejpam-6573	126	21	containing	contain	VERB
ejpam-6573	126	22	y.	y.	PROPN
ejpam-6573	126	23	thus	thus	ADV
ejpam-6573	126	24	,	,	PUNCT
ejpam-6573	126	25	y	y	PROPN
ejpam-6573	126	26	∈	∈	PROPN
ejpam-6573	126	27	v	v	ADP
ejpam-6573	126	28	⊆	⊆	NUM
ejpam-6573	126	29	(	(	PUNCT
ejpam-6573	126	30	σ1	σ1	PROPN
ejpam-6573	126	31	,	,	PUNCT
ejpam-6573	126	32	σ2)β	σ2)β	NOUN
ejpam-6573	126	33	-	-	PUNCT
ejpam-6573	126	34	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	126	35	-	-	NOUN
ejpam-6573	126	36	cl(u	cl(u	NOUN
ejpam-6573	126	37	)	)	PUNCT
ejpam-6573	126	38	)	)	PUNCT
ejpam-6573	126	39	)	)	PUNCT
ejpam-6573	126	40	and	and	CCONJ
ejpam-6573	126	41	hence	hence	ADV
ejpam-6573	126	42	f(u	f(u	PROPN
ejpam-6573	126	43	)	)	PUNCT
ejpam-6573	126	44	⊆	⊆	NUM
ejpam-6573	126	45	(	(	PUNCT
ejpam-6573	126	46	σ1	σ1	PROPN
ejpam-6573	126	47	,	,	PUNCT
ejpam-6573	126	48	σ2)β	σ2)β	NOUN
ejpam-6573	126	49	-	-	PUNCT
ejpam-6573	126	50	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	126	51	-	-	NOUN
ejpam-6573	126	52	cl(u	cl(u	NOUN
ejpam-6573	126	53	)	)	PUNCT
ejpam-6573	126	54	)	)	PUNCT
ejpam-6573	126	55	)	)	PUNCT
ejpam-6573	126	56	.	.	PUNCT
ejpam-6573	127	1	this	this	PRON
ejpam-6573	127	2	shows	show	VERB
ejpam-6573	127	3	that	that	SCONJ
ejpam-6573	127	4	f	f	PROPN
ejpam-6573	127	5	is	be	AUX
ejpam-6573	127	6	weakly	weakly	ADJ
ejpam-6573	127	7	(	(	PUNCT
ejpam-6573	127	8	τ1	τ1	NOUN
ejpam-6573	127	9	,	,	PUNCT
ejpam-6573	127	10	τ2)βopen	τ2)βopen	ADJ
ejpam-6573	127	11	.	.	PUNCT
ejpam-6573	128	1	(	(	PUNCT
ejpam-6573	128	2	1	1	X
ejpam-6573	128	3	)	)	PUNCT
ejpam-6573	128	4	⇒	⇒	NOUN
ejpam-6573	128	5	(	(	PUNCT
ejpam-6573	128	6	6	6	NUM
ejpam-6573	128	7	)	)	PUNCT
ejpam-6573	128	8	⇒	⇒	NOUN
ejpam-6573	128	9	(	(	PUNCT
ejpam-6573	128	10	7	7	NUM
ejpam-6573	128	11	)	)	PUNCT
ejpam-6573	128	12	⇒	⇒	NOUN
ejpam-6573	128	13	(	(	PUNCT
ejpam-6573	128	14	8)	8)	NUM
ejpam-6573	128	15	⇒	⇒	NOUN
ejpam-6573	128	16	(	(	PUNCT
ejpam-6573	128	17	9	9	NUM
ejpam-6573	128	18	)	)	PUNCT
ejpam-6573	128	19	⇒	⇒	NOUN
ejpam-6573	128	20	(	(	PUNCT
ejpam-6573	128	21	1	1	NUM
ejpam-6573	128	22	):	):	PUNCT
ejpam-6573	128	23	this	this	PRON
ejpam-6573	128	24	is	be	AUX
ejpam-6573	128	25	obvious	obvious	ADJ
ejpam-6573	128	26	.	.	PUNCT
ejpam-6573	129	1	theorem	theorem	NOUN
ejpam-6573	129	2	2	2	NUM
ejpam-6573	129	3	.	.	PUNCT
ejpam-6573	130	1	let	let	VERB
ejpam-6573	130	2	f	f	NOUN
ejpam-6573	130	3	:	:	PUNCT
ejpam-6573	130	4	(	(	PUNCT
ejpam-6573	130	5	x	x	NOUN
ejpam-6573	130	6	,	,	PUNCT
ejpam-6573	130	7	τ1	τ1	NOUN
ejpam-6573	130	8	,	,	PUNCT
ejpam-6573	130	9	τ2	τ2	NOUN
ejpam-6573	130	10	)	)	PUNCT
ejpam-6573	130	11	→	→	SYM
ejpam-6573	130	12	(	(	PUNCT
ejpam-6573	130	13	y	y	PROPN
ejpam-6573	130	14	,	,	PUNCT
ejpam-6573	130	15	σ1	σ1	PROPN
ejpam-6573	130	16	,	,	PUNCT
ejpam-6573	130	17	σ2	σ2	PROPN
ejpam-6573	130	18	)	)	PUNCT
ejpam-6573	130	19	be	be	VERB
ejpam-6573	130	20	a	a	DET
ejpam-6573	130	21	bijective	bijective	ADJ
ejpam-6573	130	22	function	function	NOUN
ejpam-6573	130	23	.	.	PUNCT
ejpam-6573	131	1	then	then	ADV
ejpam-6573	131	2	,	,	PUNCT
ejpam-6573	131	3	the	the	DET
ejpam-6573	131	4	following	follow	VERB
ejpam-6573	131	5	properties	property	NOUN
ejpam-6573	131	6	are	be	AUX
ejpam-6573	131	7	equivalent	equivalent	ADJ
ejpam-6573	131	8	:	:	PUNCT
ejpam-6573	131	9	(	(	PUNCT
ejpam-6573	131	10	1	1	X
ejpam-6573	131	11	)	)	PUNCT
ejpam-6573	131	12	f	f	PROPN
ejpam-6573	131	13	is	be	AUX
ejpam-6573	131	14	weakly	weakly	ADJ
ejpam-6573	131	15	(	(	PUNCT
ejpam-6573	131	16	τ1	τ1	NOUN
ejpam-6573	131	17	,	,	PUNCT
ejpam-6573	131	18	τ2)β	τ2)β	ADJ
ejpam-6573	131	19	-	-	PUNCT
ejpam-6573	131	20	open	open	ADJ
ejpam-6573	131	21	;	;	PUNCT
ejpam-6573	131	22	(	(	PUNCT
ejpam-6573	131	23	2	2	X
ejpam-6573	131	24	)	)	PUNCT
ejpam-6573	131	25	(	(	PUNCT
ejpam-6573	131	26	σ1	σ1	PROPN
ejpam-6573	131	27	,	,	PUNCT
ejpam-6573	131	28	σ2)β	σ2)β	NOUN
ejpam-6573	131	29	-	-	PUNCT
ejpam-6573	131	30	cl(f(u	cl(f(u	NOUN
ejpam-6573	131	31	)	)	PUNCT
ejpam-6573	131	32	)	)	PUNCT
ejpam-6573	132	1	⊆	⊆	NUM
ejpam-6573	132	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	132	3	-	-	PUNCT
ejpam-6573	132	4	cl(u	cl(u	NOUN
ejpam-6573	132	5	)	)	PUNCT
ejpam-6573	132	6	)	)	PUNCT
ejpam-6573	132	7	for	for	ADP
ejpam-6573	132	8	every	every	DET
ejpam-6573	132	9	τ1τ2	τ1τ2	ADJ
ejpam-6573	132	10	-	-	ADJ
ejpam-6573	132	11	open	open	ADJ
ejpam-6573	132	12	set	set	ADJ
ejpam-6573	132	13	u	u	NOUN
ejpam-6573	132	14	of	of	ADP
ejpam-6573	132	15	x	x	PRON
ejpam-6573	132	16	;	;	PUNCT
ejpam-6573	132	17	(	(	PUNCT
ejpam-6573	132	18	3	3	X
ejpam-6573	132	19	)	)	PUNCT
ejpam-6573	132	20	(	(	PUNCT
ejpam-6573	132	21	σ1	σ1	PROPN
ejpam-6573	132	22	,	,	PUNCT
ejpam-6573	132	23	σ2)β	σ2)β	ADJ
ejpam-6573	132	24	-	-	PUNCT
ejpam-6573	132	25	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	132	26	-	-	PUNCT
ejpam-6573	132	27	int(k	int(k	NUM
ejpam-6573	132	28	)	)	PUNCT
ejpam-6573	132	29	)	)	PUNCT
ejpam-6573	132	30	)	)	PUNCT
ejpam-6573	133	1	⊆	⊆	NUM
ejpam-6573	133	2	f(k	f(k	VERB
ejpam-6573	133	3	)	)	PUNCT
ejpam-6573	133	4	for	for	ADP
ejpam-6573	133	5	every	every	DET
ejpam-6573	133	6	τ1τ2	τ1τ2	ADJ
ejpam-6573	133	7	-	-	ADJ
ejpam-6573	133	8	closed	closed	ADJ
ejpam-6573	133	9	set	set	NOUN
ejpam-6573	133	10	k	k	PROPN
ejpam-6573	133	11	of	of	ADP
ejpam-6573	133	12	x.	x.	NOUN
ejpam-6573	133	13	proof	proof	NOUN
ejpam-6573	133	14	.	.	PUNCT
ejpam-6573	134	1	(	(	PUNCT
ejpam-6573	134	2	1	1	X
ejpam-6573	134	3	)	)	PUNCT
ejpam-6573	134	4	⇒	⇒	NOUN
ejpam-6573	134	5	(	(	PUNCT
ejpam-6573	134	6	3	3	NUM
ejpam-6573	134	7	):	):	PUNCT
ejpam-6573	134	8	let	let	VERB
ejpam-6573	134	9	k	k	PRON
ejpam-6573	134	10	be	be	AUX
ejpam-6573	134	11	any	any	DET
ejpam-6573	134	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	134	13	-	-	ADJ
ejpam-6573	134	14	closed	closed	ADJ
ejpam-6573	134	15	set	set	NOUN
ejpam-6573	134	16	of	of	ADP
ejpam-6573	134	17	x.	x.	NOUN
ejpam-6573	134	18	then	then	ADV
ejpam-6573	134	19	,	,	PUNCT
ejpam-6573	134	20	we	we	PRON
ejpam-6573	134	21	have	have	VERB
ejpam-6573	134	22	f(x	f(x	PROPN
ejpam-6573	134	23	−k	−k	ADV
ejpam-6573	134	24	)	)	PUNCT
ejpam-6573	135	1	=	=	SYM
ejpam-6573	135	2	y	y	PROPN
ejpam-6573	135	3	−	−	PROPN
ejpam-6573	135	4	f(k	f(k	PROPN
ejpam-6573	135	5	)	)	PUNCT
ejpam-6573	135	6	⊆	⊆	NUM
ejpam-6573	135	7	(	(	PUNCT
ejpam-6573	135	8	σ1	σ1	PROPN
ejpam-6573	135	9	,	,	PUNCT
ejpam-6573	135	10	σ2)β	σ2)β	NOUN
ejpam-6573	135	11	-	-	PUNCT
ejpam-6573	135	12	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	135	13	-	-	PUNCT
ejpam-6573	135	14	cl(x	cl(x	NOUN
ejpam-6573	135	15	−k	−k	NOUN
ejpam-6573	135	16	)	)	PUNCT
ejpam-6573	135	17	)	)	PUNCT
ejpam-6573	135	18	)	)	PUNCT
ejpam-6573	136	1	and	and	CCONJ
ejpam-6573	136	2	hence	hence	ADV
ejpam-6573	136	3	y	y	PROPN
ejpam-6573	136	4	−	−	PROPN
ejpam-6573	136	5	f(k	f(k	PROPN
ejpam-6573	136	6	)	)	PUNCT
ejpam-6573	137	1	⊆	⊆	NUM
ejpam-6573	137	2	y	y	PROPN
ejpam-6573	137	3	−	−	PROPN
ejpam-6573	137	4	(	(	PUNCT
ejpam-6573	137	5	σ1	σ1	PROPN
ejpam-6573	137	6	,	,	PUNCT
ejpam-6573	137	7	σ2)β	σ2)β	ADJ
ejpam-6573	137	8	-	-	PUNCT
ejpam-6573	137	9	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	137	10	-	-	PUNCT
ejpam-6573	137	11	int(k	int(k	NOUN
ejpam-6573	137	12	)	)	PUNCT
ejpam-6573	137	13	)	)	PUNCT
ejpam-6573	137	14	)	)	PUNCT
ejpam-6573	137	15	.	.	PUNCT
ejpam-6573	138	1	thus	thus	ADV
ejpam-6573	138	2	,	,	PUNCT
ejpam-6573	138	3	(	(	PUNCT
ejpam-6573	138	4	σ1	σ1	PROPN
ejpam-6573	138	5	,	,	PUNCT
ejpam-6573	138	6	σ2)β	σ2)β	ADJ
ejpam-6573	138	7	-	-	PUNCT
ejpam-6573	138	8	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	138	9	-	-	PUNCT
ejpam-6573	138	10	int(k	int(k	NUM
ejpam-6573	138	11	)	)	PUNCT
ejpam-6573	138	12	)	)	PUNCT
ejpam-6573	138	13	)	)	PUNCT
ejpam-6573	139	1	⊆	⊆	NUM
ejpam-6573	139	2	f(k	f(k	VERB
ejpam-6573	139	3	)	)	PUNCT
ejpam-6573	139	4	.	.	PUNCT
ejpam-6573	140	1	(	(	PUNCT
ejpam-6573	140	2	3	3	X
ejpam-6573	140	3	)	)	PUNCT
ejpam-6573	140	4	⇒	⇒	NOUN
ejpam-6573	140	5	(	(	PUNCT
ejpam-6573	140	6	2	2	NUM
ejpam-6573	140	7	):	):	PUNCT
ejpam-6573	140	8	let	let	VERB
ejpam-6573	140	9	u	u	PRON
ejpam-6573	140	10	be	be	AUX
ejpam-6573	140	11	any	any	DET
ejpam-6573	140	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	140	13	-	-	ADJ
ejpam-6573	140	14	open	open	ADJ
ejpam-6573	140	15	set	set	NOUN
ejpam-6573	140	16	of	of	ADP
ejpam-6573	140	17	x.	x.	NOUN
ejpam-6573	140	18	since	since	SCONJ
ejpam-6573	140	19	τ1τ2	τ1τ2	NOUN
ejpam-6573	140	20	-	-	NOUN
ejpam-6573	140	21	cl(u	cl(u	NOUN
ejpam-6573	140	22	)	)	PUNCT
ejpam-6573	140	23	is	be	AUX
ejpam-6573	140	24	τ1τ2	τ1τ2	NOUN
ejpam-6573	140	25	-	-	ADJ
ejpam-6573	140	26	closed	closed	ADJ
ejpam-6573	140	27	and	and	CCONJ
ejpam-6573	140	28	u	u	PRON
ejpam-6573	140	29	⊆	⊆	NUM
ejpam-6573	140	30	τ1τ2	τ1τ2	NOUN
ejpam-6573	140	31	-	-	NOUN
ejpam-6573	140	32	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	140	33	-	-	PUNCT
ejpam-6573	140	34	cl(u	cl(u	NOUN
ejpam-6573	140	35	)	)	PUNCT
ejpam-6573	140	36	)	)	PUNCT
ejpam-6573	140	37	,	,	PUNCT
ejpam-6573	140	38	by	by	ADP
ejpam-6573	140	39	(	(	PUNCT
ejpam-6573	140	40	3	3	X
ejpam-6573	140	41	)	)	PUNCT
ejpam-6573	140	42	we	we	PRON
ejpam-6573	140	43	have	have	AUX
ejpam-6573	140	44	(	(	PUNCT
ejpam-6573	140	45	σ1	σ1	PROPN
ejpam-6573	140	46	,	,	PUNCT
ejpam-6573	140	47	σ2)β	σ2)β	NOUN
ejpam-6573	140	48	-	-	PUNCT
ejpam-6573	140	49	cl(f(u	cl(f(u	NOUN
ejpam-6573	140	50	)	)	PUNCT
ejpam-6573	140	51	)	)	PUNCT
ejpam-6573	141	1	⊆	⊆	NUM
ejpam-6573	141	2	(	(	PUNCT
ejpam-6573	141	3	σ1	σ1	PROPN
ejpam-6573	141	4	,	,	PUNCT
ejpam-6573	141	5	σ2)β	σ2)β	ADJ
ejpam-6573	141	6	-	-	PUNCT
ejpam-6573	141	7	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	141	8	-	-	PUNCT
ejpam-6573	141	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	141	10	-	-	PUNCT
ejpam-6573	141	11	cl(u	cl(u	NOUN
ejpam-6573	141	12	)	)	PUNCT
ejpam-6573	141	13	)	)	PUNCT
ejpam-6573	141	14	)	)	PUNCT
ejpam-6573	141	15	)	)	PUNCT
ejpam-6573	142	1	⊆	⊆	NUM
ejpam-6573	142	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	142	3	-	-	PUNCT
ejpam-6573	142	4	cl(u	cl(u	NOUN
ejpam-6573	142	5	)	)	PUNCT
ejpam-6573	142	6	)	)	PUNCT
ejpam-6573	142	7	.	.	PUNCT
ejpam-6573	143	1	(	(	PUNCT
ejpam-6573	143	2	2	2	X
ejpam-6573	143	3	)	)	PUNCT
ejpam-6573	143	4	⇒	⇒	NOUN
ejpam-6573	143	5	(	(	PUNCT
ejpam-6573	143	6	1	1	NUM
ejpam-6573	143	7	):	):	PUNCT
ejpam-6573	143	8	let	let	VERB
ejpam-6573	143	9	u	u	PRON
ejpam-6573	143	10	be	be	AUX
ejpam-6573	143	11	any	any	DET
ejpam-6573	143	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	143	13	-	-	ADJ
ejpam-6573	143	14	open	open	ADJ
ejpam-6573	143	15	set	set	NOUN
ejpam-6573	143	16	of	of	ADP
ejpam-6573	143	17	x.	x.	NOUN
ejpam-6573	143	18	by	by	ADP
ejpam-6573	143	19	(	(	PUNCT
ejpam-6573	143	20	2	2	NUM
ejpam-6573	143	21	)	)	PUNCT
ejpam-6573	143	22	,	,	PUNCT
ejpam-6573	143	23	we	we	PRON
ejpam-6573	143	24	have	have	VERB
ejpam-6573	143	25	(	(	PUNCT
ejpam-6573	143	26	σ1	σ1	PROPN
ejpam-6573	143	27	,	,	PUNCT
ejpam-6573	143	28	σ2)β	σ2)β	NOUN
ejpam-6573	143	29	-	-	PUNCT
ejpam-6573	143	30	cl(f(x	cl(f(x	NOUN
ejpam-6573	143	31	−	−	NOUN
ejpam-6573	143	32	τ1τ2	τ1τ2	NOUN
ejpam-6573	143	33	-	-	NOUN
ejpam-6573	143	34	cl(u	cl(u	NOUN
ejpam-6573	143	35	)	)	PUNCT
ejpam-6573	143	36	)	)	PUNCT
ejpam-6573	143	37	)	)	PUNCT
ejpam-6573	144	1	⊆	⊆	NUM
ejpam-6573	144	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	144	3	-	-	PUNCT
ejpam-6573	144	4	cl(x	cl(x	NOUN
ejpam-6573	144	5	−	−	PRON
ejpam-6573	144	6	τ1τ2	τ1τ2	NOUN
ejpam-6573	144	7	-	-	NOUN
ejpam-6573	144	8	cl(u	cl(u	NOUN
ejpam-6573	144	9	)	)	PUNCT
ejpam-6573	144	10	)	)	PUNCT
ejpam-6573	144	11	)	)	PUNCT
ejpam-6573	144	12	.	.	PUNCT
ejpam-6573	145	1	since	since	SCONJ
ejpam-6573	145	2	f	f	PROPN
ejpam-6573	145	3	is	be	AUX
ejpam-6573	145	4	bijective	bijective	ADJ
ejpam-6573	145	5	,	,	PUNCT
ejpam-6573	145	6	(	(	PUNCT
ejpam-6573	145	7	σ1	σ1	PROPN
ejpam-6573	145	8	,	,	PUNCT
ejpam-6573	145	9	σ2)β	σ2)β	NOUN
ejpam-6573	145	10	-	-	PUNCT
ejpam-6573	145	11	cl(f(x	cl(f(x	NOUN
ejpam-6573	145	12	−	−	NOUN
ejpam-6573	145	13	τ1τ2	τ1τ2	NOUN
ejpam-6573	145	14	-	-	NOUN
ejpam-6573	145	15	cl(u	cl(u	NOUN
ejpam-6573	145	16	)	)	PUNCT
ejpam-6573	145	17	)	)	PUNCT
ejpam-6573	145	18	)	)	PUNCT
ejpam-6573	146	1	=	=	PUNCT
ejpam-6573	147	1	y	y	PROPN
ejpam-6573	147	2	−	−	PROPN
ejpam-6573	147	3	(	(	PUNCT
ejpam-6573	147	4	σ1	σ1	PROPN
ejpam-6573	147	5	,	,	PUNCT
ejpam-6573	147	6	σ2)β	σ2)β	NOUN
ejpam-6573	147	7	-	-	PUNCT
ejpam-6573	147	8	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	147	9	-	-	NOUN
ejpam-6573	147	10	cl(u	cl(u	NOUN
ejpam-6573	147	11	)	)	PUNCT
ejpam-6573	147	12	)	)	PUNCT
ejpam-6573	147	13	)	)	PUNCT
ejpam-6573	147	14	and	and	CCONJ
ejpam-6573	147	15	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	147	16	-	-	PUNCT
ejpam-6573	147	17	cl(x	cl(x	NUM
ejpam-6573	147	18	−	−	PRON
ejpam-6573	147	19	τ1τ2	τ1τ2	NOUN
ejpam-6573	147	20	-	-	NOUN
ejpam-6573	147	21	cl(u	cl(u	NOUN
ejpam-6573	147	22	)	)	PUNCT
ejpam-6573	147	23	)	)	PUNCT
ejpam-6573	147	24	)	)	PUNCT
ejpam-6573	148	1	=	=	SYM
ejpam-6573	148	2	f(x	f(x	PROPN
ejpam-6573	148	3	−	−	PROPN
ejpam-6573	149	1	τ1τ2	τ1τ2	NOUN
ejpam-6573	149	2	-	-	NOUN
ejpam-6573	149	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	149	4	-	-	PUNCT
ejpam-6573	149	5	cl(u	cl(u	NOUN
ejpam-6573	149	6	)	)	PUNCT
ejpam-6573	149	7	)	)	PUNCT
ejpam-6573	149	8	)	)	PUNCT
ejpam-6573	150	1	⊆	⊆	NUM
ejpam-6573	150	2	f(x	f(x	PROPN
ejpam-6573	150	3	−	−	PROPN
ejpam-6573	150	4	u	u	NOUN
ejpam-6573	150	5	)	)	PUNCT
ejpam-6573	150	6	=	=	SYM
ejpam-6573	150	7	y	y	PROPN
ejpam-6573	150	8	−	−	PROPN
ejpam-6573	150	9	f(u	f(u	PROPN
ejpam-6573	150	10	)	)	PUNCT
ejpam-6573	150	11	.	.	PUNCT
ejpam-6573	151	1	thus	thus	ADV
ejpam-6573	151	2	,	,	PUNCT
ejpam-6573	151	3	f(u	f(u	PROPN
ejpam-6573	151	4	)	)	PUNCT
ejpam-6573	151	5	⊆	⊆	NUM
ejpam-6573	151	6	(	(	PUNCT
ejpam-6573	151	7	σ1	σ1	PROPN
ejpam-6573	151	8	,	,	PUNCT
ejpam-6573	151	9	σ2)β	σ2)β	NOUN
ejpam-6573	151	10	-	-	PUNCT
ejpam-6573	151	11	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6573	151	12	-	-	NOUN
ejpam-6573	151	13	cl(u	cl(u	NOUN
ejpam-6573	151	14	)	)	PUNCT
ejpam-6573	151	15	)	)	PUNCT
ejpam-6573	151	16	)	)	PUNCT
ejpam-6573	151	17	and	and	CCONJ
ejpam-6573	151	18	hence	hence	ADV
ejpam-6573	151	19	f	f	PROPN
ejpam-6573	151	20	is	be	AUX
ejpam-6573	151	21	weakly	weakly	ADJ
ejpam-6573	151	22	(	(	PUNCT
ejpam-6573	151	23	τ1	τ1	NOUN
ejpam-6573	151	24	,	,	PUNCT
ejpam-6573	151	25	τ2)β	τ2)β	ADJ
ejpam-6573	151	26	-	-	PUNCT
ejpam-6573	151	27	open	open	ADJ
ejpam-6573	151	28	.	.	PUNCT
ejpam-6573	152	1	m.	m.	NOUN
ejpam-6573	152	2	chiangpradit	chiangpradit	PROPN
ejpam-6573	152	3	,	,	PUNCT
ejpam-6573	152	4	a.	a.	PROPN
ejpam-6573	152	5	sama	sama	PROPN
ejpam-6573	152	6	-	-	PUNCT
ejpam-6573	152	7	ae	ae	PROPN
ejpam-6573	152	8	,	,	PUNCT
ejpam-6573	152	9	c.	c.	PROPN
ejpam-6573	152	10	boonpok	boonpok	PROPN
ejpam-6573	152	11	/	/	SYM
ejpam-6573	152	12	eur	eur	PROPN
ejpam-6573	152	13	.	.	PUNCT
ejpam-6573	153	1	j.	j.	PROPN
ejpam-6573	153	2	pure	pure	PROPN
ejpam-6573	153	3	appl	appl	PROPN
ejpam-6573	153	4	.	.	PROPN
ejpam-6573	153	5	math	math	PROPN
ejpam-6573	153	6	,	,	PUNCT
ejpam-6573	153	7	18	18	NUM
ejpam-6573	153	8	(	(	PUNCT
ejpam-6573	153	9	3	3	NUM
ejpam-6573	153	10	)	)	PUNCT
ejpam-6573	153	11	(	(	PUNCT
ejpam-6573	153	12	2025	2025	NUM
ejpam-6573	153	13	)	)	PUNCT
ejpam-6573	153	14	,	,	PUNCT
ejpam-6573	153	15	6573	6573	NUM
ejpam-6573	153	16	6	6	NUM
ejpam-6573	153	17	of	of	ADP
ejpam-6573	153	18	8	8	NUM
ejpam-6573	153	19	4	4	NUM
ejpam-6573	153	20	.	.	PUNCT
ejpam-6573	154	1	characterizations	characterization	NOUN
ejpam-6573	154	2	of	of	ADP
ejpam-6573	154	3	weakly	weakly	ADJ
ejpam-6573	154	4	(	(	PUNCT
ejpam-6573	154	5	τ1	τ1	NOUN
ejpam-6573	154	6	,	,	PUNCT
ejpam-6573	154	7	τ2)β	τ2)β	ADJ
ejpam-6573	154	8	-	-	PUNCT
ejpam-6573	154	9	closed	close	VERB
ejpam-6573	154	10	functions	function	NOUN
ejpam-6573	154	11	in	in	ADP
ejpam-6573	154	12	this	this	DET
ejpam-6573	154	13	section	section	NOUN
ejpam-6573	154	14	,	,	PUNCT
ejpam-6573	154	15	we	we	PRON
ejpam-6573	154	16	introduce	introduce	VERB
ejpam-6573	154	17	the	the	DET
ejpam-6573	154	18	concept	concept	NOUN
ejpam-6573	154	19	of	of	ADP
ejpam-6573	154	20	weakly	weakly	ADJ
ejpam-6573	154	21	(	(	PUNCT
ejpam-6573	154	22	τ1	τ1	NOUN
ejpam-6573	154	23	,	,	PUNCT
ejpam-6573	154	24	τ2)β	τ2)β	ADJ
ejpam-6573	154	25	-	-	PUNCT
ejpam-6573	154	26	closed	close	VERB
ejpam-6573	154	27	functions	function	NOUN
ejpam-6573	154	28	.	.	PUNCT
ejpam-6573	155	1	furthermore	furthermore	ADV
ejpam-6573	155	2	,	,	PUNCT
ejpam-6573	155	3	some	some	DET
ejpam-6573	155	4	characterizations	characterization	NOUN
ejpam-6573	155	5	of	of	ADP
ejpam-6573	155	6	weakly	weakly	ADJ
ejpam-6573	155	7	(	(	PUNCT
ejpam-6573	155	8	τ1	τ1	NOUN
ejpam-6573	155	9	,	,	PUNCT
ejpam-6573	155	10	τ2)β	τ2)β	ADJ
ejpam-6573	155	11	-	-	PUNCT
ejpam-6573	155	12	closed	close	VERB
ejpam-6573	155	13	functions	function	NOUN
ejpam-6573	155	14	are	be	AUX
ejpam-6573	155	15	discussed	discuss	VERB
ejpam-6573	155	16	.	.	PUNCT
ejpam-6573	156	1	definition	definition	NOUN
ejpam-6573	156	2	2	2	NUM
ejpam-6573	156	3	.	.	PUNCT
ejpam-6573	157	1	a	a	DET
ejpam-6573	157	2	functions	function	NOUN
ejpam-6573	157	3	f	f	X
ejpam-6573	157	4	:	:	PUNCT
ejpam-6573	157	5	(	(	PUNCT
ejpam-6573	157	6	x	x	NOUN
ejpam-6573	157	7	,	,	PUNCT
ejpam-6573	157	8	τ1	τ1	NOUN
ejpam-6573	157	9	,	,	PUNCT
ejpam-6573	157	10	τ2	τ2	NOUN
ejpam-6573	157	11	)	)	PUNCT
ejpam-6573	157	12	→	→	SYM
ejpam-6573	157	13	(	(	PUNCT
ejpam-6573	157	14	y	y	PROPN
ejpam-6573	157	15	,	,	PUNCT
ejpam-6573	157	16	σ1	σ1	PROPN
ejpam-6573	157	17	,	,	PUNCT
ejpam-6573	157	18	σ2	σ2	PROPN
ejpam-6573	157	19	)	)	PUNCT
ejpam-6573	157	20	is	be	AUX
ejpam-6573	157	21	said	say	VERB
ejpam-6573	157	22	to	to	PART
ejpam-6573	157	23	be	be	AUX
ejpam-6573	157	24	weakly	weakly	ADJ
ejpam-6573	157	25	(	(	PUNCT
ejpam-6573	157	26	τ1	τ1	NOUN
ejpam-6573	157	27	,	,	PUNCT
ejpam-6573	157	28	τ2)β	τ2)β	NOUN
ejpam-6573	157	29	-	-	PUNCT
ejpam-6573	157	30	closed	closed	ADJ
ejpam-6573	157	31	if	if	SCONJ
ejpam-6573	157	32	(	(	PUNCT
ejpam-6573	157	33	σ1	σ1	PROPN
ejpam-6573	157	34	,	,	PUNCT
ejpam-6573	157	35	σ2)β	σ2)β	ADJ
ejpam-6573	157	36	-	-	PUNCT
ejpam-6573	157	37	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	157	38	-	-	PUNCT
ejpam-6573	157	39	int(k	int(k	NUM
ejpam-6573	157	40	)	)	PUNCT
ejpam-6573	157	41	)	)	PUNCT
ejpam-6573	157	42	)	)	PUNCT
ejpam-6573	158	1	⊆	⊆	NUM
ejpam-6573	158	2	f(k	f(k	VERB
ejpam-6573	158	3	)	)	PUNCT
ejpam-6573	158	4	for	for	ADP
ejpam-6573	158	5	every	every	DET
ejpam-6573	158	6	τ1τ2	τ1τ2	ADJ
ejpam-6573	158	7	-	-	ADJ
ejpam-6573	158	8	closed	closed	ADJ
ejpam-6573	158	9	set	set	NOUN
ejpam-6573	158	10	k	k	PROPN
ejpam-6573	158	11	of	of	ADP
ejpam-6573	158	12	x.	x.	PROPN
ejpam-6573	158	13	theorem	theorem	VERB
ejpam-6573	158	14	3	3	NUM
ejpam-6573	158	15	.	.	X
ejpam-6573	158	16	for	for	ADP
ejpam-6573	158	17	a	a	DET
ejpam-6573	158	18	function	function	NOUN
ejpam-6573	158	19	f	f	NOUN
ejpam-6573	158	20	:	:	PUNCT
ejpam-6573	158	21	(	(	PUNCT
ejpam-6573	158	22	x	x	NOUN
ejpam-6573	158	23	,	,	PUNCT
ejpam-6573	158	24	τ1	τ1	NOUN
ejpam-6573	158	25	,	,	PUNCT
ejpam-6573	158	26	τ2	τ2	NOUN
ejpam-6573	158	27	)	)	PUNCT
ejpam-6573	158	28	→	→	SYM
ejpam-6573	158	29	(	(	PUNCT
ejpam-6573	158	30	y	y	PROPN
ejpam-6573	158	31	,	,	PUNCT
ejpam-6573	158	32	σ1	σ1	PROPN
ejpam-6573	158	33	,	,	PUNCT
ejpam-6573	158	34	σ2	σ2	NOUN
ejpam-6573	158	35	)	)	PUNCT
ejpam-6573	158	36	,	,	PUNCT
ejpam-6573	158	37	the	the	DET
ejpam-6573	158	38	following	follow	VERB
ejpam-6573	158	39	properties	property	NOUN
ejpam-6573	158	40	are	be	AUX
ejpam-6573	158	41	equivalent	equivalent	ADJ
ejpam-6573	158	42	:	:	PUNCT
ejpam-6573	158	43	(	(	PUNCT
ejpam-6573	158	44	1	1	X
ejpam-6573	158	45	)	)	PUNCT
ejpam-6573	158	46	f	f	PROPN
ejpam-6573	158	47	is	be	AUX
ejpam-6573	158	48	weakly	weakly	ADJ
ejpam-6573	158	49	(	(	PUNCT
ejpam-6573	158	50	τ1	τ1	NOUN
ejpam-6573	158	51	,	,	PUNCT
ejpam-6573	158	52	τ2)β	τ2)β	NOUN
ejpam-6573	158	53	-	-	PUNCT
ejpam-6573	158	54	closed	closed	ADJ
ejpam-6573	158	55	;	;	PUNCT
ejpam-6573	158	56	(	(	PUNCT
ejpam-6573	158	57	2	2	X
ejpam-6573	158	58	)	)	PUNCT
ejpam-6573	158	59	(	(	PUNCT
ejpam-6573	158	60	σ1	σ1	PROPN
ejpam-6573	158	61	,	,	PUNCT
ejpam-6573	158	62	σ2)β	σ2)β	NOUN
ejpam-6573	158	63	-	-	PUNCT
ejpam-6573	158	64	cl(f(u	cl(f(u	NOUN
ejpam-6573	158	65	)	)	PUNCT
ejpam-6573	158	66	)	)	PUNCT
ejpam-6573	159	1	⊆	⊆	NUM
ejpam-6573	159	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	159	3	-	-	PUNCT
ejpam-6573	159	4	cl(u	cl(u	NOUN
ejpam-6573	159	5	)	)	PUNCT
ejpam-6573	159	6	)	)	PUNCT
ejpam-6573	159	7	for	for	ADP
ejpam-6573	159	8	every	every	DET
ejpam-6573	159	9	τ1τ2	τ1τ2	ADJ
ejpam-6573	159	10	-	-	ADJ
ejpam-6573	159	11	open	open	ADJ
ejpam-6573	159	12	set	set	ADJ
ejpam-6573	159	13	u	u	NOUN
ejpam-6573	159	14	of	of	ADP
ejpam-6573	159	15	x	x	PRON
ejpam-6573	159	16	;	;	PUNCT
ejpam-6573	159	17	(	(	PUNCT
ejpam-6573	159	18	3	3	X
ejpam-6573	159	19	)	)	PUNCT
ejpam-6573	159	20	(	(	PUNCT
ejpam-6573	159	21	σ1	σ1	PROPN
ejpam-6573	159	22	,	,	PUNCT
ejpam-6573	159	23	σ2)β	σ2)β	NOUN
ejpam-6573	159	24	-	-	PUNCT
ejpam-6573	159	25	cl(f(u	cl(f(u	NOUN
ejpam-6573	159	26	)	)	PUNCT
ejpam-6573	159	27	)	)	PUNCT
ejpam-6573	159	28	⊆	⊆	NUM
ejpam-6573	159	29	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	159	30	-	-	PUNCT
ejpam-6573	159	31	cl(u	cl(u	NOUN
ejpam-6573	159	32	)	)	PUNCT
ejpam-6573	159	33	)	)	PUNCT
ejpam-6573	159	34	for	for	ADP
ejpam-6573	159	35	every	every	DET
ejpam-6573	159	36	(	(	PUNCT
ejpam-6573	159	37	τ1	τ1	NOUN
ejpam-6573	159	38	,	,	PUNCT
ejpam-6573	159	39	τ2)r	τ2)r	ADJ
ejpam-6573	159	40	-	-	PUNCT
ejpam-6573	159	41	open	open	NOUN
ejpam-6573	159	42	set	set	NOUN
ejpam-6573	159	43	u	u	NOUN
ejpam-6573	159	44	of	of	ADP
ejpam-6573	159	45	x	x	PRON
ejpam-6573	159	46	;	;	PUNCT
ejpam-6573	159	47	(	(	PUNCT
ejpam-6573	159	48	4	4	X
ejpam-6573	159	49	)	)	PUNCT
ejpam-6573	159	50	for	for	ADP
ejpam-6573	159	51	each	each	DET
ejpam-6573	159	52	subset	subset	NOUN
ejpam-6573	159	53	b	b	PROPN
ejpam-6573	159	54	of	of	ADP
ejpam-6573	159	55	y	y	PROPN
ejpam-6573	159	56	and	and	CCONJ
ejpam-6573	159	57	each	each	DET
ejpam-6573	159	58	τ1τ2	τ1τ2	ADJ
ejpam-6573	159	59	-	-	ADJ
ejpam-6573	159	60	open	open	ADJ
ejpam-6573	159	61	set	set	ADJ
ejpam-6573	159	62	u	u	NOUN
ejpam-6573	159	63	of	of	ADP
ejpam-6573	159	64	x	x	PUNCT
ejpam-6573	159	65	with	with	ADP
ejpam-6573	159	66	f−1(b	f−1(b	PROPN
ejpam-6573	159	67	)	)	PUNCT
ejpam-6573	159	68	⊆	⊆	NUM
ejpam-6573	159	69	u	u	NOUN
ejpam-6573	159	70	,	,	PUNCT
ejpam-6573	159	71	there	there	PRON
ejpam-6573	159	72	exists	exist	VERB
ejpam-6573	159	73	a	a	DET
ejpam-6573	159	74	(	(	PUNCT
ejpam-6573	159	75	σ1	σ1	PROPN
ejpam-6573	159	76	,	,	PUNCT
ejpam-6573	159	77	σ2)β	σ2)β	NOUN
ejpam-6573	159	78	-	-	PUNCT
ejpam-6573	159	79	open	open	NOUN
ejpam-6573	159	80	set	set	NOUN
ejpam-6573	159	81	v	v	NOUN
ejpam-6573	159	82	of	of	ADP
ejpam-6573	159	83	y	y	PRON
ejpam-6573	159	84	such	such	ADJ
ejpam-6573	159	85	that	that	DET
ejpam-6573	159	86	b	b	PROPN
ejpam-6573	159	87	⊆	⊆	NUM
ejpam-6573	159	88	v	v	NOUN
ejpam-6573	159	89	and	and	CCONJ
ejpam-6573	159	90	f−1(v	f−1(v	NOUN
ejpam-6573	159	91	)	)	PUNCT
ejpam-6573	160	1	⊆	⊆	NUM
ejpam-6573	160	2	τ1τ2	τ1τ2	NOUN
ejpam-6573	160	3	-	-	NOUN
ejpam-6573	160	4	cl(u	cl(u	NUM
ejpam-6573	160	5	)	)	PUNCT
ejpam-6573	160	6	;	;	PUNCT
ejpam-6573	160	7	(	(	PUNCT
ejpam-6573	160	8	5	5	X
ejpam-6573	160	9	)	)	PUNCT
ejpam-6573	160	10	for	for	ADP
ejpam-6573	160	11	each	each	DET
ejpam-6573	160	12	point	point	NOUN
ejpam-6573	160	13	y	y	PROPN
ejpam-6573	160	14	∈	∈	PROPN
ejpam-6573	160	15	y	y	PROPN
ejpam-6573	160	16	and	and	CCONJ
ejpam-6573	160	17	each	each	DET
ejpam-6573	160	18	τ1τ2	τ1τ2	ADJ
ejpam-6573	160	19	-	-	ADJ
ejpam-6573	160	20	open	open	ADJ
ejpam-6573	160	21	set	set	ADJ
ejpam-6573	160	22	u	u	NOUN
ejpam-6573	160	23	of	of	ADP
ejpam-6573	160	24	x	x	PUNCT
ejpam-6573	160	25	with	with	ADP
ejpam-6573	160	26	f−1(y	f−1(y	PROPN
ejpam-6573	160	27	)	)	PUNCT
ejpam-6573	160	28	⊆	⊆	NUM
ejpam-6573	160	29	u	u	NOUN
ejpam-6573	160	30	,	,	PUNCT
ejpam-6573	160	31	there	there	PRON
ejpam-6573	160	32	exists	exist	VERB
ejpam-6573	160	33	a	a	DET
ejpam-6573	160	34	(	(	PUNCT
ejpam-6573	160	35	σ1	σ1	PROPN
ejpam-6573	160	36	,	,	PUNCT
ejpam-6573	160	37	σ2)β	σ2)β	NOUN
ejpam-6573	160	38	-	-	PUNCT
ejpam-6573	160	39	open	open	NOUN
ejpam-6573	160	40	set	set	NOUN
ejpam-6573	160	41	v	v	NOUN
ejpam-6573	160	42	of	of	ADP
ejpam-6573	160	43	y	y	PROPN
ejpam-6573	160	44	containing	contain	VERB
ejpam-6573	160	45	y	y	PRON
ejpam-6573	160	46	such	such	ADJ
ejpam-6573	160	47	that	that	DET
ejpam-6573	160	48	f−1(v	f−1(v	NOUN
ejpam-6573	160	49	)	)	PUNCT
ejpam-6573	161	1	⊆	⊆	X
ejpam-6573	161	2	τ1τ2	τ1τ2	NOUN
ejpam-6573	161	3	-	-	NOUN
ejpam-6573	161	4	cl(u	cl(u	NUM
ejpam-6573	161	5	)	)	PUNCT
ejpam-6573	161	6	;	;	PUNCT
ejpam-6573	161	7	(	(	PUNCT
ejpam-6573	161	8	6	6	NUM
ejpam-6573	161	9	)	)	PUNCT
ejpam-6573	161	10	(	(	PUNCT
ejpam-6573	161	11	σ1	σ1	PROPN
ejpam-6573	161	12	,	,	PUNCT
ejpam-6573	161	13	σ2)β	σ2)β	ADJ
ejpam-6573	161	14	-	-	PUNCT
ejpam-6573	161	15	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	161	16	-	-	PUNCT
ejpam-6573	161	17	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	161	18	-	-	PUNCT
ejpam-6573	161	19	cl(u	cl(u	NOUN
ejpam-6573	161	20	)	)	PUNCT
ejpam-6573	161	21	)	)	PUNCT
ejpam-6573	161	22	)	)	PUNCT
ejpam-6573	161	23	)	)	PUNCT
ejpam-6573	162	1	⊆	⊆	NUM
ejpam-6573	162	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	162	3	-	-	PUNCT
ejpam-6573	162	4	cl(u	cl(u	NOUN
ejpam-6573	162	5	)	)	PUNCT
ejpam-6573	162	6	)	)	PUNCT
ejpam-6573	162	7	for	for	ADP
ejpam-6573	162	8	every	every	DET
ejpam-6573	162	9	τ1τ2	τ1τ2	ADJ
ejpam-6573	162	10	-	-	ADJ
ejpam-6573	162	11	open	open	ADJ
ejpam-6573	162	12	set	set	ADJ
ejpam-6573	162	13	u	u	NOUN
ejpam-6573	162	14	of	of	ADP
ejpam-6573	162	15	x	x	PRON
ejpam-6573	162	16	;	;	PUNCT
ejpam-6573	162	17	(	(	PUNCT
ejpam-6573	162	18	7	7	X
ejpam-6573	162	19	)	)	PUNCT
ejpam-6573	162	20	(	(	PUNCT
ejpam-6573	162	21	σ1	σ1	PROPN
ejpam-6573	162	22	,	,	PUNCT
ejpam-6573	162	23	σ2)β	σ2)β	ADJ
ejpam-6573	162	24	-	-	PUNCT
ejpam-6573	162	25	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	162	26	-	-	PUNCT
ejpam-6573	162	27	int((τ1	int((τ1	NOUN
ejpam-6573	162	28	,	,	PUNCT
ejpam-6573	162	29	τ2)θ	τ2)θ	ADJ
ejpam-6573	162	30	-	-	PUNCT
ejpam-6573	162	31	cl(u	cl(u	NOUN
ejpam-6573	162	32	)	)	PUNCT
ejpam-6573	162	33	)	)	PUNCT
ejpam-6573	162	34	)	)	PUNCT
ejpam-6573	162	35	)	)	PUNCT
ejpam-6573	163	1	⊆	⊆	X
ejpam-6573	163	2	f((τ1	f((τ1	VERB
ejpam-6573	163	3	,	,	PUNCT
ejpam-6573	163	4	τ2)θ	τ2)θ	ADJ
ejpam-6573	163	5	-	-	PUNCT
ejpam-6573	163	6	cl(u	cl(u	NOUN
ejpam-6573	163	7	)	)	PUNCT
ejpam-6573	163	8	)	)	PUNCT
ejpam-6573	163	9	for	for	ADP
ejpam-6573	163	10	every	every	DET
ejpam-6573	163	11	τ1τ2	τ1τ2	ADJ
ejpam-6573	163	12	-	-	ADJ
ejpam-6573	163	13	open	open	ADJ
ejpam-6573	163	14	set	set	ADJ
ejpam-6573	163	15	u	u	NOUN
ejpam-6573	163	16	of	of	ADP
ejpam-6573	163	17	x	x	PRON
ejpam-6573	163	18	;	;	PUNCT
ejpam-6573	163	19	(	(	PUNCT
ejpam-6573	163	20	8)	8)	NUM
ejpam-6573	163	21	(	(	PUNCT
ejpam-6573	163	22	σ1	σ1	PROPN
ejpam-6573	163	23	,	,	PUNCT
ejpam-6573	163	24	σ2)β	σ2)β	NOUN
ejpam-6573	163	25	-	-	PUNCT
ejpam-6573	163	26	cl(f(u	cl(f(u	NOUN
ejpam-6573	163	27	)	)	PUNCT
ejpam-6573	163	28	)	)	PUNCT
ejpam-6573	164	1	⊆	⊆	NUM
ejpam-6573	164	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	164	3	-	-	PUNCT
ejpam-6573	164	4	cl(u	cl(u	NOUN
ejpam-6573	164	5	)	)	PUNCT
ejpam-6573	164	6	)	)	PUNCT
ejpam-6573	164	7	for	for	ADP
ejpam-6573	164	8	every	every	DET
ejpam-6573	164	9	(	(	PUNCT
ejpam-6573	164	10	τ1	τ1	NOUN
ejpam-6573	164	11	,	,	PUNCT
ejpam-6573	164	12	τ2)β	τ2)β	ADJ
ejpam-6573	164	13	-	-	PUNCT
ejpam-6573	164	14	open	open	ADJ
ejpam-6573	164	15	set	set	NOUN
ejpam-6573	164	16	u	u	NOUN
ejpam-6573	164	17	of	of	ADP
ejpam-6573	164	18	x.	x.	NOUN
ejpam-6573	164	19	proof	proof	NOUN
ejpam-6573	164	20	.	.	PUNCT
ejpam-6573	165	1	(	(	PUNCT
ejpam-6573	165	2	1	1	X
ejpam-6573	165	3	)	)	PUNCT
ejpam-6573	165	4	⇒	⇒	NOUN
ejpam-6573	165	5	(	(	PUNCT
ejpam-6573	165	6	2	2	NUM
ejpam-6573	165	7	):	):	PUNCT
ejpam-6573	165	8	let	let	VERB
ejpam-6573	165	9	u	u	PRON
ejpam-6573	165	10	be	be	AUX
ejpam-6573	165	11	any	any	DET
ejpam-6573	165	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	165	13	-	-	ADJ
ejpam-6573	165	14	open	open	ADJ
ejpam-6573	165	15	set	set	NOUN
ejpam-6573	165	16	of	of	ADP
ejpam-6573	165	17	x.	x.	NOUN
ejpam-6573	165	18	then	then	ADV
ejpam-6573	165	19	by	by	ADP
ejpam-6573	165	20	(	(	PUNCT
ejpam-6573	165	21	1	1	NUM
ejpam-6573	165	22	)	)	PUNCT
ejpam-6573	165	23	,	,	PUNCT
ejpam-6573	165	24	(	(	PUNCT
ejpam-6573	165	25	σ1	σ1	PROPN
ejpam-6573	165	26	,	,	PUNCT
ejpam-6573	165	27	σ2)β	σ2)β	NOUN
ejpam-6573	165	28	-	-	PUNCT
ejpam-6573	165	29	cl(f(u	cl(f(u	NOUN
ejpam-6573	165	30	)	)	PUNCT
ejpam-6573	165	31	)	)	PUNCT
ejpam-6573	166	1	=	=	SYM
ejpam-6573	166	2	(	(	PUNCT
ejpam-6573	166	3	σ1	σ1	PROPN
ejpam-6573	166	4	,	,	PUNCT
ejpam-6573	166	5	σ2)β	σ2)β	ADJ
ejpam-6573	166	6	-	-	PUNCT
ejpam-6573	166	7	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	166	8	-	-	PUNCT
ejpam-6573	166	9	int(u	int(u	NUM
ejpam-6573	166	10	)	)	PUNCT
ejpam-6573	166	11	)	)	PUNCT
ejpam-6573	166	12	)	)	PUNCT
ejpam-6573	167	1	⊆	⊆	X
ejpam-6573	167	2	(	(	PUNCT
ejpam-6573	167	3	σ1	σ1	PROPN
ejpam-6573	167	4	,	,	PUNCT
ejpam-6573	167	5	σ2)β	σ2)β	ADJ
ejpam-6573	167	6	-	-	PUNCT
ejpam-6573	167	7	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	167	8	-	-	PUNCT
ejpam-6573	167	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6573	167	10	-	-	PUNCT
ejpam-6573	167	11	cl(u	cl(u	NOUN
ejpam-6573	167	12	)	)	PUNCT
ejpam-6573	167	13	)	)	PUNCT
ejpam-6573	167	14	)	)	PUNCT
ejpam-6573	167	15	)	)	PUNCT
ejpam-6573	167	16	⊆	⊆	NUM
ejpam-6573	167	17	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	167	18	-	-	PUNCT
ejpam-6573	167	19	cl(u	cl(u	NOUN
ejpam-6573	167	20	)	)	PUNCT
ejpam-6573	167	21	)	)	PUNCT
ejpam-6573	167	22	.	.	PUNCT
ejpam-6573	168	1	(	(	PUNCT
ejpam-6573	168	2	2	2	X
ejpam-6573	168	3	)	)	PUNCT
ejpam-6573	168	4	⇒	⇒	NOUN
ejpam-6573	168	5	(	(	PUNCT
ejpam-6573	168	6	1	1	NUM
ejpam-6573	168	7	):	):	PUNCT
ejpam-6573	168	8	let	let	VERB
ejpam-6573	168	9	k	k	PRON
ejpam-6573	168	10	be	be	AUX
ejpam-6573	168	11	any	any	DET
ejpam-6573	168	12	τ1τ2	τ1τ2	ADJ
ejpam-6573	168	13	-	-	ADJ
ejpam-6573	168	14	closed	closed	ADJ
ejpam-6573	168	15	set	set	NOUN
ejpam-6573	168	16	of	of	ADP
ejpam-6573	168	17	x.	x.	NOUN
ejpam-6573	168	18	using	use	VERB
ejpam-6573	168	19	(	(	PUNCT
ejpam-6573	168	20	2	2	NUM
ejpam-6573	168	21	)	)	PUNCT
ejpam-6573	168	22	,	,	PUNCT
ejpam-6573	168	23	we	we	PRON
ejpam-6573	168	24	have	have	VERB
ejpam-6573	168	25	(	(	PUNCT
ejpam-6573	168	26	σ1	σ1	PROPN
ejpam-6573	168	27	,	,	PUNCT
ejpam-6573	168	28	σ2)β	σ2)β	ADJ
ejpam-6573	168	29	-	-	PUNCT
ejpam-6573	168	30	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	168	31	-	-	PUNCT
ejpam-6573	168	32	int(k	int(k	NUM
ejpam-6573	168	33	)	)	PUNCT
ejpam-6573	168	34	)	)	PUNCT
ejpam-6573	168	35	)	)	PUNCT
ejpam-6573	169	1	⊆	⊆	NUM
ejpam-6573	169	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	169	3	-	-	PUNCT
ejpam-6573	169	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6573	169	5	-	-	PUNCT
ejpam-6573	169	6	int(k	int(k	NOUN
ejpam-6573	169	7	)	)	PUNCT
ejpam-6573	169	8	)	)	PUNCT
ejpam-6573	169	9	)	)	PUNCT
ejpam-6573	170	1	⊆	⊆	NUM
ejpam-6573	170	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	170	3	-	-	PUNCT
ejpam-6573	170	4	cl(k	cl(k	NOUN
ejpam-6573	170	5	)	)	PUNCT
ejpam-6573	170	6	)	)	PUNCT
ejpam-6573	170	7	=	=	PUNCT
ejpam-6573	170	8	f(k	f(k	VERB
ejpam-6573	170	9	)	)	PUNCT
ejpam-6573	170	10	.	.	PUNCT
ejpam-6573	171	1	this	this	PRON
ejpam-6573	171	2	shows	show	VERB
ejpam-6573	171	3	that	that	SCONJ
ejpam-6573	171	4	f	f	PROPN
ejpam-6573	171	5	is	be	AUX
ejpam-6573	171	6	weakly	weakly	ADJ
ejpam-6573	171	7	(	(	PUNCT
ejpam-6573	171	8	τ1	τ1	NOUN
ejpam-6573	171	9	,	,	PUNCT
ejpam-6573	171	10	τ2)β	τ2)β	NOUN
ejpam-6573	171	11	-	-	PUNCT
ejpam-6573	171	12	closed	closed	ADJ
ejpam-6573	171	13	.	.	PUNCT
ejpam-6573	172	1	it	it	PRON
ejpam-6573	172	2	is	be	AUX
ejpam-6573	172	3	clear	clear	ADJ
ejpam-6573	172	4	that	that	SCONJ
ejpam-6573	172	5	(	(	PUNCT
ejpam-6573	172	6	1	1	X
ejpam-6573	172	7	)	)	PUNCT
ejpam-6573	172	8	⇒	⇒	NOUN
ejpam-6573	172	9	(	(	PUNCT
ejpam-6573	172	10	7	7	NUM
ejpam-6573	172	11	)	)	PUNCT
ejpam-6573	172	12	,	,	PUNCT
ejpam-6573	172	13	(	(	PUNCT
ejpam-6573	172	14	4	4	X
ejpam-6573	172	15	)	)	PUNCT
ejpam-6573	172	16	⇒	⇒	NOUN
ejpam-6573	172	17	(	(	PUNCT
ejpam-6573	172	18	5	5	NUM
ejpam-6573	172	19	)	)	PUNCT
ejpam-6573	172	20	and	and	CCONJ
ejpam-6573	172	21	(	(	PUNCT
ejpam-6573	172	22	1	1	X
ejpam-6573	172	23	)	)	PUNCT
ejpam-6573	172	24	⇒	⇒	NOUN
ejpam-6573	172	25	(	(	PUNCT
ejpam-6573	172	26	6	6	NUM
ejpam-6573	172	27	)	)	PUNCT
ejpam-6573	172	28	⇒	⇒	NOUN
ejpam-6573	172	29	(	(	PUNCT
ejpam-6573	172	30	8)	8)	NUM
ejpam-6573	172	31	⇒	⇒	NOUN
ejpam-6573	172	32	(	(	PUNCT
ejpam-6573	172	33	3	3	NUM
ejpam-6573	172	34	)	)	PUNCT
ejpam-6573	172	35	⇒	⇒	NOUN
ejpam-6573	172	36	(	(	PUNCT
ejpam-6573	172	37	1	1	NUM
ejpam-6573	172	38	)	)	PUNCT
ejpam-6573	172	39	.	.	PUNCT
ejpam-6573	173	1	to	to	PART
ejpam-6573	173	2	show	show	VERB
ejpam-6573	173	3	that	that	SCONJ
ejpam-6573	173	4	(	(	PUNCT
ejpam-6573	173	5	3	3	X
ejpam-6573	173	6	)	)	PUNCT
ejpam-6573	173	7	⇒	⇒	NOUN
ejpam-6573	173	8	(	(	PUNCT
ejpam-6573	173	9	4	4	NUM
ejpam-6573	173	10	):	):	PUNCT
ejpam-6573	173	11	let	let	VERB
ejpam-6573	173	12	b	b	X
ejpam-6573	173	13	be	be	AUX
ejpam-6573	173	14	any	any	DET
ejpam-6573	173	15	subset	subset	NOUN
ejpam-6573	173	16	of	of	ADP
ejpam-6573	173	17	y	y	PROPN
ejpam-6573	173	18	and	and	CCONJ
ejpam-6573	173	19	u	u	NOUN
ejpam-6573	173	20	be	be	VERB
ejpam-6573	173	21	any	any	DET
ejpam-6573	173	22	τ1τ2	τ1τ2	ADJ
ejpam-6573	173	23	-	-	ADJ
ejpam-6573	173	24	open	open	ADJ
ejpam-6573	173	25	set	set	NOUN
ejpam-6573	173	26	of	of	ADP
ejpam-6573	173	27	x	x	PUNCT
ejpam-6573	173	28	with	with	ADP
ejpam-6573	173	29	f−1(b	f−1(b	PROPN
ejpam-6573	173	30	)	)	PUNCT
ejpam-6573	173	31	⊆	⊆	NUM
ejpam-6573	173	32	u	u	NOUN
ejpam-6573	173	33	.	.	PUNCT
ejpam-6573	174	1	then	then	ADV
ejpam-6573	174	2	,	,	PUNCT
ejpam-6573	174	3	f−1(b	f−1(b	PROPN
ejpam-6573	174	4	)	)	PUNCT
ejpam-6573	174	5	∩	∩	NOUN
ejpam-6573	174	6	τ1τ2	τ1τ2	NOUN
ejpam-6573	174	7	-	-	NOUN
ejpam-6573	174	8	cl(x	cl(x	SYM
ejpam-6573	174	9	−	−	PRON
ejpam-6573	174	10	τ1τ2	τ1τ2	NOUN
ejpam-6573	174	11	-	-	NOUN
ejpam-6573	174	12	cl(u	cl(u	NOUN
ejpam-6573	174	13	)	)	PUNCT
ejpam-6573	174	14	)	)	PUNCT
ejpam-6573	175	1	=	=	PUNCT
ejpam-6573	175	2	∅	∅	NOUN
ejpam-6573	175	3	and	and	CCONJ
ejpam-6573	175	4	b	b	NOUN
ejpam-6573	175	5	∩	∩	ADJ
ejpam-6573	175	6	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	175	7	-	-	NOUN
ejpam-6573	175	8	cl(x	cl(x	NUM
ejpam-6573	175	9	−	−	PRON
ejpam-6573	175	10	τ1τ2	τ1τ2	NOUN
ejpam-6573	175	11	-	-	NOUN
ejpam-6573	175	12	cl(u	cl(u	NOUN
ejpam-6573	175	13	)	)	PUNCT
ejpam-6573	175	14	)	)	PUNCT
ejpam-6573	175	15	)	)	PUNCT
ejpam-6573	176	1	=	=	PUNCT
ejpam-6573	176	2	∅.	∅.	VERB
ejpam-6573	176	3	since	since	SCONJ
ejpam-6573	176	4	x	x	INTJ
ejpam-6573	176	5	−	−	NUM
ejpam-6573	176	6	τ1τ2	τ1τ2	NOUN
ejpam-6573	176	7	-	-	NOUN
ejpam-6573	176	8	cl(u	cl(u	NOUN
ejpam-6573	176	9	)	)	PUNCT
ejpam-6573	176	10	is	be	AUX
ejpam-6573	176	11	(	(	PUNCT
ejpam-6573	176	12	τ1	τ1	NOUN
ejpam-6573	176	13	,	,	PUNCT
ejpam-6573	176	14	τ2)r	τ2)r	NOUN
ejpam-6573	176	15	-	-	PUNCT
ejpam-6573	176	16	open	open	ADJ
ejpam-6573	176	17	,	,	PUNCT
ejpam-6573	176	18	b	b	NOUN
ejpam-6573	176	19	∩	∩	NOUN
ejpam-6573	176	20	(	(	PUNCT
ejpam-6573	176	21	σ1	σ1	PROPN
ejpam-6573	176	22	,	,	PUNCT
ejpam-6573	176	23	σ2)β	σ2)β	NOUN
ejpam-6573	176	24	-	-	PUNCT
ejpam-6573	176	25	cl(f(x	cl(f(x	NOUN
ejpam-6573	176	26	−	−	NOUN
ejpam-6573	176	27	τ1τ2	τ1τ2	NOUN
ejpam-6573	176	28	-	-	NOUN
ejpam-6573	176	29	cl(u	cl(u	NOUN
ejpam-6573	176	30	)	)	PUNCT
ejpam-6573	176	31	)	)	PUNCT
ejpam-6573	176	32	)	)	PUNCT
ejpam-6573	177	1	=	=	PUNCT
ejpam-6573	177	2	∅	∅	NOUN
ejpam-6573	177	3	by	by	ADP
ejpam-6573	177	4	(	(	PUNCT
ejpam-6573	177	5	3	3	NUM
ejpam-6573	177	6	)	)	PUNCT
ejpam-6573	177	7	.	.	PUNCT
ejpam-6573	178	1	m.	m.	NOUN
ejpam-6573	178	2	chiangpradit	chiangpradit	PROPN
ejpam-6573	178	3	,	,	PUNCT
ejpam-6573	178	4	a.	a.	PROPN
ejpam-6573	178	5	sama	sama	PROPN
ejpam-6573	178	6	-	-	PUNCT
ejpam-6573	178	7	ae	ae	PROPN
ejpam-6573	178	8	,	,	PUNCT
ejpam-6573	178	9	c.	c.	PROPN
ejpam-6573	178	10	boonpok	boonpok	PROPN
ejpam-6573	178	11	/	/	SYM
ejpam-6573	178	12	eur	eur	PROPN
ejpam-6573	178	13	.	.	PUNCT
ejpam-6573	179	1	j.	j.	PROPN
ejpam-6573	179	2	pure	pure	PROPN
ejpam-6573	179	3	appl	appl	PROPN
ejpam-6573	179	4	.	.	PROPN
ejpam-6573	179	5	math	math	PROPN
ejpam-6573	179	6	,	,	PUNCT
ejpam-6573	179	7	18	18	NUM
ejpam-6573	179	8	(	(	PUNCT
ejpam-6573	179	9	3	3	NUM
ejpam-6573	179	10	)	)	PUNCT
ejpam-6573	179	11	(	(	PUNCT
ejpam-6573	179	12	2025	2025	NUM
ejpam-6573	179	13	)	)	PUNCT
ejpam-6573	179	14	,	,	PUNCT
ejpam-6573	179	15	6573	6573	NUM
ejpam-6573	179	16	7	7	NUM
ejpam-6573	179	17	of	of	ADP
ejpam-6573	179	18	8	8	NUM
ejpam-6573	179	19	put	put	VERB
ejpam-6573	179	20	v	v	NOUN
ejpam-6573	179	21	=	=	SYM
ejpam-6573	179	22	y	y	PROPN
ejpam-6573	179	23	−	−	PROPN
ejpam-6573	179	24	(	(	PUNCT
ejpam-6573	179	25	σ1	σ1	PROPN
ejpam-6573	179	26	,	,	PUNCT
ejpam-6573	179	27	σ2)β	σ2)β	NOUN
ejpam-6573	179	28	-	-	PUNCT
ejpam-6573	179	29	cl(f(x	cl(f(x	NOUN
ejpam-6573	179	30	−	−	NOUN
ejpam-6573	179	31	τ1τ2	τ1τ2	NOUN
ejpam-6573	179	32	-	-	NOUN
ejpam-6573	179	33	cl(u	cl(u	NOUN
ejpam-6573	179	34	)	)	PUNCT
ejpam-6573	179	35	)	)	PUNCT
ejpam-6573	179	36	)	)	PUNCT
ejpam-6573	179	37	.	.	PUNCT
ejpam-6573	180	1	then	then	ADV
ejpam-6573	180	2	,	,	PUNCT
ejpam-6573	180	3	v	v	NOUN
ejpam-6573	180	4	is	be	AUX
ejpam-6573	180	5	a	a	DET
ejpam-6573	180	6	(	(	PUNCT
ejpam-6573	180	7	σ1	σ1	PROPN
ejpam-6573	180	8	,	,	PUNCT
ejpam-6573	180	9	σ2)β	σ2)β	NOUN
ejpam-6573	180	10	-	-	PUNCT
ejpam-6573	180	11	open	open	ADJ
ejpam-6573	180	12	set	set	NOUN
ejpam-6573	180	13	of	of	ADP
ejpam-6573	180	14	y	y	PRON
ejpam-6573	180	15	such	such	ADJ
ejpam-6573	180	16	that	that	PRON
ejpam-6573	180	17	b	b	PROPN
ejpam-6573	180	18	⊆	⊆	NUM
ejpam-6573	180	19	v	v	NOUN
ejpam-6573	180	20	and	and	CCONJ
ejpam-6573	180	21	f−1(v	f−1(v	NOUN
ejpam-6573	180	22	)	)	PUNCT
ejpam-6573	181	1	⊆	⊆	NUM
ejpam-6573	181	2	x	x	SYM
ejpam-6573	181	3	−	−	NOUN
ejpam-6573	181	4	f−1((σ1	f−1((σ1	NOUN
ejpam-6573	181	5	,	,	PUNCT
ejpam-6573	181	6	σ2)β	σ2)β	NOUN
ejpam-6573	181	7	-	-	PUNCT
ejpam-6573	181	8	cl(f(x	cl(f(x	NOUN
ejpam-6573	181	9	−	−	NOUN
ejpam-6573	181	10	τ1τ2	τ1τ2	NOUN
ejpam-6573	181	11	-	-	NOUN
ejpam-6573	181	12	cl(u	cl(u	NOUN
ejpam-6573	181	13	)	)	PUNCT
ejpam-6573	181	14	)	)	PUNCT
ejpam-6573	181	15	)	)	PUNCT
ejpam-6573	181	16	)	)	PUNCT
ejpam-6573	182	1	⊆	⊆	NUM
ejpam-6573	182	2	x	x	SYM
ejpam-6573	182	3	−	−	NOUN
ejpam-6573	182	4	f−1(f(x	f−1(f(x	NOUN
ejpam-6573	182	5	−	−	ADP
ejpam-6573	182	6	τ1τ2	τ1τ2	NOUN
ejpam-6573	182	7	-	-	NOUN
ejpam-6573	182	8	cl(u	cl(u	NOUN
ejpam-6573	182	9	)	)	PUNCT
ejpam-6573	182	10	)	)	PUNCT
ejpam-6573	182	11	)	)	PUNCT
ejpam-6573	183	1	⊆	⊆	X
ejpam-6573	183	2	τ1τ2	τ1τ2	NOUN
ejpam-6573	183	3	-	-	NOUN
ejpam-6573	183	4	cl(u	cl(u	NUM
ejpam-6573	183	5	)	)	PUNCT
ejpam-6573	183	6	.	.	PUNCT
ejpam-6573	184	1	(	(	PUNCT
ejpam-6573	184	2	7	7	X
ejpam-6573	184	3	)	)	PUNCT
ejpam-6573	184	4	⇒	⇒	NOUN
ejpam-6573	184	5	(	(	PUNCT
ejpam-6573	184	6	1	1	NUM
ejpam-6573	184	7	):	):	PUNCT
ejpam-6573	184	8	it	it	PRON
ejpam-6573	184	9	is	be	AUX
ejpam-6573	184	10	suffices	suffice	NOUN
ejpam-6573	184	11	see	see	VERB
ejpam-6573	184	12	that	that	DET
ejpam-6573	184	13	(	(	PUNCT
ejpam-6573	184	14	τ1	τ1	NOUN
ejpam-6573	184	15	,	,	PUNCT
ejpam-6573	184	16	τ2)θ	τ2)θ	ADJ
ejpam-6573	184	17	-	-	PUNCT
ejpam-6573	184	18	cl(u	cl(u	NOUN
ejpam-6573	184	19	)	)	PUNCT
ejpam-6573	184	20	=	=	PUNCT
ejpam-6573	184	21	τ1τ2	τ1τ2	NOUN
ejpam-6573	184	22	-	-	NOUN
ejpam-6573	184	23	cl(u	cl(u	NOUN
ejpam-6573	184	24	)	)	PUNCT
ejpam-6573	184	25	for	for	ADP
ejpam-6573	184	26	every	every	DET
ejpam-6573	184	27	τ1τ2	τ1τ2	ADJ
ejpam-6573	184	28	-	-	ADJ
ejpam-6573	184	29	open	open	ADJ
ejpam-6573	184	30	set	set	ADJ
ejpam-6573	184	31	u	u	NOUN
ejpam-6573	184	32	of	of	ADP
ejpam-6573	184	33	x.	x.	PROPN
ejpam-6573	184	34	(	(	PUNCT
ejpam-6573	184	35	5	5	NUM
ejpam-6573	184	36	)	)	PUNCT
ejpam-6573	184	37	⇒	⇒	NOUN
ejpam-6573	184	38	(	(	PUNCT
ejpam-6573	184	39	1	1	NUM
ejpam-6573	184	40	):	):	PUNCT
ejpam-6573	184	41	let	let	VERB
ejpam-6573	184	42	k	k	PRON
ejpam-6573	184	43	be	be	AUX
ejpam-6573	184	44	any	any	DET
ejpam-6573	184	45	τ1τ2	τ1τ2	ADJ
ejpam-6573	184	46	-	-	ADJ
ejpam-6573	184	47	closed	closed	ADJ
ejpam-6573	184	48	set	set	ADJ
ejpam-6573	184	49	u	u	NOUN
ejpam-6573	184	50	of	of	ADP
ejpam-6573	184	51	x	x	PUNCT
ejpam-6573	184	52	and	and	CCONJ
ejpam-6573	184	53	y	y	PROPN
ejpam-6573	184	54	∈	∈	PROPN
ejpam-6573	184	55	y	y	PROPN
ejpam-6573	184	56	−	−	PROPN
ejpam-6573	184	57	f(k	f(k	PROPN
ejpam-6573	184	58	)	)	PUNCT
ejpam-6573	184	59	.	.	PUNCT
ejpam-6573	185	1	since	since	SCONJ
ejpam-6573	185	2	f−1(y	f−1(y	PROPN
ejpam-6573	185	3	)	)	PUNCT
ejpam-6573	185	4	⊆	⊆	NUM
ejpam-6573	185	5	x	x	PUNCT
ejpam-6573	185	6	−k	−k	PROPN
ejpam-6573	185	7	,	,	PUNCT
ejpam-6573	185	8	there	there	PRON
ejpam-6573	185	9	exists	exist	VERB
ejpam-6573	185	10	a	a	DET
ejpam-6573	185	11	(	(	PUNCT
ejpam-6573	185	12	σ1	σ1	PROPN
ejpam-6573	185	13	,	,	PUNCT
ejpam-6573	185	14	σ2)β	σ2)β	NOUN
ejpam-6573	185	15	-	-	PUNCT
ejpam-6573	185	16	open	open	NOUN
ejpam-6573	185	17	set	set	NOUN
ejpam-6573	185	18	v	v	NOUN
ejpam-6573	185	19	of	of	ADP
ejpam-6573	185	20	y	y	PRON
ejpam-6573	185	21	such	such	ADJ
ejpam-6573	185	22	that	that	SCONJ
ejpam-6573	185	23	y	y	PROPN
ejpam-6573	185	24	∈	∈	PROPN
ejpam-6573	185	25	v	v	NOUN
ejpam-6573	185	26	and	and	CCONJ
ejpam-6573	185	27	f−1(v	f−1(v	NOUN
ejpam-6573	185	28	)	)	PUNCT
ejpam-6573	186	1	⊆	⊆	NUM
ejpam-6573	186	2	τ1τ2	τ1τ2	NOUN
ejpam-6573	186	3	-	-	ADJ
ejpam-6573	186	4	cl(x	cl(x	SYM
ejpam-6573	186	5	−k	−k	NOUN
ejpam-6573	186	6	)	)	PUNCT
ejpam-6573	186	7	=	=	PUNCT
ejpam-6573	187	1	x	x	X
ejpam-6573	187	2	−	−	ADP
ejpam-6573	187	3	τ1τ2	τ1τ2	NOUN
ejpam-6573	187	4	-	-	PUNCT
ejpam-6573	187	5	int(k	int(k	NOUN
ejpam-6573	187	6	)	)	PUNCT
ejpam-6573	187	7	by	by	ADP
ejpam-6573	187	8	(	(	PUNCT
ejpam-6573	187	9	5	5	NUM
ejpam-6573	187	10	)	)	PUNCT
ejpam-6573	187	11	.	.	PUNCT
ejpam-6573	188	1	thus	thus	ADV
ejpam-6573	188	2	,	,	PUNCT
ejpam-6573	188	3	v	v	ADP
ejpam-6573	188	4	∩	∩	ADJ
ejpam-6573	188	5	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6573	188	6	-	-	PUNCT
ejpam-6573	188	7	int(k	int(k	NOUN
ejpam-6573	188	8	)	)	PUNCT
ejpam-6573	188	9	)	)	PUNCT
ejpam-6573	189	1	=	=	NOUN
ejpam-6573	189	2	∅	∅	NOUN
ejpam-6573	190	1	and	and	CCONJ
ejpam-6573	190	2	so	so	ADV
ejpam-6573	190	3	y	y	PROPN
ejpam-6573	190	4	∈	∈	PROPN
ejpam-6573	190	5	y	y	PROPN
ejpam-6573	191	1	−	−	PROPN
ejpam-6573	191	2	(	(	PUNCT
ejpam-6573	191	3	σ1	σ1	PROPN
ejpam-6573	191	4	,	,	PUNCT
ejpam-6573	191	5	σ2)β	σ2)β	ADJ
ejpam-6573	191	6	-	-	PUNCT
ejpam-6573	191	7	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	191	8	-	-	PUNCT
ejpam-6573	191	9	int(k	int(k	NOUN
ejpam-6573	191	10	)	)	PUNCT
ejpam-6573	191	11	)	)	PUNCT
ejpam-6573	191	12	)	)	PUNCT
ejpam-6573	191	13	.	.	PUNCT
ejpam-6573	192	1	therefore	therefore	ADV
ejpam-6573	192	2	,	,	PUNCT
ejpam-6573	192	3	(	(	PUNCT
ejpam-6573	192	4	σ1	σ1	PROPN
ejpam-6573	192	5	,	,	PUNCT
ejpam-6573	192	6	σ2)β	σ2)β	ADJ
ejpam-6573	192	7	-	-	PUNCT
ejpam-6573	192	8	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	192	9	-	-	PUNCT
ejpam-6573	192	10	int(k	int(k	NUM
ejpam-6573	192	11	)	)	PUNCT
ejpam-6573	192	12	)	)	PUNCT
ejpam-6573	192	13	)	)	PUNCT
ejpam-6573	193	1	⊆	⊆	NUM
ejpam-6573	193	2	f(k	f(k	VERB
ejpam-6573	193	3	)	)	PUNCT
ejpam-6573	193	4	.	.	PUNCT
ejpam-6573	194	1	this	this	PRON
ejpam-6573	194	2	shows	show	VERB
ejpam-6573	194	3	that	that	SCONJ
ejpam-6573	194	4	f	f	PROPN
ejpam-6573	194	5	is	be	AUX
ejpam-6573	194	6	weakly	weakly	ADJ
ejpam-6573	194	7	(	(	PUNCT
ejpam-6573	194	8	τ1	τ1	NOUN
ejpam-6573	194	9	,	,	PUNCT
ejpam-6573	194	10	τ2)βclosed	τ2)βclose	VERB
ejpam-6573	194	11	.	.	PUNCT
ejpam-6573	195	1	(	(	PUNCT
ejpam-6573	195	2	7	7	X
ejpam-6573	195	3	)	)	PUNCT
ejpam-6573	195	4	⇒	⇒	NOUN
ejpam-6573	195	5	(	(	PUNCT
ejpam-6573	195	6	8)	8)	NUM
ejpam-6573	195	7	:	:	PUNCT
ejpam-6573	195	8	this	this	PRON
ejpam-6573	195	9	is	be	AUX
ejpam-6573	195	10	obvious	obvious	ADJ
ejpam-6573	195	11	since	since	SCONJ
ejpam-6573	195	12	(	(	PUNCT
ejpam-6573	195	13	τ1	τ1	NOUN
ejpam-6573	195	14	,	,	PUNCT
ejpam-6573	195	15	τ2)θ	τ2)θ	ADJ
ejpam-6573	195	16	-	-	PUNCT
ejpam-6573	195	17	cl(u	cl(u	NOUN
ejpam-6573	195	18	)	)	PUNCT
ejpam-6573	195	19	=	=	PUNCT
ejpam-6573	195	20	τ1τ2	τ1τ2	NOUN
ejpam-6573	195	21	-	-	NOUN
ejpam-6573	195	22	cl(u	cl(u	NOUN
ejpam-6573	195	23	)	)	PUNCT
ejpam-6573	195	24	for	for	SCONJ
ejpam-6573	195	25	every	every	DET
ejpam-6573	195	26	(	(	PUNCT
ejpam-6573	195	27	τ1	τ1	NOUN
ejpam-6573	195	28	,	,	PUNCT
ejpam-6573	195	29	τ2)β	τ2)β	ADJ
ejpam-6573	195	30	-	-	PUNCT
ejpam-6573	195	31	open	open	ADJ
ejpam-6573	195	32	set	set	NOUN
ejpam-6573	195	33	u	u	NOUN
ejpam-6573	195	34	of	of	ADP
ejpam-6573	195	35	x.	x.	NOUN
ejpam-6573	195	36	the	the	DET
ejpam-6573	195	37	following	follow	VERB
ejpam-6573	195	38	theorem	theorem	VERB
ejpam-6573	195	39	the	the	DET
ejpam-6573	195	40	proof	proof	NOUN
ejpam-6573	195	41	is	be	AUX
ejpam-6573	195	42	mostly	mostly	ADV
ejpam-6573	195	43	straightforward	straightforward	ADJ
ejpam-6573	195	44	and	and	CCONJ
ejpam-6573	195	45	is	be	AUX
ejpam-6573	195	46	omitted	omit	VERB
ejpam-6573	195	47	.	.	PUNCT
ejpam-6573	196	1	theorem	theorem	VERB
ejpam-6573	196	2	4	4	NUM
ejpam-6573	196	3	.	.	X
ejpam-6573	196	4	for	for	ADP
ejpam-6573	196	5	a	a	DET
ejpam-6573	196	6	function	function	NOUN
ejpam-6573	196	7	f	f	NOUN
ejpam-6573	196	8	:	:	PUNCT
ejpam-6573	196	9	(	(	PUNCT
ejpam-6573	196	10	x	x	NOUN
ejpam-6573	196	11	,	,	PUNCT
ejpam-6573	196	12	τ1	τ1	NOUN
ejpam-6573	196	13	,	,	PUNCT
ejpam-6573	196	14	τ2	τ2	NOUN
ejpam-6573	196	15	)	)	PUNCT
ejpam-6573	196	16	→	→	SYM
ejpam-6573	196	17	(	(	PUNCT
ejpam-6573	196	18	y	y	PROPN
ejpam-6573	196	19	,	,	PUNCT
ejpam-6573	196	20	σ1	σ1	PROPN
ejpam-6573	196	21	,	,	PUNCT
ejpam-6573	196	22	σ2	σ2	NOUN
ejpam-6573	196	23	)	)	PUNCT
ejpam-6573	196	24	,	,	PUNCT
ejpam-6573	196	25	the	the	DET
ejpam-6573	196	26	following	follow	VERB
ejpam-6573	196	27	properties	property	NOUN
ejpam-6573	196	28	are	be	AUX
ejpam-6573	196	29	equivalent	equivalent	ADJ
ejpam-6573	196	30	:	:	PUNCT
ejpam-6573	196	31	(	(	PUNCT
ejpam-6573	196	32	1	1	X
ejpam-6573	196	33	)	)	PUNCT
ejpam-6573	196	34	f	f	PROPN
ejpam-6573	196	35	is	be	AUX
ejpam-6573	196	36	weakly	weakly	ADJ
ejpam-6573	196	37	(	(	PUNCT
ejpam-6573	196	38	τ1	τ1	NOUN
ejpam-6573	196	39	,	,	PUNCT
ejpam-6573	196	40	τ2)β	τ2)β	NOUN
ejpam-6573	196	41	-	-	PUNCT
ejpam-6573	196	42	closed	closed	ADJ
ejpam-6573	196	43	;	;	PUNCT
ejpam-6573	196	44	(	(	PUNCT
ejpam-6573	196	45	2	2	X
ejpam-6573	196	46	)	)	PUNCT
ejpam-6573	196	47	(	(	PUNCT
ejpam-6573	196	48	σ1	σ1	PROPN
ejpam-6573	196	49	,	,	PUNCT
ejpam-6573	196	50	σ2)β	σ2)β	ADJ
ejpam-6573	196	51	-	-	PUNCT
ejpam-6573	196	52	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	196	53	-	-	PUNCT
ejpam-6573	196	54	int(k	int(k	NUM
ejpam-6573	196	55	)	)	PUNCT
ejpam-6573	196	56	)	)	PUNCT
ejpam-6573	196	57	)	)	PUNCT
ejpam-6573	197	1	⊆	⊆	NUM
ejpam-6573	197	2	f(k	f(k	VERB
ejpam-6573	197	3	)	)	PUNCT
ejpam-6573	197	4	for	for	ADP
ejpam-6573	197	5	every	every	DET
ejpam-6573	197	6	(	(	PUNCT
ejpam-6573	197	7	τ1	τ1	NOUN
ejpam-6573	197	8	,	,	PUNCT
ejpam-6573	197	9	τ2)β	τ2)β	ADJ
ejpam-6573	197	10	-	-	PUNCT
ejpam-6573	197	11	closed	close	VERB
ejpam-6573	197	12	set	set	NOUN
ejpam-6573	197	13	k	k	PROPN
ejpam-6573	197	14	of	of	ADP
ejpam-6573	197	15	x	x	PROPN
ejpam-6573	197	16	;	;	PUNCT
ejpam-6573	197	17	(	(	PUNCT
ejpam-6573	197	18	3	3	X
ejpam-6573	197	19	)	)	PUNCT
ejpam-6573	197	20	(	(	PUNCT
ejpam-6573	197	21	σ1	σ1	PROPN
ejpam-6573	197	22	,	,	PUNCT
ejpam-6573	197	23	σ2)β	σ2)β	ADJ
ejpam-6573	197	24	-	-	PUNCT
ejpam-6573	197	25	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6573	197	26	-	-	PUNCT
ejpam-6573	197	27	int(k	int(k	NUM
ejpam-6573	197	28	)	)	PUNCT
ejpam-6573	197	29	)	)	PUNCT
ejpam-6573	197	30	)	)	PUNCT
ejpam-6573	197	31	⊆	⊆	NUM
ejpam-6573	197	32	f(k	f(k	VERB
ejpam-6573	197	33	)	)	PUNCT
ejpam-6573	197	34	for	for	ADP
ejpam-6573	197	35	every	every	DET
ejpam-6573	197	36	α(τ1	α(τ1	NOUN
ejpam-6573	197	37	,	,	PUNCT
ejpam-6573	197	38	τ2)-closed	τ2)-close	VERB
ejpam-6573	197	39	set	set	NOUN
ejpam-6573	197	40	k	k	PROPN
ejpam-6573	197	41	of	of	ADP
ejpam-6573	197	42	x.	x.	PROPN
ejpam-6573	197	43	acknowledgements	acknowledgement	NOUN
ejpam-6573	197	44	this	this	DET
ejpam-6573	197	45	research	research	NOUN
ejpam-6573	197	46	project	project	NOUN
ejpam-6573	197	47	was	be	AUX
ejpam-6573	197	48	financially	financially	ADV
ejpam-6573	197	49	supported	support	VERB
ejpam-6573	197	50	by	by	ADP
ejpam-6573	197	51	mahasarakham	mahasarakham	PROPN
ejpam-6573	197	52	university	university	PROPN
ejpam-6573	197	53	.	.	PUNCT
ejpam-6573	197	54	references	reference	NOUN
ejpam-6573	197	55	[	[	X
ejpam-6573	197	56	1	1	X
ejpam-6573	197	57	]	]	PUNCT
ejpam-6573	197	58	d.	d.	PROPN
ejpam-6573	197	59	a.	a.	PROPN
ejpam-6573	197	60	rose	rise	VERB
ejpam-6573	197	61	.	.	PUNCT
ejpam-6573	198	1	on	on	ADP
ejpam-6573	198	2	weak	weak	ADJ
ejpam-6573	198	3	openness	openness	NOUN
ejpam-6573	198	4	and	and	CCONJ
ejpam-6573	198	5	almost	almost	ADV
ejpam-6573	198	6	openness	openness	NOUN
ejpam-6573	198	7	.	.	PUNCT
ejpam-6573	199	1	international	international	ADJ
ejpam-6573	199	2	journal	journal	NOUN
ejpam-6573	199	3	of	of	ADP
ejpam-6573	199	4	mathematics	mathematics	PROPN
ejpam-6573	199	5	and	and	CCONJ
ejpam-6573	199	6	mathematical	mathematical	ADJ
ejpam-6573	199	7	sciences	science	NOUN
ejpam-6573	199	8	,	,	PUNCT
ejpam-6573	199	9	7:35–40	7:35–40	NUM
ejpam-6573	199	10	,	,	PUNCT
ejpam-6573	199	11	1984	1984	NUM
ejpam-6573	199	12	.	.	PUNCT
ejpam-6573	200	1	[	[	X
ejpam-6573	200	2	2	2	X
ejpam-6573	200	3	]	]	PUNCT
ejpam-6573	200	4	d.	d.	PROPN
ejpam-6573	200	5	a.	a.	PROPN
ejpam-6573	200	6	rose	rise	VERB
ejpam-6573	200	7	and	and	CCONJ
ejpam-6573	200	8	d.	d.	PROPN
ejpam-6573	200	9	s.	s.	PROPN
ejpam-6573	201	1	janković.	janković.	PROPN
ejpam-6573	201	2	weakly	weakly	ADJ
ejpam-6573	201	3	closed	close	VERB
ejpam-6573	201	4	functions	function	NOUN
ejpam-6573	201	5	and	and	CCONJ
ejpam-6573	201	6	hausdorff	hausdorff	NOUN
ejpam-6573	201	7	spaces	space	NOUN
ejpam-6573	201	8	.	.	PUNCT
ejpam-6573	202	1	mathematische	mathematische	PROPN
ejpam-6573	202	2	nachrichten	nachrichten	PROPN
ejpam-6573	202	3	,	,	PUNCT
ejpam-6573	202	4	130:105–110	130:105–110	NUM
ejpam-6573	202	5	,	,	PUNCT
ejpam-6573	202	6	1987	1987	NUM
ejpam-6573	202	7	.	.	PUNCT
ejpam-6573	203	1	[	[	X
ejpam-6573	203	2	3	3	NUM
ejpam-6573	203	3	]	]	PUNCT
ejpam-6573	203	4	m.	m.	NOUN
ejpam-6573	203	5	caldas	caldas	PROPN
ejpam-6573	203	6	and	and	CCONJ
ejpam-6573	203	7	g.	g.	PROPN
ejpam-6573	203	8	navalagi	navalagi	PROPN
ejpam-6573	203	9	.	.	PUNCT
ejpam-6573	204	1	on	on	ADP
ejpam-6573	204	2	weak	weak	ADJ
ejpam-6573	204	3	forms	form	NOUN
ejpam-6573	204	4	of	of	ADP
ejpam-6573	204	5	preopen	preopen	ADJ
ejpam-6573	204	6	and	and	CCONJ
ejpam-6573	204	7	preclosed	preclose	VERB
ejpam-6573	204	8	functions	function	NOUN
ejpam-6573	204	9	.	.	PUNCT
ejpam-6573	205	1	archivum	archivum	PROPN
ejpam-6573	205	2	mathematicum	mathematicum	PROPN
ejpam-6573	205	3	,	,	PUNCT
ejpam-6573	205	4	40:119–128	40:119–128	PROPN
ejpam-6573	205	5	,	,	PUNCT
ejpam-6573	205	6	2004	2004	NUM
ejpam-6573	205	7	.	.	PUNCT
ejpam-6573	206	1	[	[	X
ejpam-6573	206	2	4	4	NUM
ejpam-6573	206	3	]	]	PUNCT
ejpam-6573	206	4	m.	m.	NOUN
ejpam-6573	206	5	caldas	caldas	PROPN
ejpam-6573	206	6	and	and	CCONJ
ejpam-6573	206	7	g.	g.	PROPN
ejpam-6573	206	8	navalagi	navalagi	PROPN
ejpam-6573	206	9	.	.	PUNCT
ejpam-6573	207	1	on	on	ADP
ejpam-6573	207	2	weak	weak	ADJ
ejpam-6573	207	3	forms	form	NOUN
ejpam-6573	207	4	of	of	ADP
ejpam-6573	207	5	semi	semi	ADJ
ejpam-6573	207	6	-	-	ADJ
ejpam-6573	207	7	open	open	ADJ
ejpam-6573	207	8	and	and	CCONJ
ejpam-6573	207	9	semi	semi	ADJ
ejpam-6573	207	10	-	-	ADJ
ejpam-6573	207	11	closed	closed	ADJ
ejpam-6573	207	12	functions	function	NOUN
ejpam-6573	207	13	.	.	PUNCT
ejpam-6573	208	1	missouri	missouri	PROPN
ejpam-6573	208	2	journal	journal	PROPN
ejpam-6573	208	3	of	of	ADP
ejpam-6573	208	4	mathematical	mathematical	ADJ
ejpam-6573	208	5	sciences	science	NOUN
ejpam-6573	208	6	,	,	PUNCT
ejpam-6573	208	7	18(3):165–178	18(3):165–178	PROPN
ejpam-6573	208	8	,	,	PUNCT
ejpam-6573	208	9	2006	2006	NUM
ejpam-6573	208	10	.	.	PUNCT
ejpam-6573	209	1	m.	m.	NOUN
ejpam-6573	209	2	chiangpradit	chiangpradit	PROPN
ejpam-6573	209	3	,	,	PUNCT
ejpam-6573	209	4	a.	a.	PROPN
ejpam-6573	209	5	sama	sama	PROPN
ejpam-6573	209	6	-	-	PUNCT
ejpam-6573	209	7	ae	ae	PROPN
ejpam-6573	209	8	,	,	PUNCT
ejpam-6573	209	9	c.	c.	PROPN
ejpam-6573	209	10	boonpok	boonpok	PROPN
ejpam-6573	209	11	/	/	SYM
ejpam-6573	209	12	eur	eur	PROPN
ejpam-6573	209	13	.	.	PUNCT
ejpam-6573	210	1	j.	j.	PROPN
ejpam-6573	210	2	pure	pure	PROPN
ejpam-6573	210	3	appl	appl	PROPN
ejpam-6573	210	4	.	.	PROPN
ejpam-6573	210	5	math	math	PROPN
ejpam-6573	210	6	,	,	PUNCT
ejpam-6573	210	7	18	18	NUM
ejpam-6573	210	8	(	(	PUNCT
ejpam-6573	210	9	3	3	NUM
ejpam-6573	210	10	)	)	PUNCT
ejpam-6573	210	11	(	(	PUNCT
ejpam-6573	210	12	2025	2025	NUM
ejpam-6573	210	13	)	)	PUNCT
ejpam-6573	210	14	,	,	PUNCT
ejpam-6573	210	15	6573	6573	NUM
ejpam-6573	210	16	8	8	NUM
ejpam-6573	210	17	of	of	ADP
ejpam-6573	210	18	8	8	NUM
ejpam-6573	211	1	[	[	SYM
ejpam-6573	211	2	5	5	NUM
ejpam-6573	211	3	]	]	PUNCT
ejpam-6573	211	4	t.	t.	PROPN
ejpam-6573	211	5	noiri	noiri	PROPN
ejpam-6573	211	6	.	.	PUNCT
ejpam-6573	212	1	weak	weak	ADJ
ejpam-6573	212	2	forms	form	NOUN
ejpam-6573	212	3	of	of	ADP
ejpam-6573	212	4	open	open	ADJ
ejpam-6573	212	5	and	and	CCONJ
ejpam-6573	212	6	closed	closed	ADJ
ejpam-6573	212	7	functions	function	NOUN
ejpam-6573	212	8	via	via	ADP
ejpam-6573	212	9	b	b	X
ejpam-6573	212	10	-	-	PUNCT
ejpam-6573	212	11	θ	θ	ADJ
ejpam-6573	212	12	-	-	PUNCT
ejpam-6573	212	13	open	open	ADJ
ejpam-6573	212	14	sets	set	NOUN
ejpam-6573	212	15	.	.	PUNCT
ejpam-6573	213	1	demonstratio	demonstratio	PROPN
ejpam-6573	213	2	mathematica	mathematica	PROPN
ejpam-6573	213	3	,	,	PUNCT
ejpam-6573	213	4	42(1):193–203	42(1):193–203	PROPN
ejpam-6573	213	5	,	,	PUNCT
ejpam-6573	213	6	2009	2009	NUM
ejpam-6573	213	7	.	.	PUNCT
ejpam-6573	214	1	[	[	X
ejpam-6573	214	2	6	6	NUM
ejpam-6573	214	3	]	]	PUNCT
ejpam-6573	214	4	m.	m.	NOUN
ejpam-6573	214	5	caldas	caldas	PROPN
ejpam-6573	214	6	and	and	CCONJ
ejpam-6573	214	7	g.	g.	PROPN
ejpam-6573	214	8	navalagi	navalagi	PROPN
ejpam-6573	214	9	.	.	PUNCT
ejpam-6573	215	1	on	on	ADP
ejpam-6573	215	2	weak	weak	ADJ
ejpam-6573	215	3	forms	form	NOUN
ejpam-6573	215	4	of	of	ADP
ejpam-6573	215	5	β	β	NOUN
ejpam-6573	215	6	-	-	ADJ
ejpam-6573	215	7	open	open	ADJ
ejpam-6573	215	8	and	and	CCONJ
ejpam-6573	215	9	β	β	NOUN
ejpam-6573	215	10	-	-	PUNCT
ejpam-6573	215	11	closed	closed	ADJ
ejpam-6573	215	12	functions	function	NOUN
ejpam-6573	215	13	.	.	PUNCT
ejpam-6573	216	1	analele	analele	ADP
ejpam-6573	216	2	ştiinţifice	ştiinţifice	PROPN
ejpam-6573	216	3	ale	ale	NOUN
ejpam-6573	216	4	universitǎţii	universitǎţii	PUNCT
ejpam-6573	216	5	”	"	PUNCT
ejpam-6573	216	6	alexandru	alexandru	PROPN
ejpam-6573	216	7	ioan	ioan	PROPN
ejpam-6573	216	8	cuza	cuza	PROPN
ejpam-6573	216	9	”	"	PUNCT
ejpam-6573	216	10	din	din	PROPN
ejpam-6573	216	11	lasi	lasi	PROPN
ejpam-6573	216	12	.	.	PUNCT
ejpam-6573	217	1	matematicǎ	matematicǎ	X
ejpam-6573	217	2	(	(	PUNCT
ejpam-6573	217	3	serie	serie	PROPN
ejpam-6573	217	4	nouǎ	nouǎ	PROPN
ejpam-6573	217	5	)	)	PUNCT
ejpam-6573	217	6	,	,	PUNCT
ejpam-6573	217	7	49:115–128	49:115–128	PROPN
ejpam-6573	217	8	,	,	PUNCT
ejpam-6573	217	9	2003	2003	NUM
ejpam-6573	217	10	.	.	PUNCT
ejpam-6573	218	1	[	[	X
ejpam-6573	218	2	7	7	X
ejpam-6573	218	3	]	]	X
ejpam-6573	218	4	n.	n.	NOUN
ejpam-6573	218	5	chutiman	chutiman	NOUN
ejpam-6573	218	6	and	and	CCONJ
ejpam-6573	218	7	c.	c.	PROPN
ejpam-6573	218	8	boonpok	boonpok	PROPN
ejpam-6573	218	9	.	.	PUNCT
ejpam-6573	219	1	some	some	DET
ejpam-6573	219	2	properties	property	NOUN
ejpam-6573	219	3	of	of	ADP
ejpam-6573	219	4	weakly	weakly	ADJ
ejpam-6573	219	5	b(λ	b(λ	NOUN
ejpam-6573	219	6	,	,	PUNCT
ejpam-6573	219	7	p)-open	p)-open	NOUN
ejpam-6573	219	8	functions	function	NOUN
ejpam-6573	219	9	.	.	PUNCT
ejpam-6573	220	1	international	international	ADJ
ejpam-6573	220	2	journal	journal	PROPN
ejpam-6573	220	3	of	of	ADP
ejpam-6573	220	4	mathematics	mathematic	NOUN
ejpam-6573	220	5	and	and	CCONJ
ejpam-6573	220	6	computer	computer	NOUN
ejpam-6573	220	7	science	science	NOUN
ejpam-6573	220	8	,	,	PUNCT
ejpam-6573	220	9	19(2):497–501	19(2):497–501	NUM
ejpam-6573	220	10	,	,	PUNCT
ejpam-6573	220	11	2024	2024	NUM
ejpam-6573	220	12	.	.	PUNCT
ejpam-6573	221	1	[	[	X
ejpam-6573	221	2	8	8	NUM
ejpam-6573	221	3	]	]	X
ejpam-6573	221	4	c.	c.	PROPN
ejpam-6573	221	5	boonpok	boonpok	PROPN
ejpam-6573	221	6	.	.	PUNCT
ejpam-6573	222	1	semi	semi	ADJ
ejpam-6573	222	2	-	-	ADJ
ejpam-6573	222	3	i	i	ADV
ejpam-6573	222	4	-	-	PUNCT
ejpam-6573	222	5	expandable	expandable	ADJ
ejpam-6573	222	6	ideal	ideal	ADJ
ejpam-6573	222	7	topological	topological	ADJ
ejpam-6573	222	8	spaces	space	NOUN
ejpam-6573	222	9	.	.	PUNCT
ejpam-6573	223	1	journal	journal	NOUN
ejpam-6573	223	2	of	of	ADP
ejpam-6573	223	3	mathematics	mathematic	NOUN
ejpam-6573	223	4	,	,	PUNCT
ejpam-6573	223	5	2021:9272335	2021:9272335	NUM
ejpam-6573	223	6	,	,	PUNCT
ejpam-6573	223	7	2021	2021	NUM
ejpam-6573	223	8	.	.	PUNCT
ejpam-6573	224	1	[	[	X
ejpam-6573	224	2	9	9	NUM
ejpam-6573	224	3	]	]	X
ejpam-6573	224	4	c.	c.	PROPN
ejpam-6573	224	5	klanarong	klanarong	PROPN
ejpam-6573	224	6	and	and	CCONJ
ejpam-6573	224	7	c.	c.	PROPN
ejpam-6573	224	8	boonpok	boonpok	PROPN
ejpam-6573	224	9	.	.	PUNCT
ejpam-6573	225	1	characterizations	characterization	NOUN
ejpam-6573	225	2	of	of	ADP
ejpam-6573	225	3	weakly	weakly	ADJ
ejpam-6573	225	4	s(λ	s(λ	NOUN
ejpam-6573	225	5	,	,	PUNCT
ejpam-6573	225	6	p)-open	p)-open	NOUN
ejpam-6573	225	7	functions	function	NOUN
ejpam-6573	225	8	and	and	CCONJ
ejpam-6573	225	9	weakly	weakly	ADJ
ejpam-6573	225	10	s(λ	s(λ	PROPN
ejpam-6573	225	11	,	,	PUNCT
ejpam-6573	225	12	p)-closed	p)-close	VERB
ejpam-6573	225	13	functions	function	NOUN
ejpam-6573	225	14	.	.	PUNCT
ejpam-6573	226	1	international	international	ADJ
ejpam-6573	226	2	journal	journal	PROPN
ejpam-6573	226	3	of	of	ADP
ejpam-6573	226	4	mathematics	mathematic	NOUN
ejpam-6573	226	5	and	and	CCONJ
ejpam-6573	226	6	computer	computer	NOUN
ejpam-6573	226	7	science	science	NOUN
ejpam-6573	226	8	,	,	PUNCT
ejpam-6573	226	9	19(3):809–814	19(3):809–814	PROPN
ejpam-6573	226	10	,	,	PUNCT
ejpam-6573	226	11	2024	2024	NUM
ejpam-6573	226	12	.	.	PUNCT
ejpam-6573	227	1	[	[	X
ejpam-6573	227	2	10	10	NUM
ejpam-6573	227	3	]	]	X
ejpam-6573	227	4	n.	n.	NOUN
ejpam-6573	227	5	srisarakham	srisarakham	PROPN
ejpam-6573	227	6	and	and	CCONJ
ejpam-6573	227	7	c.	c.	PROPN
ejpam-6573	227	8	boonpok	boonpok	PROPN
ejpam-6573	227	9	.	.	PUNCT
ejpam-6573	228	1	on	on	ADP
ejpam-6573	228	2	weakly	weakly	ADJ
ejpam-6573	228	3	δ(λ	δ(λ	PROPN
ejpam-6573	228	4	,	,	PUNCT
ejpam-6573	228	5	p)-open	p)-open	VERB
ejpam-6573	228	6	functions	function	NOUN
ejpam-6573	228	7	.	.	PUNCT
ejpam-6573	229	1	international	international	ADJ
ejpam-6573	229	2	journal	journal	PROPN
ejpam-6573	229	3	of	of	ADP
ejpam-6573	229	4	mathematics	mathematic	NOUN
ejpam-6573	229	5	and	and	CCONJ
ejpam-6573	229	6	computer	computer	NOUN
ejpam-6573	229	7	science	science	NOUN
ejpam-6573	229	8	,	,	PUNCT
ejpam-6573	229	9	19(2):485–489	19(2):485–489	PROPN
ejpam-6573	229	10	,	,	PUNCT
ejpam-6573	229	11	2024	2024	NUM
ejpam-6573	229	12	.	.	PUNCT
ejpam-6573	230	1	[	[	X
ejpam-6573	230	2	11	11	NUM
ejpam-6573	230	3	]	]	X
ejpam-6573	230	4	c.	c.	PROPN
ejpam-6573	230	5	klanarong	klanarong	PROPN
ejpam-6573	230	6	and	and	CCONJ
ejpam-6573	230	7	c.	c.	PROPN
ejpam-6573	230	8	boonpok	boonpok	PROPN
ejpam-6573	230	9	.	.	PUNCT
ejpam-6573	231	1	characterizations	characterization	NOUN
ejpam-6573	231	2	of	of	ADP
ejpam-6573	231	3	weakly	weakly	ADJ
ejpam-6573	231	4	δ(λ	δ(λ	PROPN
ejpam-6573	231	5	,	,	PUNCT
ejpam-6573	231	6	p)-closed	p)-close	VERB
ejpam-6573	231	7	functions	function	NOUN
ejpam-6573	231	8	.	.	PUNCT
ejpam-6573	232	1	international	international	ADJ
ejpam-6573	232	2	journal	journal	PROPN
ejpam-6573	232	3	of	of	ADP
ejpam-6573	232	4	mathematics	mathematic	NOUN
ejpam-6573	232	5	and	and	CCONJ
ejpam-6573	232	6	computer	computer	NOUN
ejpam-6573	232	7	science	science	NOUN
ejpam-6573	232	8	,	,	PUNCT
ejpam-6573	232	9	19(2):503–507	19(2):503–507	PROPN
ejpam-6573	232	10	,	,	PUNCT
ejpam-6573	232	11	2024	2024	NUM
ejpam-6573	232	12	.	.	PUNCT
ejpam-6573	233	1	[	[	X
ejpam-6573	233	2	12	12	NUM
ejpam-6573	233	3	]	]	X
ejpam-6573	233	4	c.	c.	PROPN
ejpam-6573	233	5	boonpok	boonpok	PROPN
ejpam-6573	233	6	and	and	CCONJ
ejpam-6573	233	7	m.	m.	NOUN
ejpam-6573	233	8	thongmoon	thongmoon	NOUN
ejpam-6573	233	9	.	.	PUNCT
ejpam-6573	234	1	properties	property	NOUN
ejpam-6573	234	2	of	of	ADP
ejpam-6573	234	3	weakly	weakly	ADJ
ejpam-6573	234	4	β(λ	β(λ	NOUN
ejpam-6573	234	5	,	,	PUNCT
ejpam-6573	234	6	p)-open	p)-open	NOUN
ejpam-6573	234	7	functions	function	NOUN
ejpam-6573	234	8	and	and	CCONJ
ejpam-6573	234	9	weakly	weakly	ADJ
ejpam-6573	234	10	β(λ	β(λ	NOUN
ejpam-6573	234	11	,	,	PUNCT
ejpam-6573	234	12	p)-closed	p)-close	VERB
ejpam-6573	234	13	functions	function	NOUN
ejpam-6573	234	14	.	.	PUNCT
ejpam-6573	235	1	european	european	ADJ
ejpam-6573	235	2	journal	journal	PROPN
ejpam-6573	235	3	of	of	ADP
ejpam-6573	235	4	pure	pure	ADJ
ejpam-6573	235	5	and	and	CCONJ
ejpam-6573	235	6	applied	applied	ADJ
ejpam-6573	235	7	mathematics	mathematic	NOUN
ejpam-6573	235	8	,	,	PUNCT
ejpam-6573	235	9	17(1):248–255	17(1):248–255	PROPN
ejpam-6573	235	10	,	,	PUNCT
ejpam-6573	235	11	2024	2024	NUM
ejpam-6573	235	12	.	.	PUNCT
ejpam-6573	236	1	[	[	X
ejpam-6573	236	2	13	13	NUM
ejpam-6573	236	3	]	]	PUNCT
ejpam-6573	236	4	c.	c.	PROPN
ejpam-6573	236	5	boonpok	boonpok	PROPN
ejpam-6573	236	6	and	and	CCONJ
ejpam-6573	236	7	m.	m.	NOUN
ejpam-6573	236	8	thongmoon	thongmoon	NOUN
ejpam-6573	236	9	.	.	PUNCT
ejpam-6573	237	1	weakly	weakly	ADJ
ejpam-6573	237	2	p(λ	p(λ	NOUN
ejpam-6573	237	3	,	,	PUNCT
ejpam-6573	237	4	p)-open	p)-open	NOUN
ejpam-6573	237	5	functions	function	NOUN
ejpam-6573	237	6	and	and	CCONJ
ejpam-6573	237	7	weakly	weakly	ADJ
ejpam-6573	237	8	p(λ	p(λ	NOUN
ejpam-6573	237	9	,	,	PUNCT
ejpam-6573	237	10	p)closed	p)close	VERB
ejpam-6573	237	11	functions	function	NOUN
ejpam-6573	237	12	.	.	PUNCT
ejpam-6573	238	1	international	international	ADJ
ejpam-6573	238	2	journal	journal	NOUN
ejpam-6573	238	3	of	of	ADP
ejpam-6573	238	4	analysis	analysis	NOUN
ejpam-6573	238	5	and	and	CCONJ
ejpam-6573	238	6	applications	application	NOUN
ejpam-6573	238	7	,	,	PUNCT
ejpam-6573	238	8	22:10	22:10	NUM
ejpam-6573	238	9	,	,	PUNCT
ejpam-6573	238	10	2024	2024	NUM
ejpam-6573	238	11	.	.	PUNCT
ejpam-6573	239	1	[	[	X
ejpam-6573	239	2	14	14	NUM
ejpam-6573	239	3	]	]	X
ejpam-6573	239	4	c.	c.	PROPN
ejpam-6573	239	5	boonpok	boonpok	PROPN
ejpam-6573	239	6	and	and	CCONJ
ejpam-6573	239	7	p.	p.	NOUN
ejpam-6573	239	8	pue	pue	NOUN
ejpam-6573	239	9	-	-	PUNCT
ejpam-6573	239	10	on	on	ADP
ejpam-6573	239	11	.	.	PUNCT
ejpam-6573	240	1	weakly	weakly	ADJ
ejpam-6573	240	2	θs(λ	θs(λ	NOUN
ejpam-6573	240	3	,	,	PUNCT
ejpam-6573	240	4	p)-open	p)-open	VERB
ejpam-6573	240	5	functions	function	NOUN
ejpam-6573	240	6	and	and	CCONJ
ejpam-6573	240	7	weakly	weakly	ADJ
ejpam-6573	240	8	θs(λ	θs(λ	NOUN
ejpam-6573	240	9	,	,	PUNCT
ejpam-6573	240	10	p)closed	p)close	VERB
ejpam-6573	240	11	functions	function	NOUN
ejpam-6573	240	12	.	.	PUNCT
ejpam-6573	241	1	asia	asia	PROPN
ejpam-6573	241	2	pacific	pacific	PROPN
ejpam-6573	241	3	journal	journal	PROPN
ejpam-6573	241	4	of	of	ADP
ejpam-6573	241	5	mathematics	mathematic	NOUN
ejpam-6573	241	6	,	,	PUNCT
ejpam-6573	241	7	11:13	11:13	NUM
ejpam-6573	241	8	,	,	PUNCT
ejpam-6573	241	9	2024	2024	NUM
ejpam-6573	241	10	.	.	PUNCT
ejpam-6573	242	1	[	[	X
ejpam-6573	242	2	15	15	NUM
ejpam-6573	242	3	]	]	X
ejpam-6573	242	4	c.	c.	PROPN
ejpam-6573	242	5	boonpok	boonpok	PROPN
ejpam-6573	242	6	,	,	PUNCT
ejpam-6573	242	7	c.	c.	PROPN
ejpam-6573	242	8	viriyapong	viriyapong	PROPN
ejpam-6573	242	9	,	,	PUNCT
ejpam-6573	242	10	and	and	CCONJ
ejpam-6573	242	11	m.	m.	NOUN
ejpam-6573	242	12	thongmoon	thongmoon	NOUN
ejpam-6573	242	13	.	.	PUNCT
ejpam-6573	243	1	on	on	ADP
ejpam-6573	243	2	upper	upper	ADJ
ejpam-6573	243	3	and	and	CCONJ
ejpam-6573	243	4	lower	low	ADJ
ejpam-6573	243	5	(	(	PUNCT
ejpam-6573	243	6	τ1	τ1	NOUN
ejpam-6573	243	7	,	,	PUNCT
ejpam-6573	243	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6573	243	9	multifunctions	multifunction	NOUN
ejpam-6573	243	10	.	.	PUNCT
ejpam-6573	244	1	journal	journal	PROPN
ejpam-6573	244	2	of	of	ADP
ejpam-6573	244	3	mathematics	mathematics	PROPN
ejpam-6573	244	4	and	and	CCONJ
ejpam-6573	244	5	computer	computer	NOUN
ejpam-6573	244	6	science	science	NOUN
ejpam-6573	244	7	,	,	PUNCT
ejpam-6573	244	8	18:282–293	18:282–293	NUM
ejpam-6573	244	9	,	,	PUNCT
ejpam-6573	244	10	2018	2018	NUM
ejpam-6573	244	11	.	.	PUNCT
ejpam-6573	245	1	[	[	X
ejpam-6573	245	2	16	16	NUM
ejpam-6573	245	3	]	]	X
ejpam-6573	245	4	c.	c.	PROPN
ejpam-6573	245	5	viriyapong	viriyapong	PROPN
ejpam-6573	245	6	and	and	CCONJ
ejpam-6573	245	7	c.	c.	PROPN
ejpam-6573	245	8	boonpok	boonpok	PROPN
ejpam-6573	245	9	.	.	PUNCT
ejpam-6573	246	1	(	(	PUNCT
ejpam-6573	246	2	τ1	τ1	NOUN
ejpam-6573	246	3	,	,	PUNCT
ejpam-6573	246	4	τ2)α	τ2)α	NOUN
ejpam-6573	246	5	-	-	PUNCT
ejpam-6573	246	6	continuity	continuity	NOUN
ejpam-6573	246	7	for	for	ADP
ejpam-6573	246	8	multifunctions	multifunction	NOUN
ejpam-6573	246	9	.	.	PUNCT
ejpam-6573	247	1	journal	journal	PROPN
ejpam-6573	247	2	of	of	ADP
ejpam-6573	247	3	mathematics	mathematic	NOUN
ejpam-6573	247	4	,	,	PUNCT
ejpam-6573	247	5	2020:6285763	2020:6285763	NUM
ejpam-6573	247	6	,	,	PUNCT
ejpam-6573	247	7	2020	2020	NUM
ejpam-6573	247	8	.	.	PUNCT
ejpam-6573	248	1	[	[	X
ejpam-6573	248	2	17	17	NUM
ejpam-6573	248	3	]	]	PUNCT
ejpam-6573	248	4	c.	c.	PROPN
ejpam-6573	248	5	boonpok	boonpok	PROPN
ejpam-6573	248	6	.	.	PUNCT
ejpam-6573	249	1	(	(	PUNCT
ejpam-6573	249	2	τ1	τ1	NOUN
ejpam-6573	249	3	,	,	PUNCT
ejpam-6573	249	4	τ2)δ	τ2)δ	ADJ
ejpam-6573	249	5	-	-	PUNCT
ejpam-6573	249	6	semicontinuous	semicontinuous	ADJ
ejpam-6573	249	7	multifunctions	multifunction	NOUN
ejpam-6573	249	8	.	.	PUNCT
ejpam-6573	250	1	heliyon	heliyon	NOUN
ejpam-6573	250	2	,	,	PUNCT
ejpam-6573	250	3	6	6	NUM
ejpam-6573	250	4	:	:	SYM
ejpam-6573	250	5	e05367	e05367	PROPN
ejpam-6573	250	6	,	,	PUNCT
ejpam-6573	250	7	2020	2020	NUM
ejpam-6573	250	8	.	.	PUNCT
ejpam-6573	251	1	[	[	X
ejpam-6573	251	2	18	18	NUM
ejpam-6573	251	3	]	]	X
ejpam-6573	251	4	n.	n.	PROPN
ejpam-6573	251	5	viriyapong	viriyapong	PROPN
ejpam-6573	251	6	,	,	PUNCT
ejpam-6573	251	7	s.	s.	PROPN
ejpam-6573	251	8	sompong	sompong	PROPN
ejpam-6573	251	9	,	,	PUNCT
ejpam-6573	251	10	and	and	CCONJ
ejpam-6573	251	11	c.	c.	PROPN
ejpam-6573	251	12	boonpok	boonpok	PROPN
ejpam-6573	251	13	.	.	PUNCT
ejpam-6573	252	1	(	(	PUNCT
ejpam-6573	252	2	τ1	τ1	NOUN
ejpam-6573	252	3	,	,	PUNCT
ejpam-6573	252	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6573	252	5	disconnectedness	disconnectedness	NOUN
ejpam-6573	252	6	in	in	ADP
ejpam-6573	252	7	bitopological	bitopological	ADJ
ejpam-6573	252	8	spaces	space	NOUN
ejpam-6573	252	9	.	.	PUNCT
ejpam-6573	253	1	international	international	ADJ
ejpam-6573	253	2	journal	journal	PROPN
ejpam-6573	253	3	of	of	ADP
ejpam-6573	253	4	mathematics	mathematic	NOUN
ejpam-6573	253	5	and	and	CCONJ
ejpam-6573	253	6	computer	computer	NOUN
ejpam-6573	253	7	science	science	NOUN
ejpam-6573	253	8	,	,	PUNCT
ejpam-6573	253	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6573	253	10	,	,	PUNCT
ejpam-6573	253	11	2024	2024	NUM
ejpam-6573	253	12	.	.	PUNCT
ejpam-6573	254	1	[	[	X
ejpam-6573	254	2	19	19	NUM
ejpam-6573	254	3	]	]	X
ejpam-6573	254	4	c.	c.	PROPN
ejpam-6573	254	5	viriyapong	viriyapong	PROPN
ejpam-6573	254	6	,	,	PUNCT
ejpam-6573	254	7	s.	s.	PROPN
ejpam-6573	254	8	sompong	sompong	PROPN
ejpam-6573	254	9	,	,	PUNCT
ejpam-6573	254	10	and	and	CCONJ
ejpam-6573	254	11	c.	c.	PROPN
ejpam-6573	254	12	boonpok	boonpok	PROPN
ejpam-6573	254	13	.	.	PUNCT
ejpam-6573	255	1	upper	upper	ADJ
ejpam-6573	255	2	and	and	CCONJ
ejpam-6573	255	3	lower	low	ADJ
ejpam-6573	255	4	slightly	slightly	ADV
ejpam-6573	255	5	(	(	PUNCT
ejpam-6573	255	6	τ1	τ1	NOUN
ejpam-6573	255	7	,	,	PUNCT
ejpam-6573	255	8	τ2)βcontinuous	τ2)βcontinuous	ADJ
ejpam-6573	255	9	multifunctions	multifunction	NOUN
ejpam-6573	255	10	.	.	PUNCT
ejpam-6573	256	1	asia	asia	PROPN
ejpam-6573	256	2	pacific	pacific	PROPN
ejpam-6573	256	3	journal	journal	PROPN
ejpam-6573	256	4	of	of	ADP
ejpam-6573	256	5	mathematics	mathematic	NOUN
ejpam-6573	256	6	,	,	PUNCT
ejpam-6573	256	7	11:75	11:75	NUM
ejpam-6573	256	8	,	,	PUNCT
ejpam-6573	256	9	2024	2024	NUM
ejpam-6573	256	10	.	.	PUNCT